2\\times 10^{-2}$ cm$^{-3}$ , we assume that the neutral fraction is nearly unity and take $n_{\\rm HI} = n_{\\rm H}$ ; otherwise we assume the HI density is zero.","This is approximately consistent with radiative transfer calculations [50] that take into account the metagalactic radiation background (a local radiation background would decrease the neutral gas fraction even lower, and so our bounds here are conservative).","Both the stellar mass and HI mass are calculated within the same HI column density limit we used to determine the radial extent for the rotation curves ($10^{20}$ cm$^{-2}$ ) and extend above and below the plane by 2 kpc.","The circular velocity is calculated as the average rotation velocity of the gas at a radius of 2.2 scale lengths of the stellar surface density.","To determine the scale length, we fit the stellar surface density curve with a straight line from 2 kpc to 10 kpc.","Our simulation results all lie within the grey area and scatter around the observed relation.","Although we do not have a significantly large numerical sample to make a detailed comparison, these preliminary results appear to be in good agreement, indicating that the baryonic content of our galaxies is consistent with that observed.","We note that we cannot include the purely thermal simulation (H1-0) on this plot as the systems has no systematic rotation.","Figure: Comparison between the five dwarf simulations (H1-H5) run with our standard diffusion coefficient and the baryonic Tully-Fisher relation from observations.","The solid line is the fit from .","The red crosses are simulation data.","The grey area shows the region of one sigma error in M b M_b.","The dashed line is taken from and indicates the relation between the halo circular velocity and the cosmic baryon fraction of the halo.The baryonic Tully Fisher relation compares the galaxies' baryonic mass with a measure of its total mass.","We would also like to compare the two main components of baryons: neutral hydrogen and stars.","Figure REF shows the ratio of the HI to stellar components as a function of the stellar mass, all quantities which are directly observable.","The observed relation found in [17] (their Eq.","5) can be approximated as: $\\begin{split}\\frac{M_\\mathrm {HI}}{3.36 \\times 10^9 \\rm {M}_\\odot } = &\\left(\\frac{M_\\star }{3.3 \\times 10^{10} \\rm {M}_\\odot }\\right)^{0.19}\\\\ & \\times \\left[1+\\left(\\frac{M_\\star }{3.3 \\times 10^{10} \\rm {M}_\\odot }\\right)^{0.76}\\right]\\end{split}$ In Figure REF , the solid line shows this relation.","We also estimate the observed scatter around this relation by taking a variation of +/- 0.5 log M$_\\mathrm {HI}$ /M$_\\star $ , based on a visual inspection of Figure 1 of [17]; this is shown as a grey area in our Figure REF .","The ratio from the simulations is also plotted, again using the stellar and neutral hydrogen masses within our adopted HI surface density limit, and +/- 2 kpc in vertical extent.","As can be seen from Figure REF , all of our simulation results are below the observed relation, indeed most lie below the estimated scatter in the observations (the grey band).","This demonstrates that our simulation produces consistently lower HI fraction than observed.","There is some indication that the simulated relation shows a rising gas fraction with decreasing stellar mass; however, the sample size is too small to be sure.","This mismatch in the gas to star ratio, by approximately a factor of five, when combined with the agreement in the baryonic Tully-Fisher relation (which measures the total baryonic content), implies that our galaxies contain the correct amount of baryons, but are overly efficient in their conversion of gas to stars.","One alternative way to examine just the stellar content of our systems is based on the dark matter abundance matching measurements in [17], which provides a relation between the stellar mass and halo mass under the assumption that there is a monotonic relation between halo mass and stellar mass.","Indeed, if we compare our results to their inferred M$_*$ -M$_{\\rm halo}$ relation, we find that our $M_*$ values are systematically too large by factors of 2-5, consistent with the idea that the division between stellar and gas mass is incorrect in these simulations.","Finally, we note that these relations also imply that the simulations disagree with the observed Kenicutt-Schmidt relation between surface gas density and star formation surface density.","Figure: Comparison between this work and the relation of HI mass and stellar mass inferred from observations.","The solid line is derived in .","The grey area shows an estimated scatter of +/- 0.5 log around the observed relation in the vertical direction.","The red crosses are the simulation results."],["Dark Matter Profiles","There are a number of observational indications that dwarf galaxies do not contain the cuspy dark-matter profiles predicted in dark-matter only simulations [11], [69], [3], [4], [55], [34].","A number of simulations with strong feedback have found that it is possible for the feedback to decrease the dark matter density in the center of halos [41], [20], [48], [44].","Therefore, we examine our simulation to see the impact of thermal vs. CR feedback on the core properties of the systems.","Figure REF shows the dark matter density profile of our most massive halo in the simulations with different CR physics.","Interestingly, all of the CR runs show very similar results, indicating that although they produce different distributions for the gas and stars, the presence of CR feedback leads to similar dark matter core properties.","In all cases, the profile is quite steep, with an inner density profile between $r^{-1}$ and $r^{-2}$ (similar to dark-matter only runs).","On the other hand, the run without CR diffusion shows a clear core in the dark matter with a core radius of several kpc.","The cored dark matter profile in the no-CR case might be due to the large and rapid motions of the gas, which can disturb the dark matter distribution [48] and is consistent with the observed asymmetric gas distribution and the bursty nature of the feedback in the no-CR run.","Figure: Dark matter profiles of simulations with different CR physics.We have performed a preliminary investigation into the impact of CRs on the formation and evolution of dwarf galaxies with rotation velocities in the 70-140 km/s range using cosmological simulations.","This complements similar work [53] which looked at a higher mass galaxy (with a rotation velocity above 200 km/s).","As in that work, we found that the inclusion of CR feedback had a strong impact on the gas and stellar properties.","In both cases, CR feedback led to robust outflows and rotationally supported disks with relatively flat rotation curves, but only if CR diffusion was included.","A comparison to observations showed good agreement with the baryonic Tully-Fisher relation, indicating that the total disk mass was reasonable.","This mass, converted into a fraction of the total halo mass, is well below the cosmic value, indicating that feedback was important in generating mass-loaded winds.","On the other hand, a comparison of the gas content of the simulated galaxies against observations showed that too much gas had been converted to stars.","In addition, the gas depletion times of our galaxies (computed as $M_{\\rm gas}/\\dot{M}_{\\rm SFR}$ ), are a few Gyr, well below observed values for low-mass isolated dwarfs [29].","Our star formation and feedback model is approximate, with a number of free parameters required to describe physics that is not fully modeled.","In section REF , we explored the impact of some of these for halo H1.","Remarkably, we find that the CR diffusion rate has a relatively small impact on the stellar mass produced (see Table REF ), or the gas mass.","The exception was the run with $\\kappa = 0$ , which produced a substantially higher stellar mass and did not produce a disk (this combination does not generate winds).","We also carried out an additional run of halo H1 with the standard diffusion rate (and other parameters), but a reduced CR energy efficiency (10% instead of 30%) and find that relative to the fiducial run, it produces 50% more stars.","The resulting disk rotates marginally faster such that the system stays close to the observed baryonic Tully-Fisher relation.","However, the HI to gas mass ratio is even lower the fiducial run, resulting in an an even stronger disagreement with the observed relation.","In principle, a higher CR efficiency might bring this into agreement with observations, but that would disagree with local observational and theoretical work on CR acceleration efficiencies.","Another parameter which can be varied is the star formation efficiency per free-fall time, which we took to be 1%.","We carried out another simulation much like H1 but with a star formation efficiency four times lower (with $\\kappa = 3 \\times 10^{28}$ cm$^2$ s$^{-1}$ ; this run was not included in figures and tables above).","This had almost no effect on the resulting stellar mass or gas mass, consistent with the idea that the disk is in a state of self-regulation – indeed, we verified visually that the lower efficiency resulted in a higher density gas disk and hence essentially the same net star formation rate at $z=0$ (since the star formation rate increases as $\\rho ^{1.5}$ ).","There are, of course, other parameters to vary, such as the IMF, the fraction of energy in SN, as well as other uncertainties in the subgrid model.","In addition, there is some indication from H2 and the high-mass halo explored in [53] that an additional form of feedback is required in high-density regions – possibly radiative heating or pressure.","Finally, we briefly discuss our results in comparison to [68], who recently carried out simulations of dwarf galaxies with cosmic rays.","The implementation differs somewhat in that they adopted a model in which CRs stream along magnetic field lines and heat the gas (rather than isotropically diffuse, as in our model).","Although they do not simulate cosmological volumes, their halo mass which most closely compares to ours produces approximately similar results: robust cosmic-ray driven outflows with a modest mass-loading factor.","Although we are unable to compute a mass-loading factor, the large stellar mass to halo mass we find implies a relatively low mass-loading factor."],["Feedback efficiency of the purely thermal run","Finally, we turn to a discussion of the thermal-only feedback run (H1-0).","Interestingly, the feedback in this case was actually more effective than the CR+thermal feedback, in contrast to the result for the higher mass halo with the same code and CR parameters discussed in [53].","One possibility is that the purely thermal feedback is more efficient than in more massive galaxies because the shallower potential well results in low gas densities, lowering the radiative cooling rate and permitting blast waves from SN feedback to accelerate the gas.","We explore the relative importance of cooling and heating in Figure REF , which shows a two-dimensional distribution of radiative cooling times and a measure of the possible supernova heating rate for all gas within 30 kpc of the center of this halo as $z=0$ .","We compute the SN heating rate in a way similar to that in [60], where we estimate the timescale required for our thermal energy injection mechanism to heat a cell to $T_6 = 10^6 K$ (the precise temperature selected does not affect our conclusions): $t_{\\rm heat} = \\frac{3}{2} \\frac{M_{\\rm cell} k T_6}{\\mu m_H} \\frac{t_f}{e_{\\rm SN} M_* c^2}$ where $M_{\\rm cell}$ is the gas mass cell in the cell, $\\mu \\approx 1$ , $M_* = 10^5$ M$_\\odot $ , our typical stellar particle mass, $t_f = 10$ Myr, the timescale over which a star particle returns its supernova energy to the ISM in our model, and $e_{\\rm SN} = 3 \\times 10^{-6}$ , as defined in the methods section.","We stress that not all cells are being heated at this rate, but it is the rate that a single star particle would typically heat its local gas and so is a good measure of the timescale over which feedback heating acts.","From this figure, we see that nearly all the gas in the simulation lies above the black line indicating equality.","This demonstrates that the gas densities are sufficiently low that pure thermal feedback is efficient in heating the gas and driving outflows – we focus on density because the cooling rate is proportional to the density squared.","A close examination of this figure indicates that a substantial amount of gas lies close to the line of equality (in fact, this gas is in the disk), and so we expect that the larger potential depths of higher mass halos will push gas beyond the line into a regime where purely thermal supernovae feedback cannot heat the gas.","This is consistent with the finding that our dwarf galaxy simulations do drive outflows while higher mass (e.g.","Milky Way mass) systems do not.","Note that this statement depends on details of the simulations (e.g.","see definition of $t_{\\rm heat}$ ) and so may be a reflection of numerical models rather than physical quantities.","This demonstrates that purely thermal feedback is sufficient to drive feedback but it does not answer the question why adding cosmic rays decreases the bursty outflows, as seen in Figure REF .","The answer appears to be that cosmic rays actually increase the pressure in the diffuse circumgalactic gas surrounding galaxies.","The purely thermal run has a (thermal only) CGM pressure at a radius of 10 kpc of approximately $10^{-14}$ erg cm$^{-3}$ , while even the CR run with the highest diffusion coefficient (H1), has a (mostly CR) pressure of nearly $10^{-13}$ erg cm$^{-3}$ .","This larger pressure resists the impulsive outflows driven by bursts of supernovae in the disk (although it does not appear to eliminate the slower gas outflows driven by the CR pressure gradient).","Finally, we note a consequence of the effective thermal only feedback, which is that it resulted in bursty star formation at a low-level; and was so effective that the resulting stellar system was “hot\" – that is, it was supported by velocity dispersion rather than rotation (an issue that has been raised in the context of other simulations – see the discussion in [72]."],["Summary","We simulate a series of dwarf galaxies with zoom-in cosmological simulations, and study the influence of CR feedback.","First we compare the outcome by varying the CR physics in different runs of the same galaxy halo.","The different CR physics include feedback without CR, CR feedback with zero diffusion, CR feedback with $\\kappa _\\mathrm {CR}$ equal to $3 \\times 10^{27}$ , $1 \\times 10^{28}$ , and $3 \\times 10^{28}$ cm$^2$ s$^{-1}$ .","Then we fixed the CR diffusion at $\\kappa _\\mathrm {CR} = 3 \\times 10^{28}$ cm$^2$ s$^{-1}$ and simulated five dwarf galaxies with different masses, ranging from 8 to 30 $\\times 10^{10}$ M$_\\odot $ .","We summarize the results of this work as follows: Adding CRs and some realistic level of diffusion consistently produced thin, extended disk galaxies with nearly flat rotation curves, in contrast to a case with purely thermal feedback, which produced a hot stellar systems.","The star formation rate in the CR systems was relatively constant, compared to the lower (and burstier) SFR in the purely thermal feedback model.","Simulations of our five dwarf galaxies with different masses but the same CR model matched well with the observed baryonic Tully-Fisher relation.","However, the five galaxies' neutral hydrogen fractions were always lower than what is observed, by a factor of about five and the stellar content was larger than observed (but the total baryonic content matched observations).","Simulations with CR feedback produce cuspy dark matter profiles, while our purely thermal feedback case resulted in a cored dark matter profile.","These results indicate that the impact of cosmic rays in dwarf galaxies is potentially quite important, a conclusion which is in general agreement with previous work [31].","In detail, the results are somewhat dependent on the physical model, but broadly speaking CRs with diffusion produce dwarf galaxies which are rotationally supported and have relatively flat rotation curves.","This result appears to be quite robust (and occurs even though 70% of the feedback energy is in the form of thermal energy in these models) and is probably due to the smooth pressure provided by CRs (along with the outflows they drive).","The baryonic content of the dwarf systems with CRs is in broad agreement with observations.","On the other hand, there are two areas of disagreement with observations in these models: (a) the detailed census of baryons in each system (stars vs. gas) is much too heavily weighted to the stellar side, and (b) the presence of dark matter cores may be a problem (although CR pressure may systematically support the gas at small radius and so bias the observed measures of dark matter on small scales).","Clearly, more work is required to explore and refine these results.","For example, it would be useful to explore more realistic models including magnetic fields, anisotropic CR diffusion, and the impact of field-CR streaming instabilities."],["Acknowledgments","We thank the anonymous referee for a useful report which improved the clarity of the paper.","We would like to thank Cameron Hummels, Mordecai-Mark Mac Low and Mary Putman for useful discussion related to this work.","We acknowledge financial support from NSF grants AST-1312888, and NASA grants NNX12AH41G and NNX15AB20G, as well as computational resources from NSF XSEDE, and Columbia University's Yeti cluster.","Computations described in this work were performed using the publicly-available Enzo code (http://enzo-project.org), which is the product of a collaborative effort of many independent scientists from numerous institutions around the world.","Their commitment to open science has helped make this work possible."]],"string":"[\n [\n \"Cosmological Simulations of Dwarf Galaxies with Cosmic Ray Feedback\"\n ],\n [\n \"Abstract We perform zoom-in cosmological simulations of a suite of dwarf galaxies, examining the impact of cosmic-rays generated by supernovae, including the effect of diffusion.\",\n \"We first look at the effect of varying the uncertain cosmic ray parameters by repeatedly simulating a single galaxy.\",\n \"Then we fix the comic ray model and simulate five dwarf systems with virial masses range from 8-30 $\\\\times 10^{10}$ Msun.\",\n \"We find that including cosmic ray feedback (with diffusion) consistently leads to disk dominated systems with relatively flat rotation curves and constant star formation rates.\",\n \"In contrast, our purely thermal feedback case results in a hot stellar system and bursty star formation.\",\n \"The CR simulations very well match the observed baryonic Tully-Fisher relation, but have a lower gas fraction than in real systems.\",\n \"We also find that the dark matter cores of the CR feedback galaxies are cuspy, while the purely thermal feedback case results in a substantial core.\"\n ],\n [\n \"Introduction\",\n \"Isolated dwarf galaxies are commonly observed in the local Universe, and provide an interesting way to test models of galaxy formation and evolution.\",\n \"Below a stellar mass limit of about $10^{10}$ M$_\\\\odot $ (or rotation velocity of about 100 km/s), such systems are largely star-forming systems except when they become satellites of larger halos [18], [5].\",\n \"Naively, one might expect such systems to be easy to understand, as they have small or non-existent bulges and are free from uncertainties related to AGN feedback.\",\n \"However, detailed comparisons with cosmological models have encountered two kinds of problems.\",\n \"The first problem has to do with the baryonic content of the halos – simply put, galaxy formation must become increasingly less efficient as the halo mass decreases, for standard $\\\\Lambda $ CDM to match observations.\",\n \"It has long been recognized that this is implied by the different slopes of the low-mass end of the dark matter halo mass function and the faint end of the luminosity function [73].\",\n \"More recent work has sharpened this mismatch with both semi-analytic models [24], [2], [37] and simulations [64], [20], [57].\",\n \"A closely related problem is the \\\"missing\\\" dwarf satellites around Milky-Way mass halos [40], [63].\",\n \"A second issue has to do with the inner structure of dwarf galaxies – in particular the relation between the halo mass and the rotational velocity near the core.\",\n \"Dark matter only simulations predict quite cuspy profiles with relatively high dark matter densities in the center, while observations generally find cored profiles [39], [54], [10], [70].\",\n \"A related version of this problem has recently been highlighted among the dwarf satellites as the \\\"too big to fail\\\" problem, which highlights the paucity of satellites with rotation velocities in the 30-70 km/s range [4].\",\n \"Solutions to both of these problems in an astrophysical context require efficient feedback mechanisms that can blow out a large amount of gas, suppress star formation, bring down the baryonic fraction, and possibly reduce the central dark matter density of lower mass dwarfs.\",\n \"It has long been clear that the low potential depth of dwarf systems makes them particularly susceptible to stellar feedback [12]; however, modeling feedback from supernovae is numerically challenging [42].\",\n \"A variety of numerical feedback mechanisms have been tested in different papers.\",\n \"For example, by suppressing cooling immediately after a star formation event to counteract the artificially enhanced radiative losses [21], [23], simulated galaxies produced less massive bulges.\",\n \"Supernovae feedback which injects momentum directly in to the surrounding gas has also been shown to drive mass-loaded outflows [45].\",\n \"This has been combined with other physical processes – for example, [56] combined supernovae feedback with photoionization from ultraviolet background, and produced dwarf galaxies with high mass-to-light ratios.\",\n \"More recent simulations have made progresses in reproducing the observed properties using subgrid models with strong feedback [13], [28], [58].\",\n \"For example, [62] showed that they could reproduce the star formation main sequence relation at $z = 0$ .\",\n \"Zoom-in simulations afford more resolution, and have shown much recent promise in reproducing the stellar mass content of dwarf satellites [20], [22], [28], [43].\",\n \"[6] found that energetic supernovae feedback and subsequent tidal stripping has a significant effect on reducing the dense core of Milky Way mass galaxies Here we explore the impact of a specific physical feedback mechanism on dwarf galaxies, namely dynamical cosmic ray (CR) feedback.\",\n \"CRs are high-energy, quickly diffusing particles accelerated by shock waves.\",\n \"They have a wide range of energies, with the largest contribution to the energy density coming from those particles with energies around a few GeV.\",\n \"The energy density of CRs is comparable to the magnetic energy and turbulent energy in galaxies [36].\",\n \"In this paper, we present our cosmological simulations of dwarf galaxies with CR feedback.\",\n \"We use the Enzo code, with an implementation of CR feedback by [51].\",\n \"This adopts a relatively simple two-fluid model that provides an extra pressure term and also allows for diffusion of the CRs [14].\",\n \"In [51] and [52], we applied this model to the cosmological simulation of a Milky Way mass galaxy.\",\n \"Here we examine the effect of CR feedback and diffusion on dwarf galaxies with halo masses in the range of 8 to $30 \\\\times 10^{10}$ M$_\\\\odot $ .\",\n \"In section , we describe the simulation model, the feedback scheme, initial conditions, and the selection of halos for further simulations.\",\n \"In section REF , we choose the best set of parameters by comparing the simulation result of a single system with varying CR physics.\",\n \"Then we apply this model to a series of dwarf galaxies, with halo mass spanning from 8 to $30 \\\\times 10^{10}$ M$_\\\\odot $ in section REF , and we evaluate the results by comparing to observed scaling relations.\",\n \"Finally, in Section , we discuss our results.\",\n \"Figure: Edge-on views of the same halo modeled with different CR physics.\",\n \"From left to right, the panels show a simulation with no CRs, followed by runs with CR diffusion coefficients of 0, 3×10 27 3\\\\times 10^{27}, 1×10 28 1\\\\times 10^{28}, 3×10 28 3\\\\times 10^{28} cm 2 ^2s -1 ^{-1}.\",\n \"From top to bottom, the panels show the projected (density-weighted) gas density, stellar density, and CR energy density.\",\n \"Each panel represents a 30 kpc ×30\\\\times 30 kpc area.\"\n ],\n [\n \"Method\",\n \"We begin with a brief introduction to our hydrodynamics method, and then describe the two-fluid model used to model the CRs, and finally outline the cosmological initial conditions.\"\n ],\n [\n \"Numerical code: \",\n \"Our simulations use the open-source code Enzo, which is a three-dimensional, Cartesian, grid-based hydrodynamics code that includes adaptive mesh refinement [7], [46], [30], [8].\",\n \"We include a non-equilibrium chemical model following H, H$^+$ , He, He$^+$ , He$^{++}$ , and include radiative cooling from all the above chemical species as well as metal cooling computed in a lookup table as described in [61].\",\n \"We include a metagalactic radiative background as computed in [25].\",\n \"The star formation recipe of Enzo is based on the criteria presented in [9] and described in detail in [8].\",\n \"Briefly: when a parcel of dense gas is identified to be Jeans unstable and collapsing, some of its gas may be converted into a star particle with some efficiency.\",\n \"For a more detailed discussion of the star formation (and thermal-only) feedback scheme used in this paper, see [30] – in this work we adopt an efficiency of star formation per free fall time of 1% and an energetic feedback efficiency of $e_{\\\\rm SN} = 3 \\\\times 10^{-6}$ of the stellar rest mass energy.\"\n ],\n [\n \"Cosmic Ray Feedback Scheme\",\n \"We use a model for CR feedback and diffusion as implemented, tested and described in [51].\",\n \"The model uses a two-fluid method [14], [15], [32].\",\n \"It treats the high-energy particles as an ideal gas with $\\\\gamma = 4/3$ .\",\n \"Anisotropic CR pressure and the dynamical impact of magnetic fields are assumed to be subdominant.\",\n \"The fluid equations for the thermal gas stay as usual, except that the momentum of the gas is affected by the pressure of both the gas and CR components.\",\n \"In addition, we optionally include isotropic diffusion of the cosmic rays.\",\n \"Here we simply highlight the additional equation that describes the evolution of the CR energy density $\\\\epsilon _\\\\mathrm {CR}$ : $\\\\begin{split}\\\\partial _t \\\\epsilon _\\\\mathrm {CR} + \\\\nabla \\\\cdot (\\\\epsilon _\\\\mathrm {CR} \\\\textbf {\\\\textit {u}}) =& - P_\\\\mathrm {CR} (\\\\nabla \\\\cdot \\\\textbf {\\\\textit {u}}) \\\\\\\\& + \\\\nabla \\\\cdot (\\\\kappa _\\\\mathrm {CR} \\\\nabla \\\\epsilon _\\\\mathrm {CR}) + \\\\Gamma _\\\\mathrm {CR}\\\\end{split}$ where $\\\\textbf {\\\\textit {u}}$ is the fluid velocity, P$_\\\\mathrm {CR}$ is the pressure from CR, $\\\\kappa _\\\\mathrm {CR}$ is the CR isotropic diffusion coefficient, and $\\\\Gamma _\\\\mathrm {CR}$ is the source term for CR.\",\n \"Except for the isotropic diffusion, the CRs are assumed to be tightly coupled to the gas through a tangled magnetic field (which is not explicitly simulated).\",\n \"While this model is approximate, simulations including magnetic fields and/or anisotropic diffusion have also been shown to drive winds with similar mass-loading factors [27], [19].\",\n \"We also do not include any non-adiabatic losses in the CRs; in reality, CRs lose energy primarily through collisions with the thermal plasma; this is generally subdominant to adiabatic cooling in our simulations except for runs with zero (or low) diffusion coefficients, which are not realistic in any case.\",\n \"A more detailed discussion of this point and the shortcomings of the model in general can be found in [51] and [52].\",\n \"When a Type II SNe is formed, it will eject some of its mass and energy back into the surrounding cells.\",\n \"In our model, we assume that some of the energy is in thermal form and some accelerated by blast waves (generally not resolved in the grid).\",\n \"The fraction of energy that is deposited to thermal gas and CR is 0.7 and 0.3 respectively; this CR fraction is somewhat larger than canonical values but within suggested ranges [71], [16], [33].\",\n \"Moreover, we assume a relatively steep initial mass function (our chosen $e_{\\\\rm SN}$ corresponds to 1 SN per 185 M$_\\\\odot $ masses of stars produced), and hence a relatively low SN energy injection rate, so the CR energy produced per solar mass of stars formed is within typical estimates.\",\n \"We adopt these values for consistency with previous work [30], [51], where we also explored the impact of varying them.\"\n ],\n [\n \"Initial Conditions and Halo Selection\",\n \"We are interested in modeling a number of dwarf galaxies forming out of cosmological initial conditions.\",\n \"Therefore, we use MUSIC [26] to set up the initial density and velocities fields.\",\n \"This is done using the cosmological parameters from the best fit Planck 2013 results [47], i.e.\",\n \"H$_0$ = 67.11 km/s/Mpc, $\\\\Omega _\\\\Lambda = 0.6825 $ , $\\\\Omega _\\\\mathrm {m} = 0.3175 $ , $\\\\Omega _\\\\mathrm {b} = 0.0490 $ ${\\\\sigma _8 = 0.8344}$ .\",\n \"All the simulations start at $z =100$ and stop at the present day.\",\n \"Table: Simulation parameters for fixed halo, varying CR runs.First we perform a dark matter only run in a (20 Mpc/h)$^3$ box and identify halos that are around $10^{10}$ to $10^{11}$ M$_\\\\odot $ .\",\n \"All of these halos become a little more massive in the final runs, listed in Table REF .\",\n \"To study the impact of CR feedback, we select relatively isolated halos so that the physical properties of the dwarfs are not affected by nearby massive galaxies.\",\n \"In particular, all the selected systems are at least 0.5 Mpc away from any halos that are more massive than 0.1 times of the target halo.\",\n \"Then we trace the dark matter particles currently in the halo to their positions at the beginning of the simulation.\",\n \"The selected halos are re-simulated at high-resolution with baryons and adaptive mesh refinement (AMR) is performed in the cubical region that encloses all the dark matter particles from the beginning.\",\n \"Our root grid is 128$^3$ and we use three additional levels in the initial conditions for a particle mass of M$_{DM} = 8.3\\\\times 10^5$ M$_\\\\odot $ .\",\n \"During the simulation, additional refinement is added whenever the gas (dark matter) mass in a cell exceeds $6.0\\\\times 10^5$ ($3.3\\\\times 10^6$ ) M$_\\\\odot $ .\",\n \"We allow up to 10 levels of refinement, resulting in a smallest cell size of 227 pc.\",\n \"In order to explore the impact of some of the uncertain parameters in our model, simulations with different CR physics, including a no CR run, and CR runs with diffusion coefficients of 0, $3\\\\times 10^{27}$ , $1\\\\times 10^{28}$ , and $3\\\\times 10^{28}$ cm$^2$ s$^{-1}$ are done on the most massive of the selected halos.\",\n \"The results of these simulations are listed in Table REF .\",\n \"We analyze the results from these runs (see below) and choose the coefficient of $3\\\\times 10^{28}$ cm$^2$ s$^{-1}$ to carry on the simulations with the other selected halos.\",\n \"As discussed below, this choice is somewhat arbitrary, but it is consistent with estimates from recent GALPROP models [49], [1] and with observational measurements [65], [66].\",\n \"It is also found to be consistent with galactic gamma ray emission in [52].\",\n \"The properties of all selected halos simulated with this choice of parameters are listed in Table REF .\",\n \"Table: Simulated dwarf galaxy propertiesFirst, we focus on comparisons of runs with different CR physics, using the most massive of the halos.\",\n \"Second, we discuss the physical properties of the suite of five halos with fixed CR physics.\",\n \"Third, we compare our simulation results to observations, specifically the Baryonic Tully-Fisher relation and HI stellar mass relation.\",\n \"Analysis has been carried out using the yt package [67].\"\n ],\n [\n \"Impact of Varying CR Physics\",\n \"We choose an isolated halo of around $3 \\\\times 10^{11}$ M$_\\\\odot $ and study the effect of CR feedback by varying the CR physics and fixing all the other parameters (See Table REF for the simulation parameters).\",\n \"Figure REF shows edge-on images of the gas density, stellar density and CR energy density of the dwarf galaxies produced in each run at $z=0$ , focusing on just the center of the halo where the stellar system forms.\",\n \"Since H1-0 and H1-1 are not disk-shaped, we show them in the direction of their total angular momentum within 10 kpc radius.\",\n \"The run without any CR feedback (H1-0; left-most panels) forms a diffuse, irregular cloud.\",\n \"We find that the energetic thermal feedback is actually remarkably effective in regulating the formation of dense, star-forming clumps.\",\n \"This behavior is quite different than that seen in high-mass galaxies using identical star formation and feedback prescriptions [30] and is more like the behavior observed in very low-mass galaxies [60], [59].\",\n \"We explore this point in more detail in Section REF Moving on to the simulations with cosmic-rays, we see in Figure REF that H1-1, with no CR diffusion, has an almost spherically symmetric gas distribution.\",\n \"This occurs because when the CRs are tied to the gas, they provide complete pressure support and rotational support is unimportant.\",\n \"Clearly this is unrealistic (and probably violates gamma-ray emission from dwarf irregulars – see [52]).\",\n \"Figure: Comparison of the star formation rate history in simulations of the same halo but different CR physics, as noted in the legend.The other runs, which have non-zero CR diffusion, all produce disk-shaped galaxies, generally becoming thinner as the diffusion coefficient increases.\",\n \"H1, which has the highest diffusion coefficient of $3\\\\times 10^{28}$ cm$^2$ s$^{-1}$ , results in the most extended disk.\",\n \"H1-1, which has the lowest non-zero diffusion coefficient, results in a smaller disk which is surrounded by denser gas.\",\n \"The gas density and CR energy density immediately surrounding the disk also decreases as $\\\\kappa $ increases.\",\n \"The CRs escape more easily, both because they diffuse out more rapidly and because they drive stronger outflows [51], and therefore they provide less pressure support, decreasing the weight of gas they can support in the diffuse (not-rotating) halo.\",\n \"Figure: Edge-on views of the simulated galaxies with different halo masses but a fixed diffusion coefficient (κ=3×10 28 \\\\kappa = 3 \\\\times 10^{28} cm 2 ^2 s -1 ^{-1}.\",\n \"The plot show halos H1, H2, H3, H4, H5 from left to right, and gas density, stellar density, and CR energy density from top to bottom.\",\n \"Each panel represents a 30 kpc ×30\\\\times 30 kpc area at z=0z=0.Since all the runs with non-zero CR diffusion produce disk galaxies, we can examine the rotational velocity of the gas in the disk.\",\n \"Figure REF shows the rotational velocities of the HI gas in H1-2, H1-3, and H1.\",\n \"To compute these rotation velocity curves, we look at the actual rotational velocities of the HI gas (rather than a simple dynamical measure such as $(GM/r)^{1/2}$ ).\",\n \"To do this, we adopt $10^{20}$ cm$^{-2}$ as the lowest detectable HI surface density in order to make an approximate comparison to observations – the results are not strongly sensitive to this threshold, but disks from runs with low diffusion coefficients start to pick up some CR-pressure supported gas and show falling rotation curves.\",\n \"In particular, in H1-2, the run with a diffusion coefficient of 3 $\\\\times $ 10$^{27}$ cm$^2$ s$^{-1}$ is affected by this issue: the circular velocity peak at 160 km/s at about 1.5 kpc radius and falls quickly beyond that radius.\",\n \"As can be seen from its edge-on plot, the disk in H1-2 has a small radius, but its surrounding gas is relatively dense, which makes the HI detectable radius even larger than the disk radius.\",\n \"In that run, outside the disk, the gas is actually supported in part by the pressure of the CRs, which explains why the circular velocity curve drops so quickly after the peak.\",\n \"From this figure, we see that, at least for the two higher diffusion coefficient runs, the circular velocity curves are relatively flat, in reasonable agreement with observed systems.\",\n \"For example, in the run with the highest diffusion coefficient (H1), the circular velocity is relatively flat at about 140 km/s out to 6-7 kpc.\",\n \"Next, we examine the impact of CR pressure on the star formation rate.\",\n \"This is shown in Figure REF for all the runs of halo H1 with different CR physics.\",\n \"For all of our more realistic runs, with non-zero diffusion coefficient, the SFRs are quite similar: an early rapid rise follow by a nearly constant rate that falls slowly from 1 to 0.5 M$_\\\\odot $ /yr.\",\n \"The run without diffusion shows a rate which is a factor of two larger, due to the higher gas densities in its core, while the thermal-only feedback is lower by nearly an order of magnitude and shows considerably more variation with time – consistent with the bursty feedback seen in the gas distribution.\"\n ],\n [\n \"Varying Halo Mass with Fixed CR Physics\",\n \"In this section, we fix the CR diffusion coefficient to be $3\\\\times 10^{28}$ cm$^2$ s$^{-1}$ and study the impact on dwarf galaxies with different halo masses, varying the total mass from about $8 \\\\times 10^{10}$ M$_\\\\odot $ to $3 \\\\times 10^{11}$ M$_\\\\odot $ .\",\n \"We examine how CR feedback behaves in different scales and also use this suite of runs to compare with the observational scaling relations in Section REF .\",\n \"All of these runs produce dwarf galaxies with extended disk features; Figure REF shows edge-on images of these galaxies.\",\n \"The halo masses decrease from left to right: H1 to H5 as described in Table REF .\",\n \"There is a fairly clear trend of decreasing disk extent in the stellar distribution, with the larger halos having more extended disks.\",\n \"H2 is a slight outlier in this trend with a truncated disk, particularly in the gas and CR components.\",\n \"In addition, the circumgalactic medium is less dense (with a lower CR energy density) for the lower mass halos.\",\n \"Figure: Measured circular velocity of the neutral hydrogen in the disk for halos H1 to H5.\",\n \"As in Figure , we plot the circular velocity only out to a limiting HI surface density of 10 20 10^{20} cm -2 ^{-2}.We plot the rotational velocities of the gas for these five galaxies in Figure REF .\",\n \"Within our adopted HI detection limit, the curves of H3, H4, and H5, stay flat at about 100 km/s, just above 80 km/s, and just below 80 km/s, respectively.\",\n \"The velocities are consistent with their masses, and although we do not make a detailed comparison the curves are similar to observed rotation curves – in particular, they do not posses large bulges with centrally peaked rotation curves, as seen in many simulations without effective feedback.\",\n \"H1 has a higher rotational velocity (not surprising given its larger mass) and shows some mild features at 1 and 5 kpc.\",\n \"On the other hand, the curve of H2 looks quite different: it is highly peaked, and falls sharply at about 4 kpc (again because of partial CR pressure support).\",\n \"As can be seen from the edge-on plot of H2, this dwarf galaxy does have a dense core, larger than the other dwarfs.\",\n \"Figure REF shows the star formation histories of these same halos.\",\n \"There is a clear progression of increasing SFR with increasing mass.\",\n \"Despite some bumps and wiggles, the overall histories are remarkably flat, with each profile showing typically a factor of 2-3 variation over 12-13 Gyr.\",\n \"There is a general trend for slowly falling star formation rate with time, particularly evident in H3.\",\n \"This occurs because of the relatively tight self-regulation – feedback acts to decrease the gas density, slowing star formation.\",\n \"Figure: Star formation histories of our simulated halos H1 to H5 (all with fixed CR diffusion coefficient).\"\n ],\n [\n \"Comparison with Observed Scaling Relations\",\n \"Dwarf galaxies are less efficient in star formation than normal galaxies, and the baryon fraction of galaxies decreases as the total mass goes down [38].\",\n \"In this section, we compare our simulation results with a range of observational probes, including the baryonic Tully-Fisher relation, as well as observed gas-to-star ratios and inferred stellar mass-to-halo mass relations.\",\n \"We begin with the observed relation between the disk mass (the combined stellar and gas masses) and the flat part of the rotational velocity of the disk – the baryonic Tully-Fisher relation.\",\n \"Figure REF shows the simulation results from varying halo mass with fixed CR physics overlaid on the fitted line of the Baryonic Tully-Fisher relation from [38].\",\n \"They provide a fit for disk galaxies in the range of 20 to 350 km/s given by M$_{b} = x\\\\ \\\\mathrm {log}\\\\ {V}_{c} + {A}$ with $x = 4.0$ and $A =1.65$ .\",\n \"In Figure REF , the solid line represents their best fit, and the grey area represents one standard deviation of $M_b$ .\",\n \"We estimated the scatter in the observed $M_b$ as the average for spiral and gas disk galaxies in Table 2 of [38].\",\n \"For the simulations, we determined the baryonic mass as the sum of the stellar and gas masses, with the gas mass calculated as $1/0.74$ times the HI mass.\",\n \"Although the code evolves the HI density, it is underestimated in dense gas due to the lack of self-shielding.\",\n \"To correct for this, the HI density is estimated as follows: when the total H number density $n_{\\\\rm H} > 2\\\\times 10^{-2}$ cm$^{-3}$ , we assume that the neutral fraction is nearly unity and take $n_{\\\\rm HI} = n_{\\\\rm H}$ ; otherwise we assume the HI density is zero.\",\n \"This is approximately consistent with radiative transfer calculations [50] that take into account the metagalactic radiation background (a local radiation background would decrease the neutral gas fraction even lower, and so our bounds here are conservative).\",\n \"Both the stellar mass and HI mass are calculated within the same HI column density limit we used to determine the radial extent for the rotation curves ($10^{20}$ cm$^{-2}$ ) and extend above and below the plane by 2 kpc.\",\n \"The circular velocity is calculated as the average rotation velocity of the gas at a radius of 2.2 scale lengths of the stellar surface density.\",\n \"To determine the scale length, we fit the stellar surface density curve with a straight line from 2 kpc to 10 kpc.\",\n \"Our simulation results all lie within the grey area and scatter around the observed relation.\",\n \"Although we do not have a significantly large numerical sample to make a detailed comparison, these preliminary results appear to be in good agreement, indicating that the baryonic content of our galaxies is consistent with that observed.\",\n \"We note that we cannot include the purely thermal simulation (H1-0) on this plot as the systems has no systematic rotation.\",\n \"Figure: Comparison between the five dwarf simulations (H1-H5) run with our standard diffusion coefficient and the baryonic Tully-Fisher relation from observations.\",\n \"The solid line is the fit from .\",\n \"The red crosses are simulation data.\",\n \"The grey area shows the region of one sigma error in M b M_b.\",\n \"The dashed line is taken from and indicates the relation between the halo circular velocity and the cosmic baryon fraction of the halo.The baryonic Tully Fisher relation compares the galaxies' baryonic mass with a measure of its total mass.\",\n \"We would also like to compare the two main components of baryons: neutral hydrogen and stars.\",\n \"Figure REF shows the ratio of the HI to stellar components as a function of the stellar mass, all quantities which are directly observable.\",\n \"The observed relation found in [17] (their Eq.\",\n \"5) can be approximated as: $\\\\begin{split}\\\\frac{M_\\\\mathrm {HI}}{3.36 \\\\times 10^9 \\\\rm {M}_\\\\odot } = &\\\\left(\\\\frac{M_\\\\star }{3.3 \\\\times 10^{10} \\\\rm {M}_\\\\odot }\\\\right)^{0.19}\\\\\\\\ & \\\\times \\\\left[1+\\\\left(\\\\frac{M_\\\\star }{3.3 \\\\times 10^{10} \\\\rm {M}_\\\\odot }\\\\right)^{0.76}\\\\right]\\\\end{split}$ In Figure REF , the solid line shows this relation.\",\n \"We also estimate the observed scatter around this relation by taking a variation of +/- 0.5 log M$_\\\\mathrm {HI}$ /M$_\\\\star $ , based on a visual inspection of Figure 1 of [17]; this is shown as a grey area in our Figure REF .\",\n \"The ratio from the simulations is also plotted, again using the stellar and neutral hydrogen masses within our adopted HI surface density limit, and +/- 2 kpc in vertical extent.\",\n \"As can be seen from Figure REF , all of our simulation results are below the observed relation, indeed most lie below the estimated scatter in the observations (the grey band).\",\n \"This demonstrates that our simulation produces consistently lower HI fraction than observed.\",\n \"There is some indication that the simulated relation shows a rising gas fraction with decreasing stellar mass; however, the sample size is too small to be sure.\",\n \"This mismatch in the gas to star ratio, by approximately a factor of five, when combined with the agreement in the baryonic Tully-Fisher relation (which measures the total baryonic content), implies that our galaxies contain the correct amount of baryons, but are overly efficient in their conversion of gas to stars.\",\n \"One alternative way to examine just the stellar content of our systems is based on the dark matter abundance matching measurements in [17], which provides a relation between the stellar mass and halo mass under the assumption that there is a monotonic relation between halo mass and stellar mass.\",\n \"Indeed, if we compare our results to their inferred M$_*$ -M$_{\\\\rm halo}$ relation, we find that our $M_*$ values are systematically too large by factors of 2-5, consistent with the idea that the division between stellar and gas mass is incorrect in these simulations.\",\n \"Finally, we note that these relations also imply that the simulations disagree with the observed Kenicutt-Schmidt relation between surface gas density and star formation surface density.\",\n \"Figure: Comparison between this work and the relation of HI mass and stellar mass inferred from observations.\",\n \"The solid line is derived in .\",\n \"The grey area shows an estimated scatter of +/- 0.5 log around the observed relation in the vertical direction.\",\n \"The red crosses are the simulation results.\"\n ],\n [\n \"Dark Matter Profiles\",\n \"There are a number of observational indications that dwarf galaxies do not contain the cuspy dark-matter profiles predicted in dark-matter only simulations [11], [69], [3], [4], [55], [34].\",\n \"A number of simulations with strong feedback have found that it is possible for the feedback to decrease the dark matter density in the center of halos [41], [20], [48], [44].\",\n \"Therefore, we examine our simulation to see the impact of thermal vs. CR feedback on the core properties of the systems.\",\n \"Figure REF shows the dark matter density profile of our most massive halo in the simulations with different CR physics.\",\n \"Interestingly, all of the CR runs show very similar results, indicating that although they produce different distributions for the gas and stars, the presence of CR feedback leads to similar dark matter core properties.\",\n \"In all cases, the profile is quite steep, with an inner density profile between $r^{-1}$ and $r^{-2}$ (similar to dark-matter only runs).\",\n \"On the other hand, the run without CR diffusion shows a clear core in the dark matter with a core radius of several kpc.\",\n \"The cored dark matter profile in the no-CR case might be due to the large and rapid motions of the gas, which can disturb the dark matter distribution [48] and is consistent with the observed asymmetric gas distribution and the bursty nature of the feedback in the no-CR run.\",\n \"Figure: Dark matter profiles of simulations with different CR physics.We have performed a preliminary investigation into the impact of CRs on the formation and evolution of dwarf galaxies with rotation velocities in the 70-140 km/s range using cosmological simulations.\",\n \"This complements similar work [53] which looked at a higher mass galaxy (with a rotation velocity above 200 km/s).\",\n \"As in that work, we found that the inclusion of CR feedback had a strong impact on the gas and stellar properties.\",\n \"In both cases, CR feedback led to robust outflows and rotationally supported disks with relatively flat rotation curves, but only if CR diffusion was included.\",\n \"A comparison to observations showed good agreement with the baryonic Tully-Fisher relation, indicating that the total disk mass was reasonable.\",\n \"This mass, converted into a fraction of the total halo mass, is well below the cosmic value, indicating that feedback was important in generating mass-loaded winds.\",\n \"On the other hand, a comparison of the gas content of the simulated galaxies against observations showed that too much gas had been converted to stars.\",\n \"In addition, the gas depletion times of our galaxies (computed as $M_{\\\\rm gas}/\\\\dot{M}_{\\\\rm SFR}$ ), are a few Gyr, well below observed values for low-mass isolated dwarfs [29].\",\n \"Our star formation and feedback model is approximate, with a number of free parameters required to describe physics that is not fully modeled.\",\n \"In section REF , we explored the impact of some of these for halo H1.\",\n \"Remarkably, we find that the CR diffusion rate has a relatively small impact on the stellar mass produced (see Table REF ), or the gas mass.\",\n \"The exception was the run with $\\\\kappa = 0$ , which produced a substantially higher stellar mass and did not produce a disk (this combination does not generate winds).\",\n \"We also carried out an additional run of halo H1 with the standard diffusion rate (and other parameters), but a reduced CR energy efficiency (10% instead of 30%) and find that relative to the fiducial run, it produces 50% more stars.\",\n \"The resulting disk rotates marginally faster such that the system stays close to the observed baryonic Tully-Fisher relation.\",\n \"However, the HI to gas mass ratio is even lower the fiducial run, resulting in an an even stronger disagreement with the observed relation.\",\n \"In principle, a higher CR efficiency might bring this into agreement with observations, but that would disagree with local observational and theoretical work on CR acceleration efficiencies.\",\n \"Another parameter which can be varied is the star formation efficiency per free-fall time, which we took to be 1%.\",\n \"We carried out another simulation much like H1 but with a star formation efficiency four times lower (with $\\\\kappa = 3 \\\\times 10^{28}$ cm$^2$ s$^{-1}$ ; this run was not included in figures and tables above).\",\n \"This had almost no effect on the resulting stellar mass or gas mass, consistent with the idea that the disk is in a state of self-regulation – indeed, we verified visually that the lower efficiency resulted in a higher density gas disk and hence essentially the same net star formation rate at $z=0$ (since the star formation rate increases as $\\\\rho ^{1.5}$ ).\",\n \"There are, of course, other parameters to vary, such as the IMF, the fraction of energy in SN, as well as other uncertainties in the subgrid model.\",\n \"In addition, there is some indication from H2 and the high-mass halo explored in [53] that an additional form of feedback is required in high-density regions – possibly radiative heating or pressure.\",\n \"Finally, we briefly discuss our results in comparison to [68], who recently carried out simulations of dwarf galaxies with cosmic rays.\",\n \"The implementation differs somewhat in that they adopted a model in which CRs stream along magnetic field lines and heat the gas (rather than isotropically diffuse, as in our model).\",\n \"Although they do not simulate cosmological volumes, their halo mass which most closely compares to ours produces approximately similar results: robust cosmic-ray driven outflows with a modest mass-loading factor.\",\n \"Although we are unable to compute a mass-loading factor, the large stellar mass to halo mass we find implies a relatively low mass-loading factor.\"\n ],\n [\n \"Feedback efficiency of the purely thermal run\",\n \"Finally, we turn to a discussion of the thermal-only feedback run (H1-0).\",\n \"Interestingly, the feedback in this case was actually more effective than the CR+thermal feedback, in contrast to the result for the higher mass halo with the same code and CR parameters discussed in [53].\",\n \"One possibility is that the purely thermal feedback is more efficient than in more massive galaxies because the shallower potential well results in low gas densities, lowering the radiative cooling rate and permitting blast waves from SN feedback to accelerate the gas.\",\n \"We explore the relative importance of cooling and heating in Figure REF , which shows a two-dimensional distribution of radiative cooling times and a measure of the possible supernova heating rate for all gas within 30 kpc of the center of this halo as $z=0$ .\",\n \"We compute the SN heating rate in a way similar to that in [60], where we estimate the timescale required for our thermal energy injection mechanism to heat a cell to $T_6 = 10^6 K$ (the precise temperature selected does not affect our conclusions): $t_{\\\\rm heat} = \\\\frac{3}{2} \\\\frac{M_{\\\\rm cell} k T_6}{\\\\mu m_H} \\\\frac{t_f}{e_{\\\\rm SN} M_* c^2}$ where $M_{\\\\rm cell}$ is the gas mass cell in the cell, $\\\\mu \\\\approx 1$ , $M_* = 10^5$ M$_\\\\odot $ , our typical stellar particle mass, $t_f = 10$ Myr, the timescale over which a star particle returns its supernova energy to the ISM in our model, and $e_{\\\\rm SN} = 3 \\\\times 10^{-6}$ , as defined in the methods section.\",\n \"We stress that not all cells are being heated at this rate, but it is the rate that a single star particle would typically heat its local gas and so is a good measure of the timescale over which feedback heating acts.\",\n \"From this figure, we see that nearly all the gas in the simulation lies above the black line indicating equality.\",\n \"This demonstrates that the gas densities are sufficiently low that pure thermal feedback is efficient in heating the gas and driving outflows – we focus on density because the cooling rate is proportional to the density squared.\",\n \"A close examination of this figure indicates that a substantial amount of gas lies close to the line of equality (in fact, this gas is in the disk), and so we expect that the larger potential depths of higher mass halos will push gas beyond the line into a regime where purely thermal supernovae feedback cannot heat the gas.\",\n \"This is consistent with the finding that our dwarf galaxy simulations do drive outflows while higher mass (e.g.\",\n \"Milky Way mass) systems do not.\",\n \"Note that this statement depends on details of the simulations (e.g.\",\n \"see definition of $t_{\\\\rm heat}$ ) and so may be a reflection of numerical models rather than physical quantities.\",\n \"This demonstrates that purely thermal feedback is sufficient to drive feedback but it does not answer the question why adding cosmic rays decreases the bursty outflows, as seen in Figure REF .\",\n \"The answer appears to be that cosmic rays actually increase the pressure in the diffuse circumgalactic gas surrounding galaxies.\",\n \"The purely thermal run has a (thermal only) CGM pressure at a radius of 10 kpc of approximately $10^{-14}$ erg cm$^{-3}$ , while even the CR run with the highest diffusion coefficient (H1), has a (mostly CR) pressure of nearly $10^{-13}$ erg cm$^{-3}$ .\",\n \"This larger pressure resists the impulsive outflows driven by bursts of supernovae in the disk (although it does not appear to eliminate the slower gas outflows driven by the CR pressure gradient).\",\n \"Finally, we note a consequence of the effective thermal only feedback, which is that it resulted in bursty star formation at a low-level; and was so effective that the resulting stellar system was “hot\\\" – that is, it was supported by velocity dispersion rather than rotation (an issue that has been raised in the context of other simulations – see the discussion in [72].\"\n ],\n [\n \"Summary\",\n \"We simulate a series of dwarf galaxies with zoom-in cosmological simulations, and study the influence of CR feedback.\",\n \"First we compare the outcome by varying the CR physics in different runs of the same galaxy halo.\",\n \"The different CR physics include feedback without CR, CR feedback with zero diffusion, CR feedback with $\\\\kappa _\\\\mathrm {CR}$ equal to $3 \\\\times 10^{27}$ , $1 \\\\times 10^{28}$ , and $3 \\\\times 10^{28}$ cm$^2$ s$^{-1}$ .\",\n \"Then we fixed the CR diffusion at $\\\\kappa _\\\\mathrm {CR} = 3 \\\\times 10^{28}$ cm$^2$ s$^{-1}$ and simulated five dwarf galaxies with different masses, ranging from 8 to 30 $\\\\times 10^{10}$ M$_\\\\odot $ .\",\n \"We summarize the results of this work as follows: Adding CRs and some realistic level of diffusion consistently produced thin, extended disk galaxies with nearly flat rotation curves, in contrast to a case with purely thermal feedback, which produced a hot stellar systems.\",\n \"The star formation rate in the CR systems was relatively constant, compared to the lower (and burstier) SFR in the purely thermal feedback model.\",\n \"Simulations of our five dwarf galaxies with different masses but the same CR model matched well with the observed baryonic Tully-Fisher relation.\",\n \"However, the five galaxies' neutral hydrogen fractions were always lower than what is observed, by a factor of about five and the stellar content was larger than observed (but the total baryonic content matched observations).\",\n \"Simulations with CR feedback produce cuspy dark matter profiles, while our purely thermal feedback case resulted in a cored dark matter profile.\",\n \"These results indicate that the impact of cosmic rays in dwarf galaxies is potentially quite important, a conclusion which is in general agreement with previous work [31].\",\n \"In detail, the results are somewhat dependent on the physical model, but broadly speaking CRs with diffusion produce dwarf galaxies which are rotationally supported and have relatively flat rotation curves.\",\n \"This result appears to be quite robust (and occurs even though 70% of the feedback energy is in the form of thermal energy in these models) and is probably due to the smooth pressure provided by CRs (along with the outflows they drive).\",\n \"The baryonic content of the dwarf systems with CRs is in broad agreement with observations.\",\n \"On the other hand, there are two areas of disagreement with observations in these models: (a) the detailed census of baryons in each system (stars vs. gas) is much too heavily weighted to the stellar side, and (b) the presence of dark matter cores may be a problem (although CR pressure may systematically support the gas at small radius and so bias the observed measures of dark matter on small scales).\",\n \"Clearly, more work is required to explore and refine these results.\",\n \"For example, it would be useful to explore more realistic models including magnetic fields, anisotropic CR diffusion, and the impact of field-CR streaming instabilities.\"\n ],\n [\n \"Acknowledgments\",\n \"We thank the anonymous referee for a useful report which improved the clarity of the paper.\",\n \"We would like to thank Cameron Hummels, Mordecai-Mark Mac Low and Mary Putman for useful discussion related to this work.\",\n \"We acknowledge financial support from NSF grants AST-1312888, and NASA grants NNX12AH41G and NNX15AB20G, as well as computational resources from NSF XSEDE, and Columbia University's Yeti cluster.\",\n \"Computations described in this work were performed using the publicly-available Enzo code (http://enzo-project.org), which is the product of a collaborative effort of many independent scientists from numerous institutions around the world.\",\n \"Their commitment to open science has helped make this work possible.\"\n ]\n]"},"id":{"kind":"string","value":"1605.06115"}}},{"rowIdx":1866673,"cells":{"text":{"kind":"list like","value":[["General formulation of coupled radiative and conductive heat transfer\n between compact bodies"],["Abstract We present a general framework for studying strongly coupled radiative and conductive heat transfer between arbitrarily shaped bodies separated by sub-wavelength distances.","Our formulation is based on a macroscopic approach that couples our recent fluctuating volume--current (FVC) method of near-field heat transfer to the more well known Fourier conduction transport equation.","We apply our technique to consider heat exchange between aluminum-zinc oxide nanorods and show that the presence of bulk plasmon resonances can result in extremely large radiative heat transfer rates (roughly twenty times larger than observed in planar geometries), whose interplay with conductive transport leads to nonlinear temperature profiles along the nanorods."],["3 General formulation of coupled radiative and conductive heat transfer between compact bodies Weiliang Jin Department of Electrical Engineering, Princeton University, Princeton, NJ 08544, USA Riccardo Messina Laboratoire Charles Coulomb, Université de Montpellier and CNRS, Montpellier, France Alejandro W. Rodriguez Department of Electrical Engineering, Princeton University, Princeton, NJ 08544, USA We present a general framework for studying strongly coupled radiative and conductive heat transfer between arbitrarily shaped bodies separated by sub-wavelength distances.","Our formulation is based on a macroscopic approach that couples our recent fluctuating volume–current (FVC) method of near-field heat transfer to the more well known Fourier conduction transport equation.","We apply our technique to consider heat exchange between aluminum-zinc oxide nanorods and show that the presence of bulk plasmon resonances can result in extremely large radiative heat transfer rates (roughly twenty times larger than observed in planar geometries), whose interplay with conductive transport leads to nonlinear temperature profiles along the nanorods.","Radiative heat transfer (RHT) between objects held at different temperatures can be many orders of magnitude larger in the near field (short separations $d \\ll $ thermal wavelength $\\lambda _T =\\hbar c/k_\\mathrm {B} T$ ) than for far-away objects [1], [2], [3], [4], [5].","Recently, we showed that that the interplay of near-field RHT and conduction in planar geometries can dramatically modify the temperature and thermal exchange rate at sub-micron separations [6].","Such strongly-coupled conduction–radiation (CR) phenomena are bound to play a larger role in situations involving structured materials, where RHT can be further enhanced [7], [8], [9], [10], [11], [12] and modified [13], [14], [15], [16], and in on-going experiments exploring nanometer scale gaps, where the boundary between conductive (phonon- and electron-mediated) and radiative transport begins to blurr [17], [18].","We present a general CR framework that captures the interplay of near-field RHT and thermal conduction along with the existence of large temperature gradients in arbitrary geometries.","We show that under certain conditions, i.e.","materials and structures with separations and geometric lengthscales in the nanometer range, RHT can approach and even exceed conduction, significantly changing the stationary temperature distribution of heated objects.","Our approach is based on a generalization of our recent fluctuating volume-current (FVC) formulation of electromagnetic (EM) fluctuations, which when coupled to the more standard Fourier heat equation describing conductive transport at macroscopic scales, allows studies of CR between arbitrary shapes, thereby generalizing our prior work with slabs [6].","As a proof of concept, we consider an example geometry involving aluminum-zinc oxide (AZO) nanorods separated by vacuum gaps, which exhibits more than an order of magnitude enhancement in RHT compared to planar slabs, and hence leads to even larger temperature gradients.","We find that while RHT between thin slabs is primarily mediated by surface modes, resulting in linear temperature gradients, the presence of bulk nanorod resonances leads to highly distance-dependent nonlinear temperature profiles.","Coupled radiative and conductive diffusion processes in nanostructures are becoming increasingly important [19], [18], with recent works primarily focusing on the interplay between thermal diffusion and external optical illumination such as laser-heating of plasmonic structures [20], [21], [22], [23], [24].","On the other hand, while it is known that conduction has a strong influence on RHT experiments [25], [26], the converse has thus far been largely unexplored because RHT is typically too small to result in appreciable temperature gradients [27], [28], [29].","However, our recent work [6] suggests that such an interplay can be significant at tens of nanometer separations and in fact may already have been present (though overlooked) in recent experiments involving planar systems [30], [31], [32], [33].","Moreover, since planar structures are known to exhibit highly suboptimal RHT rates [7], we expect even stronger interplays in more complex geometries, such as metasurfaces [34], hyperbolic metamaterials [35], [11], or lattices of metallic antennas [8], [7].","Figure: (a) Schematic illustration of two square lattices of nanorods(labelled aa and bb) of thickess tt, period Λ\\Lambda ,cross-sectional area l×ll\\times l, and separation dd, whosetemperature distribution and energy exchange is mediated by bothconductive ∇·[κ(𝐱)T(𝐱)]\\nabla \\cdot [\\kappa (\\mathbf {x})T(\\mathbf {x})] andradiative H(𝐱,𝐱 ' )H(\\mathbf {x},\\mathbf {x}^{\\prime }) heat transfer.","(b) Totalradiative heat transfer spectrum Φ(ω)\\Phi (\\omega ) between two AZOnanorods (solid lines) of thickness t=500t=500 nm and cross-sectionalarea A=l 2 A=l^2, separated by d=20d=20 nm and held at temperaturesT a(b) =800(300)T_{a(b)}=800(300) K. The spectrum is shown for differentcross-sections l={10,20}l=\\lbrace 10,20\\rbrace nm (blue and red lines) and in thelimit l=∞l = \\infty , corresponding to two planar slabs.","(c) Spatialradiative heat flux in nanorod aa for the case l=20l=20 nm,corresponding to the (i) first, (ii) second, and (iii) SPP plasmonresonances, respectively, annotated in (b).Formulation.— In what follows, we describe a general formulation of coupled CR applicable to arbitrary geometries.","Consider a situation involving two bodies (the same framework can be extended to multiple bodies), labelled $a$ and $b$ , subject to arbitrary temperature profiles and exchanging heat among one other, shown schematically in Fig.","REF (a).","Neglecting convection and considering bodies with lengthscales larger or of the order of their phonon mean-free path, in which case Fourier conduction is valid, the stationary temperature distribution satisfies: $\\nabla \\cdot [\\kappa (\\mathbf {x})\\nabla T(\\mathbf {x})]+\\int \\mathrm {d}^3\\mathbf {x}^{\\prime }\\,H(\\mathbf {x},\\mathbf {x}^{\\prime })=Q(\\mathbf {x})$ where $\\kappa (\\mathbf {x})$ and $Q(\\mathbf {x})$ describe the bulk Fourier conductivity and presence of some external heat source at $\\mathbf {x}$ , respectively, and $H(\\mathbf {x},\\mathbf {x}^{\\prime })$ denotes the radiative power per unit volume from $\\mathbf {x}^{\\prime }$ to $\\mathbf {x}$ .","Our ability to compute $H(\\mathbf {x},\\mathbf {x}^{\\prime })$ in full generality hinges on an extension of a recently introduced FVC method that exploits powerful EM scattering techniques [36] to enable fast calculations of RHT under arbitrary geometries and temperature distributions.","The starting point of this method is the volume-integral equation (VIE) formulation of EM, in which the scattering unknowns are 6-component polarization currents $\\xi $ in the interior of the bodies coupled via the homogeneous $6\\times 6$ Green's function $\\Gamma $ of the intervening medium [36].","Given two objects described by a susceptibility tensor $\\chi (\\mathbf {x})$ and a Galerkin decomposition of the induced currents $\\xi =\\sum _i x_i b_i$ , with $\\lbrace b_i\\rbrace $ denoting localized basis functions throughout the objects ($i$ is the global index for all bodies), the scattering of an incident field due to some fluctuating current-source $\\sigma =\\sum _is_i b_i$ can be determined via solution of a VIE equation, $x+s=Ws$ , in terms of the unknown and known expansion coefficients $\\lbrace x_i\\rbrace $ and $\\lbrace s_i\\rbrace $ , respectively, where $W^{-1}_{i,j}=\\langle b_i,(I+i\\omega \\chi G) b_j\\rangle $ and $G_{i,j}=\\langle b_{i},\\Gamma \\star b_{j}\\rangle $ are known as VIE and Green matrices [36].","Previously, we exploited this formalism to propose an efficient method for computing the total heat transfer between any two compact bodies [36], based on a simple voxel basis expansion (uniform discretization).","The solution of (REF ) requires an extension of the FVC method to include the spatially resolved heat transfer between any two voxels, which we describe below.","Consider a fluctuating current-source $\\sigma _{\\alpha }=s_{\\alpha }b_{\\alpha }$ at $\\mathbf {x}_a=b_{\\alpha }$ in a body $a$ .","Such a “dipole” source induces polarization–currents $\\xi _{\\beta }=x_{\\beta }b_{\\beta }$ and EM fields $\\phi _{\\beta }$ throughout space in body $b$ (and elsewhere), such that the heat flux at $\\mathbf {x}_b=b_{\\beta }$ is given (by Poynting's theorem) by: $\\Phi (\\omega ;\\mathbf {x}_a\\rightarrow \\mathbf {x}_b)=\\frac{1}{2}\\langle \\operatorname{Re}\\left( \\xi _{\\beta }^{*}\\phi _{\\beta } \\right)\\rangle $ where “$\\langle \\ldots \\rangle $ ” denotes a thermodynamic ensemble average.","Expressing the polarization–currents and fields in the localized basis $\\lbrace b_{\\alpha }\\rbrace $ , and exploiting the volume equivalence principle to express the field as a convolution of the incident and induced currents with the vacuum Green's function (GF), $\\phi =\\Gamma \\star (\\xi +\\sigma )$ , one finds that (REF ) can be expressed in a compact, algebraic form involving VIE matrices: $\\Phi (\\omega ;\\mathbf {x}_a\\rightarrow \\mathbf {x}_b)&=\\frac{1}{2}\\langle \\operatorname{Re}\\left\\lbrace x_{\\beta }^{*}[G(x+s^\\alpha )]_{\\beta }\\right\\rbrace \\rangle \\nonumber \\\\&=\\frac{1}{2} \\langle \\operatorname{Re}\\left\\lbrace (x+s^\\alpha )_{\\beta }^{*}[G(x+s^\\alpha )]_{\\beta }\\right\\rbrace \\rangle \\nonumber \\\\&=\\frac{1}{2}\\langle \\operatorname{Re}\\left[ (Ws^\\alpha )_{\\beta }^{*}(GWs^\\alpha )_{\\beta }\\right]\\rangle \\nonumber \\\\&=\\frac{1}{2} \\operatorname{Re}\\left[ D_{\\alpha ,\\alpha }W^{\\dagger }_{\\alpha ,\\beta }(GW)_{\\beta ,\\alpha }\\right]$ where $s^{\\alpha }$ is a vector that is zero everywhere except at the $\\alpha $ th element, denoted by $s_{\\alpha }$ , and ${D}_{\\alpha ,\\beta }=\\langle s_{\\alpha }^{*}s_{\\beta }\\rangle =\\int \\int d^3\\mathbf {x}\\,d^3\\mathbf {y}\\, b_{\\alpha }^\\ast (\\mathbf {x}) \\langle \\sigma (\\mathbf {x})\\sigma ^\\ast (\\mathbf {y}) \\rangle b_{\\beta }(\\mathbf {y})$ is a real, diagonal matrix encoding the thermodynamic and dissipative properties of each object [36] and described by the well-known fluctuation–dissipation theorem, $\\langle \\sigma _i(\\mathbf {x},\\omega ) \\sigma _j^\\ast (\\mathbf {y},\\omega ) \\rangle =\\frac{4}{\\pi } \\omega \\operatorname{Im}\\epsilon (\\mathbf {x},\\omega ) \\Theta (T_{\\mathbf {x}})\\delta (\\mathbf {x}-\\mathbf {y})\\delta _{ij}$ , where $\\Theta (T)=\\hbar \\omega /[\\mathrm {exp}(\\hbar \\omega /k_bT)-1]$ is the Planck distribution.","It follows then that the heat flux emitted or absorbed at a given position $\\mathbf {x}_a$ , the main quantity entering (REF ) through $\\int \\mathrm {d}^3\\mathbf {x}^{\\prime }\\,H(\\mathbf {x},\\mathbf {x}^{\\prime })=\\int \\mathrm {d}\\omega \\,\\Phi (\\omega ;\\mathbf {x})$ , is given by: $\\Phi (\\omega ;\\mathbf {x}_a)&=\\int _{V_{b}} \\mathrm {d}^3\\mathbf {x}_b\\left[ \\Phi (\\omega ;\\mathbf {x}_b\\rightarrow \\mathbf {x}_a)-\\Phi (\\omega ;\\mathbf {x}_a\\rightarrow \\mathbf {x}_b)\\right]\\nonumber \\\\&=\\frac{1}{2}\\text{Tr }_{\\beta | b_\\beta \\in V_b} \\operatorname{Re}\\left[D_{\\beta ,\\beta }W^{\\dagger }_{\\beta ,\\alpha }(GW)_{\\alpha ,\\beta }-(\\alpha \\leftrightarrow \\beta )\\right]\\nonumber \\\\&=\\frac{1}{2}\\operatorname{Re}\\left[\\underbrace{GWD^bW^{\\dagger }}_{\\Phi _\\mathrm {a}}-\\underbrace{DW^{\\dagger }P^bGW}_{\\Phi _\\mathrm {e}}\\right]_{\\alpha ,\\alpha }$ Here, $P^{a(b)}$ denotes the projection operator that selects only basis functions in $a(b)$ , such that $D^b=P^bDP^b$ is a diagonal matrix involving only fluctuations in object $b$ .","Furthermore, the first (second) term in (REF ) describe the absorbed (emitted) power in $\\mathbf {x}_a$ , henceforth denoted via the subscript “a(e)”.","Figure: (a) Temperature profile along the zz coordinate of a nanorod(solid lines) when it is heated from one side to a temperature of800 K, and is separated from an identical, constant- anduniform-temperature nanrod held at T=300T=300 K on the other side, by agap size d=20d=20 nm.","The nanorods have cross-sectional widthl=10l=10 nm and thicknesses t=500t=500 nm, and are made up of AZO withresults shown for multiple values of the doping concentration{2,6,11} wt %\\lbrace 2,6,11\\rbrace \\mathrm {wt}\\% (blue, red, and black lines).","Also shownare the temperature profiles of slabs (dashed lines) of the samethickness (corresponding to the limit l→∞l \\rightarrow \\infty ).","(Inset:)Temperature distribution throughout the nanorod in the case of11 wt %11\\mathrm {wt}\\%.","(b) Temperature profiles of nanorods of widthl=20l=20 nm under various separations d={5,10,20,30}d=\\lbrace 5,10,20,30\\rbrace nm (black,blue, red, and green lines).","(c.inset:) The ratio of total radiativeheat flux for nanorods of width l=20l=20 nm to that of the slabs as afunction of dd, in the presence (red dots) or absence (black dots)of temperature gradients induced by conduction and radiationinterplay, with the flux value shown in (c), for nanorods (red) andslabs (blue), also in the presence (solid lines) or absence (dashedlines) of temperature gradients induced by the interplay ofconduction and radiation.Equation REF is a generalization of our previous expression for the total heat transfer between two arbitrary inhomogeneous objects [36] in that it includes both the spatially resolved absorbed and emitted power throughout the entire geometry.","In Ref.","polimeridis2015fluctuating, we showed that the low-rank nature of the GF operator enables truncated, randomized SVD factorizations and therefore efficient evaluations of the corresponding matrix operations.","We find, however, that in this case, the inclusion of the absorption term does not permit such a factorization, except in special circumstances.","In particular, writing down the two terms separately by expanding into the subspace spanned by each object, we find: $\\Phi _\\mathrm {a}(\\omega ;\\mathbf {x}_a)&=\\frac{1}{2}\\operatorname{Re}\\left[G^{ab}W^{bb}D^{bb}W^{ab\\dagger }+G^{aa}W^{ab}D^{bb}W^{ab\\dagger }\\right] \\\\\\Phi _\\mathrm {e}(\\omega ,\\mathbf {x}_a) &= -\\frac{1}{2}\\left[ \\operatorname{Re}(D^{aa}W^{ba\\dagger } G^{ba} W^{aa})\\right.","\\nonumber \\\\&\\hspace{50.58878pt}\\left.+ D^{aa} W^{ba\\dagger } \\operatorname{sym}(G^{bb}) W^{ba} \\right]_{\\alpha ,\\alpha }$ with $X^{ij} = P^i X P^j$ denoting the sub-block of matrix $X$ connecting basis functions in object $i$ to object $j$ , and $\\operatorname{sym}X =\\frac{1}{2}(X+X^\\dagger )$ denoting the symmetric part of $X$ .","Equation , describing emission, can be evaluated efficiently because the matrices $G^{ba}$ and $\\operatorname{sym}{G^{bb}}$ are both low rank ($\\ell \\ll N$ ) [36], in which case they can be SVD factorized to allow fast matrix multiplications.","It follows that the total heat transfer, i.e.","the trace of (), can also be computed efficiently.","Unfortunately, the second term of (REF ) involves both the symmetric and anti-symmetric parts of $G^{aa}$ , the latter of which is full rank.","More conveniently, detailed balance dictates that $\\Phi (\\omega ;\\mathbf {x}_b\\rightarrow \\mathbf {x}_a)=\\Phi (\\omega ;\\mathbf {x}_a\\rightarrow \\mathbf {x}_b)$ whenever $T(\\mathbf {x}_a) = T(\\mathbf {x}_b)$ , which implies that $\\operatorname{Re}\\left[ M_{\\beta ,\\beta }W^{\\dagger }_{\\beta ,\\alpha }(GW)_{\\alpha ,\\beta }\\right]=\\operatorname{Re}\\left[M_{\\alpha ,\\alpha }W^{\\dagger }_{\\alpha ,\\beta }(GW)_{\\beta ,\\alpha }\\right]$ , where $M_{\\alpha ,\\alpha }=\\operatorname{Im}\\varepsilon (\\mathbf {x}_{\\alpha },\\omega )$ is a real, diagonal matrix encoding the dissipative properties of the bodies, leading to the following modified expression for the absorption rate: $\\Phi _\\mathrm {a}(\\omega ;\\mathbf {x}_a)&=\\frac{1}{2}\\left[\\operatorname{Re}(M^{aa}W^{ba\\dagger }K^{bb}G^{ba}W^{aa})\\right.\\nonumber \\\\&\\hspace{50.58878pt}\\left.+M^{aa}W^{ba\\dagger }\\operatorname{sym}(K^{bb}G^{bb})W^{ba}\\right]_{\\alpha ,\\alpha }$ where the real and diagonal matrix $K_{\\alpha ,\\alpha }=D_{\\alpha ,\\alpha }/M_{\\alpha ,\\alpha }$ is only relevant to the Plank function $\\Theta (T(\\mathbf {x}_{\\alpha }),\\omega )$ .","Noticeably, the symmetrized operator in the second term is full rank except whenever the temperature of object $b$ is close to uniform, in which case $\\operatorname{sym}(K^{bb} G^{bb}) \\approx K^{bb} \\operatorname{sym}G^{bb}$ .","While solution of (REF ) is feasible, it remains an open problem to find a formulation that allows fast evaluations of the spatially resolved absorbed power under arbitrary temperature distributions.","Given (REF ), one can solve the coupled CR equation in any number of ways [37].","Here, we exploit a fixed-point iteration procedure based on repeated and independent evaluations of (REF ) and (REF ), converging once both quantities approach a set of self-consistent steady-state values.","Equation REF is solved via a commercial, finite-element heat solver whereas (REF ) is solved through a free, in-house implementation of our FVC method [36].","While the above formulation is general, below we explore the computationally convenient situation in which object $b$ is kept at a constant, uniform temperature by means of a carefully chosen thermal reservoir, such that the absorbed power in object $a$ can be computed efficiently via (REF ).","Furthermore, absorption can be altogether ignored whenever one of the bodies is heated to a much larger temperature than the other (as is the case below).","The power emitted by $a$ (the heated object), obtained via (), turns out to be much more convenient to compute, since the time-consuming part of the scattering calculation can be precomputed independently from the temperature distribution and stored for repeated and subsequent evaluations of (REF ) under different temperature profiles.","Results.— As a proof of principle and to gain insights into coupled CR effects in non-planar objects, we now apply the above method to a simple geometry consisting of two metallic nanorods of cross-sectional widths $l$ and thickness $t$ ; in practice, both for easy of fabrication and to obtain even larger RHT [25], such a structure could be realized as a lattice or grating, shown schematically in Fig.","REF (a).","However, for computational convenience and conceptual simplicity, we restrict our analysis to the regime of large grating periods, in which case it suffices to consider only the transfer between nearby objects.","The strongest CR effects generally will arise in materials that exhibit large RHT, e.g.","supporting surface–plasmon polaritons (SPP) in the case of planar objects, and low thermal conductivities, including silica, sapphire, and AZO, whose typical thermal conductivities $\\sim 1$ W/m$\\cdot $ K. In the following, we take AZO as an illustrative example [38], [39].","To begin with, we show that even in the absence of CR interplay, the RHT spectrum and spatial RHT distribution inside the nanorods differ significantly from those of AZO slabs of the same thickness.","Figure REF (b) shows the RHT spectrum $\\Phi (\\omega )$ per unit area $A=l^2$ between two AZO nanorods (with doping concentration $11\\mathrm {wt}\\%$ [39]) of length $t=500$ nm and varying widths $l=\\lbrace 10,20,\\infty \\rbrace $ nm (blue solid, red solid, and black dashed lines), held at temperatures $T_{a(b)}=800(300)$ K and vacuum gap $d=20$ nm.","The limit $l\\rightarrow \\infty $ corresponds to the slab-slab geometry already explored [6], in which case the $\\Phi (\\omega )$ exhibits a single peak occuring at the SPP frequency $\\approx 3\\times 10^{14}$ rad/s.","The finite nature of the nanorods results in additional peaks at lower frequencies, corresponding to bulk/geometric plasmon resonances (red and blue solid lines) that provide additional channels of heat exchange, albeit at the expense of weaker SPP peaks, leading to a roughly 20-fold enhancement in RHT compared to slabs.","More importantly and well known, such structured antennas allow tuning and creation of bulk plasmon resonances in the near- and far-infrared spectra (much lower than many planar materials) that can more effectively transfer thermal radiation.","The contour plots in Fig.","REF (i–iii) reveal the spatial RHT distribution $\\Phi (\\omega ,\\mathbf {x})$ (in arbitrary units) at three separate frequencies $\\omega =\\lbrace 0.4,0.8,2.3\\rbrace \\times 10^{14}$ rad/s, corresponding to the first, second, and SPP resonances, respectively.","As expected, the highest-frequency resonance is primarily confined to the corners of the nanorod surface (becoming the well-known SPP resonance in the limit $l \\rightarrow \\infty $ ), with the fundamental and intermediate resonances have flux contributions stemming primarily form the bulk.","As we now show, such an enhancement results not only results in larger temperature gradients but also changes the resulting qualitative temperature distribution.","Figure REF (a) shows the temperature profile along the $z$ direction for the nanorod geometry of Fig.","REF (a), with width $l=10~$ nm and gap size $d=20$ nm, obtained via solution of (REF ).","For the purpose of generality, we show results under various doping concentrations $\\lbrace 2,6,11\\rbrace \\mathrm {wt}\\%$ (green, red, and black solid lines), corresponding to different SPP frequencies and bandwidths [39].","In particular, we consider a situation in which the boundary I of nanorod $a$ is kept at $T_\\text{I}=800$ K while the entire nanorod $b$ is held at $T_b=300$ K (through contact with a room-temperature reservoir), and assume an AZO thermal conductivity of $\\kappa =1$ W/m$\\cdot $ K [38].","The temperature along the $x$ –$y$ cross section is nearly uniform (due to the faster heat diffusion rate along the smaller dimension) and therefore only shown in the case of $11\\mathrm {wt}\\%$ (inset).","In all scenarios, the temperature gradient is significantly larger for nanorods (solid lines) than for slabs ($t \\rightarrow \\infty $ , dashed lines), becoming an order of magnitude larger in the case of $6\\mathrm {wt}\\%$ due to its larger SPP frequency compared to the peak Planck wavelength near 800 K. Furthermore, while slabs exhibit linear temperature profiles (RHT is dominated by surface emission [1]), the bulky and de-localized nature of emission in the case of nanorods results in nonlinear temperature distributions.","Figure REF (b) shows the temperature profile at various separations $d=\\lbrace 5,10,20,30\\rbrace $ nm (black, blue, red, and green lines) for nanorods of width $l=20$ nm and $11\\mathrm {wt}\\%$ , illustrating the sensitive relationship between the degree of CR interplay and gap size.","Notably, while the RHT and therefore temperature gradients increase as $d$ decreases, the profile becomes increasingly linear as the geometry approaches the slab–slab configuration.","The transition from bulk- to surface-dominated RHT and the increasing impact of the latter on conduction and vice versa is also evident from Fig.","REF (c).","The figure shows the radiative flux rate $H\\times d^2$ as a function of $d$ for slabs (black lines) of thickness $t=500$ nm and nanorods (red lines) of equal thickness and width $l=20$ nm, either including (solid lines) or excluding (dashed lines) CR interplay (with the latter involving uniform temperatures).","While the RHT between bodies of uniform temperatures is shown to scales as $1/d^2$ (dashed lines), the temperature gradients induced by CR interplay in the case of nanorods begins to change the expected powerlaw behavior at $d\\approx 15$ nm; the same occurs for slabs but at much shorter $d\\lesssim 5$ nm.","These differences are further quantified on the inset of the figure, which shows the ratio of the RHT rate between the two objects as a function of $d$ .","While the ratio remains almost a constant for uniform-temperature objects (black dots), it decreases visibly when considering CR interplay (red dots).","As shown in Ref.","riccardo2016strongly, in the limit $d\\rightarrow 0$ , RHT will asymptote to a constant (not shown) rather than a diverge.","Concluding remarks.— As experiments continue to push toward larger RHT by going to smaller vacuum gaps or through nanostructuring, accurate descriptions of CR interplay and associated effects will become increasingly important [18], [17].","Future work along these directions could focus on extending our work to periodic structures, which could potentially exhibit much larger RHT and hence CR effects.","Acknowledgements.— This work was supported by the National Science Foundation under Grant no.","DMR-1454836 and by the Princeton Center for Complex Materials, a MRSEC supported by NSF Grant DMR 1420541."]],"string":"[\n [\n \"General formulation of coupled radiative and conductive heat transfer\\n between compact bodies\"\n ],\n [\n \"Abstract We present a general framework for studying strongly coupled radiative and conductive heat transfer between arbitrarily shaped bodies separated by sub-wavelength distances.\",\n \"Our formulation is based on a macroscopic approach that couples our recent fluctuating volume--current (FVC) method of near-field heat transfer to the more well known Fourier conduction transport equation.\",\n \"We apply our technique to consider heat exchange between aluminum-zinc oxide nanorods and show that the presence of bulk plasmon resonances can result in extremely large radiative heat transfer rates (roughly twenty times larger than observed in planar geometries), whose interplay with conductive transport leads to nonlinear temperature profiles along the nanorods.\"\n ],\n [\n \"3 General formulation of coupled radiative and conductive heat transfer between compact bodies Weiliang Jin Department of Electrical Engineering, Princeton University, Princeton, NJ 08544, USA Riccardo Messina Laboratoire Charles Coulomb, Université de Montpellier and CNRS, Montpellier, France Alejandro W. Rodriguez Department of Electrical Engineering, Princeton University, Princeton, NJ 08544, USA We present a general framework for studying strongly coupled radiative and conductive heat transfer between arbitrarily shaped bodies separated by sub-wavelength distances.\",\n \"Our formulation is based on a macroscopic approach that couples our recent fluctuating volume–current (FVC) method of near-field heat transfer to the more well known Fourier conduction transport equation.\",\n \"We apply our technique to consider heat exchange between aluminum-zinc oxide nanorods and show that the presence of bulk plasmon resonances can result in extremely large radiative heat transfer rates (roughly twenty times larger than observed in planar geometries), whose interplay with conductive transport leads to nonlinear temperature profiles along the nanorods.\",\n \"Radiative heat transfer (RHT) between objects held at different temperatures can be many orders of magnitude larger in the near field (short separations $d \\\\ll $ thermal wavelength $\\\\lambda _T =\\\\hbar c/k_\\\\mathrm {B} T$ ) than for far-away objects [1], [2], [3], [4], [5].\",\n \"Recently, we showed that that the interplay of near-field RHT and conduction in planar geometries can dramatically modify the temperature and thermal exchange rate at sub-micron separations [6].\",\n \"Such strongly-coupled conduction–radiation (CR) phenomena are bound to play a larger role in situations involving structured materials, where RHT can be further enhanced [7], [8], [9], [10], [11], [12] and modified [13], [14], [15], [16], and in on-going experiments exploring nanometer scale gaps, where the boundary between conductive (phonon- and electron-mediated) and radiative transport begins to blurr [17], [18].\",\n \"We present a general CR framework that captures the interplay of near-field RHT and thermal conduction along with the existence of large temperature gradients in arbitrary geometries.\",\n \"We show that under certain conditions, i.e.\",\n \"materials and structures with separations and geometric lengthscales in the nanometer range, RHT can approach and even exceed conduction, significantly changing the stationary temperature distribution of heated objects.\",\n \"Our approach is based on a generalization of our recent fluctuating volume-current (FVC) formulation of electromagnetic (EM) fluctuations, which when coupled to the more standard Fourier heat equation describing conductive transport at macroscopic scales, allows studies of CR between arbitrary shapes, thereby generalizing our prior work with slabs [6].\",\n \"As a proof of concept, we consider an example geometry involving aluminum-zinc oxide (AZO) nanorods separated by vacuum gaps, which exhibits more than an order of magnitude enhancement in RHT compared to planar slabs, and hence leads to even larger temperature gradients.\",\n \"We find that while RHT between thin slabs is primarily mediated by surface modes, resulting in linear temperature gradients, the presence of bulk nanorod resonances leads to highly distance-dependent nonlinear temperature profiles.\",\n \"Coupled radiative and conductive diffusion processes in nanostructures are becoming increasingly important [19], [18], with recent works primarily focusing on the interplay between thermal diffusion and external optical illumination such as laser-heating of plasmonic structures [20], [21], [22], [23], [24].\",\n \"On the other hand, while it is known that conduction has a strong influence on RHT experiments [25], [26], the converse has thus far been largely unexplored because RHT is typically too small to result in appreciable temperature gradients [27], [28], [29].\",\n \"However, our recent work [6] suggests that such an interplay can be significant at tens of nanometer separations and in fact may already have been present (though overlooked) in recent experiments involving planar systems [30], [31], [32], [33].\",\n \"Moreover, since planar structures are known to exhibit highly suboptimal RHT rates [7], we expect even stronger interplays in more complex geometries, such as metasurfaces [34], hyperbolic metamaterials [35], [11], or lattices of metallic antennas [8], [7].\",\n \"Figure: (a) Schematic illustration of two square lattices of nanorods(labelled aa and bb) of thickess tt, period Λ\\\\Lambda ,cross-sectional area l×ll\\\\times l, and separation dd, whosetemperature distribution and energy exchange is mediated by bothconductive ∇·[κ(𝐱)T(𝐱)]\\\\nabla \\\\cdot [\\\\kappa (\\\\mathbf {x})T(\\\\mathbf {x})] andradiative H(𝐱,𝐱 ' )H(\\\\mathbf {x},\\\\mathbf {x}^{\\\\prime }) heat transfer.\",\n \"(b) Totalradiative heat transfer spectrum Φ(ω)\\\\Phi (\\\\omega ) between two AZOnanorods (solid lines) of thickness t=500t=500 nm and cross-sectionalarea A=l 2 A=l^2, separated by d=20d=20 nm and held at temperaturesT a(b) =800(300)T_{a(b)}=800(300) K. The spectrum is shown for differentcross-sections l={10,20}l=\\\\lbrace 10,20\\\\rbrace nm (blue and red lines) and in thelimit l=∞l = \\\\infty , corresponding to two planar slabs.\",\n \"(c) Spatialradiative heat flux in nanorod aa for the case l=20l=20 nm,corresponding to the (i) first, (ii) second, and (iii) SPP plasmonresonances, respectively, annotated in (b).Formulation.— In what follows, we describe a general formulation of coupled CR applicable to arbitrary geometries.\",\n \"Consider a situation involving two bodies (the same framework can be extended to multiple bodies), labelled $a$ and $b$ , subject to arbitrary temperature profiles and exchanging heat among one other, shown schematically in Fig.\",\n \"REF (a).\",\n \"Neglecting convection and considering bodies with lengthscales larger or of the order of their phonon mean-free path, in which case Fourier conduction is valid, the stationary temperature distribution satisfies: $\\\\nabla \\\\cdot [\\\\kappa (\\\\mathbf {x})\\\\nabla T(\\\\mathbf {x})]+\\\\int \\\\mathrm {d}^3\\\\mathbf {x}^{\\\\prime }\\\\,H(\\\\mathbf {x},\\\\mathbf {x}^{\\\\prime })=Q(\\\\mathbf {x})$ where $\\\\kappa (\\\\mathbf {x})$ and $Q(\\\\mathbf {x})$ describe the bulk Fourier conductivity and presence of some external heat source at $\\\\mathbf {x}$ , respectively, and $H(\\\\mathbf {x},\\\\mathbf {x}^{\\\\prime })$ denotes the radiative power per unit volume from $\\\\mathbf {x}^{\\\\prime }$ to $\\\\mathbf {x}$ .\",\n \"Our ability to compute $H(\\\\mathbf {x},\\\\mathbf {x}^{\\\\prime })$ in full generality hinges on an extension of a recently introduced FVC method that exploits powerful EM scattering techniques [36] to enable fast calculations of RHT under arbitrary geometries and temperature distributions.\",\n \"The starting point of this method is the volume-integral equation (VIE) formulation of EM, in which the scattering unknowns are 6-component polarization currents $\\\\xi $ in the interior of the bodies coupled via the homogeneous $6\\\\times 6$ Green's function $\\\\Gamma $ of the intervening medium [36].\",\n \"Given two objects described by a susceptibility tensor $\\\\chi (\\\\mathbf {x})$ and a Galerkin decomposition of the induced currents $\\\\xi =\\\\sum _i x_i b_i$ , with $\\\\lbrace b_i\\\\rbrace $ denoting localized basis functions throughout the objects ($i$ is the global index for all bodies), the scattering of an incident field due to some fluctuating current-source $\\\\sigma =\\\\sum _is_i b_i$ can be determined via solution of a VIE equation, $x+s=Ws$ , in terms of the unknown and known expansion coefficients $\\\\lbrace x_i\\\\rbrace $ and $\\\\lbrace s_i\\\\rbrace $ , respectively, where $W^{-1}_{i,j}=\\\\langle b_i,(I+i\\\\omega \\\\chi G) b_j\\\\rangle $ and $G_{i,j}=\\\\langle b_{i},\\\\Gamma \\\\star b_{j}\\\\rangle $ are known as VIE and Green matrices [36].\",\n \"Previously, we exploited this formalism to propose an efficient method for computing the total heat transfer between any two compact bodies [36], based on a simple voxel basis expansion (uniform discretization).\",\n \"The solution of (REF ) requires an extension of the FVC method to include the spatially resolved heat transfer between any two voxels, which we describe below.\",\n \"Consider a fluctuating current-source $\\\\sigma _{\\\\alpha }=s_{\\\\alpha }b_{\\\\alpha }$ at $\\\\mathbf {x}_a=b_{\\\\alpha }$ in a body $a$ .\",\n \"Such a “dipole” source induces polarization–currents $\\\\xi _{\\\\beta }=x_{\\\\beta }b_{\\\\beta }$ and EM fields $\\\\phi _{\\\\beta }$ throughout space in body $b$ (and elsewhere), such that the heat flux at $\\\\mathbf {x}_b=b_{\\\\beta }$ is given (by Poynting's theorem) by: $\\\\Phi (\\\\omega ;\\\\mathbf {x}_a\\\\rightarrow \\\\mathbf {x}_b)=\\\\frac{1}{2}\\\\langle \\\\operatorname{Re}\\\\left( \\\\xi _{\\\\beta }^{*}\\\\phi _{\\\\beta } \\\\right)\\\\rangle $ where “$\\\\langle \\\\ldots \\\\rangle $ ” denotes a thermodynamic ensemble average.\",\n \"Expressing the polarization–currents and fields in the localized basis $\\\\lbrace b_{\\\\alpha }\\\\rbrace $ , and exploiting the volume equivalence principle to express the field as a convolution of the incident and induced currents with the vacuum Green's function (GF), $\\\\phi =\\\\Gamma \\\\star (\\\\xi +\\\\sigma )$ , one finds that (REF ) can be expressed in a compact, algebraic form involving VIE matrices: $\\\\Phi (\\\\omega ;\\\\mathbf {x}_a\\\\rightarrow \\\\mathbf {x}_b)&=\\\\frac{1}{2}\\\\langle \\\\operatorname{Re}\\\\left\\\\lbrace x_{\\\\beta }^{*}[G(x+s^\\\\alpha )]_{\\\\beta }\\\\right\\\\rbrace \\\\rangle \\\\nonumber \\\\\\\\&=\\\\frac{1}{2} \\\\langle \\\\operatorname{Re}\\\\left\\\\lbrace (x+s^\\\\alpha )_{\\\\beta }^{*}[G(x+s^\\\\alpha )]_{\\\\beta }\\\\right\\\\rbrace \\\\rangle \\\\nonumber \\\\\\\\&=\\\\frac{1}{2}\\\\langle \\\\operatorname{Re}\\\\left[ (Ws^\\\\alpha )_{\\\\beta }^{*}(GWs^\\\\alpha )_{\\\\beta }\\\\right]\\\\rangle \\\\nonumber \\\\\\\\&=\\\\frac{1}{2} \\\\operatorname{Re}\\\\left[ D_{\\\\alpha ,\\\\alpha }W^{\\\\dagger }_{\\\\alpha ,\\\\beta }(GW)_{\\\\beta ,\\\\alpha }\\\\right]$ where $s^{\\\\alpha }$ is a vector that is zero everywhere except at the $\\\\alpha $ th element, denoted by $s_{\\\\alpha }$ , and ${D}_{\\\\alpha ,\\\\beta }=\\\\langle s_{\\\\alpha }^{*}s_{\\\\beta }\\\\rangle =\\\\int \\\\int d^3\\\\mathbf {x}\\\\,d^3\\\\mathbf {y}\\\\, b_{\\\\alpha }^\\\\ast (\\\\mathbf {x}) \\\\langle \\\\sigma (\\\\mathbf {x})\\\\sigma ^\\\\ast (\\\\mathbf {y}) \\\\rangle b_{\\\\beta }(\\\\mathbf {y})$ is a real, diagonal matrix encoding the thermodynamic and dissipative properties of each object [36] and described by the well-known fluctuation–dissipation theorem, $\\\\langle \\\\sigma _i(\\\\mathbf {x},\\\\omega ) \\\\sigma _j^\\\\ast (\\\\mathbf {y},\\\\omega ) \\\\rangle =\\\\frac{4}{\\\\pi } \\\\omega \\\\operatorname{Im}\\\\epsilon (\\\\mathbf {x},\\\\omega ) \\\\Theta (T_{\\\\mathbf {x}})\\\\delta (\\\\mathbf {x}-\\\\mathbf {y})\\\\delta _{ij}$ , where $\\\\Theta (T)=\\\\hbar \\\\omega /[\\\\mathrm {exp}(\\\\hbar \\\\omega /k_bT)-1]$ is the Planck distribution.\",\n \"It follows then that the heat flux emitted or absorbed at a given position $\\\\mathbf {x}_a$ , the main quantity entering (REF ) through $\\\\int \\\\mathrm {d}^3\\\\mathbf {x}^{\\\\prime }\\\\,H(\\\\mathbf {x},\\\\mathbf {x}^{\\\\prime })=\\\\int \\\\mathrm {d}\\\\omega \\\\,\\\\Phi (\\\\omega ;\\\\mathbf {x})$ , is given by: $\\\\Phi (\\\\omega ;\\\\mathbf {x}_a)&=\\\\int _{V_{b}} \\\\mathrm {d}^3\\\\mathbf {x}_b\\\\left[ \\\\Phi (\\\\omega ;\\\\mathbf {x}_b\\\\rightarrow \\\\mathbf {x}_a)-\\\\Phi (\\\\omega ;\\\\mathbf {x}_a\\\\rightarrow \\\\mathbf {x}_b)\\\\right]\\\\nonumber \\\\\\\\&=\\\\frac{1}{2}\\\\text{Tr }_{\\\\beta | b_\\\\beta \\\\in V_b} \\\\operatorname{Re}\\\\left[D_{\\\\beta ,\\\\beta }W^{\\\\dagger }_{\\\\beta ,\\\\alpha }(GW)_{\\\\alpha ,\\\\beta }-(\\\\alpha \\\\leftrightarrow \\\\beta )\\\\right]\\\\nonumber \\\\\\\\&=\\\\frac{1}{2}\\\\operatorname{Re}\\\\left[\\\\underbrace{GWD^bW^{\\\\dagger }}_{\\\\Phi _\\\\mathrm {a}}-\\\\underbrace{DW^{\\\\dagger }P^bGW}_{\\\\Phi _\\\\mathrm {e}}\\\\right]_{\\\\alpha ,\\\\alpha }$ Here, $P^{a(b)}$ denotes the projection operator that selects only basis functions in $a(b)$ , such that $D^b=P^bDP^b$ is a diagonal matrix involving only fluctuations in object $b$ .\",\n \"Furthermore, the first (second) term in (REF ) describe the absorbed (emitted) power in $\\\\mathbf {x}_a$ , henceforth denoted via the subscript “a(e)”.\",\n \"Figure: (a) Temperature profile along the zz coordinate of a nanorod(solid lines) when it is heated from one side to a temperature of800 K, and is separated from an identical, constant- anduniform-temperature nanrod held at T=300T=300 K on the other side, by agap size d=20d=20 nm.\",\n \"The nanorods have cross-sectional widthl=10l=10 nm and thicknesses t=500t=500 nm, and are made up of AZO withresults shown for multiple values of the doping concentration{2,6,11} wt %\\\\lbrace 2,6,11\\\\rbrace \\\\mathrm {wt}\\\\% (blue, red, and black lines).\",\n \"Also shownare the temperature profiles of slabs (dashed lines) of the samethickness (corresponding to the limit l→∞l \\\\rightarrow \\\\infty ).\",\n \"(Inset:)Temperature distribution throughout the nanorod in the case of11 wt %11\\\\mathrm {wt}\\\\%.\",\n \"(b) Temperature profiles of nanorods of widthl=20l=20 nm under various separations d={5,10,20,30}d=\\\\lbrace 5,10,20,30\\\\rbrace nm (black,blue, red, and green lines).\",\n \"(c.inset:) The ratio of total radiativeheat flux for nanorods of width l=20l=20 nm to that of the slabs as afunction of dd, in the presence (red dots) or absence (black dots)of temperature gradients induced by conduction and radiationinterplay, with the flux value shown in (c), for nanorods (red) andslabs (blue), also in the presence (solid lines) or absence (dashedlines) of temperature gradients induced by the interplay ofconduction and radiation.Equation REF is a generalization of our previous expression for the total heat transfer between two arbitrary inhomogeneous objects [36] in that it includes both the spatially resolved absorbed and emitted power throughout the entire geometry.\",\n \"In Ref.\",\n \"polimeridis2015fluctuating, we showed that the low-rank nature of the GF operator enables truncated, randomized SVD factorizations and therefore efficient evaluations of the corresponding matrix operations.\",\n \"We find, however, that in this case, the inclusion of the absorption term does not permit such a factorization, except in special circumstances.\",\n \"In particular, writing down the two terms separately by expanding into the subspace spanned by each object, we find: $\\\\Phi _\\\\mathrm {a}(\\\\omega ;\\\\mathbf {x}_a)&=\\\\frac{1}{2}\\\\operatorname{Re}\\\\left[G^{ab}W^{bb}D^{bb}W^{ab\\\\dagger }+G^{aa}W^{ab}D^{bb}W^{ab\\\\dagger }\\\\right] \\\\\\\\\\\\Phi _\\\\mathrm {e}(\\\\omega ,\\\\mathbf {x}_a) &= -\\\\frac{1}{2}\\\\left[ \\\\operatorname{Re}(D^{aa}W^{ba\\\\dagger } G^{ba} W^{aa})\\\\right.\",\n \"\\\\nonumber \\\\\\\\&\\\\hspace{50.58878pt}\\\\left.+ D^{aa} W^{ba\\\\dagger } \\\\operatorname{sym}(G^{bb}) W^{ba} \\\\right]_{\\\\alpha ,\\\\alpha }$ with $X^{ij} = P^i X P^j$ denoting the sub-block of matrix $X$ connecting basis functions in object $i$ to object $j$ , and $\\\\operatorname{sym}X =\\\\frac{1}{2}(X+X^\\\\dagger )$ denoting the symmetric part of $X$ .\",\n \"Equation , describing emission, can be evaluated efficiently because the matrices $G^{ba}$ and $\\\\operatorname{sym}{G^{bb}}$ are both low rank ($\\\\ell \\\\ll N$ ) [36], in which case they can be SVD factorized to allow fast matrix multiplications.\",\n \"It follows that the total heat transfer, i.e.\",\n \"the trace of (), can also be computed efficiently.\",\n \"Unfortunately, the second term of (REF ) involves both the symmetric and anti-symmetric parts of $G^{aa}$ , the latter of which is full rank.\",\n \"More conveniently, detailed balance dictates that $\\\\Phi (\\\\omega ;\\\\mathbf {x}_b\\\\rightarrow \\\\mathbf {x}_a)=\\\\Phi (\\\\omega ;\\\\mathbf {x}_a\\\\rightarrow \\\\mathbf {x}_b)$ whenever $T(\\\\mathbf {x}_a) = T(\\\\mathbf {x}_b)$ , which implies that $\\\\operatorname{Re}\\\\left[ M_{\\\\beta ,\\\\beta }W^{\\\\dagger }_{\\\\beta ,\\\\alpha }(GW)_{\\\\alpha ,\\\\beta }\\\\right]=\\\\operatorname{Re}\\\\left[M_{\\\\alpha ,\\\\alpha }W^{\\\\dagger }_{\\\\alpha ,\\\\beta }(GW)_{\\\\beta ,\\\\alpha }\\\\right]$ , where $M_{\\\\alpha ,\\\\alpha }=\\\\operatorname{Im}\\\\varepsilon (\\\\mathbf {x}_{\\\\alpha },\\\\omega )$ is a real, diagonal matrix encoding the dissipative properties of the bodies, leading to the following modified expression for the absorption rate: $\\\\Phi _\\\\mathrm {a}(\\\\omega ;\\\\mathbf {x}_a)&=\\\\frac{1}{2}\\\\left[\\\\operatorname{Re}(M^{aa}W^{ba\\\\dagger }K^{bb}G^{ba}W^{aa})\\\\right.\\\\nonumber \\\\\\\\&\\\\hspace{50.58878pt}\\\\left.+M^{aa}W^{ba\\\\dagger }\\\\operatorname{sym}(K^{bb}G^{bb})W^{ba}\\\\right]_{\\\\alpha ,\\\\alpha }$ where the real and diagonal matrix $K_{\\\\alpha ,\\\\alpha }=D_{\\\\alpha ,\\\\alpha }/M_{\\\\alpha ,\\\\alpha }$ is only relevant to the Plank function $\\\\Theta (T(\\\\mathbf {x}_{\\\\alpha }),\\\\omega )$ .\",\n \"Noticeably, the symmetrized operator in the second term is full rank except whenever the temperature of object $b$ is close to uniform, in which case $\\\\operatorname{sym}(K^{bb} G^{bb}) \\\\approx K^{bb} \\\\operatorname{sym}G^{bb}$ .\",\n \"While solution of (REF ) is feasible, it remains an open problem to find a formulation that allows fast evaluations of the spatially resolved absorbed power under arbitrary temperature distributions.\",\n \"Given (REF ), one can solve the coupled CR equation in any number of ways [37].\",\n \"Here, we exploit a fixed-point iteration procedure based on repeated and independent evaluations of (REF ) and (REF ), converging once both quantities approach a set of self-consistent steady-state values.\",\n \"Equation REF is solved via a commercial, finite-element heat solver whereas (REF ) is solved through a free, in-house implementation of our FVC method [36].\",\n \"While the above formulation is general, below we explore the computationally convenient situation in which object $b$ is kept at a constant, uniform temperature by means of a carefully chosen thermal reservoir, such that the absorbed power in object $a$ can be computed efficiently via (REF ).\",\n \"Furthermore, absorption can be altogether ignored whenever one of the bodies is heated to a much larger temperature than the other (as is the case below).\",\n \"The power emitted by $a$ (the heated object), obtained via (), turns out to be much more convenient to compute, since the time-consuming part of the scattering calculation can be precomputed independently from the temperature distribution and stored for repeated and subsequent evaluations of (REF ) under different temperature profiles.\",\n \"Results.— As a proof of principle and to gain insights into coupled CR effects in non-planar objects, we now apply the above method to a simple geometry consisting of two metallic nanorods of cross-sectional widths $l$ and thickness $t$ ; in practice, both for easy of fabrication and to obtain even larger RHT [25], such a structure could be realized as a lattice or grating, shown schematically in Fig.\",\n \"REF (a).\",\n \"However, for computational convenience and conceptual simplicity, we restrict our analysis to the regime of large grating periods, in which case it suffices to consider only the transfer between nearby objects.\",\n \"The strongest CR effects generally will arise in materials that exhibit large RHT, e.g.\",\n \"supporting surface–plasmon polaritons (SPP) in the case of planar objects, and low thermal conductivities, including silica, sapphire, and AZO, whose typical thermal conductivities $\\\\sim 1$ W/m$\\\\cdot $ K. In the following, we take AZO as an illustrative example [38], [39].\",\n \"To begin with, we show that even in the absence of CR interplay, the RHT spectrum and spatial RHT distribution inside the nanorods differ significantly from those of AZO slabs of the same thickness.\",\n \"Figure REF (b) shows the RHT spectrum $\\\\Phi (\\\\omega )$ per unit area $A=l^2$ between two AZO nanorods (with doping concentration $11\\\\mathrm {wt}\\\\%$ [39]) of length $t=500$ nm and varying widths $l=\\\\lbrace 10,20,\\\\infty \\\\rbrace $ nm (blue solid, red solid, and black dashed lines), held at temperatures $T_{a(b)}=800(300)$ K and vacuum gap $d=20$ nm.\",\n \"The limit $l\\\\rightarrow \\\\infty $ corresponds to the slab-slab geometry already explored [6], in which case the $\\\\Phi (\\\\omega )$ exhibits a single peak occuring at the SPP frequency $\\\\approx 3\\\\times 10^{14}$ rad/s.\",\n \"The finite nature of the nanorods results in additional peaks at lower frequencies, corresponding to bulk/geometric plasmon resonances (red and blue solid lines) that provide additional channels of heat exchange, albeit at the expense of weaker SPP peaks, leading to a roughly 20-fold enhancement in RHT compared to slabs.\",\n \"More importantly and well known, such structured antennas allow tuning and creation of bulk plasmon resonances in the near- and far-infrared spectra (much lower than many planar materials) that can more effectively transfer thermal radiation.\",\n \"The contour plots in Fig.\",\n \"REF (i–iii) reveal the spatial RHT distribution $\\\\Phi (\\\\omega ,\\\\mathbf {x})$ (in arbitrary units) at three separate frequencies $\\\\omega =\\\\lbrace 0.4,0.8,2.3\\\\rbrace \\\\times 10^{14}$ rad/s, corresponding to the first, second, and SPP resonances, respectively.\",\n \"As expected, the highest-frequency resonance is primarily confined to the corners of the nanorod surface (becoming the well-known SPP resonance in the limit $l \\\\rightarrow \\\\infty $ ), with the fundamental and intermediate resonances have flux contributions stemming primarily form the bulk.\",\n \"As we now show, such an enhancement results not only results in larger temperature gradients but also changes the resulting qualitative temperature distribution.\",\n \"Figure REF (a) shows the temperature profile along the $z$ direction for the nanorod geometry of Fig.\",\n \"REF (a), with width $l=10~$ nm and gap size $d=20$ nm, obtained via solution of (REF ).\",\n \"For the purpose of generality, we show results under various doping concentrations $\\\\lbrace 2,6,11\\\\rbrace \\\\mathrm {wt}\\\\%$ (green, red, and black solid lines), corresponding to different SPP frequencies and bandwidths [39].\",\n \"In particular, we consider a situation in which the boundary I of nanorod $a$ is kept at $T_\\\\text{I}=800$ K while the entire nanorod $b$ is held at $T_b=300$ K (through contact with a room-temperature reservoir), and assume an AZO thermal conductivity of $\\\\kappa =1$ W/m$\\\\cdot $ K [38].\",\n \"The temperature along the $x$ –$y$ cross section is nearly uniform (due to the faster heat diffusion rate along the smaller dimension) and therefore only shown in the case of $11\\\\mathrm {wt}\\\\%$ (inset).\",\n \"In all scenarios, the temperature gradient is significantly larger for nanorods (solid lines) than for slabs ($t \\\\rightarrow \\\\infty $ , dashed lines), becoming an order of magnitude larger in the case of $6\\\\mathrm {wt}\\\\%$ due to its larger SPP frequency compared to the peak Planck wavelength near 800 K. Furthermore, while slabs exhibit linear temperature profiles (RHT is dominated by surface emission [1]), the bulky and de-localized nature of emission in the case of nanorods results in nonlinear temperature distributions.\",\n \"Figure REF (b) shows the temperature profile at various separations $d=\\\\lbrace 5,10,20,30\\\\rbrace $ nm (black, blue, red, and green lines) for nanorods of width $l=20$ nm and $11\\\\mathrm {wt}\\\\%$ , illustrating the sensitive relationship between the degree of CR interplay and gap size.\",\n \"Notably, while the RHT and therefore temperature gradients increase as $d$ decreases, the profile becomes increasingly linear as the geometry approaches the slab–slab configuration.\",\n \"The transition from bulk- to surface-dominated RHT and the increasing impact of the latter on conduction and vice versa is also evident from Fig.\",\n \"REF (c).\",\n \"The figure shows the radiative flux rate $H\\\\times d^2$ as a function of $d$ for slabs (black lines) of thickness $t=500$ nm and nanorods (red lines) of equal thickness and width $l=20$ nm, either including (solid lines) or excluding (dashed lines) CR interplay (with the latter involving uniform temperatures).\",\n \"While the RHT between bodies of uniform temperatures is shown to scales as $1/d^2$ (dashed lines), the temperature gradients induced by CR interplay in the case of nanorods begins to change the expected powerlaw behavior at $d\\\\approx 15$ nm; the same occurs for slabs but at much shorter $d\\\\lesssim 5$ nm.\",\n \"These differences are further quantified on the inset of the figure, which shows the ratio of the RHT rate between the two objects as a function of $d$ .\",\n \"While the ratio remains almost a constant for uniform-temperature objects (black dots), it decreases visibly when considering CR interplay (red dots).\",\n \"As shown in Ref.\",\n \"riccardo2016strongly, in the limit $d\\\\rightarrow 0$ , RHT will asymptote to a constant (not shown) rather than a diverge.\",\n \"Concluding remarks.— As experiments continue to push toward larger RHT by going to smaller vacuum gaps or through nanostructuring, accurate descriptions of CR interplay and associated effects will become increasingly important [18], [17].\",\n \"Future work along these directions could focus on extending our work to periodic structures, which could potentially exhibit much larger RHT and hence CR effects.\",\n \"Acknowledgements.— This work was supported by the National Science Foundation under Grant no.\",\n \"DMR-1454836 and by the Princeton Center for Complex Materials, a MRSEC supported by NSF Grant DMR 1420541.\"\n ]\n]"},"id":{"kind":"string","value":"1605.05708"}}},{"rowIdx":1866674,"cells":{"text":{"kind":"list like","value":[["Effect of the polydispersity of a colloidal drop on the drying induced\n stress as measured by the buckling of a floating sheet"],["Abstract We study the stress developed during the drying of a colloidal drop of silica nanoparticles.","In particular, we use the wrinkling instability of a thin floating sheet to measure the net stress applied by the deposit on the substrate and we focus on the effect of the particle polydispersity.","In the case of a bidisperse suspension, we show that a small number of large particles substantially decreases the expected stress, which we interpret as the formation of lower hydrodynamic resistance paths in the porous material.","As colloidal suspensions are usually polydisperse, we show for different average particle sizes that the stress is effectively dominated by the larger particles of the distribution and not by the average particle size."],["Floating films","Polydimethylsiloxane (PDMS, sylgard 184, Dow Corning) is mixed and degassed at $0^\\circ $ C to slow down the reaction and to ensure a reproducible viscosity.","A silicon wafer is first coated with a 10% wt.","aqueous solution of polyvinyl alcohol (PVA, Sigma-Aldrich, $M_w = 10$ kDa).","The coating is dried until a uniform color is obtained.","Then, the PDMS is spin coated on this wafer and placed in an oven at $65^\\circ $ C for 10 hours.","The edge of the coated PDMS is removed to obtain a circular film of diameter 87 mm.","A petri dish, painted in black to enhance the contrast on the picture, is partially filled with deionized water.","The wafer is dipped in this bath to dissolve the sacrificial layer of PVA and to detach the PDMS film.","Once the PDMS film is fully detached and floats at the surface, the wafer is removed.","We used a white light spectrometer (OceanOptics USB2000+ used with a LS-1-LL tungsten halogen light source) to measure the film thicknesses, which range between $[15.8,63.9]$ $\\mu $ m. The presence of PVA lowers the surface tension of the bath, which is $\\gamma _b = 55\\pm 5$ mN/m."],["Visualisation","Our experiments of controlled drying of drops are performed in a transparent glove box, which has humidity controller presented in Fig.","REF .","The relative humidity is set to 50% in all of our experiments.","A grid is placed in the glove box on the top of the sample and a light source is placed behind a light diffuser to obtain an even illumination.","The deformation of the membrane is visualised by the distortion of the pattern made by the grid.","The time evolution of the system is recorded by a Nikon camera (D7100) mounted with a 18-55mm Nikon objective (Fig.","REF ).","Figure: Detailed experimental setup.A grid is placed between the light and the sample.The reflection of the grid on the bath interface allows the visualisation of the deformation of the elastomer."],["Dialysis","Each suspension was dialysed to ensure that they have the same properties (ionic strength and pH).","Dialysis bags (12-14 kDa, Fisherbrand) are partially filled with the colloidal solutions.","A solution denoted s1 is prepared from deionized water, the pH is raised to 9.5 by the addition of NaOH and the ionic strength is adjusted by the addition of 5 mM of NaCl.","To wash these suspensions, the bags are plunged in large baths of solution s1 for 10 days and baths are replaced twice.","Then, the diluted suspensions are concentrated by osmotic stress.","This second step consists in preparing a bath in which polyethylene glycol (Sigma-Aldrich, 35 kDa) is dissolved in a solution s1.","Mass fractions are measured and the suspensions are diluted with a solution s1 to reach a mass fraction $\\Phi _0=0.25$ , which corresponds to a volume fraction $\\phi _0=0.15$ ."],["Particle size","The particle size distributions are characterized by Dynamic Light Scattering (DLS) with a Malvern Zetasizer Nano ZS instrument.","Samples are diluted to a volume fraction of $\\approx 0.001$ .","We denote $n(a)$ the distribution function of particle number and $v(a)$ , the distribution function in volume, and satisfy $\\int _0^\\infty n(a^{\\prime }){\\rm d}a^{\\prime }=1\\\\\\int _0^\\infty v(a^{\\prime }){\\rm d}a^{\\prime }=1$ The average particle radius $a_0$ is defined as $a_0 = \\int _0^\\infty a^{\\prime } n(a^{\\prime }) {\\rm d} a^{\\prime }.$ The particle radius $a^\\star $ , representative of the large particles of the distribution is defined as follow.","We consider 30% of the tail of the particle distribution $v(a)$ , i.e.","$a>a_c$ with $\\int _0^{a_c} v(a^{\\prime }) {\\rm d} a^{\\prime } = 0.7.$ The radius $a^\\star $ is the average particle size of the distribution tail, i.e $a^\\star = \\int _{a_c}^\\infty a^{\\prime } n(a^{\\prime }) {\\rm d} a^{\\prime }.$ The suspension properties are summarized in Table REF .","Figure: NO_CAPTION$n(a)$ for the number of particles.","Table: NO_CAPTION tableProperties of the colloidal suspensions of silica nanoparticles.","For all suspensions, the pH is 9.5, the ionic strength is $[$ NaCl$]$ = 5 mM and the volume fraction is $\\phi _0=0.15$ ."],["S3. Particle organization","For blend suspensions, we define the particle volume ratio as $\\Gamma _{\\cal V} = \\frac{{\\cal V}_{\\ell }}{{\\cal V}_{s} + {\\cal V}_{\\ell }},$ and the particle number ratio as $\\Gamma _{\\cal N} = \\frac{{\\cal N}_{\\ell }}{{\\cal N}_{s} + {\\cal N}_{\\ell }} = \\left[ 1 + \\frac{a_\\ell ^3}{a_s^3} \\left(\\Gamma _{\\cal V}^{-1} - 1\\right) \\right]^{-1},$ where $a_s$ and $a_\\ell $ are the average radii of small and large particles, respectively.","SEM images of dried drops of colloidal suspensions were captured with the FEI Quanta 200 FEG Environmental-SEM in low vacuum mode.","Samples are prepared in the same conditions as presented in section S1.","To image cross-sections, a razor blade is gently pressed on the top of the deposit to nucleate a crack along the diameter.","The two pieces are then separated and placed on a sample holder.","In this section, we present additional images.","In Fig.","REF , the cross section of a dried deposit containing only SM particle is imaged and we see that the particle size is smaller than the resolution we could obtain.","In Fig.","REF , we present cross sections images for $\\Gamma _{\\cal N}=0.05$ %.","As for the pictures in the main text, we do not see a segregation of the large particles across the thickness.","Also, similar particle organisations are obtained on the top and near the contact line.","In Fig.","REF , we imaged the surface of the deposits for different $\\Gamma _{\\cal N}$ where we notice the presence of the large particles as their concentration increases.","In our experiment, we can estimate the Péclet numbers for small (${\\rm Pe}_s$ ) and large (${\\rm Pe}_\\ell $ ) particles.","The Péclet number is defined as ${\\rm } Pe = \\frac{v_e h}{D}$ where $v_e$ is the evaporation speed, $h$ the characteristic thickness and $D$ the diffusion coefficient.","We use the Stokes-Einstein diffusion coefficient $D=k_B T/(6\\pi \\eta a)$ , where $k_B$ is the Boltzmann constant, $T$ is the absolute temperature, $\\eta =10^{-3}$ Pa$\\cdot $ s is the solvent viscosity and $a$ the particle radius.","For our bidisperse suspensions and considering the average particle size of each species, we have ${\\rm Pe}_s\\approx 1.6$ and ${\\rm Pe}_\\ell \\approx 21$ for $h \\approx 1$ mm.","Fortini et al.","evidenced numerically and experimentally a segregation effect in drying films with ${\\rm Pe} \\gg 1$ for both species (Fortini et al.","PRL 116, 118301 (2016)).","However, as shown in our SEM images, we could not observe a significant segregation of the two species in the dried film.","Nevertheless, as pointed by Fortini et al., the segregation is a dynamic process that is observed after a transient state.","The duration of this transient must be function the specific interactions between the particles and the initial volume fraction.","In our system, the material becomes visco-plastic at a volume fraction about $0.3$ and our initial volume fraction is $0.15$ .","Thus, a possible explanation is that the system cannot evolve to a segregated state.","This observations would deserve further investigations, which are beyond the scope of the present study.","Figure: Scanning-electron-microscope (SEM) images in the cross-section of a dried deposit of SM particles (Γ 𝒩 =0\\Gamma _{\\cal N}=0).The different pictures show different magnifications.Figure: Scanning-electron-microscope (SEM) images in the cross-section of a dried deposit for a number ratio Γ 𝒩 =0.05\\Gamma _{\\cal N}=0.05%.Images are taken at different locations and the scale bars represent 1 μ\\mu m.Figure: Scanning-electron-microscope (SEM) images of the colloidal deposit surface after complete drying for different number ratios of particles Γ 𝒩 \\Gamma _{\\cal N}.Only the large particles can be imaged.The scale bar represents 500 nm.In our experiments, we measure the final length of the wrinkles $L_f$ .","Then, from models derived by Davidovitch et al.","[19], we calculate the inner tension $T_i$ applied by the deposit on the film.","Davidovitch et al.","predict this tension in two different domains.","The first domain is named Near Threshold (NT) limit and correspond to the limit of infinitesimal deformation amplitude $\\zeta $ and large bendability ($\\epsilon ^{-1} = R_i^2 T_o / {\\cal B}\\gg 1$ .","In that domain, the tension ratio $\\tau ^{NT} = T_i^{NT}/T_o$ is given by $\\tau ^{NT} = 1 + \\frac{L_f^2}{R_i^2},$ which corresponds to equation (1) in the Letter.","The tension ratios for this limit and for all our experiments are presented in Fig.","REF (a).","The second domain is the Far From Threashold (FFT) limit and it assumes that the deformation $\\zeta $ is finite and $\\epsilon \\rightarrow 0$ .","Therefore, the tension ratio is $\\tau ^{FFT} = \\frac{2 L_{f}}{R_i},$ which corresponds to equation (2) in the Letter.","Similarly, we plot this prediction for our data in Fig.","REF (b).","Figure: Tension ratio τ=T i /T o \\tau = T_i/T_o as a function of the bendability ϵ -1 \\epsilon ^{-1}.The points correspond to our experiments (a) where we used the expression from the NT limit to calculate the tension ratio τ NT =T i NT /T o =1+L f 2 /R i 2 \\tau ^{NT} = T_i^{NT}/T_o = 1 + L_f^2 / R_i^2,and (b), we used the expression from the FFT limit τ FFT =T i FFT /T o =2L f /R i \\tau ^{FFT} = T_i^{FFT}/T_o = 2 L_f/R_i.On both plots, the green band represents the Near Threshold (NT) domain predicted by Davidovitch et al.",".In Fig.","REF , we also represent the NT domain predicted by Davidovitch et al.","[19] and we observe that both estimations are above the NT limit.","If the NT limit would describe our results, we expect that $T_i^{NT}$ values to be in the NT green domain, but the prediction is about at least an order of magnitude higher.","Thus, the estimate from the NT limit is not self-consistent.","Let us now consider the values deduced from the FFT limit.","These values are also above the NT domain (represented in green in Fig.","REF (b)), which is self-consistent.","Also, we observe that $\\tau ^{NT} \\approx \\tau ^{FFT}$ , which further validates that the NT prediction is not valid as the prediction is significantly above the NT domain.","Consequently, for all our measurements, we considered $T_i = T_i^{FFT}$ .","Let us remark that equations (REF ) and (REF ) are valid for $R_i \\ll R_o$ .","More general equations have been established recently [24] but they do not modify significantly the estimate of the tension in the film in our experiments, which confirms this assumption."]],"string":"[\n [\n \"Effect of the polydispersity of a colloidal drop on the drying induced\\n stress as measured by the buckling of a floating sheet\"\n ],\n [\n \"Abstract We study the stress developed during the drying of a colloidal drop of silica nanoparticles.\",\n \"In particular, we use the wrinkling instability of a thin floating sheet to measure the net stress applied by the deposit on the substrate and we focus on the effect of the particle polydispersity.\",\n \"In the case of a bidisperse suspension, we show that a small number of large particles substantially decreases the expected stress, which we interpret as the formation of lower hydrodynamic resistance paths in the porous material.\",\n \"As colloidal suspensions are usually polydisperse, we show for different average particle sizes that the stress is effectively dominated by the larger particles of the distribution and not by the average particle size.\"\n ],\n [\n \"Floating films\",\n \"Polydimethylsiloxane (PDMS, sylgard 184, Dow Corning) is mixed and degassed at $0^\\\\circ $ C to slow down the reaction and to ensure a reproducible viscosity.\",\n \"A silicon wafer is first coated with a 10% wt.\",\n \"aqueous solution of polyvinyl alcohol (PVA, Sigma-Aldrich, $M_w = 10$ kDa).\",\n \"The coating is dried until a uniform color is obtained.\",\n \"Then, the PDMS is spin coated on this wafer and placed in an oven at $65^\\\\circ $ C for 10 hours.\",\n \"The edge of the coated PDMS is removed to obtain a circular film of diameter 87 mm.\",\n \"A petri dish, painted in black to enhance the contrast on the picture, is partially filled with deionized water.\",\n \"The wafer is dipped in this bath to dissolve the sacrificial layer of PVA and to detach the PDMS film.\",\n \"Once the PDMS film is fully detached and floats at the surface, the wafer is removed.\",\n \"We used a white light spectrometer (OceanOptics USB2000+ used with a LS-1-LL tungsten halogen light source) to measure the film thicknesses, which range between $[15.8,63.9]$ $\\\\mu $ m. The presence of PVA lowers the surface tension of the bath, which is $\\\\gamma _b = 55\\\\pm 5$ mN/m.\"\n ],\n [\n \"Visualisation\",\n \"Our experiments of controlled drying of drops are performed in a transparent glove box, which has humidity controller presented in Fig.\",\n \"REF .\",\n \"The relative humidity is set to 50% in all of our experiments.\",\n \"A grid is placed in the glove box on the top of the sample and a light source is placed behind a light diffuser to obtain an even illumination.\",\n \"The deformation of the membrane is visualised by the distortion of the pattern made by the grid.\",\n \"The time evolution of the system is recorded by a Nikon camera (D7100) mounted with a 18-55mm Nikon objective (Fig.\",\n \"REF ).\",\n \"Figure: Detailed experimental setup.A grid is placed between the light and the sample.The reflection of the grid on the bath interface allows the visualisation of the deformation of the elastomer.\"\n ],\n [\n \"Dialysis\",\n \"Each suspension was dialysed to ensure that they have the same properties (ionic strength and pH).\",\n \"Dialysis bags (12-14 kDa, Fisherbrand) are partially filled with the colloidal solutions.\",\n \"A solution denoted s1 is prepared from deionized water, the pH is raised to 9.5 by the addition of NaOH and the ionic strength is adjusted by the addition of 5 mM of NaCl.\",\n \"To wash these suspensions, the bags are plunged in large baths of solution s1 for 10 days and baths are replaced twice.\",\n \"Then, the diluted suspensions are concentrated by osmotic stress.\",\n \"This second step consists in preparing a bath in which polyethylene glycol (Sigma-Aldrich, 35 kDa) is dissolved in a solution s1.\",\n \"Mass fractions are measured and the suspensions are diluted with a solution s1 to reach a mass fraction $\\\\Phi _0=0.25$ , which corresponds to a volume fraction $\\\\phi _0=0.15$ .\"\n ],\n [\n \"Particle size\",\n \"The particle size distributions are characterized by Dynamic Light Scattering (DLS) with a Malvern Zetasizer Nano ZS instrument.\",\n \"Samples are diluted to a volume fraction of $\\\\approx 0.001$ .\",\n \"We denote $n(a)$ the distribution function of particle number and $v(a)$ , the distribution function in volume, and satisfy $\\\\int _0^\\\\infty n(a^{\\\\prime }){\\\\rm d}a^{\\\\prime }=1\\\\\\\\\\\\int _0^\\\\infty v(a^{\\\\prime }){\\\\rm d}a^{\\\\prime }=1$ The average particle radius $a_0$ is defined as $a_0 = \\\\int _0^\\\\infty a^{\\\\prime } n(a^{\\\\prime }) {\\\\rm d} a^{\\\\prime }.$ The particle radius $a^\\\\star $ , representative of the large particles of the distribution is defined as follow.\",\n \"We consider 30% of the tail of the particle distribution $v(a)$ , i.e.\",\n \"$a>a_c$ with $\\\\int _0^{a_c} v(a^{\\\\prime }) {\\\\rm d} a^{\\\\prime } = 0.7.$ The radius $a^\\\\star $ is the average particle size of the distribution tail, i.e $a^\\\\star = \\\\int _{a_c}^\\\\infty a^{\\\\prime } n(a^{\\\\prime }) {\\\\rm d} a^{\\\\prime }.$ The suspension properties are summarized in Table REF .\",\n \"Figure: NO_CAPTION$n(a)$ for the number of particles.\",\n \"Table: NO_CAPTION tableProperties of the colloidal suspensions of silica nanoparticles.\",\n \"For all suspensions, the pH is 9.5, the ionic strength is $[$ NaCl$]$ = 5 mM and the volume fraction is $\\\\phi _0=0.15$ .\"\n ],\n [\n \"S3. Particle organization\",\n \"For blend suspensions, we define the particle volume ratio as $\\\\Gamma _{\\\\cal V} = \\\\frac{{\\\\cal V}_{\\\\ell }}{{\\\\cal V}_{s} + {\\\\cal V}_{\\\\ell }},$ and the particle number ratio as $\\\\Gamma _{\\\\cal N} = \\\\frac{{\\\\cal N}_{\\\\ell }}{{\\\\cal N}_{s} + {\\\\cal N}_{\\\\ell }} = \\\\left[ 1 + \\\\frac{a_\\\\ell ^3}{a_s^3} \\\\left(\\\\Gamma _{\\\\cal V}^{-1} - 1\\\\right) \\\\right]^{-1},$ where $a_s$ and $a_\\\\ell $ are the average radii of small and large particles, respectively.\",\n \"SEM images of dried drops of colloidal suspensions were captured with the FEI Quanta 200 FEG Environmental-SEM in low vacuum mode.\",\n \"Samples are prepared in the same conditions as presented in section S1.\",\n \"To image cross-sections, a razor blade is gently pressed on the top of the deposit to nucleate a crack along the diameter.\",\n \"The two pieces are then separated and placed on a sample holder.\",\n \"In this section, we present additional images.\",\n \"In Fig.\",\n \"REF , the cross section of a dried deposit containing only SM particle is imaged and we see that the particle size is smaller than the resolution we could obtain.\",\n \"In Fig.\",\n \"REF , we present cross sections images for $\\\\Gamma _{\\\\cal N}=0.05$ %.\",\n \"As for the pictures in the main text, we do not see a segregation of the large particles across the thickness.\",\n \"Also, similar particle organisations are obtained on the top and near the contact line.\",\n \"In Fig.\",\n \"REF , we imaged the surface of the deposits for different $\\\\Gamma _{\\\\cal N}$ where we notice the presence of the large particles as their concentration increases.\",\n \"In our experiment, we can estimate the Péclet numbers for small (${\\\\rm Pe}_s$ ) and large (${\\\\rm Pe}_\\\\ell $ ) particles.\",\n \"The Péclet number is defined as ${\\\\rm } Pe = \\\\frac{v_e h}{D}$ where $v_e$ is the evaporation speed, $h$ the characteristic thickness and $D$ the diffusion coefficient.\",\n \"We use the Stokes-Einstein diffusion coefficient $D=k_B T/(6\\\\pi \\\\eta a)$ , where $k_B$ is the Boltzmann constant, $T$ is the absolute temperature, $\\\\eta =10^{-3}$ Pa$\\\\cdot $ s is the solvent viscosity and $a$ the particle radius.\",\n \"For our bidisperse suspensions and considering the average particle size of each species, we have ${\\\\rm Pe}_s\\\\approx 1.6$ and ${\\\\rm Pe}_\\\\ell \\\\approx 21$ for $h \\\\approx 1$ mm.\",\n \"Fortini et al.\",\n \"evidenced numerically and experimentally a segregation effect in drying films with ${\\\\rm Pe} \\\\gg 1$ for both species (Fortini et al.\",\n \"PRL 116, 118301 (2016)).\",\n \"However, as shown in our SEM images, we could not observe a significant segregation of the two species in the dried film.\",\n \"Nevertheless, as pointed by Fortini et al., the segregation is a dynamic process that is observed after a transient state.\",\n \"The duration of this transient must be function the specific interactions between the particles and the initial volume fraction.\",\n \"In our system, the material becomes visco-plastic at a volume fraction about $0.3$ and our initial volume fraction is $0.15$ .\",\n \"Thus, a possible explanation is that the system cannot evolve to a segregated state.\",\n \"This observations would deserve further investigations, which are beyond the scope of the present study.\",\n \"Figure: Scanning-electron-microscope (SEM) images in the cross-section of a dried deposit of SM particles (Γ 𝒩 =0\\\\Gamma _{\\\\cal N}=0).The different pictures show different magnifications.Figure: Scanning-electron-microscope (SEM) images in the cross-section of a dried deposit for a number ratio Γ 𝒩 =0.05\\\\Gamma _{\\\\cal N}=0.05%.Images are taken at different locations and the scale bars represent 1 μ\\\\mu m.Figure: Scanning-electron-microscope (SEM) images of the colloidal deposit surface after complete drying for different number ratios of particles Γ 𝒩 \\\\Gamma _{\\\\cal N}.Only the large particles can be imaged.The scale bar represents 500 nm.In our experiments, we measure the final length of the wrinkles $L_f$ .\",\n \"Then, from models derived by Davidovitch et al.\",\n \"[19], we calculate the inner tension $T_i$ applied by the deposit on the film.\",\n \"Davidovitch et al.\",\n \"predict this tension in two different domains.\",\n \"The first domain is named Near Threshold (NT) limit and correspond to the limit of infinitesimal deformation amplitude $\\\\zeta $ and large bendability ($\\\\epsilon ^{-1} = R_i^2 T_o / {\\\\cal B}\\\\gg 1$ .\",\n \"In that domain, the tension ratio $\\\\tau ^{NT} = T_i^{NT}/T_o$ is given by $\\\\tau ^{NT} = 1 + \\\\frac{L_f^2}{R_i^2},$ which corresponds to equation (1) in the Letter.\",\n \"The tension ratios for this limit and for all our experiments are presented in Fig.\",\n \"REF (a).\",\n \"The second domain is the Far From Threashold (FFT) limit and it assumes that the deformation $\\\\zeta $ is finite and $\\\\epsilon \\\\rightarrow 0$ .\",\n \"Therefore, the tension ratio is $\\\\tau ^{FFT} = \\\\frac{2 L_{f}}{R_i},$ which corresponds to equation (2) in the Letter.\",\n \"Similarly, we plot this prediction for our data in Fig.\",\n \"REF (b).\",\n \"Figure: Tension ratio τ=T i /T o \\\\tau = T_i/T_o as a function of the bendability ϵ -1 \\\\epsilon ^{-1}.The points correspond to our experiments (a) where we used the expression from the NT limit to calculate the tension ratio τ NT =T i NT /T o =1+L f 2 /R i 2 \\\\tau ^{NT} = T_i^{NT}/T_o = 1 + L_f^2 / R_i^2,and (b), we used the expression from the FFT limit τ FFT =T i FFT /T o =2L f /R i \\\\tau ^{FFT} = T_i^{FFT}/T_o = 2 L_f/R_i.On both plots, the green band represents the Near Threshold (NT) domain predicted by Davidovitch et al.\",\n \".In Fig.\",\n \"REF , we also represent the NT domain predicted by Davidovitch et al.\",\n \"[19] and we observe that both estimations are above the NT limit.\",\n \"If the NT limit would describe our results, we expect that $T_i^{NT}$ values to be in the NT green domain, but the prediction is about at least an order of magnitude higher.\",\n \"Thus, the estimate from the NT limit is not self-consistent.\",\n \"Let us now consider the values deduced from the FFT limit.\",\n \"These values are also above the NT domain (represented in green in Fig.\",\n \"REF (b)), which is self-consistent.\",\n \"Also, we observe that $\\\\tau ^{NT} \\\\approx \\\\tau ^{FFT}$ , which further validates that the NT prediction is not valid as the prediction is significantly above the NT domain.\",\n \"Consequently, for all our measurements, we considered $T_i = T_i^{FFT}$ .\",\n \"Let us remark that equations (REF ) and (REF ) are valid for $R_i \\\\ll R_o$ .\",\n \"More general equations have been established recently [24] but they do not modify significantly the estimate of the tension in the film in our experiments, which confirms this assumption.\"\n ]\n]"},"id":{"kind":"string","value":"1605.05915"}}},{"rowIdx":1866675,"cells":{"text":{"kind":"list like","value":[["Breaking the disc-halo degeneracy in NGC 1291 using hydrodynamic\n simulations"],["Abstract We present a pilot study on the nearby massive galaxy NGC 1291, in which we aim to constrain the dark matter in the inner regions, by obtaining a dynamical determination of the disc mass-to-light ratio (M/L).","To this aim, we model the bar-induced dust lanes in the galaxy, using hydrodynamic gas response simulations.","The models have three free parameters, the M/L of the disc, the bar pattern speed and the disc height function.","We explore the parameter space to find the best fit models, i.e.","those in which the morphology of the shocks in the gas simulations matches the observed dust lanes.","The best-fit models suggest that the M/L of NGC 1291 agrees with that predicted by stellar population synthesis models in the near-infrared ($\\approx$0.6\\,$M_{\\odot}/L_{\\odot}$), which leads to a borderline maximum disc for this galaxy.","The bar rotates fast, with corotation radius $\\leq$ 1.4 times the bar length.","Additionally, we find that the height function has a significant effect on the results, and can bias them towards lower or higher M/L."],["Introduction","The distribution of dark matter in galaxies has been a source of debate in the scientific community, with one of the main sources of uncertainty being the “disc-halo degeneracy” ([15]).","This degeneracy arises because rotation curve decompositions depend critically on the mass-to-light ratio (M/L or $\\Upsilon $ ) which is assigned to the stellar disc component.","Rotation curves can be fit equally well with barely any disc contribution, or where the disc contributes the maximum amount possible – also known as the “maximum disc hypothesis” ([15], [13]).","There is no conclusive evidence to either prove or disprove the maximum disc hypothesis and in fact there are a number of arguments both for and against it for the Milky-Way (e.g.","[13]) as well as for external galaxies (e.g.","[2], [16], ).","A way to break the degeneracy between the disc and the dark matter halo is by obtaining dynamical estimates of the M/L ratio of the disc.","One way to do this is via the non-axisymmetries induced by the bar, which shocks the gas and leads to the well known “dust lanes” observed in barred galaxies ([10]).","These shocks can be reproduced in hydrodynamic simulations ([1]), and their morphology is dependent on the M/L of the disc, the bar pattern speed and the height function of the disc.","In this study we examine the dust lanes present in the nearby galaxy NGC 1291, and aim to reproduce their morphology using hydrodynamic gas response simulations.","The major assumption of this method for obtaining the M/L is that the halo is axisymmetric, and that all the non-axisymmetry in the potential is due to the stellar component."],["Comparing the models to the observations","There are three main free parameters in the models, and we vary these parameters in order to explore the allowed parameter space.","This leads to more than 300 simulations, from which the best fit models are obtained according to how well the shock loci in the models match those in the observations.","This is done by placing pseudo-slits perpendicular to the shocks, as shown in Figure REF .","The three free parameters are varied as follows: M/L.","To obtain the potential we use a 3.6$\\mu m$ image of NGC 1291 from the S$^4$ G survey ([14]) to which we apply a M/L.","We assume a fiducial value of $\\Upsilon _{3.6}$ =0.6$M_{\\odot }/L_{\\odot }$ for the 3.6$\\mu $ m band ([9], [12]), colour corrected to account for the presence of dust by combining information from the 3.6 and 4.5$\\mu $ m images ([4], [11]).","We then vary the M/L from 1.5 to 0.25 $\\Upsilon _{3.6}$ in steps of 0.25 and add a dark matter halo which maintains the total rotation curve flat at a value of 220 km/s (see Figure REF ) as found from the Baryonic Tully-Fisher relation ([8]).","Pattern speed (Lagrangian radius).","By varying the Lagrangian radius while keeping all the other parameters of a model constant, we effectively vary the pattern speed of the bar.","The Lagrangian radius is varied between 1 and 2 times the bar semi-major axis, in steps of 0.2.","Height function and scaleheight.","We vary the height function of the disc by assuming two functional forms – isothermal and exponential – and we also vary the scaleheight from $z_0$ =0.5 to $z_0$ =1.5 kpc."],["Results","Dynamical M/L: The best fit models obtained in this study have a M/L in the range of M/L$_{3.6}$ =0.6-0.75 which falls within the range predicted by stellar population synthesis models and other approaches (i.e.","$\\Upsilon _{3.6}$ =0.6M$_\\odot $ /L$_\\odot $ by e.g.","[9] and [12]).","It is important to note that our results are independent of systematic uncertainties such as the choice of IMF.","The maximal disc of NGC 1291: The best fit models have borderline a maximal disc on average, with the disc contributing (on average) 74% of the total rotation velocity.","It is worth noting that we are using the definition of Sackett (1997) for disc maximality, which examines the rotation velocity at 2.2$h_r$ ; in this context, and because the scalelength of NGC 1291 is unusually large (5.8 kpc), we are examining the rotation velocity at a radius of almost 13 kpc.","Therefore, even for cases where the disc is not according to this definition “maximal”, the stellar disc dominates the rotation curve in the inner regions of the galaxy.","The height function of the disc: It's furthermore important to note that for this galaxy, models with exponential scaleheights lead to sub maximal discs, while models with an isothermal scaleheight lead to maximal discs.","This indicates that the choice of height function, especially for such borderline cases, can affect whether we obtain a maximal disc or not.","Additionally, the scaleheight affects the results, as was already shown in previous works, such as in [6], where we showed the important effect a boxy/peanut height function can have on the models.","Fast rotating bar: The best fit models have fast rotating bars with $\\mathcal {R}$ $\\le $ 1.4.","This is consistent with observational estimates of the bar pattern speed for early type galaxies ([5] and [3]).","The results are also in agreement with theoretical values ()."]],"string":"[\n [\n \"Breaking the disc-halo degeneracy in NGC 1291 using hydrodynamic\\n simulations\"\n ],\n [\n \"Abstract We present a pilot study on the nearby massive galaxy NGC 1291, in which we aim to constrain the dark matter in the inner regions, by obtaining a dynamical determination of the disc mass-to-light ratio (M/L).\",\n \"To this aim, we model the bar-induced dust lanes in the galaxy, using hydrodynamic gas response simulations.\",\n \"The models have three free parameters, the M/L of the disc, the bar pattern speed and the disc height function.\",\n \"We explore the parameter space to find the best fit models, i.e.\",\n \"those in which the morphology of the shocks in the gas simulations matches the observed dust lanes.\",\n \"The best-fit models suggest that the M/L of NGC 1291 agrees with that predicted by stellar population synthesis models in the near-infrared ($\\\\approx$0.6\\\\,$M_{\\\\odot}/L_{\\\\odot}$), which leads to a borderline maximum disc for this galaxy.\",\n \"The bar rotates fast, with corotation radius $\\\\leq$ 1.4 times the bar length.\",\n \"Additionally, we find that the height function has a significant effect on the results, and can bias them towards lower or higher M/L.\"\n ],\n [\n \"Introduction\",\n \"The distribution of dark matter in galaxies has been a source of debate in the scientific community, with one of the main sources of uncertainty being the “disc-halo degeneracy” ([15]).\",\n \"This degeneracy arises because rotation curve decompositions depend critically on the mass-to-light ratio (M/L or $\\\\Upsilon $ ) which is assigned to the stellar disc component.\",\n \"Rotation curves can be fit equally well with barely any disc contribution, or where the disc contributes the maximum amount possible – also known as the “maximum disc hypothesis” ([15], [13]).\",\n \"There is no conclusive evidence to either prove or disprove the maximum disc hypothesis and in fact there are a number of arguments both for and against it for the Milky-Way (e.g.\",\n \"[13]) as well as for external galaxies (e.g.\",\n \"[2], [16], ).\",\n \"A way to break the degeneracy between the disc and the dark matter halo is by obtaining dynamical estimates of the M/L ratio of the disc.\",\n \"One way to do this is via the non-axisymmetries induced by the bar, which shocks the gas and leads to the well known “dust lanes” observed in barred galaxies ([10]).\",\n \"These shocks can be reproduced in hydrodynamic simulations ([1]), and their morphology is dependent on the M/L of the disc, the bar pattern speed and the height function of the disc.\",\n \"In this study we examine the dust lanes present in the nearby galaxy NGC 1291, and aim to reproduce their morphology using hydrodynamic gas response simulations.\",\n \"The major assumption of this method for obtaining the M/L is that the halo is axisymmetric, and that all the non-axisymmetry in the potential is due to the stellar component.\"\n ],\n [\n \"Comparing the models to the observations\",\n \"There are three main free parameters in the models, and we vary these parameters in order to explore the allowed parameter space.\",\n \"This leads to more than 300 simulations, from which the best fit models are obtained according to how well the shock loci in the models match those in the observations.\",\n \"This is done by placing pseudo-slits perpendicular to the shocks, as shown in Figure REF .\",\n \"The three free parameters are varied as follows: M/L.\",\n \"To obtain the potential we use a 3.6$\\\\mu m$ image of NGC 1291 from the S$^4$ G survey ([14]) to which we apply a M/L.\",\n \"We assume a fiducial value of $\\\\Upsilon _{3.6}$ =0.6$M_{\\\\odot }/L_{\\\\odot }$ for the 3.6$\\\\mu $ m band ([9], [12]), colour corrected to account for the presence of dust by combining information from the 3.6 and 4.5$\\\\mu $ m images ([4], [11]).\",\n \"We then vary the M/L from 1.5 to 0.25 $\\\\Upsilon _{3.6}$ in steps of 0.25 and add a dark matter halo which maintains the total rotation curve flat at a value of 220 km/s (see Figure REF ) as found from the Baryonic Tully-Fisher relation ([8]).\",\n \"Pattern speed (Lagrangian radius).\",\n \"By varying the Lagrangian radius while keeping all the other parameters of a model constant, we effectively vary the pattern speed of the bar.\",\n \"The Lagrangian radius is varied between 1 and 2 times the bar semi-major axis, in steps of 0.2.\",\n \"Height function and scaleheight.\",\n \"We vary the height function of the disc by assuming two functional forms – isothermal and exponential – and we also vary the scaleheight from $z_0$ =0.5 to $z_0$ =1.5 kpc.\"\n ],\n [\n \"Results\",\n \"Dynamical M/L: The best fit models obtained in this study have a M/L in the range of M/L$_{3.6}$ =0.6-0.75 which falls within the range predicted by stellar population synthesis models and other approaches (i.e.\",\n \"$\\\\Upsilon _{3.6}$ =0.6M$_\\\\odot $ /L$_\\\\odot $ by e.g.\",\n \"[9] and [12]).\",\n \"It is important to note that our results are independent of systematic uncertainties such as the choice of IMF.\",\n \"The maximal disc of NGC 1291: The best fit models have borderline a maximal disc on average, with the disc contributing (on average) 74% of the total rotation velocity.\",\n \"It is worth noting that we are using the definition of Sackett (1997) for disc maximality, which examines the rotation velocity at 2.2$h_r$ ; in this context, and because the scalelength of NGC 1291 is unusually large (5.8 kpc), we are examining the rotation velocity at a radius of almost 13 kpc.\",\n \"Therefore, even for cases where the disc is not according to this definition “maximal”, the stellar disc dominates the rotation curve in the inner regions of the galaxy.\",\n \"The height function of the disc: It's furthermore important to note that for this galaxy, models with exponential scaleheights lead to sub maximal discs, while models with an isothermal scaleheight lead to maximal discs.\",\n \"This indicates that the choice of height function, especially for such borderline cases, can affect whether we obtain a maximal disc or not.\",\n \"Additionally, the scaleheight affects the results, as was already shown in previous works, such as in [6], where we showed the important effect a boxy/peanut height function can have on the models.\",\n \"Fast rotating bar: The best fit models have fast rotating bars with $\\\\mathcal {R}$ $\\\\le $ 1.4.\",\n \"This is consistent with observational estimates of the bar pattern speed for early type galaxies ([5] and [3]).\",\n \"The results are also in agreement with theoretical values ().\"\n ]\n]"},"id":{"kind":"string","value":"1605.05754"}}},{"rowIdx":1866676,"cells":{"text":{"kind":"list like","value":[["Applications of fidelity measures to complex quantum systems"],["Abstract We revisit the fidelity as a measure for the stability and the complexity of the quantum motion of single and many-body systems.","Within the context of cold atoms, we present on overview of applications of two fidelities which we call static and dynamical fidelity, respectively.","The static fidelity applies to quantum problems which can be diagonalized since it is defined via the eigenfunctions.","In particular, we show that the static fidelity is a highly effective practical detector of avoided crossings characterizing the complexity of the systems and their evolutions.","The dynamical fidelity is defined via the time-dependent wave functions.","Focussing on the quantum kicked rotor system, we highlight a few practical applications of fidelity measurements in order to better understand the large variety of dynamical regimes of this paradigm of a low-dimensional system with mixed regular-chaotic phase space."],["Introduction","Generally it is very useful to have a working tool for the characterization of systems and their dynamical evolution.","An experimental technique with great value for practical measurement applications is the echo method, in particular known as spin or Hahn echo and much used in nuclear magnetic resonance experiments [1].","Echos are necessarily dynamical functions characterizing the quality and the difference between forward and backward evolutions.","As a consequence, it was quite natural to suggest the echo in the context of dynamical systems as a measure for the dynamical spreading of trajectories, which was done by Asher Peres back in 1984 [2].","This quantity is defined as the overlap of two quantum mechanical wave function developed with two slightly different Hamiltonians, whose difference is quantified by the change of an appropriate control parameter.","While it turned out that a categorization of fidelity decays for different dynamical systems, with classically integrable, mixed regular-chaotic or fully chaotic phase space, is generally not simple and may depend on non universal details of the systems, it is well established that several universal decay regimes can be found characterized not so much by the systems' properties but by the perturbation strength.","These universal regimes are known as perturbation regime, Gaussian decay and Fermi Golden Rule regime [3] and they are well described in recent reviews [4], [5].","In this paper we present a compact overview of the use of the overlap function, vulgo fidelity, in the broader context of complex quantum systems, which range from quantum many-body models [6] to strongly driven dynamical quantum systems [7].","We review two concepts of overlap functions relevant for cold-atom experiments and which both are equally relevant for characterizing the response of quantum systems to perturbations as well as their dynamical evolution.","The first concept is based on the static overlap of two eigenfunction computed for two slightly different control parameters, hence it is basically a spectral characterization of the system at hand.","The second concept is the original fidelity introduced by Peres [2], which represents the overlap of two dynamical wave functions taken at equal times but evolved with two different Hamiltonians up to this time.","Both measures are useful to characterize the temporal evolution of a problem, since the latter is connected with the spectrum by the spectral theorem, of course.","What we mean is that knowing all static overlaps between eigenfunctions at different control parameters and their corresponding eigenvalues, we can in principle reconstruct the dynamical overlap function.","This is true for Hamiltonians which either do not depend explicitly on time or are periodically time-dependent (since then we can use Floquet theory to arrive at an eigenbasis).","The advantage of the purely dynamical overlap function is that it can be computed also for Hamiltonians which are explicitly time dependent, and there is no need for computing eigenvalues and eigenfunctions, which can be a true practical problem for large quantum few-body or many-body systems [9], [8]."],["Static fidelity measure","Given a control parameter $\\lambda $ and a perturbation $V$ , we are studying the following class of Hamiltonians $H(\\lambda ) = H_0 + \\lambda V$ .","Then the static fidelity between the $n$ -th eigenstates (assuming normalized states, a discrete spectrum, and an ordered spectrum such as $E_{n+1} \\ge E_n$ ), denoted by $|n\\rangle $ , of the two Hamiltonians $H(\\lambda )$ and $H(\\lambda + \\delta \\lambda )$ is defined as $f_n(\\lambda , \\delta \\lambda ) \\equiv \\vert \\langle n(\\lambda ) | n(\\lambda +\\delta \\lambda )\\rangle \\vert \\,.$ Here the change in the control parameter $\\delta \\lambda $ is assumed to be small.","Locally in the spectrum, the quantity $f_n(\\lambda ,\\delta \\lambda )$ characterizes the change of the eigenfunctions.","If the latter change adiabatically, the value of $f_n(\\lambda ,\\delta \\lambda )$ will stay close to one.","On the other hand, if we encounter avoided level crossings, at which the eigenfunctions strongly mix, the value of $f_n(\\lambda ,\\delta \\lambda )$ will substantially decrease.","To better detect and characterize avoided crossings for a given quantum level $n$ , we better investigate the fidelity change [6], also known as fidelity susceptibility in condensed-matter literature [10], defined by $S_n(\\lambda , \\delta \\lambda ) \\equiv \\frac{1 - f_n(\\lambda , \\delta \\lambda ) }{(\\delta \\lambda )^2} \\,.$ For most of the spectra so far encountered in practice, as long as $\\delta \\lambda \\ll 1$ , $S_n$ turned out to be largely independent of $\\delta \\lambda $ , i.e.","$S_n(\\lambda , \\delta \\lambda ) \\approx S_n(\\lambda )$ , and vanishingly small except in the vicinity of an avoided level crossing.","The independence of $\\delta \\lambda $ follows in the limit $\\delta \\lambda \\rightarrow 0$ directly from perturbation theory.","The independence of $\\delta \\lambda $ has a simple origin, namely that the first non-vanishing contribution to $f_n$ in the perturbation expansion of the changed state $|n(\\lambda +\\delta \\lambda )\\rangle $ is of second order in $\\delta \\lambda $ , a well known fact from quantum perturbation theory.","The fidelity measure (REF ) and the fidelity susceptibility (REF ) had found a nice application in the characterization of ground states of quantum many-body systems [11], [12], [10].","$S_n$ detects qualitative changes in the ground state very well, when they are connected to an avoided level crossing.","Since most of quantum phase transitions in finite systems are characterized by a gap, $S_n$ acts as an effective detector of these transitions and one can check its scaling with the system size, for instance, to pin down the interesting parameter value at which the transitions occurs.","Our definition above generalizes the utility of $S_n$ to gaps, e.g.","avoided crossings, lying anywhere in the spectrum.","Figure: (a) Energy spectrum of Eq.","() for x=0,y=2,z=3x=0, y =2, z=3.","All levels are coupled and the spectrum shows two close avoided crossings; (b) fidelity change S n (λ)S_n(\\lambda ) and (c) renormalized curvature C n (λ)C_n(\\lambda ) for the three levels shown in (a).","Data adapted from .Of course, the computation of $S_n$ is based on the eigenfunctions, which is a practical disadvantage for large quantum systems, where its simpler to compute just the eigenvalues (and not the eigenfunctions explicitly) with optimized numerical methods.","Yet this problem turns into an advantage in the sense that it turned out that $S_n$ detects a large percentage, close to one hundred per cent, of the total number of avoided crossings in either a region of the spectrum or the entire spectrum of complex quantum systems.","Hence, compared to other measures, e.g.","which are based solely on the eigenvalues and their change with the control parameter, our measure $S_n$ is very effective and reliable for the practical numerical detection of avoided crossings all over the spectrum.","Figure: (a) Section of the energy spectrum of the Bose-Hubbard model from Eq.","() with F=0F=0 and for U=uU = u and J=1-uJ = 1 - u for a small system 5 particles in 5 lattice sites.","One level undergoing several avoided crossings is highlighted by the thick (blue) line.","Some crossings are not resolved on the scale of the figure but marked by circles.","(b) Individual fidelity measures S n S_n of the different eigenstates (dashed lines) and the highlighted level of (a) (thick solid blue line).","In this example all avoided crossings are effectively and easily detected.There is an intrinsic connection between our measure $S_n$ and the local level curvature in the spectrum.","In order to see this connection, let us expand the eigenfunction $|n(\\lambda +\\delta \\lambda )\\rangle $ in second order in $\\delta \\lambda $ $S_n(\\lambda ) \\approx \\frac{1}{2} \\sum _{m\\ne n} \\frac{\\vert \\langle m(\\lambda ) | V | n(\\lambda )\\rangle \\vert ^2}{[E_n-E_m]^2} \\approx \\frac{\\vert \\langle n^{\\prime }(\\lambda ) | V | n(\\lambda )\\rangle \\vert ^2}{2\\;[E_n-E_{n^{\\prime }}]^2}\\,.$ In the last step, we reduced the sum near an isolated avoided crossing to the nearest neighbour level $n^{\\prime }$ .","Similarly, one can expand the renormalized curvature with the gap function $\\Delta (\\lambda )=|E_{n}-E_{n^{\\prime }}|$ , also in second order to approximate the level curvature by [13] $C_{n}(\\lambda ) \\equiv \\bigg |\\frac{1}{\\Delta (\\lambda )}\\frac{\\partial ^2 E_{n}(\\lambda )}{\\partial \\lambda ^2}\\bigg | \\approx \\bigg |\\frac{2}{\\Delta (\\lambda )} \\sum _{m\\ne n} \\frac{\\vert \\langle m(\\lambda ) | V | n(\\lambda )\\rangle \\vert ^2}{E_n-E_m}\\bigg | \\\\ \\approx 2\\frac{\\vert \\langle n^{\\prime }(\\lambda ) | V | n(\\lambda )\\rangle \\vert ^2}{[E_n-E_{n^{\\prime }}]^2}\\,.$ Within this approximation, which is obviously exact for a two level system, we have the following interesting relation between the measure $S_n$ and the level curvature $C_n(\\lambda ) \\approx 4S_n(\\lambda )$ .","This in turn justifies the use of $S_n$ for the numerical detection of avoided crossings, which are intimately connected to changes in the curvature of the eigenvalues as a function of the control parameter.","Of course, the last steps in the Eqs.","(REF ) and (REF ) are crude approximations.","Experience tells that on average over a full spectrum they are quite good though.","In any case, our numerical results and computations in general do not make use of them explicitly.","From the practical point of view it turns out the detection of avoided crossings by the fidelity is much more reliable than by using the levels.","This defines the value of $S_n$ for practical numerical work.","We are testing the measure from Eq.","(REF ) in the following.","We start with a very simple three state model described by the real symmetric matrix $H(\\lambda ) = \\left(\\begin{array}{ccc} 0 & x & y \\\\ x & 0 & z \\\\ y & z & 0 \\end{array}\\right) + \\lambda \\left(\\begin{array}{ccc} -1 & 0 & 0 \\\\ 0 & 0 & 0 \\\\ 0 & 0 & 1 \\end{array}\\right) = \\left(\\begin{array}{ccc} -\\lambda & x & y \\\\ x & 0 & z \\\\ y & z & \\lambda \\end{array}\\right) \\,.$ Figure REF nicely shows that $S_n$ is able to detect and to distinguish two nearby avoided crossings.","As an example of a quantum many-body system, we study now the class of Hamiltonians of the form $ \\hat{H}(t) = - J \\sum _{l=1}^{L} (\\text{e}^{i Ft} \\hat{a}_{l+1}^{\\dagger }\\hat{a}_{l} + \\text{h.c.}) + U \\sum _{l=1}^{L} \\hat{n}_l(\\hat{n}_l-1) \\,.$ $\\hat{a}_{l}^{\\dagger }$ and $\\hat{a}_{l}$ are creation and annihilation operators of bosons at the lattice site $l$ and $\\hat{n}_l=\\hat{a}_{l}^{\\dagger }\\hat{a}_{l}$ is the corresponding number operator.","For $F=0$ , this model is the well known Hubbard model with onsite interactions (last term) for bosonic ultracold atoms [14], [17].","Figure REF shows a scan of the many-body spectrum of a small system of five atoms in five lattice sites as a function of $U\\equiv u$ , fixing $J=1-u$ and $F=0$ .","We see that all avoided crossings are detected by our fidelity measure.","The next figure REF shows a histogram count of avoided crossings as a function of $U/J$ now.","The zone in parameter space with a large number of crossings corresponds to the spectral region which follows a Wigner-Dyson distribution for the nearest neighbour spacings in the spectrum (one possible definition of quantum chaotic system [15], [20]).","This is shown by the thick/red solid line in the figure.","In ref.","[18] further details may be found on the connection between quantum chaos, avoided crossings and also entanglement measures in time-independent Bose-Hubbard systems.","Figure: (a) Local density of avoided crossings ρ AC \\rho _{AC} (left axis) for N EL N_{EL} energy levels from the centre of the spectrum for the Bose-Hubbard model from Eq.","() as a function of U/J for 8 bosons in 7 lattice sites (F=0F=0): N EL N_{EL} = 10 (dashed), 100 (solid) and ca.","400 eigenstates(blue histogram); a χ 2 \\chi _2, see for details, test is also shown (right axis and thick/red solid line) with values close to zero for good Wigner-Dyson (quantum chaotic) statistics of the full spectrum.","The central part of the spectrum follows the overall distribution of crossings, but it has a larger local density of crossings especially around the maximum of the distribution.","Interestingly, this maximum is found close to the Mott-insulator transition in the ground state of our 1D Bose-Hubbard model , a fact that most likely has do with the reordering of the spectrum around this transition briefly discussed in .The case with finite $F$ represents a paradigm of a many-body Floquet Hamiltonian, which is analyzed in detail in references [13], [6], [8].","Ref.","[6] shows, in particular, that the transition from the regular to chaotic regime as a function of $F$ can be well captured by the analysis of the width distribution of the detected avoided crossings[19], [6] $P(c) = (1-\\gamma ) \\delta (c) + \\frac{2\\gamma ^2}{\\pi \\bar{c}}\\;\\text{exp}\\left[-\\frac{\\gamma ^2c^2}{\\pi \\bar{c}^2}\\right]\\,.$ We use the fidelity measure $S_n$ to find all avoided crossings and then just compute the difference $c$ of nearest neighbouring levels at the avoided crossings from the spectrum independently.","This distribution has a chaotic part of weight $0\\le \\gamma \\le 1$ and a finite regular component, which is visible as a strong enhancement of $P(c)$ close to zero, see the $\\delta (c)$ function.","$\\bar{c}$ is the mean value of the width in the ensemble of avoided crossings.","$P(c)$ separates better the regular from the chaotic components of a given quantum spectrum than the nearest neighbor level spacing distribution.","Using the fidelity change (REF ), the perturbation may or may not preserve symmetries intrinsic to the system.","Hence, for an analysis based on our static fidelity measure a splitting of the basis into symmetry reduced subspaces may not be strictly necessary (for symmetry breaking perturbations) in order to characterize the chaotic properties of a quantum system.","This is another advantage with respect to standard measure such as the nearest level spacing distributions [15], [20].","The static fidelity measure (REF ) is hence a very reliable technique for analysing complex quantum spectra.","Its great advantage is that it can easily be automatized and it detects almost all avoided crossings.","The measure only fails in cases where two avoided crossings get too close to each other such that their clear separation is not possible any more.","Too close means here that their distance in the control parameter becomes comparable to their width in energy.","Such close coincidences of crossings of comparable strength are from our experience rather rare in typical quantum systems.","Besides in the mentioned applications for detecting avoided crossings (see above) and, in particular, quantum phase transitions [11], the authors of [9] have recently applied our measure to characterize the complexity of an electronic two-body quantum dot systems based on heavy numerical diagonalizations.","We finally note that the concept of the static fidelity measure based on Eq.","(REF ) is easily extended to matrices of overlaps of all eigenfunctions at given control parameters, see e.g.","[21].","The disadvantage is then, however, that one obtains too much information, which may be hard to interpret, and one must compute, of course, all the eigenfunctions of a given system in order to get all the non-diagonal elements of this generalized fidelity matrix."],["Dynamical fidelity function","The stability of quantum evolution against parametric changes is a subject of broad theoretical and experimental interest.","A widely used concept is the fidelity introduced by Peres [2], and the closely related Loschmidt echo [4], [5], which is built as the interference pattern between wave functions that are obtained by propagating the same initial state under two slightly different Hamiltonians.","The behaviour of fidelity, e.g.","the quantity $F(t)=\\left| \\langle \\psi _{\\lambda +\\delta \\lambda } (t) | \\psi _ {\\lambda }(t) \\rangle \\right|^2 \\,,$ in time is known to display some universal properties that reflect also the underlying classical dynamics [4], [5].","Not much is known, however, for the temporal evolution of $F(t)$ for long times, in particular, for classically nearly integrable and mixed regular-chaotic systems.","An optimal platform for experiments on fidelity in low dimensional chaotic systems is provided by the quantum kicked rotor model.","It is realized with cold and ultracold (Bose-Einstein condensates) atoms [22], [7] and with light pulses in optical fibers [23].","The used techniques to measure fidelity range from interferometric methods in either internal atomic states [24] or in the centre-of-mass motion of the atoms [25] to the time reversal of the dynamics by exploiting the properties of the quantum resonant motion [26]."],["The quantum kicked rotor and fidelity","Experiments on the kicked rotor based on (ultra)cold atoms work with particles moving along a line, periodically kicked in time by an optical lattice, and possibly subject to a constant Stark or gravity field.","Neglecting atom-atom interactions, the quantum dynamics is described by the following Hamiltonian in dimensionless variables (such that $\\hbar =1$ ) [27], [7], [28]: $\\hat{H}(\\hat{x},\\hat{p},t)\\;=\\;\\frac{\\hat{p}^2}{2}-\\frac{\\eta }{\\tau }\\; \\hat{x}+ k\\cos ( \\hat{x}) \\sum _{{\\tt t}\\in \\mathbb {Z}}\\delta (t-{\\tt t}\\tau )\\ .$ The kick period is $\\tau $ , the kick strength is $k$ , and ${\\tt t}$ is a discrete time variable that counts the number of kicks.","The additional parameter $\\eta $ yields the change in momentum produced by the constant field over one kick period.","In the accelerated (by gravity) frame of reference, the total potential is again spatially periodic, implying the conservation of quasi-momentum $\\beta $ .","With the chosen units, $\\beta $ can take on allowed values between 0 and 1.","Using Bloch theory, the atom dynamics from immediately after the $({\\tt t}-1)$ -th kick to immediately after the next ${\\tt t}$ -th kick is then described by the unitary operator, which is derived in detail in [27]: $\\hat{\\cal U}_{\\beta ,k,\\eta }({\\tt t})\\;=\\;e^{-\\mathrm {i}k \\cos (\\hat{\\theta })}\\; e^{-\\mathrm {i}\\tau /2(\\hat{\\cal N}+ \\beta +\\eta {\\tt t} +\\eta /2)^2}\\ .$ The full evolution operator over the first ${\\tt t}$ kicks is given by $\\hat{\\cal U}_{\\beta ,k,\\eta }^{{\\tt t}}\\;\\equiv \\;\\hat{\\cal U}_{\\beta ,k,\\eta }({\\tt t-1})\\; \\hat{\\cal U}_{\\beta ,k,\\eta }({\\tt t-2})\\dots \\hat{\\cal U}_{\\beta ,k,\\eta }({\\tt 1})\\; \\hat{\\cal U}_{\\beta ,k,\\eta }({\\tt 0}) \\,.$ where $\\hat{\\cal N}= - \\mathrm {i}\\frac{\\mathrm {d}}{\\mathrm {d}\\theta }$ is the (angular) momentum operator with periodic boundary conditions.","The time-dependent Hamiltonian that generates the quantum evolution corresponding to (REF ) would be $\\hat{\\cal H}({\\hat{\\cal N}},\\hat{\\theta },\\beta ,t)= \\frac{1}{2}\\left(\\hat{\\cal N} +\\beta +\\frac{\\eta }{\\tau } t\\right)^2+ k\\cos (\\hat{\\theta }) \\sum _{{\\tt t}\\in \\mathbb {Z}}\\delta (t-{\\tt t}\\tau )\\ .$ The temporal evolution of the fidelity can now be studied with respect to changes of one of the parameter in Eq.","(REF ).","Motivated by the experiments reported in Refs.","[24], [29], we consider in the following changes in the kick strength $k$ and the gravity parameter $\\eta $ .","For a given quasimomentum $\\beta $ , the dynamical fidelity is defined as $F_\\beta (k_1,k_2,\\eta _1, \\eta _2, {\\tt t}) = \\Big | \\left< \\hat{\\cal U}_{\\beta ,k_1,\\eta _1}^{{\\tt t}} \\psi _\\beta (0) \\Big |\\hat{\\cal U}_{\\beta ,k_2,\\eta _2}^{{\\tt t}} \\psi _\\beta (0)\\right>\\Big |^2 \\,.$ In a typical experimental situations with cold atoms [32], [31], [25], [24], [7], [29], [30], the atoms are non interacting and each of them has in good approximation a well defined initial momentum, meaning that its quasimomentum is essentially fixed.","This ensemble is best approximated by a statistical density operator $\\hat{\\rho } = \\int _\\beta \\mathrm {d}\\beta \\rho (\\beta ) \\mathinner {|{\\psi _\\beta (0)}\\rangle }\\mathinner {\\langle {\\psi _\\beta (0)}|}$ .","Then the fidelity generalized to density operators [5] gives the slightly more involved formula for our system of cold atoms [33], [28]: $F(k_1,k_2,\\eta _1, \\eta _2,{\\tt t}) =\\bigg |\\int _\\beta \\rho (\\beta ) \\left< \\hat{\\cal U}_{\\beta ,k_1,\\eta _1}^{{\\tt t}} \\psi _\\beta (0) \\Big |\\hat{\\cal U}_{\\beta ,k_2,\\eta _2}^{{\\tt t}} \\psi _\\beta (0)\\right> \\mathrm {d}\\beta \\bigg |^2 \\,.$ A series of experimental investigations has looked closely at the so-called quantum resonant motion of the quantum kicked rotor over the last fifteen years, see e.g.","[7], [31] and references therein.","This interest is mainly motivated by the type of ballistic motion with fast acceleration which can be obtained in this particular region of parameters.","The quantum resonances allow one the realization of ratchet-like dynamics as well, when breaking the spatial-temporal symmetries of the kicked rotor [7], [34].","We restrict here, as most of previous works, to the main quantum resonances occurring whenever $\\tau $ is not only commensurate to $2\\pi $ but an integer multiple of it, i.e.","$\\tau =2\\pi \\ell $ , $\\ell $ integer.","Then, for specially chosen values of quasi-momentum $\\beta $ , the energy of the rotor with this fixed $\\beta $ increases quadratically as ${\\tt t}\\rightarrow +\\infty $ [35], [27].","In the presence of gravity, asymptotic quadratic growth of energy at certain values of $\\beta $ is still possible close to resonant values of the kick period[27].","From the theoretical point of view, these quantum resonances are convenient since we have analytical access to the wave functions [35], [27] and hence to analytical estimates for the development of fidelity [33], [28].","Using pseudoclassical approximation theory [35], [27], [37], [7], this is true also for small detuning from the resonance conditions.","We review here just the result of simple analytical considerations, whose details can be found in [33], [28].","The fidelity at a main quantum resonance is considered in two cases: first for the perturbation parameter $\\Delta k=k_2-k_1$ and a fixed quasimomentum $\\beta $ .","The fidelity is then computed from the overlap $\\left< \\hat{\\cal U}_{\\beta ,k_1,\\eta }^{{\\tt t}}\\psi _\\beta (0) \\Big |\\hat{\\cal U}_{\\beta ,k_2,\\eta }^{{\\tt t}} \\psi _\\beta (0)\\right>= J_0(\\Delta k|W_{{\\tt t}}|)\\ ,$ such as to yield $F_\\beta (k_1,k_2,\\eta ,{\\tt t})= \\Big |J_0(\\Delta k |W_{{\\tt t}}|)\\Big |^2 \\,.$ The argument of the zero-order Bessel function contains the following sum of phases: $W_{{\\tt t}}\\equiv W_{{\\tt t}}(\\eta ,\\beta )=\\sum _{r=0}^{{\\tt t}-1}e^{-\\mathrm {i}\\pi \\ell (2\\beta +1)r}e^{-\\mathrm {i}2\\pi \\ell r \\eta {\\tt t}+\\mathrm {i}\\pi \\ell \\eta r^2}\\ .$ Applying Eq.","(REF ) at $\\eta =0$ and resonant $\\beta =\\beta _{res}$ (e.g.","$\\beta _{res}=0$ for $\\ell =2$ ) we have simply $W_{{\\tt t}}={\\tt t}$ , which leads to $F_{\\beta _{res}}(k_1,k_2,\\eta =0,{\\tt t})= \\Big |J_0(\\Delta k {\\tt t} )\\Big |^2 \\,.$ This fidelity decays asymptotically, for large times, like a power-law with exponent one [33].","Averaging over a full ensemble of quasimomenta in a Brillouine zone, we arrive at a fidelity which asymptotically saturates to a constant value $F(\\Delta k, {\\tt t}) \\equiv \\bigg |\\int _{\\beta =0}^{\\beta =1}\\left< \\hat{\\cal U}_{\\beta ,k_1}^{{\\tt t}} \\psi _\\beta (0) \\Big |\\hat{\\cal U}_{\\beta ,k_2}^{{\\tt t}} \\psi _\\beta (0)\\right> \\mathrm {d}\\beta \\bigg |^2\\stackrel{t \\rightarrow \\infty }{\\longrightarrow } \\frac{1}{(2\\pi )^2} \\left(\\int _{0}^{2\\pi }J_0^2\\left(\\frac{\\Delta k }{2\\sin \\alpha }\\right) \\mathrm {d}\\alpha \\right)^2\\,.$ This asymptotically exact saturation of the fidelity sets in very quickly in practice for typical experimental parameters, after ${\\tt t} > z_0/\\Delta k$ , where $z_0$ is the first zero of the zero order Bessel function [33].","An interferometric cold-atom experiment at Harvard [25] nicely confirmed our theory showing this saturation of fidelity for various perturbations $\\Delta k$ and measurements up to about 60 kicks.","In the second case, we consider the motion in a gravity field for the forward motion, whilst this field is absent in the backward motion.","The kick strength does not change here.","This gives for the fidelity, following a similar reasoning as done in [28], $F_ \\beta (\\eta , {\\tt t} ) = \\left| \\langle \\psi _\\beta (0)| \\hat{\\mathcal {U}}_{\\beta , k, {\\eta =0} }^{\\dagger {\\tt t} } \\,\\hat{ \\mathcal {U} }_{\\beta , k, \\eta }^{{\\tt t} } | \\psi _\\beta (0) \\rangle \\right| ^2= \\bigg |e^{-i\\phi ({\\beta }, {\\eta }, {\\tt t})} J_0\\left(\\sqrt{({\\tt t} k)^2+k^2 |W_{{\\tt t}}|^2 -2 {\\tt t} k^2 \\text{Re} (W_{{\\tt t}}) } \\right)\\bigg |^2 \\,.$ For a single $\\beta $ , the additional phase $\\phi ({\\beta }, {\\eta }, t) = \\ell \\pi \\sum _{q=0}^{t-1} \\left({\\beta }+q{\\eta } +/ {\\eta }2\\right)^2$ is superfluous, yet it becomes important when considering contributions from different quasimomenta, see subsection REF for this case.","Figure: (left) phase space of the pseudoclassical model for η=0,k|ϵ|=0.04π\\eta =0, k|\\epsilon |=0.04\\pi (green/grey data points) as compared to its pendulum approximation , (solid lines, where the red line shows the separatrix of the pendulum).","(right) phase space of the pseudoclassical model with gravity for the parameters τ=5.97,η=0.0257,k=1.4\\tau =5.97,\\eta =0.0257,k=1.4.","(Quasi)momentum zero corresponds to the centre of both islands on the y axis, while θ\\theta is the angle parameter of the standard map as classical limit of the kicked rotor problem.","The horizon stripe in the right plot marks the typical width of an experimental quasimomentum distribution for a Bose-Einstein condensate."],["Fidelity revivals","Even if we detune the kick period $\\tau $ slightly from the mentioned resonance conditions, we may still be able to estimate the true quantum motion using a pseudoclassical model introduced in [35], [27].","In this model, the absolute value of the detuning $|\\epsilon |$ plays the role of Planck's constant, and hence the theory has a semiclassical limit at exact quantum resonance.","The $\\epsilon $ -classical dynamics is described by the following discrete map which relates the variables $I$ and $\\theta $ from immediately after the ${\\tt t}$ -th kick to immediately after the $({\\tt t}+1)$ -th one $\\begin{array}{rcl}I_{{\\tt t}+1}&=&I_{{\\tt t}}+\\tilde{k}\\sin \\theta _{{\\tt t}+1}+\\mathrm {sgn}(\\epsilon )\\tau \\eta \\\\\\theta _{{\\tt t}+1}&=&\\theta _{{\\tt t}}+\\mathrm {sgn}(\\epsilon )I_{{\\tt t}}\\;\\;\\;\\textrm { mod } 2\\pi \\end{array}\\,.$ Here, ${I}={J}+\\mathrm {sgn}(\\epsilon )[\\pi \\ell +\\tau \\beta +\\tau \\eta {\\tt t}]$ and $J=|\\epsilon | p$ is the rescaled momentum.","Similarly, the new effective kick strength is defined by $\\tilde{k}=|\\epsilon | k$ .","Since the effective kick strength is multiplied by the small detuning in this model, the classical phase space is nearly integrable and its centre corresponds to a pendulum-like resonance island [36], [38], [37], [20], see Fig.","REF .","In this subsection, we continue now to study the case without gravity, i.e.","$\\eta =0$ (see left panel of the figure).","Within the harmonic approximation, initial states starting at momentum close to zero will oscillate around the island, with an oscillation frequency depending on the coupling strength, $\\omega _i=\\sqrt{k_i |\\epsilon |}$ ($i=1,2$ ) in this case.","Hence, while the fidelity initially can decay fast at finite detuning [39], it may recover if the two frequencies – related to the two kick strength – are commensurable.","Focusing only onto resonant values of quasimomentum, the fidelity that carries over formula (REF ) to finite detuning is $F_{\\beta _{res}}(k_1,k_2, {\\tt t},\\eta =0) \\sim \\frac{\\epsilon }{2\\pi } \\left(| \\omega _2 \\cos ( \\omega _1 {\\tt t} ) \\sin ( \\omega _2 {\\tt t}) - \\omega _1 \\cos ( \\omega _2 {\\tt t} ) \\sin ( \\omega _1 {\\tt t}) | \\right)^{-1} \\,,$ where the symbol $\\sim $ means that this result is asymptotically true for small $\\epsilon $ and large times.","The predicted revivals of the fidelity are typical for integrable motion around nonlinear islands characterized by their winding numbers ($\\omega _i$ measured in number of kicks) in the Poincaré sections.","Typical results of fidelity revivals are reported in Fig.","REF .","Figure: Slightly smoothed version of the fidelity as predicted by Eq.","() (thick black line) and numerical data(grey/red curve) for k 1 =0.8π,k 2 =0.6πk_1=0.8\\pi , k_2=0.6\\pi , and detuning ϵ=0.01\\epsilon =0.01.","Please note that the fidelity does start at 1 on the yy axis,which is not visible here owing to the fast initial drop after just a few kicks.","Data adapted from ."],["Dynamical tunneling","This subsection deals now with the case of a perturbation in the kick strength and one fixed value of gravity $\\eta \\ne 0$ .","All we want to say here is essentially contained in Fig.","REF .","While initially, for short times, the fidelity suffers a fast exponential decay, this decay and the further evaluation is modulated with almost periodic oscillations.","This oscillations are the better visible, the more we average over initial conditions (in this case different values of quasimomenta).","The initial decay is arising from the part of the initial wave packet that lies in the chaotic component of either of the two dynamics (defined by the two different kick strengths) in the pseudoclassical phase space, see right panel of Fig.","REF .","While its behaviour seems to be non-uniform and strongly dependent on parameters, we can estimate the oscillation frequency by a similar argument as in the previous subsection (REF ).","The simplest form of this estimate gives an oscillation frequency of $|\\omega _1-\\omega _2|+\\Delta \\theta $ , where $\\omega _i$ ($i=1,2$ ) were defined above and $\\Delta \\theta $ is the shift of the island centre in phase space induced by the change in the kick strength at finite gravity.","This shift of the frequency can be a understood in the following way: for the elliptic motion in one island the trajectory is ahead, whereas for the elliptic motion in the other island the trajectory is behind, so they get closer to each other and rephase after a shorter period than in the case when both islands lie concentric.","The shift, which we estimate by $\\Delta \\theta $ , depends on the precise geometry of the two islands, and the overlapping area.","Already the data in the left panel of Fig.","REF shows a significant change in slope of the overall fidelity decay after some time, see the solid line at about 100 kick.","We can analyze this asymptotic behaviour better focussing on an initial wave packet with just one single initial quasimomentum, see right panel of Fig.","REF .","The fidelity is – after the initial decay – dominated by the parts of the wave packet which are trapped inside the islands, similar to the one shown in Fig.","REF (right panel).","We chose our initial states in the form of Gaussian states mostly located inside one island.","The decay from the islands into the chaotic sea is a well known phenomenon known as dynamical tunneling, see in this context [40], [30] and refs.","therein.","We will argue now that this is the main mechanism responsible of the fidelity decay at long times.","The most important contribution to fidelity comes from the parts of each factor in the scalar product in (REF ) that are trapped in the respective travelling island.","The decay of fidelity is then in turn determined by the decay of each of these two parts, which is determined by its respective tunneling rate $\\Gamma _{i}$ ($i=1,2$ ) into the chaotic sea.","Based on this semiclassical reasoning, we came up with the following ansatz for the asymptotic decay of fidelity [28] $F({\\tt t})\\;\\propto \\;{\\mu ({\\cal A}_1\\backslash {\\cal A}_2) e^{-\\Gamma _1 {\\tt t}} +\\mu ({\\cal A}_2\\backslash {\\cal A}_1) e^{-\\Gamma _2 {\\tt t}} + \\mu ({\\cal A}_2\\cap {\\cal A}_1)e^{-(\\Gamma _1+\\Gamma _2) {\\tt t}}} \\ .$ Here $\\mu $ is the classical invariant measure of phase space sets, ${\\cal A}_i$ is the island in the phase space around the fixed point associated with $k_i$ .","In our simulations, the phase-space areas appearing in (REF ) were estimated as described in the appendix of [28].","The quantum decay rates due to dynamical tunneling were computed numerically for each $k_i$ .","To do so we computed the survival probability as shown in Fig.","REF by the blue dashed line.","Our semiclassical formula (REF ) provides a simple yet powerful estimate for the fidelity decay since it is rather universal and applicable to a large variety of systems.","We finish this subsection by stating that first experimental measurements are on there way to detect signatures of dynamical tunneling of states prepared in quantum accelerator mode islands such as shown in the right panel of Fig.","REF , see ref.","[30].","The problem will remain, of course, to reach a high enough experimental stability to get access to the evolution to times longer than just a few dozens of kicks.","Figure: (left) Fidelity averaged over a broad ensemble of quasimomenta in [0,1)[0,1) (corresponding to a cold-atom experiment with wideinitial momentum distribution) for the parameters τ=5.86,η=0.01579τ,k 1 =0.8π,k 2 =0.75π\\tau = 5.86, \\eta = 0.01579\\tau , k_1 = 0.8\\pi , k_2 = 0.75\\pi (dashed line) and k 2 =0.7πk_2 = 0.7\\pi (solid line).","(right) Comparison between the survival probability (black dots and thin blue line) and the fidelity for one fixed quasimomentum(red/grey thick solid line) for τ=5.86\\tau =5.86, ϵ=τ-2π\\epsilon =\\tau -2\\pi ,η=0.01579τ\\eta =0.01579\\tau , β≈0.49\\beta \\approx 0.49, k 2 =0.8πk_2=0.8\\pi , k 1 =0.7πk_1=0.7\\pi .","The upper thin blue line corresponds to k 1 k_1 and gives thetunneling rates Γ 1 =5.1×10 -4 \\Gamma _1 = 5.1 \\times 10^{-4}, whereas the lower black dots corresponds to k 2 k_2 and gives Γ 2 =4.4×10 -5 \\Gamma _2 =4.4 \\times 10^{-5}.","The crosses represent an exponential fit with a rate Γ=3.5×10 -5 \\Gamma = 3.5 \\times 10^{-5} to guide the eye.","Data in the right paneladapted from ."],["Thermometry in ultra-cold atoms","We now turn back again at exact quantum resonance conditions, more particularly to formula (REF ) from subsection REF .","From the phase term $\\phi ({\\beta }, {\\eta }, t)$ in Eq.","(REF ) it can be seen that, when quasimomentum $\\beta $ is non zero, then the total phase induced by different values of $\\eta $ depends not only on its magnitude but also on its sign.","Experiments to investigate this situation were performed at Oklahoma with a Bose-Einstein condensate of about 40 000 rubidium atoms [29].","The atoms correspond to a distribution of quasimomenta with a FWHM of about $0.05$ in our dimensionless units.","We must average Eq.","(REF ) over these initial conditions in $\\beta $ .","The data – guided by our theory – clearly indicate that the asymmetric dependence of the averaged fidelity on gravity may turn out to be useful in determining externally applied accelerations and, in turn, the temperature (given by the kinetic energy) of ultracold atoms in the limit of negligible interatomic interactions.","The immediate value of this experimental application is highlighted by the fact that just four kicks were used in Fig.","REF , making similar experiments very fast, stable and versatile.","While the experimental data from Fig.","REF is not yet as stable as one would like for a practically useful thermometer, numerical results shown in Fig.","REF confirm the linear correlation between the normalized visibility of the asymmetry and the momentum width of the atomic cloud.","We conclude this section by remarking that the quantum kick rotor in the presence of gravity, as describe by the Hamiltonian (REF ), is for generic values of $\\eta $ not reducible to a periodically time-dependent system.","Hence, we indeed need the dynamical fidelity in order to characterize the large variety of different dynamical regimes of this paradigm system of quantum chaos.","A description by the static fidelity measure of section would instead only be possible for values of $\\eta $ commensurable with $\\tau /|\\epsilon |$ , see ref.","[40] for details.","The fidelity measure has found a plethora of applications over the last sixty years, from spin and photon echos over to low-dimensional quantum chaotic systems [1], [2], [3], [4], [5].","We reviewed here two important specific fidelities, a static overlap of eigenfunctions and a dynamical overlap of generic wave functions.","The former has been successfully used to detect avoided crossings in complex quantum systems, ranging from quantum phase transitions in the ground state [11], [10], [12] to the characterization of the chaoticity of the entire or parts of quantum spectra [21], [6], [8], [9].","The much more studied dynamical fidelity has been investigated here for the paradigm system of quantum chaos, the quantum kicked rotor.","We showed several applications, ranging from characterizing quasi-regular motion close to quantum resonance, over to dynamical tunneling and its indirect use as a thermometer for ultracold atomic gases.","We hope that this overview will inspire further searches for interesting applications of the fidelity and generalized quantum echos, possible exploiting much more the phase dependence of the quantum mechanical overlap function for new precision experiments.","Figure: Numerical simulations of an asymmetry visibility defined as [F(η - )-F(η + )]/[F(η - )+F(η + )][F(\\eta _-) - F(\\eta _+)]/[F(\\eta _-) + F(\\eta _+)] show an almost linear scaling with the momentum width of the atomic cloud for the experimentally relevant range, see previous figure, Δβ≤0.08\\Delta \\beta \\le 0.08.","This linear correlation could be used to precisely define the momentum width on the xx axis via measurements of the asymmetry (on the yy axis).","Figure taken from .The author is grateful to Italo Guarneri and Gil Summy for stimulating our work on fidelity related to the dynamics of cold atoms.","Many thanks goes to the large number of students and my former postdoc Remy Dubertrand, who all were involved in specific parts of the work reviewed here.","Support by the FIL program of the Università di Parma is kindly acknowledged.","The author thanks very much the organizers of the international conference Echoes in Complex Systems at the MPIPKS, in particular Arseni Goussev, for support and the kind invitation to Dresden."]],"string":"[\n [\n \"Applications of fidelity measures to complex quantum systems\"\n ],\n [\n \"Abstract We revisit the fidelity as a measure for the stability and the complexity of the quantum motion of single and many-body systems.\",\n \"Within the context of cold atoms, we present on overview of applications of two fidelities which we call static and dynamical fidelity, respectively.\",\n \"The static fidelity applies to quantum problems which can be diagonalized since it is defined via the eigenfunctions.\",\n \"In particular, we show that the static fidelity is a highly effective practical detector of avoided crossings characterizing the complexity of the systems and their evolutions.\",\n \"The dynamical fidelity is defined via the time-dependent wave functions.\",\n \"Focussing on the quantum kicked rotor system, we highlight a few practical applications of fidelity measurements in order to better understand the large variety of dynamical regimes of this paradigm of a low-dimensional system with mixed regular-chaotic phase space.\"\n ],\n [\n \"Introduction\",\n \"Generally it is very useful to have a working tool for the characterization of systems and their dynamical evolution.\",\n \"An experimental technique with great value for practical measurement applications is the echo method, in particular known as spin or Hahn echo and much used in nuclear magnetic resonance experiments [1].\",\n \"Echos are necessarily dynamical functions characterizing the quality and the difference between forward and backward evolutions.\",\n \"As a consequence, it was quite natural to suggest the echo in the context of dynamical systems as a measure for the dynamical spreading of trajectories, which was done by Asher Peres back in 1984 [2].\",\n \"This quantity is defined as the overlap of two quantum mechanical wave function developed with two slightly different Hamiltonians, whose difference is quantified by the change of an appropriate control parameter.\",\n \"While it turned out that a categorization of fidelity decays for different dynamical systems, with classically integrable, mixed regular-chaotic or fully chaotic phase space, is generally not simple and may depend on non universal details of the systems, it is well established that several universal decay regimes can be found characterized not so much by the systems' properties but by the perturbation strength.\",\n \"These universal regimes are known as perturbation regime, Gaussian decay and Fermi Golden Rule regime [3] and they are well described in recent reviews [4], [5].\",\n \"In this paper we present a compact overview of the use of the overlap function, vulgo fidelity, in the broader context of complex quantum systems, which range from quantum many-body models [6] to strongly driven dynamical quantum systems [7].\",\n \"We review two concepts of overlap functions relevant for cold-atom experiments and which both are equally relevant for characterizing the response of quantum systems to perturbations as well as their dynamical evolution.\",\n \"The first concept is based on the static overlap of two eigenfunction computed for two slightly different control parameters, hence it is basically a spectral characterization of the system at hand.\",\n \"The second concept is the original fidelity introduced by Peres [2], which represents the overlap of two dynamical wave functions taken at equal times but evolved with two different Hamiltonians up to this time.\",\n \"Both measures are useful to characterize the temporal evolution of a problem, since the latter is connected with the spectrum by the spectral theorem, of course.\",\n \"What we mean is that knowing all static overlaps between eigenfunctions at different control parameters and their corresponding eigenvalues, we can in principle reconstruct the dynamical overlap function.\",\n \"This is true for Hamiltonians which either do not depend explicitly on time or are periodically time-dependent (since then we can use Floquet theory to arrive at an eigenbasis).\",\n \"The advantage of the purely dynamical overlap function is that it can be computed also for Hamiltonians which are explicitly time dependent, and there is no need for computing eigenvalues and eigenfunctions, which can be a true practical problem for large quantum few-body or many-body systems [9], [8].\"\n ],\n [\n \"Static fidelity measure\",\n \"Given a control parameter $\\\\lambda $ and a perturbation $V$ , we are studying the following class of Hamiltonians $H(\\\\lambda ) = H_0 + \\\\lambda V$ .\",\n \"Then the static fidelity between the $n$ -th eigenstates (assuming normalized states, a discrete spectrum, and an ordered spectrum such as $E_{n+1} \\\\ge E_n$ ), denoted by $|n\\\\rangle $ , of the two Hamiltonians $H(\\\\lambda )$ and $H(\\\\lambda + \\\\delta \\\\lambda )$ is defined as $f_n(\\\\lambda , \\\\delta \\\\lambda ) \\\\equiv \\\\vert \\\\langle n(\\\\lambda ) | n(\\\\lambda +\\\\delta \\\\lambda )\\\\rangle \\\\vert \\\\,.$ Here the change in the control parameter $\\\\delta \\\\lambda $ is assumed to be small.\",\n \"Locally in the spectrum, the quantity $f_n(\\\\lambda ,\\\\delta \\\\lambda )$ characterizes the change of the eigenfunctions.\",\n \"If the latter change adiabatically, the value of $f_n(\\\\lambda ,\\\\delta \\\\lambda )$ will stay close to one.\",\n \"On the other hand, if we encounter avoided level crossings, at which the eigenfunctions strongly mix, the value of $f_n(\\\\lambda ,\\\\delta \\\\lambda )$ will substantially decrease.\",\n \"To better detect and characterize avoided crossings for a given quantum level $n$ , we better investigate the fidelity change [6], also known as fidelity susceptibility in condensed-matter literature [10], defined by $S_n(\\\\lambda , \\\\delta \\\\lambda ) \\\\equiv \\\\frac{1 - f_n(\\\\lambda , \\\\delta \\\\lambda ) }{(\\\\delta \\\\lambda )^2} \\\\,.$ For most of the spectra so far encountered in practice, as long as $\\\\delta \\\\lambda \\\\ll 1$ , $S_n$ turned out to be largely independent of $\\\\delta \\\\lambda $ , i.e.\",\n \"$S_n(\\\\lambda , \\\\delta \\\\lambda ) \\\\approx S_n(\\\\lambda )$ , and vanishingly small except in the vicinity of an avoided level crossing.\",\n \"The independence of $\\\\delta \\\\lambda $ follows in the limit $\\\\delta \\\\lambda \\\\rightarrow 0$ directly from perturbation theory.\",\n \"The independence of $\\\\delta \\\\lambda $ has a simple origin, namely that the first non-vanishing contribution to $f_n$ in the perturbation expansion of the changed state $|n(\\\\lambda +\\\\delta \\\\lambda )\\\\rangle $ is of second order in $\\\\delta \\\\lambda $ , a well known fact from quantum perturbation theory.\",\n \"The fidelity measure (REF ) and the fidelity susceptibility (REF ) had found a nice application in the characterization of ground states of quantum many-body systems [11], [12], [10].\",\n \"$S_n$ detects qualitative changes in the ground state very well, when they are connected to an avoided level crossing.\",\n \"Since most of quantum phase transitions in finite systems are characterized by a gap, $S_n$ acts as an effective detector of these transitions and one can check its scaling with the system size, for instance, to pin down the interesting parameter value at which the transitions occurs.\",\n \"Our definition above generalizes the utility of $S_n$ to gaps, e.g.\",\n \"avoided crossings, lying anywhere in the spectrum.\",\n \"Figure: (a) Energy spectrum of Eq.\",\n \"() for x=0,y=2,z=3x=0, y =2, z=3.\",\n \"All levels are coupled and the spectrum shows two close avoided crossings; (b) fidelity change S n (λ)S_n(\\\\lambda ) and (c) renormalized curvature C n (λ)C_n(\\\\lambda ) for the three levels shown in (a).\",\n \"Data adapted from .Of course, the computation of $S_n$ is based on the eigenfunctions, which is a practical disadvantage for large quantum systems, where its simpler to compute just the eigenvalues (and not the eigenfunctions explicitly) with optimized numerical methods.\",\n \"Yet this problem turns into an advantage in the sense that it turned out that $S_n$ detects a large percentage, close to one hundred per cent, of the total number of avoided crossings in either a region of the spectrum or the entire spectrum of complex quantum systems.\",\n \"Hence, compared to other measures, e.g.\",\n \"which are based solely on the eigenvalues and their change with the control parameter, our measure $S_n$ is very effective and reliable for the practical numerical detection of avoided crossings all over the spectrum.\",\n \"Figure: (a) Section of the energy spectrum of the Bose-Hubbard model from Eq.\",\n \"() with F=0F=0 and for U=uU = u and J=1-uJ = 1 - u for a small system 5 particles in 5 lattice sites.\",\n \"One level undergoing several avoided crossings is highlighted by the thick (blue) line.\",\n \"Some crossings are not resolved on the scale of the figure but marked by circles.\",\n \"(b) Individual fidelity measures S n S_n of the different eigenstates (dashed lines) and the highlighted level of (a) (thick solid blue line).\",\n \"In this example all avoided crossings are effectively and easily detected.There is an intrinsic connection between our measure $S_n$ and the local level curvature in the spectrum.\",\n \"In order to see this connection, let us expand the eigenfunction $|n(\\\\lambda +\\\\delta \\\\lambda )\\\\rangle $ in second order in $\\\\delta \\\\lambda $ $S_n(\\\\lambda ) \\\\approx \\\\frac{1}{2} \\\\sum _{m\\\\ne n} \\\\frac{\\\\vert \\\\langle m(\\\\lambda ) | V | n(\\\\lambda )\\\\rangle \\\\vert ^2}{[E_n-E_m]^2} \\\\approx \\\\frac{\\\\vert \\\\langle n^{\\\\prime }(\\\\lambda ) | V | n(\\\\lambda )\\\\rangle \\\\vert ^2}{2\\\\;[E_n-E_{n^{\\\\prime }}]^2}\\\\,.$ In the last step, we reduced the sum near an isolated avoided crossing to the nearest neighbour level $n^{\\\\prime }$ .\",\n \"Similarly, one can expand the renormalized curvature with the gap function $\\\\Delta (\\\\lambda )=|E_{n}-E_{n^{\\\\prime }}|$ , also in second order to approximate the level curvature by [13] $C_{n}(\\\\lambda ) \\\\equiv \\\\bigg |\\\\frac{1}{\\\\Delta (\\\\lambda )}\\\\frac{\\\\partial ^2 E_{n}(\\\\lambda )}{\\\\partial \\\\lambda ^2}\\\\bigg | \\\\approx \\\\bigg |\\\\frac{2}{\\\\Delta (\\\\lambda )} \\\\sum _{m\\\\ne n} \\\\frac{\\\\vert \\\\langle m(\\\\lambda ) | V | n(\\\\lambda )\\\\rangle \\\\vert ^2}{E_n-E_m}\\\\bigg | \\\\\\\\ \\\\approx 2\\\\frac{\\\\vert \\\\langle n^{\\\\prime }(\\\\lambda ) | V | n(\\\\lambda )\\\\rangle \\\\vert ^2}{[E_n-E_{n^{\\\\prime }}]^2}\\\\,.$ Within this approximation, which is obviously exact for a two level system, we have the following interesting relation between the measure $S_n$ and the level curvature $C_n(\\\\lambda ) \\\\approx 4S_n(\\\\lambda )$ .\",\n \"This in turn justifies the use of $S_n$ for the numerical detection of avoided crossings, which are intimately connected to changes in the curvature of the eigenvalues as a function of the control parameter.\",\n \"Of course, the last steps in the Eqs.\",\n \"(REF ) and (REF ) are crude approximations.\",\n \"Experience tells that on average over a full spectrum they are quite good though.\",\n \"In any case, our numerical results and computations in general do not make use of them explicitly.\",\n \"From the practical point of view it turns out the detection of avoided crossings by the fidelity is much more reliable than by using the levels.\",\n \"This defines the value of $S_n$ for practical numerical work.\",\n \"We are testing the measure from Eq.\",\n \"(REF ) in the following.\",\n \"We start with a very simple three state model described by the real symmetric matrix $H(\\\\lambda ) = \\\\left(\\\\begin{array}{ccc} 0 & x & y \\\\\\\\ x & 0 & z \\\\\\\\ y & z & 0 \\\\end{array}\\\\right) + \\\\lambda \\\\left(\\\\begin{array}{ccc} -1 & 0 & 0 \\\\\\\\ 0 & 0 & 0 \\\\\\\\ 0 & 0 & 1 \\\\end{array}\\\\right) = \\\\left(\\\\begin{array}{ccc} -\\\\lambda & x & y \\\\\\\\ x & 0 & z \\\\\\\\ y & z & \\\\lambda \\\\end{array}\\\\right) \\\\,.$ Figure REF nicely shows that $S_n$ is able to detect and to distinguish two nearby avoided crossings.\",\n \"As an example of a quantum many-body system, we study now the class of Hamiltonians of the form $ \\\\hat{H}(t) = - J \\\\sum _{l=1}^{L} (\\\\text{e}^{i Ft} \\\\hat{a}_{l+1}^{\\\\dagger }\\\\hat{a}_{l} + \\\\text{h.c.}) + U \\\\sum _{l=1}^{L} \\\\hat{n}_l(\\\\hat{n}_l-1) \\\\,.$ $\\\\hat{a}_{l}^{\\\\dagger }$ and $\\\\hat{a}_{l}$ are creation and annihilation operators of bosons at the lattice site $l$ and $\\\\hat{n}_l=\\\\hat{a}_{l}^{\\\\dagger }\\\\hat{a}_{l}$ is the corresponding number operator.\",\n \"For $F=0$ , this model is the well known Hubbard model with onsite interactions (last term) for bosonic ultracold atoms [14], [17].\",\n \"Figure REF shows a scan of the many-body spectrum of a small system of five atoms in five lattice sites as a function of $U\\\\equiv u$ , fixing $J=1-u$ and $F=0$ .\",\n \"We see that all avoided crossings are detected by our fidelity measure.\",\n \"The next figure REF shows a histogram count of avoided crossings as a function of $U/J$ now.\",\n \"The zone in parameter space with a large number of crossings corresponds to the spectral region which follows a Wigner-Dyson distribution for the nearest neighbour spacings in the spectrum (one possible definition of quantum chaotic system [15], [20]).\",\n \"This is shown by the thick/red solid line in the figure.\",\n \"In ref.\",\n \"[18] further details may be found on the connection between quantum chaos, avoided crossings and also entanglement measures in time-independent Bose-Hubbard systems.\",\n \"Figure: (a) Local density of avoided crossings ρ AC \\\\rho _{AC} (left axis) for N EL N_{EL} energy levels from the centre of the spectrum for the Bose-Hubbard model from Eq.\",\n \"() as a function of U/J for 8 bosons in 7 lattice sites (F=0F=0): N EL N_{EL} = 10 (dashed), 100 (solid) and ca.\",\n \"400 eigenstates(blue histogram); a χ 2 \\\\chi _2, see for details, test is also shown (right axis and thick/red solid line) with values close to zero for good Wigner-Dyson (quantum chaotic) statistics of the full spectrum.\",\n \"The central part of the spectrum follows the overall distribution of crossings, but it has a larger local density of crossings especially around the maximum of the distribution.\",\n \"Interestingly, this maximum is found close to the Mott-insulator transition in the ground state of our 1D Bose-Hubbard model , a fact that most likely has do with the reordering of the spectrum around this transition briefly discussed in .The case with finite $F$ represents a paradigm of a many-body Floquet Hamiltonian, which is analyzed in detail in references [13], [6], [8].\",\n \"Ref.\",\n \"[6] shows, in particular, that the transition from the regular to chaotic regime as a function of $F$ can be well captured by the analysis of the width distribution of the detected avoided crossings[19], [6] $P(c) = (1-\\\\gamma ) \\\\delta (c) + \\\\frac{2\\\\gamma ^2}{\\\\pi \\\\bar{c}}\\\\;\\\\text{exp}\\\\left[-\\\\frac{\\\\gamma ^2c^2}{\\\\pi \\\\bar{c}^2}\\\\right]\\\\,.$ We use the fidelity measure $S_n$ to find all avoided crossings and then just compute the difference $c$ of nearest neighbouring levels at the avoided crossings from the spectrum independently.\",\n \"This distribution has a chaotic part of weight $0\\\\le \\\\gamma \\\\le 1$ and a finite regular component, which is visible as a strong enhancement of $P(c)$ close to zero, see the $\\\\delta (c)$ function.\",\n \"$\\\\bar{c}$ is the mean value of the width in the ensemble of avoided crossings.\",\n \"$P(c)$ separates better the regular from the chaotic components of a given quantum spectrum than the nearest neighbor level spacing distribution.\",\n \"Using the fidelity change (REF ), the perturbation may or may not preserve symmetries intrinsic to the system.\",\n \"Hence, for an analysis based on our static fidelity measure a splitting of the basis into symmetry reduced subspaces may not be strictly necessary (for symmetry breaking perturbations) in order to characterize the chaotic properties of a quantum system.\",\n \"This is another advantage with respect to standard measure such as the nearest level spacing distributions [15], [20].\",\n \"The static fidelity measure (REF ) is hence a very reliable technique for analysing complex quantum spectra.\",\n \"Its great advantage is that it can easily be automatized and it detects almost all avoided crossings.\",\n \"The measure only fails in cases where two avoided crossings get too close to each other such that their clear separation is not possible any more.\",\n \"Too close means here that their distance in the control parameter becomes comparable to their width in energy.\",\n \"Such close coincidences of crossings of comparable strength are from our experience rather rare in typical quantum systems.\",\n \"Besides in the mentioned applications for detecting avoided crossings (see above) and, in particular, quantum phase transitions [11], the authors of [9] have recently applied our measure to characterize the complexity of an electronic two-body quantum dot systems based on heavy numerical diagonalizations.\",\n \"We finally note that the concept of the static fidelity measure based on Eq.\",\n \"(REF ) is easily extended to matrices of overlaps of all eigenfunctions at given control parameters, see e.g.\",\n \"[21].\",\n \"The disadvantage is then, however, that one obtains too much information, which may be hard to interpret, and one must compute, of course, all the eigenfunctions of a given system in order to get all the non-diagonal elements of this generalized fidelity matrix.\"\n ],\n [\n \"Dynamical fidelity function\",\n \"The stability of quantum evolution against parametric changes is a subject of broad theoretical and experimental interest.\",\n \"A widely used concept is the fidelity introduced by Peres [2], and the closely related Loschmidt echo [4], [5], which is built as the interference pattern between wave functions that are obtained by propagating the same initial state under two slightly different Hamiltonians.\",\n \"The behaviour of fidelity, e.g.\",\n \"the quantity $F(t)=\\\\left| \\\\langle \\\\psi _{\\\\lambda +\\\\delta \\\\lambda } (t) | \\\\psi _ {\\\\lambda }(t) \\\\rangle \\\\right|^2 \\\\,,$ in time is known to display some universal properties that reflect also the underlying classical dynamics [4], [5].\",\n \"Not much is known, however, for the temporal evolution of $F(t)$ for long times, in particular, for classically nearly integrable and mixed regular-chaotic systems.\",\n \"An optimal platform for experiments on fidelity in low dimensional chaotic systems is provided by the quantum kicked rotor model.\",\n \"It is realized with cold and ultracold (Bose-Einstein condensates) atoms [22], [7] and with light pulses in optical fibers [23].\",\n \"The used techniques to measure fidelity range from interferometric methods in either internal atomic states [24] or in the centre-of-mass motion of the atoms [25] to the time reversal of the dynamics by exploiting the properties of the quantum resonant motion [26].\"\n ],\n [\n \"The quantum kicked rotor and fidelity\",\n \"Experiments on the kicked rotor based on (ultra)cold atoms work with particles moving along a line, periodically kicked in time by an optical lattice, and possibly subject to a constant Stark or gravity field.\",\n \"Neglecting atom-atom interactions, the quantum dynamics is described by the following Hamiltonian in dimensionless variables (such that $\\\\hbar =1$ ) [27], [7], [28]: $\\\\hat{H}(\\\\hat{x},\\\\hat{p},t)\\\\;=\\\\;\\\\frac{\\\\hat{p}^2}{2}-\\\\frac{\\\\eta }{\\\\tau }\\\\; \\\\hat{x}+ k\\\\cos ( \\\\hat{x}) \\\\sum _{{\\\\tt t}\\\\in \\\\mathbb {Z}}\\\\delta (t-{\\\\tt t}\\\\tau )\\\\ .$ The kick period is $\\\\tau $ , the kick strength is $k$ , and ${\\\\tt t}$ is a discrete time variable that counts the number of kicks.\",\n \"The additional parameter $\\\\eta $ yields the change in momentum produced by the constant field over one kick period.\",\n \"In the accelerated (by gravity) frame of reference, the total potential is again spatially periodic, implying the conservation of quasi-momentum $\\\\beta $ .\",\n \"With the chosen units, $\\\\beta $ can take on allowed values between 0 and 1.\",\n \"Using Bloch theory, the atom dynamics from immediately after the $({\\\\tt t}-1)$ -th kick to immediately after the next ${\\\\tt t}$ -th kick is then described by the unitary operator, which is derived in detail in [27]: $\\\\hat{\\\\cal U}_{\\\\beta ,k,\\\\eta }({\\\\tt t})\\\\;=\\\\;e^{-\\\\mathrm {i}k \\\\cos (\\\\hat{\\\\theta })}\\\\; e^{-\\\\mathrm {i}\\\\tau /2(\\\\hat{\\\\cal N}+ \\\\beta +\\\\eta {\\\\tt t} +\\\\eta /2)^2}\\\\ .$ The full evolution operator over the first ${\\\\tt t}$ kicks is given by $\\\\hat{\\\\cal U}_{\\\\beta ,k,\\\\eta }^{{\\\\tt t}}\\\\;\\\\equiv \\\\;\\\\hat{\\\\cal U}_{\\\\beta ,k,\\\\eta }({\\\\tt t-1})\\\\; \\\\hat{\\\\cal U}_{\\\\beta ,k,\\\\eta }({\\\\tt t-2})\\\\dots \\\\hat{\\\\cal U}_{\\\\beta ,k,\\\\eta }({\\\\tt 1})\\\\; \\\\hat{\\\\cal U}_{\\\\beta ,k,\\\\eta }({\\\\tt 0}) \\\\,.$ where $\\\\hat{\\\\cal N}= - \\\\mathrm {i}\\\\frac{\\\\mathrm {d}}{\\\\mathrm {d}\\\\theta }$ is the (angular) momentum operator with periodic boundary conditions.\",\n \"The time-dependent Hamiltonian that generates the quantum evolution corresponding to (REF ) would be $\\\\hat{\\\\cal H}({\\\\hat{\\\\cal N}},\\\\hat{\\\\theta },\\\\beta ,t)= \\\\frac{1}{2}\\\\left(\\\\hat{\\\\cal N} +\\\\beta +\\\\frac{\\\\eta }{\\\\tau } t\\\\right)^2+ k\\\\cos (\\\\hat{\\\\theta }) \\\\sum _{{\\\\tt t}\\\\in \\\\mathbb {Z}}\\\\delta (t-{\\\\tt t}\\\\tau )\\\\ .$ The temporal evolution of the fidelity can now be studied with respect to changes of one of the parameter in Eq.\",\n \"(REF ).\",\n \"Motivated by the experiments reported in Refs.\",\n \"[24], [29], we consider in the following changes in the kick strength $k$ and the gravity parameter $\\\\eta $ .\",\n \"For a given quasimomentum $\\\\beta $ , the dynamical fidelity is defined as $F_\\\\beta (k_1,k_2,\\\\eta _1, \\\\eta _2, {\\\\tt t}) = \\\\Big | \\\\left< \\\\hat{\\\\cal U}_{\\\\beta ,k_1,\\\\eta _1}^{{\\\\tt t}} \\\\psi _\\\\beta (0) \\\\Big |\\\\hat{\\\\cal U}_{\\\\beta ,k_2,\\\\eta _2}^{{\\\\tt t}} \\\\psi _\\\\beta (0)\\\\right>\\\\Big |^2 \\\\,.$ In a typical experimental situations with cold atoms [32], [31], [25], [24], [7], [29], [30], the atoms are non interacting and each of them has in good approximation a well defined initial momentum, meaning that its quasimomentum is essentially fixed.\",\n \"This ensemble is best approximated by a statistical density operator $\\\\hat{\\\\rho } = \\\\int _\\\\beta \\\\mathrm {d}\\\\beta \\\\rho (\\\\beta ) \\\\mathinner {|{\\\\psi _\\\\beta (0)}\\\\rangle }\\\\mathinner {\\\\langle {\\\\psi _\\\\beta (0)}|}$ .\",\n \"Then the fidelity generalized to density operators [5] gives the slightly more involved formula for our system of cold atoms [33], [28]: $F(k_1,k_2,\\\\eta _1, \\\\eta _2,{\\\\tt t}) =\\\\bigg |\\\\int _\\\\beta \\\\rho (\\\\beta ) \\\\left< \\\\hat{\\\\cal U}_{\\\\beta ,k_1,\\\\eta _1}^{{\\\\tt t}} \\\\psi _\\\\beta (0) \\\\Big |\\\\hat{\\\\cal U}_{\\\\beta ,k_2,\\\\eta _2}^{{\\\\tt t}} \\\\psi _\\\\beta (0)\\\\right> \\\\mathrm {d}\\\\beta \\\\bigg |^2 \\\\,.$ A series of experimental investigations has looked closely at the so-called quantum resonant motion of the quantum kicked rotor over the last fifteen years, see e.g.\",\n \"[7], [31] and references therein.\",\n \"This interest is mainly motivated by the type of ballistic motion with fast acceleration which can be obtained in this particular region of parameters.\",\n \"The quantum resonances allow one the realization of ratchet-like dynamics as well, when breaking the spatial-temporal symmetries of the kicked rotor [7], [34].\",\n \"We restrict here, as most of previous works, to the main quantum resonances occurring whenever $\\\\tau $ is not only commensurate to $2\\\\pi $ but an integer multiple of it, i.e.\",\n \"$\\\\tau =2\\\\pi \\\\ell $ , $\\\\ell $ integer.\",\n \"Then, for specially chosen values of quasi-momentum $\\\\beta $ , the energy of the rotor with this fixed $\\\\beta $ increases quadratically as ${\\\\tt t}\\\\rightarrow +\\\\infty $ [35], [27].\",\n \"In the presence of gravity, asymptotic quadratic growth of energy at certain values of $\\\\beta $ is still possible close to resonant values of the kick period[27].\",\n \"From the theoretical point of view, these quantum resonances are convenient since we have analytical access to the wave functions [35], [27] and hence to analytical estimates for the development of fidelity [33], [28].\",\n \"Using pseudoclassical approximation theory [35], [27], [37], [7], this is true also for small detuning from the resonance conditions.\",\n \"We review here just the result of simple analytical considerations, whose details can be found in [33], [28].\",\n \"The fidelity at a main quantum resonance is considered in two cases: first for the perturbation parameter $\\\\Delta k=k_2-k_1$ and a fixed quasimomentum $\\\\beta $ .\",\n \"The fidelity is then computed from the overlap $\\\\left< \\\\hat{\\\\cal U}_{\\\\beta ,k_1,\\\\eta }^{{\\\\tt t}}\\\\psi _\\\\beta (0) \\\\Big |\\\\hat{\\\\cal U}_{\\\\beta ,k_2,\\\\eta }^{{\\\\tt t}} \\\\psi _\\\\beta (0)\\\\right>= J_0(\\\\Delta k|W_{{\\\\tt t}}|)\\\\ ,$ such as to yield $F_\\\\beta (k_1,k_2,\\\\eta ,{\\\\tt t})= \\\\Big |J_0(\\\\Delta k |W_{{\\\\tt t}}|)\\\\Big |^2 \\\\,.$ The argument of the zero-order Bessel function contains the following sum of phases: $W_{{\\\\tt t}}\\\\equiv W_{{\\\\tt t}}(\\\\eta ,\\\\beta )=\\\\sum _{r=0}^{{\\\\tt t}-1}e^{-\\\\mathrm {i}\\\\pi \\\\ell (2\\\\beta +1)r}e^{-\\\\mathrm {i}2\\\\pi \\\\ell r \\\\eta {\\\\tt t}+\\\\mathrm {i}\\\\pi \\\\ell \\\\eta r^2}\\\\ .$ Applying Eq.\",\n \"(REF ) at $\\\\eta =0$ and resonant $\\\\beta =\\\\beta _{res}$ (e.g.\",\n \"$\\\\beta _{res}=0$ for $\\\\ell =2$ ) we have simply $W_{{\\\\tt t}}={\\\\tt t}$ , which leads to $F_{\\\\beta _{res}}(k_1,k_2,\\\\eta =0,{\\\\tt t})= \\\\Big |J_0(\\\\Delta k {\\\\tt t} )\\\\Big |^2 \\\\,.$ This fidelity decays asymptotically, for large times, like a power-law with exponent one [33].\",\n \"Averaging over a full ensemble of quasimomenta in a Brillouine zone, we arrive at a fidelity which asymptotically saturates to a constant value $F(\\\\Delta k, {\\\\tt t}) \\\\equiv \\\\bigg |\\\\int _{\\\\beta =0}^{\\\\beta =1}\\\\left< \\\\hat{\\\\cal U}_{\\\\beta ,k_1}^{{\\\\tt t}} \\\\psi _\\\\beta (0) \\\\Big |\\\\hat{\\\\cal U}_{\\\\beta ,k_2}^{{\\\\tt t}} \\\\psi _\\\\beta (0)\\\\right> \\\\mathrm {d}\\\\beta \\\\bigg |^2\\\\stackrel{t \\\\rightarrow \\\\infty }{\\\\longrightarrow } \\\\frac{1}{(2\\\\pi )^2} \\\\left(\\\\int _{0}^{2\\\\pi }J_0^2\\\\left(\\\\frac{\\\\Delta k }{2\\\\sin \\\\alpha }\\\\right) \\\\mathrm {d}\\\\alpha \\\\right)^2\\\\,.$ This asymptotically exact saturation of the fidelity sets in very quickly in practice for typical experimental parameters, after ${\\\\tt t} > z_0/\\\\Delta k$ , where $z_0$ is the first zero of the zero order Bessel function [33].\",\n \"An interferometric cold-atom experiment at Harvard [25] nicely confirmed our theory showing this saturation of fidelity for various perturbations $\\\\Delta k$ and measurements up to about 60 kicks.\",\n \"In the second case, we consider the motion in a gravity field for the forward motion, whilst this field is absent in the backward motion.\",\n \"The kick strength does not change here.\",\n \"This gives for the fidelity, following a similar reasoning as done in [28], $F_ \\\\beta (\\\\eta , {\\\\tt t} ) = \\\\left| \\\\langle \\\\psi _\\\\beta (0)| \\\\hat{\\\\mathcal {U}}_{\\\\beta , k, {\\\\eta =0} }^{\\\\dagger {\\\\tt t} } \\\\,\\\\hat{ \\\\mathcal {U} }_{\\\\beta , k, \\\\eta }^{{\\\\tt t} } | \\\\psi _\\\\beta (0) \\\\rangle \\\\right| ^2= \\\\bigg |e^{-i\\\\phi ({\\\\beta }, {\\\\eta }, {\\\\tt t})} J_0\\\\left(\\\\sqrt{({\\\\tt t} k)^2+k^2 |W_{{\\\\tt t}}|^2 -2 {\\\\tt t} k^2 \\\\text{Re} (W_{{\\\\tt t}}) } \\\\right)\\\\bigg |^2 \\\\,.$ For a single $\\\\beta $ , the additional phase $\\\\phi ({\\\\beta }, {\\\\eta }, t) = \\\\ell \\\\pi \\\\sum _{q=0}^{t-1} \\\\left({\\\\beta }+q{\\\\eta } +/ {\\\\eta }2\\\\right)^2$ is superfluous, yet it becomes important when considering contributions from different quasimomenta, see subsection REF for this case.\",\n \"Figure: (left) phase space of the pseudoclassical model for η=0,k|ϵ|=0.04π\\\\eta =0, k|\\\\epsilon |=0.04\\\\pi (green/grey data points) as compared to its pendulum approximation , (solid lines, where the red line shows the separatrix of the pendulum).\",\n \"(right) phase space of the pseudoclassical model with gravity for the parameters τ=5.97,η=0.0257,k=1.4\\\\tau =5.97,\\\\eta =0.0257,k=1.4.\",\n \"(Quasi)momentum zero corresponds to the centre of both islands on the y axis, while θ\\\\theta is the angle parameter of the standard map as classical limit of the kicked rotor problem.\",\n \"The horizon stripe in the right plot marks the typical width of an experimental quasimomentum distribution for a Bose-Einstein condensate.\"\n ],\n [\n \"Fidelity revivals\",\n \"Even if we detune the kick period $\\\\tau $ slightly from the mentioned resonance conditions, we may still be able to estimate the true quantum motion using a pseudoclassical model introduced in [35], [27].\",\n \"In this model, the absolute value of the detuning $|\\\\epsilon |$ plays the role of Planck's constant, and hence the theory has a semiclassical limit at exact quantum resonance.\",\n \"The $\\\\epsilon $ -classical dynamics is described by the following discrete map which relates the variables $I$ and $\\\\theta $ from immediately after the ${\\\\tt t}$ -th kick to immediately after the $({\\\\tt t}+1)$ -th one $\\\\begin{array}{rcl}I_{{\\\\tt t}+1}&=&I_{{\\\\tt t}}+\\\\tilde{k}\\\\sin \\\\theta _{{\\\\tt t}+1}+\\\\mathrm {sgn}(\\\\epsilon )\\\\tau \\\\eta \\\\\\\\\\\\theta _{{\\\\tt t}+1}&=&\\\\theta _{{\\\\tt t}}+\\\\mathrm {sgn}(\\\\epsilon )I_{{\\\\tt t}}\\\\;\\\\;\\\\;\\\\textrm { mod } 2\\\\pi \\\\end{array}\\\\,.$ Here, ${I}={J}+\\\\mathrm {sgn}(\\\\epsilon )[\\\\pi \\\\ell +\\\\tau \\\\beta +\\\\tau \\\\eta {\\\\tt t}]$ and $J=|\\\\epsilon | p$ is the rescaled momentum.\",\n \"Similarly, the new effective kick strength is defined by $\\\\tilde{k}=|\\\\epsilon | k$ .\",\n \"Since the effective kick strength is multiplied by the small detuning in this model, the classical phase space is nearly integrable and its centre corresponds to a pendulum-like resonance island [36], [38], [37], [20], see Fig.\",\n \"REF .\",\n \"In this subsection, we continue now to study the case without gravity, i.e.\",\n \"$\\\\eta =0$ (see left panel of the figure).\",\n \"Within the harmonic approximation, initial states starting at momentum close to zero will oscillate around the island, with an oscillation frequency depending on the coupling strength, $\\\\omega _i=\\\\sqrt{k_i |\\\\epsilon |}$ ($i=1,2$ ) in this case.\",\n \"Hence, while the fidelity initially can decay fast at finite detuning [39], it may recover if the two frequencies – related to the two kick strength – are commensurable.\",\n \"Focusing only onto resonant values of quasimomentum, the fidelity that carries over formula (REF ) to finite detuning is $F_{\\\\beta _{res}}(k_1,k_2, {\\\\tt t},\\\\eta =0) \\\\sim \\\\frac{\\\\epsilon }{2\\\\pi } \\\\left(| \\\\omega _2 \\\\cos ( \\\\omega _1 {\\\\tt t} ) \\\\sin ( \\\\omega _2 {\\\\tt t}) - \\\\omega _1 \\\\cos ( \\\\omega _2 {\\\\tt t} ) \\\\sin ( \\\\omega _1 {\\\\tt t}) | \\\\right)^{-1} \\\\,,$ where the symbol $\\\\sim $ means that this result is asymptotically true for small $\\\\epsilon $ and large times.\",\n \"The predicted revivals of the fidelity are typical for integrable motion around nonlinear islands characterized by their winding numbers ($\\\\omega _i$ measured in number of kicks) in the Poincaré sections.\",\n \"Typical results of fidelity revivals are reported in Fig.\",\n \"REF .\",\n \"Figure: Slightly smoothed version of the fidelity as predicted by Eq.\",\n \"() (thick black line) and numerical data(grey/red curve) for k 1 =0.8π,k 2 =0.6πk_1=0.8\\\\pi , k_2=0.6\\\\pi , and detuning ϵ=0.01\\\\epsilon =0.01.\",\n \"Please note that the fidelity does start at 1 on the yy axis,which is not visible here owing to the fast initial drop after just a few kicks.\",\n \"Data adapted from .\"\n ],\n [\n \"Dynamical tunneling\",\n \"This subsection deals now with the case of a perturbation in the kick strength and one fixed value of gravity $\\\\eta \\\\ne 0$ .\",\n \"All we want to say here is essentially contained in Fig.\",\n \"REF .\",\n \"While initially, for short times, the fidelity suffers a fast exponential decay, this decay and the further evaluation is modulated with almost periodic oscillations.\",\n \"This oscillations are the better visible, the more we average over initial conditions (in this case different values of quasimomenta).\",\n \"The initial decay is arising from the part of the initial wave packet that lies in the chaotic component of either of the two dynamics (defined by the two different kick strengths) in the pseudoclassical phase space, see right panel of Fig.\",\n \"REF .\",\n \"While its behaviour seems to be non-uniform and strongly dependent on parameters, we can estimate the oscillation frequency by a similar argument as in the previous subsection (REF ).\",\n \"The simplest form of this estimate gives an oscillation frequency of $|\\\\omega _1-\\\\omega _2|+\\\\Delta \\\\theta $ , where $\\\\omega _i$ ($i=1,2$ ) were defined above and $\\\\Delta \\\\theta $ is the shift of the island centre in phase space induced by the change in the kick strength at finite gravity.\",\n \"This shift of the frequency can be a understood in the following way: for the elliptic motion in one island the trajectory is ahead, whereas for the elliptic motion in the other island the trajectory is behind, so they get closer to each other and rephase after a shorter period than in the case when both islands lie concentric.\",\n \"The shift, which we estimate by $\\\\Delta \\\\theta $ , depends on the precise geometry of the two islands, and the overlapping area.\",\n \"Already the data in the left panel of Fig.\",\n \"REF shows a significant change in slope of the overall fidelity decay after some time, see the solid line at about 100 kick.\",\n \"We can analyze this asymptotic behaviour better focussing on an initial wave packet with just one single initial quasimomentum, see right panel of Fig.\",\n \"REF .\",\n \"The fidelity is – after the initial decay – dominated by the parts of the wave packet which are trapped inside the islands, similar to the one shown in Fig.\",\n \"REF (right panel).\",\n \"We chose our initial states in the form of Gaussian states mostly located inside one island.\",\n \"The decay from the islands into the chaotic sea is a well known phenomenon known as dynamical tunneling, see in this context [40], [30] and refs.\",\n \"therein.\",\n \"We will argue now that this is the main mechanism responsible of the fidelity decay at long times.\",\n \"The most important contribution to fidelity comes from the parts of each factor in the scalar product in (REF ) that are trapped in the respective travelling island.\",\n \"The decay of fidelity is then in turn determined by the decay of each of these two parts, which is determined by its respective tunneling rate $\\\\Gamma _{i}$ ($i=1,2$ ) into the chaotic sea.\",\n \"Based on this semiclassical reasoning, we came up with the following ansatz for the asymptotic decay of fidelity [28] $F({\\\\tt t})\\\\;\\\\propto \\\\;{\\\\mu ({\\\\cal A}_1\\\\backslash {\\\\cal A}_2) e^{-\\\\Gamma _1 {\\\\tt t}} +\\\\mu ({\\\\cal A}_2\\\\backslash {\\\\cal A}_1) e^{-\\\\Gamma _2 {\\\\tt t}} + \\\\mu ({\\\\cal A}_2\\\\cap {\\\\cal A}_1)e^{-(\\\\Gamma _1+\\\\Gamma _2) {\\\\tt t}}} \\\\ .$ Here $\\\\mu $ is the classical invariant measure of phase space sets, ${\\\\cal A}_i$ is the island in the phase space around the fixed point associated with $k_i$ .\",\n \"In our simulations, the phase-space areas appearing in (REF ) were estimated as described in the appendix of [28].\",\n \"The quantum decay rates due to dynamical tunneling were computed numerically for each $k_i$ .\",\n \"To do so we computed the survival probability as shown in Fig.\",\n \"REF by the blue dashed line.\",\n \"Our semiclassical formula (REF ) provides a simple yet powerful estimate for the fidelity decay since it is rather universal and applicable to a large variety of systems.\",\n \"We finish this subsection by stating that first experimental measurements are on there way to detect signatures of dynamical tunneling of states prepared in quantum accelerator mode islands such as shown in the right panel of Fig.\",\n \"REF , see ref.\",\n \"[30].\",\n \"The problem will remain, of course, to reach a high enough experimental stability to get access to the evolution to times longer than just a few dozens of kicks.\",\n \"Figure: (left) Fidelity averaged over a broad ensemble of quasimomenta in [0,1)[0,1) (corresponding to a cold-atom experiment with wideinitial momentum distribution) for the parameters τ=5.86,η=0.01579τ,k 1 =0.8π,k 2 =0.75π\\\\tau = 5.86, \\\\eta = 0.01579\\\\tau , k_1 = 0.8\\\\pi , k_2 = 0.75\\\\pi (dashed line) and k 2 =0.7πk_2 = 0.7\\\\pi (solid line).\",\n \"(right) Comparison between the survival probability (black dots and thin blue line) and the fidelity for one fixed quasimomentum(red/grey thick solid line) for τ=5.86\\\\tau =5.86, ϵ=τ-2π\\\\epsilon =\\\\tau -2\\\\pi ,η=0.01579τ\\\\eta =0.01579\\\\tau , β≈0.49\\\\beta \\\\approx 0.49, k 2 =0.8πk_2=0.8\\\\pi , k 1 =0.7πk_1=0.7\\\\pi .\",\n \"The upper thin blue line corresponds to k 1 k_1 and gives thetunneling rates Γ 1 =5.1×10 -4 \\\\Gamma _1 = 5.1 \\\\times 10^{-4}, whereas the lower black dots corresponds to k 2 k_2 and gives Γ 2 =4.4×10 -5 \\\\Gamma _2 =4.4 \\\\times 10^{-5}.\",\n \"The crosses represent an exponential fit with a rate Γ=3.5×10 -5 \\\\Gamma = 3.5 \\\\times 10^{-5} to guide the eye.\",\n \"Data in the right paneladapted from .\"\n ],\n [\n \"Thermometry in ultra-cold atoms\",\n \"We now turn back again at exact quantum resonance conditions, more particularly to formula (REF ) from subsection REF .\",\n \"From the phase term $\\\\phi ({\\\\beta }, {\\\\eta }, t)$ in Eq.\",\n \"(REF ) it can be seen that, when quasimomentum $\\\\beta $ is non zero, then the total phase induced by different values of $\\\\eta $ depends not only on its magnitude but also on its sign.\",\n \"Experiments to investigate this situation were performed at Oklahoma with a Bose-Einstein condensate of about 40 000 rubidium atoms [29].\",\n \"The atoms correspond to a distribution of quasimomenta with a FWHM of about $0.05$ in our dimensionless units.\",\n \"We must average Eq.\",\n \"(REF ) over these initial conditions in $\\\\beta $ .\",\n \"The data – guided by our theory – clearly indicate that the asymmetric dependence of the averaged fidelity on gravity may turn out to be useful in determining externally applied accelerations and, in turn, the temperature (given by the kinetic energy) of ultracold atoms in the limit of negligible interatomic interactions.\",\n \"The immediate value of this experimental application is highlighted by the fact that just four kicks were used in Fig.\",\n \"REF , making similar experiments very fast, stable and versatile.\",\n \"While the experimental data from Fig.\",\n \"REF is not yet as stable as one would like for a practically useful thermometer, numerical results shown in Fig.\",\n \"REF confirm the linear correlation between the normalized visibility of the asymmetry and the momentum width of the atomic cloud.\",\n \"We conclude this section by remarking that the quantum kick rotor in the presence of gravity, as describe by the Hamiltonian (REF ), is for generic values of $\\\\eta $ not reducible to a periodically time-dependent system.\",\n \"Hence, we indeed need the dynamical fidelity in order to characterize the large variety of different dynamical regimes of this paradigm system of quantum chaos.\",\n \"A description by the static fidelity measure of section would instead only be possible for values of $\\\\eta $ commensurable with $\\\\tau /|\\\\epsilon |$ , see ref.\",\n \"[40] for details.\",\n \"The fidelity measure has found a plethora of applications over the last sixty years, from spin and photon echos over to low-dimensional quantum chaotic systems [1], [2], [3], [4], [5].\",\n \"We reviewed here two important specific fidelities, a static overlap of eigenfunctions and a dynamical overlap of generic wave functions.\",\n \"The former has been successfully used to detect avoided crossings in complex quantum systems, ranging from quantum phase transitions in the ground state [11], [10], [12] to the characterization of the chaoticity of the entire or parts of quantum spectra [21], [6], [8], [9].\",\n \"The much more studied dynamical fidelity has been investigated here for the paradigm system of quantum chaos, the quantum kicked rotor.\",\n \"We showed several applications, ranging from characterizing quasi-regular motion close to quantum resonance, over to dynamical tunneling and its indirect use as a thermometer for ultracold atomic gases.\",\n \"We hope that this overview will inspire further searches for interesting applications of the fidelity and generalized quantum echos, possible exploiting much more the phase dependence of the quantum mechanical overlap function for new precision experiments.\",\n \"Figure: Numerical simulations of an asymmetry visibility defined as [F(η - )-F(η + )]/[F(η - )+F(η + )][F(\\\\eta _-) - F(\\\\eta _+)]/[F(\\\\eta _-) + F(\\\\eta _+)] show an almost linear scaling with the momentum width of the atomic cloud for the experimentally relevant range, see previous figure, Δβ≤0.08\\\\Delta \\\\beta \\\\le 0.08.\",\n \"This linear correlation could be used to precisely define the momentum width on the xx axis via measurements of the asymmetry (on the yy axis).\",\n \"Figure taken from .The author is grateful to Italo Guarneri and Gil Summy for stimulating our work on fidelity related to the dynamics of cold atoms.\",\n \"Many thanks goes to the large number of students and my former postdoc Remy Dubertrand, who all were involved in specific parts of the work reviewed here.\",\n \"Support by the FIL program of the Università di Parma is kindly acknowledged.\",\n \"The author thanks very much the organizers of the international conference Echoes in Complex Systems at the MPIPKS, in particular Arseni Goussev, for support and the kind invitation to Dresden.\"\n ]\n]"},"id":{"kind":"string","value":"1605.06135"}}},{"rowIdx":1866677,"cells":{"text":{"kind":"list like","value":[["Finite intersection property and dynamical compactness"],["Abstract Dynamical compactness with respect to a family as a new concept of chaoticity of a dynamical system was introduced and discussed in [22].","In this paper we continue to investigate this notion.","In particular, we prove that all dynamical systems are dynamically compact with respect to a Furstenberg family if and only if this family has the finite intersection property.","We investigate weak mixing and weak disjointness by using the concept of dynamical compactness.","We also explore further difference between transitive compactness and weak mixing.","As a byproduct, we show that the $\\omega_{\\mathcal{F}}$-limit and the $\\omega$-limit sets of a point may have quite different topological structure.","Moreover, the equivalence between multi-sensitivity, sensitive compactness and transitive sensitivity is established for a minimal system.","Finally, these notions are also explored in the context of linear dynamics."],["Introduction","By a (topological) dynamical system $(X,T)$ we mean a compact metric space $X$ with a metric $d$ and a continuous self-surjection $T$ of $X$ .","We say it trivial if the space is a singleton.","Throughout this paper, we are only interested in a nontrivial dynamical system, where the state space is a compact metric space without isolated points.","This paper is a continuation of the research carried out in [22], where the authors discuss a dynamical property (called dynamical compactness) and examine it firstly for transitive compactness.","Some results of this paper can be considered as a contribution to dynamical topology – an area of the theory of dynamical systems in which the topological properties of maps that can be described in dynamical terms.","Let $\\mathbb {Z}_{+}$ be the set of all nonnegative integers and $\\mathbb {N}$ the set of all positive integers.","Before going on, let us recall the notion of a Furstenberg family from [1].","Denote by $\\mathcal {P} =\\mathcal {P}(\\mathbb {Z}_{+})$ the set of all subsets of $\\mathbb {Z}_{+}$ .","A subset $\\mathcal {F}\\subset \\mathcal {P}$ is a (Furstenberg) family, if it is hereditary upward, that is, $F_1 \\subset F_2$ and $F_1 \\in \\mathcal {F}$ imply $F_2 \\in \\mathcal {F}$ .","Any subset $\\mathcal {A}$ of $\\mathcal {P}$ clearly generates a family $\\lbrace F\\in \\mathcal {P}: F\\supset A\\ \\text{for some}\\ A\\in \\mathcal {A}\\rbrace $ .","Denote by $\\mathcal {B}$ the family of all infinite subsets of $\\mathbb {Z}_{+}$ , and by $\\mathcal {P}_+$ the family of all nonempty subsets of $\\mathbb {Z}_+$ .","For a family $\\mathcal {F}$ , the dual family of $\\mathcal {F}$ , denoted by $k\\mathcal {F}$ , is defined as $\\lbrace F\\in \\mathcal {P}: F\\cap F^{\\prime }\\ne \\varnothing \\mbox{ for any }F^{\\prime } \\in \\mathcal {F}\\rbrace .$ A family $\\mathcal {F}$ is proper if it is a proper subset of $\\mathcal {P}$ , that is, $\\mathbb {Z}_+\\in \\mathcal {F}$ and $\\varnothing \\notin \\mathcal {F}$ .","By a filter $\\mathcal {F}$ we mean a proper family closed under intersection, that is, $F_1, F_2\\in \\mathcal {F}$ implies $F_1\\cap F_2\\in \\mathcal {F}$ .","A filter is free if the intersection of all its elements is empty.","We extend this concept, a family $\\mathcal {F}$ is called free if the intersection of all elements of $\\mathcal {F}$ is empty.","For any $F\\in \\mathcal {P}$ , every point $x\\in X$ and each subset $G\\subset X$ , we define $T^{F} x= \\lbrace T^i x: i\\in F\\rbrace $ , $n_T(x, G)=\\lbrace n\\in \\mathbb {Z}_{+}: T^n x \\in G\\rbrace $ .","The $\\omega $ -limit set of $x$ with respect to $\\mathcal {F}$ (see [1]), or shortly the $\\omega _\\mathcal {F}$ -limit set of $x$, denoted by $\\omega _{\\mathcal {F}}(x)$Remark that the notation $\\omega _{\\mathcal {F}}(x)$ used here is different from the one used in [1] (the notation $\\omega _{\\mathcal {F}}(x)$ used here is in fact $\\omega _{k \\mathcal {F}}(x)$ introduced in [1]).","As this paper is a continuation of the research in [22], in order to avoid any confusion of notation or concept, we will follow the ones used in [22]., is defined as $\\bigcap _{F \\in \\mathcal {F}} \\overline{T^{F}x}= \\lbrace z\\in X: n_T(x,G) \\in k\\mathcal {F} \\ \\mbox{for every neighborhood} \\ G \\ \\mbox{of} \\ z \\rbrace .$ Let us remark that not always $\\omega _{\\mathcal {F}}(x)$ is a subset of the $\\omega $ -limit set $\\omega _{T}(x)$ , which is defined as $\\bigcap _{n=1}^\\infty \\overline{\\lbrace T^kx: k\\ge n\\rbrace }= \\lbrace z\\in X: N_T(x,G) \\in \\mathcal {B} \\ \\mbox{for every neighborhood} \\ G \\ \\mbox{of} \\ z \\rbrace .","$ For instance, if each element of $\\mathcal {F}$ contains 0 then any point $x\\in \\omega _{\\mathcal {F}}(x)$ .","But, as well known, a point $x\\in \\omega _{T}(x)$ if and only if $x$ is a recurrent pointA point $x\\in X$ is called recurrent if $x\\in \\omega _T(x).$ of $(X,T)$ .","Nevertheless, if a family $\\mathcal {F}$ is free, then $\\omega _{\\mathcal {F}}(x) \\subset \\omega _{T}(x)$ for any point $x\\in X$ and if $(X,T)$ has a nonrecurrent point, then the converse is true (see Proposition REF ).","A dynamical system $(X,T)$ is called compact with respect to $\\mathcal {F}$, or shortly dynamically compact, if the $\\omega _\\mathcal {F}$ -limit set $\\omega _{\\mathcal {F}}(x)$ is nonempty for all $x\\in X$ .","H. Furstenberg started a systematic study of transitive systems in his paper on disjointness in topological dynamics and ergodic theory [14], and the theory was further developed in [16] and [15].","Recall that the system $(X, T)$ is (topologically) transitive if $N_T (U_1, U_2)= \\lbrace n\\in \\mathbb {Z}_+: U_1\\cap T^{-n}U_2\\ne \\varnothing \\rbrace \\ (= \\lbrace n\\in \\mathbb {Z}_+: T^n U_1\\cap U_2\\ne \\varnothing \\rbrace )\\in \\mathcal {P}_+$ for any openeBecause we so often have to refer to open, nonempty subsets, we will call such subsets opene.","subsets $U_1, U_2\\subset X$ , equivalently, $N_T (U_1, U_2)\\in \\mathcal {B}$ for any opene subsets $U_1, U_2\\subset X$ .","In [22] the authors consider one of possible dynamical compactness — transitive compactness, and its relations with well-known chaotic properties of dynamical systems.","Let $\\mathcal {N}_T$ be the set of all subsets of $\\mathbb {Z}_+$ containing some $N_T (U, V)$ , where $U, V$ are opene subsets of $X$ .","A dynamical system $(X, T)$ is called transitive compact, if for any point $x\\in X$ the $\\omega _{\\mathcal {N}_T}$ -limit set $\\omega _{\\mathcal {N}_T}(x)$ is nonempty, in other words, for any point $x\\in X$ there exists a point $z \\in X$ such that $ n_T(x,G) \\cap N_T(U,V)\\ne \\varnothing $ for any neighborhood $G$ of $z$ and any opene subsets $U,V$ of $X$ .","Let $(X, T)$ and $(Y, S)$ be two dynamical systems and $k\\in \\mathbb {N}$ .","The product system $(X\\times Y, T\\times S)$ is defined naturally, and denote by $(X^k, T^{(k)})$ the product system of $k$ copies of the system $(X, T)$ .","Recall that the system $(X, T)$ is minimal if it does not admit a nonempty, closed, proper subset $K$ of $X$ with $T K\\subset K$ , and is weakly mixing if the product system $(X^2, T^{(2)})$ is transitive.","Any transitive compact system is obviously topologically transitive, and observe that each weakly mixing system is transitive compact ([4]).","In fact, as it was shown in [22], each of notions are different in general and equivalent for minimal systems.","Recall a very useful notion of weakly mixing subsets of a system, which was introduced in [9] and further discussed in [32] and [33].","The notion of weakly mixing subsets can be regraded as a local version of weak mixing.","Among many very interesting properties let us mention just one of them – positive topological entropy of a dynamical system implies the existence of weakly mixing sets (see [28] for details).","A nontrivial closed subset $A\\subset X$ is called weakly mixing if for every $k\\ge 2$ and any opene sets $U_1, \\dots ,U_k, V_1, \\dots , V_k$ of $X$ with $U_i\\cap A \\ne \\varnothing $ , $V_i\\cap A \\ne \\varnothing $ , for any $i=1, \\dots ,k$ , one has that $\\bigcap _{i= 1}^kN_T(U_i\\cap A, V_i)\\ne \\varnothing .$ Let $A$ be a weakly mixing subset of $X$ and let $\\mathcal {N}_T(A)$ be the set of all subsets of $\\mathbb {Z}_+$ containing some $N_T (U\\cap A, V)$ , where $U, V$ are opene subsets of $X$ intersecting $A$ .","The notion of sensitivity was first used by Ruelle [36], which captures the idea that in a chaotic system a small change in the initial condition can cause a big change in the trajectory.","According to the works by Guckenheimer [20], Auslander and Yorke [6] a dynamical system $(X,T)$ is called sensitive if there exists $\\delta > 0$ such that for every $x\\in X$ and every neighborhood $U_x$ of $x$ , there exist $y\\in U_x$ and $n\\in \\mathbb {N}$ with $d(T^n x,T^n y)> \\delta $ .","Such a $\\delta $ is called a sensitive constant of $(X, T)$ .","Recently in [30] Moothathu initiated a way to measure the sensitivity of a dynamical system, by checking how large is the set of nonnegative integers for which the sensitivity occurs (see also [29]).","For a positive $\\delta $ and a subset $U\\subset X$ define $ S_T (U, \\delta )= \\lbrace n\\in \\mathbb {Z}_+: \\mbox{ there are } x_1, x_2\\in U\\ \\mbox{such that}\\ d (T^n x_1, T^n x_2)>\\delta \\rbrace .$ A dynamical system $(X, T)$ is called multi-sensitive if there exists $\\delta > 0$ such that $\\bigcap _{i= 1}^k S_T (U_i, \\delta )\\ne \\varnothing $ for any finite collection of opene $U_1, \\dots , U_k\\subset X$ .","Such a $\\delta $ is called a constant of multi-sensitivity of $(X, T)$ .","Recall that a collection $A$ of subsets of a set $Y$ has the finite intersection property (FIP) if the intersection of any finite subcollection of $A$ is nonempty.","The FIP is useful in formulating an alternative definition of compactness of a topological space: a topological space is compact if and only if every collection of closed subsets satisfying the FIP has a nonempty intersection itself (see, for instance [13], [25]).","Obviously that a filter (say $\\mathcal {N}_T$ , when $(X,T)$ is weakly mixing), the family $\\mathcal {N}_T(A)$ for a weakly mixing subset $A$ of $(X,T)$ and the family $\\mathcal {S}_T(\\delta )$ when $(X, T)$ is a multi-sensitive system (with a constant of multi-sensitivity $\\delta > 0$ ) have FIP.","Since all of these families are also free, actually they have the strong finite intersection property (SFIP), i.e., if the intersection over any finite subcollection of the family is infinite (see Proposition 2.2).","In fact we can say more — the FIP is useful in characterizing the dynamical compactness (see Theorem REF ).","Theorem FIP All dynamical systems are dynamically compact with respect to $\\mathcal {F}$ if and only if the family $\\mathcal {F}$ has the finite intersection property.","We also introduce two new stronger versions of sensitivity: sensitive compactness and transitive sensitivity.","Denote by $\\mathcal {S}_T(\\delta )$ the set of all subsets of $\\mathbb {Z}_+$ containing $S_T (U, \\delta )$ for some $\\delta > 0$ and opene $U\\subset X$ .","We will call the system $(X, T)$ transitively sensitive if there exists $\\delta > 0$ such that $S_T (W, \\delta )\\cap N_T (U, V)\\ne \\varnothing $ for any opene subsets $U, V, W$ of $X$ ; and sensitive compact, if there exists $\\delta > 0$ such that for any point $x\\in X$ the $\\omega _{\\mathcal {S}_T(\\delta )}$ -limit set $\\omega _{\\mathcal {S}_T(\\delta )}(x)$ is nonempty, in other words, for any point $x\\in X$ there exists a point $z\\in X$ such that $ n_T(x,G)\\cap S_T (U, \\delta )\\ne \\varnothing $ for any neighborhood $G$ of $z$ and any opene $U$ of $X$ .","The paper is organized as follows.","In Section 2 we recall some basic concepts and properties used in later discussions from topological dynamics.","In Section 3 we obtain some general results concerning dynamical compactness.","In particular we show that all dynamical systems are dynamically compact with respect to a Furstenberg family if and only if this family has the finite intersection property (Theorem REF ).","In Section 4 we discuss two stronger versions of sensitivity: transitive sensitivity and sensitive compactness.","It was shown that each weakly mixing system is transitively sensitive (Proposition REF ), and in fact we can characterize transitive sensitivity of a general dynamical system in terms of dynamical compactness (Proposition REF ).","Furthermore, all of the multi-sensitivity, sensitive compactness and transitive sensitivity are equivalent for a minimal system (Theorem REF ).","Even though each minimal transitive compact system is multi-sensitive, nevertheless, there are many minimal multi-sensitive systems which are not transitive compact.","Observe that the sensitivity of a dynamical system can be lifted up from a factor to an extension by an almost open factor map between transitive systems by [17].","We prove that the transitive sensitivity can be lifted up to an extension from a factor by an almost one-to-one factor map and that the transitive sensitivity is projected from an extension to the sensitivity of a factor by a weakly almost one-to-one factor map (Lemma REF ).","In Section 5 we show that dynamical compactness can be used to characterize the weak disjointness of dynamical systems (Theorem REF ).","We also improve the result of Jian Li [27]: weak mixing implies $\\mathcal {F}_{ip}$ -point transitivity in terms of transitive compactness (Proposition REF ).","In Section 6 the further difference between weak mixing and transitive compactness is explored.","Precisely, there is a totally transitive, non weakly mixing, transitive compact system (Theorem REF ); and in fact any compact metric space can be realized as the $\\omega _{\\mathcal {N}_T}$ -limit set of a non totally transitive, transitive compact system $(X, T)$ (Theorem REF ).","As a byproduct, we show that the $\\omega _{\\mathcal {N}_T}$ -limit sets and the $\\omega $ -limit sets have quite different topological structures for a general dynamical compact system $(X,T)$ .","At the end of this section we add one more chaotical property of transitive compact systems (in additional to already known from [22]): transitive compactness implies Li-Yorke chaos (Proposition REF ).","In Section 7 we consider the dynamics of linear operators on infinite dimensional spaces in relation to the properties studied in previous sections.","In particular, we show the equivalence of the topological weak mixing property with a weak version of transitive compactness (Theorem REF ).","Some results on sensitivity are also obtained.","Acknowledgements.","The first and third authors acknowledge the hospitality of the School of Mathematical Sciences of the Fudan University, Shanghai.","The third author also acknowledges the hospitality of the Departament de Matemàtica Aplicada of the Universitat Politècnica de València and the Department of Mathematics of the Chinese University of Hong Kong.","The first author was supported by NNSF of China (11225105, 11431012); the fourth author was supported by MINECO, Project MTM2013-47093-P, and by GVA, Project PROMETEOII/2013/013; and the fifth author was supported by NNSF of China (11271078)."],["Preliminaries","In this section we recall standard concepts and results used in later discussions."],["Basic concepts in topological dynamics","Recall that $x\\in X$ is a fixed point if $T x= x$ , and an $\\mathcal {F}$ -transitive point of $(X, T)$ [27] if $n_T(x, U)\\in \\mathcal {F}$ for any opene subset $U$ of $X$ .","It is a trivial observation that if a family $\\mathcal {F}$ admits an $\\mathcal {F}$ -transitive dynamical system $(X,T)$ without isolated points, then $\\mathcal {F}$ is free.","Since $k(k\\mathcal {F})= \\mathcal {F}$ , it is easy to see that $x\\in X$ is an $\\mathcal {F}$ -transitive point of $(X, T)$ if and only if $\\omega _{k\\mathcal {F}}(x)=X$ .","Denote by ${\\mathrm {Tran}}_\\mathcal {F} (X, T)$ the set of all $\\mathcal {F}$ -transitive points of $(X, T)$ .","The system $(X, T)$ is $\\mathcal {F}$ -point transitive if ${\\mathrm {Tran}}_\\mathcal {F} (X, T)\\ne \\varnothing $ , and is $\\mathcal {F}$ -transitive if $N_T(U,V)\\in \\mathcal {F}$ for any opene subsets $U,V$ of $X$ .","Write ${\\mathrm {Tran}}(X, T)= {\\mathrm {Tran}}_{\\mathcal {P}_+} (X, T)$ for short, and we also call the point $x$ transitive if $x\\in {\\mathrm {Tran}}(X, T)$ , equivalently, its orbit ${\\mathrm {orb}}_T (x)= \\lbrace T^n x: x= 0, 1, 2, \\dots \\rbrace $ is dense in $X$ .","Since $T$ is surjective, the system $(X, T)$ is transitive if and only if ${\\mathrm {Tran}}(X, T)$ is a dense $G_\\delta $ subset of $X$ .","In general, a subset $A$ of $X$ is $T$ -invariant if $TA = A$ , and positively $T$ -invariant if $T A\\subset A$ .","If $A$ is a closed, nonempty, $T$ -invariant subset then $(A,T|_A)$ is called the associated subsystem.","A minimal subset of $X$ is a closed, nonempty, $T$ -invariant subset such that the associated subsystem is minimal.","Clearly, $(X,T)$ is minimal if and only if ${\\mathrm {Tran}}(X,T) = X$ , if and only if it admits no a proper, closed, nonempty, positively $T$ -invariant subset.","A point $x \\in X$ is called minimal if it lies in some minimal subset.","In this case, in order to emphasize the underlying system $(X, T)$ we also say that $x\\in X$ is a minimal point of $(X, T)$.","Zorn's Lemma implies that every closed, nonempty, positively $T$ -invariant set contains a minimal set.","A pair of points $x, y \\in X$ is called proximal if $\\liminf _{n\\rightarrow \\infty } d (T^n x, T^n y)= 0$ .","In this case each of points from the pair is said to be proximal to another.","Denote by ${\\mathrm {Prox}}_T (X)$ the set of all proximal pairs of points.","For each $x\\in X$ , denote by ${\\mathrm {Prox}}_T (x)$ , called the proximal cell of $x$, the set of all points which are proximal to $x$ .","Recall that a dynamical system $(X,T)$ is called proximal if ${\\mathrm {Prox}}_T (X)= X\\times X$ .","The system $(X,T)$ is proximal if and only if $(X,T)$ has the unique fixed point, which is the only minimal point of $(X,T)$ (e.g.","see [4]).","The opposition to the notion of sensitivity is the concept of equicontinuity.","Recall that $x \\in X$ is an equicontinuity point of $(X, T)$ if for every $\\varepsilon > 0$ there exists a $\\delta > 0$ such that $d (x, x^{\\prime }) < \\delta $ implies $d (T^n x, T^n x^{\\prime })< \\varepsilon $ for any $n\\in \\mathbb {Z}_+$ .","Denote by $\\text{Eq} (X, T)$ the set of all equicontinuity points of $(X, T)$ .","The system $(X, T)$ is called equicontinuous if $\\text{Eq} (X, T)= X$ .","Each dynamical system admits a maximal equicontinuous factor.","Recall that by a factor map $\\pi : (X, T)\\rightarrow (Y, S)$ between dynamical systems $(X, T)$ and $(Y, S)$ , we mean that $\\pi : X\\rightarrow Y$ is a continuous surjection with $\\pi \\circ T= S\\circ \\pi $ .","In this case, we call $\\pi : (X, T)\\rightarrow (Y, S)$ an extension; and $(X, T)$ an extension of $(Y, S)$ , $(Y, S)$ a factor of $(X, T)$ ."],["Basic concepts of Furstenberg families","In this subsection we recall from [1] basic concepts about Furstenberg families.","Let $F\\in \\mathcal {P}$ .","Recall that a subset $F$ is thick if it contains arbitrarily long runs of positive integers.","Denote by $\\mathcal {F}_{\\text{thick}}$ the set of all thick subsets of $\\mathbb {Z}_+$ , and define $\\mathcal {F}_{\\text{syn}}= k \\mathcal {F}_{\\text{thick}}$ .","Each element of $\\mathcal {F}_{\\text{syn}}$ is said to be syndetic, equivalently, $F$ is syndetic if and only if there is $N\\in \\mathbb {N}$ such that $\\lbrace i, i + 1, \\dots , i + N \\rbrace \\cap F \\ne \\varnothing $ for every $i \\in \\mathbb {Z}_{+}$ .","We say that $F$ is thickly syndetic if for every $N\\in \\mathbb {N}$ the positions where length $N$ runs begin form a syndetic set.","Denote by $\\mathcal {F}_{\\text{cof}}$ the set of all cofinite subsets of $\\mathbb {Z}_+$ .","Note that by the classic result of Gottschalk a point $x\\in X$ is minimal if and only if $n_T(x, U)= \\lbrace n\\in \\mathbb {Z}_+: T^n x\\in U\\rbrace $ is syndetic for any neighborhood $U$ of $x$ .","Hence, for any minimal system $(X, T)$ , the subset $N_T (U, V)$ is syndetic for any opene subsets $U, V$ of $X$ .","Recall that a family $\\mathcal {F}$ is proper if it is a proper subset of $\\mathcal {P}$ , that is, $\\mathbb {Z}_+\\in \\mathcal {F}$ and $\\varnothing \\notin \\mathcal {F}$ .","By a filter $\\mathcal {F}$ we mean a proper family closed under intersection, that is, $F_1, F_2\\in \\mathcal {F}$ implies $F_1\\cap F_2\\in \\mathcal {F}$ .","For families $\\mathcal {F}_1$ and $\\mathcal {F}_2$ , we define the family $\\mathcal {F}_1\\cdot \\mathcal {F}_2:= \\lbrace F_1\\cap F_2: F_1\\in \\mathcal {F}_1, F_2\\in \\mathcal {F}_2\\rbrace $ and call it the interaction of $\\mathcal {F}_1$ and $\\mathcal {F}_2$.","Thus we have $\\mathcal {F}_1 \\cup \\mathcal {F}_2 \\subset \\mathcal {F}_1 \\cdot \\mathcal {F}_2$ ; and it is easy to check that $\\mathcal {F}$ is a filter if and only if $\\mathcal {F}= \\mathcal {F}\\cdot \\mathcal {F}$ , and $\\mathcal {F}_1\\cdot \\mathcal {F}_2$ is proper if and only if $\\mathcal {F}_2 \\subset k\\mathcal {F}_1$ .","For each $i\\in \\mathbb {Z}_+$ , define $g^i: \\mathbb {Z}_+\\rightarrow \\mathbb {Z}_+, j\\mapsto i+ j$ .","Recall that a family $\\mathcal {F}$ is $+$ invariant if for every $i\\in \\mathbb {Z}_+$ , $F\\in \\mathcal {F}$ implies $g^{i} (F)\\in \\mathcal {F}$ ; $-$ invariant if for every $i\\in \\mathbb {Z}_+$ , $F\\in \\mathcal {F}$ implies $g^{- i} (F)\\in \\mathcal {F}$ , where $g^{-i}(F)=(g^i)^{-1}(F)=\\lbrace j-i:j\\in F,j\\ge i\\rbrace $ ; and translation invariant if it is both $+$ and $-$ invariant, equivalently, for every $i\\in \\mathbb {Z}_+$ , $F\\in \\mathcal {F}$ if and only if $g^{- i} (F)\\in \\mathcal {F}$ .","As $g^{- i} (g^i A)= A$ and $g^i (g^{- i} A)\\subset A$ for any $i\\in \\mathbb {Z}_{+}$ , it is easy to obtain that the family $\\mathcal {F}$ is $+$ ($-$ , translation, respectively) invariant if and only if $k \\mathcal {F}$ is $-$ ($+$ , translation, respectively) invariant (see for example [1]).","And then we have: Proposition 2.1 Let $x\\in X$ .","Then $T \\omega _\\mathcal {F} (x)\\subset \\omega _\\mathcal {F} (T x)$ .","Additionally, if $\\mathcal {F}$ is $-$ ($+$ , translation, respectively) invariant then $\\omega _\\mathcal {F} (T x)\\subset $ ($\\supset $ , $=$ , respectively) $\\omega _\\mathcal {F} (x)$ .","Since the other items are alternative versions of [1] in our notations, it suffices to prove that if $\\mathcal {F}$ is $+$ invariant then $\\omega _\\mathcal {F} (T x)\\supset \\omega _\\mathcal {F} (x)$ .","For each $y\\in \\omega _\\mathcal {F} (x)$ take an arbitrary neighborhood $U$ of $y$ , and let $F\\in \\mathcal {F}$ .","Then $g^{1} (F)= \\lbrace i+ 1\\in \\mathbb {Z}_+: i\\in F\\rbrace \\in \\mathcal {F}$ as $\\mathcal {F}$ is $+$ invariant, and hence $n_T (x, U)\\cap g^{1} (F)\\ne \\varnothing $ , thus $\\varnothing \\ne g^{- 1} (n_T (x, U)\\cap g^{1} (F))= n_T (T x, U)\\cap F$ .","It follows $y\\in \\omega _\\mathcal {F} (T x)$ from the arbitrariness of $U$ and $F$ , which finishes the proof.","Proposition 2.2 Let $(X,T)$ be a dynamical system and let $\\mathcal {F}$ be a family.","If $\\mathcal {F}$ is free, then $\\omega _{\\mathcal {F}}(x) \\subset \\omega _{T}(x)$ for any $x\\in X$ .","Moreover, if $(X,T)$ has a nonrecurrent point, then the converse is true.","If $\\mathcal {F}$ is free and has FIP then it has SFIP.","(i) Suppose $\\omega _{\\mathcal {F}}(x)\\ne \\varnothing $ and take a point $y \\in \\omega _{\\mathcal {F}} (x):=\\bigcap _{F \\in \\mathcal {F}} \\overline{T^{F}x}$ .","Let us show that $y \\in \\omega _{T} (x)$ .","Since $\\mathcal {F}$ is free, $\\bigcap _{F \\in \\mathcal {F}} T^{F}x=\\varnothing $ .","Otherwise there is $m \\in \\mathbb {Z}_+$ such that $T^mx \\in T^Fx$ for all $F \\in \\mathcal {F}$ , in other words $m \\in \\bigcap _{F \\in \\mathcal {F}}F$ , a contradiction.","It means that there is $F \\in \\mathcal {F}$ such that $y \\notin T^Fx$ .","So $y \\in \\overline{T^{F}x} \\setminus T^Fx$ , and $F$ is infinite.","Therefore there exists an infinite sequence $T^{n_1}x, \\dots , T^{n_i}x,\\dots , $ which converges to $y$ .","Hence $y \\in \\omega _T(x)$ .","Now, let $(X,T)$ be a dynamical system with a nonrecurrent point $x_0\\in X$ , let $\\mathcal {F}$ be a family and $\\omega _{\\mathcal {F}}(x) \\subset \\omega _{T}(x)$ for any $x\\in X$ .","Suppose $\\mathcal {F}$ is not free.","It means there is a $k\\in \\mathbb {Z}_{+}$ that lies in each element of $\\mathcal {F}$ .","Then obviously that $T^k(x)\\in \\omega _{\\mathcal {F}}(x)$ for any $x\\in X$ .","Since $x_0$ is nonrecurrent, $x_0\\notin \\omega _T(x_0)\\ne X$ .","It is well known that $\\omega _T(x)$ is $T$ -invariant, therefore $\\omega _T(y)=\\omega _T(x_0)$ for any $y\\in \\lbrace T^{-i}(x_0): i \\in \\mathbb {Z}_{+}\\rbrace $ , and $y\\notin \\omega _T(y)$ .","Take a point $x_k\\in X$ with $T^k(x_k)=x_0$ .","As we know $x_0 \\in \\omega _{\\mathcal {F}}(x_k)$ .","But $\\omega _{\\mathcal {F}}(x_k)\\subset \\omega _T(x_k)= \\omega _T(x_0)$ , a contradiction.","(ii) Suppose that for some $F_1, \\dots F_k \\in \\mathcal {F}$ $\\bigcap _{i=1}^k F_i=\\lbrace n_1, n_2, \\dots n_m\\rbrace $ .","Since $\\mathcal {F}$ is free for each $k \\in \\mathbb {Z}_+$ there is $G_k \\in \\mathcal {F}$ such that $k \\notin G_k$ .","Then $\\bigcap _{i=1}^k F_i \\cap \\bigcap _{j=1}^m G_{n_j}=\\varnothing $ , contradiction."],["The concept of an almost one-to-one map","Let $\\phi : X\\rightarrow Y$ be a continuous surjective map from a compact metric space $X$ onto a compact Hausdorff space $Y$ .","Recall that $\\phi $ is almost open if $\\phi (U)$ has a nonempty interior in $Y$ for any opene $U\\subset X$ .","Note that each factor map between minimal systems is almost open [5], in particular, for a minimal system $(X,T)$ the map $T:X\\rightarrow X$ is almost open [26].","Denote by $Y_0\\subset Y$ the set of all points $y\\in Y$ whose fiber is a singleton.","Then $Y_0$ is a $G_\\delta $ subset of $Y$ , because $Y_0= \\lbrace y\\in Y: \\phi ^{- 1} (y)\\ \\text{is a singleton}\\rbrace = \\bigcap _{n\\in \\mathbb {N}} \\left\\lbrace y\\in Y: {\\mathrm {diam}}(\\phi ^{- 1} (y))<\\frac{1}{n}\\right\\rbrace $ and the map $y\\mapsto {\\mathrm {diam}}(\\phi ^{- 1} (y))$ is upper semi-continuous.","Here, we denote by ${\\mathrm {diam}}(A)$ the diameter of a subset $A\\subset X$ .","Recall that the function $f: Y\\rightarrow \\mathbb {R}_+$ is upper semi-continuous if $\\limsup \\limits _{y\\rightarrow y_0} f (y)\\le f (y_0)$ for each $y_0\\in Y$ .","Denote by $X_0\\subset X$ the set of all points $x\\in X$ such that the pre-image of $\\phi (x)$ is a singleton.","Then $X_0= \\pi ^{- 1} (Y_0)$ is a $G_\\delta $ subset of $X$ .","We call $\\phi $ weakly almost one-to-one if $Y_0$ is dense in $Y$ , and almost one-to-oneHere we use the concept of almost one-to-one following [3], and the concept of almost one-to-one used in [12], [23], [26] is in fact our weakly almost one-to-one.","if $X_0$ is dense in $X$ .","It is not hard to show that: if $\\phi $ is weakly almost one-to-one, then for any $\\delta > 0$ and any opene subset $U$ of $Y$ there exists opene $V\\subset U$ with ${\\mathrm {diam}}(\\phi ^{- 1} V)< \\delta $ ; and if $\\phi $ is almost one-to-one, then for any opene subset $U^*$ of $X$ there exists an opene subset $V^*$ of $Y$ with $\\phi ^{- 1} V^*\\subset U^*$ .","Clearly almost one-to-one is much stronger than weakly almost one-to-one.","For example, let $X$ be the closed unit interval, define $T (x)= 2 x$ for $x\\in [0, \\frac{1}{2}]$ and $T (x)= 1$ for $x\\in [\\frac{1}{2}, 1]$ , and then $T: X\\rightarrow X$ is clearly not almost one-to-one but weakly almost one-to-one.","For each minimal system $(X, T)$ , the map $T: X\\rightarrow X$ is weakly almost one-to-one [26], and in fact almost one-to-one [23].","The following result characterizes the relationship between weakly almost one-to-one and almost one-to-one, which extends [23].","Proposition 2.3 Let $\\phi : X\\rightarrow Y$ be a continuous surjective map from a compact metric space $X$ onto a compact Hausdorff space $Y$ .","Then $\\phi $ is almost one-to-one if and only if it is not only almost open but also weakly almost one-to-one.","Firstly assume that $\\phi $ is almost one-to-one.","Let $U\\subset X$ be an arbitrary opene subset.","And then we can take $x_0\\in U$ such that the pre-image of $\\phi (x_0)$ is a singleton.","From this it is easy to see that $\\phi (x_0)$ is contained in the interior of $\\phi (U)$ .","This implies that $\\phi $ is almost open.","The map $\\phi $ is clearly weakly almost one-to-one.","Now assume that $\\phi $ is not only almost open but also weakly almost one-to-one.","Let $U\\subset X$ be an arbitrary opene subset.","Since $\\phi $ is almost open, $\\phi (U)$ has a nonempty interior in $Y$ , and then $\\phi ^{- 1} (y_0)$ is a singleton for some $y_0\\in \\phi (U)$ , as $\\phi $ is weakly almost one-to-one.","This shows $U\\cap X_0\\ne \\varnothing $ , which finishes the proof.","As a direct corollary, we have: Corollary 2.4 Let $\\phi : X\\rightarrow Y$ and $\\pi : Y\\rightarrow Z$ be continuous surjective maps between compact metric spaces.","Then the composition map $\\pi \\circ \\phi : X\\rightarrow Z$ is almost one-to-one if and only if both $\\phi $ and $\\pi $ are almost one-to-one.","Denote by $X_0$ ($X_1$ , respectively) the set of all points $x\\in X$ such that the pre-image of $(\\pi \\circ \\phi ) (x)$ ($\\phi (x)$ , respectively) is a singleton.","Denote by $Z_0$ ($Z_1$ , respectively) the set of all points $z\\in Z$ whose $\\pi \\circ \\phi $ -fibers ($\\pi $ -fibers, respectively) are singletons.","All of them are $G_\\delta $ subsets.","Moreover, $X_0= X_1\\cap \\phi ^{- 1} (\\pi ^{- 1} Z_1)$ .","In fact, $x\\in X_0$ if and only if $\\lbrace x\\rbrace = (\\pi \\circ \\phi )^{- 1} (\\pi \\circ \\phi (x))= \\phi ^{- 1} (\\pi ^{- 1} (\\pi (\\phi x)))$ , if and only if $\\pi ^{- 1} (\\pi (\\phi x))= \\lbrace \\phi (x)\\rbrace $ and $\\phi ^{- 1} (\\phi x)= \\lbrace x\\rbrace $ , if and only if $\\pi (\\phi x)\\in Z_1$ and $x\\in X_1$ .","First assume that $\\pi \\circ \\phi $ is almost one-to-one, and then by Proposition REF : $X_0$ is a dense subset of $X$ , $Z_0$ is a dense subset of $Z$ and the map $\\pi \\circ \\phi $ is almost open.","Note that $X_0\\subset X_1$ and $Z_0\\subset Z_1$ , we have that $X_1$ is dense in $X$ and $Z_1$ is dense in $Z$ .","Hence $\\phi $ is almost one-to-one.","Furthermore, as the map $\\pi \\circ \\phi $ is almost open, for any opene $V\\subset Y$ one has that $\\pi (V)= (\\pi \\circ \\phi ) (\\phi ^{- 1} V)$ has a nonempty interior in $Z$ , which implies that $\\pi $ is almost one-to-one by Proposition REF .","Now assume that both $\\phi $ and $\\pi $ are almost one-to-one.","Then $X_1$ is a dense $G_\\delta $ subset of $X$ and $Z_1$ is a dense $G_\\delta $ subset of $Z$ .","Moreover, by Proposition REF both $\\phi $ and $\\pi $ are almost open, and then the continuous surjection $\\pi \\circ \\phi $ is also almost open, which implies that $(\\pi \\circ \\phi )^{- 1} (Z_1)$ is also a dense $G_\\delta $ subset of $X$ .","Thus, $X_0= X_1\\cap \\phi ^{- 1} (\\pi ^{- 1} Z_1)$ is a dense $G_\\delta $ subset of $X$ , that is, the composition map $\\pi \\circ \\phi : X\\rightarrow Z$ is almost one-to-one.","This finishes the proof.","Let $\\pi : (X, T)\\rightarrow (Y, S)$ be a factor map between dynamical systems.","If the map $\\pi : X\\rightarrow Y$ is almost one-to-one (weakly almost one-to-one, respectively), then we also call $(X, T)$ an almost one-to-one extension (a weakly almost one-to-one extension, respectively) of $(Y, S)$.","The main result of [23] states that a minimal system is either multi-sensitive or a weakly almost one-to-one extension of its maximal equicontinuous factor.","This is an analog of the well-known Auslander-Yorke dichotomy theorem: a minimal system is either sensitive or equicontinuous."],["Symbolic dynamics","Let $A$ be a nonempty finite set.","We call $A$ the alphabet and elements of $A$ are symbols.","The full (one-sided) $A$ -shift is defined as $\\Sigma = \\lbrace x = \\lbrace x_i\\rbrace _{i=0}^\\infty : x_i \\in A \\text{ for all } i \\in \\mathbb {Z}_+\\rbrace ,$ where we equip $A$ with the discrete topology and $\\Sigma $ with the product topology, and the shift map $\\sigma : \\Sigma \\rightarrow \\Sigma $ is a continuous surjection given by $ x = \\lbrace x_i\\rbrace _{i=0}^\\infty \\mapsto \\sigma x = \\lbrace x_{i+1}\\rbrace _{i=0}^\\infty ,$ that is, $\\sigma (x)$ is the sequence obtained by dropping the first symbol of $x$ .","Usually we write an element of $\\Sigma $ as $x = \\lbrace x_i\\rbrace _{i=0}^\\infty = x_0 x_1 x_2 x_3 \\dots $ A block $w$ over $\\Sigma $ is a finite sequence of symbols and its length is the number of its symbols (denoted by $|w|$ ).","An $n$ -block stands for a block of length $n$ .","In general we are only interested in a block $w$ with $|w|\\ge 1$ if without any special statement, and denote by $\\Sigma ^*$ the set of all blocks over $\\Sigma $ .","The block $w$ is a subblock of a block $v= v_1 \\dots v_m$ with $v_1, \\dots , v_m\\in A$ if there exists $1\\le i\\le j\\le m$ with $w= v_i \\dots v_j$ .","The concatenation of two blocks $u = a_1 \\dots a_k$ and $v = b_1 \\dots b_l$ is the block $uv = a_1 \\dots a_k b_1 \\dots b_l$ .","We write $u^n$ for the concatenation of $n\\ge 1$ copies of a block $u$ and $u^\\infty $ for the sequence $uuu \\dots \\in \\Sigma .$ By $x_{[i,j]}$ we denote the block $x_i x_{i+1} \\dots x_j$ , where $0 \\le i \\le j$ and $x =\\lbrace x_k\\rbrace _{k=0}^\\infty \\in \\Sigma $ .","The subset $X\\subset \\Sigma $ is called a subshift if it is a closed, nonempty, $\\sigma $ -invariant subset of $\\Sigma $ .","A cylinder of an $n$ -block $w \\in \\Sigma ^*$ in a subshift $X$ is the set $C[w] = \\lbrace x \\in X: x_{[0, n-1]} = w\\rbrace $ .","The collection of all cylinders forms a basis of the topology of $X$ ."],["Dynamical compactness with respect to an arbitrary family","Recall that a family $\\mathcal {F}$ has the finite intersection property (FIP) if the intersection of any finite subcollection of $\\mathcal {F}$ is nonempty.","The following theorem shows that the FIP is useful in characterizing the dynamical compactness.","Theorem 3.1 All dynamical systems are dynamically compact with respect to the family $\\mathcal {F}$ if and only if $\\mathcal {F}$ has the finite intersection property.","Sufficiency.","Suppose that $\\mathcal {F}$ has FIP.","Take arbitrary dynamical system $(X, T)$ and let $x\\in X$ .","Obviously the family $\\lbrace \\overline{T^{F}x}: F\\in \\mathcal {F}\\rbrace $ also has FIP, and then by compactness of the space $X$ the family has a nonempty intersection itself, i.e., $\\omega _{\\mathcal {F}}(x)=\\bigcap _{F \\in \\mathcal {F}} \\overline{T^{F}x} \\ne \\varnothing $ .","Thus $(X, T)$ is dynamically compact with respect to $\\mathcal {F}$ .","Necessity.","Suppose that the family $\\mathcal {F}$ has no FIP.","And then there is a collection $\\lbrace F_1, \\dots , F_k \\rbrace \\subset \\mathcal {F}$ with $\\bigcap _{i=1}^k F_i=\\varnothing .$ Let $A=\\lbrace a_1, \\dots , a_k\\rbrace $ be an alphabet and let $(X, T):=(\\Sigma , \\sigma )$ be the full (one-sided) $A$ -shift.","We are going to define a point $x \\in X$ with $\\omega _\\mathcal {F}(x)= \\varnothing $ .","Let $x_0=a_1$ .","For any $n \\ge 1$ there is $i$ with $n \\notin F_i$ , else the intersection of $F_1, \\dots , F_k$ would be nonempty.","Then define $x_n:=a_i$ .","Finally, let $x=x_0 x_1 x_2 x_3\\dots $ and the construction is finished.","Assume the contrary that we can take $z \\in \\omega _\\mathcal {F}(x)$ , and that $z$ begins with $a_i \\in A$ .","Take $G_z=C[a_i]$ .","As $z \\in \\omega _\\mathcal {F}(x)$ we have $n_T(x, G_z) \\cap F_i \\ne \\varnothing $ .","But if $n \\in n_T(x, G_z)$ , then $x_n=a_i$ and so $n \\notin F_i$ by the construction, a contradiction.","As we have mentioned in Introduction, obviously that a filter $\\mathcal {F}$ (in particular $\\mathcal {N}_T$ , when $(X,T)$ is weakly mixing), the family $\\mathcal {N}_T(A)$ for a weakly mixing subset $A$ of $(X,T)$ and the family $\\mathcal {S}_T(\\delta )$ when $(X, T)$ is a multi-sensitive system (with a constant of multi-sensitivity $\\delta > 0$ ) have FIP.","Let $\\mathcal {F}$ has the finite intersection property.","Then there exists an ultrafilter $\\mathcal {U}$ (in $\\mathcal {P}$ ) such that $\\mathcal {F}\\subset \\mathcal {U}$ .","This result is known as Ultrafilter Lemma (see details and proof in [21]).","Recall that an ultrafilter is maximal among all proper filters.","As a consequence of this fact we have a natural open question: Question A Let $(X,T)$ be a dynamically compact system with respect to a family $\\mathcal {F}$ and $\\mathcal {F}$ has FIP.","When $\\mathcal {F}$ is a filter, or at least contains a nontrivial filter?","Especially we address this question to the family $\\mathcal {S}_T(\\delta )$ .","More precisely, when a system $(X,T)$ is dynamically compact with respect to the family $\\mathcal {N}_T$ and $\\mathcal {N}_T$ has FIP, then, as well known, the systems is weakly mixing and $\\mathcal {N}_T$ is a filter.","Now, let a system $(X,T)$ is dynamically compact with respect to the family $\\mathcal {S}_T(\\delta )$ for some $\\delta >0$ and $\\mathcal {S}_T(\\delta )$ has FIP, then the systems is multi-sensitive.","But, the following question is still open – when is $\\mathcal {S}_T(\\delta )$ a filter?","A collection $\\mathcal {H}\\subset \\mathcal {F}$ will be called a base for $\\mathcal {F}$ if for any $F\\in \\mathcal {F}$ there is $H \\in \\mathcal {H}$ with $H \\subset F$ .","We are interested in those families which have a countable base, that is, there exists a base $\\mathcal {H}$ which is countable.","Remark that not any Furstenberg family $\\mathcal {F}$ has a countable base, for example, the family $\\mathcal {B}$ .","Assume the contrary that $\\mathcal {B}$ admits a countable base $\\lbrace F_n: n\\in \\mathbb {N}\\rbrace $ .","We take $k_1\\in F_1$ , and once $k_m\\in F_m, m\\in \\mathbb {N}$ is defined we choose $k_{m+ 1}\\in F_{m+ 1}$ with $k_{m+ 1}> k_m+ m+ 1$ .","Set $E= \\lbrace k_n: n\\in \\mathbb {N}\\rbrace $ and $F= \\mathbb {Z}_+\\setminus E$ .","Then $E\\cap F_n\\ne \\varnothing $ for all $n\\in \\mathbb {N}$ , and $F\\supset \\lbrace k_m+ m: m\\in \\mathbb {N}\\rbrace $ and hence $F\\in \\mathcal {B}$ , in particular, there exists no $n\\in \\mathbb {N}$ with $F_n\\subset F$ , a contradiction.","Not so hard to show even the existence of a family with FIP, but without a countable base.","Nevertheless the families $\\mathcal {N}_T$ and $\\mathcal {S}_T (\\delta )$ have countable bases.","Indeed, we can consider a countable base $\\mathcal {U}$ of open sets for the space $X$ .","Note that $U_1 \\subset U$ , $V_1 \\subset V$ implies $N_T(U_1, V_1) \\subset N_T(U, V)$ and $S_T(U_1,\\delta ) \\subset S_T(U,\\delta )$ .","Then $\\lbrace N_T(U, V): ~U, V \\in \\mathcal {U}\\rbrace $ and $\\lbrace S_T(U,\\delta ):~U \\in \\mathcal {U}\\rbrace $ are countable bases for $\\mathcal {N}_T$ and $\\mathcal {S}_T (\\delta )$ , respectively.","The following is a general result that will be especially useful for families with countable bases.","Proposition 3.2 Let $(X,T)$ be a dynamical system and let $\\mathcal {F}$ be a family such that there exists $x\\in {\\mathrm {Tran}}_{\\mathcal {F}} (X, T)$ .","Then ${\\mathrm {orb}}_T(x)\\subset {\\mathrm {Tran}}_{\\mathcal {F}} (X, T)$ .","By assumption, given an arbitrary opene $U\\subset X$ we have that $n_T(x,U)\\in \\mathcal {F}$ .","Thus, for any $m\\in \\mathbb {N}$ , $n_T(T^mx,U)=n_T(x,T^{-m}(U))\\in \\mathcal {F},$ and we conclude that $T^mx\\in {\\mathrm {Tran}}_{\\mathcal {F}} (X, T)$ .","Proposition 3.3 Assume that $\\mathcal {F}$ admits a countable base $\\mathcal {H}$ .","Then ${\\mathrm {Tran}}_{k\\mathcal {F}} (X, T)$ is a $G_\\delta $ subset of $X$ .","Moreover, the following are equivalent: The system $(X,T)$ is $k \\mathcal {F}$ -transitive, ${\\mathrm {Tran}}_{k\\mathcal {F}} (X, T)$ is a dense $G_\\delta $ subset of $X$ , ${\\mathrm {Tran}}_{k\\mathcal {F}} (X, T)\\ne \\varnothing $ .","Let $\\mathcal {U}$ be a countable base of the family of all opene subsets of $X$ .","Then the class $\\mathcal {U} \\times \\mathcal {H}$ is countable, and we enumerate it as $\\lbrace (U_i, F_i): i \\in \\mathbb {N}\\rbrace $ .","Denote by $T^{- F} U= \\bigcup _{n\\in F} T^{- n} U$ for any $F\\subset \\mathbb {Z}_+$ and each $U\\subset X$ .","Then it is easy to obtain $ {\\mathrm {Tran}}_{k\\mathcal {F}}(X,T) = \\bigcap _{i=1}^\\infty T^{-F_i} U_i.$ In fact, given arbitrary point $x\\in X$ , $x\\in {\\mathrm {Tran}}_{k\\mathcal {F}}(X,T)$ if and only if $N_T (x, U)\\in k \\mathcal {F}$ for any opene subset $U$ of $X$ , if and only if $N_T (x, U)\\cap F\\ne \\varnothing $ for any opene subset $U$ of $X$ and each $F\\in \\mathcal {F}$ , if and only if $N_T (x, U_i)\\cap F_i\\ne \\varnothing $ for each $i\\in \\mathbb {N}$ by the construction.","In particular, ${\\mathrm {Tran}}_{k\\mathcal {F}} (X, T)$ is a $G_\\delta $ subset of $X$ .","Thus $(X, T)$ is $k \\mathcal {F}$ -transitive, if and only if for any $F\\in \\mathcal {F}$ and arbitrary opene subsets $U, V$ of $X$ we have $N_T (V, U)\\cap F\\ne \\varnothing $ and equivalently $T^{- F} U\\cap V\\ne \\varnothing $ , if and only if $T^{- F} U$ is an opene dense subset of $X$ for any $F\\in \\mathcal {F}$ and each opene subset $U$ of $X$ , if and only if ${\\mathrm {Tran}}_{k\\mathcal {F}} (X, T)$ is a dense $G_\\delta $ subset of $X$ by (REF ).","Now we assume ${\\mathrm {Tran}}_{k\\mathcal {F}} (X, T)\\ne \\varnothing $ .","Let $x\\in {\\mathrm {Tran}}_{k\\mathcal {F}} (X, T)$ .","By Proposition REF ${\\mathrm {orb}}_T (x)\\subset {\\mathrm {Tran}}_{k\\mathcal {F}} (X, T)$ , and hence ${\\mathrm {Tran}}_{k\\mathcal {F}} (X, T)$ is a dense $G_\\delta $ subset of $X$ since $x\\in {\\mathrm {Tran}}(X, T)$ .","This finishes the proof.","Remark 3.4 Observe that when the state space $X$ is a compact metric space without isolated points, $x\\in {\\mathrm {Tran}}(X, T)$ if and only if $x\\in {\\mathrm {Tran}}_\\mathcal {B} (X, T)$ .","The family $\\mathcal {F}_{\\text{cof}}$ is clearly translation invariant (and hence $+$ invariant) and admits a countable base, and $k \\mathcal {F}_{\\text{cof}}= \\mathcal {B}$ .","Thus by Proposition REF one has: $(X, T)$ is transitive if and only if ${\\mathrm {Tran}}(X, T)$ is a dense $G_\\delta $ subset of $X$ if and only if ${\\mathrm {Tran}}(X, T)\\ne \\varnothing $ ."],["Transitive sensitivity and sensitive compactness","Recall that a dynamical system $(X, T)$ is transitively sensitive if there exists $\\delta > 0$ such that $S_T (W, \\delta )\\cap N_T (U, V)\\ne \\varnothing $ for any opene subsets $U, V, W$ of $X$ ; and sensitive compact if there exists $\\delta > 0$ such that for any point $x\\in X$ the set $\\omega _{\\mathcal {S}_T(\\delta )}(x)$ is nonempty.","Sometimes in that cases we will say also $(X, T)$ is transitively sensitive with a sensitive constant $\\delta $ and $(X, T)$ is sensitive compact with a sensitive constant $\\delta $ .","The main result of this section is the following Theorem 4.1 Let $(X, T)$ be a minimal system.","Then the following conditions are equivalent: $(X, T)$ is multi-sensitive.","$(X, T)$ is sensitive compact.","There exists $\\delta > 0$ such that $\\omega _{\\mathcal {S}_T (\\delta )} (x)= X$ for each $x\\in X$ .","There exist $\\delta > 0$ and $x\\in X$ with $\\omega _{\\mathcal {S}_T (\\delta )} (x)= X$ .","$(X, T)$ is transitively sensitive.","Before proceeding, we need: Lemma 4.2 Let $\\delta > 0$ and $x\\in X$ .","If $T: X\\rightarrow X$ is almost open, then the family $\\mathcal {S}_T (\\delta )$ is $-$ invariant, and the subset $\\omega _{\\mathcal {S}_T (\\delta )} (x)$ is positively $T$ -invariant.","By Proposition REF it suffices to prove that $\\mathcal {S}_T (\\delta )$ is a $-$ invariant family.","Take arbitrary $F\\in \\mathcal {S}_T (\\delta )$ and any $i\\in \\mathbb {Z}_+$ .","Then there exists opene subset $U$ of $X$ with $S_T (U, \\delta )\\subset F$ .","As $T: X\\rightarrow X$ is almost open, $T^i: X\\rightarrow X$ is also almost open, and then we can choose opene $V \\subset T^i U$ .","One has $g^{- i} (F)\\supset g^{- i} S_T (U, \\delta )= S_T (T^i U, \\delta )\\supset S_T (V, \\delta )$ , which implies that the family $\\mathcal {S}_T (\\delta )$ is $-$ invariant.","The following result gives a characterization of transitive sensitivity for a general dynamical system in terms of dynamical compactness.","Proposition 4.3 Let $(X, T)$ be a dynamical system.","Then the family $\\mathcal {S}_T (\\delta )$ is $+$ invariant for any $ \\delta > 0$ .","Furthermore, the following conditions are equivalent: $(X, T)$ is transitively sensitive.","There exist a $\\delta > 0$ and a dense $G_\\delta $ subset $X_0\\subset X$ such that $\\omega _{\\mathcal {S}_T (\\delta )} (x)= X$ for each $x\\in X_0$ .","There exist a $\\delta > 0$ and a point $x\\in X$ with $\\omega _{\\mathcal {S}_T (\\delta )} (x)= X$ .","Firstly, we show that $\\mathcal {S}_T (\\delta )$ is a $+$ invariant family.","In fact, take any $F\\in \\mathcal {S}_T (\\delta )$ and each $i\\in \\mathbb {Z}_+$ .","We choose opene subsets $U, V$ of $X$ with $F\\supset S_T (U, \\delta )$ and $V\\subset T^{- i} U$ satisfying ${\\mathrm {diam}}(T^j V)< \\delta $ for all $j= 0, 1, \\dots , i$ .","Thus $g^i (F)\\supset g^i S_T (U, \\delta )\\supset S_T (V, \\delta )$ from the construction, and then $g^i (F)\\in \\mathcal {S}_T (\\delta )$ .","This implies the $+$ invariance of the family $\\mathcal {S}_T (\\delta )$ .","Observe that $(X, T)$ is transitively sensitive with a sensitive constant $\\delta $ , if and only if $(X, T)$ is $k \\mathcal {S}_T (\\delta )$ -transitive; and that the family $\\mathcal {S}_T (\\delta )$ has a countable base: let $\\mathcal {U}$ be a countable base of the family of all opene subsets of $X$ , then $\\lbrace S_T (U, \\delta ): U \\in \\mathcal {U}\\rbrace $ is a countable base of $\\mathcal {S}_T (\\delta )$ .","Then applying Proposition REF the equivalence of $(1)\\Leftrightarrow (2)\\Leftrightarrow (3)$ follows from the fact that $x\\in {\\mathrm {Tran}}_{k \\mathcal {S}_T (\\delta )} (X, T)$ if and only if $\\omega _{\\mathcal {S}_T (\\delta )} (x)= X$ .","Observe that by [17] the sensitivity of a dynamical system can be lifted up from a factor to an extension by an almost open factor map between transitive systems.","The following result gives the lift-up and projection property of transitive sensitivity between transitive systems.","Lemma 4.4 Let $\\pi : (X, T)\\rightarrow (Z, R)$ be a factor map between dynamical systems.","Assume that $\\pi $ is almost one-to-one.","If $(Z, R)$ is transitively sensitive with a sensitive constant $\\delta > 0$ then $(X, T)$ is also transitively sensitive.","Assume that there exists $z\\in Z$ whose fiber is a singleton.","If $(X, T)$ is transitively sensitive then $(Z, R)$ is sensitive, in particular, $\\text{Eq} (Z, R)= \\varnothing $ .","(1) We take a compatible metric $\\rho $ over $Z$ and let $\\varepsilon > 0$ such that $d (x_1, x_2)\\le \\varepsilon $ implies $\\rho (\\pi x_1, \\pi x_2)\\le \\delta $ for any $x_1, x_2\\in X$ .","Now let $U, V, W$ be arbitrary opene subsets of $X$ .","As the map $\\pi : X\\rightarrow Z$ is almost one-to-one, we may take opene subsets $U_Z, V_Z, W_Z$ of $Z$ with $\\pi ^{- 1} U_Z\\subset U$ , $\\pi ^{- 1} V_Z\\subset V$ and $\\pi ^{- 1} W_Z\\subset W$ .","Observe that if $n\\in S_R (W_Z, \\delta )\\cap N_R (U_Z, V_Z)$ , then: on one hand, there exist $z_1, z_2\\in W_Z$ with $\\rho (R^n z_1, R^n z_2)> \\delta $ , and so $d (T^n x_1, T^n x_2)> \\varepsilon $ for any $x_1\\in \\pi ^{- 1} (z_1)$ and $x_2\\in \\pi ^{- 1} (z_2)$ , hence ${\\mathrm {diam}}(T^n W)> \\varepsilon $ ; on the other hand, $U_Z\\cap R^{- n} V_Z\\ne \\varnothing $ , and then $U\\cap S^{- n} V\\supset \\pi ^{- 1} U_Z\\cap \\pi ^{- 1} (R^{- n} V_Z)\\ne \\varnothing $ .","This implies $S_T (W, \\varepsilon )\\cap N_T (U, V)\\supset S_R (W_Z, \\delta )\\cap N_R (U_Z, V_Z)\\ne \\varnothing $ , as $(Z, R)$ is transitively sensitive.","Thus, by the arbitrariness of $U, V$ and $W$ , we have that $(X, T)$ is also transitively sensitive.","(2) As $(X, T)$ is transitively sensitive (and assume with a sensitive constant $\\delta > 0$ ), it is clear that $(Z, R)$ is transitive, and then by the refined Auslander-York dichotomy the system $(Z, R)$ is sensitive if and only if $\\text{Eq} (Z, R)= \\varnothing $ (see [6], [17], [2] and the book [1]).","Thus it suffices to prove $\\text{Eq} (Z, R)= \\varnothing $ .","Let $\\rho $ be a compatible metric over $Z$ , and assume the contrary to take a point $z\\in \\text{Eq} (Z, R)$ .","By the assumption that there exists a point of $Z$ whose fiber is a singleton, we may take an opene subset $W$ in $Z$ with ${\\mathrm {diam}}(\\pi ^{- 1} W)< \\delta $ , and an opene subset $W_*\\subset W$ and $\\delta _*> 0$ such that if the distance between a point of $Z$ and $W_*$ is smaller than $\\delta _*$ then the point belongs to $W$ .","Since $z\\in \\text{Eq} (Z, R)$ , there exists an open neighborhood $U_*$ of $z$ with ${\\mathrm {diam}}(R^n U_*)< \\delta _*$ for all $n\\in \\mathbb {Z}_+$ .","As $(X, T)$ is transitively sensitive with a sensitive constant $\\delta $ , take $m\\in N_T (\\pi ^{- 1} U_*, \\pi ^{- 1} W_*)\\cap S_T (\\pi ^{- 1} U_*, \\delta )$ .","Thus $T^m (\\pi ^{- 1} U_*)\\cap \\pi ^{- 1} W_*\\ne \\varnothing $ , and then $R^m U_*\\cap W^*\\ne \\varnothing $ , which implies $R^m U_*\\subset W$ by the construction of $U_*, W_*$ and $W$ .","Hence ${\\mathrm {diam}}(T^m (\\pi ^{- 1} U_*))\\le {\\mathrm {diam}}(\\pi ^{- 1} (R^m U_*))\\le {\\mathrm {diam}}(\\pi ^{- 1} W)< \\delta ,$ a contradiction to the selection of $m\\in S_T (\\pi ^{- 1} U_*, \\delta )$ .","This finishes the proof.","Now we are ready to prove Theorem REF .","$(1)\\Rightarrow (2)$ follows directly from the definitions.","As the system $(X, T)$ is minimal, the map $T: X\\rightarrow X$ is almost open.","Observing that $\\omega _{\\mathcal {S}_T (\\delta )} (x)$ is a closed subset of $X$ for each $x\\in X$ , the implication of $(2)\\Rightarrow (3)$ follows from Lemma REF and the minimality of $(X, T)$ .","The implication of $(3)\\Rightarrow (4)\\Rightarrow (5)$ follows from Proposition REF .","Since a minimal system is either multi-sensitive or a weakly almost one-to-one extension of its maximal equicontinuous factor by [23], and then $(5)\\Rightarrow (1)$ follows from Lemma REF .","This finishes the proof.","Clearly each multi-sensitive system is sensitive compact.","Observe that each non-proximal, transitive compact system is multi-sensitive by [22].","In particular, each minimal transitive compact system is multi-sensitive, as each minimal proximal system is trivial by [4] and all dynamical systems considered are assumed to be nontrivial.","Nevertheless, there are many minimal, non transitive compact, multi-sensitive systems.","For example, consider the classical dynamical system $(X, T)$ given by $X= \\mathbb {R}^2/\\mathbb {Z}^2$ and $T: (x, y)\\mapsto (x+ \\alpha , x+ y)$ with $\\alpha \\notin \\mathbb {Q}$ (see [15]).","As commented in [22], $(X, T)$ is an invertible minimal multi-sensitive system; note that $(X, T)$ is not weakly mixing, since $(X, T)$ admits an irrational rotation as its nontrivial equicontinuous factor and any equicontinuous factor of a weakly mixing system is trivial.","Remark that by [22] for a minimal system the system is transitive compact if and only if it is weakly mixing, and then the constructed system $(X, T)$ is not transitive compact.","Proposition 4.5 Each weakly mixing system $(X, T)$ is transitively sensitive.","Observe that we are only interested a nontrivial dynamical system, and then let $0< \\delta < {\\mathrm {diam}}(X)$ .","We choose opene subsets $W_1, W_2$ of $X$ such that the distance between $W_1$ and $W_2$ is strictly larger than $\\delta $ .","Now take arbitrary opene subsets $U, V, W$ of $X$ .","As $(X, T)$ is weakly mixing, $(X^3, T^{(3)})$ is transitive by [14], and then $N_T (U, V)\\cap S_T (W, \\delta )\\supset N_{T^{(3)}} (U\\times W\\times W, V\\times W_1\\times W_2)\\ne \\varnothing $ .","This implies that $(X, T)$ is transitively sensitive with a sensitive constant $\\delta > 0$ .","We give a sufficient condition for a dynamical system being transitively sensitive (by Proposition REF ) as the end of this section.","Lemma 4.6 Assume $\\omega _{\\mathcal {S}_T (\\varepsilon )} (x)= X$ for some $x \\in X$ and $\\varepsilon >0$ .","Then there is $\\delta > 0$ such that for any opene subset $U$ of $X$ and each neighbourhood $U_x$ of $x$ there are $y\\in U_x$ and $n\\in N_T (x, U)$ with $d(T^n x, T^n y)>\\delta $ .","If in addition, the map $T: X\\rightarrow X$ is almost one-to-one, then the converse holds.","Fix an opene subset $U$ of $X$ and a neighborhood $U_x$ of $x$ .","As $\\omega _{\\mathcal {S}_T (\\varepsilon )} (x)= X$ , there is $n\\in N_T (x, U)\\cap S_T (U_x, \\varepsilon )$ , and then there are points $x_1, x_2 \\in U_x$ with $d(T^n x_1, T^n x_2)> \\varepsilon $ .","We have either $d (T^n x, T^n x_1)> \\frac{\\varepsilon }{2}$ or $d (T^n x, T^n x_2)> \\frac{\\varepsilon }{2}$ , and then obtain the desired statement for $\\delta = \\frac{\\varepsilon }{2}$ .","Now suppose that there is $\\delta > 0$ such that for any opene subset $U$ of $X$ and each neighbourhood $U_x$ of $x$ there are $y\\in U_x$ and $n\\in N_T (x, U)$ with $d(T^n x, T^n y)>\\delta $ , and that the map $T: X\\rightarrow X$ is almost one-to-one.","Let $U, W$ be arbitrary opene subsets of $X$ .","It is clear $x\\in {\\mathrm {Tran}}(X, T)$ , and so there is $k\\in N_T (x, W)$ .","Note that $T: X\\rightarrow X$ is almost one-to-one, the map $T^k: X\\rightarrow X$ is also almost one-to-one by Corollary REF , and then we may take an opene subset $V$ of $X$ with $T^{- k} V= (T^k)^{- 1} V\\subset U$ .","Observe that $T^{- k} W$ is a neighborhood of $x$ , and then by the assumption there exists a point $y\\in T^{- k} W$ and an integer $n\\in \\mathbb {Z}_+$ such that $n\\in N_T (x, V)$ and $d (T^n x, T^n y)> \\delta $ .","In fact, we may assume $n>k$ , else we can replace $T^{- k} W$ by a small enough open neighbourhood $G_x\\subset T^{- k} W$ of $x$ with ${\\mathrm {diam}}(T^i G_x)< \\delta $ for all $0\\le i\\le k$ .","Then ${\\mathrm {diam}}(T^{n- k} W)> \\delta $ as $T^k x\\in W$ and $T^k y\\in W$ , and $T^{n- k} x\\in T^{- k} (T^n x)\\subset T^{- k} V\\subset U$ .","In particular, $N_T (x, U)\\cap S_T (W, \\delta )\\ne \\varnothing $ .","Thus $\\omega _{\\mathcal {S}_T (\\delta )} (x)= X$ by the arbitrariness of $U$ and $W$ ."],["Weakly disjointness and weakly mixing","Recall that dynamical systems $(X, T)$ and $(Y, S)$ are weakly disjoint if the product system $(X\\times Y, T\\times S)$ is transitive.","The following theorem characterizes weak disjointness, which is proved firstly by Weiss [38] in some special class and then is generalized by Akin and Glasner [3].","We say that $\\mathcal {F}$ is thick if $\\tau \\mathcal {F}= \\mathcal {F}$ , where $\\tau \\mathcal {F}= \\left\\lbrace F\\subset \\mathbb {Z}_+: \\bigcap _{j= 1}^n g^{- i_j} F\\in \\mathcal {F}\\ \\text{for each}\\ n\\in \\mathbb {N}\\ \\text{and all}\\ i_1, \\dots , i_n\\in \\mathbb {Z}_+\\right\\rbrace .$ Weiss-Akin-Glasner Theorem Let $\\mathcal {F}$ be a proper, translation invariant, thick family.","A dynamical system is $k\\mathcal {F}$ -transitive if and only if it is weakly disjoint from every $\\mathcal {F}$ -transitive system.","Observe that a dynamical system is weakly mixing if and only if it is weakly disjoint from itself, and then weak disjointness is characterized by [22] in some special case.","Now we discuss weak disjointness using dynamical compactness which will be some generalization of [22].","We will need the following Lemma 5.1 Let $(X, T)$ and $(Y, S)$ be dynamical systems and let $x\\in X$ .","Then the family $\\mathcal {N}_S$ is translation invariant and $\\omega _{\\mathcal {N}_S} (x)= \\omega _{\\mathcal {N}_S} (T x)$ .","By Proposition REF it suffices to prove that $\\mathcal {N}_S$ is a translation invariant family.","We also suppose that $\\mathcal {N}_S$ is proper (i.e., $(Y,S)$ is a transitive system) since, otherwise, the result is trivial.","Take arbitrary $F\\in \\mathcal {N}_S$ and any $i\\in \\mathbb {Z}_+$ .","Then there exist opene subsets $U, V$ of $Y$ with $N_S (U, V)\\subset F$ .","As the non-singleton space $Y$ contains no isolated points, we can take suitable opene $V_1\\subset V$ and $U_1\\subset U$ such that $U_1\\cap \\bigcup _{k= 0}^i S^k \\overline{V_1}=\\varnothing $ .","One has $g^{i} N_S (U, V)\\supset N_S (S^{- i} U_1, V_1)$ , which implies that the family $\\mathcal {N}_S$ is $+$ invariant: in fact, if $n\\in N_S (S^{- i} U_1, V_1)$ then $n> i$ by the selection, and so $n- i\\in N_S (U_1, V_1)\\subset N_S (U, V)$ .","Moreover, it is clear $g^{- i} (F)\\supset g^{- i} N_S (U, V)\\supset N_S (U, S^{- i} V)$ , and then the family $\\mathcal {N}_S$ is $-$ invariant.","This finishes the proof.","Theorem 5.2 The following conditions are equivalent: The systems $(X, T)$ and $(Y, S)$ are weakly disjoint.","Both ${\\mathrm {Tran}}_{k \\mathcal {N}_S} (X, T)$ and ${\\mathrm {Tran}}_{k \\mathcal {N}_T} (Y, S)$ are dense $G_\\delta $ subsets.","The set ${\\mathrm {Tran}}_{k \\mathcal {N}_S} (X, T)$ is a dense $G_\\delta $ subset of $X$ .","Both ${\\mathrm {Tran}}_{k \\mathcal {N}_S} (X, T)$ and ${\\mathrm {Tran}}_{k \\mathcal {N}_T} (Y, S)$ are nonempty subsets.","The set ${\\mathrm {Tran}}_{k \\mathcal {N}_S} (X, T)$ is a nonempty subset of $X$ .","$(1)\\Leftrightarrow (2)\\Leftrightarrow (3)$ : It is clear from the definition that: the system $(X, T)$ is $k \\mathcal {N}_S$ -transitive, if and only if the systems $(X, T)$ and $(Y, S)$ are weakly disjoint, if and only if $(X, T)$ is $k \\mathcal {N}_S$ -transitive and $(Y, S)$ is $k \\mathcal {N}_T$ -transitive.","As both $\\mathcal {N}_T$ and $\\mathcal {N}_S$ are families admitting a countable base, it is direct to obtain the equivalence of $(1)\\Leftrightarrow (2)\\Leftrightarrow (3)$ by applying Proposition REF .","The implication $(2)\\Rightarrow (4)\\Rightarrow (5)$ is obvious.","To finish the proof, we only need to show $(5)\\Leftrightarrow (3)$ .","By Lemma REF the family $\\mathcal {N}_S$ is translation invariant, and then the family $k \\mathcal {N}_S$ is also translation invariant.","Thus the equivalence of $(5)\\Leftrightarrow (3)$ follows from Proposition REF .","Note that we have a characterization of weak mixing by using dynamical compactness [22].","Now we improve [22] as follows.","Recall that $\\mathcal {S}\\subset \\mathbb {N}$ is an IP set if there exists $\\lbrace p_k: k\\in \\mathbb {N}\\rbrace \\subset \\mathbb {N}$ with $FS\\lbrace p_i\\rbrace _{i=1}^\\infty \\subset \\mathcal {S}$ , where $FS\\lbrace p_i\\rbrace _{i=1}^\\infty =\\lbrace p_{i_1}+ \\dots + p_{i_k}: k\\in \\mathbb {N}\\ \\text{and}\\ 1\\le i_1< \\dots < i_k\\rbrace .$ Analogously, for each $n\\in \\mathbb {N}$ we define $FS\\lbrace p_i\\rbrace _{i=1}^n= \\lbrace p_{i_1}+ \\dots + p_{i_k}: k\\in \\mathbb {N}\\ \\text{and}\\ 1\\le i_1< \\dots < i_k\\le n\\rbrace .$ Denote by $\\mathcal {F}_{ip}$ the family of all IP sets.","By [27], the subset ${\\mathrm {Tran}}_{\\mathcal {F}_{ip}} (X, T)$ contains a dense $G_\\delta $ subset of $X$ for any weakly mixing system $(X, T)$ , while ${\\mathrm {Tran}}_{\\mathcal {F}_{ip}} (X, T) \\ne \\varnothing $ does not imply the weak mixing of the system $(X, T)$ by [27].","We will improve that in the following Proposition REF .","Before proceeding, we make the following Lemma 5.3 Let $(X, T)$ be a dynamical system and $\\mathcal {F}$ be a family.","Let $\\delta > 0$ .","If the family $\\mathcal {S}_T (\\delta )\\cdot \\mathcal {F}$ is proper then $\\mathcal {S}_T (\\delta )\\cdot \\mathcal {F}\\subset \\mathcal {B}$ .","If the family $\\mathcal {N}_T\\cdot \\mathcal {F}$ is proper then $\\mathcal {N}_T\\cdot \\mathcal {F}\\subset \\mathcal {B}$ .","(1) Assume the contrary that there exists an opene subset $U$ in $X$ and $F \\in \\mathcal {F}$ such that $S_T (U, \\delta )\\cap F$ is finite, and so we may choose $m \\in \\mathbb {N}$ such that $n \\notin S_T(U, \\delta )\\cap F$ for any integer $n>m$ .","Since $T:X \\rightarrow X$ is uniformly continuous one can find opene $V \\subset U$ small enough such that ${\\mathrm {diam}}(T^i V) < \\delta $ for all $0\\le i \\le m$ .","Then $S_T (V, \\delta ) \\subset S_T (U, \\delta )$ , which implies $S_T(V, \\delta ) \\cap F= \\varnothing $ .","A contradiction.","(2) Assume the contrary that there exist opene subsets $U, V$ in $X$ and $F \\in \\mathcal {F}$ such that $N_T(U, V)\\cap F$ is finite, say $N_T(U, V)\\cap F= \\lbrace n_1, \\dots , n_k\\rbrace $ .","As the non-singleton space $X$ contains no isolated points, we can take opene $U_1\\subset U$ small enough such that $V_1:= V\\setminus \\bigcup _{i= 1}^k T^{n_i} \\overline{U_1}$ is an opene subset of $X$ .","By the construction we have $N_T(U_1, V_1)\\subset N_T (U, V)$ and then $N_T(U_1, V_1)\\cap F= \\varnothing $ , a contradiction.","Proposition 5.4 The following conditions are equivalent: $(X, T)$ is weakly mixing.","There exists a dense $G_\\delta $ subset $X^{\\prime }$ of $X$ such that, for each $x\\in X^{\\prime }$ , $ n_T(x, G) \\cap N_T(U, V) \\in \\mathcal {F}_{ip}$ for all opene subsets $G, U, V$ of $X$ .","$(2)\\Rightarrow (1)$ : Just observe from the assumption that $\\omega _{\\mathcal {N}_T}(x)=X$ for all $x \\in X^{\\prime }$ , and hence the system $(X, T)$ is weakly mixing by [22].","(1) $\\Rightarrow $ (2): Since $(X, T)$ is weakly mixing, $(X^2, T^{(2)})$ is also weakly mixing by [14] and [34], and hence by [22] there is a dense $G_\\delta $ subset $Y \\subset X^2$ such that $\\omega _{\\mathcal {N}_{T^{(2)}}}((x_1, x_2))=X^2$ for each $(x_1, x_2) \\in Y$ .","Applying the well-known Ulam Lemma there is a dense $G_\\delta $ subset $X^{\\prime } \\subset X$ such that, for any $x \\in X^{\\prime }$ , $\\lbrace y: (x, y) \\in Y\\rbrace $ is a dense $G_\\delta $ subset of $X$ .","Now we show that $X^{\\prime }$ is the desired set.","Let $x \\in X^{\\prime }$ and fix any opene subsets $G, U, V$ of $X$ .","Choose $y \\in G$ with $(x, y) \\in Y$ and then $\\omega _{\\mathcal {N}_{T^{(2)}}}((x, y))=X^2$ , in particular, $(y, y) \\in \\omega _{\\mathcal {N}_{T^{(2)}}}((x, y))$ .","Thus $n_T(x, G)\\cap n_T(y, G) \\cap N_T(U, V) \\cap N_T(V, V) \\ne \\varnothing ,$ and take $p_1\\in \\mathbb {N}$ from this set by Lemma REF .","We have $p_1 \\in n_T(x, G) \\cap N_T(U, V)$ and $T^{p_1} y \\in G$ , $T^{p_1} V \\cap V \\ne \\varnothing $ .","Define opene subsets $G_1= G \\cap T^{- p_1}G\\ni y$ and $V_1=V \\cap T^{-p_1}V$ .","Now we proceed inductively.","Suppose that we are given a sequence $\\lbrace p_1, \\dots , p_k\\rbrace \\subset \\mathbb {N}$ with $FS\\lbrace p_i\\rbrace _{i=1}^{k} \\subset n_T(x, G) \\cap N_T(U, V)$ , and opene subsets $G_k=\\bigcap _{s \\in FS\\lbrace p_i\\rbrace _{i=1}^k \\cup \\lbrace 0\\rbrace } T^{-s} G\\ni y, V_k=\\bigcap _{s \\in FS\\lbrace p_i\\rbrace _{i=1}^k \\cup \\lbrace 0\\rbrace } T^{-s}V.$ As $(y, y) \\in \\omega _{\\mathcal {N}_{T^{(2)}}}((x, y))$ , we may take $p_{k+1}\\in \\mathbb {N}$ by Lemma REF from the set $n_T(x, G_k) \\cap n_T(y, G_k) \\cap N_T(U, V_k) \\cap N_T(V_k, V_k)$ .","It is not hard to check that $G_{k+1}=G_k \\cap T^{-p_{k+1}}G_k=\\bigcap _{s \\in FS\\lbrace p_i\\rbrace _{i=1}^{k+ 1} \\cup \\lbrace 0\\rbrace } T^{-s} G\\ni y,$ $V_{k+1}=V_k \\cap T^{-p_{k+1}}V_k=\\bigcap _{s \\in FS\\lbrace p_i\\rbrace _{i=1}^{k+1} \\cup \\lbrace 0\\rbrace } T^{-s}V$ are both opene subsets of $X$ , and that $FS \\lbrace p_i\\rbrace _{i=1}^{k+1} \\subset n_T(x, G) \\cap N_T(U, V)$ , which completes the induction.","Finally, $n_T(x, G) \\cap N_T(U, V)\\in \\mathcal {F}_{ip}$ with $FS \\lbrace p_i\\rbrace _{i=1}^{\\infty } \\subset n_T(x, G) \\cap N_T(U, V)$ .","This finishes the proof."],["Transitive compact (non weakly mixing) systems","Recall that the system $(X, T)$ is totally transitive if $(X, T^k)$ is transitive for each $k\\in \\mathbb {N}$ ; and is topologically mixing if $N_T(U,V)\\in \\mathcal {F}_\\text{cof}$ for any opene subsets $U,V$ in $X$ .","Note that $(X, T)$ is weakly mixing if and only if $N_T(U,V)\\in \\mathcal {F}_{\\text{thick}}$ for any opene sets $U,V$ in $X$ by [14], [34], and so any weakly mixing system is totally transitive.","It is direct to check that each weakly mixing system is transitive compact.","In [22] the authors showed the existence of non totally transitive, transitive compact systems in both proximal and non-proximal cases.","We extend it as follows: Theorem 6.1 There is a totally transitive, transitive compact system $(X, T)$ which is not weakly mixing.","Take a nontrivial proximal, topologically mixing system $(Y, S)$ and let $(\\mathbb {S}^1, R_\\alpha )$ be the standard irrational rotation on the unit circle $\\mathbb {S}^1=\\mathbb {R}/\\mathbb {Z}$ with $\\alpha \\notin \\mathbb {Q}$ .","Note that a dynamical system is proximal if and only if it contains the unique fixed point, which is the only minimal point of the system [4].","Denote by $p_Y$ the unique minimal point (fixed point) of $(Y, S)$ .","Observe that the system $(Y \\times \\mathbb {S}^1, S\\times R_\\alpha )$ is totally transitive: for each $n\\in \\mathbb {N}$ , the system $(Y, S^n)$ is topologically mixing by the definition and it is standard that the system $(\\mathbb {S}^1, R_\\alpha ^n)$ is minimal, then it is direct to see that these two systems are weakly disjoint.","Let $(X,T)$ be the quotient system $Y \\times \\mathbb {S}^1/ \\sim $ equipped with the action $T$ induced naturally from $S\\times R_\\alpha $ , where the equivalence relation $\\sim $ is defined via: given $x, y\\in X$ , $x\\sim y$ if and only if either $x=y$ or $x$ and $y$ both have $p_Y$ in the first coordinate.","In other words the space $X$ looks like a cone space, where the vertex of the cone is a point $p$ , each “horizontal” fiber spaces are the space $Y$ , the vertical fiber spaces are the circles (see Figure 1).","Clearly, $(X, T)$ is totally transitive.","calc [rotate=-90] asin(0.5/3) (h) at (0, 3); (O) at (0, 0); (A) at (-2*cos(), 3-0.5*sin()); (B) at (2*cos(), 3-0.5*sin()); [fill=gray!50] (A) – (O) – (B) – cycle; [fill=gray!30] (h) ellipse (2 and 0.5); (0,-0.2) node p; (1.15, 1.25) nodeY; (0,2.75) node$\\mathbb {S}^1$ ; Figure 1.","Denote by $q:Y \\times \\mathbb {S}^1 \\rightarrow X$ the corresponding quotient map, then $q: Y_\\infty \\times \\mathbb {S}^1\\rightarrow X \\setminus \\lbrace p\\rbrace $ is a homeomorphism, where we set $Y_\\infty = Y\\setminus \\lbrace p_Y\\rbrace $ .","It is standard that the system $(\\mathbb {S}^1, R_\\alpha )$ is not weakly mixing, and then there exist opene subsets $U_*, V_*$ of $\\mathbb {S}^1$ with $N_{R_\\alpha } (U_*, V_*)\\notin \\mathcal {F}_{\\text{thick}}$ , hence $N_{T} (q (Y_\\infty \\times U_*), q (Y_\\infty \\times V_*))= N_{S\\times R_\\alpha } (Y_\\infty \\times U_*, Y_\\infty \\times V_*)\\subset N_{R_\\alpha } (U_*, V_*)$ is not thick.","This implies that the system $(X, T)$ is not weakly mixing.","Now let $U, V$ be arbitrary opene subsets of $X$ .","We can choose opene subsets $U_1, V_1\\subset Y_\\infty $ and $U_2, V_2 \\subset \\mathbb {S}^1$ with $U_1\\times U_2 \\subset q^{- 1} U$ and $V_1\\times V_2 \\subset q^{- 1} V$ .","As $(Y, S)$ is topologically mixing and $(\\mathbb {S}^1, R_\\alpha )$ is minimal, $N_S (U_1, V_1)\\in \\mathcal {F}_\\text{cof}$ and $N_{R_\\alpha } (U_2, V_2)\\in \\mathcal {F}_\\text{syn}$ , and then $N_S (U_1, V_1)\\cap N_{R_\\alpha } (U_2, V_2)\\in \\mathcal {F}_\\text{syn}$ , thus $N_{T} (U, V)= N_{S\\times R_\\alpha } (q^{- 1} U, q^{- 1} V)\\supset N_{S\\times R_\\alpha } (U_1\\times U_2, V_1\\times V_2)$ is a syndetic set.","Observe from the construction that the system $(X, T)$ is proximal with $p$ as its unique fixed point, then $n_T (x, U_p)$ is a thickly syndetic subset for each point $x\\in X$ and any neighbourhood $U_p$ of $p$ (see [22]).","This implies $p \\in \\omega _{\\mathcal {N}_T} (x)$ for each $x\\in X$ , and then the system $(X, T)$ is transitive compact.","The following result is proved independently in [11] and [37].","Lemma 6.2 Any $\\omega $ -limit set $\\omega _T(x)$ can not be decomposed into $\\alpha $ disjoint closed, nonempty, positively $T$ -invariant subsets, where $2\\le \\alpha \\le \\aleph _0$ .","Before proceeding, we need the following example, for which we fail to find a reference and hence provide a detailed construction, as it is crucial in our arguments.","Proposition 6.3 For any given compact metric space $Z$ , there exists a topologically mixing system $(X, T)$ such that, $Z$ can be realized as the set of all its minimal points, furthermore, its each minimal point is a fixed point.","The construction is divided into two steps.","In the first step we shall construct a topologically mixing system $(Y, F)$ but with two fixed points, which are the only minimal points of the system.","Let $\\Sigma =\\lbrace 0,1\\rbrace ^{\\mathbb {Z}_+}$ and $\\sigma :\\Sigma \\rightarrow \\Sigma $ be the full (one-sided) shift.","We are going to find the system $(Y, F)$ of the form $(\\overline{{\\mathrm {orb}}_{\\sigma } (x)}, \\sigma )$ for some $x \\in \\Sigma $ .","In order to define $x\\in \\Sigma $ , firstly we represent each $W\\in \\Sigma ^*$ with $|W|\\ge 1$ in the following form: $W=a^{i} Q b^{j}$ , where $a^{i}$ and $b^{j}$ (with $i\\ge 1$ ) are the longest segments of equal digits which we can take at the beginning and at the end of $W$ , whereas $Q$ is the rest, possible the empty subblock.","Clearly, $j$ may be equal to 0 and then $b^j$ will be the empty subblock, in this case, we treat the digit $b$ as 0; in particular, if $W= a^k$ then we set $Q$ to be the empty subblock and $i= k$ , $b= 0$ , $j= 0$ .","Now we are going to define $x\\in \\Sigma $ .","Let $A_1=10$ be the first block of $x$ and define inductively the rest blocks $A_2, A_3, \\dots $ , then $x$ will be the limit of the starting blocks $A_k$ .","Suppose that we have defined $A_k, k\\in \\mathbb {N}$ .","Since $A_{k}$ has finitely many subblocks, there is a finite number of different pairs of these subblocks.","For any pair $(W_1, W_2)$ of subblocks of $A_k$ we will define a block $c (W_1, W_2)$ by using their combination.","Then we are ready to define $A_{k+1}$ : at the beginning of $A_{k+1}$ we write $A_k0^k1^k$ , and then all possible blocks $c (W_1, W_2)$ of pairs $(W_1, W_2)$ of subbloks of $A_{k}$ in any fixed order.","The definition $c (W_1, W_2)$ depends on the structure of $W_1$ and $W_2$ .","Let us write $W_1$ and $W_2$ in the form as above: $W_1=a^{i_1} Q_1 b^{j_1}$ and $W_2=c^{i_2} Q_2 d^{j_2}$ , where $a^{i_1}$ , $b^{j_1}$ , $c^{i_2}$ and $d^{j_2}$ (with $i_1, i_2\\ge 1$ ) are the longest segments of equal digits which we can take at the beginning and at the end of $W_1$ and $W_2$ , whereas $Q_1$ and $Q_2$ are the rest, possible empty subblocks ($j_1$ and $j_2$ may be equal to 0, and then we treat the corresponding digits as 0).","The combination block of the pair $W_1, W_2$ , i.e.","$c(W_1, W_2)$ , is defined as follows: $c(W_1, W_2)=a^{k+i_1} Q_1 b^{j_1+k}c^{k+i_2}Q_2d^{k+j_2}a^{k+i_1} Q_1 b^{j_1+k+1}c^{k+i_2}Q_2 d^{k+j_2} .$ We see from the construction that $A_{k+1}$ is presented as a sequence of blocks with length not longer than $|A_k|$ , which are separated from each other with some sequences of blocks of consecutive 1's or 0's of length not less than $k$ .","In fact, for all $m>k$ this property holds for $A_m$ (and hence in any subblock of $A_m$ with length more than $|A_k|+ 2k- 1$ one can find $0^k$ or $1^k$ ).","Suppose that $A_m$ may be presented in this form.","Observe that $A_{m+1}$ is obtained by adding to $A_m 0^m 1^m$ combination blocks $c(W_1^*, W_2^*)$ of all pairs $(W_1^*, W_2^*)$ of subblocks of $A_m$ glued in a proper way, and note that if the property holds for blocks $W_1^*$ and $W_2^*$ (with beginning and end parts of the blocks possible exception) then the property holds for the combination block $c(W_1^*, W_2^*)$ .","In particular, the property holds for $A_{m+ 1}$ .","Put $(Y, F):=(\\overline{{\\mathrm {orb}}_{\\sigma } (x)},\\sigma )$ .","Now let us check that it has the required properties.","Firstly, in any subblock of $x$ with length more than $|A_k|+2k-1$ one can find $0^k$ or $1^k$ .","And hence, if we increase the length of a block $w$ of $x$ the length of the biggest subblock of $w$ in form of $0^m$ or $1^m$ increases unboundedly.","Moreover, for each $k\\in \\mathbb {N}$ both $0^k$ and $1^k$ appear in $x$ .","In particular, both $0^\\infty $ and $1^\\infty $ are fixed points of $(\\overline{{\\mathrm {orb}}_{\\sigma } (x)}, \\sigma )$ , and there is no other minimal sets in it.","Recall that the base for the open sets in $\\Sigma $ is given by the collection of all cylinder sets $C[c_0c_1c_2 \\dots c_m]=\\lbrace x \\in \\Sigma :~ x_i=c_i \\text{ for } 0\\le i\\le m \\rbrace .$ Given any two subblocks $W_1$ and $W_2$ of $A_m$ , we write $W_1$ and $W_2$ in the form as above: $W_1=a^{i_1} Q_1 b^{j_1}$ and $W_2=c^{i_2} Q_2 d^{j_2}$ .","If $b= c$ , then for each $l\\ge m$ there exists a combination block $c_l (W_1, W_2)$ containing the subblock $W_1 b^l W_2$ , and hence $N_F ([W_1]\\cap Y, [W_2]\\cap Y)\\supset \\lbrace m+ |W_1|, m+ |W_1|+ 1, \\dots \\rbrace $ ; if $b\\ne c$ , then for each $l\\ge m$ there exists a combination block $c_l (W_1, W_2)$ containing the subblocks $W_1 b^k c^k W_2$ and $W_1 b^k 1 c^k W_2$ , and hence $N_F ([W_1]\\cap Y, [W_2]\\cap Y)\\supset \\lbrace 2 m+ |W_1|, 2 m+ |W_1|+ 1, \\dots \\rbrace $ .","This shows that the system $(Y, F)$ is topologically mixing.","Now we shall finish the construction by the second step.","Firstly, we take $(X^{\\prime }, T^{\\prime })$ to be the product system $\\prod _1^\\infty (Y, F)$ .","It is ready to check that the system $(X^{\\prime }, T^{\\prime })$ is topologically mixing, for which the middle-third Cantor set $C$ is the set of all its minimal points and its each minimal point is a fixed point.","Note that there exists a continuous surjection $h: C\\rightarrow Z$ (see for example [25]), and then we consider the quotient system $(X, T)$ with $X= X^{\\prime }/\\sim $ equipped with the action induced naturally from $T^{\\prime }$ , where the closed positively $T^{\\prime }\\times T^{\\prime }$ -invariant equivalence relation $\\sim $ is defined via $x\\sim y$ if and only if $x= y\\in X^{\\prime }\\setminus Z$ or $h (x)= h (y)$ for $x, y\\in Z$ .","Then the system $(X, T)$ has the required properties.","The following result shows that in general there is no a topological structure similar to Lemma REF for the $\\omega _{\\mathcal {N}_T}$ -limit sets.","Theorem 6.4 For any given compact metric space $Z$ , there exists a non totally transitive, transitive compact system $(X, T)$ such that, $Z$ can be realized as the set of all its minimal points with its each minimal point being a fixed point, furthermore, $Z$ is realized as $\\omega _{\\mathcal {N}_T} (x)$ for some $x\\in X$ .","The idea of the proof is very similar to that of the first part (proximal case) of [22].","Instead of a nontrivial proximal, topologically mixing system $(Y, F)$ there (main point is “a map with exactly one minimal point!”), we take again a topologically mixing system $(Y, F)$ , but with $Z$ realized as the set of all its minimal points where its each minimal point is a fixed point (for existence of such a dynamical system see Proposition REF ).","Then, by the wedge sum construction there, we obtain a non totally transitive system $(X, T)$ such that $Z$ is realized as the set of all its minimal points with its each minimal point being a fixed point.","Similar to arguments there, it is not hard to show $Z\\supset \\omega _{\\mathcal {N}_T} (x)$ for all $x\\in X$ .","Firstly we prove the following claim: Claim.","For $x\\in X$ , if $x= y\\in Y$ and $p\\in Z\\cap \\omega _{\\mathcal {N}_{F^2}} (y)$ then, as a point in $X$ , $p$ belongs to $\\omega _{\\mathcal {N}_T} (x, T)$ .","Let $U_p$ be an open subset of $X$ containing $p$ , and clearly $U_p$ may be also viewed as an open subset of $Y$ containing $p$ .","Now for any given opene subsets $U$ and $V$ of $X$ : if both $U$ and $V$ can be viewed as opene subsets of $Y$ , then we can take $n\\in N_{F^2} (y, U_p)\\cap N_{F^2} (U, V)$ by the assumption $p\\in \\omega _{\\mathcal {N}_{F^2}} (y)$ and hence $2 n\\in N_{T} (y, U_p)\\cap N_{T} (U, V)$ ; if both $U$ and $V$ can be viewed as opene subsets of $Y_c$ , then both $T^{- 1} U$ and $T^{- 1} V$ can be viewed as opene subsets of $Y$ and hence $N_{T} (y, U_p)\\cap N_{T} (U, V)= N_{T} (y, U_p)\\cap N_{T} (T^{- 1} U, T^{- 1} V)\\ne \\varnothing $ ; if $U$ and $V$ can be viewed as opene subsets of $Y$ and $Y_c$ , respectively, noting $p\\in Z$ and hence $T p= p$ , there is an opene subset $V_p$ of $X$ containing $p$ such that $T V_p\\subset U_p$ , and then by the above reasoning we may take $n\\in N_{T} (y, V_p)\\cap N_{T} (U, T^{- 1} V)$ , and hence $n+ 1\\in N_{T} (y, U_p)\\cap N_{T} (U, V)$ ; it can be treated similarly the other case that $U$ and $V$ can be viewed as opene subsets of $Y_c$ and $Y$ , respectively.","Now we continue our proof.","As $(Y, F)$ is topologically mixing, the system $(Y, F^2)$ is weakly mixing, and then by [22] we may choose $x^*\\in Y$ such that $\\omega _{\\mathcal {N}_{F^2}} (x^*)= Y$ , in particular, $\\omega _{\\mathcal {N}_{F^2}} (x^*)\\supset Z$ .","Thus, by the above Claim, we obtain $Z\\subset \\omega _{\\mathcal {N}_T} (x^*)$ and hence $Z= \\omega _{\\mathcal {N}_T} (x^*)$ .","Note that a dynamical system is proximal if and only if it contains the unique fixed point, which is the only minimal point of the system [4].","Thus, as a direct corollary of Lemma REF and Theorem REF , we have: Corollary 6.5 There exists a non-proximal, non totally transitive, transitive compact system $(X, T)$ and a point $x_0\\in X$ such that $\\omega _{\\mathcal {N}_T} (x_0)\\ne \\omega _T (x)$ for all $x\\in X$ .","Nevertheless is still open the following Question B Let $(X,T)$ be a weakly mixing system.","Is any $\\omega _{\\mathcal {N}_T}(x)$ undecomposable into $\\alpha $ disjoint closed, nonempty, positively $T$ -invariant subsets, where $2\\le \\alpha \\le \\aleph _0$ ?","At the end of this section let us prove one more chaotical property of transitive compact systems in additional to already known in [22].","Recall that a pair of points $x, y \\in X$ is asymptotic if $\\lim _{n\\rightarrow \\infty } d (T^n x, T^n y)= 0$ .","Denote by ${\\mathrm {Asym}}_T (X)$ the set of all asymptotic pairs of points.","Any pair $(x, y)\\in {\\mathrm {Prox}}_T (X)\\setminus {\\mathrm {Asym}}_T (X)$ is called a Li-Yorke pair.","Recall that a dynamical system $(X, T)$ is Li-Yorke chaotic if there exists an uncountable set $S\\subset X$ with $(S\\times S)\\setminus \\Delta _2 (X)\\subset {\\mathrm {Prox}}_T (X)\\setminus {\\mathrm {Asym}}_T (X)$ , where $\\Delta _2 (X)= \\lbrace (x, x): x\\in X\\rbrace $ .","Proposition 6.6 Each transitive compact system $(X, T)$ is Li-Yorke chaotic.","Clearly $(X, T)$ is transitive.","Observe that we have assumed the state space to be not a singleton and in fact a compact metric space without isolated points, then $(X, T)$ is a transitive system with $X$ infinite.","Thus, the subset ${\\mathrm {Asym}}_T (X)$ is a fist category subset of $X\\times X$ by [24].","It is easy to show that ${\\mathrm {Prox}}_T (X)$ is a $G_\\delta $ subset of $X\\times X$ , and applying [22] to the transitive compact system $(X, T)$ we have that ${\\mathrm {Prox}}_T (x)$ is a dense subset of $X$ for each $x\\in X$ .","Thus ${\\mathrm {Prox}}_T (X)$ is a dense $G_\\delta $ subset of $X\\times X$ , and then ${\\mathrm {Prox}}_T (X)\\setminus {\\mathrm {Asym}}_T (X)$ is a second category subset of $X\\times X$ .","Now applying the well-known Mycielski Theorem [31] we obtain an uncountable subset $S\\subset X$ with $(S\\times S)\\setminus \\Delta _2 (X)\\subset {\\mathrm {Prox}}_T (X)\\setminus {\\mathrm {Asym}}_T (X)$ .","That is, $(X, T)$ is Li-Yorke chaotic."],["Weak transitive compactness and sensitivity for linear operators","In this section we are considering the dynamics of linear operators on infinite dimensional spaces in relation to the properties studied in previous sections.","More precisely, we will show the equivalence of the topological weak mixing property with a weak version of transitive compactness.","We obtain some results on transitive sensitivity too.","One should keep in mind that, for a linear dynamical system $(X, T)$ , where $X$ is an infinite dimensional space, neither compactness nor even local compactness of $X$ is satisfied.","In particular, we are interested in the case where $X$ is an infinite dimensional separable Banach space and $T:X\\rightarrow X$ is a continuous linear map (in short, operator).","In this framework, we will just write $(X,T)$ is an infinite dimensional linear dynamical system.","We recall that $X$ is a Banach space if it is a vector space endowed with a norm $\\Vert \\cdot \\Vert $ such that $X$ with the associated distance $d(x,y):=\\Vert x-y\\Vert $ becomes a complete metric space.","It is well known that $T: X\\rightarrow X$ is an operator if and only if $\\Vert T\\Vert :=\\sup \\lbrace \\Vert Tx\\Vert : \\Vert x\\Vert \\le 1\\rbrace <\\infty $ .","We refer the reader to the books [7] and [19] for the theory of linear dynamics.","Note that all notations and concepts discussed in previous sections can be introduced into linear dynamics.","We also introduce a weak version of dynamical compactness.","A linear system $(X,T)$ is called weakly dynamically compact with respect to the family $\\mathcal {F}$ if there exists a dense subset $X_0\\subset X$ such that the $\\omega _\\mathcal {F}$ -limit set $\\omega _{\\mathcal {F}}(x)$ is nonempty for all $x\\in X_0$ .","In particular, $(X, T)$ is called weakly transitive compact, if there exists a dense subset $X_0\\subset X$ such that for any point $x\\in X_0$ the $\\omega _{\\mathcal {N}_T}$ -limit set $\\omega _{\\mathcal {N}_T}(x)$ is nonempty, in other words, for any point $x\\in X_0$ there exists a point $z \\in X$ such that $ n_T(x,G) \\cap N_T(U,V)\\ne \\varnothing $ for any neighborhood $G$ of $z$ and any opene subsets $U,V$ of $X$ .","Theorem 7.1 Let $(X, T)$ be an infinite dimensional linear system.","Then $(X, T)$ is weakly mixing if and only if it is weakly transitive compact.","Sufficiency.","Suppose that $(X, T)$ is weakly transitive compact.","Let $X_0\\subset X$ be a dense subset such that, for each $x\\in X_0$ , there exists $z(x) \\in X$ such that $ n_T(x,G) \\cap N_T(U,V)\\ne \\varnothing $ for any neighborhood $G$ of $z (x)$ and opene $U, V\\subset X$ .","As $(X,T)$ is obviously transitive, by [18] (see also [19]) to obtain the weak mixing property we just need to show that, for each opene $U\\subset X$ and 0-neighbourhood $W$ , there is a continuous map $S: X\\rightarrow X$ commuting with $T$ such that $ S(U)\\cap W\\ne \\varnothing \\ \\ \\ \\mbox{ and } \\ \\ \\ S(W)\\cap U\\ne \\varnothing .$ Given an opene subset $U$ of $X$ and a 0-neighborhood $W$ , we fix $x\\in U\\cap X_0$ and $z(x)\\in X$ accordingly to the weak transitive compactness of $(X,T)$ .","Since 0-neighbourhoods are absorbing, we find a scalar $\\lambda \\ne 0$ such that $\\lambda z (x)\\in W$ .","Let $G$ be a neighbourhood of $z (x)$ such that $\\lambda G\\subset W$ .","By the hypothesis we can find $m\\in n_T(x,G) \\cap N_T(\\lambda W,U).$ That is, $T^mx\\in G$ and so $\\lambda T^mx\\in W$ ; additionally, there exists $w\\in W$ with $T^m\\lambda w\\in U$ .","Now pick $S:=\\lambda T^m$ , we have that $S$ commutes with $T$ and the property (REF ) is satisfied, therefore the system is weakly mixing.","Necessity.","Conversely, under the assumption of the weak mixing property for $(X,T)$ , we know by [8] (see also [19]) that there exists an increasing sequence $\\lbrace n_k: k\\in \\mathbb {N}\\rbrace \\subset \\mathbb {N}$ and a dense subset $X_0\\subset X$ such that $T^{n_k}x\\rightarrow 0$ for each $x\\in X_0$ and, for arbitrary opene $U,V\\subset X$ , we can find $k\\in \\mathbb {N}$ such that $T^{n_k}(U)\\cap V\\ne \\varnothing $ .","Thus, we obtain easily that $(X,T)$ is weakly transitive compact by selecting $z(x)=0$ for every $x\\in X_0$ .","Concerning sensitivity, the situation is more complicated and, although we obtain some advances, three related problems are left open.","Proposition 7.2 Let $(X, T)$ be an infinite dimensional linear, topologically transitive system.","Then $(X, T)$ is thickly multi-sensitive, that is, there exists $\\delta > 0$ such that $\\bigcap _{i= 1}^k S_T (U_i, \\delta )$ is thick for any finite collection of opene $U_1, \\dots , U_k\\subset X$ .","Let $U_1, \\dots , U_k$ be opene sets, and let $m\\in \\mathbb {N}$ .","Pick points $x_1, \\dots , x_k$ such that $x_i \\in U_i$ and choose $\\varepsilon >0$ such that $B_\\varepsilon (x_i) \\subset U_i$ , where $B_\\varepsilon (x_i)$ is the open ball of radius $\\varepsilon $ centered at $x_i$ , for all $i \\in \\lbrace 1, \\dots , k\\rbrace $ .","By a hypercyclic vector we mean that its orbit is dense in the space $X$ .","Take a hypercyclic vector $u \\in B_\\varepsilon (0)$ by [19], and let $y_i=x_i+u$ .","Then $y_i \\in U_i$ by the construction.","Since $u$ is hypercyclic there is $n \\in \\mathbb {N}$ , $n>m$ , such that $\\Vert T^n u\\Vert >(\\Vert T\\Vert +1)^m$ .","Then $\\rho (T^{n-j}x_i, T^{n-j}y_i)=\\Vert T^{n-j}(x_i-y_i)\\Vert =\\Vert T^{n-j}u\\Vert >(\\Vert T\\Vert +1)^{m-j}>1$ for all $i= 1, \\dots , k$ and $j= 0, \\dots , m- 1$ .","Hence $\\lbrace n, n- 1, \\dots , n-m+1\\rbrace \\subset \\bigcap _{i=1}^k S_T(U_i, 1)$ , and therefore $(X, T)$ is thickly multi-sensitive.","Proposition 7.3 Let $(X,T)$ be an infinite dimensional linear system.","Then the following conditions are equivalent: For each $\\delta > 0$ , $(X, T)$ is transitively sensitive with a sensitive constant $\\delta $ .","There exists $\\delta _0> 0$ such that $(X, T)$ is transitively sensitive with a sensitive constant $\\delta _0$ .","There exists $\\delta _0> 0$ such that $S_T (W_0, \\delta _0)\\cap N_T (U, V)\\ne \\varnothing $ for any opene subsets $U, V$ of $X$ and any 0-neighbourhood $W_0$ .","We just need to show $(3) \\Rightarrow (1)$ .","Indeed, let $\\delta >0$ be arbitrary, and fix arbitrary opene $U, V, W$ of $X$ .","We select $\\varepsilon >0$ and $x\\in W$ such that $x+B_\\varepsilon (0)\\subset W$ .","Observing $S_T (\\lambda W_0, \\lambda \\delta _0)$ for any scalar $\\lambda \\ne 0$ , and so without loss of generality we assume $\\delta >\\delta _0$ .","Let $0<\\varepsilon ^{\\prime }< \\frac{\\delta _0 \\varepsilon }{\\delta }$ , and set $W_0=B_{\\varepsilon ^{\\prime }}(0)$ .","By the hypothesis there are $y,z\\in W_0$ and $n\\in N_T(U,V)$ such that $\\Vert T^ny-T^nz\\Vert > \\delta _0$ .","Set $y^{\\prime }=x+\\frac{\\delta }{\\delta _0} y$ and $z^{\\prime }=x+\\frac{\\delta }{\\delta _0} z$ .","We have $y^{\\prime }, z^{\\prime }\\in W$ and $\\Vert T^ny^{\\prime }-T^nz^{\\prime }\\Vert > \\delta $ .","As opene $U, V, W\\subset X$ are arbitrary, $(X, T)$ is transitively sensitive with a sensitive constant $\\delta $ .","In this framework the weak mixing property implies transitive sensitivity too.","The following result establishes a very close connection of transitivity with transitive sensitivity.","We do not know, however, whether every transitive linear system is transitively sensitive.","Proposition 7.4 Let $(X,T)$ be an infinite dimensional linear, topologically transitive system.","If $(X, T)$ is not transitively sensitive, then there exists a dense open subset $U_0\\subset X$ such that every $x\\in U_0$ has a dense orbit.","If $(X,T)$ is not transitively sensitive, by Proposition REF we find opene $U, V$ of $X$ and $\\delta >1$ such that $\\Vert T^nx\\Vert \\le \\delta $ whenever $n\\in N_T(U,V)$ and $\\Vert x\\Vert \\le 1$ .","We fix an arbitrary opene $V^{\\prime }\\subset V$ and select an opene $\\widehat{V}\\subset V^{\\prime }$ and $\\varepsilon >0$ such that $\\widehat{V}+B_\\varepsilon (0)\\subset V^{\\prime }$ .","Given $u\\in U$ , there is $\\varepsilon ^{\\prime }< \\frac{\\varepsilon }{\\delta }$ such that $U^{\\prime }:=u+B_{\\varepsilon ^{\\prime }}(0)\\subset U$ .","Since $T$ is transitive, there exists $m\\in N_T(U^{\\prime },\\widehat{V})\\subset N_T(U,V)$ .","That is, we find $u^{\\prime }=u+w\\in U^{\\prime }$ with $\\Vert w\\Vert <\\varepsilon ^{\\prime }$ and $T^mu^{\\prime }\\in \\widehat{V}$ .","By the assumption $\\Vert T^mw\\Vert \\le \\delta \\varepsilon ^{\\prime }<\\varepsilon $ .","Therefore, $T^mu=T^mu^{\\prime }-T^mw\\in \\widehat{V}+B_\\varepsilon (0)\\subset V^{\\prime }$ .","Since $u\\in U$ and opene $V^{\\prime }\\subset V$ are arbitrary, we obtain that the orbit of every element in $U$ is somewhere dense, thus everywhere dense by transitivity of the system.","Finally, the open set $U_0:=\\bigcup _{n\\in \\mathbb {N}}T^{-n}(U)$ is dense, and every element in $U_0$ has a dense orbit.","There are (very difficult) examples of linear systems $(X,T)$ such that every non-zero element has a dense orbit [35], but it seems to unknown whether every linear system that admits an open set of elements whose orbit is dense is so that every non-zero element has a dense orbit.","It is also worthy to mention that there are (also rare) examples of transitive but not weakly mixing linear systems [10] (see also [7]), but as far as we know there are no examples of transitive non-weakly mixing linear systems such that every non-zero element has a dense orbit.","Concerning weak disjointness, observe that for each separable Banach space the family of all opene subsets admits a countable base, and then it is a routine to show that Theorem REF holds true within linear systems too.","Note that the intersection of finitely many thickly syndetically sets is still thickly syndetic, and that an interesting property is that every topologically ergodic linear system $(X, T)$ (i.e., each element of $\\mathcal {N}_T$ is a syndetic set) satisfies that each element of $\\mathcal {N}_T$ is actually a thickly syndetic set (see the exercises in [19]).","Thus any finite family $(X_1,T_1), \\dots ,(X_k,T_k)$ of topologically ergodic linear systems is weakly disjoint and, moreover, the product system $(X_1\\times \\dots \\times X_k, T_1\\times \\dots \\times T_k)$ is topologically ergodic."]],"string":"[\n [\n \"Finite intersection property and dynamical compactness\"\n ],\n [\n \"Abstract Dynamical compactness with respect to a family as a new concept of chaoticity of a dynamical system was introduced and discussed in [22].\",\n \"In this paper we continue to investigate this notion.\",\n \"In particular, we prove that all dynamical systems are dynamically compact with respect to a Furstenberg family if and only if this family has the finite intersection property.\",\n \"We investigate weak mixing and weak disjointness by using the concept of dynamical compactness.\",\n \"We also explore further difference between transitive compactness and weak mixing.\",\n \"As a byproduct, we show that the $\\\\omega_{\\\\mathcal{F}}$-limit and the $\\\\omega$-limit sets of a point may have quite different topological structure.\",\n \"Moreover, the equivalence between multi-sensitivity, sensitive compactness and transitive sensitivity is established for a minimal system.\",\n \"Finally, these notions are also explored in the context of linear dynamics.\"\n ],\n [\n \"Introduction\",\n \"By a (topological) dynamical system $(X,T)$ we mean a compact metric space $X$ with a metric $d$ and a continuous self-surjection $T$ of $X$ .\",\n \"We say it trivial if the space is a singleton.\",\n \"Throughout this paper, we are only interested in a nontrivial dynamical system, where the state space is a compact metric space without isolated points.\",\n \"This paper is a continuation of the research carried out in [22], where the authors discuss a dynamical property (called dynamical compactness) and examine it firstly for transitive compactness.\",\n \"Some results of this paper can be considered as a contribution to dynamical topology – an area of the theory of dynamical systems in which the topological properties of maps that can be described in dynamical terms.\",\n \"Let $\\\\mathbb {Z}_{+}$ be the set of all nonnegative integers and $\\\\mathbb {N}$ the set of all positive integers.\",\n \"Before going on, let us recall the notion of a Furstenberg family from [1].\",\n \"Denote by $\\\\mathcal {P} =\\\\mathcal {P}(\\\\mathbb {Z}_{+})$ the set of all subsets of $\\\\mathbb {Z}_{+}$ .\",\n \"A subset $\\\\mathcal {F}\\\\subset \\\\mathcal {P}$ is a (Furstenberg) family, if it is hereditary upward, that is, $F_1 \\\\subset F_2$ and $F_1 \\\\in \\\\mathcal {F}$ imply $F_2 \\\\in \\\\mathcal {F}$ .\",\n \"Any subset $\\\\mathcal {A}$ of $\\\\mathcal {P}$ clearly generates a family $\\\\lbrace F\\\\in \\\\mathcal {P}: F\\\\supset A\\\\ \\\\text{for some}\\\\ A\\\\in \\\\mathcal {A}\\\\rbrace $ .\",\n \"Denote by $\\\\mathcal {B}$ the family of all infinite subsets of $\\\\mathbb {Z}_{+}$ , and by $\\\\mathcal {P}_+$ the family of all nonempty subsets of $\\\\mathbb {Z}_+$ .\",\n \"For a family $\\\\mathcal {F}$ , the dual family of $\\\\mathcal {F}$ , denoted by $k\\\\mathcal {F}$ , is defined as $\\\\lbrace F\\\\in \\\\mathcal {P}: F\\\\cap F^{\\\\prime }\\\\ne \\\\varnothing \\\\mbox{ for any }F^{\\\\prime } \\\\in \\\\mathcal {F}\\\\rbrace .$ A family $\\\\mathcal {F}$ is proper if it is a proper subset of $\\\\mathcal {P}$ , that is, $\\\\mathbb {Z}_+\\\\in \\\\mathcal {F}$ and $\\\\varnothing \\\\notin \\\\mathcal {F}$ .\",\n \"By a filter $\\\\mathcal {F}$ we mean a proper family closed under intersection, that is, $F_1, F_2\\\\in \\\\mathcal {F}$ implies $F_1\\\\cap F_2\\\\in \\\\mathcal {F}$ .\",\n \"A filter is free if the intersection of all its elements is empty.\",\n \"We extend this concept, a family $\\\\mathcal {F}$ is called free if the intersection of all elements of $\\\\mathcal {F}$ is empty.\",\n \"For any $F\\\\in \\\\mathcal {P}$ , every point $x\\\\in X$ and each subset $G\\\\subset X$ , we define $T^{F} x= \\\\lbrace T^i x: i\\\\in F\\\\rbrace $ , $n_T(x, G)=\\\\lbrace n\\\\in \\\\mathbb {Z}_{+}: T^n x \\\\in G\\\\rbrace $ .\",\n \"The $\\\\omega $ -limit set of $x$ with respect to $\\\\mathcal {F}$ (see [1]), or shortly the $\\\\omega _\\\\mathcal {F}$ -limit set of $x$, denoted by $\\\\omega _{\\\\mathcal {F}}(x)$Remark that the notation $\\\\omega _{\\\\mathcal {F}}(x)$ used here is different from the one used in [1] (the notation $\\\\omega _{\\\\mathcal {F}}(x)$ used here is in fact $\\\\omega _{k \\\\mathcal {F}}(x)$ introduced in [1]).\",\n \"As this paper is a continuation of the research in [22], in order to avoid any confusion of notation or concept, we will follow the ones used in [22]., is defined as $\\\\bigcap _{F \\\\in \\\\mathcal {F}} \\\\overline{T^{F}x}= \\\\lbrace z\\\\in X: n_T(x,G) \\\\in k\\\\mathcal {F} \\\\ \\\\mbox{for every neighborhood} \\\\ G \\\\ \\\\mbox{of} \\\\ z \\\\rbrace .$ Let us remark that not always $\\\\omega _{\\\\mathcal {F}}(x)$ is a subset of the $\\\\omega $ -limit set $\\\\omega _{T}(x)$ , which is defined as $\\\\bigcap _{n=1}^\\\\infty \\\\overline{\\\\lbrace T^kx: k\\\\ge n\\\\rbrace }= \\\\lbrace z\\\\in X: N_T(x,G) \\\\in \\\\mathcal {B} \\\\ \\\\mbox{for every neighborhood} \\\\ G \\\\ \\\\mbox{of} \\\\ z \\\\rbrace .\",\n \"$ For instance, if each element of $\\\\mathcal {F}$ contains 0 then any point $x\\\\in \\\\omega _{\\\\mathcal {F}}(x)$ .\",\n \"But, as well known, a point $x\\\\in \\\\omega _{T}(x)$ if and only if $x$ is a recurrent pointA point $x\\\\in X$ is called recurrent if $x\\\\in \\\\omega _T(x).$ of $(X,T)$ .\",\n \"Nevertheless, if a family $\\\\mathcal {F}$ is free, then $\\\\omega _{\\\\mathcal {F}}(x) \\\\subset \\\\omega _{T}(x)$ for any point $x\\\\in X$ and if $(X,T)$ has a nonrecurrent point, then the converse is true (see Proposition REF ).\",\n \"A dynamical system $(X,T)$ is called compact with respect to $\\\\mathcal {F}$, or shortly dynamically compact, if the $\\\\omega _\\\\mathcal {F}$ -limit set $\\\\omega _{\\\\mathcal {F}}(x)$ is nonempty for all $x\\\\in X$ .\",\n \"H. Furstenberg started a systematic study of transitive systems in his paper on disjointness in topological dynamics and ergodic theory [14], and the theory was further developed in [16] and [15].\",\n \"Recall that the system $(X, T)$ is (topologically) transitive if $N_T (U_1, U_2)= \\\\lbrace n\\\\in \\\\mathbb {Z}_+: U_1\\\\cap T^{-n}U_2\\\\ne \\\\varnothing \\\\rbrace \\\\ (= \\\\lbrace n\\\\in \\\\mathbb {Z}_+: T^n U_1\\\\cap U_2\\\\ne \\\\varnothing \\\\rbrace )\\\\in \\\\mathcal {P}_+$ for any openeBecause we so often have to refer to open, nonempty subsets, we will call such subsets opene.\",\n \"subsets $U_1, U_2\\\\subset X$ , equivalently, $N_T (U_1, U_2)\\\\in \\\\mathcal {B}$ for any opene subsets $U_1, U_2\\\\subset X$ .\",\n \"In [22] the authors consider one of possible dynamical compactness — transitive compactness, and its relations with well-known chaotic properties of dynamical systems.\",\n \"Let $\\\\mathcal {N}_T$ be the set of all subsets of $\\\\mathbb {Z}_+$ containing some $N_T (U, V)$ , where $U, V$ are opene subsets of $X$ .\",\n \"A dynamical system $(X, T)$ is called transitive compact, if for any point $x\\\\in X$ the $\\\\omega _{\\\\mathcal {N}_T}$ -limit set $\\\\omega _{\\\\mathcal {N}_T}(x)$ is nonempty, in other words, for any point $x\\\\in X$ there exists a point $z \\\\in X$ such that $ n_T(x,G) \\\\cap N_T(U,V)\\\\ne \\\\varnothing $ for any neighborhood $G$ of $z$ and any opene subsets $U,V$ of $X$ .\",\n \"Let $(X, T)$ and $(Y, S)$ be two dynamical systems and $k\\\\in \\\\mathbb {N}$ .\",\n \"The product system $(X\\\\times Y, T\\\\times S)$ is defined naturally, and denote by $(X^k, T^{(k)})$ the product system of $k$ copies of the system $(X, T)$ .\",\n \"Recall that the system $(X, T)$ is minimal if it does not admit a nonempty, closed, proper subset $K$ of $X$ with $T K\\\\subset K$ , and is weakly mixing if the product system $(X^2, T^{(2)})$ is transitive.\",\n \"Any transitive compact system is obviously topologically transitive, and observe that each weakly mixing system is transitive compact ([4]).\",\n \"In fact, as it was shown in [22], each of notions are different in general and equivalent for minimal systems.\",\n \"Recall a very useful notion of weakly mixing subsets of a system, which was introduced in [9] and further discussed in [32] and [33].\",\n \"The notion of weakly mixing subsets can be regraded as a local version of weak mixing.\",\n \"Among many very interesting properties let us mention just one of them – positive topological entropy of a dynamical system implies the existence of weakly mixing sets (see [28] for details).\",\n \"A nontrivial closed subset $A\\\\subset X$ is called weakly mixing if for every $k\\\\ge 2$ and any opene sets $U_1, \\\\dots ,U_k, V_1, \\\\dots , V_k$ of $X$ with $U_i\\\\cap A \\\\ne \\\\varnothing $ , $V_i\\\\cap A \\\\ne \\\\varnothing $ , for any $i=1, \\\\dots ,k$ , one has that $\\\\bigcap _{i= 1}^kN_T(U_i\\\\cap A, V_i)\\\\ne \\\\varnothing .$ Let $A$ be a weakly mixing subset of $X$ and let $\\\\mathcal {N}_T(A)$ be the set of all subsets of $\\\\mathbb {Z}_+$ containing some $N_T (U\\\\cap A, V)$ , where $U, V$ are opene subsets of $X$ intersecting $A$ .\",\n \"The notion of sensitivity was first used by Ruelle [36], which captures the idea that in a chaotic system a small change in the initial condition can cause a big change in the trajectory.\",\n \"According to the works by Guckenheimer [20], Auslander and Yorke [6] a dynamical system $(X,T)$ is called sensitive if there exists $\\\\delta > 0$ such that for every $x\\\\in X$ and every neighborhood $U_x$ of $x$ , there exist $y\\\\in U_x$ and $n\\\\in \\\\mathbb {N}$ with $d(T^n x,T^n y)> \\\\delta $ .\",\n \"Such a $\\\\delta $ is called a sensitive constant of $(X, T)$ .\",\n \"Recently in [30] Moothathu initiated a way to measure the sensitivity of a dynamical system, by checking how large is the set of nonnegative integers for which the sensitivity occurs (see also [29]).\",\n \"For a positive $\\\\delta $ and a subset $U\\\\subset X$ define $ S_T (U, \\\\delta )= \\\\lbrace n\\\\in \\\\mathbb {Z}_+: \\\\mbox{ there are } x_1, x_2\\\\in U\\\\ \\\\mbox{such that}\\\\ d (T^n x_1, T^n x_2)>\\\\delta \\\\rbrace .$ A dynamical system $(X, T)$ is called multi-sensitive if there exists $\\\\delta > 0$ such that $\\\\bigcap _{i= 1}^k S_T (U_i, \\\\delta )\\\\ne \\\\varnothing $ for any finite collection of opene $U_1, \\\\dots , U_k\\\\subset X$ .\",\n \"Such a $\\\\delta $ is called a constant of multi-sensitivity of $(X, T)$ .\",\n \"Recall that a collection $A$ of subsets of a set $Y$ has the finite intersection property (FIP) if the intersection of any finite subcollection of $A$ is nonempty.\",\n \"The FIP is useful in formulating an alternative definition of compactness of a topological space: a topological space is compact if and only if every collection of closed subsets satisfying the FIP has a nonempty intersection itself (see, for instance [13], [25]).\",\n \"Obviously that a filter (say $\\\\mathcal {N}_T$ , when $(X,T)$ is weakly mixing), the family $\\\\mathcal {N}_T(A)$ for a weakly mixing subset $A$ of $(X,T)$ and the family $\\\\mathcal {S}_T(\\\\delta )$ when $(X, T)$ is a multi-sensitive system (with a constant of multi-sensitivity $\\\\delta > 0$ ) have FIP.\",\n \"Since all of these families are also free, actually they have the strong finite intersection property (SFIP), i.e., if the intersection over any finite subcollection of the family is infinite (see Proposition 2.2).\",\n \"In fact we can say more — the FIP is useful in characterizing the dynamical compactness (see Theorem REF ).\",\n \"Theorem FIP All dynamical systems are dynamically compact with respect to $\\\\mathcal {F}$ if and only if the family $\\\\mathcal {F}$ has the finite intersection property.\",\n \"We also introduce two new stronger versions of sensitivity: sensitive compactness and transitive sensitivity.\",\n \"Denote by $\\\\mathcal {S}_T(\\\\delta )$ the set of all subsets of $\\\\mathbb {Z}_+$ containing $S_T (U, \\\\delta )$ for some $\\\\delta > 0$ and opene $U\\\\subset X$ .\",\n \"We will call the system $(X, T)$ transitively sensitive if there exists $\\\\delta > 0$ such that $S_T (W, \\\\delta )\\\\cap N_T (U, V)\\\\ne \\\\varnothing $ for any opene subsets $U, V, W$ of $X$ ; and sensitive compact, if there exists $\\\\delta > 0$ such that for any point $x\\\\in X$ the $\\\\omega _{\\\\mathcal {S}_T(\\\\delta )}$ -limit set $\\\\omega _{\\\\mathcal {S}_T(\\\\delta )}(x)$ is nonempty, in other words, for any point $x\\\\in X$ there exists a point $z\\\\in X$ such that $ n_T(x,G)\\\\cap S_T (U, \\\\delta )\\\\ne \\\\varnothing $ for any neighborhood $G$ of $z$ and any opene $U$ of $X$ .\",\n \"The paper is organized as follows.\",\n \"In Section 2 we recall some basic concepts and properties used in later discussions from topological dynamics.\",\n \"In Section 3 we obtain some general results concerning dynamical compactness.\",\n \"In particular we show that all dynamical systems are dynamically compact with respect to a Furstenberg family if and only if this family has the finite intersection property (Theorem REF ).\",\n \"In Section 4 we discuss two stronger versions of sensitivity: transitive sensitivity and sensitive compactness.\",\n \"It was shown that each weakly mixing system is transitively sensitive (Proposition REF ), and in fact we can characterize transitive sensitivity of a general dynamical system in terms of dynamical compactness (Proposition REF ).\",\n \"Furthermore, all of the multi-sensitivity, sensitive compactness and transitive sensitivity are equivalent for a minimal system (Theorem REF ).\",\n \"Even though each minimal transitive compact system is multi-sensitive, nevertheless, there are many minimal multi-sensitive systems which are not transitive compact.\",\n \"Observe that the sensitivity of a dynamical system can be lifted up from a factor to an extension by an almost open factor map between transitive systems by [17].\",\n \"We prove that the transitive sensitivity can be lifted up to an extension from a factor by an almost one-to-one factor map and that the transitive sensitivity is projected from an extension to the sensitivity of a factor by a weakly almost one-to-one factor map (Lemma REF ).\",\n \"In Section 5 we show that dynamical compactness can be used to characterize the weak disjointness of dynamical systems (Theorem REF ).\",\n \"We also improve the result of Jian Li [27]: weak mixing implies $\\\\mathcal {F}_{ip}$ -point transitivity in terms of transitive compactness (Proposition REF ).\",\n \"In Section 6 the further difference between weak mixing and transitive compactness is explored.\",\n \"Precisely, there is a totally transitive, non weakly mixing, transitive compact system (Theorem REF ); and in fact any compact metric space can be realized as the $\\\\omega _{\\\\mathcal {N}_T}$ -limit set of a non totally transitive, transitive compact system $(X, T)$ (Theorem REF ).\",\n \"As a byproduct, we show that the $\\\\omega _{\\\\mathcal {N}_T}$ -limit sets and the $\\\\omega $ -limit sets have quite different topological structures for a general dynamical compact system $(X,T)$ .\",\n \"At the end of this section we add one more chaotical property of transitive compact systems (in additional to already known from [22]): transitive compactness implies Li-Yorke chaos (Proposition REF ).\",\n \"In Section 7 we consider the dynamics of linear operators on infinite dimensional spaces in relation to the properties studied in previous sections.\",\n \"In particular, we show the equivalence of the topological weak mixing property with a weak version of transitive compactness (Theorem REF ).\",\n \"Some results on sensitivity are also obtained.\",\n \"Acknowledgements.\",\n \"The first and third authors acknowledge the hospitality of the School of Mathematical Sciences of the Fudan University, Shanghai.\",\n \"The third author also acknowledges the hospitality of the Departament de Matemàtica Aplicada of the Universitat Politècnica de València and the Department of Mathematics of the Chinese University of Hong Kong.\",\n \"The first author was supported by NNSF of China (11225105, 11431012); the fourth author was supported by MINECO, Project MTM2013-47093-P, and by GVA, Project PROMETEOII/2013/013; and the fifth author was supported by NNSF of China (11271078).\"\n ],\n [\n \"Preliminaries\",\n \"In this section we recall standard concepts and results used in later discussions.\"\n ],\n [\n \"Basic concepts in topological dynamics\",\n \"Recall that $x\\\\in X$ is a fixed point if $T x= x$ , and an $\\\\mathcal {F}$ -transitive point of $(X, T)$ [27] if $n_T(x, U)\\\\in \\\\mathcal {F}$ for any opene subset $U$ of $X$ .\",\n \"It is a trivial observation that if a family $\\\\mathcal {F}$ admits an $\\\\mathcal {F}$ -transitive dynamical system $(X,T)$ without isolated points, then $\\\\mathcal {F}$ is free.\",\n \"Since $k(k\\\\mathcal {F})= \\\\mathcal {F}$ , it is easy to see that $x\\\\in X$ is an $\\\\mathcal {F}$ -transitive point of $(X, T)$ if and only if $\\\\omega _{k\\\\mathcal {F}}(x)=X$ .\",\n \"Denote by ${\\\\mathrm {Tran}}_\\\\mathcal {F} (X, T)$ the set of all $\\\\mathcal {F}$ -transitive points of $(X, T)$ .\",\n \"The system $(X, T)$ is $\\\\mathcal {F}$ -point transitive if ${\\\\mathrm {Tran}}_\\\\mathcal {F} (X, T)\\\\ne \\\\varnothing $ , and is $\\\\mathcal {F}$ -transitive if $N_T(U,V)\\\\in \\\\mathcal {F}$ for any opene subsets $U,V$ of $X$ .\",\n \"Write ${\\\\mathrm {Tran}}(X, T)= {\\\\mathrm {Tran}}_{\\\\mathcal {P}_+} (X, T)$ for short, and we also call the point $x$ transitive if $x\\\\in {\\\\mathrm {Tran}}(X, T)$ , equivalently, its orbit ${\\\\mathrm {orb}}_T (x)= \\\\lbrace T^n x: x= 0, 1, 2, \\\\dots \\\\rbrace $ is dense in $X$ .\",\n \"Since $T$ is surjective, the system $(X, T)$ is transitive if and only if ${\\\\mathrm {Tran}}(X, T)$ is a dense $G_\\\\delta $ subset of $X$ .\",\n \"In general, a subset $A$ of $X$ is $T$ -invariant if $TA = A$ , and positively $T$ -invariant if $T A\\\\subset A$ .\",\n \"If $A$ is a closed, nonempty, $T$ -invariant subset then $(A,T|_A)$ is called the associated subsystem.\",\n \"A minimal subset of $X$ is a closed, nonempty, $T$ -invariant subset such that the associated subsystem is minimal.\",\n \"Clearly, $(X,T)$ is minimal if and only if ${\\\\mathrm {Tran}}(X,T) = X$ , if and only if it admits no a proper, closed, nonempty, positively $T$ -invariant subset.\",\n \"A point $x \\\\in X$ is called minimal if it lies in some minimal subset.\",\n \"In this case, in order to emphasize the underlying system $(X, T)$ we also say that $x\\\\in X$ is a minimal point of $(X, T)$.\",\n \"Zorn's Lemma implies that every closed, nonempty, positively $T$ -invariant set contains a minimal set.\",\n \"A pair of points $x, y \\\\in X$ is called proximal if $\\\\liminf _{n\\\\rightarrow \\\\infty } d (T^n x, T^n y)= 0$ .\",\n \"In this case each of points from the pair is said to be proximal to another.\",\n \"Denote by ${\\\\mathrm {Prox}}_T (X)$ the set of all proximal pairs of points.\",\n \"For each $x\\\\in X$ , denote by ${\\\\mathrm {Prox}}_T (x)$ , called the proximal cell of $x$, the set of all points which are proximal to $x$ .\",\n \"Recall that a dynamical system $(X,T)$ is called proximal if ${\\\\mathrm {Prox}}_T (X)= X\\\\times X$ .\",\n \"The system $(X,T)$ is proximal if and only if $(X,T)$ has the unique fixed point, which is the only minimal point of $(X,T)$ (e.g.\",\n \"see [4]).\",\n \"The opposition to the notion of sensitivity is the concept of equicontinuity.\",\n \"Recall that $x \\\\in X$ is an equicontinuity point of $(X, T)$ if for every $\\\\varepsilon > 0$ there exists a $\\\\delta > 0$ such that $d (x, x^{\\\\prime }) < \\\\delta $ implies $d (T^n x, T^n x^{\\\\prime })< \\\\varepsilon $ for any $n\\\\in \\\\mathbb {Z}_+$ .\",\n \"Denote by $\\\\text{Eq} (X, T)$ the set of all equicontinuity points of $(X, T)$ .\",\n \"The system $(X, T)$ is called equicontinuous if $\\\\text{Eq} (X, T)= X$ .\",\n \"Each dynamical system admits a maximal equicontinuous factor.\",\n \"Recall that by a factor map $\\\\pi : (X, T)\\\\rightarrow (Y, S)$ between dynamical systems $(X, T)$ and $(Y, S)$ , we mean that $\\\\pi : X\\\\rightarrow Y$ is a continuous surjection with $\\\\pi \\\\circ T= S\\\\circ \\\\pi $ .\",\n \"In this case, we call $\\\\pi : (X, T)\\\\rightarrow (Y, S)$ an extension; and $(X, T)$ an extension of $(Y, S)$ , $(Y, S)$ a factor of $(X, T)$ .\"\n ],\n [\n \"Basic concepts of Furstenberg families\",\n \"In this subsection we recall from [1] basic concepts about Furstenberg families.\",\n \"Let $F\\\\in \\\\mathcal {P}$ .\",\n \"Recall that a subset $F$ is thick if it contains arbitrarily long runs of positive integers.\",\n \"Denote by $\\\\mathcal {F}_{\\\\text{thick}}$ the set of all thick subsets of $\\\\mathbb {Z}_+$ , and define $\\\\mathcal {F}_{\\\\text{syn}}= k \\\\mathcal {F}_{\\\\text{thick}}$ .\",\n \"Each element of $\\\\mathcal {F}_{\\\\text{syn}}$ is said to be syndetic, equivalently, $F$ is syndetic if and only if there is $N\\\\in \\\\mathbb {N}$ such that $\\\\lbrace i, i + 1, \\\\dots , i + N \\\\rbrace \\\\cap F \\\\ne \\\\varnothing $ for every $i \\\\in \\\\mathbb {Z}_{+}$ .\",\n \"We say that $F$ is thickly syndetic if for every $N\\\\in \\\\mathbb {N}$ the positions where length $N$ runs begin form a syndetic set.\",\n \"Denote by $\\\\mathcal {F}_{\\\\text{cof}}$ the set of all cofinite subsets of $\\\\mathbb {Z}_+$ .\",\n \"Note that by the classic result of Gottschalk a point $x\\\\in X$ is minimal if and only if $n_T(x, U)= \\\\lbrace n\\\\in \\\\mathbb {Z}_+: T^n x\\\\in U\\\\rbrace $ is syndetic for any neighborhood $U$ of $x$ .\",\n \"Hence, for any minimal system $(X, T)$ , the subset $N_T (U, V)$ is syndetic for any opene subsets $U, V$ of $X$ .\",\n \"Recall that a family $\\\\mathcal {F}$ is proper if it is a proper subset of $\\\\mathcal {P}$ , that is, $\\\\mathbb {Z}_+\\\\in \\\\mathcal {F}$ and $\\\\varnothing \\\\notin \\\\mathcal {F}$ .\",\n \"By a filter $\\\\mathcal {F}$ we mean a proper family closed under intersection, that is, $F_1, F_2\\\\in \\\\mathcal {F}$ implies $F_1\\\\cap F_2\\\\in \\\\mathcal {F}$ .\",\n \"For families $\\\\mathcal {F}_1$ and $\\\\mathcal {F}_2$ , we define the family $\\\\mathcal {F}_1\\\\cdot \\\\mathcal {F}_2:= \\\\lbrace F_1\\\\cap F_2: F_1\\\\in \\\\mathcal {F}_1, F_2\\\\in \\\\mathcal {F}_2\\\\rbrace $ and call it the interaction of $\\\\mathcal {F}_1$ and $\\\\mathcal {F}_2$.\",\n \"Thus we have $\\\\mathcal {F}_1 \\\\cup \\\\mathcal {F}_2 \\\\subset \\\\mathcal {F}_1 \\\\cdot \\\\mathcal {F}_2$ ; and it is easy to check that $\\\\mathcal {F}$ is a filter if and only if $\\\\mathcal {F}= \\\\mathcal {F}\\\\cdot \\\\mathcal {F}$ , and $\\\\mathcal {F}_1\\\\cdot \\\\mathcal {F}_2$ is proper if and only if $\\\\mathcal {F}_2 \\\\subset k\\\\mathcal {F}_1$ .\",\n \"For each $i\\\\in \\\\mathbb {Z}_+$ , define $g^i: \\\\mathbb {Z}_+\\\\rightarrow \\\\mathbb {Z}_+, j\\\\mapsto i+ j$ .\",\n \"Recall that a family $\\\\mathcal {F}$ is $+$ invariant if for every $i\\\\in \\\\mathbb {Z}_+$ , $F\\\\in \\\\mathcal {F}$ implies $g^{i} (F)\\\\in \\\\mathcal {F}$ ; $-$ invariant if for every $i\\\\in \\\\mathbb {Z}_+$ , $F\\\\in \\\\mathcal {F}$ implies $g^{- i} (F)\\\\in \\\\mathcal {F}$ , where $g^{-i}(F)=(g^i)^{-1}(F)=\\\\lbrace j-i:j\\\\in F,j\\\\ge i\\\\rbrace $ ; and translation invariant if it is both $+$ and $-$ invariant, equivalently, for every $i\\\\in \\\\mathbb {Z}_+$ , $F\\\\in \\\\mathcal {F}$ if and only if $g^{- i} (F)\\\\in \\\\mathcal {F}$ .\",\n \"As $g^{- i} (g^i A)= A$ and $g^i (g^{- i} A)\\\\subset A$ for any $i\\\\in \\\\mathbb {Z}_{+}$ , it is easy to obtain that the family $\\\\mathcal {F}$ is $+$ ($-$ , translation, respectively) invariant if and only if $k \\\\mathcal {F}$ is $-$ ($+$ , translation, respectively) invariant (see for example [1]).\",\n \"And then we have: Proposition 2.1 Let $x\\\\in X$ .\",\n \"Then $T \\\\omega _\\\\mathcal {F} (x)\\\\subset \\\\omega _\\\\mathcal {F} (T x)$ .\",\n \"Additionally, if $\\\\mathcal {F}$ is $-$ ($+$ , translation, respectively) invariant then $\\\\omega _\\\\mathcal {F} (T x)\\\\subset $ ($\\\\supset $ , $=$ , respectively) $\\\\omega _\\\\mathcal {F} (x)$ .\",\n \"Since the other items are alternative versions of [1] in our notations, it suffices to prove that if $\\\\mathcal {F}$ is $+$ invariant then $\\\\omega _\\\\mathcal {F} (T x)\\\\supset \\\\omega _\\\\mathcal {F} (x)$ .\",\n \"For each $y\\\\in \\\\omega _\\\\mathcal {F} (x)$ take an arbitrary neighborhood $U$ of $y$ , and let $F\\\\in \\\\mathcal {F}$ .\",\n \"Then $g^{1} (F)= \\\\lbrace i+ 1\\\\in \\\\mathbb {Z}_+: i\\\\in F\\\\rbrace \\\\in \\\\mathcal {F}$ as $\\\\mathcal {F}$ is $+$ invariant, and hence $n_T (x, U)\\\\cap g^{1} (F)\\\\ne \\\\varnothing $ , thus $\\\\varnothing \\\\ne g^{- 1} (n_T (x, U)\\\\cap g^{1} (F))= n_T (T x, U)\\\\cap F$ .\",\n \"It follows $y\\\\in \\\\omega _\\\\mathcal {F} (T x)$ from the arbitrariness of $U$ and $F$ , which finishes the proof.\",\n \"Proposition 2.2 Let $(X,T)$ be a dynamical system and let $\\\\mathcal {F}$ be a family.\",\n \"If $\\\\mathcal {F}$ is free, then $\\\\omega _{\\\\mathcal {F}}(x) \\\\subset \\\\omega _{T}(x)$ for any $x\\\\in X$ .\",\n \"Moreover, if $(X,T)$ has a nonrecurrent point, then the converse is true.\",\n \"If $\\\\mathcal {F}$ is free and has FIP then it has SFIP.\",\n \"(i) Suppose $\\\\omega _{\\\\mathcal {F}}(x)\\\\ne \\\\varnothing $ and take a point $y \\\\in \\\\omega _{\\\\mathcal {F}} (x):=\\\\bigcap _{F \\\\in \\\\mathcal {F}} \\\\overline{T^{F}x}$ .\",\n \"Let us show that $y \\\\in \\\\omega _{T} (x)$ .\",\n \"Since $\\\\mathcal {F}$ is free, $\\\\bigcap _{F \\\\in \\\\mathcal {F}} T^{F}x=\\\\varnothing $ .\",\n \"Otherwise there is $m \\\\in \\\\mathbb {Z}_+$ such that $T^mx \\\\in T^Fx$ for all $F \\\\in \\\\mathcal {F}$ , in other words $m \\\\in \\\\bigcap _{F \\\\in \\\\mathcal {F}}F$ , a contradiction.\",\n \"It means that there is $F \\\\in \\\\mathcal {F}$ such that $y \\\\notin T^Fx$ .\",\n \"So $y \\\\in \\\\overline{T^{F}x} \\\\setminus T^Fx$ , and $F$ is infinite.\",\n \"Therefore there exists an infinite sequence $T^{n_1}x, \\\\dots , T^{n_i}x,\\\\dots , $ which converges to $y$ .\",\n \"Hence $y \\\\in \\\\omega _T(x)$ .\",\n \"Now, let $(X,T)$ be a dynamical system with a nonrecurrent point $x_0\\\\in X$ , let $\\\\mathcal {F}$ be a family and $\\\\omega _{\\\\mathcal {F}}(x) \\\\subset \\\\omega _{T}(x)$ for any $x\\\\in X$ .\",\n \"Suppose $\\\\mathcal {F}$ is not free.\",\n \"It means there is a $k\\\\in \\\\mathbb {Z}_{+}$ that lies in each element of $\\\\mathcal {F}$ .\",\n \"Then obviously that $T^k(x)\\\\in \\\\omega _{\\\\mathcal {F}}(x)$ for any $x\\\\in X$ .\",\n \"Since $x_0$ is nonrecurrent, $x_0\\\\notin \\\\omega _T(x_0)\\\\ne X$ .\",\n \"It is well known that $\\\\omega _T(x)$ is $T$ -invariant, therefore $\\\\omega _T(y)=\\\\omega _T(x_0)$ for any $y\\\\in \\\\lbrace T^{-i}(x_0): i \\\\in \\\\mathbb {Z}_{+}\\\\rbrace $ , and $y\\\\notin \\\\omega _T(y)$ .\",\n \"Take a point $x_k\\\\in X$ with $T^k(x_k)=x_0$ .\",\n \"As we know $x_0 \\\\in \\\\omega _{\\\\mathcal {F}}(x_k)$ .\",\n \"But $\\\\omega _{\\\\mathcal {F}}(x_k)\\\\subset \\\\omega _T(x_k)= \\\\omega _T(x_0)$ , a contradiction.\",\n \"(ii) Suppose that for some $F_1, \\\\dots F_k \\\\in \\\\mathcal {F}$ $\\\\bigcap _{i=1}^k F_i=\\\\lbrace n_1, n_2, \\\\dots n_m\\\\rbrace $ .\",\n \"Since $\\\\mathcal {F}$ is free for each $k \\\\in \\\\mathbb {Z}_+$ there is $G_k \\\\in \\\\mathcal {F}$ such that $k \\\\notin G_k$ .\",\n \"Then $\\\\bigcap _{i=1}^k F_i \\\\cap \\\\bigcap _{j=1}^m G_{n_j}=\\\\varnothing $ , contradiction.\"\n ],\n [\n \"The concept of an almost one-to-one map\",\n \"Let $\\\\phi : X\\\\rightarrow Y$ be a continuous surjective map from a compact metric space $X$ onto a compact Hausdorff space $Y$ .\",\n \"Recall that $\\\\phi $ is almost open if $\\\\phi (U)$ has a nonempty interior in $Y$ for any opene $U\\\\subset X$ .\",\n \"Note that each factor map between minimal systems is almost open [5], in particular, for a minimal system $(X,T)$ the map $T:X\\\\rightarrow X$ is almost open [26].\",\n \"Denote by $Y_0\\\\subset Y$ the set of all points $y\\\\in Y$ whose fiber is a singleton.\",\n \"Then $Y_0$ is a $G_\\\\delta $ subset of $Y$ , because $Y_0= \\\\lbrace y\\\\in Y: \\\\phi ^{- 1} (y)\\\\ \\\\text{is a singleton}\\\\rbrace = \\\\bigcap _{n\\\\in \\\\mathbb {N}} \\\\left\\\\lbrace y\\\\in Y: {\\\\mathrm {diam}}(\\\\phi ^{- 1} (y))<\\\\frac{1}{n}\\\\right\\\\rbrace $ and the map $y\\\\mapsto {\\\\mathrm {diam}}(\\\\phi ^{- 1} (y))$ is upper semi-continuous.\",\n \"Here, we denote by ${\\\\mathrm {diam}}(A)$ the diameter of a subset $A\\\\subset X$ .\",\n \"Recall that the function $f: Y\\\\rightarrow \\\\mathbb {R}_+$ is upper semi-continuous if $\\\\limsup \\\\limits _{y\\\\rightarrow y_0} f (y)\\\\le f (y_0)$ for each $y_0\\\\in Y$ .\",\n \"Denote by $X_0\\\\subset X$ the set of all points $x\\\\in X$ such that the pre-image of $\\\\phi (x)$ is a singleton.\",\n \"Then $X_0= \\\\pi ^{- 1} (Y_0)$ is a $G_\\\\delta $ subset of $X$ .\",\n \"We call $\\\\phi $ weakly almost one-to-one if $Y_0$ is dense in $Y$ , and almost one-to-oneHere we use the concept of almost one-to-one following [3], and the concept of almost one-to-one used in [12], [23], [26] is in fact our weakly almost one-to-one.\",\n \"if $X_0$ is dense in $X$ .\",\n \"It is not hard to show that: if $\\\\phi $ is weakly almost one-to-one, then for any $\\\\delta > 0$ and any opene subset $U$ of $Y$ there exists opene $V\\\\subset U$ with ${\\\\mathrm {diam}}(\\\\phi ^{- 1} V)< \\\\delta $ ; and if $\\\\phi $ is almost one-to-one, then for any opene subset $U^*$ of $X$ there exists an opene subset $V^*$ of $Y$ with $\\\\phi ^{- 1} V^*\\\\subset U^*$ .\",\n \"Clearly almost one-to-one is much stronger than weakly almost one-to-one.\",\n \"For example, let $X$ be the closed unit interval, define $T (x)= 2 x$ for $x\\\\in [0, \\\\frac{1}{2}]$ and $T (x)= 1$ for $x\\\\in [\\\\frac{1}{2}, 1]$ , and then $T: X\\\\rightarrow X$ is clearly not almost one-to-one but weakly almost one-to-one.\",\n \"For each minimal system $(X, T)$ , the map $T: X\\\\rightarrow X$ is weakly almost one-to-one [26], and in fact almost one-to-one [23].\",\n \"The following result characterizes the relationship between weakly almost one-to-one and almost one-to-one, which extends [23].\",\n \"Proposition 2.3 Let $\\\\phi : X\\\\rightarrow Y$ be a continuous surjective map from a compact metric space $X$ onto a compact Hausdorff space $Y$ .\",\n \"Then $\\\\phi $ is almost one-to-one if and only if it is not only almost open but also weakly almost one-to-one.\",\n \"Firstly assume that $\\\\phi $ is almost one-to-one.\",\n \"Let $U\\\\subset X$ be an arbitrary opene subset.\",\n \"And then we can take $x_0\\\\in U$ such that the pre-image of $\\\\phi (x_0)$ is a singleton.\",\n \"From this it is easy to see that $\\\\phi (x_0)$ is contained in the interior of $\\\\phi (U)$ .\",\n \"This implies that $\\\\phi $ is almost open.\",\n \"The map $\\\\phi $ is clearly weakly almost one-to-one.\",\n \"Now assume that $\\\\phi $ is not only almost open but also weakly almost one-to-one.\",\n \"Let $U\\\\subset X$ be an arbitrary opene subset.\",\n \"Since $\\\\phi $ is almost open, $\\\\phi (U)$ has a nonempty interior in $Y$ , and then $\\\\phi ^{- 1} (y_0)$ is a singleton for some $y_0\\\\in \\\\phi (U)$ , as $\\\\phi $ is weakly almost one-to-one.\",\n \"This shows $U\\\\cap X_0\\\\ne \\\\varnothing $ , which finishes the proof.\",\n \"As a direct corollary, we have: Corollary 2.4 Let $\\\\phi : X\\\\rightarrow Y$ and $\\\\pi : Y\\\\rightarrow Z$ be continuous surjective maps between compact metric spaces.\",\n \"Then the composition map $\\\\pi \\\\circ \\\\phi : X\\\\rightarrow Z$ is almost one-to-one if and only if both $\\\\phi $ and $\\\\pi $ are almost one-to-one.\",\n \"Denote by $X_0$ ($X_1$ , respectively) the set of all points $x\\\\in X$ such that the pre-image of $(\\\\pi \\\\circ \\\\phi ) (x)$ ($\\\\phi (x)$ , respectively) is a singleton.\",\n \"Denote by $Z_0$ ($Z_1$ , respectively) the set of all points $z\\\\in Z$ whose $\\\\pi \\\\circ \\\\phi $ -fibers ($\\\\pi $ -fibers, respectively) are singletons.\",\n \"All of them are $G_\\\\delta $ subsets.\",\n \"Moreover, $X_0= X_1\\\\cap \\\\phi ^{- 1} (\\\\pi ^{- 1} Z_1)$ .\",\n \"In fact, $x\\\\in X_0$ if and only if $\\\\lbrace x\\\\rbrace = (\\\\pi \\\\circ \\\\phi )^{- 1} (\\\\pi \\\\circ \\\\phi (x))= \\\\phi ^{- 1} (\\\\pi ^{- 1} (\\\\pi (\\\\phi x)))$ , if and only if $\\\\pi ^{- 1} (\\\\pi (\\\\phi x))= \\\\lbrace \\\\phi (x)\\\\rbrace $ and $\\\\phi ^{- 1} (\\\\phi x)= \\\\lbrace x\\\\rbrace $ , if and only if $\\\\pi (\\\\phi x)\\\\in Z_1$ and $x\\\\in X_1$ .\",\n \"First assume that $\\\\pi \\\\circ \\\\phi $ is almost one-to-one, and then by Proposition REF : $X_0$ is a dense subset of $X$ , $Z_0$ is a dense subset of $Z$ and the map $\\\\pi \\\\circ \\\\phi $ is almost open.\",\n \"Note that $X_0\\\\subset X_1$ and $Z_0\\\\subset Z_1$ , we have that $X_1$ is dense in $X$ and $Z_1$ is dense in $Z$ .\",\n \"Hence $\\\\phi $ is almost one-to-one.\",\n \"Furthermore, as the map $\\\\pi \\\\circ \\\\phi $ is almost open, for any opene $V\\\\subset Y$ one has that $\\\\pi (V)= (\\\\pi \\\\circ \\\\phi ) (\\\\phi ^{- 1} V)$ has a nonempty interior in $Z$ , which implies that $\\\\pi $ is almost one-to-one by Proposition REF .\",\n \"Now assume that both $\\\\phi $ and $\\\\pi $ are almost one-to-one.\",\n \"Then $X_1$ is a dense $G_\\\\delta $ subset of $X$ and $Z_1$ is a dense $G_\\\\delta $ subset of $Z$ .\",\n \"Moreover, by Proposition REF both $\\\\phi $ and $\\\\pi $ are almost open, and then the continuous surjection $\\\\pi \\\\circ \\\\phi $ is also almost open, which implies that $(\\\\pi \\\\circ \\\\phi )^{- 1} (Z_1)$ is also a dense $G_\\\\delta $ subset of $X$ .\",\n \"Thus, $X_0= X_1\\\\cap \\\\phi ^{- 1} (\\\\pi ^{- 1} Z_1)$ is a dense $G_\\\\delta $ subset of $X$ , that is, the composition map $\\\\pi \\\\circ \\\\phi : X\\\\rightarrow Z$ is almost one-to-one.\",\n \"This finishes the proof.\",\n \"Let $\\\\pi : (X, T)\\\\rightarrow (Y, S)$ be a factor map between dynamical systems.\",\n \"If the map $\\\\pi : X\\\\rightarrow Y$ is almost one-to-one (weakly almost one-to-one, respectively), then we also call $(X, T)$ an almost one-to-one extension (a weakly almost one-to-one extension, respectively) of $(Y, S)$.\",\n \"The main result of [23] states that a minimal system is either multi-sensitive or a weakly almost one-to-one extension of its maximal equicontinuous factor.\",\n \"This is an analog of the well-known Auslander-Yorke dichotomy theorem: a minimal system is either sensitive or equicontinuous.\"\n ],\n [\n \"Symbolic dynamics\",\n \"Let $A$ be a nonempty finite set.\",\n \"We call $A$ the alphabet and elements of $A$ are symbols.\",\n \"The full (one-sided) $A$ -shift is defined as $\\\\Sigma = \\\\lbrace x = \\\\lbrace x_i\\\\rbrace _{i=0}^\\\\infty : x_i \\\\in A \\\\text{ for all } i \\\\in \\\\mathbb {Z}_+\\\\rbrace ,$ where we equip $A$ with the discrete topology and $\\\\Sigma $ with the product topology, and the shift map $\\\\sigma : \\\\Sigma \\\\rightarrow \\\\Sigma $ is a continuous surjection given by $ x = \\\\lbrace x_i\\\\rbrace _{i=0}^\\\\infty \\\\mapsto \\\\sigma x = \\\\lbrace x_{i+1}\\\\rbrace _{i=0}^\\\\infty ,$ that is, $\\\\sigma (x)$ is the sequence obtained by dropping the first symbol of $x$ .\",\n \"Usually we write an element of $\\\\Sigma $ as $x = \\\\lbrace x_i\\\\rbrace _{i=0}^\\\\infty = x_0 x_1 x_2 x_3 \\\\dots $ A block $w$ over $\\\\Sigma $ is a finite sequence of symbols and its length is the number of its symbols (denoted by $|w|$ ).\",\n \"An $n$ -block stands for a block of length $n$ .\",\n \"In general we are only interested in a block $w$ with $|w|\\\\ge 1$ if without any special statement, and denote by $\\\\Sigma ^*$ the set of all blocks over $\\\\Sigma $ .\",\n \"The block $w$ is a subblock of a block $v= v_1 \\\\dots v_m$ with $v_1, \\\\dots , v_m\\\\in A$ if there exists $1\\\\le i\\\\le j\\\\le m$ with $w= v_i \\\\dots v_j$ .\",\n \"The concatenation of two blocks $u = a_1 \\\\dots a_k$ and $v = b_1 \\\\dots b_l$ is the block $uv = a_1 \\\\dots a_k b_1 \\\\dots b_l$ .\",\n \"We write $u^n$ for the concatenation of $n\\\\ge 1$ copies of a block $u$ and $u^\\\\infty $ for the sequence $uuu \\\\dots \\\\in \\\\Sigma .$ By $x_{[i,j]}$ we denote the block $x_i x_{i+1} \\\\dots x_j$ , where $0 \\\\le i \\\\le j$ and $x =\\\\lbrace x_k\\\\rbrace _{k=0}^\\\\infty \\\\in \\\\Sigma $ .\",\n \"The subset $X\\\\subset \\\\Sigma $ is called a subshift if it is a closed, nonempty, $\\\\sigma $ -invariant subset of $\\\\Sigma $ .\",\n \"A cylinder of an $n$ -block $w \\\\in \\\\Sigma ^*$ in a subshift $X$ is the set $C[w] = \\\\lbrace x \\\\in X: x_{[0, n-1]} = w\\\\rbrace $ .\",\n \"The collection of all cylinders forms a basis of the topology of $X$ .\"\n ],\n [\n \"Dynamical compactness with respect to an arbitrary family\",\n \"Recall that a family $\\\\mathcal {F}$ has the finite intersection property (FIP) if the intersection of any finite subcollection of $\\\\mathcal {F}$ is nonempty.\",\n \"The following theorem shows that the FIP is useful in characterizing the dynamical compactness.\",\n \"Theorem 3.1 All dynamical systems are dynamically compact with respect to the family $\\\\mathcal {F}$ if and only if $\\\\mathcal {F}$ has the finite intersection property.\",\n \"Sufficiency.\",\n \"Suppose that $\\\\mathcal {F}$ has FIP.\",\n \"Take arbitrary dynamical system $(X, T)$ and let $x\\\\in X$ .\",\n \"Obviously the family $\\\\lbrace \\\\overline{T^{F}x}: F\\\\in \\\\mathcal {F}\\\\rbrace $ also has FIP, and then by compactness of the space $X$ the family has a nonempty intersection itself, i.e., $\\\\omega _{\\\\mathcal {F}}(x)=\\\\bigcap _{F \\\\in \\\\mathcal {F}} \\\\overline{T^{F}x} \\\\ne \\\\varnothing $ .\",\n \"Thus $(X, T)$ is dynamically compact with respect to $\\\\mathcal {F}$ .\",\n \"Necessity.\",\n \"Suppose that the family $\\\\mathcal {F}$ has no FIP.\",\n \"And then there is a collection $\\\\lbrace F_1, \\\\dots , F_k \\\\rbrace \\\\subset \\\\mathcal {F}$ with $\\\\bigcap _{i=1}^k F_i=\\\\varnothing .$ Let $A=\\\\lbrace a_1, \\\\dots , a_k\\\\rbrace $ be an alphabet and let $(X, T):=(\\\\Sigma , \\\\sigma )$ be the full (one-sided) $A$ -shift.\",\n \"We are going to define a point $x \\\\in X$ with $\\\\omega _\\\\mathcal {F}(x)= \\\\varnothing $ .\",\n \"Let $x_0=a_1$ .\",\n \"For any $n \\\\ge 1$ there is $i$ with $n \\\\notin F_i$ , else the intersection of $F_1, \\\\dots , F_k$ would be nonempty.\",\n \"Then define $x_n:=a_i$ .\",\n \"Finally, let $x=x_0 x_1 x_2 x_3\\\\dots $ and the construction is finished.\",\n \"Assume the contrary that we can take $z \\\\in \\\\omega _\\\\mathcal {F}(x)$ , and that $z$ begins with $a_i \\\\in A$ .\",\n \"Take $G_z=C[a_i]$ .\",\n \"As $z \\\\in \\\\omega _\\\\mathcal {F}(x)$ we have $n_T(x, G_z) \\\\cap F_i \\\\ne \\\\varnothing $ .\",\n \"But if $n \\\\in n_T(x, G_z)$ , then $x_n=a_i$ and so $n \\\\notin F_i$ by the construction, a contradiction.\",\n \"As we have mentioned in Introduction, obviously that a filter $\\\\mathcal {F}$ (in particular $\\\\mathcal {N}_T$ , when $(X,T)$ is weakly mixing), the family $\\\\mathcal {N}_T(A)$ for a weakly mixing subset $A$ of $(X,T)$ and the family $\\\\mathcal {S}_T(\\\\delta )$ when $(X, T)$ is a multi-sensitive system (with a constant of multi-sensitivity $\\\\delta > 0$ ) have FIP.\",\n \"Let $\\\\mathcal {F}$ has the finite intersection property.\",\n \"Then there exists an ultrafilter $\\\\mathcal {U}$ (in $\\\\mathcal {P}$ ) such that $\\\\mathcal {F}\\\\subset \\\\mathcal {U}$ .\",\n \"This result is known as Ultrafilter Lemma (see details and proof in [21]).\",\n \"Recall that an ultrafilter is maximal among all proper filters.\",\n \"As a consequence of this fact we have a natural open question: Question A Let $(X,T)$ be a dynamically compact system with respect to a family $\\\\mathcal {F}$ and $\\\\mathcal {F}$ has FIP.\",\n \"When $\\\\mathcal {F}$ is a filter, or at least contains a nontrivial filter?\",\n \"Especially we address this question to the family $\\\\mathcal {S}_T(\\\\delta )$ .\",\n \"More precisely, when a system $(X,T)$ is dynamically compact with respect to the family $\\\\mathcal {N}_T$ and $\\\\mathcal {N}_T$ has FIP, then, as well known, the systems is weakly mixing and $\\\\mathcal {N}_T$ is a filter.\",\n \"Now, let a system $(X,T)$ is dynamically compact with respect to the family $\\\\mathcal {S}_T(\\\\delta )$ for some $\\\\delta >0$ and $\\\\mathcal {S}_T(\\\\delta )$ has FIP, then the systems is multi-sensitive.\",\n \"But, the following question is still open – when is $\\\\mathcal {S}_T(\\\\delta )$ a filter?\",\n \"A collection $\\\\mathcal {H}\\\\subset \\\\mathcal {F}$ will be called a base for $\\\\mathcal {F}$ if for any $F\\\\in \\\\mathcal {F}$ there is $H \\\\in \\\\mathcal {H}$ with $H \\\\subset F$ .\",\n \"We are interested in those families which have a countable base, that is, there exists a base $\\\\mathcal {H}$ which is countable.\",\n \"Remark that not any Furstenberg family $\\\\mathcal {F}$ has a countable base, for example, the family $\\\\mathcal {B}$ .\",\n \"Assume the contrary that $\\\\mathcal {B}$ admits a countable base $\\\\lbrace F_n: n\\\\in \\\\mathbb {N}\\\\rbrace $ .\",\n \"We take $k_1\\\\in F_1$ , and once $k_m\\\\in F_m, m\\\\in \\\\mathbb {N}$ is defined we choose $k_{m+ 1}\\\\in F_{m+ 1}$ with $k_{m+ 1}> k_m+ m+ 1$ .\",\n \"Set $E= \\\\lbrace k_n: n\\\\in \\\\mathbb {N}\\\\rbrace $ and $F= \\\\mathbb {Z}_+\\\\setminus E$ .\",\n \"Then $E\\\\cap F_n\\\\ne \\\\varnothing $ for all $n\\\\in \\\\mathbb {N}$ , and $F\\\\supset \\\\lbrace k_m+ m: m\\\\in \\\\mathbb {N}\\\\rbrace $ and hence $F\\\\in \\\\mathcal {B}$ , in particular, there exists no $n\\\\in \\\\mathbb {N}$ with $F_n\\\\subset F$ , a contradiction.\",\n \"Not so hard to show even the existence of a family with FIP, but without a countable base.\",\n \"Nevertheless the families $\\\\mathcal {N}_T$ and $\\\\mathcal {S}_T (\\\\delta )$ have countable bases.\",\n \"Indeed, we can consider a countable base $\\\\mathcal {U}$ of open sets for the space $X$ .\",\n \"Note that $U_1 \\\\subset U$ , $V_1 \\\\subset V$ implies $N_T(U_1, V_1) \\\\subset N_T(U, V)$ and $S_T(U_1,\\\\delta ) \\\\subset S_T(U,\\\\delta )$ .\",\n \"Then $\\\\lbrace N_T(U, V): ~U, V \\\\in \\\\mathcal {U}\\\\rbrace $ and $\\\\lbrace S_T(U,\\\\delta ):~U \\\\in \\\\mathcal {U}\\\\rbrace $ are countable bases for $\\\\mathcal {N}_T$ and $\\\\mathcal {S}_T (\\\\delta )$ , respectively.\",\n \"The following is a general result that will be especially useful for families with countable bases.\",\n \"Proposition 3.2 Let $(X,T)$ be a dynamical system and let $\\\\mathcal {F}$ be a family such that there exists $x\\\\in {\\\\mathrm {Tran}}_{\\\\mathcal {F}} (X, T)$ .\",\n \"Then ${\\\\mathrm {orb}}_T(x)\\\\subset {\\\\mathrm {Tran}}_{\\\\mathcal {F}} (X, T)$ .\",\n \"By assumption, given an arbitrary opene $U\\\\subset X$ we have that $n_T(x,U)\\\\in \\\\mathcal {F}$ .\",\n \"Thus, for any $m\\\\in \\\\mathbb {N}$ , $n_T(T^mx,U)=n_T(x,T^{-m}(U))\\\\in \\\\mathcal {F},$ and we conclude that $T^mx\\\\in {\\\\mathrm {Tran}}_{\\\\mathcal {F}} (X, T)$ .\",\n \"Proposition 3.3 Assume that $\\\\mathcal {F}$ admits a countable base $\\\\mathcal {H}$ .\",\n \"Then ${\\\\mathrm {Tran}}_{k\\\\mathcal {F}} (X, T)$ is a $G_\\\\delta $ subset of $X$ .\",\n \"Moreover, the following are equivalent: The system $(X,T)$ is $k \\\\mathcal {F}$ -transitive, ${\\\\mathrm {Tran}}_{k\\\\mathcal {F}} (X, T)$ is a dense $G_\\\\delta $ subset of $X$ , ${\\\\mathrm {Tran}}_{k\\\\mathcal {F}} (X, T)\\\\ne \\\\varnothing $ .\",\n \"Let $\\\\mathcal {U}$ be a countable base of the family of all opene subsets of $X$ .\",\n \"Then the class $\\\\mathcal {U} \\\\times \\\\mathcal {H}$ is countable, and we enumerate it as $\\\\lbrace (U_i, F_i): i \\\\in \\\\mathbb {N}\\\\rbrace $ .\",\n \"Denote by $T^{- F} U= \\\\bigcup _{n\\\\in F} T^{- n} U$ for any $F\\\\subset \\\\mathbb {Z}_+$ and each $U\\\\subset X$ .\",\n \"Then it is easy to obtain $ {\\\\mathrm {Tran}}_{k\\\\mathcal {F}}(X,T) = \\\\bigcap _{i=1}^\\\\infty T^{-F_i} U_i.$ In fact, given arbitrary point $x\\\\in X$ , $x\\\\in {\\\\mathrm {Tran}}_{k\\\\mathcal {F}}(X,T)$ if and only if $N_T (x, U)\\\\in k \\\\mathcal {F}$ for any opene subset $U$ of $X$ , if and only if $N_T (x, U)\\\\cap F\\\\ne \\\\varnothing $ for any opene subset $U$ of $X$ and each $F\\\\in \\\\mathcal {F}$ , if and only if $N_T (x, U_i)\\\\cap F_i\\\\ne \\\\varnothing $ for each $i\\\\in \\\\mathbb {N}$ by the construction.\",\n \"In particular, ${\\\\mathrm {Tran}}_{k\\\\mathcal {F}} (X, T)$ is a $G_\\\\delta $ subset of $X$ .\",\n \"Thus $(X, T)$ is $k \\\\mathcal {F}$ -transitive, if and only if for any $F\\\\in \\\\mathcal {F}$ and arbitrary opene subsets $U, V$ of $X$ we have $N_T (V, U)\\\\cap F\\\\ne \\\\varnothing $ and equivalently $T^{- F} U\\\\cap V\\\\ne \\\\varnothing $ , if and only if $T^{- F} U$ is an opene dense subset of $X$ for any $F\\\\in \\\\mathcal {F}$ and each opene subset $U$ of $X$ , if and only if ${\\\\mathrm {Tran}}_{k\\\\mathcal {F}} (X, T)$ is a dense $G_\\\\delta $ subset of $X$ by (REF ).\",\n \"Now we assume ${\\\\mathrm {Tran}}_{k\\\\mathcal {F}} (X, T)\\\\ne \\\\varnothing $ .\",\n \"Let $x\\\\in {\\\\mathrm {Tran}}_{k\\\\mathcal {F}} (X, T)$ .\",\n \"By Proposition REF ${\\\\mathrm {orb}}_T (x)\\\\subset {\\\\mathrm {Tran}}_{k\\\\mathcal {F}} (X, T)$ , and hence ${\\\\mathrm {Tran}}_{k\\\\mathcal {F}} (X, T)$ is a dense $G_\\\\delta $ subset of $X$ since $x\\\\in {\\\\mathrm {Tran}}(X, T)$ .\",\n \"This finishes the proof.\",\n \"Remark 3.4 Observe that when the state space $X$ is a compact metric space without isolated points, $x\\\\in {\\\\mathrm {Tran}}(X, T)$ if and only if $x\\\\in {\\\\mathrm {Tran}}_\\\\mathcal {B} (X, T)$ .\",\n \"The family $\\\\mathcal {F}_{\\\\text{cof}}$ is clearly translation invariant (and hence $+$ invariant) and admits a countable base, and $k \\\\mathcal {F}_{\\\\text{cof}}= \\\\mathcal {B}$ .\",\n \"Thus by Proposition REF one has: $(X, T)$ is transitive if and only if ${\\\\mathrm {Tran}}(X, T)$ is a dense $G_\\\\delta $ subset of $X$ if and only if ${\\\\mathrm {Tran}}(X, T)\\\\ne \\\\varnothing $ .\"\n ],\n [\n \"Transitive sensitivity and sensitive compactness\",\n \"Recall that a dynamical system $(X, T)$ is transitively sensitive if there exists $\\\\delta > 0$ such that $S_T (W, \\\\delta )\\\\cap N_T (U, V)\\\\ne \\\\varnothing $ for any opene subsets $U, V, W$ of $X$ ; and sensitive compact if there exists $\\\\delta > 0$ such that for any point $x\\\\in X$ the set $\\\\omega _{\\\\mathcal {S}_T(\\\\delta )}(x)$ is nonempty.\",\n \"Sometimes in that cases we will say also $(X, T)$ is transitively sensitive with a sensitive constant $\\\\delta $ and $(X, T)$ is sensitive compact with a sensitive constant $\\\\delta $ .\",\n \"The main result of this section is the following Theorem 4.1 Let $(X, T)$ be a minimal system.\",\n \"Then the following conditions are equivalent: $(X, T)$ is multi-sensitive.\",\n \"$(X, T)$ is sensitive compact.\",\n \"There exists $\\\\delta > 0$ such that $\\\\omega _{\\\\mathcal {S}_T (\\\\delta )} (x)= X$ for each $x\\\\in X$ .\",\n \"There exist $\\\\delta > 0$ and $x\\\\in X$ with $\\\\omega _{\\\\mathcal {S}_T (\\\\delta )} (x)= X$ .\",\n \"$(X, T)$ is transitively sensitive.\",\n \"Before proceeding, we need: Lemma 4.2 Let $\\\\delta > 0$ and $x\\\\in X$ .\",\n \"If $T: X\\\\rightarrow X$ is almost open, then the family $\\\\mathcal {S}_T (\\\\delta )$ is $-$ invariant, and the subset $\\\\omega _{\\\\mathcal {S}_T (\\\\delta )} (x)$ is positively $T$ -invariant.\",\n \"By Proposition REF it suffices to prove that $\\\\mathcal {S}_T (\\\\delta )$ is a $-$ invariant family.\",\n \"Take arbitrary $F\\\\in \\\\mathcal {S}_T (\\\\delta )$ and any $i\\\\in \\\\mathbb {Z}_+$ .\",\n \"Then there exists opene subset $U$ of $X$ with $S_T (U, \\\\delta )\\\\subset F$ .\",\n \"As $T: X\\\\rightarrow X$ is almost open, $T^i: X\\\\rightarrow X$ is also almost open, and then we can choose opene $V \\\\subset T^i U$ .\",\n \"One has $g^{- i} (F)\\\\supset g^{- i} S_T (U, \\\\delta )= S_T (T^i U, \\\\delta )\\\\supset S_T (V, \\\\delta )$ , which implies that the family $\\\\mathcal {S}_T (\\\\delta )$ is $-$ invariant.\",\n \"The following result gives a characterization of transitive sensitivity for a general dynamical system in terms of dynamical compactness.\",\n \"Proposition 4.3 Let $(X, T)$ be a dynamical system.\",\n \"Then the family $\\\\mathcal {S}_T (\\\\delta )$ is $+$ invariant for any $ \\\\delta > 0$ .\",\n \"Furthermore, the following conditions are equivalent: $(X, T)$ is transitively sensitive.\",\n \"There exist a $\\\\delta > 0$ and a dense $G_\\\\delta $ subset $X_0\\\\subset X$ such that $\\\\omega _{\\\\mathcal {S}_T (\\\\delta )} (x)= X$ for each $x\\\\in X_0$ .\",\n \"There exist a $\\\\delta > 0$ and a point $x\\\\in X$ with $\\\\omega _{\\\\mathcal {S}_T (\\\\delta )} (x)= X$ .\",\n \"Firstly, we show that $\\\\mathcal {S}_T (\\\\delta )$ is a $+$ invariant family.\",\n \"In fact, take any $F\\\\in \\\\mathcal {S}_T (\\\\delta )$ and each $i\\\\in \\\\mathbb {Z}_+$ .\",\n \"We choose opene subsets $U, V$ of $X$ with $F\\\\supset S_T (U, \\\\delta )$ and $V\\\\subset T^{- i} U$ satisfying ${\\\\mathrm {diam}}(T^j V)< \\\\delta $ for all $j= 0, 1, \\\\dots , i$ .\",\n \"Thus $g^i (F)\\\\supset g^i S_T (U, \\\\delta )\\\\supset S_T (V, \\\\delta )$ from the construction, and then $g^i (F)\\\\in \\\\mathcal {S}_T (\\\\delta )$ .\",\n \"This implies the $+$ invariance of the family $\\\\mathcal {S}_T (\\\\delta )$ .\",\n \"Observe that $(X, T)$ is transitively sensitive with a sensitive constant $\\\\delta $ , if and only if $(X, T)$ is $k \\\\mathcal {S}_T (\\\\delta )$ -transitive; and that the family $\\\\mathcal {S}_T (\\\\delta )$ has a countable base: let $\\\\mathcal {U}$ be a countable base of the family of all opene subsets of $X$ , then $\\\\lbrace S_T (U, \\\\delta ): U \\\\in \\\\mathcal {U}\\\\rbrace $ is a countable base of $\\\\mathcal {S}_T (\\\\delta )$ .\",\n \"Then applying Proposition REF the equivalence of $(1)\\\\Leftrightarrow (2)\\\\Leftrightarrow (3)$ follows from the fact that $x\\\\in {\\\\mathrm {Tran}}_{k \\\\mathcal {S}_T (\\\\delta )} (X, T)$ if and only if $\\\\omega _{\\\\mathcal {S}_T (\\\\delta )} (x)= X$ .\",\n \"Observe that by [17] the sensitivity of a dynamical system can be lifted up from a factor to an extension by an almost open factor map between transitive systems.\",\n \"The following result gives the lift-up and projection property of transitive sensitivity between transitive systems.\",\n \"Lemma 4.4 Let $\\\\pi : (X, T)\\\\rightarrow (Z, R)$ be a factor map between dynamical systems.\",\n \"Assume that $\\\\pi $ is almost one-to-one.\",\n \"If $(Z, R)$ is transitively sensitive with a sensitive constant $\\\\delta > 0$ then $(X, T)$ is also transitively sensitive.\",\n \"Assume that there exists $z\\\\in Z$ whose fiber is a singleton.\",\n \"If $(X, T)$ is transitively sensitive then $(Z, R)$ is sensitive, in particular, $\\\\text{Eq} (Z, R)= \\\\varnothing $ .\",\n \"(1) We take a compatible metric $\\\\rho $ over $Z$ and let $\\\\varepsilon > 0$ such that $d (x_1, x_2)\\\\le \\\\varepsilon $ implies $\\\\rho (\\\\pi x_1, \\\\pi x_2)\\\\le \\\\delta $ for any $x_1, x_2\\\\in X$ .\",\n \"Now let $U, V, W$ be arbitrary opene subsets of $X$ .\",\n \"As the map $\\\\pi : X\\\\rightarrow Z$ is almost one-to-one, we may take opene subsets $U_Z, V_Z, W_Z$ of $Z$ with $\\\\pi ^{- 1} U_Z\\\\subset U$ , $\\\\pi ^{- 1} V_Z\\\\subset V$ and $\\\\pi ^{- 1} W_Z\\\\subset W$ .\",\n \"Observe that if $n\\\\in S_R (W_Z, \\\\delta )\\\\cap N_R (U_Z, V_Z)$ , then: on one hand, there exist $z_1, z_2\\\\in W_Z$ with $\\\\rho (R^n z_1, R^n z_2)> \\\\delta $ , and so $d (T^n x_1, T^n x_2)> \\\\varepsilon $ for any $x_1\\\\in \\\\pi ^{- 1} (z_1)$ and $x_2\\\\in \\\\pi ^{- 1} (z_2)$ , hence ${\\\\mathrm {diam}}(T^n W)> \\\\varepsilon $ ; on the other hand, $U_Z\\\\cap R^{- n} V_Z\\\\ne \\\\varnothing $ , and then $U\\\\cap S^{- n} V\\\\supset \\\\pi ^{- 1} U_Z\\\\cap \\\\pi ^{- 1} (R^{- n} V_Z)\\\\ne \\\\varnothing $ .\",\n \"This implies $S_T (W, \\\\varepsilon )\\\\cap N_T (U, V)\\\\supset S_R (W_Z, \\\\delta )\\\\cap N_R (U_Z, V_Z)\\\\ne \\\\varnothing $ , as $(Z, R)$ is transitively sensitive.\",\n \"Thus, by the arbitrariness of $U, V$ and $W$ , we have that $(X, T)$ is also transitively sensitive.\",\n \"(2) As $(X, T)$ is transitively sensitive (and assume with a sensitive constant $\\\\delta > 0$ ), it is clear that $(Z, R)$ is transitive, and then by the refined Auslander-York dichotomy the system $(Z, R)$ is sensitive if and only if $\\\\text{Eq} (Z, R)= \\\\varnothing $ (see [6], [17], [2] and the book [1]).\",\n \"Thus it suffices to prove $\\\\text{Eq} (Z, R)= \\\\varnothing $ .\",\n \"Let $\\\\rho $ be a compatible metric over $Z$ , and assume the contrary to take a point $z\\\\in \\\\text{Eq} (Z, R)$ .\",\n \"By the assumption that there exists a point of $Z$ whose fiber is a singleton, we may take an opene subset $W$ in $Z$ with ${\\\\mathrm {diam}}(\\\\pi ^{- 1} W)< \\\\delta $ , and an opene subset $W_*\\\\subset W$ and $\\\\delta _*> 0$ such that if the distance between a point of $Z$ and $W_*$ is smaller than $\\\\delta _*$ then the point belongs to $W$ .\",\n \"Since $z\\\\in \\\\text{Eq} (Z, R)$ , there exists an open neighborhood $U_*$ of $z$ with ${\\\\mathrm {diam}}(R^n U_*)< \\\\delta _*$ for all $n\\\\in \\\\mathbb {Z}_+$ .\",\n \"As $(X, T)$ is transitively sensitive with a sensitive constant $\\\\delta $ , take $m\\\\in N_T (\\\\pi ^{- 1} U_*, \\\\pi ^{- 1} W_*)\\\\cap S_T (\\\\pi ^{- 1} U_*, \\\\delta )$ .\",\n \"Thus $T^m (\\\\pi ^{- 1} U_*)\\\\cap \\\\pi ^{- 1} W_*\\\\ne \\\\varnothing $ , and then $R^m U_*\\\\cap W^*\\\\ne \\\\varnothing $ , which implies $R^m U_*\\\\subset W$ by the construction of $U_*, W_*$ and $W$ .\",\n \"Hence ${\\\\mathrm {diam}}(T^m (\\\\pi ^{- 1} U_*))\\\\le {\\\\mathrm {diam}}(\\\\pi ^{- 1} (R^m U_*))\\\\le {\\\\mathrm {diam}}(\\\\pi ^{- 1} W)< \\\\delta ,$ a contradiction to the selection of $m\\\\in S_T (\\\\pi ^{- 1} U_*, \\\\delta )$ .\",\n \"This finishes the proof.\",\n \"Now we are ready to prove Theorem REF .\",\n \"$(1)\\\\Rightarrow (2)$ follows directly from the definitions.\",\n \"As the system $(X, T)$ is minimal, the map $T: X\\\\rightarrow X$ is almost open.\",\n \"Observing that $\\\\omega _{\\\\mathcal {S}_T (\\\\delta )} (x)$ is a closed subset of $X$ for each $x\\\\in X$ , the implication of $(2)\\\\Rightarrow (3)$ follows from Lemma REF and the minimality of $(X, T)$ .\",\n \"The implication of $(3)\\\\Rightarrow (4)\\\\Rightarrow (5)$ follows from Proposition REF .\",\n \"Since a minimal system is either multi-sensitive or a weakly almost one-to-one extension of its maximal equicontinuous factor by [23], and then $(5)\\\\Rightarrow (1)$ follows from Lemma REF .\",\n \"This finishes the proof.\",\n \"Clearly each multi-sensitive system is sensitive compact.\",\n \"Observe that each non-proximal, transitive compact system is multi-sensitive by [22].\",\n \"In particular, each minimal transitive compact system is multi-sensitive, as each minimal proximal system is trivial by [4] and all dynamical systems considered are assumed to be nontrivial.\",\n \"Nevertheless, there are many minimal, non transitive compact, multi-sensitive systems.\",\n \"For example, consider the classical dynamical system $(X, T)$ given by $X= \\\\mathbb {R}^2/\\\\mathbb {Z}^2$ and $T: (x, y)\\\\mapsto (x+ \\\\alpha , x+ y)$ with $\\\\alpha \\\\notin \\\\mathbb {Q}$ (see [15]).\",\n \"As commented in [22], $(X, T)$ is an invertible minimal multi-sensitive system; note that $(X, T)$ is not weakly mixing, since $(X, T)$ admits an irrational rotation as its nontrivial equicontinuous factor and any equicontinuous factor of a weakly mixing system is trivial.\",\n \"Remark that by [22] for a minimal system the system is transitive compact if and only if it is weakly mixing, and then the constructed system $(X, T)$ is not transitive compact.\",\n \"Proposition 4.5 Each weakly mixing system $(X, T)$ is transitively sensitive.\",\n \"Observe that we are only interested a nontrivial dynamical system, and then let $0< \\\\delta < {\\\\mathrm {diam}}(X)$ .\",\n \"We choose opene subsets $W_1, W_2$ of $X$ such that the distance between $W_1$ and $W_2$ is strictly larger than $\\\\delta $ .\",\n \"Now take arbitrary opene subsets $U, V, W$ of $X$ .\",\n \"As $(X, T)$ is weakly mixing, $(X^3, T^{(3)})$ is transitive by [14], and then $N_T (U, V)\\\\cap S_T (W, \\\\delta )\\\\supset N_{T^{(3)}} (U\\\\times W\\\\times W, V\\\\times W_1\\\\times W_2)\\\\ne \\\\varnothing $ .\",\n \"This implies that $(X, T)$ is transitively sensitive with a sensitive constant $\\\\delta > 0$ .\",\n \"We give a sufficient condition for a dynamical system being transitively sensitive (by Proposition REF ) as the end of this section.\",\n \"Lemma 4.6 Assume $\\\\omega _{\\\\mathcal {S}_T (\\\\varepsilon )} (x)= X$ for some $x \\\\in X$ and $\\\\varepsilon >0$ .\",\n \"Then there is $\\\\delta > 0$ such that for any opene subset $U$ of $X$ and each neighbourhood $U_x$ of $x$ there are $y\\\\in U_x$ and $n\\\\in N_T (x, U)$ with $d(T^n x, T^n y)>\\\\delta $ .\",\n \"If in addition, the map $T: X\\\\rightarrow X$ is almost one-to-one, then the converse holds.\",\n \"Fix an opene subset $U$ of $X$ and a neighborhood $U_x$ of $x$ .\",\n \"As $\\\\omega _{\\\\mathcal {S}_T (\\\\varepsilon )} (x)= X$ , there is $n\\\\in N_T (x, U)\\\\cap S_T (U_x, \\\\varepsilon )$ , and then there are points $x_1, x_2 \\\\in U_x$ with $d(T^n x_1, T^n x_2)> \\\\varepsilon $ .\",\n \"We have either $d (T^n x, T^n x_1)> \\\\frac{\\\\varepsilon }{2}$ or $d (T^n x, T^n x_2)> \\\\frac{\\\\varepsilon }{2}$ , and then obtain the desired statement for $\\\\delta = \\\\frac{\\\\varepsilon }{2}$ .\",\n \"Now suppose that there is $\\\\delta > 0$ such that for any opene subset $U$ of $X$ and each neighbourhood $U_x$ of $x$ there are $y\\\\in U_x$ and $n\\\\in N_T (x, U)$ with $d(T^n x, T^n y)>\\\\delta $ , and that the map $T: X\\\\rightarrow X$ is almost one-to-one.\",\n \"Let $U, W$ be arbitrary opene subsets of $X$ .\",\n \"It is clear $x\\\\in {\\\\mathrm {Tran}}(X, T)$ , and so there is $k\\\\in N_T (x, W)$ .\",\n \"Note that $T: X\\\\rightarrow X$ is almost one-to-one, the map $T^k: X\\\\rightarrow X$ is also almost one-to-one by Corollary REF , and then we may take an opene subset $V$ of $X$ with $T^{- k} V= (T^k)^{- 1} V\\\\subset U$ .\",\n \"Observe that $T^{- k} W$ is a neighborhood of $x$ , and then by the assumption there exists a point $y\\\\in T^{- k} W$ and an integer $n\\\\in \\\\mathbb {Z}_+$ such that $n\\\\in N_T (x, V)$ and $d (T^n x, T^n y)> \\\\delta $ .\",\n \"In fact, we may assume $n>k$ , else we can replace $T^{- k} W$ by a small enough open neighbourhood $G_x\\\\subset T^{- k} W$ of $x$ with ${\\\\mathrm {diam}}(T^i G_x)< \\\\delta $ for all $0\\\\le i\\\\le k$ .\",\n \"Then ${\\\\mathrm {diam}}(T^{n- k} W)> \\\\delta $ as $T^k x\\\\in W$ and $T^k y\\\\in W$ , and $T^{n- k} x\\\\in T^{- k} (T^n x)\\\\subset T^{- k} V\\\\subset U$ .\",\n \"In particular, $N_T (x, U)\\\\cap S_T (W, \\\\delta )\\\\ne \\\\varnothing $ .\",\n \"Thus $\\\\omega _{\\\\mathcal {S}_T (\\\\delta )} (x)= X$ by the arbitrariness of $U$ and $W$ .\"\n ],\n [\n \"Weakly disjointness and weakly mixing\",\n \"Recall that dynamical systems $(X, T)$ and $(Y, S)$ are weakly disjoint if the product system $(X\\\\times Y, T\\\\times S)$ is transitive.\",\n \"The following theorem characterizes weak disjointness, which is proved firstly by Weiss [38] in some special class and then is generalized by Akin and Glasner [3].\",\n \"We say that $\\\\mathcal {F}$ is thick if $\\\\tau \\\\mathcal {F}= \\\\mathcal {F}$ , where $\\\\tau \\\\mathcal {F}= \\\\left\\\\lbrace F\\\\subset \\\\mathbb {Z}_+: \\\\bigcap _{j= 1}^n g^{- i_j} F\\\\in \\\\mathcal {F}\\\\ \\\\text{for each}\\\\ n\\\\in \\\\mathbb {N}\\\\ \\\\text{and all}\\\\ i_1, \\\\dots , i_n\\\\in \\\\mathbb {Z}_+\\\\right\\\\rbrace .$ Weiss-Akin-Glasner Theorem Let $\\\\mathcal {F}$ be a proper, translation invariant, thick family.\",\n \"A dynamical system is $k\\\\mathcal {F}$ -transitive if and only if it is weakly disjoint from every $\\\\mathcal {F}$ -transitive system.\",\n \"Observe that a dynamical system is weakly mixing if and only if it is weakly disjoint from itself, and then weak disjointness is characterized by [22] in some special case.\",\n \"Now we discuss weak disjointness using dynamical compactness which will be some generalization of [22].\",\n \"We will need the following Lemma 5.1 Let $(X, T)$ and $(Y, S)$ be dynamical systems and let $x\\\\in X$ .\",\n \"Then the family $\\\\mathcal {N}_S$ is translation invariant and $\\\\omega _{\\\\mathcal {N}_S} (x)= \\\\omega _{\\\\mathcal {N}_S} (T x)$ .\",\n \"By Proposition REF it suffices to prove that $\\\\mathcal {N}_S$ is a translation invariant family.\",\n \"We also suppose that $\\\\mathcal {N}_S$ is proper (i.e., $(Y,S)$ is a transitive system) since, otherwise, the result is trivial.\",\n \"Take arbitrary $F\\\\in \\\\mathcal {N}_S$ and any $i\\\\in \\\\mathbb {Z}_+$ .\",\n \"Then there exist opene subsets $U, V$ of $Y$ with $N_S (U, V)\\\\subset F$ .\",\n \"As the non-singleton space $Y$ contains no isolated points, we can take suitable opene $V_1\\\\subset V$ and $U_1\\\\subset U$ such that $U_1\\\\cap \\\\bigcup _{k= 0}^i S^k \\\\overline{V_1}=\\\\varnothing $ .\",\n \"One has $g^{i} N_S (U, V)\\\\supset N_S (S^{- i} U_1, V_1)$ , which implies that the family $\\\\mathcal {N}_S$ is $+$ invariant: in fact, if $n\\\\in N_S (S^{- i} U_1, V_1)$ then $n> i$ by the selection, and so $n- i\\\\in N_S (U_1, V_1)\\\\subset N_S (U, V)$ .\",\n \"Moreover, it is clear $g^{- i} (F)\\\\supset g^{- i} N_S (U, V)\\\\supset N_S (U, S^{- i} V)$ , and then the family $\\\\mathcal {N}_S$ is $-$ invariant.\",\n \"This finishes the proof.\",\n \"Theorem 5.2 The following conditions are equivalent: The systems $(X, T)$ and $(Y, S)$ are weakly disjoint.\",\n \"Both ${\\\\mathrm {Tran}}_{k \\\\mathcal {N}_S} (X, T)$ and ${\\\\mathrm {Tran}}_{k \\\\mathcal {N}_T} (Y, S)$ are dense $G_\\\\delta $ subsets.\",\n \"The set ${\\\\mathrm {Tran}}_{k \\\\mathcal {N}_S} (X, T)$ is a dense $G_\\\\delta $ subset of $X$ .\",\n \"Both ${\\\\mathrm {Tran}}_{k \\\\mathcal {N}_S} (X, T)$ and ${\\\\mathrm {Tran}}_{k \\\\mathcal {N}_T} (Y, S)$ are nonempty subsets.\",\n \"The set ${\\\\mathrm {Tran}}_{k \\\\mathcal {N}_S} (X, T)$ is a nonempty subset of $X$ .\",\n \"$(1)\\\\Leftrightarrow (2)\\\\Leftrightarrow (3)$ : It is clear from the definition that: the system $(X, T)$ is $k \\\\mathcal {N}_S$ -transitive, if and only if the systems $(X, T)$ and $(Y, S)$ are weakly disjoint, if and only if $(X, T)$ is $k \\\\mathcal {N}_S$ -transitive and $(Y, S)$ is $k \\\\mathcal {N}_T$ -transitive.\",\n \"As both $\\\\mathcal {N}_T$ and $\\\\mathcal {N}_S$ are families admitting a countable base, it is direct to obtain the equivalence of $(1)\\\\Leftrightarrow (2)\\\\Leftrightarrow (3)$ by applying Proposition REF .\",\n \"The implication $(2)\\\\Rightarrow (4)\\\\Rightarrow (5)$ is obvious.\",\n \"To finish the proof, we only need to show $(5)\\\\Leftrightarrow (3)$ .\",\n \"By Lemma REF the family $\\\\mathcal {N}_S$ is translation invariant, and then the family $k \\\\mathcal {N}_S$ is also translation invariant.\",\n \"Thus the equivalence of $(5)\\\\Leftrightarrow (3)$ follows from Proposition REF .\",\n \"Note that we have a characterization of weak mixing by using dynamical compactness [22].\",\n \"Now we improve [22] as follows.\",\n \"Recall that $\\\\mathcal {S}\\\\subset \\\\mathbb {N}$ is an IP set if there exists $\\\\lbrace p_k: k\\\\in \\\\mathbb {N}\\\\rbrace \\\\subset \\\\mathbb {N}$ with $FS\\\\lbrace p_i\\\\rbrace _{i=1}^\\\\infty \\\\subset \\\\mathcal {S}$ , where $FS\\\\lbrace p_i\\\\rbrace _{i=1}^\\\\infty =\\\\lbrace p_{i_1}+ \\\\dots + p_{i_k}: k\\\\in \\\\mathbb {N}\\\\ \\\\text{and}\\\\ 1\\\\le i_1< \\\\dots < i_k\\\\rbrace .$ Analogously, for each $n\\\\in \\\\mathbb {N}$ we define $FS\\\\lbrace p_i\\\\rbrace _{i=1}^n= \\\\lbrace p_{i_1}+ \\\\dots + p_{i_k}: k\\\\in \\\\mathbb {N}\\\\ \\\\text{and}\\\\ 1\\\\le i_1< \\\\dots < i_k\\\\le n\\\\rbrace .$ Denote by $\\\\mathcal {F}_{ip}$ the family of all IP sets.\",\n \"By [27], the subset ${\\\\mathrm {Tran}}_{\\\\mathcal {F}_{ip}} (X, T)$ contains a dense $G_\\\\delta $ subset of $X$ for any weakly mixing system $(X, T)$ , while ${\\\\mathrm {Tran}}_{\\\\mathcal {F}_{ip}} (X, T) \\\\ne \\\\varnothing $ does not imply the weak mixing of the system $(X, T)$ by [27].\",\n \"We will improve that in the following Proposition REF .\",\n \"Before proceeding, we make the following Lemma 5.3 Let $(X, T)$ be a dynamical system and $\\\\mathcal {F}$ be a family.\",\n \"Let $\\\\delta > 0$ .\",\n \"If the family $\\\\mathcal {S}_T (\\\\delta )\\\\cdot \\\\mathcal {F}$ is proper then $\\\\mathcal {S}_T (\\\\delta )\\\\cdot \\\\mathcal {F}\\\\subset \\\\mathcal {B}$ .\",\n \"If the family $\\\\mathcal {N}_T\\\\cdot \\\\mathcal {F}$ is proper then $\\\\mathcal {N}_T\\\\cdot \\\\mathcal {F}\\\\subset \\\\mathcal {B}$ .\",\n \"(1) Assume the contrary that there exists an opene subset $U$ in $X$ and $F \\\\in \\\\mathcal {F}$ such that $S_T (U, \\\\delta )\\\\cap F$ is finite, and so we may choose $m \\\\in \\\\mathbb {N}$ such that $n \\\\notin S_T(U, \\\\delta )\\\\cap F$ for any integer $n>m$ .\",\n \"Since $T:X \\\\rightarrow X$ is uniformly continuous one can find opene $V \\\\subset U$ small enough such that ${\\\\mathrm {diam}}(T^i V) < \\\\delta $ for all $0\\\\le i \\\\le m$ .\",\n \"Then $S_T (V, \\\\delta ) \\\\subset S_T (U, \\\\delta )$ , which implies $S_T(V, \\\\delta ) \\\\cap F= \\\\varnothing $ .\",\n \"A contradiction.\",\n \"(2) Assume the contrary that there exist opene subsets $U, V$ in $X$ and $F \\\\in \\\\mathcal {F}$ such that $N_T(U, V)\\\\cap F$ is finite, say $N_T(U, V)\\\\cap F= \\\\lbrace n_1, \\\\dots , n_k\\\\rbrace $ .\",\n \"As the non-singleton space $X$ contains no isolated points, we can take opene $U_1\\\\subset U$ small enough such that $V_1:= V\\\\setminus \\\\bigcup _{i= 1}^k T^{n_i} \\\\overline{U_1}$ is an opene subset of $X$ .\",\n \"By the construction we have $N_T(U_1, V_1)\\\\subset N_T (U, V)$ and then $N_T(U_1, V_1)\\\\cap F= \\\\varnothing $ , a contradiction.\",\n \"Proposition 5.4 The following conditions are equivalent: $(X, T)$ is weakly mixing.\",\n \"There exists a dense $G_\\\\delta $ subset $X^{\\\\prime }$ of $X$ such that, for each $x\\\\in X^{\\\\prime }$ , $ n_T(x, G) \\\\cap N_T(U, V) \\\\in \\\\mathcal {F}_{ip}$ for all opene subsets $G, U, V$ of $X$ .\",\n \"$(2)\\\\Rightarrow (1)$ : Just observe from the assumption that $\\\\omega _{\\\\mathcal {N}_T}(x)=X$ for all $x \\\\in X^{\\\\prime }$ , and hence the system $(X, T)$ is weakly mixing by [22].\",\n \"(1) $\\\\Rightarrow $ (2): Since $(X, T)$ is weakly mixing, $(X^2, T^{(2)})$ is also weakly mixing by [14] and [34], and hence by [22] there is a dense $G_\\\\delta $ subset $Y \\\\subset X^2$ such that $\\\\omega _{\\\\mathcal {N}_{T^{(2)}}}((x_1, x_2))=X^2$ for each $(x_1, x_2) \\\\in Y$ .\",\n \"Applying the well-known Ulam Lemma there is a dense $G_\\\\delta $ subset $X^{\\\\prime } \\\\subset X$ such that, for any $x \\\\in X^{\\\\prime }$ , $\\\\lbrace y: (x, y) \\\\in Y\\\\rbrace $ is a dense $G_\\\\delta $ subset of $X$ .\",\n \"Now we show that $X^{\\\\prime }$ is the desired set.\",\n \"Let $x \\\\in X^{\\\\prime }$ and fix any opene subsets $G, U, V$ of $X$ .\",\n \"Choose $y \\\\in G$ with $(x, y) \\\\in Y$ and then $\\\\omega _{\\\\mathcal {N}_{T^{(2)}}}((x, y))=X^2$ , in particular, $(y, y) \\\\in \\\\omega _{\\\\mathcal {N}_{T^{(2)}}}((x, y))$ .\",\n \"Thus $n_T(x, G)\\\\cap n_T(y, G) \\\\cap N_T(U, V) \\\\cap N_T(V, V) \\\\ne \\\\varnothing ,$ and take $p_1\\\\in \\\\mathbb {N}$ from this set by Lemma REF .\",\n \"We have $p_1 \\\\in n_T(x, G) \\\\cap N_T(U, V)$ and $T^{p_1} y \\\\in G$ , $T^{p_1} V \\\\cap V \\\\ne \\\\varnothing $ .\",\n \"Define opene subsets $G_1= G \\\\cap T^{- p_1}G\\\\ni y$ and $V_1=V \\\\cap T^{-p_1}V$ .\",\n \"Now we proceed inductively.\",\n \"Suppose that we are given a sequence $\\\\lbrace p_1, \\\\dots , p_k\\\\rbrace \\\\subset \\\\mathbb {N}$ with $FS\\\\lbrace p_i\\\\rbrace _{i=1}^{k} \\\\subset n_T(x, G) \\\\cap N_T(U, V)$ , and opene subsets $G_k=\\\\bigcap _{s \\\\in FS\\\\lbrace p_i\\\\rbrace _{i=1}^k \\\\cup \\\\lbrace 0\\\\rbrace } T^{-s} G\\\\ni y, V_k=\\\\bigcap _{s \\\\in FS\\\\lbrace p_i\\\\rbrace _{i=1}^k \\\\cup \\\\lbrace 0\\\\rbrace } T^{-s}V.$ As $(y, y) \\\\in \\\\omega _{\\\\mathcal {N}_{T^{(2)}}}((x, y))$ , we may take $p_{k+1}\\\\in \\\\mathbb {N}$ by Lemma REF from the set $n_T(x, G_k) \\\\cap n_T(y, G_k) \\\\cap N_T(U, V_k) \\\\cap N_T(V_k, V_k)$ .\",\n \"It is not hard to check that $G_{k+1}=G_k \\\\cap T^{-p_{k+1}}G_k=\\\\bigcap _{s \\\\in FS\\\\lbrace p_i\\\\rbrace _{i=1}^{k+ 1} \\\\cup \\\\lbrace 0\\\\rbrace } T^{-s} G\\\\ni y,$ $V_{k+1}=V_k \\\\cap T^{-p_{k+1}}V_k=\\\\bigcap _{s \\\\in FS\\\\lbrace p_i\\\\rbrace _{i=1}^{k+1} \\\\cup \\\\lbrace 0\\\\rbrace } T^{-s}V$ are both opene subsets of $X$ , and that $FS \\\\lbrace p_i\\\\rbrace _{i=1}^{k+1} \\\\subset n_T(x, G) \\\\cap N_T(U, V)$ , which completes the induction.\",\n \"Finally, $n_T(x, G) \\\\cap N_T(U, V)\\\\in \\\\mathcal {F}_{ip}$ with $FS \\\\lbrace p_i\\\\rbrace _{i=1}^{\\\\infty } \\\\subset n_T(x, G) \\\\cap N_T(U, V)$ .\",\n \"This finishes the proof.\"\n ],\n [\n \"Transitive compact (non weakly mixing) systems\",\n \"Recall that the system $(X, T)$ is totally transitive if $(X, T^k)$ is transitive for each $k\\\\in \\\\mathbb {N}$ ; and is topologically mixing if $N_T(U,V)\\\\in \\\\mathcal {F}_\\\\text{cof}$ for any opene subsets $U,V$ in $X$ .\",\n \"Note that $(X, T)$ is weakly mixing if and only if $N_T(U,V)\\\\in \\\\mathcal {F}_{\\\\text{thick}}$ for any opene sets $U,V$ in $X$ by [14], [34], and so any weakly mixing system is totally transitive.\",\n \"It is direct to check that each weakly mixing system is transitive compact.\",\n \"In [22] the authors showed the existence of non totally transitive, transitive compact systems in both proximal and non-proximal cases.\",\n \"We extend it as follows: Theorem 6.1 There is a totally transitive, transitive compact system $(X, T)$ which is not weakly mixing.\",\n \"Take a nontrivial proximal, topologically mixing system $(Y, S)$ and let $(\\\\mathbb {S}^1, R_\\\\alpha )$ be the standard irrational rotation on the unit circle $\\\\mathbb {S}^1=\\\\mathbb {R}/\\\\mathbb {Z}$ with $\\\\alpha \\\\notin \\\\mathbb {Q}$ .\",\n \"Note that a dynamical system is proximal if and only if it contains the unique fixed point, which is the only minimal point of the system [4].\",\n \"Denote by $p_Y$ the unique minimal point (fixed point) of $(Y, S)$ .\",\n \"Observe that the system $(Y \\\\times \\\\mathbb {S}^1, S\\\\times R_\\\\alpha )$ is totally transitive: for each $n\\\\in \\\\mathbb {N}$ , the system $(Y, S^n)$ is topologically mixing by the definition and it is standard that the system $(\\\\mathbb {S}^1, R_\\\\alpha ^n)$ is minimal, then it is direct to see that these two systems are weakly disjoint.\",\n \"Let $(X,T)$ be the quotient system $Y \\\\times \\\\mathbb {S}^1/ \\\\sim $ equipped with the action $T$ induced naturally from $S\\\\times R_\\\\alpha $ , where the equivalence relation $\\\\sim $ is defined via: given $x, y\\\\in X$ , $x\\\\sim y$ if and only if either $x=y$ or $x$ and $y$ both have $p_Y$ in the first coordinate.\",\n \"In other words the space $X$ looks like a cone space, where the vertex of the cone is a point $p$ , each “horizontal” fiber spaces are the space $Y$ , the vertical fiber spaces are the circles (see Figure 1).\",\n \"Clearly, $(X, T)$ is totally transitive.\",\n \"calc [rotate=-90] asin(0.5/3) (h) at (0, 3); (O) at (0, 0); (A) at (-2*cos(), 3-0.5*sin()); (B) at (2*cos(), 3-0.5*sin()); [fill=gray!50] (A) – (O) – (B) – cycle; [fill=gray!30] (h) ellipse (2 and 0.5); (0,-0.2) node p; (1.15, 1.25) nodeY; (0,2.75) node$\\\\mathbb {S}^1$ ; Figure 1.\",\n \"Denote by $q:Y \\\\times \\\\mathbb {S}^1 \\\\rightarrow X$ the corresponding quotient map, then $q: Y_\\\\infty \\\\times \\\\mathbb {S}^1\\\\rightarrow X \\\\setminus \\\\lbrace p\\\\rbrace $ is a homeomorphism, where we set $Y_\\\\infty = Y\\\\setminus \\\\lbrace p_Y\\\\rbrace $ .\",\n \"It is standard that the system $(\\\\mathbb {S}^1, R_\\\\alpha )$ is not weakly mixing, and then there exist opene subsets $U_*, V_*$ of $\\\\mathbb {S}^1$ with $N_{R_\\\\alpha } (U_*, V_*)\\\\notin \\\\mathcal {F}_{\\\\text{thick}}$ , hence $N_{T} (q (Y_\\\\infty \\\\times U_*), q (Y_\\\\infty \\\\times V_*))= N_{S\\\\times R_\\\\alpha } (Y_\\\\infty \\\\times U_*, Y_\\\\infty \\\\times V_*)\\\\subset N_{R_\\\\alpha } (U_*, V_*)$ is not thick.\",\n \"This implies that the system $(X, T)$ is not weakly mixing.\",\n \"Now let $U, V$ be arbitrary opene subsets of $X$ .\",\n \"We can choose opene subsets $U_1, V_1\\\\subset Y_\\\\infty $ and $U_2, V_2 \\\\subset \\\\mathbb {S}^1$ with $U_1\\\\times U_2 \\\\subset q^{- 1} U$ and $V_1\\\\times V_2 \\\\subset q^{- 1} V$ .\",\n \"As $(Y, S)$ is topologically mixing and $(\\\\mathbb {S}^1, R_\\\\alpha )$ is minimal, $N_S (U_1, V_1)\\\\in \\\\mathcal {F}_\\\\text{cof}$ and $N_{R_\\\\alpha } (U_2, V_2)\\\\in \\\\mathcal {F}_\\\\text{syn}$ , and then $N_S (U_1, V_1)\\\\cap N_{R_\\\\alpha } (U_2, V_2)\\\\in \\\\mathcal {F}_\\\\text{syn}$ , thus $N_{T} (U, V)= N_{S\\\\times R_\\\\alpha } (q^{- 1} U, q^{- 1} V)\\\\supset N_{S\\\\times R_\\\\alpha } (U_1\\\\times U_2, V_1\\\\times V_2)$ is a syndetic set.\",\n \"Observe from the construction that the system $(X, T)$ is proximal with $p$ as its unique fixed point, then $n_T (x, U_p)$ is a thickly syndetic subset for each point $x\\\\in X$ and any neighbourhood $U_p$ of $p$ (see [22]).\",\n \"This implies $p \\\\in \\\\omega _{\\\\mathcal {N}_T} (x)$ for each $x\\\\in X$ , and then the system $(X, T)$ is transitive compact.\",\n \"The following result is proved independently in [11] and [37].\",\n \"Lemma 6.2 Any $\\\\omega $ -limit set $\\\\omega _T(x)$ can not be decomposed into $\\\\alpha $ disjoint closed, nonempty, positively $T$ -invariant subsets, where $2\\\\le \\\\alpha \\\\le \\\\aleph _0$ .\",\n \"Before proceeding, we need the following example, for which we fail to find a reference and hence provide a detailed construction, as it is crucial in our arguments.\",\n \"Proposition 6.3 For any given compact metric space $Z$ , there exists a topologically mixing system $(X, T)$ such that, $Z$ can be realized as the set of all its minimal points, furthermore, its each minimal point is a fixed point.\",\n \"The construction is divided into two steps.\",\n \"In the first step we shall construct a topologically mixing system $(Y, F)$ but with two fixed points, which are the only minimal points of the system.\",\n \"Let $\\\\Sigma =\\\\lbrace 0,1\\\\rbrace ^{\\\\mathbb {Z}_+}$ and $\\\\sigma :\\\\Sigma \\\\rightarrow \\\\Sigma $ be the full (one-sided) shift.\",\n \"We are going to find the system $(Y, F)$ of the form $(\\\\overline{{\\\\mathrm {orb}}_{\\\\sigma } (x)}, \\\\sigma )$ for some $x \\\\in \\\\Sigma $ .\",\n \"In order to define $x\\\\in \\\\Sigma $ , firstly we represent each $W\\\\in \\\\Sigma ^*$ with $|W|\\\\ge 1$ in the following form: $W=a^{i} Q b^{j}$ , where $a^{i}$ and $b^{j}$ (with $i\\\\ge 1$ ) are the longest segments of equal digits which we can take at the beginning and at the end of $W$ , whereas $Q$ is the rest, possible the empty subblock.\",\n \"Clearly, $j$ may be equal to 0 and then $b^j$ will be the empty subblock, in this case, we treat the digit $b$ as 0; in particular, if $W= a^k$ then we set $Q$ to be the empty subblock and $i= k$ , $b= 0$ , $j= 0$ .\",\n \"Now we are going to define $x\\\\in \\\\Sigma $ .\",\n \"Let $A_1=10$ be the first block of $x$ and define inductively the rest blocks $A_2, A_3, \\\\dots $ , then $x$ will be the limit of the starting blocks $A_k$ .\",\n \"Suppose that we have defined $A_k, k\\\\in \\\\mathbb {N}$ .\",\n \"Since $A_{k}$ has finitely many subblocks, there is a finite number of different pairs of these subblocks.\",\n \"For any pair $(W_1, W_2)$ of subblocks of $A_k$ we will define a block $c (W_1, W_2)$ by using their combination.\",\n \"Then we are ready to define $A_{k+1}$ : at the beginning of $A_{k+1}$ we write $A_k0^k1^k$ , and then all possible blocks $c (W_1, W_2)$ of pairs $(W_1, W_2)$ of subbloks of $A_{k}$ in any fixed order.\",\n \"The definition $c (W_1, W_2)$ depends on the structure of $W_1$ and $W_2$ .\",\n \"Let us write $W_1$ and $W_2$ in the form as above: $W_1=a^{i_1} Q_1 b^{j_1}$ and $W_2=c^{i_2} Q_2 d^{j_2}$ , where $a^{i_1}$ , $b^{j_1}$ , $c^{i_2}$ and $d^{j_2}$ (with $i_1, i_2\\\\ge 1$ ) are the longest segments of equal digits which we can take at the beginning and at the end of $W_1$ and $W_2$ , whereas $Q_1$ and $Q_2$ are the rest, possible empty subblocks ($j_1$ and $j_2$ may be equal to 0, and then we treat the corresponding digits as 0).\",\n \"The combination block of the pair $W_1, W_2$ , i.e.\",\n \"$c(W_1, W_2)$ , is defined as follows: $c(W_1, W_2)=a^{k+i_1} Q_1 b^{j_1+k}c^{k+i_2}Q_2d^{k+j_2}a^{k+i_1} Q_1 b^{j_1+k+1}c^{k+i_2}Q_2 d^{k+j_2} .$ We see from the construction that $A_{k+1}$ is presented as a sequence of blocks with length not longer than $|A_k|$ , which are separated from each other with some sequences of blocks of consecutive 1's or 0's of length not less than $k$ .\",\n \"In fact, for all $m>k$ this property holds for $A_m$ (and hence in any subblock of $A_m$ with length more than $|A_k|+ 2k- 1$ one can find $0^k$ or $1^k$ ).\",\n \"Suppose that $A_m$ may be presented in this form.\",\n \"Observe that $A_{m+1}$ is obtained by adding to $A_m 0^m 1^m$ combination blocks $c(W_1^*, W_2^*)$ of all pairs $(W_1^*, W_2^*)$ of subblocks of $A_m$ glued in a proper way, and note that if the property holds for blocks $W_1^*$ and $W_2^*$ (with beginning and end parts of the blocks possible exception) then the property holds for the combination block $c(W_1^*, W_2^*)$ .\",\n \"In particular, the property holds for $A_{m+ 1}$ .\",\n \"Put $(Y, F):=(\\\\overline{{\\\\mathrm {orb}}_{\\\\sigma } (x)},\\\\sigma )$ .\",\n \"Now let us check that it has the required properties.\",\n \"Firstly, in any subblock of $x$ with length more than $|A_k|+2k-1$ one can find $0^k$ or $1^k$ .\",\n \"And hence, if we increase the length of a block $w$ of $x$ the length of the biggest subblock of $w$ in form of $0^m$ or $1^m$ increases unboundedly.\",\n \"Moreover, for each $k\\\\in \\\\mathbb {N}$ both $0^k$ and $1^k$ appear in $x$ .\",\n \"In particular, both $0^\\\\infty $ and $1^\\\\infty $ are fixed points of $(\\\\overline{{\\\\mathrm {orb}}_{\\\\sigma } (x)}, \\\\sigma )$ , and there is no other minimal sets in it.\",\n \"Recall that the base for the open sets in $\\\\Sigma $ is given by the collection of all cylinder sets $C[c_0c_1c_2 \\\\dots c_m]=\\\\lbrace x \\\\in \\\\Sigma :~ x_i=c_i \\\\text{ for } 0\\\\le i\\\\le m \\\\rbrace .$ Given any two subblocks $W_1$ and $W_2$ of $A_m$ , we write $W_1$ and $W_2$ in the form as above: $W_1=a^{i_1} Q_1 b^{j_1}$ and $W_2=c^{i_2} Q_2 d^{j_2}$ .\",\n \"If $b= c$ , then for each $l\\\\ge m$ there exists a combination block $c_l (W_1, W_2)$ containing the subblock $W_1 b^l W_2$ , and hence $N_F ([W_1]\\\\cap Y, [W_2]\\\\cap Y)\\\\supset \\\\lbrace m+ |W_1|, m+ |W_1|+ 1, \\\\dots \\\\rbrace $ ; if $b\\\\ne c$ , then for each $l\\\\ge m$ there exists a combination block $c_l (W_1, W_2)$ containing the subblocks $W_1 b^k c^k W_2$ and $W_1 b^k 1 c^k W_2$ , and hence $N_F ([W_1]\\\\cap Y, [W_2]\\\\cap Y)\\\\supset \\\\lbrace 2 m+ |W_1|, 2 m+ |W_1|+ 1, \\\\dots \\\\rbrace $ .\",\n \"This shows that the system $(Y, F)$ is topologically mixing.\",\n \"Now we shall finish the construction by the second step.\",\n \"Firstly, we take $(X^{\\\\prime }, T^{\\\\prime })$ to be the product system $\\\\prod _1^\\\\infty (Y, F)$ .\",\n \"It is ready to check that the system $(X^{\\\\prime }, T^{\\\\prime })$ is topologically mixing, for which the middle-third Cantor set $C$ is the set of all its minimal points and its each minimal point is a fixed point.\",\n \"Note that there exists a continuous surjection $h: C\\\\rightarrow Z$ (see for example [25]), and then we consider the quotient system $(X, T)$ with $X= X^{\\\\prime }/\\\\sim $ equipped with the action induced naturally from $T^{\\\\prime }$ , where the closed positively $T^{\\\\prime }\\\\times T^{\\\\prime }$ -invariant equivalence relation $\\\\sim $ is defined via $x\\\\sim y$ if and only if $x= y\\\\in X^{\\\\prime }\\\\setminus Z$ or $h (x)= h (y)$ for $x, y\\\\in Z$ .\",\n \"Then the system $(X, T)$ has the required properties.\",\n \"The following result shows that in general there is no a topological structure similar to Lemma REF for the $\\\\omega _{\\\\mathcal {N}_T}$ -limit sets.\",\n \"Theorem 6.4 For any given compact metric space $Z$ , there exists a non totally transitive, transitive compact system $(X, T)$ such that, $Z$ can be realized as the set of all its minimal points with its each minimal point being a fixed point, furthermore, $Z$ is realized as $\\\\omega _{\\\\mathcal {N}_T} (x)$ for some $x\\\\in X$ .\",\n \"The idea of the proof is very similar to that of the first part (proximal case) of [22].\",\n \"Instead of a nontrivial proximal, topologically mixing system $(Y, F)$ there (main point is “a map with exactly one minimal point!”), we take again a topologically mixing system $(Y, F)$ , but with $Z$ realized as the set of all its minimal points where its each minimal point is a fixed point (for existence of such a dynamical system see Proposition REF ).\",\n \"Then, by the wedge sum construction there, we obtain a non totally transitive system $(X, T)$ such that $Z$ is realized as the set of all its minimal points with its each minimal point being a fixed point.\",\n \"Similar to arguments there, it is not hard to show $Z\\\\supset \\\\omega _{\\\\mathcal {N}_T} (x)$ for all $x\\\\in X$ .\",\n \"Firstly we prove the following claim: Claim.\",\n \"For $x\\\\in X$ , if $x= y\\\\in Y$ and $p\\\\in Z\\\\cap \\\\omega _{\\\\mathcal {N}_{F^2}} (y)$ then, as a point in $X$ , $p$ belongs to $\\\\omega _{\\\\mathcal {N}_T} (x, T)$ .\",\n \"Let $U_p$ be an open subset of $X$ containing $p$ , and clearly $U_p$ may be also viewed as an open subset of $Y$ containing $p$ .\",\n \"Now for any given opene subsets $U$ and $V$ of $X$ : if both $U$ and $V$ can be viewed as opene subsets of $Y$ , then we can take $n\\\\in N_{F^2} (y, U_p)\\\\cap N_{F^2} (U, V)$ by the assumption $p\\\\in \\\\omega _{\\\\mathcal {N}_{F^2}} (y)$ and hence $2 n\\\\in N_{T} (y, U_p)\\\\cap N_{T} (U, V)$ ; if both $U$ and $V$ can be viewed as opene subsets of $Y_c$ , then both $T^{- 1} U$ and $T^{- 1} V$ can be viewed as opene subsets of $Y$ and hence $N_{T} (y, U_p)\\\\cap N_{T} (U, V)= N_{T} (y, U_p)\\\\cap N_{T} (T^{- 1} U, T^{- 1} V)\\\\ne \\\\varnothing $ ; if $U$ and $V$ can be viewed as opene subsets of $Y$ and $Y_c$ , respectively, noting $p\\\\in Z$ and hence $T p= p$ , there is an opene subset $V_p$ of $X$ containing $p$ such that $T V_p\\\\subset U_p$ , and then by the above reasoning we may take $n\\\\in N_{T} (y, V_p)\\\\cap N_{T} (U, T^{- 1} V)$ , and hence $n+ 1\\\\in N_{T} (y, U_p)\\\\cap N_{T} (U, V)$ ; it can be treated similarly the other case that $U$ and $V$ can be viewed as opene subsets of $Y_c$ and $Y$ , respectively.\",\n \"Now we continue our proof.\",\n \"As $(Y, F)$ is topologically mixing, the system $(Y, F^2)$ is weakly mixing, and then by [22] we may choose $x^*\\\\in Y$ such that $\\\\omega _{\\\\mathcal {N}_{F^2}} (x^*)= Y$ , in particular, $\\\\omega _{\\\\mathcal {N}_{F^2}} (x^*)\\\\supset Z$ .\",\n \"Thus, by the above Claim, we obtain $Z\\\\subset \\\\omega _{\\\\mathcal {N}_T} (x^*)$ and hence $Z= \\\\omega _{\\\\mathcal {N}_T} (x^*)$ .\",\n \"Note that a dynamical system is proximal if and only if it contains the unique fixed point, which is the only minimal point of the system [4].\",\n \"Thus, as a direct corollary of Lemma REF and Theorem REF , we have: Corollary 6.5 There exists a non-proximal, non totally transitive, transitive compact system $(X, T)$ and a point $x_0\\\\in X$ such that $\\\\omega _{\\\\mathcal {N}_T} (x_0)\\\\ne \\\\omega _T (x)$ for all $x\\\\in X$ .\",\n \"Nevertheless is still open the following Question B Let $(X,T)$ be a weakly mixing system.\",\n \"Is any $\\\\omega _{\\\\mathcal {N}_T}(x)$ undecomposable into $\\\\alpha $ disjoint closed, nonempty, positively $T$ -invariant subsets, where $2\\\\le \\\\alpha \\\\le \\\\aleph _0$ ?\",\n \"At the end of this section let us prove one more chaotical property of transitive compact systems in additional to already known in [22].\",\n \"Recall that a pair of points $x, y \\\\in X$ is asymptotic if $\\\\lim _{n\\\\rightarrow \\\\infty } d (T^n x, T^n y)= 0$ .\",\n \"Denote by ${\\\\mathrm {Asym}}_T (X)$ the set of all asymptotic pairs of points.\",\n \"Any pair $(x, y)\\\\in {\\\\mathrm {Prox}}_T (X)\\\\setminus {\\\\mathrm {Asym}}_T (X)$ is called a Li-Yorke pair.\",\n \"Recall that a dynamical system $(X, T)$ is Li-Yorke chaotic if there exists an uncountable set $S\\\\subset X$ with $(S\\\\times S)\\\\setminus \\\\Delta _2 (X)\\\\subset {\\\\mathrm {Prox}}_T (X)\\\\setminus {\\\\mathrm {Asym}}_T (X)$ , where $\\\\Delta _2 (X)= \\\\lbrace (x, x): x\\\\in X\\\\rbrace $ .\",\n \"Proposition 6.6 Each transitive compact system $(X, T)$ is Li-Yorke chaotic.\",\n \"Clearly $(X, T)$ is transitive.\",\n \"Observe that we have assumed the state space to be not a singleton and in fact a compact metric space without isolated points, then $(X, T)$ is a transitive system with $X$ infinite.\",\n \"Thus, the subset ${\\\\mathrm {Asym}}_T (X)$ is a fist category subset of $X\\\\times X$ by [24].\",\n \"It is easy to show that ${\\\\mathrm {Prox}}_T (X)$ is a $G_\\\\delta $ subset of $X\\\\times X$ , and applying [22] to the transitive compact system $(X, T)$ we have that ${\\\\mathrm {Prox}}_T (x)$ is a dense subset of $X$ for each $x\\\\in X$ .\",\n \"Thus ${\\\\mathrm {Prox}}_T (X)$ is a dense $G_\\\\delta $ subset of $X\\\\times X$ , and then ${\\\\mathrm {Prox}}_T (X)\\\\setminus {\\\\mathrm {Asym}}_T (X)$ is a second category subset of $X\\\\times X$ .\",\n \"Now applying the well-known Mycielski Theorem [31] we obtain an uncountable subset $S\\\\subset X$ with $(S\\\\times S)\\\\setminus \\\\Delta _2 (X)\\\\subset {\\\\mathrm {Prox}}_T (X)\\\\setminus {\\\\mathrm {Asym}}_T (X)$ .\",\n \"That is, $(X, T)$ is Li-Yorke chaotic.\"\n ],\n [\n \"Weak transitive compactness and sensitivity for linear operators\",\n \"In this section we are considering the dynamics of linear operators on infinite dimensional spaces in relation to the properties studied in previous sections.\",\n \"More precisely, we will show the equivalence of the topological weak mixing property with a weak version of transitive compactness.\",\n \"We obtain some results on transitive sensitivity too.\",\n \"One should keep in mind that, for a linear dynamical system $(X, T)$ , where $X$ is an infinite dimensional space, neither compactness nor even local compactness of $X$ is satisfied.\",\n \"In particular, we are interested in the case where $X$ is an infinite dimensional separable Banach space and $T:X\\\\rightarrow X$ is a continuous linear map (in short, operator).\",\n \"In this framework, we will just write $(X,T)$ is an infinite dimensional linear dynamical system.\",\n \"We recall that $X$ is a Banach space if it is a vector space endowed with a norm $\\\\Vert \\\\cdot \\\\Vert $ such that $X$ with the associated distance $d(x,y):=\\\\Vert x-y\\\\Vert $ becomes a complete metric space.\",\n \"It is well known that $T: X\\\\rightarrow X$ is an operator if and only if $\\\\Vert T\\\\Vert :=\\\\sup \\\\lbrace \\\\Vert Tx\\\\Vert : \\\\Vert x\\\\Vert \\\\le 1\\\\rbrace <\\\\infty $ .\",\n \"We refer the reader to the books [7] and [19] for the theory of linear dynamics.\",\n \"Note that all notations and concepts discussed in previous sections can be introduced into linear dynamics.\",\n \"We also introduce a weak version of dynamical compactness.\",\n \"A linear system $(X,T)$ is called weakly dynamically compact with respect to the family $\\\\mathcal {F}$ if there exists a dense subset $X_0\\\\subset X$ such that the $\\\\omega _\\\\mathcal {F}$ -limit set $\\\\omega _{\\\\mathcal {F}}(x)$ is nonempty for all $x\\\\in X_0$ .\",\n \"In particular, $(X, T)$ is called weakly transitive compact, if there exists a dense subset $X_0\\\\subset X$ such that for any point $x\\\\in X_0$ the $\\\\omega _{\\\\mathcal {N}_T}$ -limit set $\\\\omega _{\\\\mathcal {N}_T}(x)$ is nonempty, in other words, for any point $x\\\\in X_0$ there exists a point $z \\\\in X$ such that $ n_T(x,G) \\\\cap N_T(U,V)\\\\ne \\\\varnothing $ for any neighborhood $G$ of $z$ and any opene subsets $U,V$ of $X$ .\",\n \"Theorem 7.1 Let $(X, T)$ be an infinite dimensional linear system.\",\n \"Then $(X, T)$ is weakly mixing if and only if it is weakly transitive compact.\",\n \"Sufficiency.\",\n \"Suppose that $(X, T)$ is weakly transitive compact.\",\n \"Let $X_0\\\\subset X$ be a dense subset such that, for each $x\\\\in X_0$ , there exists $z(x) \\\\in X$ such that $ n_T(x,G) \\\\cap N_T(U,V)\\\\ne \\\\varnothing $ for any neighborhood $G$ of $z (x)$ and opene $U, V\\\\subset X$ .\",\n \"As $(X,T)$ is obviously transitive, by [18] (see also [19]) to obtain the weak mixing property we just need to show that, for each opene $U\\\\subset X$ and 0-neighbourhood $W$ , there is a continuous map $S: X\\\\rightarrow X$ commuting with $T$ such that $ S(U)\\\\cap W\\\\ne \\\\varnothing \\\\ \\\\ \\\\ \\\\mbox{ and } \\\\ \\\\ \\\\ S(W)\\\\cap U\\\\ne \\\\varnothing .$ Given an opene subset $U$ of $X$ and a 0-neighborhood $W$ , we fix $x\\\\in U\\\\cap X_0$ and $z(x)\\\\in X$ accordingly to the weak transitive compactness of $(X,T)$ .\",\n \"Since 0-neighbourhoods are absorbing, we find a scalar $\\\\lambda \\\\ne 0$ such that $\\\\lambda z (x)\\\\in W$ .\",\n \"Let $G$ be a neighbourhood of $z (x)$ such that $\\\\lambda G\\\\subset W$ .\",\n \"By the hypothesis we can find $m\\\\in n_T(x,G) \\\\cap N_T(\\\\lambda W,U).$ That is, $T^mx\\\\in G$ and so $\\\\lambda T^mx\\\\in W$ ; additionally, there exists $w\\\\in W$ with $T^m\\\\lambda w\\\\in U$ .\",\n \"Now pick $S:=\\\\lambda T^m$ , we have that $S$ commutes with $T$ and the property (REF ) is satisfied, therefore the system is weakly mixing.\",\n \"Necessity.\",\n \"Conversely, under the assumption of the weak mixing property for $(X,T)$ , we know by [8] (see also [19]) that there exists an increasing sequence $\\\\lbrace n_k: k\\\\in \\\\mathbb {N}\\\\rbrace \\\\subset \\\\mathbb {N}$ and a dense subset $X_0\\\\subset X$ such that $T^{n_k}x\\\\rightarrow 0$ for each $x\\\\in X_0$ and, for arbitrary opene $U,V\\\\subset X$ , we can find $k\\\\in \\\\mathbb {N}$ such that $T^{n_k}(U)\\\\cap V\\\\ne \\\\varnothing $ .\",\n \"Thus, we obtain easily that $(X,T)$ is weakly transitive compact by selecting $z(x)=0$ for every $x\\\\in X_0$ .\",\n \"Concerning sensitivity, the situation is more complicated and, although we obtain some advances, three related problems are left open.\",\n \"Proposition 7.2 Let $(X, T)$ be an infinite dimensional linear, topologically transitive system.\",\n \"Then $(X, T)$ is thickly multi-sensitive, that is, there exists $\\\\delta > 0$ such that $\\\\bigcap _{i= 1}^k S_T (U_i, \\\\delta )$ is thick for any finite collection of opene $U_1, \\\\dots , U_k\\\\subset X$ .\",\n \"Let $U_1, \\\\dots , U_k$ be opene sets, and let $m\\\\in \\\\mathbb {N}$ .\",\n \"Pick points $x_1, \\\\dots , x_k$ such that $x_i \\\\in U_i$ and choose $\\\\varepsilon >0$ such that $B_\\\\varepsilon (x_i) \\\\subset U_i$ , where $B_\\\\varepsilon (x_i)$ is the open ball of radius $\\\\varepsilon $ centered at $x_i$ , for all $i \\\\in \\\\lbrace 1, \\\\dots , k\\\\rbrace $ .\",\n \"By a hypercyclic vector we mean that its orbit is dense in the space $X$ .\",\n \"Take a hypercyclic vector $u \\\\in B_\\\\varepsilon (0)$ by [19], and let $y_i=x_i+u$ .\",\n \"Then $y_i \\\\in U_i$ by the construction.\",\n \"Since $u$ is hypercyclic there is $n \\\\in \\\\mathbb {N}$ , $n>m$ , such that $\\\\Vert T^n u\\\\Vert >(\\\\Vert T\\\\Vert +1)^m$ .\",\n \"Then $\\\\rho (T^{n-j}x_i, T^{n-j}y_i)=\\\\Vert T^{n-j}(x_i-y_i)\\\\Vert =\\\\Vert T^{n-j}u\\\\Vert >(\\\\Vert T\\\\Vert +1)^{m-j}>1$ for all $i= 1, \\\\dots , k$ and $j= 0, \\\\dots , m- 1$ .\",\n \"Hence $\\\\lbrace n, n- 1, \\\\dots , n-m+1\\\\rbrace \\\\subset \\\\bigcap _{i=1}^k S_T(U_i, 1)$ , and therefore $(X, T)$ is thickly multi-sensitive.\",\n \"Proposition 7.3 Let $(X,T)$ be an infinite dimensional linear system.\",\n \"Then the following conditions are equivalent: For each $\\\\delta > 0$ , $(X, T)$ is transitively sensitive with a sensitive constant $\\\\delta $ .\",\n \"There exists $\\\\delta _0> 0$ such that $(X, T)$ is transitively sensitive with a sensitive constant $\\\\delta _0$ .\",\n \"There exists $\\\\delta _0> 0$ such that $S_T (W_0, \\\\delta _0)\\\\cap N_T (U, V)\\\\ne \\\\varnothing $ for any opene subsets $U, V$ of $X$ and any 0-neighbourhood $W_0$ .\",\n \"We just need to show $(3) \\\\Rightarrow (1)$ .\",\n \"Indeed, let $\\\\delta >0$ be arbitrary, and fix arbitrary opene $U, V, W$ of $X$ .\",\n \"We select $\\\\varepsilon >0$ and $x\\\\in W$ such that $x+B_\\\\varepsilon (0)\\\\subset W$ .\",\n \"Observing $S_T (\\\\lambda W_0, \\\\lambda \\\\delta _0)$ for any scalar $\\\\lambda \\\\ne 0$ , and so without loss of generality we assume $\\\\delta >\\\\delta _0$ .\",\n \"Let $0<\\\\varepsilon ^{\\\\prime }< \\\\frac{\\\\delta _0 \\\\varepsilon }{\\\\delta }$ , and set $W_0=B_{\\\\varepsilon ^{\\\\prime }}(0)$ .\",\n \"By the hypothesis there are $y,z\\\\in W_0$ and $n\\\\in N_T(U,V)$ such that $\\\\Vert T^ny-T^nz\\\\Vert > \\\\delta _0$ .\",\n \"Set $y^{\\\\prime }=x+\\\\frac{\\\\delta }{\\\\delta _0} y$ and $z^{\\\\prime }=x+\\\\frac{\\\\delta }{\\\\delta _0} z$ .\",\n \"We have $y^{\\\\prime }, z^{\\\\prime }\\\\in W$ and $\\\\Vert T^ny^{\\\\prime }-T^nz^{\\\\prime }\\\\Vert > \\\\delta $ .\",\n \"As opene $U, V, W\\\\subset X$ are arbitrary, $(X, T)$ is transitively sensitive with a sensitive constant $\\\\delta $ .\",\n \"In this framework the weak mixing property implies transitive sensitivity too.\",\n \"The following result establishes a very close connection of transitivity with transitive sensitivity.\",\n \"We do not know, however, whether every transitive linear system is transitively sensitive.\",\n \"Proposition 7.4 Let $(X,T)$ be an infinite dimensional linear, topologically transitive system.\",\n \"If $(X, T)$ is not transitively sensitive, then there exists a dense open subset $U_0\\\\subset X$ such that every $x\\\\in U_0$ has a dense orbit.\",\n \"If $(X,T)$ is not transitively sensitive, by Proposition REF we find opene $U, V$ of $X$ and $\\\\delta >1$ such that $\\\\Vert T^nx\\\\Vert \\\\le \\\\delta $ whenever $n\\\\in N_T(U,V)$ and $\\\\Vert x\\\\Vert \\\\le 1$ .\",\n \"We fix an arbitrary opene $V^{\\\\prime }\\\\subset V$ and select an opene $\\\\widehat{V}\\\\subset V^{\\\\prime }$ and $\\\\varepsilon >0$ such that $\\\\widehat{V}+B_\\\\varepsilon (0)\\\\subset V^{\\\\prime }$ .\",\n \"Given $u\\\\in U$ , there is $\\\\varepsilon ^{\\\\prime }< \\\\frac{\\\\varepsilon }{\\\\delta }$ such that $U^{\\\\prime }:=u+B_{\\\\varepsilon ^{\\\\prime }}(0)\\\\subset U$ .\",\n \"Since $T$ is transitive, there exists $m\\\\in N_T(U^{\\\\prime },\\\\widehat{V})\\\\subset N_T(U,V)$ .\",\n \"That is, we find $u^{\\\\prime }=u+w\\\\in U^{\\\\prime }$ with $\\\\Vert w\\\\Vert <\\\\varepsilon ^{\\\\prime }$ and $T^mu^{\\\\prime }\\\\in \\\\widehat{V}$ .\",\n \"By the assumption $\\\\Vert T^mw\\\\Vert \\\\le \\\\delta \\\\varepsilon ^{\\\\prime }<\\\\varepsilon $ .\",\n \"Therefore, $T^mu=T^mu^{\\\\prime }-T^mw\\\\in \\\\widehat{V}+B_\\\\varepsilon (0)\\\\subset V^{\\\\prime }$ .\",\n \"Since $u\\\\in U$ and opene $V^{\\\\prime }\\\\subset V$ are arbitrary, we obtain that the orbit of every element in $U$ is somewhere dense, thus everywhere dense by transitivity of the system.\",\n \"Finally, the open set $U_0:=\\\\bigcup _{n\\\\in \\\\mathbb {N}}T^{-n}(U)$ is dense, and every element in $U_0$ has a dense orbit.\",\n \"There are (very difficult) examples of linear systems $(X,T)$ such that every non-zero element has a dense orbit [35], but it seems to unknown whether every linear system that admits an open set of elements whose orbit is dense is so that every non-zero element has a dense orbit.\",\n \"It is also worthy to mention that there are (also rare) examples of transitive but not weakly mixing linear systems [10] (see also [7]), but as far as we know there are no examples of transitive non-weakly mixing linear systems such that every non-zero element has a dense orbit.\",\n \"Concerning weak disjointness, observe that for each separable Banach space the family of all opene subsets admits a countable base, and then it is a routine to show that Theorem REF holds true within linear systems too.\",\n \"Note that the intersection of finitely many thickly syndetically sets is still thickly syndetic, and that an interesting property is that every topologically ergodic linear system $(X, T)$ (i.e., each element of $\\\\mathcal {N}_T$ is a syndetic set) satisfies that each element of $\\\\mathcal {N}_T$ is actually a thickly syndetic set (see the exercises in [19]).\",\n \"Thus any finite family $(X_1,T_1), \\\\dots ,(X_k,T_k)$ of topologically ergodic linear systems is weakly disjoint and, moreover, the product system $(X_1\\\\times \\\\dots \\\\times X_k, T_1\\\\times \\\\dots \\\\times T_k)$ is topologically ergodic.\"\n ]\n]"},"id":{"kind":"string","value":"1605.05851"}}},{"rowIdx":1866678,"cells":{"text":{"kind":"list like","value":[["Exploiting Equitable Partitions for Efficient Block Triangularization"],["Abstract In graph theory a partition of the vertex set of a graph is called equitable if for all pairs of cells all vertices in one cell have an equal number of neighbours in the other cell.","Considering the implications for the adjacency matrix one may generalize that concept as a block partition of a complex square matrix s.t.","each block has constant row sum.","It is well known that replacing each block by its row sum yields a smaller matrix whose multiset of eigenvalues is contained in the initial spectrum.","We generalize this approach to weighted row sums and rectangular matrices and derive an efficient unitary transformation which approximately block triangularizes a matrix w.r.t.","an arbitrary partition.","Singular values and Hermiticity (if present) are preserved.","The approximation is exact in the equitable case and the error can be bounded in terms of unitarily invariant matrix norms."],["Equitable Partitions","Let $\\Gamma $ be a (multi-)graph and let $\\mathbf {A}$ be its adjacency matrix, whose entries $a_{vw}$ are the number of edges connecting vertices $v$ and $w$ .","Let $\\Pi =\\left(c_1,\\ldots ,c_k\\right)$ be a partition of the vertex set of $\\Gamma $ into $k$ cells, inducing a block partition of $\\mathbf {A}$ , i.e.","a simultaneous (disjoint and exhaustive) partition of its rows and columns.","It is convenient to define an indicator matrix of a partition as Definition 1 $\\mathbf {B}=\\left(b_{vi}\\right)\\in \\left\\lbrace 0,1\\right\\rbrace ^{N\\times k}\\text{ with }b_{vi}=\\left\\lbrace \\begin{array}{ll}1&\\text{, if item $v$ is in cell $i$}\\\\0&\\text{, else.","}\\end{array}\\right.$ The partition $\\Pi $ is called equitable if all vertices of $\\Gamma $ in the same cell have the same number of neighbours in any cell.","Equivalently, we may call it equitable if each induced submatrix of $\\mathbf {A}$ has constant row sum.","The equitable partitions of $\\mathbf {A}$ ordered by refinement form a lattice which contains the trivial equitable partition, in which every cell has size exactly one, as the minimum.","From the definition it follows that a partition of $\\mathbf {A}$ is equitable if and only if there exits a matrix $\\mathbf {\\Theta }=\\left(\\theta _{ij}\\right)$ s.t.","$\\mathbf {A}\\mathbf {B}=\\mathbf {B}\\mathbf {\\Theta }\\quad \\text{i.e.","}\\quad \\forall \\ i,j\\in \\left\\lbrace 1,\\ldots ,k\\right\\rbrace \\ \\forall \\ u\\in \\left\\lbrace 1,\\ldots ,n_i\\right\\rbrace \\ \\sum \\limits _v^{n_j}\\mathbf {A}_{ij,{uv}}=\\theta _{ij}.$ The matrix $\\mathbf {\\Theta }$ is called the quotient of the partition.","Its entries $\\theta _{ij}$ are the constant row sums of the matrix blocks $A_{ij}$ induced by the cells $c_i$ and $c_j$ , which are the number of edges connecting a fixed vertex in $c_i$ to vertices in $c_j$ ."],["Applications in Graph and Matrix Theory","The notion of equitable partitions was developed in graph theory.","In network analysis the same concept is also known as exact coloration [3] or exact role assignment [20].","It is closely related to graph fibration [2] and arises naturally in the context of graph automorphisms problems since every non trivial automorphism induces a non trival equitable partition.","As graph invariants which can be searched for quickly using quite efficient algorithms, equitable partitions are useful in attacking graph isomorphism problems.","In that context they are also known as 1-dimensional Weisfeiler-Lehman stabilizers [5].","Block partitioned matrices s.t.","each block has constant row sum are called block-stochastic matrices.","In the context of markov chains the technique of lumping exploits equitable partitions in order to reduce the number of states [4].","The quotient $\\mathbf {\\Theta }$ is also known as the front divisor [7].","Famously, the spectrum of $\\mathbf {\\Theta }$ , called the main spectrum, is a subset of the spectrum of $\\mathbf {A}$ since the columns of $\\mathbf {B}$ span an invariant subspace if (REF ) holds.","Therefore, there is a similarity transformation which $2\\times 2$ block triangularizes $\\mathbf {A}$ s.t.","one diagonal block is the quotient.","Such a transformation can be constructed and applied efficiently in a way provided in [15], [6].","The block triangularization method given below differs from that approach in order to fit in a generalized framework of equitability and provides efficient unitary transformations."],["Aim and Outline","We will generalize the notion of ordinary equitable partitions to arbitrary weighted partitions of the rows and columns of complex matrices.","According to a given partition we derive an efficient and stable unitary similarity transformation in order to $2\\times 2$ block triangularize the matrix up to an error term, which is minimized w.r.t.","to several matrix norms and vanishes if and only if exact equitability holds.","The transformation can be computed in $O\\left(N\\right)$ and applied in $O\\left(N^2\\right)$ .","It can be further generalized enabling the application to rectangular matrices while maintaining the unitarity property.","However the further generalized transformation does only preserve the singular values, but (in general) not the spectrum.","Despite offering insides into the structure of objects represented by a graph or a matrix our notion of equitability and its corresponding transformation may be used for compression and for preprocessing eigen and singular value problems.","Although describing our transformation as an efficient compression method may seem to suggest that the exploited structure is, in a sense, rare, the concept of equitable partitions, as indicated above, is rather common in various applications, where it is found directly in the studied problem or as an interesting exceptional or ideal case.","The usefulness might be increased in particular by the fact that deviations from an exact equitability may be allowed within our framework.","In order to get used to the concept and some notation, we briefly discuss in section () the special case of an ordinary unweighted equitable partition of a complex square matrix including the derivation of the associated efficient unitary block triangularization and we give an example.","In the main part, section (), we consider weighted not necessarily equitable partitions introducing the deviation matrix and give our main theorem.","In section () we consider non exact equitability as an eigenvalue perturbation, give a short overview of several other known generalizations of equation (REF ), and briefly consider the problem of finding an equitable partition.","Our further generalized version of the concept applicable to rectangular matrices can be found in the appendix.","Throughout the article we use the apostrophe to denote the complex conjugated transpose without distinguishing between real and complex operands and we utilize the following notation Definition 2 Let $n\\in \\mathbb {N}$ .","$\\mathbf {j}_n=(\\hspace{1.00006pt}\\underbrace{1,\\ldots ,1}_{\\text{$n$ times}}\\hspace{1.00006pt})^{\\prime }$ and $\\mathbf {f}_n=(1,\\hspace{-5.0pt}\\underbrace{0,\\ldots ,0}_{\\text{$(n-1)$ times}}\\hspace{-5.0pt})^{\\prime }$ ."],["Indicator Matrix and Quotient","Let $\\mathbf {A}\\in \\mathbb {C}^{N\\times N}$ and let $\\Pi =\\left(c_1,\\ldots ,c_k\\right)$ be a simultaneous (disjoint and exhaustive) partition of its rows and columns with indicator matrix $\\mathbf {B}$ as in definition (REF ).","Let $\\mathbf {A}_{ij}\\in \\mathbf {C}^{n_i\\times n_j}$ be the matrix block in $\\mathbf {A}$ induced by row cell $c_i$ and column cell $c_j$ .","Let $n_i$ be the size of the cell $c_i$ and let $\\mathbf {N}=\\left(\\mathbf {B}^{\\prime }\\mathbf {B}\\right)^{\\frac{1}{2}}=\\operatorname{diag}\\left(\\sqrt{n_1},\\ldots ,\\sqrt{n_k}\\right)$ We introduce the front quotient, the rear quotient and the Rayleigh quotient respectively as $\\mathbf {E}^{-}=\\mathbf {N}^{-2}\\mathbf {B}^{\\prime }\\mathbf {A}\\mathbf {B},\\quad \\mathbf {E}^{+}=\\mathbf {B}^{\\prime }\\mathbf {A}\\mathbf {B}\\mathbf {N}^{-2},\\quad \\mathbf {E}^{\\mathrm {0}}=\\mathbf {N}^{-1}\\mathbf {B}^{\\prime }\\mathbf {A}\\mathbf {B}\\mathbf {N}^{-1}$ We call $\\mathbf {A}$ front equitable (i) and respectively rear equitable (ii) w.r.t.","$\\mathbf {B}$ if $\\left(i\\right)\\ \\mathbf {A}\\mathbf {B}=\\mathbf {B}\\mathbf {E}^{-}\\quad ,\\quad \\left(ii\\right)\\ \\mathbf {B}^{\\prime }\\mathbf {A}=\\mathbf {E}^{+}\\mathbf {B}^{\\prime }.$ It is easy to see that for Hermitian matrices row equitability and column equitability imply each other.","For the rest of this section we assume front equitability, i.e.","$\\mathbf {A}_{ij}\\mathbf {j}_{n_j}=e^{-}_{ij}\\mathbf {j}_{n_i}\\quad \\forall \\ i,j\\in \\left\\lbrace 1,\\ldots ,k\\right\\rbrace .$"],["Block Triangularization","In order to block triangularize $\\mathbf {A}$ according to $\\mathbf {B}$ we utilize the Householder matrices $\\mathbf {H}_i=\\mathbf {I}_{n_i}-2\\frac{\\mathbf {y}_i\\mathbf {y}_i^{\\prime }}{\\mathbf {y}_i^{\\prime }\\mathbf {y}_i}\\ ,\\quad \\mathbf {y}_i=\\mathbf {j}_{i}+\\sqrt{n_i}\\mathbf {f}_{n_i}.$ The following useful relations are easily verified $\\mathbf {H}_i\\mathbf {f}_{n_i}=-\\frac{1}{\\sqrt{n_i}}\\mathbf {j}_{n_i}\\quad ,\\quad \\mathbf {H}_i^{\\prime }\\mathbf {j}_{n_i}=-\\sqrt{n_i}\\mathbf {f}_{n_i}.$ In order to simplify notations but w.l.o.g.","we assume suitable indexing which means that $\\mathbf {A}$ and $\\mathbf {B}$ are indexed in such a way that for $u$ in cell $c_i$ and $v$ in cell $c_j$ it holds that $i0$ , let $x^1=\\mathbf {f}_n^{\\prime }\\mathbf {x}$ denote its first entry and let $\\beta $ be a complex number with $\\left|\\beta \\right|=1$ , then $\\mathbf {H}\\left(\\mathbf {x},\\beta \\right)=\\left\\lbrace \\begin{array}{cc}\\mathbf {I}_n & ,\\frac{1}{\\Vert \\mathbf {x}\\Vert }\\mathbf {x}=\\overline{\\beta }\\mathbf {f}_n\\\\\\mathbf {I}_n+\\frac{\\left(\\mathbf {x}-\\Vert \\mathbf {x}\\Vert \\overline{\\beta }\\mathbf {f}_n\\right)\\left(\\mathbf {x}-\\Vert \\mathbf {x}\\Vert \\overline{\\beta }\\mathbf {f}_n\\right)^{\\prime }}{\\Vert \\mathbf {x}\\Vert \\overline{\\beta }\\left(\\overline{x^1}-\\Vert \\mathbf {x}\\Vert \\beta \\right)} & ,\\text{ else.","}\\end{array}\\right.$ Using $\\mathbf {y}=\\mathbf {x}-\\Vert \\mathbf {x}\\Vert \\overline{\\beta }\\mathbf {f}_n$ we may rewrite $\\mathbf {H}\\left(\\mathbf {x},\\beta \\right)=\\mathbf {I}_n+\\frac{\\beta }{\\Vert \\mathbf {x}\\Vert }\\left(\\mathbf {y}^{\\prime }\\mathbf {f}_n\\right)^{\\dagger }\\mathbf {y}\\mathbf {y}^{\\prime }=\\mathbf {I}_n-\\left(\\mathbf {x}^{\\prime }\\mathbf {y}\\right)^{\\dagger }\\mathbf {y}\\mathbf {y}^{\\prime }.$ And we summarize the following properties $\\mathbf {H}\\left(\\mathbf {x},\\beta \\right)\\mathbf {f}_n=\\frac{\\beta }{\\Vert \\mathbf {x}\\Vert }\\mathbf {x}\\quad \\text{and}\\quad \\mathbf {H}\\left(\\mathbf {x},\\beta \\right)^{\\prime }\\mathbf {x}=\\frac{\\Vert \\mathbf {x}\\Vert }{\\beta }\\mathbf {f}_n.$ Since $\\mathbf {H}\\left(\\mathbf {x},\\beta \\right)$ is a rank one update of $\\mathbf {I}_n$ , it can be stored with $O\\left(n\\right)$ and multiplied with a square matrix of size $n$ in $O\\left(n^2\\right)$ .","Note that $\\mathbf {H}\\left(\\mathbf {x},\\beta \\right)$ crucially depends on the ordering of the entries of $\\mathbf {x}$ , $\\mathbf {H}\\left(\\mathbf {P}^{\\prime }\\mathbf {x},\\beta \\right)\\ne \\mathbf {P}^{\\prime }\\mathbf {H}\\left(\\mathbf {x},\\beta \\right)\\mathbf {P}\\quad \\text{for general $\\mathbf {x}$ and permutation matrix $\\mathbf {P}$}.$ Although its norm is determined to be 1, the actual choice of $\\beta $ is arbitrary.","We may exploit that freedom in order to enhance the numerical properties of $\\mathbf {H}\\left(\\beta ,\\mathbf {x}\\right)$ .","Particular useful is a choice s.t.","$\\beta x^1\\in \\mathbb {R}$ , implying $\\gamma =0$ by (REF ) and leading to a Hermitian matrix.","Furthermore, for real $\\mathbf {x}$ , $\\beta \\in \\left\\lbrace -1,1\\right\\rbrace $ ensures a real matrix.","A practical recommendation might be Definition 4 $\\beta _0\\left(\\mathbf {x}\\right)=\\left\\lbrace \\begin{array}{ll}-\\frac{\\overline{x^1}}{\\left|x^1\\right|}&, x^1\\ne 0\\\\1&, x^1=0\\end{array}\\right.\\quad ,\\ \\mathbf {x}\\in \\mathbb {C}^n$ , which supports numerical stability and coincides with the usual recommendation for the numerical construction of a real Householder matrix.","In the previous section we applied $\\beta _0$ tacitly."],["Weighted Partition, Quotient and Deviation Matrix","Let $\\Pi =\\left(c_1,\\ldots ,c_k\\right)$ be a partition of $\\lbrace 1,\\ldots ,N\\rbrace $ into $k$ cells with indicator matrix $\\mathbf {B}$ .","Let $\\mathbf {w}\\in \\mathbb {C}^{N}$ and $\\mathbf {A}\\in \\mathbb {C}^{N\\times N}$ .","We introduce the weighted indicator matrix $\\mathbf {W}=\\operatorname{diag}\\left(\\mathbf {w}\\right)\\mathbf {B}$.","Definition 5 Let $\\Pi =\\left(c_1,\\ldots ,c_k\\right)$ be a partition of $\\left\\lbrace 1,\\ldots ,N\\right\\rbrace $ .","Let $\\mathbf {w}\\in \\mathbb {C}^N$ and let $w^v$ denote its $v$ -th entry.","$\\mathbf {W}=\\left(w_{vi}\\right)\\in \\mathbb {C}^{N\\times k}\\text{ with }w_{vi}=\\left\\lbrace \\begin{array}{ll}w^v&\\ v\\in c_i\\\\0&\\text{, else}\\end{array}\\right.$ A weighted indicator matrix $\\mathbf {W}$ is called admissible if $\\Vert \\mathbf {w}_i\\Vert $ for all vector blocks $\\mathbf {w}_i$ induced by $c_i$ .","This implies that $\\mathbf {W}^{\\prime }\\mathbf {W}$ is invertible and ultimately ensures that the complete spectrum of the quotient, to be defined below, is contained in the spectrum of $\\mathbf {A}$ .","For the rest of this section we assume admissibility.","We call $\\mathbf {W}$ suitably indexed if for $u\\in c_i,v\\in c_j$ it holds that $i0$ , let $x^1=\\\\mathbf {f}_n^{\\\\prime }\\\\mathbf {x}$ denote its first entry and let $\\\\beta $ be a complex number with $\\\\left|\\\\beta \\\\right|=1$ , then $\\\\mathbf {H}\\\\left(\\\\mathbf {x},\\\\beta \\\\right)=\\\\left\\\\lbrace \\\\begin{array}{cc}\\\\mathbf {I}_n & ,\\\\frac{1}{\\\\Vert \\\\mathbf {x}\\\\Vert }\\\\mathbf {x}=\\\\overline{\\\\beta }\\\\mathbf {f}_n\\\\\\\\\\\\mathbf {I}_n+\\\\frac{\\\\left(\\\\mathbf {x}-\\\\Vert \\\\mathbf {x}\\\\Vert \\\\overline{\\\\beta }\\\\mathbf {f}_n\\\\right)\\\\left(\\\\mathbf {x}-\\\\Vert \\\\mathbf {x}\\\\Vert \\\\overline{\\\\beta }\\\\mathbf {f}_n\\\\right)^{\\\\prime }}{\\\\Vert \\\\mathbf {x}\\\\Vert \\\\overline{\\\\beta }\\\\left(\\\\overline{x^1}-\\\\Vert \\\\mathbf {x}\\\\Vert \\\\beta \\\\right)} & ,\\\\text{ else.\",\n \"}\\\\end{array}\\\\right.$ Using $\\\\mathbf {y}=\\\\mathbf {x}-\\\\Vert \\\\mathbf {x}\\\\Vert \\\\overline{\\\\beta }\\\\mathbf {f}_n$ we may rewrite $\\\\mathbf {H}\\\\left(\\\\mathbf {x},\\\\beta \\\\right)=\\\\mathbf {I}_n+\\\\frac{\\\\beta }{\\\\Vert \\\\mathbf {x}\\\\Vert }\\\\left(\\\\mathbf {y}^{\\\\prime }\\\\mathbf {f}_n\\\\right)^{\\\\dagger }\\\\mathbf {y}\\\\mathbf {y}^{\\\\prime }=\\\\mathbf {I}_n-\\\\left(\\\\mathbf {x}^{\\\\prime }\\\\mathbf {y}\\\\right)^{\\\\dagger }\\\\mathbf {y}\\\\mathbf {y}^{\\\\prime }.$ And we summarize the following properties $\\\\mathbf {H}\\\\left(\\\\mathbf {x},\\\\beta \\\\right)\\\\mathbf {f}_n=\\\\frac{\\\\beta }{\\\\Vert \\\\mathbf {x}\\\\Vert }\\\\mathbf {x}\\\\quad \\\\text{and}\\\\quad \\\\mathbf {H}\\\\left(\\\\mathbf {x},\\\\beta \\\\right)^{\\\\prime }\\\\mathbf {x}=\\\\frac{\\\\Vert \\\\mathbf {x}\\\\Vert }{\\\\beta }\\\\mathbf {f}_n.$ Since $\\\\mathbf {H}\\\\left(\\\\mathbf {x},\\\\beta \\\\right)$ is a rank one update of $\\\\mathbf {I}_n$ , it can be stored with $O\\\\left(n\\\\right)$ and multiplied with a square matrix of size $n$ in $O\\\\left(n^2\\\\right)$ .\",\n \"Note that $\\\\mathbf {H}\\\\left(\\\\mathbf {x},\\\\beta \\\\right)$ crucially depends on the ordering of the entries of $\\\\mathbf {x}$ , $\\\\mathbf {H}\\\\left(\\\\mathbf {P}^{\\\\prime }\\\\mathbf {x},\\\\beta \\\\right)\\\\ne \\\\mathbf {P}^{\\\\prime }\\\\mathbf {H}\\\\left(\\\\mathbf {x},\\\\beta \\\\right)\\\\mathbf {P}\\\\quad \\\\text{for general $\\\\mathbf {x}$ and permutation matrix $\\\\mathbf {P}$}.$ Although its norm is determined to be 1, the actual choice of $\\\\beta $ is arbitrary.\",\n \"We may exploit that freedom in order to enhance the numerical properties of $\\\\mathbf {H}\\\\left(\\\\beta ,\\\\mathbf {x}\\\\right)$ .\",\n \"Particular useful is a choice s.t.\",\n \"$\\\\beta x^1\\\\in \\\\mathbb {R}$ , implying $\\\\gamma =0$ by (REF ) and leading to a Hermitian matrix.\",\n \"Furthermore, for real $\\\\mathbf {x}$ , $\\\\beta \\\\in \\\\left\\\\lbrace -1,1\\\\right\\\\rbrace $ ensures a real matrix.\",\n \"A practical recommendation might be Definition 4 $\\\\beta _0\\\\left(\\\\mathbf {x}\\\\right)=\\\\left\\\\lbrace \\\\begin{array}{ll}-\\\\frac{\\\\overline{x^1}}{\\\\left|x^1\\\\right|}&, x^1\\\\ne 0\\\\\\\\1&, x^1=0\\\\end{array}\\\\right.\\\\quad ,\\\\ \\\\mathbf {x}\\\\in \\\\mathbb {C}^n$ , which supports numerical stability and coincides with the usual recommendation for the numerical construction of a real Householder matrix.\",\n \"In the previous section we applied $\\\\beta _0$ tacitly.\"\n ],\n [\n \"Weighted Partition, Quotient and Deviation Matrix\",\n \"Let $\\\\Pi =\\\\left(c_1,\\\\ldots ,c_k\\\\right)$ be a partition of $\\\\lbrace 1,\\\\ldots ,N\\\\rbrace $ into $k$ cells with indicator matrix $\\\\mathbf {B}$ .\",\n \"Let $\\\\mathbf {w}\\\\in \\\\mathbb {C}^{N}$ and $\\\\mathbf {A}\\\\in \\\\mathbb {C}^{N\\\\times N}$ .\",\n \"We introduce the weighted indicator matrix $\\\\mathbf {W}=\\\\operatorname{diag}\\\\left(\\\\mathbf {w}\\\\right)\\\\mathbf {B}$.\",\n \"Definition 5 Let $\\\\Pi =\\\\left(c_1,\\\\ldots ,c_k\\\\right)$ be a partition of $\\\\left\\\\lbrace 1,\\\\ldots ,N\\\\right\\\\rbrace $ .\",\n \"Let $\\\\mathbf {w}\\\\in \\\\mathbb {C}^N$ and let $w^v$ denote its $v$ -th entry.\",\n \"$\\\\mathbf {W}=\\\\left(w_{vi}\\\\right)\\\\in \\\\mathbb {C}^{N\\\\times k}\\\\text{ with }w_{vi}=\\\\left\\\\lbrace \\\\begin{array}{ll}w^v&\\\\ v\\\\in c_i\\\\\\\\0&\\\\text{, else}\\\\end{array}\\\\right.$ A weighted indicator matrix $\\\\mathbf {W}$ is called admissible if $\\\\Vert \\\\mathbf {w}_i\\\\Vert $ for all vector blocks $\\\\mathbf {w}_i$ induced by $c_i$ .\",\n \"This implies that $\\\\mathbf {W}^{\\\\prime }\\\\mathbf {W}$ is invertible and ultimately ensures that the complete spectrum of the quotient, to be defined below, is contained in the spectrum of $\\\\mathbf {A}$ .\",\n \"For the rest of this section we assume admissibility.\",\n \"We call $\\\\mathbf {W}$ suitably indexed if for $u\\\\in c_i,v\\\\in c_j$ it holds that $iC_V^{\\rm max}/2$ , shown in Fig.","REF (right).","Since the width vanishes in the thermodynamic limit, a double-logarithmic plot reveals the power-law scaling Eq.","() as a straight line, here dashed black.","As previously noticed [13], an intermediate scaling regime $\\propto N^{-1}$ is clearly visible and marked with a dashed-dotted gray line.","This is consistent with the droplet as relevant system size as in this regime still a large fraction of particles contribute to the transition droplet.","However, for $N\\ge 500$ it appears that the asymptotic scaling behavior slowly manifests itself and leading-order estimates become more reliable.","This can be tested by performing a direct fit of the power-law ansatz $\\Delta T\\propto N^{-\\alpha }$ , shown in Fig.","REF for variable lower bound $N_{\\rm min}$ .","Only for $N_{\\rm min}\\ge 640$ the fit yields plausible goodness-of-fit values $Q>0.1$ .","Still, the distance to the predicted asymptotic scaling regime is noticeable.","This is consistent with results obtained for the three-dimensional lattice gas where decent results were only obtained for $N_{\\rm min}\\approx 2000$ [13].","This emphasizes once more that care has to be taken with leading-order finite-size scaling away from the asymptotic scaling regime.","While estimates of the thermodynamic limit become better with increasing system size, fit errors are difficult to judge because the underlying ansatz is not accurate enough.","The same holds true for additional higher-order corrections.","While the results are surely good estimates, the fit errors are a difficult measure, not capturing the uncertainty that comes from an incomplete fit model [24]."],["Conclusions","We have extended our previous results [13] for the three-dimensional Lennard-Jones system to larger system sizes.","This allows for more consistent estimates of the thermodynamic limit when considering leading-order and higher-order fits.","Still, the current Lennard-Jones system with up to 2048 particles at density $\\rho =10^{-2}$ is quite far away from the asymptotic scaling limit.","We notice that the rounding of the transition is a good indicator to visualize this distance from the asymptotic scaling regime.","Here, also the emerging intermediate scaling regime is best noticeable.","Figure: Results of direct fits of the ansatz ΔT∝N -α \\Delta T\\propto N^{-\\alpha } to the data inFig.","(right) with variable lower bound N min N_{\\rm min}.","Thedotted line shows the predicted asymptotic scaling exponent."],["Acknowledgments","The project was funded by the European Union, the Free State of Saxony and the Deutsche Forschungsgemeinschaft (DFG) under Grant No.","JA 483/31-1.","The authors gratefully acknowledge the computing time provided by the John von Neumann Institute for Computing (NIC) on the supercomputer JURECA at Jülich Supercomputing Centre (JSC) under Grant No.","HLZ24.","Part of this work has been financially supported by the DFG through the Leipzig Graduate School of Natural Sciences “BuildMoNa” and by the Deutsch-Französische Hochschule (DFH-UFA) through the Doctoral College “${\\mathbb {L}}^4$ ” under Grant No.","CDFA-02-07."]],"string":"[\n [\n \"Finite-size scaling of Lennard-Jones droplet formation at fixed density\"\n ],\n [\n \"Abstract We reaccess the droplet condensation-evaporation transition of a three-dimensional Lennard-Jones system upon a temperature change.\",\n \"With the help of parallel multicanonical simulations we obtain precise estimates of the transition temperature and the width of the transition for systems with up to 2048 particles.\",\n \"This allows us to supplement previous observations of finite-size scaling regimes with a clearer picture also for the case of a continuous particle model.\"\n ],\n [\n \"Introduction\",\n \"Despite the long-lasting scientific interest in droplet formation, it is still a modern problem with many open questions.\",\n \"This is partially due to the general nature of droplet formation, with relevance ranging from metastable decay over cloud formation to cluster formation in protein solutions.\",\n \"In this work, we consider equilibrium droplet formation which yields a firm basis to study the transition between a homogeneous gas and a mixed phase of a droplet in equilibrium with surrounding vapor [1], [2], [3], [4], [5].\",\n \"The problem is formulated in the canonical ensemble.\",\n \"A common control parameter is the density while the temperature is fixed.\",\n \"This allows one, in principle, to access temperature-dependent quantities such as the isothermal compressibility and the interface tension.\",\n \"At fixed temperature the theory has been verified by several computational studies, including lattice [6], [7], [8] and Lennard-Jones [9], [10], [11] systems.\",\n \"Instead, however, one can fix the density and vary the temperature which yields an equivalent but “orthogonal” finite-size scaling behavior [12], [13].\",\n \"In this case, we recently observed an intermediate scaling regime consistent with finite-size scaling results for polymer aggregation [14].\",\n \"Here, we present new results for the three-dimensional Lennard-Jones system at fixed density which extend and supplement our previous results [13].\"\n ],\n [\n \"Model and Method\",\n \"We consider $N$ Lennard-Jones particles in a three-dimensional box of length $L$ with periodic boundary conditions, see Fig.\",\n \"REF .\",\n \"The self-avoidance and short-range attraction is modeled by the pairwise Lennard-Jones potential $V_{\\\\rm LJ}(r_{ij})= 4\\\\epsilon \\\\left[\\\\left(\\\\frac{\\\\sigma }{r_{ij}}\\\\right)^{12}- \\\\left(\\\\frac{\\\\sigma }{r_{ij}}\\\\right)^6\\\\right],$ where $r_{ij}$ is the distance between particle $i$ and $j$ .\",\n \"As in Ref.\",\n \"[13], we set $\\\\epsilon =1$ and $\\\\sigma =2^{-1/6}$ .\",\n \"The computational demand can be reduced by introducing a cutoff radius $r_c=2.5\\\\sigma $ above which particles do not interact anymore.\",\n \"The potential is then shifted by $V_{\\\\rm LJ}(r_c)$ in order to be continuous, yielding $V^*_{\\\\rm LJ}(r) ={\\\\left\\\\lbrace \\\\begin{array}{ll}V_{\\\\rm LJ}(r)-V_{\\\\rm LJ}(r_c) & r < r_c \\\\\\\\0 & \\\\mathrm {else}\\\\end{array}\\\\right.\",\n \"}.$ This is in accordance with the existing literature and enables the application of a domain decomposition, where the periodic box is decomposed into equally large (cubic) domains.\",\n \"These domains have to be at least of the size $r_c$ .\",\n \"Then, the interaction of each particle is obtained by evaluating only its domain and the adjacent ones (in three dimensions this adds up to $3^3=27$ domains).\",\n \"Especially in the gas phase the simulation benefits from this procedure, where the particles are equally distributed in the full box.\",\n \"Following Ref.\",\n \"[13], we fix the density $\\\\rho =N/L^3$ and vary the temperature $T$ .\",\n \"This allows the application of multicanonical simulations [15], [16], [17], [18], which are well-suited for first-order phase transitions such as condensation.\",\n \"At a first-order transition two phases are in coexistence with suppressed transition states in between.\",\n \"This is circumvented by replacing the canonical Boltzmann weight $\\\\exp (-E/k_{\\\\rm B}T)$ with an alterable weight function, which is iteratively adapted in order to yield a flat histogram in the energy.\",\n \"Each iteration is in equilibrium, sampling the distribution according to the current weight function.\",\n \"This leads to a straightforward parallelization [19], [20] which has been shown to perform very well for the problem at hand [8].\",\n \"In the end, canonical expectation values are estimated by reweighting the data from a multicanonical production run.\",\n \"Figure: Illustration of a three-dimensional Lennard-Jones gas (left) and a dropletin equilibrium with surrounding vapor (right).\",\n \"Shown is a system with N=2048N=2048particles at density ρ=10 -2 \\\\rho =10^{-2} in a box with periodic boundaryconditions.A crucial aspect is the selection of Monte Carlo updates.\",\n \"While in Ref.\",\n \"[13] we restricted ourselves to local particles shifts, we added here particle “jumps” with a larger update range.\",\n \"The update range for a particle shift was set to $r=0.3$ and for a particle jump to $r=L/2$ .\",\n \"A new position is then selected with equal probability from a sphere of size $r$ around the old position.\",\n \"This simple non-local update significantly enhanced the sampling of the gaseous phase and increased the performance of the simulation drastically.\",\n \"Instead of 512 we are now able to sample up to 2048 particles.\"\n ],\n [\n \"Theory\",\n \"Here we only want to briefly recapture the results of many previous works [2], [3], [4], [5], [13].\",\n \"For a supersaturated gas, it was shown that the probability of intermediate-sized droplets vanishes [3], [4], which leaves us with the scenario of a homogeneous gas phase on the one side and a droplet in equilibrium with surrounding vapor on the other side, see Fig.\",\n \"REF .\",\n \"At a fixed temperature, the droplet formation may be achieved by adding more particle excess to the already supersaturated gas.\",\n \"For both sides it is possible to formulate a contribution to the free-energy in terms of fluctuations (entropy of the gaseous phase) and surface tension (energy of the droplet).\",\n \"Importantly, the finite-size dependence could be rewritten in terms of the fraction of particle excess in the droplet as a function of a dimensionless density.\",\n \"This in turn allowed us to expand the results around the infinite-size transition temperature $T_0$ to yield the leading finite-size scaling behavior of the condensation temperature $T_c$ at fixed density [13].\",\n \"Similarly, we showed that an expansion of the free-energy difference at the condensation transition yields an estimate of the leading-order finite-size scaling of the transition rounding $\\\\Delta T$ , i.e., the width of the transition.\",\n \"For details we refer to the prior literature and give here only the three-dimensional results to leading order: $T_c -T_0 &\\\\propto N^{-1/4}, \\\\\\\\\\\\Delta T &\\\\propto N^{-3/4}.\",\n \"$ A crucial observation is now that the size $R$ of the droplet at the condensation transition itself scales non-trivially with system size, namely in three dimensions $R\\\\propto N^{1/4}$ .\",\n \"Thus, the leading-order scaling may be identified in terms of powers of the linear extension of the droplet itself, $\\\\propto R^{-1}$ respectively $\\\\propto R^{-3}$ .\",\n \"This is consistent with the interpretation that the droplet size at transition is the relevant system size.\",\n \"Then, a virtual subsystem around the droplet would lead to a transition between a homogeneous liquid (droplet) phase to a homogeneous gas phase with open boundary conditions.\",\n \"The competition between finite-size contributions from volume ($\\\\propto R^3$ ) and surface ($\\\\propto R^2$ ) would in this case give rise to an intuitive finite-size correction of the order $R^{-1}$ [21], [22], [23], [13], [14].\"\n ],\n [\n \"Results\",\n \"Due to the non-local Monte Carlo update, we are now able to extend our previous results [13] for the three-dimensional Lennard-Jones system to $N=2048$ particles with improved statistics.\",\n \"The temperature scale is fixed by setting $k_{\\\\rm B}=1$ .\",\n \"The finite-size transition temperature $T_c$ is obtained as the location of the largest peak in the specific heat $C_V=k_{\\\\rm B}\\\\beta ^2\\\\left(\\\\langle E^2\\\\rangle -\\\\langle E\\\\rangle ^2\\\\right)/N$ and plotted in Fig.\",\n \"REF (left) versus the expected scaling behavior Eq.\",\n \"(REF ).\",\n \"Overall, the data qualitatively shows a linear behavior as predicted.\",\n \"A leading-order fit for $N\\\\ge 1024$ yields $T_0=0.7032(3)$ with goodness-of-fit parameter $Q\\\\approx 0.1$ , shown in the figure as dashed black line.\",\n \"Including additional higher-order corrections, i.e., $T_c=T_0 +a N^{-1/4}+b N^{-2/4}$ , yields $T_0=0.6979(2)$ for $N\\\\ge 64$ with $Q\\\\approx 0.1$ .\",\n \"Both estimates differ from our previous results outside the error bars, which is expected because the fit error underestimates systematic uncertainties.\",\n \"This is partially due to the fact that we are not yet fully in the asymptotic scaling regime, which we will discuss below.\",\n \"The general trend of the results is compatible with our previous conclusion, where we already noted that the available three-dimensional Lennard-Jones system sizes were too small for clear results.\",\n \"However, it is worth noting that the current estimates are closer to each other, implying an infinite-size limit of the transition temperature around $T_0\\\\approx 0.70$ .\",\n \"Figure: Finite-size scaling of the transition temperature (left) and rounding(right) as the maximum location and the half width of the specific-heat peak,respectively.\",\n \"Error bars are included but smaller than the symbol size.The distance from the asymptotic scaling regime is best visible in the rounding of the transition.\",\n \"This is here obtained from the half width of the specific-heat peak, i.e., the width for which $C_V(T)>C_V^{\\\\rm max}/2$ , shown in Fig.\",\n \"REF (right).\",\n \"Since the width vanishes in the thermodynamic limit, a double-logarithmic plot reveals the power-law scaling Eq.\",\n \"() as a straight line, here dashed black.\",\n \"As previously noticed [13], an intermediate scaling regime $\\\\propto N^{-1}$ is clearly visible and marked with a dashed-dotted gray line.\",\n \"This is consistent with the droplet as relevant system size as in this regime still a large fraction of particles contribute to the transition droplet.\",\n \"However, for $N\\\\ge 500$ it appears that the asymptotic scaling behavior slowly manifests itself and leading-order estimates become more reliable.\",\n \"This can be tested by performing a direct fit of the power-law ansatz $\\\\Delta T\\\\propto N^{-\\\\alpha }$ , shown in Fig.\",\n \"REF for variable lower bound $N_{\\\\rm min}$ .\",\n \"Only for $N_{\\\\rm min}\\\\ge 640$ the fit yields plausible goodness-of-fit values $Q>0.1$ .\",\n \"Still, the distance to the predicted asymptotic scaling regime is noticeable.\",\n \"This is consistent with results obtained for the three-dimensional lattice gas where decent results were only obtained for $N_{\\\\rm min}\\\\approx 2000$ [13].\",\n \"This emphasizes once more that care has to be taken with leading-order finite-size scaling away from the asymptotic scaling regime.\",\n \"While estimates of the thermodynamic limit become better with increasing system size, fit errors are difficult to judge because the underlying ansatz is not accurate enough.\",\n \"The same holds true for additional higher-order corrections.\",\n \"While the results are surely good estimates, the fit errors are a difficult measure, not capturing the uncertainty that comes from an incomplete fit model [24].\"\n ],\n [\n \"Conclusions\",\n \"We have extended our previous results [13] for the three-dimensional Lennard-Jones system to larger system sizes.\",\n \"This allows for more consistent estimates of the thermodynamic limit when considering leading-order and higher-order fits.\",\n \"Still, the current Lennard-Jones system with up to 2048 particles at density $\\\\rho =10^{-2}$ is quite far away from the asymptotic scaling limit.\",\n \"We notice that the rounding of the transition is a good indicator to visualize this distance from the asymptotic scaling regime.\",\n \"Here, also the emerging intermediate scaling regime is best noticeable.\",\n \"Figure: Results of direct fits of the ansatz ΔT∝N -α \\\\Delta T\\\\propto N^{-\\\\alpha } to the data inFig.\",\n \"(right) with variable lower bound N min N_{\\\\rm min}.\",\n \"Thedotted line shows the predicted asymptotic scaling exponent.\"\n ],\n [\n \"Acknowledgments\",\n \"The project was funded by the European Union, the Free State of Saxony and the Deutsche Forschungsgemeinschaft (DFG) under Grant No.\",\n \"JA 483/31-1.\",\n \"The authors gratefully acknowledge the computing time provided by the John von Neumann Institute for Computing (NIC) on the supercomputer JURECA at Jülich Supercomputing Centre (JSC) under Grant No.\",\n \"HLZ24.\",\n \"Part of this work has been financially supported by the DFG through the Leipzig Graduate School of Natural Sciences “BuildMoNa” and by the Deutsch-Französische Hochschule (DFH-UFA) through the Doctoral College “${\\\\mathbb {L}}^4$ ” under Grant No.\",\n \"CDFA-02-07.\"\n ]\n]"},"id":{"kind":"string","value":"1605.05844"}}},{"rowIdx":1866680,"cells":{"text":{"kind":"list like","value":[["Suppression of interference in quantum Hall Mach-Zehnder geometry by\n upstream neutral modes"],["Abstract Mach-Zehnder interferometry has been proposed as a probe for detecting the statistics of anyonic quasiparticles in fractional quantum Hall (FQH) states.","Here we focus on interferometers made of multimode edge states with upstream modes.","We find that the interference visibility is suppressed due to downstream-upstream mode entanglement; the latter serves as a \"which path\" detector to the downstream interfering trajectories.","Our analysis tackles a concrete realization of filling factor 2/3, but its applicability goes beyond that specific case, and encompasses the recent observation of ubiquitous emergence of upstream neutral modes in FQH states.","The latter, according to our analysis, goes hand in hand with the failure to observe Mach-Zehnder anyonic interference in fractional states.","We point out how charge-neutral mode disentanglement will resuscitate the interference signal."],["Supplementary Material for “Suppression of interference in quantum Hall Mach-Zehnder geometry by upstream neutral modes”","In this Supplemental Material we present some technical aspects which were omitted in the main text.","Sec.","contains a detailed calculation of the visibility in the general case.","The limit $\\min (L_d,L_u)(1/v_c+1/v_n) \\gg 1/T$ is treated in Sec.",", while the case of equal interferometer arms, $L_u=L_d=L$ is addressed in Sec.","; this leads to the results presented in Table.","II and Fig.","2 of the main text.","Sec.","presents a qualitative discussion of some general features of the results.","Finally, analysis of the almost closed QPC limit (as opposed to the nearly open QPC limit discussed in the main text) is described in Sec.","."],["Calculation of the visibility: The general case","Here we present a detailed calculation of the visibility, based on Eqs.","(3)–(6) of the main text.","This requires the evaluation of the time integral in Eq.","(5), which we will do in the complex $t$ plane.","We consider general, non-integral values of $\\kappa _c$ and $\\kappa _n$ (the values relevant for $\\nu =2/3$ are listed in Table I, where the singularities of $F(y_d,y_u,t;T)$ appearing in Eqs.","(5)–(6) are branch cuts with branch points at $y_d/v_c + i m/T + i \\delta _{d,c}$ , $y_u/v_c + i m/T + i\\delta _{u,c}$ , $-y_d/v_n + i m/T + i \\delta _{d,n}$ , and $-y_u/v_n + i m/T + i\\delta _{u,n}$ , where $m \\in \\mathbb {Z}$ and $\\delta _{d/u,c/n}$ are positive infinitesimals.","The second line of Eq.","(6) defines $F(y_d,y_u,t;T)$ on the real $t$ -axis {See e.g., Appendix H of Ref.","[38], for an explicit calculation.","The formulas appearing there require three modifications for our use: (i) the imaginary time $\\tau $ is replaced by $i t$ ; (ii) $x$ is replaced by $-y_d/v_c$ , $-y_u/v_c$ , $y_d/v_n$ , or $y_u/v_n$ , respectively; (iii) $\\sigma \\equiv \\mathrm {sgn}(\\tau )$ should be omitted, since we are interested in the greater Green function $G^>$ , rather than in the time ordered one}.","The branch cuts of $F(y_d,y_u,t;T)$ in the complex plane must be chosen in a way consistent with this value of $F(y_d,y_u,t;T)$ on the real $t$ axis but are otherwise arbitrary.","For $F(0,0,t;T)$ (i.e., $y_d=y_u=0$ ) the four sets of branch points merge into one set, at $i m/T + i \\delta $ .","It is then useful to choose the branch cuts of all the fractional-power functions appearing $F(y_d,y_u,t;T)$ (namely, $z ^{\\kappa _c^2}$ and $z ^{\\kappa _n^2}$ ) to be along the negative imaginary $z$ axis, with the argument of $z$ defined as $-\\pi /2$ and $3\\pi /2$ on the right and left sides of the cut, respectively.","We employ the rectangular contour depicted in Fig.","REF (a), which includes a “tongue” going around the $m=0$ branch cut.","Let us assume that $\\kappa _c^2,\\kappa _n^2$ are small enough so that the small circle around the branch point do not contribute; the final result can be analytically continued to the general case.","With the above choices, the part of the contour along the line $\\text{Im}(t)=1/T$ will give $-e^{i 2 \\pi (\\kappa _c^2 + \\kappa _n^2)}e^{-e^* V/T}$ times the part of the contour along the real $t$ axis.","We then get: $ j_{\\kappa _c^2,\\kappa _n^2}(0,0;V,T) & =- 2 i e^* e^{i\\pi (\\kappa _c^2 + \\kappa _n^2)} \\sin [2\\pi (\\kappa _c^2 + \\kappa _n^2)](\\pi T \\tau )^{2(\\kappa _c^2 + \\kappa _n^2)}\\frac{1 - e^{- e^* V / T} }{1 - e^{i 2 \\pi (\\kappa _c^2 + \\kappa _n^2) - e^* V/T}}\\int _{0}^\\infty \\text{d}t e^{i e^* V t }\\frac{ 1 }{\\sinh ^{2(\\kappa _c^2 + \\kappa _n^2)} ( \\pi T t ) }\\nonumber \\\\& =\\frac{ e^* \\sinh [e^* V/(2T)] \\sin [2\\pi (\\kappa _c^2 + \\kappa _n^2)]}{\\sin \\left[ \\pi (\\kappa _c^2 + \\kappa _n^2) + i \\frac{e^* V}{2T} \\right]}\\frac{(2 \\pi T \\tau )^{2(\\kappa _c^2 + \\kappa _n^2)}}{\\pi T}B \\left( 1 - 2 \\kappa _c^2 - 2 \\kappa _n^2, \\kappa _c^2 + \\kappa _n^2 - i \\frac{e^* V}{2 \\pi T} \\right)\\nonumber \\\\& =2 e^* \\tau \\sinh [e^* V/(2T)](2 \\pi T \\tau )^{2(\\kappa _c^2 + \\kappa _n^2)-1}\\frac{\\left| \\Gamma \\left( \\kappa _c^2 + \\kappa _n^2 + i \\frac{e^* V}{2 \\pi T}\\right) \\right|^2}{\\Gamma \\left[ 2(\\kappa _c^2 + \\kappa _n^2) \\right]},$ where $B(x,y)$ and $\\Gamma (z)$ are the Beta and Gamma functions, respectively [39].","Thus, for $|e^* V| \\ll T$ we have $ j_{\\kappa _c^2,\\kappa _n^2}(0,0;V,T) =2 \\pi \\tau ^2 (e^*)^2 V(2 \\pi T \\tau )^{2(\\kappa _c^2 + \\kappa _n^2 - 1 )}\\frac{\\left[ \\Gamma ( \\kappa _c^2 + \\kappa _n^2 ) \\right]^2}{\\Gamma \\left( 2\\kappa _c^2 + 2\\kappa _n^2 \\right) }\\sim V T^{2(\\kappa _c^2 + \\kappa _n^2 - 1 )},$ while for $|e^* V| \\gg T$ , remembering that $\\Gamma (z+a)/\\Gamma (z+b) \\sim z^{a-b}$ for large $|z|$ , we find $ j_{\\kappa _c^2,\\kappa _n^2}(0,0;V,T) =\\frac{(e^*)^2 V \\tau ^2}{\\Gamma (2\\kappa _c^2 + 2\\kappa _n^2)}\\left( |e^* V| \\tau \\right)^{2(\\kappa _c^2 + \\kappa _n^2 - 1)}\\sim V^{2(\\kappa _c^2 + \\kappa _n^2) - 1 }.$ Figure: Contours of integration in the complex time plane for the calculation of:(a) j κ c 2 ,κ n 2 (0,0;V,T)j_{\\kappa _c^2,\\kappa _n^2}(0,0;V,T), Eq.","();(b) j κ c 2 ,κ n 2 (L d ,L u ;V,T)j_{\\kappa _c^2,\\kappa _n^2}(L_d,L_u;V,T), Eq.","().Branch points and lines are depicted in red, the contours (traversed in a counterclockwise fashion) in dashed-blue.The infinitesimal shifts δ\\delta and δ u/d,c/n \\delta _{u/d,c/n} of the imaginary parts of the branch points from integer multiples of 1/T1/T are exaggerated for clarity.See the text for further details.For $j(L_d,L_u,t;T)$ (i.e., $y_d=L_d$ and $y_u=L_u$ ) it will be useful to choose the branch cut of the fractional power function $z ^{\\kappa _c^2}$ as before, that is to be along the negative imaginary axis, with the argument of $z$ defined as $-\\pi /2$ and $3\\pi /2$ on the right and left sides of the cut, respectively.","On the other hand, the branch cut of the fractional power function $z^{\\kappa _n^2}$ will be chosen along the positive imaginary axis, with the argument of $z$ defined as $\\pi /2$ or $-3\\pi /2$ on the right and left sides of the cut, respectively.","With this the function $F(L_d,L_u,t;T)$ will have the following branch structure [cf. Fig.","REF (b)]: The branch cuts starting at the branch points at $L_d/v_c + i m/T + i \\delta _{d,c}$ and $L_u/v_c + i m/T + i\\delta _{u,c}$ will go horizontally to the right (left) for even (odd) $m$ , and the branch cuts staring at the branch points at $-L_d/v_n + i m/T + i \\delta _{d,n}$ and $-L_u/v_n + i m/T + i\\delta _{u,n}$ to go horizontally to the left (right) for even (odd) $m$ .","We will focus on $L_u>L_d$ , and take for definiteness $\\delta _{d,c} > \\delta _{u,c}$ and $\\delta _{d,n} > \\delta _{u,n}$ .","For the right- (left-)going branches the phase of the argument of the corresponding fractional power is $\\mp \\pi /2$ below the cut and $\\pm 3\\pi /2$ above it.","Since $\\kappa _c^2,\\kappa _n^2 < 1$ (cf.","Table I), the small circles around the branch points do not contribute (by analytical continuation, the final results can be shown to be valid even without this restriction).","With the above choices, the part of he contour along the line $\\text{Im}(t)=1/T$ will give $- e^{i 2 \\pi (\\kappa _c^2 - \\kappa _n^2)}e^{-e^* V/T}$ times the part of the contour along the real $t$ axis.","We thus find $ j_{\\kappa _c^2,\\kappa _n^2}(L_d,L_u; & V,T) =- e^*\\frac{1-e^{- e^* V / T}}{1 - e^{i 2 \\pi (\\kappa _c^2 - \\kappa _n^2) - e^* V/T}}\\int _{-\\infty }^\\infty \\text{d}t e^{i e^* V (t-L_d/v_c) }\\times \\nonumber \\\\ &\\left\\lbrace e^{-i\\pi \\kappa _n^2}\\left(e^{i 3 \\pi \\kappa _c^2/2} - e^{-i \\pi \\kappa _c^2/2 }\\right)\\left[\\theta (t-L_d/v_c) \\theta (L_u/v_c-t) e^{i \\pi \\kappa _c^2/2}+ \\theta (t-L_u/v_c) \\left( e^{i 3 \\pi \\kappa _c^2/2} + e^{- i \\pi \\kappa _c^2/2} \\right)\\right]\\right.","\\nonumber \\\\ & \\left.+e^{i\\pi \\kappa _c^2}\\left(e^{-i 3 \\pi \\kappa _n^2/2} - e^{i \\pi \\kappa _n^2/2 }\\right)\\left[\\theta (t+L_u/v_n) \\theta (-L_d/v_n-t) e^{-i \\pi \\kappa _n^2/2}+ \\theta (-L_u/v_n-t) \\left( e^{-i 3 \\pi \\kappa _n^2/2} + e^{i \\pi \\kappa _n^2/2} \\right)\\right]\\right\\rbrace \\times \\nonumber \\\\ &\\left\\lbrace \\frac{ (\\pi T \\tau )^2}{\\sinh \\left[ \\pi T | t - L_d/v_c | \\right]\\sinh \\left[ \\pi T | t - L_u/v_c | \\right]}\\right\\rbrace ^{\\kappa _c^2}\\left\\lbrace \\frac{ (\\pi T \\tau )^2 }{\\sinh \\left[ \\pi T | t + L_d/v_n | \\right]\\sinh \\left[ \\pi T | t + L_u/v_n | \\right]}\\right\\rbrace ^{\\kappa _n^2}.$ This general expression can be simplified in particular cases, as detailed in the following two Sections."],["Visibility of a long-arms interferometer","An explicit expression for Eq.","(REF ) can be obtained for $\\min (L_d,L_u)(1/v_c+1/v_n) \\gg 1/T$ , allowing for arbitrary $|L_u-L_d|$ .","In this region, on the cuts corresponding to the neutral propagators the hyperbolic sines appearing in the charge propagators may be replaced by exponentials and vice versa.","The integral thus reduces to: $j_{\\kappa _c^2,\\kappa _n^2} & (L_d,L_u; V,T) =-2 i e^* e^{i\\pi (\\kappa _c^2-\\kappa _n^2)}(2 \\pi T \\tau )^{2(\\kappa _c^2 + \\kappa _n^2)}\\frac{1-e^{- e^* V / T}}{1 - e^{i 2 \\pi (\\kappa _c^2 - \\kappa _n^2) - e^* V/T}}\\int _{-\\infty }^\\infty \\text{d}t e^{i e^* V (t-L_d/v_c) }\\times \\\\ &\\left\\lbrace \\vphantom{ \\frac{ e^{-\\pi \\kappa _n^2 T (2t + L_d/v_n + L_u/v_n)} }{\\left\\lbrace 4 \\sinh \\left[ \\pi T | t - L_d/v_c | \\right] \\sinh \\left[ \\pi T | t - L_u/v_c | \\right] \\right\\rbrace ^{\\kappa _c^2} } }\\sin ( \\pi \\kappa _c^2 )\\left[ \\theta (t-L_d/v_c) \\theta (L_u/v_c-t)+ 2 \\theta (t-L_u/v_c) \\cos ( \\pi \\kappa _c^2 ) \\right]\\frac{ e^{-\\pi \\kappa _n^2 T (2t + L_d/v_n + L_u/v_n)} }{\\left\\lbrace 4 \\sinh \\left[ \\pi T | t - L_d/v_c | \\right]\\sinh \\left[ \\pi T | t - L_u/v_c | \\right] \\right\\rbrace ^{\\kappa _c^2} }\\right.","\\nonumber \\\\ & \\left.","\\nonumber - \\sin (\\pi \\kappa _n^2)\\left[\\theta (t+L_u/v_n) \\theta (-L_d/v_n-t)+ 2 \\theta (-L_u/v_n-t) \\cos (\\pi \\kappa _n^2)\\right]\\frac{ e^{\\pi \\kappa _c^2 T (2t - L_d/v_c - L_u/v_c)} }{\\left\\lbrace 4 \\sinh \\left[ \\pi T | t + L_d/v_n | \\right]\\sinh \\left[ \\pi T | t + L_u/v_n | \\right] \\right\\rbrace ^{\\kappa _n^2} }\\right\\rbrace .$ Each of the resulting integrals (on the intervals $t<-L_u/v_n$ , $-L_u/v_n < t < -L_d/v_n$ , $L_d/v_cL/v_c$ and $t<-L/v_n$ can be expressed exactly in terms of hypergeometric functions, resulting in: $ j_{\\kappa _c^2,\\kappa _n^2} (L,L;V,T) = &2 \\pi i e^* \\tau \\left( 2 \\pi T \\tau \\right)^{2(\\kappa _c^2 + \\kappa _n^2)-1}\\times \\nonumber \\\\ &\\left\\lbrace e^{- 2 \\pi \\kappa _n^2 T (L/v_c + L/v_n) }\\frac{\\sinh \\left( \\pi \\frac{e^* V}{2 \\pi T} \\right) }{\\sinh \\left( \\pi \\frac{e^* V}{2 \\pi T} - i \\pi (\\kappa _c^2-\\kappa _n^2) \\right) }\\frac{\\Gamma \\left( \\kappa _c^2 + \\kappa _n^2 - i \\frac{e^* V}{2\\pi T} \\right) }{\\Gamma (2 \\kappa _c^2) \\Gamma \\left( 1 - \\kappa _c^2 + \\kappa _n^2 - i \\frac{e^* V}{2\\pi T} \\right)}\\times \\right.","\\nonumber \\\\ &{}_2F_1 \\left( 2 \\kappa _n^2, \\kappa _c^2 + \\kappa _n^2 - i \\frac{e^* V}{2\\pi T}, 1 - \\kappa _c^2 + \\kappa _n^2 - i \\frac{e^* V}{2\\pi T}; e^{-2\\pi T (L/v_c + L/v_n) } \\right)\\nonumber \\\\ &-e^{- i e^* V (L/v_c + L/v_n) - 2 \\pi \\kappa _c^2 T (L/v_c + L/v_n)}\\frac{\\sinh \\left( \\pi \\frac{e^* V}{2 \\pi T} \\right) }{\\sinh \\left( \\pi \\frac{e^* V}{2 \\pi T} - i \\pi (\\kappa _c^2-\\kappa _n^2) \\right) }\\frac{\\Gamma \\left( \\kappa _c^2 + \\kappa _n^2 + i \\frac{e^* V}{2\\pi T} \\right) }{\\Gamma (2 \\kappa _n^2) \\Gamma \\left( 1 + \\kappa _c^2 - \\kappa _n^2 + i \\frac{e^* V}{2\\pi T} \\right)}\\times \\nonumber \\\\ &\\left.","{}_2F_1 \\left( 2 \\kappa _c^2, \\kappa _c^2 + \\kappa _n^2 + i \\frac{e^* V}{2\\pi T}, 1 + \\kappa _c^2 - \\kappa _n^2 + i \\frac{e^* V}{2\\pi T}; e^{-2\\pi T (L/v_c + L/v_n) } \\right)\\right\\rbrace .$ For $L/v_c + L/v_n \\gg 1/T$ we then find $j_{\\kappa _c^2,\\kappa _n^2}(L,L;V,T) = &2 \\pi i e^* \\tau \\left( 2 \\pi T \\tau \\right)^{2(\\kappa _c^2 + \\kappa _n^2)-1}\\times \\nonumber \\\\ &\\left\\lbrace e^{- 2 \\pi \\kappa _n^2 T (L/v_c + L/v_n) }\\frac{\\sinh \\left( \\pi \\frac{e^* V}{2 \\pi T} \\right) }{\\sinh \\left( \\pi \\frac{e^* V}{2 \\pi T} - i \\pi (\\kappa _c^2-\\kappa _n^2) \\right) }\\frac{\\Gamma \\left( \\kappa _c^2 + \\kappa _n^2 - i \\frac{e^* V}{2\\pi T} \\right) }{\\Gamma (2 \\kappa _c^2) \\Gamma \\left( 1 - \\kappa _c^2 + \\kappa _n^2 - i \\frac{e^* V}{2\\pi T} \\right)}\\right.\\nonumber \\\\ &\\left.-e^{ - i e^* V (L/v_c + L/v_n) - 2 \\pi \\kappa _c^2 T (L/v_c + L/v_n)}\\frac{\\sinh \\left( \\pi \\frac{e^* V}{2 \\pi T} \\right) }{\\sinh \\left( \\pi \\frac{e^* V}{2 \\pi T} - i \\pi (\\kappa _c^2-\\kappa _n^2) \\right) }\\frac{\\Gamma \\left( \\kappa _c^2 + \\kappa _n^2 + i \\frac{e^* V}{2\\pi T} \\right) }{\\Gamma (2 \\kappa _n^2) \\Gamma \\left( 1 + \\kappa _c^2 - \\kappa _n^2 + i \\frac{e^* V}{2\\pi T} \\right)}\\right\\rbrace .$ In this case, in the limit $T \\gg |e^* V|$ one obtains $& j_{\\kappa _c^2,\\kappa _n^2}(L,L;V,T) =- 2 \\pi (e^*)^2 V \\tau \\left( 2 \\pi T \\tau \\right)^{2(\\kappa _c^2 + \\kappa _n^2 - 1)}\\times \\nonumber \\\\ &\\left\\lbrace e^{- 2 \\pi \\kappa _n^2 T (L/v_c + L/v_n) }B(\\kappa _n^2+\\kappa _c^2,\\kappa _n^2-\\kappa _c^2)+e^{- i e^* V (L/v_c + L/v_n) - 2 \\pi \\kappa _c^2 T (L/v_c + L/v_n)}B(\\kappa _c^2+\\kappa _n^2,\\kappa _c^2-\\kappa _n^2) \\right\\rbrace ,$ and in the limit $T \\ll |e^* V|$ we have $& j_{\\kappa _c^2,\\kappa _n^2}(L,L;V,T) =2 \\pi i e^* \\tau \\left( 2 \\pi T \\tau \\right)^{2(\\kappa _c^2 + \\kappa _n^2)-1} e^{i \\pi (\\kappa _c^2-\\kappa _n^2) \\text{sgn}(e^* V)}\\times \\nonumber \\\\ &\\left\\lbrace e^{- 2 \\pi \\kappa _n^2 T (L/v_c + L/v_n) }\\frac{1}{\\Gamma (2 \\kappa _c^2)}\\left( - i \\frac{e^* V}{2\\pi T} \\right)^{2 \\kappa _c^2 - 1}-e^{- i e^* V (L/v_c + L/v_n) - 2 \\pi \\kappa _c^2 T (L/v_c + L/v_n)}\\frac{1}{\\Gamma (2 \\kappa _n^2)}\\left( i \\frac{e^* V}{2\\pi T} \\right)^{2\\kappa _n^2-1}\\right\\rbrace .$ .","For $L/v_c + L/v_n \\ll 1/T, 1/|e^* V|$ we find $j_{\\kappa _c^2,\\kappa _n^2}(L,L;V,T) =2 i e^* \\tau \\left( 2 \\pi T \\tau \\right)^{2(\\kappa _c^2 + \\kappa _n^2)-1}& \\left\\lbrace \\frac{\\sin (2 \\pi \\kappa _c^2)\\sinh \\left( \\pi \\frac{e^* V}{2 \\pi T} \\right) }{\\sinh \\left( \\pi \\frac{e^* V}{2 \\pi T} - i \\pi (\\kappa _c^2-\\kappa _n^2) \\right) }\\frac{\\Gamma (1 - 2 \\kappa _c^2 - 2 \\kappa _n^2) \\Gamma \\left( \\kappa _c^2 + \\kappa _n^2 - i \\frac{e^* V}{2\\pi T} \\right) }{ \\Gamma \\left( 1 - \\kappa _c^2 - \\kappa _n^2 - i \\frac{e^* V}{2\\pi T} \\right)}\\right.\\nonumber \\\\ &\\left.-\\frac{\\sin (2 \\pi \\kappa _n^2) \\sinh \\left( \\pi \\frac{e^* V}{2 \\pi T} \\right) }{\\sinh \\left( \\pi \\frac{e^* V}{2 \\pi T} - i \\pi (\\kappa _c^2-\\kappa _n^2) \\right) }\\frac{\\Gamma (1 - 2 \\kappa _c^2 - 2 \\kappa _n^2) \\Gamma \\left( \\kappa _c^2 + \\kappa _n^2 + i \\frac{e^* V}{2\\pi T} \\right) }{ \\Gamma \\left( 1 - \\kappa _c^2 - \\kappa _n^2 + i \\frac{e^* V}{2\\pi T} \\right)}\\right\\rbrace ,$ which can be shown {using the identity $\\Gamma (z)\\Gamma (1-z) = \\pi /\\sin (\\pi z)$ [39]} to reduce to Eqs.","(REF )–(REF ), as it should.","Finally, for $1/|e^* V| \\ll L/v_c + L/v_n \\ll 1/T$ we have: $ & j_{\\kappa _c^2,\\kappa _n^2}(L,L;V,T) =2 \\pi i e^* \\tau \\left( 2 \\pi T \\tau \\right)^{2(\\kappa _c^2 + \\kappa _n^2)-1} e^{i \\pi (\\kappa _c^2-\\kappa _n^2) \\text{sgn}(e^* V)}\\times \\nonumber \\\\ &\\left\\lbrace \\frac{1}{\\Gamma (2 \\kappa _c^2)}\\left( - i \\frac{e^* V}{2\\pi T} \\right)^{2\\kappa _c^2-1}\\left[ 2\\pi T (L/v_c + L/v_n) \\right]^{-2\\kappa _n^2}-e^{- i e^* V (L/v_c + L/v_n)}\\frac{1}{\\Gamma (2 \\kappa _n^2)}\\left( i \\frac{e^* V}{2\\pi T} \\right)^{2\\kappa _n^2-1}\\left[ 2\\pi T (L/v_c + L/v_n) \\right]^{-2\\kappa _c^2}\\right\\rbrace .$ From the above results, we can determine the visibility of the interference pattern as function of temperature and source-drain bias, using Eqs.","(3)–(4).","For $|\\gamma _a|=|\\gamma _b|$ it is the ratio $|j(L_d,L_u;V,T)/j(0,0;V,T)|^3$ , reflecting the fact that a unit charge detected at D$_2$ [Fig.","1(a)] involves the tunneling of three $e/3$ quasi-particles, each subject to a different effective AB flux.","The results are displayed in Table.","II and Fig.","2 of the main text."],["Some qualitative features of the visibility","There are some remarkable insights that follow from the results of the previous Sections.","First and foremost, the interference signal is exponentially-suppressed with increasing temperature as soon as $\\min (L_d,L_u)(1/v_c+1/v_n) \\gg 1/T$ .","This is true even even if $T \\ll |e^* V|$ (!).","At the same time, increasing the bias voltage does not lead to exponential suppression even when $|e^* V|$ is larger than any other scale, including the temperature.","This can be intuitively understood by thinking about free fermions with velocity $v$ : at zero temperature the voltage defines a window of allowed energies, and hence wavevectors, for tunneling.","Since the window is sharp, the corresponding wavepackets decay in real space at large distances $x$ only as $e^{i V x/v}/x$ , hence there is no exponential suppression of the interference.","On the other hand, introducing a finite temperature amounts to smearing the sharp energy window.","Roughly, this can be thought of as having a distribution of voltage values around some average $V_0$ .","Averaging $e^{iV x/v}$ with respect to this distribution will leave us with an oscillatory factor $e^{i V_0 x/v}$ multiplied by a suppression factor.","The latter is a function of $T x/v$ , and decays exponentially when $T x/v \\gg 1$ , independently of the value of $V_0$ .","Another noteworthy feature is that the two terms in Eq.","(REF ), and correspondingly in Eqs.","(REF ) and (REF ), originate from domains in the time integration where either the charge or the neutral mode overlap is maximal, but not both (which is the essence of the dephasing mechanism discussed in this work), cf.","Fig.","REF (b).","These two terms have a relative phase between them.","This gives rise to oscillations of the visibility as function of the voltage when the latter is high and the temperature small [so that the the two terms in Eqs.","(REF ) and (REF ) are comparable], the case described by Eqs.","(REF ) and (REF ).","This leads to the last line of Table.","II and the lobe structure seen in Fig.","2 of the main text.","Figure: An almost-closed QH MZI (compare with the almost-open case, Fig.","1 of the main text).Shown are the downstream chiral edge modes, sources, drains, and the tunneling bridges.","The schematic equivalent geometry is essentially the same as the one depicted in Fig.","1(b)–(c),with the upper and lower edges corresponding, respectively, to the inner and outer edges of the lower edge + island depicted here, instead of the island and lower edges in Fig.","1(a)."],["The case of electron tunneling","In the limit of strong tunneling bridges, the interfering paths are dominated by electron tunneling.","It is easy to obtain results for the correlation functions by noting that $L_d$ and $L_u$ are interchanged and the AB phase is flipped as compared with quasiparticle tunneling (cf. Fig.","REF ).","In addition, the values of $\\kappa _c$ and $\\kappa _n$ are now taken from the fourth, rather than the first row of Table I.","Finally, Eqs.","(3)–(4) are replaced by $ I_{D2}(V,T) =\\left(\\left| \\gamma _a \\right|^2 + \\left| \\gamma _b \\right|^2 \\right) j(0,0;V,T)+ \\gamma _a \\gamma _b^* e^{-2 \\pi i \\Phi _{AB}/\\Phi _0 } j(L_u,L_d;V,T)+ \\text{c.c.","}$ Thus, when $|\\gamma _a|=|\\gamma _b|$ the visibility is $|j(L_u,L_d;V,T)/j(0,0;V,T)|$ .","The quantities $j(L_u,L_d;V,T)$ and $j(0,0;V,T)$ were calculated in Secs. –.","The resulting behavior is summarized in Table.","II of the main text."]],"string":"[\n [\n \"Suppression of interference in quantum Hall Mach-Zehnder geometry by\\n upstream neutral modes\"\n ],\n [\n \"Abstract Mach-Zehnder interferometry has been proposed as a probe for detecting the statistics of anyonic quasiparticles in fractional quantum Hall (FQH) states.\",\n \"Here we focus on interferometers made of multimode edge states with upstream modes.\",\n \"We find that the interference visibility is suppressed due to downstream-upstream mode entanglement; the latter serves as a \\\"which path\\\" detector to the downstream interfering trajectories.\",\n \"Our analysis tackles a concrete realization of filling factor 2/3, but its applicability goes beyond that specific case, and encompasses the recent observation of ubiquitous emergence of upstream neutral modes in FQH states.\",\n \"The latter, according to our analysis, goes hand in hand with the failure to observe Mach-Zehnder anyonic interference in fractional states.\",\n \"We point out how charge-neutral mode disentanglement will resuscitate the interference signal.\"\n ],\n [\n \"Supplementary Material for “Suppression of interference in quantum Hall Mach-Zehnder geometry by upstream neutral modes”\",\n \"In this Supplemental Material we present some technical aspects which were omitted in the main text.\",\n \"Sec.\",\n \"contains a detailed calculation of the visibility in the general case.\",\n \"The limit $\\\\min (L_d,L_u)(1/v_c+1/v_n) \\\\gg 1/T$ is treated in Sec.\",\n \", while the case of equal interferometer arms, $L_u=L_d=L$ is addressed in Sec.\",\n \"; this leads to the results presented in Table.\",\n \"II and Fig.\",\n \"2 of the main text.\",\n \"Sec.\",\n \"presents a qualitative discussion of some general features of the results.\",\n \"Finally, analysis of the almost closed QPC limit (as opposed to the nearly open QPC limit discussed in the main text) is described in Sec.\",\n \".\"\n ],\n [\n \"Calculation of the visibility: The general case\",\n \"Here we present a detailed calculation of the visibility, based on Eqs.\",\n \"(3)–(6) of the main text.\",\n \"This requires the evaluation of the time integral in Eq.\",\n \"(5), which we will do in the complex $t$ plane.\",\n \"We consider general, non-integral values of $\\\\kappa _c$ and $\\\\kappa _n$ (the values relevant for $\\\\nu =2/3$ are listed in Table I, where the singularities of $F(y_d,y_u,t;T)$ appearing in Eqs.\",\n \"(5)–(6) are branch cuts with branch points at $y_d/v_c + i m/T + i \\\\delta _{d,c}$ , $y_u/v_c + i m/T + i\\\\delta _{u,c}$ , $-y_d/v_n + i m/T + i \\\\delta _{d,n}$ , and $-y_u/v_n + i m/T + i\\\\delta _{u,n}$ , where $m \\\\in \\\\mathbb {Z}$ and $\\\\delta _{d/u,c/n}$ are positive infinitesimals.\",\n \"The second line of Eq.\",\n \"(6) defines $F(y_d,y_u,t;T)$ on the real $t$ -axis {See e.g., Appendix H of Ref.\",\n \"[38], for an explicit calculation.\",\n \"The formulas appearing there require three modifications for our use: (i) the imaginary time $\\\\tau $ is replaced by $i t$ ; (ii) $x$ is replaced by $-y_d/v_c$ , $-y_u/v_c$ , $y_d/v_n$ , or $y_u/v_n$ , respectively; (iii) $\\\\sigma \\\\equiv \\\\mathrm {sgn}(\\\\tau )$ should be omitted, since we are interested in the greater Green function $G^>$ , rather than in the time ordered one}.\",\n \"The branch cuts of $F(y_d,y_u,t;T)$ in the complex plane must be chosen in a way consistent with this value of $F(y_d,y_u,t;T)$ on the real $t$ axis but are otherwise arbitrary.\",\n \"For $F(0,0,t;T)$ (i.e., $y_d=y_u=0$ ) the four sets of branch points merge into one set, at $i m/T + i \\\\delta $ .\",\n \"It is then useful to choose the branch cuts of all the fractional-power functions appearing $F(y_d,y_u,t;T)$ (namely, $z ^{\\\\kappa _c^2}$ and $z ^{\\\\kappa _n^2}$ ) to be along the negative imaginary $z$ axis, with the argument of $z$ defined as $-\\\\pi /2$ and $3\\\\pi /2$ on the right and left sides of the cut, respectively.\",\n \"We employ the rectangular contour depicted in Fig.\",\n \"REF (a), which includes a “tongue” going around the $m=0$ branch cut.\",\n \"Let us assume that $\\\\kappa _c^2,\\\\kappa _n^2$ are small enough so that the small circle around the branch point do not contribute; the final result can be analytically continued to the general case.\",\n \"With the above choices, the part of the contour along the line $\\\\text{Im}(t)=1/T$ will give $-e^{i 2 \\\\pi (\\\\kappa _c^2 + \\\\kappa _n^2)}e^{-e^* V/T}$ times the part of the contour along the real $t$ axis.\",\n \"We then get: $ j_{\\\\kappa _c^2,\\\\kappa _n^2}(0,0;V,T) & =- 2 i e^* e^{i\\\\pi (\\\\kappa _c^2 + \\\\kappa _n^2)} \\\\sin [2\\\\pi (\\\\kappa _c^2 + \\\\kappa _n^2)](\\\\pi T \\\\tau )^{2(\\\\kappa _c^2 + \\\\kappa _n^2)}\\\\frac{1 - e^{- e^* V / T} }{1 - e^{i 2 \\\\pi (\\\\kappa _c^2 + \\\\kappa _n^2) - e^* V/T}}\\\\int _{0}^\\\\infty \\\\text{d}t e^{i e^* V t }\\\\frac{ 1 }{\\\\sinh ^{2(\\\\kappa _c^2 + \\\\kappa _n^2)} ( \\\\pi T t ) }\\\\nonumber \\\\\\\\& =\\\\frac{ e^* \\\\sinh [e^* V/(2T)] \\\\sin [2\\\\pi (\\\\kappa _c^2 + \\\\kappa _n^2)]}{\\\\sin \\\\left[ \\\\pi (\\\\kappa _c^2 + \\\\kappa _n^2) + i \\\\frac{e^* V}{2T} \\\\right]}\\\\frac{(2 \\\\pi T \\\\tau )^{2(\\\\kappa _c^2 + \\\\kappa _n^2)}}{\\\\pi T}B \\\\left( 1 - 2 \\\\kappa _c^2 - 2 \\\\kappa _n^2, \\\\kappa _c^2 + \\\\kappa _n^2 - i \\\\frac{e^* V}{2 \\\\pi T} \\\\right)\\\\nonumber \\\\\\\\& =2 e^* \\\\tau \\\\sinh [e^* V/(2T)](2 \\\\pi T \\\\tau )^{2(\\\\kappa _c^2 + \\\\kappa _n^2)-1}\\\\frac{\\\\left| \\\\Gamma \\\\left( \\\\kappa _c^2 + \\\\kappa _n^2 + i \\\\frac{e^* V}{2 \\\\pi T}\\\\right) \\\\right|^2}{\\\\Gamma \\\\left[ 2(\\\\kappa _c^2 + \\\\kappa _n^2) \\\\right]},$ where $B(x,y)$ and $\\\\Gamma (z)$ are the Beta and Gamma functions, respectively [39].\",\n \"Thus, for $|e^* V| \\\\ll T$ we have $ j_{\\\\kappa _c^2,\\\\kappa _n^2}(0,0;V,T) =2 \\\\pi \\\\tau ^2 (e^*)^2 V(2 \\\\pi T \\\\tau )^{2(\\\\kappa _c^2 + \\\\kappa _n^2 - 1 )}\\\\frac{\\\\left[ \\\\Gamma ( \\\\kappa _c^2 + \\\\kappa _n^2 ) \\\\right]^2}{\\\\Gamma \\\\left( 2\\\\kappa _c^2 + 2\\\\kappa _n^2 \\\\right) }\\\\sim V T^{2(\\\\kappa _c^2 + \\\\kappa _n^2 - 1 )},$ while for $|e^* V| \\\\gg T$ , remembering that $\\\\Gamma (z+a)/\\\\Gamma (z+b) \\\\sim z^{a-b}$ for large $|z|$ , we find $ j_{\\\\kappa _c^2,\\\\kappa _n^2}(0,0;V,T) =\\\\frac{(e^*)^2 V \\\\tau ^2}{\\\\Gamma (2\\\\kappa _c^2 + 2\\\\kappa _n^2)}\\\\left( |e^* V| \\\\tau \\\\right)^{2(\\\\kappa _c^2 + \\\\kappa _n^2 - 1)}\\\\sim V^{2(\\\\kappa _c^2 + \\\\kappa _n^2) - 1 }.$ Figure: Contours of integration in the complex time plane for the calculation of:(a) j κ c 2 ,κ n 2 (0,0;V,T)j_{\\\\kappa _c^2,\\\\kappa _n^2}(0,0;V,T), Eq.\",\n \"();(b) j κ c 2 ,κ n 2 (L d ,L u ;V,T)j_{\\\\kappa _c^2,\\\\kappa _n^2}(L_d,L_u;V,T), Eq.\",\n \"().Branch points and lines are depicted in red, the contours (traversed in a counterclockwise fashion) in dashed-blue.The infinitesimal shifts δ\\\\delta and δ u/d,c/n \\\\delta _{u/d,c/n} of the imaginary parts of the branch points from integer multiples of 1/T1/T are exaggerated for clarity.See the text for further details.For $j(L_d,L_u,t;T)$ (i.e., $y_d=L_d$ and $y_u=L_u$ ) it will be useful to choose the branch cut of the fractional power function $z ^{\\\\kappa _c^2}$ as before, that is to be along the negative imaginary axis, with the argument of $z$ defined as $-\\\\pi /2$ and $3\\\\pi /2$ on the right and left sides of the cut, respectively.\",\n \"On the other hand, the branch cut of the fractional power function $z^{\\\\kappa _n^2}$ will be chosen along the positive imaginary axis, with the argument of $z$ defined as $\\\\pi /2$ or $-3\\\\pi /2$ on the right and left sides of the cut, respectively.\",\n \"With this the function $F(L_d,L_u,t;T)$ will have the following branch structure [cf. Fig.\",\n \"REF (b)]: The branch cuts starting at the branch points at $L_d/v_c + i m/T + i \\\\delta _{d,c}$ and $L_u/v_c + i m/T + i\\\\delta _{u,c}$ will go horizontally to the right (left) for even (odd) $m$ , and the branch cuts staring at the branch points at $-L_d/v_n + i m/T + i \\\\delta _{d,n}$ and $-L_u/v_n + i m/T + i\\\\delta _{u,n}$ to go horizontally to the left (right) for even (odd) $m$ .\",\n \"We will focus on $L_u>L_d$ , and take for definiteness $\\\\delta _{d,c} > \\\\delta _{u,c}$ and $\\\\delta _{d,n} > \\\\delta _{u,n}$ .\",\n \"For the right- (left-)going branches the phase of the argument of the corresponding fractional power is $\\\\mp \\\\pi /2$ below the cut and $\\\\pm 3\\\\pi /2$ above it.\",\n \"Since $\\\\kappa _c^2,\\\\kappa _n^2 < 1$ (cf.\",\n \"Table I), the small circles around the branch points do not contribute (by analytical continuation, the final results can be shown to be valid even without this restriction).\",\n \"With the above choices, the part of he contour along the line $\\\\text{Im}(t)=1/T$ will give $- e^{i 2 \\\\pi (\\\\kappa _c^2 - \\\\kappa _n^2)}e^{-e^* V/T}$ times the part of the contour along the real $t$ axis.\",\n \"We thus find $ j_{\\\\kappa _c^2,\\\\kappa _n^2}(L_d,L_u; & V,T) =- e^*\\\\frac{1-e^{- e^* V / T}}{1 - e^{i 2 \\\\pi (\\\\kappa _c^2 - \\\\kappa _n^2) - e^* V/T}}\\\\int _{-\\\\infty }^\\\\infty \\\\text{d}t e^{i e^* V (t-L_d/v_c) }\\\\times \\\\nonumber \\\\\\\\ &\\\\left\\\\lbrace e^{-i\\\\pi \\\\kappa _n^2}\\\\left(e^{i 3 \\\\pi \\\\kappa _c^2/2} - e^{-i \\\\pi \\\\kappa _c^2/2 }\\\\right)\\\\left[\\\\theta (t-L_d/v_c) \\\\theta (L_u/v_c-t) e^{i \\\\pi \\\\kappa _c^2/2}+ \\\\theta (t-L_u/v_c) \\\\left( e^{i 3 \\\\pi \\\\kappa _c^2/2} + e^{- i \\\\pi \\\\kappa _c^2/2} \\\\right)\\\\right]\\\\right.\",\n \"\\\\nonumber \\\\\\\\ & \\\\left.+e^{i\\\\pi \\\\kappa _c^2}\\\\left(e^{-i 3 \\\\pi \\\\kappa _n^2/2} - e^{i \\\\pi \\\\kappa _n^2/2 }\\\\right)\\\\left[\\\\theta (t+L_u/v_n) \\\\theta (-L_d/v_n-t) e^{-i \\\\pi \\\\kappa _n^2/2}+ \\\\theta (-L_u/v_n-t) \\\\left( e^{-i 3 \\\\pi \\\\kappa _n^2/2} + e^{i \\\\pi \\\\kappa _n^2/2} \\\\right)\\\\right]\\\\right\\\\rbrace \\\\times \\\\nonumber \\\\\\\\ &\\\\left\\\\lbrace \\\\frac{ (\\\\pi T \\\\tau )^2}{\\\\sinh \\\\left[ \\\\pi T | t - L_d/v_c | \\\\right]\\\\sinh \\\\left[ \\\\pi T | t - L_u/v_c | \\\\right]}\\\\right\\\\rbrace ^{\\\\kappa _c^2}\\\\left\\\\lbrace \\\\frac{ (\\\\pi T \\\\tau )^2 }{\\\\sinh \\\\left[ \\\\pi T | t + L_d/v_n | \\\\right]\\\\sinh \\\\left[ \\\\pi T | t + L_u/v_n | \\\\right]}\\\\right\\\\rbrace ^{\\\\kappa _n^2}.$ This general expression can be simplified in particular cases, as detailed in the following two Sections.\"\n ],\n [\n \"Visibility of a long-arms interferometer\",\n \"An explicit expression for Eq.\",\n \"(REF ) can be obtained for $\\\\min (L_d,L_u)(1/v_c+1/v_n) \\\\gg 1/T$ , allowing for arbitrary $|L_u-L_d|$ .\",\n \"In this region, on the cuts corresponding to the neutral propagators the hyperbolic sines appearing in the charge propagators may be replaced by exponentials and vice versa.\",\n \"The integral thus reduces to: $j_{\\\\kappa _c^2,\\\\kappa _n^2} & (L_d,L_u; V,T) =-2 i e^* e^{i\\\\pi (\\\\kappa _c^2-\\\\kappa _n^2)}(2 \\\\pi T \\\\tau )^{2(\\\\kappa _c^2 + \\\\kappa _n^2)}\\\\frac{1-e^{- e^* V / T}}{1 - e^{i 2 \\\\pi (\\\\kappa _c^2 - \\\\kappa _n^2) - e^* V/T}}\\\\int _{-\\\\infty }^\\\\infty \\\\text{d}t e^{i e^* V (t-L_d/v_c) }\\\\times \\\\\\\\ &\\\\left\\\\lbrace \\\\vphantom{ \\\\frac{ e^{-\\\\pi \\\\kappa _n^2 T (2t + L_d/v_n + L_u/v_n)} }{\\\\left\\\\lbrace 4 \\\\sinh \\\\left[ \\\\pi T | t - L_d/v_c | \\\\right] \\\\sinh \\\\left[ \\\\pi T | t - L_u/v_c | \\\\right] \\\\right\\\\rbrace ^{\\\\kappa _c^2} } }\\\\sin ( \\\\pi \\\\kappa _c^2 )\\\\left[ \\\\theta (t-L_d/v_c) \\\\theta (L_u/v_c-t)+ 2 \\\\theta (t-L_u/v_c) \\\\cos ( \\\\pi \\\\kappa _c^2 ) \\\\right]\\\\frac{ e^{-\\\\pi \\\\kappa _n^2 T (2t + L_d/v_n + L_u/v_n)} }{\\\\left\\\\lbrace 4 \\\\sinh \\\\left[ \\\\pi T | t - L_d/v_c | \\\\right]\\\\sinh \\\\left[ \\\\pi T | t - L_u/v_c | \\\\right] \\\\right\\\\rbrace ^{\\\\kappa _c^2} }\\\\right.\",\n \"\\\\nonumber \\\\\\\\ & \\\\left.\",\n \"\\\\nonumber - \\\\sin (\\\\pi \\\\kappa _n^2)\\\\left[\\\\theta (t+L_u/v_n) \\\\theta (-L_d/v_n-t)+ 2 \\\\theta (-L_u/v_n-t) \\\\cos (\\\\pi \\\\kappa _n^2)\\\\right]\\\\frac{ e^{\\\\pi \\\\kappa _c^2 T (2t - L_d/v_c - L_u/v_c)} }{\\\\left\\\\lbrace 4 \\\\sinh \\\\left[ \\\\pi T | t + L_d/v_n | \\\\right]\\\\sinh \\\\left[ \\\\pi T | t + L_u/v_n | \\\\right] \\\\right\\\\rbrace ^{\\\\kappa _n^2} }\\\\right\\\\rbrace .$ Each of the resulting integrals (on the intervals $t<-L_u/v_n$ , $-L_u/v_n < t < -L_d/v_n$ , $L_d/v_cL/v_c$ and $t<-L/v_n$ can be expressed exactly in terms of hypergeometric functions, resulting in: $ j_{\\\\kappa _c^2,\\\\kappa _n^2} (L,L;V,T) = &2 \\\\pi i e^* \\\\tau \\\\left( 2 \\\\pi T \\\\tau \\\\right)^{2(\\\\kappa _c^2 + \\\\kappa _n^2)-1}\\\\times \\\\nonumber \\\\\\\\ &\\\\left\\\\lbrace e^{- 2 \\\\pi \\\\kappa _n^2 T (L/v_c + L/v_n) }\\\\frac{\\\\sinh \\\\left( \\\\pi \\\\frac{e^* V}{2 \\\\pi T} \\\\right) }{\\\\sinh \\\\left( \\\\pi \\\\frac{e^* V}{2 \\\\pi T} - i \\\\pi (\\\\kappa _c^2-\\\\kappa _n^2) \\\\right) }\\\\frac{\\\\Gamma \\\\left( \\\\kappa _c^2 + \\\\kappa _n^2 - i \\\\frac{e^* V}{2\\\\pi T} \\\\right) }{\\\\Gamma (2 \\\\kappa _c^2) \\\\Gamma \\\\left( 1 - \\\\kappa _c^2 + \\\\kappa _n^2 - i \\\\frac{e^* V}{2\\\\pi T} \\\\right)}\\\\times \\\\right.\",\n \"\\\\nonumber \\\\\\\\ &{}_2F_1 \\\\left( 2 \\\\kappa _n^2, \\\\kappa _c^2 + \\\\kappa _n^2 - i \\\\frac{e^* V}{2\\\\pi T}, 1 - \\\\kappa _c^2 + \\\\kappa _n^2 - i \\\\frac{e^* V}{2\\\\pi T}; e^{-2\\\\pi T (L/v_c + L/v_n) } \\\\right)\\\\nonumber \\\\\\\\ &-e^{- i e^* V (L/v_c + L/v_n) - 2 \\\\pi \\\\kappa _c^2 T (L/v_c + L/v_n)}\\\\frac{\\\\sinh \\\\left( \\\\pi \\\\frac{e^* V}{2 \\\\pi T} \\\\right) }{\\\\sinh \\\\left( \\\\pi \\\\frac{e^* V}{2 \\\\pi T} - i \\\\pi (\\\\kappa _c^2-\\\\kappa _n^2) \\\\right) }\\\\frac{\\\\Gamma \\\\left( \\\\kappa _c^2 + \\\\kappa _n^2 + i \\\\frac{e^* V}{2\\\\pi T} \\\\right) }{\\\\Gamma (2 \\\\kappa _n^2) \\\\Gamma \\\\left( 1 + \\\\kappa _c^2 - \\\\kappa _n^2 + i \\\\frac{e^* V}{2\\\\pi T} \\\\right)}\\\\times \\\\nonumber \\\\\\\\ &\\\\left.\",\n \"{}_2F_1 \\\\left( 2 \\\\kappa _c^2, \\\\kappa _c^2 + \\\\kappa _n^2 + i \\\\frac{e^* V}{2\\\\pi T}, 1 + \\\\kappa _c^2 - \\\\kappa _n^2 + i \\\\frac{e^* V}{2\\\\pi T}; e^{-2\\\\pi T (L/v_c + L/v_n) } \\\\right)\\\\right\\\\rbrace .$ For $L/v_c + L/v_n \\\\gg 1/T$ we then find $j_{\\\\kappa _c^2,\\\\kappa _n^2}(L,L;V,T) = &2 \\\\pi i e^* \\\\tau \\\\left( 2 \\\\pi T \\\\tau \\\\right)^{2(\\\\kappa _c^2 + \\\\kappa _n^2)-1}\\\\times \\\\nonumber \\\\\\\\ &\\\\left\\\\lbrace e^{- 2 \\\\pi \\\\kappa _n^2 T (L/v_c + L/v_n) }\\\\frac{\\\\sinh \\\\left( \\\\pi \\\\frac{e^* V}{2 \\\\pi T} \\\\right) }{\\\\sinh \\\\left( \\\\pi \\\\frac{e^* V}{2 \\\\pi T} - i \\\\pi (\\\\kappa _c^2-\\\\kappa _n^2) \\\\right) }\\\\frac{\\\\Gamma \\\\left( \\\\kappa _c^2 + \\\\kappa _n^2 - i \\\\frac{e^* V}{2\\\\pi T} \\\\right) }{\\\\Gamma (2 \\\\kappa _c^2) \\\\Gamma \\\\left( 1 - \\\\kappa _c^2 + \\\\kappa _n^2 - i \\\\frac{e^* V}{2\\\\pi T} \\\\right)}\\\\right.\\\\nonumber \\\\\\\\ &\\\\left.-e^{ - i e^* V (L/v_c + L/v_n) - 2 \\\\pi \\\\kappa _c^2 T (L/v_c + L/v_n)}\\\\frac{\\\\sinh \\\\left( \\\\pi \\\\frac{e^* V}{2 \\\\pi T} \\\\right) }{\\\\sinh \\\\left( \\\\pi \\\\frac{e^* V}{2 \\\\pi T} - i \\\\pi (\\\\kappa _c^2-\\\\kappa _n^2) \\\\right) }\\\\frac{\\\\Gamma \\\\left( \\\\kappa _c^2 + \\\\kappa _n^2 + i \\\\frac{e^* V}{2\\\\pi T} \\\\right) }{\\\\Gamma (2 \\\\kappa _n^2) \\\\Gamma \\\\left( 1 + \\\\kappa _c^2 - \\\\kappa _n^2 + i \\\\frac{e^* V}{2\\\\pi T} \\\\right)}\\\\right\\\\rbrace .$ In this case, in the limit $T \\\\gg |e^* V|$ one obtains $& j_{\\\\kappa _c^2,\\\\kappa _n^2}(L,L;V,T) =- 2 \\\\pi (e^*)^2 V \\\\tau \\\\left( 2 \\\\pi T \\\\tau \\\\right)^{2(\\\\kappa _c^2 + \\\\kappa _n^2 - 1)}\\\\times \\\\nonumber \\\\\\\\ &\\\\left\\\\lbrace e^{- 2 \\\\pi \\\\kappa _n^2 T (L/v_c + L/v_n) }B(\\\\kappa _n^2+\\\\kappa _c^2,\\\\kappa _n^2-\\\\kappa _c^2)+e^{- i e^* V (L/v_c + L/v_n) - 2 \\\\pi \\\\kappa _c^2 T (L/v_c + L/v_n)}B(\\\\kappa _c^2+\\\\kappa _n^2,\\\\kappa _c^2-\\\\kappa _n^2) \\\\right\\\\rbrace ,$ and in the limit $T \\\\ll |e^* V|$ we have $& j_{\\\\kappa _c^2,\\\\kappa _n^2}(L,L;V,T) =2 \\\\pi i e^* \\\\tau \\\\left( 2 \\\\pi T \\\\tau \\\\right)^{2(\\\\kappa _c^2 + \\\\kappa _n^2)-1} e^{i \\\\pi (\\\\kappa _c^2-\\\\kappa _n^2) \\\\text{sgn}(e^* V)}\\\\times \\\\nonumber \\\\\\\\ &\\\\left\\\\lbrace e^{- 2 \\\\pi \\\\kappa _n^2 T (L/v_c + L/v_n) }\\\\frac{1}{\\\\Gamma (2 \\\\kappa _c^2)}\\\\left( - i \\\\frac{e^* V}{2\\\\pi T} \\\\right)^{2 \\\\kappa _c^2 - 1}-e^{- i e^* V (L/v_c + L/v_n) - 2 \\\\pi \\\\kappa _c^2 T (L/v_c + L/v_n)}\\\\frac{1}{\\\\Gamma (2 \\\\kappa _n^2)}\\\\left( i \\\\frac{e^* V}{2\\\\pi T} \\\\right)^{2\\\\kappa _n^2-1}\\\\right\\\\rbrace .$ .\",\n \"For $L/v_c + L/v_n \\\\ll 1/T, 1/|e^* V|$ we find $j_{\\\\kappa _c^2,\\\\kappa _n^2}(L,L;V,T) =2 i e^* \\\\tau \\\\left( 2 \\\\pi T \\\\tau \\\\right)^{2(\\\\kappa _c^2 + \\\\kappa _n^2)-1}& \\\\left\\\\lbrace \\\\frac{\\\\sin (2 \\\\pi \\\\kappa _c^2)\\\\sinh \\\\left( \\\\pi \\\\frac{e^* V}{2 \\\\pi T} \\\\right) }{\\\\sinh \\\\left( \\\\pi \\\\frac{e^* V}{2 \\\\pi T} - i \\\\pi (\\\\kappa _c^2-\\\\kappa _n^2) \\\\right) }\\\\frac{\\\\Gamma (1 - 2 \\\\kappa _c^2 - 2 \\\\kappa _n^2) \\\\Gamma \\\\left( \\\\kappa _c^2 + \\\\kappa _n^2 - i \\\\frac{e^* V}{2\\\\pi T} \\\\right) }{ \\\\Gamma \\\\left( 1 - \\\\kappa _c^2 - \\\\kappa _n^2 - i \\\\frac{e^* V}{2\\\\pi T} \\\\right)}\\\\right.\\\\nonumber \\\\\\\\ &\\\\left.-\\\\frac{\\\\sin (2 \\\\pi \\\\kappa _n^2) \\\\sinh \\\\left( \\\\pi \\\\frac{e^* V}{2 \\\\pi T} \\\\right) }{\\\\sinh \\\\left( \\\\pi \\\\frac{e^* V}{2 \\\\pi T} - i \\\\pi (\\\\kappa _c^2-\\\\kappa _n^2) \\\\right) }\\\\frac{\\\\Gamma (1 - 2 \\\\kappa _c^2 - 2 \\\\kappa _n^2) \\\\Gamma \\\\left( \\\\kappa _c^2 + \\\\kappa _n^2 + i \\\\frac{e^* V}{2\\\\pi T} \\\\right) }{ \\\\Gamma \\\\left( 1 - \\\\kappa _c^2 - \\\\kappa _n^2 + i \\\\frac{e^* V}{2\\\\pi T} \\\\right)}\\\\right\\\\rbrace ,$ which can be shown {using the identity $\\\\Gamma (z)\\\\Gamma (1-z) = \\\\pi /\\\\sin (\\\\pi z)$ [39]} to reduce to Eqs.\",\n \"(REF )–(REF ), as it should.\",\n \"Finally, for $1/|e^* V| \\\\ll L/v_c + L/v_n \\\\ll 1/T$ we have: $ & j_{\\\\kappa _c^2,\\\\kappa _n^2}(L,L;V,T) =2 \\\\pi i e^* \\\\tau \\\\left( 2 \\\\pi T \\\\tau \\\\right)^{2(\\\\kappa _c^2 + \\\\kappa _n^2)-1} e^{i \\\\pi (\\\\kappa _c^2-\\\\kappa _n^2) \\\\text{sgn}(e^* V)}\\\\times \\\\nonumber \\\\\\\\ &\\\\left\\\\lbrace \\\\frac{1}{\\\\Gamma (2 \\\\kappa _c^2)}\\\\left( - i \\\\frac{e^* V}{2\\\\pi T} \\\\right)^{2\\\\kappa _c^2-1}\\\\left[ 2\\\\pi T (L/v_c + L/v_n) \\\\right]^{-2\\\\kappa _n^2}-e^{- i e^* V (L/v_c + L/v_n)}\\\\frac{1}{\\\\Gamma (2 \\\\kappa _n^2)}\\\\left( i \\\\frac{e^* V}{2\\\\pi T} \\\\right)^{2\\\\kappa _n^2-1}\\\\left[ 2\\\\pi T (L/v_c + L/v_n) \\\\right]^{-2\\\\kappa _c^2}\\\\right\\\\rbrace .$ From the above results, we can determine the visibility of the interference pattern as function of temperature and source-drain bias, using Eqs.\",\n \"(3)–(4).\",\n \"For $|\\\\gamma _a|=|\\\\gamma _b|$ it is the ratio $|j(L_d,L_u;V,T)/j(0,0;V,T)|^3$ , reflecting the fact that a unit charge detected at D$_2$ [Fig.\",\n \"1(a)] involves the tunneling of three $e/3$ quasi-particles, each subject to a different effective AB flux.\",\n \"The results are displayed in Table.\",\n \"II and Fig.\",\n \"2 of the main text.\"\n ],\n [\n \"Some qualitative features of the visibility\",\n \"There are some remarkable insights that follow from the results of the previous Sections.\",\n \"First and foremost, the interference signal is exponentially-suppressed with increasing temperature as soon as $\\\\min (L_d,L_u)(1/v_c+1/v_n) \\\\gg 1/T$ .\",\n \"This is true even even if $T \\\\ll |e^* V|$ (!).\",\n \"At the same time, increasing the bias voltage does not lead to exponential suppression even when $|e^* V|$ is larger than any other scale, including the temperature.\",\n \"This can be intuitively understood by thinking about free fermions with velocity $v$ : at zero temperature the voltage defines a window of allowed energies, and hence wavevectors, for tunneling.\",\n \"Since the window is sharp, the corresponding wavepackets decay in real space at large distances $x$ only as $e^{i V x/v}/x$ , hence there is no exponential suppression of the interference.\",\n \"On the other hand, introducing a finite temperature amounts to smearing the sharp energy window.\",\n \"Roughly, this can be thought of as having a distribution of voltage values around some average $V_0$ .\",\n \"Averaging $e^{iV x/v}$ with respect to this distribution will leave us with an oscillatory factor $e^{i V_0 x/v}$ multiplied by a suppression factor.\",\n \"The latter is a function of $T x/v$ , and decays exponentially when $T x/v \\\\gg 1$ , independently of the value of $V_0$ .\",\n \"Another noteworthy feature is that the two terms in Eq.\",\n \"(REF ), and correspondingly in Eqs.\",\n \"(REF ) and (REF ), originate from domains in the time integration where either the charge or the neutral mode overlap is maximal, but not both (which is the essence of the dephasing mechanism discussed in this work), cf.\",\n \"Fig.\",\n \"REF (b).\",\n \"These two terms have a relative phase between them.\",\n \"This gives rise to oscillations of the visibility as function of the voltage when the latter is high and the temperature small [so that the the two terms in Eqs.\",\n \"(REF ) and (REF ) are comparable], the case described by Eqs.\",\n \"(REF ) and (REF ).\",\n \"This leads to the last line of Table.\",\n \"II and the lobe structure seen in Fig.\",\n \"2 of the main text.\",\n \"Figure: An almost-closed QH MZI (compare with the almost-open case, Fig.\",\n \"1 of the main text).Shown are the downstream chiral edge modes, sources, drains, and the tunneling bridges.\",\n \"The schematic equivalent geometry is essentially the same as the one depicted in Fig.\",\n \"1(b)–(c),with the upper and lower edges corresponding, respectively, to the inner and outer edges of the lower edge + island depicted here, instead of the island and lower edges in Fig.\",\n \"1(a).\"\n ],\n [\n \"The case of electron tunneling\",\n \"In the limit of strong tunneling bridges, the interfering paths are dominated by electron tunneling.\",\n \"It is easy to obtain results for the correlation functions by noting that $L_d$ and $L_u$ are interchanged and the AB phase is flipped as compared with quasiparticle tunneling (cf. Fig.\",\n \"REF ).\",\n \"In addition, the values of $\\\\kappa _c$ and $\\\\kappa _n$ are now taken from the fourth, rather than the first row of Table I.\",\n \"Finally, Eqs.\",\n \"(3)–(4) are replaced by $ I_{D2}(V,T) =\\\\left(\\\\left| \\\\gamma _a \\\\right|^2 + \\\\left| \\\\gamma _b \\\\right|^2 \\\\right) j(0,0;V,T)+ \\\\gamma _a \\\\gamma _b^* e^{-2 \\\\pi i \\\\Phi _{AB}/\\\\Phi _0 } j(L_u,L_d;V,T)+ \\\\text{c.c.\",\n \"}$ Thus, when $|\\\\gamma _a|=|\\\\gamma _b|$ the visibility is $|j(L_u,L_d;V,T)/j(0,0;V,T)|$ .\",\n \"The quantities $j(L_u,L_d;V,T)$ and $j(0,0;V,T)$ were calculated in Secs. –.\",\n \"The resulting behavior is summarized in Table.\",\n \"II of the main text.\"\n ]\n]"},"id":{"kind":"string","value":"1605.06060"}}},{"rowIdx":1866681,"cells":{"text":{"kind":"list like","value":[["Consensus+Innovations Distributed Kalman Filter with Optimized Gains"],["Abstract In this paper, we address the distributed filtering and prediction of time-varying random fields represented by linear time-invariant (LTI) dynamical systems.","The field is observed by a sparsely connected network of agents/sensors collaborating among themselves.","We develop a Kalman filter type consensus+innovations distributed linear estimator of the dynamic field termed as Consensus+Innovations Kalman Filter.","We analyze the convergence properties of this distributed estimator.","We prove that the mean-squared error of the estimator asymptotically converges if the degree of instability of the field dynamics is within a pre-specified threshold defined as tracking capacity of the estimator.","The tracking capacity is a function of the local observation models and the agent communication network.","We design the optimal consensus and innovation gain matrices yielding distributed estimates with minimized mean-squared error.","Through numerical evaluations, we show that, the distributed estimator with optimal gains converges faster and with approximately 3dB better mean-squared error performance than previous distributed estimators."],["Introduction","For decades, the Kalman-Bucy filter [1], [2] has played a key role in estimation, detection, or prediction of time-varying noisy signals.","The Kalman filter is found in a wide variety of applications ranging from problems in navigation to environmental studies, computer vision to bioengineering, signal processing to econometrics.","More recently, algorithms inspired by the Kalman filter have been applied to estimate random fields monitored by a network of sensors.","In these problems, we distinguish two distinct layers: (a) the physical layer of the time-varying random field; and (b) the cyber layer of sensors observing the field.","A centralized approach to field estimation poses several challenges.","It requires that all sensors communicate their measurements to a centralized fusion center.","This is fragile to central node failure and severely taxes computationally the fusion center.","Moreover, it also requires excessive communication bandwidth to and from the fusion center.","Hence, the centralized approach is inelastic to estimation of large-scale time-varying random fields, like, when estimating temperature, rainfall, or wind-speed over large geographical areas [3], [4].","In [5], we proposed the Distributed Information Kalman Filter (DIKF) that is a distributed estimator of time-varying random fields consisting of two substructures.","The first is the Dynamic Consensus on Pseudo-Observations (DCPO), a distributed estimator of the global average of the pseudo-observations (modified versions of the observations) of the agents.","The second substructure uses these average estimates of pseudo-observations to estimate the time-varying random field.","In this paper, we develop a distributed Kalman filter like estimator, the Consensus$+$ Innovations Kalman Filter (CIKF), that instead of using the pseudo-observations uses distributed estimates of the pseudo-state (modified version of a state) to estimate the field.","We show how to design optimally the gain matrices of the CIKF.","We prove that the CIKF converges in the mean-squared error (MSE) sense when the degree of instability of the dynamics of the random field is within the network tracking capacity [6], a threshold determined by the cyber network connectivity and the local observation models.","Figure: Time-scales of operation of dynamics, sensing and communications.We review related prior research on distributed estimation of time-varying random fields.","We classify prior work into two categories based on the time-scales of operation: (a) two time-scale and (b) single time-scale.","In two-time scale distributed estimators, see Fig REF , agents exchange their information multiple number of times between each dynamics/observations time-scale [7], [8], [9], [10], [11], [12], [13], [14], [15], so that average consensus occurs between observations.","In contrast, in single time-scale approaches [6], [16], [17], [18], [19], [20], [21], [22], [23], [24], [5], the agents collaborate with their neighbors only once in between each dynamics/observation evolution.","In other words, the dynamics, observation, and communication follow the same time scale as depicted in Fig REF .","The two time-scale approach demands fast communication between agents.","In most practical applications, this is not true.","Further, references [13], [14] developed distributed state estimators assuming local observability of the dynamic state in the physical layer.","Such assumption is not feasible in large-scale systems.","References [25], [26] assume a complete cyber network, which is not scalable.","We are interested in a single time-scale approach.","We consider that the time-varying system is not locally observable at each agent, and we assume that the communication network in the cyber layer is sparsely connected.","In the single time-scale category, references [19], [20] propose distributed Kalman filters where the agents communicate among themselves using the Gossip protocol [27].","Although Gossip filters require very low communication bandwidth, their MSE is higher and their convergence rate is lower than the consensus$+$ innovations based distributed approaches.","References [21], [22], [24] introduced consensus$+$ innovations type distributed estimator for parameter (static state) estimation.","This approach is extended to estimating time-varying random states in [6], [16], [17], [18].","Distributed consensus$+$ innovations dynamic state estimators converge in MSE sense if their degree of instability of the state dynamics is below the network tracking capacity, see [6].","In this paper, we derive the tracking capacity for the CIKF.","Its tracking capacity is a function of the local observation models and of the agents communication connectivity.","The single time-scale consensus$+$ innovations distributed estimators that we introduced in [28], [29], [30], [31] run a companion filter to estimate the global average of the pseudo-innovations, a modified version of the innovations.","The DIKF, we proposed in [5], uses averaged pseudo-observations (linearly transformed observations) rather than pseudo-innovations.","In contrast, this paper uses estimates of the pseudo-state, a linear transformation of the dynamic state.","In centralized information filter, the pseudo-state is directly available from the observations.","In the distributed setting, since not all the observations are available to the local sensors, the CIKF has to distributedly estimate the pseudo-state through a consensus step.","Using pseudo-state rather than pseudo-innovations [28], [29], [30], [31] or pseudo-observations [5] leads to significant better performance as we show here.","The MSE of the CIKF is lower than that of the distributed estimators in [19], [20], [28], [29], [30], [31], [5].","Developing distributed estimators for time-varying random fields (distributed Kalman filter) has gained considerable attention over the last few years.","The goal has been to achieve MSE performance as close as possible to the optimal centralized Kalman filter.","We show in this paper that the CIKF converges to a bounded MSE solution requiring minimal assumptions, namely global detectability and connected network, and not requiring the additional distributed observability assumption, as needed by the DIKF.","The distributed optimal time-varying field estimation is, in general, NP-hard.","The distributed parameter estimation [21], [32] have shown asymptotic optimality, in the sense that the distributed parameter estimator is asymptotically unbiased, consistent, and efficient converging at the same rate as the centralized optimal estimator.","However, estimation of time-varying random fields adds another degree of complication since, while information diffuses through the network, the field itself evolves.","So, this lag causes a gap in performance between distributed and centralized field estimators.","Numerical simulations show that the proposed CIKF improves the performance by 3dB over the DIKF, reducing by half the gap to the centralized (optimal) Kalman filter, while showing a faster convergence rate than the DIKF.","These improvements significantly distinguish the CIKF from the DIKF.","The rest of the paper is organized as follows.","We describe the three aspects of the problem setup in Section .","Section introduces the pseudo-state and presents the proposed optimal gain distributed Kalman filter (CIKF).","In Section , we analyze the dynamics of the error processes and their covariances.","Section includes the analysis of the tracking capacity of the proposed CIKF.","We design the optimal gain matrices to obtain distributed estimates in Section .","Numerical simulations are in Section .","We present the concluding remarks in Section .","Proofs of the proposition, lemmas and theorems are in the Appendix REF , Appendix REF -REF , and Appendix REF -REF , respectively."],["System, Observation, and Communication","The distributed estimation framework consists of three components: dynamical system, local observation, and neighborhood communications.","These three parts include two layers: the physical layer and the cyber layer.","For the sake of simplicity, we motivate the model with the example of a time-varying temperature field over a large geographical area monitored by a sensor network."],["Physical layer: dynamical system","Consider a time-varying temperature field distributed over a large geographical area, as shown in Fig REF .","A first-order approximation and discretization of the temperature field provide spatio-temporal discretized temperature variables $x^j_i, j = 1, \\cdots , M$ , of $M$ sites at discrete time indexes $i = 1, \\cdots , \\;$ .","We stack the $M$ field variables in a temperature state vector ${\\mathbf {x}}_i = \\begin{bmatrix} x^1_i, \\cdots , x^M_i \\end{bmatrix}^T \\in \\mathbb {R}^{M}$ .","The evolution of the time-varying temperature field, ${\\mathbf {x}}_i$ , can be represented by a linear time-invariant (LTI) dynamical systemEven though the random field ${\\mathbf {x}}_i$ is time-varying, the system is time-invariant because the field dynamics matrix $A$ is time-invariant.","For a time-varying dynamical system, the field dynamics matrix $A_i$ would be a function of time.","${\\mathbf {x}}_{i+1} = A {\\mathbf {x}}_i + {\\mathbf {v}}_i,$ where the first-order dynamics matrix $A \\in \\mathbb {R}^{M \\times M}$ contains the coupling effects between the $M$ temperature variables, and the residual ${\\mathbf {v}}_i = \\begin{bmatrix} v^1_i, \\cdots , v^M_i \\end{bmatrix}^T \\in \\mathbb {R}^{M}$ is the system noise driving the dynamical temperature fieldThe temperature field is a time-varying random field.","Although the field is time-varying, it is generated by a linear time-invariant system with perturbations.. At each of the $M$ sites, the input noise $v^j_i, j = 1, \\cdots , M$ accounts for the deviations in the temperature after the overall field dynamics.","The field dynamics $A$ incorporates the sparsity pattern and connectivity of the physical layer consisting of the dynamical system (REF )."],["Cyber layer: local observations","The physical layer consisting of the field dynamics (REF ) is observed by a cyber layer consisting of a network of $N$ agents (sensors).","In Fig REF , we see that there are $N$ sensors, where each sensor observes the temperatures of only a few sites.","Denote the number of sites observed by sensor $n$ by $M_n, M_n \\ll M$ , and its measurements at time $i$ by $z_i^n \\in \\mathbb {R}^{M_n}$ .","The observations of the agents in the cyber layer can be represented by a linear model: ${\\mathbf {z}}_i^n = H_n {\\mathbf {x}}_i + {\\mathbf {r}}_i^n, \\qquad \\qquad n =1, \\hdots , N,$ where, the observation matrix $H_n \\in \\mathbb {R}^{M_n \\times M}$ contains the observation pattern and strength information, and the observation noise $r_i^n \\in \\mathbb {R}^{M_n}$ reflects the inaccuracy in measurements due to sensor precision, high frequency fluctuations in temperature, and other unavoidable constraints.","To illustrate, consider that sensor 1 is observing the temperature of 3 sites $\\lbrace x^1_i, x^2_i, x^3_i \\rbrace $ , i.e., $M_1 = 3$ .","An example snapshot of the observation model of agent $n$ is $\\underbrace{\\begin{bmatrix} z^{11}_i \\\\ z^{12}_i \\\\ z^{13}_i \\end{bmatrix}}_{{\\mathbf {z}}^1_i} &=\\!\\!","\\underbrace{\\begin{bmatrix} 1 & 0 & 0 & \\hdots & 0 \\\\ 0 & 5 & 0 & \\hdots & 0 \\\\ 0 & 0 & 6 & \\hdots & 0 \\end{bmatrix}}_{H_1}\\!\\!","\\underbrace{\\begin{bmatrix} x^1_i \\\\ \\vdots \\\\ x^{\\scriptsize {M}}_i \\end{bmatrix}}_{{\\mathbf {x}}_i}\\!+\\!","\\underbrace{\\begin{bmatrix} r^{11}_i \\\\ r^{12}_i \\\\ r^{13}_i \\end{bmatrix}}_{{\\mathbf {r}}^1_i}= \\begin{bmatrix} x^1_i \\\\ 5x^2_i \\\\ 6x^3_i \\end{bmatrix} \\!+\\!","\\begin{bmatrix} r^{11}_i \\\\ r^{12}_i \\\\ r^{13}_i \\end{bmatrix}.$ Now, for the ease of analysis, we aggregate the noisy local temperature measurements, ${\\mathbf {z}}^1_i, \\cdots {\\mathbf {z}}^N_i$ , of all the $N$ agents in a global observation vector, ${\\mathbf {z}}_i \\in \\mathbb {R}^{{\\sum _{n=1}^{N} M_n}}$ : $\\underbrace{\\begin{bmatrix} {\\mathbf {z}}_i^1\\\\ \\vdots \\\\ {\\mathbf {z}}_i^N\\end{bmatrix}}_{{\\mathbf {z}}_i} &=&\\underbrace{\\begin{bmatrix}H_1 \\\\ \\vdots \\\\ H_N \\end{bmatrix}}_{H} {\\mathbf {x}}_i +\\underbrace{\\begin{bmatrix} {\\mathbf {r}}_i^1 \\\\ \\vdots \\\\ {\\mathbf {r}}_i^N\\end{bmatrix}}_{{\\mathbf {r}}_i},$ where, the global observation matrix is $H\\in \\mathbb {R}^{\\sum _{n=1}^N M_n \\times M}$ and the stacked measurement noise is $r_i\\in \\mathbb {R}^{{\\sum _{n=1}^{N} M_n}}$ .","Note that, in general, the temperature measurement model is non-linear.","For non-linear cases, refer to distributed particle filter in [33] and the references cited therein.","Here we perform a first-order approximation to obtain a linear observation sequence."],["Cyber layer: neighborhood communication","In the cyber layer, the agents exchange their temperature readings or current estimates with their neighbors.","In many applications, to reduce communications costs, neighbors communicate only with their geographically nearest agents as shown in Fig REF .","Formally, the agent communication network is defined by a simple (no self-loops nor multiple edges), undirected, connected graph $\\mathcal {G = (V,E)}$ , where $\\mathcal {V} $ is the set of sensors (nodes or agents) and $\\mathcal {E}$ is the set of local communication channels (edges or links) among the agents.","The open $\\Omega _n$ and closed $\\overline{\\Omega }_n$ neighborhoods of agent $n$ are: $\\Omega _n &= \\lbrace l|(n,l) \\in \\mathcal {E}\\rbrace .","\\nonumber \\\\\\overline{\\Omega }_n &= \\lbrace n\\rbrace \\cup \\lbrace l|(n,l) \\in \\mathcal {E}\\rbrace .","\\nonumber $ In Fig REF the open and closed neighborhoods of agent 1 are $\\Omega _1 = \\lbrace 2,3\\rbrace $ and $\\overline{\\Omega }_1 = \\lbrace 1, 2,3\\rbrace $ , respectively.","The Laplacian matrix of $\\mathcal {G}$ is denoted by $L$ .","The eigenvalues of the positive semi-definite matrix $L$ are ordered as $ 0 = \\lambda _1(L) \\le \\lambda _2(L) \\le .",".",".","\\le \\lambda _N(L)$ .","For details on graphs refer to [34].","The communication network is sparse and time-invariant."],["Modeling assumptions","We make the following assumptions.","Assumption 1 (Gaussian processes) The system noise, ${\\mathbf {v}}_i$ , the observation noise, ${\\mathbf {r}}_i$ , and the initial condition of the system, ${\\mathbf {x}}_0$ , are Gaussian sequences, with ${{\\mathbf {v}}_i \\sim \\mathcal {N}(0,V)}, \\;\\;{{\\mathbf {r}}^n_i \\sim \\mathcal {N}(0,R_n)}, \\;\\;{\\mathbf {x}}_0 \\sim \\mathcal {N}(\\bar{{\\mathbf {x}}}_0,\\Sigma _0),$ where, $V \\in \\mathbb {R}^{{M} \\times {M}} $ , $R_n \\in \\mathbb {R}^{{M_n} \\times {M_n}}$ and $\\Sigma _0 \\in \\mathbb {R}^{{M} \\times {M}}$ are the corresponding covariance matrices.","The noise covariance matrix $R$ of the global noise vector ${\\mathbf {v}}_i$ in (REF ) is block-diagonal, i.e., $R=\\textrm {blockdiag} \\lbrace R_1, \\ldots , R_N \\rbrace $ , and positive-definite, i.e., $R > 0$ .","Assumption 2 (Uncorrelated sequences) The system noise, the observation noise, and the initial condition: $\\lbrace \\lbrace {\\mathbf {v}}_i\\rbrace _i, \\lbrace {\\mathbf {r}}_i\\rbrace _i, {\\mathbf {x}}_0\\rbrace _{i \\ge 0}$ are uncorrelated random vector sequences.","Assumption 3 (Prior information) Each agent in the cyber layer knows the system dynamics model, $A$ and $V$ , the initial condition statistics, $\\bar{x}_0$ and $\\Sigma _0$ , the parameters of the observation model, $H$ and $R$ , and the communication network model, $\\mathcal {G}$ .","In large-scale system applications, the dynamics, observation, and network Laplacian matrices, $A$ , $H_n$ , and $L$ , are sparse, with $M_n \\ll M$ , and the agents communicate with only a few of their neighbors, $|\\Omega _n| \\ll N, \\forall n$ .","In the dynamics (REF ) and observations (REF ), we assume that there is no deterministic input.","The results are readily extended if there is a known deterministic input."],["Centralized Information filter","Although not practical in the context of the problem we study, we use the centralized information filter to benchmark our results on distributed estimator.","In a centralized scheme, all the agents in the cyber layer communicate their measurements to a central fusion center, as depicted in Fig REF .","The fusion center performs all needed computation tasks.","Refer to [5], [35] for the filter, gain, and error equations of the centralized information filter.","Figure: Centralized estimator"],["Distributed filtering and prediction","This section considers our single time-scale distributed solution.","We start with the introduction and derivation of the key components of our distributed estimator and then present our distributed field estimator."],["Pseudo-state model","In a centralized information filter [35], all the observations are converted into pseudo-observations [5] to obtain the optimal estimates.","Following (REF ), the pseudo-observation $\\widetilde{{\\mathbf {z}}}^n_i$ of agent $n$ is $\\widetilde{{\\mathbf {z}}}^n_i &= H^T_n R_n^{-1} {\\mathbf {z}}^n_i = \\overline{H}_n {\\mathbf {x}}_i + H^T_n R_n^{-1} {\\mathbf {r}}^n_i, \\quad \\\\\\text{where,} \\qquad \\overline{H}_n &= H^T_n R_n^{-1} H_n.$ The centralized information filter computes the sum, $\\overline{{\\mathbf {z}}}_i$ of all the pseudo-observations $\\overline{{\\mathbf {z}}}_i &= \\sum _{n=1}^{N}\\widetilde{{\\mathbf {z}}}^n_i = G{\\mathbf {x}}_i + H^T R^{-1} {\\mathbf {r}}_i \\qquad \\\\\\text{where,} \\qquad \\qquad G &= \\sum _{n=1}^{N} H^T_n R_n^{-1} H_n = \\sum _{n=1}^{N} \\overline{H}_n .$ The aggregated pseudo-observation, $\\overline{{\\mathbf {z}}}_i$ , is the key term in the centralized filter.","It provides the innovations term in the filter updates enabling the filter to converge with minimum MSE estimates.","However, in the distributed solution, each agent $n$ does not have access to all the pseudo-observations; instead it can only communicate with its neighbors.","To address this issue, in [5] we introduced a dynamic consensus algorithm to compute the distributed estimates of the averaged pseudo-observations, $\\overline{{\\mathbf {z}}}_i$ , at each agent.","In (REF ), we note that the crucial term is $G{\\mathbf {x}}_i$ which carries the information of the dynamic state, ${\\mathbf {x}}_i$ ; and the second term in $\\overline{{\\mathbf {z}}}_i$ in (REF ) is noise.","We refer to it as the pseudo-state, ${\\mathbf {y}}_i$ , ${\\mathbf {y}}_i = G{\\mathbf {x}}_i.$ The pseudo-state, ${\\mathbf {y}}_i$ , is also a random field whose time dynamics can be represented by a discrete-time linear dynamical system.","The pseudo-observations $\\widetilde{{\\mathbf {z}}}^n_i$ are its linear measurements.","We summarize the state-space model for the pseudo-state in the following proposition.","See Appendix REF for the details of the proof.","Proposition 1 The dynamics and observations of the pseudo-state ${\\mathbf {y}}_i$ are: ${\\mathbf {y}}_{i+1} &= \\widetilde{A}{\\mathbf {y}}_i + G{\\mathbf {v}}_i + \\check{A}{\\mathbf {x}}_i \\\\\\widetilde{{\\mathbf {z}}}^n_i &= \\widetilde{H}_n{\\mathbf {y}}_i + H^T_n R_n^{-1} {\\mathbf {r}}^n_i + \\check{H}_n {\\mathbf {x}}_i.$ The pseudo-dynamics matrix $\\widetilde{A}$ , pseudo-observations matrix $\\widetilde{H}_n$ , and the matrices $\\check{A}, \\check{H}_n$ , and $\\widetilde{I}$ at agent $n$ are: $\\widetilde{A} &= G A G^{\\dagger } \\\\\\widetilde{H}_n &= H^T_n R_n^{-1} H_n G^{\\dagger } = \\overline{H}_n G^{\\dagger }\\\\\\check{A} &= GA\\widetilde{I} \\\\\\check{H}_n &= H^T_n R_n^{-1} H_n \\widetilde{I} \\\\\\widetilde{I} &= I - G^{\\dagger }G,$ where, $G^{\\dagger }$ denotes the Moore-Penrose pseudo-inverse of G. In [5], the distributed information Kalman filter (DIKF) assumes distributed observability, i.e., it considers the case where $G$ is invertible.","Under this assumption, $G^{\\dagger } = G^{-1}$ and $\\widetilde{I} = 0$ .","In this paper we relax the requirement of invertibility of $G$ , proposing a distributed estimator for general dynamics-observation models under the assumption of global detectability.","In most cases $\\widetilde{I}$ is low-rank.","In (REF ), the term $\\left( G{\\mathbf {v}}_i~+~\\check{A}{\\mathbf {x}}_i = \\xi _i, \\text{say} \\right)$ can be interpreted as the pseudo-state input noise, which follows Gauss dynamics $\\xi _i &\\sim \\mathcal {N} \\left( \\check{A} \\overline{{\\mathbf {x}}}_i \\; , \\; GVG + \\check{A} \\Sigma _i \\check{A}^T\\right) \\qquad \\qquad \\\\\\text{where,} \\qquad \\Sigma _{i} &= \\mathbb {E}\\left[ ({\\mathbf {x}}_i - \\overline{{\\mathbf {x}}}_i)({\\mathbf {x}}_i - \\overline{{\\mathbf {x}}}_i)^T \\right].$ Similarly, in (), the term $\\big ( \\delta ^n_i = H^T_nR_n^{-1}{\\mathbf {r}}^n_i + \\check{H}_n {\\mathbf {x}}_i \\big )$ is the pseudo-state observation noise at agent $n$ , which is Gaussian $\\delta ^n_i \\sim \\mathcal {N} \\Big ( \\check{H}_n \\overline{{\\mathbf {x}}}_i , \\overline{H}_n + \\check{H}_n \\Sigma _{i} \\check{H}_n \\Big ).$ For the ease of analysis, we express the pseudo-state observation model in vector form by $\\widetilde{{\\mathbf {z}}}_i \\in \\mathbb {R}^{{\\sum _{n=1}^{N} M_n}}$ , the aggregate of the noisy local temperature measurements, $\\widetilde{{\\mathbf {z}}}^1_i, \\cdots \\widetilde{{\\mathbf {z}}}^N_i$ , of all the agents, $\\underbrace{\\begin{bmatrix} \\widetilde{{\\mathbf {z}}}_i^1\\\\ \\vdots \\\\ \\widetilde{{\\mathbf {z}}}_i^N\\end{bmatrix}}_{\\widetilde{{\\mathbf {z}}}_i} &=&\\underbrace{\\begin{bmatrix} \\widetilde{H}_1 \\\\ \\vdots \\\\ \\widetilde{H}_N \\end{bmatrix}}_{\\widetilde{H}} {\\mathbf {y}}_i + D_{H}^T R^{-1} {\\mathbf {r}}_i + \\underbrace{\\begin{bmatrix} \\check{H}_1 \\\\ \\vdots \\\\ \\check{H}_N \\end{bmatrix}}_{\\check{H}} {\\mathbf {x}}_i,$ where, the matrices $\\widetilde{H}, \\check{H} \\in \\mathbb {R}^{NM \\times M}$ , and, $D_{H} = \\text{blockdiag} \\lbrace H_1, \\cdots , H_N \\rbrace $ .","We have established in (REF )-() the state-space dynamics and observations model of the pseudo-state, ${\\mathbf {y}}_i$ .","The structure of (REF )-() is similar to the dynamics and observations model (REF )-(REF ) of the random field, ${\\mathbf {x}}_i$ .","In the following subsection, we develop a distributed estimator with optimized gains to obtain unbiased estimates of the pseudo-state, ${\\mathbf {y}}_i$ , and of the state, ${\\mathbf {x}}_i$ , at each agent with minimized mean-squared error.","The pseudo-state ${\\mathbf {y}}_i$ is a noise-reduced version of the sum $\\overline{{\\mathbf {z}}_i}$ of the local pseudo-observations.","We demonstrate in Section that, by sharing distributed estimates of pseudo-states, the CIKF achieves 3dB better MSE error performance than the DIKF [5], which shares distributed estimates of the summed pseudo-observations $\\overline{{\\mathbf {z}}}_i$ ."],["At time $i$ , denote the $n^{\\text{th}}$ agent's distributed filter and prediction estimates of the state ${\\mathbf {x}}_i$ by $\\widehat{{\\mathbf {x}}}_{i|i}^n$ and $\\widehat{{\\mathbf {x}}}_{i+1|i}^n$ respectively.","Similarly, its distributed filter and prediction estimates of the pseudo-state ${\\mathbf {y}}_i$ are denoted by $\\widehat{{\\mathbf {y}}}_{i|i}^n$ and $\\widehat{{\\mathbf {y}}}_{i+1|i}^n$ .","At any time $i$ , each agent $n$ has access to its own pseudo-observation $\\widetilde{z}^n_i$ and receives the prediction state pseudo-state estimates, $\\widehat{{\\mathbf {y}}}_{i|i-1}^l$ , $l \\in \\Omega _n$ , of its neighbors at the previous time $i-1$ .","Under this setup, the minimized MSE filter and prediction estimates are the conditional means, $&\\widehat{{\\mathbf {y}}}_{i|i}^n \\!=\\!","\\mathbb {E} \\!\\left[ {\\mathbf {y}}_i \\;|\\; \\widetilde{{\\mathbf {z}}}^n_i, \\lbrace \\widehat{{\\mathbf {y}}}_{i|i-1}^l \\rbrace _{l \\in \\overline{\\Omega }_n} \\right] \\\\&\\widehat{{\\mathbf {x}}}_{i|i}^n \\!=\\!","\\mathbb {E} \\left[ {\\mathbf {x}}_i \\;|\\; \\widehat{{\\mathbf {y}}}_{i|i}^n \\right] \\\\&\\!\\!\\!\\!\\!","\\widehat{{\\mathbf {y}}}_{i+1|i}^n \\!=\\!","\\mathbb {E} \\left[ {\\mathbf {y}}_{i+1} \\;|\\; \\widetilde{{\\mathbf {z}}}^n_i, \\lbrace \\widehat{{\\mathbf {y}}}_{i|i-1}^l \\rbrace _{l \\in \\overline{\\Omega }_n} \\right] \\\\&\\!\\!\\!\\!\\!","\\widehat{{\\mathbf {x}}}_{i+1|i}^n \\!=\\!","\\mathbb {E} \\left[ {\\mathbf {x}}_{i\\!+\\!1} \\;|\\; \\widehat{{\\mathbf {y}}}_{i|i}^n \\right].$ In (REF ), $\\widehat{{\\mathbf {y}}}_{i|i}^n$ is the filtered estimate of the pseudo-state ${\\mathbf {y}}_i$ given all the pseudo-observations available at agent $n$ up to time $i$ including those of its neighbors.","By the principle of recursive linear estimation, instead of storing all the pseudo-observations $ \\lbrace \\lbrace \\widetilde{{\\mathbf {z}}}^n_t \\rbrace _{t = 0, \\cdots , i}, \\lbrace \\widetilde{{\\mathbf {z}}}^{n_1}_t \\rbrace ^{n_1 \\in \\Omega _n}_{t = 0, \\cdots , i-1}, \\lbrace \\widetilde{{\\mathbf {z}}}^{n_2}_t \\rbrace ^{n_2 \\in \\Omega _{n1}, \\forall n_1}_{t = 0, \\cdots , i-2}, \\cdots \\rbrace $ , we need only the current pseudo-observation $\\widetilde{{\\mathbf {z}}}^n_i$ and the pseudo-state estimates (including those of its neighbors) from the previous time instant $\\lbrace \\widehat{{\\mathbf {y}}}_{i|i-1}^l \\rbrace _{l \\in \\Omega _n}$ .","The distributed pseudo-state estimate, $\\widehat{{\\mathbf {y}}}_{i|i}^n$ , is the expectation of the pseudo-state, ${\\mathbf {y}}_i$ , conditioned on its own pseudo-observation $\\widetilde{{\\mathbf {z}}}^n_i$ and the prior pseudo-state estimates, $\\widehat{{\\mathbf {y}}}_{i|i-1}^l, l \\in \\overline{\\Omega }_n,$ received from its neighbors.","In this paper, we formulate the distributed field estimate, $\\widehat{{\\mathbf {x}}}_{i|i}^n$ , as the expectation of the field, ${\\mathbf {x}}_i$ , conditioned on the distributed pseudo-state estimate $\\widehat{{\\mathbf {y}}}_{i|i}^n$ .","The distributed field estimates, $\\widehat{{\\mathbf {x}}}_{i|i}^n$ , are not optimal given all observations, but they are still optimal given only the pseudo-state estimates $\\widehat{{\\mathbf {y}}}_{i|i}^n$ .","The filtered estimate $\\widehat{{\\mathbf {x}}}_{i|i}^n$ of ${\\mathbf {x}}_i$ in () depends on the current pseudo-state filtered estimate $\\widehat{{\\mathbf {y}}}_{i|i}^n$ .","Similarly, the prediction estimates $\\widehat{{\\mathbf {y}}}_{i+1|i}^n$ and $\\widehat{{\\mathbf {x}}}_{i+1|i}^n$ of ${\\mathbf {y}}_{i+1}$ and ${\\mathbf {x}}_{i+1}$ in ()-(), respectively, are conditioned on the corresponding available quantities up to time $i$ .","Theorem 1 The iterative updates to compute the distributed filtered estimates with optimized gains in (REF )-() are: $ \\!\\!\\!","\\widehat{{\\mathbf {y}}}_{i|i}^n \\!=\\!","\\widehat{{\\mathbf {y}}}_{i|i-1}^n &\\!+\\!","\\sum _{l \\in \\Omega _n} \\underbrace{ \\!","B^{nl}_i \\!","\\left( \\widehat{{\\mathbf {y}}}_{i|i-1}^l \\!-\\!","\\widehat{{\\mathbf {y}}}_{i|i-1}^n \\right) }_{\\text{Consensus}} \\nonumber \\\\& \\!+\\!","B^{nn}_i \\underbrace{ \\left( \\widetilde{{\\mathbf {z}}}^n_i \\!-\\!","\\left(\\widetilde{H}_n \\widehat{{\\mathbf {y}}}_{i|i-1}^n \\!+\\!","\\check{H}_n \\widehat{{\\mathbf {x}}}_{i|i-1}^n \\right) \\right) }_{\\text{Innovations}}, \\\\ \\!\\!\\!","\\widehat{{\\mathbf {x}}}_{i|i}^n \\!=\\!","\\widehat{{\\mathbf {x}}}_{i|i-1}^n & \\!+\\!","K^{n}_i \\underbrace{ \\left( \\widehat{{\\mathbf {y}}}_{i|i}^n \\!-\\!","G \\widehat{{\\mathbf {x}}}_{i|i-1}^n \\right) }_{\\text{Innovations}}$ where, pseudo-state gain block matrix, $B_i \\in \\mathbb {R}^{MN \\times MN}$ , is $ B_i = \\begin{bmatrix} B^{nl}_i \\end{bmatrix}_{n = 1, \\cdots , N}^{l = 1, \\cdots , N}, B^{nl}_i=0 \\; \\text{if} \\; l \\notin \\Omega _n$ .","The state gain block-diagonal matrix is $K_i = \\text{blockdiag}\\lbrace K^{1}_i, \\cdots , K^{N}_i\\rbrace , K^{n}_i \\in ~\\mathbb {R}^{M \\times M}$ .","The optimized MSE prediction estimates in ()-() are: $\\widehat{{\\mathbf {y}}}_{i+1|i}^n &= \\widetilde{A}\\widehat{{\\mathbf {y}}}_{i|i}^n + \\check{A}\\widehat{{\\mathbf {x}}}_{i|i}^n, \\\\\\widehat{{\\mathbf {x}}}_{i+1|i}^n &= A \\widehat{{\\mathbf {x}}}_{i|i}^n.$ The proof of Theorem REF is presented in Appendix REF .","The update equations reflect the computation tasks of each agent $n$ at each time index $i$ .","The gain matrices, $B_i$ and $K_i$ , in (REF )-() are deterministic and can be pre-computed and saved at each agent.","We discuss the details of the design of these optimal gain matrices in Section .","With these optimal gain matrices, the Kalman type Consensus$+$ Innovations filter and prediction updates (REF )-() provide the minimized MSE distributed estimates of the dynamic states, and hence we term our solution as Consensus$+$ Innovations Kalman Filter (CIKF)."],["CIKF: Assumptions","The Consensus$+$ Innovations Kalman Filter (CIKF) achieves convergence given the following assumptions: Assumption 4 (Global detectability) The dynamic state equation (REF ) and the observations model (REF ) are globally detectable, i.e., the pair $(A, H)$ is detectable.","Assumption 5 (Connectedness) The agent communication network is connected, i.e., the algebraic connectivity $\\lambda _2(L)$ of the Laplacian matrix $L$ of the graph $\\mathcal {G}$ is strictly positive.","By Assumption REF , the state-observation model (REF )-(REF ) is globally detectable but not necessarily locally detectable, i.e., $(A, H_n), \\forall n, $ are not necessarily detectable.","Note that these two are minimal assumptions.","Assumption REF is mandatory even for a centralized system, and Assumption REF is required for consensus algorithms to converge.","Further, note that in this paper we do not consider distributed observability (invertibility of $G$ ) of the model setup, which is the strong and restrictive assumption taken in [5], [21], [22] and [36] and similar to weak detectability presented in [19]."],["CIKF: Update algorithm","In this subsection, we present the step-by-step tasks executed by each agent $n$ in the cyber layer to implement the Consensus$+$ Innovations Kalman Filter (CIKF) and thereby obtain the unbiased minimized MSE distributed estimates of the dynamic state ${\\mathbf {x}}_i$ .","Each agent $n$ runs Algorithm REF locally.","[h] Consensus$+$ Innovations Kalman Filter Input: Model parameters $A$ , $V$ , $H$ , $R$ , $G$ , $L$ , $\\overline{{\\mathbf {x}}}_0$ , $\\Sigma _0$ .","Initialize: $\\widehat{{\\mathbf {x}}}^n_{0|-1} = \\overline{{\\mathbf {x}}}_0$ , $\\widehat{{\\mathbf {y}}}^n_{0|-1} = G\\overline{{\\mathbf {x}}}_0$ .","Pre-compute: Gain matrices $B_i$ and $K_i$ using Algorithm REF .","$i \\ge 0$ $\\!\\!\\!\\!$Communications: Broadcast $\\widehat{{\\mathbf {y}}}_{i|i-1}^n$ to all neighbors $l \\in \\Omega _n$ .","Receive $\\lbrace \\widehat{{\\mathbf {y}}}_{i|i-1}^l\\rbrace _{l \\in \\Omega _n}$ from neighbors.","$\\!\\!\\!\\!$Observation: Make measurement ${\\mathbf {z}}^n_i$ of the state ${\\mathbf {x}}_i$ .","Transform ${\\mathbf {z}}^n_i$ in pseudo-observation $\\widetilde{{\\mathbf {z}}}^n_i$ using ().","$\\!\\!\\!\\!$Filter updates: Compute the estimate $\\widehat{{\\mathbf {y}}}_{i|i}^n$ of ${\\mathbf {y}}_i$ using (REF ).","Compute the estimate $\\widehat{{\\mathbf {x}}}_{i|i}^n$ of the state ${\\mathbf {x}}_i$ using ().","$\\!\\!\\!\\!$Prediction updates: Predict the estimate $\\widehat{{\\mathbf {y}}}_{i+1|i}^n$ of ${\\mathbf {y}}_{i+1}$ using (REF ).","Predict the estimate $\\widehat{{\\mathbf {x}}}_{i+1|i}^n$ of the state ${\\mathbf {x}}_{i+1}$ using ().","Later in Section , we analyze and compare the performance of CIKF with that of the distributed information Kalman filter (DKF) [5] and of the centralized Kalman filter (CKF).","The centralized filter collects measurements from all the agents in the cyber layer.","The DIKF performs distributed dynamic averaging of the pseudo-observations.","The pseudo-state can be perceived as a noise-reduced version of the global average of all the pseudo-observations or pseudo-innovations.","The distributed estimates of the pseudo-states reduce the noise of the innovations in the filtering step of field estimation and hence further improve the MSE performance of the distributed estimator.","Before going into the numerical evaluation, we do theoretical error analysis of CIKF, derive the conditions for convergence guarantees, and design the optimal consensus and innovations gains in the following sections.","We analyze the MSE performance of the CIKF and derive its error covariance matrices.","First, we define the different error processes and determine their dynamics.","Denote the filtering error processes ${\\mathbf {e}}_{i|i}^n$ and ${\\mathbf {\\epsilon }}_{i|i}^n$ of the pseudo-state and of the state at agent $n$ by ${\\mathbf {e}}_{i|i}^n &= {\\mathbf {y}}_i - \\widehat{{\\mathbf {y}}}_{i|i}^n, \\\\{\\mathbf {\\epsilon }}_{i|i}^n &= {\\mathbf {x}}_i - \\widehat{{\\mathbf {x}}}_{i|i}^n.$ Similarly, represent the prediction error processes ${\\mathbf {e}}_{i+1|i}^n$ and ${\\mathbf {\\epsilon }}_{i+1|i}^n$ of the pseudo-state and the state at agent $n$ by ${\\mathbf {e}}_{i+1|i}^n &= {\\mathbf {y}}_{i+1} - \\widehat{{\\mathbf {y}}}_{i+1|i}^n, \\\\{\\mathbf {\\epsilon }}_{i+1|i}^n &= {\\mathbf {x}}_{i+1} - \\widehat{{\\mathbf {x}}}_{i+1|i}^n.$ We establish that the CIKF provides unbiased estimates of the state and pseudo-state in the following lemma whose proof is sketched in Appendix REF .","Lemma 1 The distributed filter and prediction estimates, $\\widehat{{\\mathbf {y}}}_{i|i}^n$ , $\\widehat{{\\mathbf {x}}}_{i|i}^n$ , $\\widehat{{\\mathbf {y}}}_{i+1|i}^n$ and $\\widehat{{\\mathbf {x}}}_{i+1|i}^n$ , of the pseudo-state and state are unbiased, i.e., error processes, ${\\mathbf {e}}_{i|i}^n$ , ${\\mathbf {e}}_{i+1|i}^n$ , ${\\mathbf {\\epsilon }}_{i|i}^n$ , and ${\\mathbf {\\epsilon }}_{i+1|i}^n$ are zero-mean at all agents $n$ : $\\mathbb {E}[{\\mathbf {e}}_{i|i}^n] = 0, \\;\\mathbb {E}[{\\mathbf {e}}_{i+1|i}^n] = 0, \\;\\mathbb {E}[{\\mathbf {\\epsilon }}_{i|i}^n] = 0, \\;\\mathbb {E}[{\\mathbf {\\epsilon }}_{i+1|i}^n] = 0.$ Each agent exchanges their estimates with their neighbors, hence their error processes are correlated.","Since the error processes of the agents are coupled, we stack them in a error vector and analyze all errors together.","$\\widehat{{\\mathbf {x}}}_{i|i}&\\!=\\!","\\begin{bmatrix} \\widehat{{\\mathbf {x}}}_{i|i}^1 \\\\ \\vdots \\\\ \\widehat{{\\mathbf {x}}}_{i|i}^N \\end{bmatrix} \\!,\\widehat{{\\mathbf {x}}}_{i+1|i}\\!=\\!","\\begin{bmatrix} \\widehat{{\\mathbf {x}}}_{i+1|i}^1 \\\\ \\vdots \\\\ \\widehat{{\\mathbf {x}}}_{i+1|i}^N \\end{bmatrix}\\!,\\widehat{{\\mathbf {y}}}_{i|i}\\!=\\!","\\begin{bmatrix} \\widehat{{\\mathbf {y}}}_{i|i}^1 \\\\ \\vdots \\\\ \\widehat{{\\mathbf {y}}}_{i|i}^N \\end{bmatrix}\\!,\\widehat{{\\mathbf {y}}}_{i+1|i}\\!=\\!","\\begin{bmatrix} \\widehat{{\\mathbf {y}}}_{i+1|i}^1 \\\\ \\vdots \\\\ \\widehat{{\\mathbf {y}}}_{i+1|i}^N \\end{bmatrix}\\!, \\\\{\\mathbf {\\epsilon }}_{i|i}&\\!=\\!","\\begin{bmatrix} {\\mathbf {\\epsilon }}_{i|i}^1 \\\\ \\vdots \\\\ {\\mathbf {\\epsilon }}_{i|i}^N \\end{bmatrix}\\!,{\\mathbf {\\epsilon }}_{i+1|i}\\!=\\!","\\begin{bmatrix} {\\mathbf {\\epsilon }}_{i+1|i}^1 \\\\ \\vdots \\\\ {\\mathbf {\\epsilon }}_{i+1|i}^N \\end{bmatrix}\\!,{\\mathbf {e}}_{i|i}\\!=\\!","\\begin{bmatrix} {\\mathbf {e}}_{i|i}^1 \\\\ \\vdots \\\\ {\\mathbf {e}}_{i|i}^N \\end{bmatrix}\\!,{\\mathbf {e}}_{i+1|i}\\!=\\!","\\begin{bmatrix} {\\mathbf {e}}_{i+1|i}^1 \\\\ \\vdots \\\\ {\\mathbf {e}}_{i+1|i}^N \\end{bmatrix}\\!.$ We summarize the dynamics of the error processes in the following lemma whose proof is in Appendix REF .","Lemma 2 The error processes, ${\\mathbf {e}}_{i|i}$ , ${\\mathbf {e}}_{i+1|i}$ , ${\\mathbf {\\epsilon }}_{i|i}$ , and ${\\mathbf {\\epsilon }}_{i|i}$ are Gaussian and their dynamics are: $&\\!\\!","{\\mathbf {e}}_{i|i}\\!=\\!","\\left( I_{\\!M\\!N} \\!-\\!","B^{\\mathcal {C}}_i \\!-\\!","B^{\\mathcal {I}}_i \\widetilde{D}_{H}\\right)\\!","{\\mathbf {e}}_{i|i-1}\\!-\\!","B^{\\mathcal {I}}_i \\check{D}_{H} {\\mathbf {\\epsilon }}_{i|i-1}\\!-\\!","B^{\\mathcal {I}}_i D_{H}^T R^{-1} r_i , \\\\&{\\mathbf {\\epsilon }}_{i|i}= \\left( I_{MN} - K_i \\left( I_N \\otimes G \\right) \\right) {\\mathbf {\\epsilon }}_{i|i-1}+ K_i {\\mathbf {e}}_{i|i}, \\\\&{\\mathbf {e}}_{i+1|i}= \\left(I_{\\!N} \\!\\otimes \\!","\\widetilde{A}\\right)\\!","{\\mathbf {e}}_{i|i}+ \\!\\left( I_{\\!N} \\!\\otimes \\!","\\check{A} \\right)\\!","{\\mathbf {\\epsilon }}_{i|i}+ 1_{\\!N} \\!\\otimes \\!","\\left(G{\\mathbf {v}}_i\\right), \\\\&{\\mathbf {\\epsilon }}_{i+1|i}= \\left( I_N \\otimes A\\right) {\\mathbf {\\epsilon }}_{i|i}+ 1_N \\otimes {\\mathbf {v}}_i,$ where, $B^{\\mathcal {C}}_i$ is the consensus gain matrix and $B^{\\mathcal {I}}_i$ , $K_i$ are the innovations gain matrices for the pseudo-state and state estimation, respectively.","The block diagonal matrices are $\\widetilde{D}_{H} = \\text{blockdiag}\\lbrace \\widetilde{H}_1, \\cdots , \\widetilde{H}_N \\rbrace $ and $\\check{D}_{H} = \\text{blockdiag}\\lbrace \\check{H}_1, \\cdots , \\check{H}_N \\rbrace $ .","The symbol $\\otimes $ denotes the Kronecker matrix product.","Lemma REF established that the error processes (REF )-() are unbiased.","It then follows that the filter and prediction error covariances of the pseudo-state and state are simply: $P_{i|i} &= \\mathbb {E}\\left[{\\mathbf {e}}_{i|i}{\\mathbf {e}}_{i|i}^T \\right] \\\\P_{i+1|i} &= \\mathbb {E}\\left[{\\mathbf {e}}_{i+1|i}{\\mathbf {e}}_{i+1|i}^T \\right] \\\\\\Sigma _{i|i} &= \\mathbb {E}\\left[{\\mathbf {\\epsilon }}_{i|i}{\\mathbf {\\epsilon }}_{i|i}^T \\right] \\\\\\Sigma _{i+1|i} &= \\mathbb {E}\\left[{\\mathbf {\\epsilon }}_{i+1|i}{\\mathbf {\\epsilon }}_{i+1|i}^T \\right]$ Note that the state estimates $\\widehat{{\\mathbf {x}}}_{i|i}$ , $\\widehat{{\\mathbf {x}}}_{i+1|i}$ depend on the pseudo-state estimates $\\widehat{{\\mathbf {y}}}_{i|i}$ , $\\widehat{{\\mathbf {y}}}_{i+1|i}$ .","Hence the error process (REF )-() are not uncorrelated.","The filter and prediction cross-covariances are: $\\Pi _{i|i} &= \\mathbb {E}\\left[{\\mathbf {\\epsilon }}_{i|i}{\\mathbf {e}}_{i|i}^T \\right] \\\\\\Pi _{i+1|i} &= \\mathbb {E}\\left[{\\mathbf {\\epsilon }}_{i+1|i}{\\mathbf {e}}_{i+1|i}^T \\right] \\\\\\Gamma _{i} &= \\mathbb {E}\\left[{\\mathbf {\\epsilon }}_{i|i-1}{\\mathbf {e}}_{i|i}^T \\right]$ In the following theorem, we define and derive the evolution of the state, pseudo-state, and cross error covariances.","Figure: NO_CAPTIONTheorem 2 The filter error covariances, $P_{i|i}$ , $\\Sigma _{i|i}$ , $\\Pi _{i|i}$ , and the predictor error covariances, $P_{i+1|i}$ , $\\Sigma _{i+1|i}$ , $\\Pi _{i+1|i}$ , follow Lyapunov-type iterations ()-(), where, $J = (\\mathbf {1}_N \\mathbf {1}_N^T)\\otimes I_{\\!M}$ and the initial conditions are $\\Sigma _{0|-1} = J\\!\\otimes \\!\\Sigma _0, P_{0|-1} = J\\!\\otimes \\!\\left(G\\Sigma _0 G\\right), \\Pi _{0|-1} = J\\!\\otimes \\!\\left(\\Sigma _0 G\\right)$ .","The proof of the theorem is in Appendix REF .","The iterations () and () combined together constitute the distributed version of the discrete algebraic Riccati equation.","The MSE of the proposed CIKF is the trace of the error covariance, $\\Sigma _{i+1|i}$ in ().","The optimal design of the gain matrices, $B_i$ and $K_i$ , such that the CIKF yields minimized MSE estimates, is discussed in Section .","Before that in Section we derive the conditions under which the CIKF converges, in other words, the MSE given by the trace of $\\Sigma _{i+1|i}$ is bounded."],["Tracking Capacity","The convergence properties of the CIKF is determined by the dynamics of the pseudo-state and state error processes, ${\\mathbf {e}}_{i+1|i}$ and ${\\mathbf {\\epsilon }}_{i+1|i}$ .","If the error dynamics are asymptotically stable, then the error processes have asymptotically bounded error covariances that in turn guarantee the convergence of the CIKF.","Note that if the dynamics of the prediction error processes, ${\\mathbf {e}}_{i+1|i}$ , ${\\mathbf {\\epsilon }}_{i+1|i}$ are asymptotically stable, then the dynamics of the filter error processes, ${\\mathbf {e}}_{i|i}$ , ${\\mathbf {\\epsilon }}_{i|i}$ are also asymptotically stable.","That is why we study the dynamics of only one of the error processes and in this paper we consider the prediction error processes."],["Asymptotic stability of error processes","To analyze the stability of the error processes, we first write the evolution of the prediction error processes, combining (REF )-(), ${\\mathbf {e}}_{i+1|i}&= \\underbrace{ \\left(I_{\\!N} \\!\\otimes \\!","\\widetilde{A}\\right)\\!","\\left( I_{\\!M\\!N} \\!-\\!","B^{\\mathcal {C}}_i \\!-\\!","B^{\\mathcal {I}}_i \\widetilde{D}_{H}\\right) }_{\\widetilde{F}} {\\mathbf {e}}_{i|i-1}+ \\widetilde{\\mathbf {\\phi }}_i, \\\\{\\mathbf {\\epsilon }}_{i+1|i}&= \\underbrace{ \\left( I_{\\!N} \\!\\otimes \\!","A\\right)\\!","\\left( I_{\\!M\\!N} \\!-\\!\\!","K_i \\left( I_{\\!N} \\!\\otimes \\!","G \\right) \\right) }_{F} \\!","{\\mathbf {\\epsilon }}_{i|i-1}\\!+\\!","{\\mathbf {\\phi }}_i ,$ where, the noise processes $\\widetilde{\\mathbf {\\phi }}_i$ and ${\\mathbf {\\phi }}_i$ are $\\widetilde{\\mathbf {\\phi }}_i &= \\!\\left( I_{\\!N} \\!\\otimes \\!","\\check{A}\\right)\\!","{\\mathbf {\\epsilon }}_{i|i}- \\left(I_{\\!N} \\!\\otimes \\!","\\widetilde{A}\\right)\\!B^{\\mathcal {I}}_i \\check{D}_{H} {\\mathbf {\\epsilon }}_{i|i-1}+ 1_{\\!N} \\!\\otimes \\!","\\left(G{\\mathbf {v}}_i\\right) \\nonumber \\\\& \\qquad - \\left(I_{\\!N} \\!\\otimes \\!","\\widetilde{A}\\right)\\!","B^{\\mathcal {I}}_i D_{H}^T R^{-1} r_i, \\\\{\\mathbf {\\phi }}_i &= \\left( I_{\\!N} \\!\\otimes \\!","A\\right)\\!","K_i {\\mathbf {e}}_{i|i}+ 1_N \\otimes {\\mathbf {v}}_i.$ The statistical properties of the noises, $\\widetilde{\\mathbf {\\phi }}_i$ and ${\\mathbf {\\phi }}_i$ , of the error processes, ${\\mathbf {e}}_{i+1|i}$ and ${\\mathbf {\\epsilon }}_{i+1|i}$ , are stated in the following Lemma, and the proof is included in Appendix REF .","Lemma 3 The noise sequences $\\widetilde{\\mathbf {\\phi }}_i$ and ${\\mathbf {\\phi }}_i$ are zero-mean Gaussian that follow $\\widetilde{\\mathbf {\\phi }}_i \\in \\mathcal {N} \\left( {\\mathbf {0}}, \\widetilde{\\Phi }_i \\right)$ and ${\\mathbf {\\phi }}_i \\in \\mathcal {N} \\left( {\\mathbf {0}}, \\Phi _i \\right)$ .","The dynamics of the error processes are characterized by (REF )-() and Lemma REF .","Let $\\rho (.",")$ and $\\Vert .\\Vert _2$ denote the spectral radius and the spectral norm of a matrix, respectively.","The error processes are asymptotically stable if and only if the spectral radii of $\\widetilde{F}$ , $F$ are less than one, i.e., $\\rho \\left(\\widetilde{F}\\right) < 1, \\quad \\rho \\left(F\\right) < 1$ and the noise covariances, $\\widetilde{\\Phi }_i$ , $\\Phi _i$ are bounded, i.e., $\\Vert \\widetilde{\\Phi }_i\\Vert _2 < \\infty $ , $\\Vert \\Phi _i\\Vert _2 < \\infty , \\; \\forall i$ .","Now if (REF ) holds, then the prediction error covariances $P_{i+1|i}$ , $\\Sigma _{i+1|i}$ are bounded; this ensures the filter error covariances $P_{i|i}$ , $\\Sigma _{i|i}$ are also bounded.","Further, the model noise covariances $V$ and $R$ are bounded.","Then, by (REF )-(), the noise covariances $\\widetilde{\\Phi }_i$ and ${\\Phi }_i$ are bounded if the spectral radii are less than one.","Thus, (REF ) are the necessary and sufficient conditions for the convergence of the CIKF algorithm."],["Tracking capacity for unstable systems","The stability of the underlying dynamical system (REF ) in the physical layer is determined by the dynamics matrix $A$ .","If the system is asymptotically stable, i.e., $\\rho (A)<1$ , then there always exist gain matrices $B_i$ , $K_i$ such that (REF ) holds true.","Hence for stable systems, the CIKF always converges with a bounded MSE solution.","In contrast, for an unstable dynamical system (REF ), $\\rho (A)>1$ , it may not always be possible to find gain matrices $B_i$ , $K_i$ satisfying (REF ) conditions.","There exists an upper threshold on the degree of instability of the system dynamics, $A$ , that guarantees the convergence of the proposed CIKF.","The threshold, similar to Network Tracking Capacity in [6], is the tracking capacity of the CIKF algorithm, and it depends on the agent communication network and observation models, as summarized in the following theorem.","Theorem 3 The tracking capacity of the CIKF is, C, $C = \\max _{B^{\\mathcal {C}}_i, B^{\\mathcal {I}}_i} \\frac{\\lambda _{1}}{\\lambda _{m} \\begin{Vmatrix} I_{\\!M\\!N} \\!-\\!","B^{\\mathcal {C}}_i \\!-\\!","B^{\\mathcal {I}}_i \\widetilde{D}_{H} \\end{Vmatrix}_2 }$ where, $B^{\\mathcal {C}}_i$ has the same block sparsity pattern as the graph Laplacian $L$ , $B^{\\mathcal {I}}_i$ is a block diagonal matrix, and, $\\lambda _{1}$ and $\\lambda _{m}$ are the minimum and maximum non-zero eigenvalues of $G$ , $0<\\lambda _{1} \\le \\cdots \\le \\lambda _{m}$ .","If $\\Vert A \\Vert _2 < C$ , then there exists $B_i$ , $K_i$ such that the CIKF (REF )-() converges with bounded MSE.","The proof is in Appendix REF .","In the above theorem, the structural constraints on the gain matrices $B^{\\mathcal {C}}_i, B^{\\mathcal {I}}_i$ ensure that each agent combines its neighbors' estimates for consensus and its own pseudo-observations for the innovation part of the CIKF.","The block sparsity pattern of $B^{\\mathcal {C}}_i$ being similar to that of $L$ implies that the tracking capacity is dependent on the connectivity of the communication network.","Similarly, since $\\widetilde{D}_{H}$ is a block-diagonal matrix containing the observation matrices $H_n$ we conclude that the tracking capacity is also a function of the observation models.","The tracking capacity increases with the increase in communication graph connectivity and observation density.","For instance, the tracking capacity is infinity if all agents are connected with everyone else (complete graph) or all the agents observe the entire dynamical system (local observability).","Given the tracking capacity is satisfied for the system, observation, and communication models (REF )-(REF ), the question remains how to design the gain matrices $B_i$ and $K_i$ to minimize the MSE of the CIKF, which we discuss in the following section."],["Optimal Gain Design","The asymptotic stability of the error dynamics guarantees convergence of CIKF and bounded MSE, but here we discuss how to design the $B_i$ and $K_i$ such that the MSE is not only bounded but also minimum."],["New uncorrelated information","In CIKF Algorithm REF , at any time $i$ each agent $n$ makes pseudo-observation $\\widetilde{{\\mathbf {z}}}^n_i$ of the state and receives prior estimates $\\lbrace \\widehat{{\\mathbf {y}}}_{i|i-1}^l\\rbrace _{l \\in \\Omega _n}$ from its neighbors.","The CIKF algorithm employs this new information to compute the distributed filter estimates of the pseudo-state and state.","Denote the new information for the pseudo-state and state filtering by $\\widetilde{\\mathbf {\\theta }}^n_i$ and ${\\mathbf {\\theta }}^n_i$ , respectively, $\\widetilde{\\mathbf {\\theta }}^n_i = \\begin{bmatrix}\\widehat{{\\mathbf {y}}}_{i|i-1}^{l_1} \\\\ \\vdots \\\\ \\widehat{{\\mathbf {y}}}_{i|i-1}^{l_{d_n}} \\\\ \\widetilde{{\\mathbf {z}}}^n_i \\end{bmatrix}, \\quad {\\mathbf {\\theta }}^n_i = \\widehat{{\\mathbf {y}}}_{i|i}^n$ where, $\\lbrace l_1, \\cdots , l_{d_n}\\rbrace = \\Omega _n$ and $d_n = |\\Omega _n|$ is the degree of agent $n$ .","Note the new information, $\\widetilde{\\mathbf {\\theta }}^n_i$ and ${\\mathbf {\\theta }}^n_i$ , are Gaussian since they are linear combinations of Gaussian sequences.","However, $\\widetilde{\\mathbf {\\theta }}^n_i$ and ${\\mathbf {\\theta }}^n_i$ are correlated with the previous estimates $\\widehat{{\\mathbf {y}}}_{i|i-1}^n$ and $\\widehat{{\\mathbf {x}}}_{i|i-1}^n$ .","So, we transform them into uncorrelated new information and then combine the uncorrelated information with the previous estimates $\\widehat{{\\mathbf {y}}}_{i|i-1}^n$ and $\\widehat{{\\mathbf {x}}}_{i|i-1}^n$ to compute the current filtered estimates.","Lemma 4 The new uncorrelated information $\\widetilde{\\mathbf {\\nu }}^n_i$ and ${\\mathbf {\\nu }}^n_i$ for filtering update at agent n are, $\\widetilde{\\mathbf {\\nu }}^n_i &= \\widetilde{\\mathbf {\\theta }}^n_i - \\overline{\\widetilde{\\mathbf {\\theta }}}^n_i, \\qquad \\overline{\\widetilde{\\mathbf {\\theta }}}^n_i = \\mathbb {E} \\begin{bmatrix} \\widetilde{\\mathbf {\\theta }}^n_i | \\widetilde{\\mathbf {z}}^n_{i-1}, \\lbrace \\widehat{\\mathbf {y}}^l_{i-1|i-2} \\rbrace _{l \\in \\Omega _n} \\end{bmatrix} \\\\{\\mathbf {\\nu }}^n_i &= {\\mathbf {\\theta }}^n_i - \\overline{\\mathbf {\\theta }}^n_i, \\qquad \\overline{\\mathbf {\\theta }}^n_i = \\mathbb {E} \\begin{bmatrix} {\\mathbf {\\theta }}^n_i | \\widehat{\\mathbf {y}}^n_{i-1|i-1} \\end{bmatrix}$ that expands to $\\widetilde{\\mathbf {\\nu }}^n_i \\!=\\!","\\begin{bmatrix}\\widehat{{\\mathbf {y}}}_{i|i-1}^{l_1} - \\widehat{{\\mathbf {y}}}_{i|i-1}^n \\\\ \\vdots \\\\ \\widehat{{\\mathbf {y}}}_{i|i-1}^{l_{d_n}}-\\widehat{{\\mathbf {y}}}_{i|i-1}^n \\\\ \\widetilde{{\\mathbf {z}}}^n_i \\!\\!-\\!","\\widetilde{H}_{\\!n} \\widehat{{\\mathbf {y}}}_{i|i-1}^n \\!-\\!","\\check{H}_{\\!n} \\widetilde{I} \\widehat{{\\mathbf {x}}}_{i|i-1}^n \\end{bmatrix}\\!\\!\\!, \\;\\;{\\mathbf {\\nu }}^n_i \\!=\\!","\\widehat{{\\mathbf {y}}}_{i|i}^n \\!-\\!","G\\widehat{{\\mathbf {x}}}_{i|i-1}^n.$ The uncorrelated sequences $\\widetilde{\\mathbf {\\nu }}^n_i$ and ${\\mathbf {\\nu }}^n_i$ are zero-mean Gaussian random vectors.","Hence $\\widetilde{\\mathbf {\\nu }}^n_i$ and ${\\mathbf {\\nu }}^n_i$ are independent sequences.","The proof is in Appendix REF .","We write the CIKF filter updates (REF ) and () in terms of the new uncorrelated information $\\widetilde{\\mathbf {\\nu }}^n_i$ and ${\\mathbf {\\nu }}^n_i$ from (REF ), $ \\widehat{{\\mathbf {y}}}_{i|i}^n &= \\widehat{{\\mathbf {y}}}_{i|i-1}^n + \\widehat{B}^n_i \\widetilde{\\mathbf {\\nu }}^n_i \\\\ \\widehat{{\\mathbf {x}}}_{i|i}^n &= \\widehat{{\\mathbf {x}}}_{i|i-1}^n + K^n_i {\\mathbf {\\nu }}^n_i$ where, $\\widehat{B}^n_i$ are the building blocks of the pseudo-state gain matrix $B_i$ ."],["Consensus and innovation gains","Here, we present the methods to: (a) design the matrices $\\widehat{B}^n_i$ and $K^n_i$ ; and (b) obtain the optimal gains $B_i$ and $K_i$ from them.","These optimal gains provide the distributed minimized MSE estimates of the field.","At agent $n$ , we define the matrix $\\widehat{B}^n_i$ as $\\widehat{B}^n_i = \\begin{bmatrix} B^{n l_1}_i, \\cdots , B^{n l_{d_n}}_i, B^{n n}_i \\end{bmatrix}$ where, $\\lbrace l_1, \\cdots , l_{d_n}\\rbrace = \\Omega _n$ .","The gain matrix $B_i$ is a linear combination of $B^{\\mathcal {C}}_i$ and $B^{\\mathcal {I}}_i$ , where $B^{\\mathcal {C}}_i$ has the same block structure as the graph Laplacian $L$ and $B^{\\mathcal {I}}_i$ is a block diagonal.","The $(n,l)^{\\text{th}}$ blocks of the $n^{\\text{th}}$ row of $B^{\\mathcal {C}}_i$ are: $\\left[B^{\\mathcal {C}}_i\\right]_{nl} = \\left\\lbrace \\begin{array}{ll}- B^{n l}_i, & \\mbox{if } l \\in \\Omega _n \\\\\\sum _{j=1}^{d_n}B^{n l_j}_i, & \\mbox{if } l = n \\\\{\\mathbf {0}}, & \\text{otherwise.}","\\\\\\end{array} \\right.$ The $\\lbrace n,n\\rbrace ^{\\text{th}}$ blocks of the diagonal block matrices $B^{\\mathcal {I}}_i$ and $K_i$ are: $\\left[B^{\\mathcal {I}}_i\\right]_{nn} &= B^{n n}_i, \\\\\\left[K_i\\right]_{nn} &= K^{n}_i,$ Hence, once we design the matrices $\\widehat{B}^n_i$ and $K^{n}_i$ , it will provide the optimal gain matrices $B^{\\mathcal {C}}_i, B^{\\mathcal {I}}_i$ , and $K_i$ .","Theorem 4 The optimal gains for the CIKF algorithm are $\\widehat{B}^n_i &= \\Sigma _{{\\mathbf {y}}_i \\widetilde{\\mathbf {\\nu }}^n_i} \\left(\\Sigma _{\\widetilde{\\mathbf {\\nu }}^n_i}\\right)^{-1}, \\\\K^n_i &= \\Sigma _{{\\mathbf {x}}_i {\\mathbf {\\nu }}^n_i} \\left(\\Sigma _{{\\mathbf {\\nu }}^n_i}\\right)^{-1}$ where, $\\Sigma _{\\widetilde{\\mathbf {\\nu }}^n_i}$ , $\\Sigma _{{\\mathbf {\\nu }}^n_i}$ are the covariances of the new uncorrelated information $\\widetilde{\\mathbf {\\nu }}^n_i$ and ${\\mathbf {\\nu }}^n_i$ ; and, $\\Sigma _{{\\mathbf {y}}_i \\widetilde{\\mathbf {\\nu }}^n_i}$ , $\\Sigma _{{\\mathbf {x}}_i {\\mathbf {\\nu }}^n_i}$ are cross-covariances between ${\\mathbf {y}}_i$ , $\\widetilde{\\mathbf {\\nu }}^n_i$ and ${\\mathbf {x}}_i$ , ${\\mathbf {\\nu }}^n_i$ , respectively.","These covariance and cross-covariance matrices are related to the error covariance matrices, $P_{i|i-1}, P_{i|i}, \\Sigma _{i|i-1}, \\Pi _{i|i-1}, \\Gamma _i$ , by the following functions: $& \\!\\Sigma _{{\\mathbf {y}}_i \\widetilde{\\mathbf {\\nu }}^n_i} \\!=\\!","\\left[\\begin{array}{l;{2pt/2pt}l;{1pt/1pt}l;{2pt/2pt}l}\\!\\!\\!\\!","P_{i|i\\!-\\!1}^{nn} \\!-\\!","P_{i|i\\!-\\!1}^{nl_1} & \\cdots & P_{i|i\\!-\\!1}^{nn} \\!-\\!","P_{i|i\\!-\\!1}^{nl_{d_n}} & P_{i|i\\!-\\!1}^{nn} \\!","\\widetilde{H}^T_n \\!+\\!","\\Pi _{i|i\\!-\\!1}^{nn^T} \\!","\\check{H}^T_n \\!\\!\\!\\!\\end{array}\\right] \\\\&\\!","\\Sigma _{{\\mathbf {x}}_i {\\mathbf {\\nu }}^n_i} = \\Sigma _{i|i\\!-\\!1}^{nn} G - \\Gamma ^{nn}_i \\\\&\\begin{bmatrix} \\Sigma _{\\widetilde{\\mathbf {\\nu }}^n_i} \\end{bmatrix}_{qs} = \\\\&\\left\\lbrace \\begin{array}{ll}P_{i|i\\!-\\!1}^{nn} \\!-\\!","P_{i|i\\!-\\!1}^{nl_s} \\!-\\!","P_{i|i\\!-\\!1}^{l_q n} \\!+\\!","P_{i|i\\!-\\!1}^{l_q l_s}, & \\mbox{if } q \\le s \\le d_n \\\\\\!\\left( P_{i|i\\!-\\!1}^{nn} \\!-\\!","P_{i|i\\!-\\!1}^{l_q n} \\right)\\!\\widetilde{H}_n^T \\!+\\!","\\left( \\Pi _{i|i\\!-\\!1}^{nn} \\!-\\!","\\Pi _{i|i\\!-\\!1}^{l_q n} \\right)^T\\!\\!\\check{H}_n^T, & \\mbox{if } q < s = d_n\\!\\!+\\!1 \\\\\\widetilde{H}_n P_{i|i\\!-\\!1}^{nn}\\widetilde{H}_n^T + \\widetilde{H}_n \\Pi _{i|i\\!-\\!1}^{nn^T}\\check{H}_n^T + \\check{H}_n\\Pi _{i|i\\!-\\!1}^{nn}\\widetilde{H}_n^T & \\\\\\qquad \\qquad + \\check{H}_n\\Sigma _{i|i\\!-\\!1}^{nn}\\check{H}^T_n + \\overline{H}_n, & \\mbox{if } q = s = d_n\\!\\!+\\!1 \\\\ \\begin{bmatrix} \\Sigma _{\\widetilde{\\mathbf {\\nu }}^n_i} \\end{bmatrix}_{sq}^T, & \\mbox{if } q > s \\\\\\end{array} \\right.","\\\\&\\begin{bmatrix} \\Sigma _{{\\mathbf {\\nu }}^n_i} \\end{bmatrix}_{qs} = G\\Sigma _{i|i\\!-\\!1}^{nn}G - G \\Gamma ^{nn}_i - \\Gamma ^{nn^T}_iG + P_{i|i}^{nn}$ where, $\\left[\\Sigma _{\\widetilde{\\mathbf {\\nu }}^n_i}\\right]_{qs}$ denotes the $\\lbrace q,s\\rbrace ^{\\text{th}}$ block of the $(d_n\\!\\!+\\!1)\\times (d_n\\!\\!+\\!1)$ block matrix $\\Sigma _{\\widetilde{\\mathbf {\\nu }}^n_i}$ .","The proof is in Appendix REF .","By the Gauss-Markov theorem, the CIKF algorithm, along with this design of the consensus and innovation gain matrices, as stated in Theorem REF , results in the minimized MSE distributed estimates of the dynamic random field ${\\mathbf {x}}_i$ .","The gain matrices are deterministic.","Hence they can be precomputed offline and saved for online implementation.","In Algorithm REF , we state the steps that each agent $n$ runs to compute the optimal gain matrices.","[h] Gain Design of CIKF Input: Model parameters $A$ , $V$ , $H$ , $R$ , $G$ , $L$ , $\\Sigma _0$ .","Initialize: $\\Sigma _{0|-1} = J \\!\\otimes \\!","\\Sigma _0, \\; P_{0|-1} = J \\!\\otimes \\!","\\left(G\\Sigma _0 G\\right), \\; \\Pi _{0|-1} = J \\!\\otimes \\!","\\left(\\Sigma _0 G\\right)$ .","$i \\ge 0$ $\\!\\!\\!\\!$Optimal gains: Compute $\\widehat{B}^n_i$ and $K^n_i$ using Theorem REF .","Using (REF )-(), obtain $B^{\\mathcal {C}}_i$ , $B^{\\mathcal {I}}_i$ and $K_i$ from $\\widehat{B}^n_i$ and $K^n_i$ .","$\\!\\!\\!\\!$Prediction error covariance updates: Update $P_{i+1|i}, \\Sigma _{i+1|i}, \\Pi _{i+1|i}$ using ()-().","The offline Algorithm REF along with the online Algorithm REF completes our proposed distributed solution to obtain minimized MSE estimates of the dynamic field ${\\mathbf {x}}_i$ at each agent in the cyber network.","CIKF does not require sparsity of the matrices $A$ , $H_n$ and $L$ .","However, in most distributed applications these matrices are sparse.","For example, consider the temperature distribution over the entire United States as a time-varying random field.","It is conceivable that the temperature in a location will be directly dependent on the temperature of a few locations leading to a sparse $A$ .","Similarly a weather station will make localized measurements of the temperature (at its own location) rather than measuring the temperature all over the US.","Similarly the weather stations will exchange their data with the neighboring weather stations, thereby reducing the communication cost and leading to a sparse graph Laplacian matrix.","From a computational point of view, if the matrices $A$ , $H_n$ and $L$ are sparse, they can be stored as lists, for example, and computation of matrix products with them will involve less multiplications and/or additions operations.","Since the computation of $B_i$ and $K_i$ involve these matrices, their sparsity will reduce both the computation and storage costs."],["Numerical Evaluation","We numerically evaluate the MSE performance of the CIKF and compare it against the centralized Kalman filter (CKF) and the distributed information Kalman filter (DIKF) in [5].","To this objective, we build a time-varying random system, observation and network model that satisfies the Assumptions REF -REF .","The Algorithms REF -REF run on these model parameters.","First the Algorithm REF computes and save the gain matrices and the error covariance matrices.","The traces of the error covariance matrices provide the theoretical MSE trajectory of the CIKF with time.","Then, we Monte-Carlo simulate Algorithm REF to compute the numerical MSE of the distributed estimators, CIKF and DIKF, and the centralized estimator CKF."],["Model specifications","Here, we consider a time-varying field, ${\\mathbf {x}}_i$ , with dimension $M=50$ .","The physical layer, consisting of $M=50$ sites, is monitored by a cyber layer consisting of $N=50$ agents.","Each agent in the cyber layer observes $M_n=2$ sites of the physical layer.","We build the field dynamics matrix $A$ to be sparse and distributed.","The dynamics $A$ possess the structure of a Lattice graph, where the time evolution of a field variable depends on the neighboring field variables.","The structure of the Lattice graph is randomly generated using R library.","The degree is considered to be 4 and the non-zero entries are randomly chosen from $(0,1)$ .","For illustration, we consider an unstable field dynamics with $\\Vert A\\Vert _2=1.05$ to test the resilience of the algorithms under unstable conditions.","We scale the randomly generated $A$ such that its spectral norm becomes 1.05.","The observation matrices, $H_n \\in \\mathbb {R}^{2\\times 50}, n=1, \\cdots , 50$ , are sparse $0-1$ matrices with one non-zero element at each row corresponding to the site of ${\\mathbf {x}}_i$ observed by the $n^{\\text{th}}$ agent.","The local observations ${\\mathbf {z}}^n_i$ are $2\\times 1$ random vectors.","The mean $\\overline{x}_0$ of the initial state vector is generated at random.","The system noise covariance $V$ , the observation noise covariances $R_n$ , and the initial state covariance $\\Sigma _0$ are randomly generated symmetric positive definite matrices.","We then scale the covariance matrices such that their covariance becomes: $\\Vert V\\Vert _2=4, \\Vert R_n\\Vert _2=8$ , and $\\Vert \\Sigma _0\\Vert _2=16$ .","The agents in the cyber layer communicate among themselves following a randomly generated Erdős-Rényi graph $\\mathcal {G}$ with 50 nodes and $E=138$ edges.","The average degree of each node/agent is approximately $5.5$ .","The communication network $\\mathcal {G}$ is also sparse.","For Monte-Carlo simulations, we generate the noises, ${\\mathbf {v}}_i, {\\mathbf {r}}^n_i$ , and the initial condition, ${\\mathbf {x}}_0$ as Gaussian sequences, with ${\\mathbf {v}}_i \\sim \\mathcal {N}(\\bar{0},V), \\;{\\mathbf {r}}^n_i \\sim \\mathcal {N}(\\bar{0},R_n), \\;{\\mathbf {x}}_0 \\sim \\mathcal {N}(\\bar{{\\mathbf {x}}}_0,\\Sigma _0)$ .","The sequences $\\lbrace \\lbrace {\\mathbf {v}}_i\\rbrace _i, \\lbrace {\\mathbf {r}}_i\\rbrace _i, {\\mathbf {x}}_0\\rbrace _{i \\ge 0}$ are generated to be uncorrelated.","Each agent $n$ in the cyber layer has access to the system parameters $A, V, H, R, \\bar{{\\mathbf {x}}}_0, \\Sigma _0$ , and $\\mathcal {G}$ .","This numerical model satisfies Assumptions REF -REF .","The pair $(A,H)$ is detectable and the pairs, $(A,H_n) \\; \\forall n$ , are not detectable.","The agent communication graph $\\mathcal {G}$ is connected with the algebraic connectivity of the Laplacian $\\lambda _2(L) = 0.7 > 0$ .","Hence the Assumptions REF -REF hold true for this numerical system, observation, and network model.","Figure: Comparison of MSE performance of the proposed CIKF with CKF and DIKF."],["Optimized gains and theoretical MSE","We run Algorithm REF on the numerical model to obtain the gain matrices and the theoretical error covariances of the CIKF.","We compute the gain matrices and the error covariances of the centralized Kalman filter (CKF) and of the distributed information Kalman filter (DIKF).","In Fig.","REF , we plot the MSE (in dB), trace of the predictor error covariance matrices $\\Sigma _{i+1|i}$ , for each of these cases up to time $i=30$ .","The MSE of the optimal CKF is the smallest (recall the CKF, if feasible, would be optimal) and the objective of the distributed estimators is to achieve MSE performance as close as possible to that of the CKF.","From Fig.","REF , we see that the MSE of the proposed CIKF is 3dB more than the CKF but is 3dB less than the DIKF.","From the plot we see that the CIKF converges faster than the DIKF.","Hence, the proposed CIKF provides faster convergence and 3dB MSE performance improvement over the DIKF.","The performance of the CKF is 3dB better than the CIKF's due to the fact that the CKF has access to the observations of all the sensors at every time steps.","In contrast in CIKF, each agent has access to its own observations and the current estimates of its neighbors only; the impact of the observations from the other agents propagate through the network with delay.","As the time-varying field ${\\mathbf {x}}_i$ is evolving with input noise ${\\mathbf {v}}_i$ , lack of access to all the observations containing the driving input ${\\mathbf {v}}_i$ , combined with the network diffusion delay, causes a performance gap between the CKF and the CIKF."],["Monte-Carlo Simulations","We empirically compute the MSE of the distributed estimates given by the CIKF Algorithm REF .","We implement the algorithms using Matlab in the Microsoft Azure cloud.","Given the computation load because of the large system (M = 50) and network (N = 50) models, we run our simulation on Azure DS13 (8 cores, 56 GB memory) virtual machine (VM).","The MSE computation for the CIKF, CKF, and DIKF algorithms with 1000 Monte-Carlo runs require approximately 30 hours in the Azure DS13 VM.","Once we obtain the field prediction estimates for the three algorithms, we compute the empirical prediction error covariance matrices $\\widehat{\\Sigma }_{i+1|i}$ and then obtain the Monte-Carlo MSE (in dB) from their trace.","From the Monte-Carlo MSE plot in Fig REF , we see that the the empirical plots follow closely the theoretical plots in Fig REF .","Both the theoretical Fig REF and Monte-Carlo simulated Fig REF MSE performance confirms our CIKF analysis in Sections -.","The novel Consensus$+$ Innovations Kalman Filter (CIKF) proposed in Section along with the optimized gain designs in Section provides unbiased distributed estimates with bounded and minimized MSE for the Consensus$+$ Innovations distributed solution.","The CIKF achieves nearly 3dB better performance than the DIKF [5]."],["Conclusions","Summary: In this paper, we propose a Consensus$+$ Innovations Kalman Filter (CIKF) that obtains unbiased minimized MSE distributed estimates of the pseudo-states and real-time employs them to obtain the unbiased distributed filtering and prediction estimates of the time-varying random state at each agent.","The filter update iterations are of the Consensus$+$ Innovations type.","Using the Gauss-Markov principle, we designed the optimal gain matrices that yield approximately 3dB improvement over previous available distributed estimators like the DIKF in [5].","Contributions: The primary contributions of this paper are: (a) design of a filter and corresponding gain matrices to obtain minimized MSE distributed estimates at each agent under minimal assumptions; and (b) a theoretical characterization of the tracking capacity and distributed version of the algebraic Riccati equation.","Compared to DIKF, the contributions of this paper are: $(a)$ The CIKF is a new algorithm for distributed estimation of time-varying random fields which provides 3dB MSE performance over the DIKF and much closer to the performance of the optimal centralized Kalman filter.","$(b)$ The CIKF does not require the strict distributed observability assumption, i.e., invertibility of the matrix $G$ .","The DIKF require that $G^{-1}$ exists, which demands that all state variables be observed by at least once agent in the network.","$(c)$ In CIKF, we design the optimized gain matrices for the dynamic averaging of pseudo-states using the Gauss-Markov Theorem.","Hence, the distributed dynamic averaging step of the CIKF has lower MSE than that of the DIKF.","$(d)$ These advantages are obtained by using pseudo-state and this paper demonstrates its usefulness in distributed field estimation."],["Appendix","The Appendices prove the proposition, lemmas, and theorems stated in the paper.","Proof of Proposition"],["Proof of Proposition ","First we derive the dynamics (REF ) of the pseudo-state ${\\mathbf {y}}_i$ .","Using (REF ) and (REF ), ${\\mathbf {y}}_{i+1} &= G{\\mathbf {x}}_{i+1} \\\\&= G \\left( A{\\mathbf {x}}_i + {\\mathbf {v}}_i\\right) \\\\&= GA \\left( G^{\\dagger }G + \\widetilde{I}\\right) {\\mathbf {x}}_i + G{\\mathbf {v}}_i, \\quad \\begin{small} \\left[ \\text{by (\\ref {eqn:p_identity_mat})}, \\; I = G^{\\dagger }G + \\widetilde{I} \\right] \\end{small} \\\\&= GAG^{\\dagger }{\\mathbf {y}}_i + G{\\mathbf {v}}_i + GA\\widetilde{I}{\\mathbf {x}}_i \\\\&= \\widetilde{A}{\\mathbf {y}}_i + G{\\mathbf {v}}_i + \\check{A}{\\mathbf {x}}_i, \\qquad \\quad \\left[ \\text{by (\\ref {eqn:pstate})} \\right].$ Now, we derive the observations () of the pseudo-observations $\\widetilde{{\\mathbf {z}}}^n_i$ .","Using (REF ) and (REF ), $\\widetilde{{\\mathbf {z}}}^n_i &= H^T_n R_n^{-1} H_n{\\mathbf {x}}_i + H^T_n R_n^{-1} {\\mathbf {r}}^n_i \\\\&= H^T_nR_n^{-1}H_n \\left( G^{\\dagger }G + \\widetilde{I}\\right) {\\mathbf {x}}_i + H^T_n R_n^{-1} {\\mathbf {r}}^n_i \\\\&= H^T_nR_n^{-1}H_n G^{\\dagger }G {\\mathbf {x}}_i + H^T_n R_n^{-1} {\\mathbf {r}}^n_i + H^T_nR_n^{-1}H_n \\widetilde{I} {\\mathbf {x}}_i \\\\&= \\widetilde{H} {\\mathbf {y}}_i + H^T_n R_n^{-1} {\\mathbf {r}}^n_i + \\check{H}_n {\\mathbf {x}}_i.","\\qquad \\qquad \\qquad \\Box $ Proof of Lemmas"],["Proof of Lemma ","Consider the filtering error definitions (REF )-().","We take expectations on both sides, $\\mathbb {E}\\left[{\\mathbf {e}}_{i|i}^n\\right] &= \\mathbb {E}\\left[{\\mathbf {y}}_i - \\widehat{{\\mathbf {y}}}_{i|i}^n \\right] \\\\&= \\mathbb {E} \\left[ \\mathbb {E} \\left[ {\\mathbf {y}}_i - \\widehat{{\\mathbf {y}}}_{i|i}^n~|~\\widetilde{{\\mathbf {z}}}^n_i, \\lbrace \\widehat{{\\mathbf {y}}}_{i|i-1}^l \\rbrace _{l \\in \\Omega _n} \\right] \\right] \\\\&= \\mathbb {E} \\left[ \\widehat{{\\mathbf {y}}}_{i|i}^n - \\widehat{{\\mathbf {y}}}_{i|i}^n \\right] = 0 \\qquad \\left[ \\text{by} (\\ref {eqn:MMSE_yf}) \\right] \\\\\\mathbb {E}\\left[{\\mathbf {\\epsilon }}_{i|i}^n\\right] &= \\mathbb {E}\\left[{\\mathbf {x}}_i - \\widehat{{\\mathbf {x}}}_{i|i}^n \\right] \\\\&= \\mathbb {E} \\left[ \\mathbb {E} \\left[ {\\mathbf {x}}_i - \\widehat{{\\mathbf {x}}}_{i|i}^n~|~\\widehat{{\\mathbf {y}}}_{i|i}^n \\right] \\right] \\\\&= \\mathbb {E} \\left[ \\widehat{{\\mathbf {x}}}_{i|i}^n - \\widehat{{\\mathbf {x}}}_{i|i}^n \\right] = 0 \\qquad \\left[ \\text{by} (\\ref {eqn:epsfn}) \\right].$ Similarly, taking expectations on prediction errors (REF )-(), $\\mathbb {E}\\left[{\\mathbf {e}}_{i+1|i}^n\\right] &= \\mathbb {E}\\left[{\\mathbf {y}}_{i+1} - \\widehat{{\\mathbf {y}}}_{i+1|i}^n \\right] \\\\&= \\mathbb {E} \\left[ \\mathbb {E} \\left[ {\\mathbf {y}}_{i+1} - \\widehat{{\\mathbf {y}}}_{i+1|i}^n~|~\\widetilde{{\\mathbf {z}}}^n_i, \\lbrace \\widehat{{\\mathbf {y}}}_{i|i-1}^l \\rbrace _{l \\in \\Omega _n} \\right] \\right] \\\\&= \\mathbb {E} \\left[ \\widehat{{\\mathbf {y}}}_{i+1|i}^n - \\widehat{{\\mathbf {y}}}_{i+1|i}^n \\right] = 0 \\qquad \\left[ \\text{by} (\\ref {eqn:MMSE_yp}) \\right] \\\\\\mathbb {E}\\left[{\\mathbf {\\epsilon }}_{i+1|i}^n\\right] &= \\mathbb {E}\\left[{\\mathbf {x}}_{i+1} - \\widehat{{\\mathbf {x}}}_{i+1|i}^n \\right] \\\\&= \\mathbb {E} \\left[ \\mathbb {E} \\left[ {\\mathbf {x}}_{i+1} - \\widehat{{\\mathbf {x}}}_{i+1|i}^n~|~\\widehat{{\\mathbf {y}}}_{i|i}^n \\right] \\right] \\\\&= \\mathbb {E} \\left[ \\widehat{{\\mathbf {x}}}_{i+1|i}^n - \\widehat{{\\mathbf {x}}}_{i+1|i}^n \\right] = 0 \\qquad \\left[ \\text{by} (\\ref {eqn:epspn}) \\right].","\\qquad \\Box $"],["Proof of Lemma ","We write the pseudo-state filtering update (REF ) in vector form, $\\widehat{{\\mathbf {y}}}_{i|i}\\!=\\!","\\widehat{{\\mathbf {y}}}_{i|i-1}& \\!-\\!","B^{\\mathcal {C}}_i \\widehat{{\\mathbf {y}}}_{i|i-1}\\!+\\!","B^{\\mathcal {I}}_i \\!","\\left( \\!\\widetilde{{\\mathbf {z}}}_i \\!-\\!","\\left(\\widetilde{D}_H \\widehat{{\\mathbf {y}}}_{i|i-1}\\!+\\!","\\check{D}_H \\widehat{{\\mathbf {x}}}_{i|i-1}\\right) \\right) ,$ where, $\\left[B^{\\mathcal {C}}_i\\right]_{nl} = -B^{n,l}_i, \\; \\forall n \\ne l$ , $\\left[B^{\\mathcal {C}}_i\\right]_{nn} = \\sum _{l \\in \\Omega _n}B^{n,l}_i$ , $\\left[B^{\\mathcal {I}}_i\\right]_{nn} = B^{n,n}_i$ , $\\left[B^{\\mathcal {I}}_i\\right]_{nl} = 0, \\; \\forall n \\ne l$ .","The block-diagonal matrices: $\\widetilde{D}_H = \\text{blockdiag}\\lbrace \\widetilde{H}_1, \\cdots , \\widetilde{H}_N\\rbrace , \\; \\check{D}_H = \\text{blockdiag}\\lbrace \\check{H}_1, \\cdots , \\check{H}_N\\rbrace $ .","Note that $B^{\\mathcal {C}}_i \\left(1_N\\otimes {\\mathbf {y}}_i\\right) = \\mathbf {0}$ .","Using this relation and the vector form of $\\widehat{{\\mathbf {y}}}_{i|i}$ , we expand the pseudo-state filter error process ${\\mathbf {e}}_{i|i}$ , ${\\mathbf {e}}_{i|i}&= 1_N \\!\\otimes \\!","{\\mathbf {y}}_i - \\widehat{{\\mathbf {y}}}_{i|i}\\\\&= \\left( 1_N \\!\\otimes \\!","{\\mathbf {y}}_i - \\widehat{{\\mathbf {y}}}_{i|i-1}\\right) - B^{\\mathcal {C}}_i \\left( 1_N \\!\\otimes \\!","{\\mathbf {y}}_i - \\widehat{{\\mathbf {y}}}_{i|i-1}\\right) \\!-\\!","B^{\\mathcal {I}}_i D_{H}^TR^{-1}{\\mathbf {r}}_i \\\\& \\qquad - B^{\\mathcal {I}}_i \\widetilde{D}_H \\left(1_N \\!\\otimes \\!","{\\mathbf {y}}_i - \\widehat{{\\mathbf {y}}}_{i|i-1}\\right) - B^{\\mathcal {I}}_i\\check{D}_H \\left( 1_N \\!\\otimes \\!","{\\mathbf {x}}_i - \\widehat{{\\mathbf {x}}}_{i|i-1}\\right) \\\\&= \\left( I_{\\!M\\!N} \\!-\\!","B^{\\mathcal {C}}_i \\!-\\!","B^{\\mathcal {I}}_i \\!\\widetilde{D}_{H}\\right) {\\mathbf {e}}_{i|i-1}\\!-\\!","B^{\\mathcal {I}}_i \\check{D}_{H} {\\mathbf {\\epsilon }}_{i|i-1}\\!-\\!","B^{\\mathcal {I}}_i\\!","D_{H}^T R^{-1} r_i.$ The state filtering update (), in vector form, is $\\widehat{{\\mathbf {x}}}_{i|i}= \\widehat{{\\mathbf {x}}}_{i|i-1}+ K_i \\left( \\widehat{{\\mathbf {y}}}_{i|i}\\!-\\!","\\left(I_N\\!\\otimes \\!G\\right) \\widehat{{\\mathbf {x}}}_{i|i-1}\\right) ,$ where, $K_i = \\text{blockdiag} \\lbrace K^1_i, \\cdots , K^N_i \\rbrace $ .","Using the relation $\\widehat{{\\mathbf {y}}}_{i|i}= \\left(1_N\\otimes {\\mathbf {y}}_i\\right) - {\\mathbf {e}}_{i|i}= \\left(I_N\\!\\otimes \\!G\\right)\\left(1_N\\otimes {\\mathbf {x}}_i\\right) - {\\mathbf {e}}_{i|i}$ , we expand the state filter error process ${\\mathbf {e}}_{i|i}$ , ${\\mathbf {\\epsilon }}_{i|i}&= 1_N \\!\\otimes \\!","{\\mathbf {x}}_i - \\widehat{{\\mathbf {x}}}_{i|i}\\\\&= \\left( 1_{\\!N} \\!\\otimes \\!","{\\mathbf {x}}_i - \\widehat{{\\mathbf {x}}}_{i|i-1}\\right) \\!-\\!","K_i \\left(I_{\\!N}\\!\\otimes \\!G\\right) \\left(1_{\\!N} \\!\\otimes \\!","{\\mathbf {x}}_i \\!-\\!","\\widehat{{\\mathbf {x}}}_{i|i-1}\\right) \\!+\\!","K_i {\\mathbf {e}}_{i|i}\\\\&= \\left( I_{MN} - K^{\\mathcal {C}}_i - K_i \\left(I_N\\!\\otimes \\!G\\right) \\right) {\\mathbf {\\epsilon }}_{i|i-1}+ K_i {\\mathbf {e}}_{i|i}.$ The dynamics of the pseudo-state and state prediction errors, ${\\mathbf {e}}_{i+1|i}&= 1_N \\!\\otimes \\!","{\\mathbf {y}}_{i+1} - \\widehat{{\\mathbf {y}}}_{i+1|i}\\\\&= \\left(I_N\\!\\otimes \\!\\widetilde{A}\\right)\\left(1_N \\!\\otimes \\!","{\\mathbf {y}}_i\\right) + \\left(I_N\\!\\otimes \\!","\\check{A} \\right)\\left(1_N \\!\\otimes \\!","{\\mathbf {x}}_i\\right) \\\\&\\quad + 1_N\\!\\otimes \\!","(G{\\mathbf {v}}_i) - \\left(I_N\\!\\otimes \\!\\widetilde{A}\\right)\\widehat{{\\mathbf {y}}}_{i|i}-\\left(I_N\\!\\otimes \\!","\\check{A} \\right)\\widehat{{\\mathbf {x}}}_{i|i}\\\\&= \\left(I_N\\!\\otimes \\!\\widetilde{A}\\right){\\mathbf {e}}_{i|i}+ \\left(I_N\\!\\otimes \\!","\\check{A} \\right) {\\mathbf {\\epsilon }}_{i|i}+ 1_N\\!\\otimes \\!","(G{\\mathbf {v}}_i) \\\\{\\mathbf {\\epsilon }}_{i+1|i}&= 1_N \\!\\otimes \\!","{\\mathbf {x}}_{i+1} - \\widehat{{\\mathbf {x}}}_{i+1|i}\\\\&= \\left(I_N\\!\\otimes \\!A\\right)\\left(1_N \\!\\otimes \\!","{\\mathbf {x}}_i\\right) + 1_N\\!\\otimes \\!","{\\mathbf {v}}_i - \\left(I_N\\!\\otimes \\!A\\right)\\widehat{{\\mathbf {x}}}_{i|i}\\\\&= \\left(I_N\\!\\otimes \\!A\\right){\\mathbf {\\epsilon }}_{i|i}+ 1_N\\!\\otimes \\!","{\\mathbf {v}}_i.$ Since the state ${\\mathbf {x}}_i$ , pseudo-state ${\\mathbf {y}}_i$ , their initial condition and all the noises are Gaussian, their estimates are also Gaussian making all the filtering and prediction errors Gaussian.","$\\Box $"],["Proof of Lemma ","Lemma REF and Assumption REF guarantee that ${\\mathbf {\\epsilon }}_{i|i}, {\\mathbf {\\epsilon }}_{i+1|i}, {\\mathbf {e}}_{i|i}, {\\mathbf {v}}_i, {\\mathbf {r}}_i$ are Gaussian.","The error noises $\\widetilde{\\mathbf {\\phi }}_i$ and ${\\mathbf {\\phi }}_i$ are therefore Gaussian as they are linear combinations of the error processes and the model noises ${\\mathbf {\\epsilon }}_{i|i}, {\\mathbf {\\epsilon }}_{i+1|i}, {\\mathbf {e}}_{i|i}, {\\mathbf {v}}_i, {\\mathbf {r}}_i$ .","By Lemma REF , we have $\\mathbb {E}\\left[{\\mathbf {\\epsilon }}_{i|i}\\right] = \\mathbb {E}\\left[{\\mathbf {\\epsilon }}_{i+1|i}\\right] = \\mathbb {E}\\left[{\\mathbf {e}}_{i|i}\\right] = \\mathbf {0}$ .","From Assumption REF , we know $\\mathbb {E}\\left[{\\mathbf {v}}_i\\right] = \\mathbb {E}\\left[{\\mathbf {r}}_i\\right] = \\mathbf {0}$ .","We take expectation on both sides of (REF )-() and apply these relations $\\mathbb {E}\\left[\\widetilde{\\mathbf {\\phi }}_i\\right] &= \\!\\left( I_{\\!N} \\!\\otimes \\!","\\check{A}\\right)\\!","\\mathbb {E}\\left[{\\mathbf {\\epsilon }}_{i|i}\\right] - \\left(I_{\\!N} \\!\\otimes \\!","\\widetilde{A}\\right)\\!","B^{\\mathcal {I}}_i D_{H}^T R^{-1} \\mathbb {E}\\left[r_i\\right] \\nonumber \\\\& \\quad + 1_{\\!N} \\!\\otimes \\!","\\left(G \\mathbb {E}\\left[{\\mathbf {v}}_i\\right]\\right) - \\left(I_{\\!N} \\!\\otimes \\!","\\widetilde{A}\\right)\\!B^{\\mathcal {I}}_i \\check{D}_{H} \\mathbb {E}\\left[{\\mathbf {\\epsilon }}_{i|i-1}\\right] = \\mathbf {0}, \\\\\\mathbb {E}\\left[{\\mathbf {\\phi }}_i\\right] &= \\left( I_{\\!N} \\!\\otimes \\!","A\\right)\\!","K_i \\mathbb {E}\\left[{\\mathbf {e}}_{i|i}\\right] + 1_N \\otimes \\mathbb {E}\\left[{\\mathbf {v}}_i\\right] = \\mathbf {0}.$ Combining (), () and (REF ), we have $\\widetilde{\\mathbf {\\phi }}_i &= F_1 {\\mathbf {\\epsilon }}_{i|i-1}+ F_2 {\\mathbf {e}}_{i|i-1}- F_3 D_{H}^T R^{-1} r_i + 1_{\\!N} \\!\\otimes \\!","\\left(G {\\mathbf {v}}_i\\right)$ where, $F_1 &= \\left( I_{\\!N} \\!\\otimes \\!","\\check{A}\\!\\right)\\!","\\left( I_{\\!M\\!N} \\!-\\!","K_i \\!\\left(I_{\\!N}\\!\\otimes \\!G\\right)\\!","-\\!","K_i \\!B^{\\mathcal {I}}_i \\check{D}_{H}\\!\\!","\\right) \\!-\\!","\\left( I_{\\!N} \\!\\otimes \\!","\\widetilde{A}\\!\\right)\\!","\\!B^{\\mathcal {I}}_i \\check{D}_{H} \\\\F_2 &= \\left( I_{\\!N} \\!\\otimes \\!","\\check{A}\\!\\right)\\!","K_i \\left( I_{\\!M\\!N} \\!-\\!","B^{\\mathcal {C}}_i \\!-\\!","B^{\\mathcal {I}}_i \\!","\\widetilde{D}_{H} \\right) \\\\F_3 &= \\left( I_{\\!N} \\!\\otimes \\!","\\widetilde{A}\\!\\right)\\!","\\!B^{\\mathcal {I}}_i.$ Since $\\widetilde{\\mathbf {\\phi }}_i$ and ${\\mathbf {\\phi }}_i$ are zero-mean, the noise covariances are, $\\widetilde{\\Phi }_i &\\!=\\!","\\mathbb {E} \\!","\\left[\\!\\widetilde{\\mathbf {\\phi }}_i \\widetilde{\\mathbf {\\phi }}^T_i \\!","\\right] \\!=\\!","F_1 \\Sigma _{i|i-1} F_1^T \\!+\\!","F_2 P_{i|i-1} F_2^T \\!+\\!","F_3 \\overline{D}_H F^T_3 \\nonumber \\\\& \\qquad \\qquad + J \\!\\otimes \\!","\\left(GVG\\right) + F_1 \\Pi _{i|i-1} F_2^T + F_2 \\Pi _{i|i-1}^T F_1^T, \\\\\\Phi _i &\\!=\\!","\\mathbb {E}\\!","\\left[ \\!","{\\mathbf {\\phi }}_i {\\mathbf {\\phi }}^T_i \\!","\\right] \\!=\\!","\\left( I_{\\!N} \\!\\otimes \\!","A\\right)\\!","K_i \\Sigma _{i|i} \\!","K^T_i \\!","\\left( I_{\\!N} \\!\\otimes \\!","A^T\\right) \\!+\\!","J \\!\\otimes \\!","V,$ where, $F_1, F_2$ and $F_3$ are defined in (REF )-().","$\\Box $"],["Proof of Lemma ","We first compute the conditional means $\\overline{\\widetilde{\\mathbf {\\theta }}}^n_i$ and $\\overline{\\mathbf {\\theta }}^n_i$ of the new information $\\widetilde{\\mathbf {\\theta }}^n_i$ and ${\\mathbf {\\theta }}^n_i$ from (REF ).","The means $\\overline{\\widetilde{\\mathbf {\\theta }}}^n_i$ and $\\overline{\\mathbf {\\theta }}^n_i$ depend on the conditional means of $\\widehat{{\\mathbf {y}}}_{i|i-1}^{l}, \\widetilde{\\mathbf {z}}^n_{i-1}$ and $\\widehat{{\\mathbf {y}}}_{i|i}^n$ .","$&\\mathbb {E} \\begin{bmatrix} \\widehat{{\\mathbf {y}}}_{i|i-1}^{l} | \\widetilde{\\mathbf {z}}^n_{i-1}, \\lbrace \\widehat{\\mathbf {y}}^l_{i-1|i-2} \\rbrace _{l \\in \\Omega _n} \\end{bmatrix} \\\\&= \\mathbb {E} \\left[ \\mathbb {E} \\left[ {\\mathbf {y}}_i | \\widetilde{\\mathbf {z}}^l_{i-1}, \\lbrace \\widehat{\\mathbf {y}}^k_{i-1|i-2} \\rbrace _{k \\in \\Omega _l} \\right] \\;|\\; \\widetilde{\\mathbf {z}}^n_{i-1}, \\lbrace \\widehat{\\mathbf {y}}^l_{i-1|i-2} \\rbrace _{l \\in \\Omega _n} \\right] \\\\&= \\mathbb {E} \\left[ \\mathbb {E} \\left[ {\\mathbf {y}}_i | \\widetilde{\\mathbf {z}}^n_{i-1}, \\lbrace \\widehat{\\mathbf {y}}^l_{i-1|i-2} \\rbrace _{l \\in \\Omega _n} \\right] \\;|\\; \\widetilde{\\mathbf {z}}^l_{i-1}, \\lbrace \\widehat{\\mathbf {y}}^k_{i-1|i-2} \\rbrace _{k \\in \\Omega _l} \\right] \\\\&= \\mathbb {E} \\left[ \\widehat{{\\mathbf {y}}}_{i|i-1}^{n} \\;|\\; \\widetilde{\\mathbf {z}}^l_{i-1}, \\lbrace \\widehat{\\mathbf {y}}^k_{i-1|i-2} \\rbrace _{k \\in \\Omega _l} \\right] = \\widehat{{\\mathbf {y}}}_{i|i-1}^{n}, \\quad \\forall \\; l \\in \\Omega _n.","\\\\&\\mathbb {E} \\begin{bmatrix} \\widetilde{\\mathbf {z}}^n_i \\;|\\; \\widetilde{\\mathbf {z}}^n_{i-1}, \\lbrace \\widehat{\\mathbf {y}}^l_{i-1|i-2} \\rbrace _{l \\in \\Omega _n} \\end{bmatrix} \\\\&= \\mathbb {E} \\begin{bmatrix} \\widetilde{H}_n{\\mathbf {y}}_i + H^T_n R_n^{-1} {\\mathbf {r}}^n_i + \\check{H}_n {\\mathbf {x}}_i \\;|\\; \\widetilde{\\mathbf {z}}^n_{i-1}, \\lbrace \\widehat{\\mathbf {y}}^l_{i-1|i-2} \\rbrace _{l \\in \\Omega _n} \\end{bmatrix} \\\\&= \\widetilde{H}_n \\mathbb {E} \\begin{bmatrix} {\\mathbf {y}}_i \\;|\\; \\widetilde{\\mathbf {z}}^n_{i-1}, \\lbrace \\widehat{\\mathbf {y}}^l_{i-1|i-2} \\rbrace _{l \\in \\Omega _n} \\end{bmatrix} \\\\& \\qquad \\qquad \\qquad + \\check{H}_n \\mathbb {E} \\begin{bmatrix} {\\mathbf {x}}_i \\;|\\; \\widetilde{\\mathbf {z}}^n_{i-1}, \\lbrace \\widehat{\\mathbf {y}}^l_{i-1|i-2} \\rbrace _{l \\in \\Omega _n} \\end{bmatrix} \\\\&= \\widetilde{H}_n \\widehat{{\\mathbf {y}}}_{i|i-1}^n \\!+\\!","\\check{H}_n \\mathbb {E} \\left[ \\mathbb {E} \\left[ {\\mathbf {x}}_i | \\widehat{{\\mathbf {y}}}_{i|i}^n, \\widehat{\\mathbf {x}}^l_{i-1|i-2}, \\widetilde{\\mathbf {z}}^n_{i-1}, \\widehat{\\mathbf {y}}^l_{i-1|i-2}, \\in \\Omega _n \\right] \\right] \\\\&= \\widetilde{H}_n \\widehat{{\\mathbf {y}}}_{i|i-1}^n + \\check{H}_n \\mathbb {E} \\left[ \\widehat{{\\mathbf {x}}}_{i|i-1}^n \\;|\\; \\widetilde{\\mathbf {z}}^n_{i-1}, \\lbrace \\widehat{\\mathbf {y}}^l_{i-1|i-2} \\rbrace _{l \\in \\Omega _n}\\right] \\\\&= \\widetilde{H}_n \\widehat{{\\mathbf {y}}}_{i|i-1}^n + \\check{H}_n \\widehat{{\\mathbf {x}}}_{i|i-1}^n.","\\\\&\\mathbb {E} \\begin{bmatrix} \\widehat{{\\mathbf {y}}}_{i|i}^n | \\widehat{{\\mathbf {y}}}_{i-1|i-1}^n \\end{bmatrix} = \\mathbb {E} \\left[ {\\mathbf {y}}_i - {\\mathbf {e}}_{i|i}^n \\;|\\; \\widehat{{\\mathbf {y}}}_{i-1|i-1}^n \\right] \\\\&= \\mathbb {E} \\left[ {\\mathbf {y}}_i \\;|\\; \\widehat{{\\mathbf {y}}}_{i-1|i-1}^n \\right] - \\mathbb {E} \\left[{\\mathbf {e}}_{i|i}^n \\;|\\; \\widehat{{\\mathbf {y}}}_{i-1|i-1}^n \\right] \\\\&= G \\mathbb {E} \\left[ {\\mathbf {x}}_i \\;|\\; \\widehat{{\\mathbf {y}}}_{i-1|i-1}^n \\right] = G\\widehat{{\\mathbf {x}}}_{i|i-1}^n.$ The conditional means of $\\widehat{{\\mathbf {y}}}_{i|i-1}^{l}, \\widetilde{\\mathbf {z}}^n_{i-1}$ and $\\widehat{{\\mathbf {y}}}_{i|i}^n$ shows that the new uncorrelated information defined in (REF )-() expands to the vectors $\\widetilde{\\mathbf {\\nu }}^n_i$ and ${\\mathbf {\\nu }}^n_i$ in (REF ).","Note that $\\widetilde{\\mathbf {\\nu }}^n_i$ and ${\\mathbf {\\nu }}^n_i$ are zero mean, by definition, and, are Gaussian since they are linear combination of Gaussian vectors.","Now to prove $\\widetilde{\\mathbf {\\nu }}^n_i$ and ${\\mathbf {\\nu }}^n_i$ are sequence of uncorrelated vectors, we have to show: $\\mathbb {E}\\left[ \\widetilde{\\mathbf {\\nu }}^n_i \\widetilde{\\mathbf {\\nu }}^{n^T}_j\\right] = \\mathbb {E}\\left[ {\\mathbf {\\nu }}^n_i {\\mathbf {\\nu }}^{n^T}_j\\right] = {\\mathbf {0}}, \\; \\forall \\; i\\ne j, \\; \\forall \\; n.$ First, we write $\\widetilde{\\mathbf {\\nu }}^n_i$ and ${\\mathbf {\\nu }}^n_i$ in terms of the filtering and prediction error processes using (), (REF )-(), $\\widetilde{\\mathbf {\\nu }}^n_i \\!=\\!","\\begin{bmatrix}{\\mathbf {e}}_{i|i-1}^n \\!-\\!","{\\mathbf {e}}_{i|i-1}^{l_1} \\\\ \\vdots \\\\ {\\mathbf {e}}_{i|i-1}^n \\!-\\!","{\\mathbf {e}}_{i|i-1}^{l_{d_n}} \\\\ \\widetilde{H}_{\\!n} {\\mathbf {e}}_{i|i-1}^n \\!+\\!","\\check{H}_{\\!n} {\\mathbf {\\epsilon }}_{i|i-1}^n + H^T_n R^{-1}_n {\\mathbf {r}}^n_i\\end{bmatrix}\\!\\!\\!, \\;\\;{\\mathbf {\\nu }}^n_i \\!=\\!","G{\\mathbf {\\epsilon }}_{i|i-1}^n \\!-\\!","{\\mathbf {e}}_{i|i}^n.$ Without loss of generality, we consider $i > j$ .","The rest of the proof is similar to the proof of Lemma 5 in [5].","Here, the only difference is that we should condition on $\\lbrace \\widetilde{\\mathbf {z}}^l_{i-1}\\rbrace _{l \\in \\Omega _n}, \\lbrace \\widehat{\\mathbf {y}}^l_{i-1|i-1} \\rbrace _{l \\in \\Omega _n}$ .","$\\Box $ Proof of Theorems"],["Proof of Theorem ","In Lemma REF , we showed that $\\widetilde{\\mathbf {\\nu }}^n_i$ and ${\\mathbf {\\nu }}^n_i$ are independent Gaussian sequences.","By the Innovations Property [35], there are $1-1$ correspondence between $\\lbrace \\widetilde{{\\mathbf {z}}}^n_i, \\lbrace \\widehat{{\\mathbf {y}}}_{i|i-1}^l \\rbrace _{l \\in \\overline{\\Omega }_n} \\rbrace $ and $\\widetilde{\\mathbf {\\nu }}^n_i$ , and between $\\lbrace \\widehat{{\\mathbf {y}}}_{i|i}^n \\rbrace $ and ${\\mathbf {\\nu }}^n_i$ .","The Innovations Property guarantees that there exists a unique way to get one from the other.","$\\widehat{{\\mathbf {y}}}_{i|i}^n \\!=\\!","\\mathbb {E} \\!\\left[ {\\mathbf {y}}_i \\;|\\; \\widetilde{{\\mathbf {z}}}^n_i, \\lbrace \\widehat{{\\mathbf {y}}}_{i|i-1}^l \\rbrace _{l \\in \\overline{\\Omega }_n} \\right] ~&\\Longleftrightarrow ~ \\widehat{{\\mathbf {y}}}_{i|i}^n \\!=\\!","\\mathbb {E} \\!\\left[ {\\mathbf {y}}_i \\;|\\; \\widetilde{\\mathbf {\\nu }}^n_i \\right] \\\\\\widehat{{\\mathbf {x}}}_{i|i}^n \\!=\\!","\\mathbb {E} \\left[ {\\mathbf {x}}_i \\;|\\; \\widehat{{\\mathbf {y}}}_{i|i}^n \\right] ~&\\Longleftrightarrow ~ \\widehat{{\\mathbf {x}}}_{i|i}^n \\!=\\!","\\mathbb {E} \\left[ {\\mathbf {x}}_i \\;|\\; {\\mathbf {\\nu }}^n_i \\right]$ By the Gauss-Markov principle, $ \\widehat{{\\mathbf {y}}}_{i|i}^n &= \\widehat{{\\mathbf {y}}}_{i|i-1}^n + \\widehat{B}^n_i \\widetilde{\\mathbf {\\nu }}^n_i \\\\\\widehat{{\\mathbf {x}}}_{i|i}^n &= \\widehat{{\\mathbf {x}}}_{i|i-1}^n + K^n_i {\\mathbf {\\nu }}^n_i$ where $\\widehat{B}^n_i$ are the non-zero blocks of the $n^{\\text{th}}$ row of $B_i$ .","Now we expand the term $\\widehat{B}^n_i \\widetilde{\\mathbf {\\nu }}^n_i$ by multiplying the gain blocks $B^{nl}_i$ with the corresponding $(\\widehat{{\\mathbf {y}}}_{i|i-1}^l - \\widehat{{\\mathbf {y}}}_{i|i-1}^n)$ and the gain block $B^{nn}_i$ with $\\left( \\widetilde{{\\mathbf {z}}}^n_i \\!\\!-\\!","\\widetilde{H}_{\\!n} \\widehat{{\\mathbf {y}}}_{i|i-1}^n \\!-\\!","\\check{H}_{\\!n} \\widehat{{\\mathbf {x}}}_{i|i-1}^n \\right)$ .","This gives us the consensus$+$ innovations filtering pseudo-state update (REF ).","The pseudo-state and state prediction updates are $\\widehat{{\\mathbf {y}}}_{i+1|i}^n &\\!=\\!","\\mathbb {E} \\left[ {\\mathbf {y}}_{i+1} \\;|\\; \\widetilde{{\\mathbf {z}}}^n_i, \\lbrace \\widehat{{\\mathbf {y}}}_{i|i-1}^l \\rbrace _{l \\in \\overline{\\Omega }_n} \\right] \\\\&\\!=\\!","\\mathbb {E} \\left[ \\widetilde{A}{\\mathbf {y}}_i + G{\\mathbf {v}}_i + \\check{A}{\\mathbf {x}}_i \\;|\\; \\widetilde{{\\mathbf {z}}}^n_i, \\lbrace \\widehat{{\\mathbf {y}}}_{i|i-1}^l \\rbrace _{l \\in \\overline{\\Omega }_n} \\right] \\\\&\\!=\\!","\\widetilde{A} \\widehat{{\\mathbf {y}}}_{i|i}^n + \\check{A} \\widehat{{\\mathbf {x}}}_{i|i}^n \\\\\\widehat{{\\mathbf {x}}}_{i+1|i}^n &\\!=\\!","\\mathbb {E} \\left[ {\\mathbf {x}}_{i\\!+\\!1} \\;|\\; \\widehat{{\\mathbf {y}}}_{i|i}^n \\right] \\!=\\!","\\mathbb {E} \\left[ A {\\mathbf {x}}_i + {\\mathbf {v}}_i \\;|\\; \\widehat{{\\mathbf {y}}}_{i|i}^n \\right] = A \\widehat{{\\mathbf {x}}}_{i|i}^n.","\\quad \\Box $"],["Proof of Theorem ","By Lemma REF and Lemma REF , the error processes, ${\\mathbf {e}}_{i|i}$ , ${\\mathbf {e}}_{i+1|i}$ , ${\\mathbf {\\epsilon }}_{i|i}$ , and ${\\mathbf {\\epsilon }}_{i+1|i}$ are zero-mean Gaussian.","The Lyapunov-type iterations ()-() of the filter and predictor error covariances, $P_{i|i}$ , $\\Sigma _{i|i}$ , $\\Pi _{i|i}$ , $P_{i+1|i}$ , $\\Sigma _{i+1|i}$ , and $\\Pi _{i+1|i}$ , follow directly from the definitions (REF )-() and error dynamics (REF )-() by algebraic manipulations.","$\\Box $"],["Proof of Theorem ","For any square matrix, $\\rho (\\widetilde{F}) \\le \\Vert \\widetilde{F} \\Vert $ .","Hence if $\\Vert \\widetilde{F} \\Vert < 1$ , then it implies that $\\rho (\\widetilde{F}) < 1$ .","We derive the tracking capacity with the sufficient condition, $\\Vert \\widetilde{F} \\Vert < 1$ , $\\Vert \\widetilde{F} \\Vert &= \\begin{Vmatrix} \\left( I_{\\!N} \\!\\otimes \\!\\widetilde{A}\\!\\right)\\!","\\left( I_{\\!M\\!N} \\!-\\!","B^{\\mathcal {C}}_i \\!-\\!","B^{\\mathcal {I}}_i \\!","\\widetilde{D}_{H} \\right) \\end{Vmatrix} \\\\&\\le \\begin{Vmatrix} I_{\\!N} \\!\\otimes \\!\\widetilde{A}\\!","\\end{Vmatrix} \\begin{Vmatrix} \\left( I_{\\!M\\!N} \\!-\\!","B^{\\mathcal {C}}_i \\!-\\!","B^{\\mathcal {I}}_i \\!","\\widetilde{D}_{H} \\right) \\end{Vmatrix} \\\\&\\le \\begin{Vmatrix} GAG^{\\dagger }\\!","\\end{Vmatrix} \\begin{Vmatrix} \\left( I_{\\!M\\!N} \\!-\\!","B^{\\mathcal {C}}_i \\!-\\!","B^{\\mathcal {I}}_i \\!","\\widetilde{D}_{H} \\right) \\end{Vmatrix} \\\\&\\le \\frac{\\lambda _m}{\\lambda _1} \\begin{Vmatrix} A\\!","\\end{Vmatrix} \\begin{Vmatrix} \\left( I_{\\!M\\!N} \\!-\\!","B^{\\mathcal {C}}_i \\!-\\!","B^{\\mathcal {I}}_i \\!","\\widetilde{D}_{H} \\right) \\end{Vmatrix},$ where, $\\Vert G \\Vert _2 = \\lambda _m$ and $\\Vert G^{\\dagger } \\Vert _2 = \\frac{1}{\\lambda _1}$ .","Since $G$ is a symmetric positive semi-definite matrix, its spectral norm is its largest eigenvalue $(\\lambda _m)$ and the spectral norm of its pseudo-inverse, $G^{\\dagger }$ , is the inverse of its smallest non-zero eigenvalue $(\\lambda _1)$ .","If there exists $B^{\\mathcal {C}}_i$ and $B^{\\mathcal {I}}_i$ such that $\\frac{\\lambda _m}{\\lambda _1} \\begin{Vmatrix} A\\!","\\end{Vmatrix} \\begin{Vmatrix} \\left( I_{\\!M\\!N} \\!-\\!","B^{\\mathcal {C}}_i \\!-\\!","B^{\\mathcal {I}}_i \\!","\\widetilde{D}_{H} \\right) \\end{Vmatrix} < 1,$ then $\\Vert \\widetilde{F} \\Vert < 1$ and also $\\rho (\\widetilde{F}) < 1$ .","The bound on the spectral norm of $A$ is, $\\Vert A \\Vert &< \\frac{\\lambda _1}{\\lambda _m \\begin{Vmatrix} \\left( I_{\\!M\\!N} \\!-\\!","B^{\\mathcal {C}}_i \\!-\\!","B^{\\mathcal {I}}_i \\!","\\widetilde{D}_{H} \\right) \\end{Vmatrix} } \\\\&\\le \\max _{B^{\\mathcal {C}}_i, B^{\\mathcal {I}}_i} \\frac{\\lambda _1}{\\lambda _m \\begin{Vmatrix} \\left( I_{\\!M\\!N} \\!-\\!","B^{\\mathcal {C}}_i \\!-\\!","B^{\\mathcal {I}}_i \\!","\\widetilde{D}_{H} \\right) \\end{Vmatrix} } = C.$ Thus as long as $\\Vert A \\Vert _2 < C$ , there exists $B^{\\mathcal {C}}_i$ and $B^{\\mathcal {I}}_i$ such that $\\rho (\\widetilde{F}) < 1$ .","By global detectability Assumption REF , there exists $K_i$ such that $\\rho (F) < 1$ .","Refer to [35], for the convergence conditions of the centralized information filters.","Further, by Lemma REF the Gaussian noises processes $\\widetilde{\\mathbf {\\phi }}_i$ and ${\\mathbf {\\phi }}_i$ have bounded noise covariances.","Thus, if $\\Vert A \\Vert _2 < C$ , then from (REF )-() we conclude that the CIKF (REF )-() converges with bounded MSE.","$\\Box $"],["Proof of Theorem ","By the Innovations Property and the Gauss-Markov principle [35], the optimal gains $\\widehat{B}^n_i$ and $K^n_i$ in (REF )-() are: $\\widehat{B}^n_i &= \\Sigma _{{\\mathbf {y}}_i \\widetilde{\\mathbf {\\nu }}^n_i} \\left(\\Sigma _{\\widetilde{\\mathbf {\\nu }}^n_i}\\right)^{-1}, \\\\K^n_i &= \\Sigma _{{\\mathbf {x}}_i {\\mathbf {\\nu }}^n_i} \\left(\\Sigma _{{\\mathbf {\\nu }}^n_i}\\right)^{-1}$ that yield minimized MSE estimates $\\widehat{{\\mathbf {y}}}_{i|i}^n$ and $\\widehat{{\\mathbf {x}}}_{i|i}^n$ of the pseudo-state ${\\mathbf {y}}_i$ and of the field ${\\mathbf {x}}_i$ , respectively, at each agent $n$ .","The cross-covariances $\\Sigma _{{\\mathbf {y}}_i \\widetilde{\\mathbf {\\nu }}^n_i}$ and $\\Sigma _{{\\mathbf {x}}_i {\\mathbf {\\nu }}^n_i}$ are $&\\Sigma _{{\\mathbf {y}}_i \\widetilde{\\mathbf {\\nu }}^n_i} = \\mathbb {E} \\left[ \\left({\\mathbf {y}}_i - \\overline{{\\mathbf {y}}}_i \\right) \\widetilde{\\mathbf {\\nu }}^{n^T}_i \\right] \\\\&= \\mathbb {E} \\!\\left[ \\!\\left({\\mathbf {e}}_{i|i-1}^n \\!+\\!\\left(\\widehat{{\\mathbf {y}}}_{i|i-1}^n \\!-\\!","\\overline{{\\mathbf {y}}}_i \\right) \\right)\\!\\!","\\begin{bmatrix} {\\mathbf {e}}_{i|i-1}^n \\!-\\!","{\\mathbf {e}}_{i|i-1}^{l_1} \\\\ \\vdots \\\\ {\\mathbf {e}}_{i|i-1}^n \\!-\\!","{\\mathbf {e}}_{i|i-1}^{l_{d_n}} \\\\ \\widetilde{H}_{\\!n} {\\mathbf {e}}_{i|i-1}^n \\!\\!+\\!","\\check{H}_{\\!n} {\\mathbf {\\epsilon }}_{i|i-1}^n \\!+\\!","H^T_n R^{-1}_n {\\mathbf {r}}^n_i \\end{bmatrix}^{\\!\\!T} \\!\\right] \\\\&= \\mathbb {E}\\!","\\left[ {\\mathbf {e}}_{i|i-1}^n \\!","\\left[ \\!\\!\\begin{array}{l;{2pt/2pt}l;{1pt/1pt}l;{2pt/2pt}l}\\!","{\\mathbf {e}}_{i|i-1}^{n^T} \\!-\\!","{\\mathbf {e}}_{i|i-1}^{l^T_1}\\!","&\\!","\\cdots \\!","&\\!","{\\mathbf {e}}_{i|i-1}^{n^T} \\!-\\!","{\\mathbf {e}}_{i|i-1}^{l^T_{d_n}} \\!&\\!","\\widetilde{H}_{\\!n} {\\mathbf {e}}_{i|i-1}^{n^T} \\\\& & & \\; + \\check{H}_{\\!n}{\\mathbf {\\epsilon }}_{i|i-1}^{n^T} \\!\\!\\end{array}\\!\\!","\\right] \\!","\\right] \\\\&= \\left[\\begin{array}{l;{2pt/2pt}l;{1pt/1pt}l;{2pt/2pt}l}\\!\\!\\!\\!","P_{i|i\\!-\\!1}^{nn} \\!-\\!","P_{i|i\\!-\\!1}^{nl_1} & \\cdots & P_{i|i\\!-\\!1}^{nn} \\!-\\!","P_{i|i\\!-\\!1}^{nl_{d_n}} & P_{i|i\\!-\\!1}^{nn} \\!","\\widetilde{H}^T_n \\!+\\!","\\Pi _{i|i\\!-\\!1}^{nn^T} \\!","\\check{H}^T_n \\!\\!\\!\\!\\end{array}\\right], \\\\&\\Sigma _{{\\mathbf {x}}_i {\\mathbf {\\nu }}^n_i} = \\mathbb {E} \\left[ \\left({\\mathbf {x}}_i - \\overline{{\\mathbf {x}}}_i \\right) {\\mathbf {\\nu }}^{n^T}_i \\right] \\\\&= \\mathbb {E} \\left[ \\left({\\mathbf {\\epsilon }}_{i|i-1}^n +\\left(\\widehat{{\\mathbf {x}}}_{i|i-1}^n - \\overline{{\\mathbf {x}}}_i \\right) \\right) \\left( G{\\mathbf {\\epsilon }}_{i|i-1}^{n} \\!-\\!","{\\mathbf {e}}_{i|i}^{n} \\right)^T \\right] \\\\&= \\mathbb {E} \\left[ {\\mathbf {\\epsilon }}_{i|i-1}^n \\!","\\left( G{\\mathbf {\\epsilon }}_{i|i-1}^{n} \\!-\\!","{\\mathbf {e}}_{i|i}^{n} \\right)^T \\!","\\right] = \\Sigma _{i|i\\!-\\!1}^{nn} G - \\Gamma ^{nn}_i.$ In the above derivation, using the iterated law of expectation it can be shown that the terms $\\mathbb {E} \\left[ \\left(\\widehat{{\\mathbf {y}}}_{i|i-1}^n - \\overline{{\\mathbf {y}}}_i \\right) \\widetilde{\\mathbf {\\nu }}^{n^T}_i \\right] = \\mathbb {E} \\left[ \\left(\\widehat{{\\mathbf {x}}}_{i|i-1}^n - \\overline{{\\mathbf {x}}}_i \\right) {\\mathbf {\\nu }}^{n^T}_i \\right] = {\\mathbf {0}}$ .","Also, $\\mathbb {E} \\left[ {\\mathbf {e}}_{i|i-1}^n {\\mathbf {r}}^{n^T}_i \\right] = {\\mathbf {0}}$ due to the statistical independence of the noise sequences.","Similarly, the covariances matrices $\\Sigma _{\\widetilde{\\mathbf {\\nu }}^n_i}$ and $\\Sigma _{{\\mathbf {\\nu }}^n_i}$ are: $&\\Sigma _{\\widetilde{\\mathbf {\\nu }}^n_i} = \\mathbb {E} \\left[ \\widetilde{\\mathbf {\\nu }}^{n}_i \\widetilde{\\mathbf {\\nu }}^{n^T}_i \\right] \\\\&= \\mathbb {E} \\left[ \\begin{bmatrix} {\\mathbf {e}}_{i|i-1}^n \\!-\\!","{\\mathbf {e}}_{i|i-1}^{l_1} \\\\ \\vdots \\\\ {\\mathbf {e}}_{i|i-1}^n \\!-\\!","{\\mathbf {e}}_{i|i-1}^{l_{d_n}} \\\\ \\widetilde{H}_{\\!n} {\\mathbf {e}}_{i|i-1}^n \\!+\\!","\\check{H}_{\\!n} {\\mathbf {\\epsilon }}_{i|i-1}^n \\\\ \\qquad + H^T_n R^{-1}_n {\\mathbf {r}}^n_i \\end{bmatrix} \\begin{bmatrix} {\\mathbf {e}}_{i|i-1}^n \\!-\\!","{\\mathbf {e}}_{i|i-1}^{l_1} \\\\ \\vdots \\\\ {\\mathbf {e}}_{i|i-1}^n \\!-\\!","{\\mathbf {e}}_{i|i-1}^{l_{d_n}} \\\\ \\widetilde{H}_{\\!n} {\\mathbf {e}}_{i|i-1}^n \\!+\\!","\\check{H}_{\\!n} {\\mathbf {\\epsilon }}_{i|i-1}^n \\\\ \\qquad + H^T_n R^{-1}_n {\\mathbf {r}}^n_i \\end{bmatrix}^T \\right] \\\\&\\Sigma _{{\\mathbf {\\nu }}^n_i} = \\mathbb {E} \\left[ {\\mathbf {\\nu }}^{n}_i {\\mathbf {\\nu }}^{n^T}_i \\right] = \\mathbb {E} \\left[ \\left( G{\\mathbf {\\epsilon }}_{i|i-1}^n \\!-\\!","{\\mathbf {e}}_{i|i}^n \\right) \\left( G{\\mathbf {\\epsilon }}_{i|i-1}^n \\!-\\!","{\\mathbf {e}}_{i|i}^n \\right)^T \\right].$ The rest of the derivation of $\\Sigma _{\\widetilde{\\mathbf {\\nu }}^n_i}$ and $\\Sigma _{{\\mathbf {\\nu }}^n_i}$ in terms of the error covariance matrices is by block-by-block multiplication of the above expressions.","$\\Box $ [Figure: NO_CAPTION Since July 2016, he is a Postdoctoral Researcher at IBM T.J. Watson Research Center, Yorktown Heights, NY.","His research interests are in statistical inference, distributed estimation over multi-agent networks, model adaptation in dynamic environments and time-series analysis; broadly in the areas of statistical signal processing, machine learning and data-mining with applications in healthcare and smart infrastructure.","[Figure: NO_CAPTION He is the Philip L. and Marsha Dowd University Professor at Carnegie Mellon University (CMU).","He was on the faculty at IST and has held visiting faculty appointments at MIT and New York University (NYU).","He founded and directs a large education and research program between CMU and Portugal, www.icti.cmu.edu.","His research interests are on data science, graph signal processing, and statistical and algebraic signal and image processing.","He has published over 550 papers and holds thirteen patents issued by the US Patent Office.","The technology of two of his patents (co-inventor A. Kavčić) are in about three billion disk drives read channel chips of 60 % of all computers sold in the last 13 years worldwide and was, in 2016, the subject of the largest university verdict/settlement in the information technologies area.","Dr. Moura is the IEEE Technical Activities Vice-President (2016) and member of the IEEE Board of Directors.","He served in several other capacities including IEEE Division IX Director, member of several IEEE Boards, President of the IEEE Signal Processing Society(SPS), Editor in Chief for the IEEE Transactions in Signal Processing, interim Editor in Chief for the IEEE Signal Processing Letters.","Dr. Moura has received several awards, including the Technical Achievement Award and the Society Award from the IEEE Signal Processing.","In 2016, he received the CMU College of Engineering Distinguished Professor of Engineering Award.","He is a Fellow of the IEEE, a Fellow of the American Association for the Advancement of Science (AAAS), a corresponding member of the Academy of Sciences of Portugal, Fellow of the US National Academy of Inventors, and a member of the US National Academy of Engineering."]],"string":"[\n [\n \"Consensus+Innovations Distributed Kalman Filter with Optimized Gains\"\n ],\n [\n \"Abstract In this paper, we address the distributed filtering and prediction of time-varying random fields represented by linear time-invariant (LTI) dynamical systems.\",\n \"The field is observed by a sparsely connected network of agents/sensors collaborating among themselves.\",\n \"We develop a Kalman filter type consensus+innovations distributed linear estimator of the dynamic field termed as Consensus+Innovations Kalman Filter.\",\n \"We analyze the convergence properties of this distributed estimator.\",\n \"We prove that the mean-squared error of the estimator asymptotically converges if the degree of instability of the field dynamics is within a pre-specified threshold defined as tracking capacity of the estimator.\",\n \"The tracking capacity is a function of the local observation models and the agent communication network.\",\n \"We design the optimal consensus and innovation gain matrices yielding distributed estimates with minimized mean-squared error.\",\n \"Through numerical evaluations, we show that, the distributed estimator with optimal gains converges faster and with approximately 3dB better mean-squared error performance than previous distributed estimators.\"\n ],\n [\n \"Introduction\",\n \"For decades, the Kalman-Bucy filter [1], [2] has played a key role in estimation, detection, or prediction of time-varying noisy signals.\",\n \"The Kalman filter is found in a wide variety of applications ranging from problems in navigation to environmental studies, computer vision to bioengineering, signal processing to econometrics.\",\n \"More recently, algorithms inspired by the Kalman filter have been applied to estimate random fields monitored by a network of sensors.\",\n \"In these problems, we distinguish two distinct layers: (a) the physical layer of the time-varying random field; and (b) the cyber layer of sensors observing the field.\",\n \"A centralized approach to field estimation poses several challenges.\",\n \"It requires that all sensors communicate their measurements to a centralized fusion center.\",\n \"This is fragile to central node failure and severely taxes computationally the fusion center.\",\n \"Moreover, it also requires excessive communication bandwidth to and from the fusion center.\",\n \"Hence, the centralized approach is inelastic to estimation of large-scale time-varying random fields, like, when estimating temperature, rainfall, or wind-speed over large geographical areas [3], [4].\",\n \"In [5], we proposed the Distributed Information Kalman Filter (DIKF) that is a distributed estimator of time-varying random fields consisting of two substructures.\",\n \"The first is the Dynamic Consensus on Pseudo-Observations (DCPO), a distributed estimator of the global average of the pseudo-observations (modified versions of the observations) of the agents.\",\n \"The second substructure uses these average estimates of pseudo-observations to estimate the time-varying random field.\",\n \"In this paper, we develop a distributed Kalman filter like estimator, the Consensus$+$ Innovations Kalman Filter (CIKF), that instead of using the pseudo-observations uses distributed estimates of the pseudo-state (modified version of a state) to estimate the field.\",\n \"We show how to design optimally the gain matrices of the CIKF.\",\n \"We prove that the CIKF converges in the mean-squared error (MSE) sense when the degree of instability of the dynamics of the random field is within the network tracking capacity [6], a threshold determined by the cyber network connectivity and the local observation models.\",\n \"Figure: Time-scales of operation of dynamics, sensing and communications.We review related prior research on distributed estimation of time-varying random fields.\",\n \"We classify prior work into two categories based on the time-scales of operation: (a) two time-scale and (b) single time-scale.\",\n \"In two-time scale distributed estimators, see Fig REF , agents exchange their information multiple number of times between each dynamics/observations time-scale [7], [8], [9], [10], [11], [12], [13], [14], [15], so that average consensus occurs between observations.\",\n \"In contrast, in single time-scale approaches [6], [16], [17], [18], [19], [20], [21], [22], [23], [24], [5], the agents collaborate with their neighbors only once in between each dynamics/observation evolution.\",\n \"In other words, the dynamics, observation, and communication follow the same time scale as depicted in Fig REF .\",\n \"The two time-scale approach demands fast communication between agents.\",\n \"In most practical applications, this is not true.\",\n \"Further, references [13], [14] developed distributed state estimators assuming local observability of the dynamic state in the physical layer.\",\n \"Such assumption is not feasible in large-scale systems.\",\n \"References [25], [26] assume a complete cyber network, which is not scalable.\",\n \"We are interested in a single time-scale approach.\",\n \"We consider that the time-varying system is not locally observable at each agent, and we assume that the communication network in the cyber layer is sparsely connected.\",\n \"In the single time-scale category, references [19], [20] propose distributed Kalman filters where the agents communicate among themselves using the Gossip protocol [27].\",\n \"Although Gossip filters require very low communication bandwidth, their MSE is higher and their convergence rate is lower than the consensus$+$ innovations based distributed approaches.\",\n \"References [21], [22], [24] introduced consensus$+$ innovations type distributed estimator for parameter (static state) estimation.\",\n \"This approach is extended to estimating time-varying random states in [6], [16], [17], [18].\",\n \"Distributed consensus$+$ innovations dynamic state estimators converge in MSE sense if their degree of instability of the state dynamics is below the network tracking capacity, see [6].\",\n \"In this paper, we derive the tracking capacity for the CIKF.\",\n \"Its tracking capacity is a function of the local observation models and of the agents communication connectivity.\",\n \"The single time-scale consensus$+$ innovations distributed estimators that we introduced in [28], [29], [30], [31] run a companion filter to estimate the global average of the pseudo-innovations, a modified version of the innovations.\",\n \"The DIKF, we proposed in [5], uses averaged pseudo-observations (linearly transformed observations) rather than pseudo-innovations.\",\n \"In contrast, this paper uses estimates of the pseudo-state, a linear transformation of the dynamic state.\",\n \"In centralized information filter, the pseudo-state is directly available from the observations.\",\n \"In the distributed setting, since not all the observations are available to the local sensors, the CIKF has to distributedly estimate the pseudo-state through a consensus step.\",\n \"Using pseudo-state rather than pseudo-innovations [28], [29], [30], [31] or pseudo-observations [5] leads to significant better performance as we show here.\",\n \"The MSE of the CIKF is lower than that of the distributed estimators in [19], [20], [28], [29], [30], [31], [5].\",\n \"Developing distributed estimators for time-varying random fields (distributed Kalman filter) has gained considerable attention over the last few years.\",\n \"The goal has been to achieve MSE performance as close as possible to the optimal centralized Kalman filter.\",\n \"We show in this paper that the CIKF converges to a bounded MSE solution requiring minimal assumptions, namely global detectability and connected network, and not requiring the additional distributed observability assumption, as needed by the DIKF.\",\n \"The distributed optimal time-varying field estimation is, in general, NP-hard.\",\n \"The distributed parameter estimation [21], [32] have shown asymptotic optimality, in the sense that the distributed parameter estimator is asymptotically unbiased, consistent, and efficient converging at the same rate as the centralized optimal estimator.\",\n \"However, estimation of time-varying random fields adds another degree of complication since, while information diffuses through the network, the field itself evolves.\",\n \"So, this lag causes a gap in performance between distributed and centralized field estimators.\",\n \"Numerical simulations show that the proposed CIKF improves the performance by 3dB over the DIKF, reducing by half the gap to the centralized (optimal) Kalman filter, while showing a faster convergence rate than the DIKF.\",\n \"These improvements significantly distinguish the CIKF from the DIKF.\",\n \"The rest of the paper is organized as follows.\",\n \"We describe the three aspects of the problem setup in Section .\",\n \"Section introduces the pseudo-state and presents the proposed optimal gain distributed Kalman filter (CIKF).\",\n \"In Section , we analyze the dynamics of the error processes and their covariances.\",\n \"Section includes the analysis of the tracking capacity of the proposed CIKF.\",\n \"We design the optimal gain matrices to obtain distributed estimates in Section .\",\n \"Numerical simulations are in Section .\",\n \"We present the concluding remarks in Section .\",\n \"Proofs of the proposition, lemmas and theorems are in the Appendix REF , Appendix REF -REF , and Appendix REF -REF , respectively.\"\n ],\n [\n \"System, Observation, and Communication\",\n \"The distributed estimation framework consists of three components: dynamical system, local observation, and neighborhood communications.\",\n \"These three parts include two layers: the physical layer and the cyber layer.\",\n \"For the sake of simplicity, we motivate the model with the example of a time-varying temperature field over a large geographical area monitored by a sensor network.\"\n ],\n [\n \"Physical layer: dynamical system\",\n \"Consider a time-varying temperature field distributed over a large geographical area, as shown in Fig REF .\",\n \"A first-order approximation and discretization of the temperature field provide spatio-temporal discretized temperature variables $x^j_i, j = 1, \\\\cdots , M$ , of $M$ sites at discrete time indexes $i = 1, \\\\cdots , \\\\;$ .\",\n \"We stack the $M$ field variables in a temperature state vector ${\\\\mathbf {x}}_i = \\\\begin{bmatrix} x^1_i, \\\\cdots , x^M_i \\\\end{bmatrix}^T \\\\in \\\\mathbb {R}^{M}$ .\",\n \"The evolution of the time-varying temperature field, ${\\\\mathbf {x}}_i$ , can be represented by a linear time-invariant (LTI) dynamical systemEven though the random field ${\\\\mathbf {x}}_i$ is time-varying, the system is time-invariant because the field dynamics matrix $A$ is time-invariant.\",\n \"For a time-varying dynamical system, the field dynamics matrix $A_i$ would be a function of time.\",\n \"${\\\\mathbf {x}}_{i+1} = A {\\\\mathbf {x}}_i + {\\\\mathbf {v}}_i,$ where the first-order dynamics matrix $A \\\\in \\\\mathbb {R}^{M \\\\times M}$ contains the coupling effects between the $M$ temperature variables, and the residual ${\\\\mathbf {v}}_i = \\\\begin{bmatrix} v^1_i, \\\\cdots , v^M_i \\\\end{bmatrix}^T \\\\in \\\\mathbb {R}^{M}$ is the system noise driving the dynamical temperature fieldThe temperature field is a time-varying random field.\",\n \"Although the field is time-varying, it is generated by a linear time-invariant system with perturbations.. At each of the $M$ sites, the input noise $v^j_i, j = 1, \\\\cdots , M$ accounts for the deviations in the temperature after the overall field dynamics.\",\n \"The field dynamics $A$ incorporates the sparsity pattern and connectivity of the physical layer consisting of the dynamical system (REF ).\"\n ],\n [\n \"Cyber layer: local observations\",\n \"The physical layer consisting of the field dynamics (REF ) is observed by a cyber layer consisting of a network of $N$ agents (sensors).\",\n \"In Fig REF , we see that there are $N$ sensors, where each sensor observes the temperatures of only a few sites.\",\n \"Denote the number of sites observed by sensor $n$ by $M_n, M_n \\\\ll M$ , and its measurements at time $i$ by $z_i^n \\\\in \\\\mathbb {R}^{M_n}$ .\",\n \"The observations of the agents in the cyber layer can be represented by a linear model: ${\\\\mathbf {z}}_i^n = H_n {\\\\mathbf {x}}_i + {\\\\mathbf {r}}_i^n, \\\\qquad \\\\qquad n =1, \\\\hdots , N,$ where, the observation matrix $H_n \\\\in \\\\mathbb {R}^{M_n \\\\times M}$ contains the observation pattern and strength information, and the observation noise $r_i^n \\\\in \\\\mathbb {R}^{M_n}$ reflects the inaccuracy in measurements due to sensor precision, high frequency fluctuations in temperature, and other unavoidable constraints.\",\n \"To illustrate, consider that sensor 1 is observing the temperature of 3 sites $\\\\lbrace x^1_i, x^2_i, x^3_i \\\\rbrace $ , i.e., $M_1 = 3$ .\",\n \"An example snapshot of the observation model of agent $n$ is $\\\\underbrace{\\\\begin{bmatrix} z^{11}_i \\\\\\\\ z^{12}_i \\\\\\\\ z^{13}_i \\\\end{bmatrix}}_{{\\\\mathbf {z}}^1_i} &=\\\\!\\\\!\",\n \"\\\\underbrace{\\\\begin{bmatrix} 1 & 0 & 0 & \\\\hdots & 0 \\\\\\\\ 0 & 5 & 0 & \\\\hdots & 0 \\\\\\\\ 0 & 0 & 6 & \\\\hdots & 0 \\\\end{bmatrix}}_{H_1}\\\\!\\\\!\",\n \"\\\\underbrace{\\\\begin{bmatrix} x^1_i \\\\\\\\ \\\\vdots \\\\\\\\ x^{\\\\scriptsize {M}}_i \\\\end{bmatrix}}_{{\\\\mathbf {x}}_i}\\\\!+\\\\!\",\n \"\\\\underbrace{\\\\begin{bmatrix} r^{11}_i \\\\\\\\ r^{12}_i \\\\\\\\ r^{13}_i \\\\end{bmatrix}}_{{\\\\mathbf {r}}^1_i}= \\\\begin{bmatrix} x^1_i \\\\\\\\ 5x^2_i \\\\\\\\ 6x^3_i \\\\end{bmatrix} \\\\!+\\\\!\",\n \"\\\\begin{bmatrix} r^{11}_i \\\\\\\\ r^{12}_i \\\\\\\\ r^{13}_i \\\\end{bmatrix}.$ Now, for the ease of analysis, we aggregate the noisy local temperature measurements, ${\\\\mathbf {z}}^1_i, \\\\cdots {\\\\mathbf {z}}^N_i$ , of all the $N$ agents in a global observation vector, ${\\\\mathbf {z}}_i \\\\in \\\\mathbb {R}^{{\\\\sum _{n=1}^{N} M_n}}$ : $\\\\underbrace{\\\\begin{bmatrix} {\\\\mathbf {z}}_i^1\\\\\\\\ \\\\vdots \\\\\\\\ {\\\\mathbf {z}}_i^N\\\\end{bmatrix}}_{{\\\\mathbf {z}}_i} &=&\\\\underbrace{\\\\begin{bmatrix}H_1 \\\\\\\\ \\\\vdots \\\\\\\\ H_N \\\\end{bmatrix}}_{H} {\\\\mathbf {x}}_i +\\\\underbrace{\\\\begin{bmatrix} {\\\\mathbf {r}}_i^1 \\\\\\\\ \\\\vdots \\\\\\\\ {\\\\mathbf {r}}_i^N\\\\end{bmatrix}}_{{\\\\mathbf {r}}_i},$ where, the global observation matrix is $H\\\\in \\\\mathbb {R}^{\\\\sum _{n=1}^N M_n \\\\times M}$ and the stacked measurement noise is $r_i\\\\in \\\\mathbb {R}^{{\\\\sum _{n=1}^{N} M_n}}$ .\",\n \"Note that, in general, the temperature measurement model is non-linear.\",\n \"For non-linear cases, refer to distributed particle filter in [33] and the references cited therein.\",\n \"Here we perform a first-order approximation to obtain a linear observation sequence.\"\n ],\n [\n \"Cyber layer: neighborhood communication\",\n \"In the cyber layer, the agents exchange their temperature readings or current estimates with their neighbors.\",\n \"In many applications, to reduce communications costs, neighbors communicate only with their geographically nearest agents as shown in Fig REF .\",\n \"Formally, the agent communication network is defined by a simple (no self-loops nor multiple edges), undirected, connected graph $\\\\mathcal {G = (V,E)}$ , where $\\\\mathcal {V} $ is the set of sensors (nodes or agents) and $\\\\mathcal {E}$ is the set of local communication channels (edges or links) among the agents.\",\n \"The open $\\\\Omega _n$ and closed $\\\\overline{\\\\Omega }_n$ neighborhoods of agent $n$ are: $\\\\Omega _n &= \\\\lbrace l|(n,l) \\\\in \\\\mathcal {E}\\\\rbrace .\",\n \"\\\\nonumber \\\\\\\\\\\\overline{\\\\Omega }_n &= \\\\lbrace n\\\\rbrace \\\\cup \\\\lbrace l|(n,l) \\\\in \\\\mathcal {E}\\\\rbrace .\",\n \"\\\\nonumber $ In Fig REF the open and closed neighborhoods of agent 1 are $\\\\Omega _1 = \\\\lbrace 2,3\\\\rbrace $ and $\\\\overline{\\\\Omega }_1 = \\\\lbrace 1, 2,3\\\\rbrace $ , respectively.\",\n \"The Laplacian matrix of $\\\\mathcal {G}$ is denoted by $L$ .\",\n \"The eigenvalues of the positive semi-definite matrix $L$ are ordered as $ 0 = \\\\lambda _1(L) \\\\le \\\\lambda _2(L) \\\\le .\",\n \".\",\n \".\",\n \"\\\\le \\\\lambda _N(L)$ .\",\n \"For details on graphs refer to [34].\",\n \"The communication network is sparse and time-invariant.\"\n ],\n [\n \"Modeling assumptions\",\n \"We make the following assumptions.\",\n \"Assumption 1 (Gaussian processes) The system noise, ${\\\\mathbf {v}}_i$ , the observation noise, ${\\\\mathbf {r}}_i$ , and the initial condition of the system, ${\\\\mathbf {x}}_0$ , are Gaussian sequences, with ${{\\\\mathbf {v}}_i \\\\sim \\\\mathcal {N}(0,V)}, \\\\;\\\\;{{\\\\mathbf {r}}^n_i \\\\sim \\\\mathcal {N}(0,R_n)}, \\\\;\\\\;{\\\\mathbf {x}}_0 \\\\sim \\\\mathcal {N}(\\\\bar{{\\\\mathbf {x}}}_0,\\\\Sigma _0),$ where, $V \\\\in \\\\mathbb {R}^{{M} \\\\times {M}} $ , $R_n \\\\in \\\\mathbb {R}^{{M_n} \\\\times {M_n}}$ and $\\\\Sigma _0 \\\\in \\\\mathbb {R}^{{M} \\\\times {M}}$ are the corresponding covariance matrices.\",\n \"The noise covariance matrix $R$ of the global noise vector ${\\\\mathbf {v}}_i$ in (REF ) is block-diagonal, i.e., $R=\\\\textrm {blockdiag} \\\\lbrace R_1, \\\\ldots , R_N \\\\rbrace $ , and positive-definite, i.e., $R > 0$ .\",\n \"Assumption 2 (Uncorrelated sequences) The system noise, the observation noise, and the initial condition: $\\\\lbrace \\\\lbrace {\\\\mathbf {v}}_i\\\\rbrace _i, \\\\lbrace {\\\\mathbf {r}}_i\\\\rbrace _i, {\\\\mathbf {x}}_0\\\\rbrace _{i \\\\ge 0}$ are uncorrelated random vector sequences.\",\n \"Assumption 3 (Prior information) Each agent in the cyber layer knows the system dynamics model, $A$ and $V$ , the initial condition statistics, $\\\\bar{x}_0$ and $\\\\Sigma _0$ , the parameters of the observation model, $H$ and $R$ , and the communication network model, $\\\\mathcal {G}$ .\",\n \"In large-scale system applications, the dynamics, observation, and network Laplacian matrices, $A$ , $H_n$ , and $L$ , are sparse, with $M_n \\\\ll M$ , and the agents communicate with only a few of their neighbors, $|\\\\Omega _n| \\\\ll N, \\\\forall n$ .\",\n \"In the dynamics (REF ) and observations (REF ), we assume that there is no deterministic input.\",\n \"The results are readily extended if there is a known deterministic input.\"\n ],\n [\n \"Centralized Information filter\",\n \"Although not practical in the context of the problem we study, we use the centralized information filter to benchmark our results on distributed estimator.\",\n \"In a centralized scheme, all the agents in the cyber layer communicate their measurements to a central fusion center, as depicted in Fig REF .\",\n \"The fusion center performs all needed computation tasks.\",\n \"Refer to [5], [35] for the filter, gain, and error equations of the centralized information filter.\",\n \"Figure: Centralized estimator\"\n ],\n [\n \"Distributed filtering and prediction\",\n \"This section considers our single time-scale distributed solution.\",\n \"We start with the introduction and derivation of the key components of our distributed estimator and then present our distributed field estimator.\"\n ],\n [\n \"Pseudo-state model\",\n \"In a centralized information filter [35], all the observations are converted into pseudo-observations [5] to obtain the optimal estimates.\",\n \"Following (REF ), the pseudo-observation $\\\\widetilde{{\\\\mathbf {z}}}^n_i$ of agent $n$ is $\\\\widetilde{{\\\\mathbf {z}}}^n_i &= H^T_n R_n^{-1} {\\\\mathbf {z}}^n_i = \\\\overline{H}_n {\\\\mathbf {x}}_i + H^T_n R_n^{-1} {\\\\mathbf {r}}^n_i, \\\\quad \\\\\\\\\\\\text{where,} \\\\qquad \\\\overline{H}_n &= H^T_n R_n^{-1} H_n.$ The centralized information filter computes the sum, $\\\\overline{{\\\\mathbf {z}}}_i$ of all the pseudo-observations $\\\\overline{{\\\\mathbf {z}}}_i &= \\\\sum _{n=1}^{N}\\\\widetilde{{\\\\mathbf {z}}}^n_i = G{\\\\mathbf {x}}_i + H^T R^{-1} {\\\\mathbf {r}}_i \\\\qquad \\\\\\\\\\\\text{where,} \\\\qquad \\\\qquad G &= \\\\sum _{n=1}^{N} H^T_n R_n^{-1} H_n = \\\\sum _{n=1}^{N} \\\\overline{H}_n .$ The aggregated pseudo-observation, $\\\\overline{{\\\\mathbf {z}}}_i$ , is the key term in the centralized filter.\",\n \"It provides the innovations term in the filter updates enabling the filter to converge with minimum MSE estimates.\",\n \"However, in the distributed solution, each agent $n$ does not have access to all the pseudo-observations; instead it can only communicate with its neighbors.\",\n \"To address this issue, in [5] we introduced a dynamic consensus algorithm to compute the distributed estimates of the averaged pseudo-observations, $\\\\overline{{\\\\mathbf {z}}}_i$ , at each agent.\",\n \"In (REF ), we note that the crucial term is $G{\\\\mathbf {x}}_i$ which carries the information of the dynamic state, ${\\\\mathbf {x}}_i$ ; and the second term in $\\\\overline{{\\\\mathbf {z}}}_i$ in (REF ) is noise.\",\n \"We refer to it as the pseudo-state, ${\\\\mathbf {y}}_i$ , ${\\\\mathbf {y}}_i = G{\\\\mathbf {x}}_i.$ The pseudo-state, ${\\\\mathbf {y}}_i$ , is also a random field whose time dynamics can be represented by a discrete-time linear dynamical system.\",\n \"The pseudo-observations $\\\\widetilde{{\\\\mathbf {z}}}^n_i$ are its linear measurements.\",\n \"We summarize the state-space model for the pseudo-state in the following proposition.\",\n \"See Appendix REF for the details of the proof.\",\n \"Proposition 1 The dynamics and observations of the pseudo-state ${\\\\mathbf {y}}_i$ are: ${\\\\mathbf {y}}_{i+1} &= \\\\widetilde{A}{\\\\mathbf {y}}_i + G{\\\\mathbf {v}}_i + \\\\check{A}{\\\\mathbf {x}}_i \\\\\\\\\\\\widetilde{{\\\\mathbf {z}}}^n_i &= \\\\widetilde{H}_n{\\\\mathbf {y}}_i + H^T_n R_n^{-1} {\\\\mathbf {r}}^n_i + \\\\check{H}_n {\\\\mathbf {x}}_i.$ The pseudo-dynamics matrix $\\\\widetilde{A}$ , pseudo-observations matrix $\\\\widetilde{H}_n$ , and the matrices $\\\\check{A}, \\\\check{H}_n$ , and $\\\\widetilde{I}$ at agent $n$ are: $\\\\widetilde{A} &= G A G^{\\\\dagger } \\\\\\\\\\\\widetilde{H}_n &= H^T_n R_n^{-1} H_n G^{\\\\dagger } = \\\\overline{H}_n G^{\\\\dagger }\\\\\\\\\\\\check{A} &= GA\\\\widetilde{I} \\\\\\\\\\\\check{H}_n &= H^T_n R_n^{-1} H_n \\\\widetilde{I} \\\\\\\\\\\\widetilde{I} &= I - G^{\\\\dagger }G,$ where, $G^{\\\\dagger }$ denotes the Moore-Penrose pseudo-inverse of G. In [5], the distributed information Kalman filter (DIKF) assumes distributed observability, i.e., it considers the case where $G$ is invertible.\",\n \"Under this assumption, $G^{\\\\dagger } = G^{-1}$ and $\\\\widetilde{I} = 0$ .\",\n \"In this paper we relax the requirement of invertibility of $G$ , proposing a distributed estimator for general dynamics-observation models under the assumption of global detectability.\",\n \"In most cases $\\\\widetilde{I}$ is low-rank.\",\n \"In (REF ), the term $\\\\left( G{\\\\mathbf {v}}_i~+~\\\\check{A}{\\\\mathbf {x}}_i = \\\\xi _i, \\\\text{say} \\\\right)$ can be interpreted as the pseudo-state input noise, which follows Gauss dynamics $\\\\xi _i &\\\\sim \\\\mathcal {N} \\\\left( \\\\check{A} \\\\overline{{\\\\mathbf {x}}}_i \\\\; , \\\\; GVG + \\\\check{A} \\\\Sigma _i \\\\check{A}^T\\\\right) \\\\qquad \\\\qquad \\\\\\\\\\\\text{where,} \\\\qquad \\\\Sigma _{i} &= \\\\mathbb {E}\\\\left[ ({\\\\mathbf {x}}_i - \\\\overline{{\\\\mathbf {x}}}_i)({\\\\mathbf {x}}_i - \\\\overline{{\\\\mathbf {x}}}_i)^T \\\\right].$ Similarly, in (), the term $\\\\big ( \\\\delta ^n_i = H^T_nR_n^{-1}{\\\\mathbf {r}}^n_i + \\\\check{H}_n {\\\\mathbf {x}}_i \\\\big )$ is the pseudo-state observation noise at agent $n$ , which is Gaussian $\\\\delta ^n_i \\\\sim \\\\mathcal {N} \\\\Big ( \\\\check{H}_n \\\\overline{{\\\\mathbf {x}}}_i , \\\\overline{H}_n + \\\\check{H}_n \\\\Sigma _{i} \\\\check{H}_n \\\\Big ).$ For the ease of analysis, we express the pseudo-state observation model in vector form by $\\\\widetilde{{\\\\mathbf {z}}}_i \\\\in \\\\mathbb {R}^{{\\\\sum _{n=1}^{N} M_n}}$ , the aggregate of the noisy local temperature measurements, $\\\\widetilde{{\\\\mathbf {z}}}^1_i, \\\\cdots \\\\widetilde{{\\\\mathbf {z}}}^N_i$ , of all the agents, $\\\\underbrace{\\\\begin{bmatrix} \\\\widetilde{{\\\\mathbf {z}}}_i^1\\\\\\\\ \\\\vdots \\\\\\\\ \\\\widetilde{{\\\\mathbf {z}}}_i^N\\\\end{bmatrix}}_{\\\\widetilde{{\\\\mathbf {z}}}_i} &=&\\\\underbrace{\\\\begin{bmatrix} \\\\widetilde{H}_1 \\\\\\\\ \\\\vdots \\\\\\\\ \\\\widetilde{H}_N \\\\end{bmatrix}}_{\\\\widetilde{H}} {\\\\mathbf {y}}_i + D_{H}^T R^{-1} {\\\\mathbf {r}}_i + \\\\underbrace{\\\\begin{bmatrix} \\\\check{H}_1 \\\\\\\\ \\\\vdots \\\\\\\\ \\\\check{H}_N \\\\end{bmatrix}}_{\\\\check{H}} {\\\\mathbf {x}}_i,$ where, the matrices $\\\\widetilde{H}, \\\\check{H} \\\\in \\\\mathbb {R}^{NM \\\\times M}$ , and, $D_{H} = \\\\text{blockdiag} \\\\lbrace H_1, \\\\cdots , H_N \\\\rbrace $ .\",\n \"We have established in (REF )-() the state-space dynamics and observations model of the pseudo-state, ${\\\\mathbf {y}}_i$ .\",\n \"The structure of (REF )-() is similar to the dynamics and observations model (REF )-(REF ) of the random field, ${\\\\mathbf {x}}_i$ .\",\n \"In the following subsection, we develop a distributed estimator with optimized gains to obtain unbiased estimates of the pseudo-state, ${\\\\mathbf {y}}_i$ , and of the state, ${\\\\mathbf {x}}_i$ , at each agent with minimized mean-squared error.\",\n \"The pseudo-state ${\\\\mathbf {y}}_i$ is a noise-reduced version of the sum $\\\\overline{{\\\\mathbf {z}}_i}$ of the local pseudo-observations.\",\n \"We demonstrate in Section that, by sharing distributed estimates of pseudo-states, the CIKF achieves 3dB better MSE error performance than the DIKF [5], which shares distributed estimates of the summed pseudo-observations $\\\\overline{{\\\\mathbf {z}}}_i$ .\"\n ],\n [\n \"At time $i$ , denote the $n^{\\\\text{th}}$ agent's distributed filter and prediction estimates of the state ${\\\\mathbf {x}}_i$ by $\\\\widehat{{\\\\mathbf {x}}}_{i|i}^n$ and $\\\\widehat{{\\\\mathbf {x}}}_{i+1|i}^n$ respectively.\",\n \"Similarly, its distributed filter and prediction estimates of the pseudo-state ${\\\\mathbf {y}}_i$ are denoted by $\\\\widehat{{\\\\mathbf {y}}}_{i|i}^n$ and $\\\\widehat{{\\\\mathbf {y}}}_{i+1|i}^n$ .\",\n \"At any time $i$ , each agent $n$ has access to its own pseudo-observation $\\\\widetilde{z}^n_i$ and receives the prediction state pseudo-state estimates, $\\\\widehat{{\\\\mathbf {y}}}_{i|i-1}^l$ , $l \\\\in \\\\Omega _n$ , of its neighbors at the previous time $i-1$ .\",\n \"Under this setup, the minimized MSE filter and prediction estimates are the conditional means, $&\\\\widehat{{\\\\mathbf {y}}}_{i|i}^n \\\\!=\\\\!\",\n \"\\\\mathbb {E} \\\\!\\\\left[ {\\\\mathbf {y}}_i \\\\;|\\\\; \\\\widetilde{{\\\\mathbf {z}}}^n_i, \\\\lbrace \\\\widehat{{\\\\mathbf {y}}}_{i|i-1}^l \\\\rbrace _{l \\\\in \\\\overline{\\\\Omega }_n} \\\\right] \\\\\\\\&\\\\widehat{{\\\\mathbf {x}}}_{i|i}^n \\\\!=\\\\!\",\n \"\\\\mathbb {E} \\\\left[ {\\\\mathbf {x}}_i \\\\;|\\\\; \\\\widehat{{\\\\mathbf {y}}}_{i|i}^n \\\\right] \\\\\\\\&\\\\!\\\\!\\\\!\\\\!\\\\!\",\n \"\\\\widehat{{\\\\mathbf {y}}}_{i+1|i}^n \\\\!=\\\\!\",\n \"\\\\mathbb {E} \\\\left[ {\\\\mathbf {y}}_{i+1} \\\\;|\\\\; \\\\widetilde{{\\\\mathbf {z}}}^n_i, \\\\lbrace \\\\widehat{{\\\\mathbf {y}}}_{i|i-1}^l \\\\rbrace _{l \\\\in \\\\overline{\\\\Omega }_n} \\\\right] \\\\\\\\&\\\\!\\\\!\\\\!\\\\!\\\\!\",\n \"\\\\widehat{{\\\\mathbf {x}}}_{i+1|i}^n \\\\!=\\\\!\",\n \"\\\\mathbb {E} \\\\left[ {\\\\mathbf {x}}_{i\\\\!+\\\\!1} \\\\;|\\\\; \\\\widehat{{\\\\mathbf {y}}}_{i|i}^n \\\\right].$ In (REF ), $\\\\widehat{{\\\\mathbf {y}}}_{i|i}^n$ is the filtered estimate of the pseudo-state ${\\\\mathbf {y}}_i$ given all the pseudo-observations available at agent $n$ up to time $i$ including those of its neighbors.\",\n \"By the principle of recursive linear estimation, instead of storing all the pseudo-observations $ \\\\lbrace \\\\lbrace \\\\widetilde{{\\\\mathbf {z}}}^n_t \\\\rbrace _{t = 0, \\\\cdots , i}, \\\\lbrace \\\\widetilde{{\\\\mathbf {z}}}^{n_1}_t \\\\rbrace ^{n_1 \\\\in \\\\Omega _n}_{t = 0, \\\\cdots , i-1}, \\\\lbrace \\\\widetilde{{\\\\mathbf {z}}}^{n_2}_t \\\\rbrace ^{n_2 \\\\in \\\\Omega _{n1}, \\\\forall n_1}_{t = 0, \\\\cdots , i-2}, \\\\cdots \\\\rbrace $ , we need only the current pseudo-observation $\\\\widetilde{{\\\\mathbf {z}}}^n_i$ and the pseudo-state estimates (including those of its neighbors) from the previous time instant $\\\\lbrace \\\\widehat{{\\\\mathbf {y}}}_{i|i-1}^l \\\\rbrace _{l \\\\in \\\\Omega _n}$ .\",\n \"The distributed pseudo-state estimate, $\\\\widehat{{\\\\mathbf {y}}}_{i|i}^n$ , is the expectation of the pseudo-state, ${\\\\mathbf {y}}_i$ , conditioned on its own pseudo-observation $\\\\widetilde{{\\\\mathbf {z}}}^n_i$ and the prior pseudo-state estimates, $\\\\widehat{{\\\\mathbf {y}}}_{i|i-1}^l, l \\\\in \\\\overline{\\\\Omega }_n,$ received from its neighbors.\",\n \"In this paper, we formulate the distributed field estimate, $\\\\widehat{{\\\\mathbf {x}}}_{i|i}^n$ , as the expectation of the field, ${\\\\mathbf {x}}_i$ , conditioned on the distributed pseudo-state estimate $\\\\widehat{{\\\\mathbf {y}}}_{i|i}^n$ .\",\n \"The distributed field estimates, $\\\\widehat{{\\\\mathbf {x}}}_{i|i}^n$ , are not optimal given all observations, but they are still optimal given only the pseudo-state estimates $\\\\widehat{{\\\\mathbf {y}}}_{i|i}^n$ .\",\n \"The filtered estimate $\\\\widehat{{\\\\mathbf {x}}}_{i|i}^n$ of ${\\\\mathbf {x}}_i$ in () depends on the current pseudo-state filtered estimate $\\\\widehat{{\\\\mathbf {y}}}_{i|i}^n$ .\",\n \"Similarly, the prediction estimates $\\\\widehat{{\\\\mathbf {y}}}_{i+1|i}^n$ and $\\\\widehat{{\\\\mathbf {x}}}_{i+1|i}^n$ of ${\\\\mathbf {y}}_{i+1}$ and ${\\\\mathbf {x}}_{i+1}$ in ()-(), respectively, are conditioned on the corresponding available quantities up to time $i$ .\",\n \"Theorem 1 The iterative updates to compute the distributed filtered estimates with optimized gains in (REF )-() are: $ \\\\!\\\\!\\\\!\",\n \"\\\\widehat{{\\\\mathbf {y}}}_{i|i}^n \\\\!=\\\\!\",\n \"\\\\widehat{{\\\\mathbf {y}}}_{i|i-1}^n &\\\\!+\\\\!\",\n \"\\\\sum _{l \\\\in \\\\Omega _n} \\\\underbrace{ \\\\!\",\n \"B^{nl}_i \\\\!\",\n \"\\\\left( \\\\widehat{{\\\\mathbf {y}}}_{i|i-1}^l \\\\!-\\\\!\",\n \"\\\\widehat{{\\\\mathbf {y}}}_{i|i-1}^n \\\\right) }_{\\\\text{Consensus}} \\\\nonumber \\\\\\\\& \\\\!+\\\\!\",\n \"B^{nn}_i \\\\underbrace{ \\\\left( \\\\widetilde{{\\\\mathbf {z}}}^n_i \\\\!-\\\\!\",\n \"\\\\left(\\\\widetilde{H}_n \\\\widehat{{\\\\mathbf {y}}}_{i|i-1}^n \\\\!+\\\\!\",\n \"\\\\check{H}_n \\\\widehat{{\\\\mathbf {x}}}_{i|i-1}^n \\\\right) \\\\right) }_{\\\\text{Innovations}}, \\\\\\\\ \\\\!\\\\!\\\\!\",\n \"\\\\widehat{{\\\\mathbf {x}}}_{i|i}^n \\\\!=\\\\!\",\n \"\\\\widehat{{\\\\mathbf {x}}}_{i|i-1}^n & \\\\!+\\\\!\",\n \"K^{n}_i \\\\underbrace{ \\\\left( \\\\widehat{{\\\\mathbf {y}}}_{i|i}^n \\\\!-\\\\!\",\n \"G \\\\widehat{{\\\\mathbf {x}}}_{i|i-1}^n \\\\right) }_{\\\\text{Innovations}}$ where, pseudo-state gain block matrix, $B_i \\\\in \\\\mathbb {R}^{MN \\\\times MN}$ , is $ B_i = \\\\begin{bmatrix} B^{nl}_i \\\\end{bmatrix}_{n = 1, \\\\cdots , N}^{l = 1, \\\\cdots , N}, B^{nl}_i=0 \\\\; \\\\text{if} \\\\; l \\\\notin \\\\Omega _n$ .\",\n \"The state gain block-diagonal matrix is $K_i = \\\\text{blockdiag}\\\\lbrace K^{1}_i, \\\\cdots , K^{N}_i\\\\rbrace , K^{n}_i \\\\in ~\\\\mathbb {R}^{M \\\\times M}$ .\",\n \"The optimized MSE prediction estimates in ()-() are: $\\\\widehat{{\\\\mathbf {y}}}_{i+1|i}^n &= \\\\widetilde{A}\\\\widehat{{\\\\mathbf {y}}}_{i|i}^n + \\\\check{A}\\\\widehat{{\\\\mathbf {x}}}_{i|i}^n, \\\\\\\\\\\\widehat{{\\\\mathbf {x}}}_{i+1|i}^n &= A \\\\widehat{{\\\\mathbf {x}}}_{i|i}^n.$ The proof of Theorem REF is presented in Appendix REF .\",\n \"The update equations reflect the computation tasks of each agent $n$ at each time index $i$ .\",\n \"The gain matrices, $B_i$ and $K_i$ , in (REF )-() are deterministic and can be pre-computed and saved at each agent.\",\n \"We discuss the details of the design of these optimal gain matrices in Section .\",\n \"With these optimal gain matrices, the Kalman type Consensus$+$ Innovations filter and prediction updates (REF )-() provide the minimized MSE distributed estimates of the dynamic states, and hence we term our solution as Consensus$+$ Innovations Kalman Filter (CIKF).\"\n ],\n [\n \"CIKF: Assumptions\",\n \"The Consensus$+$ Innovations Kalman Filter (CIKF) achieves convergence given the following assumptions: Assumption 4 (Global detectability) The dynamic state equation (REF ) and the observations model (REF ) are globally detectable, i.e., the pair $(A, H)$ is detectable.\",\n \"Assumption 5 (Connectedness) The agent communication network is connected, i.e., the algebraic connectivity $\\\\lambda _2(L)$ of the Laplacian matrix $L$ of the graph $\\\\mathcal {G}$ is strictly positive.\",\n \"By Assumption REF , the state-observation model (REF )-(REF ) is globally detectable but not necessarily locally detectable, i.e., $(A, H_n), \\\\forall n, $ are not necessarily detectable.\",\n \"Note that these two are minimal assumptions.\",\n \"Assumption REF is mandatory even for a centralized system, and Assumption REF is required for consensus algorithms to converge.\",\n \"Further, note that in this paper we do not consider distributed observability (invertibility of $G$ ) of the model setup, which is the strong and restrictive assumption taken in [5], [21], [22] and [36] and similar to weak detectability presented in [19].\"\n ],\n [\n \"CIKF: Update algorithm\",\n \"In this subsection, we present the step-by-step tasks executed by each agent $n$ in the cyber layer to implement the Consensus$+$ Innovations Kalman Filter (CIKF) and thereby obtain the unbiased minimized MSE distributed estimates of the dynamic state ${\\\\mathbf {x}}_i$ .\",\n \"Each agent $n$ runs Algorithm REF locally.\",\n \"[h] Consensus$+$ Innovations Kalman Filter Input: Model parameters $A$ , $V$ , $H$ , $R$ , $G$ , $L$ , $\\\\overline{{\\\\mathbf {x}}}_0$ , $\\\\Sigma _0$ .\",\n \"Initialize: $\\\\widehat{{\\\\mathbf {x}}}^n_{0|-1} = \\\\overline{{\\\\mathbf {x}}}_0$ , $\\\\widehat{{\\\\mathbf {y}}}^n_{0|-1} = G\\\\overline{{\\\\mathbf {x}}}_0$ .\",\n \"Pre-compute: Gain matrices $B_i$ and $K_i$ using Algorithm REF .\",\n \"$i \\\\ge 0$ $\\\\!\\\\!\\\\!\\\\!$Communications: Broadcast $\\\\widehat{{\\\\mathbf {y}}}_{i|i-1}^n$ to all neighbors $l \\\\in \\\\Omega _n$ .\",\n \"Receive $\\\\lbrace \\\\widehat{{\\\\mathbf {y}}}_{i|i-1}^l\\\\rbrace _{l \\\\in \\\\Omega _n}$ from neighbors.\",\n \"$\\\\!\\\\!\\\\!\\\\!$Observation: Make measurement ${\\\\mathbf {z}}^n_i$ of the state ${\\\\mathbf {x}}_i$ .\",\n \"Transform ${\\\\mathbf {z}}^n_i$ in pseudo-observation $\\\\widetilde{{\\\\mathbf {z}}}^n_i$ using ().\",\n \"$\\\\!\\\\!\\\\!\\\\!$Filter updates: Compute the estimate $\\\\widehat{{\\\\mathbf {y}}}_{i|i}^n$ of ${\\\\mathbf {y}}_i$ using (REF ).\",\n \"Compute the estimate $\\\\widehat{{\\\\mathbf {x}}}_{i|i}^n$ of the state ${\\\\mathbf {x}}_i$ using ().\",\n \"$\\\\!\\\\!\\\\!\\\\!$Prediction updates: Predict the estimate $\\\\widehat{{\\\\mathbf {y}}}_{i+1|i}^n$ of ${\\\\mathbf {y}}_{i+1}$ using (REF ).\",\n \"Predict the estimate $\\\\widehat{{\\\\mathbf {x}}}_{i+1|i}^n$ of the state ${\\\\mathbf {x}}_{i+1}$ using ().\",\n \"Later in Section , we analyze and compare the performance of CIKF with that of the distributed information Kalman filter (DKF) [5] and of the centralized Kalman filter (CKF).\",\n \"The centralized filter collects measurements from all the agents in the cyber layer.\",\n \"The DIKF performs distributed dynamic averaging of the pseudo-observations.\",\n \"The pseudo-state can be perceived as a noise-reduced version of the global average of all the pseudo-observations or pseudo-innovations.\",\n \"The distributed estimates of the pseudo-states reduce the noise of the innovations in the filtering step of field estimation and hence further improve the MSE performance of the distributed estimator.\",\n \"Before going into the numerical evaluation, we do theoretical error analysis of CIKF, derive the conditions for convergence guarantees, and design the optimal consensus and innovations gains in the following sections.\",\n \"We analyze the MSE performance of the CIKF and derive its error covariance matrices.\",\n \"First, we define the different error processes and determine their dynamics.\",\n \"Denote the filtering error processes ${\\\\mathbf {e}}_{i|i}^n$ and ${\\\\mathbf {\\\\epsilon }}_{i|i}^n$ of the pseudo-state and of the state at agent $n$ by ${\\\\mathbf {e}}_{i|i}^n &= {\\\\mathbf {y}}_i - \\\\widehat{{\\\\mathbf {y}}}_{i|i}^n, \\\\\\\\{\\\\mathbf {\\\\epsilon }}_{i|i}^n &= {\\\\mathbf {x}}_i - \\\\widehat{{\\\\mathbf {x}}}_{i|i}^n.$ Similarly, represent the prediction error processes ${\\\\mathbf {e}}_{i+1|i}^n$ and ${\\\\mathbf {\\\\epsilon }}_{i+1|i}^n$ of the pseudo-state and the state at agent $n$ by ${\\\\mathbf {e}}_{i+1|i}^n &= {\\\\mathbf {y}}_{i+1} - \\\\widehat{{\\\\mathbf {y}}}_{i+1|i}^n, \\\\\\\\{\\\\mathbf {\\\\epsilon }}_{i+1|i}^n &= {\\\\mathbf {x}}_{i+1} - \\\\widehat{{\\\\mathbf {x}}}_{i+1|i}^n.$ We establish that the CIKF provides unbiased estimates of the state and pseudo-state in the following lemma whose proof is sketched in Appendix REF .\",\n \"Lemma 1 The distributed filter and prediction estimates, $\\\\widehat{{\\\\mathbf {y}}}_{i|i}^n$ , $\\\\widehat{{\\\\mathbf {x}}}_{i|i}^n$ , $\\\\widehat{{\\\\mathbf {y}}}_{i+1|i}^n$ and $\\\\widehat{{\\\\mathbf {x}}}_{i+1|i}^n$ , of the pseudo-state and state are unbiased, i.e., error processes, ${\\\\mathbf {e}}_{i|i}^n$ , ${\\\\mathbf {e}}_{i+1|i}^n$ , ${\\\\mathbf {\\\\epsilon }}_{i|i}^n$ , and ${\\\\mathbf {\\\\epsilon }}_{i+1|i}^n$ are zero-mean at all agents $n$ : $\\\\mathbb {E}[{\\\\mathbf {e}}_{i|i}^n] = 0, \\\\;\\\\mathbb {E}[{\\\\mathbf {e}}_{i+1|i}^n] = 0, \\\\;\\\\mathbb {E}[{\\\\mathbf {\\\\epsilon }}_{i|i}^n] = 0, \\\\;\\\\mathbb {E}[{\\\\mathbf {\\\\epsilon }}_{i+1|i}^n] = 0.$ Each agent exchanges their estimates with their neighbors, hence their error processes are correlated.\",\n \"Since the error processes of the agents are coupled, we stack them in a error vector and analyze all errors together.\",\n \"$\\\\widehat{{\\\\mathbf {x}}}_{i|i}&\\\\!=\\\\!\",\n \"\\\\begin{bmatrix} \\\\widehat{{\\\\mathbf {x}}}_{i|i}^1 \\\\\\\\ \\\\vdots \\\\\\\\ \\\\widehat{{\\\\mathbf {x}}}_{i|i}^N \\\\end{bmatrix} \\\\!,\\\\widehat{{\\\\mathbf {x}}}_{i+1|i}\\\\!=\\\\!\",\n \"\\\\begin{bmatrix} \\\\widehat{{\\\\mathbf {x}}}_{i+1|i}^1 \\\\\\\\ \\\\vdots \\\\\\\\ \\\\widehat{{\\\\mathbf {x}}}_{i+1|i}^N \\\\end{bmatrix}\\\\!,\\\\widehat{{\\\\mathbf {y}}}_{i|i}\\\\!=\\\\!\",\n \"\\\\begin{bmatrix} \\\\widehat{{\\\\mathbf {y}}}_{i|i}^1 \\\\\\\\ \\\\vdots \\\\\\\\ \\\\widehat{{\\\\mathbf {y}}}_{i|i}^N \\\\end{bmatrix}\\\\!,\\\\widehat{{\\\\mathbf {y}}}_{i+1|i}\\\\!=\\\\!\",\n \"\\\\begin{bmatrix} \\\\widehat{{\\\\mathbf {y}}}_{i+1|i}^1 \\\\\\\\ \\\\vdots \\\\\\\\ \\\\widehat{{\\\\mathbf {y}}}_{i+1|i}^N \\\\end{bmatrix}\\\\!, \\\\\\\\{\\\\mathbf {\\\\epsilon }}_{i|i}&\\\\!=\\\\!\",\n \"\\\\begin{bmatrix} {\\\\mathbf {\\\\epsilon }}_{i|i}^1 \\\\\\\\ \\\\vdots \\\\\\\\ {\\\\mathbf {\\\\epsilon }}_{i|i}^N \\\\end{bmatrix}\\\\!,{\\\\mathbf {\\\\epsilon }}_{i+1|i}\\\\!=\\\\!\",\n \"\\\\begin{bmatrix} {\\\\mathbf {\\\\epsilon }}_{i+1|i}^1 \\\\\\\\ \\\\vdots \\\\\\\\ {\\\\mathbf {\\\\epsilon }}_{i+1|i}^N \\\\end{bmatrix}\\\\!,{\\\\mathbf {e}}_{i|i}\\\\!=\\\\!\",\n \"\\\\begin{bmatrix} {\\\\mathbf {e}}_{i|i}^1 \\\\\\\\ \\\\vdots \\\\\\\\ {\\\\mathbf {e}}_{i|i}^N \\\\end{bmatrix}\\\\!,{\\\\mathbf {e}}_{i+1|i}\\\\!=\\\\!\",\n \"\\\\begin{bmatrix} {\\\\mathbf {e}}_{i+1|i}^1 \\\\\\\\ \\\\vdots \\\\\\\\ {\\\\mathbf {e}}_{i+1|i}^N \\\\end{bmatrix}\\\\!.$ We summarize the dynamics of the error processes in the following lemma whose proof is in Appendix REF .\",\n \"Lemma 2 The error processes, ${\\\\mathbf {e}}_{i|i}$ , ${\\\\mathbf {e}}_{i+1|i}$ , ${\\\\mathbf {\\\\epsilon }}_{i|i}$ , and ${\\\\mathbf {\\\\epsilon }}_{i|i}$ are Gaussian and their dynamics are: $&\\\\!\\\\!\",\n \"{\\\\mathbf {e}}_{i|i}\\\\!=\\\\!\",\n \"\\\\left( I_{\\\\!M\\\\!N} \\\\!-\\\\!\",\n \"B^{\\\\mathcal {C}}_i \\\\!-\\\\!\",\n \"B^{\\\\mathcal {I}}_i \\\\widetilde{D}_{H}\\\\right)\\\\!\",\n \"{\\\\mathbf {e}}_{i|i-1}\\\\!-\\\\!\",\n \"B^{\\\\mathcal {I}}_i \\\\check{D}_{H} {\\\\mathbf {\\\\epsilon }}_{i|i-1}\\\\!-\\\\!\",\n \"B^{\\\\mathcal {I}}_i D_{H}^T R^{-1} r_i , \\\\\\\\&{\\\\mathbf {\\\\epsilon }}_{i|i}= \\\\left( I_{MN} - K_i \\\\left( I_N \\\\otimes G \\\\right) \\\\right) {\\\\mathbf {\\\\epsilon }}_{i|i-1}+ K_i {\\\\mathbf {e}}_{i|i}, \\\\\\\\&{\\\\mathbf {e}}_{i+1|i}= \\\\left(I_{\\\\!N} \\\\!\\\\otimes \\\\!\",\n \"\\\\widetilde{A}\\\\right)\\\\!\",\n \"{\\\\mathbf {e}}_{i|i}+ \\\\!\\\\left( I_{\\\\!N} \\\\!\\\\otimes \\\\!\",\n \"\\\\check{A} \\\\right)\\\\!\",\n \"{\\\\mathbf {\\\\epsilon }}_{i|i}+ 1_{\\\\!N} \\\\!\\\\otimes \\\\!\",\n \"\\\\left(G{\\\\mathbf {v}}_i\\\\right), \\\\\\\\&{\\\\mathbf {\\\\epsilon }}_{i+1|i}= \\\\left( I_N \\\\otimes A\\\\right) {\\\\mathbf {\\\\epsilon }}_{i|i}+ 1_N \\\\otimes {\\\\mathbf {v}}_i,$ where, $B^{\\\\mathcal {C}}_i$ is the consensus gain matrix and $B^{\\\\mathcal {I}}_i$ , $K_i$ are the innovations gain matrices for the pseudo-state and state estimation, respectively.\",\n \"The block diagonal matrices are $\\\\widetilde{D}_{H} = \\\\text{blockdiag}\\\\lbrace \\\\widetilde{H}_1, \\\\cdots , \\\\widetilde{H}_N \\\\rbrace $ and $\\\\check{D}_{H} = \\\\text{blockdiag}\\\\lbrace \\\\check{H}_1, \\\\cdots , \\\\check{H}_N \\\\rbrace $ .\",\n \"The symbol $\\\\otimes $ denotes the Kronecker matrix product.\",\n \"Lemma REF established that the error processes (REF )-() are unbiased.\",\n \"It then follows that the filter and prediction error covariances of the pseudo-state and state are simply: $P_{i|i} &= \\\\mathbb {E}\\\\left[{\\\\mathbf {e}}_{i|i}{\\\\mathbf {e}}_{i|i}^T \\\\right] \\\\\\\\P_{i+1|i} &= \\\\mathbb {E}\\\\left[{\\\\mathbf {e}}_{i+1|i}{\\\\mathbf {e}}_{i+1|i}^T \\\\right] \\\\\\\\\\\\Sigma _{i|i} &= \\\\mathbb {E}\\\\left[{\\\\mathbf {\\\\epsilon }}_{i|i}{\\\\mathbf {\\\\epsilon }}_{i|i}^T \\\\right] \\\\\\\\\\\\Sigma _{i+1|i} &= \\\\mathbb {E}\\\\left[{\\\\mathbf {\\\\epsilon }}_{i+1|i}{\\\\mathbf {\\\\epsilon }}_{i+1|i}^T \\\\right]$ Note that the state estimates $\\\\widehat{{\\\\mathbf {x}}}_{i|i}$ , $\\\\widehat{{\\\\mathbf {x}}}_{i+1|i}$ depend on the pseudo-state estimates $\\\\widehat{{\\\\mathbf {y}}}_{i|i}$ , $\\\\widehat{{\\\\mathbf {y}}}_{i+1|i}$ .\",\n \"Hence the error process (REF )-() are not uncorrelated.\",\n \"The filter and prediction cross-covariances are: $\\\\Pi _{i|i} &= \\\\mathbb {E}\\\\left[{\\\\mathbf {\\\\epsilon }}_{i|i}{\\\\mathbf {e}}_{i|i}^T \\\\right] \\\\\\\\\\\\Pi _{i+1|i} &= \\\\mathbb {E}\\\\left[{\\\\mathbf {\\\\epsilon }}_{i+1|i}{\\\\mathbf {e}}_{i+1|i}^T \\\\right] \\\\\\\\\\\\Gamma _{i} &= \\\\mathbb {E}\\\\left[{\\\\mathbf {\\\\epsilon }}_{i|i-1}{\\\\mathbf {e}}_{i|i}^T \\\\right]$ In the following theorem, we define and derive the evolution of the state, pseudo-state, and cross error covariances.\",\n \"Figure: NO_CAPTIONTheorem 2 The filter error covariances, $P_{i|i}$ , $\\\\Sigma _{i|i}$ , $\\\\Pi _{i|i}$ , and the predictor error covariances, $P_{i+1|i}$ , $\\\\Sigma _{i+1|i}$ , $\\\\Pi _{i+1|i}$ , follow Lyapunov-type iterations ()-(), where, $J = (\\\\mathbf {1}_N \\\\mathbf {1}_N^T)\\\\otimes I_{\\\\!M}$ and the initial conditions are $\\\\Sigma _{0|-1} = J\\\\!\\\\otimes \\\\!\\\\Sigma _0, P_{0|-1} = J\\\\!\\\\otimes \\\\!\\\\left(G\\\\Sigma _0 G\\\\right), \\\\Pi _{0|-1} = J\\\\!\\\\otimes \\\\!\\\\left(\\\\Sigma _0 G\\\\right)$ .\",\n \"The proof of the theorem is in Appendix REF .\",\n \"The iterations () and () combined together constitute the distributed version of the discrete algebraic Riccati equation.\",\n \"The MSE of the proposed CIKF is the trace of the error covariance, $\\\\Sigma _{i+1|i}$ in ().\",\n \"The optimal design of the gain matrices, $B_i$ and $K_i$ , such that the CIKF yields minimized MSE estimates, is discussed in Section .\",\n \"Before that in Section we derive the conditions under which the CIKF converges, in other words, the MSE given by the trace of $\\\\Sigma _{i+1|i}$ is bounded.\"\n ],\n [\n \"Tracking Capacity\",\n \"The convergence properties of the CIKF is determined by the dynamics of the pseudo-state and state error processes, ${\\\\mathbf {e}}_{i+1|i}$ and ${\\\\mathbf {\\\\epsilon }}_{i+1|i}$ .\",\n \"If the error dynamics are asymptotically stable, then the error processes have asymptotically bounded error covariances that in turn guarantee the convergence of the CIKF.\",\n \"Note that if the dynamics of the prediction error processes, ${\\\\mathbf {e}}_{i+1|i}$ , ${\\\\mathbf {\\\\epsilon }}_{i+1|i}$ are asymptotically stable, then the dynamics of the filter error processes, ${\\\\mathbf {e}}_{i|i}$ , ${\\\\mathbf {\\\\epsilon }}_{i|i}$ are also asymptotically stable.\",\n \"That is why we study the dynamics of only one of the error processes and in this paper we consider the prediction error processes.\"\n ],\n [\n \"Asymptotic stability of error processes\",\n \"To analyze the stability of the error processes, we first write the evolution of the prediction error processes, combining (REF )-(), ${\\\\mathbf {e}}_{i+1|i}&= \\\\underbrace{ \\\\left(I_{\\\\!N} \\\\!\\\\otimes \\\\!\",\n \"\\\\widetilde{A}\\\\right)\\\\!\",\n \"\\\\left( I_{\\\\!M\\\\!N} \\\\!-\\\\!\",\n \"B^{\\\\mathcal {C}}_i \\\\!-\\\\!\",\n \"B^{\\\\mathcal {I}}_i \\\\widetilde{D}_{H}\\\\right) }_{\\\\widetilde{F}} {\\\\mathbf {e}}_{i|i-1}+ \\\\widetilde{\\\\mathbf {\\\\phi }}_i, \\\\\\\\{\\\\mathbf {\\\\epsilon }}_{i+1|i}&= \\\\underbrace{ \\\\left( I_{\\\\!N} \\\\!\\\\otimes \\\\!\",\n \"A\\\\right)\\\\!\",\n \"\\\\left( I_{\\\\!M\\\\!N} \\\\!-\\\\!\\\\!\",\n \"K_i \\\\left( I_{\\\\!N} \\\\!\\\\otimes \\\\!\",\n \"G \\\\right) \\\\right) }_{F} \\\\!\",\n \"{\\\\mathbf {\\\\epsilon }}_{i|i-1}\\\\!+\\\\!\",\n \"{\\\\mathbf {\\\\phi }}_i ,$ where, the noise processes $\\\\widetilde{\\\\mathbf {\\\\phi }}_i$ and ${\\\\mathbf {\\\\phi }}_i$ are $\\\\widetilde{\\\\mathbf {\\\\phi }}_i &= \\\\!\\\\left( I_{\\\\!N} \\\\!\\\\otimes \\\\!\",\n \"\\\\check{A}\\\\right)\\\\!\",\n \"{\\\\mathbf {\\\\epsilon }}_{i|i}- \\\\left(I_{\\\\!N} \\\\!\\\\otimes \\\\!\",\n \"\\\\widetilde{A}\\\\right)\\\\!B^{\\\\mathcal {I}}_i \\\\check{D}_{H} {\\\\mathbf {\\\\epsilon }}_{i|i-1}+ 1_{\\\\!N} \\\\!\\\\otimes \\\\!\",\n \"\\\\left(G{\\\\mathbf {v}}_i\\\\right) \\\\nonumber \\\\\\\\& \\\\qquad - \\\\left(I_{\\\\!N} \\\\!\\\\otimes \\\\!\",\n \"\\\\widetilde{A}\\\\right)\\\\!\",\n \"B^{\\\\mathcal {I}}_i D_{H}^T R^{-1} r_i, \\\\\\\\{\\\\mathbf {\\\\phi }}_i &= \\\\left( I_{\\\\!N} \\\\!\\\\otimes \\\\!\",\n \"A\\\\right)\\\\!\",\n \"K_i {\\\\mathbf {e}}_{i|i}+ 1_N \\\\otimes {\\\\mathbf {v}}_i.$ The statistical properties of the noises, $\\\\widetilde{\\\\mathbf {\\\\phi }}_i$ and ${\\\\mathbf {\\\\phi }}_i$ , of the error processes, ${\\\\mathbf {e}}_{i+1|i}$ and ${\\\\mathbf {\\\\epsilon }}_{i+1|i}$ , are stated in the following Lemma, and the proof is included in Appendix REF .\",\n \"Lemma 3 The noise sequences $\\\\widetilde{\\\\mathbf {\\\\phi }}_i$ and ${\\\\mathbf {\\\\phi }}_i$ are zero-mean Gaussian that follow $\\\\widetilde{\\\\mathbf {\\\\phi }}_i \\\\in \\\\mathcal {N} \\\\left( {\\\\mathbf {0}}, \\\\widetilde{\\\\Phi }_i \\\\right)$ and ${\\\\mathbf {\\\\phi }}_i \\\\in \\\\mathcal {N} \\\\left( {\\\\mathbf {0}}, \\\\Phi _i \\\\right)$ .\",\n \"The dynamics of the error processes are characterized by (REF )-() and Lemma REF .\",\n \"Let $\\\\rho (.\",\n \")$ and $\\\\Vert .\\\\Vert _2$ denote the spectral radius and the spectral norm of a matrix, respectively.\",\n \"The error processes are asymptotically stable if and only if the spectral radii of $\\\\widetilde{F}$ , $F$ are less than one, i.e., $\\\\rho \\\\left(\\\\widetilde{F}\\\\right) < 1, \\\\quad \\\\rho \\\\left(F\\\\right) < 1$ and the noise covariances, $\\\\widetilde{\\\\Phi }_i$ , $\\\\Phi _i$ are bounded, i.e., $\\\\Vert \\\\widetilde{\\\\Phi }_i\\\\Vert _2 < \\\\infty $ , $\\\\Vert \\\\Phi _i\\\\Vert _2 < \\\\infty , \\\\; \\\\forall i$ .\",\n \"Now if (REF ) holds, then the prediction error covariances $P_{i+1|i}$ , $\\\\Sigma _{i+1|i}$ are bounded; this ensures the filter error covariances $P_{i|i}$ , $\\\\Sigma _{i|i}$ are also bounded.\",\n \"Further, the model noise covariances $V$ and $R$ are bounded.\",\n \"Then, by (REF )-(), the noise covariances $\\\\widetilde{\\\\Phi }_i$ and ${\\\\Phi }_i$ are bounded if the spectral radii are less than one.\",\n \"Thus, (REF ) are the necessary and sufficient conditions for the convergence of the CIKF algorithm.\"\n ],\n [\n \"Tracking capacity for unstable systems\",\n \"The stability of the underlying dynamical system (REF ) in the physical layer is determined by the dynamics matrix $A$ .\",\n \"If the system is asymptotically stable, i.e., $\\\\rho (A)<1$ , then there always exist gain matrices $B_i$ , $K_i$ such that (REF ) holds true.\",\n \"Hence for stable systems, the CIKF always converges with a bounded MSE solution.\",\n \"In contrast, for an unstable dynamical system (REF ), $\\\\rho (A)>1$ , it may not always be possible to find gain matrices $B_i$ , $K_i$ satisfying (REF ) conditions.\",\n \"There exists an upper threshold on the degree of instability of the system dynamics, $A$ , that guarantees the convergence of the proposed CIKF.\",\n \"The threshold, similar to Network Tracking Capacity in [6], is the tracking capacity of the CIKF algorithm, and it depends on the agent communication network and observation models, as summarized in the following theorem.\",\n \"Theorem 3 The tracking capacity of the CIKF is, C, $C = \\\\max _{B^{\\\\mathcal {C}}_i, B^{\\\\mathcal {I}}_i} \\\\frac{\\\\lambda _{1}}{\\\\lambda _{m} \\\\begin{Vmatrix} I_{\\\\!M\\\\!N} \\\\!-\\\\!\",\n \"B^{\\\\mathcal {C}}_i \\\\!-\\\\!\",\n \"B^{\\\\mathcal {I}}_i \\\\widetilde{D}_{H} \\\\end{Vmatrix}_2 }$ where, $B^{\\\\mathcal {C}}_i$ has the same block sparsity pattern as the graph Laplacian $L$ , $B^{\\\\mathcal {I}}_i$ is a block diagonal matrix, and, $\\\\lambda _{1}$ and $\\\\lambda _{m}$ are the minimum and maximum non-zero eigenvalues of $G$ , $0<\\\\lambda _{1} \\\\le \\\\cdots \\\\le \\\\lambda _{m}$ .\",\n \"If $\\\\Vert A \\\\Vert _2 < C$ , then there exists $B_i$ , $K_i$ such that the CIKF (REF )-() converges with bounded MSE.\",\n \"The proof is in Appendix REF .\",\n \"In the above theorem, the structural constraints on the gain matrices $B^{\\\\mathcal {C}}_i, B^{\\\\mathcal {I}}_i$ ensure that each agent combines its neighbors' estimates for consensus and its own pseudo-observations for the innovation part of the CIKF.\",\n \"The block sparsity pattern of $B^{\\\\mathcal {C}}_i$ being similar to that of $L$ implies that the tracking capacity is dependent on the connectivity of the communication network.\",\n \"Similarly, since $\\\\widetilde{D}_{H}$ is a block-diagonal matrix containing the observation matrices $H_n$ we conclude that the tracking capacity is also a function of the observation models.\",\n \"The tracking capacity increases with the increase in communication graph connectivity and observation density.\",\n \"For instance, the tracking capacity is infinity if all agents are connected with everyone else (complete graph) or all the agents observe the entire dynamical system (local observability).\",\n \"Given the tracking capacity is satisfied for the system, observation, and communication models (REF )-(REF ), the question remains how to design the gain matrices $B_i$ and $K_i$ to minimize the MSE of the CIKF, which we discuss in the following section.\"\n ],\n [\n \"Optimal Gain Design\",\n \"The asymptotic stability of the error dynamics guarantees convergence of CIKF and bounded MSE, but here we discuss how to design the $B_i$ and $K_i$ such that the MSE is not only bounded but also minimum.\"\n ],\n [\n \"New uncorrelated information\",\n \"In CIKF Algorithm REF , at any time $i$ each agent $n$ makes pseudo-observation $\\\\widetilde{{\\\\mathbf {z}}}^n_i$ of the state and receives prior estimates $\\\\lbrace \\\\widehat{{\\\\mathbf {y}}}_{i|i-1}^l\\\\rbrace _{l \\\\in \\\\Omega _n}$ from its neighbors.\",\n \"The CIKF algorithm employs this new information to compute the distributed filter estimates of the pseudo-state and state.\",\n \"Denote the new information for the pseudo-state and state filtering by $\\\\widetilde{\\\\mathbf {\\\\theta }}^n_i$ and ${\\\\mathbf {\\\\theta }}^n_i$ , respectively, $\\\\widetilde{\\\\mathbf {\\\\theta }}^n_i = \\\\begin{bmatrix}\\\\widehat{{\\\\mathbf {y}}}_{i|i-1}^{l_1} \\\\\\\\ \\\\vdots \\\\\\\\ \\\\widehat{{\\\\mathbf {y}}}_{i|i-1}^{l_{d_n}} \\\\\\\\ \\\\widetilde{{\\\\mathbf {z}}}^n_i \\\\end{bmatrix}, \\\\quad {\\\\mathbf {\\\\theta }}^n_i = \\\\widehat{{\\\\mathbf {y}}}_{i|i}^n$ where, $\\\\lbrace l_1, \\\\cdots , l_{d_n}\\\\rbrace = \\\\Omega _n$ and $d_n = |\\\\Omega _n|$ is the degree of agent $n$ .\",\n \"Note the new information, $\\\\widetilde{\\\\mathbf {\\\\theta }}^n_i$ and ${\\\\mathbf {\\\\theta }}^n_i$ , are Gaussian since they are linear combinations of Gaussian sequences.\",\n \"However, $\\\\widetilde{\\\\mathbf {\\\\theta }}^n_i$ and ${\\\\mathbf {\\\\theta }}^n_i$ are correlated with the previous estimates $\\\\widehat{{\\\\mathbf {y}}}_{i|i-1}^n$ and $\\\\widehat{{\\\\mathbf {x}}}_{i|i-1}^n$ .\",\n \"So, we transform them into uncorrelated new information and then combine the uncorrelated information with the previous estimates $\\\\widehat{{\\\\mathbf {y}}}_{i|i-1}^n$ and $\\\\widehat{{\\\\mathbf {x}}}_{i|i-1}^n$ to compute the current filtered estimates.\",\n \"Lemma 4 The new uncorrelated information $\\\\widetilde{\\\\mathbf {\\\\nu }}^n_i$ and ${\\\\mathbf {\\\\nu }}^n_i$ for filtering update at agent n are, $\\\\widetilde{\\\\mathbf {\\\\nu }}^n_i &= \\\\widetilde{\\\\mathbf {\\\\theta }}^n_i - \\\\overline{\\\\widetilde{\\\\mathbf {\\\\theta }}}^n_i, \\\\qquad \\\\overline{\\\\widetilde{\\\\mathbf {\\\\theta }}}^n_i = \\\\mathbb {E} \\\\begin{bmatrix} \\\\widetilde{\\\\mathbf {\\\\theta }}^n_i | \\\\widetilde{\\\\mathbf {z}}^n_{i-1}, \\\\lbrace \\\\widehat{\\\\mathbf {y}}^l_{i-1|i-2} \\\\rbrace _{l \\\\in \\\\Omega _n} \\\\end{bmatrix} \\\\\\\\{\\\\mathbf {\\\\nu }}^n_i &= {\\\\mathbf {\\\\theta }}^n_i - \\\\overline{\\\\mathbf {\\\\theta }}^n_i, \\\\qquad \\\\overline{\\\\mathbf {\\\\theta }}^n_i = \\\\mathbb {E} \\\\begin{bmatrix} {\\\\mathbf {\\\\theta }}^n_i | \\\\widehat{\\\\mathbf {y}}^n_{i-1|i-1} \\\\end{bmatrix}$ that expands to $\\\\widetilde{\\\\mathbf {\\\\nu }}^n_i \\\\!=\\\\!\",\n \"\\\\begin{bmatrix}\\\\widehat{{\\\\mathbf {y}}}_{i|i-1}^{l_1} - \\\\widehat{{\\\\mathbf {y}}}_{i|i-1}^n \\\\\\\\ \\\\vdots \\\\\\\\ \\\\widehat{{\\\\mathbf {y}}}_{i|i-1}^{l_{d_n}}-\\\\widehat{{\\\\mathbf {y}}}_{i|i-1}^n \\\\\\\\ \\\\widetilde{{\\\\mathbf {z}}}^n_i \\\\!\\\\!-\\\\!\",\n \"\\\\widetilde{H}_{\\\\!n} \\\\widehat{{\\\\mathbf {y}}}_{i|i-1}^n \\\\!-\\\\!\",\n \"\\\\check{H}_{\\\\!n} \\\\widetilde{I} \\\\widehat{{\\\\mathbf {x}}}_{i|i-1}^n \\\\end{bmatrix}\\\\!\\\\!\\\\!, \\\\;\\\\;{\\\\mathbf {\\\\nu }}^n_i \\\\!=\\\\!\",\n \"\\\\widehat{{\\\\mathbf {y}}}_{i|i}^n \\\\!-\\\\!\",\n \"G\\\\widehat{{\\\\mathbf {x}}}_{i|i-1}^n.$ The uncorrelated sequences $\\\\widetilde{\\\\mathbf {\\\\nu }}^n_i$ and ${\\\\mathbf {\\\\nu }}^n_i$ are zero-mean Gaussian random vectors.\",\n \"Hence $\\\\widetilde{\\\\mathbf {\\\\nu }}^n_i$ and ${\\\\mathbf {\\\\nu }}^n_i$ are independent sequences.\",\n \"The proof is in Appendix REF .\",\n \"We write the CIKF filter updates (REF ) and () in terms of the new uncorrelated information $\\\\widetilde{\\\\mathbf {\\\\nu }}^n_i$ and ${\\\\mathbf {\\\\nu }}^n_i$ from (REF ), $ \\\\widehat{{\\\\mathbf {y}}}_{i|i}^n &= \\\\widehat{{\\\\mathbf {y}}}_{i|i-1}^n + \\\\widehat{B}^n_i \\\\widetilde{\\\\mathbf {\\\\nu }}^n_i \\\\\\\\ \\\\widehat{{\\\\mathbf {x}}}_{i|i}^n &= \\\\widehat{{\\\\mathbf {x}}}_{i|i-1}^n + K^n_i {\\\\mathbf {\\\\nu }}^n_i$ where, $\\\\widehat{B}^n_i$ are the building blocks of the pseudo-state gain matrix $B_i$ .\"\n ],\n [\n \"Consensus and innovation gains\",\n \"Here, we present the methods to: (a) design the matrices $\\\\widehat{B}^n_i$ and $K^n_i$ ; and (b) obtain the optimal gains $B_i$ and $K_i$ from them.\",\n \"These optimal gains provide the distributed minimized MSE estimates of the field.\",\n \"At agent $n$ , we define the matrix $\\\\widehat{B}^n_i$ as $\\\\widehat{B}^n_i = \\\\begin{bmatrix} B^{n l_1}_i, \\\\cdots , B^{n l_{d_n}}_i, B^{n n}_i \\\\end{bmatrix}$ where, $\\\\lbrace l_1, \\\\cdots , l_{d_n}\\\\rbrace = \\\\Omega _n$ .\",\n \"The gain matrix $B_i$ is a linear combination of $B^{\\\\mathcal {C}}_i$ and $B^{\\\\mathcal {I}}_i$ , where $B^{\\\\mathcal {C}}_i$ has the same block structure as the graph Laplacian $L$ and $B^{\\\\mathcal {I}}_i$ is a block diagonal.\",\n \"The $(n,l)^{\\\\text{th}}$ blocks of the $n^{\\\\text{th}}$ row of $B^{\\\\mathcal {C}}_i$ are: $\\\\left[B^{\\\\mathcal {C}}_i\\\\right]_{nl} = \\\\left\\\\lbrace \\\\begin{array}{ll}- B^{n l}_i, & \\\\mbox{if } l \\\\in \\\\Omega _n \\\\\\\\\\\\sum _{j=1}^{d_n}B^{n l_j}_i, & \\\\mbox{if } l = n \\\\\\\\{\\\\mathbf {0}}, & \\\\text{otherwise.}\",\n \"\\\\\\\\\\\\end{array} \\\\right.$ The $\\\\lbrace n,n\\\\rbrace ^{\\\\text{th}}$ blocks of the diagonal block matrices $B^{\\\\mathcal {I}}_i$ and $K_i$ are: $\\\\left[B^{\\\\mathcal {I}}_i\\\\right]_{nn} &= B^{n n}_i, \\\\\\\\\\\\left[K_i\\\\right]_{nn} &= K^{n}_i,$ Hence, once we design the matrices $\\\\widehat{B}^n_i$ and $K^{n}_i$ , it will provide the optimal gain matrices $B^{\\\\mathcal {C}}_i, B^{\\\\mathcal {I}}_i$ , and $K_i$ .\",\n \"Theorem 4 The optimal gains for the CIKF algorithm are $\\\\widehat{B}^n_i &= \\\\Sigma _{{\\\\mathbf {y}}_i \\\\widetilde{\\\\mathbf {\\\\nu }}^n_i} \\\\left(\\\\Sigma _{\\\\widetilde{\\\\mathbf {\\\\nu }}^n_i}\\\\right)^{-1}, \\\\\\\\K^n_i &= \\\\Sigma _{{\\\\mathbf {x}}_i {\\\\mathbf {\\\\nu }}^n_i} \\\\left(\\\\Sigma _{{\\\\mathbf {\\\\nu }}^n_i}\\\\right)^{-1}$ where, $\\\\Sigma _{\\\\widetilde{\\\\mathbf {\\\\nu }}^n_i}$ , $\\\\Sigma _{{\\\\mathbf {\\\\nu }}^n_i}$ are the covariances of the new uncorrelated information $\\\\widetilde{\\\\mathbf {\\\\nu }}^n_i$ and ${\\\\mathbf {\\\\nu }}^n_i$ ; and, $\\\\Sigma _{{\\\\mathbf {y}}_i \\\\widetilde{\\\\mathbf {\\\\nu }}^n_i}$ , $\\\\Sigma _{{\\\\mathbf {x}}_i {\\\\mathbf {\\\\nu }}^n_i}$ are cross-covariances between ${\\\\mathbf {y}}_i$ , $\\\\widetilde{\\\\mathbf {\\\\nu }}^n_i$ and ${\\\\mathbf {x}}_i$ , ${\\\\mathbf {\\\\nu }}^n_i$ , respectively.\",\n \"These covariance and cross-covariance matrices are related to the error covariance matrices, $P_{i|i-1}, P_{i|i}, \\\\Sigma _{i|i-1}, \\\\Pi _{i|i-1}, \\\\Gamma _i$ , by the following functions: $& \\\\!\\\\Sigma _{{\\\\mathbf {y}}_i \\\\widetilde{\\\\mathbf {\\\\nu }}^n_i} \\\\!=\\\\!\",\n \"\\\\left[\\\\begin{array}{l;{2pt/2pt}l;{1pt/1pt}l;{2pt/2pt}l}\\\\!\\\\!\\\\!\\\\!\",\n \"P_{i|i\\\\!-\\\\!1}^{nn} \\\\!-\\\\!\",\n \"P_{i|i\\\\!-\\\\!1}^{nl_1} & \\\\cdots & P_{i|i\\\\!-\\\\!1}^{nn} \\\\!-\\\\!\",\n \"P_{i|i\\\\!-\\\\!1}^{nl_{d_n}} & P_{i|i\\\\!-\\\\!1}^{nn} \\\\!\",\n \"\\\\widetilde{H}^T_n \\\\!+\\\\!\",\n \"\\\\Pi _{i|i\\\\!-\\\\!1}^{nn^T} \\\\!\",\n \"\\\\check{H}^T_n \\\\!\\\\!\\\\!\\\\!\\\\end{array}\\\\right] \\\\\\\\&\\\\!\",\n \"\\\\Sigma _{{\\\\mathbf {x}}_i {\\\\mathbf {\\\\nu }}^n_i} = \\\\Sigma _{i|i\\\\!-\\\\!1}^{nn} G - \\\\Gamma ^{nn}_i \\\\\\\\&\\\\begin{bmatrix} \\\\Sigma _{\\\\widetilde{\\\\mathbf {\\\\nu }}^n_i} \\\\end{bmatrix}_{qs} = \\\\\\\\&\\\\left\\\\lbrace \\\\begin{array}{ll}P_{i|i\\\\!-\\\\!1}^{nn} \\\\!-\\\\!\",\n \"P_{i|i\\\\!-\\\\!1}^{nl_s} \\\\!-\\\\!\",\n \"P_{i|i\\\\!-\\\\!1}^{l_q n} \\\\!+\\\\!\",\n \"P_{i|i\\\\!-\\\\!1}^{l_q l_s}, & \\\\mbox{if } q \\\\le s \\\\le d_n \\\\\\\\\\\\!\\\\left( P_{i|i\\\\!-\\\\!1}^{nn} \\\\!-\\\\!\",\n \"P_{i|i\\\\!-\\\\!1}^{l_q n} \\\\right)\\\\!\\\\widetilde{H}_n^T \\\\!+\\\\!\",\n \"\\\\left( \\\\Pi _{i|i\\\\!-\\\\!1}^{nn} \\\\!-\\\\!\",\n \"\\\\Pi _{i|i\\\\!-\\\\!1}^{l_q n} \\\\right)^T\\\\!\\\\!\\\\check{H}_n^T, & \\\\mbox{if } q < s = d_n\\\\!\\\\!+\\\\!1 \\\\\\\\\\\\widetilde{H}_n P_{i|i\\\\!-\\\\!1}^{nn}\\\\widetilde{H}_n^T + \\\\widetilde{H}_n \\\\Pi _{i|i\\\\!-\\\\!1}^{nn^T}\\\\check{H}_n^T + \\\\check{H}_n\\\\Pi _{i|i\\\\!-\\\\!1}^{nn}\\\\widetilde{H}_n^T & \\\\\\\\\\\\qquad \\\\qquad + \\\\check{H}_n\\\\Sigma _{i|i\\\\!-\\\\!1}^{nn}\\\\check{H}^T_n + \\\\overline{H}_n, & \\\\mbox{if } q = s = d_n\\\\!\\\\!+\\\\!1 \\\\\\\\ \\\\begin{bmatrix} \\\\Sigma _{\\\\widetilde{\\\\mathbf {\\\\nu }}^n_i} \\\\end{bmatrix}_{sq}^T, & \\\\mbox{if } q > s \\\\\\\\\\\\end{array} \\\\right.\",\n \"\\\\\\\\&\\\\begin{bmatrix} \\\\Sigma _{{\\\\mathbf {\\\\nu }}^n_i} \\\\end{bmatrix}_{qs} = G\\\\Sigma _{i|i\\\\!-\\\\!1}^{nn}G - G \\\\Gamma ^{nn}_i - \\\\Gamma ^{nn^T}_iG + P_{i|i}^{nn}$ where, $\\\\left[\\\\Sigma _{\\\\widetilde{\\\\mathbf {\\\\nu }}^n_i}\\\\right]_{qs}$ denotes the $\\\\lbrace q,s\\\\rbrace ^{\\\\text{th}}$ block of the $(d_n\\\\!\\\\!+\\\\!1)\\\\times (d_n\\\\!\\\\!+\\\\!1)$ block matrix $\\\\Sigma _{\\\\widetilde{\\\\mathbf {\\\\nu }}^n_i}$ .\",\n \"The proof is in Appendix REF .\",\n \"By the Gauss-Markov theorem, the CIKF algorithm, along with this design of the consensus and innovation gain matrices, as stated in Theorem REF , results in the minimized MSE distributed estimates of the dynamic random field ${\\\\mathbf {x}}_i$ .\",\n \"The gain matrices are deterministic.\",\n \"Hence they can be precomputed offline and saved for online implementation.\",\n \"In Algorithm REF , we state the steps that each agent $n$ runs to compute the optimal gain matrices.\",\n \"[h] Gain Design of CIKF Input: Model parameters $A$ , $V$ , $H$ , $R$ , $G$ , $L$ , $\\\\Sigma _0$ .\",\n \"Initialize: $\\\\Sigma _{0|-1} = J \\\\!\\\\otimes \\\\!\",\n \"\\\\Sigma _0, \\\\; P_{0|-1} = J \\\\!\\\\otimes \\\\!\",\n \"\\\\left(G\\\\Sigma _0 G\\\\right), \\\\; \\\\Pi _{0|-1} = J \\\\!\\\\otimes \\\\!\",\n \"\\\\left(\\\\Sigma _0 G\\\\right)$ .\",\n \"$i \\\\ge 0$ $\\\\!\\\\!\\\\!\\\\!$Optimal gains: Compute $\\\\widehat{B}^n_i$ and $K^n_i$ using Theorem REF .\",\n \"Using (REF )-(), obtain $B^{\\\\mathcal {C}}_i$ , $B^{\\\\mathcal {I}}_i$ and $K_i$ from $\\\\widehat{B}^n_i$ and $K^n_i$ .\",\n \"$\\\\!\\\\!\\\\!\\\\!$Prediction error covariance updates: Update $P_{i+1|i}, \\\\Sigma _{i+1|i}, \\\\Pi _{i+1|i}$ using ()-().\",\n \"The offline Algorithm REF along with the online Algorithm REF completes our proposed distributed solution to obtain minimized MSE estimates of the dynamic field ${\\\\mathbf {x}}_i$ at each agent in the cyber network.\",\n \"CIKF does not require sparsity of the matrices $A$ , $H_n$ and $L$ .\",\n \"However, in most distributed applications these matrices are sparse.\",\n \"For example, consider the temperature distribution over the entire United States as a time-varying random field.\",\n \"It is conceivable that the temperature in a location will be directly dependent on the temperature of a few locations leading to a sparse $A$ .\",\n \"Similarly a weather station will make localized measurements of the temperature (at its own location) rather than measuring the temperature all over the US.\",\n \"Similarly the weather stations will exchange their data with the neighboring weather stations, thereby reducing the communication cost and leading to a sparse graph Laplacian matrix.\",\n \"From a computational point of view, if the matrices $A$ , $H_n$ and $L$ are sparse, they can be stored as lists, for example, and computation of matrix products with them will involve less multiplications and/or additions operations.\",\n \"Since the computation of $B_i$ and $K_i$ involve these matrices, their sparsity will reduce both the computation and storage costs.\"\n ],\n [\n \"Numerical Evaluation\",\n \"We numerically evaluate the MSE performance of the CIKF and compare it against the centralized Kalman filter (CKF) and the distributed information Kalman filter (DIKF) in [5].\",\n \"To this objective, we build a time-varying random system, observation and network model that satisfies the Assumptions REF -REF .\",\n \"The Algorithms REF -REF run on these model parameters.\",\n \"First the Algorithm REF computes and save the gain matrices and the error covariance matrices.\",\n \"The traces of the error covariance matrices provide the theoretical MSE trajectory of the CIKF with time.\",\n \"Then, we Monte-Carlo simulate Algorithm REF to compute the numerical MSE of the distributed estimators, CIKF and DIKF, and the centralized estimator CKF.\"\n ],\n [\n \"Model specifications\",\n \"Here, we consider a time-varying field, ${\\\\mathbf {x}}_i$ , with dimension $M=50$ .\",\n \"The physical layer, consisting of $M=50$ sites, is monitored by a cyber layer consisting of $N=50$ agents.\",\n \"Each agent in the cyber layer observes $M_n=2$ sites of the physical layer.\",\n \"We build the field dynamics matrix $A$ to be sparse and distributed.\",\n \"The dynamics $A$ possess the structure of a Lattice graph, where the time evolution of a field variable depends on the neighboring field variables.\",\n \"The structure of the Lattice graph is randomly generated using R library.\",\n \"The degree is considered to be 4 and the non-zero entries are randomly chosen from $(0,1)$ .\",\n \"For illustration, we consider an unstable field dynamics with $\\\\Vert A\\\\Vert _2=1.05$ to test the resilience of the algorithms under unstable conditions.\",\n \"We scale the randomly generated $A$ such that its spectral norm becomes 1.05.\",\n \"The observation matrices, $H_n \\\\in \\\\mathbb {R}^{2\\\\times 50}, n=1, \\\\cdots , 50$ , are sparse $0-1$ matrices with one non-zero element at each row corresponding to the site of ${\\\\mathbf {x}}_i$ observed by the $n^{\\\\text{th}}$ agent.\",\n \"The local observations ${\\\\mathbf {z}}^n_i$ are $2\\\\times 1$ random vectors.\",\n \"The mean $\\\\overline{x}_0$ of the initial state vector is generated at random.\",\n \"The system noise covariance $V$ , the observation noise covariances $R_n$ , and the initial state covariance $\\\\Sigma _0$ are randomly generated symmetric positive definite matrices.\",\n \"We then scale the covariance matrices such that their covariance becomes: $\\\\Vert V\\\\Vert _2=4, \\\\Vert R_n\\\\Vert _2=8$ , and $\\\\Vert \\\\Sigma _0\\\\Vert _2=16$ .\",\n \"The agents in the cyber layer communicate among themselves following a randomly generated Erdős-Rényi graph $\\\\mathcal {G}$ with 50 nodes and $E=138$ edges.\",\n \"The average degree of each node/agent is approximately $5.5$ .\",\n \"The communication network $\\\\mathcal {G}$ is also sparse.\",\n \"For Monte-Carlo simulations, we generate the noises, ${\\\\mathbf {v}}_i, {\\\\mathbf {r}}^n_i$ , and the initial condition, ${\\\\mathbf {x}}_0$ as Gaussian sequences, with ${\\\\mathbf {v}}_i \\\\sim \\\\mathcal {N}(\\\\bar{0},V), \\\\;{\\\\mathbf {r}}^n_i \\\\sim \\\\mathcal {N}(\\\\bar{0},R_n), \\\\;{\\\\mathbf {x}}_0 \\\\sim \\\\mathcal {N}(\\\\bar{{\\\\mathbf {x}}}_0,\\\\Sigma _0)$ .\",\n \"The sequences $\\\\lbrace \\\\lbrace {\\\\mathbf {v}}_i\\\\rbrace _i, \\\\lbrace {\\\\mathbf {r}}_i\\\\rbrace _i, {\\\\mathbf {x}}_0\\\\rbrace _{i \\\\ge 0}$ are generated to be uncorrelated.\",\n \"Each agent $n$ in the cyber layer has access to the system parameters $A, V, H, R, \\\\bar{{\\\\mathbf {x}}}_0, \\\\Sigma _0$ , and $\\\\mathcal {G}$ .\",\n \"This numerical model satisfies Assumptions REF -REF .\",\n \"The pair $(A,H)$ is detectable and the pairs, $(A,H_n) \\\\; \\\\forall n$ , are not detectable.\",\n \"The agent communication graph $\\\\mathcal {G}$ is connected with the algebraic connectivity of the Laplacian $\\\\lambda _2(L) = 0.7 > 0$ .\",\n \"Hence the Assumptions REF -REF hold true for this numerical system, observation, and network model.\",\n \"Figure: Comparison of MSE performance of the proposed CIKF with CKF and DIKF.\"\n ],\n [\n \"Optimized gains and theoretical MSE\",\n \"We run Algorithm REF on the numerical model to obtain the gain matrices and the theoretical error covariances of the CIKF.\",\n \"We compute the gain matrices and the error covariances of the centralized Kalman filter (CKF) and of the distributed information Kalman filter (DIKF).\",\n \"In Fig.\",\n \"REF , we plot the MSE (in dB), trace of the predictor error covariance matrices $\\\\Sigma _{i+1|i}$ , for each of these cases up to time $i=30$ .\",\n \"The MSE of the optimal CKF is the smallest (recall the CKF, if feasible, would be optimal) and the objective of the distributed estimators is to achieve MSE performance as close as possible to that of the CKF.\",\n \"From Fig.\",\n \"REF , we see that the MSE of the proposed CIKF is 3dB more than the CKF but is 3dB less than the DIKF.\",\n \"From the plot we see that the CIKF converges faster than the DIKF.\",\n \"Hence, the proposed CIKF provides faster convergence and 3dB MSE performance improvement over the DIKF.\",\n \"The performance of the CKF is 3dB better than the CIKF's due to the fact that the CKF has access to the observations of all the sensors at every time steps.\",\n \"In contrast in CIKF, each agent has access to its own observations and the current estimates of its neighbors only; the impact of the observations from the other agents propagate through the network with delay.\",\n \"As the time-varying field ${\\\\mathbf {x}}_i$ is evolving with input noise ${\\\\mathbf {v}}_i$ , lack of access to all the observations containing the driving input ${\\\\mathbf {v}}_i$ , combined with the network diffusion delay, causes a performance gap between the CKF and the CIKF.\"\n ],\n [\n \"Monte-Carlo Simulations\",\n \"We empirically compute the MSE of the distributed estimates given by the CIKF Algorithm REF .\",\n \"We implement the algorithms using Matlab in the Microsoft Azure cloud.\",\n \"Given the computation load because of the large system (M = 50) and network (N = 50) models, we run our simulation on Azure DS13 (8 cores, 56 GB memory) virtual machine (VM).\",\n \"The MSE computation for the CIKF, CKF, and DIKF algorithms with 1000 Monte-Carlo runs require approximately 30 hours in the Azure DS13 VM.\",\n \"Once we obtain the field prediction estimates for the three algorithms, we compute the empirical prediction error covariance matrices $\\\\widehat{\\\\Sigma }_{i+1|i}$ and then obtain the Monte-Carlo MSE (in dB) from their trace.\",\n \"From the Monte-Carlo MSE plot in Fig REF , we see that the the empirical plots follow closely the theoretical plots in Fig REF .\",\n \"Both the theoretical Fig REF and Monte-Carlo simulated Fig REF MSE performance confirms our CIKF analysis in Sections -.\",\n \"The novel Consensus$+$ Innovations Kalman Filter (CIKF) proposed in Section along with the optimized gain designs in Section provides unbiased distributed estimates with bounded and minimized MSE for the Consensus$+$ Innovations distributed solution.\",\n \"The CIKF achieves nearly 3dB better performance than the DIKF [5].\"\n ],\n [\n \"Conclusions\",\n \"Summary: In this paper, we propose a Consensus$+$ Innovations Kalman Filter (CIKF) that obtains unbiased minimized MSE distributed estimates of the pseudo-states and real-time employs them to obtain the unbiased distributed filtering and prediction estimates of the time-varying random state at each agent.\",\n \"The filter update iterations are of the Consensus$+$ Innovations type.\",\n \"Using the Gauss-Markov principle, we designed the optimal gain matrices that yield approximately 3dB improvement over previous available distributed estimators like the DIKF in [5].\",\n \"Contributions: The primary contributions of this paper are: (a) design of a filter and corresponding gain matrices to obtain minimized MSE distributed estimates at each agent under minimal assumptions; and (b) a theoretical characterization of the tracking capacity and distributed version of the algebraic Riccati equation.\",\n \"Compared to DIKF, the contributions of this paper are: $(a)$ The CIKF is a new algorithm for distributed estimation of time-varying random fields which provides 3dB MSE performance over the DIKF and much closer to the performance of the optimal centralized Kalman filter.\",\n \"$(b)$ The CIKF does not require the strict distributed observability assumption, i.e., invertibility of the matrix $G$ .\",\n \"The DIKF require that $G^{-1}$ exists, which demands that all state variables be observed by at least once agent in the network.\",\n \"$(c)$ In CIKF, we design the optimized gain matrices for the dynamic averaging of pseudo-states using the Gauss-Markov Theorem.\",\n \"Hence, the distributed dynamic averaging step of the CIKF has lower MSE than that of the DIKF.\",\n \"$(d)$ These advantages are obtained by using pseudo-state and this paper demonstrates its usefulness in distributed field estimation.\"\n ],\n [\n \"Appendix\",\n \"The Appendices prove the proposition, lemmas, and theorems stated in the paper.\",\n \"Proof of Proposition\"\n ],\n [\n \"Proof of Proposition \",\n \"First we derive the dynamics (REF ) of the pseudo-state ${\\\\mathbf {y}}_i$ .\",\n \"Using (REF ) and (REF ), ${\\\\mathbf {y}}_{i+1} &= G{\\\\mathbf {x}}_{i+1} \\\\\\\\&= G \\\\left( A{\\\\mathbf {x}}_i + {\\\\mathbf {v}}_i\\\\right) \\\\\\\\&= GA \\\\left( G^{\\\\dagger }G + \\\\widetilde{I}\\\\right) {\\\\mathbf {x}}_i + G{\\\\mathbf {v}}_i, \\\\quad \\\\begin{small} \\\\left[ \\\\text{by (\\\\ref {eqn:p_identity_mat})}, \\\\; I = G^{\\\\dagger }G + \\\\widetilde{I} \\\\right] \\\\end{small} \\\\\\\\&= GAG^{\\\\dagger }{\\\\mathbf {y}}_i + G{\\\\mathbf {v}}_i + GA\\\\widetilde{I}{\\\\mathbf {x}}_i \\\\\\\\&= \\\\widetilde{A}{\\\\mathbf {y}}_i + G{\\\\mathbf {v}}_i + \\\\check{A}{\\\\mathbf {x}}_i, \\\\qquad \\\\quad \\\\left[ \\\\text{by (\\\\ref {eqn:pstate})} \\\\right].$ Now, we derive the observations () of the pseudo-observations $\\\\widetilde{{\\\\mathbf {z}}}^n_i$ .\",\n \"Using (REF ) and (REF ), $\\\\widetilde{{\\\\mathbf {z}}}^n_i &= H^T_n R_n^{-1} H_n{\\\\mathbf {x}}_i + H^T_n R_n^{-1} {\\\\mathbf {r}}^n_i \\\\\\\\&= H^T_nR_n^{-1}H_n \\\\left( G^{\\\\dagger }G + \\\\widetilde{I}\\\\right) {\\\\mathbf {x}}_i + H^T_n R_n^{-1} {\\\\mathbf {r}}^n_i \\\\\\\\&= H^T_nR_n^{-1}H_n G^{\\\\dagger }G {\\\\mathbf {x}}_i + H^T_n R_n^{-1} {\\\\mathbf {r}}^n_i + H^T_nR_n^{-1}H_n \\\\widetilde{I} {\\\\mathbf {x}}_i \\\\\\\\&= \\\\widetilde{H} {\\\\mathbf {y}}_i + H^T_n R_n^{-1} {\\\\mathbf {r}}^n_i + \\\\check{H}_n {\\\\mathbf {x}}_i.\",\n \"\\\\qquad \\\\qquad \\\\qquad \\\\Box $ Proof of Lemmas\"\n ],\n [\n \"Proof of Lemma \",\n \"Consider the filtering error definitions (REF )-().\",\n \"We take expectations on both sides, $\\\\mathbb {E}\\\\left[{\\\\mathbf {e}}_{i|i}^n\\\\right] &= \\\\mathbb {E}\\\\left[{\\\\mathbf {y}}_i - \\\\widehat{{\\\\mathbf {y}}}_{i|i}^n \\\\right] \\\\\\\\&= \\\\mathbb {E} \\\\left[ \\\\mathbb {E} \\\\left[ {\\\\mathbf {y}}_i - \\\\widehat{{\\\\mathbf {y}}}_{i|i}^n~|~\\\\widetilde{{\\\\mathbf {z}}}^n_i, \\\\lbrace \\\\widehat{{\\\\mathbf {y}}}_{i|i-1}^l \\\\rbrace _{l \\\\in \\\\Omega _n} \\\\right] \\\\right] \\\\\\\\&= \\\\mathbb {E} \\\\left[ \\\\widehat{{\\\\mathbf {y}}}_{i|i}^n - \\\\widehat{{\\\\mathbf {y}}}_{i|i}^n \\\\right] = 0 \\\\qquad \\\\left[ \\\\text{by} (\\\\ref {eqn:MMSE_yf}) \\\\right] \\\\\\\\\\\\mathbb {E}\\\\left[{\\\\mathbf {\\\\epsilon }}_{i|i}^n\\\\right] &= \\\\mathbb {E}\\\\left[{\\\\mathbf {x}}_i - \\\\widehat{{\\\\mathbf {x}}}_{i|i}^n \\\\right] \\\\\\\\&= \\\\mathbb {E} \\\\left[ \\\\mathbb {E} \\\\left[ {\\\\mathbf {x}}_i - \\\\widehat{{\\\\mathbf {x}}}_{i|i}^n~|~\\\\widehat{{\\\\mathbf {y}}}_{i|i}^n \\\\right] \\\\right] \\\\\\\\&= \\\\mathbb {E} \\\\left[ \\\\widehat{{\\\\mathbf {x}}}_{i|i}^n - \\\\widehat{{\\\\mathbf {x}}}_{i|i}^n \\\\right] = 0 \\\\qquad \\\\left[ \\\\text{by} (\\\\ref {eqn:epsfn}) \\\\right].$ Similarly, taking expectations on prediction errors (REF )-(), $\\\\mathbb {E}\\\\left[{\\\\mathbf {e}}_{i+1|i}^n\\\\right] &= \\\\mathbb {E}\\\\left[{\\\\mathbf {y}}_{i+1} - \\\\widehat{{\\\\mathbf {y}}}_{i+1|i}^n \\\\right] \\\\\\\\&= \\\\mathbb {E} \\\\left[ \\\\mathbb {E} \\\\left[ {\\\\mathbf {y}}_{i+1} - \\\\widehat{{\\\\mathbf {y}}}_{i+1|i}^n~|~\\\\widetilde{{\\\\mathbf {z}}}^n_i, \\\\lbrace \\\\widehat{{\\\\mathbf {y}}}_{i|i-1}^l \\\\rbrace _{l \\\\in \\\\Omega _n} \\\\right] \\\\right] \\\\\\\\&= \\\\mathbb {E} \\\\left[ \\\\widehat{{\\\\mathbf {y}}}_{i+1|i}^n - \\\\widehat{{\\\\mathbf {y}}}_{i+1|i}^n \\\\right] = 0 \\\\qquad \\\\left[ \\\\text{by} (\\\\ref {eqn:MMSE_yp}) \\\\right] \\\\\\\\\\\\mathbb {E}\\\\left[{\\\\mathbf {\\\\epsilon }}_{i+1|i}^n\\\\right] &= \\\\mathbb {E}\\\\left[{\\\\mathbf {x}}_{i+1} - \\\\widehat{{\\\\mathbf {x}}}_{i+1|i}^n \\\\right] \\\\\\\\&= \\\\mathbb {E} \\\\left[ \\\\mathbb {E} \\\\left[ {\\\\mathbf {x}}_{i+1} - \\\\widehat{{\\\\mathbf {x}}}_{i+1|i}^n~|~\\\\widehat{{\\\\mathbf {y}}}_{i|i}^n \\\\right] \\\\right] \\\\\\\\&= \\\\mathbb {E} \\\\left[ \\\\widehat{{\\\\mathbf {x}}}_{i+1|i}^n - \\\\widehat{{\\\\mathbf {x}}}_{i+1|i}^n \\\\right] = 0 \\\\qquad \\\\left[ \\\\text{by} (\\\\ref {eqn:epspn}) \\\\right].\",\n \"\\\\qquad \\\\Box $\"\n ],\n [\n \"Proof of Lemma \",\n \"We write the pseudo-state filtering update (REF ) in vector form, $\\\\widehat{{\\\\mathbf {y}}}_{i|i}\\\\!=\\\\!\",\n \"\\\\widehat{{\\\\mathbf {y}}}_{i|i-1}& \\\\!-\\\\!\",\n \"B^{\\\\mathcal {C}}_i \\\\widehat{{\\\\mathbf {y}}}_{i|i-1}\\\\!+\\\\!\",\n \"B^{\\\\mathcal {I}}_i \\\\!\",\n \"\\\\left( \\\\!\\\\widetilde{{\\\\mathbf {z}}}_i \\\\!-\\\\!\",\n \"\\\\left(\\\\widetilde{D}_H \\\\widehat{{\\\\mathbf {y}}}_{i|i-1}\\\\!+\\\\!\",\n \"\\\\check{D}_H \\\\widehat{{\\\\mathbf {x}}}_{i|i-1}\\\\right) \\\\right) ,$ where, $\\\\left[B^{\\\\mathcal {C}}_i\\\\right]_{nl} = -B^{n,l}_i, \\\\; \\\\forall n \\\\ne l$ , $\\\\left[B^{\\\\mathcal {C}}_i\\\\right]_{nn} = \\\\sum _{l \\\\in \\\\Omega _n}B^{n,l}_i$ , $\\\\left[B^{\\\\mathcal {I}}_i\\\\right]_{nn} = B^{n,n}_i$ , $\\\\left[B^{\\\\mathcal {I}}_i\\\\right]_{nl} = 0, \\\\; \\\\forall n \\\\ne l$ .\",\n \"The block-diagonal matrices: $\\\\widetilde{D}_H = \\\\text{blockdiag}\\\\lbrace \\\\widetilde{H}_1, \\\\cdots , \\\\widetilde{H}_N\\\\rbrace , \\\\; \\\\check{D}_H = \\\\text{blockdiag}\\\\lbrace \\\\check{H}_1, \\\\cdots , \\\\check{H}_N\\\\rbrace $ .\",\n \"Note that $B^{\\\\mathcal {C}}_i \\\\left(1_N\\\\otimes {\\\\mathbf {y}}_i\\\\right) = \\\\mathbf {0}$ .\",\n \"Using this relation and the vector form of $\\\\widehat{{\\\\mathbf {y}}}_{i|i}$ , we expand the pseudo-state filter error process ${\\\\mathbf {e}}_{i|i}$ , ${\\\\mathbf {e}}_{i|i}&= 1_N \\\\!\\\\otimes \\\\!\",\n \"{\\\\mathbf {y}}_i - \\\\widehat{{\\\\mathbf {y}}}_{i|i}\\\\\\\\&= \\\\left( 1_N \\\\!\\\\otimes \\\\!\",\n \"{\\\\mathbf {y}}_i - \\\\widehat{{\\\\mathbf {y}}}_{i|i-1}\\\\right) - B^{\\\\mathcal {C}}_i \\\\left( 1_N \\\\!\\\\otimes \\\\!\",\n \"{\\\\mathbf {y}}_i - \\\\widehat{{\\\\mathbf {y}}}_{i|i-1}\\\\right) \\\\!-\\\\!\",\n \"B^{\\\\mathcal {I}}_i D_{H}^TR^{-1}{\\\\mathbf {r}}_i \\\\\\\\& \\\\qquad - B^{\\\\mathcal {I}}_i \\\\widetilde{D}_H \\\\left(1_N \\\\!\\\\otimes \\\\!\",\n \"{\\\\mathbf {y}}_i - \\\\widehat{{\\\\mathbf {y}}}_{i|i-1}\\\\right) - B^{\\\\mathcal {I}}_i\\\\check{D}_H \\\\left( 1_N \\\\!\\\\otimes \\\\!\",\n \"{\\\\mathbf {x}}_i - \\\\widehat{{\\\\mathbf {x}}}_{i|i-1}\\\\right) \\\\\\\\&= \\\\left( I_{\\\\!M\\\\!N} \\\\!-\\\\!\",\n \"B^{\\\\mathcal {C}}_i \\\\!-\\\\!\",\n \"B^{\\\\mathcal {I}}_i \\\\!\\\\widetilde{D}_{H}\\\\right) {\\\\mathbf {e}}_{i|i-1}\\\\!-\\\\!\",\n \"B^{\\\\mathcal {I}}_i \\\\check{D}_{H} {\\\\mathbf {\\\\epsilon }}_{i|i-1}\\\\!-\\\\!\",\n \"B^{\\\\mathcal {I}}_i\\\\!\",\n \"D_{H}^T R^{-1} r_i.$ The state filtering update (), in vector form, is $\\\\widehat{{\\\\mathbf {x}}}_{i|i}= \\\\widehat{{\\\\mathbf {x}}}_{i|i-1}+ K_i \\\\left( \\\\widehat{{\\\\mathbf {y}}}_{i|i}\\\\!-\\\\!\",\n \"\\\\left(I_N\\\\!\\\\otimes \\\\!G\\\\right) \\\\widehat{{\\\\mathbf {x}}}_{i|i-1}\\\\right) ,$ where, $K_i = \\\\text{blockdiag} \\\\lbrace K^1_i, \\\\cdots , K^N_i \\\\rbrace $ .\",\n \"Using the relation $\\\\widehat{{\\\\mathbf {y}}}_{i|i}= \\\\left(1_N\\\\otimes {\\\\mathbf {y}}_i\\\\right) - {\\\\mathbf {e}}_{i|i}= \\\\left(I_N\\\\!\\\\otimes \\\\!G\\\\right)\\\\left(1_N\\\\otimes {\\\\mathbf {x}}_i\\\\right) - {\\\\mathbf {e}}_{i|i}$ , we expand the state filter error process ${\\\\mathbf {e}}_{i|i}$ , ${\\\\mathbf {\\\\epsilon }}_{i|i}&= 1_N \\\\!\\\\otimes \\\\!\",\n \"{\\\\mathbf {x}}_i - \\\\widehat{{\\\\mathbf {x}}}_{i|i}\\\\\\\\&= \\\\left( 1_{\\\\!N} \\\\!\\\\otimes \\\\!\",\n \"{\\\\mathbf {x}}_i - \\\\widehat{{\\\\mathbf {x}}}_{i|i-1}\\\\right) \\\\!-\\\\!\",\n \"K_i \\\\left(I_{\\\\!N}\\\\!\\\\otimes \\\\!G\\\\right) \\\\left(1_{\\\\!N} \\\\!\\\\otimes \\\\!\",\n \"{\\\\mathbf {x}}_i \\\\!-\\\\!\",\n \"\\\\widehat{{\\\\mathbf {x}}}_{i|i-1}\\\\right) \\\\!+\\\\!\",\n \"K_i {\\\\mathbf {e}}_{i|i}\\\\\\\\&= \\\\left( I_{MN} - K^{\\\\mathcal {C}}_i - K_i \\\\left(I_N\\\\!\\\\otimes \\\\!G\\\\right) \\\\right) {\\\\mathbf {\\\\epsilon }}_{i|i-1}+ K_i {\\\\mathbf {e}}_{i|i}.$ The dynamics of the pseudo-state and state prediction errors, ${\\\\mathbf {e}}_{i+1|i}&= 1_N \\\\!\\\\otimes \\\\!\",\n \"{\\\\mathbf {y}}_{i+1} - \\\\widehat{{\\\\mathbf {y}}}_{i+1|i}\\\\\\\\&= \\\\left(I_N\\\\!\\\\otimes \\\\!\\\\widetilde{A}\\\\right)\\\\left(1_N \\\\!\\\\otimes \\\\!\",\n \"{\\\\mathbf {y}}_i\\\\right) + \\\\left(I_N\\\\!\\\\otimes \\\\!\",\n \"\\\\check{A} \\\\right)\\\\left(1_N \\\\!\\\\otimes \\\\!\",\n \"{\\\\mathbf {x}}_i\\\\right) \\\\\\\\&\\\\quad + 1_N\\\\!\\\\otimes \\\\!\",\n \"(G{\\\\mathbf {v}}_i) - \\\\left(I_N\\\\!\\\\otimes \\\\!\\\\widetilde{A}\\\\right)\\\\widehat{{\\\\mathbf {y}}}_{i|i}-\\\\left(I_N\\\\!\\\\otimes \\\\!\",\n \"\\\\check{A} \\\\right)\\\\widehat{{\\\\mathbf {x}}}_{i|i}\\\\\\\\&= \\\\left(I_N\\\\!\\\\otimes \\\\!\\\\widetilde{A}\\\\right){\\\\mathbf {e}}_{i|i}+ \\\\left(I_N\\\\!\\\\otimes \\\\!\",\n \"\\\\check{A} \\\\right) {\\\\mathbf {\\\\epsilon }}_{i|i}+ 1_N\\\\!\\\\otimes \\\\!\",\n \"(G{\\\\mathbf {v}}_i) \\\\\\\\{\\\\mathbf {\\\\epsilon }}_{i+1|i}&= 1_N \\\\!\\\\otimes \\\\!\",\n \"{\\\\mathbf {x}}_{i+1} - \\\\widehat{{\\\\mathbf {x}}}_{i+1|i}\\\\\\\\&= \\\\left(I_N\\\\!\\\\otimes \\\\!A\\\\right)\\\\left(1_N \\\\!\\\\otimes \\\\!\",\n \"{\\\\mathbf {x}}_i\\\\right) + 1_N\\\\!\\\\otimes \\\\!\",\n \"{\\\\mathbf {v}}_i - \\\\left(I_N\\\\!\\\\otimes \\\\!A\\\\right)\\\\widehat{{\\\\mathbf {x}}}_{i|i}\\\\\\\\&= \\\\left(I_N\\\\!\\\\otimes \\\\!A\\\\right){\\\\mathbf {\\\\epsilon }}_{i|i}+ 1_N\\\\!\\\\otimes \\\\!\",\n \"{\\\\mathbf {v}}_i.$ Since the state ${\\\\mathbf {x}}_i$ , pseudo-state ${\\\\mathbf {y}}_i$ , their initial condition and all the noises are Gaussian, their estimates are also Gaussian making all the filtering and prediction errors Gaussian.\",\n \"$\\\\Box $\"\n ],\n [\n \"Proof of Lemma \",\n \"Lemma REF and Assumption REF guarantee that ${\\\\mathbf {\\\\epsilon }}_{i|i}, {\\\\mathbf {\\\\epsilon }}_{i+1|i}, {\\\\mathbf {e}}_{i|i}, {\\\\mathbf {v}}_i, {\\\\mathbf {r}}_i$ are Gaussian.\",\n \"The error noises $\\\\widetilde{\\\\mathbf {\\\\phi }}_i$ and ${\\\\mathbf {\\\\phi }}_i$ are therefore Gaussian as they are linear combinations of the error processes and the model noises ${\\\\mathbf {\\\\epsilon }}_{i|i}, {\\\\mathbf {\\\\epsilon }}_{i+1|i}, {\\\\mathbf {e}}_{i|i}, {\\\\mathbf {v}}_i, {\\\\mathbf {r}}_i$ .\",\n \"By Lemma REF , we have $\\\\mathbb {E}\\\\left[{\\\\mathbf {\\\\epsilon }}_{i|i}\\\\right] = \\\\mathbb {E}\\\\left[{\\\\mathbf {\\\\epsilon }}_{i+1|i}\\\\right] = \\\\mathbb {E}\\\\left[{\\\\mathbf {e}}_{i|i}\\\\right] = \\\\mathbf {0}$ .\",\n \"From Assumption REF , we know $\\\\mathbb {E}\\\\left[{\\\\mathbf {v}}_i\\\\right] = \\\\mathbb {E}\\\\left[{\\\\mathbf {r}}_i\\\\right] = \\\\mathbf {0}$ .\",\n \"We take expectation on both sides of (REF )-() and apply these relations $\\\\mathbb {E}\\\\left[\\\\widetilde{\\\\mathbf {\\\\phi }}_i\\\\right] &= \\\\!\\\\left( I_{\\\\!N} \\\\!\\\\otimes \\\\!\",\n \"\\\\check{A}\\\\right)\\\\!\",\n \"\\\\mathbb {E}\\\\left[{\\\\mathbf {\\\\epsilon }}_{i|i}\\\\right] - \\\\left(I_{\\\\!N} \\\\!\\\\otimes \\\\!\",\n \"\\\\widetilde{A}\\\\right)\\\\!\",\n \"B^{\\\\mathcal {I}}_i D_{H}^T R^{-1} \\\\mathbb {E}\\\\left[r_i\\\\right] \\\\nonumber \\\\\\\\& \\\\quad + 1_{\\\\!N} \\\\!\\\\otimes \\\\!\",\n \"\\\\left(G \\\\mathbb {E}\\\\left[{\\\\mathbf {v}}_i\\\\right]\\\\right) - \\\\left(I_{\\\\!N} \\\\!\\\\otimes \\\\!\",\n \"\\\\widetilde{A}\\\\right)\\\\!B^{\\\\mathcal {I}}_i \\\\check{D}_{H} \\\\mathbb {E}\\\\left[{\\\\mathbf {\\\\epsilon }}_{i|i-1}\\\\right] = \\\\mathbf {0}, \\\\\\\\\\\\mathbb {E}\\\\left[{\\\\mathbf {\\\\phi }}_i\\\\right] &= \\\\left( I_{\\\\!N} \\\\!\\\\otimes \\\\!\",\n \"A\\\\right)\\\\!\",\n \"K_i \\\\mathbb {E}\\\\left[{\\\\mathbf {e}}_{i|i}\\\\right] + 1_N \\\\otimes \\\\mathbb {E}\\\\left[{\\\\mathbf {v}}_i\\\\right] = \\\\mathbf {0}.$ Combining (), () and (REF ), we have $\\\\widetilde{\\\\mathbf {\\\\phi }}_i &= F_1 {\\\\mathbf {\\\\epsilon }}_{i|i-1}+ F_2 {\\\\mathbf {e}}_{i|i-1}- F_3 D_{H}^T R^{-1} r_i + 1_{\\\\!N} \\\\!\\\\otimes \\\\!\",\n \"\\\\left(G {\\\\mathbf {v}}_i\\\\right)$ where, $F_1 &= \\\\left( I_{\\\\!N} \\\\!\\\\otimes \\\\!\",\n \"\\\\check{A}\\\\!\\\\right)\\\\!\",\n \"\\\\left( I_{\\\\!M\\\\!N} \\\\!-\\\\!\",\n \"K_i \\\\!\\\\left(I_{\\\\!N}\\\\!\\\\otimes \\\\!G\\\\right)\\\\!\",\n \"-\\\\!\",\n \"K_i \\\\!B^{\\\\mathcal {I}}_i \\\\check{D}_{H}\\\\!\\\\!\",\n \"\\\\right) \\\\!-\\\\!\",\n \"\\\\left( I_{\\\\!N} \\\\!\\\\otimes \\\\!\",\n \"\\\\widetilde{A}\\\\!\\\\right)\\\\!\",\n \"\\\\!B^{\\\\mathcal {I}}_i \\\\check{D}_{H} \\\\\\\\F_2 &= \\\\left( I_{\\\\!N} \\\\!\\\\otimes \\\\!\",\n \"\\\\check{A}\\\\!\\\\right)\\\\!\",\n \"K_i \\\\left( I_{\\\\!M\\\\!N} \\\\!-\\\\!\",\n \"B^{\\\\mathcal {C}}_i \\\\!-\\\\!\",\n \"B^{\\\\mathcal {I}}_i \\\\!\",\n \"\\\\widetilde{D}_{H} \\\\right) \\\\\\\\F_3 &= \\\\left( I_{\\\\!N} \\\\!\\\\otimes \\\\!\",\n \"\\\\widetilde{A}\\\\!\\\\right)\\\\!\",\n \"\\\\!B^{\\\\mathcal {I}}_i.$ Since $\\\\widetilde{\\\\mathbf {\\\\phi }}_i$ and ${\\\\mathbf {\\\\phi }}_i$ are zero-mean, the noise covariances are, $\\\\widetilde{\\\\Phi }_i &\\\\!=\\\\!\",\n \"\\\\mathbb {E} \\\\!\",\n \"\\\\left[\\\\!\\\\widetilde{\\\\mathbf {\\\\phi }}_i \\\\widetilde{\\\\mathbf {\\\\phi }}^T_i \\\\!\",\n \"\\\\right] \\\\!=\\\\!\",\n \"F_1 \\\\Sigma _{i|i-1} F_1^T \\\\!+\\\\!\",\n \"F_2 P_{i|i-1} F_2^T \\\\!+\\\\!\",\n \"F_3 \\\\overline{D}_H F^T_3 \\\\nonumber \\\\\\\\& \\\\qquad \\\\qquad + J \\\\!\\\\otimes \\\\!\",\n \"\\\\left(GVG\\\\right) + F_1 \\\\Pi _{i|i-1} F_2^T + F_2 \\\\Pi _{i|i-1}^T F_1^T, \\\\\\\\\\\\Phi _i &\\\\!=\\\\!\",\n \"\\\\mathbb {E}\\\\!\",\n \"\\\\left[ \\\\!\",\n \"{\\\\mathbf {\\\\phi }}_i {\\\\mathbf {\\\\phi }}^T_i \\\\!\",\n \"\\\\right] \\\\!=\\\\!\",\n \"\\\\left( I_{\\\\!N} \\\\!\\\\otimes \\\\!\",\n \"A\\\\right)\\\\!\",\n \"K_i \\\\Sigma _{i|i} \\\\!\",\n \"K^T_i \\\\!\",\n \"\\\\left( I_{\\\\!N} \\\\!\\\\otimes \\\\!\",\n \"A^T\\\\right) \\\\!+\\\\!\",\n \"J \\\\!\\\\otimes \\\\!\",\n \"V,$ where, $F_1, F_2$ and $F_3$ are defined in (REF )-().\",\n \"$\\\\Box $\"\n ],\n [\n \"Proof of Lemma \",\n \"We first compute the conditional means $\\\\overline{\\\\widetilde{\\\\mathbf {\\\\theta }}}^n_i$ and $\\\\overline{\\\\mathbf {\\\\theta }}^n_i$ of the new information $\\\\widetilde{\\\\mathbf {\\\\theta }}^n_i$ and ${\\\\mathbf {\\\\theta }}^n_i$ from (REF ).\",\n \"The means $\\\\overline{\\\\widetilde{\\\\mathbf {\\\\theta }}}^n_i$ and $\\\\overline{\\\\mathbf {\\\\theta }}^n_i$ depend on the conditional means of $\\\\widehat{{\\\\mathbf {y}}}_{i|i-1}^{l}, \\\\widetilde{\\\\mathbf {z}}^n_{i-1}$ and $\\\\widehat{{\\\\mathbf {y}}}_{i|i}^n$ .\",\n \"$&\\\\mathbb {E} \\\\begin{bmatrix} \\\\widehat{{\\\\mathbf {y}}}_{i|i-1}^{l} | \\\\widetilde{\\\\mathbf {z}}^n_{i-1}, \\\\lbrace \\\\widehat{\\\\mathbf {y}}^l_{i-1|i-2} \\\\rbrace _{l \\\\in \\\\Omega _n} \\\\end{bmatrix} \\\\\\\\&= \\\\mathbb {E} \\\\left[ \\\\mathbb {E} \\\\left[ {\\\\mathbf {y}}_i | \\\\widetilde{\\\\mathbf {z}}^l_{i-1}, \\\\lbrace \\\\widehat{\\\\mathbf {y}}^k_{i-1|i-2} \\\\rbrace _{k \\\\in \\\\Omega _l} \\\\right] \\\\;|\\\\; \\\\widetilde{\\\\mathbf {z}}^n_{i-1}, \\\\lbrace \\\\widehat{\\\\mathbf {y}}^l_{i-1|i-2} \\\\rbrace _{l \\\\in \\\\Omega _n} \\\\right] \\\\\\\\&= \\\\mathbb {E} \\\\left[ \\\\mathbb {E} \\\\left[ {\\\\mathbf {y}}_i | \\\\widetilde{\\\\mathbf {z}}^n_{i-1}, \\\\lbrace \\\\widehat{\\\\mathbf {y}}^l_{i-1|i-2} \\\\rbrace _{l \\\\in \\\\Omega _n} \\\\right] \\\\;|\\\\; \\\\widetilde{\\\\mathbf {z}}^l_{i-1}, \\\\lbrace \\\\widehat{\\\\mathbf {y}}^k_{i-1|i-2} \\\\rbrace _{k \\\\in \\\\Omega _l} \\\\right] \\\\\\\\&= \\\\mathbb {E} \\\\left[ \\\\widehat{{\\\\mathbf {y}}}_{i|i-1}^{n} \\\\;|\\\\; \\\\widetilde{\\\\mathbf {z}}^l_{i-1}, \\\\lbrace \\\\widehat{\\\\mathbf {y}}^k_{i-1|i-2} \\\\rbrace _{k \\\\in \\\\Omega _l} \\\\right] = \\\\widehat{{\\\\mathbf {y}}}_{i|i-1}^{n}, \\\\quad \\\\forall \\\\; l \\\\in \\\\Omega _n.\",\n \"\\\\\\\\&\\\\mathbb {E} \\\\begin{bmatrix} \\\\widetilde{\\\\mathbf {z}}^n_i \\\\;|\\\\; \\\\widetilde{\\\\mathbf {z}}^n_{i-1}, \\\\lbrace \\\\widehat{\\\\mathbf {y}}^l_{i-1|i-2} \\\\rbrace _{l \\\\in \\\\Omega _n} \\\\end{bmatrix} \\\\\\\\&= \\\\mathbb {E} \\\\begin{bmatrix} \\\\widetilde{H}_n{\\\\mathbf {y}}_i + H^T_n R_n^{-1} {\\\\mathbf {r}}^n_i + \\\\check{H}_n {\\\\mathbf {x}}_i \\\\;|\\\\; \\\\widetilde{\\\\mathbf {z}}^n_{i-1}, \\\\lbrace \\\\widehat{\\\\mathbf {y}}^l_{i-1|i-2} \\\\rbrace _{l \\\\in \\\\Omega _n} \\\\end{bmatrix} \\\\\\\\&= \\\\widetilde{H}_n \\\\mathbb {E} \\\\begin{bmatrix} {\\\\mathbf {y}}_i \\\\;|\\\\; \\\\widetilde{\\\\mathbf {z}}^n_{i-1}, \\\\lbrace \\\\widehat{\\\\mathbf {y}}^l_{i-1|i-2} \\\\rbrace _{l \\\\in \\\\Omega _n} \\\\end{bmatrix} \\\\\\\\& \\\\qquad \\\\qquad \\\\qquad + \\\\check{H}_n \\\\mathbb {E} \\\\begin{bmatrix} {\\\\mathbf {x}}_i \\\\;|\\\\; \\\\widetilde{\\\\mathbf {z}}^n_{i-1}, \\\\lbrace \\\\widehat{\\\\mathbf {y}}^l_{i-1|i-2} \\\\rbrace _{l \\\\in \\\\Omega _n} \\\\end{bmatrix} \\\\\\\\&= \\\\widetilde{H}_n \\\\widehat{{\\\\mathbf {y}}}_{i|i-1}^n \\\\!+\\\\!\",\n \"\\\\check{H}_n \\\\mathbb {E} \\\\left[ \\\\mathbb {E} \\\\left[ {\\\\mathbf {x}}_i | \\\\widehat{{\\\\mathbf {y}}}_{i|i}^n, \\\\widehat{\\\\mathbf {x}}^l_{i-1|i-2}, \\\\widetilde{\\\\mathbf {z}}^n_{i-1}, \\\\widehat{\\\\mathbf {y}}^l_{i-1|i-2}, \\\\in \\\\Omega _n \\\\right] \\\\right] \\\\\\\\&= \\\\widetilde{H}_n \\\\widehat{{\\\\mathbf {y}}}_{i|i-1}^n + \\\\check{H}_n \\\\mathbb {E} \\\\left[ \\\\widehat{{\\\\mathbf {x}}}_{i|i-1}^n \\\\;|\\\\; \\\\widetilde{\\\\mathbf {z}}^n_{i-1}, \\\\lbrace \\\\widehat{\\\\mathbf {y}}^l_{i-1|i-2} \\\\rbrace _{l \\\\in \\\\Omega _n}\\\\right] \\\\\\\\&= \\\\widetilde{H}_n \\\\widehat{{\\\\mathbf {y}}}_{i|i-1}^n + \\\\check{H}_n \\\\widehat{{\\\\mathbf {x}}}_{i|i-1}^n.\",\n \"\\\\\\\\&\\\\mathbb {E} \\\\begin{bmatrix} \\\\widehat{{\\\\mathbf {y}}}_{i|i}^n | \\\\widehat{{\\\\mathbf {y}}}_{i-1|i-1}^n \\\\end{bmatrix} = \\\\mathbb {E} \\\\left[ {\\\\mathbf {y}}_i - {\\\\mathbf {e}}_{i|i}^n \\\\;|\\\\; \\\\widehat{{\\\\mathbf {y}}}_{i-1|i-1}^n \\\\right] \\\\\\\\&= \\\\mathbb {E} \\\\left[ {\\\\mathbf {y}}_i \\\\;|\\\\; \\\\widehat{{\\\\mathbf {y}}}_{i-1|i-1}^n \\\\right] - \\\\mathbb {E} \\\\left[{\\\\mathbf {e}}_{i|i}^n \\\\;|\\\\; \\\\widehat{{\\\\mathbf {y}}}_{i-1|i-1}^n \\\\right] \\\\\\\\&= G \\\\mathbb {E} \\\\left[ {\\\\mathbf {x}}_i \\\\;|\\\\; \\\\widehat{{\\\\mathbf {y}}}_{i-1|i-1}^n \\\\right] = G\\\\widehat{{\\\\mathbf {x}}}_{i|i-1}^n.$ The conditional means of $\\\\widehat{{\\\\mathbf {y}}}_{i|i-1}^{l}, \\\\widetilde{\\\\mathbf {z}}^n_{i-1}$ and $\\\\widehat{{\\\\mathbf {y}}}_{i|i}^n$ shows that the new uncorrelated information defined in (REF )-() expands to the vectors $\\\\widetilde{\\\\mathbf {\\\\nu }}^n_i$ and ${\\\\mathbf {\\\\nu }}^n_i$ in (REF ).\",\n \"Note that $\\\\widetilde{\\\\mathbf {\\\\nu }}^n_i$ and ${\\\\mathbf {\\\\nu }}^n_i$ are zero mean, by definition, and, are Gaussian since they are linear combination of Gaussian vectors.\",\n \"Now to prove $\\\\widetilde{\\\\mathbf {\\\\nu }}^n_i$ and ${\\\\mathbf {\\\\nu }}^n_i$ are sequence of uncorrelated vectors, we have to show: $\\\\mathbb {E}\\\\left[ \\\\widetilde{\\\\mathbf {\\\\nu }}^n_i \\\\widetilde{\\\\mathbf {\\\\nu }}^{n^T}_j\\\\right] = \\\\mathbb {E}\\\\left[ {\\\\mathbf {\\\\nu }}^n_i {\\\\mathbf {\\\\nu }}^{n^T}_j\\\\right] = {\\\\mathbf {0}}, \\\\; \\\\forall \\\\; i\\\\ne j, \\\\; \\\\forall \\\\; n.$ First, we write $\\\\widetilde{\\\\mathbf {\\\\nu }}^n_i$ and ${\\\\mathbf {\\\\nu }}^n_i$ in terms of the filtering and prediction error processes using (), (REF )-(), $\\\\widetilde{\\\\mathbf {\\\\nu }}^n_i \\\\!=\\\\!\",\n \"\\\\begin{bmatrix}{\\\\mathbf {e}}_{i|i-1}^n \\\\!-\\\\!\",\n \"{\\\\mathbf {e}}_{i|i-1}^{l_1} \\\\\\\\ \\\\vdots \\\\\\\\ {\\\\mathbf {e}}_{i|i-1}^n \\\\!-\\\\!\",\n \"{\\\\mathbf {e}}_{i|i-1}^{l_{d_n}} \\\\\\\\ \\\\widetilde{H}_{\\\\!n} {\\\\mathbf {e}}_{i|i-1}^n \\\\!+\\\\!\",\n \"\\\\check{H}_{\\\\!n} {\\\\mathbf {\\\\epsilon }}_{i|i-1}^n + H^T_n R^{-1}_n {\\\\mathbf {r}}^n_i\\\\end{bmatrix}\\\\!\\\\!\\\\!, \\\\;\\\\;{\\\\mathbf {\\\\nu }}^n_i \\\\!=\\\\!\",\n \"G{\\\\mathbf {\\\\epsilon }}_{i|i-1}^n \\\\!-\\\\!\",\n \"{\\\\mathbf {e}}_{i|i}^n.$ Without loss of generality, we consider $i > j$ .\",\n \"The rest of the proof is similar to the proof of Lemma 5 in [5].\",\n \"Here, the only difference is that we should condition on $\\\\lbrace \\\\widetilde{\\\\mathbf {z}}^l_{i-1}\\\\rbrace _{l \\\\in \\\\Omega _n}, \\\\lbrace \\\\widehat{\\\\mathbf {y}}^l_{i-1|i-1} \\\\rbrace _{l \\\\in \\\\Omega _n}$ .\",\n \"$\\\\Box $ Proof of Theorems\"\n ],\n [\n \"Proof of Theorem \",\n \"In Lemma REF , we showed that $\\\\widetilde{\\\\mathbf {\\\\nu }}^n_i$ and ${\\\\mathbf {\\\\nu }}^n_i$ are independent Gaussian sequences.\",\n \"By the Innovations Property [35], there are $1-1$ correspondence between $\\\\lbrace \\\\widetilde{{\\\\mathbf {z}}}^n_i, \\\\lbrace \\\\widehat{{\\\\mathbf {y}}}_{i|i-1}^l \\\\rbrace _{l \\\\in \\\\overline{\\\\Omega }_n} \\\\rbrace $ and $\\\\widetilde{\\\\mathbf {\\\\nu }}^n_i$ , and between $\\\\lbrace \\\\widehat{{\\\\mathbf {y}}}_{i|i}^n \\\\rbrace $ and ${\\\\mathbf {\\\\nu }}^n_i$ .\",\n \"The Innovations Property guarantees that there exists a unique way to get one from the other.\",\n \"$\\\\widehat{{\\\\mathbf {y}}}_{i|i}^n \\\\!=\\\\!\",\n \"\\\\mathbb {E} \\\\!\\\\left[ {\\\\mathbf {y}}_i \\\\;|\\\\; \\\\widetilde{{\\\\mathbf {z}}}^n_i, \\\\lbrace \\\\widehat{{\\\\mathbf {y}}}_{i|i-1}^l \\\\rbrace _{l \\\\in \\\\overline{\\\\Omega }_n} \\\\right] ~&\\\\Longleftrightarrow ~ \\\\widehat{{\\\\mathbf {y}}}_{i|i}^n \\\\!=\\\\!\",\n \"\\\\mathbb {E} \\\\!\\\\left[ {\\\\mathbf {y}}_i \\\\;|\\\\; \\\\widetilde{\\\\mathbf {\\\\nu }}^n_i \\\\right] \\\\\\\\\\\\widehat{{\\\\mathbf {x}}}_{i|i}^n \\\\!=\\\\!\",\n \"\\\\mathbb {E} \\\\left[ {\\\\mathbf {x}}_i \\\\;|\\\\; \\\\widehat{{\\\\mathbf {y}}}_{i|i}^n \\\\right] ~&\\\\Longleftrightarrow ~ \\\\widehat{{\\\\mathbf {x}}}_{i|i}^n \\\\!=\\\\!\",\n \"\\\\mathbb {E} \\\\left[ {\\\\mathbf {x}}_i \\\\;|\\\\; {\\\\mathbf {\\\\nu }}^n_i \\\\right]$ By the Gauss-Markov principle, $ \\\\widehat{{\\\\mathbf {y}}}_{i|i}^n &= \\\\widehat{{\\\\mathbf {y}}}_{i|i-1}^n + \\\\widehat{B}^n_i \\\\widetilde{\\\\mathbf {\\\\nu }}^n_i \\\\\\\\\\\\widehat{{\\\\mathbf {x}}}_{i|i}^n &= \\\\widehat{{\\\\mathbf {x}}}_{i|i-1}^n + K^n_i {\\\\mathbf {\\\\nu }}^n_i$ where $\\\\widehat{B}^n_i$ are the non-zero blocks of the $n^{\\\\text{th}}$ row of $B_i$ .\",\n \"Now we expand the term $\\\\widehat{B}^n_i \\\\widetilde{\\\\mathbf {\\\\nu }}^n_i$ by multiplying the gain blocks $B^{nl}_i$ with the corresponding $(\\\\widehat{{\\\\mathbf {y}}}_{i|i-1}^l - \\\\widehat{{\\\\mathbf {y}}}_{i|i-1}^n)$ and the gain block $B^{nn}_i$ with $\\\\left( \\\\widetilde{{\\\\mathbf {z}}}^n_i \\\\!\\\\!-\\\\!\",\n \"\\\\widetilde{H}_{\\\\!n} \\\\widehat{{\\\\mathbf {y}}}_{i|i-1}^n \\\\!-\\\\!\",\n \"\\\\check{H}_{\\\\!n} \\\\widehat{{\\\\mathbf {x}}}_{i|i-1}^n \\\\right)$ .\",\n \"This gives us the consensus$+$ innovations filtering pseudo-state update (REF ).\",\n \"The pseudo-state and state prediction updates are $\\\\widehat{{\\\\mathbf {y}}}_{i+1|i}^n &\\\\!=\\\\!\",\n \"\\\\mathbb {E} \\\\left[ {\\\\mathbf {y}}_{i+1} \\\\;|\\\\; \\\\widetilde{{\\\\mathbf {z}}}^n_i, \\\\lbrace \\\\widehat{{\\\\mathbf {y}}}_{i|i-1}^l \\\\rbrace _{l \\\\in \\\\overline{\\\\Omega }_n} \\\\right] \\\\\\\\&\\\\!=\\\\!\",\n \"\\\\mathbb {E} \\\\left[ \\\\widetilde{A}{\\\\mathbf {y}}_i + G{\\\\mathbf {v}}_i + \\\\check{A}{\\\\mathbf {x}}_i \\\\;|\\\\; \\\\widetilde{{\\\\mathbf {z}}}^n_i, \\\\lbrace \\\\widehat{{\\\\mathbf {y}}}_{i|i-1}^l \\\\rbrace _{l \\\\in \\\\overline{\\\\Omega }_n} \\\\right] \\\\\\\\&\\\\!=\\\\!\",\n \"\\\\widetilde{A} \\\\widehat{{\\\\mathbf {y}}}_{i|i}^n + \\\\check{A} \\\\widehat{{\\\\mathbf {x}}}_{i|i}^n \\\\\\\\\\\\widehat{{\\\\mathbf {x}}}_{i+1|i}^n &\\\\!=\\\\!\",\n \"\\\\mathbb {E} \\\\left[ {\\\\mathbf {x}}_{i\\\\!+\\\\!1} \\\\;|\\\\; \\\\widehat{{\\\\mathbf {y}}}_{i|i}^n \\\\right] \\\\!=\\\\!\",\n \"\\\\mathbb {E} \\\\left[ A {\\\\mathbf {x}}_i + {\\\\mathbf {v}}_i \\\\;|\\\\; \\\\widehat{{\\\\mathbf {y}}}_{i|i}^n \\\\right] = A \\\\widehat{{\\\\mathbf {x}}}_{i|i}^n.\",\n \"\\\\quad \\\\Box $\"\n ],\n [\n \"Proof of Theorem \",\n \"By Lemma REF and Lemma REF , the error processes, ${\\\\mathbf {e}}_{i|i}$ , ${\\\\mathbf {e}}_{i+1|i}$ , ${\\\\mathbf {\\\\epsilon }}_{i|i}$ , and ${\\\\mathbf {\\\\epsilon }}_{i+1|i}$ are zero-mean Gaussian.\",\n \"The Lyapunov-type iterations ()-() of the filter and predictor error covariances, $P_{i|i}$ , $\\\\Sigma _{i|i}$ , $\\\\Pi _{i|i}$ , $P_{i+1|i}$ , $\\\\Sigma _{i+1|i}$ , and $\\\\Pi _{i+1|i}$ , follow directly from the definitions (REF )-() and error dynamics (REF )-() by algebraic manipulations.\",\n \"$\\\\Box $\"\n ],\n [\n \"Proof of Theorem \",\n \"For any square matrix, $\\\\rho (\\\\widetilde{F}) \\\\le \\\\Vert \\\\widetilde{F} \\\\Vert $ .\",\n \"Hence if $\\\\Vert \\\\widetilde{F} \\\\Vert < 1$ , then it implies that $\\\\rho (\\\\widetilde{F}) < 1$ .\",\n \"We derive the tracking capacity with the sufficient condition, $\\\\Vert \\\\widetilde{F} \\\\Vert < 1$ , $\\\\Vert \\\\widetilde{F} \\\\Vert &= \\\\begin{Vmatrix} \\\\left( I_{\\\\!N} \\\\!\\\\otimes \\\\!\\\\widetilde{A}\\\\!\\\\right)\\\\!\",\n \"\\\\left( I_{\\\\!M\\\\!N} \\\\!-\\\\!\",\n \"B^{\\\\mathcal {C}}_i \\\\!-\\\\!\",\n \"B^{\\\\mathcal {I}}_i \\\\!\",\n \"\\\\widetilde{D}_{H} \\\\right) \\\\end{Vmatrix} \\\\\\\\&\\\\le \\\\begin{Vmatrix} I_{\\\\!N} \\\\!\\\\otimes \\\\!\\\\widetilde{A}\\\\!\",\n \"\\\\end{Vmatrix} \\\\begin{Vmatrix} \\\\left( I_{\\\\!M\\\\!N} \\\\!-\\\\!\",\n \"B^{\\\\mathcal {C}}_i \\\\!-\\\\!\",\n \"B^{\\\\mathcal {I}}_i \\\\!\",\n \"\\\\widetilde{D}_{H} \\\\right) \\\\end{Vmatrix} \\\\\\\\&\\\\le \\\\begin{Vmatrix} GAG^{\\\\dagger }\\\\!\",\n \"\\\\end{Vmatrix} \\\\begin{Vmatrix} \\\\left( I_{\\\\!M\\\\!N} \\\\!-\\\\!\",\n \"B^{\\\\mathcal {C}}_i \\\\!-\\\\!\",\n \"B^{\\\\mathcal {I}}_i \\\\!\",\n \"\\\\widetilde{D}_{H} \\\\right) \\\\end{Vmatrix} \\\\\\\\&\\\\le \\\\frac{\\\\lambda _m}{\\\\lambda _1} \\\\begin{Vmatrix} A\\\\!\",\n \"\\\\end{Vmatrix} \\\\begin{Vmatrix} \\\\left( I_{\\\\!M\\\\!N} \\\\!-\\\\!\",\n \"B^{\\\\mathcal {C}}_i \\\\!-\\\\!\",\n \"B^{\\\\mathcal {I}}_i \\\\!\",\n \"\\\\widetilde{D}_{H} \\\\right) \\\\end{Vmatrix},$ where, $\\\\Vert G \\\\Vert _2 = \\\\lambda _m$ and $\\\\Vert G^{\\\\dagger } \\\\Vert _2 = \\\\frac{1}{\\\\lambda _1}$ .\",\n \"Since $G$ is a symmetric positive semi-definite matrix, its spectral norm is its largest eigenvalue $(\\\\lambda _m)$ and the spectral norm of its pseudo-inverse, $G^{\\\\dagger }$ , is the inverse of its smallest non-zero eigenvalue $(\\\\lambda _1)$ .\",\n \"If there exists $B^{\\\\mathcal {C}}_i$ and $B^{\\\\mathcal {I}}_i$ such that $\\\\frac{\\\\lambda _m}{\\\\lambda _1} \\\\begin{Vmatrix} A\\\\!\",\n \"\\\\end{Vmatrix} \\\\begin{Vmatrix} \\\\left( I_{\\\\!M\\\\!N} \\\\!-\\\\!\",\n \"B^{\\\\mathcal {C}}_i \\\\!-\\\\!\",\n \"B^{\\\\mathcal {I}}_i \\\\!\",\n \"\\\\widetilde{D}_{H} \\\\right) \\\\end{Vmatrix} < 1,$ then $\\\\Vert \\\\widetilde{F} \\\\Vert < 1$ and also $\\\\rho (\\\\widetilde{F}) < 1$ .\",\n \"The bound on the spectral norm of $A$ is, $\\\\Vert A \\\\Vert &< \\\\frac{\\\\lambda _1}{\\\\lambda _m \\\\begin{Vmatrix} \\\\left( I_{\\\\!M\\\\!N} \\\\!-\\\\!\",\n \"B^{\\\\mathcal {C}}_i \\\\!-\\\\!\",\n \"B^{\\\\mathcal {I}}_i \\\\!\",\n \"\\\\widetilde{D}_{H} \\\\right) \\\\end{Vmatrix} } \\\\\\\\&\\\\le \\\\max _{B^{\\\\mathcal {C}}_i, B^{\\\\mathcal {I}}_i} \\\\frac{\\\\lambda _1}{\\\\lambda _m \\\\begin{Vmatrix} \\\\left( I_{\\\\!M\\\\!N} \\\\!-\\\\!\",\n \"B^{\\\\mathcal {C}}_i \\\\!-\\\\!\",\n \"B^{\\\\mathcal {I}}_i \\\\!\",\n \"\\\\widetilde{D}_{H} \\\\right) \\\\end{Vmatrix} } = C.$ Thus as long as $\\\\Vert A \\\\Vert _2 < C$ , there exists $B^{\\\\mathcal {C}}_i$ and $B^{\\\\mathcal {I}}_i$ such that $\\\\rho (\\\\widetilde{F}) < 1$ .\",\n \"By global detectability Assumption REF , there exists $K_i$ such that $\\\\rho (F) < 1$ .\",\n \"Refer to [35], for the convergence conditions of the centralized information filters.\",\n \"Further, by Lemma REF the Gaussian noises processes $\\\\widetilde{\\\\mathbf {\\\\phi }}_i$ and ${\\\\mathbf {\\\\phi }}_i$ have bounded noise covariances.\",\n \"Thus, if $\\\\Vert A \\\\Vert _2 < C$ , then from (REF )-() we conclude that the CIKF (REF )-() converges with bounded MSE.\",\n \"$\\\\Box $\"\n ],\n [\n \"Proof of Theorem \",\n \"By the Innovations Property and the Gauss-Markov principle [35], the optimal gains $\\\\widehat{B}^n_i$ and $K^n_i$ in (REF )-() are: $\\\\widehat{B}^n_i &= \\\\Sigma _{{\\\\mathbf {y}}_i \\\\widetilde{\\\\mathbf {\\\\nu }}^n_i} \\\\left(\\\\Sigma _{\\\\widetilde{\\\\mathbf {\\\\nu }}^n_i}\\\\right)^{-1}, \\\\\\\\K^n_i &= \\\\Sigma _{{\\\\mathbf {x}}_i {\\\\mathbf {\\\\nu }}^n_i} \\\\left(\\\\Sigma _{{\\\\mathbf {\\\\nu }}^n_i}\\\\right)^{-1}$ that yield minimized MSE estimates $\\\\widehat{{\\\\mathbf {y}}}_{i|i}^n$ and $\\\\widehat{{\\\\mathbf {x}}}_{i|i}^n$ of the pseudo-state ${\\\\mathbf {y}}_i$ and of the field ${\\\\mathbf {x}}_i$ , respectively, at each agent $n$ .\",\n \"The cross-covariances $\\\\Sigma _{{\\\\mathbf {y}}_i \\\\widetilde{\\\\mathbf {\\\\nu }}^n_i}$ and $\\\\Sigma _{{\\\\mathbf {x}}_i {\\\\mathbf {\\\\nu }}^n_i}$ are $&\\\\Sigma _{{\\\\mathbf {y}}_i \\\\widetilde{\\\\mathbf {\\\\nu }}^n_i} = \\\\mathbb {E} \\\\left[ \\\\left({\\\\mathbf {y}}_i - \\\\overline{{\\\\mathbf {y}}}_i \\\\right) \\\\widetilde{\\\\mathbf {\\\\nu }}^{n^T}_i \\\\right] \\\\\\\\&= \\\\mathbb {E} \\\\!\\\\left[ \\\\!\\\\left({\\\\mathbf {e}}_{i|i-1}^n \\\\!+\\\\!\\\\left(\\\\widehat{{\\\\mathbf {y}}}_{i|i-1}^n \\\\!-\\\\!\",\n \"\\\\overline{{\\\\mathbf {y}}}_i \\\\right) \\\\right)\\\\!\\\\!\",\n \"\\\\begin{bmatrix} {\\\\mathbf {e}}_{i|i-1}^n \\\\!-\\\\!\",\n \"{\\\\mathbf {e}}_{i|i-1}^{l_1} \\\\\\\\ \\\\vdots \\\\\\\\ {\\\\mathbf {e}}_{i|i-1}^n \\\\!-\\\\!\",\n \"{\\\\mathbf {e}}_{i|i-1}^{l_{d_n}} \\\\\\\\ \\\\widetilde{H}_{\\\\!n} {\\\\mathbf {e}}_{i|i-1}^n \\\\!\\\\!+\\\\!\",\n \"\\\\check{H}_{\\\\!n} {\\\\mathbf {\\\\epsilon }}_{i|i-1}^n \\\\!+\\\\!\",\n \"H^T_n R^{-1}_n {\\\\mathbf {r}}^n_i \\\\end{bmatrix}^{\\\\!\\\\!T} \\\\!\\\\right] \\\\\\\\&= \\\\mathbb {E}\\\\!\",\n \"\\\\left[ {\\\\mathbf {e}}_{i|i-1}^n \\\\!\",\n \"\\\\left[ \\\\!\\\\!\\\\begin{array}{l;{2pt/2pt}l;{1pt/1pt}l;{2pt/2pt}l}\\\\!\",\n \"{\\\\mathbf {e}}_{i|i-1}^{n^T} \\\\!-\\\\!\",\n \"{\\\\mathbf {e}}_{i|i-1}^{l^T_1}\\\\!\",\n \"&\\\\!\",\n \"\\\\cdots \\\\!\",\n \"&\\\\!\",\n \"{\\\\mathbf {e}}_{i|i-1}^{n^T} \\\\!-\\\\!\",\n \"{\\\\mathbf {e}}_{i|i-1}^{l^T_{d_n}} \\\\!&\\\\!\",\n \"\\\\widetilde{H}_{\\\\!n} {\\\\mathbf {e}}_{i|i-1}^{n^T} \\\\\\\\& & & \\\\; + \\\\check{H}_{\\\\!n}{\\\\mathbf {\\\\epsilon }}_{i|i-1}^{n^T} \\\\!\\\\!\\\\end{array}\\\\!\\\\!\",\n \"\\\\right] \\\\!\",\n \"\\\\right] \\\\\\\\&= \\\\left[\\\\begin{array}{l;{2pt/2pt}l;{1pt/1pt}l;{2pt/2pt}l}\\\\!\\\\!\\\\!\\\\!\",\n \"P_{i|i\\\\!-\\\\!1}^{nn} \\\\!-\\\\!\",\n \"P_{i|i\\\\!-\\\\!1}^{nl_1} & \\\\cdots & P_{i|i\\\\!-\\\\!1}^{nn} \\\\!-\\\\!\",\n \"P_{i|i\\\\!-\\\\!1}^{nl_{d_n}} & P_{i|i\\\\!-\\\\!1}^{nn} \\\\!\",\n \"\\\\widetilde{H}^T_n \\\\!+\\\\!\",\n \"\\\\Pi _{i|i\\\\!-\\\\!1}^{nn^T} \\\\!\",\n \"\\\\check{H}^T_n \\\\!\\\\!\\\\!\\\\!\\\\end{array}\\\\right], \\\\\\\\&\\\\Sigma _{{\\\\mathbf {x}}_i {\\\\mathbf {\\\\nu }}^n_i} = \\\\mathbb {E} \\\\left[ \\\\left({\\\\mathbf {x}}_i - \\\\overline{{\\\\mathbf {x}}}_i \\\\right) {\\\\mathbf {\\\\nu }}^{n^T}_i \\\\right] \\\\\\\\&= \\\\mathbb {E} \\\\left[ \\\\left({\\\\mathbf {\\\\epsilon }}_{i|i-1}^n +\\\\left(\\\\widehat{{\\\\mathbf {x}}}_{i|i-1}^n - \\\\overline{{\\\\mathbf {x}}}_i \\\\right) \\\\right) \\\\left( G{\\\\mathbf {\\\\epsilon }}_{i|i-1}^{n} \\\\!-\\\\!\",\n \"{\\\\mathbf {e}}_{i|i}^{n} \\\\right)^T \\\\right] \\\\\\\\&= \\\\mathbb {E} \\\\left[ {\\\\mathbf {\\\\epsilon }}_{i|i-1}^n \\\\!\",\n \"\\\\left( G{\\\\mathbf {\\\\epsilon }}_{i|i-1}^{n} \\\\!-\\\\!\",\n \"{\\\\mathbf {e}}_{i|i}^{n} \\\\right)^T \\\\!\",\n \"\\\\right] = \\\\Sigma _{i|i\\\\!-\\\\!1}^{nn} G - \\\\Gamma ^{nn}_i.$ In the above derivation, using the iterated law of expectation it can be shown that the terms $\\\\mathbb {E} \\\\left[ \\\\left(\\\\widehat{{\\\\mathbf {y}}}_{i|i-1}^n - \\\\overline{{\\\\mathbf {y}}}_i \\\\right) \\\\widetilde{\\\\mathbf {\\\\nu }}^{n^T}_i \\\\right] = \\\\mathbb {E} \\\\left[ \\\\left(\\\\widehat{{\\\\mathbf {x}}}_{i|i-1}^n - \\\\overline{{\\\\mathbf {x}}}_i \\\\right) {\\\\mathbf {\\\\nu }}^{n^T}_i \\\\right] = {\\\\mathbf {0}}$ .\",\n \"Also, $\\\\mathbb {E} \\\\left[ {\\\\mathbf {e}}_{i|i-1}^n {\\\\mathbf {r}}^{n^T}_i \\\\right] = {\\\\mathbf {0}}$ due to the statistical independence of the noise sequences.\",\n \"Similarly, the covariances matrices $\\\\Sigma _{\\\\widetilde{\\\\mathbf {\\\\nu }}^n_i}$ and $\\\\Sigma _{{\\\\mathbf {\\\\nu }}^n_i}$ are: $&\\\\Sigma _{\\\\widetilde{\\\\mathbf {\\\\nu }}^n_i} = \\\\mathbb {E} \\\\left[ \\\\widetilde{\\\\mathbf {\\\\nu }}^{n}_i \\\\widetilde{\\\\mathbf {\\\\nu }}^{n^T}_i \\\\right] \\\\\\\\&= \\\\mathbb {E} \\\\left[ \\\\begin{bmatrix} {\\\\mathbf {e}}_{i|i-1}^n \\\\!-\\\\!\",\n \"{\\\\mathbf {e}}_{i|i-1}^{l_1} \\\\\\\\ \\\\vdots \\\\\\\\ {\\\\mathbf {e}}_{i|i-1}^n \\\\!-\\\\!\",\n \"{\\\\mathbf {e}}_{i|i-1}^{l_{d_n}} \\\\\\\\ \\\\widetilde{H}_{\\\\!n} {\\\\mathbf {e}}_{i|i-1}^n \\\\!+\\\\!\",\n \"\\\\check{H}_{\\\\!n} {\\\\mathbf {\\\\epsilon }}_{i|i-1}^n \\\\\\\\ \\\\qquad + H^T_n R^{-1}_n {\\\\mathbf {r}}^n_i \\\\end{bmatrix} \\\\begin{bmatrix} {\\\\mathbf {e}}_{i|i-1}^n \\\\!-\\\\!\",\n \"{\\\\mathbf {e}}_{i|i-1}^{l_1} \\\\\\\\ \\\\vdots \\\\\\\\ {\\\\mathbf {e}}_{i|i-1}^n \\\\!-\\\\!\",\n \"{\\\\mathbf {e}}_{i|i-1}^{l_{d_n}} \\\\\\\\ \\\\widetilde{H}_{\\\\!n} {\\\\mathbf {e}}_{i|i-1}^n \\\\!+\\\\!\",\n \"\\\\check{H}_{\\\\!n} {\\\\mathbf {\\\\epsilon }}_{i|i-1}^n \\\\\\\\ \\\\qquad + H^T_n R^{-1}_n {\\\\mathbf {r}}^n_i \\\\end{bmatrix}^T \\\\right] \\\\\\\\&\\\\Sigma _{{\\\\mathbf {\\\\nu }}^n_i} = \\\\mathbb {E} \\\\left[ {\\\\mathbf {\\\\nu }}^{n}_i {\\\\mathbf {\\\\nu }}^{n^T}_i \\\\right] = \\\\mathbb {E} \\\\left[ \\\\left( G{\\\\mathbf {\\\\epsilon }}_{i|i-1}^n \\\\!-\\\\!\",\n \"{\\\\mathbf {e}}_{i|i}^n \\\\right) \\\\left( G{\\\\mathbf {\\\\epsilon }}_{i|i-1}^n \\\\!-\\\\!\",\n \"{\\\\mathbf {e}}_{i|i}^n \\\\right)^T \\\\right].$ The rest of the derivation of $\\\\Sigma _{\\\\widetilde{\\\\mathbf {\\\\nu }}^n_i}$ and $\\\\Sigma _{{\\\\mathbf {\\\\nu }}^n_i}$ in terms of the error covariance matrices is by block-by-block multiplication of the above expressions.\",\n \"$\\\\Box $ [Figure: NO_CAPTION Since July 2016, he is a Postdoctoral Researcher at IBM T.J. Watson Research Center, Yorktown Heights, NY.\",\n \"His research interests are in statistical inference, distributed estimation over multi-agent networks, model adaptation in dynamic environments and time-series analysis; broadly in the areas of statistical signal processing, machine learning and data-mining with applications in healthcare and smart infrastructure.\",\n \"[Figure: NO_CAPTION He is the Philip L. and Marsha Dowd University Professor at Carnegie Mellon University (CMU).\",\n \"He was on the faculty at IST and has held visiting faculty appointments at MIT and New York University (NYU).\",\n \"He founded and directs a large education and research program between CMU and Portugal, www.icti.cmu.edu.\",\n \"His research interests are on data science, graph signal processing, and statistical and algebraic signal and image processing.\",\n \"He has published over 550 papers and holds thirteen patents issued by the US Patent Office.\",\n \"The technology of two of his patents (co-inventor A. Kavčić) are in about three billion disk drives read channel chips of 60 % of all computers sold in the last 13 years worldwide and was, in 2016, the subject of the largest university verdict/settlement in the information technologies area.\",\n \"Dr. Moura is the IEEE Technical Activities Vice-President (2016) and member of the IEEE Board of Directors.\",\n \"He served in several other capacities including IEEE Division IX Director, member of several IEEE Boards, President of the IEEE Signal Processing Society(SPS), Editor in Chief for the IEEE Transactions in Signal Processing, interim Editor in Chief for the IEEE Signal Processing Letters.\",\n \"Dr. Moura has received several awards, including the Technical Achievement Award and the Society Award from the IEEE Signal Processing.\",\n \"In 2016, he received the CMU College of Engineering Distinguished Professor of Engineering Award.\",\n \"He is a Fellow of the IEEE, a Fellow of the American Association for the Advancement of Science (AAAS), a corresponding member of the Academy of Sciences of Portugal, Fellow of the US National Academy of Inventors, and a member of the US National Academy of Engineering.\"\n ]\n]"},"id":{"kind":"string","value":"1605.06096"}}},{"rowIdx":1866682,"cells":{"text":{"kind":"list like","value":[["Generic Demand Model Considering the Impact of Prosumers for Future Grid\n Scenario Analysis"],["Abstract The increasing uptake of residential PV-battery systems is bound to significantly change demand patterns of future power systems and, consequently, their dynamic performance.","In this paper, we propose a generic demand model that captures the aggregated effect of a large population of price-responsive users equipped with small-scale PV-battery systems, called prosumers, for market simulation in future grid scenario analysis.","The model is formulated as a bi-level program in which the upper-level unit commitment problem minimizes the total generation cost, and the lower-level problem maximizes prosumers' aggregate self-consumption.","Unlike in the existing bi-level optimization frameworks that focus on the interaction between the wholesale market and an aggregator, the coupling is through the prosumers' demand, not through the electricity price.","That renders the proposed model market structure agnostic, making it suitable for future grid studies where the market structure is potentially unknown.","As a case study, we perform steady-state voltage stability analysis of a simplified model of the Australian National Electricity Market with significant penetration of renewable generation.","The simulation results show that a high prosumer penetration changes the demand profile in ways that significantly improve the system loadability, which confirms the suitability of the proposed model for future grid studies."],["Introduction","Power systems are undergoing a major transformation driven by the increasing uptake of variable renewable energy sources (RES).","At the demand side, the emergence of cost-effective “behind-the-meter” distributed energy resources, including on-site generation, energy storage, electric vehicles, and flexible loads, and the advancement of sensor, computer, communication and energy management technologies are changing the way electricity consumers source and consume electric power.","Indeed, recent studies suggest that rooftop PV-battery systems will reach retail price parity from 2020 in the USA grids and the Australian National Electricity Market (NEM) [1].","A recent forecast by Morgan Stanley has suggested that the uptake can be even faster, by boldly predicting that up to 2 million Australian households could install battery storage by 2020 [2].","This has been confirmed by the Energy Networks Australia and the Australian Commonwealth Scientific and Industrial Research Organisation (CSIRO) who have estimated the projected uptake of solar PV and battery storage in 2050 to be 80 and 100GWh [3], which will represent between 30%–50% of total demand, a scenario called “Rise of the Prosumer” [4].","Here, the prosumer they refer to is a small-scale (residential, commercial and small industrial) electricity consumer with on-site generation.","A similar trend has been observed in Europe as well [5].","Given this, it is expected that a large uptake of demand-side technologies will significantly change demand patterns in future gridsWe interpret a future grid to mean the study of national grid type structures with the above-mentioned transformational changes for the long-term out to 2050., which will in turn affect their dynamic performance.","Existing future grid feasibility studies [6], [7], [8], [9], [10] typically use conventional demand models, possibly using some heuristics to account for the effect of emerging demand-side technologies, and the synergies that may arise between them.","They also assume specific market arrangements by which RES are integrated into grid operations.","The challenge associated with future grid planning is that the grid structure and the regulatory framework, including the market structure, cannot be simply assumed from the details of an existing one.","Instead, several possible evolution paths need to be accounted for.","Future grid planning thus requires a major departure from conventional power system planning, where only a handful of the most critical scenarios is analyzed.","To account for a wide range of possible future evolutions, scenario analysis has been proposed in many industries, e.g.","in finance and economics [11], and in energy [12], [13], [14].","As opposed to power system planning where the aim is to find an optimal transmission and/or generation expansion plan, the aim in scenario analysis is to analyze possible evolution pathways to inform policymaking.","Given the uncertainty associated with long-term projections, the focus of future grid scenario analysis is limited to the analysis of what is technically possible, although it might also consider an explicit costing [15].","Our work is part of the Future Grid Research Program funded by the CSIRO, whose aim is to explore possible future pathways for the evolution of the Australian grid out to 2050 by looking beyond simple balancing.","To this end, a comprehensive modeling framework for future grid scenario analysis has been proposed in [16], which includes a market model, power flow analysis, and stability analysis.","The demand model, however, assumes that the users are price-takers, which does not properly capture the aggregated effect of prosumers on the demand profile, as discussed later.","Due to the influence of a demand profile on power systems performance and stability, recent studies have attempted to integrate the aggregated impact of prosumers into the demand models [17], [18], [19], [20], [21], [22].","The focus, however, is usually on scheduling of particular emerging demand-side technologies, e.g.","HVAC [17], [18], [19], flexible loads [20], PV-battery systems [21], and plug-in electrical vehicles (PEVs) [22].","Most of these modeling approaches assume an existing market structure, with the impact of prosumers incorporated by allowing demand and supply to interact in some limited or predefined ways.","Specifically, this is mainly done via three different approaches: Only the supply-side is modeled physically while prosumers are considered by a simplified representation of demand-side technologies.","In [20], flexible loads' effects on reserve markets are analyzed by modeling prosumers with a tank model; however, the reserve market is greatly simplified.","In [17], prosumers are represented by a price-elasticity matrix, which is used to model changes in the aggregate demand in response to a change in the electricity price, and are acquired from the analysis of historical data.","Demand-side technologies are physically modeled while a simplified representation of supply-side is employed.","For instance, in [21], the supply-side is represented by an electricity price profile.","Both supply and demand sides can be modeled physically and optimized jointly, as in [18], [22], which can produce more realistic results.","For example, the study in [22] integrates the aggregated charging management approaches for PEVs into the market clearing process, with a simplified representation of the latter.","Although the above models have shown their merits, they are dependent on specific practical details such as the electricity price or the implementation of a mechanism for demand response (DR) aggregation, which limits their usefulness for future grid scenario analysis where the detailed market structure is potentially unknown.","Against this backdrop, the paper proposes a principled method for generic demand modeling including the aggregated effect of prosumers.","The model is formulated as a bi-level program in which the upper-level unit commitment problem minimizes the total generation cost, and the lower-level problem maximizes the aggregate prosumers' self-consumption.","In more detail, the lower-level objective is motivated by the emerging situation in Australia, where rooftop PV owners are increasingly discouraged from sending power back to the grid due to very low PV feed-in-tariffs versus increasing retail electricity prices.","In this setting, an obvious cost-minimizing strategy is to install small-scale battery storage, to maximize self-consumption of local generated energy and offset energy used in peak pricing periods.","Similar tariff settings appear likely to occur globally in the near future, as acknowledged in [23].","Moreover, self-consumption within an aggregated block of prosumers is a good approximation of many likely behaviors and responses to other future incentives and market structures, such as (peak power-based) demand charges, capacity constrained connections, virtual net metering across connection points, transactive energy and local energy trading, and a (somewhat irrational) desire for self-reliance.","A key difference from existing bi-level optimization frameworks is that in our formulation, the levels are coupled through the prosumers' demand, not through the electricity price.","In contrast, other models, which focus on the interaction between an aggregator and the prosumers [19] or the aggregator and the wholesale market [22], couple the levels through prices.","These approaches essentially define a market structure, that is, a pricing rule to support an outcome.","In contradistinction, our proposed model is market structure agnostic.","That is, it implicitly assumes that an efficient mechanism for demand response aggregation is adopted, with prices determined by that unspecified mechanism, which support the outcomes computed by our optimization framework.","The paper builds on our previous work [24], [25].","In [25], we have assumed that a prosumer aggregation represents a homogeneous group of loads; that is, we have assumed that they all behave in the same way and have the same capacity, and are allowed to send power back to the grid.","In the absence of an explicit transmission pricing, this can create perverse outcomes, such as power exchange between aggregators located in different parts of the network.","In the model proposed in this paper, the aggregators are not allowed to send the power back to the grid, which better models the assumption of self-consumption within an aggregated block of prosumers.","The model proposed in [24] is similar to the model in this paper, however, the prosumer battery storage is modeled implicitly, which requires a heuristic search to capture the prosumer behavior.","The remainder of the paper is organized as follows: Section II presents the proposed modeling framework.","In Section III, the efficacy of the proposed framework is demonstrated on a simplified 14-generator network model of the NEM with a significant RES penetration.","Finally, Section IV concludes."],["Generic Demand Model Considering the Impact of Prosumers","Generic demand models are essential for power system studies.","They are commonly used to reflect the aggregated effect of numerous physical loads [26], [27].","Conventional demand models only account for the accumulated effect of independent load changes and some relatively minor control actions, that is, they are by necessity simplified to not include all details of the loads.","In the following, we explain the motivation behind this work and the modeling assumptions."],["Research Motivation and Modeling Assumptions","The main purpose of developing generic demand models is to provide accurate dispatch decisions for balancing and stability analysis of future grid scenarios.","Given the uncertainty associated with future grid studies, the modeling framework should be market structure agnostic, and capable of easy integration of various types and penetrations of emerging demand-side technologies.","To this effect, we make the following assumptions: The loads are modeled as price anticipators.","It is well understood that price-taking load behavior, which simply responds to a given price profile, can result in load synchronization, i.e.","all users move their consumption to a low-price period, resulting in an inefficient market outcome.","In contrast, price anticipatory loads influence the electricity price by playing “a game” with the wholesale market.","The game is captured by the proposed bi-level model.","Specifically, the price-anticipating assumption implies that the aggregate effect of the prosumers is to change the market clearing prices and quantities, and, moreover, that the prosumers have a model of this effect.","Given this, the prosumer aggregation bids follow an equilibrium strategy, with an accurate expectation about the response of the market; this is the standard reasoning behind a Cournot or Stackelberg game formulation corresponding to a bi-level optimization problem.","This equilibrium strategy can be thought of as being generated by the following iterative process: First, the market operator creates a price profile by clearing the market based on the predicted demand.","Second, the prosumers (and other price-anticipatory participants) respond by shifting their consumption to cheaper time slots.","This gives a new demand profile, and the market operator clears the market again, and the process repeats until convergence.","Note that the proposed model does not specify an iterative mechanism, but rather, it encodes the optimality conditions for any price profile for the prosumers in Karush-Kuhn-Tucker (KKT) conditions, and in this way it captures the outcome resulting from the price-anticipatory prosumer behavior.","In our previous work [28], [16], we modeled the loads as price-takers, inspired by the smart home concept [29], in which the loads respond to the electricity price to minimize energy expenditure.","The study has shown that with large penetration of price-taking prosumers, the marginal benefit might become negative when secondary peaks are created due to the load synchronization (see Fig.","REF showing the operational demand with different penetrations of prosumers).","Therefore, demand response aggregators (henceforth simply called aggregators) have started to emerge to fully exploit the demand-side flexibility.","To that effect, the model implicitly assumes an efficient mechanism for demand response aggregation (an interested reader might refer to [30] for a discussion on practical implementation issues).","However, specific implementation details, like price structure or the division of the profit earning by the aggregated collection of prosumers, are not of explicit interest in the proposed model.","The demand model representing an aggregator consists of a large population of prosumers connected to an unconstrained distribution network who collectively maximize self-consumption (made possible by an efficient internal trading and balancing mechanism).","Aggregators do not alter the underlying power consumption of the prosumers.","That is, except for battery losses, the total power consumption before and after aggregation remains the same; however, the grid power intake profile does change, as a result of the prosumers using batteries to maximize self-consumption.","Prosumers have smart meters equipped with home energy management (HEM) systems for scheduling of the PV-battery systems.","Also, a communication infrastructure is assumed that allows a two-way communication between the grid, the aggregator and the prosumers, facilitating energy trading between prosumers in the aggregation.","These assumptions appear to be appropriate for scenarios arising in time frame of several decades into the future.","Figure: Operational demand with different penetrations of price-taking (top) and price-anticipating prosumers (bottom)."],["Bi-level Optimization Framework","In the model, we are specifically interested in the aggregated effect of a large prosumer population on the demand profile, assuming that the prosumers collectively maximize their self-consumption.","Given that the objective of the wholesale market is to minimize the generation cost, the problem exhibits a bi-level structure.","In game theory, such hierarchical optimization problems are known as Stackelberg games.","They can be formulated as bi-level mathematical programs of the form [19]: $\\mathop {\\operatorname{minimize}}\\limits _{ \\textbf {x}, \\textbf {y} } \\quad & \\Phi \\left( \\textbf {x}, \\textbf {y} \\right) \\\\\\operatorname{subject\\,to} \\quad & \\left( \\textbf {x}, \\textbf {y} \\right) \\in \\mathcal {Z} \\\\ & \\textbf {y} \\in \\mathcal {S} = \\mathop {\\operatorname{arg\\,min}}\\limits _{\\textbf {y}} \\lbrace \\Omega \\left( \\textbf {x}, \\textbf {y} \\right): \\textbf {y} \\in \\mathcal {C} \\left( \\textbf {x} \\right) \\rbrace $ where $\\textbf {x} \\in \\mathbb {R}^n$ , $\\textbf {y} \\in \\mathbb {R}^m$ , are decisions vectors, and $\\Phi \\left( \\textbf {x}, \\textbf {y} \\right): \\mathbb {R}^{n+m} \\rightarrow \\mathbb {R}$ and $\\Omega \\left( \\textbf {x}, \\textbf {y} \\right): \\mathbb {R}^{n+m} \\rightarrow \\mathbb {R}$ are the objective functions of the upper- and the lower-level problems, respectively.","$\\mathcal {Z}$ is the joint feasible region of the upper-level problem and $\\mathcal {C(\\textbf {x})}$ the feasible region of the lower-level problem induced by $\\textbf {x}$ .","In the existing market models that adopt a hierarchical approach, the coupling variable $\\textbf {y}$ is the electricity price (e.g.","[22], [19]).","That is, the upper-level (the wholesale market in our case) determines the price schedule, while the lower-level (the aggregator acting on behalf of the prosumers), optimizes its consumption based on this price schedule.","Fig.","REF shows the structure of the proposed modeling framework.","The demand model consists of two parts: (i) inflexible demand, $p_{\\text{d}}^{\\text{inf},m}$ , with a fixed demand profile, representing large industrial loads and loads without flexible resources; and (ii) flexible demand, $p_{\\text{d}}^{\\text{flx},m}$ , comprising a large population of prosumers who collectively maximize self-consumption.","Note that not every bus in the system has a load connected to it, hence the distinction between an aggregator $m \\in \\mathcal {M}$ and bus $i \\in \\mathcal {B}$ .","Unlike in most existing studies, the interaction between the wholesale market and the aggregators in our model is through the demand profile of the aggregator, $p_{\\text{d}}^{\\text{flx},m}$ .","Note that in contradistinction to the price-taking assumption when the electricity price is known in advance, now the collective action of the prosumers affects the wholesale market dispatch, which is the salient feature of the proposed model.","Figure: Structure of the proposed modeling framework."],["Upper-level Problem (Wholesale Market)","To emulate the market outcome, the upper-level problem is cast as a unit commitment (UC) problem aiming to minimize the generation cost: $\\mathop {\\operatorname{minimize}} \\limits _{s,u,d,p,\\theta } \\sum _{h \\in \\mathcal {H}} \\!","{\\sum _{g \\in \\mathcal {G}}} \\Big (c_g^\\text{fix} s_{g,h} \\!","+ \\!","c_g^\\text{su} u_{g,h} \\!","+ \\!","c_g^\\text{sd} d_{g,h} \\!","+ \\!","c_g^\\text{var} p_{g,h} \\Big ),$ where $s_{g,h},u_{g,h},d_{g,h} \\in \\lbrace 0,1\\rbrace $ , $p_{g,h}\\in \\mathbb {R}_{+}$ , $\\theta _{i,h}\\in \\mathbb {R}$ are the decision variables of the problem.","The problem is subject to the following constraints: $& \\sum _{g \\in \\mathcal {G}_i}{p_{g,h}} = \\!","\\sum _{m \\in \\mathcal {M}_i} \\!","(p_{\\text{d},h}^{\\text{inf},m}+p_{\\text{d},h}^{\\text{flx},m})+\\sum _{l \\in \\mathcal {L}_i}(p_{l,h}+\\Delta p_{l,h}), \\\\& |B_{i,j}(\\theta _{i,h}-\\theta _{j,h})| \\le \\overline{p}_{l}, \\\\& \\underline{p}_{g} s_{g,h} \\le p_{g,h} \\le \\overline{p}_{g} s_{g,h}, \\\\& u_{g,h}-d_{g,h}=s_{g,h}-s_{g,h-1}, \\\\& \\sum \\nolimits _{g_{\\text{synch}}\\in \\mathcal {R}}({\\overline{p}_{g}} s_{g,h} - p_{g,h}) \\ge p^{\\text{res}}_{r,h}, \\\\& u_{g,h} + \\sum \\nolimits _{\\tilde{h}=0}^{\\tau _g^{\\text{u}}-1} d_{g,h+\\tilde{h}} \\le 1, \\\\& d_{g,h} + \\sum \\nolimits _{\\tilde{h}=0}^{\\tau _g^{\\text{d}}-1} u_{g,h+\\tilde{h}} \\le 1, \\\\& -r_g^- \\le p_{g,h} - p_{g,h-1} \\le r_g^+, \\\\&\\mathop {\\operatorname{arg\\,min}} \\limits _{p_{\\text{d}}^{\\text{flx}}, p_{\\text{b}}, e_{\\text{b}} } \\left\\lbrace \\sum _{h \\in \\mathcal {H}} p_{\\text{d},h}^{\\text{flx},m} \\ \\operatorname{subject\\,to} \\ (\\ref {LC15})\\textrm {-}(\\ref {LC18}) \\right\\rbrace , $ where (REF ) is the power balance equation at each bus $i$ in the systemNote that the flexible demand of each aggregator $m$ , $p_{\\text{d},h}^{\\text{flx},m}$ , couples the upper-level (wholesale market) problem with each of the $m$ lower-level (aggregator) problems., with $\\mathcal {G}_i$ , $\\mathcal {M}_i$ , $\\mathcal {L}_i$ representing respectively the sets of generators, aggregators and lines connected to bus $i$ , and $p_{\\text{d},h}^{\\text{inf},m}$ , $p_{\\text{d},h}^{\\text{flx},m}$ , $p_{l,h}^{i,j}$ and $\\Delta p_{l,h}^{i,j}$ representing respectively the inflexible and flexible demand of aggregator $m$ , line power and line power loss (assumed to be 10% of the line flow) on each line connected to bus $i$ ; () represents line power limits; () limits the dispatch level of a generating unit between its respective minimum and maximum limits; () links the status of a generator unit to the up and down binary decision variables; () ensures sufficient spinning reserves are available in reach region of the grid; () and () ensure minimum up and minimum down times of the generators; () are the generator ramping constraints; and () is the constraint resulting from the prosumer aggregation optimization problem, as explained in Section REF ."],["Lower-level Problem (Aggregators)","The prosumer aggregation is formulated in the lower-level problem.","The loads within an aggregator's domain are assumed homogeneous, which allows us to represent the total aggregator's demand with a single load model.","The flexibility provided by the battery is only used to maximize self-consumption, with grid supply readily available.","This implies that the end-users' power consumption pattern is left unaltered, so that their comfort is not jeopardized.","The coupling between the upper-level (wholesale) and the lower-level (retail) problem in the proposed model is through the power demand.","This removes the need to define the market in terms of a pricing mechanism or rule, and it follows that the electricity price is not explicitly shown in the optimization problem.","This is why it is called a generic model.","This approach stands in contrast to other existing bi-level formulations [19],[22], in which the loads and wholesale market are coupled through the electricity price.","The prices generated by any market depend inherently on the specific pricing mechanism adopted.","However, in practice, several different pricing rules can implement a desired outcome, including the widely-used uniform price reverse auction or nodal pricing mechanisms.","For example, the electricity price could comprise the dual variables associated with the power balance constraint (REF ) and power flow constraints () of the upper-level problem, plus retail/aggregator and network charges.","Moreover, a range of different pricing rules result in different retail/aggregator and network charges, with all supporting an efficient outcome.","However, by instead coupling the upper- and lower-level problems via power demand, our proposed model avoids the need to specify a particular pricing rule, which makes it market-structure agnostic (see also Assumptions 1 and 2 in Section II-A).","Nonetheless, it is fair to assume that in any practical system, users will have access to a price forecastNote that in a HEM problem [29], a HEM system is an agent acting on behalf of a prosumer, and the electricity price is known ahead of time, resulting in a price-taking behavior.","and other bidders' historical behavior–but these are all market design-specific.","In the absence of pricing rule details, the proposed optimization model does not require such information.","Specifically, the lower-level problem is formulated as follows for each demand aggregator $m \\in \\mathcal {M}$ : $& \\mathop {\\operatorname{minimize}}\\limits _{ p_{\\text{d}}^{\\text{flx}}, p_{\\text{b}}, e_{\\text{b}} } \\sum _{h \\in \\mathcal {H}} p_{\\text{d},h}^{\\text{flx},m},$ where the decision variables of the problem are flexible demand $p_{\\text{d}}^{\\text{flx}}$ , battery power $p_{\\text{b}}$ , and battery capacity $e_{\\text{b}}$ .","The problem is subject to the following constraints: $& p_{\\text{d},h}^{\\text{flx},m} = p_{\\text{d},h}^{\\text{u},m}-p_{\\text{pv},h}^{m}+p_{\\text{b},h}^m,\\\\& e_{\\text{b},h}^m = \\eta _{\\text{b}}^m e_{\\text{b},h-1}^m + p_{\\text{b},h}^m, \\\\& \\underline{p}_{\\text{b}}^m \\le p_{\\text{b},h}^m \\le \\overline{p}_{\\text{b}}^m,\\\\& \\underline{e}_{\\text{b}}^{m} \\le {e}_{\\text{b},h}^{m} \\le \\overline{e}_{\\text{b}}^{m},$ where (REF ) is the power balance equation; and ()-() are the battery storage constraints.","Power $p_{\\text{d},h}^{\\text{u},m}$ is the underlying power demand of the prosumers.","Note that according to the Assumption 3 in Section II-B, except for battery losses, the underlying power demand does not change, however the power intake from grid can.","Finally, the KKT optimality conditions of the lower-level problem are added as the constraints to the upper-level problem, which reduces the problem to a single mixed integer linear program that can be solved using of-the-shelf solvers.","Note that because the two levels interacts through a power, not through a price, unlike in [19], no linearization is required."],["Case Studies","To showcase the efficacy of the model, we analyze steady-state voltage stability of a simplified model of the NEM with scenarios reflecting different prosumer penetrations."],["Model of the Australian National Electricity Market (NEM)","The 14-generator IEEE test system shown in Fig.","REF was initially proposed in [31] as a test bed for small-signal analysis.","The system is loosely based on the NEM, the interconnection on the Australian eastern seaboard.","The network is stringy, with large transmission distances and loads concentrated in a few load centers.","It consists of 59 buses, 28 loads, and 14 generators.","The test system consists of four areas representing the states of Queensland, New South Wales, Victoria and South Australia, and 28 aggregators, one for each load bus.","The generator technologies and modeling assumptions follow [24].","We consider two RES penetration rates.","In the business as usual (BAU) scenario, the generation portfolio includes 39.36 coal, 5.22 gas, and 2.33 hydro.","In the high-RES scenario, 40 of the total demand is covered by variable RES.","Inspired by two recent Australian 100% renewables studies [6], [10], part of coal generation is replaced with wind and utility PV using wind and solar traces from the AEMO's planning document [32], which results in 28.94 coal, 5.22 gas, 2.33 hydro, 21 wind, and 12 utility PV.","Given the deterministic nature of the model, we assume 10 reserves for each region in the system to cater for demand and RES forecast errors.","In market simulations, generators are assumed to bid according to their short-run marginal costs, while RESs bid at zero cost.","Simulations are performed using a rolling horizon approach with hourly resolution assuming a perfect foresight.","The optimization horizon is three days with a two-day overlap.","Last, wind and solar generators are assumed to operate in a voltage control mode.","Figure: Single-line diagram of the 14-generator model of the NEM with the 16 zones that define wind and solar traces ."],["Prosumer Scenarios","We assume four different prosumer penetrations: zero, low, medium and high.","With no prosumer penetration, the demand is assumed inflexible.","For the other three scenarios, we assume that part of the demand is equipped with small-scale (residential and small commercial) PV-battery systems.","The uptake of PV loosely follows a recent AEMO study [33].","The PV capacities are respectively 5, 10, and 20 for the low, medium and high uptake of prosumers (the existing penetration in the NEM is 5).","We consider three different amounts of storage: zero, 2, and 4 of storage for 1 of rooftop PV.A typical ratio in the NEM today is 2h of storage [33], however, in the future, this will likely increase due to the anticipated cost reduction.","Hourly demand and PV traces are from the AEMO's planning document [32]."],["Dispatch Results","Dispatch results for a typical summer week with high demand (12-15 January) for a few representative scenarios are shown in Figs.","REF and REF .","The figures show, respectively, generation dispatch results (top row), combined flexible demand of all aggregators (middle row), and a combined battery charging profile of all aggregators (bottom row).","Fig.","REF shows results for a medium prosumer penetration with, respectively, zero, 2h and 4h hours of storage.","Fig.","REF shows results for different prosumer penetrations (zero, medium, high) with 4h of storage.","Observe that in all six cases peak demand occurs at mid-day due to a high air-conditioning load.","After the sunset, however, the demand is still high, so gas generation is needed to cover the gap.","In the available generation mix, gas has the highest short-run marginal cost, which increases the electricity price in late afternoon/early evening.","The balancing results over the simulated year have revealed that the increased RES penetration in the renewable scenarios requires more energy from gas generation compared to the BAU scenario.","This is due to RES intermittency, and the ramp limits of conventional coal-fired generation.","An increased penetration of prosumers with higher amounts of storage, however, reduces the usage of gas due to a flatter demand profile.","Figure: Dispatch results for a typical summer week with high demand (12-15 January) for a medium prosumer penetration with different amounts of storage: zero (left), 2h (middle) and 4h (right).Figure: Dispatch results for a typical summer week with high demand (12-15 January) for different penetrations of prosumers with 4h of storage: zero (left), medium (middle), high (right).Observe in Fig.","REF how an increasing amount of storage increases prosumers' self-sufficiency.","Without storage, the load is supplied by PV during the day, and the rest is supplied from the grid.","When storage is added to the system, batteries are charged when electricity is cheap (mostly from rooftop PV during the day and from wind during the night) and discharged in late afternoon to offset the demand when the electricity is most expensive.","Note that the plots in the bottom two rows show a combined load profile of all aggregators in the system, which explains why storage is seemingly charged and discharged simultaneously.","Observe how high amounts of storage (rightmost columns in Figs.","REF and REF ) flatten the demand profile.","During the day, the flexible demand is supplied by rooftop PV, which reduces the operational demand, while during the night, with sufficient wind generation, batteries are charged, which increases the operational demand.","This has a significant beneficial effect on loadability and voltage stability, as discussed in the next section."],["Loadability and Voltage Stability Results","Dispatch results from the market simulations are used to perform a load flow analysis, which is then used in the assessment of loadability and voltage stability.","In the analysis, only scenarios with 4h of storage were considered.","The prosumer scenarios are thus called, according to the respective penetration rates, zero (ZP), low (LP), medium (MP), and high (HP).","Note that the market model only considers a simplified DC power flow with the maximum angle limit set to $30^{\\circ }$ .","This can sometimes result in a non-convergent AC power flow in scenarios with a high RES penetration.","The number of non-convergent hours is, respectively, 175, 37, 12, and 0, in scenarios ZP, LP, MP, and HP.","An increased penetration of prosumers thus improves voltage stability, as explained in more detail later.","In loadability assessment, N-1 security is considered, so a contingency screening is performed first.","We screened all credible N-1 contingencies to identify the most severe ones based on the maximum power transfer level [34].","Twenty most critical contingencies were selected for each hour of the simulated year."],["Loadability calculation","To calculate system loadability (LDB), the load in the system is progressively increased using the results of the market dispatch as the base case.","After each increase, load flow is calculated twice; first for the case after the last load increase without a contingency, and then again for each of the preselected critical contingencies.","When the load flow does not converge for a particular contingency, the last convergent load flow solution without a contingency is considered the loadability margin.","We considered two different load increase patterns, where load and generation are increased uniformly, in proportion to the base case: (i) NEM: only load and generation in the NEM are increased; (ii) SA/VIC: only load in VIC and generation in SA are increased.","The results are summarized in Table REF .","Comparing the BAU scenario and the renewable scenario with conventional demand (ZP), it can be seen that with the increased RES penetration, the average loadability margin over the simulated year in the demand increase scenario NEM is decreased from 7.8 to 2.5.","Similarly, the average loadability margin in the demand increase scenario SA/VIC is reduced from 2.2 to 0.8.","With a high RES penetration, conventional synchronous generation is replaced by inverter-based generation with inferior reactive power support capabilityFor synchronous generation, a 0.8 power factor is assumed.","For RES, we used the reactive power capability curve for the generic GE Type IV wind farm model [35], in which the reactive power generation is significantly constrained close to the nominal active power generation.","In practice, detailed planning studies are required before connection is granted, which can result in an increased requirement for reactive power support., which results in a reduced reactive power margin in the system and hence lower stability margin.","With an increased penetration of prosumers, the system loadability improves.","Observe that the average system loadability margin in both load increase scenarios, NEM and SA/VIC, is increased from 2.5 and 0.8 for the renewable scenario with no prosumers (ZP) to 7.1 and 1.7 for high penetration of prosumers (HP), respectively, which indicates a considerable improvement in the system loadability margin.","This is explained by a demand reduction when prosumer demand is supplied by rooftop PV.","In the night hours, however, even with high prosumer penetration, the loadibility can be reduced when prosumers charge their batteries, as observed in Figs.","REF and REF .","The situation is further illustrated in Fig.","REF that compares the loadability margin for the load increase scenario NEM and the results of the modal analysis, discussed next.","Table: Loadability results"],["Modal analysis","Using the market dispatch results, modal analysis of the reduced V-Q sub-matrix of the power flow Jacobian is performed to assess voltage stability.","The smallest real part of the V-Q sub-matrix' eigenvalues is used as a relative measure of the proximity to voltage instability.","Furthermore, the associated eigenvectors provide information on the critical voltage modes and the weak points in the grid, that is, the areas that are most prone to voltage instability.","The results are summarized in Table REF .","With the increased RES penetration, the average of the minimum of the real part of all eigenvalues, henceforth called the minimum eigenvalue, is reduced from 49.5Np for the BAU scenario to 42.3Np for the ZP scenario.","With an increased prosumer penetration, the average of the minimum eigenvalue over the simulated year increases from 42.3Np (ZP) to 48.8Np (HP).","Observe in Fig.","REF that the results of the modal analysis confirm the results loadability analysis.","The general trend remains the same; higher RES penetration with conventional demand reduces the minimum real value of an eigenvalue, which implies a lower voltage stability margin.","Table: Modal analysis resultsFigure: Comparison of the loadability and modal analysis results for a typical summer week with high demand (12-15 January) for the load increase scenario NEM for BAU, and low and high prosumer penetration.Another observation that can be made from the modal analysis concerns the location of the weak points in the system, that is, the buses with the highest participation factor in the critical voltage modes.","These clearly change with the increased RES penetration.","In the BAU Scenario, the weakest points are typically buses with large loads (e.g.","306 and 308, representing Melbourne).","With the increased RES penetration and no prosumers (ZP), however, the weakest part of the system become buses located close to RESs (e.g.","505 and 410, representing, respectively a large wind farm in SA and a large solar PV farm in QLD).","Further, we observed that the voltage stability margin in the system improves significantly when there are more synchronous generators in the grid, due to their superior reactive power support capability compared to RES.","The participation factor analysis of the renewable scenarios revealed that SA and QLD are the most voltage constrained regions where the penetration of WFs and utility PVs is higher compared to other regions in the NEM, which can be, to a large extent, mitigated with a sufficiently large penetration of prosumers.","This clearly illustrates that RESs and prosumers change power system stability in ways that have not been experienced before, which requires a further in-depth analysis."],["Conclusion","The emergence of demand side technologies, in particular rooftop PV, battery storage and energy management systems is changing the way electricity consumers source and consume electric power, which requires new demand models for the long-term analysis of future grids.","In this paper, we propose a generic demand model that captures the aggregate effect of a large number of prosumers on the load profile that can be used in market simulation.","The model uses a bi-level optimization framework, in which the upper level employs a unit commitment problem to minimize generation cost, and the lower level problem maximizes collective prosumers' self-consumption.","To that effect, the model implicitly assumes an efficient mechanism for demand response aggregation, for example peer-to-peer energy trading or any other form of transactive energy.","Given that any efficient mechanism for demand response aggregation that aims to minimize generation cost will utilize self-generation first, the self-consuming assumption appears reasonable at the level of abstraction assumed in long-term future grid scenario analysis.","Moreover, the model is generic in that it does not depend on specific practical implementation details that will vary in the long-run.","To showcase the efficacy of the proposed model, we study the impact of prosumers on the performance, loadability and voltage stability of the Australian NEM with a high RES penetration.","The results show that an increased prosumer penetration flattens the demand profile, which increases loadability and voltage stability, except in situations with a low underlying demand and an excess of RES generation, where the aggregate demand might increase due to battery charging.","The analysis also revealed that with a high RES penetration, the weakest points in the network move from large load centers to areas with high RES penetration.","Also, loadability and voltage stability are highly dependent on the amount of synchronous generation due to their superior reactive power capability compared to RES, which requires further analysis.","Our future work will focus on designing market mechanisms for large scale aggregation of distributed energy resources to enable the participation of prosumers in power system operation."],["MILP Formulation of the Lower-level KKT Optimality Conditions","For the sake of completeness, we provide the complete optimization problem below.","The upper-level problem (REF )-() remains the same.","The price-anticipatory prosumer behavior is captured in constraint (), defined as the solution to the lower-level prosumer aggregation problem.","To be able to incorporate it in the upper-level problem, we first need to convert it into a set of MILP constraints.","We do that by using the optimality (KKT) conditions of the underlying optimization problem.","The KKT conditions follow from the associated Lagrangian: $\\mathcal {L}(\\textbf {x}, \\lambda , \\mu ) = f(\\textbf {x})+ \\lambda ^\\top \\textbf {g}(\\textbf {x}) + \\mu ^\\top \\textbf {h}(\\textbf {x}),$ where $\\textbf {x} = \\lbrace p_{\\text{d}}^{\\text{flx}}, p_{\\text{b}}, e_{\\text{b}} \\rbrace $ is the decision vector of the lower-level problem; $f(\\textbf {x})$ is the objective function (REF ); $\\textbf {g}(\\textbf {x})$ is the vector of equality constraints (REF )-(); $\\textbf {h}(\\textbf {x})$ is the vector of inequality constraints ()-(); $\\lambda ^\\top = \\lbrace \\lambda _{h}^{m,p}, \\lambda _{h}^{m,e} \\rbrace $ is the vector of Lagrange multipliers associated with the equality constraints, and $\\mu ^\\top = \\lbrace \\mu _{h}^{\\text{flx},m}, \\mu _{h}^{m,\\underline{p}}, \\mu _{h}^{m,\\overline{p}}, \\mu _{h}^{m,\\underline{e}},\\mu _{h}^{m,\\overline{e}}\\rbrace $ is the vector of Lagrange multipliers associated with the inequality constraints.","The KKT optimality conditions consist of: primal feasibility (REF )-() with the associated Lagrange multipliers given in parentheses: $& \\qquad \\qquad p_{\\text{d},h}^{\\text{flx},m} + p_{\\text{pv},h}^{m} - p_{\\text{b},h}^{m} - p_{\\text{d},h}^{\\text{u},m} = 0, &( \\lambda _{h}^{m,p} ) \\\\& \\qquad \\qquad e_{\\text{b},h}^{m} - \\eta _\\text{b}^m e_{\\text{b},h-1}^{m} - p_{\\text{b},h}^{m} = 0, & ( \\lambda _{h}^{m,e} ) \\\\& \\qquad \\qquad -p_{\\text{d},h}^{\\text{flx},m} \\le 0, &( \\mu _{h}^{\\text{flx},m} ) \\\\& \\qquad \\qquad \\underline{p}_{\\text{b}}^{m} - p_{\\text{b},h}^{m} \\le 0, &( \\mu _{h}^{m,\\underline{p}} ) \\\\& \\qquad \\qquad p_{\\text{b},h}^{m} - \\overline{p}_{\\text{b}}^{m} \\le 0, &( \\mu _{h}^{m,\\overline{p}} ) \\\\& \\qquad \\qquad \\underline{e}_{\\text{b}}^{m} -e_{\\text{b},h}^{m} \\le 0, &( \\mu _{h}^{m,\\underline{e}} ) \\\\& \\qquad \\qquad e_{\\text{b},h}^{m} -\\overline{e}_{\\text{b}}^{m} \\le 0, &( \\mu _{h}^{m,\\overline{e}} ) $ dual feasibility: $& \\mu _{h}^{\\text{flx},m}, \\mu _{h}^{m,\\underline{p}}, \\mu _{h}^{m,\\overline{p}}, \\mu _{h}^{m,\\underline{e}}, \\mu _{h}^{m,\\overline{e}} \\ge 0,$ stationarity (REF )-(): $&\\frac{\\partial \\mathcal {L}}{\\partial p_{\\text{d},h}^{\\text{flx},m}} = 1 + \\lambda _{h}^{m,p} - \\mu _{h}^{\\text{flx},m} = 0, \\\\&\\frac{\\partial \\mathcal {L}}{\\partial p_{\\text{b},h}^{m}} = - \\lambda _{h}^{m,p} - \\lambda _{h}^{m,e} - \\mu _{h}^{m,\\underline{p}} + \\mu _{h}^{m,\\overline{p}} = 0,\\\\&\\frac{\\partial \\mathcal {L}}{\\partial e_{\\text{b},h}^{m}} = \\lambda _{h}^{m,e} - \\eta _\\text{b}^m \\lambda _{h+1}^{m,e} - \\mu _{h}^{m,\\underline{e}} + \\mu _{h}^{m,\\overline{e}} = 0,$ complementary slackness (REF )-(): $&p_{\\text{d},h}^{\\text{flx},m} \\mu _{h}^{\\text{flx},m} = 0, \\\\&(-\\underline{p}_{\\text{b}}^{m} + p_{\\text{b},h}^{m}) \\mu _{h}^{m,\\underline{p}} = 0, \\\\&(-p_{\\text{b},h}^{m} + \\overline{p}_{\\text{b}}^{m}) \\mu _{h}^{m,\\overline{p}} = 0, \\\\&(-\\underline{e}_{\\text{b}}^{m} + e_{\\text{b},h}^{m}) \\mu _{h}^{m,\\underline{e}} = 0, \\\\&(-e_{\\text{b},h}^{m} + \\overline{e}_{\\text{b}}^{m}) \\mu _{h}^{m,\\overline{e}} = 0.","$ Complementary slackness conditions (REF )-() are bilinear, however they can be linearized by introducing one binary variable $b$ and a sufficiently large and positive constant $M$ resulting in two constraints per complementarity slackness condition ()-().","Given that the KKT conditions are sufficient and necessary for optimality, the bi-level constraint () can be replaced with mixed integer linear constraints (REF )-(): $& p_{\\text{d},h}^{\\text{flx},m} + p_{\\text{pv},h}^{m} - p_{\\text{b},h}^{m} - p_{\\text{d},h}^{\\text{u},m} = 0, \\\\& e_{\\text{b},h}^{m} - \\eta _\\text{b}^m e_{\\text{b},h-1}^{m} - p_{\\text{b},h}^{m} = 0 ,\\\\& 1 - \\mu _{h}^{\\text{flx},m} + \\lambda _{h}^{m,p} = 0,\\\\& - \\lambda _{h}^{m,p} - \\mu _{h}^{m,\\underline{p}} + \\mu _{h}^{m,\\overline{p}} - \\lambda _{h}^{m,e} = 0,\\\\& - \\mu _{h}^{m,\\underline{e}} + \\mu _{h}^{m,\\overline{e}} + \\lambda _{h}^{m,e} - \\eta _\\text{b}^m \\lambda _{h-1}^{m,e} = 0,\\\\& \\underline{p}_{\\text{b}}^{m} - p_{\\text{b},h}^{m} \\le 0, \\\\& p_{\\text{b},h}^{m} -\\overline{p}_{\\text{b}}^{m} \\le 0, \\\\& \\underline{e}_{\\text{b}}^{m} - e_{\\text{b},h}^{m} \\le 0 , \\\\& e_{\\text{b},h}^{m} -\\overline{e}_{\\text{b}}^{m} \\le 0 , \\\\& p_{\\text{d},h}^{\\text{flx},m} - M{^\\text{flx}} b_{h}^{\\text{flx},m} \\le 0, \\\\& \\mu _{h}^{\\text{flx},m} - M^\\text{flx} (1-b_{h}^{\\text{flx},m}) \\le 0,\\\\& p_{\\text{b},h}^{m} - M^{\\underline{p}} b_{h}^{m,\\underline{p}} \\le 0,\\\\& \\mu _{h}^{m,\\underline{p}} - M^{\\underline{p}} (1-b_{h}^{m,\\underline{p}}) \\le 0,\\\\& -p_{\\text{b},h}^{m} -M^{\\overline{p}} b_{h}^{m,\\overline{p}} \\le 0,\\\\& \\mu _{h}^{m,\\overline{p}} - M^{\\overline{p}} (1-b_{h}^{m,\\overline{p}}) \\le 0 ,\\\\& e_{\\text{b},h}^{m} - M^{\\underline{e}} b_{h}^{m,\\underline{e}} \\le 0,\\\\& \\mu _{h}^{m,\\underline{e}} - M^{\\underline{e}} (1-b_{h}^{m,\\underline{e}}) \\le 0,\\\\& -e_{\\text{b},h}^{m} - M^{\\overline{e}} b_{h}^{m,\\overline{e}} \\le 0,\\\\& \\mu _{h}^{m,\\overline{e}} -M^{\\overline{e}} (1-b_{h}^{m,\\overline{e}}) \\le 0,$ which adds the following additional decision variables to the upper-level problem (REF )-(): $& b_{h}^{\\text{flx},m},b_{h}^{m,\\underline{p}},b_{h}^{m,\\overline{p}},b_{h}^{m,\\underline{e}},b_{h}^{m,\\overline{e}} \\in \\lbrace 0,1\\rbrace ,\\\\& p_{\\text{b},h}^{m}, \\lambda _{h}^{m,p}, \\lambda _{h}^{m,e} \\in \\mathbb {R}, \\\\& p_{\\text{d},h}^{\\text{flx},m}, e_{\\text{b},h}^{m}, \\mu _{h}^{\\text{flx},m}, \\mu _{h}^{m,\\underline{p}}, \\mu _{h}^{m,\\overline{p}}, \\mu _{h}^{m,\\underline{e}},\\mu _{h}^{m,\\overline{e}} \\in \\mathbb {R}_+.$ The resulting optimization problem is an MILP that can be solved efficiently with off-the-shelf solvers.","[Figure: NO_CAPTION [Figure: NO_CAPTION [Figure: NO_CAPTION [Figure: NO_CAPTION [Figure: NO_CAPTION Prof. Hill is a Fellow of the Society for Industrial and Applied Mathematics, the Australian Academy of Science, and the Australian Academy of Technological Sciences and Engineering.","He is also a Foreign Member of the Royal Swedish Academy of Engineering Sciences."]],"string":"[\n [\n \"Generic Demand Model Considering the Impact of Prosumers for Future Grid\\n Scenario Analysis\"\n ],\n [\n \"Abstract The increasing uptake of residential PV-battery systems is bound to significantly change demand patterns of future power systems and, consequently, their dynamic performance.\",\n \"In this paper, we propose a generic demand model that captures the aggregated effect of a large population of price-responsive users equipped with small-scale PV-battery systems, called prosumers, for market simulation in future grid scenario analysis.\",\n \"The model is formulated as a bi-level program in which the upper-level unit commitment problem minimizes the total generation cost, and the lower-level problem maximizes prosumers' aggregate self-consumption.\",\n \"Unlike in the existing bi-level optimization frameworks that focus on the interaction between the wholesale market and an aggregator, the coupling is through the prosumers' demand, not through the electricity price.\",\n \"That renders the proposed model market structure agnostic, making it suitable for future grid studies where the market structure is potentially unknown.\",\n \"As a case study, we perform steady-state voltage stability analysis of a simplified model of the Australian National Electricity Market with significant penetration of renewable generation.\",\n \"The simulation results show that a high prosumer penetration changes the demand profile in ways that significantly improve the system loadability, which confirms the suitability of the proposed model for future grid studies.\"\n ],\n [\n \"Introduction\",\n \"Power systems are undergoing a major transformation driven by the increasing uptake of variable renewable energy sources (RES).\",\n \"At the demand side, the emergence of cost-effective “behind-the-meter” distributed energy resources, including on-site generation, energy storage, electric vehicles, and flexible loads, and the advancement of sensor, computer, communication and energy management technologies are changing the way electricity consumers source and consume electric power.\",\n \"Indeed, recent studies suggest that rooftop PV-battery systems will reach retail price parity from 2020 in the USA grids and the Australian National Electricity Market (NEM) [1].\",\n \"A recent forecast by Morgan Stanley has suggested that the uptake can be even faster, by boldly predicting that up to 2 million Australian households could install battery storage by 2020 [2].\",\n \"This has been confirmed by the Energy Networks Australia and the Australian Commonwealth Scientific and Industrial Research Organisation (CSIRO) who have estimated the projected uptake of solar PV and battery storage in 2050 to be 80 and 100GWh [3], which will represent between 30%–50% of total demand, a scenario called “Rise of the Prosumer” [4].\",\n \"Here, the prosumer they refer to is a small-scale (residential, commercial and small industrial) electricity consumer with on-site generation.\",\n \"A similar trend has been observed in Europe as well [5].\",\n \"Given this, it is expected that a large uptake of demand-side technologies will significantly change demand patterns in future gridsWe interpret a future grid to mean the study of national grid type structures with the above-mentioned transformational changes for the long-term out to 2050., which will in turn affect their dynamic performance.\",\n \"Existing future grid feasibility studies [6], [7], [8], [9], [10] typically use conventional demand models, possibly using some heuristics to account for the effect of emerging demand-side technologies, and the synergies that may arise between them.\",\n \"They also assume specific market arrangements by which RES are integrated into grid operations.\",\n \"The challenge associated with future grid planning is that the grid structure and the regulatory framework, including the market structure, cannot be simply assumed from the details of an existing one.\",\n \"Instead, several possible evolution paths need to be accounted for.\",\n \"Future grid planning thus requires a major departure from conventional power system planning, where only a handful of the most critical scenarios is analyzed.\",\n \"To account for a wide range of possible future evolutions, scenario analysis has been proposed in many industries, e.g.\",\n \"in finance and economics [11], and in energy [12], [13], [14].\",\n \"As opposed to power system planning where the aim is to find an optimal transmission and/or generation expansion plan, the aim in scenario analysis is to analyze possible evolution pathways to inform policymaking.\",\n \"Given the uncertainty associated with long-term projections, the focus of future grid scenario analysis is limited to the analysis of what is technically possible, although it might also consider an explicit costing [15].\",\n \"Our work is part of the Future Grid Research Program funded by the CSIRO, whose aim is to explore possible future pathways for the evolution of the Australian grid out to 2050 by looking beyond simple balancing.\",\n \"To this end, a comprehensive modeling framework for future grid scenario analysis has been proposed in [16], which includes a market model, power flow analysis, and stability analysis.\",\n \"The demand model, however, assumes that the users are price-takers, which does not properly capture the aggregated effect of prosumers on the demand profile, as discussed later.\",\n \"Due to the influence of a demand profile on power systems performance and stability, recent studies have attempted to integrate the aggregated impact of prosumers into the demand models [17], [18], [19], [20], [21], [22].\",\n \"The focus, however, is usually on scheduling of particular emerging demand-side technologies, e.g.\",\n \"HVAC [17], [18], [19], flexible loads [20], PV-battery systems [21], and plug-in electrical vehicles (PEVs) [22].\",\n \"Most of these modeling approaches assume an existing market structure, with the impact of prosumers incorporated by allowing demand and supply to interact in some limited or predefined ways.\",\n \"Specifically, this is mainly done via three different approaches: Only the supply-side is modeled physically while prosumers are considered by a simplified representation of demand-side technologies.\",\n \"In [20], flexible loads' effects on reserve markets are analyzed by modeling prosumers with a tank model; however, the reserve market is greatly simplified.\",\n \"In [17], prosumers are represented by a price-elasticity matrix, which is used to model changes in the aggregate demand in response to a change in the electricity price, and are acquired from the analysis of historical data.\",\n \"Demand-side technologies are physically modeled while a simplified representation of supply-side is employed.\",\n \"For instance, in [21], the supply-side is represented by an electricity price profile.\",\n \"Both supply and demand sides can be modeled physically and optimized jointly, as in [18], [22], which can produce more realistic results.\",\n \"For example, the study in [22] integrates the aggregated charging management approaches for PEVs into the market clearing process, with a simplified representation of the latter.\",\n \"Although the above models have shown their merits, they are dependent on specific practical details such as the electricity price or the implementation of a mechanism for demand response (DR) aggregation, which limits their usefulness for future grid scenario analysis where the detailed market structure is potentially unknown.\",\n \"Against this backdrop, the paper proposes a principled method for generic demand modeling including the aggregated effect of prosumers.\",\n \"The model is formulated as a bi-level program in which the upper-level unit commitment problem minimizes the total generation cost, and the lower-level problem maximizes the aggregate prosumers' self-consumption.\",\n \"In more detail, the lower-level objective is motivated by the emerging situation in Australia, where rooftop PV owners are increasingly discouraged from sending power back to the grid due to very low PV feed-in-tariffs versus increasing retail electricity prices.\",\n \"In this setting, an obvious cost-minimizing strategy is to install small-scale battery storage, to maximize self-consumption of local generated energy and offset energy used in peak pricing periods.\",\n \"Similar tariff settings appear likely to occur globally in the near future, as acknowledged in [23].\",\n \"Moreover, self-consumption within an aggregated block of prosumers is a good approximation of many likely behaviors and responses to other future incentives and market structures, such as (peak power-based) demand charges, capacity constrained connections, virtual net metering across connection points, transactive energy and local energy trading, and a (somewhat irrational) desire for self-reliance.\",\n \"A key difference from existing bi-level optimization frameworks is that in our formulation, the levels are coupled through the prosumers' demand, not through the electricity price.\",\n \"In contrast, other models, which focus on the interaction between an aggregator and the prosumers [19] or the aggregator and the wholesale market [22], couple the levels through prices.\",\n \"These approaches essentially define a market structure, that is, a pricing rule to support an outcome.\",\n \"In contradistinction, our proposed model is market structure agnostic.\",\n \"That is, it implicitly assumes that an efficient mechanism for demand response aggregation is adopted, with prices determined by that unspecified mechanism, which support the outcomes computed by our optimization framework.\",\n \"The paper builds on our previous work [24], [25].\",\n \"In [25], we have assumed that a prosumer aggregation represents a homogeneous group of loads; that is, we have assumed that they all behave in the same way and have the same capacity, and are allowed to send power back to the grid.\",\n \"In the absence of an explicit transmission pricing, this can create perverse outcomes, such as power exchange between aggregators located in different parts of the network.\",\n \"In the model proposed in this paper, the aggregators are not allowed to send the power back to the grid, which better models the assumption of self-consumption within an aggregated block of prosumers.\",\n \"The model proposed in [24] is similar to the model in this paper, however, the prosumer battery storage is modeled implicitly, which requires a heuristic search to capture the prosumer behavior.\",\n \"The remainder of the paper is organized as follows: Section II presents the proposed modeling framework.\",\n \"In Section III, the efficacy of the proposed framework is demonstrated on a simplified 14-generator network model of the NEM with a significant RES penetration.\",\n \"Finally, Section IV concludes.\"\n ],\n [\n \"Generic Demand Model Considering the Impact of Prosumers\",\n \"Generic demand models are essential for power system studies.\",\n \"They are commonly used to reflect the aggregated effect of numerous physical loads [26], [27].\",\n \"Conventional demand models only account for the accumulated effect of independent load changes and some relatively minor control actions, that is, they are by necessity simplified to not include all details of the loads.\",\n \"In the following, we explain the motivation behind this work and the modeling assumptions.\"\n ],\n [\n \"Research Motivation and Modeling Assumptions\",\n \"The main purpose of developing generic demand models is to provide accurate dispatch decisions for balancing and stability analysis of future grid scenarios.\",\n \"Given the uncertainty associated with future grid studies, the modeling framework should be market structure agnostic, and capable of easy integration of various types and penetrations of emerging demand-side technologies.\",\n \"To this effect, we make the following assumptions: The loads are modeled as price anticipators.\",\n \"It is well understood that price-taking load behavior, which simply responds to a given price profile, can result in load synchronization, i.e.\",\n \"all users move their consumption to a low-price period, resulting in an inefficient market outcome.\",\n \"In contrast, price anticipatory loads influence the electricity price by playing “a game” with the wholesale market.\",\n \"The game is captured by the proposed bi-level model.\",\n \"Specifically, the price-anticipating assumption implies that the aggregate effect of the prosumers is to change the market clearing prices and quantities, and, moreover, that the prosumers have a model of this effect.\",\n \"Given this, the prosumer aggregation bids follow an equilibrium strategy, with an accurate expectation about the response of the market; this is the standard reasoning behind a Cournot or Stackelberg game formulation corresponding to a bi-level optimization problem.\",\n \"This equilibrium strategy can be thought of as being generated by the following iterative process: First, the market operator creates a price profile by clearing the market based on the predicted demand.\",\n \"Second, the prosumers (and other price-anticipatory participants) respond by shifting their consumption to cheaper time slots.\",\n \"This gives a new demand profile, and the market operator clears the market again, and the process repeats until convergence.\",\n \"Note that the proposed model does not specify an iterative mechanism, but rather, it encodes the optimality conditions for any price profile for the prosumers in Karush-Kuhn-Tucker (KKT) conditions, and in this way it captures the outcome resulting from the price-anticipatory prosumer behavior.\",\n \"In our previous work [28], [16], we modeled the loads as price-takers, inspired by the smart home concept [29], in which the loads respond to the electricity price to minimize energy expenditure.\",\n \"The study has shown that with large penetration of price-taking prosumers, the marginal benefit might become negative when secondary peaks are created due to the load synchronization (see Fig.\",\n \"REF showing the operational demand with different penetrations of prosumers).\",\n \"Therefore, demand response aggregators (henceforth simply called aggregators) have started to emerge to fully exploit the demand-side flexibility.\",\n \"To that effect, the model implicitly assumes an efficient mechanism for demand response aggregation (an interested reader might refer to [30] for a discussion on practical implementation issues).\",\n \"However, specific implementation details, like price structure or the division of the profit earning by the aggregated collection of prosumers, are not of explicit interest in the proposed model.\",\n \"The demand model representing an aggregator consists of a large population of prosumers connected to an unconstrained distribution network who collectively maximize self-consumption (made possible by an efficient internal trading and balancing mechanism).\",\n \"Aggregators do not alter the underlying power consumption of the prosumers.\",\n \"That is, except for battery losses, the total power consumption before and after aggregation remains the same; however, the grid power intake profile does change, as a result of the prosumers using batteries to maximize self-consumption.\",\n \"Prosumers have smart meters equipped with home energy management (HEM) systems for scheduling of the PV-battery systems.\",\n \"Also, a communication infrastructure is assumed that allows a two-way communication between the grid, the aggregator and the prosumers, facilitating energy trading between prosumers in the aggregation.\",\n \"These assumptions appear to be appropriate for scenarios arising in time frame of several decades into the future.\",\n \"Figure: Operational demand with different penetrations of price-taking (top) and price-anticipating prosumers (bottom).\"\n ],\n [\n \"Bi-level Optimization Framework\",\n \"In the model, we are specifically interested in the aggregated effect of a large prosumer population on the demand profile, assuming that the prosumers collectively maximize their self-consumption.\",\n \"Given that the objective of the wholesale market is to minimize the generation cost, the problem exhibits a bi-level structure.\",\n \"In game theory, such hierarchical optimization problems are known as Stackelberg games.\",\n \"They can be formulated as bi-level mathematical programs of the form [19]: $\\\\mathop {\\\\operatorname{minimize}}\\\\limits _{ \\\\textbf {x}, \\\\textbf {y} } \\\\quad & \\\\Phi \\\\left( \\\\textbf {x}, \\\\textbf {y} \\\\right) \\\\\\\\\\\\operatorname{subject\\\\,to} \\\\quad & \\\\left( \\\\textbf {x}, \\\\textbf {y} \\\\right) \\\\in \\\\mathcal {Z} \\\\\\\\ & \\\\textbf {y} \\\\in \\\\mathcal {S} = \\\\mathop {\\\\operatorname{arg\\\\,min}}\\\\limits _{\\\\textbf {y}} \\\\lbrace \\\\Omega \\\\left( \\\\textbf {x}, \\\\textbf {y} \\\\right): \\\\textbf {y} \\\\in \\\\mathcal {C} \\\\left( \\\\textbf {x} \\\\right) \\\\rbrace $ where $\\\\textbf {x} \\\\in \\\\mathbb {R}^n$ , $\\\\textbf {y} \\\\in \\\\mathbb {R}^m$ , are decisions vectors, and $\\\\Phi \\\\left( \\\\textbf {x}, \\\\textbf {y} \\\\right): \\\\mathbb {R}^{n+m} \\\\rightarrow \\\\mathbb {R}$ and $\\\\Omega \\\\left( \\\\textbf {x}, \\\\textbf {y} \\\\right): \\\\mathbb {R}^{n+m} \\\\rightarrow \\\\mathbb {R}$ are the objective functions of the upper- and the lower-level problems, respectively.\",\n \"$\\\\mathcal {Z}$ is the joint feasible region of the upper-level problem and $\\\\mathcal {C(\\\\textbf {x})}$ the feasible region of the lower-level problem induced by $\\\\textbf {x}$ .\",\n \"In the existing market models that adopt a hierarchical approach, the coupling variable $\\\\textbf {y}$ is the electricity price (e.g.\",\n \"[22], [19]).\",\n \"That is, the upper-level (the wholesale market in our case) determines the price schedule, while the lower-level (the aggregator acting on behalf of the prosumers), optimizes its consumption based on this price schedule.\",\n \"Fig.\",\n \"REF shows the structure of the proposed modeling framework.\",\n \"The demand model consists of two parts: (i) inflexible demand, $p_{\\\\text{d}}^{\\\\text{inf},m}$ , with a fixed demand profile, representing large industrial loads and loads without flexible resources; and (ii) flexible demand, $p_{\\\\text{d}}^{\\\\text{flx},m}$ , comprising a large population of prosumers who collectively maximize self-consumption.\",\n \"Note that not every bus in the system has a load connected to it, hence the distinction between an aggregator $m \\\\in \\\\mathcal {M}$ and bus $i \\\\in \\\\mathcal {B}$ .\",\n \"Unlike in most existing studies, the interaction between the wholesale market and the aggregators in our model is through the demand profile of the aggregator, $p_{\\\\text{d}}^{\\\\text{flx},m}$ .\",\n \"Note that in contradistinction to the price-taking assumption when the electricity price is known in advance, now the collective action of the prosumers affects the wholesale market dispatch, which is the salient feature of the proposed model.\",\n \"Figure: Structure of the proposed modeling framework.\"\n ],\n [\n \"Upper-level Problem (Wholesale Market)\",\n \"To emulate the market outcome, the upper-level problem is cast as a unit commitment (UC) problem aiming to minimize the generation cost: $\\\\mathop {\\\\operatorname{minimize}} \\\\limits _{s,u,d,p,\\\\theta } \\\\sum _{h \\\\in \\\\mathcal {H}} \\\\!\",\n \"{\\\\sum _{g \\\\in \\\\mathcal {G}}} \\\\Big (c_g^\\\\text{fix} s_{g,h} \\\\!\",\n \"+ \\\\!\",\n \"c_g^\\\\text{su} u_{g,h} \\\\!\",\n \"+ \\\\!\",\n \"c_g^\\\\text{sd} d_{g,h} \\\\!\",\n \"+ \\\\!\",\n \"c_g^\\\\text{var} p_{g,h} \\\\Big ),$ where $s_{g,h},u_{g,h},d_{g,h} \\\\in \\\\lbrace 0,1\\\\rbrace $ , $p_{g,h}\\\\in \\\\mathbb {R}_{+}$ , $\\\\theta _{i,h}\\\\in \\\\mathbb {R}$ are the decision variables of the problem.\",\n \"The problem is subject to the following constraints: $& \\\\sum _{g \\\\in \\\\mathcal {G}_i}{p_{g,h}} = \\\\!\",\n \"\\\\sum _{m \\\\in \\\\mathcal {M}_i} \\\\!\",\n \"(p_{\\\\text{d},h}^{\\\\text{inf},m}+p_{\\\\text{d},h}^{\\\\text{flx},m})+\\\\sum _{l \\\\in \\\\mathcal {L}_i}(p_{l,h}+\\\\Delta p_{l,h}), \\\\\\\\& |B_{i,j}(\\\\theta _{i,h}-\\\\theta _{j,h})| \\\\le \\\\overline{p}_{l}, \\\\\\\\& \\\\underline{p}_{g} s_{g,h} \\\\le p_{g,h} \\\\le \\\\overline{p}_{g} s_{g,h}, \\\\\\\\& u_{g,h}-d_{g,h}=s_{g,h}-s_{g,h-1}, \\\\\\\\& \\\\sum \\\\nolimits _{g_{\\\\text{synch}}\\\\in \\\\mathcal {R}}({\\\\overline{p}_{g}} s_{g,h} - p_{g,h}) \\\\ge p^{\\\\text{res}}_{r,h}, \\\\\\\\& u_{g,h} + \\\\sum \\\\nolimits _{\\\\tilde{h}=0}^{\\\\tau _g^{\\\\text{u}}-1} d_{g,h+\\\\tilde{h}} \\\\le 1, \\\\\\\\& d_{g,h} + \\\\sum \\\\nolimits _{\\\\tilde{h}=0}^{\\\\tau _g^{\\\\text{d}}-1} u_{g,h+\\\\tilde{h}} \\\\le 1, \\\\\\\\& -r_g^- \\\\le p_{g,h} - p_{g,h-1} \\\\le r_g^+, \\\\\\\\&\\\\mathop {\\\\operatorname{arg\\\\,min}} \\\\limits _{p_{\\\\text{d}}^{\\\\text{flx}}, p_{\\\\text{b}}, e_{\\\\text{b}} } \\\\left\\\\lbrace \\\\sum _{h \\\\in \\\\mathcal {H}} p_{\\\\text{d},h}^{\\\\text{flx},m} \\\\ \\\\operatorname{subject\\\\,to} \\\\ (\\\\ref {LC15})\\\\textrm {-}(\\\\ref {LC18}) \\\\right\\\\rbrace , $ where (REF ) is the power balance equation at each bus $i$ in the systemNote that the flexible demand of each aggregator $m$ , $p_{\\\\text{d},h}^{\\\\text{flx},m}$ , couples the upper-level (wholesale market) problem with each of the $m$ lower-level (aggregator) problems., with $\\\\mathcal {G}_i$ , $\\\\mathcal {M}_i$ , $\\\\mathcal {L}_i$ representing respectively the sets of generators, aggregators and lines connected to bus $i$ , and $p_{\\\\text{d},h}^{\\\\text{inf},m}$ , $p_{\\\\text{d},h}^{\\\\text{flx},m}$ , $p_{l,h}^{i,j}$ and $\\\\Delta p_{l,h}^{i,j}$ representing respectively the inflexible and flexible demand of aggregator $m$ , line power and line power loss (assumed to be 10% of the line flow) on each line connected to bus $i$ ; () represents line power limits; () limits the dispatch level of a generating unit between its respective minimum and maximum limits; () links the status of a generator unit to the up and down binary decision variables; () ensures sufficient spinning reserves are available in reach region of the grid; () and () ensure minimum up and minimum down times of the generators; () are the generator ramping constraints; and () is the constraint resulting from the prosumer aggregation optimization problem, as explained in Section REF .\"\n ],\n [\n \"Lower-level Problem (Aggregators)\",\n \"The prosumer aggregation is formulated in the lower-level problem.\",\n \"The loads within an aggregator's domain are assumed homogeneous, which allows us to represent the total aggregator's demand with a single load model.\",\n \"The flexibility provided by the battery is only used to maximize self-consumption, with grid supply readily available.\",\n \"This implies that the end-users' power consumption pattern is left unaltered, so that their comfort is not jeopardized.\",\n \"The coupling between the upper-level (wholesale) and the lower-level (retail) problem in the proposed model is through the power demand.\",\n \"This removes the need to define the market in terms of a pricing mechanism or rule, and it follows that the electricity price is not explicitly shown in the optimization problem.\",\n \"This is why it is called a generic model.\",\n \"This approach stands in contrast to other existing bi-level formulations [19],[22], in which the loads and wholesale market are coupled through the electricity price.\",\n \"The prices generated by any market depend inherently on the specific pricing mechanism adopted.\",\n \"However, in practice, several different pricing rules can implement a desired outcome, including the widely-used uniform price reverse auction or nodal pricing mechanisms.\",\n \"For example, the electricity price could comprise the dual variables associated with the power balance constraint (REF ) and power flow constraints () of the upper-level problem, plus retail/aggregator and network charges.\",\n \"Moreover, a range of different pricing rules result in different retail/aggregator and network charges, with all supporting an efficient outcome.\",\n \"However, by instead coupling the upper- and lower-level problems via power demand, our proposed model avoids the need to specify a particular pricing rule, which makes it market-structure agnostic (see also Assumptions 1 and 2 in Section II-A).\",\n \"Nonetheless, it is fair to assume that in any practical system, users will have access to a price forecastNote that in a HEM problem [29], a HEM system is an agent acting on behalf of a prosumer, and the electricity price is known ahead of time, resulting in a price-taking behavior.\",\n \"and other bidders' historical behavior–but these are all market design-specific.\",\n \"In the absence of pricing rule details, the proposed optimization model does not require such information.\",\n \"Specifically, the lower-level problem is formulated as follows for each demand aggregator $m \\\\in \\\\mathcal {M}$ : $& \\\\mathop {\\\\operatorname{minimize}}\\\\limits _{ p_{\\\\text{d}}^{\\\\text{flx}}, p_{\\\\text{b}}, e_{\\\\text{b}} } \\\\sum _{h \\\\in \\\\mathcal {H}} p_{\\\\text{d},h}^{\\\\text{flx},m},$ where the decision variables of the problem are flexible demand $p_{\\\\text{d}}^{\\\\text{flx}}$ , battery power $p_{\\\\text{b}}$ , and battery capacity $e_{\\\\text{b}}$ .\",\n \"The problem is subject to the following constraints: $& p_{\\\\text{d},h}^{\\\\text{flx},m} = p_{\\\\text{d},h}^{\\\\text{u},m}-p_{\\\\text{pv},h}^{m}+p_{\\\\text{b},h}^m,\\\\\\\\& e_{\\\\text{b},h}^m = \\\\eta _{\\\\text{b}}^m e_{\\\\text{b},h-1}^m + p_{\\\\text{b},h}^m, \\\\\\\\& \\\\underline{p}_{\\\\text{b}}^m \\\\le p_{\\\\text{b},h}^m \\\\le \\\\overline{p}_{\\\\text{b}}^m,\\\\\\\\& \\\\underline{e}_{\\\\text{b}}^{m} \\\\le {e}_{\\\\text{b},h}^{m} \\\\le \\\\overline{e}_{\\\\text{b}}^{m},$ where (REF ) is the power balance equation; and ()-() are the battery storage constraints.\",\n \"Power $p_{\\\\text{d},h}^{\\\\text{u},m}$ is the underlying power demand of the prosumers.\",\n \"Note that according to the Assumption 3 in Section II-B, except for battery losses, the underlying power demand does not change, however the power intake from grid can.\",\n \"Finally, the KKT optimality conditions of the lower-level problem are added as the constraints to the upper-level problem, which reduces the problem to a single mixed integer linear program that can be solved using of-the-shelf solvers.\",\n \"Note that because the two levels interacts through a power, not through a price, unlike in [19], no linearization is required.\"\n ],\n [\n \"Case Studies\",\n \"To showcase the efficacy of the model, we analyze steady-state voltage stability of a simplified model of the NEM with scenarios reflecting different prosumer penetrations.\"\n ],\n [\n \"Model of the Australian National Electricity Market (NEM)\",\n \"The 14-generator IEEE test system shown in Fig.\",\n \"REF was initially proposed in [31] as a test bed for small-signal analysis.\",\n \"The system is loosely based on the NEM, the interconnection on the Australian eastern seaboard.\",\n \"The network is stringy, with large transmission distances and loads concentrated in a few load centers.\",\n \"It consists of 59 buses, 28 loads, and 14 generators.\",\n \"The test system consists of four areas representing the states of Queensland, New South Wales, Victoria and South Australia, and 28 aggregators, one for each load bus.\",\n \"The generator technologies and modeling assumptions follow [24].\",\n \"We consider two RES penetration rates.\",\n \"In the business as usual (BAU) scenario, the generation portfolio includes 39.36 coal, 5.22 gas, and 2.33 hydro.\",\n \"In the high-RES scenario, 40 of the total demand is covered by variable RES.\",\n \"Inspired by two recent Australian 100% renewables studies [6], [10], part of coal generation is replaced with wind and utility PV using wind and solar traces from the AEMO's planning document [32], which results in 28.94 coal, 5.22 gas, 2.33 hydro, 21 wind, and 12 utility PV.\",\n \"Given the deterministic nature of the model, we assume 10 reserves for each region in the system to cater for demand and RES forecast errors.\",\n \"In market simulations, generators are assumed to bid according to their short-run marginal costs, while RESs bid at zero cost.\",\n \"Simulations are performed using a rolling horizon approach with hourly resolution assuming a perfect foresight.\",\n \"The optimization horizon is three days with a two-day overlap.\",\n \"Last, wind and solar generators are assumed to operate in a voltage control mode.\",\n \"Figure: Single-line diagram of the 14-generator model of the NEM with the 16 zones that define wind and solar traces .\"\n ],\n [\n \"Prosumer Scenarios\",\n \"We assume four different prosumer penetrations: zero, low, medium and high.\",\n \"With no prosumer penetration, the demand is assumed inflexible.\",\n \"For the other three scenarios, we assume that part of the demand is equipped with small-scale (residential and small commercial) PV-battery systems.\",\n \"The uptake of PV loosely follows a recent AEMO study [33].\",\n \"The PV capacities are respectively 5, 10, and 20 for the low, medium and high uptake of prosumers (the existing penetration in the NEM is 5).\",\n \"We consider three different amounts of storage: zero, 2, and 4 of storage for 1 of rooftop PV.A typical ratio in the NEM today is 2h of storage [33], however, in the future, this will likely increase due to the anticipated cost reduction.\",\n \"Hourly demand and PV traces are from the AEMO's planning document [32].\"\n ],\n [\n \"Dispatch Results\",\n \"Dispatch results for a typical summer week with high demand (12-15 January) for a few representative scenarios are shown in Figs.\",\n \"REF and REF .\",\n \"The figures show, respectively, generation dispatch results (top row), combined flexible demand of all aggregators (middle row), and a combined battery charging profile of all aggregators (bottom row).\",\n \"Fig.\",\n \"REF shows results for a medium prosumer penetration with, respectively, zero, 2h and 4h hours of storage.\",\n \"Fig.\",\n \"REF shows results for different prosumer penetrations (zero, medium, high) with 4h of storage.\",\n \"Observe that in all six cases peak demand occurs at mid-day due to a high air-conditioning load.\",\n \"After the sunset, however, the demand is still high, so gas generation is needed to cover the gap.\",\n \"In the available generation mix, gas has the highest short-run marginal cost, which increases the electricity price in late afternoon/early evening.\",\n \"The balancing results over the simulated year have revealed that the increased RES penetration in the renewable scenarios requires more energy from gas generation compared to the BAU scenario.\",\n \"This is due to RES intermittency, and the ramp limits of conventional coal-fired generation.\",\n \"An increased penetration of prosumers with higher amounts of storage, however, reduces the usage of gas due to a flatter demand profile.\",\n \"Figure: Dispatch results for a typical summer week with high demand (12-15 January) for a medium prosumer penetration with different amounts of storage: zero (left), 2h (middle) and 4h (right).Figure: Dispatch results for a typical summer week with high demand (12-15 January) for different penetrations of prosumers with 4h of storage: zero (left), medium (middle), high (right).Observe in Fig.\",\n \"REF how an increasing amount of storage increases prosumers' self-sufficiency.\",\n \"Without storage, the load is supplied by PV during the day, and the rest is supplied from the grid.\",\n \"When storage is added to the system, batteries are charged when electricity is cheap (mostly from rooftop PV during the day and from wind during the night) and discharged in late afternoon to offset the demand when the electricity is most expensive.\",\n \"Note that the plots in the bottom two rows show a combined load profile of all aggregators in the system, which explains why storage is seemingly charged and discharged simultaneously.\",\n \"Observe how high amounts of storage (rightmost columns in Figs.\",\n \"REF and REF ) flatten the demand profile.\",\n \"During the day, the flexible demand is supplied by rooftop PV, which reduces the operational demand, while during the night, with sufficient wind generation, batteries are charged, which increases the operational demand.\",\n \"This has a significant beneficial effect on loadability and voltage stability, as discussed in the next section.\"\n ],\n [\n \"Loadability and Voltage Stability Results\",\n \"Dispatch results from the market simulations are used to perform a load flow analysis, which is then used in the assessment of loadability and voltage stability.\",\n \"In the analysis, only scenarios with 4h of storage were considered.\",\n \"The prosumer scenarios are thus called, according to the respective penetration rates, zero (ZP), low (LP), medium (MP), and high (HP).\",\n \"Note that the market model only considers a simplified DC power flow with the maximum angle limit set to $30^{\\\\circ }$ .\",\n \"This can sometimes result in a non-convergent AC power flow in scenarios with a high RES penetration.\",\n \"The number of non-convergent hours is, respectively, 175, 37, 12, and 0, in scenarios ZP, LP, MP, and HP.\",\n \"An increased penetration of prosumers thus improves voltage stability, as explained in more detail later.\",\n \"In loadability assessment, N-1 security is considered, so a contingency screening is performed first.\",\n \"We screened all credible N-1 contingencies to identify the most severe ones based on the maximum power transfer level [34].\",\n \"Twenty most critical contingencies were selected for each hour of the simulated year.\"\n ],\n [\n \"Loadability calculation\",\n \"To calculate system loadability (LDB), the load in the system is progressively increased using the results of the market dispatch as the base case.\",\n \"After each increase, load flow is calculated twice; first for the case after the last load increase without a contingency, and then again for each of the preselected critical contingencies.\",\n \"When the load flow does not converge for a particular contingency, the last convergent load flow solution without a contingency is considered the loadability margin.\",\n \"We considered two different load increase patterns, where load and generation are increased uniformly, in proportion to the base case: (i) NEM: only load and generation in the NEM are increased; (ii) SA/VIC: only load in VIC and generation in SA are increased.\",\n \"The results are summarized in Table REF .\",\n \"Comparing the BAU scenario and the renewable scenario with conventional demand (ZP), it can be seen that with the increased RES penetration, the average loadability margin over the simulated year in the demand increase scenario NEM is decreased from 7.8 to 2.5.\",\n \"Similarly, the average loadability margin in the demand increase scenario SA/VIC is reduced from 2.2 to 0.8.\",\n \"With a high RES penetration, conventional synchronous generation is replaced by inverter-based generation with inferior reactive power support capabilityFor synchronous generation, a 0.8 power factor is assumed.\",\n \"For RES, we used the reactive power capability curve for the generic GE Type IV wind farm model [35], in which the reactive power generation is significantly constrained close to the nominal active power generation.\",\n \"In practice, detailed planning studies are required before connection is granted, which can result in an increased requirement for reactive power support., which results in a reduced reactive power margin in the system and hence lower stability margin.\",\n \"With an increased penetration of prosumers, the system loadability improves.\",\n \"Observe that the average system loadability margin in both load increase scenarios, NEM and SA/VIC, is increased from 2.5 and 0.8 for the renewable scenario with no prosumers (ZP) to 7.1 and 1.7 for high penetration of prosumers (HP), respectively, which indicates a considerable improvement in the system loadability margin.\",\n \"This is explained by a demand reduction when prosumer demand is supplied by rooftop PV.\",\n \"In the night hours, however, even with high prosumer penetration, the loadibility can be reduced when prosumers charge their batteries, as observed in Figs.\",\n \"REF and REF .\",\n \"The situation is further illustrated in Fig.\",\n \"REF that compares the loadability margin for the load increase scenario NEM and the results of the modal analysis, discussed next.\",\n \"Table: Loadability results\"\n ],\n [\n \"Modal analysis\",\n \"Using the market dispatch results, modal analysis of the reduced V-Q sub-matrix of the power flow Jacobian is performed to assess voltage stability.\",\n \"The smallest real part of the V-Q sub-matrix' eigenvalues is used as a relative measure of the proximity to voltage instability.\",\n \"Furthermore, the associated eigenvectors provide information on the critical voltage modes and the weak points in the grid, that is, the areas that are most prone to voltage instability.\",\n \"The results are summarized in Table REF .\",\n \"With the increased RES penetration, the average of the minimum of the real part of all eigenvalues, henceforth called the minimum eigenvalue, is reduced from 49.5Np for the BAU scenario to 42.3Np for the ZP scenario.\",\n \"With an increased prosumer penetration, the average of the minimum eigenvalue over the simulated year increases from 42.3Np (ZP) to 48.8Np (HP).\",\n \"Observe in Fig.\",\n \"REF that the results of the modal analysis confirm the results loadability analysis.\",\n \"The general trend remains the same; higher RES penetration with conventional demand reduces the minimum real value of an eigenvalue, which implies a lower voltage stability margin.\",\n \"Table: Modal analysis resultsFigure: Comparison of the loadability and modal analysis results for a typical summer week with high demand (12-15 January) for the load increase scenario NEM for BAU, and low and high prosumer penetration.Another observation that can be made from the modal analysis concerns the location of the weak points in the system, that is, the buses with the highest participation factor in the critical voltage modes.\",\n \"These clearly change with the increased RES penetration.\",\n \"In the BAU Scenario, the weakest points are typically buses with large loads (e.g.\",\n \"306 and 308, representing Melbourne).\",\n \"With the increased RES penetration and no prosumers (ZP), however, the weakest part of the system become buses located close to RESs (e.g.\",\n \"505 and 410, representing, respectively a large wind farm in SA and a large solar PV farm in QLD).\",\n \"Further, we observed that the voltage stability margin in the system improves significantly when there are more synchronous generators in the grid, due to their superior reactive power support capability compared to RES.\",\n \"The participation factor analysis of the renewable scenarios revealed that SA and QLD are the most voltage constrained regions where the penetration of WFs and utility PVs is higher compared to other regions in the NEM, which can be, to a large extent, mitigated with a sufficiently large penetration of prosumers.\",\n \"This clearly illustrates that RESs and prosumers change power system stability in ways that have not been experienced before, which requires a further in-depth analysis.\"\n ],\n [\n \"Conclusion\",\n \"The emergence of demand side technologies, in particular rooftop PV, battery storage and energy management systems is changing the way electricity consumers source and consume electric power, which requires new demand models for the long-term analysis of future grids.\",\n \"In this paper, we propose a generic demand model that captures the aggregate effect of a large number of prosumers on the load profile that can be used in market simulation.\",\n \"The model uses a bi-level optimization framework, in which the upper level employs a unit commitment problem to minimize generation cost, and the lower level problem maximizes collective prosumers' self-consumption.\",\n \"To that effect, the model implicitly assumes an efficient mechanism for demand response aggregation, for example peer-to-peer energy trading or any other form of transactive energy.\",\n \"Given that any efficient mechanism for demand response aggregation that aims to minimize generation cost will utilize self-generation first, the self-consuming assumption appears reasonable at the level of abstraction assumed in long-term future grid scenario analysis.\",\n \"Moreover, the model is generic in that it does not depend on specific practical implementation details that will vary in the long-run.\",\n \"To showcase the efficacy of the proposed model, we study the impact of prosumers on the performance, loadability and voltage stability of the Australian NEM with a high RES penetration.\",\n \"The results show that an increased prosumer penetration flattens the demand profile, which increases loadability and voltage stability, except in situations with a low underlying demand and an excess of RES generation, where the aggregate demand might increase due to battery charging.\",\n \"The analysis also revealed that with a high RES penetration, the weakest points in the network move from large load centers to areas with high RES penetration.\",\n \"Also, loadability and voltage stability are highly dependent on the amount of synchronous generation due to their superior reactive power capability compared to RES, which requires further analysis.\",\n \"Our future work will focus on designing market mechanisms for large scale aggregation of distributed energy resources to enable the participation of prosumers in power system operation.\"\n ],\n [\n \"MILP Formulation of the Lower-level KKT Optimality Conditions\",\n \"For the sake of completeness, we provide the complete optimization problem below.\",\n \"The upper-level problem (REF )-() remains the same.\",\n \"The price-anticipatory prosumer behavior is captured in constraint (), defined as the solution to the lower-level prosumer aggregation problem.\",\n \"To be able to incorporate it in the upper-level problem, we first need to convert it into a set of MILP constraints.\",\n \"We do that by using the optimality (KKT) conditions of the underlying optimization problem.\",\n \"The KKT conditions follow from the associated Lagrangian: $\\\\mathcal {L}(\\\\textbf {x}, \\\\lambda , \\\\mu ) = f(\\\\textbf {x})+ \\\\lambda ^\\\\top \\\\textbf {g}(\\\\textbf {x}) + \\\\mu ^\\\\top \\\\textbf {h}(\\\\textbf {x}),$ where $\\\\textbf {x} = \\\\lbrace p_{\\\\text{d}}^{\\\\text{flx}}, p_{\\\\text{b}}, e_{\\\\text{b}} \\\\rbrace $ is the decision vector of the lower-level problem; $f(\\\\textbf {x})$ is the objective function (REF ); $\\\\textbf {g}(\\\\textbf {x})$ is the vector of equality constraints (REF )-(); $\\\\textbf {h}(\\\\textbf {x})$ is the vector of inequality constraints ()-(); $\\\\lambda ^\\\\top = \\\\lbrace \\\\lambda _{h}^{m,p}, \\\\lambda _{h}^{m,e} \\\\rbrace $ is the vector of Lagrange multipliers associated with the equality constraints, and $\\\\mu ^\\\\top = \\\\lbrace \\\\mu _{h}^{\\\\text{flx},m}, \\\\mu _{h}^{m,\\\\underline{p}}, \\\\mu _{h}^{m,\\\\overline{p}}, \\\\mu _{h}^{m,\\\\underline{e}},\\\\mu _{h}^{m,\\\\overline{e}}\\\\rbrace $ is the vector of Lagrange multipliers associated with the inequality constraints.\",\n \"The KKT optimality conditions consist of: primal feasibility (REF )-() with the associated Lagrange multipliers given in parentheses: $& \\\\qquad \\\\qquad p_{\\\\text{d},h}^{\\\\text{flx},m} + p_{\\\\text{pv},h}^{m} - p_{\\\\text{b},h}^{m} - p_{\\\\text{d},h}^{\\\\text{u},m} = 0, &( \\\\lambda _{h}^{m,p} ) \\\\\\\\& \\\\qquad \\\\qquad e_{\\\\text{b},h}^{m} - \\\\eta _\\\\text{b}^m e_{\\\\text{b},h-1}^{m} - p_{\\\\text{b},h}^{m} = 0, & ( \\\\lambda _{h}^{m,e} ) \\\\\\\\& \\\\qquad \\\\qquad -p_{\\\\text{d},h}^{\\\\text{flx},m} \\\\le 0, &( \\\\mu _{h}^{\\\\text{flx},m} ) \\\\\\\\& \\\\qquad \\\\qquad \\\\underline{p}_{\\\\text{b}}^{m} - p_{\\\\text{b},h}^{m} \\\\le 0, &( \\\\mu _{h}^{m,\\\\underline{p}} ) \\\\\\\\& \\\\qquad \\\\qquad p_{\\\\text{b},h}^{m} - \\\\overline{p}_{\\\\text{b}}^{m} \\\\le 0, &( \\\\mu _{h}^{m,\\\\overline{p}} ) \\\\\\\\& \\\\qquad \\\\qquad \\\\underline{e}_{\\\\text{b}}^{m} -e_{\\\\text{b},h}^{m} \\\\le 0, &( \\\\mu _{h}^{m,\\\\underline{e}} ) \\\\\\\\& \\\\qquad \\\\qquad e_{\\\\text{b},h}^{m} -\\\\overline{e}_{\\\\text{b}}^{m} \\\\le 0, &( \\\\mu _{h}^{m,\\\\overline{e}} ) $ dual feasibility: $& \\\\mu _{h}^{\\\\text{flx},m}, \\\\mu _{h}^{m,\\\\underline{p}}, \\\\mu _{h}^{m,\\\\overline{p}}, \\\\mu _{h}^{m,\\\\underline{e}}, \\\\mu _{h}^{m,\\\\overline{e}} \\\\ge 0,$ stationarity (REF )-(): $&\\\\frac{\\\\partial \\\\mathcal {L}}{\\\\partial p_{\\\\text{d},h}^{\\\\text{flx},m}} = 1 + \\\\lambda _{h}^{m,p} - \\\\mu _{h}^{\\\\text{flx},m} = 0, \\\\\\\\&\\\\frac{\\\\partial \\\\mathcal {L}}{\\\\partial p_{\\\\text{b},h}^{m}} = - \\\\lambda _{h}^{m,p} - \\\\lambda _{h}^{m,e} - \\\\mu _{h}^{m,\\\\underline{p}} + \\\\mu _{h}^{m,\\\\overline{p}} = 0,\\\\\\\\&\\\\frac{\\\\partial \\\\mathcal {L}}{\\\\partial e_{\\\\text{b},h}^{m}} = \\\\lambda _{h}^{m,e} - \\\\eta _\\\\text{b}^m \\\\lambda _{h+1}^{m,e} - \\\\mu _{h}^{m,\\\\underline{e}} + \\\\mu _{h}^{m,\\\\overline{e}} = 0,$ complementary slackness (REF )-(): $&p_{\\\\text{d},h}^{\\\\text{flx},m} \\\\mu _{h}^{\\\\text{flx},m} = 0, \\\\\\\\&(-\\\\underline{p}_{\\\\text{b}}^{m} + p_{\\\\text{b},h}^{m}) \\\\mu _{h}^{m,\\\\underline{p}} = 0, \\\\\\\\&(-p_{\\\\text{b},h}^{m} + \\\\overline{p}_{\\\\text{b}}^{m}) \\\\mu _{h}^{m,\\\\overline{p}} = 0, \\\\\\\\&(-\\\\underline{e}_{\\\\text{b}}^{m} + e_{\\\\text{b},h}^{m}) \\\\mu _{h}^{m,\\\\underline{e}} = 0, \\\\\\\\&(-e_{\\\\text{b},h}^{m} + \\\\overline{e}_{\\\\text{b}}^{m}) \\\\mu _{h}^{m,\\\\overline{e}} = 0.\",\n \"$ Complementary slackness conditions (REF )-() are bilinear, however they can be linearized by introducing one binary variable $b$ and a sufficiently large and positive constant $M$ resulting in two constraints per complementarity slackness condition ()-().\",\n \"Given that the KKT conditions are sufficient and necessary for optimality, the bi-level constraint () can be replaced with mixed integer linear constraints (REF )-(): $& p_{\\\\text{d},h}^{\\\\text{flx},m} + p_{\\\\text{pv},h}^{m} - p_{\\\\text{b},h}^{m} - p_{\\\\text{d},h}^{\\\\text{u},m} = 0, \\\\\\\\& e_{\\\\text{b},h}^{m} - \\\\eta _\\\\text{b}^m e_{\\\\text{b},h-1}^{m} - p_{\\\\text{b},h}^{m} = 0 ,\\\\\\\\& 1 - \\\\mu _{h}^{\\\\text{flx},m} + \\\\lambda _{h}^{m,p} = 0,\\\\\\\\& - \\\\lambda _{h}^{m,p} - \\\\mu _{h}^{m,\\\\underline{p}} + \\\\mu _{h}^{m,\\\\overline{p}} - \\\\lambda _{h}^{m,e} = 0,\\\\\\\\& - \\\\mu _{h}^{m,\\\\underline{e}} + \\\\mu _{h}^{m,\\\\overline{e}} + \\\\lambda _{h}^{m,e} - \\\\eta _\\\\text{b}^m \\\\lambda _{h-1}^{m,e} = 0,\\\\\\\\& \\\\underline{p}_{\\\\text{b}}^{m} - p_{\\\\text{b},h}^{m} \\\\le 0, \\\\\\\\& p_{\\\\text{b},h}^{m} -\\\\overline{p}_{\\\\text{b}}^{m} \\\\le 0, \\\\\\\\& \\\\underline{e}_{\\\\text{b}}^{m} - e_{\\\\text{b},h}^{m} \\\\le 0 , \\\\\\\\& e_{\\\\text{b},h}^{m} -\\\\overline{e}_{\\\\text{b}}^{m} \\\\le 0 , \\\\\\\\& p_{\\\\text{d},h}^{\\\\text{flx},m} - M{^\\\\text{flx}} b_{h}^{\\\\text{flx},m} \\\\le 0, \\\\\\\\& \\\\mu _{h}^{\\\\text{flx},m} - M^\\\\text{flx} (1-b_{h}^{\\\\text{flx},m}) \\\\le 0,\\\\\\\\& p_{\\\\text{b},h}^{m} - M^{\\\\underline{p}} b_{h}^{m,\\\\underline{p}} \\\\le 0,\\\\\\\\& \\\\mu _{h}^{m,\\\\underline{p}} - M^{\\\\underline{p}} (1-b_{h}^{m,\\\\underline{p}}) \\\\le 0,\\\\\\\\& -p_{\\\\text{b},h}^{m} -M^{\\\\overline{p}} b_{h}^{m,\\\\overline{p}} \\\\le 0,\\\\\\\\& \\\\mu _{h}^{m,\\\\overline{p}} - M^{\\\\overline{p}} (1-b_{h}^{m,\\\\overline{p}}) \\\\le 0 ,\\\\\\\\& e_{\\\\text{b},h}^{m} - M^{\\\\underline{e}} b_{h}^{m,\\\\underline{e}} \\\\le 0,\\\\\\\\& \\\\mu _{h}^{m,\\\\underline{e}} - M^{\\\\underline{e}} (1-b_{h}^{m,\\\\underline{e}}) \\\\le 0,\\\\\\\\& -e_{\\\\text{b},h}^{m} - M^{\\\\overline{e}} b_{h}^{m,\\\\overline{e}} \\\\le 0,\\\\\\\\& \\\\mu _{h}^{m,\\\\overline{e}} -M^{\\\\overline{e}} (1-b_{h}^{m,\\\\overline{e}}) \\\\le 0,$ which adds the following additional decision variables to the upper-level problem (REF )-(): $& b_{h}^{\\\\text{flx},m},b_{h}^{m,\\\\underline{p}},b_{h}^{m,\\\\overline{p}},b_{h}^{m,\\\\underline{e}},b_{h}^{m,\\\\overline{e}} \\\\in \\\\lbrace 0,1\\\\rbrace ,\\\\\\\\& p_{\\\\text{b},h}^{m}, \\\\lambda _{h}^{m,p}, \\\\lambda _{h}^{m,e} \\\\in \\\\mathbb {R}, \\\\\\\\& p_{\\\\text{d},h}^{\\\\text{flx},m}, e_{\\\\text{b},h}^{m}, \\\\mu _{h}^{\\\\text{flx},m}, \\\\mu _{h}^{m,\\\\underline{p}}, \\\\mu _{h}^{m,\\\\overline{p}}, \\\\mu _{h}^{m,\\\\underline{e}},\\\\mu _{h}^{m,\\\\overline{e}} \\\\in \\\\mathbb {R}_+.$ The resulting optimization problem is an MILP that can be solved efficiently with off-the-shelf solvers.\",\n \"[Figure: NO_CAPTION [Figure: NO_CAPTION [Figure: NO_CAPTION [Figure: NO_CAPTION [Figure: NO_CAPTION Prof. Hill is a Fellow of the Society for Industrial and Applied Mathematics, the Australian Academy of Science, and the Australian Academy of Technological Sciences and Engineering.\",\n \"He is also a Foreign Member of the Royal Swedish Academy of Engineering Sciences.\"\n ]\n]"},"id":{"kind":"string","value":"1605.05833"}}},{"rowIdx":1866683,"cells":{"text":{"kind":"list like","value":[["A Robust Measurement of the Mass Outflow Rate of the Galactic Outflow\n from NGC 6090"],["Abstract To evaluate the impact of stellar feedback, it is critical to estimate the mass outflow rates of galaxies.","Past estimates have been plagued by uncertain assumptions about the outflow geometry, metallicity, and ionization fraction.","Here we use Hubble Space Telescope ultraviolet spectroscopic observations of the nearby starburst NGC 6090 to demonstrate that many of these quantities can be constrained by the data.","We use the Si~{\\sc IV} absorption lines to calculate the scaling of velocity (v), covering fraction (C$_f$), and density with distance from the starburst (r), assuming the Sobolev optical depth and a velocity law of the form: v$~\\propto(1 -\\mathrm{R}_\\mathrm{i}/\\mathrm{r} )^\\beta$ (where R$_\\mathrm{i}$ is the inner outflow radius).","We find that the velocity ($\\beta$=0.43) is consistent with an outflow driven by an r$^{-2}$ force with the outflow radially accelerated, while the scaling of the covering fraction ($C_f \\propto \\mathrm{r}^{-0.82}$) suggests that cool clouds in the outflow are in pressure equilibrium with an adiabatically expanding medium.","We use the column densities of four weak metal lines and CLOUDY photoionization models to determine the outflow metallicity, the ionization correction, and the initial density of the outflow.","Combining these values with the profile fitting, we find R$_\\mathrm{i}$ = 63 pc, with most of the mass within 300~pc of the starburst.","Finally, we find that the maximum mass outflow rate is 2.3~M$_\\odot$ yr$^{-1}$ and the mass loading factor (outflow divided by the star formation rate) is 0.09, a factor of 10 lower than the value calculated using common assumptions for the geometry, metallicity and ionization structure of the outflow."],["INTRODUCTION","Strangely, most of the gas within a galaxy is not near stars , [2], , , , .","The circum-galactic medium extends to radii greater than 150 kpc, is metal rich, and spans a range of temperatures , , .","Further, this reservoir of gas is massive, containing up to three times more mass than gas within discs .","Even though the gas extends more than 150 kpc from the stellar disc, the metal enrichment implies that the gas originated within the stellar disc.","What could loft so much mass out of discs?","High-mass stars inject energy and momentum into the ISM through high energy photons, cosmic rays, and supernovae, together commonly called stellar feedback , [63], [12], [35], , , [23], , [39], [40], [46], [47], [9].","If the stellar feedback is highly concentrated, then the combined energy and momentum drives gas out of star forming regions into a galactic outflow [35], [36], , [21].","Galactic outflows may enrich the large reservoir of cirum-galactic gas, while also removing metals from low-mass galaxies to create the mass-metallicity relation , [26], , [3], , [17], [15].","However, accurate mass outflow rates are required to constrain these relations.","Observations of galactic outflows are challenging.","First, galactic outflows are multi-phase, with emission from hot X-ray emitting plasma [31], , , to ionized hydrogen [56], [6], , , , [4], and even cold molecular gas , [62], [55].","Second, outflows are diffuse, and probing outflows with emission lines is challenging and limited to only the local universe.","Recent studies use optical [36], [59], , [11], near UV , [60], , [19], [49], [22], and far UV absorption lines , , , [32], [54], [13], [38], , [14] to study the diffuse gas within outflows.","While these studies have hinted at the role outflows play in removing gas from the discs of galaxies, the mass outflow rate calculations are plagued by uncertain assumptions.","To calculate the mass outflow rate from absorption lines, an observed column density and velocity are converted into a total mass outflow rate.","The first step converts the observed ion density to a total Hydrogen density through an assumed metallicity and ionization correction , [38].","The Hydrogen column density is then converted into a total Hydrogen mass through an assumed geometry, typically a thin shell.","These studies make four key assumptions: (1) that the absorption lines are optically thin (2) that the measured ion is the dominant ionization state, and there is no ionization correction (3) that the metallicity of the outflow is consistent with the ISM of the galaxy and (4) that the outflow is a spherical shell with a radius of 5 kpc , .","Recent observations question these assumptions.","Ionization modelling from [14] find that galactic outflows are photoionized, with the dominant ionization state, Si III, at an ionization potential near 25 eV, with only 1-5% of the gas in the neutral phase.","For neutral ions like Na I, neglecting the ionization correction can underestimate the total Hydrogen column density by a factor of 4 , [14].","Additionally, these photoionization models predict that the outflows are metal rich, and the assumption that the outflow has the metallicity of the ISM underpredicts the total Hydrogen density.","These large ionization fractions also imply that the outflow is relatively close to the ionizing source.","Stars generate outflows at distances of super-star clusters, 20-40 pc [63], , [12], but metal-enriched gas is also observed out to 10 kpc from the starburst , , , .","Even though the outflow extends over this enormous distance, studies typically calculate the mass outflow rates as if the entire mass is in a thin shell with a radius of 5 kpc .","Here, we present a new analysis of the mass outflow rate of the nearby galaxy NGC 6090 using Hubble Space Telescope ultraviolet spectroscopy.","We calculate the mass outflow rate by first measuring the distance, metallicity, and ionization fraction of the outflow.","In data we describe the data reduction, stellar continuum fitting, and measurement of the outflow properties.","We then fit for how the optical depth and covering fraction scales with velocity, using a Sobelev optical depth, a $\\beta $ velocity law, and a power-law density scaling (beta).","We apply the ionization models of [14] to the outflow of NGC 6090 to determine the metallcity (1.61 Z$_\\odot $ ), total Hydrogen density (18.73 cm$^{-3}$ ), and the ionization fractions of the outflow (ion).","In ionstruct we consider the implications of the ionization models and the metal enriched outflows, and then calculate the initial radius of the outflow (63 pc).","Using these derived quantities, we examine the covering fraction (cf), velocity (velocity), and density (density) laws with radius.","Since these relationships vary with velocity, we find that the mass outflow rate of the galaxy also varies with velocity, with a maximum mass outflow rate of 2.3 M$_\\odot $ yr$^{-1}$ .","This mass outflow rate is 10 times smaller than it would be if we used previous assumptions, and only 9% of the star formation rate of the galaxy.","In this paper we use $\\Omega _M = .28$ , $\\Omega _\\Lambda = .72$ and H$_0$ = 70 km s$^{-1}$ Mpc$^{-1}$ [43]."],["DATA","NGC 6090 is a massive star forming galaxy at a distance of 128 Mpc (see tab:ngc6090), with no indication of AGN contamination from the optical emission lines (see tab:ngc6090) and the WISE colors.","The galaxy is metal rich, with a log(O/H)+12 of 8.77 .","The galaxy is in the intermediate stages of a massive merger, as two distinct nuclei and large extend tidal tails are seen in the UV (fig:image) and optical images .","The Cosmic Origins Spectrograph [30] on the Hubble Space Telescope observed NGC 6090 during Cycle 18 [54].","The full data reductions are given in [13], and here we summarize the steps taken to measure the optical depths, covering fractions, and column densities of the outflow.","First, we describe the data reductions (reduc) and the stellar continuum fitting (cont).","We then explore the effects of the Si II fine structure and resonance emission lines (finestruct).","Finally, the optical depths, covering fractions, and column densities are measured in measure."],["Data Reduction","[54] observed NGC 6090 with the G130M grating for a total integration time of 8096 s. We downloaded the spectra from MAST and processed the data through the CalCOS pipeline, version 2.20.1.","The individual exposures were aligned and co-added following the methods outlined in .","The spectrum was deredshifted using the redshift from .","The flux was normalized to the median flux in a line free region between 1310-1320 Å in the restframe, the wavelength array was then binned by 5 pixels (10 km s$^{ -1}$ at 1400Å), and smoothed by 3 pixels.","The set-up of the G130M grating affords continuous wavelength coverage between 1125-1425 Å, in the restframe.","Our fully processed data have a median signal-to-noise ratio of 12 per pixel after rebinning."],["Stellar Continuum Fitting","Metal absorption in stellar atmospheres contaminates the ISM absorption profiles.","To remove the stellar contribution, we fit the observed spectrum with a linear combination of STARBURST99 simple stellar populations models [51], [53].","To match the resolution of the observations, and to avoid stellar libraries with Milky Way contamination, we use the fully theoretical Geneva models, with high-mass loss [64], computed using the WM-basic method [51], [53].","A linear combination of different age stellar populations is required to simultaneously recover the Si III photospheric lines near 1295 Å (signatures of B stars) and the Si IV and N V P-Cygni profiles (signatures of O stars).","Since O and B stars dominate the FUV stellar continuum, we include STARBURST99 models of ten ages between 1-20 Myr, and use MPFIT [58] to determine the linear combination of these ten models.","Five ages contribute to the stellar continuum of NGC 6090 (1, 3, 4, 5, 20 Myr), but the 4 and 5 Myr models account for 84% of the FUV light, for a weighted stellar age of 4.48 Myr (see tab:ngc6090).","The stellar continuum fit is shown in the upper panel of Figure 2 in [14].","This figure shows that the STARBURST99 models nicely reproduce the stellar P-Cygni features, and provide a continuum level for the spectrum.","Both the stellar age and metallicity determine the number of ionizing photons emitted by high-mass stars [51].","Accordingly, we fit for both the metallicity and stellar age.","STARBURST99 models have five stellar continuum metallicities – 0.05, 0.2, 0.4, 1.0, 2.0 Z$_\\odot $ – and we determine that the stellar metallicity of NGC 6090 is solar using a $\\chi ^2$ test (tab:ngc6090).","During the fitting, we include continuum reddening using a Calzetti extinction law [10], and find that the stellar continuum is extincted by E(B-V) = 0.316 mags, after accounting for foreground Milky Way extinction .","We cross-correlate the fitted stellar continuum with the observed spectrum to establish the zero-velocity of the stellar continuum.","Finally, we remove the stellar continuum by dividing the observed spectrum by the STARBURST99 model.","Strong Milky Way absorption and geocoronal emission lines are blueshifted from the metal lines of NGC 6090.","However, the Milky Way lines are masked during the stellar population fitting to avoid contaminating the fit.","Since the galaxy non-uniformly fills the COS aperture (see fig:image), the spectral resolution is degraded.","The COS spectral resolution varies from 20 km s$^{-1}$ for a point source to 200 km s$^{-1}$ for a uniformly filled aperture [27].","We fit the line width of the Milky Way absorption features, and find that the effective spectral resolution of the spectrum is 48 km s$^{ -1}$ .","fig:profile gives the metal absorption profiles from the fully reduced, stellar continuum normalized spectra.","There is a weak C I 1277Å Milky Way absorption line near the N V 1243 Å transition and the stellar continuum is slightly underestimated.","While we fit, and remove this weak C I feature, imperfect subtraction creates residual emission, contaminating the continuum near the weaker N V 1243 Å line.","Therefore, we do not draw conclusions from the N V profiles."],["Fine Structure Emission Lines","Before we measure the outflow properties from the metal absorption lines, we must understand how resonance emission lines affect the absorption profiles , , , .","Continuum photons excite ground state electrons, and the electron then transitions into a lower energy state by emitting a photon with an energy of the difference between the two states.","If the electron transitions back to the ground state through the same path as it was excited, it is called resonance emission.","Resonant photons have the same restframe wavelength as the absorption and, depending on the geometry of the outflow, the resonance emission can overlap in velocity space with the absorption.","This effectively reduces the measured absorption profile.","The amount, and the velocity, of the infilling can be estimated if fine structure splits the ground state.","The electron's angular momentum splits the ground state into multiple levels, allowing the electron to transition through either resonance or fine structure emission (fine structure emission is normally denoted by a $^\\ast $ , i.e.","as Si II$^\\ast $).","The probability of transitioning by a specific path is given as $\\mathrm {P}_\\mathrm {i} = \\mathrm {A}_\\mathrm {i} / \\Sigma \\mathrm {A}_\\mathrm {i}$ , where A$_i$ is the Einstein A coefficient corresponding to level i, and the summation is over all possible levels.","Therefore, we use the observed fine structure emission to estimate an upper limit on the in-filling of resonance profile.","There are four Si II$^\\ast $ lines in the observed wavelength regime at 1194, 1197, 1265, and 1309Å, however the 1265 line is blended with the O I geocoronal emission line.","The 1194Å line has the largest probability of fine structure emission (84% of the 1190Å absorption is re-emitted as 1194Å Si II$^\\ast $), therefore this line predicts the maximum amount of resonance emission.","The 1194 Si II$^\\ast $ line is the strongest of the four Si II$^\\ast $ lines in the bandpass, but the equivalent width is only $-0.15$ Å, a small fraction of the 1.74 Å Si II 1190 Å equivalent width.","Further, the line is redshifted by +132 km s$^{ -1}$ , with emission only extending to -80 km s$^{ -1}$ .","Below, we chiefly use the Si IV doublet, which does not have fine structure emission lines in the wavelength regime.","The Si II and Si IV profiles have similar velocity profiles [14], allowing for the Si II$^\\ast $ to be used as a proxy for the Si IV emission.","To avoid possible contributions from resonance emission, we exclude velocities greater than -80 km s$^{-1}$ when fitting the profiles in beta."],["Equivalent Width, Optical Depth, Covering Fraction, and Column Density","Once we have reduced the data, removed the stellar continuum, and explored the impact of the resonance emission features, we measure the properties of the outflow.","Here, we describe the outflow with four parameters: equivalent width (W), optical depth ($\\tau $ ), covering fraction (C$_f$ ), and column density (N).","W is measured by integrating $1-F_\\mathrm {o}$ (the observed continuum normalized flux) over the velocity ranges given in column three of tab:obs.","As discussed in [14], the maximum velocity of each transition strongly depends on the strength (the W) of the transition, not the ionization state of the transition.","The W errors are computed by bootstrapping the observed flux with the flux error array 1000 times, and the standard deviation is calculated from the resultant W distributions.","The equivalent width ratios of doublets (and triplets) can be used to diagnose the saturation of the transitions.","With an observed doublet ratio of 1.83 (and an expected ratio of 2), the Si IV 1402Å and 1392Å lines are the only doublet that does not suffer from severe saturation effects.","We also use the column densities of the weak S II 1250Å, Si II 1304Å, and O I 1302Å lines during the ionization modelling (see ion).","C$_f$ is the fraction of the continuum source covered by the foreground absorbing gas (see for a thorough description of the interpretation of C$_f$ ).","At low $\\tau $ , C$_f$ and $\\tau $ are degenerate because both impact the depth of the absorption profile.","However, this degeneracy can be broken by solving the radiative transfer equation for the C$_f$ and optical depths of the lines [33].","Here, we use the Si IV doublet because it does not suffer from strong saturation effects.","The covering fraction is calculated in each pixel to give a velocity (v) resolved C$_f$ profile as [33] $C_f(\\mathrm {v}) = \\frac{\\mathrm {F}_\\mathrm {W}(\\mathrm {v})^2-2\\mathrm {F}_\\mathrm {W}(\\mathrm {v}) + 1}{\\mathrm {F}_\\mathrm {S}(\\mathrm {v})-2\\mathrm {F}_\\mathrm {W}(\\mathrm {v})+1}$ where F$_\\mathrm {W}$ is the continuum normalized flux of the weaker doublet line (Si IV 1402Å), and F$_\\mathrm {S}$ is the continuum normalized flux of the stronger doublet line (Si IV 1394).","In this equation we use the fact that the Si IV 1402 line has half the oscillator strength of the 1392 line.","We set unphysical C$_f$ values that are less than zero to zero.","We also calculate the velocity resolved $\\tau $ as $\\tau (\\mathrm {v}) = \\ln \\left(\\frac{C_f(\\mathrm {v})}{C_f(\\mathrm {v})+F_\\mathrm {W}(\\mathrm {v})-1}\\right)$ We bootstrap the errors of C$_f$ and $\\tau $ , similar to the W errors above.","Finally, we calculate the integrated column density of each transition using the apparent optical depth method .","Since some of the weak lines are singlets, we cannot use eq:cf to calculate the column density, rather we have to assume that the lines are fully covered.","Below, we find that this assumption is fair for the line centers, which have most of the column density.","The integrated column density is calculated as $\\mathrm {N} = \\frac{3.77 \\times 10^{14}~\\text{cm}^{-2}}{\\lambda [\\text{Å}] f} \\int \\text{ln}\\frac{1}{F_\\mathrm {o}(\\mathrm {v})} \\mathrm {dv}$ where f is the oscillator strength [14], and the total N is calculated by integrating over the velocity interval given in tab:obs.","With the measured properties of the outflow, we now study how the Si IV 1402Å optical depth and covering fraction evolve with velocity."],["PROFILE FITTING","In fig:profile the metal absorption lines are broad ($\\sim 175$ km s$^{ -1}$ ) and blueshifted by $\\sim -150$ km s$^{ -1}$ .","Under the Sobolev approximation, these profiles are produced by an expanding medium, that has a steep velocity gradient (dv/dr) but a narrow intrinsic line profile, such that the expanding medium broadens the line profile by absorbing and scattering continuum photons , [50], , .","The Sobolev optical depth of the Si IV 1402Å profile is given by $\\tau \\left(\\mathrm {v}\\right) = \\frac{\\pi e^2}{\\mathrm {mc}} f \\lambda ~\\mathrm {n}_\\mathrm {4}\\left(\\mathrm {r}\\right) \\frac{\\mathrm {dr}}{\\mathrm {dv}}$ where $f$ is the oscillator strength (0.255 for Si IV), $\\lambda $ is the rest wavelength, and n$_\\mathrm {4}$ is the number density of the Si IV gas at a given radius (r).","To simplify the calculation, we introduce normalized quantities by dividing the velocity by the maximum velocity ($w = \\mathrm {v}/\\mathrm {v}_\\infty $ , v$_\\infty = -399$ km s$^{-1}$ for Si IV 1402Å) and the radius by the inner radius ($x = \\mathrm {r}/\\mathrm {R}_\\mathrm {i}$ ), which we measure in ionstruct.","v$_\\infty $ is calculated as the velocity at which the absorption reaches 90% of the continuum, similar to the method outlined in [13].","We must model how n$_\\mathrm {4}$ is distributed with radius.","Previous studies assume that the density is either constant with velocity [60], , or that the density follows the continuity equation , .","However, to remain general we assume that the density follows a power-law scaling with radius, such that $\\mathrm {n}_\\mathrm {4}(x) = \\mathrm {n}_\\mathrm {4,0} x^\\alpha $ where n$_\\mathrm {4,0}$ is the Si IV density at the initial radius, R$_\\mathrm {i}$ .","Placing this into eq:sobelv gives the relation $\\tau (x) = \\frac{\\pi e^2}{\\mathrm {mc}} f \\lambda _\\mathrm {0} \\frac{\\mathrm {R}_\\mathrm {i}}{\\mathrm {v}_\\infty } \\mathrm {n}_\\mathrm {4,0} x^\\alpha \\frac{dx}{dw} = \\tau _0 x^\\alpha \\frac{\\mathrm {d}x}{\\mathrm {d}w}$ where we have combined the constants into a single constant, $\\tau _0$ , that represents the maximum optical depth.","Additionally, we derive the covering fraction of the Si IV absorption (C$_f$ ; see eq:cf).","As we discuss in cf, the covering fraction is the proportion, expressed as a percentage, of the starburst area covered by the outflow at a given radius.","We assume that the covering fraction scales as a power-law with distance (as physically motivated in cf), such that $C_f(w) = C_{f} (\\mathrm {R}_\\mathrm {i}) \\left(\\frac{\\mathrm {r}}{\\mathrm {R}_\\mathrm {i}}\\right)^\\gamma = C_{f} (\\mathrm {R}_\\mathrm {i}) x^\\gamma $ Now we have related the two observables of the profile to the distance from the starburst.","eq:sobelvnorm and eq:cffit are given in terms of distance, while we measure the parameters in terms of velocity.","Many of the physical mechanisms for driving outflows scale the velocity with radius as a $\\beta $ velocity law [50], such that $\\mathrm {v} = \\mathrm {v}_\\infty \\left(1-\\frac{\\mathrm {R}_\\mathrm {i}}{\\mathrm {r}}\\right)^{\\beta }$ Substituting for w and x, and inverting this relation gives the normalized radius in terms of the velocity as $x = \\frac{1}{1-w^{1/\\beta }}$ Taking the derivative of this gives the velocity gradient in terms of the normalized velocity and the $\\beta $ exponent as $\\frac{\\mathrm {d}x}{\\mathrm {d}w} = \\frac{w^{1/\\beta -1}}{\\beta \\left(1-w^{1/\\beta }\\right)^2}$ We can also use the $\\beta $ -law to derive the density scaling with velocity as $\\mathrm {n}(w) = \\mathrm {n}_\\mathrm {4,0} \\left(\\frac{1}{(1-w^{1/\\beta })}\\right)^{\\alpha }$ producing an $\\tau $ (eq:sobelvnorm) and C$_f$ (eq:cffit) velocity scaling as $\\begin{aligned}\\tau \\left(w\\right) &= \\tau _\\mathrm {0} \\frac{w^{1/\\beta -1}}{\\beta (1-w^{1/\\beta })^{2+\\alpha }}\\\\C_f(w) &= \\frac{C_f (\\mathrm {R}_\\mathrm {i})}{\\left(1-w^{1/\\beta }\\right)^\\gamma }\\end{aligned}$ We then have five parameters to fit for: the maximum optical depth ($\\tau _0$ ), the covering fraction at the initial radius (C$_{f}(\\mathrm {R}_\\mathrm {i})$ ), the exponent of the beta velocity-law ($\\beta $ ), the exponent of the density law ($\\alpha $ ), and the exponent of the covering fraction law ($\\gamma $ ).","Using MPFIT [58], we simultaneously fit for the five parameters from the unsaturated Si IV 1402 Å $\\tau $ and C$_f$ distributions (see fig:beta).","At w below 0.2 ($v > -80$ km s$^{ -1}$ ) zero-velocity absorption and resonance emission may effect the distributions (see finestruct); therefore, we only fit the distributions between w of 0.2 and 1.0 (the points outside this range are coloured gray in fig:beta).","The C$_f$ and $\\tau $ fits are shown in fig:beta, and the fit parameters are given in tab:beta.","In these fits we have binned the flux array by a factor of two (20 km s$^{ -1}$ ) to increase the signal-to-noise ratio, while still Nyquist sampling the observed velocity resolution (see the bar in fig:beta).","These fits reproduce the observed $\\tau $ and C$_f$ distributions, and describe the radial velocity and density laws of the outflow through eq:beta and eq:continuity.","In discussion we discuss some implications of these trends.","To test the covering fraction fits, we can use the Si III and C II profiles.","Since these profiles are much stronger, they are strongly saturated at all velocities.","This means that we cannot measure the C$_f$ with the previous formula (eq:cf), while $\\tau $ cannot be accurately measured at all.","Rather, at large $\\tau $ , the depth of the line is completely set by C$_f$ , and is measured as C$_f(\\mathrm {v}) = 1 - F(\\mathrm {v})$ .","In the lower right panel of fig:beta we plot the C$_f$ for Si III (circles) and C II (triangles).","The Si IV C$_f$ fit is over-plotted on these measurements and provides reasonable agreement with the data.","Table: Table of the quantities derived from the profile fits of the optical depth (τ\\tau ) and covering fraction (C f _f; see eq:cfbeta).","The individual parameters are: Column 1, τ 0 \\tau _0, optical depth at line center; Column 2, β\\beta , the velocity power-law index; Column 3, α\\alpha , the column density power-law index; Column 4, C f (R i )_f (\\mathrm {R}_\\mathrm {i}), the maximum covering fraction; Column 5, γ\\gamma , the covering fraction power-law index."],["IONIZATION MODELLING","In measure we calculate the integrated column density (N) for individual metal ions, but these only describe the individual ion and not the total mass of the outflow.","While Ly-$\\alpha $ absorption traces the neutral Hydrogen in the outflow, we opt to use the metal lines to describe the outflow because (1) the strong P-Cygni Ly-$\\alpha $ profile requires detailed radiative transfer models to constrain and (2) up to 99% of the Hydrogen in the outflow may be ionized [14].","To calculate the total Hydrogen in the outflow, ionization models are required to measure the ionization fraction ($\\chi _\\mathrm {i}$ , or the fraction of the total gas within transition i) and the abundance (N$_\\mathrm {i}$ /N$_\\mathrm {H}$ , or the ratio of an element to the total Hydrogen).","In [14] we find that photoionization models describe the ionization structure of galactic outflows.","These models require the outflow metallicity (Z$_\\mathrm {o}$ ) to be greater than 0.5 Z$_\\odot $ and the ionization parameter (log(U); the ratio of the photon density to outflow density) to be between -1.5 and -2.25.","Additionally, the ionization structure depends both on the spectral energy distribution and the strength of the ionizing source, as well as the density (n$_\\mathrm {0}$ ) of the outflow.","We model the ionization structure using CLOUDY version 13.03 [25].","The CLOUDY models are not velocity resolved profiles, rather we are fitting the measured integrated column densities to the integrated values from CLOUDY.","We assume that the outflow is ionized by the observed stellar continuum, using the best-fit STRABURST99 stellar continuum model from cont.","These STARBURST99 models have an age of 4.478 Myr and a constant SFR of 25.15 M$_\\odot $ yr$^{-1}$ (see tab:ngc6090).","In the CLOUDY models, we assume an expanding spherical geometry, which is not meant to reproduce the velocity profiles, but to account for back-scattering of radiation.","We set the covering fraction to 1.00, the observed value from the Si IV transition, and scale the density with a power-law of -5.72, as measured above.","The CLOUDY models are stopped once the simulations reach 3000 K, which is lower than the default criteria to allow for higher metallicities.","These lower temperatures require a cosmic ray background to be included [42].","We use CLOUDY's default H II abundances, which are similar to the Milky Way values [5], , , , and include an Orion Nebular dust grain distribution [5].","The dust grains account for scattering and destruction of photons, as well as depletion of metals onto grains.","The abundances are scaled by a constant factor to change the outflow metallicity (Z$_\\mathrm {o}$ ).","Similarly, we vary the outflow density at the inner radius (n$_\\mathrm {0}$ ), ionization parameter (U), and the stellar continuum metallicity (Z$_\\mathrm {s}$ ).","We use a Bayesian approach to estimate Z$_\\mathrm {o}$ , n$_\\mathrm {0}$ , U, and Z$_\\mathrm {s}$ [44], [7], [24].","While [14] uses the W ratios to define the ionization structure, the relatively high signal-to-noise ratio and spectral resolution observations of NGC 6090 allow us to use the measured column densities from four weak transitions: O I 1302Å, Si II 1304Å, S II 1250Å, and Si IV 1402Å (see tab:obs).","We do not use the N V column densities due to Milky Way C I1277Å and imperfect continuum subtraction contamination near the N V line (see cont).","We create grids of CLOUDY models, with differing input parameters (tab:gridmod), and tabulate the predicted CLOUDY column densities.","We assume a uniform prior – each model is equally likely – and compute the likelihood function of each set of parameters as $\\mathrm {L} \\propto \\exp (-\\chi ^2/2).$ where $\\chi ^2$ is the chi-squared function using the observed column density (N), the errors on the observed N, and the predicted N from CLOUDY.","For each parameter (Z$_\\mathrm {o}$ , n$_\\mathrm {0}$ , U, and Z$_\\mathrm {s}$ ), the likelihood function is marginalized over the other nuisance parameters and normalized to one to create probability distribution functions (PDFs).","We calculate expectation values and standard deviations of the individual parameters from these PDFs as estimates of the parameters, and their uncertainties.","Figure: Probability density functions (PDFs) from the CLOUDY ionization modelling.","The three parameters are outflow metallicity (top panel), ionization parameter (middle panel), and the total Hydrogen density at the initial radius (bottom panel).","The PDFs are single peaked, with expectation values and standard deviations given in tab:ion.We do the fitting in two iterations.","The first iteration uses a course grid to determine the best Z$_s$ .","Since there are only five stellar continuum metallicities in the fully theoretical STARBURST99 models, we only coarsely estimate the stellar continuum metallicity.","Similar to the COS spectrum fitting (see cont), the Bayesian analysis finds that the 1 Z$_\\odot $ model best fits the ionization structure, assigning 100% of the probability to the 1 Z$_\\odot $ stellar continuum model.","Table: Grid of CLOUDY models used in the Bayesian analysis of the ionization structure.The second iteration uses a more finely spaced grid for Z$_o$ , log(U) and n$_0$ , while only using the solar metallicity stellar continuum model (see tab:gridmod).","In fig:ionpdfs we show the PDFs for the three parameters, with narrow distributions that have expectation values of $Z_\\mathrm {o} = 1.61\\pm 0.08$ Z$_\\odot $ , $n_\\mathrm {0} = 18.73\\pm 2.37$ cm$^{-3}$ , and log(U)$ =-1.85\\pm 0.02$ (see tab:ion).","Using these parameters, we create a best-fit CLOUDY model, producing the ionization fractions of the outflow (see tab:ion).","As discussed in ionstruct, these ionization models have important implications for the metallicity of the outflow, the inner radius of the outflow, and the mass outflow rate.","Figure: The velocity resolved column density plots for the Si IV (black) and O I (red) lines, normalized to their maximum values.","The velocity structure between the neutral (O I) and highly ionized gas (Si IV) stays roughly constant over the velocity range, implying that the ionization structure does not vary substantially with velocity.Table: Table of the quantities derived from the CLOUDY photoionization modelling.","Row (1) is the ionization parameter; (2) is the derived outflow metallicity; (3) is the stellar continuum metallicity; (4) is the Hydrogen density at the base of the outflow; (5) is the ionization fraction of Si IV, or the per cent of the total Si in the Si IV transition; (6) The best-fit Si to H abundance value; (7) The measured Si IV column density; (8) The total Hydrogen column density calculated from the Si IV column density, Si/H abundance, and the Si IV ionization fraction; (9) The inner radius, calculated using eq:r; (10) the H II temperature of the best-fit CLOUDY model.","The errors on χ Si IV \\chi _\\mathrm {Si {\\sc IV}} and log(Si/H) are calculated by producing CLOUDY models of the estimate plus/minus 1σ\\sigma errors of the Z o _\\mathrm {o}, n 0 _0, and log(U)."],["Ionization Structure With Velocity","In the following analysis we use an integrated ionization correction.","However, if the outflow is heated as it is accelerated, then the column density ratio of high to low ionization potential lines should increase with velocity.","fig:colfrac shows the velocity resolved column densities for the Si IV and O I transitions, showing that there is not a coherent variation of the column densities with velocity between $-400$ and 0 km s$^{ -1}$ .","This implies that a majority of the column density arises from a small radius, and the photoionization models are approximately plane-parallel."],["DISCUSSION","Here we discuss various aspects of the profile and ionization fitting from beta and ion.","In ionstruct, we first derive important parameters of the outflow.","We then study the implications for the covering fraction (cf), velocity (velocity), and density (density) radial scalings.","Finally, we combine the various relations to determine the mass outflow rate of NGC 6090 (mout)."],["Ionization Modelling","With the ionization fractions and metallicities derived in ion, we convert the measured Si IV column densities into a total Hydrogen density (see tab:ion).","The total (neutral plus ionized) Hydrogen column density is 20.89$_{-0.03}^{+0.04}$ cm$^{-2}$ , where the uncertainties for each parameter are propagated to compute the uncertainties of the total Hydrogen column density.","The UV continuum extinction provides a complimentary way to estimate the total Hydrogen column density by relating the extinction to the amount of dust through a dust-to-gas ratio.","We use the relation from [52] and [37] to calculate the total Hydrogen column density as $\\mathrm {N}_\\mathrm {H} = \\frac{3.6 \\times 10^{21} \\mathrm {E(B-V)}}{Z} \\text{cm}^{-2}$ where we use a starburst dust attenuation law and the relation for the FUV optical depth from [29] to convert the spectral slope to E(B-V).","Using the extinction measured from the UV continuum in [14] and the outflow metallicity from the ionization structure (see tab:ngc6090 and tab:ion), we expect a log(N$_\\mathrm {H}$ ) of 20.85, in agreement with the derived total N$_\\mathrm {H}$ from the ionization modelling.","The ionization model predicts that 99.3% of the Hydrogen in the outflow is ionized.","These low neutral fractions suggest that the outflow has a H I column density of $5 \\times 10^{18}$ cm$^{-2}$ .","Current radio arrays cannot detect this H I column in emission, but the upcoming SKA will detect these densities with 100 spatial resolution and 5 km s$^{-1}$ spectral resolution in 10-100 hours of integration time .","The outflow metallicity is 1.61 Z$_\\odot $ , 61% and 34% larger than the measured stellar and ISM metallicities (see tab:ngc6090).","Substantial uncertainties on log(O/H) measurements [45] means that the outflow metallicity is at least consistent with the ISM values, and likely enriched compared to the ISM of the galaxy.","Additional metals may reside in a hotter phase not probed by these observations (see below for a discussion of this phase).","Metal enriched outflows are important constraints for models using galactic outflows to explain the mass-metallicity relation , [26], , [17], [15].","For example, to match the mass-metallicity relation of [18] for galaxies with log($M_\\ast $ /M$_\\odot $ ) of 10.7, Figure 8 of suggests that the outflow metallicity is 1.58 Z$_\\odot $ .","The same figure also uses Z$_\\mathrm {o}$ to constrain the scaling of the mass outflow rate divided by the star formation rate (mass-loading factor).","In future studies, we will determine the scaling of the outflow metallicity with stellar mass and star formation rate to better constrain these studies.","The initial radius (R$_\\mathrm {i}$ ) can be determined from the fitted Si IV optical depth (see eq:sobelvnorm).","Solving for R$_\\mathrm {i}$ in terms of the observed maximum Si IV optical depth, we find that $\\mathrm {R}_\\mathrm {i} = \\frac{\\mathrm {mc}}{\\pi e^2} \\frac{1}{f \\lambda } \\frac{\\mathrm {v}_\\infty \\tau _\\mathrm {0}}{\\mathrm {n}_\\mathrm {0} \\chi _\\mathrm {Si~IV} \\mathrm {Si/H}} = 63.4~\\text{pc}$ Using the values from the $\\beta $ -profile fitting (tab:beta) and the ionization modelling (tab:ion).","We have related the initial Si IV density ($n_{4,0}$ ) to the Hydrogen density at the base of the outflow, the abundance, and the ionization corrections from the ionization models ($n_{4,0} = n_\\mathrm {0} \\chi _\\mathrm {Si IV} \\mathrm {Si/H}$ ).","The inner outflow radius and the size of the mass-loading region (maximum extent of about 150 pc; see mout and fig:mout below) is consistent with models of the mass-loading region in M 82 , , and with analytical work from [38], who assume that the inner outflow radius is twice the size of the starburst.","However, what is the physical meaning of the inner outflow radius?","The shredding of supernovae blastwaves by hydrodynamical instabilities is a possible origin for this inner radius.","Supernovae energy builds up in the ISM due to multiple impulsive events , which creates a blastwave that shocks when it encounters the ambient ISM , , , [63], , [46].","The shock compresses surrounding ISM into a thin, dense shell of relatively cool gas that travels outward.","This phase is called the snowplow phase, and the radius depends on the amount of injected energy, the density of the medium, and how much of the energy is radiated away , [63], [16], , [46], with typical values near 20-30 pc [20], [46].","Once the shell forms, Rayleigh-Taylor and Kelvin-Helmholtz instabilities rapidly destroy it, creating many small warm (10$^4$ K) cloudlets [57], [16], [28], [60], , .","The destruction of the blastwave allows the hot interior gas to escape and travel into the halo as a hot wind.","The hot, high-velocity wind, may then accelerate these clouds through ram pressure to the observed velocities [28], [60], .","Therefore, a possible origin for R$_\\mathrm {i}$ is that hydrodynamic instabilities have shredded the dense blastwave, and injected cloudlets into the hot wind.","Finally, the CLOUDY modelling estimates the temperature of the H II in the outflow to be 5470 K (see tab:ion), leading to a pressure of P/k$_\\mathrm {b}$ = $1 \\times 10^5$ K cm$^{-3}$ .","The hot wind is typically assumed to have a temperature of 10$^7$ K [12].","If the observed cloudlets and hot wind are in pressure equilibrium, then the hot wind has a density of 0.01 cm$^{-3}$ at R$_\\mathrm {i}$ , a factor of 10 lower than the density in the [12] model.","However, recent models by [9] show that the density and temperature of this hot wind can greatly vary, depending on the efficiency and mass-loading of the outflow.","Regardless of the origin of the outflow, it must be evenly distributed across the stellar continuum to produce a unity covering fraction (see tab:beta).","However, the external pressure on the outflow decreases as it accelerates out from the starburst.","The change in pressure has a dramatic effect on the size of the cloudlets in the outflow, and in turn their covering fraction.","In the next section we discuss this pressure change, and how it naturally leads to the observed scaling of the covering fraction with velocity."],["Covering Fraction","In beta, we observe that the outflow initially completely covers the background stellar continuum, but as the velocity increases the outflow covers less of the background stars (C$_f$ drops as $x^{-0.8}$ ).","To physically understand this scaling relation, we hypothesize that the absorption arises from cloudlets of gas initially a distance R$_\\mathrm {i}$ from the starburst (see the upper left panel of fig:cfnop).","At R$_\\mathrm {i}$ these clouds are large enough to completely occupy the volume along the line-of-sight to the stars: none of the stellar continuum is transmitted (see lower left panel in fig:cfnop for the face on view at the initial time, t$_0$ ).","However, the starburst imparts energy and momentum to these clouds, accelerating them radially outwards (upper right panel of fig:cfnop).","In the simplest scenario, these clouds retain their size as they move outward, and a gap appears between the clouds (see upper right panel in fig:cfnop).","The gap allows the background stellar continuum to be transmitted, reducing the covering fraction of the stellar continuum.","This is illustrated by the face on view in the lower right panel of fig:cfnop, where background stars become visible in the gaps between the clouds at higher velocities.","This physical picture can be expressed numerically as a ratio of the cloud area to the total surface area at a given radius (r) as [60] $\\frac{C_f(\\mathrm {r})}{C_f(\\mathrm {R}_\\mathrm {i})} = \\frac{\\mathrm {A}_\\mathrm {c} (\\mathrm {r})}{4 \\pi \\mathrm {r}^2} \\frac{4 \\pi \\mathrm {R}_\\mathrm {i}^2}{\\mathrm {A}_\\mathrm {c}(\\mathrm {R}_\\mathrm {i})} = \\frac{\\mathrm {A}_\\mathrm {c} (\\mathrm {r})}{\\mathrm {A}_\\mathrm {c} (\\mathrm {R}_\\mathrm {i})} x^{-2}$ Where A$_c$ is the area of the individual cloudlets and $x = \\mathrm {r}/\\mathrm {R}_\\mathrm {i}$ .","Assuming that the area of the clouds remains constant, the C$_f$ in this scenario scales as x$^{-2}$ , significantly steeper than the observed relation of x$^{-0.8}$ .","Figure: Similar to fig:cfnop, but the clouds are in an external pressure gradient (orange shaded region).","The initial pressure (P e ,0_\\mathrm {e,0}) is greater than the pressure at t 1 _1 (P e ,1_\\mathrm {e,1}).","The cloudlets adiabatically expand to remain in pressure equilibrium with the external medium (see the arrow in the upper left cloud demonstrating it's initial diameter).","Comparing the bottom right panel with the same panel in fig:cfnop, the expanded clouds now cover more area of the starburst at higher velocities.","This slows the drop in C f _f.","The scaling of the C f _f with distance depends on how the external medium changes, either isothermally (see eq:cftheoryiso) or adiabatically (see eq:cftheoryadia).However, this simple model of static clouds is not the most physical.","These cloudlets are likely in pressure equilibrium with an external pressure (P$_\\mathrm {e} = \\mathrm {P}_\\mathrm {c}$ , where P$_\\mathrm {e}$ is the external pressure and P$_\\mathrm {c}$ is the pressure of the cloud), possibly a hot wind (see fig:cfp and the discussion in ionstruct).","Changes in P$_\\mathrm {e}$ change P$_\\mathrm {c}$ , and the clouds expand adiabatically to account for these changes as P$_\\mathrm {c} \\propto V_\\mathrm {c}^{-\\gamma _\\mathrm {c}}$ , where V$_\\mathrm {c}$ is the volume of the clouds and $\\gamma _\\mathrm {c}$ is the adiabatic index (5/3 for monatomic ideal gas).","The larger cloud volume reduces the gap between the individual cloudlets [60].","The lower right panels of fig:cfnop and fig:cfp illustrate that the larger clouds cover more of the background stars than the static clouds do, and C$_f$ falls more slowly with distance.","Approximating the outflow as spherical clouds provides a relation for A$_\\mathrm {c}$ in terms of the volume of the cloudlets as A$_\\mathrm {c} \\propto \\mathrm {V}_\\mathrm {c}^{2/3}$ .","Using this approximation, eq:cfchanges, and the observed C$_f$ (C$_f \\propto x^{-0.8}$ ), the cloud volume changes with distance from the starburst as $\\frac{\\mathrm {V}_\\mathrm {c}(\\mathrm {r})}{\\mathrm {V}_\\mathrm {c}(\\mathrm {R}_\\mathrm {i})} = \\left(\\frac{\\mathrm {A}_\\mathrm {c}(\\mathrm {r})}{\\mathrm {A}_\\mathrm {c}(\\mathrm {R}_\\mathrm {i})}\\right)^{3/2} = \\left(\\frac{C_f(\\mathrm {r})}{C_f(\\mathrm {R}_\\mathrm {i})}x^2\\right)^{3/2} = x^{1.8 \\pm 0.4}$ This demonstrates that the cloud's volume increases with distance from the starburst, but what differential pressure is needed to change the volume as we observe?","Here, we consider two different pressure laws for a mass-conserving external medium: isothermal expansion and adiabatic expansion.","If the external pressure changes isothermally with radius then P$_\\mathrm {e}(\\mathrm {r})/\\mathrm {P}_\\mathrm {e}(\\mathrm {R}_\\mathrm {i}) = x^{-2}$ .","Assuming that the spherical clouds and the isothermal external medium remain in pressure equilibrium (P$_\\mathrm {c} = \\mathrm {P}_\\mathrm {e}$ ), the pressure of the outflowing clouds changes as $\\frac{\\mathrm {P}_\\mathrm {c}(\\mathrm {r})}{\\mathrm {P}_\\mathrm {c}{\\mathrm {R}_\\mathrm {i}}} = \\left(\\frac{\\mathrm {V}_\\mathrm {c}(\\mathrm {r})}{\\mathrm {V}_\\mathrm {c}(\\mathrm {R}_\\mathrm {i})}\\right)^{-\\gamma _\\mathrm {c}} = \\left(\\frac{\\mathrm {A}_\\mathrm {c}(\\mathrm {r})}{\\mathrm {A}_\\mathrm {c}(\\mathrm {R}_\\mathrm {i})}\\right)^{-3\\gamma _\\mathrm {c}/2} = x^{-2}$ Solving for $\\frac{A_\\mathrm {c}(x)}{A_\\mathrm {c}(R_i)}$ in terms of x finds that A$_\\mathrm {c}$ changes as $x^{4/(3\\gamma _\\mathrm {c})}$ .","Placing this into eq:cfchanges gives the C$_f$ scaling of $\\frac{C_f(\\mathrm {r})}{C_f(\\mathrm {R}_\\mathrm {i})} = x^{4/(3\\gamma _\\mathrm {c})} x^{-2} = x^{-1.2}$ For a $\\gamma _\\mathrm {c} = 5/3$ .","If the clouds expand adiabatically while immersed in an isothermal external medium, the covering fraction declines less steeply with height than if the clouds maintain their initial size.","eq:cftheoryiso is still too steep to match the observed scaling.","Another possibility is that the external pressure changes adiabatically.","In this situation P$_\\mathrm {e} \\propto x^{-2\\gamma _\\mathrm {e}}$ , where $\\gamma _\\mathrm {e}$ is the adiabatic index of the external medium.","Following a similar process as above, the covering fraction of an adiabatically expanding outflow in pressure equilibrium with an adiabatically expanding external medium is [60] $\\frac{C_f(\\mathrm {r})}{C_f(\\mathrm {R}_\\mathrm {i})} = x^{4\\gamma _\\mathrm {e}/(3\\gamma _\\mathrm {c}) -2} = x^{-2/3}$ where we have assumed that $\\gamma _\\mathrm {e}$ and $\\gamma _\\mathrm {c}$ are equal to each other (i.e.","both the clouds and external medium are monatomic ideal gases).","Interestingly, since diatomic gas has more degrees of freedom than monatomic gas, diatomic gas requires a larger energy change to remain in pressure equilibrium, as seen by the smaller $\\gamma _\\mathrm {c}$ of diatomic gas (7/5).","Using eq:cftheoryadia, we find that molecular clouds expand more rapidly, and consequently have a more gradually declining C$_f$ with distance, as x$^{-0.4}$ .","This has an important consequence for the survival of molecular clouds in a hot medium.","By increasing the radius of the cloud, it takes longer for a shock wave to propagate across the cloud and dissociate the molecular gas [48], .","This may extend the lifetime of molecular gas entrained in galactic outflows, while the increased C$_f$ may explain the presence of molecular gas at high velocities [62], , [55].","In beta, we find that the covering fraction scales as $C_f~=~1.0~x^{-0.8 \\pm 0.2}$ , in agreement with the adiabatic expansion of cloudlets in an adiabatically expanding external medium (eq:cftheoryadia).","However, it is an interesting theoretical question whether clouds can remain in pressure equilibrium with an external medium and still be accelerated.","Ram pressure introduces a secondary pressure term, but if the clouds are shielded from direct interaction with the hot medium then the pressure of the cloud is set by the external thermal pressure [12].","Further, ablation and ram pressure stripping create a fractal distribution into smaller cloudlets [48], , which may produce the observed change in covering fraction.","The velocity scaling of the covering fraction provides further constraints for simulations of the acceleration of clouds.","The acceleration of the cloudlets out of the star forming region decreases C$_f$ , but how are the clouds accelerated?"],["Velocity Law","The velocity law describes the acceleration of the outflowing clouds with distance.","In beta we find a $\\beta $ velocity profile of $\\mathrm {v(r)}~=~\\mathrm {v}_\\infty ~(1~-~\\frac{\\mathrm {R}_\\mathrm {i}}{\\mathrm {r}})^{0.43 \\pm 0.07}$ .","The outflow initially accelerates rapidly, but the acceleration moderates at large radii (see the black line in fig:vellaw).","Using the initial radius calculated in ionstruct, the outflow reaches 50% (90%) of the maximum velocity in 79 pc (291 pc).","Acceleration much beyond this is not well constrained by the data.","Outflows typically have a \"saw-tooth\" line profile , where the red side of the profile sharply declines and the blue side rises gradually.","The velocity and C$_f$ laws produce these profiles.","Initially, the velocity sharply increases over a short distance, keeping the C$_f$ near unity because C$_f$ scales as x$^{-0.82}$ .","Meanwhile, at higher velocities the clouds travel larger distances per velocity interval, forcing the high-velocity clouds to expand to remain in pressure equilibrium with the adiabatically expanding external medium.","This moderates the decline in C$_f$ , and produces the gentler rising blue portion of the profile.","The observed Si IV 1402Å outflows do not escape the galactic potential.","Following [36], the escape velocity is no more than three times the circular velocity [13], therefore the outflow velocity needs to exceed 651 km s$^{-1}$ for the clouds to escape the potential.","There are two possible explanations for why we do not observe gas escaping the potential: (1) the outflow does eventually exceed the escape velocity but the Si IV density drops below the detection limits at high velocities and we do not observe the escaping gas (see density) or (2) the outflow does not actually escape the galaxy, but rather recycles back into the disc as a galactic fountain .","The latter is consistent with [13], which finds most galaxies with log($M_\\ast $ /M$_\\odot $ ) greater than 10.5 cannot drive outflows faster than their escape velocity, unless they are undergoing a merger.","Observations probing lower density gas, such as Lyman-$\\alpha $ , may constrain whether these outflows are capable of escaping the gravitational potential.","Now we study how these outflows are accelerated.","There are numerous theoretical ways to accelerate galactic outflows: radiation pressure on dust grains , , cosmic rays [23], , and ram pressure of a hot wind on the cloudlets , [16], [28], [60].","Below, we use analytical expressions for how the velocity profile evolves radially to explore which mechanisms could accelerate these outflows.","In each case, we give the scaling of the normalized velocity (w = v/v$_\\infty $ ; where v$_\\infty $ is the maximum velocity) with the normalized radius (x = r/R$_\\mathrm {i}$ ; where R$_\\mathrm {i}$ is the initial radius).","We then scale the analytical expressions to the observed relations to determine the plausibility of each mechanism.","In velsum we summarize the implications for the theoretical profiles."],["Optically Thick Radiation Pressure","Radiation pressure is an attractive way to drive outflows in dusty, vigorously star-forming galaxies: the high luminosity provides a large momentum source , while the large dust optical depth scatters photons multiple times .","give the radial scaling of the velocity for optically thick radiation pressure as: $w = \\sqrt{\\frac{4\\sigma ^2}{\\mathrm {v}_\\infty ^2} (\\frac{L}{L_\\mathrm {E}}-1)~\\text{ln}(x)}$ where $\\sigma $ is the velocity dispersion of the galaxy, L is the luminosity of the galaxy, and L$_\\mathrm {E}$ is the Eddington luminosity.","We fit for the constant value that best matches the observed $\\beta $ -law using MPFIT [58], while excluding velocities less than 0.2.","The fit is shown by the blue dot-dashed line in fig:vellaw.","The optically thick radiation model poorly matches the observations."],["An r$^{-2}$ Force","A second appealing driving mechanism is ram pressure.","Supernovae thermalize ambient gas into a hot wind, which expands adiabatically out of the star forming region.","The hot wind imparts a ram pressure force on the clouds, which depends on the speed and density of the hot wind and the area of the cloud as F$_\\mathrm {ram} = \\rho _\\mathrm {h} \\mathrm {v}_\\mathrm {h}^2 \\mathrm {A}_\\mathrm {c}$ [48], .","F$_\\mathrm {ram} \\propto x^{-2}$ in a simplified case of a mass conserving, adiabatically expanding hot wind.","Ram pressure driving is appealing because the C$_f$ scaling is consistent with clouds being in pressure equilibrium with a adiabatically expanding external medium (cf), a situation that could lead to ram pressure driving.","Including the effects of gravity, calculate the velocity profile of a ram pressure driven outflow as $w(x) = \\sqrt{\\frac{v_\\mathrm {c}^2}{\\mathrm {v}_\\infty ^2} (1-\\frac{1}{x}) -\\frac{4\\sigma ^2}{\\mathrm {v}_\\infty ^2} \\mathrm {ln}(x)}$ Where v$_c$ , the characteristic velocity clouds reach before gravity dominates, is defined as v$_\\mathrm {c} = \\frac{3\\dot{M}_\\mathrm {h} \\mathrm {V}_\\mathrm {h}}{8\\pi \\rho _\\mathrm {c} \\mathrm {R}_\\mathrm {c} \\mathrm {R}_\\mathrm {i}}$ .","Defining A as v$_\\mathrm {c}^2/\\mathrm {v}_\\infty ^2$ and B as $4\\sigma ^2/\\mathrm {v}_\\infty ^2$ , we fit for the values of A (fitted value of 1.10) and B (fitted value of 0.03) that match the observed $\\beta $ velocity law, as shown by the red line in fig:vellaw.","Similarly, shockwaves from supernovae accelerate cosmic rays (CR).","These relativistic particles stream out of the star forming regions along the magnetic fields and exchange momentum with magnetized plasma.","Cosmic rays have various advantages over radiation pressure and ram pressure, including: CRs interact multiple times with the gas, magnetic fields confine CRs to the galaxy making it difficult for CRs to escape without imparting momentum, and CR feedback is independent of the distribution of ISM gas [23], .","derive the force imparted by cosmic rays as $F_\\mathrm {CR} = \\frac{\\kappa _\\mathrm {CR}}{c} \\frac{L_\\mathrm {CR}}{4 \\pi r^2}$ where $\\kappa _\\mathrm {CR}$ is the cosmic ray opacity, and L$_\\mathrm {CR}$ is the cosmic ray luminosity.","This force produces a similar velocity scaling as the ram pressure scaling in eq:ramvel.","Additionally, the scaling of the ram pressure and CR velocity profiles are similar to an optically thin radiation profile .","In fact, eq:ramvel is a general form for any r$^{-2}$ force that opposes gravity.","Therefore, the red dashed line in fig:vellaw corresponds to outflows driven by any r$^{-2}$ force (ram pressure, cosmic rays, or optically thin radiation pressure), and we cannot distinguish these mechanisms from the measured velocity profile."],["Velocity Summary","The ram pressure, cosmic rays, and optically thin radiation pressure velocity profile (any r$^{-2}$ force) matches the observed velocity relation for all observed velocities, while optically thick radiation pressure poorly matches the observed velocity profile (fig:vellaw).","In fig:vellaw, the r$^{-2}$ profile begins to decelerate at a radius of 2.5 kpc (r$_\\text{turn}$ ; marked by the vertical line in fig:vellaw).","However, at these large radii the outflow is now diffuse, and the deceleration of the outflow is challenging to study with the Si IV line profile."],["Density Law","In beta we find the density to scale with the normalized radius as $\\mathrm {n}(x) \\propto x^{-5.72}$ .","In fig:denlaw we show the measured relation for the outflow density with velocity.","At low velocities, the density remains nearly constant because the outflow accelerates over a short distance , but at higher velocities the density precipitously drops.","Stronger transitions probe lower densities than weaker transitions, causing stronger transitions to probe higher velocities than the weaker transitions.","Therefore, stronger transitions have larger measured terminal velocities and larger line widths (because they probe a wider velocity distribution) than the weaker transitions [32], [14].","This density-velocity scaling implies that the outflow traced by Si IV does not conserve mass.","Fitting the $\\tau $ distribution with a mass-conserving flow produces a poor fit because the $\\tau $ distribution declines sharply at high velocity.","In the continuity equation, the mass flux is conserved, such that the density scales as $\\mathrm {n}(w) \\propto \\frac{n_\\mathrm {0}}{x^2v} \\propto \\frac{n_\\mathrm {0}}{v_\\infty }\\frac{(1-w^{1/\\beta })^2}{w}$ In fig:denlaw we compare the mass-conserving density profile (red dashed line) with the observed profile.","Compared to the observed density profile, the density of a mass-conserving flow rapidly decreases at low velocities as the outflow accelerates over small distances, while at higher velocities the conserved density decreases more gradually because the velocity gradient is flatter (see fig:vellaw).","If the Si IV flow is not mass-conserving, what happens to the Si IV gas?","The density could decrease by: (1) increasing the volume of the outflowing clouds while keeping the number of Si IV ions constant or (2) decreasing the number of Si IV ions in the clouds.","eq:volume shows that the volume of the outflowing clouds scales as x$^{1.77}$ , implying that the number of Si IV ions in the outflow must decrease as x$^{-3.95}$ to satisfy the observed density relation.","One possibility is that the outflowing clouds lose mass through ablation, ram pressure stripping, or conduction from the hot wind [48], , , [8].","In this scenario, the hot wind rapidly destroys the clouds and incorporates them into the hot wind, making the mass undetectable through Si IV absorption at higher velocities.","A steep density scaling relation is also seen in the nearby starburst M 82.","[55] probe the surface density profiles of the molecular, neutral and ionized phases of the outflow from M 82.","They find that only the H I follows an n$ \\propto \\mathrm {r}^{-2}$ density law largely because of the large amount of tidal material present at large radii.","However, other surface density profiles along the minor axis decrease more rapidly.","The 70 $\\mu $ m emission, a tracer of warm dust, and CO emission, a tracer of diffuse molecular gas, scale with the distance from the starburst as n$ \\propto \\mathrm {r}^{-4}$ [55], roughly consistent, with the density scaling found here.","The authors use the divergence of the density profile to argue that the outflow of M82 is a galactic fountain, with gas leaving the outflow and recycling back into the disk."],["Mass Outflow Rate","Finally, we combine all of the derived relations to measure the mass outflow rate ($\\dot{M}_o$ = dM$_o$ /dt) of the photoionized galacitc outflow.","The photoionized outflow is a single component of the outflow, and other, unexplored, phases likely contribute to the total mass outflow rate.","Since the hot phase is such low density, this photoionized phases likely dominates the mass budget of the outflow.","Typically the mass outflow rate is calculated assuming the outflow is in a thin spherical shell as $\\dot{M}_\\mathrm {o} = \\Omega {C}_{f} \\mu \\mathrm {m}_\\mathrm {p} \\mathrm {N}_\\mathrm {H} \\mathrm {r} \\mathrm {v}_\\mathrm {cen}$ where $\\mu m_\\mathrm {p}$ is 1.4 times the mass of the proton, $\\Omega $ is the angular covering fraction of the wind and v$_\\mathrm {cen}$ is the centroid velocity of the outflow (132 km s$^{ -1}$ ).","Previous studies assume a radius of 5 kpc, a full opening angle of 140$^\\circ $ (and $\\Omega = 3.11\\pi $ steradians), solar metallicity, and no ionization correction , [59], , .","Using the values from the Si IV line, and these assumptions, we derive a mass outflow rate for NGC 6090 of 7.32 M$_\\odot $ yr$^{-1}$ .","However, previous studies typically use absorption lines of cooler gas like Na I, Mg II and Fe II , [59], , [60], .","If we use the Si II 1304 column density, a similar ionization potential to the previous outflow tracers, $\\dot{M}_{\\text{o}}$ rises to 22.6 M$_\\odot $ yr$^{-1}$ .","However, with our tightly constrained physical model of the outflow, now we only have to assume an angular covering fraction to calculate the mass outflow rate.","Above, we derive a radius, a metallicity, and an ionization fraction for the outflows (see tab:ion for values).","If we use the radius of peak optical depth (R$_\\mathrm {p}$ = 72.2 pc, at a w of 0.41), we calculate a total mass outflow rate of 0.81 M$_\\odot $ yr$^{-1}$ , 28 times lower than the $\\dot{M}_{\\text{o}}$ calculated using Si II.","While this calculation uses many of our derived values, it ignores the evolution of these quantities with velocity.","In velocity we find that the outflow is accelerated over short distances, in cf we find that the covering fraction decreases at large velocities, and in density we find that the outflow is not a mass conserving flow, rather it evolves as x$^{-5.7}$ .","All of these impact how mass is distributed in velocity space.","Using the scaling of the covering fraction, radius, and density with velocity, we define the mass outflow rate per velocity as $\\begin{aligned}\\dot{M}_\\mathrm {o}(\\mathrm {r}) &= \\Omega C_{f}(\\mathrm {r}) \\mathrm {v}(\\mathrm {r}) \\rho (\\mathrm {r}) \\mathrm {r}^2 \\\\\\dot{M}_\\mathrm {o}(w)&= \\Omega C_f(\\mathrm {R}_\\mathrm {i}) \\mathrm {v}_\\infty \\mu \\mathrm {m}_\\mathrm {p} \\mathrm {n}_\\mathrm {0} \\mathrm {R}_\\mathrm {i}^2 \\frac{w}{(1-w^{1/\\beta })^{2+\\gamma +\\alpha }} \\\\&= 10.1~\\text{M}_\\odot ~\\text{yr}^{-1} \\frac{w}{(1-w^{1/\\beta })^{2+\\gamma +\\alpha }}\\end{aligned}$ where we use the values from tab:beta and tab:ion for the constants and the exponents, and the only assumed parameter is that $\\Omega $ is 3.11$\\pi $ steradians.","The $\\dot{M}_{\\text{o}}$ relation is not constant with velocity, as typically assumed (fig:mout): $\\dot{M}_{\\text{o}}$ increases rapidly at low velocities as the outflow accelerates, and declines at high velocities as the density and covering fraction of the outflow decline (fig:vellaw and fig:denlaw).","At w of 0.35, the $\\dot{M}_{\\text{o}}$ relation peaks at a value of 2.3 M$_\\odot $ yr$^{-1}$ .","This is a factor of ten times smaller than the Si II value calculated with previous assumptions for geometry, ionization fractions, and metallicities.","Combining $\\dot{M}_{\\text{o}}$ with the derived outflow metallicity, we find that the maximum metal outflow rate is 0.07 M$_\\odot $ yr$^{-1}$ .","This may constrain the impact of outflows on the mass-metallicity relationship , [26], , [3], , [17], [15].","It is typical to normalize the mass outflow rate by the star formation rate to produce the \"mass-loading\" factor .","The global $\\eta $ is 0.09.","This $\\eta $ is smaller than recent measurements from [38], who find an $\\eta $ near 0.3 for galaxies with SFRs near 30 M$_\\odot $ yr$^{-1}$ .","However, [38] use the mean ISM metallicity of the sample (0.5 Z$_\\odot $ ), do not calculate ionization fractions, and use a multiple of the half-light radius (typically 0.5-1 kpc) as the radius of the outflow.","Other studies make similar assumptions about the metallicity and the ionization state, but assume constant outflow radii between 1-5 kpc , [61], .","These studies typically find $\\eta $ between 0.02 and 1 for SFRs and $M_\\ast $ similar to NGC 6090.","The method outlined here does not rely on these uncertain assumptions and allows for these quantities to vary from galaxy to galaxy, reducing the scatter in the measurements.","The relatively small inner outflow radius implies that the local SFR may be more important than the global SFR.","In [13] we calculate the fraction of the total GALEX UV flux within the COS aperture to be 22%, leading to a SFR within the COS aperture of 5.55 M$_\\odot $ yr$^{-1}$ , however this does not account for the significant spatial variation in the IR flux and is only a crude approximation for the local SFR.","Using this local SFR we calculate a local $\\eta $ within the COS aperture of 0.42.","The local and global $\\eta $ values are a factor of 5 and 22 times lower than the redshift independent $\\eta $ predicted by the FIRE simulation [39], [41], , although the high-mass FIRE galaxies at redshift 0 only have upper limits for $\\eta $ .","[34] develop an analytical model that uses the momentum from stellar feedback to regulate star formation and drive outflows.","A requirement of this model is that $\\eta $ depends on the gas fraction and the stellar mass of the host galaxy.","Making approximations for these scalings with redshift, the analytical model predicts the mass-loading for a log($M_\\ast $ /M$_\\odot $ ) galaxy of 10.7 at $z\\approx 0.03$ to be 0.05, in rough agreement with the observed global $\\eta $ .","While the outflow from NGC 6090 is weak compared to the SFR, [34] predict that 10$^9$ M$_\\odot $ galaxies drive outflows with an $\\eta $ of 10 at $z\\sim 0$ .","In a future paper we will explore the scaling of $\\dot{M}_{\\text{o}}$ and $\\eta $ with host galaxy properties to better constrain these types of studies.","Figure: The total mass within the outflow at each velocity, as given by eq:m. The total integrated outflow mass is 7.5×10 5 7.5 \\times 10^5 M ⊙ _\\odot .The total mass in the outflow (M$_o$ ) at a particular velocity is given by $\\begin{aligned}M_\\mathrm {o} (w) &= \\Omega C_f(\\mathrm {r}) \\mu \\mathrm {m}_\\mathrm {p} \\mathrm {n}_\\mathrm {0} \\mathrm {R}_\\mathrm {i}^3 \\left(\\frac{1}{1-w^{1/\\beta }}\\right)^{3+\\alpha +\\gamma } \\\\& = 1.5 \\times 10^{6} M_\\odot \\left(\\frac{1}{1-w^{1/\\beta }}\\right)^{3+\\alpha +\\gamma }\\end{aligned}$ This relation is shown in fig:m. Like the density, the total mass in the Si IV outflow declines with velocity.","The integrated mass between w of 0 and 1 is $7.5 \\times 10^{5}~M_\\odot $ .","While the current mass of outflowing gas is significantly lower than the measured H I mass of 10$^{10.2}$ M$_\\odot $ , if $\\dot{M}_{\\text{o}}$ remains constant over the $\\sim 1$ Gyr time scale of the merger, than the outflow will process (and enrich) 15% of the observed H I in NGC 6090, possibly leading to the metal enriched gas seen in the halos of galaxies , , , ."],["CONCLUSION","Here we measure a physically motivated mass outflow rate ($\\dot{M}_{\\text{o}}$ ) of the nearby starburst NGC 6090.","To calculate $\\dot{M}_{\\text{o}}$ , we first fit the optical depth with a Sobolev optical depth, and the covering fraction with a radial power-law (see eq:cfbeta and fig:beta).","We then calculate the ionization corrections and the metallicity of the outflow using a Bayesian analysis, CLOUDY models, and the measured column densities (see tab:ion and fig:ionpdfs).","The main results of this study are: The ionization model estimates the metallicity (1.61 Z$_\\odot $ ), density (18.73 cm$^{-3}$ ) and the ionization parameter (log(U) = -1.85) of the outflow.","The estimated H column density is consistent with that derived from the UV continuum extinction.","The outflow is at least as metal enriched as the ISM of the host galaxy (see ionstruct).","Using the absorption line profile and ionization models, we determine that the inner edge (R$_\\mathrm {i}$ ) of the outflow is 63 pc from the starburst (see eq:r), consistent with the absorption arising from the shredded blastwaves of supernovae remnants.","Most of the outflow is constrained within 300 pc of the starburst (ionstruct).","The covering fraction scales as C$_f = 1.0 (\\mathrm {r}/\\mathrm {R}_\\mathrm {i})^{-0.8\\pm 0.2}$ (tab:beta).","This scaling relation is consistent with predictions of clouds in pressure equilibrium with an adiabatically expanding external medium (cf).","This warrents further study of whether outflowing clouds remain in pressure equilibrium with an external medium, and if a fractal cloud distribution could produce a similar scaling relation.","The Si IV outflow velocity scales with radius as v$~=~\\mathrm {v}_\\infty (1~-~\\mathrm {R}_\\mathrm {i}/\\mathrm {r})^{0.43 \\pm 0.07}$ (fig:vellaw and tab:beta), where v$_\\infty $ is the maximum velocity.","We compare this velocity profile to models of different driving mechanisms, and find that an r$^{-2}$ force law matches the observed profile (see fig:vellaw and velocity).","The outflow density decreases with radius as n$ \\propto \\mathrm {r}^{-5.7 \\pm 1.5}$ (fig:denlaw), which declines more rapidly than a mass conserving flow (density).","This rapid density reduction could be due to interactions between the outflowing clouds and a hotter wind.","Combining all of our measurements, we derive a maximum mass outflow rate ($\\dot{M}_{\\text{o}}$ ) of 2.3 M$_\\odot $ yr$^{-1}$ (see fig:mout).","The mass-loading factor (mass outflow rate divided by star formation rate) is 0.09.","The $\\dot{M}_{\\text{o}}$ is a factor of 10 lower than the $\\dot{M}_{\\text{o}}$ calculated using common assumptions for ionization state, metallicity, and geometry (see mout).","In future work, we will continue this analysis for a larger sample of star forming galaxies, studying how the outflow properties (metallicity, initial radius, mass outflow rate) scale with host galaxy properties.","These scaling relations will constrain future models of galaxy evolution."],["Acknowledgments","We thank the anonymous referee for constructive comments that strengthened the paper.","Joseph Cassinelli inspired this work with helpful conversations and notes.","We thank Bart Wakker for help with the data reduction and discussions on the analysis.","Support for program 13239 was provided by NASA through a grant from the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5-26555.","Some of the data presented in this paper were obtained from the Mikulski Archive for Space Telescopes (MAST).","STScI is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5-26555.","All of the HST data presented in this paper were obtained from the Mikulski Archive for Space Telescopes (MAST).","STScI is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS5-26555.","Support for MAST for non-HST data is provided by the NASA Office of Space Science via grant NNX09AF08G and by other grants and contracts.","Funding for the Sloan Digital Sky Survey (SDSS) has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Aeronautics and Space Administration, the National Science Foundation, the U.S. Department of Energy, the Japanese Monbukagakusho, and the Max Planck Society.","The SDSS Web site is http: www.sdss.org/.","The SDSS is managed by the Astrophysical Research Consortium (ARC) for the Participating Institutions.","The Participating Institutions are The University of Chicago, Fermilab, the Institute for Advanced Study, the Japan Participation Group, The Johns Hopkins University, Los Alamos National Laboratory, the Max-Planck-Institute for Astronomy (MPIA), the Max-Planck-Institute for Astrophysics (MPA), New Mexico State University, University of Pittsburgh, Princeton University, the United States Naval Observatory, and the University of Washington."]],"string":"[\n [\n \"A Robust Measurement of the Mass Outflow Rate of the Galactic Outflow\\n from NGC 6090\"\n ],\n [\n \"Abstract To evaluate the impact of stellar feedback, it is critical to estimate the mass outflow rates of galaxies.\",\n \"Past estimates have been plagued by uncertain assumptions about the outflow geometry, metallicity, and ionization fraction.\",\n \"Here we use Hubble Space Telescope ultraviolet spectroscopic observations of the nearby starburst NGC 6090 to demonstrate that many of these quantities can be constrained by the data.\",\n \"We use the Si~{\\\\sc IV} absorption lines to calculate the scaling of velocity (v), covering fraction (C$_f$), and density with distance from the starburst (r), assuming the Sobolev optical depth and a velocity law of the form: v$~\\\\propto(1 -\\\\mathrm{R}_\\\\mathrm{i}/\\\\mathrm{r} )^\\\\beta$ (where R$_\\\\mathrm{i}$ is the inner outflow radius).\",\n \"We find that the velocity ($\\\\beta$=0.43) is consistent with an outflow driven by an r$^{-2}$ force with the outflow radially accelerated, while the scaling of the covering fraction ($C_f \\\\propto \\\\mathrm{r}^{-0.82}$) suggests that cool clouds in the outflow are in pressure equilibrium with an adiabatically expanding medium.\",\n \"We use the column densities of four weak metal lines and CLOUDY photoionization models to determine the outflow metallicity, the ionization correction, and the initial density of the outflow.\",\n \"Combining these values with the profile fitting, we find R$_\\\\mathrm{i}$ = 63 pc, with most of the mass within 300~pc of the starburst.\",\n \"Finally, we find that the maximum mass outflow rate is 2.3~M$_\\\\odot$ yr$^{-1}$ and the mass loading factor (outflow divided by the star formation rate) is 0.09, a factor of 10 lower than the value calculated using common assumptions for the geometry, metallicity and ionization structure of the outflow.\"\n ],\n [\n \"INTRODUCTION\",\n \"Strangely, most of the gas within a galaxy is not near stars , [2], , , , .\",\n \"The circum-galactic medium extends to radii greater than 150 kpc, is metal rich, and spans a range of temperatures , , .\",\n \"Further, this reservoir of gas is massive, containing up to three times more mass than gas within discs .\",\n \"Even though the gas extends more than 150 kpc from the stellar disc, the metal enrichment implies that the gas originated within the stellar disc.\",\n \"What could loft so much mass out of discs?\",\n \"High-mass stars inject energy and momentum into the ISM through high energy photons, cosmic rays, and supernovae, together commonly called stellar feedback , [63], [12], [35], , , [23], , [39], [40], [46], [47], [9].\",\n \"If the stellar feedback is highly concentrated, then the combined energy and momentum drives gas out of star forming regions into a galactic outflow [35], [36], , [21].\",\n \"Galactic outflows may enrich the large reservoir of cirum-galactic gas, while also removing metals from low-mass galaxies to create the mass-metallicity relation , [26], , [3], , [17], [15].\",\n \"However, accurate mass outflow rates are required to constrain these relations.\",\n \"Observations of galactic outflows are challenging.\",\n \"First, galactic outflows are multi-phase, with emission from hot X-ray emitting plasma [31], , , to ionized hydrogen [56], [6], , , , [4], and even cold molecular gas , [62], [55].\",\n \"Second, outflows are diffuse, and probing outflows with emission lines is challenging and limited to only the local universe.\",\n \"Recent studies use optical [36], [59], , [11], near UV , [60], , [19], [49], [22], and far UV absorption lines , , , [32], [54], [13], [38], , [14] to study the diffuse gas within outflows.\",\n \"While these studies have hinted at the role outflows play in removing gas from the discs of galaxies, the mass outflow rate calculations are plagued by uncertain assumptions.\",\n \"To calculate the mass outflow rate from absorption lines, an observed column density and velocity are converted into a total mass outflow rate.\",\n \"The first step converts the observed ion density to a total Hydrogen density through an assumed metallicity and ionization correction , [38].\",\n \"The Hydrogen column density is then converted into a total Hydrogen mass through an assumed geometry, typically a thin shell.\",\n \"These studies make four key assumptions: (1) that the absorption lines are optically thin (2) that the measured ion is the dominant ionization state, and there is no ionization correction (3) that the metallicity of the outflow is consistent with the ISM of the galaxy and (4) that the outflow is a spherical shell with a radius of 5 kpc , .\",\n \"Recent observations question these assumptions.\",\n \"Ionization modelling from [14] find that galactic outflows are photoionized, with the dominant ionization state, Si III, at an ionization potential near 25 eV, with only 1-5% of the gas in the neutral phase.\",\n \"For neutral ions like Na I, neglecting the ionization correction can underestimate the total Hydrogen column density by a factor of 4 , [14].\",\n \"Additionally, these photoionization models predict that the outflows are metal rich, and the assumption that the outflow has the metallicity of the ISM underpredicts the total Hydrogen density.\",\n \"These large ionization fractions also imply that the outflow is relatively close to the ionizing source.\",\n \"Stars generate outflows at distances of super-star clusters, 20-40 pc [63], , [12], but metal-enriched gas is also observed out to 10 kpc from the starburst , , , .\",\n \"Even though the outflow extends over this enormous distance, studies typically calculate the mass outflow rates as if the entire mass is in a thin shell with a radius of 5 kpc .\",\n \"Here, we present a new analysis of the mass outflow rate of the nearby galaxy NGC 6090 using Hubble Space Telescope ultraviolet spectroscopy.\",\n \"We calculate the mass outflow rate by first measuring the distance, metallicity, and ionization fraction of the outflow.\",\n \"In data we describe the data reduction, stellar continuum fitting, and measurement of the outflow properties.\",\n \"We then fit for how the optical depth and covering fraction scales with velocity, using a Sobelev optical depth, a $\\\\beta $ velocity law, and a power-law density scaling (beta).\",\n \"We apply the ionization models of [14] to the outflow of NGC 6090 to determine the metallcity (1.61 Z$_\\\\odot $ ), total Hydrogen density (18.73 cm$^{-3}$ ), and the ionization fractions of the outflow (ion).\",\n \"In ionstruct we consider the implications of the ionization models and the metal enriched outflows, and then calculate the initial radius of the outflow (63 pc).\",\n \"Using these derived quantities, we examine the covering fraction (cf), velocity (velocity), and density (density) laws with radius.\",\n \"Since these relationships vary with velocity, we find that the mass outflow rate of the galaxy also varies with velocity, with a maximum mass outflow rate of 2.3 M$_\\\\odot $ yr$^{-1}$ .\",\n \"This mass outflow rate is 10 times smaller than it would be if we used previous assumptions, and only 9% of the star formation rate of the galaxy.\",\n \"In this paper we use $\\\\Omega _M = .28$ , $\\\\Omega _\\\\Lambda = .72$ and H$_0$ = 70 km s$^{-1}$ Mpc$^{-1}$ [43].\"\n ],\n [\n \"DATA\",\n \"NGC 6090 is a massive star forming galaxy at a distance of 128 Mpc (see tab:ngc6090), with no indication of AGN contamination from the optical emission lines (see tab:ngc6090) and the WISE colors.\",\n \"The galaxy is metal rich, with a log(O/H)+12 of 8.77 .\",\n \"The galaxy is in the intermediate stages of a massive merger, as two distinct nuclei and large extend tidal tails are seen in the UV (fig:image) and optical images .\",\n \"The Cosmic Origins Spectrograph [30] on the Hubble Space Telescope observed NGC 6090 during Cycle 18 [54].\",\n \"The full data reductions are given in [13], and here we summarize the steps taken to measure the optical depths, covering fractions, and column densities of the outflow.\",\n \"First, we describe the data reductions (reduc) and the stellar continuum fitting (cont).\",\n \"We then explore the effects of the Si II fine structure and resonance emission lines (finestruct).\",\n \"Finally, the optical depths, covering fractions, and column densities are measured in measure.\"\n ],\n [\n \"Data Reduction\",\n \"[54] observed NGC 6090 with the G130M grating for a total integration time of 8096 s. We downloaded the spectra from MAST and processed the data through the CalCOS pipeline, version 2.20.1.\",\n \"The individual exposures were aligned and co-added following the methods outlined in .\",\n \"The spectrum was deredshifted using the redshift from .\",\n \"The flux was normalized to the median flux in a line free region between 1310-1320 Å in the restframe, the wavelength array was then binned by 5 pixels (10 km s$^{ -1}$ at 1400Å), and smoothed by 3 pixels.\",\n \"The set-up of the G130M grating affords continuous wavelength coverage between 1125-1425 Å, in the restframe.\",\n \"Our fully processed data have a median signal-to-noise ratio of 12 per pixel after rebinning.\"\n ],\n [\n \"Stellar Continuum Fitting\",\n \"Metal absorption in stellar atmospheres contaminates the ISM absorption profiles.\",\n \"To remove the stellar contribution, we fit the observed spectrum with a linear combination of STARBURST99 simple stellar populations models [51], [53].\",\n \"To match the resolution of the observations, and to avoid stellar libraries with Milky Way contamination, we use the fully theoretical Geneva models, with high-mass loss [64], computed using the WM-basic method [51], [53].\",\n \"A linear combination of different age stellar populations is required to simultaneously recover the Si III photospheric lines near 1295 Å (signatures of B stars) and the Si IV and N V P-Cygni profiles (signatures of O stars).\",\n \"Since O and B stars dominate the FUV stellar continuum, we include STARBURST99 models of ten ages between 1-20 Myr, and use MPFIT [58] to determine the linear combination of these ten models.\",\n \"Five ages contribute to the stellar continuum of NGC 6090 (1, 3, 4, 5, 20 Myr), but the 4 and 5 Myr models account for 84% of the FUV light, for a weighted stellar age of 4.48 Myr (see tab:ngc6090).\",\n \"The stellar continuum fit is shown in the upper panel of Figure 2 in [14].\",\n \"This figure shows that the STARBURST99 models nicely reproduce the stellar P-Cygni features, and provide a continuum level for the spectrum.\",\n \"Both the stellar age and metallicity determine the number of ionizing photons emitted by high-mass stars [51].\",\n \"Accordingly, we fit for both the metallicity and stellar age.\",\n \"STARBURST99 models have five stellar continuum metallicities – 0.05, 0.2, 0.4, 1.0, 2.0 Z$_\\\\odot $ – and we determine that the stellar metallicity of NGC 6090 is solar using a $\\\\chi ^2$ test (tab:ngc6090).\",\n \"During the fitting, we include continuum reddening using a Calzetti extinction law [10], and find that the stellar continuum is extincted by E(B-V) = 0.316 mags, after accounting for foreground Milky Way extinction .\",\n \"We cross-correlate the fitted stellar continuum with the observed spectrum to establish the zero-velocity of the stellar continuum.\",\n \"Finally, we remove the stellar continuum by dividing the observed spectrum by the STARBURST99 model.\",\n \"Strong Milky Way absorption and geocoronal emission lines are blueshifted from the metal lines of NGC 6090.\",\n \"However, the Milky Way lines are masked during the stellar population fitting to avoid contaminating the fit.\",\n \"Since the galaxy non-uniformly fills the COS aperture (see fig:image), the spectral resolution is degraded.\",\n \"The COS spectral resolution varies from 20 km s$^{-1}$ for a point source to 200 km s$^{-1}$ for a uniformly filled aperture [27].\",\n \"We fit the line width of the Milky Way absorption features, and find that the effective spectral resolution of the spectrum is 48 km s$^{ -1}$ .\",\n \"fig:profile gives the metal absorption profiles from the fully reduced, stellar continuum normalized spectra.\",\n \"There is a weak C I 1277Å Milky Way absorption line near the N V 1243 Å transition and the stellar continuum is slightly underestimated.\",\n \"While we fit, and remove this weak C I feature, imperfect subtraction creates residual emission, contaminating the continuum near the weaker N V 1243 Å line.\",\n \"Therefore, we do not draw conclusions from the N V profiles.\"\n ],\n [\n \"Fine Structure Emission Lines\",\n \"Before we measure the outflow properties from the metal absorption lines, we must understand how resonance emission lines affect the absorption profiles , , , .\",\n \"Continuum photons excite ground state electrons, and the electron then transitions into a lower energy state by emitting a photon with an energy of the difference between the two states.\",\n \"If the electron transitions back to the ground state through the same path as it was excited, it is called resonance emission.\",\n \"Resonant photons have the same restframe wavelength as the absorption and, depending on the geometry of the outflow, the resonance emission can overlap in velocity space with the absorption.\",\n \"This effectively reduces the measured absorption profile.\",\n \"The amount, and the velocity, of the infilling can be estimated if fine structure splits the ground state.\",\n \"The electron's angular momentum splits the ground state into multiple levels, allowing the electron to transition through either resonance or fine structure emission (fine structure emission is normally denoted by a $^\\\\ast $ , i.e.\",\n \"as Si II$^\\\\ast $).\",\n \"The probability of transitioning by a specific path is given as $\\\\mathrm {P}_\\\\mathrm {i} = \\\\mathrm {A}_\\\\mathrm {i} / \\\\Sigma \\\\mathrm {A}_\\\\mathrm {i}$ , where A$_i$ is the Einstein A coefficient corresponding to level i, and the summation is over all possible levels.\",\n \"Therefore, we use the observed fine structure emission to estimate an upper limit on the in-filling of resonance profile.\",\n \"There are four Si II$^\\\\ast $ lines in the observed wavelength regime at 1194, 1197, 1265, and 1309Å, however the 1265 line is blended with the O I geocoronal emission line.\",\n \"The 1194Å line has the largest probability of fine structure emission (84% of the 1190Å absorption is re-emitted as 1194Å Si II$^\\\\ast $), therefore this line predicts the maximum amount of resonance emission.\",\n \"The 1194 Si II$^\\\\ast $ line is the strongest of the four Si II$^\\\\ast $ lines in the bandpass, but the equivalent width is only $-0.15$ Å, a small fraction of the 1.74 Å Si II 1190 Å equivalent width.\",\n \"Further, the line is redshifted by +132 km s$^{ -1}$ , with emission only extending to -80 km s$^{ -1}$ .\",\n \"Below, we chiefly use the Si IV doublet, which does not have fine structure emission lines in the wavelength regime.\",\n \"The Si II and Si IV profiles have similar velocity profiles [14], allowing for the Si II$^\\\\ast $ to be used as a proxy for the Si IV emission.\",\n \"To avoid possible contributions from resonance emission, we exclude velocities greater than -80 km s$^{-1}$ when fitting the profiles in beta.\"\n ],\n [\n \"Equivalent Width, Optical Depth, Covering Fraction, and Column Density\",\n \"Once we have reduced the data, removed the stellar continuum, and explored the impact of the resonance emission features, we measure the properties of the outflow.\",\n \"Here, we describe the outflow with four parameters: equivalent width (W), optical depth ($\\\\tau $ ), covering fraction (C$_f$ ), and column density (N).\",\n \"W is measured by integrating $1-F_\\\\mathrm {o}$ (the observed continuum normalized flux) over the velocity ranges given in column three of tab:obs.\",\n \"As discussed in [14], the maximum velocity of each transition strongly depends on the strength (the W) of the transition, not the ionization state of the transition.\",\n \"The W errors are computed by bootstrapping the observed flux with the flux error array 1000 times, and the standard deviation is calculated from the resultant W distributions.\",\n \"The equivalent width ratios of doublets (and triplets) can be used to diagnose the saturation of the transitions.\",\n \"With an observed doublet ratio of 1.83 (and an expected ratio of 2), the Si IV 1402Å and 1392Å lines are the only doublet that does not suffer from severe saturation effects.\",\n \"We also use the column densities of the weak S II 1250Å, Si II 1304Å, and O I 1302Å lines during the ionization modelling (see ion).\",\n \"C$_f$ is the fraction of the continuum source covered by the foreground absorbing gas (see for a thorough description of the interpretation of C$_f$ ).\",\n \"At low $\\\\tau $ , C$_f$ and $\\\\tau $ are degenerate because both impact the depth of the absorption profile.\",\n \"However, this degeneracy can be broken by solving the radiative transfer equation for the C$_f$ and optical depths of the lines [33].\",\n \"Here, we use the Si IV doublet because it does not suffer from strong saturation effects.\",\n \"The covering fraction is calculated in each pixel to give a velocity (v) resolved C$_f$ profile as [33] $C_f(\\\\mathrm {v}) = \\\\frac{\\\\mathrm {F}_\\\\mathrm {W}(\\\\mathrm {v})^2-2\\\\mathrm {F}_\\\\mathrm {W}(\\\\mathrm {v}) + 1}{\\\\mathrm {F}_\\\\mathrm {S}(\\\\mathrm {v})-2\\\\mathrm {F}_\\\\mathrm {W}(\\\\mathrm {v})+1}$ where F$_\\\\mathrm {W}$ is the continuum normalized flux of the weaker doublet line (Si IV 1402Å), and F$_\\\\mathrm {S}$ is the continuum normalized flux of the stronger doublet line (Si IV 1394).\",\n \"In this equation we use the fact that the Si IV 1402 line has half the oscillator strength of the 1392 line.\",\n \"We set unphysical C$_f$ values that are less than zero to zero.\",\n \"We also calculate the velocity resolved $\\\\tau $ as $\\\\tau (\\\\mathrm {v}) = \\\\ln \\\\left(\\\\frac{C_f(\\\\mathrm {v})}{C_f(\\\\mathrm {v})+F_\\\\mathrm {W}(\\\\mathrm {v})-1}\\\\right)$ We bootstrap the errors of C$_f$ and $\\\\tau $ , similar to the W errors above.\",\n \"Finally, we calculate the integrated column density of each transition using the apparent optical depth method .\",\n \"Since some of the weak lines are singlets, we cannot use eq:cf to calculate the column density, rather we have to assume that the lines are fully covered.\",\n \"Below, we find that this assumption is fair for the line centers, which have most of the column density.\",\n \"The integrated column density is calculated as $\\\\mathrm {N} = \\\\frac{3.77 \\\\times 10^{14}~\\\\text{cm}^{-2}}{\\\\lambda [\\\\text{Å}] f} \\\\int \\\\text{ln}\\\\frac{1}{F_\\\\mathrm {o}(\\\\mathrm {v})} \\\\mathrm {dv}$ where f is the oscillator strength [14], and the total N is calculated by integrating over the velocity interval given in tab:obs.\",\n \"With the measured properties of the outflow, we now study how the Si IV 1402Å optical depth and covering fraction evolve with velocity.\"\n ],\n [\n \"PROFILE FITTING\",\n \"In fig:profile the metal absorption lines are broad ($\\\\sim 175$ km s$^{ -1}$ ) and blueshifted by $\\\\sim -150$ km s$^{ -1}$ .\",\n \"Under the Sobolev approximation, these profiles are produced by an expanding medium, that has a steep velocity gradient (dv/dr) but a narrow intrinsic line profile, such that the expanding medium broadens the line profile by absorbing and scattering continuum photons , [50], , .\",\n \"The Sobolev optical depth of the Si IV 1402Å profile is given by $\\\\tau \\\\left(\\\\mathrm {v}\\\\right) = \\\\frac{\\\\pi e^2}{\\\\mathrm {mc}} f \\\\lambda ~\\\\mathrm {n}_\\\\mathrm {4}\\\\left(\\\\mathrm {r}\\\\right) \\\\frac{\\\\mathrm {dr}}{\\\\mathrm {dv}}$ where $f$ is the oscillator strength (0.255 for Si IV), $\\\\lambda $ is the rest wavelength, and n$_\\\\mathrm {4}$ is the number density of the Si IV gas at a given radius (r).\",\n \"To simplify the calculation, we introduce normalized quantities by dividing the velocity by the maximum velocity ($w = \\\\mathrm {v}/\\\\mathrm {v}_\\\\infty $ , v$_\\\\infty = -399$ km s$^{-1}$ for Si IV 1402Å) and the radius by the inner radius ($x = \\\\mathrm {r}/\\\\mathrm {R}_\\\\mathrm {i}$ ), which we measure in ionstruct.\",\n \"v$_\\\\infty $ is calculated as the velocity at which the absorption reaches 90% of the continuum, similar to the method outlined in [13].\",\n \"We must model how n$_\\\\mathrm {4}$ is distributed with radius.\",\n \"Previous studies assume that the density is either constant with velocity [60], , or that the density follows the continuity equation , .\",\n \"However, to remain general we assume that the density follows a power-law scaling with radius, such that $\\\\mathrm {n}_\\\\mathrm {4}(x) = \\\\mathrm {n}_\\\\mathrm {4,0} x^\\\\alpha $ where n$_\\\\mathrm {4,0}$ is the Si IV density at the initial radius, R$_\\\\mathrm {i}$ .\",\n \"Placing this into eq:sobelv gives the relation $\\\\tau (x) = \\\\frac{\\\\pi e^2}{\\\\mathrm {mc}} f \\\\lambda _\\\\mathrm {0} \\\\frac{\\\\mathrm {R}_\\\\mathrm {i}}{\\\\mathrm {v}_\\\\infty } \\\\mathrm {n}_\\\\mathrm {4,0} x^\\\\alpha \\\\frac{dx}{dw} = \\\\tau _0 x^\\\\alpha \\\\frac{\\\\mathrm {d}x}{\\\\mathrm {d}w}$ where we have combined the constants into a single constant, $\\\\tau _0$ , that represents the maximum optical depth.\",\n \"Additionally, we derive the covering fraction of the Si IV absorption (C$_f$ ; see eq:cf).\",\n \"As we discuss in cf, the covering fraction is the proportion, expressed as a percentage, of the starburst area covered by the outflow at a given radius.\",\n \"We assume that the covering fraction scales as a power-law with distance (as physically motivated in cf), such that $C_f(w) = C_{f} (\\\\mathrm {R}_\\\\mathrm {i}) \\\\left(\\\\frac{\\\\mathrm {r}}{\\\\mathrm {R}_\\\\mathrm {i}}\\\\right)^\\\\gamma = C_{f} (\\\\mathrm {R}_\\\\mathrm {i}) x^\\\\gamma $ Now we have related the two observables of the profile to the distance from the starburst.\",\n \"eq:sobelvnorm and eq:cffit are given in terms of distance, while we measure the parameters in terms of velocity.\",\n \"Many of the physical mechanisms for driving outflows scale the velocity with radius as a $\\\\beta $ velocity law [50], such that $\\\\mathrm {v} = \\\\mathrm {v}_\\\\infty \\\\left(1-\\\\frac{\\\\mathrm {R}_\\\\mathrm {i}}{\\\\mathrm {r}}\\\\right)^{\\\\beta }$ Substituting for w and x, and inverting this relation gives the normalized radius in terms of the velocity as $x = \\\\frac{1}{1-w^{1/\\\\beta }}$ Taking the derivative of this gives the velocity gradient in terms of the normalized velocity and the $\\\\beta $ exponent as $\\\\frac{\\\\mathrm {d}x}{\\\\mathrm {d}w} = \\\\frac{w^{1/\\\\beta -1}}{\\\\beta \\\\left(1-w^{1/\\\\beta }\\\\right)^2}$ We can also use the $\\\\beta $ -law to derive the density scaling with velocity as $\\\\mathrm {n}(w) = \\\\mathrm {n}_\\\\mathrm {4,0} \\\\left(\\\\frac{1}{(1-w^{1/\\\\beta })}\\\\right)^{\\\\alpha }$ producing an $\\\\tau $ (eq:sobelvnorm) and C$_f$ (eq:cffit) velocity scaling as $\\\\begin{aligned}\\\\tau \\\\left(w\\\\right) &= \\\\tau _\\\\mathrm {0} \\\\frac{w^{1/\\\\beta -1}}{\\\\beta (1-w^{1/\\\\beta })^{2+\\\\alpha }}\\\\\\\\C_f(w) &= \\\\frac{C_f (\\\\mathrm {R}_\\\\mathrm {i})}{\\\\left(1-w^{1/\\\\beta }\\\\right)^\\\\gamma }\\\\end{aligned}$ We then have five parameters to fit for: the maximum optical depth ($\\\\tau _0$ ), the covering fraction at the initial radius (C$_{f}(\\\\mathrm {R}_\\\\mathrm {i})$ ), the exponent of the beta velocity-law ($\\\\beta $ ), the exponent of the density law ($\\\\alpha $ ), and the exponent of the covering fraction law ($\\\\gamma $ ).\",\n \"Using MPFIT [58], we simultaneously fit for the five parameters from the unsaturated Si IV 1402 Å $\\\\tau $ and C$_f$ distributions (see fig:beta).\",\n \"At w below 0.2 ($v > -80$ km s$^{ -1}$ ) zero-velocity absorption and resonance emission may effect the distributions (see finestruct); therefore, we only fit the distributions between w of 0.2 and 1.0 (the points outside this range are coloured gray in fig:beta).\",\n \"The C$_f$ and $\\\\tau $ fits are shown in fig:beta, and the fit parameters are given in tab:beta.\",\n \"In these fits we have binned the flux array by a factor of two (20 km s$^{ -1}$ ) to increase the signal-to-noise ratio, while still Nyquist sampling the observed velocity resolution (see the bar in fig:beta).\",\n \"These fits reproduce the observed $\\\\tau $ and C$_f$ distributions, and describe the radial velocity and density laws of the outflow through eq:beta and eq:continuity.\",\n \"In discussion we discuss some implications of these trends.\",\n \"To test the covering fraction fits, we can use the Si III and C II profiles.\",\n \"Since these profiles are much stronger, they are strongly saturated at all velocities.\",\n \"This means that we cannot measure the C$_f$ with the previous formula (eq:cf), while $\\\\tau $ cannot be accurately measured at all.\",\n \"Rather, at large $\\\\tau $ , the depth of the line is completely set by C$_f$ , and is measured as C$_f(\\\\mathrm {v}) = 1 - F(\\\\mathrm {v})$ .\",\n \"In the lower right panel of fig:beta we plot the C$_f$ for Si III (circles) and C II (triangles).\",\n \"The Si IV C$_f$ fit is over-plotted on these measurements and provides reasonable agreement with the data.\",\n \"Table: Table of the quantities derived from the profile fits of the optical depth (τ\\\\tau ) and covering fraction (C f _f; see eq:cfbeta).\",\n \"The individual parameters are: Column 1, τ 0 \\\\tau _0, optical depth at line center; Column 2, β\\\\beta , the velocity power-law index; Column 3, α\\\\alpha , the column density power-law index; Column 4, C f (R i )_f (\\\\mathrm {R}_\\\\mathrm {i}), the maximum covering fraction; Column 5, γ\\\\gamma , the covering fraction power-law index.\"\n ],\n [\n \"IONIZATION MODELLING\",\n \"In measure we calculate the integrated column density (N) for individual metal ions, but these only describe the individual ion and not the total mass of the outflow.\",\n \"While Ly-$\\\\alpha $ absorption traces the neutral Hydrogen in the outflow, we opt to use the metal lines to describe the outflow because (1) the strong P-Cygni Ly-$\\\\alpha $ profile requires detailed radiative transfer models to constrain and (2) up to 99% of the Hydrogen in the outflow may be ionized [14].\",\n \"To calculate the total Hydrogen in the outflow, ionization models are required to measure the ionization fraction ($\\\\chi _\\\\mathrm {i}$ , or the fraction of the total gas within transition i) and the abundance (N$_\\\\mathrm {i}$ /N$_\\\\mathrm {H}$ , or the ratio of an element to the total Hydrogen).\",\n \"In [14] we find that photoionization models describe the ionization structure of galactic outflows.\",\n \"These models require the outflow metallicity (Z$_\\\\mathrm {o}$ ) to be greater than 0.5 Z$_\\\\odot $ and the ionization parameter (log(U); the ratio of the photon density to outflow density) to be between -1.5 and -2.25.\",\n \"Additionally, the ionization structure depends both on the spectral energy distribution and the strength of the ionizing source, as well as the density (n$_\\\\mathrm {0}$ ) of the outflow.\",\n \"We model the ionization structure using CLOUDY version 13.03 [25].\",\n \"The CLOUDY models are not velocity resolved profiles, rather we are fitting the measured integrated column densities to the integrated values from CLOUDY.\",\n \"We assume that the outflow is ionized by the observed stellar continuum, using the best-fit STRABURST99 stellar continuum model from cont.\",\n \"These STARBURST99 models have an age of 4.478 Myr and a constant SFR of 25.15 M$_\\\\odot $ yr$^{-1}$ (see tab:ngc6090).\",\n \"In the CLOUDY models, we assume an expanding spherical geometry, which is not meant to reproduce the velocity profiles, but to account for back-scattering of radiation.\",\n \"We set the covering fraction to 1.00, the observed value from the Si IV transition, and scale the density with a power-law of -5.72, as measured above.\",\n \"The CLOUDY models are stopped once the simulations reach 3000 K, which is lower than the default criteria to allow for higher metallicities.\",\n \"These lower temperatures require a cosmic ray background to be included [42].\",\n \"We use CLOUDY's default H II abundances, which are similar to the Milky Way values [5], , , , and include an Orion Nebular dust grain distribution [5].\",\n \"The dust grains account for scattering and destruction of photons, as well as depletion of metals onto grains.\",\n \"The abundances are scaled by a constant factor to change the outflow metallicity (Z$_\\\\mathrm {o}$ ).\",\n \"Similarly, we vary the outflow density at the inner radius (n$_\\\\mathrm {0}$ ), ionization parameter (U), and the stellar continuum metallicity (Z$_\\\\mathrm {s}$ ).\",\n \"We use a Bayesian approach to estimate Z$_\\\\mathrm {o}$ , n$_\\\\mathrm {0}$ , U, and Z$_\\\\mathrm {s}$ [44], [7], [24].\",\n \"While [14] uses the W ratios to define the ionization structure, the relatively high signal-to-noise ratio and spectral resolution observations of NGC 6090 allow us to use the measured column densities from four weak transitions: O I 1302Å, Si II 1304Å, S II 1250Å, and Si IV 1402Å (see tab:obs).\",\n \"We do not use the N V column densities due to Milky Way C I1277Å and imperfect continuum subtraction contamination near the N V line (see cont).\",\n \"We create grids of CLOUDY models, with differing input parameters (tab:gridmod), and tabulate the predicted CLOUDY column densities.\",\n \"We assume a uniform prior – each model is equally likely – and compute the likelihood function of each set of parameters as $\\\\mathrm {L} \\\\propto \\\\exp (-\\\\chi ^2/2).$ where $\\\\chi ^2$ is the chi-squared function using the observed column density (N), the errors on the observed N, and the predicted N from CLOUDY.\",\n \"For each parameter (Z$_\\\\mathrm {o}$ , n$_\\\\mathrm {0}$ , U, and Z$_\\\\mathrm {s}$ ), the likelihood function is marginalized over the other nuisance parameters and normalized to one to create probability distribution functions (PDFs).\",\n \"We calculate expectation values and standard deviations of the individual parameters from these PDFs as estimates of the parameters, and their uncertainties.\",\n \"Figure: Probability density functions (PDFs) from the CLOUDY ionization modelling.\",\n \"The three parameters are outflow metallicity (top panel), ionization parameter (middle panel), and the total Hydrogen density at the initial radius (bottom panel).\",\n \"The PDFs are single peaked, with expectation values and standard deviations given in tab:ion.We do the fitting in two iterations.\",\n \"The first iteration uses a course grid to determine the best Z$_s$ .\",\n \"Since there are only five stellar continuum metallicities in the fully theoretical STARBURST99 models, we only coarsely estimate the stellar continuum metallicity.\",\n \"Similar to the COS spectrum fitting (see cont), the Bayesian analysis finds that the 1 Z$_\\\\odot $ model best fits the ionization structure, assigning 100% of the probability to the 1 Z$_\\\\odot $ stellar continuum model.\",\n \"Table: Grid of CLOUDY models used in the Bayesian analysis of the ionization structure.The second iteration uses a more finely spaced grid for Z$_o$ , log(U) and n$_0$ , while only using the solar metallicity stellar continuum model (see tab:gridmod).\",\n \"In fig:ionpdfs we show the PDFs for the three parameters, with narrow distributions that have expectation values of $Z_\\\\mathrm {o} = 1.61\\\\pm 0.08$ Z$_\\\\odot $ , $n_\\\\mathrm {0} = 18.73\\\\pm 2.37$ cm$^{-3}$ , and log(U)$ =-1.85\\\\pm 0.02$ (see tab:ion).\",\n \"Using these parameters, we create a best-fit CLOUDY model, producing the ionization fractions of the outflow (see tab:ion).\",\n \"As discussed in ionstruct, these ionization models have important implications for the metallicity of the outflow, the inner radius of the outflow, and the mass outflow rate.\",\n \"Figure: The velocity resolved column density plots for the Si IV (black) and O I (red) lines, normalized to their maximum values.\",\n \"The velocity structure between the neutral (O I) and highly ionized gas (Si IV) stays roughly constant over the velocity range, implying that the ionization structure does not vary substantially with velocity.Table: Table of the quantities derived from the CLOUDY photoionization modelling.\",\n \"Row (1) is the ionization parameter; (2) is the derived outflow metallicity; (3) is the stellar continuum metallicity; (4) is the Hydrogen density at the base of the outflow; (5) is the ionization fraction of Si IV, or the per cent of the total Si in the Si IV transition; (6) The best-fit Si to H abundance value; (7) The measured Si IV column density; (8) The total Hydrogen column density calculated from the Si IV column density, Si/H abundance, and the Si IV ionization fraction; (9) The inner radius, calculated using eq:r; (10) the H II temperature of the best-fit CLOUDY model.\",\n \"The errors on χ Si IV \\\\chi _\\\\mathrm {Si {\\\\sc IV}} and log(Si/H) are calculated by producing CLOUDY models of the estimate plus/minus 1σ\\\\sigma errors of the Z o _\\\\mathrm {o}, n 0 _0, and log(U).\"\n ],\n [\n \"Ionization Structure With Velocity\",\n \"In the following analysis we use an integrated ionization correction.\",\n \"However, if the outflow is heated as it is accelerated, then the column density ratio of high to low ionization potential lines should increase with velocity.\",\n \"fig:colfrac shows the velocity resolved column densities for the Si IV and O I transitions, showing that there is not a coherent variation of the column densities with velocity between $-400$ and 0 km s$^{ -1}$ .\",\n \"This implies that a majority of the column density arises from a small radius, and the photoionization models are approximately plane-parallel.\"\n ],\n [\n \"DISCUSSION\",\n \"Here we discuss various aspects of the profile and ionization fitting from beta and ion.\",\n \"In ionstruct, we first derive important parameters of the outflow.\",\n \"We then study the implications for the covering fraction (cf), velocity (velocity), and density (density) radial scalings.\",\n \"Finally, we combine the various relations to determine the mass outflow rate of NGC 6090 (mout).\"\n ],\n [\n \"Ionization Modelling\",\n \"With the ionization fractions and metallicities derived in ion, we convert the measured Si IV column densities into a total Hydrogen density (see tab:ion).\",\n \"The total (neutral plus ionized) Hydrogen column density is 20.89$_{-0.03}^{+0.04}$ cm$^{-2}$ , where the uncertainties for each parameter are propagated to compute the uncertainties of the total Hydrogen column density.\",\n \"The UV continuum extinction provides a complimentary way to estimate the total Hydrogen column density by relating the extinction to the amount of dust through a dust-to-gas ratio.\",\n \"We use the relation from [52] and [37] to calculate the total Hydrogen column density as $\\\\mathrm {N}_\\\\mathrm {H} = \\\\frac{3.6 \\\\times 10^{21} \\\\mathrm {E(B-V)}}{Z} \\\\text{cm}^{-2}$ where we use a starburst dust attenuation law and the relation for the FUV optical depth from [29] to convert the spectral slope to E(B-V).\",\n \"Using the extinction measured from the UV continuum in [14] and the outflow metallicity from the ionization structure (see tab:ngc6090 and tab:ion), we expect a log(N$_\\\\mathrm {H}$ ) of 20.85, in agreement with the derived total N$_\\\\mathrm {H}$ from the ionization modelling.\",\n \"The ionization model predicts that 99.3% of the Hydrogen in the outflow is ionized.\",\n \"These low neutral fractions suggest that the outflow has a H I column density of $5 \\\\times 10^{18}$ cm$^{-2}$ .\",\n \"Current radio arrays cannot detect this H I column in emission, but the upcoming SKA will detect these densities with 100 spatial resolution and 5 km s$^{-1}$ spectral resolution in 10-100 hours of integration time .\",\n \"The outflow metallicity is 1.61 Z$_\\\\odot $ , 61% and 34% larger than the measured stellar and ISM metallicities (see tab:ngc6090).\",\n \"Substantial uncertainties on log(O/H) measurements [45] means that the outflow metallicity is at least consistent with the ISM values, and likely enriched compared to the ISM of the galaxy.\",\n \"Additional metals may reside in a hotter phase not probed by these observations (see below for a discussion of this phase).\",\n \"Metal enriched outflows are important constraints for models using galactic outflows to explain the mass-metallicity relation , [26], , [17], [15].\",\n \"For example, to match the mass-metallicity relation of [18] for galaxies with log($M_\\\\ast $ /M$_\\\\odot $ ) of 10.7, Figure 8 of suggests that the outflow metallicity is 1.58 Z$_\\\\odot $ .\",\n \"The same figure also uses Z$_\\\\mathrm {o}$ to constrain the scaling of the mass outflow rate divided by the star formation rate (mass-loading factor).\",\n \"In future studies, we will determine the scaling of the outflow metallicity with stellar mass and star formation rate to better constrain these studies.\",\n \"The initial radius (R$_\\\\mathrm {i}$ ) can be determined from the fitted Si IV optical depth (see eq:sobelvnorm).\",\n \"Solving for R$_\\\\mathrm {i}$ in terms of the observed maximum Si IV optical depth, we find that $\\\\mathrm {R}_\\\\mathrm {i} = \\\\frac{\\\\mathrm {mc}}{\\\\pi e^2} \\\\frac{1}{f \\\\lambda } \\\\frac{\\\\mathrm {v}_\\\\infty \\\\tau _\\\\mathrm {0}}{\\\\mathrm {n}_\\\\mathrm {0} \\\\chi _\\\\mathrm {Si~IV} \\\\mathrm {Si/H}} = 63.4~\\\\text{pc}$ Using the values from the $\\\\beta $ -profile fitting (tab:beta) and the ionization modelling (tab:ion).\",\n \"We have related the initial Si IV density ($n_{4,0}$ ) to the Hydrogen density at the base of the outflow, the abundance, and the ionization corrections from the ionization models ($n_{4,0} = n_\\\\mathrm {0} \\\\chi _\\\\mathrm {Si IV} \\\\mathrm {Si/H}$ ).\",\n \"The inner outflow radius and the size of the mass-loading region (maximum extent of about 150 pc; see mout and fig:mout below) is consistent with models of the mass-loading region in M 82 , , and with analytical work from [38], who assume that the inner outflow radius is twice the size of the starburst.\",\n \"However, what is the physical meaning of the inner outflow radius?\",\n \"The shredding of supernovae blastwaves by hydrodynamical instabilities is a possible origin for this inner radius.\",\n \"Supernovae energy builds up in the ISM due to multiple impulsive events , which creates a blastwave that shocks when it encounters the ambient ISM , , , [63], , [46].\",\n \"The shock compresses surrounding ISM into a thin, dense shell of relatively cool gas that travels outward.\",\n \"This phase is called the snowplow phase, and the radius depends on the amount of injected energy, the density of the medium, and how much of the energy is radiated away , [63], [16], , [46], with typical values near 20-30 pc [20], [46].\",\n \"Once the shell forms, Rayleigh-Taylor and Kelvin-Helmholtz instabilities rapidly destroy it, creating many small warm (10$^4$ K) cloudlets [57], [16], [28], [60], , .\",\n \"The destruction of the blastwave allows the hot interior gas to escape and travel into the halo as a hot wind.\",\n \"The hot, high-velocity wind, may then accelerate these clouds through ram pressure to the observed velocities [28], [60], .\",\n \"Therefore, a possible origin for R$_\\\\mathrm {i}$ is that hydrodynamic instabilities have shredded the dense blastwave, and injected cloudlets into the hot wind.\",\n \"Finally, the CLOUDY modelling estimates the temperature of the H II in the outflow to be 5470 K (see tab:ion), leading to a pressure of P/k$_\\\\mathrm {b}$ = $1 \\\\times 10^5$ K cm$^{-3}$ .\",\n \"The hot wind is typically assumed to have a temperature of 10$^7$ K [12].\",\n \"If the observed cloudlets and hot wind are in pressure equilibrium, then the hot wind has a density of 0.01 cm$^{-3}$ at R$_\\\\mathrm {i}$ , a factor of 10 lower than the density in the [12] model.\",\n \"However, recent models by [9] show that the density and temperature of this hot wind can greatly vary, depending on the efficiency and mass-loading of the outflow.\",\n \"Regardless of the origin of the outflow, it must be evenly distributed across the stellar continuum to produce a unity covering fraction (see tab:beta).\",\n \"However, the external pressure on the outflow decreases as it accelerates out from the starburst.\",\n \"The change in pressure has a dramatic effect on the size of the cloudlets in the outflow, and in turn their covering fraction.\",\n \"In the next section we discuss this pressure change, and how it naturally leads to the observed scaling of the covering fraction with velocity.\"\n ],\n [\n \"Covering Fraction\",\n \"In beta, we observe that the outflow initially completely covers the background stellar continuum, but as the velocity increases the outflow covers less of the background stars (C$_f$ drops as $x^{-0.8}$ ).\",\n \"To physically understand this scaling relation, we hypothesize that the absorption arises from cloudlets of gas initially a distance R$_\\\\mathrm {i}$ from the starburst (see the upper left panel of fig:cfnop).\",\n \"At R$_\\\\mathrm {i}$ these clouds are large enough to completely occupy the volume along the line-of-sight to the stars: none of the stellar continuum is transmitted (see lower left panel in fig:cfnop for the face on view at the initial time, t$_0$ ).\",\n \"However, the starburst imparts energy and momentum to these clouds, accelerating them radially outwards (upper right panel of fig:cfnop).\",\n \"In the simplest scenario, these clouds retain their size as they move outward, and a gap appears between the clouds (see upper right panel in fig:cfnop).\",\n \"The gap allows the background stellar continuum to be transmitted, reducing the covering fraction of the stellar continuum.\",\n \"This is illustrated by the face on view in the lower right panel of fig:cfnop, where background stars become visible in the gaps between the clouds at higher velocities.\",\n \"This physical picture can be expressed numerically as a ratio of the cloud area to the total surface area at a given radius (r) as [60] $\\\\frac{C_f(\\\\mathrm {r})}{C_f(\\\\mathrm {R}_\\\\mathrm {i})} = \\\\frac{\\\\mathrm {A}_\\\\mathrm {c} (\\\\mathrm {r})}{4 \\\\pi \\\\mathrm {r}^2} \\\\frac{4 \\\\pi \\\\mathrm {R}_\\\\mathrm {i}^2}{\\\\mathrm {A}_\\\\mathrm {c}(\\\\mathrm {R}_\\\\mathrm {i})} = \\\\frac{\\\\mathrm {A}_\\\\mathrm {c} (\\\\mathrm {r})}{\\\\mathrm {A}_\\\\mathrm {c} (\\\\mathrm {R}_\\\\mathrm {i})} x^{-2}$ Where A$_c$ is the area of the individual cloudlets and $x = \\\\mathrm {r}/\\\\mathrm {R}_\\\\mathrm {i}$ .\",\n \"Assuming that the area of the clouds remains constant, the C$_f$ in this scenario scales as x$^{-2}$ , significantly steeper than the observed relation of x$^{-0.8}$ .\",\n \"Figure: Similar to fig:cfnop, but the clouds are in an external pressure gradient (orange shaded region).\",\n \"The initial pressure (P e ,0_\\\\mathrm {e,0}) is greater than the pressure at t 1 _1 (P e ,1_\\\\mathrm {e,1}).\",\n \"The cloudlets adiabatically expand to remain in pressure equilibrium with the external medium (see the arrow in the upper left cloud demonstrating it's initial diameter).\",\n \"Comparing the bottom right panel with the same panel in fig:cfnop, the expanded clouds now cover more area of the starburst at higher velocities.\",\n \"This slows the drop in C f _f.\",\n \"The scaling of the C f _f with distance depends on how the external medium changes, either isothermally (see eq:cftheoryiso) or adiabatically (see eq:cftheoryadia).However, this simple model of static clouds is not the most physical.\",\n \"These cloudlets are likely in pressure equilibrium with an external pressure (P$_\\\\mathrm {e} = \\\\mathrm {P}_\\\\mathrm {c}$ , where P$_\\\\mathrm {e}$ is the external pressure and P$_\\\\mathrm {c}$ is the pressure of the cloud), possibly a hot wind (see fig:cfp and the discussion in ionstruct).\",\n \"Changes in P$_\\\\mathrm {e}$ change P$_\\\\mathrm {c}$ , and the clouds expand adiabatically to account for these changes as P$_\\\\mathrm {c} \\\\propto V_\\\\mathrm {c}^{-\\\\gamma _\\\\mathrm {c}}$ , where V$_\\\\mathrm {c}$ is the volume of the clouds and $\\\\gamma _\\\\mathrm {c}$ is the adiabatic index (5/3 for monatomic ideal gas).\",\n \"The larger cloud volume reduces the gap between the individual cloudlets [60].\",\n \"The lower right panels of fig:cfnop and fig:cfp illustrate that the larger clouds cover more of the background stars than the static clouds do, and C$_f$ falls more slowly with distance.\",\n \"Approximating the outflow as spherical clouds provides a relation for A$_\\\\mathrm {c}$ in terms of the volume of the cloudlets as A$_\\\\mathrm {c} \\\\propto \\\\mathrm {V}_\\\\mathrm {c}^{2/3}$ .\",\n \"Using this approximation, eq:cfchanges, and the observed C$_f$ (C$_f \\\\propto x^{-0.8}$ ), the cloud volume changes with distance from the starburst as $\\\\frac{\\\\mathrm {V}_\\\\mathrm {c}(\\\\mathrm {r})}{\\\\mathrm {V}_\\\\mathrm {c}(\\\\mathrm {R}_\\\\mathrm {i})} = \\\\left(\\\\frac{\\\\mathrm {A}_\\\\mathrm {c}(\\\\mathrm {r})}{\\\\mathrm {A}_\\\\mathrm {c}(\\\\mathrm {R}_\\\\mathrm {i})}\\\\right)^{3/2} = \\\\left(\\\\frac{C_f(\\\\mathrm {r})}{C_f(\\\\mathrm {R}_\\\\mathrm {i})}x^2\\\\right)^{3/2} = x^{1.8 \\\\pm 0.4}$ This demonstrates that the cloud's volume increases with distance from the starburst, but what differential pressure is needed to change the volume as we observe?\",\n \"Here, we consider two different pressure laws for a mass-conserving external medium: isothermal expansion and adiabatic expansion.\",\n \"If the external pressure changes isothermally with radius then P$_\\\\mathrm {e}(\\\\mathrm {r})/\\\\mathrm {P}_\\\\mathrm {e}(\\\\mathrm {R}_\\\\mathrm {i}) = x^{-2}$ .\",\n \"Assuming that the spherical clouds and the isothermal external medium remain in pressure equilibrium (P$_\\\\mathrm {c} = \\\\mathrm {P}_\\\\mathrm {e}$ ), the pressure of the outflowing clouds changes as $\\\\frac{\\\\mathrm {P}_\\\\mathrm {c}(\\\\mathrm {r})}{\\\\mathrm {P}_\\\\mathrm {c}{\\\\mathrm {R}_\\\\mathrm {i}}} = \\\\left(\\\\frac{\\\\mathrm {V}_\\\\mathrm {c}(\\\\mathrm {r})}{\\\\mathrm {V}_\\\\mathrm {c}(\\\\mathrm {R}_\\\\mathrm {i})}\\\\right)^{-\\\\gamma _\\\\mathrm {c}} = \\\\left(\\\\frac{\\\\mathrm {A}_\\\\mathrm {c}(\\\\mathrm {r})}{\\\\mathrm {A}_\\\\mathrm {c}(\\\\mathrm {R}_\\\\mathrm {i})}\\\\right)^{-3\\\\gamma _\\\\mathrm {c}/2} = x^{-2}$ Solving for $\\\\frac{A_\\\\mathrm {c}(x)}{A_\\\\mathrm {c}(R_i)}$ in terms of x finds that A$_\\\\mathrm {c}$ changes as $x^{4/(3\\\\gamma _\\\\mathrm {c})}$ .\",\n \"Placing this into eq:cfchanges gives the C$_f$ scaling of $\\\\frac{C_f(\\\\mathrm {r})}{C_f(\\\\mathrm {R}_\\\\mathrm {i})} = x^{4/(3\\\\gamma _\\\\mathrm {c})} x^{-2} = x^{-1.2}$ For a $\\\\gamma _\\\\mathrm {c} = 5/3$ .\",\n \"If the clouds expand adiabatically while immersed in an isothermal external medium, the covering fraction declines less steeply with height than if the clouds maintain their initial size.\",\n \"eq:cftheoryiso is still too steep to match the observed scaling.\",\n \"Another possibility is that the external pressure changes adiabatically.\",\n \"In this situation P$_\\\\mathrm {e} \\\\propto x^{-2\\\\gamma _\\\\mathrm {e}}$ , where $\\\\gamma _\\\\mathrm {e}$ is the adiabatic index of the external medium.\",\n \"Following a similar process as above, the covering fraction of an adiabatically expanding outflow in pressure equilibrium with an adiabatically expanding external medium is [60] $\\\\frac{C_f(\\\\mathrm {r})}{C_f(\\\\mathrm {R}_\\\\mathrm {i})} = x^{4\\\\gamma _\\\\mathrm {e}/(3\\\\gamma _\\\\mathrm {c}) -2} = x^{-2/3}$ where we have assumed that $\\\\gamma _\\\\mathrm {e}$ and $\\\\gamma _\\\\mathrm {c}$ are equal to each other (i.e.\",\n \"both the clouds and external medium are monatomic ideal gases).\",\n \"Interestingly, since diatomic gas has more degrees of freedom than monatomic gas, diatomic gas requires a larger energy change to remain in pressure equilibrium, as seen by the smaller $\\\\gamma _\\\\mathrm {c}$ of diatomic gas (7/5).\",\n \"Using eq:cftheoryadia, we find that molecular clouds expand more rapidly, and consequently have a more gradually declining C$_f$ with distance, as x$^{-0.4}$ .\",\n \"This has an important consequence for the survival of molecular clouds in a hot medium.\",\n \"By increasing the radius of the cloud, it takes longer for a shock wave to propagate across the cloud and dissociate the molecular gas [48], .\",\n \"This may extend the lifetime of molecular gas entrained in galactic outflows, while the increased C$_f$ may explain the presence of molecular gas at high velocities [62], , [55].\",\n \"In beta, we find that the covering fraction scales as $C_f~=~1.0~x^{-0.8 \\\\pm 0.2}$ , in agreement with the adiabatic expansion of cloudlets in an adiabatically expanding external medium (eq:cftheoryadia).\",\n \"However, it is an interesting theoretical question whether clouds can remain in pressure equilibrium with an external medium and still be accelerated.\",\n \"Ram pressure introduces a secondary pressure term, but if the clouds are shielded from direct interaction with the hot medium then the pressure of the cloud is set by the external thermal pressure [12].\",\n \"Further, ablation and ram pressure stripping create a fractal distribution into smaller cloudlets [48], , which may produce the observed change in covering fraction.\",\n \"The velocity scaling of the covering fraction provides further constraints for simulations of the acceleration of clouds.\",\n \"The acceleration of the cloudlets out of the star forming region decreases C$_f$ , but how are the clouds accelerated?\"\n ],\n [\n \"Velocity Law\",\n \"The velocity law describes the acceleration of the outflowing clouds with distance.\",\n \"In beta we find a $\\\\beta $ velocity profile of $\\\\mathrm {v(r)}~=~\\\\mathrm {v}_\\\\infty ~(1~-~\\\\frac{\\\\mathrm {R}_\\\\mathrm {i}}{\\\\mathrm {r}})^{0.43 \\\\pm 0.07}$ .\",\n \"The outflow initially accelerates rapidly, but the acceleration moderates at large radii (see the black line in fig:vellaw).\",\n \"Using the initial radius calculated in ionstruct, the outflow reaches 50% (90%) of the maximum velocity in 79 pc (291 pc).\",\n \"Acceleration much beyond this is not well constrained by the data.\",\n \"Outflows typically have a \\\"saw-tooth\\\" line profile , where the red side of the profile sharply declines and the blue side rises gradually.\",\n \"The velocity and C$_f$ laws produce these profiles.\",\n \"Initially, the velocity sharply increases over a short distance, keeping the C$_f$ near unity because C$_f$ scales as x$^{-0.82}$ .\",\n \"Meanwhile, at higher velocities the clouds travel larger distances per velocity interval, forcing the high-velocity clouds to expand to remain in pressure equilibrium with the adiabatically expanding external medium.\",\n \"This moderates the decline in C$_f$ , and produces the gentler rising blue portion of the profile.\",\n \"The observed Si IV 1402Å outflows do not escape the galactic potential.\",\n \"Following [36], the escape velocity is no more than three times the circular velocity [13], therefore the outflow velocity needs to exceed 651 km s$^{-1}$ for the clouds to escape the potential.\",\n \"There are two possible explanations for why we do not observe gas escaping the potential: (1) the outflow does eventually exceed the escape velocity but the Si IV density drops below the detection limits at high velocities and we do not observe the escaping gas (see density) or (2) the outflow does not actually escape the galaxy, but rather recycles back into the disc as a galactic fountain .\",\n \"The latter is consistent with [13], which finds most galaxies with log($M_\\\\ast $ /M$_\\\\odot $ ) greater than 10.5 cannot drive outflows faster than their escape velocity, unless they are undergoing a merger.\",\n \"Observations probing lower density gas, such as Lyman-$\\\\alpha $ , may constrain whether these outflows are capable of escaping the gravitational potential.\",\n \"Now we study how these outflows are accelerated.\",\n \"There are numerous theoretical ways to accelerate galactic outflows: radiation pressure on dust grains , , cosmic rays [23], , and ram pressure of a hot wind on the cloudlets , [16], [28], [60].\",\n \"Below, we use analytical expressions for how the velocity profile evolves radially to explore which mechanisms could accelerate these outflows.\",\n \"In each case, we give the scaling of the normalized velocity (w = v/v$_\\\\infty $ ; where v$_\\\\infty $ is the maximum velocity) with the normalized radius (x = r/R$_\\\\mathrm {i}$ ; where R$_\\\\mathrm {i}$ is the initial radius).\",\n \"We then scale the analytical expressions to the observed relations to determine the plausibility of each mechanism.\",\n \"In velsum we summarize the implications for the theoretical profiles.\"\n ],\n [\n \"Optically Thick Radiation Pressure\",\n \"Radiation pressure is an attractive way to drive outflows in dusty, vigorously star-forming galaxies: the high luminosity provides a large momentum source , while the large dust optical depth scatters photons multiple times .\",\n \"give the radial scaling of the velocity for optically thick radiation pressure as: $w = \\\\sqrt{\\\\frac{4\\\\sigma ^2}{\\\\mathrm {v}_\\\\infty ^2} (\\\\frac{L}{L_\\\\mathrm {E}}-1)~\\\\text{ln}(x)}$ where $\\\\sigma $ is the velocity dispersion of the galaxy, L is the luminosity of the galaxy, and L$_\\\\mathrm {E}$ is the Eddington luminosity.\",\n \"We fit for the constant value that best matches the observed $\\\\beta $ -law using MPFIT [58], while excluding velocities less than 0.2.\",\n \"The fit is shown by the blue dot-dashed line in fig:vellaw.\",\n \"The optically thick radiation model poorly matches the observations.\"\n ],\n [\n \"An r$^{-2}$ Force\",\n \"A second appealing driving mechanism is ram pressure.\",\n \"Supernovae thermalize ambient gas into a hot wind, which expands adiabatically out of the star forming region.\",\n \"The hot wind imparts a ram pressure force on the clouds, which depends on the speed and density of the hot wind and the area of the cloud as F$_\\\\mathrm {ram} = \\\\rho _\\\\mathrm {h} \\\\mathrm {v}_\\\\mathrm {h}^2 \\\\mathrm {A}_\\\\mathrm {c}$ [48], .\",\n \"F$_\\\\mathrm {ram} \\\\propto x^{-2}$ in a simplified case of a mass conserving, adiabatically expanding hot wind.\",\n \"Ram pressure driving is appealing because the C$_f$ scaling is consistent with clouds being in pressure equilibrium with a adiabatically expanding external medium (cf), a situation that could lead to ram pressure driving.\",\n \"Including the effects of gravity, calculate the velocity profile of a ram pressure driven outflow as $w(x) = \\\\sqrt{\\\\frac{v_\\\\mathrm {c}^2}{\\\\mathrm {v}_\\\\infty ^2} (1-\\\\frac{1}{x}) -\\\\frac{4\\\\sigma ^2}{\\\\mathrm {v}_\\\\infty ^2} \\\\mathrm {ln}(x)}$ Where v$_c$ , the characteristic velocity clouds reach before gravity dominates, is defined as v$_\\\\mathrm {c} = \\\\frac{3\\\\dot{M}_\\\\mathrm {h} \\\\mathrm {V}_\\\\mathrm {h}}{8\\\\pi \\\\rho _\\\\mathrm {c} \\\\mathrm {R}_\\\\mathrm {c} \\\\mathrm {R}_\\\\mathrm {i}}$ .\",\n \"Defining A as v$_\\\\mathrm {c}^2/\\\\mathrm {v}_\\\\infty ^2$ and B as $4\\\\sigma ^2/\\\\mathrm {v}_\\\\infty ^2$ , we fit for the values of A (fitted value of 1.10) and B (fitted value of 0.03) that match the observed $\\\\beta $ velocity law, as shown by the red line in fig:vellaw.\",\n \"Similarly, shockwaves from supernovae accelerate cosmic rays (CR).\",\n \"These relativistic particles stream out of the star forming regions along the magnetic fields and exchange momentum with magnetized plasma.\",\n \"Cosmic rays have various advantages over radiation pressure and ram pressure, including: CRs interact multiple times with the gas, magnetic fields confine CRs to the galaxy making it difficult for CRs to escape without imparting momentum, and CR feedback is independent of the distribution of ISM gas [23], .\",\n \"derive the force imparted by cosmic rays as $F_\\\\mathrm {CR} = \\\\frac{\\\\kappa _\\\\mathrm {CR}}{c} \\\\frac{L_\\\\mathrm {CR}}{4 \\\\pi r^2}$ where $\\\\kappa _\\\\mathrm {CR}$ is the cosmic ray opacity, and L$_\\\\mathrm {CR}$ is the cosmic ray luminosity.\",\n \"This force produces a similar velocity scaling as the ram pressure scaling in eq:ramvel.\",\n \"Additionally, the scaling of the ram pressure and CR velocity profiles are similar to an optically thin radiation profile .\",\n \"In fact, eq:ramvel is a general form for any r$^{-2}$ force that opposes gravity.\",\n \"Therefore, the red dashed line in fig:vellaw corresponds to outflows driven by any r$^{-2}$ force (ram pressure, cosmic rays, or optically thin radiation pressure), and we cannot distinguish these mechanisms from the measured velocity profile.\"\n ],\n [\n \"Velocity Summary\",\n \"The ram pressure, cosmic rays, and optically thin radiation pressure velocity profile (any r$^{-2}$ force) matches the observed velocity relation for all observed velocities, while optically thick radiation pressure poorly matches the observed velocity profile (fig:vellaw).\",\n \"In fig:vellaw, the r$^{-2}$ profile begins to decelerate at a radius of 2.5 kpc (r$_\\\\text{turn}$ ; marked by the vertical line in fig:vellaw).\",\n \"However, at these large radii the outflow is now diffuse, and the deceleration of the outflow is challenging to study with the Si IV line profile.\"\n ],\n [\n \"Density Law\",\n \"In beta we find the density to scale with the normalized radius as $\\\\mathrm {n}(x) \\\\propto x^{-5.72}$ .\",\n \"In fig:denlaw we show the measured relation for the outflow density with velocity.\",\n \"At low velocities, the density remains nearly constant because the outflow accelerates over a short distance , but at higher velocities the density precipitously drops.\",\n \"Stronger transitions probe lower densities than weaker transitions, causing stronger transitions to probe higher velocities than the weaker transitions.\",\n \"Therefore, stronger transitions have larger measured terminal velocities and larger line widths (because they probe a wider velocity distribution) than the weaker transitions [32], [14].\",\n \"This density-velocity scaling implies that the outflow traced by Si IV does not conserve mass.\",\n \"Fitting the $\\\\tau $ distribution with a mass-conserving flow produces a poor fit because the $\\\\tau $ distribution declines sharply at high velocity.\",\n \"In the continuity equation, the mass flux is conserved, such that the density scales as $\\\\mathrm {n}(w) \\\\propto \\\\frac{n_\\\\mathrm {0}}{x^2v} \\\\propto \\\\frac{n_\\\\mathrm {0}}{v_\\\\infty }\\\\frac{(1-w^{1/\\\\beta })^2}{w}$ In fig:denlaw we compare the mass-conserving density profile (red dashed line) with the observed profile.\",\n \"Compared to the observed density profile, the density of a mass-conserving flow rapidly decreases at low velocities as the outflow accelerates over small distances, while at higher velocities the conserved density decreases more gradually because the velocity gradient is flatter (see fig:vellaw).\",\n \"If the Si IV flow is not mass-conserving, what happens to the Si IV gas?\",\n \"The density could decrease by: (1) increasing the volume of the outflowing clouds while keeping the number of Si IV ions constant or (2) decreasing the number of Si IV ions in the clouds.\",\n \"eq:volume shows that the volume of the outflowing clouds scales as x$^{1.77}$ , implying that the number of Si IV ions in the outflow must decrease as x$^{-3.95}$ to satisfy the observed density relation.\",\n \"One possibility is that the outflowing clouds lose mass through ablation, ram pressure stripping, or conduction from the hot wind [48], , , [8].\",\n \"In this scenario, the hot wind rapidly destroys the clouds and incorporates them into the hot wind, making the mass undetectable through Si IV absorption at higher velocities.\",\n \"A steep density scaling relation is also seen in the nearby starburst M 82.\",\n \"[55] probe the surface density profiles of the molecular, neutral and ionized phases of the outflow from M 82.\",\n \"They find that only the H I follows an n$ \\\\propto \\\\mathrm {r}^{-2}$ density law largely because of the large amount of tidal material present at large radii.\",\n \"However, other surface density profiles along the minor axis decrease more rapidly.\",\n \"The 70 $\\\\mu $ m emission, a tracer of warm dust, and CO emission, a tracer of diffuse molecular gas, scale with the distance from the starburst as n$ \\\\propto \\\\mathrm {r}^{-4}$ [55], roughly consistent, with the density scaling found here.\",\n \"The authors use the divergence of the density profile to argue that the outflow of M82 is a galactic fountain, with gas leaving the outflow and recycling back into the disk.\"\n ],\n [\n \"Mass Outflow Rate\",\n \"Finally, we combine all of the derived relations to measure the mass outflow rate ($\\\\dot{M}_o$ = dM$_o$ /dt) of the photoionized galacitc outflow.\",\n \"The photoionized outflow is a single component of the outflow, and other, unexplored, phases likely contribute to the total mass outflow rate.\",\n \"Since the hot phase is such low density, this photoionized phases likely dominates the mass budget of the outflow.\",\n \"Typically the mass outflow rate is calculated assuming the outflow is in a thin spherical shell as $\\\\dot{M}_\\\\mathrm {o} = \\\\Omega {C}_{f} \\\\mu \\\\mathrm {m}_\\\\mathrm {p} \\\\mathrm {N}_\\\\mathrm {H} \\\\mathrm {r} \\\\mathrm {v}_\\\\mathrm {cen}$ where $\\\\mu m_\\\\mathrm {p}$ is 1.4 times the mass of the proton, $\\\\Omega $ is the angular covering fraction of the wind and v$_\\\\mathrm {cen}$ is the centroid velocity of the outflow (132 km s$^{ -1}$ ).\",\n \"Previous studies assume a radius of 5 kpc, a full opening angle of 140$^\\\\circ $ (and $\\\\Omega = 3.11\\\\pi $ steradians), solar metallicity, and no ionization correction , [59], , .\",\n \"Using the values from the Si IV line, and these assumptions, we derive a mass outflow rate for NGC 6090 of 7.32 M$_\\\\odot $ yr$^{-1}$ .\",\n \"However, previous studies typically use absorption lines of cooler gas like Na I, Mg II and Fe II , [59], , [60], .\",\n \"If we use the Si II 1304 column density, a similar ionization potential to the previous outflow tracers, $\\\\dot{M}_{\\\\text{o}}$ rises to 22.6 M$_\\\\odot $ yr$^{-1}$ .\",\n \"However, with our tightly constrained physical model of the outflow, now we only have to assume an angular covering fraction to calculate the mass outflow rate.\",\n \"Above, we derive a radius, a metallicity, and an ionization fraction for the outflows (see tab:ion for values).\",\n \"If we use the radius of peak optical depth (R$_\\\\mathrm {p}$ = 72.2 pc, at a w of 0.41), we calculate a total mass outflow rate of 0.81 M$_\\\\odot $ yr$^{-1}$ , 28 times lower than the $\\\\dot{M}_{\\\\text{o}}$ calculated using Si II.\",\n \"While this calculation uses many of our derived values, it ignores the evolution of these quantities with velocity.\",\n \"In velocity we find that the outflow is accelerated over short distances, in cf we find that the covering fraction decreases at large velocities, and in density we find that the outflow is not a mass conserving flow, rather it evolves as x$^{-5.7}$ .\",\n \"All of these impact how mass is distributed in velocity space.\",\n \"Using the scaling of the covering fraction, radius, and density with velocity, we define the mass outflow rate per velocity as $\\\\begin{aligned}\\\\dot{M}_\\\\mathrm {o}(\\\\mathrm {r}) &= \\\\Omega C_{f}(\\\\mathrm {r}) \\\\mathrm {v}(\\\\mathrm {r}) \\\\rho (\\\\mathrm {r}) \\\\mathrm {r}^2 \\\\\\\\\\\\dot{M}_\\\\mathrm {o}(w)&= \\\\Omega C_f(\\\\mathrm {R}_\\\\mathrm {i}) \\\\mathrm {v}_\\\\infty \\\\mu \\\\mathrm {m}_\\\\mathrm {p} \\\\mathrm {n}_\\\\mathrm {0} \\\\mathrm {R}_\\\\mathrm {i}^2 \\\\frac{w}{(1-w^{1/\\\\beta })^{2+\\\\gamma +\\\\alpha }} \\\\\\\\&= 10.1~\\\\text{M}_\\\\odot ~\\\\text{yr}^{-1} \\\\frac{w}{(1-w^{1/\\\\beta })^{2+\\\\gamma +\\\\alpha }}\\\\end{aligned}$ where we use the values from tab:beta and tab:ion for the constants and the exponents, and the only assumed parameter is that $\\\\Omega $ is 3.11$\\\\pi $ steradians.\",\n \"The $\\\\dot{M}_{\\\\text{o}}$ relation is not constant with velocity, as typically assumed (fig:mout): $\\\\dot{M}_{\\\\text{o}}$ increases rapidly at low velocities as the outflow accelerates, and declines at high velocities as the density and covering fraction of the outflow decline (fig:vellaw and fig:denlaw).\",\n \"At w of 0.35, the $\\\\dot{M}_{\\\\text{o}}$ relation peaks at a value of 2.3 M$_\\\\odot $ yr$^{-1}$ .\",\n \"This is a factor of ten times smaller than the Si II value calculated with previous assumptions for geometry, ionization fractions, and metallicities.\",\n \"Combining $\\\\dot{M}_{\\\\text{o}}$ with the derived outflow metallicity, we find that the maximum metal outflow rate is 0.07 M$_\\\\odot $ yr$^{-1}$ .\",\n \"This may constrain the impact of outflows on the mass-metallicity relationship , [26], , [3], , [17], [15].\",\n \"It is typical to normalize the mass outflow rate by the star formation rate to produce the \\\"mass-loading\\\" factor .\",\n \"The global $\\\\eta $ is 0.09.\",\n \"This $\\\\eta $ is smaller than recent measurements from [38], who find an $\\\\eta $ near 0.3 for galaxies with SFRs near 30 M$_\\\\odot $ yr$^{-1}$ .\",\n \"However, [38] use the mean ISM metallicity of the sample (0.5 Z$_\\\\odot $ ), do not calculate ionization fractions, and use a multiple of the half-light radius (typically 0.5-1 kpc) as the radius of the outflow.\",\n \"Other studies make similar assumptions about the metallicity and the ionization state, but assume constant outflow radii between 1-5 kpc , [61], .\",\n \"These studies typically find $\\\\eta $ between 0.02 and 1 for SFRs and $M_\\\\ast $ similar to NGC 6090.\",\n \"The method outlined here does not rely on these uncertain assumptions and allows for these quantities to vary from galaxy to galaxy, reducing the scatter in the measurements.\",\n \"The relatively small inner outflow radius implies that the local SFR may be more important than the global SFR.\",\n \"In [13] we calculate the fraction of the total GALEX UV flux within the COS aperture to be 22%, leading to a SFR within the COS aperture of 5.55 M$_\\\\odot $ yr$^{-1}$ , however this does not account for the significant spatial variation in the IR flux and is only a crude approximation for the local SFR.\",\n \"Using this local SFR we calculate a local $\\\\eta $ within the COS aperture of 0.42.\",\n \"The local and global $\\\\eta $ values are a factor of 5 and 22 times lower than the redshift independent $\\\\eta $ predicted by the FIRE simulation [39], [41], , although the high-mass FIRE galaxies at redshift 0 only have upper limits for $\\\\eta $ .\",\n \"[34] develop an analytical model that uses the momentum from stellar feedback to regulate star formation and drive outflows.\",\n \"A requirement of this model is that $\\\\eta $ depends on the gas fraction and the stellar mass of the host galaxy.\",\n \"Making approximations for these scalings with redshift, the analytical model predicts the mass-loading for a log($M_\\\\ast $ /M$_\\\\odot $ ) galaxy of 10.7 at $z\\\\approx 0.03$ to be 0.05, in rough agreement with the observed global $\\\\eta $ .\",\n \"While the outflow from NGC 6090 is weak compared to the SFR, [34] predict that 10$^9$ M$_\\\\odot $ galaxies drive outflows with an $\\\\eta $ of 10 at $z\\\\sim 0$ .\",\n \"In a future paper we will explore the scaling of $\\\\dot{M}_{\\\\text{o}}$ and $\\\\eta $ with host galaxy properties to better constrain these types of studies.\",\n \"Figure: The total mass within the outflow at each velocity, as given by eq:m. The total integrated outflow mass is 7.5×10 5 7.5 \\\\times 10^5 M ⊙ _\\\\odot .The total mass in the outflow (M$_o$ ) at a particular velocity is given by $\\\\begin{aligned}M_\\\\mathrm {o} (w) &= \\\\Omega C_f(\\\\mathrm {r}) \\\\mu \\\\mathrm {m}_\\\\mathrm {p} \\\\mathrm {n}_\\\\mathrm {0} \\\\mathrm {R}_\\\\mathrm {i}^3 \\\\left(\\\\frac{1}{1-w^{1/\\\\beta }}\\\\right)^{3+\\\\alpha +\\\\gamma } \\\\\\\\& = 1.5 \\\\times 10^{6} M_\\\\odot \\\\left(\\\\frac{1}{1-w^{1/\\\\beta }}\\\\right)^{3+\\\\alpha +\\\\gamma }\\\\end{aligned}$ This relation is shown in fig:m. Like the density, the total mass in the Si IV outflow declines with velocity.\",\n \"The integrated mass between w of 0 and 1 is $7.5 \\\\times 10^{5}~M_\\\\odot $ .\",\n \"While the current mass of outflowing gas is significantly lower than the measured H I mass of 10$^{10.2}$ M$_\\\\odot $ , if $\\\\dot{M}_{\\\\text{o}}$ remains constant over the $\\\\sim 1$ Gyr time scale of the merger, than the outflow will process (and enrich) 15% of the observed H I in NGC 6090, possibly leading to the metal enriched gas seen in the halos of galaxies , , , .\"\n ],\n [\n \"CONCLUSION\",\n \"Here we measure a physically motivated mass outflow rate ($\\\\dot{M}_{\\\\text{o}}$ ) of the nearby starburst NGC 6090.\",\n \"To calculate $\\\\dot{M}_{\\\\text{o}}$ , we first fit the optical depth with a Sobolev optical depth, and the covering fraction with a radial power-law (see eq:cfbeta and fig:beta).\",\n \"We then calculate the ionization corrections and the metallicity of the outflow using a Bayesian analysis, CLOUDY models, and the measured column densities (see tab:ion and fig:ionpdfs).\",\n \"The main results of this study are: The ionization model estimates the metallicity (1.61 Z$_\\\\odot $ ), density (18.73 cm$^{-3}$ ) and the ionization parameter (log(U) = -1.85) of the outflow.\",\n \"The estimated H column density is consistent with that derived from the UV continuum extinction.\",\n \"The outflow is at least as metal enriched as the ISM of the host galaxy (see ionstruct).\",\n \"Using the absorption line profile and ionization models, we determine that the inner edge (R$_\\\\mathrm {i}$ ) of the outflow is 63 pc from the starburst (see eq:r), consistent with the absorption arising from the shredded blastwaves of supernovae remnants.\",\n \"Most of the outflow is constrained within 300 pc of the starburst (ionstruct).\",\n \"The covering fraction scales as C$_f = 1.0 (\\\\mathrm {r}/\\\\mathrm {R}_\\\\mathrm {i})^{-0.8\\\\pm 0.2}$ (tab:beta).\",\n \"This scaling relation is consistent with predictions of clouds in pressure equilibrium with an adiabatically expanding external medium (cf).\",\n \"This warrents further study of whether outflowing clouds remain in pressure equilibrium with an external medium, and if a fractal cloud distribution could produce a similar scaling relation.\",\n \"The Si IV outflow velocity scales with radius as v$~=~\\\\mathrm {v}_\\\\infty (1~-~\\\\mathrm {R}_\\\\mathrm {i}/\\\\mathrm {r})^{0.43 \\\\pm 0.07}$ (fig:vellaw and tab:beta), where v$_\\\\infty $ is the maximum velocity.\",\n \"We compare this velocity profile to models of different driving mechanisms, and find that an r$^{-2}$ force law matches the observed profile (see fig:vellaw and velocity).\",\n \"The outflow density decreases with radius as n$ \\\\propto \\\\mathrm {r}^{-5.7 \\\\pm 1.5}$ (fig:denlaw), which declines more rapidly than a mass conserving flow (density).\",\n \"This rapid density reduction could be due to interactions between the outflowing clouds and a hotter wind.\",\n \"Combining all of our measurements, we derive a maximum mass outflow rate ($\\\\dot{M}_{\\\\text{o}}$ ) of 2.3 M$_\\\\odot $ yr$^{-1}$ (see fig:mout).\",\n \"The mass-loading factor (mass outflow rate divided by star formation rate) is 0.09.\",\n \"The $\\\\dot{M}_{\\\\text{o}}$ is a factor of 10 lower than the $\\\\dot{M}_{\\\\text{o}}$ calculated using common assumptions for ionization state, metallicity, and geometry (see mout).\",\n \"In future work, we will continue this analysis for a larger sample of star forming galaxies, studying how the outflow properties (metallicity, initial radius, mass outflow rate) scale with host galaxy properties.\",\n \"These scaling relations will constrain future models of galaxy evolution.\"\n ],\n [\n \"Acknowledgments\",\n \"We thank the anonymous referee for constructive comments that strengthened the paper.\",\n \"Joseph Cassinelli inspired this work with helpful conversations and notes.\",\n \"We thank Bart Wakker for help with the data reduction and discussions on the analysis.\",\n \"Support for program 13239 was provided by NASA through a grant from the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5-26555.\",\n \"Some of the data presented in this paper were obtained from the Mikulski Archive for Space Telescopes (MAST).\",\n \"STScI is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5-26555.\",\n \"All of the HST data presented in this paper were obtained from the Mikulski Archive for Space Telescopes (MAST).\",\n \"STScI is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS5-26555.\",\n \"Support for MAST for non-HST data is provided by the NASA Office of Space Science via grant NNX09AF08G and by other grants and contracts.\",\n \"Funding for the Sloan Digital Sky Survey (SDSS) has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Aeronautics and Space Administration, the National Science Foundation, the U.S. Department of Energy, the Japanese Monbukagakusho, and the Max Planck Society.\",\n \"The SDSS Web site is http: www.sdss.org/.\",\n \"The SDSS is managed by the Astrophysical Research Consortium (ARC) for the Participating Institutions.\",\n \"The Participating Institutions are The University of Chicago, Fermilab, the Institute for Advanced Study, the Japan Participation Group, The Johns Hopkins University, Los Alamos National Laboratory, the Max-Planck-Institute for Astronomy (MPIA), the Max-Planck-Institute for Astrophysics (MPA), New Mexico State University, University of Pittsburgh, Princeton University, the United States Naval Observatory, and the University of Washington.\"\n ]\n]"},"id":{"kind":"string","value":"1605.05769"}}},{"rowIdx":1866684,"cells":{"text":{"kind":"list like","value":[["Computing the n-th coefficient of an algebraic power series modulo p in\n O(log n) operations"],["Abstract This is an exposition, for pedagogical purposes, of the formal power series proof of Bostan, Christol and Dumas [3] of the result stated in the title (a corollary of the Christol theorem)."],["This is an exposition, for pedagogical purposes, of the formal power series proof of Bostan, Christol and Dumas [3] of the result stated in the title (a corollary of the Christol theorem).","Introduction The generating function $c=c(x)=\\sum _{n\\ge 1}c_nx^n$ of the Catalan numbers $c_n=\\frac{1}{n}\\binom{2n-2}{n-1}=1,1,2,5,14,42,\\dots $ satisfies the quadratic equation $c=c^2+x\\;.$ Can we effectively determine parity of $c_n$ ?","Very easily: modulo 2 one has $c(x)=c(x)^2+x\\equiv c(x^2)+x$ (by Lemma C below), and therefore $c_n\\equiv c_{n/2}$ mod 2 for $n>1$ and $c_1\\equiv 1$ mod 2 ($c_{n/2}=0$ if $n/2\\notin \\mathbb {N}_0$ ).","Hence $c_n\\equiv 1$ mod 2 for $n=2^m$ , $m\\in \\mathbb {N}_0$ , and $c_n\\equiv 0$ mod 2 else .","And $c_n$ modulo 3 or modulo other prime $p$ ?","An answer to this is provided much more generally by the Christol theorem, proved by Christol [5] for $q=2$ and by Christol, Kamae, Mendès France and Rauzy [6] for any prime power $q$ (see also Allouche and Shallit [1]): a power series $f(x)\\in F_q[[x]]$ is algebraic over $F_q(x)$ if and only if one can generate its coefficients $f_0,f_1,\\dots $ by a DFAO (a deterministic finite automaton with output that reads the $l+1$ digits of the $q$ -adic expansion $n=(n_l\\dots n_1n_0)_q$ and outputs $f_n$ , we give one example in Concluding remarks).","It is immediate from the Christol theorem that one can compute the $n$ th coefficient of an algebraic power series in $F_q[[x]]$ in $O(\\log n)$ arithmetic operations because $n$ has $O(\\log n)$ $q$ -adic digits.","This write-up is an exposition of the proof of this result, stated as Theorem A below, that was given by means of power series in the recent preprint of Bostan, Christol and Dumas [3].","I promised in my course `Kombinatorické počítání' (taught in this summer semester 2015/16) to supplement my unfortunately not very clear oral presentation of (something like) Theorem A with a written one — here it is.","Algebraic power series in $F_p[[x]]$ First we review some notation and notions from algebra.","Puiseux series $K((x))_P$ and their order are only used at the end in Proposition H that is not needed to prove Theorem A.","$\\mathbb {N}=\\lbrace 1,2,\\dots \\rbrace $ , $\\mathbb {N}_0=\\lbrace 0,1,\\dots \\rbrace $ , $\\mathbb {Z}$ is the ring of integers, $F_p=(\\mathbb {Z}/p\\mathbb {Z},+,\\cdot )$ is the finite $p$ -element field where $p$ is a prime number, and $[x^n]f$ denotes the coefficient of $x^n$ in $f$ .","For $p\\in \\mathbb {N}$ , $p\\ge 2$ , the $p$ -adic expansion $m=(m_l\\dots m_1m_0)_p$ of $m\\in \\mathbb {N}_0$ is given by the unique expression $m=\\sum _{i=0}^lm_ip^i$ where $m_i\\in \\lbrace 0,1,\\dots ,p-1\\rbrace $ and (for $m>0$ ) $m_l\\ne 0$ ; $0=(0)_p$ .","Let $K$ be a field or an integral domain.","For a field we denote by $\\overline{K}$ its algebraic closure (e.g.","$\\overline{\\mathbb {R}}=\\mathbb {C}$ or $\\overline{\\mathbb {Q}}$ are the algebraic numbers).","We denote by $K[[x]]\\subset K((x))\\subset K((x))_P$ , respectively, the ring of power series and the fields of Laurent series and Puiseux series with coefficients in $K$ .","Their elements are the formal series (we treat here all power etc.","series formally) $\\sum _{n\\ge k}a_{n/d}x^{n/d}$ , $a_{n/d}\\in K$ , where $k=0$ and $d=1$ for p. s., $k\\in \\mathbb {Z}$ and $d=1$ for L. s., and $k\\in \\mathbb {Z}$ and $d\\in \\mathbb {N}$ for P. s. For $f=f(x)\\in K((x))$ we use notation $f_n:=[x^n]f(x)$ .","From the algebraic point of view, $K((x))$ is the field of fractions of $K[[x]]$ and for $K$ of characteristic 0 (e.g.","$K=\\mathbb {Q}$ ) $\\overline{K}((x))_P$ is the algebraic closure of $K((x))$ .","Thus in $\\overline{K}((x))_P$ every polynomial with coefficients in $K((x))$ has a root.","Recall that $K[x]\\subset K(x)$ is, respectively, the ring of polynomials with coefficients in $K$ and its field of fractions ($K(x)$ are the rational `functions' over $K$ ).","In fact, $K(x)$ is naturally a subfield of $K((x))$ and $K[x]$ a subring of $K[[x]]$ .","By $\\deg a$ , $a\\in K[x]$ , we denote the degree of a polynomial and by $\\deg _y a$ , $a\\in K[x,y]$ , the degree $\\deg a$ when $a$ is understood as $a\\in K[x][y]$ .","By $\\mathrm {ord\\,}f$ , $f\\in K((x))_P$ , we denote the order of $f$ , the minimum $n/d$ such that $a_{n/d}\\ne 0$ .","We set $\\deg 0=-\\infty $ and $\\mathrm {ord\\,}0=+\\infty $ .","For any $f,g\\in K((x))_P$ one has $\\mathrm {ord\\,}(fg)=\\mathrm {ord\\,}f+\\mathrm {ord\\,}g$ and $\\mathrm {ord\\,}(f+g)\\ge \\min (\\mathrm {ord\\,}f,\\mathrm {ord\\,}g)$ , with equality if $\\mathrm {ord\\,}f\\ne \\mathrm {ord\\,}g$ .","Similarly, for any $f,g\\in K[x]$ one has $\\deg (fg)=\\deg f+\\deg g$ and $\\deg (f+g)\\le \\max (\\deg f,\\deg g)$ , with equality if $\\deg f\\ne \\deg g$ .","We state the basic result on computing coefficients of an algebraic power series in $F_p[[x]]$ (this is essentially proven in $\\cite {chri_al}$ but the formulation below is ours).","Theorem A ([6], 1980).","Suppose that $p$ is prime, $P\\in \\mathbb {Z}[x,y]$ is a nonzero polynomial, $f=f_0+f_1x+f_2x^2+\\dots \\in \\mathbb {Z}[[x]]$ is a power series satisfying $P(x,f(x))=0$ , and $h=(d+1)(p^d-d+1)\\deg _xP\\ \\mbox{ where }\\ d=\\deg _yP\\ (\\ge 1)\\;.$ Then from given $p,P,f_0,f_1,\\dots ,f_h$ one can construct an algorithm (actually a DFAO) that computes $n\\mapsto f_n\\ \\mathrm {mod}\\ p$ in $O(\\log n)$ arithmetic operations ($n\\ge 2$ ).","But which operations exactly?","These are arithmetic operations ($+,-,\\times ,:$ ) in the field $F_p$ and the operation of division with remainder in the ring $\\mathbb {Z}$ .","By `one can construct' here and below we mean that the task can be performed constructively (i.e.","the algorithms or objects not merely exist but exist effectively).","Theorem A follows from the following more specific result (again the formulation is ours).","Theorem B ([3]).","Let $p$ , $P$ , $f$ , and $h$ be as in Theorem A.","We denote the mod $p$ reduction of $f$ again by $f\\in F_p[[x]]$ .","Then from given $p,P,f_0,f_1,\\dots ,f_h$ one can construct polynomials $a,c\\in F_p[x]$ , a number $e\\in \\mathbb {N}$ , a row $u\\in F_p^{1\\times e}$ , $p$ matrices $A_0,A_1,\\dots ,A_{p-1}\\in F_p^{e\\times e}$ , and a column $v\\in F_p^{e\\times 1}$ such that for any $n\\in \\mathbb {N}_0$ one has $[x^n]f(x)&=&f_n=[x^n](c(x)+a(x)h(x))\\ \\mbox{ where }\\\\&&h(x)=\\sum _{m=0}^{\\infty }(uA_{m_l}\\dots A_{m_1}A_{m_0}v)x^m\\in F_p[[x]]$ ($m=(m_l\\dots m_1m_0)_p$ is the $p$ -adic expansion of $m$ ).","Polynomial $P$ in Theorem A in general does not determine unique power series solution $y=f(x)$ of $P(x,y)=0$ , e.g.","$y^2-x^2=0$ has two solutions $y=x$ and $y=-x$ .","One can introduce restrictions, for example to require that $P(0,0)=0$ , $P_y(0,0)\\ne 0$ , and $f(0)=0$ , producing unique solution, but we want result for completely general $P$ .","So to determine $f(x)$ we need to know a few initial coefficients, which is captured by the parameter $h$ in Theorems A and B. Bounds on $h$ are considered in Corollary G and Proposition H. Let us see how Theorem B implies Theorem A.","We are given a prime $p$ , a nonzero $P\\in \\mathbb {Z}[x,y]$ , and first few coefficients $f_0,f_1,\\dots ,f_h$ of a solution $y=f(x)\\in \\mathbb {Z}[[x]]$ of the equation $P(x,y)=0$ , where $h$ is as in Theorem A.","(We assume this is an honest input, $f_0,f_1,\\dots ,f_h$ are really initial coefficients of a solution.","We leave as an exercise for the reader to design an effective check that it is true.)","Taking out the largest common factor of the coefficients of $P(x,y)$ we may assume that they are together coprime.","Then reducing $P(x,f(x))=0$ modulo $p$ we get the relation $E(x,f(x))=0$ where $E\\in F_p[x,y]$ is the nonzero mod $p$ reduction of $P$ and $f\\in F_p[[x]]$ denotes again the mod $p$ reduction of $f$ — its coefficients $f_n$ we want to compute.","We do it by Theorem B.","We construct in $O_{p,P}(1)$ operations the objects $a,c,e,u,A_0,A_1,\\dots ,A_{p-1}$ , and $v$ .","For given $n\\in \\mathbb {N}_0$ we compute $f_n$ as $f_n=[x^n]c(x)+\\sum _{m=n-\\deg a}^n[x^{n-m}]a(x)\\cdot uA_{m_l}\\dots A_{m_1}A_{m_0}v$ where $m=(m_l\\dots m_1m_0)_p$ .","For each $m$ ($>0$ ) in the summation range it takes $l+1=l(m)+1=\\lfloor \\log _pm\\rfloor +1$ operations in $\\mathbb {Z}$ (divisions by $p$ with remainder) to determine its $p$ -adic expansion.","It takes $l+1$ matrix times column multiplications, one row times column multiplication, and one scalar multiplication (in $F_p$ ) to evaluate the summand.","So we evaluate $f_n$ in $\\sum _{m=n-\\deg a}^n(l(m)+1+(l(m)+1)e(2e-1)+2e+1)=O((\\deg a)e^2\\log _pn)$ operations.","Altogether we need $O_{p,P}(1)+O((\\deg a)e^2\\log _pn)=O_{p,P}(\\log n)$ operations ($n\\ge 2$ ).","Proof of Theorem B We start with four lemmas.","Lemma C is well known and is crucial for the existence of the algorithm of Theorem A, and in fact for the all algebra (and analysis) in characteristic $p$ .","Lemma C. If $p$ is prime and $z\\in F_p((x))$ is any Laurent series then $z(x)^p=z(x^p)$ .","Consequently, for any $k\\in \\mathbb {N}_0$ one has $z(x)^{p^k}=z(x^{p^k})\\;.$ Proof.","Let $z(x)=z_rx^r+z_{r+1}x^{r+1}+\\cdots $ .","Since $a^p=a$ for any $a\\in F_p$ and $(u+v)^p=u^p+v^p$ for any $u,v\\in F_p((x))$ (because $\\binom{p}{i}\\equiv 0$ mod $p$ for $0d$ , $a_{d,n}=a_{d,n-1}a_d=a_d^{n-d+1}$ and $a_{i,n}=a_{d-1,n-1}a_i+a_{i-1,n-1}a_d$ for $0\\le i\\le d-1$ , with $a_{-1,n-1}=0$ .","Clearly, $\\max _{0\\le i\\le d}\\deg a_i=\\deg _xE$ .","Using the recurrence and induction we get the bound $\\max _i\\deg a_{i,n}=0$ for $n=0,1,\\dots ,d-1$ , $\\max _i\\deg a_{i,d}=\\deg _xE$ , and $\\max _i\\deg a_{i,n}\\le (n-d+1)\\deg _xE$ for $n\\ge d$ .","Applying Lemma E on the $d+1$ $d$ -tuples $(a_{i,n}\\;|\\;0\\le i\\le d-1)$ , $n=1,p,p^2,\\dots ,p^d$ , we get $d_i\\in F_p[x]$ in the dependency relation with $\\max _i\\deg d_i\\le (d+1)(p^d-d+1)\\deg _xE$ .","$\\Box $ The lemma is inspired by Lemma 12.2.3 in Allouche and Shallit [1].","Equations of this type are called Mahler equations, after the German–British mathematician Kurt Mahler (1903–1988) who used them to prove transcendence of various numbers.","We start the proof of Theorem B proper.","Let $0\\ne E\\in F_p[x,y]$ be the mod $p$ reduction of $P(x,y)$ (with coprime coefficients) and $d=\\deg _yE\\ge 1$ .","Since $E(x,f)=0$ , by Lemma F one can construct a Mahler equation $c_0(x)f(x)+c_1(x)f(x^p)+\\dots +c_k(x)f(x^{p^k})=0$ for $f(x)$ where $c_i\\in F_p[x]$ , $k\\le d$ , $c_0(x)\\ne 0$ , and $\\deg c_0\\le (d+1)(p^d-d+1)\\deg _xE\\le h$ ($h$ is as in Theorem A).","We change the variable $f(x)$ to $g(x)$ by $f(x)=c_0(x)g(x)$ .","Using Lemma C and dividing by $c_0(x)^2$ , we get a Mahler equation for $g(x)$ , $g(x)+c_1(x)c_0(x)^{p-2}g(x^p)+\\dots +c_k(x)c_0(x)^{p^k-2}g(x^{p^k})=0$ — now the new coefficient $c_0$ is 1.","The small price we paid is that in general $g(x)$ is not a power series but $g(x)\\in F_p((x))$ .","We write $g(x)=\\sum _{n<0}g_nx^n+\\sum _{n\\ge 0}g_nx^n=:g_-(x)+h(x)\\;,$ $g_-\\in x^{-1}F_p[x^{-1}]$ and $h\\in F_p[[x]]$ (this is the $h(x)$ of Theorem B).","Denoting the Mahler equation for $g(x)$ as $L(x,M)g(x)=0$ (here $L(x,M)$ is a skew polynomial in $x$ and the operator $M:\\;u(x)\\mapsto u(x)^p=u(x^p)$ ) we get effectively a Mahler equation for the power series $h(x)$ , $L(x,M)h(x)=-L(x,M)g_-(x)=:b(x)$ or more explicitly $h(x)+c_1(x)c_0(x)^{p-2}h(x^p)+\\dots +c_k(x)c_0(x)^{p^k-2}h(x^{p^k})=b(x)\\;.$ Necessarily $b(x)\\in F_p[x]$ because $b(x)=-L(x,M)g_-(x)\\in F_p[x^{-1},x]$ but also $b(x)=L(x,M)h(x)\\in F_p[[x]]$ .","We write the equation as $h(x)=b(x)+d_1(x)h(x^p)+\\dots +d_k(x)h(x^{p^k})$ where $b(x),d_i(x):=-c_i(x)c_0(x)^{p^i-2}\\in F_p[x]$ .","Let $D:=\\max (\\deg b,\\deg d_i,1\\le i\\le k)$ and $h_{(i)}:=h(x^{p^i})$ .","We consider the vector space $V:=\\lbrace \\alpha +\\beta _0h_{(0)}+\\dots +\\beta _kh_{(k)}\\;|\\;\\alpha ,\\beta _i\\in F_p[x],\\deg \\alpha ,\\deg \\beta _i\\le D\\rbrace $ over $F_p$ .","Clearly, $h(x)=h_{(0)}\\in V$ .","It follows that $S_r(V)\\subset V$ for each of the $p$ operators $S_r$ : using the equation for $h(x)$ and Lemma D.3 (and D.1) we have $&&S_r(\\alpha +\\beta _0h_{(0)}+\\dots +\\beta _kh_{(k)})={\\textstyle S_r(\\alpha +\\beta _0b+\\sum _{i=1}^k(\\beta _0d_i+\\beta _i)h_{(i)})}\\\\&&={\\textstyle S_r(\\alpha +\\beta _0b)+\\sum _{i=1}^kS_r(\\beta _0d_i+\\beta _i)h_{(i-1)}}$ and $\\deg S_r(\\alpha +\\beta _0b),\\deg S_r(\\beta _0d_i+\\beta _i)\\le \\deg (\\alpha +\\beta _0b)/p,\\deg (\\beta _0d_i+\\beta _i)/p\\le 2D/p\\le D$ (Lemma D.6).","The $(D+1)(k+2)$ -element set $B:=\\lbrace x^j,x^jh_{(i)}=x^jh(x^{p^i})\\;|\\;0\\le j\\le D,0\\le i\\le k\\rbrace $ is a generating set for $V$ — the linear span of $B$ over $F_p$ is $V$ (but $B$ is not a basis for $V$ , as stated in [3]; $h_{(0)}$ is a linear combination of other elements of $B$ ).","We set $e:=(D+1)(k+2)$ and by fixing an ordering of $B$ make it a tuple $B=(b_1,\\dots ,b_e)$ .","We set $v\\in F_p^{e\\times 1}$ to be the column of coordinates of $h(x)$ in $B$ (it has all 0s and one 1), $A_r\\in F_p^{e\\times e}$ for $r=0,1,\\dots ,p-1$ to be the matrices of the operators $S_r$ with respect to $B$ (the $j$ -th column of $A_r$ lists the coordinates of $S_r(b_j)$ in $B$ ), and $u:=(b_1(0),\\dots ,b_e(0))\\in F_p^{1\\times e}$ to be the row of constant terms in $B$ .","Clearly we can obtain $v,A_r$ , and $u$ effectively.","By Lemma D.4 and the definition of $v,A_r$ , and $u$ we have for any $n=(n_l\\dots n_1n_0)_p\\in \\mathbb {N}_0$ that $h_n=[x^n]h(x)=[x^0]S_{n_l}\\dots S_{n_1}S_{n_0}h(x)=uA_{n_l}\\dots A_{n_1}A_{n_0}v\\;.$ Returning to $f(x)$ we see that $f(x)=c_0(x)g(x)=c_0(x)g_-(x)+c_0(x)h(x)$ and set, finally, $c(x):=c_0(x)g_-(x)$ and $a(x):=c_0(x)$ .","It follows that $a,c\\in F_p[x]$ .","How do we get $g_-(x)$ ?","It is the negative part of $g(x)=f(x)/c_0(x)$ .","Since $\\deg c_0\\le h$ , we compute $g_-(x)$ from $c_0(x)$ and $f_0,f_1,\\dots ,f_h$ .","Let us review the computation.","We are given $p,P$ , and $f_0,f_1,\\dots ,f_h$ (initial coefficients of a solution $y=f(x)\\in \\mathbb {Z}[[x]]$ to $P(x,y)=0$ , $h$ is as given in Theorem A).","Reducing $P$ modulo $p$ we get a nonzero $E\\in F_p[x,y]$ .","From $E$ we compute by Lemma F the coefficients $c_0,\\dots ,c_k\\in F_p[x]$ , $c_0\\ne 0$ , $\\deg c_0\\le h$ , and $k\\le \\deg _yE$ , of a Mahler equation for $f(x)$ .","We compute $g_-(x)\\in x^{-1}F_p[x^{-1}]$ as the negative part of $(f_0+f_1x+\\dots +f_hx^h)/c_0(x)$ .","Thus we have $L(x,M)=M^0+\\sum _{i=1}^kc_i(x)c_0(x)^{p^i-2}M^i$ (the Mahler equation for $g(x)$ ) and compute $b(x)=-L(x,M)g_-(x)\\in F_p[x]$ .","We have $K(x,M):=-\\sum _{i=1}^kc_i(x)c_0(x)^{p^i-2}M^i$ (the relation $h(x)=b(x)+K(x,M)h(x)$ ) and set $D=\\max (\\deg b,\\deg _xK(x,M))$ and $e=(D+1)(k+2)$ .","We take the $e$ -tuple $B=(x^j,x^jh(x^{p^i})\\;|\\;0\\le j\\le D,0\\le i\\le k)$ and compute the column $v$ of coordinates of $h(x)$ in $B$ (this is easy), the matrices $A_0,\\dots ,A_{p-1}$ of the operators $S_r$ with respect to $B$ ($S_r$ acts on $x^j$ in the clear way, $S_r(x^jh(x^{p^i}))=S_r(x^j)h(x^{p^{i-1}})$ for $i>0$ , and for $i=0$ we replace $h(x)$ with $b(x)+K(x,M)h(x)$ ), and the row $u$ of constant terms in $B$ (note that the constant term $h(0)$ is the constant term in $(f_0+f_1x+\\dots +f_hx^h)/c_0(x)$ ).","Finally, we compute $c(x)=c_0(x)g_-(x),a(x)=c_0(x)\\in F_p[x]$ .","Thus we have computed $a,c,e,u,A_0,\\dots ,A_{p-1},v$ , all we need to quickly evaluate $f_n$ modulo $p$ .","We are done.","$\\Box $ Concluding remarks An interesting problem is to locate the first different coefficient of two different power series solutions of the same polynomial equation.","Theorem B yields the following corollary.","Corollary G. If $p$ is prime, $0\\ne E\\in F_p[x,y]$ , and $f,g\\in F_p[[x]]$ , $f\\ne g$ , satisfy $E(x,f(x))=E(x,g(x))=0$ , then $f_n\\ne g_n\\ \\mbox{ for some }\\ n\\le (d+1)(p^d-d+1)\\deg _xE,\\ d=\\deg _yE\\;.$ Another bound is given in Proposition H. If $H$ is any field of characteristic 0, $0\\ne P\\in H[x,y]$ , and $f,g\\in H[[x]]$ , $f\\ne g$ , satisfy $P(x,f(x))=P(x,g(x))=0$ , then $f_n\\ne g_n\\ \\mbox{ for some }\\ n\\le \\frac{d^2+d-4}{2}\\deg _xP,\\ d=\\deg _yP\\;.$ Proof.","We work with the fields $K=H(x)\\subset H((x))=L$ and the algebraic closure $M=\\overline{L}=\\overline{H}((x))_P$ , the field of Puiseux series with coefficients in $\\overline{H}$ .","There is a nonzero polynomial $Q\\in H[x,y]$ such that $Q$ divides $P$ in $H[x,y]$ , $Q(x,f(x))=Q(x,g(x))=0$ , and $Q$ as $Q(y)\\in K[y]$ has in $M$ only simple roots.","Indeed, write $P(y)=P_1(y)^{a_1}P_2(y)^{a_2}\\dots P_k(y)^{a_k}$ where each $P_i\\in K[y]$ is irreducible (in $K[y]$ ), $a_i\\in \\mathbb {N}$ , and no $P_i$ divides other $P_j$ , $i\\ne j$ .","Irreducibility of the $P_i(y)$ s implies as usual that they have in $M$ only simple roots and do not share roots.","By the Gauss lemma (Lang [7]) we may take $P_i\\in H[x][y]$ .","Hence we have $Q=P_i(y)$ where $P_i(f)=P_i(g)=0$ or $Q=P_iP_j$ , $i\\ne j$ , where $P_i(f)=P_j(g)=0$ .","Let $f_1=f,f_2=g,\\dots ,f_d\\in M$ be the (different) roots of $Q(y)=a_dy^d+\\dots +a_1y+a_0$ , $a_i\\in H[x]$ and $a_d\\ne 0$ , so $d=\\deg _yQ\\in \\mathbb {N}$ (and $d\\ge 2$ ).","Let $e=\\max _i\\deg a_i=\\deg _xQ\\in \\mathbb {N}_0$ .","From Vièta's formula $(-1)^ia_{d-i}/a_d=\\sum _{1\\le j_1<\\dots 1$ and $c_1\\\\equiv 1$ mod 2 ($c_{n/2}=0$ if $n/2\\\\notin \\\\mathbb {N}_0$ ).\",\n \"Hence $c_n\\\\equiv 1$ mod 2 for $n=2^m$ , $m\\\\in \\\\mathbb {N}_0$ , and $c_n\\\\equiv 0$ mod 2 else .\",\n \"And $c_n$ modulo 3 or modulo other prime $p$ ?\",\n \"An answer to this is provided much more generally by the Christol theorem, proved by Christol [5] for $q=2$ and by Christol, Kamae, Mendès France and Rauzy [6] for any prime power $q$ (see also Allouche and Shallit [1]): a power series $f(x)\\\\in F_q[[x]]$ is algebraic over $F_q(x)$ if and only if one can generate its coefficients $f_0,f_1,\\\\dots $ by a DFAO (a deterministic finite automaton with output that reads the $l+1$ digits of the $q$ -adic expansion $n=(n_l\\\\dots n_1n_0)_q$ and outputs $f_n$ , we give one example in Concluding remarks).\",\n \"It is immediate from the Christol theorem that one can compute the $n$ th coefficient of an algebraic power series in $F_q[[x]]$ in $O(\\\\log n)$ arithmetic operations because $n$ has $O(\\\\log n)$ $q$ -adic digits.\",\n \"This write-up is an exposition of the proof of this result, stated as Theorem A below, that was given by means of power series in the recent preprint of Bostan, Christol and Dumas [3].\",\n \"I promised in my course `Kombinatorické počítání' (taught in this summer semester 2015/16) to supplement my unfortunately not very clear oral presentation of (something like) Theorem A with a written one — here it is.\",\n \"Algebraic power series in $F_p[[x]]$ First we review some notation and notions from algebra.\",\n \"Puiseux series $K((x))_P$ and their order are only used at the end in Proposition H that is not needed to prove Theorem A.\",\n \"$\\\\mathbb {N}=\\\\lbrace 1,2,\\\\dots \\\\rbrace $ , $\\\\mathbb {N}_0=\\\\lbrace 0,1,\\\\dots \\\\rbrace $ , $\\\\mathbb {Z}$ is the ring of integers, $F_p=(\\\\mathbb {Z}/p\\\\mathbb {Z},+,\\\\cdot )$ is the finite $p$ -element field where $p$ is a prime number, and $[x^n]f$ denotes the coefficient of $x^n$ in $f$ .\",\n \"For $p\\\\in \\\\mathbb {N}$ , $p\\\\ge 2$ , the $p$ -adic expansion $m=(m_l\\\\dots m_1m_0)_p$ of $m\\\\in \\\\mathbb {N}_0$ is given by the unique expression $m=\\\\sum _{i=0}^lm_ip^i$ where $m_i\\\\in \\\\lbrace 0,1,\\\\dots ,p-1\\\\rbrace $ and (for $m>0$ ) $m_l\\\\ne 0$ ; $0=(0)_p$ .\",\n \"Let $K$ be a field or an integral domain.\",\n \"For a field we denote by $\\\\overline{K}$ its algebraic closure (e.g.\",\n \"$\\\\overline{\\\\mathbb {R}}=\\\\mathbb {C}$ or $\\\\overline{\\\\mathbb {Q}}$ are the algebraic numbers).\",\n \"We denote by $K[[x]]\\\\subset K((x))\\\\subset K((x))_P$ , respectively, the ring of power series and the fields of Laurent series and Puiseux series with coefficients in $K$ .\",\n \"Their elements are the formal series (we treat here all power etc.\",\n \"series formally) $\\\\sum _{n\\\\ge k}a_{n/d}x^{n/d}$ , $a_{n/d}\\\\in K$ , where $k=0$ and $d=1$ for p. s., $k\\\\in \\\\mathbb {Z}$ and $d=1$ for L. s., and $k\\\\in \\\\mathbb {Z}$ and $d\\\\in \\\\mathbb {N}$ for P. s. For $f=f(x)\\\\in K((x))$ we use notation $f_n:=[x^n]f(x)$ .\",\n \"From the algebraic point of view, $K((x))$ is the field of fractions of $K[[x]]$ and for $K$ of characteristic 0 (e.g.\",\n \"$K=\\\\mathbb {Q}$ ) $\\\\overline{K}((x))_P$ is the algebraic closure of $K((x))$ .\",\n \"Thus in $\\\\overline{K}((x))_P$ every polynomial with coefficients in $K((x))$ has a root.\",\n \"Recall that $K[x]\\\\subset K(x)$ is, respectively, the ring of polynomials with coefficients in $K$ and its field of fractions ($K(x)$ are the rational `functions' over $K$ ).\",\n \"In fact, $K(x)$ is naturally a subfield of $K((x))$ and $K[x]$ a subring of $K[[x]]$ .\",\n \"By $\\\\deg a$ , $a\\\\in K[x]$ , we denote the degree of a polynomial and by $\\\\deg _y a$ , $a\\\\in K[x,y]$ , the degree $\\\\deg a$ when $a$ is understood as $a\\\\in K[x][y]$ .\",\n \"By $\\\\mathrm {ord\\\\,}f$ , $f\\\\in K((x))_P$ , we denote the order of $f$ , the minimum $n/d$ such that $a_{n/d}\\\\ne 0$ .\",\n \"We set $\\\\deg 0=-\\\\infty $ and $\\\\mathrm {ord\\\\,}0=+\\\\infty $ .\",\n \"For any $f,g\\\\in K((x))_P$ one has $\\\\mathrm {ord\\\\,}(fg)=\\\\mathrm {ord\\\\,}f+\\\\mathrm {ord\\\\,}g$ and $\\\\mathrm {ord\\\\,}(f+g)\\\\ge \\\\min (\\\\mathrm {ord\\\\,}f,\\\\mathrm {ord\\\\,}g)$ , with equality if $\\\\mathrm {ord\\\\,}f\\\\ne \\\\mathrm {ord\\\\,}g$ .\",\n \"Similarly, for any $f,g\\\\in K[x]$ one has $\\\\deg (fg)=\\\\deg f+\\\\deg g$ and $\\\\deg (f+g)\\\\le \\\\max (\\\\deg f,\\\\deg g)$ , with equality if $\\\\deg f\\\\ne \\\\deg g$ .\",\n \"We state the basic result on computing coefficients of an algebraic power series in $F_p[[x]]$ (this is essentially proven in $\\\\cite {chri_al}$ but the formulation below is ours).\",\n \"Theorem A ([6], 1980).\",\n \"Suppose that $p$ is prime, $P\\\\in \\\\mathbb {Z}[x,y]$ is a nonzero polynomial, $f=f_0+f_1x+f_2x^2+\\\\dots \\\\in \\\\mathbb {Z}[[x]]$ is a power series satisfying $P(x,f(x))=0$ , and $h=(d+1)(p^d-d+1)\\\\deg _xP\\\\ \\\\mbox{ where }\\\\ d=\\\\deg _yP\\\\ (\\\\ge 1)\\\\;.$ Then from given $p,P,f_0,f_1,\\\\dots ,f_h$ one can construct an algorithm (actually a DFAO) that computes $n\\\\mapsto f_n\\\\ \\\\mathrm {mod}\\\\ p$ in $O(\\\\log n)$ arithmetic operations ($n\\\\ge 2$ ).\",\n \"But which operations exactly?\",\n \"These are arithmetic operations ($+,-,\\\\times ,:$ ) in the field $F_p$ and the operation of division with remainder in the ring $\\\\mathbb {Z}$ .\",\n \"By `one can construct' here and below we mean that the task can be performed constructively (i.e.\",\n \"the algorithms or objects not merely exist but exist effectively).\",\n \"Theorem A follows from the following more specific result (again the formulation is ours).\",\n \"Theorem B ([3]).\",\n \"Let $p$ , $P$ , $f$ , and $h$ be as in Theorem A.\",\n \"We denote the mod $p$ reduction of $f$ again by $f\\\\in F_p[[x]]$ .\",\n \"Then from given $p,P,f_0,f_1,\\\\dots ,f_h$ one can construct polynomials $a,c\\\\in F_p[x]$ , a number $e\\\\in \\\\mathbb {N}$ , a row $u\\\\in F_p^{1\\\\times e}$ , $p$ matrices $A_0,A_1,\\\\dots ,A_{p-1}\\\\in F_p^{e\\\\times e}$ , and a column $v\\\\in F_p^{e\\\\times 1}$ such that for any $n\\\\in \\\\mathbb {N}_0$ one has $[x^n]f(x)&=&f_n=[x^n](c(x)+a(x)h(x))\\\\ \\\\mbox{ where }\\\\\\\\&&h(x)=\\\\sum _{m=0}^{\\\\infty }(uA_{m_l}\\\\dots A_{m_1}A_{m_0}v)x^m\\\\in F_p[[x]]$ ($m=(m_l\\\\dots m_1m_0)_p$ is the $p$ -adic expansion of $m$ ).\",\n \"Polynomial $P$ in Theorem A in general does not determine unique power series solution $y=f(x)$ of $P(x,y)=0$ , e.g.\",\n \"$y^2-x^2=0$ has two solutions $y=x$ and $y=-x$ .\",\n \"One can introduce restrictions, for example to require that $P(0,0)=0$ , $P_y(0,0)\\\\ne 0$ , and $f(0)=0$ , producing unique solution, but we want result for completely general $P$ .\",\n \"So to determine $f(x)$ we need to know a few initial coefficients, which is captured by the parameter $h$ in Theorems A and B. Bounds on $h$ are considered in Corollary G and Proposition H. Let us see how Theorem B implies Theorem A.\",\n \"We are given a prime $p$ , a nonzero $P\\\\in \\\\mathbb {Z}[x,y]$ , and first few coefficients $f_0,f_1,\\\\dots ,f_h$ of a solution $y=f(x)\\\\in \\\\mathbb {Z}[[x]]$ of the equation $P(x,y)=0$ , where $h$ is as in Theorem A.\",\n \"(We assume this is an honest input, $f_0,f_1,\\\\dots ,f_h$ are really initial coefficients of a solution.\",\n \"We leave as an exercise for the reader to design an effective check that it is true.)\",\n \"Taking out the largest common factor of the coefficients of $P(x,y)$ we may assume that they are together coprime.\",\n \"Then reducing $P(x,f(x))=0$ modulo $p$ we get the relation $E(x,f(x))=0$ where $E\\\\in F_p[x,y]$ is the nonzero mod $p$ reduction of $P$ and $f\\\\in F_p[[x]]$ denotes again the mod $p$ reduction of $f$ — its coefficients $f_n$ we want to compute.\",\n \"We do it by Theorem B.\",\n \"We construct in $O_{p,P}(1)$ operations the objects $a,c,e,u,A_0,A_1,\\\\dots ,A_{p-1}$ , and $v$ .\",\n \"For given $n\\\\in \\\\mathbb {N}_0$ we compute $f_n$ as $f_n=[x^n]c(x)+\\\\sum _{m=n-\\\\deg a}^n[x^{n-m}]a(x)\\\\cdot uA_{m_l}\\\\dots A_{m_1}A_{m_0}v$ where $m=(m_l\\\\dots m_1m_0)_p$ .\",\n \"For each $m$ ($>0$ ) in the summation range it takes $l+1=l(m)+1=\\\\lfloor \\\\log _pm\\\\rfloor +1$ operations in $\\\\mathbb {Z}$ (divisions by $p$ with remainder) to determine its $p$ -adic expansion.\",\n \"It takes $l+1$ matrix times column multiplications, one row times column multiplication, and one scalar multiplication (in $F_p$ ) to evaluate the summand.\",\n \"So we evaluate $f_n$ in $\\\\sum _{m=n-\\\\deg a}^n(l(m)+1+(l(m)+1)e(2e-1)+2e+1)=O((\\\\deg a)e^2\\\\log _pn)$ operations.\",\n \"Altogether we need $O_{p,P}(1)+O((\\\\deg a)e^2\\\\log _pn)=O_{p,P}(\\\\log n)$ operations ($n\\\\ge 2$ ).\",\n \"Proof of Theorem B We start with four lemmas.\",\n \"Lemma C is well known and is crucial for the existence of the algorithm of Theorem A, and in fact for the all algebra (and analysis) in characteristic $p$ .\",\n \"Lemma C. If $p$ is prime and $z\\\\in F_p((x))$ is any Laurent series then $z(x)^p=z(x^p)$ .\",\n \"Consequently, for any $k\\\\in \\\\mathbb {N}_0$ one has $z(x)^{p^k}=z(x^{p^k})\\\\;.$ Proof.\",\n \"Let $z(x)=z_rx^r+z_{r+1}x^{r+1}+\\\\cdots $ .\",\n \"Since $a^p=a$ for any $a\\\\in F_p$ and $(u+v)^p=u^p+v^p$ for any $u,v\\\\in F_p((x))$ (because $\\\\binom{p}{i}\\\\equiv 0$ mod $p$ for $0d$ , $a_{d,n}=a_{d,n-1}a_d=a_d^{n-d+1}$ and $a_{i,n}=a_{d-1,n-1}a_i+a_{i-1,n-1}a_d$ for $0\\\\le i\\\\le d-1$ , with $a_{-1,n-1}=0$ .\",\n \"Clearly, $\\\\max _{0\\\\le i\\\\le d}\\\\deg a_i=\\\\deg _xE$ .\",\n \"Using the recurrence and induction we get the bound $\\\\max _i\\\\deg a_{i,n}=0$ for $n=0,1,\\\\dots ,d-1$ , $\\\\max _i\\\\deg a_{i,d}=\\\\deg _xE$ , and $\\\\max _i\\\\deg a_{i,n}\\\\le (n-d+1)\\\\deg _xE$ for $n\\\\ge d$ .\",\n \"Applying Lemma E on the $d+1$ $d$ -tuples $(a_{i,n}\\\\;|\\\\;0\\\\le i\\\\le d-1)$ , $n=1,p,p^2,\\\\dots ,p^d$ , we get $d_i\\\\in F_p[x]$ in the dependency relation with $\\\\max _i\\\\deg d_i\\\\le (d+1)(p^d-d+1)\\\\deg _xE$ .\",\n \"$\\\\Box $ The lemma is inspired by Lemma 12.2.3 in Allouche and Shallit [1].\",\n \"Equations of this type are called Mahler equations, after the German–British mathematician Kurt Mahler (1903–1988) who used them to prove transcendence of various numbers.\",\n \"We start the proof of Theorem B proper.\",\n \"Let $0\\\\ne E\\\\in F_p[x,y]$ be the mod $p$ reduction of $P(x,y)$ (with coprime coefficients) and $d=\\\\deg _yE\\\\ge 1$ .\",\n \"Since $E(x,f)=0$ , by Lemma F one can construct a Mahler equation $c_0(x)f(x)+c_1(x)f(x^p)+\\\\dots +c_k(x)f(x^{p^k})=0$ for $f(x)$ where $c_i\\\\in F_p[x]$ , $k\\\\le d$ , $c_0(x)\\\\ne 0$ , and $\\\\deg c_0\\\\le (d+1)(p^d-d+1)\\\\deg _xE\\\\le h$ ($h$ is as in Theorem A).\",\n \"We change the variable $f(x)$ to $g(x)$ by $f(x)=c_0(x)g(x)$ .\",\n \"Using Lemma C and dividing by $c_0(x)^2$ , we get a Mahler equation for $g(x)$ , $g(x)+c_1(x)c_0(x)^{p-2}g(x^p)+\\\\dots +c_k(x)c_0(x)^{p^k-2}g(x^{p^k})=0$ — now the new coefficient $c_0$ is 1.\",\n \"The small price we paid is that in general $g(x)$ is not a power series but $g(x)\\\\in F_p((x))$ .\",\n \"We write $g(x)=\\\\sum _{n<0}g_nx^n+\\\\sum _{n\\\\ge 0}g_nx^n=:g_-(x)+h(x)\\\\;,$ $g_-\\\\in x^{-1}F_p[x^{-1}]$ and $h\\\\in F_p[[x]]$ (this is the $h(x)$ of Theorem B).\",\n \"Denoting the Mahler equation for $g(x)$ as $L(x,M)g(x)=0$ (here $L(x,M)$ is a skew polynomial in $x$ and the operator $M:\\\\;u(x)\\\\mapsto u(x)^p=u(x^p)$ ) we get effectively a Mahler equation for the power series $h(x)$ , $L(x,M)h(x)=-L(x,M)g_-(x)=:b(x)$ or more explicitly $h(x)+c_1(x)c_0(x)^{p-2}h(x^p)+\\\\dots +c_k(x)c_0(x)^{p^k-2}h(x^{p^k})=b(x)\\\\;.$ Necessarily $b(x)\\\\in F_p[x]$ because $b(x)=-L(x,M)g_-(x)\\\\in F_p[x^{-1},x]$ but also $b(x)=L(x,M)h(x)\\\\in F_p[[x]]$ .\",\n \"We write the equation as $h(x)=b(x)+d_1(x)h(x^p)+\\\\dots +d_k(x)h(x^{p^k})$ where $b(x),d_i(x):=-c_i(x)c_0(x)^{p^i-2}\\\\in F_p[x]$ .\",\n \"Let $D:=\\\\max (\\\\deg b,\\\\deg d_i,1\\\\le i\\\\le k)$ and $h_{(i)}:=h(x^{p^i})$ .\",\n \"We consider the vector space $V:=\\\\lbrace \\\\alpha +\\\\beta _0h_{(0)}+\\\\dots +\\\\beta _kh_{(k)}\\\\;|\\\\;\\\\alpha ,\\\\beta _i\\\\in F_p[x],\\\\deg \\\\alpha ,\\\\deg \\\\beta _i\\\\le D\\\\rbrace $ over $F_p$ .\",\n \"Clearly, $h(x)=h_{(0)}\\\\in V$ .\",\n \"It follows that $S_r(V)\\\\subset V$ for each of the $p$ operators $S_r$ : using the equation for $h(x)$ and Lemma D.3 (and D.1) we have $&&S_r(\\\\alpha +\\\\beta _0h_{(0)}+\\\\dots +\\\\beta _kh_{(k)})={\\\\textstyle S_r(\\\\alpha +\\\\beta _0b+\\\\sum _{i=1}^k(\\\\beta _0d_i+\\\\beta _i)h_{(i)})}\\\\\\\\&&={\\\\textstyle S_r(\\\\alpha +\\\\beta _0b)+\\\\sum _{i=1}^kS_r(\\\\beta _0d_i+\\\\beta _i)h_{(i-1)}}$ and $\\\\deg S_r(\\\\alpha +\\\\beta _0b),\\\\deg S_r(\\\\beta _0d_i+\\\\beta _i)\\\\le \\\\deg (\\\\alpha +\\\\beta _0b)/p,\\\\deg (\\\\beta _0d_i+\\\\beta _i)/p\\\\le 2D/p\\\\le D$ (Lemma D.6).\",\n \"The $(D+1)(k+2)$ -element set $B:=\\\\lbrace x^j,x^jh_{(i)}=x^jh(x^{p^i})\\\\;|\\\\;0\\\\le j\\\\le D,0\\\\le i\\\\le k\\\\rbrace $ is a generating set for $V$ — the linear span of $B$ over $F_p$ is $V$ (but $B$ is not a basis for $V$ , as stated in [3]; $h_{(0)}$ is a linear combination of other elements of $B$ ).\",\n \"We set $e:=(D+1)(k+2)$ and by fixing an ordering of $B$ make it a tuple $B=(b_1,\\\\dots ,b_e)$ .\",\n \"We set $v\\\\in F_p^{e\\\\times 1}$ to be the column of coordinates of $h(x)$ in $B$ (it has all 0s and one 1), $A_r\\\\in F_p^{e\\\\times e}$ for $r=0,1,\\\\dots ,p-1$ to be the matrices of the operators $S_r$ with respect to $B$ (the $j$ -th column of $A_r$ lists the coordinates of $S_r(b_j)$ in $B$ ), and $u:=(b_1(0),\\\\dots ,b_e(0))\\\\in F_p^{1\\\\times e}$ to be the row of constant terms in $B$ .\",\n \"Clearly we can obtain $v,A_r$ , and $u$ effectively.\",\n \"By Lemma D.4 and the definition of $v,A_r$ , and $u$ we have for any $n=(n_l\\\\dots n_1n_0)_p\\\\in \\\\mathbb {N}_0$ that $h_n=[x^n]h(x)=[x^0]S_{n_l}\\\\dots S_{n_1}S_{n_0}h(x)=uA_{n_l}\\\\dots A_{n_1}A_{n_0}v\\\\;.$ Returning to $f(x)$ we see that $f(x)=c_0(x)g(x)=c_0(x)g_-(x)+c_0(x)h(x)$ and set, finally, $c(x):=c_0(x)g_-(x)$ and $a(x):=c_0(x)$ .\",\n \"It follows that $a,c\\\\in F_p[x]$ .\",\n \"How do we get $g_-(x)$ ?\",\n \"It is the negative part of $g(x)=f(x)/c_0(x)$ .\",\n \"Since $\\\\deg c_0\\\\le h$ , we compute $g_-(x)$ from $c_0(x)$ and $f_0,f_1,\\\\dots ,f_h$ .\",\n \"Let us review the computation.\",\n \"We are given $p,P$ , and $f_0,f_1,\\\\dots ,f_h$ (initial coefficients of a solution $y=f(x)\\\\in \\\\mathbb {Z}[[x]]$ to $P(x,y)=0$ , $h$ is as given in Theorem A).\",\n \"Reducing $P$ modulo $p$ we get a nonzero $E\\\\in F_p[x,y]$ .\",\n \"From $E$ we compute by Lemma F the coefficients $c_0,\\\\dots ,c_k\\\\in F_p[x]$ , $c_0\\\\ne 0$ , $\\\\deg c_0\\\\le h$ , and $k\\\\le \\\\deg _yE$ , of a Mahler equation for $f(x)$ .\",\n \"We compute $g_-(x)\\\\in x^{-1}F_p[x^{-1}]$ as the negative part of $(f_0+f_1x+\\\\dots +f_hx^h)/c_0(x)$ .\",\n \"Thus we have $L(x,M)=M^0+\\\\sum _{i=1}^kc_i(x)c_0(x)^{p^i-2}M^i$ (the Mahler equation for $g(x)$ ) and compute $b(x)=-L(x,M)g_-(x)\\\\in F_p[x]$ .\",\n \"We have $K(x,M):=-\\\\sum _{i=1}^kc_i(x)c_0(x)^{p^i-2}M^i$ (the relation $h(x)=b(x)+K(x,M)h(x)$ ) and set $D=\\\\max (\\\\deg b,\\\\deg _xK(x,M))$ and $e=(D+1)(k+2)$ .\",\n \"We take the $e$ -tuple $B=(x^j,x^jh(x^{p^i})\\\\;|\\\\;0\\\\le j\\\\le D,0\\\\le i\\\\le k)$ and compute the column $v$ of coordinates of $h(x)$ in $B$ (this is easy), the matrices $A_0,\\\\dots ,A_{p-1}$ of the operators $S_r$ with respect to $B$ ($S_r$ acts on $x^j$ in the clear way, $S_r(x^jh(x^{p^i}))=S_r(x^j)h(x^{p^{i-1}})$ for $i>0$ , and for $i=0$ we replace $h(x)$ with $b(x)+K(x,M)h(x)$ ), and the row $u$ of constant terms in $B$ (note that the constant term $h(0)$ is the constant term in $(f_0+f_1x+\\\\dots +f_hx^h)/c_0(x)$ ).\",\n \"Finally, we compute $c(x)=c_0(x)g_-(x),a(x)=c_0(x)\\\\in F_p[x]$ .\",\n \"Thus we have computed $a,c,e,u,A_0,\\\\dots ,A_{p-1},v$ , all we need to quickly evaluate $f_n$ modulo $p$ .\",\n \"We are done.\",\n \"$\\\\Box $ Concluding remarks An interesting problem is to locate the first different coefficient of two different power series solutions of the same polynomial equation.\",\n \"Theorem B yields the following corollary.\",\n \"Corollary G. If $p$ is prime, $0\\\\ne E\\\\in F_p[x,y]$ , and $f,g\\\\in F_p[[x]]$ , $f\\\\ne g$ , satisfy $E(x,f(x))=E(x,g(x))=0$ , then $f_n\\\\ne g_n\\\\ \\\\mbox{ for some }\\\\ n\\\\le (d+1)(p^d-d+1)\\\\deg _xE,\\\\ d=\\\\deg _yE\\\\;.$ Another bound is given in Proposition H. If $H$ is any field of characteristic 0, $0\\\\ne P\\\\in H[x,y]$ , and $f,g\\\\in H[[x]]$ , $f\\\\ne g$ , satisfy $P(x,f(x))=P(x,g(x))=0$ , then $f_n\\\\ne g_n\\\\ \\\\mbox{ for some }\\\\ n\\\\le \\\\frac{d^2+d-4}{2}\\\\deg _xP,\\\\ d=\\\\deg _yP\\\\;.$ Proof.\",\n \"We work with the fields $K=H(x)\\\\subset H((x))=L$ and the algebraic closure $M=\\\\overline{L}=\\\\overline{H}((x))_P$ , the field of Puiseux series with coefficients in $\\\\overline{H}$ .\",\n \"There is a nonzero polynomial $Q\\\\in H[x,y]$ such that $Q$ divides $P$ in $H[x,y]$ , $Q(x,f(x))=Q(x,g(x))=0$ , and $Q$ as $Q(y)\\\\in K[y]$ has in $M$ only simple roots.\",\n \"Indeed, write $P(y)=P_1(y)^{a_1}P_2(y)^{a_2}\\\\dots P_k(y)^{a_k}$ where each $P_i\\\\in K[y]$ is irreducible (in $K[y]$ ), $a_i\\\\in \\\\mathbb {N}$ , and no $P_i$ divides other $P_j$ , $i\\\\ne j$ .\",\n \"Irreducibility of the $P_i(y)$ s implies as usual that they have in $M$ only simple roots and do not share roots.\",\n \"By the Gauss lemma (Lang [7]) we may take $P_i\\\\in H[x][y]$ .\",\n \"Hence we have $Q=P_i(y)$ where $P_i(f)=P_i(g)=0$ or $Q=P_iP_j$ , $i\\\\ne j$ , where $P_i(f)=P_j(g)=0$ .\",\n \"Let $f_1=f,f_2=g,\\\\dots ,f_d\\\\in M$ be the (different) roots of $Q(y)=a_dy^d+\\\\dots +a_1y+a_0$ , $a_i\\\\in H[x]$ and $a_d\\\\ne 0$ , so $d=\\\\deg _yQ\\\\in \\\\mathbb {N}$ (and $d\\\\ge 2$ ).\",\n \"Let $e=\\\\max _i\\\\deg a_i=\\\\deg _xQ\\\\in \\\\mathbb {N}_0$ .\",\n \"From Vièta's formula $(-1)^ia_{d-i}/a_d=\\\\sum _{1\\\\le j_1<\\\\dots 0)$ equals either $-1$ or $+1$ .","Let us define the persistency of nonlocality of a multipartite state $\\rho $ according to Ref.","[24] as follows.","Let us take the partial trace over $02$ as well.","In particular, we have chosen a given Bell scenario ($n$ parties, $m$ settings) and by varying the $m$ angles, we maximized the value of $p_{crit}$ .","We note that this search is a heuristic one and as such it is not guaranteed to terminate in a global maximum of $p_{crit}$ .","However, the obtained value still defines a lower bound to $P^m_{NL}$ in (REF ).","The critical $p$ values obtained in function of $n$ and $m$ are displayed in Fig.","REF .","We may observe that, as $m$ increases, the affordable number of parties $n$ decreases.","For the simplest case of $m=2$ , numerically we could afford $n=14$ .","In addition, we had to resort to a symmetrization procedure of the Bell polytope introduced in [41] which considerably reduces the complexity of the problem in order to attain $n=14$ .","Figure: For ρ(n,p)\\rho (n,p) states of nn qubits (), the graph shows lower bounds on the thresholdprobability p crit p_{crit} obtained using our numerical method up tom=6m=6 measurement settings.Plugging the $p_{crit}$ values shown in Fig.","REF into formula (REF ), we get Table REF , where the computed persistencies $P_{NL}^m$ are shown for $m\\le 6$ .","This way we get entries in the table only for $N$ satisfying $N=\\lfloor n/(1-p_{crit})\\rfloor $ , but a slight modification allows us to obtain numbers for any $N$ parties: Fix $N$ , and choose the largest $k$ integer such that $p_{crit,n-k}>k/N$ , where $p_{crit,n-k}$ stands for $p_{crit}$ evaluated by the number of parties $n-k$ .","Then a lower bound on the persistency $P_{NL}^m$ for the $N$ -qubit W state is given by $P_{NL}^m=k+1$ .","Table: Lower bound values for persistency ofnonlocality P NL m P_{NL}^m for the W N W_N state with m=2,...,6m=2,\\ldots ,6binary-outcome measurement settings per party.","The results arebased on the numerical method described in the text.In Table REF , for $m=2$ , the first entries $N=2,3,\\ldots ,7$ match the numbers from previous study [24].","However, by fixing $N$ and going to higher number of settings $m$ sometimes we get a higher persistency of nonlocality.","For instance, in case of $N=6$ parties, $P_{NL}^5=3$ , which is to be compared to $P_{NL}^2=2$ .","For $m=2$ , we extracted the optimal Bell inequalities corresponding to $p_{crit}$ for various $n$ values.","For $n$ even, a common structure has been found.","In fact, they turned out to be members of a family of Bell inequalities valid for any $n$ even.","Details of this class of inequalities are presented in Sec. .","We optimized the quantum value of these inequalities in function of the two measurement angles.","Fig.","REF shows results for $p_{crit}$ up to $n=50$ of this family.","We also show results using the ansatz that the two measurement settings are $Z$ and $X$ , in which case $p_{crit}=(2n-4)/(5n-2)$ for $n$ even.","According to the figure, for larger $n$ the $(Z,X)$ pair of settings become close to optimal.","The $p_{crit}$ values have been also studied in Ref.","[28] for specific families of known Bell inequalities from the literature, with the best lower bound values coming from the WWWZB inequalities [42].","For comparison, these $p_{crit}$ values are displayed.","For $n=50$ , $p_{crit}=0.3142$ , which is suboptimal with respect to our $p_{crit}=(2n-4)/(5n-2)\\simeq 0.3871$ .","Figure: For ρ(n,p)\\rho (n,p) states of nn qubits (), thegraph shows lower bounds on the threshold probability p crit p_{crit}obtained using our numerical method for m=2m=2 measurementsettings.","These bounds coincide with the bounds coming from ourclass of Bell inequalities for 2 measurement settings up to n=14n=14and we conjecture to be the same for any even nn particles.","Weshow results for our Bell inequalities with optimized andnon-optimized settings (in the latter case, the two observablesare the Pauli operators ZZ and XX).","The lower curve reproducesthe results of Ref.","in case of the WWWZBinequalities .Finally, Fig.","REF shows the upper bound values due to section and the lower bound values for $P^2_{NL}(W_N)$ up to $N=50$ due to our Bell family.","We conjecture that the exact value for $P^2_{NL}(W_N)$ is defined by the lower curve (that is, no better Bell inequalities than our family in section exist in this respect).","For instance, the best lower bound for $P^2_{NL}(W_N)$ so far comes from the WWWZB inequalities analyzed in Ref.","[28], which lies below our curve for the lower bound, and according to the conjecture in Ref.","[28] it goes to $N/3$ in the limit of large $N$ .","On the other hand, our asymptotic lower bound value of $2N/5$ is considerable higher.","It is also worth noting that the upper bound value on $P^2_{NL}$ shown in the figure holds true for any permutationally invariant $N$ -qubit state.","Hence, Dicke states with more than one excitation, as a notable subset of permutationally invariant states, cannot perform significantly better than $W$ states in the two-setting case.","Figure: For WW states of NN qubits, the graph shows upperbounds (marked by a triangle pointing down) and lower bounds (marked by triangles pointing up) on the persistency of nonlocalityP NL m=2 (W N )P_{NL}^{m=2}(W_N).In Ref.","[28], the Dicke states (including the $W_N$ state) were analyzed in terms of two decoherence models: loss of particles and loss of excitations.","The first one relates to the measure $P_{NL}$ discussed in this paper, whereas the latter one is related to $p_{crit}$ in formula (REF ).","Indeed, if we start from an $n$ -qubit W state which is effected by a decoherence where in each mode an excitation has a probability $p$ of being lost, it brings the W state into (REF ).","Appendix A shows a derivation of this result.","Next section gives all the details, including the quantum and local bounds, of our particular family of two-setting Bell inequalities providing $p_{crit}=(2n-4)/(5n-2)$ for even $n$ .","In case of large $n$ , this goes to $2/5$ which we conjecture to be the largest achievable critical value among all two-setting Bell inequalities."],["A family of two-setting multipartite Bell inequalities","Standard form of permutationally symmetric Bell inequalities.– The Bell inequality to be considered consists of correlators (REF ) which are invariant under any permutation of the parties.","By imposing the permutational symmetry, one requires that the expectation values $\\langle \\mathcal {M}_{j_1}^{(1)}\\ldots \\mathcal {M}^{(n)}_{j_n}\\rangle $ and $\\langle \\mathcal {M}_{j_1}^{\\sigma (1)}\\ldots \\mathcal {M}^{\\sigma (n)}_{j_n}\\rangle $ are the same, where $\\sigma :\\lbrace 1,\\ldots ,n\\rbrace \\rightarrow \\lbrace \\sigma (1),\\ldots ,\\sigma (n)\\rbrace $ is an arbitrary permutation of the set $\\lbrace 1,\\ldots ,n\\rbrace $ .","Let us denote by $S_n$ all permutations of the set $\\lbrace 1,\\ldots ,n\\rbrace $ .","Then, it will be useful to define $S^o_r \\equiv \\sum _{\\sigma \\in S_n}\\langle \\mathcal {M}_{j_1}^{\\sigma (1)}\\ldots \\mathcal {M}^{\\sigma (n)}_{j_n}\\rangle ,$ where the sum ranges over all permutations of the set $\\lbrace 1,\\ldots ,n\\rbrace $ .","In $S^o_r$ above, $o$ is the order of correlators and $r$ is the number of 1's occurring in the list $\\lbrace j_1,\\ldots ,j_n\\rbrace $ .","Let us now define the symmetrized correlation vector by the above ordered real vectors as follows $\\vec{S} \\equiv \\lbrace S^o_r\\rbrace ^{o=1,\\ldots ,n}_{r=0,\\ldots ,o}.$ In a similar way, we define the vector of coefficients associated with $\\vec{S}$ as $\\vec{\\alpha }\\equiv \\lbrace \\alpha ^o_r\\rbrace ^{o=1,\\ldots ,n}_{r=0,\\ldots ,o}.$ As a result, we arrive at the general form of a permutationally symmetric Bell inequality: $I \\equiv \\vec{\\alpha }\\cdot \\vec{S} = \\sum _{\\begin{array}{c}{o=1,\\ldots ,n}\\\\{r=0,\\ldots ,o}\\end{array}}{\\alpha ^o_r S^o_r}\\le \\beta _c,$ where $\\beta _c$ is the local maximum, which holds for any local hidden variable model.","It is worth noting that the WWWZB class [42] contains only full-correlator terms ($o=n$ ), whereas our construction of Bell inequalities turns out to contain all different orders, starting from marginal terms ($o=1$ ) up to full-correlation terms ($o=n$ ).","There also exist permutationally symmetric $n$ -party Bell inequalities involving only first and second order terms ($o=1,2$ ) [43].","The usefulness of these kind of inequalities in the particle loss model is an open and interesting question in our view.","As an illustrative example, let us discuss the case of 4 particles ($n=4$ ).","In this case, $\\vec{S}$ looks as follows $\\vec{S} = \\lbrace S^1_1,S^1_0;S^2_2,S^2_1,S^2_0;S^3_3,S^3_2,S^3_1,S^3_0;S^4_4,S^4_3,S^4_2,S^4_1,S^4_0\\rbrace ,$ where we used semicolon (;) in order to separate components with different order $o$ .","By identifying $\\mathcal {M}_0^{(l)}=1$ , $l=1,\\ldots ,4$ and $\\mathcal {M}_j^{(1)}=A_j$ , $\\mathcal {M}_j^{(2)}=B_j$ , $\\mathcal {M}_j^{(3)}=C_j$ , $\\mathcal {M}_j^{(4)}=D_j$ for $j=1,2$ in Eq.","(REF ), we have the following mean values displayed for some particular cases (neglecting the mean value signs): $S^1_1 =& 6(A_1 + B_1 + C_1 + D_1),\\nonumber \\\\S^2_2 =& 4(A_1B_1 + A_1C_1 + A_1D_1 + B_1C_1 + B_1D_1 + C_1D_1),\\nonumber \\\\S^2_0 =& 4(A_2B_2 + A_2C_2 + A_2D_2 + B_2C_2 + B_2D_2 + C_2D_2),\\nonumber \\\\S^3_1 =& 2(A_1B_2C_2+A_1B_2D_2+A_1C_2D_2)\\nonumber \\\\& +2(A_2B_1C_2+A_2B_1D_2+B_1C_2D_2)\\nonumber \\\\& +2(A_2B_2C_1+A_2C_1D_2+B_2C_1D_2)\\nonumber \\\\& + 2(A_2B_2D_1+A_2C_2D_1+B_2C_2D_1),\\nonumber \\\\S^4_0 =& 24A_2B_2C_2D_2,$ where we recall that $o$ denotes the order in the superindex of $S^o_r$ , whereas $r$ is the number of occurrences of subindex 1 in $S^o_r$ .","For instance, $S^3_1$ above consists of all length-3 sequences from $A,B,C$ and $D$ letters with the occurrence of a single subindex 1.","The specific class of Bell inequalities.– Let us now give the explicit form of a Bell inequality $I$ defined by the vector of coefficients $\\vec{\\alpha }$ in Eq.","(REF ).","This family has been extracted from the numerical method of Sec. .","In particular, we introduce a family of Bell inequalities $I_n$ valid for any even number of particles $n$ as follows $I_n \\equiv \\vec{\\alpha }_n\\cdot \\vec{S}_n = \\sum _{\\begin{array}{c}{o=1,\\ldots ,n}\\\\{r=0,\\ldots ,o}\\end{array}}{\\alpha ^o_r S^o_r}\\le \\beta _{c,n}.$ The vector of coefficients $\\vec{\\alpha }=\\lbrace \\alpha ^o_r(o=1,\\ldots ,n, r=0,\\ldots ,o)\\rbrace $ reads as $\\vec{\\alpha }={\\left\\lbrace \\begin{array}{ll}\\alpha ^{2k}_0 = F_{k,n}, & k=1,\\ldots ,n/2\\\\\\alpha ^{2k-1}_1 = G_{k,n}+2w_n\\delta _{k,1}, & k=1,\\ldots ,n/2\\\\\\alpha ^2_2 = -w_n,\\\\\\alpha ^o_r = 0, & \\text{otherwise}\\end{array}\\right.","}$ where $\\delta _{i,j}$ is the Kronecker delta (1 if $i=j$ , 0 otherwise), and the functions entering eq.","(REF ) above are $F_{k,n}&=(-1)^k\\frac{n-2}{2}(n+1-2k)\\binom{n/2}{k},\\nonumber \\\\G_{k,n}&=(-1)^k\\frac{n+2}{2}n\\binom{n/2-1}{k-1},\\nonumber \\\\w_n&=\\frac{n(n-1)(n+2)}{2^{4-n}\\binom{n}{n/2}}.$ Especially for $n=4$ , our 4-party Bell inequality looks as follows: $I_4 = 12 S^1_1 - 12 S^2_2 - 6S^2_0 + 12S^3_1 + S^4_0 \\le 128,$ where the middle terms appear explicitly in (REF ) and the local bound is due to formula (REF ) given in the next subsection for the general $n$ -party case.","Local bound.– The local maximum, which is the maximum value one obtains using local resources only, is given by $\\beta _{c,n}=n!\\left(w_n-\\frac{(n-2)(n+1)}{2}\\right)$ in case of any even number of particles $n$ , where $w_n=\\frac{n(n-1)(n+2)}{2^{4-n}\\binom{n}{n/2}}.$ We have checked the validity of the above local bound $\\beta _{c,n}$ for $n\\le 16$ ($n$ even) by listing all possible deterministic strategies (i.e., vertices of the Bell polytope).","These are the strategies for which all correlators factorize (see eq.","REF ) and each single party marginal $\\langle \\mathcal {M}_{j_l}^{(l)}\\rangle $ , $(l=1,\\ldots ,n, j_l=1,2)$ equals either $-1$ or $+1$ .","One of the above deterministic strategies, due to linearity of the Bell functional, will provide the local maximum.","Notice that $\\langle \\mathcal {M}_0^{(l)}\\rangle $ , ($l=1,\\ldots ,n$ ) equals $+1$ by definition.","Below we provide an analytical proof of formula (REF ) for any even $n$ .","We checked with brute-force computations that for smaller values of $n$ the inequalities are far from tightness, that is, they do not define a facet of the Bell polytope.","We conjecture that they are not tight for larger $n$ as well.","Here follows the proof of the local limit (REF ) for even $n$ .","First let us fix notation.","For a given local deterministic strategy let us define a four-tuple ($a$ ,$b$ ,$c$ ,$d$ ) which counts the number of parties whose marginal expectations $\\lbrace \\langle \\mathcal {M}^{(l)}_1\\rangle ,\\lbrace \\langle \\mathcal {M}^{(l)}_2\\rangle \\rbrace $ are the respective pairs ($\\lbrace 1,1\\rbrace $ , $\\lbrace 1,-1\\rbrace $ , $\\lbrace -1,1\\rbrace $ , $\\lbrace -1,-1\\rbrace $ ).","By definition we have $a+b+c+d=n$ .","In case of permutationally invariance under party exchange, a four-tuple ($a,b,c,d$ ) represents uniquely a deterministic strategy.","Hence, our task is to calculate the Bell value $\\vec{\\alpha }_n\\cdot \\vec{S}_n$ for all possible integer-valued four-tuples $a,b,c,d\\ge 0$ fulfilling $a+b+c+d=n$ , and then pick the maximum value out of this set.","This defines the local maximum $\\beta _{c,n}$ attainable using classical resources.","In the proof below, all positive integer four-tuples $a,b,c,d$ are understood to sum up to $n$ , where $n$ is even.","Let us introduce the permanent of an $n\\times n$ matrix $A=(a_{i,j})$ , $(i,j=1,\\ldots ,n)$ .","It is defined as $\\text{perm}(A) = \\sum _{\\sigma \\in S_n}\\prod _{i=1}^n{a_{i,\\sigma (i)}},$ where $\\sigma $ is a permutation over the set $\\lbrace 1,\\ldots ,n\\rbrace $ .","Using the above definition for the permanent, $\\vec{S} = \\lbrace S^o_r\\rbrace $ ($o=1,\\ldots ,n$ , $r=0,\\ldots ,o$ ) is given by the components $S^o_r=\\text{perm}(M^{o,r})$ for a given deterministic strategy $a,b,c,d$ , where $M^{o,r}(a,b,c,d)$ is an $n\\times n$ , $\\pm 1$ -valued matrix whose components $u,v=1,\\ldots ,n$ are defined as follows $M^{o,r}_{u,v}(a,b,c,d)=\\nonumber $ ${\\left\\lbrace \\begin{array}{ll}-1,& \\text{if } \\quad n-o2$ .","Finding a general $N$ -party $m$ -setting family of Bell inequalities to lowerbound $P_{NL}^m$ of which the present one is a special $m=2$ member would be most welcome.","Let us mention some possible ways to generalize the persistency of nonlocality of multipartite states.","The concept of EPR steering [46] lies between entanglement and nonlocality, and EPR steering of multipartite quantum states has been investigated recently [47].","Similarly to the persistency of nonlocality, it would be interesting to study the behavior of persistency of steering for the W state or other permutationally invariant states such as Dicke states.","Finally, the question of genuine nonlocality [45] of the W state has also been left open.","Indeed, instead of studying the persistency of standard nonlocality $P_{NL}$ of the W state, we may ask as well what is the minimal number of parties $k$ to trace out from an $N$ -qubit W state, such that the reduced $N-k$ -party state lacks genuinely multipartite nonlocality.","We acknowledge financial support from the Hungarian National Research Fund OTKA (K111734)."],["Loss of excitations","Suppose that a source emits an $n$ -qubit state $\\rho $ , which can be effected by some losses, e.g., noise due to a lossy channel.","We treat channel losses in the following noise model, which is called amplitude damping.","In each mode (out of $n$ modes), there is a probability $p$ of losing an excitation.","The operator sum formalism (see e.g., Ref.","[1]) describes the transformation between the initial state $\\rho $ and the final state $\\rho _{noisy}$ in the following way $\\rho _{noisy}=\\sum _{\\vec{k}=(0,\\dots ,0)}^{(1,\\dots ,1)}\\mathcal {K}_{\\vec{k}}\\rho {\\mathcal {K}_{\\vec{k}}}^\\dagger ,$ where $\\mathcal {K}_{\\vec{k}}$ denotes the tensor product of certain combinations of the following two Kraus operators corresponding to the amplitude damping noise model $K_0 =\\begin{pmatrix}1 & 0\\\\0 & \\sqrt{1-p} \\\\\\end{pmatrix};\\quad K_1 =\\begin{pmatrix}0 & \\sqrt{p} \\\\0 & 0\\\\\\end{pmatrix},$ where $p$ stands for the probability that an excitation is lost.","With these, $\\mathcal {K}_{\\vec{k}}$ is defined by $\\mathcal {K}_{\\vec{k}} = \\otimes _{l=1,\\ldots ,n}K_{k_l}$ , where $n$ is the number of qubits and $k_l$ can take 0 or 1.","Using the W state, $\\rho _W=|W_n\\rangle \\langle W_n|$ as the initial state $\\rho $ in (REF ), the following relations turn out to hold true: $\\mathcal {K}_{0\\ldots 0}\\rho _W\\mathcal {K}_{0\\ldots 0}^\\dagger &= (1-p)\\rho _W\\nonumber \\\\\\mathcal {K}_{0\\ldots 01}\\rho _W\\mathcal {K}_{0\\ldots 01}^\\dagger &= (p/n)|0^{\\otimes n}\\rangle \\langle 0^{\\otimes n}|,$ Further, if we permute the index $\\vec{k}={0\\ldots 01}$ in all the $n$ different ways, we get the same state as that on the right hand side of the second line of (REF ), i.e., the $n$ -partite vacuum state multiplied by $p/n$ .","On the other hand, if the number of 1's in $\\vec{k}$ are at least two, we have $\\mathcal {K}_{\\vec{k}}\\rho _W\\mathcal {K}_{\\vec{k}}^\\dagger = 0$ .","Substituting into (REF ), we arrive at $\\rho _{noisy}=(1-p)|W_n\\rangle \\langle W_n| + p|0^{\\otimes n}\\rangle \\langle 0^{\\otimes n}|,$ which is the same state as (REF )."]],"string":"[\n [\n \"Bounding the persistency of the nonlocality of W states\"\n ],\n [\n \"Abstract The nonlocal properties of the W states are investigated under particle loss.\",\n \"By removing all but two particles from an $N$-qubit W state, the resulting two-qubit state is still entangled.\",\n \"Hence, the W state has high persistency of entanglement.\",\n \"We ask an analogous question regarding the persistency of nonlocality introduced in [Phys.\",\n \"Rev.\",\n \"A 86, 042113].\",\n \"Namely, we inquire what is the minimal number of particles that must be removed from the W state so that the resulting state becomes local.\",\n \"We bound this value in function of $N$ qubits by considering Bell nonlocality tests with two alternative settings per site.\",\n \"In particular, we find that this value is between $2N/5$ and $N/2$ for large $N$.\",\n \"We also develop a framework to establish bounds for more than two settings per site.\"\n ],\n [\n \"Introduction\",\n \"Entanglement and nonlocality are two manifestations of quantum correlations, both are essential ingredients of quantum theory.\",\n \"They also play an important role in the field of quantum information [1].\",\n \"For instance, entanglement is in the heart of quantum teleportation and plays a crucial role in quantum algorithms [2], [3].\",\n \"Nonlocality, on the other hand, witnessed by the violation of Bell inequalities [13], reduces communication complexity [4], [5] and enables device-independent quantum information protocols [6], [7].\",\n \"These protocols do not rely on a detailed knowledge of the internal working of the experimental devices used, thereby allowing secure cryptography involving untrusted devices [8] or expansion of secure random numbers [9], [10].\",\n \"The quantification of entanglement and nonlocality in the bipartite case (i.e., the case when two parties share an entangled state) is relatively well understood.\",\n \"However, the multipartite case is much less explored.\",\n \"This is mainly due to the rapidly growing complexity of the problem with the number of parties.\",\n \"For instance, for bipartite pure states, there is a unique measure of entanglement, however, for three or more parties, this is not true any more (see e.g., Refs.\",\n \"[11], [12]).\",\n \"Concerning nonlocality, there is a single tight Bell inequality for two binary settings per party, the Clauser-Horne-Shimony-Holt (CHSH) one [13], [14].\",\n \"However, moving to three parties, the number of tight Bell inequalities already becomes 42 [15].\",\n \"In fact, determining all Bell inequalities for growing number parties is an NP-hard problem [16], [17].\",\n \"Bipartite and tripartite quantum nonlocality are different from a fundamental point of view as well.\",\n \"While Gleason's theorem can be extended to the bipartite scenario, however, there remains a gap between the quantum and Gleason's correlations in the tripartite scenario [18].\",\n \"One of the most famous multipartite quantum states is the W state [19] playing a crucial role in the physics of the interaction between light and matter.\",\n \"Up to now, W states have been prepared in lots of experiments, e.g., photonic experiments can generate and characterize six qubit entangled states [20].\",\n \"More recently, genuine 28-particle entanglement was detected in a Dicke-like state (a generalization of the W state) in a Bose-Einstein condensate [21].\",\n \"W states are very robust to losses, hence suited to quantum information applications such as quantum memories [22].\",\n \"For instance, by tracing out all but two parties from an $N$ -party W state, the remaining two-qubit state is still entangled, no matter how big $N$ is.\",\n \"Hence, the W state shows a high persistency of entanglement against particle loss (actually, the highest possible persistency among $N$ qubit states [23]).\",\n \"On the other hand, one may wonder how robust the nonlocality of the W state is with respect to particle loss.\",\n \"In order to quantify this, Ref.\",\n \"[24] introduces the persistency of nonlocality of $N$ -party quantum states $\\\\rho $ , $P_{NL}(\\\\rho )$ , which is the minimal number of particles to be removed for nonlocal quantum correlations to vanish.\",\n \"Ref.\",\n \"[24] investigates this measure for various classes of multipartite states including W states up to $N=7$ with two settings per site.\",\n \"In this paper, we bound this value both from above and from below for any $N$ -qubit W states.\",\n \"To do this, we refine the definition of $P_{NL}(\\\\rho )$ to account for Bell nonlocality involving $m$ settings per party.\",\n \"This quantity will be called $P_{NL}^m(\\\\rho )$ .\",\n \"Clearly, in the limit of large $m$ , $P_{NL}^m(\\\\rho )$ tends to $P_{NL}(\\\\rho )$ .\",\n \"Our main result concerns the case of $m=2$ and we prove the bounds of $2N/5\\\\le P_{NL}^2(W_N)\\\\le N/2$ for $N$ large, featuring a relatively small gap between the upper and lower bounds.\",\n \"The lower bound is based on an explicit construction of a class of Bell inequalities.\",\n \"We also give a numerical framework to put reliable lower bounds on $P_{NL}^m(W_N)$ beyond two settings (up to $m=6$ ) and a tractable number of $N$ parties.\",\n \"This numerical study supports that our analytical lower bounds for $P_{NL}^2(W_N)$ are tight.\",\n \"There are recent papers which discuss the robustness of Dicke states [25] (and in particular the W state) to various types of noises [26], [28], [27].\",\n \"Lower bounds also follow from these papers for the value of $P_{NL}^2(W_N)$ .\",\n \"In particular, we obtain considerable improvement over the lower bounds presented in Ref. [28].\",\n \"Notably, the persistency of nonlocality also gives a device-independent bound on the persistency of entanglement introduced in Ref. [29].\",\n \"Other device-independent approaches to quantify multipartite entanglement including W states appeared in Refs.\",\n \"[30], [31], [32].\",\n \"On the other hand, multipartite W states are promising candidates to close the detection loophole [33] in multipartite Bell tests [34].\",\n \"Such Bell violations would complement the experimental loophole-free violations obtained recently in the bipartite case [35].\",\n \"The structure of the paper is as follows.\",\n \"Section introduces notation.\",\n \"Section proves a simple upper bound on the persistency of nonlocality $P^m_{NL}$ for W states and in general for any permutationally symmetric state with two settings per party (case $m=2$ ).\",\n \"On the other hand, section presents lower bound values based on numerical investigations.\",\n \"To this end, we first outline the numerical method, then show results for $m=2$ and also beyond $m=2$ up to $m=6$ .\",\n \"In section , a family of Bell inequalities is presented (valid for any number of parties $N$ ), which allows us to obtain good lower bounds for $P_{NL}^2(W_N)$ .\",\n \"The paper ends with a discussion in section .\"\n ],\n [\n \"Bell setup\",\n \"Let us imagine the following Bell setup [13].\",\n \"A quantum state $\\\\rho $ is shared between $n$ spatially separated systems, on which the local observers can conduct measurements.\",\n \"We focus on binary outcome measurements in which case we may define the joint correlators by the following set of expectation values: $\\\\left\\\\lbrace \\\\langle \\\\mathcal {M}_{j_1}^{(1)}\\\\ldots \\\\mathcal {M}^{(n)}_{j_n}\\\\rangle \\\\right\\\\rbrace =\\\\left\\\\lbrace \\\\mathrm {Tr}{\\\\left(\\\\rho \\\\mathcal {M}_{j_1}^{(1)}\\\\otimes \\\\ldots \\\\otimes \\\\mathcal {M}_{j_n}^{(n)}\\\\right)}\\\\right\\\\rbrace $ with $j_l=0,1,\\\\ldots ,m$ and $l=1,\\\\ldots ,n$ .\",\n \"We identify $\\\\mathcal {M}_0^{(l)}=\\\\leavevmode \\\\hbox{\\\\small 1\\\\normalsize \\\\hspace{-3.30002pt}1}$ and $\\\\mathcal {M}_{j_l}^{(l)}$ refers to the $j_l$ th $\\\\pm 1$ -valued observable of party $l$ .\",\n \"The corresponding real vector of correlators defines a point in the $(1+m)^n$ dimensional space of probabilities.\",\n \"Each member of the set (REF ) has an order, which is given by the amount of parties $o$ involving a non-trivial observable (i.e.\",\n \"not involving $\\\\mathcal {M}_0^{(l)}=\\\\leavevmode \\\\hbox{\\\\small 1\\\\normalsize \\\\hspace{-3.30002pt}1}$ ).\",\n \"In particular, those with $o=n$ are usually called full-correlators, while those with $o=1$ are called one-body correlators or marginal terms.\",\n \"A multipartite (2-outcome) Bell inequality [13] is a linear function of the above correlators (REF ), $\\\\sum _{j_1=0}^{m}\\\\ldots \\\\sum _{j_n=0}^{m}{\\\\alpha _{j_1,\\\\ldots ,j_n}}\\\\langle \\\\mathcal {M}_{j_1}^{(1)}\\\\ldots \\\\mathcal {M}^{(n)}_{j_n}\\\\rangle \\\\le \\\\beta _c,$ where we denote by $\\\\beta _c$ the bound which holds for any local hidden variable model.\",\n \"These are the correlations which the parties can simulate by merely using local strategies and some shared classical information.\",\n \"The local correlations attainable this way forms a polytope, the so-called Bell polytope, whose extremal points consist of those vectors in (REF ) in which all correlators factorize, that is, $\\\\langle \\\\mathcal {M}_{j_1}^{(1)}\\\\ldots \\\\mathcal {M}^{(n)}_{j_n}\\\\rangle =\\\\langle \\\\mathcal {M}_{j_1}^{(1)}\\\\rangle \\\\cdot \\\\ldots \\\\cdot \\\\langle \\\\mathcal {M}_{j_n}^{(n)}\\\\rangle $ and the mean value of each single party $\\\\langle \\\\mathcal {M}_{j_l}^{(l)}\\\\rangle $ , $(l=1,\\\\ldots ,n, j_l>0)$ equals either $-1$ or $+1$ .\",\n \"Let us define the persistency of nonlocality of a multipartite state $\\\\rho $ according to Ref.\",\n \"[24] as follows.\",\n \"Let us take the partial trace over $02$ as well.\",\n \"In particular, we have chosen a given Bell scenario ($n$ parties, $m$ settings) and by varying the $m$ angles, we maximized the value of $p_{crit}$ .\",\n \"We note that this search is a heuristic one and as such it is not guaranteed to terminate in a global maximum of $p_{crit}$ .\",\n \"However, the obtained value still defines a lower bound to $P^m_{NL}$ in (REF ).\",\n \"The critical $p$ values obtained in function of $n$ and $m$ are displayed in Fig.\",\n \"REF .\",\n \"We may observe that, as $m$ increases, the affordable number of parties $n$ decreases.\",\n \"For the simplest case of $m=2$ , numerically we could afford $n=14$ .\",\n \"In addition, we had to resort to a symmetrization procedure of the Bell polytope introduced in [41] which considerably reduces the complexity of the problem in order to attain $n=14$ .\",\n \"Figure: For ρ(n,p)\\\\rho (n,p) states of nn qubits (), the graph shows lower bounds on the thresholdprobability p crit p_{crit} obtained using our numerical method up tom=6m=6 measurement settings.Plugging the $p_{crit}$ values shown in Fig.\",\n \"REF into formula (REF ), we get Table REF , where the computed persistencies $P_{NL}^m$ are shown for $m\\\\le 6$ .\",\n \"This way we get entries in the table only for $N$ satisfying $N=\\\\lfloor n/(1-p_{crit})\\\\rfloor $ , but a slight modification allows us to obtain numbers for any $N$ parties: Fix $N$ , and choose the largest $k$ integer such that $p_{crit,n-k}>k/N$ , where $p_{crit,n-k}$ stands for $p_{crit}$ evaluated by the number of parties $n-k$ .\",\n \"Then a lower bound on the persistency $P_{NL}^m$ for the $N$ -qubit W state is given by $P_{NL}^m=k+1$ .\",\n \"Table: Lower bound values for persistency ofnonlocality P NL m P_{NL}^m for the W N W_N state with m=2,...,6m=2,\\\\ldots ,6binary-outcome measurement settings per party.\",\n \"The results arebased on the numerical method described in the text.In Table REF , for $m=2$ , the first entries $N=2,3,\\\\ldots ,7$ match the numbers from previous study [24].\",\n \"However, by fixing $N$ and going to higher number of settings $m$ sometimes we get a higher persistency of nonlocality.\",\n \"For instance, in case of $N=6$ parties, $P_{NL}^5=3$ , which is to be compared to $P_{NL}^2=2$ .\",\n \"For $m=2$ , we extracted the optimal Bell inequalities corresponding to $p_{crit}$ for various $n$ values.\",\n \"For $n$ even, a common structure has been found.\",\n \"In fact, they turned out to be members of a family of Bell inequalities valid for any $n$ even.\",\n \"Details of this class of inequalities are presented in Sec. .\",\n \"We optimized the quantum value of these inequalities in function of the two measurement angles.\",\n \"Fig.\",\n \"REF shows results for $p_{crit}$ up to $n=50$ of this family.\",\n \"We also show results using the ansatz that the two measurement settings are $Z$ and $X$ , in which case $p_{crit}=(2n-4)/(5n-2)$ for $n$ even.\",\n \"According to the figure, for larger $n$ the $(Z,X)$ pair of settings become close to optimal.\",\n \"The $p_{crit}$ values have been also studied in Ref.\",\n \"[28] for specific families of known Bell inequalities from the literature, with the best lower bound values coming from the WWWZB inequalities [42].\",\n \"For comparison, these $p_{crit}$ values are displayed.\",\n \"For $n=50$ , $p_{crit}=0.3142$ , which is suboptimal with respect to our $p_{crit}=(2n-4)/(5n-2)\\\\simeq 0.3871$ .\",\n \"Figure: For ρ(n,p)\\\\rho (n,p) states of nn qubits (), thegraph shows lower bounds on the threshold probability p crit p_{crit}obtained using our numerical method for m=2m=2 measurementsettings.\",\n \"These bounds coincide with the bounds coming from ourclass of Bell inequalities for 2 measurement settings up to n=14n=14and we conjecture to be the same for any even nn particles.\",\n \"Weshow results for our Bell inequalities with optimized andnon-optimized settings (in the latter case, the two observablesare the Pauli operators ZZ and XX).\",\n \"The lower curve reproducesthe results of Ref.\",\n \"in case of the WWWZBinequalities .Finally, Fig.\",\n \"REF shows the upper bound values due to section and the lower bound values for $P^2_{NL}(W_N)$ up to $N=50$ due to our Bell family.\",\n \"We conjecture that the exact value for $P^2_{NL}(W_N)$ is defined by the lower curve (that is, no better Bell inequalities than our family in section exist in this respect).\",\n \"For instance, the best lower bound for $P^2_{NL}(W_N)$ so far comes from the WWWZB inequalities analyzed in Ref.\",\n \"[28], which lies below our curve for the lower bound, and according to the conjecture in Ref.\",\n \"[28] it goes to $N/3$ in the limit of large $N$ .\",\n \"On the other hand, our asymptotic lower bound value of $2N/5$ is considerable higher.\",\n \"It is also worth noting that the upper bound value on $P^2_{NL}$ shown in the figure holds true for any permutationally invariant $N$ -qubit state.\",\n \"Hence, Dicke states with more than one excitation, as a notable subset of permutationally invariant states, cannot perform significantly better than $W$ states in the two-setting case.\",\n \"Figure: For WW states of NN qubits, the graph shows upperbounds (marked by a triangle pointing down) and lower bounds (marked by triangles pointing up) on the persistency of nonlocalityP NL m=2 (W N )P_{NL}^{m=2}(W_N).In Ref.\",\n \"[28], the Dicke states (including the $W_N$ state) were analyzed in terms of two decoherence models: loss of particles and loss of excitations.\",\n \"The first one relates to the measure $P_{NL}$ discussed in this paper, whereas the latter one is related to $p_{crit}$ in formula (REF ).\",\n \"Indeed, if we start from an $n$ -qubit W state which is effected by a decoherence where in each mode an excitation has a probability $p$ of being lost, it brings the W state into (REF ).\",\n \"Appendix A shows a derivation of this result.\",\n \"Next section gives all the details, including the quantum and local bounds, of our particular family of two-setting Bell inequalities providing $p_{crit}=(2n-4)/(5n-2)$ for even $n$ .\",\n \"In case of large $n$ , this goes to $2/5$ which we conjecture to be the largest achievable critical value among all two-setting Bell inequalities.\"\n ],\n [\n \"A family of two-setting multipartite Bell inequalities\",\n \"Standard form of permutationally symmetric Bell inequalities.– The Bell inequality to be considered consists of correlators (REF ) which are invariant under any permutation of the parties.\",\n \"By imposing the permutational symmetry, one requires that the expectation values $\\\\langle \\\\mathcal {M}_{j_1}^{(1)}\\\\ldots \\\\mathcal {M}^{(n)}_{j_n}\\\\rangle $ and $\\\\langle \\\\mathcal {M}_{j_1}^{\\\\sigma (1)}\\\\ldots \\\\mathcal {M}^{\\\\sigma (n)}_{j_n}\\\\rangle $ are the same, where $\\\\sigma :\\\\lbrace 1,\\\\ldots ,n\\\\rbrace \\\\rightarrow \\\\lbrace \\\\sigma (1),\\\\ldots ,\\\\sigma (n)\\\\rbrace $ is an arbitrary permutation of the set $\\\\lbrace 1,\\\\ldots ,n\\\\rbrace $ .\",\n \"Let us denote by $S_n$ all permutations of the set $\\\\lbrace 1,\\\\ldots ,n\\\\rbrace $ .\",\n \"Then, it will be useful to define $S^o_r \\\\equiv \\\\sum _{\\\\sigma \\\\in S_n}\\\\langle \\\\mathcal {M}_{j_1}^{\\\\sigma (1)}\\\\ldots \\\\mathcal {M}^{\\\\sigma (n)}_{j_n}\\\\rangle ,$ where the sum ranges over all permutations of the set $\\\\lbrace 1,\\\\ldots ,n\\\\rbrace $ .\",\n \"In $S^o_r$ above, $o$ is the order of correlators and $r$ is the number of 1's occurring in the list $\\\\lbrace j_1,\\\\ldots ,j_n\\\\rbrace $ .\",\n \"Let us now define the symmetrized correlation vector by the above ordered real vectors as follows $\\\\vec{S} \\\\equiv \\\\lbrace S^o_r\\\\rbrace ^{o=1,\\\\ldots ,n}_{r=0,\\\\ldots ,o}.$ In a similar way, we define the vector of coefficients associated with $\\\\vec{S}$ as $\\\\vec{\\\\alpha }\\\\equiv \\\\lbrace \\\\alpha ^o_r\\\\rbrace ^{o=1,\\\\ldots ,n}_{r=0,\\\\ldots ,o}.$ As a result, we arrive at the general form of a permutationally symmetric Bell inequality: $I \\\\equiv \\\\vec{\\\\alpha }\\\\cdot \\\\vec{S} = \\\\sum _{\\\\begin{array}{c}{o=1,\\\\ldots ,n}\\\\\\\\{r=0,\\\\ldots ,o}\\\\end{array}}{\\\\alpha ^o_r S^o_r}\\\\le \\\\beta _c,$ where $\\\\beta _c$ is the local maximum, which holds for any local hidden variable model.\",\n \"It is worth noting that the WWWZB class [42] contains only full-correlator terms ($o=n$ ), whereas our construction of Bell inequalities turns out to contain all different orders, starting from marginal terms ($o=1$ ) up to full-correlation terms ($o=n$ ).\",\n \"There also exist permutationally symmetric $n$ -party Bell inequalities involving only first and second order terms ($o=1,2$ ) [43].\",\n \"The usefulness of these kind of inequalities in the particle loss model is an open and interesting question in our view.\",\n \"As an illustrative example, let us discuss the case of 4 particles ($n=4$ ).\",\n \"In this case, $\\\\vec{S}$ looks as follows $\\\\vec{S} = \\\\lbrace S^1_1,S^1_0;S^2_2,S^2_1,S^2_0;S^3_3,S^3_2,S^3_1,S^3_0;S^4_4,S^4_3,S^4_2,S^4_1,S^4_0\\\\rbrace ,$ where we used semicolon (;) in order to separate components with different order $o$ .\",\n \"By identifying $\\\\mathcal {M}_0^{(l)}=1$ , $l=1,\\\\ldots ,4$ and $\\\\mathcal {M}_j^{(1)}=A_j$ , $\\\\mathcal {M}_j^{(2)}=B_j$ , $\\\\mathcal {M}_j^{(3)}=C_j$ , $\\\\mathcal {M}_j^{(4)}=D_j$ for $j=1,2$ in Eq.\",\n \"(REF ), we have the following mean values displayed for some particular cases (neglecting the mean value signs): $S^1_1 =& 6(A_1 + B_1 + C_1 + D_1),\\\\nonumber \\\\\\\\S^2_2 =& 4(A_1B_1 + A_1C_1 + A_1D_1 + B_1C_1 + B_1D_1 + C_1D_1),\\\\nonumber \\\\\\\\S^2_0 =& 4(A_2B_2 + A_2C_2 + A_2D_2 + B_2C_2 + B_2D_2 + C_2D_2),\\\\nonumber \\\\\\\\S^3_1 =& 2(A_1B_2C_2+A_1B_2D_2+A_1C_2D_2)\\\\nonumber \\\\\\\\& +2(A_2B_1C_2+A_2B_1D_2+B_1C_2D_2)\\\\nonumber \\\\\\\\& +2(A_2B_2C_1+A_2C_1D_2+B_2C_1D_2)\\\\nonumber \\\\\\\\& + 2(A_2B_2D_1+A_2C_2D_1+B_2C_2D_1),\\\\nonumber \\\\\\\\S^4_0 =& 24A_2B_2C_2D_2,$ where we recall that $o$ denotes the order in the superindex of $S^o_r$ , whereas $r$ is the number of occurrences of subindex 1 in $S^o_r$ .\",\n \"For instance, $S^3_1$ above consists of all length-3 sequences from $A,B,C$ and $D$ letters with the occurrence of a single subindex 1.\",\n \"The specific class of Bell inequalities.– Let us now give the explicit form of a Bell inequality $I$ defined by the vector of coefficients $\\\\vec{\\\\alpha }$ in Eq.\",\n \"(REF ).\",\n \"This family has been extracted from the numerical method of Sec. .\",\n \"In particular, we introduce a family of Bell inequalities $I_n$ valid for any even number of particles $n$ as follows $I_n \\\\equiv \\\\vec{\\\\alpha }_n\\\\cdot \\\\vec{S}_n = \\\\sum _{\\\\begin{array}{c}{o=1,\\\\ldots ,n}\\\\\\\\{r=0,\\\\ldots ,o}\\\\end{array}}{\\\\alpha ^o_r S^o_r}\\\\le \\\\beta _{c,n}.$ The vector of coefficients $\\\\vec{\\\\alpha }=\\\\lbrace \\\\alpha ^o_r(o=1,\\\\ldots ,n, r=0,\\\\ldots ,o)\\\\rbrace $ reads as $\\\\vec{\\\\alpha }={\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\alpha ^{2k}_0 = F_{k,n}, & k=1,\\\\ldots ,n/2\\\\\\\\\\\\alpha ^{2k-1}_1 = G_{k,n}+2w_n\\\\delta _{k,1}, & k=1,\\\\ldots ,n/2\\\\\\\\\\\\alpha ^2_2 = -w_n,\\\\\\\\\\\\alpha ^o_r = 0, & \\\\text{otherwise}\\\\end{array}\\\\right.\",\n \"}$ where $\\\\delta _{i,j}$ is the Kronecker delta (1 if $i=j$ , 0 otherwise), and the functions entering eq.\",\n \"(REF ) above are $F_{k,n}&=(-1)^k\\\\frac{n-2}{2}(n+1-2k)\\\\binom{n/2}{k},\\\\nonumber \\\\\\\\G_{k,n}&=(-1)^k\\\\frac{n+2}{2}n\\\\binom{n/2-1}{k-1},\\\\nonumber \\\\\\\\w_n&=\\\\frac{n(n-1)(n+2)}{2^{4-n}\\\\binom{n}{n/2}}.$ Especially for $n=4$ , our 4-party Bell inequality looks as follows: $I_4 = 12 S^1_1 - 12 S^2_2 - 6S^2_0 + 12S^3_1 + S^4_0 \\\\le 128,$ where the middle terms appear explicitly in (REF ) and the local bound is due to formula (REF ) given in the next subsection for the general $n$ -party case.\",\n \"Local bound.– The local maximum, which is the maximum value one obtains using local resources only, is given by $\\\\beta _{c,n}=n!\\\\left(w_n-\\\\frac{(n-2)(n+1)}{2}\\\\right)$ in case of any even number of particles $n$ , where $w_n=\\\\frac{n(n-1)(n+2)}{2^{4-n}\\\\binom{n}{n/2}}.$ We have checked the validity of the above local bound $\\\\beta _{c,n}$ for $n\\\\le 16$ ($n$ even) by listing all possible deterministic strategies (i.e., vertices of the Bell polytope).\",\n \"These are the strategies for which all correlators factorize (see eq.\",\n \"REF ) and each single party marginal $\\\\langle \\\\mathcal {M}_{j_l}^{(l)}\\\\rangle $ , $(l=1,\\\\ldots ,n, j_l=1,2)$ equals either $-1$ or $+1$ .\",\n \"One of the above deterministic strategies, due to linearity of the Bell functional, will provide the local maximum.\",\n \"Notice that $\\\\langle \\\\mathcal {M}_0^{(l)}\\\\rangle $ , ($l=1,\\\\ldots ,n$ ) equals $+1$ by definition.\",\n \"Below we provide an analytical proof of formula (REF ) for any even $n$ .\",\n \"We checked with brute-force computations that for smaller values of $n$ the inequalities are far from tightness, that is, they do not define a facet of the Bell polytope.\",\n \"We conjecture that they are not tight for larger $n$ as well.\",\n \"Here follows the proof of the local limit (REF ) for even $n$ .\",\n \"First let us fix notation.\",\n \"For a given local deterministic strategy let us define a four-tuple ($a$ ,$b$ ,$c$ ,$d$ ) which counts the number of parties whose marginal expectations $\\\\lbrace \\\\langle \\\\mathcal {M}^{(l)}_1\\\\rangle ,\\\\lbrace \\\\langle \\\\mathcal {M}^{(l)}_2\\\\rangle \\\\rbrace $ are the respective pairs ($\\\\lbrace 1,1\\\\rbrace $ , $\\\\lbrace 1,-1\\\\rbrace $ , $\\\\lbrace -1,1\\\\rbrace $ , $\\\\lbrace -1,-1\\\\rbrace $ ).\",\n \"By definition we have $a+b+c+d=n$ .\",\n \"In case of permutationally invariance under party exchange, a four-tuple ($a,b,c,d$ ) represents uniquely a deterministic strategy.\",\n \"Hence, our task is to calculate the Bell value $\\\\vec{\\\\alpha }_n\\\\cdot \\\\vec{S}_n$ for all possible integer-valued four-tuples $a,b,c,d\\\\ge 0$ fulfilling $a+b+c+d=n$ , and then pick the maximum value out of this set.\",\n \"This defines the local maximum $\\\\beta _{c,n}$ attainable using classical resources.\",\n \"In the proof below, all positive integer four-tuples $a,b,c,d$ are understood to sum up to $n$ , where $n$ is even.\",\n \"Let us introduce the permanent of an $n\\\\times n$ matrix $A=(a_{i,j})$ , $(i,j=1,\\\\ldots ,n)$ .\",\n \"It is defined as $\\\\text{perm}(A) = \\\\sum _{\\\\sigma \\\\in S_n}\\\\prod _{i=1}^n{a_{i,\\\\sigma (i)}},$ where $\\\\sigma $ is a permutation over the set $\\\\lbrace 1,\\\\ldots ,n\\\\rbrace $ .\",\n \"Using the above definition for the permanent, $\\\\vec{S} = \\\\lbrace S^o_r\\\\rbrace $ ($o=1,\\\\ldots ,n$ , $r=0,\\\\ldots ,o$ ) is given by the components $S^o_r=\\\\text{perm}(M^{o,r})$ for a given deterministic strategy $a,b,c,d$ , where $M^{o,r}(a,b,c,d)$ is an $n\\\\times n$ , $\\\\pm 1$ -valued matrix whose components $u,v=1,\\\\ldots ,n$ are defined as follows $M^{o,r}_{u,v}(a,b,c,d)=\\\\nonumber $ ${\\\\left\\\\lbrace \\\\begin{array}{ll}-1,& \\\\text{if } \\\\quad n-o2$ .\",\n \"Finding a general $N$ -party $m$ -setting family of Bell inequalities to lowerbound $P_{NL}^m$ of which the present one is a special $m=2$ member would be most welcome.\",\n \"Let us mention some possible ways to generalize the persistency of nonlocality of multipartite states.\",\n \"The concept of EPR steering [46] lies between entanglement and nonlocality, and EPR steering of multipartite quantum states has been investigated recently [47].\",\n \"Similarly to the persistency of nonlocality, it would be interesting to study the behavior of persistency of steering for the W state or other permutationally invariant states such as Dicke states.\",\n \"Finally, the question of genuine nonlocality [45] of the W state has also been left open.\",\n \"Indeed, instead of studying the persistency of standard nonlocality $P_{NL}$ of the W state, we may ask as well what is the minimal number of parties $k$ to trace out from an $N$ -qubit W state, such that the reduced $N-k$ -party state lacks genuinely multipartite nonlocality.\",\n \"We acknowledge financial support from the Hungarian National Research Fund OTKA (K111734).\"\n ],\n [\n \"Loss of excitations\",\n \"Suppose that a source emits an $n$ -qubit state $\\\\rho $ , which can be effected by some losses, e.g., noise due to a lossy channel.\",\n \"We treat channel losses in the following noise model, which is called amplitude damping.\",\n \"In each mode (out of $n$ modes), there is a probability $p$ of losing an excitation.\",\n \"The operator sum formalism (see e.g., Ref.\",\n \"[1]) describes the transformation between the initial state $\\\\rho $ and the final state $\\\\rho _{noisy}$ in the following way $\\\\rho _{noisy}=\\\\sum _{\\\\vec{k}=(0,\\\\dots ,0)}^{(1,\\\\dots ,1)}\\\\mathcal {K}_{\\\\vec{k}}\\\\rho {\\\\mathcal {K}_{\\\\vec{k}}}^\\\\dagger ,$ where $\\\\mathcal {K}_{\\\\vec{k}}$ denotes the tensor product of certain combinations of the following two Kraus operators corresponding to the amplitude damping noise model $K_0 =\\\\begin{pmatrix}1 & 0\\\\\\\\0 & \\\\sqrt{1-p} \\\\\\\\\\\\end{pmatrix};\\\\quad K_1 =\\\\begin{pmatrix}0 & \\\\sqrt{p} \\\\\\\\0 & 0\\\\\\\\\\\\end{pmatrix},$ where $p$ stands for the probability that an excitation is lost.\",\n \"With these, $\\\\mathcal {K}_{\\\\vec{k}}$ is defined by $\\\\mathcal {K}_{\\\\vec{k}} = \\\\otimes _{l=1,\\\\ldots ,n}K_{k_l}$ , where $n$ is the number of qubits and $k_l$ can take 0 or 1.\",\n \"Using the W state, $\\\\rho _W=|W_n\\\\rangle \\\\langle W_n|$ as the initial state $\\\\rho $ in (REF ), the following relations turn out to hold true: $\\\\mathcal {K}_{0\\\\ldots 0}\\\\rho _W\\\\mathcal {K}_{0\\\\ldots 0}^\\\\dagger &= (1-p)\\\\rho _W\\\\nonumber \\\\\\\\\\\\mathcal {K}_{0\\\\ldots 01}\\\\rho _W\\\\mathcal {K}_{0\\\\ldots 01}^\\\\dagger &= (p/n)|0^{\\\\otimes n}\\\\rangle \\\\langle 0^{\\\\otimes n}|,$ Further, if we permute the index $\\\\vec{k}={0\\\\ldots 01}$ in all the $n$ different ways, we get the same state as that on the right hand side of the second line of (REF ), i.e., the $n$ -partite vacuum state multiplied by $p/n$ .\",\n \"On the other hand, if the number of 1's in $\\\\vec{k}$ are at least two, we have $\\\\mathcal {K}_{\\\\vec{k}}\\\\rho _W\\\\mathcal {K}_{\\\\vec{k}}^\\\\dagger = 0$ .\",\n \"Substituting into (REF ), we arrive at $\\\\rho _{noisy}=(1-p)|W_n\\\\rangle \\\\langle W_n| + p|0^{\\\\otimes n}\\\\rangle \\\\langle 0^{\\\\otimes n}|,$ which is the same state as (REF ).\"\n ]\n]"},"id":{"kind":"string","value":"1605.05846"}}},{"rowIdx":1866686,"cells":{"text":{"kind":"list like","value":[["Magnetic hyperbolic metamaterial of polaritonic nanowires"],["Abstract We show that the axial component of the magnetic permeability tensor is resonant for a wire medium consisting of high-index epsilon-positive nanowires, and its real part changes the sign at a certain frequency.","At this frequency the medium experiences the topological transition from the hyperbolic to the elliptic type of dispersion.","We show that the transition regime is characterized by extremely strong dependence of the permeability on the wave vector.","This implies very high density of electromagnetic states that results in the filamentary pattern and noticeable Purcell factor for a transversely oriented magnetic dipole."],["introduction","Simple wire media, defined as optically dense arrays of parallel metal wires in a host dielectric material, have been in a considerable attention since electromagnetic metamaterials have been introduced.","Many applications of such media have been suggested from radio frequencies where the period of wire media is on the millimeter scale, to optical frequencies where this period is submicron.","Among them one can mention biological sensing [1], subwavelength imaging and endoscopy with image magnification [2], [3], [4], enhancement of quantum emitters, thermal sources, and classical dipole radiators [5], [7], [6], enhancement of radiative heat transfer [8], [9], etc.","More details can be found in overview [10].","In all these works, electromagnetic properties of wire media were described in a condensed form – via an effective permittivity tensor: $\\overline{\\overline{\\epsilon }}=\\left(\\begin{array}{ccc}\\varepsilon _{\\perp } & 0 & 0 \\\\0 &\\varepsilon _{\\perp } & 0 \\\\0& 0 & \\varepsilon _{\\parallel } \\end{array} \\right).$ Here, $\\varepsilon _\\perp $ and $\\varepsilon _\\parallel $ are the perpendicular (transversal) and parallel (axial) components with respect to the wires axis.","Therefore, standard wire media are uniaxially anisotropic materials (though not usual ones because they possess spatial dispersion – dependence of $\\overline{\\overline{\\epsilon }}$ on the wave vector ${\\bf k}$ [11], [12].","At infrared frequencies, the relative dielectric constant of the metal nanowires is highly negative.","If the fraction ratio of the metal in the effective medium is low the perpendicular component $\\varepsilon _\\perp $ of the permittivity is approximately equal to the permittivity of the host medium, whereas the parallel component $\\varepsilon _\\parallel $ has negative sign [11], [10].","For the dispersion, it implies that the transverse magnetic (TM) polarized wave has an open dispersion surface similar to a hyperboloid [10], [12].","This dispersion surface results in the high density of electromagnetic states and enhancement of a subwavelength electric dipole located in the medium orthogonally to the wires [5], [12].","The radiation of a magnetic dipole can be also enhanced but this effect is lower because such the dipole produces mainly TE-waves whose electric field is orthogonal to nanowires.","These transversely electric (TE) polarized waves weakly interact with the wire medium and their dispersion surfaces are rather similar to spheres like dispersion surfaces of free space.","Notice, that for metal nanowires the effective permeability tensor ($\\overline{\\overline{\\mu }}$ ) is practically equal to unity [10].","Recently, in Ref.","[13] one studied an anisotropic medium with artificial magnetism.","This medium is dual to the dielectric hyperbolic metamaterial described by the indefinite tensor in Eq. ().","In this magnetic metamaterial the effective permittivity tensor is definite, positive and not resonant.","Therefore, the dispersion surfaces of TM-polarized waves in this medium are similar to the spheres, whereas the dispersion surfaces of TE-waves are either hyperbolic or elliptic depending on frequency.","However, medium in Ref.","[13] is not a wire medium.","It is a racemic array of metal helices with both left-handedness and right-handedness.","Such media, in accordance to initial works [14], [15] where they were introduced, should be referred as spiral media.","Spiral medium of [13] operates at microwave frequencies, where metals are close to perfect conductors.","In the present paper, we study wire media at optical frequencies.","Figure: A medium of long parallel wires which are set in a square lattice.","The wires are made of polaritonic materials whose relative dielectric constant is much larger than unity (ε r ≫1\\varepsilon _{\\rm {r}}\\gg 1).","Here, aa and bb are the wires radius and lattice constant, respectively.","For simplicity, the host medium is free space.Below, we theoretically demonstrate that a simple wire medium (see Fig.","REF ) may have in the infrared range similar electromagnetic properties to those manifested by [13] exhibited at microwaves, however, there are also significant peculiarities which share our wire medium out from all hyperbolic metamaterials.","Nanowires of our wire medium should be made of high-index epsilon-positive materials, such as lithium tantalate, silicon carbide or hexagonal boron nitride, and some other polaritonic materials in the range 15–30 THz.","The resonance of the axial component of the effective permeability tensor arises because the dynamic magnetic polarizability of the high-index dielectric wire is resonant, and origins from the Mie resonance of a single nanowire.","We show that this axial permeability changes its sign at a certain frequency and in a narrow frequency range around this transition is a very sharp function of the wave vector.","The corresponding topological transition is related to the resonant enhancement of radiation of a subwavelength dipole located inside the medium.","It is an expected effect which is similar to that studied in [6].","However, in the present case it holds for a magnetic dipole oriented transversely to nanowires.","The sharp dependence of the permeability on the wave vector results in the unusual property of our wire medium – the radiation of the internal magnetic dipole is concentrated at the line which passes through the dipole center in parallel to the optical axis.","This property is difficult to detect from the dispersion contours (sections of the dispersion surfaces).","In the present case these contours for TE-waves do not differ very much from analogous contours for TM-waves we have studied in [9].","Moreover, these contours were analyzed for TE-waves in the spiral medium [13].","Therefore, we do not spend the volume of this paper to the demonstration and analysis of the dispersion contours in the region of topological transition and fully concentrate on two effects: pattern of the internal source and its Purcell factor.","The paper is organized as follows: In Section , we obtain analytical expressions for the magnetic polarizability of a high-index dielectric cylinder and the effective permeability of the corresponding wire medium.","In Section , we discuss the implications of the resonant and non-local behavior of this permeability and show the simulated results for the radiation of an embedded magnetic dipole.","In Section , we present the brief summary of the work."],["Effective permeability of the wire medium","Let a plane wave with electric field polarized along $\\mathbf {y}_0$ illuminate a dielectric cylinder, as it is shown in Fig.","REF .","The incident wave has perpendicular polarization and its magnetic field along $\\mathbf {z}_0$ can be written as $H_{z_{\\rm {inc}}}=H_0e^{-j(k_xx+k_zz)},$ where $H_0$ is the magnitude, $k_z$ and $k_x$ are the wave vector components in the free space.","Due to the cylindrical geometry, it is proper to expand the field into cylindrical harmonics.","Therefore, $H_{z_{\\rm {inc}}}=H_0\\sum _m\\left[(-j)^nJ_m(h_0R)e^{jm\\phi }\\right]e^{-jk_zz},$ in which $J_m(x)$ is the Bessel function (first kind, order $m$ ), $h_0=\\sqrt{k_0^2-k_z^2}$ ($k_0$ is the free-space wave number) and $(R, \\phi , z)$ are the components of the cylindrical coordinate system.","Figure: (a)–An infinite dielectric cylinder illuminated by a TE-polarized plane wave.Taking into account the scattering and penetration of the wave into the dielectric cylinder, the total electromagnetic fields outside and inside the cylinder can be described by $\\begin{split}&E_{z_{\\rm {out}}}=\\sum _m{C_mH_m^{(2)}(h_0R)e^{jm\\phi }},\\cr &H_{z_{\\rm {out}}}=\\sum _m \\left(D_mH_m^{(2)}(h_0R)+H_0(-j)^mJ_m(h_0R)\\right)e^{jm\\phi },\\end{split}$ and $\\begin{split}&E_{z_{\\rm {in}}}=\\sum _m{A_mJ_m(hR)e^{jm\\phi }},\\cr &H_{z_{\\rm {in}}}=\\sum _m{B_mJ_m(hR)e^{jm\\phi }},\\end{split}$ where $h=\\sqrt{k_0^2\\varepsilon _{\\rm {r}}-k_z^2}$ ($\\varepsilon _{\\rm {r}}$ denotes the relative dielectric constant of the cylinder) and $H^{(2)}_m(x)$ represents the Hankel function (second kind, order $m$ ).","The other components of the electromagnetic fields ($E_R$ , $H_R$ , $E_\\phi $ and $H_\\phi $ ) can be readily derived by solving the Maxwell's equations.","In Eqs.","(REF ) and (REF ), the unknown coefficients ($A_m$ , $B_m$ , $C_m$ and $D_m$ ) are related to each other through boundary conditions, i.e.","the tangential components of the electric and magnetic fields should be continuous at the radius of the cylinder ($R=a$ ).","After determining the fields inside the cylinder, we can obtain the polarization current that is equal to $\\mathbf {J}_{\\rm {p}}=j\\omega \\varepsilon _0(\\varepsilon _{\\rm {r}}-1)\\mathbf {E}_{\\rm {in}}$ .","On the other hand, we know that the magnetic moment per unit length of our cylinder is described by $\\mathbf {m}=\\frac{1}{2}\\mu _0\\int _S\\mathbf {r}\\times \\mathbf {J}_{\\rm {p}}dS,$ where $\\mathbf {r}=\\mathbf {R}+z\\mathbf {a}_z$ is the distance vector from the origin to the current element.","Therefore, the $\\mathbf {z}$ -component of the magnetic moment can be obtained as $m_z=\\frac{1}{2}j\\omega \\mu _0\\varepsilon _0(\\varepsilon _{\\rm {r}}-1)\\int _SR^2E_{\\phi _{\\rm {in}}}dRd\\phi ,$ in which the $\\phi $ -component of the electric field inside the cylinder is as follows: $\\begin{split}E_{\\phi _{\\rm {in}}}=&\\frac{j\\omega \\mu _0}{h}\\times \\cr &\\sum _m\\left(A_m\\frac{mk_z}{j\\omega \\mu _0hR}J_m(hR)+B_mJ^{\\prime }_m(hR)\\right)e^{jm\\phi }.\\end{split}$ The function $J^{\\prime }_m(x)$ is the derivative of $J_m(x)$ .","From the Eqs.","(REF ) and (REF ), we can see that the only cylindrical harmonic which is responsible for non-zero magnetic moment per unit length produced by the magnetic field of the incident wave is $m=0$ .","Then, based on Eq.","(REF ), we should only calculate the coefficient $B_0$ – $A_0$ gives the zero contribution into Eqs.","(REF ).","From imposing the boundary conditions to determine the unknown coefficients, we can achieve $B_0=\\frac{j2h}{\\pi h_0a\\left[h_0J^{\\prime }_0(ha)H^{(2)}_0(h_0a)-hJ_0(ha)H^{\\prime (2)}_0(h_0a)\\right]}H_0.$ If we substitute Eq.","() into Eq.","(REF ) and use Eq.","(REF ), finally, we derive the magnetic polarizability as $\\begin{split}\\alpha ^{zz}_{\\rm {mm}}=\\frac{m_z}{\\mu _0H_0}&=\\frac{j2k_0^2(\\varepsilon _{\\rm {r}}-1)}{h_0(k_0^2\\varepsilon _{\\rm {r}}-k_z^2)}\\times \\cr &\\frac{\\left[ha J_0(ha)-2J_1(ha)\\right]}{\\left[h_0J_1(ha)H^{(2)}_0(h_0a)-hJ_0(ha)H^{(2)}_1(h_0a)\\right]}.\\end{split}$ Figure: (a)–Real part of the magnetic polarizability versus the normalized frequency for three different incident angles.","The relative dielectric constant of the dielectric cylinder is assumed to be ε r =120\\varepsilon _{\\rm {r}}=120.","(b)—Real part of the magnetic polarizability versus the normalized frequency for three different dielectric constants.","Here, we assume that θ inc =π/11\\theta _{\\rm {inc}}=\\pi /11.To keep only the term with $m=0$ in Eq.","(REF ) is the same as to excite our cylinder by only a magnetic field of the electromagnetic wave letting the external electric field be zero at the cylinder axis.","Adding a corresponding term to Eq.","(), it is easy to see that the excitation of the cylinder by two plane waves incident on it from two opposite sides and forming the standing wave with the node of the electric field at the $z$ -axis results in the same formula ().","It is worth to note that retaining in Eq.","(REF ) the term $A_1$ we would attribute to the magnetic moment per unit length the dependence on the incident electric field, and the response of the cylinder would be bianisotropic.","Keeping only $B_0$ we follow to the approach of [16] that allows one to avoid the seeming bianisotropy in the response of a symmetric scatterer.","This bianisotropy is not physically sound and would result in the inconsistencies of the effective-medium model (see e.g.","in [17]).","Sign minus in the denominator of the second factor in () allows the resonance of the magnetic polarizability if $\\varepsilon _{\\rm {r}}$ is positive.","This is the fundamental Mie resonance of the cylinder.","Since $k_z$ enters both $h$ and $h_0$ , the polarizability is non-local i.e.","depends on the angle of incidence $\\theta _{\\rm {inc}}$ .","To show the impact of the incidence angle and that of the dielectric constant $\\varepsilon _{\\rm {r}}$ on the Mie resonance we plot the magnetic polarizability as a function of normalized frequency for several values of $\\varepsilon _{\\rm {r}}$ and $\\theta _{\\rm {inc}}$ .","In Fig.","REF it is assumed that $\\varepsilon _{\\rm {r}}=120$ .","Then the Mie resonance occurs approximately at $k_0a=0.22$ for all angles.","For small angles, the resonance is, indeed, stronger than that for large angles.","Thereby, we see that the resonant polarizability depends strongly on the incident angle or – more specifically – on the axial component $k_z=k_0\\cos (\\theta _{\\rm {inc}})$ of the wave vector.","At frequencies below the Mie resonance band the magnetic polarizability is negligibly small.","The location of the resonance on the frequency axis is a function of the dielectric constant of the cylinder, as can be seen in Fig.","REF .","When $\\varepsilon _{\\rm {r}}$ is not very high, the resonance happens at higher $k_0a$ and vice versa.","For the accuracy of the effective-medium model it is better if the resonance occurs at lower $k_0a$ .","This model requires that both the wires radius ($a$ ) and the array period ($b$ ) are sufficiently small compared to the wavelength in host medium, in other words, $k_0a0.83$ .","At $\\vert k_z/k_0\\vert =0.83$ , there is a vertical asymptote.","This resonance of $\\mu _\\parallel (k_z)$ implies that the transverse component $q$ of the wave vector experiences the similar resonance at $\\vert k_z/k_0\\vert =0.83$ .","It is clear from the dispersion equation for TE-waves in a magnetic hyperbolic medium [13] (for the case when the host medium is free space $\\varepsilon _{\\rm {h}}=1$ ): ${q^2/\\mu _\\parallel (k_z)}+{k_z^2/\\mu _{\\perp }}=k_0^2, \\quad \\mu _{\\perp }\\approx 1.$ In practical cases – when optical losses are taken into account – the resonance of $q$ means that $q$ is a very large imaginary value.","As we can see in Fig.","REF this effect keeps for propagating eigenmodes within a certain range of frequencies.","For frequencies $k_0a\\ge 0.23$ the resonances of $\\mu _\\parallel (k_z)$ still hold but correspond to very high spatial frequencies.","For so high spatial frequencies the effective-medium model is not applicable [18].","In our plot Fig.","REF the region of $k_z$ (of the order of $k_0$ or smaller) is that compatible with the model.","Imaginary $q$ for a propagating eigenmode with allowed value of $k_z$ means that the medium eigenmode propagates along the $z$ -axis, being evanescent in the transverse plane $(x-y)$ .","Very high imaginary $q$ implies the ultimate subwavelength concentration of the eigenmode in this plane.","If we embed a subwavelength source of TE-waves (such as a point magnetic dipole oriented orthogonally to the $z$ -axis) in our wire medium this mode will be excited dominating over all other modes propagating along the $z$ -axis.","This domination results from the huge slope of the curve $\\mu _\\parallel (k_z)$ which implies the huge density of electromagnetic states.","This feature was noticed for the axial permittivity of some dielectric hyperbolic metamaterials in [19].","Our magnetic hyperbolic metamaterial is dual to dielectric hyperbolic metamaterials and the roles of $\\overline{\\overline{\\epsilon }}$ and TM-waves are played by $\\overline{\\overline{\\mu }}$ and TE-waves.","Since the dominant mode strongly attenuates in both $x$ - and $y$ -directions, it means that the whole electromagnetic field generated by our magnetic dipole should be concentrated around the line which passes through the dipole center along the $z$ -axis."],["Localization of magneto-dipole radiation","This filamentary localization of magneto-dipole radiation was confirmed by using a 3D electromagnetic simulator CST Microwave Studio.","In our simulations we position a strongly subwavelength current loop whose magnetic moment is directed along the $y$ -axis in the center of the wire medium sample as shown in Fig.","REF .","The medium sample is finite-size with length $L_{\\rm {sample}}=(N-1)b+2a$ where $N=12$ is the number of the wires.","It has realistic optical losses: we suppose that the tangent of dielectric losses in the material of our wires is equal to $\\tan \\delta =0.1$ , where the complex permittivity of nanowires is $\\varepsilon _{\\rm {r}}={\\rm {Re}}[\\varepsilon _{\\rm {r}}][1-j\\tan \\delta ]$ , ${\\rm {Re}}[\\varepsilon _{\\rm {r}}]=120$ .","This value is the complex permittivity of lithium tantalate at $f=23$ THz.","The radius of the wires is $a=455$ nm and $b=4a$ as above.","The spatial distribution of the transverse component of the electric field (the $x$ -component in the $y-z$ plane) is shown in Fig.","REF for four frequencies.","We see that only at the frequency $k_0a=0.2192\\approx 0.22$ , the field so significantly decays in the transverse plane so that it is practically concentrated in one unit cell of the wire medium.","This is the maximal possible concentration in a composite material – that restricted only by the granularity of the medium.","The excited wave propagates along the $z$ -axis and experiences nearly total internal reflection at the interfaces of the medium sample.","Similar pictures correspond to other components of the electromagnetic field.","The comparison of different components confirms that the wave is TE-polarized.","At three other frequencies the waves are also TE-polarized, whereas their localization is not filamentary and even the cross-like dipole pattern inherent to usual hyperbolic metamaterials [12] can be guessed in Fig.","REF .","Figure: Distribution of the normal component of the electric-field in the wire-medium sample (consisting of 12×1212\\times 12 nanowires).","The magnetic dipole moment located in the center of the sample is along the 𝐲 0 \\mathbf {y}_0-axis.","Here, ε r =120-j12\\varepsilon _{\\rm {r}}=120-j12, f v =0.1963f_{\\rm {v}}=0.1963.The reason of the striking difference in the internal dipole patterns for our wire medium and the majority of hyperbolic materials is spatial dispersion.","Metal wire media operating at microwaves are also spatially dispersive, and the slope of $\\epsilon _\\parallel (k_z)$ (see e.g.","in [10]) at their spatial resonance $|k_z|=k_0$ is as huge as that of our $\\mu _\\parallel (k_z)$ at their resonant $k_z$ in the topological transition frequency range.","It is not surprising, therefore, that in microwave wire media the similar filamentary dipole pattern was also observed (see e.g.","in [20]).","However, in what concerns this pattern our infrared magnetic wire medium and microwave wire media differ qualitatively.","The first difference is duality: microwave wire media are hyperbolic dielectric metamaterials and the filamentary pattern in them corresponds to transverse electric dipoles.","The second difference is the effect bandwidth: microwave wire media are broadband, and the same dipole pattern is inherent for them at any frequency for which the effective-medium model is applicable.","Our wire media possess the dipole pattern which strongly varies versus frequency, that is illustrated by Fig.","REF .","The filamentary pattern of the magnetic dipole corresponds only to the range of the topological transition."],["Purcell factor","Another implication of the topological transition in our magnetic metamaterial is the strong Purcell effect.","Originally, the effect was known as the enhancement of the decay rate of a quantum emitter located in an open cavity [21], [22].","However, the role of the cavity is only to extract more power from the emitter, whereas the enhancement of its decay rate is the same as the enhancement of the radiated power.","Therefore, the notion of the Purcell effect was extended, first, to any scatterer located in the vicinity of an emitter [23], [24], [25], [26], and, finally to any active radiator whose radiation is enhanced by any environment different from free space [27], [28], [6].","Based on this general concept, we can define the magnetic Purcell factor of our wire medium as the increase of the radiated power of a subwavelength magnetic dipole due to the presence of the wire medium around it.","According to the “antenna terminology”, the radiated power of the emitter is proportional to the radiation resistance of that emitter [29].","Therefore, to calculate the Purcell factor, it is the same as to obtain the ratio of the radiation resistances in presence of the wire medium ($R_{\\rm {r}}$ ) and in absence ($R_{{\\rm {r}}_0}$ ): $P_{\\rm {F}}=\\displaystyle \\frac{R_{\\rm {r}}}{R_{{\\rm {r}}_0}}.$ For the same structure as above we have simulated the radiation resistances.","Besides of the case of the transverse magnetic dipole shown in Fig.","REF , we have studied the case when the same magnetic dipole is oriented in parallel to the $z$ -axis.","Also, we simulated the cases of the transverse and axial electric dipoles of the same subwavelength size, all centered at the same point as in Fig.","REF .","In Fig.","REF which depicts the radiation resistance versus frequency, the blue curves refer to the magnetic dipole in presence of the sample.","As one can see, the radiation resistance of the magnetic dipole oriented perpendicularly to the optical axis experiences the strong resonance at $k_0a\\approx 0.22$ , exactly where the effective-medium model predicts the topological transition.","The dashed blue curve shows weaker effect for the parallel magnetic dipole.","Red solid and dashed curves correspond to the perpendicular and parallel electric dipoles, respectively.","For comparison with the case when the host is free space, the frequency dispersion of $R_{{\\rm {r}}_0}$ is given in the same plot for both electric and magnetic dipoles.","Figure REF presents the frequency dispersion of the Purcell factor for all four dipoles.","It is clearly seen that the strongest resonance and highest $P_{\\rm {F}}$ correspond to the transversal magnetic dipole.","An axial magnetic dipole also creates the TE-polarized spatial spectrum, and therefore it also experiences the resonance at the topological transition.","However, the resonant values of $P_{\\rm {F}}$ are smaller because it mainly creates the TE-polarized radiation which weakly interacts with the nanowires i.e.","is not enhanced.","The values of $P_{\\rm {F}}$ for the axial electric dipole are low.","It is not surprising.","Recall, that in a wire medium of perfectly conducting (PC) wires the radiation of the axial electric dipole is fully suppressed [5].","The suppression results from the destructive near-field coupling of the dipole with adjacent wires.","The axial current of the dipole induces opposite axial currents in them which cancel the radiation.","This effect is similar to the suppression of radiation of a horizontal electric dipole located on the PC substrate.","Well, our wires are not PC, they are highly refractive.","However, the substrates with high positive permittivity and finite negative permittivity also suppress the radiation of the horizontal electric dipole.","This destructive interaction origins from the capacitive coupling which causes the inverse mirror image of the horizontal electric dipole in the substrate with the strong skin-effect.","A similar situation holds for the axial dipole source in wire media if the wires possess strong skin-effect.","If the wires have high or negative $\\varepsilon _{\\rm r}$ , the values of $P_{\\rm {F}}$ for the axial electric dipole should be low.","However, we can see in Figure REF that the radiation of this dipole experiences the resonance (at slightly lower frequency than that of the magnetic topological transition).","This is also not surprising, because not all radiation of an axial electric dipole is TM-polarized.","It also produces some TE-waves, and this effect is competing with the capacitive destructive interaction.","Therefore, we have $P_{\\rm {F}}=2$ at $k_0a=0.21$ , whereas far from this frequency we see $P_{\\rm {F}}\\approx 0$ .","As to the noticeable enhancement of the transverse electric dipole, it, definitely, occurs due to the constructive interaction of this dipole with the adjacent nanowires (similarly to a vertical dipole on the high-permittivity substrate).","This effect is complemented by the resonant enhancement of TE-polarized waves.","However, in the near zone of the electric dipole the magnetic field is weak.","Therefore, an electric dipole cannot interact with the magnetic medium as strongly as a magnetic dipole, and the resonant Purcell factor of the transversal magnetic dipole is higher."],["Conclusions","In this work we calculated the effective permeability of a simple wire medium whose wires are made of a material with high positive dielectric constant.","Its parallel component $\\mu _\\parallel $ is resonant due to the Mie resonance of a single wire, and is an indefinite function of both frequency and axial wave vector $k_z$ (changes the sign depending on these arguments).","This change of the sign results in a transition in the topology of the dispersion surface for TE polarized waves.","In the frequency range of the topological transition, the huge (filamentary) localization of radiation occurs for an internal magnetic source, that clearly shares our medium out from other hyperbolic metamaterials, including that introduced in [13].","Though a similar pattern is inherent to the medium of perfectly conducting wires operating at microwaves, in our case this pattern corresponds to the magnetic dipole and occurs only at the topological transition.","Our wire medium manifests a strong radiation enhancement for an internal dipole source.","Unlike dielectric hyperbolic metamaterials, including microwave wire media, and polaritonic media operating at higher frequencies [6] the maximal enhancement corresponds to magnetic dipoles.","Unlike spiral media [13] it holds at the topological transition and the maximal radiation enhancement corresponds to the transversal magnetic dipole.","What is also important: from the comparison of the present paper with [6] it is clear that the same polaritonic wire media which manifest the resonant effects related to the regime epsilon-near-zero at 35–45 THz [6] may manifest the similar (but different) resonant effects related to the regime mu-near-zero in the range 15–30 THz.","This is another argument in favor of this type of hyperbolic metamaterials, which definitely deserve more attention of theorists and an experimental implementation to confirm the claimed effects."]],"string":"[\n [\n \"Magnetic hyperbolic metamaterial of polaritonic nanowires\"\n ],\n [\n \"Abstract We show that the axial component of the magnetic permeability tensor is resonant for a wire medium consisting of high-index epsilon-positive nanowires, and its real part changes the sign at a certain frequency.\",\n \"At this frequency the medium experiences the topological transition from the hyperbolic to the elliptic type of dispersion.\",\n \"We show that the transition regime is characterized by extremely strong dependence of the permeability on the wave vector.\",\n \"This implies very high density of electromagnetic states that results in the filamentary pattern and noticeable Purcell factor for a transversely oriented magnetic dipole.\"\n ],\n [\n \"introduction\",\n \"Simple wire media, defined as optically dense arrays of parallel metal wires in a host dielectric material, have been in a considerable attention since electromagnetic metamaterials have been introduced.\",\n \"Many applications of such media have been suggested from radio frequencies where the period of wire media is on the millimeter scale, to optical frequencies where this period is submicron.\",\n \"Among them one can mention biological sensing [1], subwavelength imaging and endoscopy with image magnification [2], [3], [4], enhancement of quantum emitters, thermal sources, and classical dipole radiators [5], [7], [6], enhancement of radiative heat transfer [8], [9], etc.\",\n \"More details can be found in overview [10].\",\n \"In all these works, electromagnetic properties of wire media were described in a condensed form – via an effective permittivity tensor: $\\\\overline{\\\\overline{\\\\epsilon }}=\\\\left(\\\\begin{array}{ccc}\\\\varepsilon _{\\\\perp } & 0 & 0 \\\\\\\\0 &\\\\varepsilon _{\\\\perp } & 0 \\\\\\\\0& 0 & \\\\varepsilon _{\\\\parallel } \\\\end{array} \\\\right).$ Here, $\\\\varepsilon _\\\\perp $ and $\\\\varepsilon _\\\\parallel $ are the perpendicular (transversal) and parallel (axial) components with respect to the wires axis.\",\n \"Therefore, standard wire media are uniaxially anisotropic materials (though not usual ones because they possess spatial dispersion – dependence of $\\\\overline{\\\\overline{\\\\epsilon }}$ on the wave vector ${\\\\bf k}$ [11], [12].\",\n \"At infrared frequencies, the relative dielectric constant of the metal nanowires is highly negative.\",\n \"If the fraction ratio of the metal in the effective medium is low the perpendicular component $\\\\varepsilon _\\\\perp $ of the permittivity is approximately equal to the permittivity of the host medium, whereas the parallel component $\\\\varepsilon _\\\\parallel $ has negative sign [11], [10].\",\n \"For the dispersion, it implies that the transverse magnetic (TM) polarized wave has an open dispersion surface similar to a hyperboloid [10], [12].\",\n \"This dispersion surface results in the high density of electromagnetic states and enhancement of a subwavelength electric dipole located in the medium orthogonally to the wires [5], [12].\",\n \"The radiation of a magnetic dipole can be also enhanced but this effect is lower because such the dipole produces mainly TE-waves whose electric field is orthogonal to nanowires.\",\n \"These transversely electric (TE) polarized waves weakly interact with the wire medium and their dispersion surfaces are rather similar to spheres like dispersion surfaces of free space.\",\n \"Notice, that for metal nanowires the effective permeability tensor ($\\\\overline{\\\\overline{\\\\mu }}$ ) is practically equal to unity [10].\",\n \"Recently, in Ref.\",\n \"[13] one studied an anisotropic medium with artificial magnetism.\",\n \"This medium is dual to the dielectric hyperbolic metamaterial described by the indefinite tensor in Eq. ().\",\n \"In this magnetic metamaterial the effective permittivity tensor is definite, positive and not resonant.\",\n \"Therefore, the dispersion surfaces of TM-polarized waves in this medium are similar to the spheres, whereas the dispersion surfaces of TE-waves are either hyperbolic or elliptic depending on frequency.\",\n \"However, medium in Ref.\",\n \"[13] is not a wire medium.\",\n \"It is a racemic array of metal helices with both left-handedness and right-handedness.\",\n \"Such media, in accordance to initial works [14], [15] where they were introduced, should be referred as spiral media.\",\n \"Spiral medium of [13] operates at microwave frequencies, where metals are close to perfect conductors.\",\n \"In the present paper, we study wire media at optical frequencies.\",\n \"Figure: A medium of long parallel wires which are set in a square lattice.\",\n \"The wires are made of polaritonic materials whose relative dielectric constant is much larger than unity (ε r ≫1\\\\varepsilon _{\\\\rm {r}}\\\\gg 1).\",\n \"Here, aa and bb are the wires radius and lattice constant, respectively.\",\n \"For simplicity, the host medium is free space.Below, we theoretically demonstrate that a simple wire medium (see Fig.\",\n \"REF ) may have in the infrared range similar electromagnetic properties to those manifested by [13] exhibited at microwaves, however, there are also significant peculiarities which share our wire medium out from all hyperbolic metamaterials.\",\n \"Nanowires of our wire medium should be made of high-index epsilon-positive materials, such as lithium tantalate, silicon carbide or hexagonal boron nitride, and some other polaritonic materials in the range 15–30 THz.\",\n \"The resonance of the axial component of the effective permeability tensor arises because the dynamic magnetic polarizability of the high-index dielectric wire is resonant, and origins from the Mie resonance of a single nanowire.\",\n \"We show that this axial permeability changes its sign at a certain frequency and in a narrow frequency range around this transition is a very sharp function of the wave vector.\",\n \"The corresponding topological transition is related to the resonant enhancement of radiation of a subwavelength dipole located inside the medium.\",\n \"It is an expected effect which is similar to that studied in [6].\",\n \"However, in the present case it holds for a magnetic dipole oriented transversely to nanowires.\",\n \"The sharp dependence of the permeability on the wave vector results in the unusual property of our wire medium – the radiation of the internal magnetic dipole is concentrated at the line which passes through the dipole center in parallel to the optical axis.\",\n \"This property is difficult to detect from the dispersion contours (sections of the dispersion surfaces).\",\n \"In the present case these contours for TE-waves do not differ very much from analogous contours for TM-waves we have studied in [9].\",\n \"Moreover, these contours were analyzed for TE-waves in the spiral medium [13].\",\n \"Therefore, we do not spend the volume of this paper to the demonstration and analysis of the dispersion contours in the region of topological transition and fully concentrate on two effects: pattern of the internal source and its Purcell factor.\",\n \"The paper is organized as follows: In Section , we obtain analytical expressions for the magnetic polarizability of a high-index dielectric cylinder and the effective permeability of the corresponding wire medium.\",\n \"In Section , we discuss the implications of the resonant and non-local behavior of this permeability and show the simulated results for the radiation of an embedded magnetic dipole.\",\n \"In Section , we present the brief summary of the work.\"\n ],\n [\n \"Effective permeability of the wire medium\",\n \"Let a plane wave with electric field polarized along $\\\\mathbf {y}_0$ illuminate a dielectric cylinder, as it is shown in Fig.\",\n \"REF .\",\n \"The incident wave has perpendicular polarization and its magnetic field along $\\\\mathbf {z}_0$ can be written as $H_{z_{\\\\rm {inc}}}=H_0e^{-j(k_xx+k_zz)},$ where $H_0$ is the magnitude, $k_z$ and $k_x$ are the wave vector components in the free space.\",\n \"Due to the cylindrical geometry, it is proper to expand the field into cylindrical harmonics.\",\n \"Therefore, $H_{z_{\\\\rm {inc}}}=H_0\\\\sum _m\\\\left[(-j)^nJ_m(h_0R)e^{jm\\\\phi }\\\\right]e^{-jk_zz},$ in which $J_m(x)$ is the Bessel function (first kind, order $m$ ), $h_0=\\\\sqrt{k_0^2-k_z^2}$ ($k_0$ is the free-space wave number) and $(R, \\\\phi , z)$ are the components of the cylindrical coordinate system.\",\n \"Figure: (a)–An infinite dielectric cylinder illuminated by a TE-polarized plane wave.Taking into account the scattering and penetration of the wave into the dielectric cylinder, the total electromagnetic fields outside and inside the cylinder can be described by $\\\\begin{split}&E_{z_{\\\\rm {out}}}=\\\\sum _m{C_mH_m^{(2)}(h_0R)e^{jm\\\\phi }},\\\\cr &H_{z_{\\\\rm {out}}}=\\\\sum _m \\\\left(D_mH_m^{(2)}(h_0R)+H_0(-j)^mJ_m(h_0R)\\\\right)e^{jm\\\\phi },\\\\end{split}$ and $\\\\begin{split}&E_{z_{\\\\rm {in}}}=\\\\sum _m{A_mJ_m(hR)e^{jm\\\\phi }},\\\\cr &H_{z_{\\\\rm {in}}}=\\\\sum _m{B_mJ_m(hR)e^{jm\\\\phi }},\\\\end{split}$ where $h=\\\\sqrt{k_0^2\\\\varepsilon _{\\\\rm {r}}-k_z^2}$ ($\\\\varepsilon _{\\\\rm {r}}$ denotes the relative dielectric constant of the cylinder) and $H^{(2)}_m(x)$ represents the Hankel function (second kind, order $m$ ).\",\n \"The other components of the electromagnetic fields ($E_R$ , $H_R$ , $E_\\\\phi $ and $H_\\\\phi $ ) can be readily derived by solving the Maxwell's equations.\",\n \"In Eqs.\",\n \"(REF ) and (REF ), the unknown coefficients ($A_m$ , $B_m$ , $C_m$ and $D_m$ ) are related to each other through boundary conditions, i.e.\",\n \"the tangential components of the electric and magnetic fields should be continuous at the radius of the cylinder ($R=a$ ).\",\n \"After determining the fields inside the cylinder, we can obtain the polarization current that is equal to $\\\\mathbf {J}_{\\\\rm {p}}=j\\\\omega \\\\varepsilon _0(\\\\varepsilon _{\\\\rm {r}}-1)\\\\mathbf {E}_{\\\\rm {in}}$ .\",\n \"On the other hand, we know that the magnetic moment per unit length of our cylinder is described by $\\\\mathbf {m}=\\\\frac{1}{2}\\\\mu _0\\\\int _S\\\\mathbf {r}\\\\times \\\\mathbf {J}_{\\\\rm {p}}dS,$ where $\\\\mathbf {r}=\\\\mathbf {R}+z\\\\mathbf {a}_z$ is the distance vector from the origin to the current element.\",\n \"Therefore, the $\\\\mathbf {z}$ -component of the magnetic moment can be obtained as $m_z=\\\\frac{1}{2}j\\\\omega \\\\mu _0\\\\varepsilon _0(\\\\varepsilon _{\\\\rm {r}}-1)\\\\int _SR^2E_{\\\\phi _{\\\\rm {in}}}dRd\\\\phi ,$ in which the $\\\\phi $ -component of the electric field inside the cylinder is as follows: $\\\\begin{split}E_{\\\\phi _{\\\\rm {in}}}=&\\\\frac{j\\\\omega \\\\mu _0}{h}\\\\times \\\\cr &\\\\sum _m\\\\left(A_m\\\\frac{mk_z}{j\\\\omega \\\\mu _0hR}J_m(hR)+B_mJ^{\\\\prime }_m(hR)\\\\right)e^{jm\\\\phi }.\\\\end{split}$ The function $J^{\\\\prime }_m(x)$ is the derivative of $J_m(x)$ .\",\n \"From the Eqs.\",\n \"(REF ) and (REF ), we can see that the only cylindrical harmonic which is responsible for non-zero magnetic moment per unit length produced by the magnetic field of the incident wave is $m=0$ .\",\n \"Then, based on Eq.\",\n \"(REF ), we should only calculate the coefficient $B_0$ – $A_0$ gives the zero contribution into Eqs.\",\n \"(REF ).\",\n \"From imposing the boundary conditions to determine the unknown coefficients, we can achieve $B_0=\\\\frac{j2h}{\\\\pi h_0a\\\\left[h_0J^{\\\\prime }_0(ha)H^{(2)}_0(h_0a)-hJ_0(ha)H^{\\\\prime (2)}_0(h_0a)\\\\right]}H_0.$ If we substitute Eq.\",\n \"() into Eq.\",\n \"(REF ) and use Eq.\",\n \"(REF ), finally, we derive the magnetic polarizability as $\\\\begin{split}\\\\alpha ^{zz}_{\\\\rm {mm}}=\\\\frac{m_z}{\\\\mu _0H_0}&=\\\\frac{j2k_0^2(\\\\varepsilon _{\\\\rm {r}}-1)}{h_0(k_0^2\\\\varepsilon _{\\\\rm {r}}-k_z^2)}\\\\times \\\\cr &\\\\frac{\\\\left[ha J_0(ha)-2J_1(ha)\\\\right]}{\\\\left[h_0J_1(ha)H^{(2)}_0(h_0a)-hJ_0(ha)H^{(2)}_1(h_0a)\\\\right]}.\\\\end{split}$ Figure: (a)–Real part of the magnetic polarizability versus the normalized frequency for three different incident angles.\",\n \"The relative dielectric constant of the dielectric cylinder is assumed to be ε r =120\\\\varepsilon _{\\\\rm {r}}=120.\",\n \"(b)—Real part of the magnetic polarizability versus the normalized frequency for three different dielectric constants.\",\n \"Here, we assume that θ inc =π/11\\\\theta _{\\\\rm {inc}}=\\\\pi /11.To keep only the term with $m=0$ in Eq.\",\n \"(REF ) is the same as to excite our cylinder by only a magnetic field of the electromagnetic wave letting the external electric field be zero at the cylinder axis.\",\n \"Adding a corresponding term to Eq.\",\n \"(), it is easy to see that the excitation of the cylinder by two plane waves incident on it from two opposite sides and forming the standing wave with the node of the electric field at the $z$ -axis results in the same formula ().\",\n \"It is worth to note that retaining in Eq.\",\n \"(REF ) the term $A_1$ we would attribute to the magnetic moment per unit length the dependence on the incident electric field, and the response of the cylinder would be bianisotropic.\",\n \"Keeping only $B_0$ we follow to the approach of [16] that allows one to avoid the seeming bianisotropy in the response of a symmetric scatterer.\",\n \"This bianisotropy is not physically sound and would result in the inconsistencies of the effective-medium model (see e.g.\",\n \"in [17]).\",\n \"Sign minus in the denominator of the second factor in () allows the resonance of the magnetic polarizability if $\\\\varepsilon _{\\\\rm {r}}$ is positive.\",\n \"This is the fundamental Mie resonance of the cylinder.\",\n \"Since $k_z$ enters both $h$ and $h_0$ , the polarizability is non-local i.e.\",\n \"depends on the angle of incidence $\\\\theta _{\\\\rm {inc}}$ .\",\n \"To show the impact of the incidence angle and that of the dielectric constant $\\\\varepsilon _{\\\\rm {r}}$ on the Mie resonance we plot the magnetic polarizability as a function of normalized frequency for several values of $\\\\varepsilon _{\\\\rm {r}}$ and $\\\\theta _{\\\\rm {inc}}$ .\",\n \"In Fig.\",\n \"REF it is assumed that $\\\\varepsilon _{\\\\rm {r}}=120$ .\",\n \"Then the Mie resonance occurs approximately at $k_0a=0.22$ for all angles.\",\n \"For small angles, the resonance is, indeed, stronger than that for large angles.\",\n \"Thereby, we see that the resonant polarizability depends strongly on the incident angle or – more specifically – on the axial component $k_z=k_0\\\\cos (\\\\theta _{\\\\rm {inc}})$ of the wave vector.\",\n \"At frequencies below the Mie resonance band the magnetic polarizability is negligibly small.\",\n \"The location of the resonance on the frequency axis is a function of the dielectric constant of the cylinder, as can be seen in Fig.\",\n \"REF .\",\n \"When $\\\\varepsilon _{\\\\rm {r}}$ is not very high, the resonance happens at higher $k_0a$ and vice versa.\",\n \"For the accuracy of the effective-medium model it is better if the resonance occurs at lower $k_0a$ .\",\n \"This model requires that both the wires radius ($a$ ) and the array period ($b$ ) are sufficiently small compared to the wavelength in host medium, in other words, $k_0a0.83$ .\",\n \"At $\\\\vert k_z/k_0\\\\vert =0.83$ , there is a vertical asymptote.\",\n \"This resonance of $\\\\mu _\\\\parallel (k_z)$ implies that the transverse component $q$ of the wave vector experiences the similar resonance at $\\\\vert k_z/k_0\\\\vert =0.83$ .\",\n \"It is clear from the dispersion equation for TE-waves in a magnetic hyperbolic medium [13] (for the case when the host medium is free space $\\\\varepsilon _{\\\\rm {h}}=1$ ): ${q^2/\\\\mu _\\\\parallel (k_z)}+{k_z^2/\\\\mu _{\\\\perp }}=k_0^2, \\\\quad \\\\mu _{\\\\perp }\\\\approx 1.$ In practical cases – when optical losses are taken into account – the resonance of $q$ means that $q$ is a very large imaginary value.\",\n \"As we can see in Fig.\",\n \"REF this effect keeps for propagating eigenmodes within a certain range of frequencies.\",\n \"For frequencies $k_0a\\\\ge 0.23$ the resonances of $\\\\mu _\\\\parallel (k_z)$ still hold but correspond to very high spatial frequencies.\",\n \"For so high spatial frequencies the effective-medium model is not applicable [18].\",\n \"In our plot Fig.\",\n \"REF the region of $k_z$ (of the order of $k_0$ or smaller) is that compatible with the model.\",\n \"Imaginary $q$ for a propagating eigenmode with allowed value of $k_z$ means that the medium eigenmode propagates along the $z$ -axis, being evanescent in the transverse plane $(x-y)$ .\",\n \"Very high imaginary $q$ implies the ultimate subwavelength concentration of the eigenmode in this plane.\",\n \"If we embed a subwavelength source of TE-waves (such as a point magnetic dipole oriented orthogonally to the $z$ -axis) in our wire medium this mode will be excited dominating over all other modes propagating along the $z$ -axis.\",\n \"This domination results from the huge slope of the curve $\\\\mu _\\\\parallel (k_z)$ which implies the huge density of electromagnetic states.\",\n \"This feature was noticed for the axial permittivity of some dielectric hyperbolic metamaterials in [19].\",\n \"Our magnetic hyperbolic metamaterial is dual to dielectric hyperbolic metamaterials and the roles of $\\\\overline{\\\\overline{\\\\epsilon }}$ and TM-waves are played by $\\\\overline{\\\\overline{\\\\mu }}$ and TE-waves.\",\n \"Since the dominant mode strongly attenuates in both $x$ - and $y$ -directions, it means that the whole electromagnetic field generated by our magnetic dipole should be concentrated around the line which passes through the dipole center along the $z$ -axis.\"\n ],\n [\n \"Localization of magneto-dipole radiation\",\n \"This filamentary localization of magneto-dipole radiation was confirmed by using a 3D electromagnetic simulator CST Microwave Studio.\",\n \"In our simulations we position a strongly subwavelength current loop whose magnetic moment is directed along the $y$ -axis in the center of the wire medium sample as shown in Fig.\",\n \"REF .\",\n \"The medium sample is finite-size with length $L_{\\\\rm {sample}}=(N-1)b+2a$ where $N=12$ is the number of the wires.\",\n \"It has realistic optical losses: we suppose that the tangent of dielectric losses in the material of our wires is equal to $\\\\tan \\\\delta =0.1$ , where the complex permittivity of nanowires is $\\\\varepsilon _{\\\\rm {r}}={\\\\rm {Re}}[\\\\varepsilon _{\\\\rm {r}}][1-j\\\\tan \\\\delta ]$ , ${\\\\rm {Re}}[\\\\varepsilon _{\\\\rm {r}}]=120$ .\",\n \"This value is the complex permittivity of lithium tantalate at $f=23$ THz.\",\n \"The radius of the wires is $a=455$ nm and $b=4a$ as above.\",\n \"The spatial distribution of the transverse component of the electric field (the $x$ -component in the $y-z$ plane) is shown in Fig.\",\n \"REF for four frequencies.\",\n \"We see that only at the frequency $k_0a=0.2192\\\\approx 0.22$ , the field so significantly decays in the transverse plane so that it is practically concentrated in one unit cell of the wire medium.\",\n \"This is the maximal possible concentration in a composite material – that restricted only by the granularity of the medium.\",\n \"The excited wave propagates along the $z$ -axis and experiences nearly total internal reflection at the interfaces of the medium sample.\",\n \"Similar pictures correspond to other components of the electromagnetic field.\",\n \"The comparison of different components confirms that the wave is TE-polarized.\",\n \"At three other frequencies the waves are also TE-polarized, whereas their localization is not filamentary and even the cross-like dipole pattern inherent to usual hyperbolic metamaterials [12] can be guessed in Fig.\",\n \"REF .\",\n \"Figure: Distribution of the normal component of the electric-field in the wire-medium sample (consisting of 12×1212\\\\times 12 nanowires).\",\n \"The magnetic dipole moment located in the center of the sample is along the 𝐲 0 \\\\mathbf {y}_0-axis.\",\n \"Here, ε r =120-j12\\\\varepsilon _{\\\\rm {r}}=120-j12, f v =0.1963f_{\\\\rm {v}}=0.1963.The reason of the striking difference in the internal dipole patterns for our wire medium and the majority of hyperbolic materials is spatial dispersion.\",\n \"Metal wire media operating at microwaves are also spatially dispersive, and the slope of $\\\\epsilon _\\\\parallel (k_z)$ (see e.g.\",\n \"in [10]) at their spatial resonance $|k_z|=k_0$ is as huge as that of our $\\\\mu _\\\\parallel (k_z)$ at their resonant $k_z$ in the topological transition frequency range.\",\n \"It is not surprising, therefore, that in microwave wire media the similar filamentary dipole pattern was also observed (see e.g.\",\n \"in [20]).\",\n \"However, in what concerns this pattern our infrared magnetic wire medium and microwave wire media differ qualitatively.\",\n \"The first difference is duality: microwave wire media are hyperbolic dielectric metamaterials and the filamentary pattern in them corresponds to transverse electric dipoles.\",\n \"The second difference is the effect bandwidth: microwave wire media are broadband, and the same dipole pattern is inherent for them at any frequency for which the effective-medium model is applicable.\",\n \"Our wire media possess the dipole pattern which strongly varies versus frequency, that is illustrated by Fig.\",\n \"REF .\",\n \"The filamentary pattern of the magnetic dipole corresponds only to the range of the topological transition.\"\n ],\n [\n \"Purcell factor\",\n \"Another implication of the topological transition in our magnetic metamaterial is the strong Purcell effect.\",\n \"Originally, the effect was known as the enhancement of the decay rate of a quantum emitter located in an open cavity [21], [22].\",\n \"However, the role of the cavity is only to extract more power from the emitter, whereas the enhancement of its decay rate is the same as the enhancement of the radiated power.\",\n \"Therefore, the notion of the Purcell effect was extended, first, to any scatterer located in the vicinity of an emitter [23], [24], [25], [26], and, finally to any active radiator whose radiation is enhanced by any environment different from free space [27], [28], [6].\",\n \"Based on this general concept, we can define the magnetic Purcell factor of our wire medium as the increase of the radiated power of a subwavelength magnetic dipole due to the presence of the wire medium around it.\",\n \"According to the “antenna terminology”, the radiated power of the emitter is proportional to the radiation resistance of that emitter [29].\",\n \"Therefore, to calculate the Purcell factor, it is the same as to obtain the ratio of the radiation resistances in presence of the wire medium ($R_{\\\\rm {r}}$ ) and in absence ($R_{{\\\\rm {r}}_0}$ ): $P_{\\\\rm {F}}=\\\\displaystyle \\\\frac{R_{\\\\rm {r}}}{R_{{\\\\rm {r}}_0}}.$ For the same structure as above we have simulated the radiation resistances.\",\n \"Besides of the case of the transverse magnetic dipole shown in Fig.\",\n \"REF , we have studied the case when the same magnetic dipole is oriented in parallel to the $z$ -axis.\",\n \"Also, we simulated the cases of the transverse and axial electric dipoles of the same subwavelength size, all centered at the same point as in Fig.\",\n \"REF .\",\n \"In Fig.\",\n \"REF which depicts the radiation resistance versus frequency, the blue curves refer to the magnetic dipole in presence of the sample.\",\n \"As one can see, the radiation resistance of the magnetic dipole oriented perpendicularly to the optical axis experiences the strong resonance at $k_0a\\\\approx 0.22$ , exactly where the effective-medium model predicts the topological transition.\",\n \"The dashed blue curve shows weaker effect for the parallel magnetic dipole.\",\n \"Red solid and dashed curves correspond to the perpendicular and parallel electric dipoles, respectively.\",\n \"For comparison with the case when the host is free space, the frequency dispersion of $R_{{\\\\rm {r}}_0}$ is given in the same plot for both electric and magnetic dipoles.\",\n \"Figure REF presents the frequency dispersion of the Purcell factor for all four dipoles.\",\n \"It is clearly seen that the strongest resonance and highest $P_{\\\\rm {F}}$ correspond to the transversal magnetic dipole.\",\n \"An axial magnetic dipole also creates the TE-polarized spatial spectrum, and therefore it also experiences the resonance at the topological transition.\",\n \"However, the resonant values of $P_{\\\\rm {F}}$ are smaller because it mainly creates the TE-polarized radiation which weakly interacts with the nanowires i.e.\",\n \"is not enhanced.\",\n \"The values of $P_{\\\\rm {F}}$ for the axial electric dipole are low.\",\n \"It is not surprising.\",\n \"Recall, that in a wire medium of perfectly conducting (PC) wires the radiation of the axial electric dipole is fully suppressed [5].\",\n \"The suppression results from the destructive near-field coupling of the dipole with adjacent wires.\",\n \"The axial current of the dipole induces opposite axial currents in them which cancel the radiation.\",\n \"This effect is similar to the suppression of radiation of a horizontal electric dipole located on the PC substrate.\",\n \"Well, our wires are not PC, they are highly refractive.\",\n \"However, the substrates with high positive permittivity and finite negative permittivity also suppress the radiation of the horizontal electric dipole.\",\n \"This destructive interaction origins from the capacitive coupling which causes the inverse mirror image of the horizontal electric dipole in the substrate with the strong skin-effect.\",\n \"A similar situation holds for the axial dipole source in wire media if the wires possess strong skin-effect.\",\n \"If the wires have high or negative $\\\\varepsilon _{\\\\rm r}$ , the values of $P_{\\\\rm {F}}$ for the axial electric dipole should be low.\",\n \"However, we can see in Figure REF that the radiation of this dipole experiences the resonance (at slightly lower frequency than that of the magnetic topological transition).\",\n \"This is also not surprising, because not all radiation of an axial electric dipole is TM-polarized.\",\n \"It also produces some TE-waves, and this effect is competing with the capacitive destructive interaction.\",\n \"Therefore, we have $P_{\\\\rm {F}}=2$ at $k_0a=0.21$ , whereas far from this frequency we see $P_{\\\\rm {F}}\\\\approx 0$ .\",\n \"As to the noticeable enhancement of the transverse electric dipole, it, definitely, occurs due to the constructive interaction of this dipole with the adjacent nanowires (similarly to a vertical dipole on the high-permittivity substrate).\",\n \"This effect is complemented by the resonant enhancement of TE-polarized waves.\",\n \"However, in the near zone of the electric dipole the magnetic field is weak.\",\n \"Therefore, an electric dipole cannot interact with the magnetic medium as strongly as a magnetic dipole, and the resonant Purcell factor of the transversal magnetic dipole is higher.\"\n ],\n [\n \"Conclusions\",\n \"In this work we calculated the effective permeability of a simple wire medium whose wires are made of a material with high positive dielectric constant.\",\n \"Its parallel component $\\\\mu _\\\\parallel $ is resonant due to the Mie resonance of a single wire, and is an indefinite function of both frequency and axial wave vector $k_z$ (changes the sign depending on these arguments).\",\n \"This change of the sign results in a transition in the topology of the dispersion surface for TE polarized waves.\",\n \"In the frequency range of the topological transition, the huge (filamentary) localization of radiation occurs for an internal magnetic source, that clearly shares our medium out from other hyperbolic metamaterials, including that introduced in [13].\",\n \"Though a similar pattern is inherent to the medium of perfectly conducting wires operating at microwaves, in our case this pattern corresponds to the magnetic dipole and occurs only at the topological transition.\",\n \"Our wire medium manifests a strong radiation enhancement for an internal dipole source.\",\n \"Unlike dielectric hyperbolic metamaterials, including microwave wire media, and polaritonic media operating at higher frequencies [6] the maximal enhancement corresponds to magnetic dipoles.\",\n \"Unlike spiral media [13] it holds at the topological transition and the maximal radiation enhancement corresponds to the transversal magnetic dipole.\",\n \"What is also important: from the comparison of the present paper with [6] it is clear that the same polaritonic wire media which manifest the resonant effects related to the regime epsilon-near-zero at 35–45 THz [6] may manifest the similar (but different) resonant effects related to the regime mu-near-zero in the range 15–30 THz.\",\n \"This is another argument in favor of this type of hyperbolic metamaterials, which definitely deserve more attention of theorists and an experimental implementation to confirm the claimed effects.\"\n ]\n]"},"id":{"kind":"string","value":"1605.05701"}}},{"rowIdx":1866687,"cells":{"text":{"kind":"list like","value":[["Ultrafast Isomerization in Acetylene Dication: To Be or Not To Be"],["Abstract Experimental evidence has pointed toward the existence of ultrafast proton migration and isomerization as a key process for acetylene and its ions, however the actual mechanism for ultrafast isomerization of the acetylene [HCCH]2+ to vinylidene [H2CC]2+ dication remains nebulous.","Theoretical studies show a high potential barrier of over 2eV for the isomerization pathways on the low lying dicationic states, implying that the corresponding isomerization should take picoseconds or even longer according to transition state theory.","However a recent experiment at a femtosecond X-ray free electron laser (XFEL) [Nature Commun.","6, 8199 (2015)] suggests that large amplitude hydrogen migration proceeds on a sub-100 femtosecond time scale.","In order to resolve the contradiction, we present a complete theoretical study of the dynamics of acetylene dication produced by Auger decay after X-ray photoionization of the carbon atom K shell.","We find that isomerization does not occur on the sub-100 fs timescale and is not required to explain the time-resolved Coulomb imaging experiment.","This study resolves the seeming contradiction between experiment and theory concerning the isomerization time scale in acetylene dication.","This work calls for careful interpretation of structural information from the widely applied Coulomb momentum imaging method but also points out its strengths in mapping out momentum dispersion dynamics even when structural variation is minor."],["=1 pdfinfo= [pages=1-last]main.pdf [pages=1-last]suppl.pdf"]],"string":"[\n [\n \"Ultrafast Isomerization in Acetylene Dication: To Be or Not To Be\"\n ],\n [\n \"Abstract Experimental evidence has pointed toward the existence of ultrafast proton migration and isomerization as a key process for acetylene and its ions, however the actual mechanism for ultrafast isomerization of the acetylene [HCCH]2+ to vinylidene [H2CC]2+ dication remains nebulous.\",\n \"Theoretical studies show a high potential barrier of over 2eV for the isomerization pathways on the low lying dicationic states, implying that the corresponding isomerization should take picoseconds or even longer according to transition state theory.\",\n \"However a recent experiment at a femtosecond X-ray free electron laser (XFEL) [Nature Commun.\",\n \"6, 8199 (2015)] suggests that large amplitude hydrogen migration proceeds on a sub-100 femtosecond time scale.\",\n \"In order to resolve the contradiction, we present a complete theoretical study of the dynamics of acetylene dication produced by Auger decay after X-ray photoionization of the carbon atom K shell.\",\n \"We find that isomerization does not occur on the sub-100 fs timescale and is not required to explain the time-resolved Coulomb imaging experiment.\",\n \"This study resolves the seeming contradiction between experiment and theory concerning the isomerization time scale in acetylene dication.\",\n \"This work calls for careful interpretation of structural information from the widely applied Coulomb momentum imaging method but also points out its strengths in mapping out momentum dispersion dynamics even when structural variation is minor.\"\n ],\n [\n \"=1 pdfinfo= [pages=1-last]main.pdf [pages=1-last]suppl.pdf\"\n ]\n]"},"id":{"kind":"string","value":"1605.05707"}}},{"rowIdx":1866688,"cells":{"text":{"kind":"list like","value":[["Optical spectroscopy of Be/gamma-ray binaries"],["Abstract We report optical spectroscopic observations of the Be/gamma-ray binaries LSI+61303, MWC 148 and MWC 656.","The peak separation and equivalent widths of prominent emission lines (H-alpha, H-beta, H-gamma, HeI, and FeII) are measured.","We estimated the circumstellar disc size, compared it with separation between the components, and discussed the disc truncation.","We find that in LSI+61303 the compact object comes into contact with the outer parts of the circumstellar disc at periastron, in MWC 148 the compact object goes deeply into the disc during the periastron passage, and in MWC 656 the black hole is accreting from the outer parts of the circumstellar disc along the entire orbit.","The interstellar extinction was estimated using interstellar lines.","The rotation of the mass donors appears to be similar to the rotation of the mass donors in Be/X-ray binaries.","We suggest that X-ray/optical periodicity of about 1 day deserves to be searched for."],["Introduction","The rapid and sustained progress of high energy and very high energy astrophysics in recent years enabled the identification of a new group of binary stars emitting at TeV energies (e.g.","Paredes et al.","2013).","These objects, called $\\gamma $ -ray binaries, are high-mass X-ray binaries that consist of a compact object (neutron star or black hole) orbiting an optical companion that is an OB star.","There are five confirmed $\\gamma $ -ray binaries so far: PSR B1259-63/LS 2883 (Aharonian et al.","2005), LS 5039/V479 Sct (Aharonian et al.","2006), LSI+61$^0$ 303 (Albert et al., 2009), HESS J0632+057/MWC 148 (Aharonian et al.","2007), and 1FGL J1018.6-5856 (H.E.S.S.","Collaboration et al.","2015).","Their most distinctive fingerprint is a spectral energy distribution dominated by non-thermal photons with energies up to the TeV domain.","Recently, Eger et al.","(2016) proposed a binary nature for the $\\gamma $ -ray source HESS J1832-093/2MASS J18324516-092154 and this object probably belongs to the family of the $\\gamma $ -ray binaries as a sixth member.","The binary system, PSR B1259-63 is unique, since it is the only one where the compact object has been identified as a radio pulsar (Johnston et al.","1992, 1994).","The nature of the compact object is known in PSR B1259-63 as a neutron star, and in AGL J2241+4454/MWC 656 as a black hole (Casares et al.","2014).","Although not included in the confirmed list, MWC 656 was selected as a target here despite not having shown all the observational properties of a canonical $\\gamma $ -ray binary yet.","It was only occasionally detected by the AGILE observatory at GeV energies and not yet detected in the TeV domain (see Aleksić et al.","2015).","Nevertheless, the fact that the black hole nature of the compact companion is almost certain renders it very similar to the typical $\\gamma $ -ray binaries.","In the other systems the nature of the compact object remains unclear (e.g.","Dubus 2013).","In addition to these objects, there are several other binary systems ($\\eta $ Car, Cyg X-1, Cyg X-3, Cen X-3, and SS 433) that are detected as GeV sources, but not as TeV sources so far.","Here we report high-resolution spectral observations of LSI+61$^0$ 303, MWC 148, and MWC 656, and discuss circumstellar disc size, disc truncation, interstellar extinction, and rotation of their mass donors.","The mass donors (primaries) of these three targets are emission-line Be stars.","The Be stars are non-supergiant, fast-rotating B-type and luminosity class III-V stars which, at some point in their lives, have shown spectral lines in emission (Porter & Rivinius 2003).","The material expelled from the equatorial belt of a rapidly rotating Be star forms an outwardly diffusing gaseous, dust-free Keplerian disc (Rivinius et al.","2013).","In the optical/infrared band, the two most significant observational characteristics of Be stars and their excretion discs are the emission lines and the infrared excess.","Moving along the orbit, the compact object passes close to this disc, and sometimes may even go through it causing significant perturbations in its structure.","This circumstellar disc feeds the accretion disc around the compact object and/or interacts with its relativistic wind."],["Observations","High-resolution optical spectra of the three northern Be/$\\gamma $ -ray binaries were secured with the fibre-fed Echelle spectrograph ESpeRo attached to the 2.0 m telescope of the National Astronomical Observatory Rozhen, located in Rhodope mountains, Bulgaria.","The spectrograph uses R2 grating with 37.5 grooves/mm, Andor CCD camera 2048 x 2048 px, 13.5x13.5 $\\mu m$ px$^{-1}$ (Bonev et al.","2016).","The spectrograph provides a dispersion of 0.06 Å px$^{-1}$ at 6560 Å and 0.04 Å px$^{-1}$ at 4800 Å.","The spectra were reduced in the standard way including bias removal, flat-field correction, and wavelength calibration.","Pre-processing of data and parameter measurements are performed using various routines provided in IRAF.","The journal of observations is presented in Table REF , where the date, start of the exposure, exposure time, and signal-to-noise ratio at about $\\lambda 6600$ Å are given.","The orbital phases are calculated using $HJD_0= 2443366.775$ , $HJD_0= 2454857.5,$ and $HJD_0= 2453243.7$ for LSI+61$^0$ 303, MWC 148, and MWC 656, respectively, and orbital periods given in Sect. .","Emission line profiles of LSI+61$^0$ 303, MWC 148, and MWC 656 are plotted on Fig.REF .","Spectral line parameters equivalent width (W) and distance between the peaks ($\\Delta V$ ) for the prominent lines ($H\\alpha $ , H$\\beta $ , H$\\gamma $ , $HeI \\lambda 5876,$ and $FeII \\lambda 5316$ ) are given in Table REF .","The typical error on the equivalent width is below $\\pm 10$ % for lines with $W > 1$ Å and up to $\\pm 20$ % for lines with $W \\lesssim 1$ Å.","The typical error on $\\Delta V$ is $\\pm 10$ km s$^{-1}$ .","It is worth noting that (1) in LSI+61$^0$ 303 FeII lines are not detectable; (2) In MWC 656 on spectrum 20150705 the HeI $\\lambda 5876$ line is not visible (probably emission fills up the absorption).","In addition to the Rozhen data we use 98 spectra of MWC 148 and 68 spectra of MWC 656 (analysed in Casares et al.","2012) from the archive of the 2.0 m Liverpool TelescopeThe Liverpool Telescope is operated on the island of La Palma by Liverpool John Moores University in the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrofisica de Canarias with financial support from the UK Science and Technology Facilities Council.","(Steele et al.","2004).","These spectra were obtained using the Fibre-fed RObotic Dual-beam Optical Spectrograph (FRODOSpec; Morales-Rueda et al.","2004).","The spectrograph is fed by a fibre bundle array consisting of $12\\times 12$ lenslets of 0.82 arcsec each, which is reformatted as a slit.","The spectrograph was operated in a high-resolution mode, providing a dispersion of 0.8 Å px$^{-1}$ at 6500 Å, 0.35 Å px$^{-1}$ at 4800 Å, and typical $S/N \\gtrsim 100$ .","FRODOSpec spectra were processed using the fully automated data reduction pipeline of Barnsley et al.","(2012).","The typical error on the equivalent width is $\\pm 10$ % and on $\\Delta V$ is $\\pm 20$ km s$^{-1}$ .","Figure: Emission line profiles of LSI+61 0 ^0303, MWC 148, and MWC 656.Table: Spectral line parameters"],["Objects: System parameters","LSI+61$^0$ 303 (V615 Cas) was identified as a $\\gamma $ -ray source with the $COS B$ satellite 35 years ago (Swanenburg et al.","1981).","For the orbital period of LSI+61$^0$ 303, we adopt $P_{orb}=26.4960 \\pm 0.0028$ d, which was derived with Bayesian analysis of radio observations (Gregory 2002) and an orbital eccentricity $e=0.537$ , which was obtained on the basis of the radial velocity of the primary (Casares et al.","2005; Aragona et al.","2009).","For the primary, Grundstrom et al.","(2007) suggested a B0V star with radius $R_1=6.7 \\pm 0.9$ $R_\\odot $ .","A B0V star is expected to have on average $M_1 \\approx 15$ $M_\\odot $ (Hohle et al.","2010).","We adopt $v \\sin i = 349 \\pm 6$ km s$^{-1}$ for the projected rotational velocity of the mass donor (Hutchings & Crampton 1981, Zamanov et al.","2013).","MWC 148 (HD 259440) was identified as the counterpart of the variable TeV source HESS J0632+057 (Aharonian et al.","2007).","We adopt $P_{orb} = 315 ^{+6}_{-4}$ d derived from the X-ray data (Aliu et al.","2014), which is consistent with the previous result of $321 \\pm 5$ days (Bongiorno et al.","2011).","For this object Aragona et al.","(2010) derived $T_{eff} = 27500 - 30000$ K, $\\log g = 3.75 - 4.00$ , $M_1 = 13.2 - 19.0$ $M_\\odot $ , and $R_1 = 7.8 \\pm 1.8 $ $R_\\odot $ .","For the calculations in Sect.REF , we adopt $e=0.83$ , periastron at phase 0.967 (Casares et al.","2012), and $v \\sin i = 230 - 240$ km s$^{-1}$ (Moritani et al.","2015).","MWC 656 (HD 215227) is the emission-line Be star that lies within the positional error circle of the $AGILE$ $\\gamma $ -ray source AGL J2241+4454 (Lucarelli et al.","2010).","It is the first and until now the only detected binary composed of a Be star and a black hole (Casares et al.","2014).","For the orbital period, we adopt $P_{orb}=60.37 \\pm 0.04$ d obtained with optical photometry (Williams et al.","2010), $e=0.10 \\pm 0.04$ estimated on the basis of the radial velocity measurements and $v \\sin i = 330 \\pm 30$ km s$^{-1}$ (Casares et al.","2014).","For the primary, Williams et al.","(2010) estimated $T_{eff} = 19000 \\pm 3000$ K, $\\log g = 3.7 \\pm 0.2 $ , $M_1 = 7.7 \\pm 2.0$ $M_\\odot $ , $R_1 = 6.6 \\pm 1.9$ $R_\\odot $ .","Casares et al.","(2014) considered that the mass donor is a giant (B1.5-2 III) and give a mass range $M_1 = 10 - 16$ $M_\\odot $ .","On average a B1.5-2 III star is expected to have about $R_1 \\approx 8.3 - 8.8$ $R_\\odot $ (Straizys & Kuriliene 1981).","From newer values of the luminosity (Hohle et al.","2010), such a star is expected to have $M_1 \\approx 8.0 - 10.0$ $M_\\odot $ and radius $R_1 \\approx 9.5 - 10$ $R_\\odot $ .","We adopt $R_1 \\approx 10$ $R_\\odot $ for the calculations in Sect.REF .","Table: Disc size in different emission lines"],["Peak separation in different lines","For the Be stars, the peak separations in different lines follow approximately the relations (Hanuschik et al.","1988) $\\Delta V_\\beta \\approx 1.8 \\Delta V_\\alpha \\\\\\Delta V_\\gamma \\approx 1.2 \\Delta V_\\beta \\approx 2.2 \\Delta V_\\alpha \\\\\\Delta V_{\\rm FeII} \\approx 2.0 \\Delta V_\\alpha \\\\\\Delta V_{\\rm FeII} \\approx 1.1 \\Delta V_\\beta ,$ where Eq.","is derived from Eqs.","REF and .","For LSI+61$^0$ 303 using the measurements in Table REF , we obtain $\\Delta V_\\beta = 1.30 \\pm 0.04 \\, \\Delta V_\\alpha $ and $\\Delta V_{HeI5876}= 1.38 \\pm 0.13 \\, \\Delta V_\\alpha $ .","The ratio $\\Delta V_\\beta / \\Delta V_\\alpha $ is considerably below the average value for the Be stars (see Eq.REF ).","We obtain $\\Delta V_\\beta = 1.78 \\pm 0.06 \\, \\Delta V_\\alpha $ , $\\Delta V_\\gamma = 1.07 \\pm 0.03 \\, \\Delta V_\\beta $ , and $\\Delta V_{FeII5316} = 1.12 \\pm 0.03 \\, \\Delta V_\\beta $ , $\\Delta V_{HeI5876} = 1.47 \\pm 0.10 \\, \\Delta V_\\beta $ for MWC 148.","We use only three spectra for $H\\alpha $ (20140113, 20140217, and 20140218) when two peaks in $H\\alpha $ are visible.","The value of $\\Delta V_\\beta / \\Delta V_\\alpha \\approx 1.78$ is very similar to 1.8 in Be stars, the ratio $\\Delta V_{FeII5316} / \\Delta V_\\beta \\approx 1.07 $ is similar to 1.1 in Be stars, and the value of $\\Delta V_\\gamma / \\Delta V_\\beta \\approx 1.07$ is again similar to the value 1.2 for Be stars.","We estimate $\\Delta V_\\beta = 1.72 \\pm 0.18 \\, \\Delta V_\\alpha $ for MWC 656 (using six spectra from the Liverpool Telescope FRODOSpec, where two peaks are visible in both $H\\alpha $ and $H\\beta $ ), $\\Delta V_\\gamma = 1.22 \\pm 0.04 \\, \\Delta V_\\beta $ , $\\Delta V_{FeII5316} = 1.01 \\pm 0.10 \\, \\Delta V_\\beta $ , where all three ratios are similar to the corresponding values (Eq.","REF , , ) in Be stars.","We do not see two peaks on high-resolution Rozhen spectra of this object.","However two peaks are clearly distinguishable on a few of the LT spectra.","In every case of detection/non-detection of the double peak structure, $W\\alpha $ is very similar $19 < W\\alpha < 25$ Å.","The comparison of the peak separation of different emission lines shows that MWC 148 and MWC 656 have circumstellar disc that is similar to that of the normal Be stars.","At this stage considerable deviation from the behaviour of the Be stars is only detected in LSI+61$^0$ 303.","In this star the $H\\alpha $ -emitting disc is only 1.7 times larger than the H$\\beta $ -emitting disc, while in normal Be stars it is 3.3 times larger.","This probably is one more indication that outer parts of the disc are truncated as a result of the relatively short orbital period.","Figure: Distance between the peaks of HαH\\alpha emission line(normalized with stellar rotation and inclination)versus W α W_\\alpha on a logarithmic scale.Black empty circles indicate normal Be stars,red pluses indicate Be/γBe/\\gamma -ray binaries.The solid line denotes y=-0.592x+0.165y =- 0.592 \\: x + 0.165 (see Sect.",")."],["Disc size","For rotationally dominated profiles, the peak separation can be regarded as a measure of the outer radius ($R_{disc}$ ) of the emitting disc (Huang 1972) $\\left( \\frac{\\Delta V}{2\\,v\\,\\sin {i}} \\;\\right)= \\; \\left( \\frac{R_{disc}}{R_1}\\;\\right)^{-j} ,$ where $j=0.5$ for Keplerian rotation, $j=1$ for angular momentum conservation, $R_1$ is the radius of the primary, and $v\\,\\sin {i}$ is its projected rotational velocity.","Eq.","REF relies on the assumptions that (1) the Be star is rotating critically, and (2) that the line profile shape is dominated by kinematics and radiative transfer does not play a role.","When the two peaks are visible in the emission lines, we can estimate the disc radius using Eq.","REF .","The calculated disc size for different emission lines are given in Table REF .","In the $H\\alpha $ emission line of LSI+61$^0$ 303 two peaks are clearly visible on all of our spectra.","However in MWC 656 the $H\\alpha $ emission line seems to exhibit three peaks (see Fig.","REF ).","Two peaks in $H\\alpha $ emission of MWC 148 are clearly detectable on January-February 2014 observations.","Two peaks are not distinguishable on the spectra obtained in March 2014 (when the companion is at periastron), which probably indicates perturbations in the outer parts of the disc caused by the orbital motion of the compact object.","In the $H\\beta $ line two peaks are visible on all the Rozhen spectra.","We take this opportunity to obtain an estimation of the $R_{disc}(H\\alpha )$ using $\\Delta V_\\beta $ ; the ratios $\\Delta V_\\beta / \\Delta V_\\alpha $ (as obtained in Sect.","REF ), and Eq.REF .","The $R_{disc}(H\\alpha )$ values calculated in this way are given in Table REF and indicated with $(^b)$ .","Figure: Histograms of the disc size.The vertical dashed (red) lines indicate the distance between the componentsat periastron and apastron, a(1-e) and a(1+e), respectively.The resonances n:m are also indicated (in blue).","For LSI+61 0 ^0303 and MWC 148 only those with m=1 are given."],["Disc size and $W_\\alpha $","The disc size and $W_\\alpha $ are connected (see also Hanuschik 1989; Grundstrom & Gies 2006).","This simply expresses the fact that $R_{disc}$ grows as $W_\\alpha $ becomes larger.","In Fig.","REF , we plot $\\log \\Delta V_\\alpha / 2 $$v \\: \\sin \\: i$ versus $\\log W_\\alpha $ .","In this figure 138 data points are plotted for Be stars taken from Andrillat (1983), Hanuschik (1986), Hanuschik et al.","(1988), Dachs et al.","(1992), Slettebak et al.","(1992), and Catanzaro (2013).","In this figure, the data for Be/$\\gamma $ -ray binaries are also plotted.","The three Be/$\\gamma $ -ray binaries are inside the distribution of normal Be stars.","There is a moderate to strong correlation between the variables with Pearson correlation coefficient -0.63, Spearman's (rho) rank correlation 0.64, and $p$ -$value \\sim 10^{-15}$ .","The dependence is of the form $\\; \\; \\log \\; (\\Delta V_{\\alpha }/2 v \\sin i ) = -a \\; \\log W_\\alpha + b,$ and the slope is shallower for stars with $ W_\\alpha < 3$ Å as noted by Hanuschik et al.","(1988).","For 120 data points in the interval $3 \\le W_\\alpha \\le 50$ Å, using a least-squares approximation we calculate $a=0.592 \\pm 0.030$ and $b=0.165 \\pm 0.036$ .","This fit as well as the correlation coefficients are calculated using only normal Be stars.","Using Eq.","REF and Eq.","REF we then obtain $\\left( \\frac{R_{disc}}{R_1}\\;\\right)^{-j} = 1.462 \\; W_\\alpha ^{-0.592}.$ Having in mind that (1) the discs of the Be stars are near Keplerian (Porter & Rivinius 2003, Meilland et al.","2012); (2) the Be stars rotate at rates below the critical rate (e.g.","Chauville et al.","2001), and (3) at higher optical depths the emission line peaks are shifted towards lower velocities (Hummel & Dachs 1992), we calculate the disc radius with the following formula: $\\frac{R_{disc}}{R_1} = \\; \\epsilon \\; 0.467 \\; \\; W_\\alpha ^{1.184},$ where $\\epsilon $ is a dimensionless parameter (see also Zamanov et al.","2013), for which we adopt $\\epsilon = 0.9 \\pm 0.1$ .","The disc sizes calculated with Eq.","REF are given in Table REF and are denoted with $(^c)$ .","As can be seen, the values agree with those obtained with the conventional method.","We estimate average values of the dimensionless quantity $R_{disc} / \\epsilon R_1 = 8.7 \\pm 1.9$ (for LSI+61$^0$ 303), $R_{disc} / \\epsilon R_1 = 43 \\pm 5$ (for MWC 148), and $R_{disc} / \\epsilon R_1 = 18.0 \\pm 1.1$ (for MWC 656), respectively."],["Disc truncation","The orbit of the compact object, the average size of $H\\alpha $ disc, the average size of $H\\beta $ disc, and the Be star are plotted in Fig.","3.","The coordinates X and Y are in solar radii.","The histograms of $H\\alpha $ disc size, calculated using Eq.","REF , are plotted in Fig.","4.","For LSI+61$^0$ 303, we use our new data and published data from Paredes et al.","(1994), Steele et al.","(1996), Liu & Yan (2005), Grundstrom et al.","(2007), McSwain et al.","(2010), and Zamanov et al.","(1999, 2013).","We use Rozhen and Liverpool Telescope spectra for MWC 148 and MWC 656.","In all three stars the distribution of $R_{disc}$ values has a very well pronounced peak.","The tendency for the disc emission fluxes to cluster at specified levels is related to the truncation of the disc at specific disc radii by the orbiting compact object (e.g.","Coe et al.","2006).","Okazaki & Negueruela (2001) proposed that these limiting radii are defined by the closest approach of the companion in the high-eccentricity systems and by resonances between the orbital period and the disc gas rotational periods in the low-eccentricity systems.","The resonance radii are given by ${R_{n:m}^{3/2}} = \\frac{m \\; (G \\: M_1)^{1/2}}{2 \\: \\pi } \\: \\frac{P_{orb}}{n},$ where $G$ is the gravitational constant, $n$ is the integer number of disc gas rotational periods, and $m$ is the integer number of orbital periods.","The important resonances are not only those with $n:1$ , but can also be $n:m$ in general.","For LSI+61$^0$ 303 (assuming $M_1 \\approx 15$ $M_\\odot $ , $M_2 \\approx 1.4$ $M_\\odot $ , $e \\approx 0.537$ ), we estimate the distances between the components $a(1-e) \\approx 44$ $R_\\odot $ and $a(1+e)\\approx 146$ $R_\\odot $ for the periastron and apastron, respectively.","As can be seen from Fig.","REF , the disc size is $R_{disc} \\sim a(1-e)$ and it never goes close to $a(1+e)$ .","The resonances that correspond to disc size are between 5:1 and 1:1, and the peak on the histogram corresponds to the 2:1 resonance.","For MWC 148, with the currently available data ($M_1 \\approx 15$ $M_\\odot $ , $M_2 \\approx 4$ $M_\\odot $ , $e \\approx 0.83$ ), we estimate $a(1-e) \\approx 88$ $R_\\odot $ and $a(1+e)\\approx 951$ $R_\\odot $ for the periastron and apastron, respectively.","In Fig.","REF , it is apparent that $a(1-e) < R_{disc} < a(1+e)$ .","The 2:1 resonance is the closest to the peak of the distribution.","We note in passing that the disc size in this star could have a bi-modal distribution (a second peak with lower intensity seems to emerge close to 4:1 resonance radius).","For MWC 656 (assuming $M_1 \\approx 9$ $M_\\odot $ , $M_2 \\approx 4$ $M_\\odot $ , $e \\approx 0.1$ ), we estimate $a(1-e) \\approx 137$ $R_\\odot $ and $a(1+e)\\approx 167$ $R_\\odot $ for the periastron and apastron, respectively.","In Fig.","REF , it is seen that the 1:1 resonance is very close to the peak of the distribution and $a(1-e) \\lesssim R_{disc} \\lesssim a(1+e)$ .","The disc size rarely goes above $a(1+e)$ ."],["Interstellar extinction: Estimates of $E(B-V)$ from interstellar lines","There is a strong correlation between equivalent width of the diffuse interstellar bands (DIBs) and reddening (Herbig 1975; Puspitarini et al.","2013).","There is also a calibrated relation of reddening with the equivalent width of the interstellar line $KI \\lambda 7699$ Å (Munari & Zwitter 1997).","Aiming to estimate the interstellar extinction towards our objects, we measure equivalent widths of $KI \\lambda 7699$ Å and DIBs $\\lambda 6613, \\lambda 5780, \\lambda 5797$ .","LSI+61$^0$ 303: For this object, Hunt et al.","(1994) use $E(B-V)=0.93$ (Hutchings & Crampton 1981).","Howarth (1983) using the 2200 Å extinction bump obtained $E(B-V)=0.75 \\pm 0.1$ .","Steele et al.","(1998) estimated $E(B-V)=0.70 \\pm 0.40$ from Na I D$_2$ and $E(B-V)= 0.65 \\pm 0.25$ from diffuse interstellar bands.","For LSI+61$^0$ 303, we measure $0.17 \\le W(KI \\lambda 7699) \\le 0.19$ Å, $0.17 < W(DIB \\lambda 6613) < 0.19$ Å, $0.34 < W(DIB \\lambda 5780) < 0.41$ Å, $0.09 < W(DIB \\lambda 5797) < 0.16$ Å, which following Munari & Zwitter (1997) and Puspitarini et al.","(2013) calibrations corresponds to $E(B-V)=0.84 \\pm 0.08$ .","MWC 148: For this star, Friedemann (1992) estimated E(B-V)=0.85 from the 217 nm band.","We measure $0.14 < W(KI \\lambda 7699) < 0.20$ Å, $0.13 < W(DIB \\lambda 6613) < 0.18$ Å, $0.28 < W(DIB \\lambda 5780) < 0.34$ Å, $0.13 < W(DIB \\lambda 5797) < 0.15$ Å, which corresponds to a slightly lower value $E(B-V)=0.77 \\pm 0.06$ .","MWC 656: For this star, Williams et al.","(2010) gave a low value E(B-V)=0.02.","Casares et al.","(2014) estimated E(B-V)=0.24.","We measure $0.04 < W(DIB \\lambda 6613) < 0.06$ Å, $0.09 \\le W(DIB \\lambda 5780) \\le 0.10$ Å, $0.03 < W(DIB\\lambda 5797) < 0.05$ Å, and estimate $E(B-V)=0.25 \\pm 0.02$ .","Figure: Rotational period of the mass donor (P rot P_{rot}) vs. P orb P_{orb}.Filled (blue) circles indicate high-mass X-ray binaries with giant/supergiant mass donor, open circles indicate Be/X-ray binaries, and red plusses indicate γ\\gamma -ray binaries."],["Rotational period of the mass donors","In close binary systems the rotation of the companions of compact objects is accelerated by mass transfer and tidal forces (e.g.","Ablimit & Lü 2012).","Adopting the parameters given in Sect.",", we estimate the rotational periods of the mass donors $P_{rot} \\approx 0.92$ d (for LSI+61$^0$ 303), $P_{rot} \\approx 0.91$ d (MWC 148), and $P_{rot} \\approx 0.86$ d (MWC 656).","For the $\\gamma $ -ray binary LS 5039, the mass donor is a O6.5V((f)) star with $R_{1} = 9.3 \\pm 0.6 $ $R_\\odot $ , inclination $i \\approx 24.9^0$ , $v \\: \\sin \\: i$$= 113 \\pm 8$ km s$^{-1}$ (Casares et al.","2005).","We calculate $P_{rot} = 1.764$ d. Because of the short orbital period $P_{orb} = 3.91$ d, the rotation of the mass donor could be pseudo-synchronized with the orbital motion (Casares et al.","2005).","Radio observations of the pulsar in PSR B1259-63/LS 2883 allow the orbital parameters to be precisely established: $P_{orb} = 1236.72$ d and eccentricity of e = 0.87 (Wang et al.","2004; Shannon et al.","2014).","For the primary, Negueruela et al.","(2011) estimated $R_1 = 9.0 \\pm 1.5$ $R_\\odot $ , $v \\: \\sin \\: i$ $ = 260 \\pm 15$ km s$^{-1}$ , and $i_{orb} \\approx 23^0$ , which give $P_{rot} = 0.689$ d. In Fig.","REF we plot $P_{rot}$ versus $P_{orb}$ for a number of high-mass X-ray binaries (see also Stoyanov & Zamanov 2009).","High-mass X-ray binaries with a giant/supergiant component, Be/X-ray binaries, and Be/$\\gamma $ -ray binaries are plotted with different symbols in this figure.","The solid line represents synchronization ($P_{rot} = P_{orb}$ ).","Among the five $\\gamma $ -ray binaries with known rotational velocity of the mass donor, the rotation of the mass donor is close to synchronization with the orbital motion in only one (LS 5039).","The four Be/$\\gamma $ -ray binaries are not close to the line of synchronization.","They occupy the same region as the Be/X-ray binaries; in other words regarding the rotation of the mass donor the Be/$\\gamma $ -ray binaries are similar to the Be/X-ray binaries.","In these, the tidal force of the compact object acts to decelerate the rotation of the mass donor.","The spin-down of the Be stars due to angular momentum transport from star to disc (Porter 1998) is another source of deceleration.","In the well-known Be star $\\gamma $ Cas, Robinson & Smith (2000) found that the X-ray flux varied with a period P = 1.1 d, which they interpreted as the rotational period of the mass donor.","A similar period is detected in photometric observations (Harmanec et al.","2000; Henry & Smith 2012).","This periodicity is probably due to the interaction between magnetic field of the Be star and its circumstellar disc or the presence of some physical feature, such as a spot or cloud, co-rotating with the star.","The optical emission lines of MWC 148 are practically identical to those of $\\gamma $ Cas.","All detected lines in the observed spectral range (Balmer lines, HeI lines and FeII lines) have similar equivalent widths, intensities, profiles, and even a so-called wine-bottle structure noticeable in the $H\\alpha $ line (see Fig.","1 in Zamanov et al.","2016).","Bearing in mind the above estimations and the curious similarities between: (i) the mean 20-60 keV X-ray luminosity of $\\gamma $ Cas and LSI+61$^0$ 303 (Shrader et al.","2015) and (ii) the optical emission lines of $\\gamma $ Cas and MWC 148, we consider that periodicity $\\sim \\!1$ day could be detectable in X-ray/optical bands in the Be/$\\gamma $ -ray binaries and could provide a direct measurement of the rotational period of the mass donor."],["Discussion","The three Be/$\\gamma $ -ray binaries discussed here have non-zero eccentricities and misalignment between the spin axis of the star and the spin axis of the binary orbit could be possible (Martin et al.","2009).","The inclination of the primary star in LSI+61$^0$ 303 to the line of sight is probably $ i_{Be} \\sim 70^0$ (Zamanov et al.","2013).","Aragona et al.","(2014) derived $a_1 \\sin i_{orb} = 8.64 \\pm 0.52$ .","Assuming $M_1 = 15$$M_\\odot $ , $M_2 = 1.4$ $M_\\odot $ , we estimate $ i_{orb} \\sim 67^0 - 73^0$ .","It appears that there is no significant deviation of the orbital plane from the equatorial plane of the Be star.","The emission lines of MWC 148 are very similar to those of $\\gamma $ Cas.","The emission lines are most sensitive to the footpoint density and inclination angle (Hummel 1994).","It means that in MWC 148 the Be star inclination should be similar to that of $\\gamma $ Cas, for which the inclination is in the range $40^0 - 50^0$ (Clarke 1990, Quirrenbach et al.","1997).","For MWC 148, Casares et al.","(2012) estimated $a_1 \\sin i_{orb} = 77.6 \\pm 25.9$ , which for $M_1 = 15$ $M_\\odot $ and a 4 $M_\\odot $ black hole gives $ 45^0 \\lesssim i_{orb} \\lesssim 65^0$ .","For MWC 656, Casares et al.","(2014) give $M_1 \\sin ^3 i_{orb} = 5.83 \\pm 0.70$ .","Bearing in mind the range of 8 $M_\\odot $ $\\le M_1 \\le 10$ $M_\\odot $ , this gives $53^0 \\lesssim i_{orb} \\lesssim 59^0$ .","Inclination of the Be star can be evaluated from the full width at zero intensity of the FeII lines $FWZI/2 \\sin i = (G M_1/ R_1)^{1/2}$ .","From FWZI of FeII lines (Casares et al.","2012) and using $R_1= 9.5 - 10.0$ $R_\\odot $ we estimate $i_{Be} \\approx 53 - 61^0$ .","It appears that both planes are almost complanar.","There are no signs of considerable deviation between the two planes.","The opening half-angle of the Be stars' circumstellar disc are $\\sim \\!","10^0$ (Tycner et al.","2006; Cyr et al.","2015) and it means that the compact object is practically orbiting in the plane of the circumstellar disc.","The comparison between the orbit and circumstellar disc size (see Fig.","REF and Sect.","REF ) shows that in these three objects we have three different situations: In LSI+61$^0$ 303 the neutron star passes through the outer parts of the circumstellar disc at periastron, but it does not enter deeply in the disc; In MWC 148 the compact object goes into the innermost parts of the disc (passes through the innermost parts and penetrates deeply in the disc) during the periastron passage; In MWC 656 the black hole is constantly accreting from the outer edge of the circumstellar disc.","In MWC 656, this means that the compact object (black hole) is at the disc border at all times and as a consequence it will have a higher and stable mass accretion rate along the entire orbit.","It seems to be a very clear case of disc truncation in which the circumstellar disc is cut almost exactly at the black hole orbit.","Because the $H\\alpha $ peaks are connected with the outermost parts of the disc, the above three items probably explain the observational findings: 1.","In LSI+61$^0$ 303 the $H\\alpha $ emission line has a two-peak profile at all times (in all our spectra in the period 1987 - 2015),because the circumstellar disc size is relatively small and the neutron star passes only through the outermost parts of the circumstellar disc at periastron; 2.","In MWC 148, the big jumps in the $H\\alpha $ parameters, $W_\\alpha $ , full width at half maximum and radial velocity (see Fig.","4 of Casares 2012) occur because the compact object enters (reaches) the inner parts of the disc during the periastron passage.","3.","In MWC 656 the double-peak profile is not often visible because the black hole is at the outer edge at all times and makes perturbations exactly in the places where the $H\\alpha $ peaks are formed.","It is worth noting that when the compact object causes large-scale perturbations, distorted profiles, such as those observed in 1A 535+262 (Moritani et al.","2011, 2013), will appear.","If only a small portion of the outer disc is perturbed then it will appear in the central part of the emission line profile (e.g.","in between the peaks or even filling the central dip) because the outer parts of the disc produce the central part of the $H\\alpha $ emission line profile.","Similar additional emission is already detected in the $H\\alpha $ spectra of LSI+61$^0$ 303 (Paredes et al.","1990; Liu et al.","2000; Zamanov & Marti 2000).","Gamma-ray emission has been repeatedly observed to be periodic in the system LSI+61303 (Albert et al.","2009; Saha et al.","2016) and also very likely in MWC148, where its periodic X-ray flares are highly correlated with TeV emission (Aliu et al.","2014).","A similar situation occurs in the case of LS 5039 (Aharonian et al.","2006), 1FGL J1018.6-20135856 (H. E. S. S. Collaboration et al.","2015), and PSR B1259-63 (H.E.S.S.","Collaboration et al.","2013), which has the longest orbital period observed in a $\\gamma $ -ray binary (about 3 yr).","This is likely related to the very different physical conditions sampled by the compact companion as it revolves around the primary star in an eccentric orbit.","Non-thermal emission (e.g.","Dubus 2013) is produced from particles accelerated at the shock between the wind of the pulsar and matter flowing out from the primary star (the polar wind as well as the disc).","The X-ray and $\\gamma $ -ray light curve of MWC 148 shows two maxima at orbital phases 0.35 and 0.75 and minimum at apastron passage (Acciari et al.","2009; Aliu et al.","2014).","X-ray and $\\gamma $ -ray fluxes are correlated as mentioned above, in agreement with leptonic emission models, where relativistic electrons lose energy by synchrotron emission and inverse Compton emission (Maier & for the VERITAS Collaboration 2015).","The highly eccentric orbital geometry sketched in the central panel of Fig.","REF is also in good agreement with a periodic flaring system.","In the case of MWC 656, $\\gamma $ -rays have been detected occasionally (Williams et al.","2010), but so far never reaching the TeV energy domain.","The main differences from previous sources are the fact that the compact object has been shown to be a black hole and its orbit is only moderately eccentric (Casares et al.","2014).","Based on our spectroscopic observations, the size of the excretion disc is such that the black hole is accreting matter only from its lower density outer edges.","As mentioned before, this implies that the accretion rate is stable but at the same time low.","Indeed, the quiescent X-ray emission level of the system is as weak as $\\sim 10^{-8}$ Eddington luminosity (Munar-Adrover et al.","2014).","Therefore, episodic $\\gamma $ -ray flares such as those detected by AGILE, likely require some enhancement of mass loss from the primary Be star or clumps in its circumstellar disc.","We speculate here that the physical mechanism responsible for $\\gamma $ -ray emission in the MWC 656 black hole context could be related to the alternative microquasar-jet scenario also proposed for $\\gamma $ -ray binaries (see e.g.","Romero et al.","2007).","The fact that MWC 656 seems to adhere to the low-luminosity end of the X-ray/radio correlation for hard state compact jets also points in this direction (Dzib et al.","2015)."],["Conclusions","From the spectroscopic observations of the three Be/$\\gamma $ -ray binaries we deduce that in LSI+61$^0$ 303 the neutron star crosses the outer parts of the circumstellar disc at periastron, in MWC 148 the compact object passes deeply through the disc during the periastron passage, and in MWC 656 the black hole is accreting from the outer parts of the circumstellar disc during the entire orbital cycle.","The histograms in all three stars show that the disc size clusters at specific levels, indicating the circumstellar disc is truncated by the orbiting compact object.","We estimate the interstellar extinction towards LSI+61$^0$ 303, MWC 148, and MWC 656.","The rotation of the mass donors is similar to that of the Be/X-ray binaries.","We suggest that the three stars deserve to be searched for a periodicity of about $1.0$ day.","The authors are grateful to an anonymous referee for valuable comments and suggestions.","This work was partially supported by grant AYA2013-47447-C3-3-P from the Spanish Ministerio de Economía y Competitividad (MINECO), and by the Consejería de Economía, Innovación, Ciencia y Empleo of Junta de Andalucía as research group FQM-322, as well as FEDER funds."]],"string":"[\n [\n \"Optical spectroscopy of Be/gamma-ray binaries\"\n ],\n [\n \"Abstract We report optical spectroscopic observations of the Be/gamma-ray binaries LSI+61303, MWC 148 and MWC 656.\",\n \"The peak separation and equivalent widths of prominent emission lines (H-alpha, H-beta, H-gamma, HeI, and FeII) are measured.\",\n \"We estimated the circumstellar disc size, compared it with separation between the components, and discussed the disc truncation.\",\n \"We find that in LSI+61303 the compact object comes into contact with the outer parts of the circumstellar disc at periastron, in MWC 148 the compact object goes deeply into the disc during the periastron passage, and in MWC 656 the black hole is accreting from the outer parts of the circumstellar disc along the entire orbit.\",\n \"The interstellar extinction was estimated using interstellar lines.\",\n \"The rotation of the mass donors appears to be similar to the rotation of the mass donors in Be/X-ray binaries.\",\n \"We suggest that X-ray/optical periodicity of about 1 day deserves to be searched for.\"\n ],\n [\n \"Introduction\",\n \"The rapid and sustained progress of high energy and very high energy astrophysics in recent years enabled the identification of a new group of binary stars emitting at TeV energies (e.g.\",\n \"Paredes et al.\",\n \"2013).\",\n \"These objects, called $\\\\gamma $ -ray binaries, are high-mass X-ray binaries that consist of a compact object (neutron star or black hole) orbiting an optical companion that is an OB star.\",\n \"There are five confirmed $\\\\gamma $ -ray binaries so far: PSR B1259-63/LS 2883 (Aharonian et al.\",\n \"2005), LS 5039/V479 Sct (Aharonian et al.\",\n \"2006), LSI+61$^0$ 303 (Albert et al., 2009), HESS J0632+057/MWC 148 (Aharonian et al.\",\n \"2007), and 1FGL J1018.6-5856 (H.E.S.S.\",\n \"Collaboration et al.\",\n \"2015).\",\n \"Their most distinctive fingerprint is a spectral energy distribution dominated by non-thermal photons with energies up to the TeV domain.\",\n \"Recently, Eger et al.\",\n \"(2016) proposed a binary nature for the $\\\\gamma $ -ray source HESS J1832-093/2MASS J18324516-092154 and this object probably belongs to the family of the $\\\\gamma $ -ray binaries as a sixth member.\",\n \"The binary system, PSR B1259-63 is unique, since it is the only one where the compact object has been identified as a radio pulsar (Johnston et al.\",\n \"1992, 1994).\",\n \"The nature of the compact object is known in PSR B1259-63 as a neutron star, and in AGL J2241+4454/MWC 656 as a black hole (Casares et al.\",\n \"2014).\",\n \"Although not included in the confirmed list, MWC 656 was selected as a target here despite not having shown all the observational properties of a canonical $\\\\gamma $ -ray binary yet.\",\n \"It was only occasionally detected by the AGILE observatory at GeV energies and not yet detected in the TeV domain (see Aleksić et al.\",\n \"2015).\",\n \"Nevertheless, the fact that the black hole nature of the compact companion is almost certain renders it very similar to the typical $\\\\gamma $ -ray binaries.\",\n \"In the other systems the nature of the compact object remains unclear (e.g.\",\n \"Dubus 2013).\",\n \"In addition to these objects, there are several other binary systems ($\\\\eta $ Car, Cyg X-1, Cyg X-3, Cen X-3, and SS 433) that are detected as GeV sources, but not as TeV sources so far.\",\n \"Here we report high-resolution spectral observations of LSI+61$^0$ 303, MWC 148, and MWC 656, and discuss circumstellar disc size, disc truncation, interstellar extinction, and rotation of their mass donors.\",\n \"The mass donors (primaries) of these three targets are emission-line Be stars.\",\n \"The Be stars are non-supergiant, fast-rotating B-type and luminosity class III-V stars which, at some point in their lives, have shown spectral lines in emission (Porter & Rivinius 2003).\",\n \"The material expelled from the equatorial belt of a rapidly rotating Be star forms an outwardly diffusing gaseous, dust-free Keplerian disc (Rivinius et al.\",\n \"2013).\",\n \"In the optical/infrared band, the two most significant observational characteristics of Be stars and their excretion discs are the emission lines and the infrared excess.\",\n \"Moving along the orbit, the compact object passes close to this disc, and sometimes may even go through it causing significant perturbations in its structure.\",\n \"This circumstellar disc feeds the accretion disc around the compact object and/or interacts with its relativistic wind.\"\n ],\n [\n \"Observations\",\n \"High-resolution optical spectra of the three northern Be/$\\\\gamma $ -ray binaries were secured with the fibre-fed Echelle spectrograph ESpeRo attached to the 2.0 m telescope of the National Astronomical Observatory Rozhen, located in Rhodope mountains, Bulgaria.\",\n \"The spectrograph uses R2 grating with 37.5 grooves/mm, Andor CCD camera 2048 x 2048 px, 13.5x13.5 $\\\\mu m$ px$^{-1}$ (Bonev et al.\",\n \"2016).\",\n \"The spectrograph provides a dispersion of 0.06 Å px$^{-1}$ at 6560 Å and 0.04 Å px$^{-1}$ at 4800 Å.\",\n \"The spectra were reduced in the standard way including bias removal, flat-field correction, and wavelength calibration.\",\n \"Pre-processing of data and parameter measurements are performed using various routines provided in IRAF.\",\n \"The journal of observations is presented in Table REF , where the date, start of the exposure, exposure time, and signal-to-noise ratio at about $\\\\lambda 6600$ Å are given.\",\n \"The orbital phases are calculated using $HJD_0= 2443366.775$ , $HJD_0= 2454857.5,$ and $HJD_0= 2453243.7$ for LSI+61$^0$ 303, MWC 148, and MWC 656, respectively, and orbital periods given in Sect. .\",\n \"Emission line profiles of LSI+61$^0$ 303, MWC 148, and MWC 656 are plotted on Fig.REF .\",\n \"Spectral line parameters equivalent width (W) and distance between the peaks ($\\\\Delta V$ ) for the prominent lines ($H\\\\alpha $ , H$\\\\beta $ , H$\\\\gamma $ , $HeI \\\\lambda 5876,$ and $FeII \\\\lambda 5316$ ) are given in Table REF .\",\n \"The typical error on the equivalent width is below $\\\\pm 10$ % for lines with $W > 1$ Å and up to $\\\\pm 20$ % for lines with $W \\\\lesssim 1$ Å.\",\n \"The typical error on $\\\\Delta V$ is $\\\\pm 10$ km s$^{-1}$ .\",\n \"It is worth noting that (1) in LSI+61$^0$ 303 FeII lines are not detectable; (2) In MWC 656 on spectrum 20150705 the HeI $\\\\lambda 5876$ line is not visible (probably emission fills up the absorption).\",\n \"In addition to the Rozhen data we use 98 spectra of MWC 148 and 68 spectra of MWC 656 (analysed in Casares et al.\",\n \"2012) from the archive of the 2.0 m Liverpool TelescopeThe Liverpool Telescope is operated on the island of La Palma by Liverpool John Moores University in the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrofisica de Canarias with financial support from the UK Science and Technology Facilities Council.\",\n \"(Steele et al.\",\n \"2004).\",\n \"These spectra were obtained using the Fibre-fed RObotic Dual-beam Optical Spectrograph (FRODOSpec; Morales-Rueda et al.\",\n \"2004).\",\n \"The spectrograph is fed by a fibre bundle array consisting of $12\\\\times 12$ lenslets of 0.82 arcsec each, which is reformatted as a slit.\",\n \"The spectrograph was operated in a high-resolution mode, providing a dispersion of 0.8 Å px$^{-1}$ at 6500 Å, 0.35 Å px$^{-1}$ at 4800 Å, and typical $S/N \\\\gtrsim 100$ .\",\n \"FRODOSpec spectra were processed using the fully automated data reduction pipeline of Barnsley et al.\",\n \"(2012).\",\n \"The typical error on the equivalent width is $\\\\pm 10$ % and on $\\\\Delta V$ is $\\\\pm 20$ km s$^{-1}$ .\",\n \"Figure: Emission line profiles of LSI+61 0 ^0303, MWC 148, and MWC 656.Table: Spectral line parameters\"\n ],\n [\n \"Objects: System parameters\",\n \"LSI+61$^0$ 303 (V615 Cas) was identified as a $\\\\gamma $ -ray source with the $COS B$ satellite 35 years ago (Swanenburg et al.\",\n \"1981).\",\n \"For the orbital period of LSI+61$^0$ 303, we adopt $P_{orb}=26.4960 \\\\pm 0.0028$ d, which was derived with Bayesian analysis of radio observations (Gregory 2002) and an orbital eccentricity $e=0.537$ , which was obtained on the basis of the radial velocity of the primary (Casares et al.\",\n \"2005; Aragona et al.\",\n \"2009).\",\n \"For the primary, Grundstrom et al.\",\n \"(2007) suggested a B0V star with radius $R_1=6.7 \\\\pm 0.9$ $R_\\\\odot $ .\",\n \"A B0V star is expected to have on average $M_1 \\\\approx 15$ $M_\\\\odot $ (Hohle et al.\",\n \"2010).\",\n \"We adopt $v \\\\sin i = 349 \\\\pm 6$ km s$^{-1}$ for the projected rotational velocity of the mass donor (Hutchings & Crampton 1981, Zamanov et al.\",\n \"2013).\",\n \"MWC 148 (HD 259440) was identified as the counterpart of the variable TeV source HESS J0632+057 (Aharonian et al.\",\n \"2007).\",\n \"We adopt $P_{orb} = 315 ^{+6}_{-4}$ d derived from the X-ray data (Aliu et al.\",\n \"2014), which is consistent with the previous result of $321 \\\\pm 5$ days (Bongiorno et al.\",\n \"2011).\",\n \"For this object Aragona et al.\",\n \"(2010) derived $T_{eff} = 27500 - 30000$ K, $\\\\log g = 3.75 - 4.00$ , $M_1 = 13.2 - 19.0$ $M_\\\\odot $ , and $R_1 = 7.8 \\\\pm 1.8 $ $R_\\\\odot $ .\",\n \"For the calculations in Sect.REF , we adopt $e=0.83$ , periastron at phase 0.967 (Casares et al.\",\n \"2012), and $v \\\\sin i = 230 - 240$ km s$^{-1}$ (Moritani et al.\",\n \"2015).\",\n \"MWC 656 (HD 215227) is the emission-line Be star that lies within the positional error circle of the $AGILE$ $\\\\gamma $ -ray source AGL J2241+4454 (Lucarelli et al.\",\n \"2010).\",\n \"It is the first and until now the only detected binary composed of a Be star and a black hole (Casares et al.\",\n \"2014).\",\n \"For the orbital period, we adopt $P_{orb}=60.37 \\\\pm 0.04$ d obtained with optical photometry (Williams et al.\",\n \"2010), $e=0.10 \\\\pm 0.04$ estimated on the basis of the radial velocity measurements and $v \\\\sin i = 330 \\\\pm 30$ km s$^{-1}$ (Casares et al.\",\n \"2014).\",\n \"For the primary, Williams et al.\",\n \"(2010) estimated $T_{eff} = 19000 \\\\pm 3000$ K, $\\\\log g = 3.7 \\\\pm 0.2 $ , $M_1 = 7.7 \\\\pm 2.0$ $M_\\\\odot $ , $R_1 = 6.6 \\\\pm 1.9$ $R_\\\\odot $ .\",\n \"Casares et al.\",\n \"(2014) considered that the mass donor is a giant (B1.5-2 III) and give a mass range $M_1 = 10 - 16$ $M_\\\\odot $ .\",\n \"On average a B1.5-2 III star is expected to have about $R_1 \\\\approx 8.3 - 8.8$ $R_\\\\odot $ (Straizys & Kuriliene 1981).\",\n \"From newer values of the luminosity (Hohle et al.\",\n \"2010), such a star is expected to have $M_1 \\\\approx 8.0 - 10.0$ $M_\\\\odot $ and radius $R_1 \\\\approx 9.5 - 10$ $R_\\\\odot $ .\",\n \"We adopt $R_1 \\\\approx 10$ $R_\\\\odot $ for the calculations in Sect.REF .\",\n \"Table: Disc size in different emission lines\"\n ],\n [\n \"Peak separation in different lines\",\n \"For the Be stars, the peak separations in different lines follow approximately the relations (Hanuschik et al.\",\n \"1988) $\\\\Delta V_\\\\beta \\\\approx 1.8 \\\\Delta V_\\\\alpha \\\\\\\\\\\\Delta V_\\\\gamma \\\\approx 1.2 \\\\Delta V_\\\\beta \\\\approx 2.2 \\\\Delta V_\\\\alpha \\\\\\\\\\\\Delta V_{\\\\rm FeII} \\\\approx 2.0 \\\\Delta V_\\\\alpha \\\\\\\\\\\\Delta V_{\\\\rm FeII} \\\\approx 1.1 \\\\Delta V_\\\\beta ,$ where Eq.\",\n \"is derived from Eqs.\",\n \"REF and .\",\n \"For LSI+61$^0$ 303 using the measurements in Table REF , we obtain $\\\\Delta V_\\\\beta = 1.30 \\\\pm 0.04 \\\\, \\\\Delta V_\\\\alpha $ and $\\\\Delta V_{HeI5876}= 1.38 \\\\pm 0.13 \\\\, \\\\Delta V_\\\\alpha $ .\",\n \"The ratio $\\\\Delta V_\\\\beta / \\\\Delta V_\\\\alpha $ is considerably below the average value for the Be stars (see Eq.REF ).\",\n \"We obtain $\\\\Delta V_\\\\beta = 1.78 \\\\pm 0.06 \\\\, \\\\Delta V_\\\\alpha $ , $\\\\Delta V_\\\\gamma = 1.07 \\\\pm 0.03 \\\\, \\\\Delta V_\\\\beta $ , and $\\\\Delta V_{FeII5316} = 1.12 \\\\pm 0.03 \\\\, \\\\Delta V_\\\\beta $ , $\\\\Delta V_{HeI5876} = 1.47 \\\\pm 0.10 \\\\, \\\\Delta V_\\\\beta $ for MWC 148.\",\n \"We use only three spectra for $H\\\\alpha $ (20140113, 20140217, and 20140218) when two peaks in $H\\\\alpha $ are visible.\",\n \"The value of $\\\\Delta V_\\\\beta / \\\\Delta V_\\\\alpha \\\\approx 1.78$ is very similar to 1.8 in Be stars, the ratio $\\\\Delta V_{FeII5316} / \\\\Delta V_\\\\beta \\\\approx 1.07 $ is similar to 1.1 in Be stars, and the value of $\\\\Delta V_\\\\gamma / \\\\Delta V_\\\\beta \\\\approx 1.07$ is again similar to the value 1.2 for Be stars.\",\n \"We estimate $\\\\Delta V_\\\\beta = 1.72 \\\\pm 0.18 \\\\, \\\\Delta V_\\\\alpha $ for MWC 656 (using six spectra from the Liverpool Telescope FRODOSpec, where two peaks are visible in both $H\\\\alpha $ and $H\\\\beta $ ), $\\\\Delta V_\\\\gamma = 1.22 \\\\pm 0.04 \\\\, \\\\Delta V_\\\\beta $ , $\\\\Delta V_{FeII5316} = 1.01 \\\\pm 0.10 \\\\, \\\\Delta V_\\\\beta $ , where all three ratios are similar to the corresponding values (Eq.\",\n \"REF , , ) in Be stars.\",\n \"We do not see two peaks on high-resolution Rozhen spectra of this object.\",\n \"However two peaks are clearly distinguishable on a few of the LT spectra.\",\n \"In every case of detection/non-detection of the double peak structure, $W\\\\alpha $ is very similar $19 < W\\\\alpha < 25$ Å.\",\n \"The comparison of the peak separation of different emission lines shows that MWC 148 and MWC 656 have circumstellar disc that is similar to that of the normal Be stars.\",\n \"At this stage considerable deviation from the behaviour of the Be stars is only detected in LSI+61$^0$ 303.\",\n \"In this star the $H\\\\alpha $ -emitting disc is only 1.7 times larger than the H$\\\\beta $ -emitting disc, while in normal Be stars it is 3.3 times larger.\",\n \"This probably is one more indication that outer parts of the disc are truncated as a result of the relatively short orbital period.\",\n \"Figure: Distance between the peaks of HαH\\\\alpha emission line(normalized with stellar rotation and inclination)versus W α W_\\\\alpha on a logarithmic scale.Black empty circles indicate normal Be stars,red pluses indicate Be/γBe/\\\\gamma -ray binaries.The solid line denotes y=-0.592x+0.165y =- 0.592 \\\\: x + 0.165 (see Sect.\",\n \").\"\n ],\n [\n \"Disc size\",\n \"For rotationally dominated profiles, the peak separation can be regarded as a measure of the outer radius ($R_{disc}$ ) of the emitting disc (Huang 1972) $\\\\left( \\\\frac{\\\\Delta V}{2\\\\,v\\\\,\\\\sin {i}} \\\\;\\\\right)= \\\\; \\\\left( \\\\frac{R_{disc}}{R_1}\\\\;\\\\right)^{-j} ,$ where $j=0.5$ for Keplerian rotation, $j=1$ for angular momentum conservation, $R_1$ is the radius of the primary, and $v\\\\,\\\\sin {i}$ is its projected rotational velocity.\",\n \"Eq.\",\n \"REF relies on the assumptions that (1) the Be star is rotating critically, and (2) that the line profile shape is dominated by kinematics and radiative transfer does not play a role.\",\n \"When the two peaks are visible in the emission lines, we can estimate the disc radius using Eq.\",\n \"REF .\",\n \"The calculated disc size for different emission lines are given in Table REF .\",\n \"In the $H\\\\alpha $ emission line of LSI+61$^0$ 303 two peaks are clearly visible on all of our spectra.\",\n \"However in MWC 656 the $H\\\\alpha $ emission line seems to exhibit three peaks (see Fig.\",\n \"REF ).\",\n \"Two peaks in $H\\\\alpha $ emission of MWC 148 are clearly detectable on January-February 2014 observations.\",\n \"Two peaks are not distinguishable on the spectra obtained in March 2014 (when the companion is at periastron), which probably indicates perturbations in the outer parts of the disc caused by the orbital motion of the compact object.\",\n \"In the $H\\\\beta $ line two peaks are visible on all the Rozhen spectra.\",\n \"We take this opportunity to obtain an estimation of the $R_{disc}(H\\\\alpha )$ using $\\\\Delta V_\\\\beta $ ; the ratios $\\\\Delta V_\\\\beta / \\\\Delta V_\\\\alpha $ (as obtained in Sect.\",\n \"REF ), and Eq.REF .\",\n \"The $R_{disc}(H\\\\alpha )$ values calculated in this way are given in Table REF and indicated with $(^b)$ .\",\n \"Figure: Histograms of the disc size.The vertical dashed (red) lines indicate the distance between the componentsat periastron and apastron, a(1-e) and a(1+e), respectively.The resonances n:m are also indicated (in blue).\",\n \"For LSI+61 0 ^0303 and MWC 148 only those with m=1 are given.\"\n ],\n [\n \"Disc size and $W_\\\\alpha $\",\n \"The disc size and $W_\\\\alpha $ are connected (see also Hanuschik 1989; Grundstrom & Gies 2006).\",\n \"This simply expresses the fact that $R_{disc}$ grows as $W_\\\\alpha $ becomes larger.\",\n \"In Fig.\",\n \"REF , we plot $\\\\log \\\\Delta V_\\\\alpha / 2 $$v \\\\: \\\\sin \\\\: i$ versus $\\\\log W_\\\\alpha $ .\",\n \"In this figure 138 data points are plotted for Be stars taken from Andrillat (1983), Hanuschik (1986), Hanuschik et al.\",\n \"(1988), Dachs et al.\",\n \"(1992), Slettebak et al.\",\n \"(1992), and Catanzaro (2013).\",\n \"In this figure, the data for Be/$\\\\gamma $ -ray binaries are also plotted.\",\n \"The three Be/$\\\\gamma $ -ray binaries are inside the distribution of normal Be stars.\",\n \"There is a moderate to strong correlation between the variables with Pearson correlation coefficient -0.63, Spearman's (rho) rank correlation 0.64, and $p$ -$value \\\\sim 10^{-15}$ .\",\n \"The dependence is of the form $\\\\; \\\\; \\\\log \\\\; (\\\\Delta V_{\\\\alpha }/2 v \\\\sin i ) = -a \\\\; \\\\log W_\\\\alpha + b,$ and the slope is shallower for stars with $ W_\\\\alpha < 3$ Å as noted by Hanuschik et al.\",\n \"(1988).\",\n \"For 120 data points in the interval $3 \\\\le W_\\\\alpha \\\\le 50$ Å, using a least-squares approximation we calculate $a=0.592 \\\\pm 0.030$ and $b=0.165 \\\\pm 0.036$ .\",\n \"This fit as well as the correlation coefficients are calculated using only normal Be stars.\",\n \"Using Eq.\",\n \"REF and Eq.\",\n \"REF we then obtain $\\\\left( \\\\frac{R_{disc}}{R_1}\\\\;\\\\right)^{-j} = 1.462 \\\\; W_\\\\alpha ^{-0.592}.$ Having in mind that (1) the discs of the Be stars are near Keplerian (Porter & Rivinius 2003, Meilland et al.\",\n \"2012); (2) the Be stars rotate at rates below the critical rate (e.g.\",\n \"Chauville et al.\",\n \"2001), and (3) at higher optical depths the emission line peaks are shifted towards lower velocities (Hummel & Dachs 1992), we calculate the disc radius with the following formula: $\\\\frac{R_{disc}}{R_1} = \\\\; \\\\epsilon \\\\; 0.467 \\\\; \\\\; W_\\\\alpha ^{1.184},$ where $\\\\epsilon $ is a dimensionless parameter (see also Zamanov et al.\",\n \"2013), for which we adopt $\\\\epsilon = 0.9 \\\\pm 0.1$ .\",\n \"The disc sizes calculated with Eq.\",\n \"REF are given in Table REF and are denoted with $(^c)$ .\",\n \"As can be seen, the values agree with those obtained with the conventional method.\",\n \"We estimate average values of the dimensionless quantity $R_{disc} / \\\\epsilon R_1 = 8.7 \\\\pm 1.9$ (for LSI+61$^0$ 303), $R_{disc} / \\\\epsilon R_1 = 43 \\\\pm 5$ (for MWC 148), and $R_{disc} / \\\\epsilon R_1 = 18.0 \\\\pm 1.1$ (for MWC 656), respectively.\"\n ],\n [\n \"Disc truncation\",\n \"The orbit of the compact object, the average size of $H\\\\alpha $ disc, the average size of $H\\\\beta $ disc, and the Be star are plotted in Fig.\",\n \"3.\",\n \"The coordinates X and Y are in solar radii.\",\n \"The histograms of $H\\\\alpha $ disc size, calculated using Eq.\",\n \"REF , are plotted in Fig.\",\n \"4.\",\n \"For LSI+61$^0$ 303, we use our new data and published data from Paredes et al.\",\n \"(1994), Steele et al.\",\n \"(1996), Liu & Yan (2005), Grundstrom et al.\",\n \"(2007), McSwain et al.\",\n \"(2010), and Zamanov et al.\",\n \"(1999, 2013).\",\n \"We use Rozhen and Liverpool Telescope spectra for MWC 148 and MWC 656.\",\n \"In all three stars the distribution of $R_{disc}$ values has a very well pronounced peak.\",\n \"The tendency for the disc emission fluxes to cluster at specified levels is related to the truncation of the disc at specific disc radii by the orbiting compact object (e.g.\",\n \"Coe et al.\",\n \"2006).\",\n \"Okazaki & Negueruela (2001) proposed that these limiting radii are defined by the closest approach of the companion in the high-eccentricity systems and by resonances between the orbital period and the disc gas rotational periods in the low-eccentricity systems.\",\n \"The resonance radii are given by ${R_{n:m}^{3/2}} = \\\\frac{m \\\\; (G \\\\: M_1)^{1/2}}{2 \\\\: \\\\pi } \\\\: \\\\frac{P_{orb}}{n},$ where $G$ is the gravitational constant, $n$ is the integer number of disc gas rotational periods, and $m$ is the integer number of orbital periods.\",\n \"The important resonances are not only those with $n:1$ , but can also be $n:m$ in general.\",\n \"For LSI+61$^0$ 303 (assuming $M_1 \\\\approx 15$ $M_\\\\odot $ , $M_2 \\\\approx 1.4$ $M_\\\\odot $ , $e \\\\approx 0.537$ ), we estimate the distances between the components $a(1-e) \\\\approx 44$ $R_\\\\odot $ and $a(1+e)\\\\approx 146$ $R_\\\\odot $ for the periastron and apastron, respectively.\",\n \"As can be seen from Fig.\",\n \"REF , the disc size is $R_{disc} \\\\sim a(1-e)$ and it never goes close to $a(1+e)$ .\",\n \"The resonances that correspond to disc size are between 5:1 and 1:1, and the peak on the histogram corresponds to the 2:1 resonance.\",\n \"For MWC 148, with the currently available data ($M_1 \\\\approx 15$ $M_\\\\odot $ , $M_2 \\\\approx 4$ $M_\\\\odot $ , $e \\\\approx 0.83$ ), we estimate $a(1-e) \\\\approx 88$ $R_\\\\odot $ and $a(1+e)\\\\approx 951$ $R_\\\\odot $ for the periastron and apastron, respectively.\",\n \"In Fig.\",\n \"REF , it is apparent that $a(1-e) < R_{disc} < a(1+e)$ .\",\n \"The 2:1 resonance is the closest to the peak of the distribution.\",\n \"We note in passing that the disc size in this star could have a bi-modal distribution (a second peak with lower intensity seems to emerge close to 4:1 resonance radius).\",\n \"For MWC 656 (assuming $M_1 \\\\approx 9$ $M_\\\\odot $ , $M_2 \\\\approx 4$ $M_\\\\odot $ , $e \\\\approx 0.1$ ), we estimate $a(1-e) \\\\approx 137$ $R_\\\\odot $ and $a(1+e)\\\\approx 167$ $R_\\\\odot $ for the periastron and apastron, respectively.\",\n \"In Fig.\",\n \"REF , it is seen that the 1:1 resonance is very close to the peak of the distribution and $a(1-e) \\\\lesssim R_{disc} \\\\lesssim a(1+e)$ .\",\n \"The disc size rarely goes above $a(1+e)$ .\"\n ],\n [\n \"Interstellar extinction: Estimates of $E(B-V)$ from interstellar lines\",\n \"There is a strong correlation between equivalent width of the diffuse interstellar bands (DIBs) and reddening (Herbig 1975; Puspitarini et al.\",\n \"2013).\",\n \"There is also a calibrated relation of reddening with the equivalent width of the interstellar line $KI \\\\lambda 7699$ Å (Munari & Zwitter 1997).\",\n \"Aiming to estimate the interstellar extinction towards our objects, we measure equivalent widths of $KI \\\\lambda 7699$ Å and DIBs $\\\\lambda 6613, \\\\lambda 5780, \\\\lambda 5797$ .\",\n \"LSI+61$^0$ 303: For this object, Hunt et al.\",\n \"(1994) use $E(B-V)=0.93$ (Hutchings & Crampton 1981).\",\n \"Howarth (1983) using the 2200 Å extinction bump obtained $E(B-V)=0.75 \\\\pm 0.1$ .\",\n \"Steele et al.\",\n \"(1998) estimated $E(B-V)=0.70 \\\\pm 0.40$ from Na I D$_2$ and $E(B-V)= 0.65 \\\\pm 0.25$ from diffuse interstellar bands.\",\n \"For LSI+61$^0$ 303, we measure $0.17 \\\\le W(KI \\\\lambda 7699) \\\\le 0.19$ Å, $0.17 < W(DIB \\\\lambda 6613) < 0.19$ Å, $0.34 < W(DIB \\\\lambda 5780) < 0.41$ Å, $0.09 < W(DIB \\\\lambda 5797) < 0.16$ Å, which following Munari & Zwitter (1997) and Puspitarini et al.\",\n \"(2013) calibrations corresponds to $E(B-V)=0.84 \\\\pm 0.08$ .\",\n \"MWC 148: For this star, Friedemann (1992) estimated E(B-V)=0.85 from the 217 nm band.\",\n \"We measure $0.14 < W(KI \\\\lambda 7699) < 0.20$ Å, $0.13 < W(DIB \\\\lambda 6613) < 0.18$ Å, $0.28 < W(DIB \\\\lambda 5780) < 0.34$ Å, $0.13 < W(DIB \\\\lambda 5797) < 0.15$ Å, which corresponds to a slightly lower value $E(B-V)=0.77 \\\\pm 0.06$ .\",\n \"MWC 656: For this star, Williams et al.\",\n \"(2010) gave a low value E(B-V)=0.02.\",\n \"Casares et al.\",\n \"(2014) estimated E(B-V)=0.24.\",\n \"We measure $0.04 < W(DIB \\\\lambda 6613) < 0.06$ Å, $0.09 \\\\le W(DIB \\\\lambda 5780) \\\\le 0.10$ Å, $0.03 < W(DIB\\\\lambda 5797) < 0.05$ Å, and estimate $E(B-V)=0.25 \\\\pm 0.02$ .\",\n \"Figure: Rotational period of the mass donor (P rot P_{rot}) vs. P orb P_{orb}.Filled (blue) circles indicate high-mass X-ray binaries with giant/supergiant mass donor, open circles indicate Be/X-ray binaries, and red plusses indicate γ\\\\gamma -ray binaries.\"\n ],\n [\n \"Rotational period of the mass donors\",\n \"In close binary systems the rotation of the companions of compact objects is accelerated by mass transfer and tidal forces (e.g.\",\n \"Ablimit & Lü 2012).\",\n \"Adopting the parameters given in Sect.\",\n \", we estimate the rotational periods of the mass donors $P_{rot} \\\\approx 0.92$ d (for LSI+61$^0$ 303), $P_{rot} \\\\approx 0.91$ d (MWC 148), and $P_{rot} \\\\approx 0.86$ d (MWC 656).\",\n \"For the $\\\\gamma $ -ray binary LS 5039, the mass donor is a O6.5V((f)) star with $R_{1} = 9.3 \\\\pm 0.6 $ $R_\\\\odot $ , inclination $i \\\\approx 24.9^0$ , $v \\\\: \\\\sin \\\\: i$$= 113 \\\\pm 8$ km s$^{-1}$ (Casares et al.\",\n \"2005).\",\n \"We calculate $P_{rot} = 1.764$ d. Because of the short orbital period $P_{orb} = 3.91$ d, the rotation of the mass donor could be pseudo-synchronized with the orbital motion (Casares et al.\",\n \"2005).\",\n \"Radio observations of the pulsar in PSR B1259-63/LS 2883 allow the orbital parameters to be precisely established: $P_{orb} = 1236.72$ d and eccentricity of e = 0.87 (Wang et al.\",\n \"2004; Shannon et al.\",\n \"2014).\",\n \"For the primary, Negueruela et al.\",\n \"(2011) estimated $R_1 = 9.0 \\\\pm 1.5$ $R_\\\\odot $ , $v \\\\: \\\\sin \\\\: i$ $ = 260 \\\\pm 15$ km s$^{-1}$ , and $i_{orb} \\\\approx 23^0$ , which give $P_{rot} = 0.689$ d. In Fig.\",\n \"REF we plot $P_{rot}$ versus $P_{orb}$ for a number of high-mass X-ray binaries (see also Stoyanov & Zamanov 2009).\",\n \"High-mass X-ray binaries with a giant/supergiant component, Be/X-ray binaries, and Be/$\\\\gamma $ -ray binaries are plotted with different symbols in this figure.\",\n \"The solid line represents synchronization ($P_{rot} = P_{orb}$ ).\",\n \"Among the five $\\\\gamma $ -ray binaries with known rotational velocity of the mass donor, the rotation of the mass donor is close to synchronization with the orbital motion in only one (LS 5039).\",\n \"The four Be/$\\\\gamma $ -ray binaries are not close to the line of synchronization.\",\n \"They occupy the same region as the Be/X-ray binaries; in other words regarding the rotation of the mass donor the Be/$\\\\gamma $ -ray binaries are similar to the Be/X-ray binaries.\",\n \"In these, the tidal force of the compact object acts to decelerate the rotation of the mass donor.\",\n \"The spin-down of the Be stars due to angular momentum transport from star to disc (Porter 1998) is another source of deceleration.\",\n \"In the well-known Be star $\\\\gamma $ Cas, Robinson & Smith (2000) found that the X-ray flux varied with a period P = 1.1 d, which they interpreted as the rotational period of the mass donor.\",\n \"A similar period is detected in photometric observations (Harmanec et al.\",\n \"2000; Henry & Smith 2012).\",\n \"This periodicity is probably due to the interaction between magnetic field of the Be star and its circumstellar disc or the presence of some physical feature, such as a spot or cloud, co-rotating with the star.\",\n \"The optical emission lines of MWC 148 are practically identical to those of $\\\\gamma $ Cas.\",\n \"All detected lines in the observed spectral range (Balmer lines, HeI lines and FeII lines) have similar equivalent widths, intensities, profiles, and even a so-called wine-bottle structure noticeable in the $H\\\\alpha $ line (see Fig.\",\n \"1 in Zamanov et al.\",\n \"2016).\",\n \"Bearing in mind the above estimations and the curious similarities between: (i) the mean 20-60 keV X-ray luminosity of $\\\\gamma $ Cas and LSI+61$^0$ 303 (Shrader et al.\",\n \"2015) and (ii) the optical emission lines of $\\\\gamma $ Cas and MWC 148, we consider that periodicity $\\\\sim \\\\!1$ day could be detectable in X-ray/optical bands in the Be/$\\\\gamma $ -ray binaries and could provide a direct measurement of the rotational period of the mass donor.\"\n ],\n [\n \"Discussion\",\n \"The three Be/$\\\\gamma $ -ray binaries discussed here have non-zero eccentricities and misalignment between the spin axis of the star and the spin axis of the binary orbit could be possible (Martin et al.\",\n \"2009).\",\n \"The inclination of the primary star in LSI+61$^0$ 303 to the line of sight is probably $ i_{Be} \\\\sim 70^0$ (Zamanov et al.\",\n \"2013).\",\n \"Aragona et al.\",\n \"(2014) derived $a_1 \\\\sin i_{orb} = 8.64 \\\\pm 0.52$ .\",\n \"Assuming $M_1 = 15$$M_\\\\odot $ , $M_2 = 1.4$ $M_\\\\odot $ , we estimate $ i_{orb} \\\\sim 67^0 - 73^0$ .\",\n \"It appears that there is no significant deviation of the orbital plane from the equatorial plane of the Be star.\",\n \"The emission lines of MWC 148 are very similar to those of $\\\\gamma $ Cas.\",\n \"The emission lines are most sensitive to the footpoint density and inclination angle (Hummel 1994).\",\n \"It means that in MWC 148 the Be star inclination should be similar to that of $\\\\gamma $ Cas, for which the inclination is in the range $40^0 - 50^0$ (Clarke 1990, Quirrenbach et al.\",\n \"1997).\",\n \"For MWC 148, Casares et al.\",\n \"(2012) estimated $a_1 \\\\sin i_{orb} = 77.6 \\\\pm 25.9$ , which for $M_1 = 15$ $M_\\\\odot $ and a 4 $M_\\\\odot $ black hole gives $ 45^0 \\\\lesssim i_{orb} \\\\lesssim 65^0$ .\",\n \"For MWC 656, Casares et al.\",\n \"(2014) give $M_1 \\\\sin ^3 i_{orb} = 5.83 \\\\pm 0.70$ .\",\n \"Bearing in mind the range of 8 $M_\\\\odot $ $\\\\le M_1 \\\\le 10$ $M_\\\\odot $ , this gives $53^0 \\\\lesssim i_{orb} \\\\lesssim 59^0$ .\",\n \"Inclination of the Be star can be evaluated from the full width at zero intensity of the FeII lines $FWZI/2 \\\\sin i = (G M_1/ R_1)^{1/2}$ .\",\n \"From FWZI of FeII lines (Casares et al.\",\n \"2012) and using $R_1= 9.5 - 10.0$ $R_\\\\odot $ we estimate $i_{Be} \\\\approx 53 - 61^0$ .\",\n \"It appears that both planes are almost complanar.\",\n \"There are no signs of considerable deviation between the two planes.\",\n \"The opening half-angle of the Be stars' circumstellar disc are $\\\\sim \\\\!\",\n \"10^0$ (Tycner et al.\",\n \"2006; Cyr et al.\",\n \"2015) and it means that the compact object is practically orbiting in the plane of the circumstellar disc.\",\n \"The comparison between the orbit and circumstellar disc size (see Fig.\",\n \"REF and Sect.\",\n \"REF ) shows that in these three objects we have three different situations: In LSI+61$^0$ 303 the neutron star passes through the outer parts of the circumstellar disc at periastron, but it does not enter deeply in the disc; In MWC 148 the compact object goes into the innermost parts of the disc (passes through the innermost parts and penetrates deeply in the disc) during the periastron passage; In MWC 656 the black hole is constantly accreting from the outer edge of the circumstellar disc.\",\n \"In MWC 656, this means that the compact object (black hole) is at the disc border at all times and as a consequence it will have a higher and stable mass accretion rate along the entire orbit.\",\n \"It seems to be a very clear case of disc truncation in which the circumstellar disc is cut almost exactly at the black hole orbit.\",\n \"Because the $H\\\\alpha $ peaks are connected with the outermost parts of the disc, the above three items probably explain the observational findings: 1.\",\n \"In LSI+61$^0$ 303 the $H\\\\alpha $ emission line has a two-peak profile at all times (in all our spectra in the period 1987 - 2015),because the circumstellar disc size is relatively small and the neutron star passes only through the outermost parts of the circumstellar disc at periastron; 2.\",\n \"In MWC 148, the big jumps in the $H\\\\alpha $ parameters, $W_\\\\alpha $ , full width at half maximum and radial velocity (see Fig.\",\n \"4 of Casares 2012) occur because the compact object enters (reaches) the inner parts of the disc during the periastron passage.\",\n \"3.\",\n \"In MWC 656 the double-peak profile is not often visible because the black hole is at the outer edge at all times and makes perturbations exactly in the places where the $H\\\\alpha $ peaks are formed.\",\n \"It is worth noting that when the compact object causes large-scale perturbations, distorted profiles, such as those observed in 1A 535+262 (Moritani et al.\",\n \"2011, 2013), will appear.\",\n \"If only a small portion of the outer disc is perturbed then it will appear in the central part of the emission line profile (e.g.\",\n \"in between the peaks or even filling the central dip) because the outer parts of the disc produce the central part of the $H\\\\alpha $ emission line profile.\",\n \"Similar additional emission is already detected in the $H\\\\alpha $ spectra of LSI+61$^0$ 303 (Paredes et al.\",\n \"1990; Liu et al.\",\n \"2000; Zamanov & Marti 2000).\",\n \"Gamma-ray emission has been repeatedly observed to be periodic in the system LSI+61303 (Albert et al.\",\n \"2009; Saha et al.\",\n \"2016) and also very likely in MWC148, where its periodic X-ray flares are highly correlated with TeV emission (Aliu et al.\",\n \"2014).\",\n \"A similar situation occurs in the case of LS 5039 (Aharonian et al.\",\n \"2006), 1FGL J1018.6-20135856 (H. E. S. S. Collaboration et al.\",\n \"2015), and PSR B1259-63 (H.E.S.S.\",\n \"Collaboration et al.\",\n \"2013), which has the longest orbital period observed in a $\\\\gamma $ -ray binary (about 3 yr).\",\n \"This is likely related to the very different physical conditions sampled by the compact companion as it revolves around the primary star in an eccentric orbit.\",\n \"Non-thermal emission (e.g.\",\n \"Dubus 2013) is produced from particles accelerated at the shock between the wind of the pulsar and matter flowing out from the primary star (the polar wind as well as the disc).\",\n \"The X-ray and $\\\\gamma $ -ray light curve of MWC 148 shows two maxima at orbital phases 0.35 and 0.75 and minimum at apastron passage (Acciari et al.\",\n \"2009; Aliu et al.\",\n \"2014).\",\n \"X-ray and $\\\\gamma $ -ray fluxes are correlated as mentioned above, in agreement with leptonic emission models, where relativistic electrons lose energy by synchrotron emission and inverse Compton emission (Maier & for the VERITAS Collaboration 2015).\",\n \"The highly eccentric orbital geometry sketched in the central panel of Fig.\",\n \"REF is also in good agreement with a periodic flaring system.\",\n \"In the case of MWC 656, $\\\\gamma $ -rays have been detected occasionally (Williams et al.\",\n \"2010), but so far never reaching the TeV energy domain.\",\n \"The main differences from previous sources are the fact that the compact object has been shown to be a black hole and its orbit is only moderately eccentric (Casares et al.\",\n \"2014).\",\n \"Based on our spectroscopic observations, the size of the excretion disc is such that the black hole is accreting matter only from its lower density outer edges.\",\n \"As mentioned before, this implies that the accretion rate is stable but at the same time low.\",\n \"Indeed, the quiescent X-ray emission level of the system is as weak as $\\\\sim 10^{-8}$ Eddington luminosity (Munar-Adrover et al.\",\n \"2014).\",\n \"Therefore, episodic $\\\\gamma $ -ray flares such as those detected by AGILE, likely require some enhancement of mass loss from the primary Be star or clumps in its circumstellar disc.\",\n \"We speculate here that the physical mechanism responsible for $\\\\gamma $ -ray emission in the MWC 656 black hole context could be related to the alternative microquasar-jet scenario also proposed for $\\\\gamma $ -ray binaries (see e.g.\",\n \"Romero et al.\",\n \"2007).\",\n \"The fact that MWC 656 seems to adhere to the low-luminosity end of the X-ray/radio correlation for hard state compact jets also points in this direction (Dzib et al.\",\n \"2015).\"\n ],\n [\n \"Conclusions\",\n \"From the spectroscopic observations of the three Be/$\\\\gamma $ -ray binaries we deduce that in LSI+61$^0$ 303 the neutron star crosses the outer parts of the circumstellar disc at periastron, in MWC 148 the compact object passes deeply through the disc during the periastron passage, and in MWC 656 the black hole is accreting from the outer parts of the circumstellar disc during the entire orbital cycle.\",\n \"The histograms in all three stars show that the disc size clusters at specific levels, indicating the circumstellar disc is truncated by the orbiting compact object.\",\n \"We estimate the interstellar extinction towards LSI+61$^0$ 303, MWC 148, and MWC 656.\",\n \"The rotation of the mass donors is similar to that of the Be/X-ray binaries.\",\n \"We suggest that the three stars deserve to be searched for a periodicity of about $1.0$ day.\",\n \"The authors are grateful to an anonymous referee for valuable comments and suggestions.\",\n \"This work was partially supported by grant AYA2013-47447-C3-3-P from the Spanish Ministerio de Economía y Competitividad (MINECO), and by the Consejería de Economía, Innovación, Ciencia y Empleo of Junta de Andalucía as research group FQM-322, as well as FEDER funds.\"\n ]\n]"},"id":{"kind":"string","value":"1605.05811"}}},{"rowIdx":1866689,"cells":{"text":{"kind":"list like","value":[["Energy fluctuations of finite free-electron Fermi gas"],["Abstract We discuss the energy distribution of free-electron Fermi-gas, a problem with a textbook solution of Gaussian energy fluctuations in the limit of a large system.","We find that for a small system, characterized solely by its heat capacity $C$, the distribution can be solved analytically, and it is both skewed and it vanishes at low energies, exhibiting a sharp drop to zero at the energy corresponding to the filled Fermi sea.","The results are relevant from the experimental point of view, since the predicted non-Gaussian effects become pronounced when $C/k_B \\lesssim 10^3$ ($k_B$ is the Boltzmann constant), a regime that can be easily achieved for instance in mesoscopic metallic conductors at sub-kelvin temperatures."],["Introduction","Physical quantities of an equilibrium macroscopic system are well characterized by their average values.","However, random deviations from the average values - fluctuations - are very important since they contain important information on the system.","Under most circumstances the distribution of small fluctuations is Gaussian [1], ch. 7.","However, this is not the case for small devices, which are currently studied intensively [2], [3], see [4], [5], [6], [7] for a review.","Energy and temperature fluctuations in a single electron box [9] were considered in [8]; several works experimental and theoretical, e.g., [10], [11], [12], were devoted to temperature fluctuations.","In the present work we will derive the distribution function for a finite Fermi gas and show that its shape is determined by only one dimensionless parameter - the total heat capacity $C$ divided by the Boltzmann constant, $k_{\\text{B}}$ .","At $C/k_{\\text{B}}\\gg 1$ we recover the well-known Gaussian distribution, while at finite values of this parameter significant deviations are expected.","We will analytically derive the distribution of the energies of a finite sample of the Fermi gas kept at a given temperature and analyze its properties including moments and skewness absent in the thermodynamic limit.","Since the heat capacity of a metallic conductor of sub-micron dimensions at standard sub-kelvin experimental temperatures is of the order of $(10^2-10^3)k_{\\text{B}}$ [4], [13], our results have potential impact on sensitive bolometers [14], [4], [10].","In future, we plan to use the obtained distribution for analysis of heat exchange between a quantum device and a mesoscopic metallic calorimeter.","The paper is organized as follows.","We derive general expression for the energy distribution in Sec.","REF , which will be analyzed in Sec.","REF .","Moments of the distribution are considered in Sec.","REF .","It is convenient to calculate the Fourier transform, $F(\\lambda )$ , of the energy distribution function.","It can be expressed in the form $ F(\\lambda ) =\\left\\langle e^{i\\lambda E} \\right\\rangle =\\frac{\\sum _m e^{(i\\lambda -\\beta ) E_m} }{\\sum _m e^{ -\\beta E_m}}=\\frac{Z(\\beta - i \\lambda )}{Z(\\beta )}\\, .$ Here $Z(\\beta ) \\equiv \\sum _m e^{ -\\beta E_m}$ is the partition function, $\\beta \\equiv (k_{\\text{B}}T)^{-1}$ where $T$ is temperature, while $ E_m =\\sum _\\mathbf {k}(\\epsilon _{\\mathbf {k},m} - \\mu ) \\, n_{\\mathbf {k},m} \\, .$ A micro-state is characterized by the set of quantum numbers $\\lbrace \\mathbf {k},m\\rbrace $ where $\\mathbf {k}$ is the quasi-momentum.","In the following, concentrating on mesoscopic electronic systems, we will consider a gas of elementary particles characterized only by the quantum number $\\mathbf {k}$ .","According to the definitions (REF )-(REF ), we calculate the characteristic function of quasiparticle energy.","The energy distribution function, $P(E)$ , can be calculated as the inverse Fourier transform of $F(\\lambda )$ as $P(E)=\\int _{-\\infty }^\\infty \\frac{d\\lambda }{2\\pi } \\, e^{-i\\lambda E} F(\\lambda )\\, .$ For the Fermi gas [1], $\\ln Z(\\beta )&=& \\sum _\\mathbf {k}\\ln \\left( 1+e^{-\\beta (\\epsilon _\\mathbf {k}- \\mu )}\\right) \\\\&=&\\mathcal {V} \\int _0^\\infty d\\epsilon \\, N(\\epsilon ) \\, \\ln \\left( 1+e^{-\\beta (\\epsilon - \\mu )}\\right) \\, .$ Here $N(\\epsilon )$ is the density of states.","In the following we put $N(\\epsilon ) = \\text{const} \\equiv N_0$ .","This assumption is valid for the two-dimensional gas, while in the three-dimensional case $N(\\epsilon ) \\propto \\sqrt{\\epsilon }$ .","One can show that in the limiting case $\\beta \\mu \\gg 1$ the results differ only by renormalization of numerical constants in the expressions for the average energy and heat capacity.","For realistic experimental conditions $\\beta \\mu = T_F/T \\sim 10^6$ for a metal with Fermi temperature $T_F \\sim 10^5$ K at $T=0.1$ K. For $N(\\epsilon )= N_0$ we obtain $\\ln Z(\\beta )&=&\\frac{\\mathcal {V}N_0}{\\beta } \\int _{-\\beta \\mu }^\\infty d\\xi \\ln \\left(1+e^{- \\xi } \\right) \\nonumber \\\\&=&-\\frac{\\mathcal {V}N_0}{\\beta }\\, \\mathrm {dilog}\\, \\left(1+e^{\\beta \\mu } \\right)\\, .$ Here $\\mathrm {dilog}\\, (x) = \\int _1^x \\frac{\\ln (t)}{1-t}\\, dt$ is the dilogarithm function.","At $\\beta \\mu \\gg 1$ we expand $ - \\mathrm {dilog}\\, \\left(1+e^{\\beta \\mu } \\right) \\approx \\frac{1}{2} (\\beta \\mu )^2 + \\frac{\\pi ^2}{6}-e^{-\\beta \\mu } + \\ldots \\, .$ Then $ \\ln Z(\\beta ) \\approx \\mathcal {V}N_0 \\left( \\frac{1}{2}\\beta \\mu ^2 + \\frac{\\pi ^2}{6}\\frac{1}{\\beta }\\right)\\, .$ Note that we assume that the chemical potential is fixed rather than determined by the normalization of the single-particle distribution to the total number of particles.","This assumption is appropriate for an electronic system with contacts.","In the case when $N(\\epsilon ) \\propto \\sqrt{\\epsilon }$ we can put $N(\\epsilon ) = N_0\\sqrt{\\epsilon /\\mu }$ to obtain instead of Eq.","(REF ) $\\ln Z(\\beta ) \\approx \\mathcal {V}N_0 \\left( \\frac{2}{5}\\beta \\mu ^2 + \\frac{\\pi ^2}{4}\\frac{1}{\\beta }\\right)\\, .$ Therefore, different forms of the dependence of the density of states upon the energy indeed result only in numerical constants defining the average energy and heat capacity.","The difference in the numerical factors is compatible with the well known Sommerfeld expansion, see, e.g., [15].","Using the expansion (REF ) we obtain $\\ln F(\\lambda ) &=&\\ln \\frac{Z(\\beta - i\\lambda )}{Z(\\beta ) } \\nonumber \\\\&=&\\frac{ \\mathcal {V}N_0}{2} \\left(-i \\lambda \\mu ^2+ \\frac{\\pi ^2}{3} \\frac{i\\lambda }{\\beta (\\beta -i\\lambda )}\\right).$ The first item is responsible for the energy shift of the energy by the Fermi energy $E_0 =\\mathcal {V} N_0\\mu ^2/2$ .","In our approximation the specific heat has the standard free Fermi-gas expression (for a given chemical potential) $C=k_{\\text{B}}\\frac{\\pi ^2}{3\\beta }N_0\\mathcal {V}\\, .$ Then the quantity in the parentheses in Eq.","(REF ) can be rewritten as $ i \\lambda E_0 + \\frac{C}{2k_{\\text{B}}} \\frac{i\\lambda }{\\beta - i\\lambda }\\, .$ We rewrite $\\frac{C}{2k_{\\text{B}}} \\frac{i\\lambda }{\\beta -i\\lambda }&=&i\\lambda \\frac{C}{2k_{\\text{B}}\\beta }\\left(\\frac{i\\lambda }{\\beta -i\\lambda } +1 \\right) - i\\lambda \\frac{C}{2 k_{\\text{B}}\\beta }\\\\&\\equiv & - i\\lambda E_\\beta +\\frac{C}{2k_{\\text{B}}\\beta } \\frac{(i\\lambda )^2}{\\beta - i \\lambda }.$ Here $E_\\beta = C/2 k_{\\text{B}}\\beta $ .","Denoting $\\delta E =E-E_0- E_\\beta $ we obtain in the exponent $-i\\lambda \\, \\delta E -\\frac{C}{2 k_{\\text{B}}}\\frac{\\lambda ^2}{\\beta (\\beta - i \\lambda )}\\, .$ Let us measure the energy deviation $\\delta E$ in units of $\\beta ^{-1}\\sqrt{C/k_{\\text{B}}}$ .","Then we put $\\lambda \\equiv \\frac{\\beta }{\\sqrt{C/k_{\\text{B}}}} \\Lambda , \\ \\delta E \\equiv \\frac{ \\sqrt{C/k_{\\text{B}}}}{\\beta } u, \\ c \\equiv \\frac{C}{k_{\\text{B}}}$ to obtain for the second term $-\\frac{1}{2}\\frac{\\Lambda ^2}{1-i\\Lambda /\\sqrt{c}}= -\\frac{1}{2}\\frac{\\Lambda ^2(1+i\\Lambda /\\sqrt{c})}{1+\\Lambda ^2 /c}\\, .$ Therefore, we arrive at the expression for the distribution of dimensionless energies $u$ : $ P(u) = \\int _{-\\infty }^\\infty \\frac{d\\Lambda }{2\\pi }\\exp \\left[ i\\Lambda \\left(\\frac{1}{2}\\frac{i\\Lambda }{1-i\\Lambda /\\sqrt{c}}-u \\right)\\right]\\, .$ This is our final expression, which we have to analyze for different values of dimensionless heat capacity, $c$ ."],["Energy distributions for different dimensionless heat capacity","We compute the integral in Eq.","(REF ) by deforming the contour in the complex plane.","The integrand is analytic in the full complex plane except for (an essential) singularity at $\\Lambda =-i\\sqrt{c}$ .","For large $|\\Lambda |$ the argument of the exponential behaves as $-\\frac{1}{2}\\frac{\\Lambda ^2}{1-i\\Lambda /\\sqrt{c}}-i\\Lambda u =-\\frac{c}{2}-i\\Lambda \\left(\\frac{\\sqrt{c}}{2}+u\\right)+{\\mathcal {O}}(1).$ For thermodynamics it suffices to consider energies above the Fermi energy $E>E_0$ which means $u+\\sqrt{c}/2 >0$ .","We may then deform the contour to the lower half plane $\\mathop \\mathrm {Im}\\Lambda <0$ .","We get $P(u)=P_{0}(u)+P_\\infty (u)$ where $P_\\infty (u)$ is contribution from the semi circle at infinity in the lower half plane and $P_{0}(u)$ from a circle around $\\Lambda =-i\\sqrt{c}$ .","Writing $\\Lambda =\\rho e^{i\\phi }$ , $\\pi \\le \\phi \\le 2\\pi $ we obtain $P_\\infty (u)$ as the limit of $e^{-\\frac{_1}{^2}c}\\frac{i\\rho }{2\\pi }\\int _{2\\pi }^\\pi d\\phi e^{i(\\phi -\\rho b e^{i\\phi })}=\\frac{\\sin (b\\rho )}{\\pi b}$ as $\\rho \\rightarrow \\infty $ i.e.","$P_\\infty (u)$ where $ b=\\sqrt{c}/2+u$ .","Since we assume $b\\ne 0$ this limit vanishes.","To compute $P_{0}(u)$ change variables to $z=\\frac{1}{c}(1-i\\frac{\\Lambda }{\\sqrt{c}})$ to get $P_{-i}(u) =c^{\\frac{3}{2}}e^{-(2c+\\sqrt{c}u)}\\oint e^{\\frac{1}{z}}e^{xz}\\frac{dz}{2\\pi i}$ where $x=c(c+\\sqrt{c}u)$ and the integral is around the origin.","Then $&&\\oint e^{\\frac{1}{z}}e^{x z}\\frac{dz}{2\\pi i}=\\sum _{n=0}^\\infty \\frac{1}{n!}","\\oint z^{-n}e^{x z}\\frac{dz}{2\\pi i}\\nonumber \\\\&=&\\sum _{m=0}^\\infty \\frac{1}{(m+1)!}\\frac{1}{m!","}x^{m}=x^{-\\frac{_1}{^2}}I_1(2x^\\frac{_1}{^2})\\nonumber $ where $I_1$ is the modified Bessel function.","As a result, $P (u)& =&\\frac{c^{3/4}}{\\sqrt{\\sqrt{c}+2u}} I_1 \\left(c^{3/4}\\sqrt{\\sqrt{c}+2u}\\right)e^{-u\\sqrt{c} -c}\\nonumber $ Graphs for $P(u)$ for different heat capacities, $c$ , are shown in Fig.","REF .","One observes that at large $c$ the distribution is essentially Gaussian, while at small $c$ it becomes very asymmetric.","Figure: (Color online) Distribution P(u)P(u), where u=βδE/cu= \\beta \\delta E/\\sqrt{c}, where δE\\delta Eis the deviation of the energy of the Fermi-gas from its mean, for a few different values of c=C/k B c = C/k_B (solid lines).Dashed line is the Gaussian distribution.Let us crudely estimate the dimensionless heat capacity, $c$ .","The estimate reads as $c \\sim n_e \\mathcal {V} (k_{\\text{B}}T/\\mu )$ where $n_e$ is the electron density.","For a typical metal $n_e \\approx 10^{23} \\ \\text{cm}^{-3}$ .","Assuming the sizes of a mesoscopic system (in $\\mu \\text{m}$ ) as $(0.3-1)\\times (0.05-0.1)\\times (0.01-0.03)$ $(\\mu \\text{m})^3$ , we get $\\mathcal {V} = (1 .5\\cdot 10^{-16} - 3\\cdot 10^{-15}) \\ \\text{cm}^3\\, .","$ and the total number of electrons is $\\mathcal {N} \\equiv n_e\\mathcal {V} =1.5\\cdot 10^7 - 3\\cdot 10^8\\, .$ Putting $T_{\\text{F}} \\approx 5\\cdot 10^4~\\text{K}$ and $T=(0.05 - 0.1)~\\text{K}$ we get $k_{\\text{B}}T/\\mu =T/T_{\\text{F}}=(1-2)\\cdot 10^{-6}.$ Consequently, the range of the quantity $c$ is $15 -600$ .","As we have seen, at these values of $c$ the energy distribution is clearly non-Gaussian."],["Moments of the energy distribution","The moments of the energy distribution can be readily calculated either from the Eq.","(REF ), or explicitly from the Hamiltonian$ \\mathcal {H}=\\sum _\\mathbf {k}\\epsilon _\\mathbf {k}a^\\dag _\\mathbf {k}a_\\mathbf {k}$ .","In particular, $\\langle E \\rangle =\\sum _\\mathbf {k}\\epsilon _\\mathbf {k}\\langle a^\\dag _\\mathbf {k}a_\\mathbf {k} \\rangle =\\sum _\\mathbf {k}\\epsilon _\\mathbf {k}f(\\epsilon _\\mathbf {k})$ where $f(\\epsilon )=\\left[e^{\\beta (\\epsilon - \\mu )}+1 \\right]^{-1}$ is the Fermi function.","The second moment reads $\\langle E^2 \\rangle =\\sum _\\mathbf {k}\\epsilon _\\mathbf {k}\\epsilon _\\mathbf {l}\\langle a^\\dag _\\mathbf {k}a_\\mathbf {k}a^\\dag _\\mathbf {l}a_\\mathbf {l} \\rangle =\\langle E \\rangle ^2 + \\sum _\\mathbf {k}\\epsilon _\\mathbf {k}^2 f( \\epsilon _\\mathbf {k})[1-f( \\epsilon _\\mathbf {k})].$ Here we have decomposed the product of four Fermi operators according to the Wick theorem.","We observe that only the vicinity of the Fermi level is important for the difference $\\langle \\delta E^2 \\rangle \\equiv \\left\\langle \\left(E-\\langle E \\rangle \\right)^2\\right\\rangle =\\langle E^2 \\rangle -\\langle E \\rangle ^2\\, .","$ Therefore, while calculating $\\langle \\delta E^2 \\rangle $ we can assume constant density of states and $\\beta \\mu \\gg 1$ .","In this way we obtain $\\langle \\delta E^2 \\rangle =k_{\\text{B}}T^2 C\\, .$ In a similar way, we calculate $\\langle \\delta E^3 \\rangle =2\\sum _\\mathbf {k}\\epsilon _\\mathbf {k}^3 f( \\epsilon _\\mathbf {k})[1-f( \\epsilon _\\mathbf {k})]^2=3k_{\\text{B}}^2T^3C\\, .$ The skewness of the distribution is then $\\gamma =\\langle \\delta E^3 \\rangle /\\langle \\delta E^2 \\rangle ^{3/2} =3/\\sqrt{c}\\, .$ Obviously, the skewness vanishes as $c \\rightarrow \\infty $ which is the thermodynamic limit.","For a Gaussian distribution, the relationship between the 2nd and the 4th moments is $\\langle \\delta E^4 \\rangle =3\\langle \\delta E^2 \\rangle ^2$ .","In the general case, an additional contribution appears, such that $\\langle \\delta E^4 \\rangle =3\\langle \\delta E^2 \\rangle ^2\\left[1+\\frac{4}{\\sqrt{c}} \\right]\\, .$ The 5th moment, absent in the Gaussian approximation, is $\\langle \\delta E^5 \\rangle = 60k_{\\text{B}}^4T^5 C\\, .$ The results obtained by numerical integration of Eq.","(REF ) agree with the analytic expressions given above.","In summary, we have shown that the energy distribution of a free Fermi-gas with small heat capacity is non-Gaussian, with a sharp cut-off at low energies.","This is a natural consequence of the minimal energy of the filled Fermi sea.","We find that the heat capacities demonstrating strong non-Gaussian features can be achieved in standard metallic nanodevices at sub-kelvin temperatures.","We thank Ivan Khaymovich and Kay Schwieger for discussions.","YMG thanks Aalto University for hospitality during the preparation of this manuscript.","The work has been supported by the Academy of Finland (contracts no.","272218, 284594 and 271983)."]],"string":"[\n [\n \"Energy fluctuations of finite free-electron Fermi gas\"\n ],\n [\n \"Abstract We discuss the energy distribution of free-electron Fermi-gas, a problem with a textbook solution of Gaussian energy fluctuations in the limit of a large system.\",\n \"We find that for a small system, characterized solely by its heat capacity $C$, the distribution can be solved analytically, and it is both skewed and it vanishes at low energies, exhibiting a sharp drop to zero at the energy corresponding to the filled Fermi sea.\",\n \"The results are relevant from the experimental point of view, since the predicted non-Gaussian effects become pronounced when $C/k_B \\\\lesssim 10^3$ ($k_B$ is the Boltzmann constant), a regime that can be easily achieved for instance in mesoscopic metallic conductors at sub-kelvin temperatures.\"\n ],\n [\n \"Introduction\",\n \"Physical quantities of an equilibrium macroscopic system are well characterized by their average values.\",\n \"However, random deviations from the average values - fluctuations - are very important since they contain important information on the system.\",\n \"Under most circumstances the distribution of small fluctuations is Gaussian [1], ch. 7.\",\n \"However, this is not the case for small devices, which are currently studied intensively [2], [3], see [4], [5], [6], [7] for a review.\",\n \"Energy and temperature fluctuations in a single electron box [9] were considered in [8]; several works experimental and theoretical, e.g., [10], [11], [12], were devoted to temperature fluctuations.\",\n \"In the present work we will derive the distribution function for a finite Fermi gas and show that its shape is determined by only one dimensionless parameter - the total heat capacity $C$ divided by the Boltzmann constant, $k_{\\\\text{B}}$ .\",\n \"At $C/k_{\\\\text{B}}\\\\gg 1$ we recover the well-known Gaussian distribution, while at finite values of this parameter significant deviations are expected.\",\n \"We will analytically derive the distribution of the energies of a finite sample of the Fermi gas kept at a given temperature and analyze its properties including moments and skewness absent in the thermodynamic limit.\",\n \"Since the heat capacity of a metallic conductor of sub-micron dimensions at standard sub-kelvin experimental temperatures is of the order of $(10^2-10^3)k_{\\\\text{B}}$ [4], [13], our results have potential impact on sensitive bolometers [14], [4], [10].\",\n \"In future, we plan to use the obtained distribution for analysis of heat exchange between a quantum device and a mesoscopic metallic calorimeter.\",\n \"The paper is organized as follows.\",\n \"We derive general expression for the energy distribution in Sec.\",\n \"REF , which will be analyzed in Sec.\",\n \"REF .\",\n \"Moments of the distribution are considered in Sec.\",\n \"REF .\",\n \"It is convenient to calculate the Fourier transform, $F(\\\\lambda )$ , of the energy distribution function.\",\n \"It can be expressed in the form $ F(\\\\lambda ) =\\\\left\\\\langle e^{i\\\\lambda E} \\\\right\\\\rangle =\\\\frac{\\\\sum _m e^{(i\\\\lambda -\\\\beta ) E_m} }{\\\\sum _m e^{ -\\\\beta E_m}}=\\\\frac{Z(\\\\beta - i \\\\lambda )}{Z(\\\\beta )}\\\\, .$ Here $Z(\\\\beta ) \\\\equiv \\\\sum _m e^{ -\\\\beta E_m}$ is the partition function, $\\\\beta \\\\equiv (k_{\\\\text{B}}T)^{-1}$ where $T$ is temperature, while $ E_m =\\\\sum _\\\\mathbf {k}(\\\\epsilon _{\\\\mathbf {k},m} - \\\\mu ) \\\\, n_{\\\\mathbf {k},m} \\\\, .$ A micro-state is characterized by the set of quantum numbers $\\\\lbrace \\\\mathbf {k},m\\\\rbrace $ where $\\\\mathbf {k}$ is the quasi-momentum.\",\n \"In the following, concentrating on mesoscopic electronic systems, we will consider a gas of elementary particles characterized only by the quantum number $\\\\mathbf {k}$ .\",\n \"According to the definitions (REF )-(REF ), we calculate the characteristic function of quasiparticle energy.\",\n \"The energy distribution function, $P(E)$ , can be calculated as the inverse Fourier transform of $F(\\\\lambda )$ as $P(E)=\\\\int _{-\\\\infty }^\\\\infty \\\\frac{d\\\\lambda }{2\\\\pi } \\\\, e^{-i\\\\lambda E} F(\\\\lambda )\\\\, .$ For the Fermi gas [1], $\\\\ln Z(\\\\beta )&=& \\\\sum _\\\\mathbf {k}\\\\ln \\\\left( 1+e^{-\\\\beta (\\\\epsilon _\\\\mathbf {k}- \\\\mu )}\\\\right) \\\\\\\\&=&\\\\mathcal {V} \\\\int _0^\\\\infty d\\\\epsilon \\\\, N(\\\\epsilon ) \\\\, \\\\ln \\\\left( 1+e^{-\\\\beta (\\\\epsilon - \\\\mu )}\\\\right) \\\\, .$ Here $N(\\\\epsilon )$ is the density of states.\",\n \"In the following we put $N(\\\\epsilon ) = \\\\text{const} \\\\equiv N_0$ .\",\n \"This assumption is valid for the two-dimensional gas, while in the three-dimensional case $N(\\\\epsilon ) \\\\propto \\\\sqrt{\\\\epsilon }$ .\",\n \"One can show that in the limiting case $\\\\beta \\\\mu \\\\gg 1$ the results differ only by renormalization of numerical constants in the expressions for the average energy and heat capacity.\",\n \"For realistic experimental conditions $\\\\beta \\\\mu = T_F/T \\\\sim 10^6$ for a metal with Fermi temperature $T_F \\\\sim 10^5$ K at $T=0.1$ K. For $N(\\\\epsilon )= N_0$ we obtain $\\\\ln Z(\\\\beta )&=&\\\\frac{\\\\mathcal {V}N_0}{\\\\beta } \\\\int _{-\\\\beta \\\\mu }^\\\\infty d\\\\xi \\\\ln \\\\left(1+e^{- \\\\xi } \\\\right) \\\\nonumber \\\\\\\\&=&-\\\\frac{\\\\mathcal {V}N_0}{\\\\beta }\\\\, \\\\mathrm {dilog}\\\\, \\\\left(1+e^{\\\\beta \\\\mu } \\\\right)\\\\, .$ Here $\\\\mathrm {dilog}\\\\, (x) = \\\\int _1^x \\\\frac{\\\\ln (t)}{1-t}\\\\, dt$ is the dilogarithm function.\",\n \"At $\\\\beta \\\\mu \\\\gg 1$ we expand $ - \\\\mathrm {dilog}\\\\, \\\\left(1+e^{\\\\beta \\\\mu } \\\\right) \\\\approx \\\\frac{1}{2} (\\\\beta \\\\mu )^2 + \\\\frac{\\\\pi ^2}{6}-e^{-\\\\beta \\\\mu } + \\\\ldots \\\\, .$ Then $ \\\\ln Z(\\\\beta ) \\\\approx \\\\mathcal {V}N_0 \\\\left( \\\\frac{1}{2}\\\\beta \\\\mu ^2 + \\\\frac{\\\\pi ^2}{6}\\\\frac{1}{\\\\beta }\\\\right)\\\\, .$ Note that we assume that the chemical potential is fixed rather than determined by the normalization of the single-particle distribution to the total number of particles.\",\n \"This assumption is appropriate for an electronic system with contacts.\",\n \"In the case when $N(\\\\epsilon ) \\\\propto \\\\sqrt{\\\\epsilon }$ we can put $N(\\\\epsilon ) = N_0\\\\sqrt{\\\\epsilon /\\\\mu }$ to obtain instead of Eq.\",\n \"(REF ) $\\\\ln Z(\\\\beta ) \\\\approx \\\\mathcal {V}N_0 \\\\left( \\\\frac{2}{5}\\\\beta \\\\mu ^2 + \\\\frac{\\\\pi ^2}{4}\\\\frac{1}{\\\\beta }\\\\right)\\\\, .$ Therefore, different forms of the dependence of the density of states upon the energy indeed result only in numerical constants defining the average energy and heat capacity.\",\n \"The difference in the numerical factors is compatible with the well known Sommerfeld expansion, see, e.g., [15].\",\n \"Using the expansion (REF ) we obtain $\\\\ln F(\\\\lambda ) &=&\\\\ln \\\\frac{Z(\\\\beta - i\\\\lambda )}{Z(\\\\beta ) } \\\\nonumber \\\\\\\\&=&\\\\frac{ \\\\mathcal {V}N_0}{2} \\\\left(-i \\\\lambda \\\\mu ^2+ \\\\frac{\\\\pi ^2}{3} \\\\frac{i\\\\lambda }{\\\\beta (\\\\beta -i\\\\lambda )}\\\\right).$ The first item is responsible for the energy shift of the energy by the Fermi energy $E_0 =\\\\mathcal {V} N_0\\\\mu ^2/2$ .\",\n \"In our approximation the specific heat has the standard free Fermi-gas expression (for a given chemical potential) $C=k_{\\\\text{B}}\\\\frac{\\\\pi ^2}{3\\\\beta }N_0\\\\mathcal {V}\\\\, .$ Then the quantity in the parentheses in Eq.\",\n \"(REF ) can be rewritten as $ i \\\\lambda E_0 + \\\\frac{C}{2k_{\\\\text{B}}} \\\\frac{i\\\\lambda }{\\\\beta - i\\\\lambda }\\\\, .$ We rewrite $\\\\frac{C}{2k_{\\\\text{B}}} \\\\frac{i\\\\lambda }{\\\\beta -i\\\\lambda }&=&i\\\\lambda \\\\frac{C}{2k_{\\\\text{B}}\\\\beta }\\\\left(\\\\frac{i\\\\lambda }{\\\\beta -i\\\\lambda } +1 \\\\right) - i\\\\lambda \\\\frac{C}{2 k_{\\\\text{B}}\\\\beta }\\\\\\\\&\\\\equiv & - i\\\\lambda E_\\\\beta +\\\\frac{C}{2k_{\\\\text{B}}\\\\beta } \\\\frac{(i\\\\lambda )^2}{\\\\beta - i \\\\lambda }.$ Here $E_\\\\beta = C/2 k_{\\\\text{B}}\\\\beta $ .\",\n \"Denoting $\\\\delta E =E-E_0- E_\\\\beta $ we obtain in the exponent $-i\\\\lambda \\\\, \\\\delta E -\\\\frac{C}{2 k_{\\\\text{B}}}\\\\frac{\\\\lambda ^2}{\\\\beta (\\\\beta - i \\\\lambda )}\\\\, .$ Let us measure the energy deviation $\\\\delta E$ in units of $\\\\beta ^{-1}\\\\sqrt{C/k_{\\\\text{B}}}$ .\",\n \"Then we put $\\\\lambda \\\\equiv \\\\frac{\\\\beta }{\\\\sqrt{C/k_{\\\\text{B}}}} \\\\Lambda , \\\\ \\\\delta E \\\\equiv \\\\frac{ \\\\sqrt{C/k_{\\\\text{B}}}}{\\\\beta } u, \\\\ c \\\\equiv \\\\frac{C}{k_{\\\\text{B}}}$ to obtain for the second term $-\\\\frac{1}{2}\\\\frac{\\\\Lambda ^2}{1-i\\\\Lambda /\\\\sqrt{c}}= -\\\\frac{1}{2}\\\\frac{\\\\Lambda ^2(1+i\\\\Lambda /\\\\sqrt{c})}{1+\\\\Lambda ^2 /c}\\\\, .$ Therefore, we arrive at the expression for the distribution of dimensionless energies $u$ : $ P(u) = \\\\int _{-\\\\infty }^\\\\infty \\\\frac{d\\\\Lambda }{2\\\\pi }\\\\exp \\\\left[ i\\\\Lambda \\\\left(\\\\frac{1}{2}\\\\frac{i\\\\Lambda }{1-i\\\\Lambda /\\\\sqrt{c}}-u \\\\right)\\\\right]\\\\, .$ This is our final expression, which we have to analyze for different values of dimensionless heat capacity, $c$ .\"\n ],\n [\n \"Energy distributions for different dimensionless heat capacity\",\n \"We compute the integral in Eq.\",\n \"(REF ) by deforming the contour in the complex plane.\",\n \"The integrand is analytic in the full complex plane except for (an essential) singularity at $\\\\Lambda =-i\\\\sqrt{c}$ .\",\n \"For large $|\\\\Lambda |$ the argument of the exponential behaves as $-\\\\frac{1}{2}\\\\frac{\\\\Lambda ^2}{1-i\\\\Lambda /\\\\sqrt{c}}-i\\\\Lambda u =-\\\\frac{c}{2}-i\\\\Lambda \\\\left(\\\\frac{\\\\sqrt{c}}{2}+u\\\\right)+{\\\\mathcal {O}}(1).$ For thermodynamics it suffices to consider energies above the Fermi energy $E>E_0$ which means $u+\\\\sqrt{c}/2 >0$ .\",\n \"We may then deform the contour to the lower half plane $\\\\mathop \\\\mathrm {Im}\\\\Lambda <0$ .\",\n \"We get $P(u)=P_{0}(u)+P_\\\\infty (u)$ where $P_\\\\infty (u)$ is contribution from the semi circle at infinity in the lower half plane and $P_{0}(u)$ from a circle around $\\\\Lambda =-i\\\\sqrt{c}$ .\",\n \"Writing $\\\\Lambda =\\\\rho e^{i\\\\phi }$ , $\\\\pi \\\\le \\\\phi \\\\le 2\\\\pi $ we obtain $P_\\\\infty (u)$ as the limit of $e^{-\\\\frac{_1}{^2}c}\\\\frac{i\\\\rho }{2\\\\pi }\\\\int _{2\\\\pi }^\\\\pi d\\\\phi e^{i(\\\\phi -\\\\rho b e^{i\\\\phi })}=\\\\frac{\\\\sin (b\\\\rho )}{\\\\pi b}$ as $\\\\rho \\\\rightarrow \\\\infty $ i.e.\",\n \"$P_\\\\infty (u)$ where $ b=\\\\sqrt{c}/2+u$ .\",\n \"Since we assume $b\\\\ne 0$ this limit vanishes.\",\n \"To compute $P_{0}(u)$ change variables to $z=\\\\frac{1}{c}(1-i\\\\frac{\\\\Lambda }{\\\\sqrt{c}})$ to get $P_{-i}(u) =c^{\\\\frac{3}{2}}e^{-(2c+\\\\sqrt{c}u)}\\\\oint e^{\\\\frac{1}{z}}e^{xz}\\\\frac{dz}{2\\\\pi i}$ where $x=c(c+\\\\sqrt{c}u)$ and the integral is around the origin.\",\n \"Then $&&\\\\oint e^{\\\\frac{1}{z}}e^{x z}\\\\frac{dz}{2\\\\pi i}=\\\\sum _{n=0}^\\\\infty \\\\frac{1}{n!}\",\n \"\\\\oint z^{-n}e^{x z}\\\\frac{dz}{2\\\\pi i}\\\\nonumber \\\\\\\\&=&\\\\sum _{m=0}^\\\\infty \\\\frac{1}{(m+1)!}\\\\frac{1}{m!\",\n \"}x^{m}=x^{-\\\\frac{_1}{^2}}I_1(2x^\\\\frac{_1}{^2})\\\\nonumber $ where $I_1$ is the modified Bessel function.\",\n \"As a result, $P (u)& =&\\\\frac{c^{3/4}}{\\\\sqrt{\\\\sqrt{c}+2u}} I_1 \\\\left(c^{3/4}\\\\sqrt{\\\\sqrt{c}+2u}\\\\right)e^{-u\\\\sqrt{c} -c}\\\\nonumber $ Graphs for $P(u)$ for different heat capacities, $c$ , are shown in Fig.\",\n \"REF .\",\n \"One observes that at large $c$ the distribution is essentially Gaussian, while at small $c$ it becomes very asymmetric.\",\n \"Figure: (Color online) Distribution P(u)P(u), where u=βδE/cu= \\\\beta \\\\delta E/\\\\sqrt{c}, where δE\\\\delta Eis the deviation of the energy of the Fermi-gas from its mean, for a few different values of c=C/k B c = C/k_B (solid lines).Dashed line is the Gaussian distribution.Let us crudely estimate the dimensionless heat capacity, $c$ .\",\n \"The estimate reads as $c \\\\sim n_e \\\\mathcal {V} (k_{\\\\text{B}}T/\\\\mu )$ where $n_e$ is the electron density.\",\n \"For a typical metal $n_e \\\\approx 10^{23} \\\\ \\\\text{cm}^{-3}$ .\",\n \"Assuming the sizes of a mesoscopic system (in $\\\\mu \\\\text{m}$ ) as $(0.3-1)\\\\times (0.05-0.1)\\\\times (0.01-0.03)$ $(\\\\mu \\\\text{m})^3$ , we get $\\\\mathcal {V} = (1 .5\\\\cdot 10^{-16} - 3\\\\cdot 10^{-15}) \\\\ \\\\text{cm}^3\\\\, .\",\n \"$ and the total number of electrons is $\\\\mathcal {N} \\\\equiv n_e\\\\mathcal {V} =1.5\\\\cdot 10^7 - 3\\\\cdot 10^8\\\\, .$ Putting $T_{\\\\text{F}} \\\\approx 5\\\\cdot 10^4~\\\\text{K}$ and $T=(0.05 - 0.1)~\\\\text{K}$ we get $k_{\\\\text{B}}T/\\\\mu =T/T_{\\\\text{F}}=(1-2)\\\\cdot 10^{-6}.$ Consequently, the range of the quantity $c$ is $15 -600$ .\",\n \"As we have seen, at these values of $c$ the energy distribution is clearly non-Gaussian.\"\n ],\n [\n \"Moments of the energy distribution\",\n \"The moments of the energy distribution can be readily calculated either from the Eq.\",\n \"(REF ), or explicitly from the Hamiltonian$ \\\\mathcal {H}=\\\\sum _\\\\mathbf {k}\\\\epsilon _\\\\mathbf {k}a^\\\\dag _\\\\mathbf {k}a_\\\\mathbf {k}$ .\",\n \"In particular, $\\\\langle E \\\\rangle =\\\\sum _\\\\mathbf {k}\\\\epsilon _\\\\mathbf {k}\\\\langle a^\\\\dag _\\\\mathbf {k}a_\\\\mathbf {k} \\\\rangle =\\\\sum _\\\\mathbf {k}\\\\epsilon _\\\\mathbf {k}f(\\\\epsilon _\\\\mathbf {k})$ where $f(\\\\epsilon )=\\\\left[e^{\\\\beta (\\\\epsilon - \\\\mu )}+1 \\\\right]^{-1}$ is the Fermi function.\",\n \"The second moment reads $\\\\langle E^2 \\\\rangle =\\\\sum _\\\\mathbf {k}\\\\epsilon _\\\\mathbf {k}\\\\epsilon _\\\\mathbf {l}\\\\langle a^\\\\dag _\\\\mathbf {k}a_\\\\mathbf {k}a^\\\\dag _\\\\mathbf {l}a_\\\\mathbf {l} \\\\rangle =\\\\langle E \\\\rangle ^2 + \\\\sum _\\\\mathbf {k}\\\\epsilon _\\\\mathbf {k}^2 f( \\\\epsilon _\\\\mathbf {k})[1-f( \\\\epsilon _\\\\mathbf {k})].$ Here we have decomposed the product of four Fermi operators according to the Wick theorem.\",\n \"We observe that only the vicinity of the Fermi level is important for the difference $\\\\langle \\\\delta E^2 \\\\rangle \\\\equiv \\\\left\\\\langle \\\\left(E-\\\\langle E \\\\rangle \\\\right)^2\\\\right\\\\rangle =\\\\langle E^2 \\\\rangle -\\\\langle E \\\\rangle ^2\\\\, .\",\n \"$ Therefore, while calculating $\\\\langle \\\\delta E^2 \\\\rangle $ we can assume constant density of states and $\\\\beta \\\\mu \\\\gg 1$ .\",\n \"In this way we obtain $\\\\langle \\\\delta E^2 \\\\rangle =k_{\\\\text{B}}T^2 C\\\\, .$ In a similar way, we calculate $\\\\langle \\\\delta E^3 \\\\rangle =2\\\\sum _\\\\mathbf {k}\\\\epsilon _\\\\mathbf {k}^3 f( \\\\epsilon _\\\\mathbf {k})[1-f( \\\\epsilon _\\\\mathbf {k})]^2=3k_{\\\\text{B}}^2T^3C\\\\, .$ The skewness of the distribution is then $\\\\gamma =\\\\langle \\\\delta E^3 \\\\rangle /\\\\langle \\\\delta E^2 \\\\rangle ^{3/2} =3/\\\\sqrt{c}\\\\, .$ Obviously, the skewness vanishes as $c \\\\rightarrow \\\\infty $ which is the thermodynamic limit.\",\n \"For a Gaussian distribution, the relationship between the 2nd and the 4th moments is $\\\\langle \\\\delta E^4 \\\\rangle =3\\\\langle \\\\delta E^2 \\\\rangle ^2$ .\",\n \"In the general case, an additional contribution appears, such that $\\\\langle \\\\delta E^4 \\\\rangle =3\\\\langle \\\\delta E^2 \\\\rangle ^2\\\\left[1+\\\\frac{4}{\\\\sqrt{c}} \\\\right]\\\\, .$ The 5th moment, absent in the Gaussian approximation, is $\\\\langle \\\\delta E^5 \\\\rangle = 60k_{\\\\text{B}}^4T^5 C\\\\, .$ The results obtained by numerical integration of Eq.\",\n \"(REF ) agree with the analytic expressions given above.\",\n \"In summary, we have shown that the energy distribution of a free Fermi-gas with small heat capacity is non-Gaussian, with a sharp cut-off at low energies.\",\n \"This is a natural consequence of the minimal energy of the filled Fermi sea.\",\n \"We find that the heat capacities demonstrating strong non-Gaussian features can be achieved in standard metallic nanodevices at sub-kelvin temperatures.\",\n \"We thank Ivan Khaymovich and Kay Schwieger for discussions.\",\n \"YMG thanks Aalto University for hospitality during the preparation of this manuscript.\",\n \"The work has been supported by the Academy of Finland (contracts no.\",\n \"272218, 284594 and 271983).\"\n ]\n]"},"id":{"kind":"string","value":"1605.05877"}}},{"rowIdx":1866690,"cells":{"text":{"kind":"list like","value":[["An Apparent Dissociation Transition in Anharmonically Bound 1D Systems"],["Abstract For diatomic molecules and chains bound anharmonically by interactions such a the Lennard Jones and Morse potentials, we obtain analytical expressions for thermodynamic observables including the mean bond length, thermally averaged internal energy, and the coefficient of thermal expansion.","These results are valid across the shift from condensed to gas-like phases, a dissociation transition marked by a crossover with no singularities in thermodynamic variables for finite pressures, though singular behavior appears in the low pressure limit.","In the regime where the thermal energy $k_{\\mathrm{B}} T$ is much smaller than the dissociation energy $D$, the mean interatomic separation scales as $\\langle l \\rangle = R_{e} + {\\mathcal B} (P R_{e}/k_{\\mathrm{B}} T)^{-2} e^{-D/k_{\\mathrm{B}} T} \\left( D/k_{\\mathrm{B}} T \\right )^{1/2}$ for both the Morse and Lennard Jones potentials where $p$ is a pressure term, $R_{e}$ is the $T = 0$ bond length, and ${\\mathcal B}$ is a constant specific to the potential."],["An Apparent Dissociation Transition in Anharmonically Bound 1D Systems D. J. Priour, Jr Department of Physics & Astronomy, Youngstown State University, Youngstown, OH 44555, USA Christopher Watenpool Department of Electrical Engineering, Youngstown State University, Youngstown, OH 44555, USA For diatomic molecules and chains bound anharmonically by interactions such as the Lennard Jones and Morse potentials, we obtain analytical expressions for thermodynamic observables including the mean bond length, thermally averaged internal energy, and the coefficient of thermal expansion.","These results are valid across the shift from condensed to gas-like phases, a dissociation transition marked by a crossover with no singularities in thermodynamic variables for finite pressures, though singular behavior appears in the low pressure limit.","In the regime where the thermal energy $k_{\\mathrm {B}} T$ is much smaller than the dissociation energy $D$ , the mean interatomic separation scales as $\\langle l \\rangle = R_{e} + {\\mathcal {B}} (P R_{e}/k_{\\mathrm {B}} T)^{-2} e^{-D/k_{\\mathrm {B}} T} \\left( D/k_{\\mathrm {B}} T \\right)^{1/2}$ for both the Morse and Lennard Jones potentials where $p$ is a pressure term, $R_{e}$ is the $T = 0$ bond length, and ${\\mathcal {B}}$ is a constant specific to the potential.","64.60.De,61.43.-j,34.80.Ht,51.30.+i Potentials used to describe interactions among atoms such as the Lennard Jones potential [1] (e.g.","Van der Waals couplings among noble gas atoms such as Argon) and the Morse potential [2] (for bonding with significant covalent character) contain attractive and repulsive terms.","Whereas the attractive component decays at large distance, the repulsive piece decays even more rapidly in the large separation regime while rising sharply when atoms are in proximity where Pauli Exclusion effects play a role as atomic cores begin to overlap.","With the combination of attractive and repulsive terms, there is a potential minimum at the equilibrium separation $R_{e}$ with harmonic (parabolic) dependence in the vicinity of $R_{e}$ and increasingly asymmetric and anharmonic character for significant deviations from $R_{e}$ caused, e.g., by thermal fluctuations in the high temperature regime.","Whether due to Van der Waals coupling as in the Lennard Jones model or the sharing of charge represented by the Morse potential, entropic effects drive dissociation despite the attractive component, except at $T = 0$ where thermal fluctuations are absent.","Nevertheless, confinement of the system volume for finite temperatures is realized by taking into consideration a pressure term (incorporated in the 1D context as a finite cost per length of elongating the system, such as a chain of identical atoms), which precludes indefinite expansion of the system.","In this work, we operate in terms of three distinct energy scales relevant to both the Morse and Lennard Jones potentials; the thermal energy $k_{\\mathrm {B}} T$ , the dissociation energy $D$ needed to completely separate an interacting pair of atoms, and the pressure confinement scale $p R_{e}$ .","For high $T$ , where $k_{\\mathrm {B}} T \\gg D$ the competition among $p R_{e}$ and $k_{\\mathrm {B}} T$ dominates, and the state is well approximated as an ideal gas.","On the other hand, for low $T$ , where $k_{\\mathrm {B}} T \\ll D$ , there is a more subtle interplay among $p$ , $T$ , and $D$ with a crossover with decreasing $p$ from a condensed state ($ \\langle l \\rangle \\approx R_{e}$ ) to a diluted gas-like state ($\\langle l \\rangle \\gg R_{e}$ ).","For a more detailed description of dissociation, we obtain analytical expressions for thermodynamic variables of interest valid across the shift from the condensed to gas-like states in the low to intermediate temperature regimes.","With a unified treatment applied to both the Lennard Jones and Morse cases, we find strong qualitative similarities in molecular dissociation phenomena despite distinct bonding physics driving the potentials.","With interactions only among nearest neighbors, the system energy is $E = \\sum _{i} P_{i}^{2}/2M + \\sum _{i} V (R_{i+1} - R_{i}) + pL$ with the sum over the $N$ members of the chain, $L$ being the total system size.","Since atomic momenta and spatial coordinates are not coupled, apart from a contribution $N k_{\\mathrm {B}} T/2$ to the internal energy (and $N k_{\\mathrm {B}}/2$ to the specific heat), one need only consider site positions in sampling system configurations.","We obtain thermodynamic variables of interest from the partition function $Z$ ; though in general the rapid scaling of configuration space with the number of system components precludes an analytical calculation of $Z$ , we decouple the calculation of $Z$ by describing the system in terms of the separation among adjacent sites $\\Delta _{i} \\equiv R_{i+1} - R_{i}$ instead of the absolute atomic coordinates $R_{i}$ [3], [4].","Noting that $L = R_{N} - R_{1} = \\Delta _{N-1} + \\Delta _{N-2} + \\ldots + \\Delta _{1}$ , the partition function may be factorized as $Z = \\hat{Z}^{N-1}$ where $\\hat{Z} = \\int _{0}^{\\infty } \\exp ( -\\beta [ V(\\Delta ) + p \\Delta ] ) d \\Delta $ .","In this manner, the calculation for a chain of arbitrary length is reduced to the case of a single atomic pair separated by a distance $\\Delta $ (subsequently labeled as $R$ ), subject to an asymptotically linearly diverging effective potential $V(R) + pR$ .","It is convenient to operate in terms of dimensionless energy variables $\\varepsilon \\equiv \\beta D$ and $\\eta \\equiv p R_{e}$ as well as the ratio $\\gamma \\equiv p R_{e}/D = \\eta /\\varepsilon $ characterizing the strength of the pressure energy scale in relation to the dissociation energy.","Salient thermodynamic variables may then be calculated by differentiating with respect to $\\varepsilon $ and $\\eta $ , such as $\\langle E \\rangle = -\\partial /\\partial \\beta \\ln Z = -( \\varepsilon Z_{\\varepsilon } + \\eta Z_{\\eta } ) \\beta ^{-1} Z^{-1}$ for the internal energy where $Z_{\\varepsilon } \\equiv \\partial Z/\\partial \\varepsilon $ and $Z_{\\eta } \\equiv \\partial Z/\\partial \\eta $ .","Though having different functional forms, $V_{\\mathrm {LJ}} = B R^{-12} - A R^{-6}$ (Lennard Jones) and $V_{\\mathrm {M}} = -D + D \\lbrace -1 + \\exp [-a (R - R_{e})] \\rbrace ^{2}$ (Morse), $V_{\\mathrm {LJ}}$ and $V_{\\mathrm {M}}$ are amenable to the same analysis with qualitatively similar results in both cases.","When expressed in terms of $D$ and the reduced separation $r \\equiv R/R_{e}$ , $V_{\\mathrm {LJ}}$ and $V_{\\mathrm {M}}$ are both of the form $V(r) = -D + D[ \\chi (r) - 1]^{2}$ where $\\chi _{\\mathrm {LJ}} = r^{-6}$ and $\\chi _{\\mathrm {M}} = \\exp [- {\\mathcal {A}} (r - 1)]$ .","In terms of specific parameters, $D = -A^{2}/4B$ and $R_{e} = (2B/A)^{1/6}$ for the Lennard Jones case while ${\\mathcal {A}} = a R_{e}$ (${\\mathcal {A}} = 2.5$ for results exhibited here) for the Morse potential.","In terms of $\\chi (r)$ , the partition function for both $V_{\\mathrm {LJ}}$ and $V_{\\mathrm {M}}$ is $Z = R_{e} \\exp {\\varepsilon } \\int _{0}^{\\infty } \\exp [ \\chi (r) -1]^{2} \\exp [-\\eta r] dr$ .","Although one could in principle expand $\\chi (r) - 1$ about $r = 1$ , where the potential basin is locally parabolic, the significant departure from harmonic character with the shift from the condensed phase ($r \\approx 1$ ) to the gas-like phase ($r \\gg 1$ ) hampers a perturbative analysis of this kind.","For a treatment which offers good convergence with only a handful of terms, and which provides a good description of the condensed and dissociated states alike, we consider the substitution $u = \\chi (r)$ , obtaining $Z = -R_{e} e^{\\varepsilon } \\int _{0}^{\\infty } e^{-\\varepsilon (u - 1)^{2}} e^{-\\eta \\chi ^{-1}(u)} \\frac{1}{\\frac{d}{du} \\chi ^{-1}(u)} du$ Integrating by parts, neglecting boundary terms, and using $v = u - 1$ leads to $Z = \\frac{2 R_{e} \\varepsilon }{\\eta } e^{\\varepsilon } \\int _{-1}^{\\infty } e^{-\\varepsilon v^{2}} e^{-\\eta \\chi ^{-1}(v+1)} v dv$ which may be evaluated as a series of Gaussian integrals by Taylor expanding $\\exp [-\\eta \\chi ^{-1}(v+1)]$ about $v = 0$ with $e^{-\\eta \\chi ^{-1}(v+1)} = e^{-\\eta } \\left[ 1 - \\eta d_{1} v + \\frac{1}{2!}","(\\eta ^{2} d_{1}^{2} - \\eta d_{2})v^{2} + \\ldots \\right]$ where the $d_{i}$ are $i$ th derivatives of $\\chi ^{-1}(v+1)$ evaluated at $v = 0$ .","For $V_{\\mathrm {LJ}}$ , $d_{1} = -1/6$ , $d_{2} = 7/36$ , and $d_{3} = -91/216$ while for $V_{\\mathrm {M}}$ , $d_{1} = -1/{\\mathcal {A}}$ , $d_{2} = 1/{\\mathcal {A}}$ , and $d_{3} = -2/{\\mathcal {A}}$ .","For the sake of a quantitatively accurate description of the dissociation transition, it is important not to neglect the finiteness of the lower integration limit in evaluating the Gaussian integrals, which must be taken into consideration to account for dissociation.","In the regime of interest, $Z$ is well represented by $Z = e^{\\varepsilon - \\eta } \\bigg [ e^{-\\varepsilon } (\\eta ^{-1} + d_{1} ) - \\\\\\nonumber \\frac{\\sqrt{\\pi }}{4} \\varepsilon ^{-3/2} (4 \\varepsilon d_{1} + d_{1}^{3} \\eta ^{2} -3 \\eta d_{1} d_{2} + d_{3} ) \\bigg ]$ where we have used $\\int _{-1}^{\\infty } \\exp ( -\\varepsilon v^{2})v dv = \\exp -\\varepsilon /(2 \\varepsilon )$ , $\\int _{-1}^{\\infty } \\exp ( -\\varepsilon v^{2})v^{2} dv \\approx (\\sqrt{\\pi }/2) \\varepsilon ^{-3/2} - \\exp -\\varepsilon /(2 \\varepsilon )$ (i.e.","retaining the leading term in the asymptotic series); we have neglected $\\int _{-1}^{\\infty } \\exp ( -\\varepsilon v^{2})v^{3} dv$ , and truncated $\\int _{-1}^{\\infty } \\exp ( -\\varepsilon v^{2})v^{4} dv$ at $(3 \\sqrt{\\pi }/4) \\varepsilon ^{-5/2}$ .","For a description valid for both the condensed and gas-like phases, we obtain from $Z$ rational expressions for observables of interest where typically leading and next to leading order terms in $\\eta $ and $\\varepsilon $ are retained in the numerator and the denominator; we calculate and discuss in turn the thermally averaged bond length $\\langle l \\rangle $ , the internal energy $\\langle E \\rangle $ , the specific heat $c_{p}$ , and the thermal expansion coefficient $\\alpha $ .","With the mean interatomic separation being $-R_{e} \\partial \\ln Z/\\partial \\eta $ , the normalized thermally averaged bond length is well approximated with $\\frac{\\langle l \\rangle }{R_{e}} = 1 + \\eta ^{-1} \\left[ \\frac{e^{-\\varepsilon } - \\frac{3}{4} \\sqrt{\\pi } d_{1} d_{2} \\varepsilon ^{-3/2} \\eta ^{2} }{e^{-\\varepsilon } -\\frac{\\sqrt{\\pi }}{4} \\varepsilon ^{-3/2} \\eta \\left( 4 d_{1} \\varepsilon + d_{3} \\right)} \\right]$ readily inverted via the quadratic formula to obtain $\\eta $ in terms of $\\langle l \\rangle $ .","Asymptotically, for $\\eta \\ll \\varepsilon $ , the residual component of the mean separation (small in the condensed phase but diverging with the dissociation transition) may be represented by the simpler expression $\\tilde{\\Delta }_{l} \\equiv (\\langle l \\rangle /R_{e} - 1) = \\varepsilon ^{1/2} \\exp -\\varepsilon /(d_{1} \\sqrt{\\pi } \\eta ^{2})$ .","To more conveniently visualize $\\langle l \\rangle $ , (as well as $\\langle E \\rangle $ , $\\alpha $ , and $c_{p}$ ), we choose $\\varepsilon $ for the abscissa where only $T$ is varied while holding $p$ fixed for a given curve; $\\varepsilon \\gg 1$ and $\\varepsilon \\ll 1$ correspond to the low and high $T$ regimes respectively.","Whereas $\\gamma = p R_{e}/D$ remains constant, $\\eta = \\beta p R_{e} = \\gamma \\varepsilon $ varies with temperature.","Figure REF displays results corresponding to $V_{\\mathrm {LJ}}$ in panel (a) and $V_{\\mathrm {M}}$ in panel (b) with open symbols indicating numerical results and solid curves representing the foregoing approximations to $\\langle l \\rangle $ .","The black traces, calculated with the rational expression in Eq.","REF , are in good quantitative agreement with numerical data even for $\\gamma = 1.0$ .","On the other hand, the $\\tilde{\\Delta }_{l} + 1$ approximation is represented by lighter (red/blue for $V_{\\mathrm {LJ}}$ /$V_{\\mathrm {M}}$ ) curves, and accurately indicates the location where $\\langle l \\rangle $ begins to diverge significantly from $R_{e}$ .","Moreover, despite the simple structure, good agreement with exact numerical results is evident for $\\gamma < 0.01$ .","To estimate $\\gamma $ for physically realistic systems, we consider a chain of noble gas atoms (e.g.","Argon) in the context of the Lennard Jones Model or a covalently bonded pair of atoms (e.g.","H$_{2}$ ) in the Morse potential framework in a 1D conduit.","With typical atomic radii on the order of an Ångstrom, we assume a cross sectional area on the order of $(1~\\textrm {Å})^{2} = 10^{-20}~\\textrm {m}^{2}$ .","Using $1.01 \\times 10^{5}~\\textrm {Pa}$ for standard atmospheric pressure, find $p \\sim 10^{-15}~J/\\textrm {m}$ .","In the case of Argon, one has $D = 0.011~\\textrm {eV}$ and $R_{e} = 3.8~\\textrm {Å}$ [5], [6], [7], [8] and $\\gamma = 2.2 \\times 10^{-4}$ .","On the other hand, for H$_{2}$ , $D = 4.52~\\textrm {eV}$ [9] and we take $R_{e}$ to be the measured bond length of $0.74~\\textrm {Å}$ [10]; one obtains $\\gamma = 1.0 \\times 10^{-7}$ , with both $\\gamma $ values deep in the $\\gamma \\ll 1$ range.","Figure: (Color online) Normalized mean atomic separations for the Lennard Jones (left panel) and Morse potential (right panel)for various γ\\gamma values;solid traces are analytical approximations while open symbols are numerical data.","Black curves correspond to the Δ ˜ l +1\\tilde{\\Delta }_{l}+1relationship, while colored traces represent the rational 〈l〉\\langle l \\rangle expression.The internal energy, $\\langle E \\rangle $ , is well approximated as $\\frac{\\langle E \\rangle }{k_{\\mathrm {B}} T} = \\eta - \\varepsilon + \\frac{e^{-\\varepsilon } \\eta ^{-1} (1 + \\varepsilon ) - \\varepsilon ^{-3/2}\\frac{\\sqrt{\\pi }}{8} (4 d_{1} \\varepsilon + 3d_{3} )}{e^{-\\varepsilon }\\eta ^{-1} - \\varepsilon ^{-3/2} \\frac{\\sqrt{\\pi }}{4}(4 d_{1} \\varepsilon + d_{3}) }$ To strip away trivial dependencies, Figure REF shows the residual internal energies, $E_{\\mathrm {Res}} = \\langle E \\rangle + D - p R_{e}$ , with Lennard Jones results in the main panel and Morse potential results in the inset for a variety of $\\gamma $ values.","Solid curves obtained from the rational expression closely coincide with the open symbols representing numerical data.","Though $E_{\\mathrm {Res}}$ tends to $k_{\\mathrm {B}} T/2$ for sufficiently large $\\varepsilon $ , the expected dependence where $\\langle l \\rangle \\approx R_{e}$ where anharmonicities are negligible, the residual energy component is non-monotonic.","One sees from the main graph and the inset graph that the breadth and height of the peak separating the low and high $T$ regimes scales asymptotically as $\\log _{10}(1/\\gamma )$ Figure: (Color online) Residual internal energy results for various γ\\gamma values with the main graph showing Lennard Jones results and the insetdisplaying Morse results.","Solid traces represent analytical results, while open symbols indicate numerical data.The specific heat at constant pressure is $c_{p} = \\partial \\langle E \\rangle /\\partial T$ , which is $\\frac{c_{p}}{k_{\\mathrm {B}}} = \\frac{\\frac{1}{2} \\varepsilon ^{2} \\left( \\varepsilon + \\frac{3 d_{3}}{2d_{1}} \\right)- \\Upsilon \\varepsilon ^{1/2} \\left( 1 + \\varepsilon + \\frac{d3}{4d_{1}} \\right) + \\Upsilon ^{2}}{\\varepsilon ^{-2}\\left( \\varepsilon + \\frac{d_{3}}{2d_{1}} \\right)- 2 \\Upsilon \\varepsilon ^{-3/2} \\left( \\varepsilon + \\frac{d3}{4d_{1}} \\right) + \\Upsilon ^{2} }$ where $\\Upsilon = \\exp {-\\varepsilon }/( \\sqrt{\\pi } \\eta d_{1} )$ .","Specific heat results are displayed in the main graph (for $V_{\\mathrm {LJ}}$ ) and the inset (for $V_{\\mathrm {M}}$ ) of Fig.","REF for a range of $\\gamma $ values; as in the case of the internal energy, there is good agreement among the analytical (solid curve) and the numerical (open symbols) results.","One sees from the analytical $c_{p}$ expression in Eq.","REF and the curves in Fig.","REF that the specific heat tends to $k_{\\mathrm {B}}$ for sufficiently low $D$ (i.e.","higher $T$ ) and flattens to $k_{\\mathrm {B}}/2$ for $D \\gg 1$ (low $T$ ).","Whereas the latter is a hallmark of the condensed phase, the former is expected for a dissociated system; the transition between the two asymptotically flat regions is non-monotonic, with a peak separating the $k_{\\mathrm {B}}$ and $k_{\\mathrm {B}}/2$ regimes.","The region where $c_{p}$ rises to a peak, representing the dissociation transition, becomes taller and narrower with decreasing $\\gamma $ .","Figure: (Color online) Specific heat curves for assorted γ\\gamma values with Lennard Jones results in the main graph and Morse results in theinset.","Analytical results are shown as solid traces, while open symbols represent numerical data.The coefficient of thermal expansion $\\alpha = \\langle l \\rangle ^{-1} \\partial \\langle l \\rangle /\\partial T$ is well represented by $\\frac{\\alpha }{k_{\\mathrm {B}}} = \\frac{\\frac{3}{4} \\eta d_{2} \\varepsilon ^{-1} -\\Upsilon \\varepsilon ^{1/2} \\left( \\varepsilon + \\frac{3}{2}+ \\frac{d_{3}}{4 d_{1}} - \\frac{3 \\eta d_{2}}{2} \\right) + \\Upsilon ^{2}}{\\eta -\\Upsilon \\varepsilon ^{-1/2} \\left( \\varepsilon +2 \\eta \\varepsilon + \\frac{d_{3}}{4 d_{1}}\\right) + \\Upsilon ^{2}}$ Juxtaposed analytical (solid traces) and numerical results (open symbols) are shown in Figure 4 for various $\\gamma $ values with $D \\alpha /k_{\\mathrm {B}}$ on the vertical axis.","As in the case of the specific heat, the $\\alpha $ curves are non-monotonic, with peak heights increasing with decreasing $\\gamma $ .","Choosing $D \\alpha /k_{\\mathrm {B}}$ for the ordinate is in part to show the slow convergence to the low $T$ value of $3 d_{2}/4$ in the $\\gamma \\ll 1$ limit.","All of the cases shown correspond to experimentally realistic $\\gamma $ values, and in all but one of the curves shown, $\\alpha $ is appreciably different from the limiting value even for $\\varepsilon $ as high as 10.","Figure: (Color online) Thermal expansion coefficient results for a range of γ\\gamma values with Lennard Jones results plotted in the maingraph and corresponding results for the Morse potential in the inset.","Solid curves represent analytical results, while open symbols indicate numerical data.The peak locations calculated in the framework of the analytical approximations (solid traces) for $c_{p}$ and $\\alpha $ are in close agreement with the exact numerical results (open symbols) as may be seen in the upper panels of Fig.","REF .","In statistical mechanics singularities are not in general encountered for single component systems, with non-analytic behavior emerging only in thermodynamic limit as the number of degrees of freedom tends to infinity.","Nevertheless, the low pressure regime is atypical in the sense that singular behavior is inevitable as $p$ (or $\\gamma $ ) tends to zero due to the divergence of $\\langle l \\rangle $ for any finite $T$ as $p \\rightarrow 0$ .","Hence, the possibility of a sharp dissociation transition for $\\gamma \\ll 1$ must be examined with care.","As a measure of the extent to which the dissociation transition is singular, the sharpness of the thermal expansion coefficient and specific heat peaks is quantified in the lower panels of Fig.","REF as the relative full width half maximum, $\\Delta T_{\\mathrm {Peak}}/T_{\\mathrm {Peak}}$ .","In the case of $\\alpha $ , $\\Delta T_{\\mathrm {Peak}}/T_{\\mathrm {Peak}}$ tends to a finite value common to both $V_{\\mathrm {LJ}}$ and $V_{\\mathrm {M}}$ , indicating the $\\alpha $ peaks cease to become narrower relative to their location with decreasing $\\gamma $ .","On the other hand, the specific heat relative peak width appears to tend to zero with decreasing $\\gamma $ , a trend highlighted in the lower right panel inset of Fig.REF showing $\\Delta T_{\\mathrm {Peak}}/T_{\\mathrm {Peak}}$ relative to $1/\\log _{10} (\\gamma ^{-1})$ , The curves are asymptotically linear as $\\gamma \\rightarrow 0$ , with the relative peak width vanishing for $p \\rightarrow 0$ as singular behavior appears.","Figure: (Color online) Peak locations for the specific heat (upper right panel) and thermal expansion coefficient (upper left panel).Corresponding relative full width half maxima (i.e.","ΔT peak /T peak \\Delta T_{\\mathrm {peak}}/T_{\\mathrm {peak}}) appear in the lower right and lower left panels for thespecific heat and coefficient of thermal expansion respectively.Throughout, open squares represent Lennard Jones results and open circles Morse results.In conclusion, with a nonperturbative treatment of the anharmonicity of interatomic potentials, we provide a theoretical description of the dissociation transition valid for the condensed state as well as the gas-like phase where thermal fluctuations have driven pairs of atoms far from their equilibrium separations.","By applying a unified treatment to disparate potentials, we have obtained analytical results for salient thermodynamic observables in good agreement with precise numerical results.","Though ascribed to distinct bonding physics, there are striking similarities in results, e.g.","with $\\langle l \\rangle /R_{e} = 1 + \\varepsilon ^{1/2} \\exp -\\varepsilon /(d_{1} \\sqrt{\\pi } \\eta ^{2})$ specifying the mean bond length for $\\gamma \\ll 1$ .","Useful conversations with Michael Crescimanno are gratefully acknowledged."]],"string":"[\n [\n \"An Apparent Dissociation Transition in Anharmonically Bound 1D Systems\"\n ],\n [\n \"Abstract For diatomic molecules and chains bound anharmonically by interactions such a the Lennard Jones and Morse potentials, we obtain analytical expressions for thermodynamic observables including the mean bond length, thermally averaged internal energy, and the coefficient of thermal expansion.\",\n \"These results are valid across the shift from condensed to gas-like phases, a dissociation transition marked by a crossover with no singularities in thermodynamic variables for finite pressures, though singular behavior appears in the low pressure limit.\",\n \"In the regime where the thermal energy $k_{\\\\mathrm{B}} T$ is much smaller than the dissociation energy $D$, the mean interatomic separation scales as $\\\\langle l \\\\rangle = R_{e} + {\\\\mathcal B} (P R_{e}/k_{\\\\mathrm{B}} T)^{-2} e^{-D/k_{\\\\mathrm{B}} T} \\\\left( D/k_{\\\\mathrm{B}} T \\\\right )^{1/2}$ for both the Morse and Lennard Jones potentials where $p$ is a pressure term, $R_{e}$ is the $T = 0$ bond length, and ${\\\\mathcal B}$ is a constant specific to the potential.\"\n ],\n [\n \"An Apparent Dissociation Transition in Anharmonically Bound 1D Systems D. J. Priour, Jr Department of Physics & Astronomy, Youngstown State University, Youngstown, OH 44555, USA Christopher Watenpool Department of Electrical Engineering, Youngstown State University, Youngstown, OH 44555, USA For diatomic molecules and chains bound anharmonically by interactions such as the Lennard Jones and Morse potentials, we obtain analytical expressions for thermodynamic observables including the mean bond length, thermally averaged internal energy, and the coefficient of thermal expansion.\",\n \"These results are valid across the shift from condensed to gas-like phases, a dissociation transition marked by a crossover with no singularities in thermodynamic variables for finite pressures, though singular behavior appears in the low pressure limit.\",\n \"In the regime where the thermal energy $k_{\\\\mathrm {B}} T$ is much smaller than the dissociation energy $D$ , the mean interatomic separation scales as $\\\\langle l \\\\rangle = R_{e} + {\\\\mathcal {B}} (P R_{e}/k_{\\\\mathrm {B}} T)^{-2} e^{-D/k_{\\\\mathrm {B}} T} \\\\left( D/k_{\\\\mathrm {B}} T \\\\right)^{1/2}$ for both the Morse and Lennard Jones potentials where $p$ is a pressure term, $R_{e}$ is the $T = 0$ bond length, and ${\\\\mathcal {B}}$ is a constant specific to the potential.\",\n \"64.60.De,61.43.-j,34.80.Ht,51.30.+i Potentials used to describe interactions among atoms such as the Lennard Jones potential [1] (e.g.\",\n \"Van der Waals couplings among noble gas atoms such as Argon) and the Morse potential [2] (for bonding with significant covalent character) contain attractive and repulsive terms.\",\n \"Whereas the attractive component decays at large distance, the repulsive piece decays even more rapidly in the large separation regime while rising sharply when atoms are in proximity where Pauli Exclusion effects play a role as atomic cores begin to overlap.\",\n \"With the combination of attractive and repulsive terms, there is a potential minimum at the equilibrium separation $R_{e}$ with harmonic (parabolic) dependence in the vicinity of $R_{e}$ and increasingly asymmetric and anharmonic character for significant deviations from $R_{e}$ caused, e.g., by thermal fluctuations in the high temperature regime.\",\n \"Whether due to Van der Waals coupling as in the Lennard Jones model or the sharing of charge represented by the Morse potential, entropic effects drive dissociation despite the attractive component, except at $T = 0$ where thermal fluctuations are absent.\",\n \"Nevertheless, confinement of the system volume for finite temperatures is realized by taking into consideration a pressure term (incorporated in the 1D context as a finite cost per length of elongating the system, such as a chain of identical atoms), which precludes indefinite expansion of the system.\",\n \"In this work, we operate in terms of three distinct energy scales relevant to both the Morse and Lennard Jones potentials; the thermal energy $k_{\\\\mathrm {B}} T$ , the dissociation energy $D$ needed to completely separate an interacting pair of atoms, and the pressure confinement scale $p R_{e}$ .\",\n \"For high $T$ , where $k_{\\\\mathrm {B}} T \\\\gg D$ the competition among $p R_{e}$ and $k_{\\\\mathrm {B}} T$ dominates, and the state is well approximated as an ideal gas.\",\n \"On the other hand, for low $T$ , where $k_{\\\\mathrm {B}} T \\\\ll D$ , there is a more subtle interplay among $p$ , $T$ , and $D$ with a crossover with decreasing $p$ from a condensed state ($ \\\\langle l \\\\rangle \\\\approx R_{e}$ ) to a diluted gas-like state ($\\\\langle l \\\\rangle \\\\gg R_{e}$ ).\",\n \"For a more detailed description of dissociation, we obtain analytical expressions for thermodynamic variables of interest valid across the shift from the condensed to gas-like states in the low to intermediate temperature regimes.\",\n \"With a unified treatment applied to both the Lennard Jones and Morse cases, we find strong qualitative similarities in molecular dissociation phenomena despite distinct bonding physics driving the potentials.\",\n \"With interactions only among nearest neighbors, the system energy is $E = \\\\sum _{i} P_{i}^{2}/2M + \\\\sum _{i} V (R_{i+1} - R_{i}) + pL$ with the sum over the $N$ members of the chain, $L$ being the total system size.\",\n \"Since atomic momenta and spatial coordinates are not coupled, apart from a contribution $N k_{\\\\mathrm {B}} T/2$ to the internal energy (and $N k_{\\\\mathrm {B}}/2$ to the specific heat), one need only consider site positions in sampling system configurations.\",\n \"We obtain thermodynamic variables of interest from the partition function $Z$ ; though in general the rapid scaling of configuration space with the number of system components precludes an analytical calculation of $Z$ , we decouple the calculation of $Z$ by describing the system in terms of the separation among adjacent sites $\\\\Delta _{i} \\\\equiv R_{i+1} - R_{i}$ instead of the absolute atomic coordinates $R_{i}$ [3], [4].\",\n \"Noting that $L = R_{N} - R_{1} = \\\\Delta _{N-1} + \\\\Delta _{N-2} + \\\\ldots + \\\\Delta _{1}$ , the partition function may be factorized as $Z = \\\\hat{Z}^{N-1}$ where $\\\\hat{Z} = \\\\int _{0}^{\\\\infty } \\\\exp ( -\\\\beta [ V(\\\\Delta ) + p \\\\Delta ] ) d \\\\Delta $ .\",\n \"In this manner, the calculation for a chain of arbitrary length is reduced to the case of a single atomic pair separated by a distance $\\\\Delta $ (subsequently labeled as $R$ ), subject to an asymptotically linearly diverging effective potential $V(R) + pR$ .\",\n \"It is convenient to operate in terms of dimensionless energy variables $\\\\varepsilon \\\\equiv \\\\beta D$ and $\\\\eta \\\\equiv p R_{e}$ as well as the ratio $\\\\gamma \\\\equiv p R_{e}/D = \\\\eta /\\\\varepsilon $ characterizing the strength of the pressure energy scale in relation to the dissociation energy.\",\n \"Salient thermodynamic variables may then be calculated by differentiating with respect to $\\\\varepsilon $ and $\\\\eta $ , such as $\\\\langle E \\\\rangle = -\\\\partial /\\\\partial \\\\beta \\\\ln Z = -( \\\\varepsilon Z_{\\\\varepsilon } + \\\\eta Z_{\\\\eta } ) \\\\beta ^{-1} Z^{-1}$ for the internal energy where $Z_{\\\\varepsilon } \\\\equiv \\\\partial Z/\\\\partial \\\\varepsilon $ and $Z_{\\\\eta } \\\\equiv \\\\partial Z/\\\\partial \\\\eta $ .\",\n \"Though having different functional forms, $V_{\\\\mathrm {LJ}} = B R^{-12} - A R^{-6}$ (Lennard Jones) and $V_{\\\\mathrm {M}} = -D + D \\\\lbrace -1 + \\\\exp [-a (R - R_{e})] \\\\rbrace ^{2}$ (Morse), $V_{\\\\mathrm {LJ}}$ and $V_{\\\\mathrm {M}}$ are amenable to the same analysis with qualitatively similar results in both cases.\",\n \"When expressed in terms of $D$ and the reduced separation $r \\\\equiv R/R_{e}$ , $V_{\\\\mathrm {LJ}}$ and $V_{\\\\mathrm {M}}$ are both of the form $V(r) = -D + D[ \\\\chi (r) - 1]^{2}$ where $\\\\chi _{\\\\mathrm {LJ}} = r^{-6}$ and $\\\\chi _{\\\\mathrm {M}} = \\\\exp [- {\\\\mathcal {A}} (r - 1)]$ .\",\n \"In terms of specific parameters, $D = -A^{2}/4B$ and $R_{e} = (2B/A)^{1/6}$ for the Lennard Jones case while ${\\\\mathcal {A}} = a R_{e}$ (${\\\\mathcal {A}} = 2.5$ for results exhibited here) for the Morse potential.\",\n \"In terms of $\\\\chi (r)$ , the partition function for both $V_{\\\\mathrm {LJ}}$ and $V_{\\\\mathrm {M}}$ is $Z = R_{e} \\\\exp {\\\\varepsilon } \\\\int _{0}^{\\\\infty } \\\\exp [ \\\\chi (r) -1]^{2} \\\\exp [-\\\\eta r] dr$ .\",\n \"Although one could in principle expand $\\\\chi (r) - 1$ about $r = 1$ , where the potential basin is locally parabolic, the significant departure from harmonic character with the shift from the condensed phase ($r \\\\approx 1$ ) to the gas-like phase ($r \\\\gg 1$ ) hampers a perturbative analysis of this kind.\",\n \"For a treatment which offers good convergence with only a handful of terms, and which provides a good description of the condensed and dissociated states alike, we consider the substitution $u = \\\\chi (r)$ , obtaining $Z = -R_{e} e^{\\\\varepsilon } \\\\int _{0}^{\\\\infty } e^{-\\\\varepsilon (u - 1)^{2}} e^{-\\\\eta \\\\chi ^{-1}(u)} \\\\frac{1}{\\\\frac{d}{du} \\\\chi ^{-1}(u)} du$ Integrating by parts, neglecting boundary terms, and using $v = u - 1$ leads to $Z = \\\\frac{2 R_{e} \\\\varepsilon }{\\\\eta } e^{\\\\varepsilon } \\\\int _{-1}^{\\\\infty } e^{-\\\\varepsilon v^{2}} e^{-\\\\eta \\\\chi ^{-1}(v+1)} v dv$ which may be evaluated as a series of Gaussian integrals by Taylor expanding $\\\\exp [-\\\\eta \\\\chi ^{-1}(v+1)]$ about $v = 0$ with $e^{-\\\\eta \\\\chi ^{-1}(v+1)} = e^{-\\\\eta } \\\\left[ 1 - \\\\eta d_{1} v + \\\\frac{1}{2!}\",\n \"(\\\\eta ^{2} d_{1}^{2} - \\\\eta d_{2})v^{2} + \\\\ldots \\\\right]$ where the $d_{i}$ are $i$ th derivatives of $\\\\chi ^{-1}(v+1)$ evaluated at $v = 0$ .\",\n \"For $V_{\\\\mathrm {LJ}}$ , $d_{1} = -1/6$ , $d_{2} = 7/36$ , and $d_{3} = -91/216$ while for $V_{\\\\mathrm {M}}$ , $d_{1} = -1/{\\\\mathcal {A}}$ , $d_{2} = 1/{\\\\mathcal {A}}$ , and $d_{3} = -2/{\\\\mathcal {A}}$ .\",\n \"For the sake of a quantitatively accurate description of the dissociation transition, it is important not to neglect the finiteness of the lower integration limit in evaluating the Gaussian integrals, which must be taken into consideration to account for dissociation.\",\n \"In the regime of interest, $Z$ is well represented by $Z = e^{\\\\varepsilon - \\\\eta } \\\\bigg [ e^{-\\\\varepsilon } (\\\\eta ^{-1} + d_{1} ) - \\\\\\\\\\\\nonumber \\\\frac{\\\\sqrt{\\\\pi }}{4} \\\\varepsilon ^{-3/2} (4 \\\\varepsilon d_{1} + d_{1}^{3} \\\\eta ^{2} -3 \\\\eta d_{1} d_{2} + d_{3} ) \\\\bigg ]$ where we have used $\\\\int _{-1}^{\\\\infty } \\\\exp ( -\\\\varepsilon v^{2})v dv = \\\\exp -\\\\varepsilon /(2 \\\\varepsilon )$ , $\\\\int _{-1}^{\\\\infty } \\\\exp ( -\\\\varepsilon v^{2})v^{2} dv \\\\approx (\\\\sqrt{\\\\pi }/2) \\\\varepsilon ^{-3/2} - \\\\exp -\\\\varepsilon /(2 \\\\varepsilon )$ (i.e.\",\n \"retaining the leading term in the asymptotic series); we have neglected $\\\\int _{-1}^{\\\\infty } \\\\exp ( -\\\\varepsilon v^{2})v^{3} dv$ , and truncated $\\\\int _{-1}^{\\\\infty } \\\\exp ( -\\\\varepsilon v^{2})v^{4} dv$ at $(3 \\\\sqrt{\\\\pi }/4) \\\\varepsilon ^{-5/2}$ .\",\n \"For a description valid for both the condensed and gas-like phases, we obtain from $Z$ rational expressions for observables of interest where typically leading and next to leading order terms in $\\\\eta $ and $\\\\varepsilon $ are retained in the numerator and the denominator; we calculate and discuss in turn the thermally averaged bond length $\\\\langle l \\\\rangle $ , the internal energy $\\\\langle E \\\\rangle $ , the specific heat $c_{p}$ , and the thermal expansion coefficient $\\\\alpha $ .\",\n \"With the mean interatomic separation being $-R_{e} \\\\partial \\\\ln Z/\\\\partial \\\\eta $ , the normalized thermally averaged bond length is well approximated with $\\\\frac{\\\\langle l \\\\rangle }{R_{e}} = 1 + \\\\eta ^{-1} \\\\left[ \\\\frac{e^{-\\\\varepsilon } - \\\\frac{3}{4} \\\\sqrt{\\\\pi } d_{1} d_{2} \\\\varepsilon ^{-3/2} \\\\eta ^{2} }{e^{-\\\\varepsilon } -\\\\frac{\\\\sqrt{\\\\pi }}{4} \\\\varepsilon ^{-3/2} \\\\eta \\\\left( 4 d_{1} \\\\varepsilon + d_{3} \\\\right)} \\\\right]$ readily inverted via the quadratic formula to obtain $\\\\eta $ in terms of $\\\\langle l \\\\rangle $ .\",\n \"Asymptotically, for $\\\\eta \\\\ll \\\\varepsilon $ , the residual component of the mean separation (small in the condensed phase but diverging with the dissociation transition) may be represented by the simpler expression $\\\\tilde{\\\\Delta }_{l} \\\\equiv (\\\\langle l \\\\rangle /R_{e} - 1) = \\\\varepsilon ^{1/2} \\\\exp -\\\\varepsilon /(d_{1} \\\\sqrt{\\\\pi } \\\\eta ^{2})$ .\",\n \"To more conveniently visualize $\\\\langle l \\\\rangle $ , (as well as $\\\\langle E \\\\rangle $ , $\\\\alpha $ , and $c_{p}$ ), we choose $\\\\varepsilon $ for the abscissa where only $T$ is varied while holding $p$ fixed for a given curve; $\\\\varepsilon \\\\gg 1$ and $\\\\varepsilon \\\\ll 1$ correspond to the low and high $T$ regimes respectively.\",\n \"Whereas $\\\\gamma = p R_{e}/D$ remains constant, $\\\\eta = \\\\beta p R_{e} = \\\\gamma \\\\varepsilon $ varies with temperature.\",\n \"Figure REF displays results corresponding to $V_{\\\\mathrm {LJ}}$ in panel (a) and $V_{\\\\mathrm {M}}$ in panel (b) with open symbols indicating numerical results and solid curves representing the foregoing approximations to $\\\\langle l \\\\rangle $ .\",\n \"The black traces, calculated with the rational expression in Eq.\",\n \"REF , are in good quantitative agreement with numerical data even for $\\\\gamma = 1.0$ .\",\n \"On the other hand, the $\\\\tilde{\\\\Delta }_{l} + 1$ approximation is represented by lighter (red/blue for $V_{\\\\mathrm {LJ}}$ /$V_{\\\\mathrm {M}}$ ) curves, and accurately indicates the location where $\\\\langle l \\\\rangle $ begins to diverge significantly from $R_{e}$ .\",\n \"Moreover, despite the simple structure, good agreement with exact numerical results is evident for $\\\\gamma < 0.01$ .\",\n \"To estimate $\\\\gamma $ for physically realistic systems, we consider a chain of noble gas atoms (e.g.\",\n \"Argon) in the context of the Lennard Jones Model or a covalently bonded pair of atoms (e.g.\",\n \"H$_{2}$ ) in the Morse potential framework in a 1D conduit.\",\n \"With typical atomic radii on the order of an Ångstrom, we assume a cross sectional area on the order of $(1~\\\\textrm {Å})^{2} = 10^{-20}~\\\\textrm {m}^{2}$ .\",\n \"Using $1.01 \\\\times 10^{5}~\\\\textrm {Pa}$ for standard atmospheric pressure, find $p \\\\sim 10^{-15}~J/\\\\textrm {m}$ .\",\n \"In the case of Argon, one has $D = 0.011~\\\\textrm {eV}$ and $R_{e} = 3.8~\\\\textrm {Å}$ [5], [6], [7], [8] and $\\\\gamma = 2.2 \\\\times 10^{-4}$ .\",\n \"On the other hand, for H$_{2}$ , $D = 4.52~\\\\textrm {eV}$ [9] and we take $R_{e}$ to be the measured bond length of $0.74~\\\\textrm {Å}$ [10]; one obtains $\\\\gamma = 1.0 \\\\times 10^{-7}$ , with both $\\\\gamma $ values deep in the $\\\\gamma \\\\ll 1$ range.\",\n \"Figure: (Color online) Normalized mean atomic separations for the Lennard Jones (left panel) and Morse potential (right panel)for various γ\\\\gamma values;solid traces are analytical approximations while open symbols are numerical data.\",\n \"Black curves correspond to the Δ ˜ l +1\\\\tilde{\\\\Delta }_{l}+1relationship, while colored traces represent the rational 〈l〉\\\\langle l \\\\rangle expression.The internal energy, $\\\\langle E \\\\rangle $ , is well approximated as $\\\\frac{\\\\langle E \\\\rangle }{k_{\\\\mathrm {B}} T} = \\\\eta - \\\\varepsilon + \\\\frac{e^{-\\\\varepsilon } \\\\eta ^{-1} (1 + \\\\varepsilon ) - \\\\varepsilon ^{-3/2}\\\\frac{\\\\sqrt{\\\\pi }}{8} (4 d_{1} \\\\varepsilon + 3d_{3} )}{e^{-\\\\varepsilon }\\\\eta ^{-1} - \\\\varepsilon ^{-3/2} \\\\frac{\\\\sqrt{\\\\pi }}{4}(4 d_{1} \\\\varepsilon + d_{3}) }$ To strip away trivial dependencies, Figure REF shows the residual internal energies, $E_{\\\\mathrm {Res}} = \\\\langle E \\\\rangle + D - p R_{e}$ , with Lennard Jones results in the main panel and Morse potential results in the inset for a variety of $\\\\gamma $ values.\",\n \"Solid curves obtained from the rational expression closely coincide with the open symbols representing numerical data.\",\n \"Though $E_{\\\\mathrm {Res}}$ tends to $k_{\\\\mathrm {B}} T/2$ for sufficiently large $\\\\varepsilon $ , the expected dependence where $\\\\langle l \\\\rangle \\\\approx R_{e}$ where anharmonicities are negligible, the residual energy component is non-monotonic.\",\n \"One sees from the main graph and the inset graph that the breadth and height of the peak separating the low and high $T$ regimes scales asymptotically as $\\\\log _{10}(1/\\\\gamma )$ Figure: (Color online) Residual internal energy results for various γ\\\\gamma values with the main graph showing Lennard Jones results and the insetdisplaying Morse results.\",\n \"Solid traces represent analytical results, while open symbols indicate numerical data.The specific heat at constant pressure is $c_{p} = \\\\partial \\\\langle E \\\\rangle /\\\\partial T$ , which is $\\\\frac{c_{p}}{k_{\\\\mathrm {B}}} = \\\\frac{\\\\frac{1}{2} \\\\varepsilon ^{2} \\\\left( \\\\varepsilon + \\\\frac{3 d_{3}}{2d_{1}} \\\\right)- \\\\Upsilon \\\\varepsilon ^{1/2} \\\\left( 1 + \\\\varepsilon + \\\\frac{d3}{4d_{1}} \\\\right) + \\\\Upsilon ^{2}}{\\\\varepsilon ^{-2}\\\\left( \\\\varepsilon + \\\\frac{d_{3}}{2d_{1}} \\\\right)- 2 \\\\Upsilon \\\\varepsilon ^{-3/2} \\\\left( \\\\varepsilon + \\\\frac{d3}{4d_{1}} \\\\right) + \\\\Upsilon ^{2} }$ where $\\\\Upsilon = \\\\exp {-\\\\varepsilon }/( \\\\sqrt{\\\\pi } \\\\eta d_{1} )$ .\",\n \"Specific heat results are displayed in the main graph (for $V_{\\\\mathrm {LJ}}$ ) and the inset (for $V_{\\\\mathrm {M}}$ ) of Fig.\",\n \"REF for a range of $\\\\gamma $ values; as in the case of the internal energy, there is good agreement among the analytical (solid curve) and the numerical (open symbols) results.\",\n \"One sees from the analytical $c_{p}$ expression in Eq.\",\n \"REF and the curves in Fig.\",\n \"REF that the specific heat tends to $k_{\\\\mathrm {B}}$ for sufficiently low $D$ (i.e.\",\n \"higher $T$ ) and flattens to $k_{\\\\mathrm {B}}/2$ for $D \\\\gg 1$ (low $T$ ).\",\n \"Whereas the latter is a hallmark of the condensed phase, the former is expected for a dissociated system; the transition between the two asymptotically flat regions is non-monotonic, with a peak separating the $k_{\\\\mathrm {B}}$ and $k_{\\\\mathrm {B}}/2$ regimes.\",\n \"The region where $c_{p}$ rises to a peak, representing the dissociation transition, becomes taller and narrower with decreasing $\\\\gamma $ .\",\n \"Figure: (Color online) Specific heat curves for assorted γ\\\\gamma values with Lennard Jones results in the main graph and Morse results in theinset.\",\n \"Analytical results are shown as solid traces, while open symbols represent numerical data.The coefficient of thermal expansion $\\\\alpha = \\\\langle l \\\\rangle ^{-1} \\\\partial \\\\langle l \\\\rangle /\\\\partial T$ is well represented by $\\\\frac{\\\\alpha }{k_{\\\\mathrm {B}}} = \\\\frac{\\\\frac{3}{4} \\\\eta d_{2} \\\\varepsilon ^{-1} -\\\\Upsilon \\\\varepsilon ^{1/2} \\\\left( \\\\varepsilon + \\\\frac{3}{2}+ \\\\frac{d_{3}}{4 d_{1}} - \\\\frac{3 \\\\eta d_{2}}{2} \\\\right) + \\\\Upsilon ^{2}}{\\\\eta -\\\\Upsilon \\\\varepsilon ^{-1/2} \\\\left( \\\\varepsilon +2 \\\\eta \\\\varepsilon + \\\\frac{d_{3}}{4 d_{1}}\\\\right) + \\\\Upsilon ^{2}}$ Juxtaposed analytical (solid traces) and numerical results (open symbols) are shown in Figure 4 for various $\\\\gamma $ values with $D \\\\alpha /k_{\\\\mathrm {B}}$ on the vertical axis.\",\n \"As in the case of the specific heat, the $\\\\alpha $ curves are non-monotonic, with peak heights increasing with decreasing $\\\\gamma $ .\",\n \"Choosing $D \\\\alpha /k_{\\\\mathrm {B}}$ for the ordinate is in part to show the slow convergence to the low $T$ value of $3 d_{2}/4$ in the $\\\\gamma \\\\ll 1$ limit.\",\n \"All of the cases shown correspond to experimentally realistic $\\\\gamma $ values, and in all but one of the curves shown, $\\\\alpha $ is appreciably different from the limiting value even for $\\\\varepsilon $ as high as 10.\",\n \"Figure: (Color online) Thermal expansion coefficient results for a range of γ\\\\gamma values with Lennard Jones results plotted in the maingraph and corresponding results for the Morse potential in the inset.\",\n \"Solid curves represent analytical results, while open symbols indicate numerical data.The peak locations calculated in the framework of the analytical approximations (solid traces) for $c_{p}$ and $\\\\alpha $ are in close agreement with the exact numerical results (open symbols) as may be seen in the upper panels of Fig.\",\n \"REF .\",\n \"In statistical mechanics singularities are not in general encountered for single component systems, with non-analytic behavior emerging only in thermodynamic limit as the number of degrees of freedom tends to infinity.\",\n \"Nevertheless, the low pressure regime is atypical in the sense that singular behavior is inevitable as $p$ (or $\\\\gamma $ ) tends to zero due to the divergence of $\\\\langle l \\\\rangle $ for any finite $T$ as $p \\\\rightarrow 0$ .\",\n \"Hence, the possibility of a sharp dissociation transition for $\\\\gamma \\\\ll 1$ must be examined with care.\",\n \"As a measure of the extent to which the dissociation transition is singular, the sharpness of the thermal expansion coefficient and specific heat peaks is quantified in the lower panels of Fig.\",\n \"REF as the relative full width half maximum, $\\\\Delta T_{\\\\mathrm {Peak}}/T_{\\\\mathrm {Peak}}$ .\",\n \"In the case of $\\\\alpha $ , $\\\\Delta T_{\\\\mathrm {Peak}}/T_{\\\\mathrm {Peak}}$ tends to a finite value common to both $V_{\\\\mathrm {LJ}}$ and $V_{\\\\mathrm {M}}$ , indicating the $\\\\alpha $ peaks cease to become narrower relative to their location with decreasing $\\\\gamma $ .\",\n \"On the other hand, the specific heat relative peak width appears to tend to zero with decreasing $\\\\gamma $ , a trend highlighted in the lower right panel inset of Fig.REF showing $\\\\Delta T_{\\\\mathrm {Peak}}/T_{\\\\mathrm {Peak}}$ relative to $1/\\\\log _{10} (\\\\gamma ^{-1})$ , The curves are asymptotically linear as $\\\\gamma \\\\rightarrow 0$ , with the relative peak width vanishing for $p \\\\rightarrow 0$ as singular behavior appears.\",\n \"Figure: (Color online) Peak locations for the specific heat (upper right panel) and thermal expansion coefficient (upper left panel).Corresponding relative full width half maxima (i.e.\",\n \"ΔT peak /T peak \\\\Delta T_{\\\\mathrm {peak}}/T_{\\\\mathrm {peak}}) appear in the lower right and lower left panels for thespecific heat and coefficient of thermal expansion respectively.Throughout, open squares represent Lennard Jones results and open circles Morse results.In conclusion, with a nonperturbative treatment of the anharmonicity of interatomic potentials, we provide a theoretical description of the dissociation transition valid for the condensed state as well as the gas-like phase where thermal fluctuations have driven pairs of atoms far from their equilibrium separations.\",\n \"By applying a unified treatment to disparate potentials, we have obtained analytical results for salient thermodynamic observables in good agreement with precise numerical results.\",\n \"Though ascribed to distinct bonding physics, there are striking similarities in results, e.g.\",\n \"with $\\\\langle l \\\\rangle /R_{e} = 1 + \\\\varepsilon ^{1/2} \\\\exp -\\\\varepsilon /(d_{1} \\\\sqrt{\\\\pi } \\\\eta ^{2})$ specifying the mean bond length for $\\\\gamma \\\\ll 1$ .\",\n \"Useful conversations with Michael Crescimanno are gratefully acknowledged.\"\n ]\n]"},"id":{"kind":"string","value":"1605.06082"}}},{"rowIdx":1866691,"cells":{"text":{"kind":"list like","value":[["Ancilla-free Reversible Logic Synthesis via Sorting"],["Abstract Reversible logic synthesis is emerging as a major research component for post-CMOS computing devices, in particular Quantum computing.","In this work, we link the reversible logic synthesis problem to sorting algorithms.","Based on our analysis, an alternative derivation of the worst-case complexity of generated reversible circuits is provided.","Furthermore, a novel column-wise reversible logic synthesis method, termed RevCol, is designed with inspiration from radix sort.","Extending the principles of RevCol, we present a hybrid reversible logic synthesis framework.","The theoretical and experimental results are presented.","The results are extensively benchmarked with state-of-the-art ancilla-free reversible logic synthesis methods."],["=1 pdfinfo= Title=Ancilla-free Reversible Logic Synthesis via Sorting, Author=Anupam Chattopadhyay and Sharif MD Khairul Hossain, Subject=, Keywords= [pages=1-last]paper.pdf"]],"string":"[\n [\n \"Ancilla-free Reversible Logic Synthesis via Sorting\"\n ],\n [\n \"Abstract Reversible logic synthesis is emerging as a major research component for post-CMOS computing devices, in particular Quantum computing.\",\n \"In this work, we link the reversible logic synthesis problem to sorting algorithms.\",\n \"Based on our analysis, an alternative derivation of the worst-case complexity of generated reversible circuits is provided.\",\n \"Furthermore, a novel column-wise reversible logic synthesis method, termed RevCol, is designed with inspiration from radix sort.\",\n \"Extending the principles of RevCol, we present a hybrid reversible logic synthesis framework.\",\n \"The theoretical and experimental results are presented.\",\n \"The results are extensively benchmarked with state-of-the-art ancilla-free reversible logic synthesis methods.\"\n ],\n [\n \"=1 pdfinfo= Title=Ancilla-free Reversible Logic Synthesis via Sorting, Author=Anupam Chattopadhyay and Sharif MD Khairul Hossain, Subject=, Keywords= [pages=1-last]paper.pdf\"\n ]\n]"},"id":{"kind":"string","value":"1605.05989"}}},{"rowIdx":1866692,"cells":{"text":{"kind":"list like","value":[["Algebraically special Einstein-Maxwell fields"],["Abstract The Geroch-Held-Penrose formalism is used to re-analyse algebraically special non-null Einstein-Maxwell fields, aligned as well as non-aligned, in the presence of a possible non-vanishing cosmological constant.","A new invariant characterisation is given of the Garcia-Plebanski and Plebanski-Hacyan metrics within the family of aligned solutions and of the Griffiths metrics within the family of the non-aligned solutions.","As a corollary also the double alignment of the Debever-McLenaghan 'class D' metrics with non-vanishing cosmological constant is shown to be equivalent with the shear-free and geodesic behaviour of their Debever-Penrose vectors."],["Introduction","In the quest for exact solutions of the Einstein-Maxwell equations, $R_{ab}-\\frac{1}{2}R g_{ab}+\\Lambda g_{ab} = F_{ac} {F_b}^c-\\frac{1}{4}g_{ab}F_{cd}F^{cd},$ a large amount of research (see for example the reviews in [21], [40] has been devoted to the study of so called aligned Einstein-Maxwell fields, in which at least one of the principal null directions (PND's) of the electromagnetic field tensor ${F}$ is parallel to a PND of the Weyl tensor, also called a Debever-Penrose direction, with main emphasis on the doubly aligned Petrov type D solutions, in which both real PND's of ${F}$ are parallel to a corresponding double Weyl-PND.","One of the prominent tools in these and related activities has been the Goldberg-Sachs theorem, which in its original form [18] says that a vacuum space-time is algebraically special if and only if it contains a shear-free geodesic null congruence (in an adapted Newman-Penrose tetrad, `$\\Psi _0=\\Psi _1=0$ if and only if $\\kappa =\\sigma =0$ ').","Goldberg and Sachs also proved that, if a space-time admits a complex null tetrad $({k},{\\ell },{m},\\overline{{m}})$ such that ${k}$ is shear-free and geodesic and $R_{ab}k^a k^b = R_{ab}k^a m^b = R_{ab}m^a m^b = 0$ (as is the case when ${k}$ is a PND of the electromagnetic field tensor), then the Weyl tensor is algebraically special, with ${k}$ being a multiple Weyl-PND.","While, for a null Maxwell field [29], [38], the Maxwell and Bianchi equations imply that both conditions $\\Psi _0=\\Psi _1=0$ and $\\kappa =\\sigma =0$ are trivially satisfied (also in the presence of a possible cosmological constant), the situation is less straightforward in the non-null case.","It still is true [18], [39] that, when a PND ${k}$ of the electromagnetic field tensor is shear-free and geodesic, then the Weyl tensor is algebraically special, but the reverse no longer holds.","A key property in this respect is the Kundt-Trümper theorem [26], which says that for an algebraically special aligned non-null Einstein-Maxwell space-time (with a possible non-0 cosmological constant and with ${k}$ the PND of ${F}$ coinciding with a multiple Weyl-PND) one necessarily has $\\kappa (3 \\Psi _2-2 |\\Phi _1|^2)=0 \\textrm { and }\\sigma (3 \\Psi _2+2 |\\Phi _1|^2)=0,$ implying that the exceptional case $|\\kappa |^2+|\\sigma |^2 \\ne 0$ can only occur for Petrov types II or D with $\\frac{3}{2}\\Psi _2=\\pm |\\Phi _1|^2.$ Subsequent research has been concentrated on the doubly aligned Petrov type D cases, in which the Kundt-Trümper relations (REF ) also hold with $\\kappa $ and $\\sigma $ replaced by $\\nu $ and $\\lambda $ .","This culminated in the complete integration of the field equations for the Petrov type D doubly aligned non-null Einstein-Maxwell fields, with a possible non-0 cosmological constant: When $\\frac{9}{4}\\Psi _2^2- |\\Phi _1|^4\\ne 0$ , both real PND's are geodesic and shear-free ($\\kappa =\\nu =\\sigma =\\lambda =0$ ) and, following Debever and McLenaghan[10], I will refer to the corresponding set of solutions as the `class $\\mathcal {D}$ ' space-times.","They all admit at least a two-dimensional isometry group and count among their most famous members the Reissner-Nordström and Kerr-Newman metrics.","Several authors [5], [6], [7], [8], [9], [10], [11], [12], [14], [15], [22], [24], [33], [34] have independently contributed to the determination of class $\\mathcal {D}$ , beginning with Carter's seminal study of the separability of the Hamilton-Jacobi and Klein-Gordon equations and culminating in the discovery [33] of Plebański and Demiański's 7-parameter metric for the non-null orbit solutions and García's construction [14], [15] of a single metric form for both the non-null and null orbit solutions.","The case $\\frac{3}{2}\\Psi _2= |\\Phi _1|^2$ was fully integrated in [35].","In the resulting `Plebański-Hacyan space-times' ${k}$ and ${\\ell }$ are respectively non-geodesic ($\\kappa \\ne 0$ ) and geodesic ($\\nu =0$ ), while both are shear-free ($\\sigma =\\lambda =0$ ) and have vanishing complex divergence ($\\rho =\\mu =0$ ).","The metric is given by $\\textrm {d}s^2= 2\\textrm {d}\\zeta \\textrm {d}\\overline{\\zeta }+2 \\textrm {d}u\\textrm {d}v+[\\Lambda u^2+\\zeta \\overline{F}(v)+\\overline{\\zeta } F(v)] \\textrm {d}v^2,$ which, for an arbitrary function $F(v)$Note1 does not admit any isometries and only has an electromagnetic energy-momentum tensor of the correct sign when $\\Lambda <0$ .","The case $\\frac{3}{2}\\Psi _2= -|\\Phi _1|^2$ , which was overlooked in [35], was dealt with in [16].","Its unique solutions are the `García-Plebański metrics', $\\textrm {d}s^2 = -\\frac{1}{\\Lambda }\\left( e^{2z}{\\omega ^1}^2+e^{-2 z}{\\omega ^2}^2+\\textrm {d}z^2-4 (\\cosh z)^2{\\omega ^3}^2 \\right),$ with $\\omega ^1+i \\omega ^2 = 2\\frac{e^{iu}}{1-\\zeta \\overline{\\zeta }}\\textrm {d}\\zeta \\textrm { and } \\omega ^3 = \\textrm {d}u-i \\frac{\\zeta \\textrm {d}\\overline{\\zeta }-\\overline{\\zeta }\\textrm {d}\\zeta }{1-\\zeta \\overline{\\zeta }}.$ The corresponding space-times admit a 3 dimensional group of isometries; both ${k}$ and ${\\ell }$ are geodesic ($\\kappa =\\nu =0$ ), shearing ($\\sigma \\lambda \\ne 0$ ) and twisting, but non-expanding ($\\rho , \\mu \\in i \\mathbb {R}$ ).","Also this metric can only describe an Einstein-Maxwell space-time with an electromagnetic energy-momentum tensor of the correct sign when $\\Lambda <0$ .","In this paper I will have a closer look at the full set of algebraically special cases, both aligned (and hence by (REF ) necessarily of Petrov type II) and non-aligned ones.","First we will see that the García-Plebański metrics are the unique members of the class of algebraically special and aligned Einstein-Maxwell solutions for which the null direction ${k}$ is shearing.","While doing so, I will correct an error in [25] (cited also in [40] p. 409), claiming that, at least for $\\Lambda =0$ , the case $\\kappa =0\\ne \\sigma $ admits no solutions.","This is true indeed but, as will become clear in section 2, the proof requires a much more subtle argumentation than the one presented in [25]Note2 .","The error occurs after relation (2.6) when the case $\\overline{\\rho }=-\\rho $ is dismissed by remarking that it ”leads to $\\rho =0$ ”.","Most likely this conclusion was prematurely arrived at by inspection of the Newman-Penrose equation corresponding to our GHP equation (REF ): with $\\kappa =\\epsilon +\\overline{\\epsilon }=\\Phi _0=0$ this equation reads $D\\rho = \\rho ^2+\\sigma \\overline{\\sigma }$ , the real and imaginary parts of which only allow one to conclude that $\\rho =\\pm i |\\sigma |$ and $D \\rho =D \\sigma = 0$ .","In section 2 a correct proof of Kozarzewski's no-go claim will be provided, generalizing it to the case $\\Lambda \\geqslant 0$ and showing that for $\\Lambda <0$ the only allowed solutions are the doubly aligned García-Plebański metrics with $\\rho = \\pm i |\\sigma | \\ne 0$ .","In section 3 I will consider the algebraically special and aligned Einstein-Maxwell fields for which the null direction ${k}$ is non-geodesic.","The general solution in this family so far is not known, but, remarkably, the Plebański-Hacyan metrics exhaust the sub-family characterised by the vanishing of the complex divergence of ${k}$ .","Finally I prove in section 4 that an algebraically special Einstein-Maxwell solution possessing a shear-free and geodesic multiple Weyl-PND which is not a PND of ${F}$ necessarily has vanishing cosmological constant and I give a characterization of the sub-class of the Griffiths [20] solutions.","As a corollary of this theorem it follows that the `class $\\mathcal {D}$ ' metrics[10] are the unique Petrov type D Einstein-Maxwell solutions for which the real Weyl-PND's are both geodesic and shear-free and for which the cosmological constant is non-vanishing: in other words, the double alignment condition of the class $\\mathcal {D}$ metrics with non-vanishing cosmological constant is a consequence of their multiple Weyl-PND's being geodesic and shear-free.","Whether this property persists when $\\Lambda =0$ is at present still an open problem, with only the Kundt case ($\\rho =0$ ) so far having been dealt with.","In order to study these and related issues, linking kinematic properties of certain invariantly defined null directions (such as being geodesic and/or non-shearing) to algebraic properties of the electromagnetic field tensor or of the Weyl tensor, it is natural to use the Geroch-Held-Penrose (GHP) formalism [17].","In this formalism only a pair of null directions is singled out at each point rather than an entire null tetrad, as is the case in the Newman-Penrose [31] formalism.","The resulting formalism is covariant with respect to rotations of the spatial basis vectors and boosts of the real null directions and, as such, is `halfway' between a fully covariant approach and the NP spin-coefficient approach and leads to considerably simpler equations with fewer complex variables.","Throughout I will use the sign conventions and notations of Kramer §7.4, with the tetrad basis vectors taken as ${k},{\\ell },{m},\\overline{{m}}$ with $- k^a \\ell _a = 1 = m^a\\overline{m}_a$ .","However, in order to ease comparison with the (more familiar) Newman-Penrose formalism, I will write primed variables, such as $\\kappa ^{\\prime },\\sigma ^{\\prime },\\rho ^{\\prime }$ and $\\tau ^{\\prime }$ , as their NP equivalents $-\\nu ,-\\lambda , -\\mu $ and $-\\pi $ .","For completeness the resulting weights, commutators, GHP, Bianchi and Maxwell equations are presented in an Appendix.","Finally note that, at least when the electromagnetic field is non-null, the pair of null directions ${k},{\\ell }$ can always be invariantly defined, aligning for example ${k}$ with a PND of ${F}$ and null-rotating about ${k}$ such that $\\Phi _1$ is the only non-vanishing component of the Maxwell spinor.","Obviously this choice is not unique (for example in the non-aligned case it will be preferable to align ${k}$ with the multiple Weyl-PND and to null-rotate about ${k}$ such that $\\Phi _1=0$ and $\\Phi _0 \\Phi _2\\ne 0$ ), but it is important to realise that any ensuing well-weighted GHP relations, such as $\\kappa =0, \\rho =0, \\ldots $ are automatically geometrically invariant statements."],["Aligned electrovacs with a shearing multiple Debever-Penrose vector","Let us first, for the sake of completeness, re-confirm the well-known fact [18], [39] that if a PND ${k}$ of the electromagnetic field tensor ${F}$ is shear-free and geodesic, then the Weyl tensor is algebraically special with ${k}$ being the multiple Weyl-PND: with ${k}$ a PND of ${F}$ ($\\Phi _0=0$ ) satisfying $\\kappa =\\sigma =0$ , it follows from () that $\\Psi _0=0$ and hence ${k}$ is also a Weyl-PND.","When ${F}$ is non-null we can null-rotate about ${k}$ such that also $\\Phi _2=0$ .","The integrability conditions for the Maxwell equations (REF , ) simplify then with (, ) to $\\Phi _1 \\Psi _1=0$ .","It follows that $\\Psi _1=0$ and hence ${k}$ is a multiple Weyl-PND.","Now let us consider the reverse situation and assume that the Weyl tensor is algebraically special, with the multiple Weyl-PND ${k}$ being also a PND of the non-null electromagnetic field tensor: by means of a suitable null rotation about ${k}$ the null tetrad can be chosen such that $\\Psi _0=\\Psi _1=\\Phi _0=\\Phi _2=0$ .","The Maxwell equations (REF , ') and () become $\\textrm {Þ}\\Phi _1=2 \\rho \\Phi _1, \\ \\eth \\Phi _1 =2\\tau \\Phi _1,\\ \\eth ^{\\prime }\\Phi _1 =- 2\\pi \\Phi _1,$ while the Bianchi equations (REF , ) yield the Kundt-Trümper relations (REF ).","In the present paragraph we consider the case where ${k}$ is shearing and hence $\\kappa =0,\\ \\Psi _2 = -\\frac{2}{3} |\\Phi _1|^2 .$ Bianchi equation (REF ) simplifies now to $|\\Phi _1|^2 (\\rho +\\overline{\\rho })=0$ , implying that ${k}$ is non-expanding.","From the real part of GHP equation (REF ) it follows that $\\rho $ is related to $\\sigma $ by $\\rho = \\pm i |\\sigma |$ and hence, by (REF , , ) $\\textrm {Þ}\\rho =0,\\ \\textrm {Þ}\\sigma =0,\\ \\textrm {Þ}\\tau = \\rho (\\tau +\\overline{\\pi })+\\sigma (\\overline{\\tau }+\\pi ).$ From (,') one finds then $\\Psi _3 = -\\frac{2}{3} \\frac{|\\Phi _1|^2}{\\sigma } (2\\tau +\\overline{\\pi }),$ and $\\textrm {Þ}\\pi = 2(\\overline{\\pi }\\overline{\\sigma }-\\overline{\\tau }\\rho ) .$ Acting with the $[\\eth ,\\, \\textrm {Þ}]$ commutator on $\\Phi _1$ now leads to an expression for $\\eth \\rho $ , which, together with () and the $\\eth $ derivative of (REF ), yields $\\eth \\rho &=& \\sigma (\\overline{\\tau }+2 \\pi )+2\\rho \\tau ,\\\\\\eth ^{\\prime }\\sigma &=& \\sigma (\\overline{\\tau }+2\\pi ),\\\\\\eth \\sigma &=& \\sigma (3\\tau -2\\overline{\\pi })-2\\rho \\frac{\\sigma }{\\overline{\\sigma }}(\\overline{\\tau }+2 \\pi ).$ The remaining integrability conditions for the Maxwell equations can then be written as $\\textrm {Þ}^{\\prime }\\tau +\\eth \\mu &=& \\rho \\overline{\\nu }+\\overline{\\lambda }\\pi ,\\\\\\textrm {Þ}^{\\prime }\\pi -\\eth ^{\\prime }\\mu &=& \\lambda \\tau -\\nu \\rho -\\mu \\overline{\\tau }-\\overline{\\mu }\\pi ,\\\\\\eth \\pi +\\eth ^{\\prime }\\tau &=& \\rho (\\mu +\\overline{\\mu }),\\\\\\textrm {Þ}\\mu +\\textrm {Þ}^{\\prime }\\rho &=& \\pi \\overline{\\pi }-\\tau \\overline{\\tau }.$ Next we apply the $[\\eth ,\\, \\textrm {Þ}]$ commutator to $\\sigma $ , to obtain an expression for $\\tau $ , $\\tau = \\textstyle {\\frac{1}{3}}\\overline{\\pi }+\\frac{4}{3}\\frac{\\rho \\pi }{\\overline{\\sigma }},$ which together with its $\\textrm {Þ}$ derivative leads to $\\sigma \\pi -\\rho \\overline{\\pi }=0 .$ This suggests the definition of an auxiliary variable $S$ (by (REF ) one has $|S|=1$ , while $\\textrm {weight}(S)=[2,-2]$ ) such that $\\sigma = S \\rho $ and enabling us to conclude from (REF ) that $\\tau =-\\overline{\\pi },\\ \\overline{\\pi }= S \\pi .$ By (REF ,REF –) one has then $\\textrm {Þ}S =0,\\ \\eth S=-S (3 \\overline{\\pi }-S \\pi ),\\ \\eth ^{\\prime }S=3 S \\pi -\\overline{\\pi },$ after which (',REF ,REF ') yield $\\eth \\pi &=& S \\pi ^2 +2 \\rho (\\mu +\\overline{\\mu })-\\frac{3}{2} S \\rho ^2 \\Psi _4 |\\Phi _1|^{-2}, \\\\\\eth ^{\\prime }\\pi &=& -\\pi ^2-2 S^{-1} \\rho (\\mu +\\overline{\\mu })-\\frac{3}{2} S^{-2} \\rho ^2 \\overline{\\Psi _4} |\\Phi _1|^{-2},\\\\\\textrm {Þ}\\Psi _4 &=& \\rho (\\Psi _4 -S^{-2}\\overline{\\Psi _4})-\\frac{4}{3} S^{-1}(\\mu +\\overline{\\mu })|\\Phi _1|^2.$ Herewith () reduces to $2 (\\mu +\\overline{\\mu }){|\\Phi _1|^2}+\\rho (S^{-1}\\overline{\\Psi _4}-S\\Psi _4)= 0,$ while expressing that $\\textrm {Þ}^{\\prime }(S\\overline{S})=0$ (with $\\textrm {Þ}^{\\prime }S$ evaluated from ()) and simplifying the result by means of (REF ), one finds that $\\mu +\\overline{\\mu }= 0$ (i.e.","also ${\\ell }$ is non-expanding) and hence $\\overline{\\Psi _4} = S^2 \\Psi _4$ .","Furthermore, calculating $\\textrm {Þ}^{\\prime }\\rho $ from () and expressing that $\\textrm {Þ}^{\\prime }(\\rho +\\overline{\\rho })=0$ gives us an equation from which we can obtain $\\Psi _4, \\textrm {Þ}^{\\prime }\\rho $ and $\\textrm {Þ}^{\\prime }S$ : $\\Psi _4 &=& \\frac{|\\Phi _1|^2}{18\\rho ^2} \\left[ 24\\pi ^2+6\\rho (\\lambda -S^{-2}\\overline{\\lambda })-S^{-1} (8 |\\Phi _1|^2+12 \\rho \\mu -R) \\right], \\\\\\textrm {Þ}^{\\prime }\\rho &=& -\\frac{\\rho }{2}(S\\lambda +S^{-1}\\overline{\\lambda }),\\\\\\textrm {Þ}^{\\prime }S &=& S \\left[ -2 \\mu +S \\lambda -S^{-1}\\overline{\\lambda }+\\frac{1}{\\rho }(2 S \\pi ^2-\\textstyle {\\frac{2}{3}}|\\Phi _1|^2+\\textstyle {\\frac{1}{12}} R) \\right].$ Next we solve (REF ', ') together with () and the $[\\textrm {Þ}^{\\prime },\\, \\textrm {Þ}]\\rho ,\\ [\\eth ,\\, \\textrm {Þ}^{\\prime }] S$ commutator relations for $\\textrm {Þ}^{\\prime }\\pi $ , $\\eth \\mu $ , $\\textrm {Þ}\\mu $ , $\\textrm {Þ}\\lambda $ , $\\eth \\lambda $ and $\\eth ^{\\prime }\\lambda $ : $\\textrm {Þ}^{\\prime }\\pi &=& -\\textstyle {\\frac{3}{2}} S \\lambda \\pi +\\textstyle {\\frac{1}{2}}S^{-1}\\rho \\overline{\\nu }+S^{-1}\\overline{\\lambda }\\pi +\\textstyle {\\frac{1}{2}}\\mu \\pi -\\textstyle {\\frac{5}{2}}\\nu \\rho +( \\textstyle {\\frac{4}{3}}|\\Phi _1|^2-3 S \\pi ^2 -\\textstyle {\\frac{1}{8}} R)\\frac{\\pi }{\\rho },\\\\\\eth \\mu &=& -\\textstyle {\\frac{5}{2}} S\\rho \\nu -\\textstyle {\\frac{1}{2}}\\lambda \\pi S^2-\\textstyle {\\frac{3}{2}}S \\mu \\pi +\\textstyle {\\frac{3}{2}}\\rho \\overline{\\nu }+\\overline{\\lambda }\\pi +(\\textstyle {\\frac{2}{3}} |\\Phi _1|^2-S\\pi ^2-\\textstyle {\\frac{1}{24}} R)\\frac{S \\pi }{\\rho },\\\\\\textrm {Þ}\\mu &=& \\textstyle {\\frac{1}{2}}\\rho (S\\lambda +S^{-1} \\overline{\\lambda }),\\\\\\textrm {Þ}\\lambda &=& \\textstyle {\\frac{1}{2}}\\rho (\\lambda +S^{-1}\\overline{\\lambda })-2 \\pi ^2-\\textstyle {\\frac{1}{12}} S^{-1}(R-8 |\\Phi _1|^2), \\\\\\eth \\lambda &=& -\\textstyle {\\frac{5}{2}}(\\mu \\pi +\\nu \\rho )+\\textstyle {\\frac{3}{2}}S \\lambda \\pi +\\textstyle {\\frac{7}{2}} S^{-1}\\rho \\overline{\\nu }+(S \\pi ^2 -\\textstyle {\\frac{8}{3}}|\\Phi _1|^2+\\textstyle {\\frac{1}{24}}R)\\frac{\\pi }{\\rho },\\\\\\eth ^{\\prime }\\lambda &=& \\textstyle {\\frac{1}{2}}S^{-1}(\\mu \\pi +\\rho \\nu )+\\textstyle {\\frac{3}{2}}S^{-2}(3\\rho \\overline{\\nu }- \\overline{\\lambda }\\pi )-3\\lambda \\pi +S^{-1}(3 S\\pi ^2-\\textstyle {\\frac{8}{3}} |\\Phi _1|^2+\\textstyle {\\frac{1}{8}} R)\\frac{\\pi }{\\rho },$ after which an expression for the spin coefficient $\\mu $ follows from the $[\\eth ^{\\prime },\\, \\eth ]S$ commutator relation, $\\mu = \\frac{1}{2}(S\\lambda -S^{-1} \\overline{\\lambda })+\\frac{1}{36\\rho }(24 S \\pi ^2-32 |\\Phi _1|^2+R).$ Of the Maxwell integrability conditions there only remains now the $[\\eth ^{\\prime },\\, \\textrm {Þ}^{\\prime }]\\Phi _1$ relation, namely $6 (S \\lambda +S^{-1} \\overline{\\lambda })\\rho \\pi +\\pi (8 |\\Phi _1|^2-24 S \\pi ^2-R) -36\\rho ^2(\\nu - S^{-1}\\overline{\\nu }) = 0,$ which with GHP equation (') gets simplified to the key algebraic equation $\\pi (8|\\Phi _1|^2-24 S \\pi ^2-R)=0.$ Herewith all Bianchi equations and GHP equations (except those involving the derivatives of $\\nu $ ) are identically satisfied.","At this stage the Weyl spinor components are given by, using (REF ) to simplify (REF ), $\\Psi _0 &=& \\Psi _1=0,\\\\\\Psi _2 &=& -\\textstyle {\\frac{2}{3}}|\\Phi _1|^2,\\\\\\Psi _3 &=& \\textstyle {\\frac{2}{3}}\\frac{\\pi }{\\rho }|\\Phi _1|^2 , \\\\\\Psi _4 &=& \\frac{|\\Phi _1|^2}{27 \\rho ^2}S^{-1}(24 S \\pi ^2+4|\\Phi _1|^2+R).","$ I now show that the case $\\pi \\ne 0$ is inconsistent, while the case $\\pi =0$ leads to the García-Plebański metrics."],["$\\pi \\ne 0$","When $\\pi \\ne 0$ we can use (REF ) to rewrite (REF ) as $\\pi (S\\lambda +S^{-1}\\overline{\\lambda })-6\\rho (\\nu -S^{-1} \\overline{\\nu })=0,$ while (REF ) becomes $\\mu =\\textstyle {\\frac{1}{2}}(S\\lambda -S^{-1}\\overline{\\lambda })-\\textstyle {\\frac{2}{3}}|\\Phi _1|^2\\rho ^{-1}.$ Taking the $\\eth $ derivative of (REF ) and eliminating $\\pi $ from the resulting equation and (REF ), we also find $3\\rho (S\\lambda -\\overline{S}\\overline{\\lambda })-4|\\Phi _1|^2=0$ and hence, by (REF ), $\\mu =0$ .","The $\\textrm {Þ}$ derivative of (REF ) yields then $S\\lambda +S^{-1} \\overline{\\lambda }=0$ and hence, by (REF ), $\\lambda =\\textstyle {\\frac{2}{3}}|\\Phi _1|^2 S^{-1} \\rho ^{-1}$ .","Substituting this in the expression () for $\\eth \\lambda $ one obtains $\\nu =\\textstyle {\\frac{2}{3}}|\\Phi _1|^2\\pi \\rho ^{-2}$ , which together with GHP equation (') leads to the contradiction $|\\Phi _1|^2\\pi =0$ ."],["$\\pi =0$","Substituting $\\pi =0$ in (REF ,REF ) one obtains $R \\, (=4 \\Lambda ) =-4|\\Phi _1|^2 $ and $\\overline{\\nu }-5 S \\nu =0,$ whence $\\nu =0$ .","While (REF ) proves Kozarzewski's no-go claim for $\\Lambda =0$ , it also generalizes it to the case $\\Lambda \\geqslant 0$ .","Furthermore, when $\\Lambda < 0$ the only non-0 Weyl spinor component is $\\Psi _2=R/6$ and we are in the doubly aligned situation with both Weyl-PND's ${k}$ and ${\\ell }$ being geodesic and non-expanding.","The field equations and Maxwell equations have been completely integrated in this case by García and Plebański [16] and the resulting metric is given by (REF ), admitting a 3D isometry group."],["Aligned electrovacs with a non-geodesic multiple Debever-Penrose vector","We again consider the case where the Weyl tensor is algebraically special, with the multiple Weyl-PND ${k}$ being also a PND of the non-null electromagnetic field tensor, but now we take $\\kappa \\ne 0$ and hence, by (REF ), $\\sigma =0 \\textrm { and } \\Psi _2 = \\frac{2}{3} |\\Phi _1|^2 .$ Bianchi equation () now reduces to $|\\Phi _1|^2 (\\tau -\\overline{\\pi })=0$ , whence $\\tau =\\overline{\\pi }$ , after which (REF ) implies $\\Psi _3=\\frac{2}{3}\\frac{2\\rho -\\overline{\\rho }}{\\kappa }|\\Phi _1|^2, $ showing that solutions cannot be of Petrov type III or N when $\\rho \\ne 0$ .","GHP equations (,,) and Bianchi equation (REF ) yield then $\\eth \\kappa &=& 0 ,\\\\\\eth \\rho &=& \\overline{\\pi }(\\rho -\\overline{\\rho })+\\kappa (\\mu -\\overline{\\mu }),\\\\\\eth ^{\\prime }\\rho &=& -2 \\pi \\overline{\\rho }-2 \\overline{\\kappa }\\overline{\\mu },\\\\\\eth ^{\\prime }\\pi &=& \\pi ^2+\\lambda \\overline{\\rho }-\\nu \\overline{\\kappa }.$ The integrability conditions for the Maxwell equations furthermore give an expression for $\\textrm {Þ}\\pi $ , $\\textrm {Þ}\\pi = \\overline{\\kappa }(2\\overline{\\mu }-\\mu )+2\\pi \\overline{\\rho },$ together with four extra relations $\\eth \\mu +\\textrm {Þ}^{\\prime }\\overline{\\pi }&=& \\overline{\\nu }\\rho +\\overline{\\lambda }\\pi ,\\\\\\eth ^{\\prime }\\mu -\\textrm {Þ}^{\\prime }\\pi &=& \\pi (\\overline{\\mu }+\\mu )-\\lambda \\overline{\\pi }+\\nu \\rho ,\\\\\\eth \\pi +\\eth ^{\\prime }\\overline{\\pi }&=& \\rho \\overline{\\mu }-\\overline{\\rho }\\mu , \\\\\\textrm {Þ}\\mu +\\textrm {Þ}^{\\prime }\\rho &=& 0,$ with which GHP equations (,') imply $\\textrm {Þ}^{\\prime }\\kappa &=& \\kappa (2\\mu -\\overline{\\mu }),\\\\\\textrm {Þ}\\lambda &=& \\lambda (\\rho +\\overline{\\rho })+2\\pi ^2-2\\nu \\overline{\\kappa }.$ So far it has not been possible to complete the analysis of this case.","However, it is easy to see that the Plebański-Hacyan metrics completely exhaust the non-diverging subfamily of solutions ($\\rho =0$ ) and have $\\Lambda <0$ , whereas for $\\Lambda \\geqslant 0$ no solutions exist.","In fact, when $\\rho =0$ (REF ) and $\\kappa \\ne 0$ imply that also the second PND ${\\ell }$ of the electromagnetic field is non-diverging ($\\mu =0$ ): herewith (REF ) and (') show that $\\Psi _3=\\Psi _4=0$ , so that we are in the doubly aligned Petrov type D case.","This suffices already to conclude that the only possible solutions are the Plebański-Hacyan metrics, as this case was completely integrated in [35], modulo the error corrected in [16].","However, it is instructive to provide a short and coordinate independent proof of the essential step in [35], namely the part in which the authors prove that $\\Gamma _{423}=0$ , leading to the conclusion that ${\\ell }$ not only is shear-free ($\\lambda =0$ being an immediate consequence of (')), but also is geodesic ($\\nu =0$ ): First observe that the $[\\eth ,\\, \\textrm {Þ}^{\\prime }]$ commutator applied to $\\kappa $ implies $\\overline{\\nu }\\textrm {Þ}\\kappa =0$ .","Assuming $\\nu \\ne 0$ leads then to an inconsistency, as we would have $\\textrm {Þ}\\kappa =0, \\ \\eth ^{\\prime }\\kappa = \\overline{\\kappa }\\overline{\\pi }-\\kappa \\pi ,$ with the second relation being obtained from (REF ).","Herewith the $[\\textrm {Þ},\\, \\textrm {Þ}^{\\prime }]\\kappa $ commutator relation yields $\\kappa (\\overline{\\kappa }\\overline{\\nu }+3\\kappa \\nu -6\\pi \\overline{\\pi }+\\textstyle {\\frac{1}{6}}R-\\textstyle {\\frac{20}{3}}|\\Phi _1|^2)+2\\overline{\\kappa }\\overline{\\pi }^2=0,$ which, with $\\nu \\overline{\\kappa }=\\pi ^2$ (obtained from (')) simplifies to $|\\kappa |^2(40|\\Phi _1|^2-R)-18(\\kappa \\pi -\\overline{\\kappa }\\overline{\\pi })^2=0.$ On the other hand ('-$\\overline{\\ref {ghp5d}}$ ) reads $|\\kappa |^2(8|\\Phi _1|^2+R)-6(\\kappa \\pi -\\overline{\\kappa }\\overline{\\pi })^2=0,$ which, when added to (REF ), results in $2|\\kappa \\Phi _1|^2 - (\\kappa \\pi -\\overline{\\kappa }\\overline{\\pi })^2 =0.$ The left hand side being positive definite, this shows that the case $\\nu \\ne 0$ is inconsistent.","We therefore have $\\nu =\\pi =0$ and hence, by GHP equation (), $R = 4\\Lambda =-8|\\Phi _1|^2 < 0$ ."],["Electrovacs with a shear-free and geodesic multiple Debever-Penrose vector","We now consider algebraically special (non-conformally flat) Einstein-Maxwell fields with a possible non-zero cosmological constant for which the multiple Weyl-PND ${k}$ is geodesic and shear-free ($\\Psi _0=\\Psi _1=\\kappa =\\sigma =0$ ) and for which ${k}$ is not parallel to a PND of ${F}$ ($\\Phi _0\\ne 0$ ).","Choosing a null-rotation about ${k}$ such that $\\Phi _1=0$ , it follows that $\\Phi _2\\ne 0$ with $\\Phi _2 = 0$ ${\\ell }$ would be geodesic and shear-free; according to the generalised Goldberg-Sachs theorem we would have then $\\Psi _3=\\Psi _4=0$ and the Petrov type would be D, in which case [13], [42] the only null Einstein-Maxwell solutions are given by the (doubly aligned) Robinson-Trautman metrics.","The Maxwell equations (REF ,) and Bianchi equations (REF -) yield then $\\eth \\Phi _0 &=& 0,\\\\\\eth ^{\\prime }\\Phi _0 &=& -\\pi \\Phi _0,\\\\\\textrm {Þ}\\Phi _0 &=& 0,\\\\\\textrm {Þ}^{\\prime }\\Phi _0 &=& -\\mu \\Phi _0,\\\\\\eth \\Phi _2 &=& -\\nu \\Phi _0+\\tau \\Phi _2,\\\\\\textrm {Þ}\\Phi _2 &=& -\\lambda \\Phi _0+\\rho \\Phi _2,\\\\\\eth \\Psi _2 &=& -\\pi \\Phi _0\\overline{\\Phi _2}+3\\tau \\Psi _2,\\\\\\textrm {Þ}\\Psi _2 &=& \\mu |\\Phi _0|^2+3 \\rho \\Psi _2,$ after which the commutators $[\\eth ^{\\prime },\\, \\eth ], [\\eth ^{\\prime },\\, \\textrm {Þ}], [\\eth , \\, \\textrm {Þ}^{\\prime }]$ and $[\\textrm {Þ}^{\\prime },\\, \\textrm {Þ}]$ applied to $\\Phi _0$ give $\\eth \\pi &=& (3\\rho -\\overline{\\rho }) \\mu -2\\Psi _2+\\frac{R}{12},\\\\\\textrm {Þ}\\pi &=& 3\\rho \\pi ,\\\\ \\eth \\mu &=& \\overline{\\lambda }\\pi +3 \\mu \\tau ,\\\\\\textrm {Þ}\\mu &=& \\pi (\\overline{\\pi }+3 \\tau ) +2 \\Psi _2-\\frac{R}{12}.","$ Herewith GHP equation (') becomes a simple algebraic equation for $\\Psi _2$ , $\\Psi _2 = \\rho \\mu -\\tau \\pi +\\frac{R}{12},$ the $\\textrm {Þ}$ derivative of which (using (REF ,,REF ,)) results in $\\rho R=0$ .","As $\\rho =0$ would imply $\\Phi _0=0$ , this shows that an algebraically special Einstein-Maxwell solution possessing a shear-free and geodesic multiple Weyl-PND which is not a PND of ${F}$ necessarily has a vanishing cosmological constantNote3.","The corresponding class of solutions is non-empty: imposing the additional restriction that $\\pi =0$ one can deduce that $\\overline{\\rho }=\\rho $ and $\\overline{\\mu }=\\mu $ , together with a $[-2,0]$ –weighted relation $\\Psi _3=\\rho \\nu +\\mu \\overline{\\tau }-\\tau \\lambda $ .","It is then straightforward to construct a Newman-Penrose null-tetrad with the additional restrictions $\\alpha =\\beta =\\epsilon =0$ and $\\mu =2\\gamma $ : the unique solutions in this case are the Griffiths [20] metrics, containing as special cases the metrics [3], [4], [19], [41].","At first sight one would expect the class $\\pi \\ne 0$ to admit a larger set of solutions, but this is by no means guaranteed (compare with section 2.1): anyway no explicit examples appear to be known.","The only property which is easy to demonstrate –though somewhat tedious to be included in the present paragraph– is that no solutions exist for which ${k}$ is non-expanding ($\\Re (\\rho )\\ne 0$ ).","As a corollary of the above result we also obtain a new characterization of the class $\\mathcal {D}$ metrics[10] with non-vanishing cosmological constant: if one of the multiple Weyl-PND's a Petrov type D Einstein-Maxwell solution would be non-aligned with a PND of ${F}$ then $R=0$ .","In other words, for a non-vanishing cosmological constant, the double alignment property of the class $\\mathcal {D}$ metrics is a consequence of their Weyl-PND's being geodesic and shear-free.","It is tempting to conjecture that this same conclusion also will hold when $\\Lambda =0$ .","However it has not been possible so far to prove this, except for the special case of the Kundt space-times (i.e.","in which the Weyl-PND ${k}$ has vanishing complex divergence).","As the proof for this particular case is again quite tedious and little illuminating, I prefer to postpone this part to a possible later and more general publication."],["Discussion","Most results dealing with exact solutions for (algebraically special) Einstein-Maxwell fields have been obtained in the past introducing special coordinate systems, usually adapted to geodesic and/or shear-free null-congruences, or by imposing, sometimes haphasard looking, restrictions on the spin coefficients of a Newman-Penrose tetrad.","This not only has turned their comparison and classification into an awkward procedure, often involving sophisticated computer algebra packages, but also has made it difficult for researchers and students entering the field to recognize the blanks which remain to be filled in.","As furthermore the “Exact Solutions book”[40] touches the subject of non-aligned Einstein-Maxwell fields, or of aligned fields with a non-geodesic or shearing multiple PND, only superficially, I present below the known results, together with the ones obtained in the previous sections, schematically.","From these diagrams I exclude the conformally flat case, as it implies[36] $\\Lambda =0$ with the only non-null member being therefore[40] the Bertotti-Robinson[28], [2], [37] metric and the null members being given by a special class of plane waves[1], [30].","In the figures below labels (1), (2), ...next to vertical arrows refer to results obtained in the corresponding previous sections, $\\nexists $ indicates that no solutions are allowed, while a question mark indicates that no solutions are known to occur in the literature.","Roman capitals II, III, N, D refer to the Petrov types and $`1\\times `$ or $`2\\times `=$ indicate singly aligned or doubly aligned solutions.","Figure: NO_CAPTIONIn Fig.","1 the sub-tree corresponding to the singly aligned Einstein-Maxwell solutions with a shear-free and geodesic multiple Weyl-PND (as well as the null fields, which automatically obey this condition) is not included, as little or no progress has been made in this area since the early nineties[40], [32], with the bulk of the material contained in [40].","The double occurence of the Plebaǹski-Hacyan metric in Fig.","1 (once as PH and once as PH', the GHP-primed version of PH with the roles of ${k}$ and ${\\ell }$ interchanged) is due to the fact that PH is doubly aligned with only one null-ray being geodesic and both being shear-free.","The absence of doubly aligned Petrov type III solutions in Fig.","1 has been noticed already in [40] for $\\Lambda =0$ , but can easily be seen to hold also for $\\Lambda \\ne 0$ .","[40] also mentions that the Leroy metric[27] is the unique Einstein-Maxwell solution of Petrov type II with $\\Lambda =0$ , in which both the Weyl PND's ${k}$ and ${\\ell }$ are also PND's of ${F}$ and in which the multiple PND ${k}$ is geodesic and shear-free; this is not clear at all, as Leroy's solutions were only obtained in the non-radiative sub-case.","It furthermore remains to be checked whether this still holds –for a suitable generalisation of the Leroy metric– when $\\Lambda \\ne 0$ .","Figure: Algebraically special non-nul Einstein-Maxwell solutions for which the multiple Weyl-PND is k{k} not a PNDof F{F}.","The sub-tree under Griffiths presents in an invariant way the subcases mentioned in ,originally obtained by imposing coordinate restrictions or restrictions on the NP spin-coefficients."],["Acknowledgment","All calculations were done using the Maple symbolic algebra system[45].","I also thank Lode Wylleman for a critical reading of the manuscript."],["Appendix: GHP, Maxwell and Bianchi equations","Weights of the spin-coefficients, the Maxwell and Weyl spinor components and the GHP operators: $&\\kappa : [3, 1], \\nu : [-3, -1], \\sigma : [3, -1], \\lambda : [-3, 1], \\nonumber \\\\&\\rho : [1, 1], \\mu : [-1, -1], \\tau : [1, -1], \\pi : [-1, 1], \\nonumber \\\\&\\Phi _0 : [2, 0], \\Phi _1 : [0, 0], \\Phi _2 : [-2, 0], \\nonumber \\\\&\\Psi _0 : [4, 0], \\Psi _1 : [2, 0], \\Psi _2 : [0, 0], \\Psi _3: [-2,0], \\Psi _4 : [-4, 0] ,\\nonumber \\\\&\\eth : [1, -1], \\eth ^{\\prime }: [-1,1], \\textrm {Þ}^{\\prime }: [-1,-1], \\textrm {Þ}: [1,1].","\\nonumber $ The prime operation is an involution with $\\kappa ^{\\prime } &=& -\\nu ,\\sigma ^{\\prime }=-\\lambda ,\\rho ^{\\prime }=-\\mu , \\tau ^{\\prime }=-\\pi ,\\\\{\\Psi _0}^{\\prime } &=& \\Psi _4, {\\Psi _1}^{\\prime }=\\Psi _3, {\\Psi _2}^{\\prime }=\\Psi _2,\\\\\\Phi _0^{\\prime } &=& -\\Phi _2, \\Phi _1^{\\prime }=-\\Phi _1.$ The GHP commutators acting on $(p,q)$ weighted quantities are given by: $\\left[ \\textrm {Þ},\\textrm {Þ}^{\\prime } \\right] &=& (\\pi +\\overline{\\tau })\\eth +(\\overline{\\pi }+\\tau )\\eth ^{\\prime }+(\\kappa \\nu -\\pi \\tau + \\frac{R}{24}-\\Phi _{11}-\\Psi _2)p \\nonumber \\\\&&+(\\overline{\\kappa }\\overline{\\nu }-\\overline{\\pi }\\overline{\\tau }+ \\frac{R}{24}-\\Phi _{11}-\\overline{\\Psi }_2)q,\\\\\\left[ \\eth ,\\eth ^{\\prime }\\right] &=& (\\mu -\\overline{\\mu })\\textrm {Þ}+(\\rho -\\overline{\\rho })\\textrm {Þ}^{\\prime } +(\\lambda \\sigma -\\mu \\rho - \\frac{R}{24}-\\Phi _{11}+\\Psi _2)p \\nonumber \\\\&&-(\\overline{\\lambda \\sigma }-\\overline{\\mu }\\overline{\\rho }-\\frac{R}{24}-\\Phi _{11}+\\overline{\\Psi }_2)q,\\\\\\left[ \\textrm {Þ},\\eth \\right] &=& \\overline{\\pi }\\,\\textrm {Þ}-\\kappa \\textrm {Þ}^{\\prime } +\\overline{\\rho }\\,\\eth +\\sigma \\eth ^{\\prime }+(\\kappa \\mu -\\sigma \\pi -\\Psi _1)p + (\\overline{\\kappa \\lambda }-\\overline{\\pi }\\overline{\\rho }-\\Phi _{01})q .$ GHP equations: $\\textrm {Þ}\\rho -\\eth ^{\\prime }\\kappa &=& \\rho ^2+\\sigma \\overline{\\sigma }-\\overline{\\kappa }\\tau +\\kappa \\pi +\\Phi _{00}, \\\\\\textrm {Þ}\\sigma -\\eth \\kappa &=& (\\rho +\\overline{\\rho })\\sigma +(\\overline{\\pi }-\\tau )\\kappa +\\Psi _0, \\\\\\textrm {Þ}\\tau -\\textrm {Þ}^{\\prime }\\kappa &=& (\\tau +\\overline{\\pi })\\rho +(\\overline{\\tau }+\\pi )\\sigma +\\Phi _{01}+\\Psi _1, \\\\\\textrm {Þ}\\nu -\\textrm {Þ}^{\\prime } \\pi &=& (\\pi +\\overline{\\tau })\\mu +(\\overline{\\pi }+\\tau )\\lambda +\\Psi _3+\\overline{\\Phi _1}\\Phi _2,\\\\\\eth \\rho -\\eth ^{\\prime }\\sigma &=& (\\rho -\\overline{\\rho })\\tau +(\\mu -\\overline{\\mu })\\kappa +\\Phi _{01}-\\Psi _1,\\\\\\textrm {Þ}^{\\prime }\\sigma -\\eth \\tau &=& -\\sigma \\mu -\\overline{\\lambda }\\rho -\\tau ^2+\\kappa \\overline{\\nu }-\\Phi _{02},\\\\\\textrm {Þ}^{\\prime }\\rho -\\eth ^{\\prime }\\tau &=& -\\overline{\\mu }\\rho -\\lambda \\sigma -\\tau \\overline{\\tau }+\\kappa \\nu -\\frac{R}{12}-\\Psi _2.","$ Maxwell equations: $\\textrm {Þ}\\Phi _1-\\eth ^{\\prime }\\Phi _0 &=& \\pi \\Phi _0+2\\rho \\Phi _1-\\kappa \\Phi _2, \\\\\\textrm {Þ}\\Phi _2-\\eth ^{\\prime }\\Phi _1 &=& -\\lambda \\Phi _0+2 \\pi \\Phi _1+\\rho \\Phi _2.","$ Bianchi equations ($\\Phi _{IJ}=\\Phi _I\\overline{\\Phi _J}$ and $\\Lambda =R/4=constant$ ): ${\\eth ^{\\prime }} \\Psi _{{0}} -{\\textrm {Þ}} \\Psi _{{1}}+{\\textrm {Þ}} \\Phi _{{01}} -{\\eth } \\Phi _{{00}} &=& -\\pi \\,\\Psi _{{0}}-4\\,\\rho \\,\\Psi _{{1}}+3\\,\\kappa \\,\\Psi _{{2}}+ \\overline{\\pi }\\Phi _{{00}}+2\\,\\overline{\\rho }\\Phi _{{01}}+2\\,\\sigma \\,\\Phi _{{10}} \\nonumber \\\\&&-2\\,\\kappa \\,\\Phi _{{11}}-\\overline{\\kappa }\\Phi _{{02}}, \\\\{\\textrm {Þ}^{\\prime }} \\Psi _{{0}} -{\\eth } \\Psi _{{1}}+{\\textrm {Þ}} \\Phi _{{02}} -{\\eth } \\Phi _{{01}} &=& -\\mu \\,\\Psi _{{0}}-4\\,\\tau \\,\\Psi _{{1}}+3\\,\\sigma \\,\\Psi _{{2}}-\\overline{\\lambda }\\Phi _{{00}}+2\\, \\overline{\\pi }\\Phi _{{01}}+2\\,\\sigma \\,\\Phi _{{11}}\\nonumber \\\\&&+ \\overline{\\rho }\\Phi _{{02}}-2\\,\\kappa \\,\\Phi _{{12}}, $ $&&3\\,{\\eth ^{\\prime }} \\Psi _{{1}} -3\\,{\\textrm {Þ}} \\Psi _{{2}}+2\\,{\\textrm {Þ}} \\Phi _{{11}} -2\\,{\\eth }\\Phi _{{10}} +{\\eth ^{\\prime }} \\Phi _{{01}} -{\\textrm {Þ}^{\\prime }}\\Phi _{{00}} = 3\\,\\lambda \\,\\Psi _{{0}}-9\\,\\rho \\,\\Psi _{{2}}-6\\,\\pi \\,\\Psi _{{1}}+6\\,\\kappa \\,\\Psi _{{3}}+ (\\overline{\\mu }-2\\,\\mu ) \\Phi _{{00}}\\nonumber \\\\&& \\ \\ + 2\\,(\\pi +\\overline{\\tau }) \\Phi _{{01}}+2\\, ( \\tau + \\overline{\\pi }) \\Phi _{{10}}+2\\, ( 2\\,\\overline{\\rho }-\\rho ) \\Phi _{{11}}+2\\,\\sigma \\,\\Phi _{{20}}-\\overline{\\sigma }\\Phi _{{02}}-2\\,\\overline{\\kappa }\\Phi _{{12}}-2\\,\\kappa \\,\\Phi _{{21}}, \\\\&&3\\,{\\textrm {Þ}^{\\prime }} \\Psi _{{1}} -3\\,{\\eth } \\Psi _{{2}}+2\\,{\\textrm {Þ}} \\Phi _{{12}} -2\\,{\\eth }\\Phi _{{11}} +{\\eth ^{\\prime }} \\Phi _{{02}} -{\\textrm {Þ}^{\\prime }}\\Phi _{{01}} = 3\\,\\nu \\,\\Psi _{{0}}-6\\,\\mu \\,\\Psi _{{1}}-9\\,\\tau \\,\\Psi _{{2}}+6\\,\\sigma \\,\\Psi _{{3}}-\\overline{\\nu }\\Phi _{{00}}\\nonumber \\\\&& \\ \\ \\ +2\\, (\\overline{\\mu }-\\mu )\\Phi _{{01}} -2\\,\\overline{\\lambda }\\Phi _{{10}}+2\\, ( \\tau +2 \\overline{\\pi }) \\Phi _{{11}}+ (2\\,\\pi +\\overline{\\tau }) \\Phi _{{02}} + 2\\,(\\overline{\\rho }-\\rho ) \\Phi _{{12}}+2\\,\\sigma \\,\\Phi _{{21}}-2\\,\\kappa \\,\\Phi _{{22}}.","$"]],"string":"[\n [\n \"Algebraically special Einstein-Maxwell fields\"\n ],\n [\n \"Abstract The Geroch-Held-Penrose formalism is used to re-analyse algebraically special non-null Einstein-Maxwell fields, aligned as well as non-aligned, in the presence of a possible non-vanishing cosmological constant.\",\n \"A new invariant characterisation is given of the Garcia-Plebanski and Plebanski-Hacyan metrics within the family of aligned solutions and of the Griffiths metrics within the family of the non-aligned solutions.\",\n \"As a corollary also the double alignment of the Debever-McLenaghan 'class D' metrics with non-vanishing cosmological constant is shown to be equivalent with the shear-free and geodesic behaviour of their Debever-Penrose vectors.\"\n ],\n [\n \"Introduction\",\n \"In the quest for exact solutions of the Einstein-Maxwell equations, $R_{ab}-\\\\frac{1}{2}R g_{ab}+\\\\Lambda g_{ab} = F_{ac} {F_b}^c-\\\\frac{1}{4}g_{ab}F_{cd}F^{cd},$ a large amount of research (see for example the reviews in [21], [40] has been devoted to the study of so called aligned Einstein-Maxwell fields, in which at least one of the principal null directions (PND's) of the electromagnetic field tensor ${F}$ is parallel to a PND of the Weyl tensor, also called a Debever-Penrose direction, with main emphasis on the doubly aligned Petrov type D solutions, in which both real PND's of ${F}$ are parallel to a corresponding double Weyl-PND.\",\n \"One of the prominent tools in these and related activities has been the Goldberg-Sachs theorem, which in its original form [18] says that a vacuum space-time is algebraically special if and only if it contains a shear-free geodesic null congruence (in an adapted Newman-Penrose tetrad, `$\\\\Psi _0=\\\\Psi _1=0$ if and only if $\\\\kappa =\\\\sigma =0$ ').\",\n \"Goldberg and Sachs also proved that, if a space-time admits a complex null tetrad $({k},{\\\\ell },{m},\\\\overline{{m}})$ such that ${k}$ is shear-free and geodesic and $R_{ab}k^a k^b = R_{ab}k^a m^b = R_{ab}m^a m^b = 0$ (as is the case when ${k}$ is a PND of the electromagnetic field tensor), then the Weyl tensor is algebraically special, with ${k}$ being a multiple Weyl-PND.\",\n \"While, for a null Maxwell field [29], [38], the Maxwell and Bianchi equations imply that both conditions $\\\\Psi _0=\\\\Psi _1=0$ and $\\\\kappa =\\\\sigma =0$ are trivially satisfied (also in the presence of a possible cosmological constant), the situation is less straightforward in the non-null case.\",\n \"It still is true [18], [39] that, when a PND ${k}$ of the electromagnetic field tensor is shear-free and geodesic, then the Weyl tensor is algebraically special, but the reverse no longer holds.\",\n \"A key property in this respect is the Kundt-Trümper theorem [26], which says that for an algebraically special aligned non-null Einstein-Maxwell space-time (with a possible non-0 cosmological constant and with ${k}$ the PND of ${F}$ coinciding with a multiple Weyl-PND) one necessarily has $\\\\kappa (3 \\\\Psi _2-2 |\\\\Phi _1|^2)=0 \\\\textrm { and }\\\\sigma (3 \\\\Psi _2+2 |\\\\Phi _1|^2)=0,$ implying that the exceptional case $|\\\\kappa |^2+|\\\\sigma |^2 \\\\ne 0$ can only occur for Petrov types II or D with $\\\\frac{3}{2}\\\\Psi _2=\\\\pm |\\\\Phi _1|^2.$ Subsequent research has been concentrated on the doubly aligned Petrov type D cases, in which the Kundt-Trümper relations (REF ) also hold with $\\\\kappa $ and $\\\\sigma $ replaced by $\\\\nu $ and $\\\\lambda $ .\",\n \"This culminated in the complete integration of the field equations for the Petrov type D doubly aligned non-null Einstein-Maxwell fields, with a possible non-0 cosmological constant: When $\\\\frac{9}{4}\\\\Psi _2^2- |\\\\Phi _1|^4\\\\ne 0$ , both real PND's are geodesic and shear-free ($\\\\kappa =\\\\nu =\\\\sigma =\\\\lambda =0$ ) and, following Debever and McLenaghan[10], I will refer to the corresponding set of solutions as the `class $\\\\mathcal {D}$ ' space-times.\",\n \"They all admit at least a two-dimensional isometry group and count among their most famous members the Reissner-Nordström and Kerr-Newman metrics.\",\n \"Several authors [5], [6], [7], [8], [9], [10], [11], [12], [14], [15], [22], [24], [33], [34] have independently contributed to the determination of class $\\\\mathcal {D}$ , beginning with Carter's seminal study of the separability of the Hamilton-Jacobi and Klein-Gordon equations and culminating in the discovery [33] of Plebański and Demiański's 7-parameter metric for the non-null orbit solutions and García's construction [14], [15] of a single metric form for both the non-null and null orbit solutions.\",\n \"The case $\\\\frac{3}{2}\\\\Psi _2= |\\\\Phi _1|^2$ was fully integrated in [35].\",\n \"In the resulting `Plebański-Hacyan space-times' ${k}$ and ${\\\\ell }$ are respectively non-geodesic ($\\\\kappa \\\\ne 0$ ) and geodesic ($\\\\nu =0$ ), while both are shear-free ($\\\\sigma =\\\\lambda =0$ ) and have vanishing complex divergence ($\\\\rho =\\\\mu =0$ ).\",\n \"The metric is given by $\\\\textrm {d}s^2= 2\\\\textrm {d}\\\\zeta \\\\textrm {d}\\\\overline{\\\\zeta }+2 \\\\textrm {d}u\\\\textrm {d}v+[\\\\Lambda u^2+\\\\zeta \\\\overline{F}(v)+\\\\overline{\\\\zeta } F(v)] \\\\textrm {d}v^2,$ which, for an arbitrary function $F(v)$Note1 does not admit any isometries and only has an electromagnetic energy-momentum tensor of the correct sign when $\\\\Lambda <0$ .\",\n \"The case $\\\\frac{3}{2}\\\\Psi _2= -|\\\\Phi _1|^2$ , which was overlooked in [35], was dealt with in [16].\",\n \"Its unique solutions are the `García-Plebański metrics', $\\\\textrm {d}s^2 = -\\\\frac{1}{\\\\Lambda }\\\\left( e^{2z}{\\\\omega ^1}^2+e^{-2 z}{\\\\omega ^2}^2+\\\\textrm {d}z^2-4 (\\\\cosh z)^2{\\\\omega ^3}^2 \\\\right),$ with $\\\\omega ^1+i \\\\omega ^2 = 2\\\\frac{e^{iu}}{1-\\\\zeta \\\\overline{\\\\zeta }}\\\\textrm {d}\\\\zeta \\\\textrm { and } \\\\omega ^3 = \\\\textrm {d}u-i \\\\frac{\\\\zeta \\\\textrm {d}\\\\overline{\\\\zeta }-\\\\overline{\\\\zeta }\\\\textrm {d}\\\\zeta }{1-\\\\zeta \\\\overline{\\\\zeta }}.$ The corresponding space-times admit a 3 dimensional group of isometries; both ${k}$ and ${\\\\ell }$ are geodesic ($\\\\kappa =\\\\nu =0$ ), shearing ($\\\\sigma \\\\lambda \\\\ne 0$ ) and twisting, but non-expanding ($\\\\rho , \\\\mu \\\\in i \\\\mathbb {R}$ ).\",\n \"Also this metric can only describe an Einstein-Maxwell space-time with an electromagnetic energy-momentum tensor of the correct sign when $\\\\Lambda <0$ .\",\n \"In this paper I will have a closer look at the full set of algebraically special cases, both aligned (and hence by (REF ) necessarily of Petrov type II) and non-aligned ones.\",\n \"First we will see that the García-Plebański metrics are the unique members of the class of algebraically special and aligned Einstein-Maxwell solutions for which the null direction ${k}$ is shearing.\",\n \"While doing so, I will correct an error in [25] (cited also in [40] p. 409), claiming that, at least for $\\\\Lambda =0$ , the case $\\\\kappa =0\\\\ne \\\\sigma $ admits no solutions.\",\n \"This is true indeed but, as will become clear in section 2, the proof requires a much more subtle argumentation than the one presented in [25]Note2 .\",\n \"The error occurs after relation (2.6) when the case $\\\\overline{\\\\rho }=-\\\\rho $ is dismissed by remarking that it ”leads to $\\\\rho =0$ ”.\",\n \"Most likely this conclusion was prematurely arrived at by inspection of the Newman-Penrose equation corresponding to our GHP equation (REF ): with $\\\\kappa =\\\\epsilon +\\\\overline{\\\\epsilon }=\\\\Phi _0=0$ this equation reads $D\\\\rho = \\\\rho ^2+\\\\sigma \\\\overline{\\\\sigma }$ , the real and imaginary parts of which only allow one to conclude that $\\\\rho =\\\\pm i |\\\\sigma |$ and $D \\\\rho =D \\\\sigma = 0$ .\",\n \"In section 2 a correct proof of Kozarzewski's no-go claim will be provided, generalizing it to the case $\\\\Lambda \\\\geqslant 0$ and showing that for $\\\\Lambda <0$ the only allowed solutions are the doubly aligned García-Plebański metrics with $\\\\rho = \\\\pm i |\\\\sigma | \\\\ne 0$ .\",\n \"In section 3 I will consider the algebraically special and aligned Einstein-Maxwell fields for which the null direction ${k}$ is non-geodesic.\",\n \"The general solution in this family so far is not known, but, remarkably, the Plebański-Hacyan metrics exhaust the sub-family characterised by the vanishing of the complex divergence of ${k}$ .\",\n \"Finally I prove in section 4 that an algebraically special Einstein-Maxwell solution possessing a shear-free and geodesic multiple Weyl-PND which is not a PND of ${F}$ necessarily has vanishing cosmological constant and I give a characterization of the sub-class of the Griffiths [20] solutions.\",\n \"As a corollary of this theorem it follows that the `class $\\\\mathcal {D}$ ' metrics[10] are the unique Petrov type D Einstein-Maxwell solutions for which the real Weyl-PND's are both geodesic and shear-free and for which the cosmological constant is non-vanishing: in other words, the double alignment condition of the class $\\\\mathcal {D}$ metrics with non-vanishing cosmological constant is a consequence of their multiple Weyl-PND's being geodesic and shear-free.\",\n \"Whether this property persists when $\\\\Lambda =0$ is at present still an open problem, with only the Kundt case ($\\\\rho =0$ ) so far having been dealt with.\",\n \"In order to study these and related issues, linking kinematic properties of certain invariantly defined null directions (such as being geodesic and/or non-shearing) to algebraic properties of the electromagnetic field tensor or of the Weyl tensor, it is natural to use the Geroch-Held-Penrose (GHP) formalism [17].\",\n \"In this formalism only a pair of null directions is singled out at each point rather than an entire null tetrad, as is the case in the Newman-Penrose [31] formalism.\",\n \"The resulting formalism is covariant with respect to rotations of the spatial basis vectors and boosts of the real null directions and, as such, is `halfway' between a fully covariant approach and the NP spin-coefficient approach and leads to considerably simpler equations with fewer complex variables.\",\n \"Throughout I will use the sign conventions and notations of Kramer §7.4, with the tetrad basis vectors taken as ${k},{\\\\ell },{m},\\\\overline{{m}}$ with $- k^a \\\\ell _a = 1 = m^a\\\\overline{m}_a$ .\",\n \"However, in order to ease comparison with the (more familiar) Newman-Penrose formalism, I will write primed variables, such as $\\\\kappa ^{\\\\prime },\\\\sigma ^{\\\\prime },\\\\rho ^{\\\\prime }$ and $\\\\tau ^{\\\\prime }$ , as their NP equivalents $-\\\\nu ,-\\\\lambda , -\\\\mu $ and $-\\\\pi $ .\",\n \"For completeness the resulting weights, commutators, GHP, Bianchi and Maxwell equations are presented in an Appendix.\",\n \"Finally note that, at least when the electromagnetic field is non-null, the pair of null directions ${k},{\\\\ell }$ can always be invariantly defined, aligning for example ${k}$ with a PND of ${F}$ and null-rotating about ${k}$ such that $\\\\Phi _1$ is the only non-vanishing component of the Maxwell spinor.\",\n \"Obviously this choice is not unique (for example in the non-aligned case it will be preferable to align ${k}$ with the multiple Weyl-PND and to null-rotate about ${k}$ such that $\\\\Phi _1=0$ and $\\\\Phi _0 \\\\Phi _2\\\\ne 0$ ), but it is important to realise that any ensuing well-weighted GHP relations, such as $\\\\kappa =0, \\\\rho =0, \\\\ldots $ are automatically geometrically invariant statements.\"\n ],\n [\n \"Aligned electrovacs with a shearing multiple Debever-Penrose vector\",\n \"Let us first, for the sake of completeness, re-confirm the well-known fact [18], [39] that if a PND ${k}$ of the electromagnetic field tensor ${F}$ is shear-free and geodesic, then the Weyl tensor is algebraically special with ${k}$ being the multiple Weyl-PND: with ${k}$ a PND of ${F}$ ($\\\\Phi _0=0$ ) satisfying $\\\\kappa =\\\\sigma =0$ , it follows from () that $\\\\Psi _0=0$ and hence ${k}$ is also a Weyl-PND.\",\n \"When ${F}$ is non-null we can null-rotate about ${k}$ such that also $\\\\Phi _2=0$ .\",\n \"The integrability conditions for the Maxwell equations (REF , ) simplify then with (, ) to $\\\\Phi _1 \\\\Psi _1=0$ .\",\n \"It follows that $\\\\Psi _1=0$ and hence ${k}$ is a multiple Weyl-PND.\",\n \"Now let us consider the reverse situation and assume that the Weyl tensor is algebraically special, with the multiple Weyl-PND ${k}$ being also a PND of the non-null electromagnetic field tensor: by means of a suitable null rotation about ${k}$ the null tetrad can be chosen such that $\\\\Psi _0=\\\\Psi _1=\\\\Phi _0=\\\\Phi _2=0$ .\",\n \"The Maxwell equations (REF , ') and () become $\\\\textrm {Þ}\\\\Phi _1=2 \\\\rho \\\\Phi _1, \\\\ \\\\eth \\\\Phi _1 =2\\\\tau \\\\Phi _1,\\\\ \\\\eth ^{\\\\prime }\\\\Phi _1 =- 2\\\\pi \\\\Phi _1,$ while the Bianchi equations (REF , ) yield the Kundt-Trümper relations (REF ).\",\n \"In the present paragraph we consider the case where ${k}$ is shearing and hence $\\\\kappa =0,\\\\ \\\\Psi _2 = -\\\\frac{2}{3} |\\\\Phi _1|^2 .$ Bianchi equation (REF ) simplifies now to $|\\\\Phi _1|^2 (\\\\rho +\\\\overline{\\\\rho })=0$ , implying that ${k}$ is non-expanding.\",\n \"From the real part of GHP equation (REF ) it follows that $\\\\rho $ is related to $\\\\sigma $ by $\\\\rho = \\\\pm i |\\\\sigma |$ and hence, by (REF , , ) $\\\\textrm {Þ}\\\\rho =0,\\\\ \\\\textrm {Þ}\\\\sigma =0,\\\\ \\\\textrm {Þ}\\\\tau = \\\\rho (\\\\tau +\\\\overline{\\\\pi })+\\\\sigma (\\\\overline{\\\\tau }+\\\\pi ).$ From (,') one finds then $\\\\Psi _3 = -\\\\frac{2}{3} \\\\frac{|\\\\Phi _1|^2}{\\\\sigma } (2\\\\tau +\\\\overline{\\\\pi }),$ and $\\\\textrm {Þ}\\\\pi = 2(\\\\overline{\\\\pi }\\\\overline{\\\\sigma }-\\\\overline{\\\\tau }\\\\rho ) .$ Acting with the $[\\\\eth ,\\\\, \\\\textrm {Þ}]$ commutator on $\\\\Phi _1$ now leads to an expression for $\\\\eth \\\\rho $ , which, together with () and the $\\\\eth $ derivative of (REF ), yields $\\\\eth \\\\rho &=& \\\\sigma (\\\\overline{\\\\tau }+2 \\\\pi )+2\\\\rho \\\\tau ,\\\\\\\\\\\\eth ^{\\\\prime }\\\\sigma &=& \\\\sigma (\\\\overline{\\\\tau }+2\\\\pi ),\\\\\\\\\\\\eth \\\\sigma &=& \\\\sigma (3\\\\tau -2\\\\overline{\\\\pi })-2\\\\rho \\\\frac{\\\\sigma }{\\\\overline{\\\\sigma }}(\\\\overline{\\\\tau }+2 \\\\pi ).$ The remaining integrability conditions for the Maxwell equations can then be written as $\\\\textrm {Þ}^{\\\\prime }\\\\tau +\\\\eth \\\\mu &=& \\\\rho \\\\overline{\\\\nu }+\\\\overline{\\\\lambda }\\\\pi ,\\\\\\\\\\\\textrm {Þ}^{\\\\prime }\\\\pi -\\\\eth ^{\\\\prime }\\\\mu &=& \\\\lambda \\\\tau -\\\\nu \\\\rho -\\\\mu \\\\overline{\\\\tau }-\\\\overline{\\\\mu }\\\\pi ,\\\\\\\\\\\\eth \\\\pi +\\\\eth ^{\\\\prime }\\\\tau &=& \\\\rho (\\\\mu +\\\\overline{\\\\mu }),\\\\\\\\\\\\textrm {Þ}\\\\mu +\\\\textrm {Þ}^{\\\\prime }\\\\rho &=& \\\\pi \\\\overline{\\\\pi }-\\\\tau \\\\overline{\\\\tau }.$ Next we apply the $[\\\\eth ,\\\\, \\\\textrm {Þ}]$ commutator to $\\\\sigma $ , to obtain an expression for $\\\\tau $ , $\\\\tau = \\\\textstyle {\\\\frac{1}{3}}\\\\overline{\\\\pi }+\\\\frac{4}{3}\\\\frac{\\\\rho \\\\pi }{\\\\overline{\\\\sigma }},$ which together with its $\\\\textrm {Þ}$ derivative leads to $\\\\sigma \\\\pi -\\\\rho \\\\overline{\\\\pi }=0 .$ This suggests the definition of an auxiliary variable $S$ (by (REF ) one has $|S|=1$ , while $\\\\textrm {weight}(S)=[2,-2]$ ) such that $\\\\sigma = S \\\\rho $ and enabling us to conclude from (REF ) that $\\\\tau =-\\\\overline{\\\\pi },\\\\ \\\\overline{\\\\pi }= S \\\\pi .$ By (REF ,REF –) one has then $\\\\textrm {Þ}S =0,\\\\ \\\\eth S=-S (3 \\\\overline{\\\\pi }-S \\\\pi ),\\\\ \\\\eth ^{\\\\prime }S=3 S \\\\pi -\\\\overline{\\\\pi },$ after which (',REF ,REF ') yield $\\\\eth \\\\pi &=& S \\\\pi ^2 +2 \\\\rho (\\\\mu +\\\\overline{\\\\mu })-\\\\frac{3}{2} S \\\\rho ^2 \\\\Psi _4 |\\\\Phi _1|^{-2}, \\\\\\\\\\\\eth ^{\\\\prime }\\\\pi &=& -\\\\pi ^2-2 S^{-1} \\\\rho (\\\\mu +\\\\overline{\\\\mu })-\\\\frac{3}{2} S^{-2} \\\\rho ^2 \\\\overline{\\\\Psi _4} |\\\\Phi _1|^{-2},\\\\\\\\\\\\textrm {Þ}\\\\Psi _4 &=& \\\\rho (\\\\Psi _4 -S^{-2}\\\\overline{\\\\Psi _4})-\\\\frac{4}{3} S^{-1}(\\\\mu +\\\\overline{\\\\mu })|\\\\Phi _1|^2.$ Herewith () reduces to $2 (\\\\mu +\\\\overline{\\\\mu }){|\\\\Phi _1|^2}+\\\\rho (S^{-1}\\\\overline{\\\\Psi _4}-S\\\\Psi _4)= 0,$ while expressing that $\\\\textrm {Þ}^{\\\\prime }(S\\\\overline{S})=0$ (with $\\\\textrm {Þ}^{\\\\prime }S$ evaluated from ()) and simplifying the result by means of (REF ), one finds that $\\\\mu +\\\\overline{\\\\mu }= 0$ (i.e.\",\n \"also ${\\\\ell }$ is non-expanding) and hence $\\\\overline{\\\\Psi _4} = S^2 \\\\Psi _4$ .\",\n \"Furthermore, calculating $\\\\textrm {Þ}^{\\\\prime }\\\\rho $ from () and expressing that $\\\\textrm {Þ}^{\\\\prime }(\\\\rho +\\\\overline{\\\\rho })=0$ gives us an equation from which we can obtain $\\\\Psi _4, \\\\textrm {Þ}^{\\\\prime }\\\\rho $ and $\\\\textrm {Þ}^{\\\\prime }S$ : $\\\\Psi _4 &=& \\\\frac{|\\\\Phi _1|^2}{18\\\\rho ^2} \\\\left[ 24\\\\pi ^2+6\\\\rho (\\\\lambda -S^{-2}\\\\overline{\\\\lambda })-S^{-1} (8 |\\\\Phi _1|^2+12 \\\\rho \\\\mu -R) \\\\right], \\\\\\\\\\\\textrm {Þ}^{\\\\prime }\\\\rho &=& -\\\\frac{\\\\rho }{2}(S\\\\lambda +S^{-1}\\\\overline{\\\\lambda }),\\\\\\\\\\\\textrm {Þ}^{\\\\prime }S &=& S \\\\left[ -2 \\\\mu +S \\\\lambda -S^{-1}\\\\overline{\\\\lambda }+\\\\frac{1}{\\\\rho }(2 S \\\\pi ^2-\\\\textstyle {\\\\frac{2}{3}}|\\\\Phi _1|^2+\\\\textstyle {\\\\frac{1}{12}} R) \\\\right].$ Next we solve (REF ', ') together with () and the $[\\\\textrm {Þ}^{\\\\prime },\\\\, \\\\textrm {Þ}]\\\\rho ,\\\\ [\\\\eth ,\\\\, \\\\textrm {Þ}^{\\\\prime }] S$ commutator relations for $\\\\textrm {Þ}^{\\\\prime }\\\\pi $ , $\\\\eth \\\\mu $ , $\\\\textrm {Þ}\\\\mu $ , $\\\\textrm {Þ}\\\\lambda $ , $\\\\eth \\\\lambda $ and $\\\\eth ^{\\\\prime }\\\\lambda $ : $\\\\textrm {Þ}^{\\\\prime }\\\\pi &=& -\\\\textstyle {\\\\frac{3}{2}} S \\\\lambda \\\\pi +\\\\textstyle {\\\\frac{1}{2}}S^{-1}\\\\rho \\\\overline{\\\\nu }+S^{-1}\\\\overline{\\\\lambda }\\\\pi +\\\\textstyle {\\\\frac{1}{2}}\\\\mu \\\\pi -\\\\textstyle {\\\\frac{5}{2}}\\\\nu \\\\rho +( \\\\textstyle {\\\\frac{4}{3}}|\\\\Phi _1|^2-3 S \\\\pi ^2 -\\\\textstyle {\\\\frac{1}{8}} R)\\\\frac{\\\\pi }{\\\\rho },\\\\\\\\\\\\eth \\\\mu &=& -\\\\textstyle {\\\\frac{5}{2}} S\\\\rho \\\\nu -\\\\textstyle {\\\\frac{1}{2}}\\\\lambda \\\\pi S^2-\\\\textstyle {\\\\frac{3}{2}}S \\\\mu \\\\pi +\\\\textstyle {\\\\frac{3}{2}}\\\\rho \\\\overline{\\\\nu }+\\\\overline{\\\\lambda }\\\\pi +(\\\\textstyle {\\\\frac{2}{3}} |\\\\Phi _1|^2-S\\\\pi ^2-\\\\textstyle {\\\\frac{1}{24}} R)\\\\frac{S \\\\pi }{\\\\rho },\\\\\\\\\\\\textrm {Þ}\\\\mu &=& \\\\textstyle {\\\\frac{1}{2}}\\\\rho (S\\\\lambda +S^{-1} \\\\overline{\\\\lambda }),\\\\\\\\\\\\textrm {Þ}\\\\lambda &=& \\\\textstyle {\\\\frac{1}{2}}\\\\rho (\\\\lambda +S^{-1}\\\\overline{\\\\lambda })-2 \\\\pi ^2-\\\\textstyle {\\\\frac{1}{12}} S^{-1}(R-8 |\\\\Phi _1|^2), \\\\\\\\\\\\eth \\\\lambda &=& -\\\\textstyle {\\\\frac{5}{2}}(\\\\mu \\\\pi +\\\\nu \\\\rho )+\\\\textstyle {\\\\frac{3}{2}}S \\\\lambda \\\\pi +\\\\textstyle {\\\\frac{7}{2}} S^{-1}\\\\rho \\\\overline{\\\\nu }+(S \\\\pi ^2 -\\\\textstyle {\\\\frac{8}{3}}|\\\\Phi _1|^2+\\\\textstyle {\\\\frac{1}{24}}R)\\\\frac{\\\\pi }{\\\\rho },\\\\\\\\\\\\eth ^{\\\\prime }\\\\lambda &=& \\\\textstyle {\\\\frac{1}{2}}S^{-1}(\\\\mu \\\\pi +\\\\rho \\\\nu )+\\\\textstyle {\\\\frac{3}{2}}S^{-2}(3\\\\rho \\\\overline{\\\\nu }- \\\\overline{\\\\lambda }\\\\pi )-3\\\\lambda \\\\pi +S^{-1}(3 S\\\\pi ^2-\\\\textstyle {\\\\frac{8}{3}} |\\\\Phi _1|^2+\\\\textstyle {\\\\frac{1}{8}} R)\\\\frac{\\\\pi }{\\\\rho },$ after which an expression for the spin coefficient $\\\\mu $ follows from the $[\\\\eth ^{\\\\prime },\\\\, \\\\eth ]S$ commutator relation, $\\\\mu = \\\\frac{1}{2}(S\\\\lambda -S^{-1} \\\\overline{\\\\lambda })+\\\\frac{1}{36\\\\rho }(24 S \\\\pi ^2-32 |\\\\Phi _1|^2+R).$ Of the Maxwell integrability conditions there only remains now the $[\\\\eth ^{\\\\prime },\\\\, \\\\textrm {Þ}^{\\\\prime }]\\\\Phi _1$ relation, namely $6 (S \\\\lambda +S^{-1} \\\\overline{\\\\lambda })\\\\rho \\\\pi +\\\\pi (8 |\\\\Phi _1|^2-24 S \\\\pi ^2-R) -36\\\\rho ^2(\\\\nu - S^{-1}\\\\overline{\\\\nu }) = 0,$ which with GHP equation (') gets simplified to the key algebraic equation $\\\\pi (8|\\\\Phi _1|^2-24 S \\\\pi ^2-R)=0.$ Herewith all Bianchi equations and GHP equations (except those involving the derivatives of $\\\\nu $ ) are identically satisfied.\",\n \"At this stage the Weyl spinor components are given by, using (REF ) to simplify (REF ), $\\\\Psi _0 &=& \\\\Psi _1=0,\\\\\\\\\\\\Psi _2 &=& -\\\\textstyle {\\\\frac{2}{3}}|\\\\Phi _1|^2,\\\\\\\\\\\\Psi _3 &=& \\\\textstyle {\\\\frac{2}{3}}\\\\frac{\\\\pi }{\\\\rho }|\\\\Phi _1|^2 , \\\\\\\\\\\\Psi _4 &=& \\\\frac{|\\\\Phi _1|^2}{27 \\\\rho ^2}S^{-1}(24 S \\\\pi ^2+4|\\\\Phi _1|^2+R).\",\n \"$ I now show that the case $\\\\pi \\\\ne 0$ is inconsistent, while the case $\\\\pi =0$ leads to the García-Plebański metrics.\"\n ],\n [\n \"$\\\\pi \\\\ne 0$\",\n \"When $\\\\pi \\\\ne 0$ we can use (REF ) to rewrite (REF ) as $\\\\pi (S\\\\lambda +S^{-1}\\\\overline{\\\\lambda })-6\\\\rho (\\\\nu -S^{-1} \\\\overline{\\\\nu })=0,$ while (REF ) becomes $\\\\mu =\\\\textstyle {\\\\frac{1}{2}}(S\\\\lambda -S^{-1}\\\\overline{\\\\lambda })-\\\\textstyle {\\\\frac{2}{3}}|\\\\Phi _1|^2\\\\rho ^{-1}.$ Taking the $\\\\eth $ derivative of (REF ) and eliminating $\\\\pi $ from the resulting equation and (REF ), we also find $3\\\\rho (S\\\\lambda -\\\\overline{S}\\\\overline{\\\\lambda })-4|\\\\Phi _1|^2=0$ and hence, by (REF ), $\\\\mu =0$ .\",\n \"The $\\\\textrm {Þ}$ derivative of (REF ) yields then $S\\\\lambda +S^{-1} \\\\overline{\\\\lambda }=0$ and hence, by (REF ), $\\\\lambda =\\\\textstyle {\\\\frac{2}{3}}|\\\\Phi _1|^2 S^{-1} \\\\rho ^{-1}$ .\",\n \"Substituting this in the expression () for $\\\\eth \\\\lambda $ one obtains $\\\\nu =\\\\textstyle {\\\\frac{2}{3}}|\\\\Phi _1|^2\\\\pi \\\\rho ^{-2}$ , which together with GHP equation (') leads to the contradiction $|\\\\Phi _1|^2\\\\pi =0$ .\"\n ],\n [\n \"$\\\\pi =0$\",\n \"Substituting $\\\\pi =0$ in (REF ,REF ) one obtains $R \\\\, (=4 \\\\Lambda ) =-4|\\\\Phi _1|^2 $ and $\\\\overline{\\\\nu }-5 S \\\\nu =0,$ whence $\\\\nu =0$ .\",\n \"While (REF ) proves Kozarzewski's no-go claim for $\\\\Lambda =0$ , it also generalizes it to the case $\\\\Lambda \\\\geqslant 0$ .\",\n \"Furthermore, when $\\\\Lambda < 0$ the only non-0 Weyl spinor component is $\\\\Psi _2=R/6$ and we are in the doubly aligned situation with both Weyl-PND's ${k}$ and ${\\\\ell }$ being geodesic and non-expanding.\",\n \"The field equations and Maxwell equations have been completely integrated in this case by García and Plebański [16] and the resulting metric is given by (REF ), admitting a 3D isometry group.\"\n ],\n [\n \"Aligned electrovacs with a non-geodesic multiple Debever-Penrose vector\",\n \"We again consider the case where the Weyl tensor is algebraically special, with the multiple Weyl-PND ${k}$ being also a PND of the non-null electromagnetic field tensor, but now we take $\\\\kappa \\\\ne 0$ and hence, by (REF ), $\\\\sigma =0 \\\\textrm { and } \\\\Psi _2 = \\\\frac{2}{3} |\\\\Phi _1|^2 .$ Bianchi equation () now reduces to $|\\\\Phi _1|^2 (\\\\tau -\\\\overline{\\\\pi })=0$ , whence $\\\\tau =\\\\overline{\\\\pi }$ , after which (REF ) implies $\\\\Psi _3=\\\\frac{2}{3}\\\\frac{2\\\\rho -\\\\overline{\\\\rho }}{\\\\kappa }|\\\\Phi _1|^2, $ showing that solutions cannot be of Petrov type III or N when $\\\\rho \\\\ne 0$ .\",\n \"GHP equations (,,) and Bianchi equation (REF ) yield then $\\\\eth \\\\kappa &=& 0 ,\\\\\\\\\\\\eth \\\\rho &=& \\\\overline{\\\\pi }(\\\\rho -\\\\overline{\\\\rho })+\\\\kappa (\\\\mu -\\\\overline{\\\\mu }),\\\\\\\\\\\\eth ^{\\\\prime }\\\\rho &=& -2 \\\\pi \\\\overline{\\\\rho }-2 \\\\overline{\\\\kappa }\\\\overline{\\\\mu },\\\\\\\\\\\\eth ^{\\\\prime }\\\\pi &=& \\\\pi ^2+\\\\lambda \\\\overline{\\\\rho }-\\\\nu \\\\overline{\\\\kappa }.$ The integrability conditions for the Maxwell equations furthermore give an expression for $\\\\textrm {Þ}\\\\pi $ , $\\\\textrm {Þ}\\\\pi = \\\\overline{\\\\kappa }(2\\\\overline{\\\\mu }-\\\\mu )+2\\\\pi \\\\overline{\\\\rho },$ together with four extra relations $\\\\eth \\\\mu +\\\\textrm {Þ}^{\\\\prime }\\\\overline{\\\\pi }&=& \\\\overline{\\\\nu }\\\\rho +\\\\overline{\\\\lambda }\\\\pi ,\\\\\\\\\\\\eth ^{\\\\prime }\\\\mu -\\\\textrm {Þ}^{\\\\prime }\\\\pi &=& \\\\pi (\\\\overline{\\\\mu }+\\\\mu )-\\\\lambda \\\\overline{\\\\pi }+\\\\nu \\\\rho ,\\\\\\\\\\\\eth \\\\pi +\\\\eth ^{\\\\prime }\\\\overline{\\\\pi }&=& \\\\rho \\\\overline{\\\\mu }-\\\\overline{\\\\rho }\\\\mu , \\\\\\\\\\\\textrm {Þ}\\\\mu +\\\\textrm {Þ}^{\\\\prime }\\\\rho &=& 0,$ with which GHP equations (,') imply $\\\\textrm {Þ}^{\\\\prime }\\\\kappa &=& \\\\kappa (2\\\\mu -\\\\overline{\\\\mu }),\\\\\\\\\\\\textrm {Þ}\\\\lambda &=& \\\\lambda (\\\\rho +\\\\overline{\\\\rho })+2\\\\pi ^2-2\\\\nu \\\\overline{\\\\kappa }.$ So far it has not been possible to complete the analysis of this case.\",\n \"However, it is easy to see that the Plebański-Hacyan metrics completely exhaust the non-diverging subfamily of solutions ($\\\\rho =0$ ) and have $\\\\Lambda <0$ , whereas for $\\\\Lambda \\\\geqslant 0$ no solutions exist.\",\n \"In fact, when $\\\\rho =0$ (REF ) and $\\\\kappa \\\\ne 0$ imply that also the second PND ${\\\\ell }$ of the electromagnetic field is non-diverging ($\\\\mu =0$ ): herewith (REF ) and (') show that $\\\\Psi _3=\\\\Psi _4=0$ , so that we are in the doubly aligned Petrov type D case.\",\n \"This suffices already to conclude that the only possible solutions are the Plebański-Hacyan metrics, as this case was completely integrated in [35], modulo the error corrected in [16].\",\n \"However, it is instructive to provide a short and coordinate independent proof of the essential step in [35], namely the part in which the authors prove that $\\\\Gamma _{423}=0$ , leading to the conclusion that ${\\\\ell }$ not only is shear-free ($\\\\lambda =0$ being an immediate consequence of (')), but also is geodesic ($\\\\nu =0$ ): First observe that the $[\\\\eth ,\\\\, \\\\textrm {Þ}^{\\\\prime }]$ commutator applied to $\\\\kappa $ implies $\\\\overline{\\\\nu }\\\\textrm {Þ}\\\\kappa =0$ .\",\n \"Assuming $\\\\nu \\\\ne 0$ leads then to an inconsistency, as we would have $\\\\textrm {Þ}\\\\kappa =0, \\\\ \\\\eth ^{\\\\prime }\\\\kappa = \\\\overline{\\\\kappa }\\\\overline{\\\\pi }-\\\\kappa \\\\pi ,$ with the second relation being obtained from (REF ).\",\n \"Herewith the $[\\\\textrm {Þ},\\\\, \\\\textrm {Þ}^{\\\\prime }]\\\\kappa $ commutator relation yields $\\\\kappa (\\\\overline{\\\\kappa }\\\\overline{\\\\nu }+3\\\\kappa \\\\nu -6\\\\pi \\\\overline{\\\\pi }+\\\\textstyle {\\\\frac{1}{6}}R-\\\\textstyle {\\\\frac{20}{3}}|\\\\Phi _1|^2)+2\\\\overline{\\\\kappa }\\\\overline{\\\\pi }^2=0,$ which, with $\\\\nu \\\\overline{\\\\kappa }=\\\\pi ^2$ (obtained from (')) simplifies to $|\\\\kappa |^2(40|\\\\Phi _1|^2-R)-18(\\\\kappa \\\\pi -\\\\overline{\\\\kappa }\\\\overline{\\\\pi })^2=0.$ On the other hand ('-$\\\\overline{\\\\ref {ghp5d}}$ ) reads $|\\\\kappa |^2(8|\\\\Phi _1|^2+R)-6(\\\\kappa \\\\pi -\\\\overline{\\\\kappa }\\\\overline{\\\\pi })^2=0,$ which, when added to (REF ), results in $2|\\\\kappa \\\\Phi _1|^2 - (\\\\kappa \\\\pi -\\\\overline{\\\\kappa }\\\\overline{\\\\pi })^2 =0.$ The left hand side being positive definite, this shows that the case $\\\\nu \\\\ne 0$ is inconsistent.\",\n \"We therefore have $\\\\nu =\\\\pi =0$ and hence, by GHP equation (), $R = 4\\\\Lambda =-8|\\\\Phi _1|^2 < 0$ .\"\n ],\n [\n \"Electrovacs with a shear-free and geodesic multiple Debever-Penrose vector\",\n \"We now consider algebraically special (non-conformally flat) Einstein-Maxwell fields with a possible non-zero cosmological constant for which the multiple Weyl-PND ${k}$ is geodesic and shear-free ($\\\\Psi _0=\\\\Psi _1=\\\\kappa =\\\\sigma =0$ ) and for which ${k}$ is not parallel to a PND of ${F}$ ($\\\\Phi _0\\\\ne 0$ ).\",\n \"Choosing a null-rotation about ${k}$ such that $\\\\Phi _1=0$ , it follows that $\\\\Phi _2\\\\ne 0$ with $\\\\Phi _2 = 0$ ${\\\\ell }$ would be geodesic and shear-free; according to the generalised Goldberg-Sachs theorem we would have then $\\\\Psi _3=\\\\Psi _4=0$ and the Petrov type would be D, in which case [13], [42] the only null Einstein-Maxwell solutions are given by the (doubly aligned) Robinson-Trautman metrics.\",\n \"The Maxwell equations (REF ,) and Bianchi equations (REF -) yield then $\\\\eth \\\\Phi _0 &=& 0,\\\\\\\\\\\\eth ^{\\\\prime }\\\\Phi _0 &=& -\\\\pi \\\\Phi _0,\\\\\\\\\\\\textrm {Þ}\\\\Phi _0 &=& 0,\\\\\\\\\\\\textrm {Þ}^{\\\\prime }\\\\Phi _0 &=& -\\\\mu \\\\Phi _0,\\\\\\\\\\\\eth \\\\Phi _2 &=& -\\\\nu \\\\Phi _0+\\\\tau \\\\Phi _2,\\\\\\\\\\\\textrm {Þ}\\\\Phi _2 &=& -\\\\lambda \\\\Phi _0+\\\\rho \\\\Phi _2,\\\\\\\\\\\\eth \\\\Psi _2 &=& -\\\\pi \\\\Phi _0\\\\overline{\\\\Phi _2}+3\\\\tau \\\\Psi _2,\\\\\\\\\\\\textrm {Þ}\\\\Psi _2 &=& \\\\mu |\\\\Phi _0|^2+3 \\\\rho \\\\Psi _2,$ after which the commutators $[\\\\eth ^{\\\\prime },\\\\, \\\\eth ], [\\\\eth ^{\\\\prime },\\\\, \\\\textrm {Þ}], [\\\\eth , \\\\, \\\\textrm {Þ}^{\\\\prime }]$ and $[\\\\textrm {Þ}^{\\\\prime },\\\\, \\\\textrm {Þ}]$ applied to $\\\\Phi _0$ give $\\\\eth \\\\pi &=& (3\\\\rho -\\\\overline{\\\\rho }) \\\\mu -2\\\\Psi _2+\\\\frac{R}{12},\\\\\\\\\\\\textrm {Þ}\\\\pi &=& 3\\\\rho \\\\pi ,\\\\\\\\ \\\\eth \\\\mu &=& \\\\overline{\\\\lambda }\\\\pi +3 \\\\mu \\\\tau ,\\\\\\\\\\\\textrm {Þ}\\\\mu &=& \\\\pi (\\\\overline{\\\\pi }+3 \\\\tau ) +2 \\\\Psi _2-\\\\frac{R}{12}.\",\n \"$ Herewith GHP equation (') becomes a simple algebraic equation for $\\\\Psi _2$ , $\\\\Psi _2 = \\\\rho \\\\mu -\\\\tau \\\\pi +\\\\frac{R}{12},$ the $\\\\textrm {Þ}$ derivative of which (using (REF ,,REF ,)) results in $\\\\rho R=0$ .\",\n \"As $\\\\rho =0$ would imply $\\\\Phi _0=0$ , this shows that an algebraically special Einstein-Maxwell solution possessing a shear-free and geodesic multiple Weyl-PND which is not a PND of ${F}$ necessarily has a vanishing cosmological constantNote3.\",\n \"The corresponding class of solutions is non-empty: imposing the additional restriction that $\\\\pi =0$ one can deduce that $\\\\overline{\\\\rho }=\\\\rho $ and $\\\\overline{\\\\mu }=\\\\mu $ , together with a $[-2,0]$ –weighted relation $\\\\Psi _3=\\\\rho \\\\nu +\\\\mu \\\\overline{\\\\tau }-\\\\tau \\\\lambda $ .\",\n \"It is then straightforward to construct a Newman-Penrose null-tetrad with the additional restrictions $\\\\alpha =\\\\beta =\\\\epsilon =0$ and $\\\\mu =2\\\\gamma $ : the unique solutions in this case are the Griffiths [20] metrics, containing as special cases the metrics [3], [4], [19], [41].\",\n \"At first sight one would expect the class $\\\\pi \\\\ne 0$ to admit a larger set of solutions, but this is by no means guaranteed (compare with section 2.1): anyway no explicit examples appear to be known.\",\n \"The only property which is easy to demonstrate –though somewhat tedious to be included in the present paragraph– is that no solutions exist for which ${k}$ is non-expanding ($\\\\Re (\\\\rho )\\\\ne 0$ ).\",\n \"As a corollary of the above result we also obtain a new characterization of the class $\\\\mathcal {D}$ metrics[10] with non-vanishing cosmological constant: if one of the multiple Weyl-PND's a Petrov type D Einstein-Maxwell solution would be non-aligned with a PND of ${F}$ then $R=0$ .\",\n \"In other words, for a non-vanishing cosmological constant, the double alignment property of the class $\\\\mathcal {D}$ metrics is a consequence of their Weyl-PND's being geodesic and shear-free.\",\n \"It is tempting to conjecture that this same conclusion also will hold when $\\\\Lambda =0$ .\",\n \"However it has not been possible so far to prove this, except for the special case of the Kundt space-times (i.e.\",\n \"in which the Weyl-PND ${k}$ has vanishing complex divergence).\",\n \"As the proof for this particular case is again quite tedious and little illuminating, I prefer to postpone this part to a possible later and more general publication.\"\n ],\n [\n \"Discussion\",\n \"Most results dealing with exact solutions for (algebraically special) Einstein-Maxwell fields have been obtained in the past introducing special coordinate systems, usually adapted to geodesic and/or shear-free null-congruences, or by imposing, sometimes haphasard looking, restrictions on the spin coefficients of a Newman-Penrose tetrad.\",\n \"This not only has turned their comparison and classification into an awkward procedure, often involving sophisticated computer algebra packages, but also has made it difficult for researchers and students entering the field to recognize the blanks which remain to be filled in.\",\n \"As furthermore the “Exact Solutions book”[40] touches the subject of non-aligned Einstein-Maxwell fields, or of aligned fields with a non-geodesic or shearing multiple PND, only superficially, I present below the known results, together with the ones obtained in the previous sections, schematically.\",\n \"From these diagrams I exclude the conformally flat case, as it implies[36] $\\\\Lambda =0$ with the only non-null member being therefore[40] the Bertotti-Robinson[28], [2], [37] metric and the null members being given by a special class of plane waves[1], [30].\",\n \"In the figures below labels (1), (2), ...next to vertical arrows refer to results obtained in the corresponding previous sections, $\\\\nexists $ indicates that no solutions are allowed, while a question mark indicates that no solutions are known to occur in the literature.\",\n \"Roman capitals II, III, N, D refer to the Petrov types and $`1\\\\times `$ or $`2\\\\times `=$ indicate singly aligned or doubly aligned solutions.\",\n \"Figure: NO_CAPTIONIn Fig.\",\n \"1 the sub-tree corresponding to the singly aligned Einstein-Maxwell solutions with a shear-free and geodesic multiple Weyl-PND (as well as the null fields, which automatically obey this condition) is not included, as little or no progress has been made in this area since the early nineties[40], [32], with the bulk of the material contained in [40].\",\n \"The double occurence of the Plebaǹski-Hacyan metric in Fig.\",\n \"1 (once as PH and once as PH', the GHP-primed version of PH with the roles of ${k}$ and ${\\\\ell }$ interchanged) is due to the fact that PH is doubly aligned with only one null-ray being geodesic and both being shear-free.\",\n \"The absence of doubly aligned Petrov type III solutions in Fig.\",\n \"1 has been noticed already in [40] for $\\\\Lambda =0$ , but can easily be seen to hold also for $\\\\Lambda \\\\ne 0$ .\",\n \"[40] also mentions that the Leroy metric[27] is the unique Einstein-Maxwell solution of Petrov type II with $\\\\Lambda =0$ , in which both the Weyl PND's ${k}$ and ${\\\\ell }$ are also PND's of ${F}$ and in which the multiple PND ${k}$ is geodesic and shear-free; this is not clear at all, as Leroy's solutions were only obtained in the non-radiative sub-case.\",\n \"It furthermore remains to be checked whether this still holds –for a suitable generalisation of the Leroy metric– when $\\\\Lambda \\\\ne 0$ .\",\n \"Figure: Algebraically special non-nul Einstein-Maxwell solutions for which the multiple Weyl-PND is k{k} not a PNDof F{F}.\",\n \"The sub-tree under Griffiths presents in an invariant way the subcases mentioned in ,originally obtained by imposing coordinate restrictions or restrictions on the NP spin-coefficients.\"\n ],\n [\n \"Acknowledgment\",\n \"All calculations were done using the Maple symbolic algebra system[45].\",\n \"I also thank Lode Wylleman for a critical reading of the manuscript.\"\n ],\n [\n \"Appendix: GHP, Maxwell and Bianchi equations\",\n \"Weights of the spin-coefficients, the Maxwell and Weyl spinor components and the GHP operators: $&\\\\kappa : [3, 1], \\\\nu : [-3, -1], \\\\sigma : [3, -1], \\\\lambda : [-3, 1], \\\\nonumber \\\\\\\\&\\\\rho : [1, 1], \\\\mu : [-1, -1], \\\\tau : [1, -1], \\\\pi : [-1, 1], \\\\nonumber \\\\\\\\&\\\\Phi _0 : [2, 0], \\\\Phi _1 : [0, 0], \\\\Phi _2 : [-2, 0], \\\\nonumber \\\\\\\\&\\\\Psi _0 : [4, 0], \\\\Psi _1 : [2, 0], \\\\Psi _2 : [0, 0], \\\\Psi _3: [-2,0], \\\\Psi _4 : [-4, 0] ,\\\\nonumber \\\\\\\\&\\\\eth : [1, -1], \\\\eth ^{\\\\prime }: [-1,1], \\\\textrm {Þ}^{\\\\prime }: [-1,-1], \\\\textrm {Þ}: [1,1].\",\n \"\\\\nonumber $ The prime operation is an involution with $\\\\kappa ^{\\\\prime } &=& -\\\\nu ,\\\\sigma ^{\\\\prime }=-\\\\lambda ,\\\\rho ^{\\\\prime }=-\\\\mu , \\\\tau ^{\\\\prime }=-\\\\pi ,\\\\\\\\{\\\\Psi _0}^{\\\\prime } &=& \\\\Psi _4, {\\\\Psi _1}^{\\\\prime }=\\\\Psi _3, {\\\\Psi _2}^{\\\\prime }=\\\\Psi _2,\\\\\\\\\\\\Phi _0^{\\\\prime } &=& -\\\\Phi _2, \\\\Phi _1^{\\\\prime }=-\\\\Phi _1.$ The GHP commutators acting on $(p,q)$ weighted quantities are given by: $\\\\left[ \\\\textrm {Þ},\\\\textrm {Þ}^{\\\\prime } \\\\right] &=& (\\\\pi +\\\\overline{\\\\tau })\\\\eth +(\\\\overline{\\\\pi }+\\\\tau )\\\\eth ^{\\\\prime }+(\\\\kappa \\\\nu -\\\\pi \\\\tau + \\\\frac{R}{24}-\\\\Phi _{11}-\\\\Psi _2)p \\\\nonumber \\\\\\\\&&+(\\\\overline{\\\\kappa }\\\\overline{\\\\nu }-\\\\overline{\\\\pi }\\\\overline{\\\\tau }+ \\\\frac{R}{24}-\\\\Phi _{11}-\\\\overline{\\\\Psi }_2)q,\\\\\\\\\\\\left[ \\\\eth ,\\\\eth ^{\\\\prime }\\\\right] &=& (\\\\mu -\\\\overline{\\\\mu })\\\\textrm {Þ}+(\\\\rho -\\\\overline{\\\\rho })\\\\textrm {Þ}^{\\\\prime } +(\\\\lambda \\\\sigma -\\\\mu \\\\rho - \\\\frac{R}{24}-\\\\Phi _{11}+\\\\Psi _2)p \\\\nonumber \\\\\\\\&&-(\\\\overline{\\\\lambda \\\\sigma }-\\\\overline{\\\\mu }\\\\overline{\\\\rho }-\\\\frac{R}{24}-\\\\Phi _{11}+\\\\overline{\\\\Psi }_2)q,\\\\\\\\\\\\left[ \\\\textrm {Þ},\\\\eth \\\\right] &=& \\\\overline{\\\\pi }\\\\,\\\\textrm {Þ}-\\\\kappa \\\\textrm {Þ}^{\\\\prime } +\\\\overline{\\\\rho }\\\\,\\\\eth +\\\\sigma \\\\eth ^{\\\\prime }+(\\\\kappa \\\\mu -\\\\sigma \\\\pi -\\\\Psi _1)p + (\\\\overline{\\\\kappa \\\\lambda }-\\\\overline{\\\\pi }\\\\overline{\\\\rho }-\\\\Phi _{01})q .$ GHP equations: $\\\\textrm {Þ}\\\\rho -\\\\eth ^{\\\\prime }\\\\kappa &=& \\\\rho ^2+\\\\sigma \\\\overline{\\\\sigma }-\\\\overline{\\\\kappa }\\\\tau +\\\\kappa \\\\pi +\\\\Phi _{00}, \\\\\\\\\\\\textrm {Þ}\\\\sigma -\\\\eth \\\\kappa &=& (\\\\rho +\\\\overline{\\\\rho })\\\\sigma +(\\\\overline{\\\\pi }-\\\\tau )\\\\kappa +\\\\Psi _0, \\\\\\\\\\\\textrm {Þ}\\\\tau -\\\\textrm {Þ}^{\\\\prime }\\\\kappa &=& (\\\\tau +\\\\overline{\\\\pi })\\\\rho +(\\\\overline{\\\\tau }+\\\\pi )\\\\sigma +\\\\Phi _{01}+\\\\Psi _1, \\\\\\\\\\\\textrm {Þ}\\\\nu -\\\\textrm {Þ}^{\\\\prime } \\\\pi &=& (\\\\pi +\\\\overline{\\\\tau })\\\\mu +(\\\\overline{\\\\pi }+\\\\tau )\\\\lambda +\\\\Psi _3+\\\\overline{\\\\Phi _1}\\\\Phi _2,\\\\\\\\\\\\eth \\\\rho -\\\\eth ^{\\\\prime }\\\\sigma &=& (\\\\rho -\\\\overline{\\\\rho })\\\\tau +(\\\\mu -\\\\overline{\\\\mu })\\\\kappa +\\\\Phi _{01}-\\\\Psi _1,\\\\\\\\\\\\textrm {Þ}^{\\\\prime }\\\\sigma -\\\\eth \\\\tau &=& -\\\\sigma \\\\mu -\\\\overline{\\\\lambda }\\\\rho -\\\\tau ^2+\\\\kappa \\\\overline{\\\\nu }-\\\\Phi _{02},\\\\\\\\\\\\textrm {Þ}^{\\\\prime }\\\\rho -\\\\eth ^{\\\\prime }\\\\tau &=& -\\\\overline{\\\\mu }\\\\rho -\\\\lambda \\\\sigma -\\\\tau \\\\overline{\\\\tau }+\\\\kappa \\\\nu -\\\\frac{R}{12}-\\\\Psi _2.\",\n \"$ Maxwell equations: $\\\\textrm {Þ}\\\\Phi _1-\\\\eth ^{\\\\prime }\\\\Phi _0 &=& \\\\pi \\\\Phi _0+2\\\\rho \\\\Phi _1-\\\\kappa \\\\Phi _2, \\\\\\\\\\\\textrm {Þ}\\\\Phi _2-\\\\eth ^{\\\\prime }\\\\Phi _1 &=& -\\\\lambda \\\\Phi _0+2 \\\\pi \\\\Phi _1+\\\\rho \\\\Phi _2.\",\n \"$ Bianchi equations ($\\\\Phi _{IJ}=\\\\Phi _I\\\\overline{\\\\Phi _J}$ and $\\\\Lambda =R/4=constant$ ): ${\\\\eth ^{\\\\prime }} \\\\Psi _{{0}} -{\\\\textrm {Þ}} \\\\Psi _{{1}}+{\\\\textrm {Þ}} \\\\Phi _{{01}} -{\\\\eth } \\\\Phi _{{00}} &=& -\\\\pi \\\\,\\\\Psi _{{0}}-4\\\\,\\\\rho \\\\,\\\\Psi _{{1}}+3\\\\,\\\\kappa \\\\,\\\\Psi _{{2}}+ \\\\overline{\\\\pi }\\\\Phi _{{00}}+2\\\\,\\\\overline{\\\\rho }\\\\Phi _{{01}}+2\\\\,\\\\sigma \\\\,\\\\Phi _{{10}} \\\\nonumber \\\\\\\\&&-2\\\\,\\\\kappa \\\\,\\\\Phi _{{11}}-\\\\overline{\\\\kappa }\\\\Phi _{{02}}, \\\\\\\\{\\\\textrm {Þ}^{\\\\prime }} \\\\Psi _{{0}} -{\\\\eth } \\\\Psi _{{1}}+{\\\\textrm {Þ}} \\\\Phi _{{02}} -{\\\\eth } \\\\Phi _{{01}} &=& -\\\\mu \\\\,\\\\Psi _{{0}}-4\\\\,\\\\tau \\\\,\\\\Psi _{{1}}+3\\\\,\\\\sigma \\\\,\\\\Psi _{{2}}-\\\\overline{\\\\lambda }\\\\Phi _{{00}}+2\\\\, \\\\overline{\\\\pi }\\\\Phi _{{01}}+2\\\\,\\\\sigma \\\\,\\\\Phi _{{11}}\\\\nonumber \\\\\\\\&&+ \\\\overline{\\\\rho }\\\\Phi _{{02}}-2\\\\,\\\\kappa \\\\,\\\\Phi _{{12}}, $ $&&3\\\\,{\\\\eth ^{\\\\prime }} \\\\Psi _{{1}} -3\\\\,{\\\\textrm {Þ}} \\\\Psi _{{2}}+2\\\\,{\\\\textrm {Þ}} \\\\Phi _{{11}} -2\\\\,{\\\\eth }\\\\Phi _{{10}} +{\\\\eth ^{\\\\prime }} \\\\Phi _{{01}} -{\\\\textrm {Þ}^{\\\\prime }}\\\\Phi _{{00}} = 3\\\\,\\\\lambda \\\\,\\\\Psi _{{0}}-9\\\\,\\\\rho \\\\,\\\\Psi _{{2}}-6\\\\,\\\\pi \\\\,\\\\Psi _{{1}}+6\\\\,\\\\kappa \\\\,\\\\Psi _{{3}}+ (\\\\overline{\\\\mu }-2\\\\,\\\\mu ) \\\\Phi _{{00}}\\\\nonumber \\\\\\\\&& \\\\ \\\\ + 2\\\\,(\\\\pi +\\\\overline{\\\\tau }) \\\\Phi _{{01}}+2\\\\, ( \\\\tau + \\\\overline{\\\\pi }) \\\\Phi _{{10}}+2\\\\, ( 2\\\\,\\\\overline{\\\\rho }-\\\\rho ) \\\\Phi _{{11}}+2\\\\,\\\\sigma \\\\,\\\\Phi _{{20}}-\\\\overline{\\\\sigma }\\\\Phi _{{02}}-2\\\\,\\\\overline{\\\\kappa }\\\\Phi _{{12}}-2\\\\,\\\\kappa \\\\,\\\\Phi _{{21}}, \\\\\\\\&&3\\\\,{\\\\textrm {Þ}^{\\\\prime }} \\\\Psi _{{1}} -3\\\\,{\\\\eth } \\\\Psi _{{2}}+2\\\\,{\\\\textrm {Þ}} \\\\Phi _{{12}} -2\\\\,{\\\\eth }\\\\Phi _{{11}} +{\\\\eth ^{\\\\prime }} \\\\Phi _{{02}} -{\\\\textrm {Þ}^{\\\\prime }}\\\\Phi _{{01}} = 3\\\\,\\\\nu \\\\,\\\\Psi _{{0}}-6\\\\,\\\\mu \\\\,\\\\Psi _{{1}}-9\\\\,\\\\tau \\\\,\\\\Psi _{{2}}+6\\\\,\\\\sigma \\\\,\\\\Psi _{{3}}-\\\\overline{\\\\nu }\\\\Phi _{{00}}\\\\nonumber \\\\\\\\&& \\\\ \\\\ \\\\ +2\\\\, (\\\\overline{\\\\mu }-\\\\mu )\\\\Phi _{{01}} -2\\\\,\\\\overline{\\\\lambda }\\\\Phi _{{10}}+2\\\\, ( \\\\tau +2 \\\\overline{\\\\pi }) \\\\Phi _{{11}}+ (2\\\\,\\\\pi +\\\\overline{\\\\tau }) \\\\Phi _{{02}} + 2\\\\,(\\\\overline{\\\\rho }-\\\\rho ) \\\\Phi _{{12}}+2\\\\,\\\\sigma \\\\,\\\\Phi _{{21}}-2\\\\,\\\\kappa \\\\,\\\\Phi _{{22}}.\",\n \"$\"\n ]\n]"},"id":{"kind":"string","value":"1605.05830"}}},{"rowIdx":1866693,"cells":{"text":{"kind":"list like","value":[["Forward $J/\\psi$ production at high energy: centrality dependence and\n mean transverse momentum"],["Abstract Forward rapidity $J/\\psi$ meson production in proton-nucleus collisions can be an important constraint of descriptions of the small-$x$ nuclear wavefunction.","In an earlier work we studied this process using a dipole cross section satisfying the Balitsky-Kovchegov equation, fit to HERA inclusive data and consistently extrapolated to the nuclear case using a standard Woods-Saxon distribution.","In this paper we present further calculations of these cross sections, studying the mean transverse momentum of the meson and the dependence on collision centrality.","We also extend the calculation to backward rapidities using nuclear parton distribution functions.","We show that the parametrization is overall rather consistent with the available experimental data.","However, there is a tendency towards a too strong centrality dependence.","This can be traced back to the rather small transverse area occupied by small-$x$ gluons in the nucleon that is seen in the HERA data, compared to the total inelastic nucleon-nucleon cross section."],["Introduction","The production of $J/\\psi $ mesons at forward rapidity in high energy proton-proton and proton-nucleus collisions can provide valuable information on gluon saturation.","Indeed, the production of particles at forward rapidity probes the target at very small $x$ , where saturation effects should be enhanced.","In particular, the charm quark mass being of the same order of magnitude as the saturation scale, $J/\\psi $ production should be sensitive to these dynamics.","The charm quark mass is also large enough to provide a hard scale, making a perturbative study of this process possible.","In addition, $J/\\psi $ production has been the subject of many experimental studies at the LHC, both in proton-proton [1], [2], [3], [4], [5], [6], [7] and in proton-nucleus [8], [9], [10], [11], [12] collisions.","This provides a lot of data to confront with nuclear effects predicted by various theoretical models, both in the Color Glass Condensate (CGC) framework [13], [14], [15], [16], [17] and as constraints for nuclear parton distribution functions [18] and energy loss in cold nuclear matter [19], [20], [21], [22].","In a recent work [14] we studied, in the CGC framework, the production of forward $J/\\psi $ mesons in proton-proton and proton-nucleus collisions at the LHC.","We showed that, when using the Glauber approach to generalize the dipole cross section to nuclei, the nuclear suppression for minimum bias events is smaller than in previous CGC calculations such as [13], and much closer to experimental data.","In this paper we will study, in the same framework, other observables of interest in this process, such as $J/\\psi $ production at backward rapidity, the centrality dependence in the optical Glauber model and the mean transverse momentum of the produced ${J/\\psi }$ 's."],["Formalism","Let us briefly recall the main steps of the calculation.","For more details we refer the reader to Ref. [14].","We use the color evaporation model (CEM) to relate the cross section for $J/\\psi $ production to the $c\\bar{c}$ pair production cross section.","In this model a fixed fraction $F_{J/\\psi }$ of the $c\\bar{c}$ pairs produced below the $D$ -meson threshold are assumed to hadronize into $J/\\psi $ mesons: $\\frac{\\, \\mathrm {d}\\sigma _{J/\\psi }}{\\, \\mathrm {d}^2{{P}_\\perp }\\, \\mathrm {d}Y}=F_{J/\\psi } \\; \\int _{4m_c^2}^{4M_D^2} \\, \\mathrm {d}M^2\\frac{\\, \\mathrm {d}\\sigma _{c\\bar{c}}}{\\, \\mathrm {d}^2{{P}_\\perp }\\, \\mathrm {d}Y \\, \\mathrm {d}M^2}\\, ,$ where ${{P}_\\perp }$ , $Y$ and $M$ are the transverse momentum, the rapidity and the invariant mass of the $c\\bar{c}$ pair respectively, $M_D=1.864$ GeV is the D meson mass and $m_c$ is the charm quark mass that we will vary between 1.2 and 1.5 GeV.","Note that in this work we will focus on ratios where $F_{J/\\psi }$ cancels so we do not need to fix it to any specific value here.","The study of gluon and quark pair production in the dilute-dense limit of the CGC formalism was started some time ago [23], [24] (see also [25]) and used in several calculations such as [26], [27], [13], [28].","The physical picture is the following: an incoming gluon from the projectile can split into a quark-antiquark pair either before or after the interaction with the target.","The partons propagating trough the target are assumed to interact eikonally with it, picking up a Wilson line factor in either the adjoint (for gluons) or the fundamental (for quarks) representation.","Since we study $J/\\psi $ production at forward rapidity, where the projectile is probed at large $x$ , we will use the collinear approximation in which the incoming gluon is assumed to have zero transverse momentum.","In this approach the cross section for $c\\bar{c}$ production reads, in the large-${N_\\mathrm {c}}$ limit [13], $\\frac{\\, \\mathrm {d}\\sigma _{c\\bar{c}}}{\\, \\mathrm {d}^2{{p}_\\perp }\\, \\mathrm {d}^2{{q}_\\perp }\\, \\mathrm {d}y_p \\, \\mathrm {d}y_q}=\\frac{\\alpha _{\\mathrm {s}}^2 {N_\\mathrm {c}}}{8\\pi ^2 d_\\mathrm {A}}\\\\ \\times \\frac{1}{(2\\pi )^2}\\int \\limits _{{{k}_\\perp }}\\frac{\\Xi _{\\rm coll}({{p}_\\perp }+ {{q}_\\perp },{{k}_\\perp })}{({{p}_\\perp }+ {{q}_\\perp })^2}x_1 G_p(x_1,Q^2)\\\\\\times \\phi _{y_2=\\ln {\\frac{1}{x_2}}}^{q\\bar{q},g}({{p}_\\perp }+ {{q}_\\perp },{{k}_\\perp })\\;,$ where ${{p}_\\perp }$ and ${{q}_\\perp }$ denote the transverse momenta of the quarks, $y_p$ and $y_q$ their rapidities, $\\int _{{{k}_\\perp }} \\equiv \\int \\, \\mathrm {d}^2 {{k}_\\perp }/ (2\\pi )^2$ , and $d_\\mathrm {A}\\equiv {N_\\mathrm {c}}^2-1$ is the dimension of the adjoint representation of SU(${N_\\mathrm {c}}$ ).","The longitudinal momentum fractions $x_1$ and $x_2$ probed in the projectile and the target respectively are given by $x_{1,2}=\\frac{\\sqrt{{{P}_\\perp }^2+M^2}}{\\sqrt{s}}e^{\\pm Y} \\; .$ The explicit expression for the “hard matrix element” $\\Xi _{\\rm coll}$ is given in Ref. [13].","In Eq.","(REF ), $G_p(x_1,Q^2)$ is the gluon density in the probe and is described, in the collinear approximation that we use here, in terms of usual parton distribution functions (PDFs).","In the following we use, unless otherwise stated, the MSTW 2008 [29] parametrization for $G_p$ .","For consistency we use the leading other (LO) parton distribution since also the rest of the calculation is made at this order.","To estimate the uncertainty associated with the choice of the factorization scale $Q$ , we will vary it between $\\frac{1}{2}\\sqrt{{{P}_\\perp }^2+M^2}$ and $2\\sqrt{{{P}_\\perp }^2+M^2}$ .","The function $\\phi _{_Y}^{q \\bar{q},g}$ describes the propagation of a $q\\bar{q}$ pair in the color field of the target and reads $\\phi _{_Y}^{q \\bar{q},g}({{l}_\\perp },{{k}_\\perp })=\\int \\mathrm {d}^2 {{b}_\\perp }\\frac{N_c{{l}_\\perp }^2}{4 \\alpha _{\\mathrm {s}}} \\;\\\\ \\times S_{_Y}({{k}_\\perp },{{b}_\\perp }) \\;S_{_Y}({{l}_\\perp }-{{k}_\\perp },{{b}_\\perp }) \\;,$ where ${{b}_\\perp }$ is the impact parameter.","In this expression, the function $S_{_Y}({{k}_\\perp },{{b}_\\perp })$ contains all the information about the target.","It is the fundamental representation dipole correlator: $S_{_Y}({{k}_\\perp },{{b}_\\perp }) = \\int \\, \\mathrm {d}^2 {{r}_\\perp }e^{i{{k}_\\perp }\\cdot {{r}_\\perp }}S_{_Y}({{r}_\\perp },{{b}_\\perp }) \\;,$ with $S_{_Y}({{x}_\\perp }-{{y}_\\perp },{{b}_\\perp }) = \\frac{1}{{N_\\mathrm {c}}}\\left< \\, \\mathrm {Tr} \\, U^\\dag ({{x}_\\perp })U({{y}_\\perp })\\right>,$ where $U({{x}_\\perp })$ is a fundamental representation Wilson line in the color field of the target.","The dipole correlator $S_{_Y}({{k}_\\perp },{{b}_\\perp })$ is obtained by solving numerically the running coupling Balitsky-Kovchegov (rcBK) equation [30], [31], [32].","For the initial condition we use the 'MVe' parametrization introduced in Ref.","[33], which reads, in the case of a proton target, $S^p_{Y= \\ln \\frac{1}{x_0}}({{r}_\\perp },{{b}_\\perp }) = \\exp \\bigg [ -\\frac{{{r}_\\perp }^2 Q_\\mathrm {s0}^2}{4}\\\\ \\times \\ln \\left(\\frac{1}{|{{r}_\\perp }| \\Lambda _{\\mathrm {QCD}}}\\!+\\!e_c \\cdot e\\right)\\bigg ],$ where $x_0=0.01$ .","The running coupling is taken as: $\\alpha _{\\mathrm {s}}(r) = \\frac{12\\pi }{(33 - 2N_f) \\log \\left(\\frac{4C^2}{r^2\\Lambda _{\\mathrm {QCD}}^2} \\right)} \\;,$ where $C$ parametrizes the uncertainty related to the scale of the strong coupling in the transverse coordinate space.","The free parameters in these expressions are obtained by fitting the combined inclusive HERA DIS cross section data [34] for $Q^2<50$ GeV$^2$ and $x<0.01$ .","Their best fit values (with $\\chi ^2/\\text{d.o.f} = 1.15$ ) are $Q_\\mathrm {s0}^2= 0.060$ GeV$^2$ , $C^2= 7.2$ , $e_c=18.9$ and $\\sigma _0/2 = 16.36$ mb.","In the case of a proton target, the dipole amplitude does not have an explicit impact parameter dependence and we thus make the replacement $\\int \\mathrm {d}^2 {{b}_\\perp }\\rightarrow \\frac{\\sigma _0}{2}$ in Eq.","(REF ), $\\frac{\\sigma _0}{2}$ corresponding to the effective proton transverse area measured in DIS experiments.","The fit only includes light quarks.","In particular this leaves the charm quark mass that would be consistent with the DIS data in this model still uncertain, which is why we vary it in a rather large range for the uncertainty estimate in this work.","Figure: Nuclear modification factor R pPb R_\\text{pPb} at negative rapidity as a function of YY (left) and P ⊥ P_\\perp (right) at s NN =5\\sqrt{s_{NN}}=5 TeV.","Data from Refs.",", , .Figure: Forward to backward ratio in proton-lead collisions as a function of YY (left) and P ⊥ P_\\perp (right) at s NN =5\\sqrt{s_{NN}}=5 TeV.","Data from Refs.",", .To generalize this proton dipole correlator to the case of a nuclear target we use, as in [33], the optical Glauber model.","In this model the gluons at the initial rapidity $Y=\\ln 1/x_0$ are localized in the individual nucleons of the nucleus.","The nucleons are then taken to be distributed randomly and independently in the transverse plane according to the standard Woods-Saxon nuclear density profile.","An analytical average over the positions of the nucleons leads to the following initial condition for the rcBK evolution of a nuclear target: $S^A_{Y=\\ln \\frac{1}{x_0}}({{r}_\\perp },{{b}_\\perp }) = \\exp \\Bigg [ -A T_A({{b}_\\perp })\\frac{\\sigma _0}{2} \\frac{{{r}_\\perp }^2 Q_\\mathrm {s0}^2}{4}\\\\\\times \\ln \\left(\\frac{1}{|{{r}_\\perp }|\\Lambda _{\\mathrm {QCD}}}+e_c \\cdot e\\right) \\Bigg ] \\; .$ Here $T_A$ is the standard Woods-Saxon transverse thickness function of the nucleus: $T_A({{b}_\\perp })= \\int dz \\frac{n}{1+\\exp \\left[ \\frac{\\sqrt{{{b}_\\perp }^2 + z^2}-R_A}{d} \\right]} \\; ,$ with $d=0.54\\,\\mathrm {fm}$ and $R_A=(1.12A^{1/3}-0.86A^{-1/3})\\,\\mathrm {fm}$ , and $n$ is defined such that $T_A$ is normalized to unity.","All the other parameters in the initial condition (REF ), which is evolved using the rcBK equation for each ${{b}_\\perp }$ , are the same as in the proton case.","In this model the dipole amplitude depends on the impact parameter and we need to integrate explicitly over it.","The impact parameter dependence, which carries over to the centrality dependence, thus appears naturally in this model.","Our practical procedure for carrying out this comparison will be discussed in more detail in Sec.","."],["Backward rapidity","In our previous work [14] we only considered $J/\\psi $ production in proton-proton and proton-nucleus collisions at forward rapidity.","In these kinematics the process can be seen as the collision of a dilute proton probed at large $x$ , which can be described using well known parton distribution functions (PDFs), and a dense target described in terms of classical color fields.","The nuclear modification of $J/\\psi $ production was also measured at backward rapidity by ALICE [8], [10] and LHCb [9].","In this case the produced $J/\\psi $ is moving in the direction of the incoming nucleus and the physical picture is the same as at forward rapidity, but with the roles of the projectile and the target interchanged, i.e.","a dilute proton or nucleus interacts with a dense proton target.","The latter is described in the same way as in proton-proton collisions at forward rapidity, while the projectile is described either by a PDF in the case of a proton or by a nuclear PDF (nPDF) in the case of a nucleus.","Therefore the calculation is very similar to the case of proton-proton collisions in our previous study.","Note, however, that in general nuclear PDFs are less tightly constrained by experimental data than usual proton PDFs.","In the following we will use the leading order EPS09 nPDF parametrization [35] which provides additional error sets to estimate this uncertainty.","The LO EPS09 analysis uses the CTEQ6L1 [36] proton PDFs as a reference, therefore for consistency we use same parametrization of proton PDFs when computing the nuclear modification factor at backward rapidity.","Nevertheless, while it could be sizeable for the cross section, the difference compared to the MSTW 2008 parametrization is very small for the nuclear modification factor.","In Fig.","REF we show the nuclear modification factor $R_\\text{pA}$ , defined as $R_{\\rm pA}= \\frac{1}{A}\\frac{\\frac{\\, \\mathrm {d}\\sigma ^\\text{pA}}{\\, \\mathrm {d}^2 {{P}_\\perp }\\, \\mathrm {d}Y}}{\\frac{\\, \\mathrm {d}\\sigma ^\\text{pp}}{\\, \\mathrm {d}^2 {{P}_\\perp }\\, \\mathrm {d}Y}} \\; ,$ as a function of $Y$ and $P_\\perp $ obtained in this way at negative rapidity compared with data from ALICE [8], [10] and LHCb [9] experiments.","The uncertainty in our calculation is significantly larger than at forward rapidity [14].","This is due to the fact that we include in our uncertainty band, in addition to the variation of the charm quark mass and of the factorization scale, the nuclear PDF uncertainty obtained following the procedure described in Ref. [35].","In particular, our lower bound for the factorization scale is $Q=\\frac{1}{2}\\sqrt{{{P}_\\perp }^2+M^2}$ which can reach values of less than 2 GeV at small transverse momentum.","The nuclear PDFs are not well constrained at such small scales at the moment.","Nevertheless the general agreement with data is quite good taking into account the rather large theoretical and experimental uncertainties.","In this case deviations of $R_\\text{pPb}$ from unity are entirely due to the nuclear PDFs.","Now that we have computed the nuclear modification factor both at forward [14] and backward rapidities, we have access to the forward to backward ratio $R_\\text{FB}$ , defined as $R_\\text{FB}(P_\\perp ,Y)=\\frac{R_\\text{pA}(P_\\perp ,Y)}{R_\\text{pA}(P_\\perp ,-Y)} \\; .$ This ratio can be interesting to study because there may be an additional cancellation of some uncertainties common to the numerator and the denominator.","In particular, when determining the nuclear modification factor, experimental studies such as [8], [9] have to use an interpolation for the reference proton-proton cross section since there is no data at $\\sqrt{s}=5$ TeV.","This interpolation is not needed to study $R_\\text{FB}$ , but the final statistical uncertainty may be larger if the coverage in rapidity by the detector is not symmetric with respect to 0.","Concerning our calculation, we have seen that at negative rapidity the uncertainty on $R_\\text{pA}$ due to nuclear PDFs is rather large.","This error will remain in $R_\\text{FB}$ since the computation of $R_\\text{pA}$ at positive rapidity does not involve nPDFs.","Indeed, we see from Fig.","REF , where we show the forward to backward ratio as a function of $Y$ and $P_\\perp $ , that the uncertainty on this quantity is still quite large.","Nevertheless, within this error band the agreement with data is reasonable, although the variation at low $P_\\perp $ seems to be steeper than in the data.","In Fig.","REF (R) we only show our results for $R_\\text{FB}$ as a function of $P_\\perp $ integrated over the same $Y$ range as ALICE data [8], which is slightly smaller than for LHCb data [9], but our results for $2.5$ for the optical Glauber centrality classes compared to the values given by ALICE [12] for the experimental ones.","For central collisions the average number of binary collisions estimated by ALICE is smaller than in the optical Glauber model, while the opposite is true for peripheral collisions.","The impact parameter is not directly observable, so the experimental centrality selection has to use some other observable as a proxy for it.","In the case of the ALICE analysis [12] the observable used is the energy of the lead-going side Zero-Degree Calorimeter (ZDC).","Due to the large fluctuations in the signal for a fixed impact parameter, the values of $N_\\text{coll}$ vary less strongly with centrality in the experimental classes than in the optical Glauber ones.","Thus, for example, a relatively peripheral smaller-$N_\\text{coll}$ event can end up in the most central class in the case of a fluctuation in the ZDC signal."],["Fixed impact parameter approximation","Developing a detailed Monte Carlo Glauber model that would enable us to exactly match the experimental centrality classes would be beyond the scope of this work.","Therefore we start from the assumption that the ALICE Glauber model is correct and produces a reliable estimate for the $\\langle N_\\text{coll} \\rangle $ in each experimental class.","We then assume that the mapping between this $\\langle N_\\text{coll} \\rangle $ and the impact parameter is accurately enough described by our optical Glauber model, and use the relation $N_\\text{coll.","opt}(b)=\\langle N_\\text{coll} \\rangle _\\text{ALICE}$ to determine a mean impact parameter corresponding to the experimental centrality class.","The values of $b$ resulting from this procedure are shown in Table REF .","We then calculate the nuclear modification factor for the centrality class using this fixed value of $b$ .","We note that following this procedure for the 80-100% centrality class would lead to a value of $b$ for which our calculation would not be applicable because the saturation scale of the nucleus falls below the one of the proton.","Therefore in the following we will only show results for the five most central classes considered by ALICE.","Figure: Left: saturation scale of the lead nucleus in the MVe parametrization as a function of the impact parameter compared with the saturation scale of the proton at x=10 -4 x=10^{-4}.","Right: number of binary collisions in proton-lead collisions as a function of the impact parameter in the optical Glauber model.In Fig.","REF we show a comparison of our calculation with ALICE data for the nuclear modification factor in different centrality classes, $Q_\\text{pA}$ , defined as $Q_{\\rm pA}= \\frac{\\frac{\\, \\mathrm {d}N^\\text{pA}}{\\, \\mathrm {d}^2 {{P}_\\perp }\\, \\mathrm {d}Y}}{ A \\langle T_A \\rangle \\frac{\\, \\mathrm {d}\\sigma ^\\text{pp}}{\\, \\mathrm {d}^2 {{P}_\\perp }\\, \\mathrm {d}Y}} \\; ,$ as a function of $P_\\perp $ .","We observe that the description of experimental data is generally satisfactory in the first four bins.","For the fifth bin the value of $Q_\\text{pPb}$ obtained in the optical Glauber model is almost constant and very close to one, while the data still shows a significant variation with $P_\\perp $ .","The discrepancy with experimental data for peripheral collisions comes from the fact that in our model the saturation scale of the lead nucleus falls below the one of the proton for a value of $b$ of the order of 6.3 fm, as shown in Fig.","REF (L).","We see from Fig.","REF (R), where we show the number of binary collisions as a function of $b$ , that this corresponds to $N_\\text{coll} \\sim 4.3$ .","This is the point where, by definition, $Q_\\text{pA}$ reaches 1 in our calculation and beyond which the validity of the framework we have used is questionable.","On the other hand, ALICE data shows that $Q_\\text{pA}$ is still significantly smaller than 1 down to $N_\\text{coll} \\sim 2.1$ (see Fig.","REF ).","The strong centrality dependence is caused by the value of $\\frac{\\sigma _0}{2}$ extracted from DIS fits being much smaller than the total inelastic nucleon-nucleon cross section.","The HERA data, both the inclusive cross section fitted in Ref.","[33] and data on exclusive vector meson production (see e.g.","[37]), lead to a picture where the small-$x$ gluons that can participate in a hard process in a proton are concentrated in a rather small area in the transverse plane (see also the discussion in Ref. [38]).","This small-$x$ gluon “hot spot” is then surrounded by a larger “cloud” that only participates in soft interactions, contibuting to the total nucleon-nucleon inelastic cross section.","Our model takes this picture to the extreme, by assuming that at the initial rapidity $\\ln 1/x_0$ the gluons contributing to ${J/\\psi }$ production are concetrated in the area $\\sigma _0/2 \\sim 0.3 \\sigma _\\text{inel}$ inside the target nucleons.","Thus, for peripheral collisions, the probe proton can overlap with the soft cloud of $N_\\text{coll} \\sim 4$ target nucleons while still seeing on average only one small-$x$ gluon hot spot in the target, thus leaving a hard process like ${J/\\psi }$ production approximately unmodified.","This centrality dependence could probably be mildened by using a larger value for $\\frac{\\sigma _0}{2}$ of the order of $\\sigma _\\text{inel}$ (as is effectively done in [13]).","This would, however, lose the consistency of our description of the nucleon from HERA to the LHC.","Also, as discussed in Refs.","[33], [14], varying these parameters in an uncontrolled way could very easily, depending on how exactly it is done, lead to an excessive suppression for minimum bias collisions, or to an $R_\\text{pA}$ that is very far from unity even at high transverse momentum.","Note that the saturation scale (or gluon density) at the edge of the nucleus being smaller than that of the proton is an artefact of averaging over the transverse locations of the dense but small gluon hot spots of the nucleons in the target nucleus.","Therefore we do not use the optical Glauber parametrization in this region, but explicitly set $R_\\text{pA}$ to unity.","Nevertheless, even an explicit Monte Carlo Glauber procedure with the same parameters would not change the ordering $\\sigma _0/2 < \\sigma _\\text{inel}$ that leads to the absence of nuclear effects for peripheral collisions with $N_\\text{coll} \\lesssim 4$ .","Figure: Q pPb Q_\\text{pPb} at a function of P ⊥ P_\\perp at s NN =5\\sqrt{s_{NN}}=5 TeV in the 60-80% centrality class, both when using a fixed impact parameter and when integrating explicitly over bb using the N coll N_\\text{coll} distribution obtained in two different models.","Data from Ref.",".Figure: Nuclear modification factor Q pPb Q_\\text{pPb} as a function of N coll N_\\text{coll} at s NN =5\\sqrt{s_{NN}}=5 TeV compared with ALICE data , both when using a fixed impact parameter and when integrating explicitly over bb.Figure: Mean transverse momentum as a function of YY in proton-proton collisions at s=7\\sqrt{s}=7 TeV (left) and in proton-lead collisions at s NN =5\\sqrt{s_{NN}}=5 TeV (right).","Data from Ref.",".Figure: Average transverse momentum squared〈P ⊥ 2 〉 pPb \\langle P_\\perp ^2 \\rangle _\\text{pPb} (left) and nuclear transverse momentum broadening 〈P ⊥ 2 〉 pPb -〈P ⊥ 2 〉 pp \\langle P_\\perp ^2 \\rangle _\\text{pPb}-\\langle P_\\perp ^2 \\rangle _\\text{pp} (right) as a function of N coll N_\\text{coll} at s NN =5\\sqrt{s_{NN}}=5 TeV compared with ALICE data .","For the latter both the result using a fixed impact parameter and when integrating explicitly over bb are shown."],["Explicit integration over the impact parameter","The results we have shown in the previous section were obtained with a fixed impact parameter chosen so that the number of binary collisions in the optical Glauber model is equal to the average number of binary collisions estimated by ALICE in each centrality class.","However, the nuclear modification ratio in a given centrality bin receives contributions from a distribution of different impact parameters.","One could therefore argue that having a profile in the impact parameter space and integrating over $b$ could lead to different results.","To quantify this effect we will here use two different kinds of $N_\\text{coll}$ distributions to obtain distributions in the impact parameter space.","The first one is provided by ALICE [39] and is obtained from the Slow Nucleon Model (SNM) [40].","Since in this model the average number of binary collisions is not the same as the one obtained in the hybrid method used in Ref.","[12], we shift the distributions so that $\\langle N_\\text{coll} \\rangle $ matches the one in the third column of Table REF .","It should be noted that, contrary to the hybrid method, this method is biased [40].","In addition, this is only one possible way of extracting $N_\\text{coll}$ distributions at experiments.","Other methods could yield significantly different distributions.","To try to quantify the dependence of our results on the particular shape of the $N_\\text{coll}$ distributions, we will also use, for the 60-80% centrality bin which is the most sensitive to fluctuations, a simple linearly decreasing distribution.","The two parameters of this distribution, its height at the origin ($h$ ) and the $N_\\text{coll}$ value at which it vanishes ($N_\\text{max}$ ), are determined by imposing that it is normalized to unity and that $\\langle N_\\text{coll} \\rangle =\\langle N_\\text{coll} \\rangle _\\text{ALICE}$ : $N_\\text{max}=3 \\langle N_\\text{coll} \\rangle $ , $h=2/N_\\text{max}$ .","In Fig.","REF we show the values obtained for the nuclear modification factor as a function of $P_\\perp $ in the 60-80% centrality bin, both when using a fixed impact parameter and when integrating explicitly over $b$ using the two $N_\\text{coll}$ distributions described previously (SNM and linear).","The explicit integration over $b$ leads to a smaller $Q_\\text{pPb}$ at small transverse momentum.","The effect is more pronounced with the linear distribution.","Similar results are obtained when looking at $Q_\\text{pPb}$ integrated over $P_\\perp $ as a function of $N_\\text{coll}$ , as show in Fig.","REF .","In particular, we see that the value of $Q_\\text{pPb}$ in the 60-80% centrality bin obtained with the linear $N_\\text{coll}$ distribution is significantly closer to the ALICE data point.","On the other hand, the value obtained with the SNM distribution is very close to the fixed impact parameter result.","In conclusion, it is not possible for now to directly compare our impact parameter dependent results with the centrality dependent measurement performed by ALICE.","Indeed, for this one would need to have access to an unbiased determination of the $N_\\text{coll}$ distributions in each centrality bin, which does not exist at the moment.","Here we have tried to estimate the importance of this effect by using $N_\\text{coll}$ distributions obtained in two models.","The fact that these two models lead to significantly different results for peripheral collisions while the central bins are much less sensitive to fluctuations means that the variation of $Q_\\text{pPb}$ as a function of centrality is too model dependent to have a reliable comparison with experimental data.","Figure: Ratio of the proton-proton cross section at s=13\\sqrt{s}=13 and 8 TeV as a function of YY (left) and P ⊥ P_\\perp (right) compared with LHCb data .Figure: Nuclear modification factor R pPb R_\\text{pPb} at s NN =5\\sqrt{s_{NN}}=5 and 8 TeV as a function of YY (left) and P ⊥ P_\\perp (right).","Data from Ref.",", ,"],["Mean transverse momentum","In Ref.","[14] we found that the uncertainty on cross sections both in proton-proton and proton-nucleus collisions was rather large.","This uncertainty mostly affects the normalization and therefore quantities such as the nuclear modification factor show a smaller uncertainty.","Besides the nuclear modification factor, another observable which is not sensitive to the absolute normalization of the cross section is the mean transverse momentum of the produced $J/\\psi $ meson.","In Fig.","REF (L) we show this quantity as a function of the rapidity in proton-proton collisions at a center of mass energy of 7 TeV and compare with LHCb data [2].","We observe that our calculation is compatible with the data but it is still affected by a relatively large uncertainty.","In the collinear approximation on the proton side that we are using here, the mean transverse momentum increases slightly with rapidity, a trend not seen in the data.","One must, however, keep in mind that towards central rapidities the intrinsic transverse momentum from also the proton should increase, leading to the opposite behavior.","A matching between the collinear and $k_T$ -factorized approximations required to fully quantify this effect is beyond the scope of this paper.","In Fig.","REF (R) we show the same quantity in proton-lead collisions at a center of mass energy $\\sqrt{s_{NN}}=5$ TeV.","The ALICE collaboration has also presented results for $\\langle P_\\perp ^2~\\rangle $ as a function of $N_\\text{coll}$ .","On Fig.","REF (L) we see that our calculation agrees with this measurement within the rather large uncertainty band (except for most peripheral collisions, where our calculation is not applicable as explained previously).","When one considers the difference between $\\langle P_\\perp ^2 \\rangle $ in proton-lead and in proton-proton collisions, as shown on Fig.","REF (R), the uncertainty on our calculation shrinks and shows a too strong variation as a function of $N_\\text{coll}$ , both when using a fixed impact parameter and when integrating explicitly over $b$ using the $N_\\text{coll}$ distributions obtained in the Slow Nucleon Model.","However, as in section REF , the results obtained when integrating over $b$ depend strongly on the exact shape of the $N_\\text{coll}$ distributions used.","In particular, one can see that using a linear $N_\\text{coll}$ distribution leads to a better agreement with data for peripheral collisions."],["Proton-proton collisions","In this work we use the simple color evaporation model to describe the hadronization of $c\\bar{c}$ pairs into $J/\\psi $ mesons.","The normalization of cross sections then depends on a non perturbative constant $F_{J/\\psi }$ , see (REF ).","The uncertainty associated with this parameter can be eliminated by studying the ratio of cross sections at different center of mass energies.","In addition, from the experimental point of view, systematic uncertainties can cancel to some extent in this ratio.","Such a measurement has been made possible at the LHC for proton-proton collisions thanks to the recent increase of $\\sqrt{s}$ from 8 to 13 TeV.","In particular the ratio $\\sigma _{13 \\, \\text{TeV}}/\\sigma _{8 \\, \\text{TeV}}$ was studied as a function of $Y$ and $P_\\perp $ by the LHCb collaboration [6].","In Fig.","REF we compare these data with the results that we obtain for this ratio in our model.","The resulting uncertainty is rather small and the agreement with data is quite good, in particular at large rapidity and relatively low transverse momentum which is the kinematical domain where our calculation is expected to be the most reliable."],["Proton-nucleus collisions","Thanks to its recent upgrade, the LHC may also perform proton-lead collisions at a higher center of mass energy in the future.","Here we study how our results would be affected by a change of $\\sqrt{s_{NN}}$ from 5 to 8 TeV.","In Fig.","REF we show the nuclear modification factor $R_\\text{pPb}$ at forward rapidity as a function of $Y$ and $P_\\perp $ at these two energies, as well as existing LHC data at $\\sqrt{s_{NN}}=5$ TeV.","The values at $\\sqrt{s_{NN}}=5$ TeV shown here differ slightly from the ones shown in Figs.","8 and 10 of Ref.","[14] because we corrected a numerical problem which was causing the region of large impact parameters (where the saturation scale of the lead nucleus falls below the one of the proton, see Fig.","REF (L)) to be neglected.","Here we impose $R_\\text{pA}=1$ in this region, as in Ref. [33].","As one could expect, the higher center of mass energy leads to a stronger suppression due to the higher densities reached in the target.","However, the effect is quite small, in particular compared to the size of the uncertainties.","For this reason we only show, for $\\sqrt{s_{NN}}=8$ TeV, our results for the “central” values of the parameters ($m_c=1.29$ GeV and $Q=\\sqrt{{P}_\\perp ^2+M^2}$ ).","For $\\sqrt{s_{NN}}=5$ TeV we show both the central value and the uncertainty band corresponding to the variation of $m_c$ and $Q$ .","Therefore, while measuring forward $J/\\psi $ production in 8 TeV proton-lead collisions could help reduce experimental uncertainties by getting rid of the interpolation needed for the proton-proton reference, we do not expect significantly stronger nuclear effects at this energy.","This is not surprising since we use the same dipole cross sections as in Ref.","[33], where a weak energy dependence of the nuclear modification factor was found in single inclusive particle production."],["Conclusions","In this paper we have extended our study of forward $J/\\psi $ production in proton-nucleus collisions in the Color Glass Condensate framework to new kinematics and observables.","In particular we have studied the nuclear suppression at negative rapidities by describing the nucleus probed at large $x$ in terms of nuclear parton distribution functions.","We achieved a quite good description of experimental measurements of this quantity, even if the uncertainty is larger at backward than at forward rapidities because the nuclear PDFs are not yet very strongly constrained by data.","This allowed us to compute the forward to backward ratio, again with a good agreement with data within the rather large uncertainties.","We have also studied the centrality dependence of our calculation.","While using the optical Glauber model to extend the description from a proton target to a nucleus leads to a better agreement with experimental data for minimum bias observables than previous calculations in the same framework, it is difficult to compare directly the resulting centrality dependence to experimental data.","Indeed, using a fixed impact parameter obtained in the optical Glauber model from the average number of binary collisions estimated by ALICE leads to a too strong centrality dependence.","On the other hand, to integrate explicitly over the impact parameter one has to use $N_\\text{coll}$ distributions based on various assumptions and our calculation is very sensitive to the exact shape of these distributions.","Finally we have studied how the nuclear modification factor would be affected by an increase of the center of mass energy achievable at the LHC.","As expected nuclear effects are stronger but the change is too small to be significant given the size of theoretical uncertainties."],["Acknowledgments","T. L. and B. D. are supported by the Academy of Finland, projects 267321 and 273464.","H. M. is supported under DOE Contract No.","DE-SC0012704.","This research used computing resources of CSC – IT Center for Science in Espoo, Finland.","We would like to thank C. Hadjidakis and I. Lakomov for discussions on the ALICE data."]],"string":"[\n [\n \"Forward $J/\\\\psi$ production at high energy: centrality dependence and\\n mean transverse momentum\"\n ],\n [\n \"Abstract Forward rapidity $J/\\\\psi$ meson production in proton-nucleus collisions can be an important constraint of descriptions of the small-$x$ nuclear wavefunction.\",\n \"In an earlier work we studied this process using a dipole cross section satisfying the Balitsky-Kovchegov equation, fit to HERA inclusive data and consistently extrapolated to the nuclear case using a standard Woods-Saxon distribution.\",\n \"In this paper we present further calculations of these cross sections, studying the mean transverse momentum of the meson and the dependence on collision centrality.\",\n \"We also extend the calculation to backward rapidities using nuclear parton distribution functions.\",\n \"We show that the parametrization is overall rather consistent with the available experimental data.\",\n \"However, there is a tendency towards a too strong centrality dependence.\",\n \"This can be traced back to the rather small transverse area occupied by small-$x$ gluons in the nucleon that is seen in the HERA data, compared to the total inelastic nucleon-nucleon cross section.\"\n ],\n [\n \"Introduction\",\n \"The production of $J/\\\\psi $ mesons at forward rapidity in high energy proton-proton and proton-nucleus collisions can provide valuable information on gluon saturation.\",\n \"Indeed, the production of particles at forward rapidity probes the target at very small $x$ , where saturation effects should be enhanced.\",\n \"In particular, the charm quark mass being of the same order of magnitude as the saturation scale, $J/\\\\psi $ production should be sensitive to these dynamics.\",\n \"The charm quark mass is also large enough to provide a hard scale, making a perturbative study of this process possible.\",\n \"In addition, $J/\\\\psi $ production has been the subject of many experimental studies at the LHC, both in proton-proton [1], [2], [3], [4], [5], [6], [7] and in proton-nucleus [8], [9], [10], [11], [12] collisions.\",\n \"This provides a lot of data to confront with nuclear effects predicted by various theoretical models, both in the Color Glass Condensate (CGC) framework [13], [14], [15], [16], [17] and as constraints for nuclear parton distribution functions [18] and energy loss in cold nuclear matter [19], [20], [21], [22].\",\n \"In a recent work [14] we studied, in the CGC framework, the production of forward $J/\\\\psi $ mesons in proton-proton and proton-nucleus collisions at the LHC.\",\n \"We showed that, when using the Glauber approach to generalize the dipole cross section to nuclei, the nuclear suppression for minimum bias events is smaller than in previous CGC calculations such as [13], and much closer to experimental data.\",\n \"In this paper we will study, in the same framework, other observables of interest in this process, such as $J/\\\\psi $ production at backward rapidity, the centrality dependence in the optical Glauber model and the mean transverse momentum of the produced ${J/\\\\psi }$ 's.\"\n ],\n [\n \"Formalism\",\n \"Let us briefly recall the main steps of the calculation.\",\n \"For more details we refer the reader to Ref. [14].\",\n \"We use the color evaporation model (CEM) to relate the cross section for $J/\\\\psi $ production to the $c\\\\bar{c}$ pair production cross section.\",\n \"In this model a fixed fraction $F_{J/\\\\psi }$ of the $c\\\\bar{c}$ pairs produced below the $D$ -meson threshold are assumed to hadronize into $J/\\\\psi $ mesons: $\\\\frac{\\\\, \\\\mathrm {d}\\\\sigma _{J/\\\\psi }}{\\\\, \\\\mathrm {d}^2{{P}_\\\\perp }\\\\, \\\\mathrm {d}Y}=F_{J/\\\\psi } \\\\; \\\\int _{4m_c^2}^{4M_D^2} \\\\, \\\\mathrm {d}M^2\\\\frac{\\\\, \\\\mathrm {d}\\\\sigma _{c\\\\bar{c}}}{\\\\, \\\\mathrm {d}^2{{P}_\\\\perp }\\\\, \\\\mathrm {d}Y \\\\, \\\\mathrm {d}M^2}\\\\, ,$ where ${{P}_\\\\perp }$ , $Y$ and $M$ are the transverse momentum, the rapidity and the invariant mass of the $c\\\\bar{c}$ pair respectively, $M_D=1.864$ GeV is the D meson mass and $m_c$ is the charm quark mass that we will vary between 1.2 and 1.5 GeV.\",\n \"Note that in this work we will focus on ratios where $F_{J/\\\\psi }$ cancels so we do not need to fix it to any specific value here.\",\n \"The study of gluon and quark pair production in the dilute-dense limit of the CGC formalism was started some time ago [23], [24] (see also [25]) and used in several calculations such as [26], [27], [13], [28].\",\n \"The physical picture is the following: an incoming gluon from the projectile can split into a quark-antiquark pair either before or after the interaction with the target.\",\n \"The partons propagating trough the target are assumed to interact eikonally with it, picking up a Wilson line factor in either the adjoint (for gluons) or the fundamental (for quarks) representation.\",\n \"Since we study $J/\\\\psi $ production at forward rapidity, where the projectile is probed at large $x$ , we will use the collinear approximation in which the incoming gluon is assumed to have zero transverse momentum.\",\n \"In this approach the cross section for $c\\\\bar{c}$ production reads, in the large-${N_\\\\mathrm {c}}$ limit [13], $\\\\frac{\\\\, \\\\mathrm {d}\\\\sigma _{c\\\\bar{c}}}{\\\\, \\\\mathrm {d}^2{{p}_\\\\perp }\\\\, \\\\mathrm {d}^2{{q}_\\\\perp }\\\\, \\\\mathrm {d}y_p \\\\, \\\\mathrm {d}y_q}=\\\\frac{\\\\alpha _{\\\\mathrm {s}}^2 {N_\\\\mathrm {c}}}{8\\\\pi ^2 d_\\\\mathrm {A}}\\\\\\\\ \\\\times \\\\frac{1}{(2\\\\pi )^2}\\\\int \\\\limits _{{{k}_\\\\perp }}\\\\frac{\\\\Xi _{\\\\rm coll}({{p}_\\\\perp }+ {{q}_\\\\perp },{{k}_\\\\perp })}{({{p}_\\\\perp }+ {{q}_\\\\perp })^2}x_1 G_p(x_1,Q^2)\\\\\\\\\\\\times \\\\phi _{y_2=\\\\ln {\\\\frac{1}{x_2}}}^{q\\\\bar{q},g}({{p}_\\\\perp }+ {{q}_\\\\perp },{{k}_\\\\perp })\\\\;,$ where ${{p}_\\\\perp }$ and ${{q}_\\\\perp }$ denote the transverse momenta of the quarks, $y_p$ and $y_q$ their rapidities, $\\\\int _{{{k}_\\\\perp }} \\\\equiv \\\\int \\\\, \\\\mathrm {d}^2 {{k}_\\\\perp }/ (2\\\\pi )^2$ , and $d_\\\\mathrm {A}\\\\equiv {N_\\\\mathrm {c}}^2-1$ is the dimension of the adjoint representation of SU(${N_\\\\mathrm {c}}$ ).\",\n \"The longitudinal momentum fractions $x_1$ and $x_2$ probed in the projectile and the target respectively are given by $x_{1,2}=\\\\frac{\\\\sqrt{{{P}_\\\\perp }^2+M^2}}{\\\\sqrt{s}}e^{\\\\pm Y} \\\\; .$ The explicit expression for the “hard matrix element” $\\\\Xi _{\\\\rm coll}$ is given in Ref. [13].\",\n \"In Eq.\",\n \"(REF ), $G_p(x_1,Q^2)$ is the gluon density in the probe and is described, in the collinear approximation that we use here, in terms of usual parton distribution functions (PDFs).\",\n \"In the following we use, unless otherwise stated, the MSTW 2008 [29] parametrization for $G_p$ .\",\n \"For consistency we use the leading other (LO) parton distribution since also the rest of the calculation is made at this order.\",\n \"To estimate the uncertainty associated with the choice of the factorization scale $Q$ , we will vary it between $\\\\frac{1}{2}\\\\sqrt{{{P}_\\\\perp }^2+M^2}$ and $2\\\\sqrt{{{P}_\\\\perp }^2+M^2}$ .\",\n \"The function $\\\\phi _{_Y}^{q \\\\bar{q},g}$ describes the propagation of a $q\\\\bar{q}$ pair in the color field of the target and reads $\\\\phi _{_Y}^{q \\\\bar{q},g}({{l}_\\\\perp },{{k}_\\\\perp })=\\\\int \\\\mathrm {d}^2 {{b}_\\\\perp }\\\\frac{N_c{{l}_\\\\perp }^2}{4 \\\\alpha _{\\\\mathrm {s}}} \\\\;\\\\\\\\ \\\\times S_{_Y}({{k}_\\\\perp },{{b}_\\\\perp }) \\\\;S_{_Y}({{l}_\\\\perp }-{{k}_\\\\perp },{{b}_\\\\perp }) \\\\;,$ where ${{b}_\\\\perp }$ is the impact parameter.\",\n \"In this expression, the function $S_{_Y}({{k}_\\\\perp },{{b}_\\\\perp })$ contains all the information about the target.\",\n \"It is the fundamental representation dipole correlator: $S_{_Y}({{k}_\\\\perp },{{b}_\\\\perp }) = \\\\int \\\\, \\\\mathrm {d}^2 {{r}_\\\\perp }e^{i{{k}_\\\\perp }\\\\cdot {{r}_\\\\perp }}S_{_Y}({{r}_\\\\perp },{{b}_\\\\perp }) \\\\;,$ with $S_{_Y}({{x}_\\\\perp }-{{y}_\\\\perp },{{b}_\\\\perp }) = \\\\frac{1}{{N_\\\\mathrm {c}}}\\\\left< \\\\, \\\\mathrm {Tr} \\\\, U^\\\\dag ({{x}_\\\\perp })U({{y}_\\\\perp })\\\\right>,$ where $U({{x}_\\\\perp })$ is a fundamental representation Wilson line in the color field of the target.\",\n \"The dipole correlator $S_{_Y}({{k}_\\\\perp },{{b}_\\\\perp })$ is obtained by solving numerically the running coupling Balitsky-Kovchegov (rcBK) equation [30], [31], [32].\",\n \"For the initial condition we use the 'MVe' parametrization introduced in Ref.\",\n \"[33], which reads, in the case of a proton target, $S^p_{Y= \\\\ln \\\\frac{1}{x_0}}({{r}_\\\\perp },{{b}_\\\\perp }) = \\\\exp \\\\bigg [ -\\\\frac{{{r}_\\\\perp }^2 Q_\\\\mathrm {s0}^2}{4}\\\\\\\\ \\\\times \\\\ln \\\\left(\\\\frac{1}{|{{r}_\\\\perp }| \\\\Lambda _{\\\\mathrm {QCD}}}\\\\!+\\\\!e_c \\\\cdot e\\\\right)\\\\bigg ],$ where $x_0=0.01$ .\",\n \"The running coupling is taken as: $\\\\alpha _{\\\\mathrm {s}}(r) = \\\\frac{12\\\\pi }{(33 - 2N_f) \\\\log \\\\left(\\\\frac{4C^2}{r^2\\\\Lambda _{\\\\mathrm {QCD}}^2} \\\\right)} \\\\;,$ where $C$ parametrizes the uncertainty related to the scale of the strong coupling in the transverse coordinate space.\",\n \"The free parameters in these expressions are obtained by fitting the combined inclusive HERA DIS cross section data [34] for $Q^2<50$ GeV$^2$ and $x<0.01$ .\",\n \"Their best fit values (with $\\\\chi ^2/\\\\text{d.o.f} = 1.15$ ) are $Q_\\\\mathrm {s0}^2= 0.060$ GeV$^2$ , $C^2= 7.2$ , $e_c=18.9$ and $\\\\sigma _0/2 = 16.36$ mb.\",\n \"In the case of a proton target, the dipole amplitude does not have an explicit impact parameter dependence and we thus make the replacement $\\\\int \\\\mathrm {d}^2 {{b}_\\\\perp }\\\\rightarrow \\\\frac{\\\\sigma _0}{2}$ in Eq.\",\n \"(REF ), $\\\\frac{\\\\sigma _0}{2}$ corresponding to the effective proton transverse area measured in DIS experiments.\",\n \"The fit only includes light quarks.\",\n \"In particular this leaves the charm quark mass that would be consistent with the DIS data in this model still uncertain, which is why we vary it in a rather large range for the uncertainty estimate in this work.\",\n \"Figure: Nuclear modification factor R pPb R_\\\\text{pPb} at negative rapidity as a function of YY (left) and P ⊥ P_\\\\perp (right) at s NN =5\\\\sqrt{s_{NN}}=5 TeV.\",\n \"Data from Refs.\",\n \", , .Figure: Forward to backward ratio in proton-lead collisions as a function of YY (left) and P ⊥ P_\\\\perp (right) at s NN =5\\\\sqrt{s_{NN}}=5 TeV.\",\n \"Data from Refs.\",\n \", .To generalize this proton dipole correlator to the case of a nuclear target we use, as in [33], the optical Glauber model.\",\n \"In this model the gluons at the initial rapidity $Y=\\\\ln 1/x_0$ are localized in the individual nucleons of the nucleus.\",\n \"The nucleons are then taken to be distributed randomly and independently in the transverse plane according to the standard Woods-Saxon nuclear density profile.\",\n \"An analytical average over the positions of the nucleons leads to the following initial condition for the rcBK evolution of a nuclear target: $S^A_{Y=\\\\ln \\\\frac{1}{x_0}}({{r}_\\\\perp },{{b}_\\\\perp }) = \\\\exp \\\\Bigg [ -A T_A({{b}_\\\\perp })\\\\frac{\\\\sigma _0}{2} \\\\frac{{{r}_\\\\perp }^2 Q_\\\\mathrm {s0}^2}{4}\\\\\\\\\\\\times \\\\ln \\\\left(\\\\frac{1}{|{{r}_\\\\perp }|\\\\Lambda _{\\\\mathrm {QCD}}}+e_c \\\\cdot e\\\\right) \\\\Bigg ] \\\\; .$ Here $T_A$ is the standard Woods-Saxon transverse thickness function of the nucleus: $T_A({{b}_\\\\perp })= \\\\int dz \\\\frac{n}{1+\\\\exp \\\\left[ \\\\frac{\\\\sqrt{{{b}_\\\\perp }^2 + z^2}-R_A}{d} \\\\right]} \\\\; ,$ with $d=0.54\\\\,\\\\mathrm {fm}$ and $R_A=(1.12A^{1/3}-0.86A^{-1/3})\\\\,\\\\mathrm {fm}$ , and $n$ is defined such that $T_A$ is normalized to unity.\",\n \"All the other parameters in the initial condition (REF ), which is evolved using the rcBK equation for each ${{b}_\\\\perp }$ , are the same as in the proton case.\",\n \"In this model the dipole amplitude depends on the impact parameter and we need to integrate explicitly over it.\",\n \"The impact parameter dependence, which carries over to the centrality dependence, thus appears naturally in this model.\",\n \"Our practical procedure for carrying out this comparison will be discussed in more detail in Sec.\",\n \".\"\n ],\n [\n \"Backward rapidity\",\n \"In our previous work [14] we only considered $J/\\\\psi $ production in proton-proton and proton-nucleus collisions at forward rapidity.\",\n \"In these kinematics the process can be seen as the collision of a dilute proton probed at large $x$ , which can be described using well known parton distribution functions (PDFs), and a dense target described in terms of classical color fields.\",\n \"The nuclear modification of $J/\\\\psi $ production was also measured at backward rapidity by ALICE [8], [10] and LHCb [9].\",\n \"In this case the produced $J/\\\\psi $ is moving in the direction of the incoming nucleus and the physical picture is the same as at forward rapidity, but with the roles of the projectile and the target interchanged, i.e.\",\n \"a dilute proton or nucleus interacts with a dense proton target.\",\n \"The latter is described in the same way as in proton-proton collisions at forward rapidity, while the projectile is described either by a PDF in the case of a proton or by a nuclear PDF (nPDF) in the case of a nucleus.\",\n \"Therefore the calculation is very similar to the case of proton-proton collisions in our previous study.\",\n \"Note, however, that in general nuclear PDFs are less tightly constrained by experimental data than usual proton PDFs.\",\n \"In the following we will use the leading order EPS09 nPDF parametrization [35] which provides additional error sets to estimate this uncertainty.\",\n \"The LO EPS09 analysis uses the CTEQ6L1 [36] proton PDFs as a reference, therefore for consistency we use same parametrization of proton PDFs when computing the nuclear modification factor at backward rapidity.\",\n \"Nevertheless, while it could be sizeable for the cross section, the difference compared to the MSTW 2008 parametrization is very small for the nuclear modification factor.\",\n \"In Fig.\",\n \"REF we show the nuclear modification factor $R_\\\\text{pA}$ , defined as $R_{\\\\rm pA}= \\\\frac{1}{A}\\\\frac{\\\\frac{\\\\, \\\\mathrm {d}\\\\sigma ^\\\\text{pA}}{\\\\, \\\\mathrm {d}^2 {{P}_\\\\perp }\\\\, \\\\mathrm {d}Y}}{\\\\frac{\\\\, \\\\mathrm {d}\\\\sigma ^\\\\text{pp}}{\\\\, \\\\mathrm {d}^2 {{P}_\\\\perp }\\\\, \\\\mathrm {d}Y}} \\\\; ,$ as a function of $Y$ and $P_\\\\perp $ obtained in this way at negative rapidity compared with data from ALICE [8], [10] and LHCb [9] experiments.\",\n \"The uncertainty in our calculation is significantly larger than at forward rapidity [14].\",\n \"This is due to the fact that we include in our uncertainty band, in addition to the variation of the charm quark mass and of the factorization scale, the nuclear PDF uncertainty obtained following the procedure described in Ref. [35].\",\n \"In particular, our lower bound for the factorization scale is $Q=\\\\frac{1}{2}\\\\sqrt{{{P}_\\\\perp }^2+M^2}$ which can reach values of less than 2 GeV at small transverse momentum.\",\n \"The nuclear PDFs are not well constrained at such small scales at the moment.\",\n \"Nevertheless the general agreement with data is quite good taking into account the rather large theoretical and experimental uncertainties.\",\n \"In this case deviations of $R_\\\\text{pPb}$ from unity are entirely due to the nuclear PDFs.\",\n \"Now that we have computed the nuclear modification factor both at forward [14] and backward rapidities, we have access to the forward to backward ratio $R_\\\\text{FB}$ , defined as $R_\\\\text{FB}(P_\\\\perp ,Y)=\\\\frac{R_\\\\text{pA}(P_\\\\perp ,Y)}{R_\\\\text{pA}(P_\\\\perp ,-Y)} \\\\; .$ This ratio can be interesting to study because there may be an additional cancellation of some uncertainties common to the numerator and the denominator.\",\n \"In particular, when determining the nuclear modification factor, experimental studies such as [8], [9] have to use an interpolation for the reference proton-proton cross section since there is no data at $\\\\sqrt{s}=5$ TeV.\",\n \"This interpolation is not needed to study $R_\\\\text{FB}$ , but the final statistical uncertainty may be larger if the coverage in rapidity by the detector is not symmetric with respect to 0.\",\n \"Concerning our calculation, we have seen that at negative rapidity the uncertainty on $R_\\\\text{pA}$ due to nuclear PDFs is rather large.\",\n \"This error will remain in $R_\\\\text{FB}$ since the computation of $R_\\\\text{pA}$ at positive rapidity does not involve nPDFs.\",\n \"Indeed, we see from Fig.\",\n \"REF , where we show the forward to backward ratio as a function of $Y$ and $P_\\\\perp $ , that the uncertainty on this quantity is still quite large.\",\n \"Nevertheless, within this error band the agreement with data is reasonable, although the variation at low $P_\\\\perp $ seems to be steeper than in the data.\",\n \"In Fig.\",\n \"REF (R) we only show our results for $R_\\\\text{FB}$ as a function of $P_\\\\perp $ integrated over the same $Y$ range as ALICE data [8], which is slightly smaller than for LHCb data [9], but our results for $2.5$ for the optical Glauber centrality classes compared to the values given by ALICE [12] for the experimental ones.\",\n \"For central collisions the average number of binary collisions estimated by ALICE is smaller than in the optical Glauber model, while the opposite is true for peripheral collisions.\",\n \"The impact parameter is not directly observable, so the experimental centrality selection has to use some other observable as a proxy for it.\",\n \"In the case of the ALICE analysis [12] the observable used is the energy of the lead-going side Zero-Degree Calorimeter (ZDC).\",\n \"Due to the large fluctuations in the signal for a fixed impact parameter, the values of $N_\\\\text{coll}$ vary less strongly with centrality in the experimental classes than in the optical Glauber ones.\",\n \"Thus, for example, a relatively peripheral smaller-$N_\\\\text{coll}$ event can end up in the most central class in the case of a fluctuation in the ZDC signal.\"\n ],\n [\n \"Fixed impact parameter approximation\",\n \"Developing a detailed Monte Carlo Glauber model that would enable us to exactly match the experimental centrality classes would be beyond the scope of this work.\",\n \"Therefore we start from the assumption that the ALICE Glauber model is correct and produces a reliable estimate for the $\\\\langle N_\\\\text{coll} \\\\rangle $ in each experimental class.\",\n \"We then assume that the mapping between this $\\\\langle N_\\\\text{coll} \\\\rangle $ and the impact parameter is accurately enough described by our optical Glauber model, and use the relation $N_\\\\text{coll.\",\n \"opt}(b)=\\\\langle N_\\\\text{coll} \\\\rangle _\\\\text{ALICE}$ to determine a mean impact parameter corresponding to the experimental centrality class.\",\n \"The values of $b$ resulting from this procedure are shown in Table REF .\",\n \"We then calculate the nuclear modification factor for the centrality class using this fixed value of $b$ .\",\n \"We note that following this procedure for the 80-100% centrality class would lead to a value of $b$ for which our calculation would not be applicable because the saturation scale of the nucleus falls below the one of the proton.\",\n \"Therefore in the following we will only show results for the five most central classes considered by ALICE.\",\n \"Figure: Left: saturation scale of the lead nucleus in the MVe parametrization as a function of the impact parameter compared with the saturation scale of the proton at x=10 -4 x=10^{-4}.\",\n \"Right: number of binary collisions in proton-lead collisions as a function of the impact parameter in the optical Glauber model.In Fig.\",\n \"REF we show a comparison of our calculation with ALICE data for the nuclear modification factor in different centrality classes, $Q_\\\\text{pA}$ , defined as $Q_{\\\\rm pA}= \\\\frac{\\\\frac{\\\\, \\\\mathrm {d}N^\\\\text{pA}}{\\\\, \\\\mathrm {d}^2 {{P}_\\\\perp }\\\\, \\\\mathrm {d}Y}}{ A \\\\langle T_A \\\\rangle \\\\frac{\\\\, \\\\mathrm {d}\\\\sigma ^\\\\text{pp}}{\\\\, \\\\mathrm {d}^2 {{P}_\\\\perp }\\\\, \\\\mathrm {d}Y}} \\\\; ,$ as a function of $P_\\\\perp $ .\",\n \"We observe that the description of experimental data is generally satisfactory in the first four bins.\",\n \"For the fifth bin the value of $Q_\\\\text{pPb}$ obtained in the optical Glauber model is almost constant and very close to one, while the data still shows a significant variation with $P_\\\\perp $ .\",\n \"The discrepancy with experimental data for peripheral collisions comes from the fact that in our model the saturation scale of the lead nucleus falls below the one of the proton for a value of $b$ of the order of 6.3 fm, as shown in Fig.\",\n \"REF (L).\",\n \"We see from Fig.\",\n \"REF (R), where we show the number of binary collisions as a function of $b$ , that this corresponds to $N_\\\\text{coll} \\\\sim 4.3$ .\",\n \"This is the point where, by definition, $Q_\\\\text{pA}$ reaches 1 in our calculation and beyond which the validity of the framework we have used is questionable.\",\n \"On the other hand, ALICE data shows that $Q_\\\\text{pA}$ is still significantly smaller than 1 down to $N_\\\\text{coll} \\\\sim 2.1$ (see Fig.\",\n \"REF ).\",\n \"The strong centrality dependence is caused by the value of $\\\\frac{\\\\sigma _0}{2}$ extracted from DIS fits being much smaller than the total inelastic nucleon-nucleon cross section.\",\n \"The HERA data, both the inclusive cross section fitted in Ref.\",\n \"[33] and data on exclusive vector meson production (see e.g.\",\n \"[37]), lead to a picture where the small-$x$ gluons that can participate in a hard process in a proton are concentrated in a rather small area in the transverse plane (see also the discussion in Ref. [38]).\",\n \"This small-$x$ gluon “hot spot” is then surrounded by a larger “cloud” that only participates in soft interactions, contibuting to the total nucleon-nucleon inelastic cross section.\",\n \"Our model takes this picture to the extreme, by assuming that at the initial rapidity $\\\\ln 1/x_0$ the gluons contributing to ${J/\\\\psi }$ production are concetrated in the area $\\\\sigma _0/2 \\\\sim 0.3 \\\\sigma _\\\\text{inel}$ inside the target nucleons.\",\n \"Thus, for peripheral collisions, the probe proton can overlap with the soft cloud of $N_\\\\text{coll} \\\\sim 4$ target nucleons while still seeing on average only one small-$x$ gluon hot spot in the target, thus leaving a hard process like ${J/\\\\psi }$ production approximately unmodified.\",\n \"This centrality dependence could probably be mildened by using a larger value for $\\\\frac{\\\\sigma _0}{2}$ of the order of $\\\\sigma _\\\\text{inel}$ (as is effectively done in [13]).\",\n \"This would, however, lose the consistency of our description of the nucleon from HERA to the LHC.\",\n \"Also, as discussed in Refs.\",\n \"[33], [14], varying these parameters in an uncontrolled way could very easily, depending on how exactly it is done, lead to an excessive suppression for minimum bias collisions, or to an $R_\\\\text{pA}$ that is very far from unity even at high transverse momentum.\",\n \"Note that the saturation scale (or gluon density) at the edge of the nucleus being smaller than that of the proton is an artefact of averaging over the transverse locations of the dense but small gluon hot spots of the nucleons in the target nucleus.\",\n \"Therefore we do not use the optical Glauber parametrization in this region, but explicitly set $R_\\\\text{pA}$ to unity.\",\n \"Nevertheless, even an explicit Monte Carlo Glauber procedure with the same parameters would not change the ordering $\\\\sigma _0/2 < \\\\sigma _\\\\text{inel}$ that leads to the absence of nuclear effects for peripheral collisions with $N_\\\\text{coll} \\\\lesssim 4$ .\",\n \"Figure: Q pPb Q_\\\\text{pPb} at a function of P ⊥ P_\\\\perp at s NN =5\\\\sqrt{s_{NN}}=5 TeV in the 60-80% centrality class, both when using a fixed impact parameter and when integrating explicitly over bb using the N coll N_\\\\text{coll} distribution obtained in two different models.\",\n \"Data from Ref.\",\n \".Figure: Nuclear modification factor Q pPb Q_\\\\text{pPb} as a function of N coll N_\\\\text{coll} at s NN =5\\\\sqrt{s_{NN}}=5 TeV compared with ALICE data , both when using a fixed impact parameter and when integrating explicitly over bb.Figure: Mean transverse momentum as a function of YY in proton-proton collisions at s=7\\\\sqrt{s}=7 TeV (left) and in proton-lead collisions at s NN =5\\\\sqrt{s_{NN}}=5 TeV (right).\",\n \"Data from Ref.\",\n \".Figure: Average transverse momentum squared〈P ⊥ 2 〉 pPb \\\\langle P_\\\\perp ^2 \\\\rangle _\\\\text{pPb} (left) and nuclear transverse momentum broadening 〈P ⊥ 2 〉 pPb -〈P ⊥ 2 〉 pp \\\\langle P_\\\\perp ^2 \\\\rangle _\\\\text{pPb}-\\\\langle P_\\\\perp ^2 \\\\rangle _\\\\text{pp} (right) as a function of N coll N_\\\\text{coll} at s NN =5\\\\sqrt{s_{NN}}=5 TeV compared with ALICE data .\",\n \"For the latter both the result using a fixed impact parameter and when integrating explicitly over bb are shown.\"\n ],\n [\n \"Explicit integration over the impact parameter\",\n \"The results we have shown in the previous section were obtained with a fixed impact parameter chosen so that the number of binary collisions in the optical Glauber model is equal to the average number of binary collisions estimated by ALICE in each centrality class.\",\n \"However, the nuclear modification ratio in a given centrality bin receives contributions from a distribution of different impact parameters.\",\n \"One could therefore argue that having a profile in the impact parameter space and integrating over $b$ could lead to different results.\",\n \"To quantify this effect we will here use two different kinds of $N_\\\\text{coll}$ distributions to obtain distributions in the impact parameter space.\",\n \"The first one is provided by ALICE [39] and is obtained from the Slow Nucleon Model (SNM) [40].\",\n \"Since in this model the average number of binary collisions is not the same as the one obtained in the hybrid method used in Ref.\",\n \"[12], we shift the distributions so that $\\\\langle N_\\\\text{coll} \\\\rangle $ matches the one in the third column of Table REF .\",\n \"It should be noted that, contrary to the hybrid method, this method is biased [40].\",\n \"In addition, this is only one possible way of extracting $N_\\\\text{coll}$ distributions at experiments.\",\n \"Other methods could yield significantly different distributions.\",\n \"To try to quantify the dependence of our results on the particular shape of the $N_\\\\text{coll}$ distributions, we will also use, for the 60-80% centrality bin which is the most sensitive to fluctuations, a simple linearly decreasing distribution.\",\n \"The two parameters of this distribution, its height at the origin ($h$ ) and the $N_\\\\text{coll}$ value at which it vanishes ($N_\\\\text{max}$ ), are determined by imposing that it is normalized to unity and that $\\\\langle N_\\\\text{coll} \\\\rangle =\\\\langle N_\\\\text{coll} \\\\rangle _\\\\text{ALICE}$ : $N_\\\\text{max}=3 \\\\langle N_\\\\text{coll} \\\\rangle $ , $h=2/N_\\\\text{max}$ .\",\n \"In Fig.\",\n \"REF we show the values obtained for the nuclear modification factor as a function of $P_\\\\perp $ in the 60-80% centrality bin, both when using a fixed impact parameter and when integrating explicitly over $b$ using the two $N_\\\\text{coll}$ distributions described previously (SNM and linear).\",\n \"The explicit integration over $b$ leads to a smaller $Q_\\\\text{pPb}$ at small transverse momentum.\",\n \"The effect is more pronounced with the linear distribution.\",\n \"Similar results are obtained when looking at $Q_\\\\text{pPb}$ integrated over $P_\\\\perp $ as a function of $N_\\\\text{coll}$ , as show in Fig.\",\n \"REF .\",\n \"In particular, we see that the value of $Q_\\\\text{pPb}$ in the 60-80% centrality bin obtained with the linear $N_\\\\text{coll}$ distribution is significantly closer to the ALICE data point.\",\n \"On the other hand, the value obtained with the SNM distribution is very close to the fixed impact parameter result.\",\n \"In conclusion, it is not possible for now to directly compare our impact parameter dependent results with the centrality dependent measurement performed by ALICE.\",\n \"Indeed, for this one would need to have access to an unbiased determination of the $N_\\\\text{coll}$ distributions in each centrality bin, which does not exist at the moment.\",\n \"Here we have tried to estimate the importance of this effect by using $N_\\\\text{coll}$ distributions obtained in two models.\",\n \"The fact that these two models lead to significantly different results for peripheral collisions while the central bins are much less sensitive to fluctuations means that the variation of $Q_\\\\text{pPb}$ as a function of centrality is too model dependent to have a reliable comparison with experimental data.\",\n \"Figure: Ratio of the proton-proton cross section at s=13\\\\sqrt{s}=13 and 8 TeV as a function of YY (left) and P ⊥ P_\\\\perp (right) compared with LHCb data .Figure: Nuclear modification factor R pPb R_\\\\text{pPb} at s NN =5\\\\sqrt{s_{NN}}=5 and 8 TeV as a function of YY (left) and P ⊥ P_\\\\perp (right).\",\n \"Data from Ref.\",\n \", ,\"\n ],\n [\n \"Mean transverse momentum\",\n \"In Ref.\",\n \"[14] we found that the uncertainty on cross sections both in proton-proton and proton-nucleus collisions was rather large.\",\n \"This uncertainty mostly affects the normalization and therefore quantities such as the nuclear modification factor show a smaller uncertainty.\",\n \"Besides the nuclear modification factor, another observable which is not sensitive to the absolute normalization of the cross section is the mean transverse momentum of the produced $J/\\\\psi $ meson.\",\n \"In Fig.\",\n \"REF (L) we show this quantity as a function of the rapidity in proton-proton collisions at a center of mass energy of 7 TeV and compare with LHCb data [2].\",\n \"We observe that our calculation is compatible with the data but it is still affected by a relatively large uncertainty.\",\n \"In the collinear approximation on the proton side that we are using here, the mean transverse momentum increases slightly with rapidity, a trend not seen in the data.\",\n \"One must, however, keep in mind that towards central rapidities the intrinsic transverse momentum from also the proton should increase, leading to the opposite behavior.\",\n \"A matching between the collinear and $k_T$ -factorized approximations required to fully quantify this effect is beyond the scope of this paper.\",\n \"In Fig.\",\n \"REF (R) we show the same quantity in proton-lead collisions at a center of mass energy $\\\\sqrt{s_{NN}}=5$ TeV.\",\n \"The ALICE collaboration has also presented results for $\\\\langle P_\\\\perp ^2~\\\\rangle $ as a function of $N_\\\\text{coll}$ .\",\n \"On Fig.\",\n \"REF (L) we see that our calculation agrees with this measurement within the rather large uncertainty band (except for most peripheral collisions, where our calculation is not applicable as explained previously).\",\n \"When one considers the difference between $\\\\langle P_\\\\perp ^2 \\\\rangle $ in proton-lead and in proton-proton collisions, as shown on Fig.\",\n \"REF (R), the uncertainty on our calculation shrinks and shows a too strong variation as a function of $N_\\\\text{coll}$ , both when using a fixed impact parameter and when integrating explicitly over $b$ using the $N_\\\\text{coll}$ distributions obtained in the Slow Nucleon Model.\",\n \"However, as in section REF , the results obtained when integrating over $b$ depend strongly on the exact shape of the $N_\\\\text{coll}$ distributions used.\",\n \"In particular, one can see that using a linear $N_\\\\text{coll}$ distribution leads to a better agreement with data for peripheral collisions.\"\n ],\n [\n \"Proton-proton collisions\",\n \"In this work we use the simple color evaporation model to describe the hadronization of $c\\\\bar{c}$ pairs into $J/\\\\psi $ mesons.\",\n \"The normalization of cross sections then depends on a non perturbative constant $F_{J/\\\\psi }$ , see (REF ).\",\n \"The uncertainty associated with this parameter can be eliminated by studying the ratio of cross sections at different center of mass energies.\",\n \"In addition, from the experimental point of view, systematic uncertainties can cancel to some extent in this ratio.\",\n \"Such a measurement has been made possible at the LHC for proton-proton collisions thanks to the recent increase of $\\\\sqrt{s}$ from 8 to 13 TeV.\",\n \"In particular the ratio $\\\\sigma _{13 \\\\, \\\\text{TeV}}/\\\\sigma _{8 \\\\, \\\\text{TeV}}$ was studied as a function of $Y$ and $P_\\\\perp $ by the LHCb collaboration [6].\",\n \"In Fig.\",\n \"REF we compare these data with the results that we obtain for this ratio in our model.\",\n \"The resulting uncertainty is rather small and the agreement with data is quite good, in particular at large rapidity and relatively low transverse momentum which is the kinematical domain where our calculation is expected to be the most reliable.\"\n ],\n [\n \"Proton-nucleus collisions\",\n \"Thanks to its recent upgrade, the LHC may also perform proton-lead collisions at a higher center of mass energy in the future.\",\n \"Here we study how our results would be affected by a change of $\\\\sqrt{s_{NN}}$ from 5 to 8 TeV.\",\n \"In Fig.\",\n \"REF we show the nuclear modification factor $R_\\\\text{pPb}$ at forward rapidity as a function of $Y$ and $P_\\\\perp $ at these two energies, as well as existing LHC data at $\\\\sqrt{s_{NN}}=5$ TeV.\",\n \"The values at $\\\\sqrt{s_{NN}}=5$ TeV shown here differ slightly from the ones shown in Figs.\",\n \"8 and 10 of Ref.\",\n \"[14] because we corrected a numerical problem which was causing the region of large impact parameters (where the saturation scale of the lead nucleus falls below the one of the proton, see Fig.\",\n \"REF (L)) to be neglected.\",\n \"Here we impose $R_\\\\text{pA}=1$ in this region, as in Ref. [33].\",\n \"As one could expect, the higher center of mass energy leads to a stronger suppression due to the higher densities reached in the target.\",\n \"However, the effect is quite small, in particular compared to the size of the uncertainties.\",\n \"For this reason we only show, for $\\\\sqrt{s_{NN}}=8$ TeV, our results for the “central” values of the parameters ($m_c=1.29$ GeV and $Q=\\\\sqrt{{P}_\\\\perp ^2+M^2}$ ).\",\n \"For $\\\\sqrt{s_{NN}}=5$ TeV we show both the central value and the uncertainty band corresponding to the variation of $m_c$ and $Q$ .\",\n \"Therefore, while measuring forward $J/\\\\psi $ production in 8 TeV proton-lead collisions could help reduce experimental uncertainties by getting rid of the interpolation needed for the proton-proton reference, we do not expect significantly stronger nuclear effects at this energy.\",\n \"This is not surprising since we use the same dipole cross sections as in Ref.\",\n \"[33], where a weak energy dependence of the nuclear modification factor was found in single inclusive particle production.\"\n ],\n [\n \"Conclusions\",\n \"In this paper we have extended our study of forward $J/\\\\psi $ production in proton-nucleus collisions in the Color Glass Condensate framework to new kinematics and observables.\",\n \"In particular we have studied the nuclear suppression at negative rapidities by describing the nucleus probed at large $x$ in terms of nuclear parton distribution functions.\",\n \"We achieved a quite good description of experimental measurements of this quantity, even if the uncertainty is larger at backward than at forward rapidities because the nuclear PDFs are not yet very strongly constrained by data.\",\n \"This allowed us to compute the forward to backward ratio, again with a good agreement with data within the rather large uncertainties.\",\n \"We have also studied the centrality dependence of our calculation.\",\n \"While using the optical Glauber model to extend the description from a proton target to a nucleus leads to a better agreement with experimental data for minimum bias observables than previous calculations in the same framework, it is difficult to compare directly the resulting centrality dependence to experimental data.\",\n \"Indeed, using a fixed impact parameter obtained in the optical Glauber model from the average number of binary collisions estimated by ALICE leads to a too strong centrality dependence.\",\n \"On the other hand, to integrate explicitly over the impact parameter one has to use $N_\\\\text{coll}$ distributions based on various assumptions and our calculation is very sensitive to the exact shape of these distributions.\",\n \"Finally we have studied how the nuclear modification factor would be affected by an increase of the center of mass energy achievable at the LHC.\",\n \"As expected nuclear effects are stronger but the change is too small to be significant given the size of theoretical uncertainties.\"\n ],\n [\n \"Acknowledgments\",\n \"T. L. and B. D. are supported by the Academy of Finland, projects 267321 and 273464.\",\n \"H. M. is supported under DOE Contract No.\",\n \"DE-SC0012704.\",\n \"This research used computing resources of CSC – IT Center for Science in Espoo, Finland.\",\n \"We would like to thank C. Hadjidakis and I. Lakomov for discussions on the ALICE data.\"\n ]\n]"},"id":{"kind":"string","value":"1605.05680"}}},{"rowIdx":1866694,"cells":{"text":{"kind":"list like","value":[["Non-linear behaviour of XTE J1550-564 during its 1998-1999 outburst,\n revealed by recurrence analysis"],["Abstract We study the X-ray emission of the microquasar XTE J1550-564 and analyze the properties of its light curves using the recurrence analysis method.","The indicators for non-linear dynamics of the accretion flow are found in the very high state and soft state of this source.","The significance of deterministic variability depends on the energy band.","We discuss the non-linear dynamics of the accretion flow in the context of the disc-corona geometry and propagating oscillations in the accretion flow."],["Introduction","The microquasar XTE J1550-564 is a low-mass X-ray binary, with a companion main sequence star, and was discovered in its 1998 outburst by the All-Sky Monitor onboard the RXTE satellite [14].","The radio-jets that accompany its activity were also detected, similarly to many other black hole X-ray binaries [5].","The compact object's mass is estimated at 8-11 Solar masses and thus it is considered to be a black hole [11].","The source exhibits all the canonical spectral states typically associated with LMXBs.","The source's spectral analysis performed for the RXTE data of XTE J1550-564, that spans the period between September 23 and October 6, 1998, place this object in the very high state (VHS) category [3].","Further analysis performed for this source by [8] showed that the temperature of the accretion disk is much lower than expected for the VHS luminosity, and suggested that this should be interpreted as the signature of the disk truncation at the radius of about 30 $R_{\\rm g}$ .","The intrinsic luminosity of the source peaks above $10^{39}$ erg/s, and during both very high and soft states it remains well above $2-3 \\times 10^{38}$ erg/s.","Therefore the derived accretion rate can be in this source equal to at least 25% of the Eddington limit, and reaching even $0.8 \\dot{M}_{\\rm Edd}$ at the outburst peak.","In the standard accretion disk theory, such high values of $\\dot{M}$ lead to the intrinsic thermal and viscous instabilities caused by the dominant radiation pressure.","Such instabilities manifest themselves in the non-lienar evolution of the disk temperature and density at very short timescales, and consequently imprint a characteristic, time-dependent pattern in the emitted X-ray luminosity.","In this work, we study the properties of the XTE J1550-564 light curves.","We implement our method, recently described in [17], and we study the light curves in the X-rays as taken by the RXTE satellite during the fall 1998 and spring 1999.","The capabilities of this method were tested numerically in [18] and its dependence on parameters was discussed in [4].","Here we focus on the spectral and temporal evolution of the microquasars's properties and its possible non-linear behaviour.","We discuss the results in the context of the thermal-viscous instability of a Keplerian accretion disc, as well as coupling of its variability with the behaviour of hard X-ray emitting, quasi-spherical corona."],["Observational data","The overall light curve and hardness ratio of the outburst of XTE J1550-564 shows two separated stages of high luminosity [1].","The first one lasted from the beginning of September to the beginning of October 1998 (MJD 51063 to 51120), and the second one spanned from the mid of December 1998 to the mid of April 1999 (MJD 51160 to 51280).","From the list of the publicly available observations made by RXTE in NASA's HEASARC arxivheasarc.gsfc.nasa.gov we selected several observations, which cover the two epochs quite evenly and which are long enough for our purpose.","The data sample is summarised in Table REF ."],["Recurrence analysis and non-linear dynamics indicator","We study the properties of the light curves obtained in different energy bands by means of the time series analysis.","Our aim is to find out if the time variability of the flux contains the information about the nature of the dynamical system behind the emission of the radiation.","Mainly, we want to discover if the underlying system behaves according to deterministic non-linear dynamics, which is governed by a low number of equations rather than by stochastic variations.","In the former case, the traces of the non-linear behaviour are imprinted into the outgoing radiation.","Table: Non-linear dynamics indicator 𝒮 ¯\\bar{\\mathcal {S}} obtained for N ϵ N_\\epsilon different values of recurrence threshold ϵ\\epsilon in different energy bands for three observations 30188-06-05-00 (MJD 51068), 30191-01-12-00 (MJD 51082) and 30191-01-33-00 (MJD 51108).","The number of points used is N=32768N=32768In the preceding paper [17] we developed the method, which can reveal such traces hidden in the observational data.","The method includes performing recurrence analysis on the observational time series, which is in our case the light curve, and on the set of surrogate data series.","The results are then compared according to discriminating statistics.","Our null hypothesis is that the data are a product of a linearly autocorrelated Gaussian process, so that the data point is given by a linear combination of the preceding points and a contribution of uncorrelated Gaussian noise, $\\xi (n)$ , hence $x_n = a_0 + \\sum _{k=1}^q a_k x_{n-k} +\\xi (n)$ .","The properties of such time series are fully described by their Fourier spectrum.","The surrogate data are constructed in such a way that they share the value distribution and the Fourier spectrum with the original series [19], [13].","However, if the original time series is a product of a non-linear system, the dynamical attributes obtained by the analysis will differ from the surrogate series.","Here we use this method to study the outburst of XTE J1550-564, so we only briefly describe its main points.","We normalise the light curve to have zero mean and unit variance, then we construct a bunch of 100 surrogates using the procedure surrogates from TISEANhttp://www.mpipks-dresden.mpg.de/~tisean/[6].","The recurrence analysis is performed on the basis of the software package described by [10], [9], which also yields the cumulative histogram of diagonal lines.","We compute the recurrence quantifiers for the data and the surrogates, using the educated guess of its parameters time delay $\\Delta t$ and embedding dimension $m$ for a set of recurrence thresholds $\\epsilon $ , such that the recurrence rate is approximately in the range (1% – 25 %).","After computing the cumulative histograms of diagonal lines for the observed data and their surrogates, we estimate the Rényi's entropy $K_2$ by the linear regression of the cumulative histogram.","We follow the rules given in [17] Appendix B, in particular that the minimal length of the diagonal lines must exceed the time delay and that there must be at least five points for the regression.","As the discriminating statistic we choose the logarithm of the estimated Rényi's entropy, which we compute for each threshold for the observed time series $Q^{\\rm obs}(\\epsilon )=\\ln (K_2^{\\rm obs})$ and similarly for the set of surrogates $Q^{\\rm surr}_i(\\epsilon )$ .","We find the mean $\\bar{Q}^{\\rm surr}(\\epsilon )$ and the standard deviation $\\sigma _Q^{\\rm surr}(\\epsilon )$ of the set $Q^{\\rm surr}_i(\\epsilon )$ .","Then we define the weighted significance as $\\mathcal {S}(\\epsilon ) = \\frac{N_{\\rm sl}}{N^{\\rm surr}} \\mathcal {S}_{\\rm sl} - {\\rm sign}( Q^{\\rm obs} (\\epsilon ) - \\bar{Q}^{\\rm surr}(\\epsilon ) ) \\frac{N_{\\mathcal {S}_K}}{N^{\\rm surr}} \\mathcal {S}_{K_2}(\\epsilon ) , $ where $N_{\\rm sl}$ is the number of surrogates, which have only short diagonal lines, and $N^{\\rm surr}$ is the total number of surrogates, $\\mathcal {S}_{\\rm sl} =3$ and $\\mathcal {S}_{K_2}$ is the significance computed only from the surrogates, which have enough long lines according to the relation $\\mathcal {S}_{K_2} (\\epsilon ) = \\frac{| Q^{\\rm obs} (\\epsilon ) - \\bar{Q}^{\\rm surr}(\\epsilon ) |}{\\sigma _{Q^{\\rm surr}(\\epsilon )}}.$ Finally, we define the non-linear dynamics (NLD) indicator, $\\bar{\\mathcal {S}}$ , as the average of $\\mathcal {S}(\\epsilon )$ over the set of $N_\\epsilon $ different thresholds.","We say that the observation shows traces of non-linear behaviour if it exceeds the chosen threshold, $\\bar{\\mathcal {S}} > 1.5$ .","The accuracy of the resulting NLD indicator is quite a complicated issue, because the observational data with their uncertainness undergo several subsequent statistical processes.","However, we can treat the resulting values $\\mathcal {S}(\\epsilon )$ as a small set of $N_\\epsilon $ measurements of $\\bar{\\mathcal {S}}$ .","As such, the uncertainty of NLD indicator is given as $\\sigma = R/(2\\sqrt{N_\\epsilon })$ , where R is the spread of the data set, that is $R={\\rm max}(\\mathcal {S}(\\epsilon )) - {\\rm min}(\\mathcal {S}(\\epsilon ))$ ."],["Energy dependence of NLD indicator","We start our study with the observation 30188-06-05-00 taken on 12.9.1998 (MJD 51068).","We extract the data with the time bin ${\\rm d}t=0.025$ s in six energy bands denoted as A – F, which are listed in Table REF .","We take the longest available continuous part of the light curve up to $N_{\\rm max}=32768$ points.","We set the upper limit of the length of the studied time series in order to keep the numerical demands to a reasonable level.","The light curve in band F, the recurrence plot (RP)Recurrence plot is a visualisation of the recurrence matrix, where the recurrence points are plotted as dots.","Because the plot is symmetric with respect to the main diagonal, we plot only the triangle below/above the diagonal.","of the observation and RP of one of the surrogates are plotted in Fig.","REF .","Figure: Lightcurves in energy band F and recurrence plots for the observations (in red) and one surrogate (in green) for the three observations listed in Table :a) MJD 51068 b) MJD 51082 c) MJD 51108.We compute the NLD indicator $\\bar{\\mathcal {S}}$ with the parameters $m=9$ and $k=4$ .","The highest $\\bar{\\mathcal {S}}$ is obtained for the energy band D, which covers the energy range $7.96 - 12.99$ keV.","On the other hand in the lowest energy band A, the NLD indicator is very small ($\\bar{\\mathcal {S}}=0.45$ ).","The indicator increases from band A to band D, slightly exceeding our threshold in band B ($\\bar{\\mathcal {S}} = 1.77 > 1.5$ ) and giving quite a high value $\\bar{\\mathcal {S}} = 4.58$ in band C. In the highest energies covered by band E, the value decreases to $\\bar{\\mathcal {S}} = 2.80$ .","Figure: Dependence of 𝒮\\mathcal {S} on ϵ\\epsilon for the observation from MJD 51068 extracted in different energy bands.We also provide results for the energy band F, which includes channels 0 – 35 (< 12.99 keV).","This band corresponds to the extraction of the data from the binned mode or the single-bit mode without energy filtering, so that such a light curve has the highest count rate.","This band can also be compared with the results given in [17].","The light curve in this band yields a high value $\\bar{\\mathcal {S}} = 6.69.$ We notice that the uncertainty $\\sigma $ of NLD given in Table REF in bands D and F is very high.","The reason for this is that the significance $\\mathcal {S}(\\epsilon )$ is not independent on $\\epsilon $ , but rather grows with the threshold and the spread of the data set is high (see Fig.","REF ).","This behaviour is not so prominent in any other case, and can be explained if the underlying cumulative histograms are inspected.","The histograms of this observation for higher thresholds show an `elbow' around $l \\sim 3.5$ s and the slope of the part after the elbow is about two times smaller than for the shorter lines.","A similar elbow is not seen in the histograms of the surrogates, partly because they do not have many such long lines.","For higher $\\epsilon $ , the elbow is more prominent in the observation's histogram, lowering the estimated value of $K_2^{\\rm obs}$ and thus making the difference between the observation and the surrogates bigger.","This feature can be explained as the presence of recurrence period in the system.","The slope of the first part is higher, because it is more difficult to predict the evolution of the trajectory, when we have information about its piece shorter than the recurrence period (see [10] for more details about the properties of RPs).","Also a possible interplay between the value of the recurrence threshold and the noise in the data can play a role in this effect.","Similar analysis of the energy dependence of NLD was also made for two later observations, 30191-01-12-00 taken on MJD 51082, and 30191-01-33-00 taken on MJD 51108 (denoted as b) and c) in Fig.","REF ).","The second observation yields similar values of NLD in bands D and C and the overall band F gives the highest value.","The third observation also has its highest significance in the band F, however for this observation the NLD indicator in band C is higher then in band D. The values in all bands are much less than for the first two observations, which suggests that the non-linear dynamics in the source fades away during the outburst."],["Temporal evolution of NLD indicator during the outburst","We continue with the study of the temporal evolution of the NLD indicator during the outburst of XTE J1550-564.","We extract the light curves in energy bands D and F for each observation from Table REF .","In order to use our procedure meaningfully, we need to have a sufficiently high number of counts in each time bin.","Hence, the total count rate combined from all working proportional counter units (PCUs) in each energy band is an important quantity for setting the time bin ${\\rm d}t$ .","We provide the values and the number of working PCUs in Table REF .","The normalised count rate per PCU is given in Figs REF a) and REF b), respectively.","We quantify the hardness ratio as the count rate in band D divided by count rate in lower energies, $h = N_{\\rm D} / (N_{\\rm F} - N_{\\rm D})$ , which is given in Table REF and in Fig REF c).","Strong spectral transition from hard state, with $h \\gtrsim 0.2$ , to soft state, with $h \\approx 0.05$ , can be seen during the first stage of the outburst.","Hence, the count rate in band D drops down significantly in the later epoch.","Therefore we do not compute the significance in band D for observations after 1.11.1998 (MJD 51119).","During the second stage of the outburst, the hardness ratio $h$ is much lower, with the exception of short time period around MJD 51250 (see [8]), when it reaches $h=0.18$ .","We map this time period more closely, choosing several observations between MJD 51240 and MJD 51250.","Apart from those observations, the count rate in band D is low, so that applying our method on the light curve in band D is not possible due to the lack of counts in a short time bin.","Because of this, we provide NLD indicator only for band F. We use the time bin ${\\rm d}t^F=0.032$ s for light curves in band F and ${\\rm d}t^D=0.025$ s for light curves in band D. We find the educated guess of the embedding dimension $m$ and the time delay $\\Delta t = k {\\rm d}t$ as in [17].","The values vary in the range $m \\in [7,9]$ and $k \\in [2,9]$ .","For each observation and each energy band, $N_\\epsilon $ indicates the number of different $\\epsilon $ , for which the NLD indicator is obtained by averaging the significance.","Figure: Characteristics of the source XTE J1550-564 during both epochs of the outburst 1998/1999.a) Count rate in energy band F. b) Count rate in energy band D. c) Hardness ratio hh.Figure: NLD indicator in band D and F during the outburst of XTE J1550-564.The threshold 1.5 for significant result is indicated by the dashed horizontal line.Our results, which are summarised in Table REF and plotted in Fig.","REF , show that the NLD indicator is higher in band D than in band F at the beginning of the outburst.","This is in agreement with the result obtained for the observation 30188-06-05-00 (see Table REF ).","The traces of non-linear dynamics are most prominent in the light curves consisting of counts in the energy range of 8-13 keV.","Hence, the non-linear behaviour of the accreting system is expected to manifest mainly in the regions that emit radiation in energies 8-13 keV.","Later, this tendency is smeared and $\\bar{\\mathcal {S}}^F > \\bar{\\mathcal {S}}^D$ in some cases.","The non-linear behaviour shifts into lower energies as the outburst continues, while its total importance decreases.","This supports the trend, which we presented in section REF for the three selected observations.","The obvious exception of the NLD indicator behaviour is the observation 30191-01-02-00, taken on MJD 51075, for which the NLD indicator in band F is slightly above the threshold, but the NLD indicator in band D is non-significant.","This observation is special because it covers the peak of the outburst, during which the count rate is three times higher than in any other observation.","Remarkably, the hardness ratio does not vary much, having very similar value, $h=0.21$ , like for the other observations.","Table: Observations selected for the analysis.","Total count rate in energy bands F and D, the number of working PCUs and the hardness ratio h=N D /(N F -N D )h = N_{\\rm D} / (N_{\\rm F} - N_{\\rm D}) are shown, along with non-linear dynamics indicator 𝒮 ¯\\bar{\\mathcal {S}} in energy bands F and D, obtained for N ϵ N_\\epsilon different values of recurrence threshold ϵ\\epsilon and the time bin dt{\\rm d}t, with uncertainty σ.\\sigma .On MJD 51110, the NLD indicator is less than one in both bands and the evidence for non-linear behaviour vanishes.","The hardness ratio drops down to $h=0.13$ , while the total count rate $N^F$ remains at similar level $\\sim 14 000$ cts/s.","From that moment, not only the hardness ratio decreases further to $h\\sim 0.05$ , but the total count rate also declines significantly.","NLD indicator shows no signs of non-linear dynamics.","After the source stays at a low luminosity level for about two months, the second stage of the outburst occurs.","This epoch spans approximately from the middle of January to the beginning of April 1999.","During this epoch, the count rate in band F reaches similar values to the first stage ($\\sim 15000$ cts/s) except for the outburst peak.","Almost all observations from this stage including those with higher $h$ provide non-significant values of NLD indicator, with only slight glimpses of non-linear motion on MJD 51245 ($\\bar{\\mathcal {S}}^F=1.74$ ) and MJD 51247 ($\\bar{\\mathcal {S}}^D=1.72$ ).","Hence, there are only very weak signs of non-linear dynamics during the second epoch of the outburst and the behaviour of the source is considerably different from the first stage."],["Discussion","During the two epochs of the outburst the source XTE J1550-564 undergoes a complicated evolution of its total luminosity and the spectral and timing properties.","Existence and properties of quasi-periodic oscillations (QPOs) were studied, for example, by [12], [16], [7].","Detailed spectral study was provided by [15].","[8] discussed the spectral fitting of the data with respect to the geometry of the accretion flow.","They focus on the first epoch of the outburst, when the evolution goes from a low/hard state through a strong very high state, VHS and a weak VHS to the standard high/soft state.","Authors show that the data are hardly compatible with the constant inner radius of the accretion disc.","They propose a scenario of the evolution of the accretion flow during the initial state of the outburst, which involves the accretion disc truncated at higher radius than the ISCO, which is immersed in the hot Comptonizing corona.","With the elapsed time, the disc propagates closer to the center, eventually reaching the ISCO around MJD 51105.","Within next 12 days the remaining corona disappears and the source moves into the standard high/soft state.","The connection of the outburst time and spectral variability with the properties of companion star was shown by [20].","They proposed a scenario of the outburst, which begins with the accretion of low angular momentum gas released from a magnetic trap near the L1 point created by the companion star magnetic field.","At the same time high angular momentum accretion also begins through the Roche lobe overflow forming an accretion disc further away.","However, the low angular momentum accretion is much faster, operating near the free fall speed, and the electrons gains high energy.","Close to the black hole, where a centrifugal barrier is met, a shock in this component could create and possibly oscillate, showing the non-linear behaviour and radiating in the highest energies.","This scenario explains the fast rise of the hard X-ray component versus much slower rise of the soft component, and it is compatible with the previously presented picture of the colder Keplerian disc slowly propagating inward through the already existing hot corona [8].","This scenario is also viable in point of view of our results.","We showed that at the beginning of the outburst the non-linear behaviour is the strongest in the higher energies, particularly in band D. We note that the drop down of the NLD indicator in the highest energy band E may be caused by a small count rate, and perhaps data from different instrument would be more suitable for the analysis in the highest energies.","Later the non-linear dynamics is more prominent in lower energies (band C).","The oscillations of the accretion disc caused by the radiation pressure instability occurs typically on timescales of at least five to ten seconds.","The exact value depends on the global parameters of the source, such as the black hole mass and viscosity in the accretion disk, and is triggered by the accretion rate being a large fraction of the Eddington.","The threshold accretion rate for the instability depends weakly on the system parameters.","The truncation of the inner disc, on the other hand, could shut down the instability in XTE J1550-564.","For instance, the disc truncated at 30 Schwarzschild radii would be completely stable, if the accretion rate is below 0.7 of Eddington rate.","Therefore only the `memory' of the disc's oscillations would be kept in the hot corona, which is dynamically and radiatively coupled with the disc.","It should also be noted, that instead of the regular outbursts on large amplitudes and long timescales, which are not observed in XTE J1550-564, a kind of flickering behaviour is possible, in case of a specific configuration of system parameters for which the disc is only marginally unstable (Grzedzielski et al., 2016, in prep.).","Here, we link the non-linear behaviour of the high energy emitting regions with the oscillations inside the corona, triggered by the formation and propagation of the accretion disc in the equatorial plane.","When the disc is more deeply immersed in the corona, the non-linear behaviour shifts to the lower energies.","When the disc reaches the ISCO, and accretion rate is low, it stabilises and emits only thermal stochastic radiation.","The oscillations of the corona cease and the corona itself is depleted within several days.","The propagation of oscillations through the corona can potentially lead to the evolution of the low frequency QPOs, modelled by [2] within an outburst scenario, that invoked formation of a shock in the inner region.","We can expect that during the flare peak (MJD 51075) any non-linear oscillatory behaviour is overpowered by some rapid, very energetic event accompanied by a strong stochastic emission, so that the NLD indicator is quite low.","This can be related to the jet launching because strong radio emission, which peaked two days after the X-rays, was observed preceded by a peak in optical band that occurred about one day after the X-ray peak [20].","A few days later the apparent superluminal motion of the ejecta in the radio band was observed [5].","In the second epoch of the outburst, we detect no strong signs of non-linear behaviour, even in the short period between MJD 51244 and MJD 51250, when the hardness ratio increased significantly.","Here, however, the overall evolution of the source is different.","We first see the soft state with high luminosity and small $h$ , compatible with an accretion disc spanned down to the ISCO, and then the increase of the high energy flux $N^D$ (and $h$ ).","Obviously, the mechanism of the corona creation in this case is different than in the first epoch and the non-linear mechanism leading to the oscillations is not triggered."],["Acknowledgments","This work was supported in part by the grant DEC-2012/05/E/ST9/03914 from the Polish National Science Center."]],"string":"[\n [\n \"Non-linear behaviour of XTE J1550-564 during its 1998-1999 outburst,\\n revealed by recurrence analysis\"\n ],\n [\n \"Abstract We study the X-ray emission of the microquasar XTE J1550-564 and analyze the properties of its light curves using the recurrence analysis method.\",\n \"The indicators for non-linear dynamics of the accretion flow are found in the very high state and soft state of this source.\",\n \"The significance of deterministic variability depends on the energy band.\",\n \"We discuss the non-linear dynamics of the accretion flow in the context of the disc-corona geometry and propagating oscillations in the accretion flow.\"\n ],\n [\n \"Introduction\",\n \"The microquasar XTE J1550-564 is a low-mass X-ray binary, with a companion main sequence star, and was discovered in its 1998 outburst by the All-Sky Monitor onboard the RXTE satellite [14].\",\n \"The radio-jets that accompany its activity were also detected, similarly to many other black hole X-ray binaries [5].\",\n \"The compact object's mass is estimated at 8-11 Solar masses and thus it is considered to be a black hole [11].\",\n \"The source exhibits all the canonical spectral states typically associated with LMXBs.\",\n \"The source's spectral analysis performed for the RXTE data of XTE J1550-564, that spans the period between September 23 and October 6, 1998, place this object in the very high state (VHS) category [3].\",\n \"Further analysis performed for this source by [8] showed that the temperature of the accretion disk is much lower than expected for the VHS luminosity, and suggested that this should be interpreted as the signature of the disk truncation at the radius of about 30 $R_{\\\\rm g}$ .\",\n \"The intrinsic luminosity of the source peaks above $10^{39}$ erg/s, and during both very high and soft states it remains well above $2-3 \\\\times 10^{38}$ erg/s.\",\n \"Therefore the derived accretion rate can be in this source equal to at least 25% of the Eddington limit, and reaching even $0.8 \\\\dot{M}_{\\\\rm Edd}$ at the outburst peak.\",\n \"In the standard accretion disk theory, such high values of $\\\\dot{M}$ lead to the intrinsic thermal and viscous instabilities caused by the dominant radiation pressure.\",\n \"Such instabilities manifest themselves in the non-lienar evolution of the disk temperature and density at very short timescales, and consequently imprint a characteristic, time-dependent pattern in the emitted X-ray luminosity.\",\n \"In this work, we study the properties of the XTE J1550-564 light curves.\",\n \"We implement our method, recently described in [17], and we study the light curves in the X-rays as taken by the RXTE satellite during the fall 1998 and spring 1999.\",\n \"The capabilities of this method were tested numerically in [18] and its dependence on parameters was discussed in [4].\",\n \"Here we focus on the spectral and temporal evolution of the microquasars's properties and its possible non-linear behaviour.\",\n \"We discuss the results in the context of the thermal-viscous instability of a Keplerian accretion disc, as well as coupling of its variability with the behaviour of hard X-ray emitting, quasi-spherical corona.\"\n ],\n [\n \"Observational data\",\n \"The overall light curve and hardness ratio of the outburst of XTE J1550-564 shows two separated stages of high luminosity [1].\",\n \"The first one lasted from the beginning of September to the beginning of October 1998 (MJD 51063 to 51120), and the second one spanned from the mid of December 1998 to the mid of April 1999 (MJD 51160 to 51280).\",\n \"From the list of the publicly available observations made by RXTE in NASA's HEASARC arxivheasarc.gsfc.nasa.gov we selected several observations, which cover the two epochs quite evenly and which are long enough for our purpose.\",\n \"The data sample is summarised in Table REF .\"\n ],\n [\n \"Recurrence analysis and non-linear dynamics indicator\",\n \"We study the properties of the light curves obtained in different energy bands by means of the time series analysis.\",\n \"Our aim is to find out if the time variability of the flux contains the information about the nature of the dynamical system behind the emission of the radiation.\",\n \"Mainly, we want to discover if the underlying system behaves according to deterministic non-linear dynamics, which is governed by a low number of equations rather than by stochastic variations.\",\n \"In the former case, the traces of the non-linear behaviour are imprinted into the outgoing radiation.\",\n \"Table: Non-linear dynamics indicator 𝒮 ¯\\\\bar{\\\\mathcal {S}} obtained for N ϵ N_\\\\epsilon different values of recurrence threshold ϵ\\\\epsilon in different energy bands for three observations 30188-06-05-00 (MJD 51068), 30191-01-12-00 (MJD 51082) and 30191-01-33-00 (MJD 51108).\",\n \"The number of points used is N=32768N=32768In the preceding paper [17] we developed the method, which can reveal such traces hidden in the observational data.\",\n \"The method includes performing recurrence analysis on the observational time series, which is in our case the light curve, and on the set of surrogate data series.\",\n \"The results are then compared according to discriminating statistics.\",\n \"Our null hypothesis is that the data are a product of a linearly autocorrelated Gaussian process, so that the data point is given by a linear combination of the preceding points and a contribution of uncorrelated Gaussian noise, $\\\\xi (n)$ , hence $x_n = a_0 + \\\\sum _{k=1}^q a_k x_{n-k} +\\\\xi (n)$ .\",\n \"The properties of such time series are fully described by their Fourier spectrum.\",\n \"The surrogate data are constructed in such a way that they share the value distribution and the Fourier spectrum with the original series [19], [13].\",\n \"However, if the original time series is a product of a non-linear system, the dynamical attributes obtained by the analysis will differ from the surrogate series.\",\n \"Here we use this method to study the outburst of XTE J1550-564, so we only briefly describe its main points.\",\n \"We normalise the light curve to have zero mean and unit variance, then we construct a bunch of 100 surrogates using the procedure surrogates from TISEANhttp://www.mpipks-dresden.mpg.de/~tisean/[6].\",\n \"The recurrence analysis is performed on the basis of the software package described by [10], [9], which also yields the cumulative histogram of diagonal lines.\",\n \"We compute the recurrence quantifiers for the data and the surrogates, using the educated guess of its parameters time delay $\\\\Delta t$ and embedding dimension $m$ for a set of recurrence thresholds $\\\\epsilon $ , such that the recurrence rate is approximately in the range (1% – 25 %).\",\n \"After computing the cumulative histograms of diagonal lines for the observed data and their surrogates, we estimate the Rényi's entropy $K_2$ by the linear regression of the cumulative histogram.\",\n \"We follow the rules given in [17] Appendix B, in particular that the minimal length of the diagonal lines must exceed the time delay and that there must be at least five points for the regression.\",\n \"As the discriminating statistic we choose the logarithm of the estimated Rényi's entropy, which we compute for each threshold for the observed time series $Q^{\\\\rm obs}(\\\\epsilon )=\\\\ln (K_2^{\\\\rm obs})$ and similarly for the set of surrogates $Q^{\\\\rm surr}_i(\\\\epsilon )$ .\",\n \"We find the mean $\\\\bar{Q}^{\\\\rm surr}(\\\\epsilon )$ and the standard deviation $\\\\sigma _Q^{\\\\rm surr}(\\\\epsilon )$ of the set $Q^{\\\\rm surr}_i(\\\\epsilon )$ .\",\n \"Then we define the weighted significance as $\\\\mathcal {S}(\\\\epsilon ) = \\\\frac{N_{\\\\rm sl}}{N^{\\\\rm surr}} \\\\mathcal {S}_{\\\\rm sl} - {\\\\rm sign}( Q^{\\\\rm obs} (\\\\epsilon ) - \\\\bar{Q}^{\\\\rm surr}(\\\\epsilon ) ) \\\\frac{N_{\\\\mathcal {S}_K}}{N^{\\\\rm surr}} \\\\mathcal {S}_{K_2}(\\\\epsilon ) , $ where $N_{\\\\rm sl}$ is the number of surrogates, which have only short diagonal lines, and $N^{\\\\rm surr}$ is the total number of surrogates, $\\\\mathcal {S}_{\\\\rm sl} =3$ and $\\\\mathcal {S}_{K_2}$ is the significance computed only from the surrogates, which have enough long lines according to the relation $\\\\mathcal {S}_{K_2} (\\\\epsilon ) = \\\\frac{| Q^{\\\\rm obs} (\\\\epsilon ) - \\\\bar{Q}^{\\\\rm surr}(\\\\epsilon ) |}{\\\\sigma _{Q^{\\\\rm surr}(\\\\epsilon )}}.$ Finally, we define the non-linear dynamics (NLD) indicator, $\\\\bar{\\\\mathcal {S}}$ , as the average of $\\\\mathcal {S}(\\\\epsilon )$ over the set of $N_\\\\epsilon $ different thresholds.\",\n \"We say that the observation shows traces of non-linear behaviour if it exceeds the chosen threshold, $\\\\bar{\\\\mathcal {S}} > 1.5$ .\",\n \"The accuracy of the resulting NLD indicator is quite a complicated issue, because the observational data with their uncertainness undergo several subsequent statistical processes.\",\n \"However, we can treat the resulting values $\\\\mathcal {S}(\\\\epsilon )$ as a small set of $N_\\\\epsilon $ measurements of $\\\\bar{\\\\mathcal {S}}$ .\",\n \"As such, the uncertainty of NLD indicator is given as $\\\\sigma = R/(2\\\\sqrt{N_\\\\epsilon })$ , where R is the spread of the data set, that is $R={\\\\rm max}(\\\\mathcal {S}(\\\\epsilon )) - {\\\\rm min}(\\\\mathcal {S}(\\\\epsilon ))$ .\"\n ],\n [\n \"Energy dependence of NLD indicator\",\n \"We start our study with the observation 30188-06-05-00 taken on 12.9.1998 (MJD 51068).\",\n \"We extract the data with the time bin ${\\\\rm d}t=0.025$ s in six energy bands denoted as A – F, which are listed in Table REF .\",\n \"We take the longest available continuous part of the light curve up to $N_{\\\\rm max}=32768$ points.\",\n \"We set the upper limit of the length of the studied time series in order to keep the numerical demands to a reasonable level.\",\n \"The light curve in band F, the recurrence plot (RP)Recurrence plot is a visualisation of the recurrence matrix, where the recurrence points are plotted as dots.\",\n \"Because the plot is symmetric with respect to the main diagonal, we plot only the triangle below/above the diagonal.\",\n \"of the observation and RP of one of the surrogates are plotted in Fig.\",\n \"REF .\",\n \"Figure: Lightcurves in energy band F and recurrence plots for the observations (in red) and one surrogate (in green) for the three observations listed in Table :a) MJD 51068 b) MJD 51082 c) MJD 51108.We compute the NLD indicator $\\\\bar{\\\\mathcal {S}}$ with the parameters $m=9$ and $k=4$ .\",\n \"The highest $\\\\bar{\\\\mathcal {S}}$ is obtained for the energy band D, which covers the energy range $7.96 - 12.99$ keV.\",\n \"On the other hand in the lowest energy band A, the NLD indicator is very small ($\\\\bar{\\\\mathcal {S}}=0.45$ ).\",\n \"The indicator increases from band A to band D, slightly exceeding our threshold in band B ($\\\\bar{\\\\mathcal {S}} = 1.77 > 1.5$ ) and giving quite a high value $\\\\bar{\\\\mathcal {S}} = 4.58$ in band C. In the highest energies covered by band E, the value decreases to $\\\\bar{\\\\mathcal {S}} = 2.80$ .\",\n \"Figure: Dependence of 𝒮\\\\mathcal {S} on ϵ\\\\epsilon for the observation from MJD 51068 extracted in different energy bands.We also provide results for the energy band F, which includes channels 0 – 35 (< 12.99 keV).\",\n \"This band corresponds to the extraction of the data from the binned mode or the single-bit mode without energy filtering, so that such a light curve has the highest count rate.\",\n \"This band can also be compared with the results given in [17].\",\n \"The light curve in this band yields a high value $\\\\bar{\\\\mathcal {S}} = 6.69.$ We notice that the uncertainty $\\\\sigma $ of NLD given in Table REF in bands D and F is very high.\",\n \"The reason for this is that the significance $\\\\mathcal {S}(\\\\epsilon )$ is not independent on $\\\\epsilon $ , but rather grows with the threshold and the spread of the data set is high (see Fig.\",\n \"REF ).\",\n \"This behaviour is not so prominent in any other case, and can be explained if the underlying cumulative histograms are inspected.\",\n \"The histograms of this observation for higher thresholds show an `elbow' around $l \\\\sim 3.5$ s and the slope of the part after the elbow is about two times smaller than for the shorter lines.\",\n \"A similar elbow is not seen in the histograms of the surrogates, partly because they do not have many such long lines.\",\n \"For higher $\\\\epsilon $ , the elbow is more prominent in the observation's histogram, lowering the estimated value of $K_2^{\\\\rm obs}$ and thus making the difference between the observation and the surrogates bigger.\",\n \"This feature can be explained as the presence of recurrence period in the system.\",\n \"The slope of the first part is higher, because it is more difficult to predict the evolution of the trajectory, when we have information about its piece shorter than the recurrence period (see [10] for more details about the properties of RPs).\",\n \"Also a possible interplay between the value of the recurrence threshold and the noise in the data can play a role in this effect.\",\n \"Similar analysis of the energy dependence of NLD was also made for two later observations, 30191-01-12-00 taken on MJD 51082, and 30191-01-33-00 taken on MJD 51108 (denoted as b) and c) in Fig.\",\n \"REF ).\",\n \"The second observation yields similar values of NLD in bands D and C and the overall band F gives the highest value.\",\n \"The third observation also has its highest significance in the band F, however for this observation the NLD indicator in band C is higher then in band D. The values in all bands are much less than for the first two observations, which suggests that the non-linear dynamics in the source fades away during the outburst.\"\n ],\n [\n \"Temporal evolution of NLD indicator during the outburst\",\n \"We continue with the study of the temporal evolution of the NLD indicator during the outburst of XTE J1550-564.\",\n \"We extract the light curves in energy bands D and F for each observation from Table REF .\",\n \"In order to use our procedure meaningfully, we need to have a sufficiently high number of counts in each time bin.\",\n \"Hence, the total count rate combined from all working proportional counter units (PCUs) in each energy band is an important quantity for setting the time bin ${\\\\rm d}t$ .\",\n \"We provide the values and the number of working PCUs in Table REF .\",\n \"The normalised count rate per PCU is given in Figs REF a) and REF b), respectively.\",\n \"We quantify the hardness ratio as the count rate in band D divided by count rate in lower energies, $h = N_{\\\\rm D} / (N_{\\\\rm F} - N_{\\\\rm D})$ , which is given in Table REF and in Fig REF c).\",\n \"Strong spectral transition from hard state, with $h \\\\gtrsim 0.2$ , to soft state, with $h \\\\approx 0.05$ , can be seen during the first stage of the outburst.\",\n \"Hence, the count rate in band D drops down significantly in the later epoch.\",\n \"Therefore we do not compute the significance in band D for observations after 1.11.1998 (MJD 51119).\",\n \"During the second stage of the outburst, the hardness ratio $h$ is much lower, with the exception of short time period around MJD 51250 (see [8]), when it reaches $h=0.18$ .\",\n \"We map this time period more closely, choosing several observations between MJD 51240 and MJD 51250.\",\n \"Apart from those observations, the count rate in band D is low, so that applying our method on the light curve in band D is not possible due to the lack of counts in a short time bin.\",\n \"Because of this, we provide NLD indicator only for band F. We use the time bin ${\\\\rm d}t^F=0.032$ s for light curves in band F and ${\\\\rm d}t^D=0.025$ s for light curves in band D. We find the educated guess of the embedding dimension $m$ and the time delay $\\\\Delta t = k {\\\\rm d}t$ as in [17].\",\n \"The values vary in the range $m \\\\in [7,9]$ and $k \\\\in [2,9]$ .\",\n \"For each observation and each energy band, $N_\\\\epsilon $ indicates the number of different $\\\\epsilon $ , for which the NLD indicator is obtained by averaging the significance.\",\n \"Figure: Characteristics of the source XTE J1550-564 during both epochs of the outburst 1998/1999.a) Count rate in energy band F. b) Count rate in energy band D. c) Hardness ratio hh.Figure: NLD indicator in band D and F during the outburst of XTE J1550-564.The threshold 1.5 for significant result is indicated by the dashed horizontal line.Our results, which are summarised in Table REF and plotted in Fig.\",\n \"REF , show that the NLD indicator is higher in band D than in band F at the beginning of the outburst.\",\n \"This is in agreement with the result obtained for the observation 30188-06-05-00 (see Table REF ).\",\n \"The traces of non-linear dynamics are most prominent in the light curves consisting of counts in the energy range of 8-13 keV.\",\n \"Hence, the non-linear behaviour of the accreting system is expected to manifest mainly in the regions that emit radiation in energies 8-13 keV.\",\n \"Later, this tendency is smeared and $\\\\bar{\\\\mathcal {S}}^F > \\\\bar{\\\\mathcal {S}}^D$ in some cases.\",\n \"The non-linear behaviour shifts into lower energies as the outburst continues, while its total importance decreases.\",\n \"This supports the trend, which we presented in section REF for the three selected observations.\",\n \"The obvious exception of the NLD indicator behaviour is the observation 30191-01-02-00, taken on MJD 51075, for which the NLD indicator in band F is slightly above the threshold, but the NLD indicator in band D is non-significant.\",\n \"This observation is special because it covers the peak of the outburst, during which the count rate is three times higher than in any other observation.\",\n \"Remarkably, the hardness ratio does not vary much, having very similar value, $h=0.21$ , like for the other observations.\",\n \"Table: Observations selected for the analysis.\",\n \"Total count rate in energy bands F and D, the number of working PCUs and the hardness ratio h=N D /(N F -N D )h = N_{\\\\rm D} / (N_{\\\\rm F} - N_{\\\\rm D}) are shown, along with non-linear dynamics indicator 𝒮 ¯\\\\bar{\\\\mathcal {S}} in energy bands F and D, obtained for N ϵ N_\\\\epsilon different values of recurrence threshold ϵ\\\\epsilon and the time bin dt{\\\\rm d}t, with uncertainty σ.\\\\sigma .On MJD 51110, the NLD indicator is less than one in both bands and the evidence for non-linear behaviour vanishes.\",\n \"The hardness ratio drops down to $h=0.13$ , while the total count rate $N^F$ remains at similar level $\\\\sim 14 000$ cts/s.\",\n \"From that moment, not only the hardness ratio decreases further to $h\\\\sim 0.05$ , but the total count rate also declines significantly.\",\n \"NLD indicator shows no signs of non-linear dynamics.\",\n \"After the source stays at a low luminosity level for about two months, the second stage of the outburst occurs.\",\n \"This epoch spans approximately from the middle of January to the beginning of April 1999.\",\n \"During this epoch, the count rate in band F reaches similar values to the first stage ($\\\\sim 15000$ cts/s) except for the outburst peak.\",\n \"Almost all observations from this stage including those with higher $h$ provide non-significant values of NLD indicator, with only slight glimpses of non-linear motion on MJD 51245 ($\\\\bar{\\\\mathcal {S}}^F=1.74$ ) and MJD 51247 ($\\\\bar{\\\\mathcal {S}}^D=1.72$ ).\",\n \"Hence, there are only very weak signs of non-linear dynamics during the second epoch of the outburst and the behaviour of the source is considerably different from the first stage.\"\n ],\n [\n \"Discussion\",\n \"During the two epochs of the outburst the source XTE J1550-564 undergoes a complicated evolution of its total luminosity and the spectral and timing properties.\",\n \"Existence and properties of quasi-periodic oscillations (QPOs) were studied, for example, by [12], [16], [7].\",\n \"Detailed spectral study was provided by [15].\",\n \"[8] discussed the spectral fitting of the data with respect to the geometry of the accretion flow.\",\n \"They focus on the first epoch of the outburst, when the evolution goes from a low/hard state through a strong very high state, VHS and a weak VHS to the standard high/soft state.\",\n \"Authors show that the data are hardly compatible with the constant inner radius of the accretion disc.\",\n \"They propose a scenario of the evolution of the accretion flow during the initial state of the outburst, which involves the accretion disc truncated at higher radius than the ISCO, which is immersed in the hot Comptonizing corona.\",\n \"With the elapsed time, the disc propagates closer to the center, eventually reaching the ISCO around MJD 51105.\",\n \"Within next 12 days the remaining corona disappears and the source moves into the standard high/soft state.\",\n \"The connection of the outburst time and spectral variability with the properties of companion star was shown by [20].\",\n \"They proposed a scenario of the outburst, which begins with the accretion of low angular momentum gas released from a magnetic trap near the L1 point created by the companion star magnetic field.\",\n \"At the same time high angular momentum accretion also begins through the Roche lobe overflow forming an accretion disc further away.\",\n \"However, the low angular momentum accretion is much faster, operating near the free fall speed, and the electrons gains high energy.\",\n \"Close to the black hole, where a centrifugal barrier is met, a shock in this component could create and possibly oscillate, showing the non-linear behaviour and radiating in the highest energies.\",\n \"This scenario explains the fast rise of the hard X-ray component versus much slower rise of the soft component, and it is compatible with the previously presented picture of the colder Keplerian disc slowly propagating inward through the already existing hot corona [8].\",\n \"This scenario is also viable in point of view of our results.\",\n \"We showed that at the beginning of the outburst the non-linear behaviour is the strongest in the higher energies, particularly in band D. We note that the drop down of the NLD indicator in the highest energy band E may be caused by a small count rate, and perhaps data from different instrument would be more suitable for the analysis in the highest energies.\",\n \"Later the non-linear dynamics is more prominent in lower energies (band C).\",\n \"The oscillations of the accretion disc caused by the radiation pressure instability occurs typically on timescales of at least five to ten seconds.\",\n \"The exact value depends on the global parameters of the source, such as the black hole mass and viscosity in the accretion disk, and is triggered by the accretion rate being a large fraction of the Eddington.\",\n \"The threshold accretion rate for the instability depends weakly on the system parameters.\",\n \"The truncation of the inner disc, on the other hand, could shut down the instability in XTE J1550-564.\",\n \"For instance, the disc truncated at 30 Schwarzschild radii would be completely stable, if the accretion rate is below 0.7 of Eddington rate.\",\n \"Therefore only the `memory' of the disc's oscillations would be kept in the hot corona, which is dynamically and radiatively coupled with the disc.\",\n \"It should also be noted, that instead of the regular outbursts on large amplitudes and long timescales, which are not observed in XTE J1550-564, a kind of flickering behaviour is possible, in case of a specific configuration of system parameters for which the disc is only marginally unstable (Grzedzielski et al., 2016, in prep.).\",\n \"Here, we link the non-linear behaviour of the high energy emitting regions with the oscillations inside the corona, triggered by the formation and propagation of the accretion disc in the equatorial plane.\",\n \"When the disc is more deeply immersed in the corona, the non-linear behaviour shifts to the lower energies.\",\n \"When the disc reaches the ISCO, and accretion rate is low, it stabilises and emits only thermal stochastic radiation.\",\n \"The oscillations of the corona cease and the corona itself is depleted within several days.\",\n \"The propagation of oscillations through the corona can potentially lead to the evolution of the low frequency QPOs, modelled by [2] within an outburst scenario, that invoked formation of a shock in the inner region.\",\n \"We can expect that during the flare peak (MJD 51075) any non-linear oscillatory behaviour is overpowered by some rapid, very energetic event accompanied by a strong stochastic emission, so that the NLD indicator is quite low.\",\n \"This can be related to the jet launching because strong radio emission, which peaked two days after the X-rays, was observed preceded by a peak in optical band that occurred about one day after the X-ray peak [20].\",\n \"A few days later the apparent superluminal motion of the ejecta in the radio band was observed [5].\",\n \"In the second epoch of the outburst, we detect no strong signs of non-linear behaviour, even in the short period between MJD 51244 and MJD 51250, when the hardness ratio increased significantly.\",\n \"Here, however, the overall evolution of the source is different.\",\n \"We first see the soft state with high luminosity and small $h$ , compatible with an accretion disc spanned down to the ISCO, and then the increase of the high energy flux $N^D$ (and $h$ ).\",\n \"Obviously, the mechanism of the corona creation in this case is different than in the first epoch and the non-linear mechanism leading to the oscillations is not triggered.\"\n ],\n [\n \"Acknowledgments\",\n \"This work was supported in part by the grant DEC-2012/05/E/ST9/03914 from the Polish National Science Center.\"\n ]\n]"},"id":{"kind":"string","value":"1605.05849"}}},{"rowIdx":1866695,"cells":{"text":{"kind":"list like","value":[["Cluster-lensing: A Python Package for Galaxy Clusters & Miscentering"],["Abstract We describe a new open source package for calculating properties of galaxy clusters, including NFW halo profiles with and without the effects of cluster miscentering.","This pure-Python package, cluster-lensing, provides well-documented and easy-to-use classes and functions for calculating cluster scaling relations, including mass-richness and mass-concentration relations from the literature, as well as the surface mass density $\\Sigma(R)$ and differential surface mass density $\\Delta\\Sigma(R)$ profiles, probed by weak lensing magnification and shear.","Galaxy cluster miscentering is especially a concern for stacked weak lensing shear studies of galaxy clusters, where offsets between the assumed and the true underlying matter distribution can lead to a significant bias in the mass estimates if not accounted for.","This software has been developed and released in a public GitHub repository, and is licensed under the permissive MIT license.","The cluster-lensing package is archived on Zenodo (Ford 2016).","Full documentation, source code, and installation instructions are available at http://jesford.github.io/cluster-lensing/."],["Introduction","Clusters of galaxies are the largest gravitationally collapsed structures to have formed in the history of the universe.","As such, they are interesting both from a cosmological as well as an astrophysics perspective.","In the former case, the galaxy cluster number density as a function of mass (the cluster mass function) is a probe of cosmological parameters including the fractional matter density $\\Omega _{\\rm m}$ and the normalization of the matter power spectrum $\\sigma _8$ .","Astrophysically, the deep potential wells of galaxy clusters are environments useful for testing theories of general relativity, galaxy evolution, and gas and plasma physics, among other things [40].","The common thread among these diverse investigations is the requisite knowledge of the mass of the galaxy cluster, which is largely composed of its invisible dark matter halo.","Although many techniques exist for estimating the total mass of these systems, weak lensing has emerged as somewhat of a gold standard, since it is sensitive to the mass itself, and not to the dynamical state or other biased tracers of the underlying mass.","Scaling relations between weak lensing derived masses, and other observables, including richness, X-ray luminosity and temperature, for examples, are typically calibrated from large surveys and extrapolated to clusters for which gravitational lensing measurements are impossible or unreliable.","Since weak lensing masses are often considered the “true” masses, against which other estimates are compared [26], [41], [19], it is paramount that cluster masses from weak lensing modeling are as unbiased as possible.","For stacked weak lensing measurements of galaxy clusters, an important source of bias in fitting a mass model is the inclusion of the effect of miscentering offsets.","Miscentering occurs when the center of the mass distribution – the dark matter halo – does not perfectly coincide with the assumed center around which tangential shear (or magnification) profiles are being measured.","Candidate centers for galaxy clusters are necessarily chosen from observational proxies, and often include a single galaxy, such as the brightest or most massive member, or the centroid of some extended quantity like the peak of X-ray emission or average of galaxy positions [16].","The particular choice of center may be offset from the true center due to interesting physical processes such as recent mergers and cluster evolution, or simply due to misidentification of the proxy of interest [22].","The miscentering effect on the stacked weak lensing profile can be included in a proper modeling of the measurement, as done in [22], [27], [32], [16], [38], [31], [13], [15], [39].","The inclusion of this effect commonly assumes a form for the distribution of offsets, such as a Rayleigh distribution in radius (which represents a 2D Gaussian in the plane of the sky).","This distribution is convolved with the standard (centered) halo profile to obtain the miscentered version.","Software for calculating these miscentered weak lensing profiles was developed in order to produce results in [13], [15], and has recently been publicly released to the astronomical community [12].https://github.com/jesford/cluster-lensing When many different gravitational lenses are stacked, as is often necessary to increase signal-to-noise for weak lensing measurements, care must be taken in the interpretation of the average signal.","The issue here is that the (differential) surface mass density is not a linear function of the mass, so the average of many stacked profiles does not directly yield the average mass of the lens sample.","Care must be taken to consider the underlying distribution of cluster masses as well as the redshifts of lenses and sources, all of which affect the amplitude of the measured lensing profile.","One approach to this is to use a so-called composite-halo approach [18], [14], [13], [15], [39], where profiles are calculated for all individual lens objects and then averaged together to create a model that can be fit to the measurement.","The [style=codeintext]ClusterEnsemble() class discussed in Section REF is designed with this approach in mind.","A popular model for the dark matter distribution in a gravitationally collapsed halo, such as a galaxy cluster, is the Navarro, Frenk, and White (NFW) model.","This density profile (given in Equation REF below) was determined from numerical simulations that included the dissipationless collapse of density fluctuations under gravity [30].","The simpler Singular Isothermal Sphere density model, which only has one free parameter in contrast to the two for NFW, does not tend to fit the inner profiles of halos well and is also unphysical in that the total mass diverges [37].","Other models such as the generalized-NFW and the Einasto profile tend to better describe the full radial distribution of dark matter in halos, at the expense of adding a third parameter to characterize the inner slope of the density profiles [8].","In the software package presented in this work we only include the standard 2-parameter NFW model, but future work should make alternative models available as well."],["Description of the Code","In this section we demonstrate each of the individual modules available in the [style=codeintext]cluster-lensing package.","In Section REF we describe a class for calculating surface mass density profiles directly from NFW and cosmological parameters.","Next we outline the functions available for mass-concentration relations in Section REF .","Then in Section REF we present the class [style=codeintext]ClusterEnsemble(), and its related functions, which tie together the previously discussed functionality into a framework for easily manipulating and producing profiles for multiple galaxy clusters at once, from common observational quantities.","Much of the content of this section comes directly from the online documentation.http://jesford.github.io/cluster-lensing/ Throughout the modules, dimensionful quantities are labelled as such by means of the [style=codeintext]astropy.units package."],["The [style=codeintext]nfw.py module contains a single class called [style=codeintext]SurfaceMassDensity(), which computes the surface mass density $\\Sigma (R)$ and the differential surface mass density $\\Delta \\Sigma (R)$ using the class methods [style=codeintext]sigmanfw() and [style=codeintext]deltasigmanfw(), respectively.","These profiles are calculated according to the analytical formulas first derived by [44], assuming the spherical NFW model, and can be applied to any dark matter halo: this module is not specific to galaxy clusters.","The 3-dimensional density profile of an NFW halo is given by $\\rho (r) = \\frac{\\delta _{\\rm c} \\rho _{\\rm crit}}{(r/r_{\\rm s})(1+r/r_{\\rm s})^2},$ where $r_{\\rm s}$ is the cluster scale radius, $\\delta _{\\rm c}$ is the characteristic halo overdensity, and $\\rho _{\\rm crit} = \\rho _{\\rm crit}(z)$ is the critical energy density of the universe at the lens redshift.","These three parametersor, in the case of calculating multiple NFW halos at once, three array-like objects representing each of these parameters must be specified when instantiating the class [style=codeintext]SurfaceMassDensity(), via the arguments [style=codeintext]rs, [style=codeintext]deltac, and [style=codeintext]rhocrit, respectively.","The units on [style=codeintext]rs are assumed to be Mpc, [style=codeintext]deltac is dimensionless, and [style=codeintext]rhocrit is in $M_{\\odot }{\\rm Mpc}^{-1}{\\rm pc}^{-2}$ , although the actual inclusion of the [style=codeintext]astropy.units on these variables is optional.","The user will probably also want to choose the radial bins for the calculation, which are specified via the keyword argument [style=codeintext]rbins, in Mpc.","The surface mass density is the integral along the line-of-sight of the 3-dimensional density: $\\Sigma (R) = 2 \\int _0^{\\infty } \\rho (R,y) {\\rm d}y.$ Here $R$ is the projected radial distance (in the plane of the sky).","We can adopt the dimensionless radius $x \\equiv R/r_{\\rm s}$ and, following from [44], show that: $\\Sigma (x) = 2 r_{\\rm s} \\delta _{\\rm c} \\rho _{\\rm crit} f(x),$ where $f(x) = $ ${\\left\\lbrace \\begin{array}{ll}\\frac{1}{x^2 - 1} \\left( 1 - \\ln { \\left[ \\frac{1}{x} + \\sqrt{ \\frac{1}{x^2} - 1} \\right]} / \\sqrt{1 - x^2} \\right), & \\text{for } x < 1; \\\\1/3, & \\text{for } x = 1; \\\\\\frac{1}{x^2 - 1} \\left( 1 - \\arccos {(1/x)} / \\sqrt{x^2 - 1} \\right), & \\text{for } x > 1.\\end{array}\\right.","}$ The differential surface mass density probed by shear is calculated from the definition $\\Delta \\Sigma (x) \\equiv \\overline{\\Sigma }( 1.$ Running [style=codeintext]sigmanfw() or [style=codeintext]deltasigmanfw(), with only a specification of halo properties [style=codeintext]rs, [style=codeintext]deltac, [style=codeintext]rhocrit, and radial bins [style=codeintext]rbins, will lead to the calculation of halo profiles according to Equations REF and REF outlined above.","from clusterlensing import SurfaceMassDensity rbins = [0.1, 0.5, 1.0, 2.0, 4.0] # Mpc smd = SurfaceMassDensity(rs=[0.1], rho_crit=[0.2], delta_c=[9700.","], rbins=rbins) sigma = smd.sigma_nfw() # surface mass density with default units sigma[0] # surface mass density with no units sigma[0].value array([ 129.33333333, 11.64751032, 3.33992059, 0.89839601, 0.23327149]) These are the standard centered NFW profiles, under the assumption that the peak of the halo density distribution perfectly coincides with the identified halo center.","This may not be a good assumption, however, and the user can instead run these calculations for miscentered halos by specifying the optional input parameter [style=codeintext]offsets.","This parameter sets the width of a distribution of centroid offsets, assuming a 2-dimensional Gaussian distribution on the sky.","This offset distribution is equivalent to, and implemented in code as, a uniform distribution in angle and a Rayleigh probability distribution in radius: $P(R_{\\mathrm {off}})=\\frac{R_{\\mathrm {off}}}{\\sigma _{\\mathrm {off}}^2}\\ \\mathrm {exp}\\bigg [-\\frac{1}{2}\\bigg (\\frac{R_{\\mathrm {off}}}{\\sigma _{\\mathrm {off}}}\\bigg )^2\\ \\bigg ].$ The parameter [style=codeintext]offsets is equivalent to $\\sigma _{\\rm off}$ in this equation.","from clusterlensing import SurfaceMassDensity rbins = [0.1, 0.5, 1.0, 2.0, 4.0] # single miscentered halo profile smd = SurfaceMassDensity(rs=[0.1], rho_crit=[0.2], delta_c=[9700.","], rbins=rbins, offsets=[0.3]) sigma = smd.sigma_nfw() sigma[0] # example calculating multiple profiles smd = SurfaceMassDensity(rs=[0.1,0.2,0.2], rho_crit=[0.2,0.2,0.2], delta_c=[9700,9700,9000], rbins=rbins, offsets=[0.3,0.3,0.3]) sigma = smd.sigma_nfw() sigma The miscentered surface mass density profiles are given by the centered profiles (Equations REF and REF ), convolved with the offset distribution (Equation REF ).","We follow the offset halo formalism first written down by [45], and applied to cluster miscentering by, e.g.","[22], [16], [13], [15], [39].","Specifically, we calculate the offset surface mass density $\\Sigma ^{\\rm off}$ as follows: $\\Sigma ^{\\rm off}(R) = \\int _{0}^{\\infty } \\Sigma (R | R_{\\rm off})\\ P(R_{\\rm off})\\ {\\rm d}R_{\\rm off}$ $\\Sigma (R|R_{\\mathrm {off}})=\\frac{1}{2\\pi }\\int _{0}^{2\\pi }\\Sigma (r) \\mathrm {d}\\theta $ Here $r = \\sqrt{R^2+R_{\\mathrm {off}}^2-2RR_{\\mathrm {off}}\\cos (\\theta )}$ and $\\theta $ is the azimuthal angle [45].","The $\\Delta \\Sigma ^{\\rm off}$ profile is calculated from $\\Sigma ^{\\rm off}$ , in analogy with Equations REF and REF .","from clusterlensing import SurfaceMassDensity rbins = [0.1, 0.5, 1.0, 2.0, 4.0] # perfectly centered DeltaSigma profile smd = SurfaceMassDensity(rs=[0.1], rho_crit=[0.2], delta_c=[9700.","], rbins=rbins) deltasigma = smd.deltasigma_nfw() deltasigma[0] # miscentered DeltaSigma profile smd = SurfaceMassDensity(rs=[0.1], rho_crit=[0.2], delta_c=[9700.","], rbins=rbins, offsets=[0.3]) deltasigma = smd.deltasigma_nfw() deltasigma[0] "],["The [style=codeintext]cofm.py module currently contains three functions, each of which calculates halo concentration from mass, redshift, and cosmology, according to a prescription given in the literature.","These functions are [style=codeintext]cDuttonMaccio() [8], [style=codeintext]cDuffy() [7], and [style=codeintext]cPrada() [36].","Halo mass-concentration relations are an area of active research, and there have been discrepancies between results from different observations and simulations, and disagreement surrounding the best choice of model [8], [24].","We do not aim to join this discussion here, but focus on outlining the functionality provided by the [style=codeintext]cluster-lensing package, for calculating these different concentration values.","All three functions require two input parameters (scalars or array-like inputs), which are the halo redshift(s) [style=codeintext]z and the halo mass(es) [style=codeintext]m. Specifically, the latter is assumed to correspond to the $M_{200}$ mass definition, in units of solar masses.","$M_{200}$ is the mass interior to a sphere of radius $r_{200}$ , within which the average density is $200\\rho _{\\rm crit}(z)$ .","The default cosmology used is from the measurements by the [34], which is imported from the module [style=codeintext]astropy.cosmology.Planck13.","However, the user can specify alternative cosmological parameters.","For calculating concentration according to either the [7] or the [8] prescription, the only cosmological parameter required is the Hubble parameter, which can be passed into [style=codeintext]cDuffy() or [style=codeintext]cDuttonMaccio() as the keyword argument [style=codeintext]h. For the [36] concentration, the user would want to specify [style=codeintext]OmM and [style=codeintext]OmL (the fractional energy densities of matter and the cosmological constant) in addition to [style=codeintext]h, in the call to [style=codeintext]cPrada().","The [style=codeintext]cDuttonMaccio() calculation of concentration is done according to the power-law $\\log _{10} c_{200} = a + b \\log _{10}(M_{200} / [10^{12} h^{-1} M_{\\odot }]),$ where $a = 0.52 + 0.385\\ {\\rm exp}[-0.617\\ z^{1.21}],$ $b = -0.101 + 0.206 z.$ The above three equations map to Equations 7, 11, and 10, respectively in [8].","The values in these expressions were determined from simulations of halos between $0 < z < 5$ , spanning over 5 orders of magnitude in mass, and were shown to match observational measurements of low-redshift galaxies and clusters [8].","This concentration-mass relation is the default one used by the [style=codeintext]clusters.py module, discussed in Section REF .","from clusterlensing import cofm # single 10**14 Msun halo at z=1 cofm.c_DuttonMaccio(0.1, 1e14) array([ 5.13397936]) # example with multiple halos cofm.c_DuttonMaccio([0.1, 0.5], [1e14, 1e15]) array([ 5.13397936, 3.67907305]) The concentration calculation in [style=codeintext]cDuffy() is $c_{200} = A\\ (M_{200} / M_{\\rm pivot})^B \\ (1 + z)^C,$ where $\\lbrace A, B, C\\rbrace = \\lbrace 5.71, -0.084, -0.47\\rbrace ,$ $M_{\\rm pivot} = 2 \\times 10^{12}\\ h^{-1} M_{\\odot }.$ Equation REF above corresponds to Equation 4 in [7].","The values for $A$ , $B$ , and $C$ can be found in Table 1 of that work, where they are specific to the “full” (relaxed and unrelaxed) sample of simulated NFW halos, spanning the redshift range $0 < z < 2$ .","$M_{\\rm pivot}$ can be found in the caption of their Table 1 as well.","One caveat with this relation is that the cosmology used in creating the [7] simulations was that of the now outdated WMAP5 experiment [25].","from clusterlensing import cofm # default cosmology (h=0.6777) cofm.c_Duffy([0.1, 0.5], [1e14, 1e15]) array([ 4.06126115, 2.89302767]) # with h=1 cofm.c_Duffy([0.1, 0.5], [1e14, 1e15], h=1) array([ 3.93068341, 2.80001099]) The [style=codeintext]cPrada() concentration calculation is much more complex, and written in terms of $\\sigma (M_{200}, x_{\\rm p})$ , the rms fluctuation of the density field.","The [36] halo concentration is given bywe use the subscript “p” to distinguish some variables in the equations from [36] from those in the current work $\\begin{split}c_{200} = 2.881 B_0 (x_{\\rm p}) \\bigg [ \\left( \\frac{B_1(x_{\\rm p}) \\sigma (M_{200},x_{\\rm p})}{1.257} \\right)^{1.022} + 1 \\bigg ] \\\\\\times \\ {\\rm exp}\\left( \\frac{0.06}{[B_1(x_{\\rm p}) \\sigma (M_{200},x_{\\rm p}) ]^2} \\right).\\end{split}$ The cosmology and redshift dependence is encoded by the variable $x_{\\rm p}$ , which is $x_{\\rm p} = \\left( \\frac{\\Omega _{\\Lambda , 0}}{\\Omega _{{\\rm m}, 0}} \\right)^{1/3} (1 + z)^{-1}.$ The functions within Equation REF are as follows: $\\sigma (M_{200},x_{\\rm p}) = D(x_{\\rm p}) \\frac{16.9 y_{\\rm p}^{0.41}}{1 + 1.102 y_{\\rm p}^{0.2} + 6.22 y_{\\rm p}^{0.333}}$ $y_{\\rm p} \\equiv \\frac{10^{12} h^{-1} M_{\\odot }}{M_{200}}$ $D(x_{\\rm p}) = \\frac{5}{2} \\left( \\frac{\\Omega _{{\\rm m}, 0}}{\\Omega _{\\Lambda , 0}} \\right)^{1/3} \\frac{\\sqrt{1 + x_{\\rm p}^3}}{x_{\\rm p}^{3/2}} \\int _0^{x_{\\rm p}} \\frac{x^{3/2} {\\rm d}x}{(1 + x^3)^{3/2}}$ $B_0(x_{\\rm p}) = \\frac{c_{\\rm min}(x_{\\rm p})}{c_{\\rm min}(1.393)}$ $B_1(x_{\\rm p}) = \\frac{\\sigma ^{-1}_{\\rm min}(x_{\\rm p})}{\\sigma ^{-1}_{\\rm min}(1.393)}$ $c_{\\rm min}(x_{\\rm p}) = 3.681 + 1.352 \\bigg [ \\frac{1}{\\pi } \\arctan [6.948 (x_{\\rm p} - 0.424)] + \\frac{1}{2} \\bigg ]$ $\\sigma ^{-1}_{\\rm min}(x_{\\rm p}) = 1.047 + 0.599 \\bigg [ \\frac{1}{\\pi } \\arctan [7.386 (x_{\\rm p} - 0.526)] + \\frac{1}{2} \\bigg ]$ In order of appearance above, beginning with our Equation REF , these equations correspond to Equations 14-17, 13, 23a, 23b, 12, 18a, 18b, 19, 20 in [36].","The numerical values in these equations were obtained empirically from the simulations described in that work.","from clusterlensing import cofm cofm.c_Prada([0.1, 0.5], [1e14, 1e15]) array([ 5.06733941, 5.99897362]) cofm.c_Prada([0.1, 0.1, 0.1], [1e13, 1e14, 1e15]) array([ 5.71130928, 5.06733941, 5.30163572]) The last code example demonstrates the controversial feature of the [36] mass-concentration relation – an upturn in concentration values for the highest mass halos.","This is in opposition to the canonical view that higher mass halos have lower concentrations [29], [30], [21], [2]."],["The [style=codeintext]clusters.py module is designed to provide a catalog-level tool for calculating, tracking, and updating galaxy cluster properties and profiles, through structuring data from multiple clusters as an updatable Pandas Dataframe, and providing an intelligent interface to the other modules discussed in Sections REF and REF .","This module contains a single class [style=codeintext]ClusterEnsemble(), as well as three functions, [style=codeintext]masstorichness(), [style=codeintext]richnesstomass(), and [style=codeintext]calcdeltac().","The function [style=codeintext]calcdeltac() takes a single input parameter, the cluster concentration [style=codeintext]c200 (e.g.","as calculated by one of the functions in [style=codeintext]cofm.py), and returns the characteristic halo overdensity: $\\delta _{\\rm c} = \\left( \\frac{200}{3} \\right) \\frac{c_{200}^3}{\\ln (1 + c_{200}) - c_{200}/(1 + c_{200})}.$ Both input and output are dimensionless here.","For example, to convert a concentration value of $c_{200} = 5$ to $\\delta _{\\rm c}$ , you could do: from clusterlensing.clusters import calc_delta_c calc_delta_c(5) 8694.8101906193315 The pair of functions [style=codeintext]masstorichness() and [style=codeintext]richnesstomass(), as their names imply, perform conversions between cluster mass and richness.","The only required input parameter to [style=codeintext]masstorichness() is the [style=codeintext]mass, and likewise the only required input to [style=codeintext]richnesstomass() is [style=codeintext]richness.","The calculations assume a power-law form for the relationship between these variables: $M_{200} = M_0 \\left( \\frac{N_{200}}{20} \\right) ^ \\beta .$ Here $M_0$ is the normalization, which defaults to $2.7 \\times 10^{13}$ , but can be changed in the call to either function by setting the [style=codeintext]norm keyword argument.","The power-law slope $\\beta = 1.4$ by default, but can be set by specifying the optional [style=codeintext]slope input parameter.","When these functions are invoked by the [style=codeintext]ClusterEnsemble() class, they are applied to the particular mass definition $M_{200}$ , and assume units of $M_{\\odot }$ .","However the functions themselves do not assume a mass definition or unit, and can be generalized to any parameter (or type of richness) that has a power-law relationship with mass.","from clusterlensing.clusters import \\ mass_to_richness, richness_to_mass richness_to_mass(50) 97382243648736.9 mass_to_richness(97382243648736.9) 50.0 # specify other power-law parameters richness_to_mass(20, slope=1.5, norm=1e14) 100000000000000.0 The [style=codeintext]ClusterEnsemble() class creates, modifies and tracks a Pandas DataFrame containing the properties and attributes of many galaxy clusters at once.","When given a new or updated cluster property, it calculates and updates all dependent cluster properties, treating each cluster (row) in the DataFrame as an individual object.","This makes it easy to calculate the $\\Sigma (R)$ and $\\Delta \\Sigma (R)$ weak lensing profiles for many different mass clusters at different redshifts, with a single command.","In contrast to using the [style=codeintext]SurfaceMassDensity() class discussed in Section REF , the user only needs to specify the cluster redshifts and either of the mass or richness.","If richness is supplied, then mass is calculated from it, assuming the form of Equation REF (which is customizable); if mass is specified instead, than the inverse relation is used to calculate richness.","In either case the changes are propagated to any dependent variables.","from clusterlensing import ClusterEnsemble z = [0.1,0.2,0.3] c = ClusterEnsemble(z) n200 = [20, 20, 20] c.n200 = n200 # display cluster dataframe c.dataframe z n200 m200 r200 c200 delta_c rs 0 0.1 20 2.700000e+13 0.612222 5.839934 12421.201995 0.104834 1 0.2 20 2.700000e+13 0.591082 5.644512 11480.644557 0.104718 2 0.3 20 2.700000e+13 0.569474 5.442457 10555.781440 0.104636 # specify mass directly c.m200 = [1e13, 1e14, 1e15] c.dataframe z n200 m200 r200 c200 delta_c rs 0 0.1 9.838141 1.000000e+13 0.439664 6.439529 15599.114356 0.068276 1 0.2 50.956400 1.000000e+14 0.914520 4.979102 8612.362538 0.183672 2 0.3 263.927382 1.000000e+15 1.898248 3.886853 4947.982895 0.488377 The above examples also demonstrate that cluster masses are converted to concentrations and to characteristic halo overdensities.","This assumes the default mass-concentration relation of the [style=codeintext]cDuttonMaccio() form, or the user can instead specify another of the relations by setting the keyword [style=codeintext]cm=\"Prada\" or [style=codeintext]cm=\"Duffy\", when the [style=codeintext]ClusterEnsemble() object is instantiated.","Cosmology can also be specified upon instantiation, by setting the [style=codeintext]cosmology keyword to be any [style=codeintext]astropy.cosmology object that has an [style=codeintext]h and a [style=codeintext]Om0 attribute.","If not specified explicitly, the default cosmological model used is [style=codeintext]astropy.cosmology.Planck13.","Here is an example of creating a [style=codeintext]ClusterEnsemble() object that uses the WMAP5 cosmology [25] and the [7] concentration: from astropy.cosmology import WMAP5 as cosmo c = ClusterEnsemble(z, cm=\"Duffy\", cosmology=cosmo) c.n200 = [20, 30, 40] c.dataframe z n200 m200 r200 c200 delta_c rs 0 0.1 20 2.700000e+13 0.599910 4.520029 6920.955951 0.132723 1 0.2 30 4.763120e+13 0.702040 4.136873 5678.897592 0.169703 2 0.3 40 7.125343e+13 0.775889 3.851601 4849.836498 0.201446 Instead of using the [style=codeintext]dataframe attribute, which retrieves the Pandas DataFrame object itself, it might be useful to use the [style=codeintext]show() method, which prints additional information to the screen, including assumptions of the mass-richness relation: c.show() Cluster Ensemble: z n200 m200 r200 c200 delta_c rs 0 0.1 20 2.700000e+13 0.599910 4.520029 6920.955951 0.132723 1 0.2 30 4.763120e+13 0.702040 4.136873 5678.897592 0.169703 2 0.3 40 7.125343e+13 0.775889 3.851601 4849.836498 0.201446 Mass-Richness Power Law: M200 = norm * (N200 / 20) ^ slope norm: 2.7e+13 solMass slope: 1.4 # update the mass-richness parameters # and show the resulting table c.massrich_norm = 3e13 c.massrich_slope = 1.5 c.show() Cluster Ensemble: z n200 m200 r200 c200 delta_c rs 0 0.1 20 3.000000e+13 0.621353 4.480202 6784.805438 0.138689 1 0.2 30 5.511352e+13 0.737028 4.086481 5526.615129 0.180358 2 0.3 40 8.485281e+13 0.822406 3.795500 4696.109606 0.216679 Mass-Richness Power Law: M200 = norm * (N200 / 20) ^ slope norm: 3e+13 solMass slope: 1.5 The last example also demonstrates how the slope or normalization of the mass-richness relation can be altered, and the changes propagate from richness through to mass and other variables.","Then all the ingredients are in place to calculate halo profiles by invoking the [style=codeintext]calcnfw() method, which interfaces to the [style=codeintext]sigmanfw() and [style=codeintext]deltasigmanfw() methods of the [style=codeintext]SurfaceMassDensity() class, and passes it the required inputs $\\lbrace r_{\\rm s}, \\rho _{\\rm crit}, \\delta _{\\rm c} \\rbrace $ for all the clusters behind the scenes.","The value of $\\rho _{\\rm crit}$ is calculated at every cluster redshift using the (default [style=codeintext]astropy.cosmology.Planck13 or user-specified) cosmological model.","The user must specify the desired radial bins [style=codeintext]rbins in Mpc.","import numpy as np # create some logarithmic bins: rmin, rmax = 0.1, 5.","# Mpc rbins = np.logspace(np.log10(rmin), np.log10(rmax), num = 8) # calculate the profiles: c.calc_nfw(rbins=rbins) # profiles now exist as attributes: c.sigma_nfw c.deltasigma_nfw Similar to the direct use of [style=codeintext]SurfaceMassDensity(), discussed in Section REF , the miscentered profiles can be calculated from the [style=codeintext]calcnfw() method, by supplying the optional [style=codeintext]offsets keyword with an array-like object of length equal to the number of clusters, where each element is the width of the offset distribution in Mpc ($\\sigma _{\\rm off}$ in Equation REF ).","c.calc_nfw(rbins=rbins, offsets=[0.3,0.3,0.3]) # the offset sigma profile is now: c.sigma_nfw Although [style=codeintext]SurfaceMassDensity() from the [style=codeintext]nfw.py module, and [style=codeintext]ClusterEnsemble().calcnfw() from the [style=codeintext]clusters.py module, are both capable of computing the same $\\Sigma (R)$ and $\\Delta \\Sigma (R)$ profiles, each require different forms of input which would make sense for different use cases.","For the studies in [15], [13], and [14], the authors wanted do the profile computations for many clusters at once, while varying the mass and the miscentering offset distribution during the process of fitting the model to the data.","What was known were the redshifts and mass proxies (cluster richness in [15] and [13], and a previous mass estimate in [14]), and mass-concentration relations from the literature, so the [style=codeintext]ClusterEnsemble() framework made sense.","However, if someone wanted to simply calculate the NFW profiles according to the [44] formulation, then they might prefer to use [style=codeintext]SurfaceMassDensity() as a tool to get profiles directly from the NFW and cosmological parameters $r_{\\rm s}$ , $\\delta _{\\rm c}$ , and $\\rho _{\\rm crit}(z)$ .","As an example use case, we take the Canada-France-Hawaii Telescope Lensing Survey [17], [10] public galaxy cluster catalog, which is available on Zenodohttp://dx.doi.org/10.5281/zenodo.51291 [11].","This dataset was previously explored using a pre-release version of the [style=codeintext]cluster-lensing software in [13], [15].","The W1 field of this survey contains 10,745 galaxy cluster candidates in the redshift range $0.2 \\le z \\le 0.9$ : import numpy as np data = np.loadtxt(\"Clusters_W1.dat\") data.shape (10745, 5) data[0:4, :] # print first 4 clusters array([[ 34.8023, -7.01005, 0.3, 4.435, 10.","], [ 34.9425, -7.38996, 0.5, 4.545, 21.","], [ 34.8651, -6.69449, 0.5, 3.858, 6.","], [ 34.6224, -7.32768, 0.5, 3.619, 8.]])","redshift = data[:, 2] sig = data[:, 3] richness = data[:, 4] We select a subset of the lower redshift clusters that were detected at high significance.","Then we import [style=codeintext]clusterlensing to create a dataframe of the cluster properties, of which we just print the first several, and calculate the NFW profiles.","# select a subset here = (sig > 15) & (redshift < 0.5) sig[here].shape (15,) z = redshift[here] n200 = richness[here] import clusterlensing c = clusterlensing.ClusterEnsemble(z) c.n200 = n200 c.dataframe.head() z n200 m200 r200 c200 delta_c rs 0 0.4 181 5.897552e+14 1.531367 3.966101 5173.016417 0.386114 1 0.3 420 1.916332e+15 2.357815 3.658237 4332.615805 0.644522 2 0.4 176 5.670737e+14 1.511478 3.980218 5213.746469 0.379747 3 0.3 113 3.049521e+14 1.277703 4.341779 6324.420397 0.294281 4 0.4 162 5.049435e+14 1.454129 4.022285 5336.272412 0.361518 rbins = np.logspace(np.log10(0.1), np.log10(10.0), num=20) c.calc_nfw(rbins) Next we import the [style=codeintext]matplotlib and [style=codeintext]seaborn libraries and configure some settings to make our plots more readable.","The first plot we create with the commands below presents the $\\Sigma (R)$ profiles for every one of these 15 clusters, and is given in Figure REF .","import matplotlib.pyplot as plt import seaborn as sns; sns.set() import matplotlib matplotlib.rcParams[\"axes.labelsize\"] = 20 matplotlib.rcParams[\"legend.fontsize\"] = 20 # strings for plots raxis = \"$R\\ [\\mathrm{Mpc}]$\" sgma = \"$\\Sigma(R)$\" sgmaoff = \"$\\Sigma^\\mathrm{off}(R)$\" delta = \"$\\Delta$\" sgmaunits = \" $[M_{\\odot}\\ \\mathrm{pc}^{-2}]$\" # order from high to low richness order = c.n200.argsort()[::-1] for s, n in zip(c.sigma_nfw[order], c.n200[order]): plt.plot(rbins, s, label=str(int(n))) plt.xscale(\"log\") plt.legend(fontsize=10) plt.ylabel(sgma+sgmaunits) plt.xlabel(raxis) plt.tight_layout() plt.savefig(\"f1.eps\") plt.close() # output is Figure 1 Figure: Surface mass density profiles Σ(R)\\Sigma (R) for all 15 clusters used in this example.","These are the most significant (S/N >15>15) clusters detected at low redshifts (z<0.5z < 0.5) in the W1 field of CFHTLenS.","See the text for links to download this public dataset.","The legend gives the richness values estimated in corresponding to each of these clusters, which are assumed to scale with mass.","They are listed from highest to lowest richness, in the same order as the curves.If we had made a stacked measurement of the shear or magnification profile of these clusters, then we would want to know what the average profile of the stack looks like.","Since we already have the individual profiles, we just need to calculate the mean across the 0th axis of the [style=codeintext]sigmanfw and [style=codeintext]deltasigmanfw attribute arrays.","The plot of these average profiles is given in Figure REF .","sigma = c.sigma_nfw.mean(axis=0) dsigma = c.deltasigma_nfw.mean(axis=0) plt.plot(rbins, sigma, label=sgma) plt.plot(rbins, dsigma, \"--\", label=delta+sgma) plt.legend() plt.ylim([0., 1400.])","plt.xscale(\"log\") plt.xlabel(raxis) plt.ylabel(sgmaunits) plt.tight_layout() plt.savefig(\"f2.eps\") plt.close() # output is Figure 2 Figure: The average of all the surface mass density profiles Σ(R)\\Sigma (R) for each of the clusters shown in Figure is given in blue.","The green curve is the average of all the individual differential mass density profiles ΔΣ(R)\\Delta \\Sigma (R).","These curves assume clusters are perfectly centered on their NFW halos.Finally, we may want to investigate whether cluster miscentering has a significant effect on our sample.","We would calculate the miscentered profiles, given in Figure REF , which could be compared to the centered profiles in Figure REF to see which is a better fit to our measurement.","Below we will assume that the miscentering offset distribution peaks at 0.1 Mpc.","offsets = np.ones(c.z.shape[0]) * 0.1 c.calc_nfw(rbins, offsets=offsets) sigma_offset = c.sigma_nfw.mean(axis=0) dsigma_offset = c.deltasigma_nfw.mean(axis=0) plt.plot(rbins, sigma_offset, label=sgmaoff) plt.plot(rbins, dsigma_offset, \"--\", label=delta+sgmaoff) plt.legend() plt.ylim([0., 1400.])","plt.xscale(\"log\") plt.xlabel(raxis) plt.ylabel(sgmaunits) plt.tight_layout() plt.savefig(\"f3.eps\") plt.close() # output is Figure 3 Figure: Same as Figure but for miscentered profiles.","Cluster centroid offsets are assumed to follow a Rayleigh probability distribution (Equation , discussed in Section ), which is convolved with the perfectly centered profiles to achieve this result.The above example shows a simple application of [style=codeintext]cluster-lensing to a real dataset – a subset of the CFHTLenS cluster catalog.","For this example we kept the customizations to a minimum, but as shown in Sections REF , the user can alter the parameters in the power-law relation used to convert richness to mass, choose the form of the mass-concentration relation assumed for the NFW profiles, and specify a particular background cosmology.","When fitting a model produced by [style=codeintext]cluster-lensing to a measurement, one could iterate through parameters in this space by setting various attributes of the [style=codeintext]ClusterEnsemble() object [13], [15]."],["Relation to Existing Code","The [style=codeintext]cluster-lensing project offers some unique capabilities over other publicly-available software, most notably the cluster miscentering calculations.","Here we attempt to compare the software presented in this work with other open source tools that we are aware of, and show how [style=codeintext]cluster-lensing fits into the larger ecosystem of astronomical software.","[style=codeintext]Colossus is a Python package aimed at cosmology, halo, and large-scale structure calculations [3].","It was used in work by [4] and is made available under the MIT licensehttp://www.benediktdiemer.com/code/.","Much of the functionality of [style=codeintext]cluster-lensing appears to overlap with [style=codeintext]Colossus, including mass-concentration relations (although [style=codeintext]Colossus has the advantage of containing many more relations from the literature) and NFW surface mass density profiles.","However, [style=codeintext]cluster-lensing also provides the miscentered halo calculations, which are are lacking from [style=codeintext]Colossus.","While [3] has chosen to implement basic cosmological calculations from scratch, [style=codeintext]cluster-lensing instead relies on external modules supplied by [style=codeintext]astropy.","The only dependencies claimed by [style=codeintext]Colossus are [style=codeintext]numpy [42] and [style=codeintext]scipy [23], whereas [style=codeintext]cluster-lensing additionally requires [style=codeintext]astropy [1] and [style=codeintext]pandas [28].","Fewer dependencies might be seen as a positive feature of [style=codeintext]Colossus; on the other hand, [style=codeintext]astropy could be viewed as possibly a more robust source for standard astronomical and cosmological calculations, since it is maintained by a large community of developers.","Another related set of code is provided by Jörg Dietrich's NFW routines, archived on Zenodo [5], and available on GitHubhttps://github.com/joergdietrich/NFW.","These [style=codeintext]Python modules calculate NFW profiles for $\\Sigma (R)$ and $\\Delta \\Sigma (R)$ , as well as the 3-dimensional density profiles and total mass and projected mass inside a given radius.","[style=codeintext]cluster-lensing goes beyond the functionality of [5] by supplying means for calculating cluster miscentering, and having a built-in framework for handling many halos at once.","For [5], the user must provide the halo concentration (along with mass and redshift) to the [style=codeintext]NFW() class, but additional routines are available for converting mass to concentration, including [7] and another by [6] (a partial overlap with the mass-concentration relations provided by [style=codeintext]cluster-lensing).","[5] depends on [style=codeintext]astropy for cosmological calculations and units, similar to [style=codeintext]cluster-lensing, as well as the [style=codeintext]numpy and [style=codeintext]scipy packages."],["Future Development","Some of the future plans for [style=codeintext]cluster-lensing include adding support for different density profiles.","Currently only the NFW model is provided, and alternative mass density models would make the package more complete and useful.","The first priority will be inclusion of the Einasto profile [9], and later possibilities may include the generalized-NFW [46].","The default cosmology is currently that of [34], but should be updated to [35], since this is now available as [style=codeintext]astropy.cosmology.Planck15 (the user can currently specify this cosmology, it is just not the default).","When surface mass density profiles have to be calculated many times for many clusters, as is the case when iterating over parameters in the process of fitting a model, the processing time can become lengthy.","This issue is most pronounced for calculation of miscentered profiles, which require the convolution laid out in Equations REF and REF .","One major improvement to [style=codeintext]cluster-lensing will be the option to use parallel-processing in these computations.","The likely structure of this parallelism will be to divide the halos in a [style=codeintext]ClusterEnsemble() catalog object among the parallel threads, which will calculate the profiles for each of their assigned clusters.","All of these future developments are currently listed as issues on the GitHub repository.","This GitHub Issue trackerhttps://github.com/jesford/cluster-lensing/issues will continue to serve as the central place for listing future improvements and feature requests.","Users and potential-users alike are encouraged to submit ideas and requests through that URL."],["Summary","In this work we presented [style=codeintext]cluster-lensing, a pure-[style=codeintext]Python package for calculating galaxy cluster profiles and properties.","We described and gave worked examples of all the functionality currently available, including mass-concentration and mass-richness scaling relations, and the surface mass density profiles $\\Sigma (R)$ and $\\Delta \\Sigma (R)$ , which are relevant for gravitational lensing.","The latter density profiles are not cluster-specific, but apply to any mass halo that can be approximated by the NFW prescription.","The structure of [style=codeintext]cluster-lensing is ideal for calculating properties and profiles for many galaxy clusters at once.","This “composite-halo” approach [15], is especially useful for fitting models to a stacked sample of clusters that span a range of mass and/or redshift.","Compared to existing code, [style=codeintext]cluster-lensing stands out by seemingly being the only publicly-available software for calculating miscentered halo profiles.","Miscentering is a problem of great relevance for stacked weak lensing studies of galaxy clusters, where halo centers are imperfectly estimated from observational data or simply not well defined (as is the case for individual non-spherical halos – for example in merging systems).","The resulting offsets between the assumed and real centers change the shape of the measured shear or magnification profile and need to be accounted for in the modeling.","[style=codeintext]cluster-lensing is released under the MIT license, and archived on Zenodo [12].","It being developed in a public repository on GitHub: http://github.com/jesford/cluster-lensing/.","Contributions to the code can be made by submitting a pull request to the repository, and we welcome feedback, suggestions, and feature requests through GitHub issues, or by emailing the author.","Full documentation (including much of the content of this paper), as well as installation instructions and examples, are available in the online documentation, at http://jesford.github.io/cluster-lensing/.","If [style=codeintext]cluster-lensing is used in a research project, the authors would appreciate citations to the code [12] and this paper."],["Acknowledgements","The authors are grateful for funding from the Washington Research Foundation Fund for Innovation in Data-Intensive Discovery and the Moore/Sloan Data Science Environments Project at the University of Washington.","This project would not have been possible without packages available in Python's open scientific ecosystem, including NumPy [42], SciPy [23], Pandas [28], matplotlib [20], IPython [33], AstroPy [1], Seaborn [43], and related tools."]],"string":"[\n [\n \"Cluster-lensing: A Python Package for Galaxy Clusters & Miscentering\"\n ],\n [\n \"Abstract We describe a new open source package for calculating properties of galaxy clusters, including NFW halo profiles with and without the effects of cluster miscentering.\",\n \"This pure-Python package, cluster-lensing, provides well-documented and easy-to-use classes and functions for calculating cluster scaling relations, including mass-richness and mass-concentration relations from the literature, as well as the surface mass density $\\\\Sigma(R)$ and differential surface mass density $\\\\Delta\\\\Sigma(R)$ profiles, probed by weak lensing magnification and shear.\",\n \"Galaxy cluster miscentering is especially a concern for stacked weak lensing shear studies of galaxy clusters, where offsets between the assumed and the true underlying matter distribution can lead to a significant bias in the mass estimates if not accounted for.\",\n \"This software has been developed and released in a public GitHub repository, and is licensed under the permissive MIT license.\",\n \"The cluster-lensing package is archived on Zenodo (Ford 2016).\",\n \"Full documentation, source code, and installation instructions are available at http://jesford.github.io/cluster-lensing/.\"\n ],\n [\n \"Introduction\",\n \"Clusters of galaxies are the largest gravitationally collapsed structures to have formed in the history of the universe.\",\n \"As such, they are interesting both from a cosmological as well as an astrophysics perspective.\",\n \"In the former case, the galaxy cluster number density as a function of mass (the cluster mass function) is a probe of cosmological parameters including the fractional matter density $\\\\Omega _{\\\\rm m}$ and the normalization of the matter power spectrum $\\\\sigma _8$ .\",\n \"Astrophysically, the deep potential wells of galaxy clusters are environments useful for testing theories of general relativity, galaxy evolution, and gas and plasma physics, among other things [40].\",\n \"The common thread among these diverse investigations is the requisite knowledge of the mass of the galaxy cluster, which is largely composed of its invisible dark matter halo.\",\n \"Although many techniques exist for estimating the total mass of these systems, weak lensing has emerged as somewhat of a gold standard, since it is sensitive to the mass itself, and not to the dynamical state or other biased tracers of the underlying mass.\",\n \"Scaling relations between weak lensing derived masses, and other observables, including richness, X-ray luminosity and temperature, for examples, are typically calibrated from large surveys and extrapolated to clusters for which gravitational lensing measurements are impossible or unreliable.\",\n \"Since weak lensing masses are often considered the “true” masses, against which other estimates are compared [26], [41], [19], it is paramount that cluster masses from weak lensing modeling are as unbiased as possible.\",\n \"For stacked weak lensing measurements of galaxy clusters, an important source of bias in fitting a mass model is the inclusion of the effect of miscentering offsets.\",\n \"Miscentering occurs when the center of the mass distribution – the dark matter halo – does not perfectly coincide with the assumed center around which tangential shear (or magnification) profiles are being measured.\",\n \"Candidate centers for galaxy clusters are necessarily chosen from observational proxies, and often include a single galaxy, such as the brightest or most massive member, or the centroid of some extended quantity like the peak of X-ray emission or average of galaxy positions [16].\",\n \"The particular choice of center may be offset from the true center due to interesting physical processes such as recent mergers and cluster evolution, or simply due to misidentification of the proxy of interest [22].\",\n \"The miscentering effect on the stacked weak lensing profile can be included in a proper modeling of the measurement, as done in [22], [27], [32], [16], [38], [31], [13], [15], [39].\",\n \"The inclusion of this effect commonly assumes a form for the distribution of offsets, such as a Rayleigh distribution in radius (which represents a 2D Gaussian in the plane of the sky).\",\n \"This distribution is convolved with the standard (centered) halo profile to obtain the miscentered version.\",\n \"Software for calculating these miscentered weak lensing profiles was developed in order to produce results in [13], [15], and has recently been publicly released to the astronomical community [12].https://github.com/jesford/cluster-lensing When many different gravitational lenses are stacked, as is often necessary to increase signal-to-noise for weak lensing measurements, care must be taken in the interpretation of the average signal.\",\n \"The issue here is that the (differential) surface mass density is not a linear function of the mass, so the average of many stacked profiles does not directly yield the average mass of the lens sample.\",\n \"Care must be taken to consider the underlying distribution of cluster masses as well as the redshifts of lenses and sources, all of which affect the amplitude of the measured lensing profile.\",\n \"One approach to this is to use a so-called composite-halo approach [18], [14], [13], [15], [39], where profiles are calculated for all individual lens objects and then averaged together to create a model that can be fit to the measurement.\",\n \"The [style=codeintext]ClusterEnsemble() class discussed in Section REF is designed with this approach in mind.\",\n \"A popular model for the dark matter distribution in a gravitationally collapsed halo, such as a galaxy cluster, is the Navarro, Frenk, and White (NFW) model.\",\n \"This density profile (given in Equation REF below) was determined from numerical simulations that included the dissipationless collapse of density fluctuations under gravity [30].\",\n \"The simpler Singular Isothermal Sphere density model, which only has one free parameter in contrast to the two for NFW, does not tend to fit the inner profiles of halos well and is also unphysical in that the total mass diverges [37].\",\n \"Other models such as the generalized-NFW and the Einasto profile tend to better describe the full radial distribution of dark matter in halos, at the expense of adding a third parameter to characterize the inner slope of the density profiles [8].\",\n \"In the software package presented in this work we only include the standard 2-parameter NFW model, but future work should make alternative models available as well.\"\n ],\n [\n \"Description of the Code\",\n \"In this section we demonstrate each of the individual modules available in the [style=codeintext]cluster-lensing package.\",\n \"In Section REF we describe a class for calculating surface mass density profiles directly from NFW and cosmological parameters.\",\n \"Next we outline the functions available for mass-concentration relations in Section REF .\",\n \"Then in Section REF we present the class [style=codeintext]ClusterEnsemble(), and its related functions, which tie together the previously discussed functionality into a framework for easily manipulating and producing profiles for multiple galaxy clusters at once, from common observational quantities.\",\n \"Much of the content of this section comes directly from the online documentation.http://jesford.github.io/cluster-lensing/ Throughout the modules, dimensionful quantities are labelled as such by means of the [style=codeintext]astropy.units package.\"\n ],\n [\n \"The [style=codeintext]nfw.py module contains a single class called [style=codeintext]SurfaceMassDensity(), which computes the surface mass density $\\\\Sigma (R)$ and the differential surface mass density $\\\\Delta \\\\Sigma (R)$ using the class methods [style=codeintext]sigmanfw() and [style=codeintext]deltasigmanfw(), respectively.\",\n \"These profiles are calculated according to the analytical formulas first derived by [44], assuming the spherical NFW model, and can be applied to any dark matter halo: this module is not specific to galaxy clusters.\",\n \"The 3-dimensional density profile of an NFW halo is given by $\\\\rho (r) = \\\\frac{\\\\delta _{\\\\rm c} \\\\rho _{\\\\rm crit}}{(r/r_{\\\\rm s})(1+r/r_{\\\\rm s})^2},$ where $r_{\\\\rm s}$ is the cluster scale radius, $\\\\delta _{\\\\rm c}$ is the characteristic halo overdensity, and $\\\\rho _{\\\\rm crit} = \\\\rho _{\\\\rm crit}(z)$ is the critical energy density of the universe at the lens redshift.\",\n \"These three parametersor, in the case of calculating multiple NFW halos at once, three array-like objects representing each of these parameters must be specified when instantiating the class [style=codeintext]SurfaceMassDensity(), via the arguments [style=codeintext]rs, [style=codeintext]deltac, and [style=codeintext]rhocrit, respectively.\",\n \"The units on [style=codeintext]rs are assumed to be Mpc, [style=codeintext]deltac is dimensionless, and [style=codeintext]rhocrit is in $M_{\\\\odot }{\\\\rm Mpc}^{-1}{\\\\rm pc}^{-2}$ , although the actual inclusion of the [style=codeintext]astropy.units on these variables is optional.\",\n \"The user will probably also want to choose the radial bins for the calculation, which are specified via the keyword argument [style=codeintext]rbins, in Mpc.\",\n \"The surface mass density is the integral along the line-of-sight of the 3-dimensional density: $\\\\Sigma (R) = 2 \\\\int _0^{\\\\infty } \\\\rho (R,y) {\\\\rm d}y.$ Here $R$ is the projected radial distance (in the plane of the sky).\",\n \"We can adopt the dimensionless radius $x \\\\equiv R/r_{\\\\rm s}$ and, following from [44], show that: $\\\\Sigma (x) = 2 r_{\\\\rm s} \\\\delta _{\\\\rm c} \\\\rho _{\\\\rm crit} f(x),$ where $f(x) = $ ${\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\frac{1}{x^2 - 1} \\\\left( 1 - \\\\ln { \\\\left[ \\\\frac{1}{x} + \\\\sqrt{ \\\\frac{1}{x^2} - 1} \\\\right]} / \\\\sqrt{1 - x^2} \\\\right), & \\\\text{for } x < 1; \\\\\\\\1/3, & \\\\text{for } x = 1; \\\\\\\\\\\\frac{1}{x^2 - 1} \\\\left( 1 - \\\\arccos {(1/x)} / \\\\sqrt{x^2 - 1} \\\\right), & \\\\text{for } x > 1.\\\\end{array}\\\\right.\",\n \"}$ The differential surface mass density probed by shear is calculated from the definition $\\\\Delta \\\\Sigma (x) \\\\equiv \\\\overline{\\\\Sigma }( 1.$ Running [style=codeintext]sigmanfw() or [style=codeintext]deltasigmanfw(), with only a specification of halo properties [style=codeintext]rs, [style=codeintext]deltac, [style=codeintext]rhocrit, and radial bins [style=codeintext]rbins, will lead to the calculation of halo profiles according to Equations REF and REF outlined above.\",\n \"from clusterlensing import SurfaceMassDensity rbins = [0.1, 0.5, 1.0, 2.0, 4.0] # Mpc smd = SurfaceMassDensity(rs=[0.1], rho_crit=[0.2], delta_c=[9700.\",\n \"], rbins=rbins) sigma = smd.sigma_nfw() # surface mass density with default units sigma[0] # surface mass density with no units sigma[0].value array([ 129.33333333, 11.64751032, 3.33992059, 0.89839601, 0.23327149]) These are the standard centered NFW profiles, under the assumption that the peak of the halo density distribution perfectly coincides with the identified halo center.\",\n \"This may not be a good assumption, however, and the user can instead run these calculations for miscentered halos by specifying the optional input parameter [style=codeintext]offsets.\",\n \"This parameter sets the width of a distribution of centroid offsets, assuming a 2-dimensional Gaussian distribution on the sky.\",\n \"This offset distribution is equivalent to, and implemented in code as, a uniform distribution in angle and a Rayleigh probability distribution in radius: $P(R_{\\\\mathrm {off}})=\\\\frac{R_{\\\\mathrm {off}}}{\\\\sigma _{\\\\mathrm {off}}^2}\\\\ \\\\mathrm {exp}\\\\bigg [-\\\\frac{1}{2}\\\\bigg (\\\\frac{R_{\\\\mathrm {off}}}{\\\\sigma _{\\\\mathrm {off}}}\\\\bigg )^2\\\\ \\\\bigg ].$ The parameter [style=codeintext]offsets is equivalent to $\\\\sigma _{\\\\rm off}$ in this equation.\",\n \"from clusterlensing import SurfaceMassDensity rbins = [0.1, 0.5, 1.0, 2.0, 4.0] # single miscentered halo profile smd = SurfaceMassDensity(rs=[0.1], rho_crit=[0.2], delta_c=[9700.\",\n \"], rbins=rbins, offsets=[0.3]) sigma = smd.sigma_nfw() sigma[0] # example calculating multiple profiles smd = SurfaceMassDensity(rs=[0.1,0.2,0.2], rho_crit=[0.2,0.2,0.2], delta_c=[9700,9700,9000], rbins=rbins, offsets=[0.3,0.3,0.3]) sigma = smd.sigma_nfw() sigma The miscentered surface mass density profiles are given by the centered profiles (Equations REF and REF ), convolved with the offset distribution (Equation REF ).\",\n \"We follow the offset halo formalism first written down by [45], and applied to cluster miscentering by, e.g.\",\n \"[22], [16], [13], [15], [39].\",\n \"Specifically, we calculate the offset surface mass density $\\\\Sigma ^{\\\\rm off}$ as follows: $\\\\Sigma ^{\\\\rm off}(R) = \\\\int _{0}^{\\\\infty } \\\\Sigma (R | R_{\\\\rm off})\\\\ P(R_{\\\\rm off})\\\\ {\\\\rm d}R_{\\\\rm off}$ $\\\\Sigma (R|R_{\\\\mathrm {off}})=\\\\frac{1}{2\\\\pi }\\\\int _{0}^{2\\\\pi }\\\\Sigma (r) \\\\mathrm {d}\\\\theta $ Here $r = \\\\sqrt{R^2+R_{\\\\mathrm {off}}^2-2RR_{\\\\mathrm {off}}\\\\cos (\\\\theta )}$ and $\\\\theta $ is the azimuthal angle [45].\",\n \"The $\\\\Delta \\\\Sigma ^{\\\\rm off}$ profile is calculated from $\\\\Sigma ^{\\\\rm off}$ , in analogy with Equations REF and REF .\",\n \"from clusterlensing import SurfaceMassDensity rbins = [0.1, 0.5, 1.0, 2.0, 4.0] # perfectly centered DeltaSigma profile smd = SurfaceMassDensity(rs=[0.1], rho_crit=[0.2], delta_c=[9700.\",\n \"], rbins=rbins) deltasigma = smd.deltasigma_nfw() deltasigma[0] # miscentered DeltaSigma profile smd = SurfaceMassDensity(rs=[0.1], rho_crit=[0.2], delta_c=[9700.\",\n \"], rbins=rbins, offsets=[0.3]) deltasigma = smd.deltasigma_nfw() deltasigma[0] \"\n ],\n [\n \"The [style=codeintext]cofm.py module currently contains three functions, each of which calculates halo concentration from mass, redshift, and cosmology, according to a prescription given in the literature.\",\n \"These functions are [style=codeintext]cDuttonMaccio() [8], [style=codeintext]cDuffy() [7], and [style=codeintext]cPrada() [36].\",\n \"Halo mass-concentration relations are an area of active research, and there have been discrepancies between results from different observations and simulations, and disagreement surrounding the best choice of model [8], [24].\",\n \"We do not aim to join this discussion here, but focus on outlining the functionality provided by the [style=codeintext]cluster-lensing package, for calculating these different concentration values.\",\n \"All three functions require two input parameters (scalars or array-like inputs), which are the halo redshift(s) [style=codeintext]z and the halo mass(es) [style=codeintext]m. Specifically, the latter is assumed to correspond to the $M_{200}$ mass definition, in units of solar masses.\",\n \"$M_{200}$ is the mass interior to a sphere of radius $r_{200}$ , within which the average density is $200\\\\rho _{\\\\rm crit}(z)$ .\",\n \"The default cosmology used is from the measurements by the [34], which is imported from the module [style=codeintext]astropy.cosmology.Planck13.\",\n \"However, the user can specify alternative cosmological parameters.\",\n \"For calculating concentration according to either the [7] or the [8] prescription, the only cosmological parameter required is the Hubble parameter, which can be passed into [style=codeintext]cDuffy() or [style=codeintext]cDuttonMaccio() as the keyword argument [style=codeintext]h. For the [36] concentration, the user would want to specify [style=codeintext]OmM and [style=codeintext]OmL (the fractional energy densities of matter and the cosmological constant) in addition to [style=codeintext]h, in the call to [style=codeintext]cPrada().\",\n \"The [style=codeintext]cDuttonMaccio() calculation of concentration is done according to the power-law $\\\\log _{10} c_{200} = a + b \\\\log _{10}(M_{200} / [10^{12} h^{-1} M_{\\\\odot }]),$ where $a = 0.52 + 0.385\\\\ {\\\\rm exp}[-0.617\\\\ z^{1.21}],$ $b = -0.101 + 0.206 z.$ The above three equations map to Equations 7, 11, and 10, respectively in [8].\",\n \"The values in these expressions were determined from simulations of halos between $0 < z < 5$ , spanning over 5 orders of magnitude in mass, and were shown to match observational measurements of low-redshift galaxies and clusters [8].\",\n \"This concentration-mass relation is the default one used by the [style=codeintext]clusters.py module, discussed in Section REF .\",\n \"from clusterlensing import cofm # single 10**14 Msun halo at z=1 cofm.c_DuttonMaccio(0.1, 1e14) array([ 5.13397936]) # example with multiple halos cofm.c_DuttonMaccio([0.1, 0.5], [1e14, 1e15]) array([ 5.13397936, 3.67907305]) The concentration calculation in [style=codeintext]cDuffy() is $c_{200} = A\\\\ (M_{200} / M_{\\\\rm pivot})^B \\\\ (1 + z)^C,$ where $\\\\lbrace A, B, C\\\\rbrace = \\\\lbrace 5.71, -0.084, -0.47\\\\rbrace ,$ $M_{\\\\rm pivot} = 2 \\\\times 10^{12}\\\\ h^{-1} M_{\\\\odot }.$ Equation REF above corresponds to Equation 4 in [7].\",\n \"The values for $A$ , $B$ , and $C$ can be found in Table 1 of that work, where they are specific to the “full” (relaxed and unrelaxed) sample of simulated NFW halos, spanning the redshift range $0 < z < 2$ .\",\n \"$M_{\\\\rm pivot}$ can be found in the caption of their Table 1 as well.\",\n \"One caveat with this relation is that the cosmology used in creating the [7] simulations was that of the now outdated WMAP5 experiment [25].\",\n \"from clusterlensing import cofm # default cosmology (h=0.6777) cofm.c_Duffy([0.1, 0.5], [1e14, 1e15]) array([ 4.06126115, 2.89302767]) # with h=1 cofm.c_Duffy([0.1, 0.5], [1e14, 1e15], h=1) array([ 3.93068341, 2.80001099]) The [style=codeintext]cPrada() concentration calculation is much more complex, and written in terms of $\\\\sigma (M_{200}, x_{\\\\rm p})$ , the rms fluctuation of the density field.\",\n \"The [36] halo concentration is given bywe use the subscript “p” to distinguish some variables in the equations from [36] from those in the current work $\\\\begin{split}c_{200} = 2.881 B_0 (x_{\\\\rm p}) \\\\bigg [ \\\\left( \\\\frac{B_1(x_{\\\\rm p}) \\\\sigma (M_{200},x_{\\\\rm p})}{1.257} \\\\right)^{1.022} + 1 \\\\bigg ] \\\\\\\\\\\\times \\\\ {\\\\rm exp}\\\\left( \\\\frac{0.06}{[B_1(x_{\\\\rm p}) \\\\sigma (M_{200},x_{\\\\rm p}) ]^2} \\\\right).\\\\end{split}$ The cosmology and redshift dependence is encoded by the variable $x_{\\\\rm p}$ , which is $x_{\\\\rm p} = \\\\left( \\\\frac{\\\\Omega _{\\\\Lambda , 0}}{\\\\Omega _{{\\\\rm m}, 0}} \\\\right)^{1/3} (1 + z)^{-1}.$ The functions within Equation REF are as follows: $\\\\sigma (M_{200},x_{\\\\rm p}) = D(x_{\\\\rm p}) \\\\frac{16.9 y_{\\\\rm p}^{0.41}}{1 + 1.102 y_{\\\\rm p}^{0.2} + 6.22 y_{\\\\rm p}^{0.333}}$ $y_{\\\\rm p} \\\\equiv \\\\frac{10^{12} h^{-1} M_{\\\\odot }}{M_{200}}$ $D(x_{\\\\rm p}) = \\\\frac{5}{2} \\\\left( \\\\frac{\\\\Omega _{{\\\\rm m}, 0}}{\\\\Omega _{\\\\Lambda , 0}} \\\\right)^{1/3} \\\\frac{\\\\sqrt{1 + x_{\\\\rm p}^3}}{x_{\\\\rm p}^{3/2}} \\\\int _0^{x_{\\\\rm p}} \\\\frac{x^{3/2} {\\\\rm d}x}{(1 + x^3)^{3/2}}$ $B_0(x_{\\\\rm p}) = \\\\frac{c_{\\\\rm min}(x_{\\\\rm p})}{c_{\\\\rm min}(1.393)}$ $B_1(x_{\\\\rm p}) = \\\\frac{\\\\sigma ^{-1}_{\\\\rm min}(x_{\\\\rm p})}{\\\\sigma ^{-1}_{\\\\rm min}(1.393)}$ $c_{\\\\rm min}(x_{\\\\rm p}) = 3.681 + 1.352 \\\\bigg [ \\\\frac{1}{\\\\pi } \\\\arctan [6.948 (x_{\\\\rm p} - 0.424)] + \\\\frac{1}{2} \\\\bigg ]$ $\\\\sigma ^{-1}_{\\\\rm min}(x_{\\\\rm p}) = 1.047 + 0.599 \\\\bigg [ \\\\frac{1}{\\\\pi } \\\\arctan [7.386 (x_{\\\\rm p} - 0.526)] + \\\\frac{1}{2} \\\\bigg ]$ In order of appearance above, beginning with our Equation REF , these equations correspond to Equations 14-17, 13, 23a, 23b, 12, 18a, 18b, 19, 20 in [36].\",\n \"The numerical values in these equations were obtained empirically from the simulations described in that work.\",\n \"from clusterlensing import cofm cofm.c_Prada([0.1, 0.5], [1e14, 1e15]) array([ 5.06733941, 5.99897362]) cofm.c_Prada([0.1, 0.1, 0.1], [1e13, 1e14, 1e15]) array([ 5.71130928, 5.06733941, 5.30163572]) The last code example demonstrates the controversial feature of the [36] mass-concentration relation – an upturn in concentration values for the highest mass halos.\",\n \"This is in opposition to the canonical view that higher mass halos have lower concentrations [29], [30], [21], [2].\"\n ],\n [\n \"The [style=codeintext]clusters.py module is designed to provide a catalog-level tool for calculating, tracking, and updating galaxy cluster properties and profiles, through structuring data from multiple clusters as an updatable Pandas Dataframe, and providing an intelligent interface to the other modules discussed in Sections REF and REF .\",\n \"This module contains a single class [style=codeintext]ClusterEnsemble(), as well as three functions, [style=codeintext]masstorichness(), [style=codeintext]richnesstomass(), and [style=codeintext]calcdeltac().\",\n \"The function [style=codeintext]calcdeltac() takes a single input parameter, the cluster concentration [style=codeintext]c200 (e.g.\",\n \"as calculated by one of the functions in [style=codeintext]cofm.py), and returns the characteristic halo overdensity: $\\\\delta _{\\\\rm c} = \\\\left( \\\\frac{200}{3} \\\\right) \\\\frac{c_{200}^3}{\\\\ln (1 + c_{200}) - c_{200}/(1 + c_{200})}.$ Both input and output are dimensionless here.\",\n \"For example, to convert a concentration value of $c_{200} = 5$ to $\\\\delta _{\\\\rm c}$ , you could do: from clusterlensing.clusters import calc_delta_c calc_delta_c(5) 8694.8101906193315 The pair of functions [style=codeintext]masstorichness() and [style=codeintext]richnesstomass(), as their names imply, perform conversions between cluster mass and richness.\",\n \"The only required input parameter to [style=codeintext]masstorichness() is the [style=codeintext]mass, and likewise the only required input to [style=codeintext]richnesstomass() is [style=codeintext]richness.\",\n \"The calculations assume a power-law form for the relationship between these variables: $M_{200} = M_0 \\\\left( \\\\frac{N_{200}}{20} \\\\right) ^ \\\\beta .$ Here $M_0$ is the normalization, which defaults to $2.7 \\\\times 10^{13}$ , but can be changed in the call to either function by setting the [style=codeintext]norm keyword argument.\",\n \"The power-law slope $\\\\beta = 1.4$ by default, but can be set by specifying the optional [style=codeintext]slope input parameter.\",\n \"When these functions are invoked by the [style=codeintext]ClusterEnsemble() class, they are applied to the particular mass definition $M_{200}$ , and assume units of $M_{\\\\odot }$ .\",\n \"However the functions themselves do not assume a mass definition or unit, and can be generalized to any parameter (or type of richness) that has a power-law relationship with mass.\",\n \"from clusterlensing.clusters import \\\\ mass_to_richness, richness_to_mass richness_to_mass(50) 97382243648736.9 mass_to_richness(97382243648736.9) 50.0 # specify other power-law parameters richness_to_mass(20, slope=1.5, norm=1e14) 100000000000000.0 The [style=codeintext]ClusterEnsemble() class creates, modifies and tracks a Pandas DataFrame containing the properties and attributes of many galaxy clusters at once.\",\n \"When given a new or updated cluster property, it calculates and updates all dependent cluster properties, treating each cluster (row) in the DataFrame as an individual object.\",\n \"This makes it easy to calculate the $\\\\Sigma (R)$ and $\\\\Delta \\\\Sigma (R)$ weak lensing profiles for many different mass clusters at different redshifts, with a single command.\",\n \"In contrast to using the [style=codeintext]SurfaceMassDensity() class discussed in Section REF , the user only needs to specify the cluster redshifts and either of the mass or richness.\",\n \"If richness is supplied, then mass is calculated from it, assuming the form of Equation REF (which is customizable); if mass is specified instead, than the inverse relation is used to calculate richness.\",\n \"In either case the changes are propagated to any dependent variables.\",\n \"from clusterlensing import ClusterEnsemble z = [0.1,0.2,0.3] c = ClusterEnsemble(z) n200 = [20, 20, 20] c.n200 = n200 # display cluster dataframe c.dataframe z n200 m200 r200 c200 delta_c rs 0 0.1 20 2.700000e+13 0.612222 5.839934 12421.201995 0.104834 1 0.2 20 2.700000e+13 0.591082 5.644512 11480.644557 0.104718 2 0.3 20 2.700000e+13 0.569474 5.442457 10555.781440 0.104636 # specify mass directly c.m200 = [1e13, 1e14, 1e15] c.dataframe z n200 m200 r200 c200 delta_c rs 0 0.1 9.838141 1.000000e+13 0.439664 6.439529 15599.114356 0.068276 1 0.2 50.956400 1.000000e+14 0.914520 4.979102 8612.362538 0.183672 2 0.3 263.927382 1.000000e+15 1.898248 3.886853 4947.982895 0.488377 The above examples also demonstrate that cluster masses are converted to concentrations and to characteristic halo overdensities.\",\n \"This assumes the default mass-concentration relation of the [style=codeintext]cDuttonMaccio() form, or the user can instead specify another of the relations by setting the keyword [style=codeintext]cm=\\\"Prada\\\" or [style=codeintext]cm=\\\"Duffy\\\", when the [style=codeintext]ClusterEnsemble() object is instantiated.\",\n \"Cosmology can also be specified upon instantiation, by setting the [style=codeintext]cosmology keyword to be any [style=codeintext]astropy.cosmology object that has an [style=codeintext]h and a [style=codeintext]Om0 attribute.\",\n \"If not specified explicitly, the default cosmological model used is [style=codeintext]astropy.cosmology.Planck13.\",\n \"Here is an example of creating a [style=codeintext]ClusterEnsemble() object that uses the WMAP5 cosmology [25] and the [7] concentration: from astropy.cosmology import WMAP5 as cosmo c = ClusterEnsemble(z, cm=\\\"Duffy\\\", cosmology=cosmo) c.n200 = [20, 30, 40] c.dataframe z n200 m200 r200 c200 delta_c rs 0 0.1 20 2.700000e+13 0.599910 4.520029 6920.955951 0.132723 1 0.2 30 4.763120e+13 0.702040 4.136873 5678.897592 0.169703 2 0.3 40 7.125343e+13 0.775889 3.851601 4849.836498 0.201446 Instead of using the [style=codeintext]dataframe attribute, which retrieves the Pandas DataFrame object itself, it might be useful to use the [style=codeintext]show() method, which prints additional information to the screen, including assumptions of the mass-richness relation: c.show() Cluster Ensemble: z n200 m200 r200 c200 delta_c rs 0 0.1 20 2.700000e+13 0.599910 4.520029 6920.955951 0.132723 1 0.2 30 4.763120e+13 0.702040 4.136873 5678.897592 0.169703 2 0.3 40 7.125343e+13 0.775889 3.851601 4849.836498 0.201446 Mass-Richness Power Law: M200 = norm * (N200 / 20) ^ slope norm: 2.7e+13 solMass slope: 1.4 # update the mass-richness parameters # and show the resulting table c.massrich_norm = 3e13 c.massrich_slope = 1.5 c.show() Cluster Ensemble: z n200 m200 r200 c200 delta_c rs 0 0.1 20 3.000000e+13 0.621353 4.480202 6784.805438 0.138689 1 0.2 30 5.511352e+13 0.737028 4.086481 5526.615129 0.180358 2 0.3 40 8.485281e+13 0.822406 3.795500 4696.109606 0.216679 Mass-Richness Power Law: M200 = norm * (N200 / 20) ^ slope norm: 3e+13 solMass slope: 1.5 The last example also demonstrates how the slope or normalization of the mass-richness relation can be altered, and the changes propagate from richness through to mass and other variables.\",\n \"Then all the ingredients are in place to calculate halo profiles by invoking the [style=codeintext]calcnfw() method, which interfaces to the [style=codeintext]sigmanfw() and [style=codeintext]deltasigmanfw() methods of the [style=codeintext]SurfaceMassDensity() class, and passes it the required inputs $\\\\lbrace r_{\\\\rm s}, \\\\rho _{\\\\rm crit}, \\\\delta _{\\\\rm c} \\\\rbrace $ for all the clusters behind the scenes.\",\n \"The value of $\\\\rho _{\\\\rm crit}$ is calculated at every cluster redshift using the (default [style=codeintext]astropy.cosmology.Planck13 or user-specified) cosmological model.\",\n \"The user must specify the desired radial bins [style=codeintext]rbins in Mpc.\",\n \"import numpy as np # create some logarithmic bins: rmin, rmax = 0.1, 5.\",\n \"# Mpc rbins = np.logspace(np.log10(rmin), np.log10(rmax), num = 8) # calculate the profiles: c.calc_nfw(rbins=rbins) # profiles now exist as attributes: c.sigma_nfw c.deltasigma_nfw Similar to the direct use of [style=codeintext]SurfaceMassDensity(), discussed in Section REF , the miscentered profiles can be calculated from the [style=codeintext]calcnfw() method, by supplying the optional [style=codeintext]offsets keyword with an array-like object of length equal to the number of clusters, where each element is the width of the offset distribution in Mpc ($\\\\sigma _{\\\\rm off}$ in Equation REF ).\",\n \"c.calc_nfw(rbins=rbins, offsets=[0.3,0.3,0.3]) # the offset sigma profile is now: c.sigma_nfw Although [style=codeintext]SurfaceMassDensity() from the [style=codeintext]nfw.py module, and [style=codeintext]ClusterEnsemble().calcnfw() from the [style=codeintext]clusters.py module, are both capable of computing the same $\\\\Sigma (R)$ and $\\\\Delta \\\\Sigma (R)$ profiles, each require different forms of input which would make sense for different use cases.\",\n \"For the studies in [15], [13], and [14], the authors wanted do the profile computations for many clusters at once, while varying the mass and the miscentering offset distribution during the process of fitting the model to the data.\",\n \"What was known were the redshifts and mass proxies (cluster richness in [15] and [13], and a previous mass estimate in [14]), and mass-concentration relations from the literature, so the [style=codeintext]ClusterEnsemble() framework made sense.\",\n \"However, if someone wanted to simply calculate the NFW profiles according to the [44] formulation, then they might prefer to use [style=codeintext]SurfaceMassDensity() as a tool to get profiles directly from the NFW and cosmological parameters $r_{\\\\rm s}$ , $\\\\delta _{\\\\rm c}$ , and $\\\\rho _{\\\\rm crit}(z)$ .\",\n \"As an example use case, we take the Canada-France-Hawaii Telescope Lensing Survey [17], [10] public galaxy cluster catalog, which is available on Zenodohttp://dx.doi.org/10.5281/zenodo.51291 [11].\",\n \"This dataset was previously explored using a pre-release version of the [style=codeintext]cluster-lensing software in [13], [15].\",\n \"The W1 field of this survey contains 10,745 galaxy cluster candidates in the redshift range $0.2 \\\\le z \\\\le 0.9$ : import numpy as np data = np.loadtxt(\\\"Clusters_W1.dat\\\") data.shape (10745, 5) data[0:4, :] # print first 4 clusters array([[ 34.8023, -7.01005, 0.3, 4.435, 10.\",\n \"], [ 34.9425, -7.38996, 0.5, 4.545, 21.\",\n \"], [ 34.8651, -6.69449, 0.5, 3.858, 6.\",\n \"], [ 34.6224, -7.32768, 0.5, 3.619, 8.]])\",\n \"redshift = data[:, 2] sig = data[:, 3] richness = data[:, 4] We select a subset of the lower redshift clusters that were detected at high significance.\",\n \"Then we import [style=codeintext]clusterlensing to create a dataframe of the cluster properties, of which we just print the first several, and calculate the NFW profiles.\",\n \"# select a subset here = (sig > 15) & (redshift < 0.5) sig[here].shape (15,) z = redshift[here] n200 = richness[here] import clusterlensing c = clusterlensing.ClusterEnsemble(z) c.n200 = n200 c.dataframe.head() z n200 m200 r200 c200 delta_c rs 0 0.4 181 5.897552e+14 1.531367 3.966101 5173.016417 0.386114 1 0.3 420 1.916332e+15 2.357815 3.658237 4332.615805 0.644522 2 0.4 176 5.670737e+14 1.511478 3.980218 5213.746469 0.379747 3 0.3 113 3.049521e+14 1.277703 4.341779 6324.420397 0.294281 4 0.4 162 5.049435e+14 1.454129 4.022285 5336.272412 0.361518 rbins = np.logspace(np.log10(0.1), np.log10(10.0), num=20) c.calc_nfw(rbins) Next we import the [style=codeintext]matplotlib and [style=codeintext]seaborn libraries and configure some settings to make our plots more readable.\",\n \"The first plot we create with the commands below presents the $\\\\Sigma (R)$ profiles for every one of these 15 clusters, and is given in Figure REF .\",\n \"import matplotlib.pyplot as plt import seaborn as sns; sns.set() import matplotlib matplotlib.rcParams[\\\"axes.labelsize\\\"] = 20 matplotlib.rcParams[\\\"legend.fontsize\\\"] = 20 # strings for plots raxis = \\\"$R\\\\ [\\\\mathrm{Mpc}]$\\\" sgma = \\\"$\\\\Sigma(R)$\\\" sgmaoff = \\\"$\\\\Sigma^\\\\mathrm{off}(R)$\\\" delta = \\\"$\\\\Delta$\\\" sgmaunits = \\\" $[M_{\\\\odot}\\\\ \\\\mathrm{pc}^{-2}]$\\\" # order from high to low richness order = c.n200.argsort()[::-1] for s, n in zip(c.sigma_nfw[order], c.n200[order]): plt.plot(rbins, s, label=str(int(n))) plt.xscale(\\\"log\\\") plt.legend(fontsize=10) plt.ylabel(sgma+sgmaunits) plt.xlabel(raxis) plt.tight_layout() plt.savefig(\\\"f1.eps\\\") plt.close() # output is Figure 1 Figure: Surface mass density profiles Σ(R)\\\\Sigma (R) for all 15 clusters used in this example.\",\n \"These are the most significant (S/N >15>15) clusters detected at low redshifts (z<0.5z < 0.5) in the W1 field of CFHTLenS.\",\n \"See the text for links to download this public dataset.\",\n \"The legend gives the richness values estimated in corresponding to each of these clusters, which are assumed to scale with mass.\",\n \"They are listed from highest to lowest richness, in the same order as the curves.If we had made a stacked measurement of the shear or magnification profile of these clusters, then we would want to know what the average profile of the stack looks like.\",\n \"Since we already have the individual profiles, we just need to calculate the mean across the 0th axis of the [style=codeintext]sigmanfw and [style=codeintext]deltasigmanfw attribute arrays.\",\n \"The plot of these average profiles is given in Figure REF .\",\n \"sigma = c.sigma_nfw.mean(axis=0) dsigma = c.deltasigma_nfw.mean(axis=0) plt.plot(rbins, sigma, label=sgma) plt.plot(rbins, dsigma, \\\"--\\\", label=delta+sgma) plt.legend() plt.ylim([0., 1400.])\",\n \"plt.xscale(\\\"log\\\") plt.xlabel(raxis) plt.ylabel(sgmaunits) plt.tight_layout() plt.savefig(\\\"f2.eps\\\") plt.close() # output is Figure 2 Figure: The average of all the surface mass density profiles Σ(R)\\\\Sigma (R) for each of the clusters shown in Figure is given in blue.\",\n \"The green curve is the average of all the individual differential mass density profiles ΔΣ(R)\\\\Delta \\\\Sigma (R).\",\n \"These curves assume clusters are perfectly centered on their NFW halos.Finally, we may want to investigate whether cluster miscentering has a significant effect on our sample.\",\n \"We would calculate the miscentered profiles, given in Figure REF , which could be compared to the centered profiles in Figure REF to see which is a better fit to our measurement.\",\n \"Below we will assume that the miscentering offset distribution peaks at 0.1 Mpc.\",\n \"offsets = np.ones(c.z.shape[0]) * 0.1 c.calc_nfw(rbins, offsets=offsets) sigma_offset = c.sigma_nfw.mean(axis=0) dsigma_offset = c.deltasigma_nfw.mean(axis=0) plt.plot(rbins, sigma_offset, label=sgmaoff) plt.plot(rbins, dsigma_offset, \\\"--\\\", label=delta+sgmaoff) plt.legend() plt.ylim([0., 1400.])\",\n \"plt.xscale(\\\"log\\\") plt.xlabel(raxis) plt.ylabel(sgmaunits) plt.tight_layout() plt.savefig(\\\"f3.eps\\\") plt.close() # output is Figure 3 Figure: Same as Figure but for miscentered profiles.\",\n \"Cluster centroid offsets are assumed to follow a Rayleigh probability distribution (Equation , discussed in Section ), which is convolved with the perfectly centered profiles to achieve this result.The above example shows a simple application of [style=codeintext]cluster-lensing to a real dataset – a subset of the CFHTLenS cluster catalog.\",\n \"For this example we kept the customizations to a minimum, but as shown in Sections REF , the user can alter the parameters in the power-law relation used to convert richness to mass, choose the form of the mass-concentration relation assumed for the NFW profiles, and specify a particular background cosmology.\",\n \"When fitting a model produced by [style=codeintext]cluster-lensing to a measurement, one could iterate through parameters in this space by setting various attributes of the [style=codeintext]ClusterEnsemble() object [13], [15].\"\n ],\n [\n \"Relation to Existing Code\",\n \"The [style=codeintext]cluster-lensing project offers some unique capabilities over other publicly-available software, most notably the cluster miscentering calculations.\",\n \"Here we attempt to compare the software presented in this work with other open source tools that we are aware of, and show how [style=codeintext]cluster-lensing fits into the larger ecosystem of astronomical software.\",\n \"[style=codeintext]Colossus is a Python package aimed at cosmology, halo, and large-scale structure calculations [3].\",\n \"It was used in work by [4] and is made available under the MIT licensehttp://www.benediktdiemer.com/code/.\",\n \"Much of the functionality of [style=codeintext]cluster-lensing appears to overlap with [style=codeintext]Colossus, including mass-concentration relations (although [style=codeintext]Colossus has the advantage of containing many more relations from the literature) and NFW surface mass density profiles.\",\n \"However, [style=codeintext]cluster-lensing also provides the miscentered halo calculations, which are are lacking from [style=codeintext]Colossus.\",\n \"While [3] has chosen to implement basic cosmological calculations from scratch, [style=codeintext]cluster-lensing instead relies on external modules supplied by [style=codeintext]astropy.\",\n \"The only dependencies claimed by [style=codeintext]Colossus are [style=codeintext]numpy [42] and [style=codeintext]scipy [23], whereas [style=codeintext]cluster-lensing additionally requires [style=codeintext]astropy [1] and [style=codeintext]pandas [28].\",\n \"Fewer dependencies might be seen as a positive feature of [style=codeintext]Colossus; on the other hand, [style=codeintext]astropy could be viewed as possibly a more robust source for standard astronomical and cosmological calculations, since it is maintained by a large community of developers.\",\n \"Another related set of code is provided by Jörg Dietrich's NFW routines, archived on Zenodo [5], and available on GitHubhttps://github.com/joergdietrich/NFW.\",\n \"These [style=codeintext]Python modules calculate NFW profiles for $\\\\Sigma (R)$ and $\\\\Delta \\\\Sigma (R)$ , as well as the 3-dimensional density profiles and total mass and projected mass inside a given radius.\",\n \"[style=codeintext]cluster-lensing goes beyond the functionality of [5] by supplying means for calculating cluster miscentering, and having a built-in framework for handling many halos at once.\",\n \"For [5], the user must provide the halo concentration (along with mass and redshift) to the [style=codeintext]NFW() class, but additional routines are available for converting mass to concentration, including [7] and another by [6] (a partial overlap with the mass-concentration relations provided by [style=codeintext]cluster-lensing).\",\n \"[5] depends on [style=codeintext]astropy for cosmological calculations and units, similar to [style=codeintext]cluster-lensing, as well as the [style=codeintext]numpy and [style=codeintext]scipy packages.\"\n ],\n [\n \"Future Development\",\n \"Some of the future plans for [style=codeintext]cluster-lensing include adding support for different density profiles.\",\n \"Currently only the NFW model is provided, and alternative mass density models would make the package more complete and useful.\",\n \"The first priority will be inclusion of the Einasto profile [9], and later possibilities may include the generalized-NFW [46].\",\n \"The default cosmology is currently that of [34], but should be updated to [35], since this is now available as [style=codeintext]astropy.cosmology.Planck15 (the user can currently specify this cosmology, it is just not the default).\",\n \"When surface mass density profiles have to be calculated many times for many clusters, as is the case when iterating over parameters in the process of fitting a model, the processing time can become lengthy.\",\n \"This issue is most pronounced for calculation of miscentered profiles, which require the convolution laid out in Equations REF and REF .\",\n \"One major improvement to [style=codeintext]cluster-lensing will be the option to use parallel-processing in these computations.\",\n \"The likely structure of this parallelism will be to divide the halos in a [style=codeintext]ClusterEnsemble() catalog object among the parallel threads, which will calculate the profiles for each of their assigned clusters.\",\n \"All of these future developments are currently listed as issues on the GitHub repository.\",\n \"This GitHub Issue trackerhttps://github.com/jesford/cluster-lensing/issues will continue to serve as the central place for listing future improvements and feature requests.\",\n \"Users and potential-users alike are encouraged to submit ideas and requests through that URL.\"\n ],\n [\n \"Summary\",\n \"In this work we presented [style=codeintext]cluster-lensing, a pure-[style=codeintext]Python package for calculating galaxy cluster profiles and properties.\",\n \"We described and gave worked examples of all the functionality currently available, including mass-concentration and mass-richness scaling relations, and the surface mass density profiles $\\\\Sigma (R)$ and $\\\\Delta \\\\Sigma (R)$ , which are relevant for gravitational lensing.\",\n \"The latter density profiles are not cluster-specific, but apply to any mass halo that can be approximated by the NFW prescription.\",\n \"The structure of [style=codeintext]cluster-lensing is ideal for calculating properties and profiles for many galaxy clusters at once.\",\n \"This “composite-halo” approach [15], is especially useful for fitting models to a stacked sample of clusters that span a range of mass and/or redshift.\",\n \"Compared to existing code, [style=codeintext]cluster-lensing stands out by seemingly being the only publicly-available software for calculating miscentered halo profiles.\",\n \"Miscentering is a problem of great relevance for stacked weak lensing studies of galaxy clusters, where halo centers are imperfectly estimated from observational data or simply not well defined (as is the case for individual non-spherical halos – for example in merging systems).\",\n \"The resulting offsets between the assumed and real centers change the shape of the measured shear or magnification profile and need to be accounted for in the modeling.\",\n \"[style=codeintext]cluster-lensing is released under the MIT license, and archived on Zenodo [12].\",\n \"It being developed in a public repository on GitHub: http://github.com/jesford/cluster-lensing/.\",\n \"Contributions to the code can be made by submitting a pull request to the repository, and we welcome feedback, suggestions, and feature requests through GitHub issues, or by emailing the author.\",\n \"Full documentation (including much of the content of this paper), as well as installation instructions and examples, are available in the online documentation, at http://jesford.github.io/cluster-lensing/.\",\n \"If [style=codeintext]cluster-lensing is used in a research project, the authors would appreciate citations to the code [12] and this paper.\"\n ],\n [\n \"Acknowledgements\",\n \"The authors are grateful for funding from the Washington Research Foundation Fund for Innovation in Data-Intensive Discovery and the Moore/Sloan Data Science Environments Project at the University of Washington.\",\n \"This project would not have been possible without packages available in Python's open scientific ecosystem, including NumPy [42], SciPy [23], Pandas [28], matplotlib [20], IPython [33], AstroPy [1], Seaborn [43], and related tools.\"\n ]\n]"},"id":{"kind":"string","value":"1605.05700"}}},{"rowIdx":1866696,"cells":{"text":{"kind":"list like","value":[["Three-dimensional distribution of hydrogen fluoride gas toward NGC6334 I\n and I(N)"],["Abstract Aims.","We investigate the spatial distribution of a collection of absorbing gas clouds, some associated with the dense, massive star-forming core NGC6334 I, and others with diffuse foreground clouds.","For the former category, we aim to study the dynamical properties of the clouds in order to assess their potential to feed the accreting protostellar cores.","Methods.","We use spectral imaging from the Herschel SPIRE iFTS to construct a map of HF absorption at 243 micron in a 6x3.5 arcmin region surrounding NGC6334 I and I(N).","Results.","The combination of new, spatially fully sampled, but spectrally unresolved mapping with a previous, single-pointing, spectrally resolved HF signature yields a 3D picture of absorbing gas clouds in the direction of NGC6334.","Toward core I, the HF equivalent width matches that of the spectrally resolved observation.","The distribution of HF absorption is consistent with three of the seven components being associated with this dense star-forming envelope.","For two of the remaining four components, our data suggest that these clouds are spatially associated with the larger scale filamentary star-forming complex.","Our data also implies a lack of gas phase HF in the envelope of core I(N).","Using a simple description of adsorption onto and desorption from dust grain surfaces, we show that the overall lower temperature of the envelope of source I(N) is consistent with freeze-out of HF, while it remains in the gas phase in source I.","Conclusions.","We use the HF molecule as a tracer of column density in diffuse gas (n(H) ~ 10^2 - 10^3 cm^-3), and find that it may uniquely trace a relatively low density portion of the gas reservoir available for star formation that otherwise escapes detection.","At higher densities prevailing in protostellar envelopes (>10^4 cm^-3), we find evidence of HF depletion from the gas phase under sufficiently cold conditions."],["Introduction","The hydrogen fluoride molecule, HF, was first observed in the interstellar medium by [59] with the Infrared Space Observatory [41].","While ISO had a wavelength range that encompassed only the $J$ =2–1 rotational transition of HF, the next observatory able to observe HF – the Herschel Space Observatory [69] – covered longer far-infrared wavelengths, and it thus opened up access to the ground-state rotational transition, $J$ =1–0, at 1232.48 GHz (243.24 $\\mu $ m).","Herschel has observed HF in absorption along many lines of sight, both inside the Galaxy [56], [84], [85], [67], [42], [51], [18], [46], [27] and in nearby extragalactic objects [74], [39], [79], [52].","HF absorption has even been detected with ground-based observatories: [53] have made use of the substantial redshift of the Cloverleaf quasar at $z$ =2.56 shifting the HF 1–0 line into the submillimeter window attainable with the CSO on Mauna Kea, and [40] detect it in the $z$ =0.89 absorber toward PKS1830$-$ 211, using ALMA in the Chilean Atacama desert.","Because of its large dipole moment and high Einstein $A$ coefficient for radiative decay, rotational states $J$$\\ne $ 0 of HF only become significantly populated in highly energetic conditions.","It is for this reason that HF has been clearly detected in emission in a mere handful of cases: in the inner region of an AGB star's envelope [2], in the Orion Bar photodissociation region [90], and in an external galaxy harboring an actively accreting black hole [91].","Atomic fluoride, F, has a unique place in the interstellar chemistry of simple molecules.","It is the only element which, simultaneously, (1) is mainly neutral because of its ionization potential $>$ 13.6 eV, (2) reacts exothermically with H$_2$ – unlike any other neutral atom – to form its neutral diatomic hydride HF, and (3) lacks an efficient chemical pathway to produce its hydride cation HF$^+$ due to the strongly endothermic nature of the reaction with H${_3}^+$ .","We refer to [57], references therein, and the comprehensive review by [22] for more details on the chemistry of HF and a comparison with other hydride molecules.","For the reasons listed above, chemical models predict that essentially all interstellar F is locked in HF molecules [98], [58], which has been confirmed by observations across a wide range of atomic and molecular ISM conditions [84], [85].","With recent experimental results by [88] showing that, especially at low temperatures approaching 10 K, the reaction F + H$_2$ $\\rightarrow $ HF + H proceeds somewhat slower than earlier assumptions, chemical models are now able to reproduce HF/H$_2$ ratios of $\\sim $ 1$\\times $ 10$^{-8}$ , measured most directly by [36], and observed to be rather stable across different sightlines.","Interferometric observations show that CF$^+$ , the next most abundant F-bearing species after HF, has an abundance roughly two orders of magnitude lower than HF, both inside our Galaxy [45] and in an extragalactic absorber [54].","As for destruction of HF, the most efficient processes are UV photodissociation and reactions with C$^+$ , but both of these are unable to drive the majority of fluoride out of HF, due to shielding, already at modest depths of $A_V>0.1$ [58].","Because of the constant HF/H$_2$ abundance ratio and the high probability that HF molecules are in the rotational ground state, measurements of HF $J$ =0$\\rightarrow $ 1 absorption provide a straightforward proxy of H$_2$ column density.","This has led to the suggestion that, at least in diffuse gas, HF absorption is a more reliable tracer of total gas column density than the widely used carbon monoxide (CO) rotational emission lines, and is more sensitive than CH or H$_2$ O absorption [22].","Apart from the uncertain and variable CO abundance, local excitation conditions have a profound effect on the level populations of CO, complicating the conversion from observed line strength of a particular CO transition to H$_2$ column density [7].","The greatest gas-phase CO abundance variations occur in dense, cold regions where CO freezes out onto surfaces of dust grains, proven by observed CO abundances decreasing in the gas phase and increasing in the ice phase as conditions get colder [37], [71].","In addition, the particular fraction of the neutral ISM that is in the diffuse/translucent phase is inconspicuous in CO [7], but is detectable using hydride absorption lines.","Of course, for absorption line studies, one relies on lines of sight with sufficiently strong continuum background, for example those toward dense star-forming clouds.","Such restrictions do not apply for emission line tracers.","Besides CO rotational lines, fine structure line emission due to atomic C and the C$^+$ and N$^+$ ions has been used as a tracer of (diffuse) gas throughout the Galaxy [43], [94], [23], [28], [29].","For all these tracers, however, the conversion to H$_2$ column density depends strongly on physical properties such as ionization fraction and excitation conditions.","Based on the above arguments, HF absorption measurements are a good tracer of overall gas column density.","However, as addressed for example by [67] and [18], HF itself may suffer from freeze-out effects as occurs with other interstellar molecules.","While studies have been done on the interaction of H$_2$ O with HF as a polluting agent in the Earth's atmosphere [26], the density and temperature conditions needed for HF adsorption onto dust grains have not been studied in astrophysical contexts so far.","Any freeze-out of interstellar HF will obfuscate the direct connection between HF absorption depth and H$_2$ column density described above.","The well-known progression of pre- and protostellar stages for stars with masses similar to the Sun [83] is not applicable for high-mass stars ($\\gtrsim 8$ $M_\\odot $ ).","In the latter category, protostellar hydrogen fusion starts while accretion from the surrounding gas envelope is still ongoing [64].","In the `competitive accretion' scenario, multiple massive protostars in a clustered environment are fed from the same gas reservoir [8].","For high-mass protostars, material can continuously be added to the gravitationally bound circumstellar envelope which provides the reservoir for further accretion onto the protostar.","It is therefore important, particularly for regions of high-mass star formation, to study not just the gravitationally bound circumstellar envelopes, but also the dynamical properties of surrounding gas clouds.","Especially for the latter component, simple hydride molecules have the potential to reveal gas reservoirs to which emission lines of `traditional' tracer species, such as CO, HCO$^+$ , and CS, are insensitive due to their relatively high critical densities.","In this paper, we investigate two envelopes of (clusters of) protostars embedded in the NGC 6334 molecular cloud as well as lower density clouds surrounding the dense complex.","The filamentary, star-forming cloud NGC 6334, at a distance of 1.35 kpc [95], harbors a string of dense cores, identified in the far-infrared by roman numerals I through VI [50], with an additional source identified $\\sim $ 2 north of source I, later named `I(N)' [24].","The larger scale NGC 6334 filament has an H$_2$ column density of $>$ 2$\\times $ 10$^{22}$ $\\mathrm {cm}^{-2}$ even at positions away from the embedded cores [80].","[97] observed the velocity structure of NGC 6334 at 0.15 pc resolution.","These authors explain the velocity profile along the filament with a cylindrical model collapsing along its longest axis under the influence of gravity.","In this paper we study specifically the region of $\\sim $ 2.4$\\times $ 1.6 pc surrounding the embedded cores NGC 6334 I and NGC 6334 I(N).","Source I is host to an ultra-compact Hii region, designated source `F' in a 6 cm radio image of the cloud [77].","Based on multi-wavelength dust continuum measurements, studies by [81] and [89] have independently determined that the mass of source I(N) exceeds that of sister source I by a factor of $\\sim $ 2–5, but the ratio of their bolometric luminosities is 30–140 in favor of source I, due to the markedly lower temperature for source I(N).","As expected for a warm (up to $\\sim $ 100 K), dense, massive star-forming core, NGC 6334 I is extremely rich in molecular lines, spectacularly demonstrated by the 4300 lines detected in the 480–1907 GHz spectral survey by [96]See the introduction section of this reference for a list of earlier spectral survey work on NGC 6334 I..","The differences between the two neighboring cores all suggest that core I is in a more evolved stage of star formation than core I(N).","Both cores have been studied with radio and (sub)millimeter interferometer observatories, showing that each separates into several subcores at arcsecond resolution [34], [33], [10].","To probe gas clouds in front of the NGC 6334 complex, absorption measurements have been obtained in lines of several hydrides.","Spatially extended OH hyperfine line absorption at 1.6–1.7 GHz was observed toward the NGC 6334 filament by [11].","The spectrally resolved mapping observations from the Australia Telescope Compact Array allowed these authors to ascribe particular velocity components of the absorption to a foreground cloud close to NGC 6334 and other components to clouds with even larger angular extent.","[92] used the Heterodyne Instrument for the Far-Infrared [16] onboard Herschel to study the spectral profile of the rotational ground state lines of CH at 532 and 537 GHz, and found four distinct absorption components overlapping with the velocity range of OH absorption, and one single emission component emanating in core NGC 6334 I itself.","At 1232.5 GHz in the same spectral survey, [18] find the exact same four absorbing clouds in the HF rotational ground state, and invoke three velocity components to explain the hot core component.","While the hot core component(s) appear in emission in CH, they are in absorption in HF, because CH 1$\\rightarrow $ 0 has a lower Einstein $A$ coefficient than HF 1$\\rightarrow $ 0 (see above).","Table: SPIRE spectrometer observations used in this paper.The CH and HF signatures were observed toward NGC 6334 I with the high spectral resolution spectrometer HIFI in single point mode [16]; its single pixel receiver did not provide any spatial information.","In a map of CH or HF absorption covering the region surrounding source I, one would expect to see a disentanglement of the different spatial extent of each velocity component as illustrated in Fig.","REF .","Toward core I, the velocity resolved HF absorption signature, with a total equivalent width, $\\int (1-I_\\mathrm {norm}) \\mathrm {d}V$ , of 16 $\\mathrm {km}\\,\\mathrm {s}^{-1}$ , was modeled with seven components.","At positions away from core I, but still on the NGC 6334 filament, the equivalent width should diminish to 11 $\\mathrm {km}\\,\\mathrm {s}^{-1}$ representing the four foreground clouds, while at positions off of the cores and the filament, only the two foreground components that are more extended than the dense molecular cloud should be visible, and the equivalent width should drop to 3 $\\mathrm {km}\\,\\mathrm {s}^{-1}$ .","This paper presents results from Herschel SPIRE iFTS spectral mapping observations toward a 6$\\times $ 35 region surrounding cores I and I(N) in the NGC 6334 star-forming complex.","The observations are described in Sect.","and the resulting map of HF absorption depth is discussed in Sect. .","The signal is interpreted in Sect.",", both in the context of foreground clouds and in that of freeze-out conditions in the dense cores.","Conclusions are summarized in Sect.","."],["Observations and data reduction","The spectral mapping observations used in this work were obtained as part of the `evolution of interstellar dust' guaranteed time program [1] with the Spectral and Photometric Imaging Receiver [30] on board the Herschel space observatory [69].","SPIRE's imaging Fourier Transform Spectrometer (iFTS) provides a jiggling observing mode that uses its 54 detectors to obtain Nyquist sampled spatial maps, covering the entire frequency range of the Spectrometer Long Wavelength (SLW, 447–1018 GHz) and the Spectrometer Short Wavelength (SSW, 944–1568 GHz) bands.","The spectral resolution of 1.2 GHz corresponds to a resolving power $\\nu /\\Delta \\nu \\approx 10^3$ , roughly 300 $\\mathrm {km}\\,\\mathrm {s}^{-1}$ at the frequency of the HF 1–0 transition, 1232.5 GHz (243 $\\mu $ m).","Three partly overlapping, fully sampled SPIRE iFTS 4$\\times $ 4 jiggle observations were performed in a total of nine hours of observing time, two centered on NGC 6334 I and the third on NGC 6334 I(N).","A fourth, sparsely sampled observation, centered $\\sim $ 2 northwest of core I, has considerable overlap with the combined area covered by the three other observations.","This fourth observation is treated as an extra jiggle position in the gridding process described below.","Finally, a fifth observation, also sparsely sampled, is centered 45 northwest of core I and its footprint therefore has no overlap with our mapped area.","At each jiggle position, four repeated scans of the FTS mechanism were executed in high spectral resolution mode.","Details of the observations are summarized in Table REF .","The placement of the different pointings described here is shown in Fig.","REF in the Appendix.","After inspection of the initial observations from February 2011, some detectors were found to suffer from saturation due to the bright emission toward source I.","The observation toward source I was therefore repeated in `bright source mode' [47] in September 2012 to obtain well calibrated spectra toward the brightest position.","The majority of detector/jiggle combinations in the original observation point toward less bright regions and are therefore still useful in constructing the final map.","The above SPIRE iFTS observations are processed with the `extended source' pipeline in HIPE 12.1.0 and the spire_cal_12_3 calibration tree, which includes the outer ring of partly vignetted detectors [21].","The pipeline is interrupted at the pre-cube stage, before spectra from individual jiggle positions are gridded onto a rectangular spatial pattern.","The spectrum for each jiggle position and each detector is visually inspected.","Discarding all spectra that show excessive noise and/or irregular continuum shape (resulting from partial saturation) results in filtering 28 of the total of 919 SLW spectra (3%) and 135 of the 1696 SSW spectra (8%).","The final processing step is the combination of the individual positions from each of the 19 (SLW) or 37 (SSW) detectors of each of the 55 jiggle positions into spectral cubes with square spatial pixels in Right Ascension and Declination coordinates.","Due to the complex frequency dependence of the beam size and shape [48], the native angular resolution of SPIRE iFTS observations varies non-monotonously across its frequency range.","We use the convolution gridding scheme, which weighs each input value according to the distance from the target pixel center by means of a differential Gaussian kernel, with the aim of obtaining a cube with a constant effective reference beam of 43 FWHM.","The pixel grid is identical for SLW and SSW, with square pixels measuring 175$\\times $ 175.","Gaussian convolution only results in a completely Gaussian target beam if the original beam is also well represented by a Gaussian shape.","Such is the case for the entire SSW band and for low frequencies in SLW, but not for SLW frequencies between $\\sim $ 700 and 1018 GHz [48].","Figure: Top panel: velocity resolved spectral profile of HF absorption observed with HIFI toward core NG C6334 I (gray histogram).","The seven individual model components are the same as in and are drawn in dashed green lines, with their sum in solid gray.","Middle panel: four foreground components expected to remain visible for lines of sight away from core I, but still on the NGC 6334 molecular cloud complex (solid blue).","Bottom panel: two foreground cloud components hypothesized to be unrelated to the NGC 6334 complex (solid magenta).Figure: Illustration of the procedure adopted to fit the spectrally unresolved, Sinc-shaped absorption line feature in the SPIRE iFTS cube.","Rest frequencies of the relevant transitions are marked by vertical dashed lines.Top panel: observed spectrum (black) toward one of the pixels in our map, fit to continuum and CO lines (blue), and the resulting ratio spectrum (gray).","The rectangular box marks the axes limits of the next panel.Middle panel: zoom of continuum-normalized spectrum (gray, same as top panel) and fit to the combined signature of the HF absorption line and the nearby H 2 _2O emission line (red).Bottom panel: residual remaining after subtracting the fit shown in the middle panel.Figure: Map of equivalent width of the HF 1←\\leftarrow 0 absorption signature measured with Herschel SPIRE iFTS (colored contours).","The peak value in the center of the `12.5 km s -1 \\mathrm {km}\\,\\mathrm {s}^{-1}' contour is 16 km s -1 \\mathrm {km}\\,\\mathrm {s}^{-1}.","The grayscale represents 250 μ\\mu m dust continuum emission measured with the SPIRE photometer (Herschel observation ID 1342239909, HIPE 13 standard pipeline, color bar stretched from 4 to 250 GJy sr -1 ^{-1}).","The thin white contours are from the continuum as measured by the FTS, which closely follow the structure seen in the photometer grayscale map.","Beam sizes of the convolved spectrometer cube and the photometer map are shown in the bottom left corner in light gray and dark gray circles, respectively.","The scalebar in the top left corner indicates a projected length of 0.3 pc at a distance to the source of 1.35 kpc."],["Results","The spectral cubes, as constructed in Sect.",", show a smooth dust continuum superposed with spectrally unresolved lines.","The dominant line signal in the cube arises from the ladder of rotational transitions of CO ($J$ =4–3 to 13–12).","Early versions of the CO intensity maps of NGC 6334, based on subsets of the SPIRE iFTS observations used here, were presented in [55] and [49].","To retain the highest possible spectral resolution, we use unapodized FTS data, in which any unresolved spectral line has a Sinc-shaped profile.","It is therefore important that one carefully fits and subtracts the Sinc-shaped profiles of nearby bright lines, to avoid any remaining sidelobes of strong lines affecting the apparent profile of the weak absorption line under study.","This work focuses on absorption signatures of two hydride molecules, for which we use two spectral sections of the data cubes: 1132–1332 GHz from SSW and 760–935 GHz from SLW, chosen specifically to include the two CO emission lines closest to HF 1–0 at 1232.48 GHz [61] and CH$^+$ 1–0 at 835.08 GHz [65].","We construct a script in HIPE [63], derived from one of the post-pipeline analysis scripts provided by the SPIRE iFTS working group [70], to fit a third order polynomial for the continuum simultaneously with the following lines: CO at 1152.0 and 1267.0 GHz (SSW), and CO at 806.7 and 921.8 and [Ci] at 809.3 GHz (SLW).","Knowing that the intrinsic width of CO lines in this region is only a few $\\mathrm {km}\\,\\mathrm {s}^{-1}$ [96], lines are spectrally unresolved by SPIRE iFTS, and we adopt for each spectral line a Sinc profile with a fixed peak-to-first-zero-crossing width of 1.18 GHz.","We then divide the observed spectrum by the fit of continuum and two/three Sinc lines to obtain a continuum-normalized spectrum, $I_\\mathrm {norm}$ .","This process, illustrated in the top panel of Fig.","REF , is repeated for each spatial pixel in the cube.","In the resulting continuum-normalized cube around 1232 GHz, we fit two Sinc functions to the absorption profile of HF 0$\\leftarrow $ 1 at 1232.48 GHz and the nearby emission line of H$_2$ O 2$_{2,0}$$\\rightarrow $ 2$_{1,1}$ at 1228.79 GHz (Fig.","REF , middle panel).","In the continuum-normalized cube around 835 GHz, we fit three Sinc functions to the absorption profile of CH$^+$ 0$\\leftarrow $ 1 at 835.08 GHz and $^{13}$ CO emission lines at 771.18 and 881.27 GHz.","The maps of equivalent width of the HF and CH$^+$ absorption depth, in units of Hz, are multiplied by the ratio of the speed of light, c, and the observed frequency, $\\nu _\\mathrm {obs}$ , to convert to $\\mathrm {km}\\,\\mathrm {s}^{-1}$ units: c/$\\nu _\\mathrm {obs}$ =2.43$\\times $ 10$^{-7}$ $\\mathrm {km}\\,\\mathrm {s}^{-1}$ Hz$^{-1}$ for HF, and 3.59$\\times $ 10$^{-7}$ $\\mathrm {km}\\,\\mathrm {s}^{-1}$ Hz$^{-1}$ for CH$^+$ .","Line fits are rejected if the signal-to-noise ratio is lower than 2 and/or the fitted line center is more than half a resolution element from the line's expected frequency at $-8.3$ $\\mathrm {km}\\,\\mathrm {s}^{-1}$ $V_\\mathrm {lsr}$ [92].","The resulting map of HF equivalent width in Fig.","REF reveals the spatial distribution of the HF absorption feature, detected in 81% (signal-to-noise$>$ 2) or 72% (signal-to-noise$>$ 3) of all the pixels in the 6$\\times $ 35 map coverage.","See also the signal-to-noise map in Fig.","REF .","Uncertainties are calculated from the spectral rms noise in the continuum-normalized cubes in two 20 GHz ranges surrounding the HF absorption line.","For CH$^+$ , the frequency ranges for calculating noise are composed of a 10 GHz section below and a 30 GHz section above the CH$^+$ frequency, to avoid incorporating residual from the fit to the blended CO 7–6 and [Ci] $^3$ P$_2$ –$^3$ P$_1$ lines at 806 and 809 GHz.","The 1-$\\sigma $ noise on the equivalent width is obtained by multiplying the unitless spectral rms with the instrumental line width of 1.18 GHz $\\times $ c/$\\nu _\\mathrm {obs}$ .","Noise values are variable across the maps, in the range 1.2–3.4 $\\mathrm {km}\\,\\mathrm {s}^{-1}$ for HF and 1.4–7 $\\mathrm {km}\\,\\mathrm {s}^{-1}$ for CH$^+$ .","We do not include the following contributions to the uncertainty.","Firstly, any multiplicative effects such as those of the absolute intensity calibration [4], [87] are divided out by normalizing the spectra to the local continuum.","Secondly, additive uncertainties in the continuum level offsets are $\\sim $ 0.5$\\times $ 10$^{-19}$ W m$^{-2}$ for SLW and $\\sim $ 2$\\times $ 10$^{-19}$ W m$^{-2}$ for SSW [87], [31].","These values are negligible compared to the brightness of the continuum in our cube, which exceeds the offset uncertainty by a factor of a few hundred even in the faintest outer regions.","Spectra from the `OFF' observation, pointed just northwest of the mapped area shown in Fig.","REF , are also inspected at the HF frequency, but no convincing detections are found.","The spectra from individual detectors in the OFF observation exhibit rms noise values between 4 and 10 $\\mathrm {km}\\,\\mathrm {s}^{-1}$ , with a median of 6 $\\mathrm {km}\\,\\mathrm {s}^{-1}$ .","This noise is considerably higher than that in the cube pixels based on the other four observations combined, in which each pixel encompasses at least four, but typically more than eight individual detector pointings.","The lack of HF absorption detections in the OFF position is thus consistent at the 2-$\\sigma $ level with an HF absorption depth $\\lesssim $ 12 $\\mathrm {km}\\,\\mathrm {s}^{-1}$ in the area 3–6 northwest of source I, i.e., absorption depths could be anywhere in the range shown in our mapped area, except the central 40 around source I itself where the strongest absorption is seen.","To rule out contamination of the HF signature by other spectral lines within SPIRE's spectral resolution element, we inspect the high spectral resolution spectrum toward the position of the chemically rich core NGC 6334 I [96], observed as part of the spectral survey key program CHESS [12] using the HIFI spectrometer [16], [78].","The only spectral lines detected by HIFI in a frequency span of $\\pm $ 2 GHz around the HF frequency are four marginally detected methanol emission lines (A. Zernickel, private communication, Jun.","2014) together amounting to $<$ 0.4 $\\mathrm {km}\\,\\mathrm {s}^{-1}$ in equivalent width.","The possible methanol contaminations for the measured HF absorption depth are therefore contained within the uncertainty for our HF equivalent widths quoted above.","The effect of the H$_2$ O emission line at 1229 GHz (see also above) could be more significant: at the brightest position, toward core I, the water line is as bright as one third of the deepest HF absorption.","As described above and shown in Fig.","REF , the effects of the water line on the HF absorption profile are taken into account by applying a simultaneous fit of these two lines, separated in frequency by three times the SPIRE instrumental line width.","We also detect the signature of CH$^+$ 1$\\leftarrow $ 0 absorption at 835.08 GHz in the spectral map from the SLW array.","We refrain from interpreting its signal here for the following reasons.","Firstly, the signal-to-noise ratio in the CH$^+$ absorption map is much lower than that in the HF map (see Fig.","REF and REF ), resulting in signal-to-noise$>$ 2 detections in only 46% of the mapped pixels.","For completeness, the distribution of detected CH$^+$ absorption is shown in the Appendix in Fig.","REF .","Importantly, around the position of source I, there is no confident detection to be compared with heterodyne observations from [96].","Only a few isolated pixels near that position have detections of CH$^+$ , but at a signal-to-noise of $<$ 3.","Secondly, the spectrally resolved HIFI spectrum toward source I (A. Zernickel, private communication, Jun.","2014) show seven distinct emission line features due to methanol at frequencies within 0.6 GHz of the CH$^+$ line, i.e., half of the SPIRE spectral resolution.","The combined intensity of these lines is sufficient to compensate more than half of the CH$^+$ absorption observed by HIFI, and they therefore severely contaminate the spectrally unresolved profile of CH$^+$ absorption in the SPIRE spectrum.","In fact, these methanol emission lines could be the cause of the weak detection at the position of core I with SPIRE's modest spectral resolution.","Thirdly, the CH$^+$ transition falls in a frequency range in which the beam profile of the SPIRE iFTS is non-Gaussian in shape [48], complicating the map convolution and the interpretation of any spatial structure.","Our SPIRE spectroscopic data also show evidence for detections of NH$_2$ at 952.6 and 959.5 GHz, and NH at 974.5 GHz and at 1000 GHz.","However, compared to HF, it is more challenging for astrochemical models to explain the observed abundances of N-bearing hydrides [66], and the analysis of line features is complicated by hyperfine structure within rotational transitions [96].","For these two reasons, we refrain from interpreting the NH and NH$_2$ absorption depth maps in this paper, but for completeness they are shown in Figs.","REF and REF .","The OH$^+$ doublet at 909.05 and 909.16 GHz is also within the frequency range covered by the SPIRE iFTS, but the bright methanol emission line at 909.07 GHz apparent in the HIFI spectrum published by [96] would make analysis of the blended OH$^+$ signature in the SPIRE spectrum impossible."],["HF optical depth, equivalent width, and column density","An absorption line is saturated when its depth reaches zero, i.e., absorbing all continuum photons in any specific spectral channel.","HIFI observations seem to show that the combined absorption feature of HF due to the NGC 6334 I envelope, hot cores, and the foreground cloud at $-3.0$ $\\mathrm {km}\\,\\mathrm {s}^{-1}$ is saturated between $V_\\mathrm {lsr}$ of $-7$ and $-4$ $\\mathrm {km}\\,\\mathrm {s}^{-1}$ (Fig.","REF a).","The individual components, however, do not reach 100% absorption.","Of all seven components, the cloud at $+6.5$ $\\mathrm {km}\\,\\mathrm {s}^{-1}$ comes closest to having saturated absorption in HF.","As already noted by [18], even this component is only marginally optically thick, evidence for which is provided by the line width of the same component in the optically thin tracer CH [92] which is the same (1.5 $\\mathrm {km}\\,\\mathrm {s}^{-1}$ ) as for HF.","All other components are believed to be optically thin [18].","Absorption line depth is converted into optical depth using $I_\\mathrm {norm}=\\mathrm {e}^{-\\tau _\\nu }$ .","With the caution of one of the seven components being marginally saturated, we take the optically thin limit to relate optical depth integrated over the line profile, $\\int \\tau _\\nu \\,\\mathrm {d}V$ , to column density, $N_\\mathrm {HF}$ , following [56]: $\\int \\tau _{\\nu ,\\mathrm {HF}} \\mathrm {d}V = \\frac{A_\\mathrm {ul} g_\\mathrm {u} \\lambda ^3}{8\\pi g_\\mathrm {l}} N_\\mathrm {HF} = 4.16\\times 10^{-13} \\, [\\mathrm {cm}^2 \\, \\mathrm {km}\\, \\mathrm {s}^{-1}]\\ N_\\mathrm {HF} ,$ where $A_\\mathrm {ul}$ is the Einstein $A$ coefficient for spontaneous emission, 2.42$\\times $ 10$^{-2}$ s$^{-1}$ for HF 1–0 [68], $g_\\mathrm {u}$ =3 and $g_\\mathrm {l}$ =1 are the statistical weights of rotational levels $J$ =1 and $J$ =0, respectively, and $\\lambda $ is the wavelength of the transition, 243.24 $\\mu $ m. In addition to the optically thin limit, Eq.","(REF ) assumes that all HF molecules are in the rotational ground state, a fair assumption given its high $A_\\mathrm {ul}$ .","The conversion from $I_\\mathrm {norm}$ to $\\tau _\\nu $ follows a linear relation for low values of $\\tau _\\nu $ : $\\nonumber \\tau _\\nu & = & -\\ln (I_\\mathrm {norm}) \\\\& \\approx & 1-I_\\mathrm {norm} \\qquad [\\mathrm {for}\\ I_\\mathrm {norm} \\approx 1].$ This relation holds to within $\\sim $ 10% for $\\tau _\\nu <0.2$ (line absorbs up to 20% of the continuum), but $\\tau _\\nu /(1-I_\\mathrm {norm})$ is already 1.2 at 40% absorption, rising to 2 at 80% absorption.","The integrated optical depth is therefore systematically underestimated for a spectrally unresolved line that is smeared out over a velocity range wider than its intrinsic profile.The logarithm operator gives a disproportionally higher weight to more strongly absorbed channels.","For example, a 20 $\\mathrm {km}\\,\\mathrm {s}^{-1}$ wide feature with constant 80% absorption ($I_\\mathrm {norm}$ =0.20) and a smeared-out version of 320 $\\mathrm {km}\\,\\mathrm {s}^{-1}$ wide at 5% absorption ($I_\\mathrm {norm}$ =0.95) both have the same equivalent width, $\\int (1-I_\\mathrm {norm}) \\mathrm {d}V$ = 16 $\\mathrm {km}\\,\\mathrm {s}^{-1}$ .","The latter, however, yields a $\\int \\tau _\\nu \\mathrm {d}V$ that is smaller by a factor 2.","The effect is larger yet for line profiles that come even closer to full absorption.","Since the HF line is spectrally unresolved in our SPIRE spectra, a column density derived from these observations would merely constitute a lower limit to the true column density.","Contrary to the optical depth, the absorption depth integrated over the line profile, i.e., the equivalent width of the absorbed `area' below $I_\\mathrm {norm}$ =1, is conserved regardless of the spectral resolution.","This is confirmed by the matching equivalent width values measured by HIFI and SPIRE toward core I: $\\int (1-I_\\mathrm {norm}) \\mathrm {d}V$ is 15.7 and 16.4 $\\mathrm {km}\\,\\mathrm {s}^{-1}$ , respectively, with uncertainty margins of $\\sim $ 1 $\\mathrm {km}\\,\\mathrm {s}^{-1}$ in both cases.","In the remainder of this paper, we analyze HF absorption depth measured with SPIRE based exclusively on the conserved quantity, equivalent width, $\\int (1-I_\\mathrm {norm}) \\mathrm {d}V$.","When deriving optical depths and column densities, we rely exclusively on spectrally resolved profiles such as that in [18]."],["Distribution of HF absorbing clouds toward NGC 6334","The range of HF equivalent width values in the map shown in Fig.","REF can be divided into three regimes: (a) $>$ 12 $\\mathrm {km}\\,\\mathrm {s}^{-1}$ , only occurring toward the position of core I; (b) 8–12 $\\mathrm {km}\\,\\mathrm {s}^{-1}$ , spatially consistent with the larger scale filament in which cores I and I(N) are embedded; and (c) $\\lesssim $ 5 $\\mathrm {km}\\,\\mathrm {s}^{-1}$ , exclusively localized at projected distances $>$ 0.6 pc from the cores and the connecting filament.","The non-detection of HF in the `OFF' observation (see Sect.","), sparsely sampling the area just northwest of our map, is consistent with regimes (c) or (b)."],["Distinguishing foreground from dense star-forming gas","We interpret the three regimes in the context of the velocity resolved HF spectrum published by [18], who identified seven individual physical components responsible for the HF absorption toward NGC 6334 I: the dense envelope at $V_\\mathrm {lsr}$ =$-6.5$ $\\mathrm {km}\\,\\mathrm {s}^{-1}$ , two compact subcores at $-6.0$ and $-8.0$ $\\mathrm {km}\\,\\mathrm {s}^{-1}$ , and four foreground layers at $-3.0$ , $0.0$ , $+6.5$ , and $+8.0$ $\\mathrm {km}\\,\\mathrm {s}^{-1}$ .","The spectral signature of each of these components is reproduced in our Fig.","REF a.","Regime (a) requires all seven components to explain the total equivalent width of HF.","The two other panels in Fig.","REF represent adaptations of the model from [18] with progressively fewer absorption components taken into account.","In Fig.","REF b, regime (b) is explained by the superposition of four absorbing foreground clouds, discarding the components associated with the envelope and subcores of NGC 6334 I.","We highlight that the HF absorption depth observed toward the dense star-forming envelope I(N) is consistent with regime (b).","Differences in HF content between the dense envelopes I and I(N) are discussed in more detail in Sect.","REF .","Finally, in Fig.","REF c, we show that regime (c) is consistent with a model composed of just two specific foreground clouds, namely those at $+6.5$ and $+8.0$ $\\mathrm {km}\\,\\mathrm {s}^{-1}$ [92], [18].","In contrast with the detailed study of the HF profile in [18] and in this work, the spectral survey paper by [96], analyzing $\\sim $ 4300 individual spectral line features, uses a simplified model which explains the HF absorption with only three components.","The two approaches are not inconsistent, but the latter paper groups components together as: (1) the NGC 6334 I envelope and two subcores, (2) two foreground clouds that are kinematically close to the NGC 6334 complex, and (3) two other foreground clouds with larger $V_\\mathrm {lsr}$ offsets [18].","In this three-component model, regime (a) would be explained by groups (1)+(2)+(3), regime (b) by groups (2)+(3), and regime (c) by only group (3).","The combination of our HF absorption depth map with the previous, single-pointing, velocity resolved HF spectrum now reveals a three-dimensional picture of the layers of absorbing gas toward the NGC 6334 complex.","Our interpretation of the relative positions of the foreground layers is sketched in Fig.","REF .","With the exception of the direction toward core I, the HF absorption depth at all other positions in the map can be explained by (a subset of) the four extended foreground clouds.","Figure: Sketch of the location and extent of the foreground clouds.","Note the broken scale in the horizontal direction.","Velocity magnitudes are indicated with black arrows, shifted such that the core NGC 6334 I, at V lsr V_\\mathrm {lsr}=-6.5-6.5 km s -1 \\mathrm {km}\\,\\mathrm {s}^{-1}, is at rest in this frame.","From left to right, they represent the components referred to in the text as the +8.0+8.0, +6.5+6.5, +0.0+0.0, and -3.0-3.0 km s -1 \\mathrm {km}\\,\\mathrm {s}^{-1} clouds.","The dotted line represents the line of sight from the observer to NGC 6334 I.","The dashed lines indicate the angular extent of clouds on the far foreground.","The color differential between core I and the filament indicates the elevated temperature in core I, preventing freeze-out of HF molecules."],["Relation of foreground clouds to NGC 6334","[92] used a single-pointing Herschel HIFI observation of CH rotational line absorption, coupled with results from OH hyperfine line absorption measurements at radio wavelengths [11], to suggest that the clouds at $V_\\mathrm {lsr}$ =$-3.0$ and $+0.0$ $\\mathrm {km}\\,\\mathrm {s}^{-1}$ are associated with the NGC 6334 complex, while the remaining two velocity components, at $+6.5$ and $+8.0$ $\\mathrm {km}\\,\\mathrm {s}^{-1}$ , originate in foreground clouds farther away from the NGC 6334 complex.","This interpretation is consistent with the first two components being seen exclusively in regime (b) of the HF equivalent width map and the latter two components being spread over a more extended area of regimes (b) and (c) combined.","Therefore, the spatial distribution of HF absorption measured with SPIRE iFTS supports the previous hypothesis that the $-3.0$ and $+0.0$ $\\mathrm {km}\\,\\mathrm {s}^{-1}$ clouds are associated with NGC 6334, since they have a spatial morphology that closely follows the dense molecular cloud traced by the 250 $\\mu $ m dust emission (Fig.","REF ), i.e., a region roughly 2–3 in east-west extent and stretching along the north-south direction.","The vertical extent of the `related' foreground clouds depicted in Fig.","REF is drawn directly from the observed east-west extent of these component in the plane of the sky.","Besides the diatomic hydride species HF, CH, and OH, a subset of our absorbing clouds also exhibit absorption lines due to H$_2$ O [17], [89], H$_2$ O$^+$ [62], and H$_2$ Cl$^+$ [44].","Absorption components in OH$^+$ and H$_2$ O$^+$ detected toward both cores by [35] peak at $V_\\mathrm {lsr}$ =$-2$ and $+3$ $\\mathrm {km}\\,\\mathrm {s}^{-1}$ .","The former could be a blend of the $-3.0$ and $+0.0$ $\\mathrm {km}\\,\\mathrm {s}^{-1}$ clouds seen in CH and HF, but the latter is inconsistent with any of our components.","These lines were all detected in observations with the single-pixel, high spectral resolution Herschel HIFI spectrometer, in some cases toward both individual protostellar cores.","It is interesting that ionized species, H$_2$ O$^+$ and H$_2$ Cl$^+$ , are only detected at $V_\\mathrm {lsr}$ values that match the $-3$ and $+0$ $\\mathrm {km}\\,\\mathrm {s}^{-1}$ clouds.","Chemical models tailored to halogen hydrides [57] indicate that cation species become abundant under the influence of strong UV radiation.","We hypothesize that the presence of H$_2$ O$^+$ and H$_2$ Cl$^+$ is further evidence for the physical proximity of these two clouds to the massive protostars and the Hii region embedded in one of the dense cores.","Another ionized species that has been detected in spectrally resolved observations toward NGC 6334 is CH$^+$ .","While a velocity resolved observation of HF exists only toward the position of core I, CH$^+$ 1–0 has been observed with HIFI toward both cores I and I(N), the latter as part of the WISH program [93], [5].","Toward source I, the CH$^+$ profile looks very similar to the HF profile, amounting to a total equivalent width of $\\sim $ 20 $\\mathrm {km}\\,\\mathrm {s}^{-1}$ .","In comparison, the CH$^+$ observation toward core I(N)Estimate based on the HIFI spectrum downloaded from the Herschel Science Archive, observation ID 1342214306, processed with HIPE pipeline version 13. reveals an equivalent width of only $\\sim $ 13 $\\mathrm {km}\\,\\mathrm {s}^{-1}$ .","The majority (85%) of the reduced absorption toward core I(N) relative to core I falls in the $[-15,5]$ $\\mathrm {km}\\,\\mathrm {s}^{-1}$ range, as expected if the missing components are the envelope and subcore components at $V_\\mathrm {lsr}$ =$-6.5$ , $-6.0$ , and $-8.0$ $\\mathrm {km}\\,\\mathrm {s}^{-1}$ (cf. Fig.","REF ), i.e., those components that occur only in regime (a) in our HF absorption map.","As mentioned above in Sect.",", the SPIRE iFTS spectral cube has too low signal-to-noise near 835 GHz to make a meaningful comparison with the CH$^+$ signature detected by this instrument.","The foreground clouds at $V_\\mathrm {lsr}$ =$+6.5$ and $+8.0$ $\\mathrm {km}\\,\\mathrm {s}^{-1}$ , supposedly unrelated to the NGC 6334 dense filament [11], [92], have a combined HF equivalent width that is entirely consistent with the observation in this work that regime (c) is spatially extended beyond the dense filament.","An unrelated set of foreground cloud(s) (the two leftmost clouds in Fig.","REF ) are likely to have a larger angular extent than the background continuum source (dashed lines in Fig.","REF ), despite a possibly modest linear size.","This explains why a minimum level of HF equivalent width of $\\sim $ 3 $\\mathrm {km}\\,\\mathrm {s}^{-1}$ is detected not just toward the NGC 6334 filament, but throughout the extent of our 6$\\times $ 35 map (Figs.","REF and REF ).","All four foreground clouds detected in HF and CH are redshifted ($V_\\mathrm {lsr}$$\\ge $$-3$ $\\mathrm {km}\\,\\mathrm {s}^{-1}$ ) with respect to the $V_\\mathrm {lsr}$ of the part of NGC 6334 cloud near cores I and I(N) (around $-5$ $\\mathrm {km}\\,\\mathrm {s}^{-1}$ , based on HCO$^+$ observations by [97]).","Thus, the absorbing gas clouds are moving toward NGC 6334, instead of following the Galactic rotation, which at $\\ell =351$ yields exclusively negative line-of-sight velocities for sources between Sun and NGC 6334, i.e., approaching the standard of rest of stars in the Solar neighborhood.","We therefore conclude that, in addition to the gas flows within the dense gas along the filament's long axis [97], gas may also be accreting onto the filament in the perpendicular direction.","We also note that, whereas [35] put the OH$^+$ and H$_2$ O$^+$ absorption clouds toward both cores I and I(N) at the distance of 1.35 kpc of the NGC 6334 cloud, their location along the sight line toward NGC 6334 is in fact poorly constrained.","Recognizing that there must be peculiar motions at play, deviating from the `rigid' Galactic rotation curve, these clouds could in fact be anywhere between the local arm of the Milky Way and the Sagittarius arm that harbors the NGC 6334 complex.","For the two foreground clouds related to NGC 6334, at $-3$ and $+0$ $\\mathrm {km}\\,\\mathrm {s}^{-1}$ , we calculate their total mass by multiplying the sum of their column densities with a rough estimate of the area covered by this component of 160$\\times $ 270 (regime (b) in Fig.","REF ), corresponding to 1.5 pc$^2$ at the distance of NGC 6334.","Depending on the choice of fiducial H$_2$ tracer, either taking $N_\\mathrm {CH}$ from [92] and $N_\\mathrm {CH}$ /$N_\\mathrm {H_2}$ =(2.1–5.6)$\\times $ 10$^{-8}$ from [82], or $N$ (HF) from [18] and $N_\\mathrm {HF}$ /$N_\\mathrm {H_2}$ =(0.5–1.4)$\\times $ 10$^{-8}$ from [36], this calculation yields a mass in the range 37–98 $M_\\odot $ or 69–191 $M_\\odot $ , respectively.","From symmetry arguments, a similar reservoir of additional gas is expected to lie behind the NGC 6334 dense cloud.","This means that a significant total mass of several hundred $M_\\odot $ could be on its way to accreting onto the dense cloud NGC 6334 near ($<$ 0.3 pc) the embedded cores I and I(N).","This gas reservoir has escaped detection so far, because it does not appear in traditional gas tracers such as CO, HCO$^+$ , and CS.","While there is evidence that the $-3.0$ and $+0.0$ $\\mathrm {km}\\,\\mathrm {s}^{-1}$ foreground clouds are closer to NGC 6334 than the $+6.5$ and $+8.0$ $\\mathrm {km}\\,\\mathrm {s}^{-1}$ clouds, there is no direct metric of the geometrical distance along the line of sight from each cloud to the dense filament and cores.","Therefore, we refrain from speculating about accretion time scales of even the `related' clouds, since this would rely on unsupported assumptions on relative distances."],["Spatial distribution of HF toward other Galactic sight lines","The only other published work of a spatial map of HF absorption so far is that toward Sgr B2(M) by [19], using data also obtained with Herschel SPIRE iFTS.","A direct comparison of measured absorption line depths is complicated by the choice of [19] to present signal in terms of integrated optical depth, apparently without taking into account the systematic underestimation of optical depths derived from spectrally unresolved measurements, as discussed in Sect.","REF .","Nonetheless, it appears that the total HF absorption toward Sgr B2(M) is about an order of magnitude stronger than toward NGC 6334 I.","[19] find a variation of HF absorption depth of only a factor $\\sim $ 2 across the $\\sim $ 25 mapped area, significantly less variation than in our fully sampled map toward NGC 6334 I.","Any intrinsic variation may have been partially smoothed by the interpolation process that was applied in [19] to construct a map from the spatially undersampled SPIRE iFTS observation.","More importantly, the line of sight toward Sgr B2, close to the Galactic Center, crosses many more spiral arms than that toward NGC 6334.","Evidence of this is found for example by [73], who detect a total of 31 individual velocity components in CH rotational ground state absorption toward Sgr B2(M).","The 30 foreground clouds, associated with the various intervening Galactic arms, amount to a total CH column density of 1.7$\\times $ 10$^{15}$ $\\mathrm {cm}^{-2}$ , with Sgr B2(M) itself adding a component of only 0.5$\\times $ 10$^{15}$ $\\mathrm {cm}^{-2}$ [73].","This is consistent with many sheets of foreground gas together creating a roughly uniform cover of absorbing gas spanning at least a few arcminutes on the sky.","Comparatively little additional absorption is contributed by the massive molecular cloud Sgr B2 and the embedded cores in the background, which explains the lack of variation in HF absorption depth seen toward Sgr B2 by [19].","In contrast, our map of HF absorption toward NGC 6334 reveals a mix of components due to foreground clouds and the star-forming envelope and cores within NGC 6334, which we are able to disentangle owing to the complementary, velocity resolved spectra of CH and HF obtained with Herschel HIFI [92], [18].","In addition, we highlight the discovery by [46] of a foreground cloud toward the intermediate-mass star-forming core OMC-2 FIR 4, based on single-pointing HIFI spectra of HF and other hydride molecules.","Several oxygen-bearing hydrides show absorption exclusively at a blueshifted velocity relative to the background protostellar core.","The same foreground cloud was later also identified in H$_2$ Cl$^+$ by [38].","The HF profile shows absorption at the same blueshifted velocity, but shows additional evidence for a second absorption component that matches the $V_\\mathrm {lsr}$ of the protostellar core [46].","Instead of kinematical and morphological arguments such as those used in this work to determine the physical location of foreground clouds toward NGC 6334, [46] use detailed photochemical modeling to infer proximity of their OMC-2 foreground gas to a source of copious far-UV radiation.","With that radiation source assumed to be the trapezium cluster of OB stars, it is concluded that the absorbing slab is physically connected to OMC-1.","In an attempt to study the spatial distribution of this absorbing OMC-1 `fossil' slab, we have searched archival SPIRE iFTS data toward OMC-2 FIR 4 (Herschel observation ID 1342214847) for signatures of HF at 1232.5 GHz (and CH$^+$ at 835.1 GHz), but find no detections in any of the 37 (and 19) SSW (and SLW) detectors in the $\\sim $ 3 footprint.","The HF equivalent width of 15.9$\\pm $ 1.4 $\\mathrm {km}\\,\\mathrm {s}^{-1}$ measured at the position of NGC 6334 I in our SPIRE map is explained in Sect.","REF by invoking the superposition of four absorption components in the foreground and three associated to the dense core I itself [18].","A striking feature of our HF absorption map is the lack of additional absorption toward the position of core I(N), where the HF equivalent width of only 10.9$\\pm $ 1.1 $\\mathrm {km}\\,\\mathrm {s}^{-1}$ can be explained by the four foreground clouds alone, adding up to 10.5 $\\mathrm {km}\\,\\mathrm {s}^{-1}$ (Fig.","REF b).","In contrast, adding even just an envelope component similar to that of core I sums up to 13.7 $\\mathrm {km}\\,\\mathrm {s}^{-1}$ (not counting the two subcore components), which is inconsistent with the observation toward core I(N).","Since the envelope of core I(N) is more massive than that of core I, but has a similar size [89], the total gas column density toward core I(N) should be higher.","Therefore, the lack of HF absorption associated to the I(N) core is not due to the difference in total (H$_2$ ) column.","Instead, we hypothesize that HF is primarily frozen out onto dust grains in core I(N), while HF is in the gas phase in core I.","To support the hypothesis of HF being depleted from the gas phase in core I(N), we set up a rudimentary model based on the following ingredients.","We take the spherically symmetric physical structure, i.e., radial profiles of density and temperature, of the envelopes of NGC 6334 I and I(N) as fitted to submillimeter dust continuum maps and the far-infrared / submillimeter spectral energy distribution [89].","We then calculate, at every radial point, the timescales for adsorption (freeze-out) and desorption (evaporation) of HF molecules onto dust grains.","Following, e.g., [76] and [37], we assume that thermal desorption is the dominant mechanism that drives molecules from the grain surface back into the gas phase, and are left with the balance between adsorption rate: $\\lambda (n_\\mathrm {H}, T_\\mathrm {gas}) = 4.55\\times 10^{-18} \\left( \\frac{T_\\mathrm {gas}}{m_\\mathrm {HF}} \\right)^{0.5} n_\\mathrm {H} \\qquad [\\mathrm {s}^{-1}] ,$ and desorption rate: $\\xi (T_\\mathrm {dust}) = \\nu _\\mathrm {vib} \\exp \\left( -\\frac{E_\\mathrm {b,HF}}{k\\,T_\\mathrm {dust}} \\right) \\qquad [\\mathrm {s}^{-1}] .$ Here, $T_\\mathrm {gas}$ and $T_\\mathrm {dust}$ are the temperatures of gas and dust, assumed to be equal as in the modeling of [89], $m_\\mathrm {HF}$ is the molecular weight of HF (20), $n_\\mathrm {H}$ is the density of hydrogen nuclei, $\\nu _\\mathrm {vib}$ is the vibrational frequency of the HF molecule in its binding site, for which we adopt $10^{13}$ s$^{-1}$ , $k$ is the Boltzmann constant, and $E_\\mathrm {b,HF}$ is the binding energy of HF to the dust grain surface.","It has previously been inferred by [67] that a density of $\\sim $$10^5$ $\\mathrm {cm}^{-3}$ allows HF to condense onto dust grains, whereas densities of $\\sim $$10^3$ $\\mathrm {cm}^{-3}$ more typical for diffuse gas are too low for HF freeze-out to occur.","In our case, the density $n_\\mathrm {H}$ – which incidentally exceeds $10^5$ $\\mathrm {cm}^{-3}$ at almost all radii in the envelopes of I and I(N) – enters directly into Equation REF to govern the adsorption rate.","Figure: Radial dependence of adsorption and desorption timescales of HF from various types of grain surfaces.","The top half of the figure relates to the envelope of core I(N), the bottom half to that of core I.","The small panels above each main panel depict the physical structure (temperature in solid blue, left axis; density in dashed red, right axis) for the envelopes in question from .","The horizontal axis is logarithmic cumulative mass increasing from left to right; labels in brackets indicate the percentage of enclosed mass starting from the outer shell of the core.","Freeze-out of HF (depletion from the gas phase) occurs in the region where t desorb >t freeze t_\\mathrm {desorb} > t_\\mathrm {freeze}; light shading indicates the case of H 2 _2O ice mantles, darker shading that of CO/CO 2 _2 ice mantles.","The desorption time scale lines for CO ice and CO 2 _2 ice in the bottom panel lie to the far right, outside of the limits of the axes.In this work we consider multiple versions of the desorption timescale, because the binding energy in the exponent of Equation REF is heavily dependent on the type of grain surface.","Unlike for more common molecular species such as CO [6], [60], the desorption behavior of HF from astrophysically relevant grain surfaces has not been studied experimentally, so we rely on theoretical calculations.","Typical interstellar dust grains, especially those embedded in cold, star-forming regions, are covered in one or multiple layers of ice consisting of various molecules, mainly H$_2$ O, CO, and CO$_2$ [9].","We collect binding energy values for several types of grain surfaces in Table REF .","For CO and CO$_2$ ice covered grains we adopt calculated binding energies from the literature [13], [75], while for hydrogenated bare silicate grains and H$_2$ O ice covered grains, these values result from original ab initio chemical calculations performed for this work.","Table: Binding energies for HF onto various surfaces.To calculate the HF binding energies in the first two rows of Table REF (with a bare grain and with H$_2$ O ice), we carry out quantum calculations within the Kohn-Sham implementation of Density Functional Theory using the Quantum Espresso Simulation Package [25].","Perdew-Burke-Ernzerhof exchange-correlation functional ultrasoft pseudopotentials are used.","KS valence states are expanded in a plane-wave basis set with a cutoff at 340 eV for the kinetic energy.","The self-consistency of the electron density is obtained with the energy threshold set to $10^{-5}$ eV.","Calculations are performed using the primitive unit cell containing a total number of 46 atoms for bare hydrogenated silica, and 54 atoms for hydrogenated silica covered with one layer of H$_2$ O ice.","The geometry optimization is used within the conjugate gradients scheme, with a threshold of 0.01 eV $Å^{-1}$ on the Hellmann-Feynman forces on all atoms; the Si atoms of the bottom layers are fixed at their bulk values.","The binding energy of HF with the SiH terminus of hydrogenated crystalline silica is based on calculations for the hydroxylated alpha-quartz (001) surface.","The binding energy of HF with one layer of H$_2$ O ice on amorphous hydrogenated silicate is estimated by assuming that the the most common structure in this case is the HF molecule interacting with a H$_2$ O molecule bonded to silanol (SiOH), which is the most abundant surface group in amorphous silica [20].","The binding energies of HF with CO and CO$_2$ ice (last two rows of Table REF ) are taken from calculations by [75] and [13], respectively.","These authors performed calculations for molecules in the gas phase.","We consider the gas phase binding energies of HF with CO and CO$_2$ to be similar to those of HF with CO and CO$_2$ ices adsorbed on an inert surface such as that of hydroxylated amorphous silica.","This approximation is based on the weak interactions of these ices with hydroxylated silica and within the CO and CO$_2$ molecular solids, so that the electronic density of CO and CO$_2$ in solid form is not significantly altered with respect to their state in the gas phase.","Hence, for the aim of the present work, the binding energy of the HF molecule with CO or CO$_2$ as calculated in the gas phase is applicable for the condensed phase.","The situation is notably different for interactions with H$_2$ O in the gas or adsorbed form, because of its stronger interaction with silica and HF.","For HF interacting with H$_2$ O ice, we use binding energies from our own calculations described above.","With the physical structure of both envelopes, the equations for the adsorption/desorption balance, and the binding energy values, the `freeze-out' region within each envelope is calculated in Fig.","REF .","Defining $t_\\mathrm {freeze} = 1/\\lambda $ and $t_\\mathrm {desorb}=1/\\xi $ , HF molecules will deplete from the gas phase in the region of the envelope wherever $t_\\mathrm {desorb}$ > $t_\\mathrm {freeze}$ .","Numerical simulations by [15] suggest that, within a mixed-composition ice layer, the abundances of CO and CO$_2$ are enhanced compared to H$_2$ O in high-density ($\\gtrsim 10^5$ $\\mathrm {cm}^{-3}$ ) environments.","At such densities applicable for the protostellar envelopes studied here, we thus expect the HF binding energy to lie close to, but slightly above that of pure CO or CO$_2$ ice, and the true desorption time scale line in Fig.","REF therefore somewhat to the left of the dash-dotted line for CO ice.","In this case, HF is expected to stay frozen onto grain surfaces at a broad range of radii in core I(N): cumulative mass $\\sim $ 0.1 to 1, i.e., 90% of the mass, where the temperature is $\\lesssim $ 20 K. Core I(N) is overall colder than core I both in the envelope (blue solid lines in Fig.","REF ) and in the embedded subcores [34].","For the comparatively warmer envelope of core I, the lines for the desorption timescale of HF from CO/CO$_2$ ice fall off the scale on the right hand side of the axes, leaving no freeze-out zone in this envelope.","This could explain why HF is seen in the gas phase in core I, but not in core I(N).","In an alternative scenario in which the dust grains are covered in pure H$_2$ O ice – or the desorption characteristics of a mixed mantle are dominated by that of H$_2$ O ice [14] – the binding energy of HF with the ice would be greatly increased (see Table REF ).","In this case, the HF freeze-out zone would expand to cover $>$ 98% of the mass for both envelopes, and our observations should have revealed no gas phase HF in either of the cores.","Our observations are therefore inconsistent with pure H$_2$ O ice coating on the grains.","Instead, our interpretation relies on a significant part of the ice coating to consist of CO and/or CO$_2$ molecules.","In principle, the composition of ice coatings do not need to be the same in the two neighboring envelopes.","Particularly, given the lower temperatures of core I(N), there is higher probability that a significant amount of H$_2$ O is frozen out in that envelope.","This, again, would enhance the binding energy of HF onto the ice-covered dust grains in the envelope of core I(N), which may help to explain the lack of gas-phase HF observed toward source I(N).","Conversely, if additional mechanisms for desorption, e.g., induced by (UV) photons or cosmic rays, would be taken into account, $\\xi $ would increase, and the freeze-out region would be pushed to larger radii in the envelopes.","Particularly UV photodesorption could have a different effect in one envelope compared to the other, because core I is more evolved and contains an Hii region.","An increased total desorption rate would have no effect on HF freeze-out in the envelope of core I, in which HF is already completely in the gas phase, but would reduce the size of the freeze-out zone for envelope I(N).","If, however, thermal desorption as expressed in Equation REF is the dominant desorption mechanism and the binding energy of HF onto the ice surface is close to that of CO or CO$_2$ ice (Table REF ), our model predicts significant freeze-out of HF in core I(N), and none in core I.","It is important to note that, indeed, the temperature and density conditions under which HF remains frozen onto dust grains depend greatly on the exact composition and mixing of the ice mantle and therefore on the chemical history.","It has already been recognized for example for the CO molecule that the freeze-out temperature `threshold' can vary considerably from one object to another [72]."],["Conclusions","In this work we present a map of HF absorption toward the northern end of the molecular cloud NGC 6334, containing two well studied massive star-forming cores I and I(N).","Although in the original definition of the observing program it was not anticipated that hydride absorption lines would be found within these data, the discovery space provided by the enormous frequency coverage of the Herschel SPIRE iFTS instrument has made this study possible.","Such wide coverage in the far-infrared/submillimeter is only attainable with broadband FTS spectrometers [55].","The absorption line of HF is detected in 80% of our mapped area, although it is spectrally unresolved by SPIRE.","By complementing the new SPIRE iFTS data with existing, single pointing, high spectral resolution spectra from the Herschel HIFI instrument [92], [18], we construct a three-dimensional picture of gas clouds in front of and inside the massive star-forming filament NGC 6334.","We find that our observations are consistent with a scenario of four individual foreground clouds on the line of sight toward NGC 6334 I and I(N), two of which are unrelated to the star-forming complex (Sect.","REF ).","The other two clouds are posited to be close to the dense molecular filament based on their spatial morphology.","Their velocities are such that they are moving toward the star-forming cloud and could be adding several hundreds of solar masses of gas to the dense filament and the embedded cores in which massive star formation is already ongoing (Sect.","REF ).","This component of gas is detected in rotational lines of diatomic hydride molecules, but had been unseen in studies of traditional dense gas tracers.","In fact, using such a tracer, HCO$^+$ , [97] have inferred that the roughly cylindrically shaped NGC 6334 filament is collapsing along its longest axis.","Our work now indicates that accretion may also be ongoing in the perpendicular (radial) direction.","Future studies of (competitively) accreting high-mass star-forming cores may need to take into account this additional low-density phase of the gas reservoir.","Finally, in Sect.","REF we explain why HF is observed in the gas phase toward core I, but appears completely absent in core I(N).","For this purpose, we use a simple description of adsorption and desorption time scales for HF interacting with dust grain surfaces, depending on the (radially variable) density and temperature.","Since interactions of HF with interstellar-like dust grains have not been studied in the laboratory, we adopt binding energy values for different types of grain surfaces from theoretical calculations from the literature as well as from original work first presented in this paper.","The conclusion is that the lower temperature of core I(N) compared to core I could lead to freeze-out of HF exclusively in the former, but only if the binding energy of HF onto the grain surface is governed by that of CO or CO$_2$ ice on a silicate surface.","In this case, at the densities relevant in the envelope of source I(N) ($>$ 3$\\times $ 10$^{5}$ $\\mathrm {cm}^{-3}$ ), we find that HF freezes out in the region of the envelope where the temperature is below $\\sim $ 20 K, rather similar to the freeze-out temperature often adopted for CO.","In contrast, if H$_2$ O is the dominant constituent in the ice mantles, our model predicts that HF should have been frozen out at all radii in the envelopes of both sources I(N) and I.","Since we observe a significant amount of HF in the gas phase in source I, this scenario is inconsistent with our data.","Summarizing, this work uses HF as a sensitive tracer for (molecular) gas at relatively low densities that may be contributing mass to star forming cores.","The HF signature reveals a gas reservoir that is inconspicuous in traditional dense gas tracers such as CO.","In addition, we show that gas phase HF in higher density environments ($>$$10^5$ $\\mathrm {cm}^{-3}$ ) is extremely sensitive to interactions with dust grains and will be depleted significantly at low dust temperatures.","The research of MHDvdW at the University of Lethbridge was supported by the Canadian Space Agency (CSA) and the Natural Sciences and Engineering Research Council of Canada (NSERC), and at the University of Copenhagen by the Lundbeck Foundation.","Research at the Centre for Star and Planet Formation is funded by the Danish National Research Foundation and the University of Copenhagen's programme of excellence.","SPIRE has been developed by a consortium of institutes led by Cardiff University (UK) and including Univ.","Lethbridge (Canada); NAOC (China); CEA, LAM (France); IFSI, Univ.","Padua (Italy); IAC (Spain); Stockholm Observatory (Sweden); Imperial College London, RAL, UCL-MSSL, UKATC, Univ.","Sussex (UK); and Caltech, JPL, NHSC, Univ.","Colorado (USA).","This development has been supported by national funding agencies: CSA (Canada); NAOC (China); CEA, CNES, CNRS (France); ASI (Italy); MCINN (Spain); SNSB (Sweden); STFC, UKSA (UK); and NASA (USA).","HIPE is a joint development by the Herschel Science Ground Segment Consortium, consisting of ESA, the NASA Herschel Science Center, and the HIFI, PACS and SPIRE consortia.","This research has made use of NASA's Astrophysics Data System Bibliographic Services.","The graphical representations of the results in this paper were created using APLpy, an open-source plotting package for Python hosted at http://aplpy.github.com, Astropy, a community-developed core Python package for Astronomy (Astropy Collaboration [3]), and the matplotlib plotting library [32].","The authors are grateful to Alexander Zernickel for providing and discussing excerpts of the CHESS spectral survey data, to Raquel Monje for providing the HF spectral model component profiles, and to Floris van der Tak for providing the physical structure models of the two envelopes in electronic table format.","We thank Tommaso Grassi, Wing-Fai Thi, Jes Jørgensen, Søren Frimann, and Mihkel Kama for discussions."],["Complementary figures","Maps displaying the signal-to-noise ratio of the detections of the HF and CH$^+$ lines in each spatial pixel of our spectral cube (Sect. )","are shown in Fig.","REF and REF .","The colored contours in Fig.","REF show that CH$^+$ absorption is only confidently detected (signal-to-noise $>$ 5) in the northeastern section of the map.","In addition, although the signal from nitrogen species is not interpreted in this paper, maps of line absorption depth due to NH and NH$_2$ are shown in Figs.","REF and REF .","The continuum-normalized cube for these lines is created – analogous to those for HF and CH$^+$ in Sect.","– by fitting the continuum and the CO 9–8 line in the 945–1055 GHz section of the SSW cube.","Absorption lines of NH (at 974.47 and 999.98 GHz) and two of NH$_2$ (at 952.57 and 959.50 GHz) are then fitted simultaneously with emission lines due to $^{13}$ CO 9–8 at 991.3 GHz and H$_2$ O $2_{02}$ –$1_{11}$ at 987.9 GHz.","Figure: Placement of the SPIRE iFTS footprints on the NGC 6334 region (cf. Table ).","Solid circles indicate the fully sampled observations, two centered on cores I and one on I(N), while the sparsely sampled `INT' and `OFF' observations are marked by dashed circles.","Circles are drawn with a diameter of 31 to include the vignetted outer ring of the detector arrays .","The grayscale represents the same 250 μ\\mu m dust continuum map as in Fig.",", but with the scale bar stretching from 1 to 250 GJy sr -1 ^{-1}.","The thin white contours, the beam size indicators in the bottom left and the scale bar in the top left corners are the same as in Fig.",".Figure: Signal-to-noise maps (color scale, with gray indicating values below 2) of the absorption signature of HF 1←\\leftarrow 0.","Colored contours are at the same equivalent width levels as in Fig. .","As in Fig.",", white contours trace continuum dust emission and the beam of the SPIRE iFTS map is shown in the bottom left corner.Figure: As in Fig.",", but for CH + ^+ 1←\\leftarrow 0.","Colored contours are at 22, 19, 16 and 13 km s -1 \\mathrm {km}\\,\\mathrm {s}^{-1} in gray, blue, cyan and magenta, respectively.Figure: As in Fig.",", but for the JJ=2←\\leftarrow 1 NN=1←\\leftarrow 0 transition of NH at 974.47 GHz.","The JJ=1–1 transition at 1000.0 GHz shows a similar absorption distribution, but with detections in fewer pixels.","Colored contours are at 16.5, 13.5, and 10.5 km s -1 \\mathrm {km}\\,\\mathrm {s}^{-1} in gray, blue, and cyan, respectively.Figure: As in Fig.",", but for the 1 11 1_{11}←\\leftarrow 0 00 0_{00} JJ=3/2←\\leftarrow 1/2 transition of NH 2 _2 at 952.57 GHz.","The JJ=1/2←\\leftarrow 1/2 transition at 959.50 GHz shows a similar absorption distribution, but with detections in fewer pixels.","Colored contours are at 15.0, 11.5, and 8.0 km s -1 \\mathrm {km}\\,\\mathrm {s}^{-1} in gray, blue, and cyan, respectively."]],"string":"[\n [\n \"Three-dimensional distribution of hydrogen fluoride gas toward NGC6334 I\\n and I(N)\"\n ],\n [\n \"Abstract Aims.\",\n \"We investigate the spatial distribution of a collection of absorbing gas clouds, some associated with the dense, massive star-forming core NGC6334 I, and others with diffuse foreground clouds.\",\n \"For the former category, we aim to study the dynamical properties of the clouds in order to assess their potential to feed the accreting protostellar cores.\",\n \"Methods.\",\n \"We use spectral imaging from the Herschel SPIRE iFTS to construct a map of HF absorption at 243 micron in a 6x3.5 arcmin region surrounding NGC6334 I and I(N).\",\n \"Results.\",\n \"The combination of new, spatially fully sampled, but spectrally unresolved mapping with a previous, single-pointing, spectrally resolved HF signature yields a 3D picture of absorbing gas clouds in the direction of NGC6334.\",\n \"Toward core I, the HF equivalent width matches that of the spectrally resolved observation.\",\n \"The distribution of HF absorption is consistent with three of the seven components being associated with this dense star-forming envelope.\",\n \"For two of the remaining four components, our data suggest that these clouds are spatially associated with the larger scale filamentary star-forming complex.\",\n \"Our data also implies a lack of gas phase HF in the envelope of core I(N).\",\n \"Using a simple description of adsorption onto and desorption from dust grain surfaces, we show that the overall lower temperature of the envelope of source I(N) is consistent with freeze-out of HF, while it remains in the gas phase in source I.\",\n \"Conclusions.\",\n \"We use the HF molecule as a tracer of column density in diffuse gas (n(H) ~ 10^2 - 10^3 cm^-3), and find that it may uniquely trace a relatively low density portion of the gas reservoir available for star formation that otherwise escapes detection.\",\n \"At higher densities prevailing in protostellar envelopes (>10^4 cm^-3), we find evidence of HF depletion from the gas phase under sufficiently cold conditions.\"\n ],\n [\n \"Introduction\",\n \"The hydrogen fluoride molecule, HF, was first observed in the interstellar medium by [59] with the Infrared Space Observatory [41].\",\n \"While ISO had a wavelength range that encompassed only the $J$ =2–1 rotational transition of HF, the next observatory able to observe HF – the Herschel Space Observatory [69] – covered longer far-infrared wavelengths, and it thus opened up access to the ground-state rotational transition, $J$ =1–0, at 1232.48 GHz (243.24 $\\\\mu $ m).\",\n \"Herschel has observed HF in absorption along many lines of sight, both inside the Galaxy [56], [84], [85], [67], [42], [51], [18], [46], [27] and in nearby extragalactic objects [74], [39], [79], [52].\",\n \"HF absorption has even been detected with ground-based observatories: [53] have made use of the substantial redshift of the Cloverleaf quasar at $z$ =2.56 shifting the HF 1–0 line into the submillimeter window attainable with the CSO on Mauna Kea, and [40] detect it in the $z$ =0.89 absorber toward PKS1830$-$ 211, using ALMA in the Chilean Atacama desert.\",\n \"Because of its large dipole moment and high Einstein $A$ coefficient for radiative decay, rotational states $J$$\\\\ne $ 0 of HF only become significantly populated in highly energetic conditions.\",\n \"It is for this reason that HF has been clearly detected in emission in a mere handful of cases: in the inner region of an AGB star's envelope [2], in the Orion Bar photodissociation region [90], and in an external galaxy harboring an actively accreting black hole [91].\",\n \"Atomic fluoride, F, has a unique place in the interstellar chemistry of simple molecules.\",\n \"It is the only element which, simultaneously, (1) is mainly neutral because of its ionization potential $>$ 13.6 eV, (2) reacts exothermically with H$_2$ – unlike any other neutral atom – to form its neutral diatomic hydride HF, and (3) lacks an efficient chemical pathway to produce its hydride cation HF$^+$ due to the strongly endothermic nature of the reaction with H${_3}^+$ .\",\n \"We refer to [57], references therein, and the comprehensive review by [22] for more details on the chemistry of HF and a comparison with other hydride molecules.\",\n \"For the reasons listed above, chemical models predict that essentially all interstellar F is locked in HF molecules [98], [58], which has been confirmed by observations across a wide range of atomic and molecular ISM conditions [84], [85].\",\n \"With recent experimental results by [88] showing that, especially at low temperatures approaching 10 K, the reaction F + H$_2$ $\\\\rightarrow $ HF + H proceeds somewhat slower than earlier assumptions, chemical models are now able to reproduce HF/H$_2$ ratios of $\\\\sim $ 1$\\\\times $ 10$^{-8}$ , measured most directly by [36], and observed to be rather stable across different sightlines.\",\n \"Interferometric observations show that CF$^+$ , the next most abundant F-bearing species after HF, has an abundance roughly two orders of magnitude lower than HF, both inside our Galaxy [45] and in an extragalactic absorber [54].\",\n \"As for destruction of HF, the most efficient processes are UV photodissociation and reactions with C$^+$ , but both of these are unable to drive the majority of fluoride out of HF, due to shielding, already at modest depths of $A_V>0.1$ [58].\",\n \"Because of the constant HF/H$_2$ abundance ratio and the high probability that HF molecules are in the rotational ground state, measurements of HF $J$ =0$\\\\rightarrow $ 1 absorption provide a straightforward proxy of H$_2$ column density.\",\n \"This has led to the suggestion that, at least in diffuse gas, HF absorption is a more reliable tracer of total gas column density than the widely used carbon monoxide (CO) rotational emission lines, and is more sensitive than CH or H$_2$ O absorption [22].\",\n \"Apart from the uncertain and variable CO abundance, local excitation conditions have a profound effect on the level populations of CO, complicating the conversion from observed line strength of a particular CO transition to H$_2$ column density [7].\",\n \"The greatest gas-phase CO abundance variations occur in dense, cold regions where CO freezes out onto surfaces of dust grains, proven by observed CO abundances decreasing in the gas phase and increasing in the ice phase as conditions get colder [37], [71].\",\n \"In addition, the particular fraction of the neutral ISM that is in the diffuse/translucent phase is inconspicuous in CO [7], but is detectable using hydride absorption lines.\",\n \"Of course, for absorption line studies, one relies on lines of sight with sufficiently strong continuum background, for example those toward dense star-forming clouds.\",\n \"Such restrictions do not apply for emission line tracers.\",\n \"Besides CO rotational lines, fine structure line emission due to atomic C and the C$^+$ and N$^+$ ions has been used as a tracer of (diffuse) gas throughout the Galaxy [43], [94], [23], [28], [29].\",\n \"For all these tracers, however, the conversion to H$_2$ column density depends strongly on physical properties such as ionization fraction and excitation conditions.\",\n \"Based on the above arguments, HF absorption measurements are a good tracer of overall gas column density.\",\n \"However, as addressed for example by [67] and [18], HF itself may suffer from freeze-out effects as occurs with other interstellar molecules.\",\n \"While studies have been done on the interaction of H$_2$ O with HF as a polluting agent in the Earth's atmosphere [26], the density and temperature conditions needed for HF adsorption onto dust grains have not been studied in astrophysical contexts so far.\",\n \"Any freeze-out of interstellar HF will obfuscate the direct connection between HF absorption depth and H$_2$ column density described above.\",\n \"The well-known progression of pre- and protostellar stages for stars with masses similar to the Sun [83] is not applicable for high-mass stars ($\\\\gtrsim 8$ $M_\\\\odot $ ).\",\n \"In the latter category, protostellar hydrogen fusion starts while accretion from the surrounding gas envelope is still ongoing [64].\",\n \"In the `competitive accretion' scenario, multiple massive protostars in a clustered environment are fed from the same gas reservoir [8].\",\n \"For high-mass protostars, material can continuously be added to the gravitationally bound circumstellar envelope which provides the reservoir for further accretion onto the protostar.\",\n \"It is therefore important, particularly for regions of high-mass star formation, to study not just the gravitationally bound circumstellar envelopes, but also the dynamical properties of surrounding gas clouds.\",\n \"Especially for the latter component, simple hydride molecules have the potential to reveal gas reservoirs to which emission lines of `traditional' tracer species, such as CO, HCO$^+$ , and CS, are insensitive due to their relatively high critical densities.\",\n \"In this paper, we investigate two envelopes of (clusters of) protostars embedded in the NGC 6334 molecular cloud as well as lower density clouds surrounding the dense complex.\",\n \"The filamentary, star-forming cloud NGC 6334, at a distance of 1.35 kpc [95], harbors a string of dense cores, identified in the far-infrared by roman numerals I through VI [50], with an additional source identified $\\\\sim $ 2 north of source I, later named `I(N)' [24].\",\n \"The larger scale NGC 6334 filament has an H$_2$ column density of $>$ 2$\\\\times $ 10$^{22}$ $\\\\mathrm {cm}^{-2}$ even at positions away from the embedded cores [80].\",\n \"[97] observed the velocity structure of NGC 6334 at 0.15 pc resolution.\",\n \"These authors explain the velocity profile along the filament with a cylindrical model collapsing along its longest axis under the influence of gravity.\",\n \"In this paper we study specifically the region of $\\\\sim $ 2.4$\\\\times $ 1.6 pc surrounding the embedded cores NGC 6334 I and NGC 6334 I(N).\",\n \"Source I is host to an ultra-compact Hii region, designated source `F' in a 6 cm radio image of the cloud [77].\",\n \"Based on multi-wavelength dust continuum measurements, studies by [81] and [89] have independently determined that the mass of source I(N) exceeds that of sister source I by a factor of $\\\\sim $ 2–5, but the ratio of their bolometric luminosities is 30–140 in favor of source I, due to the markedly lower temperature for source I(N).\",\n \"As expected for a warm (up to $\\\\sim $ 100 K), dense, massive star-forming core, NGC 6334 I is extremely rich in molecular lines, spectacularly demonstrated by the 4300 lines detected in the 480–1907 GHz spectral survey by [96]See the introduction section of this reference for a list of earlier spectral survey work on NGC 6334 I..\",\n \"The differences between the two neighboring cores all suggest that core I is in a more evolved stage of star formation than core I(N).\",\n \"Both cores have been studied with radio and (sub)millimeter interferometer observatories, showing that each separates into several subcores at arcsecond resolution [34], [33], [10].\",\n \"To probe gas clouds in front of the NGC 6334 complex, absorption measurements have been obtained in lines of several hydrides.\",\n \"Spatially extended OH hyperfine line absorption at 1.6–1.7 GHz was observed toward the NGC 6334 filament by [11].\",\n \"The spectrally resolved mapping observations from the Australia Telescope Compact Array allowed these authors to ascribe particular velocity components of the absorption to a foreground cloud close to NGC 6334 and other components to clouds with even larger angular extent.\",\n \"[92] used the Heterodyne Instrument for the Far-Infrared [16] onboard Herschel to study the spectral profile of the rotational ground state lines of CH at 532 and 537 GHz, and found four distinct absorption components overlapping with the velocity range of OH absorption, and one single emission component emanating in core NGC 6334 I itself.\",\n \"At 1232.5 GHz in the same spectral survey, [18] find the exact same four absorbing clouds in the HF rotational ground state, and invoke three velocity components to explain the hot core component.\",\n \"While the hot core component(s) appear in emission in CH, they are in absorption in HF, because CH 1$\\\\rightarrow $ 0 has a lower Einstein $A$ coefficient than HF 1$\\\\rightarrow $ 0 (see above).\",\n \"Table: SPIRE spectrometer observations used in this paper.The CH and HF signatures were observed toward NGC 6334 I with the high spectral resolution spectrometer HIFI in single point mode [16]; its single pixel receiver did not provide any spatial information.\",\n \"In a map of CH or HF absorption covering the region surrounding source I, one would expect to see a disentanglement of the different spatial extent of each velocity component as illustrated in Fig.\",\n \"REF .\",\n \"Toward core I, the velocity resolved HF absorption signature, with a total equivalent width, $\\\\int (1-I_\\\\mathrm {norm}) \\\\mathrm {d}V$ , of 16 $\\\\mathrm {km}\\\\,\\\\mathrm {s}^{-1}$ , was modeled with seven components.\",\n \"At positions away from core I, but still on the NGC 6334 filament, the equivalent width should diminish to 11 $\\\\mathrm {km}\\\\,\\\\mathrm {s}^{-1}$ representing the four foreground clouds, while at positions off of the cores and the filament, only the two foreground components that are more extended than the dense molecular cloud should be visible, and the equivalent width should drop to 3 $\\\\mathrm {km}\\\\,\\\\mathrm {s}^{-1}$ .\",\n \"This paper presents results from Herschel SPIRE iFTS spectral mapping observations toward a 6$\\\\times $ 35 region surrounding cores I and I(N) in the NGC 6334 star-forming complex.\",\n \"The observations are described in Sect.\",\n \"and the resulting map of HF absorption depth is discussed in Sect. .\",\n \"The signal is interpreted in Sect.\",\n \", both in the context of foreground clouds and in that of freeze-out conditions in the dense cores.\",\n \"Conclusions are summarized in Sect.\",\n \".\"\n ],\n [\n \"Observations and data reduction\",\n \"The spectral mapping observations used in this work were obtained as part of the `evolution of interstellar dust' guaranteed time program [1] with the Spectral and Photometric Imaging Receiver [30] on board the Herschel space observatory [69].\",\n \"SPIRE's imaging Fourier Transform Spectrometer (iFTS) provides a jiggling observing mode that uses its 54 detectors to obtain Nyquist sampled spatial maps, covering the entire frequency range of the Spectrometer Long Wavelength (SLW, 447–1018 GHz) and the Spectrometer Short Wavelength (SSW, 944–1568 GHz) bands.\",\n \"The spectral resolution of 1.2 GHz corresponds to a resolving power $\\\\nu /\\\\Delta \\\\nu \\\\approx 10^3$ , roughly 300 $\\\\mathrm {km}\\\\,\\\\mathrm {s}^{-1}$ at the frequency of the HF 1–0 transition, 1232.5 GHz (243 $\\\\mu $ m).\",\n \"Three partly overlapping, fully sampled SPIRE iFTS 4$\\\\times $ 4 jiggle observations were performed in a total of nine hours of observing time, two centered on NGC 6334 I and the third on NGC 6334 I(N).\",\n \"A fourth, sparsely sampled observation, centered $\\\\sim $ 2 northwest of core I, has considerable overlap with the combined area covered by the three other observations.\",\n \"This fourth observation is treated as an extra jiggle position in the gridding process described below.\",\n \"Finally, a fifth observation, also sparsely sampled, is centered 45 northwest of core I and its footprint therefore has no overlap with our mapped area.\",\n \"At each jiggle position, four repeated scans of the FTS mechanism were executed in high spectral resolution mode.\",\n \"Details of the observations are summarized in Table REF .\",\n \"The placement of the different pointings described here is shown in Fig.\",\n \"REF in the Appendix.\",\n \"After inspection of the initial observations from February 2011, some detectors were found to suffer from saturation due to the bright emission toward source I.\",\n \"The observation toward source I was therefore repeated in `bright source mode' [47] in September 2012 to obtain well calibrated spectra toward the brightest position.\",\n \"The majority of detector/jiggle combinations in the original observation point toward less bright regions and are therefore still useful in constructing the final map.\",\n \"The above SPIRE iFTS observations are processed with the `extended source' pipeline in HIPE 12.1.0 and the spire_cal_12_3 calibration tree, which includes the outer ring of partly vignetted detectors [21].\",\n \"The pipeline is interrupted at the pre-cube stage, before spectra from individual jiggle positions are gridded onto a rectangular spatial pattern.\",\n \"The spectrum for each jiggle position and each detector is visually inspected.\",\n \"Discarding all spectra that show excessive noise and/or irregular continuum shape (resulting from partial saturation) results in filtering 28 of the total of 919 SLW spectra (3%) and 135 of the 1696 SSW spectra (8%).\",\n \"The final processing step is the combination of the individual positions from each of the 19 (SLW) or 37 (SSW) detectors of each of the 55 jiggle positions into spectral cubes with square spatial pixels in Right Ascension and Declination coordinates.\",\n \"Due to the complex frequency dependence of the beam size and shape [48], the native angular resolution of SPIRE iFTS observations varies non-monotonously across its frequency range.\",\n \"We use the convolution gridding scheme, which weighs each input value according to the distance from the target pixel center by means of a differential Gaussian kernel, with the aim of obtaining a cube with a constant effective reference beam of 43 FWHM.\",\n \"The pixel grid is identical for SLW and SSW, with square pixels measuring 175$\\\\times $ 175.\",\n \"Gaussian convolution only results in a completely Gaussian target beam if the original beam is also well represented by a Gaussian shape.\",\n \"Such is the case for the entire SSW band and for low frequencies in SLW, but not for SLW frequencies between $\\\\sim $ 700 and 1018 GHz [48].\",\n \"Figure: Top panel: velocity resolved spectral profile of HF absorption observed with HIFI toward core NG C6334 I (gray histogram).\",\n \"The seven individual model components are the same as in and are drawn in dashed green lines, with their sum in solid gray.\",\n \"Middle panel: four foreground components expected to remain visible for lines of sight away from core I, but still on the NGC 6334 molecular cloud complex (solid blue).\",\n \"Bottom panel: two foreground cloud components hypothesized to be unrelated to the NGC 6334 complex (solid magenta).Figure: Illustration of the procedure adopted to fit the spectrally unresolved, Sinc-shaped absorption line feature in the SPIRE iFTS cube.\",\n \"Rest frequencies of the relevant transitions are marked by vertical dashed lines.Top panel: observed spectrum (black) toward one of the pixels in our map, fit to continuum and CO lines (blue), and the resulting ratio spectrum (gray).\",\n \"The rectangular box marks the axes limits of the next panel.Middle panel: zoom of continuum-normalized spectrum (gray, same as top panel) and fit to the combined signature of the HF absorption line and the nearby H 2 _2O emission line (red).Bottom panel: residual remaining after subtracting the fit shown in the middle panel.Figure: Map of equivalent width of the HF 1←\\\\leftarrow 0 absorption signature measured with Herschel SPIRE iFTS (colored contours).\",\n \"The peak value in the center of the `12.5 km s -1 \\\\mathrm {km}\\\\,\\\\mathrm {s}^{-1}' contour is 16 km s -1 \\\\mathrm {km}\\\\,\\\\mathrm {s}^{-1}.\",\n \"The grayscale represents 250 μ\\\\mu m dust continuum emission measured with the SPIRE photometer (Herschel observation ID 1342239909, HIPE 13 standard pipeline, color bar stretched from 4 to 250 GJy sr -1 ^{-1}).\",\n \"The thin white contours are from the continuum as measured by the FTS, which closely follow the structure seen in the photometer grayscale map.\",\n \"Beam sizes of the convolved spectrometer cube and the photometer map are shown in the bottom left corner in light gray and dark gray circles, respectively.\",\n \"The scalebar in the top left corner indicates a projected length of 0.3 pc at a distance to the source of 1.35 kpc.\"\n ],\n [\n \"Results\",\n \"The spectral cubes, as constructed in Sect.\",\n \", show a smooth dust continuum superposed with spectrally unresolved lines.\",\n \"The dominant line signal in the cube arises from the ladder of rotational transitions of CO ($J$ =4–3 to 13–12).\",\n \"Early versions of the CO intensity maps of NGC 6334, based on subsets of the SPIRE iFTS observations used here, were presented in [55] and [49].\",\n \"To retain the highest possible spectral resolution, we use unapodized FTS data, in which any unresolved spectral line has a Sinc-shaped profile.\",\n \"It is therefore important that one carefully fits and subtracts the Sinc-shaped profiles of nearby bright lines, to avoid any remaining sidelobes of strong lines affecting the apparent profile of the weak absorption line under study.\",\n \"This work focuses on absorption signatures of two hydride molecules, for which we use two spectral sections of the data cubes: 1132–1332 GHz from SSW and 760–935 GHz from SLW, chosen specifically to include the two CO emission lines closest to HF 1–0 at 1232.48 GHz [61] and CH$^+$ 1–0 at 835.08 GHz [65].\",\n \"We construct a script in HIPE [63], derived from one of the post-pipeline analysis scripts provided by the SPIRE iFTS working group [70], to fit a third order polynomial for the continuum simultaneously with the following lines: CO at 1152.0 and 1267.0 GHz (SSW), and CO at 806.7 and 921.8 and [Ci] at 809.3 GHz (SLW).\",\n \"Knowing that the intrinsic width of CO lines in this region is only a few $\\\\mathrm {km}\\\\,\\\\mathrm {s}^{-1}$ [96], lines are spectrally unresolved by SPIRE iFTS, and we adopt for each spectral line a Sinc profile with a fixed peak-to-first-zero-crossing width of 1.18 GHz.\",\n \"We then divide the observed spectrum by the fit of continuum and two/three Sinc lines to obtain a continuum-normalized spectrum, $I_\\\\mathrm {norm}$ .\",\n \"This process, illustrated in the top panel of Fig.\",\n \"REF , is repeated for each spatial pixel in the cube.\",\n \"In the resulting continuum-normalized cube around 1232 GHz, we fit two Sinc functions to the absorption profile of HF 0$\\\\leftarrow $ 1 at 1232.48 GHz and the nearby emission line of H$_2$ O 2$_{2,0}$$\\\\rightarrow $ 2$_{1,1}$ at 1228.79 GHz (Fig.\",\n \"REF , middle panel).\",\n \"In the continuum-normalized cube around 835 GHz, we fit three Sinc functions to the absorption profile of CH$^+$ 0$\\\\leftarrow $ 1 at 835.08 GHz and $^{13}$ CO emission lines at 771.18 and 881.27 GHz.\",\n \"The maps of equivalent width of the HF and CH$^+$ absorption depth, in units of Hz, are multiplied by the ratio of the speed of light, c, and the observed frequency, $\\\\nu _\\\\mathrm {obs}$ , to convert to $\\\\mathrm {km}\\\\,\\\\mathrm {s}^{-1}$ units: c/$\\\\nu _\\\\mathrm {obs}$ =2.43$\\\\times $ 10$^{-7}$ $\\\\mathrm {km}\\\\,\\\\mathrm {s}^{-1}$ Hz$^{-1}$ for HF, and 3.59$\\\\times $ 10$^{-7}$ $\\\\mathrm {km}\\\\,\\\\mathrm {s}^{-1}$ Hz$^{-1}$ for CH$^+$ .\",\n \"Line fits are rejected if the signal-to-noise ratio is lower than 2 and/or the fitted line center is more than half a resolution element from the line's expected frequency at $-8.3$ $\\\\mathrm {km}\\\\,\\\\mathrm {s}^{-1}$ $V_\\\\mathrm {lsr}$ [92].\",\n \"The resulting map of HF equivalent width in Fig.\",\n \"REF reveals the spatial distribution of the HF absorption feature, detected in 81% (signal-to-noise$>$ 2) or 72% (signal-to-noise$>$ 3) of all the pixels in the 6$\\\\times $ 35 map coverage.\",\n \"See also the signal-to-noise map in Fig.\",\n \"REF .\",\n \"Uncertainties are calculated from the spectral rms noise in the continuum-normalized cubes in two 20 GHz ranges surrounding the HF absorption line.\",\n \"For CH$^+$ , the frequency ranges for calculating noise are composed of a 10 GHz section below and a 30 GHz section above the CH$^+$ frequency, to avoid incorporating residual from the fit to the blended CO 7–6 and [Ci] $^3$ P$_2$ –$^3$ P$_1$ lines at 806 and 809 GHz.\",\n \"The 1-$\\\\sigma $ noise on the equivalent width is obtained by multiplying the unitless spectral rms with the instrumental line width of 1.18 GHz $\\\\times $ c/$\\\\nu _\\\\mathrm {obs}$ .\",\n \"Noise values are variable across the maps, in the range 1.2–3.4 $\\\\mathrm {km}\\\\,\\\\mathrm {s}^{-1}$ for HF and 1.4–7 $\\\\mathrm {km}\\\\,\\\\mathrm {s}^{-1}$ for CH$^+$ .\",\n \"We do not include the following contributions to the uncertainty.\",\n \"Firstly, any multiplicative effects such as those of the absolute intensity calibration [4], [87] are divided out by normalizing the spectra to the local continuum.\",\n \"Secondly, additive uncertainties in the continuum level offsets are $\\\\sim $ 0.5$\\\\times $ 10$^{-19}$ W m$^{-2}$ for SLW and $\\\\sim $ 2$\\\\times $ 10$^{-19}$ W m$^{-2}$ for SSW [87], [31].\",\n \"These values are negligible compared to the brightness of the continuum in our cube, which exceeds the offset uncertainty by a factor of a few hundred even in the faintest outer regions.\",\n \"Spectra from the `OFF' observation, pointed just northwest of the mapped area shown in Fig.\",\n \"REF , are also inspected at the HF frequency, but no convincing detections are found.\",\n \"The spectra from individual detectors in the OFF observation exhibit rms noise values between 4 and 10 $\\\\mathrm {km}\\\\,\\\\mathrm {s}^{-1}$ , with a median of 6 $\\\\mathrm {km}\\\\,\\\\mathrm {s}^{-1}$ .\",\n \"This noise is considerably higher than that in the cube pixels based on the other four observations combined, in which each pixel encompasses at least four, but typically more than eight individual detector pointings.\",\n \"The lack of HF absorption detections in the OFF position is thus consistent at the 2-$\\\\sigma $ level with an HF absorption depth $\\\\lesssim $ 12 $\\\\mathrm {km}\\\\,\\\\mathrm {s}^{-1}$ in the area 3–6 northwest of source I, i.e., absorption depths could be anywhere in the range shown in our mapped area, except the central 40 around source I itself where the strongest absorption is seen.\",\n \"To rule out contamination of the HF signature by other spectral lines within SPIRE's spectral resolution element, we inspect the high spectral resolution spectrum toward the position of the chemically rich core NGC 6334 I [96], observed as part of the spectral survey key program CHESS [12] using the HIFI spectrometer [16], [78].\",\n \"The only spectral lines detected by HIFI in a frequency span of $\\\\pm $ 2 GHz around the HF frequency are four marginally detected methanol emission lines (A. Zernickel, private communication, Jun.\",\n \"2014) together amounting to $<$ 0.4 $\\\\mathrm {km}\\\\,\\\\mathrm {s}^{-1}$ in equivalent width.\",\n \"The possible methanol contaminations for the measured HF absorption depth are therefore contained within the uncertainty for our HF equivalent widths quoted above.\",\n \"The effect of the H$_2$ O emission line at 1229 GHz (see also above) could be more significant: at the brightest position, toward core I, the water line is as bright as one third of the deepest HF absorption.\",\n \"As described above and shown in Fig.\",\n \"REF , the effects of the water line on the HF absorption profile are taken into account by applying a simultaneous fit of these two lines, separated in frequency by three times the SPIRE instrumental line width.\",\n \"We also detect the signature of CH$^+$ 1$\\\\leftarrow $ 0 absorption at 835.08 GHz in the spectral map from the SLW array.\",\n \"We refrain from interpreting its signal here for the following reasons.\",\n \"Firstly, the signal-to-noise ratio in the CH$^+$ absorption map is much lower than that in the HF map (see Fig.\",\n \"REF and REF ), resulting in signal-to-noise$>$ 2 detections in only 46% of the mapped pixels.\",\n \"For completeness, the distribution of detected CH$^+$ absorption is shown in the Appendix in Fig.\",\n \"REF .\",\n \"Importantly, around the position of source I, there is no confident detection to be compared with heterodyne observations from [96].\",\n \"Only a few isolated pixels near that position have detections of CH$^+$ , but at a signal-to-noise of $<$ 3.\",\n \"Secondly, the spectrally resolved HIFI spectrum toward source I (A. Zernickel, private communication, Jun.\",\n \"2014) show seven distinct emission line features due to methanol at frequencies within 0.6 GHz of the CH$^+$ line, i.e., half of the SPIRE spectral resolution.\",\n \"The combined intensity of these lines is sufficient to compensate more than half of the CH$^+$ absorption observed by HIFI, and they therefore severely contaminate the spectrally unresolved profile of CH$^+$ absorption in the SPIRE spectrum.\",\n \"In fact, these methanol emission lines could be the cause of the weak detection at the position of core I with SPIRE's modest spectral resolution.\",\n \"Thirdly, the CH$^+$ transition falls in a frequency range in which the beam profile of the SPIRE iFTS is non-Gaussian in shape [48], complicating the map convolution and the interpretation of any spatial structure.\",\n \"Our SPIRE spectroscopic data also show evidence for detections of NH$_2$ at 952.6 and 959.5 GHz, and NH at 974.5 GHz and at 1000 GHz.\",\n \"However, compared to HF, it is more challenging for astrochemical models to explain the observed abundances of N-bearing hydrides [66], and the analysis of line features is complicated by hyperfine structure within rotational transitions [96].\",\n \"For these two reasons, we refrain from interpreting the NH and NH$_2$ absorption depth maps in this paper, but for completeness they are shown in Figs.\",\n \"REF and REF .\",\n \"The OH$^+$ doublet at 909.05 and 909.16 GHz is also within the frequency range covered by the SPIRE iFTS, but the bright methanol emission line at 909.07 GHz apparent in the HIFI spectrum published by [96] would make analysis of the blended OH$^+$ signature in the SPIRE spectrum impossible.\"\n ],\n [\n \"HF optical depth, equivalent width, and column density\",\n \"An absorption line is saturated when its depth reaches zero, i.e., absorbing all continuum photons in any specific spectral channel.\",\n \"HIFI observations seem to show that the combined absorption feature of HF due to the NGC 6334 I envelope, hot cores, and the foreground cloud at $-3.0$ $\\\\mathrm {km}\\\\,\\\\mathrm {s}^{-1}$ is saturated between $V_\\\\mathrm {lsr}$ of $-7$ and $-4$ $\\\\mathrm {km}\\\\,\\\\mathrm {s}^{-1}$ (Fig.\",\n \"REF a).\",\n \"The individual components, however, do not reach 100% absorption.\",\n \"Of all seven components, the cloud at $+6.5$ $\\\\mathrm {km}\\\\,\\\\mathrm {s}^{-1}$ comes closest to having saturated absorption in HF.\",\n \"As already noted by [18], even this component is only marginally optically thick, evidence for which is provided by the line width of the same component in the optically thin tracer CH [92] which is the same (1.5 $\\\\mathrm {km}\\\\,\\\\mathrm {s}^{-1}$ ) as for HF.\",\n \"All other components are believed to be optically thin [18].\",\n \"Absorption line depth is converted into optical depth using $I_\\\\mathrm {norm}=\\\\mathrm {e}^{-\\\\tau _\\\\nu }$ .\",\n \"With the caution of one of the seven components being marginally saturated, we take the optically thin limit to relate optical depth integrated over the line profile, $\\\\int \\\\tau _\\\\nu \\\\,\\\\mathrm {d}V$ , to column density, $N_\\\\mathrm {HF}$ , following [56]: $\\\\int \\\\tau _{\\\\nu ,\\\\mathrm {HF}} \\\\mathrm {d}V = \\\\frac{A_\\\\mathrm {ul} g_\\\\mathrm {u} \\\\lambda ^3}{8\\\\pi g_\\\\mathrm {l}} N_\\\\mathrm {HF} = 4.16\\\\times 10^{-13} \\\\, [\\\\mathrm {cm}^2 \\\\, \\\\mathrm {km}\\\\, \\\\mathrm {s}^{-1}]\\\\ N_\\\\mathrm {HF} ,$ where $A_\\\\mathrm {ul}$ is the Einstein $A$ coefficient for spontaneous emission, 2.42$\\\\times $ 10$^{-2}$ s$^{-1}$ for HF 1–0 [68], $g_\\\\mathrm {u}$ =3 and $g_\\\\mathrm {l}$ =1 are the statistical weights of rotational levels $J$ =1 and $J$ =0, respectively, and $\\\\lambda $ is the wavelength of the transition, 243.24 $\\\\mu $ m. In addition to the optically thin limit, Eq.\",\n \"(REF ) assumes that all HF molecules are in the rotational ground state, a fair assumption given its high $A_\\\\mathrm {ul}$ .\",\n \"The conversion from $I_\\\\mathrm {norm}$ to $\\\\tau _\\\\nu $ follows a linear relation for low values of $\\\\tau _\\\\nu $ : $\\\\nonumber \\\\tau _\\\\nu & = & -\\\\ln (I_\\\\mathrm {norm}) \\\\\\\\& \\\\approx & 1-I_\\\\mathrm {norm} \\\\qquad [\\\\mathrm {for}\\\\ I_\\\\mathrm {norm} \\\\approx 1].$ This relation holds to within $\\\\sim $ 10% for $\\\\tau _\\\\nu <0.2$ (line absorbs up to 20% of the continuum), but $\\\\tau _\\\\nu /(1-I_\\\\mathrm {norm})$ is already 1.2 at 40% absorption, rising to 2 at 80% absorption.\",\n \"The integrated optical depth is therefore systematically underestimated for a spectrally unresolved line that is smeared out over a velocity range wider than its intrinsic profile.The logarithm operator gives a disproportionally higher weight to more strongly absorbed channels.\",\n \"For example, a 20 $\\\\mathrm {km}\\\\,\\\\mathrm {s}^{-1}$ wide feature with constant 80% absorption ($I_\\\\mathrm {norm}$ =0.20) and a smeared-out version of 320 $\\\\mathrm {km}\\\\,\\\\mathrm {s}^{-1}$ wide at 5% absorption ($I_\\\\mathrm {norm}$ =0.95) both have the same equivalent width, $\\\\int (1-I_\\\\mathrm {norm}) \\\\mathrm {d}V$ = 16 $\\\\mathrm {km}\\\\,\\\\mathrm {s}^{-1}$ .\",\n \"The latter, however, yields a $\\\\int \\\\tau _\\\\nu \\\\mathrm {d}V$ that is smaller by a factor 2.\",\n \"The effect is larger yet for line profiles that come even closer to full absorption.\",\n \"Since the HF line is spectrally unresolved in our SPIRE spectra, a column density derived from these observations would merely constitute a lower limit to the true column density.\",\n \"Contrary to the optical depth, the absorption depth integrated over the line profile, i.e., the equivalent width of the absorbed `area' below $I_\\\\mathrm {norm}$ =1, is conserved regardless of the spectral resolution.\",\n \"This is confirmed by the matching equivalent width values measured by HIFI and SPIRE toward core I: $\\\\int (1-I_\\\\mathrm {norm}) \\\\mathrm {d}V$ is 15.7 and 16.4 $\\\\mathrm {km}\\\\,\\\\mathrm {s}^{-1}$ , respectively, with uncertainty margins of $\\\\sim $ 1 $\\\\mathrm {km}\\\\,\\\\mathrm {s}^{-1}$ in both cases.\",\n \"In the remainder of this paper, we analyze HF absorption depth measured with SPIRE based exclusively on the conserved quantity, equivalent width, $\\\\int (1-I_\\\\mathrm {norm}) \\\\mathrm {d}V$.\",\n \"When deriving optical depths and column densities, we rely exclusively on spectrally resolved profiles such as that in [18].\"\n ],\n [\n \"Distribution of HF absorbing clouds toward NGC 6334\",\n \"The range of HF equivalent width values in the map shown in Fig.\",\n \"REF can be divided into three regimes: (a) $>$ 12 $\\\\mathrm {km}\\\\,\\\\mathrm {s}^{-1}$ , only occurring toward the position of core I; (b) 8–12 $\\\\mathrm {km}\\\\,\\\\mathrm {s}^{-1}$ , spatially consistent with the larger scale filament in which cores I and I(N) are embedded; and (c) $\\\\lesssim $ 5 $\\\\mathrm {km}\\\\,\\\\mathrm {s}^{-1}$ , exclusively localized at projected distances $>$ 0.6 pc from the cores and the connecting filament.\",\n \"The non-detection of HF in the `OFF' observation (see Sect.\",\n \"), sparsely sampling the area just northwest of our map, is consistent with regimes (c) or (b).\"\n ],\n [\n \"Distinguishing foreground from dense star-forming gas\",\n \"We interpret the three regimes in the context of the velocity resolved HF spectrum published by [18], who identified seven individual physical components responsible for the HF absorption toward NGC 6334 I: the dense envelope at $V_\\\\mathrm {lsr}$ =$-6.5$ $\\\\mathrm {km}\\\\,\\\\mathrm {s}^{-1}$ , two compact subcores at $-6.0$ and $-8.0$ $\\\\mathrm {km}\\\\,\\\\mathrm {s}^{-1}$ , and four foreground layers at $-3.0$ , $0.0$ , $+6.5$ , and $+8.0$ $\\\\mathrm {km}\\\\,\\\\mathrm {s}^{-1}$ .\",\n \"The spectral signature of each of these components is reproduced in our Fig.\",\n \"REF a.\",\n \"Regime (a) requires all seven components to explain the total equivalent width of HF.\",\n \"The two other panels in Fig.\",\n \"REF represent adaptations of the model from [18] with progressively fewer absorption components taken into account.\",\n \"In Fig.\",\n \"REF b, regime (b) is explained by the superposition of four absorbing foreground clouds, discarding the components associated with the envelope and subcores of NGC 6334 I.\",\n \"We highlight that the HF absorption depth observed toward the dense star-forming envelope I(N) is consistent with regime (b).\",\n \"Differences in HF content between the dense envelopes I and I(N) are discussed in more detail in Sect.\",\n \"REF .\",\n \"Finally, in Fig.\",\n \"REF c, we show that regime (c) is consistent with a model composed of just two specific foreground clouds, namely those at $+6.5$ and $+8.0$ $\\\\mathrm {km}\\\\,\\\\mathrm {s}^{-1}$ [92], [18].\",\n \"In contrast with the detailed study of the HF profile in [18] and in this work, the spectral survey paper by [96], analyzing $\\\\sim $ 4300 individual spectral line features, uses a simplified model which explains the HF absorption with only three components.\",\n \"The two approaches are not inconsistent, but the latter paper groups components together as: (1) the NGC 6334 I envelope and two subcores, (2) two foreground clouds that are kinematically close to the NGC 6334 complex, and (3) two other foreground clouds with larger $V_\\\\mathrm {lsr}$ offsets [18].\",\n \"In this three-component model, regime (a) would be explained by groups (1)+(2)+(3), regime (b) by groups (2)+(3), and regime (c) by only group (3).\",\n \"The combination of our HF absorption depth map with the previous, single-pointing, velocity resolved HF spectrum now reveals a three-dimensional picture of the layers of absorbing gas toward the NGC 6334 complex.\",\n \"Our interpretation of the relative positions of the foreground layers is sketched in Fig.\",\n \"REF .\",\n \"With the exception of the direction toward core I, the HF absorption depth at all other positions in the map can be explained by (a subset of) the four extended foreground clouds.\",\n \"Figure: Sketch of the location and extent of the foreground clouds.\",\n \"Note the broken scale in the horizontal direction.\",\n \"Velocity magnitudes are indicated with black arrows, shifted such that the core NGC 6334 I, at V lsr V_\\\\mathrm {lsr}=-6.5-6.5 km s -1 \\\\mathrm {km}\\\\,\\\\mathrm {s}^{-1}, is at rest in this frame.\",\n \"From left to right, they represent the components referred to in the text as the +8.0+8.0, +6.5+6.5, +0.0+0.0, and -3.0-3.0 km s -1 \\\\mathrm {km}\\\\,\\\\mathrm {s}^{-1} clouds.\",\n \"The dotted line represents the line of sight from the observer to NGC 6334 I.\",\n \"The dashed lines indicate the angular extent of clouds on the far foreground.\",\n \"The color differential between core I and the filament indicates the elevated temperature in core I, preventing freeze-out of HF molecules.\"\n ],\n [\n \"Relation of foreground clouds to NGC 6334\",\n \"[92] used a single-pointing Herschel HIFI observation of CH rotational line absorption, coupled with results from OH hyperfine line absorption measurements at radio wavelengths [11], to suggest that the clouds at $V_\\\\mathrm {lsr}$ =$-3.0$ and $+0.0$ $\\\\mathrm {km}\\\\,\\\\mathrm {s}^{-1}$ are associated with the NGC 6334 complex, while the remaining two velocity components, at $+6.5$ and $+8.0$ $\\\\mathrm {km}\\\\,\\\\mathrm {s}^{-1}$ , originate in foreground clouds farther away from the NGC 6334 complex.\",\n \"This interpretation is consistent with the first two components being seen exclusively in regime (b) of the HF equivalent width map and the latter two components being spread over a more extended area of regimes (b) and (c) combined.\",\n \"Therefore, the spatial distribution of HF absorption measured with SPIRE iFTS supports the previous hypothesis that the $-3.0$ and $+0.0$ $\\\\mathrm {km}\\\\,\\\\mathrm {s}^{-1}$ clouds are associated with NGC 6334, since they have a spatial morphology that closely follows the dense molecular cloud traced by the 250 $\\\\mu $ m dust emission (Fig.\",\n \"REF ), i.e., a region roughly 2–3 in east-west extent and stretching along the north-south direction.\",\n \"The vertical extent of the `related' foreground clouds depicted in Fig.\",\n \"REF is drawn directly from the observed east-west extent of these component in the plane of the sky.\",\n \"Besides the diatomic hydride species HF, CH, and OH, a subset of our absorbing clouds also exhibit absorption lines due to H$_2$ O [17], [89], H$_2$ O$^+$ [62], and H$_2$ Cl$^+$ [44].\",\n \"Absorption components in OH$^+$ and H$_2$ O$^+$ detected toward both cores by [35] peak at $V_\\\\mathrm {lsr}$ =$-2$ and $+3$ $\\\\mathrm {km}\\\\,\\\\mathrm {s}^{-1}$ .\",\n \"The former could be a blend of the $-3.0$ and $+0.0$ $\\\\mathrm {km}\\\\,\\\\mathrm {s}^{-1}$ clouds seen in CH and HF, but the latter is inconsistent with any of our components.\",\n \"These lines were all detected in observations with the single-pixel, high spectral resolution Herschel HIFI spectrometer, in some cases toward both individual protostellar cores.\",\n \"It is interesting that ionized species, H$_2$ O$^+$ and H$_2$ Cl$^+$ , are only detected at $V_\\\\mathrm {lsr}$ values that match the $-3$ and $+0$ $\\\\mathrm {km}\\\\,\\\\mathrm {s}^{-1}$ clouds.\",\n \"Chemical models tailored to halogen hydrides [57] indicate that cation species become abundant under the influence of strong UV radiation.\",\n \"We hypothesize that the presence of H$_2$ O$^+$ and H$_2$ Cl$^+$ is further evidence for the physical proximity of these two clouds to the massive protostars and the Hii region embedded in one of the dense cores.\",\n \"Another ionized species that has been detected in spectrally resolved observations toward NGC 6334 is CH$^+$ .\",\n \"While a velocity resolved observation of HF exists only toward the position of core I, CH$^+$ 1–0 has been observed with HIFI toward both cores I and I(N), the latter as part of the WISH program [93], [5].\",\n \"Toward source I, the CH$^+$ profile looks very similar to the HF profile, amounting to a total equivalent width of $\\\\sim $ 20 $\\\\mathrm {km}\\\\,\\\\mathrm {s}^{-1}$ .\",\n \"In comparison, the CH$^+$ observation toward core I(N)Estimate based on the HIFI spectrum downloaded from the Herschel Science Archive, observation ID 1342214306, processed with HIPE pipeline version 13. reveals an equivalent width of only $\\\\sim $ 13 $\\\\mathrm {km}\\\\,\\\\mathrm {s}^{-1}$ .\",\n \"The majority (85%) of the reduced absorption toward core I(N) relative to core I falls in the $[-15,5]$ $\\\\mathrm {km}\\\\,\\\\mathrm {s}^{-1}$ range, as expected if the missing components are the envelope and subcore components at $V_\\\\mathrm {lsr}$ =$-6.5$ , $-6.0$ , and $-8.0$ $\\\\mathrm {km}\\\\,\\\\mathrm {s}^{-1}$ (cf. Fig.\",\n \"REF ), i.e., those components that occur only in regime (a) in our HF absorption map.\",\n \"As mentioned above in Sect.\",\n \", the SPIRE iFTS spectral cube has too low signal-to-noise near 835 GHz to make a meaningful comparison with the CH$^+$ signature detected by this instrument.\",\n \"The foreground clouds at $V_\\\\mathrm {lsr}$ =$+6.5$ and $+8.0$ $\\\\mathrm {km}\\\\,\\\\mathrm {s}^{-1}$ , supposedly unrelated to the NGC 6334 dense filament [11], [92], have a combined HF equivalent width that is entirely consistent with the observation in this work that regime (c) is spatially extended beyond the dense filament.\",\n \"An unrelated set of foreground cloud(s) (the two leftmost clouds in Fig.\",\n \"REF ) are likely to have a larger angular extent than the background continuum source (dashed lines in Fig.\",\n \"REF ), despite a possibly modest linear size.\",\n \"This explains why a minimum level of HF equivalent width of $\\\\sim $ 3 $\\\\mathrm {km}\\\\,\\\\mathrm {s}^{-1}$ is detected not just toward the NGC 6334 filament, but throughout the extent of our 6$\\\\times $ 35 map (Figs.\",\n \"REF and REF ).\",\n \"All four foreground clouds detected in HF and CH are redshifted ($V_\\\\mathrm {lsr}$$\\\\ge $$-3$ $\\\\mathrm {km}\\\\,\\\\mathrm {s}^{-1}$ ) with respect to the $V_\\\\mathrm {lsr}$ of the part of NGC 6334 cloud near cores I and I(N) (around $-5$ $\\\\mathrm {km}\\\\,\\\\mathrm {s}^{-1}$ , based on HCO$^+$ observations by [97]).\",\n \"Thus, the absorbing gas clouds are moving toward NGC 6334, instead of following the Galactic rotation, which at $\\\\ell =351$ yields exclusively negative line-of-sight velocities for sources between Sun and NGC 6334, i.e., approaching the standard of rest of stars in the Solar neighborhood.\",\n \"We therefore conclude that, in addition to the gas flows within the dense gas along the filament's long axis [97], gas may also be accreting onto the filament in the perpendicular direction.\",\n \"We also note that, whereas [35] put the OH$^+$ and H$_2$ O$^+$ absorption clouds toward both cores I and I(N) at the distance of 1.35 kpc of the NGC 6334 cloud, their location along the sight line toward NGC 6334 is in fact poorly constrained.\",\n \"Recognizing that there must be peculiar motions at play, deviating from the `rigid' Galactic rotation curve, these clouds could in fact be anywhere between the local arm of the Milky Way and the Sagittarius arm that harbors the NGC 6334 complex.\",\n \"For the two foreground clouds related to NGC 6334, at $-3$ and $+0$ $\\\\mathrm {km}\\\\,\\\\mathrm {s}^{-1}$ , we calculate their total mass by multiplying the sum of their column densities with a rough estimate of the area covered by this component of 160$\\\\times $ 270 (regime (b) in Fig.\",\n \"REF ), corresponding to 1.5 pc$^2$ at the distance of NGC 6334.\",\n \"Depending on the choice of fiducial H$_2$ tracer, either taking $N_\\\\mathrm {CH}$ from [92] and $N_\\\\mathrm {CH}$ /$N_\\\\mathrm {H_2}$ =(2.1–5.6)$\\\\times $ 10$^{-8}$ from [82], or $N$ (HF) from [18] and $N_\\\\mathrm {HF}$ /$N_\\\\mathrm {H_2}$ =(0.5–1.4)$\\\\times $ 10$^{-8}$ from [36], this calculation yields a mass in the range 37–98 $M_\\\\odot $ or 69–191 $M_\\\\odot $ , respectively.\",\n \"From symmetry arguments, a similar reservoir of additional gas is expected to lie behind the NGC 6334 dense cloud.\",\n \"This means that a significant total mass of several hundred $M_\\\\odot $ could be on its way to accreting onto the dense cloud NGC 6334 near ($<$ 0.3 pc) the embedded cores I and I(N).\",\n \"This gas reservoir has escaped detection so far, because it does not appear in traditional gas tracers such as CO, HCO$^+$ , and CS.\",\n \"While there is evidence that the $-3.0$ and $+0.0$ $\\\\mathrm {km}\\\\,\\\\mathrm {s}^{-1}$ foreground clouds are closer to NGC 6334 than the $+6.5$ and $+8.0$ $\\\\mathrm {km}\\\\,\\\\mathrm {s}^{-1}$ clouds, there is no direct metric of the geometrical distance along the line of sight from each cloud to the dense filament and cores.\",\n \"Therefore, we refrain from speculating about accretion time scales of even the `related' clouds, since this would rely on unsupported assumptions on relative distances.\"\n ],\n [\n \"Spatial distribution of HF toward other Galactic sight lines\",\n \"The only other published work of a spatial map of HF absorption so far is that toward Sgr B2(M) by [19], using data also obtained with Herschel SPIRE iFTS.\",\n \"A direct comparison of measured absorption line depths is complicated by the choice of [19] to present signal in terms of integrated optical depth, apparently without taking into account the systematic underestimation of optical depths derived from spectrally unresolved measurements, as discussed in Sect.\",\n \"REF .\",\n \"Nonetheless, it appears that the total HF absorption toward Sgr B2(M) is about an order of magnitude stronger than toward NGC 6334 I.\",\n \"[19] find a variation of HF absorption depth of only a factor $\\\\sim $ 2 across the $\\\\sim $ 25 mapped area, significantly less variation than in our fully sampled map toward NGC 6334 I.\",\n \"Any intrinsic variation may have been partially smoothed by the interpolation process that was applied in [19] to construct a map from the spatially undersampled SPIRE iFTS observation.\",\n \"More importantly, the line of sight toward Sgr B2, close to the Galactic Center, crosses many more spiral arms than that toward NGC 6334.\",\n \"Evidence of this is found for example by [73], who detect a total of 31 individual velocity components in CH rotational ground state absorption toward Sgr B2(M).\",\n \"The 30 foreground clouds, associated with the various intervening Galactic arms, amount to a total CH column density of 1.7$\\\\times $ 10$^{15}$ $\\\\mathrm {cm}^{-2}$ , with Sgr B2(M) itself adding a component of only 0.5$\\\\times $ 10$^{15}$ $\\\\mathrm {cm}^{-2}$ [73].\",\n \"This is consistent with many sheets of foreground gas together creating a roughly uniform cover of absorbing gas spanning at least a few arcminutes on the sky.\",\n \"Comparatively little additional absorption is contributed by the massive molecular cloud Sgr B2 and the embedded cores in the background, which explains the lack of variation in HF absorption depth seen toward Sgr B2 by [19].\",\n \"In contrast, our map of HF absorption toward NGC 6334 reveals a mix of components due to foreground clouds and the star-forming envelope and cores within NGC 6334, which we are able to disentangle owing to the complementary, velocity resolved spectra of CH and HF obtained with Herschel HIFI [92], [18].\",\n \"In addition, we highlight the discovery by [46] of a foreground cloud toward the intermediate-mass star-forming core OMC-2 FIR 4, based on single-pointing HIFI spectra of HF and other hydride molecules.\",\n \"Several oxygen-bearing hydrides show absorption exclusively at a blueshifted velocity relative to the background protostellar core.\",\n \"The same foreground cloud was later also identified in H$_2$ Cl$^+$ by [38].\",\n \"The HF profile shows absorption at the same blueshifted velocity, but shows additional evidence for a second absorption component that matches the $V_\\\\mathrm {lsr}$ of the protostellar core [46].\",\n \"Instead of kinematical and morphological arguments such as those used in this work to determine the physical location of foreground clouds toward NGC 6334, [46] use detailed photochemical modeling to infer proximity of their OMC-2 foreground gas to a source of copious far-UV radiation.\",\n \"With that radiation source assumed to be the trapezium cluster of OB stars, it is concluded that the absorbing slab is physically connected to OMC-1.\",\n \"In an attempt to study the spatial distribution of this absorbing OMC-1 `fossil' slab, we have searched archival SPIRE iFTS data toward OMC-2 FIR 4 (Herschel observation ID 1342214847) for signatures of HF at 1232.5 GHz (and CH$^+$ at 835.1 GHz), but find no detections in any of the 37 (and 19) SSW (and SLW) detectors in the $\\\\sim $ 3 footprint.\",\n \"The HF equivalent width of 15.9$\\\\pm $ 1.4 $\\\\mathrm {km}\\\\,\\\\mathrm {s}^{-1}$ measured at the position of NGC 6334 I in our SPIRE map is explained in Sect.\",\n \"REF by invoking the superposition of four absorption components in the foreground and three associated to the dense core I itself [18].\",\n \"A striking feature of our HF absorption map is the lack of additional absorption toward the position of core I(N), where the HF equivalent width of only 10.9$\\\\pm $ 1.1 $\\\\mathrm {km}\\\\,\\\\mathrm {s}^{-1}$ can be explained by the four foreground clouds alone, adding up to 10.5 $\\\\mathrm {km}\\\\,\\\\mathrm {s}^{-1}$ (Fig.\",\n \"REF b).\",\n \"In contrast, adding even just an envelope component similar to that of core I sums up to 13.7 $\\\\mathrm {km}\\\\,\\\\mathrm {s}^{-1}$ (not counting the two subcore components), which is inconsistent with the observation toward core I(N).\",\n \"Since the envelope of core I(N) is more massive than that of core I, but has a similar size [89], the total gas column density toward core I(N) should be higher.\",\n \"Therefore, the lack of HF absorption associated to the I(N) core is not due to the difference in total (H$_2$ ) column.\",\n \"Instead, we hypothesize that HF is primarily frozen out onto dust grains in core I(N), while HF is in the gas phase in core I.\",\n \"To support the hypothesis of HF being depleted from the gas phase in core I(N), we set up a rudimentary model based on the following ingredients.\",\n \"We take the spherically symmetric physical structure, i.e., radial profiles of density and temperature, of the envelopes of NGC 6334 I and I(N) as fitted to submillimeter dust continuum maps and the far-infrared / submillimeter spectral energy distribution [89].\",\n \"We then calculate, at every radial point, the timescales for adsorption (freeze-out) and desorption (evaporation) of HF molecules onto dust grains.\",\n \"Following, e.g., [76] and [37], we assume that thermal desorption is the dominant mechanism that drives molecules from the grain surface back into the gas phase, and are left with the balance between adsorption rate: $\\\\lambda (n_\\\\mathrm {H}, T_\\\\mathrm {gas}) = 4.55\\\\times 10^{-18} \\\\left( \\\\frac{T_\\\\mathrm {gas}}{m_\\\\mathrm {HF}} \\\\right)^{0.5} n_\\\\mathrm {H} \\\\qquad [\\\\mathrm {s}^{-1}] ,$ and desorption rate: $\\\\xi (T_\\\\mathrm {dust}) = \\\\nu _\\\\mathrm {vib} \\\\exp \\\\left( -\\\\frac{E_\\\\mathrm {b,HF}}{k\\\\,T_\\\\mathrm {dust}} \\\\right) \\\\qquad [\\\\mathrm {s}^{-1}] .$ Here, $T_\\\\mathrm {gas}$ and $T_\\\\mathrm {dust}$ are the temperatures of gas and dust, assumed to be equal as in the modeling of [89], $m_\\\\mathrm {HF}$ is the molecular weight of HF (20), $n_\\\\mathrm {H}$ is the density of hydrogen nuclei, $\\\\nu _\\\\mathrm {vib}$ is the vibrational frequency of the HF molecule in its binding site, for which we adopt $10^{13}$ s$^{-1}$ , $k$ is the Boltzmann constant, and $E_\\\\mathrm {b,HF}$ is the binding energy of HF to the dust grain surface.\",\n \"It has previously been inferred by [67] that a density of $\\\\sim $$10^5$ $\\\\mathrm {cm}^{-3}$ allows HF to condense onto dust grains, whereas densities of $\\\\sim $$10^3$ $\\\\mathrm {cm}^{-3}$ more typical for diffuse gas are too low for HF freeze-out to occur.\",\n \"In our case, the density $n_\\\\mathrm {H}$ – which incidentally exceeds $10^5$ $\\\\mathrm {cm}^{-3}$ at almost all radii in the envelopes of I and I(N) – enters directly into Equation REF to govern the adsorption rate.\",\n \"Figure: Radial dependence of adsorption and desorption timescales of HF from various types of grain surfaces.\",\n \"The top half of the figure relates to the envelope of core I(N), the bottom half to that of core I.\",\n \"The small panels above each main panel depict the physical structure (temperature in solid blue, left axis; density in dashed red, right axis) for the envelopes in question from .\",\n \"The horizontal axis is logarithmic cumulative mass increasing from left to right; labels in brackets indicate the percentage of enclosed mass starting from the outer shell of the core.\",\n \"Freeze-out of HF (depletion from the gas phase) occurs in the region where t desorb >t freeze t_\\\\mathrm {desorb} > t_\\\\mathrm {freeze}; light shading indicates the case of H 2 _2O ice mantles, darker shading that of CO/CO 2 _2 ice mantles.\",\n \"The desorption time scale lines for CO ice and CO 2 _2 ice in the bottom panel lie to the far right, outside of the limits of the axes.In this work we consider multiple versions of the desorption timescale, because the binding energy in the exponent of Equation REF is heavily dependent on the type of grain surface.\",\n \"Unlike for more common molecular species such as CO [6], [60], the desorption behavior of HF from astrophysically relevant grain surfaces has not been studied experimentally, so we rely on theoretical calculations.\",\n \"Typical interstellar dust grains, especially those embedded in cold, star-forming regions, are covered in one or multiple layers of ice consisting of various molecules, mainly H$_2$ O, CO, and CO$_2$ [9].\",\n \"We collect binding energy values for several types of grain surfaces in Table REF .\",\n \"For CO and CO$_2$ ice covered grains we adopt calculated binding energies from the literature [13], [75], while for hydrogenated bare silicate grains and H$_2$ O ice covered grains, these values result from original ab initio chemical calculations performed for this work.\",\n \"Table: Binding energies for HF onto various surfaces.To calculate the HF binding energies in the first two rows of Table REF (with a bare grain and with H$_2$ O ice), we carry out quantum calculations within the Kohn-Sham implementation of Density Functional Theory using the Quantum Espresso Simulation Package [25].\",\n \"Perdew-Burke-Ernzerhof exchange-correlation functional ultrasoft pseudopotentials are used.\",\n \"KS valence states are expanded in a plane-wave basis set with a cutoff at 340 eV for the kinetic energy.\",\n \"The self-consistency of the electron density is obtained with the energy threshold set to $10^{-5}$ eV.\",\n \"Calculations are performed using the primitive unit cell containing a total number of 46 atoms for bare hydrogenated silica, and 54 atoms for hydrogenated silica covered with one layer of H$_2$ O ice.\",\n \"The geometry optimization is used within the conjugate gradients scheme, with a threshold of 0.01 eV $Å^{-1}$ on the Hellmann-Feynman forces on all atoms; the Si atoms of the bottom layers are fixed at their bulk values.\",\n \"The binding energy of HF with the SiH terminus of hydrogenated crystalline silica is based on calculations for the hydroxylated alpha-quartz (001) surface.\",\n \"The binding energy of HF with one layer of H$_2$ O ice on amorphous hydrogenated silicate is estimated by assuming that the the most common structure in this case is the HF molecule interacting with a H$_2$ O molecule bonded to silanol (SiOH), which is the most abundant surface group in amorphous silica [20].\",\n \"The binding energies of HF with CO and CO$_2$ ice (last two rows of Table REF ) are taken from calculations by [75] and [13], respectively.\",\n \"These authors performed calculations for molecules in the gas phase.\",\n \"We consider the gas phase binding energies of HF with CO and CO$_2$ to be similar to those of HF with CO and CO$_2$ ices adsorbed on an inert surface such as that of hydroxylated amorphous silica.\",\n \"This approximation is based on the weak interactions of these ices with hydroxylated silica and within the CO and CO$_2$ molecular solids, so that the electronic density of CO and CO$_2$ in solid form is not significantly altered with respect to their state in the gas phase.\",\n \"Hence, for the aim of the present work, the binding energy of the HF molecule with CO or CO$_2$ as calculated in the gas phase is applicable for the condensed phase.\",\n \"The situation is notably different for interactions with H$_2$ O in the gas or adsorbed form, because of its stronger interaction with silica and HF.\",\n \"For HF interacting with H$_2$ O ice, we use binding energies from our own calculations described above.\",\n \"With the physical structure of both envelopes, the equations for the adsorption/desorption balance, and the binding energy values, the `freeze-out' region within each envelope is calculated in Fig.\",\n \"REF .\",\n \"Defining $t_\\\\mathrm {freeze} = 1/\\\\lambda $ and $t_\\\\mathrm {desorb}=1/\\\\xi $ , HF molecules will deplete from the gas phase in the region of the envelope wherever $t_\\\\mathrm {desorb}$ > $t_\\\\mathrm {freeze}$ .\",\n \"Numerical simulations by [15] suggest that, within a mixed-composition ice layer, the abundances of CO and CO$_2$ are enhanced compared to H$_2$ O in high-density ($\\\\gtrsim 10^5$ $\\\\mathrm {cm}^{-3}$ ) environments.\",\n \"At such densities applicable for the protostellar envelopes studied here, we thus expect the HF binding energy to lie close to, but slightly above that of pure CO or CO$_2$ ice, and the true desorption time scale line in Fig.\",\n \"REF therefore somewhat to the left of the dash-dotted line for CO ice.\",\n \"In this case, HF is expected to stay frozen onto grain surfaces at a broad range of radii in core I(N): cumulative mass $\\\\sim $ 0.1 to 1, i.e., 90% of the mass, where the temperature is $\\\\lesssim $ 20 K. Core I(N) is overall colder than core I both in the envelope (blue solid lines in Fig.\",\n \"REF ) and in the embedded subcores [34].\",\n \"For the comparatively warmer envelope of core I, the lines for the desorption timescale of HF from CO/CO$_2$ ice fall off the scale on the right hand side of the axes, leaving no freeze-out zone in this envelope.\",\n \"This could explain why HF is seen in the gas phase in core I, but not in core I(N).\",\n \"In an alternative scenario in which the dust grains are covered in pure H$_2$ O ice – or the desorption characteristics of a mixed mantle are dominated by that of H$_2$ O ice [14] – the binding energy of HF with the ice would be greatly increased (see Table REF ).\",\n \"In this case, the HF freeze-out zone would expand to cover $>$ 98% of the mass for both envelopes, and our observations should have revealed no gas phase HF in either of the cores.\",\n \"Our observations are therefore inconsistent with pure H$_2$ O ice coating on the grains.\",\n \"Instead, our interpretation relies on a significant part of the ice coating to consist of CO and/or CO$_2$ molecules.\",\n \"In principle, the composition of ice coatings do not need to be the same in the two neighboring envelopes.\",\n \"Particularly, given the lower temperatures of core I(N), there is higher probability that a significant amount of H$_2$ O is frozen out in that envelope.\",\n \"This, again, would enhance the binding energy of HF onto the ice-covered dust grains in the envelope of core I(N), which may help to explain the lack of gas-phase HF observed toward source I(N).\",\n \"Conversely, if additional mechanisms for desorption, e.g., induced by (UV) photons or cosmic rays, would be taken into account, $\\\\xi $ would increase, and the freeze-out region would be pushed to larger radii in the envelopes.\",\n \"Particularly UV photodesorption could have a different effect in one envelope compared to the other, because core I is more evolved and contains an Hii region.\",\n \"An increased total desorption rate would have no effect on HF freeze-out in the envelope of core I, in which HF is already completely in the gas phase, but would reduce the size of the freeze-out zone for envelope I(N).\",\n \"If, however, thermal desorption as expressed in Equation REF is the dominant desorption mechanism and the binding energy of HF onto the ice surface is close to that of CO or CO$_2$ ice (Table REF ), our model predicts significant freeze-out of HF in core I(N), and none in core I.\",\n \"It is important to note that, indeed, the temperature and density conditions under which HF remains frozen onto dust grains depend greatly on the exact composition and mixing of the ice mantle and therefore on the chemical history.\",\n \"It has already been recognized for example for the CO molecule that the freeze-out temperature `threshold' can vary considerably from one object to another [72].\"\n ],\n [\n \"Conclusions\",\n \"In this work we present a map of HF absorption toward the northern end of the molecular cloud NGC 6334, containing two well studied massive star-forming cores I and I(N).\",\n \"Although in the original definition of the observing program it was not anticipated that hydride absorption lines would be found within these data, the discovery space provided by the enormous frequency coverage of the Herschel SPIRE iFTS instrument has made this study possible.\",\n \"Such wide coverage in the far-infrared/submillimeter is only attainable with broadband FTS spectrometers [55].\",\n \"The absorption line of HF is detected in 80% of our mapped area, although it is spectrally unresolved by SPIRE.\",\n \"By complementing the new SPIRE iFTS data with existing, single pointing, high spectral resolution spectra from the Herschel HIFI instrument [92], [18], we construct a three-dimensional picture of gas clouds in front of and inside the massive star-forming filament NGC 6334.\",\n \"We find that our observations are consistent with a scenario of four individual foreground clouds on the line of sight toward NGC 6334 I and I(N), two of which are unrelated to the star-forming complex (Sect.\",\n \"REF ).\",\n \"The other two clouds are posited to be close to the dense molecular filament based on their spatial morphology.\",\n \"Their velocities are such that they are moving toward the star-forming cloud and could be adding several hundreds of solar masses of gas to the dense filament and the embedded cores in which massive star formation is already ongoing (Sect.\",\n \"REF ).\",\n \"This component of gas is detected in rotational lines of diatomic hydride molecules, but had been unseen in studies of traditional dense gas tracers.\",\n \"In fact, using such a tracer, HCO$^+$ , [97] have inferred that the roughly cylindrically shaped NGC 6334 filament is collapsing along its longest axis.\",\n \"Our work now indicates that accretion may also be ongoing in the perpendicular (radial) direction.\",\n \"Future studies of (competitively) accreting high-mass star-forming cores may need to take into account this additional low-density phase of the gas reservoir.\",\n \"Finally, in Sect.\",\n \"REF we explain why HF is observed in the gas phase toward core I, but appears completely absent in core I(N).\",\n \"For this purpose, we use a simple description of adsorption and desorption time scales for HF interacting with dust grain surfaces, depending on the (radially variable) density and temperature.\",\n \"Since interactions of HF with interstellar-like dust grains have not been studied in the laboratory, we adopt binding energy values for different types of grain surfaces from theoretical calculations from the literature as well as from original work first presented in this paper.\",\n \"The conclusion is that the lower temperature of core I(N) compared to core I could lead to freeze-out of HF exclusively in the former, but only if the binding energy of HF onto the grain surface is governed by that of CO or CO$_2$ ice on a silicate surface.\",\n \"In this case, at the densities relevant in the envelope of source I(N) ($>$ 3$\\\\times $ 10$^{5}$ $\\\\mathrm {cm}^{-3}$ ), we find that HF freezes out in the region of the envelope where the temperature is below $\\\\sim $ 20 K, rather similar to the freeze-out temperature often adopted for CO.\",\n \"In contrast, if H$_2$ O is the dominant constituent in the ice mantles, our model predicts that HF should have been frozen out at all radii in the envelopes of both sources I(N) and I.\",\n \"Since we observe a significant amount of HF in the gas phase in source I, this scenario is inconsistent with our data.\",\n \"Summarizing, this work uses HF as a sensitive tracer for (molecular) gas at relatively low densities that may be contributing mass to star forming cores.\",\n \"The HF signature reveals a gas reservoir that is inconspicuous in traditional dense gas tracers such as CO.\",\n \"In addition, we show that gas phase HF in higher density environments ($>$$10^5$ $\\\\mathrm {cm}^{-3}$ ) is extremely sensitive to interactions with dust grains and will be depleted significantly at low dust temperatures.\",\n \"The research of MHDvdW at the University of Lethbridge was supported by the Canadian Space Agency (CSA) and the Natural Sciences and Engineering Research Council of Canada (NSERC), and at the University of Copenhagen by the Lundbeck Foundation.\",\n \"Research at the Centre for Star and Planet Formation is funded by the Danish National Research Foundation and the University of Copenhagen's programme of excellence.\",\n \"SPIRE has been developed by a consortium of institutes led by Cardiff University (UK) and including Univ.\",\n \"Lethbridge (Canada); NAOC (China); CEA, LAM (France); IFSI, Univ.\",\n \"Padua (Italy); IAC (Spain); Stockholm Observatory (Sweden); Imperial College London, RAL, UCL-MSSL, UKATC, Univ.\",\n \"Sussex (UK); and Caltech, JPL, NHSC, Univ.\",\n \"Colorado (USA).\",\n \"This development has been supported by national funding agencies: CSA (Canada); NAOC (China); CEA, CNES, CNRS (France); ASI (Italy); MCINN (Spain); SNSB (Sweden); STFC, UKSA (UK); and NASA (USA).\",\n \"HIPE is a joint development by the Herschel Science Ground Segment Consortium, consisting of ESA, the NASA Herschel Science Center, and the HIFI, PACS and SPIRE consortia.\",\n \"This research has made use of NASA's Astrophysics Data System Bibliographic Services.\",\n \"The graphical representations of the results in this paper were created using APLpy, an open-source plotting package for Python hosted at http://aplpy.github.com, Astropy, a community-developed core Python package for Astronomy (Astropy Collaboration [3]), and the matplotlib plotting library [32].\",\n \"The authors are grateful to Alexander Zernickel for providing and discussing excerpts of the CHESS spectral survey data, to Raquel Monje for providing the HF spectral model component profiles, and to Floris van der Tak for providing the physical structure models of the two envelopes in electronic table format.\",\n \"We thank Tommaso Grassi, Wing-Fai Thi, Jes Jørgensen, Søren Frimann, and Mihkel Kama for discussions.\"\n ],\n [\n \"Complementary figures\",\n \"Maps displaying the signal-to-noise ratio of the detections of the HF and CH$^+$ lines in each spatial pixel of our spectral cube (Sect. )\",\n \"are shown in Fig.\",\n \"REF and REF .\",\n \"The colored contours in Fig.\",\n \"REF show that CH$^+$ absorption is only confidently detected (signal-to-noise $>$ 5) in the northeastern section of the map.\",\n \"In addition, although the signal from nitrogen species is not interpreted in this paper, maps of line absorption depth due to NH and NH$_2$ are shown in Figs.\",\n \"REF and REF .\",\n \"The continuum-normalized cube for these lines is created – analogous to those for HF and CH$^+$ in Sect.\",\n \"– by fitting the continuum and the CO 9–8 line in the 945–1055 GHz section of the SSW cube.\",\n \"Absorption lines of NH (at 974.47 and 999.98 GHz) and two of NH$_2$ (at 952.57 and 959.50 GHz) are then fitted simultaneously with emission lines due to $^{13}$ CO 9–8 at 991.3 GHz and H$_2$ O $2_{02}$ –$1_{11}$ at 987.9 GHz.\",\n \"Figure: Placement of the SPIRE iFTS footprints on the NGC 6334 region (cf. Table ).\",\n \"Solid circles indicate the fully sampled observations, two centered on cores I and one on I(N), while the sparsely sampled `INT' and `OFF' observations are marked by dashed circles.\",\n \"Circles are drawn with a diameter of 31 to include the vignetted outer ring of the detector arrays .\",\n \"The grayscale represents the same 250 μ\\\\mu m dust continuum map as in Fig.\",\n \", but with the scale bar stretching from 1 to 250 GJy sr -1 ^{-1}.\",\n \"The thin white contours, the beam size indicators in the bottom left and the scale bar in the top left corners are the same as in Fig.\",\n \".Figure: Signal-to-noise maps (color scale, with gray indicating values below 2) of the absorption signature of HF 1←\\\\leftarrow 0.\",\n \"Colored contours are at the same equivalent width levels as in Fig. .\",\n \"As in Fig.\",\n \", white contours trace continuum dust emission and the beam of the SPIRE iFTS map is shown in the bottom left corner.Figure: As in Fig.\",\n \", but for CH + ^+ 1←\\\\leftarrow 0.\",\n \"Colored contours are at 22, 19, 16 and 13 km s -1 \\\\mathrm {km}\\\\,\\\\mathrm {s}^{-1} in gray, blue, cyan and magenta, respectively.Figure: As in Fig.\",\n \", but for the JJ=2←\\\\leftarrow 1 NN=1←\\\\leftarrow 0 transition of NH at 974.47 GHz.\",\n \"The JJ=1–1 transition at 1000.0 GHz shows a similar absorption distribution, but with detections in fewer pixels.\",\n \"Colored contours are at 16.5, 13.5, and 10.5 km s -1 \\\\mathrm {km}\\\\,\\\\mathrm {s}^{-1} in gray, blue, and cyan, respectively.Figure: As in Fig.\",\n \", but for the 1 11 1_{11}←\\\\leftarrow 0 00 0_{00} JJ=3/2←\\\\leftarrow 1/2 transition of NH 2 _2 at 952.57 GHz.\",\n \"The JJ=1/2←\\\\leftarrow 1/2 transition at 959.50 GHz shows a similar absorption distribution, but with detections in fewer pixels.\",\n \"Colored contours are at 15.0, 11.5, and 8.0 km s -1 \\\\mathrm {km}\\\\,\\\\mathrm {s}^{-1} in gray, blue, and cyan, respectively.\"\n ]\n]"},"id":{"kind":"string","value":"1605.06031"}}},{"rowIdx":1866697,"cells":{"text":{"kind":"list like","value":[["Sharpness of the phase transition for continuum percolation in R^2"],["Abstract We study the phase transition of random radii Poisson Boolean percolation: Around each point of a planar Poisson point process, we draw a disc of random radius, independently for each point.","The behavior of this process is well understood when the radii are uniformly bounded from above.","In this article, we investigate this process for unbounded (and possibly heavy tailed) radii distributions.","Under mild assumptions on the radius distribution, we show that both the vacant and occupied sets undergo a phase transition at the same critical parameter $\\lambda_c$.","Moreover, - For $\\lambda < \\lambda_c$, the vacant set has a unique unbounded connected component and we give precise bounds on the one-arm probability for the occupied set, depending on the radius distribution.","- At criticality, we establish the box-crossing property, implying that no unbounded component can be found, neither in the occupied nor the vacant sets.","We provide a polynomial decay for the probability of the one-arm events, under sharp conditions on the distribution of the radius.","- For $\\lambda > \\lambda_c$, the occupied set has a unique unbounded component and we prove that the one-arm probability for the vacant decays exponentially fast.","The techniques we develop in this article can be applied to other models such as the Poisson Voronoi and confetti percolation."],["Introduction","Percolation is the branch of probability theory focused on the study of the geometry and connectivity properties of random media.","Since its foundation in the 1950s, with the work of Broadbent and Hammersley [8], the area reached new heights over the decades to come, specially in two dimensions: During the 1980s following Kesten's determination of the critical threshold [23], and at the turn of the century with Schramm's introduction of Schramm-Loewner evolution [37] and Smirnov's proof of Cardy's formula [38].","This progress has been well documented in a range of books on the subject; see for instance [24], [18], [10] and [19].","Bernoulli percolation on a symmetric planar lattice, e.g.","on the square or triangular lattices, is a cornerstone within percolation theory.","Loved for its simple yet challenging structure, Bernoulli percolation has become quintessential in the study of phase transitions and other phenomena emanating from statistical mechanics and mathematical physics.","In the site-percolation version of this model, each vertex of the lattice is independently declared `open' with probability $p$ and `closed' with probability $1-p$ .","A random graph is obtained from the initial lattice by removing the closed vertices and the connected components of this graph are called clusters.","As the parameter increases the model undergoes a sharp phase transition.","More precisely, in this planar case, it is well-known that there exists a critical parameter $p_c$ , strictly in between zero and one, such that for $pp_c$ , there exists a unique infinite open cluster.","The phase transition is said to be `sharp' because of the abrupt change in the decay of the connection probabilities, from exponentially small in the subcritical regime to uniformly bounded in supercritical.","Proofs of the above properties can be found in [18] and [10].","In this paper we investigate the sharpness of the phase transition for Poisson Boolean percolation in ${\\mathbb {R}}^2$ , establishing results analogous to the ones described above for Bernoulli percolation.","Poisson Boolean percolation is an archetype for percolation in the continuum, and shares many features of Bernoulli percolation while posing significant additional challenges.","Apart from being continuous rather than discrete, these challenges come from its asymmetrical nature (the `open' and `closed' set have different properties) and long-range dependencies.","One particular strength of the present paper is its generality, and its structure has been oriented with this in mind.","Although our results are presented for Poisson Boolean percolation, they extend to many other percolation processes in ${\\mathbb {R}}^2$ , such as Poisson Voronoi and confetti percolation, as will be described below.","Together with an accompanying paper [6], our results give a precise description of the phase transition for Poisson Boolean percolation.","There, very specific properties of Poisson Boolean percolation will be exploited, as opposed to the robust methods developed here.","An important ingredient in establishing properties (i)–(iii) for Bernoulli percolation is what we call the dual process.","It can be defined by looking at the closed vertices on a modified graph, called the matching or dual graph.","This dual graph has a critical parameter $p_c^\\star $ which was proved to satisfy the duality relation $p_c + p_c^\\star = 1.$ This relation is especially useful in the study of self-similar processes, for which the dual process at $p$ coincides with the primal process at $1-p$ .","In this case it can be linked with the equality $p_c = p_c^\\star $ , yielding $p_c = 1/2$ , see more examples in Section .","Kesten's [23] original proof of the sharpness of the phase transition for Bernoulli percolation is based on the analysis of the crossing probabilities for rectangles; a rectangle is said to be crossed if there is a path from left to right, made out of open vertices.","The proof involves three essential ingredients: Finite-size criterion – if for some $n$ , the probability to cross a $n$ by $3n$ rectangle in the short direction is smaller than some small constant $\\theta >0$ , then the two-point connection probability decays exponentially fast.","On the other side, if for some $n$ , the probability to cross a $3n$ by $n$ rectangle in the long direction is larger than $1-\\theta >0$ , then there exists an infinite cluster almost surely, see [35], [24] and [2].","Russo-Seymour-Welsh theory – relates crossing probabilities for rectangle with different aspect ratios, see [34] and [39].","Threshold phenomenon – given a positive constant $c > 0$ and a large rectangle $K$ , there is only a small interval of parameters $p$ for which the crossing probability of $K$ remains between $c$ and $1 - c$ this interval is called the critical window.","The first proof of this sharp threshold is due to Kesten in [23] and is based on a geometric interpretation of the derivative of the crossing probability for a rectangle.","A better understanding for threshold phenomena has since been obtained in work like [36], [25], [41] and elsewhere.","The strategy described above is inherently two-dimensional.","As such, the finite-size criterion extends well beyond the product structure of the Bernoulli percolation measure, and require only a minimal assumption on the decay of long-range dependencies.","The original Russo-Seymour-Welsh techniques and sharp threshold results rely in a much stronger sense on the product structure of the Bernoulli measure, and do not extend easily to more general percolation models.","These will thus be the two foremost challenges in for the present study, in which we study the sharpness of the phase transition for continuum percolation in ${\\mathbb {R}}^2$ , which may present arbitrarily slow decay of dependencies."],["The phase transition for Poisson Boolean percolation","Poisson Boolean percolation was introduced by Gilbert in [15].","In this model we start with a Poisson point process on $\\mathbb {R}^2$ with intensity parameter $\\lambda > 0$ .","We then independently associate to each of these points a disc of a randomly chosen radius, according to a fixed probability measure $\\mu $ on ${\\mathbb {R}}_+$ .","The set $\\mathcal {O} \\subseteq \\mathbb {R}^2$ of points which are covered by at least one of the above discs is called the occupied set, while its complement $\\mathcal {V}:={\\mathbb {R}}^2\\setminus \\mathcal {O}$ is referred to as the vacant set.","We denote by ${\\mathbb {P}}_\\lambda $ the measure associated with this construction, and defer a more formal definition to Section .","Denote by $[0 \\overset{\\tiny \\mathcal {O}}{\\longleftrightarrow }\\partial B_r]$ the event that there exists an occupied path from 0 to distance $r$ .","We write $[0 \\overset{\\tiny \\mathcal {O}}{\\longleftrightarrow }\\infty ]$ if $[0 \\overset{\\tiny \\mathcal {O}}{\\longleftrightarrow }\\partial B_r]$ holds for every $r \\ge 1$ .","Analogously we define $[0 \\overset{ \\mathcal {V}}{\\longleftrightarrow }\\partial B_r]$ and $[0 \\overset{ \\mathcal {V}}{\\longleftrightarrow }\\infty ]$ for the corresponding events for the vacant set.","Finally, define the critical parameters $\\begin{array}{ll}{\\lambda _{c}}:= \\sup \\big \\lbrace \\lambda \\ge 0 \\; : \\; \\mathbb {P}_\\lambda \\big [0 \\overset{\\tiny \\mathcal {O}}{\\longleftrightarrow }\\infty \\big ] = 0 \\big \\rbrace ,&\\\\{\\lambda _{c}^\\star }:= \\inf \\big \\lbrace \\lambda \\ge 0 \\; : \\; \\mathbb {P}_\\lambda \\big [0 \\overset{ \\mathcal {V}}{\\longleftrightarrow }\\infty \\big ] = 0 \\big \\rbrace .&\\end{array}$ We aim to describe the percolative properties of the model at and around these critical values.","That $\\lambda _c$ and $\\lambda _c^\\star $ are finite may be obtained via a comparison with Bernoulli percolation on ${\\mathbb {Z}}^2$ , where a standard Peierls argument may be used.","To show, on the other hand, that $\\lambda _c$ and $\\lambda _c^\\star $ are strictly positive is a different matter, and requires a condition on the radii distribution in order to be true.","The most fundamental condition in the study of Poisson Boolean percolation in ${\\mathbb {R}}^2$ is that of finite second moment on the radius distribution: $\\int _0^\\infty x^2 \\mu (\\mathop {}\\!\\mathrm {d}x) < \\infty .$ It was observed by Hall [20] that (REF ) is necessary in order to avoid the entire plane to be almost surely covered, regardless of the intensity (as long as positive) of the Poisson point process.","Gouéré [16] further showed that this condition is also sufficient for $\\lambda _c$ to be strictly positive.","Lower bounds on $\\lambda _c^\\star $ may be obtained from their comparison with $\\lambda _c$ .","Under the assumption of bounded support on $\\mu $ it is known that $\\lambda _c^\\star =\\lambda _c$ ; see [32].","At the critical point $\\lambda _c$ it is further known that there is almost surely no unbounded occupied component in the case of $\\mu $ having bounded support [32], whereas the analogous statement for an unbounded vacant component is only known to hold in the case of unit radii, see [4].","Our goal with the present and forthcoming paper [6] is to extend these results to hold under the condition (REF ).","We will in this paper make no further assumption of the above results, as they will be easy consequences of the techniques we develop.","We merely acknowledge the observation that (REF ) is necessary in order for the model to present non-trivial behavior.","Following Kesten's lead, we shall in this paper focus on the study of crossings of rectangles and build a theory around them.","Quantitative estimates on the rate of decay of connection probabilities, in the different regimes, will be obtained as consequences of this.","The first result we state will thus relate crossing probabilities to the critical parameter $\\lambda _c$ .","Let us first define, for every $r, h \\ge 1$ , the event $\\operatorname{Cross}(r, h)$ that there exists an occupied path inside the rectangle $[0, r] \\times [0, h]$ from the left side to the right side.","That is, $\\operatorname{Cross}(r, h) = \\hbox{\\;[scale=1.5]{(0, 0) rectangle (6ex, 4ex);[solid] (0, 2ex) .. controls (2.6ex, 3ex) and (3.3ex, 1ex) .. (6ex, 2ex);(3ex, .2ex) node[anchor=north] {$\\scriptstyle r$};(5.7ex, 2ex) node[anchor=west] {$\\scriptstyle h$};}}$ .","We also write $\\operatorname{Cross}^\\star (r, h)$ for the existence of a vacant crossing of the same box.","The results of this paper are based on the following theorem.","Theorem 1.1 Assume the second moment condition (REF ).","Then $\\lambda _c$ is strictly between zero and infinity, and for all $\\lambda > {\\lambda _{c}}$ and all $\\kappa > 0$ , we have $\\lim _{r \\rightarrow \\infty } \\mathbb {P}_\\lambda \\big [ \\operatorname{Cross}(\\kappa r, r) \\big ] = 1.$ for $\\lambda = {\\lambda _{c}}$ and all $\\kappa > 0$ , there exists a constant $c_{\\ref {c:thm_sharp_cross}} = c_{\\ref {c:thm_sharp_cross}}(\\kappa )>0$ such that $c_{\\ref {c:thm_sharp_cross}} < \\mathbb {P}_{{\\lambda _{c}}} \\big [ \\operatorname{Cross}(\\kappa r, r) \\big ] < 1 - c_{\\ref {c:thm_sharp_cross}}, \\text{ for every $r \\ge 1$.","}$ for all $\\lambda < {\\lambda _{c}}$ and all $\\kappa > 0$ , $\\lim _{r \\rightarrow \\infty } \\mathbb {P}_\\lambda \\big [\\operatorname{Cross}(r,\\kappa r)\\big ] = 0.$ Moreover, at $\\lambda _c$ there is almost surely no unbounded cluster of either kind.","Remark 1 Our proof gives quantitative bounds for the rate of convergence in parts (i) and (iii).","The next two theorems are intended to illustrate what can be obtained using the techniques of the present work.","They give an overview of the percolative behavior of the vacant and occupied sets in Poisson Boolean percolation.","Several of these characteristics resemble the known features of Bernoulli percolation, and these two theorems are general: Similar theorems can be proved also for other percolation processes using the methods of this paper (see Section ).","However, their general nature calls for a slightly stronger moment condition than the one in (REF ).","These two alternative hypotheses are $\\int _0^\\infty x^2\\log x\\, \\mu (\\mathop {}\\!\\mathrm {d}x) < \\infty , \\text{ and}\\\\\\text{for some $\\alpha > 0$,} \\quad \\int _0^\\infty x^{2 + \\alpha } \\,\\mu (\\mathop {}\\!\\mathrm {d}x) < \\infty .$ The three hypotheses (REF ), (REF ) and () will be useful because they imply some decorrelation inequalities discussed in Section REF .","The natural second moment assumption (REF ) already implies some spatial decorrelation properties, and the hypotheses (REF ) and () imply quantitative bounds on the spatial decorrelation.","The hypothesis (REF ) will be sufficient for most of the paper, but the hypotheses (REF ) and () will be useful to apply some renormalization methods presented in Section .","For the vacant set we have the following.","Theorem 1.2 Assume the $2+\\log $ moment condition (REF ).","Then $\\lambda _c^\\star =\\lambda _c$ and is thus strictly between zero and infinity.","Moreover, for $\\lambda = {\\lambda _{c}^\\star }$ there exists a constant $c_{\\ref {c:poly_decay}} > 0$ such that $\\mathbb {P}_{\\lambda _c^\\star }\\big [ 0 \\overset{ \\mathcal {V}}{\\longleftrightarrow }\\partial B_r \\big ] \\le \\frac{1}{c_{\\ref {c:poly_decay}}} r^{-c_{\\ref {c:poly_decay}}}.$ for all $\\lambda > {\\lambda _{c}^\\star }$ there exists a constant $c_{\\ref {c:exp_decay}} = c_{\\ref {c:exp_decay}}(\\lambda ) > 0$ such that $\\mathbb {P}_\\lambda \\big [ 0 \\overset{ \\mathcal {V}}{\\longleftrightarrow }\\partial B_r \\big ] \\le \\frac{1}{c_{\\ref {c:exp_decay}}}\\exp \\lbrace -c_{\\ref {c:exp_decay}}r\\rbrace .$ Remark 2 Condition (REF ) in Theorem REF is not sharp.","In the forthcoming paper [6], we show that the second moment condition suffices to prove all of the above.","The fact that ${\\lambda _{c}^\\star }= {\\lambda _{c}}$ is analogous to $p_c + p_c^\\star = 1$ for Bernoulli percolation.","There are important distinctions between the vacant and the occupied sets in regard to their percolation properties.","An important difference comes from the fact that parts (REF ) and (REF ) of Theorem REF do not hold in general for the occupied set, since the existence of long occupied connections can be triggered by a single disk of large radius.","For example, if one chooses the radius distribution $\\mu $ such that $\\mu [x, \\infty ) = x^{-2}(\\log x)^{-2}$ for $x$ large enough, then the hypothesis (REF ) is satisfied but for every $\\lambda > 0$ and every $r$ large enough ${\\mathbb {P}}_\\lambda \\big [0 \\overset{\\tiny \\mathcal {O}}{\\longleftrightarrow }\\partial B_r\\big ] \\,\\ge \\, c \\cdot \\lambda \\int _r^\\infty x \\,\\mu [x,\\infty )\\mathop {}\\!\\mathrm {d}x \\,\\ge \\, c \\cdot \\lambda /\\log r.$ Therefore, in this case the one-arm probability ${\\mathbb {P}}_\\lambda [0 \\overset{\\tiny \\mathcal {O}}{\\longleftrightarrow }\\partial B_r]$ cannot decay polynomially fast at criticality, just because this choice of $\\mu $ implies that the probability that 0 is contained in a ball of radius larger than $r$ does not decay polynomially.","Nevertheless, under the stronger moment assumption (), one can prove the polynomial decay of the arm exponent as shown in the next result.","Theorem 1.3 Assume that the $2+\\alpha $ moment condition () holds for some $\\alpha >0$ .","Then, for $\\lambda = {\\lambda _{c}}$ , there exists a constant $c_{\\ref {c:poly_o_decay}} = c_{\\ref {c:poly_o_decay}}(\\alpha ) > 0$ such that $\\mathbb {P}_{\\lambda _c}\\big [ 0 \\overset{\\tiny \\mathcal {O}}{\\longleftrightarrow }\\partial B_r \\big ] \\le \\frac{1}{c_{\\ref {c:poly_o_decay}}} r^{-c_{\\ref {c:poly_o_decay}}}.$ for all $\\lambda < {\\lambda _{c}}$ , there exists a constant $c_{\\ref {c:poly_decay}}=c_{\\ref {c:poly_decay}}(\\lambda ) > 0$ such that $\\mathbb {P}_\\lambda \\big [ 0 \\overset{\\tiny \\mathcal {O}}{\\longleftrightarrow }\\partial B_r \\big ] \\le c_{\\ref {c:poly_decay}} r^{-\\alpha }.$ Remark 3 The exponent in part (ii) of Theorem REF cannot be improved in general.","To see this, consider the radii distribution for which $\\mu [x,\\infty )=x^{-(2+\\alpha )}(\\log x)^{-2}$ for large $x$ .","This distribution satisfies (), but ${\\mathbb {P}}_\\lambda \\big [ 0 \\overset{\\tiny \\mathcal {O}}{\\longleftrightarrow }\\partial B_r \\big ]\\,\\ge \\,\\lambda r^2\\mu [2r,\\infty )\\,=\\,\\lambda (2r)^{-\\alpha }(\\log 2r)^{-2}$ for large $r$ .","We will now describe the main steps in the proof of the above results, which roughly speaking correspond to the three steps described for the case of Bernoulli percolation above.","These are finite-size criterion, Russo-Seymour-Welsh theory and a threshold phenomenon.","We describe each of these steps below and emphasize that their corresponding proofs can be read independently of one another."],["Finite-size criterion","In Section we prove that the crossing probabilities converge to 1 as soon as they become close enough to 1.","Roughly speaking we prove that there exists $\\theta >0$ and $r_0(\\lambda )$ such that the following are equivalent $\\mathbb {P}_\\lambda \\big [\\operatorname{Cross}(3r, r)\\big ] > 1 - \\theta \\text{ for some $r\\ge r_0$,}\\\\\\lim _{r \\rightarrow \\infty } \\mathbb {P}_\\lambda \\big [\\operatorname{Cross}(3r, r)\\big ] = 1,$ and the analogous statement holds for vacant crossings, see Proposition REF .","This motivates us to introduce $\\lambda _0 := \\sup \\big \\lbrace \\lambda \\ge 0 : \\lim _{r\\rightarrow \\infty } {\\mathbb {P}}_\\lambda \\big [ \\operatorname{Cross}(r, 3r) \\big ] = 0\\big \\rbrace ,\\\\\\lambda _1 := \\inf \\big \\lbrace \\lambda \\ge 0 : \\lim _{r\\rightarrow \\infty } {\\mathbb {P}}_\\lambda \\big [ \\operatorname{Cross}(3r, r) \\big ] = 1\\big \\rbrace .$ The first important consequence of the finite-size criterion is that $0\\,<\\,\\lambda _0\\,\\le \\,\\lambda _c\\,\\le \\, \\lambda _1\\, < \\,\\infty .$ These results are proved in Section .","Up to now, it is not clear that $\\lambda _0$ is actually equal to $\\lambda _1$ .","We call $[\\lambda _0, \\lambda _1]$ the critical regime and we can see that, for every $\\lambda \\in [\\lambda _0, \\lambda _1]$ , $\\inf _{r\\ge 1} {\\mathbb {P}}_\\lambda \\big [ \\operatorname{Cross}(r, 3r) \\big ] > 0 \\quad \\text{and} \\quad \\inf _{r\\ge 1} {\\mathbb {P}}_\\lambda \\big [ \\operatorname{Cross}^\\star (r, 3r) \\big ] > 0.$ These statements say that in the critical regime, rectangles have non-degenerate probabilities of being crossed by both the vacant and the occupied set.","However, one should notice that these crossings happen in the easy direction of the rectangles.","In order to obtain a more precise description of the behavior in the critical regime we require similar statements regarding crossing probabilities in the long direction.","This is precisely the purpose of Russo-Seymour-Welsh theory, as we describe next."],["Russo-Seymour-Welsh theory","In this step we show that (REF ) implies that $\\inf _{r\\ge 1} {\\mathbb {P}}_\\lambda \\big [ \\operatorname{Cross}(3 r, r) \\big ] > 0 \\quad \\text{and} \\quad \\inf _{r\\ge 1} {\\mathbb {P}}_\\lambda \\big [ \\operatorname{Cross}^\\star (3 r, r) \\big ] > 0.$ Note that the only difference between the above and (REF ) is that now the rectangles are crossed in the long direction.","This `box-crossing' property for the critical regime rules out the existence of unbounded clusters and provides bounds on arm probabilities.","These results are proved in Section .","Remark 4 RSW bounds were obtained separately for the occupied and vacant regimes during the 1990s by Roy [32] and Alexander [4].","However, these proofs are technical, and pose pose heavy restrictions on the radii distribution.","Recently, Tassion [42] found an argument allowing for greater generality.","His argument, presented in the setting of Voronoi percolation but extends verbatim, shows that $ \\liminf _{r\\rightarrow \\infty }{\\mathbb {P}}_\\lambda \\big [\\operatorname{Cross}(r,r)\\big ]>0\\quad \\Rightarrow \\quad \\liminf _{r\\rightarrow \\infty }{\\mathbb {P}}_\\lambda \\big [\\operatorname{Cross}(3r,r)\\big ]>0.$ While this is all that is needed in `symmetric' percolation models, such as Poisson Voronoi percolation, it is indispensable for `non-symmetric' models, which we consider here, that crossings of rectangles the easy way imply crossings also the hard way.","This extension is not straightforward."],["Sharp thresholds","The final step of the proof is to show that $\\lambda _0 = \\lambda _1$ .","The main strategy here is to show that the occupied crossing probabilities grow very fast from zero to one as we increase the density $\\lambda $ of the system.","There is a solid theory that predicts the occurrence of threshold phenomena in the setting of Boolean function on the discrete cube $\\lbrace 0, 1\\rbrace ^n$ equipped with a product structure.","Poisson point processes is the natural analogue to the discrete Bernoulli measure.","It is therefore natural to believe that these techniques should carry over also to the Poissonian continuum setting.","This is indeed the case, although doing so is not straightforward task.","We will here follow an approach introduced in [1].","This part of the argument involves the analysis of Boolean functions and is the least general part of our argument, since it involves representing the crossing probabilities in our Poisson point process as a certain function of discrete Bernoulli random variables.","This construction and the proof that $\\lambda _0 = \\lambda _1$ can be found in Section ."],["Other examples","We have chosen Poisson Boolean percolation as the main example to illustrate the techniques of this work.","However, the techniques that we develop apply in greater generality.","We emphasize this fact in Section , where we apply these techniques to Poisson Voronoi and confetti percolation.","We direct the reader to that section for a statement of the results we obtain for these models, and give here just a short description of the scope of our techniques.","The finite-size criterion described above only uses the fact that the law of our percolation process is invariant under translations and right-angle rotations, see (REF ), and that it satisfies a mixing condition, see (REF ).","The Russo-Seymour-Welsh part of our argument, we solely use the invariance of $\\mathcal {O}$ and $\\mathcal {V}$ under translations, right-angle rotations and reflections in coordinate axes, see (REF ), the mixing property (REF ), the FKG inequality (REF ) and a certain continuity property of the crossing probabilities stated in Lemma REF .","The threshold argument is the least general and uses, in addition to the assumptions made above, the fact that process is based on a Poisson point process."],["Some previous works on the model","The reference book [27] provides a general exposition of continuum percolation.","The case of uniformly bounded radius distribution has been extensively studied in [44], [45], [29], [32], [28], [4].","These works have already established much of our results in this restricted setting.","In Lemma A.2 of [17], it is proved that for sufficiently small $\\lambda > 0$ , $\\mathbb {P}_\\lambda [B_r \\leftrightarrow B_{2r}^c]$ goes to zero with $r$ under the second moment condition (REF ).","This implies that $\\lambda _0 > 0$ under the second moment condition.","In [33], Roy studied a Poisson soup of bounded sticks in the plane, proving a Russo-Seymour-Welsh theorem and establishing that ${\\lambda _{c}^\\star }= {\\lambda _{c}}$ .","Uniqueness of the unbounded components (for both vacant and occupied regions) has been established in [26]."],["Organization of the paper","In Section we provide some notation and preliminary results that are needed throughout the text.","Sections , and present the three steps of the proof that were described in the introduction, namely: the finite-size criterion, Russo-Seymour-Welsh result and the sharp threshold step.","These sections can be read independently of one another.","Section proves the continuity of the critical parameter with respect to the law $\\mu $ of the radius distribution.","The main results of the paper are then proved in Section .","We treat other models and provide some open questions in Sections and .","Acknowledgements This work began during a visit of V.T.","to IMPA, that he thanks for support and hospitality.","We thank the Centre Intradisciplinaire Bernoulli (CIB) and Stardû for hosting the authors.","D.A.","was during the course of this project financed by grant 637-2013-7302 from the Swedish Research Council.","A.T. is grateful to CNPq for its financial contribution to this work through the grants 306348/2012-8, 478577/2012-5 and 309356/2015-6 and FAPERJ through grant number 202.231/2015.","V.T.","acknowledges support from the Swiss NSF."],["Notation and preliminary results","Throughout the text we let $c$ denote positive constants which may depend on the radius distribution and may change from line to line.","However, numbered constants such as $c_0, c_1, \\dots $ refer to their first appearance in the text.","We will further write $B^\\infty (r) = [-r, r]^2$ for the ball centered at the origin in the supremum norm and let $B(x,r)$ , on the other hand, denote the closed Euclidean ball with center $x$ and radius $r$ .","When $x$ is the origin we omit it from the above notation.","Rather informally, a realization of Poisson Boolean percolation is obtained by decorating the points in a Poisson point process in ${\\mathbb {R}}^2$ by Euclidean discs with independent radii sampled from some distribution.","There are various (equivalent) ways of making this description formal.","We will start this section by describing one way which will be suitable for our purposes."],["Definition of the process","To define our process, let us first introduce a Poisson point process on the following space of point measures $\\Omega = \\Big \\lbrace \\omega = \\sum _i \\delta _{(x_i,z_i)}: (x_i,z_i) \\in \\mathbb {R}^2 \\times \\mathbb {R}_+ \\text{ and } \\omega \\big ( K \\times \\mathbb {R}_+ \\big ) < \\infty \\text{ for all $K$ compact} \\Big \\rbrace .$ We endow this space with the $\\sigma $ -algebra $\\mathcal {M}$ generated by the evaluation maps $A \\mapsto \\omega (A)$ , for $A \\in \\mathcal {B}(\\mathbb {R}^2 \\times \\mathbb {R}_+)$ , the Borel sets on ${\\mathbb {R}}^2 \\times \\mathbb {R}_+$ .","We next fix an intensity parameter $\\lambda \\ge 0$ and some probability measure $\\mu $ on $\\mathbb {R}_+$ which will give the radius distribution of our discs.","We can now define on $(\\Omega , \\mathcal {M})$ a Poisson point process with intensity $\\lambda \\cdot dx \\, \\mu (dz)$ , i.e., Lebesgue measure on $\\mathbb {R}^2$ product with $\\mu $ and multiplied by $\\lambda \\ge 0$ .","The law of this process is denoted by $\\mathbb {P}_\\lambda $ throughout the text and we complete the $\\sigma $ -algebra $\\mathcal {M}$ with respect to $\\mathbb {P}_\\lambda $ .","For each point $(x,z) \\in \\mathbb {R}^2 \\times \\mathbb {R}_+$ in the support of the measure $\\omega \\in \\Omega $ , we associate the disc $B(x, z)$ and the occupied region of the plane is consequently given by $\\mathcal {O} := \\bigcup _{(x,z) \\in \\operatorname{supp}(\\omega )} B(x, z),$ while the vacant set is given by its complement $\\mathcal {V} := \\mathbb {R}^2 \\setminus \\mathcal {O}$ .","(Below, we will often identify $\\omega $ with its support in order to ease the notation.)","From the definition of the model, it is trivial to conclude that the law of $\\mathcal {O}$ is invariant under translations, right-angle rotations, and reflection in coordinate axes.","Although the law is invariant under arbitrary rotations, and reflections in axes with other orientation, we will only use the above statement throughout the text.","There is a natural partial ordering of elements in $\\Omega $ , namely, $\\omega \\le \\omega ^{\\prime }$ if $\\omega (A) \\le \\omega ^{\\prime }(A)$ for all $A \\in \\mathcal {B}(\\mathbb {R}^2 \\times \\mathbb {R}_+)$ .","An event $A\\in \\mathcal {M}$ is said to be increasing if $\\omega \\in A$ implies that $\\omega ^{\\prime }\\in A$ for all $\\omega \\le \\omega ^{\\prime }$ .","It is decreasing if its complement is increasing.","A useful property of increasing events is that they are positively correlated.","The following proposition, known as the FKG inequality, was proved by Roy in his doctorate thesis; see also [27]: If $A_1$ and $A_2$ are increasing events, then ${\\mathbb {P}}_\\lambda (A_1\\cap A_2)\\,\\ge \\,{\\mathbb {P}}_\\lambda (A_1){\\mathbb {P}}_\\lambda (A_2).$ The above also holds when $A_1$ and $A_2$ are decreasing events.","We will also use the following standard consequence of the FKG inequality, referred to as the square-root trick.","Let $A_1,\\ldots ,A_k$ be $k$ increasing events (or $k$ decreasing events), then $\\max _{1\\le i \\le k}{\\mathbb {P}}_\\lambda (A_i)\\,\\ge \\, 1-\\big [1-{\\mathbb {P}}_\\lambda (A_1\\cup \\ldots \\cup A_k)\\big ]^{1/k}.$"],["Crossing events","Throughout the text we will often deal with crossing events of various types.","In particular, we will be interested in the following general definition.","Given subsets $A_1, A_2$ and $\\mathcal {C}$ of $\\mathbb {R}^2$ let $A_1 \\overset{\\mathcal {C}}{\\longleftrightarrow }A_2 := \\text{there is a path in $\\mathcal {C}$ connecting $A_1$ to $A_2$}.$ We are now in position to give a formal definition of the crossing event $\\operatorname{Cross}(r,h)$ , for $r, h > 0$ , as $\\operatorname{Cross}(r, h) := \\big [ A_1 {\\overset{\\mathcal {O} \\cap K}{\\longleftrightarrow }} A_2 \\big ],$ where $K$ denotes the box $[0,r] \\times [0,h]$ and $A_1 = \\lbrace 0\\rbrace \\times [0,h]$ , $A_2 = \\lbrace r\\rbrace \\times [0,h]$ stand for its left and right sides.","We also define the event corresponding to a vacant crossing as $\\operatorname{Cross}^\\star (r,h) := [A_1 {\\overset{\\mathcal {V} \\cap K}{\\longleftrightarrow }} A_2 ]$ .","It is rather straightforward to verify that events of this type are measurable.","Notice that occupied crossing events are increasing, and vacant crossing events are decreasing."],["Decay of spatial correlations","The second moment condition given in (REF ) is sufficient to imply a spatial decorrelation, in the sense of the function $\\rho _\\lambda $ introduced in Definition REF below.","First set $Y_v = \\operatorname{{\\bf 1}}_{\\lbrace v \\in \\mathcal {V}\\rbrace }$ for $v \\in \\mathbb {R}^2$ .","Definition 2.1 Given $0 < r, s < \\infty $ , let $\\rho _\\lambda (r, s) := \\sup _{f_1, f_2} \\operatorname{Cov}\\big (f_1(\\mathcal {O}), f_2(\\mathcal {O})\\big ),$ where the above suppremum is taken over all functions $f_1,f_2:\\mathcal {P}(\\mathbb {R}^2)\\rightarrow [-1,1] $ such that $f_1(\\mathcal {O}) \\in \\sigma (Y_v; v\\in B^\\infty (r))$ and $f_2(\\mathcal {O}) \\in \\sigma (Y_v; v \\notin B^\\infty (r+s))$ .","The function $\\rho _\\lambda $ has a nice geometric interpretation.","Namely, one can observe that $\\rho _\\lambda (r, r+s)$ is directly related to the probability that there exists one big occupied disk crossing the annulus $B^\\infty (r+s)\\setminus B^\\infty (r)$ .","By making this observation rigorous, and computing this crossing probability (see Lemma REF ), we prove the following upper bound.","Proposition 2.2 For any $\\lambda > 0$ and $r, s \\ge 1$ , we have $\\rho _\\lambda (r, s)\\, \\le \\, c_{\\ref {c:decorrelation}} \\lambda \\left(1+\\frac{r}{s}\\right)^2\\int _{s/2}^\\infty x^2\\mu (\\mathop {}\\!\\mathrm {d}x).$ In particular, the second moment condition (REF ) implies that $\\text{for any $\\kappa > 0$, }\\lim _{r\\rightarrow \\infty } \\rho _\\lambda (r, \\kappa r) = 0.$ and the $2 + \\alpha $ moment condition in () implies that for every $\\lambda > 0$ and $\\varepsilon >0$ $\\sup \\limits _{r\\ge 1, \\kappa \\ge \\varepsilon } r^\\alpha \\rho _\\lambda (r, \\kappa r) < \\infty .$ For $K \\subseteq \\mathbb {R}^2$ , we write $\\mathcal {O}_K = \\bigcup _{(x,z) \\in \\omega :\\, x \\in K} B(x, z).$ The next lemma says that deleting balls from $\\mathcal {O}$ which are far from a given box do not alter the configuration of $\\mathcal {O}$ inside that box.","Figure: An illustration of the sets 𝒪\\mathcal {O}, 𝒪 B ∞ (r+s/2) \\mathcal {O}_{B^\\infty (r + s/2)} and 𝒪 B ∞ (r+s/2) c \\mathcal {O}_{B^\\infty (r + s/2)^c} respectively.","Note that in this particular realization we have 𝒪 B ∞ (r+s/2) ∩B ∞ (r)=𝒪∩B ∞ (r)\\mathcal {O}_{B^\\infty (r + s/2)} \\cap B^\\infty (r) = \\mathcal {O} \\cap B^\\infty (r) and also that 𝒪 B ∞ (r+s/2) c ∩B ∞ (r+s) c =𝒪∩B ∞ (r+s) c \\mathcal {O}_{B^\\infty (r + s/2)^c} \\cap B^\\infty (r + s)^c = \\mathcal {O} \\cap B^\\infty (r + s)^c as in Lemma .Lemma 2.3 For all $r, s > 0$ , writing $K_1 = B^\\infty (r)$ and $K_2 = B^\\infty (r + s)$ , we have $\\mathbb {P}_\\lambda \\big ( \\mathcal {O}_{K_2^c} \\cap K_1 \\ne \\varnothing \\big )\\le 8\\lambda \\left(1+\\frac{r+1}{s}\\right)^2\\int _s^\\infty x^2\\mu (\\mathop {}\\!\\mathrm {d}x).$ Also, $\\mathbb {P}_\\lambda \\big ( \\mathcal {O}_{K_1} \\cap K_2^c \\ne \\varnothing \\big )\\le 8\\lambda \\left(\\frac{r}{s}\\right)^2\\int _s^\\infty x^2\\mu (\\mathop {}\\!\\mathrm {d}x).$ Remark 5 The above lemma implies in particular that, whenever $\\int _0^\\infty x^2\\mu (\\mathop {}\\!\\mathrm {d}x) < \\infty $ , if $K \\subseteq \\mathbb {R}^2$ is compact, then $\\mathbb {P}_\\lambda $ -almost surely there are only finitely many balls in $\\mathcal {O}$ that intersect $K$ .","We first derive the second bound.","Since $K_1$ has area $8r^2$ , an immediate estimate of the expected number of points in $K_1$ with radius at least $s$ gives that $\\mathbb {P}_\\lambda \\big ( \\mathcal {O}_{K_1} \\cap K_2^c \\ne \\varnothing \\big ) \\,\\le \\, 8r^2\\lambda \\int _s^\\infty \\mu (\\mathop {}\\!\\mathrm {d}x) \\,\\le \\, 8\\lambda \\left(\\frac{r}{s}\\right)^2\\int _s^\\infty x^2\\mu (\\mathop {}\\!\\mathrm {d}x),$ hence establishing (REF ).","For the first bound, we split the complement of $K_2 =B^\\infty (r+s)$ into the disjoint annuli $A_i = B^\\infty (r + s + i + 1) \\setminus B^\\infty (r + s + i)$ and write $K_2^c = \\bigcup _{i\\ge 0} A_i$ .","Using the fact that the area of $A_i$ is bounded by $8(r+s+i+1)$ , we can now estimate $\\begin{split}\\mathbb {P}_\\lambda \\big ( \\mathcal {O}_{K_2^c} \\cap K_1 \\ne \\varnothing \\big )& \\,\\le \\, \\sum _{i\\ge 0} \\mathbb {P}_\\lambda \\big ( \\mathcal {O}_{A_i} \\cap K_1 \\ne \\varnothing \\big )\\\\& \\le \\, 8\\lambda \\sum _{i\\ge 0} (r + s + i+1)\\int _{s+i}^\\infty \\mu (\\mathop {}\\!\\mathrm {d}x).\\end{split}$ Exchanging the order of summation yields the bound $8\\lambda \\int _s^\\infty \\sum _{i=0}^{x-s}(r+s+i+1)\\mu (\\mathop {}\\!\\mathrm {d}x) \\,\\le \\, 8\\lambda \\int _s^\\infty (r+x+1)^2\\mu (\\mathop {}\\!\\mathrm {d}x),$ from which (REF ) follows.","Let us now prove Proposition REF .","[Proof of Proposition REF ] Recall that $\\Vert f_1 \\Vert _\\infty , \\Vert f_2 \\Vert _\\infty \\le 1$ .","Let $K=B^\\infty (r+s/2)$ .","By the Poissonian character of the law $\\mathbb {P}_\\lambda $ , the two random variables $ f_1(\\mathcal {O}_K)$ and $f_2(\\mathcal {O}_{K^c})$ are independent.","Therefore, $\\begin{split}\\operatorname{Cov}(f_1(\\mathcal {O}), f_2(\\mathcal {O}) & \\,\\le \\, \\mathbb {P}_\\lambda \\big (f_1(\\mathcal {O}) \\ne f_1(\\mathcal {O}_K)\\big ) + \\mathbb {P}_\\lambda \\big (f_2(\\mathcal {O})\\ne f_2(\\mathcal {O}_{K^c})\\big )\\\\& \\le \\, \\mathbb {P}_\\lambda \\big ( \\mathcal {O}_{K^c} \\cap B^\\infty (r)\\ne \\varnothing \\big ) + \\mathbb {P}_\\lambda \\big ( \\mathcal {O}_K \\cap B^\\infty (r+s)^c\\ne \\varnothing \\big ),\\end{split}$ and the proof then follows from Lemma REF .","For the last conclusion (REF ), one may use Markov's inequality."],["An alternative notion of decoupling","Note that Definition REF is symmetric for $\\mathcal {O}$ and $\\mathcal {V}$ , in the sense that the roles of the occupied and vacant sets and are interchangeable.","In what follows, we will introduce another notion of decoupling that will not be symmetric for $\\mathcal {O}$ and $\\mathcal {V}$ .","This distinction will be very relevant to explain the different behavior of these sets, see Remark REF , and explains why we obtain Theorem REF under (REF ) but require (REF ) for Theorem REF with the techniques used here.","To motivate this new notion, observe from Definition REF that $\\mathbb {E}_\\lambda \\big (f_1(\\mathcal {O}) f_2(\\mathcal {O})\\big ) \\,\\le \\,\\mathbb {E}_\\lambda \\big (f_1(\\mathcal {O})\\big ) \\mathbb {E}_\\lambda \\big (f_2(\\mathcal {O})\\big ) + \\rho (r, s),$ for every functions $f_1,f_2:\\mathcal {P}(\\mathbb {R}^2)\\rightarrow [0,1]$ such that $f_1(\\mathcal {O}) \\in \\sigma (Y_v; v \\in B^\\infty (r))$ and $f_2(\\mathcal {O}) \\in \\sigma (Y_v; v \\notin B^\\infty (r + s))$ .","This will be used in several parts of the text.","However, some results will be stronger using the following modified version of the above.","Definition 2.4 Let $\\bar{\\rho }_\\lambda :\\mathbb {R}_+^2 \\rightarrow \\mathbb {R}_+$ be defined as the smallest value such that $\\mathbb {E}_\\lambda \\big (f_1(\\mathcal {O}) f_2(\\mathcal {O})\\big ) \\,\\le \\,\\mathbb {E}_\\lambda \\big (f_1(\\mathcal {O})\\big ) \\Big ( \\mathbb {E}_\\lambda \\big (f_2(\\mathcal {O})\\big ) + \\bar{\\rho }_\\lambda (r, s) \\Big ),$ for all decreasing functions $f_1, f_2:\\mathcal {P}(\\mathbb {R}^2)\\rightarrow [0,1]$ satisfying $f_1(\\mathcal {O}) \\in \\sigma (Y_v; v \\in B^\\infty (r))$ , $f_2(\\mathcal {O}) \\in \\sigma (Y_v; v \\notin B^\\infty (r + s))$ .","(A function $f:\\mathcal {P}(\\mathbb {R}^2) \\rightarrow [0,1]$ is said to be decreasing if $f(A)\\ge f(B)$ whenever $A\\subset B$ .)","Remark 6 Note that the error $\\bar{\\rho }_\\lambda $ in the above definition is being multiplied by $\\mathbb {E}_\\lambda (f_1)$ .","This represents an important improvement in some cases, as will become clear later for instance when we compare (REF ) with (REF ).","Although the error term in Definition REF is smaller than the one in (REF ), the bound (REF ) can only be used when $f_1$ and $f_2$ are decreasing functions.","This restriction was intentionally introduced in the definition of $\\bar{\\rho }_\\lambda $ and it is necessary in our proof that $\\bar{\\rho }_\\lambda $ vanishes, see Proposition REF below.","This restriction reflects the distinct behavior that the vacant and occupied sets display at criticality, see Remark REF .","We now prove an analogue of Proposition REF .","Proposition 2.5 For any $\\lambda > 0$ and $r, s \\ge 1$ we have $\\bar{\\rho }_\\lambda (r, s) \\,\\le \\, c_{\\ref {c:decorrelation}} \\lambda \\left(1+\\frac{r}{s}\\right)^2\\int _{s/2}^\\infty x^2\\mu (\\mathop {}\\!\\mathrm {d}x).$ In particular, if $\\int _0^\\infty x^2 \\mu (\\mathop {}\\!\\mathrm {d}x) < \\infty $ , then $\\text{for any } \\kappa > 0,\\, \\lim _{r\\rightarrow \\infty } \\bar{\\rho }_\\lambda (r, \\kappa r) = 0.$ We write $K_1 = B^\\infty (r)$ and $K_2 = B^\\infty (r + s)^c$ and define $\\check{\\mathcal {O}} = \\bigcup _{\\begin{array}{c}(x,z) \\in \\operatorname{supp}(\\omega ); \\\\ B(x, z) \\cap K_1 = \\varnothing \\end{array}} B(x, z).$ Note that this set is contained in $\\mathcal {O}$ and independent of $f_1(\\mathcal {O})$ .","Using the fact that $f_2$ is decreasing, we can write $\\begin{split}\\mathbb {E}_\\lambda \\big ( f_1(\\mathcal {O}) f_2(\\mathcal {O}) \\big ) & \\,\\le \\, \\mathbb {E}_\\lambda \\big ( f_1(\\mathcal {O}) f_2(\\check{\\mathcal {O}}) \\big ) \\,\\le \\, \\mathbb {E}_\\lambda \\big ( f_1(\\mathcal {O}) \\big ) \\mathbb {E}_\\lambda \\big ( f_2(\\check{\\mathcal {O}}) \\big )\\\\& \\le \\, \\mathbb {E}_\\lambda \\big ( f_1(\\mathcal {O}) \\big ) \\Big ( \\mathbb {E}_\\lambda \\big ( f_2(\\mathcal {O}) \\big ) + \\mathbb {P}_\\lambda \\big ( \\check{\\mathcal {O}} \\cap K_2 \\ne \\mathcal {O} \\cap K_2 \\big ) \\Big ),\\end{split}$ where we have used the fact that $f_1,f_2\\in [0,1]$ .","In view of the definition of $\\bar{\\rho }_\\lambda $ , all we need to do is to bound the probability above, $\\begin{split}\\mathbb {P}_\\lambda (\\check{\\mathcal {O}} & \\cap K_2 \\ne \\mathcal {O} \\cap K_2) \\,\\le \\, \\mathbb {P}_\\lambda \\Big (\\begin{array}{c}\\text{there is $(x,z) \\in \\operatorname{supp}(\\omega )$ such that}\\\\\\text{$B^\\infty (x,z)$ touches $K_1$ and $K_2$}\\end{array}\\Big )\\\\& \\le \\, \\mathbb {P}_\\lambda \\big (\\mathcal {O}_{B^\\infty (r + s/2)} \\cap K_2 \\ne \\varnothing \\big ) + \\mathbb {P}_\\lambda \\big ( \\mathcal {O}_{B^\\infty (r + s/2)^2} \\cap K_1 \\ne \\varnothing \\big ).\\end{split}$ The proof now ends in the same way as the proof of Proposition REF ."],["Local continuity","Another rather straightforward consequence of the Poissonian nature of the model is the following `local continuity' property.","In essence this property says that locally the topological properties of the set $\\mathcal {O}$ are unlikely to be affected by slight change in the radii of the discs.","Given a set $A\\subseteq {\\mathbb {R}}^2$ we define its $\\varepsilon $ -interior and $\\varepsilon $ -closure, for $\\varepsilon >0$ , as follows: $\\begin{aligned}\\operatorname{int}(A,\\varepsilon )&:=\\lbrace x\\in {\\mathbb {R}}^2:B(x,\\varepsilon )\\subseteq A\\rbrace ,\\\\\\operatorname{cl}(A,\\varepsilon )&:=\\lbrace x\\in {\\mathbb {R}}^2:B(x,\\varepsilon )\\cap A\\ne \\varnothing \\rbrace .\\end{aligned}$ We omit the proof of the following proposition.","Proposition 2.6 Let $K \\subseteq {\\mathbb {R}}^2$ be compact and convex and $A_1, A_2 \\subseteq K$ closed.","For every $\\lambda > 0$ $\\lim _{\\varepsilon \\rightarrow 0}{\\mathbb {P}}_\\lambda \\Big ( A_1 {\\overset{\\operatorname{cl}(\\mathcal {O},\\varepsilon ) \\cap K}{\\longleftrightarrow }} A_2\\text{ occurs but not }A_1 {\\overset{\\operatorname{int}(\\mathcal {O},\\varepsilon ) \\cap K}{\\longleftrightarrow }} A_2 \\Big ) \\,=\\,0.$ A consequence of the above proposition that will be used later is the following lemma.","Consider a variation of the standard crossing event defined as $\\hbox{[scale=1.5]{(0, 0) rectangle (4ex, 4ex);(3.8ex, 2ex) -- (4.2ex, 2ex);(0, 1.7ex) .. controls (1.3ex, 4ex) and (2.6ex, 1.4ex) .. (4ex, 3.2ex);(2ex, .2ex) node[anchor=north] {$\\scriptstyle r$};(.4ex, 2ex) node[anchor=east] {$\\scriptstyle h$};[decorate,decoration={brace,amplitude=.4ex,raise=.4ex},yshift=0pt] (4ex, 4ex) -- (4ex, 2ex);(4.3ex, 3ex) node[anchor=west] {$\\scriptstyle y$};}\\;}$ := [ A1 O K A2 ], where $K$ is again the box $[0,r] \\times [0,h]$ , while $A_1 = \\lbrace 0\\rbrace \\times [0,h]$ and $A_2 = \\lbrace r\\rbrace \\times [y, h]$ .","Lemma 2.7 Given $r, h$ and $\\lambda > 0$ , the function $F:[0,h] \\rightarrow [0,1]$ given by $F(y) = \\mathbb {P}_\\lambda \\Bigg [ \\smash{\\hbox{[scale=1.5]{(0, 0) rectangle (4ex, 4ex);(3.8ex, 2ex) -- (4.2ex, 2ex);(0, 1.7ex) .. controls (1.3ex, 4ex) and (2.6ex, 1.4ex) .. (4ex, 3.2ex);(2ex, .2ex) node[anchor=north] {$\\scriptstyle r$};(.4ex, 2ex) node[anchor=east] {$\\scriptstyle h$};[decorate,decoration={brace,amplitude=.4ex,raise=.4ex},yshift=0pt] (4ex, 4ex) -- (4ex, 2ex);(4.3ex, 3ex) node[anchor=west] {$\\scriptstyle y$};}\\;}}$ ] is continuous."],["Finite size criterion for percolation","The first tool in our study of the phase transition for Poisson Boolean percolation in ${\\mathbb {R}}^2$ will be the following result that bootstraps the probability of crossing a rectangle: from close to one to converging fast to one.","This will be fundamental in this paper; as a first indication of its importance we show how it relates to the existence of an unbounded component, and the non-triviality of the threshold parameters $\\lambda _0$ and $\\lambda _1$ .","Proposition 3.1 Assume the second moment condition (REF ).","Then, there exists a constant $\\theta = \\theta (\\mu ) > 0$ and an increasing function $r_0:[0,\\infty )\\rightarrow [0,\\infty )$ such that the following are equivalent: There exists $r\\ge r_0(\\lambda ) $ such that ${\\mathbb {P}}_\\lambda (\\operatorname{Cross}(3r,r))>1-\\theta $ .","$\\displaystyle \\lim _{r\\rightarrow \\infty }{\\mathbb {P}}_\\lambda (\\operatorname{Cross}(3r,r))=1$ .","Under the stronger condition (REF ), if either of (i) or (ii) holds, then $\\sum _{k\\ge 0}\\big [1-{\\mathbb {P}}_\\lambda (\\operatorname{Cross}(3^{k+1},3^k))\\big ]<\\infty $ .","These statements remains valid for $\\operatorname{Cross}$ replaced by $\\operatorname{Cross}^\\star $ .","Fix $\\theta =1/100$ and let $C$ be a sufficiently large constant so that, for all $r\\ge 1$ , $\\rho _\\lambda (5r,r)\\le g(r) := C \\lambda \\int _{r/2}^\\infty x^2\\mu (\\mathop {}\\!\\mathrm {d}x).$ The existence of the constant follows from Proposition REF .","Set $r_0:=\\min \\lbrace r\\ge 1:g(r)\\le \\theta /2\\rbrace $ .","Since $g(r)$ increases with $\\lambda $ it is clear that $r_0$ does too.","Since (ii) trivially implies (i), it will suffice to prove that (i) implies (ii) and, under the stronger assumption (REF ), that (ii) implies (iii).","Let $p(r):=1-{\\mathbb {P}}_\\lambda (\\operatorname{Cross}(3r,r))$ and let $B_k:=\\operatorname{Cross}(9r,r)$ .","Note that a $9r\\times r$ -rectangle may be tiled with seven $3r\\times r$ -rectangles, four positioned horizontally and three vertically, in such a way that if each is crossed in the `hard' direction, then the $9r\\times r$ -rectangle is crossed horizontally.","Consequently, using the union bound, we obtain that ${\\mathbb {P}}_\\lambda (B_r)\\,\\ge \\,1-7{\\mathbb {P}}_\\lambda (\\lnot \\operatorname{Cross}(3r,r))\\,=\\,1-7p(r).$ Let $B_r^{\\prime }$ denote the translate of $B_r$ along the vector $(0,2r)$ .","Since the occurrence of either of $B_r$ or $B_r^{\\prime }$ implies the occurrence of $\\operatorname{Cross}(9r,3r)$ , he have $p(3r)\\,\\le \\,{\\mathbb {P}}_\\lambda (\\lnot B_r\\cap \\lnot B_r^{\\prime })\\,\\le \\,{\\mathbb {P}}_\\lambda (\\lnot B_r)^2+\\rho _\\lambda (5r,r).$ Hence, by definition of $g$ , we obtain for every $r\\ge 1$ the bound $p(3r)\\le 49p(r)^2+g(r).$ Now, assume there exists $r\\ge r_0$ so that $p(r)<\\theta $ .","Then, via iterated use of (REF ), we find that $p(3^kr)<\\theta $ for all $k\\ge 0$ .","Consequently, further use of (REF ) gives, for $\\ell =1,2,\\ldots ,k$ $p(3^kr)\\,\\le \\,\\frac{1}{2^\\ell }p(3^{k-\\ell }r)+\\sum _{j=1}^\\ell \\frac{1}{2^{j-1}}g(3^{k-j}r)\\,\\le \\,\\frac{1}{2^\\ell }+2g(3^{k-\\ell }r).$ Hence, sending $\\ell $ to infinity with $k$ shows that $\\lim _{k\\rightarrow \\infty }p(3^kr)=0$ .","Since for every $r^{\\prime }\\in [r,3r]$ we have $p(r^{\\prime })\\le {\\mathbb {P}}_\\lambda (\\lnot B_r)\\le 7p(r)$ , it follows that $\\lim _{r\\rightarrow \\infty }p(r)=0$ , so (ii) holds.","Finally, we assume that (ii) holds and pick $k_0$ so that $3^{k_0}\\ge r_0$ and $p(3^{k_0})<\\theta $ .","Summing over $k$ in (REF ), with $r=3^{k_0}$ and $\\ell =\\lfloor k/2\\rfloor $ , leads to $\\sum _{k\\ge 1}p(3^{k+k_0})\\,\\le \\,\\sum _{k\\ge 1}\\frac{1}{2^{(k-1)/2}}+2\\sum _{k\\ge 1}g(3^{k/2+k_0})\\,\\le \\,4+2\\int _0^\\infty g(3^{y/2})\\,\\mathop {}\\!\\mathrm {d}y.$ Since $g$ is itself an integral we may use Fubini's theorem to reverse the order of integration, and obtain $\\sum _{k\\ge 1}p(3^{k+k_0})\\,\\le \\, 4+C\\lambda \\int _0^\\infty x^2\\log x\\,\\mu (\\mathop {}\\!\\mathrm {d}x),$ for a possibly larger constant $C$ , which is finite under the assumption (REF ), and (iii) follows.","Remark 7 Note that the proof of Proposition REF assumes very little about the underlying percolation model.","Indeed, the argument applies to any percolation model on ${\\mathbb {R}}^2$ satisfying the invariance assumption (REF ) and decay of correlations corresponding to conditions (REF ) or (REF ).","The following more quantitative statement sheds some light on the role of spatial correlations on the rate of convergence in Proposition REF .","Given a function $f:[1, \\infty ) \\rightarrow (0, \\infty )$ we say that it is regularly varying if for any $a \\in (0, \\infty )$ $\\frac{f(ar)}{f(r)} \\text{ converges to a non-zero limit.","}$ The following proposition says, in particular, that if the spatial dependence decays polynomially fast, then the convergence in Proposition REF occurs (at least) at the same polynomial rate.","Proposition 3.2 Assume the finite moment condition (REF ) and let $f:[1,\\infty )\\rightarrow (0,\\infty )$ be a regularly varying function satisfying $\\lim _{r\\rightarrow \\infty }f(r)=0$ and $\\rho _\\lambda (5r,r)\\le f(r)$ for all $r \\ge 1$ .","Then, if $\\lim _{r\\rightarrow \\infty }{\\mathbb {P}}_\\lambda (\\operatorname{Cross}(3r,r))=1$ , then there exists $c<\\infty $ such that ${\\mathbb {P}}_\\lambda (\\operatorname{Cross}(3r,r))\\ge 1-cf(r)\\quad \\text{for all }r\\ge 1.$ The statement remains true if $\\operatorname{Cross}$ is replaced by $\\operatorname{Cross}^\\star $ .","With $p(r):=1-{\\mathbb {P}}_\\lambda (\\operatorname{Cross}^\\star (3r,r))$ , we may repeat the proof of Proposition REF to obtain $p(3r)\\le 49p(r)^2+f(r).$ Since $f$ is regularly varying there exists a constant $c^{\\prime } \\in (0, 50)$ such that $f(3r)/f(r) \\ge c^{\\prime }$ for all $r\\ge 1$ .","Let $\\theta ^{\\prime }=c^{\\prime }/100$ and pick $r^{\\prime }$ such that $p(r)\\le \\theta ^{\\prime }$ and $f(r)\\le \\theta ^{\\prime }/2$ for all $r\\ge r^{\\prime }$ .","Let $c=\\max \\lbrace 2/c^{\\prime },1/f(r^{\\prime })\\rbrace $ .","We claim that $p(3^kr^{\\prime })\\le cf(3^kr^{\\prime })\\quad \\text{for all }k\\ge 0.$ The case $k = 0$ is immediate, and the remaining cases follows from (REF ) via an induction step.","This proves the statement of the proposition along exponentially growing sub-sequences from which one can easily derive the full statement.","Propositions REF and REF show how an observation in a finite region can be sufficient to draw conclusions regarding crossing probabilities over arbitrarily large regions.","The next well-known result connects the finite observation to the existence of unbounded components.","Proposition 3.3 The condition $\\sum _{k\\ge 0}\\big [1-{\\mathbb {P}}_\\lambda (\\operatorname{Cross}(3^{k+1},3^k))\\big ]<\\infty $ implies that the probability, at $\\lambda $ , that the origin is contained in an unbounded occupied component is positive.","The same holds when $\\operatorname{Cross}$ and `occupied' are replaced by $\\operatorname{Cross}^\\star $ and `vacant', respectively.","Denote by $R_k$ the rectangle $[0,3^{k+1}]\\times [0,3^k]$ , and by $R_k^{\\prime }$ the rectangle $[0,3^k]\\times [0,3^{k+1}]$ , resulting from a right-angle rotation of $R_k$ .","Consider the sequence, indexed by $k\\ge 0$ , alternating between $R_k$ or $R_k^{\\prime }$ , depending on whether $k$ is odd or even.","The above condition implies that the expected number of these rectangles that fail to be crossed in the `hard' direction is finite.","Hence, by Borel-Cantelli, it follows that all but finitely many contain a crossing in the `hard' direction almost surely.","However, all but finitely many of these crossings must intersect as a consequence of how the rectangles are placed.","Thus, with probability one, there is an unbounded occupied component.","Due to invariance with respect to translations it follows that the origin has positive probability to be contained in this component.","We close this section by showing that the (finite-size) phase transition is non-trivial.","Corollary 3.4 Assume the second moment condition (REF ).","Then, $0<\\lambda _0\\le \\lambda _c\\le \\lambda _1<\\infty .$ Under the stronger assumption (REF ) we also have $\\lambda _0\\le \\lambda _c^\\star \\le \\lambda _1$ .","Assume that (REF ) holds.","We first show that $\\lambda _0 > 0$ .","Let $\\theta >0$ and $r_0:[0,\\infty )\\rightarrow [0,\\infty )$ be as in Proposition REF .","Clearly ${\\mathbb {P}}_0(\\operatorname{Cross}(r, 3r))=0$ for all $r\\ge 1$ , in particular for $r=r_0(1)$ .","That the crossing probability varies continuously with respect to the intensity parameter is an easy consequence of (REF ) of Lemma REF and the fact that it is unlikely to add any point at all to a bounded region for small enough changes in the parameter.","Hence, we may pick $\\varepsilon \\in (0,1)$ such that ${\\mathbb {P}}_\\varepsilon (\\operatorname{Cross}(r,3r)) < \\theta $ , implying that $\\lim _{r\\rightarrow \\infty } {\\mathbb {P}}_\\varepsilon (\\operatorname{Cross}(r,3r)) = 0,$ and thus that $\\lambda _0 \\ge \\varepsilon > 0$ .","The fact that $\\lambda _1 < \\infty $ follows from the fact that this holds for the model with constant radius, together with a stochastic domination argument.","We next show that $\\lambda _c\\ge \\lambda _0$ and $\\lambda _c^\\star \\le \\lambda _1$ .","First, consider $\\lambda <\\lambda _0$ .","Then, since a path connecting the origin to $\\partial B(r)$ has to cross one of four rectangles of dimension $r/3\\times r$ in the easy direction, the union bound and the definition of $\\lambda _0$ gives that $\\lim _{r\\rightarrow \\infty }{\\mathbb {P}}_\\lambda (0\\overset{\\tiny \\mathcal {O}}{\\longleftrightarrow }\\partial B(r))\\,\\le \\,\\lim _{r\\rightarrow \\infty }4{\\mathbb {P}}_\\lambda (\\operatorname{Cross}(r/3,r))\\,=\\,0.$ Thus, $\\lambda \\le \\lambda _c$ , which shows that $\\lambda _0\\le \\lambda _c$ .","An analogous argument shows that $\\lambda _c^\\star \\le \\lambda _1$ , To prove that $\\lambda _c^\\star \\ge \\lambda _0$ , we make the stronger assumption that (REF ) holds and fix $\\lambda <\\lambda _0$ , in which case we have $\\lim _{r\\rightarrow \\infty }{\\mathbb {P}}_\\lambda (\\operatorname{Cross}^\\star (3r,r))=1$ .","Then, Proposition REF shows that $\\sum _{k\\ge 0}\\big [1-{\\mathbb {P}}_\\lambda (\\operatorname{Cross}^\\star (3^{k+1},3^k))\\big ]<\\infty ,$ which by Proposition REF implies the almost sure existence of an unbounded vacant component, and thus that $\\lambda \\le \\lambda _c^\\star $ .","Hence $\\lambda _c^\\star \\ge \\lambda _0$ .","An analogous argument shows, under the additional condition (REF ), that $\\lambda _c\\le \\lambda _1$ .","However, we claim that this additional assumption is unnecessary.","Our goal will therefore be to show that (REF ) holds, with $\\operatorname{Cross}^\\star $ replaced by $\\operatorname{Cross}$ , also under the weaker second moment condition (REF ).","As before, let $p(r)=1-{\\mathbb {P}}_\\lambda (\\operatorname{Cross}(3r,r))$ .","Repeating the first steps of the proof of Proposition REF , observing that the complements of occupied crossing events are decreasing, we obtain the following analogue to (REF ) $p(3r)\\,\\le \\,7p(r)\\big (7p(r)+\\bar{\\rho }_\\lambda (5r,r)\\big ).$ Since both $p(r)$ and $\\bar{\\rho }_\\lambda (5r,r)$ tend to zero as $r\\rightarrow \\infty $ , we may find $r_0$ such that $7p(r)+\\bar{\\rho }_\\lambda (5r,r)$ is bounded by $\\frac{1}{14}$ for all $r\\ge r_0$ .","So, for some $k_0\\ge 1$ , we obtain that $p(3^{k+k_0})\\,\\le \\,\\frac{1}{2}p(3^{k-1+k_0}r_0)\\,\\le \\,\\frac{1}{2^k}p(3^{k_0}),$ which is summable.","Hence (REF ) holds for $\\operatorname{Cross}$ also under the weaker condition (REF ), and $\\lambda _c\\le \\lambda _1$ follows."],["Russo-Seymour-Welsh theory","In the study of planar percolation, Russo-Seymour-Welsh (RSW) techniques play a central role and have numerous consequences.","The original proof for Bernoulli percolation ([34], [39]) is strongly based on planarity and the independence structure of the Bernoulli percolation measure, and does not extend easily to other contexts.","Considerable technicalities had to be overcome even in the extension to Poisson Boolean percolation in ${\\mathbb {R}}^2$ with fixed radii, see [32], [4], and in the case of (unbounded) random radii such a result has until this point not been obtained.","In the last few years some new arguments have been developed to prove RSW-results for dependent percolation models, e.g.","Voronoi percolation [9], [42], [3] and the random-cluster model [7], [13].","In this section we develop an RSW-theory applicable for Poisson Boolean percolation.","Our method of proof will be greatly inspired by that of [42].","However, due to the asymmetry between the vacant and occupied regions, we need a stronger version of the result proved in [42].","In that paper, the RSW statement bounds the crossing probabilities for rectangles in the long direction, assuming a bound on the crossing probabilities for squares, see (REF ).","Here we assume only that rectangles are crossed in the easier direction.","Our proof will be rather general and applies in settings far beyond Poisson Boolean percolation; see Remark REF below.","Theorem 4.1 (RSW-Theorem) Assume the finite second moment condition stated in (REF ).","Then, for any $\\lambda > 0$ , if for some $\\kappa >0$ we have $\\inf _{r\\ge 1} \\mathbb {P}_\\lambda \\big (\\operatorname{Cross}(\\kappa r, r) \\big ) > 0,$ then the same is true for all $\\kappa >0$ .","The same holds for $\\operatorname{Cross}$ replaced by $\\operatorname{Cross}^\\star $ .","Remark 8 Throughout this section, we are going to establish the above result and some of its consequences for Poisson Boolean percolation with finite second moments.","However, let us emphasize that the same proof works for several different types of percolation measures on the plane.","More precisely, the only properties of the random set $\\mathcal {O}$ that we use in this section are: $\\begin{array}{c}\\text{the translation, reflection and rotational symmetries in (\\ref {e:invariance}),}\\\\\\text{the FKG inequality (\\ref {eq:3}),}\\\\\\text{the decay of correlations stated in (\\ref {e:rho_to_zero}) and}\\\\\\text{the continuity of the crossing probabilities stated in Lemma~\\ref {l:cross_continuous}}.\\end{array}$"],["Consequences of Theorem ","The Russo-Seymour-Welsh Theorem stated above, which will be proved in Section REF , has several important consequences that we develop next.","These consequences concern `box-crossing' and `one-arm' probabilities in the critical regime $[\\lambda _0,\\lambda _1]$ .","However, we first state a `high-probability' version of Theorem REF .","All proofs of the consequences listed here will be given in Section REF below.","Corollary 4.2 Assume the finite second moment condition (REF ).","Given $\\lambda > 0$ , if for some $\\kappa >0$ we have $\\lim _{r\\rightarrow \\infty }\\mathbb {P}_\\lambda \\big (\\operatorname{Cross}(\\kappa r, r) \\big ) = 1,$ then it is true for all $\\kappa >0$ .","The same holds for $\\operatorname{Cross}$ replaced by $\\operatorname{Cross}^\\star $ .","Recall the definition of $\\lambda _0$ and $\\lambda _1$ in (REF ) and ().","Outside of the interval $[\\lambda _0,\\lambda _1]$ the probability of crossing a fixed-ratio rectangle will converge rather rapidly to either zero or one as the side length of the rectangle increases.","This is merely a matter of definition and the FKG-inequality.","One of the main consequences of the RSW Theorem is that within the interval $[\\lambda _0,\\lambda _1]$ the probability of crossing a fixed-ratio rectangle remains rather balanced.","This is sometimes referred to as the `box-crossing' property, which will also be used to provide bounds on the one-arm probabilities.","For $0 < r < r^{\\prime }$ , we define $\\operatorname{Arm}(r, r^{\\prime }) := \\big [ B^\\infty (r) \\overset{\\mathcal {O}}{\\longleftrightarrow } \\partial B^\\infty (r^{\\prime }) \\big ].$ As before we write $\\operatorname{Arm}^\\star $ for the above event, where $\\mathcal {O}$ is replaced by $\\mathcal {V}$ .","Given $r^{\\prime } > r\\ge 1$ we also introduce $\\operatorname{Circ}(r, {r^{\\prime }}) := \\hbox{[scale=1.5]{(5ex, 3.5ex) rectangle (7ex, 5.5ex);(3ex, 1.5ex) rectangle (9ex, 7.5ex);[rounded corners=.25em](4ex, 6.5ex) .. controls (6ex, 6.5ex + .4ex*rand) ..(8ex, 6.5ex) .. controls (8ex + .4ex*rand, 4.5ex) ..(8ex, 2.5ex) .. controls (6ex, 2.5ex + .4ex*rand) ..(4ex, 2.5ex) -- (4ex + .4ex*rand, 4.5ex) -- cycle;(9.4ex, 3.5ex) -- (9.4ex, 5.5ex);(9.2ex, 3.5ex) -- (9.6ex, 3.5ex) (9.2ex, 5.5ex) -- (9.6ex, 5.5ex);(9.4ex, 4.5ex) node[anchor=west] {$\\scriptstyle 2r$};(2.4ex, 1.5ex) -- (2.4ex, 7.5ex);(2.2ex, 1.5ex) -- (2.6ex, 1.5ex) (2.2ex, 7.5ex) -- (2.6ex, 7.5ex);(2ex, 4.5ex) node[anchor=east] {$\\scriptstyle 2r^{\\prime }$};}}$ := Arm(r, r')c Finally we let $\\operatorname{Circ}^\\star (r,{r^{\\prime }}) := \\operatorname{Arm}(r, r^{\\prime })^c$ .","Corollary 4.3 Assume the second moment condition in (REF ).","Then, for every $\\kappa > 0$ , there exists $c_{\\ref {c:q-bound}} = c_{\\ref {c:q-bound}}(\\kappa ) > 0$ such that for every $\\lambda \\in [\\lambda _0, \\lambda _1]$ we have ${\\mathbb {P}}_\\lambda (\\operatorname{Cross}(\\kappa r, r)) \\in (c_{\\ref {c:q-bound}}, 1 - c_{\\ref {c:q-bound}})\\quad \\text{for every $r\\ge 1$,}$ and also $\\mathbb {P}_\\lambda (\\operatorname{Arm}(r, 2r)) \\in (c^{\\prime }, 1-c^{\\prime })\\quad \\text{for every $r\\ge 1$,}$ where $c^{\\prime }=c_{\\ref {c:q-bound}}(4)^4$ .","As before, the above result also holds for the vacant set, i.e.","with $\\operatorname{Cross}$ and $\\operatorname{Arm}$ replaced by $\\operatorname{Cross}^\\star $ and $\\operatorname{Arm}^\\star $ respectively.","Another consequence of Theorem REF concerns the decay of arm probabilities at criticality.","Unlike previous results of this section, the bounds on the arm events distinguish between vacant and occupied sets.","The one-arm probability always decays polynomially fast for the vacant set.","For the occupied set, we can prove polynomial decay of the one-arm probability only under the stronger assumption () and we explain why this restriction is necessary in Remark REF .","Corollary 4.4 (Bounds on the arm-events) Assume the second moment conditions in (REF ).","There exists a function $f: (0,1) \\rightarrow (0, 1)$ such that $\\lim _{x \\rightarrow 0}f(x) = 0$ and for every $\\lambda \\in [\\lambda _0, \\lambda _1]$ we have ${\\mathbb {P}}_\\lambda (\\operatorname{Arm}(r,r^{\\prime })) \\le f \\Big ( \\frac{r}{r^{\\prime }} \\Big ).$ If $\\operatorname{Arm}$ is replaced by $\\operatorname{Arm}^\\star $ , then a stronger conclusion holds: there exists $c > 0$ such that ${\\mathbb {P}}_\\lambda ( \\operatorname{Arm}^\\star (r,r^{\\prime }) ) \\le \\frac{1}{c} \\left(\\frac{r}{r^{\\prime }}\\right)^c.$ Under the stronger assumption () the conclusion of (REF ) holds also for $\\operatorname{Arm}$ .","Remark 9 In Corollary REF , under the $2 + \\varepsilon $ -moment condition (), we can choose $f(x) = \\tfrac{1}{c} x^c$ for some constant $c > 0$ .","However, if $\\mu ([x, \\infty ))$ is $1/(x^2 \\log ^2(x))$ , then the second moment condition holds but the arm event does not decay polynomially.","To see this, consider the event that the origin is covered by a occupied disk of radius at least $r$ , and observe that the probability of this event does not decay polynomially fast in $r$ .","An immediate consequence of Corollary REF is the following observation.","Corollary 4.5 Assume the second moment conditions in (REF ).","For every $\\lambda \\in [\\lambda _0,\\lambda _1]$ there is almost surely no unbounded occupied nor vacant cluster.","Consequently, $\\lambda _c^\\star \\le \\lambda _0\\le \\lambda _1\\le \\lambda _c.$"],["Standard inequalities","Before starting the proof of the RSW Theorem and its consequences, let us recall some standard inequalities on crossing probabilities.","Lemma 4.6 For every $\\lambda >0$ , $r>0$ , $\\kappa >0$ and integer $j\\ge 1$ we have: ${\\mathbb {P}}_\\lambda [\\operatorname{Cross}((1+j\\kappa )r,r)]\\ge {\\mathbb {P}}_\\lambda [\\operatorname{Cross}((1+\\kappa )r,r)]^{2j-1}$ , ${\\mathbb {P}}_\\lambda [\\operatorname{Cross}(r,(1+\\kappa )r)]\\ge 1-\\left(1-{\\mathbb {P}}_\\lambda [\\operatorname{Cross}(r,(1+j\\kappa r)]\\right)^{1/(2j-1)}$ , ${\\mathbb {P}}_\\lambda [\\operatorname{Circ}(r,2r)]\\ge {\\mathbb {P}}_\\lambda [\\operatorname{Cross}(4r,r)]^4$ , ${\\mathbb {P}}_\\lambda [\\operatorname{Cross}(2r,r)]\\ge {\\mathbb {P}}[\\operatorname{Circ}(r,2r)]$ .","The same holds if we replace $\\operatorname{Cross}$ and $\\operatorname{Circ}$ by $\\operatorname{Cross}^\\star $ and $\\operatorname{Circ}^\\star $ , respectively.","Part (REF ) is a straightforward consequence of the FKG-inequality (REF ): A horizontal crossing of an $(1+j\\kappa )r\\times r$ -rectangle can be enforced by $j$ horizontal crossings of $(1+\\kappa )r\\times r$ -rectangles and $j-1$ vertical crossings of $r\\times (1+\\kappa )r$ -rectangles overlapping one another.","Similarly, part (REF ) is the consequence of a circuit being possible to construct out of four overlapping rectangle crossings.","Part (REF ) is a direct consequence of (REF ) since it may be rewritten as ${\\mathbb {P}}_\\lambda [\\operatorname{Cross}^\\star ((1+\\kappa )r,r)]\\le {\\mathbb {P}}_\\lambda [\\operatorname{Cross}^\\star ((1+j\\kappa )r,r)]^{1/(2j-1)}.$ Inequality (REF ) follows from the observation that any circuit in the annulus $B^\\infty (2r)\\setminus B^\\infty (r)$ must cross the rectangle $[-r,r]\\times [r,2r]$ horizontally."],["A useful circuit Lemma","For Bernoulli percolation, the bounds on crossing probabilities provide some bounds on the arm events.","This is based on a circuit argument, that uses independence.","In our case, the argument need to be adapted because of spatial dependencies.","Lemma 4.7 (Circuit Lemma) Assume the second moment conditions in (REF ).","Let $\\lambda \\in [0,\\infty )$ .","For every $c>0$ , there exists a function $f=f_{c,\\lambda }: (0,1) \\rightarrow (0,1)$ such that $\\lim _{x \\rightarrow 0} f(x) = 0$ and for all $r^{\\prime } \\ge 2r \\ge 2$ , $\\text{if }\\displaystyle \\inf _{r\\le s\\le r^{\\prime }/2} {\\mathbb {P}}_\\lambda [\\operatorname{Circ}^\\star (s,2s)]\\ge c,\\text{ then }{\\mathbb {P}}_\\lambda [ \\operatorname{Circ}^\\star (r,r^{\\prime }) ] \\ge 1- f \\left(\\tfrac{r}{r^{\\prime }}\\right).$ For every $c>0$ , there exists a constant $c^{\\prime }=c^{\\prime }(c,\\lambda )>0$ such that for all $r^{\\prime } \\ge 2r \\ge 2$ , $\\text{if }\\displaystyle \\inf _{r\\le s\\le r^{\\prime }/2} {\\mathbb {P}}_\\lambda [\\operatorname{Circ}(s,2s)]\\ge c,\\text{ then }{\\mathbb {P}}_\\lambda [ \\operatorname{Circ}(r,r^{\\prime }) ] \\ge 1-\\tfrac{1}{c^{\\prime }}\\left(\\tfrac{r}{r^{\\prime }}\\right)^{c^{\\prime }}.$ Under the $2+\\varepsilon $ moment condition (), Item $(\\ref {item:6})$ holds also for $\\operatorname{Circ}$ replaced by $\\operatorname{Circ}^\\star $ .","We begin with the proof of Item (REF ).","It suffices to prove the statement for $r^{\\prime }\\ge 16r$ .","Let $r \\ge 1$ and $r^{\\prime }\\ge 16r$ be given, and assume that $\\inf _{r\\le s\\le r^{\\prime }/2} {\\mathbb {P}}_\\lambda [\\operatorname{Circ}^\\star (s,2s)]\\ge c>0.$ Let $g(s) := \\sup _{s^{\\prime }\\ge s} \\rho _\\lambda (s^{\\prime },2s^{\\prime })$ .","By Proposition REF , we have $\\lim _{s\\rightarrow \\infty }g(s) = 0$ .","Set $t = \\sqrt{rr^{\\prime }}$ and $\\ell _i=4^it$ .","Note that if $\\operatorname{Circ}^\\star (r,r^{\\prime })$ fails to occur, then $A_i=\\operatorname{Circ}^\\star (\\ell _i,2\\ell _i)$ cannot occur for no $i=0,1,\\ldots ,k-1$ where $k=\\lfloor \\frac{1}{2}\\log _4(r^{\\prime }/r)\\rfloor $ ($k\\ge 1$ since we assumed $r^{\\prime }\\ge 16r$ ).","The hypothesis (REF ) implies that ${\\mathbb {P}}_\\lambda (A_i^c)\\le 1-c<1$ .","Therefore, ${\\mathbb {P}}_\\lambda \\Big [ {\\textstyle \\bigcap \\limits }_{i = 0}^{k-1} A_i^c \\Big ]\\,&\\le \\, {\\mathbb {P}}_\\lambda [A_{k-1}] {\\mathbb {P}}_\\lambda \\Big [ {\\textstyle \\bigcap \\limits }_{i = 0}^{k-2} A_i^c \\Big ] + \\rho _\\lambda (2\\ell _{k-2},\\ell _{k-2})\\\\&\\le \\, (1-c) {\\mathbb {P}}_\\lambda \\Big [ {\\textstyle \\bigcap \\limits }_{i = 0}^{k-2} A_i^c \\Big ] + g(t),$ which by induction gives that ${\\mathbb {P}}_\\lambda \\Big [ {\\textstyle \\bigcap \\limits }_{i = 0}^{k-1} A_i^c \\Big ] \\,\\le \\, (1-c)^k + \\frac{1}{c} g(t).$ By the choice of $k$ , and the fact that $g(t)\\le g(t/r) = g(\\sqrt{r^{\\prime }/r})$ , we finally obtain ${\\mathbb {P}}_\\lambda \\big [ \\operatorname{Circ}^\\star (r,r^{\\prime })^c \\big ] \\,\\le \\,{\\mathbb {P}}_\\lambda \\Big [ {\\textstyle \\bigcap \\limits }_{i = 0}^{k-1} A_i^c \\Big ] \\,\\le \\, \\frac{1}{1-c}\\Big ( \\frac{r^{\\prime }}{r} \\Big )^{\\frac{1}{2}\\log _4(1-c)}+ \\frac{1}{c}g\\left(\\sqrt{\\tfrac{r^{\\prime }}{r}}\\right).$ Hence, (REF ) holds with $f(x)=\\frac{1}{1-c}x^\\alpha +\\frac{1}{c}g(x^{-1/2})$ for any sufficiently small constant $\\alpha >0$ .","The stronger moment condition () implies (REF ), which readily gives that $f$ may be chosen of the form $\\frac{1}{c^{\\prime }}x^{c^{\\prime }}$ for some constant $c^{\\prime }=c^{\\prime }(c,\\lambda )>0$ , which also proves Item (REF ).","We now turn to the proof of Item (REF ) and reuse the above notation.","Set $B_i=\\operatorname{Circ}(\\ell _i,2\\ell _i)$ .","As above, by (REF ) it follows that ${\\mathbb {P}}_\\lambda \\big (B_i^c\\big )\\le 1-c$ for every $i=0,1,\\ldots ,k-1$ .","By Proposition REF we may fix $r_0$ large enough so that $\\bar{\\rho }_\\lambda (2r,r) \\le c/2\\quad \\text{for every }r\\ge r_0.$ Observe that the events $B_i$ are decreasing.","Hence, for $r\\ge r_0$ , $r^{\\prime }\\ge 16r$ and $t=\\sqrt{rr^{\\prime }}$ , we obtain $\\begin{aligned}{\\mathbb {P}}_\\lambda \\big ( \\operatorname{Circ}(r, r^{\\prime })^c \\big ) \\,& \\le \\, {\\mathbb {P}}_\\lambda \\Big ( {\\textstyle \\bigcap \\limits }_{i = 0}^{k-1}B_i^c \\Big )\\\\& \\le \\,{\\mathbb {P}}_\\lambda \\Big ( {\\textstyle \\bigcap \\limits }_{i = 1}^{k-1} B_i^c \\Big ) \\Big ( \\mathbb {P}_\\lambda (B_0^c) + \\bar{\\rho }_\\lambda (2\\ell _0, \\ell _0) \\Big )\\\\& \\le \\,\\prod _{i = 0}^{k-1} \\Big ( \\mathbb {P}_\\lambda \\big ( B_i^c \\big ) + \\bar{\\rho }_\\lambda (2\\ell _i, \\ell _i) \\Big )\\\\& \\le \\,(1-c/2)^{\\frac{1}{2}\\log _4(r^{\\prime }/r)-1}.\\end{aligned}$ By allowing for a larger constant on the right-hand side we may replace the assumption $r\\ge r_0$ with $r\\ge 1$ .","This concludes the proof of (REF )."],["Proof of the consequences of Theorem ","We now assume the validity of Theorem REF and prove Corollaries REF -REF .","[Proof of Corollary REF ] We will show that if $\\lim _{r\\rightarrow \\infty }{\\mathbb {P}}(\\operatorname{Cross}(\\kappa r,r))=1$ for some $\\kappa >0$ , then $\\lim _{r\\rightarrow \\infty }{\\mathbb {P}}(\\operatorname{Cross}(2\\kappa r,r))=1$ , from which the statement follows via iteration.","Partition the right-hand side of the rectangle $R=[0,\\kappa r]\\times [0,r]$ into $n$ equal intervals (of length $r/n$ ), and denote the event that there is a horizontal crossing of $R$ into the $k$ -th of these intervals by $A_k$ .","Using the square-root trick (REF ), we find that $\\max _{1\\le k\\le n}{\\mathbb {P}}_\\lambda (A_k)\\ge 1-[1-{\\mathbb {P}}_\\lambda (\\operatorname{Cross}(\\kappa r,r))]^{1/n},$ which by assumption tends to 1 as $r\\rightarrow \\infty $ .","By assumption, Theorem REF shows that $\\inf _{r\\ge 1}{\\mathbb {P}}_\\lambda (\\operatorname{Cross}(4r,r))>0$ .","Hence, the standard inequalities of Lemma REF show that $\\inf _{r\\ge 1}{\\mathbb {P}}_\\lambda (\\operatorname{Circ}(r,2r))>0$ .","Hence, by Lemma REF we conclude that for large enough $r$ , ${\\mathbb {P}}_\\lambda (\\operatorname{Circ}( r/n,\\kappa r/2))\\ge 1-f(\\kappa n/2),$ for some function $f$ such that $f(x)\\rightarrow 0$ as $x\\rightarrow \\infty $ .","Finally, we combine the above estimates to obtain that for every $\\varepsilon >0$ ${\\mathbb {P}}_\\lambda \\big (\\operatorname{Cross}(2\\kappa r,r)\\big )\\,\\ge \\,{\\mathbb {P}}_\\lambda \\Bigg [\\hbox{[scale=1.5]{(0, 0) rectangle (6ex, 8ex) (6ex, 0) rectangle (12ex, 8ex);(5.8ex, 4.2ex) -- (6.2ex, 4.2ex) (5.8ex, 4.8ex) -- (6.2ex, 4.8ex);[solid] (0, 3ex) .. controls (2ex, 2ex) and (4ex, 5ex) .. (6ex, 4.65ex);[solid] (6ex, 4.35ex) .. controls (8ex, 4ex) and (8ex, 7ex) .. (12ex, 6ex);[solid,decorate,decoration={brace,amplitude=.7ex,raise=.2ex},yshift=0pt] (12ex, 0ex) -- (0, 0);(5ex, 3.5ex) rectangle (7ex, 5.5ex);(3ex, 1.5ex) rectangle (9ex, 7.5ex);(3ex, 1ex) -- (9ex, 1ex);(3ex, .8ex) -- (3ex, 1.2ex) (9ex, .8ex) -- (9ex, 1.2ex);[fill, color=white] (4ex, .2ex) rectangle (8ex, 1.3ex);[solid,rounded corners=.25em](4ex, 6.5ex) .. controls (6ex, 6.5ex + .4ex*rand) ..(8ex, 6.5ex) .. controls (8ex + .4ex*rand, 4.5ex) ..(8ex, 2.5ex) .. controls (6ex, 2.5ex + .4ex*rand) ..(4ex, 2.5ex) -- (4ex + .4ex*rand, 4.5ex) -- cycle;(6ex, -.4ex) node[anchor=north] {$\\scriptstyle 2\\kappa r$};(.4ex, 4ex) node[anchor=east] {$\\scriptstyle r$};(9.4ex, 3.5ex) -- (9.4ex, 5.5ex);(9.2ex, 3.5ex) -- (9.6ex, 3.5ex) (9.2ex, 5.5ex) -- (9.6ex, 5.5ex);(9ex, 4.5ex) node[anchor=west] {$\\scriptstyle \\tfrac{r}{n}$};(6ex, 1.7ex) node[anchor=north] {$\\scriptstyle \\kappa r$};}\\;\\;}$ ] P(Ak)2 P(Circ(r/n,r/2)) > 1-, $for some choice of $ $ and $ n$, and all large $ r$, as required.$ [Proof of Corollary REF ] Take $\\theta > 0$ and $r_0 = r_0(\\lambda _1)$ so that Proposition REF is in force.","Then, ${\\mathbb {P}}_{\\lambda _1}(\\operatorname{Cross}(3r, r))\\le 1-\\theta \\text{ for all $r\\ge r_0$.","}$ Otherwise, by continuity, we could find $\\varepsilon > 0$ and $r\\ge r_0$ such that ${\\mathbb {P}}_{\\lambda _1-\\varepsilon }(\\operatorname{Cross}(3r,r))>1-\\theta $ , which by Proposition REF would imply that the probability indeed converges to 1; a contradiction to the definition of $\\lambda _1$ .","Analogously, we have ${\\mathbb {P}}_{\\lambda _0}(\\operatorname{Cross}(r, 3r)) \\ge \\theta $ for all large $r$ .","Using Theorem REF and its dual version this establishes (REF ).","Equation (REF ) follows from the fact that the complement of $\\operatorname{Arm}(r,3r)$ is $\\operatorname{Circ}^\\star (r,3r)$ and the standard inequalities of Lemma REF .","[Proof of Corollary REF ] This is an immediate consequence of Corollary REF and Lemma REF , as ${\\mathbb {P}}_\\lambda (\\operatorname{Arm}(r,r^{\\prime }))=1-{\\mathbb {P}}_\\lambda (\\operatorname{Circ}^\\star (r,r^{\\prime }))$ .","[Proof of Corollary REF ] The bounds $\\lambda _c\\ge \\lambda _1$ and $\\lambda _c^\\star \\le \\lambda _0$ follow respectively from Item (REF ) and Item (REF ) of Corollary REF ."],["Proof of Theorem ","In this section we prove Theorem REF .","We assume throughout the proof the second moment condition (REF ) and that $\\lambda >0$ is fixed.","By assumption there exists $\\kappa \\in (0,\\tfrac{1}{3}]$ such that $\\inf _{r\\ge 1}{\\mathbb {P}}_\\lambda (\\operatorname{Cross}(\\kappa r,r))>0$ .","Clearly, there is no restriction assuming the upper bound on $\\kappa $ .","Hence, we note that either also $\\inf _{r\\ge 1}{\\mathbb {P}}_\\lambda (\\operatorname{Cross}^\\star (\\kappa r,r))>0$ , or we have $\\sup _{r\\ge 1}{\\mathbb {P}}_\\lambda (\\operatorname{Cross}(3r,r))=1.$ The latter implies, via Proposition REF , that $\\lim _{r\\rightarrow \\infty }{\\mathbb {P}}_\\lambda (\\operatorname{Cross}(3r,r))=1$ , in which case there is nothing left to prove.","So, we may without loss of generality assume that for some $\\kappa >0$ $\\inf _{r\\ge 1} \\mathbb {P}_\\lambda \\big (\\operatorname{Cross}(\\kappa r, r) \\big ) > 0\\quad \\text{and}\\quad \\inf _{r\\ge 1} \\mathbb {P}_\\lambda \\big ( \\operatorname{Cross}^\\star (\\kappa r, r) \\big ) > 0.$ Moreover, by the standard inequalities of Lemma REF (part (REF ) and its dual version) this is equivalent to assuming the existence of a constant $c_{\\ref {c:easy_cross}}>0$ such that $\\inf _{r\\ge 1} \\mathbb {P}_\\lambda \\big ( \\operatorname{Cross}\\big (r, \\tfrac{5}{4}r\\big ) \\big ) \\ge c_{\\ref {c:easy_cross}} \\quad \\text{and} \\quad \\inf _{r\\ge 1} \\mathbb {P}_\\lambda \\big ( \\operatorname{Cross}^\\star \\big (r, \\tfrac{5}{4}r\\big ) \\big ) \\ge c_{\\ref {c:easy_cross}}.$ In what follows we will explore several consequences of the above assumption.","This will be done in a series of lemmas that will culminate in the proof of Theorem REF .","Throughout this section we will need to introduce a range of events that are closely related to the crossing event defined in (REF ).","We will typically depict these events graphically, since we believe the proof is better explained this way.","However, we emphasize that it is always possible to give a formal definition.","For instance, recall the definition of $[A \\overset{\\mathcal {C}}{\\leftrightarrow } B]$ in (REF ).","We start with the definition of the event of having a vacant crossing above an occupied crossing, both reaching the right-hand side on the upper half: $\\hbox{[scale=1.5]{(0, 0) rectangle (2ex, 6ex);(1.8ex, 3ex) -- (2.2ex, 3ex);[densely dotted] (0, 5ex) .. controls (.6ex, 6ex) and (1.4ex, 4ex) .. (2ex, 5ex);(0, 3.5ex) .. controls (.6ex, 5ex) and (1.4ex, 2.5ex) .. (2ex, 4ex);(1ex, .2ex) node[anchor=north] {$\\scriptstyle r$};(0ex, 3ex) node[anchor=east] {$\\scriptstyle h$};}\\;}$ := q [0, h] Q [ ( {0} [0,q] O K A ) ( {0} [q, h] V K A ) ], where $K = [0, r] \\times [0,h]$ and $A = \\lbrace r\\rbrace \\times [h/2, h]$ .","Lemma 4.8 There exist $r_0\\ge 1$ , $0 < c_{\\ref {c:prob_dual}} < 1/8$ and $c_{\\ref {c:box_height}} \\ge 23$ such that $\\inf _{r\\ge r_0} \\mathbb {P}_\\lambda \\Bigg [ \\hbox{[scale=1.5]{(0, 0) rectangle (2ex, 6ex);(1.8ex, 3ex) -- (2.2ex, 3ex);[densely dotted] (0, 5ex) .. controls (.6ex, 6ex) and (1.4ex, 4ex) .. (2ex, 5ex);(0, 3.5ex) .. controls (.6ex, 5ex) and (1.4ex, 2.5ex) .. (2ex, 4ex);(1ex, .2ex) node[anchor=north] {$\\scriptstyle r/c_{\\ref {c:box_height}}$};(0ex, 3ex) node[anchor=east] {$\\scriptstyle \\tfrac{r}{3}$};}\\;}$ ] cREF and $\\sup _{r\\ge r_0} \\mathbb {P}_\\lambda \\big ( \\operatorname{Cross}(\\tfrac{r}{3}, \\tfrac{r}{c_{\\ref {c:box_height}}}) \\big ) \\le \\frac{1}{16} \\quad \\text{and} \\quad \\sup _{r\\ge r_0} \\mathbb {P}_\\lambda \\big ( \\operatorname{Cross}^\\star (\\tfrac{r}{3},\\tfrac{r}{c_{\\ref {c:box_height}}}) \\big ) \\le \\frac{1}{16}.$ We first argue for the existence of $r_0\\ge 1$ and $c_{\\ref {c:box_height}}>18$ for which (REF ) holds.","Given $s\\ge 1$ , set $A_0:=\\operatorname{Cross}(\\tfrac{5}{4}s,s)$ and let $A_i$ denote the translate of $A_i$ along the vector $(2is,0)$ .","Translation invariance, duality and (REF ) imply that for every $i$ , ${\\mathbb {P}}_\\lambda (A_i)= 1- {\\mathbb {P}}_\\lambda \\big ( \\operatorname{Cross}^\\star \\big (s, \\tfrac{5}{4}s \\big ) \\big )\\le 1-c_{\\ref {c:easy_cross}}.$ If for some $k\\ge 1$ $\\operatorname{Cross}(2ks, s)$ occurs, then $A_i$ occur for all $i=0,1,\\ldots ,k-1$ .","Therefore, ${\\mathbb {P}}_\\lambda \\big ( \\operatorname{Cross}(2ks,s)\\big )\\,\\le \\, {\\mathbb {P}}_\\lambda \\Big (\\bigcap _{i=0}^{k-1} A_i \\Big )\\,\\le \\, {\\mathbb {P}}_\\lambda \\Big ( \\bigcap _{i=0}^{k-2}A_i \\Big )(1-c_{\\ref {c:easy_cross}}) +\\rho _\\lambda (s, \\tfrac{3}{4} s).$ Via induction in $k$ we obtain that ${\\mathbb {P}}_\\lambda \\big ( \\operatorname{Cross}(2ks,s)\\big )\\le (1-c_{\\ref {c:easy_cross}})^k+\\frac{1}{c_{\\ref {c:easy_cross}}}\\rho _\\lambda (s,\\tfrac{3}{4} s).$ For $k\\ge 4$ and $s_0\\ge 1$ large enough this expression is bounded by $1/16$ for all large $s\\ge s_0$ .","Hence, the choice $r_0=6ks_0$ and $c_{\\ref {c:box_height}}=6k\\ge 23$ guarantees that ${\\mathbb {P}}_\\lambda \\big ( \\operatorname{Cross}(\\tfrac{r}{3},\\tfrac{r}{c_{\\ref {c:box_height}}})\\big )\\le \\frac{1}{16}.$ for every $r\\ge r_0$ .","With exactly the same proof, we can show that (REF ) also holds with $\\operatorname{Cross}$ replaced by $\\operatorname{Cross}^\\star $ .","Hence (REF ) holds for this choice of $r_0$ and $c_{\\ref {c:box_height}}$ .","To conclude the proof, we will show that any choice of $c_{\\ref {c:box_height}}\\ge 23$ implies $\\inf _{r \\ge r_0} \\mathbb {P}_\\lambda \\Bigg [ \\hbox{[scale=1.5]{(0, 0) rectangle (2ex, 6ex);(1.8ex, 3ex) -- (2.2ex, 3ex);[densely dotted] (0, 5ex) .. controls (.6ex, 6ex) and (1.4ex, 4ex) .. (2ex, 5ex);(0, 3.5ex) .. controls (.6ex, 5ex) and (1.4ex, 2.5ex) .. (2ex, 4ex);(1ex, .2ex) node[anchor=north] {$\\scriptstyle r/c_{\\ref {c:box_height}}$};(0ex, 3ex) node[anchor=east] {$\\scriptstyle \\tfrac{r}{3}$};}\\;}$ ] > 0.","To prove this, observe that $\\begin{split}\\mathbb {P}_\\lambda \\Bigg [\\hbox{[scale=1.5]{(0, 0) rectangle (2ex, 6ex);(1.8ex, 3ex) -- (2.2ex, 3ex);[densely dotted] (0, 5ex) .. controls (.6ex, 6ex) and (1.4ex, 4ex) .. (2ex, 5ex);(0, 3.5ex) .. controls (.6ex, 5ex) and (1.4ex, 2.5ex) .. (2ex, 4ex);(1ex, .2ex) node[anchor=north] {$\\scriptstyle r/c_{\\ref {c:box_height}}$};(0ex, 3ex) node[anchor=east] {$\\scriptstyle \\tfrac{r}{3}$};}\\;}\\end{split} \\Bigg ] \\,& \\ge \\,{\\mathbb {P}}_\\lambda \\big ( \\operatorname{Cross}(\\tfrac{r}{c_{\\ref {c:box_height}}}, \\tfrac{r}{18})\\big ) {\\mathbb {P}}_\\lambda \\big ( \\operatorname{Cross}^\\star (\\tfrac{r}{c_{\\ref {c:box_height}}},\\tfrac{r}{18}) \\big )-\\rho _\\lambda (\\tfrac{r}{18}, \\tfrac{r}{18})\\\\& \\ge \\, c_{\\ref {c:easy_cross}}^2- \\rho _\\lambda (\\tfrac{r}{18},\\tfrac{r}{18}).$ By (REF ), this shows that the probability on the left hand side of (REF ) is larger than $ c_{\\ref {c:easy_cross}}^2/2$ for $r$ large enough, as required.","The next event we will be interested in is $\\hbox{[scale=1.5]{(0, 2ex) .. controls (1.4ex, 1ex) and (.6ex, 3ex) .. (2ex, 3.3ex);[densely dotted] (0, 4ex) .. controls (1.4ex, 5ex) and (.6ex, 3.2ex) .. (2ex, 3.7ex);(0, 0) rectangle (2ex, 6ex);(1.8ex, 2ex) -- (2.2ex, 2ex) (1.8ex, 3ex) -- (2.2ex, 3ex) (1.8ex, 4ex) -- (2.2ex, 4ex);[decorate,decoration={brace,amplitude=.5ex,raise=.4ex},yshift=0pt] (2.2ex, 4ex) -- (2.2ex, 2ex);(1ex, .2ex) node[anchor=north] {$\\scriptstyle r$};(0ex, 3ex) node[anchor=east] {$\\scriptstyle h$};(2.4ex, 3ex) node[anchor=west] {$\\scriptstyle 2a$};}}$ := q [0,h] Q [ ( {0} [0,q] O K A ) ( {0} [q, h] V K A ) ], where $K = [0,r] \\times [0,h]$ and $A = \\lbrace r\\rbrace \\times [h/2, h/2 + a]$ .","Definition 4.9 For every $r\\ge r_0$ , let $0 \\le \\alpha _r\\le r/2$ be such that $\\mathbb {P}_\\lambda \\Bigg [ \\hbox{[scale=1.5]{(0, 2ex) .. controls (1.4ex, 1ex) and (.6ex, 3ex) .. (2ex, 3.3ex);[densely dotted] (0, 4ex) .. controls (1.4ex, 5ex) and (.6ex, 3.2ex) .. (2ex, 3.7ex);(0, 0) rectangle (2ex, 6ex);(1.8ex, 2ex) -- (2.2ex, 2ex) (1.8ex, 3ex) -- (2.2ex, 3ex) (1.8ex, 4ex) -- (2.2ex, 4ex);[decorate,decoration={brace,amplitude=.5ex,raise=.4ex},yshift=0pt] (2.2ex, 4ex) -- (2.2ex, 2ex);(1ex, .2ex) node[anchor=north] {$\\scriptstyle r/c_{\\ref {c:box_height}}$};(0ex, 3ex) node[anchor=east] {$\\scriptstyle r$};(2.4ex, 3ex) node[anchor=west] {$\\scriptstyle 2 \\alpha _r$};}}$ ] = cREF 2.","Let us briefly show that $\\alpha _r\\le \\frac{r}{6}.$ By Lemma REF , the quantity $f(a)=\\mathbb {P}_\\lambda \\Bigg [ \\hbox{[scale=1.5]{(0, 2ex) .. controls (1.4ex, 1ex) and (.6ex, 3ex) .. (2ex, 3.3ex);[densely dotted] (0, 4ex) .. controls (1.4ex, 5ex) and (.6ex, 3.2ex) .. (2ex, 3.7ex);(0, 0) rectangle (2ex, 6ex);(1.8ex, 2ex) -- (2.2ex, 2ex) (1.8ex, 3ex) -- (2.2ex, 3ex) (1.8ex, 4ex) -- (2.2ex, 4ex);[decorate,decoration={brace,amplitude=.5ex,raise=.4ex},yshift=0pt] (2.2ex, 4ex) -- (2.2ex, 2ex);(1ex, .2ex) node[anchor=north] {$\\scriptstyle r/c_{\\ref {c:box_height}}$};(0ex, 3ex) node[anchor=east] {$\\scriptstyle r$};(2.4ex, 3ex) node[anchor=west] {$\\scriptstyle 2a$};}}$ ] is continuous in $a$ , and by Lemma REF , it satisfies $f(0)=0$ and $f(\\tfrac{r}{6})\\ge c_{\\ref {c:prob_dual}}$ .","Hence, there exists $0\\le \\alpha _r\\le r/6$ such that $f(\\alpha _r)=\\tfrac{c_{\\ref {c:prob_dual}}}{2}.$ The next event is what we call the fork and it is crucial in what follows $\\hbox{[scale=1.5]{(0, 0) rectangle (4ex, 4ex);(-.2ex, 1.5ex) -- (.2ex, 1.5ex) (-.2ex, 2.5ex) -- (.2ex, 2.5ex);(3.8ex, 1.5ex) -- (4.2ex, 1.5ex) (3.8ex, 2.5ex) -- (4.2ex, 2.5ex);[solid] (0, 3ex) .. controls (.5ex, 3ex) and (1ex, 2.5ex).. (1ex, 2ex) .. controls (1ex, 1.5ex) and (.5ex, 1ex) .. (0ex, 1ex);[solid] (1ex, 2ex) .. controls (1.6ex, 3ex) and (2.4ex, 1ex) .. (3ex, 2ex);[solid] (4ex, 3ex) .. controls (3.5ex, 3ex) and (3ex, 2.5ex).. (3ex, 2ex) .. controls (3ex, 1.5ex) and (3.5ex, 1ex) .. (4ex, 1ex);[decorate,decoration={brace,amplitude=.7ex,raise=.4ex},yshift=0pt] (0, 0) -- (0, 4ex);(2ex, .2ex) node[anchor=north] {$\\scriptstyle r$};(-.5ex, 2ex) node[anchor=east] {$\\scriptstyle h$};(3.8ex, 2ex) node[anchor=west] {$\\scriptstyle 2a$};}}$ := [ c ( {0} [0,h/2 - a] O K {r} [h/2 + a, h] ) ( {0} [h/2 + a, h] O K {r} [0,h/2 - a] ) ], and analogously for the dual (where the picture is traced using dashed curves).","Lemma 4.10 Assuming (REF ), for every $r\\ge r_0$ , one of the following holds $\\mathbb {P}_\\lambda \\bigg [ \\hbox{[scale=1.5]{(0, 0) rectangle (4ex, 4ex);(-.2ex, 1.5ex) -- (.2ex, 1.5ex) (-.2ex, 2.5ex) -- (.2ex, 2.5ex);(3.8ex, 1.5ex) -- (4.2ex, 1.5ex) (3.8ex, 2.5ex) -- (4.2ex, 2.5ex);[solid] (0, 3ex) .. controls (.5ex, 3ex) and (1ex, 2.5ex).. (1ex, 2ex) .. controls (1ex, 1.5ex) and (.5ex, 1ex) .. (0ex, 1ex);[solid] (1ex, 2ex) .. controls (1.6ex, 3ex) and (2.4ex, 1ex) .. (3ex, 2ex);[solid] (4ex, 3ex) .. controls (3.5ex, 3ex) and (3ex, 2.5ex).. (3ex, 2ex) .. controls (3ex, 1.5ex) and (3.5ex, 1ex) .. (4ex, 1ex);[decorate,decoration={brace,amplitude=.7ex,raise=.4ex},yshift=0pt] (0, 0) -- (0, 4ex);(2ex, .2ex) node[anchor=north] {$\\scriptstyle r$};(-.5ex, 2ex) node[anchor=east] {$\\scriptstyle r$};(3.8ex, 2ex) node[anchor=west] {$\\scriptstyle 2 \\alpha _r$};}}$ ] 2-13 or P[ ] 2-13.","By duality and invariance under right-angle's rotations, see (REF ), $\\mathbb {P}_\\lambda \\big ( \\operatorname{Cross}(r, r) \\big ) + \\mathbb {P}_\\lambda \\big ( \\operatorname{Cross}^\\star (r, r) \\big ) = 1$ The two cases in the statement of the lemma will correspond to which of the above probabilities is at least one half.","Therefore, we assume without loss of generality that $\\mathbb {P}_\\lambda \\big ( \\operatorname{Cross}(r, r) \\big ) \\ge \\frac{1}{2}.$ Using symmetry and then a simple union bound, we get $\\frac{1}{4} \\le \\mathbb {P}_\\lambda \\bigg [ \\hbox{[scale=1.5]{(0, 0) rectangle (4ex, 4ex);(3.8ex, 2ex) -- (4.2ex, 2ex);(0, 1.7ex) .. controls (1.3ex, 4ex) and (2.6ex, 1.4ex) .. (4ex, 3.2ex);(2ex, .2ex) node[anchor=north] {$\\scriptstyle r$};(.4ex, 2ex) node[anchor=east] {$\\scriptstyle r$};[decorate,decoration={brace,amplitude=.4ex,raise=.4ex},yshift=0pt] (4ex, 4ex) -- (4ex, 2ex);(4.3ex, 3ex) node[anchor=west] {$\\scriptstyle \\tfrac{r}{2}$};}\\;}$ ] P[ ] + P[ ], where the event appearing in the last term of the above sum can be rigorously defined as $ \\hbox{[scale=1.5]{(0, 0) rectangle (4ex, 4ex);(3.7ex, 1.3ex) -- (4.3ex, 1.3ex);(3.8ex, 2ex) -- (4.2ex, 2ex);(3.7ex, 2.7ex) -- (4.3ex, 2.7ex);(0, 2ex) .. controls (1.3ex, 1ex) and (2.6ex, 2ex) .. (4ex, 2.2ex);[densely dotted] (2ex, 4ex) .. controls (2ex, 3ex) and (2.3ex, 3ex) .. (4ex, 2.4ex);[decorate,decoration={brace,amplitude=.3ex,raise=.4ex},yshift=0pt] (4.2ex, 2.7ex) -- (4.2ex, 1.3ex);(.4ex, 2ex) node[anchor=east] {$\\scriptstyle r$};(2ex, .2ex) node[anchor=north] {$\\scriptstyle r$};(4.4ex, 2ex) node[anchor=west] {$\\scriptstyle 2 \\alpha _r$};}}$ = q [0, r] Q ( {0} [0,q] O K A2 ) ( A1(q) V K A2 ), $ with $ = [0, r]2$, $ A1(q) = {0} [q, r] [0, r] {r}$ and $ A2 = {r} [r/2, r/2 + r].$$ Recalling that $c_{\\ref {c:box_height}} > 1$ and (REF ), we bound the second term in the above sum by a sum of two terms.","They respectively correspond to whether the dual path depicted above stays confined in an $(r/c_{\\ref {c:box_height}})$ -wide rectangle or not.","More precisely, recalling (REF ), $\\begin{split}\\mathbb {P}_\\lambda \\bigg [ \\hbox{[scale=1.5]{(0, 0) rectangle (4ex, 4ex);(3.7ex, 1.3ex) -- (4.3ex, 1.3ex);(3.8ex, 2ex) -- (4.2ex, 2ex);(3.7ex, 2.7ex) -- (4.3ex, 2.7ex);(0, 2ex) .. controls (1.3ex, 1ex) and (2.6ex, 2ex) .. (4ex, 2.2ex);[densely dotted] (2ex, 4ex) .. controls (2ex, 3ex) and (2.3ex, 3ex) .. (4ex, 2.4ex);[decorate,decoration={brace,amplitude=.3ex,raise=.4ex},yshift=0pt] (4.2ex, 2.7ex) -- (4.2ex, 1.3ex);(.4ex, 2ex) node[anchor=east] {$\\scriptstyle r$};(2ex, .2ex) node[anchor=north] {$\\scriptstyle r$};(4.4ex, 2ex) node[anchor=west] {$\\scriptstyle 2 \\alpha _r$};}}\\end{split} \\bigg ] \\,& \\le \\, \\mathbb {P}_\\lambda \\Bigg [ \\hbox{[scale=1.5]{[color=gray!50!white](4.5ex, 2ex) -- (6ex, 2ex) (4.5ex, 4ex) -- (6ex, 4ex);(0, 0) rectangle (6ex, 6ex);(4.5ex, 0) -- (4.5ex, 6ex);(5.7ex, 2.3ex) -- (6.3ex, 2.3ex) (5.8ex, 3ex) -- (6.2ex, 3ex) (5.7ex, 3.7ex) -- (6.3ex, 3.7ex);(4.5ex, 1.5ex) .. controls (5ex, 1ex) and (5ex, 3.5ex) .. (6ex, 3.3ex);[densely dotted] (5ex, 6ex) .. controls (6ex, 5ex) and (4ex, 3.5ex) .. (6ex, 3.5ex);[decorate,decoration={brace,amplitude=.5ex,raise=.5ex},yshift=0pt] (6.2ex, 3.7ex) -- (6.2ex, 2.3ex);[decorate,decoration={brace,amplitude=.5ex,raise=.4ex},yshift=0pt] (6ex, 6ex) -- (6ex, 4ex);[decorate,decoration={brace,amplitude=.3ex,raise=.2ex},yshift=0pt] (4.5ex, 6ex) -- (6ex, 6ex);(.4ex, 3ex) node[anchor=east] {$\\scriptstyle r$};(3ex, .2ex) node[anchor=north] {$\\scriptstyle r$};(6.5ex, 3ex) node[anchor=west] {$\\scriptstyle 2 \\alpha _r$};(5.2ex, 6ex) node[anchor=south] {$\\scriptstyle r/c_{\\ref {c:box_height}}$};(6.3ex, 5ex) node[anchor=west] {$\\scriptstyle \\tfrac{r}{3}$};}}$ ] + P[ ] 116 + cREF 2 18, where we also have used (REF ), (REF ) and that $c_{\\ref {c:prob_dual}}<1/8$ .","Therefore $\\mathbb {P}_\\lambda \\bigg [ \\hbox{[scale=1.5]{(0, 0) rectangle (4ex, 4ex);(3.7ex, 1.3ex) -- (4.3ex, 1.3ex) (3.8ex, 2ex) -- (4.2ex, 2ex) (3.7ex, 2.7ex) -- (4.3ex, 2.7ex);(0, 2ex) .. controls (1.3ex, 1.4ex) and (2.6ex, 4ex) .. (4ex, 3.4ex);[decorate,decoration={brace,amplitude=.3ex,raise=.4ex},yshift=0pt] (4.2ex, 2.7ex) -- (4.2ex, 1.3ex);(.4ex, 2ex) node[anchor=east] {$\\scriptstyle r$};(2ex, .2ex) node[anchor=north] {$\\scriptstyle r$};(4.4ex, 2ex) node[anchor=west] {$\\scriptstyle \\alpha _r$};}}$ ] 18.","To finish the bound in (REF ), we are going to apply the FKG inequality, see (REF ) above.","We start by defining the events $A_1, A_2,A_3, A_4$ obtained by reflecting the above square along the vertical and horizontal axis.","Of course the above bound will remain valid by the assumed symmetry of the system.","Also, using the fact that the law is invariant with respect to rotations by right angles, the probability of observing a vertical crossing of the above box is at least one half, by (REF ).","It is not difficult to see that the fork event occurs as soon as there is a vertical crossing of the box, together with the four events $A_1, A_2, A_3, A_4$ .","Plugging the above together with the FKG-inequality leads to the bound in (REF ).","Lemma 4.11 Let $\\tilde{\\alpha }_{c_{\\ref {c:box_height}}r} := \\alpha _{c_{\\ref {c:box_height}}r} \\wedge \\big ( \\tfrac{2}{3}r\\big )$ .","Then, for some $c_{\\ref {c:hit_middle}} \\in (0, c_{\\ref {c:easy_cross}})$ , we have $\\inf _{r\\ge r_0} \\mathbb {P}_\\lambda \\bigg [ \\hbox{\\;[scale=1.5]{(0, 0) rectangle (3ex, 4ex);(2.8ex, 2ex) -- (3.2ex, 2ex) (2.8ex, 2.8ex) -- (3.2ex, 2.8ex);[solid](0, 2ex) .. controls (1ex, 3.5ex) and (2ex, 1ex) .. (3ex, 2.6ex);(1.5ex, .2ex) node[anchor=north] {$\\scriptstyle r$};(.4ex, 2ex) node[anchor=east] {$\\scriptstyle \\tfrac{4}{3}r$};(2.8ex, 2.4ex) node[anchor=west] {$\\scriptstyle \\tilde{\\alpha }_{c_{\\ref {c:box_height}} r}$};}}$ ] cREF and rr0 P[ ] cREF .","where the above event is defined as $\\lbrace 0\\rbrace \\times [0, 4r/3] \\overset{K}{\\leftrightarrow }A$ , with $K = [0,r] \\times [0,4r/3]$ and $A = \\lbrace r\\rbrace \\times [2r/3, 2r/3 + \\tilde{\\alpha }_{c_{\\ref {c:box_height}} r}]$ (and analogously for the dual).","We only show the first inequality above, since the dual case is completely analogous.","For this we split the proof in two cases.","Either $\\alpha _{c_{\\ref {c:box_height}} r} \\ge \\tfrac{2}{3}r$ , in which case $\\tilde{\\alpha }_{c_{\\ref {c:box_height}} r} = 2r/3$ and the result follows from (REF ), together with the vertical reflection symmetry of the system and a union bound.","We now treat the case $\\tilde{\\alpha }_{c_{\\ref {c:box_height}} r} =\\alpha _{c_{\\ref {c:box_height}} r} < \\tfrac{2}{3}r$ .","We first recall that (REF ) gives $\\mathbb {P}_\\lambda \\Bigg [ \\hbox{[scale=1.5]{(0, 2ex) .. controls (1.4ex, 1ex) and (.6ex, 3ex) .. (2ex, 3.3ex);[color=white] (0, 4ex) .. controls (1.4ex, 5ex) and (.6ex, 3.2ex) .. (2ex, 3.7ex);(0, 0) rectangle (2ex, 6ex);(1.8ex, 2ex) -- (2.2ex, 2ex) (1.8ex, 3ex) -- (2.2ex, 3ex) (1.8ex, 4ex) -- (2.2ex, 4ex);[decorate,decoration={brace,amplitude=.5ex,raise=.4ex},yshift=0pt] (2.2ex, 4ex) -- (2.2ex, 2ex);(1ex, .2ex) node[anchor=north] {$\\scriptstyle r$};(0ex, 3ex) node[anchor=east] {$\\scriptstyle c_{\\ref {c:box_height}}r$};(2.4ex, 3ex) node[anchor=west] {$\\scriptstyle 2 \\alpha _{c_{\\ref {c:box_height}} r}$};}}$ ] cREF 2.","We now split the interval $\\lbrace r\\rbrace \\times [c_{\\ref {c:box_height}}/2,c_{\\ref {c:box_height}}/2 + \\alpha _{c_{\\ref {c:box_height}} r}]$ into eight equal parts.","Then, using the union bound, there must exist some $h_r \\in \\lbrace c_{\\ref {c:box_height}}r/2 + i \\alpha _{c_{\\ref {c:box_height}} r}/8; i = 0, \\dots ,7\\rbrace $ , such that $\\mathbb {P}_\\lambda \\Bigg [ \\hbox{[scale=1.5]{(0, 2ex) .. controls (1.4ex, 1ex) and (.6ex, 3.3ex) .. (2ex, 3.5ex);(0, 0) rectangle (2ex, 6ex);(1.8ex, 3.3ex) -- (2.2ex, 3.3ex) (1.8ex, 3.8ex) -- (2.2ex, 3.8ex);(1ex, .2ex) node[anchor=north] {$\\scriptstyle r$};(0ex, 3ex) node[anchor=east] {$\\scriptstyle c_{\\ref {c:box_height}}r$};(2ex, 3.5ex) node[anchor=west] {$\\scriptstyle I_r$};}}$ ] cREF 16, where $I_r = \\lbrace r\\rbrace \\times [h_r, h_r + \\alpha _{c_{\\ref {c:box_height}} r}/8]$ .","Since we are assuming $\\alpha _{c_{\\ref {c:box_height}} r} < 2r/3$ , the length of $I_r$ is no larger than $r/12$ .","Thus $\\begin{split}\\mathbb {P}_\\lambda \\bigg [ \\hbox{\\;[scale=1.5]{(0, 0) rectangle (3ex, 4ex);(2.8ex, 2ex) -- (3.2ex, 2ex) (2.8ex, 2.8ex) -- (3.2ex, 2.8ex);[solid](0, 2ex) .. controls (1ex, 3.5ex) and (2ex, 1ex) .. (3ex, 2.6ex);(1.5ex, .2ex) node[anchor=north] {$\\scriptstyle r$};(.4ex, 2ex) node[anchor=east] {$\\scriptstyle \\tfrac{4}{3}r$};(2.8ex, 2.4ex) node[anchor=west] {$\\scriptstyle \\tilde{\\alpha }_{c_{\\ref {c:box_height}} r}$};}}\\end{split} \\bigg ] & \\ge \\mathbb {P}_\\lambda \\Bigg [ \\hbox{[scale=1.5]{(0, 0) rectangle (3ex, 4ex);(2.8ex, 2ex) -- (3.2ex, 2ex);(0, 2ex) .. controls (1ex, 3ex) and (2ex, 1ex) .. (3ex, 3ex);(1.5ex, .2ex) node[anchor=north] {$\\scriptstyle r$};(0ex, 2ex) node[anchor=east] {$\\scriptstyle \\tfrac{5}{4}r$};}\\;}$ ] FKG, (REF ), (REF )(cREF 2)2 cREF 16 := cREF Where above we have made a slight abuse of notation in the second term above: The first two events in the triple intersection need to be translated vertically.","This concludes the proof.","We now define $p_r:= \\mathbb {P}_\\lambda \\bigg [ \\; \\hbox{[scale=1.5]{(0,0) rectangle (4ex, 3ex);[solid] (0, 1.5ex) .. controls (1.3ex, 2.5ex) and (2.6ex, .5ex) .. (4ex, 1.5ex);(2ex, .2ex) node[anchor=north] {$\\scriptstyle 3r/2$};(3.6ex, 1.5ex) node[anchor=west] {$\\scriptstyle \\tfrac{4}{3} r$};}}$ ] and pr:= P[ ].","Lemma 4.12 If $\\alpha _{r/2} \\ge \\frac{2}{5 c_{\\ref {c:box_height}}} \\alpha _{c_{\\ref {c:box_height}}r}$ for some $r \\ge 2 r_0$ , then $\\max \\lbrace p_r, p_r^\\star \\rbrace \\ge c_{\\ref {c:rsw}} =c_{\\ref {c:rsw}}(c_{\\ref {c:box_height}}, c_{\\ref {c:hit_middle}})$ .","Using Lemma REF we can assume without loss of generality that $\\mathbb {P}_\\lambda \\bigg [ \\hbox{[scale=1.5]{(0, 0) rectangle (4ex, 4ex);(-.2ex, 1.5ex) -- (.2ex, 1.5ex) (-.2ex, 2.5ex) -- (.2ex, 2.5ex);(3.8ex, 1.5ex) -- (4.2ex, 1.5ex) (3.8ex, 2.5ex) -- (4.2ex, 2.5ex);[solid] (0, 3ex) .. controls (.5ex, 3ex) and (1ex, 2.5ex).. (1ex, 2ex) .. controls (1ex, 1.5ex) and (.5ex, 1ex) .. (0ex, 1ex);[solid] (1ex, 2ex) .. controls (1.6ex, 3ex) and (2.4ex, 1ex) .. (3ex, 2ex);[solid] (4ex, 3ex) .. controls (3.5ex, 3ex) and (3ex, 2.5ex).. (3ex, 2ex) .. controls (3ex, 1.5ex) and (3.5ex, 1ex) .. (4ex, 1ex);[decorate,decoration={brace,amplitude=.7ex,raise=.4ex},yshift=0pt] (0, 0) -- (0, 4ex);(2ex, .2ex) node[anchor=north] {$\\scriptstyle r/2$};(-.5ex, 2ex) node[anchor=east] {$\\scriptstyle r/2$};(3.8ex, 2ex) node[anchor=west] {$\\scriptstyle 2 \\smash{\\alpha _{r/2}}$};}}$ ] 2-13 and then prove that $p_r\\ge c_{\\ref {c:rsw}}$ .","If on the other hand the above bound holds for the dual, an identical proof shows that $p^\\star _r\\ge c_{\\ref {c:rsw}}$ .","Recall that we are assuming $\\alpha _{r/2} \\ge \\frac{2}{5c_{\\ref {c:box_height}}} \\alpha _{c_{\\ref {c:box_height}} r} \\ge \\frac{2}{5c_{\\ref {c:box_height}}} \\tilde{\\alpha }_{c_{\\ref {c:box_height}} r}$ .","With this, we define the intervals $I_j = \\lbrace r\\rbrace \\times [(2/3) r+ (j-1) \\alpha _{r/ 2},(2/3) r+ j \\alpha _{r/ 2}]$ , for $j = 1, \\dots , \\lceil 2 c_{\\ref {c:box_height}}/ 5 \\rceil $ , which cover the interval $A$ defined below (REF ).","Therefore, the union bound yields $\\begin{split}\\max _{j \\le \\lceil 2 c_{\\ref {c:box_height}}/5 \\rceil } \\mathbb {P}_\\lambda & \\bigg [ \\hbox{\\;[scale=1.5]{(0, 0) rectangle (3ex, 4ex);(2.8ex, 2.2ex) -- (3.2ex, 2.2ex) (2.8ex, 2.5ex) -- (3.2ex, 2.5ex);[solid](0, 2ex) .. controls (1ex, 3.5ex) and (2ex, 2ex) .. (3ex, 2.4ex);(1.5ex, .2ex) node[anchor=north] {$\\scriptstyle r$};(.4ex, 2ex) node[anchor=east] {$\\scriptstyle \\tfrac{4}{3}r$};(2.8ex, 2.4ex) node[anchor=west] {$\\scriptstyle I_j$};}}\\end{split} \\bigg ]\\, \\ge \\, \\frac{1}{\\lceil 2 c_{\\ref {c:box_height}} / 5 \\rceil } \\sum _{j \\le \\lceil 2 c_{\\ref {c:box_height}} / 5 \\rceil } \\mathbb {P}_\\lambda \\bigg [ \\hbox{\\;[scale=1.5]{(0, 0) rectangle (3ex, 4ex);(2.8ex, 2.2ex) -- (3.2ex, 2.2ex) (2.8ex, 2.5ex) -- (3.2ex, 2.5ex);[solid](0, 2ex) .. controls (1ex, 3.5ex) and (2ex, 2ex) .. (3ex, 2.4ex);(1.5ex, .2ex) node[anchor=north] {$\\scriptstyle r$};(.4ex, 2ex) node[anchor=east] {$\\scriptstyle \\tfrac{4}{3}r$};(2.8ex, 2.4ex) node[anchor=west] {$\\scriptstyle I_j$};}}$ ] 12 cREF / 5 P[ ] cREF 2 cREF / 5 =: cREF .","Let $j_o$ be the index attaining the maximum in the left hand side of the above equation.","Since the interval $I_{j_o}$ has length $\\alpha _{r/ 2}$ , it can be covered by the small interval appearing in (REF ), such as illustrated in the picture below $p_r\\,\\ge \\, \\mathbb {P}_\\lambda \\Biggl [ \\hbox{[scale=1.5]{(0, 0) rectangle (4ex, 4ex);(-4ex, -3.5ex) rectangle (4ex, 4.5ex) (0, -3.5ex) rectangle (8ex, 4.5ex);(-.2ex, 1.5ex) -- (.2ex, 1.5ex) (-.2ex, 2.5ex) -- (.2ex, 2.5ex);(3.8ex, 1.5ex) -- (4.2ex, 1.5ex) (3.8ex, 2.5ex) -- (4.2ex, 2.5ex);(0, 3ex) .. controls (.5ex, 3ex) and (1ex, 2.5ex).. (1ex, 2ex) .. controls (1ex, 1.5ex) and (.5ex, 1ex) .. (0ex, 1ex);(1ex, 2ex) .. controls (1.6ex, 3ex) and (2.4ex, 1ex) .. (3ex, 2ex);(4ex, 3ex) .. controls (3.5ex, 3ex) and (3ex, 2.5ex).. (3ex, 2ex) .. controls (3ex, 1.5ex) and (3.5ex, 1ex) .. (4ex, 1ex);(-4ex, -1ex) .. controls (0ex, -2ex) and (0ex, 1ex).. (4ex, 2ex);(0ex, 2ex) .. controls (3ex, 4ex) and (5ex, 3ex).. (8ex, 4ex);[decorate,decoration={brace,amplitude=.5ex,raise=.2ex},yshift=0pt] (4ex, -3.5ex) -- (0ex, -3.5ex);[decorate,decoration={brace,amplitude=.7ex,raise=.2ex},yshift=0pt] (-4ex, -3.5ex) -- (-4ex, 4.5ex);(-4.3ex, .5ex) node[anchor=east] {$\\scriptstyle \\tfrac{4}{3}r$};(2ex, -3.7ex) node[anchor=north] {$\\scriptstyle r/2$};[fill, color=white](4.4ex, 1.1ex) rectangle (6.2ex, 2.7ex);[decorate,decoration={brace,amplitude=.2ex,raise=.2ex},yshift=0pt] (4.2ex, 2.5ex) -- (4.2ex, 1.5ex);(4.2ex, 2ex) node[anchor=west] {$\\scriptstyle 2 \\smash{\\alpha _{r/2}}$};}}$ ] FKG2-13 cREF 2 =: cREF , finishing the proof of the lemma.","Remark 10 It is important to observe that $c_{\\ref {c:rsw}}$ is strictly smaller than $c_{\\ref {c:hit_middle}}^2$ , as one can clearly see from the definitions of $c_{\\ref {c:hit_smaller}}$ and $c_{\\ref {c:rsw}}$ .","This will be used in Lemma REF below.","Lemma 4.13 There exists a constant $c_{\\ref {c:tiny_rsw}} > 0$ such that the following holds.","Suppose that for some $r \\ge r_0$ , $p_r\\ge c_{\\ref {c:rsw}}$ (respectively $p_r^\\star \\ge c_{\\ref {c:rsw}}$ ) and that for some ${r^{\\prime }}\\ge 30 r$ we have $\\alpha _{c_{\\ref {c:box_height}}{r^{\\prime }}} \\le r$ .","Then $p_{r^{\\prime }}\\ge c_{\\ref {c:tiny_rsw}}$ (respectively $p_{r^{\\prime }}^\\star \\ge c_{\\ref {c:tiny_rsw}}$ ).","Using the fact that $p_{r} \\ge c_{\\ref {c:rsw}}$ and Inequalities (REF ) and (REF ) in Lemma REF , we can conclude that $\\mathbb {P} \\big ( \\operatorname{Circ}(2 r, 4 r) \\big ) \\ge c_{\\ref {c:rsw}}^{124},$ We now recall (REF ) and our assumption that $\\alpha _{c_{\\ref {c:box_height}}{r^{\\prime }}} \\le r$ .","Using symmetry and the FKG-inequality, we obtain $p_{{r^{\\prime }}} \\,=\\, {\\mathbb {P}}_\\lambda \\big (\\operatorname{Cross}(3 {r^{\\prime }}/2, 4 {r^{\\prime }}/3)\\big ) \\,\\ge \\, \\mathbb {P}_\\lambda \\Biggl [ \\hbox{[scale=1.5]{(0, 0) rectangle (6ex, 8ex) (6ex, 0) rectangle (12ex, 8ex);(5.8ex, 4.2ex) -- (6.2ex, 4.2ex) (5.8ex, 4.8ex) -- (6.2ex, 4.8ex);[solid] (0, 3ex) .. controls (2ex, 2ex) and (4ex, 5ex) .. (6ex, 4.65ex);[solid] (6ex, 4.35ex) .. controls (8ex, 4ex) and (8ex, 7ex) .. (12ex, 6ex);[solid,decorate,decoration={brace,amplitude=.7ex,raise=.2ex},yshift=0pt] (12ex, 0ex) -- (0, 0);(5ex, 3.5ex) rectangle (7ex, 5.5ex);(3ex, 1.5ex) rectangle (9ex, 7.5ex);(3ex, 1ex) -- (9ex, 1ex);(3ex, .8ex) -- (3ex, 1.2ex) (9ex, .8ex) -- (9ex, 1.2ex);[fill, color=white] (4ex, .2ex) rectangle (8ex, 1.3ex);[solid,rounded corners=.25em](4ex, 6.5ex) .. controls (6ex, 6.5ex + .4ex*rand) ..(8ex, 6.5ex) .. controls (8ex + .4ex*rand, 4.5ex) ..(8ex, 2.5ex) .. controls (6ex, 2.5ex + .4ex*rand) ..(4ex, 2.5ex) -- (4ex + .4ex*rand, 4.5ex) -- cycle;(6ex, -.4ex) node[anchor=north] {$\\scriptstyle 2 {r^{\\prime }}$};(.4ex, 4ex) node[anchor=east] {$\\scriptstyle \\frac{4}{3}{r^{\\prime }}$};(9.4ex, 3.5ex) -- (9.4ex, 5.5ex);(9.2ex, 3.5ex) -- (9.6ex, 3.5ex) (9.2ex, 5.5ex) -- (9.6ex, 5.5ex);(9ex, 4.5ex) node[anchor=west] {$\\scriptstyle 2 r$};(6ex, 1.7ex) node[anchor=north] {$\\scriptstyle 6 r$};}\\;\\;}$ ] cREF 2 cREF 124 =: cREF , finishing the proof of the lemma.","We have obtained in the previous lemma a condition for $p_{r^{\\prime }}\\ge c_{\\ref {c:tiny_rsw}}$ .","However, the constant is smaller than the original $c_{\\ref {c:rsw}}$ that lower bounded $p_r$ in the hypothesis of Lemma REF .","The purpose of the next lemma is to use the above in order to bootstrap the lower bound of $p_{r^{\\prime }}$ back to $c_{\\ref {c:rsw}}$ .","Lemma 4.14 There exists a constant $c_{\\ref {c:boost_rsw}} > 300$ such that the following holds.","Suppose that for some $r \\ge r_0$ , $p_r\\ge c_{\\ref {c:rsw}}$ (respectively $p_r^\\star $ ) and that for every ${r^{\\prime }}\\in [30r, c_{\\ref {c:boost_rsw}} r]$ we have $\\alpha _{c_{\\ref {c:box_height}}{r^{\\prime }}} \\le r$ .","Then $p_{2 c_{\\ref {c:boost_rsw}} {r^{\\prime }}} \\ge c_{\\ref {c:rsw}}$ (respectively $p_{2 c_{\\ref {c:boost_rsw}} {r^{\\prime }}}^\\star \\ge c_{\\ref {c:rsw}}$ ).","We wish to apply the circuit argument of Lemma REF .","For this purpose let us fix a function $f=f_{\\lambda ,c}$ as in Item (REF ) of Lemma REF , corresponding to the constant $c:=c_{\\ref {c:tiny_rsw}}^{124}$ .","Recall from Remark REF that $c_{\\ref {c:rsw}} < c_{\\ref {c:hit_middle}}^2$ , so that we can choose $c_{\\ref {c:boost_rsw}} >300$ so that $1-f(\\tfrac{300}{c_{\\ref {c:boost_rsw}}}) \\ge \\frac{c_{\\ref {c:rsw}}}{c_{\\ref {c:hit_middle}}^2}.$ We only prove the statement for the dual quantities, the same proof also works for the primal ones (except that the function $f$ need to be chosen according to Item (REF ) in Lemma REF ).","Assume that $p_r^\\star \\ge c_{\\ref {c:rsw}}$ and that for every ${r^{\\prime }}\\in [30r, c_{\\ref {c:boost_rsw}} r]$ we have $\\alpha _{c_{\\ref {c:box_height}}{r^{\\prime }}} \\le r$ .","Using Lemma REF , we know that for every $r^{\\prime \\prime } \\in [30 r, c_{\\ref {c:boost_rsw}}r]$ , we have $p_{r^{\\prime \\prime }} \\ge c_{\\ref {c:tiny_rsw}}$ .","By the standard inequalities of Lemma REF , this implies in particular $\\inf _{60r\\le r^{\\prime \\prime }\\le c_{\\ref {c:boost_rsw}}r/10}{\\mathbb {P}}_\\lambda \\big (\\operatorname{Circ}^\\star (r^{\\prime \\prime },2r^{\\prime \\prime })\\big )\\ge c_{\\ref {c:tiny_rsw}}^{124}.$ Therefore, using the circuit argument (REF ) of Lemma REF , we obtain ${\\mathbb {P}}_\\lambda \\big (\\operatorname{Circ}^\\star (60r,\\tfrac{c_{\\ref {c:boost_rsw}}r}{5})\\big )\\ge 1-f(\\tfrac{300}{c_{\\ref {c:boost_rsw}}}) \\overset{~(\\ref {e:choose_boost})}{\\ge } \\frac{c_{\\ref {c:rsw}}}{c_{\\ref {c:hit_middle}}^2}.$ We can finally proceed as in the end of the proof of Lemma REF , observing that $p_{2 c_{\\ref {c:boost_rsw}} r}^\\star \\ge {\\mathbb {P}}_\\lambda (\\operatorname{Cross}^\\star (3c_{\\ref {c:boost_rsw}} r, c_{\\ref {c:boost_rsw}} r))$ , which can be lower bounded by $\\mathbb {P}_\\lambda \\Bigg [ \\hbox{[scale=1.5]{(0, 0) rectangle (6ex, 8ex) (6ex, 0) rectangle (12ex, 8ex);(5.8ex, 4.2ex) -- (6.2ex, 4.2ex) (5.8ex, 4.8ex) -- (6.2ex, 4.8ex);[densely dotted] (0, 3ex) .. controls (2ex, 2ex) and (4ex, 5ex) .. (6ex, 4.65ex);[densely dotted] (6ex, 4.35ex) .. controls (8ex, 4ex) and (8ex, 7ex) .. (12ex, 6ex);[densely dotted,decorate,decoration={brace,amplitude=.7ex,raise=.2ex},yshift=0pt] (12ex, 0ex) -- (0, 0);(5ex, 3.5ex) rectangle (7ex, 5.5ex);(3ex, 1.5ex) rectangle (9ex, 7.5ex);(3ex, 1ex) -- (9ex, 1ex);(3ex, .8ex) -- (3ex, 1.2ex) (9ex, .8ex) -- (9ex, 1.2ex);[fill, color=white] (4ex, .2ex) rectangle (8ex, 1.3ex);[densely dotted,rounded corners=.25em](4ex, 6.5ex) .. controls (6ex, 6.5ex + .4ex*rand) ..(8ex, 6.5ex) .. controls (8ex + .4ex*rand, 4.5ex) ..(8ex, 2.5ex) .. controls (6ex, 2.5ex + .4ex*rand) ..(4ex, 2.5ex) -- (4ex + .4ex*rand, 4.5ex) -- cycle;(6ex, -.4ex) node[anchor=north] {$\\scriptstyle 3 c_{\\ref {c:boost_rsw}}r$};(.4ex, 4ex) node[anchor=east] {$\\scriptstyle c_{\\ref {c:boost_rsw}} r$};(9.4ex, 3.5ex) -- (9.4ex, 5.5ex);(9.2ex, 3.5ex) -- (9.6ex, 3.5ex) (9.2ex, 5.5ex) -- (9.6ex, 5.5ex);(9ex, 4.5ex) node[anchor=west] {$\\scriptstyle 60r$};(6ex, 1.7ex) node[anchor=north] {$\\scriptstyle c_{\\ref {c:boost_rsw}} r/5$};}\\;\\;}$ ] cREF 2 cREF cREF 2 = cREF , as required.","Lemma 4.15 Fix $r \\ge r_0$ and suppose that for some ${r^{\\prime }}\\in [30 r, c_{\\ref {c:boost_rsw}} r]$ we have $\\alpha _{c_{\\ref {c:box_height}} {r^{\\prime }}} \\ge r$ , then $\\max \\lbrace p_{r^{\\prime \\prime }}, p^\\star _{r^{\\prime \\prime }}\\rbrace \\ge c_{\\ref {c:rsw}}$ for some $r^{\\prime \\prime } \\in [{r^{\\prime }}, c_{\\ref {c:rsw_blows}} {r^{\\prime }}]$ .","Let us choose an integer $K = K(c_{\\ref {c:box_height}}, {r^{\\prime }}/r) \\ge 1$ such that $\\Big ( \\frac{5 c_{\\ref {c:box_height}}}{2} \\Big )^K r\\ge c_{\\ref {c:box_height}} (2 c_{\\ref {c:box_height}})^K {r^{\\prime }}$ and then set $c_{\\ref {c:rsw_blows}} := 2 (2 c_{\\ref {c:box_height}})^K.$ Observe now that if for some $k = 0, \\dots , K - 1$ we have $\\alpha _{c_{\\ref {c:box_height}}(2 c_{\\ref {c:box_height}})^k {r^{\\prime }}} \\ge \\big ( \\tfrac{2}{5 c_{\\ref {c:box_height}}} \\big ) \\alpha _{c_{\\ref {c:box_height}}(2 c_{\\ref {c:box_height}})^{k+1} {r^{\\prime }}},$ we can apply Lemma REF , replacing $r$ with $(2c_{\\ref {c:box_height}})^{k+1} {r^{\\prime }}$ .","This will yield the bound $\\max \\lbrace p_{(2 c_{\\ref {c:box_height}})^{k+1} {r^{\\prime }}}, p^\\star _{(2 c_{\\ref {c:box_height}})^{k+1} {r^{\\prime }}}\\rbrace \\ge c_{\\ref {c:rsw}}.$ By our choice of $c_{\\ref {c:rsw_blows}}$ , we conclude that ${r^{\\prime }}\\le (2 c_{\\ref {c:box_height}})^{k+1} {r^{\\prime }}\\le (2 c_{\\ref {c:box_height}})^K {r^{\\prime }}\\le c_{\\ref {c:rsw_blows}} {r^{\\prime }}$ , as desired.","On the other hand if (REF ) does not hold for any $k = 0, \\dots , K - 1$ , then $r \\le \\alpha _{c_{\\ref {c:box_height}} {r^{\\prime }}} \\le \\big ( \\tfrac{2}{5 c_{\\ref {c:box_height}}} \\big ) \\alpha _{c_{\\ref {c:box_height}}(2 c_{\\ref {c:box_height}}) {r^{\\prime }}} \\le \\dots \\le \\big ( \\tfrac{2}{5 c_{\\ref {c:box_height}}} \\big )^{K} \\alpha _{c_{\\ref {c:box_height}}(2 c_{\\ref {c:box_height}})^{K} {r^{\\prime }}},$ so that by our choice of $K$ we would have $\\alpha _{c_{\\ref {c:box_height}}(2 c_{\\ref {c:box_height}})^{K} {r^{\\prime }}} \\ge (5 c_{\\ref {c:box_height}} / 2)^{K} r \\ge c_{\\ref {c:box_height}}(2 c_{\\ref {c:box_height}})^{K} {r^{\\prime }}$ , which is a contradiction with (REF ).","Lemma 4.16 There exists $c_{\\ref {c:rsw_max}} > 0$ such that $\\inf _{r\\ge 1} \\big (\\max \\lbrace p_r, p^\\star _r\\rbrace \\big ) \\ge c_{\\ref {c:rsw_max}}.$ We first claim that there exists an increasing sequence $r_1\\le r_2\\le \\cdots $ such that $3\\le \\tfrac{r_{i+1}}{r_i}\\le \\max \\lbrace 2c_{\\ref {c:boost_rsw}}, c_{\\ref {c:rsw_blows}}\\rbrace $ for every $i \\ge 1$ and $\\max \\lbrace p_{r_i}, p^\\star _{r_i}\\rbrace \\ge c_{\\ref {c:rsw}}$ , for every $i \\ge 1$ .","We first construct $r_1$ .","Since $\\alpha _r\\le r/6$ for every $r\\ge 1$ (see (REF )), there must exist $k\\ge 1$ such that $\\alpha _{r_0(2 c_{\\ref {c:box_height}})^k} \\ge \\big ( \\tfrac{2}{5 c_{\\ref {c:box_height}}} \\big ) \\alpha _{r_0(2 c_{\\ref {c:box_height}})^{k+1}},$ Applying Lemma REF to $r_1:=2r_0(2c_{\\ref {c:box_height}})^{k}$ yields the bound $\\max \\lbrace p_{r_1}, p^\\star _{r_1}\\rbrace \\ge c_{\\ref {c:rsw}}$ as desired.","To finish the construction of the $(r_i)$ 's, it is enough to show that for any $r\\ge 1$ such that $p_r\\ge c_{\\ref {c:rsw}}$ , there exists $r^{\\prime \\prime } \\in [3r, \\max \\lbrace 2c_{\\ref {c:boost_rsw}}, c_{\\ref {c:rsw_blows}}\\rbrace r]$ for which $\\max \\lbrace p_{r^{\\prime \\prime }}, p^\\star _{r^{\\prime \\prime }}\\rbrace \\ge c_{\\ref {c:rsw}}$ .","We then split the proof of this claim into two cases, depending on whether or not there exists ${r^{\\prime }}\\in [3 r, c_{\\ref {c:boost_rsw}} r]$ such that $\\alpha _{c_{\\ref {c:box_height}} {r^{\\prime }}} \\ge r$ .","In this case, Lemma REF shows that the existence of $r^{\\prime \\prime } \\le c_{\\ref {c:rsw_blows}} r$ as in (REF ).","On the other hand, if (REF ) does not hold, than we can use $r^{\\prime \\prime } = 2c_{\\ref {c:boost_rsw}}$ , according to Lemma REF .","In any case we have proved (REF ), which shows the existence of $r_1,r_2, \\dots $ as above.","To end the proof we use the standard inequalities of Lemma REF to interpolate between the values $r_i$ .","We now have obtained a statement similar to the RSW result stated in Theorem REF .","However, we only know that the crossing probability is bounded below for the primal or the dual and not both at the same time.","It could be still the case that only the dual crossings have a bounded probability.","The purpose of the following lemma is to show that this cannot be the case.","Lemma 4.17 There exist constants $c_{\\ref {c:rsw_both_1}} \\ge 10$ and $c_{\\ref {c:rsw_both_2}}>0$ such that the following holds.","Given $r\\ge r_0$ , if $p_{{r^{\\prime }}}^\\star \\ge c_{\\ref {c:rsw_max}}$ for every ${r^{\\prime }}\\in [r, c_{\\ref {c:rsw_both_1}} r]$ , then $p_{c_{\\ref {c:rsw_both_1}} r} \\ge c_{\\ref {c:rsw_both_2}}$ .","Moreover, the same holds true if we swap the roles of $p^\\star $ and $p$ above.","We wish to apply the circuit argument of Lemma REF .","To this end, let us fix a function $f=f_{\\lambda ,c}$ as in Item (REF ) of Lemma REF , corresponding to the constant $c:=c_{\\ref {c:rsw_max}}^{124}$ .","We first choose $c_{\\ref {c:rsw_both_1}} = c_{\\ref {c:rsw_both_1}}(c_{\\ref {c:easy_cross}},c_{\\ref {c:rsw_max}}) \\ge 30$ large enough such that $f(\\tfrac{10}{c_{\\ref {c:rsw_both_1}}}) \\le \\frac{c_{\\ref {c:easy_cross}}}{3}$ and $c_{\\ref {c:rsw_both_2}}>0$ small enough that $\\big (1 - c_{\\ref {c:rsw_both_2}}^{1/(2c_{\\ref {c:rsw_both_1}})}\\big ) \\big ( 1 - \\frac{c_{\\ref {c:easy_cross}}}{3} \\big ) \\ge 1 - \\frac{c_{\\ref {c:easy_cross}}}{2}.$ Suppose now, for a contradiction, that $p_{c_{\\ref {c:rsw_both_1}} r} \\le c_{\\ref {c:rsw_both_2}}$ , therefore $\\mathbb {P}_\\lambda \\big [\\operatorname{Cross}^\\star \\big (\\tfrac{4}{3} c_{\\ref {c:rsw_both_1}} r, \\tfrac{3}{2} c_{\\ref {c:rsw_both_1}} r\\big ) \\big ] = 1 - p_{c_{\\ref {c:rsw_both_1}} r} \\ge 1 - c_{\\ref {c:rsw_both_2}}.$ Cover the interval $[0, \\tfrac{3}{2} c_{\\ref {c:rsw_both_1}} r]$ by at most $H =\\lceil \\tfrac{3}{2} c_{\\ref {c:rsw_both_1}} \\rceil \\le 2 c_{\\ref {c:rsw_both_1}}$ intervals of length $r$ .","The square-root trick (REF ) implies that there exists $h \\in \\lbrace 0, 1, \\dots , H-1\\rbrace $ , such that $\\mathbb {P}_\\lambda \\bigg [ \\hbox{\\;[scale=1.5]{(0, 0) rectangle (3ex, 4ex);(2.8ex, 2ex) -- (3.2ex, 2ex) (2.8ex, 2.8ex) -- (3.2ex, 2.8ex);[densely dotted](0, 2ex) .. controls (1ex, 3.5ex) and (2ex, 1ex) .. (3ex, 2.6ex);(1.5ex, .2ex) node[anchor=north] {$\\scriptstyle (4/3) c_{\\ref {c:rsw_both_1}} r$};(.4ex, 2ex) node[anchor=east] {$\\scriptstyle \\tfrac{3}{2} c_{\\ref {c:rsw_both_1}} r$};(2.8ex, 2.4ex) node[anchor=west] {$\\scriptstyle I_h$};}}$ ] 1 - cREF 1/H, where $I_h = \\lbrace \\tfrac{4}{3} c_{\\ref {c:rsw_both_1}} r\\rbrace \\times [h r, (h + 1) r]$ .","We now recall that $p_{{r^{\\prime }}}^\\star \\ge c_{\\ref {c:rsw_max}}$ for every ${r^{\\prime }}\\in [r, c_{\\ref {c:rsw_both_1}} r]$ .","Therefore, by the standard inequalities of Lemma REF , we obtain in particular $\\inf _{2r\\le r^{\\prime }\\le c_{\\ref {c:rsw_both_1}} r/10}{\\mathbb {P}}_\\lambda \\big (\\operatorname{Circ}^\\star (r^{\\prime },2r^{\\prime })\\big )\\,\\ge \\, c_{\\ref {c:rsw_max}}^{124}.$ Therefore, by the circuit argument (REF ) in Lemma REF , $\\mathbb {P} \\Big ( \\operatorname{Circ}^\\star (2 r, c_{\\ref {c:rsw_both_1}} r/ 5) \\Big ) \\,\\ge \\, 1-f(\\tfrac{10}{c_{\\ref {c:rsw_both_1}}}) \\,\\overset{(\\ref {e:choose_both_1})}{\\ge }\\, 1 - \\frac{c_{\\ref {c:easy_cross}}}{3}.$ As in (REF ), we obtain $\\mathbb {P}_\\lambda \\big ( \\operatorname{Cross}^\\star (\\tfrac{8}{3} c_{\\ref {c:rsw_both_1}} r, \\tfrac{3}{2} c_{\\ref {c:rsw_both_1}} r)\\big )\\,& \\ge \\, \\mathbb {P}_\\lambda \\Bigg [ \\hbox{[scale=1.5]{(0, 0) rectangle (6ex, 8ex) (6ex, 0) rectangle (12ex, 8ex);(5.8ex, 4.2ex) -- (6.2ex, 4.2ex) (5.8ex, 4.8ex) -- (6.2ex, 4.8ex);[solid] (0, 3ex) .. controls (2ex, 2ex) and (4ex, 5ex) .. (6ex, 4.65ex);[solid] (6ex, 4.35ex) .. controls (8ex, 4ex) and (8ex, 7ex) .. (12ex, 6ex);[solid,decorate,decoration={brace,amplitude=.7ex,raise=.2ex},yshift=0pt] (12ex, 0ex) -- (0, 0);(5ex, 3.5ex) rectangle (7ex, 5.5ex);(3ex, 1.5ex) rectangle (9ex, 7.5ex);(3ex, 1ex) -- (9ex, 1ex);(3ex, .8ex) -- (3ex, 1.2ex) (9ex, .8ex) -- (9ex, 1.2ex);[fill, color=white] (4ex, .2ex) rectangle (8ex, 1.3ex);[solid,rounded corners=.25em](4ex, 6.5ex) .. controls (6ex, 6.5ex + .4ex*rand) ..(8ex, 6.5ex) .. controls (8ex + .4ex*rand, 4.5ex) ..(8ex, 2.5ex) .. controls (6ex, 2.5ex + .4ex*rand) ..(4ex, 2.5ex) -- (4ex + .4ex*rand, 4.5ex) -- cycle;(6ex, -.4ex) node[anchor=north] {$\\scriptstyle \\tfrac{8}{3} c_{\\ref {c:rsw_both_1}} r$};(.4ex, 4ex) node[anchor=east] {$\\scriptstyle \\tfrac{3}{2} c_{\\ref {c:rsw_both_1}} r$};(9.4ex, 3.5ex) -- (9.4ex, 5.5ex);(9.2ex, 3.5ex) -- (9.6ex, 3.5ex) (9.2ex, 5.5ex) -- (9.6ex, 5.5ex);(9ex, 4.5ex) node[anchor=west] {$\\scriptstyle 2 r$};(6ex, 1.7ex) node[anchor=north] {$\\scriptstyle c_{\\ref {c:rsw_both_1}}r/5$};}\\;\\;}$ ] (1 - cREF 1/H)2 (1 - cREF 3 ) (REF ) 1 - cREF 2 which is a contradiction with (REF ).","Remark 11 In the proof of Lemmas REF and REF we have focused on the dual paths instead of primal to emphasize that we only needed the weak decoupling inequality (REF ) that has been derived from (REF ).","[Proof of Theorem REF ] We first claim that it is enough to prove that for any $r\\ge r_0$ there exists some ${r^{\\prime }}\\in [r, c_{\\ref {c:rsw_both_1}} r]$ such that $p^\\star _{{r^{\\prime }}}\\ge \\min \\lbrace c_{\\ref {c:rsw_max}}, c_{\\ref {c:rsw_both_2}}\\rbrace $ .","Indeed, if this is the case, we can use the standard inequalities in Lemma REF to show that $\\inf _{r\\ge 1} p^\\star _{r} > 0$ .","(The case $p_r$ is completely analogous.)","So, let us fix some $r\\ge r_0$ and assume that $p^\\star _{{r^{\\prime }}} < c_{\\ref {c:rsw_max}}$ for every ${r^{\\prime }}\\in [r,c_{\\ref {c:rsw_both_1}} r]$ , then by Lemma REF , we would have $p_{{r^{\\prime }}} \\ge c_{\\ref {c:rsw_max}}$ for every ${r^{\\prime }}\\in [r, c_{\\ref {c:rsw_both_1}} r]$ .","But then Lemma REF implies that $p^\\star _{c_{\\ref {c:rsw_both_1}} r} \\ge c_{\\ref {c:rsw_both_2}}$ as desired."],["Sharpness of the phase transition","We have so far showed that decay of correlations is sufficient for the two thresholds $\\lambda _0$ and $\\lambda _1$ to be non-trivial; see Section .","Moreover, we have shown that a distinctive critical behavior, namely the box-crossing property, occurs throughout the interval $[\\lambda _0,\\lambda _1]$ ; see Section .","Our goal for this section is to prove that these two thresholds $\\lambda _0$ and $\\lambda _1$ are in fact the same.","Theorem 5.1 For Poisson Boolean percolation satisfying (REF ), we have $\\lambda _0 = \\lambda _1$ .","In order to establish an equality between the two thresholds $\\lambda _0$ and $\\lambda _1$ , our aim will be to show that the crossing probabilities grow very fast from close to zero to close to one.","More precisely, we have seen that the probability of crossing a fixed-ratio rectangle is bounded away from zero and one for $\\lambda \\in [\\lambda _0, \\lambda _1]$ .","In this section, we show that this implies that the derivative of the crossing probability is large throughout the same interval.","The phenomenon of sharp thresholds is well understood in the context of Boolean functions on the discrete cube $\\lbrace 0,1\\rbrace ^n$ equipped with product measure.","In order to apply the tools from this theory in our context we will following an approach introduced in [1].","Thus, the first step of our proof will be to enhance the probability space in which we work to suit this setting, see Subsection REF .","After having defined this alternative construction, we will employ Margulis-Russo's formula to relate the derivative of crossing probabilities to influences of these auxiliary random variables.","At this point, an inequality of Talagrand will become handy in order to bound the total influence from below.","This is done in two steps, where we bound the conditional variance of the crossing probabilities from below and the individual influences from above, see Subsections REF and REF respectively.","Subsection REF combines the above bounds to prove Theorem REF .","Remark 12 Of the three main theoretical components of this paper, the one presented in this section is the least general one.","Although what we will present below does apply to some other models, it strongly uses the Poissonian nature of the process, see Section ."],["Sharp thresholds for Boolean functions","It has been known for some time that monotone events involving a large number of independent Bernoulli variables typically exhibit sharp thresholds, in the sense that the probability of the event increases sharply as the parameter of the Bernoulli variables passes a certain value.","The first evidence thereof dates back to the study of random graphs by Erdős and Rényi [14], and the proof that the critical probability for bond percolation on ${\\mathbb {Z}}^2$ equals $1/2$ by Kesten [23].","A more general understanding of these phenomena has been pursued since in work of Russo [36], Bollobás and Thomason [12], and Kahn, Kalai and Linial [25] as well as elsewhere.","See e.g.","[31] or [19].","For $p \\in [0, 1]$ let $\\textup {P}_p$ denote product measure of intensity $p$ on $\\lbrace 0,1\\rbrace ^n$ .","A Boolean function $f : \\lbrace 0, 1\\rbrace ^n \\rightarrow \\lbrace 0, 1\\rbrace $ is said to be monotone when $f(\\eta ) \\le f(\\eta ^{\\prime })$ whenever $\\eta \\le \\eta ^{\\prime }$ coordinate wise.","For monotone Boolean functions, the map $p \\mapsto \\textup {P}_p(f = 1)$ is also monotone as a function of $p$ .","Moreover, the rate of change of $\\textup {P}_p(f = 1)$ is related to the concept of influences of the individual variables, as made explicit by the Margulis-Russo formula (see [34]), $\\frac{d}{dp} \\textup {P}_p(f = 1) \\,=\\, \\sum _{i = 1}^n \\operatorname{Inf}_i^p(f),$ where $\\operatorname{Inf}_i^p(f) := \\textup {P}_p\\big (f(\\eta ) \\ne f(\\sigma _i\\eta )\\big )$ and $\\sigma _i$ is the operator that flips the value of $\\eta $ at position $i$ .","In case $f(\\eta ) \\ne f(\\sigma _i\\eta )$ , we say that the bit $i$ is pivotal for $f(\\eta )$ .","Against one's initial intuition, Russo [36] further showed that a uniform upper bound on the individual influences implies a lower bound of their sum, thus assuring a rapid increase of $\\textup {P}_p(f=1)$ .","This mystique has since become better understood with the work of Kahn, Kalai and Linial [25] and its extensions, which shows that not all influences may be too small.","The following inequality, due to Talagrand [41], illustrates this fact well and gives a precise formulation of Russo's initial observation: There exists a constant $c_{\\ref {c:talagrand}} > 0$ such that for every Boolean function $f:\\lbrace 0,1\\rbrace ^n\\rightarrow \\lbrace 0,1\\rbrace $ we have $\\sum _{i=1}^n \\operatorname{Inf}_i^p(f) \\,\\ge \\, \\frac{c_{\\ref {c:talagrand}}}{\\log \\big ( \\tfrac{2}{p (1 - p)} \\big )} \\, \\operatorname{Var}_p(f) \\log \\Big ( \\frac{1}{\\max \\lbrace \\operatorname{Inf}_i^p(f)\\rbrace } \\Big ).$ Talagrand stated this inequality for monotone functions, but this restriction is not necessary.","From (REF ) and (REF ), the strategy of the proof becomes intuitive: We first give an alternative construction of the process involving independent Bernoulli random variables, then we bound the variance of the crossing probabilities from below and the influence of individual bits from above.","These are the contents of the next subsections."],["A two-stage construction","We will for the rest of this section work on an enlarged probability space $(\\overline{\\Omega },\\overline{\\mathcal {M}},\\overline{{\\mathbb {P}}}_\\lambda )$ in order to construct our process in two stages, where the second step employs independent coin flips in order to define the final configuration.","In this way we will be able to condition on the outcome of the first step in order to arrive at a situation where tools from the discrete setting applies.","Recall that $\\Omega $ denotes the space of locally finite point measures on ${\\mathbb {R}}^2 \\times {\\mathbb {R}}_+$ .","We will by $\\overline{\\Omega }$ denote the space of locally finite point measures on ${\\mathbb {R}}^2 \\times {\\mathbb {R}}_+ \\times [0, 1] \\times [0, 1]$ , by $\\overline{\\mathcal {M}}$ we denote the corresponding Borel sigma algebra, and we let $\\overline{{\\mathbb {P}}}_\\lambda $ denote the law of a Poisson point process defined on $(\\overline{\\Omega },\\overline{\\mathcal {M}})$ with intensity $\\lambda \\,dx\\,\\mu (dz)\\,du\\,dv$ .","In this section we will use a very dense set of points and the last two coordinates $u$ and $v$ in the above construction will be used to perform a thinning of the Poisson process.","Throughout this section we will suppose that $m\\ge 1$ is a large integer and that $\\lambda =2m\\lambda _1$ is fixed.","Given $\\overline{\\omega }\\in \\overline{\\Omega }$ we denote by $\\omega $ the projection of $\\overline{\\omega }$ onto $\\Omega $ .","For $p\\in [0,1]$ and $m\\ge 1$ we define $\\omega _p:=\\sum _{\\begin{array}{c}(x,z,u,v)\\in \\overline{\\omega }\\\\ u\\le p,\\, v\\le 1/m\\end{array}}\\delta _{(x,z)}\\quad \\text{and}\\quad \\mathcal {O}_p:=\\bigcup _{(x,z)\\in \\omega _p} B(x,z).$ The usual properties of a Poisson point process imply that $\\omega $ and $\\omega _p$ are Poisson point processes on $\\Omega $ distributed according to ${\\mathbb {P}}_\\lambda $ and ${\\mathbb {P}}_{p\\lambda /m}$ , respectively.","Notice that $\\omega _p$ is a $(p/m)$ -thinning of $\\omega $ , and therefore it can be seen as a percolation process on $\\lbrace 0,1\\rbrace ^\\omega $ with density $p/m$ .","Since $\\lambda =2m\\lambda _1$ , the above construction comes with two parameters, $p$ and $m$ , of which we think of $m$ as large and fixed while $p$ will be varying.","We have for this reason suppressed $m$ in the above notation, and apply a subscript $p$ to indicate that the events $\\operatorname{Cross}_p$ , $\\operatorname{Circ}_p$ and $\\operatorname{Arm}_p$ refer to crossings in $\\mathcal {O}_p$ .","We will study the conditional probability $Z_p:=\\overline{{\\mathbb {P}}}_\\lambda \\big (\\operatorname{Cross}_p(4r,r)\\big |\\omega \\big ).$ The product structure of the above conditional measure implies that for ${\\mathbb {P}}_\\lambda $ -almost every $\\omega \\in \\Omega $ the variable $Z_p$ can be viewed as the expectation of some Boolean function $f_\\omega :\\lbrace 0,1\\rbrace ^\\omega \\rightarrow \\lbrace 0,1\\rbrace $ .","While $\\omega $ has infinite support, it follows from (REF ) that $\\overline{{\\mathbb {P}}}_\\lambda $ -almost surely, only finitely many balls touch the box $[-4r, 4r]^2$ .","Therefore the function $f_\\omega $ can be viewed as having only finitely many variables that we index by $i \\in \\lbrace 1, \\dots , i_0\\rbrace $ ($i_0$ is the number of balls touching $[-4r, 4r]^2$ ).","In this conditional setting, the tools from the discrete analysis will apply, in particular the Margulis-Russo formula and Talagrand's inequality.","With the use of these we will be able to bound the gap between $\\lambda _0$ and $\\lambda _1$ by controlling the conditional variance of $f_\\omega $ and its influences, defined, for $\\overline{{\\mathbb {P}}}_\\lambda $ -almost every $\\overline{\\omega }\\in \\Omega $ , as $\\operatorname{Inf}_i^p(f_\\omega ):=\\overline{{\\mathbb {P}}}_\\lambda \\big (f_\\omega (\\omega _p)\\ne f_\\omega (\\sigma _i\\omega _p)\\big |\\omega \\big ),$ where $\\sigma _i\\omega _p=\\omega _p+(1-2\\operatorname{{\\bf 1}}_{\\lbrace u_i\\le p,\\,v_i\\le 1/m\\rbrace })\\delta _{(x_i,z_i)}$ ."],["Quenched decoupling","In order to handle dependencies in the quenched setting we will need an alternative to the decoupling inequality proved in Proposition REF .","Given $\\overline{\\omega }\\in \\overline{\\Omega }$ , this can be done via an estimate on the influence region of each point in the support of $\\omega $ .","Since we will only consider events defined in terms of $\\mathcal {O}_p$ on $[-4r,4r]^2$ it will suffice to consider points intersecting this region.","For $r\\ge 3n\\ge 3$ , let $G = G(r,n)$ denote the event $G := \\Big [ \\max \\big \\lbrace z:(x, z)\\in \\operatorname{supp}(\\omega )\\text{ and } B(x, z) \\cap [-4r, 4r]^2 \\ne \\emptyset \\big \\rbrace \\le r/3n \\Big ].$ Lemma 5.2 Assume the second moment condition (REF ).","For every $n\\ge 1$ , $\\lim _{r\\rightarrow \\infty } \\overline{{\\mathbb {P}}}_\\lambda (G) = 1$ .","The proof follows the same steps as that of Lemma REF .","A first consequence of the above observation concerns quenched decouplings of events defined in well separated regions.","To state this precisely, fix integers $n, J \\ge 1$ and let $D_1, \\dots , D_J \\subseteq [-4 r, 4 r]^2$ be measurable sets such that $d(D_i, D_j) \\ge \\frac{r}{n}, \\text{ for every $i \\ne j$}.$ Having these well separated sets, our aim is to decouple what happens inside each of them.","For this, fix some events $A_1, \\dots , A_J$ that only look inside the sets $D_j$ , that is $A_j \\in \\sigma (Y_x, x \\in D_j)$ .","Observe that on the event $G = G(r, n)$ the events $A_j$ depend on disjoint subsets of $\\omega $ under the conditional measure $\\overline{{\\mathbb {P}}}_\\lambda (\\,\\cdot \\,|\\omega )$ .","This implies that $\\overline{{\\mathbb {P}}}_\\lambda \\big ( A_1 \\cap \\dots \\cap A_J \\big | \\omega \\big ) \\,\\le \\, \\operatorname{{\\bf 1}}_{G^c}(\\omega ) + \\prod \\nolimits _{j \\le J} \\overline{{\\mathbb {P}}}_\\lambda (A_j | \\omega ).$ For us it is important to observe that not only the events $A_j$ decouple on $G(r, n)$ , but also their conditional probabilities $\\overline{{\\mathbb {P}}}_\\lambda (A_j | \\omega )$ .","To see why this is true, let us first decompose the point measure $\\omega $ as $\\omega _1 + \\dots + \\omega _J + \\tilde{\\omega }$ , where $\\omega _j = \\sum _i \\delta _{(x_i, z_i)}$ with the sum ranging over $i \\ge 0$ such that $d(x_i, D_j) \\le r/(3n)$ and $\\tilde{\\omega }$ stands for the remainder.","Observe that on $G$ , the conditional probabilities $\\overline{{\\mathbb {P}}}_\\lambda (A_j | \\omega )$ depend on $\\omega $ through $\\omega _j$ only.","In particular, we may exhibit a function $g_j$ such that $[\\operatorname{{\\bf 1}}_G \\cdot \\overline{{\\mathbb {P}}}_\\lambda (A_j | \\omega )](\\omega ) = \\operatorname{{\\bf 1}}_G \\cdot g_j(\\omega _j)$ , and by the Poissonian nature of $\\omega $ , all the $\\omega _j$ 's are independent.","As a consequence, for any $\\delta > 0$ , $\\begin{split}\\overline{{\\mathbb {P}}}_\\lambda \\Big (G \\cap \\bigcap _{j \\le J} \\big [ \\overline{{\\mathbb {P}}}_\\lambda (A_j | \\omega ) \\le \\delta \\big ] \\Big ) \\,& \\le \\, \\prod _{j \\le J} \\overline{{\\mathbb {P}}}_\\lambda \\big [ g_j(\\omega _j) \\le \\delta \\big ]\\\\& \\le \\, \\prod _{j \\le J}\\bigg [ \\overline{{\\mathbb {P}}}_\\lambda \\big ( \\overline{{\\mathbb {P}}}_\\lambda (A_j | \\omega ) \\le \\delta \\big )+\\overline{{\\mathbb {P}}}(G^c)\\bigg ].\\end{split}$ Equations (REF ) and (REF ) give a more precise version of the decoupling of Section and will be important in bounding the conditional variance and influences in the next two sections."],["Controlling the variance","In Section we showed that the probability of crossing a rectangle is non-degenerate throughout the critical regime $[\\lambda _0, \\lambda _1]$ .","In the notation used in this section, that is to say that $\\overline{\\operatorname{{\\mathbb {E}}}}_\\lambda [Z_p]\\in (c,1-c)$ for some $c>0$ and all $r\\ge 1$ and $p\\lambda / m \\in \\big [ \\lambda _0, \\lambda _1 \\big ]$ .","Recall that $\\lambda = 2 m \\lambda _1$ .","Our goal for this section is to show that this holds with probability close to one also for $Z_p$ .","Proposition 5.3 Assume the second moment condition (REF ).","Then, for every $\\varepsilon >0$ there exists $\\delta >0$ and $c_{\\ref {c:lambda}} = c_{\\ref {c:lambda}}(\\varepsilon ) > 0$ such that for every $m\\ge c_{\\ref {c:lambda}}$ we have $\\overline{{\\mathbb {P}}}_\\lambda \\bigg (Z_p\\in (\\delta ,1-\\delta )\\text{ for all }p\\in \\big [\\tfrac{\\lambda _0}{2\\lambda _1},\\tfrac{\\lambda _1}{2\\lambda _1}\\big ]\\bigg )\\ge 1-\\varepsilon \\quad \\text{for all }r\\ge 1.$ The same holds with $\\operatorname{Cross}_p(4r, r)$ replaced by either $\\operatorname{Cross}_p^\\star (4r, r)$ , $\\operatorname{Circ}_p(r, 2r)$ or $\\operatorname{Circ}_p^\\star (r, 2r)$ .","An immediate consequence of the above proposition is that $\\overline{{\\mathbb {P}}}_\\lambda \\bigg (\\operatorname{Var}_\\lambda (\\operatorname{{\\bf 1}}_{\\operatorname{Cross}_p(4r,r)}|\\omega )\\in (\\delta ,1-\\delta )\\text{ for all }p\\in \\big [\\tfrac{\\lambda _0}{2\\lambda _1},\\tfrac{\\lambda _1}{2\\lambda _1}\\big ]\\bigg )\\ge 1/2\\quad \\text{for all }r\\ge 1.$ The first step in the proof of Proposition REF is the following estimate on the effect of observing $\\omega $ in determining whether there is a crossing of $[0,4r] \\times [0,r]$ in $\\mathcal {O}_p$ .","Lemma 5.4 For every $m\\ge 1$ and $p\\in [0,1]$ , and every measurable event $A$ , we have $\\operatorname{Var}_\\lambda \\big (\\overline{{\\mathbb {P}}}_\\lambda (A|\\omega )\\big )\\le 1/m.$ Recall that $\\omega _p$ is obtained as a subset of $\\omega $ based on the value of the third and fourth coordinates of each point in $\\overline{\\omega }$ .","Alternatively, we could obtain $\\omega _p$ in two stages: First, partition $\\overline{\\omega }$ into $\\omega _p^1,\\omega _p^2,\\ldots ,\\omega _p^m$ and some remainder $\\tilde{\\omega }$ , where $\\omega _p^j$ contains all points with third coordinate at most $p$ and fourth coordinate in $\\big [\\frac{j-1}{m},\\frac{j}{m}\\big )$ .","Second, use auxiliary randomness, independent from everything else, to determine which of the $\\omega _p^j$ 's to choose as the set $\\omega _p$ .","Since each of the $\\omega _p^j$ 's have the same distribution, the result would be the same.","Let us assume that this construction has been made, and that $J$ is the auxiliary random variable, uniformly distributed on $\\lbrace 1,2,\\ldots ,m\\rbrace $ , that determines which of the $\\omega _p^j$ 's that is chosen in the second step.","Let ${\\mathcal {F}}$ denote the sigma algebra containing the information of the partitioning $\\omega _p^1, \\ldots , \\omega _p^m$ .","Since revealing the partitioning can only increase the variance, we have that $\\operatorname{Var}_\\lambda \\big (\\overline{{\\mathbb {P}}}_\\lambda (A|\\omega )\\big )\\le \\operatorname{Var}_\\lambda \\big (\\overline{{\\mathbb {P}}}_\\lambda (A|{\\mathcal {F}})\\big ).$ Moreover, by conditioning on the outcome of $J$ , $\\overline{{\\mathbb {P}}}_\\lambda (A|{\\mathcal {F}})=\\frac{1}{m}\\sum _{j=1}^m\\operatorname{{\\bf 1}}_{\\lbrace \\omega _p^j\\in A\\rbrace }.$ Since the configurations $\\omega _p^1,\\omega _p^2,\\ldots ,\\omega _p^m$ are independent, we find that $\\operatorname{Var}_\\lambda \\big (\\overline{{\\mathbb {P}}}_\\lambda (A|\\omega )\\big )\\,\\le \\,\\operatorname{Var}_\\lambda \\bigg (\\frac{1}{m}\\sum _{j=1}^m\\operatorname{{\\bf 1}}_{\\lbrace \\omega _p^j\\in A\\rbrace }\\bigg )\\,\\le \\,\\frac{1}{m},$ as required.","[Proof of Proposition REF ] Note that $Z_p$ is monotone in $p$ , so it will suffice to consider the extremes of the interval $\\big [\\tfrac{\\lambda _0}{2\\lambda _1},\\tfrac{\\lambda _1}{2\\lambda _1}\\big ]$ .","Due to Corollary REF there exists a constant $c_{\\ref {c:q-bound}} = c_{\\ref {c:q-bound}}(4) > 0$ such that $\\overline{\\operatorname{{\\mathbb {E}}}}_\\lambda [Z_p]\\in (c_{\\ref {c:q-bound}},1-c_{\\ref {c:q-bound}})$ for all $p\\in \\big [\\tfrac{\\lambda _0}{2\\lambda _1},\\tfrac{\\lambda _1}{2\\lambda _1}\\big ]$ .","So, by Chebyshev's inequality and Lemma REF , for $b = 0, 1$ , $\\overline{{\\mathbb {P}}}_\\lambda \\Big (Z_{\\lambda _b/(2\\lambda _1)}\\notin (c_{\\ref {c:q-bound}}/2,1-c_{\\ref {c:q-bound}}/2)\\Big )\\,\\le \\,\\frac{\\operatorname{Var}_\\lambda (Z_p)}{(c_{\\ref {c:q-bound}}/2)^2}\\,\\le \\,\\frac{4}{mc_{\\ref {c:q-bound}}^2},$ which is at most $\\varepsilon /2$ for $m\\ge 8/(\\varepsilon c_{\\ref {c:q-bound}}^2)$ .","Now, the result follows via a union bound."],["Controlling the influences","The next step is to obtain an upper bound on the individual conditional influences, defined as in (REF ), allowing for Talagrand's inequality to give a lower bound on the total conditional influence.","The result proved here is the following.","Proposition 5.5 Assume the second moment condition (REF ).","There exists a constant $c_{\\ref {c:lambda2}}$ such that for every $m\\ge c_{\\ref {c:lambda2}}$ and $\\eta >0$ we have $\\overline{{\\mathbb {P}}}_\\lambda \\Big ( \\max _{i \\ge 1} \\big ( \\operatorname{Inf}^p_i (f_\\omega ) \\big ) > \\eta \\text{ for some }p\\in \\big [ \\tfrac{\\lambda _0}{2\\lambda _1}, \\tfrac{\\lambda _1}{2\\lambda _1} \\big ] \\Big ) \\le \\eta ,$ for all $r$ large enough depending only on $\\eta $ .","This will be done in two steps.","First we relate the influence of a point to arm probabilities.","Then we bound the probability of the latter.","Before stating the first lemma, consider a partitioning of the rectangle $[0, 4r] \\times [0, r]$ into $36 n^2$ smaller squares of side length $r/ 3n$ .","We write $\\operatorname{Arm}_p^\\ell (\\tfrac{r}{n}, r)$ , with $\\ell = 1, \\dots , 36 n^2$ , for the arm event defined as in (REF ), but centered around each of these boxes.","These arm events help us to bound the influences $\\operatorname{Inf}^p_i(f_\\omega )$ as follows.","Assume that $i$ is pivotal, and the corresponding ball has radius smaller or equal to $r / 3n$ .","Then there must exist some $\\ell $ for which the event $\\operatorname{Arm}_p^\\ell (r/n,r)$ must occur.","Therefore, $\\operatorname{Inf}_i^p(f_\\omega )\\le \\operatorname{{\\bf 1}}_{G^c}(\\omega ) + \\overline{{\\mathbb {P}}}_\\lambda \\big (\\operatorname{Arm}_p^\\ell (\\tfrac{r}{n},r)\\big |\\omega \\big )\\quad \\text{for some }\\ell =1,2,\\ldots , 36 n^2,$ for $\\overline{{\\mathbb {P}}}_\\lambda $ -almost every $\\overline{\\omega }$ .","Estimating influences will now boil down to estimating conditional arm probabilities.","Lemma 5.6 Assume the second moment condition (REF ).","There exist $\\gamma > 0$ and $c_{\\ref {c:lambda2}} > 0$ such that for every $m > c_{\\ref {c:lambda2}}$ and $n \\ge 16$ we have $\\overline{{\\mathbb {P}}}_\\lambda \\Big [ \\overline{{\\mathbb {P}}}_\\lambda \\big ( \\operatorname{Arm}_p\\big ( \\tfrac{r}{n}, r\\big ) \\big | \\omega \\big ) > n^{-\\gamma } \\text{ for some }p\\in \\big [\\tfrac{\\lambda _0}{2\\lambda _1}, \\tfrac{\\lambda _1}{2\\lambda _1} \\big ]\\Big ] < n^{-100},$ for all sufficiently large $r$ depending only on $n$ .","The proof follows the same structure of that of Lemma REF .","For $r\\ge n\\ge 16$ let $\\ell _i=4^i\\frac{r}{n}$ .","Note that if $\\operatorname{Circ}_p^\\star (\\frac{r}{n},r)$ fails to occur, then $A_i = \\operatorname{Circ}_p^\\star (\\ell _i, 2\\ell _i)$ must also fail for every $i = 0, 1, \\ldots , k - 1$ , where $k = \\lfloor \\frac{1}{2} \\log _4 n \\rfloor $ ($k \\ge 1$ as we assume $n \\ge 16$ ).","By (REF ) we therefore have $\\overline{{\\mathbb {P}}}_\\lambda \\big (\\operatorname{Arm}_p(\\tfrac{r}{n}, r) \\big | \\omega \\big )\\, \\le \\, \\operatorname{{\\bf 1}}_{G^c} (\\omega ) + \\prod _{i = 0}^{k - 1} \\overline{{\\mathbb {P}}}_\\lambda (A_i^c | \\omega ).$ Fix $\\varepsilon >0$ so that $2^k\\varepsilon ^{k/2}\\le \\frac{1}{2}n^{-100}$ .","By Proposition REF there exists $\\delta >0$ and $c_{\\ref {c:lambda}}(\\varepsilon ) > 0$ large enough so that for every $r \\ge 1$ and $m\\ge c_{\\ref {c:lambda}}$ we have $\\overline{{\\mathbb {P}}}_\\lambda \\Big ( \\overline{{\\mathbb {P}}}_\\lambda \\big ( \\operatorname{Circ}_p^\\star (r, 2r) \\big | \\omega \\big ) > \\delta \\text{ for every }p\\in \\big [\\tfrac{\\lambda _0}{2\\lambda _1}, \\tfrac{\\lambda _1}{2\\lambda _1} \\big ]\\Big ) > 1 - \\varepsilon .$ Fix $\\gamma > 0$ (independent of $n$ ) so that $n^{-\\gamma } \\ge (1-\\delta )^{k/2}$ .","By (REF ) we have, on $G$ , that if $\\overline{{\\mathbb {P}}}_\\lambda (A_i|\\omega )>\\delta $ for at least half of the indices $i=0,1,\\ldots ,k-1$ , then $\\overline{{\\mathbb {P}}}_\\lambda \\big (\\operatorname{Arm}_p(\\tfrac{r}{n},r)\\big |\\omega \\big )\\,\\le \\,(1-\\delta )^{k/2}\\,\\le \\, n^{-\\gamma }.$ In particular, via a union bound, $\\overline{{\\mathbb {P}}}_\\lambda \\Big (G\\cap \\big [\\overline{{\\mathbb {P}}}_\\lambda \\big (\\operatorname{Arm}_p(\\tfrac{r}{n},r)\\big |\\omega \\big )>n^{-\\gamma }\\big ]\\Big )\\,\\le \\,2^k\\sup _{I}\\overline{{\\mathbb {P}}}_\\lambda \\Big (G \\cap \\bigcap _{i \\in I}\\big [\\overline{{\\mathbb {P}}}_\\lambda (A_i|\\omega )\\le \\delta \\big ]\\Big ),$ where the supremum is taken over all $I\\subseteq \\lbrace 0,1,\\ldots ,k-1\\rbrace $ of size at least $k/2$ , of which there are at most $2^k$ .","By (REF ) and (REF ), this is bounded by $2^k\\sup _{I}\\prod _{i\\in I}\\Big [\\overline{{\\mathbb {P}}}_\\lambda \\big (\\overline{{\\mathbb {P}}}_\\lambda (A_i|\\omega )\\le \\delta \\big )+\\overline{{\\mathbb {P}}}_\\lambda (G^c)\\Big ]\\,\\le \\,2^k\\big (\\varepsilon +\\overline{{\\mathbb {P}}}_\\lambda (G^c)\\big )^{k/2}.$ For large values of $r$ , depending on $n$ , this probability is at most $n^{-100}$ by the choice of $\\varepsilon >0$ .","We have now gathered all the pieces necessary to prove the main result of this subsection.","[Proof of Proposition REF ] We start by choosing $n$ large enough so that $36 n^{-98} \\le \\eta /2$ .","Then, $\\overline{{\\mathbb {P}}}_\\lambda \\Big ( \\max _{i \\ge 1} \\big ( \\operatorname{Inf}_i^p(f_\\omega ) \\big ) > n^{-\\gamma } \\text{ for some }p \\in \\big [ \\tfrac{\\lambda _0}{2\\lambda _1}, \\tfrac{\\lambda _1}{2\\lambda _1} \\big ]\\Big )$ may via (REF ) be bounded from above by $\\overline{{\\mathbb {P}}}_\\lambda \\big ( G^c \\big ) + \\overline{{\\mathbb {P}}}_\\lambda \\Big (\\overline{{\\mathbb {P}}}_\\lambda \\big ( \\operatorname{Arm}_p^\\ell \\big ( \\tfrac{r}{n}, r\\big ) \\big | \\omega \\big ) > n^{-\\gamma }\\text{ for some }p \\in \\big [ \\tfrac{\\lambda _0}{2\\lambda _1}, \\tfrac{\\lambda _1}{2\\lambda _1} \\big ]\\text{ and }\\ell \\le 36 n^2 \\Big ).$ By Lemmas REF and REF , this is smaller than $\\eta $ by our choice of $n$ , once $r$ is taken large enough."],["Proof of Theorem ","Roughly speaking, in order to prove Theorem REF , we are going to show that for every $\\varepsilon > 0$ $Z_{\\lambda _1/(2\\lambda _1)} - Z_{\\lambda _0/(2\\lambda _1)} \\ge \\frac{\\lambda _1 - \\lambda _0}{\\varepsilon },$ with positive probability.","Since $Z_p(\\omega ) \\in [0, 1]$ almost surely, we must have that $\\lambda _0=\\lambda _1$ .","[Proof of Theorem REF ] We first give estimates on the derivatives of $Z_p$ .","Recall that, for $\\overline{{\\mathbb {P}}}_\\lambda $ almost every $\\overline{\\omega }\\in \\overline{\\Omega }$ , $Z_p$ coincides with the expectation, with respect to the conditional measure $\\overline{{\\mathbb {P}}}_\\lambda (\\,\\cdot \\,|\\omega )$ , of some function $f_\\omega :\\lbrace 0,1\\rbrace ^\\omega \\rightarrow \\lbrace 0,1\\rbrace $ .","Moreover, with probability one, only finitely many points in $\\omega $ will affect the outcome of events defined on $[-4r,4r]^2$ .","Hence, the domain of $f_\\omega $ is for $\\overline{{\\mathbb {P}}}_\\lambda $ -almost every $\\overline{\\omega }$ finite dimensional, and the Margulis-Russo formula gives that $\\frac{d Z_p}{d p} \\,=\\, \\frac{d Z_p (\\omega )}{d p} \\,=\\, \\frac{1}{m}\\sum _{i \\ge 0} \\operatorname{Inf}_i^p (f_\\omega ),$ where $\\operatorname{Inf}_i^p (f_\\omega )$ is as defined in (REF ), and the additional factor $1/m$ comes from the chain rule.","Let us now assume that $\\lambda _0 < \\lambda _1$ and fix $m\\ge c_{\\ref {c:lambda}}(1/2) \\vee c_{\\ref {c:lambda2}}$ .","Combining (REF ) with the mean-value theorem and Talagrand's inequality (REF ) we obtain, for some $q\\in \\big [\\frac{\\lambda _0}{2\\lambda _1},\\frac{\\lambda _1}{2\\lambda _1}\\big ]$ , $\\frac{Z_{\\lambda _1/(2\\lambda _1)} - Z_{\\lambda _0/(2\\lambda _1)}}{\\lambda _1 - \\lambda _0}\\, =\\, \\frac{1}{2\\lambda _1} \\frac{d Z_p(\\omega )}{d p} (q)\\, \\ge \\, \\frac{c}{2 \\lambda _1 m \\log (m)} \\operatorname{Var}_\\lambda (f_\\omega | \\omega ) \\log \\bigg ( \\frac{c}{\\max _i \\big ( \\operatorname{Inf}_i^q (f_\\omega ) \\big ) } \\bigg ).$ From Propositions REF and REF , we conclude that for $r$ large enough (depending only on $\\lambda _1 - \\lambda _0$ ) the right hand side of the above equation is larger than $2/(\\lambda _1 - \\lambda _0)$ with positive probability.","This contradicts the fact that $Z_p \\in [0,1]$ , thus finishing the proof of the theorem."],["Continuity of the critical parameter","We have in the previous section seen that $\\lambda _0=\\lambda _1$ , and thus coincides with $\\lambda _c$ , under the assumption of finite second moment of the radii distribution.","In this section we will investigate the dependence of this critical parameter with respect to the radii distribution $\\mu $ , and write $\\lambda _c=\\lambda _c(\\mu )$ throughout this section in order to emphasize this dependence.","Theorem 6.1 Let $(\\mu _m)_{m\\ge 1}$ be a sequence of radii distributions, all dominated by some distribution $\\nu $ with finite second moment.","If $\\mu _m\\rightarrow \\mu $ weakly, then ${\\lambda _{c}}(\\mu _m)\\rightarrow {\\lambda _{c}}(\\mu )$ .","Remark 13 A weaker version of Theorem REF , proved under the stronger assumption of uniformly bounded support, has been known since long (see [27]).","The argument given here is significantly simpler and shorter, but is in contrast to the aforementioned proof distinctively two-dimensional.","The convergence may fail when the assumption of uniformly bounded tails is relaxed.","For a first example of this, let $(\\mu _m)_{m\\ge 1}$ be a sequence with infinite second moments converging weakly to a point mass at 1.","The critical parameter for each $\\mu _m$ is degenerate (meaning that $\\lambda _0=\\lambda _1=0$ ), and cannot converge to the critical value of the point mass, which is strictly positive.","In the above example the convergence fails for rather obvious reasons.","A second example, which gives further insight to what can go wrong, may be obtained from a cumulative distribution function $F$ of some distribution with finite second moment, when $F_m$ is given by $F_m(x):=(1-\\tfrac{1}{m})F(x)+\\tfrac{1}{m}F(\\tfrac{x}{m}).$ Each $F_m$ has finite second moment, but there is no uniform upper bound on its value.","The dilation of $F$ by a factor $\\frac{1}{m}$ corresponds to a scaling of the radii by $m$ , and thus has an inverse quadratic effect on the critical parameter in that ${\\lambda _{c}}(F(\\frac{x}{m}))=\\frac{1}{m^2}{\\lambda _{c}}(F(x))$ .","So, although ${\\lambda _{c}}(\\mu )>0$ , we have ${\\lambda _{c}}(F_m)\\,\\le \\,{\\lambda _{c}}\\big ((1-\\tfrac{1}{m})\\operatorname{{\\bf 1}}_{\\lbrace x\\ge 0\\rbrace }+\\tfrac{1}{m}F(\\tfrac{x}{m})\\big )\\,=\\,m\\,{\\lambda _{c}}\\big (F(\\tfrac{x}{m})\\big )\\,=\\,\\tfrac{1}{m}{\\lambda _{c}}(F).$ Although $F_m\\rightarrow F$ weakly, the critical parameters diverge.","[Proof of Theorem REF ] Fro the proof, we will write $F$ , $F_m$ and $H$ for the cumulative distribution functions of $\\mu $ , $\\mu _m$ and $\\nu $ .","We first observe that if $F_m\\ge H$ for all $m\\ge 1$ and $F_m\\rightarrow F$ , then also $F\\ge H$ and $F$ has finite second moment too.","To see this assume the contrary, in which case $F(x)F(x^{\\prime })$ , contradicting the assumed convergence.","That $F$ has finite second moment in particular implies that ${\\lambda _{c}}(F)\\in (0,\\infty )$ .","We will base the proof on Propositions REF and REF .","We therefore fix $\\theta >0$ and $r_0 = r_0({\\lambda _{c}}(F) + 1)$ accordingly.","Write ${\\mathbb {P}}_\\lambda ^{F_m}$ for the measure at intensity $\\lambda $ and with radii distributed as $F_m$ .","To complete the proof, it will suffice to show that for every $\\varepsilon \\in (0,1)$ there exists $r\\ge r_0$ and $m_0$ such that for all $m\\ge m_0$ ${\\mathbb {P}}_{{\\lambda _{c}}(F)-\\varepsilon }^{F_m}\\big (\\operatorname{Cross}(r,3r)\\big )<\\theta \\quad \\text{and}\\quad {\\mathbb {P}}_{{\\lambda _{c}}(F)+\\varepsilon }^{F_m}\\big (\\operatorname{Cross}(3r,r)\\big )>1-\\theta .$ Proposition REF then implies that ${\\lambda _{c}}(F_m)\\in [{\\lambda _{c}}(F)-\\varepsilon ,{\\lambda _{c}}(F)+\\varepsilon ]$ for all $m\\ge m_0$ .","Fix $\\varepsilon >0$ .","By Corollary REF and Theorem REF , we may choose $r$ large so that ${\\mathbb {P}}_{{\\lambda _{c}}(F) - \\varepsilon }^F\\big (\\operatorname{Cross}(r, 3r)\\big ) < \\theta /4 \\quad \\text{and} \\quad {\\mathbb {P}}_{{\\lambda _{c}}(F) + \\varepsilon }^F\\big (\\operatorname{Cross}(3r, r)\\big ) > 1 - \\theta /4.$ Next, based on (REF ) of Lemma REF , fix $K\\subseteq {\\mathbb {R}}^2$ such that ${\\mathbb {P}}_{{\\lambda _{c}}(F)+\\varepsilon }^H\\big (\\mathcal {O}_{K^c}\\cap [0,r]^2=\\varnothing \\big )>1-\\theta /4.$ And, choose $\\delta >0$ such that the event in Proposition REF has ${\\mathbb {P}}_{{\\lambda _{c}}(F)\\pm \\varepsilon }^{F}$ -probability at most $\\theta /4$ to occur.","By weak convergence of $F_m$ to $F$ it follows that $F_m^{-1}(U)\\rightarrow F^{-1}(U)$ almost surely, as $m\\rightarrow \\infty $ , if $U$ is uniformly distributed on $[0,1]$ and $F^{-1}$ denotes the generalized inverse of $F$ .","Consequently, as the projection of $\\omega $ onto $K$ , denoted by $\\omega (K)$ , is almost surely finite, we will have $F_m^{-1}(z)\\in (F^{-1}(z)-\\varepsilon ,F^{-1}(z)+\\varepsilon )$ for all $(x,z)\\in \\omega (K)$ for large enough $m$ .","We may, in particular, choose $m_0$ so that for all $m\\ge m_0$ ${\\mathbb {P}}_{{\\lambda _{c}}(F) \\pm \\varepsilon }^F \\Big ( \\operatorname{int}(\\mathcal {O}_K, \\delta ) \\subseteq \\mathcal {O}_K^{F_m} \\subseteq \\operatorname{cl}(\\mathcal {O}_K, \\delta ) \\Big ) > 1 - \\theta /4.$ Combining the above estimates we conclude that (REF ) holds, as required."],["Proof of main results","In this section we complete the proofs of the three main theorems stated in the introduction.","We first prove Theorem REF , and then deduce Theorems REF and REF .","Before starting we recall the content of Corollary REF .","Assuming (REF ), we have $0<\\lambda _0\\le \\lambda _c\\le \\lambda _1<\\infty ,$ and the analogous relation holds for $\\lambda _c$ replaced by $\\lambda _c^\\star $ under the stronger condition (REF )."],["Proof of Theorem ","For any $\\lambda >\\lambda _1$ we have, by definition, that $\\lim _{r\\rightarrow \\infty }{\\mathbb {P}}_\\lambda (\\operatorname{Cross}(3r,r))=1$ .","The standard inequalities of Lemma REF and monotonicity imply that for any $\\lambda >\\lambda _1$ and $\\kappa >0$ , $\\lim _{r\\rightarrow \\infty }{\\mathbb {P}}_\\lambda (\\operatorname{Cross}(\\kappa r,r))=1.$ Similarly, for any $\\lambda <\\lambda _0$ and $\\kappa >0$ the limit in (REF ) equals 0.","Since $\\lambda _c\\in [\\lambda _0,\\lambda _1]$ , then Corollary REF shows that for any $\\kappa >0$ there exists a constant $c=c(\\kappa )>0$ such that $c<{\\mathbb {P}}_{\\lambda _c}(\\operatorname{Cross}(\\kappa r,r))<1-c\\quad \\text{for all }r\\ge 1.$ Moreover, by Corollary REF , there is at $\\lambda _c$ almost surely no unbounded component of either kind.","Finally, the proof is completed by Theorem REF , which shows that indeed $\\lambda _c=\\lambda _0=\\lambda _1$ ."],["Proof of Theorem ","Assume (REF ), in which case we by now know that $\\lambda _c^\\star =\\lambda _0=\\lambda _1=\\lambda _c$ .","Part (i) is then a consequence of Corollary REF .","The proof of part (ii) will involve a truncation of the radii distribution, allowing for a comparison with a highly supercritical 1-dependent percolation process on ${\\mathbb {Z}}^2$ , for which a standard Peierls argument gives the required exponential decay.","Fix $\\lambda >{\\lambda _{c}^\\star }$ .","For $m\\ge 1$ , let $\\mu _m$ denote the truncation of $\\mu $ at $m$ , meaning that $\\mu _m$ coincides with $\\mu $ on $[0,m)$ but that $\\mu _m([0,m])=1$ .","We shall write $\\lambda ^\\star (\\mu )$ for the critical parameter associated with radii distribution $\\mu $ .","By continuity of the critical parameter, Theorem REF , we have for large $m$ that ${\\lambda _{c}^\\star }(\\mu _m)\\in [{\\lambda _{c}^\\star }(\\mu ),\\lambda ),$ and assume for the rest of this proof that $m$ is chosen accordingly.","Given $\\gamma >0$ , we also assume that $r>2m$ is fixed so that ${\\mathbb {P}}_\\lambda ^{\\mu _m}(\\operatorname{Cross}(3r,r))>1-\\gamma .$ Now, tile the plane with overlapping $3r\\times r$ and $r\\times 3r$ rectangles as follows: For each $z\\in {\\mathbb {Z}}^2$ consider both a $3r\\times r$ and a $r\\times 3r$ rectangle positioned so their lower left corners coincide with $2rz$ .","We want to register which of these rectangles that are crossed in the hard direction by an occupied path.","Define thus, for each $z\\in {\\mathbb {Z}}^2$ , $\\eta (z,z + (1,0))$ to be the indicator of the event $\\operatorname{Cross}(3r,r)$ translated by the vector $2rz$ , and $\\eta (z,z + (0, 1))$ as the indicator of $\\operatorname{Cross}(r,3r)$ translated by the same vector.","By the choice of $m$ and $r$ , the process $\\eta $ is a 1-dependent bond percolation process on ${\\mathbb {Z}}^2$ with marginal edge probability exceeding $1 - \\gamma $ .","Therefore, by a standard Peierls argument, there is a constant $c_{\\ref {c:exp_decay}}>0$ such that the event that $\\eta $ contains an open circuit which is contained in $B^\\infty (n)$ and surrounds the origin has probability at least $1-c_{\\ref {c:exp_decay}}\\exp \\lbrace n/c_{\\ref {c:exp_decay}}\\rbrace $ .","However, by construction, an open path in $\\eta $ corresponds to an occupied crossing in $\\mathcal {O}$ , and thus that for all $n\\ge 1$ $\\mathbb {P}_\\lambda ^{\\mu _m}\\big [ 0\\overset{ \\mathcal {V}}{\\longleftrightarrow }\\partial B^\\infty (rn) \\big ] \\le c_{\\ref {c:exp_decay}}\\exp \\lbrace -n/c_{\\ref {c:exp_decay}}\\rbrace ,$ from which part (ii) of the theorem follows."],["Proof of Theorem ","Under assumption () Item (i) is a consequence of part (iii) of Corollary REF (see also Remark REF ).","So, it remains to argue for Item (ii).","However, if () holds, then by Proposition REF there exists $c > 0$ such that for all $\\lambda \\le \\lambda _c$ and $r\\ge 1$ $\\rho _\\lambda (5r,r)\\le cr^{-\\alpha }.$ Now, fix $\\lambda <\\lambda _c$ , so that $\\lim _{r\\rightarrow \\infty }{\\mathbb {P}}_\\lambda (\\operatorname{Cross}^\\star (3r,r))=1$ .","Proposition REF then gives that ${\\mathbb {P}}_\\lambda (\\operatorname{Cross}^\\star (3r,r))\\ge 1 - c^{\\prime }r^{-\\alpha }$ for all $r\\ge 1$ and some constant $c^{\\prime }$ , possibly depending on $\\lambda $ .","Consequently, for all $r\\ge 1$ , ${\\mathbb {P}}_\\lambda \\big [0 \\overset{\\tiny \\mathcal {O}}{\\longleftrightarrow }\\partial B(r)\\big ]\\,\\le \\,4\\,{\\mathbb {P}}_\\lambda (\\operatorname{Cross}(r,3r))\\,\\le \\,4c^{\\prime }r^{-\\alpha },$ as required.","This ends the proof of Theorem REF ."],["Other models","We have in Sections - provided a framework to prove the existence of a sharp phase transition and some interesting properties of the critical behavior for Poisson Boolean percolation.","In this section we will show how our techniques can be used to obtain alternative proofs for the facts that the critical probability equals $1/2$ for Poisson Voronoi percolation, earlier proved by Bollobás and Riordan [9], and Poisson confetti percolation, first proved by Hirsch [21] (for unit squares) and later Müller [30] (for unit discs).","Our results will in fact apply to some generalized versions of these models that are not self-dual.","See Figure REF for a simulation of these processes.","The finite-size criterion of Section and the RSW techniques of Section solely rely on three basic facts: That the probability measure in question is invariant (with respect to translations, right angle rotations and reflections in coordinate axis), positively associated, and features a decay of spatial correlations.","However, the argument used to prove the sharp threshold behavior in Section used the further assumption of a Poissonian structure.","All of these properties are satisfied by Poisson Voronoi and confetti percolation.","We shall below illustrate what our techniques give for these models, but emphasize that we make no attempt to sharpen the hypothesises in the results we present.","Figure: Simulations of Poisson confetti (left) and Voronoi (right) percolation.In the left picture, black confetti have random radii, while the radii of white ones are fixed.The Voronoi picture to the right uses G 0 =1G_0 = 1, G 1 =2G_1 = 2 and q=0.8q = 0.8.Note that the Voronoi cells above are determined by segments of ellipses."],["Poisson Voronoi percolation","The study of random Voronoi tessellations goes back several decades in time, yet it was only about a decade ago that Bollobàs and Riordan [9] gave the first proof for the fact that the critical probability in Poisson Voronoi percolation in the plane equals $1/2$ ; see also [3].","One of the main difficulties faced in studying the phase transition is to derive RSW techniques that apply in this setting.","In [9] a weak RSW result was provided, whereas a strong version of RSW was established later by Tassion [42], and was used to prove polynomial decay of the one-arm probability at criticality.","The RSW techniques developed here allows us to consider more general variants of the standard Voronoi model, where black and white cells do not necessarily follow the same law.","Informally speaking, to each point we associate a random `gravitational pull', which gives a bias to the size of the associated tile.","The symmetry between black and white points may be broken by considering different laws for the gravitational pull associated to them.","Consider a Poisson point process of unit density on the space of locally finite counting measures on ${\\mathbb {R}}^2 \\times [0, 1] \\times [0, 1]$ , endowed with Lebesgue measure.","The first two coordinates of a point in the support of a realization $\\omega $ will mark the location of the seed in the plane, the third and fourth coordinates will determine the gravitational pull of a given point and its color, respectively.","More precisely, given a parameter $q \\in [0, 1]$ , we say that the seed $(x, z, s) \\in \\operatorname{supp}(\\omega )$ is black if $s \\le q$ and white otherwise.","We also fix two non-decreasing functions $G_0, G_1: [0,1] \\rightarrow (0, \\infty )$ , which will be applied to the $z$ coordinate of $(x, z, s)$ to determine the gravitational pull of a given seed ($G_0$ will be used for black seeds and $G_1$ for white).","Given $q \\in [0, 1]$ , define the occupied (black) set as $\\mathcal {O}_q := \\bigcup _{\\begin{array}{c}(x, z, s) \\in \\omega ;\\\\s \\le q\\end{array}} \\Big \\lbrace y \\in {\\mathbb {R}}^2 : \\frac{|y - x|}{G_0(z)} \\le \\frac{|y - x^{\\prime }|}{G_1(z^{\\prime })} \\text{ for all $(x^{\\prime }, z^{\\prime }, s^{\\prime }) \\in \\omega $ with $s^{\\prime } > q$} \\Big \\rbrace .$ As before, let $\\mathcal {V}_q := \\mathbb {R}^2 \\setminus \\mathcal {O}_q$ denote the corresponding vacant set.","Notice that the cell corresponding to a given seed may be disconnected.","Finally, we define the critical values of the parameter $q$ as $\\begin{aligned}q_c:=&\\sup \\big \\lbrace q\\in [0,1]:{\\mathbb {P}}\\big [0\\overset{\\tiny \\mathcal {O}_q}{\\longleftrightarrow }\\infty \\big ]=0\\big \\rbrace ,\\\\q_c^\\star :=&\\inf \\big \\lbrace q\\in [0,1]:{\\mathbb {P}}\\big [0\\overset{\\tiny \\mathcal {V}_q}{\\longleftrightarrow }\\infty \\big ]=0\\big \\rbrace .\\end{aligned}$ We will here only consider the case when $G_0$ and $G_1$ take values in some interval $[a, b] \\subset (0, \\infty )$ .","It would be interesting to attempt to relax these assumptions.","When $G_0 = G_1$ , then the gravitational pull of all black and white seeds are equally distributed.","In this case the model is self-dual, and we recover the equation $q_c+q_c^\\star =1$ ; see Section REF below.","The techniques developed in this paper may easily be adapted to prove the following result for Poisson Voronoi percolation.","Theorem 8.1 Assume that $G_0$ and $G_1$ take values in some interval $[a,b]\\subset (0,\\infty )$ .","Then, $q_c=q_c^\\star $ and their common value is strictly between zero and one.","Moreover, for all $q0$ such that ${\\mathbb {P}}\\big [ 0\\overset{\\tiny \\mathcal {O}_q}{\\longleftrightarrow }\\partial B(r) \\big ] \\le \\exp \\lbrace -c r\\rbrace .$ for $q=q_c$ and all $\\kappa >0$ , there exist $c=c(\\kappa )>0$ and $\\alpha >0$ such that for every $r \\ge 1$ $c < {\\mathbb {P}}(\\operatorname{Cross}(\\kappa r,r)) < 1 - c\\quad \\text{and}\\quad \\mathbb {P} \\big [ 0 \\overset{\\tiny \\mathcal {O}_q}{\\longleftrightarrow }\\partial B(r) \\big ] \\le r^{-\\alpha }.$ for all $q>q_c$ , there exists $c=c(q)>0$ such that ${\\mathbb {P}}\\big [ 0\\overset{\\tiny \\mathcal {V}_q}{\\longleftrightarrow }\\partial B(r) \\big ] \\le \\exp \\lbrace -c r\\rbrace .$ Moreover, at $q_c$ there is almost surely no unbounded cluster of either kind.","We will for the remainder of this subsection outline the proof of this result, based on the techniques developed over the previous sections.","First of all, we note that the law of the above model is invariant with respect to translations, rotations and reflections.","The measure is further positively associated, meaning that is satisfies an FKG inequality similar to (REF ), for events that are increasing in the addition of black points and removal of white points.","This can be seen by adapting the proof of [10] for the case when $G_0=G_1$ are constant.","What remains in order for the techniques of Sections and to apply is an estimate on the spatial correlations.","Lemma 8.2 Fix a bounded measurable set $D \\subseteq \\mathbb {R}^2$ and an arbitrary function $f(\\mathcal {O}_q)$ satisfying $f \\in \\sigma (Y_v; v \\in D)$ .","Then, defining the event $G_{D, r} = \\Big \\lbrace \\text{for every $y \\in D$, there is a point $(x, z, s)$ in $\\omega $ with $d(y, x) \\le \\frac{a}{b} r$} \\Big \\rbrace ,$ we have that $\\operatorname{{\\bf 1}}_{G_{D, r}} \\cdot f$ only depends on the restriction of $\\omega $ to $B(D, r) \\times [0, 1] \\times [0, 1]$ .","Moreover, for every $\\varepsilon >0$ there exists a constant $c=c(\\varepsilon )>0$ such that for any $D\\subseteq B^\\infty (r)$ we have $\\mathbb {P}(G_{D, \\varepsilon r}) \\ge 1 - \\exp \\lbrace - c r^2 \\rbrace $ .","The proof of the first claim follows the same steps as those of Lemma 1.1 in [42].","Roughly speaking, the argument goes like this.","Given $x \\in \\mathbb {R}^2$ , if a seed in $\\omega $ is located within distance $\\ell $ from $x$ , then no other seed outside $B(x, (b/a)\\ell )$ can influence the state of $x$ .","The second claim follows from a simple large deviations bound for Poisson random variables.","As a first consequence of this fact we obtain an estimate on the decay of spatial correlations.","As in Definition REF , let $f_1, f_2:\\mathcal {P}({\\mathbb {R}}^2)\\rightarrow [-1,1]$ be two functions satisfying $f_1(\\mathcal {O}_q)\\in \\sigma \\big (Y_v;v\\in B^\\infty (r)\\big )$ and $f_2\\in \\sigma \\big (Y_v;v\\in B^\\infty (r+t)\\setminus B^\\infty (r+s)\\big )$ , for some $t>s$ .","Then, if $G=G_{B^\\infty (r),s/2}\\cap G_{B^\\infty (r+t)\\setminus B^\\infty (r+s),s/2}$ , we obtain $\\begin{split}\\mathbb {E}[f_1 f_2] \\,& \\le \\, \\mathbb {P}(G^c) + \\mathbb {E}[\\operatorname{{\\bf 1}}_G \\cdot f_1 f_2]\\\\&\\le \\,2\\mathbb {P}(G^c)+\\operatorname{{\\mathbb {E}}}[f_1^{\\prime }]\\operatorname{{\\mathbb {E}}}[f_2^{\\prime }]\\\\&\\le \\,6{\\mathbb {P}}(G^c)+\\operatorname{{\\mathbb {E}}}[f_1]\\operatorname{{\\mathbb {E}}}[f_2],\\end{split}$ where $f_1^{\\prime }$ and $f_2^{\\prime }$ denote $f_1$ and $f_2$ evaluated at the restriction of $\\omega $ to $B^\\infty (r+s/2)$ and $B^\\infty (r+t)\\setminus B^\\infty (r+s/2)$ and are hence independent.","In particular, when $s=\\varepsilon r$ and $t=\\frac{1}{\\varepsilon }r$ , which has been the case throughout this paper, the correlation between $f_1$ and $f_2$ decays as $\\exp \\lbrace - c r^2\\rbrace $ as in Lemma REF .","Consequently, the techniques of Sections and apply, and we obtain the existence of $q_0$ and $q_1$ , defined analogously as $\\lambda _0$ and $\\lambda _1$ , satisfying $0q_1$ .","To complete the proof of Theorem REF , it remains to prove that $q_0=q_1$ .","In order to show that $q_0=q_1$ we again enlarge our probability space as in Section REF , to enable a two-stage construction of our process.","We thus assume that $\\overline{\\omega }$ is a Poisson point process on the space of locally finite counting measures on $\\big ({\\mathbb {R}}^2\\times [0,1]\\times [0,1]\\big )\\times [0,1]$ .","The law of $\\overline{\\omega }$ (denoted $\\overline{\\mathbb {P}}_\\lambda $ ) is chosen with a large density $\\lambda =m\\ge 1$ and the fifth coordinate will be used to thin the process down to density one, which we then use to define $\\mathcal {O}_q$ as in (REF ).","We denote by $\\omega $ the projection of $\\overline{\\omega }$ onto $\\mathbb {R}^2 \\times [0, 1] \\times [0, 1]$ (the first four coordinates) and by $\\underline{\\omega }$ the further projection onto $\\mathbb {R}^2 \\times [0, 1]$ .","Define $W_q:=\\overline{{\\mathbb {P}}}_\\lambda (\\operatorname{Cross}(4r,r)|\\underline{\\omega }).$ As in Section REF , we note that $W_q$ for $\\overline{{\\mathbb {P}}}_\\lambda $ -almost every $\\overline{\\omega }$ coincides with the expectation of some Boolean function $f_{\\underline{\\omega }}:\\lbrace 0,1\\rbrace ^{\\underline{\\omega }}\\times \\lbrace 0,1\\rbrace ^{\\underline{\\omega }}\\rightarrow \\lbrace 0,1\\rbrace $ evaluated with respect to product measure with density $q$ and $1/m$ for choosing `color' and `presence', respectively, of each point in $\\underline{\\omega }$ .","This function is increasing in the choice of color, so we obtain via the Margulis-Russo formula that $\\frac{dW_q}{dq}=\\sum _{i}\\overline{{\\mathbb {P}}}_\\lambda \\big ( \\,\\text{point $i$ is present and its color is pivotal for }f_{\\underline{\\omega }}\\, \\big | \\,\\underline{\\omega }\\, \\big ).$ Note that the concept of pivotality arising here is with respect to color and not presence as in Section .","The crucial observation is that is that switching the color of a point has a larger potential change than switching its presence.","That is, for the existence of a black crossing, a black point is better than no point, and no point is better than a white point.","Hence, from (REF ), we obtain that $\\begin{aligned}\\frac{dW_q}{dq} \\,&=\\, \\overline{\\operatorname{{\\mathbb {E}}}}_\\lambda \\Big [ \\sum _i \\overline{{\\mathbb {P}}}_\\lambda \\big ( \\, \\text{point $i$ is present and its color is pivotal} \\, \\big | \\, \\omega \\, \\big ) \\Big | \\, \\underline{\\omega }\\, \\Big ]\\\\&\\ge \\,\\overline{\\operatorname{{\\mathbb {E}}}}_\\lambda \\Big [ \\sum _i\\overline{{\\mathbb {P}}}_\\lambda \\big ( \\, \\text{point $i$ is present and its presence is pivotal} \\, \\big | \\,\\omega \\, \\big ) \\Big |\\,\\underline{\\omega }\\,\\Big ]\\\\&=\\, \\frac{1}{m} \\overline{\\operatorname{{\\mathbb {E}}}}_\\lambda \\Big [ \\sum _i \\operatorname{Inf}^q_i(f_{\\underline{\\omega }}) \\Big |\\,\\underline{\\omega }\\,\\Big ]\\\\&\\ge \\, \\frac{1}{m} \\overline{\\operatorname{{\\mathbb {E}}}}_\\lambda \\Big [ \\frac{c}{\\log (m)} \\operatorname{Var}_\\lambda (f_{\\underline{\\omega }}|\\omega )\\log \\frac{1}{\\max _i \\big ( \\operatorname{Inf}_i^q(f_{\\underline{\\omega }}) \\big ) } \\,\\Big |\\,\\underline{\\omega }\\,\\Big ],\\end{aligned}$ where in the last step we applied (REF ), and $\\operatorname{Inf}_i^q(f_{\\underline{\\omega }})$ denotes the conditional probability, given $\\omega $ , that changing point $i$ from present to absent changes the outcome of $f_{\\underline{\\omega }}$ .","Controlling the variance $\\operatorname{Var}_\\lambda (f_{\\underline{\\omega }}|\\omega )$ can now be done exactly as in Section REF .","Also the influences $\\operatorname{Inf}_i^q(f_{\\underline{\\omega }})$ with respect to presence of a point can be estimated similarly as it was done in Section REF , but requires a slight modification of the decoupling inequalities (REF ) and (REF ).","Let $D_1,D_2,\\ldots , D_J$ be well separated sets as in (REF ), and let $A_1,A_2,\\ldots ,A_J$ be events where $A_j$ is determined by the restriction of $\\mathcal {O}_q$ to $D_j$ .","On the event $G={\\textstyle \\bigcap \\limits }_{j\\le J}G_{D_j,r/(2n)}$ the events $A_1,A_2,\\ldots ,A_J$ are, by Lemma REF , determined by restrictions of $\\overline{\\omega }$ to disjoint subsets of the plane, and are hence independent.","We obtain, therefore, the following variant of (REF ) for the Voronoi model $\\overline{{\\mathbb {P}}}_\\lambda \\big (A_1\\cap \\cdots \\cap A_J\\big |\\omega \\big )\\,\\le \\,\\overline{{\\mathbb {P}}}_\\lambda (G^c|\\omega )+\\prod _{j\\le J}\\Big [\\overline{{\\mathbb {P}}}_\\lambda (A_j|\\omega )+\\overline{{\\mathbb {P}}}_\\lambda (G^c|\\omega )\\Big ].$ Since $\\overline{{\\mathbb {P}}}_\\lambda (G^c)\\rightarrow 0$ fast, also $\\overline{{\\mathbb {P}}}_\\lambda (G^c|\\omega )\\rightarrow 0$ fast with probability tending to one.","This suffices for the application of (REF ) in the proof of Lemma REF .","In order to obtain a variant for (REF ) we observe that the difference between $\\overline{{\\mathbb {P}}}_\\lambda (A_j|\\omega )$ and the conditional probability (given $\\omega $ ) that $A_j$ occurs with respect to the restriction of $\\overline{\\omega }$ restricted to the set $B(D_j,r/(2n))$ is bounded by $\\overline{{\\mathbb {P}}}_\\lambda (G^c|\\omega )$ .","Let $H=[\\overline{{\\mathbb {P}}}_\\lambda (G^c|\\omega )\\le \\delta ]$ .","Hence, arguing similarly as for (REF ), we obtain that $\\overline{{\\mathbb {P}}}_\\lambda \\bigg (H \\cap \\bigcap _{j\\le J}\\big [\\overline{{\\mathbb {P}}}_\\lambda (A_j|\\omega )\\le \\delta \\big ]\\bigg )\\,\\le \\,\\prod _{j\\le J}\\Big [\\overline{{\\mathbb {P}}}_\\lambda \\big (\\overline{{\\mathbb {P}}}_\\lambda (A_j|\\omega )\\le 3\\delta \\big )+\\overline{{\\mathbb {P}}}_\\lambda (H^c)\\Big ].$ Since $\\overline{{\\mathbb {P}}}_\\lambda (H^c)\\rightarrow 0$ fast, as a consequence of Lemma REF , the two expressions (REF ) and (REF ) replace (REF ) and (REF ) in the proof of Proposition REF .","The proof that $q_0=q_1$ is now a straightforward adaptation of the arguments presented in Section , and this ends the outline of the proof of Theorem REF ."],["Poisson confetti percolation","Confetti percolation, or the `dead leaves' model, was introduced by Jeulin in [22].","In this model, black and white confetti `rain down' on the plane according to a Poisson point process, and each point in the plane is colored according to the color of the first confetti to cover it.","In the case of circular confetti with fixed diameter, Benjamini and Schramm [11] conjectured that the critical probability for this model equals $1/2$ .","This was later confirmed by Hirsch [21] for square shaped confetti and by Müller [30] for circular confetti.","Just as in the settings of Poisson Boolean and Voronoi percolation, our techniques allow us to handle confetti of random radii, whose laws may differ between black and white.","Consider a Poisson point process on the space of locally finite counting measures on ${\\mathbb {R}}^2 \\times {\\mathbb {R}}_+ \\times [0, 1]^2$ .","Here again the first two coordinates will assign a location in ${\\mathbb {R}}^2$ to each confetti, while the third coordinate will denote the fall-time of a confetti and will be used to order overlapping confetti.","Finally, the fourth and fifth coordinates will help us determine the radius and color of the confetti, respectively.","As before, we fix $q \\in [0, 1]$ and, given a realization $\\omega $ with density one, declare a point $(x, t, z, s) \\in \\omega $ black if $s \\le q$ and white otherwise.","We also fix non-decreasing functions $G_0, G_1 : [0, 1] \\rightarrow (0, \\infty )$ that will be applied to the $z$ coordinate, in order to determine the radius of the confetti to be placed at $x\\in {\\mathbb {R}}^2$ ($G_0$ for black points and $G_1$ for white).","More precisely, given a realization $\\omega $ of the Poisson point process, let $\\mathcal {O}_q := \\bigcup _{\\begin{array}{c}(x, t, z, s) \\in \\omega \\\\s \\le q\\end{array}} \\left\\lbrace y \\in {\\mathbb {R}}^2:\\begin{split}& |y - x| \\le G_0(z), \\text{ and } t < t^{\\prime } \\text{ for all $(x^{\\prime }, t^{\\prime }, z^{\\prime }, s^{\\prime }) \\in \\omega $}\\\\& \\text{such that $|y - x^{\\prime }| \\le G_1(z^{\\prime })$ and $s^{\\prime } > q$}\\end{split}\\right\\rbrace .$ Finally, let $\\mathcal {V}_q := \\mathbb {R}^2 \\setminus \\mathcal {O}_q$ and define the critical parameters $q_c$ and $q_c^\\star $ as in (REF ).","For the sake of simplicity, we assume here rather light tails for distribution of the radii induced by $G_0$ and $G_1$ .","If these two functions are the same, then black and white confetti have the same radii distributions, which yields $q_c+q_c^\\star =1$ by self-duality, and together with the following theorem that $q_c=1/2$ ; see Section REF below.","Theorem 8.3 Assume that $G_0$ and $G_1$ satisfy ${\\mathbb {P}}[G_i(Z) \\ge r] \\le r^{-100}$ , for $i = 0, 1$ and large $r$ , where $Z$ is uniformly distributed on $[0,1]$ .","Then, $q_c=q_c^\\star $ are strictly between zero and one, and for all $q 0$ such that ${\\mathbb {P}}\\big [ 0\\overset{\\tiny \\mathcal {O}_q}{\\longleftrightarrow }\\partial B(r) \\big ] \\le \\frac{1}{c}r^{-10}.$ for $q=q_c$ and all $\\kappa >0$ , there exist $c=c(\\kappa )>0$ and $\\alpha >0$ such that, for every $r \\ge 1$ $c<{\\mathbb {P}}(\\operatorname{Cross}(\\kappa r,r))<1-c\\quad \\text{and}\\quad \\mathbb {P}\\big [ 0\\overset{\\tiny \\mathcal {O}_q}{\\longleftrightarrow }\\partial B(r) \\big ] \\le r^{-\\alpha }.$ for all $q>q_c$ , there exists $c=c(q)>0$ such that ${\\mathbb {P}}\\big [ 0\\overset{\\tiny \\mathcal {V}_q}{\\longleftrightarrow }\\partial B(r) \\big ] \\le \\frac{1}{c}r^{-10}.$ Moreover, at $q_c$ there is almost surely no unbounded cluster of either kind.","The proof of this theorem is analogous to that of Theorem REF above.","The only distinction lies in the treatment of the spatial dependence.","The following lemma will replace Lemma REF .","Lemma 8.4 Fix a bounded measurable set $D \\subseteq \\mathbb {R}^2$ and $r \\ge 1$ , $M \\ge 1$ and define $G_{D, r, M} = \\left\\lbrace \\begin{array}{c}\\text{for every $(x, t, z, s) \\in \\omega \\cap B(D, Mr)$ with $t \\le r$, $G_0(z) \\vee G_1(z) \\le r/M$}\\\\\\text{for every $(x,t,z,s) \\in \\omega \\cap B(D,Mr)^c$ with $t\\le r$, $G_0(z) \\vee G_1(z) \\le d(x,D)$}\\\\\\text{for every $y \\in D$, there is $(x, t, z, s) \\in \\omega $ with $t \\le r$, $G_0(z) \\wedge G_1(z) \\ge |y - z|$}\\end{array}\\right\\rbrace .$ Then, $\\lim _{r \\rightarrow \\infty } r^{10} \\mathbb {P}(G_{D, r, M}) = 1$ and for an arbitrary function $f(\\mathcal {O}_q) \\in \\sigma (Y_x; x \\in D)$ , we have that $\\operatorname{{\\bf 1}}_G \\cdot f$ is measurable with respect to $\\omega $ restricted to $B(D, r/M) \\times [0, r] \\times [0, 1] \\times [0, 1]$ .","The proof of this lemma follows from a simple large deviations bound and is omitted.","As a consequence of the lemma, spatial correlations will decay as $r^{-10}$ .","This rate of decay is more than enough to follow the outlined proof of Theorem REF and obtain a proof also for Theorem REF ."],["The consequence of self-duality","For the two models considered in this section, the case when the two functions $G_0$ and $G_1$ are equal stands out due to the self-duality retained in the model.","Self-duality refers to the property that the occupied set at parameter $q$ has the same law as the vacant set has at parameter $1-q$ .","In particular, there is no unbounded occupied component at parameter $q$ if and only if there is no unbounded vacant component at parameter $1-q$ (almost surely).","As the critical parameters are defined as the supremum for which this holds it follows that $q_c+q_c^\\star =1$ , in analogy to the Bernoulli case in (REF ).","Together with the equality $q_c=q_c^\\star $ of the two parameters, obtained in Theorems REF and REF , it follows that $q_c=q_c^\\star =1/2$ , and we recover the results of [9], [21] and [30].","Remark 14 There are other Poisson percolation processes that could in principle be studied with the techniques we have employed.","These include for instance Brownian interlacements (see [40]) in ${\\mathbb {R}}^d$ intersected with a plane, and Poissonian cylinders (see [43]) in ${\\mathbb {R}}^d$ intersected with a plane.","A rather general setting for Poisson Boolean percolation but with dependence between the radii assigned to different points has been considered by Ahlberg and Tykesson [5], however that study does not address the sharpness of the threshold.","It would be interesting to investigate to what extent the techniques developed here applies in that setting."],["Open problems","We have in this paper developed techniques for the study of the sharpness of the threshold for a wide range of models.","The weakest link in this scheme is the Poissonian assumption necessary to deduce that the critical regime is indeed a single point and not an interval.","It would be interesting to develop an alternative argument for the existence of a sharp threshold beyond the Poissonian setting considered here.","It would further be interesting to pursue optimal conditions to study Poisson Voronoi and confetti percolation.","Here we only aimed at illustrating how our techniques may be applied to other settings, but we have not pursued to make these applications optimal.","Note also that the techniques we have used do not apply to Poisson Boolean percolation models where the discs are replaced by some other deterministic shape which is not sufficiently symmetric.","Besides extending the current techniques to other models, one could be interested in answering several questions that are known for Bernoulli percolation.","Examples of such problems would be to define the incipient infinite cluster, study noise sensitivity for $\\operatorname{Cross}(r, r)$ and investigate scaling relations, just to name a few."]],"string":"[\n [\n \"Sharpness of the phase transition for continuum percolation in R^2\"\n ],\n [\n \"Abstract We study the phase transition of random radii Poisson Boolean percolation: Around each point of a planar Poisson point process, we draw a disc of random radius, independently for each point.\",\n \"The behavior of this process is well understood when the radii are uniformly bounded from above.\",\n \"In this article, we investigate this process for unbounded (and possibly heavy tailed) radii distributions.\",\n \"Under mild assumptions on the radius distribution, we show that both the vacant and occupied sets undergo a phase transition at the same critical parameter $\\\\lambda_c$.\",\n \"Moreover, - For $\\\\lambda < \\\\lambda_c$, the vacant set has a unique unbounded connected component and we give precise bounds on the one-arm probability for the occupied set, depending on the radius distribution.\",\n \"- At criticality, we establish the box-crossing property, implying that no unbounded component can be found, neither in the occupied nor the vacant sets.\",\n \"We provide a polynomial decay for the probability of the one-arm events, under sharp conditions on the distribution of the radius.\",\n \"- For $\\\\lambda > \\\\lambda_c$, the occupied set has a unique unbounded component and we prove that the one-arm probability for the vacant decays exponentially fast.\",\n \"The techniques we develop in this article can be applied to other models such as the Poisson Voronoi and confetti percolation.\"\n ],\n [\n \"Introduction\",\n \"Percolation is the branch of probability theory focused on the study of the geometry and connectivity properties of random media.\",\n \"Since its foundation in the 1950s, with the work of Broadbent and Hammersley [8], the area reached new heights over the decades to come, specially in two dimensions: During the 1980s following Kesten's determination of the critical threshold [23], and at the turn of the century with Schramm's introduction of Schramm-Loewner evolution [37] and Smirnov's proof of Cardy's formula [38].\",\n \"This progress has been well documented in a range of books on the subject; see for instance [24], [18], [10] and [19].\",\n \"Bernoulli percolation on a symmetric planar lattice, e.g.\",\n \"on the square or triangular lattices, is a cornerstone within percolation theory.\",\n \"Loved for its simple yet challenging structure, Bernoulli percolation has become quintessential in the study of phase transitions and other phenomena emanating from statistical mechanics and mathematical physics.\",\n \"In the site-percolation version of this model, each vertex of the lattice is independently declared `open' with probability $p$ and `closed' with probability $1-p$ .\",\n \"A random graph is obtained from the initial lattice by removing the closed vertices and the connected components of this graph are called clusters.\",\n \"As the parameter increases the model undergoes a sharp phase transition.\",\n \"More precisely, in this planar case, it is well-known that there exists a critical parameter $p_c$ , strictly in between zero and one, such that for $pp_c$ , there exists a unique infinite open cluster.\",\n \"The phase transition is said to be `sharp' because of the abrupt change in the decay of the connection probabilities, from exponentially small in the subcritical regime to uniformly bounded in supercritical.\",\n \"Proofs of the above properties can be found in [18] and [10].\",\n \"In this paper we investigate the sharpness of the phase transition for Poisson Boolean percolation in ${\\\\mathbb {R}}^2$ , establishing results analogous to the ones described above for Bernoulli percolation.\",\n \"Poisson Boolean percolation is an archetype for percolation in the continuum, and shares many features of Bernoulli percolation while posing significant additional challenges.\",\n \"Apart from being continuous rather than discrete, these challenges come from its asymmetrical nature (the `open' and `closed' set have different properties) and long-range dependencies.\",\n \"One particular strength of the present paper is its generality, and its structure has been oriented with this in mind.\",\n \"Although our results are presented for Poisson Boolean percolation, they extend to many other percolation processes in ${\\\\mathbb {R}}^2$ , such as Poisson Voronoi and confetti percolation, as will be described below.\",\n \"Together with an accompanying paper [6], our results give a precise description of the phase transition for Poisson Boolean percolation.\",\n \"There, very specific properties of Poisson Boolean percolation will be exploited, as opposed to the robust methods developed here.\",\n \"An important ingredient in establishing properties (i)–(iii) for Bernoulli percolation is what we call the dual process.\",\n \"It can be defined by looking at the closed vertices on a modified graph, called the matching or dual graph.\",\n \"This dual graph has a critical parameter $p_c^\\\\star $ which was proved to satisfy the duality relation $p_c + p_c^\\\\star = 1.$ This relation is especially useful in the study of self-similar processes, for which the dual process at $p$ coincides with the primal process at $1-p$ .\",\n \"In this case it can be linked with the equality $p_c = p_c^\\\\star $ , yielding $p_c = 1/2$ , see more examples in Section .\",\n \"Kesten's [23] original proof of the sharpness of the phase transition for Bernoulli percolation is based on the analysis of the crossing probabilities for rectangles; a rectangle is said to be crossed if there is a path from left to right, made out of open vertices.\",\n \"The proof involves three essential ingredients: Finite-size criterion – if for some $n$ , the probability to cross a $n$ by $3n$ rectangle in the short direction is smaller than some small constant $\\\\theta >0$ , then the two-point connection probability decays exponentially fast.\",\n \"On the other side, if for some $n$ , the probability to cross a $3n$ by $n$ rectangle in the long direction is larger than $1-\\\\theta >0$ , then there exists an infinite cluster almost surely, see [35], [24] and [2].\",\n \"Russo-Seymour-Welsh theory – relates crossing probabilities for rectangle with different aspect ratios, see [34] and [39].\",\n \"Threshold phenomenon – given a positive constant $c > 0$ and a large rectangle $K$ , there is only a small interval of parameters $p$ for which the crossing probability of $K$ remains between $c$ and $1 - c$ this interval is called the critical window.\",\n \"The first proof of this sharp threshold is due to Kesten in [23] and is based on a geometric interpretation of the derivative of the crossing probability for a rectangle.\",\n \"A better understanding for threshold phenomena has since been obtained in work like [36], [25], [41] and elsewhere.\",\n \"The strategy described above is inherently two-dimensional.\",\n \"As such, the finite-size criterion extends well beyond the product structure of the Bernoulli percolation measure, and require only a minimal assumption on the decay of long-range dependencies.\",\n \"The original Russo-Seymour-Welsh techniques and sharp threshold results rely in a much stronger sense on the product structure of the Bernoulli measure, and do not extend easily to more general percolation models.\",\n \"These will thus be the two foremost challenges in for the present study, in which we study the sharpness of the phase transition for continuum percolation in ${\\\\mathbb {R}}^2$ , which may present arbitrarily slow decay of dependencies.\"\n ],\n [\n \"The phase transition for Poisson Boolean percolation\",\n \"Poisson Boolean percolation was introduced by Gilbert in [15].\",\n \"In this model we start with a Poisson point process on $\\\\mathbb {R}^2$ with intensity parameter $\\\\lambda > 0$ .\",\n \"We then independently associate to each of these points a disc of a randomly chosen radius, according to a fixed probability measure $\\\\mu $ on ${\\\\mathbb {R}}_+$ .\",\n \"The set $\\\\mathcal {O} \\\\subseteq \\\\mathbb {R}^2$ of points which are covered by at least one of the above discs is called the occupied set, while its complement $\\\\mathcal {V}:={\\\\mathbb {R}}^2\\\\setminus \\\\mathcal {O}$ is referred to as the vacant set.\",\n \"We denote by ${\\\\mathbb {P}}_\\\\lambda $ the measure associated with this construction, and defer a more formal definition to Section .\",\n \"Denote by $[0 \\\\overset{\\\\tiny \\\\mathcal {O}}{\\\\longleftrightarrow }\\\\partial B_r]$ the event that there exists an occupied path from 0 to distance $r$ .\",\n \"We write $[0 \\\\overset{\\\\tiny \\\\mathcal {O}}{\\\\longleftrightarrow }\\\\infty ]$ if $[0 \\\\overset{\\\\tiny \\\\mathcal {O}}{\\\\longleftrightarrow }\\\\partial B_r]$ holds for every $r \\\\ge 1$ .\",\n \"Analogously we define $[0 \\\\overset{ \\\\mathcal {V}}{\\\\longleftrightarrow }\\\\partial B_r]$ and $[0 \\\\overset{ \\\\mathcal {V}}{\\\\longleftrightarrow }\\\\infty ]$ for the corresponding events for the vacant set.\",\n \"Finally, define the critical parameters $\\\\begin{array}{ll}{\\\\lambda _{c}}:= \\\\sup \\\\big \\\\lbrace \\\\lambda \\\\ge 0 \\\\; : \\\\; \\\\mathbb {P}_\\\\lambda \\\\big [0 \\\\overset{\\\\tiny \\\\mathcal {O}}{\\\\longleftrightarrow }\\\\infty \\\\big ] = 0 \\\\big \\\\rbrace ,&\\\\\\\\{\\\\lambda _{c}^\\\\star }:= \\\\inf \\\\big \\\\lbrace \\\\lambda \\\\ge 0 \\\\; : \\\\; \\\\mathbb {P}_\\\\lambda \\\\big [0 \\\\overset{ \\\\mathcal {V}}{\\\\longleftrightarrow }\\\\infty \\\\big ] = 0 \\\\big \\\\rbrace .&\\\\end{array}$ We aim to describe the percolative properties of the model at and around these critical values.\",\n \"That $\\\\lambda _c$ and $\\\\lambda _c^\\\\star $ are finite may be obtained via a comparison with Bernoulli percolation on ${\\\\mathbb {Z}}^2$ , where a standard Peierls argument may be used.\",\n \"To show, on the other hand, that $\\\\lambda _c$ and $\\\\lambda _c^\\\\star $ are strictly positive is a different matter, and requires a condition on the radii distribution in order to be true.\",\n \"The most fundamental condition in the study of Poisson Boolean percolation in ${\\\\mathbb {R}}^2$ is that of finite second moment on the radius distribution: $\\\\int _0^\\\\infty x^2 \\\\mu (\\\\mathop {}\\\\!\\\\mathrm {d}x) < \\\\infty .$ It was observed by Hall [20] that (REF ) is necessary in order to avoid the entire plane to be almost surely covered, regardless of the intensity (as long as positive) of the Poisson point process.\",\n \"Gouéré [16] further showed that this condition is also sufficient for $\\\\lambda _c$ to be strictly positive.\",\n \"Lower bounds on $\\\\lambda _c^\\\\star $ may be obtained from their comparison with $\\\\lambda _c$ .\",\n \"Under the assumption of bounded support on $\\\\mu $ it is known that $\\\\lambda _c^\\\\star =\\\\lambda _c$ ; see [32].\",\n \"At the critical point $\\\\lambda _c$ it is further known that there is almost surely no unbounded occupied component in the case of $\\\\mu $ having bounded support [32], whereas the analogous statement for an unbounded vacant component is only known to hold in the case of unit radii, see [4].\",\n \"Our goal with the present and forthcoming paper [6] is to extend these results to hold under the condition (REF ).\",\n \"We will in this paper make no further assumption of the above results, as they will be easy consequences of the techniques we develop.\",\n \"We merely acknowledge the observation that (REF ) is necessary in order for the model to present non-trivial behavior.\",\n \"Following Kesten's lead, we shall in this paper focus on the study of crossings of rectangles and build a theory around them.\",\n \"Quantitative estimates on the rate of decay of connection probabilities, in the different regimes, will be obtained as consequences of this.\",\n \"The first result we state will thus relate crossing probabilities to the critical parameter $\\\\lambda _c$ .\",\n \"Let us first define, for every $r, h \\\\ge 1$ , the event $\\\\operatorname{Cross}(r, h)$ that there exists an occupied path inside the rectangle $[0, r] \\\\times [0, h]$ from the left side to the right side.\",\n \"That is, $\\\\operatorname{Cross}(r, h) = \\\\hbox{\\\\;[scale=1.5]{(0, 0) rectangle (6ex, 4ex);[solid] (0, 2ex) .. controls (2.6ex, 3ex) and (3.3ex, 1ex) .. (6ex, 2ex);(3ex, .2ex) node[anchor=north] {$\\\\scriptstyle r$};(5.7ex, 2ex) node[anchor=west] {$\\\\scriptstyle h$};}}$ .\",\n \"We also write $\\\\operatorname{Cross}^\\\\star (r, h)$ for the existence of a vacant crossing of the same box.\",\n \"The results of this paper are based on the following theorem.\",\n \"Theorem 1.1 Assume the second moment condition (REF ).\",\n \"Then $\\\\lambda _c$ is strictly between zero and infinity, and for all $\\\\lambda > {\\\\lambda _{c}}$ and all $\\\\kappa > 0$ , we have $\\\\lim _{r \\\\rightarrow \\\\infty } \\\\mathbb {P}_\\\\lambda \\\\big [ \\\\operatorname{Cross}(\\\\kappa r, r) \\\\big ] = 1.$ for $\\\\lambda = {\\\\lambda _{c}}$ and all $\\\\kappa > 0$ , there exists a constant $c_{\\\\ref {c:thm_sharp_cross}} = c_{\\\\ref {c:thm_sharp_cross}}(\\\\kappa )>0$ such that $c_{\\\\ref {c:thm_sharp_cross}} < \\\\mathbb {P}_{{\\\\lambda _{c}}} \\\\big [ \\\\operatorname{Cross}(\\\\kappa r, r) \\\\big ] < 1 - c_{\\\\ref {c:thm_sharp_cross}}, \\\\text{ for every $r \\\\ge 1$.\",\n \"}$ for all $\\\\lambda < {\\\\lambda _{c}}$ and all $\\\\kappa > 0$ , $\\\\lim _{r \\\\rightarrow \\\\infty } \\\\mathbb {P}_\\\\lambda \\\\big [\\\\operatorname{Cross}(r,\\\\kappa r)\\\\big ] = 0.$ Moreover, at $\\\\lambda _c$ there is almost surely no unbounded cluster of either kind.\",\n \"Remark 1 Our proof gives quantitative bounds for the rate of convergence in parts (i) and (iii).\",\n \"The next two theorems are intended to illustrate what can be obtained using the techniques of the present work.\",\n \"They give an overview of the percolative behavior of the vacant and occupied sets in Poisson Boolean percolation.\",\n \"Several of these characteristics resemble the known features of Bernoulli percolation, and these two theorems are general: Similar theorems can be proved also for other percolation processes using the methods of this paper (see Section ).\",\n \"However, their general nature calls for a slightly stronger moment condition than the one in (REF ).\",\n \"These two alternative hypotheses are $\\\\int _0^\\\\infty x^2\\\\log x\\\\, \\\\mu (\\\\mathop {}\\\\!\\\\mathrm {d}x) < \\\\infty , \\\\text{ and}\\\\\\\\\\\\text{for some $\\\\alpha > 0$,} \\\\quad \\\\int _0^\\\\infty x^{2 + \\\\alpha } \\\\,\\\\mu (\\\\mathop {}\\\\!\\\\mathrm {d}x) < \\\\infty .$ The three hypotheses (REF ), (REF ) and () will be useful because they imply some decorrelation inequalities discussed in Section REF .\",\n \"The natural second moment assumption (REF ) already implies some spatial decorrelation properties, and the hypotheses (REF ) and () imply quantitative bounds on the spatial decorrelation.\",\n \"The hypothesis (REF ) will be sufficient for most of the paper, but the hypotheses (REF ) and () will be useful to apply some renormalization methods presented in Section .\",\n \"For the vacant set we have the following.\",\n \"Theorem 1.2 Assume the $2+\\\\log $ moment condition (REF ).\",\n \"Then $\\\\lambda _c^\\\\star =\\\\lambda _c$ and is thus strictly between zero and infinity.\",\n \"Moreover, for $\\\\lambda = {\\\\lambda _{c}^\\\\star }$ there exists a constant $c_{\\\\ref {c:poly_decay}} > 0$ such that $\\\\mathbb {P}_{\\\\lambda _c^\\\\star }\\\\big [ 0 \\\\overset{ \\\\mathcal {V}}{\\\\longleftrightarrow }\\\\partial B_r \\\\big ] \\\\le \\\\frac{1}{c_{\\\\ref {c:poly_decay}}} r^{-c_{\\\\ref {c:poly_decay}}}.$ for all $\\\\lambda > {\\\\lambda _{c}^\\\\star }$ there exists a constant $c_{\\\\ref {c:exp_decay}} = c_{\\\\ref {c:exp_decay}}(\\\\lambda ) > 0$ such that $\\\\mathbb {P}_\\\\lambda \\\\big [ 0 \\\\overset{ \\\\mathcal {V}}{\\\\longleftrightarrow }\\\\partial B_r \\\\big ] \\\\le \\\\frac{1}{c_{\\\\ref {c:exp_decay}}}\\\\exp \\\\lbrace -c_{\\\\ref {c:exp_decay}}r\\\\rbrace .$ Remark 2 Condition (REF ) in Theorem REF is not sharp.\",\n \"In the forthcoming paper [6], we show that the second moment condition suffices to prove all of the above.\",\n \"The fact that ${\\\\lambda _{c}^\\\\star }= {\\\\lambda _{c}}$ is analogous to $p_c + p_c^\\\\star = 1$ for Bernoulli percolation.\",\n \"There are important distinctions between the vacant and the occupied sets in regard to their percolation properties.\",\n \"An important difference comes from the fact that parts (REF ) and (REF ) of Theorem REF do not hold in general for the occupied set, since the existence of long occupied connections can be triggered by a single disk of large radius.\",\n \"For example, if one chooses the radius distribution $\\\\mu $ such that $\\\\mu [x, \\\\infty ) = x^{-2}(\\\\log x)^{-2}$ for $x$ large enough, then the hypothesis (REF ) is satisfied but for every $\\\\lambda > 0$ and every $r$ large enough ${\\\\mathbb {P}}_\\\\lambda \\\\big [0 \\\\overset{\\\\tiny \\\\mathcal {O}}{\\\\longleftrightarrow }\\\\partial B_r\\\\big ] \\\\,\\\\ge \\\\, c \\\\cdot \\\\lambda \\\\int _r^\\\\infty x \\\\,\\\\mu [x,\\\\infty )\\\\mathop {}\\\\!\\\\mathrm {d}x \\\\,\\\\ge \\\\, c \\\\cdot \\\\lambda /\\\\log r.$ Therefore, in this case the one-arm probability ${\\\\mathbb {P}}_\\\\lambda [0 \\\\overset{\\\\tiny \\\\mathcal {O}}{\\\\longleftrightarrow }\\\\partial B_r]$ cannot decay polynomially fast at criticality, just because this choice of $\\\\mu $ implies that the probability that 0 is contained in a ball of radius larger than $r$ does not decay polynomially.\",\n \"Nevertheless, under the stronger moment assumption (), one can prove the polynomial decay of the arm exponent as shown in the next result.\",\n \"Theorem 1.3 Assume that the $2+\\\\alpha $ moment condition () holds for some $\\\\alpha >0$ .\",\n \"Then, for $\\\\lambda = {\\\\lambda _{c}}$ , there exists a constant $c_{\\\\ref {c:poly_o_decay}} = c_{\\\\ref {c:poly_o_decay}}(\\\\alpha ) > 0$ such that $\\\\mathbb {P}_{\\\\lambda _c}\\\\big [ 0 \\\\overset{\\\\tiny \\\\mathcal {O}}{\\\\longleftrightarrow }\\\\partial B_r \\\\big ] \\\\le \\\\frac{1}{c_{\\\\ref {c:poly_o_decay}}} r^{-c_{\\\\ref {c:poly_o_decay}}}.$ for all $\\\\lambda < {\\\\lambda _{c}}$ , there exists a constant $c_{\\\\ref {c:poly_decay}}=c_{\\\\ref {c:poly_decay}}(\\\\lambda ) > 0$ such that $\\\\mathbb {P}_\\\\lambda \\\\big [ 0 \\\\overset{\\\\tiny \\\\mathcal {O}}{\\\\longleftrightarrow }\\\\partial B_r \\\\big ] \\\\le c_{\\\\ref {c:poly_decay}} r^{-\\\\alpha }.$ Remark 3 The exponent in part (ii) of Theorem REF cannot be improved in general.\",\n \"To see this, consider the radii distribution for which $\\\\mu [x,\\\\infty )=x^{-(2+\\\\alpha )}(\\\\log x)^{-2}$ for large $x$ .\",\n \"This distribution satisfies (), but ${\\\\mathbb {P}}_\\\\lambda \\\\big [ 0 \\\\overset{\\\\tiny \\\\mathcal {O}}{\\\\longleftrightarrow }\\\\partial B_r \\\\big ]\\\\,\\\\ge \\\\,\\\\lambda r^2\\\\mu [2r,\\\\infty )\\\\,=\\\\,\\\\lambda (2r)^{-\\\\alpha }(\\\\log 2r)^{-2}$ for large $r$ .\",\n \"We will now describe the main steps in the proof of the above results, which roughly speaking correspond to the three steps described for the case of Bernoulli percolation above.\",\n \"These are finite-size criterion, Russo-Seymour-Welsh theory and a threshold phenomenon.\",\n \"We describe each of these steps below and emphasize that their corresponding proofs can be read independently of one another.\"\n ],\n [\n \"Finite-size criterion\",\n \"In Section we prove that the crossing probabilities converge to 1 as soon as they become close enough to 1.\",\n \"Roughly speaking we prove that there exists $\\\\theta >0$ and $r_0(\\\\lambda )$ such that the following are equivalent $\\\\mathbb {P}_\\\\lambda \\\\big [\\\\operatorname{Cross}(3r, r)\\\\big ] > 1 - \\\\theta \\\\text{ for some $r\\\\ge r_0$,}\\\\\\\\\\\\lim _{r \\\\rightarrow \\\\infty } \\\\mathbb {P}_\\\\lambda \\\\big [\\\\operatorname{Cross}(3r, r)\\\\big ] = 1,$ and the analogous statement holds for vacant crossings, see Proposition REF .\",\n \"This motivates us to introduce $\\\\lambda _0 := \\\\sup \\\\big \\\\lbrace \\\\lambda \\\\ge 0 : \\\\lim _{r\\\\rightarrow \\\\infty } {\\\\mathbb {P}}_\\\\lambda \\\\big [ \\\\operatorname{Cross}(r, 3r) \\\\big ] = 0\\\\big \\\\rbrace ,\\\\\\\\\\\\lambda _1 := \\\\inf \\\\big \\\\lbrace \\\\lambda \\\\ge 0 : \\\\lim _{r\\\\rightarrow \\\\infty } {\\\\mathbb {P}}_\\\\lambda \\\\big [ \\\\operatorname{Cross}(3r, r) \\\\big ] = 1\\\\big \\\\rbrace .$ The first important consequence of the finite-size criterion is that $0\\\\,<\\\\,\\\\lambda _0\\\\,\\\\le \\\\,\\\\lambda _c\\\\,\\\\le \\\\, \\\\lambda _1\\\\, < \\\\,\\\\infty .$ These results are proved in Section .\",\n \"Up to now, it is not clear that $\\\\lambda _0$ is actually equal to $\\\\lambda _1$ .\",\n \"We call $[\\\\lambda _0, \\\\lambda _1]$ the critical regime and we can see that, for every $\\\\lambda \\\\in [\\\\lambda _0, \\\\lambda _1]$ , $\\\\inf _{r\\\\ge 1} {\\\\mathbb {P}}_\\\\lambda \\\\big [ \\\\operatorname{Cross}(r, 3r) \\\\big ] > 0 \\\\quad \\\\text{and} \\\\quad \\\\inf _{r\\\\ge 1} {\\\\mathbb {P}}_\\\\lambda \\\\big [ \\\\operatorname{Cross}^\\\\star (r, 3r) \\\\big ] > 0.$ These statements say that in the critical regime, rectangles have non-degenerate probabilities of being crossed by both the vacant and the occupied set.\",\n \"However, one should notice that these crossings happen in the easy direction of the rectangles.\",\n \"In order to obtain a more precise description of the behavior in the critical regime we require similar statements regarding crossing probabilities in the long direction.\",\n \"This is precisely the purpose of Russo-Seymour-Welsh theory, as we describe next.\"\n ],\n [\n \"Russo-Seymour-Welsh theory\",\n \"In this step we show that (REF ) implies that $\\\\inf _{r\\\\ge 1} {\\\\mathbb {P}}_\\\\lambda \\\\big [ \\\\operatorname{Cross}(3 r, r) \\\\big ] > 0 \\\\quad \\\\text{and} \\\\quad \\\\inf _{r\\\\ge 1} {\\\\mathbb {P}}_\\\\lambda \\\\big [ \\\\operatorname{Cross}^\\\\star (3 r, r) \\\\big ] > 0.$ Note that the only difference between the above and (REF ) is that now the rectangles are crossed in the long direction.\",\n \"This `box-crossing' property for the critical regime rules out the existence of unbounded clusters and provides bounds on arm probabilities.\",\n \"These results are proved in Section .\",\n \"Remark 4 RSW bounds were obtained separately for the occupied and vacant regimes during the 1990s by Roy [32] and Alexander [4].\",\n \"However, these proofs are technical, and pose pose heavy restrictions on the radii distribution.\",\n \"Recently, Tassion [42] found an argument allowing for greater generality.\",\n \"His argument, presented in the setting of Voronoi percolation but extends verbatim, shows that $ \\\\liminf _{r\\\\rightarrow \\\\infty }{\\\\mathbb {P}}_\\\\lambda \\\\big [\\\\operatorname{Cross}(r,r)\\\\big ]>0\\\\quad \\\\Rightarrow \\\\quad \\\\liminf _{r\\\\rightarrow \\\\infty }{\\\\mathbb {P}}_\\\\lambda \\\\big [\\\\operatorname{Cross}(3r,r)\\\\big ]>0.$ While this is all that is needed in `symmetric' percolation models, such as Poisson Voronoi percolation, it is indispensable for `non-symmetric' models, which we consider here, that crossings of rectangles the easy way imply crossings also the hard way.\",\n \"This extension is not straightforward.\"\n ],\n [\n \"Sharp thresholds\",\n \"The final step of the proof is to show that $\\\\lambda _0 = \\\\lambda _1$ .\",\n \"The main strategy here is to show that the occupied crossing probabilities grow very fast from zero to one as we increase the density $\\\\lambda $ of the system.\",\n \"There is a solid theory that predicts the occurrence of threshold phenomena in the setting of Boolean function on the discrete cube $\\\\lbrace 0, 1\\\\rbrace ^n$ equipped with a product structure.\",\n \"Poisson point processes is the natural analogue to the discrete Bernoulli measure.\",\n \"It is therefore natural to believe that these techniques should carry over also to the Poissonian continuum setting.\",\n \"This is indeed the case, although doing so is not straightforward task.\",\n \"We will here follow an approach introduced in [1].\",\n \"This part of the argument involves the analysis of Boolean functions and is the least general part of our argument, since it involves representing the crossing probabilities in our Poisson point process as a certain function of discrete Bernoulli random variables.\",\n \"This construction and the proof that $\\\\lambda _0 = \\\\lambda _1$ can be found in Section .\"\n ],\n [\n \"Other examples\",\n \"We have chosen Poisson Boolean percolation as the main example to illustrate the techniques of this work.\",\n \"However, the techniques that we develop apply in greater generality.\",\n \"We emphasize this fact in Section , where we apply these techniques to Poisson Voronoi and confetti percolation.\",\n \"We direct the reader to that section for a statement of the results we obtain for these models, and give here just a short description of the scope of our techniques.\",\n \"The finite-size criterion described above only uses the fact that the law of our percolation process is invariant under translations and right-angle rotations, see (REF ), and that it satisfies a mixing condition, see (REF ).\",\n \"The Russo-Seymour-Welsh part of our argument, we solely use the invariance of $\\\\mathcal {O}$ and $\\\\mathcal {V}$ under translations, right-angle rotations and reflections in coordinate axes, see (REF ), the mixing property (REF ), the FKG inequality (REF ) and a certain continuity property of the crossing probabilities stated in Lemma REF .\",\n \"The threshold argument is the least general and uses, in addition to the assumptions made above, the fact that process is based on a Poisson point process.\"\n ],\n [\n \"Some previous works on the model\",\n \"The reference book [27] provides a general exposition of continuum percolation.\",\n \"The case of uniformly bounded radius distribution has been extensively studied in [44], [45], [29], [32], [28], [4].\",\n \"These works have already established much of our results in this restricted setting.\",\n \"In Lemma A.2 of [17], it is proved that for sufficiently small $\\\\lambda > 0$ , $\\\\mathbb {P}_\\\\lambda [B_r \\\\leftrightarrow B_{2r}^c]$ goes to zero with $r$ under the second moment condition (REF ).\",\n \"This implies that $\\\\lambda _0 > 0$ under the second moment condition.\",\n \"In [33], Roy studied a Poisson soup of bounded sticks in the plane, proving a Russo-Seymour-Welsh theorem and establishing that ${\\\\lambda _{c}^\\\\star }= {\\\\lambda _{c}}$ .\",\n \"Uniqueness of the unbounded components (for both vacant and occupied regions) has been established in [26].\"\n ],\n [\n \"Organization of the paper\",\n \"In Section we provide some notation and preliminary results that are needed throughout the text.\",\n \"Sections , and present the three steps of the proof that were described in the introduction, namely: the finite-size criterion, Russo-Seymour-Welsh result and the sharp threshold step.\",\n \"These sections can be read independently of one another.\",\n \"Section proves the continuity of the critical parameter with respect to the law $\\\\mu $ of the radius distribution.\",\n \"The main results of the paper are then proved in Section .\",\n \"We treat other models and provide some open questions in Sections and .\",\n \"Acknowledgements This work began during a visit of V.T.\",\n \"to IMPA, that he thanks for support and hospitality.\",\n \"We thank the Centre Intradisciplinaire Bernoulli (CIB) and Stardû for hosting the authors.\",\n \"D.A.\",\n \"was during the course of this project financed by grant 637-2013-7302 from the Swedish Research Council.\",\n \"A.T. is grateful to CNPq for its financial contribution to this work through the grants 306348/2012-8, 478577/2012-5 and 309356/2015-6 and FAPERJ through grant number 202.231/2015.\",\n \"V.T.\",\n \"acknowledges support from the Swiss NSF.\"\n ],\n [\n \"Notation and preliminary results\",\n \"Throughout the text we let $c$ denote positive constants which may depend on the radius distribution and may change from line to line.\",\n \"However, numbered constants such as $c_0, c_1, \\\\dots $ refer to their first appearance in the text.\",\n \"We will further write $B^\\\\infty (r) = [-r, r]^2$ for the ball centered at the origin in the supremum norm and let $B(x,r)$ , on the other hand, denote the closed Euclidean ball with center $x$ and radius $r$ .\",\n \"When $x$ is the origin we omit it from the above notation.\",\n \"Rather informally, a realization of Poisson Boolean percolation is obtained by decorating the points in a Poisson point process in ${\\\\mathbb {R}}^2$ by Euclidean discs with independent radii sampled from some distribution.\",\n \"There are various (equivalent) ways of making this description formal.\",\n \"We will start this section by describing one way which will be suitable for our purposes.\"\n ],\n [\n \"Definition of the process\",\n \"To define our process, let us first introduce a Poisson point process on the following space of point measures $\\\\Omega = \\\\Big \\\\lbrace \\\\omega = \\\\sum _i \\\\delta _{(x_i,z_i)}: (x_i,z_i) \\\\in \\\\mathbb {R}^2 \\\\times \\\\mathbb {R}_+ \\\\text{ and } \\\\omega \\\\big ( K \\\\times \\\\mathbb {R}_+ \\\\big ) < \\\\infty \\\\text{ for all $K$ compact} \\\\Big \\\\rbrace .$ We endow this space with the $\\\\sigma $ -algebra $\\\\mathcal {M}$ generated by the evaluation maps $A \\\\mapsto \\\\omega (A)$ , for $A \\\\in \\\\mathcal {B}(\\\\mathbb {R}^2 \\\\times \\\\mathbb {R}_+)$ , the Borel sets on ${\\\\mathbb {R}}^2 \\\\times \\\\mathbb {R}_+$ .\",\n \"We next fix an intensity parameter $\\\\lambda \\\\ge 0$ and some probability measure $\\\\mu $ on $\\\\mathbb {R}_+$ which will give the radius distribution of our discs.\",\n \"We can now define on $(\\\\Omega , \\\\mathcal {M})$ a Poisson point process with intensity $\\\\lambda \\\\cdot dx \\\\, \\\\mu (dz)$ , i.e., Lebesgue measure on $\\\\mathbb {R}^2$ product with $\\\\mu $ and multiplied by $\\\\lambda \\\\ge 0$ .\",\n \"The law of this process is denoted by $\\\\mathbb {P}_\\\\lambda $ throughout the text and we complete the $\\\\sigma $ -algebra $\\\\mathcal {M}$ with respect to $\\\\mathbb {P}_\\\\lambda $ .\",\n \"For each point $(x,z) \\\\in \\\\mathbb {R}^2 \\\\times \\\\mathbb {R}_+$ in the support of the measure $\\\\omega \\\\in \\\\Omega $ , we associate the disc $B(x, z)$ and the occupied region of the plane is consequently given by $\\\\mathcal {O} := \\\\bigcup _{(x,z) \\\\in \\\\operatorname{supp}(\\\\omega )} B(x, z),$ while the vacant set is given by its complement $\\\\mathcal {V} := \\\\mathbb {R}^2 \\\\setminus \\\\mathcal {O}$ .\",\n \"(Below, we will often identify $\\\\omega $ with its support in order to ease the notation.)\",\n \"From the definition of the model, it is trivial to conclude that the law of $\\\\mathcal {O}$ is invariant under translations, right-angle rotations, and reflection in coordinate axes.\",\n \"Although the law is invariant under arbitrary rotations, and reflections in axes with other orientation, we will only use the above statement throughout the text.\",\n \"There is a natural partial ordering of elements in $\\\\Omega $ , namely, $\\\\omega \\\\le \\\\omega ^{\\\\prime }$ if $\\\\omega (A) \\\\le \\\\omega ^{\\\\prime }(A)$ for all $A \\\\in \\\\mathcal {B}(\\\\mathbb {R}^2 \\\\times \\\\mathbb {R}_+)$ .\",\n \"An event $A\\\\in \\\\mathcal {M}$ is said to be increasing if $\\\\omega \\\\in A$ implies that $\\\\omega ^{\\\\prime }\\\\in A$ for all $\\\\omega \\\\le \\\\omega ^{\\\\prime }$ .\",\n \"It is decreasing if its complement is increasing.\",\n \"A useful property of increasing events is that they are positively correlated.\",\n \"The following proposition, known as the FKG inequality, was proved by Roy in his doctorate thesis; see also [27]: If $A_1$ and $A_2$ are increasing events, then ${\\\\mathbb {P}}_\\\\lambda (A_1\\\\cap A_2)\\\\,\\\\ge \\\\,{\\\\mathbb {P}}_\\\\lambda (A_1){\\\\mathbb {P}}_\\\\lambda (A_2).$ The above also holds when $A_1$ and $A_2$ are decreasing events.\",\n \"We will also use the following standard consequence of the FKG inequality, referred to as the square-root trick.\",\n \"Let $A_1,\\\\ldots ,A_k$ be $k$ increasing events (or $k$ decreasing events), then $\\\\max _{1\\\\le i \\\\le k}{\\\\mathbb {P}}_\\\\lambda (A_i)\\\\,\\\\ge \\\\, 1-\\\\big [1-{\\\\mathbb {P}}_\\\\lambda (A_1\\\\cup \\\\ldots \\\\cup A_k)\\\\big ]^{1/k}.$\"\n ],\n [\n \"Crossing events\",\n \"Throughout the text we will often deal with crossing events of various types.\",\n \"In particular, we will be interested in the following general definition.\",\n \"Given subsets $A_1, A_2$ and $\\\\mathcal {C}$ of $\\\\mathbb {R}^2$ let $A_1 \\\\overset{\\\\mathcal {C}}{\\\\longleftrightarrow }A_2 := \\\\text{there is a path in $\\\\mathcal {C}$ connecting $A_1$ to $A_2$}.$ We are now in position to give a formal definition of the crossing event $\\\\operatorname{Cross}(r,h)$ , for $r, h > 0$ , as $\\\\operatorname{Cross}(r, h) := \\\\big [ A_1 {\\\\overset{\\\\mathcal {O} \\\\cap K}{\\\\longleftrightarrow }} A_2 \\\\big ],$ where $K$ denotes the box $[0,r] \\\\times [0,h]$ and $A_1 = \\\\lbrace 0\\\\rbrace \\\\times [0,h]$ , $A_2 = \\\\lbrace r\\\\rbrace \\\\times [0,h]$ stand for its left and right sides.\",\n \"We also define the event corresponding to a vacant crossing as $\\\\operatorname{Cross}^\\\\star (r,h) := [A_1 {\\\\overset{\\\\mathcal {V} \\\\cap K}{\\\\longleftrightarrow }} A_2 ]$ .\",\n \"It is rather straightforward to verify that events of this type are measurable.\",\n \"Notice that occupied crossing events are increasing, and vacant crossing events are decreasing.\"\n ],\n [\n \"Decay of spatial correlations\",\n \"The second moment condition given in (REF ) is sufficient to imply a spatial decorrelation, in the sense of the function $\\\\rho _\\\\lambda $ introduced in Definition REF below.\",\n \"First set $Y_v = \\\\operatorname{{\\\\bf 1}}_{\\\\lbrace v \\\\in \\\\mathcal {V}\\\\rbrace }$ for $v \\\\in \\\\mathbb {R}^2$ .\",\n \"Definition 2.1 Given $0 < r, s < \\\\infty $ , let $\\\\rho _\\\\lambda (r, s) := \\\\sup _{f_1, f_2} \\\\operatorname{Cov}\\\\big (f_1(\\\\mathcal {O}), f_2(\\\\mathcal {O})\\\\big ),$ where the above suppremum is taken over all functions $f_1,f_2:\\\\mathcal {P}(\\\\mathbb {R}^2)\\\\rightarrow [-1,1] $ such that $f_1(\\\\mathcal {O}) \\\\in \\\\sigma (Y_v; v\\\\in B^\\\\infty (r))$ and $f_2(\\\\mathcal {O}) \\\\in \\\\sigma (Y_v; v \\\\notin B^\\\\infty (r+s))$ .\",\n \"The function $\\\\rho _\\\\lambda $ has a nice geometric interpretation.\",\n \"Namely, one can observe that $\\\\rho _\\\\lambda (r, r+s)$ is directly related to the probability that there exists one big occupied disk crossing the annulus $B^\\\\infty (r+s)\\\\setminus B^\\\\infty (r)$ .\",\n \"By making this observation rigorous, and computing this crossing probability (see Lemma REF ), we prove the following upper bound.\",\n \"Proposition 2.2 For any $\\\\lambda > 0$ and $r, s \\\\ge 1$ , we have $\\\\rho _\\\\lambda (r, s)\\\\, \\\\le \\\\, c_{\\\\ref {c:decorrelation}} \\\\lambda \\\\left(1+\\\\frac{r}{s}\\\\right)^2\\\\int _{s/2}^\\\\infty x^2\\\\mu (\\\\mathop {}\\\\!\\\\mathrm {d}x).$ In particular, the second moment condition (REF ) implies that $\\\\text{for any $\\\\kappa > 0$, }\\\\lim _{r\\\\rightarrow \\\\infty } \\\\rho _\\\\lambda (r, \\\\kappa r) = 0.$ and the $2 + \\\\alpha $ moment condition in () implies that for every $\\\\lambda > 0$ and $\\\\varepsilon >0$ $\\\\sup \\\\limits _{r\\\\ge 1, \\\\kappa \\\\ge \\\\varepsilon } r^\\\\alpha \\\\rho _\\\\lambda (r, \\\\kappa r) < \\\\infty .$ For $K \\\\subseteq \\\\mathbb {R}^2$ , we write $\\\\mathcal {O}_K = \\\\bigcup _{(x,z) \\\\in \\\\omega :\\\\, x \\\\in K} B(x, z).$ The next lemma says that deleting balls from $\\\\mathcal {O}$ which are far from a given box do not alter the configuration of $\\\\mathcal {O}$ inside that box.\",\n \"Figure: An illustration of the sets 𝒪\\\\mathcal {O}, 𝒪 B ∞ (r+s/2) \\\\mathcal {O}_{B^\\\\infty (r + s/2)} and 𝒪 B ∞ (r+s/2) c \\\\mathcal {O}_{B^\\\\infty (r + s/2)^c} respectively.\",\n \"Note that in this particular realization we have 𝒪 B ∞ (r+s/2) ∩B ∞ (r)=𝒪∩B ∞ (r)\\\\mathcal {O}_{B^\\\\infty (r + s/2)} \\\\cap B^\\\\infty (r) = \\\\mathcal {O} \\\\cap B^\\\\infty (r) and also that 𝒪 B ∞ (r+s/2) c ∩B ∞ (r+s) c =𝒪∩B ∞ (r+s) c \\\\mathcal {O}_{B^\\\\infty (r + s/2)^c} \\\\cap B^\\\\infty (r + s)^c = \\\\mathcal {O} \\\\cap B^\\\\infty (r + s)^c as in Lemma .Lemma 2.3 For all $r, s > 0$ , writing $K_1 = B^\\\\infty (r)$ and $K_2 = B^\\\\infty (r + s)$ , we have $\\\\mathbb {P}_\\\\lambda \\\\big ( \\\\mathcal {O}_{K_2^c} \\\\cap K_1 \\\\ne \\\\varnothing \\\\big )\\\\le 8\\\\lambda \\\\left(1+\\\\frac{r+1}{s}\\\\right)^2\\\\int _s^\\\\infty x^2\\\\mu (\\\\mathop {}\\\\!\\\\mathrm {d}x).$ Also, $\\\\mathbb {P}_\\\\lambda \\\\big ( \\\\mathcal {O}_{K_1} \\\\cap K_2^c \\\\ne \\\\varnothing \\\\big )\\\\le 8\\\\lambda \\\\left(\\\\frac{r}{s}\\\\right)^2\\\\int _s^\\\\infty x^2\\\\mu (\\\\mathop {}\\\\!\\\\mathrm {d}x).$ Remark 5 The above lemma implies in particular that, whenever $\\\\int _0^\\\\infty x^2\\\\mu (\\\\mathop {}\\\\!\\\\mathrm {d}x) < \\\\infty $ , if $K \\\\subseteq \\\\mathbb {R}^2$ is compact, then $\\\\mathbb {P}_\\\\lambda $ -almost surely there are only finitely many balls in $\\\\mathcal {O}$ that intersect $K$ .\",\n \"We first derive the second bound.\",\n \"Since $K_1$ has area $8r^2$ , an immediate estimate of the expected number of points in $K_1$ with radius at least $s$ gives that $\\\\mathbb {P}_\\\\lambda \\\\big ( \\\\mathcal {O}_{K_1} \\\\cap K_2^c \\\\ne \\\\varnothing \\\\big ) \\\\,\\\\le \\\\, 8r^2\\\\lambda \\\\int _s^\\\\infty \\\\mu (\\\\mathop {}\\\\!\\\\mathrm {d}x) \\\\,\\\\le \\\\, 8\\\\lambda \\\\left(\\\\frac{r}{s}\\\\right)^2\\\\int _s^\\\\infty x^2\\\\mu (\\\\mathop {}\\\\!\\\\mathrm {d}x),$ hence establishing (REF ).\",\n \"For the first bound, we split the complement of $K_2 =B^\\\\infty (r+s)$ into the disjoint annuli $A_i = B^\\\\infty (r + s + i + 1) \\\\setminus B^\\\\infty (r + s + i)$ and write $K_2^c = \\\\bigcup _{i\\\\ge 0} A_i$ .\",\n \"Using the fact that the area of $A_i$ is bounded by $8(r+s+i+1)$ , we can now estimate $\\\\begin{split}\\\\mathbb {P}_\\\\lambda \\\\big ( \\\\mathcal {O}_{K_2^c} \\\\cap K_1 \\\\ne \\\\varnothing \\\\big )& \\\\,\\\\le \\\\, \\\\sum _{i\\\\ge 0} \\\\mathbb {P}_\\\\lambda \\\\big ( \\\\mathcal {O}_{A_i} \\\\cap K_1 \\\\ne \\\\varnothing \\\\big )\\\\\\\\& \\\\le \\\\, 8\\\\lambda \\\\sum _{i\\\\ge 0} (r + s + i+1)\\\\int _{s+i}^\\\\infty \\\\mu (\\\\mathop {}\\\\!\\\\mathrm {d}x).\\\\end{split}$ Exchanging the order of summation yields the bound $8\\\\lambda \\\\int _s^\\\\infty \\\\sum _{i=0}^{x-s}(r+s+i+1)\\\\mu (\\\\mathop {}\\\\!\\\\mathrm {d}x) \\\\,\\\\le \\\\, 8\\\\lambda \\\\int _s^\\\\infty (r+x+1)^2\\\\mu (\\\\mathop {}\\\\!\\\\mathrm {d}x),$ from which (REF ) follows.\",\n \"Let us now prove Proposition REF .\",\n \"[Proof of Proposition REF ] Recall that $\\\\Vert f_1 \\\\Vert _\\\\infty , \\\\Vert f_2 \\\\Vert _\\\\infty \\\\le 1$ .\",\n \"Let $K=B^\\\\infty (r+s/2)$ .\",\n \"By the Poissonian character of the law $\\\\mathbb {P}_\\\\lambda $ , the two random variables $ f_1(\\\\mathcal {O}_K)$ and $f_2(\\\\mathcal {O}_{K^c})$ are independent.\",\n \"Therefore, $\\\\begin{split}\\\\operatorname{Cov}(f_1(\\\\mathcal {O}), f_2(\\\\mathcal {O}) & \\\\,\\\\le \\\\, \\\\mathbb {P}_\\\\lambda \\\\big (f_1(\\\\mathcal {O}) \\\\ne f_1(\\\\mathcal {O}_K)\\\\big ) + \\\\mathbb {P}_\\\\lambda \\\\big (f_2(\\\\mathcal {O})\\\\ne f_2(\\\\mathcal {O}_{K^c})\\\\big )\\\\\\\\& \\\\le \\\\, \\\\mathbb {P}_\\\\lambda \\\\big ( \\\\mathcal {O}_{K^c} \\\\cap B^\\\\infty (r)\\\\ne \\\\varnothing \\\\big ) + \\\\mathbb {P}_\\\\lambda \\\\big ( \\\\mathcal {O}_K \\\\cap B^\\\\infty (r+s)^c\\\\ne \\\\varnothing \\\\big ),\\\\end{split}$ and the proof then follows from Lemma REF .\",\n \"For the last conclusion (REF ), one may use Markov's inequality.\"\n ],\n [\n \"An alternative notion of decoupling\",\n \"Note that Definition REF is symmetric for $\\\\mathcal {O}$ and $\\\\mathcal {V}$ , in the sense that the roles of the occupied and vacant sets and are interchangeable.\",\n \"In what follows, we will introduce another notion of decoupling that will not be symmetric for $\\\\mathcal {O}$ and $\\\\mathcal {V}$ .\",\n \"This distinction will be very relevant to explain the different behavior of these sets, see Remark REF , and explains why we obtain Theorem REF under (REF ) but require (REF ) for Theorem REF with the techniques used here.\",\n \"To motivate this new notion, observe from Definition REF that $\\\\mathbb {E}_\\\\lambda \\\\big (f_1(\\\\mathcal {O}) f_2(\\\\mathcal {O})\\\\big ) \\\\,\\\\le \\\\,\\\\mathbb {E}_\\\\lambda \\\\big (f_1(\\\\mathcal {O})\\\\big ) \\\\mathbb {E}_\\\\lambda \\\\big (f_2(\\\\mathcal {O})\\\\big ) + \\\\rho (r, s),$ for every functions $f_1,f_2:\\\\mathcal {P}(\\\\mathbb {R}^2)\\\\rightarrow [0,1]$ such that $f_1(\\\\mathcal {O}) \\\\in \\\\sigma (Y_v; v \\\\in B^\\\\infty (r))$ and $f_2(\\\\mathcal {O}) \\\\in \\\\sigma (Y_v; v \\\\notin B^\\\\infty (r + s))$ .\",\n \"This will be used in several parts of the text.\",\n \"However, some results will be stronger using the following modified version of the above.\",\n \"Definition 2.4 Let $\\\\bar{\\\\rho }_\\\\lambda :\\\\mathbb {R}_+^2 \\\\rightarrow \\\\mathbb {R}_+$ be defined as the smallest value such that $\\\\mathbb {E}_\\\\lambda \\\\big (f_1(\\\\mathcal {O}) f_2(\\\\mathcal {O})\\\\big ) \\\\,\\\\le \\\\,\\\\mathbb {E}_\\\\lambda \\\\big (f_1(\\\\mathcal {O})\\\\big ) \\\\Big ( \\\\mathbb {E}_\\\\lambda \\\\big (f_2(\\\\mathcal {O})\\\\big ) + \\\\bar{\\\\rho }_\\\\lambda (r, s) \\\\Big ),$ for all decreasing functions $f_1, f_2:\\\\mathcal {P}(\\\\mathbb {R}^2)\\\\rightarrow [0,1]$ satisfying $f_1(\\\\mathcal {O}) \\\\in \\\\sigma (Y_v; v \\\\in B^\\\\infty (r))$ , $f_2(\\\\mathcal {O}) \\\\in \\\\sigma (Y_v; v \\\\notin B^\\\\infty (r + s))$ .\",\n \"(A function $f:\\\\mathcal {P}(\\\\mathbb {R}^2) \\\\rightarrow [0,1]$ is said to be decreasing if $f(A)\\\\ge f(B)$ whenever $A\\\\subset B$ .)\",\n \"Remark 6 Note that the error $\\\\bar{\\\\rho }_\\\\lambda $ in the above definition is being multiplied by $\\\\mathbb {E}_\\\\lambda (f_1)$ .\",\n \"This represents an important improvement in some cases, as will become clear later for instance when we compare (REF ) with (REF ).\",\n \"Although the error term in Definition REF is smaller than the one in (REF ), the bound (REF ) can only be used when $f_1$ and $f_2$ are decreasing functions.\",\n \"This restriction was intentionally introduced in the definition of $\\\\bar{\\\\rho }_\\\\lambda $ and it is necessary in our proof that $\\\\bar{\\\\rho }_\\\\lambda $ vanishes, see Proposition REF below.\",\n \"This restriction reflects the distinct behavior that the vacant and occupied sets display at criticality, see Remark REF .\",\n \"We now prove an analogue of Proposition REF .\",\n \"Proposition 2.5 For any $\\\\lambda > 0$ and $r, s \\\\ge 1$ we have $\\\\bar{\\\\rho }_\\\\lambda (r, s) \\\\,\\\\le \\\\, c_{\\\\ref {c:decorrelation}} \\\\lambda \\\\left(1+\\\\frac{r}{s}\\\\right)^2\\\\int _{s/2}^\\\\infty x^2\\\\mu (\\\\mathop {}\\\\!\\\\mathrm {d}x).$ In particular, if $\\\\int _0^\\\\infty x^2 \\\\mu (\\\\mathop {}\\\\!\\\\mathrm {d}x) < \\\\infty $ , then $\\\\text{for any } \\\\kappa > 0,\\\\, \\\\lim _{r\\\\rightarrow \\\\infty } \\\\bar{\\\\rho }_\\\\lambda (r, \\\\kappa r) = 0.$ We write $K_1 = B^\\\\infty (r)$ and $K_2 = B^\\\\infty (r + s)^c$ and define $\\\\check{\\\\mathcal {O}} = \\\\bigcup _{\\\\begin{array}{c}(x,z) \\\\in \\\\operatorname{supp}(\\\\omega ); \\\\\\\\ B(x, z) \\\\cap K_1 = \\\\varnothing \\\\end{array}} B(x, z).$ Note that this set is contained in $\\\\mathcal {O}$ and independent of $f_1(\\\\mathcal {O})$ .\",\n \"Using the fact that $f_2$ is decreasing, we can write $\\\\begin{split}\\\\mathbb {E}_\\\\lambda \\\\big ( f_1(\\\\mathcal {O}) f_2(\\\\mathcal {O}) \\\\big ) & \\\\,\\\\le \\\\, \\\\mathbb {E}_\\\\lambda \\\\big ( f_1(\\\\mathcal {O}) f_2(\\\\check{\\\\mathcal {O}}) \\\\big ) \\\\,\\\\le \\\\, \\\\mathbb {E}_\\\\lambda \\\\big ( f_1(\\\\mathcal {O}) \\\\big ) \\\\mathbb {E}_\\\\lambda \\\\big ( f_2(\\\\check{\\\\mathcal {O}}) \\\\big )\\\\\\\\& \\\\le \\\\, \\\\mathbb {E}_\\\\lambda \\\\big ( f_1(\\\\mathcal {O}) \\\\big ) \\\\Big ( \\\\mathbb {E}_\\\\lambda \\\\big ( f_2(\\\\mathcal {O}) \\\\big ) + \\\\mathbb {P}_\\\\lambda \\\\big ( \\\\check{\\\\mathcal {O}} \\\\cap K_2 \\\\ne \\\\mathcal {O} \\\\cap K_2 \\\\big ) \\\\Big ),\\\\end{split}$ where we have used the fact that $f_1,f_2\\\\in [0,1]$ .\",\n \"In view of the definition of $\\\\bar{\\\\rho }_\\\\lambda $ , all we need to do is to bound the probability above, $\\\\begin{split}\\\\mathbb {P}_\\\\lambda (\\\\check{\\\\mathcal {O}} & \\\\cap K_2 \\\\ne \\\\mathcal {O} \\\\cap K_2) \\\\,\\\\le \\\\, \\\\mathbb {P}_\\\\lambda \\\\Big (\\\\begin{array}{c}\\\\text{there is $(x,z) \\\\in \\\\operatorname{supp}(\\\\omega )$ such that}\\\\\\\\\\\\text{$B^\\\\infty (x,z)$ touches $K_1$ and $K_2$}\\\\end{array}\\\\Big )\\\\\\\\& \\\\le \\\\, \\\\mathbb {P}_\\\\lambda \\\\big (\\\\mathcal {O}_{B^\\\\infty (r + s/2)} \\\\cap K_2 \\\\ne \\\\varnothing \\\\big ) + \\\\mathbb {P}_\\\\lambda \\\\big ( \\\\mathcal {O}_{B^\\\\infty (r + s/2)^2} \\\\cap K_1 \\\\ne \\\\varnothing \\\\big ).\\\\end{split}$ The proof now ends in the same way as the proof of Proposition REF .\"\n ],\n [\n \"Local continuity\",\n \"Another rather straightforward consequence of the Poissonian nature of the model is the following `local continuity' property.\",\n \"In essence this property says that locally the topological properties of the set $\\\\mathcal {O}$ are unlikely to be affected by slight change in the radii of the discs.\",\n \"Given a set $A\\\\subseteq {\\\\mathbb {R}}^2$ we define its $\\\\varepsilon $ -interior and $\\\\varepsilon $ -closure, for $\\\\varepsilon >0$ , as follows: $\\\\begin{aligned}\\\\operatorname{int}(A,\\\\varepsilon )&:=\\\\lbrace x\\\\in {\\\\mathbb {R}}^2:B(x,\\\\varepsilon )\\\\subseteq A\\\\rbrace ,\\\\\\\\\\\\operatorname{cl}(A,\\\\varepsilon )&:=\\\\lbrace x\\\\in {\\\\mathbb {R}}^2:B(x,\\\\varepsilon )\\\\cap A\\\\ne \\\\varnothing \\\\rbrace .\\\\end{aligned}$ We omit the proof of the following proposition.\",\n \"Proposition 2.6 Let $K \\\\subseteq {\\\\mathbb {R}}^2$ be compact and convex and $A_1, A_2 \\\\subseteq K$ closed.\",\n \"For every $\\\\lambda > 0$ $\\\\lim _{\\\\varepsilon \\\\rightarrow 0}{\\\\mathbb {P}}_\\\\lambda \\\\Big ( A_1 {\\\\overset{\\\\operatorname{cl}(\\\\mathcal {O},\\\\varepsilon ) \\\\cap K}{\\\\longleftrightarrow }} A_2\\\\text{ occurs but not }A_1 {\\\\overset{\\\\operatorname{int}(\\\\mathcal {O},\\\\varepsilon ) \\\\cap K}{\\\\longleftrightarrow }} A_2 \\\\Big ) \\\\,=\\\\,0.$ A consequence of the above proposition that will be used later is the following lemma.\",\n \"Consider a variation of the standard crossing event defined as $\\\\hbox{[scale=1.5]{(0, 0) rectangle (4ex, 4ex);(3.8ex, 2ex) -- (4.2ex, 2ex);(0, 1.7ex) .. controls (1.3ex, 4ex) and (2.6ex, 1.4ex) .. (4ex, 3.2ex);(2ex, .2ex) node[anchor=north] {$\\\\scriptstyle r$};(.4ex, 2ex) node[anchor=east] {$\\\\scriptstyle h$};[decorate,decoration={brace,amplitude=.4ex,raise=.4ex},yshift=0pt] (4ex, 4ex) -- (4ex, 2ex);(4.3ex, 3ex) node[anchor=west] {$\\\\scriptstyle y$};}\\\\;}$ := [ A1 O K A2 ], where $K$ is again the box $[0,r] \\\\times [0,h]$ , while $A_1 = \\\\lbrace 0\\\\rbrace \\\\times [0,h]$ and $A_2 = \\\\lbrace r\\\\rbrace \\\\times [y, h]$ .\",\n \"Lemma 2.7 Given $r, h$ and $\\\\lambda > 0$ , the function $F:[0,h] \\\\rightarrow [0,1]$ given by $F(y) = \\\\mathbb {P}_\\\\lambda \\\\Bigg [ \\\\smash{\\\\hbox{[scale=1.5]{(0, 0) rectangle (4ex, 4ex);(3.8ex, 2ex) -- (4.2ex, 2ex);(0, 1.7ex) .. controls (1.3ex, 4ex) and (2.6ex, 1.4ex) .. (4ex, 3.2ex);(2ex, .2ex) node[anchor=north] {$\\\\scriptstyle r$};(.4ex, 2ex) node[anchor=east] {$\\\\scriptstyle h$};[decorate,decoration={brace,amplitude=.4ex,raise=.4ex},yshift=0pt] (4ex, 4ex) -- (4ex, 2ex);(4.3ex, 3ex) node[anchor=west] {$\\\\scriptstyle y$};}\\\\;}}$ ] is continuous.\"\n ],\n [\n \"Finite size criterion for percolation\",\n \"The first tool in our study of the phase transition for Poisson Boolean percolation in ${\\\\mathbb {R}}^2$ will be the following result that bootstraps the probability of crossing a rectangle: from close to one to converging fast to one.\",\n \"This will be fundamental in this paper; as a first indication of its importance we show how it relates to the existence of an unbounded component, and the non-triviality of the threshold parameters $\\\\lambda _0$ and $\\\\lambda _1$ .\",\n \"Proposition 3.1 Assume the second moment condition (REF ).\",\n \"Then, there exists a constant $\\\\theta = \\\\theta (\\\\mu ) > 0$ and an increasing function $r_0:[0,\\\\infty )\\\\rightarrow [0,\\\\infty )$ such that the following are equivalent: There exists $r\\\\ge r_0(\\\\lambda ) $ such that ${\\\\mathbb {P}}_\\\\lambda (\\\\operatorname{Cross}(3r,r))>1-\\\\theta $ .\",\n \"$\\\\displaystyle \\\\lim _{r\\\\rightarrow \\\\infty }{\\\\mathbb {P}}_\\\\lambda (\\\\operatorname{Cross}(3r,r))=1$ .\",\n \"Under the stronger condition (REF ), if either of (i) or (ii) holds, then $\\\\sum _{k\\\\ge 0}\\\\big [1-{\\\\mathbb {P}}_\\\\lambda (\\\\operatorname{Cross}(3^{k+1},3^k))\\\\big ]<\\\\infty $ .\",\n \"These statements remains valid for $\\\\operatorname{Cross}$ replaced by $\\\\operatorname{Cross}^\\\\star $ .\",\n \"Fix $\\\\theta =1/100$ and let $C$ be a sufficiently large constant so that, for all $r\\\\ge 1$ , $\\\\rho _\\\\lambda (5r,r)\\\\le g(r) := C \\\\lambda \\\\int _{r/2}^\\\\infty x^2\\\\mu (\\\\mathop {}\\\\!\\\\mathrm {d}x).$ The existence of the constant follows from Proposition REF .\",\n \"Set $r_0:=\\\\min \\\\lbrace r\\\\ge 1:g(r)\\\\le \\\\theta /2\\\\rbrace $ .\",\n \"Since $g(r)$ increases with $\\\\lambda $ it is clear that $r_0$ does too.\",\n \"Since (ii) trivially implies (i), it will suffice to prove that (i) implies (ii) and, under the stronger assumption (REF ), that (ii) implies (iii).\",\n \"Let $p(r):=1-{\\\\mathbb {P}}_\\\\lambda (\\\\operatorname{Cross}(3r,r))$ and let $B_k:=\\\\operatorname{Cross}(9r,r)$ .\",\n \"Note that a $9r\\\\times r$ -rectangle may be tiled with seven $3r\\\\times r$ -rectangles, four positioned horizontally and three vertically, in such a way that if each is crossed in the `hard' direction, then the $9r\\\\times r$ -rectangle is crossed horizontally.\",\n \"Consequently, using the union bound, we obtain that ${\\\\mathbb {P}}_\\\\lambda (B_r)\\\\,\\\\ge \\\\,1-7{\\\\mathbb {P}}_\\\\lambda (\\\\lnot \\\\operatorname{Cross}(3r,r))\\\\,=\\\\,1-7p(r).$ Let $B_r^{\\\\prime }$ denote the translate of $B_r$ along the vector $(0,2r)$ .\",\n \"Since the occurrence of either of $B_r$ or $B_r^{\\\\prime }$ implies the occurrence of $\\\\operatorname{Cross}(9r,3r)$ , he have $p(3r)\\\\,\\\\le \\\\,{\\\\mathbb {P}}_\\\\lambda (\\\\lnot B_r\\\\cap \\\\lnot B_r^{\\\\prime })\\\\,\\\\le \\\\,{\\\\mathbb {P}}_\\\\lambda (\\\\lnot B_r)^2+\\\\rho _\\\\lambda (5r,r).$ Hence, by definition of $g$ , we obtain for every $r\\\\ge 1$ the bound $p(3r)\\\\le 49p(r)^2+g(r).$ Now, assume there exists $r\\\\ge r_0$ so that $p(r)<\\\\theta $ .\",\n \"Then, via iterated use of (REF ), we find that $p(3^kr)<\\\\theta $ for all $k\\\\ge 0$ .\",\n \"Consequently, further use of (REF ) gives, for $\\\\ell =1,2,\\\\ldots ,k$ $p(3^kr)\\\\,\\\\le \\\\,\\\\frac{1}{2^\\\\ell }p(3^{k-\\\\ell }r)+\\\\sum _{j=1}^\\\\ell \\\\frac{1}{2^{j-1}}g(3^{k-j}r)\\\\,\\\\le \\\\,\\\\frac{1}{2^\\\\ell }+2g(3^{k-\\\\ell }r).$ Hence, sending $\\\\ell $ to infinity with $k$ shows that $\\\\lim _{k\\\\rightarrow \\\\infty }p(3^kr)=0$ .\",\n \"Since for every $r^{\\\\prime }\\\\in [r,3r]$ we have $p(r^{\\\\prime })\\\\le {\\\\mathbb {P}}_\\\\lambda (\\\\lnot B_r)\\\\le 7p(r)$ , it follows that $\\\\lim _{r\\\\rightarrow \\\\infty }p(r)=0$ , so (ii) holds.\",\n \"Finally, we assume that (ii) holds and pick $k_0$ so that $3^{k_0}\\\\ge r_0$ and $p(3^{k_0})<\\\\theta $ .\",\n \"Summing over $k$ in (REF ), with $r=3^{k_0}$ and $\\\\ell =\\\\lfloor k/2\\\\rfloor $ , leads to $\\\\sum _{k\\\\ge 1}p(3^{k+k_0})\\\\,\\\\le \\\\,\\\\sum _{k\\\\ge 1}\\\\frac{1}{2^{(k-1)/2}}+2\\\\sum _{k\\\\ge 1}g(3^{k/2+k_0})\\\\,\\\\le \\\\,4+2\\\\int _0^\\\\infty g(3^{y/2})\\\\,\\\\mathop {}\\\\!\\\\mathrm {d}y.$ Since $g$ is itself an integral we may use Fubini's theorem to reverse the order of integration, and obtain $\\\\sum _{k\\\\ge 1}p(3^{k+k_0})\\\\,\\\\le \\\\, 4+C\\\\lambda \\\\int _0^\\\\infty x^2\\\\log x\\\\,\\\\mu (\\\\mathop {}\\\\!\\\\mathrm {d}x),$ for a possibly larger constant $C$ , which is finite under the assumption (REF ), and (iii) follows.\",\n \"Remark 7 Note that the proof of Proposition REF assumes very little about the underlying percolation model.\",\n \"Indeed, the argument applies to any percolation model on ${\\\\mathbb {R}}^2$ satisfying the invariance assumption (REF ) and decay of correlations corresponding to conditions (REF ) or (REF ).\",\n \"The following more quantitative statement sheds some light on the role of spatial correlations on the rate of convergence in Proposition REF .\",\n \"Given a function $f:[1, \\\\infty ) \\\\rightarrow (0, \\\\infty )$ we say that it is regularly varying if for any $a \\\\in (0, \\\\infty )$ $\\\\frac{f(ar)}{f(r)} \\\\text{ converges to a non-zero limit.\",\n \"}$ The following proposition says, in particular, that if the spatial dependence decays polynomially fast, then the convergence in Proposition REF occurs (at least) at the same polynomial rate.\",\n \"Proposition 3.2 Assume the finite moment condition (REF ) and let $f:[1,\\\\infty )\\\\rightarrow (0,\\\\infty )$ be a regularly varying function satisfying $\\\\lim _{r\\\\rightarrow \\\\infty }f(r)=0$ and $\\\\rho _\\\\lambda (5r,r)\\\\le f(r)$ for all $r \\\\ge 1$ .\",\n \"Then, if $\\\\lim _{r\\\\rightarrow \\\\infty }{\\\\mathbb {P}}_\\\\lambda (\\\\operatorname{Cross}(3r,r))=1$ , then there exists $c<\\\\infty $ such that ${\\\\mathbb {P}}_\\\\lambda (\\\\operatorname{Cross}(3r,r))\\\\ge 1-cf(r)\\\\quad \\\\text{for all }r\\\\ge 1.$ The statement remains true if $\\\\operatorname{Cross}$ is replaced by $\\\\operatorname{Cross}^\\\\star $ .\",\n \"With $p(r):=1-{\\\\mathbb {P}}_\\\\lambda (\\\\operatorname{Cross}^\\\\star (3r,r))$ , we may repeat the proof of Proposition REF to obtain $p(3r)\\\\le 49p(r)^2+f(r).$ Since $f$ is regularly varying there exists a constant $c^{\\\\prime } \\\\in (0, 50)$ such that $f(3r)/f(r) \\\\ge c^{\\\\prime }$ for all $r\\\\ge 1$ .\",\n \"Let $\\\\theta ^{\\\\prime }=c^{\\\\prime }/100$ and pick $r^{\\\\prime }$ such that $p(r)\\\\le \\\\theta ^{\\\\prime }$ and $f(r)\\\\le \\\\theta ^{\\\\prime }/2$ for all $r\\\\ge r^{\\\\prime }$ .\",\n \"Let $c=\\\\max \\\\lbrace 2/c^{\\\\prime },1/f(r^{\\\\prime })\\\\rbrace $ .\",\n \"We claim that $p(3^kr^{\\\\prime })\\\\le cf(3^kr^{\\\\prime })\\\\quad \\\\text{for all }k\\\\ge 0.$ The case $k = 0$ is immediate, and the remaining cases follows from (REF ) via an induction step.\",\n \"This proves the statement of the proposition along exponentially growing sub-sequences from which one can easily derive the full statement.\",\n \"Propositions REF and REF show how an observation in a finite region can be sufficient to draw conclusions regarding crossing probabilities over arbitrarily large regions.\",\n \"The next well-known result connects the finite observation to the existence of unbounded components.\",\n \"Proposition 3.3 The condition $\\\\sum _{k\\\\ge 0}\\\\big [1-{\\\\mathbb {P}}_\\\\lambda (\\\\operatorname{Cross}(3^{k+1},3^k))\\\\big ]<\\\\infty $ implies that the probability, at $\\\\lambda $ , that the origin is contained in an unbounded occupied component is positive.\",\n \"The same holds when $\\\\operatorname{Cross}$ and `occupied' are replaced by $\\\\operatorname{Cross}^\\\\star $ and `vacant', respectively.\",\n \"Denote by $R_k$ the rectangle $[0,3^{k+1}]\\\\times [0,3^k]$ , and by $R_k^{\\\\prime }$ the rectangle $[0,3^k]\\\\times [0,3^{k+1}]$ , resulting from a right-angle rotation of $R_k$ .\",\n \"Consider the sequence, indexed by $k\\\\ge 0$ , alternating between $R_k$ or $R_k^{\\\\prime }$ , depending on whether $k$ is odd or even.\",\n \"The above condition implies that the expected number of these rectangles that fail to be crossed in the `hard' direction is finite.\",\n \"Hence, by Borel-Cantelli, it follows that all but finitely many contain a crossing in the `hard' direction almost surely.\",\n \"However, all but finitely many of these crossings must intersect as a consequence of how the rectangles are placed.\",\n \"Thus, with probability one, there is an unbounded occupied component.\",\n \"Due to invariance with respect to translations it follows that the origin has positive probability to be contained in this component.\",\n \"We close this section by showing that the (finite-size) phase transition is non-trivial.\",\n \"Corollary 3.4 Assume the second moment condition (REF ).\",\n \"Then, $0<\\\\lambda _0\\\\le \\\\lambda _c\\\\le \\\\lambda _1<\\\\infty .$ Under the stronger assumption (REF ) we also have $\\\\lambda _0\\\\le \\\\lambda _c^\\\\star \\\\le \\\\lambda _1$ .\",\n \"Assume that (REF ) holds.\",\n \"We first show that $\\\\lambda _0 > 0$ .\",\n \"Let $\\\\theta >0$ and $r_0:[0,\\\\infty )\\\\rightarrow [0,\\\\infty )$ be as in Proposition REF .\",\n \"Clearly ${\\\\mathbb {P}}_0(\\\\operatorname{Cross}(r, 3r))=0$ for all $r\\\\ge 1$ , in particular for $r=r_0(1)$ .\",\n \"That the crossing probability varies continuously with respect to the intensity parameter is an easy consequence of (REF ) of Lemma REF and the fact that it is unlikely to add any point at all to a bounded region for small enough changes in the parameter.\",\n \"Hence, we may pick $\\\\varepsilon \\\\in (0,1)$ such that ${\\\\mathbb {P}}_\\\\varepsilon (\\\\operatorname{Cross}(r,3r)) < \\\\theta $ , implying that $\\\\lim _{r\\\\rightarrow \\\\infty } {\\\\mathbb {P}}_\\\\varepsilon (\\\\operatorname{Cross}(r,3r)) = 0,$ and thus that $\\\\lambda _0 \\\\ge \\\\varepsilon > 0$ .\",\n \"The fact that $\\\\lambda _1 < \\\\infty $ follows from the fact that this holds for the model with constant radius, together with a stochastic domination argument.\",\n \"We next show that $\\\\lambda _c\\\\ge \\\\lambda _0$ and $\\\\lambda _c^\\\\star \\\\le \\\\lambda _1$ .\",\n \"First, consider $\\\\lambda <\\\\lambda _0$ .\",\n \"Then, since a path connecting the origin to $\\\\partial B(r)$ has to cross one of four rectangles of dimension $r/3\\\\times r$ in the easy direction, the union bound and the definition of $\\\\lambda _0$ gives that $\\\\lim _{r\\\\rightarrow \\\\infty }{\\\\mathbb {P}}_\\\\lambda (0\\\\overset{\\\\tiny \\\\mathcal {O}}{\\\\longleftrightarrow }\\\\partial B(r))\\\\,\\\\le \\\\,\\\\lim _{r\\\\rightarrow \\\\infty }4{\\\\mathbb {P}}_\\\\lambda (\\\\operatorname{Cross}(r/3,r))\\\\,=\\\\,0.$ Thus, $\\\\lambda \\\\le \\\\lambda _c$ , which shows that $\\\\lambda _0\\\\le \\\\lambda _c$ .\",\n \"An analogous argument shows that $\\\\lambda _c^\\\\star \\\\le \\\\lambda _1$ , To prove that $\\\\lambda _c^\\\\star \\\\ge \\\\lambda _0$ , we make the stronger assumption that (REF ) holds and fix $\\\\lambda <\\\\lambda _0$ , in which case we have $\\\\lim _{r\\\\rightarrow \\\\infty }{\\\\mathbb {P}}_\\\\lambda (\\\\operatorname{Cross}^\\\\star (3r,r))=1$ .\",\n \"Then, Proposition REF shows that $\\\\sum _{k\\\\ge 0}\\\\big [1-{\\\\mathbb {P}}_\\\\lambda (\\\\operatorname{Cross}^\\\\star (3^{k+1},3^k))\\\\big ]<\\\\infty ,$ which by Proposition REF implies the almost sure existence of an unbounded vacant component, and thus that $\\\\lambda \\\\le \\\\lambda _c^\\\\star $ .\",\n \"Hence $\\\\lambda _c^\\\\star \\\\ge \\\\lambda _0$ .\",\n \"An analogous argument shows, under the additional condition (REF ), that $\\\\lambda _c\\\\le \\\\lambda _1$ .\",\n \"However, we claim that this additional assumption is unnecessary.\",\n \"Our goal will therefore be to show that (REF ) holds, with $\\\\operatorname{Cross}^\\\\star $ replaced by $\\\\operatorname{Cross}$ , also under the weaker second moment condition (REF ).\",\n \"As before, let $p(r)=1-{\\\\mathbb {P}}_\\\\lambda (\\\\operatorname{Cross}(3r,r))$ .\",\n \"Repeating the first steps of the proof of Proposition REF , observing that the complements of occupied crossing events are decreasing, we obtain the following analogue to (REF ) $p(3r)\\\\,\\\\le \\\\,7p(r)\\\\big (7p(r)+\\\\bar{\\\\rho }_\\\\lambda (5r,r)\\\\big ).$ Since both $p(r)$ and $\\\\bar{\\\\rho }_\\\\lambda (5r,r)$ tend to zero as $r\\\\rightarrow \\\\infty $ , we may find $r_0$ such that $7p(r)+\\\\bar{\\\\rho }_\\\\lambda (5r,r)$ is bounded by $\\\\frac{1}{14}$ for all $r\\\\ge r_0$ .\",\n \"So, for some $k_0\\\\ge 1$ , we obtain that $p(3^{k+k_0})\\\\,\\\\le \\\\,\\\\frac{1}{2}p(3^{k-1+k_0}r_0)\\\\,\\\\le \\\\,\\\\frac{1}{2^k}p(3^{k_0}),$ which is summable.\",\n \"Hence (REF ) holds for $\\\\operatorname{Cross}$ also under the weaker condition (REF ), and $\\\\lambda _c\\\\le \\\\lambda _1$ follows.\"\n ],\n [\n \"Russo-Seymour-Welsh theory\",\n \"In the study of planar percolation, Russo-Seymour-Welsh (RSW) techniques play a central role and have numerous consequences.\",\n \"The original proof for Bernoulli percolation ([34], [39]) is strongly based on planarity and the independence structure of the Bernoulli percolation measure, and does not extend easily to other contexts.\",\n \"Considerable technicalities had to be overcome even in the extension to Poisson Boolean percolation in ${\\\\mathbb {R}}^2$ with fixed radii, see [32], [4], and in the case of (unbounded) random radii such a result has until this point not been obtained.\",\n \"In the last few years some new arguments have been developed to prove RSW-results for dependent percolation models, e.g.\",\n \"Voronoi percolation [9], [42], [3] and the random-cluster model [7], [13].\",\n \"In this section we develop an RSW-theory applicable for Poisson Boolean percolation.\",\n \"Our method of proof will be greatly inspired by that of [42].\",\n \"However, due to the asymmetry between the vacant and occupied regions, we need a stronger version of the result proved in [42].\",\n \"In that paper, the RSW statement bounds the crossing probabilities for rectangles in the long direction, assuming a bound on the crossing probabilities for squares, see (REF ).\",\n \"Here we assume only that rectangles are crossed in the easier direction.\",\n \"Our proof will be rather general and applies in settings far beyond Poisson Boolean percolation; see Remark REF below.\",\n \"Theorem 4.1 (RSW-Theorem) Assume the finite second moment condition stated in (REF ).\",\n \"Then, for any $\\\\lambda > 0$ , if for some $\\\\kappa >0$ we have $\\\\inf _{r\\\\ge 1} \\\\mathbb {P}_\\\\lambda \\\\big (\\\\operatorname{Cross}(\\\\kappa r, r) \\\\big ) > 0,$ then the same is true for all $\\\\kappa >0$ .\",\n \"The same holds for $\\\\operatorname{Cross}$ replaced by $\\\\operatorname{Cross}^\\\\star $ .\",\n \"Remark 8 Throughout this section, we are going to establish the above result and some of its consequences for Poisson Boolean percolation with finite second moments.\",\n \"However, let us emphasize that the same proof works for several different types of percolation measures on the plane.\",\n \"More precisely, the only properties of the random set $\\\\mathcal {O}$ that we use in this section are: $\\\\begin{array}{c}\\\\text{the translation, reflection and rotational symmetries in (\\\\ref {e:invariance}),}\\\\\\\\\\\\text{the FKG inequality (\\\\ref {eq:3}),}\\\\\\\\\\\\text{the decay of correlations stated in (\\\\ref {e:rho_to_zero}) and}\\\\\\\\\\\\text{the continuity of the crossing probabilities stated in Lemma~\\\\ref {l:cross_continuous}}.\\\\end{array}$\"\n ],\n [\n \"Consequences of Theorem \",\n \"The Russo-Seymour-Welsh Theorem stated above, which will be proved in Section REF , has several important consequences that we develop next.\",\n \"These consequences concern `box-crossing' and `one-arm' probabilities in the critical regime $[\\\\lambda _0,\\\\lambda _1]$ .\",\n \"However, we first state a `high-probability' version of Theorem REF .\",\n \"All proofs of the consequences listed here will be given in Section REF below.\",\n \"Corollary 4.2 Assume the finite second moment condition (REF ).\",\n \"Given $\\\\lambda > 0$ , if for some $\\\\kappa >0$ we have $\\\\lim _{r\\\\rightarrow \\\\infty }\\\\mathbb {P}_\\\\lambda \\\\big (\\\\operatorname{Cross}(\\\\kappa r, r) \\\\big ) = 1,$ then it is true for all $\\\\kappa >0$ .\",\n \"The same holds for $\\\\operatorname{Cross}$ replaced by $\\\\operatorname{Cross}^\\\\star $ .\",\n \"Recall the definition of $\\\\lambda _0$ and $\\\\lambda _1$ in (REF ) and ().\",\n \"Outside of the interval $[\\\\lambda _0,\\\\lambda _1]$ the probability of crossing a fixed-ratio rectangle will converge rather rapidly to either zero or one as the side length of the rectangle increases.\",\n \"This is merely a matter of definition and the FKG-inequality.\",\n \"One of the main consequences of the RSW Theorem is that within the interval $[\\\\lambda _0,\\\\lambda _1]$ the probability of crossing a fixed-ratio rectangle remains rather balanced.\",\n \"This is sometimes referred to as the `box-crossing' property, which will also be used to provide bounds on the one-arm probabilities.\",\n \"For $0 < r < r^{\\\\prime }$ , we define $\\\\operatorname{Arm}(r, r^{\\\\prime }) := \\\\big [ B^\\\\infty (r) \\\\overset{\\\\mathcal {O}}{\\\\longleftrightarrow } \\\\partial B^\\\\infty (r^{\\\\prime }) \\\\big ].$ As before we write $\\\\operatorname{Arm}^\\\\star $ for the above event, where $\\\\mathcal {O}$ is replaced by $\\\\mathcal {V}$ .\",\n \"Given $r^{\\\\prime } > r\\\\ge 1$ we also introduce $\\\\operatorname{Circ}(r, {r^{\\\\prime }}) := \\\\hbox{[scale=1.5]{(5ex, 3.5ex) rectangle (7ex, 5.5ex);(3ex, 1.5ex) rectangle (9ex, 7.5ex);[rounded corners=.25em](4ex, 6.5ex) .. controls (6ex, 6.5ex + .4ex*rand) ..(8ex, 6.5ex) .. controls (8ex + .4ex*rand, 4.5ex) ..(8ex, 2.5ex) .. controls (6ex, 2.5ex + .4ex*rand) ..(4ex, 2.5ex) -- (4ex + .4ex*rand, 4.5ex) -- cycle;(9.4ex, 3.5ex) -- (9.4ex, 5.5ex);(9.2ex, 3.5ex) -- (9.6ex, 3.5ex) (9.2ex, 5.5ex) -- (9.6ex, 5.5ex);(9.4ex, 4.5ex) node[anchor=west] {$\\\\scriptstyle 2r$};(2.4ex, 1.5ex) -- (2.4ex, 7.5ex);(2.2ex, 1.5ex) -- (2.6ex, 1.5ex) (2.2ex, 7.5ex) -- (2.6ex, 7.5ex);(2ex, 4.5ex) node[anchor=east] {$\\\\scriptstyle 2r^{\\\\prime }$};}}$ := Arm(r, r')c Finally we let $\\\\operatorname{Circ}^\\\\star (r,{r^{\\\\prime }}) := \\\\operatorname{Arm}(r, r^{\\\\prime })^c$ .\",\n \"Corollary 4.3 Assume the second moment condition in (REF ).\",\n \"Then, for every $\\\\kappa > 0$ , there exists $c_{\\\\ref {c:q-bound}} = c_{\\\\ref {c:q-bound}}(\\\\kappa ) > 0$ such that for every $\\\\lambda \\\\in [\\\\lambda _0, \\\\lambda _1]$ we have ${\\\\mathbb {P}}_\\\\lambda (\\\\operatorname{Cross}(\\\\kappa r, r)) \\\\in (c_{\\\\ref {c:q-bound}}, 1 - c_{\\\\ref {c:q-bound}})\\\\quad \\\\text{for every $r\\\\ge 1$,}$ and also $\\\\mathbb {P}_\\\\lambda (\\\\operatorname{Arm}(r, 2r)) \\\\in (c^{\\\\prime }, 1-c^{\\\\prime })\\\\quad \\\\text{for every $r\\\\ge 1$,}$ where $c^{\\\\prime }=c_{\\\\ref {c:q-bound}}(4)^4$ .\",\n \"As before, the above result also holds for the vacant set, i.e.\",\n \"with $\\\\operatorname{Cross}$ and $\\\\operatorname{Arm}$ replaced by $\\\\operatorname{Cross}^\\\\star $ and $\\\\operatorname{Arm}^\\\\star $ respectively.\",\n \"Another consequence of Theorem REF concerns the decay of arm probabilities at criticality.\",\n \"Unlike previous results of this section, the bounds on the arm events distinguish between vacant and occupied sets.\",\n \"The one-arm probability always decays polynomially fast for the vacant set.\",\n \"For the occupied set, we can prove polynomial decay of the one-arm probability only under the stronger assumption () and we explain why this restriction is necessary in Remark REF .\",\n \"Corollary 4.4 (Bounds on the arm-events) Assume the second moment conditions in (REF ).\",\n \"There exists a function $f: (0,1) \\\\rightarrow (0, 1)$ such that $\\\\lim _{x \\\\rightarrow 0}f(x) = 0$ and for every $\\\\lambda \\\\in [\\\\lambda _0, \\\\lambda _1]$ we have ${\\\\mathbb {P}}_\\\\lambda (\\\\operatorname{Arm}(r,r^{\\\\prime })) \\\\le f \\\\Big ( \\\\frac{r}{r^{\\\\prime }} \\\\Big ).$ If $\\\\operatorname{Arm}$ is replaced by $\\\\operatorname{Arm}^\\\\star $ , then a stronger conclusion holds: there exists $c > 0$ such that ${\\\\mathbb {P}}_\\\\lambda ( \\\\operatorname{Arm}^\\\\star (r,r^{\\\\prime }) ) \\\\le \\\\frac{1}{c} \\\\left(\\\\frac{r}{r^{\\\\prime }}\\\\right)^c.$ Under the stronger assumption () the conclusion of (REF ) holds also for $\\\\operatorname{Arm}$ .\",\n \"Remark 9 In Corollary REF , under the $2 + \\\\varepsilon $ -moment condition (), we can choose $f(x) = \\\\tfrac{1}{c} x^c$ for some constant $c > 0$ .\",\n \"However, if $\\\\mu ([x, \\\\infty ))$ is $1/(x^2 \\\\log ^2(x))$ , then the second moment condition holds but the arm event does not decay polynomially.\",\n \"To see this, consider the event that the origin is covered by a occupied disk of radius at least $r$ , and observe that the probability of this event does not decay polynomially fast in $r$ .\",\n \"An immediate consequence of Corollary REF is the following observation.\",\n \"Corollary 4.5 Assume the second moment conditions in (REF ).\",\n \"For every $\\\\lambda \\\\in [\\\\lambda _0,\\\\lambda _1]$ there is almost surely no unbounded occupied nor vacant cluster.\",\n \"Consequently, $\\\\lambda _c^\\\\star \\\\le \\\\lambda _0\\\\le \\\\lambda _1\\\\le \\\\lambda _c.$\"\n ],\n [\n \"Standard inequalities\",\n \"Before starting the proof of the RSW Theorem and its consequences, let us recall some standard inequalities on crossing probabilities.\",\n \"Lemma 4.6 For every $\\\\lambda >0$ , $r>0$ , $\\\\kappa >0$ and integer $j\\\\ge 1$ we have: ${\\\\mathbb {P}}_\\\\lambda [\\\\operatorname{Cross}((1+j\\\\kappa )r,r)]\\\\ge {\\\\mathbb {P}}_\\\\lambda [\\\\operatorname{Cross}((1+\\\\kappa )r,r)]^{2j-1}$ , ${\\\\mathbb {P}}_\\\\lambda [\\\\operatorname{Cross}(r,(1+\\\\kappa )r)]\\\\ge 1-\\\\left(1-{\\\\mathbb {P}}_\\\\lambda [\\\\operatorname{Cross}(r,(1+j\\\\kappa r)]\\\\right)^{1/(2j-1)}$ , ${\\\\mathbb {P}}_\\\\lambda [\\\\operatorname{Circ}(r,2r)]\\\\ge {\\\\mathbb {P}}_\\\\lambda [\\\\operatorname{Cross}(4r,r)]^4$ , ${\\\\mathbb {P}}_\\\\lambda [\\\\operatorname{Cross}(2r,r)]\\\\ge {\\\\mathbb {P}}[\\\\operatorname{Circ}(r,2r)]$ .\",\n \"The same holds if we replace $\\\\operatorname{Cross}$ and $\\\\operatorname{Circ}$ by $\\\\operatorname{Cross}^\\\\star $ and $\\\\operatorname{Circ}^\\\\star $ , respectively.\",\n \"Part (REF ) is a straightforward consequence of the FKG-inequality (REF ): A horizontal crossing of an $(1+j\\\\kappa )r\\\\times r$ -rectangle can be enforced by $j$ horizontal crossings of $(1+\\\\kappa )r\\\\times r$ -rectangles and $j-1$ vertical crossings of $r\\\\times (1+\\\\kappa )r$ -rectangles overlapping one another.\",\n \"Similarly, part (REF ) is the consequence of a circuit being possible to construct out of four overlapping rectangle crossings.\",\n \"Part (REF ) is a direct consequence of (REF ) since it may be rewritten as ${\\\\mathbb {P}}_\\\\lambda [\\\\operatorname{Cross}^\\\\star ((1+\\\\kappa )r,r)]\\\\le {\\\\mathbb {P}}_\\\\lambda [\\\\operatorname{Cross}^\\\\star ((1+j\\\\kappa )r,r)]^{1/(2j-1)}.$ Inequality (REF ) follows from the observation that any circuit in the annulus $B^\\\\infty (2r)\\\\setminus B^\\\\infty (r)$ must cross the rectangle $[-r,r]\\\\times [r,2r]$ horizontally.\"\n ],\n [\n \"A useful circuit Lemma\",\n \"For Bernoulli percolation, the bounds on crossing probabilities provide some bounds on the arm events.\",\n \"This is based on a circuit argument, that uses independence.\",\n \"In our case, the argument need to be adapted because of spatial dependencies.\",\n \"Lemma 4.7 (Circuit Lemma) Assume the second moment conditions in (REF ).\",\n \"Let $\\\\lambda \\\\in [0,\\\\infty )$ .\",\n \"For every $c>0$ , there exists a function $f=f_{c,\\\\lambda }: (0,1) \\\\rightarrow (0,1)$ such that $\\\\lim _{x \\\\rightarrow 0} f(x) = 0$ and for all $r^{\\\\prime } \\\\ge 2r \\\\ge 2$ , $\\\\text{if }\\\\displaystyle \\\\inf _{r\\\\le s\\\\le r^{\\\\prime }/2} {\\\\mathbb {P}}_\\\\lambda [\\\\operatorname{Circ}^\\\\star (s,2s)]\\\\ge c,\\\\text{ then }{\\\\mathbb {P}}_\\\\lambda [ \\\\operatorname{Circ}^\\\\star (r,r^{\\\\prime }) ] \\\\ge 1- f \\\\left(\\\\tfrac{r}{r^{\\\\prime }}\\\\right).$ For every $c>0$ , there exists a constant $c^{\\\\prime }=c^{\\\\prime }(c,\\\\lambda )>0$ such that for all $r^{\\\\prime } \\\\ge 2r \\\\ge 2$ , $\\\\text{if }\\\\displaystyle \\\\inf _{r\\\\le s\\\\le r^{\\\\prime }/2} {\\\\mathbb {P}}_\\\\lambda [\\\\operatorname{Circ}(s,2s)]\\\\ge c,\\\\text{ then }{\\\\mathbb {P}}_\\\\lambda [ \\\\operatorname{Circ}(r,r^{\\\\prime }) ] \\\\ge 1-\\\\tfrac{1}{c^{\\\\prime }}\\\\left(\\\\tfrac{r}{r^{\\\\prime }}\\\\right)^{c^{\\\\prime }}.$ Under the $2+\\\\varepsilon $ moment condition (), Item $(\\\\ref {item:6})$ holds also for $\\\\operatorname{Circ}$ replaced by $\\\\operatorname{Circ}^\\\\star $ .\",\n \"We begin with the proof of Item (REF ).\",\n \"It suffices to prove the statement for $r^{\\\\prime }\\\\ge 16r$ .\",\n \"Let $r \\\\ge 1$ and $r^{\\\\prime }\\\\ge 16r$ be given, and assume that $\\\\inf _{r\\\\le s\\\\le r^{\\\\prime }/2} {\\\\mathbb {P}}_\\\\lambda [\\\\operatorname{Circ}^\\\\star (s,2s)]\\\\ge c>0.$ Let $g(s) := \\\\sup _{s^{\\\\prime }\\\\ge s} \\\\rho _\\\\lambda (s^{\\\\prime },2s^{\\\\prime })$ .\",\n \"By Proposition REF , we have $\\\\lim _{s\\\\rightarrow \\\\infty }g(s) = 0$ .\",\n \"Set $t = \\\\sqrt{rr^{\\\\prime }}$ and $\\\\ell _i=4^it$ .\",\n \"Note that if $\\\\operatorname{Circ}^\\\\star (r,r^{\\\\prime })$ fails to occur, then $A_i=\\\\operatorname{Circ}^\\\\star (\\\\ell _i,2\\\\ell _i)$ cannot occur for no $i=0,1,\\\\ldots ,k-1$ where $k=\\\\lfloor \\\\frac{1}{2}\\\\log _4(r^{\\\\prime }/r)\\\\rfloor $ ($k\\\\ge 1$ since we assumed $r^{\\\\prime }\\\\ge 16r$ ).\",\n \"The hypothesis (REF ) implies that ${\\\\mathbb {P}}_\\\\lambda (A_i^c)\\\\le 1-c<1$ .\",\n \"Therefore, ${\\\\mathbb {P}}_\\\\lambda \\\\Big [ {\\\\textstyle \\\\bigcap \\\\limits }_{i = 0}^{k-1} A_i^c \\\\Big ]\\\\,&\\\\le \\\\, {\\\\mathbb {P}}_\\\\lambda [A_{k-1}] {\\\\mathbb {P}}_\\\\lambda \\\\Big [ {\\\\textstyle \\\\bigcap \\\\limits }_{i = 0}^{k-2} A_i^c \\\\Big ] + \\\\rho _\\\\lambda (2\\\\ell _{k-2},\\\\ell _{k-2})\\\\\\\\&\\\\le \\\\, (1-c) {\\\\mathbb {P}}_\\\\lambda \\\\Big [ {\\\\textstyle \\\\bigcap \\\\limits }_{i = 0}^{k-2} A_i^c \\\\Big ] + g(t),$ which by induction gives that ${\\\\mathbb {P}}_\\\\lambda \\\\Big [ {\\\\textstyle \\\\bigcap \\\\limits }_{i = 0}^{k-1} A_i^c \\\\Big ] \\\\,\\\\le \\\\, (1-c)^k + \\\\frac{1}{c} g(t).$ By the choice of $k$ , and the fact that $g(t)\\\\le g(t/r) = g(\\\\sqrt{r^{\\\\prime }/r})$ , we finally obtain ${\\\\mathbb {P}}_\\\\lambda \\\\big [ \\\\operatorname{Circ}^\\\\star (r,r^{\\\\prime })^c \\\\big ] \\\\,\\\\le \\\\,{\\\\mathbb {P}}_\\\\lambda \\\\Big [ {\\\\textstyle \\\\bigcap \\\\limits }_{i = 0}^{k-1} A_i^c \\\\Big ] \\\\,\\\\le \\\\, \\\\frac{1}{1-c}\\\\Big ( \\\\frac{r^{\\\\prime }}{r} \\\\Big )^{\\\\frac{1}{2}\\\\log _4(1-c)}+ \\\\frac{1}{c}g\\\\left(\\\\sqrt{\\\\tfrac{r^{\\\\prime }}{r}}\\\\right).$ Hence, (REF ) holds with $f(x)=\\\\frac{1}{1-c}x^\\\\alpha +\\\\frac{1}{c}g(x^{-1/2})$ for any sufficiently small constant $\\\\alpha >0$ .\",\n \"The stronger moment condition () implies (REF ), which readily gives that $f$ may be chosen of the form $\\\\frac{1}{c^{\\\\prime }}x^{c^{\\\\prime }}$ for some constant $c^{\\\\prime }=c^{\\\\prime }(c,\\\\lambda )>0$ , which also proves Item (REF ).\",\n \"We now turn to the proof of Item (REF ) and reuse the above notation.\",\n \"Set $B_i=\\\\operatorname{Circ}(\\\\ell _i,2\\\\ell _i)$ .\",\n \"As above, by (REF ) it follows that ${\\\\mathbb {P}}_\\\\lambda \\\\big (B_i^c\\\\big )\\\\le 1-c$ for every $i=0,1,\\\\ldots ,k-1$ .\",\n \"By Proposition REF we may fix $r_0$ large enough so that $\\\\bar{\\\\rho }_\\\\lambda (2r,r) \\\\le c/2\\\\quad \\\\text{for every }r\\\\ge r_0.$ Observe that the events $B_i$ are decreasing.\",\n \"Hence, for $r\\\\ge r_0$ , $r^{\\\\prime }\\\\ge 16r$ and $t=\\\\sqrt{rr^{\\\\prime }}$ , we obtain $\\\\begin{aligned}{\\\\mathbb {P}}_\\\\lambda \\\\big ( \\\\operatorname{Circ}(r, r^{\\\\prime })^c \\\\big ) \\\\,& \\\\le \\\\, {\\\\mathbb {P}}_\\\\lambda \\\\Big ( {\\\\textstyle \\\\bigcap \\\\limits }_{i = 0}^{k-1}B_i^c \\\\Big )\\\\\\\\& \\\\le \\\\,{\\\\mathbb {P}}_\\\\lambda \\\\Big ( {\\\\textstyle \\\\bigcap \\\\limits }_{i = 1}^{k-1} B_i^c \\\\Big ) \\\\Big ( \\\\mathbb {P}_\\\\lambda (B_0^c) + \\\\bar{\\\\rho }_\\\\lambda (2\\\\ell _0, \\\\ell _0) \\\\Big )\\\\\\\\& \\\\le \\\\,\\\\prod _{i = 0}^{k-1} \\\\Big ( \\\\mathbb {P}_\\\\lambda \\\\big ( B_i^c \\\\big ) + \\\\bar{\\\\rho }_\\\\lambda (2\\\\ell _i, \\\\ell _i) \\\\Big )\\\\\\\\& \\\\le \\\\,(1-c/2)^{\\\\frac{1}{2}\\\\log _4(r^{\\\\prime }/r)-1}.\\\\end{aligned}$ By allowing for a larger constant on the right-hand side we may replace the assumption $r\\\\ge r_0$ with $r\\\\ge 1$ .\",\n \"This concludes the proof of (REF ).\"\n ],\n [\n \"Proof of the consequences of Theorem \",\n \"We now assume the validity of Theorem REF and prove Corollaries REF -REF .\",\n \"[Proof of Corollary REF ] We will show that if $\\\\lim _{r\\\\rightarrow \\\\infty }{\\\\mathbb {P}}(\\\\operatorname{Cross}(\\\\kappa r,r))=1$ for some $\\\\kappa >0$ , then $\\\\lim _{r\\\\rightarrow \\\\infty }{\\\\mathbb {P}}(\\\\operatorname{Cross}(2\\\\kappa r,r))=1$ , from which the statement follows via iteration.\",\n \"Partition the right-hand side of the rectangle $R=[0,\\\\kappa r]\\\\times [0,r]$ into $n$ equal intervals (of length $r/n$ ), and denote the event that there is a horizontal crossing of $R$ into the $k$ -th of these intervals by $A_k$ .\",\n \"Using the square-root trick (REF ), we find that $\\\\max _{1\\\\le k\\\\le n}{\\\\mathbb {P}}_\\\\lambda (A_k)\\\\ge 1-[1-{\\\\mathbb {P}}_\\\\lambda (\\\\operatorname{Cross}(\\\\kappa r,r))]^{1/n},$ which by assumption tends to 1 as $r\\\\rightarrow \\\\infty $ .\",\n \"By assumption, Theorem REF shows that $\\\\inf _{r\\\\ge 1}{\\\\mathbb {P}}_\\\\lambda (\\\\operatorname{Cross}(4r,r))>0$ .\",\n \"Hence, the standard inequalities of Lemma REF show that $\\\\inf _{r\\\\ge 1}{\\\\mathbb {P}}_\\\\lambda (\\\\operatorname{Circ}(r,2r))>0$ .\",\n \"Hence, by Lemma REF we conclude that for large enough $r$ , ${\\\\mathbb {P}}_\\\\lambda (\\\\operatorname{Circ}( r/n,\\\\kappa r/2))\\\\ge 1-f(\\\\kappa n/2),$ for some function $f$ such that $f(x)\\\\rightarrow 0$ as $x\\\\rightarrow \\\\infty $ .\",\n \"Finally, we combine the above estimates to obtain that for every $\\\\varepsilon >0$ ${\\\\mathbb {P}}_\\\\lambda \\\\big (\\\\operatorname{Cross}(2\\\\kappa r,r)\\\\big )\\\\,\\\\ge \\\\,{\\\\mathbb {P}}_\\\\lambda \\\\Bigg [\\\\hbox{[scale=1.5]{(0, 0) rectangle (6ex, 8ex) (6ex, 0) rectangle (12ex, 8ex);(5.8ex, 4.2ex) -- (6.2ex, 4.2ex) (5.8ex, 4.8ex) -- (6.2ex, 4.8ex);[solid] (0, 3ex) .. controls (2ex, 2ex) and (4ex, 5ex) .. (6ex, 4.65ex);[solid] (6ex, 4.35ex) .. controls (8ex, 4ex) and (8ex, 7ex) .. (12ex, 6ex);[solid,decorate,decoration={brace,amplitude=.7ex,raise=.2ex},yshift=0pt] (12ex, 0ex) -- (0, 0);(5ex, 3.5ex) rectangle (7ex, 5.5ex);(3ex, 1.5ex) rectangle (9ex, 7.5ex);(3ex, 1ex) -- (9ex, 1ex);(3ex, .8ex) -- (3ex, 1.2ex) (9ex, .8ex) -- (9ex, 1.2ex);[fill, color=white] (4ex, .2ex) rectangle (8ex, 1.3ex);[solid,rounded corners=.25em](4ex, 6.5ex) .. controls (6ex, 6.5ex + .4ex*rand) ..(8ex, 6.5ex) .. controls (8ex + .4ex*rand, 4.5ex) ..(8ex, 2.5ex) .. controls (6ex, 2.5ex + .4ex*rand) ..(4ex, 2.5ex) -- (4ex + .4ex*rand, 4.5ex) -- cycle;(6ex, -.4ex) node[anchor=north] {$\\\\scriptstyle 2\\\\kappa r$};(.4ex, 4ex) node[anchor=east] {$\\\\scriptstyle r$};(9.4ex, 3.5ex) -- (9.4ex, 5.5ex);(9.2ex, 3.5ex) -- (9.6ex, 3.5ex) (9.2ex, 5.5ex) -- (9.6ex, 5.5ex);(9ex, 4.5ex) node[anchor=west] {$\\\\scriptstyle \\\\tfrac{r}{n}$};(6ex, 1.7ex) node[anchor=north] {$\\\\scriptstyle \\\\kappa r$};}\\\\;\\\\;}$ ] P(Ak)2 P(Circ(r/n,r/2)) > 1-, $for some choice of $ $ and $ n$, and all large $ r$, as required.$ [Proof of Corollary REF ] Take $\\\\theta > 0$ and $r_0 = r_0(\\\\lambda _1)$ so that Proposition REF is in force.\",\n \"Then, ${\\\\mathbb {P}}_{\\\\lambda _1}(\\\\operatorname{Cross}(3r, r))\\\\le 1-\\\\theta \\\\text{ for all $r\\\\ge r_0$.\",\n \"}$ Otherwise, by continuity, we could find $\\\\varepsilon > 0$ and $r\\\\ge r_0$ such that ${\\\\mathbb {P}}_{\\\\lambda _1-\\\\varepsilon }(\\\\operatorname{Cross}(3r,r))>1-\\\\theta $ , which by Proposition REF would imply that the probability indeed converges to 1; a contradiction to the definition of $\\\\lambda _1$ .\",\n \"Analogously, we have ${\\\\mathbb {P}}_{\\\\lambda _0}(\\\\operatorname{Cross}(r, 3r)) \\\\ge \\\\theta $ for all large $r$ .\",\n \"Using Theorem REF and its dual version this establishes (REF ).\",\n \"Equation (REF ) follows from the fact that the complement of $\\\\operatorname{Arm}(r,3r)$ is $\\\\operatorname{Circ}^\\\\star (r,3r)$ and the standard inequalities of Lemma REF .\",\n \"[Proof of Corollary REF ] This is an immediate consequence of Corollary REF and Lemma REF , as ${\\\\mathbb {P}}_\\\\lambda (\\\\operatorname{Arm}(r,r^{\\\\prime }))=1-{\\\\mathbb {P}}_\\\\lambda (\\\\operatorname{Circ}^\\\\star (r,r^{\\\\prime }))$ .\",\n \"[Proof of Corollary REF ] The bounds $\\\\lambda _c\\\\ge \\\\lambda _1$ and $\\\\lambda _c^\\\\star \\\\le \\\\lambda _0$ follow respectively from Item (REF ) and Item (REF ) of Corollary REF .\"\n ],\n [\n \"Proof of Theorem \",\n \"In this section we prove Theorem REF .\",\n \"We assume throughout the proof the second moment condition (REF ) and that $\\\\lambda >0$ is fixed.\",\n \"By assumption there exists $\\\\kappa \\\\in (0,\\\\tfrac{1}{3}]$ such that $\\\\inf _{r\\\\ge 1}{\\\\mathbb {P}}_\\\\lambda (\\\\operatorname{Cross}(\\\\kappa r,r))>0$ .\",\n \"Clearly, there is no restriction assuming the upper bound on $\\\\kappa $ .\",\n \"Hence, we note that either also $\\\\inf _{r\\\\ge 1}{\\\\mathbb {P}}_\\\\lambda (\\\\operatorname{Cross}^\\\\star (\\\\kappa r,r))>0$ , or we have $\\\\sup _{r\\\\ge 1}{\\\\mathbb {P}}_\\\\lambda (\\\\operatorname{Cross}(3r,r))=1.$ The latter implies, via Proposition REF , that $\\\\lim _{r\\\\rightarrow \\\\infty }{\\\\mathbb {P}}_\\\\lambda (\\\\operatorname{Cross}(3r,r))=1$ , in which case there is nothing left to prove.\",\n \"So, we may without loss of generality assume that for some $\\\\kappa >0$ $\\\\inf _{r\\\\ge 1} \\\\mathbb {P}_\\\\lambda \\\\big (\\\\operatorname{Cross}(\\\\kappa r, r) \\\\big ) > 0\\\\quad \\\\text{and}\\\\quad \\\\inf _{r\\\\ge 1} \\\\mathbb {P}_\\\\lambda \\\\big ( \\\\operatorname{Cross}^\\\\star (\\\\kappa r, r) \\\\big ) > 0.$ Moreover, by the standard inequalities of Lemma REF (part (REF ) and its dual version) this is equivalent to assuming the existence of a constant $c_{\\\\ref {c:easy_cross}}>0$ such that $\\\\inf _{r\\\\ge 1} \\\\mathbb {P}_\\\\lambda \\\\big ( \\\\operatorname{Cross}\\\\big (r, \\\\tfrac{5}{4}r\\\\big ) \\\\big ) \\\\ge c_{\\\\ref {c:easy_cross}} \\\\quad \\\\text{and} \\\\quad \\\\inf _{r\\\\ge 1} \\\\mathbb {P}_\\\\lambda \\\\big ( \\\\operatorname{Cross}^\\\\star \\\\big (r, \\\\tfrac{5}{4}r\\\\big ) \\\\big ) \\\\ge c_{\\\\ref {c:easy_cross}}.$ In what follows we will explore several consequences of the above assumption.\",\n \"This will be done in a series of lemmas that will culminate in the proof of Theorem REF .\",\n \"Throughout this section we will need to introduce a range of events that are closely related to the crossing event defined in (REF ).\",\n \"We will typically depict these events graphically, since we believe the proof is better explained this way.\",\n \"However, we emphasize that it is always possible to give a formal definition.\",\n \"For instance, recall the definition of $[A \\\\overset{\\\\mathcal {C}}{\\\\leftrightarrow } B]$ in (REF ).\",\n \"We start with the definition of the event of having a vacant crossing above an occupied crossing, both reaching the right-hand side on the upper half: $\\\\hbox{[scale=1.5]{(0, 0) rectangle (2ex, 6ex);(1.8ex, 3ex) -- (2.2ex, 3ex);[densely dotted] (0, 5ex) .. controls (.6ex, 6ex) and (1.4ex, 4ex) .. (2ex, 5ex);(0, 3.5ex) .. controls (.6ex, 5ex) and (1.4ex, 2.5ex) .. (2ex, 4ex);(1ex, .2ex) node[anchor=north] {$\\\\scriptstyle r$};(0ex, 3ex) node[anchor=east] {$\\\\scriptstyle h$};}\\\\;}$ := q [0, h] Q [ ( {0} [0,q] O K A ) ( {0} [q, h] V K A ) ], where $K = [0, r] \\\\times [0,h]$ and $A = \\\\lbrace r\\\\rbrace \\\\times [h/2, h]$ .\",\n \"Lemma 4.8 There exist $r_0\\\\ge 1$ , $0 < c_{\\\\ref {c:prob_dual}} < 1/8$ and $c_{\\\\ref {c:box_height}} \\\\ge 23$ such that $\\\\inf _{r\\\\ge r_0} \\\\mathbb {P}_\\\\lambda \\\\Bigg [ \\\\hbox{[scale=1.5]{(0, 0) rectangle (2ex, 6ex);(1.8ex, 3ex) -- (2.2ex, 3ex);[densely dotted] (0, 5ex) .. controls (.6ex, 6ex) and (1.4ex, 4ex) .. (2ex, 5ex);(0, 3.5ex) .. controls (.6ex, 5ex) and (1.4ex, 2.5ex) .. (2ex, 4ex);(1ex, .2ex) node[anchor=north] {$\\\\scriptstyle r/c_{\\\\ref {c:box_height}}$};(0ex, 3ex) node[anchor=east] {$\\\\scriptstyle \\\\tfrac{r}{3}$};}\\\\;}$ ] cREF and $\\\\sup _{r\\\\ge r_0} \\\\mathbb {P}_\\\\lambda \\\\big ( \\\\operatorname{Cross}(\\\\tfrac{r}{3}, \\\\tfrac{r}{c_{\\\\ref {c:box_height}}}) \\\\big ) \\\\le \\\\frac{1}{16} \\\\quad \\\\text{and} \\\\quad \\\\sup _{r\\\\ge r_0} \\\\mathbb {P}_\\\\lambda \\\\big ( \\\\operatorname{Cross}^\\\\star (\\\\tfrac{r}{3},\\\\tfrac{r}{c_{\\\\ref {c:box_height}}}) \\\\big ) \\\\le \\\\frac{1}{16}.$ We first argue for the existence of $r_0\\\\ge 1$ and $c_{\\\\ref {c:box_height}}>18$ for which (REF ) holds.\",\n \"Given $s\\\\ge 1$ , set $A_0:=\\\\operatorname{Cross}(\\\\tfrac{5}{4}s,s)$ and let $A_i$ denote the translate of $A_i$ along the vector $(2is,0)$ .\",\n \"Translation invariance, duality and (REF ) imply that for every $i$ , ${\\\\mathbb {P}}_\\\\lambda (A_i)= 1- {\\\\mathbb {P}}_\\\\lambda \\\\big ( \\\\operatorname{Cross}^\\\\star \\\\big (s, \\\\tfrac{5}{4}s \\\\big ) \\\\big )\\\\le 1-c_{\\\\ref {c:easy_cross}}.$ If for some $k\\\\ge 1$ $\\\\operatorname{Cross}(2ks, s)$ occurs, then $A_i$ occur for all $i=0,1,\\\\ldots ,k-1$ .\",\n \"Therefore, ${\\\\mathbb {P}}_\\\\lambda \\\\big ( \\\\operatorname{Cross}(2ks,s)\\\\big )\\\\,\\\\le \\\\, {\\\\mathbb {P}}_\\\\lambda \\\\Big (\\\\bigcap _{i=0}^{k-1} A_i \\\\Big )\\\\,\\\\le \\\\, {\\\\mathbb {P}}_\\\\lambda \\\\Big ( \\\\bigcap _{i=0}^{k-2}A_i \\\\Big )(1-c_{\\\\ref {c:easy_cross}}) +\\\\rho _\\\\lambda (s, \\\\tfrac{3}{4} s).$ Via induction in $k$ we obtain that ${\\\\mathbb {P}}_\\\\lambda \\\\big ( \\\\operatorname{Cross}(2ks,s)\\\\big )\\\\le (1-c_{\\\\ref {c:easy_cross}})^k+\\\\frac{1}{c_{\\\\ref {c:easy_cross}}}\\\\rho _\\\\lambda (s,\\\\tfrac{3}{4} s).$ For $k\\\\ge 4$ and $s_0\\\\ge 1$ large enough this expression is bounded by $1/16$ for all large $s\\\\ge s_0$ .\",\n \"Hence, the choice $r_0=6ks_0$ and $c_{\\\\ref {c:box_height}}=6k\\\\ge 23$ guarantees that ${\\\\mathbb {P}}_\\\\lambda \\\\big ( \\\\operatorname{Cross}(\\\\tfrac{r}{3},\\\\tfrac{r}{c_{\\\\ref {c:box_height}}})\\\\big )\\\\le \\\\frac{1}{16}.$ for every $r\\\\ge r_0$ .\",\n \"With exactly the same proof, we can show that (REF ) also holds with $\\\\operatorname{Cross}$ replaced by $\\\\operatorname{Cross}^\\\\star $ .\",\n \"Hence (REF ) holds for this choice of $r_0$ and $c_{\\\\ref {c:box_height}}$ .\",\n \"To conclude the proof, we will show that any choice of $c_{\\\\ref {c:box_height}}\\\\ge 23$ implies $\\\\inf _{r \\\\ge r_0} \\\\mathbb {P}_\\\\lambda \\\\Bigg [ \\\\hbox{[scale=1.5]{(0, 0) rectangle (2ex, 6ex);(1.8ex, 3ex) -- (2.2ex, 3ex);[densely dotted] (0, 5ex) .. controls (.6ex, 6ex) and (1.4ex, 4ex) .. (2ex, 5ex);(0, 3.5ex) .. controls (.6ex, 5ex) and (1.4ex, 2.5ex) .. (2ex, 4ex);(1ex, .2ex) node[anchor=north] {$\\\\scriptstyle r/c_{\\\\ref {c:box_height}}$};(0ex, 3ex) node[anchor=east] {$\\\\scriptstyle \\\\tfrac{r}{3}$};}\\\\;}$ ] > 0.\",\n \"To prove this, observe that $\\\\begin{split}\\\\mathbb {P}_\\\\lambda \\\\Bigg [\\\\hbox{[scale=1.5]{(0, 0) rectangle (2ex, 6ex);(1.8ex, 3ex) -- (2.2ex, 3ex);[densely dotted] (0, 5ex) .. controls (.6ex, 6ex) and (1.4ex, 4ex) .. (2ex, 5ex);(0, 3.5ex) .. controls (.6ex, 5ex) and (1.4ex, 2.5ex) .. (2ex, 4ex);(1ex, .2ex) node[anchor=north] {$\\\\scriptstyle r/c_{\\\\ref {c:box_height}}$};(0ex, 3ex) node[anchor=east] {$\\\\scriptstyle \\\\tfrac{r}{3}$};}\\\\;}\\\\end{split} \\\\Bigg ] \\\\,& \\\\ge \\\\,{\\\\mathbb {P}}_\\\\lambda \\\\big ( \\\\operatorname{Cross}(\\\\tfrac{r}{c_{\\\\ref {c:box_height}}}, \\\\tfrac{r}{18})\\\\big ) {\\\\mathbb {P}}_\\\\lambda \\\\big ( \\\\operatorname{Cross}^\\\\star (\\\\tfrac{r}{c_{\\\\ref {c:box_height}}},\\\\tfrac{r}{18}) \\\\big )-\\\\rho _\\\\lambda (\\\\tfrac{r}{18}, \\\\tfrac{r}{18})\\\\\\\\& \\\\ge \\\\, c_{\\\\ref {c:easy_cross}}^2- \\\\rho _\\\\lambda (\\\\tfrac{r}{18},\\\\tfrac{r}{18}).$ By (REF ), this shows that the probability on the left hand side of (REF ) is larger than $ c_{\\\\ref {c:easy_cross}}^2/2$ for $r$ large enough, as required.\",\n \"The next event we will be interested in is $\\\\hbox{[scale=1.5]{(0, 2ex) .. controls (1.4ex, 1ex) and (.6ex, 3ex) .. (2ex, 3.3ex);[densely dotted] (0, 4ex) .. controls (1.4ex, 5ex) and (.6ex, 3.2ex) .. (2ex, 3.7ex);(0, 0) rectangle (2ex, 6ex);(1.8ex, 2ex) -- (2.2ex, 2ex) (1.8ex, 3ex) -- (2.2ex, 3ex) (1.8ex, 4ex) -- (2.2ex, 4ex);[decorate,decoration={brace,amplitude=.5ex,raise=.4ex},yshift=0pt] (2.2ex, 4ex) -- (2.2ex, 2ex);(1ex, .2ex) node[anchor=north] {$\\\\scriptstyle r$};(0ex, 3ex) node[anchor=east] {$\\\\scriptstyle h$};(2.4ex, 3ex) node[anchor=west] {$\\\\scriptstyle 2a$};}}$ := q [0,h] Q [ ( {0} [0,q] O K A ) ( {0} [q, h] V K A ) ], where $K = [0,r] \\\\times [0,h]$ and $A = \\\\lbrace r\\\\rbrace \\\\times [h/2, h/2 + a]$ .\",\n \"Definition 4.9 For every $r\\\\ge r_0$ , let $0 \\\\le \\\\alpha _r\\\\le r/2$ be such that $\\\\mathbb {P}_\\\\lambda \\\\Bigg [ \\\\hbox{[scale=1.5]{(0, 2ex) .. controls (1.4ex, 1ex) and (.6ex, 3ex) .. (2ex, 3.3ex);[densely dotted] (0, 4ex) .. controls (1.4ex, 5ex) and (.6ex, 3.2ex) .. (2ex, 3.7ex);(0, 0) rectangle (2ex, 6ex);(1.8ex, 2ex) -- (2.2ex, 2ex) (1.8ex, 3ex) -- (2.2ex, 3ex) (1.8ex, 4ex) -- (2.2ex, 4ex);[decorate,decoration={brace,amplitude=.5ex,raise=.4ex},yshift=0pt] (2.2ex, 4ex) -- (2.2ex, 2ex);(1ex, .2ex) node[anchor=north] {$\\\\scriptstyle r/c_{\\\\ref {c:box_height}}$};(0ex, 3ex) node[anchor=east] {$\\\\scriptstyle r$};(2.4ex, 3ex) node[anchor=west] {$\\\\scriptstyle 2 \\\\alpha _r$};}}$ ] = cREF 2.\",\n \"Let us briefly show that $\\\\alpha _r\\\\le \\\\frac{r}{6}.$ By Lemma REF , the quantity $f(a)=\\\\mathbb {P}_\\\\lambda \\\\Bigg [ \\\\hbox{[scale=1.5]{(0, 2ex) .. controls (1.4ex, 1ex) and (.6ex, 3ex) .. (2ex, 3.3ex);[densely dotted] (0, 4ex) .. controls (1.4ex, 5ex) and (.6ex, 3.2ex) .. (2ex, 3.7ex);(0, 0) rectangle (2ex, 6ex);(1.8ex, 2ex) -- (2.2ex, 2ex) (1.8ex, 3ex) -- (2.2ex, 3ex) (1.8ex, 4ex) -- (2.2ex, 4ex);[decorate,decoration={brace,amplitude=.5ex,raise=.4ex},yshift=0pt] (2.2ex, 4ex) -- (2.2ex, 2ex);(1ex, .2ex) node[anchor=north] {$\\\\scriptstyle r/c_{\\\\ref {c:box_height}}$};(0ex, 3ex) node[anchor=east] {$\\\\scriptstyle r$};(2.4ex, 3ex) node[anchor=west] {$\\\\scriptstyle 2a$};}}$ ] is continuous in $a$ , and by Lemma REF , it satisfies $f(0)=0$ and $f(\\\\tfrac{r}{6})\\\\ge c_{\\\\ref {c:prob_dual}}$ .\",\n \"Hence, there exists $0\\\\le \\\\alpha _r\\\\le r/6$ such that $f(\\\\alpha _r)=\\\\tfrac{c_{\\\\ref {c:prob_dual}}}{2}.$ The next event is what we call the fork and it is crucial in what follows $\\\\hbox{[scale=1.5]{(0, 0) rectangle (4ex, 4ex);(-.2ex, 1.5ex) -- (.2ex, 1.5ex) (-.2ex, 2.5ex) -- (.2ex, 2.5ex);(3.8ex, 1.5ex) -- (4.2ex, 1.5ex) (3.8ex, 2.5ex) -- (4.2ex, 2.5ex);[solid] (0, 3ex) .. controls (.5ex, 3ex) and (1ex, 2.5ex).. (1ex, 2ex) .. controls (1ex, 1.5ex) and (.5ex, 1ex) .. (0ex, 1ex);[solid] (1ex, 2ex) .. controls (1.6ex, 3ex) and (2.4ex, 1ex) .. (3ex, 2ex);[solid] (4ex, 3ex) .. controls (3.5ex, 3ex) and (3ex, 2.5ex).. (3ex, 2ex) .. controls (3ex, 1.5ex) and (3.5ex, 1ex) .. (4ex, 1ex);[decorate,decoration={brace,amplitude=.7ex,raise=.4ex},yshift=0pt] (0, 0) -- (0, 4ex);(2ex, .2ex) node[anchor=north] {$\\\\scriptstyle r$};(-.5ex, 2ex) node[anchor=east] {$\\\\scriptstyle h$};(3.8ex, 2ex) node[anchor=west] {$\\\\scriptstyle 2a$};}}$ := [ c ( {0} [0,h/2 - a] O K {r} [h/2 + a, h] ) ( {0} [h/2 + a, h] O K {r} [0,h/2 - a] ) ], and analogously for the dual (where the picture is traced using dashed curves).\",\n \"Lemma 4.10 Assuming (REF ), for every $r\\\\ge r_0$ , one of the following holds $\\\\mathbb {P}_\\\\lambda \\\\bigg [ \\\\hbox{[scale=1.5]{(0, 0) rectangle (4ex, 4ex);(-.2ex, 1.5ex) -- (.2ex, 1.5ex) (-.2ex, 2.5ex) -- (.2ex, 2.5ex);(3.8ex, 1.5ex) -- (4.2ex, 1.5ex) (3.8ex, 2.5ex) -- (4.2ex, 2.5ex);[solid] (0, 3ex) .. controls (.5ex, 3ex) and (1ex, 2.5ex).. (1ex, 2ex) .. controls (1ex, 1.5ex) and (.5ex, 1ex) .. (0ex, 1ex);[solid] (1ex, 2ex) .. controls (1.6ex, 3ex) and (2.4ex, 1ex) .. (3ex, 2ex);[solid] (4ex, 3ex) .. controls (3.5ex, 3ex) and (3ex, 2.5ex).. (3ex, 2ex) .. controls (3ex, 1.5ex) and (3.5ex, 1ex) .. (4ex, 1ex);[decorate,decoration={brace,amplitude=.7ex,raise=.4ex},yshift=0pt] (0, 0) -- (0, 4ex);(2ex, .2ex) node[anchor=north] {$\\\\scriptstyle r$};(-.5ex, 2ex) node[anchor=east] {$\\\\scriptstyle r$};(3.8ex, 2ex) node[anchor=west] {$\\\\scriptstyle 2 \\\\alpha _r$};}}$ ] 2-13 or P[ ] 2-13.\",\n \"By duality and invariance under right-angle's rotations, see (REF ), $\\\\mathbb {P}_\\\\lambda \\\\big ( \\\\operatorname{Cross}(r, r) \\\\big ) + \\\\mathbb {P}_\\\\lambda \\\\big ( \\\\operatorname{Cross}^\\\\star (r, r) \\\\big ) = 1$ The two cases in the statement of the lemma will correspond to which of the above probabilities is at least one half.\",\n \"Therefore, we assume without loss of generality that $\\\\mathbb {P}_\\\\lambda \\\\big ( \\\\operatorname{Cross}(r, r) \\\\big ) \\\\ge \\\\frac{1}{2}.$ Using symmetry and then a simple union bound, we get $\\\\frac{1}{4} \\\\le \\\\mathbb {P}_\\\\lambda \\\\bigg [ \\\\hbox{[scale=1.5]{(0, 0) rectangle (4ex, 4ex);(3.8ex, 2ex) -- (4.2ex, 2ex);(0, 1.7ex) .. controls (1.3ex, 4ex) and (2.6ex, 1.4ex) .. (4ex, 3.2ex);(2ex, .2ex) node[anchor=north] {$\\\\scriptstyle r$};(.4ex, 2ex) node[anchor=east] {$\\\\scriptstyle r$};[decorate,decoration={brace,amplitude=.4ex,raise=.4ex},yshift=0pt] (4ex, 4ex) -- (4ex, 2ex);(4.3ex, 3ex) node[anchor=west] {$\\\\scriptstyle \\\\tfrac{r}{2}$};}\\\\;}$ ] P[ ] + P[ ], where the event appearing in the last term of the above sum can be rigorously defined as $ \\\\hbox{[scale=1.5]{(0, 0) rectangle (4ex, 4ex);(3.7ex, 1.3ex) -- (4.3ex, 1.3ex);(3.8ex, 2ex) -- (4.2ex, 2ex);(3.7ex, 2.7ex) -- (4.3ex, 2.7ex);(0, 2ex) .. controls (1.3ex, 1ex) and (2.6ex, 2ex) .. (4ex, 2.2ex);[densely dotted] (2ex, 4ex) .. controls (2ex, 3ex) and (2.3ex, 3ex) .. (4ex, 2.4ex);[decorate,decoration={brace,amplitude=.3ex,raise=.4ex},yshift=0pt] (4.2ex, 2.7ex) -- (4.2ex, 1.3ex);(.4ex, 2ex) node[anchor=east] {$\\\\scriptstyle r$};(2ex, .2ex) node[anchor=north] {$\\\\scriptstyle r$};(4.4ex, 2ex) node[anchor=west] {$\\\\scriptstyle 2 \\\\alpha _r$};}}$ = q [0, r] Q ( {0} [0,q] O K A2 ) ( A1(q) V K A2 ), $ with $ = [0, r]2$, $ A1(q) = {0} [q, r] [0, r] {r}$ and $ A2 = {r} [r/2, r/2 + r].$$ Recalling that $c_{\\\\ref {c:box_height}} > 1$ and (REF ), we bound the second term in the above sum by a sum of two terms.\",\n \"They respectively correspond to whether the dual path depicted above stays confined in an $(r/c_{\\\\ref {c:box_height}})$ -wide rectangle or not.\",\n \"More precisely, recalling (REF ), $\\\\begin{split}\\\\mathbb {P}_\\\\lambda \\\\bigg [ \\\\hbox{[scale=1.5]{(0, 0) rectangle (4ex, 4ex);(3.7ex, 1.3ex) -- (4.3ex, 1.3ex);(3.8ex, 2ex) -- (4.2ex, 2ex);(3.7ex, 2.7ex) -- (4.3ex, 2.7ex);(0, 2ex) .. controls (1.3ex, 1ex) and (2.6ex, 2ex) .. (4ex, 2.2ex);[densely dotted] (2ex, 4ex) .. controls (2ex, 3ex) and (2.3ex, 3ex) .. (4ex, 2.4ex);[decorate,decoration={brace,amplitude=.3ex,raise=.4ex},yshift=0pt] (4.2ex, 2.7ex) -- (4.2ex, 1.3ex);(.4ex, 2ex) node[anchor=east] {$\\\\scriptstyle r$};(2ex, .2ex) node[anchor=north] {$\\\\scriptstyle r$};(4.4ex, 2ex) node[anchor=west] {$\\\\scriptstyle 2 \\\\alpha _r$};}}\\\\end{split} \\\\bigg ] \\\\,& \\\\le \\\\, \\\\mathbb {P}_\\\\lambda \\\\Bigg [ \\\\hbox{[scale=1.5]{[color=gray!50!white](4.5ex, 2ex) -- (6ex, 2ex) (4.5ex, 4ex) -- (6ex, 4ex);(0, 0) rectangle (6ex, 6ex);(4.5ex, 0) -- (4.5ex, 6ex);(5.7ex, 2.3ex) -- (6.3ex, 2.3ex) (5.8ex, 3ex) -- (6.2ex, 3ex) (5.7ex, 3.7ex) -- (6.3ex, 3.7ex);(4.5ex, 1.5ex) .. controls (5ex, 1ex) and (5ex, 3.5ex) .. (6ex, 3.3ex);[densely dotted] (5ex, 6ex) .. controls (6ex, 5ex) and (4ex, 3.5ex) .. (6ex, 3.5ex);[decorate,decoration={brace,amplitude=.5ex,raise=.5ex},yshift=0pt] (6.2ex, 3.7ex) -- (6.2ex, 2.3ex);[decorate,decoration={brace,amplitude=.5ex,raise=.4ex},yshift=0pt] (6ex, 6ex) -- (6ex, 4ex);[decorate,decoration={brace,amplitude=.3ex,raise=.2ex},yshift=0pt] (4.5ex, 6ex) -- (6ex, 6ex);(.4ex, 3ex) node[anchor=east] {$\\\\scriptstyle r$};(3ex, .2ex) node[anchor=north] {$\\\\scriptstyle r$};(6.5ex, 3ex) node[anchor=west] {$\\\\scriptstyle 2 \\\\alpha _r$};(5.2ex, 6ex) node[anchor=south] {$\\\\scriptstyle r/c_{\\\\ref {c:box_height}}$};(6.3ex, 5ex) node[anchor=west] {$\\\\scriptstyle \\\\tfrac{r}{3}$};}}$ ] + P[ ] 116 + cREF 2 18, where we also have used (REF ), (REF ) and that $c_{\\\\ref {c:prob_dual}}<1/8$ .\",\n \"Therefore $\\\\mathbb {P}_\\\\lambda \\\\bigg [ \\\\hbox{[scale=1.5]{(0, 0) rectangle (4ex, 4ex);(3.7ex, 1.3ex) -- (4.3ex, 1.3ex) (3.8ex, 2ex) -- (4.2ex, 2ex) (3.7ex, 2.7ex) -- (4.3ex, 2.7ex);(0, 2ex) .. controls (1.3ex, 1.4ex) and (2.6ex, 4ex) .. (4ex, 3.4ex);[decorate,decoration={brace,amplitude=.3ex,raise=.4ex},yshift=0pt] (4.2ex, 2.7ex) -- (4.2ex, 1.3ex);(.4ex, 2ex) node[anchor=east] {$\\\\scriptstyle r$};(2ex, .2ex) node[anchor=north] {$\\\\scriptstyle r$};(4.4ex, 2ex) node[anchor=west] {$\\\\scriptstyle \\\\alpha _r$};}}$ ] 18.\",\n \"To finish the bound in (REF ), we are going to apply the FKG inequality, see (REF ) above.\",\n \"We start by defining the events $A_1, A_2,A_3, A_4$ obtained by reflecting the above square along the vertical and horizontal axis.\",\n \"Of course the above bound will remain valid by the assumed symmetry of the system.\",\n \"Also, using the fact that the law is invariant with respect to rotations by right angles, the probability of observing a vertical crossing of the above box is at least one half, by (REF ).\",\n \"It is not difficult to see that the fork event occurs as soon as there is a vertical crossing of the box, together with the four events $A_1, A_2, A_3, A_4$ .\",\n \"Plugging the above together with the FKG-inequality leads to the bound in (REF ).\",\n \"Lemma 4.11 Let $\\\\tilde{\\\\alpha }_{c_{\\\\ref {c:box_height}}r} := \\\\alpha _{c_{\\\\ref {c:box_height}}r} \\\\wedge \\\\big ( \\\\tfrac{2}{3}r\\\\big )$ .\",\n \"Then, for some $c_{\\\\ref {c:hit_middle}} \\\\in (0, c_{\\\\ref {c:easy_cross}})$ , we have $\\\\inf _{r\\\\ge r_0} \\\\mathbb {P}_\\\\lambda \\\\bigg [ \\\\hbox{\\\\;[scale=1.5]{(0, 0) rectangle (3ex, 4ex);(2.8ex, 2ex) -- (3.2ex, 2ex) (2.8ex, 2.8ex) -- (3.2ex, 2.8ex);[solid](0, 2ex) .. controls (1ex, 3.5ex) and (2ex, 1ex) .. (3ex, 2.6ex);(1.5ex, .2ex) node[anchor=north] {$\\\\scriptstyle r$};(.4ex, 2ex) node[anchor=east] {$\\\\scriptstyle \\\\tfrac{4}{3}r$};(2.8ex, 2.4ex) node[anchor=west] {$\\\\scriptstyle \\\\tilde{\\\\alpha }_{c_{\\\\ref {c:box_height}} r}$};}}$ ] cREF and rr0 P[ ] cREF .\",\n \"where the above event is defined as $\\\\lbrace 0\\\\rbrace \\\\times [0, 4r/3] \\\\overset{K}{\\\\leftrightarrow }A$ , with $K = [0,r] \\\\times [0,4r/3]$ and $A = \\\\lbrace r\\\\rbrace \\\\times [2r/3, 2r/3 + \\\\tilde{\\\\alpha }_{c_{\\\\ref {c:box_height}} r}]$ (and analogously for the dual).\",\n \"We only show the first inequality above, since the dual case is completely analogous.\",\n \"For this we split the proof in two cases.\",\n \"Either $\\\\alpha _{c_{\\\\ref {c:box_height}} r} \\\\ge \\\\tfrac{2}{3}r$ , in which case $\\\\tilde{\\\\alpha }_{c_{\\\\ref {c:box_height}} r} = 2r/3$ and the result follows from (REF ), together with the vertical reflection symmetry of the system and a union bound.\",\n \"We now treat the case $\\\\tilde{\\\\alpha }_{c_{\\\\ref {c:box_height}} r} =\\\\alpha _{c_{\\\\ref {c:box_height}} r} < \\\\tfrac{2}{3}r$ .\",\n \"We first recall that (REF ) gives $\\\\mathbb {P}_\\\\lambda \\\\Bigg [ \\\\hbox{[scale=1.5]{(0, 2ex) .. controls (1.4ex, 1ex) and (.6ex, 3ex) .. (2ex, 3.3ex);[color=white] (0, 4ex) .. controls (1.4ex, 5ex) and (.6ex, 3.2ex) .. (2ex, 3.7ex);(0, 0) rectangle (2ex, 6ex);(1.8ex, 2ex) -- (2.2ex, 2ex) (1.8ex, 3ex) -- (2.2ex, 3ex) (1.8ex, 4ex) -- (2.2ex, 4ex);[decorate,decoration={brace,amplitude=.5ex,raise=.4ex},yshift=0pt] (2.2ex, 4ex) -- (2.2ex, 2ex);(1ex, .2ex) node[anchor=north] {$\\\\scriptstyle r$};(0ex, 3ex) node[anchor=east] {$\\\\scriptstyle c_{\\\\ref {c:box_height}}r$};(2.4ex, 3ex) node[anchor=west] {$\\\\scriptstyle 2 \\\\alpha _{c_{\\\\ref {c:box_height}} r}$};}}$ ] cREF 2.\",\n \"We now split the interval $\\\\lbrace r\\\\rbrace \\\\times [c_{\\\\ref {c:box_height}}/2,c_{\\\\ref {c:box_height}}/2 + \\\\alpha _{c_{\\\\ref {c:box_height}} r}]$ into eight equal parts.\",\n \"Then, using the union bound, there must exist some $h_r \\\\in \\\\lbrace c_{\\\\ref {c:box_height}}r/2 + i \\\\alpha _{c_{\\\\ref {c:box_height}} r}/8; i = 0, \\\\dots ,7\\\\rbrace $ , such that $\\\\mathbb {P}_\\\\lambda \\\\Bigg [ \\\\hbox{[scale=1.5]{(0, 2ex) .. controls (1.4ex, 1ex) and (.6ex, 3.3ex) .. (2ex, 3.5ex);(0, 0) rectangle (2ex, 6ex);(1.8ex, 3.3ex) -- (2.2ex, 3.3ex) (1.8ex, 3.8ex) -- (2.2ex, 3.8ex);(1ex, .2ex) node[anchor=north] {$\\\\scriptstyle r$};(0ex, 3ex) node[anchor=east] {$\\\\scriptstyle c_{\\\\ref {c:box_height}}r$};(2ex, 3.5ex) node[anchor=west] {$\\\\scriptstyle I_r$};}}$ ] cREF 16, where $I_r = \\\\lbrace r\\\\rbrace \\\\times [h_r, h_r + \\\\alpha _{c_{\\\\ref {c:box_height}} r}/8]$ .\",\n \"Since we are assuming $\\\\alpha _{c_{\\\\ref {c:box_height}} r} < 2r/3$ , the length of $I_r$ is no larger than $r/12$ .\",\n \"Thus $\\\\begin{split}\\\\mathbb {P}_\\\\lambda \\\\bigg [ \\\\hbox{\\\\;[scale=1.5]{(0, 0) rectangle (3ex, 4ex);(2.8ex, 2ex) -- (3.2ex, 2ex) (2.8ex, 2.8ex) -- (3.2ex, 2.8ex);[solid](0, 2ex) .. controls (1ex, 3.5ex) and (2ex, 1ex) .. (3ex, 2.6ex);(1.5ex, .2ex) node[anchor=north] {$\\\\scriptstyle r$};(.4ex, 2ex) node[anchor=east] {$\\\\scriptstyle \\\\tfrac{4}{3}r$};(2.8ex, 2.4ex) node[anchor=west] {$\\\\scriptstyle \\\\tilde{\\\\alpha }_{c_{\\\\ref {c:box_height}} r}$};}}\\\\end{split} \\\\bigg ] & \\\\ge \\\\mathbb {P}_\\\\lambda \\\\Bigg [ \\\\hbox{[scale=1.5]{(0, 0) rectangle (3ex, 4ex);(2.8ex, 2ex) -- (3.2ex, 2ex);(0, 2ex) .. controls (1ex, 3ex) and (2ex, 1ex) .. (3ex, 3ex);(1.5ex, .2ex) node[anchor=north] {$\\\\scriptstyle r$};(0ex, 2ex) node[anchor=east] {$\\\\scriptstyle \\\\tfrac{5}{4}r$};}\\\\;}$ ] FKG, (REF ), (REF )(cREF 2)2 cREF 16 := cREF Where above we have made a slight abuse of notation in the second term above: The first two events in the triple intersection need to be translated vertically.\",\n \"This concludes the proof.\",\n \"We now define $p_r:= \\\\mathbb {P}_\\\\lambda \\\\bigg [ \\\\; \\\\hbox{[scale=1.5]{(0,0) rectangle (4ex, 3ex);[solid] (0, 1.5ex) .. controls (1.3ex, 2.5ex) and (2.6ex, .5ex) .. (4ex, 1.5ex);(2ex, .2ex) node[anchor=north] {$\\\\scriptstyle 3r/2$};(3.6ex, 1.5ex) node[anchor=west] {$\\\\scriptstyle \\\\tfrac{4}{3} r$};}}$ ] and pr:= P[ ].\",\n \"Lemma 4.12 If $\\\\alpha _{r/2} \\\\ge \\\\frac{2}{5 c_{\\\\ref {c:box_height}}} \\\\alpha _{c_{\\\\ref {c:box_height}}r}$ for some $r \\\\ge 2 r_0$ , then $\\\\max \\\\lbrace p_r, p_r^\\\\star \\\\rbrace \\\\ge c_{\\\\ref {c:rsw}} =c_{\\\\ref {c:rsw}}(c_{\\\\ref {c:box_height}}, c_{\\\\ref {c:hit_middle}})$ .\",\n \"Using Lemma REF we can assume without loss of generality that $\\\\mathbb {P}_\\\\lambda \\\\bigg [ \\\\hbox{[scale=1.5]{(0, 0) rectangle (4ex, 4ex);(-.2ex, 1.5ex) -- (.2ex, 1.5ex) (-.2ex, 2.5ex) -- (.2ex, 2.5ex);(3.8ex, 1.5ex) -- (4.2ex, 1.5ex) (3.8ex, 2.5ex) -- (4.2ex, 2.5ex);[solid] (0, 3ex) .. controls (.5ex, 3ex) and (1ex, 2.5ex).. (1ex, 2ex) .. controls (1ex, 1.5ex) and (.5ex, 1ex) .. (0ex, 1ex);[solid] (1ex, 2ex) .. controls (1.6ex, 3ex) and (2.4ex, 1ex) .. (3ex, 2ex);[solid] (4ex, 3ex) .. controls (3.5ex, 3ex) and (3ex, 2.5ex).. (3ex, 2ex) .. controls (3ex, 1.5ex) and (3.5ex, 1ex) .. (4ex, 1ex);[decorate,decoration={brace,amplitude=.7ex,raise=.4ex},yshift=0pt] (0, 0) -- (0, 4ex);(2ex, .2ex) node[anchor=north] {$\\\\scriptstyle r/2$};(-.5ex, 2ex) node[anchor=east] {$\\\\scriptstyle r/2$};(3.8ex, 2ex) node[anchor=west] {$\\\\scriptstyle 2 \\\\smash{\\\\alpha _{r/2}}$};}}$ ] 2-13 and then prove that $p_r\\\\ge c_{\\\\ref {c:rsw}}$ .\",\n \"If on the other hand the above bound holds for the dual, an identical proof shows that $p^\\\\star _r\\\\ge c_{\\\\ref {c:rsw}}$ .\",\n \"Recall that we are assuming $\\\\alpha _{r/2} \\\\ge \\\\frac{2}{5c_{\\\\ref {c:box_height}}} \\\\alpha _{c_{\\\\ref {c:box_height}} r} \\\\ge \\\\frac{2}{5c_{\\\\ref {c:box_height}}} \\\\tilde{\\\\alpha }_{c_{\\\\ref {c:box_height}} r}$ .\",\n \"With this, we define the intervals $I_j = \\\\lbrace r\\\\rbrace \\\\times [(2/3) r+ (j-1) \\\\alpha _{r/ 2},(2/3) r+ j \\\\alpha _{r/ 2}]$ , for $j = 1, \\\\dots , \\\\lceil 2 c_{\\\\ref {c:box_height}}/ 5 \\\\rceil $ , which cover the interval $A$ defined below (REF ).\",\n \"Therefore, the union bound yields $\\\\begin{split}\\\\max _{j \\\\le \\\\lceil 2 c_{\\\\ref {c:box_height}}/5 \\\\rceil } \\\\mathbb {P}_\\\\lambda & \\\\bigg [ \\\\hbox{\\\\;[scale=1.5]{(0, 0) rectangle (3ex, 4ex);(2.8ex, 2.2ex) -- (3.2ex, 2.2ex) (2.8ex, 2.5ex) -- (3.2ex, 2.5ex);[solid](0, 2ex) .. controls (1ex, 3.5ex) and (2ex, 2ex) .. (3ex, 2.4ex);(1.5ex, .2ex) node[anchor=north] {$\\\\scriptstyle r$};(.4ex, 2ex) node[anchor=east] {$\\\\scriptstyle \\\\tfrac{4}{3}r$};(2.8ex, 2.4ex) node[anchor=west] {$\\\\scriptstyle I_j$};}}\\\\end{split} \\\\bigg ]\\\\, \\\\ge \\\\, \\\\frac{1}{\\\\lceil 2 c_{\\\\ref {c:box_height}} / 5 \\\\rceil } \\\\sum _{j \\\\le \\\\lceil 2 c_{\\\\ref {c:box_height}} / 5 \\\\rceil } \\\\mathbb {P}_\\\\lambda \\\\bigg [ \\\\hbox{\\\\;[scale=1.5]{(0, 0) rectangle (3ex, 4ex);(2.8ex, 2.2ex) -- (3.2ex, 2.2ex) (2.8ex, 2.5ex) -- (3.2ex, 2.5ex);[solid](0, 2ex) .. controls (1ex, 3.5ex) and (2ex, 2ex) .. (3ex, 2.4ex);(1.5ex, .2ex) node[anchor=north] {$\\\\scriptstyle r$};(.4ex, 2ex) node[anchor=east] {$\\\\scriptstyle \\\\tfrac{4}{3}r$};(2.8ex, 2.4ex) node[anchor=west] {$\\\\scriptstyle I_j$};}}$ ] 12 cREF / 5 P[ ] cREF 2 cREF / 5 =: cREF .\",\n \"Let $j_o$ be the index attaining the maximum in the left hand side of the above equation.\",\n \"Since the interval $I_{j_o}$ has length $\\\\alpha _{r/ 2}$ , it can be covered by the small interval appearing in (REF ), such as illustrated in the picture below $p_r\\\\,\\\\ge \\\\, \\\\mathbb {P}_\\\\lambda \\\\Biggl [ \\\\hbox{[scale=1.5]{(0, 0) rectangle (4ex, 4ex);(-4ex, -3.5ex) rectangle (4ex, 4.5ex) (0, -3.5ex) rectangle (8ex, 4.5ex);(-.2ex, 1.5ex) -- (.2ex, 1.5ex) (-.2ex, 2.5ex) -- (.2ex, 2.5ex);(3.8ex, 1.5ex) -- (4.2ex, 1.5ex) (3.8ex, 2.5ex) -- (4.2ex, 2.5ex);(0, 3ex) .. controls (.5ex, 3ex) and (1ex, 2.5ex).. (1ex, 2ex) .. controls (1ex, 1.5ex) and (.5ex, 1ex) .. (0ex, 1ex);(1ex, 2ex) .. controls (1.6ex, 3ex) and (2.4ex, 1ex) .. (3ex, 2ex);(4ex, 3ex) .. controls (3.5ex, 3ex) and (3ex, 2.5ex).. (3ex, 2ex) .. controls (3ex, 1.5ex) and (3.5ex, 1ex) .. (4ex, 1ex);(-4ex, -1ex) .. controls (0ex, -2ex) and (0ex, 1ex).. (4ex, 2ex);(0ex, 2ex) .. controls (3ex, 4ex) and (5ex, 3ex).. (8ex, 4ex);[decorate,decoration={brace,amplitude=.5ex,raise=.2ex},yshift=0pt] (4ex, -3.5ex) -- (0ex, -3.5ex);[decorate,decoration={brace,amplitude=.7ex,raise=.2ex},yshift=0pt] (-4ex, -3.5ex) -- (-4ex, 4.5ex);(-4.3ex, .5ex) node[anchor=east] {$\\\\scriptstyle \\\\tfrac{4}{3}r$};(2ex, -3.7ex) node[anchor=north] {$\\\\scriptstyle r/2$};[fill, color=white](4.4ex, 1.1ex) rectangle (6.2ex, 2.7ex);[decorate,decoration={brace,amplitude=.2ex,raise=.2ex},yshift=0pt] (4.2ex, 2.5ex) -- (4.2ex, 1.5ex);(4.2ex, 2ex) node[anchor=west] {$\\\\scriptstyle 2 \\\\smash{\\\\alpha _{r/2}}$};}}$ ] FKG2-13 cREF 2 =: cREF , finishing the proof of the lemma.\",\n \"Remark 10 It is important to observe that $c_{\\\\ref {c:rsw}}$ is strictly smaller than $c_{\\\\ref {c:hit_middle}}^2$ , as one can clearly see from the definitions of $c_{\\\\ref {c:hit_smaller}}$ and $c_{\\\\ref {c:rsw}}$ .\",\n \"This will be used in Lemma REF below.\",\n \"Lemma 4.13 There exists a constant $c_{\\\\ref {c:tiny_rsw}} > 0$ such that the following holds.\",\n \"Suppose that for some $r \\\\ge r_0$ , $p_r\\\\ge c_{\\\\ref {c:rsw}}$ (respectively $p_r^\\\\star \\\\ge c_{\\\\ref {c:rsw}}$ ) and that for some ${r^{\\\\prime }}\\\\ge 30 r$ we have $\\\\alpha _{c_{\\\\ref {c:box_height}}{r^{\\\\prime }}} \\\\le r$ .\",\n \"Then $p_{r^{\\\\prime }}\\\\ge c_{\\\\ref {c:tiny_rsw}}$ (respectively $p_{r^{\\\\prime }}^\\\\star \\\\ge c_{\\\\ref {c:tiny_rsw}}$ ).\",\n \"Using the fact that $p_{r} \\\\ge c_{\\\\ref {c:rsw}}$ and Inequalities (REF ) and (REF ) in Lemma REF , we can conclude that $\\\\mathbb {P} \\\\big ( \\\\operatorname{Circ}(2 r, 4 r) \\\\big ) \\\\ge c_{\\\\ref {c:rsw}}^{124},$ We now recall (REF ) and our assumption that $\\\\alpha _{c_{\\\\ref {c:box_height}}{r^{\\\\prime }}} \\\\le r$ .\",\n \"Using symmetry and the FKG-inequality, we obtain $p_{{r^{\\\\prime }}} \\\\,=\\\\, {\\\\mathbb {P}}_\\\\lambda \\\\big (\\\\operatorname{Cross}(3 {r^{\\\\prime }}/2, 4 {r^{\\\\prime }}/3)\\\\big ) \\\\,\\\\ge \\\\, \\\\mathbb {P}_\\\\lambda \\\\Biggl [ \\\\hbox{[scale=1.5]{(0, 0) rectangle (6ex, 8ex) (6ex, 0) rectangle (12ex, 8ex);(5.8ex, 4.2ex) -- (6.2ex, 4.2ex) (5.8ex, 4.8ex) -- (6.2ex, 4.8ex);[solid] (0, 3ex) .. controls (2ex, 2ex) and (4ex, 5ex) .. (6ex, 4.65ex);[solid] (6ex, 4.35ex) .. controls (8ex, 4ex) and (8ex, 7ex) .. (12ex, 6ex);[solid,decorate,decoration={brace,amplitude=.7ex,raise=.2ex},yshift=0pt] (12ex, 0ex) -- (0, 0);(5ex, 3.5ex) rectangle (7ex, 5.5ex);(3ex, 1.5ex) rectangle (9ex, 7.5ex);(3ex, 1ex) -- (9ex, 1ex);(3ex, .8ex) -- (3ex, 1.2ex) (9ex, .8ex) -- (9ex, 1.2ex);[fill, color=white] (4ex, .2ex) rectangle (8ex, 1.3ex);[solid,rounded corners=.25em](4ex, 6.5ex) .. controls (6ex, 6.5ex + .4ex*rand) ..(8ex, 6.5ex) .. controls (8ex + .4ex*rand, 4.5ex) ..(8ex, 2.5ex) .. controls (6ex, 2.5ex + .4ex*rand) ..(4ex, 2.5ex) -- (4ex + .4ex*rand, 4.5ex) -- cycle;(6ex, -.4ex) node[anchor=north] {$\\\\scriptstyle 2 {r^{\\\\prime }}$};(.4ex, 4ex) node[anchor=east] {$\\\\scriptstyle \\\\frac{4}{3}{r^{\\\\prime }}$};(9.4ex, 3.5ex) -- (9.4ex, 5.5ex);(9.2ex, 3.5ex) -- (9.6ex, 3.5ex) (9.2ex, 5.5ex) -- (9.6ex, 5.5ex);(9ex, 4.5ex) node[anchor=west] {$\\\\scriptstyle 2 r$};(6ex, 1.7ex) node[anchor=north] {$\\\\scriptstyle 6 r$};}\\\\;\\\\;}$ ] cREF 2 cREF 124 =: cREF , finishing the proof of the lemma.\",\n \"We have obtained in the previous lemma a condition for $p_{r^{\\\\prime }}\\\\ge c_{\\\\ref {c:tiny_rsw}}$ .\",\n \"However, the constant is smaller than the original $c_{\\\\ref {c:rsw}}$ that lower bounded $p_r$ in the hypothesis of Lemma REF .\",\n \"The purpose of the next lemma is to use the above in order to bootstrap the lower bound of $p_{r^{\\\\prime }}$ back to $c_{\\\\ref {c:rsw}}$ .\",\n \"Lemma 4.14 There exists a constant $c_{\\\\ref {c:boost_rsw}} > 300$ such that the following holds.\",\n \"Suppose that for some $r \\\\ge r_0$ , $p_r\\\\ge c_{\\\\ref {c:rsw}}$ (respectively $p_r^\\\\star $ ) and that for every ${r^{\\\\prime }}\\\\in [30r, c_{\\\\ref {c:boost_rsw}} r]$ we have $\\\\alpha _{c_{\\\\ref {c:box_height}}{r^{\\\\prime }}} \\\\le r$ .\",\n \"Then $p_{2 c_{\\\\ref {c:boost_rsw}} {r^{\\\\prime }}} \\\\ge c_{\\\\ref {c:rsw}}$ (respectively $p_{2 c_{\\\\ref {c:boost_rsw}} {r^{\\\\prime }}}^\\\\star \\\\ge c_{\\\\ref {c:rsw}}$ ).\",\n \"We wish to apply the circuit argument of Lemma REF .\",\n \"For this purpose let us fix a function $f=f_{\\\\lambda ,c}$ as in Item (REF ) of Lemma REF , corresponding to the constant $c:=c_{\\\\ref {c:tiny_rsw}}^{124}$ .\",\n \"Recall from Remark REF that $c_{\\\\ref {c:rsw}} < c_{\\\\ref {c:hit_middle}}^2$ , so that we can choose $c_{\\\\ref {c:boost_rsw}} >300$ so that $1-f(\\\\tfrac{300}{c_{\\\\ref {c:boost_rsw}}}) \\\\ge \\\\frac{c_{\\\\ref {c:rsw}}}{c_{\\\\ref {c:hit_middle}}^2}.$ We only prove the statement for the dual quantities, the same proof also works for the primal ones (except that the function $f$ need to be chosen according to Item (REF ) in Lemma REF ).\",\n \"Assume that $p_r^\\\\star \\\\ge c_{\\\\ref {c:rsw}}$ and that for every ${r^{\\\\prime }}\\\\in [30r, c_{\\\\ref {c:boost_rsw}} r]$ we have $\\\\alpha _{c_{\\\\ref {c:box_height}}{r^{\\\\prime }}} \\\\le r$ .\",\n \"Using Lemma REF , we know that for every $r^{\\\\prime \\\\prime } \\\\in [30 r, c_{\\\\ref {c:boost_rsw}}r]$ , we have $p_{r^{\\\\prime \\\\prime }} \\\\ge c_{\\\\ref {c:tiny_rsw}}$ .\",\n \"By the standard inequalities of Lemma REF , this implies in particular $\\\\inf _{60r\\\\le r^{\\\\prime \\\\prime }\\\\le c_{\\\\ref {c:boost_rsw}}r/10}{\\\\mathbb {P}}_\\\\lambda \\\\big (\\\\operatorname{Circ}^\\\\star (r^{\\\\prime \\\\prime },2r^{\\\\prime \\\\prime })\\\\big )\\\\ge c_{\\\\ref {c:tiny_rsw}}^{124}.$ Therefore, using the circuit argument (REF ) of Lemma REF , we obtain ${\\\\mathbb {P}}_\\\\lambda \\\\big (\\\\operatorname{Circ}^\\\\star (60r,\\\\tfrac{c_{\\\\ref {c:boost_rsw}}r}{5})\\\\big )\\\\ge 1-f(\\\\tfrac{300}{c_{\\\\ref {c:boost_rsw}}}) \\\\overset{~(\\\\ref {e:choose_boost})}{\\\\ge } \\\\frac{c_{\\\\ref {c:rsw}}}{c_{\\\\ref {c:hit_middle}}^2}.$ We can finally proceed as in the end of the proof of Lemma REF , observing that $p_{2 c_{\\\\ref {c:boost_rsw}} r}^\\\\star \\\\ge {\\\\mathbb {P}}_\\\\lambda (\\\\operatorname{Cross}^\\\\star (3c_{\\\\ref {c:boost_rsw}} r, c_{\\\\ref {c:boost_rsw}} r))$ , which can be lower bounded by $\\\\mathbb {P}_\\\\lambda \\\\Bigg [ \\\\hbox{[scale=1.5]{(0, 0) rectangle (6ex, 8ex) (6ex, 0) rectangle (12ex, 8ex);(5.8ex, 4.2ex) -- (6.2ex, 4.2ex) (5.8ex, 4.8ex) -- (6.2ex, 4.8ex);[densely dotted] (0, 3ex) .. controls (2ex, 2ex) and (4ex, 5ex) .. (6ex, 4.65ex);[densely dotted] (6ex, 4.35ex) .. controls (8ex, 4ex) and (8ex, 7ex) .. (12ex, 6ex);[densely dotted,decorate,decoration={brace,amplitude=.7ex,raise=.2ex},yshift=0pt] (12ex, 0ex) -- (0, 0);(5ex, 3.5ex) rectangle (7ex, 5.5ex);(3ex, 1.5ex) rectangle (9ex, 7.5ex);(3ex, 1ex) -- (9ex, 1ex);(3ex, .8ex) -- (3ex, 1.2ex) (9ex, .8ex) -- (9ex, 1.2ex);[fill, color=white] (4ex, .2ex) rectangle (8ex, 1.3ex);[densely dotted,rounded corners=.25em](4ex, 6.5ex) .. controls (6ex, 6.5ex + .4ex*rand) ..(8ex, 6.5ex) .. controls (8ex + .4ex*rand, 4.5ex) ..(8ex, 2.5ex) .. controls (6ex, 2.5ex + .4ex*rand) ..(4ex, 2.5ex) -- (4ex + .4ex*rand, 4.5ex) -- cycle;(6ex, -.4ex) node[anchor=north] {$\\\\scriptstyle 3 c_{\\\\ref {c:boost_rsw}}r$};(.4ex, 4ex) node[anchor=east] {$\\\\scriptstyle c_{\\\\ref {c:boost_rsw}} r$};(9.4ex, 3.5ex) -- (9.4ex, 5.5ex);(9.2ex, 3.5ex) -- (9.6ex, 3.5ex) (9.2ex, 5.5ex) -- (9.6ex, 5.5ex);(9ex, 4.5ex) node[anchor=west] {$\\\\scriptstyle 60r$};(6ex, 1.7ex) node[anchor=north] {$\\\\scriptstyle c_{\\\\ref {c:boost_rsw}} r/5$};}\\\\;\\\\;}$ ] cREF 2 cREF cREF 2 = cREF , as required.\",\n \"Lemma 4.15 Fix $r \\\\ge r_0$ and suppose that for some ${r^{\\\\prime }}\\\\in [30 r, c_{\\\\ref {c:boost_rsw}} r]$ we have $\\\\alpha _{c_{\\\\ref {c:box_height}} {r^{\\\\prime }}} \\\\ge r$ , then $\\\\max \\\\lbrace p_{r^{\\\\prime \\\\prime }}, p^\\\\star _{r^{\\\\prime \\\\prime }}\\\\rbrace \\\\ge c_{\\\\ref {c:rsw}}$ for some $r^{\\\\prime \\\\prime } \\\\in [{r^{\\\\prime }}, c_{\\\\ref {c:rsw_blows}} {r^{\\\\prime }}]$ .\",\n \"Let us choose an integer $K = K(c_{\\\\ref {c:box_height}}, {r^{\\\\prime }}/r) \\\\ge 1$ such that $\\\\Big ( \\\\frac{5 c_{\\\\ref {c:box_height}}}{2} \\\\Big )^K r\\\\ge c_{\\\\ref {c:box_height}} (2 c_{\\\\ref {c:box_height}})^K {r^{\\\\prime }}$ and then set $c_{\\\\ref {c:rsw_blows}} := 2 (2 c_{\\\\ref {c:box_height}})^K.$ Observe now that if for some $k = 0, \\\\dots , K - 1$ we have $\\\\alpha _{c_{\\\\ref {c:box_height}}(2 c_{\\\\ref {c:box_height}})^k {r^{\\\\prime }}} \\\\ge \\\\big ( \\\\tfrac{2}{5 c_{\\\\ref {c:box_height}}} \\\\big ) \\\\alpha _{c_{\\\\ref {c:box_height}}(2 c_{\\\\ref {c:box_height}})^{k+1} {r^{\\\\prime }}},$ we can apply Lemma REF , replacing $r$ with $(2c_{\\\\ref {c:box_height}})^{k+1} {r^{\\\\prime }}$ .\",\n \"This will yield the bound $\\\\max \\\\lbrace p_{(2 c_{\\\\ref {c:box_height}})^{k+1} {r^{\\\\prime }}}, p^\\\\star _{(2 c_{\\\\ref {c:box_height}})^{k+1} {r^{\\\\prime }}}\\\\rbrace \\\\ge c_{\\\\ref {c:rsw}}.$ By our choice of $c_{\\\\ref {c:rsw_blows}}$ , we conclude that ${r^{\\\\prime }}\\\\le (2 c_{\\\\ref {c:box_height}})^{k+1} {r^{\\\\prime }}\\\\le (2 c_{\\\\ref {c:box_height}})^K {r^{\\\\prime }}\\\\le c_{\\\\ref {c:rsw_blows}} {r^{\\\\prime }}$ , as desired.\",\n \"On the other hand if (REF ) does not hold for any $k = 0, \\\\dots , K - 1$ , then $r \\\\le \\\\alpha _{c_{\\\\ref {c:box_height}} {r^{\\\\prime }}} \\\\le \\\\big ( \\\\tfrac{2}{5 c_{\\\\ref {c:box_height}}} \\\\big ) \\\\alpha _{c_{\\\\ref {c:box_height}}(2 c_{\\\\ref {c:box_height}}) {r^{\\\\prime }}} \\\\le \\\\dots \\\\le \\\\big ( \\\\tfrac{2}{5 c_{\\\\ref {c:box_height}}} \\\\big )^{K} \\\\alpha _{c_{\\\\ref {c:box_height}}(2 c_{\\\\ref {c:box_height}})^{K} {r^{\\\\prime }}},$ so that by our choice of $K$ we would have $\\\\alpha _{c_{\\\\ref {c:box_height}}(2 c_{\\\\ref {c:box_height}})^{K} {r^{\\\\prime }}} \\\\ge (5 c_{\\\\ref {c:box_height}} / 2)^{K} r \\\\ge c_{\\\\ref {c:box_height}}(2 c_{\\\\ref {c:box_height}})^{K} {r^{\\\\prime }}$ , which is a contradiction with (REF ).\",\n \"Lemma 4.16 There exists $c_{\\\\ref {c:rsw_max}} > 0$ such that $\\\\inf _{r\\\\ge 1} \\\\big (\\\\max \\\\lbrace p_r, p^\\\\star _r\\\\rbrace \\\\big ) \\\\ge c_{\\\\ref {c:rsw_max}}.$ We first claim that there exists an increasing sequence $r_1\\\\le r_2\\\\le \\\\cdots $ such that $3\\\\le \\\\tfrac{r_{i+1}}{r_i}\\\\le \\\\max \\\\lbrace 2c_{\\\\ref {c:boost_rsw}}, c_{\\\\ref {c:rsw_blows}}\\\\rbrace $ for every $i \\\\ge 1$ and $\\\\max \\\\lbrace p_{r_i}, p^\\\\star _{r_i}\\\\rbrace \\\\ge c_{\\\\ref {c:rsw}}$ , for every $i \\\\ge 1$ .\",\n \"We first construct $r_1$ .\",\n \"Since $\\\\alpha _r\\\\le r/6$ for every $r\\\\ge 1$ (see (REF )), there must exist $k\\\\ge 1$ such that $\\\\alpha _{r_0(2 c_{\\\\ref {c:box_height}})^k} \\\\ge \\\\big ( \\\\tfrac{2}{5 c_{\\\\ref {c:box_height}}} \\\\big ) \\\\alpha _{r_0(2 c_{\\\\ref {c:box_height}})^{k+1}},$ Applying Lemma REF to $r_1:=2r_0(2c_{\\\\ref {c:box_height}})^{k}$ yields the bound $\\\\max \\\\lbrace p_{r_1}, p^\\\\star _{r_1}\\\\rbrace \\\\ge c_{\\\\ref {c:rsw}}$ as desired.\",\n \"To finish the construction of the $(r_i)$ 's, it is enough to show that for any $r\\\\ge 1$ such that $p_r\\\\ge c_{\\\\ref {c:rsw}}$ , there exists $r^{\\\\prime \\\\prime } \\\\in [3r, \\\\max \\\\lbrace 2c_{\\\\ref {c:boost_rsw}}, c_{\\\\ref {c:rsw_blows}}\\\\rbrace r]$ for which $\\\\max \\\\lbrace p_{r^{\\\\prime \\\\prime }}, p^\\\\star _{r^{\\\\prime \\\\prime }}\\\\rbrace \\\\ge c_{\\\\ref {c:rsw}}$ .\",\n \"We then split the proof of this claim into two cases, depending on whether or not there exists ${r^{\\\\prime }}\\\\in [3 r, c_{\\\\ref {c:boost_rsw}} r]$ such that $\\\\alpha _{c_{\\\\ref {c:box_height}} {r^{\\\\prime }}} \\\\ge r$ .\",\n \"In this case, Lemma REF shows that the existence of $r^{\\\\prime \\\\prime } \\\\le c_{\\\\ref {c:rsw_blows}} r$ as in (REF ).\",\n \"On the other hand, if (REF ) does not hold, than we can use $r^{\\\\prime \\\\prime } = 2c_{\\\\ref {c:boost_rsw}}$ , according to Lemma REF .\",\n \"In any case we have proved (REF ), which shows the existence of $r_1,r_2, \\\\dots $ as above.\",\n \"To end the proof we use the standard inequalities of Lemma REF to interpolate between the values $r_i$ .\",\n \"We now have obtained a statement similar to the RSW result stated in Theorem REF .\",\n \"However, we only know that the crossing probability is bounded below for the primal or the dual and not both at the same time.\",\n \"It could be still the case that only the dual crossings have a bounded probability.\",\n \"The purpose of the following lemma is to show that this cannot be the case.\",\n \"Lemma 4.17 There exist constants $c_{\\\\ref {c:rsw_both_1}} \\\\ge 10$ and $c_{\\\\ref {c:rsw_both_2}}>0$ such that the following holds.\",\n \"Given $r\\\\ge r_0$ , if $p_{{r^{\\\\prime }}}^\\\\star \\\\ge c_{\\\\ref {c:rsw_max}}$ for every ${r^{\\\\prime }}\\\\in [r, c_{\\\\ref {c:rsw_both_1}} r]$ , then $p_{c_{\\\\ref {c:rsw_both_1}} r} \\\\ge c_{\\\\ref {c:rsw_both_2}}$ .\",\n \"Moreover, the same holds true if we swap the roles of $p^\\\\star $ and $p$ above.\",\n \"We wish to apply the circuit argument of Lemma REF .\",\n \"To this end, let us fix a function $f=f_{\\\\lambda ,c}$ as in Item (REF ) of Lemma REF , corresponding to the constant $c:=c_{\\\\ref {c:rsw_max}}^{124}$ .\",\n \"We first choose $c_{\\\\ref {c:rsw_both_1}} = c_{\\\\ref {c:rsw_both_1}}(c_{\\\\ref {c:easy_cross}},c_{\\\\ref {c:rsw_max}}) \\\\ge 30$ large enough such that $f(\\\\tfrac{10}{c_{\\\\ref {c:rsw_both_1}}}) \\\\le \\\\frac{c_{\\\\ref {c:easy_cross}}}{3}$ and $c_{\\\\ref {c:rsw_both_2}}>0$ small enough that $\\\\big (1 - c_{\\\\ref {c:rsw_both_2}}^{1/(2c_{\\\\ref {c:rsw_both_1}})}\\\\big ) \\\\big ( 1 - \\\\frac{c_{\\\\ref {c:easy_cross}}}{3} \\\\big ) \\\\ge 1 - \\\\frac{c_{\\\\ref {c:easy_cross}}}{2}.$ Suppose now, for a contradiction, that $p_{c_{\\\\ref {c:rsw_both_1}} r} \\\\le c_{\\\\ref {c:rsw_both_2}}$ , therefore $\\\\mathbb {P}_\\\\lambda \\\\big [\\\\operatorname{Cross}^\\\\star \\\\big (\\\\tfrac{4}{3} c_{\\\\ref {c:rsw_both_1}} r, \\\\tfrac{3}{2} c_{\\\\ref {c:rsw_both_1}} r\\\\big ) \\\\big ] = 1 - p_{c_{\\\\ref {c:rsw_both_1}} r} \\\\ge 1 - c_{\\\\ref {c:rsw_both_2}}.$ Cover the interval $[0, \\\\tfrac{3}{2} c_{\\\\ref {c:rsw_both_1}} r]$ by at most $H =\\\\lceil \\\\tfrac{3}{2} c_{\\\\ref {c:rsw_both_1}} \\\\rceil \\\\le 2 c_{\\\\ref {c:rsw_both_1}}$ intervals of length $r$ .\",\n \"The square-root trick (REF ) implies that there exists $h \\\\in \\\\lbrace 0, 1, \\\\dots , H-1\\\\rbrace $ , such that $\\\\mathbb {P}_\\\\lambda \\\\bigg [ \\\\hbox{\\\\;[scale=1.5]{(0, 0) rectangle (3ex, 4ex);(2.8ex, 2ex) -- (3.2ex, 2ex) (2.8ex, 2.8ex) -- (3.2ex, 2.8ex);[densely dotted](0, 2ex) .. controls (1ex, 3.5ex) and (2ex, 1ex) .. (3ex, 2.6ex);(1.5ex, .2ex) node[anchor=north] {$\\\\scriptstyle (4/3) c_{\\\\ref {c:rsw_both_1}} r$};(.4ex, 2ex) node[anchor=east] {$\\\\scriptstyle \\\\tfrac{3}{2} c_{\\\\ref {c:rsw_both_1}} r$};(2.8ex, 2.4ex) node[anchor=west] {$\\\\scriptstyle I_h$};}}$ ] 1 - cREF 1/H, where $I_h = \\\\lbrace \\\\tfrac{4}{3} c_{\\\\ref {c:rsw_both_1}} r\\\\rbrace \\\\times [h r, (h + 1) r]$ .\",\n \"We now recall that $p_{{r^{\\\\prime }}}^\\\\star \\\\ge c_{\\\\ref {c:rsw_max}}$ for every ${r^{\\\\prime }}\\\\in [r, c_{\\\\ref {c:rsw_both_1}} r]$ .\",\n \"Therefore, by the standard inequalities of Lemma REF , we obtain in particular $\\\\inf _{2r\\\\le r^{\\\\prime }\\\\le c_{\\\\ref {c:rsw_both_1}} r/10}{\\\\mathbb {P}}_\\\\lambda \\\\big (\\\\operatorname{Circ}^\\\\star (r^{\\\\prime },2r^{\\\\prime })\\\\big )\\\\,\\\\ge \\\\, c_{\\\\ref {c:rsw_max}}^{124}.$ Therefore, by the circuit argument (REF ) in Lemma REF , $\\\\mathbb {P} \\\\Big ( \\\\operatorname{Circ}^\\\\star (2 r, c_{\\\\ref {c:rsw_both_1}} r/ 5) \\\\Big ) \\\\,\\\\ge \\\\, 1-f(\\\\tfrac{10}{c_{\\\\ref {c:rsw_both_1}}}) \\\\,\\\\overset{(\\\\ref {e:choose_both_1})}{\\\\ge }\\\\, 1 - \\\\frac{c_{\\\\ref {c:easy_cross}}}{3}.$ As in (REF ), we obtain $\\\\mathbb {P}_\\\\lambda \\\\big ( \\\\operatorname{Cross}^\\\\star (\\\\tfrac{8}{3} c_{\\\\ref {c:rsw_both_1}} r, \\\\tfrac{3}{2} c_{\\\\ref {c:rsw_both_1}} r)\\\\big )\\\\,& \\\\ge \\\\, \\\\mathbb {P}_\\\\lambda \\\\Bigg [ \\\\hbox{[scale=1.5]{(0, 0) rectangle (6ex, 8ex) (6ex, 0) rectangle (12ex, 8ex);(5.8ex, 4.2ex) -- (6.2ex, 4.2ex) (5.8ex, 4.8ex) -- (6.2ex, 4.8ex);[solid] (0, 3ex) .. controls (2ex, 2ex) and (4ex, 5ex) .. (6ex, 4.65ex);[solid] (6ex, 4.35ex) .. controls (8ex, 4ex) and (8ex, 7ex) .. (12ex, 6ex);[solid,decorate,decoration={brace,amplitude=.7ex,raise=.2ex},yshift=0pt] (12ex, 0ex) -- (0, 0);(5ex, 3.5ex) rectangle (7ex, 5.5ex);(3ex, 1.5ex) rectangle (9ex, 7.5ex);(3ex, 1ex) -- (9ex, 1ex);(3ex, .8ex) -- (3ex, 1.2ex) (9ex, .8ex) -- (9ex, 1.2ex);[fill, color=white] (4ex, .2ex) rectangle (8ex, 1.3ex);[solid,rounded corners=.25em](4ex, 6.5ex) .. controls (6ex, 6.5ex + .4ex*rand) ..(8ex, 6.5ex) .. controls (8ex + .4ex*rand, 4.5ex) ..(8ex, 2.5ex) .. controls (6ex, 2.5ex + .4ex*rand) ..(4ex, 2.5ex) -- (4ex + .4ex*rand, 4.5ex) -- cycle;(6ex, -.4ex) node[anchor=north] {$\\\\scriptstyle \\\\tfrac{8}{3} c_{\\\\ref {c:rsw_both_1}} r$};(.4ex, 4ex) node[anchor=east] {$\\\\scriptstyle \\\\tfrac{3}{2} c_{\\\\ref {c:rsw_both_1}} r$};(9.4ex, 3.5ex) -- (9.4ex, 5.5ex);(9.2ex, 3.5ex) -- (9.6ex, 3.5ex) (9.2ex, 5.5ex) -- (9.6ex, 5.5ex);(9ex, 4.5ex) node[anchor=west] {$\\\\scriptstyle 2 r$};(6ex, 1.7ex) node[anchor=north] {$\\\\scriptstyle c_{\\\\ref {c:rsw_both_1}}r/5$};}\\\\;\\\\;}$ ] (1 - cREF 1/H)2 (1 - cREF 3 ) (REF ) 1 - cREF 2 which is a contradiction with (REF ).\",\n \"Remark 11 In the proof of Lemmas REF and REF we have focused on the dual paths instead of primal to emphasize that we only needed the weak decoupling inequality (REF ) that has been derived from (REF ).\",\n \"[Proof of Theorem REF ] We first claim that it is enough to prove that for any $r\\\\ge r_0$ there exists some ${r^{\\\\prime }}\\\\in [r, c_{\\\\ref {c:rsw_both_1}} r]$ such that $p^\\\\star _{{r^{\\\\prime }}}\\\\ge \\\\min \\\\lbrace c_{\\\\ref {c:rsw_max}}, c_{\\\\ref {c:rsw_both_2}}\\\\rbrace $ .\",\n \"Indeed, if this is the case, we can use the standard inequalities in Lemma REF to show that $\\\\inf _{r\\\\ge 1} p^\\\\star _{r} > 0$ .\",\n \"(The case $p_r$ is completely analogous.)\",\n \"So, let us fix some $r\\\\ge r_0$ and assume that $p^\\\\star _{{r^{\\\\prime }}} < c_{\\\\ref {c:rsw_max}}$ for every ${r^{\\\\prime }}\\\\in [r,c_{\\\\ref {c:rsw_both_1}} r]$ , then by Lemma REF , we would have $p_{{r^{\\\\prime }}} \\\\ge c_{\\\\ref {c:rsw_max}}$ for every ${r^{\\\\prime }}\\\\in [r, c_{\\\\ref {c:rsw_both_1}} r]$ .\",\n \"But then Lemma REF implies that $p^\\\\star _{c_{\\\\ref {c:rsw_both_1}} r} \\\\ge c_{\\\\ref {c:rsw_both_2}}$ as desired.\"\n ],\n [\n \"Sharpness of the phase transition\",\n \"We have so far showed that decay of correlations is sufficient for the two thresholds $\\\\lambda _0$ and $\\\\lambda _1$ to be non-trivial; see Section .\",\n \"Moreover, we have shown that a distinctive critical behavior, namely the box-crossing property, occurs throughout the interval $[\\\\lambda _0,\\\\lambda _1]$ ; see Section .\",\n \"Our goal for this section is to prove that these two thresholds $\\\\lambda _0$ and $\\\\lambda _1$ are in fact the same.\",\n \"Theorem 5.1 For Poisson Boolean percolation satisfying (REF ), we have $\\\\lambda _0 = \\\\lambda _1$ .\",\n \"In order to establish an equality between the two thresholds $\\\\lambda _0$ and $\\\\lambda _1$ , our aim will be to show that the crossing probabilities grow very fast from close to zero to close to one.\",\n \"More precisely, we have seen that the probability of crossing a fixed-ratio rectangle is bounded away from zero and one for $\\\\lambda \\\\in [\\\\lambda _0, \\\\lambda _1]$ .\",\n \"In this section, we show that this implies that the derivative of the crossing probability is large throughout the same interval.\",\n \"The phenomenon of sharp thresholds is well understood in the context of Boolean functions on the discrete cube $\\\\lbrace 0,1\\\\rbrace ^n$ equipped with product measure.\",\n \"In order to apply the tools from this theory in our context we will following an approach introduced in [1].\",\n \"Thus, the first step of our proof will be to enhance the probability space in which we work to suit this setting, see Subsection REF .\",\n \"After having defined this alternative construction, we will employ Margulis-Russo's formula to relate the derivative of crossing probabilities to influences of these auxiliary random variables.\",\n \"At this point, an inequality of Talagrand will become handy in order to bound the total influence from below.\",\n \"This is done in two steps, where we bound the conditional variance of the crossing probabilities from below and the individual influences from above, see Subsections REF and REF respectively.\",\n \"Subsection REF combines the above bounds to prove Theorem REF .\",\n \"Remark 12 Of the three main theoretical components of this paper, the one presented in this section is the least general one.\",\n \"Although what we will present below does apply to some other models, it strongly uses the Poissonian nature of the process, see Section .\"\n ],\n [\n \"Sharp thresholds for Boolean functions\",\n \"It has been known for some time that monotone events involving a large number of independent Bernoulli variables typically exhibit sharp thresholds, in the sense that the probability of the event increases sharply as the parameter of the Bernoulli variables passes a certain value.\",\n \"The first evidence thereof dates back to the study of random graphs by Erdős and Rényi [14], and the proof that the critical probability for bond percolation on ${\\\\mathbb {Z}}^2$ equals $1/2$ by Kesten [23].\",\n \"A more general understanding of these phenomena has been pursued since in work of Russo [36], Bollobás and Thomason [12], and Kahn, Kalai and Linial [25] as well as elsewhere.\",\n \"See e.g.\",\n \"[31] or [19].\",\n \"For $p \\\\in [0, 1]$ let $\\\\textup {P}_p$ denote product measure of intensity $p$ on $\\\\lbrace 0,1\\\\rbrace ^n$ .\",\n \"A Boolean function $f : \\\\lbrace 0, 1\\\\rbrace ^n \\\\rightarrow \\\\lbrace 0, 1\\\\rbrace $ is said to be monotone when $f(\\\\eta ) \\\\le f(\\\\eta ^{\\\\prime })$ whenever $\\\\eta \\\\le \\\\eta ^{\\\\prime }$ coordinate wise.\",\n \"For monotone Boolean functions, the map $p \\\\mapsto \\\\textup {P}_p(f = 1)$ is also monotone as a function of $p$ .\",\n \"Moreover, the rate of change of $\\\\textup {P}_p(f = 1)$ is related to the concept of influences of the individual variables, as made explicit by the Margulis-Russo formula (see [34]), $\\\\frac{d}{dp} \\\\textup {P}_p(f = 1) \\\\,=\\\\, \\\\sum _{i = 1}^n \\\\operatorname{Inf}_i^p(f),$ where $\\\\operatorname{Inf}_i^p(f) := \\\\textup {P}_p\\\\big (f(\\\\eta ) \\\\ne f(\\\\sigma _i\\\\eta )\\\\big )$ and $\\\\sigma _i$ is the operator that flips the value of $\\\\eta $ at position $i$ .\",\n \"In case $f(\\\\eta ) \\\\ne f(\\\\sigma _i\\\\eta )$ , we say that the bit $i$ is pivotal for $f(\\\\eta )$ .\",\n \"Against one's initial intuition, Russo [36] further showed that a uniform upper bound on the individual influences implies a lower bound of their sum, thus assuring a rapid increase of $\\\\textup {P}_p(f=1)$ .\",\n \"This mystique has since become better understood with the work of Kahn, Kalai and Linial [25] and its extensions, which shows that not all influences may be too small.\",\n \"The following inequality, due to Talagrand [41], illustrates this fact well and gives a precise formulation of Russo's initial observation: There exists a constant $c_{\\\\ref {c:talagrand}} > 0$ such that for every Boolean function $f:\\\\lbrace 0,1\\\\rbrace ^n\\\\rightarrow \\\\lbrace 0,1\\\\rbrace $ we have $\\\\sum _{i=1}^n \\\\operatorname{Inf}_i^p(f) \\\\,\\\\ge \\\\, \\\\frac{c_{\\\\ref {c:talagrand}}}{\\\\log \\\\big ( \\\\tfrac{2}{p (1 - p)} \\\\big )} \\\\, \\\\operatorname{Var}_p(f) \\\\log \\\\Big ( \\\\frac{1}{\\\\max \\\\lbrace \\\\operatorname{Inf}_i^p(f)\\\\rbrace } \\\\Big ).$ Talagrand stated this inequality for monotone functions, but this restriction is not necessary.\",\n \"From (REF ) and (REF ), the strategy of the proof becomes intuitive: We first give an alternative construction of the process involving independent Bernoulli random variables, then we bound the variance of the crossing probabilities from below and the influence of individual bits from above.\",\n \"These are the contents of the next subsections.\"\n ],\n [\n \"A two-stage construction\",\n \"We will for the rest of this section work on an enlarged probability space $(\\\\overline{\\\\Omega },\\\\overline{\\\\mathcal {M}},\\\\overline{{\\\\mathbb {P}}}_\\\\lambda )$ in order to construct our process in two stages, where the second step employs independent coin flips in order to define the final configuration.\",\n \"In this way we will be able to condition on the outcome of the first step in order to arrive at a situation where tools from the discrete setting applies.\",\n \"Recall that $\\\\Omega $ denotes the space of locally finite point measures on ${\\\\mathbb {R}}^2 \\\\times {\\\\mathbb {R}}_+$ .\",\n \"We will by $\\\\overline{\\\\Omega }$ denote the space of locally finite point measures on ${\\\\mathbb {R}}^2 \\\\times {\\\\mathbb {R}}_+ \\\\times [0, 1] \\\\times [0, 1]$ , by $\\\\overline{\\\\mathcal {M}}$ we denote the corresponding Borel sigma algebra, and we let $\\\\overline{{\\\\mathbb {P}}}_\\\\lambda $ denote the law of a Poisson point process defined on $(\\\\overline{\\\\Omega },\\\\overline{\\\\mathcal {M}})$ with intensity $\\\\lambda \\\\,dx\\\\,\\\\mu (dz)\\\\,du\\\\,dv$ .\",\n \"In this section we will use a very dense set of points and the last two coordinates $u$ and $v$ in the above construction will be used to perform a thinning of the Poisson process.\",\n \"Throughout this section we will suppose that $m\\\\ge 1$ is a large integer and that $\\\\lambda =2m\\\\lambda _1$ is fixed.\",\n \"Given $\\\\overline{\\\\omega }\\\\in \\\\overline{\\\\Omega }$ we denote by $\\\\omega $ the projection of $\\\\overline{\\\\omega }$ onto $\\\\Omega $ .\",\n \"For $p\\\\in [0,1]$ and $m\\\\ge 1$ we define $\\\\omega _p:=\\\\sum _{\\\\begin{array}{c}(x,z,u,v)\\\\in \\\\overline{\\\\omega }\\\\\\\\ u\\\\le p,\\\\, v\\\\le 1/m\\\\end{array}}\\\\delta _{(x,z)}\\\\quad \\\\text{and}\\\\quad \\\\mathcal {O}_p:=\\\\bigcup _{(x,z)\\\\in \\\\omega _p} B(x,z).$ The usual properties of a Poisson point process imply that $\\\\omega $ and $\\\\omega _p$ are Poisson point processes on $\\\\Omega $ distributed according to ${\\\\mathbb {P}}_\\\\lambda $ and ${\\\\mathbb {P}}_{p\\\\lambda /m}$ , respectively.\",\n \"Notice that $\\\\omega _p$ is a $(p/m)$ -thinning of $\\\\omega $ , and therefore it can be seen as a percolation process on $\\\\lbrace 0,1\\\\rbrace ^\\\\omega $ with density $p/m$ .\",\n \"Since $\\\\lambda =2m\\\\lambda _1$ , the above construction comes with two parameters, $p$ and $m$ , of which we think of $m$ as large and fixed while $p$ will be varying.\",\n \"We have for this reason suppressed $m$ in the above notation, and apply a subscript $p$ to indicate that the events $\\\\operatorname{Cross}_p$ , $\\\\operatorname{Circ}_p$ and $\\\\operatorname{Arm}_p$ refer to crossings in $\\\\mathcal {O}_p$ .\",\n \"We will study the conditional probability $Z_p:=\\\\overline{{\\\\mathbb {P}}}_\\\\lambda \\\\big (\\\\operatorname{Cross}_p(4r,r)\\\\big |\\\\omega \\\\big ).$ The product structure of the above conditional measure implies that for ${\\\\mathbb {P}}_\\\\lambda $ -almost every $\\\\omega \\\\in \\\\Omega $ the variable $Z_p$ can be viewed as the expectation of some Boolean function $f_\\\\omega :\\\\lbrace 0,1\\\\rbrace ^\\\\omega \\\\rightarrow \\\\lbrace 0,1\\\\rbrace $ .\",\n \"While $\\\\omega $ has infinite support, it follows from (REF ) that $\\\\overline{{\\\\mathbb {P}}}_\\\\lambda $ -almost surely, only finitely many balls touch the box $[-4r, 4r]^2$ .\",\n \"Therefore the function $f_\\\\omega $ can be viewed as having only finitely many variables that we index by $i \\\\in \\\\lbrace 1, \\\\dots , i_0\\\\rbrace $ ($i_0$ is the number of balls touching $[-4r, 4r]^2$ ).\",\n \"In this conditional setting, the tools from the discrete analysis will apply, in particular the Margulis-Russo formula and Talagrand's inequality.\",\n \"With the use of these we will be able to bound the gap between $\\\\lambda _0$ and $\\\\lambda _1$ by controlling the conditional variance of $f_\\\\omega $ and its influences, defined, for $\\\\overline{{\\\\mathbb {P}}}_\\\\lambda $ -almost every $\\\\overline{\\\\omega }\\\\in \\\\Omega $ , as $\\\\operatorname{Inf}_i^p(f_\\\\omega ):=\\\\overline{{\\\\mathbb {P}}}_\\\\lambda \\\\big (f_\\\\omega (\\\\omega _p)\\\\ne f_\\\\omega (\\\\sigma _i\\\\omega _p)\\\\big |\\\\omega \\\\big ),$ where $\\\\sigma _i\\\\omega _p=\\\\omega _p+(1-2\\\\operatorname{{\\\\bf 1}}_{\\\\lbrace u_i\\\\le p,\\\\,v_i\\\\le 1/m\\\\rbrace })\\\\delta _{(x_i,z_i)}$ .\"\n ],\n [\n \"Quenched decoupling\",\n \"In order to handle dependencies in the quenched setting we will need an alternative to the decoupling inequality proved in Proposition REF .\",\n \"Given $\\\\overline{\\\\omega }\\\\in \\\\overline{\\\\Omega }$ , this can be done via an estimate on the influence region of each point in the support of $\\\\omega $ .\",\n \"Since we will only consider events defined in terms of $\\\\mathcal {O}_p$ on $[-4r,4r]^2$ it will suffice to consider points intersecting this region.\",\n \"For $r\\\\ge 3n\\\\ge 3$ , let $G = G(r,n)$ denote the event $G := \\\\Big [ \\\\max \\\\big \\\\lbrace z:(x, z)\\\\in \\\\operatorname{supp}(\\\\omega )\\\\text{ and } B(x, z) \\\\cap [-4r, 4r]^2 \\\\ne \\\\emptyset \\\\big \\\\rbrace \\\\le r/3n \\\\Big ].$ Lemma 5.2 Assume the second moment condition (REF ).\",\n \"For every $n\\\\ge 1$ , $\\\\lim _{r\\\\rightarrow \\\\infty } \\\\overline{{\\\\mathbb {P}}}_\\\\lambda (G) = 1$ .\",\n \"The proof follows the same steps as that of Lemma REF .\",\n \"A first consequence of the above observation concerns quenched decouplings of events defined in well separated regions.\",\n \"To state this precisely, fix integers $n, J \\\\ge 1$ and let $D_1, \\\\dots , D_J \\\\subseteq [-4 r, 4 r]^2$ be measurable sets such that $d(D_i, D_j) \\\\ge \\\\frac{r}{n}, \\\\text{ for every $i \\\\ne j$}.$ Having these well separated sets, our aim is to decouple what happens inside each of them.\",\n \"For this, fix some events $A_1, \\\\dots , A_J$ that only look inside the sets $D_j$ , that is $A_j \\\\in \\\\sigma (Y_x, x \\\\in D_j)$ .\",\n \"Observe that on the event $G = G(r, n)$ the events $A_j$ depend on disjoint subsets of $\\\\omega $ under the conditional measure $\\\\overline{{\\\\mathbb {P}}}_\\\\lambda (\\\\,\\\\cdot \\\\,|\\\\omega )$ .\",\n \"This implies that $\\\\overline{{\\\\mathbb {P}}}_\\\\lambda \\\\big ( A_1 \\\\cap \\\\dots \\\\cap A_J \\\\big | \\\\omega \\\\big ) \\\\,\\\\le \\\\, \\\\operatorname{{\\\\bf 1}}_{G^c}(\\\\omega ) + \\\\prod \\\\nolimits _{j \\\\le J} \\\\overline{{\\\\mathbb {P}}}_\\\\lambda (A_j | \\\\omega ).$ For us it is important to observe that not only the events $A_j$ decouple on $G(r, n)$ , but also their conditional probabilities $\\\\overline{{\\\\mathbb {P}}}_\\\\lambda (A_j | \\\\omega )$ .\",\n \"To see why this is true, let us first decompose the point measure $\\\\omega $ as $\\\\omega _1 + \\\\dots + \\\\omega _J + \\\\tilde{\\\\omega }$ , where $\\\\omega _j = \\\\sum _i \\\\delta _{(x_i, z_i)}$ with the sum ranging over $i \\\\ge 0$ such that $d(x_i, D_j) \\\\le r/(3n)$ and $\\\\tilde{\\\\omega }$ stands for the remainder.\",\n \"Observe that on $G$ , the conditional probabilities $\\\\overline{{\\\\mathbb {P}}}_\\\\lambda (A_j | \\\\omega )$ depend on $\\\\omega $ through $\\\\omega _j$ only.\",\n \"In particular, we may exhibit a function $g_j$ such that $[\\\\operatorname{{\\\\bf 1}}_G \\\\cdot \\\\overline{{\\\\mathbb {P}}}_\\\\lambda (A_j | \\\\omega )](\\\\omega ) = \\\\operatorname{{\\\\bf 1}}_G \\\\cdot g_j(\\\\omega _j)$ , and by the Poissonian nature of $\\\\omega $ , all the $\\\\omega _j$ 's are independent.\",\n \"As a consequence, for any $\\\\delta > 0$ , $\\\\begin{split}\\\\overline{{\\\\mathbb {P}}}_\\\\lambda \\\\Big (G \\\\cap \\\\bigcap _{j \\\\le J} \\\\big [ \\\\overline{{\\\\mathbb {P}}}_\\\\lambda (A_j | \\\\omega ) \\\\le \\\\delta \\\\big ] \\\\Big ) \\\\,& \\\\le \\\\, \\\\prod _{j \\\\le J} \\\\overline{{\\\\mathbb {P}}}_\\\\lambda \\\\big [ g_j(\\\\omega _j) \\\\le \\\\delta \\\\big ]\\\\\\\\& \\\\le \\\\, \\\\prod _{j \\\\le J}\\\\bigg [ \\\\overline{{\\\\mathbb {P}}}_\\\\lambda \\\\big ( \\\\overline{{\\\\mathbb {P}}}_\\\\lambda (A_j | \\\\omega ) \\\\le \\\\delta \\\\big )+\\\\overline{{\\\\mathbb {P}}}(G^c)\\\\bigg ].\\\\end{split}$ Equations (REF ) and (REF ) give a more precise version of the decoupling of Section and will be important in bounding the conditional variance and influences in the next two sections.\"\n ],\n [\n \"Controlling the variance\",\n \"In Section we showed that the probability of crossing a rectangle is non-degenerate throughout the critical regime $[\\\\lambda _0, \\\\lambda _1]$ .\",\n \"In the notation used in this section, that is to say that $\\\\overline{\\\\operatorname{{\\\\mathbb {E}}}}_\\\\lambda [Z_p]\\\\in (c,1-c)$ for some $c>0$ and all $r\\\\ge 1$ and $p\\\\lambda / m \\\\in \\\\big [ \\\\lambda _0, \\\\lambda _1 \\\\big ]$ .\",\n \"Recall that $\\\\lambda = 2 m \\\\lambda _1$ .\",\n \"Our goal for this section is to show that this holds with probability close to one also for $Z_p$ .\",\n \"Proposition 5.3 Assume the second moment condition (REF ).\",\n \"Then, for every $\\\\varepsilon >0$ there exists $\\\\delta >0$ and $c_{\\\\ref {c:lambda}} = c_{\\\\ref {c:lambda}}(\\\\varepsilon ) > 0$ such that for every $m\\\\ge c_{\\\\ref {c:lambda}}$ we have $\\\\overline{{\\\\mathbb {P}}}_\\\\lambda \\\\bigg (Z_p\\\\in (\\\\delta ,1-\\\\delta )\\\\text{ for all }p\\\\in \\\\big [\\\\tfrac{\\\\lambda _0}{2\\\\lambda _1},\\\\tfrac{\\\\lambda _1}{2\\\\lambda _1}\\\\big ]\\\\bigg )\\\\ge 1-\\\\varepsilon \\\\quad \\\\text{for all }r\\\\ge 1.$ The same holds with $\\\\operatorname{Cross}_p(4r, r)$ replaced by either $\\\\operatorname{Cross}_p^\\\\star (4r, r)$ , $\\\\operatorname{Circ}_p(r, 2r)$ or $\\\\operatorname{Circ}_p^\\\\star (r, 2r)$ .\",\n \"An immediate consequence of the above proposition is that $\\\\overline{{\\\\mathbb {P}}}_\\\\lambda \\\\bigg (\\\\operatorname{Var}_\\\\lambda (\\\\operatorname{{\\\\bf 1}}_{\\\\operatorname{Cross}_p(4r,r)}|\\\\omega )\\\\in (\\\\delta ,1-\\\\delta )\\\\text{ for all }p\\\\in \\\\big [\\\\tfrac{\\\\lambda _0}{2\\\\lambda _1},\\\\tfrac{\\\\lambda _1}{2\\\\lambda _1}\\\\big ]\\\\bigg )\\\\ge 1/2\\\\quad \\\\text{for all }r\\\\ge 1.$ The first step in the proof of Proposition REF is the following estimate on the effect of observing $\\\\omega $ in determining whether there is a crossing of $[0,4r] \\\\times [0,r]$ in $\\\\mathcal {O}_p$ .\",\n \"Lemma 5.4 For every $m\\\\ge 1$ and $p\\\\in [0,1]$ , and every measurable event $A$ , we have $\\\\operatorname{Var}_\\\\lambda \\\\big (\\\\overline{{\\\\mathbb {P}}}_\\\\lambda (A|\\\\omega )\\\\big )\\\\le 1/m.$ Recall that $\\\\omega _p$ is obtained as a subset of $\\\\omega $ based on the value of the third and fourth coordinates of each point in $\\\\overline{\\\\omega }$ .\",\n \"Alternatively, we could obtain $\\\\omega _p$ in two stages: First, partition $\\\\overline{\\\\omega }$ into $\\\\omega _p^1,\\\\omega _p^2,\\\\ldots ,\\\\omega _p^m$ and some remainder $\\\\tilde{\\\\omega }$ , where $\\\\omega _p^j$ contains all points with third coordinate at most $p$ and fourth coordinate in $\\\\big [\\\\frac{j-1}{m},\\\\frac{j}{m}\\\\big )$ .\",\n \"Second, use auxiliary randomness, independent from everything else, to determine which of the $\\\\omega _p^j$ 's to choose as the set $\\\\omega _p$ .\",\n \"Since each of the $\\\\omega _p^j$ 's have the same distribution, the result would be the same.\",\n \"Let us assume that this construction has been made, and that $J$ is the auxiliary random variable, uniformly distributed on $\\\\lbrace 1,2,\\\\ldots ,m\\\\rbrace $ , that determines which of the $\\\\omega _p^j$ 's that is chosen in the second step.\",\n \"Let ${\\\\mathcal {F}}$ denote the sigma algebra containing the information of the partitioning $\\\\omega _p^1, \\\\ldots , \\\\omega _p^m$ .\",\n \"Since revealing the partitioning can only increase the variance, we have that $\\\\operatorname{Var}_\\\\lambda \\\\big (\\\\overline{{\\\\mathbb {P}}}_\\\\lambda (A|\\\\omega )\\\\big )\\\\le \\\\operatorname{Var}_\\\\lambda \\\\big (\\\\overline{{\\\\mathbb {P}}}_\\\\lambda (A|{\\\\mathcal {F}})\\\\big ).$ Moreover, by conditioning on the outcome of $J$ , $\\\\overline{{\\\\mathbb {P}}}_\\\\lambda (A|{\\\\mathcal {F}})=\\\\frac{1}{m}\\\\sum _{j=1}^m\\\\operatorname{{\\\\bf 1}}_{\\\\lbrace \\\\omega _p^j\\\\in A\\\\rbrace }.$ Since the configurations $\\\\omega _p^1,\\\\omega _p^2,\\\\ldots ,\\\\omega _p^m$ are independent, we find that $\\\\operatorname{Var}_\\\\lambda \\\\big (\\\\overline{{\\\\mathbb {P}}}_\\\\lambda (A|\\\\omega )\\\\big )\\\\,\\\\le \\\\,\\\\operatorname{Var}_\\\\lambda \\\\bigg (\\\\frac{1}{m}\\\\sum _{j=1}^m\\\\operatorname{{\\\\bf 1}}_{\\\\lbrace \\\\omega _p^j\\\\in A\\\\rbrace }\\\\bigg )\\\\,\\\\le \\\\,\\\\frac{1}{m},$ as required.\",\n \"[Proof of Proposition REF ] Note that $Z_p$ is monotone in $p$ , so it will suffice to consider the extremes of the interval $\\\\big [\\\\tfrac{\\\\lambda _0}{2\\\\lambda _1},\\\\tfrac{\\\\lambda _1}{2\\\\lambda _1}\\\\big ]$ .\",\n \"Due to Corollary REF there exists a constant $c_{\\\\ref {c:q-bound}} = c_{\\\\ref {c:q-bound}}(4) > 0$ such that $\\\\overline{\\\\operatorname{{\\\\mathbb {E}}}}_\\\\lambda [Z_p]\\\\in (c_{\\\\ref {c:q-bound}},1-c_{\\\\ref {c:q-bound}})$ for all $p\\\\in \\\\big [\\\\tfrac{\\\\lambda _0}{2\\\\lambda _1},\\\\tfrac{\\\\lambda _1}{2\\\\lambda _1}\\\\big ]$ .\",\n \"So, by Chebyshev's inequality and Lemma REF , for $b = 0, 1$ , $\\\\overline{{\\\\mathbb {P}}}_\\\\lambda \\\\Big (Z_{\\\\lambda _b/(2\\\\lambda _1)}\\\\notin (c_{\\\\ref {c:q-bound}}/2,1-c_{\\\\ref {c:q-bound}}/2)\\\\Big )\\\\,\\\\le \\\\,\\\\frac{\\\\operatorname{Var}_\\\\lambda (Z_p)}{(c_{\\\\ref {c:q-bound}}/2)^2}\\\\,\\\\le \\\\,\\\\frac{4}{mc_{\\\\ref {c:q-bound}}^2},$ which is at most $\\\\varepsilon /2$ for $m\\\\ge 8/(\\\\varepsilon c_{\\\\ref {c:q-bound}}^2)$ .\",\n \"Now, the result follows via a union bound.\"\n ],\n [\n \"Controlling the influences\",\n \"The next step is to obtain an upper bound on the individual conditional influences, defined as in (REF ), allowing for Talagrand's inequality to give a lower bound on the total conditional influence.\",\n \"The result proved here is the following.\",\n \"Proposition 5.5 Assume the second moment condition (REF ).\",\n \"There exists a constant $c_{\\\\ref {c:lambda2}}$ such that for every $m\\\\ge c_{\\\\ref {c:lambda2}}$ and $\\\\eta >0$ we have $\\\\overline{{\\\\mathbb {P}}}_\\\\lambda \\\\Big ( \\\\max _{i \\\\ge 1} \\\\big ( \\\\operatorname{Inf}^p_i (f_\\\\omega ) \\\\big ) > \\\\eta \\\\text{ for some }p\\\\in \\\\big [ \\\\tfrac{\\\\lambda _0}{2\\\\lambda _1}, \\\\tfrac{\\\\lambda _1}{2\\\\lambda _1} \\\\big ] \\\\Big ) \\\\le \\\\eta ,$ for all $r$ large enough depending only on $\\\\eta $ .\",\n \"This will be done in two steps.\",\n \"First we relate the influence of a point to arm probabilities.\",\n \"Then we bound the probability of the latter.\",\n \"Before stating the first lemma, consider a partitioning of the rectangle $[0, 4r] \\\\times [0, r]$ into $36 n^2$ smaller squares of side length $r/ 3n$ .\",\n \"We write $\\\\operatorname{Arm}_p^\\\\ell (\\\\tfrac{r}{n}, r)$ , with $\\\\ell = 1, \\\\dots , 36 n^2$ , for the arm event defined as in (REF ), but centered around each of these boxes.\",\n \"These arm events help us to bound the influences $\\\\operatorname{Inf}^p_i(f_\\\\omega )$ as follows.\",\n \"Assume that $i$ is pivotal, and the corresponding ball has radius smaller or equal to $r / 3n$ .\",\n \"Then there must exist some $\\\\ell $ for which the event $\\\\operatorname{Arm}_p^\\\\ell (r/n,r)$ must occur.\",\n \"Therefore, $\\\\operatorname{Inf}_i^p(f_\\\\omega )\\\\le \\\\operatorname{{\\\\bf 1}}_{G^c}(\\\\omega ) + \\\\overline{{\\\\mathbb {P}}}_\\\\lambda \\\\big (\\\\operatorname{Arm}_p^\\\\ell (\\\\tfrac{r}{n},r)\\\\big |\\\\omega \\\\big )\\\\quad \\\\text{for some }\\\\ell =1,2,\\\\ldots , 36 n^2,$ for $\\\\overline{{\\\\mathbb {P}}}_\\\\lambda $ -almost every $\\\\overline{\\\\omega }$ .\",\n \"Estimating influences will now boil down to estimating conditional arm probabilities.\",\n \"Lemma 5.6 Assume the second moment condition (REF ).\",\n \"There exist $\\\\gamma > 0$ and $c_{\\\\ref {c:lambda2}} > 0$ such that for every $m > c_{\\\\ref {c:lambda2}}$ and $n \\\\ge 16$ we have $\\\\overline{{\\\\mathbb {P}}}_\\\\lambda \\\\Big [ \\\\overline{{\\\\mathbb {P}}}_\\\\lambda \\\\big ( \\\\operatorname{Arm}_p\\\\big ( \\\\tfrac{r}{n}, r\\\\big ) \\\\big | \\\\omega \\\\big ) > n^{-\\\\gamma } \\\\text{ for some }p\\\\in \\\\big [\\\\tfrac{\\\\lambda _0}{2\\\\lambda _1}, \\\\tfrac{\\\\lambda _1}{2\\\\lambda _1} \\\\big ]\\\\Big ] < n^{-100},$ for all sufficiently large $r$ depending only on $n$ .\",\n \"The proof follows the same structure of that of Lemma REF .\",\n \"For $r\\\\ge n\\\\ge 16$ let $\\\\ell _i=4^i\\\\frac{r}{n}$ .\",\n \"Note that if $\\\\operatorname{Circ}_p^\\\\star (\\\\frac{r}{n},r)$ fails to occur, then $A_i = \\\\operatorname{Circ}_p^\\\\star (\\\\ell _i, 2\\\\ell _i)$ must also fail for every $i = 0, 1, \\\\ldots , k - 1$ , where $k = \\\\lfloor \\\\frac{1}{2} \\\\log _4 n \\\\rfloor $ ($k \\\\ge 1$ as we assume $n \\\\ge 16$ ).\",\n \"By (REF ) we therefore have $\\\\overline{{\\\\mathbb {P}}}_\\\\lambda \\\\big (\\\\operatorname{Arm}_p(\\\\tfrac{r}{n}, r) \\\\big | \\\\omega \\\\big )\\\\, \\\\le \\\\, \\\\operatorname{{\\\\bf 1}}_{G^c} (\\\\omega ) + \\\\prod _{i = 0}^{k - 1} \\\\overline{{\\\\mathbb {P}}}_\\\\lambda (A_i^c | \\\\omega ).$ Fix $\\\\varepsilon >0$ so that $2^k\\\\varepsilon ^{k/2}\\\\le \\\\frac{1}{2}n^{-100}$ .\",\n \"By Proposition REF there exists $\\\\delta >0$ and $c_{\\\\ref {c:lambda}}(\\\\varepsilon ) > 0$ large enough so that for every $r \\\\ge 1$ and $m\\\\ge c_{\\\\ref {c:lambda}}$ we have $\\\\overline{{\\\\mathbb {P}}}_\\\\lambda \\\\Big ( \\\\overline{{\\\\mathbb {P}}}_\\\\lambda \\\\big ( \\\\operatorname{Circ}_p^\\\\star (r, 2r) \\\\big | \\\\omega \\\\big ) > \\\\delta \\\\text{ for every }p\\\\in \\\\big [\\\\tfrac{\\\\lambda _0}{2\\\\lambda _1}, \\\\tfrac{\\\\lambda _1}{2\\\\lambda _1} \\\\big ]\\\\Big ) > 1 - \\\\varepsilon .$ Fix $\\\\gamma > 0$ (independent of $n$ ) so that $n^{-\\\\gamma } \\\\ge (1-\\\\delta )^{k/2}$ .\",\n \"By (REF ) we have, on $G$ , that if $\\\\overline{{\\\\mathbb {P}}}_\\\\lambda (A_i|\\\\omega )>\\\\delta $ for at least half of the indices $i=0,1,\\\\ldots ,k-1$ , then $\\\\overline{{\\\\mathbb {P}}}_\\\\lambda \\\\big (\\\\operatorname{Arm}_p(\\\\tfrac{r}{n},r)\\\\big |\\\\omega \\\\big )\\\\,\\\\le \\\\,(1-\\\\delta )^{k/2}\\\\,\\\\le \\\\, n^{-\\\\gamma }.$ In particular, via a union bound, $\\\\overline{{\\\\mathbb {P}}}_\\\\lambda \\\\Big (G\\\\cap \\\\big [\\\\overline{{\\\\mathbb {P}}}_\\\\lambda \\\\big (\\\\operatorname{Arm}_p(\\\\tfrac{r}{n},r)\\\\big |\\\\omega \\\\big )>n^{-\\\\gamma }\\\\big ]\\\\Big )\\\\,\\\\le \\\\,2^k\\\\sup _{I}\\\\overline{{\\\\mathbb {P}}}_\\\\lambda \\\\Big (G \\\\cap \\\\bigcap _{i \\\\in I}\\\\big [\\\\overline{{\\\\mathbb {P}}}_\\\\lambda (A_i|\\\\omega )\\\\le \\\\delta \\\\big ]\\\\Big ),$ where the supremum is taken over all $I\\\\subseteq \\\\lbrace 0,1,\\\\ldots ,k-1\\\\rbrace $ of size at least $k/2$ , of which there are at most $2^k$ .\",\n \"By (REF ) and (REF ), this is bounded by $2^k\\\\sup _{I}\\\\prod _{i\\\\in I}\\\\Big [\\\\overline{{\\\\mathbb {P}}}_\\\\lambda \\\\big (\\\\overline{{\\\\mathbb {P}}}_\\\\lambda (A_i|\\\\omega )\\\\le \\\\delta \\\\big )+\\\\overline{{\\\\mathbb {P}}}_\\\\lambda (G^c)\\\\Big ]\\\\,\\\\le \\\\,2^k\\\\big (\\\\varepsilon +\\\\overline{{\\\\mathbb {P}}}_\\\\lambda (G^c)\\\\big )^{k/2}.$ For large values of $r$ , depending on $n$ , this probability is at most $n^{-100}$ by the choice of $\\\\varepsilon >0$ .\",\n \"We have now gathered all the pieces necessary to prove the main result of this subsection.\",\n \"[Proof of Proposition REF ] We start by choosing $n$ large enough so that $36 n^{-98} \\\\le \\\\eta /2$ .\",\n \"Then, $\\\\overline{{\\\\mathbb {P}}}_\\\\lambda \\\\Big ( \\\\max _{i \\\\ge 1} \\\\big ( \\\\operatorname{Inf}_i^p(f_\\\\omega ) \\\\big ) > n^{-\\\\gamma } \\\\text{ for some }p \\\\in \\\\big [ \\\\tfrac{\\\\lambda _0}{2\\\\lambda _1}, \\\\tfrac{\\\\lambda _1}{2\\\\lambda _1} \\\\big ]\\\\Big )$ may via (REF ) be bounded from above by $\\\\overline{{\\\\mathbb {P}}}_\\\\lambda \\\\big ( G^c \\\\big ) + \\\\overline{{\\\\mathbb {P}}}_\\\\lambda \\\\Big (\\\\overline{{\\\\mathbb {P}}}_\\\\lambda \\\\big ( \\\\operatorname{Arm}_p^\\\\ell \\\\big ( \\\\tfrac{r}{n}, r\\\\big ) \\\\big | \\\\omega \\\\big ) > n^{-\\\\gamma }\\\\text{ for some }p \\\\in \\\\big [ \\\\tfrac{\\\\lambda _0}{2\\\\lambda _1}, \\\\tfrac{\\\\lambda _1}{2\\\\lambda _1} \\\\big ]\\\\text{ and }\\\\ell \\\\le 36 n^2 \\\\Big ).$ By Lemmas REF and REF , this is smaller than $\\\\eta $ by our choice of $n$ , once $r$ is taken large enough.\"\n ],\n [\n \"Proof of Theorem \",\n \"Roughly speaking, in order to prove Theorem REF , we are going to show that for every $\\\\varepsilon > 0$ $Z_{\\\\lambda _1/(2\\\\lambda _1)} - Z_{\\\\lambda _0/(2\\\\lambda _1)} \\\\ge \\\\frac{\\\\lambda _1 - \\\\lambda _0}{\\\\varepsilon },$ with positive probability.\",\n \"Since $Z_p(\\\\omega ) \\\\in [0, 1]$ almost surely, we must have that $\\\\lambda _0=\\\\lambda _1$ .\",\n \"[Proof of Theorem REF ] We first give estimates on the derivatives of $Z_p$ .\",\n \"Recall that, for $\\\\overline{{\\\\mathbb {P}}}_\\\\lambda $ almost every $\\\\overline{\\\\omega }\\\\in \\\\overline{\\\\Omega }$ , $Z_p$ coincides with the expectation, with respect to the conditional measure $\\\\overline{{\\\\mathbb {P}}}_\\\\lambda (\\\\,\\\\cdot \\\\,|\\\\omega )$ , of some function $f_\\\\omega :\\\\lbrace 0,1\\\\rbrace ^\\\\omega \\\\rightarrow \\\\lbrace 0,1\\\\rbrace $ .\",\n \"Moreover, with probability one, only finitely many points in $\\\\omega $ will affect the outcome of events defined on $[-4r,4r]^2$ .\",\n \"Hence, the domain of $f_\\\\omega $ is for $\\\\overline{{\\\\mathbb {P}}}_\\\\lambda $ -almost every $\\\\overline{\\\\omega }$ finite dimensional, and the Margulis-Russo formula gives that $\\\\frac{d Z_p}{d p} \\\\,=\\\\, \\\\frac{d Z_p (\\\\omega )}{d p} \\\\,=\\\\, \\\\frac{1}{m}\\\\sum _{i \\\\ge 0} \\\\operatorname{Inf}_i^p (f_\\\\omega ),$ where $\\\\operatorname{Inf}_i^p (f_\\\\omega )$ is as defined in (REF ), and the additional factor $1/m$ comes from the chain rule.\",\n \"Let us now assume that $\\\\lambda _0 < \\\\lambda _1$ and fix $m\\\\ge c_{\\\\ref {c:lambda}}(1/2) \\\\vee c_{\\\\ref {c:lambda2}}$ .\",\n \"Combining (REF ) with the mean-value theorem and Talagrand's inequality (REF ) we obtain, for some $q\\\\in \\\\big [\\\\frac{\\\\lambda _0}{2\\\\lambda _1},\\\\frac{\\\\lambda _1}{2\\\\lambda _1}\\\\big ]$ , $\\\\frac{Z_{\\\\lambda _1/(2\\\\lambda _1)} - Z_{\\\\lambda _0/(2\\\\lambda _1)}}{\\\\lambda _1 - \\\\lambda _0}\\\\, =\\\\, \\\\frac{1}{2\\\\lambda _1} \\\\frac{d Z_p(\\\\omega )}{d p} (q)\\\\, \\\\ge \\\\, \\\\frac{c}{2 \\\\lambda _1 m \\\\log (m)} \\\\operatorname{Var}_\\\\lambda (f_\\\\omega | \\\\omega ) \\\\log \\\\bigg ( \\\\frac{c}{\\\\max _i \\\\big ( \\\\operatorname{Inf}_i^q (f_\\\\omega ) \\\\big ) } \\\\bigg ).$ From Propositions REF and REF , we conclude that for $r$ large enough (depending only on $\\\\lambda _1 - \\\\lambda _0$ ) the right hand side of the above equation is larger than $2/(\\\\lambda _1 - \\\\lambda _0)$ with positive probability.\",\n \"This contradicts the fact that $Z_p \\\\in [0,1]$ , thus finishing the proof of the theorem.\"\n ],\n [\n \"Continuity of the critical parameter\",\n \"We have in the previous section seen that $\\\\lambda _0=\\\\lambda _1$ , and thus coincides with $\\\\lambda _c$ , under the assumption of finite second moment of the radii distribution.\",\n \"In this section we will investigate the dependence of this critical parameter with respect to the radii distribution $\\\\mu $ , and write $\\\\lambda _c=\\\\lambda _c(\\\\mu )$ throughout this section in order to emphasize this dependence.\",\n \"Theorem 6.1 Let $(\\\\mu _m)_{m\\\\ge 1}$ be a sequence of radii distributions, all dominated by some distribution $\\\\nu $ with finite second moment.\",\n \"If $\\\\mu _m\\\\rightarrow \\\\mu $ weakly, then ${\\\\lambda _{c}}(\\\\mu _m)\\\\rightarrow {\\\\lambda _{c}}(\\\\mu )$ .\",\n \"Remark 13 A weaker version of Theorem REF , proved under the stronger assumption of uniformly bounded support, has been known since long (see [27]).\",\n \"The argument given here is significantly simpler and shorter, but is in contrast to the aforementioned proof distinctively two-dimensional.\",\n \"The convergence may fail when the assumption of uniformly bounded tails is relaxed.\",\n \"For a first example of this, let $(\\\\mu _m)_{m\\\\ge 1}$ be a sequence with infinite second moments converging weakly to a point mass at 1.\",\n \"The critical parameter for each $\\\\mu _m$ is degenerate (meaning that $\\\\lambda _0=\\\\lambda _1=0$ ), and cannot converge to the critical value of the point mass, which is strictly positive.\",\n \"In the above example the convergence fails for rather obvious reasons.\",\n \"A second example, which gives further insight to what can go wrong, may be obtained from a cumulative distribution function $F$ of some distribution with finite second moment, when $F_m$ is given by $F_m(x):=(1-\\\\tfrac{1}{m})F(x)+\\\\tfrac{1}{m}F(\\\\tfrac{x}{m}).$ Each $F_m$ has finite second moment, but there is no uniform upper bound on its value.\",\n \"The dilation of $F$ by a factor $\\\\frac{1}{m}$ corresponds to a scaling of the radii by $m$ , and thus has an inverse quadratic effect on the critical parameter in that ${\\\\lambda _{c}}(F(\\\\frac{x}{m}))=\\\\frac{1}{m^2}{\\\\lambda _{c}}(F(x))$ .\",\n \"So, although ${\\\\lambda _{c}}(\\\\mu )>0$ , we have ${\\\\lambda _{c}}(F_m)\\\\,\\\\le \\\\,{\\\\lambda _{c}}\\\\big ((1-\\\\tfrac{1}{m})\\\\operatorname{{\\\\bf 1}}_{\\\\lbrace x\\\\ge 0\\\\rbrace }+\\\\tfrac{1}{m}F(\\\\tfrac{x}{m})\\\\big )\\\\,=\\\\,m\\\\,{\\\\lambda _{c}}\\\\big (F(\\\\tfrac{x}{m})\\\\big )\\\\,=\\\\,\\\\tfrac{1}{m}{\\\\lambda _{c}}(F).$ Although $F_m\\\\rightarrow F$ weakly, the critical parameters diverge.\",\n \"[Proof of Theorem REF ] Fro the proof, we will write $F$ , $F_m$ and $H$ for the cumulative distribution functions of $\\\\mu $ , $\\\\mu _m$ and $\\\\nu $ .\",\n \"We first observe that if $F_m\\\\ge H$ for all $m\\\\ge 1$ and $F_m\\\\rightarrow F$ , then also $F\\\\ge H$ and $F$ has finite second moment too.\",\n \"To see this assume the contrary, in which case $F(x)F(x^{\\\\prime })$ , contradicting the assumed convergence.\",\n \"That $F$ has finite second moment in particular implies that ${\\\\lambda _{c}}(F)\\\\in (0,\\\\infty )$ .\",\n \"We will base the proof on Propositions REF and REF .\",\n \"We therefore fix $\\\\theta >0$ and $r_0 = r_0({\\\\lambda _{c}}(F) + 1)$ accordingly.\",\n \"Write ${\\\\mathbb {P}}_\\\\lambda ^{F_m}$ for the measure at intensity $\\\\lambda $ and with radii distributed as $F_m$ .\",\n \"To complete the proof, it will suffice to show that for every $\\\\varepsilon \\\\in (0,1)$ there exists $r\\\\ge r_0$ and $m_0$ such that for all $m\\\\ge m_0$ ${\\\\mathbb {P}}_{{\\\\lambda _{c}}(F)-\\\\varepsilon }^{F_m}\\\\big (\\\\operatorname{Cross}(r,3r)\\\\big )<\\\\theta \\\\quad \\\\text{and}\\\\quad {\\\\mathbb {P}}_{{\\\\lambda _{c}}(F)+\\\\varepsilon }^{F_m}\\\\big (\\\\operatorname{Cross}(3r,r)\\\\big )>1-\\\\theta .$ Proposition REF then implies that ${\\\\lambda _{c}}(F_m)\\\\in [{\\\\lambda _{c}}(F)-\\\\varepsilon ,{\\\\lambda _{c}}(F)+\\\\varepsilon ]$ for all $m\\\\ge m_0$ .\",\n \"Fix $\\\\varepsilon >0$ .\",\n \"By Corollary REF and Theorem REF , we may choose $r$ large so that ${\\\\mathbb {P}}_{{\\\\lambda _{c}}(F) - \\\\varepsilon }^F\\\\big (\\\\operatorname{Cross}(r, 3r)\\\\big ) < \\\\theta /4 \\\\quad \\\\text{and} \\\\quad {\\\\mathbb {P}}_{{\\\\lambda _{c}}(F) + \\\\varepsilon }^F\\\\big (\\\\operatorname{Cross}(3r, r)\\\\big ) > 1 - \\\\theta /4.$ Next, based on (REF ) of Lemma REF , fix $K\\\\subseteq {\\\\mathbb {R}}^2$ such that ${\\\\mathbb {P}}_{{\\\\lambda _{c}}(F)+\\\\varepsilon }^H\\\\big (\\\\mathcal {O}_{K^c}\\\\cap [0,r]^2=\\\\varnothing \\\\big )>1-\\\\theta /4.$ And, choose $\\\\delta >0$ such that the event in Proposition REF has ${\\\\mathbb {P}}_{{\\\\lambda _{c}}(F)\\\\pm \\\\varepsilon }^{F}$ -probability at most $\\\\theta /4$ to occur.\",\n \"By weak convergence of $F_m$ to $F$ it follows that $F_m^{-1}(U)\\\\rightarrow F^{-1}(U)$ almost surely, as $m\\\\rightarrow \\\\infty $ , if $U$ is uniformly distributed on $[0,1]$ and $F^{-1}$ denotes the generalized inverse of $F$ .\",\n \"Consequently, as the projection of $\\\\omega $ onto $K$ , denoted by $\\\\omega (K)$ , is almost surely finite, we will have $F_m^{-1}(z)\\\\in (F^{-1}(z)-\\\\varepsilon ,F^{-1}(z)+\\\\varepsilon )$ for all $(x,z)\\\\in \\\\omega (K)$ for large enough $m$ .\",\n \"We may, in particular, choose $m_0$ so that for all $m\\\\ge m_0$ ${\\\\mathbb {P}}_{{\\\\lambda _{c}}(F) \\\\pm \\\\varepsilon }^F \\\\Big ( \\\\operatorname{int}(\\\\mathcal {O}_K, \\\\delta ) \\\\subseteq \\\\mathcal {O}_K^{F_m} \\\\subseteq \\\\operatorname{cl}(\\\\mathcal {O}_K, \\\\delta ) \\\\Big ) > 1 - \\\\theta /4.$ Combining the above estimates we conclude that (REF ) holds, as required.\"\n ],\n [\n \"Proof of main results\",\n \"In this section we complete the proofs of the three main theorems stated in the introduction.\",\n \"We first prove Theorem REF , and then deduce Theorems REF and REF .\",\n \"Before starting we recall the content of Corollary REF .\",\n \"Assuming (REF ), we have $0<\\\\lambda _0\\\\le \\\\lambda _c\\\\le \\\\lambda _1<\\\\infty ,$ and the analogous relation holds for $\\\\lambda _c$ replaced by $\\\\lambda _c^\\\\star $ under the stronger condition (REF ).\"\n ],\n [\n \"Proof of Theorem \",\n \"For any $\\\\lambda >\\\\lambda _1$ we have, by definition, that $\\\\lim _{r\\\\rightarrow \\\\infty }{\\\\mathbb {P}}_\\\\lambda (\\\\operatorname{Cross}(3r,r))=1$ .\",\n \"The standard inequalities of Lemma REF and monotonicity imply that for any $\\\\lambda >\\\\lambda _1$ and $\\\\kappa >0$ , $\\\\lim _{r\\\\rightarrow \\\\infty }{\\\\mathbb {P}}_\\\\lambda (\\\\operatorname{Cross}(\\\\kappa r,r))=1.$ Similarly, for any $\\\\lambda <\\\\lambda _0$ and $\\\\kappa >0$ the limit in (REF ) equals 0.\",\n \"Since $\\\\lambda _c\\\\in [\\\\lambda _0,\\\\lambda _1]$ , then Corollary REF shows that for any $\\\\kappa >0$ there exists a constant $c=c(\\\\kappa )>0$ such that $c<{\\\\mathbb {P}}_{\\\\lambda _c}(\\\\operatorname{Cross}(\\\\kappa r,r))<1-c\\\\quad \\\\text{for all }r\\\\ge 1.$ Moreover, by Corollary REF , there is at $\\\\lambda _c$ almost surely no unbounded component of either kind.\",\n \"Finally, the proof is completed by Theorem REF , which shows that indeed $\\\\lambda _c=\\\\lambda _0=\\\\lambda _1$ .\"\n ],\n [\n \"Proof of Theorem \",\n \"Assume (REF ), in which case we by now know that $\\\\lambda _c^\\\\star =\\\\lambda _0=\\\\lambda _1=\\\\lambda _c$ .\",\n \"Part (i) is then a consequence of Corollary REF .\",\n \"The proof of part (ii) will involve a truncation of the radii distribution, allowing for a comparison with a highly supercritical 1-dependent percolation process on ${\\\\mathbb {Z}}^2$ , for which a standard Peierls argument gives the required exponential decay.\",\n \"Fix $\\\\lambda >{\\\\lambda _{c}^\\\\star }$ .\",\n \"For $m\\\\ge 1$ , let $\\\\mu _m$ denote the truncation of $\\\\mu $ at $m$ , meaning that $\\\\mu _m$ coincides with $\\\\mu $ on $[0,m)$ but that $\\\\mu _m([0,m])=1$ .\",\n \"We shall write $\\\\lambda ^\\\\star (\\\\mu )$ for the critical parameter associated with radii distribution $\\\\mu $ .\",\n \"By continuity of the critical parameter, Theorem REF , we have for large $m$ that ${\\\\lambda _{c}^\\\\star }(\\\\mu _m)\\\\in [{\\\\lambda _{c}^\\\\star }(\\\\mu ),\\\\lambda ),$ and assume for the rest of this proof that $m$ is chosen accordingly.\",\n \"Given $\\\\gamma >0$ , we also assume that $r>2m$ is fixed so that ${\\\\mathbb {P}}_\\\\lambda ^{\\\\mu _m}(\\\\operatorname{Cross}(3r,r))>1-\\\\gamma .$ Now, tile the plane with overlapping $3r\\\\times r$ and $r\\\\times 3r$ rectangles as follows: For each $z\\\\in {\\\\mathbb {Z}}^2$ consider both a $3r\\\\times r$ and a $r\\\\times 3r$ rectangle positioned so their lower left corners coincide with $2rz$ .\",\n \"We want to register which of these rectangles that are crossed in the hard direction by an occupied path.\",\n \"Define thus, for each $z\\\\in {\\\\mathbb {Z}}^2$ , $\\\\eta (z,z + (1,0))$ to be the indicator of the event $\\\\operatorname{Cross}(3r,r)$ translated by the vector $2rz$ , and $\\\\eta (z,z + (0, 1))$ as the indicator of $\\\\operatorname{Cross}(r,3r)$ translated by the same vector.\",\n \"By the choice of $m$ and $r$ , the process $\\\\eta $ is a 1-dependent bond percolation process on ${\\\\mathbb {Z}}^2$ with marginal edge probability exceeding $1 - \\\\gamma $ .\",\n \"Therefore, by a standard Peierls argument, there is a constant $c_{\\\\ref {c:exp_decay}}>0$ such that the event that $\\\\eta $ contains an open circuit which is contained in $B^\\\\infty (n)$ and surrounds the origin has probability at least $1-c_{\\\\ref {c:exp_decay}}\\\\exp \\\\lbrace n/c_{\\\\ref {c:exp_decay}}\\\\rbrace $ .\",\n \"However, by construction, an open path in $\\\\eta $ corresponds to an occupied crossing in $\\\\mathcal {O}$ , and thus that for all $n\\\\ge 1$ $\\\\mathbb {P}_\\\\lambda ^{\\\\mu _m}\\\\big [ 0\\\\overset{ \\\\mathcal {V}}{\\\\longleftrightarrow }\\\\partial B^\\\\infty (rn) \\\\big ] \\\\le c_{\\\\ref {c:exp_decay}}\\\\exp \\\\lbrace -n/c_{\\\\ref {c:exp_decay}}\\\\rbrace ,$ from which part (ii) of the theorem follows.\"\n ],\n [\n \"Proof of Theorem \",\n \"Under assumption () Item (i) is a consequence of part (iii) of Corollary REF (see also Remark REF ).\",\n \"So, it remains to argue for Item (ii).\",\n \"However, if () holds, then by Proposition REF there exists $c > 0$ such that for all $\\\\lambda \\\\le \\\\lambda _c$ and $r\\\\ge 1$ $\\\\rho _\\\\lambda (5r,r)\\\\le cr^{-\\\\alpha }.$ Now, fix $\\\\lambda <\\\\lambda _c$ , so that $\\\\lim _{r\\\\rightarrow \\\\infty }{\\\\mathbb {P}}_\\\\lambda (\\\\operatorname{Cross}^\\\\star (3r,r))=1$ .\",\n \"Proposition REF then gives that ${\\\\mathbb {P}}_\\\\lambda (\\\\operatorname{Cross}^\\\\star (3r,r))\\\\ge 1 - c^{\\\\prime }r^{-\\\\alpha }$ for all $r\\\\ge 1$ and some constant $c^{\\\\prime }$ , possibly depending on $\\\\lambda $ .\",\n \"Consequently, for all $r\\\\ge 1$ , ${\\\\mathbb {P}}_\\\\lambda \\\\big [0 \\\\overset{\\\\tiny \\\\mathcal {O}}{\\\\longleftrightarrow }\\\\partial B(r)\\\\big ]\\\\,\\\\le \\\\,4\\\\,{\\\\mathbb {P}}_\\\\lambda (\\\\operatorname{Cross}(r,3r))\\\\,\\\\le \\\\,4c^{\\\\prime }r^{-\\\\alpha },$ as required.\",\n \"This ends the proof of Theorem REF .\"\n ],\n [\n \"Other models\",\n \"We have in Sections - provided a framework to prove the existence of a sharp phase transition and some interesting properties of the critical behavior for Poisson Boolean percolation.\",\n \"In this section we will show how our techniques can be used to obtain alternative proofs for the facts that the critical probability equals $1/2$ for Poisson Voronoi percolation, earlier proved by Bollobás and Riordan [9], and Poisson confetti percolation, first proved by Hirsch [21] (for unit squares) and later Müller [30] (for unit discs).\",\n \"Our results will in fact apply to some generalized versions of these models that are not self-dual.\",\n \"See Figure REF for a simulation of these processes.\",\n \"The finite-size criterion of Section and the RSW techniques of Section solely rely on three basic facts: That the probability measure in question is invariant (with respect to translations, right angle rotations and reflections in coordinate axis), positively associated, and features a decay of spatial correlations.\",\n \"However, the argument used to prove the sharp threshold behavior in Section used the further assumption of a Poissonian structure.\",\n \"All of these properties are satisfied by Poisson Voronoi and confetti percolation.\",\n \"We shall below illustrate what our techniques give for these models, but emphasize that we make no attempt to sharpen the hypothesises in the results we present.\",\n \"Figure: Simulations of Poisson confetti (left) and Voronoi (right) percolation.In the left picture, black confetti have random radii, while the radii of white ones are fixed.The Voronoi picture to the right uses G 0 =1G_0 = 1, G 1 =2G_1 = 2 and q=0.8q = 0.8.Note that the Voronoi cells above are determined by segments of ellipses.\"\n ],\n [\n \"Poisson Voronoi percolation\",\n \"The study of random Voronoi tessellations goes back several decades in time, yet it was only about a decade ago that Bollobàs and Riordan [9] gave the first proof for the fact that the critical probability in Poisson Voronoi percolation in the plane equals $1/2$ ; see also [3].\",\n \"One of the main difficulties faced in studying the phase transition is to derive RSW techniques that apply in this setting.\",\n \"In [9] a weak RSW result was provided, whereas a strong version of RSW was established later by Tassion [42], and was used to prove polynomial decay of the one-arm probability at criticality.\",\n \"The RSW techniques developed here allows us to consider more general variants of the standard Voronoi model, where black and white cells do not necessarily follow the same law.\",\n \"Informally speaking, to each point we associate a random `gravitational pull', which gives a bias to the size of the associated tile.\",\n \"The symmetry between black and white points may be broken by considering different laws for the gravitational pull associated to them.\",\n \"Consider a Poisson point process of unit density on the space of locally finite counting measures on ${\\\\mathbb {R}}^2 \\\\times [0, 1] \\\\times [0, 1]$ , endowed with Lebesgue measure.\",\n \"The first two coordinates of a point in the support of a realization $\\\\omega $ will mark the location of the seed in the plane, the third and fourth coordinates will determine the gravitational pull of a given point and its color, respectively.\",\n \"More precisely, given a parameter $q \\\\in [0, 1]$ , we say that the seed $(x, z, s) \\\\in \\\\operatorname{supp}(\\\\omega )$ is black if $s \\\\le q$ and white otherwise.\",\n \"We also fix two non-decreasing functions $G_0, G_1: [0,1] \\\\rightarrow (0, \\\\infty )$ , which will be applied to the $z$ coordinate of $(x, z, s)$ to determine the gravitational pull of a given seed ($G_0$ will be used for black seeds and $G_1$ for white).\",\n \"Given $q \\\\in [0, 1]$ , define the occupied (black) set as $\\\\mathcal {O}_q := \\\\bigcup _{\\\\begin{array}{c}(x, z, s) \\\\in \\\\omega ;\\\\\\\\s \\\\le q\\\\end{array}} \\\\Big \\\\lbrace y \\\\in {\\\\mathbb {R}}^2 : \\\\frac{|y - x|}{G_0(z)} \\\\le \\\\frac{|y - x^{\\\\prime }|}{G_1(z^{\\\\prime })} \\\\text{ for all $(x^{\\\\prime }, z^{\\\\prime }, s^{\\\\prime }) \\\\in \\\\omega $ with $s^{\\\\prime } > q$} \\\\Big \\\\rbrace .$ As before, let $\\\\mathcal {V}_q := \\\\mathbb {R}^2 \\\\setminus \\\\mathcal {O}_q$ denote the corresponding vacant set.\",\n \"Notice that the cell corresponding to a given seed may be disconnected.\",\n \"Finally, we define the critical values of the parameter $q$ as $\\\\begin{aligned}q_c:=&\\\\sup \\\\big \\\\lbrace q\\\\in [0,1]:{\\\\mathbb {P}}\\\\big [0\\\\overset{\\\\tiny \\\\mathcal {O}_q}{\\\\longleftrightarrow }\\\\infty \\\\big ]=0\\\\big \\\\rbrace ,\\\\\\\\q_c^\\\\star :=&\\\\inf \\\\big \\\\lbrace q\\\\in [0,1]:{\\\\mathbb {P}}\\\\big [0\\\\overset{\\\\tiny \\\\mathcal {V}_q}{\\\\longleftrightarrow }\\\\infty \\\\big ]=0\\\\big \\\\rbrace .\\\\end{aligned}$ We will here only consider the case when $G_0$ and $G_1$ take values in some interval $[a, b] \\\\subset (0, \\\\infty )$ .\",\n \"It would be interesting to attempt to relax these assumptions.\",\n \"When $G_0 = G_1$ , then the gravitational pull of all black and white seeds are equally distributed.\",\n \"In this case the model is self-dual, and we recover the equation $q_c+q_c^\\\\star =1$ ; see Section REF below.\",\n \"The techniques developed in this paper may easily be adapted to prove the following result for Poisson Voronoi percolation.\",\n \"Theorem 8.1 Assume that $G_0$ and $G_1$ take values in some interval $[a,b]\\\\subset (0,\\\\infty )$ .\",\n \"Then, $q_c=q_c^\\\\star $ and their common value is strictly between zero and one.\",\n \"Moreover, for all $q0$ such that ${\\\\mathbb {P}}\\\\big [ 0\\\\overset{\\\\tiny \\\\mathcal {O}_q}{\\\\longleftrightarrow }\\\\partial B(r) \\\\big ] \\\\le \\\\exp \\\\lbrace -c r\\\\rbrace .$ for $q=q_c$ and all $\\\\kappa >0$ , there exist $c=c(\\\\kappa )>0$ and $\\\\alpha >0$ such that for every $r \\\\ge 1$ $c < {\\\\mathbb {P}}(\\\\operatorname{Cross}(\\\\kappa r,r)) < 1 - c\\\\quad \\\\text{and}\\\\quad \\\\mathbb {P} \\\\big [ 0 \\\\overset{\\\\tiny \\\\mathcal {O}_q}{\\\\longleftrightarrow }\\\\partial B(r) \\\\big ] \\\\le r^{-\\\\alpha }.$ for all $q>q_c$ , there exists $c=c(q)>0$ such that ${\\\\mathbb {P}}\\\\big [ 0\\\\overset{\\\\tiny \\\\mathcal {V}_q}{\\\\longleftrightarrow }\\\\partial B(r) \\\\big ] \\\\le \\\\exp \\\\lbrace -c r\\\\rbrace .$ Moreover, at $q_c$ there is almost surely no unbounded cluster of either kind.\",\n \"We will for the remainder of this subsection outline the proof of this result, based on the techniques developed over the previous sections.\",\n \"First of all, we note that the law of the above model is invariant with respect to translations, rotations and reflections.\",\n \"The measure is further positively associated, meaning that is satisfies an FKG inequality similar to (REF ), for events that are increasing in the addition of black points and removal of white points.\",\n \"This can be seen by adapting the proof of [10] for the case when $G_0=G_1$ are constant.\",\n \"What remains in order for the techniques of Sections and to apply is an estimate on the spatial correlations.\",\n \"Lemma 8.2 Fix a bounded measurable set $D \\\\subseteq \\\\mathbb {R}^2$ and an arbitrary function $f(\\\\mathcal {O}_q)$ satisfying $f \\\\in \\\\sigma (Y_v; v \\\\in D)$ .\",\n \"Then, defining the event $G_{D, r} = \\\\Big \\\\lbrace \\\\text{for every $y \\\\in D$, there is a point $(x, z, s)$ in $\\\\omega $ with $d(y, x) \\\\le \\\\frac{a}{b} r$} \\\\Big \\\\rbrace ,$ we have that $\\\\operatorname{{\\\\bf 1}}_{G_{D, r}} \\\\cdot f$ only depends on the restriction of $\\\\omega $ to $B(D, r) \\\\times [0, 1] \\\\times [0, 1]$ .\",\n \"Moreover, for every $\\\\varepsilon >0$ there exists a constant $c=c(\\\\varepsilon )>0$ such that for any $D\\\\subseteq B^\\\\infty (r)$ we have $\\\\mathbb {P}(G_{D, \\\\varepsilon r}) \\\\ge 1 - \\\\exp \\\\lbrace - c r^2 \\\\rbrace $ .\",\n \"The proof of the first claim follows the same steps as those of Lemma 1.1 in [42].\",\n \"Roughly speaking, the argument goes like this.\",\n \"Given $x \\\\in \\\\mathbb {R}^2$ , if a seed in $\\\\omega $ is located within distance $\\\\ell $ from $x$ , then no other seed outside $B(x, (b/a)\\\\ell )$ can influence the state of $x$ .\",\n \"The second claim follows from a simple large deviations bound for Poisson random variables.\",\n \"As a first consequence of this fact we obtain an estimate on the decay of spatial correlations.\",\n \"As in Definition REF , let $f_1, f_2:\\\\mathcal {P}({\\\\mathbb {R}}^2)\\\\rightarrow [-1,1]$ be two functions satisfying $f_1(\\\\mathcal {O}_q)\\\\in \\\\sigma \\\\big (Y_v;v\\\\in B^\\\\infty (r)\\\\big )$ and $f_2\\\\in \\\\sigma \\\\big (Y_v;v\\\\in B^\\\\infty (r+t)\\\\setminus B^\\\\infty (r+s)\\\\big )$ , for some $t>s$ .\",\n \"Then, if $G=G_{B^\\\\infty (r),s/2}\\\\cap G_{B^\\\\infty (r+t)\\\\setminus B^\\\\infty (r+s),s/2}$ , we obtain $\\\\begin{split}\\\\mathbb {E}[f_1 f_2] \\\\,& \\\\le \\\\, \\\\mathbb {P}(G^c) + \\\\mathbb {E}[\\\\operatorname{{\\\\bf 1}}_G \\\\cdot f_1 f_2]\\\\\\\\&\\\\le \\\\,2\\\\mathbb {P}(G^c)+\\\\operatorname{{\\\\mathbb {E}}}[f_1^{\\\\prime }]\\\\operatorname{{\\\\mathbb {E}}}[f_2^{\\\\prime }]\\\\\\\\&\\\\le \\\\,6{\\\\mathbb {P}}(G^c)+\\\\operatorname{{\\\\mathbb {E}}}[f_1]\\\\operatorname{{\\\\mathbb {E}}}[f_2],\\\\end{split}$ where $f_1^{\\\\prime }$ and $f_2^{\\\\prime }$ denote $f_1$ and $f_2$ evaluated at the restriction of $\\\\omega $ to $B^\\\\infty (r+s/2)$ and $B^\\\\infty (r+t)\\\\setminus B^\\\\infty (r+s/2)$ and are hence independent.\",\n \"In particular, when $s=\\\\varepsilon r$ and $t=\\\\frac{1}{\\\\varepsilon }r$ , which has been the case throughout this paper, the correlation between $f_1$ and $f_2$ decays as $\\\\exp \\\\lbrace - c r^2\\\\rbrace $ as in Lemma REF .\",\n \"Consequently, the techniques of Sections and apply, and we obtain the existence of $q_0$ and $q_1$ , defined analogously as $\\\\lambda _0$ and $\\\\lambda _1$ , satisfying $0q_1$ .\",\n \"To complete the proof of Theorem REF , it remains to prove that $q_0=q_1$ .\",\n \"In order to show that $q_0=q_1$ we again enlarge our probability space as in Section REF , to enable a two-stage construction of our process.\",\n \"We thus assume that $\\\\overline{\\\\omega }$ is a Poisson point process on the space of locally finite counting measures on $\\\\big ({\\\\mathbb {R}}^2\\\\times [0,1]\\\\times [0,1]\\\\big )\\\\times [0,1]$ .\",\n \"The law of $\\\\overline{\\\\omega }$ (denoted $\\\\overline{\\\\mathbb {P}}_\\\\lambda $ ) is chosen with a large density $\\\\lambda =m\\\\ge 1$ and the fifth coordinate will be used to thin the process down to density one, which we then use to define $\\\\mathcal {O}_q$ as in (REF ).\",\n \"We denote by $\\\\omega $ the projection of $\\\\overline{\\\\omega }$ onto $\\\\mathbb {R}^2 \\\\times [0, 1] \\\\times [0, 1]$ (the first four coordinates) and by $\\\\underline{\\\\omega }$ the further projection onto $\\\\mathbb {R}^2 \\\\times [0, 1]$ .\",\n \"Define $W_q:=\\\\overline{{\\\\mathbb {P}}}_\\\\lambda (\\\\operatorname{Cross}(4r,r)|\\\\underline{\\\\omega }).$ As in Section REF , we note that $W_q$ for $\\\\overline{{\\\\mathbb {P}}}_\\\\lambda $ -almost every $\\\\overline{\\\\omega }$ coincides with the expectation of some Boolean function $f_{\\\\underline{\\\\omega }}:\\\\lbrace 0,1\\\\rbrace ^{\\\\underline{\\\\omega }}\\\\times \\\\lbrace 0,1\\\\rbrace ^{\\\\underline{\\\\omega }}\\\\rightarrow \\\\lbrace 0,1\\\\rbrace $ evaluated with respect to product measure with density $q$ and $1/m$ for choosing `color' and `presence', respectively, of each point in $\\\\underline{\\\\omega }$ .\",\n \"This function is increasing in the choice of color, so we obtain via the Margulis-Russo formula that $\\\\frac{dW_q}{dq}=\\\\sum _{i}\\\\overline{{\\\\mathbb {P}}}_\\\\lambda \\\\big ( \\\\,\\\\text{point $i$ is present and its color is pivotal for }f_{\\\\underline{\\\\omega }}\\\\, \\\\big | \\\\,\\\\underline{\\\\omega }\\\\, \\\\big ).$ Note that the concept of pivotality arising here is with respect to color and not presence as in Section .\",\n \"The crucial observation is that is that switching the color of a point has a larger potential change than switching its presence.\",\n \"That is, for the existence of a black crossing, a black point is better than no point, and no point is better than a white point.\",\n \"Hence, from (REF ), we obtain that $\\\\begin{aligned}\\\\frac{dW_q}{dq} \\\\,&=\\\\, \\\\overline{\\\\operatorname{{\\\\mathbb {E}}}}_\\\\lambda \\\\Big [ \\\\sum _i \\\\overline{{\\\\mathbb {P}}}_\\\\lambda \\\\big ( \\\\, \\\\text{point $i$ is present and its color is pivotal} \\\\, \\\\big | \\\\, \\\\omega \\\\, \\\\big ) \\\\Big | \\\\, \\\\underline{\\\\omega }\\\\, \\\\Big ]\\\\\\\\&\\\\ge \\\\,\\\\overline{\\\\operatorname{{\\\\mathbb {E}}}}_\\\\lambda \\\\Big [ \\\\sum _i\\\\overline{{\\\\mathbb {P}}}_\\\\lambda \\\\big ( \\\\, \\\\text{point $i$ is present and its presence is pivotal} \\\\, \\\\big | \\\\,\\\\omega \\\\, \\\\big ) \\\\Big |\\\\,\\\\underline{\\\\omega }\\\\,\\\\Big ]\\\\\\\\&=\\\\, \\\\frac{1}{m} \\\\overline{\\\\operatorname{{\\\\mathbb {E}}}}_\\\\lambda \\\\Big [ \\\\sum _i \\\\operatorname{Inf}^q_i(f_{\\\\underline{\\\\omega }}) \\\\Big |\\\\,\\\\underline{\\\\omega }\\\\,\\\\Big ]\\\\\\\\&\\\\ge \\\\, \\\\frac{1}{m} \\\\overline{\\\\operatorname{{\\\\mathbb {E}}}}_\\\\lambda \\\\Big [ \\\\frac{c}{\\\\log (m)} \\\\operatorname{Var}_\\\\lambda (f_{\\\\underline{\\\\omega }}|\\\\omega )\\\\log \\\\frac{1}{\\\\max _i \\\\big ( \\\\operatorname{Inf}_i^q(f_{\\\\underline{\\\\omega }}) \\\\big ) } \\\\,\\\\Big |\\\\,\\\\underline{\\\\omega }\\\\,\\\\Big ],\\\\end{aligned}$ where in the last step we applied (REF ), and $\\\\operatorname{Inf}_i^q(f_{\\\\underline{\\\\omega }})$ denotes the conditional probability, given $\\\\omega $ , that changing point $i$ from present to absent changes the outcome of $f_{\\\\underline{\\\\omega }}$ .\",\n \"Controlling the variance $\\\\operatorname{Var}_\\\\lambda (f_{\\\\underline{\\\\omega }}|\\\\omega )$ can now be done exactly as in Section REF .\",\n \"Also the influences $\\\\operatorname{Inf}_i^q(f_{\\\\underline{\\\\omega }})$ with respect to presence of a point can be estimated similarly as it was done in Section REF , but requires a slight modification of the decoupling inequalities (REF ) and (REF ).\",\n \"Let $D_1,D_2,\\\\ldots , D_J$ be well separated sets as in (REF ), and let $A_1,A_2,\\\\ldots ,A_J$ be events where $A_j$ is determined by the restriction of $\\\\mathcal {O}_q$ to $D_j$ .\",\n \"On the event $G={\\\\textstyle \\\\bigcap \\\\limits }_{j\\\\le J}G_{D_j,r/(2n)}$ the events $A_1,A_2,\\\\ldots ,A_J$ are, by Lemma REF , determined by restrictions of $\\\\overline{\\\\omega }$ to disjoint subsets of the plane, and are hence independent.\",\n \"We obtain, therefore, the following variant of (REF ) for the Voronoi model $\\\\overline{{\\\\mathbb {P}}}_\\\\lambda \\\\big (A_1\\\\cap \\\\cdots \\\\cap A_J\\\\big |\\\\omega \\\\big )\\\\,\\\\le \\\\,\\\\overline{{\\\\mathbb {P}}}_\\\\lambda (G^c|\\\\omega )+\\\\prod _{j\\\\le J}\\\\Big [\\\\overline{{\\\\mathbb {P}}}_\\\\lambda (A_j|\\\\omega )+\\\\overline{{\\\\mathbb {P}}}_\\\\lambda (G^c|\\\\omega )\\\\Big ].$ Since $\\\\overline{{\\\\mathbb {P}}}_\\\\lambda (G^c)\\\\rightarrow 0$ fast, also $\\\\overline{{\\\\mathbb {P}}}_\\\\lambda (G^c|\\\\omega )\\\\rightarrow 0$ fast with probability tending to one.\",\n \"This suffices for the application of (REF ) in the proof of Lemma REF .\",\n \"In order to obtain a variant for (REF ) we observe that the difference between $\\\\overline{{\\\\mathbb {P}}}_\\\\lambda (A_j|\\\\omega )$ and the conditional probability (given $\\\\omega $ ) that $A_j$ occurs with respect to the restriction of $\\\\overline{\\\\omega }$ restricted to the set $B(D_j,r/(2n))$ is bounded by $\\\\overline{{\\\\mathbb {P}}}_\\\\lambda (G^c|\\\\omega )$ .\",\n \"Let $H=[\\\\overline{{\\\\mathbb {P}}}_\\\\lambda (G^c|\\\\omega )\\\\le \\\\delta ]$ .\",\n \"Hence, arguing similarly as for (REF ), we obtain that $\\\\overline{{\\\\mathbb {P}}}_\\\\lambda \\\\bigg (H \\\\cap \\\\bigcap _{j\\\\le J}\\\\big [\\\\overline{{\\\\mathbb {P}}}_\\\\lambda (A_j|\\\\omega )\\\\le \\\\delta \\\\big ]\\\\bigg )\\\\,\\\\le \\\\,\\\\prod _{j\\\\le J}\\\\Big [\\\\overline{{\\\\mathbb {P}}}_\\\\lambda \\\\big (\\\\overline{{\\\\mathbb {P}}}_\\\\lambda (A_j|\\\\omega )\\\\le 3\\\\delta \\\\big )+\\\\overline{{\\\\mathbb {P}}}_\\\\lambda (H^c)\\\\Big ].$ Since $\\\\overline{{\\\\mathbb {P}}}_\\\\lambda (H^c)\\\\rightarrow 0$ fast, as a consequence of Lemma REF , the two expressions (REF ) and (REF ) replace (REF ) and (REF ) in the proof of Proposition REF .\",\n \"The proof that $q_0=q_1$ is now a straightforward adaptation of the arguments presented in Section , and this ends the outline of the proof of Theorem REF .\"\n ],\n [\n \"Poisson confetti percolation\",\n \"Confetti percolation, or the `dead leaves' model, was introduced by Jeulin in [22].\",\n \"In this model, black and white confetti `rain down' on the plane according to a Poisson point process, and each point in the plane is colored according to the color of the first confetti to cover it.\",\n \"In the case of circular confetti with fixed diameter, Benjamini and Schramm [11] conjectured that the critical probability for this model equals $1/2$ .\",\n \"This was later confirmed by Hirsch [21] for square shaped confetti and by Müller [30] for circular confetti.\",\n \"Just as in the settings of Poisson Boolean and Voronoi percolation, our techniques allow us to handle confetti of random radii, whose laws may differ between black and white.\",\n \"Consider a Poisson point process on the space of locally finite counting measures on ${\\\\mathbb {R}}^2 \\\\times {\\\\mathbb {R}}_+ \\\\times [0, 1]^2$ .\",\n \"Here again the first two coordinates will assign a location in ${\\\\mathbb {R}}^2$ to each confetti, while the third coordinate will denote the fall-time of a confetti and will be used to order overlapping confetti.\",\n \"Finally, the fourth and fifth coordinates will help us determine the radius and color of the confetti, respectively.\",\n \"As before, we fix $q \\\\in [0, 1]$ and, given a realization $\\\\omega $ with density one, declare a point $(x, t, z, s) \\\\in \\\\omega $ black if $s \\\\le q$ and white otherwise.\",\n \"We also fix non-decreasing functions $G_0, G_1 : [0, 1] \\\\rightarrow (0, \\\\infty )$ that will be applied to the $z$ coordinate, in order to determine the radius of the confetti to be placed at $x\\\\in {\\\\mathbb {R}}^2$ ($G_0$ for black points and $G_1$ for white).\",\n \"More precisely, given a realization $\\\\omega $ of the Poisson point process, let $\\\\mathcal {O}_q := \\\\bigcup _{\\\\begin{array}{c}(x, t, z, s) \\\\in \\\\omega \\\\\\\\s \\\\le q\\\\end{array}} \\\\left\\\\lbrace y \\\\in {\\\\mathbb {R}}^2:\\\\begin{split}& |y - x| \\\\le G_0(z), \\\\text{ and } t < t^{\\\\prime } \\\\text{ for all $(x^{\\\\prime }, t^{\\\\prime }, z^{\\\\prime }, s^{\\\\prime }) \\\\in \\\\omega $}\\\\\\\\& \\\\text{such that $|y - x^{\\\\prime }| \\\\le G_1(z^{\\\\prime })$ and $s^{\\\\prime } > q$}\\\\end{split}\\\\right\\\\rbrace .$ Finally, let $\\\\mathcal {V}_q := \\\\mathbb {R}^2 \\\\setminus \\\\mathcal {O}_q$ and define the critical parameters $q_c$ and $q_c^\\\\star $ as in (REF ).\",\n \"For the sake of simplicity, we assume here rather light tails for distribution of the radii induced by $G_0$ and $G_1$ .\",\n \"If these two functions are the same, then black and white confetti have the same radii distributions, which yields $q_c+q_c^\\\\star =1$ by self-duality, and together with the following theorem that $q_c=1/2$ ; see Section REF below.\",\n \"Theorem 8.3 Assume that $G_0$ and $G_1$ satisfy ${\\\\mathbb {P}}[G_i(Z) \\\\ge r] \\\\le r^{-100}$ , for $i = 0, 1$ and large $r$ , where $Z$ is uniformly distributed on $[0,1]$ .\",\n \"Then, $q_c=q_c^\\\\star $ are strictly between zero and one, and for all $q 0$ such that ${\\\\mathbb {P}}\\\\big [ 0\\\\overset{\\\\tiny \\\\mathcal {O}_q}{\\\\longleftrightarrow }\\\\partial B(r) \\\\big ] \\\\le \\\\frac{1}{c}r^{-10}.$ for $q=q_c$ and all $\\\\kappa >0$ , there exist $c=c(\\\\kappa )>0$ and $\\\\alpha >0$ such that, for every $r \\\\ge 1$ $c<{\\\\mathbb {P}}(\\\\operatorname{Cross}(\\\\kappa r,r))<1-c\\\\quad \\\\text{and}\\\\quad \\\\mathbb {P}\\\\big [ 0\\\\overset{\\\\tiny \\\\mathcal {O}_q}{\\\\longleftrightarrow }\\\\partial B(r) \\\\big ] \\\\le r^{-\\\\alpha }.$ for all $q>q_c$ , there exists $c=c(q)>0$ such that ${\\\\mathbb {P}}\\\\big [ 0\\\\overset{\\\\tiny \\\\mathcal {V}_q}{\\\\longleftrightarrow }\\\\partial B(r) \\\\big ] \\\\le \\\\frac{1}{c}r^{-10}.$ Moreover, at $q_c$ there is almost surely no unbounded cluster of either kind.\",\n \"The proof of this theorem is analogous to that of Theorem REF above.\",\n \"The only distinction lies in the treatment of the spatial dependence.\",\n \"The following lemma will replace Lemma REF .\",\n \"Lemma 8.4 Fix a bounded measurable set $D \\\\subseteq \\\\mathbb {R}^2$ and $r \\\\ge 1$ , $M \\\\ge 1$ and define $G_{D, r, M} = \\\\left\\\\lbrace \\\\begin{array}{c}\\\\text{for every $(x, t, z, s) \\\\in \\\\omega \\\\cap B(D, Mr)$ with $t \\\\le r$, $G_0(z) \\\\vee G_1(z) \\\\le r/M$}\\\\\\\\\\\\text{for every $(x,t,z,s) \\\\in \\\\omega \\\\cap B(D,Mr)^c$ with $t\\\\le r$, $G_0(z) \\\\vee G_1(z) \\\\le d(x,D)$}\\\\\\\\\\\\text{for every $y \\\\in D$, there is $(x, t, z, s) \\\\in \\\\omega $ with $t \\\\le r$, $G_0(z) \\\\wedge G_1(z) \\\\ge |y - z|$}\\\\end{array}\\\\right\\\\rbrace .$ Then, $\\\\lim _{r \\\\rightarrow \\\\infty } r^{10} \\\\mathbb {P}(G_{D, r, M}) = 1$ and for an arbitrary function $f(\\\\mathcal {O}_q) \\\\in \\\\sigma (Y_x; x \\\\in D)$ , we have that $\\\\operatorname{{\\\\bf 1}}_G \\\\cdot f$ is measurable with respect to $\\\\omega $ restricted to $B(D, r/M) \\\\times [0, r] \\\\times [0, 1] \\\\times [0, 1]$ .\",\n \"The proof of this lemma follows from a simple large deviations bound and is omitted.\",\n \"As a consequence of the lemma, spatial correlations will decay as $r^{-10}$ .\",\n \"This rate of decay is more than enough to follow the outlined proof of Theorem REF and obtain a proof also for Theorem REF .\"\n ],\n [\n \"The consequence of self-duality\",\n \"For the two models considered in this section, the case when the two functions $G_0$ and $G_1$ are equal stands out due to the self-duality retained in the model.\",\n \"Self-duality refers to the property that the occupied set at parameter $q$ has the same law as the vacant set has at parameter $1-q$ .\",\n \"In particular, there is no unbounded occupied component at parameter $q$ if and only if there is no unbounded vacant component at parameter $1-q$ (almost surely).\",\n \"As the critical parameters are defined as the supremum for which this holds it follows that $q_c+q_c^\\\\star =1$ , in analogy to the Bernoulli case in (REF ).\",\n \"Together with the equality $q_c=q_c^\\\\star $ of the two parameters, obtained in Theorems REF and REF , it follows that $q_c=q_c^\\\\star =1/2$ , and we recover the results of [9], [21] and [30].\",\n \"Remark 14 There are other Poisson percolation processes that could in principle be studied with the techniques we have employed.\",\n \"These include for instance Brownian interlacements (see [40]) in ${\\\\mathbb {R}}^d$ intersected with a plane, and Poissonian cylinders (see [43]) in ${\\\\mathbb {R}}^d$ intersected with a plane.\",\n \"A rather general setting for Poisson Boolean percolation but with dependence between the radii assigned to different points has been considered by Ahlberg and Tykesson [5], however that study does not address the sharpness of the threshold.\",\n \"It would be interesting to investigate to what extent the techniques developed here applies in that setting.\"\n ],\n [\n \"Open problems\",\n \"We have in this paper developed techniques for the study of the sharpness of the threshold for a wide range of models.\",\n \"The weakest link in this scheme is the Poissonian assumption necessary to deduce that the critical regime is indeed a single point and not an interval.\",\n \"It would be interesting to develop an alternative argument for the existence of a sharp threshold beyond the Poissonian setting considered here.\",\n \"It would further be interesting to pursue optimal conditions to study Poisson Voronoi and confetti percolation.\",\n \"Here we only aimed at illustrating how our techniques may be applied to other settings, but we have not pursued to make these applications optimal.\",\n \"Note also that the techniques we have used do not apply to Poisson Boolean percolation models where the discs are replaced by some other deterministic shape which is not sufficiently symmetric.\",\n \"Besides extending the current techniques to other models, one could be interested in answering several questions that are known for Bernoulli percolation.\",\n \"Examples of such problems would be to define the incipient infinite cluster, study noise sensitivity for $\\\\operatorname{Cross}(r, r)$ and investigate scaling relations, just to name a few.\"\n ]\n]"},"id":{"kind":"string","value":"1605.05926"}}},{"rowIdx":1866698,"cells":{"text":{"kind":"list like","value":[["Active Learning On Weighted Graphs Using Adaptive And Non-adaptive\n Approaches"],["Abstract This paper studies graph-based active learning, where the goal is to reconstruct a binary signal defined on the nodes of a weighted graph, by sampling it on a small subset of the nodes.","A new sampling algorithm is proposed, which sequentially selects the graph nodes to be sampled, based on an aggressive search for the boundary of the signal over the graph.","The algorithm generalizes a recent method for sampling nodes in unweighted graphs.","The generalization improves the sampling performance using the information gained from the available graph weights.","An analysis of the number of samples required by the proposed algorithm is provided, and the gain over the unweighted method is further demonstrated in simulations.","Additionally, the proposed method is compared with an alternative state of-the-art method, which is based on the graph's spectral properties.","It is shown that the proposed method significantly outperforms the spectral sampling method, if the signal needs to be predicted with high accuracy.","On the other hand, if a higher level of inaccuracy is tolerable, then the spectral method outperforms the proposed aggressive search method.","Consequently, we propose a hybrid method, which is shown to combine the advantages of both approaches."],["introduction","This paper studies the problem of binary label prediction on a graph.","In this problem, we are given a graph $G = (V,E)$ , where the edges $E$ (which can be weighted) capture the similarity relationship between the objects represented by the nodes $V$ .","Each node has an initially unknown label associated with it, given by a signal $f:V \\rightarrow \\lbrace -1,+1\\rbrace $ .","The goal is to reconstruct the entire signal by sampling its values on a small subset of the nodes.","This is achievable when the signal bears some degree of smoothness over the graph, which means that similar objects are more likely to have the same label.","Active learning aims to minimize the number of samples needed by selecting the most informative nodes.","This problem arises in many machine learning applications, where there is an abundance of unlabeled data but labeled data is scarce and expensive to obtain, for example, requiring human expertise or elaborate experiments.","Active learning is an effective way to minimize the cost of labeling in such scenarios [1].","A graph based approach to this problem starts by creating a graph where the nodes correspond to the data points ${\\cal X}= \\lbrace {\\bf x}_1, \\ldots , {\\bf x}_n\\rbrace $ and the edges capture the similarity between them.","Typically, a sparse graph which connects each data point to few of its most similar neighbors is used.","The unknown labels $f_i \\in \\lbrace -1,+1\\rbrace $ associated with the data points define a binary function on the nodes.","In datasets of interest, the signal is often notably smooth on the graph.","There are two approaches to active learning on graphs.","The first approach focuses on identifying the nodes near the boundary region, where the signal changes from $+1$ to $-1$ .","The methods with this approach [2], [3] sample nodes sequentially, i.e., the nodes to be sampled next are chosen based on the graph structure as well as previously observed signal values.","The second approach [4], [5], [6], in contrast, utilizes global properties of the graph in order to identify the most informative nodes, and sample them all at once.","Such global approaches usually focus on providing a good approximation of the signal, rather than exact recovery.","It is also possible to combine the two approaches, for example, as in [7].","The first contribution of this paper is a new sampling algorithm, called weighted $S^2$ which takes the boundary sampling approach.","The weighted $S^2$ algorithm is a generalization of a recently proposed algorithm called $S^2$ [2], which is defined only for the case of unweighted edges.","The purpose of the generalized algorithm is to take advantage of the additional information available in the form of the edge weights, in order to reduce the sampling complexity.","We characterize the sampling budget required for signal recovery by the weighted $S^2$ algorithm, as a function of the complexity of the signal with respect to the graph.","We explain how this generalization can be useful in reducing sampling complexity, and demonstrate a significant reduction (nearly $25\\%$ in one dataset), when neighboring nodes of opposite labels are considerably less similar to each other than identically labeled neighbors.","We further compare the sampling complexity of the weighted $S^2$ algorithm with an alternative state-of-the-art method called the cutoff maximization method [4].","Unlike the $S^2$ methods, which aim for a complete recovery of the signal by aggressively searching for the boundary nodes, the cutoff maximization method is focused only on providing a good approximation of the signal by ensuring that the unsampled nodes are well-connected to the sampled nodes [8].","This method finds a sampling set by optimizing a spectral function defined using the graph Laplacian.","We perform the comparison on three realistic data sets, and observe two interesting results: 1.","The cutoff maximization method does not discover the entire boundary (i.e, nodes with oppositely labeled neighbors) unless the sampling set contains almost all the nodes.","In contrast, the number of nodes required by the $S^2$ methods to discover the boundary is not considerably larger than the number of boundary nodes.","2.","When the sampling budget is quite limited, the cutoff maximization method provides a much better approximation of the signal.","There exists a threshold in terms of the sampling budget that determines which method offers better accuracy.","Conversely, the tolerable degree of inaccuracy determines which of the methods offer lower sampling complexity.","Motivated by the second observation, we propose a hybrid approach (similar in spirit to [7]) which samples the first few nodes with the cutoff maximization method to approximate the boundary and then switches to the weighted $S^2$ method to refine the approximation.","The experiments suggest that the hybrid approach combines the advantages of both methods."],["$S^2$ Algorithm for Weighted Graphs"," The $S^2$ algorithm was proposed in [2] for unweighted graphs.","In this section we describe the principle of the algorithm, and then generalize the algorithm to weighted graphs.","In the next section we analyze the query complexity of the generalized algorithm."],["Original $S^2$ Algorithm","The goal of the algorithm is to find the signal $f$ .","To do this, the algorithm operates by finding the edges that connect oppositely labeled nodes.","These edges are called cut edges, and together the set of cut edges is called the cut.","The algorithm incrementally identifies the cut, with the rationale that once the entire cut is identified, the signal is completely recovered.","To find the cut, the algorithm maintains a copy of the graph $G$ , and each time it samples a node that neighbors previously sampled nodes of opposite label, it removes the newly discovered cut edges from the graph copy.","This way, the remaining graph copy contains only the undiscovered part of the cut, and the algorithm can more easily focus on discovering these edges.","The main idea of the algorithm is to look for a pair of oppositely labeled nodes, and then to find the cut edge on the shortest path between the nodes, using a binary search procedure.","The algorithm begins with a random sampling phase.","At this phase the algorithm queries a random node, according to the uniform distribution.","After each sample, the algorithm checks whether the sampled node has any previously sampled neighbors of opposite label.","If such neighbors exist, then the connecting edges are newly discovered cut edges, and they are removed from the graph.","After checking and potentially removing newly discovered cut edges, the algorithm checks whether the remaining graph contains any pair of connected nodes of opposite labels.","If no such pair exists, the algorithm proceeds with a random sample.","If pairs of connected, oppositely labeled nodes do exist, the algorithm looks for the shortest path among all the paths connecting such pairs, and sample the node in the middle of that path (breaking ties arbitrarily).","After each sampling operation, either a random sample or a bisecting sample, the algorithm again removes all newly discovered cut edges.","The $S^2$ algorithm is described more formally in Algorithm REF .","The algorithm is given a budget, which determines the number of queries to perform.","Once the budget is exhausted, the algorithm calls a label completion function, which predicts the labels of the unsampled nodes.","Several such label completion algorithms are known, such as the POCS method in [9].","The $S^2$ algorithm uses a function called Middle Shortest Path (MSP), which returns the node in the middle of the shortest path among all the paths connecting a pair of oppositely labels nodes in the remaining graph.","$S^2$ Algorithm [1] Inputs: Graph $G$ , BUDGET $\\le n$ .","$L\\leftarrow \\emptyset $ 1 $x\\leftarrow $ Randomly chosen unlabeled node.","Add $(x,f(x))$ to $L$ .","Remove newly discovered cut edges from $G$ .","$|L| =\\text{BUDGET}$ LabelCompletion$(G,L)$ $x\\leftarrow \\text{MSP(G,L)}$ exists"],["Generalization for Weighted Graphs","The $S^2$ algorithm in [2] is defined only for unweighted graphs.","Since many learning scenarios provide a weighted graph, we extend the algorithm to exploit the additional available information by modifying the MSP function in the algorithm.","Our modification is based on the assumption that the signal is smooth, which means that high-weight edges connect mostly nodes of the same label.","Therefore, the weight of cut edges is generally low.","In the unweighted $S^2$ algorithm, each MSP query reduces the number of nodes in the shortest of the paths between any two oppositely labeled nodes by approximately one half.","The main idea in our generalization is to take advantage of the low weights of cut edges in order to reduce this number by more than a half with each query.","To do this, we first switch our perspective, for convenience, from the edge weights to the distances associated with the edges, which are inversely proportional to the weights.","The distance between non-neighboring nodes is defined as the sum of the lengths of the edges in the shortest path connecting the nodes.","Since the weights are a decreasing function of the distances, it follows that cut edges are typically longer than other edges.","We take advantage of this fact by modifying the MSP function to sample the node closest to the midpoint of the path, where the midpoint is computed in terms of length, rather than the number of edges in the path.","With each query, the proposed sampling rule can potentially reduce the number of nodes along the shortest of the paths between any two oppositely labeled nodes by more than half if the cut edges contribute significantly more than the non-cut edges to the length of the path.","Thus, ultimately it requires less samples to discover a cut edge.","This intuition is demonstrated in Figure REF .","In this example, the nodes labeled $+1$ are connected with an edge of length $l/2$ , the nodes labeled $-1$ are connected with an edge of length $l$ and the cut edge is of length $3l$ .","Figure: An illustration of advantage of weighted S 2 S^2 over unweighted S 2 S^2.Given the labels of the end nodes of this path, the binary search phase of the unweighted $S^2$ algorithm needs to sample labels of 3 extra nodes to discover the cut edge.","The weighted $S^2$ algorithm, on the other hand, finds the cut edge with only 2 samples.","This type of situation arises more prominently in an unbalanced data set, where the number of nodes in one class is much larger than the other.","The advantage of weighted $S^2$ algorithm in such a case is experimentally verified in Section ."],["Notation","Note that $f$ partitions the vertices of $G$ into a collection of connected components with identically labeled vertices.","Let $V_1,V_2,\\dots ,V_k$ be these $k$ connected components.","Notice that the first node that $S^2$ queries in each collection $V_i$ is often queried randomly, and not by a bisection query.","Define $\\beta \\triangleq \\min _{1\\le i\\le k}\\frac{|V_i|}{n}.$ If $\\beta $ is small, more random queries are required by $S^2$ .","Let $C$ be the set of cut edges in $G$ .","The length of the shortest cut edge in $G$ is denoted by $l_{\\text{cut}}$ .","Let $\\partial C$ be the set of nodes which share an edge with at least one oppositely labeled node.","The nodes in $\\partial C$ are called boundary nodes.","For $1\\le i 0$ , the weighted $S^2$ will recover $C$ with probability at least $1-\\epsilon $ if the BUDGET is at least $\\small m\\left\\lceil 2\\log _2 \\left(\\frac{l_n}{l_{\\text{cut}}}\\right)\\right\\rceil +(|\\partial C|-m)\\left\\lceil 2\\log _2 \\left(\\frac{l_{\\kappa }}{l_{\\text{cut}}}\\right)\\right\\rceil +\\frac{\\log (1/(\\beta \\epsilon ))}{\\log (1/(1-\\beta ))}.$ The proof of Theorem REF uses the fact that after a pair of connected and oppositely labeled nodes is found, the number of queries until a boundary node is sampled is at most logarithmic in $(l/l_{\\text{cut}})$ , where $l$ is the length of the path between the nodes.","We show this fact in the following lemma.","Lemma 2 The cut-edge of length $l_\\text{cut}$ is found after no more than $r = \\left\\lceil {2\\log _2\\left(\\frac{l}{l_\\text{cut}}\\right)}\\right\\rceil $ aggressive steps.","Figure: The interval of interest is at least halved after two queries.In Figure REF , let $C$ and $D$ denote the queried nodes, such that $C$ is sampled first.","Let $E$ denote the midpoint of the interval between $A$ and $B$ , where there may not be a node.","By considering all the cases based on labels of $C$ and $D$ and their positions relative to $E$ , it can be shown that after two queries, the length of the interval of interest (i.e., the interval containing the cut-edge) is at least halved.","The details are omitted in the interest of space.","After $i$ (where $i$ is even) queries, the length of the interval of interest is at most $\\frac{l}{2^{i/2}}$ .","Note that the cut-edge is found when the length of the interval of interest is less than or equal to $l_\\text{cut}$ .","If $r$ is the maximum number of queries required to locate the cut-edge, then $\\frac{l}{2^{r/2}} = l_\\text{cut} \\Rightarrow r = \\left\\lceil {2\\log _2\\left(\\frac{l}{l_\\text{cut}}\\right)}\\right\\rceil $ [Proof of Theorem REF ] The random sampling phase follows the same argument as in [2], which gives the term $\\frac{\\log (1/(\\beta \\epsilon ))}{\\log (1/(1-\\beta ))}$ .","A sequence of bisection queries that commences after a random query or an edge removal, and terminates with an edge removal, is call a run.","In each run, at least one boundary node is being queried.","Therefore, the number of runs is no greater than $|\\partial C|$ .","In each cut component, after the first cut edge and boundary node are discovered, the rest of the boundary nodes are discovered in $\\left\\lceil {2\\log _2\\left(\\frac{l_{\\kappa }}{l_\\text{cut}}\\right)}\\right\\rceil $ queries each, according to Lemmas REF and REF .","For discovering the first cut edge in each cut components, we trivially bound the number of queries by $\\left\\lceil {2\\log _2\\left(\\frac{l_{n}}{l_\\text{cut}}\\right)}\\right\\rceil $ .","Since there are $m$ cut components, we bound the total number of bisection queries by $m\\left\\lceil 2\\log _2 \\left(\\frac{l_n}{l_{\\text{cut}}}\\right)\\right\\rceil +(|\\partial C|-m)\\left\\lceil 2\\log _2 \\left(\\frac{l_{\\kappa }}{l_{\\text{cut}}}\\right)\\right\\rceil .$"],["Experiments","We consider the following graph based classification problems: 1.","USPS handwritten digit recognition [10]: For our experiments, we consider two binary classification sub-problems, namely, 7 vs. 9 and 2 vs. 4, consisting of 200 randomly selected images of each class represented as vectors of dimension 256.","The distance between two data points is $d(i,j) = ||{\\bf x}_i-{\\bf x}_j||$ .","An unweighted graph $G$ is constructed by connecting a pair of nodes $(i,j)$ if $j$ is a $k$ -nearest neighbor ($k$ -nn with $k = 4$ ) Due to lack of space, we do not study the effect of $k$ in detail.","of $i$ or vice versa.","A weighted dissimilarity graph $G_d$ is defined to have the same topology as $G$ but the weight associated with edge $(i,j)$ is set to $d(i,j)$ .","A weighted similarity graph $G_w$ is defined to have the same topology as $G$ but the weight associated with edge $(i,j)$ is set to $w(i,j) = \\exp \\left(- {d(i,j)^2}/{2\\sigma ^2} \\right)$ .","The parameter $\\sigma $ is set to be $1/3$ -rd of the average distance to the $k$ -th nearest neighbor for all datapoints.","2.","Newsgroups text classification [11]: For our experiments, we consider a binary classification sub-problem Baseball vs. Hockey, where each class contains 200 randomly selected documents.","Each document $i$ is represented by a 3000 dimensional vector ${\\bf x}_i$ whose elements are the tf-idf statistics of the 3000 most frequent words in the dataset [8].","The cosine similarity between a pair data points $(i,j)$ is given by $w(i,j) = \\frac{\\left\\langle {{\\bf x}_i,{\\bf x}_j}\\right\\rangle }{||{\\bf x}_i||||{\\bf x}_j||}$ .","The distance between them is defined as $d(i,j) = \\sqrt{1-w(i,j)^2}$ .","The $k$ -nn unweighted graph $G$ (with $k=4$ ), the dissimilarity graph $G_d$ and the similarity graph $G_w$ are constructed using these distance and similarity measures as in the previous example.","In addition to the above datasets, we generate a synthetic two circles dataset, shown in Figure REF , in order to demonstrate the advantage of weighted $S^2$ over unweighted $S^2$ .","It contains 900 points in one class (marked red) evenly distributed on the inner circle of mean radius 1 and variance $0.05$ and 100 points in the second class (marked blue) on the outer circle of mean radius $1.1$ and variance $0.45$ .","A 4-nn graph is constructed using the Euclidean distance between the coordinates of the points.","Figure: A synthetic two circles datasetWe compare the performance of the following active learning methods: (1) unweighted $S^2$ method [2] with graph $G$ , (2) weighted $S^2$ method with dissimilarity graph $G_d$ , (3) cutoff maximization method [8] with similarity graph $G_w$ and (4) a hybrid approach combining cutoff maximization and weighted $S^2$ method.","After the nodes selected by each method have been sampled, we reconstruct the unknown label signal using the approximate POCS based bandlimited reconstruction scheme [8] to get the soft labels.","We threshold these soft labels to get the final label predictions.","The hybrid approach uses the non-adaptive cutoff maximization approach in the beginning and switches to the weighted $S^2$ method after sampling a certain number of nodes $n_\\text{switch}$ .","In order to determine $n_\\text{switch}$ , after sampling the $i$ -th node with the cutoff method, we compute $1-\\frac{\\mathinner {\\langle {\\hat{{\\bf f}}_i,\\hat{{\\bf f}}_{i-1}}\\rangle }}{\\Vert \\hat{{\\bf f}}_{i}\\Vert \\Vert \\hat{{\\bf f}}_{i-1}\\Vert }$ , where $\\hat{{\\bf f}}_i$ denotes the vector of predicted soft labels.","Once this value falls below below $0.001$ , indicating that the newly added label only marginally changed the predictions, hybrid approach switches to weighted $S^2$ .","Table REF lists the number of samples required by each of the sampling methods to discover all the cut edges using the observed labels.","It shows that weighted $S^2$ can reduce the sample complexity significantly (by $25\\%$ ) compared to unweighted $S^2$ if the ratio of mean length of cut edges and mean length of non-cut edges is high as is the case in the unbalanced two circles dataset.","In rest of the datasets, the gain offered by weighted $S^2$ is negligible since the cut edges are only slightly longer than non-cut edges and as a result, taking lengths into account in the bisection phase does not offer much advantage.","We also observe that the number of samples required by the weighted $S^2$ method is close to the size of the cut $|\\partial C|$ in most of the datasets.","Table REF also shows that the adaptive methods $S^2$ and weighted $S^2$ are very efficient at recovering the entire cut exactly, compared to the non-adaptive cutoff maximization method.","In practice, it is not necessary to reconstruct the signal exactly and some reconstruction error is allowed.","Figure REF plots the classification error against the number of sampled nodes.","It shows that the classification error of the cutoff maximization method decreases rapidly in the early stages of sampling when very few samples are observed.","However, the decrease is slow in later stages of sampling.","$S^2$ methods, on the other hand, are good at reducing the error in the later stages, but performs poorly with only a few samples.","The figure also shows that the hybrid method performs as well as the better method in each region."],["Conclusions"," The paper generalizes the $S^2$ algorithm for the case of weighted graphs.","The sampling complexity of the generalized algorithm is analyzed, and the gain over the unweighted version is demonstrated by simulation.","Additional experiments identify the region of tolerable reconstruction error in which the $S^2$ algorithms outperforms a graph frequency based global approach.","A hybrid approach is proposed with the advantages of both methods.","It remains open to analytically characterize of the gain of the weighted $S^2$ method over the unweighted version.","Another interesting avenue for future work is to provide a performance analysis for the spectral sampling method which can suggest an optimal switching criterion for the hybrid method."]],"string":"[\n [\n \"Active Learning On Weighted Graphs Using Adaptive And Non-adaptive\\n Approaches\"\n ],\n [\n \"Abstract This paper studies graph-based active learning, where the goal is to reconstruct a binary signal defined on the nodes of a weighted graph, by sampling it on a small subset of the nodes.\",\n \"A new sampling algorithm is proposed, which sequentially selects the graph nodes to be sampled, based on an aggressive search for the boundary of the signal over the graph.\",\n \"The algorithm generalizes a recent method for sampling nodes in unweighted graphs.\",\n \"The generalization improves the sampling performance using the information gained from the available graph weights.\",\n \"An analysis of the number of samples required by the proposed algorithm is provided, and the gain over the unweighted method is further demonstrated in simulations.\",\n \"Additionally, the proposed method is compared with an alternative state of-the-art method, which is based on the graph's spectral properties.\",\n \"It is shown that the proposed method significantly outperforms the spectral sampling method, if the signal needs to be predicted with high accuracy.\",\n \"On the other hand, if a higher level of inaccuracy is tolerable, then the spectral method outperforms the proposed aggressive search method.\",\n \"Consequently, we propose a hybrid method, which is shown to combine the advantages of both approaches.\"\n ],\n [\n \"introduction\",\n \"This paper studies the problem of binary label prediction on a graph.\",\n \"In this problem, we are given a graph $G = (V,E)$ , where the edges $E$ (which can be weighted) capture the similarity relationship between the objects represented by the nodes $V$ .\",\n \"Each node has an initially unknown label associated with it, given by a signal $f:V \\\\rightarrow \\\\lbrace -1,+1\\\\rbrace $ .\",\n \"The goal is to reconstruct the entire signal by sampling its values on a small subset of the nodes.\",\n \"This is achievable when the signal bears some degree of smoothness over the graph, which means that similar objects are more likely to have the same label.\",\n \"Active learning aims to minimize the number of samples needed by selecting the most informative nodes.\",\n \"This problem arises in many machine learning applications, where there is an abundance of unlabeled data but labeled data is scarce and expensive to obtain, for example, requiring human expertise or elaborate experiments.\",\n \"Active learning is an effective way to minimize the cost of labeling in such scenarios [1].\",\n \"A graph based approach to this problem starts by creating a graph where the nodes correspond to the data points ${\\\\cal X}= \\\\lbrace {\\\\bf x}_1, \\\\ldots , {\\\\bf x}_n\\\\rbrace $ and the edges capture the similarity between them.\",\n \"Typically, a sparse graph which connects each data point to few of its most similar neighbors is used.\",\n \"The unknown labels $f_i \\\\in \\\\lbrace -1,+1\\\\rbrace $ associated with the data points define a binary function on the nodes.\",\n \"In datasets of interest, the signal is often notably smooth on the graph.\",\n \"There are two approaches to active learning on graphs.\",\n \"The first approach focuses on identifying the nodes near the boundary region, where the signal changes from $+1$ to $-1$ .\",\n \"The methods with this approach [2], [3] sample nodes sequentially, i.e., the nodes to be sampled next are chosen based on the graph structure as well as previously observed signal values.\",\n \"The second approach [4], [5], [6], in contrast, utilizes global properties of the graph in order to identify the most informative nodes, and sample them all at once.\",\n \"Such global approaches usually focus on providing a good approximation of the signal, rather than exact recovery.\",\n \"It is also possible to combine the two approaches, for example, as in [7].\",\n \"The first contribution of this paper is a new sampling algorithm, called weighted $S^2$ which takes the boundary sampling approach.\",\n \"The weighted $S^2$ algorithm is a generalization of a recently proposed algorithm called $S^2$ [2], which is defined only for the case of unweighted edges.\",\n \"The purpose of the generalized algorithm is to take advantage of the additional information available in the form of the edge weights, in order to reduce the sampling complexity.\",\n \"We characterize the sampling budget required for signal recovery by the weighted $S^2$ algorithm, as a function of the complexity of the signal with respect to the graph.\",\n \"We explain how this generalization can be useful in reducing sampling complexity, and demonstrate a significant reduction (nearly $25\\\\%$ in one dataset), when neighboring nodes of opposite labels are considerably less similar to each other than identically labeled neighbors.\",\n \"We further compare the sampling complexity of the weighted $S^2$ algorithm with an alternative state-of-the-art method called the cutoff maximization method [4].\",\n \"Unlike the $S^2$ methods, which aim for a complete recovery of the signal by aggressively searching for the boundary nodes, the cutoff maximization method is focused only on providing a good approximation of the signal by ensuring that the unsampled nodes are well-connected to the sampled nodes [8].\",\n \"This method finds a sampling set by optimizing a spectral function defined using the graph Laplacian.\",\n \"We perform the comparison on three realistic data sets, and observe two interesting results: 1.\",\n \"The cutoff maximization method does not discover the entire boundary (i.e, nodes with oppositely labeled neighbors) unless the sampling set contains almost all the nodes.\",\n \"In contrast, the number of nodes required by the $S^2$ methods to discover the boundary is not considerably larger than the number of boundary nodes.\",\n \"2.\",\n \"When the sampling budget is quite limited, the cutoff maximization method provides a much better approximation of the signal.\",\n \"There exists a threshold in terms of the sampling budget that determines which method offers better accuracy.\",\n \"Conversely, the tolerable degree of inaccuracy determines which of the methods offer lower sampling complexity.\",\n \"Motivated by the second observation, we propose a hybrid approach (similar in spirit to [7]) which samples the first few nodes with the cutoff maximization method to approximate the boundary and then switches to the weighted $S^2$ method to refine the approximation.\",\n \"The experiments suggest that the hybrid approach combines the advantages of both methods.\"\n ],\n [\n \"$S^2$ Algorithm for Weighted Graphs\",\n \" The $S^2$ algorithm was proposed in [2] for unweighted graphs.\",\n \"In this section we describe the principle of the algorithm, and then generalize the algorithm to weighted graphs.\",\n \"In the next section we analyze the query complexity of the generalized algorithm.\"\n ],\n [\n \"Original $S^2$ Algorithm\",\n \"The goal of the algorithm is to find the signal $f$ .\",\n \"To do this, the algorithm operates by finding the edges that connect oppositely labeled nodes.\",\n \"These edges are called cut edges, and together the set of cut edges is called the cut.\",\n \"The algorithm incrementally identifies the cut, with the rationale that once the entire cut is identified, the signal is completely recovered.\",\n \"To find the cut, the algorithm maintains a copy of the graph $G$ , and each time it samples a node that neighbors previously sampled nodes of opposite label, it removes the newly discovered cut edges from the graph copy.\",\n \"This way, the remaining graph copy contains only the undiscovered part of the cut, and the algorithm can more easily focus on discovering these edges.\",\n \"The main idea of the algorithm is to look for a pair of oppositely labeled nodes, and then to find the cut edge on the shortest path between the nodes, using a binary search procedure.\",\n \"The algorithm begins with a random sampling phase.\",\n \"At this phase the algorithm queries a random node, according to the uniform distribution.\",\n \"After each sample, the algorithm checks whether the sampled node has any previously sampled neighbors of opposite label.\",\n \"If such neighbors exist, then the connecting edges are newly discovered cut edges, and they are removed from the graph.\",\n \"After checking and potentially removing newly discovered cut edges, the algorithm checks whether the remaining graph contains any pair of connected nodes of opposite labels.\",\n \"If no such pair exists, the algorithm proceeds with a random sample.\",\n \"If pairs of connected, oppositely labeled nodes do exist, the algorithm looks for the shortest path among all the paths connecting such pairs, and sample the node in the middle of that path (breaking ties arbitrarily).\",\n \"After each sampling operation, either a random sample or a bisecting sample, the algorithm again removes all newly discovered cut edges.\",\n \"The $S^2$ algorithm is described more formally in Algorithm REF .\",\n \"The algorithm is given a budget, which determines the number of queries to perform.\",\n \"Once the budget is exhausted, the algorithm calls a label completion function, which predicts the labels of the unsampled nodes.\",\n \"Several such label completion algorithms are known, such as the POCS method in [9].\",\n \"The $S^2$ algorithm uses a function called Middle Shortest Path (MSP), which returns the node in the middle of the shortest path among all the paths connecting a pair of oppositely labels nodes in the remaining graph.\",\n \"$S^2$ Algorithm [1] Inputs: Graph $G$ , BUDGET $\\\\le n$ .\",\n \"$L\\\\leftarrow \\\\emptyset $ 1 $x\\\\leftarrow $ Randomly chosen unlabeled node.\",\n \"Add $(x,f(x))$ to $L$ .\",\n \"Remove newly discovered cut edges from $G$ .\",\n \"$|L| =\\\\text{BUDGET}$ LabelCompletion$(G,L)$ $x\\\\leftarrow \\\\text{MSP(G,L)}$ exists\"\n ],\n [\n \"Generalization for Weighted Graphs\",\n \"The $S^2$ algorithm in [2] is defined only for unweighted graphs.\",\n \"Since many learning scenarios provide a weighted graph, we extend the algorithm to exploit the additional available information by modifying the MSP function in the algorithm.\",\n \"Our modification is based on the assumption that the signal is smooth, which means that high-weight edges connect mostly nodes of the same label.\",\n \"Therefore, the weight of cut edges is generally low.\",\n \"In the unweighted $S^2$ algorithm, each MSP query reduces the number of nodes in the shortest of the paths between any two oppositely labeled nodes by approximately one half.\",\n \"The main idea in our generalization is to take advantage of the low weights of cut edges in order to reduce this number by more than a half with each query.\",\n \"To do this, we first switch our perspective, for convenience, from the edge weights to the distances associated with the edges, which are inversely proportional to the weights.\",\n \"The distance between non-neighboring nodes is defined as the sum of the lengths of the edges in the shortest path connecting the nodes.\",\n \"Since the weights are a decreasing function of the distances, it follows that cut edges are typically longer than other edges.\",\n \"We take advantage of this fact by modifying the MSP function to sample the node closest to the midpoint of the path, where the midpoint is computed in terms of length, rather than the number of edges in the path.\",\n \"With each query, the proposed sampling rule can potentially reduce the number of nodes along the shortest of the paths between any two oppositely labeled nodes by more than half if the cut edges contribute significantly more than the non-cut edges to the length of the path.\",\n \"Thus, ultimately it requires less samples to discover a cut edge.\",\n \"This intuition is demonstrated in Figure REF .\",\n \"In this example, the nodes labeled $+1$ are connected with an edge of length $l/2$ , the nodes labeled $-1$ are connected with an edge of length $l$ and the cut edge is of length $3l$ .\",\n \"Figure: An illustration of advantage of weighted S 2 S^2 over unweighted S 2 S^2.Given the labels of the end nodes of this path, the binary search phase of the unweighted $S^2$ algorithm needs to sample labels of 3 extra nodes to discover the cut edge.\",\n \"The weighted $S^2$ algorithm, on the other hand, finds the cut edge with only 2 samples.\",\n \"This type of situation arises more prominently in an unbalanced data set, where the number of nodes in one class is much larger than the other.\",\n \"The advantage of weighted $S^2$ algorithm in such a case is experimentally verified in Section .\"\n ],\n [\n \"Notation\",\n \"Note that $f$ partitions the vertices of $G$ into a collection of connected components with identically labeled vertices.\",\n \"Let $V_1,V_2,\\\\dots ,V_k$ be these $k$ connected components.\",\n \"Notice that the first node that $S^2$ queries in each collection $V_i$ is often queried randomly, and not by a bisection query.\",\n \"Define $\\\\beta \\\\triangleq \\\\min _{1\\\\le i\\\\le k}\\\\frac{|V_i|}{n}.$ If $\\\\beta $ is small, more random queries are required by $S^2$ .\",\n \"Let $C$ be the set of cut edges in $G$ .\",\n \"The length of the shortest cut edge in $G$ is denoted by $l_{\\\\text{cut}}$ .\",\n \"Let $\\\\partial C$ be the set of nodes which share an edge with at least one oppositely labeled node.\",\n \"The nodes in $\\\\partial C$ are called boundary nodes.\",\n \"For $1\\\\le i 0$ , the weighted $S^2$ will recover $C$ with probability at least $1-\\\\epsilon $ if the BUDGET is at least $\\\\small m\\\\left\\\\lceil 2\\\\log _2 \\\\left(\\\\frac{l_n}{l_{\\\\text{cut}}}\\\\right)\\\\right\\\\rceil +(|\\\\partial C|-m)\\\\left\\\\lceil 2\\\\log _2 \\\\left(\\\\frac{l_{\\\\kappa }}{l_{\\\\text{cut}}}\\\\right)\\\\right\\\\rceil +\\\\frac{\\\\log (1/(\\\\beta \\\\epsilon ))}{\\\\log (1/(1-\\\\beta ))}.$ The proof of Theorem REF uses the fact that after a pair of connected and oppositely labeled nodes is found, the number of queries until a boundary node is sampled is at most logarithmic in $(l/l_{\\\\text{cut}})$ , where $l$ is the length of the path between the nodes.\",\n \"We show this fact in the following lemma.\",\n \"Lemma 2 The cut-edge of length $l_\\\\text{cut}$ is found after no more than $r = \\\\left\\\\lceil {2\\\\log _2\\\\left(\\\\frac{l}{l_\\\\text{cut}}\\\\right)}\\\\right\\\\rceil $ aggressive steps.\",\n \"Figure: The interval of interest is at least halved after two queries.In Figure REF , let $C$ and $D$ denote the queried nodes, such that $C$ is sampled first.\",\n \"Let $E$ denote the midpoint of the interval between $A$ and $B$ , where there may not be a node.\",\n \"By considering all the cases based on labels of $C$ and $D$ and their positions relative to $E$ , it can be shown that after two queries, the length of the interval of interest (i.e., the interval containing the cut-edge) is at least halved.\",\n \"The details are omitted in the interest of space.\",\n \"After $i$ (where $i$ is even) queries, the length of the interval of interest is at most $\\\\frac{l}{2^{i/2}}$ .\",\n \"Note that the cut-edge is found when the length of the interval of interest is less than or equal to $l_\\\\text{cut}$ .\",\n \"If $r$ is the maximum number of queries required to locate the cut-edge, then $\\\\frac{l}{2^{r/2}} = l_\\\\text{cut} \\\\Rightarrow r = \\\\left\\\\lceil {2\\\\log _2\\\\left(\\\\frac{l}{l_\\\\text{cut}}\\\\right)}\\\\right\\\\rceil $ [Proof of Theorem REF ] The random sampling phase follows the same argument as in [2], which gives the term $\\\\frac{\\\\log (1/(\\\\beta \\\\epsilon ))}{\\\\log (1/(1-\\\\beta ))}$ .\",\n \"A sequence of bisection queries that commences after a random query or an edge removal, and terminates with an edge removal, is call a run.\",\n \"In each run, at least one boundary node is being queried.\",\n \"Therefore, the number of runs is no greater than $|\\\\partial C|$ .\",\n \"In each cut component, after the first cut edge and boundary node are discovered, the rest of the boundary nodes are discovered in $\\\\left\\\\lceil {2\\\\log _2\\\\left(\\\\frac{l_{\\\\kappa }}{l_\\\\text{cut}}\\\\right)}\\\\right\\\\rceil $ queries each, according to Lemmas REF and REF .\",\n \"For discovering the first cut edge in each cut components, we trivially bound the number of queries by $\\\\left\\\\lceil {2\\\\log _2\\\\left(\\\\frac{l_{n}}{l_\\\\text{cut}}\\\\right)}\\\\right\\\\rceil $ .\",\n \"Since there are $m$ cut components, we bound the total number of bisection queries by $m\\\\left\\\\lceil 2\\\\log _2 \\\\left(\\\\frac{l_n}{l_{\\\\text{cut}}}\\\\right)\\\\right\\\\rceil +(|\\\\partial C|-m)\\\\left\\\\lceil 2\\\\log _2 \\\\left(\\\\frac{l_{\\\\kappa }}{l_{\\\\text{cut}}}\\\\right)\\\\right\\\\rceil .$\"\n ],\n [\n \"Experiments\",\n \"We consider the following graph based classification problems: 1.\",\n \"USPS handwritten digit recognition [10]: For our experiments, we consider two binary classification sub-problems, namely, 7 vs. 9 and 2 vs. 4, consisting of 200 randomly selected images of each class represented as vectors of dimension 256.\",\n \"The distance between two data points is $d(i,j) = ||{\\\\bf x}_i-{\\\\bf x}_j||$ .\",\n \"An unweighted graph $G$ is constructed by connecting a pair of nodes $(i,j)$ if $j$ is a $k$ -nearest neighbor ($k$ -nn with $k = 4$ ) Due to lack of space, we do not study the effect of $k$ in detail.\",\n \"of $i$ or vice versa.\",\n \"A weighted dissimilarity graph $G_d$ is defined to have the same topology as $G$ but the weight associated with edge $(i,j)$ is set to $d(i,j)$ .\",\n \"A weighted similarity graph $G_w$ is defined to have the same topology as $G$ but the weight associated with edge $(i,j)$ is set to $w(i,j) = \\\\exp \\\\left(- {d(i,j)^2}/{2\\\\sigma ^2} \\\\right)$ .\",\n \"The parameter $\\\\sigma $ is set to be $1/3$ -rd of the average distance to the $k$ -th nearest neighbor for all datapoints.\",\n \"2.\",\n \"Newsgroups text classification [11]: For our experiments, we consider a binary classification sub-problem Baseball vs. Hockey, where each class contains 200 randomly selected documents.\",\n \"Each document $i$ is represented by a 3000 dimensional vector ${\\\\bf x}_i$ whose elements are the tf-idf statistics of the 3000 most frequent words in the dataset [8].\",\n \"The cosine similarity between a pair data points $(i,j)$ is given by $w(i,j) = \\\\frac{\\\\left\\\\langle {{\\\\bf x}_i,{\\\\bf x}_j}\\\\right\\\\rangle }{||{\\\\bf x}_i||||{\\\\bf x}_j||}$ .\",\n \"The distance between them is defined as $d(i,j) = \\\\sqrt{1-w(i,j)^2}$ .\",\n \"The $k$ -nn unweighted graph $G$ (with $k=4$ ), the dissimilarity graph $G_d$ and the similarity graph $G_w$ are constructed using these distance and similarity measures as in the previous example.\",\n \"In addition to the above datasets, we generate a synthetic two circles dataset, shown in Figure REF , in order to demonstrate the advantage of weighted $S^2$ over unweighted $S^2$ .\",\n \"It contains 900 points in one class (marked red) evenly distributed on the inner circle of mean radius 1 and variance $0.05$ and 100 points in the second class (marked blue) on the outer circle of mean radius $1.1$ and variance $0.45$ .\",\n \"A 4-nn graph is constructed using the Euclidean distance between the coordinates of the points.\",\n \"Figure: A synthetic two circles datasetWe compare the performance of the following active learning methods: (1) unweighted $S^2$ method [2] with graph $G$ , (2) weighted $S^2$ method with dissimilarity graph $G_d$ , (3) cutoff maximization method [8] with similarity graph $G_w$ and (4) a hybrid approach combining cutoff maximization and weighted $S^2$ method.\",\n \"After the nodes selected by each method have been sampled, we reconstruct the unknown label signal using the approximate POCS based bandlimited reconstruction scheme [8] to get the soft labels.\",\n \"We threshold these soft labels to get the final label predictions.\",\n \"The hybrid approach uses the non-adaptive cutoff maximization approach in the beginning and switches to the weighted $S^2$ method after sampling a certain number of nodes $n_\\\\text{switch}$ .\",\n \"In order to determine $n_\\\\text{switch}$ , after sampling the $i$ -th node with the cutoff method, we compute $1-\\\\frac{\\\\mathinner {\\\\langle {\\\\hat{{\\\\bf f}}_i,\\\\hat{{\\\\bf f}}_{i-1}}\\\\rangle }}{\\\\Vert \\\\hat{{\\\\bf f}}_{i}\\\\Vert \\\\Vert \\\\hat{{\\\\bf f}}_{i-1}\\\\Vert }$ , where $\\\\hat{{\\\\bf f}}_i$ denotes the vector of predicted soft labels.\",\n \"Once this value falls below below $0.001$ , indicating that the newly added label only marginally changed the predictions, hybrid approach switches to weighted $S^2$ .\",\n \"Table REF lists the number of samples required by each of the sampling methods to discover all the cut edges using the observed labels.\",\n \"It shows that weighted $S^2$ can reduce the sample complexity significantly (by $25\\\\%$ ) compared to unweighted $S^2$ if the ratio of mean length of cut edges and mean length of non-cut edges is high as is the case in the unbalanced two circles dataset.\",\n \"In rest of the datasets, the gain offered by weighted $S^2$ is negligible since the cut edges are only slightly longer than non-cut edges and as a result, taking lengths into account in the bisection phase does not offer much advantage.\",\n \"We also observe that the number of samples required by the weighted $S^2$ method is close to the size of the cut $|\\\\partial C|$ in most of the datasets.\",\n \"Table REF also shows that the adaptive methods $S^2$ and weighted $S^2$ are very efficient at recovering the entire cut exactly, compared to the non-adaptive cutoff maximization method.\",\n \"In practice, it is not necessary to reconstruct the signal exactly and some reconstruction error is allowed.\",\n \"Figure REF plots the classification error against the number of sampled nodes.\",\n \"It shows that the classification error of the cutoff maximization method decreases rapidly in the early stages of sampling when very few samples are observed.\",\n \"However, the decrease is slow in later stages of sampling.\",\n \"$S^2$ methods, on the other hand, are good at reducing the error in the later stages, but performs poorly with only a few samples.\",\n \"The figure also shows that the hybrid method performs as well as the better method in each region.\"\n ],\n [\n \"Conclusions\",\n \" The paper generalizes the $S^2$ algorithm for the case of weighted graphs.\",\n \"The sampling complexity of the generalized algorithm is analyzed, and the gain over the unweighted version is demonstrated by simulation.\",\n \"Additional experiments identify the region of tolerable reconstruction error in which the $S^2$ algorithms outperforms a graph frequency based global approach.\",\n \"A hybrid approach is proposed with the advantages of both methods.\",\n \"It remains open to analytically characterize of the gain of the weighted $S^2$ method over the unweighted version.\",\n \"Another interesting avenue for future work is to provide a performance analysis for the spectral sampling method which can suggest an optimal switching criterion for the hybrid method.\"\n ]\n]"},"id":{"kind":"string","value":"1605.05710"}}},{"rowIdx":1866699,"cells":{"text":{"kind":"list like","value":[["Declarative Machine Learning - A Classification of Basic Properties and\n Types"],["Abstract Declarative machine learning (ML) aims at the high-level specification of ML tasks or algorithms, and automatic generation of optimized execution plans from these specifications.","The fundamental goal is to simplify the usage and/or development of ML algorithms, which is especially important in the context of large-scale computations.","However, ML systems at different abstraction levels have emerged over time and accordingly there has been a controversy about the meaning of this general definition of declarative ML.","Specification alternatives range from ML algorithms expressed in domain-specific languages (DSLs) with optimization for performance, to ML task (learning problem) specifications with optimization for performance and accuracy.","We argue that these different types of declarative ML complement each other as they address different users (data scientists and end users).","This paper makes an attempt to create a taxonomy for declarative ML, including a definition of essential basic properties and types of declarative ML.","Along the way, we provide insights into implications of these properties.","We also use this taxonomy to classify existing systems.","Finally, we draw conclusions on defining appropriate benchmarks and specification languages for declarative ML."],["Introduction","Large-scale machine learning (ML) leverages large data collections for advanced analytics in order to find interesting patterns and train robust predictive models.","Traditional frameworks and tools like R, Matlab, Weka, SPSS, or SAS provide rich functionality but—except for dedicated packages—struggle to provide scalable analytics.","Due to the data-intensive characteristics, increasingly often data-parallel frameworks like MapReduce [14], Spark [41], or Flink [3] are used for cost-effective parallelization on commodity hardware.","However, large-scale computation inherently increases the complexity of specifying ML algorithms, especially with regard to efficient and scalable execution.","Large-Scale ML Libraries: Large-scale ML libraries like MLlib (aka SparkML) [30], Mahout [37], and MADlib [12], [23] are currently the predominant tools for large-scale ML.","These libraries provide algorithms with fixed distributed runtime plans and often expose the underlying physical data representation.","Although such libraries are very valuable tools for end-users, it takes substantial effort to write new or customize existing algorithms because it requires knowledge of ML algorithms, their distributed implementation, and the underlying data-parallel framework.","Similarly, improvements often require a modification of all individual algorithms to exploit these improvements.","Declarative ML: Declarative ML aims at a high-level specification of ML tasks or algorithms to simplify the usage and/or development of ML algorithms by separating application or algorithm semantics from the underlying data representations and execution plans.","Table REF categorizes types of declarative ML and delineates them from ML libraries.","Overall, the major benefits of declarative ML are: Simple, Analysis-Centric Specification, Physical Data Independence, Automatic Execution Plan Generation (optimization, platform independence, data-size independence), Ease of Deployment (platform independence, adaptivity of “packaged” applications), and Separation of Concerns (skill sets of users/devs).","Over time, systems at different levels of abstraction have been proposed by industry and academia.","Example systems range from UDF-centric ML extensions of data-parallel frameworks, to domain-specific languages (DSLs) for ML tasks or ML algorithms.","This broad spectrum of systems aiming for declarative ML naturally led to a controversy regarding the scope of declarative ML.","Not surprisingly—as many ML algorithms are iterative—the discussion centers around the syntax for specifying loops and control flow in general.","Various projects adopt an R/Python-like syntax [1], [15], [22], [40], [42], [43] inheriting the full flexibility of loops, branches, and functions.","Others support loops with (1) more restrictive iteration constructs [13], [17], (2) model updates with implicit convergence checks [8], or encapsulating entire algorithm classes as ML tasks [27], [33], [42].","We argue that the specific language-level syntax is actually irrelevant if the ML task or algorithm specification conforms to a set of basic properties required for declarative ML.","Table: Delineation of Types of Declarative ML.Contributions and Structure: The primary contribution of this paper is a systematic analysis and classification of declarative machine learning.","We first define—in an syntax-independent manner—a set of basic properties in Section , that any system for declarative ML should satisfy.","Subsequently, we describe the types of declarative ML in Section .","Finally, we use this taxonomy to classify existing systems in Section , and draw conclusions for defining appropriate benchmarks and languages in Section ."],["Basic Properties","As a foundation for discussing types of declarative ML, we define essential, basic properties in the three categories of data, operations, and result correctness.","We also discuss the implications of the individual properties and provide examples of how Apache SystemML—as a representative system for declarative ML—realizes these properties."],["Physical Data Independence","The most significant goal of declarative ML is data independence because it decouples the high-level specification of ML tasks or algorithms from the underlying data representations and related runtime plans and operations.","Property 1 Independence of Data Structures: The data types of inputs, intermediate results, and outputs like matrices or scalars are exposed as abstract data types without access to the underlying physical data representations.","In the context of declarative ML, independence of data structures serves two major purposes.","First, abstract data types like matrix hide the decision on distributed vs local data representations.","Accordingly, specified tasks or algorithms become independent of data size and deployment context (e.g., distributed computation vs streaming, different runtime backends).","Second, abstract data types also hide the physical data representation (e.g., dense/sparse matrices, lossless compression), which allow internal improvements of storage and operation efficiency.","Property 2 Independence of Data Flow Properties: Data flow properties are not exposed, i.e., the user has, at specification level, no explicit control over properties like partitioning, caching, and blocking configurations.","The property of independence of data flow properties further restricts the notion of abstract data types, disallowing the explicit specification of interesting data flow properties.","Examples are (1) caching and checkpointing (e.g., local and distributed caching, with certain storage levels), (2) logical and physical partitioning (e.g., row/column/block partitioning and range/hash partitioning of distributed data sets), (3) blocking configurations (row/column block sizes, fixed or variable logical/physical block sizes), as well as (4) data formats (text or binary cell/block formats).","Note, however, that it is valid to allow the specification of ordering because it is both a logical and physical data flow property.","Example SystemML: SystemML satisfies both properties by exposing only the abstract data types frame, matrix, and scalar without their physical data structures or interesting data flow properties as shown in Figure REF , as an example of a valid specification.","The decisions on physical data flow properties are, however, crucial for performance.","Hence, the system automatically injects, for example, caching and partitioning directives via rewrites.","Overall, data independence allowed us to evolve and rebase SystemML without changing a single ML algorithm.","Examples are extensions such as the support for different sparse representations, compression, and additional backends like Spark or GPUs.","In contrast, for example, Mahout Samsara [15] does not satisfy the properties of data independence because decisions on dense/sparse and distributed/local matrices (e.g., drmFromHDFS, collect) as well as data flow properties like partitioning (e.g., par) and caching (e.g., checkpoint) are exposed to the user as shown in Figure REF .","Figure: Mahout Samsara"],["Operation Semantics","The second major goal of declarative ML is to specify ML tasks or algorithms using domain-specific, high-level operations with well-defined semantics to simplify algorithm usage or development, and enable efficient evaluation plans.","Property 3 Analysis-Centric Operation Primitives: Basic operation primitives, common in the target analytics domain, are supported.","For ML algorithms, this includes linear algebra and statistical functions, whereas for ML tasks, this includes task-specific primitives and models.","In order to allow declarative ML with simple specification, there is a need for operation primitives that closely resemble a natural description of ML tasks or algorithms at a conceptual level.","For ML algorithms this includes linear algebra, aggregations, and statistical functions but specific domains like deep learning might require additional domain-specific operations like convolution.","Similarly, for ML tasks this includes task-specific abstractions, operations, and models.","For example, for specifying a task like classify, common classification algorithms, loss functions, and parameters should be supported to describe candidates and the optimization objective.","The same applies for the task of general-purpose optimization optimize, where one would expect a way to specify gradient/loss functions, and termination conditions.","Note that this property excludes systems that are declarative but unrelated to ML.","Property 4 Known Semantics of Operation Primitives: The semantics of operation primitives used to specify ML tasks or algorithms are known to the system in terms of knowledge of operation characteristics and equivalences.","We require operational semantics, where there exists at least one naïve evaluation plan or straightforward mapping.","Knowledge of operation semantics is essential for generating efficient evaluation plans—for example, via rewrites and operator selection—from high-level specifications.","In the context of ML, operation semantics also cover characteristics like commutativity and associativity, sparse-safeness (correctness of processing only non-zero cells), value repositioning (e.g., reorg operations like transpose, or order), symmetry properties, as well as an understanding of composite operations (e.g., sum-product for matrix multiplication).","This meta information might be built into the system or annotated in case of extensible systems.","Overall operation semantics allow to reason about equivalences, alternative execution strategies, and costs of these alternatives.","Property 5 Implementation-Agnostic Operations: The specification of ML tasks or algorithms is independent of the underlying runtime operations.","This property prohibits user-defined execution strategies and parameterization.","Specifying implementation-agnostic operations—i.e., independent of runtime backends, distributed vs local operations, and execution strategies—is related to the properties for data independence (see Subsection REF ) but with a focus on operations to ensure the flexibility of alternative or hybrid runtime backends, alternative deployments, and optimizations like rewrites and operator selection.","Furthermore, avoiding low level parameterization like degree of parallelism or cache blocking ensures independence of the ML tasks or algorithms from workload characteristics (e.g., data size) and underlying hardware infrastructure.","Property 6 Well-Defined Plan Optimization Objective: ML tasks or algorithms specify their expected results unambiguously, using a well-defined (potentially multi-criteria) objective for execution plan optimization.","To specify ML tasks or algorithms in an unambiguous manner, the specification must exhibit (implicitly or explicitly) a well-defined plan optimization objective.","This property differentiates major types of declarative ML.","If the specification relies on unambiguous operations, the implicit optimization objective is efficiency (runtime or resource requirements).","In case of multi-objective optimizations, we further need to define a primary dimension and constraints for the other dimensions.","For example, if we aim to optimize for both efficiency and accuracy, we might want to optimize accuracy in terms of a quality measure (e.g., L2 loss on a holdout dataset) as the primary dimension along with constraints on efficiency (e.g., time budget, number of models).","Example SystemML: SystemML satisfies these properties by minimizing execution time (under memory budget constraints per execution context) of specified ML algorithms, composed of linear algebra and statistical operations with well-defined semantics.","The known operation semantics are used to propagate dimension and sparsity information through the entire algorithm, compute memory estimates, apply static (size-independent) and dynamic (size-dependent) rewrites, and eventually decide upon alternative physical operators.","Implementation-agnostic operations allow fine-grained optimization decisions per operation.","Note that sequences of operations like t(X) %*% X %*% p from Figure REF or specifications like independent foreach loops (parfor) [6] are assertions on semantics and properties of the algorithm rather than imperative execution strategies."],["Result Correctness","The properties of data independence and operation semantics are necessary but not sufficient for declarative ML.","We further need to define the notion of result correctness.","With regard to practicability for distributed computing, we define operation results as equivalent if they are essentially the same, i.e., they are algebraically (logically) equivalent, which ignores round-off errors (e.g., due to partial aggregation or alternative evaluation orders of operations).","Property 7 Implementation-Agnostic Results: The results of ML tasks or algorithms as well as individual operations are equivalent (essentially the same), independent of type and location of underlying runtime operations.","This property further qualifies implementation-agnostic operations (P5).","In order to produce correct results, independent of optimization decisions, alternative execution strategies need to produce equivalent results no matter if they are executed locally or as distributed operations.","To accomplish that for an operation like rand with fixed seed, both local and distributed operations need to consistently generate seeds for fixed-sized blocks from the initially given input seed.","Furthermore, this property prohibits, for example, lossy compression to reduce communication overhead.","Property 8 Deterministic Results: A given ML task or algorithm yields equivalent (essentially the same) results for multiple executions over the same input data and configuration.","Randomized tasks or algorithms achieve this using pseudorandom number generators.","Deterministic results of operations and ML tasks or algorithms is an important property, especially with regard to fault tolerance, where the same operation might be executed multiple times.","Furthermore, it is also the basis for benchmarking ML systems in a systematic manner.","Example SystemML: In SystemML, the properties of result correctness are satisfied via consistent local and distributed as well as deterministic operations.","ML algorithms are composed of these operations, lifting the properties of result correctness to algorithm level too.","Furthermore, we bound the round-off errors via numerically stable operations (based on Kahan+) [38] for descriptive statistics and aggregations.","SystemML also provides configuration knobs to disable rewrites and operator selection to force strict computation.","However, other than for debugging, we have not seen data scientists or end-users making use of that.","Table: Classification of Existing Systems wrt Declarative ML Algorithms (Type 1).So far we discussed general properties of declarative machine learning, which apply to all types of declarative ML.","We now create a taxonomy of types of declarative ML, namely declarative ML algorithms and declarative ML tasks.","These types refer to fundamentally different concepts and thus, also differ in their scope of specification."],["Declarative ML Algorithms (Type 1)","Declarative ML algorithms allow data scientists to write and customize ML algorithms in a declarative manner.","This scope requires fine-grained semantics including control flow and data flow, where the core operation primitives are often based on linear algebra or statistical functions and the common optimization objective is to minimize execution time but other objectives such as resource consumption are possible.","The algorithm-centric specification defines precise semantics but leaves substantial freedom regarding data representations and execution plan optimization.","This abstraction level allows data scientists to encode algorithms as they are most naturally expressed and thus, to quickly exploit the latest algorithmic advances.","End users also benefit from simply calling these algorithms in terms of automatic optimization, adaptivity, and portability.","Example system categories are DSL-centric, SQL-centric, and UDF-centric systems."],["Declarative ML Tasks (Type 2)","In contrast to declarative ML algorithms, declarative ML tasks allow end users (without ML background), to specify ML tasks like classify, factorize, optimize independent of ML algorithm specifics.","This coarse-grained scope includes automatic feature and model selection and allows for the optimization of both model accuracy and runtime.","Core operation primitives are task-specific, i.e., depending on the task at hand, alternative candidate algorithms, loss functions (measure for goodness of fit), and hyper parameters are supported.","Operation semantics are either built-in or annotated at the level of used algorithms or loss functions.","The properties of declarative ML need to apply to the core optimization problem of the given ML task not the entire stack of used ML algorithms, as long as they are annotated with relevant properties that allow reasoning about alternative plans and costs.","However, relying on declarative ML algorithms provides additional flexibility.","Example system categories for declarative ML tasks are general-purpose optimization, model selection, and feature selection."],["Systems Classification","Given the taxonomy of basic properties and types of declarative ML, we now classify existing systems.","There has been some related work on similar classifications, most notably, Kumar et al.","defined a notion of a model selection management system [28], along with categories of ML systems, but primarily focused on coverage of industrial systems rather than specifics of declarative machine learning.","Tables REF and REF classify—in the scope of declarative ML algorithms and tasks—existing systems with regard to the defined basic properties and types of declarative machine learning.","This classification also indicates distributed vs local operations and the used optimization objective.","Table: Classification of Existing Systems wrt Declarative ML Tasks (Type 2)."],["Declarative ML Algorithms","DSL-Centric Systems: The class of DSL-centric systems focuses on domain-specific languages (DSLs) for ML to simplify the writing of ML algorithms.","Early examples of declarative systems are RIOT [43] and OptiML [35], which provide R and Scala DSLs, respectively.","Both focus on single-node computation only which makes it easier to adhere to basic properties of declarative machine learning.","SystemML [7], [22] covers both single-node and distributed computation (on MapReduce and Spark) and satisfies all eight properties of declarative ML as described throughout this paper.","More recent systems like Cumulon [24], [25], Mahout Samsara [15], DMac [40], and TensorFlow [1] similarly aim for declarative, large-scale ML but struggle to satisfy all properties of declarative ML.","Cumulon [24], [25] can be seen as a declarative system.","However, in a strict sense, the optimization objective is ill-defined (P6) as the hard runtime constraint cannot be satisfied without knowing the number of iterations until convergence.","Cumulon still allows users to explore per-iteration trade-offs of monetary cost and runtime in cloud environments.","Mahout Samsara [15], Distributed R [39], and DMac [40] are not declarative because they expose physical data structures and distributed operations to the user (P1, P5).","Mahout Samsara and Distributed R require the user to decide between local and distributed matrices and expose data flow properties like caching and partitioning (P2), while DMac exposes dense/sparse data structures.","Additionally, Distributed R executes arbitrary user-defined R functions—i.e., with unknown operation semantics (P4)—per partition.","TensorFlow [1] is a compelling system but focuses more on extensibility than declarative specification.","Accordingly, operations are handled as black-box kernels, i.e., with unknown operation semantics (P4).","Also, deep-learning-centric optimizations like lossy compression for communication work very well for noisy data but do not satisfy the property of implementation-agnostic results (P7) required for general-purpose declarative ML.","SQL-Centric Systems: The class of SQL-centric systems complements the class of DSL-centric systems as both aim for custom analysis algorithms.","Common SQL-centric systems are array databases and SQL-like ML (e.g., for Bayesian ML).","These systems often follow the compelling argument of integrating advanced analytics with traditional query processing to simplify data pre-processing and leverage well-understood abstractions for data and operations.","At the same time, linear algebra operations are either directly supported or emulated.","A prime example of array databases is SciDB [9], [34] which indeed satisfies all basic properties for declarative ML.","Furthermore, also SimSQL [11]—as an example of Bayesian ML or more broadly stochastic analysis—satisfies all basic properties of declarative ML.","UDF-Centric Systems: There is also a class of UDF-centric systems that evolved bottom up from existing data-parallel frameworks like MapReduce, Spark, or Flink as well as compiler frameworks to simplify large-scale ML.","Examples are ScalOps [8] (which compiles Scala UDF workflows to datalog programs), Tupleware [13] (which compiles workflows of UDFs in various frontend languages to custom distributed programs of native code via LLVM), and Emma [4] (which compiles Scala UDF workflows to Spark and Flink programs).","These systems cover a great variety of use cases, but they systematically fail to satisfy several properties of declarative ML.","First, data independence (P1) is not satisfied as the UDFs are implemented against custom data structures which makes it hard to efficiently support dense/sparse or compressed datasets.","Second, the operation semantics of UDFs are by definition unknown (P4), miss support of analysis-centric operations (P3), and the focus on large-scale computation requires the UDF workflows to realize distributed algorithm implementations (P5)."],["Declarative ML Tasks","As discussed before, systems for declarative ML tasks mostly target end users (not algorithm developers) and automatically optimize for runtime and accuracy of general-purpose optimization or model and feature selection tasks.","General-Purpose Optimization: Bismarck [18] provides in-database general-purpose optimization via incremental gradient decent, where users provide UDFs for initialization, transition, and termination.","This abstraction covers many algorithms over existing relational data.","Similar to UDF-centric systems, however, it does not satisfy the independence of data structures (P1) and known operation semantics (P4).","Furthermore, effective parallelization requires either violations of implementation-agnostic and deterministic results (P7, P8) for the pure UDA approach or modifications of the UDFs (P5) for the shared-memory UDA approach.","The latter also does not satisfy P7/P8 due to Hogwild!-style [31] model updates.","TensorFlow [1] also provides primitives for general-purpose optimization via different optimization algorithms.","Similar to Bismarck, users provide UDFs for loss and gradient computation.","However, as TensorFlow provides sufficient data abstraction via so-called placeholders, it satisfies the properties of data independence.","Furthermore, the operation semantics of inference, loss, and training are known and UDFs can leverage the existing built-in functions which also provide abstractions for gradient computation.","Thus, although TensorFlow did not qualify for declarative ML algorithms, it satisfies the properties at the level of declarative ML tasks.","Model and Feature Selection: MLbase [27] with its TUPAQ [33] component for automatic model search allows users to specify candidate model configurations, a quality measure, and runtime constraints (e.g., models considered, number of scans, etc) and returns the best model along with tuned hyperparameters.","Disregarding the underlying library of ML algorithms, MLbase can be classified as a system for declarative ML tasks, as the model search problem is well-defined, independent of the underlying data and operations, with annotated algorithm characteristics, and deterministic results (unless the runtime constraint is a runtime budget).","Columbus [42] allows users to specify feature engineering workflows in R including data preparations and model building with existing R packages.","The optimization objective is to minimize runtime, but an error tolerance allows for more aggressive reuse by leveraging runtime-accuracy trade-offs.","Columbus also satisfies the properties for declarative ML tasks.","DeepDive [32] enables knowledge base construction via statistical inference.","In a first step, users specify SQL queries and UDFs for feature extraction to populate a factor graph.","In a second step, candidate mappings are specified via SQL queries to express rules for entities and relations.","Finally, marginal probabilities are learned via statistical inference over this factor graph.","As data and operations are abstracted via SQL and the actual inference algorithms and details are not exposed, DeepDive can be classified as a declarative system for feature selection.","Other System Categories: Finally, there are also declarative systems for more specific ML tasks like time series forecasting (e.g., Fa [16], a skip-list approach [20], or F2DB [19]), which also consider the trade-off between accuracy and runtime (model selection) but are not subject to this classification for general-purpose declarative ML."],["Benchmarking ML Systems","Having discussed the individual types of declarative ML, it is clear that there cannot exist a single benchmark to cover them all.","Existing benchmarks for large-scale computation like BigBench [5], [21], SparkBench [2], [29], or HiBench [26] do cover machine learning but often simply refer to reference implementations of large-scale ML libraries.","This is fine to evaluate underlying Hadoop or Spark implementations but simply cannot serve as a benchmark for declarative ML.","A Case for Type-Specific Benchmarks: We argue that industry and academia is best served with benchmarks specific to types of declarative machine learning (see Section ).","In order to properly reflect common workload characteristics, the benchmarks should be further tailored to major subcategories of systems.","For example, systems for declarative ML algorithms cover the major sub-categories of DSL-centric, UDF-centric, and SQL-centric systems, which—despite some overlap of operation primitives—all target different primary usage scenarios.","This calls for specific benchmarks that allow fair comparisons and foster system advancements via challenging workloads.","Interestingly, exactly that already happened for SQL-centric systems as both, the SciDB [9], [34] and SimSQL [11] projects published benchmarks covering the main characteristics of array databases [36] and Bayesian ML [10].","Defining simple yet challenging ML benchmarks that also cover common workload characteristics is not easy, so we—as a community—should highly appreciate contributions in this area.","Note on Specification Languages: The same reasoning as with system-type-specific benchmarks also applies to specification languages.","In order to satisfy the property of analysis-centric operation support (P4), systems need to support the operations of their primary analysis use case, which motivates tailor-made specification languages.","However, as the properties for declarative ML are syntax independent, we could establish a common syntax for a specification language to cover multiple types of declarative ML."],["Conclusion","To summarize, we introduced a taxonomy of declarative ML in terms of basic properties of data, operations, and result correctness as well as types of systems for declarative ML.","The classification of existing systems has shown that this taxonomy is indeed a useful tool for qualifying systems characteristics in a systematic manner.","Fundamentally, this makes a case for a syntax-independent classification of declarative ML, which disqualifies the philosophical argument against loops and control flow in general.","We are at the beginning of an exciting era of declarative ML, with a good understanding of various aspects but also lots of open research challenges.","As advanced analytics become ubiquitous and technology environments are changing at an increasing rate, a declarative specification of ML tasks or algorithms becomes increasingly important.","Accordingly, we encourage the research community to participate in this discussion on basic properties of declarative ML in order to eventually converge to a common understanding."]],"string":"[\n [\n \"Declarative Machine Learning - A Classification of Basic Properties and\\n Types\"\n ],\n [\n \"Abstract Declarative machine learning (ML) aims at the high-level specification of ML tasks or algorithms, and automatic generation of optimized execution plans from these specifications.\",\n \"The fundamental goal is to simplify the usage and/or development of ML algorithms, which is especially important in the context of large-scale computations.\",\n \"However, ML systems at different abstraction levels have emerged over time and accordingly there has been a controversy about the meaning of this general definition of declarative ML.\",\n \"Specification alternatives range from ML algorithms expressed in domain-specific languages (DSLs) with optimization for performance, to ML task (learning problem) specifications with optimization for performance and accuracy.\",\n \"We argue that these different types of declarative ML complement each other as they address different users (data scientists and end users).\",\n \"This paper makes an attempt to create a taxonomy for declarative ML, including a definition of essential basic properties and types of declarative ML.\",\n \"Along the way, we provide insights into implications of these properties.\",\n \"We also use this taxonomy to classify existing systems.\",\n \"Finally, we draw conclusions on defining appropriate benchmarks and specification languages for declarative ML.\"\n ],\n [\n \"Introduction\",\n \"Large-scale machine learning (ML) leverages large data collections for advanced analytics in order to find interesting patterns and train robust predictive models.\",\n \"Traditional frameworks and tools like R, Matlab, Weka, SPSS, or SAS provide rich functionality but—except for dedicated packages—struggle to provide scalable analytics.\",\n \"Due to the data-intensive characteristics, increasingly often data-parallel frameworks like MapReduce [14], Spark [41], or Flink [3] are used for cost-effective parallelization on commodity hardware.\",\n \"However, large-scale computation inherently increases the complexity of specifying ML algorithms, especially with regard to efficient and scalable execution.\",\n \"Large-Scale ML Libraries: Large-scale ML libraries like MLlib (aka SparkML) [30], Mahout [37], and MADlib [12], [23] are currently the predominant tools for large-scale ML.\",\n \"These libraries provide algorithms with fixed distributed runtime plans and often expose the underlying physical data representation.\",\n \"Although such libraries are very valuable tools for end-users, it takes substantial effort to write new or customize existing algorithms because it requires knowledge of ML algorithms, their distributed implementation, and the underlying data-parallel framework.\",\n \"Similarly, improvements often require a modification of all individual algorithms to exploit these improvements.\",\n \"Declarative ML: Declarative ML aims at a high-level specification of ML tasks or algorithms to simplify the usage and/or development of ML algorithms by separating application or algorithm semantics from the underlying data representations and execution plans.\",\n \"Table REF categorizes types of declarative ML and delineates them from ML libraries.\",\n \"Overall, the major benefits of declarative ML are: Simple, Analysis-Centric Specification, Physical Data Independence, Automatic Execution Plan Generation (optimization, platform independence, data-size independence), Ease of Deployment (platform independence, adaptivity of “packaged” applications), and Separation of Concerns (skill sets of users/devs).\",\n \"Over time, systems at different levels of abstraction have been proposed by industry and academia.\",\n \"Example systems range from UDF-centric ML extensions of data-parallel frameworks, to domain-specific languages (DSLs) for ML tasks or ML algorithms.\",\n \"This broad spectrum of systems aiming for declarative ML naturally led to a controversy regarding the scope of declarative ML.\",\n \"Not surprisingly—as many ML algorithms are iterative—the discussion centers around the syntax for specifying loops and control flow in general.\",\n \"Various projects adopt an R/Python-like syntax [1], [15], [22], [40], [42], [43] inheriting the full flexibility of loops, branches, and functions.\",\n \"Others support loops with (1) more restrictive iteration constructs [13], [17], (2) model updates with implicit convergence checks [8], or encapsulating entire algorithm classes as ML tasks [27], [33], [42].\",\n \"We argue that the specific language-level syntax is actually irrelevant if the ML task or algorithm specification conforms to a set of basic properties required for declarative ML.\",\n \"Table: Delineation of Types of Declarative ML.Contributions and Structure: The primary contribution of this paper is a systematic analysis and classification of declarative machine learning.\",\n \"We first define—in an syntax-independent manner—a set of basic properties in Section , that any system for declarative ML should satisfy.\",\n \"Subsequently, we describe the types of declarative ML in Section .\",\n \"Finally, we use this taxonomy to classify existing systems in Section , and draw conclusions for defining appropriate benchmarks and languages in Section .\"\n ],\n [\n \"Basic Properties\",\n \"As a foundation for discussing types of declarative ML, we define essential, basic properties in the three categories of data, operations, and result correctness.\",\n \"We also discuss the implications of the individual properties and provide examples of how Apache SystemML—as a representative system for declarative ML—realizes these properties.\"\n ],\n [\n \"Physical Data Independence\",\n \"The most significant goal of declarative ML is data independence because it decouples the high-level specification of ML tasks or algorithms from the underlying data representations and related runtime plans and operations.\",\n \"Property 1 Independence of Data Structures: The data types of inputs, intermediate results, and outputs like matrices or scalars are exposed as abstract data types without access to the underlying physical data representations.\",\n \"In the context of declarative ML, independence of data structures serves two major purposes.\",\n \"First, abstract data types like matrix hide the decision on distributed vs local data representations.\",\n \"Accordingly, specified tasks or algorithms become independent of data size and deployment context (e.g., distributed computation vs streaming, different runtime backends).\",\n \"Second, abstract data types also hide the physical data representation (e.g., dense/sparse matrices, lossless compression), which allow internal improvements of storage and operation efficiency.\",\n \"Property 2 Independence of Data Flow Properties: Data flow properties are not exposed, i.e., the user has, at specification level, no explicit control over properties like partitioning, caching, and blocking configurations.\",\n \"The property of independence of data flow properties further restricts the notion of abstract data types, disallowing the explicit specification of interesting data flow properties.\",\n \"Examples are (1) caching and checkpointing (e.g., local and distributed caching, with certain storage levels), (2) logical and physical partitioning (e.g., row/column/block partitioning and range/hash partitioning of distributed data sets), (3) blocking configurations (row/column block sizes, fixed or variable logical/physical block sizes), as well as (4) data formats (text or binary cell/block formats).\",\n \"Note, however, that it is valid to allow the specification of ordering because it is both a logical and physical data flow property.\",\n \"Example SystemML: SystemML satisfies both properties by exposing only the abstract data types frame, matrix, and scalar without their physical data structures or interesting data flow properties as shown in Figure REF , as an example of a valid specification.\",\n \"The decisions on physical data flow properties are, however, crucial for performance.\",\n \"Hence, the system automatically injects, for example, caching and partitioning directives via rewrites.\",\n \"Overall, data independence allowed us to evolve and rebase SystemML without changing a single ML algorithm.\",\n \"Examples are extensions such as the support for different sparse representations, compression, and additional backends like Spark or GPUs.\",\n \"In contrast, for example, Mahout Samsara [15] does not satisfy the properties of data independence because decisions on dense/sparse and distributed/local matrices (e.g., drmFromHDFS, collect) as well as data flow properties like partitioning (e.g., par) and caching (e.g., checkpoint) are exposed to the user as shown in Figure REF .\",\n \"Figure: Mahout Samsara\"\n ],\n [\n \"Operation Semantics\",\n \"The second major goal of declarative ML is to specify ML tasks or algorithms using domain-specific, high-level operations with well-defined semantics to simplify algorithm usage or development, and enable efficient evaluation plans.\",\n \"Property 3 Analysis-Centric Operation Primitives: Basic operation primitives, common in the target analytics domain, are supported.\",\n \"For ML algorithms, this includes linear algebra and statistical functions, whereas for ML tasks, this includes task-specific primitives and models.\",\n \"In order to allow declarative ML with simple specification, there is a need for operation primitives that closely resemble a natural description of ML tasks or algorithms at a conceptual level.\",\n \"For ML algorithms this includes linear algebra, aggregations, and statistical functions but specific domains like deep learning might require additional domain-specific operations like convolution.\",\n \"Similarly, for ML tasks this includes task-specific abstractions, operations, and models.\",\n \"For example, for specifying a task like classify, common classification algorithms, loss functions, and parameters should be supported to describe candidates and the optimization objective.\",\n \"The same applies for the task of general-purpose optimization optimize, where one would expect a way to specify gradient/loss functions, and termination conditions.\",\n \"Note that this property excludes systems that are declarative but unrelated to ML.\",\n \"Property 4 Known Semantics of Operation Primitives: The semantics of operation primitives used to specify ML tasks or algorithms are known to the system in terms of knowledge of operation characteristics and equivalences.\",\n \"We require operational semantics, where there exists at least one naïve evaluation plan or straightforward mapping.\",\n \"Knowledge of operation semantics is essential for generating efficient evaluation plans—for example, via rewrites and operator selection—from high-level specifications.\",\n \"In the context of ML, operation semantics also cover characteristics like commutativity and associativity, sparse-safeness (correctness of processing only non-zero cells), value repositioning (e.g., reorg operations like transpose, or order), symmetry properties, as well as an understanding of composite operations (e.g., sum-product for matrix multiplication).\",\n \"This meta information might be built into the system or annotated in case of extensible systems.\",\n \"Overall operation semantics allow to reason about equivalences, alternative execution strategies, and costs of these alternatives.\",\n \"Property 5 Implementation-Agnostic Operations: The specification of ML tasks or algorithms is independent of the underlying runtime operations.\",\n \"This property prohibits user-defined execution strategies and parameterization.\",\n \"Specifying implementation-agnostic operations—i.e., independent of runtime backends, distributed vs local operations, and execution strategies—is related to the properties for data independence (see Subsection REF ) but with a focus on operations to ensure the flexibility of alternative or hybrid runtime backends, alternative deployments, and optimizations like rewrites and operator selection.\",\n \"Furthermore, avoiding low level parameterization like degree of parallelism or cache blocking ensures independence of the ML tasks or algorithms from workload characteristics (e.g., data size) and underlying hardware infrastructure.\",\n \"Property 6 Well-Defined Plan Optimization Objective: ML tasks or algorithms specify their expected results unambiguously, using a well-defined (potentially multi-criteria) objective for execution plan optimization.\",\n \"To specify ML tasks or algorithms in an unambiguous manner, the specification must exhibit (implicitly or explicitly) a well-defined plan optimization objective.\",\n \"This property differentiates major types of declarative ML.\",\n \"If the specification relies on unambiguous operations, the implicit optimization objective is efficiency (runtime or resource requirements).\",\n \"In case of multi-objective optimizations, we further need to define a primary dimension and constraints for the other dimensions.\",\n \"For example, if we aim to optimize for both efficiency and accuracy, we might want to optimize accuracy in terms of a quality measure (e.g., L2 loss on a holdout dataset) as the primary dimension along with constraints on efficiency (e.g., time budget, number of models).\",\n \"Example SystemML: SystemML satisfies these properties by minimizing execution time (under memory budget constraints per execution context) of specified ML algorithms, composed of linear algebra and statistical operations with well-defined semantics.\",\n \"The known operation semantics are used to propagate dimension and sparsity information through the entire algorithm, compute memory estimates, apply static (size-independent) and dynamic (size-dependent) rewrites, and eventually decide upon alternative physical operators.\",\n \"Implementation-agnostic operations allow fine-grained optimization decisions per operation.\",\n \"Note that sequences of operations like t(X) %*% X %*% p from Figure REF or specifications like independent foreach loops (parfor) [6] are assertions on semantics and properties of the algorithm rather than imperative execution strategies.\"\n ],\n [\n \"Result Correctness\",\n \"The properties of data independence and operation semantics are necessary but not sufficient for declarative ML.\",\n \"We further need to define the notion of result correctness.\",\n \"With regard to practicability for distributed computing, we define operation results as equivalent if they are essentially the same, i.e., they are algebraically (logically) equivalent, which ignores round-off errors (e.g., due to partial aggregation or alternative evaluation orders of operations).\",\n \"Property 7 Implementation-Agnostic Results: The results of ML tasks or algorithms as well as individual operations are equivalent (essentially the same), independent of type and location of underlying runtime operations.\",\n \"This property further qualifies implementation-agnostic operations (P5).\",\n \"In order to produce correct results, independent of optimization decisions, alternative execution strategies need to produce equivalent results no matter if they are executed locally or as distributed operations.\",\n \"To accomplish that for an operation like rand with fixed seed, both local and distributed operations need to consistently generate seeds for fixed-sized blocks from the initially given input seed.\",\n \"Furthermore, this property prohibits, for example, lossy compression to reduce communication overhead.\",\n \"Property 8 Deterministic Results: A given ML task or algorithm yields equivalent (essentially the same) results for multiple executions over the same input data and configuration.\",\n \"Randomized tasks or algorithms achieve this using pseudorandom number generators.\",\n \"Deterministic results of operations and ML tasks or algorithms is an important property, especially with regard to fault tolerance, where the same operation might be executed multiple times.\",\n \"Furthermore, it is also the basis for benchmarking ML systems in a systematic manner.\",\n \"Example SystemML: In SystemML, the properties of result correctness are satisfied via consistent local and distributed as well as deterministic operations.\",\n \"ML algorithms are composed of these operations, lifting the properties of result correctness to algorithm level too.\",\n \"Furthermore, we bound the round-off errors via numerically stable operations (based on Kahan+) [38] for descriptive statistics and aggregations.\",\n \"SystemML also provides configuration knobs to disable rewrites and operator selection to force strict computation.\",\n \"However, other than for debugging, we have not seen data scientists or end-users making use of that.\",\n \"Table: Classification of Existing Systems wrt Declarative ML Algorithms (Type 1).So far we discussed general properties of declarative machine learning, which apply to all types of declarative ML.\",\n \"We now create a taxonomy of types of declarative ML, namely declarative ML algorithms and declarative ML tasks.\",\n \"These types refer to fundamentally different concepts and thus, also differ in their scope of specification.\"\n ],\n [\n \"Declarative ML Algorithms (Type 1)\",\n \"Declarative ML algorithms allow data scientists to write and customize ML algorithms in a declarative manner.\",\n \"This scope requires fine-grained semantics including control flow and data flow, where the core operation primitives are often based on linear algebra or statistical functions and the common optimization objective is to minimize execution time but other objectives such as resource consumption are possible.\",\n \"The algorithm-centric specification defines precise semantics but leaves substantial freedom regarding data representations and execution plan optimization.\",\n \"This abstraction level allows data scientists to encode algorithms as they are most naturally expressed and thus, to quickly exploit the latest algorithmic advances.\",\n \"End users also benefit from simply calling these algorithms in terms of automatic optimization, adaptivity, and portability.\",\n \"Example system categories are DSL-centric, SQL-centric, and UDF-centric systems.\"\n ],\n [\n \"Declarative ML Tasks (Type 2)\",\n \"In contrast to declarative ML algorithms, declarative ML tasks allow end users (without ML background), to specify ML tasks like classify, factorize, optimize independent of ML algorithm specifics.\",\n \"This coarse-grained scope includes automatic feature and model selection and allows for the optimization of both model accuracy and runtime.\",\n \"Core operation primitives are task-specific, i.e., depending on the task at hand, alternative candidate algorithms, loss functions (measure for goodness of fit), and hyper parameters are supported.\",\n \"Operation semantics are either built-in or annotated at the level of used algorithms or loss functions.\",\n \"The properties of declarative ML need to apply to the core optimization problem of the given ML task not the entire stack of used ML algorithms, as long as they are annotated with relevant properties that allow reasoning about alternative plans and costs.\",\n \"However, relying on declarative ML algorithms provides additional flexibility.\",\n \"Example system categories for declarative ML tasks are general-purpose optimization, model selection, and feature selection.\"\n ],\n [\n \"Systems Classification\",\n \"Given the taxonomy of basic properties and types of declarative ML, we now classify existing systems.\",\n \"There has been some related work on similar classifications, most notably, Kumar et al.\",\n \"defined a notion of a model selection management system [28], along with categories of ML systems, but primarily focused on coverage of industrial systems rather than specifics of declarative machine learning.\",\n \"Tables REF and REF classify—in the scope of declarative ML algorithms and tasks—existing systems with regard to the defined basic properties and types of declarative machine learning.\",\n \"This classification also indicates distributed vs local operations and the used optimization objective.\",\n \"Table: Classification of Existing Systems wrt Declarative ML Tasks (Type 2).\"\n ],\n [\n \"Declarative ML Algorithms\",\n \"DSL-Centric Systems: The class of DSL-centric systems focuses on domain-specific languages (DSLs) for ML to simplify the writing of ML algorithms.\",\n \"Early examples of declarative systems are RIOT [43] and OptiML [35], which provide R and Scala DSLs, respectively.\",\n \"Both focus on single-node computation only which makes it easier to adhere to basic properties of declarative machine learning.\",\n \"SystemML [7], [22] covers both single-node and distributed computation (on MapReduce and Spark) and satisfies all eight properties of declarative ML as described throughout this paper.\",\n \"More recent systems like Cumulon [24], [25], Mahout Samsara [15], DMac [40], and TensorFlow [1] similarly aim for declarative, large-scale ML but struggle to satisfy all properties of declarative ML.\",\n \"Cumulon [24], [25] can be seen as a declarative system.\",\n \"However, in a strict sense, the optimization objective is ill-defined (P6) as the hard runtime constraint cannot be satisfied without knowing the number of iterations until convergence.\",\n \"Cumulon still allows users to explore per-iteration trade-offs of monetary cost and runtime in cloud environments.\",\n \"Mahout Samsara [15], Distributed R [39], and DMac [40] are not declarative because they expose physical data structures and distributed operations to the user (P1, P5).\",\n \"Mahout Samsara and Distributed R require the user to decide between local and distributed matrices and expose data flow properties like caching and partitioning (P2), while DMac exposes dense/sparse data structures.\",\n \"Additionally, Distributed R executes arbitrary user-defined R functions—i.e., with unknown operation semantics (P4)—per partition.\",\n \"TensorFlow [1] is a compelling system but focuses more on extensibility than declarative specification.\",\n \"Accordingly, operations are handled as black-box kernels, i.e., with unknown operation semantics (P4).\",\n \"Also, deep-learning-centric optimizations like lossy compression for communication work very well for noisy data but do not satisfy the property of implementation-agnostic results (P7) required for general-purpose declarative ML.\",\n \"SQL-Centric Systems: The class of SQL-centric systems complements the class of DSL-centric systems as both aim for custom analysis algorithms.\",\n \"Common SQL-centric systems are array databases and SQL-like ML (e.g., for Bayesian ML).\",\n \"These systems often follow the compelling argument of integrating advanced analytics with traditional query processing to simplify data pre-processing and leverage well-understood abstractions for data and operations.\",\n \"At the same time, linear algebra operations are either directly supported or emulated.\",\n \"A prime example of array databases is SciDB [9], [34] which indeed satisfies all basic properties for declarative ML.\",\n \"Furthermore, also SimSQL [11]—as an example of Bayesian ML or more broadly stochastic analysis—satisfies all basic properties of declarative ML.\",\n \"UDF-Centric Systems: There is also a class of UDF-centric systems that evolved bottom up from existing data-parallel frameworks like MapReduce, Spark, or Flink as well as compiler frameworks to simplify large-scale ML.\",\n \"Examples are ScalOps [8] (which compiles Scala UDF workflows to datalog programs), Tupleware [13] (which compiles workflows of UDFs in various frontend languages to custom distributed programs of native code via LLVM), and Emma [4] (which compiles Scala UDF workflows to Spark and Flink programs).\",\n \"These systems cover a great variety of use cases, but they systematically fail to satisfy several properties of declarative ML.\",\n \"First, data independence (P1) is not satisfied as the UDFs are implemented against custom data structures which makes it hard to efficiently support dense/sparse or compressed datasets.\",\n \"Second, the operation semantics of UDFs are by definition unknown (P4), miss support of analysis-centric operations (P3), and the focus on large-scale computation requires the UDF workflows to realize distributed algorithm implementations (P5).\"\n ],\n [\n \"Declarative ML Tasks\",\n \"As discussed before, systems for declarative ML tasks mostly target end users (not algorithm developers) and automatically optimize for runtime and accuracy of general-purpose optimization or model and feature selection tasks.\",\n \"General-Purpose Optimization: Bismarck [18] provides in-database general-purpose optimization via incremental gradient decent, where users provide UDFs for initialization, transition, and termination.\",\n \"This abstraction covers many algorithms over existing relational data.\",\n \"Similar to UDF-centric systems, however, it does not satisfy the independence of data structures (P1) and known operation semantics (P4).\",\n \"Furthermore, effective parallelization requires either violations of implementation-agnostic and deterministic results (P7, P8) for the pure UDA approach or modifications of the UDFs (P5) for the shared-memory UDA approach.\",\n \"The latter also does not satisfy P7/P8 due to Hogwild!-style [31] model updates.\",\n \"TensorFlow [1] also provides primitives for general-purpose optimization via different optimization algorithms.\",\n \"Similar to Bismarck, users provide UDFs for loss and gradient computation.\",\n \"However, as TensorFlow provides sufficient data abstraction via so-called placeholders, it satisfies the properties of data independence.\",\n \"Furthermore, the operation semantics of inference, loss, and training are known and UDFs can leverage the existing built-in functions which also provide abstractions for gradient computation.\",\n \"Thus, although TensorFlow did not qualify for declarative ML algorithms, it satisfies the properties at the level of declarative ML tasks.\",\n \"Model and Feature Selection: MLbase [27] with its TUPAQ [33] component for automatic model search allows users to specify candidate model configurations, a quality measure, and runtime constraints (e.g., models considered, number of scans, etc) and returns the best model along with tuned hyperparameters.\",\n \"Disregarding the underlying library of ML algorithms, MLbase can be classified as a system for declarative ML tasks, as the model search problem is well-defined, independent of the underlying data and operations, with annotated algorithm characteristics, and deterministic results (unless the runtime constraint is a runtime budget).\",\n \"Columbus [42] allows users to specify feature engineering workflows in R including data preparations and model building with existing R packages.\",\n \"The optimization objective is to minimize runtime, but an error tolerance allows for more aggressive reuse by leveraging runtime-accuracy trade-offs.\",\n \"Columbus also satisfies the properties for declarative ML tasks.\",\n \"DeepDive [32] enables knowledge base construction via statistical inference.\",\n \"In a first step, users specify SQL queries and UDFs for feature extraction to populate a factor graph.\",\n \"In a second step, candidate mappings are specified via SQL queries to express rules for entities and relations.\",\n \"Finally, marginal probabilities are learned via statistical inference over this factor graph.\",\n \"As data and operations are abstracted via SQL and the actual inference algorithms and details are not exposed, DeepDive can be classified as a declarative system for feature selection.\",\n \"Other System Categories: Finally, there are also declarative systems for more specific ML tasks like time series forecasting (e.g., Fa [16], a skip-list approach [20], or F2DB [19]), which also consider the trade-off between accuracy and runtime (model selection) but are not subject to this classification for general-purpose declarative ML.\"\n ],\n [\n \"Benchmarking ML Systems\",\n \"Having discussed the individual types of declarative ML, it is clear that there cannot exist a single benchmark to cover them all.\",\n \"Existing benchmarks for large-scale computation like BigBench [5], [21], SparkBench [2], [29], or HiBench [26] do cover machine learning but often simply refer to reference implementations of large-scale ML libraries.\",\n \"This is fine to evaluate underlying Hadoop or Spark implementations but simply cannot serve as a benchmark for declarative ML.\",\n \"A Case for Type-Specific Benchmarks: We argue that industry and academia is best served with benchmarks specific to types of declarative machine learning (see Section ).\",\n \"In order to properly reflect common workload characteristics, the benchmarks should be further tailored to major subcategories of systems.\",\n \"For example, systems for declarative ML algorithms cover the major sub-categories of DSL-centric, UDF-centric, and SQL-centric systems, which—despite some overlap of operation primitives—all target different primary usage scenarios.\",\n \"This calls for specific benchmarks that allow fair comparisons and foster system advancements via challenging workloads.\",\n \"Interestingly, exactly that already happened for SQL-centric systems as both, the SciDB [9], [34] and SimSQL [11] projects published benchmarks covering the main characteristics of array databases [36] and Bayesian ML [10].\",\n \"Defining simple yet challenging ML benchmarks that also cover common workload characteristics is not easy, so we—as a community—should highly appreciate contributions in this area.\",\n \"Note on Specification Languages: The same reasoning as with system-type-specific benchmarks also applies to specification languages.\",\n \"In order to satisfy the property of analysis-centric operation support (P4), systems need to support the operations of their primary analysis use case, which motivates tailor-made specification languages.\",\n \"However, as the properties for declarative ML are syntax independent, we could establish a common syntax for a specification language to cover multiple types of declarative ML.\"\n ],\n [\n \"Conclusion\",\n \"To summarize, we introduced a taxonomy of declarative ML in terms of basic properties of data, operations, and result correctness as well as types of systems for declarative ML.\",\n \"The classification of existing systems has shown that this taxonomy is indeed a useful tool for qualifying systems characteristics in a systematic manner.\",\n \"Fundamentally, this makes a case for a syntax-independent classification of declarative ML, which disqualifies the philosophical argument against loops and control flow in general.\",\n \"We are at the beginning of an exciting era of declarative ML, with a good understanding of various aspects but also lots of open research challenges.\",\n \"As advanced analytics become ubiquitous and technology environments are changing at an increasing rate, a declarative specification of ML tasks or algorithms becomes increasingly important.\",\n \"Accordingly, we encourage the research community to participate in this discussion on basic properties of declarative ML in order to eventually converge to a common understanding.\"\n 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"Coxeter Groups and Abstract Elementary Classes: The Right-Angled Case"
],
[
"Abstract We study classes of right-angled Coxeter groups with respect to the strong submodel relation of parabolic subgroup.",
"We show that the class of all right-angled Coxeter group is not smooth, and establish some general combinatorial criteria for such classes to be abstract elementary classes, for them to be finitary, and for them to be tame.",
"We further prove two combinatorial conditions ensuring the strong rigidity of a right-angled Coxeter group of arbitrary rank.",
"The combination of these results translate into a machinery to build concrete examples of $\\mathrm{AECs}$ satisfying given model-theoretic properties.",
"We exhibit the power of our method constructing three concrete examples of finitary classes.",
"We show that the first and third class are non-homogeneous, and that the last two are tame, uncountably categorical and axiomatizable by a single $L_{\\omega_{1}, \\omega}$-sentence.",
"We also observe that the isomorphism relation of any countable complete first-order theory is $\\kappa$-Borel reducible (in the sense of generalized descriptive set theory) to the isomorphism relation of the theory of right-angled Coxeter groups whose Coxeter graph is an infinite random graph."
],
[
"Introduction",
"Abstract elementary classes ($\\mathrm {AECs}$ ) [17] are pairs $(\\mathbf {K}, \\preccurlyeq )$ such that $\\mathbf {K}$ is a class of structures of the same similarity type, and $\\preccurlyeq $ is a partial order on $\\mathbf {K}$ , often referred to as a strong submodel relation, satisfying a certain set of axioms, which generalise some of the properties of the relation of elementary submodel of first-order logic.",
"Although $\\mathrm {AECs}$ generalize the first-order setting, the situation in $\\mathrm {AECs}$ is very different from the one in elementary model theory.",
"In fact, in the latter setting the strong submodel relation is always fixed.",
"The same remark holds for the model theory of infinitary languages, since also in this context one tends to use the canonical strong submodel relations (which in this case depend on what is the formula defining the class under study).",
"On the other hand, in the theory of abstract elementary classes we are free to choose any strong submodel relation, as long as the $\\mathrm {AECs}$ axioms are satisfied.",
"This choice determines very strongly the model-theoretic properties of the class under analysis.",
"A classical example is when we consider as $\\mathbf {K}$ the class of all abelian groups.",
"In this case, letting $\\preccurlyeq _0$ to be the subgroup relation, and $\\preccurlyeq _{1}$ to the pure subgroup relation, we have that $(\\mathbf {K}, \\preccurlyeq _0)$ is $\\omega $ -stable, while $(\\mathbf {K}, \\preccurlyeq _{1})$ is not even superstable.",
"In the context of $\\mathrm {AECs}$ , when one tries to find examples of various model-theoretic properties, one tends to start from a class $\\mathbf {K}$ of structures, and then search for a suitable or natural strong submodel relation $\\preccurlyeq $ .",
"In this paper we make an experiment, and reverse this process.",
"That is, we first choose the relation $\\preccurlyeq $ and then we try to find $\\mathbf {K}$ so that $(\\mathbf {K}, \\preccurlyeq )$ satisfies certain given model-theoretic properties.",
"We hope that in this way we are able to increase our understanding of the vast number of dividing lines that currently dominate the universe of $\\mathrm {AECs}$ , and to generate new (counter-)examples for the theory.",
"A similar approach has been pioneered in [13], where several well-behaved classes of geometric lattices have been found in this way, when considering as $\\preccurlyeq $ the strong submodel relation of principal extension of a combinatorial geometry, arising from the work of Crapo [7].",
"In this case study, we consider the strong submodel relation of parabolic subgroup, from geometric group theory.",
"The beginning of our study is the search for groups which together with the parabolic subgroup relation are $\\mathrm {AECs}$ (i.e.",
"the first property we test is the property of being an $\\mathrm {AEC}$ ).",
"We very quickly restricted our attention to classes consisting of so-called right-angled Coxeter groups.",
"These groups are in fact the most well-understood structures in geometric group theory.",
"In particular, they satisfy a crucial requirement known as rigidity [8] (see belowNotice that here rigidity does not mean what it usually means in model theory.).",
"However, it turns out that rigidity alone is not enough for our purposes.",
"In fact, we will see that the Smoothness Axiom fails in general and, without additional assumptions, we do not even know whether $\\preccurlyeq $ is transitive or not.",
"We get out of this empasse assuming a stronger property, known as strong rigidity.",
"While in the case of finitely generated right-angled Coxeter groups clear necessary and sufficient conditions are known for strong rigidity, not much is known about infinitely generated ones.",
"What is known is basically just that in this more general setting these conditions are only necessary, but not sufficient.",
"Thus, we start our study by giving two combinatorial conditions ensuring the strong rigidity of an arbitrary right-angled Coxeter group.",
"These results will be used to construct three concrete examples of $\\mathrm {AECs}$ : $(\\mathbf {K}_0, \\preccurlyeq )$ , $(\\mathbf {K}_1, \\preccurlyeq )$ and $(\\mathbf {K}_2, \\preccurlyeq )$ .",
"We continue our study by giving some general criteria for a class of strongly rigid right-angled Coxeter groups to be an abstract elementary class, and for it to satisfy the usual sufficient conditions for the construction of a monster model, i.e.",
"amalgamation, joint embedding and arbitrarly large models.",
"We then turn to notions that describe the behaviour of Galois-types, namely homogeneity, finitarity and tameness (we will also point out that excluding the class of infinite vector spaces over the two element field, classes of infinite right-angled Coxeter groups are not first-order axiomatizable).",
"Also in this case we give general criteria for the satisfaction of these properties, under the assumption of strong rigidity.",
"The underlying theme of these general results is the reduction of model-theoretic properties of a class of right-angled Coxeter groups to combinatorial conditions on the associated graphs, the so-called Coxeter graphs.",
"These conditions are often easy to realize, and, paired with our two general results on the strong rigidity of right-angled Coxeter groups, they translate into a machinery to build concrete examples of $\\mathrm {AECs}$ .",
"The classes $(\\mathbf {K}_0, \\preccurlyeq ~)$ , $(\\mathbf {K}_1, \\preccurlyeq )$ and $(\\mathbf {K}_2, \\preccurlyeq )$ should be considered under this perspective, as explicit examples of this machinery.",
"We conclude the paper with a close analysis of these classes.",
"First, we show that $(\\mathbf {K}_0, \\preccurlyeq )$ , $(\\mathbf {K}_1, \\preccurlyeq )$ and $(\\mathbf {K}_2, \\preccurlyeq )$ are finitary.",
"Then, we show that $(\\mathbf {K}_0, \\preccurlyeq )$ has the independence property (and thus it is unstable), while $(\\mathbf {K}_1, \\preccurlyeq )$ and $(\\mathbf {K}_2, \\preccurlyeq )$ are both tame and uncountably categorical (and thus stable in every infinite cardinality).",
"Finally, we show that $(\\mathbf {K}_0, \\preccurlyeq )$ and $(\\mathbf {K}_2, \\preccurlyeq )$ are not homogeneous.",
"We leave the tameness of $(\\mathbf {K}_0, \\preccurlyeq )$ and the homogeneity of $(\\mathbf {K}_1, \\preccurlyeq )$ as open questions.",
"John Baldwin pointed out to us that by combining our results with results from [14], various definability results can be obtained.",
"E.g.",
"the classes $(\\mathbf {K}_1, \\preccurlyeq )$ and $(\\mathbf {K}_2, \\preccurlyeq )$ are axiomatizable by a single $L_{\\omega _{1}, \\omega }$ -sentence, and over strong submodels Galois types and $L_{\\omega _{1}, \\omega }$ -types coincide in both $(\\mathbf {K}_1, \\preccurlyeq )$ and $(\\mathbf {K}_2, \\preccurlyeq )$ .",
"On the way of writing this paper, we also observed that right-angled Coxeter groups provide a way of finding a group whose first-order theory is maximal in the order of complexity that was introduced in the theory of generalized descriptive set theory [9].",
"We will point out how one can see this."
],
[
"Coxeter Groups",
"Let $S$ be a set.",
"A matrix $m: S \\times S \\rightarrow \\lbrace 1, 2, .",
".",
".",
", \\infty \\rbrace $ is called a Coxeter matrix if it satisfies $m(s, s^{\\prime }) = m(s^{\\prime } , s);$ $m(s, s^{\\prime }) = 1 \\Leftrightarrow s = s^{\\prime }.$ Equivalently, $m$ can be represented by a labelled graph $\\Gamma $ , called a Coxeter graph, whose node set is $S$ and whose edges are the unordered pairs $\\lbrace s, s^{\\prime } \\rbrace $ such that $m(s, s^{\\prime }) < \\infty $ , with label $m(s, s^{\\prime })$ .",
"(Notice that some authors refer to the Coxeter graph as the graph $\\Gamma $ such that $s$ and $s^{\\prime }$ are adjacent iff $m(s, s ) > 2$ .)",
"Let $S^2_{fin} = \\lbrace (s, s^{\\prime }) \\in S^2 : m(s, s^{\\prime } ) < \\infty \\rbrace $ .",
"A Coxeter matrix $m$ determines a group $W$ with presentation ${\\left\\lbrace \\begin{array}{ll} \\text{Generators}: S \\\\\\text{Relations}: (ss^{\\prime })^{m(s,s^{\\prime })} = e, \\text{ for all } (s, s^{\\prime } ) \\in S^2_{fin}.\\end{array}\\right.",
"}$ If a group $W$ has a presentation such as (2.1), then the pair $(W, S)$ is called a Coxeter system of type $m = m_{(W, S)}$ or of type $\\Gamma = \\Gamma _{(W, S)}$ .",
"The group $W = W_{\\Gamma }$ is called a Coxeter group and the set $S$ a Coxeter basis (or Coxeter generating set) for $W$ .",
"The cardinality of $S$ is called the rank of $(W, S)$ .",
"Notice that in the present paper we do not assume that our Coxeter groups are of finite rank, as it is done in most of the literature on the subject.",
"As well-known, the isomorphism type of $\\Gamma _{(W, S)}$ is not determined by the group $W$ alone (see e.g.",
"[5]).",
"This motivates the following definition.",
"Definition 2.1 Let $W$ be a Coxeter group.",
"We say that $W$ is rigid if for any two Coxeter bases $S$ and $S^{\\prime }$ for $W$ there is an automorphism $\\alpha \\in Aut(W)$ such that $\\alpha (S) = S^{\\prime }$ .",
"We say that $W$ is strongly rigid if for any two Coxeter bases $S$ and $S^{\\prime }$ for $W$ there is an inner automorphism $\\alpha \\in Inn(W)$ such that $\\alpha (S) = S^{\\prime }$ .",
"That is, $W$ is rigid if and only if for any two Coxeter bases $S$ and $S^{\\prime }$ for $W$ there exists an isomorphism of labelled graphs between $\\Gamma _{(W, S)}$ and $\\Gamma _{(W, S^{\\prime })}$ .",
"The problem of deciding whether two non-isomorphic Coxeter graphs determine isomorphic Coxeter groups is known as the isomorphism problem for Coxeter groups.",
"This problem is highly non-trivial, and it has been solved only partially [1].",
"The most well understood class of Coxeter groups in this respect (and any other respect) is the class of so-called right-angled Coxeter groups.",
"Definition 2.2 We say that a Coxeter system $(W, S)$ is right-angled if $m_{(W, S)}$ has coefficients in $\\lbrace 1, 2, \\infty \\rbrace $ , and that a Coxeter group $W$ is right-angled if there exists a right-angled Coxeter system for $W$ .",
"Theorem 2.3 (Castella [8]) The right-angled Coxeter groups are rigid.",
"Thus, in the case of right-angled Coxeter systems $(W, S)$ the group $W$ alone determines the isomorphism type of $\\Gamma _{(W, S)}$ .",
"Consequently, given a right-angled Coxeter group $W$ we denote by $\\Gamma _{W}$ (or simply $\\Gamma $ ) its associated Coxeter graph (unique modulo graph isomorphisms).",
"Given a Coxeter group $W$ there is a special class of subgroups of $W$ , which are called the parabolic subgroups of $W$ .",
"These subgroups (and the subgroup relation which they induce) will be the main ingredient in our model-theoretic analysis of right-angled Coxeter groups.",
"Definition 2.4 Let $W$ be a Coxeter group.",
"Given a Coxeter basis $S$ for $W$ , we say that $W^{\\prime }$ is an $S$ -parabolic subgroup of $W$ if $W^{\\prime } = \\langle S^{\\prime } \\rangle _W$ for some $S^{\\prime } \\subseteq S$ , i.e.",
"$W^{\\prime }$ is generated by a subset of $S$ .",
"In this case, we denote the subgroup $W^{\\prime }$ as $W_{S^{\\prime }}$ .",
"We say that $W^{\\prime }$ is a parabolic subgroup of $W$ , denoted as $W^{\\prime } \\preccurlyeq W$ , if $W^{\\prime }$ is an $S$ -parabolic subgroup of $W$ for some Coxeter basis $S$ of $W$ .",
"A parabolic subgroup $W^{\\prime } = \\langle S^{\\prime } \\rangle _W$ of a Coxeter group $W = (W, S)$ is Coxeter group in its own right, with as Coxeter generating set the induced subgraph determined by $S^{\\prime }$ (see e.g.",
"[5]).",
"As evident from the definition, the parabolic subgroup relation depends on the particular choice of Coxeter basis $S$ for $W$ .",
"This generates some difficulties in the analysis of this relation, e.g.",
"in the proof of very basic properties such as transitivity.",
"To this end, the notion of strong-rigidity (cfr.",
"Definition REF ) is of great help (notice for example that in the presence of strong-rigidity the transitivity of the parabolic subgroup relation is essentially trivial, see the proof of Theorem REF ).",
"For this reason we are interested in sufficient (and possibly necessary) conditions for strong rigidity.",
"The problem of (strong) rigidity of a Coxeter group $W$ is of course strictly related to our understanding of the corresponding group of automorphisms $Aut(W)$ .",
"In the case of right-angled Coxeter groups a fundamental result of Tits [19] gives an explicit description of $Aut(W)$ as a semidirect product of “tame” subgroups of $Aut(W)$ .",
"We describe these two subgroups.",
"Given a right-angled Coxeter group $W$ with Coxeter graph $\\Gamma = (S, E)$ , let $F(\\Gamma )$ be the collection of the $S$ -spheric subgroups of $W$ , i.e.",
"the $S$ -parabolic subgroups $W_{S^{\\prime }}$ of $W$ , with $S^{\\prime }$ a finite clique of $\\Gamma _{(W, S)}$ (i.e.",
"$m_{(W, S)}(s, s^{\\prime }) \\in \\lbrace 1, 2 \\rbrace $ ).",
"Let then $Aut(W, F(\\Gamma ))$ be the subgroup of $Aut(W)$ which stabilizes $F(\\Gamma )$ , and $Spe(W)$ the subgroup of $Aut(W)$ which stabilizes the conjugacy class of every $s \\in S$ .",
"Theorem 2.5 (Tits [19]) Let $W$ be a right-angled Coxeter group.",
"Then $Aut(W) = Spe(W) \\rtimes Aut(W, F(\\Gamma )).$ Evidently, $Inn(W) \\subseteq Spe(W) \\text{ and } Aut(\\Gamma ) \\subseteq Aut(W, F(\\Gamma )),$ where $Aut(\\Gamma )$ denotes the automorphism group of the graph $\\Gamma $ , which is naturally thought as a subgroup of $Aut(W)$ , since every automorphism of $\\Gamma $ extends canonically to an automorphism of $Aut(W)$ .",
"The next proposition shows the connection between $Inn(W)$ and $Aut(\\Gamma )$ , and the strong rigidity of $W$ .",
"Proposition 2.6 Let $W$ be a right-angled Coxeter group.",
"Then $W \\text{ is strongly rigid } \\Leftrightarrow Inn(W) = Spe(W) \\text{ and } Aut(\\Gamma ) = Aut(W, F(\\Gamma )).$ [8].",
"We are then interested in criteria which ensure that the two containments in (2.2) are equalities.",
"The next theorem recapitulates what is known on the subject.",
"We first introduce some definitions which will be useful for the statement of the theorem.",
"Definition 2.7 Let $\\Gamma = (V, E)$ be a graph.",
"For $v \\in \\Gamma $ , we let $N(v) = \\left\\lbrace v^{\\prime } \\in \\Gamma : v E v^{\\prime } \\right\\rbrace $ and $st(v) = N(v) \\cup \\left\\lbrace v \\right\\rbrace $ .",
"We say that $\\Gamma $ is star-connected if for every $v \\in \\Gamma $ we have that $\\Gamma - st(v)$ is connected.",
"We say that $\\Gamma $ has the star property if for every $v \\ne v^{\\prime } \\in \\Gamma $ we have that $st(v) \\lnot \\subseteq st(v^{\\prime })$ .",
"Theorem 2.8 Let $W$ be a right-angled Coxeter group.",
"$ Aut(W, F(\\Gamma )) = Aut(\\Gamma )$ if and only if $\\Gamma _{W}$ has the star property (cfr.",
"[8], Proposition 7).",
"If $W$ is of finite rank, then $Spe(W) = Inn(W)$ if and only if $\\Gamma _W$ is star-connected (cfr.",
"[16], corollary to the main theorem).",
"If $W$ is of arbitrary rank, then the star-connectedness of $\\Gamma _W$ is a necessary but not sufficient condition for $Spe(W) = Inn(W)$ .",
"(c) For the necessity of the condition see [19].",
"The non-sufficiency of the condition is claimed in [19], in the final remark of Section 3, but the exhibited map is not surjective.",
"We thus show the non-sufficiency of the condition.",
"Let $\\Gamma = \\bigcup _{i < \\omega } \\Gamma _i$ be a countably infinite star-connected graph such that for each $i < \\omega $ we have that $\\Gamma _i$ is finite and there exists $a_i \\ne b_i \\in \\Gamma _i - \\Gamma _{i-1}$ such that $a_i$ is not adjacent to $b_i$ and $a_i$ is adjacent to every element in $\\Gamma _{i-1}$ .",
"Such a $\\Gamma = \\bigcup _{i < \\omega } \\Gamma _i$ can easily be found, take e.g.",
"the countably infinite random graph.",
"For every $i < \\omega $ , let $\\alpha _i \\in Spe(W_{\\Gamma _i})$ be such for every $x \\in \\Gamma _i$ we have $\\alpha _i(x) = a_0 \\cdots a_i x a_i \\cdots a_0$ .",
"Then for every $i \\leqslant j < \\omega $ we have that $\\alpha _j$ restricted to $W_{\\Gamma _i}$ equals to $\\alpha _i$ , and so $\\alpha = \\bigcup _{i < \\omega }\\alpha _i \\in Spe(W_{\\Gamma })$ .",
"But obviously $\\alpha \\notin Inn(W_{\\Gamma })$ .",
"Point (c) above was already observed in [19], and also noticed in [8], where it is also shown that the star property is equivalent to one of the two conditions used in [6] to characterize strong rigidity in the finite rank case.",
"In the case of right-angled Coxeter groups of arbitrary rank a necessary and sufficient condition on $\\Gamma _W$ ensuring $Spe(W) = Inn(W)$ is not known.",
"In the next two theorems, relying on technology from [18] and [19], we establish two sufficient conditions for $Spe(W) = Inn(W)$ .",
"We first need to develop some combinatorics of right-angled Coxeter groups.",
"Let $(W, S)$ be a Coxeter system.",
"Each element $w \\in W$ can be written as a product of generators: $w = s_1 s_2 \\cdots s_k,$ with $s_i \\in S$ .",
"(The identity element $e$ is represented by the empty word.)",
"If $k$ is minimal among all such expressions for $w$ , then $k$ is called the length of $w$ (written as $|w| = k$ ) and the word $s_1 s_2 \\cdots s_k$ is called a normal form (or reduced word) for $w$ .",
"We denote by $sp(w)$ the set of letters appearing in any normal form for $w$ , and call it the support of $W$ with respect to the Coxeter basis $S$ .",
"This is well-defined, since if $s_1 s_2 \\cdots s_k$ and $s^{\\prime }_1 s^{\\prime }_2 \\cdots s^{\\prime }_k$ are two normal forms for $w$ , then the set of letters appearing in the word $s_1 s_2 \\cdots s_k$ equals to the set of letters appearing in $s^{\\prime }_1 s^{\\prime }_2 \\cdots s^{\\prime }_k$ (cfr.",
"e.g.",
"[5]).",
"We now describe two “moves” which take a word $s_1 s_2 \\cdots s_k$ in $(W, S)$ and change it into another word in $(W, S)$ that represents the same elements of $W$ and which is at most as long: if $s_i = s_{i+1}$ cancel the letters $s_i$ and $s_{i+1}$ ; if $m(s_i, s_{i+1}) = 2$ exchange $s_i$ and $s_{i+1}$ .",
"Theorem 2.9 (Tits [20]) Let $(W, S)$ be a right-angled Coxeter system.",
"If $s_1 s_2 \\cdots s_n$ and $s^{\\prime }_1 s^{\\prime }_2 \\cdots s^{\\prime }_m$ are two words representing the same element $w \\in W$ , then $s_1 s_2 \\cdots s_n$ and $s^{\\prime }_1 s^{\\prime }_2 \\cdots s^{\\prime }_m$ can be reduced to identical normal forms using moves $(M_1)$ and $(M_2)$ .",
"Proposition 2.10 Let $s_1 \\cdots s_n$ be a word in the right-angled Coxeter system $(W,S)$ .",
"Then $s_1 \\cdots s_n$ is a normal form if and only if for every $1 \\leqslant i < j \\leqslant k$ with $s_i = s_j$ , there exists $i < l < j$ such that $s_l \\notin st(s_i)$ .",
"See e.g.",
"[3].",
"Proposition 2.11 Let $s_1 \\cdots s_n$ be a word in the right-angled Coxeter system $(W,S)$ , and suppose that $s_i$ and $s_j$ can be brought next to each other using $(M_2)$ moves in order to use the move $(M_1)$ to shorten the word $s_1 \\cdots s_n$ .",
"Then $s_i$ and $s_j$ can be brought together using only moves each of which involves either $s_i$ or $s_j$ .",
"See e.g.",
"[3] (where it is proved more).",
"We now prove some facts about reflections (see definition below) in right-angled Coxeter groups.",
"In this section we will only use Corollary REF , but the rest will be crucial in what follows.",
"Specifically, Lemma REF will be the main ingredient in the proof of Theorem REF .",
"Definition 2.12 Let $(W, S)$ be a Coxeter system.",
"We define the set of reflections of $(W, S)$ to be the set $R(W, S) = \\lbrace wsw^{-1} : s \\in S, w \\in W \\rbrace $ .",
"Lemma 2.13 Let $(W, S)$ be a right-angled Coxeter system, $wsw^{-1} \\in R(W,S)$ and $a_1 \\cdots a_k$ a normal form for $w$ .",
"If $a_1 \\cdots a_k s a_k \\cdots a_1$ is not a normal form for $wsw^{-1}$ , then there exists $1 \\leqslant i \\leqslant k$ such that: $wsw = a_1 \\cdots a_{i-1} a_{i+1} \\cdots a_k s a_k \\cdots a_{i+1}a_{i-1} \\cdots a_1$ ; $a_i$ commmutes with $a_j$ for every $i < j \\leqslant k$ ; $a_i$ commmutes with $s$ ; $a_1 \\cdots a_{i-1} a_{i+1} \\cdots a_k$ is a normal form.",
"If $a_1 \\cdots a_k s a_k \\cdots a_1 = b_1 \\cdots b_{2k+1}$ is not a normal form for $wsw^{-1}$ , then because of Theorem REF and the fact that $a_k \\cdots a_1$ is normal, it must be the case that in any reduction of $a_1 \\cdots a_k s a_k \\cdots a_1$ to a normal form at some point we use the move $(M_1)$ for the pair $(b_x,b_y)$ , where $x < y$ and either $b_x = b_i$ for $i \\leqslant k$ and $b_y = b_{k+1} = s$ , or $b_y = b_i$ for $k+2 \\leqslant i \\leqslant 2k+1$ and $b_x = b_{k+1} = s$ , or $b_x = b_i$ for $i \\leqslant k$ and $b_y = b_j$ for $k+2 \\leqslant j \\leqslant 2k+1$ .",
"Furthermore, because of Proposition REF , we can assume that in this reduction we only use moves that involve either $b_x$ or $b_y$ .",
"Now, if we are in case (iii), then it is clear that $i$ is as wanted.",
"In fact it must be the case that $j = (2k+1)-(i-1)$ , otherwise $a_1 \\cdots a_k$ is not normal, and so we satisfy condition (a) because of our assumption that we use only moves that involve either $b_x$ or $b_y$ .",
"Furthermore, conditions (b) and (c) are satisfied because of Proposition REF .",
"Finally, it is easy to see that also (d) is satisfied, because otherwise $a_1 \\cdots a_k$ is not normal.",
"Case (i) and (ii) are symmetric, and so it suffices to analyse case (i).",
"But this is essentially as in case (iii), since after deleting the pair $(b_x,b_y)$ we can move $b_{(2k+1)-(i-1)} = s$ where $b_y = b_{k+1} = s$ was, i.e.",
"in the middle of the word.",
"Lemma 2.14 Let $(W, S)$ be a right-angled Coxeter system, $T \\subseteq S$ and $wsw^{-1} \\in R(W, S) \\cap W_T$ .",
"Let $a_1 \\cdots a_k$ be a normal form for $w$ , and $a_{q_{1}} \\cdots a_{q_n}$ be the subword of $a_1 \\cdots a_k$ obtained by deleting all the occurrences of letters in $S - T$ .",
"Then $wsw^{-1} = a_{q_{1}} \\cdots a_{q_n} s a_{q_n} \\cdots a_{q_{1}}.$ Iterating Lemma REF , we get $l \\leqslant k$ and a sequence of words $(w_i)_{i \\leqslant l}$ such that: $w_0 = a_1 \\cdots a_k$ ; for every $i < l$ , the word $w_{i+1}$ is a subword of $w_i$ of length $|w_i| - 1$ ; for every $i \\leqslant l$ , the word $w_i$ is normal; for every $i \\leqslant l$ , $w_isw_i^{-1} = wsw$ ; $w_lsw_l^{-1}$ is normal (and so $sp(w_l), sp(s) \\subseteq T$ ); $w_l$ is a subword of $a_{q_{1}} \\cdots a_{q_n}$ .",
"For $i < l$ , let $a_i$ be the letter witnessing that $w_{l-i}$ is a subword of $w_{l-(i+1)}$ of length $|w_{l-(i+1)}| - 1$ , and consider the sequence $((a_i, a_i))_{i < l}$ .",
"Then, because of conditions (b) and (c) of Lemma REF , for every $X = \\lbrace i_1 < \\cdots < i_{m} \\rbrace \\subseteq l$ , the pairs $((a_i, a_i))_{i \\in X}$ can be put back into the word $w_lsw_l^{-1}$ following the order $(a_{i_1}, a_{i_1}) < \\cdots < (a_{i_{m}}, a_{i_{m}})$ .",
"This suffices, since $w_l$ is a subword of $a_{q_{1}} \\cdots a_{q_n}$ .",
"The following corollary is immediate from Lemma REF .",
"This is fact is known for any Coxeter group, see e.g.",
"[10].",
"Corollary 2.15 Let $(W, S)$ be a Coxeter system and $T \\subseteq S$ .",
"Then $R(W, S) \\cap W_T = R(W_T, T).$ We also need an explicit description of centralizers of Coxeter generators.",
"Lemma 2.16 (Tits [19]) Let $W$ be a right-angled Coxeter group and $v \\in \\Gamma _W$ .",
"Then the centralizer $C_W(v)$ of $v$ in $W$ is the parabolic subgroup $W_{st(v)}$ .",
"[19].",
"We now go back to the main theme of this section, i.e.",
"strong rigidity.",
"To this end, we need two lemmas.",
"These lemmas are essentially Theorem 3 and Lemma 4 of [18] proved in the context of Coxeter groups ([18] proves this fact for Artin groups (a.k.a graph groups)).",
"Lemma 2.17 Let $W$ be a right-angled Coxeter group, $\\alpha \\in Spe(W)$ , $v \\in \\Gamma _W$ and $Y$ a connected component of $\\Gamma _W - st(v)$ .",
"Then if $v \\in sp(\\alpha (y))$ for some $y \\in Y$ , then $v \\in sp(\\alpha (x))$ for every $x \\in Y$ .",
"We show that $v \\in sp(\\alpha (x))$ for any $x$ adjacent to $y$ and not adjacent to $v$ , the result follows by the connectedness of $Y$ .",
"Now, $\\alpha (y) = wyw^{-1}$ for some $w \\in W$ , because $\\alpha \\in Spe(W)$ , and $sp(wyw^{-1}) \\subseteq sp(w) \\cup sp(y)$ (cfr.",
"Theorem REF ).",
"By hypothesis $v \\in sp(\\alpha (y))$ , and evidently $v \\notin sp(y) = \\lbrace y \\rbrace $ , thus $v \\in sp(w)$ .",
"Consider now $\\alpha (x)$ .",
"As for $\\alpha (y)$ , there exists $p \\in W$ such that $\\alpha (x) = pxp^{-1}$ .",
"By the choice of $x$ , the element $y$ commutes with $x$ , and so $\\alpha (y)$ commutes with $\\alpha (x)$ .",
"That is, $\\alpha (x) \\in C_W(\\alpha (y))$ .",
"By Lemma REF $C_W(wyw^{-1}) = wC_W(y)w^{-1} = wW_{st(y)}w^{-1},$ and so $\\alpha (x) \\in wW_{st(y)}w^{-1}$ , i.e.",
"$\\alpha (x) = wy^{\\prime }w^{-1}$ for some $y^{\\prime } \\in st(y)$ .",
"Furthermore, being $\\alpha (x)$ conjugate to $x$ , we have $x \\in sp(\\alpha (x)) = sp(wy^{\\prime }w^{-1})$ .",
"We distinguish two cases.",
"Case 1.",
"$x \\in sp(y^{\\prime })$ .",
"If this is the case, then $v \\in sp(wy^{\\prime }w^{-1})$ , because $x$ is not adjacent to $v$ (cfr.",
"Theorem REF ).",
"Case 2.",
"$x \\notin sp(y^{\\prime })$ .",
"We show that this case is not possible.",
"If $x \\notin sp(y^{\\prime })$ , then $x \\in sp(w) -sp(y^{\\prime })$ .",
"Thus, for any normal form $w_1 \\cdots w_k$ and $y^{\\prime }_1 \\cdots y^{\\prime }_m$ for $w$ and $y^{\\prime }$ , respectively, we have that $x$ occurs an even number of times in $w_1 \\cdots w_ky^{\\prime }_1 \\cdots y^{\\prime }_mw_k \\cdots w_1.$ Hence, $x$ occurs an even number of times also in $p_1 \\cdots p_l x p_l \\cdots p_1$ , for $p_1 \\cdots p_l$ a normal form for $p$ (see e.g.",
"[15]), but this is obviously absurd.",
"Lemma 2.18 Let $W$ be a right-angled Coxeter group such that $\\Gamma _W$ satisfies the following conditions: $\\Gamma _W$ is star-connected; $\\Gamma _W$ is triangle-free; $\\Gamma _W$ contains a copy of $P_4$ (the path of length 4) as a subgraph (not necessarily induced).",
"Then for every $\\alpha \\in Spe(W)$ there exists $w \\in W$ such that $w \\alpha w^{-1}$ fixes $P_4$ pointwise.",
"Let $P_4 = aEbEcEd$ and $\\alpha \\in Spe(W)$ .",
"Then $\\alpha (a) = pap^{-1}$ and so conjugating $\\alpha $ by $p^{-1}$ we get $\\alpha _1 \\in Spe(W)$ such that $\\alpha _1(a) = a$ .",
"Now, $a$ and $b$ commute and so we have $\\alpha _1(b) = qbq^{-1}$ with $sp(q) \\subseteq N(a)$ (cfr.",
"Lemma REF ).",
"Thus, conjugating $\\alpha _1$ by $q^{-1}$ we get $\\alpha _2 \\in Spe(W)$ such that $\\alpha _2(a) = a$ and $\\alpha _2(b) = b$ .",
"Similarly, $b$ and $c$ commute and so we have $\\alpha _2(c) = rcr^{-1}$ with $sp(r) \\subseteq N(b)$ .",
"Let $x \\in N(b) - \\lbrace a, c\\rbrace $ , then by the triangle-freeness of $\\Gamma _W$ , $x$ is adjacent neither to $a$ nor to $c$ , and so $a, c \\in \\Gamma _W - st(x)$ .",
"By the star-connectedness of $\\Gamma _W$ , $a$ and $c$ are connected in $\\Gamma _W - st(x)$ , and so given that $x \\notin sp(\\alpha _2(a)) = sp(a) = \\lbrace a \\rbrace $ , by Lemma REF we have $x \\notin sp(\\alpha _2(c))$ .",
"Hence $sp(\\alpha _2(c)) \\subseteq \\lbrace a, c \\rbrace $ .",
"Then $\\langle \\alpha _2(a) = a, \\alpha _2(c) \\rangle _W \\subseteq \\langle a, c \\rangle _W$ .",
"On the other hand, $\\alpha _2^{-1} \\in Spe(W)$ , $\\alpha _2^{-1}(a) = a$ and $\\alpha _2^{-1}(b) = b$ , and so the same argument used for $\\alpha _2$ shows that $sp(\\alpha _2^{-1}(c)) \\subseteq \\lbrace a, c \\rbrace $ .",
"Thus, $\\alpha _2^{-1}(c) \\in \\langle a, c \\rangle _W$ , from which it follows that $c \\in \\alpha _2(\\langle a, c \\rangle _W) = \\langle \\alpha _2(a) = a, \\alpha _2(c) \\rangle _W,$ i.e.",
"$\\langle a, c \\rangle _W \\subseteq \\langle a, \\alpha _2(c) \\rangle _W$ .",
"Hence, $\\langle a, c \\rangle _W = \\langle a, \\alpha _2(c) \\rangle _W.$ That is, $\\alpha _2$ restricted to $\\langle a, c \\rangle _W = W_{\\lbrace a, c \\rbrace } \\in Aut(W_{\\lbrace a, c \\rbrace })$ .",
"Furthermore, because of Corollary REF we see that $\\alpha _2 \\in Spe(W_{\\lbrace a, c \\rbrace })$ .",
"Also, $(\\lbrace a, c \\rbrace , E) = (\\lbrace a, c \\rbrace , \\emptyset )$ is star-connected, and so by Theorem REF (b) we have $\\alpha _2 \\in Inn(W_{\\lbrace a, c \\rbrace })$ .",
"But then obviously it must be the case that $\\alpha _2(c)$ is either $c$ or $aca$ , because otherwise $\\alpha _2(a) \\ne a$ .",
"It follows that $sp(r) \\subseteq \\lbrace a \\rbrace $ , and so conjugating $\\alpha _2$ by $r^{-1}$ we get $\\alpha _3 \\in Spe(W)$ such that $\\alpha _3(a) = a$ , $\\alpha _3(b) = b$ and $\\alpha _3(c) = c$ .",
"Using the same argument for $\\alpha _3(d) = tdt^{-1}$ , we see that $sp(t) \\subseteq \\lbrace b \\rbrace $ , and so conjugating $\\alpha _3$ by $t^{-1}$ we get $\\alpha _4 \\in Spe(W)$ such that $\\alpha _4(a) = a$ , $\\alpha _4(b) = b$ , $\\alpha _4(c) = c$ and $\\alpha _4(d) = d$ .",
"We now arrive at the first sufficient condition for $Spe(W) = Inn(W)$ .",
"This theorem takes inspiration from [18], where his use of Theorem 3 and Lemma 4 is replaced by our Lemmas REF and REF .",
"Theorem 2.19 Let $W$ be a right-angled Coxeter group such that $\\Gamma _W$ satisfies the following conditions: $\\Gamma _W$ is star-connected; $\\Gamma _W$ is triangle-free; $\\Gamma _W$ contains $P_4$ as a subgraph.",
"Then $Spe(W) = Inn(W)$ .",
"Let $\\alpha \\in Spe(W)$ , then by Lemma REF there exists $w \\in W$ such that $w \\alpha w^{-1}$ fixes $P_4 = aEbEcEd$ pointwise.",
"We show that $\\alpha _{1} = w \\alpha w^{-1}$ is the identity $id_W$ on $W$ .",
"This of course suffices, since then $\\alpha = w^{-1}w \\alpha w^{-1}w = w^{-1}id_Ww = Inn(w^{-1}),$ where, for $x \\in W$ , $Inn(x)$ denotes the inner automorphism determined by $x$ .",
"To this end, let $y \\notin P_4$ and suppose that $\\alpha _1(y) \\ne g y g^{-1}$ .",
"Then there is $v \\in sp(g)$ such that $v \\ne y$ and $v$ is not adjacent to $y$ .",
"By the triangle-freeness of $\\Gamma _W$ there exists $e \\in \\lbrace a, b, c, d \\rbrace - \\lbrace v \\rbrace $ such that $e$ is not adjacent to $v$ .",
"It follows that $\\Gamma - st(v)$ contains $y$ and $e$ .",
"Furthermore, $v \\in sp(\\alpha _1(y))$ and so by Lemma REF we have $v \\in sp(\\alpha _1(e)) = sp(e) = \\lbrace e\\rbrace ,$ which is a contradiction.",
"Thus, we must have $\\alpha _1(y) = y$ .",
"It follows that $\\alpha _1 = id_W$ .",
"Corollary 2.20 Let $W$ be as in Theorem REF and suppose that in addition $\\Gamma _W$ has the star property.",
"Then $W$ is strongly rigid.",
"Immediate from Proposition REF , Theorem REF and Theorem REF .",
"Finally, we arrive at the second sufficient condition for $Spe(W) = Inn(W)$ .",
"This theorem takes inspiration from [19], although the setting of the reference is quite different from the one in the theorem.",
"Theorem 2.21 Let $W$ be a right-angled Coxeter group such that $\\Gamma _W$ satisfies the following conditions: $\\Gamma _W$ is star-connected; and either $\\Gamma _W$ is finite or there exists $s, s^{\\prime } \\in \\Gamma _W$ such that: $st(s) \\cup st(s^{\\prime })$ is finite and star-connected (as an induced subgraph); for every $v \\in \\Gamma _W$ there exists $a \\in st(s) \\cup st(s^{\\prime })$ such that $a \\ne v$ and $a$ is not adjacent to $v$ .",
"Then $Spe(W) = Inn(W)$ .",
"If $\\Gamma _W$ is finite, then we know that star-connectedness suffices for $Spe(W) = Inn(W)$ .",
"Suppose then that $\\Gamma _W$ is infinite (and so conditions (b) and (c) hold).",
"Let $s, s^{\\prime }$ be as in the statement of the theorem and $\\alpha \\in Spe(W)$ .",
"We will show that there exists $w \\in W$ such that $w \\alpha w^{-1}$ is the identity on $W$ .",
"By assumption $\\alpha (s) = psp^{-1}$ and so conjugating $\\alpha $ by $p^{-1}$ we get $\\alpha _1 \\in Spe(W)$ such that $\\alpha _1(s) = s$ .",
"Now, $s$ and $s^{\\prime }$ commute and so we have $\\alpha _1(s^{\\prime }) = qs^{\\prime }q^{-1}$ with $sp(q) \\subseteq N(a)$ (cfr.",
"Lemma REF ).",
"Thus, conjugating $\\alpha _1$ by $q^{-1}$ we get $\\alpha _2 \\in Spe(W)$ such that $\\alpha _2(s) = s$ and $\\alpha _2(s^{\\prime }) = s^{\\prime }$ .",
"Given that $C_W(s) = W_{st(s)}$ and $C_W(s^{\\prime }) = W_{st(s^{\\prime })}$ (cfr.",
"Lemma REF ) we must have that $\\alpha _2$ fixes $W_{st(s) \\cup st(s^{\\prime })}$ setwise, i.e.",
"$\\alpha _2$ restricted to $W_{st(s) \\cup st(s^{\\prime })}$ is in $Aut(W_{st(s) \\cup st(s^{\\prime })})$ .",
"Furthermore, because of Corollary REF we see that $\\alpha _2 \\in Spe(W_{st(s) \\cup st(s^{\\prime })})$ .",
"Also, by assumption $st(s) \\cup st(s^{\\prime })$ is finite and star-connected, and so we have $\\alpha _2 \\in Inn(W_{st(s) \\cup st(s^{\\prime })})$ (cfr.",
"Theorem REF (b)).",
"Thus, composing $\\alpha _2$ with an inner automorphism, we get $\\alpha _3 \\in Spe(W)$ which fixes $W_{st(s) \\cup st(s^{\\prime })}$ pointwise.",
"We show that $\\alpha _3$ fixes every element of $\\Gamma _W$ .",
"To this end, let $y \\notin st(s) \\cup st(s^{\\prime })$ and suppose that $\\alpha _3(y) = g y g^{-1}$ is not fixed.",
"Then there is $v \\in sp(g)$ such that $v \\ne y$ and $v$ is not adjacent to $y$ .",
"Notice that because of (c) there exists $a \\ne v \\in st(s) \\cup st(s^{\\prime })$ such that $v$ is not adjacent to $a$ .",
"It follows that $\\Gamma - st(v)$ contains $y$ and $a$ .",
"Furthermore, $v \\in sp(\\alpha _3(y))$ , and so by Lemma REF we have $v \\in sp(\\alpha _3(a)) = sp(a) = \\lbrace a\\rbrace ,$ which is a contradiction.",
"Thus, we must have $\\alpha _3(y) = y$ .",
"It follows that $\\alpha _3 = id_W$ .",
"Corollary 2.22 Let $W$ be as in Theorem REF and suppose that in addition $\\Gamma _W$ has the star property.",
"Then $W$ is strongly rigid.",
"Immediate from Proposition REF , Theorem REF and Theorem REF .",
"We will refer to groups satisfying the conditions of Corollary REF as centered right-angled Coxeter groups (centered because of the $s$ and $s^{\\prime }$ )."
],
[
"Random Right-Angled Coxeter Groups",
"Let $T_{rg}$ be the first-order theory of random graphs, and $T_{racg}$ be $Th(A)$ for $A$ any right-angled Coxeter group such that $\\Gamma _A \\models T_{rg}$ .",
"This does not depend on $A$ , since for every right-angled Coxeter groups $B$ and $C$ such that $\\Gamma _B, \\Gamma _C \\models T_{rg}$ the two groups $B$ and $C$ are elementary equivalent.",
"This can be seen using e.g.",
"the Ehrenfeucht-Fraïssé game $EF_{\\omega }(B, C)$ of length $\\omega $ (this definitely suffices, since it shows that $B$ and $C$ are elementary equivalent in the infinitary logic $L_{\\infty , \\omega }$ ).",
"We sketch the idea.",
"If in the game $EF_{\\omega }(B, C)$ Player I plays an element $b_0 \\in B$ with normal form $s^0_1 \\cdots s^0_n$ , then Player II plays the element $c_0 = t^0_1 \\cdots t^0_n$ , for $t^0_1 \\cdots t^0_n$ the answer of Player II to the move $s^0_1 \\cdots s^0_n$ of Player I in the game $EF_{\\omega }(\\Gamma _B, \\Gamma _C)$ , in which, as well-known, Player II has a winning strategy, since $\\Gamma _B, \\Gamma _C \\models T_{rg}$ .",
"(Notice that in a game of length $\\omega $ playing elements or tuples does not matter.)",
"The other moves are played in the same fashion.",
"We now fix a cardinal $\\kappa > \\omega $ such that $\\kappa ^{<\\kappa } = \\kappa $ and code models $A$ of cardinalilty $\\kappa $ in a universal countable language $L^*$ (countably many relation symbols for any arity) as elements $\\eta (A)$ of $2^{\\kappa }$ in the usual fashion (see e.g.",
"[9]).",
"Given a complete first-order theory $T$ in the language $L^*$ , we define the isomorphism relation $\\cong _T$ on $2^{\\kappa } \\times 2^{\\kappa }$ as the relation $ \\lbrace (\\eta (A), \\eta (B)) \\in 2^{\\kappa } \\times 2^{\\kappa } : A, B \\models T, A \\cong B \\rbrace \\cup \\lbrace (\\eta (A), \\eta (B)) \\in 2^{\\kappa } \\times 2^{\\kappa } : A, B \\lnot \\models T \\rbrace .$ Finally, given two complete first-order theories $T_0$ and $T_1$ in the language $L^*$ we can say that the isomorphism relation of $T_0$ reduces to the isomorphism relation of $T_1$ , denoted as $\\cong _{T_0} \\; \\leqslant _B \\; \\cong _{T_1}$ , if the relation $\\cong _{T_0}$ is Borel reducible to $\\cong _{T_1}$ in the usual sense of generalized descriptive set theory (cfr.",
"e.g.",
"[9]).",
"Clearly any (complete) countable first-order theory can be thought canonically as a (complete) theory in the language $L^*$ (in particular $T_{rg}$ and $T_{racg}$ can be thought so).",
"We denote by $\\cong _{RACG}$ the isomorphism relation $\\cong _{T_{racg}}$ .",
"Given a graph $\\Gamma =( V, E)$ and $X \\subseteq V$ we say that $V$ is a clique (resp.",
"an indepenent set) if for every $x \\ne y \\in X$ we have $xEy$ (resp.",
"$x$ is not adjacent to $y$ ).",
"Theorem 3.1 For any countable complete first-order theory $T$ , $\\cong _T \\; \\leqslant _B \\; \\cong _{T_{rg}}.$ This is folklore, we sketch a proof for completeness of exposition.",
"As well-known, it suffices to do the following: for every graph $\\Gamma $ of power $\\kappa $ we define a random graph $R_{\\Gamma }$ of power $\\kappa $ such that $\\Gamma \\cong \\Gamma ^{\\prime }$ iff $R_{\\Gamma } \\cong R_{\\Gamma ^{\\prime }}$ .",
"We do this.",
"Let $\\Gamma = (V, E)$ be a graph of power $\\kappa $ with $V \\cap E = \\emptyset $ (without loss of generality).",
"Define a graph $R^0_{\\Gamma }$ on $V \\cup E$ by letting $a$ and $b$ be adjacent to $\\lbrace a, b\\rbrace $ , for every $\\lbrace a, b\\rbrace \\in E$ .",
"Now, for every $a \\in V$ add a clique $K_a$ of size $\\omega _1$ such that $a$ is adjacent to co-countably many $x \\in K_a$ , i.e.",
"$K_a - N(a)$ has size $\\omega $ .",
"Similarly, for every $\\lbrace a,b\\rbrace \\in E$ add a clique of size $\\omega _1$ such that $K_{a,b} \\cap N(\\lbrace a,b\\rbrace )$ and $K_{a,b} - N(\\lbrace a,b\\rbrace )$ have both size $\\omega _1$ .",
"Let $R^1_{\\Gamma }(0)$ be the resulting graph, and define $R^2_{\\Gamma }(i+1)$ by closing $R^2_{\\Gamma }(i)$ under the following condition: for every finite $X$ there exists $a_X$ such that $N(a_X) = X$ .",
"Then $\\bigcup _{i < \\omega } R^2_{\\Gamma }(i) = R_{\\Gamma } \\models T_{rg}$ is as wanted.",
"Theorem 3.2 For any countable complete first-order theory $T$ , $\\cong _T \\; \\leqslant _B \\; \\cong _{RACG}.$ Because of Theorem REF , it suffices to show that $\\cong _T \\; \\leqslant _B \\; \\cong _{RACG}$ for $T = T_{rg}$ the theory of random graphs.",
"But this is immediate since we can define $F: 2^{\\kappa } \\rightarrow 2^{\\kappa }$ by setting $F(\\eta (\\Gamma )) = {\\left\\lbrace \\begin{array}{ll}\\eta (A) \\;\\;\\; \\text{ if } \\Gamma \\lnot \\models T_{rg} \\\\\\eta (A_{\\Gamma }) \\; \\text{ if } \\Gamma \\models T_{rg},\\end{array}\\right.",
"}$ where in the first clause $A$ denotes any fixed right-angled Coxeter group $A$ such that $A \\lnot \\models T_{racg}$ , and in the second clause $A_{\\Gamma }$ is the right-angled Coxeter group of type $\\Gamma $ .",
"The function $F$ is evidently Borel.",
"The following result shows that no non-trivial class of right-angled Coxeter groups can be treated from the perspective of first-order model theory.",
"This motivates our use of abstract elementary classes.",
"Theorem 3.3 Let $\\mathbf {K}$ be a class of right-angled Coxeter groups such that there exists $A \\in \\mathbf {K}$ with $\\Gamma _A$ containing two non-adjacent vertices $a$ and $b$ .",
"Then $\\mathbf {K}$ is not first-order axiomatizable.",
"Let $A$ , $a$ and $b$ be as in the statement of the theorem.",
"Then for every positive integer $n$ the element $c_n = (ab)^n \\in A$ is divisible by $n$ .",
"It follows that in the ultrapower $\\prod _{i < \\omega } A_i/U$ ($U$ non-principal ultrafilter) there exists a divisible element $c$ (i.e.",
"an element divisible by every positive integer $n$ ), but a Coxeter group can not contain such an element $c$ .",
"Thus, $\\prod _{i < \\omega } A_i/U \\notin \\mathbf {K}$ (and so $\\mathbf {K}$ is not first-order)."
],
[
"Abstract Elementary Classes",
"In this section we introduce the basics of abstract elementary classes (see e.g.",
"[17] and [11]).",
"This machinery will be used in later sections in order to study various classes of right-angled Coxeter groups.",
"As usual in this context, type means Galois type (cf.",
"e.g.",
"[2]).",
"Given a class $\\mathbf {K}$ of structures in the vocabulary $L$ , we denote by $\\leqslant $ the $L$ -submodel relation on structures in $\\mathbf {K}$ .",
"Definition 4.1 (Abstract Elementary Class [17]) Let $\\mathbf {K}$ be a class of structures in the vocabulary $L$ .",
"We say that $(\\mathbf {K}, \\preccurlyeq )$ is an abstract elementary class ($\\mathrm {AEC}$ ) if the following conditions are satisfied.",
"$\\mathbf {K}$ and $\\preccurlyeq $ are closed under isomorphism.",
"If ${A} \\preccurlyeq {B}$ , then ${A}$ is a substructure of ${B}$ (${A} \\leqslant {B}$ ).",
"The relation $\\preccurlyeq $ is a partial order on $\\mathbf {K}$ .",
"If $({A}_i)_{i < \\delta }$ is an increasing continuous $\\preccurlyeq $ -chain, then: $\\bigcup _{i < \\delta } {A}_i \\in \\mathbf {K}$ ; for each $j < \\delta $ , ${A}_j \\preccurlyeq \\bigcup _{i < \\delta } {A}_i$ ; if each ${A}_j \\preccurlyeq {B}$ , then $\\bigcup _{i < \\delta } {A}_i \\preccurlyeq {B}$ (Smoothness Axiom).",
"If ${A}, {B}, {C} \\in \\mathbf {K}$ , ${A} \\preccurlyeq {C}$ , ${B} \\preccurlyeq {C}$ and ${A} \\leqslant {B}$ , then ${A} \\preccurlyeq {B}$ (Coherence Axiom).",
"There is a Löwenheim-Skolem number $\\mathrm {LS}(\\mathbf {K}, \\preccurlyeq )$ such that if ${A} \\in \\mathbf {K}$ and $B \\subseteq A$ , then there is ${C} \\in \\mathbf {K}$ such that $B \\subseteq C$ , ${C} \\preccurlyeq {A}$ and $|C| \\leqslant |B| + |L| + \\mathrm {LS}(\\mathbf {K}, \\preccurlyeq )$ (Existence of LS-number).",
"Definition 4.2 If ${A}, {B} \\in \\mathbf {K}$ and $f: {A} \\rightarrow {B}$ is an embedding such that $f({A}) \\preccurlyeq {B}$ , then we say that $f$ is a $\\preccurlyeq $ -embedding.",
"Let $\\lambda $ be a cardinal.",
"We let $\\mathbf {K}_{\\lambda } = \\left\\lbrace {A} \\in \\mathbf {K} \\; | \\; |A| = \\lambda \\right\\rbrace $ .",
"Definition 4.3 Let $(\\mathbf {K}, \\preccurlyeq )$ be an $\\mathrm {AEC}$ .",
"We say that $(\\mathbf {K}, \\preccurlyeq )$ has the amalgamation property $(\\mathrm {AP})$ if for any ${A}, {B}_0, {B}_1 \\in \\mathbf {K}$ with ${A} \\preccurlyeq {B}_i$ , for $i < 2$ , there are ${C} \\in \\mathbf {K}$ and $\\preccurlyeq $ -embeddings $f_i: {B}_i \\rightarrow {C}$ , for $i < 2$ , such that $f_0 \\upharpoonright A = f_1 \\upharpoonright A$ .",
"We say that $(\\mathbf {K}, \\preccurlyeq )$ has the joint embedding property $(\\mathrm {JEP})$ if for any ${B}_0, {B}_1 \\in \\mathbf {K}$ there are ${C} \\in \\mathbf {K}$ and $\\preccurlyeq $ -embeddings $f_i: {B}_i \\rightarrow {C}$ , for $i < 2$ .",
"We say that $(\\mathbf {K}, \\preccurlyeq )$ has arbitrarily large models $(\\mathrm {ALM})$ if for every $\\lambda \\geqslant \\!\\mathrm {LS}(\\mathbf {K}, \\!\\preccurlyeq )$ , $\\mathbf {K}_{\\lambda } \\ne \\emptyset $ .",
"As well-known, given an $\\mathrm {AEC}$ , say $(\\mathbf {K}, \\preccurlyeq )$ , with $\\mathrm {AP}$ , $\\mathrm {JEP}$ and $\\mathrm {ALM}$ , we can construct a monster model $\\mathfrak {M} = \\mathfrak {M}(\\mathbf {K}, \\preccurlyeq )$ for $(\\mathbf {K}, \\preccurlyeq )$ , i.e.",
"a $\\kappa $ -model homogeneous and $\\kappa $ -universal (for $\\kappa $ large enough) structure in $\\mathbf {K}$ .",
"We say that a subset $A$ of $\\mathfrak {M}$ is bounded if its cardinality is smaller than $\\kappa $ .",
"Given bounded $A \\subseteq \\mathfrak {M}$ and $n < \\omega $ , we denote by $S_n(A)$ the set of Galois typesFor a definition of Galois type see e.g.",
"[14].",
"over $A$ of length $n$ , and by $S(A)$ the set $\\bigcup _{n < \\omega } S_n(A)$ .",
"Definition 4.4 Let $(\\mathbf {K}, \\preccurlyeq )$ be an $\\mathrm {AEC}$ with $\\mathrm {AP}$ , $\\mathrm {JEP}$ and $\\mathrm {ALM}$ .",
"We say that $(\\mathbf {K}, \\preccurlyeq )$ has the independence property if there exists finite $A \\subseteq \\mathfrak {M}$ and $P \\subseteq S(A)$ such that for every ordinal $\\alpha < |\\mathfrak {M}|$ there exist $(a_i)_{i < \\alpha } \\in \\mathfrak {M}$ such that for every $X \\subseteq \\alpha $ there exists $b_X \\in \\mathfrak {M}$ such that $tp(b_Xa_i/A) \\in P$ if and only if $i \\in X$ .",
"Definition 4.5 Let $(\\mathbf {K}, \\preccurlyeq )$ be an $\\mathrm {AEC}$ with $\\mathrm {AP}$ , $\\mathrm {JEP}$ and $\\mathrm {ALM}$ .",
"We say that $(\\mathbf {K}, \\preccurlyeq )$ is homogeneous if for every ordinal $\\alpha < |\\mathfrak {M}|$ and $(a_i)_{i< \\alpha }, (b_i)_{i< \\alpha } \\in \\mathfrak {M}$ , if $tp(a_X) = tp(b_X)$ for every $X \\subseteq _{fin} \\alpha $ , then $tp((a_i)_{i< \\alpha }) = tp((b_i)_{i< \\alpha })$ .",
"Definition 4.6 ([12] and [14]) Let $(\\mathbf {K}, \\preccurlyeq )$ be an $\\mathrm {AEC}$ .",
"We say that $(\\mathbf {K}, \\preccurlyeq )$ has finite character if whenever $A \\leqslant B$ and for every $X \\subseteq _{fin} A$ there exists $\\preccurlyeq $ -embedding $f_X: A \\rightarrow B$ such that $f \\upharpoonright X = id_X$ , then $A \\preccurlyeq B$ .",
"Definition 4.7 ([12]) Let $(\\mathbf {K}, \\preccurlyeq )$ be an $\\mathrm {AEC}$ .",
"We say that $(\\mathbf {K}, \\preccurlyeq )$ is finitary if the following are satisfied: $\\mathrm {LS}(\\mathbf {K}, \\preccurlyeq ) = \\omega $ ; $(\\mathbf {K}, \\preccurlyeq )$ has arbitrarily large models; $(\\mathbf {K}, \\preccurlyeq )$ has the amalgamation property; $(\\mathbf {K}, \\preccurlyeq )$ has the joint embedding property; $(\\mathbf {K}, \\preccurlyeq )$ has finite character.",
"Definition 4.8 Let $(\\mathbf {K}, \\preccurlyeq )$ be an $\\mathrm {AEC}$ with $\\mathrm {AP}$ , $\\mathrm {JEP}$ and $\\mathrm {ALM}$ .",
"For $\\mathrm {LS}(\\mathbf {K} ,\\preccurlyeq ~) \\leqslant \\kappa \\leqslant \\lambda $ , we say that $(\\mathbf {K}, \\preccurlyeq )$ is $(\\kappa , \\lambda )$ -tame if for every $B \\in \\mathbf {K}$ of power $\\lambda $ and $a, b \\in \\mathfrak {M}^{< \\omega }$ , if $tp(a/B) \\ne tp(b/B)$ , then there is $A \\preccurlyeq B$ of power $\\kappa $ such that $tp(a/A) \\ne tp(b/A)$ .",
"We say that $(\\mathbf {K}, \\preccurlyeq )$ is tame if it is $(\\mathrm {LS}(\\mathbf {K}, \\preccurlyeq ), \\lambda )$ -tame for every $\\lambda \\geqslant \\mathrm {LS}(\\mathbf {K}, \\preccurlyeq )$ .",
"As usual, we say that $(\\mathbf {K}, \\preccurlyeq )$ is uncountably categorical if for every uncountable cardinal $\\kappa $ there exists only one model of power $\\kappa $ , up to isomorphism.",
"In later sections we will use the following classical result on abstract elementary classes.",
"Theorem 4.9 (see e.g.",
"[2]) Let $(\\mathbf {K}, \\preccurlyeq )$ be an $\\mathrm {AEC}$ with $\\mathrm {AP}$ , $\\mathrm {JEP}$ and $\\mathrm {ALM}$ .",
"If $(\\mathbf {K}, \\preccurlyeq )$ is uncountably categorical, then $(\\mathbf {K}, \\preccurlyeq )$ is stableThe notion of stability in this context is the exact analogous of the notion of stability in the classical context of first-order logic, where we replace the notion of type with the notion of Galois type.",
"For an explicit definition see e.g.",
"[2].",
"in every infinite cardinality $\\lambda \\geqslant \\mathrm {LS}(\\mathbf {K}, \\preccurlyeq )$ .",
"We will also use the following results connecting finitary abstract elementary classes with infinitary logic.",
"Given $\\theta \\in L_{\\infty , \\omega }$ , we let $Mod(\\theta ) = \\lbrace A : A \\models \\theta \\rbrace $ .",
"Theorem 4.10 (Kueker [14]) Let $(\\mathbf {K}, \\preccurlyeq )$ be a finitary $\\mathrm {AEC}$ with countable vocabulary.",
"If $\\mathbf {K}$ contains at most $\\lambda $ models of cardinality $\\lambda $ for some infinite $\\lambda $ , then $\\mathbf {K} = Mod(\\theta )$ for some $\\theta \\in L_{\\infty , \\omega }$ .",
"If in addition $\\mathbf {K}$ contains at most $\\lambda $ models of cardinality $< \\lambda $ , then we can find $\\theta \\in L_{\\lambda ^+, \\omega }$ .",
"Definition 4.11 Let $(\\mathbf {K}, \\preccurlyeq )$ be a finitary $\\mathrm {AEC}$ with monster model $\\mathfrak {M}$ .",
"Let also $a \\in \\mathfrak {M}^{< \\omega }$ and $A \\preccurlyeq \\mathfrak {M}$ .",
"Then $ tp_{\\omega _{1}, \\omega }(a/A) = \\lbrace \\phi (x, b) : \\phi (x, y) \\in L_{\\omega _1, \\omega }, \\; b \\in A^{< \\omega } \\text{ and } \\mathfrak {M} \\models \\phi (a, b) \\rbrace .$ Theorem 4.12 (Kueker [14]) Let $(\\mathbf {K}, \\preccurlyeq )$ be a finitary and tame $\\mathrm {AEC}$ with countable vocabulary.",
"Assume also that $(\\mathbf {K}, \\preccurlyeq )$ is $\\omega $ -stable.",
"Then for every $A \\preccurlyeq \\mathfrak {M}$ we have $tp_{\\omega _{1}, \\omega }(a/A) = tp_{\\omega _{1}, \\omega }(b/A) \\text{ iff } tp(a/A) = tp(b/A)$ ."
],
[
"Triangle-Free Right-Angled Coxeter Groups",
"From now till the end of the paper we denote by $\\mathbf {K}$ the class of right-angled Coxeter groups, and by $\\preccurlyeq $ the parabolic subgroup relation on $\\mathbf {K}$ (cfr.",
"Definition REF ), i.e.",
"$A \\preccurlyeq B$ if and only if there exists a Coxeter basis $S$ for $B$ such that $A \\cap S$ is a Coxeter basis for $A$ .",
"Also, we denote by $\\leqslant $ both the subgroup and the induced subgraph relation.",
"Finally, we simply talk of bases instead of Coxeter bases.",
"The next theorem shows that $(\\mathbf {K}, \\preccurlyeq )$ does not give rise to an abstract elementary class.",
"In the rest of the paper we will see that restricting to particular classes of strongly rigid right-angled Coxeter groups we do get abstract elementary classes, and actually finitary ones (and in some cases also tame).",
"Theorem 5.1 The Smoothness Axiom fails for $(\\mathbf {K}, \\preccurlyeq )$ .",
"Let $(B, S)$ be the Coxeter system with $S = \\lbrace a_i : i < \\omega \\rbrace \\cup \\lbrace b_i : i < \\omega _1 \\rbrace $ such that $\\lbrace a_i : i < \\omega \\rbrace $ is an independent set, $\\lbrace b_i : i < \\omega _1 \\rbrace $ is a clique, and $a_i$ commutes with $b_j$ iff $j < i$ , for every $i < \\omega $ .",
"For $n < \\omega $ , let $c_n = a_0 \\cdots a_n$ , $e_n = c_nb_nc_n^{-1}$ and $A_n = \\langle e_i : i < n \\rangle _B$ .",
"Notice that for every $i \\leqslant j < \\omega $ we have $c_jb_ic_j^{-1} = c_ib_ic_i^{-1}$ .",
"It follows that for every $m < n < \\omega $ , we have $A_m \\preccurlyeq A_n \\preccurlyeq B$ , as witnessed by the bases $ \\lbrace e_i : i < m \\rbrace \\subseteq \\lbrace e_i : i < n \\rbrace \\subseteq c_nSc_n^{-1}$ .",
"We claim that $\\bigcup _{n < \\omega } A_n = A \\lnot \\preccurlyeq B$ .",
"Suppose not, and let $S^*$ be a basis of $A$ that extends to a basis $S^{\\prime }$ of $B$ .",
"Let $\\alpha \\in Aut(B)$ be such that $\\alpha (S^{\\prime }) = S$ .",
"Then $\\alpha (S^*) \\subseteq \\lbrace b_i : i < \\omega _1\\rbrace $ , and so there exists $x \\in S-\\alpha (S^*)$ such that $x$ commutes with every element of $\\alpha (S^*)$ .",
"Let $y = \\alpha ^{-1}(x)$ , then $y$ commutes with every element of $A$ .",
"Let $n < \\omega $ be such that if $b_i$ or $a_i$ is in the $S$ -support of $y$ , then either $i \\geqslant \\omega $ or $i < n$ .",
"Also, let $z = c_n^{-1}yc_n$ .",
"Now, $y$ commutes with every element of $A$ , and so in particular it commutes with $e_n$ .",
"Thus, $z = c_n^{-1}yc_n$ commutes with $c_n^{-1}e_nc_n=b_n$ .",
"Now, if for some $i \\geqslant n$ , $b_i$ is in the $S$ -support of $z$ , then also $a_n$ is there and so $z$ does not commutes with $b_n$ (cfr.",
"Lemma REF ).",
"Similarly, for every $i < \\omega $ , $a_i$ is not in the $S$ -support of $z$ .",
"Thus, $z \\in \\langle b_i : i < n \\rangle _B$ and so $c_nzc_n^{-1} = y \\in \\langle c_nb_ic_n^{-1} : i < n \\rangle _B = A_n$ , which is a contradiction, since $y = \\alpha ^{-1}(x)$ , for $x \\in S-\\alpha (S^*)$ .",
"Theorem 5.2 Let $\\mathbf {K}^{\\prime }_{*}$ be a class of graphs such that $(\\mathbf {K}^{\\prime }_{*}, \\leqslant )$ is closed under limits and every $B \\in \\mathbf {K}_{*} = \\left\\lbrace A \\in \\mathbf {K} : \\Gamma _A \\in \\mathbf {K}^{\\prime }_{*} \\right\\rbrace $ is strongly rigid.",
"Then $(\\mathbf {K}_{*}, \\preccurlyeq )$ satisfies conditions (1), (2), (3), (4.1), (4.2) and (5) of Definition REF .",
"Furthermore, $\\mathrm {LS}(\\mathbf {K}_{*}, \\preccurlyeq ) = \\mathrm {LS}(\\mathbf {K}^{\\prime }_{*}, \\leqslant )$ , and if $(\\mathbf {K}^{\\prime }_{*}, \\leqslant )$ has $\\mathrm {AP}$ , $\\mathrm {JEP}$ and $\\mathrm {ALM}$ , then $(\\mathbf {K}_{*}, \\preccurlyeq )$ does.",
"The furthermore part is immediate.",
"For amalgamation, let $A, B, C \\in \\mathbf {K}_{*}$ be such that $C \\preccurlyeq A, B$ and $A \\cap B = C$ (without loss of generality).",
"Then there exists basis $S^{\\prime }$ for $A$ and $T^{\\prime }$ for $B$ such that $S = S^{\\prime }\\cap A$ and $T = T^{\\prime } \\cap B$ are bases for $C$ .",
"Thus, there exists $g \\in C$ such that $gTs^{-1} = S$ , and so $gT^{\\prime }s^{-1} = S^{\\prime \\prime }$ is a basis for $B$ such that $S^{\\prime } \\cap S^{\\prime \\prime } = S$ .",
"Hence, any amalgam for $(S,E) \\leqslant (S^{\\prime }, E), (S^{\\prime \\prime }, E)$ is an amalgam for $C \\preccurlyeq A, B$ .",
"Items (1) and (2) of Definition REF are clear.",
"We prove (3).",
"Let $A \\preccurlyeq B \\preccurlyeq C$ .",
"Then there exists a basis $S^{\\prime }$ for $B$ such that $S = S^{\\prime } \\cap A$ is a basis for $A$ , and a basis $T^{\\prime \\prime }$ for $C$ such that $T^{\\prime } = T^{\\prime \\prime } \\cap B$ is a basis for $B$ .",
"Thus, because of strong rigidity, there exists $g \\in B$ such that $S^{\\prime } = gT^{\\prime }g^{-1}$ , and so $S^{\\prime \\prime } = gT^{\\prime \\prime }g^{-1}$ is a basis for $C$ containing $S$ , i.e.",
"$A \\preccurlyeq C$ .",
"We prove (4.1) and (4.2).",
"Let $({A}_i)_{i < \\delta }$ be an increasing continuous $\\preccurlyeq $ -chain.",
"Using strong rigidity, without loss of generality we can assume that $(\\Gamma _{A_i} = (S_i, E))_{i < \\alpha }$ is an increasing continuous chain of graphs under the induced subgraph relation.",
"Using the Universality Property for Coxeter groups (see e.g.",
"[5]) it is immediate to see that $\\bigcup _{i < \\delta } {A}_i = A$ is the Coxeter group of type $\\bigcup _{i < \\alpha }\\Gamma _{A_i}$ , and so $A \\in \\mathbf {K}$ .",
"This establishes (4.1) and (4.2) at once.",
"We prove (5).",
"Let ${A} \\preccurlyeq {C}$ , $B \\preccurlyeq {C}$ and ${A} \\leqslant {B}$ .",
"Let $S^{\\prime \\prime }$ be a witness for ${A} \\preccurlyeq {C}$ and $S = S^{\\prime \\prime } \\cap A$ .",
"Let also $T^{\\prime \\prime }$ be a witness for ${B} \\preccurlyeq {C}$ and $T^{\\prime } = T^{\\prime \\prime } \\cap B$ .",
"Now, $S^{\\prime \\prime }$ and $T^{\\prime \\prime }$ are two bases for $C$ and so we can find $g \\in C$ such that $S^{\\prime \\prime } = gT^{\\prime \\prime }g^{-1}$ , i.e.",
"for every $s \\in S^{\\prime \\prime }$ there exists $t_s \\in T^{\\prime \\prime }$ such that $s = gt_sg^{-1}$ .",
"Let $a_1 \\cdots a_k$ be a $T^{\\prime \\prime }$ -normal form for $g$ .",
"Notice that $S \\subseteq A \\subseteq B$ and $S \\subseteq S^{\\prime \\prime }$ , and so for every $s \\in S$ we have $s = gt_sg^{-1} \\in B$ .",
"Thus $sp(gt_sg^{-1}) \\subseteq T^{\\prime },$ where the support is taken in the basis $T^{\\prime \\prime }$ .",
"Let $a_{q_1} \\cdots a_{q_n}$ be the subword of $a_1 \\cdots a_k$ obtained by deleting all the occurrences of letters in $T^{\\prime \\prime } - T^{\\prime }$ .",
"Then because of (REF ) and Lemma REF we have that $a_{q_1} \\cdots a_{q_n} = h \\in B$ is such that $s = gt_sg^{-1} = ht_sh^{-1},$ for every $s \\in S$ .",
"Thus, $hT^{\\prime }h^{-1} = S^{\\prime }$ is a basis for $B$ such that $S \\subseteq S^{\\prime }$ , and so $A \\preccurlyeq B$ .",
"Lemma 5.3 Let $B$ be a strongly rigid right-angled Coxeter group, and $T_0$ and $T_1$ bases for $B$ .",
"If $T_0 \\cap T_1$ contains $P_4 = s_0Es_1Es_2Es_3$ , $s_0$ is not adjacent to $s_2$ , $s_1$ is not adjacent to $s_3$ and there is no $t \\in T_1$ such that $s_0EtEs_1$ , then $T_0 = T_1$ .",
"Let $T_0$ , $T_1$ and $P_4 = s_0Es_1Es_2Es_3$ be as in the statement of the theorem.",
"Then there exists $g \\in B$ such that $T_1 = gT_0g^{-1}$ .",
"Let $s \\in P_4$ , then $g s g^{-1} = s$ , because otherwise we would have $s \\ne g s g^{-1}$ both in $T_1$ , contradicting the fact that $T_1$ is a basis for $B$ (cfr.",
"[4]).",
"Suppose now that there exists $t \\in sp(g) - \\lbrace s_0, s_1 \\rbrace $ , where the support is taken in the basis $T_1$ .",
"Then $t$ commutes with $s_0$ because otherwise by Theorem REF we have $s_0 \\ne gs_0g^{-1}$ .",
"Similarly, $t$ commutes with $s_1$ because otherwise $s_1 \\ne gs_1g^{-1}$ .",
"Thus, $s_0EtEs_1$ , which is a contradiction.",
"Hence, $sp(g) \\subseteq \\lbrace s_0, s_1 \\rbrace $ .",
"On the other hand, $s_0 \\notin sp(g)$ and $s_1 \\notin sp(g)$ , because otherwise $s_2 \\ne gs_2g^{-1}$ or $s_3 \\ne gs_3g^{-1}$ .",
"It follows that $g = 1$ , i.e.",
"$T_1 = T_0$ .",
"Theorem 5.4 Let $\\mathbf {K}_{*}$ be a class of strongly rigid right-angled Coxeter groups such that for every $A \\in \\mathbf {K}_{*}$ we have that $\\Gamma _A$ is triangle-free.",
"Suppose further that whenever $A \\preccurlyeq B \\in \\mathbf {K}_{*}$ and $T$ is a basis for $B$ such that $S = T \\cap A$ is a basis for $A$ , then the basis $S$ contains a copy of $P_4 = s_0Es_1Es_2Es_3$ such that $s_0$ is not adjacent to $s_2$ and $s_1$ is not adjacent to $s_3$ .",
"Then $(\\mathbf {K}_{*}, \\preccurlyeq )$ satisfies the Smoothness Axiom and it has finite character.",
"We show that $(\\mathbf {K}_{*}, \\preccurlyeq )$ is smooth.",
"Let $({A}_i)_{i < \\alpha }$ be an increasing continuous $\\preccurlyeq $ -chain such that each ${A}_i \\preccurlyeq {B}$ .",
"Using strong rigidity, without loss of generality we can assume that $(\\Gamma _{A_i} = (S_i, E))_{i < \\alpha }$ is an increasing continuous chain of graphs under the induced subgraph relation, and that there are $(T_i)_{i < \\alpha }$ bases for $B$ such that $T_i \\cap A_i = S_i$ , for every $i < \\alpha $ .",
"Let $i < \\alpha $ , then using the assumption of the theorem for $T_i$ and $S_0$ we have that $T_0 \\cap T_i$ contains $P_4 = s_0Es_1Es_2Es_3$ , $s_0$ is not adjacent to $s_1$ , $s_1$ is not adjacent to $s_3$ and there is no $t \\in T_i$ such that $s_0EtEs_1$ .",
"Thus, by Lemma REF , we have that $T_i = T_0$ .",
"Hence, $\\bigcup _{i < \\alpha } S_i \\subseteq T_0$ , witnessing that $\\bigcup _{i< \\alpha } A_i \\preccurlyeq B$ .",
"We show that $(\\mathbf {K}_{*}, \\preccurlyeq )$ has finite character.",
"Suppose that $A \\leqslant B$ and for every $X \\subseteq _{fin} A$ there exists $\\preccurlyeq $ -embedding $f_X: A \\rightarrow B$ such that $f \\upharpoonright X = id_X$ .",
"Let $S$ be a basis for $A$ .",
"For every $X \\subseteq A$ we have $A \\cong f_X(A)$ , and so $f_X(S)$ is a basis for $f_X(A)$ .",
"It follows that: $\\text{\\qquad \\mathrm {(\\star )}$\\forall X \\subseteq _{fin} S$, $\\exists \\, T_X$ basis of $B$ such that $T_X$ extends $f_X(S)$ and $X \\subseteq T_X$},$ this is because $f_X(A) \\preccurlyeq B$ , of course.",
"Fix $Y \\subseteq _{fin} S$ , then $f_Y(A) \\preccurlyeq B$ , and so using the assumption of the theorem for $T_Y$ and $f_Y(S)$ we get $P^{\\prime }_4 = s^{\\prime }_0Es^{\\prime }_1Es^{\\prime }_2Es^{\\prime }_3$ in $f_Y(S)$ , such that $s^{\\prime }_0$ is not adjacent to $s^{\\prime }_2$ and $s^{\\prime }_1$ is not adjacent to $s^{\\prime }_3$ .",
"Let now $f^{-1}_Y(P^{\\prime }_4) = P_4 = s_0Es_1Es_2Es_3$ .",
"Then, noticing that $P_4 \\subseteq S$ , and recalling $(\\star )$ and that $\\Gamma _B$ is triangle free we have that $T_{P_4}$ is a basis of $B$ such that $s_0$ is not adjacent to $s_1$ , $s_1$ is not adjacent to $s_3$ and there is no $t \\in T_{P_4}$ such that $s_0EtEs_1$ .",
"Thus, by Lemma REF , for every $P_4 \\subseteq X \\subseteq _{fin} S$ we have that $T_X = T_{P_4}$ .",
"Hence, for every $X \\subseteq _{fin} S$ we have $X \\subseteq T_{P_4}$ , and so $S \\subseteq T_{P_4}$ , i.e.",
"$A \\preccurlyeq B$ .",
"Let $\\mathbf {K}^{\\prime }_{*}$ be a class of graphs such that $(\\mathbf {K}^{\\prime }_{*}, \\leqslant )$ is an $\\mathrm {AEC}$ with $\\mathrm {AP}$ , $\\mathrm {JEP}$ and $\\mathrm {ALM}$ .",
"Suppose that $\\mathbf {K}_{*} = \\left\\lbrace A \\in \\mathbf {K} \\!",
": \\Gamma _A \\!\\in \\mathbf {K}^{\\prime }_{*} \\right\\rbrace $ is a class of strongly rigid right-angled Coxeter groups, and that $(\\mathbf {K}_{*}, \\preccurlyeq )$ is also an $\\mathrm {AEC}$ (and thus, by Theorem REF , it has $\\mathrm {AP}$ , $\\mathrm {JEP}$ and $\\mathrm {ALM}$ ).",
"Notice that under these conditions, modifying a little the construction of $\\mathfrak {M}(\\mathbf {K}_{*}, \\preccurlyeq )$ we can assume that $\\Gamma _{\\mathfrak {M}(\\mathbf {K}_{*}, \\preccurlyeq )} = \\mathfrak {M}(\\mathbf {K}^{\\prime }_{*}, \\leqslant )$ .",
"In the following theorem we will use this assumption crucially.",
"Theorem 5.5 Let $\\mathbf {K}^{\\prime }_{*}$ be a class of graphs such that $(\\mathbf {K}^{\\prime }_{*}, \\leqslant )$ is an $\\mathrm {AEC}$ with $\\mathrm {AP}$ , $\\mathrm {JEP}$ and $\\mathrm {ALM}$ .",
"Suppose that $\\mathbf {K}_{*} = \\left\\lbrace A \\in \\mathbf {K} \\!",
": \\Gamma _A \\!\\in \\mathbf {K}^{\\prime }_{*} \\right\\rbrace $ is a class of strongly rigid right-angled Coxeter groups, and that $(\\mathbf {K}_{*}, \\preccurlyeq )$ is also an $\\mathrm {AEC}$ (and thus, by Theorem REF , it has $\\mathrm {AP}$ , $\\mathrm {JEP}$ and $\\mathrm {ALM}$ ) with $\\mathrm {LS}(\\mathbf {K}_{*}, \\preccurlyeq ) = \\omega $ .",
"Suppose further that for every $A \\in \\mathbf {K}$ , $Aut(\\mathfrak {M}/A) \\leqslant Aut(\\Gamma _{\\mathfrak {M}})$ .",
"Then if $(\\mathbf {K}^{\\prime }_{*}, \\leqslant )$ is tame, so is $(\\mathbf {K}_{*}, \\preccurlyeq )$ .",
"We show the tameness of $(\\mathbf {K}_{*}, \\preccurlyeq )$ for elements, the argument generalizes to tuples.",
"Let $B \\in \\mathbf {K}_{*}$ and $a, b$ elements in $\\mathfrak {M}(\\mathbf {K}_{*}, \\preccurlyeq )$ , and suppose that $tp(a/B) \\ne tp(b/B)$ .",
"Notice that for every $\\alpha \\in Aut(\\Gamma _{\\mathfrak {M}})$ the following are equivalent: $\\alpha (a) = b$ ; $\\alpha $ restricted to $sp(a)$ is a bijection from $sp(a)$ into $sp(b)$ such that if $a_1 \\cdots a_k$ is a normal form for $a$ , then $\\alpha (a_1) \\cdots \\alpha (a_k)$ is a normal form form $b$ ; $\\alpha $ restricted to $sp(a)$ is a bijection from $sp(a)$ into $sp(b)$ , and there exists a normal form $a_1 \\cdots a_k$ for $a$ , such that $\\alpha (a_1) \\cdots \\alpha (a_k)$ is a normal form form $b$ .",
"Now, if $|sp(a)| \\ne |sp(b)|$ then for any countable $A \\preccurlyeq B$ we have that $tp(a/A) \\ne tp(b/A)$ , since by assumption $Aut(\\mathfrak {M}/A) \\leqslant Aut(\\Gamma _{\\mathfrak {M}})$ .",
"Suppose then that $|sp(a)| = |sp(b)|$ , fix a normal form $a_1 \\cdots a_k$ for $a$ and let $\\lbrace b_1^j \\cdots b_k^j : j < n \\rbrace $ be the set of normal forms for $b$ .",
"For every $j < n$ we must have that $tp((a_i)_{0 < i \\leqslant k}/\\Gamma _B) \\ne tp((b_i^j))_{0 < i \\leqslant k}/\\Gamma _B),$ where types are in the sense of $(\\mathbf {K}^{\\prime }_{*}, \\leqslant )$ .",
"In fact, otherwise there is $\\alpha \\in Aut(\\mathfrak {M}(\\mathbf {K}^{\\prime }_{*}, \\leqslant )/ \\Gamma _B) = Aut(\\Gamma _{\\mathfrak {M}(\\mathbf {K}_{*}, \\preccurlyeq )}/B)$ such that $\\alpha (sp(a)) = sp(b)$ and $\\alpha (a_1) \\cdots \\alpha (a_k)$ is a normal form form $b$ , and so $tp(a/B) = tp(b/B)$ .",
"Thus, by the tameness of $(\\mathbf {K}^{\\prime }_{*}, \\leqslant )$ , for every $j < n$ there is countable $\\Gamma _{A_j} \\leqslant \\Gamma _B$ such that $tp((a_i)_{0 < i \\leqslant k}/\\Gamma _{A_j}) \\ne tp((b_i^j))_{0 < i \\leqslant k}/\\Gamma _{A_j}).$ Let $A \\preccurlyeq B$ be such that $\\bigcup _{j < n} A_j \\subseteq A$ .",
"Then $tp(a/A) \\ne tp(b/A)$ .",
"In fact, otherwise there exists $\\alpha \\in Aut(\\Gamma _{\\mathfrak {M}}/A)$ such that $\\alpha (sp(a)) = sp(b)$ and $\\alpha (a_1) \\cdots \\alpha (a_k)$ is a normal form form $b$ , and so there exists $j<n$ and $\\alpha \\in Aut(\\mathfrak {M}(\\mathbf {K}^{\\prime }_{*}, \\leqslant )/ \\Gamma _{A_j})$ mapping $(a_i)_{0 < i \\leqslant k}$ to $(b_i^j)_{0 < i \\leqslant k}$ , which is a contradiction.",
"Let $\\mathbf {K}^{\\prime }_0$ be the class of graphs satisfying the following requirements: $\\Gamma $ has the star property; $\\Gamma $ is star-connected; $\\Gamma $ is triangle-free; $\\Gamma $ contains $C_4$ (the cycle of length 4) as an (induced) subgraph.",
"Let then $\\mathbf {K}_0 = \\left\\lbrace A \\in \\mathbf {K} : \\Gamma _A \\in \\mathbf {K}^{\\prime }_0 \\right\\rbrace $ .",
"Notice that because of Corollary REF , every $A \\in \\mathbf {K}_0$ is strongly rigid.",
"We ask that $\\Gamma $ contains $C_4$ instead of simply $P_4$ because $C_4$ has the star property, while $P_4$ does not.",
"The fact that $C_4$ embeds as an induced subgraph in every structure in $\\mathbf {K}^{\\prime }_0$ will be useful in proving joint embedding from amalgamation.",
"We need a lemma before proving the main theorem of this section.",
"Lemma 5.6 Let $\\Gamma $ be triangle-free and such that it contains $C_4$ as an induced subgraph.",
"By induction on $i < \\omega $ , define $\\Gamma _i$ such that: $\\Gamma _0 = \\Gamma $ ; $\\Gamma _{i+1}$ is the extension of $\\Gamma _i$ following the condition: for every $a \\ne b \\in \\Gamma _i$ if $a$ is not adjacent to $b$ , then add $c$ such that $N(c) = \\lbrace a, b \\rbrace $ .",
"Then $\\Gamma \\leqslant \\bigcup _{i < \\omega } \\Gamma _i = \\Gamma ^* \\in \\mathbf {K}^{\\prime }_0$ .",
"Obviously, $C_4 \\leqslant \\Gamma \\leqslant \\Gamma ^*$ and $\\Gamma ^*$ is triangle-free.",
"Regarding the star-property, let $a \\ne b \\in \\Gamma ^*$ , we show that $st(a) \\lnot \\subseteq st(b)$ .",
"Assume $a, b \\in \\Gamma _i$ .",
"Then $\\Gamma _{i+1} - \\Gamma _i$ contains an element $x$ which is not adjacent to $a$ (since $C_4$ contains two adjacent vertices different from $a$ ).",
"Now, $\\Gamma _{i+2} - \\Gamma _{i+1}$ contains an element $c$ which is adjacent to $a$ and $x$ , but not to $b$ .",
"Hence, $c \\in st(a) - st(b)$ , as wanted.",
"Regarding star-connectedness, let $v \\in \\Gamma ^*$ and $a \\ne b \\in \\Gamma ^* - st(v)$ .",
"Assume that $v,a,b \\in \\Gamma _i$ .",
"If $a$ and $b$ are adjacent in $\\Gamma ^*$ , then they are connected in $\\Gamma ^* - st(v)$ (since $a \\ne b \\in \\Gamma ^* - st(v)$ ).",
"If $a$ and $b$ are not adjacent in $\\Gamma ^*$ , then they are not adjacent in $\\Gamma _i$ either, and so at stage $\\Gamma _{i+1}$ we have added $c$ such that $N(c) = \\lbrace a, b \\rbrace $ , witnessing the connectedness of $a$ and $b$ in $\\Gamma ^* - st(v)$ .",
"Theorem 5.7 $(\\mathbf {K}_{0}, \\preccurlyeq )$ is a finitary $\\mathrm {AEC}$ .",
"As already noticed, because of Corollary REF , every $A \\in \\mathbf {K}_0$ is strongly rigid.",
"Furthermore, obviously $(\\mathbf {K}^{\\prime }_0, \\leqslant )$ is closed under limits and $\\mathrm {LS}(\\mathbf {K}^{\\prime }_{0}, \\leqslant ) = \\omega $ .",
"Also, every $A \\in \\mathbf {K}_0$ is such that $\\Gamma _A$ is triangle-free and contains $C_4$ as an induced subgraph, and so we can always a $P_4$ as in Theorem REF .",
"Thus, by Theorems REF and REF , in order to conclude it suffices to show that $(\\mathbf {K}^{\\prime }_{0}, \\leqslant )$ has joint embedding and amalgamation.",
"Now, $C_4 \\in \\mathbf {K}^{\\prime }_{0}$ and $C_4$ embeds as an induced subgraph in every $A \\in \\mathbf {K}^{\\prime }_0$ , thus it suffices to prove amalgamation.",
"Let then $A, B, C \\in \\mathbf {K}^{\\prime }_{0}$ be such that $C \\leqslant A, B$ and $A \\cap B = C$ (without loss of generality), and consider $D = (A \\cup B)^*$ .",
"Then is it easy to see that $D$ is an amalgam of $A$ and $B$ over $C$ .",
"Theorem 5.8 $(\\mathbf {K}_{0}, \\preccurlyeq )$ is not homogeneous.",
"$(\\mathbf {K}_0, \\preccurlyeq )$ has the independence property, and thus it is unstable.",
"We prove (a).",
"Let $(t_i)_{i < \\omega }$ and $(a_i)_{i < \\omega }$ in $\\Gamma _{\\mathfrak {M}}$ , for $\\mathfrak {M}$ the monster model of $(\\mathbf {K}_{0}, \\preccurlyeq )$ , be such that the following conditions are met: $(t_i)_{i < \\omega }$ is an independent set; $(a_i)_{i < \\omega }$ is an independent set; for every $i < \\omega $ , $a_i$ is adjacent to $t_j$ iff $j \\leqslant i$ .",
"Such sequences $(t_i)_{i < \\omega }$ and $(a_i)_{i < \\omega }$ can be found in $\\Gamma _{\\mathfrak {M}}$ , e.g.",
"using Lemma REF .",
"For $i < \\omega $ , let $c_i = a_0 \\cdots a_{i-1}t_ia_{i-1} \\cdots a_{0}.$ Then for every $X \\subseteq _{fin} \\omega $ we have $tp(t_X/\\emptyset ) = tp(c_X/\\emptyset )$ , as witnesses by the inner automorphism determined by $a_0 \\cdots a_{k-1}$ , for $k = max\\lbrace i<\\omega : i \\in X\\rbrace $ .",
"On the other hand, $tp((t_i)_{i < \\omega }/\\emptyset ) \\ne tp((c_i)_{i < \\omega }/\\emptyset )$ because there is no automorphism of $\\mathfrak {M}$ such that $t_i \\mapsto c_i$ for every $i < \\omega $ , as this would contradict the strong rigidity of $\\mathfrak {M}$ , in fact no inner automorphism $gxg^{-1}$ (for $g \\in \\mathfrak {M}$ ) could serve as witness for this candidate automorphism, since $sp(g)$ is finite.",
"We prove (b).",
"Let $P = \\lbrace p \\in S_2(\\emptyset ) : \\forall a, b \\in \\mathfrak {M}, \\text{ if } (a,b) \\models p \\text{ then } ab = ba \\rbrace ,$ $\\alpha < |{\\mathfrak {M}}|,$ and $(t_i)_{i < \\alpha }$ and $(a_X)_{X \\subseteq \\alpha }$ in $\\Gamma _{\\mathfrak {M}}$ be such that the following conditions are met: $(t_i)_{i < \\alpha }$ is an independent set; $(a_X)_{X \\subseteq \\alpha }$ is an independent set; for every $X \\subseteq \\alpha $ , $a_X$ is adjacent to $t_i$ iff $i \\in X$ .",
"Such sequences $(t_i)_{i < \\alpha }$ and $(a_X)_{X \\subseteq \\alpha }$ can be found in $\\Gamma _{\\mathfrak {M}}$ , e.g.",
"using Lemma REF .",
"Evidently, $tp(a_Xt_i/\\emptyset ) \\in P$ if and only if $i \\in X$ .",
"Remark 5.9 The first configuration used in the proof of Theorem REF will play a crucial role also in the proof of Theorem REF (where a similar non-homogeneity result is proved).",
"It is interesting to notice that the existence of this configuration (on tuples of elements), also known as the half-graph, can always be find in a definable way in the monster model of an unstable theory.",
"Thus, we here have an analogy between non-homogeneity in AEC's and unstability in first-order theories.",
"Given a graph $\\Gamma = (V, E)$ we define the barycentric subdivision of $\\Gamma $ , denoted $\\hat{\\Gamma }$ , to be the graph whose node set is the disjoint union of $V$ and $\\lbrace c_{a,b} : a,b \\in \\Gamma , aEb \\rbrace $ , and so that $N(c_{a,b}) = \\lbrace a,b \\rbrace $ and, for $a \\in V$ , $N(a) = \\lbrace c_{a,b} : b \\in \\Gamma , aEb \\rbrace $ .",
"Let $\\mathbf {K}^{\\prime }_1$ be the class of barycentric subdivisions of clique with at least four elements, and $\\mathbf {K}_1 = \\left\\lbrace A \\in \\mathbf {K} : \\Gamma (A) \\in \\mathbf {K}^{\\prime }_1 \\right\\rbrace $ .",
"Theorem 5.10 $(\\mathbf {K}_{1}, \\preccurlyeq )$ is a finitary $\\mathrm {AEC}$ .",
"Obviously, $(\\mathbf {K}^{\\prime }_{1}, \\leqslant )$ is closed under limits, it has $\\mathrm {AP}$ , $\\mathrm {JEP}$ and $\\mathrm {ALM}$ , and $\\mathrm {LS}(\\mathbf {K}^{\\prime }_{1}, \\preccurlyeq ) = \\omega $ .",
"Also, it is immediate to see that every $\\Gamma \\in \\mathbf {K}^{\\prime }_1$ is star-connected, it has the star property and it contains $P_4$ , and so, by Corollary REF , every $A \\in \\mathbf {K}_1$ is strongly rigid.",
"Finally, it is obvious from the definition that for any graph $\\Gamma $ the graph $\\hat{\\Gamma }$ is bipartite (and thus triangle-free).",
"Hence, by Theorems REF and REF we are done.",
"Let $\\mathbf {K}^{\\prime \\prime }_{1}$ be the class of infinite structures in $\\mathbf {K}^{\\prime }_{1}$ .",
"It is immediate to see that the class $\\mathbf {K}^{\\prime \\prime }_{1}$ is axiomatizable by the following first-order theory $T$ : there are infinitely many elements; every $x$ has either exactly two neighbours or at least three neighbours; if $x$ has exactly two neighbours $y$ and $z$ , then $y$ and $z$ have at least three neighbours; if $x$ has at least three neighbours, then each neighbour of $x$ has exactly two neighbours; if $x \\ne y$ have at least three neighbours, then there exists unique $z$ such that $xEzEy$ .",
"Proposition 5.11 $T$ is complete and it is model complete.",
"Standard.",
"Theorem 5.12 $(\\mathbf {K}_{1}, \\preccurlyeq )$ is tame.",
"Obviously, $(\\mathbf {K}^{\\prime }_{1}, \\leqslant )$ is an $\\mathrm {AEC}$ with $\\mathrm {AP}$ , $\\mathrm {JEP}$ and $\\mathrm {ALM}$ .",
"Furthermore, by Lemma REF , for every $A \\in \\mathbf {K}_1$ , $Aut(\\mathfrak {M}/A) \\leqslant Aut(\\Gamma _{\\mathfrak {M}})$ .",
"Thus, by Theorem REF , it suffices to show that $(\\mathbf {K}^{\\prime }_{1}, \\leqslant )$ is tame.",
"Clearly, it suffices to prove tameness for the class $\\mathbf {K}^{\\prime \\prime }_{1}$ of infinite structures in $\\mathbf {K}^{\\prime }_{1}$ .",
"By Proposition REF , the class $\\mathbf {K}^{\\prime \\prime }_{1}$ is axiomatizable by a complete first-order theory which is model complete.",
"Thus, $(\\mathbf {K}^{\\prime \\prime }_{1}, \\leqslant ) = (\\mathbf {K}^{\\prime \\prime }_{1}, \\preccurlyeq ^*)$ , where $\\preccurlyeq ^*$ denotes the elementary submodel relation of first-order logic, and clearly $(\\mathbf {K}^{\\prime \\prime }_{1}, \\preccurlyeq ^*)$ is tame.",
"Theorem 5.13 $(\\mathbf {K}_{1}, \\preccurlyeq )$ is uncountably categorical.",
"For uncountable $A, B \\in \\mathbf {K}_{1}$ , letting $\\Gamma _A = \\hat{\\Gamma }_0$ and $\\Gamma _B = \\hat{\\Gamma }_1$ (for $\\Gamma _0$ and $\\Gamma _1$ cliques), we have $|A| = |B|$ iff $|\\Gamma _A| = |\\Gamma _B|$ iff $|\\Gamma _0| = |\\Gamma _1|$ iff $\\Gamma _0 \\cong \\Gamma _1$ iff $\\Gamma _A \\cong \\Gamma _B$ iff $A \\cong B$ .",
"Corollary 5.14 $(\\mathbf {K}_{1}, \\preccurlyeq )$ is stable in every infinite cardinality.",
"This is a consequence of Theorems REF , REF and REF .",
"Corollary 5.15 $\\mathbf {K}_1 = Mod(\\theta )$ for some $\\theta \\in L_{\\omega _1, \\omega }$ .",
"Furthermore, for every $A \\preccurlyeq \\mathfrak {M}$ we have $tp_{\\omega _{1}, \\omega }(a/A) = tp_{\\omega _{1}, \\omega }(b/A) \\text{ iff } tp(a/A) = tp(b/A)$ .",
"This is an immediate consequence of Theorems REF , REF , REF , REF and REF together with the easy observation that $\\mathbf {K}_1$ has at most countably many countable models."
],
[
"Centered Right-Angled Coxeter Groups",
"Theorems REF and REF leave open the question of finding classes of right-angled Coxeter groups which are stable and non-homogeneous.",
"In this section we use Corollary REF to achieve this.",
"Let $C^*$ be the graph on vertex set $\\lbrace s,s^{\\prime }\\rbrace \\cup \\lbrace t_i : i < 4 \\rbrace $ , with the following edge relation: $t_0Et_2Et_3Et_1Et_0$ , $t_0EsEt_2$ , $t_1Es^{\\prime }Et_3$ and $sEs^{\\prime }$ (cf.",
"Figure ).",
"For every $B_T \\models T = Th(\\mathbb {N}, s, 0)$ (where $s$ denotes the successor function) we define a graph $\\Gamma _{B_{T}} = (C^* \\cup B_T, E)$ in the following way (without loss of generality we assume $s^n(0) = n$ in $B_T$ ): $C^*$ is an induced subgraph of $\\Gamma _{B_{T}}$ ; $t_0$ and $t_2$ are adjacent to all the even numbers in $\\mathbb {N}$ ; $t_1$ is adjacent to 2 and to all the odd numbers in $\\mathbb {N}$ ; $t_3$ is adjacent to all the odd numbers in $\\mathbb {N}$ ; $N(0) - \\mathbb {N} = \\lbrace 1 \\rbrace $ and, for every $0 < n \\in \\mathbb {N}$ , $N(n) \\cap \\mathbb {N} = \\lbrace n-1, n+1 \\rbrace $ ; for every copy $Z$ of $\\mathbb {Z}$ in $B_T$ and $b \\in Z$ , $N(b) \\cap Z = \\lbrace b-1, b+1 \\rbrace $ ; for every copy $Z$ of $\\mathbb {Z}$ in $B_T$ , there exists $0_Z \\in Z$ such that for every $\\pm n_Z = 0_Z \\pm n$ we have $N(\\pm n_Z) \\cap \\mathbb {N} = \\lbrace 0, ..., n \\rbrace $ ; for every copy $Z$ of $\\mathbb {Z}$ in $B_T$ , $0_Z$ , $-1_Z$ and $1_Z$ are adjacent to $t_3$ .",
"Let $\\mathbf {K}^{\\prime }_2$ be the class of graphs $\\Gamma ^{\\prime }$ isomorphic to one of the graphs $\\Gamma = (C^* \\cup B_T, E)$ described above, and $\\mathbf {K}_2 = \\left\\lbrace A \\in \\mathbf {K} : \\Gamma _A \\in \\mathbf {K}^{\\prime }_2 \\right\\rbrace $ .",
"Remark 6.1 The proof of the theorem below is straightforward, but the details are tiresome.",
"We include them for completeness of exposition.",
"Figure: The graph C * C^*.Theorem 6.2 $(\\mathbf {K}_{2}, \\preccurlyeq )$ is a finitary $\\mathrm {AEC}$ .",
"Notice that for every $\\Gamma = (C^* \\cup B_T, E) \\in \\mathbf {K}^{\\prime }_2$ , the structure $B_T$ can be recovered from $\\Gamma $ , and so $(\\mathbf {K}^{\\prime }_{2}, \\leqslant )$ is closed under limits, it has $\\mathrm {AP}$ , $\\mathrm {JEP}$ and $\\mathrm {ALM}$ , and $\\mathrm {LS}(\\mathbf {K}^{\\prime }_{2}, \\preccurlyeq ) = \\omega $ .",
"Thus, by Theorems REF and REF we are left to show that every $A \\in \\mathbf {K}_2$ is strongly rigid, and that the assumptions of Theorem REF are met.",
"The latter is immediate, since for every $A \\in \\mathbf {K}_{2}$ and basis $T$ of $A$ , the elements $sEs^{\\prime }Et_1Et_0 \\in C^*$ (without loss of generality $C^*$ is in $T$ ) are such that $s$ is not adjacent to $t_1$ , $s^{\\prime }$ is not adjacent to $t_0$ and there is no $t \\in T$ such that $sEtEs^{\\prime }$ .",
"To see strong rigidity we use Corollary REF .",
"Let $A \\in \\mathbf {K}_{2}$ , then the elements $s, s^{\\prime } \\in C^*$ are such $s$ is adjacent to $s^{\\prime }$ in $\\Gamma _A$ , and $st(s) \\cup st(s^{\\prime }) = C^*$ is finite and star-connected, since for every $x \\in C^*$ we have $C^* - st(x) = \\lbrace y, z \\rbrace $ , for some $y,z \\in C^*$ such that $y$ is adjacent to $z$ .",
"Furthermore, clearly for every $v \\in \\Gamma _A$ there exists $v \\ne a \\in st(s) \\cup st(s^{\\prime })$ such that $v$ is not adjacent to $a$ .",
"Thus, we are left to show that $\\Gamma _A$ is star-connected and it has the star property.",
"For ease of notation, we assume that in $\\Gamma _A$ the copies of $C^*$ and $\\mathbb {N}$ are actually $C^*$ and $\\mathbb {N}$ (we already did this for $C^*$ above).",
"Also, we denote by $Z$ , $Z^{\\prime }$ , etc.",
"the copies of $\\mathbb {Z}$ possibly present in $\\Gamma _A$ .",
"We first show that $\\Gamma _A$ has the star property.",
"Let $a \\ne b \\in \\Gamma _A$ .",
"Case 1.",
"$a,b \\in C^*$ .",
"Clear.",
"Case 2.",
"$a,b \\in \\mathbb {N}$ .",
"Without loss of generality $a < b$ .",
"If $a = 0$ and $b = 1$ , then $t_0 \\in st(a) - st(b)$ , and $t_1 \\in st(b) - st(a)$ .",
"If $a=0$ and $b=2$ , then $3 \\in st(2) - st(0)$ and $t_1 \\in st(2) - st(0)$ .",
"If $a = 0$ and $b > 2$ , then $b+1 \\in st(b) - st(0)$ and $1 \\in st(0) - st(b)$ .",
"If $a > 0$ , then $b+1 \\in st(b) - st(a)$ and $a-1 \\in st(a) - st(b)$ .",
"Case 3.",
"$a,b \\in Z$ .",
"Without loss of generality $a < b$ .",
"We have $b+1 \\in st(b) - st(a)$ and $a-1 \\in st(a) - st(b)$ .",
"Case 4.",
"$a \\in C^*$ and $b \\in \\mathbb {N}$ .",
"If $a = s$ or $a = s^{\\prime }$ , then $a$ is not adjacent to $b$ .",
"Let $a = t_i$ , for $i < 4$ .",
"If $i$ is even and $b$ is odd, then $a$ is not adjacent to $b$ .",
"If $i$ is odd and $b$ is even, then $a$ is not adjacent to $b$ .",
"If $i$ is even and $b$ is even, then $s \\in st(a) - st(b)$ and $b + 1 \\in st(b) - st(a)$ .",
"If $i$ is odd and $b$ is odd, then $s^{\\prime } \\in st(a) - st(b)$ and $b + 3 \\in st(b) - st(a)$ (in the case $b = 1$ we have $1+1 = 2Et_1$ ).",
"Case 5.",
"$a \\in C^*$ and $b \\in Z$ .",
"In this case $a$ is not adjacent to $b$ , unless $a = t_3$ and $b \\in \\lbrace 0_Z, -1_Z, 1_Z \\rbrace $ .",
"In this case we have $t_2 \\in st(a) - st(b)$ and $3_Z \\in st(b) - st(a)$ .",
"Case 6.",
"$a \\in \\mathbb {N}$ and $b \\in Z$ .",
"Let $b = \\pm n_Z$ .",
"If $a > n$ , then $a$ is not adjacent to $b$ .",
"If $0 < a \\leqslant n$ , then $n+2 \\in st(a) -st(b)$ and $a-1 \\in st(b) - st(a)$ .",
"If $a = 0 = n$ , then $n+2 \\in st(a) -st(b)$ and $t_3 \\in st(b) - st(a)$ .",
"If $a = 0 < n$ , then $n+2 \\in st(a) -st(b)$ and $1 \\in st(b) - st(a)$ .",
"Case 7.",
"$a \\in Z$ and $b \\in Z^{\\prime }$ .",
"In this case $a$ is not adjacent to $b$ .",
"We now show that $\\Gamma _A$ is star-connected.",
"Let $v \\in \\Gamma _A$ and $a \\ne b \\in \\Gamma _A - st(v)$ .",
"Case A.",
"$v \\in C^*$ .",
"If $v = s$ or $v = s^{\\prime }$ , then it is clear that $a$ is connected to $b$ .",
"Suppose then that $v = t_i$ , for $i < 4$ .",
"Case A.1.",
"$a,b \\in C^*$ .",
"Clear.",
"Case A.2.",
"$a,b \\in \\mathbb {N}$ .",
"Then either both $a$ and $b$ are even, or both $a$ and $b$ are odd.",
"In either cases we are fine.",
"Case A.3.",
"$a,b \\in Z$ .",
"If $i < 3$ , we have $(\\Gamma _A - st(v)) \\cap Z = Z$ .",
"If $i = 3$ , we have $(\\Gamma _A - st(v)) \\cap Z = Z - \\lbrace 0_Z, -1_Z, 1_Z\\rbrace $ .",
"In either cases we are fine.",
"Case A.4.",
"$a \\in C^*$ and $b \\in \\mathbb {N}$ .",
"If $a = s$ or $a = s^{\\prime }$ , then $a$ is not adjacent to $b$ .",
"Suppose then that $a \\notin \\lbrace s,s^{\\prime }\\rbrace $ .",
"If $i$ is even (for $v = t_i$ , remember) then $a = t_j$ is such that $j$ is odd and $b$ is odd, and so we are fine.",
"If $i$ is odd, then $a = t_j$ is such that $j$ is even and $b$ is even, and so we are fine.",
"Case A.5.",
"$a \\in C^*$ and $b \\in Z$ .",
"If $i$ is even, then we can find an odd number $n \\in \\Gamma _A - st(v)$ that connects what is left of $C^*$ to $nEn_ZEb$ .",
"If $i$ is odd, then we can find an even number that does the same.",
"Case A.6.",
"$a \\in \\mathbb {N}$ and $b \\in Z$ .",
"If $i$ is even, then we can find an odd number $n \\in \\Gamma _A - st(v)$ such that $aEnEn_ZEb$ .",
"If $i$ is odd, then we can find an even number that does the same.",
"Case A.7.",
"$a \\in Z$ and $b \\in Z^{\\prime }$ .",
"If $i$ is even, then we can find an odd number $n \\in \\Gamma _A - st(v)$ such that $aEn_ZEnEn_{Z^{\\prime }}Eb$ .",
"If $i$ is odd, then we can find an even number that does the same.",
"Case B.",
"$v \\in \\mathbb {N}$ .",
"Case B.1.",
"$a,b \\in C^*$ .",
"If $v = 2$ , then $(\\Gamma _A - st(v)) \\cap C^* = sEs^{\\prime }Et_3$ .",
"If $v \\ne 2$ , then $(\\Gamma _A - st(v)) \\cap C^*$ is either $sEs^{\\prime }Et_1Et_3Es^{\\prime }$ or $s^{\\prime }EsEt_0Et_2Es$ .",
"In all of these cases we are fine.",
"Case B.2.",
"$a,b \\in \\mathbb {N}$ .",
"If $v = 0$ , then $(\\Gamma _A - st(v)) \\cap \\mathbb {N} = \\lbrace 1 \\rbrace $ , and so this case is not possible, since we are assuming that $a \\ne b$ .",
"Thus, we must have that $v = n \\ne 0$ , and so $n-1 = aEb = n+1$ .",
"Case B.3.",
"$a,b \\in Z$ .",
"If $v = 0$ or $v = 1$ , then $(\\Gamma _A - st(v)) \\cap Z$ is either $\\emptyset $ or $\\lbrace 0_Z \\rbrace $ , and so this case is not possible, since we are assuming that $a \\ne b$ .",
"If $v = 2$ , then $(\\Gamma _A - st(v)) \\cap Z = \\lbrace 0_Z, -1_Z, 1_Z \\rbrace $ , but $-1_ZEt_3E0_ZEt_3E1_Z$ and $t_3 \\in \\Gamma _A - st(2)$ , and so we are fine.",
"If $v > 2$ , then $(\\Gamma _A - st(v)) \\cap Z = \\lbrace -(v_Z-1), ..., 0_Z , ..., (v_Z-1)\\rbrace $ is “long enough”, and so it is connected.",
"Case B.4.",
"$a \\in C^*$ and $b \\in \\mathbb {N}$ .",
"If $a = s$ or $a = s^{\\prime }$ , then $a$ is not adjacent to $b$ .",
"Suppose then that $a \\notin \\lbrace s,s^{\\prime }\\rbrace $ .",
"If $v$ is odd, then $b$ is even and $a \\in \\lbrace t_0, t_2\\rbrace $ , and so we are fine.",
"If $v$ is even, then $b$ is odd and $a \\in \\lbrace t_1, t_3 \\rbrace $ , and so we are fine.",
"Case B.5.",
"$a \\in C^*$ and $b \\in Z$ .",
"If $v = 0$ , then $(\\Gamma _A - st(v)) \\cap Z = \\emptyset $ , and so this case is not possible.",
"If $v = 1$ , then $(\\Gamma _A - st(v)) \\cap Z = \\lbrace 0_Z \\rbrace $ and $(\\Gamma _A - st(v)) \\cap C^* = \\lbrace t_0, s, s^{\\prime }, t_2 \\rbrace $ , but then we are fine because $0_ZE0Et_0Et_2EsEs^{\\prime }$ and $0 \\in \\Gamma _A - st(1)$ .",
"If $v = 2$ , then $(\\Gamma _A - st(v)) \\cap Z = \\lbrace 0_Z, -1_Z, 1_Z \\rbrace $ and $(\\Gamma _A - st(v)) \\cap C^* = \\lbrace t_3, s, s^{\\prime } \\rbrace $ , but $-1_ZEt_3E0_ZEt_3E1_Z$ , and so we are fine.",
"If $v > 2$ , then $(\\Gamma _A - st(v)) \\cap Z$ is connected, and so we can connect it to what is left of $C^*$ via $v-1 \\notin (\\Gamma _A - st(v))$ .",
"Case B.6.",
"$a \\in \\mathbb {N}$ and $b \\in Z$ .",
"If $v = 0$ , then $(\\Gamma _A - st(v)) \\cap Z = \\emptyset $ , and so this case is not possible.",
"If $v = 1$ , then $(\\Gamma _A - st(v)) \\cap Z = \\lbrace 0_Z \\rbrace $ and $(\\Gamma _A - st(v)) \\cap \\mathbb {N} = \\lbrace 0, 2 \\rbrace $ , and $0_ZE0E2$ .",
"If $v = 2$ , then $(\\Gamma _A - st(v)) \\cap Z = \\lbrace 0_Z, -1_Z, 1_Z \\rbrace $ and $(\\Gamma _A - st(v)) \\cap \\mathbb {N} = \\lbrace 1, 3 \\rbrace $ , and for $x \\in \\lbrace 0_Z, -1_Z, 1_Z \\rbrace $ and $y \\in \\lbrace 1, 3 \\rbrace $ we have $xEt_3Ey$ , and so we are fine because $t_3 \\in \\Gamma _A -st(2)$ .",
"If $v > 2$ , then $(\\Gamma _A - st(v)) \\cap Z$ is connected, and $v_Z-1Ev-iEv+1$ .",
"Case B.7.",
"$a \\in Z$ and $b \\in Z^{\\prime }$ .",
"If $v = 0$ , then $(\\Gamma _A - st(v)) \\cap Z = \\emptyset = (\\Gamma _A - st(v)) \\cap Z^{\\prime }$ , and so this case is not possible.",
"If $v = 1$ , then $(\\Gamma _A - st(v)) \\cap Z = \\lbrace 0_Z \\rbrace $ and $(\\Gamma _A - st(v)) \\cap Z = \\lbrace 0_{Z^{\\prime }} \\rbrace $ , and $0_ZE0E0_{Z^{\\prime }}$ , and so we are fine because $0 \\notin \\Gamma _A - st(1)$ .",
"If $v = 2$ , then $(\\Gamma _A - st(v)) \\cap Z = \\lbrace 0_Z, -1_Z, 1_Z \\rbrace $ and $(\\Gamma _A - st(v)) \\cap Z^{\\prime } = \\lbrace 0_{Z^{\\prime }}, -1_{Z^{\\prime }}, 1_{Z^{\\prime }} \\rbrace $ , and for $x \\in \\lbrace 0_Z, -1_Z, 1_Z \\rbrace $ and $y \\in \\lbrace 0_{Z^{\\prime }}, -1_{Z^{\\prime }}, 1_{Z^{\\prime }} \\rbrace $ we have $xEt_3Ey$ , and so we are fine because $t_3 \\in \\Gamma _A -st(2)$ .",
"If $v > 2$ , then $(\\Gamma _A - st(v)) \\cap Z$ and $(\\Gamma _A - st(v)) \\cap Z^{\\prime }$ are connected, and so we can connect them via $v-1 \\notin \\Gamma _A - st(v)$ .",
"Case C. $v \\in Z$ .",
"Let $v = \\pm n_Z$ .",
"Case C.1.",
"$a,b \\in C^*$ .",
"We have $(\\Gamma _A - st(v)) \\cap C^* \\subseteq C^* - \\lbrace t_3 \\rbrace $ , and so we are fine.",
"Case C.2.",
"$a,b \\in \\mathbb {N}$ .",
"We have $(\\Gamma _A - st(v)) \\cap \\mathbb {N} = \\lbrace m \\in \\mathbb {N} : n < m \\rbrace $ , and so we are fine.",
"Case C.3.",
"$a,b \\in Z$ .",
"In this case $v-1 = aEb = v+1$ .",
"Case C.4.",
"$a \\in C^*$ and $b \\in \\mathbb {N}$ .",
"We have $(\\Gamma _A - st(v)) \\cap C^* \\subseteq C^* - \\lbrace t_3 \\rbrace $ and $(\\Gamma _A - st(v)) \\cap \\mathbb {N} = \\lbrace m \\in \\mathbb {N} : n < m \\rbrace $ , and so we are fine.",
"Case C.5.",
"$a \\in C^*$ and $b \\in Z$ .",
"We have $(\\Gamma _A - st(v)) \\cap C^* \\subseteq C^* - \\lbrace t_3 \\rbrace $ and $(\\Gamma _A - st(v)) \\cap Z = \\lbrace \\pm n_Z-1, \\pm n_Z+1 \\rbrace $ .",
"Now, $\\pm n_Z-1E\\pm n_Z+1En+1Et_j$ , for some $j < 3$ , and so we are fine because $n+1 \\in \\Gamma _A - st(v)$ and $(\\Gamma _A - st(v)) \\cap C^*$ is connected.",
"Case C.6.",
"$a \\in \\mathbb {N}$ and $b \\in Z$ .",
"We have $(\\Gamma _A - st(v)) \\cap \\mathbb {N} = \\lbrace m \\in \\mathbb {N} : n < m \\rbrace $ and $(\\Gamma _A - st(v)) \\cap Z = \\lbrace \\pm n_Z-1, \\pm n_Z+1 \\rbrace $ .",
"Now, $\\pm n_Z-1E\\pm n_Z+1En+1$ , and $n+1$ is connected in $\\Gamma _A - st(v)$ to every $x \\in \\lbrace m \\in \\mathbb {N} : n < m \\rbrace $ .",
"Case C.7.",
"$a \\in Z$ and $b \\in Z^{\\prime }$ .",
"We have $(\\Gamma _A - st(v)) \\cap Z = \\lbrace \\pm n_Z-1, \\pm n_Z+1 \\rbrace $ and $(\\Gamma _A - st(v)) \\cap Z^{\\prime } = Z^{\\prime }$ , and so $\\pm n_Z-1E\\pm n_Z+1En+1En_{Z^{\\prime }}+1$ , and $n_{Z^{\\prime }}+1$ is connected in $\\Gamma _A - st(v)$ to every $x \\in Z^{\\prime }$ .",
"Theorem 6.3 $(\\mathbf {K}_{2}, \\preccurlyeq )$ is tame.",
"Obviously, $(\\mathbf {K}^{\\prime }_{2}, \\leqslant )$ is an $\\mathrm {AEC}$ with $\\mathrm {AP}$ , $\\mathrm {JEP}$ and $\\mathrm {ALM}$ .",
"Furthermore, by Lemma REF , for every $A \\in \\mathbf {K}_2$ , $Aut(\\mathfrak {M}/A) \\leqslant Aut(\\Gamma _{\\mathfrak {M}})$ .",
"Thus, by Theorem REF , it suffices to show that $(\\mathbf {K}^{\\prime }_{2}, \\leqslant )$ is tame.",
"We show tameness of $(\\mathbf {K}^{\\prime }_{2}, \\leqslant )$ for elements, the argument generalizes to tuples.",
"Let $B \\in \\mathbf {K}^{\\prime }_{2}$ and assume that in $B$ the copies of $C^*$ and $\\mathbb {N}$ are actually $C^*$ and $\\mathbb {N}$ .",
"Let $a, b \\in \\mathfrak {M}(\\mathbf {K}^{\\prime }_{2}, \\leqslant ) - B$ .",
"Then $a$ and $b$ lie in some of the copies of $\\mathbb {Z}$ not in $B$ , say $a$ is in $Z$ and $b$ is in $Z^{\\prime }$ .",
"Let $a = 0_Z \\pm n$ and $b = 0_{Z^{\\prime }} \\pm m$ .",
"Notice that there is $\\alpha \\in Aut(\\mathfrak {M}/B)$ mapping $a$ to $b$ iff $n =m$ iff there is $\\alpha \\in Aut(\\mathfrak {M}/C^* \\cup \\mathbb {N})$ mapping $a$ to $b$ .",
"In fact, for every copy $Z^{\\prime \\prime }$ of $\\mathbb {Z}$ we have that $0_{Z^{\\prime \\prime }} \\pm n$ is adjacent to exactly $n+1$ elements from $\\mathbb {N}$ .",
"It follows that $tp(a/B) \\ne tp(b/B)$ iff $tp(a/C^* \\cup \\mathbb {N}) \\ne tp(b/C^* \\cup \\mathbb {N})$ , and so $(\\mathbf {K}^{\\prime }_{2}, \\leqslant )$ is tame, because $C^* \\cup \\mathbb {N} \\leqslant B$ .",
"Theorem 6.4 $(\\mathbf {K}_{2}, \\preccurlyeq )$ is not homogeneous.",
"$(\\mathbf {K}_{2}, \\preccurlyeq )$ is uncountably categorical.",
"The proof of (a) is as in the proof of Theorem REF (a).",
"In fact letting $t^{\\prime }_i = i$ and $a_i = i_Z$ , for $Z$ a copy of $\\mathbb {Z}$ , we have that the argument used in the proof of Theorem REF (a) works also in this case (where the role of the $t_i$ 's there is played by the $t^{\\prime }_i$ 's here).",
"Uncountable categoricity is also immediate, since for $C,D \\in \\mathbf {K}_{2}$ we have $(A \\cup B_T, E) = \\Gamma _C \\cong \\Gamma _D = (A^{\\prime } \\cup B^{\\prime }_T, E^{\\prime })$ iff $B_T \\cong B^{\\prime }_T$ (in the language $\\lbrace 0, s \\rbrace $ ), and $T = Th(\\mathbb {N}, s, 0)$ is well-known to be uncountably categorical.",
"Corollary 6.5 $(\\mathbf {K}_{2}, \\preccurlyeq )$ is stable in every infinite cardinality.",
"This is a consequence of Theorems REF , REF and REF .",
"Corollary 6.6 $\\mathbf {K}_2 = Mod(\\theta )$ for some $\\theta \\in L_{\\omega _1, \\omega }$ .",
"Furthermore, for every $A \\preccurlyeq \\mathfrak {M}$ we have $tp_{\\omega _{1}, \\omega }(a/A) = tp_{\\omega _{1}, \\omega }(b/A) \\text{ iff } tp(a/A) = tp(b/A)$ .",
"This is an immediate consequence of Theorems REF , REF , REF , REF and REF , together with the easy observation that $\\mathbf {K}_2$ has at most countably many countable models.",
"We conclude the paper with the following open problem.",
"Open Problem 6.7 Find combinatorial conditions on $\\Gamma _A$ which are necessary and sufficient for the strong rigidity of an arbitrary right-angled Coxeter group $A$ , and use them to develop the model theory of strongly rigid right-angled Coxeter groups, in the style of the present paper."
]
] | 1605.06003 |
[
[
"A minimal coupled fluid-discrete element model for bedload transport"
],
[
"Abstract A minimal Lagragian two-phase model to study turbulent bedload transport focusing on the granular phase is presented, and validated with experiments.",
"The model intends to describe bedload transport of massive particles in fully rough flows at relatively low Shields numbers, for which no suspension occurs.",
"A discrete element method for the granular phase is coupled with a one dimensional volume-averaged two-phase momentum equation for the fluid phase.",
"The coupling between the discrete granular phase and the continuous fluid phase is discussed, and a consistent averaging formulation adapted to bedload transport is introduced.",
"An original simple discrete random walk model is proposed to account for the fluid velocity fluctuations.",
"The model is compared with experiments considering both classical sediment transport rate as a function of the Shields number, and depth profiles of solid velocity, volume fraction, and transport rate density, from existing bedload transport experiments in inclined flume.",
"The results successfully reproduce the classical 3/2 power law, and more importantly describe well the depth profiles of the granular phase, showing that the model is able to reproduce the particle scale mechanisms.",
"From a sensitivity analysis, it is shown that the fluctuation model allows to reproduce a realistic critical Shields number, and that the influence of the granular parameters on the macroscopic results are weak.",
"Nevertheless, the analysis of the corresponding depth profiles reveals an evolution of the depth structure of the granular phase with varying restitution and friction coefficients, which denotes the non-trivial underlying physical mechanisms."
],
[
"Introduction",
"Historically studied by hydraulic engineers in relation to the management of river waterways [1], bedload represents the main contribution of sediment transport to the evolution of riverbeds.",
"As such, it has major implications for environmental flows and associated risks like floods for example.",
"In contrast to suspension, bedload transport is characterized by sediment transport for which the vertical gravity force is on average stronger than the upward fluid force, i.e.",
"sediments rolling, sliding or in saltation over the bed.",
"The paper focuses on bedload transport in turbulent flow conditions, which is the most common case in nature.",
"In this phenomenon, one of the main challenges is to link the sediment transport rate to the fluid flow rate.",
"By making the problem dimensionless, it is equivalent to linking the dimensionless sediment transport rate $Q_s^* = \\frac{Q_s}{d\\sqrt{(\\rho ^p/\\rho ^f -1)gd}} $ , to the Shields number which compares the fluid bed shear stress $\\tau ^f_b$ to the buoyant weight $\\theta = \\frac{\\tau ^f_b}{(\\rho ^p-\\rho ^f) g d}$ ; where $Q_s$ is the sediment transport rate per unit width, $d$ is the particle diameter, $\\rho ^p$ and $\\rho ^f$ are respectively the particle and fluid density, and g is the acceleration of gravity.",
"The usual semi-empirical formulas established in this framework such as the Meyer-Peter and Müller [2] one, can differ by two orders of magnitude from what is observed in the field [3].",
"This difference is usually explained by the difficulty of measurements, the complexity of the physical processes and the great variability of the situations encountered in the field (e.g.",
"grain size segregation, particle shape, channel geometry)[4].",
"Bedload transport can be viewed as a granular medium in interaction with a fluid flow.",
"Following this two-phase decomposition, there are two major possibilities for numerical modelling: a continuous description for the two phases (Euler/Euler) or a continuous description for the fluid phase and a discrete one for the granular phase (Euler/Lagrange).",
"The former considers the momentum conservation of the two phases viewed as two continua in interaction, and is based on the two-phase Reynolds Averaged Navier-Stokes (RANS) equations [5], [6].",
"The averaged equations require different closures, and the main differences between the models pertain to the Reynolds stress tensor and the constitutive law for the intergranular stress.",
"The Reynolds stress tensor models the effect of turbulence on the mean fluid flow, and ranges from simple descriptions such as mixing length formulations, to more complex ones such as $k-\\epsilon $ .",
"In the case of intense bedload transport, also termed sheet flow, a substantial number of particle layers are in motion.",
"The Euler/Euler description has therefore been mainly used for this regime, with closures for the granular stress tensor according to the main theories for granular media, i.e.",
"Bagnold formulation [7], the $\\mu (I)$ rheology [8], [9], or the kinetic theory [10], [11].",
"The continuous approximation breaks down for the granular phase when considering bedload transport closer to the threshold of motion which is the common situation in mountain streams and the focus of this paper.",
"Moreover, the constitutive equation for granular media is still a matter of debate and thus limits the analysis of the results of such models.",
"Euler/Lagrange models overcome these two limitations by resolving the motion of each grain.",
"For bedload transport, the high concentration of particles inside the bed requires modelling the contact between grains, this is today commonly handled with the Discrete Element Method (DEM).",
"The different scales of fluid description range from large scale average description, to Direct Numerical Simulation (DNS) resolving the fluid locally around the particles down to the smallest turbulence length scale.",
"Euler/Lagrange approaches have been intensively developed in recent years for problems with particles in fluids such as fluidized bed [12], [13], particle suspension [14], or sheet flow [15], [16], [17], [18], [19], [20].",
"Focusing on bedload transport, up to now only a few contributions have taken advantage of the Eulerian/Lagrangian approach.",
"The work of Duran et al.",
"[21] used a simple average description of the fluid with a two dimensional discrete element method for the particles, to numerically study the dependence of sediment transport on the solid-fluid density ratio.",
"Bedload transport was considered in this paper as an extreme case of low density ratio, the closures of the model being more adapted to aeolian transport.",
"Using a DNS/DEM model, Ji et al.",
"[22] focused on the influence of particle transport on the turbulence.",
"With a similar model, Fukuoka et al.",
"[23] studied particle shape influence and size-segregation effects.",
"Bedload transport has mainly been studied focusing on the fluid phase.",
"It is however clear that the granular behavior is important in this phenomenon and should be studied further [24], [25].",
"The idea is therefore to analyze bedload transport at the particle scale in order to understand the behavior of the bed as a granular medium.",
"Considering the complexity of the experimental technique for particle-scale three dimensional (3D) bedload transport analysis (e.g.",
"index matching [9], or Magnetic Resonance Imaging [26]), there are interests in developing a numerical approach of the problem.",
"Focusing on the granular phase, the paper presents a model for bedload transport using a DEM Lagrangian description of the granular phase coupled with a one dimensional volume-averaged two-phase momentum equation for the fluid phase.",
"Although this type of model is common, to our knowledge, no previous contribution focused on bedload transport at relatively low Shields number.",
"Moreover, the usual experimental validations are limited to the classical macroscopic results of dimensionless transport rate as a function of the Shields number.",
"In this paper, we present a model adapted to subaqueous bedload transport (section ) and perform a new particle-scale experimental comparison with solid depth profiles in quasi-2D bedload transport cases [27] (section ).",
"In addition, the classical experimental comparison of the sediment transport rate as a function of the Shields number is considered in a more general 3D framework (section ).",
"The influence of the different model contributions is considered in terms of sediment transport rate and solid depth profiles."
],
[
"Numerical model formulation",
"The proposed model is based on a DEM Lagrangian description for the solid phase and an Eulerian description for the fluid phase.",
"In the present approach, the fluid flow is not solved at the particle scale and the momentum coupling is ensured in an averaged sense via semi-empirical correlations.",
"After presenting briefly the Discrete Element Method (section REF ) and the fluid phase description (section REF ), the coupling between both phases is discussed by detailing in particular the averaging procedure and the coupling forces in the framework of bedload transport (section REF )."
],
[
"Solid phase",
"The DEM, originally introduced by Cundall & Strack [28] for granular media, is based on the explicit resolution of Newton's equation of motion for each individual particle considering nearest neighbor contact forces $\\vec{f_c^p}$ .",
"For each particle $p$ at position $\\vec{x}^p$ the equation of motion reads: $m \\frac{d^2 \\vec{x}^p}{d t^2} = \\vec{f}_c^p + \\vec{f}_g^p + \\vec{f}_f^p,$ where $\\vec{f_g^p}$ is the force due to gravity and $\\vec{f}_f^p$ represents the forces applied by the fluid on particle $p$ .",
"This last term arises from the DEM-fluid coupling and will be detailed in subsection REF .",
"The application of the gravity force is straightforward.",
"The contact forces are determined from the relative displacement of the neighboring particles using a defined contact law.",
"In bedload transport, there is a sharp transition between rapidly sheared particles at the interface with the fluid, and almost quasi-static motion inside the bed.",
"The so-called spring-dashpot contact law used in this paper, allows description of these two types of behavior and is classical in granular flows modelling.",
"The contact law is based on a spring of stiffness $k_n$ in parallel with a viscous damper of coefficient $c_n$ for the normal contact, coupled with a spring of stiffness $k_s$ associated with a slider of friction coefficient $\\mu $ for the tangential contact.",
"For normal contact, the linear elastic spring and viscous damping define a constant restitution coefficient $e_n$ characteristic of the energy loss at collision, which can be evaluated experimentally."
],
[
"Fluid phase",
"The fluid phase model is based on spatially averaged two-phase Navier-Stokes equations [5], and inspired from the one-dimensional Euler-Euler model proposed by Revil-Baudard & Chauchat [8] to deal with turbulent unidirectional sheet-flows.",
"The simplifications of the general fluid phase equations [5] due to the incompressible and unidirectional character of the present bulk flow lead to the resolution of the same fluid phase momentum equation: $\\epsilon \\ \\rho ^f \\frac{\\partial \\left<u_x\\right>^f}{\\partial t} = \\frac{d~(\\epsilon \\left<\\tau _{xz}\\right>^f)}{dz} + \\frac{d R_{xz}^f}{dz} + \\epsilon \\rho ^f g \\ \\sin \\alpha - n \\left<f_x\\right>^s,$ where $\\epsilon $ is the fluid phase volume fraction, $\\rho ^f$ is the fluid density, $ \\left< u_x \\right>^f$ is the averaged fluid velocity, $\\left<\\tau _{xz}\\right>^f$ is the averaged fluid viscous shear stress, $R_{xz}^f$ is the Reynolds shear stress, $\\alpha $ is the channel inclination angle, and $n\\left<f_x \\right>^s$ is the averaged fluid-particle general interaction term.",
"A schematic picture representing the main fluid model variables is shown in figure REF .",
"The operator $\\left< .\\right>^s$ denotes a spatial averaging over the solid phase while the operator $\\left< .\\right>^f$ denotes a spatial averaging over the fluid phase.",
"The major difference with the continuous two-phase model proposed by Revil-Baudard & Chauchat [8] and with Euler/Euler models in general, stands in the average fluid-particle interaction $n\\left<f_x \\right>^s$ and solid volume fraction $\\phi = 1- \\epsilon $ , obtained in the present model from a spatial averaging of the DEM solution, whereas otherwise obtained by solving a continuous momentum balance equation.",
"All the details concerning the averaging process and the fluid-particle interaction term will be given in subsection REF .",
"In equation (REF ), omitting the fluid-particle interaction term, closure laws for the viscous shear stress $\\left<\\tau _{xz}\\right>^f$ and the Reynolds shear stress $R_{xz}^f$ need to be prescribed.",
"In the present model, the fluid is considered as Newtonian, so that: $\\left<\\tau _{xz}\\right>^f = \\rho ^f \\nu ^f \\frac{d \\left<u_x\\right>^f}{dz},$ where $\\nu ^f$ is the clear fluid kinematic viscosity.",
"The Reynolds shear stress, representing the vertical turbulent mixing of horizontal momentum, is modeled based on the eddy viscosity concept ($\\nu ^t$ ) with a mixing length formulation: $R_{xz}^f = \\rho ^f ~ \\nu ^t \\frac{d \\left<u_x\\right>^f}{dz} \\ \\ \\text{with}\\ \\ \\nu ^t = \\epsilon \\ l_m^2 \\left|\\frac{d \\left< u_x \\right>^f}{dz}\\right|,$ in which the mixing length $l_m$ formulation proposed by Li & Sawamoto[29] is used: $l_m(z) = \\kappa \\int _0^z{\\frac{\\phi ^{max} - \\phi (\\zeta )}{\\phi ^{max}} ~d\\zeta },$ where $\\kappa =0.41$ represents the von Karman constant.",
"This simple formulation allows recovery of the two expected asymptotic behaviors: the mixing length is linear with $z$ when the solid phase volume fraction vanishes (i.e.",
"clear fluid), as in the law of the wall [30], and the mixing length is zero when the solid phase is at its maximum packing fraction, i.e.",
"the turbulence is fully damped inside the dense sediment bed.",
"As explained in Revil-Baudard & Chauchat [8], this formulation is well adapted for boundary layer flow above mobile rough beds.",
"Indeed, the integral of the solid volume fraction predicts a non zero mixing length at the transition between the granular dominated and turbulent dominated layers.",
"Also, with this formulation no virtual origin for the mixing length has to be prescribed."
],
[
"DEM-fluid coupling",
"The key point in continuous-discrete models consists in the coupling of the two phases, which involves an averaging procedure and a parametrization of the fluid forces applied on the particles."
],
[
"Averaging procedure",
"For this purpose, the spatial averaging operator for the solid phase needs to be defined consistently with the spatial averaging operator for the fluid phase [5], [31].",
"According to Jackson [5], the solid phase volume fraction $\\phi (\\vec{x_1})$ at a given position $\\vec{x_1}$ is defined as: $\\phi (\\vec{x_1}) = \\sum _p \\int _{V_p} \\mathcal {G}(|\\vec{x_1}-\\vec{x}|) dV$ where the sum is over all the particles, $V_p$ is the volume of particle $p$ , and $\\mathcal {G}(\\vec{x})$ is a weighting function that must be normalized on the whole physical domain.",
"Providing that the weighting function $\\mathcal {G}$ is defined, the solid phase spatial average of a scalar quantity $\\gamma $ at a given position $\\vec{x_1}$ is defined as: $\\left< \\gamma \\right>^s = \\frac{1}{\\phi (\\vec{x_1})}\\sum _p \\int _{V_p} \\gamma (\\vec{x}) \\mathcal {G}(|\\vec{x_1}-\\vec{x}|) dV,$ In the general case, Jackson [5] uses a radial weighting function $\\mathcal {G}$ .",
"However in the present case, to match the discretization of the fluid resolution it is more convenient to define a cuboid weighting function.",
"To fulfill the normalization property, a three-dimensional step function is chosen for the weighting function: $\\displaystyle \\mathcal {G}(\\vec{x}) = \\left\\lbrace \\begin{array}{ll}\\displaystyle \\frac{1}{l_x ~ l_y ~ l_z} & \\mbox{for } |x| \\le l_x/2 \\mbox{, } |y| \\le l_y/2 \\mbox{, } |z| \\le l_z/2 \\\\[12pt]0 & \\mbox{otherwise}\\\\\\end{array}\\right.$ In order to properly define the spatial averaging, the average should be independent from the length scales chosen for the weighting function: $l_x$ , $l_y$ and $l_z$ [5], [31].",
"This is only possible if a separation of scales exists between the macroscopic length scale of the problem $L$ , the length scales associated with the weighting function $l_x,l_y,l_z$ and the particle diameter $d$ , i.e.",
"$L>>l_x\\text{, }l_y\\text{, }l_z>>d$ .",
"Due to the sharp transition occurring at the sediment bed interface in the wall-normal direction, the wall-normal macroscopic length scale of the problem $L$ is lower than the particle diameter $d$ .",
"Therefore the vertical length scale of the weighting function $l_z$ should be lower than the particle diameter in order to accurately resolve the vertical gradients of the averaged solid phase variables.",
"We postulate that this limited vertical length scale $l_z$ , can be compensated statistically by larger complementary horizontal length scales, $l_x$ and $l_y$ .",
"The convergence analysis of the numerical results on the length scale $l_x$ presented in appendix lends credibility to this hypothesis."
],
[
"Fluid forces",
"The force applied by the fluid on a single particle $\\vec{f}_f^p$ introduced in equation (REF ) is defined as the integral of the total fluid stress, pressure and shear stress, acting on the particle surface [5].",
"In the present model, the fluid flow is not resolved at the particle scale so that the hydrodynamic forces cannot be computed explicitly, and need to be prescribed through semi-empirical formulas based on average fluid variables.",
"The main hydrodynamic forces in bedload transport reduce to the buoyancy, the drag and the shear-induced lift.",
"Ji et al.",
"[22] numerical results exhibit a non-negligible importance of the lift force with respect to the other two.",
"However, Schmeeckle et al.",
"[32] showed experimentally that the usual formulation of the lift [33], derived using the inviscid flow assumption, is not valid close to the threshold of motion.",
"Based on this observation and the absence of alternative formulation, it has been decided not to include the lift force at this stage.",
"Therefore, the force $\\vec{f}_f^p$ induced by the fluid on a particle p appearing in the DEM model (equation (REF )), reduces to buoyancy $\\vec{f}_{b}^p$ and drag $\\vec{f}_{D}^p$ : $\\vec{f}_f^p = \\vec{f}_{b}^p + \\vec{f}_{D}^p.$ According to Jackson[5] the generalized buoyancy force is defined as: $\\vec{f}_{b}^p = V^p ~\\left( -\\vec{\\nabla }\\left<P \\right>^f + \\vec{\\nabla }.\\left<\\overline{\\overline{{\\tau ^f}}} \\right>\\right),$ where $\\left<P\\right>^f$ is the average fluid pressure and $\\left<\\overline{\\overline{{\\tau }}} \\right>^f$ is the average viscous shear stress tensor taken at a larger scale than the particle scale.",
"This definition generalizes the so-called Archimedes buoyancy force for hydrostatic problems to cases where the fluid volume is submitted to a macroscopic deformation at a scale much larger than the particle scale i.e.",
"the fluid deformation viewed by the particles can be considered as constant.",
"Similarly to Revil-Baudard & Chauchat [8], we found that the viscous stress tensor contribution is however negligible with respect to the pressure contribution in bedload transport.",
"The force applied on each particle can then be approximated by the usual buoyancy expression, which is equivalent to apply the buoyant weight in the vertical direction.",
"The drag force exerted by the fluid flow on a single particle is classically expressed as: $\\displaystyle \\vec{f}_{D}^p = \\frac{1}{2}\\rho ^f \\frac{\\pi d^2}{4} ~ C_D ~ \\left|\\left| \\left<\\vec{u}\\right>^f_{\\vec{x^p}} - \\vec{v^p} \\right|\\right|\\left(\\left<\\vec{u}\\right>^f_{\\vec{x^p}} - \\vec{v^p}\\right),$ where $C_D$ is the drag coefficient, and $\\left<\\vec{u}\\right>^f_{\\vec{x^p}} - \\vec{v^p}$ is the relative velocity between the particle and the average fluid velocity taken at the position of the particle center.",
"The Dallavalle formulation [34] together with a Richardson-Zaki correction [35] is used in the present model for the drag coefficient: $C_D = \\left(0.4+ \\frac{24.4}{Re_p}\\right) (1-\\phi )^{-\\zeta },$ where $Re_p=|| \\left<\\vec{u}\\right>^f_{\\vec{x^p}} - \\vec{v^p} || d / \\nu ^f$ is the particulate Reynolds number for particle $p$ .",
"This simple formulation has been used in different two-phase flow models for sediment transport applications [8], [10], [11].",
"The Richardson-Zaki correction $(1-\\phi )^{-\\zeta }$ accounts for the hindrance effect induced by the local particle concentration, and allows to recover realistic fluid velocity in the particle bed.",
"The exponent has been set to $\\zeta = 3.1$ in reference to Jenkins & Hanes[10].",
"Equations (REF ) and (REF ) are used to compute the drag force on each individual particles in the DEM model (equation (REF )), while the effect of buoyancy is taken into account through the vertical buoyant weight.",
"The average effect of the particles on the fluid momentum balance does not simply consist in the solid averaging of the momentum transfer associated with the hydrodynamic forces.",
"It also includes higher-order correlations which appear in the averaging process, and are due to perturbations of the flow by the presence of the particles.",
"For the case of Stokesian particles at low concentration, Jackson showed analytically [36] that these higher-order correlations lead to a modification of the viscosity in the average viscous fluid stress tensor formulation, which takes the form of Einstein's effective viscosity [37].",
"In the model, the clear fluid viscosity in equation (REF ) has been replaced by Einstein's effective viscosity $\\nu ^e$ to take this effect into account: $\\displaystyle \\nu ^e = \\nu ^f \\left( 1 + \\frac{5}{2} \\phi \\right).\\\\$ The phase interaction term in the fluid momentum balance (eq.",
"REF ) reduces then to the momentum transfer associated with the hydrodynamic forces.",
"In the present 1D fluid resolution, it is expressed as the average number of particles $n = \\phi /V_p = 6 \\phi /\\pi d^3$ multiplied by the solid-phase average streamwise associated force.",
"For drag force, it gives: $n\\left<{f_D}_x\\right>^s = \\frac{3}{4}~\\frac{\\phi ~ \\rho ^f}{d} \\left< C_D ~ \\left|\\left| \\left<\\vec{u}\\right>^f_{\\vec{x^p}} - \\vec{v^p} \\right|\\right| ~ \\left( \\left<u_x\\right>^f - v^p_x\\right) \\right>^s.$ The drag coefficient $C_D$ depends on the relative velocity through the particle Reynolds number, so that it should be included in the spatial averaging."
],
[
"Velocity fluctuation model",
"The proposed average model for the fluid phase does not account for the fluid turbulent velocity fluctuations, which are known in particular to influence the particle threshold of motion.",
"In order to account for these turbulent processes in the average fluid model, a Discrete Random Walk (DRW) model for the fluid velocity fluctuations inspired from Zannetti[38] is therefore introduced.",
"It consists in associating a random velocity fluctuation with each particle for a given duration, as a function of the local turbulent intensity and turbulent time scale.",
"The fluctuations are not correlated in space, nor in time, and the model is built so that the Reynolds shear stress definition is consistent between the average fluid model and the DRW model: $\\displaystyle \\overline{ {u_x^f}^{\\prime } {u_z^f}^{\\prime } } = - \\frac{R_{xz}^f}{\\rho ^f \\epsilon },$ where the $\\overline{\\bullet }$ represents an averaging operator in time.",
"From experimental measurements in open-channel flows [39], , it has been observed that the magnitude of the fluctuations in the streamwise direction is roughly two times larger than in the vertical direction.",
"With this constraint the following DRW model for the streamwise component $({u_x^f}^{\\prime })^p$ and the normal component $({u_z^f}^{\\prime })^p$ of the fluid velocity fluctuation associated with each particle $p$ is proposed: $({u_z^f}^{\\prime })^p &= \\lambda _1\\\\({u_x^f}^{\\prime })^p &= - ({u_z^f}^{\\prime })^p + \\lambda _2,$ where $\\lambda _1$ and $\\lambda _2$ are two Gaussian random numbers of zero mean and of standard deviation $\\sigma $ .",
"This standard deviation is obtained from the local value of the Reynolds shear stress at the position of the particle center $\\sigma = \\sqrt{\\frac{R_{xz}^f}{\\rho ^f \\epsilon }(\\vec{x^p})}$ .",
"The velocity fluctuations are updated every $\\tau _t$ , defined as the turbulent eddy turn over time, which can be estimated as $\\tau _t = w_d/U^f$ where $w_d$ is the water depth, and $U^f$ is the average fluid velocity.",
"These velocity fluctuations are added to the average fluid velocity in the drag force expression (equation (REF ))."
],
[
"Numerical resolution strategy",
"The resolution of the fluid equation still needs to be clarified.",
"For numerical reasons, it is necessary to express equation (REF ) linearly as a function of the average fluid velocity.",
"The numerical treatment of the drag force is then handled as follows: $n\\left<{f_D}_x\\right> = \\beta ~ \\left( \\left<u_x\\right>^f - \\left<v^p_x\\right>^s\\right),$ where $\\beta $ is computed according to equation (REF ) as: $\\displaystyle \\beta = \\frac{3}{4}~\\frac{\\phi ~ \\rho ^f}{d} ~ \\frac{\\left<C_D ~ \\left|\\left| \\left<\\vec{u}\\right>^f_{\\vec{x^p}} - \\vec{v^p} \\right|\\right|\\left(\\left<u_x\\right>^f_{\\vec{x^p}} - v^p_x\\right)\\right>^s}{{\\left<u_x\\right>^f}_{\\vec{x^p}} - \\left<v^p_x\\right>^s}$ This formulation allows strictly the same average momentum transfer in the discrete solid phase problem and the continuous fluid phase one.",
"With this definition the fluid phase momentum equation to be solved can be rewritten as: $\\epsilon \\rho ^f \\frac{\\partial \\left<u_x \\right>^f}{\\partial t} = \\rho ^f\\frac{\\partial }{\\partial z} \\left[ \\left(\\epsilon ~ \\nu ^e + \\nu ^t\\right) \\frac{\\partial \\left<u_x\\right>^f}{\\partial z} \\right]\\\\+ \\epsilon \\rho ^f g \\ \\sin \\alpha - \\beta \\left(\\left<u_x\\right>^f - \\left<v^p_x\\right>^s\\right)$ This equation is discretized using implicit finite differences for the diffusion and the drag terms.",
"The resulting tridiagonal system is solved using a double-sweep algorithm .",
"The fluid phase resolution period $\\tau _f$ should be small enough compared with the particle relaxation time $\\tau _{D} = \\beta ^{-1}$ .",
"This characteristic time corresponds to the time needed by a particle initially at rest to reach its steady state velocity in a constant fluid flow.",
"The DEM solid phase model is solved using the open-source code Yade .",
"The time integration is explicit with a second order centered scheme to ensure energy conservation.",
"The time step has been estimated with a method similar to Catalano (pp.",
"84, see also ), considering the rigidity of the system of springs and dampers as decoupled."
],
[
"Experimental comparison",
"The model is to be compared against experimental data.",
"The declared aim of the present model is to focus on the granular phase behavior.",
"We therefore reproduce the quasi-2D experiments of Frey [27], in which particle tracking allowed to obtain average solid depth profiles of bedload transport.",
"After a brief presentation of the experiment, the numerical set-up and the comparison with the experimental results will be presented."
],
[
"Description of the experiment",
"The experiment of Frey [27] consisted in a quasi-2D ideal case of mountain stream bedload transport on a steep slope.",
"The setup is depicted in figure REF , it is composed of a $2~m$ long inclined channel of slope $S_0 = 0.1$ , and width $6.5d/6$ .",
"Water ($\\rho ^f = 1000~kg/m^3$ ) flows inside the open-channel and entrains the spherical glass particles ($\\rho ^p = 2500~kg/m^3$ ) of diameter $d = 6~mm$ .",
"Particles are introduced at the inlet and create an erodible bed thanks to the obstacle placed at the outlet.",
"The number of particle layers is controlled by the height of this obstacle.",
"The channel bottom is made of metal half-cylinders of diameter $d$ , fixed at a random elevation between $-2.75~mm$ and $2.75~mm$ with steps of $0.5~mm$ to break clusterization.",
"The particle feeding rate is controlled, and the flow rate is adjusted in order to reach transport equilibrium, i.e.",
"feeding rate equal to the sediment transport rate at the outlet without having aggradation and degradation of the bed.",
"The free-surface fluid flow is turbulent ($Re = U^f w_d/\\nu ^f \\sim 10^4$ ), hydraulically rough ($Re_p \\sim 10^3$ ), and supercritical ($Fr = U^f/\\sqrt{g w_d} \\gtrsim 1$ ).",
"The particle settling velocity ($w_s = 0.54 ~ m/s$ ) and the suspension number $S^* = w_s/u_*$ are high, meaning that the particles are weakly influenced by the turbulent structures.",
"A camera is placed perpendicular to the sidewall, filming a window of $25\\text{x}8~cm^2$ at 131.2 frames per second.",
"Due to the one particle diameter width of the channel, image processing enables particle trajectories to be followed inside the measurement window, and the average free-surface elevation to be evaluated.",
"In each experiment, once bedload transport is at equilibrium, data acquisition time lasted $60~s$ .",
"Experimental data are averaged in the same way as in the model using the definition of section REF .",
"For more details on the experimental setup, refer to the original experimental article of Frey [27].",
"The order of magnitude of the main dimensionless numbers associated with the experiment are shown in table REF .",
"The Stokes number comparing the inertia of the particle and the viscosity of the fluid is given by $St = \\rho ^p v^p d/(9\\eta ^f)$ .",
"Table: Characteristic values of the main dimensionless numbers.Figure: Experimental setup scheme, modified from Böhm et al and Hergault et al.",
"The inclined channel width of 6.5d/66.5d/6 implies a quasi-2D bead flow, permitting particle tracking of each spherical bead in the observation window filmed by the camera placed perpendicularly."
],
[
"Application to the model",
"To compare the model with the experiments, the simulation needs to match the experimental set-up.",
"To focus on the bulk equilibrium properties of bedload transport, periodic boundary conditions are considered in the streamwise direction for the present 2D case.",
"The periodic characteristic of the granular phase, does not enable us to impose a feeding rate.",
"To have a situation equivalent to the experiment, the density of beads per unit length (equivalent to the number of layers of particle $N_l$ ) and the free-surface position $h$ are instead imposed.",
"Indeed, there is a unique couple, slope-water depth, corresponding to the transport equilibrium and it can be reproduced by fixing $h$ , $N_l$ and the slope $S_0$ for a periodic sample.",
"In the simulation, for the solid phase, the number of particles is therefore imposed as a function of the length of the periodic cell $l_x$ , which has been fixed to $l_x = 1000~d$ to define a consistent and convergent averaging (see appendix ).",
"The bottom made of fixed particles is randomly generated with the experimental characteristics described in the previous subsection.",
"The boundary conditions for the fluid resolution are imposed considering a fixed boundary at the channel bottom, and forcing the derivative of the fluid velocity to zero at the fixed free-surface elevation measured in the experiment.",
"The other experimental parameters such as the bead size, density and material, or the width of the channel, are set in the simulation at their known experimental values.",
"In the experiment the width to depth ratio is low, and we expect in consequence the fluid flow to have a complex 3D structure.",
"However, experimental flow measurements in this particular channel showed that it still has a typical logarithmic profile .",
"This, together with the stated aim of the model to focus on the granular phase, made us consider only a correction for the fluid dissipation at the smooth lateral walls.",
"The correction was included as a source term in the fluid averaged momentum balance resolution (eq.",
"REF ), taking the form of a dissipation term evaluated from the classical Einstein method with a Graf and Altinakar friction factor .",
"The method description can be found in Frey et al.",
"For each run the channel bottom is newly generated randomly, and particles are deposited under gravity.",
"Once the system with fluid resolution is at equilibrium, the simulations last 100 seconds and measurements are made each 0.1 second.",
"The latter corresponds to the particle relaxation time to the fluid velocity $\\tau _D = \\beta ^{-1}$ (eq.",
"REF ), and is characteristic from the evolution of the system.",
"For the post-processing of both experimental and numerical results, the averaging definition is taken consistently with the numerical resolution from equation (REF ).",
"Table: Model input parameters for the contact law and the fluid resolution.",
"k n k_n and k s k_s are respectively the normal and tangential contact stiffness, e n e_n and μ\\mu denote the restitution and friction coefficient, κ\\kappa is the Von Karman constant, ζ\\zeta the Richardson-Zaki exponent, φ max \\phi ^{max} the bed solid volume fraction, and τ f \\tau _f the fluid resolution period.To study the stability of the coupling, we performed a sensitivity analysis on the fluid resolution period, shown in appendix , and set it to $\\tau _f = 10^{-2}s$ regarding the results obtained.",
"In agreement with Revil-Baudard & Chauchat[8], it has been found that the fluid effective rheology does not influence the fluid behavior as it is dominated by the turbulent shear stress.",
"We therefore used a clear fluid viscosity.",
"The restitution coefficient was set to $e_n = 0.5$ based on measurements previously made in the experimental channel considered .",
"In the present situation, the limited particle pressure allows artificial softening of the spheres stiffness in order to reduce computational costs.",
"$k_n$ was set to $5.10^3N/m$ which leads to an acceptable average overlap of the order of $10^{-4}d$ and allows to be in the rigid grain limit .",
"The tangential stiffness was set as a function of the normal one $k_s = k_n/2$ .",
"The friction coefficient was taken as the classical value for glass in the dry case $\\mu = 0.4$ .",
"The main parameters values of the simulation are summarized in table REF .",
"The simulation results correspond to the application of the experimental conditions, and are not fitted with any parameter afterwards.",
"A summary of the main characteristics of the experimental (Exp) and numerical (Sim) runs is shown in table REF with respectively the positions of the free-surface $h$ and the number of layers of particle $N_l$ (both measured in the experiments and imposed in the simulation), the measured sediment transport rate expressed in beads per second (b/s) $\\dot{n}$ , and the measured Shields numbers $\\theta $ and $\\theta ^*$ .",
"The latter is defined based on the turbulent shear stress tensor, and can be evaluated only in the simulation, we will come back on the different definition in the next subsection REF ."
],
[
"Results",
"In bedload transport, one of the main challenges lies in the prediction of the integrated transport rate as a function of the flow rate.",
"The experiment of Frey [27] was designed to give more insight into the granular behavior, and to focus on the depth profile of bedload transport at the particle scale.",
"It has been noted previously [27], that the integrated transport rate per unit width $Q_s$ can be expressed as a function of the average transport rate density $q_s$ , the product of the average solid velocity $\\left<v_p\\right>$ and solid volume fraction $\\phi $ : $Q_s = \\int {\\left<q_s\\right>^s dz} = \\int {\\left<v^p\\right>^s\\phi dz}.$ Considering bulk equilibrium properties of bedload transport, $\\left<v^p\\right>^s$ , $\\phi $ , and $\\left<q_s\\right>^s$ depends only on the depth $z$ .",
"The experimental comparison will then focus on the depth profiles of the solid volume fraction, the average solid velocity, and the transport rate density, which will be called for simplicity transport rate in the following.",
"To complete this decomposition, we evaluated also the Shields number for each case.",
"This was done with two different methods: from macroscopic parameters following Frey [27], $\\theta = \\rho ^f Rh_b S_0/[(\\rho ^p - \\rho ^f)d]$ , with $Rh_b$ the corrected water depth; and from the fluid bottom shear stress defined by the friction velocity $u_*$ : $\\theta ^* = \\rho ^f u_*^2/[(\\rho ^p - \\rho ^f)gd]$ , where $u_*$ is given by the maximum turbulent shear stress $u_* = max(R_{xz}^f(z))$ .",
"$\\theta ^*$ was evaluated only in the simulation.",
"This formulation avoids use of the macroscopic determination of the Shields numbers, which is sensitive to the water depth evaluation and the type of wall correction used.",
"Table: Experimental and numerical run characteristics.",
"The free surface position hh and the number of bead layers N l N_l are both measured in the experiment and imposed in the simulation.",
"n ˙\\dot{n} is the measured transport rate, θ\\theta and θ * \\theta ^* the Shields numbers respectively based on macroscopic flow parameters and turbulent shear stress profile.",
"The latter has only been determined in the simulations.In the previous subsections, we did not introduce any experimental or numerical error.",
"It appears that the dispersion is dominated by the limited measurement window length of the experiment ($40~d$ ).",
"The order of magnitude of this dispersion has been evaluated numerically.",
"Figure REF exhibits the depth profiles of the solid volume fraction, the solid streamwise velocity, and the solid transport rate for a single simulation with different post-processing averaging properties.",
"The fluid mechanics convention is used, where the depth is represented on the y-axis while the quantities of interest are represented on the x-axis.",
"The simulation corresponds to the case Sim20 in table REF , considering a periodic length cell of $l_x = 1000~d$ .",
"The figure shows the variability of the results when the averaging cell length is taken equal to the experimental one at different position in the channel.",
"This dispersion is much greater than the evaluated experimental dispersion and the numerical variability due to the size of the periodic cell simulated.",
"These latter two will be consequently ignored in the comparison, and the variability observed will be taken as error bars.",
"Figure: Experimental comparison for the different cases presented in table : (a) Case 20, (b) Case 14, (c) Case 6.",
"The figure shows for each case the depth profiles of the average streamwise solid velocity (m/sm/s), solid volume fraction, and sediment transport rate density (m/sm/s).",
"The full symbol (red•\\color {red}{\\bullet }, blue⧫\\color {blue}{\\blacklozenge },green▪\\color {green}{\\blacksquare }) represents the experimental results from Frey while the empty linked one represents the simulation (red+\\color {red}{+}, bluex\\color {blue}{\\text{x}},green·\\color {green}{\\cdot }).",
"The black line represents the imposed free surface position.",
"The error bars show the variability of the experimental results as evaluated from figure .",
"The results show a good general agreement for the three profiles with well-reproduced trends in each case.The three different experiments detailed in table REF are considered for experimental comparison.",
"The slope is the same and equilibrium transport rate ranges from 6 to $20~beads/s$ .",
"The differences in the input parameters between the runs lie in the water surface position $h$ and the number of layers of particle $N_l$ .",
"The different experimental cases represent a good test to evaluate the sensitivity to the parameters and the ability of the model to reproduce different experimental conditions.",
"The macroscopic results presented in table REF show that the integrated transport rates $\\dot{n}$ are in good agreement with the experiment even if slightly overestimated.",
"Considering the Shields number, the two different methods of evaluation lead to an over-estimation using the macroscopic formulation $\\theta $ , and an under-estimation using the formulation based on the turbulent shear stress $\\theta ^*$ .",
"This underlines the complexity to measure the Shields number in the present case, especially when using the macroscopic formulation which is very sensitive to the small water depth.",
"The trends observed with both formulations are good, and the values have the same order of magnitude than the experiment.",
"In the following, we will use $\\theta ^*$ in order to avoid the somehow arbitrary determination of the water depth.",
"Using this definition, the value observed for case 6 is below the classical critical Shields number.",
"It should however be kept in mind that the present quasi-2D mono-disperse bed is less resistant, and the value of the critical Shields number is accordingly lowered.",
"To summarize, the general trends observed for the macroscopic parameters are good and these results show that we are able to reproduce the experimental sensitivity to the free surface position and the number of bead layers.",
"To go further, figure REF shows the solid depth profiles of velocity, volume fraction and transport rate density, for the three experimental comparisons.",
"The global trends from one case to the other are well reproduced, and the shape of the simulated curves are close to the experimental ones.",
"Focusing on the transport rate density profile, for each case the value of the peak is slightly overestimated, while the rest of the curve is in very good agreement with experiments.",
"We note an overestimation of the exponential decrease in the bed, in a part weakly affecting the total sediment transport rate density.",
"The oscillations present in each experimental solid volume fraction profile, are representative of the limited size of the experimental averaging window, and impact the sediment transport rate density profile.",
"They are therefore not reproduced in the simulation, and the comparison should be considered with respect to the average value around which it is oscillating.",
"For the solid volume fraction profile, the agreement between simulation and experiments is excellent for case 6 and 20, while we note a slight discrepancy for case 14 at the interface.",
"The solid velocity profiles show a good estimation of the maximal velocity, and of the depth structure.",
"The underestimation of the sediment transport rate peak is shown to correspond to an overestimation of the solid volume fraction in case 14, and of solid velocity in case 6 and 20.",
"For completeness, the fluid velocity profiles are presented in figure REF .",
"No experimental data are available for comparison so that the simulated solid velocity profiles have been added for reference.",
"In the fixed bed, the fluid velocity exhibits some oscillations around a finite constant value and the solid velocity is negligible.",
"The oscillations are due to the layering observed in the solid volume fraction profile (figure REF ) that makes the drag coefficient oscillate accordingly (eq.",
"REF -REF ).",
"In the upper part, the velocity difference between the solid and the fluid phases is of the order of an isolated particle settling velocity $w_s\\sim 0.54m/s$ .",
"The results are consistent with the drag coefficient formulation adopted and cannot be interpreted further in the absence of experimental data.",
"Figure: Simulated solid and fluid (–) velocity profiles for the three cases presented in table .Considering the comparison for the three different cases, with respect to the simplicity of the fluid description and the goal of describing the average solid behavior, the agreement with the experiments is good.",
"The values of the integrated transport rate are close to the experimental ones and the sensitivity to the experimental parameters such as the free-surface position or the number of bead layers has been well reproduced.",
"The comparison of the averaged depth profile of the solid velocity, the solid volume fraction and the transport rate showed that the model is able to reproduce the particle-scale trends observed experimentally, and the variation between the three different runs."
],
[
"Discussion",
"The experimental comparison gives credits to the model presented, and shows that the depth structure of the phenomenon is well reproduced.",
"Starting from this point, the effect of the grains parameters (restitution and friction coefficients) and the fluid velocity fluctuations model are analyzed over a wide range of Shields numbers, in terms of dimensionless sediment transport rate versus Shields number, completed by solid depth profiles.",
"The analysis aims at characterizing the influence of these parameters, in order to both study the influence of the different terms on the phenomenon, and the robustness of the experimental comparison.",
"To extend the generality, a 3D bi-periodic (streamwise, spanwise) situation is considered.",
"The random fixed bottom is generated from a gravity deposition, fixing all the particles contained in a slice of height $d$ at a given elevation in the granular bed.",
"The size of the periodic cell has been chosen from a convergence analysis similar to the one undertaken for the 2D case (see appendix ) and a cell size of $l_x = l_y = 30d$ has been chosen to ensure statistical convergence and numerical stability.",
"For each run, the DEM results are averaged over 100 seconds."
],
[
"Macroscopic considerations",
"The dimensionless sediment transport rate as a function of the Shields number is presented in figure REF .",
"The model results are compared with experimental data from Meyer-Peter and Müller [2] (+), and Wilson (x).",
"Simulation parameters for the reference configuration, represented as black squares ($\\blacksquare $ ), are the same as the one used for the experimental comparison (see table REF ).",
"The 3/2 power law is recovered by the numerical simulations and the results show a good agreement with experimental data for Shields number $\\theta >0.1$ .",
"Near the threshold of motion, the model results differ from experimental measurements.",
"The linear inset shows that the transition around the critical Shields number, characteristic of the onset of motion, is sharper in the numerical simulation results than in the experimental measurements.",
"Also, the critical Shields number is slightly lower: around 0.04 in the model, against 0.047 for Meyer-Peter and Müller [2] data.",
"The underestimation is consistent with the use of spheres in the numerical simulations, which present smaller imbrication, and consequently smaller resistance to entrainment than the natural sediment used in the experiments.",
"It is also worth noting that the scatter of the experimental data is usually very important close to the threshold of motion due to different definitions of the onset of motion and difficulties in shear stress measurements .",
"In particular, the present choice, based on the maximum turbulent shear stress, is less arbitrary than classical momentum balance estimates based on the water depth measurement, but most probably underestimates the Shields number.",
"Considering the whole range of Shields number investigated, the results are in good agreement with literature data, and this shows that the numerical model is able to reproduce almost quantitatively the sediment transport rate.",
"The results of the model without the fluid velocity fluctuations are shown in figure REF as empty squares ($\\square $ ).",
"At high Shields number negligible differences are observed, while the influence is important close to the threshold of motion.",
"It is consistent with the present conditions, where the suspension number is relatively high ($S^*=w_s/u_* \\in [1.7 ; 10 ]$ ) and the fluid velocity fluctuations are expected to mostly play a role close to the threshold of motion.",
"Focusing on the linear plot, it is observed that the critical Shields number is changed from around 0.04 to around 0.09 in the case without fluid velocity fluctuations.",
"The former is in the range of observed values under turbulent flow conditions , while the latter is close to the value observed under laminar flow conditions .",
"The influence of the fluctuations on the critical Shields number can be associated with turbulent coherent structures (e.g.",
", ).",
"However, the present simple fluid velocity fluctuation model does not account for the space-time correlations induced by turbulent boundary layer coherent structures.",
"This partly explains that the fluctuations model does not allow to describe well the evolution of the sediment transport rate with Shields number close to the threshold of motion (figure REF ).",
"Nevertheless it permits to successfully reproduce the onset of sediment transport motion in the turbulent regime, resulting in a good comparison with experimental depth profiles (section REF ).",
"In the rigid grains limit, the granular interactions are characterized by the restitution and the friction coefficients.",
"The restitution coefficient is representative of the energy loss during collisions, and has been shown experimentally to decrease with the impact velocity following a power law exponent lower than 1/4 .",
"For a limited range of impact velocity and in a first approximation it can be considered as constant.",
"As the fluid flow model does not allow to resolve the fluid at the particle scale, the local lubrication effect is included in the effective restitution coefficient $e_n$ .",
"Following Gondret et al.",
", the effective restitution coefficient depends on the local Stokes number comparing the grain inertia to fluid viscous forces: $St = \\rho ^p v d/(9\\eta ^f)$ where $\\eta ^f$ is the fluid dynamic viscosity, and $v$ is the impact velocity.",
"In the region where the collisions are dominant, the Stokes number is of order $10^2-10^3$ , corresponding to effective restitution coefficient in the range $e_n \\in [0.6e_n^d-0.9e_n^d]$ , respectively, where $e_n^d$ is the restitution coefficient for dry grains .",
"In our model a constant restitution coefficient is adopted, taking into account the lubrication effect globally.",
"It is therefore a characteristic of the material and of the lubrication effect.",
"The influence of the restitution coefficient is shown in figure REF for two different Shields numbers, by keeping the free-surface position and number of particle layers constant.",
"The restitution coefficient has been varied in the range $e_n \\in [0.01,1]$ .",
"It corresponds to a realistic range $e_n \\in [0.25,0.75]$ , complemented by two extreme cases: no rebounds ($e_n = 0.01$ ) and no dissipation at contact ($e_n = 1$ ).",
"Focusing on the realistic range at high Shields number ($\\theta \\sim 0.45$ ), the effect on the sediment transport rate is negligible.",
"A slight trend is observed, the sediment transport rate and the Shields number being respectively increasing and decreasing function of the restitution coefficient.",
"The extreme case without dissipation at contact ($e_n=1$ ) follows the same trend but exhibits a more important transport rate increase.",
"Quite surprisingly, the case $e_n = 0.01$ shows an increase in transport rate with respect to case $e_n = 0.25$ .",
"For the lower Shields number value ($\\theta \\sim 0.1$ ), while the dependency in restitution coefficient is limited in the realistic range, there is no associated clear trend.",
"The non-monotonous dependencies observed show non-trivial coupling between the granular phase characteristics and the sediment transport rate.",
"The global weak dependency on the restitution coefficient is consistent with results obtained by Drake & Calantoni [17] under oscillatory flow conditions, and show that there is no need to include a lubrication model in the present condition.",
"However, the relatively low importance of the restitution coefficient at such a high Shields number value is surprising.",
"It is usually thought that collisional interactions are the dominant mechanism of momentum diffusion for such inertial particles [10], .",
"The effect of the particle friction coefficient is also shown in figure REF , represented by circles: $\\mu = 0.2$ ($\\color {grey1}{\\bullet }$ ), $\\mu = 0.6$ ($\\color {grey3}{\\bullet }$ ), $\\mu = 0.8$ ($\\bullet $ ).",
"Unlike for the restitution coefficient, the trend observed is monotonous for all values, and similar at low ($\\theta \\sim 0.09$ ) and high Shields number ($\\theta \\sim 0.35$ ).",
"The sediment transport rate and the Shields number decrease with increasing friction coefficient.",
"The effect appears to be non-linear as the observed influence for a variation from $\\mu = 0.2$ to $0.4$ is much greater than the one observed for a variation from $\\mu = 0.4$ and $0.8$ .",
"This type of dependency is characteristic of dry dense granular flows .",
"As a partial conclusion, the present analysis shows that (i) the 3/2 power law for the sediment transport versus Shields number relationship is well captured by the proposed model; (ii) the fluid velocity fluctuations model is essential to capture a realistic value for the critical Shields number under turbulent flow conditions (iii) the influence of the granular interaction parameters is low, when taken in a realistic range.",
"These results underline the robustness of the model and strengthen the experimental validation.",
"Extreme value of particle friction and restitution coefficient affects the results, and show complex behaviors.",
"In order to understand better the mechanisms at work, the sensitivity to granular interaction parameters will be further discussed by analyzing the results in terms of depth profiles."
],
[
"Depth profiles analysis",
"Figure REF a shows the solid depth profiles for a Shields number value $\\theta \\sim 0.45$ , and for the different restitution coefficient values.",
"Such a high Shields number value enhances the effect of restitution coefficient.",
"From the profile, a clear trend appears, the sediment transport rate density profile is broader with increasing restitution coefficient.",
"Excepting case $e_n = 0.01$ , this is associated with an overall increase of particle velocity throughout the depth, the velocity profile being shifted with almost the same shape.",
"The solid volume fraction profile shows an increase of the mobile layer thickness when the restitution coefficient is increased: the solid volume fraction is lowered close to the quasi-static bed while it is increased in the upper part of the flow.",
"This can be explained considering predictions of the kinetic theory of granular flows, where particle phase normal stress is an increasing function of the restitution coefficient [10].",
"The increase in particle normal stress is logically associated with a decompaction of the bed, which is submitted only to gravity.",
"The case $e_n = 0.01$ is peculiar, the global trend is observed in the lower part of the flow while a higher particle velocity is predicted in the upper part.",
"This reflects the coupling with the fluid phase, and features the complex mechanisms at work.",
"Figure REF b shows the influence at high Shields number $\\theta \\sim 0.45$ of the friction coefficient over the range $\\mu \\in [0.2 ; 0.8]$ with $0.2$ steps.",
"Interestingly, the solid volume fraction profile is not affected by the variation in friction coefficient.",
"On the contrary, the particle velocity and thus the sediment transport rate density profiles are increasing when the friction coefficient is decreased.",
"The increase of the velocity throughout the depth is mainly affecting the lower part of the sediment transport rate density profile, where the solid volume fraction is maximum.",
"It corresponds to the denser part of the granular flow, for which the frictional interactions are dominant.",
"This analysis shows that, while affecting weakly the macroscopical results, the friction and restitution coefficient impact the depth structure of the granular flow differently.",
"In addition, the non-monotonous behavior observed suggests the presence of non-trivial coupling between the solid and the fluid phases."
],
[
"Conclusions",
"The model presented is a step toward a description of the granular processes of steady bedload transport.",
"By adapting the closures to this particular case, it has been shown that the model is able to reproduce the classical macroscopic validation in term of sediment transport rate and Shields number.",
"In addition, an original detailed validation with existing bedload transport experiments has been performed, comparing simultaneous measurements of average solid velocity and volume fraction.",
"The good agreement with experiments together with the rather low sensitivity of the results to the granular parameters show the relevancy and the robustness of the proposed model, which reproduces not only the evolution of the sediment transport rate as a function of the Shields number, but also the depth structure of the granular phase.",
"The influence of the different model contributions have been studied.",
"In particular, the discrete random walk fluid velocity fluctuations model has been shown to be sufficient to reproduce the reduction of the critical Shields number due to turbulent fluctuations.",
"A weak impact of the restitution and friction coefficients variations has been observed on the macroscopic sediment transport rate versus Shields number curve.",
"The analysis of the depth profiles variations shows however that the granular parameters influence the depth structure of the granular flows and induce non-trivial coupling with the fluid phase.",
"The rigorous development of the model and the experimental validations demonstrate the potential of this modeling approach to deal with granular processes in bedload transport.",
"Future work will take advantage of the description of the depth structure to analyze the effective granular rheology under bedload transport conditions."
],
[
"Averaging expression and convergence analysis",
"The averaging of the solid phase takes a central part in the coupling between the Lagrangian solid phase and the Eulerian fluid phase.",
"The important wall-normal gradient requires the length scale of the weighting function in this direction to be lower than particle diameter ($l_z \\simeq d/30$ for the lowest Shields number) in order to define a rigorous averaging.",
"We postulated that the complementary length scales $l_x$ and $l_y$ can statistically compensate the limited $l_z$ .",
"In the present model, the average fluid description is 1D so that it depends only on the wall-normal component, $z$ .",
"The solid averaging can therefore be performed on the full width and length cell.",
"With the cuboidal formulation of the weighting function defined previously (eq.",
"REF ), the solid averaging of a scalar particle quantity $\\gamma $ at a wall-normal position $z$ can be rewritten as: $\\left<\\gamma \\right>^s(z) = \\frac{1}{\\phi (z)} \\sum _{ \\lbrace p | z^p \\in [z- l_z/2 ; z+ l_z/2]\\rbrace } \\tilde{V}^p \\gamma ^p$ Where $l_z$ is the defined wall-normal weighting function length scale, and $\\tilde{V}^p$ is the fraction of the particle volume contained in the slice of height $l_z$ at elevation $z$ .",
"We recover here the averaging formulation of Hill et al , which is convenient to compute since the volume of a slice of spheres can be evaluated analytically.",
"The averaging box height $l_z$ is imposed by the vertical grid size of the fluid problem and no overlapping between the different slices is allowed.",
"For each fluid resolution, so at each given time step, the statistical representativity of the averages is a requirement for a consistent definition of the averaging process (section REF ).",
"The spatial convergence of the averages with increasing complementary length scales $l_x$ and $l_y$ includes the effect of the bottom boundary conditions and particles arrangement, in addition to the statistical representativity.",
"The results are required to be independent from the three effects, and consequently the spatial convergence of the results with respect to $l_x$ and $l_y$ is analyzed in the present appendix.",
"There are two different convergence scale in the problem.",
"The first one is associated to the spatial convergence at each given time step, and the second one is associated to the temporal convergence of a simulation with a given cell size.",
"In the present analysis, as the paper focus on steady equilibrium results, we consider time-averaged results which are converged in time.",
"The convergence analysis is conducted with respect to a large reference cell size for which we consider that the results at each fluid resolution are spatially converged.",
"Indeed, a convergence with respect to this case ensures that the error made in the spatial averaging along the simulation are compensating each other.",
"The analysis focuses on the transport rate depth profile.",
"For both 2D and 3D cases, $l_z$ is taken at its minimal value in the problem $l_z = d/30$ ."
],
[
"quasi-2D convergence analysis",
"We present here the results of the time-averaged spatial convergence analysis for the quasi-2D case Sim20 detailed in section REF .",
"In this configuration $l_y$ is fixed at the channel width, and the problem is considered only as a function of the streamwise length $l_x$ which determines the size of the averaging cells.",
"The convergence analysis is made with respect to the reference state chosen as $l_x = 10000d$ , corresponding to a periodic length cell of $60m$ for particles of $6mm$ and about 80000 particles in the simulation.",
"We performed different simulations with a periodic streamwise length cell $l_x$ of respectively 50, 100, 200, 300, 500, 1000, 2500 and $5000~d$ .",
"To quantitatively analyze the transport rate profile differences, an indicator representative of the deviation with respect to a reference case is defined.",
"It is given as the root mean square (RMS) of the difference between the considered transport rate profile and the reference one: $\\displaystyle \\frac{{{\\Delta Q}_{rms}}^i}{\\left<Q^{ref}\\right>} = \\frac{\\sqrt{\\frac{1}{N}\\sum _{z = 0}^{N}{(\\left<Q\\right>_z^i - \\left<Q\\right>_z^{ref})^2}}}{\\frac{1}{N}\\sum _{z = 0}^{N}{\\left<Q\\right>_z^{ref}}},$ where the RMS ${{\\Delta Q}_{rms}}^i$ is normalized by the average transport rate of the reference configuration $\\left<Q\\right>^{ref}$ , $N$ is the number of averaging cell in the depth, $\\left<Q\\right>_z^i$ and $\\left<Q\\right>_z^{ref}$ are the values of transport rate in the cell z for the considered case and the reference case respectively.",
"This variable effectively measures how close the results considered is from the reference case.",
"Figure: Convergence of the average sediment transport rate profile as a function of the periodic cell size considered for the quasi-2D case.",
"The vertical axis represents the deviation with respect to the reference configuration (l x =10000dl_x = 10000d) as defined in equation ().",
"The inset shows that the convergence is slightly superior to l x -0.5 l_x^{-0.5} (–).Figure REF shows the normalized deviation with respect to the reference configuration defined by equation (REF ) as a function of the streamwise periodic cell length of the simulation considered.",
"The time-averaged results show a convergence as a function of cell length $l_x$ of the order of $l_x^{-\\frac{1}{2}}$ .",
"The size of the periodic cell used for the simulations presented in the experimental comparison of the paper, was chosen as $l_x = 1000~d$ , as it gives the best trade off between computational time and deviation observed.",
"A similar analysis has been undertaken for the three dimensional bi-periodic configuration at a Shields number around $0.1$ .",
"The reference case has been chosen as $l_x = l_y = 100d$ , i.e.",
"the relative cell size $V = l_x~l_y~l_z$ being the same as the quasi-2D case.",
"Different simulations have been performed with a cell of $l_x = l_y = 10~d$ , $20~d$ , $30~d$ ,$50~d$ , and three cases with $l_x \\ne l_y$ : $(l_x,l_y) = (500~d,5~d)$ , $(300~d,5~d)$ , and $(10~d,100~d)$ .",
"It has not been possible to consider smaller cell sizes, as the coupled model becomes unstable.",
"The main results are summarized in figure REF , expressing the RMS deviation with respect to the reference case as a function of the product $l_x~l_y/d^2$ .",
"The latter reflects the statistical representativity as it directly determines the size of the averaging cell $V = l_x~l_y~l_z$ ($l_z = dz$ fixed).",
"The figure shows that cases with $l_x = l_y$ give better convergence than the ones with $l_x \\ne l_y$ .",
"There does not seem to be a convergence with increasing cell area.",
"To our opinion, this reflects the fact that the results are already converged.",
"Interestingly, the relative periodic cell size for convergence is less important in the 3D configuration, as a cell size of $l_x = l_y = 10d$ (to compare with $l_x = 1000d$ and $l_y = 1d$ ) is almost already converged with respect to $l_x = l_y = 100d$ .",
"This can be explained by the better randomness of the 3D packing, and suggests that the statistical representativity was not the limiting parameter in the quasi-2D convergence analysis.",
"For 3D cases, a higher Shields number increases the numerical instability of the coupling, so that it has been necessary to consider cell sizes up to $l_x = l_y = 50d$ for the highest Shields numbers simulations presented in the paper."
],
[
"fluid resolution period",
"The DEM time step needs to be particularly low and the evolution of the granular medium over this time is limited.",
"Consequently the fluid resolution period $\\tau _D$ does not need necessarily to be equal to the solid time step.",
"The stability of the coupling however depends on this period of resolution and it is important to study this parameter in order to have a meaningful model.",
"The fluid resolution period should be defined to be smaller than the characteristic time of evolution of the granular medium.",
"The fluid resolution is 1D and the equation is influenced only by the streamwise particle velocity and the wall-normal particle position (through respectively $\\left<f^p_{D,x}\\right>$ and $\\phi $ ).",
"For a single particle the evolution of these properties depends on the collisions and the entrainment by the fluid.",
"As seen previously (section ), collisions do not significantly influence the phenomenon so that we consider only the characteristic time of entrainment.",
"As explained in section REF , the characteristic time of relaxation to accelerate a particle to the fluid velocity is given by $\\tau _D = \\beta ^{-1}$ , with $\\beta $ expressed from equation (REF ).",
"In the present case it is of order $\\tau _D \\sim 10^{-1}s$ .",
"However, the characteristic time associated with each independent particle is not in general representative of the evolution of a complex many-body problem.",
"We therefore performed a sensitivity analysis on the period of fluid resolution.",
"The results are shown in table REF in term of RMS deviation (as defined by equation (REF )) with respect to the reference configuration Sim20 for which $\\tau _f = 10^{-2}s$ .",
"It includes $\\tau _f = 10^{-3}s$ , $10^{-1}s$ , and $1s$ .",
"The results exhibit no dependence on the fluid period resolution in the range $10^{-3}s$ to $1s$ .",
"The values of the RMS deviation with respect to the reference configuration $\\tau _f = 10^{-2}s$ is for all cases below $0.1$ , i.e.",
"below the reproducibility deviation value.",
"The fluid resolution period therefore does not have an influence on the averaged equilibrium results within the range considered.",
"This means that the solid average quantities transmitted to the fluid do not vary importantly during the simulation after reaching equilibrium.",
"These results confirm that the simulations considered are at transport equilibrium.",
"The fluid resolution period could however be important for unsteady conditions.",
"We are grateful to Michael Church for general remarks and English corrections.",
"This research was supported by Irstea (formerly Cemagref) and the French Institut National des Sciences de l'Univers program EC2CO-BIOHEFECT, and EC2CO-LEFE « MODSED »."
]
] | 1605.06085 |
[
[
"Capacity of the State-Dependent Wiretap Channel: Secure Writing on Dirty\n Paper"
],
[
"Abstract In this paper we consider the State-Dependent Wiretap Channel (SD-WC).",
"As the main idea, we model the SD-WC as a Cognitive Interference Channel (CIC), in which the primary receiver acts as an eavesdropper for the cognitive transmitter's message.",
"By this point of view, the Channel State Information (CSI) in SD-WC plays the role of the primary user's message in CIC which can be decoded at the eavesdropper.",
"This idea enables us to use the main achievability approaches of CIC, i.~e., Gel'fand-Pinsker Coding (GPC) and Superposition Coding (SPC), to find new achievable equivocation-rates for the SD-WC.",
"We show that these approaches meet the capacity under some constraints on the rate of the channel state.",
"Similar to the dirty paper channel, extending the results to the Gaussian case shows that the GPC lead to the capacity of the Gaussian SD-WC which is equal to the capacity of the wiretap channel without channel state.",
"Hence, we achieve the capacity of the Gaussian SD-WC using the dirty paper technique.",
"Moreover, our proposed approaches provide the capacity of the Binary SD-WC.",
"It is shown that the capacity of the Binary SD-WC is equal to the capacity of the Binary wiretap channel without channel state."
],
[
"Introduction",
"Secure communication from an information theoretic perspective attracts some attentions nowadays [1], [2], [3], [4], [5].",
"There are a lot of works to study the secrecy problem in different channel models, which are inspired of the wiretap channel as a basic physical layer model [6].",
"All these attempts are based on two principal elements: the Equivocation as a measurement to evaluate the secrecy level at the eavesdropper which is introduced by Shannon [7], and the Random Coding [6] as a coding scheme which leads to the secrecy condition in the wiretap channels (see [5] and the references therein).",
"Using the Channel State Information (CSI) in information theoretic communication models was initiated by Shannon [8] in which he considered the availability of CSI at the Transmitter (CSIT).",
"Gel'fand and Pinsker obtained the capacity of the discrete memoryless channel with non-causal CSIT [9].",
"Their main result was based on a binning scheme named Gel'fand- Pinsker Coding (GPC), and it was shown that the capacity of the Gaussian state-dependent channel is equal to the capacity of a channel with no channel state [10].",
"This means that the transmitter, who knows the CSI non-causally, can adapt its signal to the channel state such that the receiver senses no interference [10], as if the receiver had knowledge of the interference and could subtract it out.",
"The name of the Dirty Paper channel [10] is inspired of a spotted paper, one wish to write on such that the reader can easily find the text.",
"Despite of all these works which are trying to cancel out the channel state, the authors in [11] and [12] deal with the CSIT in an innovative manner.",
"In these works, the CSIT assumed as the user's signal which is wished to be sent through the channel.",
"In [12], the transmitter wishes to mask the CSI at the legitimate receiver, whereas in [11] the transmitter wishes to forward the CSI to the legitimate receiver.",
"The achievable rate in each case is derived and results to the trade-off between the rates of the transmitter's information and the CSI.",
"The State-Dependent Wiretap Channel (SD-WC) was studied in [1], [2].",
"In [1], the authors derived an achievable equivocation-rate for the SD-WC in general case, and in [2] the Gaussian SD-WC was considered.",
"The main ideas of these papers were based on using a combination of GPC and random coding.",
"In these works, it was assumed that the eavesdropper has no access to the CSI.",
"Therefore, the CSI can be used potentially to improve the secrecy rate of the SD-WC.",
"Specially in Gaussian case, under some constraints, the capacity was derived as the capacity of a channel [1], [2] with no state.",
"Afterward, [13] studied the SD-WC from the secret-key sharing aspect.",
"In this model, the channel state was assumed as a key to achieve the secrecy rate.",
"El-Gamal et.",
"al., improved the equivocation-rate for the SD-WC in the case that the channel state is known causally at both sides [14].",
"In this paper we study the SD-WC (see Fig.",
"REF ).",
"We consider the channel state sequence as a random sequence with elements drawn from a finite alphabet, known non-causally at the transmitter.",
"The transmitter wishes to keep its message secret from an external eavesdropper.",
"A well known concept is to cancel out the CSI using the GPC and use the random coding to achieve the secrecy rate as [1].",
"We know from the random coding idea [6] that we should randomize a part of the message to confuse the eavesdropper which has less channel capacity than the main channel.",
"In this paper, we model the CSI of SD-WC as the message of the primary user at CIC and we use it at the transmitter to randomize a part of the message.",
"Our main idea relies on modeling the CSI with a (primary) message, since both are known at the (cognitive) transmitter [15], [16].",
"In this model, the transmitter of SD-WC plays the role of the cognitive transmitter of CIC and the CSI is considered as the message of the primary one.",
"The transmitter, similar to the cognitive one in CIC, tries to cancel out the CSI, equivalently the primary message, at its destination with low enough information leakage of its message to the eavesdropper, i. e., the primary receiver in CIC.",
"Then, we use the rate splitting and the interference cancelation schemes as CICs in [15], [16], i. e., GPC and Superposition Coding (SPC), to achieve the secrecy rate for the SD-WC.",
"At first, using these two coding schemes, we derive two new achievable equivocation-rates for the SD-WC.",
"Then, we prove that these achievable equivocation-rates meet the secrecy capacity under some constraints.",
"As an example, we show that the derived equivocation-rates meet the capacity in the binary SD-WC.",
"Afterward, we use these results in the Gaussian SD-WC.",
"We prove that the achievable equivocation-rates for the SD-WC lead to the secrecy capacity using the Dirty Paper Coding (DPC) [10] approach.",
"On account of this result, we call this approach as secure writing on dirty paper to point to the problem of writing a message on a sheet of paper covered with independent dirt spots, such that the reader can distinguish the message, but the eavesdropper cannot do that.",
"The rest of the paper is organized as follows.",
"In Section II the channel model is introduced.",
"In Section III the encoding strategy for the point to point state-dependent communication channel is explained.",
"The main results on the achievable equivocation-rates, the capacity of the SD-WC and the binary example are presented in Section IV.",
"In Section V, using DPC, the capacity of the Gaussian SD-WC is derived and the paper is concluded with some discussions in the last section.",
"Figure: The State-Dependent Wiretap Channel (SD-WC) in which the channelstate is assumed to be known non-causally at the transmitter."
],
[
"Channel Model, Definitions and Preliminaries",
"First, we clear our notation in this paper.",
"Let $\\mathcal {X}$ be a finite set.",
"Denote its cardinality by $|\\mathcal {X}|$ .",
"If we consider $\\mathcal {X}^N$ , its members are $x^N = (x_1, x_2,\\ldots , x_N)$ , where subscripted letters denote the components and superscripted letters denote the vector.",
"A similar convention applies to random vectors and random variables, which are denoted by uppercase letters.",
"Denote the zero mean Gaussian random variable with variance $P$ by $\\mathcal {N}(0, P)$ .",
"A Bernoulli random variable $X$ with $Pr\\lbrace X = 1\\rbrace = P$ is denoted by $X \\sim \\mathcal {B}(P)$ .",
"Consider the channel model shown in Fig.",
"REF .",
"Assume that the state information of the channel, i.e., $S_i,~ 1\\le i \\le N$ is known at the transmitter non-causally.",
"The transmitter sends the message $W$ which is uniformly distributed on the set $\\mathcal {W} \\in \\lbrace 1,\\ldots , M\\rbrace $ to the legitimate receiver in $N$ channel uses.",
"Based on the $w$ and $s^N$ , the transmitter generates the codeword $x^N$ to transmit through the channel.",
"The decoder at the legitimate receiver makes an estimation of the transmitted message $\\hat{w}$ based on the channel output $y^N$ ; and $z^N$ is the corresponding output at the eavesdropper.",
"The channel is memoryless, i.e., rCl p (yN, zN|xN, sN) =i=1N p(yi, zi|xi, si).",
"The secrecy level of the transmitter's message at the eavesdropper is measured by normalized equivocation: rCl Re(N)=1N H(W|ZN, SN), in which we assume that the eavesdropper can decode the CSI.",
"The average probability of error $P_e$ is given by rCl Pe=1Mi=1MPr(w(YN)w).",
"We define the rate of the transmission to the intended receiver to be rCl R=MN.",
"The rate-pair $(R,R_e)$ is achievable if for any $0\\le \\epsilon \\le 1$ there exists an $(M,N, P_e)$ code such that $P_e\\le \\epsilon $ , and the secrecy rate $R_e$ is rCl 0 ReN Re(N).",
"We define the perfect secrecy condition as the case in which we have $R \\le R_e$ .",
"Thus, under this condition, the sent message to the legitimate receiver must be secure."
],
[
"Overview of Encoding Strategy",
"Consider SD-WC in Fig.",
"REF .",
"Assume that the channel state $S$ plays the role of the codebook of rate $R_S =I(S; Y )$ interfering with the communication of message $ W$ at the rate $ R = I(X; Y )$ .",
"As proposed in [17], two coding strategy can be used for this scenario: GPC scheme and SPC depending on the interference's rate $R_S$ .",
"When $R_S$ is small, it can be exploited for achieving the higher rates using the SPC.",
"Using the GPC, the interference is considered as a codebook with rate $R_S$ .",
"The following lemma expresses the result of using these two coding schemes.",
"This lemma is used to derive the achievable equivocation-rate for the SD-WC in the next section.",
"Lemma 1 [17] The following rate is achievable for a point to point communicating system with non-causal CSIT rCl R PU|S, x(u,s){ I(X; Y |S), {I(U, S; Y ) - RS, I(U; Y ) - I(U; S)}}.",
"[Outline Of the Proof] This lemma is proved in [17, Appendix B].",
"For the case $I(S;U,Y) \\le R_S \\le H(S)$ , the second term of (REF ) is achievable using the GPC.",
"For $R_S \\le I(S;U, Y )$ , the first term in (REF ) is achievable using the SPC.",
"Remark 1 Interestingly, in the case $R_S \\le I(S;U, Y )$ , the receiver decodes both the messages and the channel state, i. e., the interference.",
"A related model in which both data and the channel state are decoded at the receiver was considered by [11]."
],
[
"Results on The Capacity of The SD-WC",
"In this section, we provide two achievable equivocation-rates for the SD-WC, shown in Fig.",
"REF , in two cases, i. e., GPC and SPC.",
"Figure: Gel'fand-Pinsker Coding (GPC) scheme for thecase min{I(S;U;Y);I(S;V;Y|U)}≤R S ≤H(S)\\min \\lbrace I(S;U; Y ); I(S; V; Y |U)\\rbrace \\le R_S \\le H(S)."
],
[
"Gel'fand-Pinsker Coding (GPC)",
"Assume that $V$ and $U$ are two random variables to construct the private and the common messages, respectively.",
"In the case that $\\min \\lbrace I(S;U, Y ), I(S; V, Y |U)\\rbrace \\le R_S \\le H(S)$ , using GPC and the rate splitting, we have the following lemma for the achievable equivocation-rate of the SD-WC.",
"Lemma 2 (Achievable equivocation-rate using GPC) The set of rates $(R,R_1,R_2,R_e)$ satisfying rCl RGPC=P(T)P(S,U,V,X|T)P(Y,Z|X,S) {l (R,R1,R2,Re) : R = R1 + R2 R1I(V ; Y |U, T) - I(V ; S|U, T), R I(V,U; Y |T) - I(V,U; S|T), ReI(V ; Y | U, T) - I(V ; S,Z | U, T), }, is achievable for SD-WC, in which the equations in the right-hand-sides of (REF ) are non-negative and $T$ is a time-sharing random variable.",
"[Outline of the Proof] The details of the proof are relegated to the Appendix A based on the result of [15].",
"As an outline, the message $W$ is split into two messages $W_1$ and $W_2$ with denoted rates $R_1$ and $R_2$ , respectively.",
"In our scheme, $W_2$ can be decoded at the eavesdropper and does not contribute to the secrecy level of $W$ at the eavesdropper.",
"Therefore, $W_1$ may be hidden from the unintended receiver.",
"We model SD-WC with a CIC with a confidential message [15], in which the message of the primary transmitter plays the role of CSI and the transmitter who knows CSI, can be considered as the cognitive transmitter.",
"By this setting, the proof of Lemma REF is deduced from the proof of [15] in which a cognitive radio tries to communicate with a related destination through a main channel which belongs to a primary transmitter-receiver pair.",
"Therefore, the transmitter in SD-WC (similar to the cognitive transmitter in CIC), can cancel the channel state out at the corresponding receiver, using the known CSI.",
"The transmitter uses rate splitting scheme and GPC by binning $U$ against the channel state $S$ , and binning $V$ against $S$ conditioned on $U$ .",
"The GPC structure is shown in Fig.",
"REF .",
"The perfect secrecy condition using the GPC scheme is derived as follows.",
"Theorem 1 (Perfect secrecy condition using GPC) In the case that $\\min \\lbrace I(S;U, Y ), I(S; V, Y |U)\\rbrace \\le R_S \\le H(S)$ , the perfect secrecy rate $R^{GPC}_{ps}$ satisfying rCl RGPCps=PU,V,SPX|U,V,SPY,Z|X,S {Re {I(V,U; Y ) - I(V,U; S), I(V ; Y |U) - I(V ; S,Z|U)}}, is achievable for the SD-WC.",
"The proof of the theorem is strictly deduced from Lemma 2 applying perfect secrecy condition and using Fourier-Motzkin elimination [3].",
"Corollary 1 The following rate rCl R= PU,V,SPX|U,V,SPY,Z|X,S I(V ; Y |S) - {I(V ;Z|S),H(S|Y )}, is an achievable secrecy rate for the SD-WC.",
"Substituting $U = S$ in the proof of the Corollary 1 means that the transmitter sends the channel state $S$ as a part of its message.",
"Thus, the channel state can be decoded at the legitimate receiver.",
"This setting is similar to the one proposed by [11], in which the transmitter sends its data and the information about the channel state to the receiver, without secrecy issue.",
"Remark 2 The proof of the corollary is deduced directly from Theorem 1 by substituting $U = S$ and using Fourier-Motzkin elimination [3].",
"Note that the transmitter which knows CSI non-causally can use the channel state ransom variable $S$ as part of its message, i. e., the common one.",
"Figure: Superposition Coding (SPC) scheme for the case R S ≤min{I(S;U;Y);I(S;V;Y|U)}R_S \\le \\min \\lbrace I(S;U; Y ); I(S; V; Y |U)\\rbrace ."
],
[
"Superposition Coding (SPC)",
"Consider the case $R_S \\le \\min \\lbrace I(S;U, Y ), I(S; V, Y |U)\\rbrace $ .",
"From Lemma 1, we can superimpose $U^N$ and $V^N$ against $S^N$ instead of using GPC.",
"To derive the perfect secrecy rate for the SD-WC in this case, first we present the following lemma.",
"Lemma 3 (Achievable equivocation-rate using SPC) In the case that $R_S \\le \\min \\lbrace I(S;U, Y ), I(S; V, Y |U)\\rbrace $ , the following equivocation-rate is achievable for the SD-WC: rCl RSPC=PU,V,SPX|U,V,SPY,Z|X,S {l (R, Re) : R I(U, V ; Y |S) Re I(V ; Y |U, S)- I(V ;Z|U, S) }.",
"[Outline of the Proof] The transmitter splits its message $W$ into two components $W_1$ and $W_2$ with denoted rates $R_1$ and $R_2$ , respectively.",
"Modeling SD-WC with a CIC with confidential message [16], the proof of the lemma is deduced from the proof of the [16] in which the cognitive transmitter superimposes its message on the primary transmitter's message.",
"In our approach, the transmitter superimposes its message on the channel state sequence $S$ to derive the achievable equivocation-rate for the SD-WC.",
"The encoding structure in this case is shown in Fig.",
"REF .",
"More details on the proof are relegated to the Appendix B.",
"Thus, using the superposition coding scheme, the following theorem presents the perfect secrecy condition for the SD-WC.",
"Theorem 2 (Perfect secrecy condition using SPC) In the case $R_S \\le \\min \\lbrace I(S;U,Y), I(S; V, Y |U)\\rbrace $ , the perfect secrecy rate $R^{SPC}_{ps}$ satisfying rCl RSPCps=PU,V,SPX|U,V,SPY,Z|X,S {Re I(V ; Y |U, S) - I(V ;Z|U, S)}}, is achievable for SD-WC.",
"The proof is strictly deduced from the Lemma REF applying perfect secrecy condition and using Fourier- Motzkin elimination.",
"Fig.",
"REF shows the conditions of using GPC and SPC in theorems 1 and 2 respect to the CSI rate $R_S$ .",
"Corollary 2 The rate rCl R=PU,V,SPX|U,V,SPY,Z|X,S {I(V ; Y |S) - I(V ;Z|S)}, is an achievable secrecy rate for the SD-WC in the case that $R_S\\le \\min \\lbrace I(S;U, Y ), I(S; V, Y |U)\\rbrace $ .",
"The proof of the corollary is deduced directly from Theorem 2 by substituting $U =\\emptyset $ and using Fourier-Motzkin elimination [3].",
"Figure: The results of the theorems 1 and 2, respect to the channel stateinformation rate R S R_S."
],
[
"Capacity of the SD-WC",
"Now, we have the following theorem to explain the secrecy capacity of the SD-WC in general case.",
"Theorem 3 (Secrecy Capacity) In the cases that rCl RS {I(S;U, Y ), I(S; V, Y |U)}, or rCl RS{{I(S;U, Y ), I(S; V, Y |U)}, I(V,U; S) - I(U; Y |S) - I(V ;Z|S)}, the secrecy capacity $(C_{S})$ of the SD-WC is as follows rCl CS=PSPU,V |SPX|U,V,SPY,Z|X,S {I(V ; Y |U, S) - I(V ;Z|U, S)}.",
"The achievability of (REF ) is derived directly from Theorem 1 and Theorem 2.",
"In more detail, in the case that $R_S \\le \\min \\lbrace I(S;U, Y ), I(S; V, Y |U)\\rbrace $ , the rate (REF ) is achievable for the SD-WC by using SPC as Theorem REF .",
"In the case that $\\max \\lbrace \\min \\lbrace I(S;U, Y ), I(S; V, Y |U)\\rbrace , I(V,U; S) -I(U; Y |S)-I(V ;Z|S)\\rbrace \\le R_S$ , GPC achieves the rate (REF ), by using Theorem REF and substituting $U = (U, S)$ .",
"The converse proof is relegated to the Appendix C. Remark 3 The capacity of the SD-WC in Theorem REF , in the case that $U = S = \\emptyset $ , is reduced to the capacity of the wiretap channel without channel state information.",
"Corollary 3 In the case that $Y$ is more capable than $Z$ , i. e., $I(X; Y|U)\\ge I(X; Z|U)$ for all $p(x)$ , and under the conditions (REF )–(REF ), the secrecy capacity of the SD-WC is as follows rCl CS=PSPX|SPY,Z|X,S {I(X ; Y | S)- I(X ;Z| S)}.",
"The achievability of (REF ) is directly proved by substituting $U =\\emptyset , V = X $ in Theorem REF .",
"To prove the converse, we have rCl I(V ; Y |U, S)- I(V ;Z|U, S) = I(X, V ; Y |U, S)- I(X, V ;Z|U, S) -[I(X ; Y |U, S, V)- I(X ;Z|U, S, V)] = I(X ; Y |U, S)- I(X ;Z|U, S) +[I(V;Y|U, S, X)-I(V;Z|U, S, X)] -[I(X ; Y |U, S, V)- I(X ;Z|U, S, V)] (a)= I(X ; Y |U, S)- I(X ;Z|U, S) -[I(X ; Y |U, S, V)- I(X ;Z|U, S, V)] (b) I(X ; Y |U, S)- I(X ;Z|U, S), where $(a)$ is due to the Markov chain relationship $(U,V)\\rightarrow (X,S)\\rightarrow (Y,Z)$ which implies that $[I(V;Y|U, S, X)-I(V;Z|U, S, X)]= 0$ ; and $(b)$ is derived by using the more capable condition.",
"Now, we have rCl I(X ; Y |U, S)- I(X ;Z|U, S) = I(U, X ; Y | S)- I(U, X ;Z|S) - [I(U, X ; Y | S)- I(U, X ;Z|S)] = I(X ; Y | S)- I(X ;Z|S) + [I(U ; Y | S)- I(U ;Z|S)] - [I(U, X ; Y | S)- I(U, X ;Z|S)] (c) I(X ; Y | S)- I(X ;Z| S), where $(c)$ is derived by using the the Markov chain relationship and the more capable condition."
],
[
"An Example (Binary SD-WC)",
"As an example, consider the Binary SD-WC (BSD-WC), in which the channel outputs are described as rCl Y = X S 1, Z = X S 2, where $\\eta _1\\sim \\mathcal {B}(N_1)$ , $\\eta _2\\sim \\mathcal {B}(N_2)$ and $S\\sim \\mathcal {B}(Q)$ .",
"For this channel we have the following theorem.",
"Theorem 4 When rCl RS I(S; Y ), or rCl RS {I(S; Y ), I(X; S) - I(X;Z|S)}, the secrecy capacity of the BSD-WC is rCl CSBSD-WC=[H(N2)- H(N1)]+, in which $[x]^+ = \\max \\lbrace 0, x\\rbrace $ .",
"Substituting $U =\\emptyset , V = X $ in Theorem REF leads to the secrecy capacity as following under the conditions (REF )-(REF ) rCl CS'=PSPX|SPY,Z|X,S I(X; Y |S) - I(X;Z|S).",
"Let $X \\sim \\mathcal {B}(P)$ which is independent of $S$ .",
"Note that the nature of the BSC forces us to choose Bernoulli distribution function for the channel input.Now, we have rCl I(X; Y |S)- I(X;Z|S) = H(Y |S) - H(Y |X, S) - H(Z|S) + H(Z|X, S) = H(X 1) - H(1) - H(X 2) + H(2) = H(N2) - H(N1) - [H(P * N2) - H(P * N1)], in which $P * u = P(1 - u) + (1 - P)u$ and $H(N_i) =-N_i \\log (N_i) - (1- N_i) \\log (1 - N_i)$ , $i = \\lbrace 1, 2\\rbrace $ is the binary entropy function.",
"For $P < \\frac{1}{2}$ , the function $P *u$ is monotonically increasing in $u \\in [0, 1/2]$ .",
"Hence, setting $P = 1/2$ we have $H(P *N_2)-H(P *N_1) = 0$ which achieves the maximum of $I(X; Y |S) - I(X;Z|S)$ as $H(N_2)-H(N_1)$ .",
"In the case that $N_1 \\le N_2$ the right hand side of (REF ) is negative.",
"It means that the eavesdropper can decode any message intended for the receiver.",
"The proof of the converse is directly derived from Corollary REF which implies that the right-hand-side of (REF ) is an outer bound on the capacity of the model in which the legitimate receiver is more capable than the eavesdropper.",
"This completes the proof.",
"Remark 4 The capacity of the BSD-WC under the conditions (REF ) and (REF ), is equal to the capacity of the binary wiretap channel without channel state.",
"It means that the coding schemes used in the theorems 1 and 2 cancels the channel state out to meet the capacity."
],
[
"Gaussian SD-WC: Dirty Paper Scheme",
"In this section, we extend the results of the theorems 1 and 2 to the Gaussian SD-WC (GSD-WC).",
"First, consider the GSD-WC (Fig.",
"REF ) which is described as follows.",
"rCl Y = X + S + 1, Z = X + S + 2, where $X$ denotes the channel input and $Y$ and $Z$ denote the channel outputs at the legitimate receiver and eavesdropper, respectively.",
"$\\eta _i \\sim N(0,N_i), i \\in \\lbrace 1, 2\\rbrace $ is Additive White Gaussian Noise (AWGN), and we assume the channel state random variable as $S \\sim N(0,Q)$ .",
"Now, we have the following theorem for the GSD-WC.",
"Figure: Gaussian state-dependent wiretap channel.Theorem 5 The secrecy capacity of the GSD-WC is rCl CSGSD-WC=[C(PN1)- C(PN2)]+, in which $\\mathcal {C}(x) = \\frac{1}{2} \\log (1 + x)$ .",
"The proof the achievability of the rate (REF ) is derived from Theorem 1 and 2 as follows: First, we assume the channel input as $X\\sim N(0, P)$ which is independent of the channel state sequence and the AWGNs.",
"Then, we use the Dirty Paper approach [10] on theorems 1 and 2, directly.",
"For this, we split the channel input as $X =X_1 + X_2 +\\sqrt{\\bar{\\frac{\\alpha }{P}}{Q}}$ , in which $X_1$ is related to the confidential message and $X_2$ is related to the common message which reduces the interference at the legitimate receiver, and we have rCl X1 N(0, P), X2 N(0, P), where $0\\le \\alpha , \\beta \\le 1$ are the power coefficients for sending the confidential and the common messages in the transmitter, respectively.",
"Also we define $\\bar{\\alpha }=1-\\alpha , \\bar{\\beta }=1-\\beta $ .",
"Let the axillary random variables $V$ and $U$ as rCl V = X1 + 1 S, U = X2 + 2 S in which the transmitter uses $0\\le \\lambda _1, \\lambda _2 \\le 1$ to bin its message against the state of the channel as GPC.",
"Note that in the SPC the transmitter does not forward the channel state, and the channel state is not contained in $V$ .",
"Thus, in SPC case, we should substitute $\\lambda _1 =0$ .",
"Now, we find the variables $\\alpha , \\beta , \\lambda _1, \\lambda _2$ which maximize the achievable secrecy rate of Theorem 1 leading to the capacity of GSD-WC.",
"Based on Theorem 1, for the GPC case, we can define rCl Re1 I(V,U; Y ) - I(V,U; S) = H(V,U|S) - H(V,U|Y ), Re2 I(V ; Y |U) - I(V ; S,Z|U) = H(V |U) - H(V |Y,U) -H(S,Z|U) + H(S,Z|U, V ).",
"Now, we calculate each term of () using the standard approach [18] yielding (REF )-(REF ) in top of the next page.",
"Figure: NO_CAPTIONTaking derivatives of $R_{e_1}$ with respect to $\\lambda _1$ and $\\lambda _2$ and setting to zero yields rCl 1= P (K+1)P+N1 1=P (K+1)P+N1 Then, we substitute these optimal variables in () to get rCl Re1(1, 2)=C[PN1], Re2(1, 2)=C[PN1]-C[PN2].",
"Next, we optimize ()-() with respect to $\\alpha $ and $\\beta $ .",
"In the case $N_1 > N_2$ , the derived result of $R_{e_2}$ is a decreasing function with respect to $\\alpha $ and $\\bar{\\beta }$ .",
"Thus, this function is maximized at $\\alpha =\\bar{\\beta }= 0$ , and $ \\lambda _1 = \\lambda _2 = 0$ .",
"Therefore, the achievable rates are equal to zero for this case, i.e., $R_{e_1} = R_{e_2} = 0$ .",
"In the case that $N_1 < N_2$ , $R_{e_2}$ is an increasing function with respect to $\\alpha $ and $\\bar{\\beta }$ .",
"Thus, this function is maximized at $\\alpha ^*=\\bar{\\beta }^*=1$ .",
"Finally, the variables $\\alpha ^*=1, \\beta ^*=0,\\lambda _1^*=\\frac{P}{P+N_1}, \\lambda _2^*=0$ leads the secrecy achievable rate of Theorem 1 to rCl Re* = {Re1(1, 2), Re2(1, 2)} = C(PN1)- C(PN2).",
"We conclude that for the GPC for all cases of $N_1$ and $N_2$ the secrecy achievable rate is as follows rCl Re-GPC*= [C(PN1)- C(PN2)]+.",
"Note that the necessary condition, in Theorem 1, under which the GPC gives the secrecy achievable rate, is satisfied and discussed in Remark 5.",
"Now, based on Theorem 2 for the SPC case, we can define rCl V = X, U= .",
"substituting these parameters in (REF ) we have rCl I(V ; Y |U, S) -I(V ;Z|U, S) = I(X; Y |S)- I(X;Z|S) = H(Y |S) - H(Y |X, S) - H(Z|S) + H(Z|X, S) = H(X + 1) - H(1) - H(X + 2) + H(2) = C(PN1)- C(PN2), which means that the chosen parameters in ()-(), lead Theorem 2 to the following secrecy achievable rate rCl Re-SPC*= [C(PN1)- C(PN2)]+.",
"Note that the necessary condition, in Theorem 2, under which the SPC gives the secrecy achievable rate, is satisfied and discussed in Remark 5.",
"For the converse proof, using the fact that the legitimate receiver is more capable than the wiretapper, we can use the result of Corollary REF .",
"Thus, we should prove that choosing the Gaussian distribution for the channel input, maximizes the achievable rate to the capacity of the wiretap cannel without channel state.",
"Without loss of generality, we assume that the channel is physically degraded, i. e., $\\eta _2=\\eta _1+ \\eta _2^{^{\\prime }}$ , in which $\\eta _2^{^{\\prime }} \\sim N(0,N_2-N_1)$ .",
"For the outer bound on the capacity of the channel we have rCl I(X ; Y | S)- I(X ;Z| S) = I(X ; X+1)- I(X ; X+2) = H(X+1)-H(1)- H(X+2)+ H(2) = 12(N2N1)-[H(X+2)-H(X+1)].",
"Then, by substituting $M=X+\\eta _1 $ we have rCl H(X+2)-H(X+1) = H(M+2')-H(M) (c) 12(22H(2')+ 22H(M))- H(M) = 12(2e (N2 - N1)+ 22H(M))- H(M), where $(c)$ is derived by the entropy power inequality (EPI) [3].",
"Moreover, $\\frac{1}{2}\\log (2\\pi e (N_2 - N_1)+ 2^{2H(u)})- H(u)$ is a monotonic increasing function with respect to $u$ and $H(M) \\le \\frac{1}{2}\\log (2\\pi e (P+ N_1))$ .",
"Thus, we have rCl H(X+2)-H(X+1) 12(2e (N2 - N1)+ 2e (P+ N1)) - 12(2e (P+ N1)) = 12(P+N2P+N1).",
"Finally, we have rCl I(X ; Y | S)- I(X ;Z| S) 12(N2N1)- 12(P+N2P+N1) = C(PN1)- C(PN2), and the equality is attained by choosing $X\\sim \\mathcal {N} (0, P)$ .",
"Remark 5 We should note that GPC is used in the case that $ \\min \\lbrace I(S;U, Y ), I(S; V, Y |U)\\rbrace \\le R_S \\le H(S)$ .",
"By substituting the optimal parameters $\\alpha ^*=1, \\beta ^*=0,\\lambda _1^*=\\frac{P}{P+N_1}, \\lambda _2^*=0$ , this condition is reduced to $I(S; Y ) \\le R_S \\le H(S)$ .",
"On the other hand, SPC is used when $R_S\\le \\min \\lbrace I(S;U, Y ), I(S; V, Y |U)\\rbrace $ , which by substituting R.V.s as ()-(), we have $R_S\\le I(S; Y )$ .",
"As shown in Fig.",
"REF , our proposed coding schemes meet the capacity for any $R_S$ .",
"Note that in the Gaussian case, $I(S; Y ) = \\mathcal {C}(\\frac{Q}{P+N_1})$ .",
"Remark 6 In GPC scheme, choosing $\\alpha ^* = \\bar{\\beta }^*=1$ and $\\lambda _1^*, \\lambda _2^*$ results in rCl V= X+(PP+N1)S, U=, and it is noticeable that the parameter $\\lambda ^*_1= \\frac{P}{P+N_1}$ is similar to the one chosen in dirty paper channel [10] to achieve the capacity in the state-dependent channel.",
"This inspired the authors to name the proposed method Secure Dirty Paper Coding (SDPC).",
"Figure: The conditions under which GPC (by substituting R.V.s as V=X+(P P+N 1 )SV = X+(\\frac{P}{P+N_1})S) and SPC (by substituting R.V.s V=X;U=∅V = X; U = \\emptyset )are used in Gaussian state-dependent wiretap channel."
],
[
"Discussions and Conclusions",
"In this paper we derived two equivocation-rates for the state-dependent wiretap channel in which the channel state information is assumed to be known non-causally at the transmitter.",
"These equivocation-rates are derived from the equivocation-rate regions reported for the cognitive interference channel [15], [16].",
"Comparing our model to the cognitive interference channel, the channel state plays the role of the message of the primary user.",
"The transmitter uses the coding schemes previously used by the cognitive transmitter, i.e.",
"Gel'fand-Pinsker coding and superposition coding.",
"By this point of view, we derived new achievable equivocation-rates for the state-dependent wiretap channel.",
"Then, we showed that the derived equivocation-rates meet the capacity of the state-dependent wiretap channel under some conditions.",
"As an example, the secure capacity of a state-dependent binary symmetric channel was considered which confirms the general results.",
"Afterward, the state-dependent Gaussian wiretap channel was studied, and our achievable equivocation-rates lead to the capacity in Gaussian case.",
"It was shown that the capacity of the state-dependent Gaussian wiretap channel is equal to the capacity of the Gaussian wiretap channel without channel state.",
"This result was derived using dirty paper coding approach [10], by maximizing the equivocation-rates.",
"The authors called this coding scheme Secure Dirty Paper Coding.",
"To compare our model with the one presented in [1], we should note that in [1], the transmitter which non-causally knows the CSI, uses this information to increase its secrecy rate to $R^{Chen-Vinck}=I(V; Y)-\\max \\lbrace I(V;S), I(V;Z)\\rbrace $ .",
"Therein, it is assumed that the CSI is not known at the eavesdropper.",
"Thus, in the case that $I(V;S)\\ge I(V;Z)$ , binning the codewords into $2^{nI(V;S)}$ bins, overcomes the channel state, and in this case the message will be kept secure by the random coding scheme.",
"In Gaussian case, when $I(V;S)\\ge I(V;Z)$ , this approach leads to the capacity of the point to point channel, i. e., $\\frac{1}{2}\\log (1+\\frac{P}{N_1})$ .",
"Furthermore, the output of the channel at the eavesdropper assumed to be a degraded version of the one at the legitimate receiver, i. e., the $U \\rightarrow (X,S) \\rightarrow Y \\rightarrow Z$ forms a Markov chain in [1].",
"But in the model considered in this paper, the CSI can be decoded at the eavesdropper.",
"Thus, the CSI cannot be used to improve the secrecy rate in SD-WC.",
"It is noticeable that due to the capability of the eavesdropper to estimate the channel, the assumption that the CSI can be decoded at the eavesdropper sounds a little realistic.",
"Moreover, in our model it is not necessary to assume the channel output at the eavesdropper to be a degraded version of the legitimate receiver's one."
],
[
"Proof of the Lemma 2",
"The SD-WC is modeled with a CIC, i. e., the CSI is considered as a primary transmitter's message and thus is transmitted through the channel; and the transmitter in SD-WC plays the role of a cognitive transmitter who has the message of the primary one non-causally.",
"On the other hand, the transmitted message in SD-WC must be confidential at the primary receiver who acts as an eavesdropper for the cognitive transmitter's message.",
"First, note that in [15] two confidential messages are considered in CIC model, i. e., the primary and the cognitive receivers act as eavesdroppers for each other's message.",
"Here, we just consider the cognitive transmitter's message to be confidential at the primary receiver.",
"Hence, we reduce the rate region of [15] to the CIC with one confidential message by excluding extra secrecy condition.",
"Thus, we have the following lemma.",
"Lemma 4 [15] The set of the rates $(R_1,R_{2_a},R_c,R_{e_2} )$ satisfying rCl R1 I(X1; U, Y1 | T), R2a I(V ; Y2 | U, T) - I(V ;X1 | U, T), R2 I(U, V ; Y2 | T) - I(U, V ;X1 | T), R1 + Rc I(X1,U; Y1 | T), Re2 I(V ; Y2 | U, T) - I(V ;X1, Y1 | U, T), is achievable for CIC with a confidential message.",
"Now, comparing the SD-WC (Fig.",
"REF ) with the CIC (Fig.",
"REF ), we can derive a new achievable rate for the SD-WC.",
"We should note that the message of the primary transmitter plays the role of the channel sate in SD-WC.",
"Since the eavesdropper is not forced to decode the channel state in SD-WC, we should relax the terms contain $R_1$ which is related to the rate of the primary transmitter's message (the channel state in SD-WC).",
"Thus, by setting rCl W2 = W,R1 = RS,R2a = R1, Rc = R2,Re2 = Re, R = R1 + R2, Y1 = Z, Y2 = Y, X1 = S,X2 = X, and relaxing the rates (48) and (51), we derive the equivocation-rate of Lemma REF .",
"Finally, we remark that we can derive the equivocation-rate of Lemma REF directly by introducing the codebook generation, encoding and decoding schemes, error analysis and equivocation computation similar to the one presented in [15].",
"Figure: Cognitive interference channel with a confidential messages."
],
[
"Proof of the Lemma 3",
"Consider the CIC with one confidential message (Fig.",
"REF ).",
"Using the SPC, the achievable equivocation-rate region for this channel is derived by [16] as follows.",
"Lemma 5 [16] The set of the rates $(R_1,R_2,R_e)$ satisfying rCl R1 {I(U,X1; Y ), I(U,X1;Z)} R2 I(U, V ;Z | X1), R1 + R2 {I(U,X1; Y ), I(U,X1;Z)} + I(V ;Z | U,X1), Re I(V ;Z | U,X1) - I(V ; Y | U,X1), is achievable for CIC with a secret message.",
"Now, by substituting $R_2 = R,X_1 = S$ and relaxing the rates (REF ) and (REF ) which correspond to the primary user (channel state in our setting) the achievability of the equivocation-rate of Lemma REF is proved."
],
[
"The Converse Proof of Theorem 3",
"The converse proof of Theorem REF is derived as follows.",
"Consider the rate-pair $(R,R_e)$ to be achievable.",
"Then, we have rCl NR (a) I(W; YN) - I(W;ZN, SN) + I(W; YN| SN) - I(W;ZN|SN) + i=1N I(W; Yi|Yi-1, SN) - I(W;Zi|Zi+1N, SN) + (b) i=1N I(W, Zi+1N; Yi|Yi-1, SN) - I(W, Yi-1;Zi|Zi+1N, SN) + (c) i=1N I(W; Yi|Yi-1, Zi+1N, SN) - I(W;Zi| Yi-1, Zi+1N, SN) + (d)= i=1N I(Vi; Yi|Ui, Si) - I(Vi;Zi|Ui, Si) + (e)=I(V; Y|U, S) - I(V;Z|U, S) + in which $(a)$ follows from the Fano's inequality and the fact that $I(W;Z^N, S^N$ ) tends to zero for $N\\rightarrow \\infty $ ; $(b)$ and $(c)$ follow from the Csiszár sum identity [3]; $ (d)$ is derived by substituting the random variables $ U_i = (Y^{i-1},Z^N_{i+1}, S^{i-1}, S^N_{i+1}), V_i =(W,U_i)$ , and $ (e)$ follows by defining a time-sharing random variable $ Q$ and defining $ U = (U_Q,Q), V = (V_Q,Q), Y = YQ,$ and $ Z = Z_Q$ .",
"This completes the proof."
]
] | 1605.05693 |
[
[
"Expected values of parameters associated with the minimum rank of a\n graph"
],
[
"Abstract We investigate the expected value of various graph parameters associated with the minimum rank of a graph, including minimum rank/maximum nullity and related Colin de Verdi\\`ere-type parameters.",
"Let $G(v,p)$ denote the usual Erd\\H{o}s-R\\'enyi random graph on $v$ vertices with edge probability $p$.",
"We obtain bounds for the expected value of the random variables ${\\rm mr}(G(v,p))$, ${\\rm M}(G(v,p))$, $\\nu(G(v,p))$ and $\\xi(G(v,p))$, which yield bounds on the average values of these parameters over all labeled graphs of order $v$."
],
[
"Introduction",
"The set of $v\\times v$ real symmetric matrices will be denoted by $\\mathbb {R}^{(v)}$ .",
"For $A\\in \\mathbb {R}^{(v)}$ , the graph of $A$ , denoted $\\mathcal {G}(A)$ , is the graph with vertices $\\lbrace 1,\\dots ,v \\rbrace $ and edges $\\lbrace \\lbrace i,j \\rbrace : ~a_{ij} \\ne 0, 1 \\le i <j \\le v \\rbrace $ .",
"Note that the diagonal of $A$ is ignored in determining $\\mathcal {G}(A)$ .",
"The minimum rank of a graph $G$ on $v$ vertices is $\\operatorname{mr}(G)=\\min \\lbrace \\operatorname{rank}A : A \\in \\mathbb {R}^{(v)}, \\mathcal {G}(A)=G\\rbrace .$ The maximum nullity or maximum corank of a graph $G$ is $\\operatorname{M}(G)=\\max \\lbrace \\operatorname{null}A : A \\in \\mathbb {R}^{(v)}, \\mathcal {G}(A)=G\\rbrace .$ Note that $\\operatorname{mr}(G)+\\operatorname{M}(G)=v.$ Here a graph is a pair $G=(V(G),E(G))$ , where $V$ is the (finite, nonempty) set of vertices and $E$ is the set of edges (an edge is a two-element subset of vertices); what we call a graph is sometimes called a simple undirected graph.",
"We use the notation $v(G)=|V(G)|$ and $e(G)=|E(G)|$ .",
"The minimum rank problem (of a graph, over the real numbers) is to determine $\\operatorname{mr}(G)$ for any graph $G$ .",
"See [12] for a survey of known results and discussion of the motivation for the minimum rank problem; an extensive bibliography is also provided there.",
"The minimum rank problem was a focus of the 2006 workshop “Spectra of families of matrices described by graphs, digraphs, and sign patterns\" held at the American Institute of Mathematics [2].",
"One of the questions raised during the workshop was: Question 1.1 What is the average minimum rank of a graph on $v$ vertices?",
"Formally, we define the average minimum rank of graphs of order $v$ to be the sum over all labeled graphs of order $v$ of the minimum ranks of the graphs, divided by the number of (labeled) graphs of order $v$ .",
"That is, $\\operatorname{amr}(v)=\\frac{\\sum _{v(G) = v} \\operatorname{mr}(G)}{2^{v \\atopwithdelims ()2}}.$ Let $G(v,p)$ denote the Erdős-Rényi random graph on $v$ vertices with edge probability $p$ .",
"That is, every pair of vertices is adjacent, independently, with probability $p$ .",
"Note that for $G(v,1/2)$ , every labeled $v$ -vertex graph is equally likely (each labeled graph is chosen with probability $2^{-\\binom{v}{2}}$ ), so $ \\operatorname{amr}(v)=E\\left[\\operatorname{mr}(G(v,1/2))\\right] .",
"$ Our goal in this paper is to determine statistics about the random variable $\\operatorname{mr}(G(v,p))$ and other related parameters.",
"We highlight the two main results of this paper by focusing on the $p=1/2$ case: Theorem 1.2 Given $\\operatorname{amr}(v)=E\\left[\\operatorname{mr}(G(v,1/2))\\right]$ , then for $v$ sufficiently large, $\\displaystyle \\left|\\operatorname{mr}(G(v,1/2))-\\operatorname{amr}(v)\\right|<\\sqrt{v\\ln \\ln v}$ with probability approaching 1 as $v\\rightarrow \\infty $ , and $0.146907v<\\operatorname{amr}(v)<0.5v+\\sqrt{7v\\ln v}$ .",
"In general, we show that the random variable $\\operatorname{mr}(G(v,p))$ is tightly concentrated around its mean (Section ), and establish lower and upper bounds for its expected value in Sections and .",
"We also establish an upper bound on the Colin de Verdière type parameter $\\xi (G)$ , which is related to $\\operatorname{M}(G)$ , in Section (the definition of $\\xi $ is given in that section).",
"This bound is used in Section to establish bounds on the expected value of the random variable $\\xi (G(v,p))$ .",
"The upper bound on $\\xi (G(v,p))$ may lead to a better upper bound on the expected value of $\\operatorname{M}(G(v,p))$ and hence a better lower bound on the expected value of $\\operatorname{mr}(G(v,p))$ ."
],
[
"Tight concentration of expected minimum rank",
"Although we are unable to determine precisely the mean of $\\operatorname{mr}(G(v,p))$ , in this section we show that this random variable is tightly concentrated around its mean, and thus $\\operatorname{mr}(G(v,1/2))$ is tightly concentrated around the average minimum rank.",
"A martingale is a sequence of random variables $X_0,\\dots ,X_{v-1}$ such that ${\\bf E}[X_{i+1} | X_i,X_{i-1},\\dots ,X_1]=X_i.$ The martingale we use is the vertex exposure martingale (as described on pages 94-95 of [1]) for the graph parameter $f(G)=\\frac{1}{2} \\operatorname{mr}(G)$ (the factor $\\frac{1}{2}$ is needed because deletion of a vertex may change the minimum rank by 2; see Corollary REF below).",
"$G(v,p)$ is sampled to obtain a specific graph $H$ , and $X_i$ is the expected value of the graph parameter $f(G)=\\frac{1}{2} \\operatorname{mr}(G)$ when the neighbors of vertices $v_1,\\dots ,v_i$ are known.",
"Since nothing is known for $X_0$ , $X_0={\\bf E}[f(G(v,p))]=\\frac{1}{2}{\\bf E}[\\operatorname{mr}(G(v,p))]$ .",
"Since the entire graph $H$ is revealed at stage $v-1$ , $X_{v-1}=\\frac{1}{2}\\operatorname{mr}(H)$ .",
"The method for showing tight concentration uses Azuma's inequality for martingales (see Section 7.2 of [1]) and was pioneered by Shamir and Spencer [19].",
"The following corollary of Azuma's inequality is used.",
"Theorem 2.1 [1] Let $b=X_0,\\dots ,X_v$ be a martingale with $| X_{i+1}-X_i |\\le 1$ for all $0\\le i\\le v$ .",
"Then $\\Pr [|X_v-b|>\\beta \\sqrt{v}]<2e^{-\\beta ^2/2}.$ The proof that derives the tight concentration of the chromatic number of the random graph [1] from [1] via the vertex exposure martingale remains valid for any graph parameter $f(G)$ such that when $G$ and $H$ differ only in the exposure of a single vertex, then $| f(G) - f(H) | \\le 1$ .",
"Theorem 2.2 Let $p\\in (0,1)$ .",
"Let $f$ be a graph invariant such that for any graphs $G$ and $H$ , if $x\\in V(G)=V(H)$ and $G-x = H-x$ , then $| f(G) - f(H) | \\le 1$ .",
"Let $\\mu ={\\bf E}\\left[f(G(v,p))\\right]$ .",
"Then, for any $\\beta >0$ , $ \\Pr \\left[\\left|f(G(v,p))-\\mu \\right|>\\beta \\sqrt{v-1}\\right]<2e^{-\\beta ^2/2} .$ Corollary 2.3 Let $p\\in (0,1)$ be fixed and let $\\mu ={\\bf E}\\left[\\operatorname{mr}(G(v,p))\\right]$ .",
"For any $\\beta >0$ , $ \\Pr \\left[\\left|\\operatorname{mr}(G(v,p))-\\mu \\right|>2\\beta \\sqrt{v-1}\\right]<2e^{-\\beta ^2/2} .",
"$ In particular, $ \\left|\\operatorname{mr}(G(v,p))-\\mu \\right|<\\sqrt{v\\ln \\ln v} $ with probability approaching 1 as $v\\rightarrow \\infty $ .",
"It is well-known that for any graph $G$ and any vertex $x\\in V(G)$ , $ 0\\le \\operatorname{mr}(G)-\\operatorname{mr}(G-x)\\le 2.$ Thus if $V(H)=V(G)$ and $G-x = H-x$ , then $| \\operatorname{mr}(G) - \\operatorname{mr}(H) | \\le 2$ .",
"For the first statement, apply Theorem REF with $f(G)=\\frac{1}{2} \\operatorname{mr}(G)$ .",
"For the second statement, let $\\beta =\\frac{1}{2} \\sqrt{\\ln \\ln v}$ and conclude $ \\Pr \\left[\\left|\\operatorname{mr}(G(v,p))-\\mu \\right|>\\sqrt{v\\ln \\ln v}\\right]<2\\left(\\frac{1}{{\\ln v}}\\right)^{1/8}.$ Note that Corollary REF gives the result in Theorem REF (REF )."
],
[
"Observations on parameters of random graphs",
"Large deviation bounds easily show that the degree sequence of the random graph is tightly concentrated.",
"In this section, we provide some well-known results that will be used later.",
"The version of the Chernoff-Hoeffding bound that we use is given in [1].",
"Theorem 3.1 [1] Let $X_i$ , $1\\le i\\le n$ , be mutually independent random variables with all ${\\bf E}[X_i]=0$ and all $|X_i|\\le 1$ .",
"Set $S=X_1+\\cdots +X_n$ .",
"Then for any $a>0$ , $\\Pr [S>a]<\\exp \\lbrace -a^2/(2n)\\rbrace .$ It is well-known that Theorem REF can be applied to the number of edges in a random graph: Theorem 3.2 Let $p$ be fixed and let $G$ be distributed according to $G(v,p)$ .",
"Then, $ e(G)\\le p\\binom{v}{2}+v\\sqrt{2\\ln v}, $ with probability at least $1-v^{-2}$ .",
"In addition, $e(G)\\ge p\\binom{v}{2}-v\\sqrt{2\\ln v}$ with probability at least $1-v^{-2}$ .",
"Let $G$ be distributed according to the random variable $G(v,p)$ .",
"We may regard $\\left\\lbrace \\lbrace x,y\\rbrace \\in E(G) : x\\ne y\\right\\rbrace $ to be $\\binom{v}{2}$ mutually independent indicator random variables.",
"Subtract $p$ from each and they become random variables with mean 0 and magnitude at most 1.",
"Using Theorem REF , we see that $\\Pr \\left[e(G)-p\\binom{v}{2}>a\\right]<\\exp \\left\\lbrace -a^2/\\left(2\\binom{v}{2}\\right)\\right\\rbrace $ .",
"Choose $a=v\\sqrt{2\\ln v}$ ; we see that $ e(G)-p\\binom{v}{2}\\le v\\sqrt{2\\ln v} , $ with probability at least $1-v^{-2}$ .",
"By multiplying the random variables above by $-1$ , we obtain $ e(G)-p\\binom{v}{2}\\ge -v\\sqrt{2\\ln v}, $ with probability at least $1-v^{-2}$ .",
"Let $\\delta (G)$ (respectively, $\\Delta (G)$ ) denote the minimum (maximum) degree of a vertex of $G$ .",
"Theorem REF can also be applied to the neighborhood of each vertex to give bounds on $\\delta (G)$ and $\\Delta (G)$ .",
"Theorem 3.3 Let $p$ be fixed and let $G$ be distributed according to $G(v,p)$ .",
"Then, $ pv-\\sqrt{6v\\ln v} \\le \\delta (G)\\le \\Delta (G)\\le pv+\\sqrt{6v\\ln v}$ with probability at least $1-2v^{-2}$ .",
"Let $G$ be distributed according to the random variable $G(v,p)$ .",
"For each $x\\in V(G)$ , we may regard $\\left\\lbrace \\lbrace x,y\\rbrace \\in E(G) : y\\ne x\\right\\rbrace $ to be $v-1$ mutually independent indicator random variables.",
"Using Theorem REF , we see that $\\Pr \\left[\\left|\\deg (x)-p(v-1)\\right|>a\\right]<2\\exp \\lbrace -a^2/(2(v-1))\\rbrace $ .",
"Thus, the probability that there exists a vertex with degree that deviates by more than $a$ from $p(v-1)$ is at most $ v\\times 2\\exp \\lbrace -a^2/(2(v-1))\\rbrace .",
"$ Choose $a=\\sqrt{6v\\ln v}$ and we see that, simultaneously for all $x\\in V(G)$ , $ \\left|\\deg (x)-pv\\right|\\le \\sqrt{6v\\ln v}, $ with probability at least $1-2v^{-2}$ ."
],
[
"A lower bound for expected minimum rank",
"In this section we show that if $v$ is sufficiently large, then the expected value of $\\operatorname{mr}(G(v,p))$ is at least $c(p) v + o(v)$ , where $c(p)$ is the solution to equation (REF ) below.",
"In the case $p=1/2$ , $c(p)\\approx 0.1469077$ , so the average minimum rank is greater than $0.146907v$ for $v$ sufficiently large.",
"The zero-pattern $\\zeta ({\\bf x})$ of the real vector ${\\bf x}= (x_1,\\ldots , x_\\ell )$ is the $(0, *)$ -vector obtained from ${\\bf x}$ by replacing its nonzero entries by $*$ .",
"The support of the zero pattern ${\\bf z}=(z_1,\\dots ,z_\\ell )$ is the set $S({\\bf z})=\\lbrace i : z_i\\ne 0\\rbrace $ .",
"We modify the proof of Theorem 4.1 from [18] to obtain the following result.",
"Theorem 4.1 If ${\\bf f}({\\bf x}) = (f_1({\\bf x}), f_2({\\bf x}),\\ldots , f_m({\\bf x}))$ is an $m$ -tuple of polynomials in $n$ variables over a field $F$ with $m\\ge n$ , each $f_i$ of degree at most $d$ , then the number of zero-patterns ${\\bf z}=\\zeta ({\\bf f}({\\bf x}))$ with $| S({\\bf z}) |\\le s$ is at most $ {n+sd \\atopwithdelims ()n} .$ We follow the proof in [18].",
"Assume that the $m$ -tuple ${\\bf f}=(f_1,\\ldots ,f_m)$ of polynomials over field $F$ has the $M$ zero-patterns ${\\bf z}_1,\\dots ,{\\bf z}_M$ .",
"Choose ${\\bf u}_1,\\ldots ,{\\bf u}_M\\in F^n$ such that $\\zeta ({\\bf f}({\\bf u}_i))={\\bf z}_i$ .",
"Set $ g_i=\\prod _{k\\in S({\\bf z}_i)}f_k .",
"$ Note that $ g_i({\\bf u}_j)\\ne 0\\qquad \\mbox{if and only if}\\qquad S({\\bf z}_i)\\subseteq S({\\bf z}_j).",
"$ We show that polynomials $g_1,\\ldots ,g_M$ are linearly independent.",
"Assume on the contrary that there is a nontrivial linear combination $\\sum _{i=1}^M\\beta _ig_i=0$ , where each $\\beta _i\\in F$ .",
"Let $j$ be a subscript such that $|S({\\bf z}_j)|$ is minimal among the $S({\\bf z}_i)$ with $\\beta _i\\ne 0$ , so for every $i$ such that $i\\ne j$ and $\\beta _i\\ne 0$ , $S({\\bf z}_i)\\lnot \\subseteq S({\\bf z}_j)$ .",
"So substituting ${\\bf u}_j$ into the linear combination gives $\\beta _j g_j({\\bf u}_j)= 0$ , a contradiction.",
"Thus, $g_1,\\ldots ,g_M$ are linearly independent over $F$ .",
"Each $g_i$ has degree at most $sd$ and the dimension of the space of polynomials of degree $\\le D$ is exactly $\\binom{n+D}{n}$ .",
"By Sylvester's Law of Inertia, every real symmetric $v\\times v$ matrix of rank at most $r$ can be expressed in the form $X^TD_iX$ for some $i$ such that $0\\le i \\le r$ , where $D_i={\\rm diag}(1,\\dots ,1,-1,\\dots ,-1)$ is an $r\\times r$ diagonal matrix with $i$ diagonal entries equal to 1 and $r-i$ equal to $-1$ and $X$ is an $r\\times v$ real matrix.",
"There are $r+1$ diagonal matrices $D_i$ .",
"Let each entry of $X$ be a variable; the total number of variables is $rv$ and each entry of the matrix $X^TD_iX$ is a polynomial of degree at most 2.",
"Let $c(p)$ be the solution to $\\frac{(c+p)^{2c+2p}}{(c)^{2c}(p)^{2p}}p^p(1-p)^{(1-p)}=1$ for a fixed value of $p (0<p<1)$ .",
"This equation has a unique solution, because it is equivalent to $\\frac{(c+p)^{2 c+2 p}}{c^{2 c} }=\\frac{p^p}{(1-p)^{(1-p)}}$ , and for a fixed $p$ and $c\\ge 0$ , $\\frac{(c+p)^{2 c+2 p}}{c^{2 c} }$ is a strictly increasing function of $c$ and $p^{2p}<\\frac{p^p}{(1-p)^{(1-p)}}$ .",
"The values of $c(p), 0<p<1$ are graphed in Figure REF .",
"Figure: The graph of c(p)c(p)Theorem 4.2 Let $G$ be distributed according to $G(v,p)$ for a fixed $p$ , $0<p<1$ .",
"For any $c<c(p)$ , the expectation $ {\\bf E}\\left[\\operatorname{mr}(G)\\right]$ satisfies $ {\\bf E}\\left[\\operatorname{mr}(G)\\right]>cv$ for $v$ sufficiently large.",
"Furthermore, for any such $c$ , $\\Pr \\left[\\operatorname{mr}(G(v,p))\\le cv\\right]\\rightarrow 0$ as $v\\rightarrow \\infty $ .",
"Let $G$ be distributed according to $G(v,p)$ .",
"Let $\\mathcal {E}$ be the event that $\\left|e(G)-p\\binom{v}{2}\\right|\\le v\\sqrt{2\\ln v}$ .",
"By the law of total expectation, ${\\bf E}[\\operatorname{mr}(G)] & = & {\\bf E}[\\operatorname{mr}(G)\\mid {\\mathcal {E}}]\\Pr [{\\mathcal {E}}]+{\\bf E}[\\operatorname{mr}(G)\\mid {\\mathcal {E}}^c]\\Pr [{\\mathcal {E}}^c] \\\\& \\ge & {\\bf E}[\\operatorname{mr}(G) \\mid {\\mathcal {E}}]\\Pr \\left[{\\mathcal {E}}\\right] \\\\& \\ge & (r+1)\\Pr [\\operatorname{mr}(G)>r\\mid {\\mathcal {E}}]\\Pr [{\\mathcal {E}}] \\\\& = & (r+1)\\left(1-\\Pr [\\operatorname{mr}(G)\\le r\\mid {\\mathcal {E}}]\\right)\\left(1-\\Pr [{\\mathcal {E}^c}]\\right) \\\\& \\ge & (r+1)-(r+1)\\Pr [\\operatorname{mr}(G)\\le r\\mid {\\mathcal {E}}]-(r+1)\\Pr [{\\mathcal {E}}^c] \\\\& \\ge & (r+1)-v\\Pr [\\operatorname{mr}(G)\\le r\\mid {\\mathcal {E}}]-v\\Pr [{\\mathcal {E}}^c]$ Theorem REF shows that $v\\Pr [{\\mathcal {E}}^c]\\le v^{-1}$ .",
"It remains to bound $\\Pr [\\operatorname{mr}(G)\\le r\\mid {\\mathcal {E}}]$ .",
"$\\Pr \\left[\\operatorname{mr}(G)\\le r\\mid {\\mathcal {E}}\\right]& = & \\sum _{\\scriptsize \\begin{array}{l} G : v(G)=v, \\operatorname{mr}(G)\\le r \\\\ \\left|e(G)-p\\binom{v}{2}\\right|\\le v\\sqrt{2\\ln v}\\end{array}}\\Pr [G\\in G(v,p)] \\\\& = & \\sum _{\\scriptsize \\begin{array}{l} G : v(G)=v, \\operatorname{mr}(G)\\le r \\\\ \\left|e(G)-p\\binom{v}{2}\\right|\\le v\\sqrt{2\\ln v}\\end{array}}p^{e(G)}(1-p)^{\\binom{v}{2}-e(G)} \\\\& = & \\sum _{\\scriptsize \\begin{array}{l} G : v(G)=v, \\operatorname{mr}(G)\\le r \\\\ \\left|e(G)-p\\binom{v}{2}\\right|\\le v\\sqrt{2\\ln v}\\end{array}}\\left(\\frac{p}{1-p}\\right)^{e(G)}(1-p)^{\\binom{v}{2}}$ If $p<1/2$ , then we use a lower bound for $e(G)$ , given $\\mathcal {E}$ ; if $p>1/2$ , an upper bound.",
"So, we can bound the term inside the summation as $ \\left(\\frac{p}{1-p}\\right)^{e(G)}(1-p)^{\\binom{v}{2}}\\le \\left(\\frac{\\max \\lbrace p,1-p\\rbrace }{\\min \\lbrace p,1-p\\rbrace }\\right)^{v\\sqrt{2\\ln v}}\\left(p^p(1-p)^{(1-p)}\\right)^{\\binom{v}{2}} .",
"$ Hence, ${\\Pr \\left[\\operatorname{mr}(G)\\le r\\mid {\\mathcal {E}}\\right]} \\\\& \\le & \\left(\\frac{\\max \\lbrace p,1-p\\rbrace }{\\min \\lbrace p,1-p\\rbrace }\\right)^{v\\sqrt{2\\ln v}}\\left(p^p(1-p)^{(1-p)}\\right)^{\\binom{v}{2}}\\left|\\left\\lbrace G : v(G)=v, \\left|e(G)-p\\binom{v}{2}\\right|\\le v\\sqrt{2\\ln v} , \\operatorname{mr}(G)\\le r\\right\\rbrace \\right| .$ The number of $v$ vertex graphs with between $p\\binom{v}{2}-v\\sqrt{2\\ln v}$ and $p\\binom{v}{2}+v\\sqrt{2\\ln v}$ edges and minimum rank at most $r$ is at most the number of $v\\times v$ symmetric pattern matrices obtained as $X^TD_iX, i=0,\\dots ,r$ with $X$ an $r\\times v$ matrix for which the cardinality of the support of the superdiagonal entries is at most $p\\binom{v}{2}+v\\sqrt{2\\ln v}$ .",
"We can apply Theorem REF with $n=rv$ , $d=2$ and $s\\le p\\binom{v}{2}+v\\sqrt{2\\ln v}$ .",
"Therefore, because there are $r+1$ diagonal matrices, ${\\Pr \\left[\\operatorname{mr}(G)\\le r\\mid {\\mathcal {E}}\\right]} \\\\& \\le & \\left(\\frac{\\max \\lbrace p,1-p\\rbrace }{\\min \\lbrace p,1-p\\rbrace }\\right)^{v\\sqrt{2\\ln v}}\\left(p^p(1-p)^{(1-p)}\\right)^{\\binom{v}{2}}(r+1)\\binom{rv+2p\\binom{v}{2}+2v\\sqrt{2\\ln v}}{rv} .$ By Corollary REF in Appendix , for fixed $c$ and $p$ with $r=cv$ , $ \\binom{rv+2p\\binom{v}{2}+2v\\sqrt{2\\ln v}}{rv}\\le \\left((1+o(1))\\left(\\frac{(c+p)^{c+p}}{c^cp^p}\\right)\\right)^{v^2} .",
"$ Thus $ \\Pr [\\operatorname{mr}(G)\\le cv\\mid {\\mathcal {E}}]\\le \\left((1+o(1))\\frac{(c+p)^{2c+2p}}{(c)^{2c}(p)^{2p}}p^p(1-p)^{(1-p)}\\right)^{v^2/2} .",
"$ As long $c<c(p)$ and $v$ is sufficiently large, the quantity $v\\Pr \\left[\\operatorname{mr}(G)\\le r\\mid {\\mathcal {E}}\\right]$ is less than 1, giving $ {\\bf E}[\\operatorname{mr}(G)]\\ge (r+1)-v\\Pr [\\operatorname{mr}(G)\\le r\\mid {\\mathcal {E}}]-v\\Pr [{\\mathcal {E}}^c]>r+1-o(1)\\ge r. $ Furthermore, as long as $c<c(p)$ , $ \\Pr \\left[\\operatorname{mr}(G)\\le cv\\mid {\\mathcal {E}}\\right]\\rightarrow 0$ as $v\\rightarrow \\infty $ , and by Theorem REF , $\\Pr [{\\mathcal {E}}^c]\\rightarrow 0$ as $v\\rightarrow \\infty $ .",
"Since $\\Pr [\\operatorname{mr}(G)\\le cv] \\le \\Pr [\\operatorname{mr}(G)\\le cv\\mid {\\mathcal {E}}] + \\Pr [{\\mathcal {E}}^c],$ $\\Pr [\\operatorname{mr}(G)\\le cv]\\rightarrow 0$ as $v\\rightarrow \\infty $ .",
"Corollary 4.3 For $v$ sufficiently large, the average minimum rank over all labeled graphs of order $v$ satisfies $ \\operatorname{amr}(v)>0.146907v.$ Furthermore, if $G$ is chosen at random from all labeled graphs of order $v$ , $\\Pr [\\operatorname{mr}(G)\\le 0.146907v]\\rightarrow 0$ as $v\\rightarrow \\infty $ .",
"For $p=1/2$ , $ {\\bf E}[\\operatorname{mr}(G)]=\\operatorname{amr}(v)$ and $0.146907<c(p)$ .",
"Note that Corollary REF gives the lower bound in Theorem REF (REF ).",
"We note further the lack of symmetry with respect to $p$ .",
"The value $c(p)$ approaches zero as $p$ approaches zero, which is not the case with the upper bound that we describe in the next section."
],
[
"An upper bound for expected minimum rank",
"In this section we show that if $v$ is sufficiently large, then the expected value of $\\operatorname{mr}(G(v,p))$ is at most $ (1-p)v+\\sqrt{7v\\ln v}$ .",
"Thus the average minimum rank for graphs of order $v$ is at most $0.5v+\\sqrt{7v\\ln v}$ .",
"Let $\\kappa (G)$ denote the vertex connectivity of $G$ .",
"That is, if $G$ is not complete, it is the smallest number $k$ such that there is a set of vertices $S$ , with $|S|=k$ , for which $G-S$ is disconnected.",
"By convention, $\\kappa (K_v)=v-1$ .",
"Following the terminology of [15], for a graph $G$ an orthogonal representation of $G$ of dimension $d$ is a set of vectors in $\\mathbb {R}^d$ , one corresponding to each vertex, such that if two vertices are nonadjacent, then their corresponding vectors are orthogonal.",
"Every graph has an orthogonal representation in any dimension by associating the zero vector with every vertex.",
"A faithful orthogonal representation of $G$ of dimension $d$ is a set of vectors in $\\mathbb {R}^d$ , one corresponding to each vertex, such that two (distinct) vertices are nonadjacent if and only if their corresponding vectors are orthogonal.",
"Note that in the minimum rank literature, the term “orthogonal representation\" is customarily used for what is here called a faithful orthogonal representation.",
"The following result of Lovász, Saks and Schrijver [15] (see also the note on errata, [16] Theorem 1.1) is the basis for an upper bound for minimum rank.",
"Theorem 5.1 [15] Every graph $G$ on $v$ vertices has a faithful orthogonal representation of dimension $v-\\kappa (G)$ .",
"Let $\\operatorname{mr}_{+}(G)$ denote the minimum rank among all symmetric positive semidefinite matrices $A$ such that $\\mathcal {G}(A)=G$ , and let $\\operatorname{M}_{+}(G)$ denote the maximum nullity among all such matrices.",
"It is well known (and easy to see) that every faithful orthogonal representation of dimension $d$ gives rise to a positive semidefinite matrix of rank $d$ and vice versa.",
"Corollary 5.2 For any graph $G$ on $v$ vertices, $ \\operatorname{mr}(G)\\le \\operatorname{mr}_{+}(G)\\le v-\\kappa (G),$ or equivalently, $ \\kappa (G)\\le \\operatorname{M}_{+}(G)\\le \\operatorname{M}(G).$ Our proof of the upper bound on the expected value of $\\operatorname{mr}(G(v,p))$ uses the bound (REF ) and the relationship (on average) between the connectivity $\\kappa (G)$ and the minimum degree $\\delta (G)$ .",
"At the AIM workshop [2] it was conjectured that for any graph $G$ , $\\delta (G)\\le M(G)$ , or equivalently $\\operatorname{mr}(G)\\le v(G)-\\delta (G)$ [9].",
"The conjecture was proved for bipartite graphs in [4] but remains open in general.",
"In [15] it is reported that in 1987, Maehara made a conjecture equivalent to $\\operatorname{mr}_{+}(G)\\le v(G)-\\delta (G) $ , which would imply $\\operatorname{mr}(G)\\le v(G)-\\delta (G) $ .",
"Theorem 5.3 Let $G$ be distributed according to $G(v,p)$ .",
"For $v$ sufficiently large, the expected value of minimum rank satisfies ${\\bf E}[\\operatorname{mr}(G)]\\le (1-p)v+\\sqrt{7v\\ln v}$ .",
"For $v$ sufficiently large, the average minimum rank over all labeled graphs of order $v$ satisfies $ \\operatorname{amr}(v)\\le 0.5v+\\sqrt{7v\\ln v}.$ In [8] (see also section 7.2 of [7]), Bollobás and Thomason prove that if $G$ is distributed according to $G(v,p)$ , then $\\Pr [\\kappa (G)=\\delta (G)]\\rightarrow 1$ as $v\\rightarrow \\infty $ , without any restriction on $p$ .",
"Lemma REF in Appendix shows that for $p$ fixed and $v$ large enough, $\\Pr [\\kappa (G)<\\delta (G)]\\le 3v^{-2}$ .",
"Let $\\mathcal {E}$ be the event that $\\kappa (G)=\\delta (G)$ and $\\delta (G)\\ge pv-\\sqrt{6v\\ln v}$ .",
"For $G$ distributed according to $G(v,p)$ , the law of total expectation gives ${\\bf E}[\\kappa (G)] & = & {\\bf E}[\\kappa (G)\\mid {\\mathcal {E}}]\\Pr [{\\mathcal {E}}]+{\\bf E}[\\kappa (G)\\mid {\\mathcal {E}}^c]\\Pr [{\\mathcal {E}}^c] \\\\& \\ge & \\left(pv-\\sqrt{6v\\ln v}\\right)\\left(1-\\Pr [{\\mathcal {E}}^c]\\right) \\\\& \\ge & pv-\\sqrt{6v\\ln v}-v\\left(\\Pr [\\delta (G)<pv-\\sqrt{6v\\ln v}]+\\Pr [\\kappa (G)<\\delta (G)]\\right).$ We use Theorem REF and the result that $v\\Pr [\\kappa (G)<\\delta (G)]\\le 3v^{-1}$ to see that $ {\\bf E}\\left[\\kappa (G)\\right]\\ge pv-\\sqrt{6v\\ln v}-2v^{-1}-3v^{-1}\\ge pv-\\sqrt{7v\\ln v} , $ for $v$ sufficiently large.",
"By (REF ), ${\\bf E}[\\operatorname{mr}(G)]\\le (1-p)v+\\sqrt{7v\\ln v}$ .",
"Theorem REF gives the upper bound in Theorem REF (REF ).",
"Note that Theorem REF actually establishes ${\\bf E}[\\operatorname{mr}_+(G)]\\le (1-p)v+\\sqrt{7v\\ln v}$ .",
"Since $\\operatorname{mr}(G)\\le \\operatorname{mr}_+(G)$ for any graph $G$ , the lower bound in Theorem REF in Section certainly bounds ${\\bf E}[\\operatorname{mr}_+(G)]$ from below."
],
[
"Bounds for $\\nu (G)$ and {{formula:d41639fc-0ba3-4625-855c-e7a526c061f2}}",
"In this section we discuss the the Colin de Verdière type parameters $\\nu (G)$ and $\\xi (G)$ , and establish an upper bound on $\\xi (G)$ in terms of the number of edges of the graph.",
"This upper bound, and a known lower bound for $\\nu (G)$ , have implications for the average value of $\\nu $ and $\\xi $ (see Section ).",
"In 1990 Colin de Verdière ([10] in English) introduced the graph parameter $\\mu $ that is equal to the maximum multiplicity of eigenvalue 0 among all matrices satisfying several conditions including the Strong Arnold Hypothesis (defined below).",
"The parameter $\\mu $ , which is used to characterize planarity, is the first of several parameters that require the Strong Arnold Hypothesis and bound the maximum nullity from below (called Colin de Verdière type parameters).",
"All the Colin de Verdière type parameters we discuss have been shown to be minor monotone.",
"A contraction of $G$ is obtained by identifying two adjacent vertices of $G$ , deleting any loops that arise in this process, and replacing any multiple edges by a single edge.",
"A minor of $G$ arises by performing a sequence of deletions of edges, deletions of isolated vertices, and/or contractions of edges.",
"A graph parameter $\\beta $ is minor monotone if for any minor $G^{\\prime }$ of $G$ , $\\beta (G^{\\prime }) \\le \\beta (G)$ .",
"A symmetric real matrix $M$ is said to satisfy the Strong Arnold Hypothesis (SAH) provided there does not exist a nonzero real symmetric matrix $X$ satisfying $AX = 0$ , $A\\circ X = 0$ , and $I\\circ X=0$ , where $\\circ $ denotes the Hadamard (entrywise) product and $I$ is the identity matrix.",
"The SAH is equivalent to the requirement that certain manifolds intersect transversally.",
"Specifically, for $A=[a_{ij}]\\in \\mathbb {R}^{(v)}$ let $\\mathcal {R}_A=\\lbrace B\\in \\mathbb {R}^{(v)}: \\operatorname{rank}B = \\operatorname{rank}A\\rbrace ,$ and $\\mathcal {S}_A = \\lbrace B\\in \\mathbb {R}^{(v)}: \\mathcal {G}(B)=\\mathcal {G}(A)\\rbrace .$ Then $\\mathcal {R}_A$ and $\\mathcal {S}_A$ intersect transversally at $A$ if and only if $A$ satisfies the SAH (see [14]).",
"Another minor monotone parameter, introduced by Colin de Verdière in [11], is denoted by $\\nu (G)$ and defined to be the maximum nullity among matrices $A$ that satisfy: $\\mathcal {G}(A )=G$ ; $A$ is positive semidefinite; $A$ satisfies the Strong Arnold Hypothesis.",
"Clearly $\\nu (G)\\le \\operatorname{M}_+(G)$ .",
"The parameter $\\xi (G)$ was introduced in [3] as a Colin de Verdière type parameter intended for use in computing maximum nullity and minimum rank, by removing any unnecessary restrictions while preserving minor monotonicity.",
"Define $\\xi (G)$ to be the maximum multiplicity of 0 as an eigenvalue among matrices $A\\in \\mathbb {R}^{(v)}$ that satisfy: $\\mathcal {G}(A )=G$ .",
"$A$ satisfies the Strong Arnold Hypothesis.",
"Clearly, $\\nu (G) \\le \\xi (G)\\le \\operatorname{M}(G)$ .",
"The following lower bound on $\\nu (G)$ has been established by van der Holst using the results of Lovász, Saks and Schrijver.",
"Theorem 6.1 [13] For every graph $G$ , $\\kappa (G)\\le \\nu (G)\\le \\xi (G).$ The following bound on the Colin de Verdière number $\\mu $ in terms of the number of edges $e(G)$ is given in [17] for any connected graph $G\\ne K_{3,3}:$ $e(G) \\ge \\frac{\\mu (G)(\\mu (G)+1)}{2}.$ We will show that for any connected graph $G$ , $e(G) + b\\ge \\frac{\\xi (G)(\\xi (G)+1)}{2}$ where $b=1$ if $G$ is bipartite and $b=0$ otherwise.",
"For a manifold $\\mathcal {M}$ and matrix $A\\in \\mathcal {M}$ , let $\\mathcal {T}_{\\mathcal {M}_A}$ be the tangent space in $\\mathbb {R}^{(v)}$ to $\\mathcal {M}$ at $A$ and let $\\mathcal {N}_{\\mathcal {M}_A}$ be the normal (orthogonal complement) to $\\mathcal {T}_{\\mathcal {M}_A}$ .",
"Observation 6.2 [14] $\\hbox{}$ $\\mathcal {T}_{\\mathcal {S}_A}=\\lbrace B: \\forall i\\ne j, a_{ij}=0\\Rightarrow b_{ij}=0\\rbrace $ .",
"$\\mathcal {N}_{\\mathcal {S}_A}=\\lbrace X: \\forall i, x_{ii}=0 \\mbox{ and } \\forall i\\ne j, a_{ij}\\ne 0\\Rightarrow x_{ij}=0\\rbrace $ .",
"$\\mathcal {T}_{\\mathcal {R}_A}=\\lbrace WA+AW^T: W\\in \\mathbb {R}^{n\\times n}\\rbrace =\\lbrace B\\in \\mathbb {R}^{(v)}: {\\bf v}^TB{\\bf v}=0~ \\forall {\\bf v}\\in \\ker A\\rbrace $ .",
"$\\mathcal {N}_{\\mathcal {R}_A}=\\operatorname{span}(\\lbrace {\\bf v}{\\bf v}^T: {\\bf v}\\in \\ker A\\rbrace )=\\lbrace X\\in \\mathbb {R}^{(v)}:AX = 0\\rbrace $ .",
"Clearly $\\dim \\mathcal {T}_{\\mathcal {S}_A}= e(G)+v$ .",
"These observations can also be used to provide the exact dimension of $\\mathcal {N}_{\\mathcal {R}_A}$ and thus of $\\mathcal {T}_{\\mathcal {R}_A}$ .",
"Proposition 6.3 Let $A\\in \\mathbb {R}^{(v)}$ and let ${\\bf u}_1,\\dots ,{\\bf u}_q$ be an orthonormal basis for $\\ker A$ .",
"Then $U=\\lbrace {\\bf u}_i{\\bf u}_i^T:1\\le i\\le q\\rbrace \\cup \\lbrace {\\bf u}_i{\\bf u}_j^T+{\\bf u}_j{\\bf u}_i^T:1\\le i<j\\le q\\rbrace $ is a basis for $ \\operatorname{span}(\\lbrace {\\bf v}{\\bf v}^T: {\\bf v}\\in \\ker A\\rbrace ).$ Thus $\\dim \\mathcal {N}_{\\mathcal {R}_A}=\\frac{q(q+1)}{2}$ .",
"Let $\\mathcal {N}=\\operatorname{span}(\\lbrace {\\bf v}{\\bf v}^T: {\\bf v}\\in \\ker A\\rbrace )$ .",
"Since ${\\bf u}_i{\\bf u}_j^T+{\\bf u}_j{\\bf u}_i^T=({\\bf u}_i+{\\bf u}_j)({\\bf u}_i+{\\bf u}_j)^T-{\\bf u}_i{\\bf u}_i^T-{\\bf u}_j{\\bf u}_j^T$ , $U\\subset \\mathcal {N}.$ Show $U$ spans $\\mathcal {N}$ : $\\left(\\sum _{i=1}^q s_i{\\bf u}_i\\right)\\left(\\sum _{j=1}^q s_j{\\bf u}_i\\right)^T=\\sum _{i=1}^q\\sum _{j=1}^q s_is_j{\\bf u}_i{\\bf u}_j^T=\\sum _{i=1}^q s_i^2{\\bf u}_i{\\bf u}_i^T+\\sum _{1\\le i<j\\le q} s_is_j({\\bf u}_i{\\bf u}_j^T+{\\bf u}_j{\\bf u}_i^T)$ Show $U$ is linearly independent: Let $Y=\\sum _{i=1}^q s_i{\\bf u}_i{\\bf u}_i^T+\\sum _{1\\le i<j\\le q} s_{ij}({\\bf u}_i{\\bf u}_j^T+{\\bf u}_j{\\bf u}_i^T)$ and suppose $Y=0.$ For any $k$ , $0={\\bf u}_k^TY{\\bf u}_k=s_k$ and for $\\ell <k$ , $0={\\bf u}_\\ell Y{\\bf u}_k=s_{\\ell k}$ , and $U$ is linearly independent.",
"Corollary 6.4 $\\dim \\mathcal {T}_{\\mathcal {R}_A}=v\\operatorname{rank}A-\\frac{\\operatorname{rank}A(\\operatorname{rank}A-1)}{2}$ .",
"Let $\\operatorname{rank}A=r$ .",
"By Observation REF and Proposition REF , $\\dim \\mathcal {T}_{\\mathcal {R}_A}=\\dim \\mathbb {R}^{(v)}-\\dim \\mathcal {N}_{\\mathcal {R}_A}= \\frac{v(v+1)}{2}-\\frac{(v-r)(v-r+1)}{2}.",
"$ An optimal matrix for $\\xi (G)$ is a matrix $A$ such that $\\mathcal {G}(A)=G, \\operatorname{null}A=\\xi (A)$ , and $A$ has the Strong Arnold Hypothesis.",
"Theorem 6.5 Let $G$ be a connected graph.",
"$e(G) + b\\ge \\frac{\\xi (G)(\\xi (G)+1)}{2}$ where $b=1$ if $G$ is bipartite and every optimal matrix for $\\xi (G)$ has zero diagonal, and $b=0$ otherwise.",
"Let $A$ be an optimal matrix for $\\xi (G)$ , chosen to have at least one nonzero diagonal entry if there is such an optimal matrix.",
"Let $\\operatorname{rank}A=r$ .",
"The Strong Arnold Hypothesis for $A$ is $ \\mathcal {N}_{\\mathcal {R}_A}\\cap \\mathcal {N}_{\\mathcal {S}_A}=\\lbrace 0\\rbrace $ , which is equivalent by taking orthogonal complements to $\\mathcal {T}_{\\mathcal {R}_A}+ \\mathcal {T}_{\\mathcal {S}_A}= \\mathbb {R}^{(v)}$ Therefore $\\dim \\mathcal {T}_{\\mathcal {R}_A}+ \\dim \\mathcal {T}_{\\mathcal {S}_A}-\\dim (\\mathcal {T}_{\\mathcal {R}_A}\\cap \\mathcal {T}_{\\mathcal {S}_A}) &= & \\dim \\mathbb {R}^{(v)}\\\\vr-\\frac{r(r-1)}{2} + e(G) + v-\\dim (\\mathcal {T}_{\\mathcal {R}_A}\\cap \\mathcal {T}_{\\mathcal {S}_A}) &= & \\frac{v(v+1)}{2}$ $e(G) & = & \\frac{v(v+1)}{2} - vr+\\frac{r(r-1)}{2}- v + \\dim (\\mathcal {T}_{\\mathcal {R}_A}\\cap \\mathcal {T}_{\\mathcal {S}_A}) \\\\& = & \\frac{1}{2}((v-r)^2 +(v-r)) - v + \\dim (\\mathcal {T}_{\\mathcal {R}_A}\\cap \\mathcal {T}_{\\mathcal {S}_A}) \\\\& = & \\frac{\\xi (G)(\\xi (G)+1)}{2} - v + \\dim (\\mathcal {T}_{\\mathcal {R}_A}\\cap \\mathcal {T}_{\\mathcal {S}_A})$ Thus $\\frac{\\xi (G)(\\xi (G)+1)}{2} = e(G) + v - \\dim (\\mathcal {T}_{\\mathcal {R}_A}\\cap \\mathcal {T}_{\\mathcal {S}_A}).$ Let $D={\\rm diag}(d_1,\\dots ,d_n)$ be a diagonal matrix.",
"Then by Observation REF .REF , $DA+AD\\in \\mathcal {T}_{\\mathcal {R}_A}$ .",
"Clearly, $DA+AD\\in \\mathcal {T}_{\\mathcal {S}_A}$ , so $DA+AD\\in \\mathcal {T}_{\\mathcal {R}_A}\\cap \\mathcal {T}_{\\mathcal {S}_A}$ .",
"Let ${\\bf e}_k$ be the $k$ th standard basis vector of $\\mathbb {R}^{v}$ .",
"Define $D_k={\\rm diag}({\\bf e}_k)$ and $B_k=D_kA+AD_k$ .",
"Note that $(B_k)_{ij}=(\\delta _{ki}+\\delta _{kj}) a_{ij}$ , where $\\delta _{ii}=1$ and $\\delta _{ij}=0$ for $i\\ne j$ .",
"We show first that if $\\displaystyle \\sum _{k=1}^{v}c_k B_k=0 \\mbox{ and } c_t=0 \\mbox{ for some } t \\mbox{ such that } 1\\le t\\le v,$ then $c_k=0$ for all $ 1\\le k\\le v$ .",
"For every neighbor $y$ of $t$ , $ 0=\\left(\\sum _{k=1}^{v}c_k B_k\\right)_{ty}=\\sum _{k=1}^{v}c_k (\\delta _{kt}+\\delta _{ky}) a_{ty}=c_y a_{ty}.$ Since $\\lbrace t,y\\rbrace $ is an edge of $G$ , $a_{ty}\\ne 0$ , and so $c_y=0$ .",
"Since $G$ is connected, every vertex can be reached by a path from $t$ , and so $c_1=\\dots =c_{v}=0$ .",
"Since $\\sum _{k=1}^{v-1}c_k B_k = \\sum _{k=1}^{v}c_k B_k \\mbox{ with } c_v=0,$ it follows that for every graph $G$ and $\\xi (G)$ -optimal matrix $A$ (without any assumption about the diagonal), the matrices $B_k, k=1,\\dots ,v-1,$ are linearly independent, and thus $\\dim (\\mathcal {T}_{\\mathcal {R}_A}\\cap \\mathcal {T}_{\\mathcal {S}_A})\\ge v-1\\qquad \\mbox{ and }\\qquad \\frac{\\xi (G)(\\xi (G)+1)}{2} \\le e(G) +1,$ Now suppose that $A$ has a nonzero diagonal entry or $G$ is not bipartite.",
"We show that the matrices $B_k, k=1,\\dots ,v$ are linearly independent, so $\\dim (\\mathcal {T}_{\\mathcal {R}_A}\\cap \\mathcal {T}_{\\mathcal {S}_A})\\ge v\\qquad \\mbox{ and }\\qquad \\frac{\\xi (G)(\\xi (G)+1)}{2} \\le e(G)$ Let $\\sum _{k=1}^vc_k B_k=0.$ If $A$ has a nonzero diagonal entry $a_{tt}$ , then $ 0=\\left(\\sum _{k=1}^{v}c_k B_k\\right)_{tt}=2c_t a_{tt}$ , and so $c_t=0$ .",
"If $G$ is not bipartite, there is an odd cycle; without loss of generality let this odd cycle be $(1,\\dots ,t)$ .",
"Then for $i=1,\\dots , t-1$ , $ 0=\\left(\\sum _{k=1}^{v}c_k B_k\\right)_{i,i+1}=(c_i+c_{i+1}) a_{i,i+1};$ similarly $0=(c_t+c_{1}) a_{t,1}$ .",
"Since $ \\lbrace t,1\\rbrace $ and $\\lbrace i,i+1\\rbrace , i=1,\\dots ,t-1$ are edges of $G$ , $ c_i+c_{i+1}=0,i=1,\\dots , t-1, \\mbox{ and } c_t+c_{1}=0.$ By adding equations $(-1)^{i} (c_i+c_{i+1}=0), i=1,\\dots ,t-1$ to $c_t+c_{1}=0$ , we obtain $ 2 c_t=0.$ If $G$ is the disjoint union of its connected components $G_1,\\dots ,G_h$ , then $\\xi (G)=\\max _{i=1,\\dots ,h}\\lbrace \\xi (G_i)\\rbrace $ [3].",
"Corollary 6.6 For every graph $G$ , $ \\frac{\\xi (G)(\\xi (G)+1)}{2} \\le e(G) + 1.$ Example 6.7 The complete bipartite graph $K_{3,3}$ demonstrates that $b=1$ is sometimes necessary in the bound (REF ), because $\\xi (K_{3,3})=4$ , so $\\frac{\\xi (G)(\\xi (G)+1)}{2}=10$ , and $e(K_{3,3}) = 9$ ."
],
[
"Bounds for the expected value of $\\xi $",
"In this section we show that if $v$ is sufficiently large, then the expected value of $\\xi (G(v,p))$ is asymptotically at most $\\sqrt{p} v$ .",
"It follows that the average value of $\\xi $ for graphs of order $v$ is asymptotically at most $\\frac{1}{\\sqrt{2}} v$ .",
"We will make the notion of asymptotic expected value more precise, both for minimum rank and for $\\xi $ .",
"Define $\\overline{\\operatorname{mr}}(p)=\\limsup _{v\\rightarrow \\infty } \\frac{{\\bf E}\\left[\\operatorname{mr}(G(v,p))\\right]}{v}.$ (This is a careful definition, as the lim sup is almost certainly a limit.)",
"In previous sections we have shown that for $0<p<1$ , $c(p)\\le \\overline{\\operatorname{mr}}(p)\\le 1-p.$ Now define $\\bar{\\xi }(p)=\\liminf _{v\\rightarrow \\infty } \\frac{{\\bf E}\\left[\\xi (G(v,p))\\right]}{v}.$ The quantity $\\bar{\\xi }(p)$ should be compared to $1-\\overline{\\operatorname{mr}}(p)$ rather than $\\overline{\\operatorname{mr}}(p)$ , since $\\xi (G)$ measures a nullity rather than a rank.",
"Our starting point is an immediate consequence of Corollary REF .",
"Corollary 7.1 For every graph $G$ , $\\xi (G) & \\le & \\frac{1}{2}(-1+\\sqrt{9+8e(G) }).$ Corollary 7.2 For $0 < p < 1$ , $p \\le \\bar{\\xi }(p)\\le \\sqrt{p}.$ The proof that $p \\le \\bar{\\xi }(p)$ follows from Theorem REF by exactly the same reasoning that showed that $\\overline{\\operatorname{mr}}(p) \\le 1-p$ .",
"From inequality (REF ), if $e(G)\\ge 2$ , $\\xi (G) \\le \\sqrt{2e(G)}.$ For a fixed $\\epsilon >0$ , as $v\\rightarrow \\infty $ , almost all graphs sampled from $G(v,p)$ satisfy $ e(G) \\le (1+\\epsilon )\\frac{p}{2} v(v-1),$ so almost all graphs satisfy $\\xi (G)\\le \\sqrt{2e(G)} \\le \\sqrt{2(1+\\epsilon )\\frac{p}{2} v(v-1)}\\le \\sqrt{1+\\epsilon }\\sqrt{p}v.$ This completes the proof of the second inequality $\\bar{\\xi }(p)\\le \\sqrt{p}.$ Since for every graph $G$ , $\\xi (G)\\le \\operatorname{M}(G)$ and for every $v>1$ there exists a graph $H$ such that $\\xi (H)<\\operatorname{M}(H)$ , ${\\bf E}\\left[\\xi (G(v,p))\\right]$ is strictly less than ${\\bf E}\\left[\\operatorname{M}(G(v,p))\\right]$ for $v>1$ and $0 < p < 1$ .",
"However it is quite possible that taking the limit gives $\\bar{\\xi }(p) + \\overline{\\operatorname{mr}}(p) = 1$ , in which case Corollary REF would provide a better asymptotic lower bound for expected minimum rank than that given in Corollary REF .",
"The graphs of these bounds are shown in Figure REF .",
"Figure: The graphs of 1-c(p)>p>p1-c(p)> \\sqrt{p}> p for 0<p<10<p<1"
],
[
"Appendix: Estimation of the binomial coefficient",
"Lemma A.1 Let $N$ be a positive integer and $\\alpha ,\\beta ,\\gamma $ be real numbers with $\\alpha ,\\beta \\in (0,1)$ and $\\gamma \\in [0,1]$ .",
"Then, $ \\binom{(\\alpha +\\beta +\\gamma )N}{\\alpha N}\\le E(\\alpha ,\\beta ,\\gamma ,N)\\left(\\frac{(\\alpha +\\beta )^{\\alpha +\\beta }}{\\alpha ^\\alpha \\beta ^\\beta }\\right)^N , $ where $ E(\\alpha ,\\beta ,\\gamma ,N)=\\sqrt{\\frac{\\alpha +\\beta }{2\\pi \\alpha \\beta N}}\\exp \\left\\lbrace \\frac{1}{12(\\alpha +\\beta )N}+\\gamma \\left(1+\\frac{\\alpha }{\\beta }\\right)N\\right\\rbrace .",
"$ We use Stirling's formula as given in [6]: $ \\sqrt{2\\pi n}\\left(\\frac{n}{e}\\right)^n\\le n!\\le e^{1/(12n)}\\sqrt{2\\pi n}\\left(\\frac{n}{e}\\right)^n .",
"$ From this formula, ${\\binom{(\\alpha +\\beta +\\gamma )N}{\\alpha N}} \\\\ & \\le &e^{1/(12(\\alpha +\\beta +\\gamma )N)} \\frac{\\sqrt{2\\pi (\\alpha +\\beta +\\gamma )N}}{\\sqrt{2\\pi \\alpha N}\\sqrt{2\\pi (\\beta +\\gamma )N}}\\left(\\frac{(\\alpha +\\beta +\\gamma )N}{e}\\right)^{(\\alpha +\\beta +\\gamma )N}\\left(\\frac{e}{\\alpha N}\\right)^{\\alpha N}\\left(\\frac{e}{(\\beta +\\gamma )N}\\right)^{(\\beta +\\gamma )N} \\\\& \\le &e^{1/(12(\\alpha +\\beta )N)} \\sqrt{\\frac{\\alpha +\\beta +\\gamma }{2\\pi \\alpha (\\beta +\\gamma )N}} \\frac{\\left(\\alpha +\\beta +\\gamma \\right)^{(\\alpha +\\beta +\\gamma )N}}{\\alpha ^{\\alpha N}\\left(\\beta +\\gamma \\right)^{(\\beta +\\gamma )N}}$ Since $\\frac{\\alpha +\\beta +\\gamma }{\\alpha (\\beta +\\gamma )}\\le \\frac{\\alpha +\\beta }{\\alpha \\beta }$ , ${\\binom{(\\alpha +\\beta +\\gamma )N}{\\alpha N}} \\\\& \\le &e^{1/(12(\\alpha +\\beta )N)} \\sqrt{\\frac{\\alpha +\\beta }{2\\pi \\alpha \\beta N}} \\left(\\frac{\\left(\\alpha +\\beta +\\gamma \\right)^{(\\alpha +\\beta +\\gamma )}}{\\alpha ^{\\alpha }\\left(\\beta +\\gamma \\right)^{(\\beta +\\gamma )}} \\right)^N\\\\& \\le &e^{1/(12(\\alpha +\\beta )N)} \\sqrt{\\frac{\\alpha +\\beta }{2\\pi \\alpha \\beta N}} \\left(\\frac{\\left(\\alpha +\\beta \\right)^{\\alpha +\\beta }}{\\alpha ^{\\alpha }\\beta ^{\\beta }}\\right)^N\\left(\\left(\\frac{\\alpha +\\beta +\\gamma }{\\alpha +\\beta }\\right)^{\\alpha +\\beta }\\left(\\frac{\\beta }{\\beta +\\gamma }\\right)^\\beta \\left(\\frac{\\alpha +\\beta +\\gamma }{\\beta +\\gamma }\\right)^\\gamma \\right)^N \\\\& \\le &\\left(\\frac{\\left(\\alpha +\\beta \\right)^{\\alpha +\\beta }}{\\alpha ^{\\alpha }\\beta ^{\\beta }}\\right)^N\\sqrt{\\frac{\\alpha +\\beta }{2\\pi \\alpha \\beta N}} e^{1/(12(\\alpha +\\beta )N)}\\left(\\left(1+\\frac{\\gamma }{\\alpha +\\beta }\\right)^{\\alpha +\\beta }\\left(1+\\frac{\\alpha }{\\beta }\\right)^\\gamma \\right)^N \\\\$ Because $1+x\\le e^x$ , $\\binom{(\\alpha +\\beta +\\gamma )N}{\\alpha N} \\le \\left(\\frac{\\left(\\alpha +\\beta \\right)^{\\alpha +\\beta }}{\\alpha ^{\\alpha }\\beta ^{\\beta }}\\right)^N\\sqrt{\\frac{\\alpha +\\beta }{2\\pi \\alpha \\beta N}} e^{1/(12(\\alpha +\\beta )N)} \\exp \\left\\lbrace \\gamma N+\\gamma \\frac{\\alpha }{\\beta }N\\right\\rbrace $ Corollary A.2 Let $p,c$ be fixed and let $r=cv$ , with $v\\rightarrow \\infty $ .",
"$ \\binom{rv+2p\\binom{v}{2}+2v\\sqrt{2\\ln v}}{rv}\\le \\left((1+o(1))\\left(\\frac{(c+p)^{c+p}}{c^cp^p}\\right)\\right)^{v^2} .",
"$ $ \\binom{rv+2p\\binom{v}{2}+2v\\sqrt{2\\ln v}}{rv}=\\binom{c v^2+pv^2 -pv+2v\\sqrt{2\\ln v}}{cv^2} .",
"$ Let $N=v^2$ , $\\alpha =c$ , $\\beta =p$ , and $\\gamma =\\frac{1}{v^2}(-pv +2v\\sqrt{2\\ln v})$ .",
"With $p$ and $c$ fixed, by Lemma REF we see that $ \\binom{rv+2p\\binom{v}{2}+2v\\sqrt{2\\ln v}}{rv}\\le \\left((1+o(1))\\left(\\frac{(c+p)^{c+p}}{c^cp^p}\\right)\\right)^{v^2} .",
"$"
],
[
"Appendix: Connectivity is minimum degree",
"Bollobás and Thomason [8] proved that for $G\\sim G(v,p)$ , regardless of $p$ , then $\\Pr [\\kappa (G)<\\delta (G)]\\rightarrow 0$ as $v\\rightarrow \\infty $ .",
"Bollobás [5] proved the result for $p$ in a restricted interval, but the statement of his theorem is much more general.",
"For our result, we need to bound the probability that $\\kappa (G)=\\delta (G)$ where $G\\sim G(v,p)$ , but need the result only for a fixed $p$ .",
"Lemma B.1 Let $p\\in (0,1)$ be fixed and $G$ be distributed according to $G(v,p)$ .",
"If $v$ is sufficiently large, then $ \\Pr [\\kappa (G)<\\delta (G)]\\le 3v^{-2} .",
"$ Let $\\delta =\\delta (G)$ .",
"By Theorem REF we see that, with probability at least $1-v^{-2}$ , $\\delta \\le \\frac{2e(G)}{v}\\le \\frac{2\\left(p\\binom{v}{2}+v\\sqrt{2\\ln v}\\right)}{v}=p(v-1)+2\\sqrt{2\\ln v}\\le pv+2\\sqrt{2\\ln v} .",
"$ For the remainder of the proof we assume $\\delta \\le pv+2\\sqrt{2\\ln v} .$ If $\\kappa (G)<\\delta $ , then there exists a partition $V(G)=V_1\\cup S\\cup V_2$ such that $|S|<\\delta $ , $2\\le |V_1|\\le |V_2|$ and there is no edge between $V_1$ and $V_2$ .",
"Let the closed neighborhood of vertex $x$ be denoted $N[x]$ and be equal to $\\lbrace x\\rbrace \\cup N(x)$ .",
"We will show first that there is an integer $t$ such that the probability that $2\\le |V_1|\\le t$ is at most $v^{-2}$ (we will determine the value of $t$ later).",
"By a different calculation, we will then show that the probability that $t<|V_1|\\le (v+\\delta )/2$ is also at most $v^{-2}$ .",
"Note that we don't attempt to optimize the probability or to give a range of $p$ over which these conditions hold.",
"A total probability of $2v^{-2}$ is sufficient for our purposes and results in an easier proof.",
"The event $\\lbrace 2\\le |V_1|\\le t\\rbrace $ can occur only if there are two distinct vertices, $x_1$ and $x_2$ , such that the cardinality of the union of their closed neighborhoods is less than $t+\\delta $ .",
"For vertices $y_i\\in V(G)\\backslash \\lbrace x_1,x_2\\rbrace $ , let $Y_i$ be independent indicator variables for $y_i\\in N[x_1]\\cup N[x_2]$ .",
"Since the probability $y_i\\notin N[x_1]\\cup N[x_2]$ is $(1-p)^2$ , ${\\bf E}[|N[x_1]\\cup N[x_2]|]={\\bf E}[2+Y_1+\\dots + Y_{v-2}]]=2+(v-2)(2p-p^2).$ Hence, assuming $(t+\\delta )-\\left(2+(2p-p^2)(v-2)\\right)<0$ , by the negative version of Theorem REF , $\\Pr \\left[2\\le |V_1|\\le t\\right] & \\le &\\binom{v}{2}{\\Pr \\left[\\left|N[x_1]\\cup N[x_2]\\right|<t+\\delta \\right]} \\\\& = & \\binom{v}{2}\\Pr \\left[\\left|N[x_1]\\cup N[x_2]\\right|-\\left(2+(2p-p^2)(v-2)\\right)<(t+\\delta )-\\left(2+(2p-p^2)(v-2)\\right)\\right] \\\\& \\le & \\exp \\left\\lbrace 2\\ln v-\\frac{1}{2(v-2)}\\left((t+\\delta )-2-(2p-p^2)(v-2)\\right)^2\\right\\rbrace .$ Thus if $t+\\delta \\le (2p-p^2)(v-2)+2-3\\sqrt{v\\ln v}$ , then $\\Pr \\left[2\\le |V_1|\\le t\\right]<v^{-2}$ .",
"Since $v\\ge 2$ and we have assumed $\\delta \\le pv+2\\sqrt{2\\ln v}$ , we may set $ t=(2p-p^2)v-3\\sqrt{v\\ln v}-\\delta \\ge (2p-p^2)v-3\\sqrt{v\\ln v}-pv-2\\sqrt{2\\ln v}\\ge p(1-p)v-5\\sqrt{v\\ln v} .",
"$ We will use the bound $\\binom{v}{i}\\le \\left(\\frac{ev}{i}\\right)^i$ which is true for all $1\\le i\\le v$ [6], and the trivial bound $\\binom{v-i}{\\delta }\\le v^{\\delta }$ , which is true for all $v-i,\\delta \\ge 0$ .",
"The event $\\lbrace t<|V_1|\\le (v+\\delta )/2\\rbrace $ has a probability which is bounded as follows: $\\Pr \\left[t<|V_1|\\le (v+\\delta )/2\\right] & \\le & \\sum _{i=\\lfloor t+1\\rfloor }^{\\lfloor (v+\\delta )/2\\rfloor } \\binom{v}{i}\\binom{v-i}{\\delta }(1-p)^{i(v-i-\\delta )} \\\\& \\le & \\sum _{i=\\lfloor t+1\\rfloor }^{\\lfloor (v+\\delta )/2\\rfloor } \\left(\\frac{ev}{i}\\right)^iv^{\\delta }(1-p)^{i(v-\\delta )/2} \\\\& \\le & v^{\\delta }\\sum _{i=\\lfloor t+1\\rfloor }^{\\lfloor (v+\\delta )/2\\rfloor } \\left[ev(1-p)^{(v-\\delta )/2}\\right]^i$ If $v$ is large enough, then $ev(1-p)^{(v-\\delta )/2}<ev(1-p)^{(v-pv-2\\sqrt{2\\ln v})/2}<1$ .",
"Using this in our calculation, along with the bound $1-p\\le e^{-p}$ , $\\Pr \\left[t<|V_1|\\le (v+\\delta )/2\\right] & \\le &v^{\\delta }\\sum _{i=\\lfloor t+1\\rfloor }^{\\lfloor (v+\\delta )/2\\rfloor } \\left[ev(1-p)^{(v-\\delta )/2}\\right]^i \\nonumber \\\\& \\le & v^{\\delta +1}\\left[ev(1-p)^{(v-pv-2\\sqrt{2\\ln v})/2}\\right]^t \\nonumber \\\\& \\le & v^{\\delta +1+t}e^t\\exp \\left\\lbrace -\\frac{pvt}{2}+\\frac{p^2vt}{2}+pt\\sqrt{2\\ln v}\\right\\rbrace \\nonumber \\\\& \\le & v^ve^v\\exp \\left\\lbrace -\\frac{p(1-p)vt}{2}+pv\\sqrt{2\\ln v}\\right\\rbrace \\nonumber \\\\& = & \\exp \\left\\lbrace v\\ln v+v+pv\\sqrt{2\\ln v}-\\frac{p(1-p)}{2}vt\\right\\rbrace $ Since $t\\ge p(1-p)v-5\\sqrt{v\\ln v}$ , the expression in (REF ) is easily bounded above by $\\exp \\lbrace -2\\ln v\\rbrace $ for $v$ large enough.",
"Summarizing, if $v$ is large enough, then with probability at least $1-3v^{-2}$ , there is no set $S$ of size less than $\\delta $ such that $V(G)-S$ is disconnected."
]
] | 1605.05692 |
[
[
"Irreducible Projective Representations and Their Physical Applications"
],
[
"Abstract An eigenfunction method is applied to reduce the regular projective representations (Reps) of finite groups to obtain their irreducible projective Reps. Anti-unitary groups are treated specially, where the decoupled factor systems and modified Schur's lemma are introduced.",
"We discuss the applications of irreducible Reps in many-body physics.",
"It is shown that in symmetry protected topological phases, geometric defects or symmetry defects may carry projective Rep of the symmetry group; while in symmetry enriched topological phases, intrinsic excitations (such as spinons or visons) may carry projective Rep of the symmetry group.",
"We also discuss the applications of projective Reps in problems related to spectrum degeneracy, such as in search of models without sign problem in quantum Monte Carlo Simulations."
],
[
"Introduction",
"Group theory has wide applications in contemporary physics, including particle physics, quantum field theory, gravitational theory and condensed matter physics.",
"There are mainly two types of groups in physics, the symmetry groups and the gauge groups, where the group elements correspond to global or local operations respectively.",
"The linear Rep theory of groups is one of the fundamental mathematical tools in quantum physics.",
"For example, the Hilbert space of orbital (or integer spin) angular momentum forms linear Rep space of $SO(3)$ rotational symmetry group, while charged particles carry Reps of $U(1)$ gauge group.",
"On the other hand, projective Reps of groups were less known to physicists , , , , , , , , , , , .",
"In projective Reps, the representing matrices of group elements obey the group multiplication rule up to $U(1)$ phase factors.",
"These $U(1)$ phase factors are called factor systems of the corresponding projective Rep (see REF ).",
"In some sense, the theory of projective Reps has closer relationship to quantum theory since quantum states are defined up to global $U(1)$ phase factors.",
"The well known Kramers degeneracy owning to time reversal symmetry is actually a typical example of projective Rep, .",
"When acting on operators, the square of time-reversal operator is equal to identity $T^2=E$ and thus defines a two-element group $Z_2^T=\\lbrace E,T\\rbrace $ ; however, when acting on a Hilbert space of odd number of electrons, the square of (the Rep of) the time reversal operator is no longer identity: $\\widehat{T}^2=-1$ .",
"This nontrivial phase factor $-1$ stands for a nontrivial projective Rep of $Z_2^T$ , which guarantees that each energy level is (at least) doubly degenerate.",
"Another typical example is that half-integer spins carry projective representations of $SO(3)$ group.",
"Projective Rep is a natural tool to describe symmetry fractionalization and is widely used in recently devoleped theories such as Symmetry Protected Topological (SPT) phases, , and Symmetry Enriched Topological (SET) phases , .",
"These exotic quantum phases are beyond the Laudau symmetry breaking paradigm because they exhibit no long-range correlations of local order parameters.",
"The SPT states are short range entangled and thus carry trivial topological order, while the SET states are long-range entangled and thus carry nontrivial topological orders.",
"In these novel quantum states, certain defects (such as boundaries or symmetry fluxes) or elementary excitations (called anyons) carry projective Reps of the symmetry group.",
"Especially, one-dimensional SPT phases with an on-site symmetry group $G$ are characterized by projective Reps of $G$ and therefore classified by the second group cohomology $\\mathcal {H}^2(G, U(1))$ , .",
"Most of the groups in quantum physics are unitary.",
"A group is unitary if every group element stands for a unitary operator which keep the inner product of any two states $\\langle \\psi _1|\\psi _2\\rangle $ invariant, namely, $\\langle g\\psi _1|g\\psi _2\\rangle =\\langle \\psi _1|\\psi _2\\rangle .$ where $|g\\psi _{1,2}\\rangle =\\hat{g}|\\psi _{1,2}\\rangle $ for any group element $g$ .",
"On the other hand, a group is called anti-unitary (also called `antiunitary', or `nonunitary') if it contains at least one anti-unitary element $g$ (such as the time reversal operator $T$ ) which transforms the inner product of two states into its complex conjugate, $\\langle g\\psi _1|g \\psi _2\\rangle =\\langle \\psi _1|\\psi _2\\rangle ^*=\\langle \\psi _2|\\psi _1\\rangle .$ Generally, the anti-unitary operator $\\hat{g}$ corresponding to the anti-unitary element $g$ can be written as $\\hat{g}=\\widehat{U} K$ , where $\\widehat{U}$ is a unitary operator and $K$ is the complex-conjugate operator satisfying $ Ka=a^{\\ast }K$ for any complex number $a$ .",
"Anti-unitary elements also differ from unitary ones by their nontrivial module as defined later in Eq.",
"(REF ).",
"In the case that an anti-unitary group has the same group table with that of a unitary group, one can distinguish them by checking the existence of anti-unitary elements (such as the anti-unitary time-reversal group $Z_2^T=\\lbrace E,T\\rbrace $ and unitary spatial-inversion group $Z_2^P=\\lbrace E, P\\rbrace $ ).",
"The simplest anti-unitary group is the two-element group $Z_2^T=\\lbrace E,T\\rbrace $ , where the time reversal operator $T$ is anti-unitary.",
"Anti-unitary groups play important roles in quantum theory.",
"For example, the Schrödinger equations and the Dirac equations for free particles are invariant under the time-reversal symmetry group $Z_2^T$ if there are no magnetic fields.",
"Under time-reversal transformation $T$ , in addition to $t\\rightarrow -t$ , the equations should be transformed into their complex conjugation.",
"The importance of anti-unitary groups can also be seen from the well known CPT theorem.",
"In condensed matter physics, anti-unitary symmetry groups are also important.",
"For instance, the magnetic point groups and magnetic space groups are anti-unitary groups; the well studied topological insulators, , , , , are SPT phases protected by $U(1)$ charge conservation symmetry and time reversal symmetry.",
"Many properties of unitary groups cannot be straightforwardly generalized to anti-unitary groups.",
"For example, the classification of SPT phases with anti-unitary groups in three dimensions is different from that with unitary groups, .",
"Comparing to unitary groups , it is also difficult to gauge a global anti-unitary symmetry into a local symmetry.",
"Since anti-unitary groups are special, a systematic method of obtaining their linear Reps (also called co-representations ) and projective Reps (also called co-ray representations) is urgent.",
"Similar to the linear Rep theory, only irreducible projective Reps are of physical interest.",
"So it is an important issue to find a systematic way to obtain all the irreducible projective Reps.",
"The irreducible projective Reps of a group $G$ can be obtained from the linear Reps of its covering groups (also called representation groups).",
"The covering groups are central extensions of $G$ and are also classified by the second group cohomology.",
"For example, the double point group in spin-orbit coupled systems is a covering group of the corresponding point group.",
"For a given factor system, the corresponding covering group can be constructed in the following steps: (1) transform the factor system into its standard form where all the $U(1)$ phase factors are the $N$ th roots of 1, here $N$ is the period of the factor system (see Sec.REF ); (2) split each group element $g$ in $G$ into $N$ elements, $\\eta g, \\eta ^2 g, ... , \\eta ^N g$ , where $\\eta ^N=1$ ; (3) taking into account the factor system, the new elements obey a new multiplication rule under which the $G\\times N$ elements (here $G$ is the order of group $G$ ) form a new group $G^{\\prime }$ satisfying $G^{\\prime }/Z_N=G$ .",
"This extended group $G^{\\prime }$ is the covering group corresponding to the given factor system.",
"Then from the irreducible linear Rep of $G^{\\prime }$ and the many-to-one mapping from $G^{\\prime }$ to $G$ , one can obtain irreducible projective Rep of the quotient group $G$ with the given factor system.",
"Since the covering group $G^{\\prime }$ is larger than $G$ , this method is very indirect and will not be discussed in detail.",
"Alternatively, we can obtain the irreducible projective Reps from the regular projective Reps without referring to the covering group.",
"Useful knowledge can be learned from linear Reps of groups, where all irreducible Reps can be obtained by reducing the regular Reps. A remarkable eigenfunction method was introduced by J.-Q.",
"Chen, to reduce the regular Reps.",
"In this approach the Rep theory is handled in a physical way.",
"The main idea is to label the irreducible bases by non-degenerate quantum numbers, namely, the eigenvalues of a set of commuting operators.",
"So the main task is to find the Complete Set of Commuting Operators (CSCO) of the Rep space.",
"By making use of the class operators of the group $G$ and those of the canonical subgroup chain of $G$ , together with the class operators of the canonical subgroup chain of the intrinsic group $\\bar{G}$ , the regular Reps of finite groups and the tensor Reps of compact Lie groups (such as $U(n)$ ) are successfully reduced.",
"Chen applied this method to obtain irreducible Reps of space groups, and as a byproduct, part of the projective Reps of point groups were obtained.",
"In this paper, we will generalize the eigenfunction method to reduce the regular projective Reps of finite groups, especially anti-unitary groups.",
"The factor systems of projective Reps are classified by group cohomology and can be obtained by solving the 2-cocycle equations.",
"For anti-unitary groups we show that the factor system can be decoupled into two parts, one contains the information of the quotient group $Z_2^T$ while other part contains the information of the unitary normal subgroup.",
"The introduction of decoupled factor systems is an important step which greatly simplifies the calculations.",
"The regular projective Reps are then obtained by acting the group elements on the group space itself, and the matrix elements are the 2-cocycles.",
"After finding out the CSCO and their eigenfunctions of the regular projective Reps, all the irreducible projective Reps are obtained.",
"In this approach, we only need to treat matrices with maximum dimension $G$ , so it is much simpler than the covering group method.",
"The physical applications of projective Reps are then discussed.",
"Some new viewpoints are presented, for instance, the Majorana zero modes in topological superconductors are explained as projective Reps of some symmetry groups; some models which are free of sign problem under quantum Monte Carlo simulations are interpreted as the properties of projective Reps of anti-unitary groups, and generalizations (i.e.",
"possible new classes of sign-free models) are proposed.",
"The paper is organized as follows.",
"In section , we introduce the projective Reps and the factor systems, and the regular projective Reps of finite groups.",
"In section , we apply the eigenfunction method to reduce the regular projective Reps into irreducible ones.",
"Unitary groups and anti-unitary groups are discussed separately and the results of some finite groups are listed in Table REF .",
"Readers who are more interested in the applications of projective Reps in concrete physical problems can skip this part and go to section directly, where symmetry fractionalizations in topological phases of matter, sign problems in quantum Monte Carlo simulations, and other topics related to spectrum degeneracy are discussed.",
"Section is devoted to the conclusion and discussion.",
"Projective Reps and the factor systems.",
"We first consider $U(1)$ -coefficient projective Reps of a finite unitary group $G$ .",
"Later we will also discuss $\\mathcal {A}$ -coefficient projective Reps, where $\\mathcal {A}$ is a finite subgroup of $U(1)$ .",
"A projective Rep of $G$ is a map from the element $g\\in G$ to a matrix $M(g)$ such that for any pair of elements $g_1, g_2\\in G$ , $M(g_1)M(g_2)=M(g_1g_2)e^{i\\theta _2(g_1,g_2)},$ where the $U(1)$ phase factor $\\omega _2(g_1,g_2) = e^{i\\theta _2(g_1,g_2)}$ is a function of two group variables and is called the factor system.",
"If $\\omega _2(g_1,g_2) = 1$ for any $g_1,g_2\\in G$ , then above projective Rep is trivial, namely, it is a linear Rep.",
"The associativity relation of matrix multiplication yields constraints on the factor system.",
"For any three elements $g_1,g_2,g_3\\in G$ , $ M(g_1)M(g_2)M(g_3)&=&[M(g_1)M(g_2)]M(g_3)\\\\&=&M(g_1g_2g_3)e^{i\\theta _2(g_1,g_2)}e^{i\\theta _2(g_1g_2,g_3)}\\\\&=&M(g_1)[M(g_2)M(g_3)]\\\\&=&M(g_1g_2g_3)e^{i\\theta _2(g_2,g_3)}e^{i\\theta _2(g_1,g_2g_3)},$ which requires that $ \\omega _2(g_1,g_2)\\omega _2(g_1g_2,g_3) = \\omega _2(g_2,g_3)\\omega _2(g_1,g_2g_3),$ or equivalently $ \\theta _2(g_1,g_2) + \\theta _2(g_1g_2,g_3) = \\theta _2(g_2,g_3) + \\theta _2(g_1,g_2g_3),$ where the equal sign means equal mod $2\\pi $ .",
"The factor system of a projective Rep must satisfy equation (REF ).",
"Conversely, all the solutions of above equation correspond to the factor system of a projective Rep.",
"If we introduce a `gauge transformation' to the projective Rep, $ M^{\\prime }(g) = M(g)e^{i\\theta _1(g)},$ where the phase factor $\\Omega _1(g)=e^{i\\theta _1(g)}$ depends on a single group variable, then $ M^{\\prime }(g_1)M^{\\prime }(g_2) &=& M(g_1g_2)e^{i\\theta _2(g_1,g_2)}e^{i\\theta _1(g_1)+i\\theta _1(g_2)}\\\\&=& M^{\\prime }(g_1g_2)e^{i\\theta ^{\\prime }_2(g_1,g_2)},$ where the factor system transforms into $ e^{i\\theta ^{\\prime }_2(g_1,g_2)}=e^{i\\theta _2(g_1,g_2)}{e^{i\\theta _1(g_1)+i\\theta _1(g_2)}\\over e^{i\\theta _1(g_1g_2)}},$ namely, $ \\omega ^{\\prime }_2(g_1,g_2)=\\omega _2(g_1,g_2)\\Omega _2(g_1,g_2),$ with $ \\Omega _2(g_1,g_2) = {\\Omega _1(g_1)\\Omega _1(g_2)\\over \\Omega _1(g_1g_2)}$ where $\\Omega _1(g)=e^{i\\theta _1(g)}$ .",
"Since $M^{\\prime }(g)$ can be adiabatically transformed into $M(g)$ by continuously adjusting the phase $\\Omega _1(g)$ , the two projective Reps are considered to be equivalent.",
"Generally, if the factor systems of any two projective Reps $M(g)$ and $M^{\\prime }(g)$ are related by the relation (REF ), then $M(g)$ and $M^{\\prime }(g)$ are considered to belong to the same class of projective Reps (even if their dimensions are different).",
"Now we consider anti-unitary groups.",
"If a group $G$ is anti-unitary, then half of its group elements are anti-unitary, and the remaining unitary elements form a normal subgroup $H$ , $G/H\\simeq Z_2^T$ with $Z_2^T=\\lbrace E,\\mathbb {T}\\rbrace $ and $\\mathbb {T}^2=E$ .",
"There are two types of anti-unitary groups.",
"In Type-I anti-unitary groups, $\\mathbb {T}$ can be chosen as an element of $G$ , in this case the group $G$ is either a product group $G=H\\times Z_2^T$ or a semi-product group $G=H\\rtimes Z_2^T$ .",
"In type-II anti-unitary groups, $\\mathbb {T}\\notin G$ and the period of any anti-unitary elements in $G$ is at least 4.",
"More rigorous definitions of type-I and type-II anti-unitary groups are given in appendix .",
"In this work, we will mainly focus on type-I anti-unitary groups.",
"Only some simple fermionic groups of type-II will be discussed.",
"Supposing $g$ is a group element in $G$ , then it is represented by $M(g)$ if $g$ is a unitary element and represented by $M(g)K$ if $g$ is anti-unitary, where $K$ is the complex-conjugate operator shown in Eq.",
"(REF ).The multiplication of projective Reps of $g_1, g_2$ depends on if they are unitary or anti-unitary.",
"There are four cases: (A) both $g_1,g_2$ are unitary, then we obtain $M(g_1)M(g_2)=M(g_1g_2)e^{i\\theta _2(g_1,g_2)};$ which is the same as Eq.",
"(REF ); (B) both $g_1,g_2$ are anti-unitary, then the result is $M(g_1)KM(g_2)K=M(g_1)M^*(g_2)=M(g_1g_2)e^{i\\theta _2(g_1,g_2)};$ (C) if $g_1$ is unitary while $g_2$ is anti-unitary, then the result is $M(g_1)M(g_2)K=M(g_1g_2)e^{i\\theta _2(g_1,g_2)}K;$ (D) if $g_1$ is anti-unitary while $g_2$ is unitary, then the result is $M(g_1)KM(g_2) = M(g_1)M^*(g_2)K= M(g_1g_2)e^{i\\theta _2(g_1,g_2)}K.$ If we define $s(g)=\\left\\lbrace \\begin{aligned}&1,& &{\\ \\rm if} \\ g \\ {\\rm is\\ unitary,\\ \\ \\ } \\\\&-1,& &{\\ \\rm if}\\ g \\ {\\rm is\\ antiunitary,\\ \\ \\ }\\end{aligned}\\right.$ and define the corresponding operator $K_{s(g)}$ as $K_{s(g)}=\\left\\lbrace \\begin{aligned}&I,& &{\\ \\rm if\\ } s(g)=1,\\ \\ \\\\&K,& &{\\ \\rm if\\ }s(g)=-1,\\end{aligned}\\right.$ then the above four cases (A)$\\sim $ (D) can be unified as a single equation $M(g_1)K_{s(g_1)}M(g_2)K_{s(g_2)} = M(g_1g_2)e^{i\\theta _2(g_1,g_2)}K_{s(g_1g_2)}.$ Substituting above results into the associativity relation of the sequence of operations $g_1\\times g_2\\times g_3$ , similar to (REF ), we can obtain $ &&M(g_1)K_{s(g_1)}M(g_2)K_{s(g_2)}M(g_3)K_{s(g_3)} \\ \\\\&=& M(g_1g_2g_3)\\omega _2(g_1,g_2)\\omega _2(g_1g_2,g_3) K_{s(g_1g_2g_3)}\\\\&=& M(g_1g_2g_3)\\omega _2(g_1,g_2g_3)\\omega _2^{s(g_1)}(g_2,g_3)K_{s(g_1g_2g_3)},$ namely, $ \\omega _2(g_1,g_2)\\omega _2(g_1g_2,g_3) = \\omega _2^{s(g_1)}(g_2,g_3)\\omega _2(g_1,g_2g_3).$ Eq.",
"(REF ) is the general relation that the factor systems of any finite group (no matter unitary or anti-unitary) should satisfy.",
"Similar to Eq.",
"(REF ), if we introduce a gauge transformation $M^{\\prime }(g)K_{s(g)}=M(g)\\Omega _1(g)K_{s(g)}$ , then the factor system changes into $ \\omega ^{\\prime }_2(g_1,g_2)=\\omega _2(g_1,g_2)\\Omega _2(g_1,g_2),$ with $ \\Omega _2(g_1,g_2) = {\\Omega _1(g_1)\\Omega _1^{s(g_1)}(g_2)\\over \\Omega _1(g_1g_2)}.$ The equivalent relations (REF ) and (REF ) define the equivalent classes of the solutions of (REF ).",
"The number of equivalent classes for a finite group is usually finite.",
"The 2nd group cohomology and the 2-cocycles.",
"Actually, the equivalent classes of the factor systems associated with the projective Reps of group $G$ form a group, called the second group cohomology.",
"The group cohomology $\\lbrace {\\rm Kernel}\\ d/{\\rm Image}\\ d\\rbrace $ is defined by the coboundary operator $d$ (for details see appendix ), $& & (d\\omega _{n})(g_{1},\\ldots ,g_{n+1}) \\nonumber \\\\&& =[g_{1}\\cdot \\omega _{n}(g_{2},\\ldots ,g_{n+1})]\\omega ^{(-1)^{n+1}}_{n}(g_{1},\\ldots ,g_{n})\\times \\nonumber \\\\&&\\prod ^{n}_{i=1}\\omega ^{(-1)^{i}}_{n}(g_{1},\\ldots ,g_{i-1},g_{i}g_{i+1},g_{i+2},\\ldots ,g_{n+1}).$ where $g_1,...,g_{n+1}\\in G$ and the variables $\\omega _{n}(g_{1},\\ldots ,g_{n})$ take value in an Abelian group $\\mathcal {A}$ [usually $\\mathcal {A}$ is a subgroup of $U(1)$ ].",
"The set of variables $\\omega _{n}(g_{1},\\ldots ,g_{n})$ is called a $\\mathcal {A}$ -cochain.",
"The module $g\\cdot $ is defined by $g\\cdot \\omega _{n}(g_{1},\\ldots ,g_{n}) =\\omega _{n}^{s(g)}(g_{1},\\ldots ,g_{n}).$ With this notation, Eq.",
"(REF ) can be rewritten as $ (d\\omega _2)(g_1,g_2,g_3)=1,$ the solutions of above equations are called 2-cocycles with $\\mathcal {A}$ -coefficient.",
"Similarly, Eq.",
"(REF ) can be rewritten as $ \\Omega _2(g_1,g_2)=(d\\Omega _1)(g_1,g_2),$ where $\\Omega _1(g_1),\\Omega _1(g_2)\\in \\mathcal {A}$ and thus defined $\\Omega _2(g_1,g_2)$ are called 2-coboundaries.",
"Two 2-cocycles $\\omega _2^{\\prime }(g_1,g_2)$ and $\\omega _2(g_1,g_2)$ are equivalent if they differ by a 2-coboundary, see Eq.",
"(REF ).",
"The equivalent classes of the 2-cocycles $\\omega _2(g_1,g_2)$ form the second group cohomology $\\mathcal {H}^2(G,\\mathcal {A})$ .",
"A projective Rep whose factor system is a $\\mathcal {A}$ -coefficient 2-cocycle is called $\\mathcal {A}$ -coefficient projective Rep. By default, projective Reps are defined with $U(1)$ coefficient, namely, $\\mathcal {A}=U(1)$ .",
"In this case, $\\omega _{2}(g_{1},g_{2})\\in U(1)$ so we can write $\\omega _{2}(g_{1},g_{2})=e^{i\\theta _{2}(g_{1},g_{2})}$ , where $\\theta _{2}(g_{1},g_{2})\\in [0,2\\pi )$ .",
"The cocycle equations $(d\\omega _{2})(g_{1},g_2,g_{3})=1$ can be written in terms of linear equations, $&&s(g_{1})\\theta _{2}(g_{2},g_{3})-\\theta _{2}(g_{1}g_{2},g_{3})+\\theta _{2}(g_{1},g_{2}g_{3})\\nonumber \\\\&& -\\theta _{2}(g_{1},g_{2})=0.$ Similarly, if we write $\\Omega _1(g_1)=e^{i\\theta _1(g_1)}$ and $\\Omega _2(g_1,g_2)=e^{i\\Theta _2(g_1,g_2)}$ , then the 2-coboundary (REF ) can be written as $ \\Theta _{2}(g_{1},g_{2})= s(g_{1})\\theta _{1}(g_{2})-\\theta _{1}(g_{1}g_{2})+\\theta _{1}(g_{1}) .$ The equal sign in (REF ) and (REF ) means equal mod $2\\pi $ .",
"From these linear equations, we can obtain the classification and the cocycles solutions of each class (for details see appendix ).",
"The classification of the factor system also gives a constraint on the dimensions of corresponding projective Reps, as discussed in appendix REF .",
"Gauge fixing.",
"For each class of 2-cocyles satisfying (REF ), there are infinite number of solutions which differ by 2-coboundaries.",
"We will adopt the canonical gauge by fixing $\\omega _2(E,g)=\\omega _2(g,E)=1.$ However, above canonical gauge condition doesn't completely fix the coboundaries and there are still infinite number of solutions.",
"In later discussion, we will select one of the 2-cocycle solutions in each class as the factor system of the corresponding projective Rep.",
"In that case, the gauge degrees of freedom (i.e.",
"the 2-coboundaries) are completely fixed.",
"The case where $\\mathcal {A}$ is a finite abelian group.",
"Mathematically, the second group cohomology $\\mathcal {H}^2(G,\\mathcal {A})$ with $\\mathcal {A}$ -coefficient classifies the central extensions of $G$ by $\\mathcal {A}$ .",
"In the default case, $\\mathcal {A}=U(1)$ and the projective Reps of $G$ are classified by $\\mathcal {H}^2(G,U(1))$ and correspond to linear Reps of $U(1)$ extensions of $G$ .",
"Physically, we are also interested in the cases where $\\mathcal {A}$ is a finite abelian group, such as $Z_N$ .",
"For example, the topological phases with $Z_2$ topological order and symmetry group $G$ are (partially) classified by $\\mathcal {H}^2(G,Z_2)$ (see section REF ).",
"The classification of $Z_N$ coefficient projective Reps is usually different from the $U(1)$ coefficient projective Reps (see appendix ), for instance, certain $U(1)$ coboundaries may be nontrivial $Z_N$ cocycles.",
"Once the factor systems are obtained by solving the cocycle equaitons and coboundary equations (see appendix ), the method of obtaining $\\mathcal {A}$ -coefficient irreducible projective Reps is the same with the cases of $U(1)$ -coefficient.",
"In the following we will discuss the method of obtaining $U(1)$ coefficient irreducible projective Reps by reducing the regular projective Reps."
],
[
"Regular projective Reps",
"Regular projective Reps. For a given factor system, we can easily construct the corresponding regular projective Rep (the regular Rep twisted by the 2-cocycles) using the group space as the Rep space.",
"The group element $g$ is not only an operator $\\hat{g}$ , but also a basis $|g\\rangle $ .",
"The operator $\\hat{g}_{1}$ acts on the basis $|g_{2}\\rangle $ as the following, $\\hat{g}_{1}|g_{2}\\rangle =e^{i\\theta _2(g_{1},g_{2})}K_{s(g_1)}|g_{1}g_{2}\\rangle ,$ or in matrix form $ \\hat{g}_1=M(g_1)K_{s(g_1)},$ with matrix element $M(g_{1})_{g,g_{2}}=\\langle g|\\hat{g}_1|g_2\\rangle =e^{i\\theta _2(g_{1},g_{2})}\\delta _{g,g_{1}g_{2}}.$ For any group element $g_{3}$ , we have $\\hat{g}_{1}\\hat{g}_{2}|g_{3}\\rangle &=& \\hat{g}_{1}\\left[e^{i\\theta _2(g_{2},g_{3})}K_{s(g_2)}|g_{2}g_{3}\\rangle \\right] \\\\&=&e^{i\\theta _2(g_{1},g_{2}g_{3})}K_{s(g_1)}\\left[e^{i\\theta _2(g_{2},g_{3})}K_{s(g_2)}|g_{1}g_{2}g_{3}\\rangle \\right] \\\\&=&e^{i\\theta _2(g_{1},g_{2}g_{3})}e^{is(g_1)\\theta _2(g_{2},g_{3})}K_{s(g_1g_2)}|g_{1}g_{2}g_{3}\\rangle ,$ and $ \\widehat{g_{1}g_{2}}|g_{3}\\rangle &=&e^{i\\theta _2(g_{1}g_{2},g_{3})}K_{s(g_1g_2)}|g_{1}g_{2}g_{3}\\rangle .$ Comparing with the 2-cocycle equation (REF ), it is easily obtained that $\\hat{g}_1\\hat{g}_2=e^{i\\theta _2(g_1,g_2)} \\widehat{g_1g_2}$ .",
"In matrix form, this relation reads $M(g_{1})K_{s(g_1)}M(g_{2})K_{s(g_2)}= e^{i\\theta _2(g_{1},g_{2})}M(g_{1}g_{2})K_{s(g_1g_2)}.\\nonumber \\\\$ Eq.",
"(REF ) indicates that $M(g)K_{s(g)}$ is indeed a projective Rep of the group $G$ .",
"For the trivial 2-cocycle where $e^{i\\theta _2(g_{1},g_{2})}=1$ for all $g_1, g_2\\in G$ , $M(g)K_{s(g)}$ reduces to the regular linear Rep of $G$ .",
"Intrinsic Regular projective Reps.",
"In order to reduce the regular projective Rep of $G$ , we will make use of the intrinsic group $\\bar{G}$ .",
"Each element $g$ in $G$ corresponds to an intrinsic group element $\\bar{g}$ in $\\bar{G}$ , and the intrinsic elements obey right-multiplication rule with $\\bar{g}_1 g_2=g_2g_1$ and $\\bar{g}_1\\bar{g}_2=\\overline{g_2g_1}$ .",
"Obviously, the intrinsic group $\\bar{G}$ commutes with $G$ since $(\\bar{g}_1g_2)g_3 = (g_2\\bar{g}_1) g_3 = g_2g_3g_1$ .",
"Corresponding to $\\bar{G}$ , we can define the intrinsic regular projective Rep, $\\hat{\\bar{g}}_{1} |g_{2}\\rangle = e^{i\\theta _2(g_{2},g_{1})}|g_{2}g_{1}\\rangle ,$ or in matrix form $\\hat{\\bar{g}}_1=M(\\bar{g}_1)$ with $M(\\bar{g}_{1})_{g,g_{2}}=e^{i\\theta _2(g_{2},g_{1})}\\delta _{g,g_{2}g_{1}}.$ The complex-conjugate operator $K$ is absent in (REF ), indicating that the intrinsic projective Reps for anti-unitary groups are essentially unitary.",
"For any group element $g_{3}$ , we have $\\hat{\\bar{g}}_{1}\\hat{\\bar{g}}_{2} |g_{3}\\rangle &=& \\hat{\\bar{g}}_{1}e^{i\\theta _2(g_{3},g_{2})}|g_{3}g_{2}\\rangle \\\\&=& e^{i\\theta _2(g_{3},g_{2})}e^{i\\theta _2(g_{3}g_{2},g_{1})}|g_{3}g_{2}g_{1}\\rangle $ and $ \\widehat{\\overline{g_{2}g_{1}}} |g_{3}\\rangle &=& e^{i\\theta _2(g_{3},g_{2}g_{1})}|g_{3}g_{2}g_{1}\\rangle $ Comparing with the 2-cocycle equation (REF ), we find that $ \\hat{\\bar{g}}_1\\hat{\\bar{g}}_2|g_3\\rangle =\\widehat{\\overline{g_2g_1}}e^{is(g_{3})\\theta _2(g_{2},g_{1})}|g_3\\rangle ,$ which means that if the state $|g_3\\rangle $ is anti-unitary (unitary), then the coefficients of the operators on its left will take a complex conjugate (remain unchanged).",
"In matrix form, above relation can be written as $M(\\bar{g}_{1})M(\\bar{g}_{2}) = M(\\overline{g_{2}g_{1}})\\left[e^{i\\theta _2(g_{2},g_{1})}P_u + e^{-i\\theta _2(g_{2},g_{1})}P_a\\right],$ where $P_u (P_a)$ is the projection operator projecting onto the Hilbert space formed by unitary (anti-unitary) bases.",
"Now we show that the regular projective Rep of $G$ commutes with its intrinsic regular projective Rep. On the one hand, $ \\hat{\\bar{g}}_1\\hat{g}_2 |g_3\\rangle &=& \\hat{\\bar{g}}_1 \\omega _2(g_2,g_3)K_{s(g_2)}|g_2g_3\\rangle \\\\&=&\\omega _2(g_2g_3,g_1) \\omega _2(g_2,g_3)K_{s(g_2)}|g_2g_3g_1\\rangle .$ On the other hand, $ \\hat{g}_2\\hat{\\bar{g}}_1|g_3\\rangle &=& \\hat{g}_2\\omega _2(g_3,g_1)|g_3g_1\\rangle \\\\&=&\\omega _2^{s(g_2)}(g_3,g_1)\\omega _2(g_2,g_3g_1)K_{s(g_2)}|g_2g_3g_1\\rangle .$ Comparing with the cocycle equation $(d\\omega _2)(g_2,g_3,g_1)=1$ , we have $\\hat{\\bar{g}}_1\\hat{g}_2 =\\hat{g}_2\\hat{\\bar{g}}_1.$ In matrix form, above equation reads $M(\\bar{g}_1)M(g_2)K_{s(g_2)}=M(g_2)K_{s(g_2)}M(\\bar{g}_1).$"
],
[
"CSCO and irreducible projective Reps ",
"Supposing a group element $g\\in G$ is represented by $M(g)K_{s(g)}$ , we assume that all the Rep matrices $M(g)$ are unitary (for finite groups all the Reps can be transformed into this form).",
"If a projective Rep can not be reduced by unitary transformations into direct sum of lower dimensional Reps, then it is called irreducible projective Rep.",
"If two irreducible projective Reps $M(g)K_{s(g)}$ and $M^{\\prime }(g)K_{s(g)}$ can be transformed into each other by a unitary matrix $U$ , $ M^{\\prime }(g)K_{s(g)}=U^\\dag M(g)K_{s(g)} U,$ then $M^{\\prime }(g)K_{s(g)}$ and $M(g)K_{s(g)}$ are said to be the same projective Rep; otherwise, they are two different Reps. On the other hand, if two different irreducible projective Reps have the same (class of) factor systems, they belong to the same class.",
"Obviously, any one-dimensional Rep of a group must be a trivial projective Rep since it is gauge equivalent to the identity Rep (except for the cases where the coefficient group $\\mathcal {A}$ is a finite group).",
"If $M(g)K_{s(g)}$ is a projective Rep of group $G$ with factor system $\\omega _2(g_1,g_2)$ , then its complex conjugate $M^*(g)K_{s(g)}$ is also a projective Rep whose factor system is $\\omega _2^*(g_1,g_2)=\\omega _2^{-1}(g_1,g_2)$ .",
"It is known that all the irreducible linear Reps of a finite group $G$ can be obtained by reducing its regular Rep.",
"Similar idea can be applied for projective Reps.",
"In section , we have constructed the regular projective Rep with a given factor system(or 2-cocycle).",
"In this section we will generalize the eigenfunction method to reduce the regular projective Reps into irreducible ones.",
"As mentioned, we will completely fix the 2-coboundary by selecting one solution in each class of 2-cocyles (for anti-unitary groups we will transform the 2-cocyles into the decoupled factor system as defined in Appendix ).",
"Suppose that we choose a different 2-coboundary [see(REF ) and (REF )], then the corresponding regular projective Rep is given by $\\hat{g}_1=W(g_1)K_{s(g_1)}$ with $ W(g_1)_{g,g_2}=\\delta _{g,g_1g_2}\\omega _2(g_1,g_2){\\Omega _1(g_1)\\Omega _1^{s(g_1)}(g_2)\\over \\Omega _1(g_1g_2)}.$ Above $W(g_1)K_{s(g_1)}$ is equivalent to $M(g_1)K_{s(g_1)}$ defined in (REF ) since they are related by a unitary transformation $U$ followed by a gauge transformation $\\Omega _1(g_1)$ , $ W(g_1)K_{s(g_1)}=\\Omega _1(g_1)[U^\\dag M(g_1)K_{s(g_1)}U],$ where $U$ is a diagonal matrix with entries $U_{g,g^{\\prime }}=\\delta _{g,g^{\\prime }}\\Omega _1(g)$ .",
"Similarly, the intrinsic regular projective Rep (REF ) becomes $\\hat{\\bar{g}}_1=W(\\bar{g}_1)$ with $ W(\\bar{g}_1) = U^\\dag M(\\bar{g}_1) U\\left[\\Omega _1(g_1) P_u +\\Omega _1^*(g_1) P_a\\right] .$ Therefore, we can safely select one 2-cocyle in each class to construct the regular projective Rep.",
"Some tools used in linear Reps, such as class operators and characters, can be introduced for the projective Reps. For unitary groups, the character $\\chi _i^{(\\nu )}$ of an irreducible (projective) Rep is a function of class operators (each conjugate class gives rise to a class operator) $ C_i = \\sum _{g\\in G}g^{-1}g_i g,$ here we have ignored a normalization constant.",
"On the other hand, for a given class operator $C_i$ the character $\\chi _i^{(\\nu )}$ is a function of irreducible (projective) Reps $(\\nu )$ since the unitary transformation (REF ) will not change the character.",
"The characters $\\chi _i^{(\\nu )}$ form complete bases for both the class space $\\lbrace C_i\\rbrace $ and the irreducible (projective) Rep space $\\lbrace (\\nu )\\rbrace $ .",
"As a result, the number of different irreducible (projective) Reps is equal to $N_c$ , the number of independent class operators.",
"Group theory also tells us that for a finite group $G$ , a $n_\\nu $ dimensional irreducible Rep appears $n_\\nu $ times in the reduced regular Rep. Consequently, $\\sum _{\\nu =1}^{N_c} n_\\nu ^2=G,$ where $G$ is the order of the group $G$ .",
"This result also holds for the projective Reps of unitary groups.",
"However, we should carefully use these tools for anti-unitary groups.",
"For example, the `character' of the anti-unitary element $g$ may be changed under unitary transformation $ {\\rm Tr}\\ [M(g)K]\\ne {\\rm Tr}\\ [U^\\dag M(g)KU]={\\rm Tr} [U^\\dag M(g)U^*]$ for arbitrary unitary matrix $U$ .",
"Similarly, the two unitary elements $g$ and $\\tilde{g}=T^{-1}gT$ , which belong to the same class, may have different characters.",
"Therefore we need to redefine the conjugate classes and the class operators such that the number of different irreducible Reps (of the same class) is still equal to the number of independent class operators.",
"Similarly, some other conclusions of unitary groups should be modified for anti-unitary groups (see Sec.REF )."
],
[
"Brief review of the reduction of regular linear Reps for unitary groups",
"First we introduce Schur's lemma and its corollary for unitary groups without proving them.",
"They are also valid for irreducible projective Reps of unitary groups.",
"Schur's lemma: If a nonzero matrix $C$ commutes with all the irreducible Rep matrices $M^{(\\nu )}(g)$ of a unitary group $G$ , namely, $CM^{(\\nu )}(g)=M^{(\\nu )}(g)C,$ for $g\\in G$ , then $C$ must be a constant matrix $C=\\lambda I$ .",
"Corollary: If a unitary operator $\\widehat{C}$ commutes with a unitary group $G$ with $[\\hat{g},\\widehat{C}]=0$ for all $g\\in G$ , then an eigenspace of $C$ (or the surporting space of Jordan blocks with the same eigenvalue) is a Rep space of $G$ .",
"In most cases, the operator $C$ (such as the class operators discussed below) is diagonalizable and the Hilbert space can be reduced as direct sum of its eigenspaces.",
"However, it is possible that $C$ cannot be completely diagonalized, in this case $C$ can be transformed into its Jordan normal form and we can just replace each `eigenspace' by a `supporting space of Jordan blocks with the same eigenvalue'.",
"Reducing the regular Rep is equivalent to identifying all the bases of the irreducible Reps.",
"In Ref.",
"ChenJQ02, the authors use a series of quantum numbers, i.e.",
"the eigenvalues of the complete set of commuting operators (CSCO), to distinguish the irreducible bases.",
"This idea comes from quantum mechanics.",
"For example, for spin systems we use two quantum numbers $|S,m\\rangle $ to label a state, where $S$ is the main quantum number labeling the irreducible Rep and $m$ is the magnetic quantum number labeling different bases.",
"Noticing that $S(S+1)$ and $m$ are eigenvalues of the operators $(\\hat{S}^2, \\hat{S}_z)$ respectively, where $\\hat{S}^2$ is the invariant quantity (Casimir operator) of $SO(3)$ and $\\hat{S}_z$ is that of its subgroup $SO(2)$ .",
"The two commuting operators $(\\hat{S}^2, \\hat{S}_z)$ form the CSCO of the Hilbert space of a spin, so we can use their eigenvalues to label different bases.",
"Similarly, for any unitary group $G$ , we can use invariant quantities (that commute with the group $G$ ) to provide the main quantum numbers to label different irreducible Reps.",
"The class operators defined in (REF ) are ideal candidates since they commute with each other and commute with all the group elements in $G$ .",
"According to the corollary of Schur's lemma, every irreducible Rep space $(\\nu )$ is an eigenspace of all the class operators $C_i$ and can be labeled by the eigenvalues (it can be shown that the eigenvalues are proportional to the corresponding characters $\\chi _i^{(\\nu )}$ ).",
"It turns out that the $N_c$ independent class operators can provide exactly $N_c$ sets of different quantum numbers and can completely identify the $N_c$ different irreducible Reps.",
"The reason is that the $N_c$ linearly independent class operators form an algebra since they are closed under multiplication.",
"The natural Rep of the class algebra provides $N_c$ different sets of eigenvalues (corresponding to the eigenstates in the class space).",
"These $N_c$ different eigenvalues are nothing but the quantum numbers labeling the irreducible Reps. Generally, these quantum numbers are degenerate and can not distinguish each of the irreducible bases.",
"Therefore, the class operators of $G$ form a subset of the commuting operators and are called the CSCO-I.",
"Actually, we can linearly combine the $N_c$ independent class operators to form a single operator $C=\\sum _ik_iC_i$ (where $k_i$ are some real constants) as the CSCO-I, as long as the operator $C$ has $N_c$ different eigenvalues.",
"Secondly, we need some `magnetic' quantum numbers to distinguish different bases in one copy of irreducible Rep. We will make use of the chain of subgroups $G(s)=G_1\\supset G_2\\supset ...$ , the set of their CSCO-I $C(s)=(C(G_1),C(G_2),...)$ commute with the CSCO-I of $G$ and provide the required quantum numbers.",
"The set $\\left(C,C(s)\\right)$ formed by $C$ (the CSCO-I of $G$ ) and $C(s)$ is called the CSCO-II .",
"Finally, since the $n_\\nu $ dimensional irreducible Rep $(\\nu )$ appears $n_\\nu $ times in the regular Rep, the quantum numbers of CSCO-II are still degenerate.",
"To lift this degeneracy, we need more quantum numbers (i.e.",
"the multiplicity quantum numbers).",
"Noticing that the intrinsic group $\\bar{G}$ commutes with $G$ , we can use the CSCO-Is of the subgroup chain $\\bar{G}(s)=\\bar{G}_1\\supset \\bar{G}_2\\supset ...$ , namely, $\\bar{C}(s)=(\\bar{C}(G_1), \\bar{C}(G_2), ...)$ .",
"The operators $\\bar{C}(s)$ commute with the CSCO-II of $G$ and can completely lift the degeneracy.",
"So we obtain the complete set $(C, C(s), \\bar{C}(s))$ , called the CSCO-III.",
"The CSCO-III provides $G$ sets of non-degenerate quantum numbers which completely label all the irreducible bases in the regular Rep space.",
"As an example, we apply this method to reduce the regular Rep of the permutation group $S_3$ .",
"For $S_3$ , there are 6 group elements and the canonical subgroup chain is $S_3\\supset S_2$ .",
"The group elements $P=(12),Q=(23),R=(13)$ belong to the same class.",
"So we may choose class operator $C=(12)+(23)+(13)$ , the class operator of subgroup $S_2$ $C(s)=(12)$ and the class operator of intrinsic subgroup $\\overline{S_2}$ $\\bar{C}(s)=(\\overline{12})$ to construct CSCO.",
"In the regular Rep space (group space), the Rep matrices of these class operators can be written as,respectively $C&=&\\left(\\begin{array}{cccccc}0&1&1&1&0&0\\\\1&0&0&0&1&1\\\\1&0&0&0&1&1\\\\1&0&0&0&1&1\\\\0&1&1&1&0&0\\\\0&1&1&1&0&0\\end{array}\\right),C(s) =\\left(\\begin{array}{cccccc}0&1&0&0&0&0\\\\1&0&0&0&0&0\\\\0&0&0&0&0&1\\\\0&0&0&0&1&0\\\\0&0&0&1&0&0\\\\0&0&1&0&0&0\\end{array}\\right),\\nonumber \\\\\\bar{C}(s)&=&\\left(\\begin{array}{cccccc}0&1&0&0&0&0\\\\1&0&0&0&0&0\\\\0&0&0&0&1&0\\\\0&0&0&0&0&1\\\\0&0&1&0&0&0\\\\0&0&0&1&0&0\\end{array}\\right).$ The eigenvalues of $C$ are $3,0,-3$ , in which 3 and $-3$ are non-degenerate but 0 is four-fold degenerate.",
"We may use the complete set $(C,C(s),\\bar{C}(s))$ to lift the degeneracy.",
"Then a unitary transformation matrix $U$ can be formed via the common eigenvectors of $(C,C(s),\\bar{C}(s))$ and the regular linear Rep matrices of all group elements can be block diagonalized simultaneously.",
"We only give the results of two generators: $ U^{\\dag }M(P)U&=&\\left(\\begin{array}{cccccc}-1&0&0&0&0&0\\\\0&-1&0&0&0&0\\\\0&0&1&0&0&0\\\\0&0&0&-1&0&0\\\\0&0&0&0&1&0\\\\0&0&0&0&0&1\\end{array}\\right),\\\\U^{\\dag }M(Q)U&=&\\left(\\begin{array}{cccccc}-1&0&0&0&0&0\\\\0&\\frac{1}{2}&\\frac{\\sqrt{3}}{2}&0&0&0\\\\0&\\frac{\\sqrt{3}}{2}&-\\frac{1}{2}&0&0&0\\\\0&0&0&\\frac{1}{2}&\\frac{\\sqrt{3}}{2}&0\\\\0&0&0&\\frac{\\sqrt{3}}{2}&-\\frac{1}{2}&0\\\\0&0&0&0&0&1\\end{array}\\right).$ From above results, we find two inequivalent one-dimensional Reps occur only once, but a two-dimensional Rep occurs twice.",
"The quantum numbers of $\\bar{C}(s)$ are used to distinguish the two equivalent irreducible Reps."
],
[
"Reduction of regular projective Reps",
"Now we generalize the eigenfunction method to reduce the regular projective Reps. Unitary groups and anti-unitary groups will be discussed separately."
],
[
" Unitary groups",
"First of all, we need to define the class operators.",
"For a regular projective Rep of any unitary group $G$ , the class operator $C_1$ corresponding to $g_1\\in G$ is defined as $C_1=\\sum _{g_{2}\\in G}\\hat{ g}^{-1}_{2}\\hat{g}_{1}\\hat{ g}_{2}=\\sum _{g_{2}\\in G}M^{-1}(g_{2})M(g_{1})M(g_{2}).$ The class operator $C_1$ commutes with all the regular projective Rep matrices since $ C_1\\hat{ g}_3&=&\\sum _{g_{2}\\in G}\\hat{ g}_3[\\hat{ g}_2\\hat{ g}_3]^{-1}\\hat{ g}_1\\hat{ g}_2\\hat{ g}_3\\\\&=&\\hat{ g}_3\\sum _{g_{2}\\in G}\\widehat{ {g_2g_3}}^{-1}\\hat{ g}_1\\widehat{ {g_2g_3}}\\\\&=&\\hat{ g}_3C_1$ for all $g_3\\in G$ .",
"It can be shown that if $g_1^{\\prime }=g^{-1}g_1g$ then $C_{g_1^{\\prime }}\\propto C_{g_1}$ , namely, each conjugate class gives at most one linearly independent class operator.",
"Since a nontrivial projective Rep $M(g)$ is not a faithful Rep of $G$ , the class operators for some conjugate classes may be zero.",
"Consequently, the number of linearly independent class operators is generally less than the number of classes of the group $G$ .",
"Since character is still a good quantity, the number of different irreducible projective Reps (of the same class) is still equal to the number of independent class operators.",
"Due to the corollary of Schur's lemma, we can use the eigenvalues of the class operators to label different Rep spaces.",
"Therefore, the class operators (or their linear combination $C=\\sum _i k_i C_i$ ) form CSCO-I of the regular projective Rep.",
"The eigenvalues of CSCO-I are degenerate.",
"To distinguish the bases of the same irreducible Rep, we can make use of the class operators of the subgroup chain $G(s)=G_1\\supset G_2\\supset ...$ .",
"For instance, the class operators of the subgroup $G_1$ are defined as $C_1(G_1)=\\sum _{g_{2}\\in G_1}M^{-1}(g_{2})M(g_{1})M(g_{2}) ,\\ \\ \\ g_{1}\\in G_1,$ and $C(G_1)=\\sum _i k_i C_i(G_1)$ is the CSCO-I of $G_1$ .",
"Repeating the procedure we obtain the set of CSCO-Is for the subgroup chain $C(s)=(C(G_1),C(G_2),...)$ .",
"The operator set $(C,C(s))$ is called CSCO-II, which can be used to distinguish all the bases if every irreducible Rep occurs only once in the reduced projective Rep.",
"However, a $n_\\nu $ -dimensional irreducible projective Rep $(\\nu )$ may occur more than once and this will cause degeneracy in the eigenvalues of CSCO-II.",
"To lift this degeneracy we need more quantum numbers to label the mutiplicity.",
"Similar to the linear Reps, we can use the class operators of the intrinsic group $\\bar{G}$ .",
"It can be shown that the class operators of $\\bar{G}$ is identical to those of $G$ : for any $g_3\\in G$ , we have $ \\bar{C}_1|g_3\\rangle &=& \\left[ \\sum _{g_2}\\hat{\\bar{g}}_2\\hat{\\bar{g}}_1\\hat{\\bar{g}}_2^{-1}\\right] \\hat{\\bar{g}}_3|E\\rangle \\nonumber \\\\&=&\\sum _{g_2}e^{-i\\theta _2(g_2,g_2^{-1})}\\left\\lbrace \\left[(\\hat{\\bar{g}}_2\\hat{\\bar{g}}_1)\\widehat{\\overline{g_2^{-1}}}\\right] \\hat{\\bar{g}}_3\\right\\rbrace |E\\rangle \\nonumber \\\\&=&\\sum _{g_2}e^{-i\\theta _2(g_2,g_2^{-1})}e^{i\\theta _2(g_1,g_2)}e^{i\\theta _2(g_2^{-1},g_1g_2)}\\nonumber \\\\&&\\times e^{i\\theta _2(g_3,g_2^{-1}g_1g_2)}|g_3g_2^{-1}g_1g_2\\rangle ,$ where we have used $\\widehat{\\overline{g_2^{-1}}}=e^{i\\theta _2(g_2,g_2^{-1})}\\hat{\\bar{g}}_2^{-1}$ .",
"On the other hand, $ C_1|g_3\\rangle &=& \\left[ \\sum _{g_2}\\hat{g}^{-1}_2\\hat{ g}_1\\hat{ g}_2\\right] \\hat{ g}_3|E\\rangle \\nonumber \\\\&=& \\hat{ g}_3\\left[ \\sum _{g_2} e^{-i\\theta _2(g_2,g_2^{-1})}\\widehat{g^{-1}_2}\\hat{ g}_1\\hat{ g}_2\\right]|E\\rangle \\nonumber \\\\&=&\\sum _{g_2}e^{-i\\theta _2(g_2,g_2^{-1})}e^{i\\theta _2(g_1,g_2)}e^{i\\theta _2(g_2^{-1},g_1g_2)}\\nonumber \\\\&&\\times e^{i\\theta _2(g_3,g_2^{-1}g_1g_2)}|g_3g_2^{-1}g_1g_2\\rangle ,$ where $\\widehat{g^{-1}_2}=e^{i\\theta _2(g_2,g_2^{-1})}\\hat{g}^{-1}_2$ .",
"Therefore, $\\bar{C}_1=C_1$ , namely, the class operators of $\\bar{G}$ do not provide any new invariant quantities.",
"However, the class operators $\\bar{C}(s)$ [$\\bar{C}(s)$ can be obtained from (REF ) by replacing $g$ with $\\bar{g}$ ] of the chain of subgroups $\\bar{G}(s)=\\bar{G}_1\\supset \\bar{G}_2\\supset ...$ are different from $C(s)$ and can lift the degeneracy of the eigenvalues of CSCO-II.",
"Now we obtain the complete set of class operators $\\left(C,C(s),\\bar{C}(s)\\right)$ , called CSCO-III.",
"The eigenvalues of $\\bar{C}(s)$ are the same as those of $C(s)$ , consequently the number of times an irreducible projective Rep $(\\nu )$ occurs is equal to its dimension $n_\\nu $ , and $\\sum _\\nu n_\\nu ^2=G$ is still valid for a given class of irreducible projective Reps. For unitary groups, the common eigenvectors of the operators in CSCO-III are the orthonormal bases of the irreducible projective Reps, and each eigenvector has unique `quantum numbers'.",
"These eigenvectors form a unitary matrix $U$ which block diagonalizes all regular projective Rep matrices $M(g)$ simultaneously.",
"Now we summarize the steps to obtain all the irreducible projective Reps with the eigenfunction method: 1, Solve the 2-cocycle equations, obtain the solutions and their classification (see appendix ); 2, Select one solution of a given class as the factor system and obtain the corresponding regular projective Rep (see section REF ); 3, Construct the CSCO-III with $\\left(C,C(s),\\bar{C}(s)\\right)$ , where $C, C(s), \\bar{C}(s)$ are class operators of $G$ , of the subgroups $G(s)\\subset G$ , and of the subgroups $\\bar{G}(s)\\subset \\bar{G}$ respectively.",
"Then reduce the regular projective Reps into irreducible ones; 4, Change the class and repeat above procedure, until the irreducible projective Reps of all classes are obtained."
],
[
" Anti-unitary groups",
"Now denote the time reversal conjugate of $g$ as $\\tilde{g}$ , namely, $T^{-1}gT=\\tilde{g}$ for $g\\in G$ .",
"Since $G/H=Z_2^T$ , any group element in $G$ either belongs to the unitary normal subgroup $H$ or belongs to its coset $TH=HT$ .",
"This means that any anti-unitary group element must be written in forms of $hT$ or $T\\tilde{h}$ with $h,\\tilde{h}\\in H$ .",
"Some conclusions of unitary groups should be modified for anti-unitary groups.",
"We first generalize Schur's lemma to anti-unitary groups.",
"Generalized Schur's lemma: If a nonzero matrix $C$ commutes with the irreducible (projective or linear) Rep $(\\nu )$ of an anti-unitary group $G$ with $C\\hat{g}=\\hat{g}C$ , namely, $CM^{(\\nu )}(g)K_{s(g)}=M^{(\\nu )}(g)K_{s(g)}C,$ for all $g\\in G$ , then $C$ has at most two eigenvalues which are complex conjugate to each other, including the special case that $C$ is a real constant matrix $C=\\lambda I$ with a single real eigenvalue $\\lambda $ .",
"The proof is simple.",
"No matter $C$ is completely diagonalized or not, $C$ has at least one eigenstate $|\\lambda \\rangle $ with $ C|\\lambda \\rangle &=& \\lambda |\\lambda \\rangle .$ For any unitary element $h\\in H$ and anti-unitary element $Th$ , we have $ &&C\\hat{h}|\\lambda \\rangle = \\hat{h}C|\\lambda \\rangle =\\lambda \\left(\\hat{h}|\\lambda \\rangle \\right),\\\\&&C\\widehat{Th}|\\lambda \\rangle =\\widehat{Th}C|\\lambda \\rangle =\\widehat{Th}(\\lambda |\\lambda \\rangle )= \\lambda ^* \\left(\\widehat{Th}|\\lambda \\rangle \\right).$ Above relations indicate that unitary operators reserve the eigenvalue of $C$ , while anti-unitary operators switch the eigenspace of $\\lambda $ to the eigenspace of its complex conjugate $\\lambda ^*$ .",
"If $(\\nu )$ is irreducible, then $C$ is completely diagonalizable and has at most two eigenvalues $\\lambda $ and $\\lambda ^*$ .",
"Specially, if $C$ is Hermitian, then it must be a real constant matrix $C=\\lambda I$ with a single real eigenvalue $\\lambda $ .",
"The generalized Schur's lemma yields the following corollary: Corollary: If a linear operator $\\widehat{C}$ commutes with an anti-unitary group $G$ with $[\\hat{g},\\widehat{C}]=0$ for all $g\\in G$ , namely, $M(C)M(g)K_{s(g)}=M(g)K_{s(g)}M(C),$ then a direct sum of two eigenspaces whose eigenvalues are mutually complex conjugate is a Rep space of $G$ (If $C$ is not completely diagonalizable, we can transform it into Jordan normal form and replace each `eigenspace' by a `supporting space of Jordan blocks with the same eigenvalue').",
"Our aim is to use the eigenvalues of a set of linear operators to label the irreducible bases of anti-unitary groups.",
"Decoupled factor systems.",
"For type-I anti-unitary groups, we can adopt the following decoupled factor system (see Appendix for details of tuning the coboundary), which satisfy $ &&\\omega _2(E,g_1)=\\omega _2(g_1,E)= 1,\\nonumber \\\\&&\\omega _2(T,T)=\\pm 1,\\nonumber \\\\ &&\\omega _2(T,g_1)=\\omega _2(g_1,T)=1,\\nonumber \\\\&&\\omega _2(Tg_1,g_2)=\\omega _2^{-1}(g_1,g_2),\\nonumber \\\\&&\\omega _2(g_1,Tg_2)=\\omega _2(g_1,\\tilde{g}_2), \\nonumber \\\\&&\\omega _2(Tg_1,Tg_2)=\\omega _2(\\tilde{g}_1,g_2)\\omega _2(T,T),$ for any $g_1,g_2\\in H$ .",
"The first equation is the canonical gauge condition.",
"When above relations are satisfied, it can be shown that $\\omega _2(g_1, g_2)\\omega _2(\\tilde{g}_1,\\tilde{g}_2)=1$ is automatically satisfied.",
"From (REF ), only the cocycles $\\omega _2(T,T)$ and $\\omega _2(g_1,g_2)$ with $g_1,g_2\\in H$ are important, therefore we only need to focus on $\\widehat{T}$ and the normal subgroup $H$ .",
"Once the representation matrices of $T$ and the unitary elements in $H$ are reduced, we then obtain the irreducible Rep for the full group $G$ .",
"For type-II anti-unitary groups, above discussion of decoupled factor system should be slightly modified.",
"In Appendix REF , we provide the procedure to obtain the decoupled factor system in the fermionic anti-unitary groups.",
"Since the discussions for the two types of anti-unitary groups are similar, we will mainly focus on type-I anti-unitary groups in the following.",
"Class operators and CSCO-I.",
"In order to reserve the relation between irreducible representations and classes, we should redefine the classes and class operators for an anti-unitary group $G$ .",
"The class corresponds to unitary element $g_i$ is defined as $\\mathrm {Class}(g_i)=\\lbrace h^{-1}g_ih, (Th)^{-1} g_i^{-1} (Th); h\\in H\\rbrace ,$ while for anti-unitary element $g_iT$ , the conjugate class is defined as $\\mathrm {Class}(g_iT)=\\lbrace h^{-1}(g_iT)\\tilde{h}, (Th)^{-1} ( \\tilde{g}_iT)^{-1}T^2(T\\tilde{h}); h\\in H\\rbrace .$ Obviously, $\\lbrace T\\rbrace $ forms a class itself.",
"Before defining class operators, we define the following operators, $ &&D_{g_i}=\\sum _{h\\in H}\\left( \\hat{h}^{-1}\\hat{g}_i\\hat{h} + \\widehat{Th}^{-1} \\hat{g}_i^{-1} \\widehat{Th}\\right),\\nonumber \\\\&&D_{g_i^{-1}}=\\sum _{h\\in H}\\left( \\hat{h}^{-1}\\hat{g}_i^{-1}\\hat{h} + \\widehat{Th}^{-1} \\hat{g}_i \\widehat{Th}\\right),\\nonumber \\\\&&D_{g_iT}=\\sum _{h\\in H}\\left( \\hat{h}^{-1}\\widehat{g_iT}\\hat{\\tilde{h}} + \\widehat{Th}^{-1} \\widehat{\\tilde{g}_i T}^{-1}\\widehat{T}^2 \\widehat{T\\tilde{h}} \\right),\\nonumber \\\\&&D_{(g_iT)^{-1}}=\\sum _{h\\in H}\\left( \\hat{h}^{-1}\\widehat{\\tilde{g}_i T}^{-1}\\widehat{T}^2\\hat{\\tilde{h}} + \\widehat{Th}^{-1} \\widehat{g_iT}\\widehat{T\\tilde{h}} \\right).\\nonumber \\\\$ For type-I anti-unitary groups $T^2=E$ , and $\\omega _2(g_i,T)=\\omega _2(T,\\tilde{g}_i)=1$ , we have $\\hat{g}_i\\widehat{T} = \\widehat{T}\\hat{\\tilde{g}}_i$ .",
"Therefore, $ &&D_{g_i}=\\sum _{h\\in H}\\left( \\hat{h}^{-1}\\hat{g}_i\\hat{h} + \\hat{h}^{-1}\\hat{\\tilde{g}}_i^{-1}\\hat{h} \\right)= C_{g_i}^H + C_{\\tilde{g}_i^{-1}}^H,\\nonumber \\\\&&D_{g_i^{-1}}=\\sum _{h\\in H}\\left( \\hat{h}^{-1}\\hat{g}_i^{-1}\\hat{h} + \\hat{h}^{-1}\\hat{\\tilde{g}}_i\\hat{h} \\right)= C_{g_i^{-1}}^H + C_{\\tilde{g}_i}^H,\\nonumber \\\\&&D_{g_iT}=\\sum _{h\\in H}\\left( \\hat{h}^{-1}\\hat{g}_i\\hat{h} + \\hat{h}^{-1}\\hat{\\tilde{g}}_i^{-1}\\hat{h} \\right)\\widehat{T}=D_{g_i}\\widehat{T},\\nonumber \\\\&&D_{(g_iT)^{-1}}=\\sum _{h\\in H}\\left( \\hat{h}^{-1}\\hat{g}_i^{-1}\\hat{h} + \\hat{h}^{-1}\\hat{\\tilde{g}}_i\\hat{h} \\right)\\widehat{T}=D_{g_i^{-1}}\\widehat{T},\\nonumber \\\\$ where $C_{g_i}^{H}$ , $C_{\\tilde{g}_i^{-1}}^{H}$ , $C_{g_i^{-1}}^{H}$ and $C_{\\tilde{g}_i}^{H}$ denote the class operators of the normal subgroup $H$ corresponding to unitary elements $g_i$ , $\\tilde{g}_{i}^{-1}$ , $g_i^{-1}$ and $\\tilde{g}_{i}$ respectively.",
"It is easy to prove that $\\widehat{T}^{-1}D_{g_i}\\widehat{T}=D_{g_i^{-1}},$ or $D_{g_i}\\widehat{T}=\\widehat{T} D_{g_i^{-1}}$ .",
"Namely, the operators $D_{g_i}$ do not commute with all the group elements.",
"We can define the following class operators to solve this problem: 1) if $g_i$ and $g_i^{-1}$ belong to different classes, namely, if $g_i\\ne g_i^{-1}$ and $g_i\\ne \\tilde{g}_i$ , then we define two class operators corresponding to the classes of $g_i$ and $g_i^{-1}$ , $ && C_{i^+}=D_{g_i}+D_{g_i^{-1}}=C_{g_i}^H +C_{\\tilde{g}_i^{-1}}^H+C_{g_i^{-1}}^H +C_{\\tilde{g}_i}^H , \\\\&&C_{i^-}=i(D_{g_i}-D_{g_i^{-1}})=i(C_{g_i}^H +C_{\\tilde{g}_i^{-1}}^H-C_{g_i^{-1}}^H -C_{\\tilde{g}_i}^H ).",
"\\nonumber \\\\$ Obviously $C_{i^{\\pm }}$ commute with $\\hat{g}_j$ for any $g_j\\in H$ , $C_{i^{\\pm }}\\hat{g}_j=\\hat{g}_j C_{i^{\\pm }}$ .",
"Since $ &&\\widehat{T}C_{g_i}^H=C_{\\tilde{g} _i}^H\\widehat{T} ~,~\\widehat{T}C_{\\tilde{g} _i}^H=C_{g_i}^H\\widehat{T},\\\\&&\\widehat{T}C_{g_i^{-1}}^H=C_{\\tilde{g} _i^{-1}}^H\\widehat{T} ~,~\\widehat{T}C_{\\tilde{g} _i^{-1}}^H=C_{g_i^{-1}}^H\\widehat{T},$ $C_{i^{\\pm }}$ commute with the time reversal operator, $ &&\\widehat{T}C_{i^+}=(C_{\\tilde{g}_i}^H +C_{g_i^{-1}}^H+C_{\\tilde{g}_i^{-1}}^H +C_{ g_i}^H)\\widehat{T}=C_{i^+}\\widehat{T},\\\\&&\\widehat{T}C_{i^-}=-i(C_{\\tilde{g}_i}^H +C_{ g_i^{-1}}^H-C_{\\tilde{g}_i^{-1}}^H -C_{ g_i}^H )\\widehat{T}=C_{i^-}\\widehat{T}.$ Therefore, $C_{i^{\\pm }}$ commute with all the operators in $G$ .",
"2) if $g_i$ and $g_i^{-1}$ belong to the same class, namely, if $g_i=g_i^{-1}$ or $g_i=\\tilde{g}_i$ , then the class operator corresponding to $g_i$ is $C_i=C_{i^+}=D_{g_i}$ (obviously $C_{i^-}=0$ in this case).",
"It can be easily checked that above class operators $C_{i^\\pm }$ are Hermitian if the Rep is unitary.",
"Owing to relation (REF ), the class operators for anti-unitary elements $g_iT$ and $(g_iT)^{-1}$ are $C_{i^{+}T}&=&D_{g_i T}+D_{(g_i T)^{-1}}\\nonumber \\\\&=&(C_{g_i}^H +C_{\\tilde{g}_i}^H+C_{g_i^{-1}}^H+C_{\\tilde{g}_i^{-1}}^H)\\widehat{T} =C_{i^{+}}\\widehat{T}, \\\\C_{i^{-}T}&=&i(D_{g_i T}-D_{(g_i T)^{-1}})\\nonumber \\\\&=&i[(C_{g_i}^H -C_{\\tilde{g}_i}^H)-(C_{g_i^{-1}}^H-C_{\\tilde{g}_i^{-1}}^H )]\\widehat{T}=C_{i^{-}}\\widehat{T}.",
"\\nonumber \\\\$ This gives a one-to-one correspondence between the anti-unitary class operators $C_{i^{\\pm }T}$ and the unitary class operators $C_{i^{\\pm }}$ , where the unitary class operators can be obtained solely from the subgroup $H$ by the equations (REF ) and ().",
"However, it can be shown that the anti-unitary class operators do not provide any meaningful quantum numbers for $U(1)$ coefficient projective Reps. To see why the anti-unitary class operators do not correspond to more irreducible Reps, we focus on the class operator $\\widehat{T}$ first.",
"Noticing that $M(T)$ is a real matrix, so $\\widehat{T}^2=[M(T)]^2=\\omega _2(T,T)=\\pm 1$ .",
"If $\\widehat{T}^2=1$ , then $M(T)$ has eigenvalues $\\pm 1$ with eigenstates $|\\phi ^+\\rangle $ and $|\\phi ^-\\rangle $ satisfying $\\widehat{T}|\\phi ^\\pm \\rangle =\\pm |\\phi ^\\pm \\rangle $ respectively.",
"Under the unitary transformation $|\\phi ^+\\rangle ^{\\prime }=i|\\phi ^+\\rangle $ , $\\widehat{T}|\\phi ^+\\rangle ^{\\prime }=-i|\\phi ^+\\rangle =-|\\phi ^+\\rangle ^{\\prime },$ which gives $M^{+^{\\prime }}(T)=-1=M^-(T)$ .",
"So the two one-dimensional Reps $(+)$ and $(-)$ are equivalent, and the quantum numbers corresponding to the class operator $\\widehat{T}$ are redundant (The two Reps are non-equivalent when one considers $Z_2$ coefficient projective Reps).",
"On the other hand, if $\\widehat{T}^2=-1$ , the bases form Kramers doublets and $\\widehat{T}$ cannot reduce to one dimensional Reps.",
"In other words, the operator $\\widehat{T}$ has no `eigenvalues' at all.",
"In this case, the class operator $\\widehat{T}$ doesn't contribute any quantum numbers to label different Reps.",
"Since $C_{i^\\pm T}=C_{i^\\pm }\\widehat{T}=\\widehat{T} C_{i^\\pm }$ , the eigenvalues of $C_{i^\\pm T}$ , if exist, are equal to the product of eigenvalues of $C_{i^\\pm }$ and $\\widehat{T}$ .",
"As a result, all the anti-unitary class operators do not contribute useful quantum numbers.",
"Therefore, the number of different irreducible projective Reps is determined by the unitary class operators $C=\\lbrace C_{i^\\pm }\\rbrace $ .",
"Namely, $C=\\lbrace C_{i^\\pm }\\rbrace $ are the CSCO-I of the anti-unitary group $G$ .",
"It can be shown that unitary elements belonging to the same class have the same character in an irreducible projective Rep.",
"Therefore, the characters of the unitary group elements as functions of the irreducible Reps, are also functions of the unitary class operators.",
"As a consequence, the number of different irreducible projective Reps is equal to the number of linearly independent unitary class operators in $\\lbrace C_{i^\\pm }\\rbrace $ .",
"This result is similar to the unitary groups.",
"Since all the class operators corresponding to unitary classes are Hermitian, their eigenvalues are all real numbers, each set of eigenvalues corresponds to an irreducible Rep space.",
"CSCO-II and CSCO-III.",
"The class operators $C(s)$ of the subgroups $G(s)\\subset H$ are defined as usual unitary groups.",
"Then we obtain the CSCO-II $(C, C(s))$ , where the quantum numbers of $C(s)$ can be used to distinguish different bases of one copy of irreducible Rep.",
"Since an irreducible Rep may occur more than once, we need to make use of the class operators of the intrinsic group to label the multiplicity.",
"For the unitary elements $\\bar{g}_i$ and $\\bar{g}^{-1}_{i}$ , we define the following operators $ &&\\bar{D}_{g_i}=\\sum _{\\bar{h}\\in \\overline{H}}\\left( \\hat{\\bar{h}}^{-1}\\hat{\\bar{g}}_i\\hat{\\bar{h}} + \\hat{\\bar{h}}^{-1}\\hat{\\bar{\\tilde{g}}}_i^{-1}\\hat{\\bar{h}} \\right)= \\bar{C}_{g_i}^H + \\bar{C}_{\\tilde{g}_i^{-1}}^H,\\nonumber \\\\&&\\bar{D}_{g_i^{-1}}=\\sum _{\\bar{h}\\in \\overline{H}}\\left( \\hat{\\bar{h}}^{-1}\\hat{\\bar{g}}_i^{-1}\\hat{ \\bar{h}} + \\hat{\\bar{h}}^{-1}\\hat{\\bar{\\tilde{g}}}_i\\hat{\\bar{h}}\\right)=\\bar{C}_{g_i^{-1}}^H + \\bar{C}_{\\tilde{g}_i}^H\\nonumber \\\\$ For the anti-unitary elements $\\overline{Tg_i }$ the corresponding conjugate class is defined as $\\mathrm {Class}(\\overline{Tg_i})=\\lbrace \\bar{h}^{-1}\\overline{Tg_i}\\bar{\\tilde{h}}, \\overline{hT}^{-1} \\overline{T\\tilde{g}_i}^{-1}\\overline{T}^2\\overline{\\tilde{h} T}; \\overline{h}\\in \\overline{H}\\rbrace ,$ we have $ \\bar{D}_{Tg_i}&=&\\sum _{\\bar{h}\\in \\overline{H}}\\left( \\hat{\\bar{h}}^{-1}\\widehat{\\overline{Tg_i}}\\hat{\\bar{\\tilde{h}}} + \\widehat{\\overline{hT}}^{-1} \\widehat{ \\overline{T\\tilde{g}_i}}^{-1}\\widehat{ \\overline{T}}^2 \\widehat{\\overline{\\tilde{h} T}} \\right)\\nonumber \\\\&=&\\sum _{\\bar{h}\\in \\overline{H}}\\left( \\hat{\\bar{h}}^{-1}\\hat{\\bar{g}}_i\\hat{\\bar{h}} + \\hat{\\bar{h}}^{-1}\\hat{\\bar{\\tilde{g}}}_i^{-1}\\hat{\\bar{h}} \\right)\\widehat{\\overline{T}}=\\bar{D}_{g_i}\\widehat{\\overline{T}},\\nonumber $ $ \\bar{D}_{(Tg_i)^{-1}}&=&\\sum _{\\bar{h}\\in \\overline{H}}\\left( \\hat{\\bar{h}}^{-1}\\widehat{ \\overline{T\\tilde{g}_i}}^{-1}\\widehat{ \\overline{T}}^2\\hat{\\bar{\\tilde{h}}} + \\widehat{\\overline{hT}}^{-1} \\widehat{\\overline{Tg_i}} \\widehat{\\overline{\\tilde{h} T}} \\right)\\nonumber \\\\&=&\\sum _{\\bar{h}\\in \\overline{H}}\\left( \\hat{\\bar{h}}^{-1}\\hat{\\bar{g}}_i^{-1}\\hat{ \\bar{h}} + \\hat{\\bar{h}}^{-1}\\hat{\\bar{\\tilde{g}}}_i\\hat{\\bar{h}}\\right)\\widehat{\\overline{T}}=\\bar{D}_{g_i^{-1}}\\widehat{\\overline{T}}.\\nonumber \\\\$ The class operators corresponding to $\\bar{g}_i, \\bar{g}^{-1}_{i}\\in \\overline{H}$ , $\\overline{Tg_i }$ and $\\overline{Tg_{i}}^{-1}$ are defined following (REF ) and () as $ \\bar{C}_{i^+} &=&\\bar{D}_{g_i}+\\bar{D}_{g_i^{-1}}\\nonumber \\\\&=&\\bar{C}_{g_i}^{H} + \\bar{C}_{\\tilde{g}_i^{-1}}^{H} + \\bar{C}_{g_i^{-1}}^{H} + \\bar{C}_{\\tilde{g}_i}^{H},\\\\\\bar{C}_{i^-} &=&\\left(\\bar{D}_{g_i}-\\bar{D}_{g_i^{-1}}\\right)S\\nonumber \\\\&=&(\\bar{C}_{g_i}^{H} + \\bar{C}_{\\tilde{g}_i^{-1}}^{H} - \\bar{C}_{g_i^{-1}}^{H} - \\bar{C}_{\\tilde{g}_i}^{H})S,$ where Eq.",
"(REF ) has been used, and $S=i(P_u-P_a)=\\left(\\begin{matrix}iI&0\\\\0&-iI \\end{matrix}\\right)$ (here $I$ stands for a $H$ -dimensional identity matrix, $H$ is the order of normal subgroup $H$ ) is a diagonal matrix with entries $i$ for unitary bases or $-i$ for anti-unitary bases, which satisfies $S^*=-S, S^2=-1$ and $ &&M(g_i) S=SM(g_i),\\ \\ M(g_iT)S^*=SM(g_iT),\\\\&&M(\\bar{g}_i)S=SM(\\bar{g}_i),\\ \\ M(\\overline{g_i T})S^\\ast =SM(\\overline{g_i T}).$ Above relations can be easily verified since $M(g_i)$ and $M(\\bar{g}_i)$ are block diagonalized with the form $\\left(\\begin{matrix}A&0\\\\0&B \\end{matrix}\\right)$ , where the blocks $A$ and $B$ are $H\\times H$ nonzero matrices, while $M(g_iT)$ and $M(\\overline{g_iT})$ are block-off-diagonalized with the form $\\left(\\begin{matrix}0&A\\\\B&0 \\end{matrix}\\right)$ .",
"Furthermore, it can be checked (see Appendix ) that $ &&\\bar{C}_{i^+}=C_{i^+}, \\ \\ \\ \\ \\ \\ \\ \\ \\bar{C}_{i^-}=C_{i^-},\\nonumber \\\\&&\\bar{C}_{i^+ T}=\\bar{C}_{i^+}\\widehat{\\overline{T}}, \\ \\ \\ \\ \\bar{C}_{i^- T}=\\bar{C}_{i^-}\\widehat{\\overline{T}},$ where $\\bar{C}_{i^+ T}=\\bar{D}_{Tg_i}+\\bar{D}_{(Tg_i)^{-1}}$ and $\\bar{C}_{i^- T}=S(\\bar{D}_{Tg_i}-\\bar{D}_{(Tg_i)^{-1}})=(\\bar{D}_{Tg_i}-\\bar{D}_{(Tg_i)^{-1}})S^{\\ast }$ .",
"Again, $\\widehat{\\overline{T}}$ forms a class operator itself.",
"And for any $g\\in G$ we have $ \\widehat{\\overline{T}}\\hat{\\bar{g}}=\\hat{\\bar{\\tilde{g}}}\\widehat{\\overline{T}},\\ \\ \\ \\widehat{\\overline{T}}\\hat{g}=\\hat{ g}\\widehat{\\overline{T}},$ or equivalently, $ &&M(\\overline{T})M(\\bar{g})=M(\\bar{\\tilde{g}})M(\\overline{T}),\\nonumber \\\\&&M(\\overline{T})M(g)K_{s(g)}=M(g)K_{s(g)}M(\\overline{T})\\nonumber \\\\&&=M(g)M(\\overline{T})K_{s(g)},$ where $M(\\overline{T})$ is a real matrix.",
"$\\widehat{\\overline{T}}$ doesn't contain complex-conjugate operator $K$ and commutes with all the Rep matrices $M(g)$ of $G$ .",
"Noticing that $\\widehat{T}|g\\rangle =|Tg\\rangle $ , $\\widehat{\\overline{T}}|g\\rangle =|gT\\rangle =|T\\tilde{g}\\rangle $ , and $\\tilde{g}$ is generally different from $g$ (except that $G=H\\times Z_2^T$ ), so generally $M(T)\\ne M({\\overline{T}})$ .",
"This implies $\\bar{C}_{i^{+}T}\\ne C_{i^{+}T}$ , namely, the anti-unitary class operator is generally different from its intrinsic partner.",
"This is a difference between unitary and anti-unitary groups.",
"Since $G$ is anti-unitary while all the operators in CSCO-II are obtained from the unitary normal subgroup $H$ (and its subgroups), we need to use at least one anti-unitary class operator of $\\bar{G}$ (or its subgroup chain $\\bar{G}(s^{\\prime })$ ) to construct CSCO-III.",
"In most cases, we can adopt the class operator $\\widehat{\\overline{T}}$ as a member of $\\bar{C}(s^{\\prime })$ in CSCO-III (an exception is given in Appendix REF ).",
"If an intrinsic subgroup $\\bar{G}_1\\in \\bar{G}(s)$ is anti-unitary, and $\\bar{G}_1/\\bar{H}_1=\\overline{Z_2^T}$ where $\\bar{H}_1$ is the unitary normal subgroup with $\\bar{H}_1\\subset \\bar{H}$ , then the corresponding class operators are given by $ &&\\bar{C}_{g_i}^{H_1} + \\bar{C}_{\\tilde{g}_i^{-1}}^{H_1}+ \\bar{C}_{ g_i^{-1}}^{H_1}+ \\bar{C}_{\\tilde{g}_i}^{H_1}, \\nonumber \\\\&& (\\bar{C}_{g_i}^{H_1} + \\bar{C}_{\\tilde{g}_i^{-1}}^{H_1}- \\bar{C}_{ g_i^{-1}}^{H_1}- \\bar{C}_{\\tilde{g}_i}^{H_1}) S,$ where $\\bar{C}_{g_i}^{H_1}$ (or $\\bar{C}_{\\tilde{g}_i}^{H_1}$ ) is the class operator of $\\bar{H}_1$ corresponding to the element $\\bar{g}_i$ (or $\\bar{\\tilde{g}}_i$ ).",
"On the other hand, if an intrinsic subgroup $\\bar{G}_2\\in \\bar{G}(s)$ is unitary, then its (intrinsic) class operators are defined in the same way as usual unitary groups, the only constraint is that all these class operators should commute with all the class operators in CSCO-II and should be mutually commuting.",
"After carefully choosing the operators of $\\bar{C}(s^{\\prime })$ such that the degeneracy of the quantum numbers are completely lifted, we obtain the CSCO-III $\\left(C,C(s), \\bar{C}(s^{\\prime }) \\right)$ , where $C=\\lbrace C_{i^\\pm }\\rbrace $ .",
"Before going to examples, we summarize some special properties of anti-unitary groups which are different from unitary groups: 1, The anti-unitary class operators do not contribute any meaningful quantum numbers to CSCO-I.",
"The simplest example is $Z_2^T$ , which has only one 1-dimensional linear Rep When acting on a Hilbert space, the 1-dimensional Reps of $Z_2^T$ is classified by $\\protect \\mathcal {H}^1(Z_2^T, U(1))=\\protect \\mathbb {Z}_1$ , so there is only one 1-dimensional Rep.",
"However, when acting on Hermitian operators, then the 1-dimensional Reps is classified by $\\protect \\mathcal {H}^1(Z_2^T, Z_2)=\\protect \\mathbb {Z}_2$ , namely, there are TWO different Reps characterized by $T\\protect {widehat}05EOT^{-1} =\\pm \\protect {widehat}05EO$ , where $\\protect {widehat}05EO$ is an Hermitian operator.",
"This result can be generalized to any anti-unitary groups.",
"and one 2-dimensional irreducible projective Rep. 2, An irreducible Rep (after lifting the multiplicity) may be either labeled by a real quantum number, or labeled by a pair of complex conjugating quantum numbers.",
"Since we redefined the conjugate classes for anti-unitary groups and all the class operators in CSCO-I are Hermitian, the main quantum numbers are real.",
"But the multiplicity quantum numbers are generally complex numbers, so an irreducible Rep is generally labeled by a pair of complex conjugating quantum numbers."
],
[
"Some examples",
"In this section, we list the nontrivial irreducible projective Reps of a few finite groups.",
"The group elements, the generators, the classification labels and the coboundary variables are given below.",
"The results of the projective Reps are shown in Table REF .",
"For simplicity, we only list the irreducible projective Rep with the lowest dimension in each nontrivial class, and only the Rep matrices of the generators are given.",
"Unitary groups: $Z_2\\times Z_2=\\lbrace E, P\\rbrace \\times \\lbrace E, Q\\rbrace $ with $P^2=Q^2=E, QP=PQ$ .",
"There are two generators $P, Q$ .",
"The classification is labeled by $\\omega _{2}(Q,PQ)$ .",
"The coboundary variables ($11,15,16$ ) are set to be 1.",
"$Z_2\\times Z_2\\times Z_2 = \\lbrace E, P\\rbrace \\times \\lbrace E, Q\\rbrace \\times \\lbrace E, R\\rbrace $ with three generators $P, Q,R$ .",
"The classification is labeled by $\\omega _{2}(PR,QR)$ , $\\omega _{2}(PR,PQR)$ , $\\omega _{2}(QR,PQR)$ .The coboundary variables ($46,54,55,61\\sim 64$ ) are set to be 1.",
"$Z_3\\times Z_3=\\lbrace E, P,P^2\\rbrace \\times \\lbrace E, Q,Q^2\\rbrace $ with two generators $P, Q$ .",
"The classification is labeled by $\\omega _{2}(PQ^{2},P^{2}Q^{2})$ .",
"The coboundary variables ($70,71,76\\sim 81$ ) are set to be 1.",
"$Z_3\\times Z_3\\times Z_3=\\lbrace E, P,P^2\\rbrace \\times \\lbrace E, Q,Q^2\\rbrace \\times \\lbrace E, R,R^2\\rbrace $ with three generators $P, Q, R$ .",
"The classification is labeled by $\\omega _{2}(P^{2}QR^{2},PQ^{2}R^{2})$ , $\\omega _{2}(P^{2}QR^{2},P^{2}Q^{2}R^{2})$ , $\\omega _{2}(PQ^{2}R^{2},P^{2}Q^{2}R^{2})$ .",
"The coboundary variables ($642,645,695,696,698,699\\sim 701,712 \\sim 729$ ) are set to be 1.",
"$Z_4\\times Z_8 = \\lbrace E, P,P^2,P^3\\rbrace \\times \\lbrace E,Q,Q^2, Q^3,Q^4, Q^5,Q^6, Q^7\\rbrace $ , with two generators $P, Q$ .",
"The classification is labeled by $\\omega _{2}(P^{2}Q^{7},P^{3}Q^{7})$ .",
"The coboundary variables ($989,990,991,997\\sim 1024$ ) are set to be 1.",
"$A_4=\\lbrace E,(123),(132),(124),(142),(134), (143), (234),\\\\ (243), (12)(34),(13)(24),(14)(23)\\rbrace $ , the normal subgroup of $S_4=\\lbrace E,\\ (123),\\ (132),\\ (124),\\ (142), ...,\\ (12),\\ (13),\\\\ (23), (14), (24), ... \\rbrace $ formed by even-parity permutations with $S_4/A_4\\simeq Z_2=\\lbrace E,(12)\\rbrace $ .",
"The group $A_4$ has generators $P=(123), Q=(124)$ with $P^3=Q^3=E,\\ PQ=Q^{2}P^{2}$ .",
"The classification is labeled by $\\omega _{2}(P^2Q^2,P^2Q^2)$ .",
"The coboundary variables ($104,105,107,108,128,129,136,138,140,141,143$ ) are set to be 1.",
"$Z_4\\rtimes Z_2=\\lbrace E, P, P^2, P^3\\rbrace \\rtimes \\lbrace E, Q\\rbrace $ with $P^mQ=QP^{4-m}$ , there are two generators $P$ and $Q$ .",
"The classification is labeled by $\\omega _{2}(P^{2}Q,Q)$ .",
"The coboundary variables ($54\\sim 56, 61\\sim 64$ ) are set to be 1.",
"$Z_2\\times Z_2\\times Z_2\\times Z_2=\\lbrace E, P\\rbrace \\times \\lbrace E, Q\\rbrace \\times \\lbrace E, R\\rbrace \\times \\lbrace E, S\\rbrace $ with four generators $P, Q, R, S$ .",
"The classification is labeled by $\\omega _{2}(PQS,PRS),\\ \\omega _{2}(PQS,QRS),\\ \\omega _{2}(PQS,PQRS)$ , $\\omega _{2}(PRS,QRS),\\ \\omega _{2}(PRS,PQRS),\\ \\omega _{2}(QRS,PQRS)$ .",
"The coboundary variables ($205, 221,222,234, 237\\sim 239, 249\\sim 256$ ) are set to be 1.",
"Anti-unitary groups: $Z_2\\times Z_2^T=\\lbrace E, P\\rbrace \\times \\lbrace E, T\\rbrace $ with two generators $P$ and $T$ .",
"The classification is labeled by ($\\omega _{2}(T,T)$ , $\\omega _{2}(PT,PT)$ ).",
"The coboundary variables ($12,15$ ) are set to be 1.",
"$Z_2\\times Z_2\\times Z_2^T = \\lbrace E, P\\rbrace \\times \\lbrace E, Q\\rbrace \\times \\lbrace E, T\\rbrace $ with three generators $P, Q,T$ .",
"The classification is labeled by $\\omega _{2}(PT,PT)$ , $\\omega _{2}(PT,QT)$ , $\\omega _{2}(QT,QT)$ , $\\omega _{2}(PQT,PQT)$ .",
"The coboundary variables ($48,54,56,61\\sim 63$ ) are set to be 1.",
"$(Z_2\\times Z_2)\\rtimes Z_2^T = (\\lbrace E, P\\rbrace \\times \\lbrace E, Q\\rbrace )\\rtimes \\lbrace E, T\\rbrace $ with $PQ=QP,\\ TP=PT$ and $TQ=PQT$ .",
"This group can be generated by two generators $Q,T$ since $(TQ)^2=(QT)^2=P$ .",
"The classification is labeled by $\\omega _{2}(PT,PT)$ , $\\omega _{2}(QT,Q)$ .",
"The coboundary variables ($48,60\\sim 64$ ) are set to be 1.",
"$Z_4\\times Z_2^T=\\lbrace E, P, P^2, P^3\\rbrace \\times \\lbrace E, T\\rbrace $ , there are two generators $P$ and $T$ .",
"The classification is labeled by $\\omega _{2}(P^{2}T,P^{2}T)$ , $\\omega _{2}(P^{3}T,P^{2})$ .",
"The coboundary variables ($56,60\\sim 64$ ) are set to be 1.",
"$Z_4\\rtimes Z_2^T=\\lbrace E, P, P^2, P^3\\rbrace \\rtimes \\lbrace E, T\\rbrace $ with $P^mT=TP^{4-m}$ , there are two generators $P$ and $T$ .",
"The classification is labeled by $\\omega _{2}(P^{2}T,P^{2}T)$ , $\\omega _{2}(P^{3}T,P^{3}T)$ .",
"The coboundary variables ($53,54,56,61\\sim 63$ ) are set to be 1.",
"$Z_{3}\\times (Z_{3}\\rtimes Z_{2}^{T}) \\simeq (Z_3\\times Z_3)\\rtimes Z_2^T=\\lbrace E,P,P^2\\rbrace \\times (\\lbrace E,Q,Q^2\\rbrace \\rtimes \\lbrace E,T\\rbrace )$ with $TP=PT, TQ=Q^2T,PQ=QP$ , there are three generators $P,Q,T$ .",
"The classification is labeled by $\\omega _{2}(P^{2}QT,PQ^{2}T)$ .",
"The coboundary variables ($264,270,309,311,312,314\\sim 324$ ) are set to be 1.",
"$S_4^T=A_4\\rtimes Z_2^T$ , there are three generators $P=(123), Q=(124)$ and $T=(12)K$ with $P^3=Q^3=T^2=E$ and $TP=P^{2}T, TQ=Q^2T,PQ=Q^{2}P^{2}$ .",
"The classification is labeled by $\\omega _{2}(PQT,QP^{2}T)$ , $\\omega _{2}(P^{2}QT,PQ)$ .",
"The coboundary variables ($548\\sim 552, 557, 559\\sim 561, 564\\sim 576$ ) are set to be 1.",
"$A_4\\times Z_2^T$ , there are three generators $P=(123), Q=(124)$ and $T$ with $P^3=Q^3=T^2=E$ and $TP=PT, TQ=QT,PQ=Q^{2}P^{2}$ .",
"The classification is labeled by $\\omega _{2}(P^2,P)$ , $\\omega _{2}(P^{2},Q^{2}T)$ .",
"The coboundary variables ($34,35,37,45,84,93,148, 152,260,286,296,334,351,373$ , $390,399,408,427,459,477,537,549$ ) are set to be 1.",
"Fermionic Anti-unitary groups: $Z_4^T=\\lbrace E,P_f,T,P_f T\\rbrace $ with $T^4=E$ and $T^2=P_f$ .",
"There is only one generator $T$ .",
"The classification is labeled by $\\omega _{2}(P_f T,P_f)$ .",
"The coboundary variables ($15,16$ ) are set to be 1.",
"The projective Reps can also be characterized by the invariant $[M(T)K]^4=\\pm 1$ [see eqs.",
"(REF ) and (REF )].",
"$Z_2\\times Z_4^T=\\lbrace E,P\\rbrace \\times \\lbrace E,P_f,T,P_f T\\rbrace $ with two generators $P$ and $T$ .",
"The classification is labeled by $\\omega _{2}(P_f T,P_f T)$ , $\\omega _{2}(PP_f T,P_f)$ .",
"The coboundary variables ($56,60\\sim 64$ ) are set to be 1.",
"$Z_2\\ltimes Z_4^T=\\lbrace E,P\\rbrace \\ltimes \\lbrace E,T,T^2,T^3\\rbrace $ with $TP=PT^3$ is a type-I anti-unitary group since $(TP)^2=E$ .",
"It is easy to verify that $Z_2\\ltimes Z_4^T$ is isomorphic to $D_{2d}^T \\simeq (Z_2\\times Z_2)\\rtimes Z_2^T$ .",
"$G_-^+(Z_4,T)$ with two generators $P$ and $T$ where $Z_4=\\lbrace E,P,P^2,P^3\\rbrace $ and $P^2=T^2=P_f$ , $PT=TP$ .",
"Since $(TP)^2=E$ and $P^4=E$ , $G_-^+(Z_4,T)\\simeq Z_4\\times Z_2^T$ , they have the same representations.",
"$G_-^-(Z_4,T)$ with two generators $P$ and $T$ where $Z_4=\\lbrace E,P,P^2,P^3\\rbrace $ and $P^2=T^2=P_f$ , $P^mT=TP^{4-m}$ .",
"The classification is labeled by $\\omega _{2}(P_f T,P_f T)$ .",
"The coboundary variables ($56,60\\sim 64$ ) are set to be 1.",
"Remarks: 1) Some of the above groups are isomorphic to point groups.",
"For example,$Z_2\\times Z_2 \\simeq D_2$ ; $Z_2\\times Z_2\\times Z_2\\simeq D_{2h}$ ; $Z_4\\rtimes Z_2 \\simeq C_{4v}$ (or $D_{2d}$ ); $A_4\\simeq T$ (the symmetry group of tetrahedron); $S_4\\simeq T_d$ (or $O$ ).",
"For anti-unitary groups, we interpret the operations containing mirror reflection as anti-unitary elements, for example, we regard the horizontal mirror reflection in the group $Z_2\\times Z_2^T\\simeq C_{2h}^T$ or the vertical mirror reflection in the group $(Z_2\\times Z_2)\\rtimes Z_2^T\\simeq D_{2d}^T$ , as the generator $T$ of $Z_2^T$ .",
"It should be noted that although $C_{4v}\\simeq D_{2d}$ , the anti-unitary groups $C_{4v}^T$ and $D_{2d}^T$ are NOT isomorphic since their unitary normal subgroups are different.",
"2) When solving the cocycle equations (REF ) to obtain the factor systems, we have set some coboundary variables to be 1.",
"In order to label these variables, we first label the $G$ group elements as $1, 2, ..., G$ .",
"For direct product (or semi-direct product) groups $G_1\\times G_2$ , the group elements are sorted by the coset of the first group $G_1$ , for example, for the $Z_2\\times Z_2$ group the elements are sorted by $\\lbrace E, P\\rbrace \\times \\lbrace E, Q\\rbrace = \\lbrace \\lbrace E, P\\rbrace , \\lbrace E, P\\rbrace Q\\rbrace =\\lbrace E,P, Q,PQ\\rbrace $ .",
"Then we sort the $G\\times G$ variables of the 2-cocycle with the order $\\omega _2(1,1), \\omega _2(1,2),..., \\omega _2(1,G),\\omega _2(2,1),...,\\omega _2(G,G)$ and further label them by numbers $1,2, ..., G^2$ .",
"The values of the classification labels and the coboundary variables completely fix the factor system (see appendix ).",
"3) For anti-unitary groups, we adopt the decoupled factor system (REF ) by multiplying a coboundary $\\Omega _{1}(g)$ (see Appendix ).",
"After the reduction, we divide the irreducible Rep matrices $M(g)$ by the coboundary $\\Omega _{1}(g)$ to go back to the original factor system.",
"4) In Table REF in the appendix, we list all the irreducible linear Reps of several anti-unitary groups.",
"We also give the number of independent unitary class operators and the multiplicity of each irreducible Rep. From the table we can see that different from unitary groups, the number of times an irreducible Rep occurs in the regular Rep is not always equal to its dimension.",
"Table: Irreducible projective Reps of some simple finite groups, we only give the lowest dimensional Rep, and only list the representation matrices of the generators.",
"The symbols σ x,y,z \\sigma _{x,y,z} denote the Pauli matrices, and ω=e i2π 3 \\omega =e^{i\\frac{2\\pi }{3}}, Ω=e i2π 9 \\Omega =e^{i\\frac{2\\pi }{9}}, ω 1 2 =e iπ 3 \\omega ^{\\frac{1}{2}}=e^{i\\frac{\\pi }{3}}.Table: NO_CAPTIONTable: NO_CAPTIONTable: NO_CAPTIONTable: NO_CAPTION"
],
[
"CG coefficients for projective Reps",
"The direct product of two projective Reps $(\\nu _1)$ and $(\\nu _2)$ of group $G$ is usually a reducible (projective) Rep. Once the direct product Rep is reduced into irreducible (projective) Reps, $C^\\dag M^{(\\nu _1)}(g) \\otimes M^{(\\nu _2)}(g) K_{s(g) }C =\\bigoplus _{\\nu _3} M^{(\\nu _3)}(g)K_{s(g)}$ we can obtain the corresponding Clebsch-Gordan (CG) coefficients $C$ .",
"The eigenfunction method can also be applied to calculate the CG coefficients.",
"Here we will not give details of the calculations.",
"In this section, we will summarize some physical applications of irreducible projective Reps."
],
[
" 1-D SPT phases",
"The well studied spin-1 Haldane phase, , was known for its disordered gapped ground state and its spin-1/2 edge states at the open boundaries, .",
"The exotic properties of the Haldane phase is protected by $Z_2\\times Z_2$ spin rotation symmetry or time reversal symmetry, .",
"The edge states vary projectively under the action of the symmetry group.",
"The Haldane phase was later generalized to other 1D SPT phases with different symmetries classified by the second group cohomology, .",
"In Ref.CGLW2, topological nonlinear sigma model (NLSM) was generalized to finite symmetry group $G$ in discrete space-time to describe SPT phases in any spatial dimensions.",
"In the following, based on irreducible projective Reps we will give a minimal field theory description of 1D SPT phases.",
"Traditionally, the spin-1 Haldane phase is described by the $O(3)$ NLSM with the following topological $\\theta $ -term , , $\\mathcal {L}_{\\theta }[n(x, \\tau )] =\\theta {i\\over 4\\pi } n\\cdot (\\partial _xn\\times \\partial _\\tau n),$ where $\\theta = {2\\pi }$ and $|n\\rangle $ is the spin coherent state $\\langle n|\\hat{S}|n\\rangle = n$ which varies under rotation in the following way, $\\hat{R}|n\\rangle =e^{i\\varphi }|Rn\\rangle ,$ the phase factor $e^{i\\varphi }$ depends on the axis of the rotation $R$ and the gauge choice of the bases $|n\\rangle $ .",
"The collection of the end points of the vector $n$ form a sphere, i.e.",
"the symmetric space of the $SO(3)$ group $S^2=SO(3)/SO(2)$ .",
"The Lagrange density ${i\\theta \\over 4\\pi } n\\cdot (\\partial _xn\\times \\partial _\\tau n)dxd\\tau $ describes the `Berry phase' of a spin-$1/2$ particle evolving among three states at $(x,\\tau )$ , $(x+dx,\\tau )$ and $(x,\\tau +d\\tau )$ .",
"If the space-time is closed, then the action amplitude $e^{-\\int d {x} d\\tau \\mathcal {L}_{\\theta }[n(x,\\tau )]}$ equals 1.",
"If spacial boundary condition is open, then the Berry phase on the boundary explains the existence of spin-1/2 edge state.",
"The spin-1/2 edge state varries as $M[R_{m}(\\theta )] =e^{-im\\cdot \\sigma \\over 2 \\theta }$ under $SO(3)$ spin rotation of angle $\\theta $ along direction $m$ .",
"Since $M[R_{m}(\\pi )]^2=M[R_{m}(2\\pi )]=-1$ and the minus sign can not be gauged away, the edge state carries a nontrivial projective Rep of the symmetry group $SO(3)$ .",
"Above picture can be generalized even if the symmetry is a finite group $G$ .",
"Similar to the spin coherent state, we introduce the following group element labeled bases $ &&|g^{r}\\rangle = \\hat{g}^r|\\alpha _1^{(\\nu )}\\rangle ,\\\\&&|g^{l}\\rangle = \\hat{g}^l|\\beta _1^{( {\\nu }^*)}\\rangle ,$ where $|\\alpha _1^{(\\nu )}\\rangle $ (or $|\\beta _1^{( {\\nu }^*)}\\rangle $ ) is one of the irreducible bases of projective Rep $(\\nu )$ [or $( {\\nu }^*)$ , the complex conjugate Rep of $(\\nu )$ ], $g^r$ and $g^l$ are different group elements with $ &&\\hat{g} |g_1^{r}\\rangle =\\omega _2(g ,g_1^r)|g g_1^{r}\\rangle ,\\\\&&\\hat{g} |g_1^{l}\\rangle =\\omega ^{-1}_2(g ,g_1^l)| g g_1^{l}\\rangle .$ The states $\\lbrace |g^{r}\\rangle \\rbrace $ (or $\\lbrace |g^{l}\\rangle \\rbrace $ ) are not necessarily orthogonal, but they form over complete bases for the Rep $(\\nu )$ [or $( {\\nu }^*)$ ] since (see appendix REF ) $\\sum _g |g^{r}\\rangle \\langle g^{r}|\\propto I =\\sum _{i=1}^{n_\\nu }|\\alpha _i^{(\\nu )}\\rangle \\langle \\alpha _i^{(\\nu )}|.$ So summing over $|g^{r}\\rangle $ is equivalent to summing over all the $n_\\nu $ bases $\\alpha _1^{(\\nu )},...,\\alpha _{n_\\nu }^{(\\nu )}$ of the irreducible Rep space $(\\nu )$ (and similar result holds for $|g^{l}\\rangle $ ) .",
"The physical degrees of freedom at site $i$ are combinations of the bases $|g^{l}_i\\rangle $ and $|g^{r}_i\\rangle $ $|g^l_i,g^r_i\\rangle ^{\\rm p} = \\omega _2(g^l_i, (g_i^l) ^{-1}g_i^r )|g_i^l,g_i^r\\rangle ,$ such that $|g^l_i,g^r_i\\rangle ^{\\rm p}$ varies under group action in a way similar to $|n\\rangle $ varies under rotation (the difference is that there is no phase factor $e^{i\\varphi }$ here), $\\hat{g}|g^l_i,g^r_i\\rangle ^{\\rm p} = \\omega _2(g^l_i,(g_i^l)^{-1}g_i^r ){\\omega _2(g,g_i^r)\\over \\omega _2(g,g_i^l)}|gg_i^l,gg_i^r\\rangle =|gg_i^l, gg_i^r\\rangle ^{\\rm p}.$ In the bond between neighboring sites $i$ and $i+1$ , the degrees of freedom $g^r_i$ and $g^l_{i+1}$ are locked by the constraint $g^r_i=g^l_{i+1}$ owning to strong interactions.",
"We discretize the space-time and put a variable $g_i^r$ at each space-time point $i$ and omit $g_i^l$ since it is the same as $g_{i-1}^r$ .",
"(Without causing confusion, we can eliminate the superscript $r$ in the following.)",
"The action amplitude for a space-time unit $(ijk)$ is defined as the Aharonov-Bohm phase of the basis $|g_i\\rangle $ evolving between the two paths $\\overline{(g_j^{-1}g_k)}\\ \\overline{(g_i^{-1}g_j)}|g_i\\rangle $ and $\\overline{(g_i^{-1}g_k)}|g_i\\rangle $ , see Fig.REF (A), $ \\varphi (|{g}_i \\rangle , |{g} _j\\rangle , |{g} _k\\rangle ) &=& M(\\overline{g_j^{-1}g_k})M(\\overline{g_i^{-1}g_j})\\left[M(\\overline{g_i^{-1}g_k})\\right]^{-1}\\\\&=&\\omega _2^{s(g_i)}(g_i^{-1}g_j, g_j^{-1}g_k).$ The total action amplitude stands for the topological phase factor (the `Berry phase') of the physical degrees of freedom and can be used to describe the low energy effective field theory of the system.",
"Neglecting dynamic terms, we obtain the fixed point partition function of the corresponding SPT phase, $Z\\propto \\sum _{\\lbrace g_{i}\\rbrace }e^{ - S(\\lbrace g_{i}\\rbrace ) }&=&\\sum _{\\lbrace g_{i}\\rbrace }\\prod _{\\lbrace ijk\\rbrace }\\varphi ^{s_{ijk}}(|{g} _{i}\\rangle ,|{g} _{j}\\rangle ,|{g} _{k}\\rangle )\\nonumber \\\\$ where $s_{ijk}$ is the orientation of the triangle $(ijk)$ , which is equal to 1 if it is pointing outside and $-1$ otherwise.",
"Figure: (A)Aharonov-Bohm phase as the topological term; (B)Quantized topological action amplitude on closed space-time.Similar to the $\\theta $ -term of the $O(3)$ NLSM, the total action amplitude is normalized if the space-time is closed (the simplest case is the surface of a tetrahedron, see Fig.REF (B)), $ &&\\varphi ^{s_{012}}(|{g} _0\\rangle , |{g}_1 \\rangle , |{g}_2 \\rangle ) \\varphi ^{s_{013}}(|{g}_0 \\rangle , |{g}_1 \\rangle , |{g}_3 \\rangle ) \\\\&&\\times \\varphi ^{s_{023}}(|{g}_0 \\rangle , |{g}_2 \\rangle , |{g}_3 \\rangle ) \\varphi ^{s_{123}}(|{g}_1 \\rangle , |{g}_2 \\rangle , |{g}_3 \\rangle ) \\\\&&=1$ owning to the cocycle equation $ (d\\omega _2)(g_0^{-1}g_1, g_1^{-1}g_2, g_2^{-1}g_3)=1.$ The partition function can be regarded as imaginary time evolution operator $Z=U(0,\\tau )=|\\psi ^\\tau \\rangle \\langle \\psi ^0|$ , where $|\\psi ^\\tau \\rangle $ is the ground state at time $\\tau $ .",
"After some calculation we can write out the ground state (under periodic boundary condition) as $ |\\psi \\rangle &=&\\sum _{\\lbrace g_ig_jg_k\\rbrace } \\prod _{\\lbrace ijk\\rbrace }B^{-1}(|E\\rangle ,|{g}^r _{i}\\rangle ,|{g}^r _{j}\\rangle )|{g} _{i}^l{g} _{i}^r{g} _{j}^l{g} _j^r{g} _{k}^l{g} _k^r...\\rangle ^{\\rm p}\\\\&=&\\sum _{\\lbrace g_ig_jg_k\\rbrace } \\prod _{\\lbrace ijk\\rbrace }\\omega _2^{-1} ({g}_i^r,(g_i^r)^{-1}g_j^r)|{g} _{i}^l{g} _{i}^r{g} _{j}^l{g} _j^r{g} _{k}^l{g} _k^r...\\rangle ^{\\rm p}.$ Noticing that in 1D $j=i+1,\\ k=j+1,\\ ....$ , the wave function $\\omega _2^{-1} ({g}_i^r,(g_i^r)^{-1}g_j^r)=\\omega _2^{-1} ({g}_j^l,(g_j^l)^{-1}g_j^r)$ is the CG coefficient that fractionalize the physical degrees of freedom into two projective Reps $|g_j^lg_j^r\\rangle =\\omega _2^{-1}({g}_j^l,(g_j^l)^{-1}g_j^r)|g_j^lg_j^r\\rangle ^{\\rm p},$ the ground state wave function can also be written in forms of product of dimers $ |\\psi \\rangle =\\sum _{\\lbrace g_i^lg_i^r...\\rbrace } |...{g} _{i}^l)({g} _{i}^r{g} _{j}^l)({g} _j^r{g} _{k}^l)({g} _k^r...\\rangle ,$ where $g_i^r=g_j^l, g_j^r=g_k^l,...$ and each bracket means a singlet (or a dimer) on a bond between neighboring sites.",
"The dangling degrees of freedom on the ends stand for the edge states which carry projective Reps of the symmetry group.",
"From the above ground state wave function of SPT phase, it is easily seen that the fixed point parent Hamiltonian is constructed by projector onto the bond singlets.",
"If we further project the physical degrees of freedom to its subspace, then we can obtain an AKLT-type state, and the parent Hamiltonian can also be constructed using projection operators."
],
[
"Defects in 2D topological phases",
"Except for 1D SPT phases, projective Reps also have applications in 2D topological phases, including SPT phases and intrinsic topological phases.",
"In the following we will give several examples."
],
[
"Vortices in topological superconductors (fermionic SPT phases)",
"It is known that the vortices of $p+ip$ topological superconductor carry Majorana zero modes and the degeneracy of the wave function depends on the number of spatially separated vortices.",
"In the following, we will show that if we interpret each Majorana zero mode as a `symmetry' operation, then the set of symmetry operations form an Abelian group, and the degeneracy of the Majorana Hilbert space can be understood as the projective Rep of this group.",
"Supposing the majorana zero mode $\\gamma _1$ is an eigen state of a Hamiltonian $H$ with $[\\gamma _1, H]=0$ , then we can define an operation $\\hat{\\Gamma }_1$ corresponding to the majorana mode $\\gamma _1$ , $\\hat{\\Gamma }_1(O)=\\gamma _1O\\gamma _1,$ where $O$ is an arbitrary operator.",
"Obviously, $ &&\\hat{\\Gamma }_1(\\gamma _1)=\\gamma _1 \\gamma _1 \\gamma _1 = \\gamma _1,\\\\&&\\hat{\\Gamma }_1(\\gamma _i)=\\gamma _1 \\gamma _i \\gamma _1 = -\\gamma _i, \\ \\ {\\rm for \\ any\\ }\\langle \\gamma _1|\\gamma _i\\rangle =0.$ Since $\\hat{\\Gamma }_1(H)=\\gamma _1 H \\gamma _1 = H$ , the operation $\\hat{\\Gamma }_1$ is a `symmetry operation' of the system.",
"The eigenvalues of $\\hat{\\Gamma }_1$ are $\\pm 1$ and $\\hat{\\Gamma }_1^2=I$ , so $\\hat{\\Gamma }_1$ generates a $Z_2$ `symmetry group'.",
"It should be mentioned that the form of $\\hat{\\Gamma }_i$ ( or $\\gamma _1$ ) depends on the details of $H$ , this $Z_2$ `symmetry group' is not a symmetry in the usual sense.",
"The existence of this special $Z_2$ symmetry is a consequence of the nontrivial winding number of the $p+ip$ superconductor and the presence of topological defect (i.e.",
"vortex) where the Majorana mode $\\gamma _1$ locates.",
"If $\\gamma _2$ is another majorana zero mode of the Hamiltonian, then it defines another $Z_2$ symmetry group generated by $\\hat{\\Gamma }_2(O)=\\gamma _2O\\gamma _2,$ and it is easily checked that $\\hat{\\Gamma }_1\\hat{\\Gamma }_2(O)=\\hat{\\Gamma }_2\\hat{\\Gamma }_1(O)$ , namely, $\\hat{\\Gamma }_1\\hat{\\Gamma }_2=\\hat{\\Gamma }_2\\hat{\\Gamma }_1.$ So, if $H$ contains two Majorana zero modes $\\gamma _1$ and $\\gamma _2$ , then it has a $Z_2\\times Z_2$ `symmetry group'.",
"The two majorana zero modes result in two-fold degeneracy of the ground states.",
"In the ground state subspace, the operators $\\hat{\\Gamma }_{1}, \\hat{\\Gamma }_{2}$ act projectively and their Rep matrices are $\\hat{\\Gamma }_1\\rightarrow M(\\Gamma _1)= \\sigma _x,\\ \\ \\ \\hat{\\Gamma }_2\\rightarrow M( \\Gamma _2)=\\sigma _y,$ with $M(\\Gamma _1)^2=M(\\Gamma _2)^2=1$ and the fermionic anti-commuting relation $\\lbrace M(\\Gamma _1),M(\\Gamma _2)\\rbrace =0$ .",
"Now suppose the system has four Majorana zero modes $\\gamma _1,\\gamma _2,\\gamma _3, \\gamma _4$ , then the corresponding symmetry operations $\\hat{\\Gamma }_1,\\hat{\\Gamma }_2,\\hat{\\Gamma }_3, \\hat{\\Gamma }_4$ generate a $Z_2\\times Z_2\\times Z_2\\times Z_2$ symmetry group.",
"The degenerate ground states carry an irreducible projective Rep of the class $(-1,-1,-1,-1,-1,-1)$ (up to a gauge transformation comparing with Table REF ) $ M(\\Gamma _{1})&=&\\sigma _{z}\\otimes \\sigma _{z} ,\\\\M(\\Gamma _{2})&=&I\\otimes \\sigma _{y} , \\\\M(\\Gamma _{3})&=&I\\otimes \\sigma _{x} ,\\\\M(\\Gamma _{4})&=&\\sigma _{x}\\otimes \\sigma _{z},$ which satisfy the relation $\\lbrace M(\\Gamma _{j}),M(\\Gamma _{k})\\rbrace =2\\delta _{jk}$ .",
"Similar results can be generalized to the cases with more majorana zero modes, and the degeneracy of the ground states can be understood as the existence of projective Rep of the corresponding `symmetry group'.",
"Above picture is still valid even if there are interactions in the Hamiltonian $H$ ."
],
[
"Monodromy defects in bosonic SPT phases",
"In 2D fermionic topological insulators, a magnetic $\\pi $ -flux (a symmetry defect) gives rise to extra degeneracy.",
"Actually, this picture also holds for bosonic SPT phases.",
"Generally, symmetry defects can not be created locally if they carry fractional symmetry flux.",
"These defects are the end points of strings and are called the Monodromy defects in literature.",
"In some SPT phases, monodromy defects may carry projective Reps of the symmetry group, , , .",
"Bosonic topological insulators.",
"Bosonic topological insulators are SPT phases protected by $U(1)\\rtimes Z_2^T$ symmetry.",
"In 2D bosonic topological insulators, a $\\pi $ -flux (gauging the $U(1)$ symmetry) carries projective Rep of the remaining symmetry $Z_2^T$ which gives rise to Kramers' degeneracy.",
"This nontrivial result comes from the fact that in the bosonic topological insulator (which is a nontrivial SPT phase), the vortex creation operator reverses its sign under time reversal $\\widehat{T} b_v\\widehat{T}^{-1} = -b_v^\\dag $ , then $\\widehat{T}|\\pi \\rangle = |-\\pi \\rangle = \\hat{b}_v |\\pi \\rangle ,$ $\\widehat{T} |-\\pi \\rangle = \\widehat{T}\\hat{b}_v \\widehat{T}^{-1}\\widehat{T}|\\pi \\rangle =-b_v^\\dag |-\\pi \\rangle =-|\\pi \\rangle ,$ so $\\widehat{T}^2=-1,$ which is a projective Rep of $Z_2^T$ .",
"On the other hand, in the trivial bosonic insulator $\\widehat{T} b_v\\widehat{T}^{-1} = b_v^\\dag $ , which yields a linear Rep $\\widehat{T}^2=1$ .",
"In this case creating $\\pi $ fluxes will not cause degeneracy.",
"$Z_{N_1} \\times Z_{N_2}\\times Z_{N_3}$ SPT phases.",
"Another example is the 2D bosonic SPT phases with symmetry group $G=Z_{N_1} \\times Z_{N_2}\\times Z_{N_3}$ (where $N_1,N_2,N_3$ are integers) with generator $h_1,h_2,h_3$ respectively.",
"The SPT phases are classified by the third group cohomology $\\mathcal {H}^3(G,U(1))$ and each nontrivial phase corresponds to a nontrivial class of 3-cocycles, .",
"We will focus on the type-III SPT phases described by the type-III cocycles .",
"If we label a group element $A$ as $A=(a_1,a_2,a_3)=(h_1^{a_1},h_2^{a_2},h_3^{a_3})$ with $0\\le a_i\\le N_i-1$ , then a type-III cocycle takes the following form, $\\omega _3(A,B,C)=e^{2\\pi i{p^{III}\\over N_{123}} a_1b_2c_3}, \\ \\ 1\\le p^{III}\\le N_{123},$ where $N_{123}$ is the greatest common devisor of $N_1, N_2, N_3$ .",
"From above 3-cocyle, one can construct a 2-cocyle from the slant product $i_A$ : $ \\chi _A(B,C) &=& (i_A\\omega _3)(B,C)\\\\&=&\\omega _3(A,B,C)\\omega _3^{-1}(B,A,C)\\omega _3(B,C,A)\\\\&=&e^{2\\pi i{p^{III}\\over N_{123}}(a_1b_2c_3-b_1a_2c_3+b_1c_2a_3)}.$ Since $i_A$ commutes with the coboundary operator $d$ , $\\chi _A(B,C)$ is a 2-cocycle: $(d \\chi _A)(D,B,C)=d i_A \\omega _3(D,B,C)=i_A d\\omega _3(D,B,C)=1$ .",
"Furthermore, if $p^{III}\\ne 0$ then $\\chi _A(B,C)$ is a nontrivial 2-cocyle since it agrees with the analytic solution in Eq.",
"(REF ).",
"Specially, if $A\\in Z_{N_1}$ with $a_2=a_3=0$ , then $\\chi _A(B,C)$ is a nontrivial 2-cocyle of the subgroup $Z_{N_2}\\times Z_{N_3}$ .",
"The physical meaning of above result is that in the SPT phases corresponding to nonzero $p^{III}$ , the symmetry fluxes of $Z_{N_1}$ carry nontrivial projective Reps (thus give rise to degeneracy of the energy spectrum) of the group $Z_{N_2}\\times Z_{N_3}$, .",
"Similarly, the $Z_{N_2}$ (or $Z_{N_3}$ ) fluxes carry projective Reps of the group $Z_{N_1}\\times Z_{N_3}$ (or $Z_{N_1}\\times Z_{N_2}$ ).",
"Similar conclusion also holds for certain SPT phases with non-Abelian symmetry groups."
],
[
"2D topological order/SET phases",
"We have shown that in SPT phases, Monodromy defects may carry projective Reps of the symmetry group (or subgroup of the symmetry).",
"If the system carries intrinsic topological order, then point-like excitations, such as spinons (`electronic' excitations) or visons (`magnetic' excitations), can carry projective Reps of the symmetry group.",
"Topologically ordered phases enriched by symmetry are called symmetry enriched topological (SET) phases.",
"An example is the projective symmetry group (PSG), in quantum spin liquid with lattice space group symmetry.",
"In the parton construction of spin-1/2 quantum spin liquids, the system has an $SU(2)$ gauge symmetry.",
"At mean field level, this gauge symmetry may break down to $U(1)$ or $Z_2$ (called invariant gauge group, or IGG in brief) owning to paring between the spinons.",
"Generally the space group is no longer the symmetry of the mean field ground state, but the mean field state is invariant under space group operation followed by a gauge transformation.",
"The symmetry elements, which are combinations of space group operations and the corresponding gauge transformations, form the PSG.",
"In PSG, the multiplication rule of two space group operations are twisted by gauge transformations in the IGG.",
"As a result, the PSG can be classified by the second group cohomology $\\mathcal {H}^2(SG, IGG)$ (for details, see Refs.",
"WenRMP06, Ran13).",
"PSG actually describes how the spinons (the `electronic' excitations) carry fractional charge of the symmetry group.",
"On the other hand, visons (the `magnetic' excitations), or bond states of vison and spinon, may also carry projective Reps of the symmetry group, .",
"For example, in the $SO(3)$ non-abelian chiral spin liquid, a vison not only carries non-Abelian anyon, but also carries spin-1/2 degrees of freedom, which is a projective Rep of the symmetry group $SO(3)$ .",
"As a result, a more complete classification of SET phases with symmetry group $G$ is $\\mathcal {H}^2(G,\\mathcal {A})$, , where $\\mathcal {A}$ is the fusion group of abelian anyons in the raw topological order.",
"A subtle issue is that obstructions may exist in realizing all the phases classified by $\\mathcal {H}^2(G,\\mathcal {A})$ in pure 2D, .",
"QMC simulations.",
"The generalized Schur's lemma corresponding to projective Reps of anti-unitary groups can also be applied to search for models which are free of sign problems in quantum Monte Carlo (QMC) simulations, , , , .",
"By introducing the Hubbard-Stratonovich field $\\xi (x,\\tau )$ , interacting fermion models can be mapped to free fermion Hamiltonians.",
"The Boltzmann weight $\\rho (\\xi )$ of the auxiliary field configuration $\\xi $ can be obtained by tracing out the fermions.",
"Sign problem occurs when $\\rho (\\xi )<0$ , in which case one have trouble to simulate the quantum system in a Markov process.",
"When the fermion number is conserved in the Hubbard-Stratonovich Hamiltonian $ H(\\xi )=C^\\dag H(\\xi )C$ where $C=(c_1,...,c_N)^T$ are fermion bases, then the Boltzmann weight in the QMC simulation is given by $\\rho (\\xi ) = {\\rm Tr}\\ [{\\mathbf {T}}e^{-\\int _0^\\beta H(\\xi ) d\\tau }] = \\det [I+B(\\xi )], $ where $B(\\xi )={\\mathbf {T}}e^{-\\int _0^\\beta H(\\xi ) d\\tau }$ , $\\mathbf {T}$ means time ordered integration, $I$ is an identity matrix, $1/\\beta $ is the temperature and $\\xi $ is space-time dependent Hubbard-Stratonovich field.",
"If the matrix $H$ , and then the matrix $(I+B)$ , are commuting with the projective representation of an anti-unitary symmetry group $Z_2^T=\\lbrace E,T\\rbrace $ with $\\widehat{T}^2=-1$ , then according to the generalized Schur's lemma and its corollary, the eigenvalues of $(I+B)$ are either pairs of complex conjugate numbers, or even-fold degenerate (the Kramers degeneracy) real numbers, or a mixture of them.",
"As a consequence, the determinant must be a non-negative number, i.e.",
"$\\rho (\\xi )$ is non-negative and the model is free of sign problem under QMC simulation.",
"If fermion pairing terms exist after the Hubbard-Stratonovich decomposition, one can use Majorana bases to write the decoupled Hamiltonian as $H(\\xi )=\\Gamma ^T H(\\xi )\\Gamma $ , where $\\Gamma =(\\gamma _1,...,\\gamma _{2N})^T$ are Majorana bases and $H$ is a skew symmetric matrix $H^T=-H$ at half-filling (consequently all the nonzero eigenvalues of $H$ appear in pairs of $\\pm \\lambda $ ) .",
"In this case, the Boltzmann weigh is given by $\\rho (\\xi ) = [\\det (B^{-1}+B)]^{1\\over 2},$ where $B(\\xi )=\\mathbf {T}e^{-2\\int _0^\\beta H(\\xi )d\\tau } = e^{-\\beta {\\bar{H}(\\xi )}}$ , ${\\bar{H}}(\\xi )$ is a skew symmetric matrix.",
"The Boltzmann weigh can also be written in forms of the eigenvalues $\\lambda _k$ of ${\\bar{H}}$ as $\\rho (\\xi ) = \\prod _k(e^{2\\beta \\lambda _k} +e^{-2\\beta \\lambda _k})$ , where any pair of $(\\lambda _k,-\\lambda _k)$ are counted only once in $\\prod _k$ .",
"If $H$ (and then ${\\bar{H}}$ ) commute with the projective Rep of $Z_2^T$ symmetry with $\\widehat{T}^2=-1$ , both $(\\lambda _k, -\\lambda _k)$ and $(\\lambda _k^*, -\\lambda _k^*)$ are eigenvalues of ${\\bar{H}}$ when $\\lambda _k$ is not purely imaginary.",
"However, if $\\lambda _k$ is purely imaginary, then $\\lambda _k^*=-\\lambda _k$ and it is possible that only $(\\lambda _k, -\\lambda _k)$ appear in the eigenvalues.",
"In that case, sign problem appears.",
"To avoid this problem, one can enlarge the symmetry group.",
"The simplest case is to enlarge the symmetry group as $G=Z_2\\times Z_2^T=\\lbrace E, P, T, PT\\rbrace $ and let $H$ commute with a non-trivial projective Rep of $G$ .",
"In Majorana bases, the representations are required to be real orthogonal matrices such that after the group action the bases are still Majorana fermions.",
"Under this constraint, the projective Reps are classified by $\\mathcal {H}^2(G, Z_2)=\\mathbb {Z}_2^3$ , which contains 7 nontrivial classes as shown in Table REF (it should be noted that real orthogonality is a sufficient but not necessary condition for $Z_2$ coefficient projective Reps, namely, real orthogonal projective Reps are classified by $\\mathcal {H}^2(G, Z_2)$ but a $Z_2$ coefficient projective Rep is not necessarily real orthogonal).",
"In the class $(-1,+1,-1)$ irreducible projective Rep where $\\lbrace \\widehat{T}, \\widehat{PT}\\rbrace =0$ and $\\widehat{T}^2=-1$ , $(\\widehat{ PT})^2=1$ , one cannot find a 2-dimensional (diagonal) block of $\\bar{H}$ which satisfies the following conditions: it is skew-symmetric and commutes with the Rep of $G$ .",
"The smallest block satisfying above conditions is at least 4-dimensional, i.e.",
"it spans the direct sum space of at least two block of irreducible projective Reps $I\\otimes M(g)K_{s(g)}$ .",
"The 4-dimensional block takes the form $i\\sigma _y\\otimes (a I+b i\\sigma _z)$ which has eigenvalues $b\\pm ia$ and $-b\\pm ia$ , where $a,b$ are real numbers.",
"This will not cause sign problem even if all the eigenvalues are purely imaginary.",
"Another way to see the absence of sign problem is to use the eigenstates of $P$ to block diagonalize $\\bar{H}$ into two parts that are mutually complex conjugate to each other.",
"The class $(+1,-1,-1)$ gives the same result since it defers from the class $(-1,+1,-1)$ only by switching the roles of $T$ and $PT$ .",
"These symmetry classes are called the Majorana class.",
"In the class $(-1,-1,-1)$ where $\\lbrace \\widehat{T}, \\widehat{PT}\\rbrace =0$ and $\\widehat{T}^2=-1$ , $(\\widehat{ PT})^2=-1$ , the Rep is 4-dimensional.",
"It can be reduced into 2-dimensional irreducible Reps by unitary transformations but cannot be reduced by real orthogonal transformations.",
"Since $\\widehat{P}$ is skew-symmetric and anti-commutes with $\\widehat{T}$ , we can use the eigenstates of $P$ to transform the Hamiltonian in forms of (REF ).",
"It is free of sign problem owning to the $\\widehat{T}^2=-1$ symmetry.",
"This is called the Kramers class.",
"Table: Real orthogonal projective Reps of the group G=Z 2 ×Z 2 T ={E,P,T,PT}G=Z_2\\times Z_2^T=\\lbrace E,P,T,PT\\rbrace have classification ℋ 2 (G,Z 2 )=ℤ 2 3 \\mathcal {H}^{2}(G,Z_2)=\\mathbb {Z}_2^3, which is labeled by ω 2 (T,T),ω 2 (PT,PT),ω 2 (T,PT)\\left(\\omega _2(T,T), \\omega _2(PT,PT),\\omega _2(T,PT)\\right) respectively.",
"We have fixed ω 2 (TP,T)=1\\omega _2(TP,T)=1, therefore ω 2 (T,PT)=±1\\omega _2(T,PT)=\\pm 1 means that T ^\\widehat{T} and TP ^\\widehat{TP} are commuting/anti-commuting, respectively.Above results can be generalized.",
"If particle number is conserved in $H(\\xi )$ , the condition $\\widehat{T}^2=-1$ can be relaxed.",
"The essential point of avoiding the sign problem is that the Hubbard-Stratonovich Hamiltonian $H(\\xi )$ commutes with a nontrivial projective Rep of an anti-unitary group $G$ , where all the irreducible projective Reps of this class are even-dimensional.",
"Even if for any anti-unitary operator $Tg$ (where $g\\in G$ is unitary) $[M(Tg)K]^2\\ne -1$ , the sign problem can still be avoided.",
"Here we give two examples.",
"For the type-I anti-unitary groups, the class $(+1,-1)$ irreducible projective Rep of the group $(Z_2\\times Z_2)\\rtimes Z_2^T\\simeq D_{2d}^T$ in Table REF is an even dimensional Rep, where $[M(T)K]^2=[M(PT)K]^2=1$ and $[M(QT)K]^2=-[M(PQT)K]^2=-i\\sigma _z$ .",
"Especially, as an example of the type-II anti-unitary groups, the nontrivial irreducible projective Rep of the group $Z_4^T$ is an even dimensional Rep and is characterized by $[M(T)K]^4=-1$ .",
"Similar even-dimensional projective Reps can also be found in other anti-unitary groups.",
"If particle number is not conserved in $H(\\xi )$ , the real orthogonal condition of the Reps of the symmetry group $G$ (which is imposed in the Majorana class and Kramers symmetry class) can be relaxed.",
"If an anti-unitary symmetry group $G$ has a 4-dimensional (or $4n$ -dimensional, $n\\in Z$ ) irreducible unitary projective Rep, and if $H(\\xi )$ commutes with it, then the model is free of sign problem.",
"For example, the group $Z_2\\times Z_2\\times Z_2^T\\simeq D_{2h}^T$ has four classes of 4-dimensional irreducible projective Reps (see Table REF ).",
"If ${\\bar{H}}$ [or $(B^{-1}+B)$ ] commutes with one of these projective Reps, the degeneracy of the eigenvalues of $(B^{-1}+B)$ is increased: if an eigenvalue $\\Lambda $ is real, then it is 4-fold degenerate; if $\\Lambda $ is complex, then both $\\Lambda $ and $\\Lambda ^*$ are 2-fold degenerate.",
"Resultantly, the Boltzmann weight is non-negative.",
"These generalizations provide hint to find new classes of models which are free of sign problem in quantum Monte Carlo simulation."
],
[
"Space groups, Spectrum degeneracy and other applications",
"Projective Reps can also be applied in many other fields, such as space groups, spectrum degeneracy and so on.",
"Non-symmorphic space groups.",
"In the Rep theory of non-symmorphic space groups, the `little co-group' (subgroup of the point group) is represented projectively at some symmetric wave vectors, where the factor system comes naturally from fractional translations.",
"The representation theory of space groups was thoroughly studied in literature, for instance in Ref.",
"ChenJQRMP85 and references therein, so we will not discuss in more detail here.",
"Spectrum degeneracy.",
"Similar to the Kramers degeneracy in time reversal symmetric systems with odd number of electrons, in some cases projective Reps can also explain the degeneracy of the ground states or excited states.",
"For example, consider a spin-1/2 system respecting $Z_2\\times Z_2=\\lbrace E,R_x\\rbrace \\times \\lbrace E,R_y\\rbrace $ spin-rotation symmetry, where the two $Z_2$ subgroups are generated by $\\pi $ rotation along $x$ and $y$ directions respectively, $ R_x(S_m)=e^{-i\\pi S_x}S_m e^{i\\pi S_x},\\ \\ R_y(S_m)=e^{-i\\pi S_y}S_m e^{i\\pi S_y}.$ More explicitly, $ &&R_x(S_x)=S_x,\\ R_x(S_y)=-S_y,\\ R_x(S_z)=-S_z,\\\\&&R_y(S_x)=-S_x,\\ R_y(S_y)= S_y,\\ R_y(S_z)=-S_z.$ If the system contains an odd number of spins, then the operators carry linear Rep of the symmetry group $Z_2\\times Z_2$ but the total Hilbert space forms a (reducible) projective Rep. As a consequence all the energy levels are at least doubly degenerate as long as the $Z_2\\times Z_2$ symmetry is not broken explicitly."
],
[
"Conclusion and Discussion",
"In summary, we generalized the eigenfunction method to obtain irreducible projective Reps of finite unitary or anti-unitary groups by reducing the regular projective Reps. To this end we first solved the 2-cocyle equations of a group $G$ to obtain the factor systems and their classification, specially if $G$ is anti-unitary we introduced the decoupled factor systems for convenience.",
"Therefore regular projective representations are obtained.",
"Then using class operators we constructed the complete set of commuting operators, and using their common eigenfunctions we transformed the regular projective representations into irreducible ones.",
"Anti-unitary groups are generally complicated than unitary groups in the reduction process, where modified Schur's lemmas were used.",
"We applied this method to a few familiar finite groups and gave their irreducible Reps. We then discussed applications of projective Reps in many-body physics, for instance, the edge states of one-dimensional Symmetry Protected Topological (SPT) phases, symmetry fluxes in 2-dimensional SPT phases and anyons in Symmetry Enriched Topological (SET) phases carry projective Reps of the symmetry groups.",
"We also showed that recently discovered models which are free of sign problem under quantum Monte Carlo simulations are commuting with nontrivial projective Reps of certain anti-unitary groups.",
"Other applications related to spectrum degeneracy were also summarized.",
"Our approach can be generalized to obtain irreducible projective Reps of infinite groups, such as space groups, magnetic space groups and Lie groups.",
"Furthermore, it may shed light on the reduction of symmetry operations corresponding to higher order group cocycles, namely, in the boundary of 2D SPT phases which are classified by the third group cohomology.",
"We thank Zheng-Cheng Gu, Xie Chen, Meng Cheng, Peng Ye, Xiong-Jun Liu, Shao-Kai Jian, Zhong-Chao Wei, Cong-Jun Wu, Tao Li and Li You for helpful discussions.",
"We especially thank Xiao-Gang Wen for discussion about the numerical calculation of group cohomology, and thank Hong Yao and Zi-Xiang Li for comments and helpful discussion about the application of projective Reps in QMC.",
"This work is supported by NSFC (Grant Nos.",
"11574392), the Ministry of Science and Technology of China (Grant No.",
"2016YFA0300504), and the Fundamental Research Funds for the Central Universities, and the Research Funds of Renmin University of China (No.",
"15XNLF19).",
"J.Y.",
"is supported by MOST (No.",
"2013CB922004) of the National Key Basic Research Program of China, by NSFC (No.",
"91421305) and by National Basic Research Program of China (2015CB921104)."
],
[
"Group cohomology",
"In this section, we will briefly introduce the group cohomology theoryFor an introduction to group cohomology, see wiki http://en.wikipedia.org/wik/Group_cohomology and Romyar Sharifi, AN INTRODUCTION TO GROUP COHOMOLOGY.",
"Let $\\omega _{n}(g_{1},\\ldots ,g_{n})$ be a $U(1)$ valued function of $n$ group elements.",
"In other words, $\\omega _{n}: G^{n}\\rightarrow U(1)$ .",
"Let $\\mathcal {C}^{n}(G,U(1))=\\lbrace \\omega _{n}\\rbrace $ be the space of such functions.",
"Note that $\\mathcal {C}^{n}(G,U(1))$ is an Abelian group under the function multiplication $\\omega ^{\\prime \\prime }_{n}(g_{1},\\ldots ,g_{n})=\\omega _{n}(g_{1},\\ldots ,g_{n})\\omega ^{\\prime }_{n}(g_{1},\\ldots ,g_{n})$ .",
"We define a map $d$ from $\\mathcal {C}^{n}(G,U(1))$ to $\\mathcal {C}^{n+1}(G,U(1))$ : $& & (d\\omega _{n})(g_{1},\\ldots ,g_{n+1}) \\nonumber \\\\&& =[g_{1}\\cdot \\omega _{n}(g_{2},\\ldots ,g_{n+1})]\\omega ^{(-1)^{n+1}}_{n}(g_{1},\\ldots ,g_{n})\\times \\nonumber \\\\&&\\prod ^{n}_{i=1}\\omega ^{(-1)^{i}}_{n}(g_{1},\\ldots ,g_{i-1},g_{i}g_{i+1},g_{i+2},\\ldots ,g_{n+1}),$ where $g\\cdot \\omega _{n}(g_{2},\\ldots ,g_{n+1})=\\omega _{n}(g_{2},\\ldots ,g_{n+1})$ if $g$ is unitary and $g\\cdot \\omega _{n}(g_{2},\\ldots ,g_{n+1})=\\omega _{n}^{-1}(g_{2},\\ldots ,g_{n+1})$ if $g$ is anti-unitary.",
"In other words, $g\\cdot \\omega _{n}(g_{2},\\ldots ,g_{n+1})=\\omega _{n}^{s(g)}(g_{2},\\ldots ,g_{n+1})$ , where $s(g)=1$ if $g$ is unitary and $s(g)=-1$ otherwise.",
"From above definition (REF ), it is easily checked that $d^2\\equiv 1$ .",
"Let $\\mathcal {B}^{n}(G,M)=\\lbrace \\omega _{n}\\mid \\omega _{n}=d\\omega _{n-1}\\mid \\omega _{n-1}\\in \\mathcal {C}^{n-1}(G,M)\\rbrace $ and $\\mathcal {Z}^{n}(G,M)=\\lbrace \\omega _{n}\\mid d\\omega _{n}=1,\\omega _{n}\\in \\mathcal {C}^{n}(G,M)\\rbrace $ $\\mathcal {B}^{n}(G,M)$ and $\\mathcal {Z}^{n}(G,M)$ are also Abelian groups which satisfy $\\mathcal {B}^{n}(G,M)\\subset \\mathcal {Z}^{n}(G,M)$ .",
"The $n$ th-cohomology of $G$ is defined as $\\mathcal {H}^{n}(G,M)=\\mathcal {Z}^{n}(G,M)/\\mathcal {B}^{n}(G,M).$ Now we give some examples.",
"From $(d\\omega _{1})(g_{1},g_{2})=\\frac{\\omega _{1}(g_{1})\\omega _{1}^{s(g_{1})}(g_{2})}{\\omega _{1}(g_{1}g_{2})},$ we find $\\mathcal {Z}^{1}(G,U(1))=\\lbrace \\omega _{1}\\mid \\frac{\\omega _{1}^{s(g_{1})}(g_{2})\\omega _{1}(g_{1})}{\\omega _{1}(g_{1}g_{2})}=1\\rbrace ,$ and $\\mathcal {B}^{1}(G,U(1))=\\lbrace \\omega _{1}\\mid \\omega _{1}(g_{1})=\\frac{\\omega _{0}^{s(g_{1})}}{\\omega _{0}}\\rbrace .$ $\\mathcal {H}^{1}(G,U(1))=\\frac{\\mathcal {Z}^{1}(G,U(1))}{\\mathcal {B}^{1}(G,U(1))}$ is the set of all inequivalent $1D$ Reps of $G$ .",
"From $(d_{2}\\omega _{2})(g_{1},g_{2},g_{3})=\\frac{\\omega _{2}^{s(g_{1})}(g_{2},g_{3})\\omega _{2}(g_{1},g_{2}g_{3})}{\\omega _{2}(g_{1}g_{2},g_{3})\\omega _{2}(g_{1},g_{2})},$ we find $\\mathcal {Z}^{2}(G,U(1))=\\lbrace \\omega _{2}\\mid \\frac{\\omega _{2}^{s(g_{1})}(g_{2},g_{3})\\omega _{2}(g_{1},g_{2}g_{3})}{\\omega _{2}(g_{1}g_{2},g_{3})\\omega _{2}(g_{1},g_{2})}=1\\rbrace $ and $\\mathcal {B}^{2}(G,U(1))=\\lbrace \\omega _{2}\\mid \\omega _{2}(g_{1},g_{2})=\\frac{\\omega _{1}(g_{1})\\omega _{1}^{s(g_{1})}(g_{2})}{\\omega _{1}(g_{1}g_{2})}\\rbrace .$ The 2-cohomology of group $\\mathcal {H}^{2}(G,U(1))=\\frac{\\mathcal {Z}^{2}(G,U(1))}{\\mathcal {B}^{2}(G,U(1))}$ can classify the projective Reps of symmetry group $G$ .",
"Correspondingly, it can classify 1-dimensional SPT phases with on-site symmetry group $G$ ."
],
[
"Canonical gauge choice and solutions of cocycles",
"Once the cocycle equations and the coboundary functions are written in terms of linear equations (see section REF ), the classification of cocycles is equivalent to solving some linear algebra.",
"In the following, we will focus on 2-cocycles.",
"For a general group $G$ , the 2-cocycle $\\omega _2(g_1,g_2)=e^{i\\theta _2(g_1,g_2)}$ has $G^{2}$ components ($G$ is the order of group) which satisfy the following equations ($g_1,g_2,g_3\\in G$ ): $s(g_{1})\\theta _{2}(g_{2},g_{3})+\\theta _{2}(g_{1},g_{2}g_{3})-\\theta _{2}(g_{1}g_{2},g_{3})-\\theta _{2}(g_{1},g_{2})=0 .$ Notice that `=0' in above equation means ${\\rm mod\\ } 2\\pi =0$ (the same below).",
"Above equations can be written in matrix form as $\\sum _n \\left(C^{mn}\\right)\\theta _2^n=0,$ where $C$ is an $G^3\\times G^2$ matrix (the matrix elements are given above), and $\\theta _2$ is a $G^2$ -component vector.",
"We do not specify the coefficient space at the beginning, and will go back to it when discussing the classification of the cocycles.",
"The coboundaries $\\Omega _2(g_1,g_2)=e^{i\\Theta _2(g_1,g_2)}$ are given by: $\\Theta _2(g_1,g_2)=s(g_{1})\\theta _{1}(g_{2})+\\theta _{1}(g_{1})-\\theta _{1}(g_{1}g_{2}).$ The $\\theta _1$ contains $G$ variables.",
"Above equations can also be written in matrix form $\\Theta _2^m= \\sum _n \\left(B^{mn}\\right)\\theta _1^n,$ where $B$ is an $G^2\\times G$ matrix.",
"The $C$ matrix in (REF ) defines a set of equations while the $B$ matrix in (REF ) defines a set of functions.",
"All the functions defined by B are solutions of the equations defined by $C$ .",
"If one can impose some constraints on both $C$ and $B$ , the classification will be easier to obtain.",
"For instance, in the extreme case, if there exists a certain `gauge' condition such that there is only one trivial function defined by $B$ (namely, all the 2-coboundaries $\\Theta _2(g_1,g_2)$ equal zero), then each solution of $C$ will stand for an equivalent class, in which case, the classification reduces to solving the equations under the special gauge condition.",
"Actually, we can really fix some gauge degrees of freedom.",
"It has been proved that the canonical gauge is a valid gauge condition by setting $ \\theta _{2}(E,g_{1})=\\theta _{2}(g_{1},E)=0,$ where $E$ is the unit of group $G$ .",
"Under this convention, there are only $(G-1)^2$ components remaining nonzero in $\\theta _2(g_1,g_2)$ and the size of the new matrix $C$ is $(G-1)^3\\times (G-1)^2$ .",
"Similarly, for coboundaries, we have $ \\Theta _2(E,g_1)=\\theta _1(E)=0.$ As a result, we only have $(G-1)^2$ nonzero variables for coboundaries and the size of the new matrix $B$ is $(G-1)^2\\times (G-1)$ .",
"Although this canonical convention does not fix all the `gauge' degrees of freedom, it does simplify the calculation to solve the cocycle equations.",
"After this simplification, the rank of 2-cocycle equations (REF ) is denoted as $R_{C}$ , and $R_{B}$ is the rank of 2-coboundary equations (REF ).",
"For finite groups, the identity $ (G-1)^{2}-R_{C}=R_{B}$ is always satisfied.",
"As usual linear problems, the cocycle equations can be solved by elimination method.",
"To this end, we need to transform the matrix $C$ (and $B$ ) into partially diagonal form $C^{\\prime }$ ( and $B^{\\prime }$ ) by linear operations, namely, by adding or subtracting multiple of one row (or column) to some other row (or column).",
"However, since the equation is defined mod $2\\pi $ , in each step of linear operations, the coefficients must be integer numbers.",
"As a result, in the final matrix all the entries are integer numbers.",
"The (partially) diagonal matrix with integer entries is called Smith normal formSee wiki http://en.wikipedia.org/wik/Smith_normal_form.",
"The diagonal entries of $C^{\\prime }$ and $B^{\\prime }$ completely determine the classification of cocycles.",
"As an example, we consider the group $D_{2}=Z_2\\times Z_2=\\lbrace E,P,Q,PQ\\rbrace $ where $P^2=Q^2=E$ .",
"If we adopt the canonical gauge condition, there are 9 nonzero variables in $\\theta _{2}(g_1,g_2)$ and 3 nonzero ones in $\\theta _{1}(g_1)$ .",
"The cocycle equations and the coboundary functions correspond to two matrices $C$ ($27\\times 9$ , with rank 6) and $B$ ($9\\times 3$ , with rank 3), respectively: $ C=\\left(\\begin{array}{ccccccccc}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\-1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\\\-1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & -1 & 1 & 1 & 0 & 0 & -1 & 0 & 0\\\\0 & -1 & 0 & 0 & 1 & 0 & 0 & -1 & 0\\\\1 & -1 & 0 & 0 & 0 & 1 & 0 & 0 & -1\\\\0 & 1 & -1 & -1 & 0 & 0 & 1 & 0 & 0\\\\1 & 0 & -1 & 0 & -1 & 0 & 0 & 1 & 0\\\\0 & 0 & -1 & 0 & 0 & -1 & 0 & 0 & 1\\\\1 & 0 & 0 & -1 & 0 & 0 & -1 & 0 & 0\\\\0 & 1 & 0 & -1 & 0 & 1 & 0 & -1 & 0\\\\0 & 0 & 1 & -1 & 1 & 0 & 0 & 0 & -1\\\\0 & 0 & 0 & 1 & -1 & 1 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 1 & -1 & 1 & 0 & 0 & 0\\\\-1 & 0 & 0 & 0 & 1 & -1 & 1 & 0 & 0\\\\0 & -1 & 0 & 1 & 0 & -1 & 0 & 1 & 0\\\\0 & 0 & -1 & 0 & 0 & -1 & 0 & 0 & 1\\\\1 & 0 & 0 & -1 & 0 & 0 & -1 & 0 & 0\\\\0 & 1 & 0 & 0 & -1 & 0 & -1 & 0 & 1\\\\0 & 0 & 1 & 0 & 0 & -1 & -1 & 1 & 0\\\\-1 & 0 & 0 & 1 & 0 & 0 & 0 & -1 & 1\\\\0 & -1 & 0 & 0 & 1 & 0 & 0 & -1 & 0\\\\0 & 0 & -1 & 0 & 0 & 1 & 1 & -1 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & -1\\\\0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & -1\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\end{array}\\right),$ $ B=\\left(\\begin{array}{ccc}2 & 0 & 0\\\\1 & 1 & -1\\\\1 & -1 & 1\\\\1 & 1 & -1\\\\0 & 2 & 0\\\\-1 & 1 & 1\\\\1 & -1 & 1\\\\-1 & 1 & 1\\\\0 & 0 & 2\\end{array}\\right).$ After some linear operations as mentioned above, the matrices $C$ and $B$ can be written in the Smith normal form $C^{\\prime }&=&\\left( \\begin{array}{ccccccccc}1 & 0 & 0 & 0 & 0 & 1 & -1 & 1& -1\\\\ 0 & 1 & 0 & 0 & 0 & 0 & -1& 1&0\\\\0 & 0 & 1 & 0 & 0 & 1 & 0 & 0& -1\\\\ 0 & 0& 0 & 1 & 0 & -1 & -1 & 2&0\\\\0 & 0 & 0 & 0 & 1 & 0 & 0 & 1&-1\\\\ 0 & 0 & 0 & 0 & 0 & 2 & 0 & -2&0\\end{array} \\right),\\\\B^{\\prime }&=&\\left( \\begin{array}{ccc}1 & -1 & -1 \\\\ 0 & 2 & 0\\\\0 & 0 & 2\\end{array} \\right).$ Notice that the number of variables (number of columns) in () is greater than the number of equations (number of rows), and the difference is 3.",
"If the coefficient space is $U(1)$ , the solutions form a 3-dimensional continuous space $U(1)\\times U(1)\\times U(1)$ .",
"Furthermore, the last row of () indicates the two variables corresponding to 2 is determined upto $\\frac{2m\\pi }{2}(m=0,1)$ , which gives an extra $Z_2$ group.",
"So the solution space of $\\theta _{2}(g_1,g_2)$ is $U(1)\\times U(1)\\times U(1) \\times Z_2$ ,while the coboundary space is $U(1)\\times U(1)\\times U(1)$ .",
"The quotient gives the classification of cocycles: $\\mathcal {H}^{2}(D_2,U(1))=\\mathbb {Z}_2.$ If the coefficient space is $Z$ , then the common factor 2 in the last row of () can be eliminated, and the solution space is $Z\\times Z\\times Z$ .",
"On the other hand, the common factor 2 in the last two rows of () indicates that the coboundary space is $Z\\times 2Z\\times 2Z$ .",
"Therefore, the classification is $\\mathcal {H}^{2}(D_2,Z)=\\mathbb {Z}_2\\times \\mathbb {Z}_2.$ If the coefficient space is $Z_2$ , then the solution space of $\\theta _{2}(g_1,g_2)$ is $Z_2\\times Z_2\\times Z_2\\times Z_2$ .",
"The last two rows of () give trivial functions, so the coboundary space is $Z_2$ .",
"The quotient gives the classification of cocycles: $\\mathcal {H}^2(D_2, Z_2)=\\mathbb {Z}_2^3$ .",
"Similarly, if the coefficient space is $Z_N$ , then the classification of cocycles is $\\mathcal {H}^2(D_2, Z_N)=\\mathbb {Z}_{(N,2)}\\times \\mathbb {Z}_{(N,2)}\\times \\mathbb {Z}_{(N,2)}$ , where $(N,2)$ stands for the greatest common divisor of $N$ and 2.",
"The first $\\mathbb {Z}_{(N,2)}$ is owing to the factor 2 in the Smith normal form $C^{\\prime }$ , and the last two $\\mathbb {Z}_{(N,2)}$ are owing to the two 2s in the Smith normal form of $B^{\\prime }$ .",
"This rule of classification is also true for $\\mathcal {H}^2(G, Z_N)$ with a general symmetry group $G$ .",
"As another example, we consider the anti-unitary symmetry group $Z_2\\times Z_2^T$ .",
"Under canonical gauge condition, there are 9 nonzero variables in $\\theta _{2}(g_1,g_2)$ and 3 nonzero variables in $\\theta _{1}(g_1)$ .",
"The cocycle matrix $C$ and coboundary matrix $B$ can be respectively transformed into the Smith normal form $C^{\\prime }$ and $B^{\\prime }$ after some linear operations, $C^{\\prime }&=&\\left( \\begin{array}{ccccccccc}1 & 0 & 0 & 0 & 0 & -1 & -1 & 1& 1\\\\ 0 & 1 & 0 & 0 & 0 & -1 & 0 & 0&1\\\\0 & 0 & 1 & 0 & 0 & 0 & -1 & 1& 0\\\\ 0 & 0& 0 & 1 & 0 & -1 & 0 & -1&0\\\\0 & 0 & 0 & 0 & 1 & 0 & -1 & 0&-1\\\\ 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0&0\\\\0 & 0 & 0 & 0 & 0 & 0 & 2 & 0&0\\end{array} \\right), \\\\B^{\\prime }&=&\\left( \\begin{array}{ccc}1 & -1 & 1 \\\\ 0 & 2 & -2\\end{array} \\right).$ If the coefficient space is $U(1)$ , it can be seen that the solution space of $\\theta _{2}(g_1,g_2)$ is $U(1)\\times U(1) \\times Z_2\\times Z_2$ from the last two rows of (REF ).",
"And the coboundary space is still $U(1) \\times U(1)$ .",
"The quotient gives the classification of cocycles: $\\mathcal {H}^{2}(Z_2\\times Z_2^T,U(1))=\\mathbb {Z}_2^2$ .",
"If the coefficient space is $Z$ , the solution space of $\\theta _{2}(g_1,g_2)$ is $Z\\times Z$ , while the coboundary space is $Z\\times 2Z$ .",
"The quotient gives the classification of cocycles: $\\mathcal {H}^2(Z_2\\times Z_2^T, Z)=\\mathbb {Z}_2$ .",
"Similarly, if the coefficient space is $Z_N$ , then $\\mathcal {H}^2(Z_2\\times Z_2^T, Z_N)=\\mathbb {Z}_{(N,2)}^3$ .",
"From the Smith normal form, we can also obtain the solutions of the cocycles.",
"For example, in the Smith normal form (REF ) of the cocycle equations of group $D_2=Z_2\\times Z_2=\\lbrace E,P,Q,PQ\\rbrace $ , the columns correspond to variables $\\theta _2(P,P)$ , $\\theta _2(P,Q)$ , $\\theta _2(P,PQ)$ , $\\theta _2(Q,P)$ , $\\theta _2(PQ,P)$ , $\\theta _2(Q,PQ)$ , $\\theta _2(Q,Q)$ , $\\theta (PQ,Q)$ , $\\theta _2(PQ,PQ)$ , respectively.",
"We note these variables as $X_1,X_2,..., X_9$ .",
"The last three variables are coboundary degrees of freedom and can be fixed as $X_7=X_8=X_9=0$ .",
"Under this condition, the last equation becomes $2X_6=0$ mod $2\\pi $ , which has two solutions $X_6=0$ and $X_6=\\pi $ .",
"The first solution is trivial, from which we can obtain that all other components are zero; the second solution is nontrivial, from (REF ) we can easily obtain that $X_1=-\\pi , X_2=0, X_3=-\\pi , X_4=\\pi , X_5=0$ .",
"Equivalently, the nontrivial solution is given by $\\omega _2(P,P)=\\omega _2(P,PQ)=\\omega _2(Q,P)=\\omega _2(Q,PQ)=-1$ , other components are equal to 1.",
"In this paper we solve all the 2-cocycle equations using above method.",
"However, for abelian unitary groups there exist analytic cocyle solutions.",
"For example, for the group $G=Z_{N_1}\\times Z_{N_2}\\times ...\\times Z_{N_k}$ with generators $h_1,...,h_k$ , if we label the group elements as $A=(a_1,a_2,...,a_k)=(h_1^{a_1},h_2^{a_2},...,h_k^{a_k})=h_1^{a_1}h_2^{a_2}...h_k^{a_k}$ , then the analytic 2-cocycle solutions take the following form: $ \\omega _2 (A,B)=e^{2\\pi i\\sum _{i<j}{p^{ij}\\over N_{ij}}a_ib_j},$ where $N_{ij}$ is the greatest common devisor of $N_i$ and $N_j$ and $0\\le p^{ij}\\le N_{ij}-1$ .",
"There are totally $\\prod _{i<j} N_{ij}$ different classes of solutions."
],
[
"Anti-unitary groups: decoupled cocycles ",
"In this section, we consider anti-unitary symmetry group $G$ which contains a unitary subgroup $H$ as its invariant subgroup, and the quotient group is the time reversal group, $G/H\\simeq Z_2^T,$ where $Z_2^T=\\lbrace E,\\mathbb {T}\\rbrace $ with $\\mathbb {T}^2=E$ .",
"It should be noted that $\\mathbb {T}$ is not necessarily an element of $G$.",
"Supposing $T$ is the anti-unitary group element in $G$ with the smallest order $2^m$ with $m\\in \\mathbb {Z}$ and $m\\ge 1$ , namely, $T^{2^m}=E$ (the order of $T$ should not contain odd factors.",
"The reason is that if $T^{2^p(2q+1)}=E$ where $p,q\\in \\mathbb {Z}$ and $p\\ge 1,q\\ge 1$ , then we can find another anti-unitary group element $T^{\\prime }=T^{2q+1}$ with $T^{\\prime 2^p}=E$ such that the order of $T^{\\prime }$ is less than $T$ ), then there are roughly two types of anti-unitary groups: Type-I: $T^2=E$ , namely, $m=1$ .",
"In this case, we can identify $\\mathbb {T}$ with $T$ , such that $\\mathbb {T}\\in G$ .",
"Type-II: $T^{2^m}=E$ with $m\\ge 2$ .",
"In this case, $\\mathbb {T}$ can never be identified with any element of $G$ , namely, $\\mathbb {T}\\notin G$ .",
"The simplest example with $m=2$ is the fermionic time reversal group $Z_4^T=\\lbrace E,T^2=P_f,T,T^3\\rbrace $ , where $P_f$ is the fermion parity.",
"We will only discuss the fermionic symmetry groups, and will not discuss the more complicated cases with $m>2$ ."
],
[
"Type-I anti-unitary groups: $T^2=E$",
"This kind of groups can be written in forms of direct product group $G=H\\times Z_2^T$ or semi-direct product group $G=H\\rtimes Z_2^T$ .",
"In the following we will note the generator of $Z_2^T$ as $T=\\mathbb {T}$ .",
"We can decouple the cocycles into two parts, namely, the $Z_2^T$ part $\\omega _2(T,T)$ and the $H$ part $\\omega _2(g_1,g_2)$ , and illustrate that they determine the classification.",
"The crossing terms, such as $\\omega _2(Tg_1,g_2),\\omega _2(g_1,Tg_2),\\omega _2(Tg_1,Tg_2)$ , can be either fixed or expressed in terms of $\\omega _2(T,T)$ and $\\omega _2(g_1,g_2)$ .",
"This can be shown in the following four steps: (1) No gauge degrees of freedom in $\\omega _2(T,T)=\\pm 1$ .",
"Let us see $\\omega _2(T,T)$ first.",
"From the cocycle equation $(d\\omega _2)(T,T,T)&=&\\omega _2^{-1}(T,T)\\omega _2^{-1}(E,T)\\omega _2(T,E)\\omega _2^{-1}(T,T) \\\\&=&1,$ we get $\\omega _2^2(T,T)=1$ , so $ \\omega _2(T,T)=\\pm 1,$ here we have adopted the canonical gauge.",
"Since $(d\\Omega _1)(T,T)=\\Omega _1^{-1}(T)\\Omega _1^{-1}(E)\\Omega _1(T)=1$ , there are no gauge degrees of freedom for $\\omega _2(T,T)$ , $\\omega _2(T,T)=1$ and $\\omega _2(T,T)=-1$ stand for different classes.",
"(2) Tune coboundaries to fix $\\omega _2(T, g)=1$ and $\\omega _2(g,T)=1$ .",
"Further, for unitary elements $g\\in H$ , we will show that the following gauge can be fixed: $&&\\omega _2(T, g)=\\omega _2(T, \\tilde{g})=1,\\nonumber \\\\&&\\omega _2(g, T)=\\omega _2(\\tilde{g},T)=1,$ where $Tg=\\tilde{g}T$ .",
"If $\\tilde{g}=g$ , then we can tune the coboundary $&&(d\\Omega _1) (T,g)=\\Omega _1^{-1}(g) \\Omega _1^{-1}(Tg)\\Omega _1(T),\\\\&&(d\\Omega _1) ( g,T)=\\Omega _1(T) \\Omega _1^{-1}(gT)\\Omega _1( g),$ such that $\\omega _2^{\\prime }(T,g)=\\omega _2(T,g)(d\\Omega _1)(T,g)=1$ , and $\\omega _2^{\\prime }(g,T)=\\omega _2(g,T)(d\\Omega _1)(g,T)=1$ .",
"This can be done by setting $&&\\Omega _1(g)=\\sigma \\left({\\omega _2(T,g)\\over \\omega _2(g,T)}\\right)^{1/2},\\\\&&\\Omega _1(gT)=\\sigma \\Omega _1(T)\\left[\\omega _2(T,g)\\omega _2(g,T)\\right]^{1/2}.",
"\\nonumber $ where $\\sigma =\\pm 1$ is a $Z_2$ variable and $\\Omega _1(T)$ is a $U(1)$ variable which can be set as 1.",
"In the square root, $-1$ should be treated as $e^{i\\pi }$ such that $\\sqrt{(-1)\\times (-1)}=-1$ and $\\sqrt{(-1)/(-1)}=1$ .",
"If $\\tilde{g}\\ne g$ , then the illustration of the gauge choice (REF ) is a little complicated.",
"From the cocycle equation $(d\\omega _2)(\\tilde{g}, T,T)=1$ , we have $\\omega _2(Tg,T)\\omega _2(\\tilde{g},T)=\\omega _2(T,T).$ Similarly, from $(d\\omega _2)(T, T, \\tilde{g})=1$ we have ${\\omega _2(T,gT)\\over \\omega _2(T,\\tilde{g})}=\\omega _2(T,T).$ On the other hand, $(d\\omega _2)(T, g, T)=1$ gives ${\\omega _2(T,gT)\\over \\omega _2(Tg,T)}=\\omega _2(T,g)\\omega _2(g,T).$ Comparing (REF ),(REF ) and (REF ), we obtain $\\omega _2(T,g)\\omega _2(g,T)=\\omega _2(T,\\tilde{g})\\omega _2(\\tilde{g},T).$ Owing to this relation, the following gauge fixing equations have solutions: $\\omega _2^{\\prime }(T,g)=\\omega _2(T,g)(d\\Omega _1)(T,g)=1$ , $\\omega _2^{\\prime }(T,\\tilde{g})=\\omega _2(T,\\tilde{g})(d\\Omega _1)(T,\\tilde{g})=1$ , $\\omega _2^{\\prime }(g, T)=\\omega _2(g,T)(d\\Omega _1)(g, T)=1$ , $\\omega _2^{\\prime }(\\tilde{g}, T)=\\omega _2(\\tilde{g},T)(d\\Omega _1)(\\tilde{g}, T)=1$ , where $&&(d\\Omega _1) (T,g)=\\Omega _1^{-1}(g) \\Omega _1^{-1}(Tg)\\Omega _1(T),\\\\&&(d\\Omega _1) (T, \\tilde{g})=\\Omega _1^{-1}(\\tilde{g}) \\Omega _1^{-1}(gT)\\Omega _1(T),\\\\&&(d\\Omega _1) (\\tilde{g},T)=\\Omega _1(T) \\Omega _1^{-1}(Tg)\\Omega _1(\\tilde{g}),\\\\&&(d\\Omega _1) ( g,T)=\\Omega _1(T) \\Omega _1^{-1}(gT)\\Omega _1( g),$ and the solutions are $&&\\Omega _1(\\tilde{g})=\\Omega _1^{-1}(g)\\omega _2(T,g)\\omega _2^{-1}(\\tilde{g},T),\\nonumber \\\\&&\\Omega _1(Tg)=\\Omega _1(T)\\Omega _1^{-1}(g)\\omega _2(T,g),\\\\&&\\Omega _1(gT)=\\Omega _1(T)\\Omega _1(g)\\omega _2(g,T), \\nonumber $ where $\\Omega _1(g)$ is an $U(1)$ variable and $\\Omega _1(T)$ can be set as 1.",
"Once the gauge fixing condition (REF ) is satisfied, then in the remaining gauge degrees of freedom for each pair of $g$ ,$\\tilde{g}$ with $\\tilde{g}\\ne g$ , $\\Omega _{1}(g)\\Omega _{1}(\\tilde{g})=1$ is required and there remains one free coboundary variable $\\Omega _{1}(g)$ .",
"In later discussion we will fix $\\omega _2(T,g)=\\omega _2(g,T)=1$ for any $g\\in H$ .",
"(3) The values of $\\omega _2(Tg_1,Tg_2)$ are determined.",
"From the equation $ (d\\omega _2)(T,T,g)=\\omega _2^{-1}(T,g)\\omega _2^{-1}(E,g)\\omega _2(T,Tg)\\omega _2^{-1}(T,T)=1, $ we get $\\omega _2(T,Tg)=\\omega _2(T,g)\\omega _2(T,T)=\\omega _2(T,T)$ .",
"Similarly, from $(d\\omega _2)(T,g,T)=1$ we have $\\omega _2(Tg, T)=\\omega _2(\\tilde{g}T, T)=\\omega _2(T,T).$ That is to say, $\\omega _2(T,Tg)$ and $\\omega _2(Tg,T)$ can be fixed as real $ \\omega _2(T,Tg)= \\omega _2(Tg,T)=\\omega _2(T,T)=\\pm 1,$ for any $g\\in H$ .",
"Furthermore, from the cocycle equation $(d\\omega _2)(Tg_1,T,g_2)=\\omega _2^{-1}(T,g_2)\\omega _2^{-1}(\\tilde{g}_1,g_2) \\omega _2(Tg_1,Tg_2)\\omega _2^{-1}(Tg_1,T)=1$ with $g_1,g_2\\in H$ , we obtain $ \\omega _2(Tg_1,Tg_2)=\\omega _2(\\tilde{g}_1,g_2)\\omega _2(T,T).$ (4) The values of $\\omega _2(Tg_1,g_2)$ and $\\omega _2(g_1,Tg_2)$ are determined.",
"Finally, considering the following cocycle equations $& &(d\\omega _2)(T, g_1,g_2)\\nonumber \\\\& & =\\omega _2^{-1}(g_1,g_2)\\omega _2^{-1}(Tg_1,g_2)\\omega _2(T,g_1g_2)\\omega _2^{-1}(T,g_1)\\nonumber \\\\& & =\\omega _2^{-1}(g_1,g_2)\\omega _2^{-1}(Tg_1,g_2)=1, \\\\& & (d\\omega _2)(g_1,T,g_2)\\nonumber \\\\& & =\\omega _2(T,g_2)\\omega _2^{-1}(g_1T,g_2)\\omega _2(g_1,Tg_2)\\omega _2^{-1}(g_1,T)\\nonumber \\\\&& =\\omega _2^{-1}(g_1T,g_2)\\omega _2(g_1,Tg_2)=1,\\\\& & (d\\omega _2)( g_1,g_2,T)\\nonumber \\\\&& =\\omega _2(g_2,T)\\omega _2^{-1}(g_1g_2,T)\\omega _2(g_1,g_2T)\\omega _2^{-1}(g_1,g_2)\\nonumber \\\\&& =\\omega _2(g_1,g_2T)\\omega _2^{-1}(g_1,g_2)=1,$ we obtain the following relations $ &&\\omega _2(Tg_1,g_2)=\\omega _2^{-1}(g_1,g_2),\\nonumber \\\\&&\\omega _2(g_1,Tg_2)=\\omega _2(g_1,\\tilde{g}_2),$ and an important constraint $ \\omega _2(g_1,g_2)\\omega _2(\\tilde{g}_1,\\tilde{g}_2)=1, $ where $g_1,g_2\\in H$ .",
"We call the cocycles satisfying relations (REF ), (REF ), (REF ),(REF ) as `decoupled cocyles' or decoupled factor systems, in which case only $\\omega _2(T,T)$ and $\\omega _2(g_1,g_2)$ are important.",
"Since $\\omega _2(T,T)$ contributes $Z_2$ classification to $\\mathcal {H}^2(G,U(1))$ , in appendix we will see how the constraint (REF ) influences the classification of $\\omega _2(g_1,g_2)$ ."
],
[
"Type-II anti-unitary groups: $T^{2^m}=E, m\\ge 2$",
"We focus on the case $m=2$ , which corresponds to the anti-unitary symmetry group of fermions with half-integer spin.",
"Fermionic time reversal group $Z_4^T$ .",
"The simplest example of the type-II anti-unitary group is the fermionic time reversal group $Z_4^T=\\lbrace E,P_f,T,P_f T\\rbrace $ with $T^4=E$ and $T^2=P_f$ , $P_f$ is the fermion parity.",
"The unitary normal subgroup of $Z_4^T$ is $H=Z_2^f=\\lbrace E,P_f\\rbrace $ with $Z_4^T/H=Z_2^T=\\lbrace E,\\mathbb {T}\\rbrace $ .",
"Obviously $\\mathbb {T}$ is not an element of $Z_4^T$ .",
"In the following we will figure out the classification of 2-cocyles of $Z_4^T$ by gauge fixing procedure.",
"Firstly we adopt the canonical gauge by setting $\\omega _{2}(E,g)=\\omega _{2}(g,E)=1$ .",
"We still define the time reversal conjugation as $T^{-1}gT=\\tilde{g}$ , since $T^{-1}=TP_f$ , we have $T\\tilde{g}T^{-1}=T^{-1}\\tilde{g}T=g$ , namely, $\\tilde{\\tilde{g}}=g$ .",
"Since $\\tilde{P}_f=P_f$ , similar to (REF ) we can fix $ \\omega _2(T,P_f)=\\omega _2(P_f,T)=1.$ After above gauge fixing, from (REF ), there are still a $Z_2$ degree of freedom $\\sigma $ and an U(1) degree of freedom $\\Omega _1(T)$ .",
"If we set $\\Omega _1(T)=1$ , then there remains a $Z_2$ gauge degree of freedom $\\Omega _1(P_f)=\\sigma =\\pm 1$ .",
"Since $(d\\Omega _1)(P_f,P_f)=\\Omega _1(P_f)^2=1$ , under the gauge fixing (REF ), $\\omega _2(P_f,P_f)$ is completely fixed without any gauge degrees of freedom.",
"In other words, different values of $\\omega _2(P_f,P_f)$ stand for different classes of 2-cocycles.",
"From $(d\\omega _2)(T,T,P_f)=1$ we have $\\omega _2(T,TP_f)={\\omega _2(P_f,P_f) \\omega _2(T,T)}.$ On the other hand, $(d\\omega _2)(TP_f,T,TP_f)=1$ gives $\\omega _2(TP_f,T)\\omega _2(T,TP_f)=1;$ while $(d\\omega _2)(T,P_f,T)=1$ yields $\\omega _2(T,TP_f)=\\omega _2(TP_f,T).$ Comparing these relations, we obtain $\\left[{\\omega _2(P_f,P_f) \\omega _2(T,T)}\\right]^2=\\omega _2(P_f,P_f)^2=1,$ namely, $\\omega _2(P_f,P_f)=\\pm 1$ .",
"Here we have used $\\omega _2(T,T)^2={\\omega _2(T,P_f)\\over \\omega _2(P_f,T)}=1$ , which results from $(d\\omega _2)(T,T,T)=1$ .",
"We have shown previously that $\\omega _2(P_f,P_f)$ has no gauge degrees of freedom, therefore $\\omega _2(P_f,P_f)=1$ (trivial) and $\\omega _2(P_f,P_f)=-1$ (nontrivial) stand for two different classes.",
"In the following we will show that all other components are dependent on $\\omega _2(P_f,P_f)$ or can be fixed to be 1.",
"From $(d\\omega _2)(P_f,P_f,T)=1$ and $(d\\omega _2)(P_f,T,P_f)=1$ we obtain that $\\omega _2(P_f,TP_f)=\\omega _2(TP_f,P_f)=\\omega _2(P_f,P_f)$ .",
"Since $\\omega _2(T,T)^2=1$ and $(d\\Omega _1)(T,T)=\\Omega _1(P_f)^{-1}=\\sigma =\\pm 1$ , the value of $\\omega _2(T,T)$ only depends on the gauge degree of freedom $\\sigma $ .",
"We can tune $\\sigma $ such that $\\omega _2(T,T)=\\omega _2(P_f, P_f)$ and consequently $\\omega _2(T,TP_f)=\\omega _2(TP_f,T)=1$ .",
"Finally, the value of $\\omega _2(TP_f,TP_f)=1$ can be derived from $(d\\omega _2)(TP_f,TP_f,T)=1$ .",
"Above we showed that the value of $\\omega _2(P_f,P_f)=\\pm 1$ completely determines the classification $\\mathcal {H}^2(Z_4^T,U(1))=\\mathbb {Z}_2$ .",
"The nontrivial factor system is given as $ &&\\omega _2(P_{f},P_{f})=-1,\\\\&&\\omega _2(P_{f},TP_{f})=\\omega _2(TP_{f},P_{f})=\\omega _2(P_f,P_f)=-1,\\\\&&\\omega _2(T,T)=-1,$ and others are fixed to be 1.",
"Similar to the invariant $[M(T)K]^2=\\pm 1$ for the projective Reps of $Z_2^T$ , the projective Reps of $Z_4^T$ also have an invariant $ [M(T)K]^4=\\pm 1.$ Noticing $[M(T)K]^4=\\omega _2(T,T)^2\\omega _2(P_f,P_f)$ , it is invariant under gauge transformation $M^{\\prime }(g)K_{s(g)}=M(g)\\Omega _1(g)K_{s(g)}$ since $ [M^{\\prime }(T)K]^4&=&\\omega _2^{\\prime }(T,T)^2\\omega _2^{\\prime }(P_f,P_f) \\nonumber \\\\&=&\\omega _2(T,T)^2\\omega _2(P_f,P_f) \\Omega _2(T,T)^2\\Omega _2(P_f,P_f)\\nonumber \\\\&=&\\omega _2(T,T)^2\\omega _2(P_f,P_f) {\\Omega _1(P_f)^2\\over \\Omega _1(P_f)^2}\\nonumber \\\\&=&[M(T)K]^4.$ General fermionic anti-unitary groups.",
"In the following, we will discuss general fermionic groups $G_f$ .",
"Since physical operations usually conserve fermion parity (the fermion creation or annihilation operators do not conserve fermion parity, but they are not allowed in the Hamiltonian), we assume that $[P_f,g]=0$ for any $g\\in G_f$ .",
"Therefore, the subgroup $Z_2^f=\\lbrace E,P_f\\rbrace $ is a center of $G_f$ and we have $ G_f/Z_2^f=G_b,$ where $G_b$ is the bosonic (namely, type-I) anti-unitary group.",
"If we note $H_f$ as the unitary normal subgroup of $G_f$ such that $G_f/H_f=Z_2^T$ , and note the unitary normal subgroup of $G_b$ as $H_b$ such that $G_b/H_b=Z_2^T$ , then $H_f/Z_2^f=H_b$ , there are two cases: (1)$H_f$ is a direct product group of $H_b$ and $Z_2^f$ , $H_f=H_b\\times Z_2^f$ , then the group $G_f$ can be written as $G_f=H_b\\times Z_4^T$ or $G_f=H_b\\rtimes Z_4^T$ ; (2) $H_f$ is not a direct product group of $H_b$ and $Z_2^f$ , then the group can be noted as $G_f=G_-^\\pm (H_f,T)$ , where the subscript $-$ means $T^2=P_f$ , and the superscript $\\pm $ means that $Th=h^{\\pm 1}T$ for $h\\in H_f$ .",
"After slight modification, the discussion about decoupled factor system in section REF can also be applied for the fermionic anti-unitary groups.",
"For the cocycles of the $Z_4^T$ subgroup of $G$ , we can adopt the previous gauge fixing procedure.",
"In the following we try to decouple the factor system as we have done previously.",
"Supposing $g\\in H_f$ , from the cocycle equations $(d\\omega _2)(P_f,P_f,g)=1$ and $(d\\omega _2)(g,P_f,P_f)=1$ , we get ${\\omega _2(g,P_f)\\over \\omega _2(P_f,g)}={\\omega _2(P_f,P_f g)\\over \\omega _2(gP_f,P_f)}.$ On the other hand, from the cocycle equation $(d\\omega _2) (P_f,g,P_f)=1$ we have ${\\omega _2(g,P_f)\\over \\omega _2(P_f,g)}={\\omega _2(P_fg,P_f)\\over \\omega _2(P_f,gP_f)}.$ Since $gP_f=P_fg$ , comparing above two equations we obtain $\\left({\\omega _2(g,P_f)\\over \\omega _2(P_f,g)}\\right)^2=1,$ namely, ${\\omega _2(g,P_f)\\over \\omega _2(P_f,g)}=\\pm 1$ is a $Z_2$ variable, we note it as ${\\omega _2(g,P_f)\\over \\omega _2(P_f,g)}=\\sigma _{g},\\ \\ \\ g\\ne P_f.$ We can fix the coboundary $\\Omega _1(P_fg)=\\omega _2(P_f,g)\\Omega _1(g)\\Omega _1(P_f)$ such that $\\omega _2^{\\prime }(P_f,g)=\\omega _2(P_f,g)(d\\Omega _1)(P_f,g)=1$ .",
"After this gauge fixing, we have ($g\\ne P_f$ ) $ &&\\omega _2(P_f,g)=1,\\\\&&\\omega _2(g,P_f)=\\sigma _g.$ Similar to (REF ), (REF ), (REF ), from the cocycle equations $(d\\omega _2)(g,T,T)=1$ , $(d\\omega _2)(T,T,g)=1$ , and $(d\\omega _2)(T, \\tilde{g},T)=1$ , we have $ {\\omega _2( g,T) \\omega _2(T, g) \\over \\omega _2(T,\\tilde{g}) \\omega _2(\\tilde{g},T) } = \\sigma _{ g} =\\sigma _{\\tilde{g}}^{-1}=\\sigma _{\\tilde{g}}=\\omega _2(\\tilde{g},P_f).$ In the following we fix the values of $\\omega _2(T, g)$ and $\\omega _2(g,T)$ and decouple the factor system as discussed in section REF .",
"(1), If $\\tilde{g}=g$ , namely, $Tg=gT$ , then $\\sigma _g=1$ .",
"Similar to (REF ), we can fix $\\omega _2(T,g) = \\omega _2(g,T)=1$ by tuning the values of $\\Omega _1(g)$ and $\\Omega _1(Tg)$ .",
"(2), If $\\tilde{g}\\ne g$ , the gauge fixing depends on the value of $\\sigma _g$ : A) $\\sigma _g=1$ .",
"In this case, the discussion in equations (REF ), (REF ), (REF ) is still valid, we can simultaneously fix $\\omega _2(T,g)=\\omega _2(g,T)=\\omega _2(T,\\tilde{g})=\\omega _2(\\tilde{g},T)=1$ by adopting the gauge in (REF ).",
"B) $\\sigma _g=-1$ .",
"Owing to (REF ), the values of $\\omega _2(T,g), \\omega _2(g,T), \\omega _2(T,\\tilde{g}), \\omega _2(\\tilde{g},T)$ cannot be simultaneously set to be 1 (in this sense the factor system is not completely decoupled), and the coboundary in (REF ) should be slightly modified.",
"We can set $\\omega _2(T,g)=\\omega _2(T,\\tilde{g})=1,$ and set one of $\\omega _2(g,T), \\omega _2(\\tilde{g},T)$ to be 1.",
"For example, we can choose $ &&\\omega _2(\\tilde{g},T)=1,\\\\&&\\omega _2(g,T)=\\sigma _g.$ Since $g$ and $\\tilde{g}$ are mutually time-reversal conjugate, there is still a gauge degree of freedom in above equation.",
"Above gauge fixing can be carried out by adopting the following coboundary $ &&\\Omega _1(\\tilde{g})=\\Omega _1^{-1}(g)\\omega _2(T,g)\\omega _2^{-1}(\\tilde{g},T),\\nonumber \\\\&&\\Omega _1(Tg)=\\Omega _1(T)\\Omega _1^{-1}(g)\\omega _2(T,g),\\\\&&\\Omega _1(gT)=\\sigma _g\\Omega _1(T)\\Omega _1(g)\\omega _2(g,T).",
"\\nonumber $ We further discuss the gauge fixing of other components.",
"The equations (REF ),(REF ) should be modified as follows: $ \\omega _2(Tg_1,Tg_2)=\\omega _2(g_1,T) \\omega _2(\\tilde{g}_1,g_2)\\omega _2(T,T) \\sigma _{\\tilde{g}_1}\\sigma _{g_2}\\sigma _{\\tilde{g}_1g_2},$ (here we have used $\\omega _2(\\tilde{g}_1P_f,g_2)=\\omega _2(\\tilde{g}_1,g_2)\\sigma _{\\tilde{g}_1}\\sigma _{g_2}\\sigma _{\\tilde{g}_1g_2}$ ) and $ &&\\omega _2(Tg_1,g_2) = \\omega _2^{-1}(g_1,g_2),\\\\&&\\omega _2(g_1,Tg_2) =\\omega _2(g_1,\\tilde{g}_2)\\omega _2^{-1}(\\tilde{g}_2,T)\\omega _2(g_1\\tilde{g}_2,T).$ And the constraint (REF ) becomes $\\omega _2(g_1,g_2)\\omega _2(\\tilde{g}_1,\\tilde{g}_2) = \\sigma _{g_1}\\sigma _{g_2}\\sigma _{g_1g_2}.$ It can be proved that $\\sigma _{g_1}\\sigma _{g_2}\\sigma _{g_1g_2}=1$ and $\\sigma _{\\tilde{g}_1}\\sigma _{g_2}\\sigma _{\\tilde{g}_1g_2}=1$ , so $\\sigma _{g}$ carries a $Z_2$ valued linear representation of $H_f$ .",
"After the factor system being decoupled, the reduction of regular projective Reps can be performed following the procedure of type-I anti-unitary groups (slight modification may be necessary)."
],
[
"Classification of some simple anti-unitary groups",
"In this section we only discuss type-I anti-unitary groups.",
"In subsections REF and REF , we discuss the groups where all the 2-cocycles can be set as real numbers.",
"In subsection REF , we give some examples where the 2-cocycles cannot be set as real numbers."
],
[
"Direct product group $H\\times Z_2^T$",
"If the group is $G=H\\times Z_2^T$ , then $Tg=gT$ and $\\tilde{g}=g.$ From () we have $\\omega _2^2(g_1,g_2)=1$ which constraints $ \\omega _2(g_1,g_2)=\\pm 1.$ Owing to the relations (REF ) $\\sim $ (), we have $\\omega _2(Tg_1,g_2)=\\omega _2(g_1,Tg_2)=\\omega _2(Tg_1,Tg_2)=\\omega _2(g_1,g_2),$ so all the cocycles (and the remaining coboundary degrees of freedom) are fixed as real numbers.",
"Consequently, the classification of $H\\times Z_2^T$ are determined by two parts, $\\omega _2(T,T)=\\pm 1$ which is given by $\\mathcal {H}^2(Z_2^T, U(1))$ , and $\\omega _2(g_1,g_2)=\\pm 1$ which is given by $\\mathcal {H}^2(H, Z_2)$ .",
"In other words, $ \\mathcal {H}^2(H\\times Z_2^T, U(1))&=&\\mathcal {H}^2(Z_2^T, U(1))\\times \\mathcal {H}^2(H, Z_2)\\nonumber \\\\&=& \\mathbb {Z}_2\\times \\mathcal {H}^2(H, Z_2)\\\\&=& \\mathbb {Z}_2^n, \\ \\ \\ {\\rm with\\ } n\\ge 1.\\nonumber $"
],
[
"Semidirect product group $H\\rtimes Z_2^T$ with {{formula:fa305c3b-6399-463b-8449-0dd203514c16}}",
"The general case of semidirect product groups $H\\rtimes Z_2^T$ is a little complicated.",
"Here we only consider the simple case where $H$ is abelian and $Tg=g^{-1}T$ , or equivalently $\\tilde{g}=g^{-1}$ .",
"Noticing that $T\\tilde{g}_1\\tilde{g}_2 = T(\\tilde{g}_1\\tilde{g}_2) =T\\widetilde{g_2g_1}= (g_2g_1 )T,$ on the other hand, $T\\tilde{g}_1\\tilde{g}_2 = g_1T\\tilde{g}_2= g_1g_2T=(g_1g_2)T ,$ so we have $g_1g_2 = g_2g_1,$ namely, the subgroup $H$ is essentially Abelian.",
"We first show that $\\omega _2(g, \\tilde{g})$ must be real.",
"From equation $(d\\omega _2)(\\tilde{g}, g,\\tilde{g})=\\omega _2(g,\\tilde{g}) \\omega _2^{-1}(E,\\tilde{g}) \\omega _2(\\tilde{g},E) \\omega _2^{-1} (\\tilde{g},g)=1$ we obtain $\\omega _2(g,\\tilde{g})=\\omega _2(\\tilde{g},g)$ .",
"On the other hand, from (), $\\omega _2(g,\\tilde{g})\\omega _2(\\tilde{g}, g)=1$ , so we have $\\omega _2^2(g,\\tilde{g})=1,$ which yields the constraint, $\\omega _2(g,\\tilde{g})=\\pm 1.$ Notice that there are no gauge degrees of freedom for $\\omega _2(g,\\tilde{g})$ , because the value of the coboundary $(d\\Omega _{1})(g,\\tilde{g})=\\Omega _{1}(g)\\Omega _{1}(\\tilde{g})=1$ has been fixed after (REF ).",
"Furthermore, from the relations $&&M(g_1)M(g_2) = M(g_1g_2) \\omega _2(g_1,g_2),\\\\&&M(\\tilde{g}_2)M(\\tilde{g}_1) = M(\\widetilde{g_1g_2}) \\omega _2(\\tilde{g}_2, \\tilde{g}_1)\\\\$ and $M(g)M(\\tilde{g})=\\omega _2(g,\\tilde{g})$ , we can obtain $\\omega _2(g_1,g_2) \\omega _2(\\tilde{g}_2,\\tilde{g}_1)=\\omega _2(g_1,\\tilde{g}_1)\\omega _2(g_2,\\tilde{g}_2)\\omega _2(g_1g_2,\\widetilde{g_1g_2}).$ Comparing with (), we have $\\omega _2(\\tilde{g}_1,\\tilde{g}_2) =\\omega _2(\\tilde{g}_2,\\tilde{g}_1)\\omega _2(g_1,\\tilde{g}_1)\\omega _2(g_2,\\tilde{g}_2)\\omega _2(g_1g_2,\\widetilde{g_1g_2}),$ or equivalently $\\omega _2(g_1,g_2) &=&\\omega _2(g_2,g_1)\\omega _2(g_1,\\tilde{g}_1)\\nonumber \\\\&&\\times \\omega _2(g_2,\\tilde{g}_2)\\omega _2(g_1g_2,\\widetilde{g_1g_2}).$ Since $H$ is abelian, it must be a cyclic group or a direct product of cyclic groups.",
"We will consider the two cases separately."
],
[
"When $H=Z_N$",
"We first consider the case $H$ is a cyclic group $Z_N$ generated by $g$ with $g^N=E$ .",
"Since $Z_2\\rtimes Z_2^T=Z_2\\times Z_2^T$ , we assume $N\\ge 3$ .",
"Suppose that in the projective Rep of $H\\rtimes Z_2^T$ , the Rep matrix of $g$ is $M(g)$ .",
"We tune the phase factor of $M(g)$ such that $[M(g)]^N=1.$ Obviously, $M(g^n)$ is proportional to $[M(g)]^n$ by a phase factor.",
"Suppose $M(g^n)= [M(g)]^n \\mu (n),$ where $\\mu (n)$ is an $U(1)$ phase factor and obviously $\\mu (0)=\\mu (N)=1$ .",
"From $M(g^m)M(g^n)&=&M(g^n)M(g^m)\\\\&=&M(g)^{m+n}\\mu (m)\\mu (n)\\\\&=&M(g^{m+n})\\mu (m)\\mu (n)/\\mu (m+n)$ we have $\\omega _2(g^m,g^n)=\\omega _2(g^n,g^m)=\\frac{\\mu (m)\\mu (n)}{\\mu (m+n)}.$ From $\\omega _2(g^m,g^n)=\\omega _2(g^n,g^m)$ and (REF ) we have $\\omega _2(g^m,\\tilde{g}^m)\\omega _2(g^n,\\tilde{g}^n)\\omega _2(g^{m+n},\\tilde{g}^{m+n})=1$ From (REF ) and (REF ), we can see that $\\omega _2(g^m,\\tilde{g}^m)$ form a $Z_2$ Rep of the group $H$ , namely, they are real numbers and are classified by $\\mathcal {H}^1(H,Z_2)$ .",
"In the following we discuss $N$ =even and $N$ =odd separately.",
"1), $N$ is odd: $N=2Q+1, \\ Q\\in Z$ Since $\\mathcal {H}^{1}(Z_{2Q+1},Z_2)=Z_1$ , there are no nontrivial solutions for $\\omega _2(g^m,\\tilde{g}^m)$ , we can safely set $\\omega _2(g^m,\\tilde{g}^m)=1$ in the following discussion.",
"Since $\\mu (m)\\mu (-m)=\\omega _2(g^m,\\tilde{g}^{m})=1$ , for each pair of $g^m,\\tilde{g}^m$ , only one $\\mu (m)$ is free.",
"Remembering that for each pair of $g^m, \\tilde{g}^m$ we have a free coboundary degrees of freedom, so we can further gauge $\\mu (m)$ away by introducing $&&M^{\\prime }(g^m)=M(g^m)\\Omega _1(g^m),\\\\&&M^{\\prime }(g^{-m})=M(g^{-m})\\Omega _1^{-1}(g^m)$ with $\\Omega _1(g^m)=\\mu ^{-1}(m)$ for $1\\le m\\le Q$ , such that $&&\\mu ^{\\prime }(m)=\\mu (m)\\Omega _1(g^m)=1,\\\\&&\\mu ^{\\prime }(-m)=\\mu (-m)\\Omega _1^{-1}(g^m)=\\omega _2(g^m,\\tilde{g}^m)=1,\\\\&&\\mu ^{\\prime }(0)=\\mu (0)=1,\\\\&&\\mu ^{\\prime }(2Q+1)=\\mu (2Q+1)=1,$ then we have $M^{\\prime }(g^m)M^{\\prime }(g^n)=M^{\\prime }(g^{m+n})\\omega ^{\\prime }_2(g^m,g^n),$ $\\omega ^{\\prime }_2(g^m,g^n)=\\mu ^{\\prime }(m)\\mu ^{\\prime }(n)/\\mu ^{\\prime }(m+n),$ where $\\omega ^{\\prime }_2(g^m,g^n)=1$ is a trivial cocycle.",
"So $M^{\\prime }(g^m)$ form a linear Rep of $H=Z_{2Q+1}$ .",
"Until now, all the coboundary degrees of freedom are fixed (except for $\\Omega _1(T)$ , which is not important since it does not change any cocycle values, so it can be set as 1).",
"So we conclude that the classification of projective Reps of $Z_{2Q+1}\\rtimes Z_2^T$ is purely determined by $\\mathcal {H}^2(Z_2^T,U(1))=\\mathbb {Z}_2$ .",
"Namely, $\\mathcal {H}^2(Z_{2Q+1}\\rtimes Z_2^T,U(1))= \\mathcal {H}^2(Z_2^T,U(1))=\\mathbb {Z}_2.$ 2), $N$ is even: $N=4Q, \\ Q\\in Z$ .",
"Since $&&M(g^m)M(g^m)M(\\tilde{g}^m)M(\\tilde{g}^m)\\\\&\\ \\ &= \\omega _2(g^m,g^m)\\omega _2(\\tilde{g}^m,\\tilde{g}^m)\\omega _2(g^{2m},\\tilde{g}^{2m})\\\\&\\ \\ &=\\omega _2(g^m,\\tilde{g}^m)^2=1,$ owing to () we have $\\omega _2(g^{2m},\\tilde{g}^{2m})=1.$ Further, from (REF ), since $g^{N}=E$ and $g^{N/2}=\\tilde{g}^{N/2}$ , $\\mu (N)=\\mu (N/2)^2/\\omega _2(g^{N/2},\\tilde{g}^{N/2})=1$ , we have $\\mu ^2(N/2)=\\mu (2Q)^2=1$ (here we have used $\\omega _2(g^{N/2},\\tilde{g}^{N/2})=\\omega _2(g^{2Q},\\tilde{g}^{2Q})=1$ ), which yields $\\mu (N/2)=\\mu (2Q)=\\pm 1.$ Since there is remaining $Z_2$ coboundary degrees of freedom for $\\Omega _1(g^{2Q})$ after fixing $\\omega _2(T,g^{2Q})=\\omega _2(g^{2Q},T)=1$ (see (REF )), we can choose $\\Omega _1(g^{2Q})=\\mu ^{-1}(2Q)$ and redefine $M^{\\prime }(g^{2Q})=M(g^{2Q})\\Omega _1(g^{2Q})=M(g)^{2Q}$ such that $\\mu ^{\\prime }(2Q)=\\mu (2Q)\\Omega _1(g^{2Q})=1$ .",
"Similar to the discussion in case (1), we can introduce the following conboundary to gauge away $\\mu (m)$ , here we constrain $-2Q< m< 2Q$ : $&&M^{\\prime }(g^m)=M(g^m)\\Omega _1(g^m),\\\\&&M^{\\prime }(g^{-m})=M(g^{-m})\\Omega _1^{-1}(g^m)$ with $\\Omega _1(g^m)=\\mu ^{-1}(m)$ such that $&&\\mu ^{\\prime }(m)=\\mu (m)\\Omega _1(g^m)=1,\\\\&&\\mu ^{\\prime }(-m)=\\mu (-m)\\Omega _1^{-1}(g^m)=\\omega _2(g^m,\\tilde{g}^m),\\\\&&\\mu ^{\\prime }(0)=\\mu ^{\\prime }(4Q)=1,\\\\&&\\mu ^{\\prime }(2Q)=1,$ then $M^{\\prime }(g^m)M^{\\prime }(g^n)=M^{\\prime }(g^{m+n})\\omega ^{\\prime }_2(g^m,g^n),$ where $\\omega ^{\\prime }_2(g^m,g^n)=\\mu ^{\\prime }(m)\\mu ^{\\prime }(n)/\\mu ^{\\prime }(m+n)$ with $\\mu ^{\\prime }(m\\pm 4Q)=\\mu ^{\\prime }(m)$ .",
"Until now, all the cocycles are set to be real and all the coboundary degrees of freedom are fixed (except for $\\Omega _1(T)$ , which is not important since it does not change any cocycle values, so it can be set as 1).",
"This means that the $H$ part of the 2-cocycles are classified by $\\mathcal {H}^2(H,Z_2)$ .",
"Notice that all the nontrivial 2-cocycles are related to $\\omega _2(g,\\tilde{g})$ , which is classified by $\\mathcal {H}^1(H,Z_2)$ .",
"So we conclude that for the group $Z_{4Q}\\rtimes Z_2^T$ , $\\mathcal {H}^2(H\\rtimes Z_2^T, U(1))&=&\\mathcal {H}^2(Z_2^T, U(1))\\times \\mathcal {H}^2(H, Z_2)\\nonumber \\\\&=& \\mathbb {Z}_2\\times \\mathcal {H}^1(H, Z_2)\\\\&=& \\mathbb {Z}_2^2.\\nonumber $ Remark: When $\\omega _2(T,T)=1$ , the nontrivial sign $\\omega _2(g^m,\\tilde{g}^m)=-1$ stands for a Kramers doublet.",
"The reason is the following.",
"Notice that $(Tg^m)^2=E$ and $\\omega _2(Tg^m,Tg^m)=\\omega _2(Tg^m,\\tilde{g}^m T) =\\omega _2^{-1}(g^m,\\tilde{g}^m)=-1,$ we have $[M(Tg^m)K]^2 = - M[(Tg^m)^2]=-1.$ So the nontrivial sign $\\omega _2(g^m,\\tilde{g}^m)=-1$ corresponds to a Kramers doublet.",
"3), $N$ is even: $N=4Q+2, \\ Q\\in Z.$ The discussion is similar to case (2), except that $\\mu (N/2)=\\mu (2Q+1)$ is not necessarily $\\pm 1$ , it can be complex: $\\mu ^2(2Q+1)=\\omega _2(g^{2Q+1},\\tilde{g}^{2Q+1})=\\pm 1.$ But the conclusion is the same, since we can introduce the following coboundary for $-Q\\le m\\le Q$ : with $\\Omega _1(g^{2m+1})=\\mu ^{-1}(2m+1)\\mu (2Q+1)$ and $\\Omega _1(g^{2m})=\\mu ^{-1}(2m)$ such that $&&\\mu ^{\\prime }(2m+1)=\\mu (2m+1)\\Omega _1(g^{2m+1})=\\mu (2Q+1),\\\\&&\\mu ^{\\prime }(-2m-1)=\\mu (-2m-1)\\Omega _1^{-1}(g^{2m+1})\\\\&&\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ =\\omega _2(g^{2m+1},\\tilde{g}^{2m+1})\\mu ^{-1}(2Q+1),\\\\&&\\mu ^{\\prime }(2m)=\\mu (2m)\\Omega _1(g^{2m})=1,\\\\&&\\mu ^{\\prime }(-2m)=\\mu (-2m)\\Omega _1^{-1}(g^{2m})=\\omega _2(g^{2m},\\tilde{g}^{2m}),\\\\&&\\mu ^{\\prime }(0)=\\mu ^{\\prime }(4Q+2)=1,\\\\&&\\mu ^{\\prime }(2Q+1)=\\mu (2Q+1),$ then $M^{\\prime }(g^m)M^{\\prime }(g^n)=M^{\\prime }(g^{m+n})\\omega ^{\\prime }_2(g^m,g^n),$ where $\\omega ^{\\prime }_2(g^m,g^n)=\\mu ^{\\prime }(m)\\mu ^{\\prime }(n)/\\mu ^{\\prime }(m+n)$ with $\\mu ^{\\prime }(m\\pm (4Q+2))=\\mu ^{\\prime }(m)$ .",
"Notice that all $\\omega _2(g^m,g^n)$ are real numbers which are classified by $\\mathcal {H}^{2}(H,Z_{2})$ .",
"So, similar to case (2), we conclude that, for the group $Z_{4Q+2}\\rtimes Z_2^T$ , $\\mathcal {H}^2(H\\rtimes Z_2^T, U(1))&=&\\mathcal {H}^2(Z_2^T, U(1))\\times \\mathcal {H}^2(H, Z_2)\\nonumber \\\\&=& \\mathbb {Z}_2\\times \\mathcal {H}^1(H, Z_2)\\\\&=& \\mathbb {Z}_2^2.\\nonumber $ Remark: When approaching $U(1)$ by taking the limit $\\lim _{N\\rightarrow \\infty }Z_{N}=U(1)$ , the result depends on if $N$ is even or odd.",
"Actually, careful analysis shows that $\\mathcal {H}^{2}(U(1)\\rtimes Z_{2}^{T},U(1))=\\mathbb {Z}_{2}$ .",
"The reason is the following.",
"Suppose that the generator of projective Rep of $U(1)$ is an Hermitian matrix $N$ , then we can assume $M(U_{\\theta })=e^{iN\\theta }f(\\theta )$ where $U_{\\theta }$ , $\\theta \\in [0,2\\pi )$ is an element in $U(1)$ , and $|f(\\theta )|=1$ to ensure that $M(U_{\\theta })$ is unitary.",
"Obviously $f(0)=1$ .",
"We further assume that $T$ is represented by $M(T)K$ , then the gauge condition $\\omega _{2}(T,\\theta )=\\omega _{2}(-\\theta ,T)=1$ and $TU_{\\theta }=U_{-\\theta }T$ indicate that $M(T)Ke^{iN\\theta }f(\\theta )&=&M(T)e^{-iN^{\\ast }\\theta }f^{\\ast }(\\theta )K \\\\&=&e^{-iN\\theta }f(-\\theta )M(T)K.$ Above equation is true for arbitrary $\\theta $ , so we have $M(T)N^{\\ast }=NM(T)~,~f^{\\ast }(\\theta )=f(-\\theta ).$ Noticing that $\\omega _{2}(\\theta ,-\\theta )=\\frac{f(\\theta )f(-\\theta )}{f(0)}=f^{\\ast }(\\theta )f(\\theta )=1,$ which is under the same condition with the group $Z_{2Q+1}$ .So the classification is purely determined by $\\mathcal {H}^{2}(Z_{2}^{T},U(1))=\\mathbb {Z}_{2}$ .",
"However, for the group $U(1)\\times Z_{2}^{T}$ , we have $TU_{\\theta }=U_{\\theta }T$ and $M(T)Ke^{iN\\theta }f(\\theta )&=&M(T)e^{-iN^{\\ast }\\theta }f^{\\ast }(\\theta )K \\\\&=&e^{iN\\theta }f(\\theta )M(T)K,$ which means $-M(T)N^{\\ast }=NM(T)~,~f^{\\ast }(\\theta )=f(\\theta ).$ So $f(\\theta )$ is real $f(\\theta )=\\pm 1$ , and $\\omega _{2}(\\theta ,\\theta ^{\\prime })=\\frac{f(\\theta )f(\\theta ^{\\prime })}{f(\\theta +\\theta ^{\\prime })}=\\pm 1$ .",
"This is consistent with the discussion in section REF ,where it is shown that all the cocycles $\\omega _{2}(\\theta ,\\theta ^{\\prime })$ can be fixed as real and is classified by $\\mathcal {H}^2(U(1),Z_2)$ .",
"The complete classification of $U(1)\\times Z_{2}^{T}$ is then given by $\\mathcal {H}^2(U(1)\\times Z_{2}^{T},U(1))=\\mathcal {H}^2(Z_{2}^{T},U(1))\\times \\mathcal {H}^2(U(1),Z_2)=\\mathbb {Z}_2^2.$"
],
[
"When $H$ is a direct product of cyclic groups",
"Now suppose the group $H=Z_M\\times Z_N$ has more than one generator, noted as $h_1,h_2, ...$ .",
"For each cyclic subgroup, the argument in the previous section applies, and all cocycles $\\omega _2(h_1^m, h_1^n)$ and $\\omega _2(h_2^m, h_2^n)$ are constrained to be $\\pm 1$ by introducing gauge transformation $\\Omega _1(h_1^m)$ and $\\Omega _1(h_2^m)$ as discussed previously.",
"In the following we will discuss about the cocycles $\\omega _2(h_1^m,h_2^n)$ .",
"In the coboundary $(d\\Omega _1)(h_2^n,h_1^m) = \\Omega _1(h_1^m) \\Omega _1(h_2^n)\\Omega _1^{-1}(h_2^n h_1^m),$ if we set $\\Omega _1(h_2^n h_1^m)= \\Omega _1(h_2^n) \\Omega _1(h_1^m)\\omega _2( h_2^n,h_1^m)$ , where the values of $\\Omega _1(h_2^n)$ and $\\Omega _1(h_1^m)$ have been fixed in previous gauge fixing processes, then $\\omega _2( h_2^n,h_1^m)$ can be fixed: $\\omega _2^{\\prime }( h_2^n,h_1^m) =\\omega _2( h_2^n,h_1^m) (d\\Omega _1)(h_2^n,h_1^m) =1.$ From Eqs.",
"(REF ) and (REF ), we have $\\omega _2(h_1^m, h_2^n)&=&\\omega _2( h_2^n,h_1^m) \\omega _2(h_1^m,\\tilde{h}_1^m) \\omega _2(h_2^n,\\tilde{h}_2^n) \\\\&& \\cdot \\omega _2(h_1^mh_2^n,\\widetilde{h_1^mh_2^n})\\\\&=&\\omega _2(h_1^m,\\tilde{h}_1^m) \\omega _2(h_2^n,\\tilde{h}_2^n) \\omega _2(h_1^mh_2^n,\\widetilde{h_1^mh_2^n})\\\\&=&\\pm 1.$ Thus all the components with the form $\\omega _2(h_1^m, h_2^n)$ and $\\omega _2(h_2^m, h_1^n)$ are set to be real numbers.",
"Owing to the cocycle equations $(d\\omega _2)(h_1^m,h_2^n,h_2^p)={\\omega _2(h_2^n,h_2^p)\\omega _2(h_1^m,h_2^{n+p})\\over \\omega _2(h_1^mh_2^n,h_2^p)\\omega _2(h_1^m,h_2^n)}=1$ , we have $\\omega _2(h_1^mh_2^n,h_2^p)={\\omega _2(h_2^n,h_2^p)\\omega _2(h_1^m,h_2^{n+p})\\over \\omega _2(h_1^m,h_2^n)}$ , which is a real number.",
"Similarly, owning to cocycle equations, all the components $\\omega _2(h_1^mh_2^n,h_1^qh_2^p)$ take values $\\pm 1$ under above gauge fixing.",
"Now it is clear that the cocycles $\\omega _2(g_1,g_2)$ (where $g_1,g_2\\in H$ ) are classified by $\\mathcal {H}^2(H,Z_2)$ .",
"This conclusion can be generalized to the case where $H$ is a direct product of more cyrclic groups $H=Z_M\\times Z_N\\times ...$ .",
"In conclusion, the complete classification is given by $\\mathcal {H}^2(H\\rtimes Z_2^T, U(1))&=&\\mathcal {H}^2(Z_2^T, U(1))\\times \\mathcal {H}^2(H, Z_2)\\nonumber \\\\&=& \\mathbb {Z}_2\\times \\mathcal {H}^2(H, Z_2)$"
],
[
"The case of complex cocycles",
"For a general group $G=H\\rtimes Z_2^T$ , the cocycles may not be tuned into real numbers.",
"For example, $\\mathcal {H}^2(Z_3\\times (Z_3\\rtimes Z_2^T), U(1))=\\mathbb {Z}_6$ , 2 classes of the cocycles can be set as real numbers, and the remaining 4 classes are complex numbers.",
"It can be shown that for $G=Z_m\\times (Z_n\\rtimes Z_2^T)$ , $ &&\\mathcal {H}^2(G,U(1)) = \\mathcal {H}^2(Z_m,Z_2)\\times \\mathcal {H}^2(Z_n,Z_2)\\\\&&\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\times \\mathcal {H}^2(H,U(1))\\times \\mathcal {H}^2(Z_2^T,U(1)),$ where $H=Z_m\\times Z_n$ ."
],
[
"The Dimension of irreducible projective Rep is a multiple of the period of its factor system",
"For a symmetry group $G$ , its irreducible projective Rep matrix is $M(g)$ , then $M(g_{1})M_{s(g_{1})}(g_{2})=\\omega _{2}(g_{1},g_{2})M(g_{1}g_{2}).$ We denote the determinant of Rep matrix as following $ &&\\det [ M(g_{1})]=\\varepsilon _{1},\\\\&&\\det [M(g_{2})]=\\varepsilon _{2},\\\\&&\\det [M(g_{1}g_{2})]=\\varepsilon _{12}.$ From (REF ) we have $\\varepsilon _{1}\\cdot (\\varepsilon _{2})_{s(g_1)} = \\omega _{2}^{D}(g_{1},g_{2})\\varepsilon _{12}$ , where $(\\varepsilon _{2})_{s(g_1)}= \\varepsilon _{2}$ if $g_1$ is unitary and $(\\varepsilon _{2})_{s(g_1)} = \\varepsilon _{2}^*$ if $g_1$ is anti-unitary.",
"This yields $\\omega _{2}(g_{1},g_{2}) = e^{i\\frac{2\\pi n}{D}}\\times \\left[\\frac{\\varepsilon _{1}\\cdot (\\varepsilon _{2})_{s(g_1)}}{\\varepsilon _{12}}\\right]^{\\frac{1}{D}},$ where $D$ is the lowest dimension of the irreducible projective Reps corresponding to the factor system $\\omega _2(g_1,g_2)$ and $n=0,1,\\cdots ,D-1$ .",
"We may choose a coboundary $\\Omega _{2}(g_{1},g_{2})=\\left[\\frac{\\varepsilon _{12}}{\\varepsilon _{1}\\cdot (\\varepsilon _{2})_{s(g_1)}}\\right]^{\\frac{1}{D}}$ to get a different factor system $\\omega _{2}^{\\prime }(g_{1},g_{2})= \\Omega _{2}(g_{1},g_{2})\\omega _{2}(g_{1},g_{2})=e^{i\\frac{2\\pi n}{D}},$ which satisfies $ \\omega _{2}^{\\prime }(g_{1},g_{2})^D=1.$ Now we define the period $P$ of a factor system $\\omega _2(g_1,g_2)$ as the smallest positive integer number such that $\\omega _2(g_1,g_2)^P$ is a trivial class of 2-cocycle.",
"$P$ can be read out from the group structure of the classification of 2-cocycles, i.e.",
"the second group cohomology of $G$.",
"For example, if the second group cohomology of $G$ is $\\mathcal {H}^2(G, U(1))=\\mathbb {Z}_m\\times \\mathbb {Z}_n$ , and the cocycle $\\omega _2(g_1,g_2)$ belongs to the $(a,b)$ th class, where $a,b\\in Z$ and $1\\le a\\le m$ and $1\\le b\\le n$ and the $(m,n)$ th class stands for the trivial class.",
"Then the period of cocycle $\\omega _2(g_1,g_2)$ is $P=\\left[ {[a,m]\\over a}, {[b,n]\\over b}\\right],$ where $[a,m]$ is the least common multiple of $a$ and $m$ .",
"From the relation (REF ), we conclude that the dimension $D$ of any projective Rep with factor system $\\omega _2(g_1,g_2)$ must be integer times of $P$ , $D= NP,$ where $N\\in Z$ and $N\\ge 1$ ."
],
[
"Completeness of irreducible bases",
"Since $ |g^r\\rangle =\\hat{g}|\\alpha _i^{(\\nu )}\\rangle =\\sum _{j=1}^{n_\\nu }M(g)_{ji}K_{s(g)}|\\alpha _j^{(\\nu )}\\rangle ,$ the complete relation $\\sum _g |g^{r}\\rangle \\langle g^{r}|\\propto I$ can be written as $\\sum _{g}\\sum _{k,j=1}^{n_\\nu }M(g)_{ki}|\\alpha _{k}^{\\nu }\\rangle \\langle \\alpha _{j}^{\\nu }|M^{\\dagger }(g)_{ij}\\propto 1,$ which is equivalent to $\\sum _{g}M(g)_{ki}M^{\\dag }(g)_{ij}\\propto \\delta _{kj},$ where we assume $M(g)$ represents equivalent irreducible unitary projective Reps. To prove (REF ), we introduce the matrix $ Y&=&\\sum _{g}M(g)K_{s(g)}X K_{s(g)}M^\\dag (g)\\nonumber \\\\&=&\\sum _{g}M(g)X M^\\dag (g),$ where the matrix $X$ is real.",
"For any $g_n\\in G$ , from (REF ), the matrix $Y$ satisfies $ &&M(g_{n})K_{s(g_n)}Y\\nonumber \\\\&=&\\sum _{g}M(g_{n})K_{s(g_n)}M(g)K_{s(g)}X K_{s(g)}M^\\dag (g) \\nonumber \\\\&=&\\sum _{g}M(g_{n}g)\\omega _2(g_n,g)K_{s(g_n g)}X K_{s(g)}M^\\dag (g) K_{s(g_n)}\\nonumber \\\\& &\\cdot M^\\dag (g_n)M(g_n)K_{s(g_n)}\\nonumber \\\\&=&\\sum _{g}M(g_{n}g)\\omega _2(g_n,g)K_{s(g_n g)}X\\nonumber \\\\& &\\cdot [M(g_n)K_{s(g_n)}M(g)K_{s(g)}]^{\\dag }M(g_n)K_{s(g_n)}\\nonumber \\\\&=&\\sum _{g}M(g_{n}g)\\omega _2(g_n,g)K_{s(g_n g)}X \\nonumber \\\\& &\\cdot [\\omega _2(g_n,g)M(g_ng)K_{s(g_n g)}]^\\dag M(g_n)K_{s(g_n)}\\nonumber \\\\&=&\\sum _{g}M(g_{n}g)\\omega _2(g_n,g)K_{s(g_n g)}X K_{s(g_n g)}\\omega _2^\\ast (g_n,g)\\nonumber \\\\& &\\cdot M^\\dag (g_n g)M(g_n)K_{s(g_n)}\\nonumber \\\\&=&\\sum _{g}M(g_{n}g)K_{s(g_n g)}X K_{s(g_n g)}M^\\dag (g_n g)\\cdot M(g_n)K_{s(g_n)}\\nonumber \\\\&=&YM(g_{n})K_{s(g_n)},$ which means $ [\\hat{g}_n, Y]=0$ .",
"Supposing $X$ is a diagonal matrix with only one nonzero element $X_{ii}=1$ , then from Eq.",
"(REF ), $Y$ is an Hermitian matrix with $ Y_{kj}&=&\\sum _{g}M(g)_{ki}M^{\\dag }(g)_{ij},\\\\$ Since the projective representation $M(g_n)K_{s(g_n)}$ is irreducible, according to the generalized Schur lemma, $Y$ must be a constant matrix, i.e.",
"$Y=\\lambda I$ , where $\\lambda $ is a real number.",
"Therefore, $Y_{kj}=\\lambda \\delta _{kj}$ , and the relation (REF ) is proved."
],
[
"Two relations between class operators of anti-unitary groups",
"Here we prove the following two relations between the class operators.",
"(1) $C_i=\\bar{C}_i$ for anti-unitary groups, where $C_i$ is the class operator corresponding to the unitary element $g_i$ .",
"Suppose $G$ is anti-unitary, $g_3\\in G$ is an arbitrary element, and $C_1$ is a class operator corresponding to a unitary element $\\hat{g}_1$ .",
"Then $ \\bar{C}_1|g_3\\rangle &=& \\left[ \\sum _{g_2\\in G}\\hat{\\bar{g}}_2\\hat{\\bar{g}}_1\\hat{\\bar{g}}_2^{-1}\\right] \\hat{\\bar{g}}_3|E\\rangle \\\\&=&\\sum _{g_2\\in G}e^{-is({g_3})\\theta _2(g_2^{-1},g_2)}\\left\\lbrace \\left[(\\hat{\\bar{g}}_2\\hat{\\bar{g}}_1)\\widehat{\\overline{g_2^{-1}}}\\right] \\hat{\\bar{g}}_3\\right\\rbrace |E\\rangle $ $ &=&\\sum _{g_2\\in G}e^{-is({g_3})\\theta _2(g_2^{-1},g_2)}e^{is({g_2g_3})\\theta _2(g_1,g_2)}\\nonumber \\\\&&\\times e^{is({g_3})\\theta _2(g_2^{-1},g_1g_2)} e^{i\\theta _2(g_3,g_2^{-1}g_1g_2)}|g_3g_2^{-1}g_1g_2\\rangle ,$ here we have used $\\widehat{\\overline{g_2^{-1}}}|g_3\\rangle =e^{is({g_3})\\theta _2(g_2^{-1},g_2)} \\hat{\\bar{g}}_2^{-1}|g_3\\rangle $ .",
"On the other hand, from $ \\widehat{g_{2}^{-1}}\\hat{g}_{2}=M(g_{2}^{-1})M_{s(g_2)}(g_2)=e^{i\\theta _{2}(g_{2}^{-1},g_2)},\\hat{g}_{2}=M(g_2)K_{s(g_2)},$ we get $ \\hat{g}_{2}^{-1}&=&M_{s(g_2)}^{-1}(g_2)K_{s(g_2)}=e^{-i\\theta _{2}(g_{2}^{-1},g_2)}M(g_{2}^{-1})K_{s(g_2)}\\\\&=&e^{-i\\theta _{2}(g_{2}^{-1},g_2)}\\widehat{g_{2}^{-1}}.$ So, we have $ C_1|g_3\\rangle &=& \\left[ \\sum _{g_2\\in G}\\hat{g}^{-1}_2\\hat{ g}_1\\hat{ g}_2\\right] \\hat{ g}_3|E\\rangle \\nonumber \\\\&=& \\hat{ g}_3\\left[ \\sum _{g_2\\in G}\\hat{g}^{-1}_2 \\hat{ g}_1\\hat{ g}_2\\right]|E\\rangle \\nonumber \\\\ &=& \\hat{ g}_3\\left[ \\sum _{g_2\\in G} e^{-i\\theta _2(g_2^{-1},g_2)}\\widehat{g^{-1}_2}\\hat{ g}_1\\hat{ g}_2\\right]|E\\rangle \\\\&=&\\sum _{g_2\\in G}e^{-is({g_3})\\theta _2(g_2^{-1},g_2)}e^{i\\theta _2(g_3,g_2^{-1}g_1g_2)}\\nonumber \\\\&&\\times e^{is({g_3})\\theta _2(g_2^{-1},g_1g_2)} e^{is({g_2g_3})\\theta _2(g_1,g_2)}|g_3g_2^{-1}g_1g_2\\rangle .$ Therefore, $C_1=\\bar{C}_1$ .",
"(2) $i(C_i^H-C_{\\tilde{i}}^H) = (\\bar{C}_i^H-\\bar{C}_{\\tilde{i}}^H)S$ , where $S=i(P_u-P_a)$ .",
"Under the gauge transformation $\\Omega _1(g)=e^ { i{\\pi \\over 2}(\\delta _{g,g_{1}} -\\delta _{g,\\tilde{g}_{1}})} $ ($g_1$ is unitary), from the relations (REF ) and (REF ), the regular projective Reps will be changed into $ &&M(g_1) \\rightarrow U^\\dag iM(g_1) U, \\\\&&M(\\tilde{g}_1) \\rightarrow -U^\\dag iM(\\tilde{g}_1) U, \\\\&&M(\\bar{g}_1) \\rightarrow U^\\dag M(\\bar{g}_1) [i(P_u-P_a)] U=U^\\dag M(\\bar{g}_1) S U, \\\\&&M(\\bar{\\tilde{g}}_1) \\rightarrow U^\\dag M(\\bar{\\tilde{g}}_1) [-i(P_u-P_a)]U= -U^\\dag M(\\bar{\\tilde{g}}_1) SU, \\\\&&M(g_2) \\rightarrow U^\\dag M(g_2) U, \\ \\ \\ g_2\\ne g_1,\\tilde{g}_1\\\\&&M(\\bar{g}_2) \\rightarrow U^\\dag M(\\bar{g}_2) U, \\ \\ \\ g_2\\ne g_1,\\tilde{g}_1$ where $U_{g_i,g_j}=\\delta _{g_i,g_j}\\Omega _1(g_i)$ is a diagonal matrix.",
"Consequently, the class operators $C_1= C_1^H + C_{\\tilde{1}}^H,\\ \\bar{C}_1=\\bar{C}_1^H + \\bar{C}_{\\tilde{1}}^H$ are transformed into $ &&C_1\\rightarrow U^\\dag i(C_1^H - C_{\\tilde{1}}^H)U,$ $ &&\\bar{C}_1\\rightarrow U^\\dag (\\bar{C}_1^H - \\bar{C}_{\\tilde{1}}^H)SU.$ After the gauge transformation, above two quantities are still equal, namely, $U^\\dag i(C_1^H - C_{\\tilde{1}}^H)U= U^\\dag (\\bar{C}_1^H - \\bar{C}_{\\tilde{1}}^H)SU$ .",
"This yields $i(C_1^H - C_{\\tilde{1}}^H) = (\\bar{C}_1^H - \\bar{C}_{\\tilde{1}}^H)S.$ Table: Non-Abelian anti-unitary group S 4 T S_{4}^{T}"
]
] | 1605.05805 |
[
[
"Full characterization of generalized bent functions as (semi)-bent\n spaces, their dual, and the Gray image"
],
[
"Abstract In difference to many recent articles that deal with generalized bent (gbent) functions $f:\\mathbb{Z}_2^n \\rightarrow \\mathbb{Z}_q$ for certain small valued $q\\in \\{4,8,16 \\}$, we give a complete description of these functions for both $n$ even and odd and for any $q=2^k$ in terms of both the necessary and sufficient conditions their component functions need to satisfy.",
"This enables us to completely characterize gbent functions as algebraic objects, namely as affine spaces of bent or semi-bent functions with interesting additional properties, which we in detail describe.",
"We also specify the dual and the Gray image of gbent functions for $q=2^k$.",
"We discuss the subclass of gbent functions which corresponds to relative difference sets, which we call $\\mathbb{Z}_q$-bent functions, and point out that they correspond to a class of vectorial bent functions.",
"The property of being $\\mathbb{Z}_q$-bent is much stronger than the standard concept of a gbent function.",
"We analyse two examples of this class of functions."
],
[
"Introduction",
"Let ${\\mathbb {V}}_n$ be an $n$ -dimensional vector space over ${\\mathbb {F}}_2$ , and let $\\mathcal {B}_n$ denote the set of Boolean functions from ${\\mathbb {V}}_n$ to ${\\mathbb {F}}_2$ .",
"The Walsh-Hadamard transform of $f\\in \\mathcal {B}_n$ at a point $u \\in {\\mathbb {V}}_n$ is defined by $\\mathcal {W}_{f}(u)=\\sum _{x\\in {\\mathbb {V}}_n}(-1)^{f(x)\\oplus u\\cdot x},$ where “$\\cdot $ ” denotes an inner product on ${\\mathbb {V}}_n$ .",
"When ${\\mathbb {V}}_n = {\\mathbb {F}}_2^n$ we may take the dot product, in the case of a finite field with $2^n$ elements ${\\mathbb {F}}_{2^n}$ , the standard inner product is $ x \\cdot y = {\\rm Tr_n}(xy)$ , where ${\\rm Tr_n}(z)$ denotes the absolute trace of $z$ .",
"Note that the characters of ${\\mathbb {V}}_n\\times {\\mathbb {F}}_2$ are $\\chi _{a,u}(x,y) = (-1)^{ay \\oplus u \\cdot x}$ , $a\\in \\lbrace 0,1\\rbrace $ , $u\\in {\\mathbb {V}}_n$ .",
"Hence $\\mathcal {W}_{f}(u) = \\chi _{1,u}(D)$ , where $D=\\lbrace (x,f(x))\\,:\\,x\\in {\\mathbb {V}}_n\\rbrace $ is the graph of $f$ .",
"A function $f\\in \\mathcal {B}_n$ is called bent if $\\mathcal {W}_{f}(u)$ has absolute value $2^{n/2}$ for all $u\\in {\\mathbb {V}}_n$ , i.e., $|\\chi _{a,u}(D)| = 2^{n/2}$ for all characters which are nontrivial on $\\lbrace 0\\rbrace \\times {\\mathbb {F}}_2$ ($a\\ne 0$ ), hence $f$ can be identified with a relative difference set $D$ in ${\\mathbb {V}}_n\\times {\\mathbb {F}}_2$ , see [14], [23].",
"Since $\\mathcal {W}_{f}(u)$ is an integer, for a bent function we have $\\mathcal {W}_{f}(u) = 2^{n/2}(-1)^{f^*(u)}$ for a Boolean function $f^*\\in \\mathcal {B}$ , called the dual of $f$ , which then is also bent.",
"Obviously, Boolean bent functions only exist when $n$ is even.",
"When $n$ is odd, a semi-bent function is defined as a function $f\\in \\mathcal {B}_n$ for which $\\mathcal {W}_f(u) \\in \\lbrace \\pm 2^{\\frac{n+1}{2}}, 0\\rbrace $ for all $u\\in {\\mathbb {V}}_n$ .",
"A function $f\\in \\mathcal {B}_n$ is called $s$ -plateaued if its Walsh spectrum only takes three values 0 and $\\pm 2^{\\frac{n+s}{2}}$ ($0\\le s\\le n$ ).",
"Note that $n$ and $s$ must have the same parity.",
"Let $K$ be a subset of ${\\mathbb {V}}_n$ .",
"By $\\phi _{K}$ we denote a Boolean function in $\\mathcal {B}_n$ whose value in $x$ is 1 if $x\\in K$ and 0 if $x\\notin K$ .",
"The function $\\phi _E$ is called the indicator of $K$.",
"Many more variants of bent functions, like bent functions in odd characteristic, vectorial bent functions from ${\\mathbb {F}}_p^n$ to ${\\mathbb {F}}_p^m$ , negabent functions, bent$_4$ functions, all corresponding to relative difference sets in respective groups, have been investigated.",
"The reader is referred to, for instance, the articles [4], [6], [13], [16], [19], [25] and the recent survey article [17].",
"For a very general viewpoint considering bent functions over arbitrary abelian groups, we refer to [15].",
"For a positive integer $q$ let ${\\mathbb {Z}}_q$ be the ring of integers modulo $q$ .",
"We call a function $f$ from ${\\mathbb {V}}_n$ to ${\\mathbb {Z}}_q$ a generalized Boolean function, and denote the set of generalized Boolean functions from ${\\mathbb {V}}_n$ to ${\\mathbb {Z}}_q$ by $\\mathcal {GB}_n^q$ .",
"If $q=2$ , then $f$ is Boolean, and $\\mathcal {GB}_n^2 = \\mathcal {B}_n$ .",
"Having applications of functions from ${\\mathbb {V}}_n$ to ${\\mathbb {Z}}_4$ in code-division multiple access systems in mind, in [18] Schmidt introduced a class of functions which further on were called generalized bent (gbent).",
"A function $f\\in \\mathcal {GB}^q_n$ for which the generalized Walsh-Hadamard transform (GWHT) at a point $u\\in {\\mathbb {V}}_n$ defined as the complex valued function $\\mathcal {H}^{(q)}_{f}(u)=\\sum _{x\\in {\\mathbb {V}}_n}\\zeta _q^{f(x)}(-1)^{ u\\cdot x},$ where $\\zeta _q=e^{2\\pi i/q}$ (or any other complex $q$ th-primitive root of unity), has absolute value $2^{n/2}$ for all $u\\in {\\mathbb {V}}_n$ , is called a generalized bent function.",
"Note that when $f$ is Boolean, then $\\mathcal {H}^{(2)}_{f}(u) = \\mathcal {W}_f(u)$ .",
"Currently there is a lot of research activity regarding the construction and analysis of gbent functions, see for instance [5], [7], [8], [9], [12], [18], [20], [22].",
"The quaternary $q=4$ and octal case $q=8$ were investigated in [18], [20] and [12], [21], respectively.",
"Quite recently [8], a complete characterization of gbent functions was also given for $q=16$ .",
"In these cases gbent functions were fully characterized by specifying both the related necessary and sufficient conditions.",
"The first general characterization of gbent functions, in terms of the coordinate functions for any $q$ even and $n$ both even/odd, is given in [5].",
"More precisely, for $2^{k-1}<q\\le 2^k,$ to any generalized function $f:\\mathbb {Z}^n_2\\rightarrow \\mathbb {Z}_q,$ one may associate a unique sequence of Boolean functions $a_i\\in \\mathcal {B}_n$ ($i=0,1,\\ldots ,h-1$ ) such that $f(x)=a_0(x)+2a_1(x)+2^2a_2(x)+\\ldots +2^{k-1}a_{k-1}(x),\\; \\forall x\\in \\mathbb {Z}^n_{2}$ , where $a_i$ are then called the coordinate functions of $f$ .",
"The gbent conditions derived in [5] were only sufficient and whether these are also necessary was left as an open problem.",
"Necessary conditions for a function $f\\in \\mathcal {GB}_n^{2^k}$ to be gbent are in [8] when $n$ is even and also for odd $n$ in [10].",
"In this paper we are interested in gbent functions in $\\mathcal {GB}_n^{q}$ where $q=2^k$ and $k>1$ is a positive integer.",
"In difference to the above mentioned results that only considered gbent functions for $q \\in \\lbrace 4,8,16\\rbrace $ , we give a complete characterization of these functions for both $n$ even and odd and for any $q=2^k$ .",
"We start with with both, necessary and sufficient conditions their coordinate functions need to satisfy.",
"These conditions are equivalent to those very recently published online in [24].",
"Notably we then describe gbent functions as algebraic objects, a characterization which goes far beyond the conventional descriptions in terms the Walsh transforms of linear combinations of the coordinate functions, which in accordance with the terminilogy for vectorial bent function we call the component functions of the gbent function.",
"We show that gbent functions correspond to affine spaces of bent functions when $n$ is even and semi-bent functions when $n$ is odd, with certain interesting additional properties, which we precisely describe.",
"Employing conventional equivalence, we show that gbent functions and affine spaces of bent (semi-bent) functions with these properties, are identical objects.",
"These results essentially completely resolve the case of gbent functions from ${\\mathbb {V}}_n$ to ${\\mathbb {Z}}_{2^k}$ , using the approach based on Hadamard matrices introduced in [5].",
"We recall that in the case of $q=2^k$ we always have $\\mathcal {H}_f^{(2^k)}(u) = 2^{n/2}\\zeta _{2^k}^{f^*(u)}$ , (except for the case that $n$ is odd and $q=4$ ), for a function $f^*\\in \\mathcal {GB}_n^{2^k}$ , which we call the dual of $f$ , see [8].",
"As pointed out in [9], $f^*$ is also a gbent function.",
"In this direction, we completely specify the dual and Grey image of any gbent function when $n$ is even and for any $q=2^k$ .",
"The case $n$ being odd appears to be harder to deal with the approach based on Hadamard matrices (as used in this article) and it is left as an open problem.",
"Regarding a natural connection of gbent functions and the so-called Gray images, it is shown that the Gray map of a gbent function $f \\in \\mathcal {GB}_n^{2^k}$ is $(k-1)$ -plateaued if $n$ is even, and $(k-2)$ -plateaued if $n$ is odd.",
"This generalizes the results on the Gray map given in [8], [20] for $k=2,3$ and 4.",
"We emphasize here that a gbent function conceptually does not correspond to a bent function, since in the definition of GWHT not all characters of ${\\mathbb {V}}_n\\times {\\mathbb {Z}}_{2^k}$ are considered.",
"Thus, in general, a gbent function does not give rise to a relative difference set.",
"For this reason we extend the definition and introduce the term of a ${\\mathbb {Z}}_q$ -bent function.",
"We call a function $f\\in \\mathcal {GB}_n^{2^k}$ a ${\\mathbb {Z}}_q$ -bent function if $\\mathcal {H}^{(q)}_{f}(a,u)=\\sum _{x\\in {\\mathbb {V}}_n}\\zeta _q^{af(x)}(-1)^{ u\\cdot x}$ has absolute value $2^{n/2}$ for all $u\\in {\\mathbb {V}}_n$ and all nonzero $a\\in {\\mathbb {Z}}_{2^k}$ .",
"Whereas there are several constructions of gbent functions, a class of ${\\mathbb {Z}}_q$ -bent functions seems not to be easy to obtain.",
"For a construction using (partial) spreads, we refer to [9].",
"This article is organized as follows.",
"In Section we recall some preliminary results which are used later.",
"A necessary and sufficient condition, given in terms of Hadamard matrices, for a function $f\\in \\mathcal {GB}_n^{2^k}$ to be gbent is given in Section .",
"In Section REF , we use these conditions to completely characterize gbent functions as affine (semi-)bent spaces with certain properties.",
"We introduce the term ${\\mathbb {Z}}_q$ -bent functions for those gbent functions that correspond to relative difference sets in ${\\mathbb {V}}_n\\times {\\mathbb {Z}}_q$ , $q=2^k$ , in Section .",
"We describe some of their properties and analyse two explicit examples of this class of functions.",
"In Section we specify the dual and Gray map of gbent functions $f\\in \\mathcal {GB}_n^{2^k}$ ."
],
[
"Preliminaries",
"A $(1,-1)$ -matrix $H$ of order $p$ is called a Hadamard matrix if $HH^{T}=pI_p,$ where $H^{T}$ is the transpose of $H$ , and $I_p$ is the $p\\times p$ identity matrix.",
"A special kind of Hadamard matrix is the Sylvester-Hadamard or Walsh-Hadamard matrix, denoted by $H_{2^{k}},$ which is constructed recursively using Kronecker product $H_{2^{k}}=H_2\\otimes H_{2^{k-1}},$ where $H_1=(1);\\hspace{11.38092pt}H_2=\\left(\\begin{array}{cc}1 & 1 \\\\1 & -1 \\\\\\end{array}\\right);\\hspace{11.38092pt}H_{2^k}=\\left(\\begin{array}{cc}H_{2^{k-1}} & H_{2^{k-1}} \\\\H_{2^{k-1}} & -H_{2^{k-1}} \\\\\\end{array}\\right).$ For technical reasons we start the row and column index of $H_{2^k}$ with 0, and we denote the $r$ -th row of $H_{2^k}$ by $H_{2^k}^{(r)}$ , $0\\le r\\le 2^k-1$ .",
"To an integer $j=\\sum _{i=0}^{k-1}j_i2^i$ , $0\\le j\\le 2^k-1$ , we assign $z_j = (j_0,j_1,\\ldots ,j_{k-1})\\in {\\mathbb {F}}_2^k$ , which also implies an ordering of the elements of ${\\mathbb {F}}_2^k$ .",
"We summarize some properties of the Sylvester-Hadamard matrix in the following lemma.",
"The first one follows from the recursive definition of $H_{2^k}$ , the second is the well-known property that each row of $H_{2^k}$ is the evaluation of some linear function.",
"The third one may be less well known, hence we provide the proof of this property.",
"Lemma 1 (i) Each row of $H_{2^k}$ is uniquely determined by the signs of the entries at positions $2^s$ , $s=0,1,\\ldots ,k-1$ .",
"(ii) Let $z_j = (j_0,j_1,\\ldots ,j_{k-1})\\in {\\mathbb {F}}_2^k$ , where $j=\\sum _{i=0}^{k-1}j_i2^i$ , $0\\le j\\le 2^k-1$ .",
"Then $ H_{2^k}^{(r)} = ((-1)^{z_0\\cdot z_r}, (-1)^{z_1\\cdot z_r}, \\ldots , (-1)^{z_{2^k-1}\\cdot z_r}).",
"$ (iii) Let $W = (w_0,w_1,\\ldots ,w_{2^k-1})$ , where $w_i = \\pm 1$ , $0\\le i\\le 2^k-1$ .",
"Then $W = \\pm H_{2^k}^{(r)}$ for some $r\\in \\lbrace 0,\\ldots ,2^k-1\\rbrace $ if and only if for any four distinct integers $j,c,l,v \\in \\lbrace 0,\\ldots ,2^k-1\\rbrace $ such that $z_j \\oplus z_c\\oplus z_l\\oplus z_v = {\\bf 0}$ we have $w_jw_c = w_lw_v.$ Proof of (iii).",
"Let $W = (w_0,w_1,\\ldots ,w_{2^k-1}) = \\pm H_{2^k}^{(r)}$ for some (fixed) $r\\in \\lbrace 0,\\ldots ,2^k-1\\rbrace $ , and let $j,c,l,v\\in \\lbrace 0,\\ldots ,2^k-1\\rbrace $ be arbitrary distinct integers such that $z_j\\oplus z_c\\oplus z_l\\oplus z_v={\\bf 0}$ .",
"By (ii), $ H^{(r)}_{2^k}=((-1)^{z_r\\cdot z_0},(-1)^{z_r\\cdot z_1},\\ldots ,(-1)^{z_r}\\cdot z_{2^k-1}).",
"$ Hence relation $(\\ref {wwww})$ can be written as $ (-1)^{z_r\\cdot z_j}(-1)^{z_r\\cdot z_c}=(-1)^{z_r\\cdot z_{l}}(-1)^{z_r\\cdot z_{v}}, $ or equivalently $ (-1)^{z_r\\cdot (z_j\\oplus z_c \\oplus z_l \\oplus z_v)} = 1, $ which is satisfied for $z_j,z_c,z_l,z_v$ with $z_j\\oplus z_c\\oplus z_l\\oplus z_v={\\bf 0}$ .",
"Suppose conversely that $(\\ref {wwww})$ holds for all $j,c,l,z,v$ with $z_j\\oplus z_c\\oplus z_l\\oplus z_v={\\bf 0}$ .",
"Then, to show that $W = \\pm H_{2^k}^{(r)}$ for some $r$ , $0\\le r\\le 2^k-1$ , we proceed by induction on $k$ .",
"Trivially it holds for $k=1$ , since $\\pm H_2^{(r)}$ , $r=0,1$ , covers all possible combinations for $(w_0,w_1)$ .",
"For $k=2$ , we first notice that all solutions of the equality $w_0w_1=w_2w_3$ with $w_i = \\pm 1, i=0,1,2,3$ , are the quadruples $(w_0,w_1,w_2,w_3)$ containing an even number of $-1$ s. As it is easy to see, all such quadruples $W$ are of the form $W = (w_0,w_1,\\pm (w_0,w_1))$ , hence equal to $\\pm H_4^{(r)}$ for some $r\\in \\lbrace 0,1,2,3\\rbrace $ .",
"Before we continue with the induction proof, we also add the argument for $k=3$ .",
"With the above argument applied to the quadruples $(4,5,6,7)$ and $(0,1,4,5)$ , we get $(w_6,w_7) = \\pm (w_4,w_5)$ and $(w_4,w_5) = \\pm (w_0,w_1)$ .",
"Consequently, $ \\nonumber (w_0,\\ldots ,w_7) &=& (w_0,w_1,\\pm (w_0,w_1),\\pm (w_0,w_1,\\pm (w_0,w_1))) \\\\ \\nonumber &=&\\pm (H^{(d)}_{2^2},\\pm H^{(d)}_{2^2})=\\pm H^{(r)}_{2^3},$ for some $0 \\le r\\le 7$ .",
"Now suppose that the following holds for a tuple $W = (w_0,w_1,\\ldots ,w_{2^{k-1}-1})$ of length $2^{k-1}$ with entries in $\\lbrace -1,1\\rbrace $ : If for all $0\\le j < c < l < v\\le 2^{k-1}-1$ with $z_j \\oplus z_c \\oplus z_l \\oplus z_v = {\\bf 0}$ we have $w_jw_c = w_lw_v$ , then $W = \\pm H_{2^{k-1}}^{(r)}$ for some $r \\in \\lbrace 0,1,\\ldots ,2^{k-1}-1\\rbrace $ .",
"Let now $W = (w_0,w_1,\\ldots ,w_{2^k-1})$ , $w_i = \\pm 1$ , $i=0,1,\\ldots ,2^k-1$ , such that $w_jw_c = w_lw_v$ for all $0\\le j < c < l < v\\le 2^k-1$ with $z_j \\oplus z_c \\oplus z_l \\oplus z_v = {\\bf 0}$ .",
"By induction hypothesis, we then have $(w_0,w_1,\\ldots ,w_{2^{k-1}-1}) = \\pm H_{2^{k-1}}^{(r)}$ and $(w_{2^{k-1}},w_{2^{k-1}+1},\\ldots ,w_{2^k-1}) = \\pm H_{2^{k-1}}^{(\\bar{r})}$ for some $r,\\bar{r} \\in \\lbrace 0,1,\\ldots ,2^{k-1}-1\\rbrace $ .",
"We have to show that $\\bar{r} = r$ , or equivalently $w_{2^{k-1}+j} = w_j$ , $j=0,1,\\ldots , 2^{k-1}-1$ , or $w_{2^{k-1}+j} = -w_j$ , $j=0,1,\\ldots , 2^{k-1}-1$ .",
"By (i), it is sufficient to show that $w_{2^{k-1}+2^s} = w_{2^s}$ , $s=0,1,\\ldots ,k-2$ , or $w_{2^{k-1}+2^s} = -w_{2^s}$ , $s=0,1,\\ldots ,k-2$ .",
"We consider the quadruples $(j,c,l,v) = (0,2^s,2^{k-1},2^{k-1}+2^s)$ , $s=0,1,\\ldots ,k-2$ , for which $z_j \\oplus z_c \\oplus z_l \\oplus z_v = 0$ always holds.",
"Since they satisfy $(\\ref {wwww})$ , either $w_0,w_{2^s},w_{2^{k-1}},w_{2^{k-1}+2^s}$ have all the same sign, or exactly two of them are negative.",
"Consequently, if $w_0 = w_{2^{k-1}}$ , then we must have $w_{2^s} = w_{2^{k-1}+2^s}$ , $s=0,1,\\ldots ,k-2$ , and if $w_0 = -w_{2^{k-1}}$ , then $w_{2^s} = -w_{2^{k-1}+2^s}$ , $s=0,1,\\ldots ,k-2$ .",
"$\\Box $ In what follows we derive and recall some basic results on gbent functions, which are proved useful in the sequel.",
"Lemma 2 Let $k\\ge 3$ .",
"Then $\\sqrt{2}\\zeta _{2^k}^j$ is uniquely represented in ${\\mathbb {Q}}(\\zeta _{2^k})$ as $ \\sqrt{2}\\zeta _{2^k}^j = \\pm \\zeta _{2^k}^{J_1} \\pm \\zeta _{2^k}^{J_2} \\in {\\mathbb {Q}}(\\zeta _{2^k}).",
"$ for some $0\\le J_1 < J_2 \\le 2^{k-1}-1$ with $J_2-J_1 = 2^{k-2}$ .",
"Proof.",
"W.l.o.g.",
"let $\\zeta _{2^3} = \\zeta _{2^k}^{2^{k-3}} = (1+i)/\\sqrt{2}$ , and hence $ \\sqrt{2}\\zeta _{2^k}^j = (\\zeta _{2^k}^j + i\\zeta _{2^k}^j)/\\zeta _{2^k}^{2^{k-3}} = \\zeta _{2^k}^{j-2^{k-3}} + \\zeta _{2^k}^{2^{k-2}}\\zeta _{2^k}^{j-2^{k-3}} =\\zeta _{2^k}^{j-2^{k-3}} + \\zeta _{2^k}^{j+2^{k-3}}.",
"$ As $\\zeta _{2^k}^j = -\\zeta _{2^k}^{j-2^{k-1}}$ we can assume that $0\\le j\\le 2^{k-1}-1$ .",
"Again using that $\\zeta _{2^k}^{2^{k-1}} = -1$ , we can then write $\\sqrt{2}\\zeta _{2^k}^j$ as $ \\sqrt{2}\\zeta _{2^k}^j =\\left\\lbrace \\begin{array}{ll}-\\zeta _{2^k}^{j-2^{k-3}+2^{k-1}} + \\zeta _{2^k}^{j+2^{k-3}} & \\mbox{if}\\; j-2^{k-3}<0, \\\\\\zeta _{2^k}^{j-2^{k-3}} + \\zeta _{2^k}^{j+2^{k-3}} & \\mbox{if}\\; 0\\le j-2^{k-3} < j+2^{k-3} < 2^{k-1}, \\\\\\zeta _{2^k}^{j-2^{k-3}} - \\zeta _{2^k}^{j+2^{k-3}-2^{k-1}} & \\mbox{if}\\; j+2^{k-3} \\ge 2^{k-1}.\\end{array}\\right.$ In either case $\\sqrt{2}\\zeta ^j$ is of the form $\\pm \\zeta _{2^k}^{J_1} \\pm \\zeta _{2^k}^{J_2}$ for some $0\\le J_1 < J_2 \\le 2^{k-1}-1$ with $J_2-J_1 = 2^{k-2}$ .",
"Since $\\lbrace 1,\\zeta _{2^k},\\ldots ,\\zeta _{2^k}^{2^{k-1}-1}\\rbrace $ is a basis of ${\\mathbb {Q}}(\\zeta _{2^k})$ , this representation is unique.",
"$\\Box $ To any generalized Boolean function $f:{\\mathbb {V}}_n\\rightarrow {\\mathbb {Z}}_{2^k},$ we may associate the sequence of Boolean functions $a_j\\in \\mathcal {B}_n$ , $j=0,1,\\ldots ,k-1$ , for which $f(x)=a_0(x)+2a_1(x)+2^2a_2(x)+\\cdots +2^{k-1}a_{k-1}(x),\\; \\forall x\\in {\\mathbb {V}}_n.$ For an integer $i$ , $0\\le i\\le 2^{k-1}-1$ , with $i = \\sum _{j=0}^{2^{k-1}}i_j2^j$ , $i_j\\in \\lbrace 0,1\\rbrace $ , we define the $i$ -th component function $g_i\\in \\mathcal {B}_n$ of $f$ as $g_i(x) = a_{k-1}(x)\\oplus i_0a_0(x)\\oplus \\cdots \\oplus i_{k-2}a_{k-2}(x).$ For an element $u\\in {\\mathbb {V}}_n$ , let $\\mathbf {\\mathcal {W}}(u) = (\\mathcal {W}_{g_0}(u),$ $\\mathcal {W}_{g_1}(u),\\ldots ,\\mathcal {W}_{g_{2^{k-1}-1}}(u))$ and let $\\mathbf {S}(u) = (S_0,S_1,\\ldots ,S_{2^{k-1}-1})$ be the vector defined by $\\mathbf {S}(u)=\\left(\\begin{array}{c}S_0 \\\\S_1 \\\\\\vdots \\\\S_{2^{k-1}-1} \\\\\\end{array}\\right):=H_{2^{k-1}}\\left(\\begin{array}{c}\\mathcal {W}_{g_0}(u) \\\\\\mathcal {W}_{g_1}(u) \\\\\\vdots \\\\\\mathcal {W}_{g_{2^{k-1}-1}}(u) \\\\\\end{array}\\right).$ In [5] the following proposition has been shown.",
"Proposition 1 [5] Let $f\\in \\mathcal {GB}_n^{2^k}$ and $u\\in {\\mathbb {V}}_n$ .",
"Then $ 2^{k-1}\\mathcal {H}_f(u) = (1,\\zeta _{2^k},\\ldots ,\\zeta _{2^k}^{2^{k-1}-1})\\cdot \\mathbf {S}(u) = S_0+S_1\\zeta _{2^k}+\\cdots +S_{2^{k-1}-1}\\zeta _{2^k}^{2^{k-1}-1}.",
"$"
],
[
"Necessary and sufficient conditions",
"In this section we present necessary and sufficient conditions for the gbentness of functions $f\\in \\mathcal {GB}_n^{2^k}$ given as in $(\\ref {eq:1})$ .",
"We provide an equivalent form of these conditions in terms of certain spectral properties of the component functions of $f$ .",
"In the next section, we will use these conditions to completely characterize gbent functions as algebraic objects, which are shown to possess a lot of structure and to have some interesting properties.",
"Theorem 1 Let $f(x) = a_0(x)+\\cdots +2^{k-2}a_{k-2}(x)+2^{k-1}a_{k-1}(x) \\in \\mathcal {GB}_n^{2^k}$ , and let $g_i(x) = a_{k-1}(x) \\oplus i_0a_0(x)\\oplus i_1a_1(x)\\oplus \\cdots \\oplus i_{k-2}a_{k-2}(x)$ , $0\\le i\\le 2^{k-1}-1$ , where $i=\\sum _{j=0}^{k-2}i_j2^j$ and $i_j \\in \\lbrace 0,1\\rbrace $ .",
"(i) If $n$ is even, then $f$ is gbent if and only if $g_i$ is bent for all $0\\le i\\le 2^{k-1}-1$ , such that for all $u\\in {\\mathbb {V}}_n$ , $\\mathbf {\\mathcal {W}}(u) = (\\mathcal {W}_{g_0}(u),\\mathcal {W}_{g_1}(u),\\ldots ,\\mathcal {W}_{g_{2^{k-1}-1}}(u)) = \\pm 2^{\\frac{n}{2}}H_{2^{k-1}}^{(r)}$ for some $r$ , $0\\le r\\le 2^{k-1}-1$ , depending on $u$ .",
"(ii) If $n$ is odd, then $f$ is gbent if and only if $g_i$ is semi-bent for all $0\\le i\\le 2^{k-1}-1$ , such that for all $u\\in {\\mathbb {V}}_n$ , $\\mathbf {\\mathcal {W}}(u) = (\\pm 2^{\\frac{n+1}{2}}H^{(r)}_{2^{k-2}},\\textbf {0}_{2^{k-2}})\\quad \\mbox{or}\\quad \\mathbf {\\mathcal {W}}(u) = (\\textbf {0}_{2^{k-2}},\\pm 2^{\\frac{n+1}{2}} H^{(r)}_{2^{k-2}})$ for some $r$ , $0\\le r\\le 2^{k-2}-1$ , depending on $u$ ($\\textbf {0}_{2^{k-2}}$ is the all-zero vector of length $2^{k-2}$ ).",
"First we consider the case (i) when $n$ is even.",
"The sufficiency of $(\\ref {W=S})$ has been shown in [5], though in a more general context for $f\\in \\mathcal {GB}_n^q$ , where $q$ is an arbitrary even integer.",
"For the sake of completeness we include the proof arguments here.",
"Suppose that $(\\ref {W=S})$ holds, which also implies that all $g_i$ are bent.",
"By the definition of $S_t$ , $0\\le t\\le 2^{k-1}-1$ , we then have $S_t = 0$ if $t\\ne r$ , and $S_r = \\pm 2^{n/2}2^{k-1}$ .",
"Proposition REF then yields $2^{k-1}\\mathcal {H}_h(u) = \\pm 2^{n/2}2^{k-1}\\zeta _{2^k}^r$ , hence $f$ is gbent.",
"Now, conversely, suppose that $f$ is gbent.",
"By Proposition REF , we then have $ S_0 + S_1\\zeta _{2^k} + \\cdots + S_{2^{k-1}-1}\\zeta _{2^k}^{2^{k-1}-1} = 2^{k-1}\\mathcal {H}_f(u) = \\pm 2^{k-1}2^{\\frac{n}{2}}\\zeta _{2^k}^r $ for some $r$ , $0\\le r\\le 2^{k-1}-1$ .",
"Since $\\lbrace 1,\\zeta _{2^k},\\ldots ,\\zeta _{2^k}^{2^{k-1}-1}\\rbrace $ is a basis of ${\\mathbb {Q}}(\\zeta _{2^k})$ , this implies that $S_t = 0$ , $0\\le t\\le 2^{k-1}-1$ , $t \\ne r$ , and $S_r = \\pm 2^{k-1}2^{\\frac{n}{2}}$ .",
"By the invertibility of $H_{2^{k-1}}$ , the only solution for $\\mathbf {\\mathcal {W}}(u)$ in the resulting linear system is $\\mathbf {\\mathcal {W}}(u) = \\pm 2^{\\frac{n}{2}}H_{2^{k-1}}^{(r)}$ .",
"Hence $(\\ref {W=S})$ holds, also implying that all $g_i$ are bent.",
"For the case (ii), when $n$ is odd, the sufficiency of $(\\ref {oddW=S})$ has also been shown in [5].",
"Again, for the sake of completeness, we include the proof arguments.",
"If $(\\ref {oddW=S})$ holds, then by $(\\ref {Sk2})$ , for $j \\in \\lbrace r,r+2^{k-2}\\rbrace $ we have $S_j = \\pm 2^{k-2}2^{\\frac{n+1}{2}}$ , and $S_j = 0$ if $j\\ne r,r+2^{k-2}$ .",
"Hence, from Proposition REF , we get $ \\mathcal {H}_f(u) = \\pm 2^{\\frac{n+1}{2}}\\zeta _{2^k}^r \\pm 2^{\\frac{n+1}{2}}\\zeta _{2^k}^{r+2^{k-2}} = 2^{\\frac{n+1}{2}}\\zeta _{2^k}^r(\\pm 1 \\pm i) = 2^{\\frac{n}{2}}\\zeta _{2^k}^r\\zeta _8^j, $ for some $j\\in \\lbrace 1,3,5,7\\rbrace $ .",
"Therefore, $f$ is gbent.",
"If conversely $f$ is gbent, then by Proposition REF we have $ S_0 + S_1\\zeta _{2^k} + \\cdots + S_{2^{k-1}-1}\\zeta _{2^k}^{2^{k-1}-1} = 2^{k-1}\\mathcal {H}_f(u) = 2^{k-1}2^{\\frac{n-1}{2}}\\sqrt{2}\\zeta _{2^k}^j, $ for some $0\\le j\\le 2^{k-1}-1$ .",
"By Lemma REF , there exists (a unique) $r$ , $0\\le r\\le 2^{k-2}-1$ , such that $ \\sqrt{2}\\zeta _{2^k}^j = \\pm \\zeta _{2^k}^r \\pm \\zeta _{2^k}^{r+2^{k-2}}.",
"$ Combining the two above relations, we have $ S_0 + S_1\\zeta _{2^k} + \\cdots + S_{2^{k-1}-1}\\zeta _{2^k}^{2^{k-1}-1} = 2^{k-1}2^{\\frac{n-1}{2}}(\\pm \\zeta _{2^k}^r \\pm \\zeta _{2^k}^{r+2^{k-2}}).",
"$ Therefore, $S_r = \\pm 2^{k-2}2^{\\frac{n+1}{2}}$ , $S_{r+2^{k-2}} = \\pm 2^{k-2}2^{\\frac{n+1}{2}}$ , and $S_t = 0$ for $t \\ne r,r+2^{k-2}$ , i.e., $ \\mathbf {S}(u)=\\left(\\begin{array}{c}S_0 \\\\\\vdots \\\\S_r \\\\\\vdots \\\\S_{r+2^{k-2}} \\\\\\vdots \\\\S_{2^{k-1}-1} \\\\\\end{array}\\right)= 2^{k-2}2^{\\frac{n+1}{2}}\\left(\\begin{array}{c}0 \\\\\\vdots \\\\(-1)^{e_1} \\\\\\vdots \\\\(-1)^{e_2} \\\\\\vdots \\\\0 \\\\\\end{array}\\right),\\quad e_1,e_2\\in \\lbrace 0,1\\rbrace .",
"$ By the invertibility of $H_{2^{k-1}}$ , the linear system $(\\ref {Sk2})$ has a unique solution for all four possibilities of $\\mathbf {S}(u)$ .",
"As now easily observed, these solutions are $(2^{\\frac{n+1}{2}}H^{(r)}_{2^{k-2}},\\textbf {0}_{2^{k-2}})$ , $(-2^{\\frac{n+1}{2}}H^{(r)}_{2^{k-2}},\\textbf {0}_{2^{k-2}})$ , $(\\textbf {0}_{2^{k-2}},2^{\\frac{n+1}{2}} H^{(r)}_{2^{k-2}})$ , and $(\\textbf {0}_{2^{k-2}},-2^{\\frac{n+1}{2}} H^{(r)}_{2^{k-2}})$ for $(e_1,e_2) = (1,1), (-1,1), (1,-1)$ , and $(-1,-1)$ , respectively.",
"Combining Theorem REF and Lemma REF (iii) gives a characterization of the gbent property in terms of the Walsh spectral values of the component functions.",
"More precisely, the quadruples (four vectors) of a suitable vector space ${\\mathbb {V}}_{k-1}$ which build a 2-dimensional flat specify the component functions whose spectra satisfy certain conditions as described below.",
"In other words, the characterization in Theorem REF , which relates the spectral values of component functions to the rows of Hadamard matrices, turns out to be equivalent to a particular relation of the Walsh spectral values for the above defined quadruples.",
"Proposition 2 (i) Let $n$ be even, $k \\ge 3$ , and represent $i=\\sum _{j=0}^{k-2}i_j2^j$ for $0\\le i\\le 2^{k-1}-1$ , with $i_j \\in \\lbrace 0,1\\rbrace $ .",
"Assume $g_i(x) = a_{k-1}(x) \\oplus i_0a_0(x)\\oplus i_1a_1(x)\\oplus \\cdots \\oplus i_{k-2}a_{k-2}(x)$ are bent functions, for $0\\le i\\le 2^{k-1}-1$ .",
"For $u\\in {\\mathbb {V}}_n$ , the condition in Theorem REF $\\mathbf {\\mathcal {W}}(u) = (\\mathcal {W}_{g_0}(u),\\mathcal {W}_{g_1}(u),\\ldots ,\\mathcal {W}_{g_{2^{k-1}-1}}(u)) = \\pm 2^{\\frac{n}{2}}H_{2^{k-1}}^{(r)}$ holds for some $r\\in \\lbrace 0,\\ldots ,2^{k-1}-1\\rbrace $ , if and only if for any four distinct integers $j,c,l,v\\in \\lbrace 0,\\ldots ,2^{k-1}-1\\rbrace $ such that $z_j\\oplus z_c\\oplus z_l\\oplus z_v={\\bf 0}$ , the integers $\\mathcal {W}_{g_j}(u),\\mathcal {W}_{g_c}(u),\\mathcal {W}_{{g_l}}(u),\\mathcal {W}_{{g_v}}(u)\\in \\lbrace -2^{\\frac{n}{2}},2^{\\frac{n}{2}}\\rbrace $ satisfy the equality $\\mathcal {W}_{g_j}(u)\\mathcal {W}_{g_c}(u)=\\mathcal {W}_{g_l}(u)\\mathcal {W}_{{g_v}}(u).$ (ii) Similarly, when $n$ be odd, let us assume that $g_i(x) = a_{k-1}(x) \\oplus i_0a_0(x)\\oplus i_1a_1(x)\\oplus \\cdots \\oplus i_{k-2}a_{k-2}(x)$ are semi-bent functions, for any $0\\le i\\le 2^{k-1}-1$ .",
"Then, $\\mathbf {\\mathcal {W}}(u) = (\\pm 2^{\\frac{n+1}{2}}H^{(r)}_{2^{k-2}},\\textbf {0}_{2^{k-2}})$ for some $0\\le r\\le 2^{k-2}-1$ , if and only if $\\mathcal {W}_{g_j}(u) = 0$ for all $2^{k-2}\\le j\\le 2^{k-1}-1$ and $\\mathcal {W}_{g_j}(u) \\ne 0$ for all $0 \\le j\\le 2^{k-2}-1$ such that for any four distinct integers $j,c,l,v\\in \\lbrace 0,\\ldots ,2^{k-2}-1\\rbrace $ with $z_j\\oplus z_c\\oplus z_l\\oplus z_v={\\bf 0}$ , the integers $\\mathcal {W}_{g_j}(u),\\mathcal {W}_{g_c}(u),\\mathcal {W}_{{g_l}}(u),\\mathcal {W}_{{g_v}}(u)\\in \\lbrace -2^{\\frac{n+1}{2}},2^{\\frac{n+1}{2}}\\rbrace $ satisfy the equality $\\mathcal {W}_{g_j}(u)\\mathcal {W}_{g_c}(u)=\\mathcal {W}_{g_l}(u)\\mathcal {W}_{{g_v}}(u).$ A similar statement is valid for $\\mathbf {\\mathcal {W}}(u) = (\\textbf {0}_{2^{k-2}},\\pm 2^{\\frac{n+1}{2}} H^{(r)}_{2^{k-2}})$ .",
"The proposition follows from Theorem REF and Lemma REF (iii)."
],
[
"Gbent conditions in terms of affine (semi-)bent spaces",
"In the previous section we have provided two different characterizations of gbent property, though both are closely related to certain properties of the component functions.",
"The derived conditions essentially also capture the inherent properties of the affine spaces of (semi-)bent functions that correspond to gbent functions.",
"In this section we specify these affine spaces of (semi-)bent functions and also address the affine equivalence of gbent functions in a rigour manner.",
"We first develop equivalent gbent conditions in terms of affine bent spaces for even $n$ .",
"In this case, by the definition of the dual $g^*$ of a bent function $g$ , the relation (REF ) in Proposition REF is equivalent to $ (-1)^{g^*_j(u)}(-1)^{g^*_c(u)}=(-1)^{g^*_l(u)}(-1)^{g^*_v(u)},$ for all $u\\in {\\mathbb {V}}_n$ .",
"Hence $g^*_j\\oplus g^*_c\\oplus g^*_l\\oplus g^*_v=0,$ if $j,c,l,v$ satisfy $z_j\\oplus z_c\\oplus z_l\\oplus z_v={\\bf 0}$ .",
"Observing that $g_j\\oplus g_c\\oplus g_l\\oplus g_v=0$ if and only if $z_j\\oplus z_c\\oplus z_l\\oplus z_v={\\bf 0}$ , we obtain the following corollary from Theorem REF and Proposition REF .",
"Corollary 1 A function $f:{\\mathbb {V}}_n\\rightarrow {\\mathbb {Z}}_{2^k}$ , $n$ even, given as $f(x) = a_0(x) + 2a_1(x) + \\cdots + 2^{k-1}a_{k-1}(x)$ is gbent if and only if $ \\mathcal {A} = a_{k-1} \\oplus \\langle a_0,a_1,\\ldots ,a_{k-2}\\rangle $ is an affine vector space of bent functions such that for any $h_0,h_1,h_2,h_3 \\in \\mathcal {A}$ with $h_0\\oplus h_1\\oplus h_2\\oplus h_3 = 0$ we have $h_0^*\\oplus h_1^*\\oplus h_2^*\\oplus h_3^* = 0$ .",
"Equivalently, if $h_3 = h_0\\oplus h_1\\oplus h_2$ , then $h^*_3 = h^*_0\\oplus h^*_1\\oplus h^*_2$ .",
"Corollary REF generalizes an observation in [12], where the relations between octal gbent functions and a secondary construction of bent functions proposed by Carlet [2] were investigated.",
"We state the version of this construction [2] given by Mesnager in [11].",
"Proposition 3 [11] Let $g_0,g_1,g_2,g_3$ be bent functions from ${\\mathbb {V}}_n$ to ${\\mathbb {F}}_2$ such that $g_0\\oplus g_1\\oplus g_2\\oplus g_3 = 0$ .",
"Then the function $ g_0g_1 \\oplus g_0g_2 \\oplus g_1g_2 $ is bent if and only if $g_0^*\\oplus g_1^*\\oplus g_2^*\\oplus g_3^* = 0$ , and its dual is $g_0^*g_1^* \\oplus g_0^*g_2^* \\oplus g_1^*g_2^*$ .",
"Combining Corollary REF and Proposition REF we get interesting alternative conditions for gbent functions in $\\mathcal {GB}_n^{2^k}$ when $n$ is even.",
"Corollary 2 Let $n$ be even.",
"A function $f(x) = a_0(x) + 2a_1(x) + \\cdots + 2^{k-1}a_{k-1}(x) \\in \\mathcal {GB}_n^{2^k}$ is a gbent function if and only if $\\mathcal {A} = a_{k-1} \\oplus \\langle a_0,a_1,\\ldots ,a_{k-2}\\rangle $ is an affine vector space of bent functions such that for every (pairwise distinct) $g_i,g_j,g_l\\in \\mathcal {A}$ the function $g_ig_j \\oplus g_ig_l \\oplus g_jg_l$ is bent.",
"Remark 1 Note that if the bent functions $g_i,g_j,g_l$ are not pairwise distinct, then $g_ig_j \\oplus g_ig_l \\oplus g_jg_l$ is trivially bent.",
"To address the case when $n$ is odd, we show the following analog of Proposition REF for semi-bent functions.",
"Proposition 4 Let $g_0,g_1,g_2,g_3$ be semi-bent functions from ${\\mathbb {V}}_n$ to ${\\mathbb {F}}_2$ such that $g_0\\oplus g_1\\oplus g_2\\oplus g_3 = 0$ .",
"Then, the function $ g_0g_1 \\oplus g_0g_2 \\oplus g_1g_2 $ is semibent if and only if for all $u\\in {\\mathbb {V}}_n$ , $\\mathcal {W}_{g_i}(u) = 0$ for an even number of $i\\in \\lbrace 0,1,2,3\\rbrace $ , and if $\\mathcal {W}_{g_i}(u) \\ne 0$ for all $i\\in \\lbrace 0,1,2,3\\rbrace $ , then $\\mathcal {W}_{g_0}(u)\\mathcal {W}_{g_1}(u)=\\mathcal {W}_{g_2}(u)\\mathcal {W}_{{g_3}}(u),$ or $|\\lbrace i\\,:\\,\\mathcal {W}_{g_i}(u) = 2^{(n+1)/2}\\rbrace | = 1,3,\\quad \\mbox{but not}\\quad \\mathcal {W}_{g_0}(u)=\\mathcal {W}_{g_1}(u)=\\mathcal {W}_{g_2}(u).$ Proof.",
"By [2] (see also Proposition 2 in [11]), for (pairwise distinct) Boolean functions $g_0,g_1,g_2,g_3$ such that $g_0\\oplus g_1\\oplus g_2\\oplus g_3 = 0$ , the Walsh-Hadamard transform of $g=g_0g_1 \\oplus g_0g_2 \\oplus g_1g_2$ satisfies $ \\mathcal {W}_g(u) = \\frac{1}{2}(\\mathcal {W}_{g_0}(u)+\\mathcal {W}_{g_1}(u)+\\mathcal {W}_{g_2}(u)-\\mathcal {W}_{g_3}(u)) $ for all $u\\in {\\mathbb {V}}_n$ .",
"The correctness of the proposition follows then easily by checking all possible combinations of $\\mathcal {W}_{g_i}(u)$ , $i\\in \\lbrace 0,1,2,3\\rbrace $ .",
"Note that $(\\ref {wquad1})$ is equivalent to $\\mathcal {W}_{g_i}(u) = -2^{(n+1)/2}$ for an even number of $i\\in \\lbrace 0,1,2,3\\rbrace $ .",
"$\\Box $ Remark 2 If for any $u\\in {\\mathbb {V}}_n$ for which $\\mathcal {W}_{g_i}(u) \\ne 0$ , $i=0,1,2,3$ , the condition $(\\ref {wquad1})$ always applies, then $g = g_ig_j\\oplus g_ig_l\\oplus g_jg_l$ is semi-bent for any $\\lbrace i,j,l\\rbrace \\subset \\lbrace 0,1,2,3\\rbrace $ .",
"If for some of $u \\in {\\mathbb {V}}_n$ we have $(\\ref {wquad2})$ , then this is not true.",
"Corollary 3 Let $n$ be odd.",
"If $f(x) = a_0(x) + 2a_1(x) + \\cdots + 2^{k-1}a_{k-1}(x) \\in \\mathcal {GB}_n^{2^k}$ is a gbent function, then $\\mathcal {A} = a_{k-1} \\oplus \\langle a_0,a_1,\\ldots ,a_{k-2}\\rangle =a_{k-1} \\oplus \\mathcal {L}$ is an affine vector space of semi-bent functions such that for every (pairwise distinct) $g_i,g_j,g_l\\in \\mathcal {A}$ the function $G=g_ig_j \\oplus g_ig_l \\oplus g_jg_l$ is semi-bent.",
"Moreover, for every $u\\in {\\mathbb {V}}_n$ we have $\\nonumber & \\mathcal {W}_g(u) = 0\\;\\;\\mbox{if and only if}\\;\\; g\\in a_{k-1} \\oplus \\langle a_0,a_1,\\ldots ,a_{k-3}\\rangle ,\\; \\mbox{or} \\\\& \\mathcal {W}_g(u) \\ne 0\\;\\;\\mbox{if and only if}\\;\\; g\\in a_{k-1} \\oplus \\langle a_0,a_1,\\ldots ,a_{k-3}\\rangle .$ Conversely, if $\\mathcal {A} = a_{k-1} \\oplus \\mathcal {L}$ is an affine vector space of semibent functions such that for every (pairwise distinct) $g_i,g_j,g_l\\in \\mathcal {A}$ the function $G=g_ig_j \\oplus g_ig_l \\oplus g_jg_l$ is semibent, and $\\mathcal {A} = a_{k-1} \\oplus \\langle a_{k-2},\\mathcal {L}_1\\rangle $ for some subspace $\\mathcal {L}_1$ of $\\mathcal {L}$ and $a_{k-2}\\notin \\mathcal {L}_1$ , with the property that for all $u \\in {\\mathbb {V}}_n$ we have $\\nonumber & \\mathcal {W}_g(u) = 0\\;\\;\\mbox{if and only if}\\;\\; g\\in a_{k-1} \\oplus \\mathcal {L}_1,\\; \\mbox{or}\\; \\\\& \\mathcal {W}_g(u) \\ne 0\\;\\;\\mbox{if and only if}\\;\\; g\\in a_{k-1} \\oplus \\mathcal {L}_1,$ then $f(x) = a_0(x) + 2a_1(x) + \\cdots + 2^{k-3}a_{k-3}(x) + 2^{k-2}a_{k-2}(x) + 2^{k-1}a_{k-1}(x)$ , $a_i\\in \\mathcal {L}_1$ , $0\\le i\\le k-3$ , is gbent.",
"Let $f(x) = a_0(x)+2a_1(x)+\\cdots +2^{k-1}a_{k-1}(x)$ be a gbent function from ${\\mathbb {V}}_n$ to ${\\mathbb {Z}}_{2^k}$ , $n$ odd, and for $0\\le i\\le 2^{k-1}-1$ let $g_i(x) = i_0a_0(x) \\oplus i_1a_1(x) \\oplus \\cdots \\oplus i_{k-2}a_{k-2}(x) \\oplus a_{k-1}(x)$ , where $(i_0,i_1,\\ldots ,i_{k-2})=z_i$ is the binary representation of $i$ .",
"By Theorem REF (ii), $g_i$ is semi-bent for all $0\\le i\\le 2^{k-1}-1$ and $\\mathbf {\\mathcal {W}}(u) = (\\pm 2^{\\frac{n+1}{2}}H^{(r)}_{2^{k-2}},\\textbf {0}_{2^{k-2}})\\;\\;\\mbox{or}\\;\\;\\mathbf {\\mathcal {W}}(u) = (\\textbf {0}_{2^{k-2}},\\pm 2^{\\frac{n+1}{2}} H^{(r)}_{2^{k-2}})$ for some $0\\le r\\le 2^{k-2}-1$ .",
"Let $0 \\le i < j < l < v \\le 2^{k-1}-1$ be such that $z_i \\oplus z_j \\oplus z_l \\oplus z_v = {\\bf 0}$ , where $z_i = (i_0,i_1,\\ldots ,i_{k-2}) \\in {\\mathbb {V}}_{k-1}$ is the binary representation of $i$ .",
"Since $i_{k-2}\\oplus j_{k-2} \\oplus l_{k-2} \\oplus v_{k-2} = 0$ , the following situations can then occur: (i) $0 \\le i,j,l,v \\le 2^{k-2}-1$ : In this case, either $\\mathcal {W}_h(u)=0$ for all $h\\in \\lbrace g_i,g_j,g_l,g_v\\rbrace $ , or $\\mathcal {W}_h(u)\\ne 0$ for all $h\\in \\lbrace g_i,g_j,g_l,g_v\\rbrace $ .",
"In the latter case, by Proposition REF , $\\mathcal {W}_{g_i}(u)\\mathcal {W}_{g_j}(u)=\\mathcal {W}_{g_l}(u)\\mathcal {W}_{{g_v}}(u)$ .",
"In both cases, by Proposition REF the function $G$ is semi-bent.",
"(ii) $2^{k-2} \\le i,j,l,v \\le 2^{k-1}-1$ : The same argument as for (i) applies to this case.",
"(iii) $0 \\le i,j \\le 2^{k-2}-1$ , $2^{k-2} \\le l,v \\le 2^{k-1}-1$ : In this case, exactly two of $\\mathcal {W}_{g_i}(u)$ , $\\mathcal {W}_{g_j}(u)$ , $\\mathcal {W}_{g_l}(u)$ $\\mathcal {W}_{g_v}(u)$ are zero, hence by Proposition REF the function $G$ is semi-bent.",
"Finally, $(\\ref {half=0})$ follows directly from $(\\ref {0HH0})$ .",
"To show the converse, we first note that $(\\ref {0ne0})$ implies that exactly $2^{k-2}$ entries of $\\mathcal {W}(u)$ are zero, all of them being located either at the first or at the second half of $\\mathcal {W}(u)$ .",
"Since we suppose that $g_ig_j \\oplus g_ig_l \\oplus g_jg_l$ is semibent for all (pairwise distinct) $g_i,g_j,g_l\\in \\mathcal {A}$ , by Proposition REF and Remark REF , the nonzero half of $\\mathcal {W}(u)$ equals to $\\pm 2^{\\frac{n+1}{2}}H_{2^{k-2}}^{(r)}$ for some $0\\le r\\le 2^{k-2}-1$ .",
"As a consequence, $f$ is gbent by Theorem REF (ii)."
],
[
"Equivalence of gbent functions",
"We now give the complete characterization of gbent functions, both for even and odd $n$ , as an algebraic object.",
"Similarly to the case of standard bent functions we discuss the concept of affine equivalence of gbent functions.",
"As already demonstrated, a gbent function $f(x) = a_0(x)+2a_1(x) + \\cdots + 2^{k-2}a_{k-2}(x) + 2^{k-1}a_{k-1}(x)$ gives rise to $\\mathcal {A} = a_{k-1} \\oplus \\langle a_0,\\ldots ,a_{k-2}\\rangle $ which is an affine space of bent functions (semi-bent functions) with certain properties.",
"Thus, it is natural to investigate its correspondence to apparently similar class of functions, namely to vectorial bent functions.",
"Recall that a vectorial bent function $F:{\\mathbb {V}}_n\\rightarrow {\\mathbb {V}}_k$ , $n$ even, $k\\le n/2$ , is a function $F(x) = (a_0(x),a_1(x),\\ldots ,a_{k-1}(x)),\\quad a_i\\in \\mathcal {B}_n,\\;0\\le i\\le k-1,$ for which every (nontrivial) component function $i_0a_0 \\oplus i_1a_1 \\oplus \\cdots \\oplus i_{k-1}a_{k-1}$ , $i_j\\in \\lbrace 0,1\\rbrace $ , $0\\le j\\le k-1$ , is bent.",
"Equivalently, $F$ is a $k$ -dimensional vector space of bent functions with a basis $\\lbrace a_0,a_1,\\ldots ,a_{k-1}\\rbrace $ .",
"Changing the basis, that is, performing a coordinate transformation on ${\\mathbb {V}}_k$ does not change the vector space.",
"It is rather the representation in the form $(\\ref {vecbent})$ that changes.",
"In spite of a different appearance, the functions are considered to be the same.",
"Furthermore, it is well known that a coordinate transformation on ${\\mathbb {V}}_n$ also results in a vectorial bent function, which is said to be equivalent and is not seen as a different object.",
"For these reasons a discussion about the equivalence of gbent functions seems to be in place.",
"Let $f(x) = a_0(x)+2a_1(x) + \\cdots + 2^{k-2}a_{k-2}(x) + 2^{k-1}a_{k-1}(x)\\in \\mathcal {GB}_n^{2^k}$ be a gbent function, $b\\in \\langle a_0,a_1,\\ldots ,a_{k-2}\\rangle $ , and let $B$ be an invertible $(k-1)\\times (k-1)$ -matrix over ${\\mathbb {V}}_2$ .",
"Set ${\\bf a}= (a_0(x),a_1(x),\\ldots ,a_{k-2}(x))$ and let $B{\\bf a}^T = (b_0(x),b_1(x),\\ldots ,b_{k-2}(x))$ .",
"Then, $ a_{k-1} \\oplus \\langle a_0,a_1,\\ldots ,a_{k-2}\\rangle \\quad \\mbox{and}\\quad (a_{k-1} \\oplus b) \\oplus \\langle b_0,b_1,\\ldots ,b_{k-2}\\rangle $ define the same affine space of bent functions respectively semi-bent functions.",
"In particular, when $n$ is even, the function $ f_1(x) = b_0(x)+2b_1(x)+\\cdots +2^{k-2}b_{k-2}(x)+2^{k-1}(a_{k-1}(x) \\oplus b(x)) $ is also a gbent function, describing the same object as $f$ does.",
"One has to be little bit more careful when $n$ is odd, since then the vector space $\\mathcal {L}=\\langle a_0,a_1,\\ldots ,a_{k-2}\\rangle $ contains a subspace $\\mathcal {L}_1$ as described in Corollary REF .",
"Thus, for our standard representation, when $f$ is of the form $(\\ref {eq:1})$ , $a_{k-2}$ has to be chosen from $\\mathcal {L}\\setminus \\mathcal {L}_1$ .",
"As for (vectorial) bent functions one can obtain seemingly new gbent functions from a given one by applying a coordinate transformation on ${\\mathbb {V}}_n$ .",
"Let $f:{\\mathbb {V}}_n \\rightarrow {\\mathbb {Z}}_{2^k}$ and let $A$ be an invertible $n \\times n$ -matrix over ${\\mathbb {F}}_2$ .",
"Then for $u\\in {\\mathbb {V}}_n$ , $\\mathcal {H}_{f(Ax)}(u) & = \\sum _{x\\in {\\mathbb {V}}_n}\\zeta _{2^k}^{f(Ax)}(-1)^{u\\cdot x} = \\sum _{x\\in {\\mathbb {V}}_n}\\zeta _{2^k}^{f(x)}(-1)^{u\\cdot A^{-1}x} \\\\& = \\sum _{x\\in {\\mathbb {V}}_n}\\zeta _{2^k}^{f(x)}(-1)^{(A^{-1})^Tu\\cdot x} = \\mathcal {H}_{f}((A^{-1})^Tu).$ Hence $f(Ax)$ is gbent if and only if $f$ is gbent.",
"Consequently, gbentness is invariant under linear coordinate transformations on ${\\mathbb {V}}_n$ .",
"From the above discussion, when $n$ is even, we may say that $f(x) = a_0(x)+2a_1(x)+\\cdots +2^{k-2}a_{k-2}(x)+2^{k-1}a_{k-1}(x)$ and $f_1(x)$ are equivalent if there exist $A\\in GL(n,{\\mathbb {F}}_2)$ , $B\\in GL(k-1,{\\mathbb {F}}_2)$ and $b\\in \\langle a_0,a_1,\\ldots ,a_{k-2}\\rangle $ , such that $ f_1(x) = b_0(Ax)+2b_1(Ax)+\\cdots +2^{k-2}b_{k-2}(Ax)+2^{k-1}b_{k-1}(Ax) $ with $(b_0(x),b_1(x),\\ldots ,b_{k-2}(x)) = B{\\bf a}^T$ and $b_{k-1} = a_{k-1}\\oplus b$ .",
"When $n$ is odd, we require that the coordinate transformation induced by $B$ leaves the subspace $\\mathcal {L}_1$ invariant.",
"We notice that the gbent property does not require require that $a_0,a_1,\\ldots ,a_{k-2}$ are linearly independent.",
"Hence, the vector space $\\mathcal {L}=\\langle a_0,a_1,\\ldots ,a_{k-2}\\rangle $ may not have “full” dimension $k-1$ .",
"When $n$ is even, in the extreme case $\\dim (\\mathcal {L})=0$ , and $f(x) = 2^{k-1}a_{k-1}(x)$ is a gbent function if $a_{k-1}$ is a bent function.",
"Then the image set of $f$ is two-valued taking the values in $\\lbrace 0,2^{k-1}\\rbrace $ , but certainly one will not consider $a_{k-1}$ and $2^{k-1}a_{k-1}$ as different objects.",
"In general, it is easily verified that if $ f(x) = a_0(x)+2a_1(x)+\\cdots +2^{k-2}a_{k-2}(x)+2^{k-1}a_{k-1}(x) $ is a gbent function in $\\mathcal {GB}_n^{2^k}$ , then $ \\tilde{f}(x) = a_0(x)+2a_1(x)+\\cdots +2^{k-2}a_{k-2}(x)+2^{r-1}a_{r-1}(x) $ is a gbent function in $\\mathcal {GB}_n^{2^r}$ for any $r\\ge k$ .",
"However, this quite artificial lifted version of $f$ with a quite restricted image set, is essentially identified with $f\\in \\mathcal {GB}_n^{2^k}$ .",
"When $n$ is odd, if $ f(x) = a_0(x)+2a_1(x)+\\cdots +2^{k-2}a_{k-2}(x)+2^{k-1}a_{k-1}(x) $ is a gbent function in $\\mathcal {GB}_n^{2^k}$ , then $ \\tilde{f}(x) = a_0(x)+2a_1(x)+\\cdots +2^{k-3}a_{k-3}(x)+2^{r-2}a_{r-2}(x)+2^{r-1}a_{r-1}(x) $ is also a gbent function in $\\mathcal {GB}_n^{2^r}$ , for any $r\\ge k$ .",
"Again, we identify this lifted version $\\tilde{f}$ with $f$ .",
"Let $n$ be even and suppose that the vector space $\\langle a_0,a_1,\\ldots ,a_{r-2}\\rangle $ has dimension $k-1$ for some $k\\le r$ .",
"Then there exists a matrix $B\\in GL(r-1,{\\mathbb {F}}_2)$ such that $B (a_0,a_1,\\ldots ,a_{r-2})^T = (b_0,b_1,\\ldots ,b_{k-2},0\\ldots ,0)$ for some linearly independent $b_0,b_1,\\ldots ,b_{k-2}$ .",
"Hence $ \\tilde{f}_1(x) = a_0(x)+2a_1(x)+\\cdots +2^{r-1}a_{r-1}(x) $ is equivalent to $ \\tilde{f} = b_0(x)+2b_1(x)+\\cdots +2^{k-2}b_{k-2}(x)+2^{r-1}a_{r-1}(x), $ which is the lifted version of $ f = b_0(x)+2b_1(x)+\\cdots +2^{k-2}b_{k-2}(x)+2^{k-1}a_{r-1}(x) \\in \\mathcal {GB}_n^{2^k}.",
"$ As a consequence of the above discussion, we can restrict ourselves to gbent functions $f(x) = a_0(x)+2a_1(x)+\\cdots +2^{k-2}a_{k-2}(x)+2^{k-1}a_{k-1}(x)$ for which $a_0,a_1,\\ldots ,a_{k-2}$ are linearly independent.",
"The same argument also applies to the $n$ odd case.",
"The following summary of our discussion is fundamental for the characterization and possibly a classification of gbent functions.",
"For a gbent function $f(x)=a_0(x)+2a_1(x)+\\cdots +2^{k-2}a_{k-2}(x)+2^{k-1}a_{k-1}(x) \\in \\mathcal {GB}_n^{2^k}$ , the set $\\lbrace a_0,a_1,\\ldots ,a_{k-2}\\rbrace $ is always linearly independent (otherwise it reduces to a gbent function in $\\mathcal {GB}_n^{2^{k^\\prime }}$ for some $k^\\prime < k$ ).",
"A gbent function is independent from its representation of the form $(\\ref {eq:1})$ via a basis of $\\mathcal {L} = \\langle a_0,a_1,\\ldots ,a_{k-2}\\rangle $ , and the choice of the coset leader $a_{k-1}$ (for odd $n$ the existence of the distinguised subspace $\\mathcal {L}_1$ of $\\mathcal {L}$ has to be respected in the representation).",
"We can now state the main theorems, the characterization of gbent functions in terms of affine (semi-)bent spaces.",
"Theorem 2 Let $n$ be even.",
"A gbent function in $\\mathcal {GB}_n^{2^k}$ is a $(k-1)$ -dimensional affine vector space $\\mathcal {A}$ of bent functions such that for every $g_i,g_j,g_l\\in \\mathcal {A}$ the function $g_ig_j \\oplus g_ig_l \\oplus g_jg_l$ is bent.",
"Theorem 3 Let $n$ be odd.",
"A gbent function in $\\mathcal {GB}_n^{2^k}$ is a $(k-1)$ -dimensional affine vector space $\\mathcal {A} = a_{k-1} \\oplus \\mathcal {L}$ of semibent functions for which $g_ig_j \\oplus g_ig_l \\oplus g_jg_l$ is semibent for every $g_i,g_j,g_l\\in \\mathcal {A}$ , and for all $u\\in {\\mathbb {V}}_n$ we have $& \\mathcal {W}_g(u) = 0\\;\\mbox{if and only if}\\; g\\in a_{k-1} \\oplus \\mathcal {L}_1,\\; \\mbox{or} \\\\& \\mathcal {W}_g(u) \\ne 0\\;\\mbox{if and only if}\\; g\\in a_{k-1} \\oplus \\mathcal {L}_1,$ for some $(k-2)$ -dimensional subspace $\\mathcal {L}_1$ of $\\mathcal {L}$ ."
],
[
"${\\mathbb {Z}}_q$ -bent functions, vectorial bent functions and relative difference sets",
"In this section, $n$ is always even, $q=2^k$ .",
"We recall that a ${\\mathbb {Z}}_q$ -bent function is a function from an $n$ -dimensional vector space ${\\mathbb {V}}_n$ over ${\\mathbb {V}}_2$ to ${\\mathbb {Z}}_q$ , for which $ \\mathcal {H}_f(a,u) = \\sum _{x\\in {\\mathbb {V}}_n}\\zeta _q^{af(x)}(-1)^{u\\cdot x} $ has absolute value $2^{n/2}$ for every $u\\in {\\mathbb {V}}_n$ and nonzero $a\\in {\\mathbb {Z}}_q={\\mathbb {Z}}_{2^k}$ .",
"Equivalently, a ${\\mathbb {Z}}_q$ -bent function given by its graph $D = \\lbrace (x,f(x))\\,:\\,x\\in {\\mathbb {V}}_n\\rbrace $ is a $(2^n,2^k,2^n,2^{n-k})$ -relative difference set in ${\\mathbb {V}}_n\\times {\\mathbb {Z}}_{2^k}$ .",
"Clearly, a ${\\mathbb {Z}}_q$ -bent function is always gbent.",
"In [9] more general vectorial ${\\mathbb {Z}}_q$ -bent functions are considered.",
"We focus on the most interesting case where the codomain is cyclic.",
"Proposition 5 A function $f(x)=a_0(x)+2a_1(x)+\\cdots +2^{k-1}a_{k-1}(x)\\in \\mathcal {GB}_n^{2^k}$ , $n$ even, is ${\\mathbb {Z}}_q$ -bent if and only if $2^tf(x) = 2^ta_0(x)+2^{t+1}a_1(x)+\\cdots +2^{k-1}a_{k-t-1}(x) \\sim a_0(x)+2a_1(x)+\\cdots +2^{k-t-1}a_{k-t-1}(x)$ is a gbent function with dimension $k-1-t$ for every $t=0,1,\\ldots ,k-1$ .",
"If $f$ is ${\\mathbb {Z}}_q$ -bent, then $|\\mathcal {H}^{(2^k)}_f(2^t,u)| = 2^{n/2}$ for every $u\\in {\\mathbb {V}}_n$ and $t=0,1,\\ldots ,k-1$ by definition.",
"For the converse, let $a = 2^tz$ , for an odd integer $z$ and $0\\le t\\le k-1$ .",
"We have to show that $|\\mathcal {H}_f(a,u)| = 2^{n/2}$ for all $u\\in {\\mathbb {V}}_n$ .",
"Let $f_t(x) = a_0(x)+2a_1(x)+\\cdots +2^{k-1-t}a_{k-1-t}(x)$ .",
"By assumption, for all $u\\in {\\mathbb {V}}_n$ , $& \\mathcal {H}^{(2^k)}_f(2^t,u) = \\sum _{x\\in {\\mathbb {V}}_n}\\zeta _{2^k}^{2^ta_0(x)+2^{t+1}a_1(x)+\\cdots +2^{k-1}a_{k-t-1}(x)}(-1)^{u\\cdot x} = \\\\& \\sum _{x\\in {\\mathbb {V}}_n}\\zeta _{2^{k-t}}^{a_0(x)+2a_1(x)+\\cdots +2^{k-t-1}a_{k-t-1}(x)}(-1)^{u\\cdot x} = \\mathcal {H}^{(2^{k-t})}_{f_t}(u),$ has absolute value $2^{n/2}$ .",
"Equivalently, $f_t$ is gbent, hence $\\mathcal {H}^{(2^{k-t})}_{f_t}(u) = 2^{n/2}\\zeta _{2^{k-t}}^r$ for some $0\\le r\\le 2^{k-t}-1$ (depending on $u$ ).",
"Observing that $\\mathcal {H}^{(2^k)}_f(2^tz,u) = \\mathcal {H}^{(2^{k-t})}_{f_t}(z,u)$ is obtained from $\\mathcal {H}^{(2^{k-t})}_{f_t}(u)$ by exchanging the primitive $2^{k-t}$ -th complex root of unity $\\zeta _{2^{k-t}}$ with the primitive $2^{k-t}$ -th complex root of unity $\\zeta ^z_{2^{k-t}}$ , we conclude the proof.",
"Remark 3 As for a ${\\mathbb {Z}}_q$ -bent function in $\\mathcal {GB}_n^{2^k}$ we require that both $f = a_0(x)+2a_1(x)+\\cdots +2^{k-1}a_{k-1}(x)\\in \\mathcal {GB}_n^{2^k}$ and $f_1(x) = a_0(x)+2a_1(x)+\\cdots +2^{k-2}a_{k-2}(x)\\in \\mathcal {GB}_n^{2^{k-1}}$ are gbent, thus $\\langle a_0,a_1,\\ldots ,a_{k-1}\\rangle $ is a vector space of bent functions, i.e., a vectorial bent function.",
"We continue with two examples of ${\\mathbb {Z}}_q$ -bent functions.",
"For the first example, we employ the fact that $h(x,y) = {\\rm Tr}_m(x\\pi (y))$ is a (Maiorana-McFarland) bent function from ${\\mathbb {F}}_{2^m}\\times {\\mathbb {F}}_{2^m}$ to ${\\mathbb {F}}_2$ if and only if $\\pi $ is a permutation of ${\\mathbb {F}}_{2^m}$ .",
"Example 1 This example is based on the result in [12].",
"Let $m$ be an integer divisible by 4 but not by 5, let $b,c\\in {\\mathbb {F}}_{2^m}^*$ with $b^4+b+1 = 0$ , and let $d$ be the multiplicative inverse of 11 modulo $2^m-1$ .",
"Then the function $f:{\\mathbb {F}}_{2^m}\\times {\\mathbb {F}}_{2^m}\\rightarrow {\\mathbb {Z}}_{2^3}$ $ f(x,y) = {\\rm Tr}_m(c(1+b)y^dx) + 2{\\rm Tr}_m(c(1+b^{-1})y^dx) + 4{\\rm Tr}_m(cy^dx) $ is gbent [12].",
"Now observe that ${\\rm Tr}_m(c(1+b^{-1})y^dx)$ and ${\\rm Tr}_m(c(1+b)y^dx) \\oplus {\\rm Tr}_m(c(1+b^{-1})y^dx)$ are both Maiorana-McFarland bent functions.",
"Hence the function $f_1(x,y) = {\\rm Tr}_m(c(1+b)y^dx) + 2{\\rm Tr}_m(c(1+b^{-1})y^dx)$ is gbent in $\\mathcal {GB}_{2m}^4$ by [20].",
"The function $f_2(x,y) = {\\rm Tr}_m(c(1+b)y^dx)$ is bent, thus formally in $\\mathcal {GB}_{2m}^2={\\mathcal {B}}_{2m}$ .",
"Therefore, by Proposition REF , $f(x,y)$ is ${\\mathbb {Z}}_8$ -bent.",
"As our second example, we analyse the ${\\mathbb {Z}}_q$ -bent function given in Theorem 12 in [9] for $t=1$ .",
"The function is defined via spreads and it is not given in the form $(\\ref {eq:1})$ .",
"We start by recalling that a spread of ${\\mathbb {V}}_n$ , $n=2m$ , is a family $S$ of $2^m+1$ subspaces $U_0,U_1,\\ldots ,U_{2^m}$ of ${\\mathbb {V}}_n$ , whose pairwise intersection is trivial.",
"The classical example is the regular spread, which for ${\\mathbb {V}}_n = {\\mathbb {F}}_{2^m}\\times {\\mathbb {F}}_{2^m}$ is represented by the family $S = \\bigcup _{s\\in {\\mathbb {F}}_{2^m}} \\lbrace (x,sx)\\,:\\,x\\in {\\mathbb {F}}_{2^m}\\rbrace \\cup \\lbrace (0,y)\\,:\\,y\\in {\\mathbb {F}}_{2^m}\\rbrace $ .",
"For the regular spread in ${\\mathbb {V}}_n={\\mathbb {F}}_{2^n}$ we can take the family $S = \\lbrace \\alpha _i{\\mathbb {F}}_{2^m}\\,:\\,i = 1,\\ldots ,2^m+1\\rbrace $ , where $\\lbrace \\alpha _i\\,:\\,i = 1,\\ldots ,2^m+1\\rbrace $ is a set of representatives of the cosets of the subgroup ${\\mathbb {F}}_{2^m}^*$ of the multiplicative group ${\\mathbb {F}}_{2^n}^*$ (one may take the set of the $(2^m+1)$ -th roots of unity).",
"Example 2 Let $U_0,U_1,\\ldots ,U_{2^m}$ be the elements of a spread of ${\\mathbb {V}}_n$ , $n=2m$ .",
"We first construct a vectorial bent function $F$ , and thereafter a ${\\mathbb {Z}}_q$ -bent function $f$ .",
"We notice that $F$ and $f$ are connected as discussed in Remark REF .",
"Let $\\phi :\\lbrace 1,2,\\ldots ,2^{n/2}\\rbrace \\rightarrow {\\mathbb {F}}_2^k$ be a balanced map, thus any $y \\in {\\mathbb {F}}_2^k$ has exactly $2^{n/2-k}$ preimages in the set $\\lbrace 1,2,\\ldots ,2^{n/2}\\rbrace $ .",
"Then the function $F:{\\mathbb {V}}_n\\rightarrow {\\mathbb {F}}_2^k$ given by $ F(x) = \\left\\lbrace \\begin{array}{l@{\\quad :\\quad }l}\\phi (s) & x\\in U_s,\\,1\\le s\\le 2^m,\\;\\mbox{and}\\;x\\ne 0, \\\\0 & x\\in U_0,\\end{array}\\right.$ is a vectorial bent function, see e.g.",
"Theorem 4 in [3].",
"If $a_i\\in \\mathcal {B}_n$ , $0\\le i\\le k-1$ , are the coordinate functions of $F$ , i.e.",
"if $F(x) = (a_0(x),a_1(x),$ $\\ldots ,a_{k-1}(x))$ , then $F$ is the vector space of bent functions given as $\\langle a_0,a_1,\\ldots ,a_{k-1}\\rangle $ .",
"We now proceed with the construction of the ${\\mathbb {Z}}_q$ -bent function given as in [9].",
"From the balanced map $\\phi $ , we obtain in a natural way a balanced map $\\bar{\\phi }$ from $\\lbrace 1,2,\\ldots ,2^{n/2}\\rbrace $ to ${\\mathbb {Z}}_{2^k}$ defined as $\\bar{\\phi }(s) = y_0+2y_1+\\cdots +2^{k-1}y_{k-1}$ if $\\phi (s) = (y_0,y_1,\\ldots ,y_{k-1})$ .",
"By Theorem 12 in [9], the function $ f(x) = \\left\\lbrace \\begin{array}{l@{\\quad :\\quad }l}\\bar{\\phi }(s) & x\\in U_s,\\,1\\le s\\le 2^m,\\;\\mbox{and}\\;x\\ne 0, \\\\0 & x\\in U_0,\\end{array}\\right.$ from ${\\mathbb {V}}_n$ to ${\\mathbb {Z}}_{2^k}$ is ${\\mathbb {Z}}_q$ -bent.",
"Then, written in the form $(\\ref {eq:1})$ , $f$ is represented as $f(x) = a_0(x) + 2a_1(x) + \\cdots + 2^{k-1}a_{k-1}(x)$ , with the Boolean functions $a_i$ , $0\\le i\\le k-1$ , given as above.",
"We can change the representation of the vectorial bent function $F$ by changing the basis from $\\lbrace a_0,a_1,\\ldots ,a_{k-1}\\rbrace $ to $\\lbrace a_0^\\prime ,a_1^\\prime ,\\ldots ,a^\\prime _{k-1}\\rbrace $ .",
"The same vectorial bent function has then the representation $F(x) = \\lbrace a_0^\\prime (x),a_1^\\prime (x),\\ldots ,a^\\prime _{k-1}(x)\\rbrace $ .",
"This change of the basis implies a modification of $\\phi $ and $\\bar{\\phi }$ , and results also in an alternative formal expression for the ${\\mathbb {Z}}_q$ -bent function.",
"We emphasize that the property of being ${\\mathbb {Z}}_q$ -bent is much stronger than the property of being vectorial bent.",
"${\\mathbb {Z}}_q$ -bent functions are very interesting vectorial bent functions since they correspond to two relative difference sets with parameters $(2^n,2^k,2^n,2^{n-k})$ : First of all, being vectorial bent, they correspond to the relative difference set $D=\\lbrace (x,a_0(x),a_1(x),\\ldots ,a_{k-1}(x))\\;:\\;x\\in {\\mathbb {V}}_n\\rbrace $ in ${\\mathbb {V}}_n\\times {\\mathbb {F}}_2^k$ , and secondly, to the relative difference set $R=\\lbrace (x,a_0(x)+2a_1(x)+\\cdots +a_{k-1}(x))\\;:\\;x\\in {\\mathbb {V}}_n\\rbrace $ in ${\\mathbb {V}}_n\\times {\\mathbb {Z}}_{2^k}$ .",
"Moreover, further relative difference sets are enclosed in such a vector space of bent functions, the relative difference sets of the bent functions of the form $g_ig_j \\oplus g_ig_l \\oplus g_jg_l$ for some component functions $g_i,g_j,g_l$ .",
"These bent functions are in general not component functions of the vectorial bent function, hence their relative difference sets are not projections of $D$ .",
"Here we have provided a first systematic description of this class of vectorial bent functions.",
"There are many questions on analysis and construction of such functions which one can investigate.",
"We are convinced that these functions are an interesting target for future research."
],
[
"The dual and Gray map of gbent functions",
"In this section we firstly attempt to describe the dual $f^*$ of an arbitrary gbent function $f\\in \\mathcal {GB}^{2^k}_n$ .",
"Furthermore, the Gray map of gbent functions is considered."
],
[
"The dual of a gbent function",
"We start recalling a result of [5], which there is given more general for functions in $\\mathcal {GB}_n^q$ , $q$ even.",
"Theorem 4 [5] Let $f\\in \\mathcal {GB}_n^{2^k}$ be given as in $(\\ref {eq:1})$ , and $g_i(x) = a_{k-1}(x) \\oplus i_0a_0(x)\\oplus i_1a_1(x)\\oplus \\cdots \\oplus i_{k-2}a_{k-2}(x)$ , $0\\le i\\le 2^{k-1}-1$ , where $i=\\sum _{j=0}^{k-2}i_j2^j$ and $i_j \\in \\lbrace 0,1\\rbrace $ .",
"Then $\\zeta ^{f(x)}_{2^k} = \\sum ^{2^{k-1}-1}_{i=0}\\alpha _i (-1)^{g_i(x)},$ where $\\alpha _i$ is given by $\\alpha _i=2^{-(k-1)}H^{(i)}_{2^{k-1}}B,$ for the column matrix $B$ defined as $B=[\\zeta ^i_{2^k}]^{2^{k-1}-1}_{i=0}$ .",
"Consequently, for any $u\\in {\\mathbb {V}}_n$ we have $\\mathcal {H}_f(u)=\\sum ^{2^{k-1}-1}_{i=0}\\alpha _i \\mathcal {W}_{g_i}(u).$ For even $n$ we will describe the dual $f^*$ of a gbent function $f\\in \\mathcal {GB}_n^{2^k}$ via the duals of the component functions of $f$ .",
"Theorem 5 Let $n$ be even and $f\\in \\mathcal {GB}^{2^k}_n$ be a gbent function given as $f(x)=a_0(x)+2a_1(x)+\\cdots +2^{k-2}a_{k-2}(x)+2^{k-1}a_{k-1}(x),$ for some $a_i\\in \\mathcal {B}_n$ $(i=0,\\ldots ,k-1)$ , with component functions $g_j$ , $0\\le j\\le 2^{k-1}-1$ .",
"Then the dual $f^*\\in \\mathcal {GB}^{2^k}_n$ of the function $f$ is given as follows: $f^*(x)=b_0(x)+2b_1(x)+\\ldots +2^{k-2}b_{k-2}(x)+2^{k-1}b_{k-1}(x),\\;\\;x\\in {\\mathbb {V}}_n,$ where $b_{k-1}(x)=a^*_{k-1}(x)$ , $b_j(x)=a^*_{k-1}(x)\\oplus (a_{k-1}\\oplus a_{2^j})^*(x),$ $j=0,\\ldots ,k-2.$ $(i)$ From Theorem REF and the regularity of a gbent function $f$ , we have $ \\mathcal {H}_f(u) = \\sum _{i=0}^{2^{k-1}-1}\\alpha _i\\mathcal {W}_{g_i}(u) = 2^\\frac{n}{2}\\sum _{i=0}^{2^{k-1}-1}\\alpha _i(-1)^{g_i^*(u)}=2^\\frac{n}{2}\\zeta ^{f^*(u)}_{2^{k}} .",
"$ Suppose that $f^*(x) = b_0(x) + 2b_1(x) + \\cdots + 2^{k-1}b_{k-1}(x)$ and denote the component functions of $f^*$ by $h_i=b_{k-1}\\oplus i_0 b_0\\oplus \\ldots i_{k-2}b_{k-2}$ , $0\\le i\\le 2^{k-1}-1$ ($i=\\sum ^{k-2}_{j=0}i_j 2^{j}$ ).",
"By Theorem REF , $ \\zeta ^{f^*(x)}_{2^{k}}= \\sum _{i=0}^{2^{k-1}-1}\\alpha _i(-1)^{h_i(x)}.",
"$ Combining we get $ \\sum _{i=0}^{2^{k-1}-1}\\alpha _i(-1)^{h_i(u)} = \\sum _{i=0}^{2^{k-1}-1}\\alpha _i(-1)^{g_i^*(u)}.",
"$ Observing that $\\alpha _0,\\alpha _1,\\ldots ,\\alpha _{2^{k-1}-1}$ are linearly independent ${\\mathbb {Q}}(\\zeta )$ (invertible matrix times $(\\zeta _0,\\zeta _1,$ $\\ldots ,\\zeta _{2^{k-1}-1})$ ), we obtain $h_i(x) = g_i^*(x)$ , $i=0,1,\\ldots ,2^{k-1}-1$ (and all $x\\in {\\mathbb {V}}_n$ ).",
"Finally, $b_{k-1}=g^*_0=a^*_{k-1}$ , and with $g^*_{2^j} = b_{k-1}\\oplus b_j$ and $g_{2^j} = a_{k-1}\\oplus a_j$ , $j=1,\\ldots ,k-2$ , we get $ b_j = a_{k-1}^* \\oplus (a_{k-1}\\oplus a_j)^*,\\,j=1,\\ldots ,k-2.",
"$ Theorem REF generalizes the results in [12] where a similar conclusion was stated for $k=2,3$ only.",
"If $n$ is odd, then the component functions of $f$ are semi-bent, hence the description of the dual of $f$ for $n$ even cannot transfer to $n$ odd in a straightforward manner.",
"We leave the description of the gual od a gbent function $f\\in \\mathcal {GB}_n^{2^k}$ , $n$ odd, as an open problem."
],
[
"The Gray map of gbent functions",
"In this section we specify the Gray image of any gbent function by showing that its Gray map is a $(k-1)$ -plateaued function if $n$ is even, and $(k-2)$ -plateaued function if $n$ is odd.",
"This again generalizes the existing results that was derived in given in [8], [20] for $k=2,3$ and 4.",
"Let $f:{\\mathbb {V}}_n\\rightarrow {\\mathbb {Z}}_{2^k}$ be a generalized Boolean function given as $ f(x)=a_0(x)+2a_1(x)+2^2a_2(x)+\\cdots +2^{k-1}a_{k-1}(x),\\; \\forall x\\in {\\mathbb {V}}_n.",
"$ The generalized Gray map $\\psi (f):\\mathcal {GB}^{2^k}_n\\rightarrow \\mathcal {B}_{n+k-1}$ of $f$ is defined by, cf.",
"[1], $\\psi (f)(x,y_0,\\ldots ,y_{k-2})=\\bigoplus ^{k-2}_{i=0}a_i(x)y_i\\oplus a_{k-1}(x).$ We start with the following result.",
"Lemma 3 [8] Let $n, k-1\\ge 2$ be positive integers and $F:{\\mathbb {V}}_n\\times {\\mathbb {V}}_{k-1}\\rightarrow {\\mathbb {F}}_2$ be defined by $F(x,y_0,\\ldots ,y_{k-2})=a_{k-1}(x)\\oplus \\bigoplus ^{k-2}_{i=0}y_ia_i(x),\\;\\;\\; x\\in {\\mathbb {V}}_n,$ where $a_i\\in \\mathcal {B}_n$ , $0\\le i\\le k-1$ .",
"Denote by $A(x)$ the vectorial Boolean function $A=(a_0(x),\\ldots ,a_{k-2}(x))$ and let $u\\in {\\mathbb {V}}_n$ and $z_r\\in {\\mathbb {V}}_{k-1}$ .",
"The Walsh-Hadamard transform of $F$ at point $(u,z_r)\\in {\\mathbb {V}}_n\\times {\\mathbb {V}}_{k-1}$ is then $\\mathcal {W}_F(u,z_r)=\\sum _{z_j\\in {\\mathbb {V}}_{k-1}}(-1)^{z_j\\cdot z_r}\\mathcal {W}_{a_{k-1}\\oplus z_j\\cdot A}(u)=H^{(r)}_{2^{k-1}}\\mathcal {W}^T(u),$ where $\\mathcal {W}(u)$ is the row vector defined by (REF ), i.e., $\\mathcal {W}(u)=(\\mathcal {W}_0(u),\\ldots ,\\mathcal {W}_{2^{k-1}-1}(u))$ and $\\mathcal {W}_j(u)=\\mathcal {W}_{a_{k-1}\\oplus z_j\\cdot A}(u),$ $j=0,\\ldots ,2^{k-1}-1$ .",
"We now can show that the Gray map of a gbent function in $\\mathcal {GB}_n^{2^k}$ is a certain plateaued function, thus generalizing the results on Gray maps in [22] and [8] which were only given for $q =4,8,16$ .",
"Proposition 6 Let $f\\in \\mathcal {GB}_n^{2^k}$ , $n$ even, be a gbent function.",
"Then $\\psi (f)$ is a $(k-1)$ -plateaued function in $\\mathcal {B}_{n+k-1}$ , thus $\\mathcal {W}_{\\psi (f)} \\in \\lbrace 0,\\pm 2^{n/2 + k-1}\\rbrace $ .",
"By Theorem REF , for any $u\\in {\\mathbb {V}}_n$ we have $\\mathcal {W}(u)=\\pm 2^{\\frac{n}{2}}H^{(r)}_{2^{k-1}}$ ($f$ is gbent), for some $r\\in \\lbrace 0,\\ldots ,2^{k-1}-1\\rbrace .$ Then for arbitrary $(u,z_j)\\in {\\mathbb {V}}_n\\times {\\mathbb {V}}_{k-1}$ , where $z_j\\in {\\mathbb {V}}_{k-1}$ , from Lemma REF we have $F=\\psi (f)$ and thus $\\mathcal {W}_{\\psi (f)}(u,z_j)&=&H^{(j)}_{2^{k-1}}\\mathcal {W}^T(u)=H^{(j)}_{2^{k-1}}(\\pm 2^{\\frac{n}{2}}H^{(r)}_{2^{k-1}})^T=\\pm 2^{\\frac{n}{2}}H^{(j)}_{2^{k-1}}(H^{(r)}_{2^{k-1}})^T\\\\&=&\\left\\lbrace \\begin{array}{cc}\\pm 2^{\\frac{n}{2}+k-1}, & r=j \\\\0 & r\\ne j\\end{array}\\right.,$ since $H^{(j)}_{2^{k-1}}(H^{(r)}_{2^{k-1}})^T=\\left\\lbrace \\begin{array}{cc}2^{k-1}, & r=j \\\\0, & r\\ne j\\end{array}\\right.$ , where $H^{(j)}_{2^{k-1}}, H^{(r)}_{2^{k-1}}$ are considered as row vectors.",
"Clearly, for $k\\ge 1$ we have $\\mathcal {W}_{\\psi (f)}(u,z_j)\\in \\lbrace 0,\\pm 2^{\\frac{n}{2}+k-1}\\rbrace $ , which means that $\\psi (f)$ is a $(k-1)$ -plateaued function in $\\mathcal {B}_{n+k-1}$ .",
"Proposition 7 Let $f$ (defined by (REF )) be a gbent function in $\\mathcal {GB}^{2^k}_n$ , $n$ odd.",
"Then $\\psi (f)$ is a $(k-2)$ -plateaued function in $\\mathcal {B}_{n+k-1}$ , thus $\\mathcal {W}_{\\psi (f)} \\in \\lbrace 0,\\pm 2^{\\frac{n+1}{2} + k-2}\\rbrace $ .",
"Recall that for any $u\\in {\\mathbb {V}}_n$ we have $\\mathcal {W}(u)=(\\pm 2^{\\frac{n+1}{2}}H^{(r)}_{2^{k-2}},\\textbf {0}_{2^{k-2}})\\;\\; \\text{or}\\;\\;\\mathcal {W}(u)=(\\textbf {0}_{2^{k-2}},\\pm 2^{\\frac{n+1}{2}}H^{(r)}_{2^{k-2}}),$ for some $r\\in \\lbrace 0,\\ldots ,2^{k-2}-1\\rbrace .$ Consequently, for any $(u,z_j)\\in {\\mathbb {V}}_n\\times {\\mathbb {V}}_{k-1}$ , $W_{\\psi (f)}(u,z_j)=H^{(j)}_{2^{k-1}}\\mathcal {W}^T(u)=\\left\\lbrace \\begin{array}{cc}\\pm 2^{\\frac{n+1}{2}+k-2}, & r\\in \\lbrace j,j+2^{k-2}\\rbrace \\\\0 & r\\notin \\lbrace j,j+2^{k-2}\\rbrace \\end{array}\\right.,$ what completes the proof.",
"Remark 4 Note that Proposition REF and Proposition REF hold for any $q$ if $f$ is constructed by [5].",
"Acknowledgement.",
"Samir Hodžić is supported in part by the Slovenian Research Agency (research program P3-0384 and Young Researchers Grant).",
"Wilfried Meidl is supported by the Austrian Science Fund (FWF) Project no.",
"M 1767-N26.",
"Enes Pasalic is partly supported by the Slovenian Research Agency (research program P3-0384 and research project J1-6720)."
]
] | 1605.05713 |
[
[
"Test of the Einstein equivalence principle with spectral distortions in\n the cosmic microwave background"
],
[
"Abstract The Einstein Equivalence Principle~(EEP) can be verified by the measurement of the spectral distortions of the Cosmic Microwave Background (CMB).",
"The existence of energy-dependency in the cosmological redshift effect means the EEP violation.",
"Introducing the energy-dependent Friedmann-Robertson-Walker metric motivated by rainbow gravity, we show that the energy-dependent redshift effect causes the CMB spectral distortions.",
"Assuming the simple energy-dependent form of the metric, we evaluate the distortions.",
"From the COBE/FIRAS bound, we find that the deviation degree from the EEP, which is comparable to the difference of the parameterized-post-Newtonian parameter \"gamma\" in energy, is less than 10^{-9} at the CMB energy scale."
],
[
"Introduction",
"Observations of the cosmic microwave background (CMB) have become essential tools in modern cosmology.",
"Precise measurements of the CMB temperature and polarization anisotropies provide valuable information about the Universe [1].",
"Recently, the measurement of CMB spectral distortions, that is, the deviation of the CMB frequency spectrum from a blackbody spectrum, has been expected as a new cosmological probe.",
"COBE/FIRAS has obtained the almost perfect blackbody spectrum of the CMB with the temperature $T_0 = 2.726$ K [2].",
"Although they have not been detected yet, CMB distortions can be generated within the standard cosmological model as well as with new physics (for reviews, see [3], [4], [5]).",
"Currently, the observational bound on the spectral distortions is given in terms of two types of distortions, $\\mu $ and $y$ -type distortions [6], [7].",
"The $\\mu $ -type distortion is described with a nonvanishing chemical potential $\\mu $ and created at $10^6 \\gtrsim z \\gtrsim 5 \\times 10^4$ where, even if the CMB spectral distortionsarise, Compton scattering is efficient enough to maintain the kinetic equilibrium of CMB photons.",
"The $y$ -type distortion is parametrized by the Compton $y$ parameter and generated in lower redshifts $z<5 \\times 10^4$ where, once the CMB spectrum is distorted, the kinetic equilibrium of CMB photons is no longer maintained.",
"The current constraints on the distortion parameters are provided by COBE/FIRAS as $|y| < 1.5\\times 10^{-5}$ and $|\\mu |<9\\times 10^{-5}$ [8].",
"To improve these bounds, next-generation CMB spectrometers are being discussed [9], [10].",
"The future measurements or constraints on the CMB distortions allow us to access the properties of primordial fluctuations [11], the nature of dark matter [12], the abundance of primordial black holes [13], the existence of primordial magnetic fields [14] and other high-energy physics [15].",
"In this paper, we discuss that the measurement of CMB distortions can also test general relativity (GR), in particular, the Einstein equivalence principle (EEP).",
"Since GR was proposed as the theory of gravity by Einstein, the theory has passed almost all tests such as ground-based and Solar System experiments [16].",
"And furthermore, the gravitational wave detection by LIGO proves the accuracy of the theory even in a strong gravitational field [17], [18].",
"However, it still leaves room to verify GR at the cosmological scales.",
"Since the first evidence was presented by the type-Ia supernova observations [19], [20], independent cosmological observations strongly support the accelerating expansion of the Universe.",
"As an origin of this acceleration, GR requires the existence of unknown dark energy.",
"Alternatively, the modification of GR on cosmological scales is suggested to explain the acceleration as an effect of gravity [21], [22], [23].",
"Therefore, it is still quite important to verify GR in the cosmological context, and we pay attention to the validity of the EEP which is one of the fundamental principles in GR.",
"The EEP is tested from laboratory to Solar System scales by many authors (for reference, see Ref. [24]).",
"In these studies, the validity of the EEP has been obtained from the travels of a test particle through the gravitational potential.",
"Therefore, as the constraint on the EEP, these studies have provided the constraint on the energy dependency of parametrized-post-Newtonian (PPN) parameter $\\gamma $ .",
"Recently, this energy dependency has been also tested by using high-energy photons emitted by gamma ray bursts, fast radio bursts, and TeV blazers with the gravitational potential of the Milky Way [25], [26], [27], [28], [29], [30], [31], [32].",
"In this paper, we focus on the independency of the cosmological redshift effect on the energy of a test particle.",
"Because this energy independency is one of the consequences of the EEP on cosmological scales, it is important to test the independency of the redshift effect by cosmological observations.",
"We show that the independency of the redshift effect can be verified by measurement of the CMB distortion.",
"After submitting our paper, Ref.",
"[33] appeared.",
"They have investigated the energy dependence of the cosmological redshift effect using the emission lines over the 3700–6800 Å range in SDSS spectroscopic data at $0.1 <z <0.25$ .",
"Their conclusion is that they cannot find any energy-dependence of the redshift with a precision of $10^{-6}$ at $z<0.1$ and $10^{-5}$ at $0.1<z<0.25$ .",
"Our method is complementary with theirs because probing energy is different.",
"Besides, CMB distortion can verify the EEP up to redshifts larger than $z\\sim 1000$ .",
"To demonstrate the test of the EEP through the CMB distortion, we introduce a simple energy dependence of the Friedmann-Robertson-Walker (FRW) metric.",
"Generally, when a metric depends on energy, the EEP is violated in this metric theory of gravity.",
"In other words, the existence of the energy dependence of the metric means that the structure of spacetime felt by a test particle depends on its own energy.",
"In GR, although CMB photons are redshifted due to the cosmic expansion, the blackbody spectrum of the CMB is hold during their free-streaming because the EEP ensures that redshift effect is independent of the photon energy.",
"However, when the redshift effect depends on the photon energy, the deviation from the blackbody spectrum arises even in the free streaming regime.",
"We evaluate the CMB distortion and obtain the constraints on the accuracy of the EEP on cosmological time and length scales through a comparison with the COBE/FIRAS data."
],
[
"energy-dependent FRW metric",
"Since the energy dependency of the metric violates the EEP, we first consider the energy-dependent FRW metric.",
"Taking into account the cosmological principle, we can be allowed to introduce two energy dependent functions, $f(E)$ and $g(E)$ , in the FRW metric as $ds^2 = -\\frac{dt^2}{f^2(E)}+\\frac{a^2(t)}{g^2(E)}\\delta _{ij}dx^idx^j\\,, $ where $E$ denotes the energy of a photon observed by a free-falling observer in this metric, and $f(E)$ and $g(E)$ are arbitrary functions of $E$ .",
"Although, without assuming any certain gravity theory, we determine the form of Eq.",
"(REF ) based on the cosmological principle, the same energy dependence is often discussed in rainbow gravity, which is one of the gravity theories without the EEP [34], [35], [36].",
"Since the FRW metric depends on the energy, the redshift effect due to the cosmological expansion also has energy dependence.",
"To derive the redshift effect, we consider the geodesic equation for a photon with energy $E$ .",
"In the metric given by Eq.",
"(REF ), nonvanishing Christoffel symbols are $&\\Gamma ^{0}_{00} = -\\frac{\\dot{f}}{f}\\,,\\quad ~\\Gamma ^{0}_{ij} = \\left(\\frac{f}{g}\\right)^2\\left(a\\dot{a}-a^2\\frac{\\dot{g}}{g}\\right)\\delta _{ij}\\,,\\nonumber \\\\&\\Gamma ^{i}_{0j} = \\left(\\frac{\\dot{a}}{a}-\\frac{\\dot{g}}{g}\\right)\\delta ^i_j\\,,$ where the dot denotes the derivative with respect to time.",
"Therefore, the geodesic equation provides the modified redshift effect, $\\dot{E} = -\\frac{\\dot{a}}{a} \\left({1 -\\frac{d \\log g}{d \\log E}} \\right)^{-1} E.$ When $f$ and $g$ are constant, the redshift effect is the same as in GR."
],
[
"CMB distortions due to the energy-dependent redshift effect",
"After the epoch of recombination, the universe becomes transparent for photons and they are free to stream out.",
"During such a free-streaming regime, the evolution of the CMB photon energy distribution is given by the collisionless Boltzmann equation.",
"Assuming the homogeneity and isotropy of the Universe, the collisionless Boltzmann equation in the metric by Eq.",
"(REF ) can be described as $\\frac{\\partial {n_E}}{\\partial {t}} -\\frac{\\dot{a}}{a} E\\left( 1 - \\frac{d \\log g}{d \\log E} \\right)^{-1}\\frac{\\partial {n_E}}{\\partial {E}} = 0\\,.$ Although the general solution of Eq.",
"(REF ) is provided in a function of the combination value, $a E/g$ , we need the initial condition of the energy distribution to solve Eq.",
"(REF ).",
"Well before the epoch of recombination, the time scale of thermal equilibrium for CMB photons is much shorter than the cosmological time scale.",
"In this regime, when the deviation from a blackbody spectrum arises, the deviation is quickly erased and the blackbody spectrum is maintained.",
"Therefore, for simplicity, we assume that the energy distribution of the CMB is a blackbody spectrum, $[\\exp (E/T_{\\rm re})-1]^{-1}$ , at the epoch of recombination, where $T_{\\rm re}$ is the temperature at that epoch.",
"However, as mentioned above, CMB distortions can be generated below $z \\sim 10^6$ , which is well before the epoch of the recombination.",
"During this regime, the evolution of the CMB distortions is provided by the collisional Boltzmann equation.",
"We will discuss this issue later.",
"With this assumption, the solution of Eq.",
"(REF ) is given by $n_E=\\frac{1}{\\exp [\\eta (E, z) E/T_z] -1},$ where $T_z = T_{\\rm re} (1+z)/(1+ z_{\\rm re})$ with the redshift for the epoch of recombination $z_{\\rm re}$ , and $\\eta (E,z)$ is provided by $\\eta (E,z) =\\frac{g(E_{\\rm re}(E,z) )}{g(E)},$ where the function $E_{\\rm re}(E,z)$ represents the energy at $z_{\\rm re}$ for a photon whose energy is redshifted to $E$ at the redshift $z$ .",
"We can obtain $E_{\\rm re}(E,z)$ from Eq.",
"(REF ).",
"Since various tests support the validity of GR, we assume that $g^{-1}$ can be approximated in $g^{-1} \\approx 1 + h(E)$ with $h(E) \\ll 1$ .",
"In the leading order of $h(E)$ , the function $\\eta (E,z)$ can be expanded in $\\eta (E,z) \\approx 1+h(E) - h \\left(\\frac{1+ z_{\\rm re}}{1+z} E\\right).$ The aim of this paper is to obtain the constraint on $h(E)$ from the measurement of the CMB distortions.",
"Here we demonstrate two simple cases of the function $h(E)$ .",
"In the first case, $h(E)$ is a linear function of $E$ .",
"In the second case, $h(E) $ is proportional to $E^{-1}$ ."
],
[
" The case with $h(E) \\propto E $",
"We assume that the form of $h(E)$ is given by $h(E) = \\delta _{T_0} E/T_0,$ with $\\delta _{T_0} \\ll 1 $ .",
"Here the parameter $\\delta _{T_0}$ represents the deviation degree from the EEP at the energy scale $T_0$ .",
"The CMB photon energy distribution at the present epoch is given from Eqs.",
"(REF ) and (REF ).",
"Expanding the photon energy distribution up to the linear order of $\\delta _{T_0}$ , we obtain $n_E \\approx \\frac{1}{\\exp (E/T_0) -1 } +\\frac{\\exp (E/T_0)}{ [\\exp (E/T_0) -1]^2}\\left(\\frac{E}{T_0} \\right)^2z_{\\rm re} \\delta _{T_0}.$ The first term represents the blackbody spectrum with $T_0$ and the second term provides the deviation from the blackbody spectrum.",
"We define the relative deviation from the blackbody spectrum as $\\Delta _E = (n_E - n_{{\\rm BB},E}) /n_{{\\rm BB},E}$ where $n_{{\\rm BB},E}$ is the blackbody spectrum with $T_0$ .",
"According to Eq.",
"(REF ) $\\Delta _E$ is given by $\\Delta _E = \\frac{\\exp (E/T_0)}{ \\exp (E/T_0) -1}\\left(\\frac{E}{T_0} \\right)^2z_{\\rm re} \\delta _{T_0}.$ We show $\\Delta _E$ as a function of $E$ in Fig.",
"REF .",
"Here we set $\\delta _{T_0}=10^{-9}$ and $z_{\\rm re}=1100$ .",
"COBE/FIRAS has provided the possible residual from the blackbody spectrum [8].",
"We plot the residual as blue points in Fig.",
"REF .",
"From the figure, we conclude that COBE/FIRAS gives the upper bound, $|\\delta _{T_0}| \\lesssim 10^{-9}.$ Figure: The relative deviation from the blackbody spectrum,Δ E \\Delta _E.",
"Here weadopt h(E)=δ T 0 E/T 0 h(E) = \\delta _{T_0 } E/T_0 with δ T 0 =10 -9 \\delta _{T_0} = 10^{-9}.",
"Thexx axis is theenergy of CMB photons in units of Kelvin.",
"Blue points represent theresidual measured by COBE/FIRAS ."
],
[
" The case with $h(E) \\propto E^{-1}$",
"Next we consider the case where $h(E)$ is represented as $h(E) = \\delta _{T_0} T_0/E .$ Similarly to the previous case, we can obtain the CMB photon distribution from Eqs.",
"(REF ) and (REF ).",
"The CMB photon distribution can be approximated to $n_E \\approx \\left(\\exp \\left[\\frac{E}{T_0} \\left(1+ \\frac{z_{\\rm re}}{1+z_{\\rm re}}\\frac{T_0}{E}\\delta _{T_0} \\right)\\right] -1\\right)^{-1}.$ This corresponds to the Bose-Einstein distribution, $n_E = (\\exp (E/T_0 + \\mu ) -1 )^{-1} $ , with the dimensionless chemical potential $\\mu =z_{\\rm re} \\delta _{T_0}/(1+z_{\\rm re})$ .",
"COBE/FIRAS provides the constraint on $\\mu $ for CMB photons, $|\\mu | < 9\\times 10^{-5}$ .",
"Therefore we obtain the limit, $|\\delta _{T_0}| \\lesssim 9 \\times 10^{-5}.$ Currently, PIXIE is designed to be 3 orders of magnitude better than COBE/FIRAS in the sensitivity [9].",
"The sensitivity of PIXIE is expected to be close to that required to measure the distortions arising from the dissipation of the scale-invariant primordial fluctuations, $\\mu \\sim 10^{-8}$ , which is one of unavoidable cosmological sources for CMB distortions.",
"When PIXIE provides the constraint $\\mu \\lesssim 10^{-8}$ , the constraint on the EEP reaches $|\\delta _{T_0}| \\lesssim 10^{-7}$ in the case of Eq.",
"(REF )."
],
[
"Conclusions",
"In this paper, we have proposed that the measurement of CMB spectral distortions can test the accuracy of the EEP on cosmological scales.",
"The energy independence of the cosmological redshift effect is one of consequences of the EEP.",
"When the FRW metric has energy dependence, the EEP is violated on cosmological scales.",
"As a result, the geodesic equation of a photon is modified and the redshift effect depends on its energy.",
"We have shown that, in the energy-dependent FRW metric, CMB distortions are generated even in the free-streaming regime through the energy-dependent redshift effect.",
"The shape and amplitude of the distortion depends on the form of the energy dependency.",
"To parametrize the validity of the EEP in the FRW metric, we have introduced the deviation parameter $\\delta _{T_0}$ representing the deviation from the EEP on the CMB energy scale.",
"We have analytically evaluated the CMB distortions in two simple power-law cases of the energy-dependent deviation in the FRW metric with the power law indices $n=1$ and $n=-1$ .",
"In the first case with $n=1$ , we have found that the COBE/FIRAS bound indicates that the EEP is valid within the degree of the deviation, $|\\delta _{T_0}| \\lesssim 10^{-9}$ , on the CMB energy scale, $0.0001$ –1 eV.",
"When $n>0$ , the deviations at higher energy scales are larger than at lower energy scales.",
"This means that, as $n$ becomes larger, the deviation increases at higher redshifts.",
"Therefore, the constraint on $\\delta _{T_0}$ becomes tighter when $n$ increases.",
"In the second case with $n=-1$ , the generated distortion is represented as the $\\mu $ -type distortion and the COBE/FIRAS bound provides the constraint $|\\delta _{T_0}| \\lesssim 10^{-5}$ .",
"When $n<0$ , the deviations at lower energy scales are larger than at higher energy scales.",
"Therefore, we obtain $|\\delta _{T_0}| \\lesssim 10^{-5}$ for $n<0$ .",
"Depending on the energy dependence of the FRW metric, the spectral shape is different from the ordinary CMB distortions, $\\mu $ - and $y$ -type distortions.",
"Therefore, the precise measurement of the distortion shape can provide us with a strong constraint on the EEP violation.",
"It is worth summarizing previous works about the test of the EEP and providing comments on the relevance of our study.",
"In previous studies, the accuracy of the EEP is investigated with the energy dependency of the PPN parameter $\\gamma $ .",
"Using the gamma ray observations, the constraint is provided as $\\gamma _{\\rm GeV} -\\gamma _{\\rm MeV} \\lesssim 10^{-8}$ and $\\gamma _{\\rm eV} -\\gamma _{\\rm MeV} \\lesssim 10^{-7}$ [27].",
"In the radio frequency range, the energy difference of $\\gamma $ is less than $10^{-8}$ , which is comparable to our results, from the observations of fast radio bursts [28], [32].",
"Since the constraints on the energy difference of $\\gamma $ is related to the gravitational potential, these constraints are valid for the Schwarzschild metric.",
"Therefore, these constraints cannot be directly applicable to the FRW metric without taking a theory of gravity.",
"In Ref.",
"[37], the authors have discussed that the measurement of CMB distortions can provide the constraint on the time variation of the fine structure constant due to the EEP violation in the electromagnetic sector.",
"Our constraint is completely independent of these limits.",
"In more detail, we have provided a bound on the EEP in the FRW metric for the cosmological time scale from the epoch of recombination to the present time.",
"Additionally, upcoming observations with PIXIE provide 3 orders of magnitude stronger constraints than that of COBE/FIRAS.",
"Recently Ref.",
"[33] has investigated the energy dependence of the cosmological redshift with SDSS data.",
"Their result is consistent with no energy dependence of the redshift effect with a precision of $10^{-6}$ at $z<0.1$ and $10^{-5}$ at $0.1<z<0.25$ .",
"In this work, they used the spectral lines over the 3700–6800 Å range whose energy range is higher than in the CMB observation frequencies.",
"Although their investigated redshifts are not so high, their results are complementary with our results.",
"According to both results, the violation of the EEP in the FRW metric is not found in the range from microwave to optical frequencies in the current observation precision.",
"In this paper, we have demonstrated that the measurement of the CMB distortions can test the EEP in the FRW metric by taking some assumptions.",
"In particular, to evaluate the CMB distortions analytically, we neglected the evolution of the CMB distortions before the epoch of recombination.",
"Although the distortions can be generated in the energy-dependent FRW metric before this epoch, it is necessary to solve the collisional Boltzmann equation numerically.",
"Because the next-generation CMB spectrometers are being planned to measure CMB distortions precisely, further detailed calculation is required.",
"We will address these issues for the EEP bound in our future works.",
"The spectral distortions of the CMB can be generated by other physical mechanisms, in particular, the processes related to the thermal history of the Universe.",
"Therefore, it is difficult to solve these degeneracies to point out the effect of the EEP violation by only the CMB distortion measurement.",
"However, although we have only studied the CMB distortions of the CMB, neutrinos and gravitons also suffer an energy-dependent redshift effect and their spectra are modified from ones in the standard cosmology when the EEP is violated in the FRW metric.",
"Therefore, the frequency spectral measurement of not only CMB but also neutrinos and gravitational waves can allow us to obtain the observational suggestion to the EEP violation.",
"We thank Naoshi Sugiyama, Yuko Urakawa, and Jens Chluba for the useful discussions.",
"S.A. and D.N.",
"are in part supported by the Ministry of Education, Culture, Sports, Science and Technology, Japan (MEXT) Grant-in-Aid for Scientific Research on Innovative Areas, No.",
"15H05890, H.T.",
"is supported by Japan Society for the Promotion of Science (JSPS) KAKENHI Grant No.",
"15K17646 and MEXT's Program for Leading Graduate Schools “ Ph.D. professionals, Gateway to Success in Frontier Asia.\""
]
] | 1605.05852 |
[
[
"Cosmological backreaction in higher-derivative gravity expansions"
],
[
"Abstract We calculate a general effective stress-energy tensor induced by cosmological inhomogeneity in effective theories of gravity where the action is Taylor-expandable in the Riemann tensor and covariant derivatives of the Riemann tensor.",
"This is of interest as an effective fluid that might provide an alternative to the cosmological constant, but it also applies to gravitational waves.",
"We use an adaptation of Green and Wald's weak-averaging framework, which averages over perturbations in the field equation where the perturbation length scales are small compared to the averaging scale.",
"In this adaptation, the length scale of the effective theory, $1/M$, is also taken to be small compared with the averaging scale.",
"This ensures that the perturbation length scales remain in fixed proportion to the length scale of the effective theory as the cosmological averaging scale is taken to be large.",
"We find that backreaction from higher-derivative terms in the effective action can continue to be important in the late universe, given a source of sufficiently high-frequency metric perturbations.",
"This backreaction might also provide a window on exotic particle physics in the far ultraviolet."
],
[
"Introduction",
"The standard cosmological concordance model is the $\\Lambda $ CDM model.",
"Assumed by this is the Cosmological Principle, that the universe is homogeneous and isotropic at sufficiently large distance scales and can thus the metric at large scales is accurately approximated by a Friedmann-Lemaître-Robertson-Walker (FLRW) metric.",
"This is supported by astronomical observations, which indicate that the universe transitions to homogeneity at length scales of order 100 $h^{-1}$ Mpc, see for example the analyses in [1], [2], [3], [4], [5], [6].",
"Homogeneity has also been verified at a much larger volume scale of 14 $h^{-3}$ Gpc in [7].",
"The $\\Lambda $ CDM model considers a universe that consists of homogeneous fluids, which are the cosmological constant, cold dark matter, a small amount of baryonic matter and radiation, the latter being unimportant at late times.",
"At small distance scales, the universe is obviously not homogeneous.",
"The density of the Earth is a factor of $10^{31}$ greater than the cosmological average and nucleons are a factor of $10^{46}$ more dense than the cosmological average.",
"The standard cosmological model is a simplified picture in which it is assumed that these variations can be averaged out at large scales without introducing significant changes to the dynamics or expansion rate of the universe.",
"A problem with this view is that Einstein's field equations for General Relativity (GR) are non-linear, with the result that performing an averaging procedure on the equations does not merely return the same equations with an averaged metric, but rather includes extra terms that could be interpreted as additional effective fluids.",
"This effect is called cosmological backreaction.",
"The backreaction effects in Einstein's GR are typically considered to be small, radiation-like and unimportant in late-universe cosmology, as argued in [8] and [9].",
"Much of the modern interest in backreaction comes from applying Buchert's averaging scheme, see for example [10], [11], [12], [13].",
"Buchert's approach has also been applied to modified theories of gravity in [14] and [15].",
"An extensive literature as accumulated on the rôle of inhomogeneity in cosmology, see for example [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29].",
"An elegant and rigorous framework for studying cosmological inhomogeneity has been proposed by Green and Wald in [30].",
"They demonstrated their framework against specific examples in [31].",
"In this framework, an effective stress-energy tensor is calculated for the backreaction.",
"The method uses a generalization of Burnett's work on gravitational waves [32] to the non-vacuum case where there exists a stress-energy tensor that satisfies the weak energy condition.",
"That in turn is a mathematically rigorous formulation of Isaacson's high-frequency approximation for the stress-energy tensor of gravitational waves from a distant source [33], [34].",
"The effective stress-energy tensor obtained, given Einstein's theory of gravity, has a vanishing trace, indicating that it is radiation-like.",
"Such a form for the backreaction cannot be important in late-universe cosmology.",
"This conclusion was contested in [35], the criticisms were responded to in [36] and a simpler argument was provided in [37].",
"If one accepts that the metric does converge under a suitable averaging procedure to a FLRW background at large distance scales, as argued in [38], this conclusion is rigorous, given Einstein's gravity theory.",
"Extending Einstein's GR to include higher-derivative terms can be motivated theoretically via constructing low-energy effective thoeries.",
"Effective theories can be constructed via Renormalization Group (RG) flows by integrating out high-energy modes down to some cutoff scale [39].",
"The effective action will then appear as a series expansion in local operators with coefficients carrying mass dimensions as powers of the cutoff.",
"Locality is an essential feature, since RG is fundamentally underpinned by Kadanoff blocking [40], [41].",
"A useful introduction can be found in [42].",
"When constructing an effective theory of gravity, it is usually required that the effective action is diffeomorphism-invariant, ensuring that its description is independent of our choice of spacetime coordinates.",
"In practice, this means that the effective action expansion is in the Riemann tensor and covariant derivative operators, with higher mass dimension operators being suppressed by higher powers of the cutoff.",
"A manifestly diffeomorphism-invariant Exact RG has recently been constructed for gravity at the classical level [43].",
"In addition to the advantage of manifest diffeomorphism invariance, its background-independent construction allows for very easy implementation in this study of backreaction.",
"For an alternative exploration of background independence in the Exact RG for gravity, see [44].",
"Higher-derivative terms are also motivated phenomenologically for constructing cosmological models.",
"The Starobinsky $R^2$ term [45], [46], provides a mechanism for early-universe inflation that is currently favoured by observations of the Cosmic Microwave Background, given a scale for $M\\sim 10^{13}$ GeV: see the results from WMAP [47] and Planck [48], [49].",
"The action for this can be written as $S = \\int d^4 x\\sqrt{-g}\\left[\\frac{1}{16\\pi G}\\left(R+\\frac{R^2}{6M^2}-2\\Lambda \\right)+\\mathcal {L}_{\\rm Matter}\\right].$ The “natural” scale for quantum gravity is usually taken from Newton's gravitational constant in the natural units commonly used in particle physics to be of order $M \\sim 10^{19}$ GeV, but a closer scale where interesting physics is anticipated is the Grand Unified Theory (GUT) scale of $10^{16}$ GeV, where a unification of the Standard Model gauge interactions is commonly expected.",
"This paper further generalizes work on backreaction in $R+R^2 /6M^2$ gravity developed in [50], using an adaptation of the Green and Wald framework.",
"An alternative approach that uses the Green and Wald framework under the approximation that the stress-energy tensor is set to its background form can be found in [51].",
"Another study based on Green and Wald's formalism has considered the case where gravity is coupled to a massless scalar field [52].",
"The stress-energy tensor for gravitational waves in higher-derivative gravity has also been studied using Isaacson's formulation in [53] and [54].",
"Both of these studies agree with this work that those Lagrangian terms that are of cubic order or higher in the Riemann tensor do not contribute to the effective stress-energy tensor, as will be discussed in Section REF .",
"Both studies differ from this work in that they both fix a gauge rather than maintaining diffeomorphism invariance, which will be demonstrated for this work in Section REF .",
"Another difference is that [53] evaluates the stress-energy tensor at asymptotically-flat future null infinity, and [54] sets the background to Minkowski, whereas in this work we consider a general cosmological spacetime background.",
"The extension to higher-derivative gravity presents the challenge of incorporating the additional “scalaron” mass scale, $M$ , into Green and Wald's existing framework.",
"As will be discussed also in this paper, the result found in [50] for the simple $R+R^2/6M^2$ theory was that the effective stress-energy tensor due to backreaction was no longer traceless, but rather possessed a negative-definite pure trace component.",
"This exciting result opened the possibility that cosmological backreaction in a theory of gravity with higher-derivative terms could effectively mimic a positive cosmological constant, offering an alternative to the standard $\\Lambda $ CDM model.",
"Since the scalaron mass would be expected to be large, corresponding to high-energy modes integrated out of a more fundamental theory of gravity, the effect of this term would be strongly suppressed unless the length scale of the perturbations were also of some very short distance scale.",
"To emphasise this point, let us fix a generous value for $M$ at $3\\times 10^{13}$ GeV, as motivated by Starobinsky inflation, and let us consider the inhomogeneity as a single Fourier mode in the metric with wavelength similar to the radius of the Earth, i.e.",
"$L$ where $1/L\\sim 3\\times 10^{-23}$ GeV in natural units.",
"An operator of the form $\\nabla ^2/M^2$ would then introduce a suppression by a factor of $\\sim 1/(ML)^2$ , i.e.",
"72 orders of magnitude.",
"Larger length scales for the inhomogeneity, e.g.",
"galaxy clusters, result in even larger suppression of higher derivatives.",
"To obtain a significant effect from the higher derivatives, we need to consider more high-frequency sources of inhomogeneity that would be related to high-energy particle physics, rather than the large structures ordinarily considered in astrophysics.",
"Suggested in the conclusions of [50] were two examples of exotic sources for such inhomogeneity.",
"The first suggestion was WIMPzillas [55], [56], however the dilution of the WIMPzilla density as the universe expands makes this suggestion phenomenologically unattractive.",
"The second, even more speculative, suggestion was that fluctuations in quantum spacetime might average to smooth classical perturbations at a length scale to which the effective gravity theory is sensitive.",
"Such an idea would risk problems with naturalness; it would, however, scale appropriately as the universe expands.",
"For an example of spontaneous breaking of translational symmetry in Planck-scale quantum gravity, see [57].",
"A recent study proposed that the required inhomogeneity could be sourced from a vacuum that breaks translational symmetry in a non-Abelian gauge theory that is in a sector of particle physics disconnected from the Standard Model [58].",
"For a choice of $M\\sim 10^{13}$ GeV, they concluded that an inhomogeneity wavelength at the electroweak scale would mimic a cosmological constant of the correct energy scale, which is $\\sim 10^{-12}$ GeV.",
"More specifically, the scaling of the effective vacuum energy in $R+R^2/6M^2$ gravity was estimated to be $E_{\\rm vac} \\sim \\frac{\\Lambda ^2_{\\rm stripe}}{\\sqrt{MM_{\\rm Planck}}},$ where $E_{\\rm vac}$ is the energy scale for the effective vacuum energy from backreaction, $\\Lambda _{\\rm stripe}\\sim 100$ GeV is the energy scale that sets the amplitude and wavenumber of the translational symmetry violation and $M_{\\rm Planck}\\sim 10^{18}$ GeV is the reduced Planck mass.",
"The reason for disconnecting this sector from the Standard Model is to avoid introducing violations of Lorentz symmetry that would have already been observed, see for example [59], [60].",
"Ordinarily, new physics of this kind would be inaccesible to experiment, however its effect on backreaction would be cosmologically observable.",
"Put another way, one can constrain extensions of this kind to high-energy physics via the observable backreaction effect they would have if physically realized.",
"An attraction of linking the inhomogeneity with the vacuum is that it would have the correct scaling as the universe expands.",
"In this paper, we will further generalize the Green and Wald framework to calculate the general form for the stress-energy tensor for a diffeomorphism-invariant higher-derivative gravity expansion.",
"To ensure that the averaging procedure converges in the weak limit, we will require that the field equations are Taylor-expandable in metric perturbations.",
"Together with diffeomorphism invariance, this translates into requiring that the action is Taylor-expandable in the Riemann tensor and covariant derivatives of the Riemann tensor.",
"This can be intuitively viewed as a locality requirement, as is reasonable for an averaging scheme.",
"This paper is structured as follows.",
"Section outlines the notation conventions used in this paper.",
"Section summarizes the features of the Green and Wald formalism and the extension to $f(R)$ models that are important for this paper.",
"In Section , we calculate the generalization of the effective stress-energy tensor for local, manifestly diffeomorphism-invariant effective theories of gravity parametrized by a large mass scale.",
"The discussion and conclusions are given in Section .",
"Appendix contains the results of applying the averaging procedure to individual field equation terms.",
"Appendix gives a consistency demonstration for the trace of the effective stress-energy tensor."
],
[
"Notation",
"We adopt Landau-Lifshitz spacelike sign conventions, (+,+,+), where the metric signature is mostly positive, a Ricci tensor defined as $R_{\\mu \\nu } := R^{\\alpha }_{\\ \\mu \\alpha \\nu }$ , and $R^{\\alpha }_{\\ \\beta \\gamma \\delta } = 2\\partial _{[\\gamma }\\Gamma ^{\\alpha }_{\\ \\delta ]\\beta } + 2\\Gamma ^{\\alpha }_{\\ \\lambda [\\gamma }\\Gamma ^{\\lambda }_{\\ \\delta ]\\beta }.$ We use the torsionless metric connection: $\\Gamma ^{\\alpha }_{\\ \\beta \\gamma } = \\frac{1}{2}g^{\\alpha \\lambda }\\left(\\partial _\\beta g_{\\gamma \\lambda }+\\partial _\\gamma g_{\\beta \\lambda }-\\partial _\\lambda g_{\\beta \\gamma }\\right).$ We will use a covariant derivative $D_\\mu $ associated with the full metric $g_{\\rho \\sigma }$ such that when it acts on some tensor $T_{\\alpha _1\\cdots \\alpha _m}^{\\ \\ \\ \\ \\ \\ \\ \\beta _1\\cdots \\beta _n}$ , we have $D_\\mu T_{\\alpha _1\\cdots \\alpha _m}^{\\ \\ \\ \\ \\ \\ \\ \\beta _1\\cdots \\beta _n} = \\partial _\\mu T_{\\alpha _1\\cdots \\alpha _m}^{\\ \\ \\ \\ \\ \\ \\ \\beta _1\\cdots \\beta _n} - \\sum _{i=1}^{m}\\Gamma ^{\\lambda }_{\\ \\ \\mu \\alpha _i}T_{\\alpha _1\\cdots \\lambda \\cdots \\alpha _m}^{\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\beta _1\\cdots \\beta _n} + \\sum _{i=1}^{n}\\Gamma ^{\\beta _i}_{\\ \\ \\mu \\lambda }T_{\\alpha _1\\cdots \\alpha _m}^{\\ \\ \\ \\ \\ \\ \\ \\beta _1\\cdots \\lambda \\cdots \\beta _n}.$ A shorthand notation is used such that $D^2 = g^{\\alpha \\beta }D_\\alpha D_\\beta $ .",
"We will also use a covariant derivative $\\nabla _\\mu $ that is associated with a background metric $g^{(0)}_{\\mu \\nu }$ such that $D_\\mu T_{\\alpha _1\\cdots \\alpha _m}^{\\ \\ \\ \\ \\ \\ \\ \\beta _1\\cdots \\beta _n} = \\nabla _\\mu T_{\\alpha _1\\cdots \\alpha _m}^{\\ \\ \\ \\ \\ \\ \\ \\beta _1\\cdots \\beta _n} - \\sum _{i=1}^{m}C^{\\lambda }_{\\ \\ \\mu \\alpha _i}T_{\\alpha _1\\cdots \\lambda \\cdots \\alpha _m}^{\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\beta _1\\cdots \\beta _n} + \\sum _{i=1}^{n}C^{\\beta _i}_{\\ \\ \\mu \\lambda }T_{\\alpha _1\\cdots \\alpha _m}^{\\ \\ \\ \\ \\ \\ \\ \\beta _1\\cdots \\lambda \\cdots \\beta _n},$ where $C^{\\alpha }_{\\ \\beta \\gamma }$ is the difference between full and background connections: $C^{\\alpha }_{\\ \\beta \\gamma } = \\frac{1}{2}g^{\\alpha \\lambda }\\left(\\nabla _\\beta h_{\\gamma \\lambda }+\\nabla _\\gamma h_{\\beta \\lambda }-\\nabla _\\lambda h_{\\beta \\gamma }\\right).$ The full Ricci tensor can be split into background and perturbation parts by $R_{\\alpha \\beta } = R^{(0)}_{\\alpha \\beta } - 2\\nabla _{[\\alpha }C^{\\gamma }_{\\ \\gamma ]\\beta } + 2C^{\\gamma }_{\\ \\beta [\\alpha }C^{\\delta }_{\\ \\delta ]\\gamma }.$ We will refer to the linear part of the perturbation to the Riemann tensor as $R^{(1)}_{\\alpha \\beta \\gamma \\delta } := -2\\nabla _{[\\alpha |}\\nabla _{[\\gamma }h_{\\delta ]|\\beta ]}.$ Using the notation that $\\nabla ^2=g^{(0)\\alpha \\beta }\\nabla _{\\alpha }\\nabla _{\\beta }$ and $h=g^{(0)\\alpha \\beta }h_{\\alpha \\beta }$ , the linearized Ricci tensor is $R^{(1)}_{\\alpha \\beta } := \\frac{1}{2}\\left(2\\nabla _\\lambda \\nabla _{(\\alpha }h_{\\beta )}^{\\ \\ \\lambda } - \\nabla ^2 h_{\\alpha \\beta } - \\nabla _\\alpha \\nabla _\\beta h\\right).$ Finally, the linearized Ricci scalar is $R^{(1)} := \\nabla _\\alpha \\nabla _\\beta h^{\\alpha \\beta } - \\nabla ^2 h.$ The linearized Riemann tensor and its contractions are invariant under linearized diffeomorphisms, given by $\\delta h_{\\alpha \\beta } = 2\\nabla _{(\\alpha }\\xi _{\\beta )}.$ Throughout this paper, we will be working in “natural units”, as commonly used in high-energy physics, i.e.",
"$c=\\hbar =1$ ."
],
[
"Weak-limit averaging method",
"In this section, we will summarize how the weak-averaging procedure developed by Green and Wald in [30] has been adapted to higher-derivative gravity in [50].",
"The weak-limit averaging procedure considers a one-parameter family of metrics whose inhomogeneity parameter, $\\lambda $ , is related to the wavelength of metric perturbations.",
"A tensor $A_{\\alpha _1\\cdots \\alpha _n}(\\lambda )$ converges in the weak limit to its “average” tensor, $B_{\\alpha _1\\cdots \\alpha _n}$ , if $\\lim _{\\lambda \\rightarrow 0}\\int d^4 x \\sqrt{-g^{(0)}}f^{\\alpha _1\\cdots \\alpha _n}A_{\\alpha _1\\cdots \\alpha _n}(\\lambda )=\\int d^4 x\\sqrt{-g^{(0)}}f^{\\alpha _1\\cdots \\alpha _n}B_{\\alpha _1\\cdots \\alpha _n}$ for any smooth test field $f^{\\alpha _1\\cdots \\alpha _n}$ of compact support.",
"The physical interpretation of $\\lambda \\rightarrow 0$ is that the averaging scale is much larger than the length scale of fluctuations, i.e.",
"the ratio of the wavelength of any perturbation mode to the averaging scale tends to zero as the averaging scale is taken to be large.",
"It will be convenient to denote equality under weak limit averaging as $\\underset{\\rm weak}{=}$ , such that if both $A_{\\alpha _1\\cdots \\alpha _n}(\\lambda )$ and another tensor, $C_{\\alpha _1\\cdots \\alpha _n}(\\lambda )$ , converge in weak limit to $B_{\\alpha _1\\cdots \\alpha _n}$ , we can write $A_{\\alpha _1\\cdots \\alpha _n} \\underset{\\rm weak}{=}B_{\\alpha _1\\cdots \\alpha _n} \\underset{\\rm weak}{=}C_{\\alpha _1\\cdots \\alpha _n}.$ The metric, $g_{\\mu \\nu }(x,\\lambda )$ , can be separated into a $\\lambda $ -independent background metric, $g^{(0)}_{\\mu \\nu }(x)$ , and a perturbation defined by $h_{\\mu \\nu }(x,\\lambda ) := g_{\\mu \\nu }(x,\\lambda ) - g^{(0)}_{\\mu \\nu }(x).$ Note that we do not need to specify a particular choice of background.",
"The metric converges in the weak limit to the background metric, i.e.",
"the metric perturbation vanishes in weak limit.",
"More specifically, the metric perturbation is of $\\mathcal {O}(\\lambda )$ , by which it is meant that $h_{\\alpha \\beta }(x,\\lambda )$ is uniformly bounded by a constant times $\\lambda $ , for sufficiently small $\\lambda $ .",
"Similarly, an $\\mathcal {O}(\\lambda ^n)$ term is uniformly bounded by a constant times $\\lambda ^n$ .",
"The choice of background metric that is best motivated by cosmology is the FLRW metric, but note that none of our calculations are specific to any particular choice of background, which can be chosen freely.",
"Applying a background covariant derivative to the metric perturbation lowers the order in $\\lambda $ by one: $\\nabla _{\\alpha _1}\\cdots \\nabla _{\\alpha _n}h_{\\beta \\gamma } \\sim \\lambda ^{1-n}.$ In this respect, we see that $\\lambda $ is proportional to the length scales of perturbation modes.",
"However, this comes with a caveat.",
"Total derivatives, i.e.",
"derivatives that act on the entire tensor that we perform the weak limit averaging on, do not change the order in $\\lambda $ .",
"This can be seen by applying (REF ) to a total derivative term and performing an integration by parts: $\\int d^4 x\\sqrt{-g^{(0)}}f^{\\lambda \\alpha _1\\cdots \\alpha _n}\\nabla _\\lambda A_{\\alpha _1\\cdots \\alpha _n}(\\lambda ) = -\\int d^4x\\sqrt{-g^{(0)}}\\left(\\nabla _\\lambda f^{\\lambda \\alpha _1\\cdots \\alpha _n}\\right)A_{\\alpha _1\\cdots \\alpha _n}(\\lambda ).$ Since $f^{\\alpha _1\\cdots \\alpha _n}$ is independent of $\\lambda $ , so are its derivatives, $\\nabla _{\\beta _1}\\cdots \\nabla _{\\beta _n}f^{\\alpha _1\\cdots \\alpha _n}$ .",
"Thus we can see that the weak limit of a term consisting solely of total derivatives of $A_{\\alpha _1\\cdots \\alpha _n}(\\lambda )$ is of the same order in $\\lambda $ as $A_{\\alpha _1\\cdots \\alpha _n}(\\lambda )$ itself.",
"This prescription will be applied to the field equations for a given gravity theory.",
"The field equation can be written as $\\mathcal {G}_{\\mu \\nu } := \\frac{2\\kappa }{\\sqrt{-g}}\\frac{\\delta S_{\\rm grav}}{\\delta g^{\\mu \\nu }} = \\kappa T_{\\mu \\nu },$ where $S_{\\rm grav}$ is the gravitational part of the action, i.e.",
"the part constructed from a series expansion in the Riemann tensor and covariant derivatives of it, $\\kappa =8\\pi G$ , and $T_{\\mu \\nu }$ is the stress-energy tensor.",
"We will wish to split $\\mathcal {G}_{\\mu \\nu }$ into a background part, $\\mathcal {G}_{\\mu \\nu }^{(0)}$ i.e.",
"the value of $\\mathcal {G}_{\\mu \\nu }$ for $g_{\\mu \\nu }=g^{(0)}_{\\mu \\nu }$ , and a perturbation part, $\\delta \\left[\\mathcal {G}_{\\mu \\nu }\\right]$ .",
"The field equation is then written as $\\mathcal {G}^{(0)}_{\\mu \\nu } + \\delta \\left[\\mathcal {G}_{\\mu \\nu }\\right] = \\kappa T_{\\mu \\nu }.$ Supposing that the perturbation term is non-vanishing in the weak limit, we see that, moving it to the right hand side, we obtain an effective stress-energy tensor induced by inhomogeneity: $\\mathcal {G}^{(0)}_{\\mu \\nu } \\underset{\\rm weak}{=}\\kappa T_{\\mu \\nu }^{(0)} + \\kappa t^{(0)}_{\\mu \\nu },$ where $\\delta \\left[\\mathcal {G}_{\\mu \\nu }\\right] \\underset{\\rm weak}{=}-\\kappa t^{(0)}_{\\mu \\nu }.$ As discussed, linear terms in $h$ are of $\\mathcal {O}(\\lambda )$ , because total derivatives do not change the order in $\\lambda $ .",
"Quadratic terms in $h$ , on the other hand, can be of $\\mathcal {O}(1)$ , i.e.",
"being of the zeroth order in $\\lambda $ , they converge to finite values in the weak limit: $h_{\\rho \\sigma }\\nabla _{\\alpha }\\nabla _{\\beta }h_{\\mu \\nu } \\underset{\\rm weak}{=}- \\nabla _{\\alpha }h_{\\rho \\sigma }\\nabla _{\\beta }h_{\\mu \\nu } \\sim \\mathcal {O}(1).$ Terms of this form then provide non-vanishing contributions to the effective stress-energy tensor, $t_{\\mu \\nu }^{(0)}$ .",
"This is not an assertion that such terms are necessarily of the same order of magnitude as the background, rather that they are able to converge to give non-zero values in the limit where the averaging scale becomes large.",
"This is subject to a constraint explored in the next section that can still result in some such terms vanishing, especially in Einstein gravity."
],
[
"Zero tensors",
"There exists an additional constraint on $t_{\\mu \\nu }^{(0)}$ that is used in the papers by Isaacson [33], [34], Burnett [32] and Green and Wald [30] for the Einstein gravity case.",
"This was referred to in [50] as the “zero tensor”.",
"Since it will be used frequently here also, it is convenient to continue referring to it as the zero tensor.",
"In the Green and Wald paper, it is proven that $A(\\lambda )B(\\lambda ) \\underset{\\rm weak}{=}A(0)B(0),$ provided that $A(\\lambda )$ is a smooth tensor field converging uniformly on compact sets to $A(0)$ and that $B(\\lambda )$ is a non-negative smooth function converging to $B(0)$ in the weak limit.",
"Thus, provided that the stress-energy tensor $T_{\\mu \\nu }$ satisfies the weak energy condition, i.e.",
"given any timelike vector $t^{\\alpha }(x,\\lambda )$ , $T_{\\alpha \\beta }(x,\\lambda )t^{\\alpha }(x,\\lambda )t^{\\beta }(x,\\lambda ) \\ge 0,$ it is also proven that $h_{\\rho \\sigma }T_{\\mu \\nu } \\underset{\\rm weak}{=}0.$ Let us consider taking the field equation, as written in (REF ), and multiplying by $h_{\\rho \\sigma }$ : $\\underbrace{h_{\\rho \\sigma }\\mathcal {G}^{(0)}_{\\mu \\nu }}_{\\text{vanishes in weak limit}} + h_{\\rho \\sigma }\\delta \\left[\\mathcal {G}_{\\mu \\nu }\\right] = \\underbrace{h_{\\rho \\sigma }\\kappa T_{\\mu \\nu }}_{\\text{vanishes in weak limit}}.$ Thus the form of the zero tensor is given by $h_{\\rho \\sigma }\\delta \\left[\\mathcal {G}_{\\mu \\nu }\\right] \\underset{\\rm weak}{=}0,$ where, this time, it is the part of $\\delta \\left[\\mathcal {G}_{\\mu \\nu }\\right]$ that is linear in $h$ that gives us non-vanishing contributions.",
"This zero tensor constraint retains the information from the linear order in $h$ , which the weak limit would otherwise discard."
],
[
"Application to Einstein gravity",
"The action for Einstein's theory for gravity can be written as $\\int d^4 x \\sqrt{-g}\\frac{1}{2\\kappa }\\left(R -2\\Lambda \\right) + S_{\\rm matter}.$ The field equation is $R_{\\mu \\nu } -\\frac{1}{2}g_{\\mu \\nu }R + \\Lambda g_{\\mu \\nu } = \\kappa T_{\\mu \\nu }.$ The cosmological constant term converges in the weak limit to its background value: $\\Lambda g_{\\mu \\nu } \\underset{\\rm weak}{=}\\Lambda g^{(0)}_{\\mu \\nu }.$ This is clear because the metric converges in the weak limit to its background, and one is at liberty to multiply both sides by a constant.",
"Weak limits of the other two terms are given in Appendix , along with the weak limits of more terms that appear in higher-derivative gravity.",
"The form of the effective stress-energy tensor, before applying the zero tensor constraint, is then $\\kappa t^{E}_{\\mu \\nu } \\underset{\\rm weak}{=}\\frac{1}{2}h^{\\alpha \\beta }R_{\\mu \\alpha \\nu \\beta }^{(1)}+\\frac{3}{4}h_{\\mu \\nu }R^{(1)} - R^{(1)}_{\\alpha (\\mu }h_{\\nu )}^{\\ \\ \\alpha } - \\frac{1}{8}g_{\\mu \\nu }^{(0)}\\left(hR^{(1)}+2h^{\\alpha \\beta }R_{\\alpha \\beta }^{(1)}\\right).$ The zero tensor can be powerfully expressed for Einstein gravity as $R_{\\alpha \\beta }^{(1)}h_{\\gamma \\delta } \\underset{\\rm weak}{=}0.$ This immediately simplifies (REF ) to $\\kappa t^{E}_{\\mu \\nu } \\underset{\\rm weak}{=}\\frac{1}{2}h^{\\alpha \\beta }R_{\\mu \\alpha \\nu \\beta }^{(1)},$ which is then traceless under the constraint from the zero tensor, implying that the backreaction is radiation-like.",
"The implication of this is that cosmological backreaction is unable to account for accelerating expansion and is not important in late-time cosmology, which was the conclusion in [30]."
],
[
"Extension to higher derivatives",
"We will review the application of weak-limit averaging to a Taylor-expandable $f(R)$ model whose action has its own length scale, $M$ .",
"This procedure was peformed explicitly for the $R+R^2 /6M^2$ case in [50].",
"As we will see, the results are already general for the complete local $f(R)$ expansion.",
"Consider the following action: $S = \\int d^4 x\\sqrt{-g}\\left(f(R)-2\\Lambda \\right) + S_{\\rm matter},$ where $f(R) = R + \\frac{R^2}{6M^2} + {\\rm const}\\times \\frac{R^3}{M^4} + \\cdots .$ The inclusion of a new length scale, $1/M$ , presents us with the challenge of how to rigorously incorporate it.",
"Recall that the limit $\\lambda \\rightarrow 0$ corresponds to choosing a cosmological averaging length scale that is much greater than the length scale of perturbations.",
"The naïve suggestion that $M$ should be independent of $\\lambda $ presents us with problems.",
"Firstly, consider the ratio of the scale $1/M$ and the perturbation scale, which goes like $\\lambda $ , see (REF ).",
"The ratio between these two scales should be fixed physically, since it is the cosmological averaging scale that we are tending to be large.",
"Leaving $1/M$ independent of $\\lambda $ would correspond to taking the perturbation length scale to be much smaller than $1/M$ in the limit where we take the cosmological averaging scale to be large.",
"This would be outside the validity of the effective theory.",
"The perturbations would have a length scale that is much shorter than $1/M$ , which is the scale to which high-frequency modes in the more fundamental theory have been integrated out.",
"This would manifest as each higher-derivative term in the action giving increasingly divergent field equation contributions in the weak limit.",
"The naïve suggestion that the latter problem could be fixed by rescaling to $h\\sim \\lambda ^2$ would fail once one also allows the action to use explicit covariant derivative operators, e.g.",
"in Lagrangian terms like $RD^2 R/M^4$ .",
"All of these problems are resolved by setting $M\\sim \\lambda ^{-l}$ where $l=1$ .",
"In particular, $l=1$ is the only scaling that keeps a fixed ratio of the perturbation wavelength and the scale $1/M$ in the weak limit.",
"Setting $l<1$ would result in divergences in the field equation caused by higher derivatives.",
"Setting $l>1$ would cause all contributions from Lagrangian terms with more than two derivatives to vanish, trivially leaving us with only the Einstein gravity contributions.",
"For the reasons discussed, we set $l=1$ , with the result that $\\frac{1}{M^{k-2}}h_{\\mu \\nu }\\nabla _{\\alpha _1}\\cdots \\nabla _{\\alpha _k}h_{\\rho \\sigma } \\underset{\\rm weak}{=}\\mathcal {O}(1),$ and other non-zero orders in $h$ vanish in the weak limit.",
"The field equation for $f(R)$ gravity can be written as $\\left(R_{\\mu \\nu } -D_\\mu D_\\nu + g_{\\mu \\nu }D^2 \\right)f^{\\prime }(R) -\\frac{1}{2}g_{\\mu \\nu }f(R) + \\Lambda g_{\\mu \\nu } = \\kappa T_{\\mu \\nu }.$ All contributions to $t_{\\mu \\nu }^{(0)}$ from the $R^3$ and higher action terms vanish in the weak limit.",
"To see this, note that, for $m\\ge 0$ and $n>0$ , $\\frac{1}{M^{2(m+n)}}h_{\\alpha \\beta }\\nabla _\\gamma \\nabla _\\delta R^{(1)n}R^{(0)m}\\sim \\lambda ^{2m+n-1},$ thus our contributions from higher derivative terms are non-vanishing for $n=1$ and $m=0$ .",
"Alternatively, note that (REF ) tells us that the only non-vanishing perturbation contributions (at any order in derivatives) are from the quadratic order in the metric perturbation where all of the covariant derivative operators are acting on an instance of the metric perturbation not as a total derivative.",
"More generally, any field equation contribution from a Lagrangian term at cubic order or higher in the Riemann tensor vanishes in weak limit.",
"The $f(R)$ field equation expands up to $\\mathcal {O}(R^2)$ as $R_{\\mu \\nu } + \\Lambda g_{\\mu \\nu } -\\frac{1}{2}g_{\\mu \\nu }\\left(R+\\frac{R^2}{6M^2}\\right)+\\frac{1}{3M^2}\\left(RR_{\\mu \\nu }-D_\\mu D_\\nu R + g_{\\mu \\nu }D^2 R\\right)\\cdots = \\kappa T_{\\mu \\nu }$ Once again, upon taking a weak limit, we find that the Einstein (2-derivative) part of $\\kappa t^{(0)}_{\\mu \\nu }$ is given by (REF ).",
"The Starobinsky (4-derivative) part of the field equation also gives a contribution to $\\kappa t^{(0)}_{\\mu \\nu }$ , which we will denote by $\\kappa t^{S}_{\\mu \\nu }$ .",
"As before, the field equation in the weak limit can be written as $R_{\\mu \\nu }^{(0)} -\\frac{1}{2}g_{\\mu \\nu }^{(0)}R^{(0)} \\underset{\\rm weak}{=}\\kappa T_{\\mu \\nu } + \\kappa t^{(0)}_{\\mu \\nu },$ where the higher-derivative contributions to $\\mathcal {G}^{(0)}_{\\mu \\nu }$ vanish in the weak limit because of the scaling of $M$ with $\\lambda $ , i.e.",
"the higher-derivative background terms become unimportant in the limit where the averaging scale is very large.",
"However, this is not true of $t_{\\mu \\nu }^{S}$ , which does not vanish in the weak limit.",
"For this reason, backreaction provides a rare opportunity for these terms to have an influence on cosmology at very large scales.",
"Weak limits of the individual field equation terms in (REF ) are given in Appendix .",
"Putting these ingredients together, we find the contributions to the effective stress-energy tensor from the Starobinsky parts of the field equation: $\\kappa t_{\\mu \\nu }^{S} \\underset{\\rm weak}{=}\\frac{R^{(1)}}{3M^2}\\left(\\frac{1}{2}g_{\\mu \\nu }^{(0)}\\nabla ^2 h- \\nabla ^2 h_{\\mu \\nu }+\\frac{1}{2}\\nabla _\\mu \\nabla _\\nu h + \\frac{1}{4}g_{\\mu \\nu }^{(0)}R^{(1)}\\right).$ There are two particularly useful forms of the zero tensor for this theory: $h_{\\alpha \\beta }R^{(1)}_{\\gamma \\delta } -\\frac{g_{\\gamma \\delta }^{(0)}}{6M^2}R^{(1)}\\nabla ^2 h_{\\alpha \\beta }-\\frac{1}{3M^2}\\left(\\nabla _\\gamma \\nabla _\\delta h_{\\alpha \\beta }\\right)R^{(1)} \\underset{\\rm weak}{=}0$ and $h_{\\alpha \\beta }R^{(1)}_{\\gamma \\delta } - \\frac{g^{(0)}_{\\gamma \\delta }}{6}h_{\\alpha \\beta }R^{(1)}-\\frac{1}{3M^2}\\left(\\nabla _\\gamma \\nabla _\\delta h_{\\alpha \\beta }\\right)R^{(1)} \\underset{\\rm weak}{=}0.$ These are both related via the trace over $\\gamma \\delta $ : $R^{(1)}\\left(1-\\frac{\\nabla ^2}{M^2}\\right)h_{\\alpha \\beta } \\underset{\\rm weak}{=}0.$ Equation (REF ), having only a single very generic 2-derivative term, is useful for converting 2-derivative terms into 4-derivative terms.",
"This is especially true when considering the trace of the effective stress-energy tensor, $t^{(0)}$ , where, as we will see, it is possible to rewrite every term in 4-derivative form.",
"This is because $t^{(0)}$ would vanish if not for the 4-derivative extension of the action, i.e.",
"the Einstein gravity case has a radiation-like backreaction.",
"Equation (REF ), conversely, having only a single very generic 4-derivative term, is useful for converting 4-derivative terms into 2-derivative terms.",
"In fact, (REF ) can be used to rephrase $t^{(0)}_{\\mu \\nu }$ entirely in terms of 2-derivative terms.",
"This is because, in a pure $R^2$ theory, $t_{\\mu \\nu }^{(0)}$ vanishes completely.",
"Thus we see that the rôle of the zero tensor is not as powerful as in the Einstein-gravity case, but it is still able to rewrite $t_{\\mu \\nu }^{(0)}$ into more convenient forms.",
"The 2-derivative form of the Starobinsky part of $t_{\\mu \\nu }^{(0)}$ is $\\kappa t^{S}_{\\mu \\nu } \\underset{\\rm weak}{=}\\frac{1}{2}hR_{\\mu \\nu }^{(1)} + \\frac{g^{(0)}_{\\mu \\nu }}{4}h^{\\alpha \\beta }R_{\\alpha \\beta }^{(1)}-\\frac{1}{3}h_{\\mu \\nu }R^{(1)}-\\frac{g_{\\mu \\nu }^{(0)}}{24}hR^{(1)}.$ Putting the pieces together, the effective stress-energy tensor for this simple model in 2-derivative form is $\\kappa t^{(0)}_{\\mu \\nu } \\underset{\\rm weak}{=}\\frac{1}{2}h^{\\alpha \\beta }R^{(1)}_{\\mu \\alpha \\nu \\beta }-R^{(1)}_{\\alpha (\\mu }h_{\\nu )}^{\\ \\ \\alpha }+\\frac{5}{12}h_{\\mu \\nu }R^{(1)}+\\frac{1}{2}hR_{\\mu \\nu }^{(1)}-\\frac{1}{6}g_{\\mu \\nu }^{(0)}hR^{(1)},$ for which the trace is $\\kappa t^{(0)} \\underset{\\rm weak}{=}\\frac{1}{4}hR^{(1)}-\\frac{1}{2}h^{\\alpha \\beta }R_{\\alpha \\beta }^{(1)}.$ The 4-derivative form of the trace is $\\kappa t^{(0)} \\underset{\\rm weak}{=}-\\frac{R^{(1)2}}{6M^2}.$ from this, we can immediately see that the inclusion of the Starobinsky $R^2$ term has given us a form for $t_{\\mu \\nu }^{(0)}$ that is not purely radiation-like.",
"Splitting the stress-energy tensor into its traceless and pure trace components, i.e.",
"$t^{(0)}_{\\mu \\nu } = \\underbrace{t^{(0)}_{\\mu \\nu } - \\frac{1}{4}g_{\\mu \\nu }^{(0)}t^{(0)}}_{\\text{traceless}}+\\underbrace{\\frac{1}{4}g^{(0)}_{\\mu \\nu }t^{(0)}}_{\\text{pure trace}},$ we see that the pure trace component of $t_{\\mu \\nu }^{(0)}$ has the correct sign and order in $\\lambda $ to mimic a positive cosmological constant, given an appropriate cosmological context.",
"The diffeomorphism invariance of this construction was demonstrated in [50], but we will postpone the derivation here until Section REF , where we demonstrate diffeomorphism invariance for the fully generalized case.",
"Also discussed in [50] was the equivalent scalar-tensor construction, which is formed via a Legendre transform, $f(R)=\\phi R - V(\\phi )$ , where we would require that $V(\\phi )$ is Taylor-expandable in $\\phi $ so that it converges in the weak limit.",
"Perturbations in $\\phi $ would only contribute to $t_{\\mu \\nu }^{(0)}$ up to their quadratic order, beyond which they would be suppressed by $\\lambda $ .",
"Naïvely, one might also think to scale the cosmological constant dimensionally, i.e.",
"$\\Lambda \\sim \\lambda ^{-2}$ .",
"Naïvely again, one might think that the cosmological constant would then give an $\\mathcal {O}(1)$ contribution to the zero tensor of the form $h_{\\rho \\sigma }h_{\\mu \\nu }\\Lambda $ .",
"However, this would give us a $\\lambda ^{-2}$ divergence in the field equation.",
"For the sake of the convergence of the weak-limit and the overall consistency of the formalism, we must leave $\\Lambda $ constant.",
"This does not introduce the pathology of changing the ratio of physical scales in the weak limit because the cosmological constant does not mix with other scales in the field equation.",
"To see this, note that (REF ) does not have any implications for the relative sizes of $\\Lambda ^{-1/2}$ , the perturbation wavelength and $1/M$ .",
"This is unlike the higher-derivative terms, which effectively take a ratio of $1/M$ to the perturbation wavelength via the $M$ -suppressed higher derivative operators.",
"A cosmological constant term that converges in the weak limit, as given in (REF ), does not contribute to the zero tensor.",
"To see this, note that a positive cosmological constant term satisfies the weak energy condition and therefore (REF ) applies here: $h_{\\rho \\sigma }g_{\\mu \\nu }\\Lambda \\underset{\\rm weak}{=}0.$ This result is also true for a negative cosmological constant.",
"Thus the cosmological constant does not have any influence on the backreaction."
],
[
"Contributing action terms",
"As discussed in Section REF , especially via equations (REF ) and (REF ), the only local action terms that give non-vanishing contributions to $t_{\\mu \\nu }^{(0)}$ are those that are linear or quadratic in the Riemann tensor.",
"Since the Gauss-Bonnet term is a topological invariant in four dimensions, $\\frac{\\delta }{\\delta g^{\\mu \\nu }}\\int d^4x \\sqrt{-g}\\left(R_{\\alpha \\beta \\gamma \\delta }R^{\\alpha \\beta \\gamma \\delta } -4 R_{\\alpha \\beta }R^{\\alpha \\beta } + R^2\\right) = 0,$ we can write a local expansion in the Riemann tensor up to the quadratic order as $S = \\int d^4 x \\sqrt{-g}\\frac{1}{2\\kappa }\\left(-2\\Lambda + R + \\frac{a}{M^2}R_{\\mu \\nu }R^{\\mu \\nu } + \\frac{b}{M^2}R^2 + \\cdots \\right)+S_{\\rm matter}.$ We can further generalize this action while maintaining diffeomorphism invariance and a suitable notion of locality by also introducing explicit covariant derivative operators in the action, again in a Taylor-expandable structure.",
"The first instance in the action where we can introduce these covariant derivative operators in a non-trivial way is at the quadratic order in the Riemman tensor e.g.",
"in a Lagrangian term like $R_{\\mu \\nu }a\\left(\\frac{D^2}{M^2}\\right)R^{\\mu \\nu }$ .",
"Since, as already discussed, our expression for $t^{(0)}_{\\mu \\nu }$ is unaffected by cubic and higher terms in the Riemann tensor, we only need to consider explicit covariant derivatives in terms at quadratic order in the Riemann tensor.",
"We are free to rearrange the order of these covariant derivatives, since the commutators only introduce corrections at higher order in the Riemann tensor, e.g.",
": $\\left[D_{\\mu },D_{\\nu }\\right]v_{\\alpha } = R^{\\lambda }_{\\ \\alpha \\nu \\mu }v_{\\lambda }.$ Naïvely, one would expect that there would exist a large number of independent index structures at quadratic order in the Riemann tensor.",
"However, using the (anti-)symmetry properties of the Riemann tensor and the (second) Bianchi identity: $D_{\\lambda }R_{\\alpha \\beta \\gamma \\delta } + D_{\\gamma }R_{\\alpha \\beta \\delta \\lambda } + D_{\\delta }R_{\\alpha \\beta \\lambda \\gamma } = 0,$ a particularly useful specialization of which is $g^{\\alpha \\beta }D_{\\alpha }R_{\\beta \\gamma } = \\frac{1}{2}D_{\\gamma }R,$ we are able to rearrange all possible index structures, up to the quadratic order in the Riemann tensor, to $S & = & \\int d^4x \\sqrt{-g}\\frac{1}{2\\kappa }\\left(R -2\\Lambda + \\frac{1}{M^2}R_{\\alpha \\beta }a\\left(\\frac{D^2}{M^2}\\right)R^{\\alpha \\beta } + \\frac{1}{M^2}Rb\\left(\\frac{D^2}{M^2}\\right)R \\right.",
"\\nonumber \\\\ && \\left.+ \\frac{1}{M^2}R_{\\alpha \\beta \\gamma \\delta }c\\left(\\frac{D^2}{M^2}\\right)R^{\\alpha \\beta \\gamma \\delta }+ \\cdots \\right) + S_{\\rm matter}.$ However, not all of these terms give independent contributions to $t_{\\mu \\nu }^{(0)}$ .",
"As already discussed, the three quadratic terms without extra covariant derivative operators are related via the Gauss-Bonnet topological invariant (REF ).",
"Although the introduction of the covariant derivatives breaks the topological invariance, the effect of this is not apparent in the weak limit, i.e.",
"there are no new corrections of the form (REF ).",
"To see this, one can consult equations (REF ) and (REF ), noting that all the terms that contribute in the weak limit are of the same structure as in the case of keeping $a$ and $b$ as constant numbers, except for the final terms with $a^{\\prime }$ and $b^{\\prime }$ .",
"Even these extra terms cancel if we choose the Gauss-Bonnet structure because $\\frac{1}{M^4}\\left(R_{\\alpha \\beta \\gamma \\delta }^{(1)}a^{\\prime } \\nabla _\\mu \\nabla _\\nu R^{\\alpha \\beta \\gamma \\delta (1)} -4R_{\\alpha \\beta }^{(1)}a^{\\prime } \\nabla _\\mu \\nabla _\\nu R^{\\alpha \\beta (1)} + R^{(1)}a^{\\prime }\\nabla _\\mu \\nabla _\\nu R^{(1)}\\right) \\underset{\\rm weak}{=}0.$ Thus the topological invariance of the Gauss-Bonnet term given in (REF ) is effectively inherited by its higher-derivative generalization in the weak limit.",
"More precisely, we mean that $\\frac{\\delta }{\\delta g^{\\mu \\nu }}\\int d^4x \\sqrt{-g}\\frac{1}{M^2}\\left(R_{\\alpha \\beta \\gamma \\delta }a\\left(\\frac{D^2}{M^2}\\right)R^{\\alpha \\beta \\gamma \\delta } -4 R_{\\alpha \\beta }a\\left(\\frac{D^2}{M^2}\\right)R^{\\alpha \\beta } + Ra\\left(\\frac{D^2}{M^2}\\right)R\\right) \\underset{\\rm weak}{=}0.$ For this reason, when calculating $t_{\\mu \\nu }^{(0)}$ , we lose nothing by rearranging the three quadratic terms in the action to a basis of just two of them.",
"Finally, we are left with a concise, closed form for the contributing terms in the effective action that still gives us a completely general result for the form of $t_{\\mu \\nu }^{(0)}$ : $S = \\int d^4x \\sqrt{-g}\\frac{1}{2\\kappa }\\left(R -2\\Lambda + \\frac{1}{M^2}R_{\\alpha \\beta }a\\left(\\frac{D^2}{M^2}\\right)R^{\\alpha \\beta } + \\frac{1}{M^2}Rb\\left(\\frac{D^2}{M^2}\\right)R + \\cdots \\right) + S_{\\rm matter}.$"
],
[
"Calculation of the effective stress-energy tensor",
"Let us consider the action given in (REF ).",
"To obtain the field equation, the ingredients we need are $\\frac{\\delta }{\\delta g^{\\mu \\nu }}\\int d^4x \\sqrt{-g}R_{\\alpha \\beta }aR^{\\alpha \\beta } & = & \\sqrt{-g}\\left(-\\frac{1}{2}g_{\\mu \\nu }R_{\\alpha \\beta }aR^{\\alpha \\beta }+2R_{\\mu \\alpha }aR_{\\nu }^{\\ \\alpha } \\right.\\nonumber \\\\ & &\\left.",
"+D^2 aR_{\\mu \\nu }+\\frac{1}{2}g_{\\mu \\nu }D^2 aR-2D_\\alpha D_{(\\mu } aR_{\\nu )}^{\\ \\ \\alpha } \\right.\\nonumber \\\\ & &\\left.",
"+ \\frac{1}{M^2}R_{\\alpha \\beta }D_{\\mu }D_{\\nu }a^{\\prime }R^{\\alpha \\beta }+\\cdots \\right),$ $\\frac{\\delta }{\\delta g^{\\mu \\nu }}\\int d^4x \\sqrt{-g}RbR & = & \\sqrt{-g}\\left(-\\frac{1}{2}g_{\\mu \\nu }RbR + 2RbR_{\\mu \\nu }-2D_{\\mu }D_{\\nu }bR + 2g_{\\mu \\nu }D^2 bR \\right.",
"\\nonumber \\\\ & & \\left.",
"+ \\frac{1}{M^2}R D_{\\mu }D_{\\nu }b^{\\prime }R+\\cdots \\right),$ where $a^{\\prime }$ and $b^{\\prime }$ are the derivatives of $a$ and $b$ with respect to their arguments, and we have neglected terms that vanish entirely in the weak limit.",
"The vanishing terms come from varying with respect to the inverse metric the connections from applying the explicit covariant derivative operators that appear in $a$ and $b$ .",
"As before, we can take the weak limit to find the effective stress-energy tensor from metric perturbations.",
"In all of the terms above, the functions $a\\left(\\frac{D^2}{M^2}\\right)$ and $b\\left(\\frac{D^2}{M^2}\\right)$ converge simply in the weak limit to $a\\left(\\frac{\\nabla ^2}{M^2}\\right)$ and $b\\left(\\frac{\\nabla ^2}{M^2}\\right)$ , respectively.",
"This is because they do not appear as total derivatives in any of the field equation terms.",
"By this, we mean, for example, $\\frac{1}{M^2}Rb\\left(\\frac{D^2}{M^2}\\right)R_{\\mu \\nu } \\underset{\\rm weak}{=}\\frac{1}{M^2} R^{(1)}b\\left(\\frac{\\nabla ^2}{M^2}\\right)R^{(1)}_{\\mu \\nu }.$ We have already calculated the weak limits of the field equation terms that are also found in $R+R^2 /6M^2$ gravity.",
"The generalized action now gives us new terms whose forms did not appear in that model, these are also given in Appendix .",
"Putting together all the pieces, the effective stress-energy tensor in the weak limit is $\\kappa t^{(0)}_{\\mu \\nu } & = & \\frac{1}{2}h^{\\alpha \\beta }R^{(1)}_{\\mu \\alpha \\nu \\beta } + \\frac{3}{4}h_{\\mu \\nu }R^{(1)}-R^{(1)}_{\\alpha (\\mu }h_{\\nu )}^{\\ \\ \\alpha }-\\frac{1}{8}g^{(0)}_{\\mu \\nu }\\left(hR^{(1)}+2h^{\\alpha \\beta }R^{(1)}_{\\alpha \\beta }\\right) \\nonumber \\\\ &&+\\frac{1}{M^2}\\left(\\frac{1}{2}g^{(0)}_{\\mu \\nu }R^{(1)}bR^{(1)}+h\\nabla _{\\mu }\\nabla _{\\nu }bR^{(1)}-2h_{\\mu \\nu }\\nabla ^2 bR^{(1)}+g^{(0)}_{\\mu \\nu }h\\nabla ^2 bR^{(1)} \\right.",
"\\nonumber \\\\ &&+ \\frac{1}{2}g_{\\mu \\nu }^{(0)}R^{(1)}_{\\alpha \\beta }aR^{(1)\\alpha \\beta }-2R_{\\mu \\alpha }^{(1)}aR_{\\nu }^{(0)\\alpha }+ \\frac{1}{2}h\\nabla ^2 a R_{\\mu \\nu }^{(1)}-2h^{\\alpha }_{\\ (\\mu }\\nabla ^2aR_{\\nu )\\alpha }^{(1)}+h^{\\alpha \\beta }\\nabla _\\mu \\nabla _\\nu aR_{\\alpha \\beta }^{(1)} \\nonumber \\\\ &&+ h^\\alpha _{\\ (\\mu }\\nabla _{\\nu )}\\nabla _\\alpha a R^{(1)} + \\frac{1}{4}g_{\\mu \\nu }^{(0)}h\\nabla ^2 a R^{(1)}-\\frac{1}{2}h\\nabla _\\mu \\nabla _\\nu a R^{(1)} - \\frac{1}{2}h_{\\mu \\nu }\\nabla ^2 aR^{(1)}\\nonumber \\\\ &&\\left.",
"-R^{(1)}\\frac{\\nabla _\\mu \\nabla _\\nu }{M^2}b^{\\prime }R^{(1)} -R^{(1)}_{\\alpha \\beta }\\frac{\\nabla _\\mu \\nabla _\\nu }{M^2}a^{\\prime }R^{(1)\\alpha \\beta }\\right).$ To gain more insight into this stress-energy tensor, we derive the corresponding zero tensor in the next section."
],
[
"The zero tensor",
"As before, we derive the zero tensor by multiplying the field equation by $h_{\\rho \\sigma }$ and taking a weak limit.",
"We get $0 & \\underset{\\rm weak}{=}& h_{\\rho \\sigma }R^{(1)}_{\\mu \\nu } - \\frac{1}{2}g^{(0)}_{\\mu \\nu }h_{\\rho \\sigma }R^{(1)} + \\frac{1}{M^2}\\left(-2h_{\\rho \\sigma }\\nabla _\\mu \\nabla _\\nu bR^{(1)} + 2g_{\\mu \\nu }^{(0)}h_{\\rho \\sigma }\\nabla ^2 bR^{(1)} \\right.",
"\\nonumber \\\\ &&\\left.",
"h_{\\rho \\sigma }\\nabla ^2 aR_{\\mu \\nu }^{(1)}+\\frac{1}{2}g_{\\mu \\nu }^{(0)}h_{\\rho \\sigma }\\nabla ^2 aR^{(1)} - h_{\\rho \\sigma }\\nabla _\\mu \\nabla _\\nu aR^{(1)}\\right).$ Taking the trace over $\\mu $ and $\\nu $ , we get $h_{\\rho \\sigma }R^{(1)} \\underset{\\rm weak}{=}\\frac{2}{M^2}h_{\\rho \\sigma }(a+3b)\\nabla ^2 R^{(1)}.$ Substituting this back into (REF ) allows us to write the zero tensor in other forms, the most useful of which is $h_{\\rho \\sigma }R^{(1)}_{\\mu \\nu } \\underset{\\rm weak}{=}\\frac{1}{M^2}\\left(h_{\\rho \\sigma }\\nabla _\\mu \\nabla _\\nu (a+2b)R^{(1)} + \\frac{1}{2}g_{\\mu \\nu }^{(0)}h_{\\rho \\sigma }\\nabla ^2 (a+2b)R^{(1)}-h_{\\rho \\sigma }\\nabla ^2 a R_{\\mu \\nu }^{(1)}\\right).$ A useful contraction of this form is $h_{\\alpha \\beta }R^{(1)\\alpha \\beta } \\underset{\\rm weak}{=}\\frac{1}{M^2}\\left(h^{\\alpha \\beta }\\nabla _\\alpha \\nabla _\\beta (a+2b)R^{(1)}+\\frac{1}{2}h\\nabla ^2 (a+2b)R^{(1)}-h^{\\alpha \\beta }\\nabla ^2 a R^{(1)}_{\\alpha \\beta }\\right).$"
],
[
"Trace of the effective stress-energy tensor",
"In this section, we will take the trace of the field equation and perform the weak limit to find the trace of the effective stress-energy tensor.",
"This is equivalent to taking the trace of (REF ), as demonstrated in Appendix .",
"Taking the trace of the field equation, we get $-R + \\frac{2}{M^2}D^2 (a+3b)R + \\frac{1}{M^4}\\left(R_{\\alpha \\beta }D^2 a^{\\prime }R^{\\alpha \\beta } + RD^2 b^{\\prime }R\\right)+\\cdots = \\kappa T.$ Noting that $-R^{(2)} & \\underset{\\rm weak}{=}& \\frac{1}{4}hR^{(1)}+\\frac{1}{2}h^{\\alpha \\beta }R^{(1)}_{\\alpha \\beta } \\nonumber \\\\ & \\underset{\\rm weak}{=}&\\frac{1}{2M^2}\\left(h^{\\alpha \\beta }\\nabla _\\alpha \\nabla _\\beta (a+2b)R^{(1)}+\\frac{1}{2}h\\nabla ^2 (3a+8b)R^{(1)}-h^{\\alpha \\beta }\\nabla ^2 a R^{(1)}_{\\alpha \\beta }\\right),$ and $\\frac{1}{M^2}\\delta \\left[D^2 (a+3b)R\\right] \\underset{\\rm weak}{=}-\\frac{1}{2M^2}h\\nabla ^2 (a+3b)R^{(1)},$ we can derive the form of $\\kappa t^{(0)}$ in the weak limit to be $\\kappa t^{(0)} & \\underset{\\rm weak}{=}& -\\frac{1}{2M^2}\\left(h^{\\alpha \\beta }\\nabla _\\alpha \\nabla _\\beta (a+2b)R^{(1)}-h\\nabla ^2 (\\frac{a}{2}+2b)R^{(1)}-h^{\\alpha \\beta }\\nabla ^2 aR^{(1)}_{\\alpha \\beta }\\right.",
"\\nonumber \\\\ &&\\left.",
"+2R^{(1)}_{\\alpha \\beta }a^{\\prime }\\frac{\\nabla ^2}{M^2}R^{(1)\\alpha \\beta } + 2R^{(1)}b^{\\prime }\\frac{\\nabla ^2}{M^2}R^{(1)}\\right).$ A more elegant way of writing this can be found by separating the first three terms into two expressions.",
"Firstly, $\\frac{1}{M^2}\\left(h^{\\alpha \\beta }\\nabla _{\\alpha }\\nabla _{\\beta }(a+2b)R^{(1)} - h\\nabla ^2 (a+2b)R^{(1)}\\right) \\underset{\\rm weak}{=}\\frac{1}{M^2}R^{(1)}(a+2b)R^{(1)},$ but also, less obviously, $\\frac{1}{M^2}\\left(\\frac{1}{2}h\\nabla ^2 aR^{(1)}-h^{\\alpha \\beta }\\nabla ^2 a R^{(1)}_{\\alpha \\beta }\\right) \\underset{\\rm weak}{=}\\frac{1}{M^2}\\left(2R^{(1)}_{\\alpha \\beta }aR^{(1)\\alpha \\beta } - R^{(1)}aR^{(1)}\\right).$ Now we can simplify (REF ) to a more elegant, more clearly diffeomorphism-invariant form: $\\kappa t^{(0)} \\underset{\\rm weak}{=}-\\frac{1}{M^2}\\left(R^{(1)}_{\\alpha \\beta }\\left(a+ a^{\\prime }\\frac{\\nabla ^2}{M^2}\\right)R^{(1)\\alpha \\beta } + R^{(1)}\\left(b+b^{\\prime }\\frac{\\nabla ^2}{M^2}\\right)R^{(1)}\\right).$ We can extract from this the previous result for $R+R^2 /6M^2$ gravity by inserting $a=0$ , $b=1/6$ .",
"As before, we see a non-zero trace for a gravity theory with higher-order derivatives."
],
[
"Diffeomorphism invariance",
"Demonstrating diffeomorphism invariance is useful both as a consistency check of our derivation and to show that these results do not depend on any choice of coordinates.",
"Let us apply the diffeomorphism transformation to the effective stress-energy tensor in (REF ).",
"The diffeomorphism transformation of a metric perturbation is given via the Lie derivative of the metric: $\\delta h_{\\alpha \\beta } = £_\\xi g_{\\alpha \\beta } = 2g_{\\lambda (\\alpha }\\nabla _{\\beta )}\\xi ^\\lambda + g^{\\gamma \\delta }\\xi _\\gamma \\nabla _\\delta h_{\\alpha \\beta },$ where we have chosen to use the covariant derivative associated with the background metric $\\nabla _\\mu $ , but we could have chosen to use a different covariant derivative.",
"Knowing that $h_{\\alpha \\beta }(\\lambda )\\sim \\lambda $ , we require that $\\xi _\\alpha (\\lambda )\\sim \\lambda ^2$ such that $\\nabla _\\alpha \\xi _\\beta (\\lambda ) \\sim \\lambda $ .",
"Thus, in the weak limit, only the linearized diffeomorphisms are non-vanishing here.",
"Before making use of the zero tensor, the result of varying (REF ) under diffeomorphisms is $\\kappa \\delta t^{(0)}_{\\mu \\nu } & \\underset{\\rm weak}{=}& -\\xi _{(\\mu }\\nabla _{\\nu )}R^{(1)}-\\xi \\cdot \\nabla R^{(1)}_{\\mu \\nu } + 2\\xi _{\\alpha }\\nabla _{(\\mu }R^{(1)\\alpha }_{\\nu )}+\\frac{1}{2}g^{(0)}_{\\mu \\nu }\\xi \\cdot \\nabla R^{(1)} \\nonumber \\\\ &&+\\frac{2}{M^2}\\left(-\\xi \\cdot \\nabla \\nabla _\\mu \\nabla _\\nu bR^{(1)} + 2\\xi _{(\\mu }\\nabla _{\\nu )}\\nabla ^2 bR^{(1)} - g^{(0)}_{\\mu \\nu }\\xi \\cdot \\nabla \\nabla ^2 bR^{(1)}\\right) \\nonumber \\\\ &&+\\frac{1}{M^2}\\left(-\\xi \\cdot \\nabla \\nabla ^2 aR^{(1)}_{\\mu \\nu }+2\\xi ^{\\alpha }\\nabla _{(\\mu }\\nabla ^2 a R^{(1)}_{\\nu )\\alpha }-\\frac{1}{2}g^{(0)}_{\\mu \\nu }\\xi \\cdot \\nabla \\nabla ^2 aR^{(1)}\\right.\\nonumber \\\\ &&\\left.-\\xi \\cdot \\nabla \\nabla _{\\mu }\\nabla _{\\nu }aR^{(1)}+\\xi _{(\\mu }\\nabla _{\\nu )}\\nabla ^2 aR^{(1)}\\right).$ We take the form of the zero tensor given in (REF ) and perform a diffeomorphism transformation to get $\\xi _{(\\rho }\\nabla _{\\sigma )}R^{(1)}_{\\mu \\nu } & \\underset{\\rm weak}{=}& \\frac{1}{M^2}\\left(\\xi _{(\\rho }\\nabla _{\\sigma )}\\nabla _\\mu \\nabla _\\nu (a+2b)R^{(1)} + \\frac{1}{2}g^{(0)}_{\\mu \\nu }\\xi _{(\\rho }\\nabla _{\\sigma )}\\nabla ^2 (a+2b)R^{(1)} \\right.\\nonumber \\\\ &&\\left.-\\xi _{(\\rho }\\nabla _{\\sigma )}\\nabla ^2 aR^{(1)}_{\\mu \\nu }\\right).$ The top line of (REF ) has four terms at the third order in derivatives that we want to convert into higher-derivative expressions using the zero tensor.",
"The contracted forms of the zero tensor that we need are $\\xi _{(\\mu }\\nabla _{\\nu )}R^{(1)} \\underset{\\rm weak}{=}\\frac{1}{M^2}\\xi _{(\\mu }\\nabla _{\\nu )}\\nabla ^2 (2a+6b)R^{(1)},$ $\\xi \\cdot \\nabla R^{(1)}_{\\mu \\nu } \\underset{\\rm weak}{=}\\frac{1}{M^2}\\left(\\xi \\cdot \\nabla \\nabla _\\mu \\nabla _\\nu (a+2b)R^{(1)}+\\frac{1}{2}g^{(0)}_{\\mu \\nu }\\xi \\cdot \\nabla \\nabla ^2 (a+2b)R^{(1)}-\\xi \\cdot \\nabla \\nabla ^2aR^{(1)}_{\\mu \\nu }\\right),$ $-2\\xi _{\\alpha }\\nabla _{(\\mu }R^{(1)\\alpha }_{\\nu )} & \\underset{\\rm weak}{=}& \\frac{1}{M^2}\\left(-2\\xi \\cdot \\nabla \\nabla _\\mu \\nabla _\\nu (a+2b)R^{(1)}-\\xi _{(\\mu }\\nabla _{\\nu )}\\nabla ^2 (a+2b)R^{(1)} \\right.",
"\\nonumber \\\\ &&\\left.+2\\xi _{\\alpha }\\nabla _{(\\mu }\\nabla ^2 aR_{\\nu )}^{\\ \\ \\alpha }\\right),$ $-\\frac{1}{2}g^{(0)}_{\\mu \\nu }\\xi \\cdot \\nabla R^{(1)} = -\\frac{1}{M^2}g^{(0)}_{\\mu \\nu }\\xi \\cdot \\nabla \\nabla ^2 (a+3b)R^{(1)}.$ Putting these terms together, we can see that, with the help of these zero tensor relations, the top line of (REF ) cancels the rest of the terms exactly.",
"Thus the effective stress-energy tensor given in (REF ) is diffeomorphism-invariant.",
"This is an important check that gives us confidence in that result."
],
[
"Discussion and conclusion",
"As reviewed in Section REF , adding a Starobinsky $R^2$ term changes the result of the weak-limit calculation of the effective stress-energy tensor for backreaction such that the trace no longer vanishes in the weak limit.",
"This is true even though the background contribution of the $R^2$ term to the field equation vanishes in the weak limit.",
"This tells us the $R^2$ term can still give a cosmologically important contribution to the backreaction after the universe has grown to a sufficient size that the $R^2$ term is no longer important in the pure background case.",
"If the $R^2$ term has a positive coefficient, $\\kappa t^{(0)}$ converges to a negative value, as required for a candidate to mimic a positive cosmological constant.",
"Intutitively, this pure trace component can be attributed to there existing a “scalaron” mode in the $f(R)$ model that is most clearly apparent after performing a Legendre transformation into the equivalent scalar-tensor description.",
"This was the conclusion found in [50] for $R+R^2 /6M^2$ gravity.",
"This offered a motivation for alternative cosmological models that use exotic sources of inhomogeneity, such as the one discussed in [58].",
"In this paper we have noted that this result is general for an $f(R)$ expansion that begins with $R+R^2 /6M^2$ .",
"We then further generalized the procedure to find a general form for $\\kappa t_{\\mu \\nu }^{(0)}$ in effective theories of gravity whose actions are expressible as a Taylor expansion in the Riemann tensor and covariant derivatives of the Riemann tensor, i.e.",
"local, manifestly diffeomorphism-invariant gravity theories.",
"The higher derivatives are balanced by powers of a mass scale for the effective theory, $M$ .",
"We have argued that the physical consistency of the formalism requires us to scale this mass as $M\\sim \\lambda ^{-1}$ , as was also performed in [50].",
"This is because the weak limit formalism describes the limit of a large averaging scale via $\\lambda \\rightarrow 0$ , where $\\lambda $ can be interpreted as a parameter proportional to the length scale for perturbations via (REF ).",
"Thus the derivative operator $\\nabla _\\mu /M$ effectively reads the ratio of the length scale of the theory to the perturbation length scale, which, being physical, should remain fixed as we tend the averaging scale to be large.",
"If we left $M$ as a constant in $\\lambda $ , not only would the ratio change, but the length scale of the perturbation would be driven below the cutoff scale of the effective field theory, which would be extremely pathological.",
"This scaling also ensures that we are comparing perturbation terms of the same order in $h$ , such that $t^{(0)}_{\\mu \\nu }$ is written purely at $\\mathcal {O}(h^2)$ .",
"By this method, we wrote the complete set of action terms that contribute to $t_{\\mu \\nu }^{(0)}$ in closed form in (REF ).",
"We derived the general form of $t_{\\mu \\nu }^{(0)}$ in closed form, as given in (REF ).",
"Unlike in (REF ), the new result in (REF ) contains two independent structures whose coefficients have been left arbitrary in the general case.",
"Relating this to a specific example, we can look to the manifestly diffeomorphism-invariant classical Exact RG [43], where we can specialize (REF ) to the effective action derived in the “Einstein scheme” simply by setting $a=-2b$ .",
"For the choice of inhomogeneity model discussed in [58], the effective vacuum energy from backreaction was estimated to be of the form given in (REF ) in $R+R^2/6M^2$ gravity.",
"The new result in (REF ) has two new features.",
"Firstly, it now has a potentially infinite expansion in higher-derivative operators, since $a$ and $b$ are both functions of $\\nabla ^2/M^2$ .",
"For each additional $\\nabla ^2/M^2$ operator found in a term, its additive contribution to $E_{\\rm vac}^4$ would be suppressed by an extra factor of $(\\Lambda _{\\rm stripe}/M)^2$ .",
"Secondly, the trace now has two independent structures at each order in $M$ , which have independent coefficients with unspecified signs.",
"However, (REF ) would still be expected to be a sensible estimate for the magnitude of the backreaction in a generic case.",
"Although our effective stress-energy tensor has been derived with cosmological backreaction in mind, the method uses formalism that was originally constructed to describe gravitational waves from distant sources in [33], [34], [32].",
"This effective stress-energy tensor equally well applies to high-frequency gravitational waves.",
"Although it is necessary to introduce a background metric, we have not required any specific choice of background.",
"The only requirement imposed by the formalism for the matter content in the physical stress-energy tensor is that it satisfies the weak energy condition, as discussed in [30].",
"As demonstrated in Section REF , the effective stress-energy tensor in (REF ) is diffeomorphism-invariant.",
"Our new result includes the previous result as a special case where only the $R+R^2 /6M^2$ part is significant.",
"The non-zero trace found in (REF ) motivates further research into the possible cosmological significance of backreaction in higher-derivative gravity models.",
"It also raises the possibility of using backreaction as a window on exotic ultraviolet physics that might otherwise be inaccessible."
],
[
"Acknowledgements",
"I acknowledge support from the University of Southampton through a Mayflower scholarship.",
"I thank Tim Morris for helpful comments."
],
[
"Weak limits of individual field equation terms",
"The weak limits of the individual field equation perturbations in Einstein gravity are $\\delta \\left[R_{\\mu \\nu }\\right] \\underset{\\rm weak}{=}-\\frac{1}{2}h^{\\alpha \\beta }R^{(1)}_{\\mu \\alpha \\nu \\beta } - \\frac{1}{4}h_{\\mu \\nu }R^{(1)}+R^{(1)}_{\\alpha (\\mu }h_{\\nu )}^{\\ \\ \\alpha },$ $\\delta \\left[g_{\\mu \\nu }R\\right] \\underset{\\rm weak}{=}h_{\\mu \\nu }R^{(1)} - \\frac{1}{4}g_{\\mu \\nu }^{(0)}\\left(hR^{(1)}+2h^{\\alpha \\beta }R_{\\alpha \\beta }^{(1)}\\right).$ For local $f(R)$ gravity, we must consider some additional field equation contributions: $\\delta \\left[R_{\\mu \\nu }R\\right]/M^2 \\underset{\\rm weak}{=}R^{(1)}_{\\mu \\nu }R^{(1)}/M^2,$ $\\delta \\left[g_{\\mu \\nu }R^2\\right]/M^2 \\underset{\\rm weak}{=}g_{\\mu \\nu }^{(0)}R^{(1)2}/M^2,$ $\\delta \\left[D_\\mu D_\\nu R\\right]/M^2 \\underset{\\rm weak}{=}\\frac{1}{2}\\left(2R_{\\mu \\nu }^{(1)}+\\nabla _\\mu \\nabla _\\nu h\\right)R^{(1)}/M^2,$ $\\delta \\left[g_{\\mu \\nu } D^2 R\\right]/M^2 \\underset{\\rm weak}{=}\\left(\\nabla ^2 h_{\\mu \\nu }-\\frac{1}{2}g_{\\mu \\nu }^{(0)}\\nabla ^2 h\\right)R^{(1)}/M^2.$ The fully generalized gravity theory introduces new contributions again: $\\frac{1}{M^4}\\delta \\left[R_{\\alpha \\beta }D_{\\mu }D_{\\nu }a^{\\prime }\\left(\\frac{D^2}{M^2}\\right)R^{\\alpha \\beta }\\right] \\underset{\\rm weak}{=}\\frac{1}{M^4}R_{\\alpha \\beta }^{(1)}\\nabla _{\\mu }\\nabla _{\\nu }a^{\\prime }\\left(\\frac{\\nabla ^2}{M^2}\\right)R^{(1)\\alpha \\beta },$ $\\frac{1}{M^4}\\delta \\left[R D_{\\mu }D_{\\nu }b^{\\prime }\\left(\\frac{D^2}{M^2}\\right)R\\right] \\underset{\\rm weak}{=}\\frac{1}{M^4}R^{(1)} \\nabla _{\\mu }\\nabla _{\\nu }b^{\\prime }\\left(\\frac{\\nabla ^2}{M^2}\\right)R^{(1)},$ $\\frac{1}{M^2}\\delta \\left[g_{\\mu \\nu }R_{\\alpha \\beta }a\\left(\\frac{D^2}{M^2}\\right)R^{\\alpha \\beta }\\right] \\underset{\\rm weak}{=}\\frac{g^{(0)}_{\\mu \\nu }}{M^2} R^{(1)}_{\\alpha \\beta }a\\left(\\frac{\\nabla ^2}{M^2}\\right)R^{(1)\\alpha \\beta },$ $\\frac{1}{M^2}\\delta \\left[R_{\\mu \\alpha }a\\left(\\frac{D^2}{M^2}\\right)R_{\\nu }^{\\ \\alpha }\\right] \\underset{\\rm weak}{=}\\frac{1}{M^2}R^{(1)}_{\\mu \\alpha }a\\left(\\frac{\\nabla ^2}{M^2}\\right)R_{\\nu }^{(1) \\alpha },$ $\\frac{1}{M^2}\\delta \\left[D^2 a\\left(\\frac{D^2}{M^2}\\right)R_{\\mu \\nu }\\right] & \\underset{\\rm weak}{=}& \\frac{1}{M^2}\\left(-\\frac{1}{2}ha\\left(\\frac{\\nabla ^2}{M^2}\\right)\\nabla ^2 R^{(1)}_{\\mu \\nu } + h_{\\alpha (\\mu }a\\left(\\frac{\\nabla ^2}{M^2}\\right)\\nabla ^2 R_{\\nu )}^{(1)\\alpha }\\right.",
"\\nonumber \\\\ &&\\left.+ h^{\\alpha \\beta }a\\left(\\frac{\\nabla ^2}{M^2}\\right)\\nabla _{\\alpha }\\nabla _{(\\mu } R^{(1)}_{\\nu )\\beta }\\right.",
"\\nonumber \\\\ && \\left.",
"-\\frac{1}{2}h^{\\alpha }_{\\ (\\mu }\\nabla _{\\nu )}\\nabla _{\\alpha } a\\left(\\frac{\\nabla ^2}{M^2}\\right)R^{(1)}\\right),$ $\\frac{1}{M^2}\\delta \\left[D_{\\alpha }D_{(\\mu }a\\left(\\frac{D^2}{M^2}\\right)R_{\\nu )}^{\\ \\ \\alpha }\\right] & \\underset{\\rm weak}{=}& \\frac{1}{M^2}\\left(\\frac{1}{4}h_{\\ (\\mu }^{\\alpha }\\nabla _{\\nu )}\\nabla _{\\alpha }a\\left(\\frac{\\nabla ^2}{M^2}\\right)R^{(1)} + \\frac{1}{2}h^{\\alpha \\beta }a\\left(\\frac{\\nabla ^2}{M^2}\\right)\\nabla _{\\alpha }\\nabla _{(\\mu }R^{(1)}_{\\nu )\\beta }\\right.",
"\\nonumber \\\\ &&- \\frac{1}{2}h^{\\alpha }_{\\ (\\mu |}a\\left(\\frac{\\nabla ^2}{M^2}\\right)\\nabla ^2 R^{(1)}_{|\\nu )\\alpha }+\\frac{1}{2}h^{\\alpha \\beta }a\\left(\\frac{\\nabla ^2}{M^2}\\right)\\nabla _{\\mu }\\nabla _\\nu R^{(1)}_{\\alpha \\beta } \\nonumber \\\\ &&\\left.- \\frac{1}{4}ha\\left(\\frac{\\nabla ^2}{M^2}\\right)\\nabla _\\mu \\nabla _\\nu R^{(1)}\\right).$"
],
[
"Consistency of the trace of the effective stress-energy tensor",
"There are two ways to evaluate the trace of the effective stress-energy tensor in weak limit, $t^{(0)}$ .",
"Firstly, we can begin with the complete expression for $\\kappa t^{(0)}_{\\mu \\nu }$ , given in (REF ), and take the trace using the background metric.",
"Alternatively, we can take the trace of the field equation using the full metric first and then perform the weak limit to extract $t^{(0)}$ .",
"To demonstrate that these two approaches give the same answer, consider the weak limit of the field equation perturbation: $\\kappa t^{(0)}_{\\mu \\nu } \\underset{\\rm weak}{=}-\\delta \\left[\\frac{2\\kappa }{\\sqrt{g}}\\frac{\\delta S_{\\rm grav}}{\\delta g^{\\mu \\nu }}\\right].$ The zero tensor takes the form $0 \\underset{\\rm weak}{=}h_{\\rho \\sigma }\\delta \\left[\\frac{2\\kappa }{\\sqrt{g}}\\frac{\\delta S_{\\rm grav}}{\\delta g^{\\mu \\nu }}\\right].$ Pursuing the first approach, we can take the trace of (REF ) using the background metric to get $\\kappa t^{(0)} \\underset{\\rm weak}{=}-g^{(0)\\mu \\nu }\\delta \\left[\\frac{2\\kappa }{\\sqrt{g}}\\frac{\\delta S_{\\rm grav}}{\\delta g^{\\mu \\nu }}\\right] \\underset{\\rm weak}{=}-\\delta \\left[g^{\\mu \\nu }\\frac{2\\kappa }{\\sqrt{g}}\\frac{\\delta S_{\\rm grav}}{\\delta g^{\\mu \\nu }}\\right]-\\underbrace{h^{\\mu \\nu }\\delta \\left[\\frac{2\\kappa }{\\sqrt{g}}\\frac{\\delta S_{\\rm grav}}{\\delta g^{\\mu \\nu }}\\right]}_{\\text{zero, via the zero tensor}}.$ The right-hand side is the form for $\\kappa t^{(0)}$ expected from the alternative approach of taking the trace of the field equation first and the weak limit second.",
"Thus both methods must give the same answer."
]
] | 1605.06121 |
[
[
"Bayesian Spatial Monotonic Multiple Regression"
],
[
"Abstract We consider monotonic, multiple regression for a set of contiguous regions (lattice data).",
"The regression functions permissibly vary between regions and exhibit geographical structure.",
"We develop new Bayesian non-parametric methodology which allows for both continuous and discontinuous functional shapes and which are estimated using marked point processes and reversible jump Markov Chain Monte Carlo techniques.",
"Geographical dependency is incorporated by a flexible prior distribution; the parametrisation allows the dependency to vary with functional level.",
"The approach is tuned using Bayesian global optimization and cross-validation.",
"Estimates enable variable selection, threshold detection and prediction as well as the extrapolation of the regression function.",
"Performance and flexibility of our approach is illustrated by simulation studies and an application to a Norwegian insurance data set."
],
[
"Introduction",
"Geospatial data are considered in several areas, including ecology [23], forestry [31] and epidemiology [42].",
"Data in a locally aggregated form, lattice data [17], are common due to practicality or confidentiality concerns and are typically over an irregular lattice.",
"Statistical methods for such area-level data aim to model associations between a response variable and a set of explanatory variables, whilst accounting for potential geographical dependency in the model parameters.",
"To introduce geographical dependence, a neighbourhood structure, often based upon the arrangement of the areal units (regions) on a map, is typically introduced.",
"Although widely applied, most approaches only consider geographical variation in the baseline or random effect and are otherwise usually limited to the regression function being linear [41].",
"From this perspective, a modelling framework enabling both geographical variation and non-linearity in the regression surface would increase flexibility.",
"In this paper, the constraint of linearity is thus relaxed and substituted by one of monotonicity, an important assumption in several applications [36], [27], [19], [44].",
"Indeed, tests of monotonicity for the underlying process are introduced by [10] and [21].",
"Conditional on the monotonicity constraint, we develop methodology which allows for estimation of the association between the response and explanatory variables for each region; whilst exploiting any neighbourhood structure.",
"A brief review of the motivating work, with limitations, follows.",
"The estimation of a multivariate, monotonic function is considered in several statistical areas and is usually referred to as isotonic regression.",
"Early publications discuss the inference on parameter values under monotonic constraints [2], [11], [4].",
"Algorithms for solving these problems are available in the optimization literature [12], [32], [29] with the derived solutions being of piecewise constant form.",
"Isotonic regression is further considered for additive [3], [40], [15] and high-dimensional models [18], [5], functional data analysis [33], [8] and Bayesian non-parametrics [25], [39], [37], [28].",
"Despite this variety of approaches, isotonic regression is rarely applied to geospatial data.",
"One of the few examples is the work by [30] using additive modelling with univariate monotonic functions in an epidemiological setting.",
"Note, that possible geographical auto-correlation between functions is not allowed for in their work.",
"Geographical variation of the regression function, on the other hand, is usually considered in a generalized linear, or additive, modelling framework.",
"Geographically weighted regression (GWR) [13], [20] is the dominant approach and, for instance, applied in forestry [45] and social science [14].",
"Whilst GWR is based on weighted least-squares methodology, the geographically varying coefficient (GVC) model [1] 'borrows' information locally via a Bayesian specification and conditional autoregressive (CAR) modelling [6], [7].",
"[43] find that both GWR and GVC are qualitatively similar by applying them to alcohol and violence data.",
"[38] introduce an alternative approach which borrows statistical information locally for variable selection rather than for the estimation of covariate effects.",
"For the more flexible class of generalized additive models, [16] proposes the decomposition into local and geographically filtered effects.",
"However, none of these methods offer flexibility in terms of recovering discontinuities.",
"More precisely, abrupt changes in the regression surface are not captured unless these are explicitly included.",
"Negligence of such effects may result in a bias due to oversmoothing [9].",
"In the rest of this paper, we refer to these discontinuities as threshold effects.",
"We introduce a novel Bayesian, non-parametric, methodology, Bayesian Spatial Monotonic Multiple Regression (BSMMR), to facilitate analysis of lattice data under the sole assumption of response function monotonicity in the covariates.",
"Extending the approach of [37], the regional (areal) monotonic functions are each represented by a set of marked point processes.",
"The point process formulation is highly flexible and permits both smooth contours and threshold effects in the regression surface.",
"Beliefs on the geographical similarity of the monotonic functions are incorporated by a joint prior distribution.",
"In particular, the prior is constructed based upon a pair-wise discrepancy measure, resulting in a Gibbs distribution on functional spaces.",
"The defined prior is flexible in the sense that dependency between functions may be either constant, increasing or decreasing with an increasing functional mean.",
"In order to tune the prior, we propose a new algorithm, EGO-CV, which combines the concepts of cross-validation and Bayesian global optimization.",
"Realizations of the posterior are obtained by a reversible jump MCMC (RJMCMC) algorithm [22].",
"Stored samples facilitate the analysis of the regression surface with regard to threshold effects, variable selection, prediction and extrapolation.",
"The remainder of this paper is organized as follows: Section 2 details the statistical framework for the BSMMR approach, focusing on the point process representation of the regional monotonic regression functions and the constructed prior density.",
"The algorithm for estimating the functions is outlined at the end of Section 2 and further details are provided in Appendix A.",
"In order to assess the performance and sensitivity of the RJMCMC algorithm, BSMMR is then applied to simulated data in Section 3.",
"In Section 4, our methodology is applied to Norwegian insurance and meteorological data, with a view to investigating weather related claim dynamics over the region.",
"Finally, the paper concludes with a summary and discussion of our approach in Section 5."
],
[
"Modelling and Inference",
"BSMMR is derived from several subcomponents which are explained in this section.",
"Subsection REF outlines the areal data construct and also the notation used.",
"More specifically, the modelling framework for geographical variation in the functional relationship between the response and explanatory variables is formalised for a finite set of regions.",
"Geographical proximity of the regions is specified, for instance, by an adjacency matrix.",
"Without loss of generality, the monotonic functions are set to be isotonic: any monotonic function can be made isotonic by reversing some of the coordinate axis.",
"Subsection REF summarises the marked point process representation for a single monotonic function (as introduced by [37]) and which is extended in this work to lattice data: allowing for geographical variation.",
"This representation of the monotonic functions is then embedded in a Bayesian framework that facilitates borrowing statistical information between regions.",
"Construction, with motivation, of the joint prior on the monotonic functional spaces is then detailed in Subsection REF .",
"Posterior realisations and estimates of the model parameters are obtained by the algorithm described in Subsection REF ."
],
[
"Probability model and Notation",
"Suppose data are available in the form of a lattice (regular or irregular) for a set of $K$ contiguous regions.",
"Let $y_k\\in \\mathbb {R}$ and $\\mathbf {x}_k = \\left(x_{k,1},\\ldots , x_{k,m}\\right)\\in X_k\\subset \\mathbb {R}^m$ denote the response and explanatory variables, respectively, for region $k$ , $k=1,\\ldots ,K$ .",
"The closed set $X_k$ is the regional covariate space which is permissibly different across regions.",
"The associated probability model is then formally defined as $f(y_k~|~\\lambda _k(\\mathbf {x}_k), \\mathbf {\\theta }_k),$ where $\\lambda _k(\\cdot )$ refers to the monotonic function for region $k$ .",
"In this work, each $\\lambda _k(\\cdot )$ is assumed to lie within a prespecified interval $\\left[\\delta _{\\min }, \\delta _{\\max }\\right]$ .",
"Monotonicity is defined in terms of the partial Euclidean ordering, that is, for any two covariate values $\\mathbf {u},\\mathbf {v}\\in X_k$ with $\\mathbf {u}\\le \\mathbf {v}$ component-wise, $\\lambda _k(\\mathbf {u})\\le \\lambda _k(\\mathbf {v})$ .",
"The vector $\\mathbf {\\theta }_k$ contains additional model parameters which may also vary geographically.",
"Note, the probability model (REF ) contains generalisations of several modelling frameworks, for example, the generalised linear modelling family with the linear predictor being replaced here by $\\lambda _k$ ."
],
[
"A marked point process model formulation",
"The formulation by [37] is applied respectively to each of the $K$ monotonic functions.",
"Hence, $\\lambda _k$ , $k=1,\\ldots ,K$ , is postulated to be piecewise constant and is represented by a marked point process, $\\Delta _k$ , on $X_k$ .",
"We assume $X_k = \\left[0,1\\right]^m$ for notational simplicity in the rest of this subsection.",
"The points in $\\Delta _k$ define the locations in $X_k$ of the changes in the functional levels of $\\lambda _k$ .",
"Consequently, the estimation problem is shifted from the space of monotonic functions to one of marked point processes with monotonic constraints.",
"The point process approach does not restrict the space of potential solutions, as any bounded function can be approximated up to a desired degree of accuracy by increasing the number of support points in $\\Delta _k$ .",
"[37] further propose a partition of $\\Delta _k$ into a set $\\mathbf {\\Delta }_k = \\left\\lbrace \\Delta _{k,i},~i=1,\\ldots ,I\\right\\rbrace $ , with the marked point processes being defined on disjoint subspaces of $X_k$ .",
"The marked point process $\\Delta _{k,i}$ on subspace $i$ for region $k$ is then formally denoted by $\\Delta _{k,i} = \\left\\lbrace \\left(\\mathbf {\\xi }_{k,i,j}, \\delta _{k,i,j}\\right) ~ : ~ j = 1,\\ldots ,n(\\Delta _{k,i})\\right\\rbrace ,$ where $\\mathbf {\\xi }_{k,i,j}$ and $\\delta _{k,i,j}$ refer to a location and associated mark, respectively, and $n(\\Delta _{k,i})$ is the number of points in process $\\Delta _{k,i}$ .",
"In this paper, the marked point processes are defined on the non-empty subsets of the covariate set.",
"For instance, $\\Delta _{k,1}$ is based on the first covariate only and contains the locations with all but the first component being 0, $\\mathbf {\\xi }_{k,1,j} = \\left(\\cdot ,0,\\ldots ,0\\right)$ , $j = 1,\\ldots ,n\\left(\\Delta _{k,1}\\right)$ .",
"Assumed $m=2$ , this partitioning results in $I=3$ subprocesses, see Figure REF , two for the one-dimensional subsets and one for the full covariate set.",
"Figure: Partition of Δ k \\Delta _k (left) into Δ k =Δ k,1 ,Δ k,2 ,Δ k,3 \\mathbf {\\Delta }_k = \\left\\lbrace \\Delta _{k,1}, \\Delta _{k,2} , \\Delta _{k,3}\\right\\rbrace (right) for the bivariate covariate space [0,1] 2 [0,1]^2.",
"In the right panel, the marked point processes Δ k,1 \\Delta _{k,1} (triangle) and Δ k,2 \\Delta _{k,2} (square) are defined on the one-dimensional covariate subsets, 1\\left\\lbrace 1\\right\\rbrace and 2\\left\\lbrace 2\\right\\rbrace , respectively.",
"Δ k,3 \\Delta _{k,3} (diamond) is defined on the full covariate set 1,2\\left\\lbrace 1,2\\right\\rbrace .The functional level $\\lambda _k(\\mathbf {x})$ is then defined by $\\mathbf {\\Delta }_k$ as the highest mark $\\delta _{k,i,j}$ such that $\\mathbf {x}$ imposes a monotonic constraint on the associated location $\\mathbf {\\xi }_{k,i,j}$ .",
"Formally, $\\lambda _k(\\mathbf {x})$ results in $\\lambda _k(\\mathbf {x}) = \\max _{i,j} \\left\\lbrace \\delta _{k,i,j}: \\mathbf {\\xi }_{k,i,j}\\preceq \\mathbf {x}\\right\\rbrace ,$ where $\\preceq $ denotes the partial Euclidean ordering.",
"This leads indeed to a monotonic function as shown by [37].",
"The reader is referred to [37] for a discussion on other potential choices for the definition of $\\lambda _k$ .",
"The partition $\\mathbf {\\Delta }_k$ in (REF ) simplifies the analysis of the monotonic function.",
"Most importantly, in the context of parsimony, investigating the estimated marked point processes allows for variable selection.",
"Suppose that, for instance, the first explanatory variable for region $k$ , $x_{k,1}$ , is redundant.",
"Consequently, the functional level $\\lambda _k(\\mathbf {x})$ will be constant with increasing $x_{k,1}$ , i.e.",
"$\\lambda _k(\\mathbf {x}) = \\lambda _k(\\mathbf {x} + \\mathbf {\\epsilon })$ , $\\forall ~\\mathbf {x}\\in X_k$ , where $\\mathbf {\\epsilon } = \\left(\\epsilon , 0,\\ldots ,0\\right)$ has positive first component and is zero otherwise.",
"The points in $\\Delta _k$ are hence 0 in the first component, as they represent the locations of the changes in the functional level.",
"In the bivariate case, this redundancy implies that all locations lie on the vertical $x_2$ -axis in Figure REF .",
"Therefore, all points are contained in $\\Delta _{k,2}$ after the partition of $\\Delta _k$ into $\\mathbf {\\Delta }_k$ .",
"More generally, for higher dimensions, subprocesses considering point locations with non-zero first component are empty conditional on the partitioning."
],
[
"Modelling geographical dependency",
"Beliefs on geographical dependencies in $\\lambda _1,\\ldots ,\\lambda _K$ are accommodated by a joint prior distribution.",
"Little research exists on such models for both monotonic functions and marked point processes with monotonic constraints.",
"A prior is therefore constructed in this subsection which focuses on monotonic functions.",
"This choice is based upon the assumption that the main interest lies in the functional shapes.",
"More precisely, the prior model penalises discrepancies in the functional levels of $\\lambda _k$ and $\\lambda _{k^{\\prime }}$ , $k\\ne k^{\\prime }$ , $k,k^{\\prime }\\in \\left\\lbrace 1,\\ldots , K\\right\\rbrace $ , and not the number and location of support points.",
"In the first step, a pair-wise discrepancy measure $D$ for two functions $\\lambda _k$ and $\\lambda _{k^{\\prime }}$ is introduced.",
"For notational simplicity, the levels of both functions are assumed to be non-negative since in general, one would naturally consider $\\lambda _k(\\mathbf {x})-\\delta _{\\min }$ instead of $\\lambda _k(\\mathbf {x})$ .",
"Also, $D$ should be minimal, if and only if, the functions are equal.",
"In certain applications, differences in the lower or higher functional levels should be particularly avoided.",
"These considerations result in the discrepancy measure $D$ formally being defined by $D_{p,q}(\\lambda _{k}, \\lambda _{k^{\\prime }}) = \\int _{A_{k,k^{\\prime }}} \\Big |\\left[1+\\lambda _k(\\mathbf {x})\\right]^p - \\left[1+\\lambda _{k^{\\prime }}(\\mathbf {x})\\right]^p\\Big | \\left|\\lambda _k(\\mathbf {x}) - \\lambda _{k^{\\prime }}(\\mathbf {x})\\right|^q ~\\mbox{d}\\mathbf {x}~,~ p\\in \\mathbb {R}~,~q\\ge 0.$ The integral in (REF ) can be computed efficiently in our setting as $\\lambda _k$ and $\\lambda _{k^{\\prime }}$ are taken to be piecewise constant (Subsection REF ).",
"The functional levels in the first modulus term are increased by 1 in order to ensure numerical stability for the case $p<0$ as $\\lambda _k(\\mathbf {x})$ can be close to 0.",
"Figure REF illustrates that the discrepancy increases with increasing difference $\\lambda _{k^{\\prime }}(\\mathbf {x})- \\lambda _{k}(\\mathbf {x})$ in the functional levels regardless of the values for $p$ and $q$ .",
"Further, Figure REF shows that the discrepancy for a difference in the functional levels increases with increasing functional levels for $p>1$ while it decreases for $p<1$ and remains constant for $p=1$ .",
"For instance, in Figure REF , the setting $p=2, q=1$ leads to a five-fold discrepancy increase for a level difference $\\lambda _{k^{\\prime }}(\\mathbf {x}) - \\lambda _{k}(\\mathbf {x})=0.1$ when $\\lambda _k(\\mathbf {x}) = 4$ compared to when $\\lambda _k(\\mathbf {x})=0$ .",
"In contrast, a value of $p<0$ leads to a reduction in the discrepancy for higher values of $\\lambda _k(\\mathbf {x})$ .",
"Figure: Pointwise behaviour of the discrepancy D p,q D_{p,q} for different values of pp and qq with (a) increasing difference λ k ' (𝐱)-λ k (𝐱)\\lambda _{k^{\\prime }}(\\mathbf {x})- \\lambda _{k}(\\mathbf {x}) in the functional means but constant functional level for λ k (𝐱)\\lambda _k(\\mathbf {x}) and (b) constant difference λ k ' (𝐱)-λ k (𝐱)=0.1\\lambda _{k^{\\prime }}(\\mathbf {x})- \\lambda _{k}(\\mathbf {x}) = 0.1 but increasing functional level of λ k (𝐱)\\lambda _k(\\mathbf {x}).Sensitivity on $p$ and $q$ is explored via simulations in Section REF .",
"The domain $A_{k,k^{\\prime }}$ depends on the covariate spaces for regions $k$ and $k^{\\prime }$ and two possible settings are considered in Section REF .",
"The first defines $A_{k,k^{\\prime }}$ as the set of values contained in both covariate spaces, so that $A_{k,k^{\\prime }}$ is the intersection of $X_k$ and $X_{k^{\\prime }}$ : $A_{k,k^{\\prime }}=X_k\\cap X_{k^{\\prime }}$ , while the second defines it as the union of $X_k$ and $X_{k^{\\prime }}$ : $A_{k,k^{\\prime }}=X_k\\cup X_{k^{\\prime }}$ .",
"The latter facilitates the extrapolation of $\\lambda _k$ to covariate values contained in $X_{k^{\\prime }}$ and vice versa.",
"The joint prior on $\\lambda _1, \\ldots , \\lambda _K$ is then defined as a Gibbs distribution with the discrepancy measure $D_{p,q}$ in (REF ) as a per-potential.",
"In order to avoid overfitting, the model is extended to accommodate model complexity via a Geometric prior on the total number of points in $\\mathbf {\\Delta }_k$ , $n(\\mathbf {\\Delta }_{k})$ .",
"Formally, the joint prior specification for the $K$ -set of monotonic functions is given by $\\pi \\left(\\lambda _1, \\ldots , \\lambda _K|\\omega , \\eta \\right) \\propto \\prod _{k<k^{\\prime }} \\exp \\Big [-\\omega \\cdot d_{k,k^{\\prime }} \\cdot D_{p,q}\\left(\\lambda _k, \\lambda _{k^{\\prime }}\\right)\\Big ] \\times \\prod _{k=1}^K\\left(1 -\\frac{1}{\\eta }\\right)^{n(\\mathbf {\\Delta }_{k})},\\omega \\ge 0,\\eta >1.$ The non-negative constant $d_{k,k^{\\prime }}$ describes our belief on the degree of similarity of regions $k$ and $k^{\\prime }$ .",
"Many applications using conditional autoregressive models set $d_{k,k^{\\prime }}=1$ if regions $k$ and $k^{\\prime }$ are adjacent (share a border) and 0 otherwise.",
"Such a choice leads to a decrease in the computational complexity as the integral need not to be evaluated for each pair of functions.",
"This setting is considered in Sections 3 and 4.",
"The parameter $\\eta $ in (REF ) refers to the model complexity with the penalty for adding a new point decreasing in $\\eta $ .",
"Finally, the degree of geographical dependency increases in $\\omega $ with $\\omega =0$ corresponding to the functions being independent.",
"Hence, the prior $\\pi \\left(\\lambda _1, \\ldots , \\lambda _K|\\omega , \\eta \\right)$ takes it mode if all functions are equal and constant as $D_{p,q}$ , $\\omega $ , $d_{k,k^{\\prime }}$ and $\\eta $ are non-negative.",
"The probability model (REF ) and the constructed joint prior (REF ) fully specify a posterior distribution for the monotonic functions; see Appendix .",
"In summary, the posterior distribution of $\\lambda _1,\\ldots ,\\lambda _K$ is determined by three components: (a) the specified likelihood in (REF ), (b) the geographical dependency induced by $D_{p,q}$ , $\\omega $ and $d_{k,k^{\\prime }}$ in (REF ) and (c) the model complexity parameter $\\eta $ ."
],
[
"Inference and Analysis",
"The statistical framework defined in Subsections REF to REF permits efficient estimation of the underlying regression functions.",
"Inference has to be performed for both the monotonic functions and the smoothing parameter $\\omega $ .",
"Subsection REF outlines the estimation of the monotonic functions $\\lambda _1,\\ldots ,\\lambda _K$ while Subsection REF details the estimation of the smoothing parameter $\\omega $ via cross-validation and Bayesian global optimization.",
"Finally, Subsection REF considers the analysis of the regression functions $\\lambda _1,\\ldots ,\\lambda _K$ based upon the realisations sampled from the posterior."
],
[
"Estimation of the monotonic function $\\lambda _1,\\ldots ,\\lambda _K$",
"The $K$ monotonic functions are estimated by MCMC techniques, exploiting the marked point process formulation in Section REF .",
"Each location $\\mathbf {\\xi }_{k,i,j}$ with associated mark $\\delta _{k,i,j}$ is considered as one parameter with the number of points, hence the dimension of the parameter space, unknown.",
"All functions are initially constant with predefined level $\\delta _{min}$ and each point process $\\Delta _{k,i}$ contains no point.",
"Similarly to [37], the marked point processes are then updated sequentially by RJMCMC.",
"More precisely, one of three moves, implying local changes in the regression surface, is proposed on one of the processes $\\Delta _{k,1}, \\ldots , \\Delta _{k,I}$ for region $k$ , $k=1,\\ldots ,K$ , in turn.",
"The first move, Birth, adds a point $\\left(\\mathbf {\\xi }^*,\\delta ^*\\right)$ to the process with the level $\\delta ^*$ being sampled such that monotonicity is preserved.",
"The sample space for $\\mathbf {\\xi }^*$ may, for instance, be $X_k$ or an extended space; the latter facilitating extrapolation of $\\lambda _k$ .",
"A Death removes a point from the current process, maintaining reversibility.",
"Finally, a Shift leads to a 'local' change in both the location and level of an existing support point, subject to the monotonic structure of the locations being maintained.",
"For more details on the RJMCMC algorithm see Appendix .",
"In addition to the three moves defined above, the RJMCMC approach requires the specification of a maximum possible number of points, $n_{max}$ .",
"However, this does not limit the statistical rigour as the number of points is unlikely to exceed the number of data points.",
"Consequently, integrating (REF ) over the set of potential monotonic functions leads to a finite value due to the boundaries on the covariate spaces, levels (Subsection REF ) and the number of support points.",
"Further, any data based likelihood function implies a proper posterior, even though the prior is improper, in the sense that it has no mean.",
"Therefore, updating the monotonic functions by the proposed RJMCMC algorithm is feasible and approximates, after convergence, the true posterior distribution."
],
[
"Estimation of the smoothing parameter $\\omega $",
"Performance of our approach relies on a suitable value for the smoothing parameter $\\omega $ in (REF ).",
"If $\\omega $ is too high, the prior dominates the posterior distribution and geographical variation in the regression function is oversmoothed.",
"Otherwise, the data may be overfitted by the estimated function if $\\omega $ is too small.",
"The normalising constant of (REF ) can, however, not be calculated analytically.",
"Even though approaches to handle intractable normalising constants are available, these cannot be easily applied in this setting as efficient sampling from the specified prior distribution is hard.",
"Approximate Bayesian computation, for instance, would require sampling multiple times from (REF ) for each update of $\\omega $ .",
"Hence, here the parameter $\\omega $ is not updated while running the RJMCMC algorithm but, rather, is estimated before.",
"In this work, a suitable value for $\\omega $ is found by $s$ -fold cross-validation.",
"The data for each of the $K$ regions are split into $s$ subsets of equal size and the RJMCMC algorithm is performed $s$ times with varying training and test data for each considered value of $\\omega $ .",
"In order to reduce dependency on the split, multiple repetitions of the $s$ -fold cross validation with the same value for $\\omega $ are performed.",
"Parameter values are assessed and compared by the mean squared error (MSE) of the posterior predictive functional mean of the test data points derived by Monte Carlo integration.",
"Model comparison may alternatively be considered in terms of the posterior predictive densities.",
"Nevertheless, the number of evaluated values for $\\omega $ should be as small as possible since the RJMCMC algorithm described above is computationally expensive.",
"We propose to reduce the number of cross-validations by applying Bayesian optimization, in particular, the efficient global optimization (EGO) algorithm by [26].",
"Despite having potential to reduce the number of evaluations substantially, this concept has, to the best of our knowledge, never been applied in combination with cross-validation.",
"Hence, we outline a new algorithm, termed EGO-CV, in the following which combines the two concepts and aims to reduce the computational time.",
"The EGO concept postulates a sequential design strategy to detect global extrema of black-box functions.",
"EGO is widely applied in simulations if the objective function is costly to evaluate and the parameter space is relatively small [35].",
"The rationale is to model the unknown function by a Gaussian process which is updated sequentially after each evaluation and proposals are then based on the expected improvement criterion.",
"More formally, the expected improvement at an arbitrary point $z$ , for a Gaussian process $G$ , and given the current optimal value, $f_{opt}$ , of the unknown function is defined as $\\mathbb {E}\\left[\\max \\left(f_{opt}-G(z),0\\right)\\right]$ and represents the potential for $z$ to improve upon the current optimal value.",
"Proposals are considered until the expected improvement falls below a critical value which corresponds to the current solution being close to the unknown optimum.",
"As EGO balances between a local exploration of the areas likely to provide 'good model fit' and a global search (in order to avoid a local but not global minima), a suitable parameter value is generally found after a reasonable number of evaluations.",
"In the context of estimating $\\omega $ , interest lies in the global minimum of the unknown cross-validation function, CV($\\omega $ ), and a general layout of our $\\textit {EGO-CV}$ approach is given in Algorithm REF .",
"Prior to performing Bayesian optimization, an upper bound is derived as EGO can only be applied to a closed set.",
"Hence, an initial bound $\\omega _u$ is increased until the associated MSE is sufficiently greater than the one obtained for $\\omega =0$ .",
"More clarity is provided in lines 2 to 7 in Algorithm REF .",
"In this paper, an upper bound based upon $\\beta =1.1$ in Algorithm REF proved reasonable in all simulations.",
"Once the bound is fixed, an initial proposal $\\omega ^*\\in \\left[0, \\omega _u\\right]$ is made, guaranteeing that the Gaussian process is fitted with at least 3 data points.",
"After performing cross-validation for $\\omega ^*$ , the EGO algorithm is performed until the expected improvement falls below the critical value $\\alpha $ .",
"The value $\\omega _{opt}$ providing the lowest MSE is finally used as the smoothing parameter $\\omega $ in the conclusive RJMCMC algorithm.",
"In this work, EGO is performed by the DiceOptim package implemented in R by [35].",
"In addition to the MSE, its variance across the repetitions is derived too, as the DiceOptim package allows to account for uncertainty in the function evaluations.",
"The parameter $\\eta $ may be estimated similarly by investigating the regions separately or setting $\\omega =0$ .",
"The simulations in Section 3 focus, however, solely on the estimation of $\\omega $ .",
"Efficient Global Optimization within Cross-Validation (EGO-CV) [1] Observations and parameter settings for RJMCMC algorithm Cross-validation parameters: number of folds and number of repetitions Initial upper bound $\\omega _u$ , critical value $\\alpha $ , factor $\\beta $ Initialize cross validation results cv_MSE and expected improvement max_EI$>\\alpha $ Perform cross-validation for $\\omega =0$ and store CV(0) in cv_MSE Perform cross-validation for $\\omega _u$ and store CV($\\omega _u$ ) in cv_MSE CV($\\omega _u$ ) $<$ $\\beta $ CV(0) Increase upper bound $\\omega _u$ Perform cross-validation for new $\\omega _u$ and store CV($\\omega _u$ ) Set initial proposal $\\omega ^*$ , e.g.",
"$\\omega ^* = \\omega _u/2$ max_EI $>\\alpha $ Perform cross-validation for $\\omega ^*$ and store CV($\\omega ^*$ ) in cv_MSE Perform EGO on the interval $[0,\\omega _u]$ and update $\\omega ^*$ and max_EI Parameter value $\\omega _{opt}$ which provides smallest mean squared error in cv_MSE"
],
[
"Analysis",
"The RJMCMC algorithm in Subsection REF and Appendix runs for a fixed number of iterations and thinning is performed in order to reduce autocorrelation of the samples which is high as the functions change locally only.",
"Convergence is checked by sampling uniformly a fixed number of points from the covariate space and investigating the associated trace plots and auto-correlation functions.",
"Realisations sampled from the posterior distribution are rich and facilitate detailed analysis of the estimated monotonic functions.",
"Posterior estimates for $\\lambda _k$ are obtained by averaging over stored realisations sampled in the Bayesian framework.",
"Both smooth and discontinuous functional forms can be recovered by averaging over a large number of realisations with varying number, locations and levels of the points [24].",
"The posterior mean and quantiles are accessible for any covariate value $\\mathbf {x}_k\\in X_k$ by deriving the associated functional level $\\lambda _k^{(r)}(\\mathbf {x}_k)$ for each sample $r,~r=1,\\ldots ,R$ .",
"Plots of the posterior functional mean are obtained by evaluating the estimated posterior mean for a finite set of covariate values, e.g.",
"by defining a regular grid on $X_k$ .",
"Finally, the samples also facilitate the detection of threshold effects in the regression surface.",
"This detection requires distinguishing the points in the sampled marked point processes into those representing a threshold effect and those approximating a continuous shape.",
"In general, threshold effects are expected to occur in most of the samples, i.e.",
"they are removed with low probability and a shift is only likely to be accepted if it changes the point marginally.",
"Here, threshold effects are additionally defined in terms of representing a large functional level change in the regression surface.",
"Based on these considerations, each sampled point is classified.",
"Points across samples are considered as the same threshold effect if both points are very close in the covariate space and have similar functional levels.",
"Based on this classification, potential threshold effects are listed and their empirical occurrence rate across samples is derived."
],
[
"Simulation Study",
"This section aims to demonstrate that BSMMR is highly flexible, in terms of reconstructing a wide range of regression surfaces, and appraise the value for sharing statistical information geographically between regions.",
"Multiple simulations studies are performed in order to Illustrate that BSMMR improves estimates if similarities between functional shapes exist, and is also robust if the functions are dissimilar Verify that EGO-CV, Algorithm REF in Section REF , yields a sensible value for $\\omega $ Examine sensitivity on the prior parameters $p$ , $q$ and $\\eta $ in (REF ).",
"The first two goals are considered by comparing results to those for $\\omega =0$ which corresponds to imposing no geographical dependency between functions.",
"In order to facilitate visualisation of the estimated posterior mean functions, covariate spaces are bivariate in all simulations.",
"Estimates are assessed by the mean absolute error (MAE) and the standard deviation of the difference between posterior mean and true underlying function on a regular $100\\times 100$ grid of the covariate space.",
"The EGO-CV algorithm is applied with 5 repetitions of a 10-fold cross-validation.",
"Estimates for each fold are obtained by performing 50,000 iterations and storing every 100th draw after a burn-in of 25,000.",
"The initial bound $\\omega _u$ is set to 50 and increased by factor 10 until CV($\\omega _u$ ) is at least 10% higher than CV(0).",
"The critical value $\\alpha $ is set to 0.01% of the current minimum, i.e.",
"$\\alpha = \\min ($cv_MSE$)/10000$ .",
"Additionally, the algorithm also stops if 30 values for $\\omega $ have been considered.",
"Since smoothing is more sensitive on lower than upper values for $\\omega $ , EGO is performed on a transformed scale with $\\widetilde{\\omega } = \\sqrt{\\omega /50}$ which provided increased robustness.",
"Alternatively, EGO may also be applied on a transformed log scale, etc.",
"The RJMCMC algorithm described in Section REF and Appendix then runs with the derived parameter value $\\omega _{opt}$ for 2,500,000 iteration steps after a burn-in period of 500,000 and every 1000th iteration is considered for analysis.",
"The maximum number of points, $n_{max}$ , is fixed to 200 and Birth, Death and Shift are proposed with probabilities 0.3, 0.3 and 0.4, respectively.",
"Convergence of the algorithm is checked by investigating the trace plots of the functional levels for ten random points for each region $k$ , $k=1,\\ldots ,K$ ; examples are provided in [34].",
"Section REF considers two contiguous regions with Gaussian response data and illustrates sensitivity analysis on the model complexity parameter $\\eta $ and the prior parameters $p$ and $q$ in Subsections REF and REF , respectively.",
"Subsection REF also compares BSMMR to a geographically varying coefficient (GVC) model with an unique CAR prior on each covariate effect.",
"More complex geographical networks with Binomial response data and varying covariate spaces are considered in Section REF ."
],
[
"Gaussian Data",
"Observations for region $k=1,2$ are simulated independently from a Normal distribution $y_k\\sim \\mathcal {N}\\left(~\\alpha _k + \\lambda _k(\\mathbf {x}_k),~ \\sigma _k^2~\\right),$ with the monotonic functions $\\lambda _1$ and $\\lambda _2$ both defined on the unit square: $X_1=X_2=[0,1]^2$ .",
"The distribution of the covariate values varies across the two sets of simulations and is described in the respective subsections.",
"Functional levels $\\alpha _k + \\lambda _k(\\mathbf {x}_k)$ lie between 0 and 2 across simulations, facilitating comparability of the different settings.",
"The variances $\\sigma _k^2$ , $k=1,2$ , are treated as unknown, with Inverse-Gamma priors, and are updated by Gibbs sampling.",
"In Subsection REF , a CAR prior [7] is placed on the the baseline levels $\\alpha _k$ , $k=1,2$ , and these are estimated using Metropolis-within-Gibbs.",
"For comparison they are fixed to $\\alpha _k=0$ in Subsection REF ."
],
[
"Sensitivity analysis for model complexity $\\eta $",
"To explore the flexibility of BSMMR to fit a wide range of functional shapes, various pairs of monotonic functions are considered.",
"The true underlying functions $\\lambda _1$ and $\\lambda _2$ , illustrated in the first two columns of Figure REF , range from smooth curves through to discontinuous surfaces with several threshold effects.",
"Data are sampled such that 1,000 data points are observed for region 1 while 100 are observed for region 2.",
"Hence, $\\lambda _1$ can be estimated more precisely than $\\lambda _2$ .",
"This scenario facilitates, in particular, examination of the potential benefits of estimating $\\lambda _2$ when borrowing statistical information from region 1.",
"Covariate values are sampled uniformly across the unit square, $\\mathbf {x}_k \\sim \\mathcal {U}([0,1]^2)$ , and $\\sigma _k^2=0.05^2$ , $k=1,2$ .",
"Three values are considered for the model complexity parameter, $\\eta =(2,10,1000)$ , corresponding respectively, to high, moderate and low penalties for adding a new point.",
"The remaining parameters in the prior density in (REF ) are set to $p=q=1$ , so that $D_{p,q}$ simplifies to the integrated squared difference.",
"The considered GVC model consists of one intercept and two covariate effects for region $k=1,2$ and parameters are estimated by a Metropolis-within-Gibbs algorithm with 100,000 iterations.",
"Table: Mean absolute errors×10 -2 \\times 10^{-2} and standard deviations×10 -2 \\times 10^{-2} of the difference between the true function and posterior mean for different values of the model complexity parameter η\\eta for the five considered pairs of monotonic functions in Studies 1 to 5.",
"The first row for each study refers to region 1.",
"The first column gives the results for performing BSMMR with the derived parameter value ω opt \\omega _{opt} while the second column gives the results for ω=0\\omega =0.",
"The last column contains the results for an estimated GVC model.Study 1 and 2 in Figure REF and Table REF consider cases where the regional functions are identical: $\\lambda _1 = \\lambda _2$ .",
"Both the MAE and the standard deviation of the difference decrease, in particular, for region 2 by performing BSMMR with $\\omega _{opt}$ .",
"The estimated posterior mean for $\\lambda _2$ in the final column of Figure REF illustrates that both the smooth surfaces and threshold effects are captured well.",
"The estimated GVC models perform worse due to the non-linearity of the true underlying functions.",
"Study 3 and 4 consider cases where $\\lambda _1$ and $\\lambda _2$ are similar and the conclusions align with those for Study 1 and 2.",
"The improvement is also visible in the posterior mean plots for $\\omega =0$ and $\\omega =\\omega _{opt}$ in Study 1 and 4, see Appendix for details.",
"With respect to $\\eta $ , the results show only slight differences in the model fit.",
"One exception is that the model fit in Study 3 improves consistently with increasing $\\eta $ .",
"Finally, Study 5 applies the algorithm to a case where the functions are different.",
"The results show no worsening for both region 1 and region 2 as the estimated smoothing parameter $\\omega _{opt}$ is indeed equal 0 for $\\eta =10$ and $\\eta =1000$ .",
"Posterior mean plots for all settings are provided in [34].",
"The five simulation studies illustrate that BSMMR may be used effectively to improve estimates by borrowing statistical information from adjacent regions.",
"Results for region 2 improve for all cases with similar shapes which indicates that BSMMR is able to exploit neighbourhood information regardless of the underlying functions.",
"The proposed EGO-CV algorithm returns a suitable and robust value $\\omega _{opt}$ which does not oversmooth even if the functional shapes show no similarities.",
"Slight, or no, variations are found with respect to $\\eta $ , with only Study 3 showing a consistent improvement of the model fit with increasing $\\eta $ .",
"As higher values for $\\eta $ allow on average for a higher number of process points, the smooth surfaces are fitted better due to the posterior mean having more but smaller jumps.",
"Since the simulations indicate little, or no, sensitivity, the parameter value is fixed to $\\eta =2$ in the following simulations."
],
[
"Sensitivity analysis for prior parameters $p$ and {{formula:244b3e38-b93e-4905-9c24-9421b0dc3bd9}}",
"Subsection REF explored the performance of BSMMR for different functional shapes but only considered uniformly distributed covariate values.",
"In the following simulations, interest lies in exploring the flexibility of BSMMR for handling non-uniform distributions of $\\mathbf {x}$ .",
"This setting also allows a more general sensitivity analysis on $p$ and $q$ .",
"In particular, performance may depend on whether relatively more data points are observed in areas with similar functional levels.",
"The first column in Figure REF illustrates the true pair of underlying monotonic functions which is fixed across Studies 1 to 3.",
"Both functions exhibit a threshold effect at $(0.5, 0.5)$ and have similar lower functional levels.",
"For each of the two regions, 200 data points are simulated with $\\sigma _1^2=0.2^2$ and $\\sigma _2^2=0.3^2$ .",
"The number of data points sampled above the threshold effect changes across simulations.",
"Study 1 considers the case with 150 covariate values sampled uniformly above the threshold effect $(0.5, 0.5)$ .",
"Study 3 considers the case where 25 covariate values are sampled for the upper functional levels.",
"The remaining data points, 50 and 175, respectively, are sampled uniformly below the threshold effect.",
"Study 2 considers the case with the covariate values being sampled uniformly from the unit square, $\\mathbf {x}\\sim \\mathcal {U}\\left[0,1\\right]^2$ .",
"Five different settings for $p$ and $q$ are applied and compared.",
"The first two settings: (1) $p=1,q=1$ and (2) $p=1,q=2$ impose a constant degree of dependency between $\\lambda _1$ and $\\lambda _2$ , independent of the functional levels.",
"Settings (3) $p=0.5,q=1$ and (4) $p=-1,q=1$ , allow for stronger dependency in the lower functional levels while (5) $p=3,q=1$ imposes increased dependency for higher levels.",
"Figure: True functions and posterior mean plots for the parameter settings of pp and qq providing the lowest combined MAE by performing BSMMR with the proposed value ω opt \\omega _{opt}for Study 1 to 3.",
"Plots for region 1 are given in the first row while the second row refers to region 2.",
"The threshold effect is at (0.5,0.5)(0.5, 0.5) in the true functions.",
"The 200 covariate values for each region are sampled uniformly on [0,1] 2 [0,1]^2 in Study 2 and in the ratios 150:50150:50 and 25:17525:175, above:below the threshold, respectively, in Studies 1 and 3.Table REF shows that the settings with $p<1$ perform generally best and improve the average MAE by up to $17\\%$ compared to $\\omega =0$ .",
"The degree of improvement by sharing information geographically decreases from Study 1 to 3 due to the number of data points being available to fit the lower functional levels.",
"Figure REF illustrates that the threshold effect is captured correctly in each study.",
"Finally, the model fit for region 1 is better than for region 2 across simulations due to the reduced variability in the observations.",
"Posterior mean plots for all settings are provided in [34] In Study 1, all settings for $p$ and $q$ improve the model fit in terms of the averaged MAE and standard deviation compared to the setting $\\omega =0$ .",
"The overall improvement is due to the higher concentration of data points above the threshold.",
"In particular, a value $\\omega _{opt}$ is found such that the prior contributes beneficially to the estimation of the lower functional levels without causing a large bias on the upper functional levels.",
"The settings with $p<1$ perform best as these impose a very small penalty for differences in the upper levels, hence allowing for increased dependency in the lower functional levels.",
"The posterior mean plots in the second column of Figure REF illustrate that some features in the lower functional levels are not captured well due to the low number of data points available.",
"For Study 2, Table REF shows that the settings $p\\ge 1$ perform similarly to $\\omega =0$ whilst $p<1$ leads to substantial improvements for uniformly distributed data.",
"As the intensity of data points is similar across the covariate space, values for $\\omega $ leading to improvements for the lower functional levels affect the estimation of the upper functional levels more strongly than in Study 1.",
"As such, settings with $p\\ge 1$ cause some bias in the upper functional levels while improving estimates for the lower functional levels.",
"Figure REF also shows that the lower functional levels are fitted better when compared to Study 1.",
"Finally, all settings except for $p=-1,q=1$ perform similarly to $\\omega =0$ in Study 3.",
"As the intensity of observations is increased on lower levels, compared to upper levels, $\\omega _{opt}$ often implies a bias for the upper levels whilst improving model fit on the lower levels.",
"This does not occur for $p=-1, q=1$ as dependency decreases relatively quickly with increasing functional level.",
"In summary, the simulations performed in this subsection illustrate two important aspects of the BSMMR approach.",
"Firstly, the chosen values for $p$ and $q$ affect the performance of BSMMR quite strongly.",
"As such, $p$ should be chosen to be smaller than 1 if functions are presumably similar in their lower functional levels only.",
"Conversely, $p$ should be set to be greater than 1 if the upper levels are more similar.",
"Also, the appropriateness of the choice for $p$ and $q$ depends not only on functional similarities but also on the distribution of the covariate values.",
"Table: Mean absolute errors×10 -2 \\times 10^{-2} and standard deviation×10 -2 \\times 10^{-2} of the difference between true function and posterior mean for various settings of the prior parameters pp and qq.",
"The first row for each study refers to region 1 and the last column gives the results for ω=0\\omega =0."
],
[
"Binomial Data",
"In this subsection, BSMMR is applied to two geographical networks of 5 regions in order to explore its performance in more complex settings.",
"Observations of the response variable are taken to be Binomially distributed, $y_k\\sim \\mbox{Binomial}(A, p_k)$ , with the number of trials $A=100$ fixed and the success probability for region $k$ , $p_k$ , on the logit scale being modelled by a monotonic function $\\lambda _k$ .",
"Formally, responses $y_k$ are simulated from $y_k \\sim \\mbox{Binomial}\\left(100, \\frac{\\exp {\\left[\\lambda _k\\left(\\mathbf {x}_k\\right)\\right]}}{1+\\exp {\\left[\\lambda _k\\left(\\mathbf {x}_k\\right)\\right]}}\\right), ~k=1,\\ldots ,5,$ where $\\lambda _k\\left(\\cdot \\right)$ takes values between 0 and 3, i.e.",
"$p_k$ is assumed to lie between 0.5 and 0.95.",
"In the first simulation study, covariate values $\\mathbf {x}_k$ are sampled uniformly on $[0,1]^2$ .",
"In the second study, the covariate spaces $X_1,\\ldots ,X_5$ vary.",
"The variation of the covariate spaces is considered to facilitate analysis with respect to $A_{k,k^{\\prime }}$ in (REF ).",
"Figure: Network 2Study 1 considers the geographical network illustrated in Figure REF with region 2 being neighbour of regions 1 and 3, region 3 being neighbour of 2 and 4, etc.",
"The true underlying functions are constructed such that (i) neighbouring functions share similarities and (ii) region 1 and 5 are quite different.",
"Figure REF illustrates the true functions (Column 1) and shows, for instance, an increase in the maximum level from region 1 through to 5.",
"The covariate spaces $X_1,\\ldots ,X_5$ are set equal to the unit square and 300 covariate values are sampled for each region.",
"BSMMR is performed with two settings, (1) $p=1,q=1$ and (2) $p=-1,q=1$ , and results are again compared to those for $\\omega =0$ .",
"Table REF shows an overall improvement in both the MAE and the standard deviation.",
"However, the degree of improvement is not consistent across all five regions.",
"In particular, the model fit improves for regions 2 to 4 while regions 1 and 5 show only small, if any, improvement.",
"The setting $p=-1,q=1$ performs best, based on the average MAE across the five regions, which aligns with the results in Subsection REF since the lower functional levels are more similar than are the upper ones.",
"The larger improvement for regions 2 through 4 is due to their functional shape being close to the average of their neighbours and thus increased smoothing is preferred.",
"In contrast, higher values for $\\omega $ imply that region 1 and region 5 are overly smoothed.",
"With respect to the posterior mean plots, Figure REF indicates only slight differences for the settings.",
"Figure: True underlying functions (first column) and posterior mean estimates obtained for ω=0\\omega =0 (second column) and p=-1,q=1p=-1,q=1 (third column) for the five regions with neighbourhood structure as detailed in Network 1.The geographical structure for Network 2 is illustrated in Figure REF with regions having between 1 and 4 neighbours.",
"Covariate spaces vary respective to the first covariate, in particular, taking values between 0.0 and 0.7 for regions 1, 3 and 5, 0.1 and 0.9 for region 4 and 0.2 to 1.0 for region 2.",
"This setting facilitates examination of BSMMR with respect to extrapolation to both lower and upper functional levels.",
"The number of observations generated for regions 1 through 5 are 100, 500, 200, 300 and 200, respectively.",
"Figure REF shows that the true underlying functions, plotted on the unit square, are similar in their functional levels over the whole covariate space.",
"In this Study 2, BSMMR is applied with $p=1,q=1$ and two different settings for the sets $A_{k,k^{\\prime }}$ in (REF ).",
"Table: Mean absolute errors×10 -2 \\times 10^{-2} and standard deviation×10 -2 \\times 10^{-2} of the residuals obtained by BSMMR for networks of five regions with different settings in the prior density.",
"The last column gives the average mean absolute error of the five regions.",
"The last row in Study 2 provides the mean absolute error on the extrapolated space.In the first setting, $A_{k,k^{\\prime }}$ is set equal to the intersection of $X_k$ and $X_{k^{\\prime }}$ , $A_{k,k^{\\prime }} = X_k\\cap X_{k^{\\prime }}$ .",
"Table REF indicates a reduction in the overall MAE, compared to $\\omega =0$ , with larger improvements being achieved for region 3 to 5.",
"In conclusion, region 3 through 5 borrow statistical strength from region 2 as a consequence of the higher number of observations for region 2.",
"Further, estimates for $\\lambda _2$ are more similar to $\\lambda _3$ to $\\lambda _5$ than to $\\lambda _1$ due to the weights $d_{k,k^{\\prime }}$ , implying that only small amounts of statistical information can be borrowed when estimating $\\lambda _1$ .",
"Additionally, both MAE and standard deviation are quite high for region 1 due to the paucity of data points and the high variance.",
"Figure REF also indicates limitations in fitting the functional shapes around the threshold effect due to the small number of observations.",
"The second setting defines $A_{k,k^{\\prime }}$ as the union of the covariates spaces $X_k$ and $X_{k^{\\prime }}$ , $A_{k,k^{\\prime }} = X_k\\cup X_{k^{\\prime }}$ .",
"This setting allows to extrapolate the functions $\\lambda _1, \\ldots , \\lambda _5$ by borrowing information from adjacent regions.",
"For the original covariate spaces, Table REF indicates that the setting $A_{k,k^{\\prime }} = X_k\\cup X_{k^{\\prime }}$ performs similarly to $A_{k,k^{\\prime }} = X_k\\cap X_{k^{\\prime }}$ in terms of the overall MAE but not with respect to the single regions.",
"In particular, the setting $A_{k,k^{\\prime }} = X_k\\cap X_{k^{\\prime }}$ provides the lowest MAE for regions 1 and 2 while $A_{k,k^{\\prime }} = X_k\\cup X_{k^{\\prime }}$ reduces the MAE for region 5.",
"Since statistical information for extrapolating $\\lambda _2$ and $\\lambda _4$ to the lower values of Covariate 1 is borrowed from regions 1, 3 and 5, the model fit on the extrapolated spaces depends on the degree of similarity.",
"Similar arguments hold for the extrapolation of $\\lambda _1$ , $\\lambda _3$ and $\\lambda _5$ to the higher values of the first covariate.",
"This effect is observable in the last column of Figure REF .",
"The higher values for the MAE in Table REF on the extrapolated spaces are hence expected.",
"Figure: True underlying functions (first column) and posterior mean estimates for the settings in Study 2.",
"The second corresponds to A k,k ' A_{k,k^{\\prime }} being the intersection of X k X_k and X k ' X_{k^{\\prime }} while the last column gives the extrapolated functions for A k,k ' A_{k,k^{\\prime }} being defined as the union of X k X_k and X k ' X_{k^{\\prime }}.In summary, all simulations performed in this section demonstrated that BSMMR leads to improved model fit, irrespective of the functional shapes, if similarities between neighbouring regions exists.",
"This conclusion is found for both continuous and discrete observations and independent of the variance of the data process.",
"The proposed EGO-CV algorithm, Algorithm REF , proved to be suitable and robust for both similar and dissimilar neighbouring functions.",
"While simulations showed a clear sensitivity with respect to $p$ and $q$ , the results show little, or no, sensitivity to the model complexity parameter $\\eta $ .",
"Further, sensitivity with respect to $p$ and $q$ also depends on the distribution of the covariate observations over the associated covariate spaces $X_1,\\ldots ,X_K$ .",
"Finally, Study 2 in Section 3.2 clearly demonstrated the potential of BSMMR to extrapolate monotonic functions."
],
[
"Case Study",
"BSMMR is applied in order to investigate the weather dynamics leading to property insurance claims in Norway.",
"The data provide the daily number of claims per Norwegian municipality from 1997 to 2006 due to precipitation, surface water, snow melt, undermined drainage, sewage back-flow or blocked pipes.",
"The monthly number of policies and daily observations of multiple weather covariates are also available for each municipality.",
"While [38] consider several covariates, we focus analysis on the amount of precipitation on the current and previous days.",
"This selection is based on [38] finding that these are the most important covariates.",
"Intuitively, the claim risk per property increases with the amount of precipitation.",
"Therefore, the monotonicity assumption appears reasonable and BSMMR is applicable; the assumption could be verified based on the test by [10].",
"Analysis is performed for a contiguous set of $K=11$ municipalities around the Oslofjord and Figure REF illustrates the neighbourhood structure.",
"Figure: Map of the 11 municipalities considered.The applied modelling framework is formalised in the following.",
"Let $N_{k,t}$ denote the number of claims recorded on day $t$ for municipality $k$ .",
"Further, $R_{k,t}$ and $R_{k,t-1}$ refer to the amount of precipitation on the current and previous day respectively for municipality $k$ .",
"Interest lies in the association between $N_{k,t}$ and the two covariates $R_{k,t}$ and $R_{k,t-1}$ .",
"Due to the number of policies $A_{k,t}$ being known, $N_{k,t}$ is modelled by a Binomial distribution with the claim probability $p_{k,t}$ changing monotonically in $R_{k,t}$ and $R_{k,t-1}$ .",
"Formally, the claim model is given by $\\begin{split}N_{k,t} &\\sim \\mbox{Binomial}\\left(A_{k,t}, p_{k,t}\\right)\\\\\\mbox{logit}\\left(p_{k,t}\\right) &= \\alpha _k + \\lambda _k\\left(R_{k,t}, R_{k,t+1}\\right),\\end{split}$ where $\\lambda _k$ and $\\alpha _k$ , $k=1,\\ldots ,11$ , are, respectively, the unknown regional monotonic functions and baseline levels.",
"As in the simulation study in Subsection REF , a CAR prior is set on the regional intercepts $\\alpha _1, \\ldots , \\alpha _{11}$ .",
"BSMMR is then applied with prior parameters $p=-1$ , $q=1$ , $\\eta =2$ and $d_{k,k^{\\prime }}=1$ if municipalities $k$ and $k^{\\prime }$ share a border and 0, otherwise.",
"The selection $p=-1$ is due to the high occurrence of days with little or no precipitation.",
"More specifically, relatively more data points are available to model the lower functional levels compared to the number of days with high amount of precipitation.",
"Hence, $p=-1$ is a preferable choice based on the simulation findings in Subsection REF .",
"The set $A_{k,k^{\\prime }}$ in the prior density is defined as the union of the covariate spaces for municipality $k$ and $k^{\\prime }$ .",
"In order to obtain more uniformly distributed covariate values, BSMMR is applied to the transformed covariates $\\sqrt{R_{k,t}}$ and $\\sqrt{R_{k,t-1}}$ .",
"Observations for 2001 and 2003 are stored as test data in order to assess and compare predictive performance.",
"The model parameters are then estimated based on the remaining 8 years.",
"BSMMR is compared to two competing models: Average dailly number of claims over the training data set Geographically varying coefficient (GVC) model with unique CAR priors on each covariate effect.",
"The EGO-CV algorithm, Algorithm 1, is applied with 3 repetitions of 10-fold cross-validation.",
"The initial baseline level is set to -9.0 for all municipalities, based on the GVC model fit for Oslo.",
"The boundaries on the functional levels, $\\delta _{\\min }$ and $\\delta _{\\max }$ , are set to 0.0 and 6.0, respectively.",
"After selecting an appropriate smoothing parameter $\\omega _{opt}$ , the final RJMCMC algorithm runs for 1,000,000 iteration steps and every 500th sample is stored for analysis after a burn-in of 200,000.",
"The GVC model is fitted with the two covariate effects by performing 10,000 iteration steps with a burn-in of 1,000.",
"Table: Sum of squared prediction errors for 2001 and 2003 based on the model fitted with explanatory variables R t R_t, R t+1 R_{t+1} for the remaining years between 1997 and 2006.Table REF shows that BSMMR performs the best in terms of the overall predictive error, denoted $\\Sigma $ , reducing the value from 1620 for the GVC fit to 1490.",
"In case of Baerum, BSMMR leads to a much smaller predictive error than the GVC fit which indicates that the underlying regression function is non-linear.",
"The difference for Asker is due to one day with a high number of claims that is not captured well as days with similar covariate values in the training data show no occurrence of this magnitude.",
"No substantial differences are found for several municipalities which corresponds to zero high-claim days being observed over the test period.",
"The occurrence of zero high-claim days can be seen in the results as the predictive squared error obtained for the average daily number of claims is low for most municipalities.",
"Finally, the results show that estimates improve by accounting for geographical dependency.",
"The small level of improvement from $\\omega =0$ to $\\omega =\\omega _{opt}$ can be explained by the high number of training data points ($\\approx $ 3000) for each municipality.",
"Hence, important structures in the regression surface are likely to be detected without using statistical information from neighbouring municipalities.",
"In conclusion, the application of BSMMR improved the model fit."
],
[
"Discussion",
"We have developed new non-parametric Bayesian methodology which facilitates the modelling and estimation of geographically varying monotonic regression functions.",
"Each regional function is defined to be of piecewise constant form and is represented by a set of marked point processes.",
"Statistical information is 'shared' geographically by a prior which also includes a penalty for model complexity.",
"The prior is constructed based on a pair-wise discrepancy measure which penalizes differences in the functional levels.",
"The discrepancy measure is flexible and allows the geographical dependency to vary with the functional levels.",
"As the normalising constant of the prior was intractable, we developed the EGO-CV algorithm, which combines cross-validation and Bayesian global optimization, in order to optimise the smoothing parameter $\\omega $ .",
"Our simulation and case studies have illustrated that BSMMR has the potential to improve estimates if similarities between neighbouring functions exist.",
"These conclusions were irrespective of the functional shapes and the distribution of the covariate values.",
"From a general perspective, BSMMR provides a useful modelling approach which allows for both smooth and discontinuous functional forms which may not be captured if a linear or additive form is assumed.",
"The approach may be applied generally for network and dependency modelling and is not limited to a geographical context.",
"BSMMR is well suited for covariate spaces of low dimensions and offers great flexibility.",
"More caution is, however, recommended for higher dimensions, as it is for many flexible modelling approaches.",
"This is due to the computational cost for calculating the integral in the prior distribution scales exponentially with the dimension of the covariate space.",
"Additionally, the monotonic constraint becomes less restrictive with increasing dimensions, leading to a potential overfit of the data.",
"Hence, the considered monotonic functions should preferably be defined on covariate spaces of dimension two to five.",
"Nevertheless, it is possible to estimate an additive model of one monotonic and one linear function or to combine several monotonic functions.",
"This may be in particular useful if two covariate subsets affect the probability model independently from each other.",
"Consequently, the methodology introduced in this paper may be used for higher dimensional covariate spaces but requires a pre-analysis in order to achieve optimal results.",
"Computationally, the approach is demanding, depending on the geographical network structure, the dimension of the covariate space and the number of data points per region.",
"As mentioned in the previous paragraph, the calculation of the prior ratio is computationally expensive, as the evaluation of the integral is non-trivial.",
"We reduce the computational time by firstly deriving the area of the covariate space affected by the proposal and then evaluating $D_{p,q}$ over this subspace.",
"However, further splits into smaller subspaces are usually required as the neighbouring functions are likely to vary over this subspace.",
"Another computational step is the update of the likelihood function as monotonicity has to be checked respectively for each data point.",
"Hence, each $s$ -fold cross-validation requires a long time.",
"The combination of cross-validation and Bayesian optimization reduced the computational time as the EGO algorithm is fast.",
"C++ code and R files used in Section can be downloaded from www.lancaster.ac.uk/pg/rohrbeck/BSMMR.",
"The work presented in this paper can be extended in several ways.",
"From a theoretical perspective, an approach for estimating suitable values for $p$ and $q$ is of interest and will be considered in future research.",
"Alternatively, the discrepancy measure in (REF ) may be defined differently, e.g.",
"based on the Kullback-Leiber divergence.",
"Further, the decision to perform 10-fold cross-validation in all studies was an arbitrary selection.",
"As the value for $\\omega $ depends also on the number of data points, it may be better to have more folds in order to obtain more accurate estimates.",
"The additional computational time may be tackled using parallelised computing techniques.",
"We plan to implement this in future versions of our software package.",
"Finally, the claim model in Section is not the most sophisticated as [38] find, for instance, that drainage also effects the claim probability positively.",
"The model also takes no temporal variation into account.",
"The modelling framework is, however, extendible to a spatio-temporal setting with a function being estimated for each municipality for each year."
],
[
"Acknowledgements",
"Rohrbeck gratefully acknowledges funding of the EPSRC funded STOR-i centre for doctoral training (grant number EP/H023151/1).",
"This research was also financially supported by the Norwegian Research Council.",
"This paper greatly benefited from discussions with Jonathan Tawn, Elija Arjas, Christopher Nemeth, Sylvia Richardson, Lawrence Bardwell, Jamie Fairbrother, David Hofmeyr and Ida Scheel.",
"We also thank Ida Scheel for providing access to the Norwegian insurance and weather data."
],
[
"Details of the RJMCMC algorithm",
"The RJMCMC algorithm in Section REF outlines our approach to sample realizations from the posterior $\\pi \\left(\\lambda _1, \\ldots , \\lambda _K, \\mathbf {\\theta }_1, \\ldots , \\mathbf {\\theta }_K|\\mathcal {D}, \\omega , \\eta \\right) \\propto \\prod _{k=1}^K \\prod _{t=1}^{T_k} f\\left(y_{k,t}| \\lambda _k\\left(\\mathbf {x}_{k,t}\\right), \\mathbf {\\theta }_k \\right) \\times \\pi \\left(\\lambda _1, \\ldots , \\lambda _K|\\omega , \\eta \\right) \\times \\pi (\\mathbf {\\theta }_1, \\ldots , \\mathbf {\\theta }_K),$ where $\\mathcal {D}$ denotes the observations of the explanatory and response variables, $\\mathbf {x}_{k,t}$ and $y_{k,t}$ respectively, for the $K$ regions.",
"The number $T_k$ refers to the number of observations for region $k$ .",
"In the following, the update of one of the monotonic functions is described in detail.",
"Function $\\lambda _k$ , $k=1,\\ldots ,K$ , is updated by selecting one of the subprocesses $\\Delta _{k,i}$ , $i=1,\\ldots ,I$ with equal probability and sampling one of the predefined moves.",
"For simplicity in the notation, the probability of proposing Birth and Death are set to be equal in the following.",
"Here, Death or Shift are rejected instantly if the process $\\Delta _{k,i}$ is empty and, similarly, Birth results in an unchanged process if the maximum number of points, $n_{max}$ , is reached.",
"The Jacobian in the acceptance probability is equal to 1 as the mapping for adding a point is equal to the identity function.",
"Hence, the acceptance probability for accepting the new function $\\lambda _k^*$ conditional on all other functions, $\\lambda _{-k}$ , is given by $\\begin{split}\\alpha \\left(\\lambda _k,\\lambda _k^{*}\\right) &= \\min \\left\\lbrace 1,~R\\left(\\lambda _k,\\lambda _k^{*}\\right)\\right\\rbrace \\\\&= \\min \\left\\lbrace 1,~\\prod _{t=1}^{T_k}\\frac{f\\left(\\mathbf {y}_{k,t}~|~\\lambda _k^{*}(\\mathbf {x}_{k,t}), \\mathbf {\\theta }_k\\right)}{f\\left(\\mathbf {y}_{k,t}~|~\\lambda _k(\\mathbf {x}_{k,t}), \\mathbf {\\theta }_k\\right)} \\times \\frac{\\pi \\left(\\lambda _{k}^{*}~|~\\lambda _{-k},\\omega ,\\eta \\right)}{\\pi \\left(\\lambda _{k}~|~ \\lambda _{-k},\\omega ,\\eta \\right)}\\times \\frac{q\\left(\\lambda _{k}|\\lambda _{k}^{*}\\right)}{q\\left(\\lambda _{k}^{*}|\\lambda _{k}\\right)}\\right\\rbrace ,\\\\\\end{split}$ where $\\mathbf {y}_k=\\left(y_{k,1}, \\ldots ,y_{k,T_k}\\right)$ and $\\mathbf {x}_k=\\left(\\mathbf {x}_{k,1}, \\ldots ,\\mathbf {x}_{k,T_k}\\right)$ refer to the observed responses and covariate values for region $k$ respectively.",
"In case of a Birth, the addition of a support point $\\left(\\mathbf {\\xi }^*, \\delta ^*\\right)$ is proposed with the location $\\mathbf {\\xi }^*$ being sampled uniformly from the space $X_{k,i}$ associated to $\\Delta _{k,i}$ .",
"The level $\\delta ^*$ is then sampled uniformly on the interval $\\left[b_l,b_u\\right]\\subseteq \\left[\\delta _{\\min }, \\delta _{\\max }\\right]$ of possible levels such that monotonicity is preserved.",
"The reverse move Death selects one of the $n\\left(\\Delta _{k,i}\\right)$ existing points with equal probability and proposes to remove it.",
"Hence, $R\\left(\\lambda _k,\\lambda _k^{*}\\right)$ in the acceptance probability result in $R\\left(\\lambda _k,\\lambda _k^{*}\\right) = \\prod _{t=1}^{T_k}\\frac{f\\left(\\mathbf {y}_{k,t}~|~ \\lambda _k^{*}(\\mathbf {x}_k), \\mathbf {\\theta }_k\\right)}{f\\left(\\mathbf {y}_{k,t}~ |~\\lambda _k(\\mathbf {x}_k), \\mathbf {\\theta }_k\\right)} \\times \\prod _{\\begin{array}{c}k^{\\prime }=1\\\\k^{\\prime }\\ne k\\end{array}}^K \\frac{\\exp \\left[-\\omega \\cdot d_{k,k^{\\prime }} \\cdot D_{p,q}\\left(\\lambda _k^{*}, \\lambda _{k^{\\prime }}\\right)\\right]}{\\exp \\left[-\\omega \\cdot d_{k,k^{\\prime }} \\cdot D_{p,q}\\left(\\lambda _k, \\lambda _{k^{\\prime }}\\right)\\right]}\\left(1-\\frac{1}{\\eta }\\right)\\times \\frac{\\left|X_{k,i}\\right|\\left(b_u-b_l\\right)}{n\\left(\\Delta _{k,i}\\right)+1},$ for a new support point $\\left(\\mathbf {\\xi }^*, \\delta ^*\\right)$ , where $\\left|X_{k,i}\\right|$ denotes the volume of $X_{k,i}$ .",
"Equivalently, it yields to $R\\left(\\lambda _k,\\lambda _k^{*}\\right) = \\prod _{t=1}^{T_k}\\frac{f\\left(\\mathbf {y}_{k,t}~|~ \\lambda _k^{*}(\\mathbf {x}_k), \\mathbf {\\theta }_k\\right)}{f\\left(\\mathbf {y}_{k,t}~|~\\lambda _k(\\mathbf {x}_k), \\mathbf {\\theta }_k\\right)}\\times \\prod _{\\begin{array}{c}k^{\\prime }=1\\\\k^{\\prime }\\ne k\\end{array}}^K \\frac{\\exp \\left[-\\omega \\cdot d_{k,k^{\\prime }} \\cdot D_{p,q}\\left(\\lambda _k^{*}, \\lambda _{k^{\\prime }}\\right)\\right]}{\\exp \\left[-\\omega \\cdot d_{k,k^{\\prime }} \\cdot D_{p,q}\\left(\\lambda _k, \\lambda _{k^{\\prime }}\\right)\\right]} \\frac{1}{\\left(1-\\frac{1}{\\eta }\\right)}\\times \\frac{n\\left(\\Delta _{k,i}\\right)}{|X_{k,i}|\\left(b_u-b_l\\right)}$ for removing a current support point.",
"Finally, a Shift proposes a local change of $\\lambda _k$ by shifting both the location and level of an existing support point but without changing the current partial ordering of the support points.",
"In this work, the index $j^*$ of the support point $\\left(\\mathbf {\\xi }_{k,i,j^*}, \\delta _{k,i,j^*}\\right)$ to be moved is selected with probability proportional to the current level in order to improve the convergence for the higher levels.",
"The new location $\\mathbf {\\xi }^*_{k,i,j^*}$ is then sampled uniformly with the lower and upper bounds in each covariate being given by the next higher and lower covariate values; see Appendix of [37] for details.",
"The proposed level $\\delta _{k,i,j^*}^*$ is then sampled uniformly on the set of possible values which preserve the monotonic constraint.",
"Therefore, $R\\left(\\lambda _k,\\lambda _k^{*}\\right) = \\prod _{t=1}^{T_k}\\frac{f\\left(\\mathbf {y}_{k,t}~|~ \\lambda _k^{*}(\\mathbf {x}_k), \\mathbf {\\theta }_k\\right)}{f\\left(\\mathbf {y}_{k,t}~|~\\lambda _k(\\mathbf {x}_k), \\mathbf {\\theta }_k\\right)}\\times \\prod _{\\begin{array}{c}k^{\\prime }=1\\\\k^{\\prime }\\ne k\\end{array}}^K \\frac{\\exp \\left[-\\omega \\cdot d_{k,k^{\\prime }} \\cdot D_{p,q}\\left(\\lambda _k^{*}, \\lambda _{k^{\\prime }}\\right)\\right]}{\\exp \\left[-\\omega \\cdot d_{k,k^{\\prime }} \\cdot D_{p,q}\\left(\\lambda _k, \\lambda _{k^{\\prime }}\\right)\\right]} \\times \\frac{\\delta ^*_{k,i,j^*}\\sum _{j=1}^{n(\\Delta _{k,i})} \\delta _{k,i,j}}{\\delta _{k,i,j^*}\\sum _{j=1}^{n(\\Delta _{k,i})} \\delta _{k,i,j}^*},$ where $\\sum _{j=1}^{n(\\Delta _{k,i})}\\delta _{k,i,j}$ denotes the sum of levels of the current support points in process $\\Delta _{k,i}$ and $\\sum _{j=1}^{n(\\Delta _{k,i})} \\delta _{k,i,j}^*$ is the updated sum given the proposal."
],
[
"Details of the posterior mean plots for Study 1 and 4 in Subsection ",
"The improvements in Table REF for Region 2 can also be visualised for Study 1 and 4.",
"Study 1 considered the case of both monotonic functions being identical and continuous.",
"Figure REF illustrates that the posterior mean for $\\lambda _2$ appears smoother for $\\omega =\\omega _{opt}$ than for $\\omega =0$ , in particular, for the higher functional levels of $\\lambda _2$ .",
"For Study 4 with $\\lambda _1$ and $\\lambda _2$ being similar, Figure REF shows an improved modelling of the threshold effect.",
"The posterior mean for $\\omega =0$ shows no clear location of the threshold effect and the surface appears slightly continuous.",
"For $\\omega =\\omega _{opt}$ , the threshold effect is fitted better and its location is much better visible.",
"In summary, the posterior mean plots illustrate that the estimates obtained for $\\omega =\\omega _{opt}$ may improve both the estimation of smooth surfaces as well as threshold effects.",
"Figure: True functions (first column) and posterior mean for Region 2 in Study 1 (top row) and 4 (bottom row) obtained by BSMMR with η=10\\eta =10 and, ω=ω opt \\omega =\\omega _{opt} (second column) and ω=0\\omega =0 (third column), respectively."
]
] | 1605.06025 |
[
[
"Direct Search Implications for a Custodially-Embedded Composite Top"
],
[
"Abstract We assess current experimental constraints on the bi-doublet + singlet model of top compositeness previously proposed in the literature.",
"This model extends the standard model's spectrum by adding a custodially-embedded vector-like electroweak bi-doublet of quarks and a vector-like electroweak singlet quark.",
"While either of those states alone would produce a model in tension with constraints from precision electroweak data, in combination they can produce a viable model.",
"We show that current precision electroweak data, in the wake of the Higgs discovery, accommodate the model and we explore the impact of direct collider searches for the partners of the top quark."
],
[
"Introduction",
"In the context of the Standard Model (SM), the top quark's large mass is an enduring mystery.",
"All SM fermions are chiral: their left-handed components are weak doublets while their right-handed components are weak singlets.",
"This property forbids SM fermion mass terms of the Dirac form, because they would violate the electroweak gauge symmetry.",
"The Standard Model solves this dilemma via Yukawa interactions, which couple left-handed and right-handed components of fermions to the Higgs doublet.",
"When the electroweak symmetry spontaneously breaks to electromagnetism and the Higgs doublet spontaneously acquires a nonzero vacuum expectation value, all of those Yukawa interactions yield mass terms for the fermions.",
"Hence, the SM implies a tree-level proportionality between the Yukawa coupling and mass of each fermion.",
"The top quark's large mass ($173\\text{ GeV}$ ) demands its Yukawa coupling be nearly one: $y_t \\sim \\sqrt{2}m_t/v \\lesssim 1$ , where $v=246\\text{ GeV}$ is the electroweak scale.",
"Consequently, the Yukawa coupling of the top quark is far larger than the Yukawa couplings of the other SM fermions.",
"The available evidence implies the SM is an excellent description of nature, including its hypothesis that all fermions are chiral.",
"However, there could exist heavier, undiscovered fermions that disrupt this pattern.",
"In fact, electroweak precision data suggest there is more room in theories beyond the Standard Model for fermions that are vectorial under the electroweak gauge group than for new chiral fermions [1].",
"Notably, vector-like fermions can possess Dirac mass terms without violating the electroweak gauge symmetry; these Dirac masses are not tied to the size of the Higgs boson's vacuum expectation value or Yukawa couplings.",
"The existence of vector-like quarks may provide an alternative explanation for the top quark's large mass.",
"Suppose there exists some vector-like gauge-eigenstate quark (with a Dirac mass) whose quantum numbers allow it to mix with a chiral gauge-eigenstate quark (with a Yukawa coupling to the Higgs boson).",
"The mass eigenstates of the system would each be a superposition of the two gauge eigenstates and the masses would arise from a combination of Dirac and Yukawa terms.",
"If the top quark is composite in this manner, then only a small fraction of its mass need be due to Yukawa interactions while the majority could come from a Dirac mass term.",
"The Yukawa coupling of the top quark then must be larger than the SM prediction to reproduce the measured mass of the top quark.",
"As a result, we can attribute the size of $y_t$ to top compositeness and consider $y_t$ naturally large [2], [3], [4].",
"This work explores the possibility that vector-like fermions of this kind exist and play a role in the composition and phenomenology of the top quarkWhile the other quarks and leptons could also be superpositions of vector-like and chiral fermions, they are so light relative to the top quark that the degree of mixing would be far smaller.",
"Furthermore, data significantly constrains the degree to which the couplings of most other fermions deviate from the SM values, severely limiting the degree of mixing [5].",
"Therefore we ignore possible compositeness of the light fermions..",
"Restricting the Higgs sector to consist of only the observed Higgs doublet constrains these vector-like “top partner\" states to be EW singlets, doublets, or triplets [6].",
"In minimal effective theories, the triplet case generates phenomenologically interesting effects that are already included in the singlet and doublet cases [7]; hence we will focus on a model employing the singlet and doublet states directly.",
"If the physical top quark contains a substantial superposition of vectorial and chiral fermions, its couplings to other particles are altered from their SM predictions.",
"This has immediate consequences for the electroweak precision data, which are heavily influenced by the top quark's couplings to electroweak gauge bosons.",
"Models with composite top quarks are expected to deviate from the SM through the introduction of tree-level flavor-changing neutral currents, right-handed charged currents, and significant weak isospin violation.",
"In particular, in the flavor-conserving sector, these models are significantly constrained by the oblique parameters $\\hat{S}$ and $\\hat{T}$ and corrections to the $Zb_L\\overline{b}_L$ coupling $g_{Lb}$ .",
"An analysis [7] of composite top models performed before the 2012 Higgs discovery [8]-[10] suggested that including a particular combination of new vector-like fermions could lead to a “bi-doublet + singlet model\" consistent with electroweak data.",
"In this paper, we revisit the model in light of the Higgs boson discovery and other data from LHC Run I.",
"We establish that the model remains viable, demonstrate how the new data has further constrained the open parameter space, and suggest the likely impact of further searches for new physics at the LHC.",
"In section 2, we review the essential features of the bi-doublet + singlet model from Ref [7].",
"Section 3 details how this model influences the relevant experimental parameters.",
"Section 4 presents a new analysis of the model that includes updated electroweak precision constraints, the measured value of the Higgs boson's mass, and the impact of direct searches for vector-like top partner fermions.",
"Finally, section 5 summarizes our conclusions and addresses the implications of prospective LHC Run-II direct limits.",
"Table: The third-generation fermion field content of the bi-doublet + singlet model.",
"The SU(2) L SU(2)_L, U(1) Y U(1)_Y, and U(1) Q U(1)_Q quantum number of each field is listed, respectively, in the first, second, and third column."
],
[
"The Model",
"We will now review the essential features of the bi-doublet + singlet model [7].",
"The gauge sector is identical to that of the Standard Model, as are the Lagrangian terms for the leptons, the first and second generation quarks, and the Higgs potential.",
"New physics enters through the top-quark sector, via the following Lagrangian terms: $\\mathcal {L}_{bi\\text{-}doublet+singlet} &= \\overline{q}_{0L}i{D}q_{0L}+\\overline{t}_{0R}i{D}t_{0R}+\\overline{b}_{0R}i{D}b_{0R}+\\overline{t}_1i{D}t_1 +\\text{Tr}\\left(\\overline{Q}_1i{D}Q_1\\right) \\nonumber \\\\&\\hspace{20.0pt} - M_t\\overline{t}_1t_1-M_q\\text{Tr}\\left(\\overline{Q}_1Q_1\\right) -\\mu _q\\left(\\overline{q}_{0L}q_{1R}+\\text{h.c.}\\right)-\\mu _t\\left(\\overline{t}_{1L}t_{0R}+\\text{h.c.}\\right) \\nonumber \\\\&\\hspace{20.0pt}-y_q\\left[\\text{Tr}\\left(\\overline{Q}_{1L}\\Phi \\right)t_{0R}+\\text{h.c.}\\right]-y_t\\left[\\text{Tr}\\left(\\overline{Q}_{1L}\\Phi \\right)t_{1R}+\\text{h.c.}\\right]$ where $Q_1\\equiv \\left(q_{1}, \\Psi _{1}\\right) \\equiv \\left(\\hspace{-5.0pt}\\begin{tabular}{ c c } {t_1^q} & {\\Omega _1} \\\\ {b_1} & {t^\\Psi _1} \\end{tabular}\\hspace{-5.0pt}\\right)\\hspace{40.0pt}\\Phi \\equiv \\left(\\tilde{\\varphi },\\varphi \\right)\\equiv \\left(i\\sigma _2\\varphi ^*,\\varphi \\right) $ and $\\varphi $ is the usual Higgs doublet.",
"The field content of the top-sector Lagrangian is summarized in Table 1.",
"The Lagrangian features a left-handed SM-like EW doublet $q_{0L}$ consisting of the SM top and bottom quarks $(t_{0L}, b_{0L})$ , and right-handed SM-like EW singlets $t_{0R}$ and $b_{0R}$ .",
"By comparison with the SM, this model possesses 5 new fields: three top-like fields ($t^q_1$ , $t^\\Psi _1$ , $t_1$ ), a bottom-like field ($b_1$ ), and an exotic quark field with $+5/3$ charge ($\\Omega _1$ ).",
"The field combinations $Q_1$ and $\\Phi $ are constructed to (approximately) preserve a global $O(4)\\sim SU(2)_L\\times SU(2)_R\\times P_{LR}$ symmetry.",
"The $P_{LR}$ symmetry acts on the fields as follows: $Q_1 \\rightarrow -\\epsilon Q_{1}^T \\epsilon \\hspace{23.0pt}\\Phi \\rightarrow -\\epsilon \\Phi ^T \\epsilon \\hspace{23.0pt}q_{0L}\\rightarrow q_{0L} \\hspace{23.0pt}t_{0R}\\rightarrow t_{0R} \\hspace{23.0pt} t_1\\rightarrow t_1$ where $\\epsilon $ is an $SU(2)$ parity operation, $\\epsilon \\equiv \\left(\\hspace{-5.0pt}\\begin{tabular}{ c c } {0} & {1} \\\\ {-1} & {0} \\end{tabular}\\hspace{-5.0pt}\\right) $ Qualitatively, we expect cancellations in corrections to the $Z\\rightarrow b\\bar{b}$ amplitude [11] between the $t^q_1$ and $t^\\Psi _1$ fields because $t^q_1$ carries isospin $T_{L3}=\\frac{1}{2}$ while $t^\\Psi _1$ carries isospin $T_{L3}=-\\frac{1}{2}$ .",
"Under the $P_{LR}$ transformation, most of the terms in the Lagrangian are invariant.",
"Exceptions are the $\\mu _q$ and $\\mu _t$ terms, which impose soft breaking of the global $O(4)$ down to a global $O(3)\\sim SU(2)_V \\times P_{LR}$ .",
"We assume the SM bottom Yukawa coupling satisfies $y_b v \\ll \\mu _q$ and $y_b v\\ll M_q$ so we can neglect $y_b$ in our model.",
"The four top-like fermions will mix with one another to produce four top-like physical states.",
"Similarly, the two bottom-like fermions will mix with one another to produce two bottom-like physical states.",
"The bottom-like states mix in the same way as described for the SM + EW-doublet model of Ref [7].",
"The exotic $+5/3$ fermion is unique in its quantum numbers and hence does not mix with any other particles.",
"The SM limit of the bi-doublet + singlet model results from taking $\\mu _q\\gg M_q\\rightarrow \\infty $ and $M_t=\\mu _t\\rightarrow \\infty $ .",
"In this limit, the left-handed EW doublet $q_{1L}$ becomes the SM left-handed top and bottom quarks, while the EW singlets $t_{1R}$ and $b_{0R}$ become the SM right-handed top and bottom quarks.",
"All of the other fields listed in Table 1 mix into heavy states that decouple from the remainder of the theory.",
"To preserve the same qualitative behavior with fewer degrees of freedom, we will follow the analysis in Ref [7] and consider the $\\mu _q \\rightarrow \\infty $ limit of the bi-doublet + singlet model.",
"This effectively eliminates $q_{0L}$ and $q_{1R}$ , and reduces the top Lagrangian to $\\mathcal {L}_{top} &= \\overline{t}_{0R}i{D}t_{0R}+\\overline{t}_1i{D}t_1 +\\text{Tr}\\left(\\overline{Q}_{1L}i{D}Q_{1L}\\right)+\\overline{\\Psi }_{1R}i{D}\\Psi _{1R}- M_t\\overline{t}_1t_1\\nonumber \\\\&\\hspace{8.0pt} -M_q\\left(\\overline{\\Psi }_{1L}\\Psi _{1R}+\\text{h.c.}\\right)-\\mu _t\\left(\\overline{t}_{1L}t_{0R}+\\text{h.c.}\\right)-y_t\\left[\\text{Tr}\\left(\\overline{Q}_{1L}\\Phi \\right)t_{1R}+\\text{h.c.}\\right]$ With $q_{0L}$ removed, $q_{1L}$ now plays the role of the left-handed SM-like top-bottom weak doublet.",
"Note that this is precisely the doublet-extended standard model (DESM) of Ref.",
"[12] with an added vector-like EW singlet.",
"As such, we will refer to this limit as the DESM + singlet model.",
"The DESM + singlet Lagrangian contains four parameters, one of which is fixed by demanding that the lightest top-like mass eigenstate correspond to the physical top quark.",
"Furthermore, we can rearrange the other parameters into a more convenient combination, and define, $\\tan \\beta \\equiv \\dfrac{\\mu _t}{M_t}$ such that our three free parameters are $\\sin \\beta $ , $M_q$ , and $M_t$ .",
"(Because $\\mu _t$ and $M_t$ are positive, $\\beta $ must be between 0 and $\\pi /2$ , so that $\\sin \\beta $ ranges between 0 and 1.)",
"Electroweak symmetry breaking proceeds in the usual way, $\\varphi = \\dfrac{1}{\\sqrt{2}} \\left(\\hspace{-5.0pt}\\begin{tabular}{ c } {\\phi ^1+i\\phi ^2} \\\\ {\\phi ^0 +i\\phi ^3} \\end{tabular}\\hspace{-5.0pt}\\right) \\hspace{10.0pt}\\rightarrow \\hspace{10.0pt}\\dfrac{1}{\\sqrt{2}}\\left(\\hspace{-5.0pt}\\begin{tabular}{ c } {0} \\\\ {v} \\end{tabular}\\hspace{-5.0pt}\\right)+\\dfrac{1}{\\sqrt{2}}\\left(\\hspace{-5.0pt}\\begin{tabular}{ c } {\\phi ^1+i\\phi ^2} \\\\ {H+i\\phi ^3} \\end{tabular}\\hspace{-5.0pt}\\right), $ where $H$ is the Higgs boson.",
"The resulting mass terms for the third-generation quarks are, $\\mathcal {L}_{top} &\\supset -\\left(\\hspace{-5.0pt}\\begin{tabular}{ c c c }{\\overline{t}^q_{1L}} & {\\overline{t}^\\Psi _{1L}}& {\\overline{t}_{1L}}\\end{tabular}\\hspace{-5.0pt}\\right)\\left(\\hspace{-5.0pt}\\begin{tabular}{ c c c } {0} & {0}& {\\hat{m}_t} \\\\ {0} & {M_q} & {\\hat{m}_t}\\\\{\\mu _t}& {0}& {M_t}\\end{tabular}\\hspace{-5.0pt}\\right)\\left(\\hspace{-5.0pt}\\begin{tabular}{ c } {t_{0R}} \\\\ {t^\\Psi _{1R}} \\\\ {t_{1R}} \\end{tabular}\\hspace{-5.0pt}\\right)-M_q\\overline{\\Omega }_{1L}\\Omega _{1R}\\\\&\\hspace{20.0pt}-{\\lim _{\\mu _q\\rightarrow \\infty }}\\left(\\hspace{-5.0pt}\\begin{tabular}{ c c }{\\overline{b}_{0L}} & {\\overline{b}_{1L}}\\end{tabular}\\hspace{-5.0pt}\\right)\\left(\\hspace{-5.0pt}\\begin{tabular}{ c c } {0} & {\\mu _q} \\\\ {0} & {M_q} \\end{tabular}\\hspace{-5.0pt}\\right)\\left(\\hspace{-5.0pt}\\begin{tabular}{ c } {b_{0R}} \\\\ {b_{1R}} \\end{tabular}\\hspace{-5.0pt}\\right)+\\text{h.c.},$ where the quantity $\\hat{m}_t \\equiv y_t v/\\sqrt{2}$ is the tree-level standard model prediction for the top quark mass (in the limit of negligible CKM mixing).",
"Note that $\\Omega =\\Omega _1$ is a mass eigenstate with mass $M_q$ .",
"We diagonalize the top-related mass matrix in the usual way with unitary matrices $L_t$ , $R_t$ , defined by, $\\left(\\hspace{-5.0pt}\\begin{tabular}{ c } {{t}_{1L}^q} \\\\ {{t}_{1L}^\\Psi } \\\\ {{t}_{1L}} \\end{tabular}\\hspace{-5.0pt}\\right)\\equiv L_t\\left(\\hspace{-5.0pt}\\begin{tabular}{ c } {t_L} \\\\ {T_L^\\Psi } \\\\ {T_L} \\end{tabular}\\hspace{-5.0pt}\\right)\\hspace{25.0pt} \\left(\\hspace{-5.0pt}\\begin{tabular}{ c } {t_{0R}} \\\\ {t_{1R}^\\Psi } \\\\ {t_{1R}} \\end{tabular}\\hspace{-5.0pt}\\right)\\equiv R_t\\left(\\hspace{-5.0pt}\\begin{tabular}{ c } {t_R} \\\\ {T_R^\\Psi } \\\\ {T_R} \\end{tabular}\\hspace{-5.0pt}\\right)$ where $t$ denotes the physical top quark and $T^\\Psi $ , $T$ are the physical top partners.",
"We define $T$ to be the lighter of the two top partners.",
"In the limit with $M_t,M_q\\gg m_t$ , we find, to second order in $m_t$ , $m_T = \\min \\left\\lbrace \\dfrac{M_t}{\\cos \\beta },M_q\\right\\rbrace \\hspace{25.0pt}m_\\Psi = \\max \\left\\lbrace \\dfrac{M_t}{\\cos \\beta },M_q\\right\\rbrace $ where $m_T$ is the mass of the lighter top partner and $m_\\Psi $ is the mass of the heavier top partner.",
"In the full bi-doublet + singlet model, diagonalization of the bottom-related mass matrix occurs according to, $\\left(\\hspace{-5.0pt}\\begin{tabular}{ c } {b_{0L}} \\\\ {b_{1L}} \\end{tabular}\\hspace{-5.0pt}\\right)=\\left(\\hspace{-5.0pt}\\begin{tabular}{ c c } {\\cos \\alpha } & {\\sin \\alpha } \\\\ {-\\sin \\alpha } & {\\cos \\alpha } \\end{tabular}\\hspace{-5.0pt}\\right)\\left(\\hspace{-5.0pt}\\begin{tabular}{ c } {b_L} \\\\ {B_L} \\end{tabular}\\hspace{-5.0pt}\\right)\\hspace{20.0pt}\\left(\\hspace{-5.0pt}\\begin{tabular}{ c } {b_{0R}} \\\\ {b_{1R}} \\end{tabular}\\hspace{-5.0pt}\\right)=\\left(\\hspace{-5.0pt}\\begin{tabular}{ c } {b_R} \\\\ {B_R} \\end{tabular}\\hspace{-5.0pt}\\right) $ where $\\tan \\alpha \\equiv \\mu _q/M_q$ .",
"In the DESM + singlet model, we take $\\mu _q\\rightarrow \\infty $ (such that $\\alpha \\rightarrow \\pi /2$ ).",
"This drives $m_b\\rightarrow 0$ and $m_B\\rightarrow \\infty $ [7].",
"Therefore, the DESM + singlet model predicts a massless bottom quark and an infinitely massive bottom partner that decouples from the theory.",
"The SM limit of the DESM + singlet model results from additionally taking $M_q\\rightarrow \\infty $ , $M_t\\rightarrow \\infty $ , and $\\sin \\beta \\rightarrow 1/\\sqrt{2}$ .",
"As we discuss in Section 3.1, the Standard Model predicts $g_{Lb}$ to be approximately two standard deviations below the experimentally measured value.",
"Therefore, for the purposes of this analysis, the SM limit of the DESM + singlet model lies in disfavored regions of parameter space."
],
[
"Relevant experimental parameters",
"Mixing among vector-like and chiral fermions leads to non-SM couplings of fermions to the electroweak gauge bosons.",
"These altered couplings can induce sizable deviations from SM predictions.",
"Of the SM quarks, the top quark is the least experimentally constrained, and so it presents us with the widest window for discovering new effects without violating current experimental limits [5].",
"This simultaneously affords an opportunity to explain the large size of the top quark mass.",
"We expect the extensions of the top sector discussed here to be constrained by the Peskin-Takeuchi parameters, $\\hat{S}\\equiv \\hat{\\alpha }(M_Z) S$ and $\\hat{T}\\equiv \\hat{\\alpha }(M_Z) T$ [13].",
"Corrections to the $\\hat{U}$ -parameter due to new physics are anticipated to be small compared to those for $\\hat{S}$ and $\\hat{T}$ .",
"Furthermore, our model generates more significant corrections to $\\hat{T}$ than to $\\hat{S}$ because we only introduce vector-like fermions.",
"The current experimental limits on $S$ and $T$ at the $Z$ -pole corresponding to a Higgs boson with mass $m_H=125\\text{ GeV}$ are $S=-0.03\\pm 0.10$ and $T=0.01\\pm 0.12$ [1],[14].",
"Using $\\hat{\\alpha }(M_Z)^{-1}=127.940\\pm 0.014$ , $M_Z=91.1876\\pm 0.0021$ GeV, and $m_t=173.24\\pm 0.81$ GeV [1], we will compare our theoretical predictions against $\\hat{S}$ and $\\hat{T}$ .",
"Note that both measured values are consistent with the SM prediction $S=T=0$ , so any content beyond the SM must generate largely self-canceling corrections.",
"The $Zb_L\\overline{b}_L$ coupling $g_{Lb}$ can also receive large corrections when one augments the top sector.",
"Experimentally, $g_{Lb}=-0.4182\\pm 0.0015$ , while the SM predicts a value of $g_{Lb,SM}=-0.42114\\pm \\begin{array}{c}0.00045 \\\\ 0.00024\\end{array}$ [14].",
"Because the SM value sits approximately two standard deviations below the experimental value, experiment favors a positive value of $\\delta g_{Lb}\\equiv g_{Lb}-g_{Lb,SM}\\approx 0.003$ .",
"In order to compare our model with experiment, we have calculated $g_{Lb}$ in the gaugeless limit, as discussed in Ref [7].",
"We are interested in models that reproduce all electroweak precision data to within $1\\sigma $ of the experimentally measured values.",
"Consequently, the Standard Model is disfavored in our analysis."
],
[
"Applications to top partner models",
"The simplest top partner models that add a single kind of vector-like quark to the SM spectrum struggle to simultaneously satisfy constraints from $\\hat{T}$ and $g_{Lb}$ .",
"Explicit analysis demonstrates that $\\hat{S}$ provides a weaker constraint than $\\hat{T}$ ; therefore we will discuss $\\hat{S}$ no further.",
"For instance, adding an EW vector-like singlet moves $\\hat{T}$ and $g_{Lb}$ in the positive direction such that by the time $g_{Lb}$ is within $1\\sigma $ of its experimental value, $\\hat{T}$ is already well outside of the experimental $1\\sigma $ band [7].",
"Incorporating an EW vector-like doublet, instead, yields a qualitatively similar conclusion, with even larger discrepancies.",
"The addition of an EW vector-like bi-doublet (imbued with additional symmetry structures) generates negative corrections to $\\hat{T}$ and preserves the SM value of $g_{Lb}$ .",
"Adding just one kind of vector-like quark to the SM spectrum does not produce a viable model.",
"However, Ref [7] constructed an experimentally consistent theory by combining vector-like bi-doublet and vector-like singlet extensions of the SM spectrum; this is the bi-doublet + singlet model described in Section 2.",
"Relative to the separate EW bi-doublet extension and EW singlet extension, the bi-doublet + singlet model reproduces the measured $\\hat{T}$ to within $1\\sigma $ across a larger region of parameter space.",
"This is accomplished through cancellations between positive corrections due to the EW singlet and negative corrections due to the EW bi-doublet.",
"The bi-doublet + singlet model also controls positive $\\delta g_{Lb}$ corrections by imposing symmetries on the bi-doublet.",
"Specifically, $\\hat{T}$ corrections are diminished by imposing a global $SU(2)_L\\times SU(2)_R$ symmetry on the symmetry-breaking sector and collapsing it to the usual custodial $SU(2)_c$ upon electroweak symmetry breaking [15],[16].",
"Agashe et al.",
"noted that this symmetry could also protect $g_{Lb}$ by imposing an additional parity symmetry $P_{LR}$ between the $SU(2)_L$ and $SU(2)_R$ and demanding that $b_L$ be an eigenstate of $P_{LR}$ [11].",
"The DESM + singlet limit of the bi-doublet + singlet model includes $b_L$ as an odd eigenstate of $P_{LR}$ with quantum numbers $T_L=T_R=1/2$ and $T_L^3=T_R^3=-1/2$ (e.g., $b_L$ is embedded in a bi-doublet $(2,2)_{2/3}$ of $SU(2)_L\\times SU(2)_R\\times U(1)_X$ ) so that its coupling to the $Z$ -boson will receive suppressed corrections [11].",
"This is equivalent to embedding $b_L$ in the vector portion of $SO(3)\\sim SU(2)_c$ , such that $\\delta g_{Lb} = \\delta g_{Rb}$ .",
"Because $b_L$ is odd under $P_{LR}$ , one finds that $\\delta g_{Lb} = -\\delta g_{Rb}$ , and we conclude that $\\delta g_{Lb}=\\delta g_{Rb}=0$ .",
"This would only be approximately true in the full theory because the global symmetries are not exact.",
"For example, the mass splitting $m_t\\ne m_b$ breaks $SU(2)_c$ even in the limit of zero hypercharge.",
"Regardless, embedding $b_L$ in an EW bi-doublet protects $g_{Lb}$ from large corrections.",
"This allows the vector-like singlet quark to provide the primary corrections to $g_{Lb}$ , making agreement with experiment feasible.",
"Figure 1 shows the shaded regions of parameter space that are excluded by present constraints on $\\hat{T}$ (coarse hatching) and $g_{Lb}$ (fine hatching), with the measured value of the Higgs mass taken into account; the white region in each pane of fixed $\\sin \\beta $ shows the area in the $M_t$ v.s.",
"$M_q$ plane that remains viable in light of precision EW data."
],
[
"Incorporating Top Partner Searches",
"Since 2011, the LHC experiments have performed a number of direct searches for top partners [1].",
"Each search assumes that the top partners have particular decay properties and reports lower limits on their masses based on the absence of signal events.",
"In particular, a top partner $T$ is often assumed to decay in only three ways: $T\\rightarrow Zt$ , $T\\rightarrow Wb$ , and $T\\rightarrow Ht$ .",
"Since this assumption holds for the lighter top partner in our model, we can apply the existing search results to determine constraints on the top partner's mass.",
"There are three top-like physical quarks in the DESM + singlet model.",
"By construction, the lightest of these mass eigenstates is the physical top quark.",
"The remaining two mass eigenstates are top partners, where we defined $T$ ($T^\\Psi $ ) to be the lighter (heavier) top partner.",
"The heavier top partner's mass varies between $0.80$ and 21 TeV in regions of parameter space allowed by $\\hat{T}$ and $g_{Lb}$ , which overlaps direct limits; however, any point of parameter space for which the heavier top partner is eliminated by this constraint is already eliminated from direct search constraints on the lighter top partner.",
"Therefore, we focus on the constraints the data place upon the lighter top partner.",
"The labeled curves within the white region of Figure 1 illustrate how $m_T$ varies as a function of $(M_t,M_q)$ , and for different values of $\\sin \\beta $ , across the regions allowed by the electroweak precision data ($\\hat{T}$ , $\\delta g_{Lb}$ ).",
"Different points in the $(M_q,M_t,\\sin \\beta )$ parameter space of our model generally correspond to different coupling values, and yield different values of $Br(T\\rightarrow Zt)$ , $Br(T\\rightarrow Wb)$ , and $Br(T\\rightarrow Ht)$ .",
"Therefore, applying the experimental limits from direct searches requires careful analysis of $(M_q,M_t,\\sin \\beta )$ space.",
"CMS and ATLAS performed inclusive vector-like top partner searches that incorporate all values of top partner branching fractions into $bW$ , $tZ$ , and $tH$ (assuming only these decay modes).",
"The ATLAS search finds its strongest limit (950 GeV) when $Br(T\\rightarrow Ht)\\approx 1$ [17].",
"Because the DESM + singlet model predicts $Br(T\\rightarrow Ht)\\le 0.125$ in regions of parameter space consistent with electroweak precision data, this limit is inapplicable to the DESM + singlet model.",
"CMS provides a stronger direct limit of 920 GeV for top partners $T$ such that $Br(T\\rightarrow Wb)\\approx 1$ [18].",
"Because the DESM + singlet model predicts a large $Br(T\\rightarrow Wb)$ throughout most of the parameter space consistent with electroweak precision data, we will designate regions satisfying $m_T>920$ GeV as viable.",
"The separate panes of Figure 2 plot the viable regions of the DESM + singlet model when constrained by $\\hat{T}$ , $\\delta g_{Lb}$ , and $m_T>920$ GeV for several values of $\\sin \\beta $ .",
"The region of downward sloping (from left-to-right) stripes in Figure 2 is where the lightest top partner would be lighter than the physical top quark.",
"By construction, that region is excluded, since the top partners are to be heavier than the top quark.",
"The upward sloping (darker) striped region is where the top Yukawa coupling becomes non-perturbative; this lies outside the realm of our analysis.",
"The coarsely gridded regions correspond to points where $\\hat{T}$ lies outside of the experimentallly allowed limits $\\hat{T}=0.01\\pm 0.12$ .",
"The more finely cross-hatched regions correspond to points where $g_{Lb}$ lies outside of the experimentally allowed values $g_{Lb}=-0.4182\\pm 0.0015$ .",
"The light grey curves intersecting within the white region indicate the loci of the central experimental values of $\\hat{T}$ and ${g}_{Lb}$ .",
"Finally, the checkerboard black-and-gray region is excluded by the approximate direct top partner search bound, $m_T > 920$ GeV.",
"The remaining white regions correspond to viable parts of parameter space.",
"As indicated in Figure 2, the values of $\\sin \\beta $ where precision electroweak tests are weakest ($\\sin \\beta \\lesssim 0.2$ ) remain untouched by even the strongest limits from direct searches for top partners.",
"As $\\sin \\beta $ increases, the direct searches have increasing impact.",
"Interestingly, the limit ($m_T > 920$ GeV) impacts parameter space most significantly for $\\sin \\beta $ values where $m_T$ was already tightly constrained by precision data (as illustrated in Figure 1).",
"For instance, the limit first makes contact with the available parameter space for $\\sin \\beta $ values slightly below $\\sin \\beta =0.3$ .",
"At $\\sin \\beta =0.3$ , the mass of the lightest top partner ranges from $0.81\\text{ TeV}$ to $2.4\\text{ TeV}$ .",
"Compare this to the case at $\\sin \\beta =0.1$ , where the limit has no impact on parameter space points that reproduce electroweak precision data, and where the lightest top partner mass ranges from $1.5\\text{ TeV}$ to $17\\text{ TeV}$ .",
"The mass range at $\\sin \\beta =0.1$ is nearly ten times as large as the mass range at $\\sin \\beta =0.3$ .",
"As we move toward higher values of $\\sin \\beta $ , the range of allowed $m_T$ values contracts and moves toward lower masses.",
"Consequently, we find that for values of $\\sin \\beta $ above $0.55$ , there is no viable region of parameter space left.",
"The model also predicts an exotic quark $\\Omega $ with an electric charge of $+5/3$ and a tree-level mass $m_\\Omega = M_q$ .",
"Between ATLAS and CMS, the strongest direct limit on charge $+5/3$ quarks is 960 GeV [19],[20].",
"Any point of parameter space which is excluded by this constraint (i.e., points for which $m_\\Omega < 960$ GeV) is already excluded in Figure 2.",
"So this constraint does not presently add to our understanding of the phenomenology.",
"Our analysis establishes that the DESM + singlet model remains viable for $0.05\\lesssim \\sin \\beta \\lesssim 0.55$ , and strongly suggests that the full bi-doublet + singlet model is similarly viable across a significant swath of parameter space.",
"Figure: Illustration of how the mass (m T m_T) of the lighter top partner varies within the region of parameter space allowed by precision EW data.",
"Several contours of constant m T m_T are labeled in each plot.",
"Each panel shows the M q M_q vs. M t M_t plane for a particular value of sinβ\\sin \\beta .",
"In the white regions, the DESM + singlet model produces T ^\\hat{T} and δg Lb \\delta g_{Lb} values consistent to within 1σ1\\sigma of experimental bounds.",
"We exclude the more coarsely (finely) cross-hatched regions because those points produce values of T ^\\hat{T} (g Lb {g}_{Lb}) outside the 1σ1\\sigma experimental band.",
"The region excluded by T ^\\hat{T} that lies above (below) the white region consists of points for which T ^\\hat{T} is greater (less) than the allowed range of values.",
"The region excluded by g Lb g_{Lb} that lies left (right) of the white region consists of points for which g Lb g_{Lb} is greater (less) than the allowed range of values.",
"The thin contour in the panes for sinβ≥0.3\\sin \\beta \\ge 0.3 indicates the lowest value of the top partner mass now allowed by CMS direct searches; nearly all m T m_T values in the white region for sinβ=0.55\\sin \\beta = 0.55 are below that limit.",
"Note that the ranges of M t M_t and M q M_q on the axes vary between plots.Figure: The DESM + singlet model is viable within the thick-bordered white regions which survive precision EW tests and a conservative estimate of the direct search limits on top partners.",
"Each panel shows the M q M_q vs. M t M_t plane for a particular value of sinβ\\sin \\beta .",
"We exclude the L-shaped region of downward-sloping stripes by requiring the top partner masses to exceed the physical top quark mass.",
"We exclude the L-shaped region of darker, upward-sloping stripes by keeping the Yukawa coupling of the top quark less than 4π4\\pi .",
"The cross-hatched regions are excluded by precision EW data as described in Figure 1.",
"The checkerboard black-and-gray region is excluded by CMS direct searches for top partners ; points in this region produce a top partner with mass below 920 GeV.",
"No viable region survives for sinβ≥0.56\\sin \\beta \\ge 0.56.",
"Note that the ranges of M t M_t and M q M_q on the axes vary between plots."
],
[
"Conclusion",
"In this paper, we assessed the bi-doublet + singlet model for viability against current electroweak precision data ($\\hat{T}$ , $g_{Lb}$ ) and direct searches for top-quark partners.",
"This model of top compositeness extends the standard model's spectrum by adding a custodially-symmetric vector-like electroweak bi-doublet and a vector-like electroweak singlet.",
"Hence, it introduces three top partners, one bottom partner, and an exotic quark with electric charge $+5/3$ .",
"Our analysis focused on the $\\mu _q\\rightarrow \\infty $ limit of this model, which eliminated the bottom partner and one top partner, resulting in the DESM + singlet model; we expect the phenomenology to be qualitatively similar to that of the bi-doublet + singlet model.",
"In Figure 2, we illustrate the area of model parameter space that is consistent with electroweak precision data and direct searches for top partners (the thick-bordered white region) in the $M_q$ vs. $M_t$ plane.",
"Overall, the DESM + singlet model (and hence the bi-doublet + singlet model) remains viable against constraints due to $\\hat{T}$ , $g_{Lb}$ , and direct searches for top partners.",
"Backovic̀ et.",
"al.",
"estimate that searches at Run-II of the LHC could be sensitive to top partner masses up to $2\\text{ TeV}$ [21].",
"If the bi-doublet + singlet model is a correct description of nature, a top partner discovery might well be made within the next decade.",
"On the other hand, if no top partner is seen up to masses of 2 TeV, our conservative analysis documented in Figure 1 suggests this would eliminate parameter space points for which $\\sin \\beta \\gtrsim 0.30$ .",
"A detailed analysis, similar to what was reported above at the end of section 4, would then be needed to establish the exact range of model parameter space that remains."
],
[
"Acknowledgments",
"The work of.",
"R.S.C., D.F., and E.H.S.",
"was supported by the National Science Foundation under Grants PHY-0854889 and PHY-1519045.",
"R.S.C.",
"and E.H.S.",
"also acknowledge the support of NSF Grant PHYS-1066293 and the hospitality of the Aspen Center for Physics during work on this paper."
]
] | 1605.06088 |
[
[
"Raman spectroscopy of K_xCo_{2-y}Se_2 single crystals near the\n ferromagnet-paramagnet transition"
],
[
"Abstract Polarized Raman scattering spectra of the K_xCo_{2-y}Se_2 single crystals reveal the presence of two phonon modes, assigned as of the A_{1g} and B_{1g} symmetry.",
"Absence of additional modes excludes the possibility of vacancy ordering, unlike in K_xFe_{2-y}Se_2.",
"The ferromagnetic (FM) phase transition at T_c\\approx 74 K leaves a clear fingerprint on the temperature dependence of the Raman mode energy and linewidth.",
"For T>T_c the temperature dependence looks conventional, driven by the thermal expansion and anharmonicity.",
"The Raman modes are rather broad due to the electron-phonon coupling increased by the disorder and spin fluctuation effects.",
"In the FM phase the phonon frequency of both modes increases, while an opposite trend is seen in their linewidth: the A_{1g} mode narrows in the FM phase, whereas the B_{1g} mode broadens.",
"We argue that the large asymmetry and anomalous frequency shift of the B_{1g} mode is due to the coupling of spin fluctuations and vibration.",
"Our density functional theory (DFT) calculations for the phonon frequencies agree rather well with the Raman measurements, with some discrepancy being expected since the DFT calculations neglect the spin fluctuations."
],
[
"Introduction",
"In the last few years considerable attention was focused on the iron-based superconductors in an effort to gain deeper insight into their physical properties and to determine the origin of high-$T_c$ superconductivity [1], [2], [3], [4].",
"Discovery of superconductivity in alkali-doped iron chalcogenides, together with its uniqueness among the iron based superconductors, challenged the physical picture of the superconducting mechanism in iron pnictides [5].",
"Absence of hole pockets even suggested the possibility for the different type of pairing mechanism [6].",
"Another striking feature in K$_x$ Fe$_{2-y}$ Se$_2$ was the presence of the intrinsic nano to mesoscale phase separation between an insulating phase and a metallic/superconducting phase [7], [8], [9], [10].",
"Insulating phase hosts antiferromagnetically, $\\sqrt{5} \\times \\sqrt{5}$ ordered iron vacancies, whereas superconducting stripe-like phase is free of vacancies [7].",
"Theoretical study of Huang et al.",
"[11] revealed that proximity effects of the two phases result in the Fermi surface deformation due to interlayer hopping and, consequently, suppression of superconductivity.",
"On the other hand, large antiferromagnetic order protects the superconductivity against interlayer hopping, thus explaining relatively high $T_c$ in K$_x$ Fe$_{2-y}$ Se$_2$ [11].",
"However, the correlation between the two phases and its impact on superconductivity are still not fully understood.",
"Although the absolute values of resistivity are much smaller for the Ni-member of the K$_x$ M$_{2-y}$ Se$_2$ (M = transition metal) series than for the iron member, this material does not exhibit superconductivity down to 0.3 K [12].",
"As opposed to K$_x$ Fe$_{2-y}$ Se$_2$ , vacancy ordering has not been observed in the K$_x$ Ni$_{2-y}$ Se$_2$ single crystal [13].",
"These materials, together with the Co- and Ni-doped K$_x$ Fe$_{2-y}$ Se$_2$ single crystals, have very rich structural, magnetic and transport phase diagrams.",
"This opens a possibility for fine tuning of their physical properties by varying the sample composition [14], [15].",
"First results obtained on K$_x$ Co$_{2-y}$ Se$_2$ single crystal revealed the ferromagnetic ordering below $T_c \\sim $ 74 K, as well as the absence of the superconducting phase [16].",
"Raman spectroscopy is a valuable tool not only for measuring vibrational spectra, but also helps in the analysis of structural, electronic and magnetic properties, and phase transitions.",
"There are several recent studies of the influence of the antiferromagnetic order, [17], [18] ferromagnetism, [19], [20] and magnetic fluctuations [21] on the Raman spectra.",
"In this paper the Raman scattering study of K$_x$ Co$_{2-y}$ Se$_2$ single crystal (x=0.3, y=0.1), together with the lattice dynamics calculations of KCo$_2$ Se$_2$ , is presented.",
"The polarized Raman scattering measurements were performed in the temperature range from 20 K up to 300 K. The observation of only two Raman active modes when measuring from the (001)-oriented samples suggests that K$_x$ Co$_{2-y}$ Se$_2$ single crystal has no ordered vacancies.",
"The temperature dependence of the energy and linewidth of the observed Raman modes reveals a clear fingerprint of the phase transition.",
"Large linewidth of the B$_{1g}$ mode and its Fano line shape indicate the importance of spin fluctuations.",
"The rest of the manuscript is organized as follows.",
"Section II contains a brief description of the experimental and numerical methods, Section III are the results, and Section IV contains a discussion of the phonon frequencies and linewidths and their temperature dependencies.",
"Section V summarizes the results."
],
[
"Experiment and numerical method",
"Single crystals of K$_x$ Co$_{2-y}$ Se$_2$ were grown by the self-flux method, as described in Ref. [12].",
"The elemental analysis was performed using energy-dispersive x-ray spectroscopy (EDX) in a JEOL JSM-6500 scanning electron microscope.",
"Raman scattering measurements were performed on freshly cleaved (001)-oriented samples with size up to 3$\\times $ 3$\\times $ 1 mm$^3$ , using a TriVista 557 Raman system equipped with a nitrogen-cooled CCD detector, in backscattering micro-Raman configuration.",
"The 514.5 nm line of an Ar$^+$ /Kr$^+$ ion gas laser was used as an excitation source.",
"A microscope objective with $50 \\times $ magnification was used for focusing the laser beam.",
"All measurements were carried out at low laser power, in order to minimize local heating of the sample.",
"Low temperature measurements were performed using KONTI CryoVac continuous flow cryostat with 0.5 mm thick window.",
"Spectra were corrected for the Bose factor.",
"The electronic structure of the ferromagnetic (FM) and paramagnetic (PM) phases is calculated within the density functional theory (DFT), and the phonon frequencies at the $\\Gamma $ -point are obtained within the density functional perturbation theory (DFPT) [22].",
"All calculations are performed using the QUANTUM ESPRESSO package [23].",
"We have used Projector Augmented-Wave (PAW) pseudo-potentials with Perdew-Burke-Ernzerhof (PBE) exchange-correlation functional with nonlinear core correction and Gaussian smearing of 0.005 Ry.",
"The electron wave-function and the density energy cutoffs are 40 Ry and 500 Ry, respectively.",
"The Brillouin zone is sampled with 16$\\times $ 16$\\times $ 8 Monkhorst-Pack k-space mesh.",
"The phonon frequencies were calculated with relaxed unit cell parameters and, for comparison, with the unit cell size taken from the experiments and the relaxed positions of only Se atoms.",
"The forces acting on individual atoms in the relaxed configuration were smaller than $10^{-4}$ Ry/a.u.",
"and the pressure smaller than 0.5 kbar."
],
[
"Results",
"KCo$_2$ Se$_2$ crystallizes in the tetragonal crystal structure of ThCr$_2$ Si$_2$ -type, $I4/mmm$ space group, which is shown in Figure REF .",
"The experimental values of the unit cell parameters are $a=$ 3.864(2) $Å$ and $c=$ 13.698(2) $Å$ [24].",
"The potassium atoms are at $2a$ :$(0,0,0)$ , Co atoms at $4d$ :$(0,\\frac{1}{2},\\frac{1}{4})$ , and Se atoms at $4e$ :$(0,0,z)$ Wyckoff positions, with the experimental value $z=0.347$ .",
"KCo$_2$ Se$_2$ single crystal consists of alternatively stacked K ions and CoSe layers, isostructural to the KFe$_2$ Se$_2$ [25].",
"Factor group analysis for the $I4/mmm$ space group yields a normal mode distribution at the Brillouin-zone center which is shown in Table REF .",
"According to the selection rules, when measuring from the (001)-plane of the sample, only two modes (A$_{1g}$ and B$_{1g}$ ) are expected to be observed in the Raman scattering experiment.",
"Displacement patterns of the experimentally observable Raman modes are illustrated in Fig.",
"REF .",
"The A$_{1g}$ (B$_{1g}$ ) mode represents the vibrations of the Se (Co) ions along the $c$ -axis, whereas the E$_g$ modes (which are not observable for our scattering configuration) involve the vibration of both Co and Se ions within the (001)-plane.",
"Table: Atomic types with their Wyckoff positions and the contribution of the each site to the Γ\\Gamma -point phonons, the Raman tensors and the selection rules for the K x _xCo 2-y _{2-y}Se 2 _2 single crystal (I4/mmmI4/mmm space group).Figure: (Color online) Unit cell of KCo 2 _2Se 2 _2 single crystal, together with the displacement patterns of the A 1g _{1g} and B 1g _{1g} Raman modes.Figure: (Color online) Upper panel: Integrated intensity of the observed Raman modes as a function of the crystal orientation with respect to the laboratory axes 𝐱 0 \\mathbf {x_0} and 𝐲 0 \\mathbf {y_0}.",
"In order to estimate the intensity of the modes, phonon at 198 cm -1 ^{-1} was fitted with Lorentzian, whereas asymmetric Raman mode appearing at 187 cm -1 ^{-1} was fitted with Fano line shape.",
"Lower panel: Raman scattering spectra of K x _xCo 2-y _{2-y}Se 2 _2 single crystal measured at room temperature, in various sample orientations (𝐱=[100],𝐲=[010]\\mathbf {x}=[100], \\mathbf {y}=[010]).Figure REF shows polarized Raman scattering spectra of K$_x$ Co$_{2-y}$ Se$_2$ single crystal, measured from the (001)-plane of the sample at room temperature, in different sample orientations.",
"Only two modes, at about 187 and 198 cm$^{-1}$ , are observed, which is in agreement with the selection rules for (001)-oriented samples.",
"In some iron-chalcogenide compounds, the appearance of additional Raman active modes due to the iron vacancy ordering and, consequently, symmetry lowering, has been observed [8], [26].",
"Absence of additional phonon modes in Fig.",
"2 suggests that in K$_x$ Co$_{2-y}$ Se$_2$ single crystals vacancy ordering does not occur in our samples.",
"Selection rules imply that the A$_{1g}$ mode may be observed for any sample orientation, provided that the polarization vector of the incident light $\\mathbf {e}_i$ is parallel to the scattered light polarization vector $\\mathbf {e}_s$ , whereas it vanishes if these vectors are perpendicular.",
"On the other hand, the intensity of the B$_{1g}$ mode strongly depends on the sample orientation ($I_{B_{1g}} \\sim |c|^2 \\cos ^2 (\\theta +2\\beta )$ , where $\\theta =\\angle (\\mathbf {e}_i,\\mathbf {e}_s)$ and $\\beta = \\angle (\\mathbf {e}_i,\\mathbf {x})$ ) [8].",
"This implies that, in parallel polarization configuration ($\\theta =0^{\\circ }$ ), the intensity of the B$_{1g}$ mode is maximal when the sample is oriented so that $\\mathbf {e}_i$$\\Vert $$\\mathbf {x}$ , gradually decreases with increasing $\\beta $ and finally vanishes for $\\beta =45^{\\circ }$ .",
"In crossed polarization configuration ($\\theta =90^{\\circ }$ ), B$_{1g}$ mode intensity decreases from its maximal value for $\\beta =45^{\\circ }$ to zero, which reaches when $\\beta =0^{\\circ }$ .",
"From Fig.",
"REF it can be seen that the intensity of the Raman mode at about 187 cm$^{-1}$ coincides with theoretically predicted behavior for the B$_{1g}$ mode; thereby, this phonon mode is assigned accordingly.",
"Phonon mode at $\\sim $ 198 cm$^{-1}$ , which is present in Raman spectra only for parallel polarization configuration ($\\theta =0^{\\circ }$ ) and whose intensity is independent on the sample orientation, can be assigned as the A$_{1g}$ mode.",
"The intensity ratio of the two Raman modes can be obtained from the spectrum measured in ($\\theta =0^{\\circ }, \\beta =0^{\\circ }$ ) scattering geometry as $I_{B_{1g}}/I_{A_{1g}} \\approx 1.38$ .",
"Having in mind that the A$_{1g}$ mode intensity is given by [8] $I_{A_{1g}} \\sim |a|^2 \\cos ^2 \\theta $ , the ratio of the appropriate Raman tensor components can be estimated as $|c|/|a| \\approx 1.17$ .",
"Table: Optimized lattice constants and internal coordinate z Se z_{_{Se}} in the FM and PM solution.",
"Next two rows give the relaxed z Se z_{_{Se}} when the unit cell size is taken from the experiment, and the last row contains the atomic positions from the experiment Figure: (Color online) Temperature dependent Raman spectra of K x _xCo 2-y _{2-y}Se 2 _2 single crystal in parallel (left panel) and crossed (right panel) polarization configuration (𝐱 0 =1 2[110],𝐲 0 =1 2[1 ¯10]\\mathbf {x_0}=\\frac{1}{\\sqrt{2}}[110], \\mathbf {y_0} = \\frac{1}{\\sqrt{2}}[\\bar{1}10]).",
"The solid lines represent fits of the experimental spectra with the Lorentzian (A 1g _{1g} mode) and the Fano profile (B 1g _{1g} mode).Table: The experimental phonon energies measured at 20 K in the FM phase and the extrapolated value to 0 K from the PM phase (see the text).",
"The phonon frequencies at the Γ\\Gamma point are calculated with fully relaxed atomic positions.",
"The frequencies obtained with relaxed only internal coordinate are given in parenthesis.Figure: (Color online) Temperature dependence of the energy and linewidth for the A 1g _{1g} (a, b) and B 1g _{1g} (c, d) Raman modes of K x _xCo 2-y _{2-y}Se 2 _2 single crystal.",
"Solid lines are a theoretical fit (see the text) and the dotted lines are the extrapolation to the FM phase.",
"Upper inset: Temperature dependence of the B 1g _{1g} mode frequency, compared with (M(T)/M(0)) 2 (M(T)/M(0))^2 curve.",
"Lower inset: Measure of the electron-mediated photon-phonon coupling (1/q1/q) of the B 1g _{1g} mode as a function of temperature.The experimentally determined frequencies are compared with those obtained with DFT numerical calculations.",
"The experimental lattice constants [24] are shown in Table II, together with their values from the DFT calculation which relaxes or keeps fixed the unit cell size.",
"The DFPT phonon frequencies obtained using the fully relaxed atomic positions in both FM and PM phases are given in Table III, with the corresponding values obtained with the fixed unit cell size and relaxed only fractional coordinate $z_{_{Se}}$ given in the parenthesis.",
"The equilibrium atomic positions in the FM solution are given by $a=3.893$ Å, $c=13.269$ Å, and $z_{_{Se}}=0.350$ .",
"The corresponding phonon frequencies are 199.5 $\\mathrm {cm}^{-1}$ for A$_{1g}$ mode and 171.2 $\\mathrm {cm}^{-1}$ for B$_{1g}$ mode.",
"When we enforce the PM solution, we obtain $a=3.766$ Å, $c=13.851$ Å, and $z_{_{Se}}=0.368$ , and 212.6 $\\mathrm {cm}^{-1}$ , 176.6 $\\mathrm {cm}^{-1}$ for the frequencies of the A$_{1g}$ and B$_{1g}$ mode, respectively.",
"These values agree rather well with the experimental data, and agree with recently published numerical results [27], [28].",
"They can be used to confirm the experimental assignment of the modes, but cannot resolve subtle changes of the phonon frequencies near the FM-PM transition.",
"This level of discrepancy is expected for metallic materials with magnetic ordering since the DFT calculations neglect spin fluctuations, as discussed in some detail in the next Section (see also Ref. [21]).",
"A rather large difference between the calculated frequencies in the two phases is due the relatively large change in the unit cell size.",
"This difference between the unit cell sizes in the FM and PM phases is overestimated in the calculation which neglects spin fluctuations.",
"For comparison, we also calculated the frequencies keeping the experimental values of the unit cell size, and relaxing just the coordinate $z_{_{Se}}$ of the Se atoms, which is often done in the case of iron based superconductors and related compounds [21].",
"This gives $z_{_{Se}}=0.3486$ in the FM solution and $z_{_{Se}}=0.3496$ in the PM solution, while the change in the phonon frequencies between the two solutions is much smaller, see Table III and a discussion in Section IV.",
"Polarized Raman scattering spectra of K$_x$ Co$_{2-y}$ Se$_2$ single crystals, measured at various temperatures from the (001)-plane of the sample, are presented in Figure REF .",
"Orientation of the sample is chosen so that each of the observable modes appears in different polarization configuration.",
"A pronounced feature in the spectra is an asymmetric Fano profile of the B$_{1g}$ mode, persisting down to low temperatures, as well as its large linewidth compared to isostructural K$_x$ Fe$_{2-y}$ Se$_2$ [8], [29].",
"This feature should by mainly due to the spin fluctuations influencing the B$_{1g}$ vibrational mode which modulates the distances between the magnetic Co atoms.",
"A detailed discussion of the frequency and linewidth temperature dependence is given in the next Section."
],
[
"Discussion",
"There are several factors that affect the phonon frequencies (energies) and linewidths, and their changes across the FM-PM transition.",
"In general, the temperature dependence of the phonon frequency of the mode $i$ , $\\omega _i(T)$ , is influenced by thermal expansion and magnetostriction, anharmonicity effects, electron-phonon and magnetic exchange interaction (spin-phonon coupling) [30], [31] $\\omega _i(T) - \\omega _i(T_0) &=& \\Delta \\omega _i(T) = (\\Delta \\omega _i)_{latt} + (\\Delta \\omega _i)_{anh} \\nonumber \\\\&+& (\\Delta \\omega _i)_{el-ph} + (\\Delta \\omega _i)_{sp-ph}.$ The first term is the frequency shift due to the change of the unit cell size caused by the thermal effects and magnetostriction.",
"$(\\Delta \\omega _i)_{anh}$ is the anharmonic frequency shift.",
"$(\\Delta \\omega _i)_{el-ph}$ appears due to the change in the electron-phonon interaction primarily influenced by changes in the electronic spectrum near the Fermi level, and $(\\Delta \\omega _i)_{sp-ph}$ is the spin-phonon contribution caused by the modulation of exchange interaction by lattice vibrations.",
"In our case of K$_x$ Co$_{2-y}$ Se$_2$ , for temperatures above $T_c$ , $\\omega _i(T)$ decreases and $\\Gamma _i(T)$ (full width at half-maximum, FWHM) increases with increasing temperature for $A_{1g}$ and $B_{1g}$ modes, similar as in the Raman spectra of many other materials.",
"However, they show anomalous behavior near $T_c$ , see Figure REF .",
"In the following, we analyze $\\omega _i(T)$ and $\\Gamma _i(T)$ more closely."
],
[
"Phonon frequencies",
"The frequencies of the A$_{1g}$ and B$_{1g}$ modes change by less than 2 percent in the temperature range between 20 K and 250 K. The red solid lines in Figs.",
"REF (a),(c) represent the fits of the phonon energy temperature dependence (see below), following the frequencies of the two modes in the high-temperature PM phase.",
"The red dotted line is the extrapolation to $T=0$ .",
"For $T>T_c$ , the temperature dependence of the frequency looks conventional for both modes: the frequency decreases with increasing temperature.",
"This behavior is expected both due to the thermal expansion and the anharmonicity.",
"These two effects can be standardly analyzed as follows.",
"The temperature dependent frequency of the vibrational mode $i$ is given by $\\omega _i(T)=\\omega _{0,i}+\\Delta _i(T),$ where $\\omega _{0,i}$ denotes the temperature independent term and $\\Delta _i(T)$ can be decomposed as [32], [33], [19] $\\Delta _i(T)=\\Delta _i^V+\\Delta _i^A.$ $\\Delta _i^V$ describes a change of the Raman mode energy as a consequence of the lattice thermal expansion and can be expressed with [32] $\\Delta _i^V = \\omega _{0,i}\\left(e^{-3\\gamma _i \\int _{0}^{T} \\alpha (T^{\\prime })dT^{\\prime }}-1\\right),$ where $\\gamma _i$ is the Grüneisen parameter of the Raman mode $i$ and $\\alpha (T)$ is the thermal expansion coefficient of a considered single crystal.",
"$\\Delta _i^A$ represents the anharmonic contribution to the Raman mode energy.",
"If we assume, for simplicity, that anharmonic effects are described by three-phonon processes, this term is given by [32], [34] $\\Delta _i^A = -C_i \\left(1+\\frac{2 \\lambda _{p-p,i}}{e^{\\hbar \\omega _{0,i}/2k_BT}-1}\\right),$ where $C$ is the anharmonic constant and $\\lambda _{p-p,i}$ is a fitting parameter which describes the phonon-phonon coupling, including the nonsymmetric phonon decay processes.",
"The relative importance of the thermal expansion and anharmonicity to frequency changes is, to the best of our knowledge, not yet firmly established for pnictides and chalcogenides.",
"In several cases [17], [13] the anharmonic formula, Eq.",
"(REF ), is used for the $\\omega (T)$ fit.",
"We follow here the arguments from Refs.",
"[35], [19], [29] that $\\omega (T)$ is dominated by the thermal expansion.",
"To the best of our knowledge, the thermal expansion coefficient $\\alpha (T)$ of K$_x$ Co$_{2-y}$ Se$_2$ single crystal is unknown.",
"For estimating the lattice thermal expansion contribution to the phonon energy change, the coefficient $\\alpha (T)$ for FeSe, given in Ref.",
"[36], is used.",
"The best fit shown in our Fig.",
"4 is obtained with $\\omega _{0,A_{1g}} = 201.3$ cm$^{-1}$ , $\\gamma _{A_{1g}} =1.23$ and $\\omega _{0,B_{1g}} = 194.2$ cm$^{-1}$ , $\\gamma _{B_{1g}} =1.7$ .",
"There exists a shift in phonon frequencies as the temperature is lowered below $T_c$ .",
"This shift does not show clear discontinuity (as well as the corresponding shift in the linewidths) and no additional modes are registered in the Raman spectra, which suggest that the FM-PM transition is continuous, without structural changes.",
"There are several causes of the sudden frequency change as the sample gets magnetized.",
"It can change due to the magnetostriction, modulation of the magnetic exchange by lattice vibrations (spin-phonon coupling), and due to the changes in the electron-phonon interaction due to spin polarization and changes in the electronic spectrum.",
"The effect of spin-phonon interactions, caused by the modulation of magnetic exchange interaction by lattice vibrations, may be quantitatively examined within the framework developed in Ref.",
"[30] for insulating magnets, and recently applied also to several itinerant ferromagnets [37], [38], [39], [40].",
"In this model, the shift of the Raman mode energy due to the spin-phonon interaction is proportional to the spin-spin correlation function $\\langle S_i|S_j \\rangle $ between nearest magnetic ions.",
"This term should have the same temperature dependence as $(M(T)/M_0)^2$ , where $M(T)$ is the magnetization per magnetic ion at a temperature $T$ and $M_0$ is the saturation magnetization, $\\Delta \\omega (T) = \\omega _{{exp}}(T)-\\omega _{{fit}}(T) \\propto \\pm \\left(\\frac{M(T)}{M_0}\\right)^2,$ where $\\omega _{fit}(T)$ is the extrapolation from the high-temperature data.",
"This model does not predict the sign of the phonon energy shift - softening or hardening.",
"From the inset in Fig.",
"REF (c) it can be seen that the B$_{1g}$ mode energy renormalization scales well with the $(M(T)/M_0)^2$ curve.",
"However, the effect of the magnetostriction (change of the unit cell size due to the magnetization) cannot be excluded based just on this plot, especially since the A$_{1g}$ mode corresponding to the vibrations of nonmagnetic Se ions also shows a similar shift in frequency.",
"The DFT calculations can give us some guidance for understanding of the changes of the phonon frequencies and linewidths, but one has to be aware of its limitations.",
"The DFT calculations (see Table II) give a rather large magnetostriction, i.e.",
"rather large change in the size of the unit cell between the FM and PM phases ($a$ changes by 3.2% and $c$ by 4.3%).",
"This leads to very large changes in the phonon frequencies, see Table III.",
"The calculated frequencies are lower in the FM phase, as opposed to the experimental data.",
"This already points to the limitations of the DFT calculations, which is expected near the phase transition.",
"Similar conclusion is also present in Ref. [21].",
"The DFT ignores spin fluctuations which often leads to quantitative discrepancy in various physical quantities [41] and, in some cases, even predicts wrong phases.",
"In the case of K$_x$ Co$_{2-y}$ Se$_2$ , the DFT calculations correctly predict the FM ground state, but the calculated magnetic moment $m=0.947 \\mu _B$ is much larger than the experimental value $m\\approx 0.72 \\mu _B$ [16].",
"This already shows the importance of correlations and quantum fluctuations which are neglected within the DFT.",
"Strong correlation effects can be captured using screened hybrid functional [42] or within the dynamical mean field theory combined with DFT (LDA+DMFT) [43], which is beyond our present work.",
"Since the magnetostriction effects are overestimated in the DFT calculations with relaxed unit cell size, we repeated the DFT (DFPT) calculations keeping the experimental value for the unit cell size and relaxing only the fractional coordinate (positions of the Se atoms).",
"This is often done in the literature on iron based superconductors and related compounds [21].",
"Our calculated frequencies are given in the parenthesis in Table III.",
"We see that the frequency changes between the two phases are small, in better agreement with the experiment."
],
[
"Phonon linewidths",
"The phonon linewidths of the A$_{1g}$ and B$_{1g}$ modes are very large, $\\Gamma _{i,A_{1g}} \\sim 10$ $\\mathrm {cm}^{-1}$ and $\\Gamma _{i,B_{1g}} \\sim 20$ $\\mathrm {cm}^{-1}$ , which implies the importance of disorder (impurities, nonstoichiometry, lattice imperfections) in measured samples.",
"In general, the broadening of the phonon lines can be a consequence of the electron-phonon interaction, disorder, spin fluctuations and anharmonicity effects.",
"The temperature dependence of the linewidth in the PM phase is, however, very weak, which indicates that the anharmonicity effects are small.",
"The DFT calculation of the linewidth is usually based on the Allen's formula, [44] $\\Gamma _{{\\bf q},i} = \\pi N(E_F)\\lambda _{{\\bf q},i} \\omega _{{\\bf q},i}^2$ .",
"Here, $N(E_F)$ is the density of states (DOS) at the Fermi level, $\\lambda _{{\\bf q},i}$ is the electron-phonon coupling constant, and $\\omega _{{\\bf q},i}^2$ is the phonon frequency of the mode $i$ and wavevector ${\\bf q}$ .",
"A straightforward implementation of Allen's formula in the ${\\bf q} \\rightarrow 0$ limit corresponding to the $\\Gamma $ point is, however, unjustified, as explained for example in Refs.",
"[45], [46].",
"In addition, structural disorder and impurities break the conservation of the momentum, which means that phonons with finite wave vectors also contribute to the Raman scattering spectra.",
"The standard DFT calculation for the Brillouin zone averaged electron-phonon coupling constant $\\lambda $ gives too small value to explain the large width of the Raman lines in pnictides and chalcogenides,[34] and several other metallic systems like MgB$_2$ [45] and fullerides [47].",
"A correct estimate of the phonon linewidth can be obtained only by explicitly taking into account the disorder and electron scattering which enhances the electron-phonon interaction,[45], [47] which is beyond the standard DFT approach and scope of the present work.",
"The Raman mode linewidth is not directly affected by the lattice thermal expansion.",
"Assuming that the three-phonon processes represent leading temperature dependent term in the paramagnetic phase, full width at half-maximum, $\\Gamma _i(T)$ , is given by $\\Gamma _i(T) = \\Gamma _{0,i} \\left(1+\\frac{2\\lambda _{p-p,i}}{e^{\\hbar \\omega _{0,i}/2k_BT}-1}\\right)+A_i.$ The first term represents the anharmonicity induced effects, where $\\Gamma _{0,i}$ is the anharmonic constant.",
"The second term $A_i$ includes the contributions from other scattering channels, i.e.",
"structural disorder and/or coupling of phonons with other elementary excitations, like particle-hole and spin excitations.",
"These effects, typically, depend very weakly on temperature, but can become important near the phase transition.",
"The best fit parameters are $\\lambda _{p-p,i}=0.2$ for both modes, $A_{A_{1g}}=6.6$ cm$^{-1}$ and $A_{B_{1g}}=17.3$ cm$^{-1}$ .",
"The value $\\Gamma _{0,i}=2$ cm$^{-1}$ is adopted from Ref.",
"[29] for related compound K$_x$ Fe$_{2-y}$ Se$_2$ , where the anharmonic effects dominate the temperature dependence.",
"We see that $\\lambda _{p-p,i}$ assume values much smaller than 1.",
"Small and sometimes irregular changes in $\\Gamma _i(T)$ are also observed in other materials whose Raman spectra are considered to be dominated by spin fluctuations [34], [21].",
"Therefore, we believe that a simple separation of $\\Gamma _i(T)$ to the anharmonic and temperature independent term, which works well in many systems, is not appropriate for itinerant magnetic systems like K$_x$ Co$_{2-y}$ Se$_2$ .",
"We conclude that the spin fluctuations and electron-phonon coupling are likely to affect the linewidth even above $T_c$ .",
"The electron-phonon interaction strength is proportional to the density of states at the Fermi level $N(E_F)$ .",
"Our DFT calculations for the DOS agree with those in Ref. [48].",
"The calculated DOS in the FM phase, $N(E_F)=3.69$ /eV, is smaller than, $N(E_F)=5.96$ /eV, in the PM phase.",
"(Though, in reality, it is possible that the DOS significantly differs from the one given by the DFT calculations due to the spin fluctuations and disorder effects.)",
"Therefore, one expects that the phonon line is narrower in the FM phase than in the PM phase.",
"This is indeed the case for the A$_{1g}$ mode, but the opposite is observed for the B$_{1g}$ mode.",
"It is also interesting to note that the B$_{1g}$ mode is much more asymmetric than the A$_{1g}$ mode and almost twice broader.",
"These two observations are in striking similarity with the Raman spectra in the quasi-one-dimensional superconductor K$_2$ Cr$_3$ As$_3$ [21].",
"In this material the vibrational mode that modulates the distance between the magnetic Cr atoms also features large asymmetry and linewidth.",
"In our case, the distances between the magnetic Co ions are modulated by the vibrations of the B$_{1g}$ mode, see Fig. 1.",
"This leads us to the conclusion that the anomalous features of the B$_{1g}$ mode are the consequence of spin fluctuations coupled to the electronic structure via lattice vibrations (in addition to the magnetostriction and spin polarization, which change the electronic spectrum near the Fermi level and, therefore, affect the electron-phonon interaction for both modes).",
"It should be noted that similar anomalous properties of B$_{1g}$ phonon were experimentally observed in cuprate high-temperature superconductor YBa$_2$ Cu$_3$ O$_7$ [49], [50], and explained as a consequence of the out-of-phase nature of this mode which couples to oxygen-oxygen in-plane charge fluctuations [51], [52], [53] In the case of iron-based superconductors and related compounds, the chalcogen atoms and Fe (or Co) are not in the same plane and phonons of A$_{1g}$ symmetry can also directly couple with the electrons.",
"A satisfactory agreement of theory and Raman experiments remains to be established [54].",
"The asymmetric B$_{1g}$ phonon line can be described by the Fano profile [55], [21], [37], [56] $I(\\omega )=I_0 \\frac{(\\epsilon + q)^2}{1+\\epsilon ^2},$ where $\\epsilon = 2(\\omega - \\omega _0)/\\Gamma $ , $\\omega _0$ is the bare phonon frequency, $\\Gamma $ is the linewidth.",
"$I_0$ is a constant and $q$ is the Fano asymmetry parameter.",
"It serves as a measure of a strength of the electron-phonon coupling: an increase in $|1/q|$ indicates an increase in the electron-phonon interaction, more precisely, an increase in the electron-mediated photon-phonon coupling function [52], [54].",
"From the inset of Fig.",
"REF (d) it can be seen that $|1/q|$ increases as the temperature is lowered and reaches the highest values around $T_c$ , when the spin fluctuations are the strongest.",
"Spin fluctuations increase the electron-phonon scattering, similar as does the disorder.",
"Technically, the electronic Green function acquires an imaginary component of the self energy due to the spin fluctuations, and this implies the increase in the damping term in the phonon self-energy, as explained in, e.g.",
"Ref. [45].",
"This leads us to conclude that the spin fluctuations strongly enhance the electron-phonon interaction for the B$_{1g}$ vibrational mode affecting its frequency and linewidth near $T_c$ ."
],
[
"Conclusion",
"In summary, the Raman scattering study of the K$_x$ Co$_{2-y}$ Se$_2$ $(x=0.3, y=0.1)$ single crystals and lattice dynamics calculations of the KCo$_2$ Se$_2$ , have been presented.",
"Two out of four Raman active phonons are experimentally observed and assigned.",
"The lack of any additional modes indicates the absence of vacancy ordering.",
"The Raman spectra show sudden changes in the phonon energy and linewidth near the FM-PM phase transition.",
"Above $T_c$ the energy and linewidth temperature dependence of the A$_{1g}$ and B$_{1g}$ modes look conventional, as expected from the thermal expansion and anharmonicity effects.",
"The linewidth, though, has very weak temperature dependence even above $T_c$ which may the consequence of the proximity of the phase transition and spin fluctuations.",
"The B$_{1g}$ vibrational mode has particularly large linewidth and features a Fano profile, which is likely the consequence of the magnetic exchange coupled to the vibrations of the Co atoms.",
"Interestingly, the A$_{1g}$ mode linewidth decreases below $T_c$ , whereas the linewidth of the B$_{1g}$ mode increases.",
"The DFT calculations generally agree with the measured phonon frequencies.",
"However, fine frequency differences in the two phases cannot be correctly predicted since the DFT calculations do not account for the spin fluctuation effects.",
"We gratefully acknowledge discussions with R. Hackl.",
"This work was supported by the Serbian Ministry of Education, Science and Technological Development under Projects ON171032, III45018 and ON171017, by the European Commission under H2020 project VI-SEEM, Grant No.",
"675121, as well as by the DAAD through the bilateral Serbian-German project (PPP Serbien, grant-no.",
"56267076) \"Interplay of Fe-vacancy ordering and spin fluctuations in iron-based high temperature superconductors.\"",
"Work at Brookhaven is supported by the U.S. DOE under Contract No.",
"DE-SC0012704 and in part by the Center for Emergent Superconductivity, an Energy Frontier Research Center funded by the U.S. DOE, Office for Basic Energy Science (C.P.).",
"Numerical simulations were run on the PARADOX supercomputing facility at the Scientific Computing Laboratory of the Institute of Physics Belgrade.",
"M.M.R also acknowledges the support by the Deutsche Forschungsgemeinschaft through Transregio TRR 80 and Research Unit FOR 1346."
]
] | 1605.05871 |
[
[
"Stability of ion acoustic nonlinear waves and solitons in magnetized\n plasmas"
],
[
"Abstract Early results concerning the shape and stability of ion acoustic waves are generalized to propagation at an angle to the magnetic field lines.",
"Each wave has a critical angle for stability.",
"Known soliton results are recovered as special cases.",
"A historical overview of the problem concludes the paper."
],
[
"Introduction",
"Some time ago the problem of stability of waves described by the Zakharov Kuznetsov equation (ZK, 1974) was solved for arbitrary shape of the wave.",
"Its propagation, however, was limited to clinging to the magnetic field (Infeld 1985).",
"As a check, the soliton case, further limited to perpendicular instabilities, was recovered as found by Laedke and Spatschek (1982).",
"Since then several authors have looked at the propagation and some at stability in various configurations (Infeld and Frycz, 1987 & 1989, Allen and Rowlands, 1993, 1995 & 1997, Munro & Parks 1999, Nawaz et a1 2013, Murawski and Edwin 1992, Bas and Bulent 2010, Mothibi 2015) and several others.",
"Here we present a small $K$ stability analysis of a nonlinear wave propagating at an angle to the magnetic field.",
"Its shape is treated exactly and a cubic is obtained for the frequency of the perturbation (or growth rate).",
"For zero angle, the cubic found in Infeld 1985 is recovered, (see also the book by Infeld and Rowlands 2000).",
"The ZK equation is taken in the form $\\frac{{\\partial }n}{{\\partial }t}+\\frac{1}{3}\\left({\\partial }_x^2+{\\partial }_y^2+3n\\right)\\frac{{\\partial }n}{{\\partial }x} = 0.$ Here $x$ is along the uniform field $B$ .",
"The wave is propagating at an angle $\\theta $ and it proves convenient to rotate the system by this angle.",
"Thus $&&x = \\cos \\theta \\,\\xi +\\sin \\theta \\,\\eta \\nonumber \\\\&&y = \\sin \\theta \\,\\xi -\\cos \\theta \\,\\eta ,$ and ZK is now $\\frac{{\\partial }n}{{\\partial }t}+\\left(n+\\frac{{\\partial }_\\xi ^2+{\\partial }_\\eta ^2}{3}\\right)\\left(\\cos \\theta \\,n_\\xi + \\sin \\theta \\, n_\\eta \\right) = 0$"
],
[
"Shape of the wave and possible perturbation",
"We assume the nonlinear wave or soliton to be a function of $\\xi -Ut$ We now look for stationary nonlinear solutions of the form $n_0^\\prime (\\xi ^\\prime )= n_0+U/\\cos \\,\\theta ,\\quad \\xi ^\\prime = \\xi -Ut$ thus adding a constant to $n_0$ which we simply include in zero order.",
"Thus in the new variables, dropping the primes and integrating twice $&& \\frac{1}{3}\\,{\\partial }^2_\\xi n_0 + \\frac{1}{2} n_0^2 = C\\nonumber \\\\&&\\frac{1}{6}({\\partial }_\\xi n_0)^2 = C n_0-\\frac{1}{6} n_0^3+D.$ By rescaling the variables and the constants we may always reduce the number of parameters putting $C = 1/6,\\quad D(C = 1/6) = \\zeta /6.$ To obtain positive $n_{0\\xi }^2=n_0-n_0^3+\\zeta $ in a finite interval of $n_0$ , the parameter $\\zeta $ has to satisfy $-2/\\sqrt{27}<\\zeta < 2/\\sqrt{27}$ and the stationary solution $n_0(\\xi )$ is periodic with a period $\\lambda $ , which may be defined in terms of complete elliptic integrals.",
"Suppose a periodic wave solution is perturbed such that the wave vector $K$ of the perturbation forms an angle $\\psi $ with the direction $\\xi $ of the nonlinear wave.",
"In the coordinate system of the basic wave we have $n&=&n_0(\\xi )+\\delta n = n_0(\\xi )+\\tilde{\\delta } n(\\xi ) e^{i[(K\\,\\cos \\, \\psi )\\xi +(K\\,\\sin \\,\\psi )\\eta - \\Omega t]}\\\\K &=& K\\,(\\cos \\,\\psi ,~\\sin \\,\\psi )$ and $\\tilde{\\delta } n(\\xi )$ is $\\lambda $ periodic.",
"We now assume $K$ small and expand: $\\Omega &=&\\Omega _1(\\psi ) K+\\Omega _2(\\psi ) K^2 +...\\\\\\tilde{\\delta }n&=&\\delta n_0+K\\delta n_1+K^2 \\delta n_2+...$ Consistency in second order will yield a relationship of the form $G(\\Omega ,K,\\zeta )=0,$ generalizing a dispersion relation theory.",
"Introducing $L = \\frac{1}{3}{\\partial }_\\xi ^2+n_0\\nonumber $ we find $&&{\\partial }_\\xi L\\tilde{\\delta } n=\\frac{1}{3} {\\partial }_\\xi ^3\\tilde{\\delta } n+{\\partial }_\\xi n_0\\tilde{\\delta } n =-i \\,\\Omega \\tilde{\\delta } n \\nonumber \\\\ && -i K \\left[\\cos \\,\\theta \\, \\cos \\,\\psi \\,( n_0\\tilde{\\delta } n +\\tilde{\\delta } n_{\\xi \\xi } )+\\sin \\,\\theta \\, \\sin \\,\\psi \\left(n_0\\tilde{\\delta } n +\\frac{1}{3}\\tilde{\\delta } n_{\\xi \\xi }\\right)\\right]\\nonumber \\\\&&+K^2 \\left[\\cos \\,\\theta \\left(\\cos ^2\\psi +\\frac{1}{3}\\sin ^2\\psi \\right)+\\frac{2}{3} \\sin \\theta \\, \\sin \\,\\psi \\,\\cos \\,\\psi \\right]\\tilde{\\delta } n_\\xi $"
],
[
"Dispersion relation cubic for $\\Omega /K$",
"In second order in $K$ , a dispersion relation is obtained.",
"$&&\\left(\\Omega ^*/K\\right)^3+\\frac{2}{9} \\frac{[4 + 3 Y (Y + 6 \\zeta )] (3 \\cos \\psi - \\tan \\theta \\sin \\psi )}{Y^3 - 4 Y - 8 \\zeta }\\left(\\Omega ^*/K\\right)^2\\nonumber \\\\&&+\\frac{2}{15}\\frac{(Y + 3 \\zeta ) (Y^2 - 4/3) (1 - 5 \\tan ^2\\theta ) \\sin ^2\\psi +5 [4 \\zeta + Y (8/3 + 3 Y \\zeta )] \\sin 2 \\psi \\tan \\theta }{Y^3 - 4 Y - 8 \\zeta }\\left(\\Omega ^*/K\\right)\\nonumber \\\\&&+\\frac{8}{27}\\frac{(27 \\zeta ^2 - 4) \\cos ^2\\psi (\\cos \\psi + \\sin \\psi \\tan \\theta )}{Y^3 - 4 Y - 8 \\zeta }\\nonumber \\\\&&+\\frac{8}{45}\\frac{(Y + 3 \\zeta )^2 \\sin ^2\\psi (\\cos \\psi + \\sin \\psi \\tan \\theta )}{Y^3 - 4 Y - 8 \\zeta } =0$ where $\\Omega ^* = \\Omega - U K \\sin \\psi \\tan \\theta $ .",
"Here $n_1$ , $n_2$ , $n_3$ , are roots of $n_x^2$ in increasing order.",
"Also $n_0$ is contained between $n_2$ , and $n_3$ .",
"Other definitions are $s = \\frac{n_3-n_2}{n_3-n_1}\\le 1,\\quad Y=2\\langle n_0\\rangle =2\\left[n_1+(n_3-n_1)\\frac{E(s)}{K(s)}\\right]$ $E(s)$ and $K(s)$ are complete elliptic integrals of modulus $s$ .",
"For $\\theta =0$ we regain the result of the Infeld and Rowlands book (eq.",
"(8.3.90) in ref.",
"[8]).",
"We limit further analysis to the most unstable angle of perturbation $\\psi =\\pi /2$ .",
"There is an instability for $\\theta =0$ , and $\\theta >0$ up to a critical angle.",
"The angle may easily be obtained e.g.",
"by an analysis of the discriminant of the cubic.",
"For $\\zeta =0$ ($n_1=-1,~n~_2=0,~n_3=1$ ), the critical angle is equal $0.792946~\\mathrm {rad}\\approx 45.4325$$^\\circ $ (Fig.1).",
"Figure: The cubic for θ=0,θ=45\\theta =0,~\\theta =45 ∘ ^\\circ and θ=46\\theta =46 ∘ ^\\circ , while ψ=π/2\\psi =\\pi /2 .",
"Single θ\\theta -intercept means that two roots are complex conjugate; one of them corresponds to the unstable mode.",
"When the cubic has 3 real roots, no instability occurs.",
"The critical angle apparently lies between 45 ∘ ^\\circ and 46 ∘ ^\\circ .The growth rate $\\Gamma /K$ increases from its value at $\\theta =0$ to a maximum, then falls to zero at the critical $\\theta $ (Fig.2).",
"For all acute angles $\\theta $ above the critical one the system is stable to first order in $K$ .",
"For $\\theta >\\pi /2$ the situation is symmetric with respect to $\\theta \\rightleftarrows \\pi -\\theta $ .",
"Figure: The growth-rate Γ/K\\Gamma /K as a function of θ\\theta for ζ=0\\zeta =0.",
"The instability vanishes above the critical angle θ≈0.793\\theta \\approx 0.793.Figure: The growth-rate Γ/K\\Gamma /K as a function of θ\\theta for the soliton case ζ=2/27\\zeta =2/\\sqrt{27}.",
"The growth-rate decreases from 0.298 at θ=0\\theta =0 to zero at the critical angle θ≈0.659\\theta \\approx 0.659.Another limit of interest is for a soliton propagating at an angle to $B$ .",
"For it $\\zeta = 2/\\sqrt{27}$ and the cubic dispersion relation reads $(\\Omega ^*/K)^3 -\\frac{\\sin \\theta }{3 \\sqrt{3}} (\\Omega ^*/K)^2 +\\frac{4}{45}(\\cos ^2\\theta - 5 \\sin ^2\\theta ) (\\Omega ^*/K)+ \\frac{4}{45 \\sqrt{3}} \\cos ^2\\theta \\sin \\theta =0$ Here the growth rate is greatest for $\\theta =0$ , where it is equal $2/(3 \\sqrt{5})\\approx 0.298142$ .",
"Then it decreases to zero at the critical angle, which is $\\mathrm {arctan}\\sqrt{3/5}\\approx 0.659058~\\mathrm {rad}\\approx 37.7612$$^\\circ $ (Fig.3), in agreement with Allen and Rowlands (1995).",
"To first order in $K$ the system is stable for acute angles greater than this angle.",
"The behaviour for angles above $\\pi /2$ again follows from the symmetry $\\theta \\rightleftarrows \\pi -\\theta $ ."
],
[
"A bit of history",
"Forty years ago Infeld and Rowlands pointed out flaws in the way people were calculating the stability of solitons.",
"The perturbations introduced failed to vanish at infinity (1977).",
"Many scientists took the problem seriously.",
"The two authors looked at the problem differently.",
"Rowlands pointed out that the soliton problem involves four different kinds of secularity and removed all by introducing multi variables.",
"Infeld pointed out that removing secular terms is simpler for periodic structures, so lets treat a soliton train and take it to the $\\lambda \\rightarrow \\infty $ limit.",
"(This effort is in that spirit.)",
"Lowest order results of the two methods so far agree, but the transition is not understood.",
"All this notwithstanding the nonlinear wave problems' importance."
]
] | 1605.05726 |
[
[
"The Asymptotic Number of Simple Singular Vector Tuples of a Cubical\n Tensor"
],
[
"Abstract S. Ekhad and D. Zeilberger recently proved that the multivariate generating function for the number of simple singular vector tuples of a generic $m_1 \\times \\cdots \\times m_d$ tensor has an elegant rational form involving elementary symmetric functions, and provided a partial conjecture for the asymptotic behavior of the cubical case $m_1 = \\cdots = m_d$.",
"We prove this conjecture and further identify completely the dominant asymptotic term, including the multiplicative constant.",
"Finally, we use the method of differential approximants to conjecture that the subdominant connective constant effect observed by Ekhad and Zeilberger for a particular case in fact occurs more generally."
],
[
"1.1em S. Ekhad and D. Zeilberger recently proved that the multivariate generating function for the number of simple singular vector tuples of a generic $m_1 \\times \\cdots \\times m_d$ tensor has an elegant rational form involving elementary symmetric functions, and provided a partial conjecture for the asymptotic behavior of the cubical case $m_1 = \\cdots = m_d$ .",
"We prove this conjecture and further identify completely the dominant asymptotic term, including the multiplicative constant.",
"Finally, we use the method of differential approximants to conjecture that the subdominant connective constant effect observed by Ekhad and Zeilberger for a particular case in fact occurs more generally."
],
[
"Introduction",
"In this note, we confirm a conjecture of Ekhad and Zeilberger [3] regarding the number of simple singular vector tuples of a generic $m_1 \\times \\cdots \\times m_d$ complex tensor.",
"We refer the reader to the work of Friedland and Ottaviani [4] for the definitions of these terms, as they are not important to the content of this article.",
"Therein, the authors prove the following theorem.",
"Theorem 2.1 (Friedland and Ottaviani [4]) The number of simple singular vector tuples of a generic $m_1 \\times \\cdots \\times m_d$ complex tensor is equal to the coefficient of $t_1^{m_1-1}\\cdots t_d^{m_d-1}$ in the expression $\\prod _{i=1}^d \\frac{\\widehat{t_i}^{m_i} - {t_i}^{m_i}}{\\widehat{t_i}-t_i}, \\qquad \\text{where \\;\\; } \\widehat{t_i} = \\left(\\sum _{j=1}^d t_j\\right) - t_i.$ Denoting by $a_d(m_1, \\ldots , m_d)$ the quantity described in Theorem REF , Ekhad and Zeilberger [3] derived a rational generating function for this multi-indexed sequence.",
"Theorem 2.2 (Ekhad and Zeilberger [3]) Let $e_i(x_1, \\ldots , x_d)$ be the $i$ th elementary symmetric function $e_i(x_1, \\ldots , x_d) = \\displaystyle \\sum _{1 \\le r_1 < \\cdots < r_i \\le d} x_{r_1}\\cdots x_{r_i}.$ Then, the multivariate generating function $A_d(x_1, \\ldots , x_d)$ for the sequence $a_d(m_1, \\ldots , m_d)$ is $A_d(x_1, \\ldots , x_d) = \\sum _{{m_1,\\ldots , m_d \\ge 0}} a_d(m_1, \\ldots , m_d) x_1^{m_1} \\ldots x_d^{m_d} = \\left(1 - \\displaystyle \\sum _{i=2}^d (i-1)e_i(x_1, \\ldots , x_d)\\right)^{-1}\\prod _{i=1}^d \\frac{x_i}{1-x_i}.$ We are primarily interested in the number of simple singular vector tuples of tensors for which $m_1 = \\cdots = m_d$ , known as cubical tensors.",
"Denote $C_d(n) = a_d(\\underbrace{n, \\ldots , n}_{d\\text{ times}}),$ and observe that Theorem REF implies that the generating function $F_d(x)$ of the sequence $\\lbrace C_d(n)\\rbrace _{n \\ge 0}$ is the diagonal of $A_d(x_1, \\ldots , x_d)$ ; that is, $F_d(x) = \\displaystyle \\sum _{n \\ge 0} C_d(n)x^n = \\displaystyle \\sum _{n \\ge 0} [x_1^n \\cdots x_d^n] A_d(x_1, \\ldots , x_d)x^n.$ Here $[x_1^n\\cdots x_d^n]A(x)$ denotes the coefficient of $x_1^n\\cdots x_d^n$ in $A(x)$ .",
"A univariate generating function $A(x)$ is said to be D-finite if it is the solution of a nontrivial linear differential equation with polynomial coefficients (in $x$ ), and a sequence $a(n)$ is said to be P-recursive if it satisfies a recurrence relation of the form $p_0(n)a(n) + p_1(n)a(n-1) + \\cdots + p_k(n)a(n-k) = 0$ where each $p_i(n)$ is a polynomial and $p_0(n) \\ne 0$ .",
"These two notions are in fact equivalent—a generating function $A(x)$ is D-finite if and only if the coefficients of its power series expansion are P-recursive.",
"The theory of D-finite functions (see, e.g., Zeilberger [13], Christol [2], and Lipshitz [7]) guarantees that each of the functions $F_d(x)$ is D-finite, as they are diagonals of rational functions.",
"Unfortunately, current implementations of constructive approaches to finding $F_d(x)$ cannot handle even $d=5$ .",
"Ekhad and Zeilberger [3] provide the recurrence relation for $C_3(n)$ and use this to find that the asymptotic behavior of the sequence is $C_3(n) \\sim \\frac{2}{\\pi \\sqrt{3}} \\,8^n\\,n^{-1}.$ The exponential growth rate 8 (sometimes called the connective constant) and the polynomial exponent $-1$ are derived rigorously from the recurrence relation for $C_3(n)$ , while the multiplicative constant $2/(\\pi \\sqrt{3})$ is estimated through the calculation of many initial terms.",
"After calculating 160 initial terms of $C_4(n)$ , Ekhad and Zeilberger further conjecture that $C_4(n) \\sim \\alpha _4 \\, 81^n \\, n^{-3/2},$ for an unknown constant $\\alpha _4$ .",
"Combining these with other numerical calculations, Ekhad and Zeilberger ultimately conjecture that $C_d(n) \\sim \\alpha _d \\, ((d-1)^d)^n\\,n^{(1-d)/2}.$ In Section we confirm this conjecture, and more, by using multivariate asymptotic methods to prove the following theorem.",
"Theorem 2.3 For $d \\ge 3$ , the number $C_d(n)$ of simple singular vector tuples of a $d$ -dimensional $n \\times \\cdots \\times n$ cubical tensor is asymptotically $C_d(n) = \\frac{(d-1)^{d-1}}{(2\\pi )^{(d-1)/2}d^{(d-2)/2}(d-2)^{(3d-1)/2}} \\, ((d-1)^d)^n \\, n^{(1-d)/2} \\left( 1 + O\\left(\\frac{1}{n}\\right)\\right).$ In Section we discuss the intractability of the computational problem of determining $F_d(x)$ exactly for small $d$ , and we apply the method of differential approximants to explore the phenomenon of subdominant connective constants."
],
[
"The Asymptotic Behavior of $C_d(n)$",
"Only recently have the techniques of analytic combinatorics been reliably extended to the multivariate case.",
"In this section we appeal primarily to two articles of Raichev and Wilson [12], [11] and one of Pemantle and Wilson [10].",
"We start by repeating the necessary definitions and theorems from these articles.",
"For a $d$ -dimensional complex vector $\\textbf {x}$ , define $G_d(\\textbf {x}) &= \\prod _{i=1}^d x_i,\\\\H_d(\\textbf {x}) &= \\left(\\prod _{i=1}^d (1-x_i)\\right) \\left(1 - \\displaystyle \\sum _{i=2}^d (i-1)e_i(\\textbf {x})\\right),$ so that $A_d(\\textbf {x}) = G_d(\\textbf {x})/H_d(\\textbf {x})$ is the generating function whose main diagonal asymptotic behavior we wish to compute.",
"Going forward, we will drop the subscript when the context is clear.",
"Let $\\mathcal {V}$ be the variety defined by $H(\\textbf {x}) = 0$ .",
"For complex $\\textbf {x}$ , define the polydisk $\\textbf {D}(\\textbf {x})$ and the torus $\\textbf {T}(\\textbf {x})$ by $\\textbf {D}(\\textbf {x}) &= \\left\\lbrace \\textbf {x}^{\\prime } : |x_i^{\\prime }| \\le |x_i| \\text{ for all } i\\right\\rbrace ,\\\\\\textbf {T}(\\textbf {x}) &= \\left\\lbrace \\textbf {x}^{\\prime } : |x_i^{\\prime }| = |x_i| \\text{ for all } i\\right\\rbrace .$ A point $\\textbf {x}\\in \\mathcal {V}$ is said to be minimal if all of its coordinates are non-zero and $\\mathcal {V}\\cap \\textbf {D}(\\textbf {x}) \\subset \\textbf {T}(\\textbf {x})$ .",
"Further, $\\textbf {x}$ is strictly minimal if it is the only point of $\\mathcal {V}$ in $\\textbf {T}(\\textbf {x})$ .",
"For many practical examples, the primary obstacle in computing the asymptotic expansion of the diagonal (or more generally, the asymptotic expansion in any direction) is detecting which points of $\\mathcal {V}$ contribute to the asymptotic behavior.",
"We will use a variety of direct calculations to show that for each $d$ , the asymptotic behavior of the sequence $C_d(n)$ is governed by a single strictly minimal point in the positive orthant $\\mathbb {R}_+^d$ .",
"Theorem REF will then be proved by applying a theorem of Raichev and Wilson [12].",
"The theorem is stated below; we have simplified it to apply only to asymptotic behavior along the main diagonal $F_{n\\textbf {1}}$ of a $d$ -variate generating function $F(\\textbf {x}) = G(\\textbf {x})/H(\\textbf {x})$ .More specifically, we have substituted $\\alpha = \\textbf {1}$ and $p=1$ .",
"Definitions of new terms are given after the statement of the theorem, and we use the short-hand $\\partial _i H(\\textbf {y})$ to denote the partial derivative of $H(\\textbf {x})$ with respect to $x_i$ , evaluated at $\\textbf {x}= \\textbf {y}$ .",
"We also use $\\widehat{\\textbf {x}_i}$ to denote $\\textbf {x}$ with the $i$ th coordinate deleted and $\\widehat{\\textbf {x}_{i,j}}$ to denote $\\textbf {x}$ with the $i$ th and $j$ th coordinates deleted.",
"Theorem 3.1 (Raichev and Wilson [12]) Let $d \\ge 2$ .",
"If $\\textbf {c}\\in \\mathcal {V}$ is strictly minimal, smooth with $c_d\\partial _d H(\\textbf {c}) \\ne 0$ , critical and isolated, and nondegenerate, then for all $N \\in \\mathbb {N}$ , $F_{n\\textbf {1}} = \\textbf {c}^{-n\\textbf {1}}\\left[\\left(\\left(2\\pi n\\right)^{d-1}\\det \\widetilde{g}^{\\prime \\prime }(0)\\right)^{-1/2} \\sum _{k < N} n^{-k}L_k(\\widetilde{u}_0,\\widetilde{g}) + O\\left(n^{-(d-1)/2-N}\\right)\\right]$ as $n \\rightarrow \\infty $ .",
"The quantities $\\det \\widetilde{g}^{\\prime \\prime }(0)$ , $L_k$ , $\\widetilde{u}_0$ , and $\\widetilde{g}$ will be defined later, as needed.",
"For now it suffices to remark that they can all be computed and hence Theorem REF permits the computation of the asymptotic behavior of the main diagonal to arbitrary precision.",
"There are a number of hypotheses that must be verified to apply Theorem REF .",
"We have already stated what it means for a point in $\\mathcal {V}$ to be strictly minimal.",
"Further, $\\textbf {c}\\in \\mathcal {V}$ is smooth if $\\partial _i H(\\textbf {c}) \\ne 0$ for some $i$ , $\\textbf {c}\\in \\mathcal {V}$ is critical if it's smooth and $c_1\\partial _1H(\\textbf {c}) = c_2\\partial _2H(\\textbf {c}) = \\cdots = c_d\\partial _d H(\\textbf {c}),$ $\\textbf {c}$ is isolated if there is a neighborhood around $\\textbf {c}$ in which it is the only critical point, and $\\textbf {c}$ is nondegenerate if $\\det \\widetilde{g}^{\\prime \\prime }(0) \\ne 0$ .",
"We claim that $\\textbf {c}= \\bigg (\\underbrace{\\frac{1}{d-1}, \\ldots , \\frac{1}{d-1}}_{d\\text{ times}}\\bigg ),$ satisfies the hypotheses of Theorem REF and therefore is the sole contributing point to the asymptotic behavior of $C_d(n)$ .",
"The verification of this claim relies on tedious computation using several properties of the symmetric functions $e_i(\\textbf {x})$ ; these will be stated as they are required.",
"To simplify notation, denote $P(\\textbf {x}) &= \\prod _{i=1}^d (1-x_i),\\\\S(\\textbf {x}) &= 1 - \\displaystyle \\sum _{i=2}^d (i-1)e_i(\\textbf {x}).$ Proposition 3.2 The point $\\textbf {c}$ lies in the variety $\\mathcal {V}$ .",
"It suffices to show that $S(\\textbf {c}) = 0$ .",
"Observe that $e_i(k\\textbf {1}) = {d \\atopwithdelims ()i}k^i,$ and therefore $S(\\textbf {c}) = 1 - \\displaystyle \\sum _{i=2}^d(i-1)e_i(\\textbf {c}) = 1 - \\displaystyle \\sum _{i=2}^d (i-1){d \\atopwithdelims ()i}\\left(\\frac{1}{d-1}\\right)^i = 0,$ the final equality being verified by the computer algebra system Maple (which itself employs an algorithm of Zeilberger [14]).",
"Proposition 3.3 The point $\\textbf {c}$ is strictly minimal in $\\mathcal {V}$ .",
"The variety $\\mathcal {V}$ can be written as the union $\\mathcal {V}= \\left\\lbrace \\textbf {x}: x_i = 1\\text{ for some $i$} \\right\\rbrace \\; \\cup \\; \\left\\lbrace \\textbf {x}: S(\\textbf {x}) = 0 \\right\\rbrace .$ This union is not disjoint.",
"Suppose that $\\textbf {c}$ were not minimal.",
"Then, there would exist a minimal point $\\textbf {y}\\in \\mathcal {V}\\cap D(\\textbf {c})$ different from $\\textbf {c}$ .",
"Since $|y_i| \\le 1/(d-1)$ for all $i$ , we must have $S(\\textbf {y}) = 0$ .",
"Consider the variety $\\mathcal {V}^{\\prime }$ defined by $S(\\textbf {y}) = 0$ .",
"We say that a polynomial $P$ is aperiodic if the set of integer combinations of the exponent vectors of its monomials is all of $\\mathbb {Z}^d$ .",
"For example, $x_1 + x_1^2x_2$ is aperiodic because the the $\\mathbb {Z}$ -span of $\\lbrace (1, 0), (2, 1)\\rbrace $ is $\\mathbb {Z}^2$ , while $x_1^2 + x_2^2$ is not aperiodic.",
"Proposition 3.17 from Pemantle and Wilson [10] states that if $H = 1 - P$ for an aperiodic polynomial $P$ , then every minimal point of the variety defined by $H=0$ is strictly minimal and lies in the positive orthant.The proof of Proposition 3.17 in [10] does not rely on their Assumption 3.6.",
"Applying this to $S$ and $\\mathcal {V}^{\\prime }$ , we conclude that $0 < y_i \\le 1/(d-1)$ for all $i$ .",
"It follows that $\\displaystyle \\sum _{i=2}^d (i-1)e_i(\\textbf {y}) \\le \\sum _{i=2}^d (i-1){d \\atopwithdelims ()i}\\left(\\frac{1}{d-1}\\right)^i = 1,$ with equality only when $\\textbf {y}= \\textbf {c}$ .",
"Therefore $\\textbf {c}$ is minimal, and again by Proposition 3.17 from [10], $\\textbf {c}$ is in fact strictly minimal.",
"Proposition 3.4 The point $\\textbf {c}$ is smooth.",
"Observe first that $\\frac{\\partial }{\\partial x_j} e_i(\\textbf {x}) = e_{i-1}(\\widehat{\\textbf {x}_j}).$ Therefore, $\\partial _d H(\\textbf {c}) &= (\\partial _d P(\\textbf {c}))S(\\textbf {c}) + P(\\textbf {c})(\\partial _d S(\\textbf {c}))\\\\&= -\\left(\\prod _{i=1}^{d-1} (1-c_i)\\right)S(\\textbf {c}) + P(\\textbf {c})\\left(-\\displaystyle \\sum _{i=2}^d(i-1)e_{i-1}(\\widehat{\\textbf {c}_d})\\right)\\\\&= -\\left(\\frac{d-2}{d-1}\\right)^{d}\\left(\\frac{d}{d-1}\\right)^{d-2}\\\\&= -\\frac{(d-2)^{d}d^{d-2}}{(d-1)^{2d-2}} \\ne 0.$ Thus $\\textbf {c}$ is a smooth point.",
"Proposition 3.5 The point $\\textbf {c}$ is critical.",
"To prove the criticality of $\\textbf {c}$ we must verify that $c_j \\partial _j H(\\textbf {c}) = c_k\\partial _kH(\\textbf {c})$ for all $j,k$ .",
"As both $\\textbf {c}$ and $H$ are symmetric, this is trivially true.",
"Proposition 3.6 The point $\\textbf {c}$ is isolated.",
"Let $\\epsilon > 0$ be small and let $B$ be the $\\epsilon $ -neighborhood around $\\textbf {c}$ .",
"Suppose $\\textbf {y}\\in B$ is critical.",
"It must then be true that $y_1 \\partial _1 H(\\textbf {y}) = y_d \\partial _d H(\\textbf {y}).$ Using the calculation performed in Proposition REF along with the fact that $S(\\textbf {y}) = 0$ , it follows that $y_1 \\sum _{i=2}^d (i-1)e_{i-1}(\\widehat{\\textbf {y}_1}) = y_d \\sum _{i=2}^d (i-1)e_{i-1}(\\widehat{\\textbf {y}_d}).$ Using the identity $e_i(x_1, \\ldots , x_d) = x_1e_{i-1}(x_2, \\ldots , x_d) + e_i(x_2, \\ldots , x_d)$ (with the convention that $e_d(x_2, \\ldots , x_d) = 0$ ), the equality becomes $y_1 \\sum _{i=2}^d (i-1)\\left(y_de_{i-1}(\\widehat{\\textbf {y}_{1,d}}) + e_i(\\widehat{\\textbf {y}_{1,d}})\\right) = y_d \\sum _{i=2}^d (i-1)\\left(y_1e_{i-1}(\\widehat{\\textbf {y}_{1,d}}) + e_i(\\widehat{\\textbf {y}_{1,d}})\\right).$ By canceling like terms, we see $y_1 \\sum _{i=2}^d (i-1)e_i(\\widehat{\\textbf {y}_{1,d}}) = y_d \\sum _{i=2}^d (i-1)e_i(\\widehat{\\textbf {y}_{1,d}}).$ Since the function $U(\\textbf {x}) = \\displaystyle \\sum _{i=2}^d (i-1)e_i(\\widehat{\\textbf {x}_{1,d}})$ is nonzero and continuous at $\\textbf {x}= \\textbf {c}$ , $\\epsilon $ can be chosen small enough to ensure that $U(\\textbf {y}) \\ne 0$ .",
"Dividing both sides by $U(\\textbf {y})$ yields $y_1 = y_d,$ and symmetry implies that $\\textbf {y}$ has the form $y\\textbf {1}$ for some $y \\in .$ Noting that $H(y\\textbf {1}) = 1 - \\displaystyle \\sum _{i=2}^d (i-1){d \\atopwithdelims ()i}y^i = (y+1)^{d-1}((1-d)y+1),$ we find that $\\textbf {y}= \\textbf {c}$ .",
"Therefore, $\\textbf {c}$ is isolated.",
"The last hypothesis to check is that $\\textbf {c}$ is nondegenerate.",
"This amounts to checking that $\\det \\widetilde{g}^{\\prime \\prime }(0) \\ne 0$ .",
"In non-symmetric cases, the definition of $\\widetilde{g}^{\\prime \\prime }(0)$ is quite cumbersome.",
"Thankfully, Proposition 4.2 of [12] proves that in cases where $H$ and $\\textbf {c}$ are symmetric, $\\det \\widetilde{g}^{\\prime \\prime }(0) = dq^{d-1},$ where $q = 1 + \\frac{c_1}{\\partial _d H(\\textbf {c})}\\left(\\partial _{dd}H(\\textbf {c}) - \\partial _{1d}H(\\textbf {c})\\right).$ Proposition 3.7 The point $\\textbf {c}$ is nondegenerate.",
"We showed in Proposition REF that $\\partial _d H(\\textbf {x}) = -\\left(\\prod _{i=1}^{d-1} (1-x_i)\\right)S(\\textbf {x}) + P(\\textbf {x})\\left(-\\displaystyle \\sum _{i=2}^d(i-1)e_{i-1}(\\widehat{\\textbf {x}_d})\\right)\\\\$ and $\\partial _d H(\\textbf {c}) = -\\frac{(d-2)^{d}d^{d-2}}{(d-1)^{2d-2}}.$ Additionally, noting that the first term in the left summand and the second term in the right summand are independent of $x_d$ , we have $\\partial _{dd} H(\\textbf {x}) &= -\\left(\\prod _{i=1}^{d-1} (1-x_i)\\right) \\partial _d S(\\textbf {x}) + (\\partial _d P(\\textbf {x}))\\left(-\\displaystyle \\sum _{i=2}^d(i-1)e_{i-1}(\\widehat{\\textbf {x}_d})\\right)\\\\&= 2\\left(\\prod _{i=1}^{d-1} (1-x_i)\\right)\\left(\\displaystyle \\sum _{i=2}^d(i-1)e_{i-1}(\\widehat{\\textbf {x}_d})\\right),$ proving $\\partial _{dd} H(\\textbf {c}) = 2\\left(\\frac{d-2}{d-1}\\right)^{d-1}\\left(\\frac{d}{d-1}\\right)^{d-2} = \\frac{2d^{d-2}(d-2)^{d-1}}{(d-1)^{2d-3}}.$ Furthermore, $\\partial _{1d} H(\\textbf {x}) &= (\\partial _{1d} P(\\textbf {x}))S(\\textbf {x}) + (\\partial _d P(\\textbf {x}))(\\partial _1 S(\\textbf {x})) + (\\partial _1 P(\\textbf {x}))(\\partial _d S(\\textbf {x})) + P(\\textbf {x})(\\partial _{1d} S(\\textbf {x}))\\\\&= \\left(\\prod _{i=2}^{d-1} (1-x_i)\\right)\\left(S(\\textbf {x}) + (1-x_1)\\left(\\sum _{i=2}^d(i-1)e_{i-1}(\\widehat{\\textbf {x}_1})\\right)\\right.\\\\& \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad + (1-x_d)\\left(\\sum _{i=2}^d(i-1)e_{i-1}(\\widehat{\\textbf {x}_d})\\right)\\\\& \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\left.- (1-x_1)(1-x_d)\\left(\\sum _{i=2}^d(i-1)e_{i-2}(\\widehat{\\textbf {x}_{1,d}})\\right)\\right),$ proving, $\\partial _{1d} H(\\textbf {c}) = \\left(\\frac{d-2}{d-1}\\right)^{d-1}\\left(2\\left(\\frac{d}{d-1}\\right)^{d-2} - 2\\left(\\frac{d-2}{d-1}\\right)\\left(\\frac{d}{d-1}\\right)^{d-3}\\right) = \\frac{4d^{d-3}(d-2)^{d-1}}{(d-1)^{2d-3}}.$ We can now compute $q$ : $q = 1 + \\frac{c_1}{\\partial _d H(\\textbf {c})}\\left(\\partial _{dd}H(\\textbf {c}) - \\partial _{1d}H(\\textbf {c})\\right) = \\frac{d-2}{d}.$ Finally, $\\det \\widetilde{g}^{\\prime \\prime }(0) = dq^{d-1} = d\\left(\\frac{d-2}{d}\\right)^{d-1} = \\frac{(d-2)^{d-1}}{d^{d-2}} \\ne 0.$ Hence, $\\textbf {c}$ is nondegenerate.",
"Having verified the hypotheses of Theorem REF for $\\textbf {c}$ , we will now define and compute several of the quantities in its conclusion.",
"To find the first-order asymptotic behavior, we consider the case $N=1$ in the theorem.",
"Applying the appropriate simplifications to the definitions of $L_k$ , $\\widetilde{u}_0$ , and $\\widetilde{g}$ in [12], we find that $L_0(\\widetilde{u}_0, \\widetilde{g}) = \\frac{G(\\textbf {c})}{-c_d\\partial _dH(\\textbf {c})} = \\left(\\frac{1}{d-1}\\right)^d\\left(\\frac{(d-1)^{2d-1}}{d^{d-2}(d-2)^{d}}\\right) = \\frac{(d-1)^{d-1}}{d^{d-2}(d-2)^{d}}.$ Assembling all computed quantities into the conclusion of Theorem REF yields $C_d(n) = \\frac{L_0(\\widetilde{u}_0, \\widetilde{g})}{\\sqrt{(2\\pi )^{d-1} \\det \\widetilde{g}^{\\prime \\prime }(0)}} ((d-1)^d)^n n^{(1-d)/2}\\left( 1 + O\\left(\\frac{1}{n}\\right)\\right)$ and so $C_d(n) = \\frac{(d-1)^{d-1}}{(2\\pi )^{(d-1)/2}d^{(d-2)/2}(d-2)^{(3d-1)/2}} ((d-1)^d)^n n^{(1-d)/2}\\left( 1 + O\\left(\\frac{1}{n}\\right)\\right),$ proving Theorem REF .",
"The computation of the asymptotic behavior of off-diagonal sequences can also be performed using the same techniques.",
"In this case, however, the loss of symmetry will complicate some of the necessary calculations."
],
[
"Computational Aspects and Subdominant Connective Constants",
"All known automatic methods for computing diagonals of rational functions, either exactly or asymptotically, suffer from large run-times.",
"Recent advances have improved the situation, though such calculations still remain out of reach for even reasonably sized rational functions in more than a few variables.",
"We comment on two such implementations.",
"Apagodu and Zeilberger [1] provide an algorithm that produces a linear recurrence with polynomial coefficients (in $n$ ) for the diagonal coefficients of a rational function.",
"Applying the algorithm to $C_3(n)$ returns, after a few hours, a recurrence of order 6 with polynomial coefficients of degree at most 7.",
"We did not attempt to apply the algorithm to $C_4(n)$ .",
"Ekhad and Zeilberger [3] note that it is much faster to generate terms of the sequence $C_3(n)$ and guess a linear recurrence.",
"More recently, Lairez [6] has provided a Magma implementation to produce the differential equation satisfied by the diagonal of a rational function.",
"It finds the generating function for $C_3(n)$ in a few seconds and the generating function for $C_4(n)$ in about 40 minutes.",
"On the asymptotic side, recent work of Melczer and Salvy [8] provides an improved algorithm to rigorously compute the asymptotic behavior of diagonals of rational functions.",
"Their implementation provides the correct asymptotic behavior for $C_3(n)$ and $C_4(n)$ in a few seconds, and that of $C_5(n)$ in a few minutes.",
"Upon the calculation of the linear recurrence satisfied by $C_3(n)$ , Ekhad and Zeilberger note that for the correct initial conditions the connective constant (better known in some circles as the exponential growth rate) is 8.",
"However, for most other initial conditions, the resulting sequence would have connective constant 9.",
"Though we are not able to find linear recurrences for $C_d(n)$ for $d \\ge 5$ , we can provide some evidence that this phenomenon of subdominant connective constants persists for all values of $d$ .",
"We employ the method of differential approximants, pioneered by Guttmann and Joyce [5] and a favorite tool of statistical mechanists.",
"It allows for experimental estimation of the asymptotic behavior of a sequence given only a finite number of known initial terms.",
"A forthcoming article [9] by the present author will explore the inner workings of the method, its usefulness to enumerative combinatorics, and provide an open-source implementation.",
"Using the first 100 terms of $C_3(n)$ , the method of differential approximants predicts that the generating function $F_3(x)$ has, as expected, a singularity located at $x \\approx 0.12500000000000000000000000000001 \\pm (2\\cdot 10^{-32})$ corresponding to the known connective constant 8.",
"More interestingly, it also detects a singularity located at $x \\approx 0.11111111111113 \\pm (4 \\cdot 10^{-14}).$ In most cases, this would imply a connective constant 9.",
"Being in that case the dominant singularity, we would expect it to be estimated more accurately than the singularities further from the origin, not less.",
"In our experience, this indicates that a sequence has a subdominant connective constant, as is known to be true in this case.",
"Applying the same process to the first 100 terms of $C_4(n)$ yields estimates for the location of singularities of $F_4(x)$ at $x &\\approx 0.0123456790123456790123456790123456790123456790123457 \\pm 2\\cdot 10^{-52}\\text{, and}\\\\x &\\approx 0.00799999999999999 \\pm (5 \\cdot 10^{-17}).$ The first indicates the known connective constant 81, while the second indicates that this connective constant is subdominant to a connective constant 125.",
"The first 70 terms of $C_5(n)$ are sufficient to predict the location of singularities of $F_5(x)$ to be $x &\\approx 0.000976562499999999999999999999999999996 \\pm 4\\cdot 10^{-39}\\text{, and}\\\\x &\\approx 0.0004164930 \\pm (2 \\cdot 10^{-10}),$ implying that the known connective constant 1024 is subdominant to a connective constant 2401.",
"This evidence leads us to conjecture that the known connective constants $(d-1)^d$ of all $C_d(n)$ are subdominant to the connective constants $(2d-3)^{d-1}$ for generic solutions to the linear recurrence for $C_d(n)$ .",
"Acknowledgements: The author would like to thank the referees for their careful reading and feedback, which significantly improved this article."
]
] | 1605.06099 |
[
[
"Convergence of many-body wavefunction expansions using a plane wave\n basis in the thermodynamic limit"
],
[
"Abstract Basis set incompleteness error and finite size error can manifest concurrently in systems for which the two effects are phenomenologically well-separated in length scale.",
"When this is true, we need not necessarily remove the two sources of error simultaneously.",
"Instead, the errors can be found and remedied in different parts of the basis set.",
"This would be of great benefit to a method such as coupled cluster theory since the combined cost of $n_{occ}^6 n_{virt}^4$ could be separated into $n_{occ}^6$ and $n_{virt}^4$ costs with smaller prefactors.",
"In this Communication, we present analysis on a data set due to Baardsen and coworkers, containing coupled cluster doubles energies for the 2DEG for $r_s=$ 0.5, 1.0 and 2.0 a.u.~at a wide range of basis set sizes and particle numbers.",
"In obtaining complete basis set limit thermodynamic limit results, we find that within a small and removable error the above assertion is correct for this simple system.",
"This approach allows for the combination of methods which separately address finite size effects and basis set incompleteness error."
],
[
"0pt 0pt hyperindex,breaklinks,colorlinks=true,linkcolor=blue,citecolor=blue,urlcolor=blue,filecolor=blue Convergence of many-body wavefunction expansions using a plane wave basis in the thermodynamic limit James J. Shepherd [email protected] Department of Chemistry, Massachusetts Institute of Technology, 77 Massachusetts Ave, Cambridge MA, 02139 71.10.Ca, 71.15.Ap Basis set incompleteness error and finite size error can manifest concurrently in systems for which the two effects are phenomenologically well-separated in length scale.",
"When this is true, we need not necessarily remove the two sources of error simultaneously.",
"Instead, the errors can be found and remedied in different parts of the basis set.",
"This would be of great benefit to a method such as coupled cluster theory since the combined cost of $n_{\\text{occ}}^6 n_{\\text{virt}}^4$ could be separated into $n_{\\text{occ}}^6$ and $n_{\\text{virt}}^4$ costs with smaller prefactors.",
"In this Communication, we present analysis on a data set due to Baardsen and coworkers, containing coupled cluster doubles energies for the 2DEG for $r_s=$ 0.5, 1.0 and 2.0 a.u.",
"at a wide range of basis set sizes and particle numbers.",
"In obtaining complete basis set limit thermodynamic limit results, we find that within a small and removable error the above assertion is correct for this simple system.",
"This approach allows for the combination of methods which separately address finite size effects and basis set incompleteness error.",
"Introduction.– Since it is extremely challenging to devise methods to simulate an infinite solid directly, a common approach to addressing solid state problems is to use a supercell with a finite particle number and a judicious choice of boundary conditions.",
"The error made in such an approach is termed finite size error, and represents a substantial road block in the development of realistic wavefunction descriptions of solids.",
"This is because the error is both substantial and slowly-decaying; for the total energy it commonly falls away as the inverse of the system size, $1/N$ .",
"Figure: The complete basis set limit, thermodynamic limit, coupled cluster doubles correlation energy is obtained for a 2D electron gas at r s =1.0r_s=1.0 for comparison with Green's function Monte Carlo.",
"In (a), the raw data (originally fromRefs.",
"Baardsen1 and Baardsen2)are shown plotted against basis set size MM for various particle numbers NN.",
"In (b), an axis transformation M→1/m 2 M\\rightarrow 1/m^2 with m=M/Nm=M/N has been performed.",
"It can be seen the the basis set convergence is actually consistent with particle number NN.",
"In (c), the fluctuation in particle number due to finite size effects are shown to have relatively small variation when de-coupled from basis set energy.",
"In (d), the average value of (c) is added to each line to re-scale each set of points.",
"Now each NN value yields a consistent estimate of the final thermodynamic limit, complete basis set limit energy.",
"The final value is -0.198-0.198 (±0.004\\pm 0.004) Ha.On account of finite size effects being such a large source of error, many methods have been developed which alleviate them.",
"A review is well beyond the scope of this work (however, see e.g.",
"Ref.",
"mullerincrementally2013), but these methods include the hierarchical [2], [3] and incremental schemes [4], [1], [5], progressive downsampling [6], and embedding theories for density functionals [7], [8], [9], [10], [11] and density matrices [12], [13], [14], [15].",
"Local orbitals [16], [17], [18], [19], local interactions [20], [21], [22], [23], [24], and length scale (including range separation) schemes [25], [26], [21] have also been developed that exploit a length scale separation between correlations within a unit cell and correlations between unit cells.",
"Finally, some many-body methods can be directly integrated to find the thermodynamic limit [27], [28], [29], [30], [31], [32], [33] from which analytic corrections can be derived [34].",
"Even for a finite system, a finite number of basis functions yields a different type of finite size effect in wavefunction calculations.",
"A finite set of smooth functions is unable to correctly describe interelectron cusps, and the resultant errors in the energy also have slow convergence in the basis set size.",
"In a recent study, convergence of the many-body wave function expansion using a plane wave basis was analyzed for the electron gas and lithium hydride solid [35] drawing inspiration from a large body of literature on this subject for molecular systems [36].",
"When analyzed, the basis set incompleteness error fell off as the inverse of the number of basis functions used: $1/M$ .",
"The power-laws derived were used in the years leading up to that study and then subsequently to achieve complete basis set results for a variety of systems [37], [38], [39] and in particular the uniform electron gas [40], [41], [42], [43], [44], [45], [46], [47], [48], [49], [50], [51].",
"In recent times, several further major developments addressing plane wave basis set incompleteness error have been made.",
"In particular, explicit correlation has been applied to a plane-wave basis, including F12 methods [52], [53], [54] and transcorrelation [55]; corrections have been derived for a semi-analytical correction has been found for the direct term MP2 and for dRPA [56]; and hybrid basis sets of plane-wave derived occupied orbitals and Gaussian virtual orbitals have been implemented [57].",
"Converging finite basis sets and finite particle numbers simultaneously to their respective limits is challenging, and a $1/(MN)$ scaling of the error makes even simple methods such as coupled cluster doubles prohibitively expensive in computational cost (scaling as $\\mathcal {O} [ N^6 M^4 ]$ see Ref. ohnishihybrid2011).",
"A method to alleviate this cost is to undertake calculations with small particle numbers for large basis sets and vice versa, and combining the two results to estimate the result of taking both limits.",
"Instead of converging the coupled error brute force (at $\\mathcal {O} [ N^6 M^4 ]$ cost) this would mean that the two can be removed separately at $\\mathcal {O} [ N^6 ] +\\mathcal {O} [ M^4 ]$ cost at the penalty of a small, controllable, and analyzable error.",
"It is difficult to track the origin of such approaches, with perhaps the earliest mention coming from Nozières and Pines [30]; recent authors give attribution to work due to Hirata [58], [59], [60], [61], [62], [63], [64], [6], [65], Kresse [66], and colleagues.",
"Similar physics can also be found in neighboring fields, such as during the construction of Jastrow functions [67], frequency theshholding and renormalization [68], or removal of finite size effects using DFT corrections [69].",
"The aim of this Communication is to analyze the finite size effects present in the basis set incompleteness error for coupled cluster doubles correlation energies (CCD).",
"To this end, we perform numerical analysis on the convergence properties of a large data set due to Baardsen and coworkers [70], [71] which contains CCD energies for the 2D electron gas for $r_s=0.5$ , 1.0, 2.0 at a wide range of basis set sizes (up to $M=500$ ) and particle numbers ($26 \\le N \\le 138$ ).",
"By analyzing the convergence of the basis set incompleteness error as the particle number changes, we find that for these systems basis set incompleteness error and finite size effects are effectively decoupled in the energy and can be removed independently from one another.",
"There are physical reasons to believe these two limits would not be strongly coupled, however, since they arise from two different limits of the interelectron interaction ($1/r_{12}$ ).",
"Basis set incompleteness error arises from the difficulty in describing electron coallescence points, i.e.",
"interactions as $r_{12} \\rightarrow 0$ , whereas finite size error arises from the long-range interactions of electrons which are being improperly truncated (or mirrored) by periodic supercell approaches.",
"This being the case, this overall error is strongly system dependent and requires that the physics of the system naturally separates these two effects in range.",
"This allows for the benchmarking of CCD against Green's function Monte Carlo methods which are in the complete basis set and thermodynamic limits.",
"The strategy presented here works directly with the pure plane wave basis suitable for infinite systems.",
"This includes the electron gas, but can also be applied to other infinite systems of interest to condensed matter theorists and nuclear physicists.",
"In these systems, finite size effects are captured by an electron number, and extreme pathologies can arise from shell-filling finite size errors.",
"We rely particularly on work achieved in this area by Drummond and coworkers [23].",
"It is anticipated that these additional effects will arise in the treatment of complex materials, and we hope that the developments presented here prove of interest to this community as well.",
"Deriving consistent basis sets with variation in particle number.– Coupled cluster doubles (CCD) energies are presented in Fig.",
"REF for a 2D electron gas at $r_s=1.0$ ; these derive from Ref. Baardsen1,Baardsen2.",
"We wish to compare the number with the more exact Green's function Monte Carlo (GFMC) result (also shown) which is already both at the complete basis set limit and the thermodynamic limit.",
"The data are in neither limit and show signs that this might be difficult to achieve.",
"Concretely, this is because convergence to the complete basis set limit seems to be at different speeds depending on $N$ , and once at the complete basis set limit the value of this also changes with $N$ .",
"In other words, the basis set incompleteness error and the finite size effects seem inextricably coupled.",
"This effect arises because, when varying particle number $N$ , the basis set index $M$ changes in energy due to changes in the Fermi energy.",
"A more consistent measure for basis set size is one which does not change in energy, which, in general, is a function of the number of basis functions per electron $m=M/N$ ; this is in common with papers which use the energy itself to converge the basis set incompleteness error, but, as we examined in a previous paper, the $M$ gives better convergence properties for the systems shown here [35].",
"It is also possible to derive relationships for how basis set incompleteness error behaves on approach to the complete basis set limit [35], [37], [38], [56], which for 2D is $1/m^2$ .",
"When the correlation energies in our data set are plotted in this way, in Fig.",
"REF , we can see that the convergence to the CBS is parallel in $m$ for different particle numbers.",
"Linear interpolation for particle-number fluctuations.– By transforming the data set to $(1/m)^2$ , energies converge with basis set in a manner that is consistent and invariant with $N$ .",
"Looking at Fig.",
"REF , we can see the lines are still offset from one another along the $y$ -axis due to finite size effects.",
"The trend is non-monotonic due to the data coming from $\\Gamma $ -point calculations.",
"We can measure this $N$ -dependent shift in the energy.",
"Observing that in Fig.",
"REF that the lines no longer share $1/m^2$ -dependent points, this can be measured by linear interpolation.",
"For a specific target value of $1/m^2$ , we can find the discrete points on either side $(1/m_1^2,E_1)$ and $(1/m_2^2,E_2)$ and then the shift value can be computed as: $E_\\Delta (N,m)=E_2+\\frac{1/m^2-1/m_2^2}{1/m_1^2-1/m_2^2} ( E_1-E_2).$ Since the lines are parallel, it should not matter unduly what shift we ultimately choose.",
"In general, the larger the value then the worse the finite size effects, and the smaller the value the more expensive the calculations required.",
"A plot of this shift value is shown in Fig.",
"REF for $m=0.0009$ .",
"This was chosen for convenience, since all the $N$ values have this basis set size within their ranges.",
"The energy shift value $E_\\Delta (N,m)$ can be seen to fall with $N$ , but in a manner that fluctuates substantially and although we expect the overall limiting scaling to be $N^{-\\frac{5}{4}}$ , these data do not support an extrapolation.",
"This might be resolved by twist-averaging the data [72], but for this work we make do with a relatively crude estimate: the mean and associated error, which comes out at $-0.1866 \\pm 0.0041$ Ha.",
"This may seem like a large value to be computing, but recall that the absolute value of the quantity depends on the value of a target $1/m^2$ .",
"Instead, we should estimate the size of the effect by the variation with $N$ of this quantity which is 0.02 Ha in range, decreasing to within our error bar for higher $N$ .",
"Crucially, we found that this is not qualitatively sensitive to values of $m = 0.001, 0.1, 0.2, 0.25$ .",
"In other words, it is possible to use far smaller basis sets to obtain these shift estimates.",
"We are using a data set from elsewhere, so it is beyond the scope of this work to return to do further calculations in $N$ .",
"That said, this identifies that this data set requires a greater resolution in $N$ , which we now know can be estimated at low $m$ and at minimal cost.",
"Furthermore, twist-averaging would improve the trend in $N$ of the energy shift value due to alleviating the shell-filling effects we see.",
"Final extrapolation.– We are now in a position to modify our original graph to show extrapolation to the complete basis set and thermodynamic limits.",
"By shifting the lines by the quantity $E_\\Delta (N)-E_\\Delta (\\infty )$ , we can overlay them and show the complete basis set extrapolation to a thermodynamic limit quantity (Fig.",
"REF ).",
"Extrapolating to the combined complete basis set and thermodynamic limits, we find: $\\begin{split}E =-0.198 +0.057 \\left(N/M\\right) + 0.690 \\left(N/M\\right)^2,\\end{split}$ and so our extrapolated energy is 90$\\pm $ 2 % of the GFMC energy, which is in agreement with similar previous findings for 3D electron gases [46], [43].",
"Repeating this procedure gives $E =-0.251 \\pm 0.003$ and $E =-0.1394 \\pm 0.0007$ for $r_s=0.5$ and 2.0 respectively.",
"Figure: In (a), the conventional strategy is shown where a single particle number (Fermi vector k f k_f) is converged to the complete basis set limit m c →∞m_c \\rightarrow \\infty .",
"In (b), one value of mm is chosen, and the area around the Fermi sphere is gridded finely.",
"Then the basis set limit m c →∞m_c\\rightarrow \\infty is found for a small system.Discussion & Concluding remarks.– Analyzing Baardsen's coupled cluster doubles data set in the manner presented above reveals that, to a good approximation, the complete basis set and the thermodynamic limits are decoupled.",
"The schematic shown in Fig.",
"REF shows the physical interpretation of such an approach in $k$ -space.",
"In $k$ -space, the thermodynamic limit corresponds to an infinitesimally small grid spacing; the complete basis set limit is reached when an infinite expanse of $k$ -space is included.",
"When we separate both limits, we effectively say we can grid the area around the Fermi surface more finely to converge finite size effects up to a specific basis set limit of $m=M/N$ .",
"In contrast, the area outside of the $m$ cut-off is treated with a more coarse grid and this limit sent out to infinity.",
"Provided the finite-$m$ error can be brought under control in the coarse grained part of the space, we can now explore the complete basis set limit at a smaller $N$ and the thermodynamic limit at a smaller $m$ than before.",
"This results in substantially improved scaling.",
"The remaining source of error is two-fold.",
"The first is that there are still finite size effects in the jagged extrapolation of $N$ .",
"This could be partially resolved by twist-averaging, and a method for this has been described elsewhere [73].",
"Further, the data presented here lead us to the expectation that twist-averaging can also be performed at small basis set sizes.",
"The second source of error comes from $m$ , the inner cutoff, not being large enough/ This causes shell-filling errors and errors caused by coupling between complete basis set limit and the thermodynamic limit.",
"The finite-$m$ approximation is not as severe as it might look at first glance.",
"For sufficiently large basis set sizes (here, $m$ ) the Fermi sphere becomes point-like compared with the length of the momentum transfer vector which couples the correlated state determinant with the reference.",
"We can also argue that there is a limit in which the inter-electron coalescence is not affected by the additional electrons that are provided to the system.",
"The additional electrons mediate the fineness of the $k$ -point grid, so what this says is that beyond a certain point the fineness of the mesh saturates.",
"This is reasonable: the main purpose of a finer grid is to describe the low-lying excitations in the spectrum and to resolve the Fermi sphere.",
"In any case, the approximation is less severe as $m$ is raised; in the $m \\rightarrow \\infty $ limit the expression returns to the original extrapolation scheme without our approximation.In order words, we have a controlled approximation with systematic improvability.",
"The extent of the coupling between the two regimes and the size of $m$ will depend entirely on physics of the system but is reasonable to think that these two effects are well-separated in length scale [23].",
"We believe that this provides numerical evidence describing phenomenology that would be of considerable excitement for the community of people examining solid state problems with wavefunction methods.",
"Acknowledgements.– JJS acknowledges Royal Commission for the Exhibition of 1851 for a Research Fellowship.",
"Thanks is given to Morten Hjorth-Jensen et.",
"al.",
"for early access to this data set [70], [71]."
]
] | 1605.05699 |
[
[
"Simulations of Particle Impact at Lunar Magnetic Anomalies and\n Comparison with Spectral Observations"
],
[
"Abstract Ever since the Apollo era, a question has remained as to the origin of the lunar swirls (high albedo regions coincident with the regions of surface magnetization).",
"Different processes have been proposed for their origin.",
"In this work we test the idea that the lunar swirls have a higher albedo relative to surrounding regions because they deflect incoming solar wind particles that would otherwise darken the surface.",
"3D particle tracking is used to estimate the influence of five lunar magnetic anomalies on incoming solar wind.",
"The regions investigated include Mare Ingenii, Gerasimovich, Renier Gamma, Northwest of Apollo and Marginis.",
"Both protons and electrons are tracked as they interact with the anomalous magnetic field and impact maps are calculated.",
"The impact maps are then compared to optical observations and comparisons are made between the maxima and minima in surface fluxes and the albedo and optical maturity of the regions.",
"Results show deflection of slow to typical solar wind particles on a larger scale than the fine scale optical, swirl, features.",
"It is found that efficiency of a particular anomaly for deflection of incoming particles does not only scale directly with surface magnetic field strength, but also is a function of the coherence of the magnetic field.",
"All anomalous regions can also produce moderate deflection of fast solar wind particles.",
"The anomalies' influence on ~ 1 GeV SEP particles is only apparent as a slight modification of the incident velocities."
],
[
"Introduction",
"Lunar swirls are high albedo regions on the lunar surface which appear to correspond to surface magnetic anomalies.",
"(See reviews by [22] and [3]).",
"While the origin of the lunar swirls is not yet resolved, one of the main theories is that the anomalous magnetic field deflects incoming solar wind, which would otherwise impact the surface and chemically weather, or darken, the lunar regolith through the creation of nanophase iron ([16], [17],[18], [15], [21], and [20]).",
"These incoming particles may be completely deflected away from the surface or they may be deflected to other regions on the surface.",
"It is thought that the dark lanes, regions of very low albedo adjacent to swirls, may correspond to locations of enhanced particle flux and, thus weathering, due to nearby particle deflection (For a more in-depth discussion see [21]).",
"The idea that the lunar magnetic anomalies may deflect incoming solar wind was first proposed during the Apollo era to explain compression of the anomalous magnetic field, or amplifications of the magnetic field near the limb (called “limb compression”) as observed from orbit [[5] and [37]].",
"Observations by Lunar Prospector gave the first conclusive evidence that the magnetic anomalies could deflect the solar wind, forming mini-magnetospheres [[23]].",
"Subsequent observations by Lunar Prospector [e.g.",
"[8] and [9]], Nozomi [[6]], SELENE/Kaguya [e.g.",
"[39], [45] and [26]], Chandrayaan-1 [[14]], and Chang'E-2 [[44]] further confirmed that the magnetic anomalies can not only deflect incoming solar wind particles but also modify the distribution [[38]].",
"Different processes may be involved in deflecting incoming particles, and the relative importance of each process will be a function of the surface magnetic field strength and the scale size of the anomalous region.",
"One process for deflecting particles is called magnetic mirroring.",
"Charged particles move with both a spiral motion perpendicular, and parallel to the magnetic field.",
"If the particles move into a region with magnetic field that increases in magnitude (or converges), the kinetic energy of the particle parallel to the magnetic field will be converted into kinetic energy perpendicular to the magnetic field.",
"Eventually, if the magnetic field magnitude is strong enough for a given incident kinetic energy, all of the energy will be converted to perpendicular to the magnetic field, and the particle will be reflected.",
"The initial observations of deflection around the Imbrium antipode region by Lunar Prospector [[23]] suggested that if the anomalous region is large enough, the incident solar wind plasma has a collective behavior, and the plasma behaves in a fluid manner (see also [8]).",
"The dynamic pressure of the incident plasma was balanced by the magnetic pressure of the anomaly, slowing the plasma, producing signatures of a shock region forming, and mini-magnetosphere, around the anomaly.",
"Subsequent observations of the solar wind interacting with a wider range of anomalous regions for a variety of solar wind conditions has lead to a complex picture of the interaction.",
"Observations by Chandrayaan-1, in the vicinity of both strong and weak anomalies [[26]], revealed a plasma interaction in which the electrons behaved in a fluid manner, while the protons became demagnetized.",
"This would lead to charge separation and the generation of an ambipolar electric field, which would act to both accelerate electrons and slow protons.",
"More recent observations have also confirmed the presence of density cavities around magnetic anomalies (e.g.",
"[38], [45]).",
"While these observations help refine our understanding of the physics governing the formation of mini-magnetospheres, there is still uncertainty with regard to how the plasma deflection seen above the surface may be connected to the high albedo swirls seen on the surface.",
"Particle tracking by [17] suggested that the Reiner Gamma region, modeled as a collection of dipoles, has sufficient magnetic field strengths to deflect solar wind ions.",
"Particle tracking was also employed by [11] in order to investigate why backscatter ENAs were typically observed by Chandrayaan-1 over the magnetic anomalies in the southern hemisphere while in the solar wind but not in the terrestrial plasma sheet.",
"They found that when a nearly monoenergetic, monodirectional population of protons (analogous to the solar wind) interacted with a subsurfacemagnetic field, a density cavity formed near the surface.",
"Conversely, when an isotropic, Maxwellian distribution of particles (analogous to plasma sheet protons) interacted with the subsurface dipole, the density cavity near the surface was greatly reduced in size, due to the increased spread in incident particle directions.",
"Self consistent 2.5D fluid and particle simulations of the solar wind interacting with a small dipole on the surface of the Moon by [12] showed that neither the small scale size of a magnetic anomaly nor kinetic effects from the different behavior of ions and electrons prevent a shock-like region from forming.",
"Additional fluid simulations modeling the anomaly as a collection of dipoles [[13]] showed that the nature of the shock would be highly dependent on the structure of the magnetic field and the orientation of the IMF.",
"3D Hall-MHD simulations by [46] have produced a similar result.",
"[19] used the combination of a 3D hybrid model and a 1D Particle-in-a-Cell (PIC) model to look at the kinetic effects of the solar wind interacting with a dipole field both locally (PIC) and globally (hybrid).",
"They found that the central portion of the dipole field could completely block access of protons to the surface while the surrounding regions showed enhanced density and flux.",
"[32] used a 1.5D PIC model to investigate the solar wind interacting with cusp-like structures that may be present at some magnetic anomalies, and possibly be co-located with the dark lanes.",
"They looked at a variety of surface magnetic field strengths and found significant modification of the interacting plasma relative to the incoming distribution.",
"They saw ion deceleration and electron acceleration similar to that observed by Kaguya [[38]].",
"[2] used a vacuum chamber to look at plasma incident on two different dipole magnetic fields (a strong and a weak magnet) and compared the deflection with that seen by theory and satellite observations.",
"They found that a shock-like structure formed around both magnetic fields, and that the general shape of the structure that formed around both magnets was similar, even though the structure around the weak magnet was considerably smaller.",
"In this work, we present results from 3D particle tracking studies following the interaction of protons and electrons using 3D vector magnetic fields models of five different lunar magnetic anomalies, generated from satellite observations [[34]].",
"This work is unique in that it looks at how solar wind particles may interact with realistic anomalous magnetic fields over an extended region and attempts to correlated the particle response with observations of the lunar swirls.",
"The results presented here, begin to address the open question of why not all magnetic anomalies are associated with lunar swirls and why some anomalies with weak or moderate magnetic field strengths have more extensive swirl regions than other, comparatively stronger magnetic anomalies.",
"The five magnetic anomaly regions investigated are Mare Ingenii, Reiner Gamma, Gerasimovich, Marginis, and Northwest (NW) of Apollo,.",
"The first three were selected for study as they are classified among the strongest anomalies and have observable swirl characteristics [[3]].",
"Marginis was selected as it is classified as a weak anomaly but is co-located with a complex swirl pattern, similar in general nature to the swirls at Mare Ingenii, a strong anomaly region.",
"NW of Apollo was selected as it is one of the moderate anomalies but does not have an easily identifiable swirl region.",
"The goal for NW of Apollo is to test the ability of the particle tracking to help guide the search for swirl regions.",
"Particle tracking was employed as a way to both investigate particle deflections at a wide selection of anomalous regions for a variety of incident particle energies in a feasible time frame, and assess how effective just the anomalous magnetic field is alone in deflecting incoming particles.",
"As discussed above, it is difficult to resolve from observations alone the relative importance of anomalous magnetic field, ambipolar electric fields and kinetic effects in the formation and structure of mini-magnetospheres.",
"This study allows for the quantification of the effect of the anomalous field alone in influencing the incident plasma.",
"[1], presents results of fully self-consistent particle simulations for the Reiner Gamma region, assuming a dipole magnetic field.",
"A comparison of the results in this paper with those in [1] quantify how much the incident plasma in additionally influenced by the development of charge separation and the resulting ambipolar electric field.",
"That paper also discusses the similarities and differences with the results from 3D PIC simulations by [4].",
"As part of this work, impact maps for each simulated anomalous region are generated and co-located with both optical and maturity observations of the same regions.",
"The results presented here focus on solar wind regime incident particle energies, but do look at the possibility for deflection of solar energetic particle (SEP) events associated with solar activity.",
"Protons were selected as the ion species as solar wind hydrogen is considered responsible for the creation of nanophase iron, which causes darkening and reddening of the surface spectra as a soil matures ([21],[20]).",
"Simulated anomalous magnetic field were generated at www.planet-mag.net/index.html, using the Correlative model described in [34].",
"The model magnetic fields were generated from observations, typically with passes separated by $1^{o}$ , at altitudes down to 30 km.",
"The model vector magnetic fields were generated at a given altitude, over a range of latitudes and longitudes appropriate for each case.",
"The resolution of the model magnetic field in latitude and longitude ranged from $0.1^{o}$ to $0.15^{o}$ .",
"This was selected as it would create a magnetic field model with a resolution similar to the optical images ($\\sim $ 100m/pixel).",
"In reality, the resolution of the magnetic field model scales with the lowest altitudes of the observations made by Lunar Prospector (> 10s of km).",
"Planes of magnetic field were generated at 0.5 km slices up to approximately 100 km.",
"The upper bound for each case was determined by where the anomalous magnetic field could not be distinguished from the background of $\\sim $ 0.1 nT.",
"The constant altitude slices were stitched together to make a three dimensional grid with vector magnetic fields at each grid point.",
"The simulated region size varied for each case, but for each included the central anomalous region plus several degrees surrounding.",
"This allowed incident particles to be deflected without undergoing an interaction with the simulation side boundaries.",
"Some of the magnetic anomalies studied form extended regions.",
"In these cases, only the central, peak magnetic field region was of interest.",
"The basic characteristics for the five regions studied are given in Table REF .",
"The total magnetic field simulated included four cases: just the anomalous magnetic field, and the anomalous magnetic field plus a superposition of three different interplanetary magnetic fields: $\\rm {B}_{\\rm {vertical}} = \\pm 2 \\rm {nT,}\\rm {B}_{\\rm {horizontal}} = 2 \\rm {nT}$ , where horizontal and vertical are relative to the surface with the magnetic anomalies.",
"Table: Values in columns 2-4 taken from .",
"Magnetic field strengths in column 4 areestimates at 30 km altitude from satellite observations.",
"Surface magnetic field strengths in column 5are from the model used as input for the particle tracking.For the particle tracking studies, 400,000 non-interacting protons or electrons were launched at the magnetized surface for the variety of total magnetic field configurations.",
"Initial locations, within the launch region, were randomly assigned.",
"Particle trajectories were computed using the Lorenz force law until all the particles either impacted the surface or left the simulation area.",
"The particles were not forced to move only on grid points where the magnetic field was defined, but rather could have arbitrary locations within the simulation region.",
"The magnetic field at each particle location was computed in a weighted fashion from the magnetic fields defined at nearest grid points.",
"For all the anomalous regions simulated, the particle flux maps showed little difference between the cases with an interplanetary magnetic field (IMF) and the case without an IMF, and little difference among the different IMF cases.",
"The results shown here will focus on the cases with the IMF equal to $\\rm {B}_{\\rm {vertical}} = -2 \\rm {nT,}$ .",
"For all cases, runs were also completed that kept all aspects of the parameter space the same, except the anomalous magnetic field was removed.",
"This was done to allow for an estimate of the level of uncertainty in the results when calculating variations caused by the anomalous magnetic field, for a specific case, and verify that any features seen in the density and flux maps are in fact associated with modification by the anomalous magnetic field and not an artifact of non-randomness in the initial random location of the particles.",
"The velocity distributions for the baseline cases have a mean of 200 km $\\rm {s}^{-1}$ , and a Gaussian distribution with a thermal speed of 75 km $\\rm {s}^{-1}$ .",
"This speed represents either a slow solar wind speed or a high speed flow in the terrestrial magnetotail.",
"Plasma from two sources will impact the lunar surface and contribute to weathering - the solar wind composed primarily of hydrogen ions, and terrestrial plasma sheet plasma composed of varying concentrations of hydrogen and oxygen ions.",
"The plasma in the terrestrial plasma sheet will typically have a much broader range of thermal speeds than the solar wind.",
"The narrow thermal distribution was retained for this slowest speed to facilitate comparison among cases.",
"For each case, accumulated densities and accumulated fluxes (i.e.",
"density of the impacting particles at a grid point times the average speed of the particles at that grid point) for the five anomaly regions were calculated to compare with the optical and (if available) OMAT images.",
"Swirl mappings from the observations are compared to the density and flux maps.",
"It is important to note that as these are results from particle tracking, the proton impact maps represent a maximum impact model.",
"In all cases the trajectories for electrons were also determined (but not all are shown).",
"As the electrons are much more easily deflected by the anomalous magnetic field, many of the incident electrons do not impact the surface, and instead are completely reflected back, away from the surface.",
"The electron simulations were run until all of the particles initially launched towards the surface either impacted the surface or were deflected away from the surface (either towards one of the simulations walls or back upstream).",
"Although both hydrogen ions and electrons were tracked in our simulations, only the trajectories of hydrogen ions were used for comparison with optical imagery as only they can impact with sufficient energy to both break bonds and be utilized as the reducing agent to create nanophase iron.",
"[35] indicates that the energy required to break the FeO bond is $\\sim $ 3-5 eV over a range of several Kelvin to a couple thousand Kelvin.",
"This energy is equivalent to the kinetic energy of a 30 km $\\rm {s}^{-1}$ proton.",
"[42], on the other hand, indicates that an energy of 50 eV is required to break the FeO bond at approximately 300 K. This energy is equivalent to a 100 km $\\rm {s}^{-1}$ proton.",
"Realistically though, some percentage of incident protons will scatter off other minerals within the regolith, loosing energy, before they encounter an FeO molecule, and not all of the kinetic energy from the incident proton will necessarily be transferred to breaking the bond.",
"For comparison, 250-300 km $\\rm {s}^{-1}$ is the bulk speed at which it is estimated that protons, with a temperature of 5-10 eV, will produce a maximum sputtering yield from the lunar surface ([33] and references therein).",
"The case of 200 km $\\rm {s}^{-1}$ (or a proton with a kinetic energy of $\\sim $ 200 eV) is therefore treated in this paper to be near the real minimum energy needed to weather the lunar regolith but not necessarily produce sputtering.",
"Knowing an exact minimum in a realistic setting would require more extensive modeling and experiments to determine.",
"Additional cases for the $\\rm {B}_{\\rm {vertical}}$ = -2 nT case were run with a mean proton velocity of 400 km $\\rm {s}^{-1}$ (typical solar wind velocity), 2000 km $\\rm {s}^{-1}$ (i.e.",
"fast solar wind) or 40,000 km $\\rm {s}^{-1}$ (i.e.",
"$\\sim $ 1 GeV SEPs - relativistic effects not included).",
"This IMF case was also run for incident electrons with a mean velocity of 200 km $\\rm {s}^{-1}$ or 400 km $\\rm {s}^{-1}$ at each anomaly.",
"Total densities and fluxes at the surface were computed by distributing the particles, in a weighted manner, on to a grid with the same resolution as the magnetic field data, and summing over the collected particles.",
"Densities and fluxes were normalized so that the super-particle density in the initial launch region corresponded to 5 particles $\\rm {cm}^{-3}$ , nominal solar wind densities at 1 AU."
],
[
"Swirl Identification and Mapping",
"Mapping and spectroscopic analysis of the swirls used data from Clementine, Lunar Reconnaissance Orbiter (LRO) cameras and Global Lunar Digital terrain model (GLD100) [[40]].",
"This data was supplemented, during analysis, with OH abundances measured by the Moon Mineralogy Mapper (M$^3$ ) on Chandrayaan-1 to ensure consistency with previous results [[20]].",
"Clementine ultraviolet-visible (UV-VIS) and near-infrared (NIR) DIMs [[30]] were resampled to 100 m/pixel, combined into seamless 11-band images cubes, and the empirically-derived correction factors of [25] (USGS Clementine NIR global mosaic, available at http:// astrogeology.usgs.gov/Projects/ClementineNIR/) were applied.",
"The cubes were then were processed, mosaiced, projected (simple cylindrical), and co-registered to match the magnetic field maps using the Environment for Visualizing Images (ENVI).",
"With the exception of Marginis, the basemap, upon which the swirls are outlined for each swirl region (Figures REF g, REF g, REF g, REF e), is a simulated true (sim-true) color image generated from Clementine data (red = 900 nm, green = 750 nm, blue = 415 nm).",
"The Clementine data for Marginis suffered too many gaps in coverage.",
"Therefore, the basemap for Marginis (Figure REF e) used data from LRO's Wide Angle Camera (WAC).",
"Lunar swirls can be difficult to unambiguously identify, and more so to assign boundaries to, owing to their often diffuse nature, range in shapes and sizes, and contrast against the terrain on which they occur.",
"Although the swirls are high albedo, so usually easily distinguished against the dark background of the maria, they tend to blend into the surrounding terrain when they overly bright highlands material.",
"Therefore, in addition to band albedo images, we generated spectral parameter (SP) images from both Clementine and M$^3$ data to facilitate mapping the swirls and to identify broad spectral characteristics of the surface that coincide with the modeling results.",
"A spectral parameter utilizes the spectral features that are specific to an attribute of scientific interest in an algorithm in order to accentuate that attribute.",
"For example, the wavelength and depth of absorption features can be used to identify specific minerals; the peak albedo and slope of the entire spectral continuum can be used to estimate maturity.",
"A spectral parameter image is created by applying such an algorithm to each pixel of a spectral image to derive the spatial context of the desired feature, or parameter.",
"Several different SP maps were needed to outline the swirls, because different SPs accentuate different spectral attributes of swirls, as well as other geologic features that share that spectral attribute.",
"No single SP has been found that is unique to the swirls.",
"For example, the swirls are optically immature so appear bright in an optical maturity parameter (OMAT) map [[24]], however, so does ejecta from fresh impact craters.",
"In addition, [20] showed that the swirls are depleted in OH relative to their surroundings using OH abundance maps generated from M$^3$ mosaics.",
"This makes the relative OH abundance parameter a strong swirl identifier, although not absolute, since OH abundance also varies as a function of the angle between the Sun and the surface (time of day, slope, latitude, etc.)",
"[[31], [27]].",
"However, considered collectively the various SP maps aid in identifying swirls better than any individual image."
],
[
"Mare Ingenii",
"Figure REF shows the results for the swirl region at Mare Ingenii.",
"The swirls are outlined in light cyan based on optical imagery in Figure REF g. These outlines are shown, in black, overlain on a proton density map (Figure REF c), and an accumulated proton flux map (Figure REF e), for the baseline case to compare model results with the locations of the high concentration of swirls.",
"The strongest surface magnetic field ($\\sim 35^{o}$ S and $160^{o}$ E, Figure REF a) is seen to correspond with low particle densities at the surface (Figure REF d) and a void in the flux (Figure REF e).",
"Surrounding the void regions, for both density and flux, are regions with enhanced density and flux.",
"The flux and density at the surface when no magnetic field is present is approximately $2.5 \\times 10^{10}$ particles $\\rm {cm}^{-2}$ $\\rm {s}^{-1}$ and 18 particles $\\rm {cm}^{-3}$ .",
"The reduction in the flux in the void regions is primarily due to the reduction in density.",
"And while the speed of the particles that do impact in and around this portion of the anomaly is reduced by approximately 5%, the velocity provides a more complete picture.",
"In the central portion of the anomaly, the component of the velocity perpendicular to the surface decreases by approximately 75 - 100 km $\\rm {s}^{-1}$ while the magnitude of the components parallel to the surface increase from approximately zero to 75 km $\\rm {s}^{-1}$ (as compared to when no magnetic field is present).",
"Central to the void region at $\\sim 35^{o}$ S and $160^{o}$ E is the portion of the lunar swirl with the highest optical albedo (Figure REF g).",
"This corresponds with the brightest, bluest (flat spectral continuum), and most optically immature swirl surface at Ingenii.",
"Both to the north and south of this void are regions of enhanced surface flux and density.",
"It is harder to correlate these regions of enhanced flux with dark lanes purely from the optical image alone in part because these locations are coincident with the rims of the mare-filled craters Thompson and Thompson M, which, being rich in the plagioclase, cannot darken like the dark lanes on the maria.",
"The simulations begin to describe the interactions that occur between the particles and the magnetic field that are pattern manifested as complex patterns of bright and dark on the surface.",
"This is demonstrated by comparing the simulation results and swirl outlines within Thompson Crater, where the void in the proton flux is coincident with a group of swirls.",
"Unfortunately, the simulations stops short of describing the intricacy of the dark lanes observable in the optical images due to variations from high to low flux/density regions with scale sizes much larger than the scale sizes of the swirls.",
"This is a consequence of the coarser resolution of the model magnetic field data, as the resolution of the optical image in Figure REF g (100m/pixel) is much smaller than the resolution of the observed magnetic field (> 10s of km).",
"The highest surface impact density and flux is associated with the region of moderate magnetic field around $\\sim 25^{o}$ S and $170^{o}$ E. This peak magnetic field in this region is about 70% that of the anomaly at $\\sim 35^{o}$ S and $165^{o}$ E, but it is less localized.",
"It is a region in which the components of the magnetic field both perpendicular and parallel to the surface are fairly uniform over the whole extended region.",
"This means that particles incident from above this whole region will be deflected around it, translating into a higher density, as particles accumulate from over a more extended than that for the stronger anomaly.",
"The region of highest density and flux is co-located with a darkened region, surrounded by swirls (Figure REF g).",
"Figures REF b and REF d show a comparison of the density impact maps for electrons and protons, with the same incident velocity.",
"On the order of 80% of the electrons were deflected away from the surface, while, at most, a few percent of the protons did not eventually impact the surface somewhere.",
"What this means, when considering the system as a whole, is that as the solar wind approaches the magnetic anomaly, the electrons will begin to be deflected or reflected, due to their lower mass (Figure REF b).",
"The ions, with their heavier mass, will continue towards the surface.",
"An electric field will then be created by this charge separation.",
"This electric field will be in the opposite direction of the incident flow, and will slow the ions as a positive charge will want to move in the direction of the electric field, thus enhancing the deflection caused by the anomalous magnetic field.",
"The electrons will also feel the effects of the electric field and be pulled closer.",
"The net effect though will be stronger deflection of the ions than when looking at the individual particle tracking alone, and those ions that do impact the surface will have a lower velocity than when the influence of the the electrons is ignored.",
"This has been observed by Kaguya, measuring both protons and alpha particles being slowed, heated and reflected by the magnetic anomalies, while the electrons were accelerated towards the surface [[38]].",
"We can still use the particle tracking to get an estimate of both the influence of the magnetic field alone (helping to gauge how much of a role the electric fields play in deflecting particles at magnetic anomalies) and the level of complexity in the vector magnetic field data required to explain the fine detail seen in the swirls, from actual swirls patterns to dark lanes near the swirls.",
"For most of the Moon, our only measurement of surface magnetic fields comes from electron reflectometery measurements, which only return total magnitude, not vector fields [[28]].",
"Besides being able to deflect particles for slow solar wind speeds, the results suggest that the Mare Ingenii can also deflect nominal and fast solar wind particles.",
"Figures REF f and REF h show the surface density maps for nominal solar wind speed and a fast solar wind, respectively.",
"The surface density map for mid-range SEPs is shown in Figure REF a. Deflection of the faster solar wind particles around the strongest portion of the anomalous region still occurs ($\\sim 35^{o}$ S and $160^{o}$ E, of Figure REF h), but not as effectively as the baseline case, with a density on the order of 10 particles $\\rm {cm}^{-3}$ in the main void region, as opposed to approximately 2-4 particles $\\rm {cm}^{-3}$ in the baseline case (Figure REF d).",
"The density of solar wind particles impacting the surface around the strongest region is higher for both faster solar wind cases (Figures REF f and REF h) than the baseline case as those particles get much closer to the surface before they begin to be deflected, and are thus localized in their impact.",
"The density of particles impacting the surface surrounding the secondary anomaly, at $\\sim 25^{o}$ S and $170^{o}$ E, exhibits this same characteristic.",
"As the initial speed of the particles increases from 200 km $\\rm {s}^{-1}$ to 400 km $\\rm {s}^{-1}$ to 2000 km $\\rm {s}^{-1}$ , the regions with the highest impact densities transitions to regions more adjacent to the anomalies.",
"This is also associated with the fact that, with increased speed (and those kinetic energy), the particles are being deflected less before they impact the surface.",
"This behavior is in agreement with the idea that the dark lanes are locations of increased weathering as particles are preferentially deflected into those regions [[21]].",
"The impact density (Figure REF a) and flux for SEP particles show no apparent influence by the anomalies on the incident particles, when compared to a run with the same incident velocities but no anomalous magnetic field present (not shown).",
"Examination of the components of the particle velocity show some influence though by the anomalies.",
"Figures REF b and REF c show the components of the velocity tangential to the surface for SEP particles.",
"The two strongest anomaly regions cause some deflection in the incident particles by converting some of the kinetic energy directed toward the surface into kinetic energy parallel to the surface, as is evident by magnitudes parallel to the surface that are non-zero.",
"As the velocity components tangential to the surface are, at most, 0.75% of the incident velocity, it is not enough to be noticeable in the density or flux maps."
],
[
"Reiner Gamma",
"Figure REF shows the results for the Reiner Gamma anomaly region.",
"As for Ingenii, the Reiner Gamma swirls are outlined in cyan based on optical imagery in Figure REF g. These outlines are shown, in black, overlain on the proton density map (Figure REF c), and proton flux map (Figure REF e) for the baseline velocity, to compare model results with the locations of the high concentration of swirls.",
"For this case, spectral imagery indicating optical maturity (OMAT) from Clementine is also available and shown in Figure REF h, with the swirls marked in white.",
"For the Reiner Gamma case, there is a reduced density and flux near the strongest portion of the magnetic field and an enhanced density and flux surrounding the anomaly (Figures REF c and REF e).",
"The flux to the surface in the region of peak field strength at Reiner Gamma is not close to zero though, like at the regions of strongest magnetic fields of the Mare Ingenii anomaly.",
"Instead the flux at the peak field region at Reiner Gamma is on the order of $5 \\times 10^{9}$ particles $\\rm {cm}^{-2}$ $\\rm {s}^{-1}$ , approximately 1/5th the flux when no anomalous field is present.",
"The density within this region is reduced by approximately an order of magnitude.",
"Like that for Mare Ingenii, the regions surrounding the central magnetic anomaly experience an enhanced flux, evident by the regions of yellow and orange to the north and south of the strong central magnetic field.",
"This deflection of particles around the anomaly also leads to a higher density adjacent to this same region.",
"These densities on the order of 25 particles $\\rm {cm}^{-3}$ are an enhancement above what is seen when no anomaly is present.",
"This behavior of decreased density near the central eye of the Reiner Gamma anomaly and an increased flux surrounding the eye, is qualitatively similar to that seen for the full particle simulations presented in the companion paper [[1]], in which the central eye of the Reiner Gamma anomaly was modeled as both a single dipole with the moment in three different orientations relative to the surface.",
"For the cases with the moment parallel to the surface, the enhancement in the density surrounding the eye in the full particle simulations is a factor of 3-5 times background, as opposed to slightly less than a factor of 2 for the results in Figure REF c. Another difference is that by using a model of the full anomaly for the particle tracking, density enhancements are only seen to the north and south of the eye, whereas the assuming a dipole magnetic field for the full particle simulations results in density enhancements surrounding the entire eye, albeit not symmetrically.",
"With a magnitude of 58 nT at the center of the magnetic anomaly, the Reiner Gamma anomaly is weaker than the strongest anomaly region at Mare Ingenii (at 74 nT) but comparable to the secondary region at Mare Ingenii (at 54 nT).",
"The impact density at the center of the secondary region at Mare Ingenii (2 - 5 particles $\\rm {cm}^{-3}$ ) is comparable to that at the center of the Reiner Gamma anomaly, as is the flux.",
"In this case, the speed of the deflected particles is modified only negligibly, while the components of the velocity towards the surface and along the East-West direction show some modification.",
"Particles are deflected up and to the left or down and to the right.",
"The very low albedo of the whole region surrounding the swirl means it is not possible to determine if enhanced weathering occurs around the anomaly from the optical images alone, as would be predicted by the particle tracking.",
"Figure REF h shows the OMAT image.",
"The regions of highest flux and density for the particle tracking are noted by the red arrows in Figure REF h. Enhanced weathering is not apparent in those regions, on the same scale as the maturity of the highlands material in the lower left corner.",
"The dark lanes contained within the eye of the central anomaly (obscured by the swirl mappings in Figure REF h, but visible in the supplemental images) have an OMAT appearance similar to the regions surrounding the central eye of the anomaly.",
"The behavior of the particles as the speed increases in very similar to that seen for Mare Ingenii.",
"The density of impacting particles surrounding the anomaly, for the 400 km $\\rm {s}^{-1}$ case is higher than for the 200 km $\\rm {s}^{-1}$ , at 35-40 particles $\\rm {cm}^{-3}$ ) (Figure REF d).",
"With the reduced efficiency of the Reiner Gamma region in deflecting the slowest solar wind speeds relative to Mare Ingenii (evident in that even at the location of the strongest fields at Reiner Gamma, the flux of particles to the surface is non-zero), it is not surprising that particle tracking indicates that the Reiner Gamma anomaly has only a modest influence on the fast solar wind particles (Figure REF f).",
"The influence of the Reiner Gamma anomaly on the SEP range particles is only apparent when looking at the components of the velocity (not shown).",
"Electrons impacting the surface are confined to surrounding the anomaly region (Figure REF b).",
"The efficiency of the Reiner Gamma anomaly in deflecting electrons is indicated by the low peak densities, suggesting this region would significant would experience even greater deflection of the protons from the surface."
],
[
"Gerasimovich",
"The particle tracking results show that the Gerasimovich magnetic anomaly is also able to deflect incident protons, but not as effectively as Mare Ingenii or Reiner Gamma (Figure REF ).",
"The format of Figure REF is similar to Figure REF with both optical (Figure REF g) and spectral information from Clementine available (Figure REF h).",
"The swirl contours are shown in cyan on Figures REF g and REF h, and in black on Figures REF c and REF e. At Gerasimovich, the flux at the region of strongest magnetic field is on the order of $5-7 \\times 10^{9}$ particles $\\rm {cm}^{-2}$ $\\rm {s}^{-1}$ .",
"Similarly, the density at the locations of peak magnetic field is reduced, but to a lesser extent than at Mare Ingenii, as it does not approach zero.",
"The density of approximately 7-10 particles $\\rm {cm}^{-3}$ is only 50-60% that of when no anomalous magnetic field is present.",
"Using observations of energetic neutrals coming from the lunar surface made by the Chandrayaan-1 mission, [43] estimated the shielding efficiency the Gerasimovich anomaly to be between 5% and 50% over regions with magnetic field strength at 30 km between 5 nT and 13 nT, respectively, at low solar wind dynamic pressures.",
"The reduction in surface flux over the strongest portions of the Gerasimovich anomaly seen in the particle tracking is thus comparable to observed the upper end of observed values.",
"That the minimum density seen at the surface is greater than that seen for the Mare Ingenii cases can not be explained by surface magnetic field strength alone.",
"The peak magnetic field strengths at Gerasimovich are comparable to those at Mare Ingenii, where the strongest anomalous magnetic field resulted in a near zero flux of particles at the location of peak magnetic field.",
"The peak magnetic field strength at Gerasimovich is also approximately 20% stronger than the peak field strength at Reiner Gamma, which has a comparable flux and slightly lower density at the center of the anomaly.",
"The swirls at Gerasimovich are strongly co-located with the peak magnetic field strengths and minimums in the impact density and flux (Figures REF c and REF e).",
"In the OMAT image for Gerasimovich (Figure REF h), an extended, diffuse region can be observed surrounding the swirls (one portion of which is indicated by a red arrow).",
"The lighter coloring in OMAT indicates that the diffuse region is of higher maturity than the swirls, but lower maturity than the broader, surrounding area.",
"This region matches the shape of the extended magnetic field (Figure REF a) and is surrounded by regions of high impact density and flux (with a red arrow indicating those region in-between the two main swirl groups in Figures REF c and REF e).",
"The high flux and density region between the two peaks in the magnetic field magnitude is coincident with a dark region in the OMAT map, which would likely appear even darker were it not for the occurrence of a fresh impact crater in that location (also red arrow in Figure REF h).",
"The reduced efficiency deflecting the slowest particles translates to a reduced influence on faster incident particles.",
"Gerasimovich can deflect 400 km $\\rm {s}^{-1}$ (Figure REF d) but the peak densities adjacent to the anomaly, at 30 particles $\\rm {cm}^{-3}$ , are the smaller than for Mare Ingenii and Reiner Gamma.",
"With correspondingly higher densities in the anomalous regions, this means that the particles are experiencing less deflection by the magnetic field.",
"Gerasimovich has little influence on the impact density or flux for the fast solar wind case (Figure REF f), and no influence on the impact density or flux for the SEP case.",
"Like the other anomalies though, modification in the velocity components parallel to the surface does occur for both the fast solar wind and SEP cases, but not to the same extent as more efficient anomalies."
],
[
"Marginis",
"With a peak surface magnetic field magnitude of approximately 30 nT, the Marginis anomaly is the weakest of the five regions investigated in this study.",
"The results is that while, particles are deflected, the anomaly is much less effective in doing so.",
"For all of the other four regions studied, at least some portion of the anomalous region experiences a surface flux and density of at least half that compared to an unaltered region.",
"This is not true for Marginis.",
"Figure REF shows the results for the slow (200 km $\\rm {s}^{-1}$ ) particle case as compared to the optical and OMAT measurements.",
"For Marginis, the difference in peak densities and reduced densities is a factor of two.",
"For the flux, the relative difference is also a factor of two.",
"The reductions are only a by a factor of 1.3, as compared to when no anomalous magnetic field is present.",
"The reduced surface density and flux are co-located with regions of magnetic field strengths in excess of 15 nT.",
"Only the X component of the velocity of the incident particles shows any modification by the anomalies.",
"This component is perpendicular to the largest tangential component of magnetic field.",
"Because of the inefficiency of the region in deflecting the slow solar wind plasma, results for higher incident plasma speeds are not shown.",
"Mare Marginis is a region of significant interest because of its prominent lunar swirls both on mare and highland soils.",
"Marginis was chosen for this study to compare results of a similar region of complex swirl patterns, Mare Ingenii, which was also analyzed in [21].",
"It cannot be ignored that the pattern of swirls at Marginis appear to emanate from Goddard A, a fresh 11 km crater, which ejected bright highlands material over the surrounding mare and highlands regions.",
"The initial mapping of swirls at Marginis also used the quasi-slope map generated from the LRO WAC 643 nm normalized reflectance map and the LRO GLD100 map.",
"The intricate pattern of the swirls at Marginis necessitated the use of LRO’s Narrow Angle Camera (NAC) [[36]] in some locations to map the swirls in detail.",
"Although some swirls continue in the highlands east of this region, mapping was restricted to the mare and nearby highlands for the purpose of this study (Figures REF e and REF f).",
"Outlines of the mare and swirl overlaid on both the surface flux (Figure REF c) and density (Figure REF d).",
"Swirls were found across the highlands and some of the mare regions, in locations where the particle tracking shows a decreased proton flux.",
"Regions of increased proton flux tended to be lacking in swirls, while regions of decreased proton flux tended to be rich in swirls.",
"The swirls were quite obvious in the regions of reduced proton flux in both the high-FeO mare and low-FeO highlands.",
"Locations of significant interest are Goddard basin (at 14.8N and 89.0E) and Ibn Yunus basin (at 14.1N and 91.1E), both of which are vast regions that lack any observable swirls despite being within the same radial distance from Goddard A as other locations that exhibit prominent swirl patterns (Figure REF e).",
"The particle tracking predicts a high proton flux across both mare-filled basins, which explains the lack of swirls within the basins (Figure REF c).",
"The target mare soils are rich in FeO with which to create nanophase iron, causing the Goddard and Ibn Yunus basin floors to be weathered more efficiently than the nearby highlands.",
"This effect is observable in the OMAT map (Figure REF f) where Goddard and Ibn Yunus appear dark (i.e.",
"mature), while the surrounding highlands appear brighter (i.e.",
"immature) and swirled.",
"Although some of its ejecta is confused with the swirls, across Goddard and Ibn Yunus basins, as well as a few locations in the highlands, where Goddard A ejecta can be observed to have a typical impact ejecta pattern, that is, not swirled, and the ejecta appear more mature than radially proximate swirls.",
"This is strong supporting evidence that the ejecta, which landed where the simulations predict the proton flux is high, are being weathered at an accelerated rate while those that landed on highland swirls are being preserved, further supporting the solar wind magnetic deflection model, even at this weaker anomaly.",
"When comparing the optical images for Ingenii and Marginis, it would be tempting to conclude that the magnetic fields would be similar, based upon a comparison of swirl extent and contrast with surrounding terrain.",
"As the above analysis shows though, it is difficult to explain the swirls based upon the limited efficiency of the magnetic field alone in deflecting incident particles, as compared to other anomalies."
],
[
"Northwest of Apollo",
"The NW of Apollo region is comprised of two anomalies with comparable surface magnetic field strength.",
"Both anomalous regions lead to reduced density and flux when compared to the case with no magnetic field present.",
"The anomaly in the upper right corner ($\\sim 15^{o}$ S and $155^{o}$ W) (Figure REF a), while weaker in total magnetic field magnitude than the anomaly at $\\sim 25^{o}$ S and $165^{o}$ W, is more efficient in reducing the flux of particles to the surface (Figure REF c).",
"In this region the particle density is approximately 75% less than when no magnetic field is present, as opposed to the 60% reduction for the region at the center of the strongest anomaly.",
"The components of the magnetic field show that the North-South components of the magnetic field at the secondary anomaly are 1.5 to 2.0 times larger than the North-South components for the strongest anomaly.",
"The East-West components at the primary and secondary anomaly are comparable.",
"The stronger magnetic field magnitude at the primary anomaly comes from a larger vertical component.",
"This may partially explain why that portion of NW of Apollo region is less effective in deflecting the incoming solar wind.",
"The east-west component of the magnetic field for the strongest anomaly in the Mare Ingenii region are comparable in magnitude to the radial component.",
"This aspect will be discussed further in the next section.",
"The high albedo of the ejecta rays from the crater Crookes, to the north of the anomaly region, make swirl identification difficult from the optical images, but the particle tracking can assist with refining the regions to look at.",
"The white ovals in Figure REF c mark the regions of low impact flux.",
"They are replicated in the optical image (Figure REF e).",
"The albedo inside the two ovals between $\\sim 20^{o}$ S and $25^{o}$ S does appear to be lighter than the surrounding regions.",
"That this higher albedo is perpendicular to the ejecta rays suggests it may be associated with the anomalous region instead.",
"The extended structure of the magnetic field throughout the region means the density of particles impacting the surface is more complex than the previous cases.",
"The near equivalent magnitude of the primary and secondary anomalies deflects particles in toward the region between the two, but a weak anomaly near the center of the region is still strong enough to lead to moderate deflection.",
"Thus the highest densities and fluxes are in the region straddling the three anomalies.",
"This structure holds for the faster solar wind cases as well (Figure REF f and Figure REF a) It is also visible in the modification of the tangential velocity components for the SEP case (Figures REF b and REF c) but not the density or flux.",
"The electrons show more deflection than protons with a similar speed, but are still able to impact near the central portion of the anomalous region, in between the two regions of strongest magnetic fields (Figure REF b).",
"The surface density of impacting electrons is also the highest for NW of Apollo, when compared to the previous three anomalies, investigated.",
"That the electrons can access the central portion of NW of Apollo is likely associated with the more complicated magnetic field, which is also manifest in the complex impact patterns seen for the protons.",
"With more access by the electrons, the electric fields generated by the charge separation between protons and electrons (which is a function of the separation distance) should be smaller than for regions where the particle tracking shows little access by the electrons.",
"The protons will be slowed less by the smaller electric field, potentially increasing access to the surface.",
"Swirls in NW of Apollo have been identified (e.g.",
"[3]), but had not been mapped, likely because this heavily cratered highlands region has little contrast between swirl and background albedo, and the complicated topography causes albedo anomalies across the region.",
"Mapping swirls for NW Apollo proved more difficult than other regions so we generated a quasi-slope map by contrasting the LRO WAC 643 nm normalized reflectance map with the LRO GLD100 topographical map [[40]].",
"This provided high albedo swirls.",
"Even so, only about half of the swirls shown here were found with this method.",
"The rest were found using particle tracking as a guide, doubling the number of identified swirls.",
"All of the swirls mapped using the combined set of techniques are shown in in Figure REF e. This highlights that quick particle tracking (as opposed to more computationally expensive full particle simulations) can be a useful tool in helping refine a search area when mapping swirls at other anomalies in which swirl identification is complicated by the surrounding material."
],
[
"Discussion and Conclusions",
"One problem in understanding the plasma physics occurring near the swirls from observations is that it is not possible to deconvolve all of the different processes to understand the relative importance of each, and ultimately resolve the origin of the swirls - is it deflection of the weathering solar wind or the transport of charged dust by the subsequent electric fields [7], or are both occurring?",
"This has left open questions with regard to how important the electric fields are in deflecting incident plasma and if the anomalous magnetic field alone can deflect ions, due to the small scale size of the anomalous regions relative to the proton gyroradius.",
"By using particle tracking only, the results presented in this paper can shed light on how effective the anomalous magnetic field alone is in deflecting incident protons of varying energies and what aspects of that anomalous magnetic field are most important.",
"The results from this study further highlight that the small-scale size of the lunar magnetic anomalies do not prevent them from at least partially deflecting incoming solar wind particles, from slow to fast solar wind speeds.",
"And while none of the anomaly regions investigated could deflect moderate SEP-energy particles in such a way produce local impact density variations, all of the anomalies could influence the velocities of moderate SEP particles.",
"The effectiveness of each anomaly region in deflecting particles does not scale exactly with peak surface magnetic field strength, as one might expect from magnetic mirroring.",
"It is important to note that while the process of magnetic mirroring does not depend on the direction of the converging magnetic field, just the magnitude, it does assume that the scale size of the magnetic mirror region is larger than the gyro-radius of the incident particle (which is a function of the kinetic energy of the incident particle).",
"The peak surface magnetic field strengths on the order of 50-70 nT, correspond to a gyro-radius of 30-40 km for the slowest speed investigated and 300-400 km for the fast solar wind case.",
"A magnetic field strength of 25 nT corresponds to a gyro-radius of 84 km for the slowest case and 1670 km for the fast solar wind case.",
"The anomaly region most effective in deflecting particles at Mare Ingenii, has magnetic field 25 nT or greater spread out over approximately 40 km by 30 km.",
"The secondary anomaly at Mare Ingenii, while weaker in peak magnitude, has magnetic field 25 nT or greater spread out over approximately 55 km by 30 km.",
"While Gerasimovich has a peak surface magnetic field strength comparable to Mare Ingenii, the anomalous magnetic field is more localized to single anomaly region approximately 50 km by 40 km.",
"The Reiner Gamma region is the most localized, with a region of magnetic field 25 nT or greater that is circular in shape, with a diameter of approximately 20 km.",
"The local anomaly in the NW of Apollo region that is most effective in deflecting solar wind particles has magnetic field 25 nT or greater over a 30 km by 25 km region.",
"The other anomaly in the region, that is less effective in deflecting particles, is larger at approximately 50 km by 35 km.",
"Marginis has a region of magnetic field on the order of 15 nT spread out over approximately 60 km by 50 km, but only a very small area exceeding 25 nT.",
"That 1) the effectiveness of Gerasimovich in deflecting particles is comparable to Reiner Gamma, 2) the larger anomaly in NW of Apollo is less effective than the smaller anomaly, and 3) Marginis can deflect particles at all with sufficient efficiency to result in a complex region of swirls, all indicate that another quality of the magnetic field, namely coherence, is important as well.",
"The coherence of the magnetic field can be thought of as a measure of how much the magnetic field changes orientation over a given spatial distance.",
"For example, Reiner Gamma has a peak surface magnetic field strength $\\sim $ 80% that of Gerasimovich but is highly localized and the region is circular in shape.",
"The size of the gyro-radius of the incident particles will sense both the coherence of the anomalous magnetic field and the scale size of the magnetic anomaly, as an incoherent anomaly region will not have large regions of converging magnetic field, due to the field changing orientation over small distances.",
"Coherence can be assessed by looking at the coefficients of the spherical harmonic expansion used to model the anomalous magnetic field.",
"The larger the dipole term is relative to the higher order terms indicates a more coherent magnetic field.",
"Unfortunately the model used to recreate the anomalous magnetic field did not allow for this type of analysis.",
"Instead other aspects of the magnetic field were used to estimate how coherent the magnetic field is in each region - fall off with altitude and component analysis (Figures REF and REF ).",
"The higher the order the moment in a spherical harmonic expansion, the faster the field, from that term, decreases with distance from the source.",
"The first column in Figures REF and REF shows the magnitude of the anomalous magnetic field in each region at 40 km.",
"The analysis of the impact maps indicates that Mare Ingenii is the most effective in deflecting incoming solar wind, Marginis and Gerasimovich the least effective, with Reiner Gamma and NW of Apollo somewhere in between.",
"The ranking of NW of Apollo depends on which portion of the anomalous region one looks at.",
"This ranking can be partially explained by looking at the magnetic field above the surface.",
"While Gerasimovich has some of the largest surface magnetic field strengths, the magnetic field for Gerasimovich is the weakest at 40 km.",
"And while Marginis has the lowest surface magnetic field magnitudes, the field strength at 40 km is still 15% that of the surface field strength (for comparison, the peak magnitude at 40 km above Gerasimovich is 5% that of the peak surface field strength).",
"The fast fall-off of the field at Gerasimovich, helps explain why the anomaly is not as effective as Reiner Gamma, in deflecting incoming solar wind, even though the surface field strength at Gerasimovich is stronger.",
"NW of Apollo, in contrast, has the strongest fields at 40 km, even though it has the second weakest surface field of the five regions studied.",
"To explain why it is not the most effective in deflecting particles, and why Reiner Gamma seems more effective than it should be given its weak magnetic field strength at 40 km, requires looking at the details of the magnetic field.",
"When looking at the components of the magnetic field, a perfect dipole with the moment perpendicular to the surface would look very similar to the images also for Reiner Gamma (Figures REF f - REF h) - radial magnetic field exiting at the center, with oppositely directed radial field at the edge (Figure REF h), and anti-symmetric tangential magnetic fields centered about the radial magnetic field (Figures REF f - REF g).",
"The very dipole-like nature of the Reiner Gamma anomaly helps explain why its effectiveness in deflecting particles is similar to Gerasimovich, even through its peak surface field strength is weaker.",
"It also explains why the field magnitude at 40 km above Reiner Gamma is larger than above Gerasimovich.",
"Figures REF b - REF d show that the strongest anomaly in the Mare Ingenii region has dipole-like characteristics while the field within Gerasimovich is more complex.",
"At NW of Apollo, only the anomaly around $\\sim 15^{o}$ S and $155^{o}$ W, is dipole-like, and that region is the most effective in deflecting particles.",
"Thus the more dipole-like (or coherent) the anomalous magnetic field (both in terms in the components and the decrease in strength with distance), the more effective it is at deflecting particles, when all other aspects are the same.",
"One of the tangential components (Y) of the magnetic field at Marginis is comparable in strength to the same component at Ingenii (Figure REF g vs. REF c), but the other components at Marginis are much weaker than those same components at all the other anomalies.",
"That the magnetic field at Marginis does not fall off as quickly as Gerasimovich and that the surface magnetic field is primarily parallel to the surface helps explain the existence of the swirl patterns at Marginis, but not the extensive nature.",
"It is important to note that one issue the results highlight is that the resolution of the model magnetic field used in this study is typically much coarser than the optical images.",
"This becomes most apparent when analyzing the Reiner Gamma and Gerasimovich anomalies.",
"The swirl regions in both of these cases span only a few degrees in lateral extent, and corresponds to only 20 grid points in the simulation magnetic field.",
"The true resolution of the magnetic field model is comparable to the lowest altitude of the satellite making the observations.",
"The data used to generate the magnetic field models used for this work came from Lunar Prospector, which produced global magnetic field maps down to 30 km and made local magnetic field measurements down to 20 km [M. Purucker, 2014 private communication].",
"This means that the actual magnetic field observations are even courser than the model magnetic field.",
"The lower resolution of the magnetic field data will act to average out more complex (i.e.",
"small-scale) structure, making the magnetic field appear to be more coherent than it actually is.",
"Recent modeling using low altitude ( 10 km) observations by Kaguya [41], in addition to the low altitude Lunar Prospector observations, has allowed for the generation of surface vector magnetic field maps with resolution of $0.2^{o}$ for some regions.",
"Observable differences on the order of many 10s of nT were seen for the Reiner Gamma region.",
"This still may not be sufficient to explain some of the smallest swirl features.",
"The work presented here shows that impact maps for 3D particle tracking at lunar magnetic anomalies can be correlated with observations of the the lunar swirls in the same regions.",
"Although the small-scale swirl features cannot be matched with the results of the simulations due to the disparity in the spatial resolutions of the imaging data and the magnetic field data, the simulations do show that protons are consistent with the spatial pattern of the swirls; that is, protons are deflected away from the locations of the high-albedo swirls and onto inter-swirls or swirl-adjacent locations.",
"This is consistent with the conclusions of [21], [20] that solar wind ions are the dominant agents responsible for the creation of nanophase iron, which is largely responsible for the spectral characteristics of space weathering and optical maturation [e.g.",
"[10], [29]].",
"On the swirls, the decreased proton flux slows the spectral effects of space weathering (relative to non-swirl regions) by limiting the nanophase iron production mechanism almost exclusively to micrometeoroid impact vaporization/deposition.",
"Immediately adjacent to the swirls, maturation is accelerated by the increased flux of protons deflected from the swirls.",
"Our results show that the shape and strength of the magnetic anomalies, independent of an induced electric field, can explain the deflection and focusing of incident protons at solar wind velocities at the distance of the Earth-Moon system.",
"Although this may not fully represent the intricacies of the interaction, it is an important result for understanding and further refinement of relevant models.",
"More work needs to be done though to correlate the small-scale details of the swirls and dark lanes with impact maps.",
"While the next step in the process involve conducting fully self-consistent 3D particle simulations of the solar wind interacting with realistic anomalous magnetic fields, the above analysis suggests that may not be enough to explain the details of the features seen at swirls.",
"A companion paper [[1]] shows results from full 3D particle simulations using realistic proton to electron mass ratios for both a single dipole and a double dipole system.",
"Due to the heavy computational load associated with such simulations, only the central portion of the Reiner Gamma region was investigated, but this technique works well for the central portion of the Reiner Gamma region as the anomalous magnetic field appears to be very coherent and dipole-like.",
"The above analysis indicates that even with full particle simulations of more extended regions around anomalies, the results will most likely show a disconnect with the intricate features seen at swirls until much higher resolution vector magnetic field measurements are made near the surface."
],
[
"Acknowledgments",
"The authors would like to thank Dr. Michael Purucker and Dr. Joseph Nicholas for providing us with the source code to calculate the lunar magnetic fields.",
"This research was supported by the NASA LASER grant #NNX12AK02G."
]
] | 1605.05778 |
[
[
"Linearized GMM Kernels and Normalized Random Fourier Features"
],
[
"Abstract The method of \"random Fourier features (RFF)\" has become a popular tool for approximating the \"radial basis function (RBF)\" kernel.",
"The variance of RFF is actually large.",
"Interestingly, the variance can be substantially reduced by a simple normalization step as we theoretically demonstrate.",
"We name the improved scheme as the \"normalized RFF (NRFF)\".",
"We also propose the \"generalized min-max (GMM)\" kernel as a measure of data similarity.",
"GMM is positive definite as there is an associated hashing method named \"generalized consistent weighted sampling (GCWS)\" which linearizes this nonlinear kernel.",
"We provide an extensive empirical evaluation of the RBF kernel and the GMM kernel on more than 50 publicly available datasets.",
"For a majority of the datasets, the (tuning-free) GMM kernel outperforms the best-tuned RBF kernel.",
"We conduct extensive experiments for comparing the linearized RBF kernel using NRFF with the linearized GMM kernel using GCWS.",
"We observe that, to reach a comparable classification accuracy, GCWS typically requires substantially fewer samples than NRFF, even on datasets where the original RBF kernel outperforms the original GMM kernel.",
"The empirical success of GCWS (compared to NRFF) can also be explained from a theoretical perspective.",
"Firstly, the relative variance (normalized by the squared expectation) of GCWS is substantially smaller than that of NRFF, except for the very high similarity region (where the variances of both methods are close to zero).",
"Secondly, if we make a model assumption on the data, we can show analytically that GCWS exhibits much smaller variance than NRFF for estimating the same object (e.g., the RBF kernel), except for the very high similarity region."
],
[
"Introduction",
"It is popular in machine learning practice to use linear algorithms such as logistic regression or linear SVM.",
"It is known that one can often improve the performance of linear methods by using nonlinear algorithms such as kernel SVMs, if the computational/storage burden can be resolved.",
"In this paper, we introduce an effective measure of data similarity termed “generalized min-max (GMM)” kernel and the associated hashing method named “generalized consistent weighted sampling (GCWS)”, which efficiently converts this nonlinear kernel into linear kernel.",
"Moreover, we will also introduce what we call “normalized random Fourier features (NRFF)” and compare it with GCWS.",
"We start the introduction with the basic linear kernel.",
"Consider two data vectors $u,v\\in \\mathbb {R}^D$ .",
"It is common to use the normalized linear kernel (i.e., the correlation): $\\rho = \\rho (u,v) = \\frac{\\sum _{i=1}^D u_i v_i }{\\sqrt{\\sum _{i=1}^D u_i^2}\\sqrt{\\sum _{i=1}^Dv_i^2}}$ This normalization step is in general a recommended practice.",
"For example, when using LIBLINEAR or LIBSVM packages [6], it is often suggested to first normalize the input data vectors to unit $l_2$ norm.",
"In addition to packages such as LIBLINEAR which implement batch linear algorithms, methods based on stochastic gradient descent (SGD) become increasingly important especially for truly large-scale industrial applications [2].",
"In this paper, the proposed GMM kernel is defined on general data types which can have both negative and positive entries.",
"The basic idea is to first transform the original data into nonnegative data and then compute the min-max kernel [20], [9], [12] on the transformed data."
],
[
"Data Transformation",
"Consider the original data vector $u_i$ , $i=1$ to $D$ .",
"We define the following transformation, depending on whether an entry $u_i$ is positive or negative:This transformation can be generalized by considering a “center vector” $\\mu _i$ , $i=1$ to $D$ , such that $\\left\\lbrace \\begin{array}{cc}\\tilde{u}_{2i-1} = u_i - \\mu _i,\\hspace{7.22743pt} \\tilde{u}_{2i} = 0&\\text{if } \\ u_i >\\mu _i\\\\\\tilde{u}_{2i-1} = 0,\\hspace{7.22743pt} \\tilde{u}_{2i} = -u_i+\\mu _i &\\text{if } \\ u_i \\le \\mu _i\\end{array}\\right.$ In this paper, we always use $\\mu _i=0,\\ \\forall i$ .",
"Note that the same center vector $\\mu $ should be used for all data vectors.",
"$\\left\\lbrace \\begin{array}{cc}\\tilde{u}_{2i-1} = u_i,\\hspace{7.22743pt} \\tilde{u}_{2i} = 0&\\text{if } \\ u_i >0\\\\\\tilde{u}_{2i-1} = 0,\\hspace{7.22743pt} \\tilde{u}_{2i} = -u_i &\\text{if } \\ u_i \\le 0\\end{array}\\right.$ For example, when $D=2$ and $u = [-5\\ \\ 3]$ , the transformed data vector becomes $\\tilde{u} = [0\\ \\ 5\\ \\ 3\\ \\ 0]$ ."
],
[
"Generalized Min-Max (GMM) Kernel",
"Given two data vectors $u, v\\in \\mathbb {R}^D$ , we first transform them into $\\tilde{u}, \\tilde{v}\\in \\mathbb {R}^{2D}$ according to (REF ).",
"Then the generalized min-max (GMM) similarity is defined as $&GMM(u,v) = \\frac{\\sum _{i=1}^{2D} \\min (\\tilde{u}_i,\\ \\tilde{v}_i)}{\\sum _{i=1}^{2D} \\max (\\tilde{u}_i,\\ \\tilde{v}_i)}$ We will show in Section that GMM is indeed an effective measure of data similarity through an extensive experimental study on kernel SVM classification.",
"It is generally nontrivial to scale nonlinear kernels for large data [3].",
"In a sense, it is not practically meaningful to discuss nonlinear kernels without knowing how to compute them efficiently (e.g., via hashing).",
"In this paper, we focus on the generalized consistent weighted sampling (GCWS)."
],
[
"Generalized Consistent Weighted Sampling (GCWS)",
"Algorithm REF summarizes the “generalized consistent weighted sampling” (GCWS).",
"Given two data vectors $u$ and $v$ , we transform them into nonnegative vectors $\\tilde{u}$ and $\\tilde{v}$ as in (REF ).",
"We then apply the original “consistent weighted sampling” (CWS) [20], [9] to generate random tuples: $\\left(i^*_{\\tilde{u},j}, t^*_{\\tilde{u},j}\\right)\\ \\text{ and }\\ \\left(i^*_{\\tilde{v},j}, t^*_{\\tilde{v},j}\\right),\\ \\ j = 1, 2, ..., k$ where $i^*\\in [1,\\ 2D]$ and $t^*$ is unbounded.",
"Following [20], [9], we have the basic probability result.",
"Theorem 1 $\\mathbf {Pr}\\left\\lbrace \\left(i^*_{\\tilde{u},j}, t^*_{\\tilde{u},j}\\right) = \\left(i^*_{\\tilde{v},j}, t^*_{\\tilde{v},j}\\right)\\right\\rbrace = GMM({u},{v})$ Input: Data vector $u$ = ($i=1$ to $D$ ) Transform: Generate vector $\\tilde{u}$ in $2D$ -dim by (REF ) Output: Consistent uniform sample ($i^*$ , $t^*$ ) For $i$ from 1 to $2D$ $r_i\\sim Gamma(2, 1)$ , $c_i\\sim Gamma(2, 1)$ , $\\beta _i\\sim Uniform(0, 1)$ $t_i\\leftarrow \\lfloor \\frac{\\log \\tilde{u}_i }{r_i}+\\beta _i\\rfloor $ , $a_i\\leftarrow \\log (c_i)- r_i(t_i+1-\\beta _i)$ End For $i^* \\leftarrow arg\\min _i \\ a_i$ , $t^* \\leftarrow t_{i^*}$ Generalized Consistent Weighted Sampling (GCWS).",
"Note that we slightly re-write the expression for $a_i$ compared to [9].",
"With $k$ samples, we can simply use the averaged indicator to estimate $GMM(u,v)$ .",
"By property of the binomial distribution, we know the expectation ($E$ ) and variance ($Var$ ) are $&E\\left[1\\lbrace i^*_{\\tilde{u},j} = i^*_{\\tilde{v},j} \\ \\text{and} \\ t^*_{\\tilde{u},j} = t^*_{\\tilde{v},j}\\rbrace \\right] = GMM(u,v),\\\\&Var\\left[1\\lbrace i^*_{\\tilde{u},j} = i^*_{\\tilde{v},j} \\ \\text{and} \\ t^*_{\\tilde{u},j} = t^*_{\\tilde{v},j}\\rbrace \\right] =(1- GMM(u,v))GMM(u,v)$ The estimation variance, given $k$ samples, will be $\\frac{1}{k}(1- GMM)GMM$ , which vanishes as GMM approaches 0 or 1, or as the sample size $k\\rightarrow \\infty $ ."
],
[
"0-bit GCWS for Linearizing GMM Kernel SVM",
"The so-called “0-bit” GCWS idea is that, based on intensive empirical observations [12], one can safely ignore $t^*$ (which is unbounded) and simply use $\\mathbf {Pr}\\left\\lbrace i^*_{\\tilde{u},j} = i^*_{\\tilde{v},j}\\right\\rbrace \\approx GMM({u},{v})$ For each data vector $u$ , we obtain $k$ random samples $i^*_{\\tilde{u},j}$ , $j=1$ to $k$ .",
"We store only the lowest $b$ bits of $i^*$ , based on the idea of [18].",
"We need to view those $k$ integers as locations (of the nonzeros) instead of numerical values.",
"For example, when $b=2$ , we should view $i^*$ as a vector of length $2^b=4$ .",
"If $i^*=3$ , then we code it as $[1\\ 0\\ 0\\ 0]$ ; if $i^*=0$ , we code it as $[0\\ 0\\ 0\\ 1]$ .",
"We can concatenate all $k$ such vectors into a binary vector of length $2^b\\times k$ , with exactly $k$ 1's.",
"For linear methods, the computational cost is largely determined by the number of nonzeros in each data vector, i.e., the $k$ in our case.",
"For the other parameter $b$ , we recommend to use $b\\ge 4$ .",
"The natural competitor of the GMM kernel is the RBF (radial basis function) kernel, and the competitor of the GCWS hashing method is the RFF (random Fourier feature) algorithm."
],
[
" RBF Kernel and Normalized Random Fourier Features (NRFF)",
"The radial basis function (RBF) kernel is widely used in machine learning and beyond.",
"In this study, for convenience (e.g., parameter tuning), we recommend the following version: $RBF(u,v;\\gamma ) = e^{-\\gamma (1-\\rho )}$ where $\\rho =\\rho (u,v)$ is the correlation defined in (REF ) and $\\gamma >0$ is a crucial tuning parameter.",
"Based on Bochner's Theorem [24], it is known [22] that, if we sample $w\\sim uniform(0,2\\pi )$ , $r_{i}\\sim N(0,1)$ i.i.d., and let $x = \\sum _{i=1}^D u_i r_{ij}$ , $y = \\sum _{i=1}^D v_i r_{ij}$ , where $\\Vert u\\Vert _2=\\Vert v\\Vert _2=1$ , then we have $E\\left(\\sqrt{2}\\cos (\\sqrt{\\gamma } x+w)\\sqrt{2}\\cos (\\sqrt{\\gamma } y+w)\\right)= e^{-\\gamma (1-\\rho )}$ This provides a nice mechanism for linearizing the RBF kernel and the RFF method has become popular in machine learning, computer vision, and beyond, e.g., [21], [27], [1], [7], [5], [28], [8], [25], [4], [23].",
"Theorem 2 Given $x\\sim N(0,1)$ , $y\\sim N(0,1)$ , $E(xy) = \\rho $ , and $w\\sim uniform(0,2\\pi )$ , we have $&E\\left[\\sqrt{2}\\cos (\\sqrt{\\gamma }x+w)\\sqrt{2}\\cos (\\sqrt{\\gamma }y+w)\\right]=e^{-\\gamma (1-\\rho )}\\\\&E\\left[\\cos (\\sqrt{\\gamma }x)\\cos (\\sqrt{\\gamma }y)\\right]=\\frac{1}{2}e^{-\\gamma (1-\\rho )} + \\frac{1}{2}e^{-\\gamma (1+\\rho )}\\\\&Var\\left[\\sqrt{2}\\cos (\\sqrt{\\gamma }x+w)\\sqrt{2}\\cos (\\sqrt{\\gamma }y+w)\\right] = \\frac{1}{2} + \\frac{1}{2}\\left(1-e^{-2\\gamma (1-\\rho )}\\right)^2$ The proof for () can also be found in [26].",
"One can see that the variance of RFF can be large.",
"Interestingly, the variance can be substantially reduced if we normalize the hashed data, a procedure which we call “normalized RFF (NRFF)”.",
"The theoretical results are presented in Theorem REF .",
"Theorem 3 Consider $k$ iid samples ($x_j, y_j, w_j$ ) where $x_j\\sim N(0,1)$ , $y_j\\sim N(0,1)$ , $E(x_jy_j) = \\rho $ , $w_j\\sim uniform(0,2\\pi )$ , $j =1, 2, ..., k$ .",
"Let $X_j = \\sqrt{2}\\cos \\left(\\sqrt{\\gamma }x_j + w_j\\right)$ and $Y_j = \\sqrt{2}\\cos \\left(\\sqrt{\\gamma }y_j + w_j\\right)$ .",
"As $k\\rightarrow \\infty $ , the following asymptotic normality holds: $&\\sqrt{k}\\left(\\frac{\\sum _{j=1}^k X_j Y_j}{\\sqrt{\\sum _{j=1}^k X_j^2}\\sqrt{\\sum _{j=1}^k Y_j^2}} - e^{-\\gamma (1-\\rho )}\\right)\\overset{D}{\\Longrightarrow }N\\left(0,V_{n,\\rho ,\\gamma }\\right)\\\\\\text{where}\\hspace{14.45377pt}&\\\\&V_{n,\\rho ,\\gamma } = V_{\\rho ,\\gamma }- \\frac{1}{4}e^{-2\\gamma (1-\\rho )} \\left[3-e^{-4\\gamma (1-\\rho )} \\right]\\\\&V_{\\rho ,\\gamma } = \\frac{1}{2}+\\frac{1}{2}\\left(1-e^{-2\\gamma (1-\\rho )}\\right)^2$ Obviously, $V_{n,\\rho ,\\gamma } < V_{\\rho ,\\gamma }$ (in particular, $V_{n,\\rho ,\\gamma }=0$ at $\\rho =1$ ), i.e., the variance of the normalized RFF is (much) smaller than that of the original RFF.",
"Figure REF plots $\\frac{V_{n,\\rho ,\\gamma }}{V_{\\gamma ,\\gamma }}$ to visualize the improvement due to normalization, which is most significant when $\\rho $ is close to 1.",
"Figure: The ratio V n,ρ,γ V γ,γ \\frac{V_{n,\\rho ,\\gamma }}{V_{\\gamma ,\\gamma }} from Theorem for visualizing the improvement due to normalization.Note that the theoretical results in Theorem REF are asymptotic (i.e., for larger $k$ ).",
"With $k$ samples, the variance of the original RFF is exactly $\\frac{V_{\\rho ,\\gamma }}{k}$ , however the variance of the normalized RFF (NRFF) is written as $\\frac{V_{n,\\rho ,\\gamma }}{k} + O\\left(\\frac{1}{k^2}\\right)$ .",
"It is important to understand the behavior when $k$ is not large.",
"For this purpose, Figure REF presents the simulated mean square error (MSE) results for estimating the RBF kernel $e^{-\\gamma (1-\\rho )}$ , confirming that a): the improvement due to normalization can be substantial, and b): the asymptotic variance formula () becomes accurate for merely $k>10$ .",
"Figure: A simulation study to verify the asymptotic theoretical results in Theorem .",
"With kk samples, we estimate the RBF kernel e -γ(1-ρ) e^{-\\gamma (1-\\rho )}, using both the original RFF and the normalized RFF (NRFF).",
"With 10 5 10^5 repetitions at each kk, we can compute the empirical mean square error: MSE = Bias 2 ^2+Var.",
"Each panel presents the MSEs (solid curves) for a particular choice of (ρ,γ)(\\rho ,\\gamma ), along with the theoretical variances: V ρ,γ k\\frac{V_{\\rho ,\\gamma }}{k} and V n,ρ,γ k\\frac{V_{n,\\rho ,\\gamma }}{k} (dashed curves).",
"The variance of the original RFF (curves above, or red if color is available) can be substantially larger than the MSE of the normalized RFF (curves below, or blue).",
"When k>10k>10, the normalized RFF provides an unbiased estimate of the RBF kernel and its empirical MSE matches the theoretical asymptotic variance.Next, we attempt to compare RFF with GCWS.",
"While ultimately we can rely on classification accuracy as a metric for performance, here we compare their variances ($Var$ ) relative to their expectations ($E$ ) in terms of $Var/E^2$ , as shown in Figure REF .",
"For GCWS, we know $Var/E^2 = E(1-E)/E^2=(1-E)/E$ .",
"For the original RFF, we have $Var/E^2 = \\left[ \\frac{1}{2}+\\frac{1}{2}\\left(1-E^2\\right)^2\\right]/E^2$ , etc.",
"Figure REF shows that the relative variance of GCWS is substantially smaller than that of the original RFF and the normalized RFF (NRFF), especially when $E$ is not large.",
"For the very high similarity region (i.e., $E\\rightarrow 1$ ), the variances of both GCWS and NRFF approach zero.",
"Figure: Ratio of the variance over the squared expectation, denoted as Var/E 2 {Var}/{E^2}, for the convenience of comparing RFF/NRFF with GCWS.",
"Smaller (lower) is better.The results from Figure REF provide one explanation why later we will observe that, in the classification experiments, GCWS typically needs substantially fewer samples than the normalized RFF, in order to achieve similar classification accuracies.",
"Note that for practical data, the similarities among most data points are usually small (i.e., small $E$ ) and hence it is not surprising that GCWS may perform substantially better.",
"Also see Section and Figure REF for a comparison from the perspective of estimating RBF using GCWS based on a model assumption.",
"In a sense, this drawback of RFF is expected, due to nature of random projections.",
"For example, as shown in [16], [17], the linear estimator of the correlation $\\rho $ using random projections has variance $\\frac{1+\\rho ^2}{k}$ , where $k$ is the number of projections.",
"In order to make the variance small, one will have to use many projections (i.e., large $k$ ).",
"Proof of Theorem REF: The following three integrals will be useful in our proof: $\\int _{-\\infty }^\\infty \\cos (cx)e^{-x^2/2}dx = \\sqrt{2\\pi } e^{-c^2/2}$ $\\int _{-\\infty }^\\infty \\cos (c_1x)\\cos (c_2x)e^{-x^2/2}dx=&\\frac{1}{2}\\int _{-\\infty }^\\infty \\left[\\cos ((c_1+c_2)x) + \\cos ((c_1-c_2)x)\\right]e^{-x^2/2}dx \\\\=& \\frac{\\sqrt{2\\pi }}{{2}}\\left[ e^{-(c_1+c_2)^2/2} + e^{-(c_1-c_2)^2/2} \\right]$ $&\\int _{-\\infty }^\\infty \\sin (c_1x)\\sin (c_2x)e^{-x^2/2}dx = \\frac{\\sqrt{2\\pi }}{{2}}\\left[ e^{-(c_1-c_2)^2/2} - e^{-(c_1+c_2)^2/2} \\right]$ Firstly, we consider integers $b_1, b_2 = 1, 2, 3, ..., $ and evaluate the following general integral: $&E\\left(\\cos (c_1x+b_1w)\\cos (c_2y+b_2w)\\right)\\\\=&\\frac{1}{2\\pi }\\int _0^{2\\pi }E(\\cos (c_1x+b_1t)\\cos (c_2y+b_2t))dt\\\\=&\\frac{1}{2\\pi }\\int _0^{2\\pi }\\int _{-\\infty }^\\infty \\int _{-\\infty }^\\infty (\\cos (c_1x+b_1t)\\cos (c_2y+b_2t)) \\frac{1}{2\\pi }\\frac{1}{\\sqrt{1-\\rho ^2}}e^{-\\frac{x^2+y^2-2\\rho xy}{2(1-\\rho ^2)}} dxdydt\\\\=&\\frac{1}{2\\pi }\\int _0^{2\\pi }\\int _{-\\infty }^\\infty \\int _{-\\infty }^\\infty (\\cos (c_1x+b_1t)\\cos (c_2y+b_2t)) \\frac{1}{2\\pi }\\frac{1}{\\sqrt{1-\\rho ^2}}e^{-\\frac{x^2+y^2-2\\rho xy+\\rho ^2x^2-\\rho ^2x^2}{2(1-\\rho ^2)}} dxdydt\\\\=&\\frac{1}{2\\pi }\\int _0^{2\\pi }\\int _{-\\infty }^\\infty \\frac{1}{2\\pi }\\frac{1}{\\sqrt{1-\\rho ^2}}e^{-\\frac{x^2}{2}} \\cos (c_1x+b_1t) dx\\int _{-\\infty }^\\infty \\cos (c_2y+b_2t)e^{-\\frac{(y-\\rho x)^2}{2(1-\\rho ^2)}} dydt\\\\=&\\frac{1}{2\\pi }\\int _0^{2\\pi }\\int _{-\\infty }^\\infty \\frac{1}{2\\pi }e^{-\\frac{x^2}{2}} \\cos (c_1x+b_1t) dx\\int _{-\\infty }^\\infty \\cos (c_2y\\sqrt{1-\\rho ^2}+c_2\\rho x+b_2t)e^{-y^2/2} dydt\\\\=&\\frac{1}{2\\pi }\\int _0^{2\\pi }\\int _{-\\infty }^\\infty \\frac{1}{2\\pi }e^{-\\frac{x^2}{2}} \\cos (c_1x+b_1t)\\cos (c_2\\rho x+b_2t) dx\\int _{-\\infty }^\\infty \\cos (c_2y\\sqrt{1-\\rho ^2})e^{-y^2/2} dydt\\\\=&\\frac{1}{2\\pi }\\int _0^{2\\pi }\\int _{-\\infty }^\\infty \\frac{1}{2\\pi }e^{-\\frac{x^2}{2}} \\cos (c_1x+b_1t)\\cos (c_2\\rho x+b_2t) \\sqrt{2\\pi }e^{-\\frac{c_2^2(1-\\rho ^2)}{2}} dxdt\\\\=&\\frac{1}{2\\pi }\\frac{1}{\\sqrt{2\\pi }}e^{-\\frac{c_2^2(1-\\rho ^2)}{2}}\\int _0^{{2\\pi }} \\int _{-\\infty }^\\infty e^{-\\frac{x^2}{2}} \\cos (c_1x+b_1t)\\cos (c_2\\rho x+b_2t) dxdt$ Note that $&\\int _0^{2\\pi }\\cos (c_1x+b_1t)\\cos (c_2\\rho x+b_2t)dt \\\\=&\\int _0^{2\\pi }\\cos (c_1x)\\cos (b_1t)\\cos (c_2\\rho x)\\cos (b_2t)dt + \\int _0^{2\\pi }\\sin (c_1x)\\sin (b_1t)\\sin (c_2\\rho x)\\sin (b_2t)dt\\\\-&\\int _0^{2\\pi }\\cos (c_1x)\\cos (b_1t)\\sin (c_2\\rho x)\\sin (b_2t)dt - \\int _0^{2\\pi }\\sin (c_1x)\\sin (b_1t)\\cos (c_2\\rho x)\\cos (b_2t)dt$ When $b_1\\ne b_2$ , we have $&\\int _0^{2\\pi }\\cos (b_1t)\\cos (b_2t)dt = \\frac{1}{2}\\int _0^{2\\pi }\\cos (b_1t-b_2t) + \\cos (b_1t+ b_2t)dt = 0\\\\&\\int _0^{2\\pi }\\sin (b_1t)\\sin (b_2t)dt = \\frac{1}{2}\\int _0^{2\\pi }\\cos (b_1t-b_2t) - \\cos (b_1t+ b_2t)dt = 0$ If $b_1 = b_2$ , then $&\\int _0^{2\\pi }\\cos (b_1t)\\cos (b_2t)dt = \\int _0^{2\\pi }\\sin (b_1t)\\sin (b_2t)dt = \\pi $ In addition, for any $b_1, b_2 = 1, 2, 3, ...$ , we always have $&\\int _0^{2\\pi }\\sin (b_1t)\\cos (b_2t)dt = \\frac{1}{2}\\int _0^{2\\pi }\\sin (b_1t-b_2t) + \\sin (b_1t+ b_2t)dt = 0\\\\$ Thus, only when $b_1=b_2$ we have $&\\int _0^{2\\pi }\\cos (c_1x+b_1t)\\cos (c_2\\rho x+b_2t)dt = \\pi \\cos (c_1x) \\cos (c_2\\rho x) + \\pi \\sin (c_1x) \\sin (c_2\\rho x) = \\pi \\cos ((c_1-c_2\\rho )x)$ Otherwise, $\\int _0^{2\\pi }\\cos (c_1x+b_1t)\\cos (c_2\\rho x+b_2t)dt =0$ .",
"Therefore, when $b_1=b_2$ , we have $&E\\left(\\cos (c_1x+b_1w)\\cos (c_2y+b_2w)\\right)\\\\=&\\frac{1}{2\\pi }\\frac{1}{\\sqrt{2\\pi }}e^{-\\frac{c_2^2(1-\\rho ^2)}{2}}\\int _0^{{2\\pi }} \\int _{-\\infty }^\\infty e^{-\\frac{x^2}{2}} \\cos (c_1x+b_1t)\\cos (c_2\\rho x+b_2t) dxdt\\\\=&\\frac{1}{2\\pi }\\frac{1}{\\sqrt{2\\pi }}e^{-\\frac{c_2^2(1-\\rho ^2)}{2}} \\int _{-\\infty }^\\infty e^{-\\frac{x^2}{2}} \\pi \\cos ((c_1-c_2\\rho )x) dx \\\\=&\\frac{1}{2\\pi }\\frac{1}{\\sqrt{2\\pi }}e^{-\\frac{c_2^2(1-\\rho ^2)}{2}} \\pi \\sqrt{2\\pi } e^{-(c_1-c_2\\rho )^2/2}\\\\=&\\frac{1}{2}e^{-\\frac{c_1^2+c_2^2-2c_1c_2\\rho }{2}} \\\\=&\\frac{1}{2}e^{-c^2(1-\\rho )},\\hspace{21.68121pt} \\text{when } \\ c_1 = c_2 = c$ This completes the proof of the first moment.",
"Next, using the following fact $E \\cos (2cx+2w)=&\\frac{1}{2\\pi }\\int _0^{2\\pi } \\frac{1}{\\sqrt{2\\pi }}\\int _{-\\infty }^\\infty \\cos (2cx+2t) e^{-x^2/2} dx dt\\\\=&\\frac{1}{2\\pi }\\int _0^{2\\pi } \\frac{1}{\\sqrt{2\\pi }}\\frac{1}{2}\\sin 2t\\int _{-\\infty }^\\infty \\cos (2cx) e^{-x^2/2} dx dt\\\\=&\\frac{1}{4\\pi } e^{-2c^2}\\int _0^{2\\pi }\\sin 2t dt = 0$ we are ready to compute the second moment $&E\\left[\\cos (cx+w)\\cos (cy+w)\\right]^2\\\\=&\\frac{1}{4}E\\left[\\cos (2cx+2w)\\cos (2cy+2w) + \\cos (2cx+2w) +\\cos (2cy+2w)\\right] + \\frac{1}{4}\\\\=&\\frac{1}{4}E\\left[\\cos (2cx+2w)\\cos (2cy+2w)\\right] + \\frac{1}{4}\\\\=&\\frac{1}{8}e^{-4c^2(1-\\rho )} + \\frac{1}{4}$ and the variance $&Var\\left[\\cos (cx+w)\\cos (cy+w)\\right] = \\frac{1}{8}e^{-4c^2(1-\\rho )} + \\frac{1}{4} - \\frac{1}{4}e^{-2c^2(1-\\rho )}$ Finally, we prove the first moment without the “$w$ ” random variable: $E\\left(\\cos (c x)\\cos (c y)\\right)=&\\int _{-\\infty }^\\infty \\int _{-\\infty }^\\infty \\cos (c x)\\cos (c y) \\frac{1}{2\\pi }\\frac{1}{\\sqrt{1-\\rho ^2}}e^{-\\frac{x^2+y^2-2\\rho xy+\\rho ^2x^2-\\rho ^2x^2}{2(1-\\rho ^2)}} dxdy\\\\=&\\int _{-\\infty }^\\infty \\frac{1}{2\\pi }\\frac{1}{\\sqrt{1-\\rho ^2}}e^{-\\frac{x^2}{2}} \\cos (c x) dx\\int _{-\\infty }^\\infty \\cos (c y)e^{-\\frac{(y-\\rho x)^2}{2(1-\\rho ^2)}} dy\\\\=&\\int _{-\\infty }^\\infty \\frac{1}{2\\pi }e^{-\\frac{x^2}{2}} \\cos (c x) dx\\int _{-\\infty }^\\infty \\cos (c y\\sqrt{1-\\rho ^2}+c \\rho x)e^{-y^2/2} dy\\\\=&\\int _{-\\infty }^\\infty \\frac{1}{2\\pi }e^{-\\frac{x^2}{2}} \\cos (c x)\\cos (c \\rho x) dx\\int _{-\\infty }^\\infty \\cos (c y\\sqrt{1-\\rho ^2})e^{-y^2/2} dy\\\\=&\\int _{-\\infty }^\\infty \\frac{1}{2\\pi }e^{-\\frac{x^2}{2}} \\cos (c x)\\cos (c \\rho x) \\sqrt{2\\pi }e^{-c^2\\frac{1-\\rho ^2}{2}} dx\\\\=&\\frac{1}{\\sqrt{2\\pi }}e^{-c^2\\frac{1-\\rho ^2}{2}}\\int _{-\\infty }^\\infty e^{-\\frac{x^2}{2}} \\cos (c x)\\cos (c\\rho x) dx\\\\=&\\frac{1}{\\sqrt{2\\pi }}e^{-c^2\\frac{1-\\rho ^2}{2}}\\frac{\\sqrt{2\\pi }}{2} \\left[e^{-c^2\\frac{(1-\\rho )^2}{2}} + e^{-c^2\\frac{(1+\\rho )^2}{2}}\\right]\\\\=&\\frac{1}{2}e^{-c^2(1-\\rho )}+\\frac{1}{2}e^{-c^2(1+\\rho )}$ This completes the proof of Theorem REF .$\\hfill \\Box $ Proof of Theorem REF: We will use some of the results from the proof of Theorem REF .",
"Define $X_j = \\sqrt{2}\\cos (\\sqrt{\\gamma }x_j+w_j),\\hspace{14.45377pt}Y_j = \\sqrt{2}\\cos (\\sqrt{\\gamma }y_j+w_j),\\hspace{14.45377pt}Z_k = \\frac{\\sum _{j=1}^k X_j Y_j}{\\sqrt{\\sum _{j=1}^k X_j^2}\\sqrt{\\sum _{j=1}^k Y_j^2}}$ From Theorem 2, it is easy to see that, as $k\\rightarrow \\infty $ , we have $&\\frac{1}{k}\\sum _{j=1}^k X_j^2 \\rightarrow E\\left(X_j^2\\right) = e^{-\\gamma (1-1)} = 1, \\ \\ a.s. \\hspace{36.135pt} \\frac{1}{k}\\sum _{j=1}^k Y_j^2 \\rightarrow 1, \\ \\ a.s.\\\\&Z_k = \\frac{\\frac{1}{k}\\sum _{j=1}^k X_j Y_j}{\\sqrt{\\frac{1}{k}\\sum _{j=1}^k X_j^2}\\sqrt{\\frac{1}{k}\\sum _{j=1}^k Y_j^2}} \\rightarrow e^{-\\gamma (1-\\rho )} = Z_\\infty , \\ \\ a.s.$ We express the deviation $Z_k - Z_\\infty $ as $Z_k - Z_\\infty =& \\frac{\\frac{1}{k}\\sum _{j=1}^k X_j Y_j - Z_\\infty + Z_\\infty }{\\sqrt{\\frac{1}{k}\\sum _{j=1}^k X_j^2}\\sqrt{\\frac{1}{k}\\sum _{j=1}^k Y_j^2}} - Z_\\infty \\\\=& \\frac{\\frac{1}{k}\\sum _{j=1}^k X_j Y_j - Z_\\infty }{\\sqrt{\\frac{1}{k}\\sum _{j=1}^k X_j^2}\\sqrt{\\frac{1}{k}\\sum _{j=1}^k Y_j^2}} + Z_\\infty \\frac{1-\\sqrt{\\frac{1}{k}\\sum _{j=1}^k X_j^2}\\sqrt{\\frac{1}{k}\\sum _{j=1}^k Y_j^2} }{\\sqrt{\\frac{1}{k}\\sum _{j=1}^k X_j^2}\\sqrt{\\frac{1}{k}\\sum _{j=1}^k Y_j^2}}\\\\=& \\frac{1}{k}\\sum _{j=1}^k X_j Y_j - Z_\\infty + Z_\\infty \\frac{1-\\frac{1}{k}\\sum _{j=1}^k X_j^2\\frac{1}{k}\\sum _{j=1}^k Y_j^2}{2} + O_P(1/k)\\\\=& \\frac{1}{k}\\sum _{j=1}^k X_j Y_j - Z_\\infty + Z_\\infty \\frac{1-\\frac{1}{k}\\sum _{j=1}^k X_j^2}{2}+ Z_\\infty \\frac{1-\\frac{1}{k}\\sum _{j=1}^k Y_j^2}{2} + O_P(1/k)$ Note that if $a\\approx 1$ and $b\\approx 1$ , then $1-ab = 1-(1-(1-a))(1-(1-b)) =(1-a)+(1-b) -(1-a)(1-b)$ and we can ignore the higher-order term.",
"Therefore, to analyze the asymptotic variance, it suffices to study the following expectation $&E\\left( XY - Z_\\infty + Z_\\infty \\frac{1-X^2}{2}+ Z_\\infty \\frac{1-Y^2}{2}\\right)^2\\\\=& E\\left(XY-Z_\\infty (X^2+Y^2)/2\\right)^2\\\\=&E(X^2Y^2)+Z_\\infty ^2E(X^4+Y^4+2X^2Y^2)/4 - Z_\\infty E(X^3Y) - Z_\\infty E(XY^3)$ which can be obtained from the results in the proof of Theorem 2.",
"In particular, if $b_1=b_2$ , then $&E\\left(\\cos (c_1x+b_1w)\\cos (c_2y+b_2w)\\right) =\\frac{1}{2}e^{-\\frac{c_1^2+c_2^2-2c_1c_2\\rho }{2}}$ Otherwise $E\\left(\\cos (c_1x+b_1w)\\cos (c_2y+b_2w)\\right) = 0$ .",
"We can now compute $&E\\left[\\cos (cx+w)^3\\cos (cy+w)\\right]\\\\=&E\\left[\\frac{1}{4}\\cos (3(cx+w))\\cos (cy+w) +\\frac{3}{4}\\cos (cx+w)\\cos (cy+w) \\right]\\\\=&\\frac{3}{8}e^{-c^2(1-\\rho )}$ $&E\\left[\\cos (cx+w)\\cos (cy+w)\\right]^2 =\\frac{1}{8}e^{-4c^2(1-\\rho )} + \\frac{1}{4}$ $&E\\left[\\cos (cx+w)\\right]^4 = \\frac{1}{8} + \\frac{1}{4} = \\frac{3}{8}$ $V_{n,\\rho ,\\gamma } =& E\\left( XY - Z_\\infty + Z_\\infty \\frac{1-X^2}{2}+ Z_\\infty \\frac{1-Y^2}{2}\\right)^2\\\\=&E(X^2Y^2)+Z_\\infty ^2E(X^4+Y^4+2X^2Y^2)/4 - Z_\\infty E(X^3Y) - Z_\\infty E(XY^3)\\\\=&\\frac{1}{2}e^{-4c^2(1-\\rho )} +1+e^{-2c^2(1-\\rho )}\\left(\\frac{3}{8}+\\frac{3}{8} + \\frac{1}{4}e^{-4c^2(1-\\rho )} + \\frac{1}{2}\\right) -e^{-c^2(1-\\rho )}\\left( \\frac{3}{2}e^{-c^2(1-\\rho )}+\\frac{3}{2}e^{-c^2(1-\\rho )} \\right)\\\\=&\\frac{1}{2}e^{-4c^2(1-\\rho )} +1+e^{-2c^2(1-\\rho )}\\left(\\frac{5}{4} + \\frac{1}{4}e^{-4c^2(1-\\rho )}\\right) -3e^{-2c^2(1-\\rho )} \\\\=&\\frac{1}{2}e^{-4c^2(1-\\rho )} +1+\\frac{1}{4}e^{-6c^2(1-\\rho )}-\\frac{7}{4}e^{-2c^2(1-\\rho )} \\\\=& V_{\\rho ,\\gamma }- \\frac{1}{4}e^{-2c^2(1-\\rho )} \\left[3-e^{-4c^2(1-\\rho )} \\right]$ where $V_{\\rho ,\\gamma }$ is the corresponding variance factor without using normalization: $V_{\\rho ,\\gamma } = \\frac{1}{2}+\\frac{1}{2}\\left(1-e^{-2c^2(1-\\rho )}\\right)^2$ This completes the proof of Theorem REF .$\\hfill \\Box $"
],
[
"Another Comparison Based on Asymptotic of GMM",
"As proved in a technical report following this paper [19], under mild model assumption, as the dimension $D$ becomes large, the GMM kernel converges to a function of the true data correlation: $GMM \\rightarrow \\frac{1-\\sqrt{(1-\\rho )/2}}{1+\\sqrt{(1-\\rho )/2}} = g$ The convergence holds almost surely for data with bounded first moment.",
"Using the expression of $g$ we can express RBF $e^{-\\gamma (1-\\rho )}$ in terms of $g$ : $\\rho = 1-2\\left(\\frac{1-g}{1+g}\\right)^2,\\hspace{21.68121pt}e^{-\\gamma (1-\\rho )} = e^{-2\\gamma \\left(\\frac{1-g}{1+g}\\right)^2}$ For the convenience of conducting theoretical analysis, we assume $GMM =\\frac{1-\\sqrt{(1-\\rho )/2}}{1+\\sqrt{(1-\\rho )/2}} = g$ , exactly instead of asymptotically.",
"Then we have another estimator of the RBF kernel from GCWS.",
"Note that with $k$ hashes, the estimate of GMM follows a binomial distribution $binomial(k,g)$ .",
"Theorem 4 Assume $g = \\frac{1-\\sqrt{(1-\\rho )/2}}{1+\\sqrt{(1-\\rho )/2}}$ and $X\\sim binomial(k,g)$ .",
"Then, denoting $\\bar{X} = \\frac{1}{k}\\sum _{i=1}^k X_i$ , we have $&E\\left( e^{-2\\gamma \\left(\\frac{1-\\bar{X}}{1+\\bar{X}}\\right)^2}\\right) = e^{-\\gamma (1-\\rho )} + O\\left(\\frac{1}{k}\\right)\\\\&Var\\left( e^{-2\\gamma \\left(\\frac{1-\\bar{X}}{1+\\bar{X}}\\right)^2}\\right) = \\frac{V_{g,\\gamma }}{k} + O\\left(\\frac{1}{k^2}\\right)\\\\\\text{where } \\ &V_{g,\\gamma } =e^{-2\\gamma (1-\\rho )}\\frac{g(1-g)^3}{(1+g)^6}64\\gamma ^2$ Proof of Theorem REF : For an asymptotic analysis with large $k$ , it suffices to consider $Z = \\frac{1-\\hat{X}}{1+\\hat{X}}$ as a normal random variable, whose mean and variance can be calculated to be $\\mu = \\frac{1-g}{1+g},\\ \\ \\sigma ^2 = \\frac{1}{k}\\frac{4g(1-g)}{(1+g)^4}$ .",
"Thus, it suffices to compute $E\\left(e^{X^2t}\\right) =& \\int _{-\\infty }^\\infty e^{x^2t}\\frac{1}{\\sqrt{2\\pi }\\sigma } e^{-\\frac{(x-\\mu )^2}{2\\sigma ^2}}dx= \\int _{-\\infty }^\\infty \\frac{1}{\\sqrt{2\\pi }\\sigma } e^{-\\frac{(x-\\mu )^2-2\\sigma ^2x^2t}{2\\sigma ^2}}dx\\\\=&\\int _{-\\infty }^\\infty \\frac{1}{\\sqrt{2\\pi }\\sigma } e^{-\\frac{(x-\\mu )^2-2\\sigma ^2x^2t}{2\\sigma ^2}}dx=\\int _{-\\infty }^\\infty \\frac{1}{\\sqrt{2\\pi }\\sigma } e^{-\\frac{(1-2\\sigma ^2t)x^2 - 2\\mu x + \\mu ^2}{2\\sigma ^2}}dx\\\\=&\\int _{-\\infty }^\\infty \\frac{1}{\\sqrt{2\\pi }\\sigma } e^{-\\frac{x^2 - 2\\mu /c^2 x + \\mu ^2/c^2}{2\\sigma ^2/c^2}}dx,\\hspace{21.68121pt}\\text{where } c^2 = 1-2\\sigma ^2t\\\\=&\\frac{1}{c}\\int _{-\\infty }^\\infty \\frac{1}{\\sqrt{2\\pi }\\sigma /c} e^{-\\frac{(x - \\mu /c^2)^2 - \\mu ^2/c^4 + \\mu ^2/c^2}{2\\sigma ^2/c^2}}dx\\\\=&\\frac{1}{c}e^{\\frac{\\mu ^2(1-c^2)}{2\\sigma ^2c^2}} = \\frac{1}{c}e^{\\frac{\\mu ^2}{c^2}t} = \\frac{1}{\\sqrt{1-2\\sigma ^2t}}e^{\\frac{\\mu ^2t}{1-2\\sigma ^2t}}$ from which we can compute the variance (letting $\\sigma ^2 = \\frac{1}{k}\\lambda ^2$ ) $Var\\left(e^{X^2t}\\right)=& E\\left(e^{X^22t}\\right) - E^2\\left(e^{X^2t}\\right)= \\frac{1}{\\sqrt{1-2\\sigma ^22t}}e^{\\frac{\\mu ^22t}{1-2\\sigma ^22t}} - \\frac{1}{{1-2\\sigma ^2t}}e^{\\frac{\\mu ^22t}{1-2\\sigma ^2t}}\\\\=&\\left(1+\\frac{2\\lambda ^2t}{k}+O\\left(\\frac{1}{k^2}\\right)\\right)e^{2\\mu ^2t\\left(1+\\frac{4\\lambda ^2t}{k} +O\\left(\\frac{1}{k^2}\\right) \\right)} -\\left(1+\\frac{2\\lambda ^2t}{k}+O\\left(\\frac{1}{k^2}\\right)\\right)e^{2\\mu ^2t\\left(1+\\frac{2\\lambda ^2t}{k} +O\\left(\\frac{1}{k^2}\\right) \\right)}\\\\=&\\left(1+O\\left(\\frac{1}{k}\\right)\\right)e^{2\\mu ^2t}\\left(1+\\frac{8\\mu ^2\\lambda ^2t^2}{k} +O\\left(\\frac{1}{k^2}\\right)\\right) -\\left(1+O\\left(\\frac{1}{k}\\right)\\right)e^{2\\mu ^2t}\\left(1+\\frac{4\\mu ^2\\lambda ^2t^2}{k} +O\\left(\\frac{1}{k^2}\\right) \\right)\\\\=&\\frac{4\\mu ^2\\lambda ^2t^2}{k} e^{2\\mu ^2t}+O\\left(\\frac{1}{k^2}\\right)$ Plugging in $t = -2\\gamma $ , $\\mu = \\frac{1-g}{1+g}$ , and $\\lambda ^2 = \\frac{4g(1-g)}{(1+g)^4}$ , yields $Var\\left(e^{-2\\gamma \\left(\\frac{1-\\bar{X}}{1+\\bar{X}}\\right)^2}\\right)=\\frac{64\\gamma ^2}{k}\\frac{g(1-g)^3}{(1+g)^6}e^{-4\\gamma \\left(\\frac{1-g}{1+g}\\right)^2}+O\\left(\\frac{1}{k^2}\\right)=\\frac{64\\gamma ^2}{k}\\frac{g(1-g)^3}{(1+g)^6}e^{-2\\gamma (1-\\rho )}+O\\left(\\frac{1}{k^2}\\right)$ $\\hfill \\Box $ This theoretical result provides a direct comparison of GCWS with NRFF for estimating the same object, by visualizing the variance ratio: $\\frac{V_{n,\\rho ,\\gamma }}{V_{g,\\gamma }}$ , using results from Theorem REF .",
"As shown in Figure REF , for estimating the RBF kernel, the variance of GCWS is substantially smaller than the variance of NRFF, except for the very high similarity region (depending on $\\gamma $ ).",
"At high similarity, the variances of both methods approach zero.",
"This provides another explanation for the superb empirical performance of GCWS compared to NRFF, as will be reported later in the paper.",
"Figure: The variance ratio: V n,ρ,γ V g,γ \\frac{V_{n,\\rho ,\\gamma }}{V_{g,\\gamma }} provides another comparison of GCWS with NRFF.",
"V g,γ V_{g,\\gamma } is derived in Theorem and V n,ρ,γ V_{n,\\rho ,\\gamma } is derived in Theorem .",
"The ratios are significantly larger than 1 except for the very high similarity region (where the variances of both methods are close to zero)."
],
[
"An Experimental Study on Kernel SVMs",
"Table REF lists datasets from the UCI repository.",
"Table REF presents datasts from the LIBSVM website as well as datasets which are fairly large.",
"Table REF contains datasets used for evaluating deep learning and trees [10], [11].",
"Except for the relatively large datasets in Table REF , we also report the classification accuracies for the linear SVM, kernel SVM with RBF, and kernel SVM with GMM, at the best $l_2$ -regularization $C$ values.",
"More detailed results (for all regularization $C$ values) are available in Figures REF , REF , REF , and REF .",
"To ensure repeatability, we use the LIBSVM pre-computed kernel functionality.",
"This also means we can not (easily) test nonlinear kernels on larger datasets.",
"For the RBF kernel, we exhaustively experimented with 58 different values of $\\gamma \\in \\lbrace $ 0.001, 0.01, 0.1:0.1:2, 2.5, 3:1:20 25:5:50, 60:10:100, 120, 150, 200, 300, 500, 1000$\\rbrace $ .",
"Basically, Tables REF , REF , and REF reports the best RBF results among all $\\gamma $ and $C$ values in our experiments.",
"The classification results indicate that, on these datasets, kernel (GMM and RBF) SVM classifiers improve over linear classifiers substantially.",
"For more than half of the datasets, the GMM kernel (which has no tuning parameter) outperforms the best-tuned RBF kernel.",
"For a small number of datasets (e.g., “SEMG1”), even though the RBF kernel performs better, we will show in Section that the GCWS hashing can still be substantially better than the NRFF hashing.",
"Table: Public (UCI) classification datasets and l 2 l_2-regularized kernel SVM results.",
"We report the test classification accuracies for the linear kernel, the best-tuned RBF kernel (and the best γ\\gamma ), and the GMM kernel, at their individually- best SVM regularization CC values.Table: Datasets in group 1 and group 3 are from the LIBSVM website.",
"Datasets in group 2 are from the UCI repository.",
"Datasets in group 2 and 3 are too large for LIBSVM pre-computed kernel functionality and are thus only used for testing hashing methods.Table: Datasets from , .",
"See the technical report on “tunable GMM kernels” for substantially improved results, by introducing tuning parameters in the GMM kernel.Figure: Test classification accuracies using kernel SVMs.",
"Both the GMM kernel and RBF kernel substantially improve linear SVM.",
"CC is the l 2 l_2-regularization parameter of SVM.",
"For the RBF kernel, we report the result at the best γ\\gamma value for every CC value.Figure: Test classification accuracies using kernel SVMs.",
"Both the GMM kernel and RBF kernel substantially improve linear SVM.",
"CC is the l 2 l_2-regularization parameter of SVM.",
"For the RBF kernel, we report the result at the best γ\\gamma value for every CC value.Figure: Test classification accuracies using kernel SVMs.",
"Both the GMM kernel and RBF kernel substantially improve linear SVM.",
"CC is the l 2 l_2-regularization parameter of SVM.",
"For the RBF kernel, we report the result at the best γ\\gamma value for every CC value.Figure: Test classification accuracies using kernel SVMs.",
"Both the GMM kernel and RBF kernel substantially improve linear SVM.",
"CC is the l 2 l_2-regularization parameter of SVM.",
"For the RBF kernel, we report the result at the best γ\\gamma value for every CC value.For the datasets in Table REF , since [10] also conducted experiments on the RBF kernel, the polynomial kernel, and neural nets, we assembly the (error rate) results in Figure REF and Table REF .",
"Figure: Error rates on 6 datasets: M-Noise1 to M-Noise6 as in Table .",
"In this figure, the curve labeled as “SVM” represents the results on RBF kernel SVM conducted by , while the curve labeled as “RBF” presents our own experiments.",
"The small discrepancies might be caused by the fact that we always use normalized data (i.e., ρ\\rho ).Table: Summary of test error rates of various algorithms on other datasets used in , .",
"Results in group 1 are reported by for using RBF kernel, polynomial kernel, and neural nets.",
"Results in group 2 are from our own experiments.",
"Also, see the technical report on “tunable GMM kernels” for substantially improved results, by introducing tuning parameters in the GMM kernel."
],
[
"Hashing for Linearizing Nonlinear Kernels",
"It is known that a straightforward implementation of nonlinear kernels can be difficult for large datasets [3].",
"For example, for a small dataset with merely $100,000$ data points, the $100,000 \\times 100,000$ kernel matrix has $10^{10}$ entries.",
"In practice, being able to linearize nonlinear kernels becomes very beneficial, as that would allow us to easily apply efficient linear algorithms especially online learning [2].",
"Randomization (hashing) is a popular tool for kernel linearization.",
"In the introduction, we have explained how to linearize both the RBF kernel and the GMM kernel.",
"From practitioner's perspective, while the kernel classification results in Tables REF , REF , and REF are informative, they are not sufficient for guiding the choice of kernels.",
"For example, as we will show, for some datasets, even though the RBF kernel outperform the GMM kernel, the linearization algorithm (i.e., the normalized RFF) requires substantially more samples (i.e., larger $k$ ).",
"Note that in our SVM experiments, we always normalize the input features to the unit $l_2$ norm (i.e., we will always use NRFF instead of RFF).",
"We will report detailed experimental results on 6 datasets.",
"As shown in Table REF , on the first two datasets, the original RBF and GMM kernels perform similarly; in the second group, the GMM kernel noticeably outperforms the RBF kernel; in the last group, the RBF kernel noticeably outperforms the GMM kernel.",
"We will show on all these 6 datasets, the GCWS hashing is substantially more accurate than the NRFF hashing at the same number of sample size ($k$ ).",
"We will then present less detailed results on other datasets.",
"Table: 6 datasets used for presenting detailed experimental results on GCWS and NRFF.Figure REF reports the test classification accuracies on the Letter dataset, for both linearized GMM kernel with GCWS and linearized RBF kernel (at the best $\\gamma $ ) with NRFF, using LIBLINEAR.",
"From Table REF , we can see that the original RBF kernel slightly outperforms the GMM kernel.",
"Obviously, the results obtained by GCWS hashing are noticeably better than the results of NRFF hashing, especially when the number of samples ($k$ ) is not too large (i.e., the left panels).",
"Figure: Letter: Test classification accuracies of the linearized GMM kernel (solid, GCWS) and linearized RBF kernel (dashed, NRFF), using LIBLINEAR, averaged over 10 repetitions.",
"In each panel, we report the results on 4 different kk (sample size) values: 128, 256, 1024, 4096 (right panels), and 16, 32, 64, 128 (left panels).",
"We can see that the linearized qRBF (using NRFF) would require substantially more samples in order to reach the same accuracies as the linearized GMM kernel (using GCWS).",
"Two interesting points: (i) Although the original (best-tuned) RBF kernel slightly outperforms the original GMM kernel, the results of GCWS are still more accurate than the results of RFF even at k=4096k = 4096, which is very large, considering the original data dimension is merely 16.",
"(ii) With merely k=16k=16 samples (b≥4b\\ge 4), GCWS already produces better results than linear SVM based on the original dataset (the solid curve marked by *).For the “Letter” dataset, the original dimension is merely 16.",
"It is known that, for modern linear algorithms, the computational cost is largely determined by the number of nonzeros.",
"Hence the number of samples (i.e., $k$ ) is a crucial parameter which directly controls the training complexity.",
"From the left panels of Figure REF , we can see that with merely $k=16$ samples, GCWS already produces better results than the original linear method.",
"This phenomenon is exciting, because in industrial practice, the goal is often to produce better results than linear methods without consuming much more resources.",
"Figure: Webspam20k: Test classification accuracies of the linearized GMM kernel (solid, GCWS) and linearized RBF kernel (dashed, NRFF), using LIBLINEAR, averaged over 10 repetitions.",
"In each panel, we report the results on 4 different kk (sample size) values: 128, 256, 1024, 4096 (right panels), and 16, 32, 64, 128 (left panels).",
"We can see that the linearized RBF (using NRFF) would require substantially more samples in order to reach the same accuracies as the linearized GMM kernel (using GCWS).",
"The linear SVM results are represented by solid curves marked by *.Figure REF reports the test classification accuracies on the Webspam20k dataset.",
"Again, the results obtained by GCWS hashing and linear classification are noticeably better than the results of NRFF hashing and linear classification, especially when the number of samples ($k$ ) is not too large (i.e., the left panels).",
"For this dataset, the original dimension is 254.",
"With GCWS hashing and merely $k=128$ , we can achieve higher accuracy than using linear classifier on the original data.",
"However, with NRFF hashing, we need almost $k=1024$ in order to outperform linear classifier on the original data.",
"Also, note that it is sufficient to use $b=4$ for GCWS hashing on this dataset.",
"Figure REF and Figure REF report the test classification accuracies on the DailySports dataset and the RobotNavi dataset, respectively.",
"For both datasets, the original GMM kernel noticeably outperforms the original RBF kernel.",
"Not surprisingly, NRFF hashing requires substantially more samples in order to reach similar accuracy as GCWS hashing, on both datasets.",
"The results also illustrate that the parameter $b$ (i.e., the number of bits we store for each GCWS hashed value $i^*$ ) does matter, but nevertheless, as long as $b\\ge 4$ , the results do not differ much.",
"Figure REF and Figure REF report the test classification accuracies on the SEMG1 dataset and M-Rotate dataset, respectively.",
"For both datasets, the original RBF kernel considerably outperforms the original GMM kernel.",
"Nevertheless, NRFF hashing still needs substantially more samples than GCWS hashing on both datasets.",
"Again, for GCWS, the results do not differ much once we use $b\\ge 4$ .",
"These results again confirm the advantage of GCWS hashing.",
"Figure REF reports the test classification accuracies on more datasets, only for $b=8$ and $k\\ge 128$ .",
"Figure REF presents the hashing results on 6 larger datasets for which we can not directly train kernel SVMs.",
"We report only for $b=8$ and $k$ up to 1204.",
"All these results confirm that linearization via GCWS works well for the GMM kernel.",
"In contrast, the normalized random Fourier feature (NRFF) approach typically requires substantially more samples (i.e., much larger $k$ ).",
"This phenomenon can be largely explained by the theoretical results in Theorem REF and Theorem REF , which conclude that GCWS hashing is more (considerably) accurate than NRFF hashing, unless the similarity is high.",
"At high similarity, the variances of both hashing methods become very small.",
"We should mention that the original (tuning-free) GMM kernel can be modified by introducing tuning parameters.",
"The original GCWS algorithm can be slightly modified to linearize the new (and tunable) GMM kernel.",
"As shown in [15], on many datasets, the tunable GMM kernel can be a strong competitor compared to computationally expensive algorithms such as deep nets or trees.",
"Figure: DailySports: Test classification accuracies of the linearized GMM kernel (solid) and linearized RBF kernel (dashed) , using LIBLINEAR.",
"In each panel, we report the results on 4 different kk (sample size) values: 128, 256, 1024, 4096 (right panels), and 16, 32, 64, 128 (left panels).",
"We can see that the linearized RBF (using NRFF) would require substantially more samples in order to reach the same accuracies as the linearized GMM kernel (using GCWS).Figure: RobotNavi: Test classification accuracies of the linearized GMM kernel (solid) and linearized RBF kernel (dashed), using LIBLINEAR.",
"In each panel, we report the results on 4 different kk (sample size) values: 128, 256, 1024, 4096 (right panels), and 16, 32, 64, 128 (left panels).",
"We can see that the linearized RBF (using NRFF) would require substantially more samples in order to reach the same accuracies as the linearized GMM kernel (using GCWS).Training time: For linear algorithms, the training cost is largely determined by the number of nonzero entries per input data vector.",
"In other words, at the same $k$ , the training times of GCWS and NRFF will be roughly comparable.",
"For GCWS and batch algorithms (such as LIBINEAR), a larger $b$ will increase the training time but not much.",
"See Figure REF for an example, which actually shows that NRFF will consume more time at high $C$ (for achieving a good accuracy).",
"Note that, with online learning, it would be more obvious that the training time is determined by the number of nonzeros and number of epoches.",
"For industrial practice, typically only one epoch or a few epoches are used.",
"Figure: PAMAP105Large: Training times of GCWS (solid curves) and NRFF (dashed curves), for four sample sizes k∈{128,256,512,1024}k\\in \\lbrace 128, 256, 512, 1024\\rbrace , and b∈{2,4,8}b\\in \\lbrace 2, 4,8\\rbrace .Data storage: For GCWS, the storage cost per data vector is $b\\times k$ while the cost for NRFF would be $k\\times $ number of bits per hashed value (which might be a large value such as 32).",
"Therefore, at the same sample size $k$ , GCWS will likely need less space to store the hashed data than NRFF."
],
[
"Conclusion",
"Large-scale machine learning has become increasingly important in practice.",
"For industrial applications, it is often the case that only linear methods are affordable.",
"It is thus practically beneficial to have methods which can provide substantially more accurate prediction results than linear methods, with no essential increase of the computation cost.",
"The method of “random Fourier features” (RFF) has been a popular tool for linearizing the radial basis function (RBF) kernel, with numerous applications in machine learning, computer vision, and beyond, e.g., [21], [27], [1], [7], [5], [28], [8], [25], [4], [23].",
"In this paper, we rigorously prove that a simple normalization step (i.e., NRFF) can substantially improve the original RFF procedure by reducing the estimation variance.",
"In this paper, we also propose the “generalized min-max (GMM)” kernel as a measure of data similarity, to effectively capture data nonlinearity.",
"The GMM kernel can be linearized via the generalized consistent weighted sampling (GCWS).",
"Our experimental study demonstrates that usually GCWS does not need too many samples in order to achieve good accuracies.",
"In particular, GCWS typically requires substantially fewer samples to reach the same accuracy as the normalized random Fourier feature (NRFF) method.",
"This is practically important, because the training (and testing) cost and storage cost are determined by the number of nonzeros (which is the number of samples in NRFF or GCWS) per data vector of the dataset.",
"The superb empirical performance of GCWS can be largely explained by our theoretical analysis that the estimation variance of GCWS is typically much smaller than the variance of NRFF (even though NRFF has improved the original RFF).",
"By incorporating tuning parameters, [15] demonstrated that the performance of the GMM kernel and GCWS hashing can be further improved, in some datasets remarkably so.",
"See [15] for the comparisons with deep nets and trees.",
"Lastly, we should also mention that GCWS can be naturally applied in the context of efficient near neighbor search, due to the discrete nature of the samples, while NRFF or samples from Nystrom method can not be directly used for building hash tables."
]
] | 1605.05721 |
[
[
"Quantum dress for a naked singularity"
],
[
"Abstract We investigate semiclassical backreaction on a conical naked singularity space-time with a negative cosmological constant in (2+1)-dimensions.",
"In particular, we calculate the renormalized quantum stress-energy tensor for a conformally coupled scalar field on such naked singularity space-time.",
"We then obtain the backreacted metric via the semiclassical Einstein equations.",
"We show that, in the regime where the semiclassical approximation can be trusted, backreaction dresses the naked singularity with an event horizon, thus enforcing cosmic censorship."
],
[
"Introduction",
"Naked Singularities (NS) in gravitation theory are irksome: the curvature tensor and the energy density can `blow up'; the space-time fabric may fail to resemble a smooth manifold and it may not be possible to continue geodesics past them; the laws of physics and standard features like causality may be violated [1].",
"If singularities are hidden behind an event horizon, however, one can safely ignore the problem because no causal signal can reach an outside observer from the troublesome region.",
"It is the spirit of Penrose's Cosmic Censorship hypothesis that NSs do not occur in nature [2].",
"In its weak version, this hypothesis essentially states that, generically, no `naked' (i.e., without an event horizon) space-time singularities can form in Nature.",
"NSs have been seen to form in some settings in $(3+1)$ -dimensions, e.g., [3], [4], although how `natural' and `generic' these settings are may be a matter of debate; in higher dimensions, NSs have been seen to form in [5].",
"However, in none of these works, quantum effects were taken into account.",
"That naturally leads to the question of whether NSs are stable under quantum effects or else, for example, these effects lead to the formation of an event horizon.",
"Quantum effects on a curved background space-time, however, are notoriously difficult to calculate.",
"One way to incorporate quantum effects is to include them in the energy momentum tensor and to solve the `semiclassical' Einstein equations (Eq.",
"(REF ) below) for the backreaction on the metric.",
"The quantized stress-energy tensor for matter fields suffers from well-known ultraviolet divergences and so it must be appropriately renormalized (see, e.g., [6]).",
"Such renormalization and obtention of the corresponding backreacted gravitational field, however, is very hard to perform in practice in $(3+1)$ -dimensions unless the background space-time is highly-symmetric –such as pure de Sitter or pure anti de Sitter, (A)dS, space-times–, which is not the case for a black hole or NS space-times in $(3+1)$ -dimensions.",
"On the other hand, in a space-time with one dimension less it is possible to make significant analytical progress while the results still yield an important insight into the physical processes that take place and into what one might expect there to happen in similar settings in $(3+1)$ -dimensions.",
"In this paper we will investigate conical defects/excesses in ($2+1$ )-AdS space-time.",
"These are a particular class of NSs that do not seem to give rise to catastrophic phenomena.",
"Like in an ordinary cone, the curvature singularity is a Dirac delta distribution at the tip.",
"The source that produces this curvature can be identified with a point particle, which can also be understood as a removed point from the manifold [7].",
"The geometry with the conical singularity is obtained by identification under a Killing vector in the universal covering of anti de Sitter space-time, CAdS$_3$ , in complete analogy with the 2+1 (BTZ) black hole [8], [9].",
"Since the identification does not change the local geometry, the conical singularity is a locally AdS space-time.",
"The static circularly symmetric metric in Schwarzschild-like coordinates, $-\\infty <t <\\infty , 0 \\le r <\\infty , 0\\le \\theta \\le 2\\pi \\approx 0$ , is given by $ ds^2=-\\left(\\frac{r^2}{\\ell ^2}-M\\right)dt^2+\\left(\\frac{r^2}{\\ell ^2}-M\\right)^{-1}dr^2+r^2d\\theta ^2,$ where the mass $M$ is an integration constant and the cosmological constant is given by $\\Lambda = -\\ell ^{-2}$ [8].",
"This metric corresponds to a family of extrema of the vacuum Einstein-Hilbert action in $(2+1)$ -dimensions.",
"In three dimensions, black holes and conical singularities are just different parts of the spectrum of pure gravity, with black holes occupying the mass range $M>0$ and naked conical singularities corresponding to $0> M \\ne -1$ .",
"The case $M=-1$ corresponds to $\\text{AdS}_3$ .",
"The naked singularity is of a conical type at $r=0$ , with deficit/excess angle $\\Delta \\equiv 2\\pi (1-\\sqrt{-M})$ : for $0 > M > -1$ there is an angular defect, while for $-1> M$ there is an angular excess.",
"For $M\\rightarrow 0^-$ the conical deficit approaches $2\\pi $ and the NS undergoes a topological transition: the cone becomes a cylinder.",
"On the other side of the transition there is a black hole of vanishing mass $M=0^+$ .",
"As shown in [10], conical singularities can also carry angular momentum $J$ , with $M\\le -|J|$ .",
"In the extreme case $M=-|J|$ , these spinning particles, like the extreme black holes counterparts ($M=|J|$ ), are BPS states, admitting a supersymmetric extension and enjoying perturbative stability [11].",
"The identification vector $\\xi $ in CAdS$_3$ that produces the black hole has norm $\\xi \\cdot \\xi = r^2$ .",
"Thus, the region where $\\xi $ is spacelike ($r^2>0$ ) is defined as the BTZ space-time.",
"The region where the vector is timelike is excised in order to avoid the closed timelike curves produced by the identification, generating a causal boundary at $r=0$ .",
"On the other hand, the conical singularity is produced by identifying with a rotation Killing vector $\\eta \\equiv \\Delta \\partial _\\theta $ , in AdS$_3$ , where $\\theta $ is the azimuthal angle and $\\Delta $ is the conical deficit around $r=0$ .",
"This Killing vector is spacelike everywhere and therefore does not produce closed timelike curves.",
"However, the identification gives rise to a conical singularity at $r=0$ , the fixed point of $\\eta $ .",
"The opposite of a conical defect, an angular excess, is also a NS with a “negative angular deficit\", which is not produced by an identification, but by an insertion of an angular wedge.",
"These features make conical singularities in AdS$_3$ as acceptable as black holes.",
"The question we wish to address, then, is, what happens in the geometry of a conical singularity when one includes vacuum fluctuations of some matter field: does the conical defect of the NS grow?",
"what is the fate of the singularity?",
"In this paper we investigate precisely this issue on a non-rotating, naked conical singularity space-time in $(2+1)$ -dimensions with a negative cosmological constant and find that quantum effects create an event horizon surrounding a curvature singularity.",
"This paper is organized as follows.",
"In Sec., we calculate the renormalized expectation value of the stress-energy tensor for a scalar field in a NS space-time after reviewing the corresponding literature result in a black hole space-time.",
"In Sec.REF we analytically calculate the quantum-backreacted metric.",
"We finish in Sec.",
"with a discussion of our results.",
"We use units $c=1$ , $G=1/8$ throughout."
],
[
"Renormalized Stress-energy Tensor",
"In [12] it was shown that quantum fluctuations of a scalar field with periodic boundary conditions around a black hole make its horizon radius growThis result was extended to non-conformal coupling for the massless black hole in [13] and to the four-dimensional planar massless black hole metric, in the conformal case, in [14]..",
"Hence, a black hole will remain a black hole if the quantum fluctuations are included.",
"Here we explore the effect of quantum fluctuations on a conical singularity: does the conical singularity remain naked, or do the quantum corrections dress this singularity with an event horizon?",
"Our analysis shows that the latter is the case.",
"A similar question in flat (zero cosmological constant) $(2+1)$ -dimensional space-time was raised by Souradeep and Sahni [15] and by Soleng [16], who showed that quantum effects on a conical singularity in flat space turn it into a $(2+1)$ -dimensional `Schwarzschild-like' space-time with gravitational attraction.",
"In flat 2+1 space-time, an analogous question was also addressed in the context of an accelerated C-metric in [17].",
"In order to address the above question, we consider the semiclassical Einstein equations $ G_{\\mu \\nu } - \\ell ^{-2} g_{\\mu \\nu }= \\kappa \\left\\langle \\hat{T}_{\\mu \\nu } \\right\\rangle _{\\text{ren}} \\,,$ where $G_{\\mu \\nu }$ is the Einstein tensor for the metric $g_{\\mu \\nu }$ and $\\kappa =8\\pi G$ .",
"These equations determine the perturbed metric via the renormalized expectation value of the stress-energy tensor (RSET), $ \\left\\langle \\hat{T}_{\\mu \\nu } \\right\\rangle _{\\text{ren}}$ , of the matter field in some quantum state.",
"We consider as quantum source a conformally coupled scalar field without a mass parameter, whose (unrenormalized) expectation value of the stress-energy tensor is given by ([6], [20]): $\\langle \\hat{T}_{\\mu \\nu }(x)\\rangle =&\\lim _{x^{\\prime } \\rightarrow x}\\frac{\\hbar }{4}\\left[ 3\\nabla ^x_\\mu \\nabla ^{x^{\\prime }}_\\nu - g_{\\mu \\nu } g^{\\alpha \\beta } \\nabla ^x_\\alpha \\nabla ^{x^{\\prime }}_\\beta \\right.\\nonumber \\\\ &\\left.- \\nabla ^x_\\mu \\nabla ^x_\\nu - \\frac{1}{4\\ell ^2}g_{\\mu \\nu } \\right] G(x,x^{\\prime }),$ where $x$ and $x^{\\prime }$ are space-time points.",
"Here, $G(x,x^{\\prime })$ is Hadamard's elementary two-point function, i.e., the anticommutator $\\langle \\lbrace \\hat{\\Phi }(x),\\hat{\\Phi }(x^{\\prime })\\rbrace \\rangle $ , where $\\hat{\\Phi }(x)$ is the quantum scalar field.",
"The quantum state of the field where the expectation values of the stress-energy tensor and of the two-point function are evaluated is determined by imposing boundary conditions on the solutions of the field equations.",
"In the present analysis, we choose for the two-point function $G(x,x^{\\prime })$ for the scalar field to satisfy `transparent' boundary conditions [12].",
"Imposing transparent boundary conditions corresponds to quantizing the scalar field using modes which are smooth on the entire Einstein static universe [18], [19].",
"We first review the calculation of the RSET existing in the literature in the case of the black hole and afterwards we derive our new results in the case of the NS."
],
[
"Black hole case ($M>0$ )",
"In the BTZ black hole case, the RSET in Eq.",
"(REF ) when the scalar field satisfies `transparent' boundary conditions takes the form ([12], [20]): $\\kappa \\langle \\hat{T}^{\\mu }{}_{\\nu }\\rangle _{\\text{ren}} =\\frac{l_P}{r^3}F_{BH}(M)\\text{diag}(1,1,-2),$ in $\\lbrace t,r,\\theta \\rbrace $ coordinates, where $l_P= \\hbar G$ is the Planck length and $F_{BH}(M)$ is a function that we give in Eq.",
"(REF ) below.",
"The two-point function in the BTZ black hole case can be calculated via the method of images from the two-point function in $\\text{AdS}_3$ , taking advantage of the fact that the black-hole manifold $\\mathcal {M}_{BTZ}$ is obtained by an identification of the universal covering space $\\text{CAdS}_3$ of $\\text{AdS}_3$ under one of its isometries.",
"Explicitly, $\\mathcal {M}_{BTZ} \\approx \\text{AdS}_3/H_\\xi $ , where $H_\\xi $ is the discrete group obtained by applying a Lorentz boost $\\xi $ [9].",
"The method of images then leads to an expression for the two-point function in $\\mathcal {M}_{BTZ}$ in terms of the two-point function in $\\text{CAdS}_3$ for a scalar field $\\Phi $ with periodic boundary conditions $\\Phi (\\theta )=\\Phi (\\theta +2\\pi )$ (where the coordinates other than $\\theta $ are suppressed): $ G_{BH}(x,x^{\\prime })=\\sum _{n\\in \\mathbb {Z}}G_{CAdS_3}(x, H^n_\\xi x^{\\prime }),$ The two-point function in $\\text{CAdS}_3$ is (e.g., [20], [21],[22],[23]), $ G_{CAdS_3}(x, x^{\\prime })=\\frac{1}{4\\pi }\\frac{1}{|x-x^{\\prime }|},$ where $|x-x^{\\prime }|=\\sqrt{(x-x^{\\prime })^a(x-x^{\\prime })_a}$ is the geodesic distance between $x$ and $x^{\\prime }$ in the embedding space $\\mathbb {R}^{2,2}$ .",
"Representing the points in $\\text{CAdS}_3$ as embedded in flat $\\mathbb {R}^{2,2}$ as $\\left(x^a\\right)=\\left(T_1,X_1,T_2,X_2\\right)^T$ , where $T_{1,2}$ are time coordinates and $X_{1,2}$ spatial ones, the identification takes the form $H_\\xi =\\left( \\begin{array}{cccc}\\cosh (2\\pi \\sqrt{M}) & \\sinh (2\\pi \\sqrt{M}) & 0 & 0 \\\\\\sinh (2\\pi \\sqrt{M}) & \\cosh (2\\pi \\sqrt{M}) & 0 & 0 \\\\0 & 0 & 1 & 0 \\\\0 & 0 & 0 & 1\\end{array} \\right).$ By noting that the space-time is locally AdS, it follows that the two-point function in Eq.",
"(REF ) is renormalized by subtracting the two-point function in $\\text{CAdS}_3$ , Eq.",
"(REF ); this amounts to merely subtracting the $n=0$ term in Eq.",
"(REF ) [24], [15], [21].",
"Finally, combining Eqs.",
"(REF ), (REF ) and (REF ), the RSET for a black hole is obtained as in Eq.",
"(REF ), where [20] $ F_{BH}(M)= \\frac{M^{3/2}}{2\\sqrt{2}} \\sum _{n=1}^{\\infty }\\frac{\\cosh \\left(2n\\pi \\sqrt{M}\\right)+3}{\\left(\\cosh \\left(2n\\pi \\sqrt{M}\\right)-1\\right)^{3/2}}.$"
],
[
"Naked singularity case ($M<0$ )",
"It is possible to find the expressions corresponding to Eqs.",
"(REF ), (REF ), (REF ) and (REF ) for the NS space-time with $-1<M<0$ , because the conical defect space-time is also obtained by an identification in the AdS$_3$ geometry given by (REF ) with $M=-1$ .",
"The only difference is that the identification Killing vector $\\eta $ is not along a boost, but a spatial rotation that generates the angular deficit.",
"The coordinate transformation between the coordinates in $\\mathbb {R}^{2,2}$ and $\\bar{t}\\equiv \\sqrt{-M} t$ , $\\bar{r}\\equiv r/\\sqrt{-M}$ and $\\bar{\\theta }\\equiv \\sqrt{-M} \\theta $ in the NS space-time is [10]: $&T_1=\\sqrt{\\bar{r}^2+\\ell ^2}\\cos \\left(\\bar{t}/\\ell \\right),\\quad X_1=\\bar{r}\\cos \\bar{\\theta },\\nonumber \\\\ &T_2=\\sqrt{\\bar{r}^2+\\ell ^2}\\sin \\left(\\bar{t}/\\ell \\right),\\quad X_2=\\bar{r}\\sin \\bar{\\theta }.$ Note that the angle $\\theta $ is now on the $X_1-X_2$ plane (as opposed to on the $X_1-T_1$ plane in the BTZ black hole case).",
"In these barred coordinates, Eq.",
"(REF ) becomes $ ds^2=-\\left(\\frac{\\bar{r}^2}{\\ell ^2}+1\\right)d\\bar{t}^2+\\left(\\frac{\\bar{r}^2}{\\ell ^2}+1\\right)^{-1}d\\bar{r}^2+\\bar{r}^2d\\bar{\\theta }^2.$ As $\\theta $ has a period of $2\\pi $ , then $\\bar{\\theta }$ has a period of $2\\pi \\sqrt{-M}$ , which clearly represents a conical singularity.",
"This NS is obtained from $\\text{CAdS}_3$ under the identification of $\\bar{\\theta }$ and $\\bar{\\theta }+2\\pi \\sqrt{-M}$ , which is obtained by the matrix $H_{\\eta } =\\left( \\begin{array}{cccc}\\cos (2\\pi \\sqrt{-M}) & \\sin (2\\pi \\sqrt{-M}) & 0 & 0 \\\\-\\sin (2\\pi \\sqrt{-M}) & \\cos (2\\pi \\sqrt{-M}) & 0 & 0 \\\\0 & 0 & 1 & 0 \\\\0 & 0 & 0 & 1\\end{array} \\right),$ in coordinates $\\left(x^a\\right)=\\left(X_1,X_2,T_1,T_2\\right)^T$ , where the angular deficit is $\\Delta \\equiv 2\\pi (1-\\sqrt{-M}) \\in (0,2\\pi )$ .",
"The resulting NS metric is given by Eq.",
"(REF ), where $-1<M<0$ .",
"The two-point function for a conformally-coupled and massless scalar field satisfying `transparent' boundary conditions in NS can be given by the method of images (cf.",
"[15], [12], [20]) in the case of an angular deficit $\\Delta =2\\pi (1-1/N)$ (i.e., $M=-1/N^2$ ), with $N=1, 2, 3, ...$ In this case, the two-point function is given by $G_{NS}(x,x^{\\prime })=\\sum _{n=0}^{N-1}G_{CAdS_3}(x, H^n_{\\eta } x^{\\prime })= \\frac{1}{4\\pi } \\sum _{n=0}^{N-1} \\frac{1}{|x- H^n_{\\eta } x^{\\prime }|}.$ In the conical geometry this sum contains a finite number of terms because for two points on the surface of a cone there is a finite number $N$ of geodesics connecting them.This can be easily understood considering a cone in $\\mathbb {R}^2$ : For $0<\\Delta <\\pi $ there is a unique geodesic connecting two points ($N=1$ ).",
"For $\\Delta =\\pi $ , $N=2$ ; for $\\Delta = 4\\pi /3$ , $N=3$ ; in general, for $\\Delta = 2\\pi (k-1)/k$ , $N=k$ .",
"Finally, for $\\Delta \\rightarrow 2\\pi $ the cone approaches a cylinder and the number of geodesics becomes infinite [25].",
"For $\\Delta \\ne 2\\pi (1-1/N)$ , the method of images does not apply and the two-point function must be computed as a sum over the field modes on the cone.",
"The construction in this case follows the procedure of [15], where instead of the Bessel function that appears in the expression for the two-point function in flat conical space, one finds associated Legendre functions with continuous degree and order [26].",
"In this way, the two-point function is found to be continuous in $M$ and coincides with the expression in Eq.",
"(REF ) when $M=-1/N^2$ (i.e., $\\Delta = 2\\pi (1-1/N)$ ).",
"In the case of angular excesses ($M < -1$ , $\\Delta <0$ ), Eqs.",
"(REF ), (REF ) and (REF ) also apply, there is a unique geodesic joining two space-time points and the method of images fails to be adequate as well.",
"The rationale is the same as in electrostatics: the method of images for a point charge between two conducting plates forming an angle $\\theta =2\\pi /p$ produces a finite number of images for rational $p$ , otherwise the required images are infinitely many and densely distributed.",
"This does not happen in the black hole case, for the same reason that the method of images for two parallel plates does not depend on the separation between the conductors, and also requires an infinite countable number of images, as in Eq.",
"(REF ).",
"In analogy with the black hole case, the two-point function (REF ) is to be renormalized by dropping the $n=0$ term.",
"The RSET in this case is then given by $\\kappa \\langle \\hat{T}^{ \\mu }{}_{ \\nu }\\rangle _{\\text{ren}} =\\frac{l_P}{ r^3} F_{NS}(M)\\text{diag}(1,1,-2),$ in $\\lbrace t,r,\\theta \\rbrace $ coordinates, with $F_{NS}(M)= \\frac{(-M)^{3/2}}{2\\sqrt{2}} \\sum _{n=1}^{N-1} \\frac{\\cos (2n\\pi \\sqrt{-M} )+3}{\\left(1-\\cos (2n\\pi \\sqrt{-M})\\right)^{3/2}}.$ We note that $F_{NS}(M)$ can be obtained from $F_{BH}(M)$ by analytic continuation, except for the important difference that Eq.",
"(REF ) possesses a finite sum (as opposed to Eq.",
"(REF )) and consequently $F_{NS}$ is manifestly finite.",
"We also note that both $F_{NS}(M)$ and $F_{BH}(M)$ are positive definite within their respective mass ranges.",
"The value of $F_{NS}(0)$ may be easily obtained by taking the limit $M=-1/N^2 \\rightarrow 0^-$ (i.e., $N\\rightarrow \\infty $ and $\\Delta \\rightarrow 2\\pi $ ) in Eq.",
"(REF ) and applying L'Hôpital rule.",
"One readily finds $ F_{NS}(0^-)=\\frac{\\zeta (3)}{2\\pi ^3},$ where $\\zeta $ is the Riemann zeta function.",
"This value is the same as the limit $M\\rightarrow 0^+$ in Eq.",
"(REF )."
],
[
"Backreacted Metric",
"In [12], the backreaction on the geometry produced by a correction of the form $\\kappa \\langle \\hat{T}^{\\mu }{}_{\\nu }\\rangle _{\\text{ren}} =\\frac{A(M)}{r^3}\\text{diag}(1,1,-2),$ as in Eq.",
"(REF ), was computed, showing that the metric takes the form of the exact solution in the presence of a conformally coupled scalar field.",
"Following the same steps, in our case we find $ ds^2=&-\\left(\\frac{r^2}{\\ell ^{2}}-M-\\frac{2l_P F_{NS}(M)}{r}\\right) dt^2\\nonumber \\\\ &+ \\left(\\frac{r^2}{\\ell ^{2}}-M-\\frac{2l_P F_{NS}(M)}{r}\\right)^{-1} dr^2 + r^2 d \\theta ^2.$ This metric now has an event horizon since the equation $\\frac{r^2}{\\ell ^{2}}-M=\\frac{2l_P F_{NS}(M)}{r}$ has one positive root for any value of $M$ since $F_{NS}>0$ (see Fig.1).The other two roots are complex.",
"Figure: Schematic description of the roots of Eq.().",
"The black (straight) curve is the right hand side of Eq.",
"() and the blue (dotted) and red (dashed) curves are the left hand side of Eq.",
"() for, respectively, M<0M<0 and M>0M>0.",
"The radius r + r_{+} where the black and blue curves meet corresponds to the horizon of the backreacted NS geometry;the radius r +BH r_{+ \\text{BH}} where the black and red curves meet corresponds to the horizon of the backreacted black hole geometry;the radius r BH r_{\\text{BH}} where the red curve crosses the horizontal axis corresponds to the horizon of the classical BTZ black hole.For $M<0$ the horizon radius is given by $ &r_{+}=\\frac{b}{3}+\\frac{M \\ell ^2}{b}, \\quad \\mbox{with}$ $b = \\left(27 F_{NS}(M) \\ell ^2 l_P+ 3\\ell ^2\\sqrt{81 F_{NS}(M)^2 l_P^2-3M ^3 \\ell ^2}\\right)^{\\frac{1}{3}} >0.$ The above metric has a curvature singularity at $r=0$ , as shown by the Kretschmann scalar $K= R_{\\alpha \\beta \\gamma \\delta }R^{\\alpha \\beta \\gamma \\delta } =12 \\left(\\frac{1}{\\ell ^4} + \\frac{2 (l_P F_{NS}(M))^2}{r^6}\\right).$ Thus, similarly to the backreacted black hole [12], a curvature singularity at $r=0$ is generated.",
"However, this singularity would lie inside a horizon with a radius that vanishes for $l_P \\rightarrow 0$ .",
"For $r \\gg r_{+}$ , the geometry approaches AdS space-time.",
"It is interesting to consider two different limits.",
"The first one consists of taking $l_P\\rightarrow 0$ with finite $M$ .",
"From Eq.",
"(REF ) one obtains $ r_{+}= \\frac{2l_P F_{NS}(M)}{-M}+O(l_P^3/\\ell ^3).$ Note that in the classical limit, $l_P \\rightarrow 0$ , $r_{+}$ goes to zero, while the two complex roots of Eq.",
"(REF ) approach $\\pm i \\ell \\sqrt{-M}$ .",
"This means that the positive real root is a purely quantum effect that enforces cosmic censorship.",
"No matter how large the conical defect is, quantum corrections of the vacuum would dress up the naked singularity.",
"For the expansion on the right hand side of Eq.",
"(REF ) to make sense one cannot take $M\\rightarrow 0^-$ .",
"This limit, with small but finite $l_P$ , can be explored from Eq.",
"(REF ), which gives $ r_{+}= \\frac{(l_P \\ell ^2\\zeta (3))^{\\frac{1}{3}}}{\\pi } +O(M).$"
],
[
"Discussion",
"The backreacted metric Eq.",
"(REF ) shows a horizon forming at a finite radius $r_{+}$ .",
"However, for finite $M$ the horizon radius is of order $l_P$ .",
"This indicates that, in order to resolve a region of size $r \\sim r_{+}$ , it would be required to go into the strong quantum gravity regime, casting doubt on the meaning of the classical description of space-time.",
"Classical notions like metric, distance and curvature are meaningful for coarser resolutions, which can be applied for large distances and far away from the horizon.",
"Thus, for finite $M$ the classical theory breaks down for $r \\rightarrow 0$ and there is no ground to claim that a space-time singularity –naked or otherwise– exists, because we have no theoretical framework to describe the space-time for $r \\lesssim r_{+}$ .",
"For small $M$ , on the other hand, Eq.",
"(REF ) implies $r_{+}/\\ell \\sim (l_P/\\ell )^{1/3}$ , which means $r_{+}^3 \\sim \\ell ^2 l_P$ and, therefore, in the semiclassical approximation, $r_{+}\\gg l_P$ .",
"This gives support to the interpretation of $r_{+}$ as a classical notion so that the claim that a horizon forms may be trusted.",
"Moreover, as mentioned earlier, in the limit $M\\rightarrow 0^+$ Eq.",
"(REF ) coincides with Eq.",
"(REF ), $ F_{BH}(0^+)=\\frac{\\zeta (3)}{2\\pi ^3}=F_{NS}(0^-)\\, .$ Hence, one can expect that quantum corrections on a conical singularity with a deficit angle approaching $2\\pi $ , turn it into a state indistinguishable from a small mass black hole.",
"The Kretschmann invariant Eq.",
"(REF ) shows a strong curvature singularity forming at $r=0$ , which seems to make matters worse than in the original conical singularity.",
"However, as we saw in the previous discussion, for finite $M$ the semiclassical approximation is inadequate for describing the central region of the space-time.",
"In the small $M$ approximation, however, the semiclassical approximation can be trusted and the central singularity –if any– would be hidden by a horizon.",
"In this latter case, substituting Eq.",
"(REF ) in Eq.",
"(REF ) gives $K(r_{+}) \\sim 18/\\ell ^4$ , which is 50% greater than the classical value.",
"In [27], brane-localised black holes in $AdS_4$ were interpreted, via AdS/CFT, as quantum-corrected BTZ black holes.",
"In the classical NS regime, it was noted that no calculation of the stress energy tensor of a conformal field in conical ($M<0$ ) $AdS_3$ space-time was found in the literature to compare with, nor of its backreaction.",
"Our paper fills this gap in the literature and confirms, in agreement with the analysis of [27], that quantum effects can censor singularities.",
"The analysis presented here can be extended to include angular momentum $M<-|J|$ .",
"Among the spinning cases, the extremal ones, $M=-|J|$ , are particularly interesting because they are BPS configurations and not expected to receive quantum corrections [10].",
"We leave these questions for future work.",
"Another interesting case to study is the conformally coupled scalar field system in 2+1 gravity, where a soliton with negative mass exists, which represents a non-trivial vacuum for hairy black hole sector [28]."
],
[
"Acknowledgments",
"We would like to thank F. Canfora, A. C. Ottewill, P. Taylor, D. Tempo, R. Troncoso and T. Zojer for valuable comments and observations.",
"This work has been partially funded throught grants 1130658, 1140155 and 1161311 from Fondecyt.",
"The Centro de Estudios Científicos (CECs) is funded by the Chilean Government through the Centers of Excellence Base Financing Program of Conicyt.",
"M.C.",
"acknowledges partial financial support by CNPq (Brazil), process number 308556/2014-3.",
"A.F.",
"acknowledges partial financial support from the Spanish MINECO through the grant FIS2014-57387-C3-1-P and the Severo Ochoa Excellence Center Project SEV-2014-0398."
]
] | 1605.06078 |
[
[
"XMMSL1J063045.9-603110: a tidal disruption event fallen into the back\n burner"
],
[
"Abstract Black holes at the centre of quiescent galaxies can be switched on when they accrete gas that is gained from stellar tidal disruptions.",
"A star approaching a black hole on a low angular momentum orbit may be ripped apart by tidal forces, which triggers raining down of a fraction of stellar debris onto the compact object through an accretion disc and powers a bright flare.",
"In this paper we discuss XMMSL1J063045.9-603110 as a candidate object for a tidal disruption event.",
"The source has recently been detected to be bright in the soft X-rays during an XMM-Newton slew and later showed an X-ray flux decay by a factor of about 10 in twenty days.",
"We analyse XMM-Newton and Swift data.",
"XMMSL1J063045.9-603110 shows several features typical of tidal disruption events: the X-ray spectrum shows the characteristics of a spectrum arising from a thermal accretion disc, the flux decay follows a t^-5/3 law, and the flux variation is > 350.",
"Optical observations testify that XMMSL1J063045.9-603110 is probably associated with an extremely small galaxy or even a globular cluster, which suggests that intermediate-mass black holes are located in the cores of (at least) some of them."
],
[
"Introduction",
"The formation and evolution processes of supermassive black holes (SMBHs), which live in the centre of massive galaxies (Kormendy & Richstone 1995; Kormendy & Gebhardt 2001), are one of the main riddles in astrophysics today.",
"It is commonly accepted that when these SMBHs are settled at the centre of their host galaxies, they grow mainly by accretion of the surrounding gas and merging with smaller black holes (Soltan 1982; Yu & Tremaine 2002).",
"Major inflows of gas drive the emission of huge amounts of energy that power active galactic nuclei (AGNs), even though AGNs typically have short duty cycles and galactic central SMBHs are mostly in a low luminous state (Ho 2008).",
"The lighting engine of a quiescent SMBH can also be the tidal disruption (TD) of a star orbiting it (Rees 1988).",
"Dynamical encounters in the nuclear star cluster increase the probability for a star to be scattered close to the SMBH on a low angular momentum orbit, in which it would experience the SMBH tidal field (Alexander 2012).",
"When the stellar self-gravity is no longer able to counteract the SMBH tidal force, the star is disrupted.",
"A fraction of the resulting stellar debris is bound to the SMBH and accretes onto it through an accretion disc.",
"This triggers a peculiar flaring emission (e.g.",
"Rees 1988; Phinney 1989).",
"The critical pericentre distance of a star for TD is the black hole (BH) tidal radius $r_{\\rm t}\\sim R_{\\rm *}\\Bigl (\\frac{M_{\\rm BH}}{M_{\\rm *}} \\Bigr )^{1/3}\\sim 10^2 \\rm R_{\\rm \\odot } \\Bigl (\\frac{\\it R_{\\rm *}}{1 \\rm R_{\\rm \\odot }}\\Bigr )\\Bigl (\\frac{\\it M_{\\rm BH}}{10^6 \\rm M_{\\rm \\odot }}\\Bigr )^{1/3}\\Bigl (\\frac{1 \\rm M_{\\rm \\odot }}{\\it M_{\\rm *}}\\Bigr )^{1/3} ,$ with $R_{\\rm *}$ and $M_{\\rm *}$ being the star radius and mass and $M_{\\rm BH}$ the BH mass (Hills 1975; Frank & Rees 1976).",
"If $r_{\\rm t}<r_{\\rm s}, $ where $r_{\\rm s}= \\frac{\\rm x\\it GM_{\\rm BH}}{c^2}\\sim 5\\rm R_{\\rm \\odot } \\Bigl (\\frac{\\it M_{\\rm BH}}{10^6 \\rm M_{\\rm \\odot }}\\Bigr )\\Bigl (\\frac{x}{2}\\Bigr ) $ is the BH event horizon radius (x=2 for non-rotating BHs), the star enters the BH horizon before it is tidally disrupted and no flares are observable.",
"For non-rotating BHs (Kesden 2012), Equation REF implies that solar-type stars, white dwarfs, and giant stars are swallowed entirely if the $M_{\\rm BH}$ is greater than $10^8 \\rm M_{\\rm \\odot }$ , $10^5 \\rm M_{\\rm \\odot }$ and $10^{10} \\rm M_{\\rm \\odot }$ , respectively.",
"In contrast, less massive BHs in quiescent (or low-luminous) galaxies can be inferred from the observation of tidal disruption flares.",
"TDs are observationally estimated to occur at a rate of about $10^{-5} \\rm galaxy^{-1} \\rm yr^{-1}$ (Donley et al.",
"2002).",
"Although they are rare and observations are sparse, several candidate TDs have been discovered in the optical-UV (Gezari 2012 and references therein) and soft X-ray bands (Komossa 2012; Komossa 2015 and references therein), where the peak of an accretion disc emission lies (Strubbe & Quataert 2009), and in the hard X-ray-radio band, where they are accompanied by a jetted emission (e.g.",
"Bloom et al.",
"2011; Burrows et al.",
"2011; Cenko et al.",
"2012; Hryniewicz & Walter 2016).",
"The BH mass can sometimes be estimated for candidate TDs.",
"Tidal disruption candidates have recently also been reported in dwarf galaxies, which suggests that an intermediate-mass BH (IMBH; $10^2 \\rm M_{\\rm \\odot } \\le \\it M_{\\rm BH} \\le \\rm 10^6 \\rm M_{\\rm \\odot }$ ; Ghosh et al.",
"2006; Maksym et al.",
"2013; Maksym et al.",
"2014a; Donato et al.",
"2014; Maksym et al.",
"2014b) is located in their nuclei.",
"The formation process of IMBHs is still an open question (e.g.",
"Madau & Rees 2001; Miller & Hamilton 2002; Portegies Zwart & McMillan 2002; Begelman et al.",
"2006; Latif et al.",
"2013), but confirming their existence, detecting them, and obtaining a mass estimate are extremely important, as they could fill in the current gap in mass distribution between stellar-mass BHs and SMBHs (Merloni & Heinz 2013) and also explain the origin of SMBHs through mergers of small galaxies hosting IMBHs (e.g.",
"Volonteri 2010).",
"TDs in dwarf galaxies are an opportunity for achieving all this.",
"In this paper we discuss XMMSL1J063045.9-603110, a recently discovered bright soft X-ray source whose X-ray activity might be attributable to a TD.",
"We briefly describe the fundamental structure of TDs (Sect. )",
"and summarise what is known about XMMSL1J063045.9-603110 (Sect.",
").",
"We discuss the possible TD nature of the source in an extremely small galaxy or even in a globular cluster, reducing (Sect. )",
"and exploring X-ray data from spectral analysis (Sect. )",
"to flux variability (Sect. )",
"and also investigate the activity of XMMSL1J063045.9-603110 at lower energies (Sect. )",
"and evaluate the probably host absolute magnitude (Sect.",
").",
"Our results are summarised in Sect.",
"."
],
[
"Tidal disruption",
"A star approaching the central BH of a galaxy on a low angular momentum orbit may be swallowed whole by the compact object or be tidally disrupted outside the BH horizon.",
"The latter event occurs if the BH tidal radius $r_{\\rm t}$ (Eq.",
"REF ) is greater than the BH event horizon radius $r_{\\rm s}$ (Eq.",
"REF ), that is, if $M_{\\rm BH}< \\frac{c^3}{(\\rm x \\it G)^{\\rm 3/2}}\\frac{R_{\\rm *}^{3/2}}{M_{\\rm *}^{1/2}}\\sim 10^8 \\rm M_{\\rm \\odot } \\Bigl (\\frac{\\it R_{\\rm *}}{1 \\rm R_{\\rm \\odot }}\\Bigr )^{3/2}\\Bigl (\\frac{1 \\rm M_{\\rm \\odot }}{\\it M_{\\rm *}}\\Bigr )^{1/2}\\Bigl (\\frac{2}{\\rm x}\\Bigr )^{3/2}, $ when the BH tidal force overcomes the star self-gravity, which prevents the star from remaining assembled.",
"In this regime, the star is completely disrupted if its pericentre distance $r_{\\rm p}$ is shorter than about $r_{\\rm t}$ .For $r_{\\rm p}\\gtrsim r_{\\rm t}$ the star is only partially disrupted (MacLeod et al.",
"2012; Guillochon & Ramirez-Ruiz 2013, 2015a).",
"We do not describe this case here in detail because the physics of partial TDs recovers that of total TDs to a first approximation.",
"When the star is on a parabolic orbit, nearly half of the resulting stellar debris is scattered onto highly eccentric orbits bound to the BH with a spread in the specific orbital energies of $\\Delta E\\sim GM_{\\rm BH}R_{\\rm *}/r_{\\rm t}^2$ (Lacy et al.",
"1982).",
"The other half of stellar debris leaves the system on hyperbolic orbits.",
"The first returning time at pericentre of the bound debris depends on their new orbital energy through Kepler's third law, and for the most bound material it is $t_{\\rm min}=\\frac{\\pi }{\\sqrt{2}}\\frac{GM_{\\rm BH}}{\\Delta E^{3/2}} \\sim 40 \\rm d \\Bigl (\\frac{\\it R_{\\rm *}}{1 \\rm R_{\\rm \\odot }}\\Bigr )^{3/2}\\Bigl (\\frac{\\it M_{\\rm BH}}{10^6 \\rm M_{\\rm \\odot }}\\Bigr )^{1/2}\\Bigl (\\frac{1\\rm M_{\\rm \\odot }}{\\it M_{\\rm *}}\\Bigr ).",
"$ The returned debris then gradually circularise and form an accretion disc, and the subsequent fallback of the material onto the BH is thus driven by the viscous time.",
"The rate of material returning at pericentre is $\\dot{M} (t) =[(2\\pi GM_{\\rm BH})^{2/3}/3](dM/dE)t^{-5/3}$ .",
"Here, $dM/dE$ , which is the distribution of the bound debris per unit of specific orbital energy $E$ as a function of $E$ (i.e.",
"$t$ , the time since disruption) is not exactly uniform (Lodato et al.",
"2009; Guillochon & Ramirez-Ruiz 2013, 2015a), but, to a first approximation, we can consider it as such (Rees 1988; Evans & Kochanek 1989).",
"Considering that half of the stellar debris is bound to the BH after disruption and using Eq.",
"REF , the returning rate trend starting from $t_{\\rm peak}\\sim 1.5 t_{\\rm min}$ (Evans & Kochanek 1989; Lodato et al.",
"2009) reads $\\dot{M} (t) \\sim \\frac{(2\\pi GM_{\\rm BH})^{2/3}}{3}\\frac{M_{\\rm *}/2}{\\Delta E} t^{-5/3}\\sim \\frac{2}{3} \\frac{M_{\\rm *}/2}{t_{\\rm min}}\\Bigl (\\frac{\\it t}{t_{\\rm min}}\\Bigr )^{-5/3}\\sim \\\\\\sim 3.04 \\rm M_{\\rm \\odot } \\rm yr^{-1} \\Bigl (\\frac{\\it R_{\\rm *}}{1 \\rm R_{\\rm \\odot }}\\Bigr )\\Bigl (\\frac{\\it M_{\\rm BH}}{10^6 \\rm M_{\\rm \\odot }}\\Bigr )^{1/3} \\Bigl (\\frac{\\it M_{\\rm *}}{1 \\rm M_{\\rm \\odot }}\\Bigr )^{1/3}\\Bigl (\\frac{\\it t}{0.109 \\rm yr}\\Bigr )^{\\rm -5/3}.$ If the viscous time $t_{\\rm \\nu }\\sim (2^{5/2} R_{\\rm *}^{3/2})/(\\sqrt{G}M_{\\rm *}^{1/2}\\alpha h^2)(r_{\\rm t}/r_{\\rm p})^{-3/2}$ (Li et al.",
"2002), where $\\alpha $ is the viscous disc parameter and $h$ the disc half-height divided by its radius, is smaller than $t_{\\rm min}$ , meaning that if their ratio $t_{\\rm \\nu }/t_{\\rm min}\\sim 0.025 (\\alpha /0.1)^{-1} h^{-2} (10^6 \\rm M_{\\rm \\odot }/\\it M_{\\rm BH})^{\\rm 1/2} (M_{\\rm *}/ \\rm 1 M_{\\rm \\odot })^{1/2}(\\it r_{\\rm t}/r_{\\rm p})^{\\rm -3/2}$ is low, the rate of debris returning at pericentre $\\dot{M} (t)$ coincides to a first approximationFor very deep encounters, relativistic effects may change the debris evolution from the one discussed here (e.g.",
"Dai et al.",
"2015).",
"For example, in-plane relativistic precession probably causes the stream of debris to self-cross, which speeds up its circularisation (Shiokawa et al.",
"2015; Bonnerot et al.",
"2016), while nodal relativistic precession deflects the debris out of their orbital plane, which delays self-intersection and circularisation (Guillochon & Ramirez-Ruiz 2015b).",
"with the rate of material accreting onto the BH.",
"The thicker the accretion disc (i.e.",
"$h\\sim 1$ ), the better this approximation.",
"We also note that when the parameter $\\beta =r_{\\rm t}/r_{\\rm p}$ approaches unity (it is $\\gtrsim 1$ for total disruptions), this ratio is accordingly reduced.",
"The luminosity produced by accretion can therefore be evaluated as $\\nonumber L (t) \\sim \\eta \\dot{M} (t) c^2\\sim 1.7\\times 10^{46} \\frac{\\rm erg}{\\rm s} \\frac{\\eta }{0.1}\\Bigl (\\frac{\\it R_{\\rm *}}{1 \\rm R_{\\rm \\odot }}\\Bigr ) \\Bigl (\\frac{\\it M_{\\rm BH}}{10^6 \\rm M_{\\rm \\odot }}\\Bigr )^{1/3} \\times \\\\ \\times \\Bigl (\\frac{\\it M_{\\rm *}}{1 \\rm M_{\\rm \\odot }}\\Bigr )^{1/3} \\Bigl (\\frac{\\it t}{0.109 \\rm yr}\\Bigr )^{\\rm -5/3}$ (where $\\eta $ is the radiation efficiency), again starting from $t_{\\rm peak}$ .",
"The peak luminosity $L_{\\rm peak}$ turns out to be a factor of $\\sim 130$ super-Eddington for $M_{\\rm BH}=10^6 \\rm M_{\\rm \\odot }$ , $M_{\\rm *}=1 \\rm M_{\\rm \\odot }$ , $R_{\\rm *}=1 \\rm R_{\\odot }$ and $\\eta =0.1$ , which means that probably a fraction of the accreted mass is ejected in a wind from the disc (Strubbe & Quataert 2009; Lodato & Rossi 2010).",
"If we assume that $L_{\\rm peak}$ roughly coincides with the Eddington luminosity and know the unabsorbed peak flux $F_{\\rm peak}$ , then we can evaluate the host galaxy distance $d$ as a function of $M_{\\rm BH}$ as $d\\sim 0.1 \\rm Mpc \\Bigl (\\frac{\\it M_{\\rm BH}}{1 \\rm M_{\\rm \\odot }}\\Bigr )^{1/2}\\Bigl (\\frac{10^{-10}\\rm erg/ \\rm s/ \\rm cm^{2}}{\\it F_{\\rm peak}}\\Bigr )^{1/2} .$ In addition to using methods which require optical spectra (e.g.",
"Nelson 2000; Marconi & Hunt 2003; Sani et al.",
"2010), $M_{\\rm BH}$ may then be approximately estimated from Eq.",
"REF if $d$ is known."
],
[
"XMMSL1J063045.9-603110",
"On December 1, 2011, the new point-like source XMMSL1J063045.9-603110 (hereafter XMMSL1J063045) was detected to be bright in the X-ray sky probed by the XMM-Newton Slew Survey (Saxton et al.",
"2008), at RA=06:30:45.9, DEC=-60:31:10 ($8^{\\prime \\prime }$ error circle, 1$\\sigma $ confidence level).",
"The source was soft, with essentially no emission above 2keV.",
"Fitting the Slew X-ray spectrum, Read et al.",
"(2011a) estimated an absorption of $N_{\\rm H}=0.11\\times 10^{22} \\rm cm^{-2}$ ($\\sim $ 2.1$N_{\\rm H_{\\rm Gal}}$ ), a blackbody temperature of $T_{\\rm bb}=85 \\rm eV$ , and an absorbed 0.2-2keV EPIC-PN flux of $4.0\\times 10^{-11} \\rm erg$ $\\rm s^{-1}$ $\\rm cm^{-2}$ , starting from a count rate of $32.6 \\rm ct$ $\\rm s^{-1}$ .",
"When the same spectral model is assumed, the upper limits obtained from two previous XMM-Newton slews over this position, which are $< 0.52 \\rm ct$ $\\rm s^{-1}$ (14/08/2002) and $< 1.76 \\rm ct$ $\\rm s^{-1}$ (18/11/2008), give absorbed 0.2-2keV EPIC-PN fluxes of $< 6.4\\times 10^{-13} \\rm erg$ $\\rm s^{-1}$ $\\rm cm^{-2}$ and $< 2.2 \\times 10^{-12} \\rm erg$ $\\rm s^{-1}$ $\\rm cm^{-2}$ .",
"These are factors of more than 63 and 18 below the bright Slew detection, respectively.",
"This flux gap, together with the non-detection of previous lower energy counterparts (Fig.",
"REF , left panel), led Read et al.",
"to suggest that XMMSL1J063045 might be a new nova.",
"On December 18, 2011, Kann et al.",
"(2011) identified an object at RA=06:30:45.45, DEC=-60:31:12.8 with an error of $\\pm 0.3^{\\prime \\prime }$ , which is fully within the XMMSL1J063045 Slew error circle, based on simultaneous filtered observations of the XMMSL1J063045 field with the optical telescope GROND.",
"The authors suggested it might be the counterpart of XMMSL1J063045.",
"Table REF summarises the AB magnitudes they measured with different filters.",
"The faint brightness (the Galactic reddening at this position is only E(B-V)=0.07; Schlegel et al.",
"1998), coupled with the very blue g'-r' colour evaluated by Kann et al., is atypical for a nova, which discards the classification suggested by Read et al.",
"(2011a) and favours an accretion disc hypothesis.",
"Fitting the source spectrum with a -2 power law, Kann et al.",
"found a deviation in the g' band ($4000-5400 \\rm Å$ ), which they interpreted as due to a strong HeII emission ($\\lambda _{\\rm HeII}=4685 \\rm Å$ ).",
"On December 20, 2011, the Swift satellite also revealed a soft X-ray source coincident with the object detected with GROND.",
"To be specific, the Swift/UVOT UVW1 ($2000-3300 \\rm Å$ ) source position is RA=06:30:45.42, DEC=-60:31:12.54 ($0.44^{\\prime \\prime }$ error circle, $90\\%$ confidence level).",
"From fitting the XRT spectrum with $N_{\\rm H}\\equiv \\it N_{\\rm H_{\\rm Gal}}=\\rm 5.11\\times 10^{20} \\rm cm^{-2}$ , Read et al.",
"(2011b) found a blackbody temperature of $T_{\\rm bb}=48\\pm 5 \\rm eV$ and an absorbed 0.2-2keV flux of $3.4^{+0.8}_{-1.2}\\times 10^{-12} \\rm erg$ $\\rm s^{-1}$ $\\rm cm^{-2}$ , which is a factor of about 12 below the XMM-Newton Slew bright flux.",
"Despite the peculiar features of this source (soft X-ray thermal spectrum, blackbody temperature decrease, high and rapid X-ray flux decay, accretion-disc-like optical-UV spectrum), nothing else can be found in the literature.",
"Table REF lists the whole of the XMMSL1J063045 X-ray observations, also including four observations that were not previously reported in the literature (in italics) and another that was specifically required to check the current state of the source (in bold italics).",
"In the following sections we present our X-ray data analysis and discuss the possible nature of the source."
],
[
"XMM-Newton Slew Survey",
"XMMSL observations were carried out with all the three imaging EPIC cameras (PN, MOS1, and MOS2) onboard XMM-Newton, but the high Slew speed and the slow readout time of MOS1 and MOS2 (Turner et al.",
"2001) prevent MOS data from being analysed.",
"Therefore Read et al.",
"(2011a) analysed only EPIC-PN data (Str$\\rm \\ddot{u}$ der et al.",
"2001) of XMMSL1J063045.",
"XMMSL data are very difficult to analyse, and for this reason we rely on the analysis of Read et al.",
"(Sect.",
")."
],
[
"Swift",
"The composite XRT spectrum of XMMSL1J063045, obtained by grouping the four Swift XRT observations close in time listed in Table REF , can be directly downloaded from the online Swift source catalogue (Evans et al.",
"2014).http://www.swift.ac.uk/1SXPS/spec.php?sourceID=1SXPS$_$ J063045.2-603110 No emission above 2keV is observed.",
"Hence, the source count rates reported in Table REF for each XRT observationhttp://www.swift.ac.uk/1SXPS/1SXPS%20J063045.2-603110 can be approximately associated with the 0.3-2keV energy band.",
"We binned spectral data with the grppha tool of HEASoft (v.6.17) to a minimum of one photon per channel of energy, given the low number of photons, and we adopted Cash-statistics when fitting data.",
"The last XRT observation reported in bold italics in Table REF was reprocessed using xrtpipeline (v.0.13.2), and its corresponding upper limit on the source count rate was evaluated using the XIMAGE (v.4.5.1) task sosta."
],
[
"XMM-Newton",
"The XMM-Newton pointed observation of XMMSL1J063045 (Table REF ) was carried out with all the three XMM-Newton EPIC cameras using the thin filter.",
"We reprocessed data using SAS (v. 13.5.0).",
"We filtered them for periods of high background flaring activity, setting the maximum threshold on the source light curve count rates at 0.4 (0.35)ct $\\rm s^{-1}$ for the PN (MOS) camera.",
"Data were also filtered with the FLAG==0 option, and only events with pattern $\\le $ 4 ($\\le $ 12) were retained.",
"For all the three cameras, we extracted the source$+$ background spectrum from a circular region of radius $40^{\\prime \\prime }$ , centred on the source.",
"We cleaned these spectra of the background, extracted from a circular region of radius $40^{\\prime \\prime }$ on the same CCD, free of sources and bad columns.",
"RMF and ARF files were produced using the appropriate tasks.",
"We binned the obtained source spectra to a minimum of 20 photons per channel of energy.",
"Data were accumulated in the 0.2-2keV (0.3-2keV) energy band for the PN (MOS) camera."
],
[
"X-ray spectral analysis",
"From the XMM-Newton pointed observation of XMMSL1J063045 reported in Table REF we obtained three distinct soft X-ray spectra, one for each EPIC camera (Sect.",
"REF ).We also inspected the XMM-Newton RGS data, but failed to find emission or absorption lines.",
"We fit them together with an absorbed (using TBABS) power-law model from the package XSPEC (v.12.9.0), tying together all the column densities and all the photon indexes.",
"The photon index is extremely high, with $\\Gamma =9.8\\pm 0.2$ (all errors are determined at the $90\\%$ confidence level), and the column density $N_{\\rm H}=(17.41\\pm 0.31)\\times 10^{20} \\rm cm^{-2}$ significantly exceeds the Galactic value $N_{\\rm H_{\\rm Gal}}=5.11\\times 10^{20} \\rm cm^{-2}$ .",
"We obtained a $\\chi ^{2}$ -statistics value of 359.7 with 200 degrees of freedom (dof) and a corresponding null hypothesis probability (nhp) of $10^{-11}$ .",
"An absorbed bremsstrahlung model provides significantly better results: $\\chi ^2=248.7$ with 200 dof and a corresponding nhp of $1.1\\%$ .",
"Even better results are obtained with an absorbed thermal accretion disc (diskbb) model, which also unifies the thermal nature of the XMMSL1J063045 X-ray emission, as identified by Read et al.",
"(2011a), and the accretion disc appearance inferred by Kann et al.",
"(2011) from the source optical emission (Sect.",
").",
"This model, with column densities $N_{\\rm H}$ and diskbb temperatures $T$ tied together, gives $N_{\\rm H}=7.79^{+0.55}_{-0.53}\\times 10^{20} \\rm cm^{-2}$ , somewhat in excess of the Galactic value, and $T=59\\pm 1 \\rm eV$ , returning a $\\rm \\chi ^{2}$ -statistics value of 237.5 with 200 dof and a corresponding nhp of $3.7\\%$ (Fig.",
"REF ).",
"Figure: PN (black), MOS1 (red) and MOS2 (green) soft X-ray spectra obtained from the XMM-Newton pointed observation of XMMSL1J063045 reported in Table , fitted with an absorbed thermal accretion disc model, in agreement with Read et al.",
"(2011a) and Kann et al.",
"(2011; Sect.",
").",
"Residuals in terms of Δχ\\Delta \\chi are plotted in the lower panel with corresponding colours and are well distributed throughout the concerned energy band.Given the low number of photons that appear in the Swift X-ray observations of XMMSL1J063045, we again fit the composite XRT spectrum of the source with the diskbb model, fixing the column density to $N_{\\rm H}=7.79\\times 10^{20} \\rm cm^{-2}$ .",
"This fit gives $T=58^{+6}_{-5} \\rm eV$ , returning a $\\chi ^{2}$ -statistics value of 20.86 with 37 dof, assessed using the Churazov-weighted $\\chi ^{2}$ -statistics (Churazov et al.",
"1996) applied to the best fit with Cash-statistics."
],
[
"X-ray flux variability",
"We now modelled the XMMSL1J063045 (soft) X-ray emission.",
"To do this, we converted the count rates associated with the source X-ray observations into unabsorbed fluxes (Table REF ).",
"From the XMMSL observations, we considered the 0.2-2keV absorbed fluxes that result from the analysis of Read et al.",
"(2011a; Sect.",
").",
"The conversion factor aimed at obtaining the corresponding 0.2-2keV unabsorbed fluxes can be easily estimated based on the Read et al.",
"best spectral fit of the more recent XMMSL data, setting $N_{\\rm H}=0$ .",
"By applying this correction factor to all the three XMMSL observations, we obtained the 0.2-2keV unabsorbed fluxes reported in Table REF .",
"The assumed relative uncertainty on fluxes for XMMSL observations is $10\\%$ .",
"The 0.2-2keV unabsorbed fluxes associated with Swift XRT count rates were computed by means of the conversion factor $(1.83^{+0.31}_{-0.27})\\times 10^{-10} \\rm erg$ $\\rm s^{-1}$ $\\rm cm^{-2}$ ($\\rm ct$ $\\rm s^{-1}$ )$^{-1}$ extracted from the unabsorbed thermal accretion disc spectral model applied to the composite XRT spectrum.",
"In particular, we applied the conversion factor obtained by summing four XRT observations to each XRT observation, assuming the same spectral model also for the last observation in Table REF .",
"Uncertainties on the unabsorbed fluxes result from error propagation.",
"We used the same method to compute the 0.2-2keV (EPIC-PN) unabsorbed flux corresponding to the source XMM-Newton pointed observation and its uncertainty.",
"A relative systematic uncertainty of $10\\%$ , as for XMMSL fluxes, was also considered for all observations according to error propagation, thus justifying the comparison of data carried out with different satellites.",
"Figure REF shows the XMMSL1J063045 X-ray flux light curve without the XMMSL and XRT upper limits.",
"The right top panel shows the light curve with these limits.",
"We fit the unabsorbed fluxes with a $(t_{\\rm MJD}-t_{\\rm 0})^{-5/3}$ power law, typical of a tidal disruption event (Sect.",
"), with $t_{\\rm 0}$ being a characteristic parameter (red solid line).",
"The obtained $\\chi ^2$ is 9.9 with 4 dof ($\\chi ^2_{\\rm red}=2.5$ ) and the corresponding nhp is $4.2\\%$ .",
"A fit with a free power-law index (blue dashed line) gives $\\chi ^2=9.6$ with 3 dof ($\\chi ^2_{\\rm red}=3.2$ ) and a corresponding nhp of $2.2\\%$ .",
"Moreover, the power-law index is $-1.71\\pm 0.04$ , fully in agreement with $-5/3$ .",
"The XRT upper limit is lower than the last fitted flux value, which means that the source is still quiescent in the X-ray band today.",
"Figure: XMMSL1J063045 X-ray flux light curve fitted with a -5/3-5/3 power law (red solid line; χ red 2 =2.5\\chi ^2_{\\rm red}=2.5, nhp =4.2%\\rm nhp=4.2\\%), as for tidal disruption events, and with a free power law (blue dashed line; χ red 2 =3.2\\chi ^2_{\\rm red}=3.2, nhp =2.2%\\rm nhp=2.2\\%), which gives a decay index of -1.71±0.4-1.71\\pm 0.4, fully in agreement with -5/3.",
"The right top panel also includes the upper limits on flux.In addition to the XMMSL1J063045 X-ray light curve, we found a downward trend in the temperatures derived from spectral analysis.",
"In particular, we simulated the XMMSL source spectrum analysed by Read et al.",
"(Sect.",
"; 01/12/2011) using the fakeit option from the package XSPEC and grouped it to a minimum of 20 photons per channel of energy.",
"Fitting it with an absorbed thermal accretion disc model (TBABS*diskbb from XSPEC), we obtained a diskbb temperature of 97eV.",
"The following Swift composite spectrum (Dec. 2011/Jan.",
"2012) and XMM-Newton pointed observation (22/12/2011) show diskbb lower temperatures of $58^{+6}_{-5} \\rm eV$ and $59\\pm 1 \\rm eV$ , respectively.",
"The soft X-ray thermal accretion disc emission of the source together with its temperature decrease and its high and rapid $t^{-5/3}$ flux decay (a factor of about 115 in only a month and a half) are all evidence of the probable TD nature of XMMSL1J063045."
],
[
"Swift UVOT data",
"A further comment on XMMSL1J063045 concerns its activity at lower energies.",
"The left panel of Fig.",
"REF shows no lower energy counterparts of the source before it lights up in X-rays.",
"Swift XRT observations reported in Table REF , subsequent to the source X-ray ignition, are all coupled with Swift UVOT observations, each one carried out using only one filter (uvw1, uvuu, uvw2, uvm2, and uvvv).",
"The central panel of Fig.",
"REF shows the XMMSL1J063045 uvw1 filtered field and the lighting up of a probably lower energy counterpart of the source, immediately after its X-ray activity.",
"The right panel of Fig.",
"REF shows no lower energy counterparts of the source in its uvvv filtered field, about four years after the source detection in X-rays.",
"Table REF also collects the source counterpart AB magnitudes associated with the five differently filtered UVOT observations, assessed with the uvotsource tool of HEASoft without correcting for Galactic extinction.",
"This clearly is a soft X-ray/ optical-UV transient."
],
[
"XMMSL1J063045 host galaxy",
"The main factor in stellar tidal disruptions is the destroyer BH, whose mass can be approximately related to the source luminosity distance $d$ through Eq.",
"REF .",
"For XMMSL1J063045, the unabsorbed peak bolometric flux $F_{\\rm peak}$ that appears in Eq.",
"REF can be inferred by fitting our simulated XMMSL spectrum (Sect. )",
"with the best spectral model of Read et al.",
"(2011a) by setting $N_{\\rm H}=0$ and extrapolating data in the 0.01-10keV energy band.",
"The flux is $2.8\\times 10^{-10} \\rm erg$ $\\rm s^{-1}$ $\\rm cm^{-2}$ , so that $d\\sim 0.06 \\rm Mpc \\Bigl (\\frac{\\it M_{\\rm BH}}{1 \\rm M_{\\rm \\odot }}\\Bigr )^{1/2}, $ which is 60Mpc (redshift $z$ =0.014; $H_{\\rm 0}=69.6 \\rm km$ $\\rm s^{-1}$ $\\rm Mpc^{-1}$ , $\\Omega _{\\rm m}=0.286$ ) for a BH of mass $10^6 \\rm M_{\\rm \\odot }$ .",
"An upper limit on XMMSL1J063045 distance ($z$ ) can be assessed by imposing a maximum value for $M_{\\rm BH}$ of $10^8 \\rm M_{\\rm \\odot }$ , as for tidally disrupted solar-type stars.",
"This limit is $600\\rm Mpc$ ($z$ =0.13).",
"We here assumed that the observed peak luminosity of the source coincides with its Eddington limit.",
"On one hand, we are aware that this limit can be exceeded by a factor of several, but on the other hand, we note that the actual outburst peak is probably brighter than the value we infer from the bright Slew detection.",
"Hence we consider the Eddington limit as an acceptable compromise between these two competing instances.",
"The XMMSL1J063045 redshift and, consequently, luminosity distance might be inferred from its host galaxy spectroscopy, provided that there is a host galaxy, which should be in the case of TDs.",
"No signs of it can be found in the DSS image (Fig.",
"REF , left panel) or in the Swift UVOT uvvv filtered image (Fig.",
"REF , right panel).",
"On January 9, 2016, we obtained a deep 300 s V-band ESO-NTT image of the field of XMMSL1J063045.",
"This was carried out with the EFOSC2 instrument.",
"The observations were taken as part of the Public ESO Spectroscopic Survey of Transient Objects (PESSTOwww.pessto.org), and details of data products and reductions can be found in Smartt et al.",
"(2015).",
"Calibrating it through the identification of four objects in the UVOT uvvv filtered image using the uvotsource tool of HEASoft and the GAIA software (v. 4.4.1), we find an object of apparent V magnitude $m_{\\rm V}\\sim 23.26\\pm 0.27$ at the UVOT position of XMMSL1J063045 (Fig.",
"REF , green circle).",
"This probably is the XMMSL1J063045 (dim) host galaxy.",
"Figure: Zoom of a reduced 300 s ESO-NTT EFOSC2 recent image of the XMMSL1J063045 field in the V filter (09/01/2016, about four years after XMMSL1J063045 X-ray detection).",
"The image is 0.9 ' ×0.9 ' 0.9^{\\prime } \\times 0.9^{\\prime }, north is up, east is left.",
"A dim extended source, possibly the XMMSL1J063045 host galaxy, is visible within 2.5 '' 2.5^{\\prime \\prime } from XMMSL1J063045 UVOT position (green circle).Currently, the dimmest (and smallest) galaxy associated with a TD is WINGS J134849.88+263557.7 in Abell1795 (Maksym et al.",
"2013; Maksym et al.",
"2014a; Donato et al.",
"2014).",
"It is a faint ($m_{\\rm V}=22.46$ ) dwarf ($r\\sim 300 \\rm pc$ ) galaxy lying at the same redshift of the cluster ($z=0.062$ ; $M_{\\rm V}\\sim -14.8$ ), possibly hosting an IMBH ($M_{\\rm BH}=(2-5)\\times 10^5 \\rm M_{\\rm \\odot }$ ).",
"The problem for XMMSL1J063045 is that no spectra deep enough of its dim host galaxy are currently available in which absorption lines might be identified to evaluate $z$ .",
"Using Eq.",
"REF to estimate the source luminosity distance $d_{\\rm pc}$ (in parsec) from a BH of mass $10^4-10^6 \\rm M_{\\rm \\odot }$ and considering the relation between the absolute magnitude ($M_{\\rm V}$ ) and the apparent magnitude ($m_{\\rm V}$ ), the dim host ends up with $M_{\\rm V}\\sim -5.7\\div -10.7$ .",
"When $L_{\\rm peak}$ is assumed to be ten times the Eddington luminosity (which is very high), $M_{\\rm V}$ lies in the range $-8.2\\div -13.2$ .",
"This value is at the level of the faintest dwarf spheroidal galaxies in our Milky Way (Sculptor has $M_{\\rm V}=-10.7$ ) or of the brightest globular clusters (NGC5139 has $M_{\\rm V}=-10.2$ ), opening the possibility of observing the first TD in a globular cluster and suggesting that IMBHs are present in the cores of at least some of them."
],
[
"Summary and conclusions",
"In a galactic nucleus, a star that approaches to the central black hole too closely may be tidally disrupted before it is fully absorbed into the black hole horizon.",
"A fraction of the produced stellar debris is bound to the compact object, accreting onto it through an accretion disc and lighting it up through a characteristic flare (Rees 1988; Phinney 1989).",
"Up to now, a limited number ($\\sim 65$ ) of tidal disruption candidates have been observationally identified,https://astrocrash.net/resources/tde-catalogue/ mainly in the optical-UV and soft X-ray bands (e.g.",
"Gezari 2012 and references therein; Komossa 2015 and references therein), both involving massive galaxies, that is, supermassive black holes (Kormendy & Richstone 1995; Kormendy & Gebhardt 2001), and dwarf galaxies, which possibly host intermediate-mass black holes at their centres (Ghosh et al.",
"2006; Maksym et al.",
"2013; Maksym et al.",
"2014a; Donato et al.",
"2014; Maksym et al.",
"2014b).",
"The discovery of new tidal disruption candidates would certainly improve our understanding of the physics behind them, and if they were detected in dwarf galaxies, we might be able to determine plausible destroyer intermediate-mass black holes.",
"This class of black holes is currently under study (e.g.",
"Ptak & Griffiths 1999; Davis & Mushotzky 2004; Wolter et al.",
"2006; Greene & Ho 2007; Farrell et al.",
"2009; Irwin et al.",
"2010; Jonker et al.",
"2010; Krolik & Piran 2011; Jonker et al.",
"2013) as the connecting bridge between stellar-mass and supermassive black holes and the raw material that makes up supermassive black holes (e.g.",
"Volonteri 2010).",
"On December 1, 2011 the new point-like source XMMSL1J063045.9-603110 was detected to be bright in the soft X-rays, with an underlying thermal emission (Read et al.",
"2011a).",
"An accretion-disc nature was suggested (Kann et al.",
"2011).",
"After about twenty days, XMMSL1J063045.9-603110 was also observed by the Swift satellite, again producing a soft X-ray thermal emission, a factor of about 10 below its first detection (Read et al.",
"2011b).",
"We reported here a comprehensive data analysis of all the available X-ray (Table REF ) and lower energy data (Sect. )",
"of XMMSL1J063045.9-603110.",
"We suggest that the source is a tidal disruption event.",
"It showed an accretion-disc-like thermal spectrum in the soft X-rays (Fig.",
"REF ) together with a high and rapid flux decay (a factor of $\\sim 115$ in only a month and a half) that is well modelled by a power law of index -5/3 (Fig.",
"REF ).",
"Moreover, the source also blazed up at lower energies (Fig.",
"REF ), even if it slightly lags behind the X-ray flaring.",
"We reject the hypothesis that XMMSL1J063045.9-603110 is a Galactic nova (Galactic latitude $b=-26$ ).",
"The softness of the source X-ray spectrum would require such a nova to be in a super-soft state (SSS).",
"To reach this state, the source would need to radiate at Eddington ($\\sim 1.3\\times 10^{38} \\rm erg$ $\\rm s^{-1}$ ) and would have to lie at $d \\sim 65 \\rm kpc$ .",
"Furthermore, the source magnitude variation is about 5 mag, which is too small for a typical nova, and its quiescent magnitude (Sect. )",
"is too high for a typical super-soft nova (the dimmest one, GQ Mus, has a quiescent V magnitude of $\\sim 18$ ; e.g.",
"Warner 2002).",
"The reported magnitudes are not enhanced by Galactic extinction, given that this is very low at the source position (Schlegel et al.",
"1998).",
"We also discard the idea that XMMSL1J063045.9-603110 is an AGN because it would have been detected in all the observations if that were the case (see also the discussion in Campana et al.",
"2015).",
"Based on the hypothesis that XMMSL1J063045.9-603110 is a candidate tidal disruption event, the low diskbb temperature that characterises the source ($\\sim 100 \\rm eV$ ; Sect. )",
"would call for a $\\sim 10^4 \\rm M_{\\rm \\odot }$ destroyer black hole, assuming that it accretes at the Eddington rate, or a $\\sim 10^5 \\rm M_{\\rm \\odot }$ black hole, assuming that it accretes at ten times the Eddington rate.",
"It might be a tidal disruption event in a very dim dwarf galaxy of even in a very bright globular cluster ($M_{\\rm V}\\sim -10$ ), which then could host a black hole at their centre.",
"Globular clusters typically do not wander alone in the cosmos, but are associated with a parent galaxy.",
"Figure REF shows that the field of XMMSL1J063045.9-603110 is sparsely crowded, but there is something around it, possibly also a parent galaxy.",
"Spectroscopic observations of XMMSL1J063045.9-603110 host will provide a clearer answer.",
"This work is based (in part) on observations collected at the European Organisation for Astronomical Research in the Southern Hemisphere, Chile as part of PESSTO (the Public ESO Spectroscopic Survey of Transient Objects) ESO program 188.D-3003, 191.D-0935.",
"We thank P. D'Avanzo and S. J. Smartt for useful comments and discussions.",
"We acknowledge M. C. Baglio for useful comments about image calibration.",
"We also thank the anonymous referee for valuable comments on the manuscript and constructive suggestions."
]
] | 1605.06133 |
[
[
"Unlimited-Power Reflectors, Absorbers, and Emitters with Conjugately\n Matched Layers"
],
[
"Abstract In order to ensure the fastest wireless energy transfer from a source to the user one needs to maximize the channel capacity for power transport.",
"In communications technologies, the concept of MIMO (multiple input, multiple output) exploits the idea of sending signals via many different rays which may reach the receiver.",
"However, if we are concerned with the task of energy transfer, still only one mode is exploited, even if multiple antennas are used to send power to the receiver.",
"In the near-field scenario, this is the magnetic dipole mode of receiving coil antennas.",
"In the far-field scenario, this is the propagating plane wave TEM mode.",
"Recently, it was shown that using special artificial materials it is possible to ensure that all electromagnetic modes of free space are conjugately matched to the modes of a material body and, thus, all modes deliver power to the body in the most effective way.",
"Such a fascinating feature is acquired because the conjugate matching does not concern only the propagating modes but, most importantly, is applied to all evanescent modes.",
"However, coupling to higher-order (mostly evanescent) modes is weak and disappears in the limit of an infinite planar boundary.",
"We show that properly perturbing the surface of the receiving or emitting body with, for example, randomly distributed small particles we can open up channels for super-radiation into far zone.",
"The currents induced in the small particles act as secondary sources which send the energy to travel far away from the surface and, reciprocally, receive power from far-located sources.",
"We theoretically predict about 20-fold power transfer enhancement between the conjugately matched power-receiving body (as compared with the ideal black body) and far-zone sources.",
"Reciprocally, the proposed structure radiates about 20 times more power into far zone as compared with the same source over a perfect reflector."
],
[
"Introduction",
"The problem of optimizing and maximizing absorption and emission of electromagnetic energy is relevant for a broad variety or applications such as antennas, radar absorbers, thermal emitters and accumulators, photovoltaic devices and more.",
"For macroscopic bodies (having sizes large compared to the wavelength of electromagnetic radiation), it is usually assumed that the ultimate absorber and emitter is the ideal black body that completely absorbs all incident rays.",
"Conceptually ideal black body, introduced by Kirchhoff [1], is totally opaque and has zero reflection coefficient for any propagating plane wave (any incidence angle and any polarization), and in this sense it is the perfect absorber of electromagnetic energy.",
"Following the Planck theory of thermal radiation [2], the ideal black body appears to be also the ultimate thermal emitter for radiating heat into free space.",
"Practical realization of bodies whose properties mimic those of ideal black bodies is a scientific and technical challenge, see e.g.",
"[3], [4].",
"However, it has been recently demonstrated that, in principle, it is possible to engineer bodies which can absorb power not only from incident propagating waves (incident rays in the Kirchhoff black-body concept), but also from external evanescent fields or high-order spherical harmonics of the incident-wave spectrum [5], [6].",
"Due to the resonant nature of surface modes of these superabsorbing and superemitting bodies, their absorption cross section grows without limit when the medium parameters approach the ideal values, and the thermal spectral emissivity at the resonance frequency becomes arbitrarily high compared to Planck's black body of the same size and the same temperature.",
"The material structures proposed in [5], [6] realize the ideas of conjugate matching of all modes of free space to all modes of the absorbing/emitting body [7], [8], [9], and we call them conjugately matched bodies or layers (CML).",
"The material realizations proposed in [5], [6] are based on the use of double-negative (DNG) isotropic or uniaxial media which obey the uniaxial perfectly matched layer (PML) conditions [10].",
"High-order modes of conjugately matched bodies [5], [6] resonate with all modes of free space and most effectively exchange energy with them.",
"However, to effectively absorb or emit all the modes, it is necessary that the modes of the body are sufficiently coupled with the corresponding modes of free space.",
"In [5], it is shown that if a conjugately matched body fills a half space (a planar infinite interface with free space), its properties are the same as of the conventional ideal black body and no absorption or radiation enhancement over the ideal black-body limit can take place.",
"To couple with higher-order free-space modes, we can make the surface not planar, and in [5] it is shown that for bodies of finite sizes, on an example of a sphere, unlimited power exchange power is indeed possible.",
"An alternative scenario was explored in paper [6], where the conjugately matched body filled a half space with an infinite planar interface, but the sources in free space were positioned close to the interface and created large evanescent fields which directly couple to the resonant surface modes of the conjugately matched layer.",
"Also in this case it was seen that the absorption in the body was dramatically stronger than in an ideal black body at the same position.",
"In this paper, we show that it is possible to realize an infinite and planar surface which can absorb and emit more power than the ideal black body by perturbing the surface of a conjugately matched layer, introduced in [6].",
"In this scenario, small subwavelength scatterers randomly distributed over the body surface, offer necessary coupling between high-order resonant surface modes and the far-zone fields, opening channels for extra absorption or emission of energy.",
"In the limit of ideal material parameters, this planar interface not only absorbs or reflects all incident propagating waves, but does the same also for all evanescent harmonics.",
"We show that a perturbed interface with a low-loss conjugately matched body acts as a “super-reflector” of fields created by a small antenna in its vicinity by launching the energy stored in the antenna near field into space.",
"In particular, we introduce a random grid of electrically thin cylinders close to a resonant interface with a conjugately matched layer, where huge reactive energy is stored.",
"Inevitably, currents induced in thin conductive cylinders radiate into far zone as linear antennas, and we say that these cylinders act as radiation “vessels”.",
"A random and sparse enough distribution of cylinders ensures that diffuse radiation survives in the far zone and is not coherently combined into a plane wave.",
"We test the effects of this cluster of particles on the radiation from various conjugately matched layers and conclude that for a realizable passive structure one can achieve a stable 20-30-fold enhancement of the far-field power.",
"Such super-reflectors are extremely strongly coupled to evanescent fields of external sources and can extract power from them in the most efficient way.",
"In the antenna terminology, the effective area of the CML reflector is larger than the geometrical one, although the reflector size is very large compared with the wavelength.",
"Basically, we aim at realization of a surface which (at its resonant frequency) would be “more reflective than the ideal perfectly conducting mirror”, and this property would hold even in the limit of the infinite planar reflector.",
"If the perturbing elements are lossy, instead of enhancing reflection we can enhance absorption in a planar absorbing layer or enhance thermal radiation from a planar hot surface into far zone beyond the Planck limit of the ideal black body.",
"We expect that perturbing the surface can be a more effective mean to couple to far-zone field as compared to curved surfaces.",
"In the study [5] it was expectedly found that for large spherical bodies, when the curvature of the surface becomes small, one needs extreme (low loss) values of material parameters in order to realize enough effective coupling to high-order harmonics.",
"Surface perturbation approach, introduced here, does not have this limitation."
],
[
"Conjugately Matched Layer (CML)",
"We begin the study by a brief explanation of the concept of the conjugately matched layer, introduced in [6].",
"It has been recently reported [5], [6] that there can exist material bodies which optimally absorb energy of electromagnetic fields, by achieving conjugate matching for every free-space mode.",
"In the theoretical limit of negligible losses in the absorbing body, such an optimally designed finite-sized body can absorb the whole infinite energy of an incident propagating plane wave [5].",
"In [6], a half-space and a cylinder filled with a uniaxial medium with special values for its constituent parameters have been suggested as possible realizations.",
"The permittivities and permeabilities (transversal with subscript $t$ and normal to the material sample boundary with subscript $n$ ) satisfy the uniaxial perfectly matched layer (PML) conditions [10], [11], [12], [13], [14] and simultaneously possess negative real parts as in a double negative (DNG) medium [15], contrary to the conventional uniaxial PML choice.",
"For planar interfaces and TM polarization (sole magnetic component parallel to the half-space boundary), the material parameters satisfy $\\varepsilon _{rt}=\\mu _{rt}=\\frac{1}{\\varepsilon _{rn}}=a-jb,$ where $a$ and $b$ are real parameters and $a<0$ .",
"The parameter $b>0$ corresponds to losses for propagating plane waves, and it is easy to show [10], [20] that sufficiently thick slabs of such materials behave as perfect absorbers for arbitrary incident propagating plane waves.",
"We assume harmonic time dependence $\\exp (+j\\omega t)$ , where $\\omega $ is the angular operating frequency.",
"From duality, a similar expression for the parameters of uniaxial perfect absorbers holds for the fields of the TE polarization: $\\varepsilon _{rt}=\\mu _{rt}=\\frac{1}{\\mu _{rn}}=a-jb$ .",
"To ensure that the thought properties hold for both orthogonal polarizations, we can require that $\\varepsilon _{rn}=\\mu _{rn}$ .",
"For compactness, in the following we present only formulas for the TM polarization, without compromising the generality.",
"Any uniaxial medium characterized by the constituent parameters $(\\varepsilon _{rt},\\mu _{rt},\\varepsilon _{rn})$ , has the following TM wave impedance $Z$ (e.g., [20]): $Z=-j\\frac{\\eta _0}{k_0\\varepsilon _{rt}}\\sqrt{\\frac{\\varepsilon _{rt}}{\\varepsilon _{rn}}k_t^2-\\varepsilon _{rt}\\mu _{rt}k_0^2},$ which relates the tangential to the interface components of electric and magnetic fields of plane waves in the medium.",
"The notations $\\eta _0=\\sqrt{\\mu _0/\\varepsilon _0}$ and $k_0=\\omega \\sqrt{\\varepsilon _0\\mu _0}=2\\pi /\\lambda _0$ correspond to the free-space wave impedance and wavenumber, respectively ($\\varepsilon _0$ and $\\mu _0$ are the permittivity and permeability of vacuum, while $\\lambda _0$ is the operational wavelength in free space).",
"The symbol $k_t$ is used for the transversal wavenumber of the incident plane wave.",
"Vector $\\textbf {k}_t$ is parallel to the boundary of the half space and normal to the sole component of magnetic field.",
"The basic property of a material with the constituent parameters given by (REF ) when $a<0$ , is that its wave impedance $Z$ is the complex conjugate of the TM wave impedance of vacuum $Z^*=Z_0=-j\\frac{\\eta _0}{k_0}\\sqrt{k_t^2-k_0^2}$ [6].",
"Most importantly, this equality is valid for every real wavenumber $k_t\\in \\mathbb {R}$ , either of a propagating wave in free space ($|k_t|<k_0$ ) or of an evanescent mode ($|k_t|>k_0$ ).",
"Therefore, the use of such conjugate matched layers (CLM), as we call them, leads to maximal power transfer from arbitrary incident fields into the medium since they optimally use every possible way (mode) available from sources outside of the material sample.",
"In particular, the TM impedance in vacuum $Z_0$ is real for propagating fields and accordingly $Z=Z_0$ , which means that no power is reflected from the half-space surface, exactly as in the conventional uniaxial PML case [10].",
"However, for evanescent fields we have a purely imaginary $Z_0$ , which means that $Z=Z_0^*=-Z_0$ .",
"Thus, the maximal power transfer is achieved (in the limit of very small but non-zero losses in the medium) and simultaneously maximal reflections are taking place.",
"Actually, in this ideal case of overall lossless conjugately matched medium, fields tends to infinity at the material surface.",
"Assuming infinitesimally small losses in the CML, infinite power can be delivered to the medium, provided that the incident evanescent field is created by an antenna fed by an ideal voltage or current source, capable of supplying infinite power.",
"In other words, the CML structure is identical to ordinary PML for $|k_t|<k_0$ but operates totally differently for $|k_t|>k_0$ aiming not at zero reflection but at the maximal power transfer.",
"Figure: (a) The test-bed configuration of a grounded electrically thick slab filled with a uniaxial material (ε rt ,μ rt ,ε rn )(\\varepsilon _{rt},\\mu _{rt},\\varepsilon _{rn}) with the thickness LL, excited by a TM electric dipole line inclined by the angle θ\\theta located at a small distance gg from the air-medium interface.",
"(b) The same structure in the presence of a cluster of NN electrically thin (the radius rr) circular perfect magnetic conductor (PMC) cylinders randomly distributed in the vicinity of the air-slab interface with arbitrary coordinates (x n ,y n )(x_n,y_n) for n=1,⋯,Nn=1,\\cdots ,N.By inspection of (REF ) one can directly infer that if $b>0$ , the transversal relative constituent parameters $\\varepsilon _{rt},\\mu _{rt}$ , are lossy; however, the normal component of the permittivity $\\varepsilon _{rn}$ is an active one.",
"In order to identify the overall character of the uniaxial medium, we consider a perturbed version of the ideal material parameter values by using a small additional parameter $\\delta $ controlling the imaginary part of the normal permittivity $\\varepsilon _{rn}=\\frac{1}{\\varepsilon _{rt}}-j\\delta =\\frac{1}{a-jb}-j\\delta $ , which tends to the ideal CML medium parameters for $\\delta \\rightarrow 0$ .",
"To study the properties of such a quasi-CML medium, we use the test-bed configuration illustrated in Fig.",
"REF .",
"For simplicity of analytical considerations, we assume that there is no dependence on one of the tangential coordinates ($z$ ).",
"A grounded slab of the thickness $L$ , filled with a uniaxial material with the constituent parameters $(\\varepsilon _{rt},\\mu _{rt},\\varepsilon _{rn})$ , is excited by an infinite electric-dipole line located at the distance $g$ from the air-medium interface.",
"The exciting dipole moments are orthogonal to the axis $z$ and inclined by the angle $\\theta $ with respect to the axis $x$ (Fig.",
"REF ).",
"We choose such a source since the spectral content for the evanescent modes is much more significant as compared to the common cylindrical wave (in the same way that has been used in [16]).",
"In this configuration, the magnetic field has only one non-zero component (along $z$ ) and the fields are TM-polarized.",
"In [6], an approximate analytical formula for the absorbed power (per unit length along the $\\hat{\\textbf {z}}$ axis) has been derived.",
"It shows that the absorbed power is a sum of two terms.",
"The first term corresponds to the power delivered by the propagating modes $P_{\\rm prop}=\\mu _0\\omega ^3q^2/16$ , and it is independent from the angle $\\theta $ .",
"Here $q$ is the electric dipole moment per unit length of the line (measured in $Coulomb$ ).",
"The second term gives the power absorbed from the evanescent-modes fields and is given by [6] $P_{\\rm evan}\\cong P_{\\rm prop} \\frac{8|a|}{k_0^2\\pi } \\int _{k_0}^{+\\infty }\\frac{k_t^2\\left(k_t^2-k_0^2\\sin ^2\\theta \\right)}{\\left(k_t^2-k_0^2\\right)^{3/2}}e^{-2g\\sqrt{k_t^2-k_0^2}}\\frac{\\delta }{\\left[1+\\operatorname{sgn}(a)\\right]^2+\\delta ^2\\left[\\frac{k_t^2 |\\varepsilon _{rt}|}{2\\left(k_t^2-k_0^2\\right)}\\right]^2}dk_t,$ which is an approximate expression valid under the assumption that $\\delta \\rightarrow 0$ .",
"It is remarkable that in the CML case ($a<0\\Rightarrow \\operatorname{sgn}(a)=-1$ ), the absorbed power from the evanescent-mode fields behaves as $P_{\\rm evan}=O\\left(1/\\delta \\right)$ for small $\\delta $ , namely, we obtain extremely high values of $P=P_{\\rm prop}+P_{\\rm evan}$ , whose sign is the same as the sign of the perturbation variable $\\delta $ .",
"In other words, the CML slab acts as an ultra-efficient passive absorber ($P\\rightarrow +\\infty $ ) of the incoming illumination for $\\delta >0$ and as an infinite-power active emitter ($P\\rightarrow -\\infty $ ) for $\\delta <0$ .",
"In the limit of $\\delta \\rightarrow 0^+$ , both the field strength at the surface and the absorbed power diverge and tend to infinity.",
"Therefore, it would be meaningful to inspect the field distributions leading to such unbounded field concentrations."
],
[
"Excitation of CML",
"Let us examine the fields created by a small source in the vicinity of an infinite and homogeneous CML slab within the test-bed setup shown in Fig.",
"REF .",
"The corresponding boundary-value problem is scalar, and the magnetic field possesses a sole component parallel to $\\hat{\\textbf {z}}$ axis ($\\textbf {H}=\\hat{\\textbf {z}}H(x,y)$ ).",
"The used Cartesian coordinate system $(x,y,z)$ is also defined in Fig.",
"REF , with the primary dipole line source positioned at $(x,y)=(0,0)$ .",
"The incident magnetic field from that electric-dipole line (existing in vacuum) can be expressed in the following integral form [16]: $H_{\\rm inc}(x,y)=-\\frac{\\omega q}{4\\pi }\\int _{-\\infty }^{+\\infty }e^{-|x|\\kappa _0(k_t)}\\left[\\frac{k_t}{\\kappa _0(k_t)}\\cos \\theta +j\\sin \\theta \\operatorname{sgn}(x)\\right]e^{-jk_ty}dk_t,$ where the normal to the interface component of the wavenumber $\\kappa _0(k_t)=\\sqrt{k_t^2-k_0^2}$ is evaluated with a positive real part and in case when the real part is zero, as a positive imaginary number.",
"Analytical expression for the incident field involving Hankel function [16], [17] is also available but not given here for brevity, since all the field quantities are expressed as spectral integrals.",
"The formulated boundary-value problem can be solved analytically.",
"As a result, we find that the secondary field developed due to the presence of the uniaxial slab and the PEC plane in free space ($x<g$ ) is given by: $H_{\\rm sec}(x,y)=\\int _{-\\infty }^{+\\infty }S_{\\rm sec}(k_t)e^{\\kappa _0(k_t)x-jk_ty}dk_t$ , where $S_{\\rm sec}(k_t)=\\frac{\\omega q}{4\\pi }e^{-2g\\kappa _0(k_t)}\\left(\\frac{k_t}{\\kappa _0(k_t)}\\cos \\theta +j\\sin \\theta \\right)\\frac{\\kappa (k_t)-\\varepsilon _{rt}\\coth [\\kappa (k_t)L]\\kappa _0(k_t)}{\\kappa (k_t)+\\varepsilon _{rt}\\coth [\\kappa (k_t)L]\\kappa _0(k_t)}.$ The value of $\\kappa (k_t)=\\sqrt{k_t^2\\frac{\\varepsilon _{rt}}{\\varepsilon _{rn}}-k_0^2\\varepsilon _{rt}\\mu _{rt}}$ is the normal component of the plane-wave propagation constant in the CML slab.",
"The total field in vacuum equals to $H_{\\rm back}(x,y)=H_{\\rm inc}(x,y)+H_{\\rm sec}(x,y)$ .",
"Figure: The magnitude of the spatial spectral density function of the secondary field S sec (k t )S_{\\rm sec}(k_t) at x=0x=0 with respect to the normalized wavenumber k t /k 0 k_t/k_0 for various inclination angles θ\\theta and: (a) a=2a=2 (DPS-PML) and (b) a=-2a=-2 (CML).",
"Common plot parameters: b=0.1b=0.1, δ=0.001\\delta =0.001, g=0.03λ 0 g=0.03\\lambda _0, L=3λ 0 L=3\\lambda _0.",
"The represented quantity is normalized by S inc (90 ∘ )=jωq/(4π)S_{\\rm inc}(90^{\\circ })=j\\omega q/(4\\pi ).In Figs.",
"REF we present the magnitude of the integrand in the formula of the secondary magnetic field for $x=0$ (very close to the air-CML boundary located at $x=g$ ), equal to $|S_{\\rm sec}(k_t)|$ as a function of the normalized transversal wavenumber $k_t/k_0$ for various inclination angles of the source $\\theta $ .",
"The presented quantity is normalized by the (constant) magnitude of the integrand of the incident field (REF ) for $\\theta =90^{\\circ }$ : $S_{\\rm inc}(90^{\\circ })=\\frac{j\\omega q}{4\\pi }$ , which is independent from $k_t$ and gives us a metric of the incident power.",
"Figure REF corresponds to a double-positive (DPS) conventional uniaxial PML [10] and it is directly observed that $|S_{\\rm sec}|$ vanishes exponentially for evanescent waves ($|k_t|>k_0$ ).",
"On the contrary, for the CML case (Fig.",
"REF ) the integrand values have huge magnitudes for $|k_t|>k_0$ regardless of the angle $\\theta $ .",
"These graphs verify the aforementioned theoretical expectation that unbounded absorbed power of (REF ) in the CML case ($a<0$ and $|\\delta |\\rightarrow 0$ ) is due to the extremely large magnitudes of the evanescent fields developed in the vicinity of the interface, as demonstrated by Fig.",
"REF .",
"Note the different scale in Figs.",
"REF and REF : the values in the region $-1<k_t/k_0<1$ , which correspond to the propagating-wave part of the spectrum, are the same in both figures.",
"Figure: The spatial distributions of the total magnetic field H back (x,y)H_{\\rm back}(x,y) normalized by H inc (g,0)H_{\\rm inc}(g,0) expressed in dBdB for: (a) a=2a=2 (DPS-PML) and (b) a=-2a=-2 (CML).",
"Common plot parameters: b=0.1b=0.1, δ=0.001\\delta =0.001, g=0.03λ 0 g=0.03\\lambda _0, L=3λ 0 L=3\\lambda _0, θ=90 ∘ \\theta =90^{\\circ }.Figures REF show the spatial distribution of the total magnetic field $|H_{\\rm back}(x,y)|$ for the two cases of Figs.",
"REF (with $\\theta =90^{\\circ }$ ).",
"The represented quantity is normalized by the value of the incident field at $(x,y)=(g,0)$ and is expressed in $dB$ .",
"We again observe how more efficient is the CML medium (Fig.",
"REF ) in exciting fields along its boundary compared to the conventional PML case (Fig.",
"REF ).",
"However, since the nature of these fields is evanescent, they are rapidly decaying with increasing the distance from the surface ($x\\rightarrow -\\infty $ ).",
"It should be stressed that the concentration of the fields in the vicinity of the CML interface is always huge regardless of the sign of $\\delta $ , both for overall passive ($\\delta >0$ ) or active ($\\delta <0$ ) structures.",
"In this paper, we propose to make use of this concentration of fields along the boundary of the two regions ($x=g$ ) to create an “antenna,” which would “launch” the energy stored in this region into the far-zone region $x\\rightarrow -\\infty $ .",
"This is not an easy task, though.",
"It is well-known that resonant surface modes at infinite and regular surfaces do not radiate energy into far zone.",
"In other words, despite the huge difference of the two systems (PML versus CML slab) in the near field, the behavior of the field radiated in the far region is similar.",
"Actually, with the use of the stationary phase method, one can directly evaluate the azimuthal profiles of the incident and the secondary fields in the far field as follows: $h_{\\rm inc}(\\varphi )\\sim \\frac{k_0j\\omega q}{4}\\sin (\\varphi -\\theta ),~~~x\\rightarrow -\\infty ,$ $h_{\\rm sec}(\\varphi )\\sim -\\pi k_0 S_{\\rm sec}(k_0\\sin \\varphi )\\cos \\varphi ,~~~x\\rightarrow -\\infty ,$ for $90^{\\circ }<\\varphi <270^{\\circ }$ .",
"We notice that the expression of the secondary component, which describes the effect of the grounded slab, is proportional to a specific value of the function $S_{\\rm sec}(k_t)$ : the one corresponding to $k_t=k_0\\sin \\varphi $ .",
"Since this value is always smaller in magnitude than $k_0$ ($\\varphi \\in \\mathbb {R}$ ), namely corresponding to a propagating and not to an evanescent mode, it is clear that huge reactive fields of Fig.",
"REF do not to contribute to far-zone radiation.",
"We need something that can act as a radiation “vessel” to allow the field energy stored in resonant surface modes to propagate far away from the source."
],
[
"Circuit Theory Approach",
"In an attempt to find a way to exploit this huge field concentration and transform the sizeable magnitude of evanescent modes (developed close to $x=g$ ) into radiative fields, we consider the configuration of Fig.",
"REF .",
"Let us randomly distribute small cylindrical scatterers in the vicinity of the air-CML slab interface.",
"It is expected that the large evanescent fields would excite currents along these wires, which will act as radiation vessels, and their own field would be expressed as cylindrical modes which are always partially propagating and survive in the far region.",
"Figure: Sketch of the equivalent circuit for excitation by a particular evanescent TM plane-wave component.",
"The wave impedance of TM waves in vacuum Z 0 Z_0 corresponds to the capacitance C VAC C_{\\rm VAC}, while the impedance ZZ of the CML contains a loss resistor R CML R_{\\rm CML} and an inductive L CML L_{\\rm CML} component.",
"The cylindrical vessel in the near field of vacuum-CML interface is characterized by a radiation resistance R RAD R_{\\rm RAD} and a reactive impedance X RAD X_{\\rm RAD}.The idea of perturbing the surface with tiny scatterers can be understood from the equivalent circuit corresponding to the fields of a particular evanescent plane-wave component exciting the CML slab in presence of a small scatterer, shown in Fig.",
"REF .",
"The ideal voltage source $V$ represents the primary radiator (an electric dipole source in this configuration) and capacitance $C_{\\rm VAC}$ expresses the wave impedance of free space for a specific value of $k_t$ , given by (REF ) for $\\varepsilon _{rt}=\\mu _{rt}=\\varepsilon _{rn}=1$ .",
"The inductive complex impedance $(R_{\\rm CML}+j\\omega L_{\\rm CML})$ is given by (REF ) with the parameters of the CML layer.",
"Resistance $R_{\\rm CML}$ models the dissipative losses in the CML slab and the current flown through the primary circuit is denoted by $I$ .",
"The small scatterer (radiation “vessel”) in the vicinity of the interface is modeled by the radiation vessel circuit, of current $I_{\\rm RAD}$ , formed by a non-resonant reactive element $(jX_{\\rm RAD})$ (capacitive or inductive) and a small resistor of the radiator $R_{\\rm RAD}$ .",
"If the scatterer is lossless, $R_{\\rm RAD}$ corresponds to the radiation resistance, and in case when it is absorptive, the resistance is the sum of the radiative and dissipative term.",
"Near-field coupling between the scatterer and the resonant surface mode of the interface is modeled for simplicity by mutual inductance $j\\omega M$ .",
"In general, the mutual impedance $j\\omega M$ can be a complex number with any sign of the imaginary part; however, here we confine our analysis to a very closely positioned particle, in which case the mutual impedance is predominantly reactive ($M\\in \\mathbb {R}$ ) and inductive for the considered TM polarization.",
"The circuit in the absence of the radiation vessels $(M=0)$ has been analyzed in [6], and it is clear that the power delivered to the loss resistor $R_{\\rm CML}$ tends to infinity when the series $LC$ circuit works at resonance and under the additional condition $R_{\\rm CML}\\rightarrow 0$ .",
"In other words, the absorbed power is limited only by the energy available from the primary source, while there is no radiation towards the far zone $(P_{\\rm RAD}=0)$ .",
"In the presence of the vessels, however, the systems behaves dramatically different.",
"Considering the circuit of Fig.",
"REF , we can easily find the current amplitude both in the directly fed branch ($I$ ) and in the circuit of the radiating vessel ($I_{\\rm RAD}$ ): $I=\\frac{V\\left(R_{\\rm RAD}+jX_{\\rm RAD}\\right)}{\\omega ^2M^2+\\left(R_{\\rm RAD}+jX_{\\rm RAD}\\right)\\left(R_{\\rm CML}+j\\omega L_{\\rm CML}+\\frac{1}{j\\omega C_{\\rm VAC}}\\right)},$ $I_{\\rm RAD}=-\\frac{j\\omega M V}{\\omega ^2M^2+\\left(R_{\\rm RAD}+jX_{\\rm RAD}\\right)\\left(R_{\\rm CML}+j\\omega L_{\\rm CML}+\\frac{1}{j\\omega C_{\\rm VAC}}\\right)}.$ If we assume that there are no particles playing the role of radiation vessels ($M=0$ ), we notice that at the resonant frequency of the mode $j\\omega L_{\\rm CML}+\\frac{1}{j\\omega C_{\\rm VAC}}=0$ and in the limit of negligible losses into the CML slab $R_{\\rm CML}\\rightarrow 0$ , the current $I$ increases without bound.",
"However, the radiated power $P_{\\rm RAD}$ is zero because the resistance $R_{\\rm CML}$ represents the dissipative process, not the radiative operation.",
"In this way, we come again to the aforementioned conclusion that the power is accumulated in the near field and does not reach the far zone.",
"On the contrary, when the pin comes close to the vacuum-CML interface, the radiated power equals to that delivered to $R_{\\rm RAD}$ since it models the function of the vessel as antenna.",
"Therefore, $P_{\\rm RAD}=\\frac{|V|^2 }{2}\\frac{\\omega ^2 M^2 R_{\\rm RAD}}{\\left(\\omega ^2 M^2+R_{\\rm RAD}R_{\\rm CML}\\right)^2+\\left(X_{\\rm RAD}R_{\\rm CML}\\right)^2},$ under the assumption that the system works at CML resonant frequency, namely $\\omega = 1/\\sqrt{L_{\\rm CML}C_{\\rm VAC}}$ .",
"For resonant and low-loss CML ($R_{\\rm CML}\\rightarrow 0$ ), the expression for the radiated power simplifies to: $P_{\\rm RAD}=\\frac{R_{\\rm RAD}|V|^2}{2\\omega ^2 M^2 }$ .",
"It is clear that in order to enhance radiation, we need to bring the CML to resonance and reduce its losses, while the vessels can be small and non-resonant.",
"Coupling between the scatterers and the surface modes should be weak in the scenario.",
"In the reciprocal situation of excitation by far-zone sources, we see that it is possible to enhance absorption beyond the ideal black-body full absorption of propagating plane waves by making the small scatterers lossy.",
"In this case, assuming that the scatterers do not create a significant shadow for the propagating modes, the propagating plane waves deliver nearly all their power to the CML body, while the evanescent waves (high-order cylindrical harmonics) couple to the resonant surface modes via the small scatterers and deliver additional power into the loss resistors of the scatterers."
],
[
"Electromagnetic Theory Approach",
"Having understood the basic operational principle from an equivalent circuit, which is by default an approximation for every single mode $k_t$ , we will next solve the problem rigorously for the entire spectrum of $k_t$ .",
"A spectrum integral formulation is feasible if we assume random but specific positions of a finite number of scatterers $(x_n,y_n)$ on the $xy$ plane, where $n=1,\\cdots , N$ (as shown in Fig.",
"REF ).",
"For the sake of simplicity of test calculations, we assume that particles are circular cylinders of a small radius $r$ and of perfectly magnetically conducting (PMC) material.",
"We chose perfect magnetic conductor pins as a simple model of lossless scatterers supporting magnetic currents, as appropriate for the considered TM polarization.",
"Conceptual results will not change for any other small lossless scatterers at the same positions.",
"Green's function of the considered configuration for both source $(\\chi ,\\psi )$ and observation points $(x,y)$ in vacuum is comprised of two components.",
"The singular component is just a cylindrical wave [17]: $G_{\\rm singular}(x,y,\\chi ,\\psi )=-\\frac{j}{4}H_0^{(2)}\\left(k_0\\sqrt{(x-\\chi )^2+(y-\\psi )^2}\\right),$ where $H_0^{(2)}$ is the Hankel function of zero order and second type.",
"The smooth component of Green's function describes the effect of the grounded CML slab on the free-space field and is found as follows: $G_{\\rm smooth}(x,y,\\chi ,\\psi )=\\int _{-\\infty }^{+\\infty }S_{\\rm gre}(k_t)e^{\\kappa _0(k_t)(x+\\chi )}e^{-jk_t(y-\\psi )}dk_t,$ where the spatial spectral density is given by [18] $S_{\\rm gre}(k_t)=\\frac{1}{4\\pi }\\frac{e^{-2\\kappa _0(k_t)g}}{\\kappa _0(k_t)}\\frac{\\varepsilon _{rt}\\coth [\\kappa (k_t)L]\\kappa _0(k_t)-\\kappa (k_t)}{\\varepsilon _{rt}\\coth [\\kappa (k_t)L]\\kappa _0(k_t)+\\kappa (k_t)}.$ If we use the symbol $M_n$ ($n=1,\\cdots , N$ ) for the magnetic currents (measured in $Volt/meter$ ) induced along the axes of the cylinders, the scattered magnetic field produced due to the presence of them is given as the following integral [19]: $H_{\\rm scat}(x,y)=-\\frac{jk_0}{\\eta _0}\\sum _{n=1}^N\\int _{(C_n)}M_n(l)\\left[G_{\\rm singular}(x,y,\\chi (l), \\psi (l))+G_{\\rm smooth}(x,y,\\chi (l), \\psi (l))\\right]dl.$ The notation $C_n$ is used for the contours of cylinder's surfaces.",
"Since the cylinder radius is electrically small ($k_0r\\ll 1$ ), the magnetic currents can be assumed to be uniformly distributed over the cylinder perimeter, and modeled by line currents along the cylinder axes, namely $M_n(l)\\cong M_n$ .",
"In this way, the approximate boundary condition for zero total magnetic field at the centers of the cylinders $H_{\\rm back}(x_m,y_m)+H_{\\rm scat}(x_m,y_m)=0$ for $m=1,\\cdots ,N$ can be enforced to formulate the following $N\\times N$ linear system of equations with respect to the unknown magnetic currents $M_n$ : $\\sum _{n=1}^N M_n \\left[I_{mn}+2\\pi r G_{\\rm smooth}(x_m,y_m,x_n,y_n)\\right]=\\frac{\\eta _0}{jk_0}H_{\\rm back}(x_m,y_m).$ The quantity $I_{mn}$ is the following approximately evaluated integral: $I_{mn}=\\int _{(C_n)}G_{\\rm singular}(x_m,y_m,\\chi (l), \\psi (l))dl=-\\frac{j\\pi r}{2}\\left\\lbrace \\begin{array}{cc}H_0^{(2)}(k_0r) &, \\; m=n\\\\H_0^{(2)}(k_0d_{mn}) &, \\; m\\ne n\\end{array}\\right.,$ where: $d_{mn}=\\sqrt{(x_m-x_n)^2+(y_m-y_n)^2}$ is the distance between the centers of the $n$ -th and the $m$ -th particle.",
"In this way, the induced magnetic currents can be found and the scattered field in the far region ($\\varphi $ -dependent profile) takes the form: $h_{\\rm scat}(\\varphi )\\sim -\\frac{j2\\pi k_0r}{\\eta _0}\\sum _{n=1}^NM_n\\left[e^{jk_0\\rho _n\\cos (\\varphi -\\varphi _n)}-\\pi k_0 \\cos \\varphi S_{\\rm gre}(k_0\\sin \\varphi )e^{-jk_0\\rho _n\\cos (\\varphi +\\varphi _n)}\\right],~~~x\\rightarrow -\\infty ,$ where $\\rho _n=\\sqrt{x_n^2+y_n^2}$ and $\\varphi _n=\\arctan (x_n,y_n)$ are the polar coordinates of the cylindrical radiation vessels.",
"Thus, we have obtained the analytical solution for the far field of the CML slab in the presence of numerous radiation vessels.",
"In the following, we are going to use both approaches (circuit analysis and electromagnetic analysis) in order to study, interpret and quantify the radiation enhancement achieved when the pins are located in the vicinity of the vacuum-CML interface."
],
[
"Numerical Results",
"In the following examples, we use a large number of vessels which are positioned neither very close to each other, to avoid building effective PMC walls which will block the incident illumination, nor too far since we want a strong background field at their positions.",
"In particular, we locate $N=80$ random points $(x_n,y_n)$ for $n=1,\\cdots , N$ belonging to a narrow vertical strip $\\left\\lbrace -\\lambda _0/20<x<\\lambda _0/20~,-10\\lambda _0<y<10\\lambda _0\\right\\rbrace $ .",
"The distance between every couple of centers of the cylinders $d_{mn}$ is kept larger than $\\lambda _0/5$ , so that the lattice is inhomogeneous at the wavelength scale and there is strong diffuse scattering into the far zone [20].",
"As referred above, we confine ourselves to uniaxial media (under TM illumination) with: $\\varepsilon _{rt}=\\mu _{rt}=a-jb~~~,~~~\\varepsilon _{rn}=\\frac{1}{\\varepsilon _{rt}}-j\\delta =\\frac{1}{a-jb}-j\\delta ,$ and we are mainly interested in the CML cases with $a<0$ .",
"Obviously, the presented results are dependent on the random distribution of the radiation vessels; however, our studies of a number of particular realizations of the pins distribution show that the obtained conclusions are valid regardless of the spatial distribution of the PMC pins in the vicinity of the CML slab."
],
[
"Radiation Enhancement",
"A metric of how strong is the effect of the radiation vessels on the radiated far field strength should be definitely related with the energy of the azimuthal field profiles: $\\left\\lbrace h_{\\rm inc}(\\varphi ), h_{\\rm sec}(\\varphi ), h_{\\rm scat}(\\varphi )\\right\\rbrace $ .",
"In particular, we can define the radiation enhancement ratio $R$ as the ratio of the far-zone power radiated in the presence of the near-field scatterers and the corresponding quantity in the absence of them: $R=\\frac{\\int _{\\pi /2}^{3\\pi /2}|h(\\varphi )|^2d\\varphi }{\\int _{\\pi /2}^{3\\pi /2}|h_{\\rm inc}(\\varphi )+h_{\\rm sec}(\\varphi )|^2d\\varphi }\\equiv \\frac{\\int _{\\pi /2}^{3\\pi /2}|h_{\\rm inc}(\\varphi )+h_{\\rm sec}(\\varphi )+h_{\\rm scat}(\\varphi )|^2d\\varphi }{\\int _{\\pi /2}^{3\\pi /2}|h_{\\rm inc}(\\varphi )+h_{\\rm sec}(\\varphi )|^2d\\varphi }.$ Here we evaluate and analyze the radiation enhancement factor $R$ when certain parameters of the analyzed configuration vary.",
"We are seeking for combinations of the structure, the material parameters, and the excitation which lead to $R\\gg 1$ , namely a substantial improvement of the radiated power when one puts a random cluster of cylinders in the near region of the resonant surface.",
"Figure: The radiation enhancement ratio RR as a function of the real part of the transversal components a= Re [ε rt ]= Re [μ rt ]a={\\rm Re}[\\varepsilon _{rt}]={\\rm Re}[\\mu _{rt}] for: (a) several perturbation parameters δ= Im [1/ε rt -ε rn ]\\delta ={\\rm Im}[1/\\varepsilon _{rt}-\\varepsilon _{rn}] of the normal component of the material parameters (with b=0.1b=0.1) and (b) several values of the imaginary part b=- Im [ε rt ]=- Im [μ rt ]b=-{\\rm Im}[\\varepsilon _{rt}]=-{\\rm Im}[\\mu _{rt}] (with δ=0.03\\delta =0.03).",
"Plot parameters: r=λ 0 /200r=\\lambda _0/200, N=80N=80, g=λ 0 /20g=\\lambda _0/20, L=3λ 0 L=3\\lambda _0, θ=90 ∘ \\theta =90^{\\circ }.In Fig.",
"REF we show the ratio $R$ as a function of the real part $a$ of the relative transversal permittivities/permeabilities ($a={\\rm Re}[\\varepsilon _{rt}]={\\rm Re}[\\mu _{rt}]$ ) for various perturbation parameters $\\delta $ .",
"One directly observes a huge change in the magnitude of $R$ taking place when the material parameters transit from the double-negative (CML) slabs ($a<0$ ) to double-positive, conventional PML slabs ($a>0$ ).",
"This feature is explained by the resonant nature of the CML with $a<0$ .",
"That is why we are focusing on the case of CML ($a<0$ ) rather than the conventional uniaxial PML ($a>0$ ).",
"With the purple dots we show (in the DNG cases $a<0$ ) the points on each curve for which the normal permittivity becomes lossless (it is lossy on the left side of the dots and active on the right side).",
"In other words, the dots indicate the equality: $\\delta =\\frac{b}{a^2+b^2}\\Rightarrow a=-\\sqrt{\\frac{b}{\\delta }-b^2}$ , which is valid (within the considered ranges of $a$ ) only for the three of the four curves of Fig.",
"REF (and for none of Fig.",
"REF ).",
"It is clear that when one moves along that “ultimate passivity boundary” (where none of the permittivity/permeability components is active) defined by the aforementioned successive purple dots, takes a decreasing $R$ both for increasing $\\delta $ and for increasing $a<0$ (when $|a<0|$ becomes smaller).",
"We can also conclude that even when the medium is lossy for any direction of the fields, the radiation enhancement is significant.",
"Most importantly, these results prove that the radiation enhancement due to strong coupling of resonant surface modes to the far-field modes is orders of magnitude stronger than possible reduction of radiation of propagating modes into far zone.",
"Recall that in the absence of the scatterers and $\\delta \\rightarrow 0$ the CML slab is perfectly matched to free space.",
"For small values of $b$ , all propagating modes are fully reflected, and we clearly see that adding pins makes the reflected fields more than two orders of magnitude stronger than reflected from a conventional perfect reflector.",
"Furthermore, $R$ is larger when $\\delta $ is closer to zero which is anticipated by the limiting expression of (REF ) in the DNG case.",
"The best results are recorded when $a$ is negative but close to zero (much larger than $-1$ , namely for $-1\\ll a<0$ ) where the radiation enhancement is giant and practically independent from $\\delta $ .",
"In Fig.",
"REF , we represent $R$ as a function of $a$ for several loss parameters $b$ .",
"Again, we note that the cluster works only in the CML case, where it does a very good job ($R>50$ on average).",
"Finally, a smaller imaginary part $b$ (with fixed $\\delta >0$ ) favors the increase in the radiated power achieved with the cylindrical vessels.",
"Figure: The decimal logarithm of the quality factor of the equivalent circuit logQ\\log Q with respect to the real part of the transversal components a= Re [ε rt ]= Re [μ rt ]a={\\rm Re}[\\varepsilon _{rt}]={\\rm Re}[\\mu _{rt}] and the inverse perturbation parameter 1/δ=1/ Im [1/ε rt -ε rn ]1/\\delta =1/{\\rm Im}[1/\\varepsilon _{rt}-\\varepsilon _{rn}].",
"Plot parameters: k t =1.5k 0 k_t=1.5k_0, b=0.1b=0.1.The singular behavior of the radiated power in the limit $a\\rightarrow 0^-$ can be explained by considering the quality factor of the resonating surface modes.",
"To do that, we calculate the equivalent inductance $L_{\\rm CML}$ and resistance $R_{\\rm CML}$ considering the wave impedance $Z$ of the CML medium (REF ); similarly one can find expressions for the capacitive effect of free space $C_{\\rm VAC}$ by evaluating $Z_0$ .",
"If one assumes that $\\delta >0$ (to ensure overall passivity), evanescent modes $|k_t|>k_0$ (for which the interesting phenomena happen) and $a<0$ (to have resonance), the quality factor of the equivalent $RLC$ series circuit takes the form: $Q=\\frac{1}{R_{\\rm CML}}\\sqrt{\\frac{L_{\\rm CML}}{C_{\\rm VAC}}}\\Rightarrow Q\\cong \\frac{\\sqrt{2(k_t^2-k_0^2)}}{k_t^2}\\frac{\\sqrt{2(k_t^2-k_0^2)+b\\delta k_t^2}}{(-a)\\delta }~,~~~\\delta \\rightarrow 0^+.$ It is easy to see that the loss parameter $R_{\\rm CML}$ is proportional to $(-a)\\delta $ in this case.",
"Thus, for a fixed level of losses in the CML slab (measured by $\\delta $ ), the quality factor behaves as $1/a$ for $a\\rightarrow 0^-$ .",
"Figure REF shows the values of the quality factor on the plane $(a,1/\\delta )$ in the region $-2<a<0.2$ , $1<1/\\delta <100$ .",
"It is clear that $Q$ obtains huge magnitudes when $|a|, \\delta $ are very small for the CML scenario, namely under the assumption of $a<0$ .",
"The ultimate passivity boundary, along which we have a nonactive $\\varepsilon _{rn}$ (${\\rm Im}[\\varepsilon _{rn}]=0$ ), is indicated by a white line with purple dots.",
"It divides the map $(a,1/\\delta )$ into two regions: one upper right which corresponds to active normal permittivity (${\\rm Im}[\\varepsilon _{rn}]>0$ ) and one lower left which concerns a passive normal permittivity (${\\rm Im}[\\varepsilon _{rn}]<0$ ).",
"Figure: The radiation enhancement ratio RR as function of the perturbation parameter δ= Im [1/ε rt -ε rn ]\\delta ={\\rm Im}[1/\\varepsilon _{rt}-\\varepsilon _{rn}] for: (a) various real part of transversal components a= Re [ε rt ]= Re [μ rt ]a={\\rm Re}[\\varepsilon _{rt}]={\\rm Re}[\\mu _{rt}] (with b=0.2b=0.2) and (b) various imaginary parts b=- Im [ε rt ]=- Im [μ rt ]b=-{\\rm Im}[\\varepsilon _{rt}]=-{\\rm Im}[\\mu _{rt}] (with a=-1a=-1).",
"Plot parameters: r=λ 0 /200r=\\lambda _0/200, N=80N=80, g=λ 0 /20g=\\lambda _0/20, L=3λ 0 L=3\\lambda _0, θ=90 ∘ \\theta =90^{\\circ }.In Fig.",
"REF , we depict the variations of the radiation enhancement $R$ with respect to the perturbation parameter $\\delta ={\\rm Im}\\left[\\frac{1}{\\varepsilon _{rt}}-\\varepsilon _{rn}\\right]={\\rm Im}\\left[\\frac{1}{\\mu _{rt}}-\\varepsilon _{rn}\\right]$ for several values of the real parts $a$ of the transversal constituent components.",
"When $\\delta >0$ , namely, when the structure is overall passive, the effect of the radiation vessels becomes weaker and weaker for increasing $\\delta $ , which is also obvious from Fig.",
"REF .",
"Note, however, that when the real part of the transverse permittivity approaches zero remaining negative, radiated power enhancement remains huge even for rather large positive $\\delta $ , that is, for rather high overall losses in the system.",
"On the other hand, when $\\delta <0$ , the structure is overall active and the whole slab acts as an additional power source; that is why the variations are sharper and more parameter-dependent.",
"In particular, $R$ possesses substantially oscillating and, on the average, much higher values when $\\delta <0$ .",
"The fluctuations are weaker and the output more stable when $a$ is negative but close to zero.",
"Again one can observe the behavior of the system along the “ultimate passivity limit” (purple dots) which indicate once again that the functions $R=R(\\delta >0)$ and $R=R(a<0)$ are decreasing.",
"In Fig.",
"REF the change of $R=R(\\delta )$ is shown for various $b={\\rm Im}[\\varepsilon _{rt}]={\\rm Im}[\\mu _{rt}]$ .",
"The radiation enhancement is almost independent from the imaginary part $b$ in the passive scenario, while, similarly to Fig.",
"REF , shaky response is observed when $\\delta <0$ .",
"It appears that when the system is overall active, one can find specific narrow intervals of $\\delta $ where extremely high radiation is achieved regardless of the inherent losses $b$ along the transversal directions.",
"Figure: The radiation enhancement ratio RR for various values of the real part of transversal components a= Re [ε rt ]= Re [μ rt ]a={\\rm Re}[\\varepsilon _{rt}]={\\rm Re}[\\mu _{rt}] as a function of the electrical distance of the source from the interface g/λ 0 g/\\lambda _0 (L=3λ 0 L=3\\lambda _0).",
"(a) Overall passive CML with δ=0.02\\delta =0.02; (b) Overall active CML with δ=-0.02\\delta =-0.02.",
"Plot parameters: b=0.2b=0.2, r=λ 0 /200r=\\lambda _0/200, N=80N=80, θ=90 ∘ \\theta =90^{\\circ }.In Figs.",
"REF , we identify the influence of the location of the primary source in representative passive and active scenarios ($\\delta =\\pm 0.02$ ).",
"In Fig.",
"REF we can see that the radiation falls rapidly as the primary source gets distant from the air-CML slab interface, because the evanescent part of the exciting field gets weaker.",
"However, especially in the active case shown in Fig.",
"REF , radiation enhancement remains significant even when the distance to the source is much larger than the wavelength.",
"As indicated above, in the active case the enhancement factor $R$ takes, on the average, higher values and exhibits a less monotonic behavior as a function of the geometrical and material parameters of the configurations."
],
[
"Radiation Patterns",
"Apart from the macroscopic insight offered by the radiation-enhancement metric $R$ , one can understand many features by observing the azimuthal variations of the far-field patterns for the introduced radiation-enhancing mirrors.",
"The represented quantities are normalized by $|h_{\\rm inc}(0)|=k_0\\omega q/4$ .",
"The incident field in the far region $h_{\\rm inc}($ is evaluated by (REF ), the background field $h_{\\rm inc}(+h_{\\rm sec}($ is given by (REF ), and the total field in the presence of the cylinders $h_{\\rm inc}(+h_{\\rm sec}(+h_{\\rm scat}($ is computed using (REF ).",
"Figure: The azimuthal profiles |h(| 2 |h(|^2 (normalized by |h inc (0)| 2 =k 0 2 ω 2 q 2 /16|h_{\\rm inc}(0)|^2=k_0^2\\omega ^2q^2/16) of the incident field, the background field and the total field as functions of angle for:(a)apassivescenario( for: (a) a passive scenario (=+0.01)and(b)anactivescenario() and (b) an active scenario (=-0.01).Plotparameters:).",
"Plot parameters: a=-2,, b=0,, L=30,, g0/10,, r=0/100,, N=80,, =90..In Figs.",
"REF , we illustrate two characteristic cases: one passive (Fig.",
"REF ) and one active (Fig.",
"REF ).",
"The far-field patterns are represented in the case that the grounded CML slab is nearly fully reflecting propagating waves (we select the dissipation parameter $b=0$ and the CML loss factor $\\delta $ is small).",
"The three curves compare the far-field pattern of the primary source into free space (green), the pattern for the CML slab without perturbing pins (blue), and the CML slab with radiation enhancing vessels (red).",
"The green curve is simply the pattern of a dipole line source, with the maximum in the broadside direction (in both Figs.",
"REF ).",
"CML slab without pins basically acts as a reflector for the propagating part of the incident spectrum and its response gets substantially enhanced along the grazing-angle directions ($90^{\\circ }, 270^{\\circ }$ ) for the active scenario.",
"We can see that for the passive mirror the maximum enhancement of the field strength equals 2 (four-fold in terms of power), which takes place for directions along which the reflected field sums up in phase with the field of the primary source.",
"This value is the maximal possible value of the reflected field from any ideally reflecting planar mirror (with arbitrary reflection phase).",
"We clearly see that the radiation vessels provide an additional radiation channel via the evanescent part of the spectrum, and the radiated power is strongly enhanced, well above the fundamental limit for any lossless mirror.",
"Figure: The azimuthal profiles |h(| 2 |h(|^2 (normalized by |h inc (0)| 2 =k 0 2 ω 2 q 2 /16|h_{\\rm inc}(0)|^2=k_0^2\\omega ^2q^2/16) of the incident field, the background field and the total field for: (a) a=-0.25a=-0.25, δ=0.02\\delta =0.02, L=2.557λ 0 L=2.557\\lambda _0, g=λ 0 /20g=\\lambda _0/20 (R≅84R\\cong 84) and (b) a=-1a=-1, δ=-0.02\\delta =-0.02, L=3λ 0 L=3\\lambda _0, g=0.0665λ 0 g=0.0665\\lambda _0 (R≅525R\\cong 525).",
"Plot parameters: b=0.2b=0.2, r=λ 0 /200r=\\lambda _0/200, N=80N=80, θ=90 ∘ \\theta =90^{\\circ }.In Figs.",
"REF , we represent the results also for a passive (Fig.",
"REF ) and an active (Fig.",
"REF ) case which correspond to high radiation enhancement $R$ .",
"Losses in the CML are present ($b=0.2$ ), so that the reflections of the propagating part of incident waves are weak.",
"That is why we see that the blue and green curves nearly coincide for the passive CML.",
"The chosen value of $\\delta =0.02$ leads to an enhancement in radiation by a factor of $R\\cong 84$ .",
"In this case, it is apparent that the far-field response of the CML slab without the radiation enhancing cluster is almost identical to the incident field, which is anticipated from (REF ).",
"However, when one puts the randomly distributed vessels in the near field, the output power of the antenna gets significantly amplified (and the pattern becomes asymmetric with respect to $180^{\\circ }$ ).",
"In Fig.",
"REF , corresponding to an active CML, we use $\\delta =-0.02$ and the enhancement is huge in all directions (the overall radiated power enhancement factor $R\\cong 525$ )."
],
[
"Conclusions",
"It is well known that infinite homogeneous planar surfaces can fully reflect electromagnetic waves in the limit of negligible losses.",
"In this case, the amplitude of waves radiated by a source near the mirror can be doubled as compared with the incident waves.",
"This surface is thought to be “ideally shiny.\"",
"On the other hand, the reflection coefficient from planar surfaces can be in principle made zero for all incident propagating waves (any polarization and any incident angle).",
"In this case, all power of incident propagating plane waves is absorbed and the surface is “ideally black,\" absorbing maximum power and, reciprocally, emitting maximal heat power according to the Plank law.",
"In both these scenarios, evanescent waves do not participate in power exchange between far-zone external sources and the material body.",
"In this paper, we have shown that perturbing the surface of an infinite planar surface which maintain resonant surface modes we can in principle realize a planar reflector which reflects more power than any ideally reflecting planar surface.",
"Due to perturbations, surface modes couple to propagating plane waves and create additional channels for power exchange via evanescent fields.",
"Such perturbed resonant surface extracts extra power from a near sources and sends that into space.",
"The amplitude of the reflected field can be orders of magnitude larger than the maximal value of 2 for any usual lossless mirror.",
"Making the perturbation lossy, it can become possible to overcome the black-body limit even for planar surfaces.",
"In this scenario, the black body absorbs nearly all propagating waves, while the perturbations provide coupling between the resonant surface modes and an additional energy sink via evanescent modes.",
"The perturbations are random at the wavelength scale and non-resonant.",
"In this configuration, the surface-averaged currents induced in the perturbation objects (which could partially reflect propagating waves and compromise their absorption in the CML body) are small because they couple to non-resonant propagating modes of the absorbing or reflecting body.",
"On the other hand, current components which vary fast on the wavelength scale can be huge because they couple to highly resonant CML body.",
"These spatially inhomogeneous resonant currents on the perturbations provide additional channels for power exchange between the body and free-space wave modes, allowing stronger reflections than from an ideal reflector or more absorption than in the ideal black body.",
"Analyzing the far-zone radiation patterns we see that there is an analogy between the revealed phenomena and superdirectivity of antennas [21].",
"Superdirective radiators can create a narrow beam with the directivity higher than that of the same configuration which is uniformely excited [22].",
"However, in the configuration which we have introduced here, it appears that the planar reflector sends superdirective beams nearly everywhere (pattern oscillations are determined by random positions of radiation vessels).",
"There is also a connection of the revealed phenomena to the concept of perfect lens as a slab of a lossless double-negative material [23].",
"The perfect lens operation also exploits resonance of surface modes at an interface between free space and a double-negative material.",
"In the perfect lens concept, high-amplitude reactive fields at the entry interface are focused behind the lens thanks to interactions between resonant modes of the two parallel surfaces of the lens.",
"In papers [5], [6] it was shown how the reactive energy of the resonant surface modes can be fully absorbed.",
"Here we have shown that this energy can be launched into space, creating super-reflectors and far-field superemitters.",
"Although in this paper we have considered a particular realization of surface perturbations in form of a random array of thin cylinders, the concept is general and the surface can be perturbed in many various ways, for instance simply making the surface rough at the appropriate wavelength scale.",
"Likewise, the surface does not have to be planar nor infinite: properly perturbing the surface of a finite-size conjugately matched body we can dramatically enhance its coupling to electromagnetic fields in space.",
"Discussed super-emission and super-absorption phenomena can potentially enable new approaches to optimizing wireless transfer of energy or information and in radiative heat transfer management."
]
] | 1605.06040 |
[
[
"Privacy-Related Consequences of Turkish Citizen Database Leak"
],
[
"Abstract Personal data is collected and stored more than ever by the governments and companies in the digital age.",
"Even though the data is only released after anonymization, deanonymization is possible by joining different datasets.",
"This puts the privacy of individuals in jeopardy.",
"Furthermore, data leaks can unveil personal identifiers of individuals when security is breached.",
"Processing the leaked dataset can provide even more information than what is visible to naked eye.",
"In this work, we report the results of our analyses on the recent \"Turkish citizen database leak\", which revealed the national identifier numbers of close to fifty million voters, along with personal information such as date of birth, birth place, and full address.",
"We show that with automated processing of the data, one can uniquely identify (i) mother's maiden name of individuals and (ii) landline numbers, for a significant portion of people.",
"This is a serious privacy and security threat because (i) identity theft risk is now higher, and (ii) scammers are able to access more information about individuals.",
"The only and utmost goal of this work is to point out to the security risks and suggest stricter measures to related companies and agencies to protect the security and privacy of individuals."
],
[
"0pt*33.5pt"
],
[
"0pt*3.53.5pt"
],
[
"0pt*1.50pt Personal data is collected and stored more than ever by the governments and companies in the digital age.",
"Even though the data is only released after anonymization, deanonymization is possible by joining different datasets.",
"This puts the privacy of individuals in jeopardy.",
"Furthermore, data leaks can unveil personal identifiers of individuals when security is breached.",
"Processing the leaked dataset can provide even more information than what is visible to naked eye.",
"In this work, we report the results of our analyses on the recent \"Turkish citizen database leak\", which revealed the national identifier numbers of close to fifty million voters, along with personal information such as date of birth, birth place, and full address.",
"We show that with automated processing of the data, one can uniquely identify (i) mother's maiden name of individuals and (ii) landline numbers, for a significant portion of people.",
"This is a serious privacy and security threat because (i) identity theft risk is now higher, and (ii) scammers are able to access more information about individuals.",
"The only and utmost goal of this work is to point out to the security risks and suggest stricter measures to related companies and agencies to protect the security and privacy of individuals."
],
[
"Introduction",
"Personal data is crucial for companies, governments, and service providers.",
"Data collected by the service providers about the individuals increases the utility of the services they provide.",
"As a result, today, an adult's records exists in more than 700 databases around the World [1].",
"Companies or governments either (publicly) share personal data after anonymization (e.g., for research purposes) or they keep it locally for their own records.",
"In today's interconnected world, even if a dataset is shared after anonymization (i.e., removal of personal identifiers from the data such as the name or telephone number), in many cases, it is possible to deanonymize the dataset participants by using auxiliary information (that is publicly available).",
"Data leaks are also major privacy threats for individuals.",
"Leaked datasets alone may include considerable amount of information about the individuals (especially, if it is not anonymized).",
"On top of this, an attacker can infer sensitive information about the individuals in the leaked dataset (that cannot be directly observed from the dataset).",
"Even worse, an attacker can use auxiliary information (e.g., online social networks or publicly available datasets) to infer more about the individuals that are in the leaked dataset (and about their family members, friends, etc.).",
"In this work, we report our findings about the privacy-related threats of the “Turkish citizen database leak”.",
"Personal records of 49,611,709 Turkish citizens have publicly become available a few months ago [2].",
"Experts confirmed this leak as one of the biggest public leaks of personal data ever seen, as it puts almost two-thirds of the country's population at risk of fraud and identity theft.",
"The leak actually happened before 2010, but it recently became publicly available.",
"The leaked dataset includes voter registration data of the citizens for the 2009 local elections and reveals sensitive information about the citizens including their personal details and national identifiers (equivalent of the SSN in the US).",
"Potential attacks by directly using such information of the citizens include (i) generating fake identity cards, (ii) loaning money from a bank, (iii) getting phone lines, (iv) founding a company, and (v) being a guarantor.",
"We mainly study two important aspects of this leaked database: (i) the uniqueness of the citizens (inspired by the well-known study of Latanya Sweeney on the US census data [20]) to show the comparative threat of disclosing various demographic data, and (ii) potential further inference about unrevealed sensitive data of the individuals.",
"For the latter, we focus on determining “mother's maiden name” and “personal landline number” of each individual in the dataset.",
"Mother's maiden name has been a security question since 1882 [3].",
"It is commonly used as a remote authentication question by many service providers.",
"Especially financial institutions use this type of “knowledge-based authentication” very commonly because the answer does not change over time, the question has an answer for all the users, users can easily remember the answer, and the institutions think that it is safe (i.e., hard to guess by an attacker).",
"However, in today's internet era, people share vast amount of information on numerous online services, and hence much of this knowledge-based information becomes available to the attackers via online social networks.",
"If an attacker gets to learn a victim's mother's maiden name, for example, he can convince the customer service representative of a bank to wire funds from the victim's bank account to his account, a simple consequence that recently happened in the US [4].",
"Note that even if a customer service representative gets suspicious, the attacker can always call again to try a different one.",
"Similarly, if the attacker is unsure about the sensitive information (e.g., if the attacker can narrow down the potential candidates for the mother's maiden name to 2), he can also try more than once.",
"Learning one's mother's maiden name through social networks, even though is possible, is a targeted attack.",
"In other words, it is not scalable (the attacker needs to spend a lot of effort to learn this information for thousands or millions of individuals).",
"However, as we will show in this work, with this leaked dataset, it is possible to infer the mother's maiden name of thousands of individuals.Note that this information is not directly available in the dataset, but we use an inference technique to infer this information for each individual.",
"Similarly, learning the personal landline numbers of individuals (whose personal information is available in the leaked dataset) also poses a significant threat, especially considering the widespread phone scams.",
"In such phone scams, scammers use the terms “you won a prize”, “you have some dept for your health insurance”, “we are calling from the law enforcement, it seems that some criminal activity has occurred in your bank account”, or “there is a court order against you” on the phone and they request money or credit card information from the individuals.",
"Of course, they, especially if they claim that they are calling from the law enforcement or the court, provide more information about the victim to make their scenario stronger.",
"Using sensitive information that is directly available from the leaked dataset (e.g., national identifier or full address) or inferred from the leaked dataset (e.g., mother's maiden name, as we will show in this work) throughout the call makes the victim less suspicious about the attacker, and hence the victim becomes more prone to giving what the attacker wants.",
"In this work, we will also show how the scammers can learn the landline numbers of the individuals in the database systematically by using other publicly available sources on the Internet.",
"The rest of this paper is organized as follows.",
"In the next section, we summarize the known database leaks and inference attacks.",
"In Section , we describe the format of the leaked dataset.",
"In Section , we discuss the threat model.",
"In Section , we present our findings about the uniqueness of the citizens based on different demographics.",
"In Section , we describe two inference attacks by using the leaked dataset.",
"In Section , we discuss the potential countermeasures and future work.",
"Finally, in Section , we conclude the paper."
],
[
"Related Work",
"There have been many data beaches (intentional or unintentional release of secure information) from well-known companies in the recent years.",
"In 2014, a collection of about 500 photos of several celebrities have been leaked from iCloud service of Apple [5].",
"The main reasons of this leak were (i) weak passwords of the users, (ii) easy-to-guess security questions, and (iii) a bug in Apple's photo backup service (so that hackers were able to force their way into celebrities' photo collections by repeatedly guessing passwords).",
"Similarly, a recent data breach exposed user account information of around 38 million Adobe users [6].",
"As a result of a well-known data breach from Target, hackers obtained credit card information of around 40 million Target customers [7].",
"Target had to reach a $39 million settlement with several US banks due to this breach [8].",
"Furthermore, according to Office of Civil Rights (OCR), there have been 253 health data breaches that affected 500 individuals or more with a combined loss of over 112 million health records in just 2015 [9].",
"Thus, Turkish citizen data leak is not the only data leak in the World, but it is one of the most serious leaks to date.",
"Hackers or attackers have many techniques to obtain unauthorized information from the leaked or publicly shared (anonymized) data.",
"The most well-known technique that can be used to learn more about individuals is profile matching (or deanonymization).",
"Studies show that in today's digital world, anonymization is not an effective way of protecting sensitive data.",
"For example, Latanya Sweeney showed that it is possible to de-anonymize individuals by using publicly available anonymized health records and other auxiliary information that can be publicly accessed on the Internet [21].",
"It has been shown that anonymization is also an ineffective technique for sharing genomic data [13], [14].",
"For instance, genomic variants on the Y chromosome are correlated with the last name (for males).",
"This last name can be inferred using public genealogy databases.",
"With further effort (e.g., using voter registration forms) the complete identity of the individual can also be revealed [14].",
"Also, unique features in patient-location visit patterns in a distributed health care environment can be used to link the genomic data to the identity of the individuals in publicly available records [18].",
"Furthermore, Narayanan and Shmatikov proposed an efficient graph-based deanonymization framework based on seed and extend technique [19].",
"Ji et.",
"al proposed another structural data deanonymization framework that does not require the seed information [17].",
"Ji et.",
"al also proposed an evaluation system for graph anonymization and deanonymization named SecGraph [16].",
"Note that in this work, we do not need a deanonymization technique, as the leaked data already includes the identifiers of the individuals.",
"But, this dataset can be used as auxiliary data to deanonymize the records of individuals in other datasets."
],
[
"Database Format",
"As mentioned, the leaked database includes the records of nearly 50 million citizens (approximately 7GB of data).",
"For each citizen, the database includes the fields below: National Identifier First Name Last Name Mother's First Name Father's First Name Gender City of Birth Date of Birth ID Registration City and District Full Address (door number, street, neighborhood, district, and city) Note that all individuals in the dataset are older than 18 as the dataset includes voter registration data.",
"In the following sections, we will show how one can exploit the data in these fields to infer more information about the citizens in the database."
],
[
"Threat Model",
"In this work, we consider a passive attacker.",
"That is, the attacker can only access publicly available information.",
"This includes the leaked dataset itself, and other public resources that are available on the Internet.",
"We do not consider any other background information for the attacker (e.g., the attacker personally knowing some individuals from the dataset, or the attacker buying information about some individuals that are in the dataset).",
"We also assume that the attacker is computationally bounded.",
"That is, the attacker can only run polynomial-time algorithms to infer data about the individuals."
],
[
"Uniqueness of Turkish Citizens",
"Latanya Sweeney showed that 87$\\%$ of the US population can be uniquely identified by gender, ZIP code, and full date of birth based on 1990 US census data [20].",
"Later, using the more recent 2000 census data, another study showed that these 3 pseudo-identifiers can only uniquely identify 63$\\%$ of the US population [15].",
"Such studies are in particularly important for researchers and data collectors that need estimates of the threat of disclosing anonymized datasets along with pseudo-identifiers (e.g., demographic data).",
"In this part, we do a similar study for the Turkish citizens using the leaked dataset and compare our results with the previous work.",
"The result of this study shows the fraction of Turkish citizens that can be uniquely identified by using combinations of different pseudo-identifiers, and hence it gives some insight about how to release anonymized datasets of Turkish citizens.",
"For this study, we use the following pseudo-identifiers (and their generalizations) that are available in the dataset: (i) gender ($G$ ), (ii) full date of birth ($D$ ), (iii) city of birth ($C$ ), and (i) full address ($A$ ).",
"The pseudo-identifier set we use in this study can be represented as $\\mathbf {P}=\\lbrace G_i^j,D_i^j,C_i^j,A_i^j\\rbrace $ , where $i$ represents the generalization level of the corresponding pseudo-identifier, and $j$ denotes whether the corresponding pseudo-identifier is used in a given experiment ($j\\in \\lbrace 0,1\\rbrace $ , 1 means the pseudo-identifier is used, and 0 means otherwise).",
"For a given pseudo-identifier, $i=0$ represents the ground level, at which no generalization made.",
"Next, we briefly discuss our generalization assumptions.",
"Full address includes door number, street, neighborhood, district, and city when $i=0$ .",
"One level generalization ($i=1$ ) only includes street, neighborhood, district, and city.",
"Two level generalization ($i=2$ ) only includes neighborhood, district, and city.",
"Three level generalization ($i=3$ ) only includes district, and city.",
"Four level generalization ($i=4$ ) only includes the city.",
"Full date of birth includes day, month, and year when $i=0$ .",
"One level generalization ($i=1$ ) only includes the month and the year.",
"Two level generalization ($i=2$ ) only includes the year.",
"We do not generalize gender and the city of birth ($i$ values is always 0 for $G$ and $C$ ).",
"In the following, we report some of our notable findings.",
"When $\\mathbf {P}=\\lbrace G_0^1,D_0^1,C_0^1,A_0^1\\rbrace $ , we could uniquely identify $93.6\\%$ of the Turkish citizens.",
"When $\\mathbf {P}=\\lbrace G_0^1,D_0^1,C_0^0,A_0^1\\rbrace $ , we could uniquely identify $93.4\\%$ of the Turkish citizens.",
"When $\\mathbf {P}=\\lbrace G_0^1,D_0^1,C_0^0,A_2^1\\rbrace $ (the most similar case to the previous studies on US census data), we could uniquely identify $67.9\\%$ of the Turkish citizens.",
"The rest of the results for (i) $\\mathbf {P}=\\lbrace G_0^1,D_0^1,C_0^0,A_i^1\\rbrace $ for $i=\\lbrace 0,1,2,3,4\\rbrace $ and (ii) $\\mathbf {P}=\\lbrace G_0^1,D_i^1,C_0^0,A_0^1\\rbrace $ for $i=\\lbrace 0,1,2\\rbrace $ are also shown in Table REF .",
"From these results, we conclude that both the address and the date of birth should be generalized for at least 2 levels when disclosing anonymized datasets (that include these pseudo-identifiers).",
"Table: Uniqueness of the citizens with respect to the generalization level for date of birth and full address."
],
[
"Attacks and Results",
"In this section, we show two potential attacks via further inference from the revealed dataset about unrevealed sensitive data of the individuals."
],
[
"Inferring Mother's Maiden Name",
"As discussed, one's mother's maiden name is a very popular knowledge-based authentication technique that is used by several service providers (especially the financial institutions).",
"This information is not directly available in the leaked dataset that we consider.",
"In this section, we show that using the leaked dataset (or using similar publicly available datasets), it is possible to infer this information for individuals without using any other auxiliary information.",
"The attacker does not need to uniquely determine the mother's maiden name of a victim via this attack.",
"As we mentioned, if the attacker is unsure about the sensitive information, he can easily try multiple times (e.g., through different customer service representatives).",
"Thus, in this attack, the ultimate goal of the attacker is to reduce the size of the anonymity set as much as possible for a victim, and hence we evaluate the power of this attack by analyzing the size of the anonymity set.",
"We denote the victim as $v$ (i.e., individual whose mother's maiden name is inferred by the algorithm) and the full address of the individual $i$ as $\\mathrm {Addr_i}$ .",
"The attacker runs Algorithm 1 (which either calls Algorithm 2 or 3 depending on the type of the victim) for each victim.",
"As a result, the attacker gets the result set (i.e., anonymity set) for each victim.",
"Algorithm 1 (finding the mother's maiden name of an individual) Initialize the result set $G$ and intermediate result set $R$ as empty Get the victim $v$ If the victim is female, infer the marital status using the address $\\mathrm {Addr_v}$If living alone or living with her parents, we assume the victim is single; if living with her husband and no parents, we assume the victim is married.",
"If the victim is male or single female go to Algorithm 2 If the victim is a married female go to Algorithm 3 Return the result set $G$ Algorithm 2 (finding the mother of a male or single female citizen) Assumption 1: Male or single female victim has the same last name as his/her father Assumption 2: The mother is at least $H$ years older than the victimWe run the algorithm for three different values of $H$ (15, 25, and 35).",
"Input: victim $v$ and set $R$ Initialize sets $F$ and $M$ as empty Set target person $t$ to $v$ Get the last name of the target: $\\mathrm {last\\_name}$ Get the first name of target's father: $\\mathrm {father\\_first}$ Get the first name of target's mother: $\\mathrm {mother\\_first}$ Get the address of the target: $\\mathrm {Addr_t}$ //If father and mother are married Find all males with $\\mathrm {father\\_first}$ & $\\mathrm {last\\_name}$ and construct $F$ Find all females with $\\mathrm {mother\\_first}$ & $\\mathrm {last\\_name}$ and construct $M$ Compare $\\mathrm {Addr_t}$ with the addresses of individuals in $F$ and $M$ If there is an individual in $M$ with the same address as $\\mathrm {Addr_t}$ , add her to $R$ (target living with his/her parents in the same house) Else if there is an individual in $M$ with a different address than $\\mathrm {Addr_t}$ , but with the same address with an individual in $F$ , add her into $R$ (target living away from his/her parents, parents are living together) If $R$ is non-empty, go to Algorithm 4 using the females in $R$ and victim $v$ as input // If father and mother are divorced or not married If $R$ is still an empty set Find all males with $\\mathrm {father\\_first}$ & $\\mathrm {last\\_name}$ and construct $F$ Find all females with $\\mathrm {mother\\_first}$ & $\\mathrm {any\\_last\\_name}$ and construct $M$ Compare $\\mathrm {Addr_t}$ with the addresses of individuals in $F$ and $M$ If there is an individual in $M$ with a different address than $\\mathrm {Addr_t}$ , but with the same address with an individual in $F$ , add her into $R$ (unmarried couples with children) Else if there is an individual in $M$ with the same address as $\\mathrm {Addr_t}$ , add her to $R$ (divorced parents, child living with the mother) If $R$ is non-empty Set $G=R$ (females in $R$ already have their maiden names) If $R$ is still empty return unsuccessful search Algorithm 3 (finding the mother of a married female citizen) Assumption 1: Female victim has the last name of her husband; not the last name as her father Assumption 2: The mother is at least $H$ years older than the victimWe run the algorithm for three different values of $H$ (15, 25, and 35).",
"Input: victim $v$ and set $R$ Initialize sets $F$ and $M$ as empty Set target person $t$ to $v$ Get the first name of target's father: $\\mathrm {father\\_first}$ Get the first name of target's mother: $\\mathrm {mother\\_first}$ Get the address of the target: $\\mathrm {Addr_t}$ //If father and mother are married Find all males with $\\mathrm {father\\_first}$ & $\\mathrm {any\\_last\\_name}$ and construct $F$ Find all females with $\\mathrm {mother\\_first}$ & $\\mathrm {any\\_last\\_name}$ and construct $M$ Compare $\\mathrm {Addr_t}$ with the addresses of individuals in $F$ and $M$ If there is an individual in $M$ with the same address as $\\mathrm {Addr_t}$ , add her to $R$ (target living with her mother in the same house) Else if there is an individual in $F$ with the same address as $\\mathrm {Addr_t}$ , add the corresponding individual from M to $R$ (target living with her father in the same house) Else if there is an individual in $M$ with a different address than $\\mathrm {Addr_t}$ , but with the same address with an individual in $F$ , add her into $R$ (target living away from her parents, parents are living together) If $R$ is non-empty, go to Algorithm 4 using the females in $R$ and victim $v$ as input If $R$ is empty, return unsuccessful search Algorithm 4 (finding the father of victim's mother) Assumption 1: Victim's mother has the last name of her husband; not the last name as her father Assumption 2: The father is at least $H$ years older than the daughterWe run the algorithm for three different values of $H$ (15, 25, and 35).",
"Input: victim $v$ and set $R$ Initialize set $R^{\\prime }$ as empty Initialize sets $F$ and $M$ as empty For each individual $i$ in set $R$ , set the target person $t$ to $R(i)$ Get the first name of target's father: $\\mathrm {father\\_first}$ Get the first name of target's mother: $\\mathrm {mother\\_first}$ Get the address of the target: $\\mathrm {Addr_t}$ //If father and mother are married Find all males with $\\mathrm {father\\_first}$ & $\\mathrm {any\\_last\\_name}$ and construct $F$ Find all females with $\\mathrm {mother\\_first}$ & $\\mathrm {any\\_last\\_name}$ and construct $M$ Compare $\\mathrm {Addr_t}$ with the addresses of individuals in $F$ and $M$ If there is an individual in $M$ with the same address as $\\mathrm {Addr_t}$ , add the corresponding individual from $F$ to $R^{\\prime }$ (target living with her mother in the same house) Else if there is an individual in $F$ with the same address as $\\mathrm {Addr_t}$ , add him to $R^{\\prime }$ (target living with her father in the same house) Else if there is an individual in $F$ with a different address than $\\mathrm {Addr_t}$ , but with the same address with an individual in $M$ , add him into $R^{\\prime }$ (target living away from her parents, parents are living together) $G = G \\cup R^{\\prime }$ and go to 4 If $R^{\\prime }$ is still an empty set return unsuccessful search For the evaluation, we randomly sampled around 20000 individuals from the dataset and set them as victims.",
"Among the sampled individuals, we made sure that the gender distribution and the age distribution are uniform.",
"We ran Algorithm 1 on all the sampled individuals and constructed the anonymity set for each victim.",
"We call a search unsuccessful when (i) the result set $G$ is empty or (ii) the size of the anonymity set is above 100.",
"In Figure REF , we show the distribution of unsuccessful search for different age intervals and for each gender.",
"As expected, the threat is stronger for males as it is easier to find the parents of a male compared to a married female (as married females mostly adopt the last names of their husband).",
"Therefore, the inference algorithm returns a successful search more for males compared to females.",
"Furthermore, algorithm returns unsuccessful searches mostly for older victims as the algorithm needs to trace one or two generations to find the result and such information is mostly unavailable for older individuals (e.g., due to deceased parents that do not exist in the database).",
"Figure: Distribution of unsuccessful search for different age intervals and for each gender.In Figures REF and REF , we only focus on the successful searches and show the size of the obtained anonymity set for different age intervals and for each gender (numbers in these figures are also shown in Table REF ).",
"In general, we observe that on the average, the size of the anonymity set is smaller for males and for younger victims.",
"Also, we observe that for $69.25\\%$ of the sampled individuals that are male and in the age range of $18-30$ the size of the anonymity set is below 10.",
"Figure: Size of the anonymity set for males with different age ranges.Figure: Size of the anonymity set for females with different age ranges.Note that we did this study by only using the dataset and without using any other auxiliary information.",
"Inference can be even stronger by using publicly available data on online social networks.Married female users prefer declaring their maiden names on social networks (e.g., to find their friends from school).",
"We will also consider such auxiliary information in future work and will show how the size of the anonymity set changes with such auxiliary information.",
"Table: Fractions of individuals in each anonymity set based on their demographics (i.e., age range and gender)."
],
[
"Inferring Landline Numbers",
"Turkish terror phone scams has been a major issue in the recent years and have attracted global attention [10].",
"Typically, scammers call people, claim they are police officers and tell them that their (or a relative's) name has showed up in a terror investigation.",
"They also say that they know the victim is innocent but need their help and they are supposed to deposit a certain amount of money into a specified bank account to clear their names.",
"They even play police radio sounds in the background to convince people that they are calling from the police station.",
"Even a famous professor in Turkey has been a victim of this scam as well [10].",
"Landline phone numbers and addresses of individuals were used to be released in thick hard-copy books called “Golden Directory” in the 90s in Turkey.",
"Even though slightly out-dated, such directories are still publicly available.",
"Today, the main provider of landline phones in Turkey, Turk Telekom, provides an online lookup service named “White Pages” [11].",
"This is a captcha protected service in which a user can query a last name in conjunction with an area code.",
"Up to 20 individuals along with their landline numbers and addresses (number, neighborhood, district and city) are listed.",
"To see 20 more individuals, user is requested to pass another captcha test.",
"If there are too many matching last names, users are requested to enter additional information to narrow the query result (e.g., first name).",
"Thus, scammers are able to address individuals by their names on the phone, which gives them credibility to a certain degree that they are who they claim to be.",
"So far, the scammers were not able to provide any other credentials such as national identifier, date of birth, or birth place.",
"However, this has potential to change with the recent leak.",
"Now, a scammer has access to various extra private information about potential targets.",
"Attackers can link the landline number and address of an individual (which are public) to the leaked dataset.",
"The only obstacle for an attacker is the possibly large number of matching records in the phone directory due to namesakes.",
"In order to determine uniquely identifiable phone numbers given a name, we have randomly retrieved 983 phone numbers from the White Pages.",
"Then, we checked the number of unique entries found in the leaked dataset for each landline number to determine the percentage of uniquely identifiable individuals.",
"Out of the retrieved 983 records, we could find 702 of them in the leaked dataset.281 entries could not be found possibly due to change of address.",
"White Pages includes the recent addresses of individuals where data in the leaked dataset is from 2009.",
"Out of these 702 records, we observed that 60.11% of the individuals are uniquely identifiable.",
"Also, for 79.49% of the individuals, we observed that the size of the anonymity set is smaller than or equal to 2 and for 85.47% of the individuals, the size of the anonymity set is smaller than equal to 3.",
"These results mean that a scammer can randomly pick an individual from the public phone directory, call him and in addition to addressing him with his name, the scammer can provide personal information such as national identifier, date of birth, and birth place.",
"Even if the scammer is unsuccessful at the first attempt for the people in 2-anonymity sets, he will be able to hang-up the phone and call the next victim.",
"Moreover, as shown in Section REF , it is possible to (i) find relatives of individuals in the leaked dataset, and (ii) with high probability, determine his mother's maiden name.",
"Thus, a scammer can enumerate the relatives of the victim and can even tell their mother's maiden name.",
"Given that scammers were very successful at convincing people and stealing their money with information limited to only first and last name, current situation poses a huge security risk."
],
[
"Discussion and Future Work",
"The recent Turkish citizen dataset leak has major privacy risks as the identities of around 50 million individuals can now be associated with their national identifier numbers, date of births, and birth places.",
"We show in this paper that with simple automated processing of the data, we can infer more information about the individuals, and the situation is even worse than currently anticipated.",
"Strict measures must be taken to ensure the privacy and security of Turkish citizens because (i) the authentication systems of various institutions such as banks or government agencies are currently vulnerable, and (ii) very detailed information about each individual is now available for the scammers.",
"Mother's maiden name is a convenient authentication mechanism.",
"It is usually not shared with other people because it is out of context and only known by the individual.",
"It is also easy to remember.",
"However, it is also known by close relatives and this means it is still not a personal secret.",
"This was a problem during the pre-internet era.",
"In today's world where so many people use social media as a means of communication, it is very easy to track someone's relatives and access their mothers' maiden names.",
"For mothers who retain their maiden name, this is even as simple as learning the target's mother's full name.",
"This makes mother's maiden name even more accessible.",
"Having that said, in Turkey (and also in the World) mother's maiden name is still commonly used as an authentication scheme.",
"The situation is now much worse for Turkish citizens due to the leak.",
"We showed that it is possible to pinpoint the maiden name of the mother's of around 30% of the individuals (in the age range of 18-30, regardless of the gender).",
"We were able to do this in an automated way, and only by using the leaked dataset itself.",
"An attacker can still find out the mother's maiden name of other targets by linking more external information to this leaked dataset such as Google search results or social media accounts.",
"Therefore, we suggest the institutions (especially the financial institutions), which use mother's maiden name as an authentication scheme, to basically stop using it.",
"One alternative is to use 2-factor authentication for telephone banking.",
"2-factor authentication is widely used to obtain temporary passwords for internet banking.",
"The idea is to use an independent communication channel that is known to be private to the user (e.g., a smart phone running with a specific simcard).",
"For telephone banking, on the other hand, mother's maiden name is preferred, as it is easier to verify verbally.",
"Given today all individuals own at least one cell phone, 2-factor authentication is reasonable to use for telephone banking and it should replace mother's maiden name as the security mechanism.",
"Another alternative can be the Verint Identity Authentication [12].",
"Verint Identity Authentication and Verint Fraud Detection work by instantly comparing the caller's voice to the voice prints of an extensive database of known fraud perpetrators.",
"This comparison is done silently in the background of a call.",
"The solution combines a new generation of passive voice biometrics with unique predictive analysis that can help accurately detect fraudsters and authenticate customers without caller interruption.",
"Voice recognition technology offers a powerful combination of reduced customer effort and fraud deterrence.",
"We expect this adoption will rapidly increase, not just among financial institutions but also among insurance, e-commerce, utilities, and other security-sensitive businesses.",
"As stated earlier, scammers take advantage of naive citizens by convincing them that they are involved in a legal investigation regarding terrorists and they need to deposit money into a bank account to clear their names.",
"The online phone directory provided by Turk Telekom enables scammers to address individuals by name and a generalized address only.",
"We showed that by automatically linking the leaked dataset to a subset of Turk Telekom phone directory, we are able uniquely identify more than 60% of the individuals.",
"Hence, now an attacker can address individuals by their name plus all the information about them in the leaked dataset.",
"Moreover, they can infer their mothers' maiden names and also gather information about their close relatives.",
"Scammers have been able to trick a significant number of people by just using their names (in some cases even without using their names).",
"Now with all the information publicly available, we expect the number of events to rise as scammers will now have more credibility.",
"We recommend the authorities (e.g., law enforcement and the media) to warn Turkish citizens by all means against this attack.",
"Also, the online phone directory should be more difficult to query.",
"For instance, a query with only a last name should be disabled.",
"In future work, we will further study the privacy implications of the leaked dataset.",
"For instance, linking publicly available social network data can yield more information about individuals and their relatives.",
"This will require obtaining social network data, pre-processing it, and linking individuals with public profiles in the social network data with the leaked dataset.",
"We will also analyze the threat smartphone apps pose (along with the information in the leaked dataset).",
"Users tend to approve any data access request by apps if they really need the functionality app provides.",
"Attackers, using these apps as trojan horses, can have access to valuable personal information about the users.",
"Attackers can even have access to the text messages of the user, which may contain 2-factor authentication passwords.",
"One other aspect of our analysis will be the threat analysis for e-government accounts in relation to this data leak.",
"E-government accounts only use the national identifier number and a user defined password.",
"Should the password be weak, with the current leak many e-government accounts could be compromised.",
"Based on the results of our future analyses, we are going to propose new security measures to protect individuals from possible threats."
],
[
"Conclusion",
"In this work, we demonstrated that the Turkish citizen dataset leak poses a larger threat than currently anticipated.",
"The fact that the mother's maiden name is not directly provided has been a relief for the Turkish community.",
"We aimed to perform automated attacks that show without any manual intervention we can extract valuable personal information.",
"We showed that even in this scenario, disclosed information is much more than just national identifier numbers.",
"We also showed how to uniquely determine the landline numbers of the individuals in the leaked dataset and proposed basic countermeasures for the presented attacks.",
"Finally, we studied the uniqueness of the Turkish citizens based on their demographics.",
"We believe that the outcome of this study will guide the researchers that need estimates of the threat of disclosing anonymized datasets along with demographics."
]
] | 1605.05847 |
[
[
"Benchmarking NFV Software Dataplanes"
],
[
"Abstract A key enabling technology of NFV is software dataplane, which has attracted much attention in both academia and industry recently.",
"Yet, till now there is little understanding about its performance in practice.",
"In this paper, we make a benchmark measurement study of NFV software dataplanes in terms of packet processing capability, one of the most fundamental and critical performance metrics.",
"Specifically, we compare two state-of-the-art open-source NFV dataplanes, SoftNIC and ClickOS, using commodity 10GbE NICs under various typical workloads.",
"Our key observations are that (1) both dataplanes have performance issues processing small (<=128B) packets; (2) it is not always best to put all VMs of a service chain on one server due to NUMA effect.",
"We propose resource allocation strategies to remedy the problems, including carefully adding CPU cores and vNICs to VMs, and spreading VMs of a service chain to separate servers.",
"To fundamentally address these problems and scale their performance, SoftNIC and ClickOS could improve the support for NIC queues and multiple cores."
],
[
"Introduction",
"Middleboxes are ubiquitous in today's networks and provide important network functions to operators [24].",
"Traditionally, middleboxes are deployed as dedicated proprietary hardware.",
"Network function virtualization (NFV) is now emerging to replace hardware boxes with virtual software instances running on commodity servers.",
"NFV holds great promises to improve flexibility and efficiency of network function management.",
"Thus it is quickly gaining momentum in the industry [11].",
"The ETSI Industry Specification Group for NFV has attracted over 290 individual companies as members, including major service providers such as AT&T, NTT, Sprint, China Mobile, China Unicom, and IT vendors such as Cisco, Huawei, IBM, and VMware [4].",
"The NFV market is projected to grow more than 5-fold through 2019 to reach $11.6 billion [6].",
"A key enabling technology to NFV is software dataplane.",
"It provides a virtualized platform for hosting software middleboxes with high performance packet I/O to userspace.",
"A number of NFV dataplanes have been developed recently.",
"For example SoftNIC [13] and NetVM [15] use KVM for virtualization and Intel DPDK for packet I/O.",
"ClickOS [19] relies on Xen and netmap [21] instead for virtualization and packet I/O, respectively.",
"Despite the progress, there is a lack of understanding on NFV dataplane performance in the community.",
"A software dataplane desires high-performance packet processing, flexible programming interfaces, security/isolation between colocating VNFs, and so forth [19], [15].",
"Among these performance metrics, packet processing capability is fundamental and critical since it determines the basic usability of a software dataplane.",
"Hence it becomes the focus of our study.",
"Specifically, we ask the question, how well do these software dataplanes perform packet processing in practice?",
"Existing work and their evaluation do not address this question well.",
"Most work (e.g.",
"SoftNIC [13]) focuses on raw packet I/O without software middleboxes running as VMs on top.",
"Some (e.g.",
"ClickOS [19]) report performance of different software middleboxes only when deployed individually.",
"More importantly, no performance comparison is done across NFV dataplanes under the same environment.",
"Thus, it is unclear whether these NFV dataplanes can achieve line rate with different packet processing logic in software middleboxes, what are their bottlenecks if any, and how they perform against each other in various deployment settings such as NF chaining and colocation?",
"In this paper, we present arguably the first measurement study of NFV dataplanes that provides initial answers to the above question.",
"We strategically choose two popular open source NFV dataplanes, SoftNIC [13] and ClickOS [19], that differ widely in virtualization and packet I/O technologies.",
"As a first step we focus on their packet processing throughput running on commodity servers and 10GbE NICs.",
"We use two basic virtual network functions (VNFs): L3 forwarding and firewall.",
"Our measurements reveal several major findings: Both SoftNIC and ClickOS can achieve line rate with medium to large packets ($>$ 128B), even when CPU is clocked down to 1.2GHz.",
"In a practical setting with mixed packet sizes and low network utilization, both dataplanes can handle typical traffic.",
"Both dataplanes cannot achieve line rate processing small packets ($\\le $ 128B) on a 2.6GHz CPU.",
"For SoftNIC the bottleneck is due to the lack of multi-queue support at vNIC.",
"We observe that adding more vNICs and correspondingly more vCPUs achieves line rate for 64B packets.",
"For ClickOS we believe the bottleneck is its high CPU usage, which cannot be resolved due to its lack of SMP support.",
"Performance also degrades in the NF chaining scenario, with ClickOS being more sensitive to chain length.",
"Perhaps surprisingly, placing all VNFs of the chain on the same server does not necessarily lead to best performance, because the NUMA effect may further degrade performance when there are too many VNFs to be put to the same CPU socket.",
"In this case simply assigning VNFs to different servers and using NICs to chain them can eliminate the NUMA effect.",
"The results provide useful implications for efficient resource management of NFV deployment in practice.",
"For a telecom or ISP that deploys NFV to run her middleboxes, our results suggest that a dynamic resource allocation strategy can be adopted to opportunistically adjust the CPU speed or number of cores of the VNF and save energy without sacrificing performance.",
"Most production networks are mildly utilized, suggesting that significant savings of electricity cost can be realized using this approach.",
"Our results on NF chaining also shed light on VNF placement, an important management task of an NFV cluster.",
"We show that it is better to place VNFs on separate servers (on the same CPU socket) and chain them up using NICs in order to eliminate the NUMA effect, when it is impossible to assign them to one CPU socket.",
"Our study also provides helpful implications for the research community on performance optimization of software dataplane.",
"The results consistently suggest that an important research direction is the support for multiple cores and NIC queues, which can fundamentally scale the performance of software dataplane in demanding scenarios, especially as the network evolves to 40G and beyond."
],
[
"Background",
"We start by providing background of SoftNIC and ClickOS."
],
[
"SoftNIC",
"The SoftNIC software dataplane composes of three components: Intel DPDK [3] as the high-performance userspace packet I/O framework, SoftNIC [13] as the programmable dataplane, and KVM as the hypervisor to isolate the VNFs.",
"DPDK.",
"The Intel DPDK framework allows applications to poll data directly from the NIC without kernel involvement, thus providing high-performance userspace network I/O.",
"To achieve line rate, a DPDK process occupies the CPU core and constantly polls the NIC for packets.",
"SoftNIC.",
"SoftNIC [13] is a programmable dataplane abstraction layer that allows developers to flexibly build software that leverages NIC features with minimal performance loss.",
"One can develop her own packet processing pipeline with a series of modules.",
"A module can interact with a physical NIC (pNIC) and/or a vNIC of a VM.",
"When two modules are connected in a pipeline, a traffic class (TC) is created.",
"A TC is assigned with a unique worker thread running on a dedicated core to move packets between the modules.",
"A worker may be assigned to multiple TCs.",
"The open source version of SoftNIC has recently been renamed BESS [2].",
"We do not use the name here to avoid confusion.",
"KVM.",
"SoftNIC provides a backend vNIC driver based on vhost-net [7] which allows it to interact with KVM.",
"We thus choose KVM as the hypervisor environment for it."
],
[
"ClickOS",
"ClickOS.",
"ClickOS [19] is another popular NFV platform.",
"It composes of netmap [21] and VALE [22] as the packet I/O framework, Click [17] as the programmable dataplane, and Xen as the hypervisor.",
"By redesigning the virtual network drivers in Xen, ClickOS achieves very high packet processing performance.",
"Meanwhile, by leveraging Click users can flexibly build software middleboxes.",
"VALE and netmap.",
"VALE is a virtual software switch for packet I/O between VMs based on netmap [21].",
"ClickOS modifies VALE to support pNIC directly.",
"A key difference between VALE and TCs in SoftNIC (or any DPDK based software dataplane) is that VALE does not use dedicated threads/cores to move packets between modules; the sending thread does the work of copying packets into the Rx queue."
],
[
"Methodology",
"We explain our measurement methodology in detail here."
],
[
"Hardware Setup",
"We conduct our measurements using physical machines rent from Aptlab[1].",
"We use two c6220 nodes with 2 Xeon E5-2650v2 processors (8 cores each, 2.6Ghz), 64GB DDR3 1.86GHz Memory and an Intel X520 10GbE PCIe dual port NIC.",
"For most of the experiments, one node runs a packet generator to send packets of different sizes to the other node, which serves as the hypervisor hosting VNFs to process packets.",
"Packets are sent back through another NIC of the hypervisor to the first node, which is also our vantage point.",
"We disable DFVS and fix the CPU at 2.6Ghz for both nodes unless otherwise stated.",
"For NICs, we disable auto-negotiation, TSO, and GSO as recommended by ClickOS[19]."
],
[
"Software Dataplane Settings",
"We met some difficulties in deploying SoftNIC and ClickOS on our testbed.",
"Since their components are independently maintained and some have evolved, we were unable to build them in the same environment reported in the original papers or the available online documentation.",
"Some components, such as netmap, VALE, and Xen, have strict dependencies on the kernel version, NIC models, and hardware features which further complicate the problem.",
"With the help of developers for both dataplanes, we experimented with over a dozen different environments, and found settings that yield the best performance.For ClickOS, we found only one environment where we can successfully build all the components as in [19] in our testbed.",
"For SoftNIC, we use Linux kernel 3.19.0, QEMU/KVM 2.2.1, DPDK 2.20, and the latest SoftNIC source code [2].",
"For ClickOS, we use Linux kernel 3.9.10, Xen 4.4.2[10], ClickOS 0.1 [9] and netmap commit 3ccdada [5] respectively.",
"We use an older version of netmap that works with the modified Xennet library [8] provided by ClickOS for Xen backend and frontend NIC drivers.",
"We confirmed our settings with the authors of SoftNIC and ClickOS to ensure validity of our measurements.",
"We also verified that the baseline performance of L2 forwarding is consistent with or no worse than those reported previously in [13] and [19].",
"Figure REF illustrates the packet I/O pipeline we use in our measurements with a single VNF.",
"Each VNF has two vNICs, vNIC0 as the ingress NIC and vNIC1 as the egress NIC to its next hop.",
"In general, for both SoftNIC and ClickOS, first a packet is moved by the pNIC0 driver to the backend of vNIC0, which then sends it to the frontend driver in the VNF.",
"After being processed by the VNF, the packet is sent to the frontend and then backend of vNIC1.",
"Finally it is sent to an output NIC in the hypervisor.",
"The difference between the two software dataplanes is that, SoftNIC uses a TC to connect a pNIC and a VNF (or two VNFs), while ClickOS uses a VALE switch instead.",
"NF chaining can be realized by having the TC or VALE connecting the vNIC1 backend driver of the previous VNF to the vNIC0 backend of the next VNF in the chain.",
"Figure: Packet I/O pipeline with a single VNF in our measurement.For maximum performance, we pin vCPU(s) of VNFs into fixed physical cores by taskset in KVM and xl vcpu-pin in Xen.",
"As mentioned in sec:softnic TCs in SoftNIC need to be assigned with workers which also must be pinned into dedicated cores.",
"We pin vCPUs to cores in the same socket whenever possible to avoid severe performance penalty caused by NUMA [18] (more in sec:resultchain)."
],
[
"Network Functions",
"We use two NFs in this study, L3 forwarding (L3FWD), and firewall.",
"We cannot use generic software such as Snort or Bro because they are not written with DPDK or Click to exploit software dataplane for packet processing.",
"We assign 1 vCPU and 1GB memory to each VNF unless stated otherwise.",
"L3FWD.",
"We use the ip_pipeline example code provided by DPDK as the L3FWD implementation in SoftNIC.",
"The ClickOS implementation is done by concatenating FromDevice, StaticIPLookup, and ToDevice elements.",
"In both implementations we insert 10 entries to the routing table.",
"Firewall.",
"We build the firewall implementation based on the same ip_pipeline in SoftNIC.",
"In ClickOS, the implementation uses FromDevice, IPFilter, and ToDevice elements.",
"We use 10 rules to filter packets.",
"We also use the simple L2 forwarding in the NF chaining experiment only though.",
"We do not consider it a NF as it does not have any packet processing logic.",
"L2FWD.",
"We directly use the L2FWD provided out-of-the-box from ip_pipeline of DPDK for SoftNIC.",
"In ClickOS, we implement L2FWD by connecting the FromDevice and ToDevice elements between two vNICs."
],
[
"Miscellaneous ",
"We use the DPDK pkt-gen module to generate packets for experiments with SoftNIC, and the netmap pkt-gen for ClickOS.",
"We use different packet sizes: 64B, 128B, 256B, 512B, 1024B, and 1500B.",
"We also use an empirical packet size distribution from Facebook's data center network [23] to see how the dataplanes perform in a practical environment.",
"In all scenarios, we verify that the pkt-gens can achieve line rate of 10Gbps.",
"We mainly use throughput in both million packets per second (Mpps) and Gbps as the performance metrics.",
"We investigate the performance of SoftNIC and ClickOS in different scenarios in this section.",
"The thesis of the evaluation is simple: can these NFV dataplanes achieve line rate, and if not, what are the bottlenecks?",
"We first look at the baseline scenario with a single software middlebox, running different NFs with varying CPU speed (sec:frequency).",
"Based on the results we analyze and identify performance bottlenecks of both dataplanes (sec:cores).",
"We then deploy multiple NFs in two scenarios that are commonplace in practice: NF chaining where packets go through the middleboxes sequentially for processing (sec:resultchain), and NF colocation where multiple NFs colocate on the same server and work independently (sec:colocation)."
],
[
"Baseline Performance",
"We start with just a single VNF.",
"Since software packet processing is CPU-intensive, we want to see if CPU speed is the bottleneck here.",
"In this set of experiments, we vary the CPU speed from the configurable range of 1.2GHz to 2.6GHz for our CPU without Turbo Boost, and investigate the throughput with different packet sizes.",
"We modify the CPU frequency using cpufreq-set and xenpm set-scaling-speed for SoftNIC and ClickOS, respectively.",
"Figure REF demonstrates the performance of L3FWD.",
"We observe the following.",
"First, performance increases with CPU speed for small packets (64B–256B), which is expected—a faster CPU can process more instructions and thus more packets.",
"For 64B and 128B packets, performance improvement is commensurate with CPU speed-up between 1.2GHz to 2.4GHz.",
"With 64B packet for instance, at 1.2GHz throughput of SoftNIC and ClickOS is 4.31Mpps and 2.10Mpps, respectively, while at 2.4GHz it roughly doubles at 8.60Mpps and 4.26Mpps, respectively.",
"The improvement is smaller at 2.6GHz.",
"Second, both NFV dataplanes achieve line rate for packets bigger than 128B, but have problems dealing with smaller packets even at 2.6GHz.",
"SoftNIC can achieve 10Gbps with 128B packets at 2.4GHz, and ClickOS can process 256B packets at 10Gbps at 2.6GHz.",
"Yet for 64B packets, even at 2.6GHz, neither achieves line rate: SoftNIC tops at 9.34Mpps and ClickOS 5.34Mpps.",
"Third, SoftNIC outperforms ClickOS in all cases, especially for small packets.",
"We compare our results with those reported in the ClickOS original paper [19] and confirm they are similar.",
"For example, in Figure 13 of [19], throughput of L3FWD and Firewall with 64B packets are 4.25Mpps and 5.40Mpps, respectively, and ours are 5.34Mpps and 5.03Mpps, respectively.",
"Finally, we find that the performance difference between L3FWD and firewall is minimal, as shown in Table REF .",
"We thus only show L3FWD results hereafter for brevity.",
"Table: Throughput of different NFs with varying CPU speed.",
"Packet size is 64 bytes.We also use an empirical packet size distribution from Facebook's web server cluster [23] to see how the software dataplanes perform in a practical environment.",
"The median packet size is $\\sim $ 120B, and most packets are less than 256B.",
"We configure pkt-gen to sample the trace and generate packets first at 10Gbps, and observe that the average throughput in SoftNIC is 8.662Mpps.",
"However a production network is rarely fully utilized.",
"Facebook reports their median link utilization is 10%–20% [23].",
"This implies that the median packet processing requirement is 0.87Mpps–1.73Mpps.",
"Both SoftNIC and ClickOS are able to provide such capability in lowest CPU frequency of 1.2GHz.",
"These observations have interesting implications to NFV resource allocation.",
"They suggest that there are ample opportunities for the operator to downclock the CPU in order to save energy and electricity cost in the average case.",
"Care has to be taken though, of course, to ensure performance does not suffer when there are sudden bursts of small packets.",
"This may be a useful resource allocation strategy for operators as well as meaningful research directions to look into for the networking community.",
"Our observations also motivate us to identify performance bottlenecks in both NFV dataplanes for small packets, which we explain next."
],
[
"Performance Bottlenecks",
"One may argue that the performance deficiency of software dataplanes in small packet regime is acceptable in practice, since small packets may be less common.",
"However, these systems may not be able to achieve line rates in the emerging 40G or 100G networks [16], even for large packets.",
"Therefore we believe it is important for us to understand the performance bottleneck and improve performance.",
"To identify bottlenecks, we conduct the following analysis.",
"For SoftNIC, we observe from using monitor port command that about 5Mpps 64B packets are lost in the pipeline between pNIC0 and vNIC0.",
"To see if VMs are the bottleneck, we allocate more vCPUs and memory and observe the L3FWD throughput with 64B, 96B, and 128B packets.",
"As shown in Figure REF , however, this results in little improvement.",
"After discussing with SoftNIC authors, we suspect that the bottleneck is the vhost-net queue in vNIC0.",
"SoftNIC currently does not support multiple vhost-net queues for vNIC, which explains why adding resources to the VM does not help.",
"To verify the analysis, we conduct another experiment by adding a round-robin module (RR) and another two vNICs to the L3FWD VM as shown in Figure REF .",
"Traffic is evenly split between the two input vNICs.",
"This time throughput reaches line rates for all packet sizes as shown in Figure REF .",
"Figure: Small packet throughput of SoftNIC with different resource allocations.",
"For example “SoftNIC-1C-1G-2vNICs” means we use 1 vCPU, 1GB RAM, and 2vNICs (one for input and one for output) for the VM.",
"For the last setting, we use 2 vCPUs, 1GB RAM, and 4vNICs (two for input and two for output).Figure: The pipeline of 2 NFs with a round-robin module in SoftNIC.For ClickOS, we analyze the CPU utilization of the L3FWD instance with the CPU at the highest 2.6GHz.",
"As shown in Table REF , ClickOS uses 100% CPU when processing 64B and 128B packets, and larger packets lead to much lower CPU utilization.",
"This implies that more CPU resource may be needed here.",
"However, ClickOS currently does not have SMP support [19], preventing us from adding more cores to the VM.",
"This also means adding more vNICs does not help without more CPU.",
"Another possible solution is to use multiple VNFs working in parallel.",
"This naturally requires a load balancer (LB) to split the traffic.",
"VALE is a simple L2 switch without any load balancing capability [22].",
"Adding a LB VM does not work either since the LB itself becomes the bottleneck.",
"Therefore we are unable to resolve the bottleneck without modification to ClickOS itself.",
"Table: The CPU utilization of L3FWD in ClickOS.",
"CPU is at 2.6GHz.To summarize, the results here verify that SoftNIC's bottleneck is the vhost-net queue of the vNIC.",
"This can be resolved by sending traffic to two vNICs of one VNF in parallel to fully utilize multiple vCPUs.",
"A complete fix requires SoftNIC to add support for multiple vhost-net queues.",
"We have confirmed our analysis with the SoftNIC team already.",
"We also present evidence to suggest that ClickOS should add SMP support that allows it to utilize multiple CPU cores.",
"In any case, we note that it is imperative for the NFV software dataplane architecture to provide horizontal scaling of its performance, in order to better utilize multiple cores and physical NIC queues.",
"We believe this is an interesting open research area as the NICs evolves to 40Gbps and beyond."
],
[
"NF Chaining",
"It is common to deploy multiple software middleboxes on the same machine.",
"In this section we look into the NF chaining scenario, where the processing pipeline consists of a chain of different middleboxes.",
"We are interested to see if the performance of an NF chain can match that of just a single NF.",
"We compose chains of different lengths: the 1-NF chain uses only a L3FWD; the 2-NF chain uses a firewall followed by a L3FWD; and the 3-NF chain adds a L2FWD to the end of the 2-NF chain.",
"Figure: Throughput with varying length of NF chain.The results are shown in Figure REF with all vCPUs running on the same physical CPU.",
"The performance of SoftNIC suffers mild degradation for 64B–128B packets, as can be seen from the overlapping lines of 2-NF chain and 3-NF chain.",
"We suspect there is a bottleneck in chaining the vNICs of different VMs because both firewall and L3FWD can achieve higher performance individually as shown in sec:frequency.",
"Performance of ClickOS also degrades as the chain grows, especially for small packets.",
"A ClickOS L3FWD achieves line rate with 256B packets, but a 2-NF chain or 3-NF chain cannot.",
"A 3-NF chain cannot even reach line rate with 512B packets.",
"Note that here we use multiple VALE switches with independent vCPUs pinning to different cores to chain the VMs as suggested by [19].",
"We believe the overhead of copying packets in VALE attributes to the performance penalty.",
"When deploying a NF chain, an important factor we must consider is the affinity of vCPUs and the effect of NUMA.",
"As an example Figure REF depicts two possibilities of vCPU settings with a 3-NF chain.",
"We can pin each vCPU to the same physical CPU, or pin them to CPUs in different sockets.",
"The latter is unavoidable sometimes as the commodity CPUs have limited cores per CPU, and DPDK-based NFV dataplanes like SoftNIC require many dedicated cores as mentioned in sec:softnic and sec:software.",
"Figure: Pipeline of NF-ChainsWe perform another measurement to evaluate the effect of NUMA on NF chaining.",
"Figure REF shows the result for SoftNIC as a case study.",
"NUMA has a significant impact on performance.",
"For the 2-NF chain, assigning two vCPUs and TCs to different sockets cuts the throughput of 64B packet by nearly half.",
"For the 3-NF chain (the third VNF runs in a different socket than the first two), line rate is only reached for 512B and larger packets.",
"The performance discrepancy is mainly because operations between different NUMA sockets can cause cache misses and ping pong effect [18].",
"To mitigate NUMA effect, we attempt to bridge NFs in a chain via NICs across servers.",
"For example, in a 3-NF chain, NF1 and NF2 are located on server A on the same NUMA socket, and NF3 is located in server B.",
"The two servers are connected by 10GbE NIC.",
"We observe that this eliminates the NUMA effect: throughput of the chain is identical to the case when all 3 NFs share the same CPU socket as shown in Figure REF .",
"Figure: Throughput of NF chaining with the NUMA effect.",
"“*” here denotes the case when the vCPUs belong to different sockets.",
"“^” here denotes the case when NFs are chained up via NICs of different servers to avoid penalty from NUMA.To summarize, the results here show that SoftNIC works adequately with small performance drop in a NF chain, while ClickOS's throughput becomes lower with longer chains.",
"They also demonstrate the importance of carefully assigning cores to VMs of the chain due to NUMA, which implies that it is not always best to colocate VNFs of a chain on the same server.",
"A practical strategy is to place them on different servers to avoid NUMA effect.",
"These observations are useful for real NFV deployment."
],
[
"NF Colocation",
"We also measure the performance when multiple NFs colocate on the same server.",
"This is another common deployment scenario of NFV.",
"In the experiments here, we instantiate multiple VMs, bundling each of them to an independent packet generator in the same server, and measure the aggregated throughput.",
"For SoftNIC, we build pkt-gen in separate VMs and connect them to L3FWD VNFs by independent TCs.",
"On the other hand for ClickOS, we directly use pkt-gen to generate packets on the VALE switch connected to the VM.",
"We pin pkt-gen and the corresponding VM to the same CPU socket for better performance.",
"Note this is the only scenario where we generate packets at the hypervisor.",
"We scale to at most 3 bundles beyond which the number of cores on our CPU is not enough (our CPU has 16 cores: SoftNIC needs 4 cores for each bundle, and the hypervisor needs cores too).",
"Figure: Aggregated throughput of multiple colocating L3FWD.As we observe in Figure REF , both NFV dataplanes perform very well.",
"In almost all results, throughput scales linearly as we colocate more bundles of VMs and packet generators.",
"This demonstrates that current technologies provide satisfactory performance isolation and guarantee with multi-core CPUs for realizing NFV."
],
[
"Related Work",
"We introduce related work on NFV dataplane other than SoftNIC and ClickOS now.",
"NetVM [15] is another NFV platform based on DPDK and KVM, similar to SoftNIC.",
"It provides high-speed inter-VM communication with zero-copy through shared huge pages.",
"We plan to study its performance when the code becomes available.",
"Systems such as ptnetmap [12] and mSwitch [14] based on netmap address efficient transfer between VMs in a single server.",
"E2 [20] is a general NFV management framework focusing on NF placement, scheduling, scaling, etc.",
"Its dataplane uses SoftNIC.",
"Our measurement study provides performance comparison across solutions with actual VNFs and complements existing work that evaluates their own system with mostly L2 forwarding.",
"There is little measurement study on NFV in general.",
"Wu et al.",
"design PerfSight [25] as a diagnostic tool for extracting comprehensive low-level information regarding packet processing performance of the various elements.",
"It focuses on virtualization layer (KVM) without integrating with any NFV dataplane such as SoftNIC and ClickOS."
],
[
"Conclusion",
"In this paper, we conducted a measurement study on the performance of SoftNIC and ClickOS.",
"Both dataplanes are capable of achieving 10G line rate with medium and large packets, and scaling performance with multiple colocating VNFs.",
"They have performance issues in the small packet regime and NF chaining scenario, which may become more severe in high speed networks.",
"We proposed to fundamentally address the limitation by architecturing the software dataplane for horizontal performance scaling, in order to better utilize multiple cores and NIC queues.",
"Our study can be extended in many directions.",
"One possibility is to consider more complex NFs such as NAT, VPN, etc., and more metrics such as processing delay.",
"We also plan to further investigate the chaining scenario and identify ways to improve performance."
],
[
"Acknowledgment",
"We are very thankful to members of SoftNIC and ClickOS teams, especially to Sangjin Han from UC Berkeley, and Filipe Manco from NEC Laboratories Europe."
]
] | 1605.05843 |
[
[
"Cyclic and heteroclinic flows near general static spherically symmetric\n black holes: Semi-cyclic flows -- Addendum and corrigendum"
],
[
"Abstract We present new accretion solutions of a polytropic perfect fluid onto an f(R)-gravity de Sitter-like black hole.",
"We consider two f(R)-gravity models and obtain finite-period cyclic flows oscillating between the event and cosmological horizons as well as semi-cyclic critical flows executing a two-way motion from and back to the same horizon.",
"Besides the generalizations and new solutions presented in this work, a corrigendum to Eur.",
"Phys.",
"J.",
"C (2016) 76:280 is provided."
],
[
"Accretion of perfect fluids onto static spherically symmetric black holes",
"This work is based on our previous paper [1] where we set the general dynamical-system formalism for accretion of perfect fluids onto static spherically symmetric black holes.",
"We keep using the same notation for the thermodynamic functions of the fluid and the Hamiltonian $\\mathcal {H}$ .",
"So, $n$ , $h$ , $e$ , $s$ , $p$ , $T$ , and $u^{\\mu }$ are the baryon number density, specific enthalpy (enthalpy per particle), energy density, specific entropy, pressure, temperature, and four-velocity vector, respectively.",
"In a locally inertial frame, the three-dimensional speed of sound $a$ is given by $a^2=(\\partial p/\\partial e)_s$ .",
"When the entropy $s$ is constant, which is the case for accretion of perfect fluids onto static spherically symmetric black holes, this reduces to $a^2=dp/de$ .",
"The aim of this short work is two-fold: (1) Generalize the dynamical-system formalism for accretion of perfect fluids to metrics of the form ${\\rm d}s^{2}= -A(r){\\rm d}t^2+\\frac{{\\rm d}r^{2}}{B(r)}+C(r)({\\rm d}\\theta ^2+\\sin ^2\\theta {\\rm d}\\phi ^2).$ This will allow us to generalize the properties of the accreting fluid intended for future use.",
"(2) Obtain new interesting solutions not discussed so far in the scientistic literature.",
"In (REF ), ($A,\\,B,\\,C$ ) are any functions of the radial coordinate $r$ assumed to be well-defined and positive-definite in the regions where the Killing vector $\\xi ^{\\mu }=(1,0,0,0)$ is timelike and their ratio $D\\equiv A/B$ is positive-definite on the horizons too.",
"For the metric (REF ) to describe a black hole solution, the equations $B(r)=0$ and $A(r)=0$ should have the same set of solutions with same multiplicities.",
"In our applications we restrict ourselves to the cases where the global structure of the spacetime is well-determined by ($A,\\,B,\\,C$ ).",
"To keep the analysis general, we do not assume asymptotic flatness of the metric as we intend to apply it to the de Sitter and anti-de Sitter-like black holes.",
"This metric form generalizes the one used in Refs.",
"[1], [2], [3]; In Ref.",
"[1] we restricted ourselves to the case $A=B=f(r)$ and $C=r^2$ .",
"It is easy to show that equations (5), (7), (11), (23), (24), (25) of Ref.",
"[1] generalize respectively to $&u_t=\\pm \\sqrt{A+Du^2}, &\\sqrt{D}~Cnu=C_1\\ne 0,\\\\&h\\sqrt{A+Du^2}=C_2, &v^2=\\frac{Du^2}{A+Du^2},\\\\&u^2=\\frac{Bv^2}{1-v^2}, &\\frac{AC^2n^2v^2}{1-v^2}=C_1^2,$ where ($C_1,C_2$ ) are constants of motion, $u\\equiv u^r$ , and $-1<v<1$ is the three-velocity of a fluid element as measured by a locally static observer.",
"The second line in () expresses the law of particle conservation, $\\nabla _{\\mu }(n u^{\\mu })=0$ , and $C_2$ is the constant of motion $hu_{\\mu }\\xi ^{\\mu }$ [$\\xi ^{\\mu }=(1,0,0,0)$ is a timelike Killing vector].",
"This constant is the inertial-equivalent generalization of the energy conservation equation $mu_{\\mu }\\xi ^{\\mu }$ [4].",
"The constant $C_1^2$ in () can be written as $A_0C_0^2n_0^2v_0^2/(1-v_0^2)$ where “0\" denotes any reference point ($r_0,\\,v_0$ ) from the phase portrait; this could be a CP, if there is any, spatial infinity ($r_{\\infty },\\,v_{\\infty }$ ), or any other reference point.",
"We can thus write $\\frac{n^2}{n_0^2}=\\frac{A_0C_0^2v_0^2}{1-v_0^2}~\\frac{1-v^2}{AC^2v^2}=C_1^2~\\frac{1-v^2}{AC^2v^2}.$ The equation of state (EoS) of the fluid may be given in the form $e=F(n)$ or equivalently in the form $p=G(n)$ [1].",
"It has been shown [1] that ($F,\\,G$ ), the specific enthalpy, and the three-dimensional speed of sound satisfy $&nF^{\\prime }(n)-F(n)=G(n),\\\\&h=F^{\\prime }(n),\\\\&a^2=n(\\ln F^{\\prime })^{\\prime },$ where the prime denotes derivative with respect to $n$ .",
"Equations (REF ) and () imply that the specific enthalpy $h$ depends explicitly on ($A,\\,C,\\,v$ ) only.",
"There is no explicit dependence on $B$ .",
"As we shall see in the next subsection, this implies that the Hamiltonian $\\mathcal {H}$ , which defines the dynamical system, will also depend explicitly on ($A,\\,C,\\,v$ ) only.",
"This fact will have consequences on the location of the critical points (CP's).",
"If the fluid had further properties, say, being isothermal or polytropic, $h$ and $\\mathcal {H}$ can be expressed explicitly in terms of ($A,C$ ), as this is done in the following section.",
"Horizons $r_h$ are defined by $A(r_h)=0$ and $B(r_h)=0$ or simply by $B(r_h)=0$ since the equations $A(r_h)=0$ and $B(r_h)=0$ are assumed to have the same set of solutions with same multiplicities.",
"The zeros of $B(r_h)=0$ determine the regions in three-space where the fluid flow takes place: These are the regions where $\\xi ^{\\mu }=(1,0,0,0)$ is timelike.",
"Note that the case $C_1\\equiv 0$ , corresponding either to $n=0$ (REF ) (no fluid) or to $v=0$ () and any $n$ (no flow), is not interesting.",
"So, we assume $C_1\\ne 0$ .",
"As the fluid approaches, or emanates from, any horizon ($r\\rightarrow r_h$ ), $A$ approaches 0.",
"Three cases emerge from (REF ): $v\\rightarrow \\pm 1^{\\mp }$ and $n$ may converge or diverge there ($r\\rightarrow r_h$ ); $v\\rightarrow 0$ and $n$ diverges there.",
"The fluid cumulates near the horizon resulting in a divergent pressure which repulses the fluid backwards [1]; $|v|$ assumes any value between 0 and 1 there.",
"This yields a divergent $n$ as $r\\rightarrow r_h$ .",
"Since $0<|v|<1$ there is no reason that the fluid cumulates near the horizon: The flow continues until all fluid particles have crossed the horizon.",
"We rule out this case for it is not physical.",
"This conclusion, due to the law of particle conservation, is general and it does not depend on the fluid characteristics.",
"Since a non-perfect simple fluid (containing a single particle species) also obeys the law of particle conservation (), these conclusions remain valid for real fluids too."
],
[
"Dynamical system - Critical points",
"If the fluid had a uniform pressure, that is, if the fluid were not subject to acceleration, the specific enthalpy $h$ reduces to the particle mass $m$ and the first equation in () reduces to $mu_{\\mu }\\xi ^{\\mu }=C_2$ along the fluidlines.",
"This is the well know energy conservation law which stems from the fact that the fluid flow is in this case geodesic.",
"Now, if the pressure throughout the fluid is not uniform, acceleration develops through the fluid and the fluid flow becomes non-geodesic; the energy conservation equation $mu_{\\mu }\\xi ^{\\mu }=cst$ , which is no longer valid, generalizes to its relativistic equivalent [4] $hu_{\\mu }\\xi ^{\\mu }=C_2$ as expressed in the first equation in ().",
"Let the Hamiltonian $\\mathcal {H}$ of the dynamical system be proportional to $C_2^2$ (), which is a constant of motion.",
"Substituting the first equation in () into the first equation in () yields $\\mathcal {H}(r,v)=\\frac{h(r,v)^2A(r)}{1-v^2}.$ With $\\mathcal {H}$ given by (REF ), the dynamical system reads $\\dot{r}=\\mathcal {H}_{,v}\\,, \\quad \\quad \\dot{v}=-\\mathcal {H}_{,r}.$ (here the dot denotes the $\\bar{t}$ derivative where $\\bar{t}$ is the time variable of the Hamiltonian dynamical system).",
"In (REF ) it is understood that $r$ is kept constant when performing the partial differentiation with respect to $v$ in $\\mathcal {H}_{,v}$ and that $v$ is kept constant when performing the partial differentiation with respect to $r$ in $\\mathcal {H}_{,r}$ .",
"The critical points (CPs) of the dynamical system are the points ($r_c,v_c$ ) where the rhs's in (REF ) are zero.",
"To take advantage of the calculations made in [1] we introduce the radial coordinate $\\rho $ and the notation $f$ defined by $\\rho ^2(r)\\equiv C(r),\\quad \\quad f(\\rho )\\equiv A(r).$ The Hamiltonian takes the form $\\mathcal {H}(\\rho ,v)=\\frac{h(\\rho ,v)^2f(\\rho )}{1-v^2}.$ The derivative $\\mathcal {H}(\\rho ,v)_{,\\rho }$ has been evaluated in Ref.",
"[1] by [Eq.",
"(40) of Ref.",
"[1]]: $\\mathcal {H}_{,\\rho }=\\frac{h^2}{1-v^2}\\Big [\\frac{{\\rm d}f}{{\\rm d}\\rho }-2a^2f~\\frac{{\\rm d}\\ln (\\sqrt{f}\\rho ^2)}{{\\rm d}\\rho }\\Big ].$ Using $\\mathcal {H}_{,r}=\\frac{{\\rm d}\\rho }{{\\rm d}r}~\\mathcal {H}_{,\\rho }=\\frac{h^2}{1-v^2}\\Big [\\frac{{\\rm d}f}{{\\rm d}r}-2a^2f~\\frac{{\\rm d}\\ln (\\sqrt{f}\\rho ^2)}{{\\rm d}r}\\Big ]\\\\=\\frac{h^2}{1-v^2}\\Big [\\frac{{\\rm d}A}{{\\rm d}r}-2a^2A~\\frac{{\\rm d}\\ln (\\sqrt{A}C)}{{\\rm d}r}\\Big ],$ we obtain $&\\dot{r}=\\frac{2h^2A}{v(1-v^2)^2}~(v^2-a^2),\\\\&\\dot{v}=-\\frac{h^2}{1-v^2}\\Big [\\frac{{\\rm d}A}{{\\rm d}r}-2a^2A~\\frac{{\\rm d}\\ln (\\sqrt{A}C)}{{\\rm d}r}\\Big ].$ Introducing the notation $g_c=g(r)|_{r=r_c}$ and $g_{c,r_c}=g_{,r}|_{r=r_c}$ where $g$ is any function of $r$ , the following equations provide a set of CPs that are solutions to $\\dot{r}=0$ and $\\dot{v}=0$ : $v_c^2=a_c^2\\quad \\text{ and }\\\\ a_c^2=\\frac{C_cA_{c,r_c}}{C_cA_{c,r_c}+2AC_{c,r_c}}=\\frac{C^2A_{,r}}{(C^2A)_{,r}}\\Big |_{r=r_c},$ where $a_c$ is the three-dimensional speed of sound evaluated at the CP.",
"The first equation states that at a CP the three-velocity of the fluid equals the speed of sound.",
"The second equation determines $r_c$ once the EoS, $e=F(n)$ or $p=G(n)$ (REF ), is known.",
"From the set of equations (REF ) we see that the metric function $B$ does not enter explicitly in the determination of the CP's; however, it does that implicitly via the successive dependence of $h$ on $n$ , of $n$ on $v$ , and of $v$ on $D$ .",
"Other sets of CPs, solutions to $\\dot{r}=0$ and $\\dot{v}=0$ , may exist too.",
"For instance, we may have (1) $A_c=0$ and $A_{c,r_c}=0$ which, by (REF ) and (), yield $\\dot{r}=0$ and $\\dot{v}=0$ without having to impose the constraint $v_c^2=a_c^2$ at the CP.",
"This corresponds to a double-root horizon of an extremal black hole.",
"When this is the case, the accretion becomes transonic well before the fluid reaches the CP, which is the horizon itself (recall that the three-velocity $v$ , as the fluid approaches the horizon, tends to $-1$ ).",
"However, extremal black holes are unstable and whatever accretes onto the hole modifies its mass making it non-extremal so that $A_{c,r_c}=0$ no longer holds.",
"We may also have (2) $h=0$ at some point where $\\xi ^{\\mu }=(1,0,0,0)$ is timelike, which may hold only for non-ordinary (dark, phantom, or else) accreting matter."
],
[
"Hamiltonian system for test isothermal perfect fluids",
"The isothermal EoS is of the form $p=ke=kF(n)$ with $G(n)=kF(n)$ where $k$ , the so-called state parameter, obeys the constraints $0<k\\le 1$ .",
"The differential equation (REF ) reads $nF^{\\prime }(n)-F(n)=kF(n),$ yielding $e=F=\\frac{e_0}{n_0^{k+1}}\\,n^{k+1},\\quad h=\\frac{(k+1)e_0}{n_0}\\Big (\\frac{n^2}{n_0^2}\\Big )^{k/2},$ where we have used ().",
"Using this and (REF ) in (REF ) we obtain $\\mathcal {H}(r,v)=\\frac{A(r)^{1-k}}{C(r)^{2 k} v^{2 k}(1-v^2)^{1-k}},$ where all the constant factors have been absorbed into the redefinition of the time $\\bar{t}$ and the Hamiltonian $\\mathcal {H}$ ."
],
[
"Hamiltonian system for test polytropic perfect fluids",
"The polytropic equation of state is $p=G(n)=\\mathcal {K}n^{\\gamma },$ where $\\mathcal {K}$ and $\\gamma >1$ are constants.",
"Inserting (REF ) into the differential equation (REF ), it is easy to determine the specific enthalpy by integration [1] $h=m+\\frac{\\mathcal {K}\\gamma n^{\\gamma -1}}{\\gamma -1},$ where we have introduced the baryonic mass $m$ .",
"Introducing the constant $Y\\equiv \\frac{\\mathcal {K}\\gamma (C_1n_0)^{\\gamma -1}}{m(\\gamma -1)}=\\text{ const.",
"}>0,$ then using (REF ), $h$ takes the form $h=m\\Big [1+Y\\Big (\\frac{1-v^2}{AC^2v^2}\\Big )^{(\\gamma -1)/2}\\Big ].$ Finally, the Hamiltonian (REF ) reduces to $\\mathcal {H}=\\frac{A}{1-v^2}~\\Big [1+Y\\Big (\\frac{1-v^2}{AC^2v^2}\\Big )^{(\\gamma -1)/2}\\Big ]^2,$ where $m^2$ has been absorbed into a re-definition of ($\\bar{t},\\mathcal {H}$ ).",
"The three-dimensional speed of sound is obtained from () $a^2=\\frac{(\\gamma -1)X}{m(\\gamma -1)+X}\\qquad (X\\equiv \\mathcal {K}\\gamma n^{\\gamma -1}),$ Since $\\gamma >1$ , this implies $a^2<\\gamma -1$ and, particularly, $v_c^2<\\gamma -1$ if there are CPs to the Hamiltonian system.",
"Equation (REF ) bears a striking similarity with Eq.",
"(2.249) on page 119 of Ref.",
"[4]."
],
[
"Corrigendum",
"Using the second equation in () and (REF ), we rewrite $X$ (REF ) as $X=m(\\gamma -1)Y\\Big (\\frac{1-v^2}{AC^2v^2}\\Big )^{(\\gamma -1)/2}.$ Substituting into (REF ), we arrive at $a^2=Y(\\gamma -1-a^2)\\Big (\\frac{1-v^2}{AC^2v^2}\\Big )^{(\\gamma -1)/2}.$ This equation along with the second line in (REF ) take the following expressions at the CPs $&v_c^2=Y(\\gamma -1-v_c^2)\\Big (\\frac{1-v_c^2}{A_cC_c^2v_c^2}\\Big )^{(\\gamma -1)/2},\\\\&v_c^2=\\frac{r_cA_{c,r_c}}{r_cA_{c,r_c}+4A_c}.$ For a given value of the positive constant $Y$ , the resolution of this system of equations in ($r_c,v_c$ ) provides all the CPs, if there are any.",
"In Ref.",
"[1] we worked with $A=B=f$ and $C=r^2$ reducing (REF ) to $v_c^2=Y(\\gamma -1-v_c^2)\\Big (\\frac{1-v_c^2}{r_c^4f_cv_c^2}\\Big )^{(\\gamma -1)/2},$ which is the correct expression of Eq.",
"(112) of Ref. [1].",
"In both equations (111) and (112) of Sec.",
"VI of Ref.",
"[1], one should replace the constant factor $\\frac{n_c}{Y}~\\Big (\\frac{r_c^5f_{c,r_c}}{4}\\Big )^{1/2}$ by 1.",
"The presence of this extra factor did not affect the results, solutions, and conclusions made in Ref.",
"[1]; however, some new interesting solutions have been missed in its Sec.",
"VI on polytropic fluids.",
"Besides the results and conclusions we have discussed so far in the two first sections of this work, we aim (1) to re-derive the same solutions derived in Sec.",
"VI of Ref.",
"[1], using the correct expression (REF ), and (2) to construct new solutions."
],
[
"f(R)-gravity model of Ref. {{cite:9ec045ce6aea138aeb06c5dfdc2a58c745f02488}}",
"In Ref.",
"[1], we considered three models of f(R) gravity [5], [6], [7].",
"For the model of Ref.",
"[5] the black hole solution is of the form $A=B=f$ and $C=r^2$ with $f\\equiv 1-\\frac{2M}{r}+\\beta r-\\frac{\\Lambda r^{2}}{3}.$ Following the notation of Ref.",
"[1], we employ in this work the symbol “$f$ \" for the metric component, $-g_{tt}$ , and the symbol “f\" for the function f(R) defining the f(R)-gravity model.",
"The solutions shown in Fig.",
"5 of Ref.",
"[1], which depict the accretion of a polytropic perfect fluid onto an anti-de Sitter-like $\\text{f}(R)$ black hole (REF ), are re-derived using the same values of the parameters: $M=1$ , $\\beta =0.05$ , $\\Lambda =-0.04$ , $\\gamma =1/2$ , and $Y=-1/8$ .",
"The re-derived solutions using the correct expressions (REF ) and () are plotted in Fig.",
"REF of this work.",
"Figure: Left panel is a contour plot of ℋ\\mathcal {H} () for an anti-de Sitter-like f(R)\\text{f}(R) black hole () with M=1M=1, β=0.05\\beta =0.05, Λ=-0.04\\Lambda =-0.04, γ=1/2\\gamma =1/2, Y=-1/8Y=-1/8.",
"The parameters are r h ≃1.76955r_h\\simeq 1.76955, r c ≃3.68101r_c\\simeq 3.68101, v c ≃0.498889v_c\\simeq 0.498889.",
"Black plot: the solution curve through the CPs (r c ,v c )(r_{c},v_c) and (r c ,-v c )(r_{c},-v_c) for which ℋ=ℋ c ≃0.487469\\mathcal {H}=\\mathcal {H}_c\\simeq 0.487469.",
"Red plot: the solution curve for which ℋ=ℋ c -0.09\\mathcal {H}=\\mathcal {H}_c - 0.09.",
"Magenta plot: the solution curve for which ℋ=ℋ c +0.09\\mathcal {H}=\\mathcal {H}_c + 0.09.",
"Right panel is a contour plot of ℋ\\mathcal {H} () for an anti-de Sitter-like f(R)\\text{f}(R) black hole () with M=1M=1, β=0.05\\beta =0.05, Λ=-0.04\\Lambda =-0.04, γ=5.5/3\\gamma =5.5/3, Y=1/8Y=1/8.",
"The horizon is at r h ≃1.76955r_h\\simeq 1.76955 and there are no CPs.",
"Continuous black plot: the solution curve corresponding to ℋ=0.94447\\mathcal {H}=0.94447.",
"Dashed black plot: the solution curve corresponding to ℋ=0.443809\\mathcal {H}=0.443809.",
"For the clarity of the plot, we have partially removed the branches v<0v<0.Figure: Contour plot of ℋ\\mathcal {H} () for a de Sitter-like f(R)\\text{f}(R) black hole () with M=1M=1, β=0.05\\beta =0.05, Λ=0.04\\Lambda =0.04, γ=1.7\\gamma =1.7, Y=1/8Y=1/8.",
"The parameters are r eh ≃1.91048r_{eh}\\simeq 1.91048, r ch ≃9.8282r_{ch}\\simeq 9.8282, r c ≃4.66942r_{c}\\simeq 4.66942, v c ≃0.19387v_{c}\\simeq 0.19387.",
"Black plot: the solution curve through the CPs (r c ,v c )(r_{c},v_{c}) and (r c ,-v c )(r_{c},-v_{c}) for which ℋ=ℋ c ≃0.59691\\mathcal {H}=\\mathcal {H}_{c}\\simeq 0.59691.",
"This solution is new and it was not discovered in Ref. .",
"Magenta plot: the solution curve corresponding to ℋ=ℋ c +0.005\\mathcal {H}=\\mathcal {H}_{c}+ 0.005.",
"Blue plot: the solution curve corresponding to ℋ=ℋ c +0.05\\mathcal {H}=\\mathcal {H}_{c}+ 0.05.",
"The solutions depicted by the magenta and blue plots are not new and were discovered in Ref.",
".The first thing we note is that the re-derived solution corresponding to $\\gamma =5.5/3$ , right panel of Fig.",
"REF , has no CPs.",
"Apart from this, the re-derived solutions have the same characteristics as those shown in Fig.",
"5 of Ref. [1].",
"Figure: Contour plot of ℋ\\mathcal {H} () for a de Sitter-like f(R)\\text{f}(R) black hole with f(R)\\text{f}(R) given by Hu-Sawicki formula ().",
"We worked with M=1M=1, Q=0.01Q =0.01, R 0 =0.16R_0 =0.16, γ=1.7\\gamma =1.7, Y=1/8Y=1/8, q 1 =41q_1=41, q 2 =19q_2=19, and c 1 c_1 and c 2 c_2 are given by () taking the lower sign corresponding to the physical solution |f ' (R 0 )|≪1|\\text{f}\\,^{\\prime }(R_0)|\\ll 1.",
"Left plot: Critical flow corresponding to ℋ=ℋ c ≃0.35067\\mathcal {H}=\\mathcal {H}_{c}\\simeq 0.35067.",
"Here f ' (R 0 )≃0.0372803\\text{f}\\,^{\\prime }(R_0) \\simeq 0.0372803, r eh ≃2.12854r_{eh}\\simeq 2.12854, r ch ≃7.3975r_{ch}\\simeq 7.3975, r c ≃4.03815r_{c}\\simeq 4.03815, v c ≃0.223489v_{c}\\simeq 0.223489.",
"This solution is new and it was not discovered in Ref. .",
"Right plot: Corresponds to ℋ=ℋ c +0.01\\mathcal {H}=\\mathcal {H}_{c}+0.01.",
"This solution is not new and was discovered in Ref.",
".The solutions depicting the accretion of a polytropic perfect fluid onto a de Sitter-like $\\text{f}(R)$ black hole (REF ), which are shown in Fig.",
"REF of this work, have been constructed using the same values of the parameters used in Fig.",
"6 of Ref.",
"[1]: $M=1$ , $\\beta =0.05$ , $\\Lambda =0.04$ , $\\gamma =1.7$ , and $Y=1/8$ .",
"The magenta and blue solutions were discovered in Ref. [1].",
"The new solutions are the semi-cyclic critical black plots of Fig.",
"REF .",
"The first semi-cyclic solution represents a supersonic accretion from the cosmological horizon, where the initial three-velocity is almost $-1$ , then it becomes subsonic passing the CP, and it vanishes on the event horizon.",
"The accretion is followed by a flowout back to the cosmological horizon reversing all the details.",
"The second semi-cyclic solution is a flowout from the event horizon with an initial three-velocity in the vicinity of 1, which decreases gradually until it is sonic at the CP then zero at the cosmological horizon.",
"This flowout is then followed by an accretion back to the event horizon.",
"Notice that on the horizons, $r_h=r_{eh}$ (event horizon) or $r_h=r_{ch}$ (cosmological horizon), the pressure of the fluid diverges as [1] $p\\propto |r-r_h|^{\\frac{-\\gamma }{2(\\gamma -1)}}\\qquad (1<\\gamma <2).$ if there $v=0$ .",
"This explains why the fluid, once it reaches any horizon with vanishing three-velocity, it is repulsed backward under the effect of its own pressure.",
"As the value of the Hamiltonian exceeds the critical value $\\mathcal {H}_{c}\\equiv \\mathcal {H}(r_c,v_c)$ , cyclic flows between the two horizons form.",
"These flows are sandwiched by two separate branches corresponding to supersonic accretion and flowout, as depicted by the magenta and blue plots of Fig.",
"REF .",
"The separation between these supersonic branches increases with the value of the Hamiltonian resulting in faster accretion and flowout while the cyclic flow tends to become more and more nonrelativistic."
],
[
"f(R)-gravity model of Ref. {{cite:57ff44aaad29d75b79c7186af475372d0df2875b}}",
"For the f(R) model of Ref.",
"[6] we constructed the constant-curvature black hole solution [1] $f(r)=1-\\frac{2M}{r}+\\frac{Q^2}{[1+\\text{f}\\,^{\\prime }(R_0)]r^2}-\\frac{R_0}{12}~r^2,$ where $\\text{f}(R)=-\\mathcal {M}^2\\frac{c_1(R/\\mathcal {M}^2)^n}{c_2(R/\\mathcal {M}^2)^n+1}.$ Here $n>0$ , ($c_1,c_2$ ) are proportional constants [6] $\\frac{c_1}{c_2}\\equiv q_2\\approx 6~\\frac{\\Omega _{\\Lambda }}{\\Omega _{m}}=6~\\frac{0.76}{0.24}=19,$ and the mass scale $\\mathcal {M}^2=(8315\\text{Mpc})^{-2}\\Big (\\frac{\\Omega _{m}h^2}{0.13}\\Big ).$ At the present epoch [6] $\\frac{R_0}{\\mathcal {M}^2}\\equiv q_1\\approx \\frac{12}{\\Omega _{m}}-9=41.$ Taking $n=2$ , we found [1] $c_1=q_2c_2,\\qquad c_2=-\\frac{1}{q_1^{3/2}(\\sqrt{q_1}\\pm \\sqrt{2q_2})}.$ The physical solution corresponds to $|\\text{f}\\,^{\\prime }(R_0)|\\ll 1$ .",
"Using the correct equation (REF ) and Eq.",
"(), which takes the form $\\hspace{-5.69054pt}v_c^2=\\frac{(1+\\text{f}\\,^{\\prime }(R_0)) (R_0 r_c^3-12 M) r_c+12 Q^2}{3 [(1+\\text{f}\\,^{\\prime }(R_0)) (R_0 r_c^3-8 r_c+12 M) r_c-4 Q^2]},$ we construct the new solutions, shown in Fig.",
"REF , using the same values of the parameters used in Fig.",
"7 of Ref.",
"[1]: $M=1$ , $Q =0.01$ , $R_0 =0.16$ , $\\gamma =1.7$ , $Y=1/8$ , $q_1=41$ , $q_2=19$ , and $c_1$ and $c_2$ are given by (REF ) taking the lower sign corresponding to the physical solution $|\\text{f}\\,^{\\prime }(R_0)|\\ll 1$ .",
"These new solutions have the same characteristics of those depicted in Fig.",
"REF .",
"Only the semi-cyclic critical black plots represent new solutions: The solution depicted by the magenta plot of Fig.",
"REF is not new and was discovered in Ref.",
"[1]."
],
[
"Conclusion",
"In this addendum we have first generalized the dynamical-system procedure describing the accretion/flowout of perfect fluids to all black holes endowed with spherical symmetry.",
"This is needed for many future investigations [8].",
"In our dynamical-system procedure we took the radial coordinate and the three-velocity as dynamical variables of the Hamiltonian, which is proportional to the square of the constant of motion $hu_{\\mu }\\xi ^{\\mu }$ [$\\xi ^{\\mu }=(1,0,0,0)$ is a timelike Killing vector].",
"This constant is the relativisitc equivalent generalization of the energy conservation equation $mu_{\\mu }\\xi ^{\\mu }$ [4].",
"We have shown that the de Sitter-like black holes, of different f(R)-gravity models, present cyclic non-critical flows and semi-cyclic critical flows all characterized by a vanishing three-velocity on either horizon or a luminal three-velocity there: no situations where the fluid reaches, or emanates from, either horizon with intermediate three-velocity occur.",
"This is due to the law of particle conservation and it is not related to the nature of the fluid.",
"This conclusion remains valid for real fluids too as they approach any horizon from within a region where $\\xi ^{\\mu }=(1,0,0,0)$ is timelike."
]
] | 1605.06063 |
[
[
"Stringent theoretical and experimental bounds on graviton mass"
],
[
"Abstract We show from theoretical considerations, that if the graviton is massive, its mass is constrained to be about $10^{-32}~eV/c^2$.",
"This estimate is consistent with those obtained from experiments, including the recent gravitational wave detection in advanced LIGO."
],
[
"Upper bound on graviton mass",
"Exactly how long ranged is gravity?",
"It has been established beyond reasonable doubt that GR is valid at least in the present epoch, until the far reaches of our Universe, likely until its edge e.g.",
"as seen from the supernovae 1a data [2], [3], and also from recent gravitational wave observations, the latter in fact imposing strong constraints on deviations from GR [4], although not ruling them out completely [5].",
"However, the fact remains that its range has not been, and in fact cannot be tested beyond the cosmological horizon at any epoch, regions outside being causally disconnected from the inside.",
"This horizon is approximately equal to the Hubble radius $L=c/H$ , where $H=$ Hubble parameter.",
"At present, $H\\equiv H_0 \\approx 68~km/s/Mpc$ and $L \\equiv L_0 \\approx 10^{9}$ Mpc.",
"Consequently, from the Yukawa type potential for massive gravity, $V \\propto -\\frac{e^{-cmr/\\hbar }}{r}~,$ it follows that graviton mass $m \\le \\hbar /cL_0~\\approx 10^{-32}~eV/c^2$ , cannot be ruled out.",
"Any any earlier epoch, $L=L_0a$ , where $a=$ the scale factor ($a=1$ in the current epoch, and $\\ll 1$ in earlier epochs), and a weaker bound $m \\le (10^{-32}/a)~eV/c^2$ , with $a<1$ , would have been obtained."
],
[
"Lower bound on graviton mass",
"It was shown long back that we live in an expanding [6] (and in fact accelerating [2], [3]) Universe.",
"It was also shown rigorously in [7] that if one attempts to formulate a quantum field theory of a massive gravitational field in an accelerating Universe, then the graviton annihilation and creation operators satisfy the commutator $M$ and $m$ of [7] are $m$ and $\\Lambda /3$ of this article respectively.",
"$[b(k), b(k^{\\prime })^\\dagger ] = \\frac{4 (k^2)^2}{3 m^2 (m^2 - \\frac{2h^2\\Lambda }{3c^2} )} \\delta ^{(3)} (k-k^{\\prime })~,$ where $k,k^{\\prime }$ are the momenta.",
"Thus in a dark energy dominated Universe at late times, when one can replace $\\Lambda \\approx 1/L_0^2$ , one concludes from Eq.",
"(REF ), that to avoid negative norm states (i.e.",
"ghosts), equivalently negative probabilities when one introduced interactions, the graviton mass is bounded from below by $m \\ge \\hbar /cL~\\approx 10^{-32}~eV/c^2$ .",
"This with the previous upper bound implies that the graviton mass is strongly constrained at $m = 10^{-32}~eV$ .",
"Other theoretical considerations also lead to a graviton mass which is consistent with the above estimate [8], [9], [10], [11], [12].",
"Once again, in any earlier epoch, this lower bound would have remained the same (since $\\Lambda $ is a constant), and perfectly consistent with the upper bound $m\\le (10^{-32}/a)~eV/c^2$ referred to earlier."
],
[
"Consistency with observations",
"Estimates of the graviton mass have been made from various observations, such as estimates of Hubble radius, from orbital decay of binary pulsars and other astrophysical observations [13].",
"Recently, detection of a transient gravitational wave signal from a black hole merger was reported [1].",
"If gravitons were massive, this signal would have reached a time $\\Delta t$ seconds after the arrival of the corresponding light signal of the same event (assuming they were produced at the same time).",
"This was used in [1] (and using the results of [14]) to examine a bound on the graviton mass $m$ $m < 7 \\times 10^{-23} \\left( \\frac{D}{200} \\frac{100}{f} \\frac{1}{f\\Delta t} \\right)^{-1/2} eV/c^2~.$ where the distance from the source $D$ is in Mpc and the (average) frequency of gravitational wave $f$ in Hz.",
"$f\\Delta t$ in the above is bounded from below by $1/\\sigma $ , where $\\rho $ is the signal to noise ratio of the apparatus [14] The signal to noise ratio was denoted by $\\rho $ in [1], [14].. For the detected gravity wave GW150914, using $D=410$ Mpc, $f=100$ Hz and $\\sigma =24$ , one gets $m<10^{-22}$ eV, which is weaker than the observational upper bound mentioned earlier, but consistent with it."
],
[
"Gravitational wave astronomy",
"A massive graviton would imply that there would potentially be advance notice for the arrival of gravitational waves - light from astrophysical events such as supernovae or black hole mergers would arrive $\\Delta t$ seconds before the gravitational wave from the same event, where $\\Delta t > \\frac{2D}{f^2} \\left( 10^{21}~m \\right)^2~.$ Therefore, for $m = 10^{-32}~eV/c^2$ , one gets $&& \\Delta t > \\frac{2 \\times 10^{-22} D}{f^2}~s~.$ Although the above appears tiny, with continued progress in the measurement of smaller and smaller time intervals, observation of such a time difference may not only be possible, but would constitute one of the strongest experimental tests of graviton mass."
],
[
"Source of Dark Matter and Dark Energy",
"It was shown in [15] that a gravitons with mass $m$ , being Bosons, can form a Bose-Einstein condensate (BEC), at temperatures less than the critical temperature, given by $T_c = \\frac{\\hbar c}{k_B} \\left( \\frac{\\rho \\pi ^2}{5 m\\zeta (3)} \\right)^{1/3}$ Furthermore, identifying the density of the condensate, $\\rho $ with the dark matter density of our universe, i.e.",
"$\\rho =0.25\\rho _{crit}/a^3$ , where $\\rho _{crit}=10^{-26} kg/m^3$ is the current critical density of our Universe and $m$ expressed in $eV/c^2$ , Eq.",
"(REF ) becomes [15], [16] $T_c = \\frac{3}{m^{1/3} a}~K$ From Eq.",
"(REF ) it follows that for $m\\le 1~eV/c^2$ , that the background temperature of our Universe at any epoch, $T(a) = 2.7/a < T_c (a)$ .",
"That is the critical temperature exceeds the background temperature of the Universe $T(a)$ at all times, and a giant BEC consisting of gravitons in their ground state would have formed at a very early epoch.",
"This BEC may thus account for the dark matter content in our Universe (with its density variations accounting for the dark matter density profiles).",
"Furthermore, under the assumption of large scale homogeneity and isotropy, the quantum potential associated with this wavefunction can also account for its dark energy content [17], [18].",
"If true, this would provide an unifying picture of dark matter and dark energy.",
"This would also resolve the initial (Big Bang) singularity, since it can be shown that the `quantal trajectories' of points within this BEC, when extrapolated back in time, come very close, yet do not meet at a point.",
"This follows from the well-known no-crossing property of these trajectories [19], [18] There has been some criticism of the quantum potential and its application to the Quantum Raychaudhuri Equation, which is related to some of the results of this section [20].",
"As pointed out in [21], the authors of [20] start with incorrect assumptions, and arrive at equally flawed and irrelevant conclusions.",
"Regardless of their incorrect expressions for the quantum corrections, as explained in [21], [19], the absence of conjugate points and the resultant invalidity of the singularity theorems in a fixed classical background, the everlasting nature of these trajectories, and the no-requirement of quantal trajectories to remain timelike throughout, simply follow from the first order nature of the guiding equations, and the equivalence between quantum wave equations and the equations for quantal trajectories.",
"Furthermore the BEC critical temperature (Eqs.",
"(REF -REF )) depend on the graviton mass alone.",
"None of the above depend on the precise form of the quantum corrections..",
"In other words, a small graviton mass may be able to address all of the above issues at one stroke."
],
[
"Conclusions",
"We showed in this essay that not only is a tiny graviton mass allowed by theory as well as experiments, but that if non-zero, then it is tightly constrained to be about $10^{-32}~eV/c^2$ , no more, no less.",
"While it is still possible that graviton mass is strictly zero (like that of the photon), a non-zero graviton mass would have some distinct advantages, such as forming a cosmic BEC accounting for dark matter and dark energy, and assisting in the resolution of the initial cosmological singularity.",
"Furthermore, it would also open up the possibility of observing astrophysical events sequentially, first via light and then via gravitational waves emitted from them.",
"Such an advance warning for observing weak gravitational waves would clearly be useful.",
"Note that graviton with above mass would mean gravity would be rendered `short ranged' in the future, when the cosmological horizon $L\\gg L_0$ , and deviations from GR should result.",
"In this way, the current epoch appears to play a special role, simply because it marks the era in which the cosmological constant or dark energy starts dominating the matter content of our Universe.",
"Thus there is all the more reason for studying various implications of a small graviton mass, both from the theoretical as well as the observational side.",
"Although there has been some progress in incorporating graviton mass in a covariant theory of gravity (i.e.",
"extension of general relativity) [10], [22], [23], much more needs to be done to have a satisfactory theory.",
"We hope that there will be progress in the above directions in the near future.",
"Acknowledgment This work is supported by the Natural Sciences and Engineering Research Council of Canada."
]
] | 1605.05928 |
[
[
"Simulation and understanding of quantum crystals"
],
[
"Abstract Quantum crystals abound in the whole range of solid-state species.",
"Below a certain threshold temperature the physical behavior of rare gases (4He and Ne), molecular solids (H2 and CH4), and some ionic (LiH), covalent (graphite), and metallic (Li) crystals can be only explained in terms of quantum nuclear effects (QNE).",
"A detailed comprehension of the nature of quantum solids is critical for achieving progress in a number of fundamental and applied scientific fields like, for instance, planetary sciences, hydrogen storage, nuclear energy, quantum computing, and nanoelectronics.",
"This review describes the current physical understanding of quantum crystals and the wide variety of simulation techniques that are used to investigate them.",
"Relevant aspects in these materials such as phase transformations, energy and structural properties, elasticity, and the effects of crystalline defects and dimensionality, are discussed thoroughly.",
"An introduction to quantum Monte Carlo techniques, which in the present context are the simulation methods of choice, and other quantum simulation approaches (e. g., path-integral molecular dynamics and quantum thermal baths) is provided.",
"The overarching objective of this article is twofold.",
"First, to clarify in which crystals and physical situations the disregard of QNE may incur in important bias and erroneous interpretations.",
"And second, to promote the study and appreciation of QNE, a topic that traditionally has been treated in the context of condensed matter physics, within the broad and interdisciplinary areas of materials science."
],
[
"Quantum crystals: definition and interests",
"Quantum crystals are characterised by light-weight particles interacting through weak long-range forces.",
"At low temperatures, the kinetic energy per particle in a quantum crystal, $E_{\\rm k}$ , is much larger than $k_{B}T$ , where $k_{B}$ is the Boltzmann constant, and the spatial fluctuations about the equilibrium lattice sites are up to 30% of the distance to the neighboring lattice sites, that is, much larger than in any classical solid.",
"These qualities can be understood only in terms of quantum mechanical arguments.",
"Consider, for instance, the quantum expression of the atomic kinetic energy for a system of $N$ indistinguishable particles with mass $m$ : $E_{\\rm k} = -\\frac{\\hbar ^{2}}{2m} \\left< \\frac{{\\bf \\nabla }^{2} \\Psi }{\\Psi } \\right>~,$ where $\\hbar $ is the Planck constant, $\\Psi $ the ground-state wave function of the system, and $\\langle \\cdots \\rangle $ denotes expected value.",
"If the light-weight particles were to rest immobile on the positions of the crystal arrangement, ${\\bf R}_{0}$ , that minimises their potential energy, $E_{\\rm p}$ , that is, $\\Psi \\propto \\sqrt{\\delta \\left( {\\bf R}-{\\bf R}_{0} \\right)}$ , $E_{\\rm k}$ would diverge.",
"Rather, particles in a quantum crystal remain fluctuating around such equilibrium lattice sites in order to minimise their total energy $E = E_{\\rm p} + E_{\\rm k}$ .",
"The corresponding degree of spatial delocalisation is determined by a subtle balance between the accompanying gains in kinetic and potential energies.",
"Examples of quantum solids include, Wigner crystals (Wigner, 1934; Ceperley and Alder, 1980; Drummond et al., 2004; Militzer and Graham, 2006; Drummond and Needs, 2009), vortex lattices (Safar et al., 1992; Cooper, Wilkin, and Gunn, 2001; Abo-Shaeer et al., 2001), dipole systems (Astrakharchik et al., 2007; Matveeva and Giorgini, 2012; Boninsegni, 2013a; Moroni and Boninsegni, 2014), rare-gases, molecular solids, light metals, and many other similar systems (see the next paragraphs).",
"For the sake of focus, however, in this review we will concentrate on quantum crystals formed by atoms and small molecules.",
"A quantitative indicator of the degree of quantumness of a system is given by the de Boer parameter, $\\Lambda ^{\\ast }$ (Sevryuk et al., 2010).",
"This is defined as the ratio of the corresponding de Broglie wavelength, $\\lambda (\\epsilon )$ , and a typical interatomic distance, $r_{0}$ , namely: $\\Lambda ^{\\ast } = \\frac{\\lambda (\\epsilon )}{r_{0}} = \\frac{\\hbar }{r_{0}\\sqrt{m \\epsilon }}~,$ where $\\epsilon $ is an energy scale characterising the interactions between particles.",
"The smaller $m$ and $\\epsilon $ are, the larger $\\Lambda ^{\\ast }$ results.",
"Figure REF shows the de Boer parameter estimated in a series of crystals that are representative of the broad spectrum of solid-state species.",
"$\\Lambda ^{\\ast }$ values were calculated using reported Lennard-Jones potential parameters for “Rare-gas” and “Molecular” solids.",
"In the rest of cases, we used reported experimental cohesive energies as $\\epsilon $ 's and equilibrium lattice parameters as $r_{0}$ 's.",
"The crystals in which quantum nuclear effects (QNE) are expected to be large, somehow arbitrarily defined here as $\\Lambda ^{\\ast } \\ge 0.012$ (which in the limiting case coincides with graphite), are indicated with red dots.",
"As it is observed, most rare gases and light-weight molecular solids, among which we highlight helium, hydrogen, and methane, are quantum crystals.",
"An important number of quantum specimens also are found in the remnant of solid-state categories like, for instance, metal hydride (ionic), carbon-based (covalent), and alkali metal (metallic) compounds.",
"Quantum paraelectrics, although not included in Fig.",
"REF , also conform to an intriguing class of quantum crystals.",
"Quantum paraelectrics are materials in which the onset of ferroelectricity, that is, the appearance of a spontaneous and externally switchable electrical polarisation, is suppressed by quantum nuclear fluctuations (Müller and Burkard, 1979; Rytz, Höchli, and Bilz, 1980; Conduit and Simons, 2010).",
"Examples of quantum paraelectrics include SrTiO$_{3}$ and KTaO$_{3}$ , which normally are classified as complex oxide perovskites (Ohtomo and Hwang, 2004; Cazorla and Stengel, 2012).",
"Quantum paraelectrics do not follow the conventional definition of a quantum crystal since they contain heavy atomic species that interact through strong covalent and ionic forces.",
"Actually, the size of QNE in these materials should be rather small (i. e., $\\Lambda ^{\\ast } \\ll 0.01$ ).",
"At low temperatures, however, quantum paraelectrics are on the verge of a phase transition involving crystal structures with very similar energies and thus the impact of QNE in these and other related materials is large (see Sec. ).",
"The study of quantum solids is very important to understand nature.",
"Hydrogen and helium, for instance, are the most abundant elements in the universe; they represent the $\\sim 70-95$ % of the mass of giant planets in our Solar System such as Jupiter and Saturn (Fortney, 2004; Baraffe, Chabrier, and Barman, 2010).",
"An exhaustive knowledge of their condensed matter phases at extreme thermodynamic conditions is then crucial for understanding the chemical composition and past and future evolution of planetary bodies.",
"Quantum solids are also sought after for technological applications.",
"Rare-gases alloys, for instance, are intensively employed as pressure-transmitting media in high-load compression experiments and synthesis processes, due to their intriguing elastic properties (Errandonea et al., 2006; Dewaele et al., 2008; Cazorla, Errandonea, and Sola 2009).",
"Other examples of scientific fields in which quantum crystals are important include nuclear energy, gas storage, quantum computing, and nanoelectronics.",
"For instance, lithium hydride (LiH) and deuteride (LiD) are thoroughly used in the nuclear industry either as shielding agents or fuel in energy reactors (Welch, 1974; Veleckis, 1977).",
"Metal hydrides are also promising for hydrogen storage applications (Grochala and Edwards, 2004; Shevlin and Guo, 2009) since they can supply large amounts of gas upon thermodynamic destabilisation.",
"Likewise, carbon-based nanostructures (e. g., graphene, nanotubes, and fullerenes) exhibit large gas uptake capacities (Cazorla, Shevlin, and Guo, 2011; Gadipelli, 2015; Cazorla, 2015) as a consequence of their large surface-to-volume ratio, light atomic weight, and great thermodynamic stability.",
"Diamonds with negatively charged nitrogen-vacancy centers, another type of carbonaceous nanomaterial, are playing a crucial role on the development of scalable quantum computing components (Fuchs et al., 2011; Nemoto et al., 2014).",
"This class of crystals also can be employed as tunable quantum simulators that, in analogy to ultracold atom gases trapped in optical lattices (Lewestein et al., 2007), can be used to answer fundamental questions in the fields of condensed matter, biology, and high energy physics (Georgescu et al., 2014; Wang et al., 2015).",
"Finally, quantum paraelectrics find numerous applications in nanoelectronics as varistors, supercapacitor electrodes, and substrates on which to grow epitaxial films of other perovskite compounds (Lawless, 1974; Schlom et al., 2007; Cazorla and Stengel, 2012).",
"Figure: (Color online) De Boer parameter estimated in a series of crystals spanning over thewhole range of solid-state species.",
"The cases in which Λ * \\Lambda ^{\\ast } adoptsa value larger than 0.0120.012 are indicated with red dots.Besides of their fundamental and applied interests, quantum crystals are also very important in the framework of development and testing of new theories.",
"The interactions between particles in quantum crystals typically are of dispersion, hydrogen bond, and multipole-multipole types, which are long-ranged and weak.",
"The cohesion between atoms in solid helium, for instance, is so weak that to a first approximation this crystal can be described by a system of hard spheres (Kalos, Levesque, and Verlet, 1974).",
"Nevertheless, the description of long-ranged and very weak interactions poses a serious challenge to some families of first-principles methods (also known as ab initio because do not rely on any predetermined knowledge of the atomic forces) as the analytical expression of the corresponding electronic exchange and correlation energies are intricate and difficult to approximate for computational purposes (Klimeš and Michaelides, 2012; Cazorla, 2015).",
"This circumstance converts quantum solids into an ideal playground in which to perform benchmark calculations for assessing the performance of standard and advanced electronic band-structure first-principles methods like, for instance, density functional theory (DFT) and electronic quantum Monte Carlo (eQMC) [Driver et al., 2010; Henning et al., 2010; Clay III et al., 2014; Clay III et al., 2016] (see Sec.",
"REF ).",
"Likewise, quantum nuclear effects (QNE) must be fully accounted for in any study dealing with quantum solids since they may affect noticeably the most fundamental properties of crystals like, for instance, atomic structure, vibrational phonon excitations, magnetic spin order, and electronic energy band gap.",
"This fact leads to the situation in which approaches describing QNE only at a qualitative or approximate level (e. g., the Debye model and quasi-harmonic approximation) normally are inadequate for investigating genuine quantum crystals (see, for instance, Morales et al., 2013; Monserrat et al., 2014; Cazorla and Boronat, 2015); instead, full quantum approaches based on the solution to the Schrödinger equation or path-integral formulation of quantum mechanics due to Feynman (1948) must be employed (the fundamentals of these and others quantum simulation techniques will be reviewed in Sec.",
")."
],
[
"A bit of history and theory",
"The experimental study of quantum solids was initiated with the solidification of $^{4}$ He at the Kamerlingh Onnes Laboratories in Leiden, by W. H. Keesom on June 1926 (Keesom, 1942; Domb and Dugdale, 1957).",
"It was not until the late 1960's and early 1970's, however, that with the establishment of neutron inelastic scattering techniques solid helium started being investigated thoroughly (Klein and Venables, 1974).",
"The aim of those early neutron inelastic scattering experiments (Lipschultz et al., 1967; Minkiewicz et al., 1968) was to understand the phonon dynamics in such a highly anharmonic solid.",
"Actually, harmonic calculations render a mechanically unstable solid (that is, with imaginary lattice phonon frequencies) at low densities (Wette and Nijboer, 1965), hence it was very appealing to rationalise the real dynamics of the crystal.",
"An interest on understanding how hydrostatic pressure could modify the physical properties of quantum solids started to develop also at that time (Eckert et al., 1977; Stassis et al., 1978).",
"Likewise, the initial theoretical efforts were concentrated in finding a theory that could describe correctly the dynamical stability observed in highly anharmonic crystals.",
"This was accomplished with the development of the self-consistent phonon (SCP) theory (Koehler, 1966; Glyde, 1994).",
"In the SCP approach, one essentially assumes an harmonic solid with force constants that best represent the real anharmonic crystal, which are determined on the basis of a variational principle.",
"We note that in recent years variants of the SCP approach have been successfully applied to the study of highly anharmonic metallic, molecular, and superconductor materials, in the context of electronic first-principles calculations (Errea, Rousseau, and Bergara, 2011; Errea, Calandra, and Mauri, 2013; Errea, Calandra, and Mauri, 2014; Monserrat et al., 2014; Engel, Monserrat, and Needs, 2015; Errea, et al., 2015).",
"In quantum crystals, due to the large excursions of the atoms around the equilibrium positions, a good treatment of the short-range correlations is necessary.",
"The need for considering such microscopic effects, which are beyond the extent of harmonic and quasi-harmonic approaches, led to the development of the variational theory of quantum solids (Nosanow, 1966; Koehler, 1967).",
"Nosanow proposed a general wave function model for a quantum solid of the form: $\\Psi \\left( {\\bf r}_{1}, \\cdots , {\\bf r}_{N} ; \\lbrace {\\bf R}_{I} \\rbrace \\right) =\\prod _{j < k} f(r_{jk}) \\prod _{i = 1}^{N} g\\left( |{\\bf r}_{i} - {\\bf R}_{i}| \\right)~,$ where ${\\bf R}_{I}$ are the position vectors defining the equilibrium crystal lattice, ${\\bf r}_{i}$ the position vectors of the particles, $r_{jk} \\equiv |{\\bf r}_{j} - {\\bf r}_{k}|$ , and $g(r)$ and $f(r)$ Gaussian and two-body correlation factors, respectively.",
"The second factor in Eq.",
"(REF ) localises each particle around a particular equilibrium lattice site while the first accounts for the interparticle correlations introduced by the atomic interactions.",
"After McMillan's and other authors' works on liquid $^{4}$ He (McMillan, 1965; Schiff and Verlet, 1967), the two-body correlation factors in $\\Psi $ are frequently expressed as $f(r) = \\exp {\\left[ -\\frac{1}{2}\\left(\\frac{b}{r}\\right)^{5} \\right]}$ .",
"This function corresponds to the asymptotic solution of the Schrödinger equation in the $r \\rightarrow 0$ limit of a two-body problem in which the interparticle interaction is of the Lennard-Jones type.",
"With such a relatively simple analytical model of $\\Psi $ and by employing Monte Carlo multidimensional integration techniques (Metropolis et al., 1953; Wood and Parker, 1957), it was then possible to perform variational calculations of the ground state of solid helium (Hansen and Levesque, 1968) and other quantum solids (Hansen, 1968; Bruce, 1972).",
"These advancements set the foundations of the variational Monte Carlo (VMC) method as applied to the study of quantum solids (see Sec.",
"REF ).",
"Despite that variational approaches may be very insightful from a physical point of view, they rarely provide the exact quantitative answer in realistic problems.",
"In order to obtain the precise solution to a quantum many-body problem one, for instance, may deal explicitly with the corresponding Schrödinger equation.",
"To this end, more sophisticated techniques than VMC, albeit related, were developed during the 1970s, among which we highlight the Green's function Monte Carlo (GFMC) method due to Kalos and co-workers (Kalos, 1962; Kalos, Levesque, and Verlet, 1974; Ceperley et al., 1976; Whitlock and Kalos, 1979; Whitlock et al., 1979; Schmidt and Kalos, 1984).",
"The basic idea behind GFMC is to employ Monte Carlo sampling techniques to solve the time-independent Schrödinger equation of a many-body system, when that is expressed as an integral equation containing a Green's function.",
"Although the exact form of the Green's function normally is not known, this can be reproduced with stochastic sampling techniques involving probability distribution functions that are generated with the help of Trotter's product formula (Trotter, 1959) [see Sec.",
"REF ].",
"An intimately related method to GFMC is diffusion Monte Carlo (DMC), in which the imaginary time-dependent Schrödinger equation, rather than the time-independent, is integrated by using a analytical short-time approximation to the Green's function (Ceperley and Alder, 1980; Reynolds et al., 1982; Guardiola, 1986; Hammond et al., 1994).",
"Both GFMC and DMC are exact ground-state methods, in the sense that provide results for the energy that in principle are affected only by statistical errors.",
"These two methods belong to the family known as “projection techniques”, in which a projector operator is iteratively applied in order to cast out the ground state of the targeted quantum many-body system [in this latter category we also find, for instance, the reptation Monte Carlo method due to Baroni and Moroni (1999)].",
"Nonetheless, the DMC method is more efficient in dealing with arbitrary boundary conditions and potential energy functions (Anderson, 2002), hence the use of GFMC is very infrequent nowadays.",
"In Sec.",
"REF , we will review the fundamentals of the DMC method as applied to the study of quantum bosonic crystals.",
"Quantum nuclear effects are also crucial to understand quantum solids at finite temperature (i. e., $T\\ne 0$ ).",
"The threshold temperature below which QNE are important can be considered to be equal to the Debye temperature $\\Theta _{D}$ (Born and Huang, 1954).",
"$\\Theta _{D}$ is defined as $\\hbar \\omega _{m} /k_{B}$ , where $\\omega _{m}$ is the largest vibrational frequency in the crystal (that is, at $\\Theta _{D} \\le T$ all phonon modes in the solid are excited).",
"This threshold temperature can be obtained directly from neutron inelastic scattering or specific heat measurements, and in the particular case of rare-gases $\\Theta _{D}$ ranges from 25 to 85 K. It is important to note that $\\Theta _{D}$ can increase dramatically under the application of external pressure, hence making unavoidable the consideration of QNE in the study of highly compressed quantum crystals.",
"In molecular hydrogen, for instance, the Debye temperature at normal pressure conditions amounts to $\\sim 100$ K whereas at $P = 20$ GPa turns out to be $\\sim 1,000$ K (Diatschenko et al., 1985).",
"The theoretical method of choice for simulation of quantum solids at $T \\ne 0$ is path-integral Monte Carlo (PIMC).",
"The PIMC method is based on Feynman's formulation of non-relativistic quantum mechanics, which can be thought of as a generalisation of the action principle of classical mechanics (Feynman, 1948; Feynman and Hibbs, 1965).",
"In Feynman's path integral theory, however, a functional integral over an infinity of possible trajectories (that is, a path integral) replaces the notion of probability amplitude.",
"From a computational perspective, Feynman's formalism allows one to map out the atomic quantum system of interest onto a classical model of interacting polymers that evolve in imaginary time.",
"This idea, which is known as the “classical isomorphism” (Feynman, 1972; Barker, 1979; Chandler and Wolynes, 1981; Ceperley, 1995), makes it possible to sample the corresponding space of possible configurations with stochastic techniques, laying the foundations of the PIMC method.",
"PIMC relies exclusively on the knowledge of the many-body Hamiltonian and, in contrast to other simulation techniques like for instance GFMC and DMC, does not comprise the use of projector operators (this method will be explained in detail in Sec.",
"REF ).",
"Interestingly, the PIMC approach can be generalised to zero-temperature calculations by exploiting the formal similarities between imaginary time propagators and thermal density matrices (Sarsa et al., 2000).",
"This methodological extension is named path-integral ground state (PIGS) and will be reviewed in Sec.",
"REF .",
"The isomorphism between classical and quantum systems also allows to employ molecular dynamics simulation techniques for sampling of path integrals (Chakravarty, 1997; Tuckerman and Hughes, 1998).",
"In this last framework, generally known as path-integral molecular dynamics (PIMD), the atoms are treated as distinguisable particles.",
"Consequently, genuine quantum statistical effects, that in liquids and disordered systems at low temperatures may give rise to intriguing quantum phenomena like Bose-Einstein condensation and superfluidity, are neglected.",
"Nonetheless, in situations in which atomic quantum exchanges are not relevant (i. e., high temperatures) PIMD becomes a very powerful method that can be used, for instance, to compute quantum time-correlation functions and transition state rates very efficiently (Gillan, 1990; Habershon et al., 2013; Herrero and Ramírez, 2014).",
"Recently, an alternative to path-integral quantum simulation approaches has been proposed that relies on the combined action of “quantum thermal baths” (QTB) and molecular dynamics (Wang, 2007; Buyukdagli et al., 2008; Dammak et al., 2009; Hernández-Rojas, Calvo, and González-Noya, 2015).",
"In the QTB formalism, the dynamics of the system is governed by a Langevin-type equation including dissipative and Gaussian random forces that mimics the power spectral density given by the quantum fluctuation-dissipation theorem (Callen and Welton, 1951).",
"Although quantum statistical effects are also neglected in QTB approaches, these methods are becoming increasingly more popular in the last years due to their reduced computational expense as compared to path-integral based techniques.",
"Meanwhile, hybrid PIMD and QTB schemes have been developed recently that exhibit improved convergence and scalability as compared to PIMD (Ceriotti, Bussi, and Parrinello, 2009; Ceriotti, Manolopoulos, and Parrinello, 2011; Ceriotti and Manolopoulos, 2012; Brieuc, Dammak, and Hayoun, 2016).",
"In Sec.",
"REF , we shall provide a brief introduction to these emergent quantum simulation techniques."
],
[
"Quantum vs. classical solids",
"Let us take a deeper look into the main differences between classical and quantum solids in the zero-temperature limit (see Fig.",
"REF ).",
"Atoms in a classical solid remain practically immobile on the positions of the periodic arrangement that minimises their potential energy (i. e., $E_{\\rm k} \\ll k_{B}T$ ), whereas in a quantum solid particles remain loosely localised around those sites (i. e., $E_{\\rm k} \\gg k_{B}T$ ).",
"As a consequence, large Lindemann ratios (i. e., $\\gamma \\equiv \\sqrt{\\langle u^{2} \\rangle }/a$ , where $\\langle u^{2} \\rangle $ represents the atomic mean squared displacement and $a$ the lattice parameter) of the order of $\\sim 0.1$ are observed in the latter case.",
"Also, the radial pair distribution function, $g(r)$ (i. e., the average number density at a distance $r$ from an atom divided by the overall particle density), presents different features in the two types of crystals.",
"In the zero-temperature limit, the $g(r)$ of a classical solid exhibits a series of sharp peaks signaling the radial distances between crystal lattice sites.",
"By contrast, in a quantum solid $g(r)$ is continuous and displays a pattern of peaks-and-valleys that oscillates around unity at large distances (see Fig.",
"REF a).",
"Likewise, the structure factor in a quantum solid, which is related to the Fourier transform of $g(r)$ , presents some broadening and depletion of the main scattering amplitudes as compared to that in a classical solid (Whitlock et al., 1979; Draeger and Ceperley, 2000).",
"A further difference between quantum and classical crystals is provided by the momentum distribution, $n({\\bf k})$ .",
"In classical solids, $n({\\bf k})$ is always (that is, independently of the interactions between the atoms) equal to the Maxwell-Boltzmann distribution: $n({\\bf k})^{\\rm class} = \\left( \\frac{1}{2 \\pi \\hat{\\alpha _{2}}} \\right)^{3/2} \\exp {\\Big [-\\frac{{\\bf k}^{2}}{2\\hat{\\alpha _{2}}}\\Big ]}~,$ where by the equipartition theorem $\\hat{\\alpha _{2}} \\equiv \\frac{m k_{B} T}{\\hbar ^{2}}$ .",
"In quantum solids, however, the momenta and positions of the atoms are not independent and consequently $n({\\bf k})$ may depart significantly from $n({\\bf k})^{\\rm class}$ .",
"In solid $^{4}$ He, for instance, $n({\\bf k})$ is non-Gaussian as it has a larger occupation of low momentum states as compared to a Maxwell-Boltzmann distribution (Diallo et al., 2004; Rota and Boronat, 2011).",
"The atomic momentum distribution of condensed matter systems can be measured by inelastic neutron scattering performed at high momentum transfer (Glyde, 1994; Diallo et al., 2007).",
"In this case, the Compton profile of the longitudinal momentum distribution, $J(y)$ , is the quantity that is directly measured, that in the impulse approximation is related to $n({\\bf k})$ through the expression (Withers and Glyde, 2007): $J(y) = 2 \\pi \\int _{|y|}^{\\infty } dk \\, k n({\\bf k})~,$ where $y$ is a scaling variable.",
"Compton profile experiments can provide a wealth of information about the nature of quantum solids (Glyde, 1994).",
"For instance, recent neutron scattering measurements have found that the atomic kinetic energy in solid helium at $T \\approx 0$ amounts to $24.25(0.30)$ K (the number within parentheses represents the accompanying uncertainty) [Diallo et al., 2007], which is in very good agreement with quantum Monte Carlo estimations (Ceperley et al., 1996; Cazorla and Boronat, 2008a; Vitiello, 2011).",
"Figure: (Color online) Sketch of the main differences between quantum and classical solids in the zero-temperaturelimit.",
"(a) Radial pair distribution function, g(r)g(r).",
"(b) Localisation of the atoms around their equilibriumlattice positions; in the quantum case, lines represent the evolution of the atomic positions along time andshow the occurrence of atomic quantum exchanges.",
"(c) Compton profile of the longitudinal momentum distribution, J(y)J(y).Kinetic isotopic effects, which attribute different kinetic energies to the isotopes of a same chemical element [see Eq.",
"(REF )], are indicators of the existence of QNE.",
"The magnitude of these effects can be inferred from inspection of functions $g(r)$ and $n({\\bf k})$ (Mao and Hemley, 1994; Boninsegni et al., 1994; Cazorla and Boronat, 2005).",
"For instance, narrowing (widening) of the peaks in $g(r)$ may be caused by the presence of heavier (lighter) species.",
"Kinetic isotopic effects can also manifest in the thermal expansion of quantum solids (Pamuk et al., 2012; Herrero and Ramírez, 2011a) and corresponding $P-T$ boundaries delimiting the thermodynamic stability regions of different phases (Lorenzana, Silvera, and Goettel, 1990; Goncharov, Hemley, and Mao, 2011).",
"In quantum mechanics, atoms of a same species are indistinguishable, that is, they can exchange positions while leaving the configuration of the system (namely, the square of the wave function) invariant.",
"These atomic exchanges can occur as pairwise interchanges, three-particle, four-particle, and so on cyclic permutations.",
"When the particles involved in such permutations are bosons and their number grows to infinity, the system becomes superfluid and the atoms on it can flow coherently without any resistance (Feynman, 1972; Ceperley, 1995).",
"In quantum solids, as opposed to classical crystals, atoms can swap their positions and further delocalise in configurational space (see Fig.",
"REF b for a schematic representation).",
"An illustrative example of this class of QNE is given by solid $^{3}$ He.",
"$^{3}$ He atoms, which are fermions, have a non-zero magnetic moment and at low pressure the stable phase is a cubic bcc crystal.",
"At temperatures below $1.5$ mK, this system adopts an exotic magnetic order that consists of two planes of up spins followed by two planes of down spins (Roger, Hetherington, and Delrieu, 1983).",
"In terms of classical interaction arguments, that is, if only nearest-neighbor pair exchanges were important, the magnetic order in this crystal should be antiferromagnetic.",
"However, quantum exchanges between more than two $^{3}$ He atoms are very frequent and as a result a strong competition between ferromagnetism and antiferromagnetism appears in the crystal that leads to the observed magnetic order (Ceperley, 1995).",
"It has been theoretically shown that in commensurate $^{4}$ He crystals (i. e., crystals with exactly two atoms per hcp unit cell, without any point or line defects such as vacancies, dislocations, or grain boundaries) typical cyclic permutations occurring at few tenths of K only involve a small number of atoms.",
"Consequently, the superfluid density in perfect quantum solids is null (Ceperley and Bernu, 2004; Bernu and Ceperley, 2005; Boninsegni, Prokof'ev, and Svistunov, 2006b).",
"This conclusion appears to be consistent with the results of most recent and conclusive torsional oscillator experiments performed by Kim and Chan (2012, 2014).",
"Meanwhile, in the presence of crystalline defects or atomic disorder quantum Monte Carlo calculations agree in predicting that the length of ring quantum exchanges increases noticeably, and thus the possibility of realising superfluidity starts to depart from zero (Boninsegni, Prokof'ev, and Svistunov, 2006b; Boninsegni et al., 2007; Rota and Boronat, 2012).",
"We must note, however, that convincing experimental evidence of superfluid-like manifestations in quantum crystals are yet elusive (Chan et al., 2013; Hallock, 2015).",
"We will discuss these topics in more detail in Sec.",
"."
],
[
"Incomplete understanding of quantum crystals",
"Although a lot is already known on the physics of quantum crystals, there are still few puzzling and controversial aspects that urge for an improved understanding.",
"One of these aspects is related to the interactions between different types of crystalline defects, their formation energy, and transport properties.",
"In a seminal work, Day and Beamish (2007) reported the experimental dependence of the shear modulus, $\\mu $ , in solid $^{4}$ He as a function of temperature.",
"They found that $\\mu $ increased with decreasing $T$ below a certain temperature of $0.15$ K. The observed increase in stiffness was rationalised in terms of line defects mobility: below a particular temperature threshold the dislocations present in the crystal could be pinned by $^{3}$ He impurities, in spite of the incredibly small concentration of the latter (i. e., just 200 parts per billion of $^{4}$ He atoms).",
"This argument has been subsequently ratified by a number of compelling experimental works carried out by the groups of Beamish, in the University of Alberta, and Balibar, in the Ecole Normale Supérieure de Paris (see, for instance, Haziot et al., 2013a; Haziot et al., 2013b; Fefferman et al., 2014; Souris et al., 2014a).",
"Remarkably, Haziot et al.",
"(2013c) have recently shown that in ultra-pure single crystals of $^{4}$ He the resistance to shear along one particular direction nearly vanishes at around $T = 0.1$ K, whereas normal elastic behavior is observed in the others.",
"The exact origins of this intriguing effect, which has been termed as “giant plasticity”, however, are still under debate (Zhou et al., 2013; Haziot et al., 2013d), and the exact ways in which dislocations and isotopic impurities interact remain not fully understood (see Fig.",
"REF ).",
"Recent theoretical arguments put forward by Kuklov et al.",
"(2014) suggest that quantum crystals might constitute a unique kind of materials in which topological lattice defects, that is, dislocations, could display quantum behavior like, for instance, quantum tunneling of kinks and jogs.",
"Kuklov's hypotheses appear to be sustained by recent experimental observations in $^{4}$ He and $^{3}$ He crystals (see, for instance, Ray and Hallock, 2008; Lisunov et al., 2015).",
"Verifying such a possible quantum scenario, however, turns out to be very challenging in practice due to the difficulties encountered both in the experiments and atomistic simulations.",
"For instance, according to recent reports it appears to be extremely challenging to grow perfect helium crystals totally free of dislocations (Souris et al., 2014b).",
"Concerning the calculations, a detailed and reliable simulation of line defects entails the use of large systems containing up to several thousands of atoms (Bulatov and Cai, 2006; Proville, Rodney, and Marinica, 2012), which currently is in the edge of quantum simulations.",
"Due to these issues, many fundamental questions remain yet unanswered like for instance: What is the magnitude of the formation energy of dislocations in quantum solids?",
"Can dislocations really behave as quantum entities so that they delocalise in space?",
"Through which exact mechanisms quantum impurities like $^{3}$ He atoms, which are extremely mobile, interact with dislocations?",
"Solving these and other similar puzzles is crucial for advancing the field of quantum solids; this knowledge could have also an impact on the areas of materials science in which plasticity has a central role (e. g., fatigue in crystals and amorphous and martensitic transformations) [Proville, Rodney, and Marinica, 2012].",
"We will comment further on these points in Sec. .",
"Figure: (Color online) Shear modulus in an extremely pure 4 ^{4}He crystal expressed as a functionof temperature (adapted from Haziot et al., 2013c).",
"Three different regimes areobserved that can be explained in terms of dislocation dynamics (see text).",
"The unsolvedproblem about which are the interactions between dislocations and isotopic impurities isnoted schematically.Another source of unawareness in quantum crystals is posed by their behavior at extreme thermodynamic conditions.",
"When a crystal is compressed the bonds between atoms normally are shortened so that particles become more localised in order to avoid increasing their (highly repulsive) potential energy.",
"At the same time, the kinetic energy of the solid increases due to Heisenberg's uncertainty principle.",
"In quantum crystals, such a pressure-induced energy gain may be compensated in part by quantum atomic exchanges and quantum tunneling, which tend to favor the delocalisation of particles (Kosevich, 2005).",
"The existence of proton quantum tunneling, for instance, has been demonstrated in solid hydrogen and ice under pressure, a QNE that is key to understand their corresponding phase diagrams and vibrational properties (Benoit, Marx, and Parrinello, 1998; Hemley, 2000; Howie et al., 2012a; Drechsel-Grau and Marx, 2014).",
"On the other hand, quantum fluctuations in highly compressed solid hydrogen are known to hinder molecular rotation, which counterintuitively leads to some kind of atomic localisation (Kitamura et al., 2000; Li et al., 2013).",
"The ways in which QNE manifest and affect the physical properties of highly compressed quantum crystals actually seem to be quite unpredictable.",
"Simulation of QNE phenomena at high pressures is technically difficult and demands intensive computational resources.",
"The main reason for this is that the interactions between atoms cannot longer be described correctly with semi-empirical approaches like, for instance, pairwise potentials, and thereby the treatment of both the electronic and ionic degrees of freedom needs to be done quantum mechanically (see Sec. ).",
"Likewise, carrying out high-$P$ high-$T$ experiments in the laboratory is extremely challenging due to the occurrence of unwanted chemical reactions between the samples and containers (Dewaele et al., 2010).",
"In addition to this, it is complicated to determine the exact atomic structure in highly compressed solids with low $Z$ numbers because their x-ray scattering cross sections are very small (Goncharov, Howie, and Gregoryanz, 2013; Dzyabura et al., 2014).",
"Due to all these difficulties, the $P-T$ phase diagram of many quantum solids remain contentious and a complete understanding of the accompanying QNE features (e. g., quantum atomic exchanges and kinetic energy) is still pending.",
"Further progress in this field is essential for advancing our knowledge in condensed matter physics and earth and planetary sciences (see Sec.",
"for more details)."
],
[
"Aims and organisation of this review",
"This review is concerned with the simulation and understanding of quantum solids formed by atoms and small molecules under broad $P-T$ conditions.",
"Important aspects in these systems like, for instance, their energetic and structural properties, phase transitions and elasticity, are discussed in detail.",
"The effects that crystalline defects and reduced dimensionality have on the physical properties of archetypal quantum solids (i. e., $^{4}$ He and H$_{2}$ ), are also reviewed.",
"Special emphasis is put on identifying those systems and physical situations in which QNE must be considered in order to avoid likely misconceptions.",
"In fact, QNE have been traditionally analysed in the field of condensed matter physics, however, comprehension of this class of effects is crucial for advancing in many other research areas such as planetary and materials sciences.",
"We start by explaining the basics of the simulation methods that are used most frequently in the study of quantum crystals (Secs.",
"and ).",
"In Secs.",
"-, we describe the phenomenology and current understanding of quantum solids by surveying a large number of experimental and theoretical studies on archetypal and other less popular quantum crystals (e. g., H$_{2}$ O, N$_{2}$ , CH$_{4}$ , LiH, and BaTiO$_{3}$ ).",
"Finally, we comment on promising research directions involving quantum solids and summarise our general conclusions in Sec.",
"."
],
[
"Quantum simulation methods",
"We review the basics of customary quantum simulations methods that are employed for the investigation of quantum crystals.",
"We classify them into two major categories, namely, ground-state ($T = 0$ ) and finite-temperature ($T \\ne 0$ ) methods.",
"In the zero-temperature case, we differentiate between “approximate” and “exact” techniques.",
"Depending on the nature of the problem that is going to be investigated and the amount of computational resources that are available, one may opt for using one or another."
],
[
"Quasi-harmonic approximation.",
"In the quasi-harmonic approach (QHA) one assumes that the potential energy of a crystal can be approximated with a quadratic expansion around the equilibrium atomic configuration of the form (Born and Huang, 1954; Kittel, 2005): $E_{\\rm qh} = E_{\\rm eq} + \\frac{1}{2}\\sum _{l\\kappa \\alpha ,l^{\\prime }\\kappa ^{\\prime }\\alpha ^{\\prime }}\\Phi _{l\\kappa \\alpha ,l^{\\prime }\\kappa ^{\\prime }\\alpha ^{\\prime }} u_{l\\kappa \\alpha }u_{l^{\\prime }\\kappa ^{\\prime }\\alpha ^{\\prime }}~,$ where $E_{\\rm eq}$ is the total energy of the perfect lattice, $\\Phi $ the corresponding force-constant matrix, and $u_{l\\kappa \\alpha }$ is the displacement along Cartesian direction $\\alpha $ of atom $\\kappa $ at lattice site $l$ .",
"Normally, this dynamical problem is solved by introducing: $u_{l\\kappa \\alpha }(t) = \\sum _{q} u_{q\\kappa \\alpha } \\exp { \\left[ i\\left(\\omega t - q \\cdot (l+\\tau _{\\kappa } \\right) \\right] }~,$ where $q$ is a wave vector in the first Brillouin zone (BZ) that is defined by the equilibrium unit cell; $l+\\tau _{\\kappa }$ is the vector that locates atom $\\kappa $ at cell $l$ in the equilibrium structure.",
"The normal modes are then found by diagonalizing the dynamical matrix: $\\begin{split}& D_{q;\\kappa \\alpha ,\\kappa ^{\\prime }\\alpha ^{\\prime }} =\\\\ &\\frac{1}{\\sqrt{m_{\\kappa }m_{\\kappa ^{\\prime }}}} \\sum _{l^{\\prime }}\\Phi _{0\\kappa \\alpha ,l^{\\prime }\\kappa ^{\\prime }\\alpha ^{\\prime }} \\exp {\\left[iq\\cdot (\\tau _{\\kappa }-l^{\\prime }-\\tau _{\\kappa ^{\\prime }})\\right]}~,\\end{split}$ and thus the crystal can be treated as a collection of non-interacting harmonic oscillators with frequencies $\\omega _{qs}$ (positively defined and non-zero) and energy levels: $E^{n}_{qs} = \\left( \\frac{1}{2} + n \\right)\\omega _{qs}~,$ where $0 \\le n < \\infty $ .",
"In this approximation, the Helmholtz free energy of a crystal with volume $V$ at temperature $T$ is given by: $F_{\\rm qh} (V,T) = \\frac{1}{N_{q}}~k_{B} T \\sum _{qs}\\ln \\left[ 2\\sinh \\left(\\frac{\\hbar \\omega _{qs}(V)}{2k_{\\rm B}T} \\right) \\right]~,$ where $N_{q}$ is the total number of wave vectors used for integration over the BZ, and the $V$ -dependence of the vibrational frequencies has been noted explicitly.",
"In the zero-temperature limit, Eq.",
"(REF ) transforms into: $F_{\\rm qh} (V,0) = \\frac{1}{N_{\\rm q}} \\sum _{qs}\\frac{1}{2}\\hbar \\omega _{qs}(V)~,$ which is usually referred to as the “zero-point energy” (ZPE).",
"We note that despite quasi-harmonic approaches may not be adequate for the study of archetypal quantum solids (Morales et al., 2013; Monserrat et al., 2014; Cazorla and Boronat, 2015), QHA ZPE corrections normally are decisive in predicting accurate phase transitions in other materials (Cazorla, Alfè, and Gillan, 2008; Shevlin, Cazorla, and Guo, 2012; Cazorla and ${\\rm \\acute{I}}$${\\rm \\tilde{n}}$ iguez, 2013)."
],
[
"Variational Monte Carlo.",
"Variational theory has been one of the most fruitful computational approaches to study quantum fluids and solids.",
"The strong repulsive interaction at short distances between particles produce a failure of conventional perturbation methods.",
"The variational principle of quantum mechanics states that the expectation value of a Hamiltonian, $\\hat{H}$ , obtained with a model wave function, $|\\Psi \\rangle $ , provides an upper bound to the true ground-state energy of the system, $E_0$ , namely: $E=\\frac{\\langle \\Psi | \\hat{H} | \\Psi \\rangle }{\\langle \\Psi | \\Psi \\rangle } \\ge E_0~.$ In a many-body system, the evaluation of $E$ is not an easy task because one has to deal with a $3N$ -dimensional integral.",
"In this context, Monte Carlo integration techniques emerge as one of the most efficient computational methods.",
"In variational Monte Carlo (VMC, named so because of its variational nature), one defines the multivariate probability density function (p.d.f.",
"): $f(\\mathbf {R})=\\frac{|\\Psi (\\mathbf {R})|^2}{\\int d \\mathbf {R} ~ |\\Psi (\\mathbf {R})|^2}~,$ which is normalised and positively defined.",
"Meanwhile, the local energy, which adopts the form: $E_{\\text{L}}(\\mathbf {R})= \\frac{1}{\\Psi (\\mathbf {R})} \\, \\hat{H} \\Psi (\\mathbf {R})~,$ allows one to express the expectation value of the Hamiltonian in the integral form: $\\langle \\hat{H} \\rangle _\\Psi = \\int d \\mathbf {R}~ E_{\\text{L}}(\\mathbf {R}) f(\\mathbf {R})~.$ In the two expressions above $\\mathbf {R}$ stands for a multidimensional point (also called “walker”), $\\mathbf {R} \\equiv \\lbrace \\mathbf {r}_1,\\ldots ,\\mathbf {r}_N\\rbrace $ .",
"The expected value of the Hamiltonian then is calculated as the mean value of $E_{\\text{L}}(\\mathbf {R})$ , evaluated in a series of points, $n_s$ , that are generated according to the p.d.f.",
"$f(\\mathbf {R})$ , namely: $\\langle \\hat{H} \\rangle _\\Psi = \\frac{1}{n_s} \\sum _{i=1}^{n_s}E_{\\text{L}}(\\mathbf {R}_i)~.$ Effective sampling of the multidimensional p.d.f.",
"$f(\\mathbf {R})$ can be done with the Metropolis method (Metropolis et al., 1953; Wood and Parker, 1957).",
"Given a trial wave function, $\\Psi (\\mathbf {R})$ , the VMC method provides the exact value of $\\langle \\hat{H} \\rangle _\\Psi $ to within statistical errors.",
"Trial wave functions normally contain a set of parameters that are optimised in order to find the absolute minimum of $\\langle \\hat{H} \\rangle _\\Psi $ .",
"Alternatively, one can search for the parameter values that minimise the variance of the energy, whose lower bound a priori is known to be zero (Hammond, Lester, and Reynolds, 1994).",
"With regard to Bose crystals (that is, formed by boson particles), the most widely used wave function is the Nosanow-Jastrow (NJ) model: $\\Psi _{\\rm {NJ}}({\\mathbf {r}}_1,\\ldots ,{\\mathbf {r}}_N) = \\prod _{i<j}^{N} f(r_{ij}) \\,\\prod _{i,I=1}^{N} g(r_{iI})~,$ where $N$ is the number of particles and lattice sites, $f(r)$ a two-body Jastrow correlation function, and $g(r)$ a one-body localization factor that links particle $i$ to site $I$ (see Sec.",
"REF ).",
"The Jastrow factor takes into account, at the lowest order, the dynamical correlations between particles induced by the interatomic potential, whereas the one-body term introduces the symmetry of the crystal.",
"Wave function $ \\Psi _{\\rm {NJ}}$ leads to an excellent description of the equation of state and structural properties of atomic quantum solids.",
"However, it cannot be used to calculate properties that depend directly on the Bose-Einstein statistics (e. g., superfluidity and off-diagonal long-range order) because it is not symmetric under the exchange of particles.",
"The latter symmetry requirement can be formally written as: $\\Psi _{\\rm {PNJ}}({\\mathbf {r}}_1,\\ldots ,{\\mathbf {r}}_N) = \\prod _{i<j}^{N} f(r_{ij}) \\,\\left( \\sum _{P(J)} \\prod _{i=1}^{N} g(r_{iJ}) \\right)~,$ where $P(J)$ indicates a sum over all possible particle permutations involving the lattice sites.",
"This wave function model, however, presents some practical issues since the number of configurations that needs to be sampled in order to reach convergence grows exponentially with the number of particles.",
"Effective calculations involving a symmetric NJ wave function can be performed with the model: $\\Psi _{\\rm {SNJ}}({\\mathbf {r}}_1,\\ldots ,{\\mathbf {r}}_N) = \\prod _{i<j}^{N} f(r_{ij}) \\,\\prod _{J=1}^{N} \\left( \\sum _{i=1}^{N} g(r_{iJ}) \\right)~,$ which has been introduced recently by Cazorla et al.",
"(2009).",
"This symmetric wave function possesses a localization factor that suppresses lattice voids arising from double site occupancy, a desirable feature that also is reproduced by wave function $\\Psi _{\\rm {PNJ}}$ .",
"Other symmetric wave functions have been proposed in the context of quantum solids that do not rely on the symmetrization of $\\Psi _{\\rm {NJ}}$ .",
"These include a Bloch-like function (Ceperley, Chester, and Kalos, 1978), inspired in the band theory of electrons, and the shadow wave function (Galli, Rossi, and Reatto, 2005).",
"The first model was introduced in a VMC study of the Yukawa system (Ceperley, Chester, and Kalos, 1978); the resulting variational energies, however, were significantly higher than those estimated with the non-symmetric NJ wave function, and the creation of vacancies or double occupancy of a same lattice site in the crystal could not be prevented.",
"Consequently, this model has been overlooked in posterior studies.",
"A more realistic symmetric model is provided by the shadow wave function (Vitiello, Runge, and Kalos, 1988; MacFarland et al., 1994), which is defined as: $\\Psi _{\\rm {sh}}({\\mathbf {r}}_1,\\ldots ,{\\mathbf {r}}_N) = \\Phi _p({\\mathbf {R}}) \\, \\int d {\\mathbf {S}} \\Theta ({\\mathbf {R}},{\\mathbf {S}}) \\Phi _s({\\mathbf {S}})~,$ in which auxiliary variables ${\\mathbf {S}}$ (also called “shadows”) are introduced in order to avoid the explicit definition of any particular atomic arrangement.",
"In Eq.",
"REF , $\\Phi _p({\\mathbf {R}})$ and $\\Phi _s({\\mathbf {S}})$ are Jastrow factors that correlate particles and shadows separately; function $\\Theta ({\\mathbf {R}},{\\mathbf {S}})$ , on the other hand, introduces a coupling between particles and shadows.",
"The shadow variables finally are integrated out of $\\Psi _{\\rm {sh}}$ in order to remove any explicit dependence on them.",
"Despite that variational methods may provide qualitatively correct results, it is not possible to determine their accuracy in absolute terms.",
"Green's function Monte Carlo (GFMC) methods eliminate any variational constraints by solving directly the Schrödinger equation for a $N$ -body problem.",
"The most advanced of these methods is domain GFMC, in which the corresponding Green's function is time-independent (Kalos, 1962; Kalos, Levesque, and Verlet, 1974; Ceperley et al., 1976; Whitlock and Kalos, 1979; Whitlock et al., 1979; Schmidt and Kalos, 1984).",
"A related method is diffusion Monte Carlo (DMC), which is time-dependent and nowadays widely used (Ceperley and Alder, 1980; Reynolds et al., 1982; Hammond, Lester, and Reynolds, 1994; Anderson, 2002).",
"DMC is a projector method that, by working in imaginary time, is able to retrieve exact energy results for the ground state of a many-particle system.",
"In imaginary time, $\\tau $ , the Schrödinger equation becomes: $- \\frac{\\partial \\Psi ({\\mathbf {R}},\\tau )}{\\partial \\tau } = (\\hat{H}-E_{0}) \\Psi ({\\mathbf {R}},\\tau )~,$ where ${\\mathbf {R}}=\\lbrace {\\mathbf {r}}_1, \\ldots ,{\\mathbf {r}}_N \\rbrace $ and time is expressed in units of $\\hbar $ .",
"The time-dependent wave function of the system, $\\Psi ({\\mathbf {R}},\\tau )$ , can be expanded in terms of the complete set of eigenfunctions of the Hamiltonian, $\\phi _i({\\mathbf {R}})$ , namely: $\\Psi ({\\mathbf {R}},\\tau )=\\sum _{n}c_n \\, \\exp \\left[\\, -(E_i-E_{0}) \\tau \\, \\right]\\,\\phi _i({\\bf R})\\ ,$ where $E_i$ is the eigenvalue associated to $\\phi _i({\\mathbf {R}})$ .",
"The asymptotic solution of Eq.",
"(REF ) in the $\\tau \\rightarrow \\infty $ limit then corresponds to $\\phi _0({\\mathbf {R}})$ , provided that there is a non-zero overlap between $\\Psi ({\\mathbf {R}},\\tau =0)$ and the true ground-state wave function, $\\phi _0({\\mathbf {R}})$ .",
"Direct application of Eq.",
"(REF ) to condensed matter problems is hindered by the repulsive interactions that atoms experience at short distances, which translates into large energy variances.",
"To overcome this problem, one introduces importance sampling, a technique that is widely used in MC calculation of integrals.",
"Importance sampling as applied to Eq.",
"(REF ) consists in rewriting the Schrödinger equation in terms of the p.d.f.",
": $f({\\mathbf {R}},\\tau )\\equiv \\psi ({\\mathbf {R}})\\,\\Psi ({\\mathbf {R}},\\tau )~,$ where $\\psi ({\\mathbf {R}})$ is a time-independent trial wave function that at the variational level describes the ground state of the crystal correctly.",
"By considering a Hamiltonian of the form: $\\hat{H} = -\\frac{\\hbar ^2}{2\\,m} \\, {\\mbox{$\\nabla $}}^2_{{\\mathbf {R}}}+ \\hat{V} ({\\mathbf {R}})~,$ Eq.",
"(REF ) turns into: $-\\frac{\\partial f({\\mathbf {R}},t)}{\\partial \\tau } & = & -D\\,{\\mbox{$\\nabla $}}^2 f({\\mathbf {R}},\\tau ) + \\nonumber \\\\& & D\\,{\\mbox{$\\nabla $}} \\left[{\\mathbf {F}}({\\mathbf {R}}) \\,f({\\mathbf {R}},\\tau )\\right] + \\nonumber \\\\& & \\left[ E_L({\\mathbf {R}})-E_{0} \\right] \\cdot f({\\mathbf {R}},\\tau )~,$ where $D \\equiv \\hbar ^2 /(2m)$ , $E_L({\\bf R}) \\equiv \\psi ({\\mathbf {R}})^{-1} H \\psi ({\\mathbf {R}})$ is the local energy, and ${\\mathbf {F}}({\\mathbf {R}}) \\equiv 2\\, \\psi ({\\mathbf {R}})^{-1}{\\mbox{$\\nabla $}} \\psi ({\\mathbf {R}})$ is the so-called drift or quantum force.",
"${\\mathbf {F}}({\\mathbf {R}})$ acts as an external force that guides the diffusion process rendered by the first term in the right-hand side of Eq.",
"(REF ).",
"In particular, walkers are attracted towards regions in which the value of $\\psi ({\\mathbf {R}})$ is large, thus avoiding the repulsive core of the interaction that produces large fluctuations in the energy.",
"The right-hand side of Eq.",
"(REF ) can be written as the action of three operators, $\\hat{A_i}$ , acting on the p.d.f.",
"$f({\\mathbf {R}},\\tau )$ , namely: $-\\frac{\\partial f({\\mathbf {R}},\\tau )}{\\partial \\tau } = (\\hat{A}_1+ \\hat{A}_2+ \\hat{A}_3)\\,f({\\mathbf {R}},\\tau ) \\equiv \\hat{A}\\, f({\\mathbf {R}},\\tau )~.$ Operator $\\hat{A}_1$ corresponds to a free diffusion process with coefficient $D$ , $\\hat{A}_2$ to a driving force produced by an external potential, and $\\hat{A}_3$ to a birth/death branching term.",
"In quantum Monte Carlo, the Schrödinger equation is most manageable when expressed in a integral form.",
"This is achieved by introducing a Green's function, $G({\\mathbf {R}}^{\\prime },{\\mathbf {R}},\\tau )$ , that describes the transition probability to move from an initial state, ${\\mathbf {R}}$ , to a final state, ${\\mathbf {R}}^{\\prime }$ , during the time interval $\\Delta \\tau $ , that is: $f({\\mathbf {R}}^{\\prime },\\tau +\\Delta \\tau ) =\\int G({\\mathbf {R}}^{\\prime },{\\mathbf {R}},\\Delta \\tau )\\, f({\\mathbf {R}},\\tau )\\, d{\\mathbf {R}}~.$ More explicitly, the Green's function can be expressed in terms of the $\\hat{A}$ operator as: $G({\\mathbf {R}}^{\\prime },{\\mathbf {R}}, \\Delta \\tau ) =\\left\\langle \\,{\\mathbf {R}}^{\\prime }\\, | \\, \\exp (-\\Delta \\tau \\hat{A})\\, |\\, {\\mathbf {R}}\\, \\right\\rangle ~,$ and be approximated in practice with Trotter's product formula (Trotter, 1959): $e^{- \\tau \\left( \\hat{A}_{1} + \\hat{A}_{2} \\right)} = \\lim _{n \\rightarrow \\infty } \\left( e^{-\\frac{\\tau }{n} \\hat{A}_{1}} e^{-\\frac{\\tau }{n} \\hat{A}_{2}} \\right)^{n}~.$ DMC algorithms rely on reasonable approximations to the propagator $G({\\mathbf {R}}^{\\prime },{\\mathbf {R}},\\Delta \\tau )$ in the $\\Delta \\tau \\rightarrow 0$ limit, which are iterated repeatedly until reaching the asymptotic regime $f({\\mathbf {R}},\\tau \\rightarrow \\infty )$ (that is, when the ground state is effectively sampled).",
"The order of the employed $G({\\mathbf {R}}^{\\prime },{\\mathbf {R}},\\Delta \\tau )$ approximation introduces a certain time-step bias on the results, that needs to be removed in order to provide perfectly converged solutions (Boronat and Casulleras, 1994).",
"In DMC, the sampling of an operator, $\\hat{O}$ , is performed according to the mixed distribution $f({\\mathbf {R}},\\tau )$ [see Eq.",
"(REF )], rather than to $\\phi _0({\\mathbf {R}})$ .",
"Therefore, the standard DMC output corresponds to the so-called “mixed” estimator.",
"Mixed estimators in general are biased by the trial wave function that is used for importance sampling.",
"Only when $\\hat{O}$ is the Hamiltonian of the system or an operator that commutes with it, the mixed estimator and exact result coincide.",
"A simple scheme that is employed to remove partially the bias introduced by $\\psi ({\\mathbf {R}})$ is: $\\langle \\hat{O}({\\mathbf {R}}) \\rangle _e=2 \\, \\langle \\hat{O}({\\mathbf {R}})\\rangle _m - \\langle \\hat{O}({\\mathbf {R}}) \\rangle _v~,$ which is built from the mixed ($m$ ) and variational ($v$ ) estimators, and is known as the “extrapolated” estimator ($e$ ) [Ceperley and Kalos, 1979].",
"Nevertheless, expectation values obtained with the extrapolation approach never are totally free of bias, and it is difficult to estimate a priori the size of the accompanying errors.",
"In order to overcome such limitations, one can calculate “pure” expectation values (that is, exact to within the statistical errors) by using the forward walking technique (Casulleras and Boronat, 1995)."
],
[
"Path-integral ground-state Monte Carlo",
"An interesting alternative to the DMC method has been put forward by Sarsa et al.",
"(2000), based on a previous proposal by Ceperley (1975).",
"This method is termed path integral ground-state method (PIGS) and it is directly related to the path integral Monte Carlo (PIMC) method used at finite temperature (see next Sec.",
"REF ).",
"The integral version of the Schrödinger equation can be written in terms of the Green's function as: $\\Psi ({\\mathbf {R}},\\tau )=\\int d {\\mathbf {R}}^\\prime \\ G({\\mathbf {R}},{\\mathbf {R}}^\\prime ; \\tau -\\tau _0)\\Psi ({\\mathbf {R}}^\\prime ,\\tau _0)~.$ In the PIGS method one exploits the formal identity between the Green's function at imaginary time $\\tau $ , $G({\\mathbf {R}},{\\mathbf {R}}^\\prime ; \\tau -\\tau _0)$ , and the statistical density matrix operator at an inverse temperature $\\beta \\equiv 1/T$ , $\\rho ({\\mathbf {R}},{\\mathbf {R}}^\\prime ; \\beta )$ .",
"The convolution property of the density matrix permits to estimate $\\rho ({\\mathbf {R}},{\\mathbf {R}}^\\prime ; \\beta )$ through a convolution of density matrices calculated at smaller values, $\\beta /N_b$ , namely: $&\\rho ({\\mathbf {R}},{\\mathbf {R}}^\\prime ; \\beta ) & = \\int d{\\mathbf {R}}_1 \\ldots d{\\mathbf {R}}_{N_b-1} \\nonumber \\\\\\times & \\rho ({\\mathbf {R}},{\\mathbf {R}}_1; \\beta /N_b) & \\ldots \\rho ({\\mathbf {R}}_{N_b-1},{\\mathbf {R}}^\\prime ; \\beta /N_b)~.$ In PIMC ($T \\ne 0$ ), one has to deal with the trace of the density matrix operator and hence the boundary condition ${\\mathbf {R}} = {\\mathbf {R}}^\\prime $ is imposed; this makes a closed path.",
"In the context of the “classical isomorphism” (Feynman, 1972; Barker, 1979; Chandler and Wolynes, 1981; Ceperley, 1995), a path is interpreted as a polymer in which first neighbors are connected with springs; moving a quantum particle is equivalent to evolve such a polymer.",
"In PIGS ($T = 0$ ), at difference with PIMC, one truncates the path by imposing that the end points, ${\\mathbf {R}}^\\prime $ , terminate in a trial wave function, $\\psi $ ; the path then is open.",
"In this case, the expectation value of an operator, $\\hat{O}$ , is determined by: $\\hat{O}=\\frac{\\langle \\psi | G(\\tau /2) \\hat{O} G(\\tau /2) | \\psi \\rangle }{\\langle \\psi | G(\\tau ) | \\psi \\rangle }~,$ where $\\tau $ is the total imaginary time that the system takes to move from the initial to the end point.",
"The most remarkable aspect of this method is that in the middle of the path, $\\tau /2$ , the sampling of any operator is exact, independently of whether $\\hat{O}$ commutes or not with the Hamiltonian of the system.",
"In other words, calculation of “pure” estimators is the standard output in PIGS, contrarily to what occurs in DMC (although for operators that commute with the Hamiltonian both methods shall provide equivalent results).",
"Actually, for non-diagonal operators, like for instance the one-body density matrix, only PIGS is able to provide unbiased zero-temperature results in an efficient manner.",
"In order to perform PIGS calculations as efficiently as possible in practice, it is necessary to develop approximations for the propagator operator that are accurate to within a certain order in the time step.",
"To this regard, significant progress has been achieved in recent years by developing splitting schemes for the exponential of the Hamiltonian operator, $\\hat{H} = \\hat{K} + \\hat{V}$ (where $\\hat{K}$ and $\\hat{V}$ are the kinetic and potential energy operators, respectively), of the form: $\\exp [\\varepsilon (\\hat{T}+\\hat{V})] = \\prod _{i=1}^{N} \\exp (t_i\\varepsilon \\hat{T}) \\exp (v_i \\varepsilon \\hat{V})~,$ where the value of the parameters $\\lbrace t_i\\rbrace $ and $\\lbrace v_i\\rbrace $ are selected in a way that satisfies forward propagation (Chin and Chen, 2002).",
"Under this constraint, one can write algorithms that are accurate up to fourth-order (Rota et al., 2010) and which produce very consistent convergence towards the ground state (see next Sec.",
"REF ).",
"Recent applications of the PIGS method involving high-order decomposition methods have shown that it is actually possible to obtain results that are completely independent of the trial wave function that is used as boundary condition (Rossi et al., 2009; Rota et al., 2010).",
"Even in the limiting case of considering only the symmetry requirement of the system (e. g., $\\psi ({\\mathbf {R}})=1$ in the bosonic case) the PIGS method works reliably, with the only penalty of producing slightly larger variances.",
"These methodological advancements permit to achieve accurate zero-temperature results in systems for which is difficult to find a good trial wave function.",
"A related method to PIGS is the reptation quantum Monte Carlo method (RQMC) due to Baroni and Moroni (1999); the starting point in RQMC is the same than in PIGS, that is, Eq.",
"(REF ).",
"The main difference relies on the approximation that is used to the Green's function: RQMC adopts a short-time expression similar to the one used in DMC [see Eq.",
"REF ] consisting of a drifted Gaussian that incorporates importance sampling.",
"The ways in which the paths are sampled are also different in the two methods.",
"In the case of knowing a good trial wave function, RQMC may be advantageous as the resulting energy variance can be reduced significantly; otherwise, for the reasons explained in the paragraph above, PIGS may turn out to be a more reliable method (Rossi et al., 2009; Rota et al., 2010)."
],
[
"Path-integral Monte Carlo",
"PIMC is based on the convolution property of the thermal density matrix shown in Eq.",
"(REF ).",
"This allows one to estimate the density matrix at low temperature from its knowledge at higher temperatures, the latter being described by classical statistical mechanics.",
"The partition function, $Z$ , of a quantum system becomes then a multidimensional integral with a distribution law that resembles that of a closed classical polymer with an inter-bead harmonic coupling.",
"If one assumes that all particles are bosons, the corresponding quantum statistical distribution is then positively defined and can be interpreted as a probability distribution function that can be sampled with standard Metropolis Monte Carlo techniques.",
"The finite-$T$ mapping of a quantum system into a classical one composed of polymers was first proposed by Feynman (Feynman, 1972) and subsequently applied by Barker (1979), and Chandler and Wolynes (1981) to condensed-matter simulations.",
"The quantum partition function of a general Hamiltonian, $\\hat{H}$ , at temperature $T$ is: $Z = \\text{Tr} \\, e^{-\\beta \\hat{H}}~.$ The non-commutativity of operators $\\hat{K}$ and $\\hat{V}$ makes it impractical a direct calculation of $Z$ in the quantum regime.",
"Nevertheless, one can exploit the convolution property: $e^{-\\beta (\\hat{K} + \\hat{V})} = \\left( e^{-\\varepsilon ( \\hat{K} + \\hat{V})} \\right)^M ~,$ where $\\varepsilon = \\beta /M$ , since now each of the terms in the right-hand side of the equality effectively corresponds to a higher temperature, that is, $T^\\prime = M \\cdot T$ .",
"In the lowest order approximation, known as the primitive action (PA), the kinetic and potential contributions are factorised as: $e^{-\\varepsilon ( \\hat{K} + \\hat{V})} \\simeq e^{-\\varepsilon \\hat{K}} \\,e^{-\\varepsilon \\hat{V}}~,$ and the convergence to the exact result is guaranteed by Trotter's product formula (Trotter, 1959): $e^{-\\beta (\\hat{K} + \\hat{V})} = \\lim _{M \\rightarrow \\infty } \\left(e^{-\\varepsilon \\hat{K}} \\, e^{-\\varepsilon \\hat{V}} \\right)^M ~.$ The PA approximation, however, is not accurate enough to reach proper convergence at very low temperatures, when the number of terms involved, also called “beads”, is large.",
"In recent years, there has been relevant progress in achieving better convergence by using high-order splitting schemes of the exponential operator.",
"Fourth-order algorithms can be developed by introducing double commutators (Chin and Chen, 2002; Sakkos, Casulleras, and Boronat, 2009) of the form: $[[\\hat{V},\\hat{K}],\\hat{V}] = \\frac{\\hbar ^2}{m} \\, \\sum _{i=1}^{N}|\\mathbf {F}_i|^2 ~,$ where $\\mathbf {F}_i$ is the “force” acting on particle $i$ , namely: $\\mathbf {F}_i = \\sum _{j \\ne i}^{N} \\mathbf {\\nabla }_i V(r_{ij})~.$ One of the most efficient splitting schemes corresponds to: $e^{-\\varepsilon \\hat{H}} &\\simeq & e^{- v_1 \\varepsilon \\hat{W}_{a_1}}e^{- t_1 \\varepsilon \\hat{K}} e^{- v_2 \\varepsilon \\hat{W}_{1-2 a_1}} \\times \\\\ & & e^{- t_1 \\varepsilon \\hat{K}} e^{- v_1 \\varepsilon \\hat{W}_{a_1}}e^{- 2 t_0 \\varepsilon \\hat{K}}~, \\nonumber $ where $W(r)$ is a generalized potential that includes the double commutator in Eq.",
"(REF ).",
"We note that by optimising the value of the parameters in the expansion above, convergence with nearly sixth-order accuracy in $\\varepsilon $ can be achieved.",
"From the knowledge of the quantum partition function one can access the total and kinetic energies using the well-known thermodynamic relations: $E = \\langle \\hat{H} \\rangle & = & - \\frac{1}{Z} \\, \\frac{\\partial Z}{\\partial \\beta } \\\\E_{\\rm k} = \\langle \\hat{K} \\rangle & = & \\frac{m}{\\beta Z} \\, \\frac{\\partial Z}{\\partial m}~,$ where the potential energy comes from the difference $E_{\\rm p} = E - E_{\\rm k}$ .",
"The potential energy also can be computed using the expression: $\\hat{O} (\\mathbf {R}) = - \\frac{1}{\\beta } \\, \\frac{1}{Z(\\hat{V})} \\, \\left.",
"\\frac{d Z(\\hat{V} + \\lambda \\hat{O})}{d \\lambda } \\right|_{\\lambda =0}~,$ that in general is suitable for estimating operators that depend only on particle coordinates.",
"We note that the kinetic energy expression in Eq.",
"(), which is known as the “thermodynamic” estimator, presents some technical drawbacks like for instance a diverging variance when the number of beads is large.",
"Several solutions have been proposed to overcome this limitation, among which we highlight the “virial” estimator introduced by Cao and Berne (1989).",
"An alternative to the discussed decomposition scheme of the exponential operator, is to use a pair product approximation (Ceperley, 1995).",
"In this case, one approximates the density matrix by a factorization of correlations up to second order, resembling the Jastrow approximation used for the ground state, namely: $\\rho ({\\mathbf {R}},{\\mathbf {R}}^\\prime ;\\varepsilon )= \\prod _{i=1}^{N} \\rho _1({\\mathbf {r}}_j,{\\mathbf {r}}_j^\\prime ;\\varepsilon )\\prod _{i<j}^{N} \\hat{\\rho }_2({\\mathbf {r}}_{ij},{\\mathbf {r}}_{ij}^\\prime ;\\varepsilon )~.$ In Eq.",
"(REF ), $\\rho _1$ represents the density matrix for a non-interacting system and $\\hat{\\rho }_2$ the normalized pair density matrix, that is: $\\hat{\\rho }_2({\\mathbf {r}}_{ij},{\\mathbf {r}}_{ij}^\\prime ;\\varepsilon ) =\\frac{\\rho _2({\\mathbf {r}}_{ij},{\\mathbf {r}}_{ij}^\\prime ;\\varepsilon )}{\\rho _2^0({\\mathbf {r}}_{ij},{\\mathbf {r}}_{ij}^\\prime ;\\varepsilon )}~,$ in which $\\rho _2$ and $\\rho _2^0$ are the relative density matrices of the interacting and non-interacting systems, respectively.",
"The pair action is specially useful when the pair density matrix is known analytically or an accurate approximation of it is at hand.",
"Application of this approach is particularly suitable for the study of central potentials, although it is not restricted to this type of interactions (Pierleoni and Ceperley, 2006).",
"The formalism already explained in this section applies only to distinguishable particles (i. e., “boltzmanons”), since the symmetry requirement under exchange of particles has been neglected systematically.",
"In order to describe correctly quantum Bose crystals one needs to symmetrize the corresponding thermal density matrix, namely: $\\rho _s({\\mathbf {R}},{\\mathbf {R}}^\\prime ;\\varepsilon )= \\frac{1}{N!}",
"\\sum _{\\cal P}\\rho ({\\mathbf {R}},{\\cal P} {\\mathbf {R}}^\\prime ;\\varepsilon )~,$ where the summation runs over all possible $N!$ permutations involving the particles in the system.",
"In contrast to the boltzmanon case, in which the number of closed polymers equals the number of particles (${\\mathbf {R}}_{M+1}={\\mathbf {R}}_1$ , with $M$ the number of beads), the new boundary condition ${\\mathbf {R}}_{M+1}={\\cal P}{\\mathbf {R}}_1$ implies that each closed polymer can represent more than one particle.",
"The acceptance rate for the proposed permutations then increases with the inverse of the temperature; when the thermal wavelength $\\lambda _T$ is comparable to the mean interparticle distance the size of closed polymers becomes macroscopic, originating Bose-Einstein condensation and superfluidity (see Sec.",
"REF ).",
"The fraction of particles occupying the lowest momentum state in a bosonic system, i. e., the condensate fraction $n_{0} \\equiv n(k=0)$ , can be obtained from the long-range behavior of the one-body density matrix, defined as: $\\varrho _{1}(r_{1 1^\\prime })= \\frac{V}{Z} \\int d{\\mathbf {r}}_2 \\ldots d{\\mathbf {r}}_N ~\\rho _s({\\mathbf {R}}, {\\mathbf {R}}^\\prime ;\\beta )~,$ namely, $n_{0} = \\lim _{r \\rightarrow \\infty } \\varrho _{1}(r)$ .",
"In practice, $\\varrho _{1}(r)$ is estimated by calculating frequency histograms over distances between ${\\mathbf {r}}_1$ and ${\\mathbf {r}}_1^\\prime $ .",
"Sampling the space of permutations is technically involved because one has to guarantee ergodicity.",
"In recent years, the introduction of the worm algorithm has improved significantly the efficiency in this type of calculations (Boninsegni, Prokof'ev, and Svistunov, 2006a).",
"The idea behind the worm algorithm is to work in an extended configuration space with two sectors.",
"In the diagonal sector, termed $Z$ , all paths are closed, which corresponds to conventional PIMC simulations.",
"In the second sector, termed $Z_{G}$ , all paths are closed except one, which is called the worm; this latter sector, therefore, is non-diagonal.",
"The generalized partition function then can be written as: $Z_{W} = Z + C Z_{G} ~,$ where $C > 0$ is a dimensionless parameter that is fixed during the simulation.",
"Parameter $C$ controls the relative statistics between sectors $Z$ and $Z_{G}$ .",
"In the non-diagonal sector one proposes swap movements that generate multi-particle permutations (i. e., by single pair permutations between the worm and closed paths), whereas in the diagonal sector particles evolve as boltzmanons."
],
[
"Path-integral molecular dynamics",
"In the PIMD formalism, the partition function of a quantum system is approximated with the Maxwell-Boltzmann expression: $Z &&\\approx \\frac{1}{N!}",
"\\left(\\frac{mL}{2 \\pi \\beta \\hbar ^{2}} \\right)^{3NL/2} \\nonumber \\\\&& \\times \\int \\prod _{j=1}^{N} \\prod _{i=1}^{L} d{\\bf r}_{ij}~ e^{-\\beta \\left( E_{\\rm k} + E_{\\rm p} \\right)}~.$ The equation above completely disregards possible quantum atomic exchanges stemming from the indistinguishability of the atoms (in contrast to the PIGS and PIMC methods, see Secs.",
"REF and REF ); that is, particles are treated as boltzmanons.",
"Nevertheless, in the case of quantum crystals it is well-known that the role of quantum statistics is secondary at moderate and high temperatures (e. g., $T > 100$ K in hydrogen at $P \\sim 100$ GPa, see McMahon et al., 2012).",
"In those situations, PIMD can be used to compute, for instance, quantum time-correlation functions and transition state rates in a very efficient manner (Gillan, 1990; Habershon et al., 2013; Herrero and Ramírez, 2014).",
"The key idea behind PIMD is to formulate a Hamiltonian framework in which new space coordinates and momenta, $\\left( {\\bf u}_{ij} , {\\bf p}_{ij} \\right)$ , are introduced for sampling the integral in Eq.",
"(REF ) with molecular dynamics techniques.",
"In particular, the new space coordinates and momenta are referred to the staging modes, ${\\bf u}_{ij}$ , that diagonalize the harmonic energy term, namely: $E_{\\rm k} &&= \\frac{m L}{2 \\beta ^{2} \\hbar ^{2}} \\sum _{j=1}^{N} \\sum _{i=1}^{L} \\left({\\bf r}_{ij}-{\\bf r}_{(i+1)j}\\right)^{2} \\nonumber \\\\&&= \\sum _{j=1}^{N} \\sum _{i=2}^{L} \\frac{m_{i} L}{2 \\beta ^{2} \\hbar ^{2}} {\\bf u}_{ij}^{2}~.$ For a given atom $j$ , the staging mode coordinates are defined as ${\\bf u}_{1j} = {\\bf r}_{1j}$ , and ${\\bf u}_{ij} = {\\bf r}_{ij} - \\frac{i-1}{i} {\\bf r}_{(i+1)j} - \\frac{1}{i} {\\bf r}_{1j} $ in the rest of cases; the corresponding staging mode masses are $m_{1} = 0$ , and $m_{i} = \\frac{i}{i-1}m$ in the rest of cases.",
"The momentum variables that are required for the molecular dynamics algorithm to work, are introduced through the substitution of the pre-factor in the partition function by a Gaussian integral of the form: $\\left( \\frac{mL}{2 \\pi \\beta \\hbar ^{2}} \\right)^{3NL/2} =C \\int \\prod _{j=1}^{N} \\prod _{i=1}^{L} d{\\bf p}_{ij}~ e^{\\left( \\frac{-\\beta {\\bf p}_{ij}^{2}}{2 \\chi _{i}} \\right)}~,$ where $C$ is a constant that depends on the staging momentum masses, but which has no influence on the calculation of the equilibrium properties; ${\\bf p}_{ij}$ is the $i$ staging momentum of particle $j$ .",
"Masses, $\\chi _{i}$ , in Eq.",
"(REF ) can be defined as $\\chi _{1} = m$ , and $\\chi _{i} = m_{i}$ in the rest of cases; essentially, these must be chosen so that all $i > 1$ staging modes evolve in the same time scale.",
"In either $(N,V,T)$ or $(N,P,T)$ PIMD simulations, control of the temperature is achieved through a massive thermostatting of the system that implies a chain of Nosé-Hoover thermostats coupled to each staging variable ${\\bf u}_{ij}$ (Tuckerman and Hughes, 1998).",
"The involved thermostats introduce friction terms in the corresponding dynamic equations and thus the dynamics of the quantum system is not longer Hamiltonian.",
"Nevertheless, it is always possible to define a quantity with units of energy that is well conserved during the simulation and that can be used to check whether integration of the equations of motion is being done correctly (Martyna, Tuckerman, and Hughes, 1999).",
"Finally, we note that equivalent estimators in the PIMD and PIMC frameworks may present some differences, but only in the terms involving momentum variables.",
"Nevertheless, in those cases in which quantum atomic exchanges can be safely neglected, both PIMD and PIMC approaches should provide identical expectation values, as it follows from the equipartition theorem (Herrero and Ramírez, 2014).",
"A detailed account of the path-integral molecular dynamics (PIMD) method certainly is out of the scope of the present review.",
"The details of this technique have been described thoroughly by Tuckerman and Hughes (1998) and Martyna, Tuckerman, and Hughes (1999), hence we refer the interested reader to those works."
],
[
"Quantum thermal baths",
"The key idea behind quantum thermal baths (QTB) is to use a Langevin-type approach in which a dissipative force and a Gaussian random force are adjusted to have the power spectral density given by the quantum fluctuation-dissipation theorem (Dammak et al., 2009; Ceriotti, Bussi, and Parrinello, 2009; Barrat and Rodney, 2011).",
"In doing this, the internal energy of the system can be mapped into that of an ensemble of harmonic oscillators whose vibrational modes follow a Bose-Einstein distribution.",
"It is worth noticing that while such a quantum discretisation is applied to the energy, the atoms in the system are invariably treated as distinguisable particles.",
"Consequently, QTB are not well suited for describing physical phenomena in which quantum atomic exchanges are important, which typically occur in disordered and incommensurate systems at low temperatures (in contrast to the PIGS and PIMC methods, see Secs.",
"REF and REF ).",
"In analogy to the classical Langevin thermostat method, each particle is coupled to a fictitious bath by introducing a random force and a dissipation term in the equations of motion of the form: $m \\frac{d^{2}{\\bf r}}{dt^{2}} = {\\bf F}({\\bf r}) - \\gamma m \\frac{d{\\bf r}}{dt} + \\sqrt{2m\\gamma }\\Theta (t)~,$ where ${\\bf r}$ and ${\\bf F}$ represent the atomic positions and total forces exerted by the rest of particles, respectively.",
"Function $\\Theta (t)$ is a colored noise with a power spectral density that follows the Bose-Einstein distribution, namely: $\\tilde{\\Theta }(\\omega ) &&= \\int e^{-i \\omega t} \\langle \\Theta (t) \\Theta (t^{\\prime }) \\rangle dt \\nonumber \\\\&&= \\hbar \\omega \\left( \\frac{1}{2} + \\frac{1}{e^{\\hbar \\omega /k_{\\rm B}T}-1} \\right)~,$ which takes into account the zero-point energy of the system as given by the quasi-harmonic approximation (see Sec.",
"REF ).",
"In practice, $\\tilde{\\Theta }(\\omega )$ can be generated by using a signal-processing method based on filtering of white noise (Barrat and Rodney, 2011).",
"The implementation of QTB in a discrete MD algorithm then is quite straightforward.",
"QTB neither slow down the calculations appreciably nor are detrimental in terms of memory requirements.",
"For these reasons, the use of QTB for simulation of QNE is becoming increasingly more popular in recent years (Hernández-Rojas, Calvo, and González-Noya, 2015).",
"A word of caution, however, must be added here.",
"QTB alone fail to reproduce the correct quantum behavior in highly anharmonic systems and processes (Ceriotti, Bussi, and Parrinello, 2009; Barrozo and de Koning, 2011; Bedoya-Martínez, Barrat, and Rodney, 2014).",
"Consequently, the conclusions attained with QTB-based methods should be always validated against results obtained with more accomplished quantum approaches (e. g., PIMC and PIMD).",
"Meanwhile, it has been recently demonstrated that QTB can be used to accelerate noticeably the convergence in PIMD calculations (Ceriotti, Manolopoulos, and Parrinello, 2011; Ceriotti and Manolopoulos, 2012; Brieuc, Dammak, and Hayoun, 2016).",
"In particular, generalized Langevin thermostats allow to sample the canonical distribution more efficiently by reducing the usual ergodic problems encountered in path-integral simulations performed with a large number of beads.",
"It is probably in this latter context that QTB techniques can be particularly useful.",
"Table: List of computer simulation packages that allow to simulate quantum nuclear effects in periodic systems.PIMD, PIMC, VMC, DMC, and PIGS in the “Capability” row stand for, path-integral molecular dynamics,path-integral Monte Carlo, variational Monte Carlo, diffusion Monte Carlo, and ground-state path-integralMonte Carlo, respectively.",
"CPU and GPU in the “Parallelisation” row stand for central and graphicalprocessing units.While the number of classical simulation packages, either commercially or freely available, is practically countless, the number of computer packages that allow to simulate QNE is very limited.",
"In Table I, we list those computer packages that, to the best of our knowledge, are publicly available and can be used to simulate QNE in periodic systems, along with a brief description of their basic capabilities.",
"In total, they amount to a bit more than ten.",
"We note that PIMD (see Sec.",
"REF ) is the method that is implemented most frequently.",
"On the other hand, quantum Monte Carlo techniques (i. e., VMC and DMC) are available in fewer codes.",
"Although it is not indicated in Table I, most of the listed simulation packages also allow to describe the interactions between atoms through ab initio methods (see next Sec.",
"REF ).",
"In addition to this, they are all designed to run in high-performance computing architectures and can be downloaded free of charge from internet or made available on request.",
"A likely reason behind the scarcity of studies considering QNE may be, apart from the increased computational and technical burdens, the limited number of available quantum simulation packages.",
"We note that most of the codes in Table I are relatively new, hence until recently any researcher interested in simulating QNE had to craft his/her own quantum implementation.",
"Nevertheless, we expect that due to the steady growth in computing power and the increasing awareness of the importance of QNE in condensed matter systems and materials, the availability and user-friendliness of quantum simulation packages will increase in the next years.",
"The simulation techniques that are used to describe the atomic interactions in quantum crystals, and materials in general, can be classified in two major categories: “semi-empirical” and “first-principles”.",
"In semi-empirical approaches, the interparticle forces are typically modeled with analytical functions, known as force fields or classical potentials, that are devised to reproduce a particular set of experimental data or the results of highly accurate calculations.",
"The inherent simplicity of classical potentials makes it possible to address the study of quantum solids within ample thermodynamic intervals and large length/time scales, with well-established quantum simulation techniques like the ones discussed in Sec. .",
"By using semi-empirical potentials and exploiting the current computational power and algorithm development, quantum simulations of condensed matter systems can be routinely performed nowadays in multi-core processors.",
"Nevertheless, in spite of their great versatility, classical potentials may sometimes present some impeding transferability issues.",
"Transferability issues are related to the impossibility of mimicking the targeted systems at conditions different from those in which the setup of the corresponding force field was performed.",
"An illustrative example of such a failure is given by the unreliable description of highly compressed rare-gas crystals with pairwise potentials (Cazorla and Boronat, 2008a; Cazorla and Boronat, 2015a; Cazorla and Boronat, 2015b).",
"In addition to this, there are many physical phenomena that simply cannot be reproduced accurately with straightforward force fields (e. g., magnetic spin interactions, electronic screening effects, and oxidation-state changes, to cite just a few examples).",
"In this context, the output of first-principles calculations, also known as ab initio, turns out to be crucial.",
"In first-principles approaches, as the name indicates, no empirical information is assumed on the derivation of the atomic interactions: these are directly obtained from applying the principles of quantum mechanics to the electrons and nuclei.",
"Transferability issues, therefore, are absent.",
"First-principles approaches are in general very accurate, but they can be also very demanding in terms of computational expense.",
"This circumstance makes the full ab initio study of quantum crystals, that is, in which both the electronic and nuclear degrees of freedom are treated quantum mechanically, intricate and computationally very demanding (see, for instance, Pierleoni, Ceperley, and Holzmann, 2004; Pierleoni and Ceperley, 2005; Pierleoni and Ceperley, 2006; McMahon et al., 2012).",
"Common acceleration schemes within first-principles schemes involve the use of pseudopotentials (see, for instance, Vanderbilt, 1990; Troullier and Martins, 1991), which avoids to treat explicitly the core electrons.",
"This approximation is based on the fact that many materials properties can be predicted by focusing exclusively on the behavior of valence electrons.",
"Nonetheless, pseudopotentials can actually be the source of potential errors.",
"Fortunately, some strategies can be used to minimize the impact of the approximations introduced by pseudopotentials like, for instance, the projector augmented wave method (Blöchl, 1994) and linearized augmented plane waves (Andersen, 1975).",
"Next, we concisely explain some basic aspects of first-principles and semi-empirical methods as related to the study of quantum solids."
],
[
"First-principles methods",
"In solids, the dynamics of electrons and nuclei can be decoupled to a good approximation because their respective masses differ by several orders of magnitude.",
"The wave function of the corresponding many-electron system, $\\Psi ({\\bf r}_{1}, {\\bf r}_{2},...,{\\bf r}_{N})$ , therefore can be determined by solving the Schrödinger equation involving the non-relativistic Born-Oppenheimer Hamiltonian: $H = -\\frac{1}{2} \\sum _{i} {\\bf \\nabla }^{2}_{i} - \\sum _{I} \\sum _{i} \\frac{Z_{I}}{|{\\bf R}_{I} - {\\bf r}_{i}|} \\nonumber \\\\+ \\frac{1}{2} \\sum _{i} \\sum _{j \\ne i} \\frac{1}{|{\\bf r}_{i} - {\\bf r}_{j}|}~,$ where $Z_{I}$ are the nuclear charges, ${\\bf r}_{i}$ the positions of the electrons, and ${\\bf R}_{I}$ the positions of the nuclei, which are considered fixed.",
"(We note that non-adiabatic effects beyond the Born-Oppenheimer approximation in principle can be also treated within first-principles methods by using wave functions that explicitly depend on the electronic and nuclear degrees of freedom; see, for instance, Ceperley and Alder, 1987; Tubman et al., 2014; Yang et al., 2015.)",
"In real materials $\\Psi $ is a complex mathematical function that in most cases is unknown.",
"Electrons are fermion particles, hence their wave function must change sign when two of them exchange orbital states.",
"This quantum antisymmetry leads to an effective repulsion between electrons, called the Pauli repulsion, that helps in lowering their total Coulomb energy.",
"At the heart of any first-principles method is to find a good approximation to $\\Psi $ , or an equivalent solution, that is manageable enough to perform calculations and simultaneously describes the system of interest correctly.",
"Examples of ab initio methods include density functional theory (DFT), Møller-Plesset perturbation theory (MP2), the coupled-cluster method with single, double and perturbative triple excitations [CCSD(T)], and electronic quantum Monte Carlo (eQMC), to cite just a few.",
"From these, DFT and eQMC have been most intensively applied to the study of quantum solids and for this reason we summarise their foundations in what follows.",
"In 1965, Kohn and Sham developed a pioneering theory to effectively calculate the energy and properties of many-electron systems without the need of explicitly knowing $\\Psi $ (Kohn and Sham, 1965; Sham and Kohn, 1966).",
"The main idea underlying this theory, called density functional theory (DFT), is that the exact ground-state energy, $E$ , and electron density, $n({\\bf r})$ , can be determined by solving an effective one-electron Schrödinger equation of the form: $H_{eff} \\psi _{i \\sigma } = \\epsilon _{i \\sigma } \\psi _{i \\sigma }~,$ where index $i$ labels different one-electron orbitals and $\\sigma $ the corresponding spin state.",
"In particular, $H_{eff} = -\\frac{1}{2}\\nabla ^{2} + V_{ext}({\\bf r}) + \\int \\frac{n({\\bf r^{\\prime }})}{|{\\bf r} - {\\bf r^{\\prime }}|} d{\\bf r^{\\prime }} + V_{xc}({\\bf r})~,$ and $n({\\bf r}) = \\sum _{i \\sigma } |\\psi _{i \\sigma } ({\\bf r})|^{2}~,$ where $V_{ext}$ represents an external field and $V_{xc} ({\\bf r}) = \\delta E_{xc} / \\delta n ({\\bf r})$ is the exchange-correlation potential.",
"The exchange-correlation energy has a purely quantum mechanical origin and can be defined as the interaction energy difference between a quantum many-electron system and its classical counterpart.",
"Despite $E_{xc}$ represents a relatively small fraction of the total energy, this contribution is extremely crucial for all materials and molecules because it acts directly on the bonding between atoms.",
"In general, $E_{xc} [n]$ is unknown and needs to be approximated.",
"This is the only source of fundamental error in DFT methods.",
"The exact form of the exchange-correlation energy can be readily expressed through the adiabatic connection fluctuation-dissipation theorem as (Langreth and Perdew, 1975; Nguyen and de Gironcoli, 2009): $E_{xc} [n] = \\int n({\\bf r})~d{\\bf r} \\int \\frac{n_{xc}({\\bf r} , {\\bf r^{\\prime }})}{|{\\bf r} - {\\bf r^{\\prime }}|}~d{\\bf r^{\\prime }}~,$ where $n_{xc}({\\bf r} , {\\bf r^{\\prime }}) = n_{x} ({\\bf r} , {\\bf r^{\\prime }}) + n_{c} ({\\bf r} , {\\bf r^{\\prime }})$ is the exchange-correlation hole density at position ${\\bf r^{\\prime }}$ surrounding an electron at position r. Some important constraints on $n_{xc}({\\bf r} , {\\bf r^{\\prime }})$ are already known.",
"For instance, $n_{x} ({\\bf r}, {\\bf r^{\\prime }})$ must be non-positive everywhere and its space integral is equal to $-1$ .",
"Also, the space integral of the correlation hole density is zero.",
"These constraints can be employed in the construction of approximate $E_{xc} [n]$ functionals.",
"In standard DFT approaches $E_{xc} [n]$ is approximated with the expression: $E_{xc}^{approx}[n] = \\int \\epsilon _{xc}^{approx}({\\bf r}) n({\\bf r}) d{\\bf r}~,$ where $\\epsilon _{xc}^{approx}$ is made to depend on $n({\\bf r})$ , $\\nabla n({\\bf r})$ , and/or the electronic kinetic energy $\\tau ({\\bf r}) = \\frac{1}{2} \\sum _{i \\sigma } |\\nabla \\psi _{i \\sigma } ({\\bf r})|^{2}$ .",
"Next, we summarise the basic aspects of the most popular $E_{xc} [n]$ functionals found in computational studies of condensed matter systems and materials.",
"Additional details on these topics can be found in recent and more specialized reviews (see, for instance, Perdew, 2013; Klimeš and Michaelides 2012; Dobson and Gould, 2012; Cazorla, 2015).",
"We note that the current number of commercially available and open-source DFT computer packages is huge (at least in comparison to that of eQMC codes); a reference to some of them can be found, for instance, in Cazorla (2015).",
"In local approaches (e. g., local density approximation -LDA-), $E_{xc}$ is approximated with Eq.",
"(REF ) and the exchange-correlation energy is taken to be equal to that in an uniform electron gas of density $n ({\\bf r})$ , namely $\\epsilon _{xc}^{unif}$ .",
"The exact $\\epsilon _{xc}^{unif} [n]$ functional is known numerically from quantum Monte Carlo calculations (Perdew and Zunger, 1981; Ceperley and Alder, 1980).",
"In order to deal with the non-uniformity in real electronic systems, the space is partitioned into infinitesimal volume elements that are considered to be locally uniform.",
"In semi-local approaches (e. g., generalized gradient approximation -GGA-), $E_{xc}$ is approximated also with Eq.",
"(REF ) but $\\epsilon _{xc}^{approx}$ is made to depend on $n({\\bf r})$ and its gradient $\\nabla n({\\bf r})$ (Perdew et al., 1992; Perdew et al., 1996).",
"Both local and semi-local approximations satisfy certain exact $E_{xc}$ constraints (e. g., some exact scaling relations and the exchange-correlation hole sum rules) and can work notably well for systems in which the electronic density varies slowly over the space (e. g., bulk crystals at equilibrium conditions).",
"However, by construction local and semi-local functionals cannot account for long-range electronic correlations, otherwise known as dispersion interactions, which certainly are ubiquitous in quantum crystals."
],
[
"Hybrid Exchange Functionals.",
"Hybrid functionals comprise a combination of non-local exact Hartree-Fock and local exchange energies, together with semi-local correlation energies.",
"The proportion in which both non-local and local exchange densities are mixed generally relies on empirical rules.",
"The popular B3LYP approximation (Becke, 1993), for instance, takes a 20 % of the exact HF exchange energy and the rest from the GGA and LDA functionals.",
"Other well-known hybrid functionals are the HSE proposed by Heyd-Scuseria-Ernzerhof (Heyd et al., 2003), PBE0 (Adamo and Barone, 1999), and the family of Minnesota meta hybrid GGA (Zhao et al., 2005).",
"In contrast to local and semi-local functionals, hybrids can describe to some extent the delocalisation of the exchange-correlation hole around an electron.",
"This characteristic is specially useful when dealing with strongly correlated systems containing $d$ and $f$ electronic orbitals (e. g., perovskite oxides).",
"Hybrid functionals, however, do not account for the long range part of the correlation hole energy and thus cannot reproduce dispersion forces.",
"Effective ways to correct for these drawbacks have been proposed by several authors (Chai and Head-Gordon, 2008; Lin et al., 2013; Mardirossian and Head-Gordon, 2014)."
],
[
"Dispersion-Corrected Functionals.",
"DFT-based dispersion schemes reproduce the asymptotic $1/r^{6}$ interaction between two particles separated by a distance $r$ in a gas.",
"The most straightforward way of achieving this consists in adding an attractive energy term to the exchange-correlation energy of the form $E_{\\rm disp} =-\\sum _{i,j} C_{ij} / r^{6}_{ij}$ (indexes $i$ and $j$ label different particles).",
"This approximation represents the core of a suite of methods named DFT-D that, due to their simplicity and low computational cost, are being employed widely (Grimme, 2004).",
"Nevertheless, DFT-D methods present some inherent limitations.",
"For instance, many-body dispersion effects and faster decaying terms such as the $B_{ij}/ r^{8}_{ij}$ and $C_{ij} / r^{10}_{ij}$ interactions are completely disregarded.",
"Also, it is not totally clear from where one should obtain the optimal $C_{ij}$ coefficients.",
"Several improvements on DFT-D methods have been proposed, in which the value of the dispersion coefficients are made to depend somehow on the specific atomic environment.",
"Examples of those include the DFT-D3 method by Grimme (Grimme et al., 2010), the vdW(TS) approach by Tkatchenko and Scheffler (2009), and the BJ model by Becke and Johnson (2007).",
"A further degree of elaboration exists in which no external input parameters are needed and the dispersion interactions are directly computed from the electron density.",
"In this context, the exchange-correlation energy is expressed as $E_{xc} = E_{x}^{\\rm GGA} + E_{c}^{\\rm LDA} + E_{c}^{\\rm nl}$ , where $E_{c}^{\\rm nl}$ is the non-local correlation energy.",
"$E_{c}^{\\rm nl}$ can be calculated as a double space integral involving the electron density and a two-position integration kernel.",
"This approach, introduced by Dion et al.",
"(2004), represents a key development in DFT methods as it combines all types of interaction ranges within a same formula.",
"Refinements of this scheme have been proposed recently in which the original two-position integration kernel is modified (Vydrov and Voorhis, 2012), or the exchange term in $E_{xc}$ is replaced with other more accurate functionals (Lee et al., 2010; Carrasco et al., 2011)."
],
[
"Electronic quantum Monte Carlo",
"Here we explain the basics of the diffusion Monte Carlo (DMC) method (see Sec.",
"REF ) as applied to the study of many-electron systems [for a more technical and complete discussion on this topic see, for instance, Foulkes et al.",
"(2001) and Towler (2006)].",
"In electronic quantum Monte Carlo (eQMC) methods one deals explicitly with the solution to the imaginary-time dependent Schrödinger equation (in contrast to DFT methods).",
"The quantum antisymmetry of the electrons leads to the so-called “sign problem”, that is related to the fact that the probability distribution function $f = \\Psi _{T} \\Psi _{0}$ is not positive definite everywhere (see Sec.",
"REF ).",
"If the nodes of the guiding and true ground-state wave functions [that is, the $3N-1$ -dimensional surfaces at which $\\Psi ({\\bf r}_{1}, {\\bf r}_{2},...,{\\bf r}_{N}) = 0$ ] were coincident, the sign problem would disappear.",
"However, in most many-electron problems this condition is never satisfied.",
"Several approaches have been proposed in the literature to tackle the sign problem, among which we highlight the “fixed-node” and “released-node” methods."
],
[
"Fixed-node method.",
"In this method the nodes of the ground-state function $\\Psi _{0}$ are forced to be equal to those of the guiding wave function $\\Psi _{T}$ [see, Anderson (1975) and (1976)].",
"As a result, the probability distribution function that is asymptotically sampled is always positive because a change of sign in $\\Psi _{T}$ is replicated by a change of sign in $\\Psi _{0}$ .",
"By using this approximation, however, one always obtains results that are upper bounds to the exact ground-state energy (Reynolds et al., 1982).",
"When dealing with fermionic systems, therefore, it is critically important to choose guiding wave functions with high quality nodal surfaces.",
"This requirement is also necessary for guaranteeing numerical stability in the simulations, since the divergence of the drift force, ${\\bf F} = 2 {\\bf \\nabla } \\Psi _{T}/ \\Psi _{T}$ , close to a node cannot always be counteracted by the replication of energetically favorable configurations.",
"The fixed-node electronic DMC (FN-DMC) method was first applied to the electron gas by Ceperley and Alder (1980).",
"Subsequently, it was employed to study solid hydrogen (Ceperley and Alder, 1987) and other crystals containing heavier atoms (Fahy et al., 1988, 1990; Li et al., 1991).",
"Wigner crystals in two and three dimensions have been also investigated thoroughly with eQMC methods (Tanatar and Ceperley, 1989; Drummond et al., 2004; Drummond and Needs, 2009).",
"Important FN-DMC developments include the introduction of variance minimization techniques to optimize wave functions (Umrigar et al., 1988) and the use of non-local pseudopotentials (Hammond et al., 1987; Hurley and Christiansen, 1987; Fahy et al., 1988; Mitas et al., 1991; Trail and Needs, 2013; Lloyd-Williams, Needs, and Conduit, 2015; Trail and Needs, 2015).",
"We also highlight the generalisation of eQMC methods to systems with broken time-reversal symmetry (e. g., interacting electrons in an applied magnetic field or states with non-zero angular momentum), which is known as the “fixed-phase” approximation (Ortiz, Ceperley, and Martin, 1993).",
"These improvements, together with a certain availability of commercial and open source simulation packages (see Sec.",
"REF ), have stimulated the study of a wide range of electronic systems with DMC like, for instance, strongly correlated oxide materials (Huihuo and Wagner, 2015; Wagner, 2015), hydrates (Alfè et al., 2013; Cox et al., 2014), and organic molecules (Purwanto et al., 2011; Jiang et al., 2012)."
],
[
"Released-node method.",
"In the released node (RN) method the nodal constraints imposed by the guiding function are relaxed in order to adapt to those of the exact wave function (Ceperley and Alder, 1980; Ceperley and Alder, 1984; Hammond et al., 1994; Tubman et al., 2011).",
"As we explain next, this technique provides a solution that is not stable in imaginary time and thus its use is restricted to systems in which (i) the nodes of the guiding function are relatively accurate, or (ii) the Pauli principle is relatively unimportant (that is, the energy difference between the many-fermion system and its many-boson counterpart is small).",
"An arbitrary antisymmetric wave function, $\\Psi _{A}$ , can always be expressed as a linear combination of two positive functions like: $\\Psi _{A} \\left( {\\bf r}, \\tau \\right) = \\phi _{+} \\left( {\\bf r}, \\tau \\right) - \\phi _{-} \\left( {\\bf r}, \\tau \\right)~.$ As both $\\phi _{+}$ and $\\phi _{-}$ are positively defined, each one can be interpreted as a probability density and be propagated individually.",
"A convenient definition of $\\phi _{\\pm }$ at $\\tau = 0$ is: $\\phi _{\\pm } \\left( \\tau = 0 \\right) = \\frac{1}{2} \\left( |\\Psi _{A}| \\pm \\Psi _{A} \\right)~,$ since at large imaginary time the corresponding projected states are: $\\phi _{\\pm } \\left( \\tau \\rightarrow \\infty \\right) = \\pm C_{F} \\Psi _{0}^{F} + C_{B} \\Psi _{0}^{B} e^{\\left( E_{0}^{F} - E_{0}^{B} \\right) \\tau }~,$ where $\\Psi _{0}^{F}$ and $\\Psi _{0}^{B}$ are the ground-state fermion and boson wave functions of the Hamiltonian, respectively.",
"$\\Psi _{A}$ consistently renders the ground-state energy of the fermionic system, namely: $E_{0}^{RN} = \\frac{\\int \\Psi _{A} \\left(\\tau \\rightarrow \\infty \\right) {\\rm H} \\Psi _{0} d{\\bf r}}{\\int \\Psi _{A} \\left(\\tau \\rightarrow \\infty \\right) \\Psi _{0} d{\\bf r}} =E_{0}^{F}~.$ However, since the $E_{0}^{F} - E_{0}^{B}$ energy difference is always positive, the bosonic parts in $\\phi _{\\pm }$ grow exponentially with imaginary time (see Eq.",
"REF ), leading to an energy variance of: $\\sigma \\left( E_{0}^{RN} \\right) \\propto e^{\\left( E_{0}^{F} - E_{0}^{B} \\right) \\tau }~,$ that is divergent.",
"For this reason the RN method is classified as a “transient estimator” approach.",
"Cases in which application of the RN method is judicious, numerical weighting techniques can be used to reduce considerably the variance of the energy and other quantities (Hammond et al., 1994; Tubman et al., 2011).",
"The RN approach can be employed also as a measure of the quality of the upper bounds provided by the fixed-node method (Casulleras and Boronat, 2000; Sola, Casulleras, and Boronat, 2006)."
],
[
"Electronic guiding wave functions.",
"In eQMC methods, the choice of the guiding function is particularly important because it determines the degree of accuracy in the calculations.",
"The most widely used $\\Psi _{T}$ model is the Slater-Jastrow wave function, that is expressed as: $\\Psi _{T}({\\bf X}) = e^{J({\\bf X})} \\sum _{j} c_{j}D_{j}({\\bf X})~,$ where ${\\bf X} = \\left( {\\bf x}_{1}, {\\bf x}_{2}, \\cdots , {\\bf x}_{N} \\right)$ and ${\\bf x}_{i} = \\lbrace {\\bf r}_{i}, \\sigma _{i} \\rbrace $ represent the space and spin coordinates of electron $i$ , $e^{J}$ is the Jastrow factor, $c_{j}$ are coefficients, and $D_{i}$ Slater determinants of single-particle orbitals of the form: $D_{j}({\\bf X}) = \\left| \\begin{array}{cccc}\\phi _{1}^{j}({\\bf x}_{1}) & \\phi _{1}^{j}({\\bf x}_{2}) & \\ldots & \\phi _{1}^{j}({\\bf x}_{N}) \\\\\\phi _{2}^{j}({\\bf x}_{1}) & \\phi _{2}^{j}({\\bf x}_{2}) & \\ldots & \\phi _{2}^{j}({\\bf x}_{N}) \\\\\\vdots & \\vdots & \\vdots & \\\\\\phi _{N}^{j}({\\bf x}_{1}) & \\phi _{N}^{j}({\\bf x}_{2}) & \\ldots & \\phi _{N}^{j}({\\bf x}_{N}) \\end{array} \\right|~.$ The orbitals $\\lbrace \\phi _{i}^{j} \\rbrace $ often are obtained from DFT or Hartree-Fock calculations, and are assumed to be products of factors that depend either on the space or spin coordinates.",
"It is common practice in eQMC calculations to replace $D_{j}$ with products of separate up- and down-spin determinants, since this improves computational efficiency (Foulkes et al., 2001).",
"The Jastrow factor in Eq.",
"(REF ) normally contains one and two-body terms, namely: $J({\\bf X}) = \\sum _{i}^{N} \\chi ({\\bf x}_{i}) - \\frac{1}{2}\\sum _{i}^{N}\\sum _{j \\ne i}^{N} u({\\bf x}_{i},{\\bf x}_{j})~,$ where functions $u$ describe the electron-electron correlations and $\\chi $ the electron-nuclear correlations.",
"The two-electron terms in Eq.",
"(REF ) reduce the value of the wave function whenever two electrons approach each to the other, hence reducing the repulsive electron-electron interaction energy.",
"However, the introduction of $u$ terms also has the unwanted effect of pushing electrons away from regions of high-charge density into regions of low-charge density, thus depleting the electronic density in the atomic bonds.",
"By introducing the one-body functions $\\chi $ in the Jastrow factor this problem is overcome.",
"Another approach that can be employed to improve the description of electron-electron correlations is to consider backflow correlations within the Slater determinants.",
"Backflow correlations were originally derived from a current conservation argument due to Feynman and Cohen (1956) to provide a picture of excitations in liquid $^{4}$ He; they represent the characteristic flow pattern in a quantum fluid where particles in front of a moving one go on filling the space left behind it.",
"The introduction of backflow correlations may relax in a practical way the constraints associated to the fixed-node approximation.",
"For instance, it has been demonstrated that the use of backflow wave functions in homogeneous electron systems reduces significantly the corresponding VMC and DMC energies (Kwon, Ceperley, and Martin, 1993; Kwon, Ceperley, and Martin, 1998; López-Ríos et al., 2006).",
"In the last decade, important methodological progress has been made in the context of DFT calculations that allow now to describe the electronic features of many materials adequately.",
"Examples of these advancements are explained in Sec.",
"REF and essentially are related to the construction of accurate and computationally efficient hybrid exchange and dispersion-corrected functionals.",
"A pending challenge in DFT methods, however, is posed by the difficulties encountered in the reproduction of many-body and Coulomb screening effects.",
"This type of shortcomings stems from the pairwise additivity that is assumed in the construction of most DFT functionals.",
"Essentially, the interaction energy between two atoms completely neglects the effects introduced by the medium that separates them (Misquitta et al., 2010; Tkatchenko, Alfè, and Kim, 2012; Gobre and Tkatchenko, 2013).",
"In this context, the adiabatic connection fluctuation-dissipation theorem has been exploited to calculate correlation DFT energies that incorporate many-body terms beyond pairwise.",
"This is the case of the random phase approximation to DFT (Dobson, White, and Rubio, 2006) and DFT+MBD methods (Ruiz et al., 2012; Tkatchenko et al., 2012; Ambrosetti et al., 2014), which at the moment are receiving the highest attention.",
"In the latest DFT+MBD versions, for instance, the Schrödinger equation of a set of fluctuating and interacting quantum harmonic oscillators is solved directly within the dipole approximation, and the resulting many-body energy is coupled to an approximate semilocal DFT functional (Tkatchenko et al., 2012; Ambrosetti et al., 2014).",
"Many-body DFT-based methods, however, are still in their infancy and the associated computational expenses are elevated, hence their applicability yet is limited.",
"Electronic QMC methods, on the other hand, are inherently exact as they account for any type of electronic correlation, exchange, or many-body screening effect (although they are affected by the “sign problem” explained in previous sections).",
"A further advantage of using eQMC methods is that it is possible to treat the zero-point motion of the nuclei beyond the Born-Oppenheimer approximation, that is, considering non-adiabatic effects (Ceperley and Alder, 1987; Tubman et al., 2014; Yang et al., 2015).",
"This can be done by using wave functions that explicitly depend on both the electronic and nuclear degrees of freedom in projector MC schemes (e. g., DMC and GFMC).",
"Another interesting feature of eQMC, in contraposition to DFT methods, is that in the case of light atoms the use of pseudopotentials can be avoided; this aspect is specially desirable for the study of quatum solids like H$_{2}$ and $^{4}$ He, since core electrons then can be simulated without any constraints (Morales, Pierleoni, and Ceperely, 2009; Morales et al., 2013).",
"On the down side, eQMC methods present some technical difficulties that are absent in DFT calculations.",
"For instance, the periodic Ewald sum that is used to estimate the electron-electron interactions introduces a finite-size error in the exchange-correlation energy, since it depends on the size and shape of the simulation cell (Foulkes et al., 2001).",
"Consequently, the use of either increasingly large simulation cells or effective correction schemes (Fraser et al., 1996; Hood et al., 1997; Chiesa et al., 2006) is necessary to guarantee proper convergence.",
"For a detailed description of finite-size errors treatment in eQMC methods we refer the interested reader to the recent and specialised articles by Drummondet al.",
"(2008), Ma, Zhang, and Krakauer (2011), and Holzmann et al.",
"(2016).",
"Another intricacy is found in the calculation of the atomic forces.",
"Calculating forces using a stochastic algorithm turns out to be very difficult because straightforward derivation of the total energy with respect to the atomic positions, as it follows from the Hellmann-Feynman principle, leads to estimators with very large variances.",
"Correlated sampling techniques have been proposed to make the statistical errors in the relative energy of different geometries much smaller than the errors in the separate energies (Filippi and Umrigar, 2000).",
"Finite difference methods, however, become already impractical when considering systems containing a few tens of atoms.",
"Alternative approaches based on “zero-variance” Hellmann-Feynman estimators (Assaraf and Caffarel, 2000; Chiesa, Ceperley, and Zhang, 2005; Per, Russo, and Snook, 2008; Clay III et al., 2016) and sampling of “pure” probability distributions (Badinski et al., 2010) have been introduced more recently; nevertheless, the central problem of calculating accurate forces in extended systems efficiently yet remains.",
"The great accuracy of eQMC methods neither comes free of cost.",
"Although the scaling with respect to the number of electrons is the same than in DFT methods, namely $N^{3}$ in standard cases, the pre-factors in eQMC are considerably larger (e. g., roughly 10 and 100 times larger in VMC and DMC, respectively; Foulkes et al., 2001; Towler, 2006).",
"Also, the convergence of the total energy is achieved more slowly than in DFT due to the usual MC propagation and sampling procedures.",
"In spite of this, thanks to the escalating increase in computing efficiency and recent algorithmic advances, the use of eQMC methods is transitioning from that of benchmark calculations in few-atoms systems to that of production runs in hundreds-of-atoms systems (Kim et al., 2012; Esler et al., 2012; Wagner, 2014).",
"Actually, efficient QMC-based methods have been already developed that allow to simulate both the electrons and nuclei in crystals quantum mechanically (Grossman and Mitas, 2005; Wagner and Grossman, 2010).",
"Among those, we highlight the coupled electron-ion Monte Carlo method due to Pierleoni, Ceperley, and collaborators (Pierleoni, Ceperley, and Holzmann, 2004; Pierleoni and Ceperley, 2005; Pierleoni and Ceperley, 2006), for its special relevance to the field of quantum solids (see, for instance, Sec.",
"REF ).",
"In view of this progress, we foresee that in the next years the use of eQMC techniques will become more popular within the community of computational condensed matter scientists."
],
[
"Effective interaction models",
"Using first-principles methods to describe the interactions between atoms in quantum crystals normally requires intensive computational resources.",
"Fortunately, the interactions between particles sometimes are so simple that they can be approximated with analytical functions known as classical interatomic potentials or force fields.",
"In those particular cases one can concentrate in solving the quantum mechanical equations for the nuclear degrees of freedom only, hence accelerating the calculations dramatically.",
"Classical interaction models are constructed by following physical knowledge and intuition; they normally contain a set of parameters that are adjusted to reproduce experimental or ab initio data.",
"The force matching method due to Ercolessi and Adams (1994), for instance, is a well-established force field fitting technique that is widely employed in computational physics and materials science (Masia, Guàrdia, and Nicolini, 2014).",
"Nevertheless, the ways in which classical interatomic potentials are constructed are neither straightforward nor uniquely defined, and the thermodynamic intervals over which they remain reliable are not known a priori.",
"In situations where the use of first-principles methods is prohibitive and the available classical potentials are not versatile enough to reproduce the physical phenomena of interest, machine learning techniques can be very useful.",
"Machine learning (ML) is a subfield of artificial intelligence that exploits the systematic identification of correlation in data sets, to make predictions and analysis (Behler, 2010; Rupp, 2015).",
"Effective potentials resulting from ML are not built on physically motivated functional forms but obtained from purely mathematical fitting techniques that reproduce a set of reference data as closely as possible.",
"Some of these fitting procedures strongly rely on the concept of artificial neural networks, which can “learn” the topology of a potential-energy surface from a set of reference points.",
"ML techniques are common tools in mathematics and computer science, and are starting to be applied with confidence in Chemistry (Raghunathan et al., 2015) and Physics (Manzhos, Yamashita, and Carrington, 2009; Li, Kermode, and De Vita, 2015)."
],
[
"Classical potentials",
"The interactions between atoms in quantum solids have been traditionally modeled with two-body potentials.",
"The most popular of all them is the Lennard-Jones (LJ) potential, which is expressed as: $V_{2}^{\\rm LJ} (r) = 4 \\epsilon \\left[ \\left( \\frac{\\sigma }{r}\\right)^{12} - \\left(\\frac{\\sigma }{r}\\right)^{6} \\right]~,$ where $\\epsilon $ and $\\sigma $ are free parameters, and $r$ is the distance between two particles.",
"The first term in Eq.",
"(REF ) represents repulsive short-ranged electrostatic and Pauli-like interactions acting between electrons; the second term represents the attractive long-ranged van der Waals interactions resulting from instantaneous electronic dipoles.",
"In spite of its simplicity, the LJ potential has been used in the study of condensed matter systems with great success; it was the first interaction model to be systematically employed in variational Monte Carlo simulations of quantum solids (Hansen and Levesque, 1968; Hansen, 1968; Bruce, 1972).",
"The LJ potential is convenient also for simulating atomic systems composed of several chemical species for which the corresponding $\\sigma $ 's and $\\epsilon $ 's are already known; the resulting crossed interactions then can be approximated to a good extent with the LJ parameters given by the Lorentz-Berthelot rules: $\\sigma _{ij} = \\frac{\\sigma _{ii} + \\sigma _{jj}}{2}$ and $\\epsilon _{ij} = \\sqrt{\\epsilon _{ii}\\epsilon _{jj}}$ .",
"When two atoms are brought together, however, the value of the repulsive LJ term in general increases too rapidly.",
"In quantum solids particles can be close to each other due to their zero point motion, hence an accurate description of the atomic interactions at short distances is necessary even at low densities (Ceperley and Partridge, 1986; Boronat and Casulleras, 1994).",
"In this context, the pairwise interaction model originally proposed by Ahlrichs, Penco and Scoles (1977) is more appropriate since it reproduces ab initio results for the repulsive interactions between closed shell atoms (Hepburn, Scoles, and Penco, 1975).",
"The form of this potential is: $V_{2}^{\\rm Aziz} (r) = A e^{-a r + b r^{2}} -f(r) \\sum _{i=6,8,10} \\frac{C_{i}}{r^{i}}~,$ where $A$ , $a$ , $b$ , and $C_{i}$ are free parameters, and $f(r)$ is an exponential damping function that is introduced to avoid the divergence of the $1/r^{n}$ terms at small distances.",
"Aziz and collaborators worked extensively in this model to deliver an accurate description of the atomic interactions in many rare-gas systems (Aziz et al., 1979; Aziz, Meath, and Allnatt, 1983; Aziz, McCourt, and Wong, 1989), hence the notation employed.",
"Equation (REF ) also yields an improved description of the long-range dispersion forces as compared to the LJ model, since it contains several types of multi-pole interactions.",
"In some situations, a well balanced description of solids cannot be attained with pairwise potentials only.",
"This is the case, for instance, of crystals at extreme thermodynamic conditions (Loubeyre, 1987; Cazorla and Boronat, 2008a; Cazorla and Errandonea, 2014; Cazorla and Boronat, 2015b).",
"A possible solution to overcome this modeling difficulty is to go beyond pairwise additivity, that is, to consider higher order terms in the approximation to the atomic interactions.",
"Several three-body interatomic potentials have been proposed in the literature (Axilrod and Teller, 1943; Cencek, Patkowski, and Szalewicz, 2009), and the most popular in the context of quantum solids is (Bruch and McGee, 1973): $V_{3} \\left({\\bf x}, {\\bf y}, {\\bf z} \\right) &&= \\frac{\\nu }{x^{3} y^{3} z^{3}} - B e^{-c ( x + y + z )} \\nonumber \\\\&& \\times \\left( 1 + 3 \\cos {\\alpha } \\cos {\\beta } \\cos {\\gamma } \\right)~,$ where $\\nu $ , $B$ , $c$ are free parameters, $\\lbrace x,y,z \\rbrace $ the distance between particles in a trimer, and $\\lbrace \\alpha ,\\beta ,\\gamma \\rbrace $ the corresponding interior angles.",
"$V_{3}$ is an interatomic potential that renders triple dipole and exchange interactions; inclusion of this type of forces appears to be necessary for obtaining a realistic description of the energy and elastic properties of very dense quantum solids (Grimsditch, Loubeyre, and Polian, 1986; Pechenik, Kelson, and Makov, 2008; Cazorla and Boronat, 2015b).",
"Figure: (Color online) Sketch of the key idea behind machine learning for the modeling ofatomic interactions.",
"Ab initio results, which are obtainedat high computational cost, are approximated by interpolating withML potentials between selected reference configurations (blue dots)."
],
[
"Machine learning",
"When calculations are performed in a series of similar systems or a number of configurations involving a same system, the results contain redundant information.",
"One example is to run a molecular dynamics simulation in which the total internal energy and atomic forces are calculated at each time step; after a sufficiently long time, points which are close in configurational space and have similar energies are visited during the sampling of the potential-energy surface.",
"Such a redundancy can be exploited to perform computationally intensive calculations (that is, of first-principles type) only in few selected configurations and to use machine learning (ML) to interpolate between those, obtaining so approximate solutions for the remaining of configurations (see Fig.",
"REF ).",
"The success of this approach depends on a balance between incurred errors due to interpolation and invested computational effort.",
"ML modeling tools can provide both the energy and atomic forces directly from the atomic positions, hence they can be regarded as a particular class of atomistic potentials.",
"ML potentials, however, rely on very flexible analytic functions rather than on physically motivated functionals.",
"Promising analytic approaches that have been recently proposed to construct ML potentials include permutation invariant polynomials (Brown et al., 2003), the modified Shepard method using Taylor expansions (Bowman, Czakó, and Fu, 2011), Gaussian processes (Bartók et al., 2010; Bartók et al., 2013), interpolating moving least squares (Dawes et al., 2007), and artificial neural networks (Lorenz, Groß, and Scheffler, 2004).",
"Artificial neural networks, for instance, have been demonstrated to be “universal approximators” (Behler, 2015) since they allow to approximate unknown multidimensional functions to within arbitrary accuracy given a set of known function values.",
"To the best of our knowledge, ML potentials have not been applied yet to the study of quantum solids.",
"However, the great versatility of ML approaches (Behler, 2010; Rupp, 2015; Behler, 2015) could be exploited to describe such systems in specially challenging situations like, for instance, molecular solids (e. g., H$_{2}$ , N$_{2}$ , and CH$_{4}$ ) under extreme thermodynamic conditions.",
"Classical interaction models normally disregard the orientational degrees of freedom in molecules and require the specification of bond connectivity between atoms.",
"Therefore, they are not able to describe the orientational phase transitions and breaking/formation of atomic bonds occurring at high-$P$ and high-$T$ conditions (see Sec. ).",
"ML potentials could represent an intermediate solution between classical potentials and first-principles methods, both in terms of numerical accuracy and computational burden."
],
[
"Archetypal quantum crystals",
"Helium and hydrogen are the lightest elements in Nature and the paradigm of quantum solids.",
"The classical picture of a crystal at low temperature, with all the atoms strongly localised around their equilibrium lattice positions, breaks completely in solid helium and hydrogen.",
"In archetypal quantum crystals atoms move noticeably around the equilibrium lattice positions even in the limit of zero temperature, and exchanges between few particles occur with frequency.",
"Consequently, the degree of anharmonicity in these systems is very high.",
"Quantum simulation methods beyond the harmonic approximation (see Sec. )",
"in fact are necessary for describing archetypal quantum solids correctly."
],
[
"Helium",
"Helium has two stable isotopes, $^4$ He and $^3$ He, which are a bosonic and a fermionic particles, respectively.",
"Both isotopes solidify under moderate pressures in the $T \\rightarrow 0$ limit, namely at $P \\simeq 25$ bar in $^4$ He and 30 bar in $^3$ He.",
"$^4$ He solidifies in the hexagonal hcp phase except for a small region at low pressures in which the stable phase is cubic bcc (see Fig.",
"REF ).",
"Meanwhile, $^3$ He solidifies in the cubic bcc phase with a relatively large molar volume of $V \\simeq 24.5$ cm$^3$ mol$^{-1}$ .",
"Under specific $P-T$ conditions, both isotopes transform into the cubic fcc phase (Glyde, 1994).",
"Figure: Phase diagram of 4 ^4He at low pressures and temperatures.At low pressure ($P < 1$ GPa), the thermodynamic properties of solid $^{4}$ He are well known from experiments and accurately reproduced by QMC methods.",
"In Fig.",
"REF , we compare experimental and computational results for the dependence of the energy per particle on density.",
"The theoretical results correspond to DMC simulations performed by Vranješ et al.",
"(2005) using a semi-empirical pair potential (Aziz, McCourt, and Wong, 1987); the agreement between observations and theory is excellent.",
"Likewise, accurate results have been obtained also with the PIGS method (Rossi et al., 2012).",
"From function $E/N(\\rho )$ one can easily work out the pressure, $P(\\rho )=\\rho ^2d(E/N)/d \\rho $ , obtaining so the corresponding equation of state (e.o.s.).",
"Excellent agreement between theory and experiment has been demonstrated also for this quantity [see, for instance, Cazorla and Boronat (2008a)].",
"The quantum nature of solid $^4$ He is thermodynamically reflected on its high compressibility; for instance, the corresponding experimental molar volume is reduced from 21 cm$^3$ mol$^{-1}$ at 25 bar to 9 cm$^3$ mol$^{-1}$ at 5 kbar.",
"The location of the first-order liquid-solid phase transition also is accurately reproduced by QMC calculations.",
"Recent DMC estimations provide a transition pressure of $27.3$ atm, with freezing and melting densities equal to $\\rho _f = 0.437~\\sigma ^{-3}$ and $\\rho _m = 0.481~\\sigma ^{-3}$ ($\\sigma =2.556$ Å), respectively (Vranješ et al., 2005).",
"The corresponding experimental values are, $P_{t}^{\\rm expt} = 25$ atm, $\\rho _f^{\\rm expt} = 0.434~\\sigma ^{-3}$ , and $\\rho _m^{\\rm expt} = 0.479~\\sigma ^{-3}$ (Glyde, 1994).",
"Figure: Energy per particle in solid 4 ^4He expressed as a function ofdensity.",
"Open triangles represent experimental data from Edwards and Pandorf(1965); other symbols and lines correspond to the DMC results.",
"Solid circlesand line represent results in which the bias introduced by finite size effectshas been reduced significantly.",
"Solid triangles and the dashed line representresults which have been corrected only partially for the same type of bias.Adapted from Vranješ et al.",
"(2005).Valuable information on the quantum nature of a solid is obtained from its Lindemann ratio: $\\gamma = \\frac{1}{a} \\left[ \\frac{1}{N} \\sum _{i=1}^N ({\\mathbf {r}}_i-{\\mathbf {R}}_i)^2\\right]=\\frac{\\langle {\\mathbf {u}}^2 \\rangle ^{1/2}}{a}~,$ where ${\\mathbf {R}}_i $ represent the coordinates of the perfect lattice sites, and $a$ the corresponding lattice constant.",
"Parameter $\\gamma $ quantifies the displacement of particles around their equilibrium positions.",
"The quantum character of a solid can be said to be proportional to the value of its Lindemann ratio.",
"In solid $^{4}$ He and $^{3}$ He at ultra-low temperatures, for instance, $\\gamma $ amounts to $\\sim 0.3$ (Glyde, 1994), which are the largest values known in any material at those thermodynamic conditions.",
"The large excursions of helium atoms around their lattice positions allow them to explore the non-harmonic part of the potential energy surface, leading to high anharmonicity.",
"Another singular aspect in solid helium is the large kinetic energy per particle.",
"At $P = 50$ atm, for instance, $E_{\\rm k}$ amounts to $\\sim 24$ K (Diallo et al., 2007), which is of the same order of magnitude than the corresponding potential energy, namely $\\sim -31$ K (resulting from a cancellation between large repulsive and attractive terms).",
"Figure: One-body density matrix calculated in hcp 4 ^4Heat density ρ=0.0294\\rho = 0.0294 Å -3 ^{-3}.",
"Circles, squares, anddiamonds stand for results obtained at T=0T = 0, 1, and 2 K,respectively.",
"Adapted from Rota and Boronat (2012).The influence of Bose-Einstein statistics on the energy and structural properties of solid $^{4}$ He is negligible [$\\sim 1~\\mu $ K/atom, Clark and Ceperley (2006)].",
"In fact, many of the results just presented have been obtained with non-symmetric wave functions and as it has been explained the agreement with the experiments is excellent.",
"However, quantum atomic exchanges play a pivotal role in other intriguing properties like, for instance, Bose-Einstein condensation and superfluidity (Ceperley, 1995).",
"These phenomena occur in liquid $^{4}$ He at ultra-low temperatures and, due to the extreme quantum nature of helium, it was wondered long time ago whether the same effects could be observed also in the crystal phase (see Sec.",
"for a historical overview of this topic).",
"In recent years, there have been several theoretical works aimed at clarifying these questions.",
"In particular, the one-body density matrix, $\\varrho _{1}(r)$ [Eq.",
"(REF )], of solid $^{4}$ He has been calculated with different methods.",
"Initial zero-temperature estimations based on symmetrized wave functions (Cazorla et al., 2009; Galli and Reatto, 2006) provided a non-zero but small plateau at long distances.",
"However, unbiased $\\varrho _{1}(r)$ results obtained with the PIGS and PIMC methods have unequivocally demonstrated that the condensate fraction in perfect solid $^{4}$ He is actually zero (Ceperley and Bernu, 2004; Bernu and Ceperley, 2005; Clark and Ceperley, 2006; Boninsegni, Prokof'ev, and Svistunov, 2006b).",
"In particular, the tail of $\\varrho _{1}(r)$ decays exponentially at long distances, as it is illustrated in Fig.",
"REF (Rota and Boronat, 2012).",
"Actually, the exchange frequency between particles at different lattice sites is very small as compared to that in the liquid phase [for instance, the exchange frequency for 2, 3, and 4 atom exchanges is of $\\sim 3~\\mu $ K/atom, Ceperley and Bernu (2004); Clark and Ceperley (2006)], and long permutation cycles able to trigger superfluidity are highly improbable.",
"Nevertheless, we note that when a finite and stable concentration of defects is assumed to exist in the crystal these conclusions change drastically (see Sec. ).",
"Figure: Zero-temperature equation of state of hcp 4 ^4He at highpressures calculated with several effective three-body interactionmodels fitted to reproduce ab initio data, and the DFT-Dmethod (see text).",
"Experimental data from Loubeyre et al.",
"(1993)are shown for comparison.",
"Inset: P(V)P(V) curves in the high-PPregion are magnified in order to appreciate better the differences.From Cazorla and Boronat (2015b).At high pressures, the atoms in a crystal experience strong short-range repulsions due to electrostatic forces and the Pauli exclusion principle.",
"Customary semi-empirical potentials that at low densities provide a good description of the crystal, start then to be unreliable due to severe transferability issues (see Sec.",
"REF ).",
"This is the case of the Aziz potential for $^4$ He (Aziz, McCourt, and Wong, 1987), which possesses a too steep repulsive core and leads to inaccurate results at pressures $P \\ge 1$ GPa (Cazorla and Boronat, 2008a).",
"Recently, dispersion-corrected density functional theory (DFT) has been used in combination with the DMC method to study the quantum behavior of solid helium at pressures up to $\\sim 150$ GPa (Cazorla and Boronat, 2015a; Cazorla and Boronat, 2015b).",
"Essentially, analytical potentials have been constructed to reproduce sets of atomic energies and forces calculated with first-principles methods.",
"In a first approximation (Cazorla and Boronat, 2015a), the effective pair interaction has been obtained by fitting the static compression curve calculated with DFT-D to an analytical function based on the Aziz potential [see Eq.",
"(REF )] and an attenuation repulsion factor proposed by Moraldi (Moraldi, 2012).",
"This has allowed for a sizable improvement in the description of the high-$P$ e.o.s.",
"as compared to the available experimental data.",
"However, it has been shown that such a simple approach provides unphysical results for the elastic constants and pressure dependence of the kinetic energy.",
"In a posterior work, Cazorla and Boronat (2015b) have introduced a family of three-body interaction potentials based on Eq.",
"(REF ) that allow to overcome (in part) these modeling shortcomings while still providing an accurate e.o.s.",
"up to $\\sim 60$ GPa (see Fig.",
"REF ).",
"With regard to solid $^3$ He, the number of related studies is very limited.",
"Besides some old variational calculations, the most recent and accurate investigation of its thermodynamic properties has been performed by Moroni et al.",
"(2000) with the DMC method.",
"In Moroni et al.",
"'s (2000) work the quantum antisymmetry of the system is neglected, that is, particles are treated as bosons rather than as fermions.",
"Nevertheless, since the exchange energy in the crystal is very small (of the order of mK, see Ceperley and Jacucci, 1987; C${\\rm \\hat{a}}$ ndido, Hai, and Ceperley, 2011) it can be expected that quantum symmetry effects will play an insignificant role on the energy.",
"The results obtained for the dependence of the energy on density show a discrepancy with the experimental data, which Moroni et al.",
"(2000) have attributed to a wrong reference in the integration of the experimental equation of state.",
"After correction of such an error, the agreement between theory and experiments becomes excellent, namely, of the same quality than achieved in solid $^4$ He."
],
[
"Hydrogen",
"Bulk molecular hydrogen (deuterium) at zero pressure, in contrast to $^{4}$ He, solidifies at a temperature of $\\sim 14$ K ($\\sim 19$ K) due to the stronger attractive interactions between particles.",
"${\\rm H_{2}}$ (${\\rm D_{2}}$ ) molecules are composed of two hydrogen (deuterium) atoms joined by a covalent bond, which in the para-hydrogen (ortho-deuterium) state have zero angular momentum and spherically symmetric wave functions.",
"Both types of particles, therefore, are bosons and the interactions between molecules of the same species can be modelled with radial pairwise potentials (at high pressures, however, the molecular angular momentum is not longer zero and thereby pairwise approximations to the intermolecular interactions become invalid, see Sec.",
"REF ).",
"Actually, in most quantum simulation studies of ${\\rm H_{2}}$ and ${\\rm D_{2}}$ crystals at low pressure (i. e., $P \\le 0.1$ GPa) the intermolecular forces have been modelled with the semi-empirical Silvera-Goldman (Silvera and Goldman, 1978) and Buck (Buck et al., 1983; Norman et al., 1984) pair potentials.",
"The role of three-body forces on the corresponding low-$P$ equation of state has been explored, but their net effects have been found to be negligible (Operetto and Pederiva, 2006).",
"Meanwhile, anisotropic corrections to the pair potential have been tested against experiments and found to be significant only at pressures higher than $\\sim 10$ GPa (Cui et al., 1997).",
"Figure: DMC energy per H 2 _2 molecule as a function of density.",
"Squaresand circles correspond to the liquid and solid phases, respectively.",
"Solidand dashed lines are polynomial fits to the DMC energies for the liquidand solid phases, respectively.",
"The diamond represents the experimental energy ofhcp molecular hydrogen from Schnepp (1970).",
"From Osychenko, Rota, and Boronat(2012).Figure: Equation of state of liquid (solid line) and solid (dashed line) H 2 _2.Experimental points for the solid phase are represented with solid circles(Driessen, de Waal, and Silvera, 1979).",
"From Osychenko, Rota, and Boronat (2012).In Fig.",
"REF , we show the energy per molecule in hexagonal hcp H$_2$ calculated in the limit of zero temperature with the DMC method and the Silvera-Goldman potential (Osychenko, Rota, and Boronat, 2012).",
"We note that the experimental energy per particle is $E/N=-89.9$ K, which is underestimated (overestimated) by the Silvera-Goldman (Buck) potential model.",
"Close to the equilibrium density the H$_2$ kinetic energy is $89.5$ K, which roughly amounts to half of the potential energy.",
"In comparison to solid $^4$ He, in which both types of energies are nearly equal, quantum nuclear effects in solid hydrogen turn out to be smaller (see also Fig.",
"REF ).",
"The energy curve of the metastable liquid is shown in Fig.",
"REF for comparison.",
"Results for the equation of state of solid and liquid H$_2$ are enclosed in Fig.",
"REF .",
"The agreement between theory and experiments (Driessen, de Waal, and Silvera, 1979) is quite satisfactory in the solid phase, although at pressures beyond $\\sim 100$ MPa this starts to worsen due to the limitations of the employed intermolecular potential (Moraldi, 2012; Omiyinka and Boninsegni, 2013).",
"As a by-product of the recent experimental activity on the search for a supersolid state of matter (see Sec.",
"), an interest has developed in studying highly disordered solids like, for instance, amorphous or glassy systems.",
"A glassy state in solid $^4$ He, termed as “superglass”, has been predicted to exhibit superfluid behavior by Boninsegni, Prokof'ev, and Svistunov (2006b).",
"An analogous study has been carried out more recently in solid H$_2$ by Osychenko, Rota, and Boronat (2012).",
"In this case, PIMC simulations have shown that glassy molecular hydrogen eventually becomes superfluid at temperatures below $\\sim 1$ K. The critical temperature for this transition, however, is so small that it is unlikely to be observed in experiments (Kühnel et al., 2011).",
"Interestingly, the free surface of bulk H$_{2}$ has been also investigated with experiments (Brewer et al., 1990; Vilches, 1992; Kinder, Bouwen, and Schoemaker, 1995) and PIMC simulations (Wagner and Ceperley, 1994; Wagner and Ceperley, 1996).",
"It has been found that the melting temperature of the bare hydrogen surface is reduced down to $\\sim 6$ K and that zero-point molecular fluctuations are considerably enhanced with respect to bulk.",
"For instance, at low temperatures the corresponding Lindemann ratio increases from $\\sim 0.1$ in the inner layers up to $\\sim 0.2$ in the outer surface (Wagner and Ceperley, 1996).",
"Yet, the corresponding melting temperature is still too high to expect that liquid H$_{2}$ will become superfluid (that is, well above the predicted critical temperature $T_{c} \\sim 1-2$ K; Apenko, 1999)."
],
[
"Neon",
"Solid neon behaves more “classically” than solid helium but more “quantumly” than the rest of rare-gas species (see Fig.",
"REF ).",
"The study of this crystal is useful to understand the transition from the quantum regime to the classical one in solid-state systems.",
"The interest in solid neon as a case study of a moderate quantum system dates back to the 1960's.",
"Bernades (1958) and Nosanow and Shaw (1962) were the first to attempt an estimation of the kinetic energy in solid neon using theoretical methods.",
"By relying on variational and self-consistent Hartree calculations performed with uncorrelated single-particle wave functions, they reported ground-state $E_{\\rm k}$ values of $\\sim 41$ K. However, the binding energies reported in those early works were in strong disagreement with contemporary experiments, evidencing the need to go beyond uncorrelated microscopic approaches.",
"Few years later, Koehler (1966) applied the self-consistent phonon approach to the same system and obtained results for the cohesive energy that were in better agreement with the experiments; Koehler's estimation of the kinetic energy was $42.6$ K. Figure: (Color online) Quantum nuclear effects in solid neon at low temperatures.Top panel Excess kinetic energy expressed as a function of temperature.",
"Experimental data from Timms,Simmons, and Mayer (2003) are represented with ; measurements from Peek et al.",
"(1992)with •\\bullet ; PIMC calculations from Timms et al.",
"(1996) with ▴\\blacktriangle ; DMC ground-statecalculations from Cazorla and Boronat (2008b) with ▾\\blacktriangledown .",
"The lines in the plot are linearfits to the experimental data.",
"Bottom panel Ground-state momentum distribution in solid Ne calculatedwith the DMC method (green dots).",
"The solid line (red) is a Gaussian fit to the results, the width of whichrepresents its uncertainty.",
"Adapted from Cazorla and Boronat (2008b).It was not until the 1990's that, with the development of the deep inelastic neutron scattering technique, the kinetic energy in quantum crystals could be measured precisely.",
"Peek et al.",
"(1992) were the first to perform those measurements in solid neon, reporting a ground-state kinetic energy of $49.1~(2.8)$ K. In view of the large discrepancies found with respect to previous estimations based on harmonic models, the authors of that study suggested that solid neon was highly anharmonic.",
"Later on, Timms et al.",
"(1996) carried out a new series of neutron scattering experiments in which higher momentum and energy transfers were considered.",
"They found that in the temperature interval $4-20$ K their excess kinetic energy measurements, defined as $E_{\\rm exc} \\equiv E_{\\rm k} - \\frac{3}{2}k_{\\rm B}T$ , were systematically lower than Peek's results by few Kelvin (see Fig.",
"REF ).",
"The validity of Timms et al.",
"'s results (1996) is supported by the outcomes of several PIMC studies based on classical interatomic potentials (Timms et al., 1996; Cuccoli et al., 2001; Neumann and Zoppi, 2002).",
"More recently, Timms, Simmons, and Mayer (2003) have performed additional neutron scattering measurements and reported that the ground-state kinetic energy in solid neon is $41(2)$ K. The accuracy of this result has been confirmed by recent DMC calculations performed by Cazorla and Boronat (2008b), which provide $E_{\\rm k} = 41.51(6)$ K in the $T \\rightarrow 0$ limit (see Fig.",
"REF ).",
"We note that the Lindemann ratio (see Sec.",
"REF ) in solid neon is approximately three times smaller than in helium, namely $\\gamma _{\\rm Ne} \\sim 0.08$ (Cazorla and Boronat, 2008b), pointing to a moderate degree of quantumness.",
"An interesting topic in the study of solid neon is related to the shape of its momentum distribution, $n({\\bf k})$ (see Fig.",
"REF ).",
"Withers and Glyde (2007) have shown that when a crystal has a momentum distribution that is not well described by a Gaussian, it may be that the system is highly anharmonic or that quantum atomic exchanges occur frequently in it.",
"While quantum atomic exchanges are very likely to be negligible, anharmomic effects seem to be fairly important in solid neon.",
"This has been demonstrated by Cazorla and Boronat (2008b), who have performed different types of harmonic-based calculations and compared their results to those obtained with full anharmonic methods.",
"For instance, the ground-state kinetic energy that is predicted with the self-consistent average phonon approach (Shukla et al., 1981) amounts to $\\sim 47$ K and the corresponding Lindemann ratio to $0.06$ , which are appreciably different from the experimental and DMC results.",
"Nevertheless, there is widespread agreement among experimentalists and theorists in that the momentum distribution in solid Ne is well approximated by a Gaussian function (Timms et al., 1996; Neumann and Zoppi, 2002; Cazorla and Boronat, 2008b).",
"The question then arises about how anharmonic (or quantum) a crystal must be for its $n({\\bf k})$ to differ appreciably from a Gaussian.",
"The momentum distribution in solid $^{4}$ He is non-Gaussian as it has a large occupation of low momentum states as compared to a Maxwell-Boltzmann distribution [Diallo et al., 2004; Rota and Boronat, 2011].",
"However, little is known about the relation between anharmonicity and $n({\\bf k})$ in other quantum crystals lying in between helium and neon, in terms of quantumness, which are mostly molecular systems (see Fig.",
"REF ).",
"In the case of solid para-H$_{2}$ , for instance, inelastic neutron scattering experiments have provided a translational momentum distribution that is Gaussian (Langel et al., 1988); however, it is not clear whether this result can be attributed to the intensive use of Gaussian approximations during the refinement of experimental data (Colognesi et al., 2015).",
"Is it then solid helium the only quantum crystal with a non-Gaussian $n({\\bf k})$ ?",
"The answer, so far, seems to be yes.",
"Free-energy calculations based on path-integral simulations and semi-empirical pairwise potentials have been employed also to study the quantum phase diagram of neon up to pressures of $2-3$ kbar and temperatures of 50 K (Ramírez and Herrero, 2008; Ramírez et al., 2008; Brito and Antonelli, 2012).",
"Significant QNE have been found in the solid-gas and liquid-gas $P-T$ coexistence lines, which consist in a shift of about $1.5$ K towards lower temperatures as compared to the classical phase diagram.",
"Moderate quantum isotopic effects have been observed also in the triple solid-liquid-gas coexistence point both in experiments (Furukawa, 1972) and path-integral calculations (Ramírez and Herrero, 2008)."
],
[
"Elasticity and mechanical properties",
"The free energy of a crystal subjected to a homogeneous elastic deformation is: $F(V, T, \\epsilon ) = F_{0}(V,T) + \\frac{1}{2} V \\sum _{ij} C_{ij} \\epsilon _{i} \\epsilon _{j}~,$ where $F_{0}$ is the free energy of the undeformed solid, $C_{ij}$ the corresponding elastic constants, and $\\epsilon _{i}$ a general strain deformation (the latter two quantities are expressed in Voigt notation and the subscripts indicate Cartesian directions).",
"The symmetry of the crystal determines the number of elastic constants that are inequivalent and non-zero.",
"For a crystal to be dynamically stable, its change in free energy due to an arbitrary strain deformation must be always positive; this requirement leads to a number of mechanical stability conditions that need to be fulfilled at any stable or metastable state, and which depend on the particular symmetry of the crystal (Born and Huang, 1954; Grimvall et al., 2012).",
"The elastic constants of a solid can be measured with ultrasonic techniques since the velocity of density waves depends on the elastic properties of the medium in which they propagate.",
"Brillouin scattering spectroscopy and synchrotron x-ray diffraction techniques can be employed also to this end.",
"Likewise, the calculation of elastic constants with quantum simulation methods is a well established technique.",
"At zero temperature, one can calculate the energy of the solid as a function of strain, using for instance the DMC or PIGS method (see Sec.",
"REF ), and then simply compute the value of its second derivative numerically (Cazorla, Lutsyshyn, and Boronat, 2012; Cazorla and Boronat, 2013).",
"At $T \\ne 0$ , the calculation of $C_{ij}$ 's is not so straightforward since one has to consider also the effects of thermal excitations.",
"Schöffel and Müser (2001) were to first to undertake such a type of calculation by using the path-integral Monte Carlo method.",
"They estimated the elastic constants in solid Ar and $^{3}$ He through direct derivation of the partition function with respect to the strain components.",
"More recently, Pe${\\rm \\tilde{n}}$ a-Ardila, Vitiello, and de Koning (2011) have proposed an alternative path-integral approach in which a suitable expression for the estimation of the stress-tensor is worked out.",
"In this section, we review the elastic properties of perfect quantum solids, that is, free of crystalline defects.",
"Crystalline defects can affect considerably the elastic behavior of quantum (and also classical) crystals, so that we leave those aspects for Sec. .",
"Our analysis here is divided into low and high pressures because the fundamental character of elasticity in quantum crystals changes when moving from one regime to the other.",
"Figure: (Color online) Elastic constants in solid 4 ^{4}He at moderate pressures.Experimental data from Crepeau et al.",
"(1971) and Greywall (1977) are represented with▴\\blacktriangle and ▿\\triangledown , respectively; C 44 C_{44} measurements from Syshchenko, Day, and Beamish(2009) with ▾\\blacktriangledown ; VMC calculations from Pessoa, Vitiello, and de Koning (2010) with ;DMC ground-state calculations from Cazorla, Lutsyshyn, and Boronat (2012) with •\\bullet .",
"The dashedlines (green) in the plots represent linear fits to the DMC results.",
"The freezing pressure in the crystalis marked with a vertical (magenta) line.",
"Adapted from Cazorla, Lutsyshyn, and Boronat (2012)."
],
[
"Low-pressure regime",
"The elastic properties of traditional quantum solids like helium (Crepeau et al., 1971; Greywall, 1977) and hydrogen (Nielsen and Møller, 1971; Wanner and Meyer, 1973; Nielsen, 1973), have been measured extensively.",
"In experiments, however, it is difficult to determine the exact contribution of quantum nuclear effects (QNE) to elasticity.",
"In this context, the outcomes of first-principles studies can be very valuable.",
"For instance, Schöffel and Müser (2001) performed a thorough PIMC study on the elastic properties of solid $^{3}$ He in the hexagonal hcp and cubic bcc and fcc phases, considering low temperatures and pressures.",
"Their results were in good agreement with the reported experimental data, and they concluded that QNE accounted for about 30 % of the $C_{ij}$ values.",
"A similar quantum influence on the elasticity of solid Ar was also reported (namely, $\\sim 20$ %), a crystal that is considered to behave much more classically than helium.",
"More recently, the elastic properties of solid $^{4}$ He have been studied in detail by several authors using different QMC techniques.",
"Cazorla, Lutsyshyn, and Boronat (2012) have employed the DMC method to calculate the zero-temperature elastic constants, Grüneisen parameters, sound velocities, and Debye temperature over a wide pressure interval of $\\sim 100$ bar.",
"The computed $C_{ij}$ values are in overall good agreement (i. e., discrepancies to less than 5 % in most cases) with the reported experimental data and results obtained by other authors using the VMC (Pessoa, Vitiello, and de Koning, 2010; Pessoa, de Koning, and Vitiello, 2013) and PIMC (Pe${\\rm \\tilde{n}}$ a-Ardila, Vitiello, and de Koning, 2011) methods (see Fig.",
"REF ).",
"It has been found that the pressure dependence of all five elastic costants close to equilibrium is practically linear (see Fig.",
"REF ).",
"Interestingly, the contribution of QNE to the elastic constants in hcp $^{4}$ He has been shown to be $\\sim 30$ %, which roughly coincides with the results obtained by Schöffel and Müser (2001) in solid $^{3}$ He.",
"In essence, all these theoretical studies conclude that QNE profoundly affect the elastic properties of quantum crystals at low pressures (that is, $P \\le 0.01$ GPa).",
"A fundamental question that can be easily addressed with simulations but not with experiments is: what is the limit of mechanical stability in a quantum crystal?",
"When the density in a system is reduced progressively, eventually this becomes unstable against long wavelength density fluctuations.",
"This limit, also known as the spinodal point, has been analysed comprehensively in liquid $^{4}$ He and $^{3}$ He (Boronat, Casulleras, and Navarro, 1994; Maris, 1995; Maris and Edwards, 2002); however, it has not been until recently that has been estimated directly in the crystal phase (Cazorla and Boronat, 2015c).",
"Theoretically, the spinodal point in a crystal is identified with the thermodynamic state at which any of the mechanical stability conditions involving the elastic constants is not satisfied.",
"One can expect that, due to the presence of QNE and inherent structural softness, the limit of mechanical stability in quantum crystals lies very low in density.",
"Based on $C_{ij}$ calculations performed with the DMC method and a semi-empirical pairwise potential, Cazorla and Boronat (2015c) have estimated that the ground-state spinodal pressure in solid $^{4}$ He is $P_{s} = -33.8(1)$ bar, which corresponds to a volume of $V_{s} = 50.81(5)$ Å$^{3}$ .",
"In particular, it has been found that the mechanical stability condition $(C_{33} - P)(C_{11} + C_{12}) - 2(C_{13} + P)^{2} > 0$ is violated at $P_{s}$ .",
"Regarding the propagation of density waves, previous calculations based on phenomenological models (Maris, 2009; Maris, 2010) had suggested that, in analogy to the liquid phase, the sound velocities in hcp $^{4}$ He near the spinodal density could follow a power law of the form $\\propto (P - P_{s})^{1/3}$ .",
"However, Cazorla and Boronat (2015c) have shown that quantum solids and liquids behave radically different in the vicinity of their mechanical stability limits; in particular, none of the sound velocity components, either propagating along the $c$ -axis or in the basal plane, follow the previously proposed “1/3” power law."
],
[
"High-pressure regime",
"The elastic properties of archetypal quantum solids under high pressure (i. e., $P > 1$ GPa) have been thoroughly investigated with experiments (Zha et al., 1993; Zha, Mao, and Hemley, 2004).",
"Surprisingly, the results of first-principles DFT studies in which QNE are completely or partially neglected, show very good agreement with the measured $C_{ij}$ and sound velocity data (Nabi et al., 2005; Freiman et al., 2013; Grechnev et al., 2015).",
"In view of the importance of QNE on the elastic properties of quantum crystals at low pressures (see previous section), such a good agreement could be explained in terms of (i) a systematic error cancellation involving the disregard of QNE, on one hand, and an inaccurate description of the system provided by standard DFT functionals, on the other, or (ii) a steady diminishing of the importance of QNE on elasticity as pressure is increased.",
"To the best of our knowledge, there are not fully ab initio studies (i. e., works in which both the electronic and ionic degrees of freedom are described with quantum mechanical methods) on the elastic properties of highly compressed quantum crystals.",
"The reason for this is likely to be the large computational expense involved in the calculation of partition function derivatives or the stress-tensor with sufficient accuracy (Schöffel and Müser, 2001; Pe${\\rm \\tilde{n}}$ a-Ardila, Vitiello, and de Koning, 2011).",
"On the other hand, the semi-empirical two-body potentials that at low pressures describe the interactions between atoms in quantum crystals correctly, become unreliable at high pressures (see Sec. ).",
"In addition to this, pairwise interaction models are in general not well suited for the study of elasticity in very dense crystals since they inevitably lead to zero values of the Cauchy relations (Wallace, 1972; Pechenik, Kelson, and Makov, 2008), which is in contrast to the observations (Zha, Mao, and Hemley, 2004).",
"An estimation of the importance of QNE in the elasticity of quantum crystals using such unrealistic interatomic potentials, therefore, could be misleading.",
"Cazorla and Boronat (2015b) have recently introduced a set of effective three-body potentials based on Eq.",
"(REF ), to simulate solid $^{4}$ He at high pressures realistically and with affordable computational effort (see Sec.",
"REF ).",
"The new parametrisations have been obtained from fits to ab initio energies and atomic forces calculated with a dispersion-corrected DFT functional (see Sec.",
"REF ).",
"It has been shown that an overall improvement in the description of $^{4}$ He elasticity at zero temperature and pressures $0 \\le P \\le 25$ GPa can be achieved with some of the proposed three-body interaction models.",
"Interestingly, Cazorla and Boronat (2015b) have found that the role of QNE on helium elastic constants becomes secondary at very large densities.",
"For instance, the inclusion of QNE makes the value of the shear modulus, $C_{44}$ , to decrease by less than 4 % at a pressure of $\\sim 50$ GPa (to be compared with $\\sim 30$ % found near equilibrium conditions).",
"This conclusion appears to be consistent with the results of a recent PIMC study performed by Landinez-Borda, Cai, and de Koning (2014), in which the ideal shear strength on the basal plane of solid helium (that is, the maximum stress that the crystal can resist without yielding irreversibly) has been found to behave quite similarly to that in classical hcp solids.",
"Consequently, it can be expected that by treating QNE with approximate methods (e. g., the Debye model or quasi-harmonic approximations) one can obtain a reasonably good description of elasticity in quantum solids at high pressures.",
"It will be very interesting to test whether this is also valid in quantum molecular crystals, the elastic properties of which remain largely unexplored at high densities."
],
[
"Crystalline defects",
"A crystal is characterised by a periodic arrangement of atoms or molecules defined by an unit cell.",
"The regular pattern in a solid, however, normally is interrupted by crystalline defects, which can be classified into point, line, and planar types.",
"Point defects occur only at or around one lattice site, and typical examples include vacancies, impurities, and interstitials (Kittel, 2005; see Fig.",
"REF ).",
"Line defects entail entire rows of atoms that are misaligned with respect to the others; common examples of line defects are dislocations, which in turn are classified into “edge” and “screw” (Bulatov and Cai, 2006; Hull and Bacon, 2011).",
"An edge dislocation, for instance, is created by introducing an extra half-plane of atoms in mid-way through the crystal (see Fig.",
"REF ).",
"Planar defects can occur in single crystals or in the boundaries between single crystals, and include grain and twin boundaries, steps, and stacking faults (Kittel, 2005).",
"The study of crystalline defects is very important since these can affect considerably the mechanical, electrical, structural, and adsorption properties of materials.",
"Dislocations, for example, are key to understand the microscopic origins of plasticity, that is, the regime in which mechanical deformations in a crystal become non-reversible (Kosevich, 2005).",
"In 1969, Andreev and Lifshitz proposed that a state of matter in which crystalline order and Bose-Einstein condensation coexisted could occur in a quantum crystal, the so-called supersolid.",
"For this supersolid to exist, the presence of crystal vacancies was a necessary condition.",
"At that time some experimentalists got attracted by the possibility of realising such an exotic state of matter, and several mass flow and torsional oscillator experiments were carried out in solid $^{4}$ He at ultra-low temperatures.",
"In all those experiments, however, no evidence for a supersolid state was found.",
"Several decades later, a renewed interest in supersolids blossomed after the torsional oscillator experiments performed by Kim and Chan (2004a, 2004b).",
"In their experiments, Kim and Chan observed a shift in the period of the torsional oscillator in solid helium as the temperature was lowered below $\\sim 0.1$ K. This sign was interpreted as the mass decoupling between the normal and superfluid fractions in the crystal.",
"Meanwhile, few years later Day and Beamish (2007) measured the shear modulus in solid helium and found a striking resemblance with the temperature dependence of the oscillation period reported by Kim and Chan: the shear modulus increased as the temperature was lowered below $\\sim 0.1$ K. Day and Beamish attributed that increase in stiffness to the temperature dependence of the mobility of dislocations in the solid, which could be pinned by static $^{3}$ He impurities.",
"Day and Beamish's findings motivated a series of subsequent theoretical and experimental studies which have demonstrated that a change in the moment of inertia of the experimental torsional cell can be correlated to a change in the structure of the solid inside of it (Reppy, 2010; Maris, 2012; Shin et al., 2016).",
"In 2012, Kim and Chan completely redesigned their torsional oscillator setup making it stiffer, and the original mass-decoupling signal disappeared to within the experimental errors (see also, Kim and Chan, 2014).",
"Thus, any convincing evidence of the existence of a supersolid yet is to be found.",
"Figure: (Color online) Common types of defects in crystals.",
"(a) Representation ofpoint defects; “V”, “Im”, and “I” respectively stand for vacancy,impurity, and interstitial.",
"(b) Representation of an edge dislocation, aclass of line defect, in a solid with cubic symmetry.As a by-product of the frustrated investigations on supersolids, an interest in the plastic behavior of quantum solids has emerged.",
"Recently, Haziot et al.",
"(2013c) have shown that in ultra-pure single crystals of hcp $^{4}$ He the resistance to shear along one particular direction nearly vanishes at around $T = 0.1$ K due to free gliding of dislocations within the basal planes.",
"This intriguing effect has been termed as “giant plasticity” and disappears in the presence of numerous $^{3}$ He impurities or when the temperature is raised.",
"In this section, we review the current understanding of crystalline defects in quantum crystals.",
"Our analysis is focused on vacancies and dislocations since these are the two types of defects that have been studied in more detail in solid $^{4}$ He.",
"Special emphasis is put on identifying those aspects that remain unknown or controversial."
],
[
"Vacancies",
"Both experiments and theory agree in that the vacancy formation enthalpy, $\\Delta H_{\\rm v}$ , in solid $^{4}$ He at ultra-low temperatures amounts to $\\sim 15$ K (Fraass, Granfors, and Simmons, 1989; Galli and Reatto, 2004; Lutsyshyn et al., 2010).",
"The general understanding then is that vacancies cannot be activated thermally in this crystal at temperatures as low as $0.1-1.0$ K. In fact, the classical equilibrium concentration of vacancies in a crystal is given by the expression $x_{\\rm v}^{\\rm class} = \\exp {\\left( -\\Delta G_{\\rm v} / T \\right)}$ , where $\\Delta G_{\\rm v}$ is the Gibbs free energy difference between the perfect and incommensurate (that is, defective) system.",
"$\\Delta G_{\\rm v}$ is equal to $\\Delta H_{\\rm v} - T \\Delta S_{\\rm v}$ , where $\\Delta S_{\\rm v}$ is the entropy change induced by the presence of vacancies.",
"In turn, $\\Delta S_{\\rm v}$ can be estimated as the sum of a vibrational and a configurational contribution.",
"The vibrational contribution corresponds to the variation of the lattice phonon frequencies as a result of the local relaxation occurring around the vacancy; in the limit of very small $x_{\\rm v}$ , this contribution can be safely neglected.",
"The configurational entropy stems from the equivalency between lattice sites when creating a vacancy; this contribution is $\\Delta S_{\\rm v}^{\\rm conf} = -\\ln {\\left( x_{\\rm v} \\right)}$ and cannot be disregarded in the $x_{\\rm v} \\ll 1$ limit.",
"By neglecting vibrational contributions to $\\Delta G_{\\rm v}$ and substituting the value of $\\Delta S_{\\rm v}^{\\rm conf}$ in $x_{\\rm v}^{\\rm class}$ , one has that the classical equilibrium concentration of vacancies in a crystal is: $x_{\\rm v}^{\\rm class} = \\exp {\\left( -\\frac{\\Delta H_{\\rm v}}{2T} \\right)}~.$ In the case of solid $^{4}$ He at $T = 0.1$ K, for instance, it follows that $x_{\\rm v}^{\\rm class} \\sim 10^{-22}$ when considering $\\Delta H_{\\rm v} \\sim 10$ K. In fact, such a classical equilibrium concentration of vacancies is so extremely small that in principle it is physically irrelevant.",
"Interestingly, Rossi et al.",
"(2008) and Pessoa, de Koning, and Vitiello (2009a, 2009b) have recently estimated, by using a reversible-work approach that exploits a quantum-classical isomorphism, that the zero-point vacancy concentration in solid $^{4}$ He is $x_{v} \\sim 10^{-3}$ .",
"Actually, this result is many orders of magnitude larger than the classical result obtained with Eq.",
"(REF ), and it follows from assuming that the crystal is correctly described with a shadow wave function (Vitiello, Runge, and Kalos, 1988; MacFarland et al., 1994).",
"An hypothetical equilibrium vacancy concentration of $\\sim 10^{-3}$ , although probably still is not experimentally detectable, would start being relevant to understand the physical behavior of defective solid $^{4}$ He.",
"Nevertheless, since Rossi and Vitiello's estimations ultimately rely on a variational model the large $x_{\\rm v}^{\\rm class} - x_{v}$ difference cannot be rigorously ascribed to the quantum nature of the crystal.",
"Even when assuming that the equilibrium concentration of vacancies in solid helium is practically null, it cannot be discarded that in the process of growing a crystal from the liquid phase a small concentration of point defects is created.",
"A possible question to answer next then is: do vacancies in a quantum crystal clusterise or keep dispersed?",
"If vacancies clusterised, then they would segregate from the perfect system and become irrelevant.",
"On the contrary, if vacancies remained separated, they could affect the general properties of the quantum crystal quite noticeably (Rota et al., 2012).",
"Unfortunately, there is not a general consensus between theorists about how vacancies interact and distribute in solid $^{4}$ He.",
"Pollet et al.",
"(2008) have estimated from thermodynamic arguments that the binding energy of a divacancy is $E_{\\rm div}^{\\rm bind} = 1.4(5)$ K; we note that this result is about two times larger than the energy found by Clark and Ceperley (2008) using the PIMC method and a semi-empirical pairwise potential.",
"It has been argued then that if vacancies existed they would separate into a vacancy-rich region and segregate from the perfect crystal.",
"However, as we have noted earlier, at finite temperatures is crucial to consider also the entropic contributions to the Gibbs free energy, which cannot be obtained directly from the simulations.",
"Actually, as we show next, it turns out to be much more favorable for the configurational entropy of the crystal to have two independent vacancies rather than a bound divacancy state.",
"By completely ignoring vibrational effects, the resulting entropy gain can be estimated as: $\\delta S_{\\rm 2v-div}^{\\rm conf} \\approx 2 \\cdot \\Delta S_{\\rm v}^{\\rm conf} - \\Delta S_{\\rm div}^{\\rm conf} = -\\ln {\\left( 2 \\cdot x_{\\rm v} \\right)}~,$ where the constraint $x_{\\rm div} = x_{\\rm v}/2$ is employed.",
"By considering the temperature and concentration of vacancies employed in PIMC simulations (Clark and Ceperley, 2008), namely $0.2$ K and $\\sim 10^{-2}$ , one obtains that $T \\cdot \\delta S_{\\rm 2v-div}^{\\rm conf} \\sim 1$ K, which actually is of the same order of magnitude than the estimated $E_{\\rm div}^{\\rm bind}$ .",
"We note that the same conclusion also holds when considering smaller $T$ 's and $x_{\\rm v}$ 's.",
"Therefore, an attractive interaction between vacancies does not necessarily implies the existence of vacancy clusters or vacancy segregation.",
"An alternative analysis to discern whether $^{4}$ He vacancies coalescence or not, consists in monitoring their spatial correlations in quantum Monte Carlo simulations.",
"For instance, if a multiple-vacancy bound state was to exist then an exponential decay in the corresponding vacancy-vacancy pair-correlation function should appear at separations beyond a specific interaction distance.",
"Following this approach, Lutsyshyn, Cazorla, and Boronat (2010) and Lutsyshyn, Rota, and Boronat (2011) have not found any evidence for the existence of a multiple-vacancy bound state at zero temperature.",
"In particular, at the freezing point and also at higher densities the tail in the vacancy-vacancy pair-correlation function always exhibits an asymptotic plateau.",
"Pessoa, de Koning, and Vitiello (2009b) have arrived at a similar conclusion by means of VMC calculations performed with a shadow wave function model.",
"Contrarily, Rossi et al.",
"(2010) have reported, based on the results of PIGS simulations, that when vacancies are present in large concentrations ($x_{\\rm v} \\sim 1$ %) they tend to form bound states.",
"We note that other possible processes involving vacancies, appart from clusterising or dispersing in bulk, have been also suggested; these include nucleation of dislocations (Rossi et al., 2010) and annealing towards the interface regions with the system container (Rossi, Reatto, and Galli, 2012).",
"Test of these hypotheses in realistic crystals with first-principles methods, however, appears to be challenging due to the large system-size and relaxation-time scales involved in the simulations.",
"In spite of the ongoing controversy about the possible existence of vacancies, the effects that hypothetically dispersed vacancies would have on the physical properties of solid helium have already been investigated thoroughly.",
"For instance, the elastic properties of the incommensurate crystal in the limit of zero temperature have been analysed by Cazorla, Lutsyshyn, and Boronat (2013); it has been shown that when considering large vacancy concentrations ($x_{\\rm v} \\sim 1$ %) the shear modulus of the solid undergoes a small reduction of just few percent with respect to the perfect crystal case.",
"Based on PIGS simulations and fundamental arguments, Galli and Reatto have demonstrated (2006) that Bose-Einstein condensation (BEC) occurs in the ground-state of incommensurate solid $^{4}$ He, that is, $n_{0} \\ne 0$ (see Sec.",
"REF ), and that the corresponding critical temperature follows the relation $T_{0} \\propto x_{\\rm v}^{2/3}$ .",
"Recently, Rota and Boronat (2012) have corroborated the occurrence of vacancy-induced BEC in solid helium at low temperatures by means of PIMC simulations.",
"It has been shown that below $T_{0}$ vacancies become quantum entities that completely delocalise in space; they have found also that the dependence of the critical temperature on $x_{\\rm v}$ is best represented by a power law with coefficient $1.57$ (rather than of $2/3$ ), suggesting that the correlations between vacancies are stronger than previously thought.",
"Interestingly, recent experiments performed by Benedek et al.",
"(2016) appear to support the possibility of a vacancy-induced BEC scenario in solid helium under non-equilibrium conditions."
],
[
"Dislocations",
"Since the seminal work by Day and Beamish (2007), there is little doubt that dislocations play a pivotal role on interpreting the mechanical behavior of solid $^{4}$ He.",
"If in the case of point defects it can be said that theory has led the way to their (partial) understanding, in the case of dislocations is the other way around.",
"At present, most of what we know about dislocations in quantum solids comes from recent experiments performed by the groups of Beamish, in the University of Alberta, and Balibar, in the Ecole Normale Supérieure de Paris (see, for instance, Haziot et al., 2013a; Haziot et al., 2013b; Fefferman et al., 2014; Souris et al., 2014a).",
"Such a gap between theory and experiments is due to several reasons.",
"First, in order to simulate dislocations reliably, large simulation cells containing at least several thousands of atoms need to be considered (Bulatov and Cai, 2006; Proville, Rodney, and Marinica, 2012); this system size is actually too large to be handled efficiently in quantum simulations.",
"And second, dislocations are complex topological objects that until recently were not studied in depth in the context of low temperature physics, as a preponderant interest in ground-state properties leads to consider perfect crystals by default.",
"However, as we describe next, quantum simulation of dislocations is critical for advancing our understanding of quantum crystals.",
"Figure: (Color online) The measured distribution, N(L N )N(L_{N}), of lengths, L N L_{N}, between dislocationnetwork nodes in a 4 ^{4}He crystal at T=27T = 27 mK; the contribution to the shear modulus fromeach dislocation length is also indicated.",
"From Fefferman et al., 2014.Dislocations are created during the growth process of solid helium (e. g., due to thermal contraction of the samples during cooling) as rough estimations of their formation energy amount to several thousands of K, hence they cannot be thermally activated at low $T$ .",
"Consider the classical elastic contribution to the formation energy per unit length of an edge dislocation (Cotterill and Doyama, 1966): $E_{\\rm disl}^{\\rm elast}/L = \\frac{\\mu b^{2}}{4 \\pi \\left( 1 - \\nu \\right)} \\ln {\\left( \\frac{r_{\\rm d}}{r_{\\rm c}} \\right)} + E_{\\rm core}^{\\rm elast}~,$ where $\\mu $ is the shear modulus of the crystal, $\\nu $ its Poisson ratio, $b$ the length of the Burgers vector describing the dislocation, $r_{\\rm d}$ the dislocation radius, $r_{\\rm c}$ the dislocation core radius, and $E_{\\rm core}^{\\rm elast}$ the elastic energy of the dislocation core.",
"Since we are interested in obtaining an approximate order of magnitude for $E_{\\rm disl}^{\\rm elast}$ , we can neglect the second term in Eq.",
"(REF ), which is always positive, and assume that $\\ln {\\left( r_{\\rm d} / r_{\\rm c} \\right)} \\sim 1$ .",
"Using the elastic data reported for perfect solid $^{4}$ He by Pessoa, Vitiello and de Koning (2010), that is, $\\mu = 17.1$ MPa and $\\nu = 0.15$ , and adopting an usual Burgers vector of modulus $b = a / \\sqrt{3} = 2.1$ Å, one obtains that $E_{\\rm disl}^{\\rm elast}/L \\sim 1$ K/Å .",
"Considering now that dislocations in solid $^{4}$ He typically are several $\\mu $ m long (see Fig REF ), one finally obtains that, at least, $E_{\\rm disl}^{\\rm elast} \\sim 10^{4}$ K. We note that although this rough estimation of the elastic formation energy of line defects is several orders of magnitude larger than the cost of creating, for instance, a vacancy (see Sec.",
"REF ), in principle it is not possible to grow $^{4}$ He crystals free of dislocations with current state-of-the-art synthesis methods (Souris et al., 2014b).",
"The apparently inevitable presence of dislocations in solid helium near the zero temperature limit already poses a puzzle to the theorist's mind.",
"In a series of compelling experimental works, Balibar, Beamish and collaborators have characterised the energy, structural, and dynamic properties of dislocations in solid $^{4}$ He (for a recent review, see Balibar et al., 2016).",
"The usual experimental setup in those studies consists in a measurement cell supplied with two piezoelectric shear plates that are placed facing each to the other with a separation of few millimeters; the narrow gap that is formed between the transducers then is filled with a crystal that is oriented in a particular direction.",
"By applying a voltage to one of the piezoelectric plates a shear strain is induced in the crystal, and the resulting stress is measured by the opposite shear plate.",
"This process is done repeatedly by using alternating currents.",
"The theory underlying most of Balibar and Beamish's results is that due to Granato and Lücke (1956), in which an analogy is made between the vibration of a dislocation pinned down by impurity particles under an alternating stress field and the classical (that is, not quantum mechanical) problem of the forced damped vibration of a string.",
"In Granato and Lücke's classical theory it is assumed that at high temperatures dislocations interact with thermal lattice phonons, and that as a consequence a maximum shear modulus change of: $\\delta _{\\mu } \\equiv \\frac{\\Delta C_{44}}{C_{44}} = \\frac{\\alpha \\Lambda L^{2}}{1 + \\alpha \\Lambda L^{2}}$ and a dissipation (that is, the phase difference between the applied strain and resulting stress) of: $\\frac{1}{Q} = \\delta _{\\mu } B L^{2} \\omega T^{3}~,$ occur in the crystal.",
"In the context of solid $^{4}$ He, “high temperatures” are considered to be $T \\ge 0.3$ K (Balibar et al., 2016).",
"In the equations above $\\alpha $ and $B$ represent two thermal phonon damping parameters (which in solid $^{4}$ He are equal to $0.019$ and 905 s$\\cdot $ m$^{-2}$ K$^{-3}$ , respectively, see Souris et al., 2014a), $\\Lambda $ the density of dislocation lines per surface unit, $L$ a typical length between nodes in the dislocation network, and $\\omega $ the frequency of the alternating strain field.",
"Using Eqs.",
"(REF ) and (REF ) and from direct measurements of $\\delta _{\\mu }$ and $1/Q$ , Haziot et al.",
"(2013a) have found that typical values of $\\Lambda $ and $L$ in solid helium are $10^{4}-10^{6}$ cm$^{-2}$ and $100-230$ $\\mu $ m (see Fig.",
"REF ), which in the latter case turn out to be macroscopic.",
"In very high quality crystals (Souris et al., 2014b), it has been observed that dislocations avoid crossing each other by forming two-dimensional arrays of parallel lines called “sub-boundaries”, and that they glide together parallel to the basal planes.",
"Remarkably, in the limit of zero temperature the dissipation associated to the gliding of dislocations in the basal plane vanishes (Fefferman et al., 2014), an effect that has been interpreted as evidence of quantum behavior.",
"Nevertheless, whether such an observation implies that the formation energy of dislocation kinks and jogs (that is, defect perturbations that affect the straightness of the dislocation line) also vanishes at ultra-low temperatures, or that dislocation kinks and jogs are able to quantum tunneling through small energy barriers, yet needs to be clarified (Kuklov et al., 2014).",
"In this context, the outcomes of quantum simulations could be highly valuable.",
"At temperatures below $0.2$ K, it is found that the dynamics of dislocations is greatly influenced by the presence of isotopic $^{3}$ He impurities.",
"When the concentration of $^{3}$ He atoms, $x_{3}$ , is large enough (i. e., $\\sim 10^{-7}$ or larger) and $T$ is progressively reduced, the impurities start to bind to the dislocations with an energy that, according to Souris et al.",
"'s (2014a) measurements, is of $0.7~(0.1)$ K. At those conditions, the mobility of the dislocations depends also on the frequency of the applied strain.",
"At high frequencies, that is, at high dislocation speeds of $> 45$ $\\mu $ m/s, the impurities cannot move fast enough to follow the line defects so that they end up anchoring them.",
"However, at lower frequencies, and always considering Souris et al.",
"'s (2014a) arguments, dislocations can actually move dressed with $^{3}$ He atoms.",
"A pertinent comment has to be made here.",
"Several nuclear magnetic resonance studies have shown that at low temperatures $^{3}$ He atoms in solid $^{4}$ He behave as quantum quasi-particles that can move through the lattice at velocities as high as $\\sim 1$ mm/s (Allen, Richards, and Schratter, 1982; Kim et al., 2013), that is, significantly higher than 45 $\\mu $ m/s.",
"It has been argued then that near the dislocation the mobility of isotopic impurities could be reduced considerably by the existing local strain (Balibar et al., 2016); however, there is not quantitative evidence showing that such a huge variation of about three orders of magnitude in the mobility of $^{3}$ He impurities could be actually possible.",
"Clearly, a microscopic understanding of what are the interactions between dislocations and quantum isotopic impurities, and the factors that can affect the mobility of the latter, is necessary.",
"The outcomes of quantum simulation studies again could be very useful in clarifying these issues.",
"With regard to theory, Boninsegni et al.",
"(2007) have shown using PIMC simulations that the core of screw dislocations with Burgers vectors oriented perpendicular to the basal plane in solid $^{4}$ He are superfluid.",
"Boninsegni et al.",
"'s predictions have led to a number of hypotheses about possible new phenomena involving quantum dislocations like, for instance, “superclimb” (Söyler et al., 2009; Aleinikava, Dedits, and Kuklov, 2011) and superfluidity in dislocation networks (Boninsegni et al., 2007).",
"In a recent PIMC study by Landinez-Borda, Cai, and de Koning (2016) on solid helium, it has been reported that either screw or edge dislocations with Burgers vectors along the basal plane are not superfluid.",
"In particular, both types of dislocations are predicted to dissociate into non-superfluid Shockley partial dislocations separated by ribbons of stacking fault, as it normally occurs in classical hcp crystals (Bulatov and Cai, 2006; Hull and Bacon, 2011).",
"Landinez-Borda, Cai, and de Koning (2016) have also concluded that in solid helium the resistance to flow of partial dislocations is negligible (that is, the corresponding Peierls stress is nominally zero) mostly due to zero-point fluctuations.",
"The results presented in this latter simulation work provide valuable insight into the physical origins of the observed “giant plasticity” effect (Haziot et al., 2013c; Zhou et al., 2013; Haziot et al., 2013d).",
"Apparently, there seems to be some inconsistencies among the conclusions presented by Boninsegni et al.",
"(2007) and Landinez-Borda, Cai, and de Koning (2016) as to what concerns the superfluid properties of dislocation cores.",
"Nevertheless, we must note that the linear defects analysed in those two studies are different as their respective Burgers vectors are either oriented along the $c$ -axis or contained in the basal plane.",
"Further quantum simulation studies indeed appear to be necessary for clarifying the role of Burgers vector orientation on the transport properties of dislocation cores in solid $^{4}$ He.",
"Finally, recent experiments done in the group of Hallock in the University of Massachusetts have shown unexpected mass flow through $^{4}$ He crystals at low temperatures ($T < 600$ mK) when sandwiched between two regions of superfluid liquid in which a pressure gradient is applied (Ray and Hallock, 2008; Ray and Hallock, 2010; Vekhov and Hallock, 2012).",
"This phenomenon has been dubbed as “giant isochoric compressibility” or the “syringe effect”.",
"The observed mass flow has been interpreted in terms of two possible scenarios (Hallock, 2015), namely the climbing (i. e., the passing of an obstacle to start moving again) of either superfluid dislocations (Söyler et al., 2009) or grain boundaries (Burovski et al., 2005; Sasaki et al., 2006; Pollet et al., 2007; Cheng and Beamish, 2016).",
"Recently, analogous mass flow phenomena have been observed also in an inverted solid-superfluid-solid setup by Cheng et al.",
"(2015), in which the effects of $^{3}$ He impurities concentration and distribution have been analysed in detail.",
"The exact atomistic mechanisms underlying the inverted and direct syringe effects, however, still remain an open question.",
"New systematic experiments and quantum simulation studies certainly are necessary to achieve a more accomplished knowledge of mass transport along quantum linear and planar defects (i. e., dislocations and grain boundaries)."
],
[
"The role of dimensionality",
"Quantum crystals at reduced dimensionality have been the object of numerous experimental and theoretical studies.",
"The interplay between quantum correlations and structural confinement opens a series of interesting new prospects that since the beginning of the quantum Monte Carlo era have been investigated meticulously with theory.",
"The search for novel phases and physical phenomena in quantum gases adsorbed on graphitic and metallic substrates or on the surface of carbon nanostructures and the interior of narrow silica pores, represent well-known examples."
],
[
"Quantum films",
"We focus here on helium and hydrogen since in these species QNE are most pronounced.",
"In very thin films one can expect that two-dimensional effects become dominant, and for this reason many works have concentrated in studying the thermodynamic, structural, and dynamical properties of purely 2D quantum many-body systems.",
"At zero temperature and zero pressure 2D $^4$ He is a liquid with an estimated equilibrium density of $\\sigma _{0} = 0.043$ Å$^{-2}$ and binding energy of $E/N = -0.90$ K (Giorgini, Boronat, and Casulleras, 1996).",
"By increasing the density, the liquid solidifies into a triangular lattice (Whitlock, Chester, and Kalos, 1988).",
"The liquid and solid are in equilibrium at densities $\\rho _{f} = 0.068$ Å$^{-2}$ (freezing) and $\\rho _{m} = 0.072$ Å$^{-2}$ (melting), respectively.",
"On the other hand, $^3$ He at low densities remains in the gas phase due to its lower mass, and more importantly, fermionic character (Grau, Boronat, and Casulleras, 2002).",
"Upon steady increase in density the gas eventually transforms into a triangular solid, although the critical point associated to this transition has not been characterised yet with precision.",
"Figure: Energy per particle in two-dimensional solid D 2 {\\rm D_{2}} (solidline and filled circles), and two-dimensional solid H 2 {\\rm H_{2}} (dottedline and empty triangles).",
"From Cazorla and Boronat (2008c).The ground-state of two-dimensional molecular hydrogen and deuterium have been investigated also with QMC methods (Cazorla and Boronat, 2008c; Boninsegni, 2004).",
"The primary interest of these studies was to discern whether by reducing the dimensionality it was possible to stabilise the liquid phase.",
"Those theoretical works, however, have shown that the fluid is never stable, neither when considering negative pressures (Cazorla and Boronat, 2008c; Boninsegni, 2004).",
"In Fig.",
"REF , we enclose the density dependence of the energy calculated in 2D H$_{2}$ and D$_{2}$ with the DMC method and Silvera-Goldman potential (Cazorla and Boronat, 2008c).",
"At zero pressure, both crystals stabilise in a triangular lattice of density $\\sigma _0=0.0673$ Å$^{-2}$ and $0.0785$ Å$^{-2}$ , respectively.",
"The corresponding energy per particle at those conditions are $-23.45$ K in H$_2$ and $-42.30$ K in D$_2$ .",
"This large energy difference indicates that the presence of quantum isotopic effects is also significant when considering only two dimensions.",
"With regard to the possibility of realising H$_{2}$ superfluidity (see Sec.",
"REF ), several strategies have been explored also in reduced dimensionality.",
"It was first proposed by Gordillo and Ceperley (1997) that the intercalation of alkali atoms could frustrate the formation of the solid due to the weaker interaction between impurities and H$_2$ than between hydrogen molecules.",
"K and Cs were considered as the likely candidates to induce H$_{2}$ melting in a PIMC study by Gordillo and Ceperley (1997).",
"Large superfluid fractions of $\\rho _{s}/\\rho \\sim 0.2-0.5$ were reported in the resulting hydrogen-alkali thin films.",
"However, subsequent quantum simulation studies performed with a larger number of particles have found very small values of $\\rho _{s}/\\rho $ in equivalent systems (Boninsegni, 2005; Cazorla and Boronat, 2004).",
"More recently, Cazorla and Boronat (2013) have predicted by using the DMC method and semi-empirical pairwise potentials that frustration of 2D solid H$_{2}$ could be achieved with sodium atoms arranged in a triangular lattice of constant 10 Å .",
"The main reason for this is that the forces between Na atoms and hydrogen molecules are weaker than those considered in previous studies, hence a significant reduction of the equilibrium density is induced that favors stabilization of the liquid phase.",
"We note, however, that in a posterior PIMC study by Boninsegni (2016) this conclusion has been disputed by arguing that the system remains in the solid phase independently of its density and type of alkali impurity that is considered.",
"Experimental realisation of quasi-two dimensional quantum solids is achieved through adsorption of quantum gases on attractive substrates.",
"In this context, one of the most extensively investigated substrates is graphite.",
"The physics of gas-adsorption phenomena in graphite is very rich (Bruch, Cole, and Zaremba, 1997) as a large sequence of transitions have been experimentally observed and described with microscopic theory (Clements et al., 1993; Clements, Krotscheck, and Saarela, 1997).",
"We concentrate here on describing the properties of the first adsorbed layer and other related crystalline phases.",
"It is worth noticing that when corrugation effects are included in the simulations (that is, the spatial distribution of carbon atoms in the underlying substrate is explicitly taken into consideration), denser commensurate phases are normally obtained.",
"According to recent quantum simulation studies performed with semi-empirical pairwise potentials, the ground state of $^4$ He adsorbed on graphite (and graphene) is a $\\sqrt{3}\\times \\sqrt{3}$ commensurate phase with a surface density of $0.0636$ Å$^{-2}$ (Gordillo, Cazorla, and Boronat, 2011).",
"The liquid phase is metastable with respect to the crystal.",
"As the density is increased, the commensurate crystal transforms into a triangular incommensurate solid of density $\\sim 0.08$ Å$^{-2}$ (Gordillo and Boronat, 2009b; Pierce and Manousakis, 2000; Corboz et al., 2008).",
"This description is in excellent agreement with the available experimental data (Bruch, Cole, and Zaremba, 1997).",
"By increasing further the density, a second layer develops on top of the first with an equilibrium density of $0.12$ Å$^{-2}$ .",
"Recent QMC studies of the registered phases of H$_2$ adsorbed on graphite and graphene provide a description that is identical to that obtained in $^4$ He, and which is in very good agreement with the experiments (Gordillo and Boronat, 2010).",
"In particular, the ground state is a commensurate $\\sqrt{3}\\times \\sqrt{3}$ phase that undergoes a first-order transition towards an incommensurate triangular crystal at $\\rho =0.077$ Å$^{-2}$ .",
"The phase diagram of D$_2$ on graphite has been investigated thoroughly in experiments (Bruch, Cole, and Zaremba, 1997) but not yet with theory.",
"It is known that this is richer than its H$_2$ counterpart since at least two additional commensurate phases appear in the first adsorbed layer: the $\\varepsilon $ phase, which is a $4\\times 4$ structure ($0.0835$ Å$^{-2}$ ), and the $\\delta $ one, corresponding to a $5\\sqrt{3}\\times 5\\sqrt{3}$ lattice ($0.0789$ Å$^{-2}$ ).",
"According to DMC calculations none of these latter commensurate phases are stable in H$_2$ (Gordillo and Boronat, 2010)."
],
[
"One-dimensional systems",
"Carbon nanotubes and nanopores embedded in solid matrices have opened the possibility of studying quantum systems in quasi-one dimensional geometries (Calbi et al., 2001).",
"Investigating individual carbon nanotubes, however, has proved challenging due to the fact that they are normally capped and adsorption of atoms on their interior is energetically unfavored.",
"Alternatively, adsorption on the intersites and grooves formed between adjacent nanotubes have been observed.",
"A topic of interest in these systems is the study of the crossover between three-dimensional and one-dimensional physics (Gordillo and Boronat, 2009a).",
"For instance, changes on the superfluid fraction and condensate fraction of liquid $^{4}$ He upon variation of the nanopore radius have been systematically studied by Vranješ Markić and Glyde (2015) with PIMC simulations.",
"Strictly one-dimensional quantum systems possess unique features as compared to the rest of low-dimensional systems.",
"One of the most relevant aspects is that the presence of a hard core in the interatomic interactions makes quantum statistics irrelevant.",
"This means that simulation of a Fermi system (e. g., $^3$ He) can be made exactly because the nodes of the corresponding wave function are known a priori, hence one can get rid of the sign problem in practice (see Sec.",
"REF ).",
"Another important characteristic is the absence of continuous phase transitions (that is, as described by Landau's theory), although the limiting $T=0$ case may be an exception.",
"Finally, if the excitation spectrum of the system is gapless, i. e., $E_{k} = \\hbar |k| c$ , Luttinger theory applies and consequently the behavior of correlation functions at large-distance (or small-momenta) is known analytically (Giamarchi, 2004; Imambekov, Schmidt, and Glazman, 2012).",
"In this latter case, the behavior of the system is universal in terms of the Luttinger parameter, $K$ , which in a homogeneous system is directly related to the Fermi velocity, $v_{F}$ , and speed of sound, $c$ , as $K = v_{F}/c$ .",
"The Fermi velocity $v_{F}$ is completely fixed by the linear density, $\\rho $ , whereas the speed of sound $c$ depends on the many-body interactions.",
"In Luttinger's theory, the height of the $l$ -th peak in the static structure factor, $S(k)$ , is given by (Haldane, 1981): $S(k = 2l k_{F}) = A_{l} N^{1-2l^{2}K}~,$ which diverges when $K < 1/2l^{2}$ .",
"If the first peak in $S (k)$ diverges, that is, $K<1/2$ , the system is called a “quasi-crystal”.",
"There is a number of important differences between quasi and real crystals.",
"A three-dimensional crystal possesses diagonal long-range order since the density oscillations in the two-body distribution function remain in phase over long distances.",
"In one dimension, on the contrary, that order is lost according to a power-law decay.",
"The height of the first peak diverges in both cases, however in a true crystal the Bragg peak grows linearly with the number of particles, $S(k_{peak}) \\propto N$ , whereas in a quasi-crystal the corresponding exponent is smaller than unity [see Eq.",
"(REF )].",
"Figure: (Color online) Luttinger parameter, KK, in one-dimensional 3 ^3He expressed as afunction of the linear density, ρ\\rho .",
"The corresponding speed of sound, as extracted fromthe phononic part of the static structure factor (symbols) and thermodynamic compressibility(line), is also shown.",
"Adapted from Astrakharchik and Boronat (2014).QMC calculations of 1D $^4$ He at equilibrium have shown that this is a self-bound system with a tiny binding energy of $\\sim -4$ mK (Gordillo, Boronat, and Casulleras, 2000a; Boninsegni and Moroni, 2000).",
"When the density is increased, the system eventually becomes a quasi-crystal.",
"Recently, the ground state of one-dimensional $^3$ He has been studied thoroughly with the DMC method (Astrakharchik and Boronat, 2014).",
"The lower mass of the isotope makes the system to be non self-bound, and thus it remains in the gas phase down to zero pressure.",
"Through calculation of the corresponding Luttinger parameter one can appreciate the richness of its behavior as a function of density (see Fig.",
"REF ).",
"As the interaction between hydrogen molecules is more attractive than between helium atoms, H$_{2}$ is also self bound in the one-dimensional limit.",
"When H$_2$ molecules, or helium atoms, are adsorbed inside of a nanopore the resulting phases depend strongly on the amount of space that is available.",
"In very narrow nanotubes, for instance, one observes the existence of real quasi-1D systems, that is, in the Luttinger sense (Gordillo and Boronat, 2009a).",
"On the contrary, if the nanopore diameter is wide enough, particles migrate towards the nanopore walls due to the strong attractive interactions.",
"Eventually, if the nanopore interior is further enlarged, nucleation of a narrow channel containing a liquid may occur (Rossi, Galli,and Reatto, 2005).",
"In the case of molecular hydrogen, however, the possible stabilisation of a 1D fluid remains controversial (Gordillo, Boronat, and Casulleras, 2000b; Boninsegni, 2013b; Omiyinka and Boninsegni, 2016; Rossi and Ancilotto, 2016).",
"Recently, the adsorption of quantum gases on the external surface of a single nanotube has drawn some attention.",
"State-of-the-art resonance experiments on a single suspended carbon nanotube have been able to determine the phase diagram of the deposited rare gases with high precision (Wang et al., 2010; Tavernarakis et al., 2014).",
"For instance, in the $T = 0$ limit one can identify either a registered $\\sqrt{3}\\times \\sqrt{3}$ phase, already known from adsorption on planar substrates, or incommensurate phases, depending on the chemical species.",
"Theoretical predictions on these systems (Gordillo and Boronat, 2011) agree well with the experimental findings."
],
[
"Clusters",
"Helium and hydrogen drops can be generated in the laboratory by means of free jet expansions from a stagnation source chamber that go through a thin walled nozzle (Grebenev, Toennies, and Vilesov, 1998).",
"Helium drops are the most clean example of inhomogeneous quantum liquids with either boson ($^4$ He) or fermion ($^3$ He) quantum statistics.",
"In recent years, the relevance on He drops has been reinforced by the increasing interest in studying the behavior of small molecules placed in their interior.",
"In fact, quantum clusters can act as ideal matrices in which to carry out accurate spectroscopy analysis of the embedded molecules.",
"When the guest molecule is surrounded by $^4$ He atoms, the corresponding rotational spectrum presents a peaked structure that has been attributed to the superfluid nature of helium.",
"By contrast, in $^3$ He drops a broad peak is recorded.",
"This phenomena, termed as microscopic superfluidity, has been the object of many QMC studies in the last years (Sindzingre, Klein, and Ceperley, 1989; Sola, Boronat, and Casulleras, 2006).",
"H$_2$ clusters have been produced also in the laboratory with jet expansion techniques (Tejeda et al., 2004).",
"The behavior of H$_{2}$ drops is richer than that of $^4$ He since they can be either liquid or solid depending on the number of constituent particles.",
"The first PIMC study on H$_2$ clusters was carried out by Sindzingre, Ceperley, and Klein (1991), and it was found that clusters comprising a number of molecules up to $N \\simeq 18$ were superfluid at temperatures below $T=2$ K. In a subsequent PIMC work (Khairallah et al., 2007) the limiting number of molecules exhibiting superfluid behavior has been raised to $N \\simeq 26$ .",
"The results reported by Khairallah et al.",
"(2007) appear to show superfluidity mostly localised in the surface of the cluster, which points to an inhomogeneous structure formed by a solid core surrounded by a liquid skin, that at low temperatures becomes superfluid.",
"This interpretation, however, has been challenged in a posterior PIMC work in which it has been argued that, in spite of the local variation in molecular order, superfluidity remains a global property of the entire cluster (Mezzacapo and Boninsegni, 2008).",
"Figure: (Color online) Optimal distribution of equilibrium sites in solid H 2 _2clusters with N=18N=18, 19, and 20 molecules at zero temperature.",
"From Sola andBoronat (2011).The structure and energy of small H$_2$ clusters in the limit of zero temperature have been studied accurately with both the DMC (Guardiola and Navarro, 2008) and PIGS (Cuervo and Roy, 2006) methods.",
"The presence of magic-cluster sizes, identified with a kink in the chemical potential, have been reported in those studies.",
"The number of molecules contained in the smallest and energetically most stable clusters, appear to coincide with the results of Raman spectroscopy measurements (Tejeda et al., 2004).",
"A combination of the DMC and stochastic optimization (i. e., simulated annealing) techniques has allowed to determine the equilibrium structure in most stable solid H$_2$ clusters (Sola and Boronat, 2011).",
"Examples of optimal molecular arrangements obtained in those clusters are shown in Fig.",
"REF ."
],
[
"Molecular crystals",
"Molecular systems are of critical importance in astronomy, biology, and environmental science.",
"Hydrogen is the most abundant element in the universe and over wide thermodynamic conditions is most stable in molecular form (see Sec.",
"REF ).",
"Water is vital to all known forms of life and it covers around three quarter parts of the Earth's surface.",
"Nitrogen and methane are found in the interior and crust of many celestial bodies and also in organic substances.",
"When all these species are compressed eventually they become crystals in which, due to the light weight of their atoms and weak interparticle interactions, QNE play a pivotal role at low temperatures (see Fig.",
"REF ).",
"Next, we briefly review the knowledge of the phase diagram of these compounds and highlight the aspects that remain contentious.",
"Due to their intrinsically rich and complex nature, it is not possible to provide here a detailed description of H$_{2}$ , H$_{2}$ O, N$_{2}$ , and CH$_{4}$ , hence we address the interested reader to other recent and more specialised articles (see, for instance, McMahon et al., 2012; Goncharov, Howie, and Gregoryanz, 2013; Herrero and Ramírez, 2014).",
"For the sake of focus, only those aspects related to the crystalline phases are considered in this section."
],
[
"H$_{2}$ at extreme {{formula:3cf6a70a-55c8-47a4-9c5b-392d6e74a16e}} conditions",
"Due to its low $Z$ number, hydrogen's x-ray scattering cross section is very low.",
"This means that it is extremely challenging to determine with accuracy its atomic structure under extreme thermodynamic conditions in the laboratory.",
"Infrared (IR) and Raman spectroscopy techniques have been used to monitor the changes in the vibrational properties of the crystal that can be ascribed to a phase transition.",
"However, due to the high reactivity, mobility and diffusion of the molecules already at moderate temperatures (i. e., $\\ge 250$ K), this type of measurements turn out to be very difficult (Goncharov, Howie,and Gregoryanz, 2013).",
"Here is where the inputs of theory and computer simulations become critical.",
"By comparing the vibrational phonon spectra of low-energy structures obtained in first-principles searches with experimental data, candidate atomic structures can be identified for each of the detected transformations.",
"Unfortunately, for the reasons highlighted in Sec.",
"REF , the theoretical study of hot and dense solid hydrogen is technically difficult and very sensitive to the employed method (that is, the free energy differences between phases normally are very small, of the order of few meV, which coincide with the typical accuracy threshold in first-principles calculations).",
"As a consequence, the description of hydrogen-based systems obtained with various levels of theory may differ (Morales et al., 2013; Drummond et al., 2015), complicating even further the characterisation of solid hydrogen.",
"The H$_{2}$ crystal phases that are experimentally well established are denoted by I, II, III, and IV (see Fig.",
"REF ).",
"Phase I corresponds to the close-packed hcp structure, in which para-H$_{2}$ molecules have zero angular momentum and spherically symmetric wave functions (Silvera, 1980).",
"At low temperatures and as pressure is increased, breaking of rotational symmetry eventually occurs and the crystal stabilises in phase II (Lorenzana, Silvera, and Goettel, 1990); the boundary between phases I and II is strongly dependent on isotope type (see Fig.",
"REF ), which indicates the presence of important QNE.",
"Around 150 GPa, molecular hydrogen undergoes another phase transformation into phase III (Hemley and Mao, 1988; Lorenzana, Silvera, and Goettel, 1989), which has been shown to extend up to pressures of $\\sim 300$ GPa and temperatures of $\\sim 300$ K (Zha, Liu, and Hemley, 2012).",
"Experiments have been able also to provide constraints on the molecular orientation in phases II and III although, for the reasons specified above, not full structural characterisations (Goncharov et al., 1998).",
"There have been many attempts to identify the structure of phases II and III using theoretical methods.",
"Due to the technical difficulties encountered in the treatment of weak dispersive intermolecular interactions and of the indispensable consideration of QNE, however, there is not yet general agreement on this matter.",
"For phase II, there is a number of candidate structures including the orthorhombic $Cmc2_{1}$ (Kitamura et al., 2000), monoclinic $P2_{1}/c$ (Zhang et al., 2007), and orthorhombic $Pca2_{1}$ (Kohanoff et al., 1997; Städele and Martin, 2000).",
"From all these structures, $Pca2_{1}$ emerges as one of the most likely molecular models (Moraldi, 2009; McMahon et al., 2012).",
"Raman experiments, however, indicate that phase II possesses one vibrational mode whereas the $Pca2_{1}$ phase has four (Cui, Chen, and Silvera, 1995).",
"More recently, a new monoclinic $P2_{1}/c$ phase containing 24 atoms in the primitive cell has been proposed also as a likely candidate for phase II (Pickard and Needs, 2009).",
"This structure has been obtained through the ab initio random structure searching method (Pickard and Needs, 2006) and its vibrational phonon features appear to be consistent with Raman experiments (Drummond et al., 2015).",
"With regard to phase III, it was initially proposed that a hcp lattice with molecules tilted roughly $60^{\\circ }$ with respect to the $c$ axis could be a strong candidate (Natoli, Martin, and Ceperley, 1995).",
"This suggestion is consistent with the reported spectroscopy data, in which intense IR activity is appreciated (Cui, Chen, and Silvera, 1995), and with a recent x-ray diffraction study by Akahama et al.",
"(2010).",
"Subsequently, Pickard and Needs (2007) proposed, again relying on the outcomes of DFT-based random structure searches, a different candidate structure consisting of 12 molecules per unit cell with the centers close to those in a distorted hcp lattice.",
"The symmetry of this phase is $C2/c$ (monoclinic) and its vibrational phonon features are also consistent with the available experimental data.",
"More recently, a hexagonal structure with $P6_{1}22$ symmetry has been introduced as another possible candidate for phase III (Monserrat et al., 2016).",
"In 1995, Goncharov et al.",
"found in deuterium a small discontinuity in the vibron mode (that is, the intramolecular stretching mode) and a change in the slope of the corresponding I-III phase boundary at pressures around 150 GPa and temperatures above 175 K. This small vibron discontinuity practically disappeared at $T \\ge 250$ K. These observations suggest the possible existence of a new phase denoted by I' (see Fig.",
"REF ), that is isostructural to phase III, and of a critical I-I'-III point.",
"PIMC calculations by Surh et al.",
"(1997) on a system of quantum rotors interacting through an effective LDA model, provide some support to this hypothesis.",
"In subsequent spectroscopy experiments, Baer, Evans, and Yoo (2007, 2009) have found vibron signatures that are also consistent with the existence of phase I'.",
"However, for these latter observations to be consistent with those by Goncharov et al.",
"(1995), the slope of the I-I' phase boundary needs to be negative, a feature that was not reported in the earliest work.",
"In a recent study Goncharov, Hemley and Mao (2011) have performed a refined vibrational spectroscopy analysis and concluded that the new data do not support the existence of phase I'.",
"Further systematic investigations appear to be necessary to clarify these issues.",
"Recently, room-temperature static diamond-anvil-cell (DAC) experiments have been performed in which pressures of up to 300 GPa have been reached (Eremets and Troyan, 2011; Howie et al., 2012a; Howie et al., 2012b; Loubeyre, Occelli, and Dumas, 2013).",
"Eremets and Troyan (2011) have reported that solid hydrogen becomes metallic at a pressure of 265 GPa.",
"Subsequent experimental studies (Howie et al., 2012a; Howie et al., 2012b; Loubeyre, Occelli, and Dumas, 2013), however, do not appear to support the validity of this result.",
"The pressure threshold for the insulator to metal transition in hydrogen still is believed to lie between 325 (Goncharov et al., 2001) and 450 GPa (Loubeyre, Occelli, and LeToullec, 2002).",
"These recent room-temperature DAC studies, on the other hand, agree all in that hydrogen transforms to a new phase, denoted by IV (see Fig.",
"REF ), at a pressure near 220 GPa.",
"During the III-IV transformation, a large vibron Raman frequency discontinuity and the appearance of two IR and two Raman vibron modes are observed.",
"The existence of phase IV, therefore, now is regarded as well established.",
"Again, several candidate structures have been proposed for phase IV.",
"Howie et al.",
"(2012a, 2012b) have tentatively indexed it as $Pbcn$ , based on the results of the DFT-based random structure searches carried out by Pickard and Needs (2007) and their experimental spectroscopy analysis.",
"This new orthorhombic structure presents a quite peculiar molecular arrangement in which consecutive graphene-like layers alternate between ordered and disordered structures.",
"Chiefly, proton tunneling occurs within the graphene-like disordered layers and the corresponding frequency increases under pressure (Howie et al., 2012a).",
"Pickard, Martinez-Canales, and Needs (2012a, 2012b), however, have shown using DFT-based methods that the $Pbcn$ phase is vibrationally unstable at zero temperature.",
"The same authors have proposed a monoclinic $Pc$ structure to represent phase IV.",
"This monoclinic phase is dynamically stable and contains 96 atoms in its unit cell; it consists of alternating layers of weakly coupled molecules with short intra-molecular bonds, and strongly coupled molecules forming graphene-like sheets with long intra-molecular bonds.",
"Recent synchrotron infrared measurements by Loubeyre, Occelli, and Dumas (2013) appear to support the validity of this structural layered model.",
"By relying also on the results of first-principles simulations, Liu et al.",
"(2012) have proposed a monoclinic $Cc$ structure as a new possible candidate for phase IV; this phase is vibrationally stable and structurally very similar to the $Pc$ structure proposed by Pickard, Martinez-Canales, and Needs (2012a, 2012b), although is thermodynamically less stable and has no orientational order.",
"Further systematic investigations appear to be necessary to determine with precision the molecular structure of phase IV.",
"Several other phases have been predicted to exist in solid hydrogen at low temperatures and pressures beyond 250 GPa.",
"Most of those phases have been predicted based on the results of first-principles crystal structure searches that incorporate QNE through the quasi-harmonic approximation (see Sec.",
"REF ).",
"Pickard, Martinez-Canales, and Needs (2012a, 2012b) have proposed that their candidate structure for phase III, that is, monoclinic $C2/c$ , transforms into an orthorhombic $Cmca-12$ phase containing 12 atoms per unit cell at $P = 285$ GPa, and that this subsequently transforms into another $Cmca$ phase with a smaller number of atoms at $P = 385$ GPa.",
"Liu, Wang, and Ma (2012), have also predicted that at pressures higher than $\\sim 500$ GPa hydrogen transforms into a new monoclinic $C2/c$ phase that possesses two types of intramolecular bonds with different lengths.",
"In fact, new crystal phases (e. g., IV' and V) have been observed in DAC experiments performed at pressures beyond $\\sim 300$ GPa (Howie et al., 2012a; Dalladay-Simpson, Howie, and Gregoryanz, 2016); however, their precise molecular arrangements still remain unknown.",
"The possibility of stabilising an atomic, rather than a molecular, phase in solid hydrogen by means of pressure has been also explored by several authors with theory.",
"This possibility is very interesting from a fundamental point of view as it could render a metallic system (Wigner and Huntington, 1935).",
"Considering the orthorhombic $Cmca$ phase originally proposed by Pickard and Needs (2007) and relying on ab initio random structure searches, McMahon and Ceperley (2011) have proposed that molecular hydrogen dissociates into a monoatomic body-centered tetragonal structure near 500 GPa.",
"Labet et al.",
"(2012) and Labet, Hoffmann, and Ashcroft (2012a, 2012b, 2012c) have also analysed in the detail the process of molecular dissocation by focusing on the structures predicted by Pickard and Needs (2007); they have found a discontinuous shift in the distances between protons when transitioning from the orthorhombic $Cmca$ to the atomic phase, and have proposed an intermediate phase that would allow for a continuous dissociation.",
"More recently, Azadi et al.",
"(2014) have concluded, based on electronic QMC methods (see Sec.",
"REF ) and considering nuclear anharmonic contributions to the enthalpy through DFT, that a transition from the orthorhombic $Cmca-12$ to an atomic $I4_{1}/amd$ phase could occur at $P = 374$ GPa.",
"Interestingly, Dalladay-Simpson, Howie, and Gregoryanz (2016) have just reported experimental evidence for a new phase in hydrogen, denoted by V, which at room temperature is stabilised at a pressure of 325 GPa.",
"The experimental evidence consist of a substantial weakening of the vibrational Raman activity, a change in the pressure dependence of the vibron, and a partial loss of the low-frequency excitations.",
"Whether this new phase could be identified with the debated $P$ -induced atomisation of solid hydrogen, is still not well established.",
"As it has been explained in this section, many complex and controversial aspects still need to be solved in solid hydrogen under extreme thermodynamic conditions.",
"On the theoretical side, most of the predictions on phases II, III, and IV rely on techniques that incorporate QNE only approximately (e. g., quasi-harmonic approaches) and on standard DFT methods.",
"Using such approaches to reproduce the thermodynamic stability of highly compressed hydrogen, however, seems to be inadequate.",
"For instance, Chen et al.",
"(2014) have recently shown in a thorough PIMC benchmark study on H$_{2}$ that those cases in which good agreement between standard DFT calculations and experiments is obtained, large error cancellations are likely to be affecting the simulations.",
"Similar conclusions have been attained also by Geneste et al.",
"(2012), Morales et al.",
"(2013), and Drummond et al.",
"(2015) by using non-standard computational approaches (e. g., non-harmonic simulation methods in combination with electronic QMC).",
"In order to provide more conclusive estimations in solid H$_{2}$ , therefore, is necessary to employ quantum simulation methods that simultaneously describe QNE (e. g., PIMD, PIMC and PIGS, see Sec. )",
"and long-range intermolecular forces (e. g., non-standard DFT functionals and eQMC, see Sec. )",
"accurately."
],
[
"Solid water",
"QNE are unquestionably important for understanding the physical properties of ice.",
"Due to the small moment of inertia of the H$_{2}$ O molecule and relatively low strength of the intermolecular hydrogen bonds, QNE persist in ice up to temperatures of $\\sim 100$ K (Gai, Schenter, and Garrett, 1996a; Ceriotti, Bussi, and Parrinello, 2009; Vega et al., 2010; Moreira and de Koning, 2015).",
"Numerous examples of these effects can be found in the literature.",
"For example, incoherent single-particle tunneling has been disclosed in cubic ice at Mbar pressures, explaining so the origins of the measured H/D isotopic effects on the antiferroelectric ice VIII $\\rightarrow $ paraelectric VII phase transformation (Benoit, Marx, and Parrinello, 1998; Benoit, Romero, and Marx, 2002).",
"A recent neutron scattering study has also revealed an anomalous $T$ -dependent dynamic effect in normal (hexagonal) ice I$_{h}$ (Bove et al., 2009), that has been explained in terms of collective tunneling of protons (up to six) within locally ordered rings (Drechsel-Grau and Marx, 2014).",
"These findings suggest that quantum many-body tunneling could be important also in a variety of related H-bonded systems, including other phases of ice and cyclic water clusters on metal surfaces (Drechsel-Grau and Marx, 2014).",
"In analogy to solid helium and hydrogen, an interest has developed in understanding the features of the momentum distribution in ice.",
"Both inelastic neutron scattering experiments (Reiter et al., 2004; Flammini et al., 2012) and advanced path-integral calculations (Morrone and Car, 2008; Lin et al., 2010; Lin et al., 2011) agree in describing the corresponding $n({\\bf k})$ with an anisotropic Gaussian.",
"This result implies that the protons experience an anisotropic quasi-harmonic effective potential with distinct principal frequencies that reflect the possible molecular orientation.",
"According to both neutron scattering experiments and path-integral calculations (Flammini et al., 2012) the excess kinetic energy in ice I$_{h}$ at low temperatures amounts to $\\sim 150$ meV, which evidences a marked quantum character (see Sec.",
"REF ).",
"The presence of quantum isotopic effects is also notable in solid water.",
"The effects of hydrogen isotope substitution on the structural, kinetic energy and atomic delocalization properties of ice, have been investigated in detail with experiments and path-integral calculations.",
"For example, quantum simulations of D$_{2}$ O in the I$_{h}$ phase at $T = 100$ K have found a decrease in the crystal volume and intramolecular O-D distance of $0.6$ % and $0.4$ %, respectively, as compared to H$_{2}$ O (Herrero and Ramírez, 2011a).",
"An increase of $\\sim 6$ % in the melting temperature of D$_{2}$ O at ambient pressure has been also predicted with path-integral simulations (Ramírez and Herrero, 2010).",
"Similarly, the presence of quantum isotopic effects in highly compressed amorphous ice have been reported by several groups (Gai, Schenter, and Garret, 1996b; Herrero and Ramírez, 2012).",
"Interestingly, an anomalous thermal expansion isotopic effect has been observed in ice; the volume of solid D$_{2}$ O is larger than that of solid H$_{2}$ O (Röttger et al., 1994), in contrast to what occurs in other crystals upon substitution with heavier species.",
"This quantum nuclear effect has been rationalised recently by Pamuk et al.",
"(2012) with ab initio calculations based on the quasi-harmonic approximation.",
"The importance of QNE on the phase diagram of ice has been determined quantitatively with path-integral Monte Carlo simulations based on the TIP4PQ/2005 force field by McBride et al.",
"(2012) [see Fig.",
"REF ].",
"It is worth noting that although the intermolecular potential model employed by McBride et al.",
"(2012) is non-flexible and non-polarisable, the agreement obtained with the experiments is fairly good.",
"In particular, quantum simulations provide phase boundaries that are shifted $\\sim 20$ K to lower temperatures as compared to the observations (see Fig.",
"3 in McBride et al., 2012).",
"As it is shown in Fig.",
"REF , QNE play a significant role on the thermodynamic stability of the different phases of ice: the melting lines are shifted to higher temperatures and the solid-solid transitions to higher pressures.",
"Another important difference is that the region of thermodynamic stability of phase II is significantly reduced in the classical phase diagram, as this phase appears there only at temperatures below 80 K (that is, in the classical phase diagram shown in Fig.",
"REF phase II is missing).",
"The origins of these quantum effects have been rationalised in terms of the tetrahedral angular order ascribed to each polymorph and the volume change involved in the phase transformations.",
"The pressure dependence of the crystal volume, bulk modulus, interatomic distances, atomic delocalisation, and kinetic energy in hexagonal ice (I$_{h}$ ) under pressure, have been also analysed thoroughly with similar computational techniques by Herrero and Ramírez (2011b).",
"Despite of the mounting experimental and theoretical evidence showing the importance of QNE, these effects are normally disregarded in most computational studies of water and ice at $T \\ne 0$ conditions.",
"This is due in part to the difficulties encountered in the description of molecular interactions in H$_{2}$ O.",
"Different types of computationally inexpensive empirical potentials, which either assume rigid or flexible molecules and polarisable or non-polarisable ions (e. g., the so-called SCP, TIP4P, q-TIP4P/F and TIP4PQ/2005 force fields), have been employed in most simulation studies of ice at finite temperatures.",
"Some of those force fields have been fitted to reproduce experimental data, to be used subsequently in classical simulation studies, hence they already incorporate QNE effectively.",
"Quantum calculations based on those interaction models, therefore, may provide in some cases worse agreement with the experiments than classical simulation studies due to double counting of quantum nuclear effects (Herrero and Ramírez, 2014).",
"In other words, the inaccurracies affecting common empirical interaction models may disguise to some extent the real influence of QNE by providing reasonably good agreement with the experiments.",
"In some cases it has been actually demonstrated that the combined description of molecular interactions and ionic effects at the quantum level is necessary for reproducing correctly the experimental findings in ice.",
"Examples include the anomalous volume expansion observed in ice isotopes (Pamuk et al., 2012) and the interpretation of measured x-ray absorption spectra (Kong, Wu, and Car, 2012; Kang et al., 2013).",
"Ab initio treatment of the molecular interactions in ice has been mostly done with DFT methods.",
"However, the presence of hydrogen bonds and dispersive long-range forces makes the description of this crystal difficult, demanding the use of computational methods going beyond standard DFT (see Sec.",
"REF ).",
"For a detailed description of the strengths and weaknesses of different exchange-correlation DFT approximations in describing H$_{2}$ O-based systems, we refer the interested reader to Morales et al.",
"(2014) and the recent review by Gillan, Alfè, and Michaelides (2016).",
"We note in passing that application of electronic quantum Monte Carlo methods to the study of bulk ice is very rare (Santra et al., 2011), mainly due to computational affordability issues and the likely bias associated to the use of pseudopotentials (Driver and Militzer, 2012).",
"An advantageous aspect in the simulation of ice, as compared to that of solid helium or hydrogen, is that QNE in principle can be described correctly with the quasi-harmonic approximation (QHA, see Sec.",
"REF ).",
"This conclusion has been attained by several authors based on comparisons provided between ab initio QHA results and neutron scattering experiments (see, for instance, Senesi et al., 2013) and, more convincingly, between QHA and path-integral simulations performed with a same effective interaction model (Pamuk et al., 2012; Ramírez et al., 2012).",
"Consequently, the study of the low-temperature phase diagram and thermodynamic properties of H$_{2}$ O in principle is affordable with first-principles methods.",
"Nevertheless, a note of caution must be added here.",
"Very recently, Engel, Monserrat, and Needs (2015) have shown in a DFT-based study that anharmonic contributions to the free energy can turn out to be decisive for describing correctly the thermodynamic stability of different ice polymorph with very close quasi-harmonic free energies.",
"In particular, it has been shown that anharmonic quantum nuclear effects are decisive in stabilizing the hexagonal I$_{h}$ phase with respect to cubic I$_{c}$ , the latter being a rare form of ice that presents a different stacking of layers of tetrahedrally coordinated water molecules.",
"As noted by the authors of that study, treatment of anharmonicity in general could be crucial for correctly describing the energy differences between similar polymorph in hydrogen-bonded molecular crystals (which can be relevant, for instance, to the pharmaceutical sciences)."
],
[
"Nitrogen and Methane",
"QNE are more pronounced in molecular nitrogen (N$_{2}$ ) and methane (CH$_{4}$ ) than in H$_{2}$ O (see Fig.",
"REF ).",
"This is due to the fact that the intermolecular interactions in the two former systems are dominated by long-range dipole-dipole (CH$_{4}$ ), dipole-quadrupole (CH$_{4}$ ), and quadrupole-quadrupole (N$_{2}$ and CH$_{4}$ ) forces, which are weaker than hydrogen bonds (Cazorla, 2015).",
"Certainly, under normal thermodynamic conditions H$_{2}$ O is a liquid whereas N$_{2}$ and CH$_{4}$ are gases.",
"However, the study of QNE in solid nitrogen and methane is very marginal in comparison to that in ice (or hydrogen).",
"Improving our quantitative understanding of solid N$_{2}$ and CH$_{4}$ is actually important for planetary and energy materials sciences.",
"For example, these species are believed to abound in the surface and interior of Uranus, Neptune, and Pluto (Hubbard et al., 1991; Protopapa et al., 2008).",
"Meanwhile, under high pressures ($\\ge 110$ GPa) molecular nitrogen dissociates into singly bonded polymeric nitrogen, the so-called cubic gauge phase, that is being considered as a potential high-energy-density material because can exist in a metastable form at ambient pressure (Eremets et al., 2004).",
"The $P-T$ phase diagrams of compressed nitrogen and methane are very complex, as it occurs in most molecular systems, due to the prominence of the orientational degrees of freedom.",
"N$_{2}$ exhibits five solid molecular phases at pressures below $\\sim 10$ GPa and temperatures $T \\le 300$ K (Gregoryanz et al., 2007; Tomasino et al., 2014).",
"The low-temperature phases in molecular nitrogen are governed by quadrupole-quadrupole interactions and in moving from zero to higher pressures the crystal first transforms from an orientationally disordered cubic ($\\alpha $ ) to an ordered tetragonal ($\\gamma $ ) phase, and then to an ordered rhombohedral phase ($\\epsilon $ ); when increasing $T$ , a disordered hexagonal phase ($\\beta $ ) first appears at $2.4$ GPa that subsequently transforms into a cubic phase ($\\delta $ ) with orientational disorder by effect of pressure.",
"It is worth noting that large isotopic effects have been observed in the $P$ -induced $\\alpha \\rightarrow \\gamma $ transformation occurring at low temperatures (Scott, 1976), which indicates the presence of important QNE.",
"Some other phases have been observed to stabilise at higher pressures in the experiments, the structures of which are unknown in most cases.",
"This lack of knowledge has motivated an intense theoretical activity in solid nitrogen.",
"Over a dozen of different structures have been predicted to be stable in the pressure range $0 \\le P \\le 400$ GPa; among these we highlight the layered $Pba2$ or $Iba2$ ($188-320$ GPa) and helical tunnel $P2_{1}2_{1}2_{1}$ structures ($>320$ GPa)[Ma et al., 2009], and the cluster form of nitrogen diamondoid ($>350$ GPa) [Wang et al., 2012], which have been obtained through systematic crystal structure searches based on DFT methods.",
"Unquestionably, the results of DFT-based studies on molecular nitrogen are invaluable for advancing in the knowledge of its phase diagram; however, we must note that most of the first-principles investigations presented thus far systematically neglect two basic aspects in N$_{2}$ crystals: long-range dispersion interactions and QNE (i. e., they have been performed with standard LDA and GGA DFT exchange-correlation functionals and disregarding likely zero-point motion effects even through the quasi-harmonic approximation).",
"It could be argued that the importance of these two elements become secondary at high pressures or that somehow they cancel out when comparing the enthalpy of different phases.",
"However, by taking into consideration all the similarities between N$_{2}$ and H$_{2}$ in terms of intermolecular interactions and degree of quantumness, one can suspect that this is not the case (i. e., as it has been explicitly shown in solid hydrogen, see Sec.",
"REF ).",
"Therefore, it is reasonable to think that the transition pressures and phase boundaries reported in standard DFT studies of N$_{2}$ are likely to be inaccurate.",
"With regard to this last point, it was first predicted from standard DFT calculations that molecular nitrogen transforms into a polymeric phase (i. e., cubic gauche $cg$ -N) prior to metallization at a pressure of $\\sim 50$ GPa (Mailhiot, Yang, and McMahan, 1992); this transformation has been observed subsequently in experiments, however, at thermodynamic conditions much higher than the predicted ones, namely, $P \\ge 110$ GPa and $T \\ge 2000$ K (Eremets et al., 2004).",
"Whether the causes of the discrepancies between theory and experiments lie on the use of inaccurate DFT functionals and neglecting of QNE, or the use of incorrect molecular structures in the calculations, or the existence of large kinetic barriers for the dissociation of N$_{2}$ molecules that complicate the measurements, or a combination of all these factors, is not clear yet.",
"Systematic computational studies analysing the importance of QNE and benchmarking the description of intermolecular interactions in highly compressed nitrogen, are very desirable for clarifying these issues.",
"To the best of our knowledge, there is only one computational study by Presber et al.",
"(1998) in which the importance of QNE on the orientational phase transitions in bulk solid N$_{2}$ at low $P$ and low $T$ has been assessed.",
"By using the PIMC method and a classical N$_{2}$ -N$_{2}$ interaction potential, Presber et al.",
"(1998) found that the transition temperature corresponding to the $\\alpha \\rightarrow \\gamma $ transformation is reduced by about 11 % with respect to the result obtained with classical methods.",
"We note that, in spite of the simplicity of the employed interaction model, Presber et al.",
"'s quantum predictions show reasonably good agreement with the experiments.",
"Similarly, the impact of quantum nuclear effects on the orientational ordering of N$_{2}$ molecules adsorbed on graphite has been investigated with PIMC methods by Marx and co-workers in a series of works (Marx et al., 1993; Marx, Sengupta, and Nielaba, 1993; Marx and Müser, 1999).",
"To this end, rigid rotors with their centers of mass pinned on a triangular lattice commensurate with the graphite basal plane, and molecule-molecule and molecule-surface interactions treated with atomistic models and point charges, were analysed.",
"The main conclusions from those studies can be summarised as that quantum fluctuations lead to “10 % effects” on the physical properties of N$_{2}$ films (Marx et al., 1993; Marx, Sengupta, and Nielaba, 1993).",
"For example, the temperature corresponding to the so-called “2-in” herringbone orientational transition that occurs at low temperatures and low densities is shifted down by about 10 % as a result of zero-point motion, in good agreement with the experiments.",
"These results imply that in order to make quantitatively correct predictions in N$_{2}$ crystals QNE must be taken into account.",
"Regarding methane, i. e., CH$_{4}$ , the ground-state phase at low pressures is a cubic structure that can be thought of two molecular sublattices, one of which is orientationally ordered and the other disordered (James and Keenan, 1959).",
"A first-order phase transition between this cubic and a partially ordered phase occurs at a temperature of $20.4$ K (Press and Kollmar, 1975); in CD$_{4}$ , a similar transition occurs but at a higher temperature of $27.4$ K (Press, 1972).",
"This large isotopic effect, again, marks the presence of significant QNE.",
"In fact, by using the PIMC method and a model potential based on ab initio results, Müser and Berne (1996) were able to replicate such a large isotopic shift in the transition temperature, otherwise not reproducible with classical methods.",
"The phase diagram of methane at high pressures, on the other hand, remains contentious.",
"Up to nine different phases have been observed in CH$_{4}$ at pressures below $\\sim 10$ GPa and temperatures $0 \\le T \\le 300$ K (Bini and Pratesi, 1997; Maynard-Casely et al., 2010), and only three of them have been determined.",
"For instance, based on neutron scattering measurements Maynard-Casely et al.",
"(2010) have proposed that the so-called phase A, which appears at pressures about 1 GPa and temperatures above $\\sim 100$ K, consists of 21 molecules in a rhombohedral unit cell that is strongly distorted with respect to the cubic ground-state.",
"Using systematic crystal structure searches based on a genetic algorithm and dispersion-corrected DFT methods, Zhu et al.",
"(2012) have predicted a similar candidate structure for phase A that, in contrast to the experimentally determined one, presents orientationally disordered molecules.",
"At pressures beyond $\\sim 100$ GPa, CH$_{4}$ is expected to become chemically unstable and to decompose (Gao et al., 2010).",
"Unfortunately, possibly due in part to the lack of knowledge on the molecular phases that appear below that pressure limit, the impact of QNE on the high-$P$ and low-$T$ phase diagram of solid methane remains largely unexplored.",
"Recent simulation studies by Goldman, Reed, and Fried (2009) and Qi and Reed (2012), have shown that quantum nuclear effects in fact are crucial for understanding the behavior of solid CH$_{4}$ at high-$P$ and high-$T$ conditions.",
"By adopting a quantum thermal bath scheme to treat QNE (see Sec.",
"REF ) and a multi-scale simulation approach to model the molecular interactions, Qi and Reed (2012) have quantified the impact of QNE on the Hugoniot of compressed methane.",
"It has been found that quantum nuclear effects are responsible for a huge shift of $\\sim 40$ % to lower pressures in the onset of decomposition.",
"The primary factor behind such a tremendous effect has been ascribed to the large variation in the heat capacity that occurs when QNE are considered.",
"In a previous work, some of those authors had already shown that quantum temperature corrections to classical DFT calculations on the Hugoniot of methane were as large as $20-30$ %, and that these improved the agreement with the experiments (Goldman, Reed, and Fried, 2009).",
"Analogously to the situation explained for solid N$_{2}$ , there is a pressing need for unravelling the influence of QNE on the thermodynamic and structural properties of methane at low and high pressures.",
"From this knowledge, our description and understanding of quantum molecular crystals could be improved significantly."
],
[
"Quantum materials science",
"Here we explain the physical properties of crystals that are technologically relevant and in which at the same time the influence of QNE is significant.",
"In this category we include from light-weight and metallic crystals (e. g., Li) to heavy-weight and insulator compounds (e. g., BaTiO$_{3}$ ).",
"The former systems respond to the traditional definition of a quantum crystal, that is, solids composed of low-$Z$ atoms that interact through relatively weak forces.",
"The latter systems are better described as highly anharmonic crystals with a shallow multi-well potential energy surface (PES).",
"In this case, disparate phases, identified with local minima in the PES, are energetically very competitive and thus QNE play a crucial role on stabilising one or another (see Fig.",
"REF a-b)."
],
[
"Perovskite oxides",
"Perovskite oxides have the general formula ABO$_{3}$ in which A and B are cations, the latter being a transition metal element with a smaller radius than A.",
"The ideal perovskite structure is cubic with space group $Pm\\overline{3}m$ , where the B cation is 6-fold coordinated with the oxygen anions and A 12-fold coordinated.",
"Perovskite oxides display many interesting physical properties like, for instance, ferroelectricity (e. g., BaTiO$_{3}$ ), ferromagnetism (e. g., SrRuO$_{3}$ ), multiple coupled ferroic orders (e. g., BiFeO$_{3}$ ), and insulator-to-metallic transitions (e. g., LaCoO$_{3}$ ).",
"All these properties are very sensitive to the chemistry, crystalline defects, electrical boundary conditions, and applied stress, so that they can be tuned externally.",
"For this reason, pervoskite oxides normally are referred to as functional materials in the literature (Ohtomo and Hwang, 2004; Schlom et al., 2007; Cazorla and Stengel, 2012).",
"BaTiO$_{3}$ is an archetypal ferroelectric.",
"At room temperature this material adopts a rhombohedral (R) phase that displays a spontaneous and switchable ferroelectric polarisation.",
"As temperature is raised from zero to $\\sim 300$ K, BaTiO$_{3}$ goes through the series of phase transformations R $\\rightarrow $ O $\\rightarrow $ T $\\rightarrow $ C, where “O” stands for an orthorhombic phase, “T” for a tetragonal, and “C” for a cubic.",
"The same sequence of phases is observed under pressure (see Fig.",
"REF ).",
"The high-$T$ cubic phase corresponds to the ideal perovskite structure, which is non-polar (that is, has null ferroelectric polarisation).",
"The appearance of ferroelectricity in perovskite oxides is originated by a delicate balance between long-range Coulomb interactions, that favor the ferroelectric state, and short-range repulsive forces, that favor the cubic non-polar state; the hybridisation between B cation $d$ electronic orbitals and oxygen $2p$ plays an essential role on that equilibrium (Cohen, 1992).",
"The ferroelectric phase transition in BaTiO$_{3}$ is considered to be an example of displacive transition, in which a zone-center vibrational mode, called “soft”, has a vanishing frequency at the phase transition and its eigenvector is similar to the atomic displacements observed in the ferroelectric state.",
"Therefore, BaTiO$_{3}$ is a highly anharmonic crystal in which several phases are energetically very competitive.",
"In 2002, ${\\rm \\acute{I}}$${\\rm \\tilde{n}}$ iguez and Vanderbilt estimated the impact of QNE on the $P-T$ phase diagram and ferroelectric properties of BaTiO$_{3}$ .",
"Using the PIMC approach based on an effective Hamiltonian model fitted to DFT results, the authors of that study found that the $P-T$ boundaries separating the regions of stability of different phases varied considerably when considering QNE.",
"As it is shown in Fig REF , the phase boundaries are noticeably shifted towards lower pressures and temperatures as compared to those found in the classical phase diagram.",
"As a result, the agreement between theory and experiments was improved significantly.",
"${\\rm \\acute{I}}$${\\rm \\tilde{n}}$ iguez and Vanderbilt (2002) also found that the electrical polarisation in BaTiO$_{3}$ shrinks by about 10 % when considering quantum fluctuations, and that the same quantity exhibits null variation in the $T \\rightarrow 0$ limit [as it is expected from quantum arguments (Hayward and Salje, 1998)].",
"More recently, Geneste et al.",
"(2013) have analysed the influence of QNE on the dielectric permittivity and piezoelectric constants of rhombohedral BaTiO$_{3}$ .",
"Using a path-integral molecular dynamic approach based on the same effective Hamiltonian model than employed by ${\\rm \\acute{I}}$${\\rm \\tilde{n}}$ iguez and Vanderbilt (2002), Geneste et al.",
"(2013) have found that inclusion of quantum nuclear effects systematically enhances the dielectric response and piezoelectric constants of the crystal by approximately a factor of 2.",
"This huge effect has been explained in terms of the strong anharmonicity of BaTiO$_{3}$ , which is retained by the crystal down to zero temperature.",
"Geneste et al.",
"'s findings suggest that quasi-harmonic approaches are not adequate for describing the behavior of displacive ferroelectrics at low temperatures.",
"QNE can influence strongly the low-$T$ response of a system when is near to a structural phase transition.",
"In most ferroelectrics, the stability limit of the polar phase, $T_{C}$ , falls within a region governed by classical Boltzmann fluctuations.",
"In few crystals known as “quantum paraelectrics”, however, $T_{C}$ is very close to the zero-temperature limit and thus quantum statistical fluctuations play a dominant role in the transformation (Müller and Burkard, 1979; Rytz, Höchli, and Bilz, 1980; Conduit and Simons, 2010).",
"Examples of quantum paraelectrics include the perovskite oxides SrTiO$_{3}$ , KTaO$_{3}$ , and KTaO$_{3}$ -NaTaO$_{3}$ and KTaO$_{3}$ -KNbO$_{3}$ solid solutions (Höchli and Boatner, 1979; Samara, 1988).",
"At low temperatures, the dielectric properties of a quantum paraelectric are appreciably different from those of a classical material.",
"For instance, the Curie-Weiss law describing the variation of the static dielectric constant, $\\epsilon $ , near $T_{C}$ , namely, $ \\propto (T - T_{C})^{-1}$ , is not longer fulfilled; instead $\\epsilon $ follows a $\\propto T^{-2}$ relation (Höchli and Boatner, 1979).",
"As the temperature is raised, or as a specific tuning parameter that induces atomic displacements is varied (e. g., pressure), the dielectric behavior in quantum and classical polar materials eventually become analogous.",
"Therefore, a classical-quantum crossover regime exists in quantum paraelectrics in which intriguing quantum phenomena can be expected to occur (see sketch in Fig REF c).",
"This is the case, for instance, of ferroelectric quantum criticality, which have been recently observed in SrTiO$_{3}$ and KTaO$_{3}$ crystals (Rowley et al., 2014).",
"In particular, the inverse of the dielectric constant in these materials, which below 50 K follow the non-classical $\\propto T^{2}$ dependence, experiences an anomalous upturn at very low temperatures that extends into the millikelvin range.",
"This unexpected effect has been rationalised in terms of quantum criticality theory, after considering the influence of long-range dipolar interactions and of the coupling of the electrical polarisation with acoustic phonons (Rowley et al., 2014).",
"The quantum critical regime associated to quantum paraelectrics is significantly different (e. g., in terms of the collective dynamics and tuning parameter) from the better known quantum regime occurring in quantum ferromagnetic materials (e. g., Ni$_{3}$ Ga and ZrZn$_{2}$ ); interesting new prospects in the field of quantum phase transitions, therefore, appear to be opened.",
"Quantum paraelectrics are also important from a technological point of view.",
"Currently, there is a great interest in exploiting magnetoelectric (ME) effects, which are responsible for the coupling between the electrical and magnetic degrees of freedom in multiferroic crystals, for nanoelectronics applications.",
"ME effects could be used to induce the reversal of the magnetisation in a material with an electric field, making it possible to store information in advanced electronic devices with minimal power consumption (i. e., creating magnetic fields generally involves higher energy expenses than electric fields).",
"For ME effects to be practical, the value of the magnetic and electrical susceptibilities need to be large around a same transition temperature.",
"Unfortunately, this rarely occurs in any material.",
"Recently, Shvartsman et al.",
"(2010) have measured a large ME effect in EuTiO$_{3}$ near $T_{N}= 5$ K, a quantum paraelectric that undergoes an anti-ferromagnetic to paramagnetic phase transition at very low temperatures.",
"The magnetoelectric moments revealed at the magnetic phase transition are comparable to those found in benchmark multiferroic crystals such as TbPO$_{4}$ .",
"Shvartsman et al.",
"'s findings suggest that quantum paraelectrics could be promising for nanoelectronic applications.",
"Nonetheless, for the realisation of practical devices based on quantum paraelectrics the observed magnetoelectric activity should be brought closer to room temperature.",
"High compression could represent a solution to this problem as it can extend the regime in which QNE remain important while simultaneously shifting $T_{N}$ towards higher temperatures.",
"In this last regard, the outcomes of quantum simulation studies based on first-principles methods could be very insightful.",
"Actually, recent phonon calculations by Evarestov et al.",
"(2011) performed with hybrid DFT methods (see Sec.",
"REF ) have accurately reproduced the experimental $T$ -dependence of the heat capacity in SrTiO$_{3}$ .",
"On the other hand, theoretical approaches that allow to estimate $T$ -renormalised phonon modes and frequencies are already well established (e. g., velocity auto-correlation and self-consistent harmonic methods, see: Teweldeberhan, Dubois, and Bonev, 2010; Errea, Rousseau, and Bergara, 2011; Errea, Calandra, and Mauri, 2014).",
"Nevertheless, to the best of our knowledge, full ab initio studies of quantum paraelectrics under pressure are absent in the literature."
],
[
"H-bond ferroelectrics",
"H-bond ferroelectrics normally consist of polar stacks of sheets of hydrogen-bonded molecules.",
"Hydrogen bonding can create electrical dipoles in crystals among hydrogen-donating molecules, which become partially negative, and hydrogen-accepting molecules, which become partially positive.",
"Upon application of an electric field, protons associated with one molecule shift cooperatively towards a hydrogen-bonded neighbor, switching the molecular dipole and thus producing a large electrical polarisation (Horiuchi and Tokura, 2008).",
"Examples of H-bond ferroelectrics include the molecular compounds KH$_{2}$ PO$_{4}$ (KDP) and C$_{4}$ H$_{2}$ O$_{4}$ (H2SQ), which present some well-characterised crystal structures and can be synthesised both in standard and deuterated forms.",
"A key aspect in the ferroelectric behavior of most H-bonded ferroelectrics is the motion of H atoms in correlated double well potentials.",
"This correlation consists of H atoms in neighboring H-bonds being strongly coupled due to the energetic requirement for satisfying the “ice rules” (Singer et al., 2005); in the KDP and H2SQ systems this condition implies that each molecule participates in 4 different H-bonds, two of which have donating character and the rest accepting.",
"H-bond ferroelectric materials currently are attracting a lot of attention because polar order near room-temperature has been revealed in some organic species (Horiuchi et al., 2005; Horiuchi et al., 2010).",
"This finding opens the possibility for manufacturing cheaper and more environment friendly nanoelectronic components and devices.",
"Interestingly, the Curie transition temperature, $T_{C}$ , in H-bond ferroelectrics can increase by about 100 K upon deuteration.",
"The origins of such an enormous isotopic effect however, remain contentious.",
"Originally, a simple quantum model consisting of proton quantum-tunneling in a double-well potential was proposed to rationalise the $T_{C}$ observations (Blinc and Svetina, 1966); however, this simple model failed to explain the so-called Ubbelohde effect, which relates the experimentally observed elongation of H bonds upon deuteration to a purely geometric origin (Ubbelohde and Gallagher, 1955).",
"Subsequently, models involving coupled vibrational lattice modes and proton dynamics were proposed (Dalal, Klymachyov, and Bussmann-Holder, 1998) that led to the conviction that quantum-tunneling effects were not necessary for explaining the isotopic influence on $T_{C}$ (McMahon et al., 1990).",
"More recently, however, neutron Compton scattering experiments performed on KDP have found evidence for coherent proton quantum tunneling occurring at temperatures above the ferroelectric transition ($T \\sim 125$ K), whereas no such evidence is found in the analogous deuterated system (Reiter, Mayers, and Platzman, 2002; Reiter et al., 2008).",
"On the theoretical side, some authors have attempted to reconcile the differing interpretations by arguing that a mechanism behind the Ubbelohde effect itself might be collective quantum tunneling in atomic clusters (Koval et al., 2002).",
"Quantum ab initio studies of H-bond ferroelectrics are very desirable to help in clarifying the controversy about the relevance of QNE on the observed $T_{C}$ isotope dependence.",
"Nevertheless, due to the large size of the unit cells involved and complex collective dynamics of hydrogen/deuterium atoms, quantum simulation of H-bond ferroelectrics turns out to be very challenging; to the best of our knowledge, the number of published works on this topic can be counted with one hand.",
"Srinivasan and Sebastiani (2011) have performed ab initio PIMD simulations in KDP (i. e., KH$_{2}$ PO$_{4}$ ) and DKDP crystals, in order to estimate the degree of quantum-mechanical localisation of hydrogen and deuterium atoms in the paraelectric phase.",
"In both systems, they have found that proton quantum delocalisation in the OH$\\cdots $ O hydrogen bond is necessary for stabilisation of the disordered state.",
"The only difference between KDP and DKDP is that quantum tunneling occurs coherently in the former system whereas incoherently in the latter, an effect that has been linked by the authors to the observed $T_{C}$ dependence on the isotope.",
"Recently, Wikfeldt and Michaelides (2014) have employed ab initio PIMD simulations based on DFT to investigate the importance of QNE on the atomic ordering and structure of H2SQ (i. e., C$_{4}$ H$_{2}$ O$_{4}$ ).",
"We note that in this case the authors have explicitly considered long-range van der Waals interactions by using a dispersion-corrected DFT functional (see Sec.",
"REF ).",
"It has been found that concerted proton jumps along H-bond chains are facilitated dramatically by quantum tunneling of several protons occurring at the same time.",
"According to Wikfeldt and Michaelides' results, QNE are crucial in this order-disorder phase transition (that is, ferroelectric to paraelectric).",
"The same phenomenology has been observed also in the analogous deuterated crystal but in a smaller magnitude, leading to an Ubbelohde effect that is in good agreement with the experiments (i. e., elongation of the oxygen-oxygen distances by $\\sim 0.02$ Å).",
"Subsequently, Wikfeldt (2014) has introduced a simple model for a coupled one-dimensional H-bond chain that has been parametrised to DFT calculations performed in H2SQ.",
"Such an effective model allows for an efficient exploration of QNE in larger systems over longer simulation times.",
"The PIMC results obtained with Wikfeldt's H2SQ model in fact appear to be consistent with the conclusions presented in a previous full ab initio work (Wikfeldt and Michaelides, 2014).",
"Further systematic studies are necessary to determine exactly how important QNE are for the understanding of proton dynamics and proton order in H-bond ferroelectrics.",
"The computational evidence gathered to date appears to indicate that quantum nuclear effects certainly are crucial.",
"Reassuringly, in a recent ab initio PIMC study Li, Walker, and Michaelides (2011) have shown that the quantum nature of the H-bond manifests appreciably in most hydrogen-bonded materials."
],
[
"Lithium and related compounds",
"Lithium (Li) is the lightest metallic element; at ambient conditions it is most stable in a cubic bcc crystal.",
"Li represents the prototype of a simple metal with a Fermi surface that is nearly spherical.",
"As pressure is increased, however, this material undergoes a series of symmetry-breaking structural transformations that provoke an increase in complexity on its electronic band-structure (Guillaume et al., 2011).",
"The presence of QNE in solid Li under pressure is notable.",
"Experimentally, large isotopic effects have been observed in the equation of state and elastic properties of the bulk crystal at low temperatures ($T \\le 77$ K) and pressures up to $\\sim 2$ GPa (Gromnitskaya, Stal'gorova, and S. M. Stishov, 1999).",
"For instance, differences of about 7 % have been reported for the transversal and longitudinal sound-wave velocities in solid $^{6}$ Li and $^{7}$ Li at low and high pressures.",
"On the theoretical side, Filippi and Ceperley (1998) have analysed the influence of quantum nuclear effects on the kinetic energy of the crystal with PIMC simulations based on a pairwise interaction potential.",
"It has been found that the excess kinetic energy in Li decreases from about $10.4$ % of the classical value at 300 K to $4.5$ % at 450 K, hence QNE are important all the way up to melting.",
"The role of QNE in the structural and electronic properties of small Li clusters has been estimated also as very influential by Rousseau and Marx (1998).",
"One of the effects of applying pressure in a crystal is to increase the kinetic energy of the atoms.",
"It has been argued theoretically that if the increment in zero-point motion due to compression is higher than that attained in the potential energy, eventually the crystal could melt at low temperatures.",
"This possibility has attracted a lot of attention in hydrogen since according to some theoretical arguments and effective models a metallic liquid with exotic properties could be stabilised in the regime of Mbar pressures (Babaev, Sudbø, and Ashcroft, 2004).",
"In solid lithium it has been experimentally observed (Lazicki, Fei, and Hemley, 2010) and calculated with first-principles methods (Hernández et al., 2010) that at a pressure of $\\sim 10$ GPa the corresponding melting line develops a negative slope (in analogy to what occurs in sodium at $P \\sim 30$ GPa).",
"This finding appears to open an alternative for the possible realisation of a ground-state metallic liquid at high pressures (although possibly in the Mbar regime, or even at much higher pressures).",
"The role of QNE on the sudden drop observed in the melting line of Li, however, remains controversial.",
"Guillaume et al.",
"(2011) have measured a melting temperature of $\\sim 190$ K at a pressure of $\\sim 40$ GPa, which represents by far the lowest melting temperature observed for any material at such pressures.",
"The authors of this work have suggested that QNE play an important role in shaping the phase diagram of Li.",
"This suggestion seems to be consistent with the fact that classical first-principles simulations (Hernández et al., 2010) provide melting temperatures which are about 100 K higher than the experimental points obtained by Guillaume et al.",
"(2011).",
"A more recent experiment by Schaeffer et al.",
"(2012), however, has revealed a totally different scenario in which excellent agreement with the classical ab initio results by Hernández et al.",
"(2010) is obtained.",
"Then, what is the real extent of QNE on the melting properties of Li?",
"Very recently, Feng et al.",
"have performed a systematic first-principles PIMD study aimed at answering this question.",
"These authors have found that the net effect of considering QNE on the melting temperature of Li is minimal (e. g., a small shift of 15 K towards lower temperatures at $P = 45$ GPa).",
"Interestingly, QNE influence noticeably the free energy of the solid and liquid phases by separate, however there is a strong QNE compensation effect between the two phases at melting (Feng et al., 2015).",
"Light-weight materials based on lithium are important from a technological point of view, and thus so are QNE.",
"Two interesting examples are lithium hydride (LiH) and lithium imide (Li$_{2}$ NH).",
"LiH is used in the nuclear industry either as a shielding agent or fuel in energy reactors (Welch, 1974; Veleckis, 1977).",
"LiH is an ionic crystal that stabilises in the rocksalt structure at ambient conditions; Li is the cation (positively charged ion) and H the anion (negatively charged ion).",
"The presence of large QNE in LiH has been reported both in experiments and quantum simulations (Boronat et al., 2004).",
"At room temperature the experimental Lindemann ratio of the hydrogen ion amounts to $0.12$ (Vidal and Vidal-Valat, 1986), which lies in between those measured for solid H$_{2}$ and Ne (i. e., $0.18$ and $0.09$ , respectively), thus indicating very strong quantum character.",
"Large quantum isotopic effects have been reported in lithium hydride for numerous quantities including the kinetic energy, Lindemann ratio, and lattice parameter.",
"For example, Cazorla and Boronat (2005) have estimated by means of VMC calculations based on classical interatomic potentials that the kinetic energy of the hydrogen anion at zero temperature changes from $84(1)$ in LiH to $67(1)$ meV in LiD.",
"Meanwhile, the corresponding Lindemann ratio is reduced by about 24 % in LiD as compared to that in LiH.",
"More recently, Dammak et al.",
"(2012) have found by using a quantum thermostat approach in combination with first-principles methods, that the lattice parameter difference between LiH and LiD amounts to $0.019$ and $0.016$ Å at 0 and 300 K, respectively, in fairly good agreement with the experiments (namely, $0.016$ and $0.014$ Å).",
"QNE can also affect considerably the electronic properties of a quantum solid, in particular the electronic energy band gap, $E_{g}$ , due to the presence of electron-phonon couplings.",
"By using a first-principles approach that consistently takes into account anharmonic and zero-point motion effects, Monserrat, Drummond and Needs (2013) have calculated the quantum-mechanical expectation value of $E_{g}$ in LiH and LiD over the temperature interval $0 \\le T \\le 800$ K. They have found that the isotopic effect in $E_{g}$ roughly amounts to $4-7$ %, with LiD exhibiting always the largest energy band gap.",
"Interestingly, QNE at zero temperature account for a $E_{g}$ variation of $\\sim 2$ % as compared to the value calculated with classical methods, which is $\\sim 3.00$ eV.",
"Lithium imide (Li$_{2}$ NH) is a very promising hydrogen storage material due to its low molecular weight and central role played on the decomposition reaction (Shevlin and Guo, 2009): ${\\rm LiNH_{2} + 2LiH} && \\leftrightarrow {\\rm Li_{2}NH + LiH + H_{2}} \\nonumber \\\\&& \\leftrightarrow {\\rm Li_{3}N + 2H_{2}}~,$ where in the first stage a total of $5.5$ wt% H$_{2}$ is released and of $5.2$ wt% H$_{2}$ in the second.",
"A lot of research, both of computational and experimental nature, has been devoted to understand the atomic structure and phase transitions occurring in Li$_{2}$ NH.",
"Due to the light mass of the atoms and relatively weak interactions between particles, QNE are likely to affect the fundamental properties of this material.",
"Zhang, Dyer, and Alavi (2005) have solved the Schrödinger equation of a proton in the potential energy surface of Li$_{2}$ NH calculated with DFT methods, to analyse the influence of QNE on its dynamics.",
"It has been found that the quantum character of H atoms is very strong, leading to partial delocalisation of the proton around certain N centers through quantum tunneling.",
"The origin of this effect has been traced back to the relatively flat potential energy landscape of the system.",
"The results of a more recent computational study by Ludue${\\rm \\tilde{n}}$ a and Sebastiani (2010) based on ab initio PIMD simulations, appear to support the validity of these results.",
"The proton momentum distribution in Li$_{2}$ NH has been experimentally measured with inelastic neutron scattering techniques, and calculated with quantum thermostatted ab initio molecular dynamics (Ceriotti et al., 2010b).",
"The reported experimental and computational $n({\\bf k})$ results are in good agreement, providing a large average kinetic energy of 415 K for Li atoms, of 410 K for N, and of 858 K for H. The presence of proton quantum tunneling and large zero-point motion in Li$_{2}$ NH has several implications.",
"First, the conventional treatment of quantum nuclear effects through the quasi-harmonic approximation should not be adequate in this system.",
"And second, the real energy barrier for hydrogen diffusion, a key parameter for understanding and designing new H-storage materials, must be lower than predicted with classical simulation methods.",
"In fact, Zhang, Dyer, and Alavi (2005) have estimated that at room temperature the H diffusion coefficient in Li$_{2}$ NH is about 4 orders of magnitude higher than the one expected from classical theory.",
"In the light of these results, full quantum treatment of hydrogen atoms in crystals similar to Li$_{2}$ NH (e. g., LiBH$_{4}$ ) may allow for an improved rational engineering of H-storage materials."
],
[
"Carbon-based crystals and nanomaterials",
"Carbon atoms are found in a large number of technologically relevant materials, including diamond and the prolific family of carbon nanostructures (e. g., graphene, nanotubes, and nanohorns).",
"Diamonds, which due to their strong covalent atomic bonds possess superlative hardness and thermal conductivity, are used as anvil cells to study condensed matter systems over wide $P-T$ ranges, and also have a major industrial application as cutting and polishing tools.",
"At low temperatures, diamond is an archetypal quasi-harmonic crystal (Ceriotti, Bussi, and Parrinello, 2009) in which the presence of QNE has a profound impact on its structural, elastic, and electronic band-structure features.",
"Herrero and Ramírez (2000) have studied the influence of zero-point motion on the thermodynamic properties of this solid with PIMC simulations based on an empirical interatomic potential.",
"They have found that QNE account for an increase of $0.5$ % in the lattice parameter and a decrease of 5 % in the bulk modulus with respect to the values obtained with classical simulation methods.",
"More recently, Monserrat, Drummond, and Needs (2013) have estimated with a fully anharmonic first-principles approach that the zero-point motion renormalisation of the electronic energy band gap in diamond amounts to $-461$ meV.",
"The origins of this large effect have been discussed in detail by Monserrat and Needs (2014), in terms of important electron-phonon interactions that affect differently valence and conduction band electrons (e. g., the latter being specially sensitive to the size of the Lindemann ratio in the crystal).",
"Diamondoids, namely nanocages with formula C$_{x}$ H$_{y}$ in which the carbon atoms are $sp^{3}$ bonded like in diamond, are biocompatible and superhard molecules.",
"These organic nanoparticles are found in large concentrations in petroleum fluids and currently are attracting a lof of attention due to their potential use in drug delivery and nanotechnology applications (Mochalin et al., 2012).",
"In analogy to diamond, a strong electron-phonon coupling is expected to occur in diamondoids.",
"Recently, Patrick and Giustino (2013) have demonstrated by means of first-principles simulations combined with Monte Carlo sampling techniques that the role of QNE on the “photophysics” of these molecules is pivotal.",
"In particular, for the theoretically calculated optical absorption spectra of diamondoids to be in quantitative agreement with the experiments, the zero-point motion of the atoms must be taken into account.",
"Also, the accompanying renormalisation of the electronic energy band gaps amounts to $0.4-0.6$ eV, depending on the selected C$_{x}$ H$_{y}$ species, which coincides with the results obtained by Monserrat, Drummond, and Needs (2013) in diamond.",
"QNE can affect significantly the gas adsorption and transport properties of carbon-based nanostructures from zero up to room temperature.",
"A case study that has been thoroughly investigated both with theory and experiments is the adsorption and diffusion of hydrogen molecules and atoms on graphene and other related nanomaterials.",
"To understand this problem correctly is critical from a technological point of view [e. g., for the design of improved hydrogen storage materials (Cazorla, 2015)] and also for fundamental reasons [e. g., to rationalise the formation of molecular hydrogen in the interstellar medium and improve the astrophysical models of star evolution (Bromley et al., 2014)].",
"Experimental evidence of the importance of QNE in hydrogenated carbon-based surfaces and cavities is abundant.",
"Tanaka et al.",
"(2005) have measured the adsorption of H$_{2}$ and D$_{2}$ on single-wall carbon nanohorns at $T = 77$ K and reported appreciably different behaviors in the two cases.",
"In particular, around $6-7$ % more of deuterium molecules are adsorbed on the interior of the nanoparticles.",
"The observed kinetic isotope effect has been ascribed, on basis to the results of path-integral grand canonical MC simulations performed with semi-empirical potentials, to the presence of QNE that favor the localisation of D$_{2}$ in the cone part of the nanohorns.",
"A similar adsorption isotope effect has been reported also for graphene, which has been interpreted in terms of similar quantum-mechanically nuclear arguments (Paris et al., 2013).",
"Lovell et al.",
"(2009) have studied the room-temperature adsorption of H$_{2}$ in the graphite intercalation compound KC$_{24}$ with inelastic neutron scattering techniques.",
"By comparing their experimental data to the results of quantum first-principles simulations, they have concluded that QNE are responsible for a tremendous reduction of $\\sim 60$ % in the amount of taken gas.",
"On the purely computational side, Kowalczyk et al.",
"(2007) have described the physical adsorption of molecular hydrogen in slit-like carbon nanopores at low temperatures and high gas densities, using classical and path-integral grand canonical Monte Carlo simulations based on semi-empirical interatomic potentials.",
"It has been found that classical simulations overestimate the amount of hydrogen in carbon nanopores due to neglecting of QNE (although the differences between the classical and quantum predictions are ameliorated when the size of the slit-carbon pore diameter is wider than $\\sim 6$ Å ).",
"Herrero and Ramírez (2010) have studied with PIMD simulations and a tight-binding potential fitted to DFT calculations, the finite-temperature properties of H$_{2}$ molecules adsorbed in graphite.",
"It has been shown that H$_{2}$ molecules are disposed parallel to the graphite-layer plane and that they can rotate freely about their center of mass in that plane.",
"The stretching mode of the hydrogen molecule is found to change considerably under graphitic confinement by reducing its frequency $\\sim 3.5$ % with respect to the isolated molecule.",
"Herrero and Ramírez (2010) have also reported strong quantum isotopic effects in this system; for instance, at room temperature the ratio between the kinetic energy of H$_{2}$ and D$_{2}$ amounts to $1.31$ , where $E_{\\rm k}$ (H$_{2}$ ) is equal to $0.238$ eV.",
"Kowalczyk et al.",
"(2015) have also investigated the structural and dynamical properties of hydrogen and deuterium molecules adsorbed in the interior of carbon-based nanotubes at low temperatures, using PIMD techniques and classical force fields.",
"A large isotope effect caused by QNE has been revealed that consists of H$_{2}$ molecules diffusing seven to eight times faster than D$_{2}$ on the inner H$_{2}$ /D$_{2}$ monolayer that coats the carbon atoms.",
"This effect, which is quantum in nature, could be exploited in light-weight isotope separation processes employing nanoporous molecular sieves.",
"Several quantum studies involving a variety of simulation techniques have been performed to investigate also the chemisorption and diffusivity of H atoms on graphene (see, for instance, Herrero and Ramírez, 2009; Garashchuk et al., 2013; Karlický, Lepetit, and Lemoine, 2014; Bonfanti et al., 2015).",
"The general picture deriving from all these works is that QNE appreciably facilitate both the adsorption and posterior diffusion of hydrogen atoms on the carbon surface.",
"Consequently, hydrogenation of large areas of graphene could be achieved more easily in practice than previously inferred from classical simulation studies.",
"Interestingly, Davidson et al.",
"(2014) have pointed to the need of explicitly considering van der Waals forces in this type of quantum simulation studies; the estimated energetic barriers for the chemisorption and diffusion of H atoms then are reduced further, in some cases as much as $\\sim 25$ % (depending on the employed DFT functional).",
"In view of the results presented in the last part of this section, we can conclude that inclusion of QNE and long-range dispersive interactions in modeling of hydrogenated carbon-based nanomaterials is necessary for providing a realistic estimation of gas-adsorption capacities and transition states at low temperatures."
],
[
"Summary and Outlook",
"We have presented an overview of the current understanding of quantum crystals formed by atoms and small molecules over wide thermodynamic intervals, focusing on the insights provided by quantum simulations.",
"We have described the fundamentals of the computational methods that are used to study QNE in quantum solids including variational, projector and path-integral Monte Carlo techniques, among others.",
"Also, we have explained the basic notions of popular first-principles electronic band-structure methods (e. g., DFT and eQMC) as applied to the description of atomic interactions in crystals.",
"Our analysis shows that consideration of QNE in computer simulation studies of rare-gases, molecular solids, H-bond ferroelectrics, light-weight ionic compounds, carbon-based nanomaterials, and even some perovskite oxides, is crucial for understanding the origins of their energy, structural and functional properties at low temperatures.",
"In most quantum crystals (e. g., $^{4}$ He, H$_{2}$ , Li$_{2}$ NH, and BaTiO$_{3}$ ) quasi-harmonic approaches turn out to be inadequate for describing their thermodynamic stability and the energy differences between energetically competitive phases; one instead has to consider methods that fully take into account anharmonicity.",
"Meanwhile, the interatomic interactions in quantum solids normally are not described correctly by standard first-principles (LDA and GGA DFT functionals) or semi-empirical approaches.",
"Combination of these two factors makes the simulation of quantum solids very challenging.",
"QNE are important in a large number of systems and processes that are relevant to materials science.",
"These include, hydrogen storage (e. g., Li$_{2}$ NH and LiH), perovskite oxides (e. g., BaTiO$_{3}$ and SrTiO$_{3}$ ), ferroelectricity (e. g., H-bonded polar compounds), solid plasticity (e. g., $^{4}$ He), and high-energy density materials (e. g., N$_{2}$ ).",
"It is also likely that QNE are more influential than previously assumed in systems that are relevant to the pharmaceutical industry (molecular crystal polymorph) and catalysis (diffusion and adsorption of small molecules on carbon-based and metallic surfaces).",
"We hope that our review will motivate new investigations in the context of materials science that will take into consideration the quantum nature of atoms in particular systems and processes.",
"In spite of all the insight gathered in quantum solids, there are still a few remaining aspects that need to be better understood.",
"These are essentially related to comprehension of (i) the behavior of different types of crystalline defects and the interactions between them, and (ii) the energy, structural, and dynamical properties of quantum crystals under extreme $P-T$ conditions.",
"Advancing in the first of these two challenges is crucial for substantiating the microscopic arguments that have been proposed to explain the intriguing plastic phenomena observed in solid $^{4}$ He at ultra-low temperatures.",
"In particular, a quantitative description of dislocations at the atomic scale and of their interactions with isotopic $^{3}$ He impurities is still pending.",
"Quantum simulations could contribute significantly to this endeavor.",
"Nevertheless, due to the large size of the simulation cells involved ($\\sim 10^{4}-10^{5}$ atoms) and inherent structural complexity of line defects, this progress is slow at the moment (see, for instance, Boninsegni et al., 2007 and Landinez-Borda, Cai, and de Koning, 2016).",
"Meanwhile, the crystal structures appearing in the phase diagram of most molecular solids at high pressures either are vaguely characterised or unknown.",
"The outcomes of systematic structural searches based on first-principles methods in fact have been very useful to better identify them.",
"Nevertheless, the influence of QNE on the thermodynamic stability of different high-$P$ polymorph generally are disregarded in computational studies (see, for instance, the case of N$_{2}$ and CH$_{4}$ ), or considered straightforwardly within the quasi-harmonic approximation.",
"It is worth stressing once again that a consequence of applying pressure in a crystal is to extend the temperature range over which QNE are relevant; therefore, the presence of quantum nuclear effects like zero-point motion, quantum atomic exchanges and quantum tunneling, the majority of which are not reproduced correctly by harmonic-based approaches, are key aspects for understanding the properties of molecular solids under extreme thermodynamic conditions.",
"Such a comprehension is crucial to advance our knowledge in planetary sciences.",
"Common to these challenges is the underlying problem about how to describe the interactions between atoms in quantum crystals correctly.",
"As we have explained before, these interactions require to go beyond standard first-principles approaches which, in addition to the unavoidable task of treating QNE, sometimes makes the simulation of quantum solids prohibitive in terms of computational expense.",
"To this regard, the outcomes of systematic benchmark studies involving non-standard DFT and eQMC methods are crucial for rigorously establishing acceptable balances between numerical accuracy and computational load.",
"Further progress in current electronic-band structure algorithms, on one hand, and improvement on the availability of quantum computer packages that allow to simulate QNE, on the other, would facilitate enormously this task.",
"As a final reflection, we would like to mention that in not a few situations QNE are “put under the rug” by arguing that they should play a minor role or somehow cancel out.",
"This is normally supported by a reasoning of the type “good agreement with the experiments” obtained in classical studies.",
"Nevertheless, several authors have demonstrated that the causes behind such a good agreement sometimes can be traced back to an inaccurate representation of the atomic forces, which can disguise the real magnitude of QNE [see, for instance, the case of the predicted atomisation transition in solid H$_{2}$ under pressure, Chen et al.",
"(2014)].",
"Therefore, tests on the influence of QNE in light-weight and highly anharmonic crystals should not be avoided but instead performed systematically.",
"As expressed by Miller (2005), “If one performs only classical simulations, one will never know whether quantum effects are important.",
"One must have the ability to include quantum effects into a simulation, even if only approximately, to know when they are important and when they are not.” This research was supported under the Australian Research Council's Future Fellowship funding scheme (project number FT140100135), and MICINN-Spain (Grants No.",
"MAT2010-18113, CSD2007-00041, and FIS2014-56257-C2-1-P).",
"Computational resources and technical assistance were provided by the Australian Government through Magnus under the National Computational Merit Allocation Scheme."
]
] | 1605.05820 |
[
[
"More Lepton Flavor Violating Observables for LHCb's Run 2"
],
[
"Abstract The R_K measurement by LHCb suggests non-standard lepton non-universality (LNU) to occur in b -> s l+ l- transitions, with effects in muons rather than electrons.",
"A number of other measurements of b -> s l+ l- transitions by LHCb and B-factories display disagreement with the SM predictions and, remarkably, these discrepancies are consistent in magnitude and sign with the R_K effect.",
"Non-standard LNU suggests non-standard lepton flavor violation (LFV) as well, for example in B -> K l l' and B_s -> l l'.",
"There are good reasons to expect that the new effects may be larger for generations closer to the third one.",
"In this case, the B_s -> mu e decay may be the most difficult to reach experimentally.",
"We propose and study in detail the radiative counterpart of this decay, namely B_s -> mu e gamma, whereby the chiral-suppression factor is replaced by a factor of order alpha/pi.",
"A measurement of this mode would be sensitive to the same physics as the purely leptonic LFV decay and, depending on experimental efficiencies, it may be more accessible.",
"A realistic expectation is a factor of two improvement in statistics for either of the B_{d,s} modes."
],
[
"Acknowledgements",
"The work of DG is partially supported by the CNRS grant PICS07229.",
"The authors acknowledge K. Lane for discussions on related subjects, and J.-F. Marchand for remarks on the manuscript."
]
] | 1605.05718 |
[
[
"Terminal-Pairability in Complete Graphs"
],
[
"Abstract We investigate terminal-pairability properties of complete graphs and improve the known bounds in two open problems.",
"We prove that the complete graph $K_n$ on $n$ vertices is terminal-pairable if the maximum degree $\\Delta$ of the corresponding demand graph $D$ is at most $2\\lfloor\\frac{n}{6}\\rfloor-2$.",
"We also verify the terminal-pairability property when the number of edges in $D$ does not exceed $2n-5$ and $\\Delta\\leq n-1$ holds."
],
[
"Introduction",
"We discuss a graph theoretic concept of terminal-pairability emerging from a practical networking problem introduced by Csaba, Faudree, Gyárfás, Lehel, and Shelp [1] and further studied by Faudree, Gyárfás, and Lehel [2], [3], [4] and by Kubicka, Kubicki and Lehel [5].",
"We revisit two open problems presented in [1] and [5].",
"Let $G$ be a graph with vertex set $V(G) = T(G)\\cup I(G)$ where the set $T(G)$ consists of $t$ ($t$ even) vertices of degree 1.",
"We call $G$ a terminal-pairable network if for any pairing of the vertices of $T(G)$ there exist edge-disjoint paths in $G$ between the paired vertices.",
"$T(G)$ is referred to as the set of terminal nodes or terminals and $I(G)$ is called the set of interior nodes of the network.",
"Given a particular pairing of the terminals, the pairs of terminals in the pairing are simply called pairs.",
"For an inner vertex $v$ we denote the number of terminal and interior vertices incident to $v$ by $d_{T(G)}(v)$ and $d_{I(G)}(v)$ , respectively.",
"In a terminal-pairable network pairs of vertices of a graph are to be connected with edge-disjoint paths, thus the notion is clearly related to multicommodity flow problems.",
"The concept is also related to weakly-linked (in our case weakly-$t/2$ -linked) graphs: a graph $G$ is weakly $k$ -linked if, for every pair of $k$ -element sets, $X = \\lbrace x_1,\\dots ,x_k\\rbrace $ and $Y = \\lbrace y_1,\\dots ,y_k\\rbrace $ , there exist edge-disjoint paths $P_1,\\dots ,P_k$ , such that each $P_i$ is an $x_i y_i$ -path.",
"Observe that joining terminal vertices (leaves) to the vertices of a weakly-$k$ -linked graph $G$ results in a terminal-pairable graph as long as every vertex of $G$ receives at most $k$ terminals.",
"On the other hand, note that terminal-pairable graphs are not necessarily highly-weakly-linked.",
"The stars (complete bipartite graph $K_{1,n}$ where one class is formed by a singleton) give a very illustrative example of terminal-pairable graphs with many terminal vertices that are not even weakly-2-linked.",
"Given a terminal-pairable network $G$ with a particular pairing $\\mathcal {P}$ of the terminals the demand multigraph $D=(V(D),E(D))$ is defined as follows: we set $V(D)=I(G)$ and join two vertices $u,v\\in V(D)$ by as many copies of the edge $uv$ as there are pairs of terminals in $\\mathcal {P}$ s.t.",
"one vertex of the pair is joined to $u$ and the other is joined to $v$ in $G$ .",
"Obviously, $|E(D)|=\\frac{|T(G)|}{2}$ and $d_D(v)= d_{T(G)}(v)$ for every $v\\in V(D)$ , thus in fact $\\Delta (D)= \\max \\lbrace d_{T(G)}(v)\\ |\\ v\\in I(G)\\rbrace $ .",
"For convenience, demand multigraphs are referred to simply as demand graphs from now on.",
"Observe that a terminal pairing problem is fully described by the underlying network $G$ and the demand graph $D$ .",
"We call the process of substituting the demand edges by disjoint paths in $G$ the resolution of the demand graph.",
"Given a simple graph $G$ , one central question in the topic of terminal-pairability is the maximum value of $t$ for which an arbitrary extension of $G$ by $t$ terminal nodes results in a terminal-pairable graphs.",
"As at a given vertex $v\\in I(G)$ at most $d_{I(G)}(v)$ edge-disjoint paths can start, the minimum degree $\\delta _{I(G)}$ of the graph induced by the interior vertices provides an obvious upper bound on $t$ .",
"However, with a balanced placement of the terminals with restriction on $\\Delta (D)$ of the corresponding demand graph (resembling the structure of weakly-linked graphs), the $\\delta _{I(G)}$ bound on the extremal value of $t$ can be greatly improved.",
"Csaba, Faudree, Gyárfás, Lehel, and Shelp [1] studied above extremal value for the complete graph $K_n$ and investigated the following question: Problem 1 ([1]) Let $K_n^q$ denote the graph obtained from the complete graph $K_n$ ($n$ even) by adding $q$ terminal vertices to every initial vertex.",
"What is the highest value of $q$ (in terms of $n$ ) for which $K_n^q$ is terminal-pairable?",
"One can easily verify that the parameter $q$ cannot exceed $\\frac{n}{2}$ .",
"Indeed, take the demand graph $D$ obtained by replacing every edge in a one-factor on $n$ vertices by $q$ parallel edges.",
"In order to create edge-disjoint paths most paths need to use at least two edges in $K_n$ , thus a rather short calculation implies the indicated upper bound.",
"The so far best result on the lower bound is due to Csaba, Faudree, Gyárfás, Lehel, and Shelp: Theorem 2 (Csaba, Faudree, Gyárfás, Lehel, Shelp [1]) If $q\\le \\frac{n}{4+2\\sqrt{3}}$ , then $K_n^q$ is terminal-pairable.",
"We improve their result by proving the following theorem: Theorem 3 If $q\\le 2\\lfloor \\frac{n}{6}\\rfloor -4$ , then $K_n^q$ is terminal-pairable.",
"Kubicka, Kubicki and Lehel [5] investigated terminal-pairability properties of the Cartesian product of complete graphs.",
"In their paper the following “Clique-Lemma” was proved and frequently used: Lemma 4 (Kubicka, Kubicki, Lehel [5]) Let $G$ be a complete graph on $n$ vertices, where $n\\ge 5$ .",
"If every vertex of $G$ has at most $n-1$ adjacent terminals and the total number of terminals is $2n$ , then for every pairing of terminals, there are edge disjoint paths for all pairs.",
"In the same paper the following related problem was raised about the possible strengthening of Lemma REF : Problem 5 ([5]) Find the largest value of $\\alpha $ such that $K_n$ with $\\alpha \\cdot n$ terminals (at most $n-1$ at each vertex) has the above property for all $n$ larger than some constant $n_0$ .",
"Obviously, $2\\le \\alpha $ due to Lemma REF .",
"It is also easy to see that $\\alpha < 4$ .",
"Let $D$ be a demand graph on $n\\ge 4$ vertices, in which two pairs of vertices, $U,V$ and $X,Y$ are both joined by $(n-2)$ parallel edges ($2n-4$ edges or equivalently $4n-8$ terminals in total; $d_D(W)=0$ for $W\\notin \\lbrace X,Y,U,V\\rbrace $ ).",
"Observe that to resolve the demand graph any disjoint path system must contain a path from $X$ to $Y$ passing through $U$ or $V$ .",
"However, there are also $n-2$ disjoint paths connecting $U$ and $V$ , meaning that $U$ or $V$ is incident to at least $2+(n-2)=n$ disjoint edges, which is clearly a contradiction.",
"This implies that the number of terminals in $G$ cannot exceed $4n-10$ .",
"We show that this bound is sharp by proving the following theorem: Theorem 6 Let $D$ be a demand graph with at most $2n-5$ edges such that no vertex is incident to more than $n-1$ edges.",
"Then $D$ can be resolved.",
"Before the proofs we fix further notation and terminology.",
"For convenience, we call a pair of edges joining the same two vertices a $C_2$ .",
"For $k>2$ , $C_k$ denotes the cycle on $k$ vertices.",
"For a subset $S\\subset V(G)$ of vertices let $d(S,V(G)-S)$ denote the number of edges with exactly one endpoint in $S$ .",
"Let $[S]$ denote the subgraph induced by the subset of vertices $S$ .",
"We call a pair of vertices joined by $k$ parallel edges a $k$ -bundle.",
"For a vertex $v$ we denote the set of neighbors by $\\Gamma (v)$ and use $\\gamma (v)=|\\Gamma (v)|$ .",
"We define the multiplicity $m(v)$ of a vertex $v$ as follows: $m(v)=d(v)-\\gamma (v)$ .",
"Observe that $m(v)$ is the minimal number of direct edges that need to be replaced by longer paths in the graph to guarantee an edge-disjoint path-system for the terminals of $v$ .",
"Figure: Lifting 2 edges of uvuv to zz and wwWe define an operation that we will subsequently use in our proofs: given an edge ${uv}\\in E$ we say that we lift ${uv}$ to a vertex $w$ when substituting the edge ${uv}$ by a path of consecutive edges ${uw}$ and ${wv}$ .",
"Note that this operation increases the degree of $w$ by 2, but does not affect the degree of any other vertex (including $u$ and $v$ ).",
"Also, as a by-product of the operation, if $w$ is already joined by an edge to $u$ or $v$ , the multiplicity of the appropriate pair increases by one (see Figure REF ).",
"Finally, note that if a graph $G$ has $n$ vertices and $d(v)\\le n-1$ , all multiplicities of $v$ can be easily resolved by subsequent liftings.",
"Indeed, $v$ has $n-1-\\gamma (v)$ non-neighbors and $m(v)=d(v)-\\gamma (v)\\le n-1-\\gamma (v)$ multiplicities, thus we can assign every edge of $v$ causing a multiplicity to a non-neighbor to which that particular edge can be lifted.",
"We call this resolution of the multiplicities of $v$ (see Figure REF ).",
"Figure: Resolving the multiplicities at vv"
],
[
"Proof of Theorem ",
"We show that if $D=(V,E)$ is a demand multigraph on $n$ vertices and $\\Delta (G)\\le 2\\lfloor \\frac{n}{6}\\rfloor -4$ , then $D$ can be transformed into a simple graph by replacing parallel edges by paths of $D$ .",
"We prove the statement by induction on $n$ .",
"Observe first that the statement is obvious for $n < 18$ .",
"For $18\\le n < 24$ , note that the demand graph $D$ is the disjoint union of 2-bundles, circles, paths, and isolated vertices.",
"It is easy to see that multiplicities in these demand graphs can be resolved; we leave the verification of the statement to the reader.",
"From now on assume $n\\ge 24$ .",
"We may assume without loss of generality that $D$ is an $\\big (2\\lfloor \\frac{n}{6}\\rfloor -4\\big )$ -regular multigraph; if necessary, additional parallel edges may be added to $D$ .",
"Should a single vertex $v$ fail to meet the degree requirement, we bump up its degree by further lifting operations as follows: as the deficit $\\big (2\\lfloor \\frac{n}{6}\\rfloor -4\\big )-d(v)$ must be even, we can lift an arbitrary edge $e\\in E([V(D)-v])$ to $v$ .",
"We remind the reader that lifting $e$ to $v$ increases $d(v)$ by two while it does not affect the degree of the rest of the vertices.",
"We will use the well known 2-Factor-Theorem of Petersen [6].",
"Be aware that a 2-factor of a multigraph may contain several $C_2$ 's (however, this is the only way parallel edges may appear in it).",
"Theorem 7 ([6]) Let $G$ be a $2k$ -regular multigraph.",
"Then $E(G)$ can be decomposed into the union of $k$ edge-disjoint 2-factors.",
"Some operations, which are performed later in the proof, are featured in the following definition, claim, and lemma.",
"Definition 8 (Lifting coloring) Let $F$ be a multigraph, and $c:E(F)\\cup V(F)\\rightarrow \\lbrace 1,2,3\\rbrace $ be a coloring of the edges and vertices of $F$ .",
"We call $c$ a lifting coloring of $F$ if and only if for any edge $e=uv\\in E(F)$ , $c(u)\\ne c(e)$ and $c(v)\\ne c(e)$ , and for any two edges $e_1,e_2\\in E(F)$ incident to a common vertex we have $c(e_1)\\ne c(e_2)$ .",
"Moreover, if the number of vertices in different color classes differ by either 0, 1, or 2, then we call $c$ a balanced lifting coloring of $F$ .",
"Claim 9 Let $F$ be a multigraph such that $\\forall v\\in V(F)$ we have $d_F(v)\\le 2$ .",
"If $w_1,w_2,w_3\\in V(F)$ are three pairwise non-adjacent different vertices, then $F$ has a balanced lifting coloring where $w_i$ gets color $i$ .",
"The proof is easy but its complete presentation requires a rather lengthy (but straightforward) casework.",
"We leave the verification of the statement to the reader.",
"Figure REF shows an example output of this lemma.",
"Figure: A balanced lifting coloring, where x 1 ,x 2 ,x 3 x_1,x_2,x_3 get pairwise different colors.Lemma 10 Let $D$ be a demand graph on $n$ vertices, such that $\\Delta (D)\\le \\lfloor \\frac{n}{3}\\rfloor -4$ .",
"Furthermore, let $X=\\lbrace x_1,x_2,x_3\\rbrace $ be a subset of $V(D)$ of cardinality 3, such that $|E(D[X])|=0$ .",
"Let $B$ be an at most 3 element subset of $V(D)\\setminus X$ .",
"Let $F$ be an $\\le 2$ -factor of $D$ .",
"Then there exists a demand graph $H$ which satisfies $V(H)=V(D)\\setminus X$ , $E(H)\\supset E(D[V(H)])\\setminus F$ , $\\lbrace e\\in E(H):\\ e\\text{ is incident to at least one of }B\\rbrace \\subset E(D)$ , and for any $v\\in V(H)$ we have $d_H(v)\\le d_D(v)-d_F(v)(+1\\text{ if } v\\notin B)$ .",
"Moreover, if $H$ has a resolution, then so does $D$ .",
"We will perform a series of liftings in $D$ in two phases, obtaining $D^{\\prime }$ and $D^{\\prime \\prime }$ .",
"At the end of the second phase, we will achieve that $X$ has no parallel edges in $D^{\\prime \\prime }$ .",
"Therefore setting $H=D^{\\prime \\prime }-X$ will satisfy the second claim of the lemma.",
"First, we determine the series of liftings to be executed in the first phase.",
"Notice that Claim REF implies the existence of a balanced lifting coloring $c$ of $F$ such that $c(x_i)\\equiv i+1 \\pmod {3}$ .",
"Lift each edge $f\\in F$ to $x_{c(f)}$ , except if $f$ is incident to $x_{c(f)}$ .",
"Let $F^{\\prime }$ be the set of lifted edges, that is $F^{\\prime }=\\bigcup _{\\begin{array}{c}f\\in F,\\\\ x_{c(f)}\\notin f\\end{array}}\\Big \\lbrace \\text{two edges joining $x_{c(f)}$ to the two vertices of $f$}\\Big \\rbrace ,$ where $\\dot{\\cup }$ denotes the disjoint union.",
"Let the multigraph $D^{\\prime }$ be defined on the same vertex set as $D$ , and let its edge set be $E(D^{\\prime })=\\lbrace e\\in E(D):\\ e\\notin F\\text{ or }x_{c(e)}\\notin e\\rbrace \\dot{\\cup }F^{\\prime }.$ In other words, $D^{\\prime }$ is the demand graph into which $D$ is transformed by lifting the elements of $F$ .",
"Let $Y=V(D)\\setminus X$ .",
"Observe that $d_{D^{\\prime }}(y)=d_{D}(y)$ for $y\\in Y$ .",
"Let $Y_i=\\lbrace y\\in Y\\setminus B\\ |\\ c(y)=i\\rbrace $ be the color $i$ vertices in $Y\\setminus B$ .",
"The balancedness of $c$ guarantees that $|Y_i|=|c^{-1}(i)\\setminus \\lbrace x_{i-1}\\rbrace \\setminus B|\\ge |c^{-1}(i)|-1-|B|\\ge \\left\\lfloor \\frac{n}{3}\\right\\rfloor -5.$ In the second phase, our task is to resolve all multiplicities of $x_i$ in $D^{\\prime }$ .",
"Observe that as edges of $F$ of the same color formed a matching, out of every two parallel edges that are incident to $x_i$ in $D^{\\prime }$ at least one of them must be an initial edge in $E(D^{\\prime })\\setminus F^{\\prime }$ .",
"The vertex $x_i$ is incident to $d_{D^{\\prime }}(x_i)-d_{F^{\\prime }}(x_i)$ edges of $E(D^{\\prime })\\setminus F^{\\prime }$ ; we plan to lift these edges to the elements of $Y_i$ by using every vertex in $Y_i$ for lifting at most once.",
"Since $d_{D^{\\prime }}(x_i)-d_{F^{\\prime }}(x_i)\\le d_{D}(x_i)-1=\\Delta (D)-1\\le \\left\\lfloor \\frac{n}{3}\\right\\rfloor -5\\le |Y_i|,$ and elements of $Y_i$ are not incident to edges of color $i$ , the set $Y_i$ offers enough space to carry out the liftings.",
"That being said, note that neighbors of $x_i$ in $Y_i$ cannot be used for lifting as they would create additional multiplicities.",
"On the other hand, if $v\\in Y_i$ and $e={vx_i}\\in E(D)$ then $e$ is an initial edge of $x_i$ that either generates no multiplicity at all or it is part of a bundle of parallel edges, one of which we do not lift.",
"In other words, for every vertex of $Y_i$ that is excluded from the lifting we mark an initial edge of $x_i$ that we do not need to lift.",
"As a result of this, resolution of the remaining multiplicities at $x_i$ can be performed in $Y_i-\\Gamma (x_i)$ .",
"Let $D^{\\prime \\prime }$ denote the demand graph obtained after resolving all of the multiplicities of $x_1$ , $x_2$ , and $x_3$ .",
"At most 1 element of $E(D^{\\prime })\\setminus E(F^{\\prime })$ has been lifted to each $y\\in Y$ , therefore there are no multiple edges between the sets $X$ and $Y$ in the demand graph $D^{\\prime \\prime }$ .",
"Moreover, $D^{\\prime \\prime }[X]=D^{\\prime }[X]$ is a subgraph of a triangle, which emerges as we lift the at most one edge of color $i+2$ of $x_i$ to $x_{i+1}$ (take the indices cyclically), for $i=1,2,3$ .",
"Any vertex $y\\in Y$ of color $i$ has at most two incident edges in $F^{\\prime }$ , joining $y$ to a subset of $\\lbrace x_{i+1},x_{i+2}\\rbrace $ .",
"If an edge has been lifted to $y\\in Y$ of color $i$ , then $y$ is adjacent to $x_i$ and $d_{D^{\\prime \\prime }}(y)=d_{D^{\\prime }}(y)+2$ .",
"Thus $y$ is joined to at least $d_{F^{\\prime }}(y)+1$ elements of $X$ in $D^{\\prime \\prime }$ .",
"As no edge of color $i$ can be incident to $y$ , we have $d_{F^{\\prime }}(y)=d_F(y)$ .",
"Therefore $d_{D^{\\prime \\prime }[Y]}(y)\\le d_{D^{\\prime \\prime }}(y)-d_{F^{\\prime }}(y)-1=d_{D^{\\prime }}(y)-d_{F}(y)+1=d_{D}(y)-d_{F}(y)+1.$ If no edges have been lifted to $y\\in Y$ , then $d_{D^{\\prime \\prime }}(y)=d_{D^{\\prime }}(y)$ and $y$ is adjacent to at least $d_{F}(y)$ elements of $X$ in $D^{\\prime \\prime }$ .",
"Therefore $d_{D^{\\prime \\prime }[Y]}(y)=d_{D^{\\prime \\prime }}(y)-d_F(y)\\le d_{D^{\\prime }}(y)-d_F(y)=d_{D}(y)-d_F(y).$ As elements of $B$ are excluded from $Y_i$ , 0 edges are lifted to them, and so we proved the statement of the lemma.",
"Let $X_1=\\lbrace x_1,x_2,x_3\\rbrace $ be a subset of 3 elements of $V(D)$ , such that $D[X]$ has 0 edges.",
"Such a set trivially exists, as any two non-adjacent vertices have $(n-2)-2\\Delta (D)\\ge \\frac{n}{3}+2$ common non-neighbors.",
"Since the degree in $D$ is at least $2\\cdot (24/6)-4=4$ , Theorem REF implies the existence of two disjoint 2-factors, $A_1$ and $A_2$ of $D$ .",
"Notice that $A_2-X$ has 3 path components (as a special case, an isolated vertex is a path on one vertex).",
"Extend $A_2-X$ to a maximal $\\le 2$ -factor $F_2$ of $D-A_1$ .",
"It is easy to see that there exists a 3-element subset $B_1$ of $V(D)\\setminus X$ such that $B_1$ induces 0 edges in $D-A_1$ , $\\lbrace v\\in V(D)\\setminus X:\\ d_{F_2}(v)= 0\\rbrace \\subset B_1$ , and $B_2=\\lbrace v\\in V(D)\\setminus X:\\ d_{F_2}(v)=1\\rbrace \\setminus B_1$ has cardinality at most 3.",
"We are ready to use Lemma REF .",
"First, apply it to $D$ , where we lift $F=A_1$ to elements of $X=X_1$ , while not creating new edges incident to $B=B_1$ .",
"Let the obtained graph be $H_1$ .",
"We have $\\Delta (H_1)\\le \\Delta (D)-\\delta (A_1)+1=\\Delta (D)-1$ .",
"Furthermore, $E(H_1[B_1])\\subseteq E(D[B_1])=\\emptyset $ .",
"We apply Lemma REF once more.",
"Now $H_1$ is our base demand graph, $F_2$ is the $\\le 2$ -factor to be lifted to elements of $B_1$ , and we avoid lifting to elements of $B_2$ .",
"Let the resulting demand graph be $H_2$ , whose vertex set is $V(D)\\setminus X\\setminus B_1$ of cardinality $n-6$ .",
"We have $d_{H_2}(v)&\\le \\left\\lbrace \\begin{array}{ll}d_{H_1}(v)-d_{F_2}(v)+1 & \\text{ if }v\\notin B_2, \\\\d_{H_1}(v)-d_{F_2}(v) & \\text{ if }v\\in B_2.\\end{array}\\right.\\le \\\\&\\le \\left\\lbrace \\begin{array}{ll}(\\Delta (D)-1)-2+1 & \\text{ if }v\\notin B_2, \\\\(\\Delta (D)-1)-1 & \\text{ if }v\\in B_2.\\end{array}\\right.\\le \\\\&\\le \\Delta (D)-2=2\\left\\lfloor \\frac{n-6}{6}\\right\\rfloor -4.$ By induction on $n$ , we know that $H_2$ has a resolution, implying that $H_1$ has a resolution, which in turn implies that $D$ has a resolution."
],
[
"Proof of Theorem ",
"We prove our statement by induction on $n$ .",
"For $n\\le 4$ the statement is straightforward, the cases $n=5,6$ require a somewhat cumbersome casework.",
"Note that if $n\\ge 4$ we may assume $D$ has exactly $2n-5$ edges, otherwise we join two non-neighbors whose degree is smaller than $n-1$ .",
"For the inductive step, we choose a vertex $x$ , resolve all of its multiplicities, and delete it from the demand graph.",
"There are two additional conditions to assert as the number of vertices decreases from $n$ to $n-1$ : i) We need to delete at least 2 edges from $D$ .",
"These edges can be either already incident to $x$ or can be lifted to $x$ .",
"ii) Let $B$ denote the set of vertices of degree greater than or equal to $n-1$ .",
"Obviously, to apply induction we need to decrease the degree $d(v)$ of every vertex $v\\in B$ by at least one.",
"Decreasing $d(v)$ can be performed by lifting an edge incident to $v$ to $x$ .",
"Note that this operation might create additional multiplicities that need to be resolved before the deletion of $x$ .",
"In addition, observe that we can lift at least one edge to a vertex $v$ without its degree exceeding the degree bound for $n^{\\prime }=n-1$ if and only if $d(v)< n-2$ .",
"Let $B=\\lbrace z_1,\\dots ,z_{|B|}\\rbrace =\\lbrace v\\in V(D): d(v)\\ge n-2\\rbrace .$ As $\\sum \\limits _{v\\in V(D)}d(v) = 4n-10$ , it follows that $|B|\\le 3$ .",
"We perform a casework on $|B|$ .",
"$|B|=0$ : If $B$ is empty, then the only condition we need to guarantee is the deletion of at least two edges in $D$ .",
"We have two cases.",
"If there is an $x\\in V(D)$ with $\\gamma (x)\\ge 2$ : we have $n-1-\\gamma (x)$ vertices for lifting to resolve the $d(x)-\\gamma (x)$ multiplicities of $x$ .",
"Obviously, $d(x)-\\gamma (x)\\le n-3 -\\gamma (x)$ thus we have enough space to resolve all multiplicities of $x$ .",
"After the deletion of $x$ , the graph has lost $\\gamma (x)\\ge 2$ edges, and the maximum degree is still two less then the number of vertices.",
"If $\\forall x\\in V(D)$ we have $\\gamma (x)\\le 1$ , then $D$ is the disjoint union of bundles and isolated vertices, which is trivial to resolve.",
"$|B|=1$ : We perform the same operation as in the previous case with the choice $x=z_1$ .",
"Observe that our inequality becomes $d(z_1)-\\gamma (z_1)\\le n - 1 -\\gamma (z_1)$ thus we have enough vertices in the multigraph to perform all the necessary liftings.",
"$|B|=2$ : Observe first that $z_1$ and $z_2$ are joined by an edge $e$ or else $2n-5\\ge d(B,V(D)-B)= d(z_1)+d(z_2)\\ge 2n-4,$ a contradiction.",
"Let us first assume that $z_1$ or $z_2$ (say, $z_1$ ) has an edge ending in a vertex different from $z_2$ (i.e.",
"$d(B,V(D)-B)>0$ ).",
"Observe that in this case $m(z_1)=d(z_1)-\\gamma (z_1)\\le (n-1)-\\gamma (z_1)$ , thus all multiplicities of $z_1$ can be resolved by lifting the appropriate edges to $V(D)-\\lbrace z_1\\rbrace -\\Gamma (z_1)$ .",
"In the remaining case $z_1$ and $z_2$ form a bundle of at most $n-1$ edges.",
"We can lift $n-2$ of these edges to $V(D)-B$ without difficulties, delete one of the vertices in $B$ , and proceed by induction.",
"$|B|=3$ : Observe that any two vertices of $\\lbrace z_1,z_2,z_3\\rbrace $ must be joined by and edge else the same reasoning as above leads to contradiction.",
"Note also that a simple average degree calculation guarantees the existence of an isolated vertex $x$ .",
"We distinguish two cases: i) If $d(B,V(D)-B)=0$ , we may assume that $V(D)-B$ contains an edge, otherwise $3(n-3)\\ge 4n-10\\Rightarrow n\\le 7$ and all edges are contained in $B$ .",
"For $n=5,6,7$ that leads to 4 possible demand graphs whose resolution can be easily completed; a case for $n=6$ is shown in Figure REF .",
"Figure: A solutionLet $f$ denote an arbitrary edge in $V(D)-B$ .",
"We lift two edges of $B$ not belonging to the same pair as well as $f$ to $x$ ; observe that the degrees of all vertices in $B$ dropped by at least 1.",
"As $n\\ge 7$ , the multiple edge created at vertex $x$ can be lifted to a vertex of $V(D)-B$ that was not incident to $f$ .",
"ii) If $d(B,V(D)-B) > 0$ let $f$ be an edge between $B$ and $V(D)-B$ .",
"Without loss of generality we may assume $f$ is incident to $z_3$ .",
"We lift $f$ as well as an edge $e$ between $z_1$ and $z_2$ ; as $e$ and $f$ are disjoint, no new multiplicity is created, thus we can proceed by induction."
],
[
"Acknowledgment",
"We would like to heartfully thank Professor Jenő Lehel for drawing our attention to the above discussed problems.",
"The third author in particular wishes to express his gratitude for the intriguing discussions about terminal-pairability problems.",
"The first and third authors of this current paper consider themselves especially fortunate to have worked with and have been inspired by the work of Professor Ralph Faudree.",
"We would like to dedicate our paper to the memory of Professor Faudree as a humble contribution to one of his favorite research topics."
]
] | 1605.05857 |
[
[
"High Power Characterization of Piezoelectric Ceramics Using the\n Burst/Transient Method with Resonance and Antiresonance Analysis"
],
[
"Abstract In this paper, a comprehensive methodology for characterizing the high power resonance behavior of bulk piezoelectric ceramics using the burst method is described.",
"In the burst method, the sample is electrically driven at its resonance frequency, and then either a short circuit or an open circuit condition is imposed, after which the vibration decays at the resonance or antiresonance frequency, respectively.",
"This decay can be used to measure the quality factor in either of these conditions.",
"The resulting current in the short circuit vibration condition is related to the vibration velocity through the \"force factor.\"",
"The generated voltage in the open circuit vibration condition corresponds to the displacement by the \"voltage factor.\"",
"The force factor and the voltage factor are related to material properties and physical dimensions of the sample.",
"Using this method, the high power behavior of the permittivity, compliance, piezoelectric charge constant, electromechanical coupling factor, and material losses can be determined directly by measuring the resonance (short circuit) and antiresonance (open circuit) frequencies, their corresponding quality factors, the force factor $A$, and the voltage factor $B$.",
"The experimental procedure to apply this method is described and demonstrated on commercially available hard and semi-hard PZT materials of $k_{31}$"
],
[
"Introduction",
"Piezoelectric materials are used in a variety of high power applications such as ultrasonic motors and underwater sonar transducers.",
"The properties of these materials are subject to the conditions in which they are applied in.",
"Therefore, the properties must be measured in comparable high power testing environments in order to achieve relevant measurements.",
"This paper uses the burst/transient method for measuring the properties of piezoelectric materials in high power conditions using a comprehensive approach with resonance and antiresonance analysis.",
"The burst method offers several advantages over other high power measurement methods: appropriate application of linear theory, short measurement time, data collection over large range of vibration levels in single measurement, simplicity in experimental application, and no heat generation/temperature rise during measurement.",
"Two methods are commonly used to determine the material properties in piezoelectric ceramics in high power resonance conditions: resonance electrical spectroscopy and the burst/transient method.",
"In resonance impedance spectroscopy, the sample is continuously driven electrically, and its electrical impedance or admittance is measured across its resonance and antiresonance frequency.",
"By fitting an equivalent circuit or by utilizing a power bandwidth approach, the loss factors and material properties for a particular vibration mode can be calculated.",
"Using the resonance frequency, the elastic compliance can be measured.",
"Utilizing the relative difference between the resonance and antiresonance frequencies, the electromechanical coupling factor can be calculated.",
"[1], [2] The second method of measuring properties in high power conditions is through the burst or transient method.",
"In this method, the ceramic is driven using a large excitation voltage at its resonance frequency for a set number of cycles.",
"Then, a short circuit condition is imposed and the sample's oscillation rings down at its resonance frequency.",
"In this case, the short circuit current is proportional to the vibration amplitude.",
"If instead, an open circuit condition is imposed, the sample’s oscillation rings down at its antiresonance frequency.",
"In this case, the open circuit voltage is proportional to the displacement.",
"By measuring the rate of signal decay, the loss factors at resonance and antiresonance can be calculated using an incremental time constant formulation.",
"Using the resonance and antiresonance frequencies, the elastic compliance and electromechanical coupling factor can be calculated similarly to the process of the impedance method.",
"The burst method can be considered as a mechanical excitation method because electrical stimulation is not applied during the measurement period.",
"[1], [3], [4].",
"In the burst/transient method experiment, temperature rise is not generated due to low driving times (often less than 10ms).",
"Also, data for a wide range of vibration levels can be obtained from the decaying oscillation, whereas data must be collected for a single vibration level at a time using the continuous drive method.",
"It has been shown that results from the burst method show higher quality factors (lower losses), and that this is not completely due to temperature rise difference [1], [4].",
"However, to the authors’ knowledge, a rigorous comparison of these techniques has not been made accounting for temperature distribution in the ceramic and number drive cycles.",
"This may lead to comparable results between the two methods.",
"The burst method was developed by Umeda et al.",
"for determining the equivalent circuit parameters of a piezoelectric transducer [3].",
"It was thereafter adapted to measure the properties of piezoelectric ceramic samples [4], [5].",
"The analysis has proved useful to many researchers studying the change of properties with vibration velocity and also to compare property values between materials, especially between PZT and lead-free materials [6], [7], [8], [9], [10].",
"However, almost all the analyses have been done at the resonance condition (short circuit).",
"Chang and coworkers have characterized the response of a Langevin Transducer at antiresonance by introducing an open circuit condition to evaluate the equivalent circuit [11], [12], [13].",
"Also, the open circuit/antiresonance condition has been applied to a hard $k_{31}$ PZT material to characterize its antiresonance quality factor [1], [14].",
"However, the analytical formulation to derive material properties from the voltage factor and the method to analyze the dielectric permittivity from it will be discussed in this paper for the first time.",
"The objective of this paper is to provide a comprehensive characterization approach for the $k_{31}$ resonator using the burst method.",
"Some of the techniques have already been presented by previous researchers, but this work adds new analytical methods, supporting experimental techniques, and discussion of physical significance of the high power measurement results.",
"Firstly, the derivation of the force factor $A_{31}$ and voltage factor $B_{31}$ in terms of material properties and sample geometry will be derived from the constitutive equations for the $k_{31}$ piezoelectric sample.",
"The force factor $A_{31}$ is the relationship between current and vibration in resonance and it is related to the effective piezoelectric stress coefficient $e_{31}^{*}$ .",
"The force factor analysis for the $k_{31}$ has been presented previously by Takahashi [4], but its explicit derivation has not.",
"The voltage factor $B_{31}$ is the relationship between open circuit voltage and displacement in resonance, related with the effective piezoelectric stiffness coefficient $h_{31}^{*}$ .",
"We believe it has not been applied nor analyzed in bulk piezoceramics in resonance conditions.",
"Additionally, use of the resonance-antiresonance frequency separation to calculate the electromechanical coupling factor using the burst method will be presented for the first time.",
"Also, the theory for measuring the permittivity directly in resonance conditions will be outlined and demonstrated.",
"Secondly, the experimental results of the high power resonance response of a hard PZT and semi-hard PZT material of $k_{31}$ geometry will be discussed.",
"The quality factors and the various properties will be measured using application of resonance and antiresonance.",
"The experimental methods to apply the burst method will be described, along with supporting data analysis techniques."
],
[
"Derivation of the force factor and the voltage factor for $k_{31}$ \nresonators",
"In this section, the force factor and the voltage factor for the $k_{31}$ resonator will be developed and the approach for calculating material properties will be summarized.",
"Also, the method to characterize the dielectric permittivity in resonance conditions will be described.",
"Fig.",
"1 shows the geometry of the sample and Tab.",
"1 defines the symbols used in the derivation as they will appear.",
"The derivations assume a rectangular plate with $a\\ll b\\ll L$ , fully electroded, and poled along the thickness ($a$ ).",
"Therefore, the criteria for a $k_{31}$ piezoelectric resonator are met.",
"The relationship between current and vibration in a resonating piezoelectric ceramic under short circuit and open circuit conditions will be derived for the $k_{31}$ mode.",
"These relationships will be used to experimentally evaluate the behavior of piezoelectric samples tested using the burst method in the next section.",
"For the discussion in this paper, the resonance frequency $\\omega _{A}$ will be described as A-type resonance and antiresonance frequency $\\omega _{B}$ will be described as B-type resonance .",
"The $k_{31}$ mode undergoes mechanical resonance during electrical resonance, corresponding to $s_{11}^{E}$ .",
"This mode undergoes an electromechanical coupling resonance at its antiresonance frequency, effectively electrically coupling the motional and damped branch of the resonator (Fig.",
"2) [15].",
"Electrical antiresonance is achieved by an open circuit ($D$ constant) condition and electrical resonance is achieved by a short circuit condition ($E$ constant).",
"As long as the sample is symmetric, both in its geometry and boundary conditions, the mode shape will be symmetric about the center of the sample.",
"In general, the mode shape of a piezoelectric resonator with stress free boundary conditions, undergoing vibration in one dimension, with losses, and having finite displacement can be described as $u(x,t)=u_{0}f(x)\\sin (\\omega t),$ where $f(x)$ is a function symmetric about the origin normalized to the displacement at the ends of the piezoelectric resonator, where $f(0)=0$ .",
"Strain is defined as $\\partial u/\\partial x=u_{0}f^{\\prime }(x).$ Then, according to the fundamental theorem of calculus $\\int _{-L/2}^{L/2}\\frac{\\partial u}{\\partial x}\\,\\mathrm {dx}=u(L/2,t)-u(-L/2,t)=2u_{0}\\sin (\\omega t).$ In general, the vibration or displacement distribution (mode shape) for a $k_{31}$ resonator and the $k_{33}$ resonator is sinusoidal.",
"In general, its mode shape as a function of frequency and normalized to the edge displacement is : for the $k_{31}$ mode [15] $u(x)=u_{0}\\frac{\\sin \\Omega _{31}x}{\\sin \\Omega _{31}L/2}.$ where $\\Omega _{31}=\\omega /\\nu _{11}^{E}$ and $\\nu $ is the speed of sound in the material $\\nu _{11}^{E}=1/\\sqrt{\\rho s_{11}^{E}}$ .",
"These mode shape functions define the displacement distribution at resonance modes (e.g.",
"1st, 2nd, and 3rd mode) and also at frequencies in between.",
"The normalized mode shapes at the first three mechanical resonance modes of the $k_{33}$ and $k_{31}$ resonators are given in Fig.",
"3.",
"The displacement distribution of frequencies in between resonance modes have distributions of partial wavelengths, unlike mechanical resonance modes, whose distributions of wavelengths of $\\frac{1}{2}+(n-1)$ of the geometry in the direction of vibration.",
"According to Eq.",
"(REF ), Eqs.",
"(REF ) can be written as $\\int _{-L/2}^{L/2}\\frac{\\partial \\left(u_{0}\\frac{\\sin \\Omega _{31}x}{\\sin \\Omega _{31}L/2}\\right)}{\\partial x}\\,\\mathrm {dx}=2u_{0}.$ Eq.",
"(REF ) does not assume a particular frequency, so the results are true at the mechanical resonance frequency and at other frequencies.",
"Depending on the mode type, either the electrical antiresonance frequency or electrical resonance frequency is the mechanical resonance frequency determined by the speed of sound in the direction of vibration.",
"However, because the derivation is general, the specific vibration distribution in question will not need to be explicitly handled in the derivation because they have been solved in the general case in Eq.",
"(REF )."
],
[
"Short circuit analysis of the force factor",
"The constitutive equation describing the electric displacement of a piezoelectric $k_{31}$ resonator is [15] $D_{3}(t)=d_{31}X_{1}+\\varepsilon _{33}^{X}\\varepsilon _{0}E_{3}(t).$ By using the electromechanical coupling factor, this equation can be rewritten as $D_{3}(t)=e_{31}^{*}\\frac{\\partial u}{\\partial x}+\\varepsilon _{33}^{x_{1}}\\varepsilon _{0}E_{3}(t),$ where $e_{31}^{*}$ is the effective piezoelectric stress coefficient defined as $e_{31}^{*}=d_{31}/s_{11}^{E}$ and $\\varepsilon _{33}^{x_{1}}$ is the relative permittivity having clamping in the length (1-direction).",
"The actual piezoelectric stress coefficient $e_{31}$ (non-star) is defined from clamped boundary conditions in the 2-direction and the 3-direciton, whereas for the $k_{31}$ resonator these boundaries are stress free.",
"Thus, the effective piezoelectric stress coefficient is used as defined above.",
"For the electrical boundary condition of zero electric potential case (short circuit), the external field is equal to zero.",
"Therefore, $D_{3}(t)=e_{31}^{*}\\frac{\\partial u}{\\partial x},$ and $\\dot{D}_{3}=e_{31}^{*}\\frac{\\partial ^{2}u}{\\partial x\\partial t}.$ The current can be written as $i(t)=\\int _{A_{e}}\\dot{D}_{3}\\mathrm {\\,dA_{e}},$ where $\\mathrm {dA_{e}}=b\\,\\mathrm {dx}.$ Therefore, $i(t)=b\\int _{-L/2}^{L/2}\\dot{D}_{3}\\,\\mathrm {dx}.$ Assuming that the sample is undergoing free vibration at the resonance frequency (constant E/short circuit conditions), we can apply Eq.",
"(REF ) such that the current can be written as $i_{0}=-2e_{31}^{*}u_{0}b\\omega _{A}.$ This equation can also be written in terms of vibration velocity at the plate edge ($x=\\pm L/2$ ), given $\\omega u_{0}=v_{0}$ for sinusoidal time varying displacement $i_{0}=-2e_{31}^{*}bv_{0}.$ The force factor $A_{31}$ , defined as the ratio between short circuit current and edge vibration velocity, can then be written as $A_{31}=\\frac{i_{0}}{v_{0}}=-2e_{31}^{*}b=-2b\\frac{d_{31}}{s_{11}^{E}}.$"
],
[
"Open circuit analysis of the voltage factor",
"For open circuit conditions the total electric displacement is equal to zero $\\int _{-L/2}^{L/2}D_{3}(t)\\,\\mathrm {dx_{1}}=0.$ Therefore, the constitutive equation described in Eq.",
"(REF ) can be written as $\\int _{-L/2}^{L/2}D_{3}(t)\\,\\mathrm {dx_{1}}=\\int _{-L/2}^{L/2}\\left(e_{31}^{*}\\frac{\\partial u}{\\partial x}+\\varepsilon _{33}^{x_{1}}\\varepsilon _{0}E_{3}(t)\\right)\\,\\mathrm {dx_{1}}.$ Assuming the variation of strain in thickness is negligible, the electric field across the thickness is uniform.",
"Therefore, the $E_{3}(x,t)=-V(t)/a$ .",
"Integrating across the length of the resonator, $\\int _{-L/2}^{L/2}E_{3}(x,t)\\,\\mathrm {dx}=-\\int _{-L/2}^{L/2}V(t)/a\\,\\mathrm {dx}=\\frac{e_{31}^{*}}{\\varepsilon _{33}^{x_{1}}\\varepsilon _{0}}\\int _{-L/2}^{L/2}\\frac{\\partial u}{\\partial x}\\,\\mathrm {dx}.$ This equation can be rewritten using Eq.",
"(REF ) assuming natural vibration at the antiresonance frequency in open circuit conditions $LV_{0}/a=\\frac{e_{31}^{*}}{\\varepsilon _{33}^{x_{1}}\\varepsilon _{0}}2u_{0}.$ Thus, the relationship between displacement and generated open circuit voltage for a $k_{31}$ resonator in its antiresonance mode can be written as $V_{0}=\\frac{2ae_{31}^{*}}{L\\varepsilon _{33}^{x_{1}}\\varepsilon _{0}}u_{0},$ and the voltage factor ($B_{31}$ ), the ratio between open circuit voltage and edge displacement, can be written as $B_{31}=\\frac{V_{0}}{u_{0}}=\\frac{2a}{L}\\frac{e_{31}^{*}}{\\varepsilon _{33}^{x_{1}}\\varepsilon _{0}}=\\frac{2a}{L}\\frac{g_{31}}{s_{11}^{D}}=\\frac{2a}{L}h_{31}^{*},$ given negligible cross coupling.",
"The effective piezoelectric stiffness coefficient $h_{31}^{*}$ the electric field generated in the 3 direction (polarization direction) for an applied strain in the 1 direction under constant $D$ conditions (open circuit) having free boundary conditions in the 2-direciton and the 3-direction: $E_{3}=h_{31}^{*}\\frac{\\partial u_{1}}{\\partial x_{1}},$ the relationship between stress and strain under constant $D$ $x_{1}=s_{11}^{D}X_{1},$ and the relationship between generated electric field under constant $D$ and stress $X$ is $E_{3}=g_{31}X_{1}.$"
],
[
"Analysis of elastic compliance, piezoelectric charge coefficient,\nand the electromechanical coupling factor ",
"It is well known that the sound velocity in a $k_{31}$ resonator propagating in the 1-direction occurs is governed by the density and the elastic compliance under constant electric field [15].",
"The first resonance frequency in the $k_{31}$ resonator corresponds to the $s_{11}^{E}$ according to the equation $s_{11}^{E}=1/((2Lf_{A})^{2}\\rho ).$ By utilizing the measurement of the force factor, the piezoelectric charge coefficient can be computed $d_{31}=-A_{31}s_{11}^{E}/2b.$ A more common approach to calculate this coefficient, frequently used in electrical resonance spectroscopy, is as follows: the off-resonance permittivity and resonance elastic compliance can be used to separate the piezoelectric charge coefficient from the coupling coefficient.",
"$k_{31}$ is calculated from a trigonometric function whose variables are the resonance and antiresonance frequencies [15].",
"Therefore, $d_{31}$ can be expressed as $d_{31}=-k_{31}\\sqrt{s_{11}^{E}\\varepsilon _{33}^{X}\\varepsilon _{0}}.$ This approach assumes that the $\\varepsilon _{33}^{X}$ does not change in resonance conditions.",
"The calculation of $d_{31}$ using the force factor does not make this assumption, so it is expected to be more accurate.",
"A detailed experimental analysis will follow in the next section.",
"By using the piezoelectric stress coefficient calculated at resonance (from the force factor) and the converse piezoelectric constant calculated at antiresonance (from the newly derived voltage factor), the clamped permittivity can be calculated in resonance conditions directly.",
"Then, $\\varepsilon _{33}^{X}$ can be calculated using the $k_{31}^{2}$ .",
"$\\varepsilon _{0}\\varepsilon _{33}^{X}(1-k_{31}^{2})=\\varepsilon _{0}\\varepsilon _{33}^{x_{1}}=\\frac{e_{31}^{*}}{h_{31}^{*}}=\\frac{A_{31}}{B_{31}}\\frac{a}{Lb}$ Permittivity has never been calculated directly in resonance conditions according to the authors’ knowledge.",
"Takahashi et al.",
"has reported permittivity in resonance conditions, but assumes that only the motional capacitance changes and the clamped capacitance in resonance does not change.",
"The validity of this assumption will be evaluated from the data in the next section."
],
[
"Excitation of resonance and antiresonance using burst drive",
"In the burst mode experiment, the sample is first driven with an oscillating voltage in order to build up vibration.",
"Then, the driving signal is removed, and thus the sample vibration decays at the system's natural frequency.",
"Fig.",
"4a demonstrates the application of the burst mode using resonance methods (constant $E$ ).",
"The sample is driven at its resonance frequency for a set number of cycles, after which a short circuit condition is imposed, inducing natural vibration at the electrical resonance frequency.",
"Fig.",
"4b demonstrates the case where an open circuit condition is imposed, generating antiresonance vibration.",
"For the open circuit condition, a bias voltage (not shown) can be generated along with the decaying oscillating voltage as reported by Chang [11].",
"This bias voltage is generated from the charge on the electrode at the time of introducing the open circuit condition.",
"Depending on the instantaneous charge before the open circuit condition is imposed, the bias voltage can be positive, negative, or zero.",
"By driving the sample near the antiresonance frequency before the open circuit is introduced, the resulting bias voltage is greatly reduced because of the small current in this condition.",
"This was the unique approach taken in this research for measuring the antiresonance burst response.",
"By applying the burst mode at resonance (short circuit) and antiresonance (open circuit) conditions, the loss factors and the real properties of the material can be measured.",
"For a damped linear system oscillating at its natural frequency, the quality factor can be described using the relative rate of decay of vibration amplitude.",
"In general, [3] $Q=\\frac{2\\pi f}{2\\ln (\\frac{v_{1}}{v_{2}})/(t_{2}-t_{1})}.$ This equation is true at both resonance and antiresonance.",
"At resonance, the current is proportional to the vibration velocity; therefore, its decay can be used.",
"Similarly, the voltage decay can be used at antiresonance to determine the quality factor at antiresonance."
],
[
"Sample selection",
"The burst mode experiment was performed on $k_{31}$ samples of commercially available PZT compositions: three PIC 184 (PI Ceramic, Germany) $k_{31}$ samples and three PIC 144 (PI Ceramic, Germany) $k_{31}$ samples, whose low power properties are listed in Tab.",
"1.",
"All the samples have a geometry of $\\mathrm {40\\times 6\\times 1mm}$$^{3}$ .",
"Commercially available materials were chosen because they provide the most relevant results from a practical perspective.",
"The samples were supported from the center (nodal point) by adjustable spring loaded electrodes.",
"Each sample in each condition (resonance and antiresonance) was measured twice.",
"After every measurement, the sample was removed, turned over, and reloaded into the sample holder.",
"Therefore, each data point presented is an average of six measurements and the error bars are the standard deviation of these measurements.",
"Previous researchers either did not present multiple measurements or did not discuss the effect of reloading the sample on the standard deviation [3], [5], .",
"By measuring one sample several times without removing and reloading it, the standard deviation is significantly reduced, but the true value of the material property remains more ambiguous.",
"Therefore, the fact that the samples were removed and reloaded in the sample holder is highlighted.",
"The measurement of the properties of piezoelectric material having low losses, such as hard PZT, are more sensitive to the sample holder positioning."
],
[
"Experimental setup and procedure",
"The experimental setup diagram can be seen in Fig.",
"5.",
"The sample was driven by a function generator (Siglent SDO342) through a power amplifier (NF4010).",
"The sample was held at the nodal point.",
"The drive period was set at 10 cycles.",
"For the short circuit measurement, the sample was driven at its resonance frequency at 50$\\mathrm {V/mm}$ using the burst mode of the function generator.",
"After the burst signal finished, the applied voltage to the sample was 0$\\mathrm {V}$ , which is equivalent to a short circuit condition.",
"The current and vibration then decayed.",
"The measured voltage during the short circuit condition indicated an added effective inductive load on the order of 1$\\Omega $ .",
"This was due to the current flow through the BNC cable from the amplifier.",
"After applying the burst signal, antiresonance vibration was induced by isolating the sample from the driving signal using an electromechanical relay.",
"The electromechanical relay was triggered with a precise time delay from the burst signal using the second channel of the function generator.",
"For the open circuit experiments, the sample was driven close to its antiresonance frequency (minimum current), and for the short circuit experiments, the sample was driven close to its resonance frequency (maximum current).",
"This is because after imposing the open circuit or short condition, the sample’s oscillating frequency immediately shifts to the frequency particular to the electrical boundary condition, antiresonance or resonance, respectively.",
"If this frequency shift is significant, transient higher harmonics are generated and they cause the vibration decay to become irregular and difficult to analyze.",
"By driving the system at its resonance frequency prior to applying the short circuit and the antiresonance frequency prior to applying open circuit, transient higher harmonics were minimized and random error was reduced significantly.",
"The current was measured by a 10x wire loop and a current probe (Tektronix TCPA300 w/ TCP 305 probe).",
"The current probe was verified to produce accurate readings for the current levels tested by measuring the current produced in a 1$\\mathrm {k}\\Omega $ precision resistor near the experimental resonance frequency.",
"For the open circuit experiments, a electromechanical relay was used to isolate the sample from the signal.",
"The function generator used has two channels, allowing for a programmable delay between them.",
"One of the channels was responsible for providing the burst signal (10 sinusoidal cycles) through the amplifier, and the other channel was responsible for driving the relay directly.",
"The channel powering the relay was normally on at 10V, and it decreased to 0V after the trigger was initiated.",
"The delay for the turn off time of the relay was approximately 0.45ms.",
"This delay time was added to the burst signal channel along with the delay time needed to allow for the 10 drive cycles to the sample to build up vibration.",
"The open circuit voltage was measured using a 100x voltage probe (Tektronix P1500).",
"The voltage probe had a resistance of 10M$\\Omega $ , much higher than the antiresonance impedance of the samples, and the capacitance of the probe was more than a thousand times less than that of the samples measured.",
"Additional measurement issues should be taken into consideration when selecting the sample geometry, in addition to isolating the vibration mode of analysis.",
"A sufficient width for the sample should be chosen to generate the level of current needed according to the force factor for the equipment to make accurate measurements.",
"Similarly, for antiresonance, the $a/L$ ratio changes the voltage factor.",
"Usually, voltage can be measured with high accuracy using an oscilloscope without special considerations.",
"Depending on the thickness and length, over a 400$\\mathrm {V}$ can be generated, as reported by Uchino [1], so therefore the $a/L$ ratio may need to be considered for the sample fabrication in order to avoid voltage levels which may damage measurement equipment."
],
[
"Data collection method",
"A crucial factor in the attainment of repeatable and accurate results was the data collection and analysis method used, and the storage capacity of the oscilloscope.",
"A LabVIEW program utilizing a function which estimates the frequency and amplitude of the dominant harmonic signal from a Fourier transform analysis was applied.",
"It was found that the built-in FFT analysis function of the oscilloscope lacks sufficient resolution, both in frequency and amplitude measurements, to provide systematic results; analyzing the waveforms on the PC provided better final results.",
"Small instability in the signal can cause large error in frequency measured from zero crossing of the oscillating signals and amplitude characterization using maximum values.",
"Therefore, a FFT analysis is the most appropriate.",
"The name of the LabVIEW function used to accomplish this was called “Extract Single Tone,” by which the frequency and amplitude of the signal were measured by interpolating the FFT spectrum of the waveform.",
"The oscilloscope time base was initially set to be large enough to allow the vibration velocity to decrease ten times from the maximum value within a single capture of the oscilloscope.",
"After this acquisition, a smaller time base was used to approximate the capture of data at specific instances in time.",
"By moving the viewing window forward (with the small time base), data for different vibration levels could be measured.",
"For the hard PZT measured in this study, the smaller time base used was 4 cycles, and for the semi-hard PZT the time base used was 2 cycles.",
"A smaller number of cycles must be used for the semi-hard PZT because its vibration signal decays more rapidly.",
"The initial use of the large time base for the waveform acquisition and then the subsequent decrease in the time base showed no apparent difference in measurement from using a small time base for the initial acquisition.",
"The ability of the FFT function used over several cycles increased the measurement integrity significantly over using the peak values of the waveforms to determine signal amplitude, decay, and frequency."
],
[
"Measurement of the force factor and voltage factor",
"Regarding resonance characterization, the ratio between the short circuit current and edge vibration velocity was used to calculate the force factor and the piezoelectric stress constant, $e_{31}^{*}=d_{31}/s_{11}^{E}$ , according to Eq. (14).",
"Using the resonance frequency, the compliance was calculated, and the piezoelectric charge coefficient as well using the piezoelectric stress coefficient.",
"The resonance characterization for PIC 144 and PIC 184 is shown in Figs.",
"6a and 6b.",
"PIC 144 shows lower property values ($d_{31}$ , $s_{11}^{E}$ , $e_{31}^{*}$ ) than PIC 184, which is expected because PIC 144 is a hard PZT composition and has a higher quality factor judging from its low power properties (Tab.",
"2).",
"The properties $d_{31}$ , $s_{11}^{E}$ , and $e_{31}^{*}$ of PIC 184 and PIC 144 change linearly with vibration velocity.",
"However, PIC 144 has more stable properties with vibration velocity but a larger deviation from a linear change in material properties with vibration velocity.",
"That being said, the change in properties of PIC 144 with respect to the low power properties (lowest vibration level) is significantly less than that of PIC 184.",
"By utilizing the displacement and open circuit voltage at antiresonance, the voltage factor $B_{31}$ and the effective piezoelectric stiffness coefficient $h_{31}^{*}$ were calculated using Eq.",
"(20) (Fig.",
"7).",
"The coupling factor was calculated using the resonance and antiresonance frequencies, and the trigonometric function defined for the $k_{31}$ resonator [15].",
"The coupling factor increases with the vibration velocity; the change in the coupling factor with vibration velocity (slope) of PIC 184 is three times larger than that of PIC 144.",
"$h_{31}^{*}$ decreases with increasing vibration velocity, contrary to the trend of the other properties.",
"That being said, it has a much smaller dependence on vibration velocity than the other properties, namely those determined at resonance."
],
[
"Characterization of losses at resonance and antiresonance ",
"Using the decay of vibration at resonance and antiresonance, the quality factors were calculated.",
"Each data point used amplitude data from two vibration measurements; therefore, the scale was readjusted as an average of the vibration velocity.",
"Fig.",
"8 shows the results; a log-log plot was used to easily distinguish and compare the trends between the two compositions.",
"PIC 144 shows stable characteristics of the quality factor, until about 150 mm/s RMS, after which a sharp degradation in the quality factors occurred.",
"PIC 184, however, showed an immediate decrease in its quality factors.",
"$Q_{B}$ was larger than $Q_{A}$ for both the materials."
],
[
"Off resonance approximation of the permittivity ",
"Traditionally, the dielectric permittivity is measured by applying an oscillating electric potential at an off-resonance frequency and measuring the resulting charge or current [15].",
"The dielectric permittivity has not been measured at resonance high power conditions because the dielectric response in this condition does not display a distinct characteristic which can be measured to compute it.",
"Therefore, researchers have used one of the two approaches to estimate the permittivity in high power conditions: Assume the permittivity measured in off-resonance conditions applies to resonance conditions.",
"This approach is problematic because the stress conditions and the frequency is different at resonance, and therefore the property is expected to change, similar to other properties.",
"The other approach is to assume that the permittivity in high power conditions can be considered as a perturbation of the off-resonance permittivity using a variation in the motional capacitance, which is proportional to $d_{31}^{2}/s_{11}^{E}$ for the $k_{31}$ mode.",
"The first approach is flawed because the elastic and piezoelectric properties of the material are known to change with applied stress; therefore, the dielectric response must also follow a similar tendency because $s^{E},d,$ and $\\varepsilon ^{X}$ all have extrinsic contributions from non-$180^{\\circ }$ domain walls which are strongly affected by applied stress.",
"Researchers using the second approach assume that clamped permittivity $\\varepsilon ^{x}$ of the material is the same in high power resonance conditions and in low power off-resonance conditions.",
"The permittivity under constant stress ($\\varepsilon _{33}^{X}$ ) is a combination of the motional and clamped dielectric response of the material.",
"Assuming $\\varepsilon ^{x}$ from off-resonance measurements, adjusting the motional contribution to $\\varepsilon ^{X}$ according the piezoelectric charge constant, the compliance can determine the permittivity $\\varepsilon ^{X}$ in these conditions.",
"The basis for the assumption of the equivalency of the lower power and high power clamped permittivity is as follows: The dielectric permittivity $\\varepsilon $ in PZT thin films are much lower than the permittivity $\\varepsilon ^{X}$ in bulk PZT.",
"This is because the substrate effectively clamps the PZT film, and thereby significantly reduces the non-$180^{\\circ }$ domain wall motion and its contribution to the permittivity.",
"The $180^{\\circ }$ domain wall motion contribution to the permittivity, however, is relatively undisturbed because it does not result in change in strain and therefore is relatively unaffected by the substrate [16].",
"In the $k_{31}$ resonator examined, $\\varepsilon ^{x}$ refers to the dielectric response of the material with an applied field in the 3-direction and polarization in the 3-direction with clamping in the 1-direction.",
"This clamping results is far less clamping than the substrate clamping in PZT thin films.",
"Because the effective stresses applied due to resonance vibration of the $k_{31}$ resonator are in the 1-direction, and the $\\varepsilon ^{x}$ represents the permittivity after “clamping out” the domain wall contributing to domain wall motion the 1-direction, the following equality should be true: $\\varepsilon _{low\\,power}^{x}=\\varepsilon _{high\\,power}^{x}$ .",
"This was the approach taken by Takahashi et.",
"al.",
"They have reported permittivity in resonance conditions, but assume that only the motional capacitance changes and the clamped capacitance in resonance does not [4].",
"For the burst method, they have reported the motional capacitance ($d_{31}^{2}/s_{11}^{E}$ ) to remain constant, and therefore the permittivity, which is calculated partially from off-resonance measurements, to be constant as well.",
"That being said, there may exist other phenomena which also affect the clamped permittivity.",
"This may include the frequency response.",
"The clamped permittivity may change with increasing frequency due to the dependence of domain wall motion spectral response.",
"Therefore, this assumption may be invalid for this or other reasons, and it is preferred to calculate the permittivity using only data from resonance measurements in order to ensure compatibility between measurements to resolve a final property."
],
[
"Determination of the permittivity using the voltage factor and force\nfactor in resonance conditions ",
"By using the piezoelectric stress coefficient calculated at resonance (from the force factor) and the converse piezoelectric constant calculated at antiresonance (from the newly derived voltage factor), the clamped permittivity can be calculated in resonance conditions directly.",
"Then from Eq.",
"REF $\\varepsilon _{33}^{X}$ can be calculated using the $k_{31}^{2}$ : $\\varepsilon _{0}\\varepsilon _{33}^{X}(1-k_{31}^{2})=\\varepsilon _{0}\\varepsilon _{33}^{x_{1}}=\\frac{e_{31}^{*}}{h_{31}^{*}}=\\frac{A_{31}}{B_{31}}\\frac{a}{Lb}.$ The change in permittivity with vibration velocity can be seen on Fig.",
"9.",
"The off-resonance permittivity measured for the samples is in good agreement with the low vibration velocity permittivity measured through the burst technique as seen in this figure.",
"The off-resonance permittivity is represented by a star symbol.",
"The clamped and free permittivity are both changing with increasing vibration velocity.",
"The permittivity of PIC 184 is larger than that of PIC 144, and this is to be expected because PIC 184 is a semi-hard PZT with larger off-resonance permittivity.",
"From the low vibration state to the high one, the permittivity of both compositions increase.",
"However, the increase in PIC 184 is larger, demonstrating that its properties have a larger dependence on vibration conditions.",
"As mentioned earlier in this section, we expect $\\varepsilon _{low\\,power}^{x}=\\varepsilon _{high\\,power}^{x}$ because of the clamping out of non-$180^{\\circ }$ domain walls.",
"However, the result shown in this study demonstrates that a majority of the change seen in the free permittivity can actually be attributed to the clamped permittivity change.",
"The open circuit voltage generated during antiresonance may have caused the property changes being reported in this study.",
"For the PIC 184 samples, the largest voltage generated was 85$\\mathrm {V/mm\\,RMS}$ @ 450$\\mathrm {mm/s\\,RMS}$ , and for the PIC 144 samples the largest voltage was 120$\\mathrm {V/mm\\,RMS}$ @ 600$\\mathrm {mm/s\\,RMS}$ .",
"The open circuit voltage generated may cause domain reorientation in the 180$^{\\mathrm {o}}$ domain wall regions.",
"The motional capacitance (proportional to the difference, $\\varepsilon _{33}^{X}-\\varepsilon _{33}^{x_{1}}$ ) remains fairly constant, so it cannot explain the change in permittivity.",
"The difference in the $\\varepsilon _{low\\,power}^{x}$ and $\\varepsilon _{high\\,power}^{x}$ may also be due to the details of the clamping applied.",
"As mentioned earlier, in the $k_{31}$ resonator examined, the clamping considered is the 1-direction only.",
"Therefore, it should not be expected that all of non-$180^{\\circ }$ domain walls are clamped because the clamping is dissimilar from the substrate clamping effect seen in thin films.",
"From the data in Fig.",
"9, it can be said that the change in clamped permittivity represents the stresses in the 1-direction affects non-$180^{\\circ }$ domain wall motion in the 2 and 3 direction."
],
[
"Summary",
"Traditionally, characterization of piezoelectric materials in resonance is accomplished using continuous drive methods with impedance spectroscopy.",
"However, the burst/transient method can be used instead.",
"It affords simple experimental application, reasonable application of linear theories, and naturally lends itself toward high power measurements.",
"In this chapter, a comprehensive measurement approach toward characterization of the $k_{31}$ resonator was demonstrated using resonance and antiresonance drive methods.",
"The experimental procedure to achieve both of these conditions was described.",
"The measurement incorporated the use of the force factor analysis to characterize the elastic compliance and piezoelectric charge coefficient as used by other researchers.",
"Antiresonance analysis using the voltage factor was performed for the first time.",
"In antiresonance, the ratio between open circuit voltage and displacement gives the voltage factor, which is proportional to the converse piezoelectric coefficient.",
"With increasing vibration level, the elastic compliance (constant electric field), piezoelectric charge constant, and piezoelectric stress constant increased.",
"At the same time, the converse piezoelectric constant and the quality factors decreased.",
"Also, the calculation of the permittivity in resonance conditions only using measurements from resonance conditions was demonstrated.",
"The results show that the change in the clamped and free permittivity for the $k_{31}$ resonators tested were of similar levels; therefore, the change in motional capacitance was small and change in clamped capacitance dominated the dependence of $\\varepsilon _{33}^{X}$ on vibration level."
],
[
"Acknowledgment",
"The authors would also like to acknowledge the Office of Naval Research for sponsoring this research under grant number: ONR N00014-12-1-1044.",
"Figure: NO_CAPTION Figure 1 Geometry of a $k_{31}$ resonator Figure: NO_CAPTION Figure 2 Simplified equivalent circuit of a $k_{31}$ piezoelectric resonator (material property equivalencies are presented for illustration) Figure: NO_CAPTION Figure 3 Displacement distribution for the first three resonance modes for the $k_{31}$ resonator Figure: NO_CAPTIONFigure: NO_CAPTION Figure 4 Qualitative waveforms describing voltage, current, and vibration (a) before and after short circuit and (b) before and after open circuit Figure: NO_CAPTION Figure 5 Experimental setup to apply burst method measurement Figure: NO_CAPTIONFigure: NO_CAPTION Figure 6 (a) Resonance characterization of PIC 184 $k_{31}$ and (b) resonance characterization of PIC 144 $k_{31}$ Figure: NO_CAPTIONFigure: NO_CAPTION Figure 7 (a) Antiresonance characterization of PIC 184 $k_{31}$ and (b) antiresonance characterization of PIC 144 Figure: NO_CAPTION Figure 8 Change in quality factors with vibration velocity for PIC 184 and PIC 144 $k_{31}$ samples Figure: NO_CAPTION Figure 9 Change in dielectric permittivity with increasing vibration velocity Table: Low power properties of PIC 184 and PIC 144 as calculated from theburst mode"
]
] | 1605.05736 |
[
[
"Slip avalanches in metallic glasses and granular matter reveal universal\n dynamics"
],
[
"Abstract Universality in materials deformation is of intense interest: universal scaling relations if exist would bridge the gap from microscopic deformation to macroscopic response in a single material-independent fashion.",
"While recent agreement of the force statistics of deformed nanopillars, bulk metallic glasses, and granular materials with mean-field predictions supports the idea of universal scaling relations, here for the first time we demonstrate that the universality extends beyond the statistics, and applies to the slip dynamics as well.",
"By rigorous comparison of two very different systems, bulk metallic glasses and granular materials in terms of both the statistics and dynamics of force fluctuations, we clearly establish a material-independent universal regime of deformation.",
"We experimentally verify the predicted universal scaling function for the time evolution of individual avalanches, and show that both the slip statistics and dynamics are universal, i.e.",
"independent of the scale and details of the material structure and interactions.",
"These results are important for transferring experimental results across scales and material structures in a single theory of deformation."
],
[
"Introduction",
"The notion of universality represents a longstanding question in materials deformation, which has been traditionally described by material-specific relations and mechanisms.",
"The existence of universal scaling relations, if confirmed experimentally, would provide a novel means to connect microscopic rearrangements to macroscopic stress-strain response in a single theory of deformation across a wide range of solid materials.",
"Recently, power-law distributions measured in the stress signals of slowly deformed single crystals [1], [2], bulk metallic glasses (BMGs) [3], [4], rocks [5], [6], granular materials [7], [8], [9], [10], [11] and even earthquakes [12], [13], [14], [15] reveal very similar strongly correlated deformation, suggesting underlying universal scaling relations in the slow deformation of solids.",
"These distributions are also well described by a mean-field model of elasto-plastic deformation [16], in which the material's elasticity causes coupling between locally yielding regions resulting in slip avalanches with intermittency as observed in the experiments.",
"A recent comparison of widely different systems showed that indeed the fluctuations of the applied stress follow very similar power-law distributions across a wide range of length scales from nanometers to kilometers [6], as adequately described by the mean-field model, lending credence to the idea of an underlying universal mechanism of deformation.",
"Yet, unlike equilibrium critical phenomena, where universality has been rigorously demonstrated by meticulous measurements of scaling relations, such measurements are lacking in the deformation of materials.",
"More importantly, no measurement has yet elucidated the applicability of universal scaling relations to the dynamics of the slip avalanches.",
"Establishing the universality not only of the statistics, but also of the dynamics of the slip avalanches would provide important grounds for the claim of universality.",
"In this paper we provide the first rigorous investigation of universal scaling behavior by comparing avalanche statistics and dynamics in two very different systems, bulk metallic glasses [17] and granular materials [7], in which high resolution stress measurements are possible.",
"We show that despite the large differences in the nature of the two materials in terms of the size, interactions, and dynamics of the constituent particles, they share a regime with identical rescaled stress fluctuations, temporal profiles, and dynamics, all accurately described by mean-field theory.",
"This universal regime results from the system-independent elastic coupling of yielding regions, leading to long-range correlated avalanches of deformation as described by the mean-field model.",
"Besides this universal regime, we also delineate a non-universal regime with system-specific power-law statistics, governed by boundary conditions and finite size effects.",
"This first rigorous comparison of slip statistics and dynamics in two disparate systems corroborates the existence of a universal scaling regime and suggests a universal theory of deformation."
],
[
"Experimental setup and mean-field model",
"Due to their very different nature of hard versus soft solids, metallic glasses and granular materials differ greatly in mechanical properties such as modulus, ductility and elastic strain.",
"To nevertheless resolve and compare the fine fluctuations of the applied stress in the two systems, we developed specific experimental protocols for each [3], [7].",
"For the metallic glass specimens we applied uniaxial compression tests using a precisely aligned load train with a fast-response load cell and high-rate data acquisition, see Fig.",
"REF (a).",
"We used a constant displacement rate with a nominal strain rate of $10^{-4} s^{-1}$ , and a bulk metallic glass with composition $Zr_{45}Hf_{12}Nb_{5}Cu_{15.4}Ni_{12.6}Al_{10}$ , and specimens 6 mm long along the loading direction with a cross section of 1.5 mm x 2 mm.",
"During compression, the specimen deforms elastically until a shear band or slip event initiates.",
"This causes the displacement rate to temporarily exceed the displacement rate imposed on the specimen, resulting in a stress drop as shown in Fig.",
"REF (c), inset [17].",
"The size of the stress drop is proportional to the slip size.",
"For the granular system, we used a shear cell with built-in pressure sensors to record the force fluctuations on the tilting walls, as shown in Fig.",
"REF (b) [7].",
"The granular particles, around $3\\cdot 10^5$ spheres with a diameter of $d=1.5$ mm and a polydispersity of $\\sim 5\\%$ , are confined by a top plate, subjected to confining normal pressure between 4 and 10 kPa, resulting in a particle volume fraction of $55-60\\%$ .",
"The granular material is sheared at a constant rate $\\dot{\\gamma }=9.1 \\cdot 10^{-4}$ to a total strain of $\\gamma =20\\%$ , and force drops are identified around the monotonically increasing average force, as shown in Fig.",
"REF (d), inset.",
"The number of granular particles is large compared to typical laboratory granular studies, but of course many orders of magnitude smaller than the number of atoms in the metallic glass specimens.",
"The granular linear system size of $\\sim 70$ particle diameters across can lead to significant truncation of large avalanches and hence to more pronounced finite size effects than for the metallic glass.",
"To describe the stress fluctuations in both systems, we use a simple analytic model that predicts the slip statistics for elasto-plastic solids [17], [16].",
"The model assumes that real solids have elastically coupled weak spots, which are known as shear transformation zones in a metallic glass.",
"Each weak spot slips by a random amount when the local stress exceeds a threshold.",
"A slipping weak spot can trigger another weak spot to slip as well in a slip avalanche, causing the intermittent response that is observed in experiments.",
"For bulk metallic glasses and granular materials the model assumes that a recently slipped weak spot is slightly weaker than before, due to dilation [17].",
"As a result, the model predicts a universal power-law scaling for slip avalanches in a range of sizes that is not affected by finite-size effects of the specimen.",
"Large avalanche slips have different dynamics.",
"They recur almost periodically and span a macroscopic fraction of the system.",
"The model predicts how the average slip avalanche size for the smaller slips grows as a large slip is approached.",
"From the average slip size it is in principle possible to extract at what stress the next catastrophically large slip will take place.",
"The model also predicts a large number of scaling laws, for example how the statistics changes with applied strain rate and stress, allowing us to extrapolate from one loading condition to another.",
"Figure: Metallic glass and granular setups and measurements.",
"(a) Schematic of the bulk metallic glass measurement setup.",
"Two tungsten carbide platens that are constrained by a steel sleeve compress the metallic glass specimen.",
"See Ref.",
"for details.",
"(Drawing is courtesy of Adrienne Beaver, Bucknell University.",
")(b) Schematic of the granular shear cell setup with force sensors in the walls.",
"Loads imposed on top exert a constant confining pressure.",
"See Ref.",
".",
"(c) and (d) Metallic glass and granular data - the main panels show applied stress or force versus time, insets show magnifications of the data.",
"This is the same data as shown in and ."
],
[
"Results",
"The raw stress-time data of the two systems in Fig.",
"REF c and d show pronounced stress fluctuations; rapid stress drops demarcate stress relaxation events, during which the displacement rate temporarily exceeds the displacement rate applied to the specimen.",
"We define the size of these avalanche events $S$ from the magnitude of sharp stress drops $\\Delta \\sigma $ and force drops $\\Delta F$ in the metallic glass and granular systems, respectively (see insets of Figs.",
"REF c and d).",
"Due to the very different particle size (angstroms for atoms in the metallic glass opposed to millimeters for the granular particles), and the different nature of interaction (atomic potential versus frictional contacts), the magnitude of stress fluctuations differs greatly, being several ten megapascals for the metallic glass, opposed to several hundred pascals for the granular material.",
"Nevertheless, we can collapse the stress drop distributions by simple rescaling that accounts for the different stress magnitudes of the hard metallic and soft granular materials.",
"To show this, we plot selected rescaled distributions of stress drop magnitudes and durations in Fig.",
"REF .",
"The probability of stress drops larger than size $S$ , known as the complementary cumulative size distribution, is shown in Fig.",
"REF a.",
"The metallic glass and granular distributions show excellent overlap for avalanche sizes $S$ in the range $S_{\\min }^{GRN}<S<S_{\\max }^{GRN}$ that are not affected by the sample boundaries, and exhibit significant deviations for larger avalanches.",
"In the small-avalanche regime, both distributions closely follow a power law $C(S)\\sim S^{-(\\tau - 1)}$ with exponent $\\tau - 1 = 1/2$ , in excellent agreement with predictions by the mean field model (dashed line).",
"In particular, the granular data approaches that of the metallic glass and the model predictions with increasing confining pressure that pushes the granulate deeper into the jammed solid regime.",
"This is because, unlike the metallic glass that is held together by attractive molecular interactions, the granular particles are repulsive and held together merely by the applied confining pressure.",
"For larger sizes, the granular power-law distributions are truncated due to finite size effects: they extend over significantly shorter ranges than those of the bulk metallic glass that follows the mean-field prediction up to larger avalanches.",
"In contrast, in the small-avalanche regime not affected by finite size effects, which is the scaling regime of the model, the overlap of the distributions and model predictions is surprisingly good.",
"In this scaling regime, each weak spot slips only once in an avalanche; these slips are small enough to not be affected by the sample boundaries.",
"In contrast, large avalanches for $S>S_{\\max }^{GRN}$ have very different dynamics: they behave similarly to a crack cutting through a macroscopic fraction of the sample, and feel the boundaries of the sample.",
"In the model they are unstoppable or runaway events, where each weak spot slips many times during the same avalanche.",
"The resulting time evolution of slip is very smooth, while for the small avalanches the time evolution of slip is very jerky [18].",
"Small avalanches are power law distributed because they are always close to stopping, while large avalanches are runaway events that only stop when they have cut through a macroscopic fraction of the system.",
"The similarity of the avalanche statistics is further confirmed in Fig.",
"REF b where we show the avalanche duration versus size.",
"Again, for the small avalanches with size $S_{\\min }^{GRN}<S<S_{\\max }^{GRN}$ , the agreement between the metallic glass and granular data is remarkable: both exhibit identical scaling of the size-dependent duration according to $t(S)\\sim S^{\\sigma \\nu z}$ with $\\sigma \\nu z\\simeq 1/2$ as predicted by the model.",
"Again, the scaling regime extends to larger avalanches for the metallic glass due to its larger system size.",
"For the granular material, the data crosses over to a second scaling regime $t(S)\\sim S^{\\tau _{L}}$ with a much smaller exponent $\\tau _{L}\\simeq 0.1$ .",
"This much shallower growth of avalanche duration indeed indicates large slip events, common for shear bands or cracks, for which uniform sliding occurs along the entire shear plane.",
"Indeed, it is known that granular materials always shear band [19], while metallic glasses exhibit a regime of homogeneous deformation at high temperature and low shear rates.",
"We further explore the correspondence of avalanches by plotting their duration distribution in Fig.",
"REF c. We again find good agreement in the small-avalanche regime: the metallic glass and granular systems exhibit identical power-law distributions with the predicted slope of $-1$ .",
"Similar to the avalanche size in Fig.",
"REF a, the scaling regime extends to large avalanches for the metallic glass, while for the granular system a second scaling regime emerges that clearly changes with the applied confining pressure, and is thus a non-universal regime that depends on the system details.",
"Another characteristic property of the force drops is the rate of stress release, which gives insight into the propagation dynamics of individual avalanches.",
"Plotting the rate of stress release as a function of avalanche size, we find excellent agreement between the metallic glass and granular data over the entire avalanche regime (Fig.",
"REF d), signifying that the underlying avalanche propagation dynamics for small avalanches may be the same in both systems.",
"Yet the scaling range of the stress drop sizes that can be compared to mean field theory is limited by finite size effects.",
"Figure: Avalanche statistics.",
"Four scaling parameters are compared for the metallic glass (green dashed line) and the granular material (colored solid lines with color denoting confining pressure):(a) Complimentary cumulative distribution C(S)C(S) of avalanche size,(b) Avalanche duration versus avalanche size,(c) Complimentary cumulative distribution C(t)C(t) for avalanche duration, and(d) Stress drop rate versus avalanche size.",
"In each plot the solid black line shows the portion of the granular data corresponding to the scaling regime (S min GRN <S<S max GRN S_{\\min }^{GRN}<S<S_{\\max }^{GRN}) for data collected at 9.6 kPa.",
"The dashed black line shows the slope expected from the prediction of the mean field model.",
"The legends show the slope value of the granular curves in the scaling regime for pressures 4.0, 6.8 and 9.6 kPa.We now take advantage of the finely resolved signals to compare the full time evolution of individual avalanches.",
"We show the rate of force release as a function of time for avalanches in the scaling regime in Fig.",
"REF (for data collected at maximum pressure 9.6 kPa).",
"As expected for the scaling regime, we can indeed collapse all granular avalanche profiles with different durations onto a single master curve as shown in Fig.",
"REF a.",
"A similar collapse for the metallic glass avalanches has been shown in [17].",
"This self-similarity lends credence to the idea that in this regime the system is indeed described by robust scaling relations.",
"We compare the granular data with that of the metallic glass and mean-field predictions in Fig.",
"REF b.",
"While for the metallic glass, the avalanches show symmetric profiles, in good agreement with mean-field predictions [17], for the granular material, the avalanche profiles exhibit a slight asymmetry.",
"We associate this asymmetry with delayed damping effects [20] similar to earthquakes [21].",
"Delay effects can originate with time scales inherent to the friction between the particles.",
"As the granular particles are relatively soft, their elastic relaxation time that sets the microscopic delay time for the onset of slip is considerable.",
"A similar explanation has been suggested for earthquakes [21], and for Barkhausen noise in magnetic materials, where the delay is due to eddy currents in the material [20], [22].",
"In contrast, in metallic glasses there is no friction between the atoms inside the alloy, and consequently no significant microscopic delay time to yield a noticeable skewing of the avalanche shapes.",
"In either case, the asymmetry of the velocity profile does not affect the scaling exponents; they are still given by the mean field model predictions as shown in [23].",
"Another difference between the avalanche shapes is that they are not as flattened for the granular material as they are for the metallic glass.",
"The reason for this difference is the limited machine stiffness.",
"For the metallic glass experiments the machine stiffness was chosen to be large [17], leading to a broadening of the avalanche shape [17], [20].",
"In the granular experiments the walls are of similar stiffness to the granular particles, thus not significantly flattening the temporal profile.",
"By fitting the predicted form with limited machine stiffness to the granular data we find good agreement between model predictions and experiments as shown in Fig.",
"REF b.",
"In summary, our finely resolved measurements of the avalanche profiles reveal the details of machine stiffness and avalanche delay effects due to interparticle friction [20] within the same scaling and universality class.",
"Figure: Avalanche dynamics: Temporal avalanche profiles in the universal scaling regime.",
"(a) Average stress-drop rate of the granular system, normalized by the maximum rate.",
"Profiles are averaged over avalanches from small bins of their durations.",
"Error bars are calculated as standard error of the mean.",
"(b) Stress-drop rates compared for metallic glass (brown) and granular material (green).",
"Solid lines show averages over avalanches in the scaling regime, and dash-dotted curves show mean-field predictions.",
"(c) Granular stress-drop rates for fixed avalanche sizes SS in the scaling regime (S min GRN <S<S max GRN S_{\\min }^{GRN}<S<S_{\\max }^{GRN}).",
"Inset shows original data, and main panel shows collapsed profiles scaled along both axes by S -1/2 S^{-1/2}.",
"(d) Comparison of the averaged collapsed profiles for granular data (c) and metallic glass data .",
"The black curve shows scaling functions predicted by the mean-field model for both granular materials (bottom axis) and metallic glasses (top axes), which perfectly overlap with each other, further corroborating the similarity of the slip avalanche statistics of metallic glasses and granular materials.",
"The granular fitting constants for the scaling function Axexp(-Bx 2 )Ax\\exp (-Bx^2) are A=1.46A=1.46 and B=6.6×10 -4 B=6.6\\times 10^{-4}, the metallic glass constants are A=3.98×10 11 A=3.98\\times 10^{11} and B=2.18×10 11 B=2.18\\times 10^{11}.We can further test the scaling relation between avalanche size and duration by collapsing the avalanche profiles as a function of avalanche size.",
"To do so, we sort avalanches according to their size, focusing on the small-avalanche scaling regime.",
"Individual profiles are shown in the inset of Fig.",
"REF c. These profiles indeed collapse onto a single master curve when we rescale both axes by $S^{-1/2}$ (main panel of Fig.",
"REF c) giving yet another confirmation of the validity of the scaling relation.",
"The average of these profiles also agrees well with that of the metallic glass, and closely matches the prediction by mean-field theory, as shown in Fig.",
"REF d. The small difference between metallic glass and granular data for small values of $tS^{-1/2}<0.02$ appears due to differences in particle softness and machine stiffness, similar to Fig.",
"REF b. Consequently, the metallic glass profiles can be also fitted with the mean-field theory using slightly different values of the non-universal parameters $A$ and $B$ described in [17], but the form of the two scaling functions $Ax\\exp (-Bx^2)$ for granular and metallic glass data can be perfectly overlapped with each other when plotted in their corresponding axes (Fig.",
"REF d).",
"While some small deviations for the granular avalanches occur at large values of $tS^{-1/2}>0.06$ due to poor statistics, the agreement of the granular and metallic glass data with the mean field model is striking.",
"We thus conclude that this small-avalanche regime, while limited for the granular system due to finite size effects, has hallmarks of a universal scaling regime.",
"In contrast, the second scaling regime for large avalanches $S>S_{\\max }^{GRN}$ shows non-universal properties that depend on the system geometry and boundary conditions, consistent with the model predictions.",
"Indeed, Figs.",
"REF a and c already suggest that the granular scaling in this regime depends on the applied confining pressure that drives the system into different jammed states.",
"We can still collapse the avalanche profiles for the granular system using data collected at maximum pressure 9.6 kPa; starting with the raw profiles (inset in Fig.",
"REF a), we achieve excellent collapse by scaling the vertical axis by $S^{-1.1}$ , as shown in the main panel of Fig.",
"REF a.",
"Scaling along the horizontal axis is not required due to the almost constant avalanche duration (Fig.",
"REF b).",
"Although the predictions that mean field theory makes for the small avalanches are not expected to necessarily extend all the way to the large avalanches that are affected by system boundaries and loading geometries, still the collapsed granular profiles are also in good agreement with the small-avalanche mean-field scaling function $Ax\\exp (-Bx^2)$ indicated by the black line.",
"Interestingly, the details of these large-avalanche profiles allow us to elucidate the connection to the small-avalanche regime: initially ($t<8\\cdot 10^{-3}$ $s$ ), these profiles exhibit a \"foot\" that corresponds precisely to the profile of the small avalanches shown in the inset of Fig.",
"REF c. When reaching their maximum $\\langle dF/dt \\rangle \\approx 10$ $N/s$ at $t\\approx 7\\cdot 10^{-3}$ $s$ , some of these small avalanches nucleate into larger ones, upon which the avalanche size grows faster, as clearly seen in the profiles in Fig.",
"REF a, inset.",
"This nucleation picture of large avalanches is consistent with the mean field model [13], [16], [24].",
"For the metallic glass, this foot is very long [18] (see inset of Fig.",
"REF b where profiles are centered at the peak positions).",
"Neglecting the foot, we can achieve a reasonably good collapse along the vertical axis by scaling with $S^{-1.6}$ , which is quite far from the mean-field scaling $S^{-0.5}$ for the small avalanches.",
"This is not surprising, since the mean-field theory predicts that the $S^{-0.5}$ scaling only applies to the small avalanches but not for the large ones.",
"We associate this difference in the granular and the metallic glass behavior with the difference in boundary effects and loading conditions of the two systems: because the large avalanches feel the system boundaries, and the boundary conditions in both experiments are different, it is expected that the large avalanche profiles in these systems look different.",
"We hence identify this empirical scaling behavior in the large-avalanche regime as non-universal and system-specific, and even changing with confining pressure for the granular system.",
"The nucleation of large avalanches from the small ones and its dependence on the internal and external conditions is an interesting topic for further studies.",
"Figure: Temporal profiles of large granular avalanches.",
"Stress-drop rate profiles of large avalanches for the granular (a) and metallic glass system (b).",
"Profiles averaged by size are shown in the insets, and collapsed data is shown in the main panels.",
"Error bars are calculated as standard error of the mean.",
"(a) Large granular avalanches show good collapse when scaling the vertical axis by S -1.1 S^{-1.1}.",
"Scaling along the horizontal axis is not required.",
"The mean-field scaling function Axexp(-Bx 2 )Ax\\exp (-Bx^2) can be fitted with A=7.4A=7.4 and B=3×10 -3 B=3\\times 10^{-3}.",
"The nucleation point at t≈7·10 -3 t\\approx 7\\cdot 10^{-3} ss and 〈dF/dt〉≈10\\langle dF/dt \\rangle \\approx 10 N/sN/s when the small avalanche turns into a large one is very close to a maximum point of the small avalanche profiles shown in Fig.",
"c.(b) Large metallic-glass avalanches show collapse when the vertical axis is scaled by S -1.6 S^{-1.6}.",
"Along the horizontal axis, the profiles have been centered manually at the peak positions.",
"In the large-avalanche regime, the scaling of the avalanches is not universal, as it is affected by the different boundary conditions of the systems, in agreement with mean-field predictions."
],
[
"Conclusion",
"We have demonstrated universal features of slip avalanches in metallic glasses and granular systems.",
"Due to the very different particle size and interaction of the hard atomic and soft granular amorphous solids, the stress fluctuations are orders of magnitude different.",
"Yet, their distributions reveal strikingly similar statistics and dynamics with identical power-law behavior of avalanche sizes and durations.",
"For the metallic glass this scaling regime spans a relatively broad range; for the granular material, besides the universal regime, we also observe a non-universal scaling regime, characterized by large system-spanning avalanches that appear to have almost identical duration.",
"We attribute the large avalanches to shear bands or crack-like slip.",
"Most importantly, for the scaling regime, we observe clear universal behavior in the scaling exponents and avalanche dynamics of the small avalanches that do not span across the entire sample.",
"The detected slight differences in the avalanche profiles arise from delayed damping effects in the granular materials due to friction, which differs from the particle interactions in bulk metallic glasses.",
"In this way the large, asymmetric granular avalanche profiles are similar to those of large-scale earthquakes.",
"These results provide an important step towards a universal understanding of the deformation of amorphous materials.",
"While previous studies showed that slowly-compressed single crystals, bulk metallic glasses, rocks, granular materials, and the earth all deform via intermittent slips or \"quakes\" [6], the current study for the first time compares not only scaling exponents but also scaling functions for the dynamics of slip across two systems with completely different scales, structures, and interactions.",
"The good agreement between the systems, and between the experimental data and mean-field predictions for avalanche statistics and dynamics significantly expands the claim that these systems may be described by a unifying theory not only with respect to their slip statistics, but also with respect to the slip dynamics."
],
[
"Acknowledgements",
"This work is part of the research program of FOM (Stichting voor Fundamenteel Onderzoek der Materie), which is financially supported by NWO (Nederlandse Organisatie voor Wetenschappelijk Onderzoek); NSF DMR 1042734 (WJW); NSF DMS 1069224, NSF DMR 1005209 and NSF CBET 1336634 (KD).",
"TCH acknowledges support from the National Science Foundation under grant DMR 1408686.",
"We also thank the Kavli Institute for Theoretical Physics and the Aspen Center of Physics for hospitality and support via grants NSF PHY 1125915 and NSF PHY 1066293 respectively."
],
[
"Author contribution",
"D.V.D.",
"and P.S.",
"designed the granular research, D.V.D.",
"performed the granular measurements, D.V.D.",
"and K.A.L.",
"analyzed the granular data.",
"W.J.W.",
"designed the metallic glass experiments, X.J.G.",
"performed the metallic glass measurements, A.N.",
"analyzed the metallic glass data, D.V.D compared the data for the metallic glass and granular systems.",
"J.T.U.",
"and K.A.D.",
"derived the theoretical predictions and guided the metallic glass data analysis and comparison to the model predictions.",
"P.S.",
"and D.V.D wrote major parts of the manuscript, with contributions from W.J.W, K.D, and T.C.H."
],
[
"Competing financial interests",
"The authors declare no competing financial interests."
]
] | 1605.05896 |
[
[
"Minimal blow-up masses and existence of solutions for an asymmetric\n sinh-Poisson equation"
],
[
"Abstract For a sinh-Poisson type problem with asymmetric exponents of interest in hydrodynamic turbulence, we establish the optimal lower bounds for the blow-up masses.",
"We apply this result to construct solutions of mountain pass type on two-dimensional tori."
],
[
"Introduction and main results",
"Motivated by the equations introduced in the context of hydrodynamic turbulence by Onsager in [18] and by Sawada and Suzuki in [27], we study the following sinh-Poisson type problem: $\\left\\lbrace \\begin{aligned}-\\Delta _g v=&\\lambda _1\\frac{e^v}{\\int _\\Sigma e^v\\,dV_g}-\\lambda _2\\frac{e^{-\\gamma v}}{\\int _\\Sigma e^{-\\gamma v}\\,dV_g}-\\kappa &&\\mbox{on\\ }\\Sigma \\\\\\int _\\Sigma v\\,dV_g=&0.\\end{aligned}\\right.$ Here, the unknown function $v$ corresponds to the stream function of the turbulent Euler flow, $(\\Sigma ,g)$ is a compact, orientable, Riemannian 2-manifold without boundary, the constant $\\gamma \\in (0,1]$ describes the intensity of the negatively oriented vortices, and $\\lambda _1,\\lambda _2>0$ are constants related to the inverse temperature, $\\kappa \\in \\mathbb {R}$ is given by $\\kappa =\\frac{\\lambda _1-\\lambda _2}{|\\Sigma |},$ where $|\\Sigma |=\\int _\\Sigma dV_g$ is the volume of $\\Sigma $ .",
"The constant $\\kappa $ ensures that the right hand side in (REF ) has zero mean value.",
"The trivial solution $v\\equiv 0$ always exists for (REF ).",
"Problem (REF ) admits a variational structure.",
"Indeed, (REF ) is the Euler-Lagrange equation of the functional: $J(v)=\\frac{1}{2}\\int _\\Sigma |\\nabla v|^2\\,dV_g-\\lambda _1 \\log \\int _\\Sigma e^vdV_g-\\frac{\\lambda _2}{\\gamma } \\log \\int _\\Sigma e^{-\\gamma v}\\,dV_g$ defined on the space $\\mathcal {E}=\\left\\lbrace v\\in H^1(\\Sigma ):\\ \\int _\\Sigma v\\,dV_g=0\\right\\rbrace .$ With this notation, $v$ is a classical solution for (REF ) if and only if $v$ is a critical point for $J$ in $\\mathcal {E}$ .",
"When $\\gamma =1$ , problem (REF ) reduces to the mean field equation for equilibrium turbulence derived in [13], [20], under the assumption that the point vortices have equal intensities and arbitrary orientation.",
"This case has received a considerable attention in recent years.",
"In particular, existence of saddle point solutions was obtained in [9], [23], [31]; such results exploit the blow-up analysis derived in [12], [17] in order to establish the compactness of solution sequences.",
"A detailed blow-up analysis is also contained in [10].",
"By the Lyapunov-Schmidt approach introduced in [6], sign-changing concentrating solutions for a related local sinh-Poisson type equation were constructed in [1] and sign-changing bubble-tower solutions were constructed in [8].",
"The case $\\gamma \\ne 1$ corresponds to the assumption where all positively oriented point vortices have unit intensity and all negatively oriented point vortices have intensity equal to $\\gamma $ .",
"Such a case already appears in an unpublished manuscript of Onsager [7], although the rigorous derivation of (REF ) is due to [27].",
"Actually, the mean field equation derived in [27] concerns the very general case where the point vortex intensity is described by a Borel probability measure $\\mathcal {P}\\in \\mathcal {M}([-1,1])$ ; it is given by $\\left\\lbrace \\begin{aligned}-\\Delta _g v=&\\lambda \\int _{[-1,1]}\\alpha \\left(\\frac{e^{\\alpha v}}{\\int _\\Sigma e^{\\alpha v}\\,dV_g}-\\frac{1}{|\\Sigma |}\\right)\\,\\mathcal {P}(d\\alpha )&&\\mbox{on\\ }\\Sigma \\\\\\int _\\Sigma v\\,dV_g=&0,\\end{aligned}\\right.$ where $\\lambda >0$ is a constant related to the inverse temperature.",
"The variational functional for (REF ) takes the form $J_{\\mathcal {P}}(v)=\\frac{1}{2}\\int _\\Sigma |\\nabla v|^2\\,dV_g-\\lambda \\int _{[-1,1]}\\log \\left(\\int _\\Sigma e^{\\alpha v}\\,dV_g\\right)\\,\\mathcal {P}(d\\alpha ), \\qquad v\\in \\mathcal {E}.$ In this context, problem (REF ) corresponds to the case $\\mathcal {P}=\\tau \\delta _1+(1-\\tau )\\delta _{-\\gamma }, \\qquad \\lambda _1=\\lambda \\tau ,\\ \\lambda _2=\\lambda \\gamma (1-\\tau ).$ Problem (REF ) was analyzed in [15], [24], where concentration properties for non-compact solution sequences is studied in the spirit of [2].",
"The optimal Moser-Trudinger constant for $J_\\mathcal {P}$ is computed in [21].",
"More precisely, it is shown in [21] that $J_{\\mathcal {P}}$ is bounded from below if $\\lambda <\\bar{\\lambda }$ , where $\\bar{\\lambda }=8\\pi \\inf \\left\\lbrace \\frac{\\mathcal {P}(K_\\pm )}{\\left(\\int _{K_\\pm }\\alpha \\,\\mathcal {P}(d\\alpha )\\right)^2}:\\ K_\\pm \\subset I_\\pm \\cap \\mathrm {supp}\\mathcal {P}\\right\\rbrace ,$ where $I_+:=[0,1]$ and $I_-=[-1,0)$ , $K_\\pm $ are Borel sets.",
"On the other hand, $J_{\\mathcal {P}}$ is unbounded from below if $\\lambda >\\bar{\\lambda }$ .",
"In the case where $\\mathcal {P}$ is discrete, an equivalent form of (REF ) was derived in [28] in the context of Liouville systems.",
"However, particularly with respect to the blow-up behavior of solutions, it appears complicated to further analyze (REF ) in its full generality.",
"Therefore, recent effort has been devoted to study cases where $\\mathcal {P}$ is the sum of two Dirac masses, such as (REF ).",
"In particular, for small values of $\\gamma $ , existence results for (REF ) in planar domains with holes were obtained in [22]; sign-changing blow-up solutions for a local Dirichlet problem related to (REF ) are constructed in [19].",
"The case $\\gamma =1/2$ is equivalent to the Tzitzéica equation describing surfaces of constant affine curvature, and was recently analyzed in [11].",
"Here, we are concerned with the existence and blow-up properties of (REF ) for general values of $\\gamma \\in (0,1]$ .",
"It is known [2], [15], [24] that unbounded solution sequences for problem (REF ) necessarily concentrate on a finite set $\\mathcal {S}\\subset \\Sigma $ .",
"Our first aim in this note is to derive the optimal lower bounds for the blow-up masses, see Theorem REF below.",
"Then, we apply the blow-up analysis results to the construction of a non-zero solution for (REF ) via a mountain pass argument, provided the compact surface $\\Sigma $ satisfies a suitable condition concerning the first eigenvalue, see Theorem REF below.",
"In order to state our results precisely, we introduce some notation.",
"Let $(\\lambda _{1,n},\\lambda _{2,n},v_n)$ be a solution sequence to (REF ).",
"If $v_n$ is unbounded in $L^\\infty (\\Sigma )$ , then, up to subsequences, we have $\\begin{aligned}&\\lambda _{1,n}\\frac{e^{v_n}}{\\int _\\Sigma e^{v_n}\\,dV_g}\\,dV_g\\stackrel{\\ast }{\\rightharpoonup }\\sum _{p\\in \\mathcal {S}_1}m_1(p)\\delta _{p}+r_1\\,dV_g,\\\\&\\lambda _{2,n}\\frac{e^{-\\gamma v_n}}{\\int _\\Sigma e^{-\\gamma v_n}\\,dV_g}\\,dV_g\\stackrel{\\ast }{\\rightharpoonup }\\sum _{p\\in \\mathcal {S}_2}m_2(p)\\delta _{p}+r_2\\,dV_g,\\end{aligned}$ weakly in the sense of measures, where the blow-up sets $\\mathcal {S}_1,\\mathcal {S}_2\\subset \\Sigma $ are finite, the “blow-up masses\" $m_i(p)$ satisfy the lower bound $m_i(p)\\ge 4\\pi $ for all $p\\in \\mathcal {S}_1\\cup \\mathcal {S}_2$ , $r_i\\in L^1(\\Sigma )$ , $i=1,2$ .",
"See Lemma REF below for a more detailed statement.",
"Our first aim is to improve the lower bound for the blow-up masses $m_i$ , $i=1,2$ .",
"Theorem 1.1 Let $(\\lambda _{1,n},\\lambda _{2,n},v_n)$ be a concentrating solution sequence for (REF ) and suppose that (REF ) holds.",
"Then, the blow-up masses satisfy the following lower bounds: $\\begin{aligned}&m_1(p)\\ge 8\\pi \\ \\forall p\\in \\mathcal {S}_1;&&m_2(p)\\ge \\frac{8\\pi }{\\gamma }\\ \\forall p\\in \\mathcal {S}_2.\\end{aligned}$ We apply Theorem REF to derive the existence of non-zero mountain-pass solutions to problem (REF ).",
"In order to state the existence result, we denote by $\\mu _1(\\Sigma )$ the first positive eigenvalue of $-\\Delta _g$ on $\\Sigma $ , namely $\\mu _{1}(\\Sigma ):=\\inf _{\\phi \\in \\mathcal {E}\\setminus \\lbrace 0\\rbrace }\\frac{\\int _{\\Sigma }\\left|\\nabla \\phi \\right|^{2}dV_g}{\\int _{\\Sigma }\\phi ^{2} \\,dV_g}.$ Our existence result holds for surfaces $\\Sigma $ satisfying the condition $8\\pi <\\mu _1(\\Sigma )|\\Sigma |< 16\\pi (1+\\gamma ).$ Condition (REF ) is satisfied, e.g., when $\\Sigma $ is the flat torus ${\\mathbb {R}}^2/\\mathbb {Z}^2$ , since in this case $\\mu _1(\\Sigma )|\\Sigma |=4\\pi ^2$ .",
"Let $\\Lambda :=\\left\\lbrace (\\lambda _1,\\lambda _2)\\subset \\mathbb {R}^2:\\begin{aligned}&(i)\\quad &&\\lambda _1,\\lambda _2\\ge 0, \\max \\lbrace \\lambda _1,\\gamma \\lambda _2\\rbrace >8\\pi \\\\&(ii)\\quad &&\\lambda _1\\notin 8\\pi \\mathbb {N}, \\ \\lambda _2\\notin \\frac{8\\pi }{\\gamma }\\mathbb {N}\\\\&(ii)\\quad \\ &&\\lambda _1+\\gamma \\lambda _2<\\mu _1(\\Sigma )|\\Sigma |\\end{aligned}\\right\\rbrace .$ Our existence result is the following.",
"Theorem 1.2 Suppose $(\\Sigma ,g)$ satisfies (REF ).",
"Then, there exists a nontrivial solution to (REF ) for all $(\\lambda _1,\\lambda _2)\\in \\Lambda $ .",
"We note that condition (REF ) implies $\\Lambda \\ne \\emptyset $ .",
"More precisely, $\\Lambda $ is a union of two (possibly overlapping) triangles $T_1$ , $T_2$ , where $T_1$ has vertices $\\lbrace (8\\pi ,0),\\,(\\mu _1(\\Sigma )|\\Sigma | ,0),\\,(8\\pi , (\\mu _1(\\Sigma )|\\Sigma |-8\\pi )/\\gamma )\\rbrace $ , and $T_2$ has vertices $\\lbrace (0,8\\pi /\\gamma ),\\,(0, \\mu _1(\\Sigma )|\\Sigma |/\\gamma ),\\,( \\mu _1(\\Sigma )|\\Sigma |-8\\pi , 8\\pi /\\gamma )\\rbrace $ .",
"The remaining part of this note is devoted to the proof of Theorem REF and Theorem REF .",
"The proof of Theorem REF is in the spirit of [16], [26], where the case $\\gamma =1$ is considered.",
"The case $\\gamma \\ne 1$ introduces an asymmetry in the problem, which requires some careful modifications in the proof.",
"The proof of Theorem REF relies on the variational setting introduced in [30], [14], [23], based on Struwe's monotonicity trick [29].",
"In view of the improved lower bounds for blow-up masses, as stated in Theorem REF , Theorem REF with $\\gamma =1$ is more general than the corresponding result in [23]."
],
[
"Notation",
"We denote by $C>0$ a general constant whose actual value may vary from line to line.",
"When the integration variable is clear, we may omit it.",
"We take subsequences without further notice.",
"We set $\\Vert \\cdot \\Vert =\\Vert \\cdot \\Vert _{\\mathcal {E}}$ ."
],
[
"Proof of Theorem ",
"This section is devoted to the proof of Theorem REF .",
"We denote by $(\\lambda _{1,n},\\lambda _{2,n},v_n)$ a solution sequence to (REF ) with $\\lambda _{1,n}\\rightarrow \\lambda _{1,0}\\in \\mathbb {R}$ , $\\lambda _{2,n}\\rightarrow \\lambda _{2,0}\\in \\mathbb {R}$ , as $n\\rightarrow \\infty $ .",
"We define the measures $\\mu _{1,n},\\mu _{2,n}\\in \\mathcal {M}(\\Sigma )$ by $&\\mu _{1,n}=\\lambda _{1,n}\\frac{e^{v_n}}{ \\int _\\Sigma e^{v_n}}\\,dV_g,&&\\mu _{2,n}=\\lambda _{2,n}\\frac{e^{-\\gamma v_n}}{\\int _\\Sigma e^{-\\gamma v_n}}\\,dV_g.$ We may assume that $\\mu _{i,n}\\stackrel{*}{\\rightharpoonup }\\mu _i\\in \\mathcal {M}(\\Sigma )$ weakly in the sense of measures, $i=1,2$ .",
"We define the blow-up sets: $\\mathcal {S}_1=&\\lbrace p\\in \\Sigma :\\exists \\,p_n\\rightarrow p\\ \\mathrm {s.t.\\ } v_n(p_n) \\rightarrow +\\infty )\\rbrace \\\\\\mathcal {S}_2=&\\lbrace p\\in \\Sigma :\\exists \\,p_n\\rightarrow p\\ \\mathrm {s.t.\\ } v_n(p_n) \\rightarrow -\\infty )\\rbrace $ and we denote $\\mathcal {S}=\\mathcal {S}_1\\cup \\mathcal {S}_2$ .",
"For the reader's convenience and in order to fix notation, we collect in the following lemma the necessary known blow-up results for (REF ).",
"Lemma 2.1 ([15], [24]) For the solution sequence $ (\\lambda _{1,n},\\lambda _{2,n},v_n)$ the following alternative holds.",
"Compactness: $\\limsup _{n\\rightarrow \\infty }\\Vert v_n\\Vert _\\infty <+\\infty $ .",
"We have $\\mathcal {S}=\\emptyset $ and there exist a solution $v \\in \\mathcal {E}$ to (REF ) with $\\lambda _{1}=\\lambda _{1,0}$ and $\\lambda _2=\\lambda _{2,0}$ and a subsequence $\\lbrace v_{n_k}\\rbrace $ such that $v_{n_k}\\rightarrow v$ in $\\mathcal {E}$ .",
"Concentration: $\\limsup _{n\\rightarrow \\infty }\\Vert v_n\\Vert _\\infty =+\\infty $ .",
"We have $\\mathcal {S}\\ne \\emptyset $ and $&\\mu _1=\\sum _{p\\in \\mathcal {S}_1}m_1(p)\\delta _p+r_1\\,dV_g\\\\&\\mu _2=\\sum _{p\\in \\mathcal {S}_2}m_2(p)\\delta _p+r_2\\,dV_g$ where $\\delta _p$ denotes the Dirac delta centered at $p\\in \\mathcal {S}$ .",
"The constants $m_1(p),\\,m_2(p)$ satisfy the lower bound $m_i(p)\\geqslant 4 \\pi ,\\qquad i=1,2$ and $r_i\\in L^1(\\Sigma )\\cap L^\\infty _{loc}(\\Sigma \\setminus \\mathcal {S})$ .",
"Moreover, for every $p\\in \\mathcal {S}_1\\cap \\mathcal {S}_2$ , we have $8\\pi \\left[m_1(p)+\\frac{m_2(p)}{\\gamma }\\right]= [m_1(p)-m_2(p)]^2.$ We note that (REF ) implies that for all $p\\in \\mathcal {S}_1\\setminus \\mathcal {S}_2$ we have $m_2(p)=0$ and therefore $m_1(p)=8\\pi $ .",
"Similarly, for every $p\\in \\mathcal {S}_2\\setminus \\mathcal {S}_1$ we have $m_1(p)=0$ and therefore $m_2(p)=\\frac{8\\pi }{\\gamma }$ .",
"In particular, Theorem REF is satisfied for $p\\in (\\mathcal {S}_1\\setminus \\mathcal {S}_2)\\cup (\\mathcal {S}_2\\setminus \\mathcal {S}_1)$ .",
"Therefore, henceforth we assume $p\\in \\mathcal {S}_1\\cap \\mathcal {S}_2$ .",
"Let $\\mathcal {O}\\subset \\mathbb {R}^2$ be a smooth bounded open set.",
"We shall need the following version of the Brezis-Merle type alternatives for Liouville systems.",
"Lemma 2.2 ([14]) Suppose $(w_{1,n},w_{2,n})$ is a solution sequence to the Liouville system $\\left\\lbrace \\begin{split}&-\\Delta w_{1,n}=aV_{1,n}e^{w_{1,n}}-bV_{2,n}e^{ w_{2,n}}&&\\mathrm {in\\ }\\mathcal {O}\\\\&-\\Delta w_{2,n}=-cV_{1,n}e^{w_{1,n}}+dV_{2,n}e^{w_{2,n}}&&\\mathrm {in\\ }\\mathcal {O},\\end{split}\\right.$ where $V_{i,n}\\in L^\\infty (\\mathcal {O})$ , $i=1,2$ , are given functions, $a,b,c,d>0$ are fixed constants and $0\\leqslant V_{i,n}\\le C,\\quad \\int _\\mathcal {O}e^{w_{i,n}}\\le C,\\qquad i=1,2.$ Then, up to subsequences, exactly one of the following alternatives holds true.",
"1.",
"Both $w_{1,n}$ and $w_{2,n}$ are locally uniformly bounded in $\\mathcal {O}$ .",
"2.",
"There is $i\\in \\lbrace 1,2\\rbrace $ such that $w_{i,n}$ is uniformly bounded in $\\mathcal {O}$ and $w_{j,n}\\rightarrow -\\infty $ locally uniformly in $\\mathcal {O}$ for $j\\ne i.$ 3.",
"Both $w_{1,n}\\rightarrow -\\infty $ and $w_{2,n}\\rightarrow -\\infty $ locally uniformly in $\\mathcal {O}$ .",
"4.",
"For the blow-up sets $\\mathcal {S}_1^0$ , $\\mathcal {S}_2^0$ defined for this subsequence, we have $\\mathcal {S}_1^0\\cup \\mathcal {S}_2^0\\ne \\emptyset $ and $\\sharp (\\mathcal {S}_1^0\\cup \\mathcal {S}_2^0)< +\\infty .$ Furthermore, for each $i\\in \\lbrace 1,2\\rbrace $ , either $w_{i,n}$ is locally uniformly bounded in $\\mathcal {O}\\setminus (S_1^0\\cup \\mathcal {S}_2^0)$ or $w_{i,n}\\rightarrow -\\infty $ locally uniformly in $\\mathcal {O}\\setminus (S_1^0\\cup \\mathcal {S}_2^0).$ Here, if $\\mathcal {S}_i^0\\setminus (S_1^0\\cap \\mathcal {S}_2^0) \\ne \\emptyset $ then $w_{i,n}\\rightarrow -\\infty $ locally uniformly in $\\mathcal {O}\\setminus (S_1^0\\cup \\mathcal {S}_2^0),$ and each $x_0 \\in \\mathcal {S}_i^0$ satisfies $m_i(x_0 )\\ge 4\\pi $ such that $V_{i,n}(x)e^{w_{i,n}}\\rightharpoonup \\sum _{x_0\\in \\mathcal {S}_i^0}m_i(x_0) \\delta _{x_0} \\qquad *\\mbox{-weakly in }\\mathcal {M}(\\mathcal {O}),$ $i=1,2$ .",
"For the case $a=c=2$ , $b=d=1$ , corresponding to the case of the Toda system, Lemma REF was established in [14], Theorem 4.2.",
"However, by carefully inspecting the proof in [14], it is clear that Theorem 4.2 in [14] holds true for general $a,b,c,d>0$ .",
"We also use the following known result.",
"Lemma 2.3 Let $p_0\\in \\mathcal {S}_1\\cap \\mathcal {S}_2 $ .",
"There exists a sequence $x_{1,n}\\rightarrow p_0$ and a sequence $x_{2,n} \\rightarrow p_{0}$ such that: i) $v_n(x_{1,n})\\rightarrow +\\infty ,\\qquad \\qquad v_n(x_{1,n})-\\log \\int _\\Sigma e^{v_n}\\rightarrow +\\infty $ , ii) $-v_n(x_{2,n})\\rightarrow +\\infty ,\\qquad \\quad - v_n(x_{2,n})-\\frac{1}{\\gamma }\\log \\int _\\Sigma e^{-\\gamma v_n}\\rightarrow +\\infty .$ We can now complete the proof of our first result.",
"We denote by $(\\Psi , \\mathcal {U})$ an isothermal chart satisfying $\\bar{\\mathcal {U}}\\cap \\mathcal {S}=\\lbrace p_0\\rbrace $ , $\\Psi (\\mathcal {U})=\\mathcal {O}\\subset \\mathbb {R}^2$ and $\\Psi (p_0)=0,\\qquad g(X)=e^{\\xi (X)}(dX_1^2+dX^2_2), \\qquad \\xi (0)=0,$ where $X=(X_1,X_2)$ denotes a coordinate system on $\\mathcal {O}$ .",
"In particular, identifying $\\varphi (X)=v(\\Psi ^{-1}(X))$ for any function $\\varphi $ defined on $\\Sigma $ , we have that a solution $v$ to equation (REF ) satisfies $-\\Delta v=\\lambda _1e^\\xi \\frac{e^v}{\\int _\\Sigma e^v}-\\lambda _2e^\\xi \\frac{e^{-\\gamma v}}{\\int _\\Sigma e^{-\\gamma v}}-\\kappa e^\\xi \\qquad \\mathrm {in\\ }\\mathcal {O}.$ We define $h_\\xi $ by $\\left\\lbrace \\begin{split}-\\Delta h_\\xi =&e^{\\xi }\\qquad \\mathrm {in\\ }\\mathcal {O},\\\\h_\\xi =&0\\qquad \\mathrm {on\\ }\\partial \\mathcal {O}.\\end{split}\\right.$ For the solution sequence $ (\\lambda _{1,n},\\lambda _{2,n},v_n)$ to (REF ) such that $\\lambda _{1,n}\\rightarrow \\lambda _{1,0}$ and $\\lambda _{2,n}\\rightarrow \\lambda _{2,0}$ , we define $w_{i,n}:\\mathcal {O}\\rightarrow \\mathbb {R}$ , $i=1,2$ , by setting $&w_{1,n}:=v_n-\\log \\int _\\Sigma e^{v_n} +\\kappa h_\\xi ,\\\\&w_{2,n}:=-\\gamma v_n-\\log \\int _\\Sigma e^{-\\gamma v_n}-\\gamma \\kappa h_\\xi ,$ where, as before, we identify $v_n(X)=v_n(\\Psi ^{-1}(X))$ .",
"Then, the above definitions imply that $&\\frac{e^{v_n}}{\\int _\\Sigma e^{v_n}}=e^{w_{1,n}-\\kappa h_\\xi },&&\\frac{e^{-\\gamma v_n}}{\\int _\\Sigma e^{-\\gamma v_n}}=e^{w_{2,n}+\\gamma \\kappa h_\\xi }.$ Setting $&V_{1,n}:=\\lambda _{1,n} e^{\\xi -\\kappa h_\\xi },&&V_{2,n}:=\\lambda _{2,n}e^{\\xi +\\gamma \\kappa h_\\xi },$ we may write $\\begin{aligned}V_{1,n}e^{w_{1,n}}=&\\lambda _{1,n} e^{\\xi -\\kappa h_\\xi } \\frac{e^{v_n}}{\\int _\\Sigma e^{v_n}}e^{\\kappa h_\\xi }=\\lambda _{1,n} e^\\xi \\frac{e^v}{\\int _\\Sigma e^v}\\\\V_{2,n}e^{w_{2,n}}=&\\lambda _{2,n}e^{\\xi +\\gamma \\kappa h_\\xi }\\frac{e^{-\\gamma v_n}}{\\int _\\Sigma e^{-\\gamma v_n}}e^{-\\gamma \\kappa h_\\xi }=\\lambda _{2,n}e^\\xi \\frac{e^{-\\gamma v_n}}{\\int _\\Sigma e^{-\\gamma v_n}}\\end{aligned}$ By definition of $w_{1,n}$ , $w_{2,n}$ and $h_\\xi $ , we thus have $-\\Delta w_{1,2}=-\\Delta v_n+\\kappa e^\\xi =\\lambda _{1,n} e^\\xi \\frac{e^v}{\\int _\\Sigma e^v}-\\lambda _{2,n}e^\\xi \\frac{e^{-\\gamma v_n}}{\\int _\\Sigma e^{-\\gamma v_n}}$ and $\\Delta w_{2,n}=\\gamma \\Delta v_n-\\gamma \\kappa e^\\xi =-\\gamma \\lambda _{1,n} e^\\xi \\frac{e^v}{\\int _\\Sigma e^v}+\\gamma \\lambda _{2,n}e^\\xi \\frac{e^{-\\gamma v_n}}{\\int _\\Sigma e^{-\\gamma v_n}}.$ In particular, $(w_{1,n}, w_{2,n})$ is a solution to the Liouville system $\\left\\lbrace \\begin{split}&-\\Delta w_{1,n}=V_{1,n}e^{w_{1,n}}-V_{2,n}e^{\\gamma w_{2,n}}&&\\mathrm {in\\ }\\mathcal {O}\\\\&-\\Delta w_{2,n}=-\\gamma V_{1,n}e^{w_{1,n}}+\\gamma V_{2,n}e^{\\gamma w_{2,n}}&&\\mathrm {in\\ }\\mathcal {O},\\end{split}\\right.$ which is of the form (REF ) with $a=b=1$ , $c=d=\\gamma $ .",
"We claim that solutions to (REF ) satisfy the estimates (REF ) in view of (REF ).",
"Indeed, we have $0\\le V_{i,n}\\le \\sup _n\\lambda _{i,n}e^{\\Vert \\xi \\Vert _\\infty +\\kappa \\Vert h_\\xi \\Vert _\\infty }\\le C,\\qquad i=1,2.$ Moreover, $&\\int _{\\mathcal {O}}e^{w_{1,n}}=\\int _{\\mathcal {O}}\\frac{e^{v_n}}{\\int _\\Sigma e^{v_n}}e^{\\kappa h_\\xi }\\le e^{\\kappa \\Vert h_\\xi \\Vert _\\infty }\\\\&\\int _{\\mathcal {O}} e^{w_{2,n}}=\\int _{\\mathcal {O}}\\frac{e^{-\\gamma v_n}}{\\int _\\Sigma e^{-\\gamma v_n}}e^{-\\gamma \\kappa h_\\xi }\\le e^{\\gamma \\kappa \\Vert h_\\xi \\Vert _\\infty }.$ For later use, we also note that $w_{1,n}+\\frac{1}{\\gamma }w_{2,n}=-\\log \\int _\\Sigma e^v- \\frac{1}{\\gamma }\\log \\int _\\Sigma e^{-\\gamma v}\\leqslant C.$ We define $\\begin{split}V_1=\\lambda _{1,0}\\,e^{\\xi -\\kappa h_\\xi },\\qquad V_2=\\lambda _{2,0}\\,e^{\\xi +\\gamma \\kappa h_\\xi },\\end{split}$ so that $V_{i,n}\\rightarrow V_i$ , $i=1,2$ , uniformly on $\\overline{\\mathcal {O}}$ .",
"We also define $\\mathcal {S}_i^0= \\left\\lbrace X\\in \\mathcal {O} : \\exists X_n\\rightarrow X\\ \\mathrm {s.t.\\ } w_{i,n}(X_n)\\rightarrow +\\infty \\right\\rbrace ,\\quad i=1,2.$ In view of Lemma REF there exist $x_{1,n}$ and $x_{2,n}$ such that $x_{1,n}\\rightarrow p_0$ , $v_n(x_{1,n})\\rightarrow +\\infty $ , $x_{2,n}\\rightarrow p_0$ and $-v_n(x_{2,n})\\rightarrow +\\infty $ , and furthermore $\\begin{array}{lll}X_{1,n}= \\Psi (x_{1,n})\\rightarrow 0& \\mbox{ and }& w_{1,n}(X_{1,n})\\rightarrow +\\infty \\\\X_{2,n}= \\Psi (x_{2,n})\\rightarrow 0& \\mbox{ and }& w_{2,n}(X_{2,n})\\rightarrow +\\infty .\\end{array}$ In particular, $0\\in \\mathcal {S}_1^0\\cap \\mathcal {S}_2^0$ and $\\mathcal {S}_1^0=\\Psi (\\mathcal {U}\\cap \\mathcal {S}_1)=\\left\\lbrace 0\\right\\rbrace =\\Psi (\\mathcal {U}\\cap \\mathcal {S}_2)=\\mathcal {S}_2^0.$ On the other hand, in view of (REF ) and recalling that $\\xi (0)=0$ , we note that $\\begin{split}&V_{1,n}e^{w_{1,n}}\\,dX\\stackrel{*}{\\rightharpoonup }m_1(p_0)\\delta _{X=0}+s_1(X)\\,dX,\\\\&V_{2,n}e^{w_{2,n}}\\,dX\\stackrel{*}{\\rightharpoonup }m_2(p_0)\\delta _{X=0}+s_2(X)\\,dX,\\\\\\end{split}$ $\\ast $ -weakly in $\\mathcal {M} (\\bar{\\mathcal {O})}$ , where $s_i(X)=r_i(\\Psi ^{-1}(X))e^{\\xi (X)}$ , $s_i\\in L^1(\\mathcal {O})\\cap L^\\infty _{loc}(\\bar{\\mathcal {O}} \\setminus \\lbrace 0\\rbrace )$ and $m_i(p_0)\\ge 4\\pi $ , $i=1,2$ .",
"In view of (REF ) there exist $Y_{1,n},Y_{2,n}\\in \\mathcal {O}$ , $Y_{1,n},Y_{2,n}\\rightarrow 0$ such that $w_{i,n}(Y_{i,n})=&\\sup _{\\mathcal {O}}w_{i,n}\\rightarrow +\\infty , \\quad i=1,2.$ In order to conclude the proof, we rescale the Liouville system (REF ) twice.",
"Proof of $m_1(p_0)\\ge 8\\pi $ .",
"We first rescale (REF ) around $Y_{1,n}$ with respect the rescaling parameter $\\varepsilon _{1,n}=&e^{-w_{1,n}(Y_{1,n})/2},$ that is $w_{1,n}(Y_{1,n})=\\sup _{\\mathcal {O}}w_{1,n}=-2\\log \\varepsilon _{1,n}$ .",
"Namely, we define the expanding domain $\\mathcal {O}_n^1=\\lbrace X\\in \\mathbb {R}^2:\\ Y_{1,n}+\\varepsilon _{1,n}X\\in \\mathcal {O}\\rbrace $ and we define $\\widetilde{w}_{1,n}^1,\\widetilde{w}_{2,n}^1:\\mathcal {O}_n^1\\rightarrow \\mathbb {R}$ by setting $\\widetilde{w}_{1,n}^1(X)=&w_{1,n}(Y_{1,n}+\\varepsilon _{1,n}X)-w_{1,n}(Y_{1,n})\\\\\\widetilde{w}_{2,n}^1(X)=&w_{2,n}(Y_{1,n}+\\varepsilon _{1,n}X)-w_{1,n}(Y_{1,n}).$ Then, $\\widetilde{w}_{1,n}^1,\\widetilde{w}_{2,n}^1$ is a solution for the Liouville system $\\left\\lbrace \\begin{split}-\\Delta \\widetilde{w}_{1,n}^1=&\\widetilde{V}_{1,n}^1e^{\\widetilde{w}_{1,n}^1}-\\widetilde{V}_{2,n}^1e^{\\widetilde{w}_{2,n}^1}\\\\-\\Delta \\widetilde{w}_{2,n}^1=&-\\gamma \\widetilde{V}_{1,n}^1e^{\\widetilde{w}_{1,n}^1}+\\gamma \\widetilde{V}_{2,n}^1e^{\\widetilde{w}_{2,n}^1}\\end{split}\\right.$ in $\\mathcal {O}_n^1$ , where $\\widetilde{V}_{1,n}^1(X)=V_{1,n}(Y_{1,n}+\\varepsilon _{1,n}X)$ and $\\widetilde{V}_{2,n}^1(X)=V_{2,n}(Y_{1,n}+\\varepsilon _{1,n}X)$ , which is of the form (REF ) with $a=b=1$ , $c=d=\\gamma $ .",
"Moreover, (REF ) is satisfied since $0\\le \\widetilde{V}_{i,n}^1\\le C$ and $\\int _{\\mathcal {O}_n^1}e^{\\widetilde{w}_{i,n}^1}\\,dX=\\int _{\\mathcal {O}} e^{w_{i,n}}\\,dX\\le C.$ We also note that (REF ) implies that $\\widetilde{w}_{1,n}^1(X)\\le \\widetilde{w}_{1,n}^1(0)=0,\\qquad \\widetilde{w}_{1,n}^1+\\frac{1}{\\gamma }\\widetilde{w}_{2,n}^1\\rightarrow -\\infty .$ We apply Lemma REF on a ball $B_R\\subset \\mathbb {R}^2$ of fixed large radius $R>0$ .",
"In view of (REF ) we rule out Alternative 1 and Alternative 3.",
"Suppose Alternative 2. holds.",
"Then, $\\widetilde{w}_{1,n}^1$ is uniformly bounded in $B_R$ and $\\widetilde{w}_{2,n}^1\\rightarrow -\\infty $ locally uniformly in $B_R$ .",
"In particular, there exists $\\widetilde{w}_{1}^1\\in C_{\\mathrm {l}oc}^{1,\\alpha }(\\mathbb {R}^2)$ such that $\\widetilde{w}_{1,n}^1\\rightarrow \\widetilde{w}_{1}^1$ , $e^{\\widetilde{w}_{2,n}^1}\\rightarrow 0$ , locally uniformly in $\\mathbb {R}^2$ .",
"Consequently, we derive that $\\widetilde{w}_{1}^1$ satisfies the Liouville equation $-\\Delta \\widetilde{w}_{1}^1=V_1e^{\\widetilde{w}_{1}^1}\\qquad \\mathrm {on\\ }\\mathbb {R}^2.$ Since $\\int _{B_R}e^{\\widetilde{w}_{1}^1}=\\lim _n\\int _{B_R}e^{\\widetilde{w}_{1,n}^1}\\le \\int _{\\mathcal {O}_n^1}e^{\\widetilde{w}_{1,n}^1}\\le C$ we also have $\\int _{\\mathbb {R}^2}e^{\\widetilde{w}_{1}^1}\\le C$ .",
"Now the classification theorem in [4] implies that $\\int _{\\mathbb {R}^2}V_1e^{\\widetilde{w}_{1}^1}=8\\pi $ and consequently $m_1(p_0)=8\\pi $ .",
"This established the asserted lower bound $m_1(p_0)\\ge 8\\pi $ in the case where $\\widetilde{w}_{1,n}^1,\\widetilde{w}_{2,n}^1$ satisfy Alternative 2.",
"Suppose Alternative 4 holds.",
"Then $\\widetilde{w}_{1,n}^1$ is locally uniformly bounded,$\\mathcal {S}_1^0=\\emptyset $ , $\\mathcal {S}_2^0\\ne \\emptyset $ and $\\widetilde{w}_{2,n}^1\\rightarrow -\\infty $ locally uniformly.",
"Moreover, $\\widetilde{V}_{2,n}e^{\\widetilde{w}_{2,n}^1}\\,dX\\stackrel{\\ast }{\\rightharpoonup }m_2(x_0)\\delta _0.$ We conclude that there exists $\\widetilde{w}_{1}^1\\in C_{\\mathrm {l}oc}^{1,\\alpha }(\\mathbb {R}^2)$ such that $\\widetilde{w}_{1,n}^1\\rightarrow \\widetilde{w}_{1}^1$ locally uniformly, $-\\Delta \\widetilde{w}_{1}^1=V_1e^{\\widetilde{w}_{1}^1}-m_2(x_0)\\delta _{x_0}.$ In view of [5] we conclude that $\\int _{\\mathbb {R}^2}V_1e^{\\widetilde{w}_{1}^1}>4\\pi +m_2(x_0)>8\\pi ,$ and the asserted lower bound $m_1(x_0)\\ge 8\\pi $ holds true in this case as well.",
"Proof of $m_2(p_0)\\ge \\frac{8\\pi }{\\gamma }$ .",
"Similarly as above, we rescale (REF ) around $Y_{2,n}$ with respect to $\\varepsilon _{2,n}$ given by $\\varepsilon _{2,n}=&e^{-{w_{2,n}(Y_{2,n})} /2},$ that is $w_{2,n}(Y_{2,n})=\\sup _{\\mathcal {O}}w_{2,n}=-2\\log \\varepsilon _{2,n}$ .",
"We define the expanding domain $\\mathcal {O}_n^2=\\lbrace X\\in \\mathbb {R}^2:\\ Y_{2,n}+\\varepsilon _{2,n}X\\in \\mathcal {O}\\rbrace $ and we define $\\widetilde{w}_{1,n}^2,\\widetilde{w}_{2,n}^2:\\mathcal {O}_n^2\\rightarrow \\mathbb {R}$ by setting $\\widetilde{w}_{1,n}^2(X)=&w_{1,n}(Y_{2,n}+\\varepsilon _{2,n}X)-w_{2,n}(Y_{2,n})\\\\\\widetilde{w}_{2,n}^2(X)=&w_{2,n}(Y_{2,n}+\\varepsilon _{2,n}X)-w_{2,n}(Y_{2,n}).$ Then, $\\widetilde{w}_{1,n}^2,\\widetilde{w}_{2,n}^2$ is a solution for the Liouville system $\\left\\lbrace \\begin{split}-\\Delta \\widetilde{w}_{1,n}^2=&\\widetilde{V}_{1,n}^2e^{\\widetilde{w}_{1,n}^2}-\\widetilde{V}_{2,n}^2e^{\\widetilde{w}_{2,n}^2}\\\\-\\Delta \\widetilde{w}_{2,n}^2=&-\\gamma \\widetilde{V}_{1,n}^2e^{\\widetilde{w}_{1,n}^2}+\\gamma \\widetilde{V}_{2,n}^2e^{\\widetilde{w}_{2,n}^2}\\end{split}\\right.$ in $\\mathcal {O}_n^2$ , where $\\widetilde{V}_{1,n}^2(X)=V_{1,n}(Y_{2,n}+\\varepsilon _{2,n}X)$ and $\\widetilde{V}_{2,n}^2(X)=V_{2,n}(Y_{2,n}+\\varepsilon _{2,n}X)$ .",
"Furthermore, as above, $\\widetilde{w}_{1,n}^2+\\frac{1}{\\gamma }\\widetilde{w}_{2,n}^2\\rightarrow -\\infty .$ We observe that $0\\leqslant \\widetilde{V}_{i,n}^2(X)\\leqslant C,\\qquad \\int _{\\mathcal {O}_n^2}e^{\\widetilde{w}_{i,n}^2}\\,dX\\leqslant C$ for $i,=1,2$ , for some $C>0$ .",
"Therefore, Lemma REF may be applied locally to the Liouville systems (REF ) and (REF ).",
"By analogous arguments as above, we conclude that there exists $\\widetilde{w}_{2}^1$ such that $\\widetilde{w}_{2,n}^2\\rightarrow \\widetilde{w}_{2}^1$ locally uniformly in $\\mathbb {R}^2$ with $\\gamma \\int _{\\mathbb {R}^2}\\widetilde{V}_2^2 e^{\\widetilde{w}_{2}^1}\\ge 8\\pi .$ That is, $m_2(p_0)=\\int _{\\mathbb {R}^2}\\widetilde{V}_2^2 e^{\\widetilde{w}_{2}^1}\\ge \\frac{8\\pi }{\\gamma },$ as desired.",
"Proof of Theorem REF We recall from Section that the variational functional for problem (REF ) is given by: $J(u)=\\frac{1}{2}\\int _\\Sigma |\\nabla _g v|^2-\\lambda _1\\log \\int _\\Sigma e^v-\\frac{\\lambda _2}{\\gamma }\\log \\int _\\Sigma e^{-\\gamma v},\\quad v\\in \\mathcal {E}.$ We begin by checking that $J$ admits a mountain pass geometry for all $(\\lambda _1,\\lambda _2)\\in \\Lambda $ .",
"Lemma 3.1 (Existence of a local minimum) Fix $\\lambda _1,\\lambda _2 \\ge 0.$ If $\\lambda _1+\\gamma \\lambda _2 <{\\mu _{1}(\\Sigma )\\left|\\Sigma \\right|},$ then $v\\equiv 0$ is a strict local minimum for $J$ .",
"We set $G_1(v):=\\log \\left( \\int _{\\Sigma }e^{ v}\\,dV_g\\right),\\quad G_2(v):=\\log \\left( \\int _{\\Sigma }e^{ -\\gamma v}\\,dV_g\\right),$ so that we may write $J(v)=\\frac{1}{2}\\int _{\\Sigma }\\left|\\nabla _g v\\right|^{2}-\\lambda _1G_1(v)-\\frac{\\lambda _2}{\\gamma }G_2(v).$ For every $\\phi \\in \\mathcal {E}$ , we compute: $G_1^{\\prime }(v)\\phi =\\int _{\\Sigma }\\frac{e^{ v}\\phi }{\\int _{\\Sigma }e^{v}};\\qquad G_2^{\\prime }(v)\\phi =\\int _{\\Sigma }\\frac{-\\gamma e^{\\gamma v}\\phi }{\\int _{\\Sigma }e^{-\\gamma v}}.$ Moreover, for every $\\phi ,\\psi \\in \\mathcal {E}$ , $\\left\\langle G_1^{\\prime \\prime }(v)\\phi ,\\psi \\right\\rangle =\\frac{\\left(\\int _{\\Sigma }e^{v}\\phi \\psi \\right)\\left(\\int _{\\Sigma }e^{ v}\\right) - \\left( \\int _{\\Sigma } e^{ v}\\phi \\right)\\left(\\int _{\\Sigma } e^{ v}\\psi \\right)}{\\left(\\int _{\\Sigma } e^{ v}\\right)^2 }.$ $\\left\\langle G_2^{\\prime \\prime }(v)\\phi ,\\psi \\right\\rangle =\\gamma ^2 \\frac{\\left(\\int _{\\Sigma }e^{-\\gamma v}\\phi \\psi \\right)\\left(\\int _{\\Sigma }e^{-\\gamma v}\\right) - \\left( \\int _{\\Sigma } e^{-\\gamma v}\\phi \\right)\\left(\\int _{\\Sigma } e^{-\\gamma v}\\psi \\right)}{\\left(\\int _{\\Sigma } e^{-\\gamma v}\\right)^2 }.$ In particular, for every $\\phi \\in \\mathcal {E}$ , we derive that $\\begin{split}G_1^{\\prime }(0)\\phi =G_2^{\\prime }(0)\\phi =0;\\quad \\left\\langle G_1^{\\prime \\prime }(0)\\phi ,\\phi \\right\\rangle =\\frac{1}{|\\Sigma |}\\int _\\Sigma \\phi ^2;\\quad \\left\\langle G_2^{\\prime \\prime }(0)\\phi ,\\phi \\right\\rangle =\\frac{\\gamma ^2}{|\\Sigma |}\\int _\\Sigma \\phi ^2.\\end{split}$ Consequently, for every $ \\phi \\in \\mathcal {E}$ , $J^\\prime (0)\\phi =0, \\qquad \\qquad \\left\\langle J^{\\prime \\prime }(0)\\phi ,\\phi \\right\\rangle =\\int _\\Sigma |\\nabla \\phi |^2-\\frac{{\\lambda _1}+\\gamma \\lambda _2}{|\\Sigma |} \\int _\\Sigma \\phi ^2.$ Recalling the Poincaré inequality $\\int _\\Sigma \\phi ^2 \\le \\frac{1}{\\mu _1(\\Sigma )} \\int _\\Sigma |\\nabla \\phi |^2,\\qquad \\forall \\phi \\in \\mathcal {E},$ where $\\mu _1(\\Sigma )$ is the first eigenvalue defined in (REF ), we deduce by Taylor expansion at 0 that $\\begin{split}J(u)-J(0)&= J^\\prime (0)u +\\frac{1}{2} \\left\\langle J^{\\prime \\prime }(0)u,u\\right\\rangle +o(\\Vert u\\Vert ^2)\\\\&\\ge \\frac{1}{2} \\left( 1 -\\frac{\\lambda _1 + \\gamma \\lambda _2 }{\\mu _1(\\Sigma ) |\\Sigma | } \\right) \\Vert u\\Vert +o (\\Vert u\\Vert ^2),\\end{split}$ for all $u\\in \\mathcal {E}$ .",
"Hence, 0 is a local minimum for $J$ , as asserted.",
"In the next lemma we check that there exists a direction for $J$ along which $J$ is unbounded from below.",
"Lemma 3.2 If $\\max \\lbrace \\lambda _1,\\gamma \\lambda _2 \\rbrace >8\\pi $ , then there exists $v_1\\in \\mathcal {E}$ such that $J(v_1)<0$ and $\\Vert v_1\\Vert \\ge 1$ .",
"The unboundedness from below of $J$ , i.e., the existence of $v_1$ , may be derived simply inserting the special form of $\\mathcal {P}$ given by (REF ) into formula (REF ).",
"However, since the Struwe monotonicity argument also requires a control on $\\Vert v_1\\Vert $ , we use a family of test functions derived from the classical “Liouville bubbles\".",
"Namely, we fix $p_0\\in \\Sigma $ and $r_0>0$ a constant smaller than the injectivity radius of $\\Sigma $ at $p_0$ .",
"Let $\\mathcal {B}_{r_0}=\\lbrace p\\in \\Sigma :\\ d_g(p,p_0)<r_0\\rbrace $ denote the geodesic ball of radius $r_0$ centered at $p_0$ .",
"For every $\\varepsilon >0$ let $U_\\varepsilon $ be the function defined by $U_\\varepsilon (p)={\\left\\lbrace \\begin{array}{ll}\\log \\frac{\\varepsilon ^2}{(\\varepsilon ^2+d_g(p,p_0)^2)^2}&\\mbox{in\\ }\\mathcal {B}_{r_0}\\\\\\log \\frac{\\varepsilon ^2}{(\\varepsilon ^2+r_0^2)^2}&\\mbox{in\\ }\\Sigma \\setminus \\mathcal {B}_{r_0}.\\end{array}\\right.",
"}$ Let $v_\\varepsilon \\in \\mathcal {E}$ be defined by $v_\\varepsilon =U_\\varepsilon -\\frac{1}{|\\Sigma |}\\int _\\Sigma U_\\varepsilon \\,dV_g.$ By elementary computations we have $\\begin{aligned}&\\int _\\Sigma |\\nabla _gU_\\varepsilon |^2\\,dV_g=16\\pi \\log \\frac{1}{\\varepsilon ^2}+O(1);\\\\&\\frac{1}{|\\Sigma |}\\int _{\\Sigma }U_\\varepsilon \\,dV_g=\\ln \\varepsilon ^2+O(1);\\end{aligned}$ and therefore $\\begin{aligned}&\\int _\\Sigma |\\nabla _gv_\\varepsilon |^2\\,dV_g=16\\pi \\log \\frac{1}{\\varepsilon ^2}+O(1);\\\\&\\log \\int _\\Sigma e^{v_\\varepsilon }\\,dV_g=\\log \\frac{1}{\\varepsilon ^2}+O(1);\\\\&\\log \\int _\\Sigma e^{-av_\\varepsilon }\\,dV_g=O(1),\\quad \\forall a>0.\\end{aligned}$ By assumption we have $\\lambda _1>8\\pi $ or $\\lambda _2>8\\pi /\\gamma $ .",
"Suppose $\\lambda _1>8\\pi $ .",
"Then, in view of the expansions above, we have: $J(v_\\varepsilon )=(8\\pi -\\lambda _1)\\log \\frac{1}{\\varepsilon ^2}+O(1),$ and therefore there exists $\\varepsilon _1>0$ such that $v_1:=v_{\\varepsilon _1}$ satisfies the desired properties.",
"Suppose $\\lambda _2>8\\pi /\\gamma $ .",
"Then, $J(-\\frac{v_\\varepsilon }{\\gamma })=\\frac{1}{\\gamma }(\\frac{8\\pi }{\\gamma }-\\lambda _2)\\log \\frac{1}{\\varepsilon ^2}+O(1),$ and the statement holds true in this case as well.",
"In particular, Lemma REF and Lemma REF imply that the functional $J$ admits a mountain-pass geometry whenever $(\\lambda _1,\\lambda _2)\\in \\Lambda $ .",
"Finally, in order to conclude the proof of Theorem REF we need the following consequences of Theorem REF .",
"Lemma 3.3 The following properties hold.",
"For every $p\\in \\mathcal {S}_1\\cap \\mathcal {S}_2$ we have $m_1+\\gamma m_2\\ge 16\\pi (1+{\\gamma }).$ If $\\mathcal {S}_1\\setminus \\mathcal {S}_2\\ne \\emptyset $ , then $r_1\\equiv 0$ ; in particular, in this case $\\lambda _{1,0}= 8\\pi \\,\\sharp \\mathcal {S}_1$ .",
"Similarly, if $\\mathcal {S}_2\\setminus \\mathcal {S}_1\\ne \\emptyset $ , then $r_2\\equiv 0$ and $\\lambda _{2,0}= \\frac{8\\pi }{\\gamma }\\,\\sharp \\mathcal {S}_2$ .",
"Proof of Part (i).",
"In view of Theorem REF and identity (REF ) it suffices to show that $\\alpha _\\gamma :=\\min \\left\\lbrace x+\\gamma y:\\ x\\ge 8\\pi ,\\ y\\ge \\frac{8\\pi }{\\gamma },\\ (x-y)^2=8\\pi (x+\\frac{y}{\\gamma })\\right\\rbrace =16\\pi (1+\\gamma ).$ Let $\\mathbf {p}:=\\lbrace (x,y)\\in \\mathbb {R}^2:\\ (x-y)^2=8\\pi (x+\\frac{y}{\\gamma })\\rbrace $ be the parabola defined by the quadratic relation (REF ).",
"Let $\\overline{x},\\overline{y}>0$ be such that $(\\overline{x},\\frac{8\\pi }{\\gamma })\\in \\mathbf {p}$ and $(8\\pi ,\\overline{y})\\in \\mathbf {p}$ .",
"In view of the geometric properties of $\\mathbf {p}$ we find that $\\alpha _\\gamma =\\min \\lbrace \\overline{x}+8\\pi ,8\\pi +\\gamma \\overline{y}\\rbrace .$ Solving the equation $(x-y)^2=8\\pi (x+\\frac{y}{\\gamma })$ with respect to $x$ , we have $x(y)=x_\\pm (y)=y+4\\pi \\pm \\sqrt{8\\pi (1+\\frac{1}{\\gamma })y+16\\pi ^2}.$ We deduce that $\\overline{x}=x_+(\\frac{8\\pi }{\\gamma })=8\\pi (1+\\frac{2}{\\gamma }).$ Similarly, solving $(x-y)^2=8\\pi (x+\\frac{y}{\\gamma })$ with respect to $y$ , we have $y=y_\\pm (x)=x+\\frac{4\\pi }{\\gamma }\\pm \\sqrt{8\\pi (1+\\frac{1}{\\gamma }x+\\frac{16\\pi ^2}{\\gamma ^2})}.$ We conclude that $\\overline{y}=y_+(8\\pi )=8\\pi (2+\\frac{1}{\\gamma }).$ It follows that $\\alpha _\\gamma =\\min \\lbrace 16\\pi (1+\\frac{1}{\\gamma }),\\,16\\pi (1+\\gamma )\\rbrace =16\\pi (1+\\gamma ).$ Proof of Part (ii).",
"Let $&u_{1,n}=G\\ast \\lambda _{1,n}\\frac{e^{v_n}}{\\int _\\Sigma e^{v_n}},&&u_{2,n}=G\\ast \\lambda _{2,n}\\frac{e^{-\\gamma v_n}}{\\int _\\Sigma e^{-\\gamma v_n}},$ where $G=G(x,p)$ denotes the Green's function defined by ${\\left\\lbrace \\begin{array}{ll}-\\Delta _{x}G(\\cdot ,p)=\\delta _p-\\frac{1}{|\\Sigma |}&\\mbox{on\\ }\\Sigma \\\\\\int _\\Sigma G(\\cdot ,p)\\,dV_g=0\\end{array}\\right.",
"}$ and where $\\ast $ denotes convolution.",
"Then, $v_n=u_{1,n}-u_{2,n}$ .",
"For a measurable function $f$ defined on a subset of $\\Sigma $ and for any $T>0$ we set $f^T=\\min \\lbrace f,T\\rbrace $ .",
"Suppose $p\\in \\mathcal {S}_1\\setminus \\mathcal {S}_2$ .",
"We claim that $\\int _\\Sigma e^{v_n}\\rightarrow +\\infty .$ Indeed, we have, using (REF ) and Theorem REF : $u_{1,n}(x)\\ge G^T\\ast \\lambda _{1,n}\\frac{e^{v_n}}{\\int _\\Sigma e^{v_n}}\\rightarrow G^T\\ast \\sum _{p\\in \\mathcal {S}_1}(m_1(p)+r_1)\\ge 8\\pi G^T(x,p)-C.$ In local coordinates centered at $p$ we obtain $u_{1,n}(X)\\ge \\ln [\\frac{1}{|X|^4}]^T-C.$ Since $p\\notin \\mathcal {S}_2$ , we have that $\\Vert u_{2,n}\\Vert _{L^\\infty (B_\\rho (p))}$ is uniformly bounded with respect to $n$ , for some sufficiently small $\\rho >0$ .",
"Now, Fatou's lemma implies that $\\liminf _{n\\rightarrow \\infty }\\int _\\Sigma e^{v_n}=\\liminf _{n\\rightarrow \\infty }\\int _\\Sigma e^{u_{1,n}-u_{2,n}}\\ge c_0\\int _{B_\\rho (p)}[\\frac{1}{|X|^4}]^T$ for some $c_0>0$ independent of $n$ .",
"Letting $T\\rightarrow +\\infty $ , we deduce that (REF ) holds true, and consequently $r_1\\equiv 0$ .",
"Now, we assume that $p\\in \\mathcal {S}_2\\setminus \\mathcal {S}_1$ .",
"Similarly as above, we claim that $\\int _\\Sigma e^{-\\gamma v_n}\\rightarrow +\\infty .$ Indeed, we note that $u_{2,n}(x)\\ge G^T\\ast \\lambda _{2,n}\\frac{e^{-\\gamma v_n}}{\\int _\\Sigma e^{-\\gamma v_n}}\\rightarrow G^T\\ast \\sum _{p\\in \\mathcal {S}_2}(m_2(p)\\delta _p+r_2)\\ge \\frac{8\\pi }{\\gamma }\\sum _{p\\in \\mathcal {S}_2}G^T(x,p)-C,$ for some $C>0$ independent of $n$ .",
"In particular, in local coordinates $X$ near $p$ , $u_{2,n}(X)\\ge \\frac{4}{\\gamma }\\ln [\\frac{1}{|X|}]^T-C.$ On the other hand, since $p\\notin \\mathcal {S}_1$ , we have that $\\Vert u_{1,n}\\Vert _{L^\\infty (B_\\rho (p))}$ is bounded independently of $n$ in a ball centered at $p$ provided $\\rho >0$ is sufficiently small.",
"We conclude that $\\int _\\Sigma e^{-\\gamma v_n}\\ge c_0\\int _{B_\\rho (p)}[\\frac{1}{|X|^4}]^T.$ In view of Fatou's lemma and letting $T\\rightarrow +\\infty $ , we derive (REF ).",
"Therefore, $r_2\\equiv 0$ , as asserted.",
"Finally, we provide the proof of Theorem REF .",
"In view of Struwe's Monotonicity Trick, for almost every $(\\lambda _1,\\lambda _2) \\in \\Lambda $ there exists a nontrivial solution to (REF ).",
"As this argument is by now standard, we refer to [29], [30] for a proof.",
"A detailed proof in this specific context may also be found in [24].",
"Let $(\\lambda _1^0,\\lambda _2^0)\\in \\Lambda $ be fixed.",
"Let $(\\lambda _{1,n},\\lambda _{2,n},v_n)$ be a solution sequence for (REF ) with $\\lambda _{1,n}\\rightarrow \\lambda _{1}^0$ and $\\lambda _{2,n}\\rightarrow \\lambda _{2}^0$ , $(\\lambda _{1,n},\\lambda _{2,n})\\in \\Lambda $ .",
"We claim that $v_n$ converges to some $v\\in \\mathcal {E}$ , solution to (REF ).",
"To this aim we exploit the blow up analysis as derived in Theorem REF .",
"By contradiction we assume that $p_0\\in \\mathcal {S}$ .",
"If $p_0\\in \\mathcal {S}_1\\setminus \\mathcal {S}_2$ , then in view of Lemma REF –(ii) we have $\\lambda _1^0\\in 8\\pi \\mathbb {N}$ , a contradiction to (REF ).",
"If $p_0\\in \\mathcal {S}_2\\setminus \\mathcal {S}_1$ , then in view of Lemma REF –(ii) we have $\\lambda _2^0\\in \\frac{8\\pi }{\\gamma }\\mathbb {N}$ , a contradiction to (REF ).",
"Suppose that $p_0\\in \\mathcal {S}_1\\cap \\mathcal {S}_2$ .",
"In this case, using (REF ), $\\lambda _1+\\gamma \\lambda _2 \\ge m_1(p_0)+\\gamma m_2 (p_0)\\ge 16 \\pi (1+\\gamma )$ and we again obtain a contradiction to (REF ).",
"Hence, the compactness for the solution sequence $(\\lambda _{1,n},\\lambda _{2,n},v_n)$ holds and Theorem REF is completely established.",
"Acknowledgments.",
"This work is supported by Progetto GNAMPA-INDAM 2015: Alcuni aspetti di equazioni ellittiche non lineari.",
"The first author is also supported by PRIN Aspetti variazionali e perturbativi nei problemi differenziali nonlineari.",
"The second author is also supported by Progetto STAR 2015: Variational Analysis and Equilibrium Models in Physical and Socio-Economic Phenomena."
],
[
"Proof of Theorem ",
"We recall from Section that the variational functional for problem (REF ) is given by: $J(u)=\\frac{1}{2}\\int _\\Sigma |\\nabla _g v|^2-\\lambda _1\\log \\int _\\Sigma e^v-\\frac{\\lambda _2}{\\gamma }\\log \\int _\\Sigma e^{-\\gamma v},\\quad v\\in \\mathcal {E}.$ We begin by checking that $J$ admits a mountain pass geometry for all $(\\lambda _1,\\lambda _2)\\in \\Lambda $ .",
"Lemma 3.1 (Existence of a local minimum) Fix $\\lambda _1,\\lambda _2 \\ge 0.$ If $\\lambda _1+\\gamma \\lambda _2 <{\\mu _{1}(\\Sigma )\\left|\\Sigma \\right|},$ then $v\\equiv 0$ is a strict local minimum for $J$ .",
"We set $G_1(v):=\\log \\left( \\int _{\\Sigma }e^{ v}\\,dV_g\\right),\\quad G_2(v):=\\log \\left( \\int _{\\Sigma }e^{ -\\gamma v}\\,dV_g\\right),$ so that we may write $J(v)=\\frac{1}{2}\\int _{\\Sigma }\\left|\\nabla _g v\\right|^{2}-\\lambda _1G_1(v)-\\frac{\\lambda _2}{\\gamma }G_2(v).$ For every $\\phi \\in \\mathcal {E}$ , we compute: $G_1^{\\prime }(v)\\phi =\\int _{\\Sigma }\\frac{e^{ v}\\phi }{\\int _{\\Sigma }e^{v}};\\qquad G_2^{\\prime }(v)\\phi =\\int _{\\Sigma }\\frac{-\\gamma e^{\\gamma v}\\phi }{\\int _{\\Sigma }e^{-\\gamma v}}.$ Moreover, for every $\\phi ,\\psi \\in \\mathcal {E}$ , $\\left\\langle G_1^{\\prime \\prime }(v)\\phi ,\\psi \\right\\rangle =\\frac{\\left(\\int _{\\Sigma }e^{v}\\phi \\psi \\right)\\left(\\int _{\\Sigma }e^{ v}\\right) - \\left( \\int _{\\Sigma } e^{ v}\\phi \\right)\\left(\\int _{\\Sigma } e^{ v}\\psi \\right)}{\\left(\\int _{\\Sigma } e^{ v}\\right)^2 }.$ $\\left\\langle G_2^{\\prime \\prime }(v)\\phi ,\\psi \\right\\rangle =\\gamma ^2 \\frac{\\left(\\int _{\\Sigma }e^{-\\gamma v}\\phi \\psi \\right)\\left(\\int _{\\Sigma }e^{-\\gamma v}\\right) - \\left( \\int _{\\Sigma } e^{-\\gamma v}\\phi \\right)\\left(\\int _{\\Sigma } e^{-\\gamma v}\\psi \\right)}{\\left(\\int _{\\Sigma } e^{-\\gamma v}\\right)^2 }.$ In particular, for every $\\phi \\in \\mathcal {E}$ , we derive that $\\begin{split}G_1^{\\prime }(0)\\phi =G_2^{\\prime }(0)\\phi =0;\\quad \\left\\langle G_1^{\\prime \\prime }(0)\\phi ,\\phi \\right\\rangle =\\frac{1}{|\\Sigma |}\\int _\\Sigma \\phi ^2;\\quad \\left\\langle G_2^{\\prime \\prime }(0)\\phi ,\\phi \\right\\rangle =\\frac{\\gamma ^2}{|\\Sigma |}\\int _\\Sigma \\phi ^2.\\end{split}$ Consequently, for every $ \\phi \\in \\mathcal {E}$ , $J^\\prime (0)\\phi =0, \\qquad \\qquad \\left\\langle J^{\\prime \\prime }(0)\\phi ,\\phi \\right\\rangle =\\int _\\Sigma |\\nabla \\phi |^2-\\frac{{\\lambda _1}+\\gamma \\lambda _2}{|\\Sigma |} \\int _\\Sigma \\phi ^2.$ Recalling the Poincaré inequality $\\int _\\Sigma \\phi ^2 \\le \\frac{1}{\\mu _1(\\Sigma )} \\int _\\Sigma |\\nabla \\phi |^2,\\qquad \\forall \\phi \\in \\mathcal {E},$ where $\\mu _1(\\Sigma )$ is the first eigenvalue defined in (REF ), we deduce by Taylor expansion at 0 that $\\begin{split}J(u)-J(0)&= J^\\prime (0)u +\\frac{1}{2} \\left\\langle J^{\\prime \\prime }(0)u,u\\right\\rangle +o(\\Vert u\\Vert ^2)\\\\&\\ge \\frac{1}{2} \\left( 1 -\\frac{\\lambda _1 + \\gamma \\lambda _2 }{\\mu _1(\\Sigma ) |\\Sigma | } \\right) \\Vert u\\Vert +o (\\Vert u\\Vert ^2),\\end{split}$ for all $u\\in \\mathcal {E}$ .",
"Hence, 0 is a local minimum for $J$ , as asserted.",
"In the next lemma we check that there exists a direction for $J$ along which $J$ is unbounded from below.",
"Lemma 3.2 If $\\max \\lbrace \\lambda _1,\\gamma \\lambda _2 \\rbrace >8\\pi $ , then there exists $v_1\\in \\mathcal {E}$ such that $J(v_1)<0$ and $\\Vert v_1\\Vert \\ge 1$ .",
"The unboundedness from below of $J$ , i.e., the existence of $v_1$ , may be derived simply inserting the special form of $\\mathcal {P}$ given by (REF ) into formula (REF ).",
"However, since the Struwe monotonicity argument also requires a control on $\\Vert v_1\\Vert $ , we use a family of test functions derived from the classical “Liouville bubbles\".",
"Namely, we fix $p_0\\in \\Sigma $ and $r_0>0$ a constant smaller than the injectivity radius of $\\Sigma $ at $p_0$ .",
"Let $\\mathcal {B}_{r_0}=\\lbrace p\\in \\Sigma :\\ d_g(p,p_0)<r_0\\rbrace $ denote the geodesic ball of radius $r_0$ centered at $p_0$ .",
"For every $\\varepsilon >0$ let $U_\\varepsilon $ be the function defined by $U_\\varepsilon (p)={\\left\\lbrace \\begin{array}{ll}\\log \\frac{\\varepsilon ^2}{(\\varepsilon ^2+d_g(p,p_0)^2)^2}&\\mbox{in\\ }\\mathcal {B}_{r_0}\\\\\\log \\frac{\\varepsilon ^2}{(\\varepsilon ^2+r_0^2)^2}&\\mbox{in\\ }\\Sigma \\setminus \\mathcal {B}_{r_0}.\\end{array}\\right.",
"}$ Let $v_\\varepsilon \\in \\mathcal {E}$ be defined by $v_\\varepsilon =U_\\varepsilon -\\frac{1}{|\\Sigma |}\\int _\\Sigma U_\\varepsilon \\,dV_g.$ By elementary computations we have $\\begin{aligned}&\\int _\\Sigma |\\nabla _gU_\\varepsilon |^2\\,dV_g=16\\pi \\log \\frac{1}{\\varepsilon ^2}+O(1);\\\\&\\frac{1}{|\\Sigma |}\\int _{\\Sigma }U_\\varepsilon \\,dV_g=\\ln \\varepsilon ^2+O(1);\\end{aligned}$ and therefore $\\begin{aligned}&\\int _\\Sigma |\\nabla _gv_\\varepsilon |^2\\,dV_g=16\\pi \\log \\frac{1}{\\varepsilon ^2}+O(1);\\\\&\\log \\int _\\Sigma e^{v_\\varepsilon }\\,dV_g=\\log \\frac{1}{\\varepsilon ^2}+O(1);\\\\&\\log \\int _\\Sigma e^{-av_\\varepsilon }\\,dV_g=O(1),\\quad \\forall a>0.\\end{aligned}$ By assumption we have $\\lambda _1>8\\pi $ or $\\lambda _2>8\\pi /\\gamma $ .",
"Suppose $\\lambda _1>8\\pi $ .",
"Then, in view of the expansions above, we have: $J(v_\\varepsilon )=(8\\pi -\\lambda _1)\\log \\frac{1}{\\varepsilon ^2}+O(1),$ and therefore there exists $\\varepsilon _1>0$ such that $v_1:=v_{\\varepsilon _1}$ satisfies the desired properties.",
"Suppose $\\lambda _2>8\\pi /\\gamma $ .",
"Then, $J(-\\frac{v_\\varepsilon }{\\gamma })=\\frac{1}{\\gamma }(\\frac{8\\pi }{\\gamma }-\\lambda _2)\\log \\frac{1}{\\varepsilon ^2}+O(1),$ and the statement holds true in this case as well.",
"In particular, Lemma REF and Lemma REF imply that the functional $J$ admits a mountain-pass geometry whenever $(\\lambda _1,\\lambda _2)\\in \\Lambda $ .",
"Finally, in order to conclude the proof of Theorem REF we need the following consequences of Theorem REF .",
"Lemma 3.3 The following properties hold.",
"For every $p\\in \\mathcal {S}_1\\cap \\mathcal {S}_2$ we have $m_1+\\gamma m_2\\ge 16\\pi (1+{\\gamma }).$ If $\\mathcal {S}_1\\setminus \\mathcal {S}_2\\ne \\emptyset $ , then $r_1\\equiv 0$ ; in particular, in this case $\\lambda _{1,0}= 8\\pi \\,\\sharp \\mathcal {S}_1$ .",
"Similarly, if $\\mathcal {S}_2\\setminus \\mathcal {S}_1\\ne \\emptyset $ , then $r_2\\equiv 0$ and $\\lambda _{2,0}= \\frac{8\\pi }{\\gamma }\\,\\sharp \\mathcal {S}_2$ .",
"Proof of Part (i).",
"In view of Theorem REF and identity (REF ) it suffices to show that $\\alpha _\\gamma :=\\min \\left\\lbrace x+\\gamma y:\\ x\\ge 8\\pi ,\\ y\\ge \\frac{8\\pi }{\\gamma },\\ (x-y)^2=8\\pi (x+\\frac{y}{\\gamma })\\right\\rbrace =16\\pi (1+\\gamma ).$ Let $\\mathbf {p}:=\\lbrace (x,y)\\in \\mathbb {R}^2:\\ (x-y)^2=8\\pi (x+\\frac{y}{\\gamma })\\rbrace $ be the parabola defined by the quadratic relation (REF ).",
"Let $\\overline{x},\\overline{y}>0$ be such that $(\\overline{x},\\frac{8\\pi }{\\gamma })\\in \\mathbf {p}$ and $(8\\pi ,\\overline{y})\\in \\mathbf {p}$ .",
"In view of the geometric properties of $\\mathbf {p}$ we find that $\\alpha _\\gamma =\\min \\lbrace \\overline{x}+8\\pi ,8\\pi +\\gamma \\overline{y}\\rbrace .$ Solving the equation $(x-y)^2=8\\pi (x+\\frac{y}{\\gamma })$ with respect to $x$ , we have $x(y)=x_\\pm (y)=y+4\\pi \\pm \\sqrt{8\\pi (1+\\frac{1}{\\gamma })y+16\\pi ^2}.$ We deduce that $\\overline{x}=x_+(\\frac{8\\pi }{\\gamma })=8\\pi (1+\\frac{2}{\\gamma }).$ Similarly, solving $(x-y)^2=8\\pi (x+\\frac{y}{\\gamma })$ with respect to $y$ , we have $y=y_\\pm (x)=x+\\frac{4\\pi }{\\gamma }\\pm \\sqrt{8\\pi (1+\\frac{1}{\\gamma }x+\\frac{16\\pi ^2}{\\gamma ^2})}.$ We conclude that $\\overline{y}=y_+(8\\pi )=8\\pi (2+\\frac{1}{\\gamma }).$ It follows that $\\alpha _\\gamma =\\min \\lbrace 16\\pi (1+\\frac{1}{\\gamma }),\\,16\\pi (1+\\gamma )\\rbrace =16\\pi (1+\\gamma ).$ Proof of Part (ii).",
"Let $&u_{1,n}=G\\ast \\lambda _{1,n}\\frac{e^{v_n}}{\\int _\\Sigma e^{v_n}},&&u_{2,n}=G\\ast \\lambda _{2,n}\\frac{e^{-\\gamma v_n}}{\\int _\\Sigma e^{-\\gamma v_n}},$ where $G=G(x,p)$ denotes the Green's function defined by ${\\left\\lbrace \\begin{array}{ll}-\\Delta _{x}G(\\cdot ,p)=\\delta _p-\\frac{1}{|\\Sigma |}&\\mbox{on\\ }\\Sigma \\\\\\int _\\Sigma G(\\cdot ,p)\\,dV_g=0\\end{array}\\right.",
"}$ and where $\\ast $ denotes convolution.",
"Then, $v_n=u_{1,n}-u_{2,n}$ .",
"For a measurable function $f$ defined on a subset of $\\Sigma $ and for any $T>0$ we set $f^T=\\min \\lbrace f,T\\rbrace $ .",
"Suppose $p\\in \\mathcal {S}_1\\setminus \\mathcal {S}_2$ .",
"We claim that $\\int _\\Sigma e^{v_n}\\rightarrow +\\infty .$ Indeed, we have, using (REF ) and Theorem REF : $u_{1,n}(x)\\ge G^T\\ast \\lambda _{1,n}\\frac{e^{v_n}}{\\int _\\Sigma e^{v_n}}\\rightarrow G^T\\ast \\sum _{p\\in \\mathcal {S}_1}(m_1(p)+r_1)\\ge 8\\pi G^T(x,p)-C.$ In local coordinates centered at $p$ we obtain $u_{1,n}(X)\\ge \\ln [\\frac{1}{|X|^4}]^T-C.$ Since $p\\notin \\mathcal {S}_2$ , we have that $\\Vert u_{2,n}\\Vert _{L^\\infty (B_\\rho (p))}$ is uniformly bounded with respect to $n$ , for some sufficiently small $\\rho >0$ .",
"Now, Fatou's lemma implies that $\\liminf _{n\\rightarrow \\infty }\\int _\\Sigma e^{v_n}=\\liminf _{n\\rightarrow \\infty }\\int _\\Sigma e^{u_{1,n}-u_{2,n}}\\ge c_0\\int _{B_\\rho (p)}[\\frac{1}{|X|^4}]^T$ for some $c_0>0$ independent of $n$ .",
"Letting $T\\rightarrow +\\infty $ , we deduce that (REF ) holds true, and consequently $r_1\\equiv 0$ .",
"Now, we assume that $p\\in \\mathcal {S}_2\\setminus \\mathcal {S}_1$ .",
"Similarly as above, we claim that $\\int _\\Sigma e^{-\\gamma v_n}\\rightarrow +\\infty .$ Indeed, we note that $u_{2,n}(x)\\ge G^T\\ast \\lambda _{2,n}\\frac{e^{-\\gamma v_n}}{\\int _\\Sigma e^{-\\gamma v_n}}\\rightarrow G^T\\ast \\sum _{p\\in \\mathcal {S}_2}(m_2(p)\\delta _p+r_2)\\ge \\frac{8\\pi }{\\gamma }\\sum _{p\\in \\mathcal {S}_2}G^T(x,p)-C,$ for some $C>0$ independent of $n$ .",
"In particular, in local coordinates $X$ near $p$ , $u_{2,n}(X)\\ge \\frac{4}{\\gamma }\\ln [\\frac{1}{|X|}]^T-C.$ On the other hand, since $p\\notin \\mathcal {S}_1$ , we have that $\\Vert u_{1,n}\\Vert _{L^\\infty (B_\\rho (p))}$ is bounded independently of $n$ in a ball centered at $p$ provided $\\rho >0$ is sufficiently small.",
"We conclude that $\\int _\\Sigma e^{-\\gamma v_n}\\ge c_0\\int _{B_\\rho (p)}[\\frac{1}{|X|^4}]^T.$ In view of Fatou's lemma and letting $T\\rightarrow +\\infty $ , we derive (REF ).",
"Therefore, $r_2\\equiv 0$ , as asserted.",
"Finally, we provide the proof of Theorem REF .",
"In view of Struwe's Monotonicity Trick, for almost every $(\\lambda _1,\\lambda _2) \\in \\Lambda $ there exists a nontrivial solution to (REF ).",
"As this argument is by now standard, we refer to [29], [30] for a proof.",
"A detailed proof in this specific context may also be found in [24].",
"Let $(\\lambda _1^0,\\lambda _2^0)\\in \\Lambda $ be fixed.",
"Let $(\\lambda _{1,n},\\lambda _{2,n},v_n)$ be a solution sequence for (REF ) with $\\lambda _{1,n}\\rightarrow \\lambda _{1}^0$ and $\\lambda _{2,n}\\rightarrow \\lambda _{2}^0$ , $(\\lambda _{1,n},\\lambda _{2,n})\\in \\Lambda $ .",
"We claim that $v_n$ converges to some $v\\in \\mathcal {E}$ , solution to (REF ).",
"To this aim we exploit the blow up analysis as derived in Theorem REF .",
"By contradiction we assume that $p_0\\in \\mathcal {S}$ .",
"If $p_0\\in \\mathcal {S}_1\\setminus \\mathcal {S}_2$ , then in view of Lemma REF –(ii) we have $\\lambda _1^0\\in 8\\pi \\mathbb {N}$ , a contradiction to (REF ).",
"If $p_0\\in \\mathcal {S}_2\\setminus \\mathcal {S}_1$ , then in view of Lemma REF –(ii) we have $\\lambda _2^0\\in \\frac{8\\pi }{\\gamma }\\mathbb {N}$ , a contradiction to (REF ).",
"Suppose that $p_0\\in \\mathcal {S}_1\\cap \\mathcal {S}_2$ .",
"In this case, using (REF ), $\\lambda _1+\\gamma \\lambda _2 \\ge m_1(p_0)+\\gamma m_2 (p_0)\\ge 16 \\pi (1+\\gamma )$ and we again obtain a contradiction to (REF ).",
"Hence, the compactness for the solution sequence $(\\lambda _{1,n},\\lambda _{2,n},v_n)$ holds and Theorem REF is completely established.",
"Acknowledgments.",
"This work is supported by Progetto GNAMPA-INDAM 2015: Alcuni aspetti di equazioni ellittiche non lineari.",
"The first author is also supported by PRIN Aspetti variazionali e perturbativi nei problemi differenziali nonlineari.",
"The second author is also supported by Progetto STAR 2015: Variational Analysis and Equilibrium Models in Physical and Socio-Economic Phenomena."
]
] | 1605.05895 |
[
[
"Volumes of representations and birationality of the peripheral holonomy"
],
[
"Abstract We discuss here a generalization of a theorem by Dunfield stating that the peripheral holonomy map, from the character variety of a 3-manifold to the A-polynomial is birational.",
"Dunfield's proof involves the rigidity of maximal volume.",
"The volume is still an important ingredient in this paper.",
"Unfortunately at this point no complete proof is done.",
"Instead, a conjecture is stated about the volume function on the character variety that would imply the generalized birationality result."
],
[
"Introduction",
"Let $M$ be an orientable hyperbolic manifold with one cusp (e.g.",
"a knot complement) and $\\Gamma $ its fundamental group.",
"Choose an embedding $\\mathbf {Z}^2\\rightarrow \\Gamma $ of the fundamental group of the peripheral torus.",
"Let $X(\\Gamma ,\\mathrm {PGL}(2,)$ be the character variety: $ X(\\Gamma ,\\mathrm {PGL}(2,) = \\mathrm {Hom}(\\Gamma ,\\mathrm {PGL}(2,)// \\mathrm {PGL}(2, $ and $X_2$ be the component of the hyperbolic monodromy $\\rho _\\mathrm {hyp} : \\Gamma \\rightarrow \\mathrm {PGL}(2,$ .",
"Moreover, let $\\mathrm {Hol}_\\mathrm {periph}$ denote, as in [11], [9], the peripheral holonomy.",
"It is the map $X(\\Gamma ,\\mathrm {PGL}(2,)\\rightarrow X(\\mathbf {Z}^2,\\mathrm {PGL}(2,)$ naturally induced by the restriction of a representation $\\rho :\\Gamma \\rightarrow \\mathrm {PGL}(2,$ to the peripheral $\\mathbf {Z}^2$ .",
"Note that the image of $X_2$ by the map $\\mathrm {Hol}_\\mathrm {periph}$ is the usually computed $A$ -polynomial when $M$ is a knot complement.",
"In his paper [7], Dunfield proves the following theorem: Theorem 1 The map $\\mathrm {Hol}_\\mathrm {periph}$ , from $X_2$ to its image, is a birational map.",
"We discuss in this paper a possible generalization of this result to the case of target group $\\mathrm {PGL}(n,$ for $n\\ge 2$ and multicusped $M$ .",
"The generalization for $n=2$ and $M$ multicusped is proven by Klaff-Tillmann [14].",
"Dunfield's proof uses in a crucial way the properties of the volume of representations.",
"We will review the needed facts about this function.",
"A major obstacle for a generalization is that the proof also uses – as does [14] – the fact that the set of points in $X_2$ corresponding to the hyperbolic monodromy of a Dehn surgery of $M$ is Zariski-dense in $X_2$ .",
"Rigidity of volume maximality for these points then grants the theorem.",
"Our main problem is that this Zariski density does not hold for $n>2$ .",
"We present in this paper a possible workaround, still using properties of the volume.",
"Indeed we will state a conjecture, of geometric flavour, that implies the birationality in Dunfield's theorem.",
"Hopefully, the conjecture may be easier to tackle in the general case than the birationality problem.",
"We will recall in section some facts about the character variety and the map $\\mathrm {Hol}_\\mathrm {periph}$ following mainly the presentation of [9].",
"One important result for the present paper is the local rigidity theorem (see theorem REF ).",
"We define the geometric representation as the composition $\\rho _\\mathrm {geom}= r_n\\circ \\rho _\\mathrm {hyp}$ , where $r_n$ is the unique irreducible representation from $\\mathrm {PGL}(2,$ to $\\mathrm {PGL}(n,$ .",
"The component $X_n$ of the character variety $X(\\Gamma ,\\mathrm {PGL}(n,)$ containing the geometric representation is called the geometric component.",
"We proceed in section with the definition of the volume of representations using bounded cohomology, following Bucher-Burger-Iozzi [3].",
"Moreover we recall another approach to this function: the more combinatorial notion of volume defined originally by Thurston for $n=2$ , then by Bergeron-Falbel-Guilloux [1] for $n=3$ and fully generalized both by Dimofte-Gabella-Goncharov and Garoufalidis-Goerner-Zickert [6], [10].",
"Both approaches to the volume function yield valuable informations: first, through bounded cohomology one gets the volume rigidity of the geometric representation.",
"Indeed, as proven in [3], the volume map has a unique maximum on the geometric component $X_n$ , which corresponds to $\\rho _\\mathrm {geom}$ .",
"An information given by the combinatorial approach to the volume is a formula for its derivative which is only expressed in terms of the peripheral representations (see theorem REF ).",
"In other words, the volume map on $X_n$ – as in the case $n=2$ – always factors through $\\mathrm {Hol}_\\mathrm {periph}$ (see proposition REF ) on a Zariski-open set.",
"A problem that seems rather natural is to explicitly study the volume as a function defined on $X_n$ and try to retrieve geometrical information from its behaviour.",
"For example, it raises the following question, which will be relevant for us is the following: if a representation has almost the maximal volume, is it almost the geometric representation?",
"We state the following conjecture: Conjecture 1 Let $M$ be an orientable cusped hyperbolic 3-manifold, $X_n$ the geometric component of the character variety with target group $\\mathrm {PGL}(n,$ and $\\mathrm {Hol}_\\mathrm {periph}$ the peripheral holonomy map.",
"Then, outside of a neighbourhood of $[\\rho _\\mathrm {geom}]$ , the volume is bounded away from its maximum on $X_n$ .",
"This conjecture raises an interesting and natural question per se.",
"But it also turns out that this conjecture leads to a generalization of Dunfield's theorem.",
"Indeed, we prove the following result (see section ): Theorem 2 Let $M$ be an orientable cusped hyperbolic manifold, $X_n$ the geometric component of the character variety with target group $\\mathrm {PGL}(n,$ and $\\mathrm {Hol}_\\mathrm {periph}$ the peripheral holonomy map.",
"Suppose that outside of a neighborhood of $[\\rho _\\mathrm {geom}]$ , the volume is bounded away from its maximum on $X_n$ .",
"Then the map $\\mathrm {Hol}_\\mathrm {periph}$ is a birational isomorphism between $X_n$ and its image.",
"Some experimental evidences for this conjecture have been gathered and are presented in the last section"
],
[
"Character variety and local rigidity",
"Most of the material of this section is already reviewed, with the same notations, in [9].",
"Let $n\\ge 2$ be an integer.",
"Let $N$ be a compact, oriented 3-manifold, with non-empty boundary $\\partial N$ , such that its interior $M$ is an oriented cusped hyperbolic 3-manifold.",
"Denote by $\\Gamma $ the fundamental group of $M$ ."
],
[
"Character variety and geometric representation",
"Let $G$ be a finitely generated group.",
"The character variety $X(G,\\mathrm {PGL}(n,)$ is the GIT quotient: $X(G,\\mathrm {PGL}(n,) = \\mathrm {Hom}(G,\\mathrm {PGL}(n,)//\\mathrm {PGL}(n,.$ We refer to a paper bu Sikora [19] for a general exposition of this object.",
"We will restrict to two cases: first when $G=\\Gamma =\\pi _1(M)$ and second when $G=\\mathbf {Z}^2$ .",
"We know in this setting that the character variety is an affine algebraic variety.",
"There is always a distinguished point in $X(\\Gamma ,\\mathrm {PGL}(n,)$ : the class $[\\rho _\\mathrm {geom}]$ of the geometric representation.",
"This representation is defined as the composition of the hyperbolic monodromy $\\rho _\\mathrm {hyp}: \\Gamma \\rightarrow \\mathrm {PGL}(2,$ of $M$ with the (unique) irreducible representationRecall for example that when $n=3$ , the representation $r_n$ is also known as the adjoint representation.",
"$r_n : \\mathrm {PGL}(2, \\rightarrow \\mathrm {PGL}(n,$ : $\\rho _\\mathrm {geom}= r_n \\circ \\rho _\\mathrm {hyp}.$ The character variety is not irreducible and we will not study it entirely.",
"The main object of interest in this paper will be the geometric component $X_n$ : it is the unique component of $X(\\Gamma ,\\mathrm {PGL}(n,)$ that contains $[\\rho _\\mathrm {geom}]$ ."
],
[
"Peripheral holonomy and local rigidity",
"Let $t$ be the number of peripheral toriThe reader may as well assume $t=1$ and restricts to the case of a knot complement.",
"It will not really interfere, and may simplify notations.",
"of $M$ .",
"Let $(T_i)_{1\\le i\\le t}$ be the collection of these tori and, for each $i$ , $\\Delta _i\\simeq \\mathbf {Z}^2$ be (a choice of) an injection of $\\pi _1(T_i)$ inside $\\Gamma $ .",
"For any representation $\\rho :\\Gamma \\rightarrow \\mathrm {PGL}(n,$ , one may consider its restrictions $\\rho _{\\Delta _i}$ to the various subgroups $\\Delta _i$ .",
"This restriction defines a natural algebraic map, called peripheral holonomy: $\\mathrm {Hol}_\\mathrm {periph}: X(\\Gamma ,\\mathrm {PGL}(n,) \\rightarrow \\prod _{i=1}^t X(\\Delta _i,\\mathrm {PGL}(n,).$ This map is, together with the volume, a main character of this paper.",
"Dunfield's theorem, indeed, states that (in the case $n=2$ , $t=1$ ) it is a birational isomorphism between $X_n$ and its image.",
"This map has already been studied through different points of view [11], [15], [2].",
"For the scope of this paper, we need to recall the result of Menal-Ferrer and Porti [15] (see also [2], [12]): Theorem 3 (Menal-Ferrer – Porti) On a neighborhood of $[\\rho _\\mathrm {geom}]$ in $X_n$ , the map $\\mathrm {Hol}_\\mathrm {periph}$ is a bijection on its image.",
"The theorem, as proven in the references, is more precise than this statement, giving local parameter for $X_n$ around $[\\rho _\\mathrm {geom}]$ .",
"We will not need the enhanced version.",
"As a corollary, one may note that $\\mathrm {Hol}_\\mathrm {periph}$ is a ramified covering (it is of finite degree).",
"We would like to prove that its degree is indeed 1."
],
[
"Bounded cohomology and rigidity",
"Recall the important definitions and results from the work of Bucher-Burger-Iozzi [3] about the notion of volume of a representation $\\rho : \\Gamma \\rightarrow \\mathrm {PGL}(n,$ .",
"In the cited paper, the volume map – there called Borel invariant – is defined for any $[\\rho ]$ in the character variety.",
"It is the evaluation on the fundamental class of $N$ in $H_3(N,\\partial N)$ of a suitably constructed bounded cocycle on $\\mathrm {PGL}(n,$ pulled back by $\\rho $ .",
"We will not review here the precise definition.",
"For the present work, the approach of the cited article gives a crucial theorem, namely the volume rigidity result: the maximal volume is only attained once on the whole character variety, at $[\\rho _\\mathrm {geom}]$ .",
"This rigidity theorem is a key point for the present paper.",
"Recall that $\\mathrm {Vol}_\\mathrm {hyp}$ is the hyperbolic volume of $M$ Theorem 4 (Bucher-Burger-Iozzi) The map $\\mathrm {Vol}: X(\\Gamma ,\\mathrm {PGL}(n,)\\rightarrow \\mathbf {R}$ is onto $[-\\frac{n(n^2-1)}{6}\\mathrm {Vol}_\\mathrm {hyp};\\frac{n(n^2-1)}{6}\\mathrm {Vol}_\\mathrm {hyp}]$ .",
"Moreover, for any point $[\\rho ]$ in $X(\\Gamma ,\\mathrm {PGL}(n,)$ , we have $\\mathrm {Vol}([\\rho ])=\\frac{n(n^2-1)}{6}\\mathrm {Vol}_\\mathrm {hyp}$ iff $[\\rho ]=[\\rho _\\mathrm {geom}]$ , $\\mathrm {Vol}([\\rho ])=-\\frac{n(n^2-1)}{6}\\mathrm {Vol}_\\mathrm {hyp}$ iff $[\\bar{\\rho }]=[\\rho _\\mathrm {geom}]$ .",
"The stated conjecture REF comes from a question arisen during the work here presented: is it possible to \"perturbate\" the previous theorem.",
"In other terms, does it holds that a representation has almost the hyperbolic volume if and only if it is almost the geometric one.",
"Some experimentations to check this conjecture in the case $n=2$ are presented in the last section"
],
[
"Combinatorics and computation of the derivative",
"Another approach for the volume function (and historically the first one) was proposed by Thurston first, then in [1] in the case $n=3$ and generalized for any $n$ in [10], [6].",
".",
"It is combinatorial and goes through a triangulation of $M$ .",
"The idea is to work with representations decorated by flags.",
"For a decorated representation, each tetrahedron of the triangulation becomes a tetrahedron of flags (a hyperbolic ideal tetrahedron in the case $n=2$ ).",
"For these tetrahedra, a notion of volume is defined by sums of Bloch-Wigner dilogarithms of cross-ratios.",
"We will not describe more precisely this approach.",
"Still, two valuable informations are: The combinatorial notion of volume (defined on a Zariski-open subset of the character variety) coincide with the map $\\mathrm {Vol}$ defined in the previous section, as explained in [3].",
"As a consequence, the map $\\mathrm {Vol}$ is real analytic on a Zariski open subset of $X(\\Gamma ,\\mathrm {PGL}(n,)$ and we know a formula for its first derivative.",
"Indeed, an important achievement of the combinatorial approach is that it yields a formula for the derivative of the volume.",
"And the crucial point for us is that this formula only depends on the peripheral holonomy.",
"Indeed, the following theorem is proven by Neumann-Zagier [16] for $n=2$ , in [1] for $n=3$ and its generalization to any $n$ is given in [12].",
"Theorem 5 There is a 1-form $\\mathrm {wp}$ on $\\prod _{i=1}^t X(\\Delta _i,\\mathrm {PGL}(n,)$ such that $d\\mathrm {Vol}$ is the pullback by $\\mathrm {Hol}_\\mathrm {periph}$ of $\\mathrm {wp}$ on a Zariski-open subset of $X_n$ ."
],
[
"The birationality result: a conditional proof",
"We prove in this section theorem REF .",
"Let $M$ be, as before, an oriented cusped hyperbolic manifold, $t$ the number of its cusps.",
"Recall that $X_n$ is the geometric component of its character variety for $\\mathrm {PGL}(n,$ .",
"Throughout this section, we assume the conjecture REF holds for this particular $M$ : outside a neighborhood of $[\\rho _\\mathrm {geom}]$ in $X_n$ , the volume function $\\mathrm {Vol}$ is bounded away from its maximum on $X_n$ .",
"Under this assumption, we prove theorem REF stating that $\\mathrm {Hol}_\\mathrm {periph}$ is a birational isomorphism between $X_n$ and its image.",
"The first step of the proof is already proven – and crucial – when $n=2$ in [7] and [14]: Proposition 1 There is a real-analytic map $V$ from a Zariski-open subset of $\\mathrm {Hol}_\\mathrm {periph}(X_n)$ in $\\prod _{i=1}^t X(\\Delta _i,\\mathrm {PGL}(n,)$ to $\\mathbf {R}$ such that for any $[\\rho ]$ in a Zariski-open subset of $X_n$ , $\\mathrm {Vol}([\\rho ])=V(\\mathrm {Hol}_\\mathrm {periph}([\\rho ]))$ .",
"The point is to prove that $\\mathrm {wp}$ is exact on a Zariski-open subset of the image of $X_n$ .",
"$V$ is then one primitive.",
"Consider a loop $l$ in $\\mathrm {Hol}_\\mathrm {periph}(X_n)$ and assume it avoids the ramification locus of $\\mathrm {Hol}_\\mathrm {periph}$ .",
"Let $\\bar{l}$ be a lift in $X_n$ .",
"The two ends of $\\bar{l}$ have a volume differing by the integral $\\int _l \\mathrm {wp}$ .",
"If $\\bar{l}$ is not a loop, we may continue the lifting of $l$ to construct a sequence of points in $X_n$ such that two consecutive points always have volumes differing by this integral.",
"As the volume is bounded on $X_n$ , this forces the integral to vanish.",
"Hence $\\mathrm {wp}$ is exact on a Zariski-open subset of the image of $X_n$ and the function $V$ is its primitive whose value at $\\mathrm {Hol}_\\mathrm {periph}([\\rho _\\mathrm {geom}])$ is $\\frac{n(n^2-1)}{6}\\mathrm {Vol}_\\mathrm {hyp}$ .",
"Let us proceed with the proof of theorem 2.",
"Let $U$ be a neighborhood of $[\\rho _\\mathrm {geom}]$ in $X_n$ such that: Restricted to $U$ , the map $\\mathrm {Hol}_\\mathrm {periph}$ is a bijection onto its image (local rigidity, see thm REF ).",
"$\\mathrm {Vol}^{-1}(\\mathrm {Vol}(U))=U$ .",
"The fact that $U$ exists is a consequence of the conjecture: $\\mathrm {Vol}(U)$ is a small neighbourhood of $\\frac{n(n^2-1)}{6}\\mathrm {Vol}_\\mathrm {hyp}$ and it has no preimage far from $[\\rho _\\mathrm {geom}]$ .",
"Now, let $z$ be in $\\mathrm {Hol}_\\mathrm {periph}(U)$ such that $V$ is defined at $z$ .",
"We want to prove that $z$ has a unique preimage by $\\mathrm {Hol}_\\mathrm {periph}$ .",
"Let $[\\rho _0]\\in U$ be such that $z=\\mathrm {Hol}_\\mathrm {periph}([\\rho _0])$ .",
"By definition $V(z)=\\mathrm {Vol}([\\rho _0])$ .",
"Let $[\\rho ]$ be any point in $X_n$ such that $\\mathrm {Hol}_\\mathrm {periph}([\\rho ])=z$ .",
"As $\\mathrm {Vol}([\\rho ])=V(z)$ by the previous proposition, we get that $[\\rho ]$ belongs to $U$ .",
"By definition of $U$ , it implies that $[\\rho ]=[\\rho _0]$ .",
"Hence there is an open subset in the image $\\mathrm {Hol}_\\mathrm {periph}(X_n)$ on which the preimages of points are singletons.",
"This means that the degree of $\\mathrm {Hol}_\\mathrm {periph}$ , from $X_n$ to its image, is 1."
],
[
"Experimental evidences",
"We have implemented, using Sage 7.1 [5] and Snappy [4], a threefold test to explore the validity of the conjecture in the case $n=2$ .",
"Snappy contains a census of 61 911 cusped oriented manifold build as the gluing of at most 9 tetrahedra.",
"We focus on the 1263 manifolds with at most 6 tetrahedra.",
"Here is a description of the test built from three different tests of increasing complexity and power.",
"Let $M$ be an orientable manifold and $V_M$ its hyperbolic volume.",
"We assume that $M$ is ideally triangulated by $\\nu $ tetrahedra.",
"Let $v_0 \\simeq 1.015$ be the volume of the regular ideal hyperbolic tetrahedron.",
"It is the maximal volume for an ideal hyperbolic tetrahedron.",
"The computation will be done at the natural level for a Snappy object: the deformation variety, defined by the famous gluing equations of Thurston (see for example [20]).",
"We work with decorated representations, i.e.",
"monodromies of the gluing of hyperbolic realisations for the $n$ tetrahedra.",
"The deformation variety is seen as an affine algebraic subset in $\\nu $ (often written in ${3\\nu }$ to keep tracks of the symmetries of the tetrahedra): each tetrahedron is described by the cross-ratio of its vertices.",
"A crucial point is that the volume is the sum of Bloch-Wigner dilogarithm of the cross-ratios and hence extend to the compactification of the character variety in $(\\mathbf {CP}^1)^\\nu $ , as the dilogarithm is well-defined and continuous on $\\mathbf {CP}^1$ .",
"We can check the conjecture for $M$ by proving that, at ideals points in this compactification, the generalized volume is bounded away from $V_M$ .",
"Now define $k$ to be $\\nu - \\lceil \\frac{V_M}{v_0}\\rceil + 1$ .",
"The meaning of $k$ is: if we know for an (ideal) decorated representation that the volume of $k$ tetrahedra among the $\\nu $ vanish, then the volume of this (ideal) representation is less than $V_M$ : indeed, even putting the remaining $\\nu -k$ tetrahedra to the maximal volume $v_0$ , we do not reach $V_M$ .",
"First test The first test is very crude: if $k\\le 2$ then the conjecture holds for $M$ .",
"Indeed, at an ideal point at least a tetrahedron degenerates and has volume 0.",
"But, one may further assume that another tetrahedron is non-positively oriented, which implies it has of volume $\\le 0$ .",
"This assumption is licit because on the subset of the deformation variety where each tetrahedron is positively oriented, the volume function is convex in suitable coordinates.",
"Hence, at the boundary of the positively oriented part, the volume of ideal points is strictly less than the maximum $V_M$ .",
"We need only to check the volume of ideal points outside of this boundary.",
"This first test grants that the conjecture holds for 1144 among the 1263 manifolds.",
"Moreover, as it is straightforward to compute, one may check that 25986 out of the 61911 pass the test.",
"Second test For the 119 manifolds left undecided by the first test, we may use the logarithmic limit set to determine the minimal number $l$ of tetrahedra degenerating at an ideal point.",
"The reader may find in [20] a presentation of the logarithmic limit set for the deformation variety.",
"Recall that any point in the logarithmic limit set corresponds to ideal points for the deformation variety.",
"Snappy may be used to recover the gluing equations defining the deformation variety.",
"Then SageMath, through the software Gfan [13], is able to explicitly compute the points in the logarithmic limit set.",
"Remark that any tetrahedron whose coordinate in the logarithmic limit set does not vanish does indeed degenerate at the corresponding ideal points.",
"Thus, we compute the minimum $l$ of degenerating tetrahedra at an ideal point.",
"If $l\\ge k$ , then the manifold pass the test.",
"It is a crude test, as each non degenerating tetrahedron is set to the maximal volume.",
"Still further 47 manifolds pass the test.",
"Note that this test, as written, may not be executed for manifolds with more tetrahedra.",
"Indeed for more than 7 tetrahedra, Gfan does not compute the logarithmic limit.",
"The system defining the deformation variety seems too big ($\\ge 21$ variables and equations).",
"The remaining 72 manifolds all have 6 tetrahedra, and have all a single cusp.",
"We try on them a third test, more involved computationally.",
"Third test We now try to compute explicitly the ideal points in $(\\mathbf {CP}^1)^\\nu $ .",
"This problem is hard in general (recall that we have a system with 18 variables, 18 equations and of degree around 10).",
"A partly hand-driven computation is still often possible.",
"We compute Gröbner basis (with the Giac Gröbner engine [17] which appears to be the most effective for this problem), for the ideal defining the deformation variety and then the ideal $I$ defining its ideal points in $(\\mathbf {CP}^1)^\\nu $ .",
"This part of the computation is done in a spirit similar to [9] (trying to project on few variables and then reconstructing the whole ideal).",
"When $I$ is 0-dimensional, we may then use the rational univariate representation [18], as for example in [8] through a call to the relevant Maple function to get a parametrization of ideal points.",
"We are able to compute compute explicitly an approximated volume at ideal points by computing dilogarithms.",
"This procedure works for 29 from the 72 manifolds.",
"And in each successful case we check that the maximal volume for ideal points is indeed less than the hyperbolic volume $V_M$ .",
"In fact, the volumes computed never exceed $10^{-13}$ .",
"There are two risks of failure for this procedure: it happens that the ideal $I$ is not 0-dimensional.",
"It also happens that the computation is too long.",
"In the first case of failure, some further study has been done on one example, enabling to check the conjecture for this example.",
"But such a study is not at all automatic.",
"At the end of this threefold test, 1221 out of 1263 pass the test, giving hope for the conjecture.",
"For the remaining 42 ones and manifolds with more than 7 tetrahedra, the computations as presented here are too complicated."
]
] | 1605.05917 |
[
[
"The definition of the thermodynamic entropy in statistical mechanics"
],
[
"Abstract A definition of the thermodynamic entropy based on the time-dependent probability distribution of the macroscopic variables is developed.",
"When a constraint in a composite system is released, the probability distribution for the new equilibrium values goes to a narrow peak.",
"Defining the entropy by the logarithm of the probability distribution automatically makes it a maximum at the equilibrium values, so it satisfies the Second Law.",
"It is also satisfies the postulates of thermodynamics.",
"Objections to this definition by Dieks and Peters are discussed and resolved."
],
[
"Introduction",
"Thermodynamics is an extremely successful phenomenological theory of macroscopic experiments.",
"The entropy plays a central role in this theory because it is a unique function for each system that determines all thermodynamic information.",
"The calculation of the form of the entropy lies in the microscopic description given by statistical mechanics.",
"In this paper, I present a simple derivation of the entropy using reasonable assumptions about the probability distributions of macroscopic variables and approximations based on the large number of particles in macroscopic systems.",
"The basic task of thermodynamics is the prediction of the values of the macroscopic variables after the release of one or more constraints and the subsequent relaxation to a new equilibrium.",
"This appears in the key thermodynamic postulate that is a particular form of the second law.",
"[1], [2], [3].",
"The values assumed by the extensive parameters of an isolated composite system in the absence of an internal constraint are those that maximize the entropy over the set of all constrained macroscopic states[3].",
"I will show that the solution to this problem in statistical mechanics leads to a function that satisfies this postulate, as well as satisfying the rest of the postulates of thermodynamics.",
"Since these postulates are sufficient to generate all of thermodynamics, and since the thermodynamic entropy is unique[4], this function can be identified as the entropy.",
"I have presented other derivations in the past that are equivalent, though perhaps not as direct[5], [6], [7], [8], [9], [10].",
"They have been criticized by Dieks[11], [12] and Peters[13], [14], whose arguments will be discussed in Sections and ."
],
[
"The prediction of equilibrium values from statistical mechanics",
"Thermodynamics is a description of the properties of systems containing many particles (macroscopic systems), for which the fluctuations can be ignored because they are smaller than the experimental resolution.",
"The basic problem of thermodynamics is to predict the equilibrium values of the extensive variables after the release of a constraint in a composite system.",
"I will first consider this as a problem is statistical mechanics, without using any thermodynamic concepts.",
"Consider a composite system of $M \\ge 2$ subsystems, with a total energy $E_T$ , volume $V_T$ , and particle number $N_T$[15].",
"Denote the total phase space for this composite system (in three dimensions) by $\\lbrace p,q\\rbrace $ , where $p$ is the $3N_T$ -dimensional momentum space, and $q$ is the $3N_T$ -dimensional configuration space.",
"Define the probability distribution in the phase space of the composite system as $\\phi _T \\left( \\lbrace p, q\\rbrace , t \\right)$ , where $t$ is the time.",
"I'll assume that the composite system is initially in equilibrium at time $t=0$ , and that the initial conditions are given by setting $\\phi _T$ equal to a constant, subject to all information available about the system at that time.",
"Assume that interactions between subsystems are sufficiently short-ranged that they may be neglected[16].",
"Then, we can write the total Hamiltonian as a sum of contributions from each system.",
"$H_T=\\sum _{j=1}^MH_j (E_j, V_j, N_j)$ The energy, volume, and particle number of subsystem $j$ are denoted as $E_j$ , $V_j$ , and $N_j$ , subject to the conditions on the sums.",
"$\\sum _{j=1}^{M} E_j = E_T ; \\,\\sum _{j=1}^{M} V_j = V_T ; \\,\\sum _{j=1}^{M} N_j = N_T$ In keeping with the idea that we are describing macroscopic experiments, assume that no measurements are made that might identify individual particles, whether or not they are formally indistinguishable[17].",
"This means that there are $N_T!/ \\left( \\prod _{j=1}^{M} N_j!\\right)$ different permutations for assigning particles to subsystems, and all permutations may be regarded as equally probable.",
"The probability distribution in the phase space of the composite system is given by $\\phi _T \\left( \\lbrace p, q\\rbrace , t =0 \\right)&=&\\frac{1}{\\Omega _T}\\left( \\frac{N_T!",
"}{ \\prod _{j=1}^{M} N_j! }",
"\\right) \\nonumber \\\\&& \\times \\prod _{k=1}^{M}\\delta \\left( E_k - H_k( \\lbrace p_k, q_k \\rbrace ) \\right) ,$ where $\\lbrace p_k, q_k \\rbrace $ is the phase space for the particles in subsystem $k$ , and $\\Omega _T$ is a normalization factor.",
"The constraint that the $N_k$ particles in subsystem $k$ are restricted to a volume $V_k$ is left implicit in Eq.",
"(REF ).",
"The probability distribution for the macroscopic observables can then be written as $W \\left(\\lbrace E_j, V_j,N_j \\rbrace \\right)&=&\\frac{N_T!",
"}{\\Omega _T}\\left(\\frac{1}{\\prod _j N_j!",
"}\\right)\\nonumber \\\\&&\\times \\int dp \\int dq\\prod _{j=1}^M\\delta ( E_j - H_j ) ,$ or $W(\\lbrace E_j, V_j, N_j\\rbrace )=\\frac{\\prod _{j=1}^M\\Omega _j ( E_j, V_j, N_j )}{\\Omega _T / N_T!",
"h^{3N_T}} ,$ where $\\Omega _j=\\frac{1}{h^{3N_j} N_j!",
"}\\int _{-\\infty }^{\\infty } dp_j \\int _{V_j} dq_j \\,\\delta ( E_j - H_j ) .$ The factor of $1/h^{3N_j}$ , where $h$ is Planck's constant, is not necessary for classical mechanics.",
"It has been included to ensure that the final answer agrees with the classical limit from quantum statistical mechanics[3].",
"There is no requirement that the Hamiltonians $H_j$ are the same, so there is also no requirement that the individual $\\Omega _j$ 's have the same functional form.",
"Long-range interactions within a system are allowed.",
"If one or more constraints are now released, the probability $\\phi _T \\left( \\lbrace p, q\\rbrace , t \\right)$ will become time dependent.",
"After sufficient time has passed, the probability distribution will have spread throughout the available phase space, although it will still be non-uniform on the finest scale due to Liouville's theorem.",
"The probability distribution for the macroscopic variables will again be given by $W(\\lbrace E_j, V_j, N_j\\rbrace )$ , but now without the constraints on the variables that have been released[18].",
"The functional dependence of $W$ on the variables $\\lbrace E_j, V_j, N_j\\rbrace $ does not change when a constraint is released.",
"An important advantage of working with the probability distributions for macroscopic observables is that they converge to the equilibrium probability distributions at the end of an irreversible process[18].",
"Although it is not necessary, the introduction of coarse graining[19] or the modification of the microscopic probability distribution by invoking typicality[20], [21] leads to the same results.",
"Usually, $W$ is a very narrow function of the released variables.",
"The main exception is the case of a first-order phase transition, in which it can be a very broad function of the relevant variable[16].",
"This situation is discussed in Ref.",
"[16], and I will ignore it for the present discussion.",
"The location of the narrow peak in $W$ as a function of the variable describing a released constraint gives the final equilibrium value of that variable at the end of the irreversible process.",
"For example, if subsystems 1 and 2 are brought in thermal contact so that energy transfer is possible, the final value of $E_1$ would be given by the location of the maximum of $W$ to within thermal fluctuations.",
"This characterizes the equilibrium values as the mode of the probability distribution, not the mean.",
"The difference between the mean and the mode is of the order of $1/N$ , which is very small and far less than the assumed experimental accuracy.",
"Indeed, it is not even measurable for macroscopic systems[22].",
"When subsystems are separated, the probability $W$ remains unchanged.",
"The constraint is restored, and the variable that was being exchanged keeps its value, which is known to within the very small fluctuations.",
"The normalization constant, $\\Omega _T$ , is dependent on exactly which constraints might be released, but the other factors are not.",
"Since the only property of the function $W( \\lbrace E_j, V_j, N_j \\rbrace )$ that is needed is that it has a very narrow peak at the equilibrium value(s) after the release of constraint(s), the value of $\\Omega _T$ does not affect the argument.",
"Now that the probability distribution for the equilibrium variables has been determined, we can turn to the definition of entropy."
],
[
"The definition of the thermodynamic entropy",
"Following Boltzmann[23], [24], [6], the thermodynamic entropy may be identified as the logarithm of the probability distribution $W$ , plus an arbitrary constant.",
"$S_T ( \\lbrace E_j, V_j, N_j\\rbrace )=k_B \\ln W+ X$ Since the probability is a maximum at equilibrium, the entropy is also with this definition.",
"Although Boltzmann considered a dimensionless entropy and never used the “Boltzmann constant,” $k_B$ , which was introduced by Planck[25], [26], I have included a factor of $k_B$ to be consistent with physical units.",
"Combining Eqs.",
"(REF ), (REF ) and (REF ), the total entropy can be written as a sum of $M$ terms, each of which depends only on the properties of a single subsystem, plus a constant.",
"$S_T=\\sum _{j=1}^MS_j ( E_j, V_j, N_j )-k_B\\ln \\left[ \\frac{ \\Omega _T }{ N_T!",
"h^{3N_T} } \\right]+X$ The entropy of the $j$ -th subsystem in Eq.",
"(REF ) is given by $S_j ( E_j, V_j, N_j )=k_B \\ln \\Omega _j ( E_j, V_j, N_j ) ,$ or $S_j =k_B \\ln \\left[\\frac{1}{h^{3N_j} N_j!",
"}\\int _{-\\infty }^{\\infty } dp_j \\int _{V_j} dq_j \\,\\delta ( E_j - H_j )\\right] .$ The entropy of subsystem $j$ contains the factor $1/N_j!$ , which arises from the multinomial factor in Eq.",
"(REF ).",
"It would be possible to add an arbitrary constant $X_j$ to $S_j$ in Eq.",
"(REF ), but I have chosen to set $X_j=0$ for all $j$ , which is the usual convention[9].",
"$S_j$ only depends on the properties of system $j$ , which means that the total entropy is separable.",
"This is just the usual thermodynamic property of additivity, but viewed from the perspective of dividing up a composite system, rather than assembling one.",
"Since $\\Omega _T$ has been defined to be a normalization constant, if all chosen constraints are released, the value of $S_T$ after the composite system has returned to equilibrium is given entirely by the additive constant (neglecting terms of the order of the logarithm of the particle numbers).",
"$S_T (\\text{after release})\\rightarrow X$ This will be true regardless of which constraints have been chosen to determine $\\Omega _T$ , as long as all of those constraints are released.",
"A convenient choice of $X$ is $k_B \\ln \\left[ \\Omega _T / N_T!",
"h^{3N_T} \\right]$ .",
"Then the total entropy of the composite system is just given by the sum of the subsystem entropies.",
"But this choice is not required."
],
[
"The application of the entropy equations",
"Eqs.",
"(REF ), (REF ), and (REF ) are intended to be applied to the set of all systems in the world that can be regarded as classical.",
"That includes not only systems in a particular laboratory, but also those in a different city or continent.",
"Most systems will not interact with each other because of physical separation, and the constraints of their not exchanging energy, volume, or particles are expected to remain indefinitely.",
"The entropy of a single system is given by Eq.",
"(REF ).",
"For experiments involving only a local group of systems (or subsystems of the overall composite system), the existence of many other (sub)systems can be safely ignored, because their properties do not affect the local thermodynamic variables.",
"Similarly, the value of the additive constants in Eq.",
"(REF ) will not affect the predictions of any experiment.",
"Eqs.",
"(REF ) and (REF ) allow us to find the non-negative change in total entropy ($\\Delta S_T \\ge 0$ ) during any irreversible process between equilibrium states that occurs after the release of a constraint, as well as the final equilibrium values of thermodynamic observables.",
"Dieks has criticized this derivation of the entropy.",
"I discuss his views in the next section."
],
[
"Dieks' objection",
"Dieks' criticism rests on the claim that the choice of additive constant, $X$ in Eq.",
"(REF ), is essential for obtaining my results for the entropy[11].",
"This claim is untenable, since I have derived the entropy of an arbitrary subsystem [Eq.",
"(REF )] without fixing the value of $X$ , and the value of $X$ has no physical consequences.",
"Looking further, we can see that Dieks means something different.",
"He is interested in the value of the entropy of the entire composite system of $M$ subsystems for the case in which all constraints have been released.",
"As shown above in Eq.",
"(REF ), the release of all constraints leads to a constant $S_T \\rightarrow X$ , where $X$ is arbitrary.",
"Dieks is concerned about the determination of a particular form of this constant.",
"Since there are no physical consequences for any value of $X$ , I fail to see the importance of the issue.",
"Dieks explicitly recognizes that this issue is without importance.",
"Writing $N$ for what I have called $N_T$ , he says in a footnote: A more detailed discussion should also take into account that the division by $N!$ is without significance anyway as long as $N$ is constant[11].",
"However, he still uses the value of this constant to frame his objection to my definition.",
"The reason for this contradiction might lie in his incorrect description of my definition of entropy, which he claims amounts simply to dividing the traditional expression by $N!$ .",
"I will consider his argument in detail."
],
[
"Two simple subsystems",
"Dieks considered an isolated composite system consisting of only two ideal gases ($M=2$ ), and simplified his analysis by ignoring the energy dependence.",
"In discussing his argument, I will depart from Dieks' notation[27] by using $N_T=N_1+N_2$ as the constant total number of particles to be consistent with the notation I used in previous sections.",
"For clarity, I will also retain an arbitrary value of the additive constant $X$ (see Eq.",
"(REF ), above) until the end of the discussion, although Dieks makes the specific choice of $X=k_B \\ln \\left( V_T^{N_T}/N_T!",
"\\right)$ , “for reasons of convenience,” early in his argument[11].",
"For Dieks' two subsystems of classical ideal gases, my Eq.",
"(REF ) becomes his Eq.",
"(2), $S_T (N_1,V_1;N_2,V_2)=&k_B\\ln \\left(\\frac{N_T!}{N_1!",
"N_2!",
"}\\frac{ V_1^{N_1}V_2^{N_2} }{ V_T^N}\\right)+X \\nonumber \\\\=&k_B\\ln \\left(\\frac{V_1^{N_1}}{N_1!",
"}\\right)+k_B\\ln \\left(\\frac{V_2^{N_2}}{ N_2!",
"}\\right) \\nonumber \\\\&-k_B\\ln \\left(\\frac{ V_T^{N_T} }{ N_T!",
"}\\right)+X ,$ where $N_T=N_1+N_2$ and $V_T=V_1+V_2$ are constants.",
"Note that Dieks' choice for the value of the constant $X$ means that the last two terms in Eq.",
"(REF ) cancel in his Eq. (2).",
"Since Eq.",
"(REF ) is valid for all values of $N_1$ , $N_2$ , $V_1$ , and $V_2$ , we immediately have the (partial) entropies, $S_j (V_j,N_j )=k_B\\ln \\left(\\frac{V_j^{N_j}}{ N_j!",
"}\\right) ,$ where $j=1$ or 2.",
"I claim that this is a proper derivation of the factors $1/N_j!$ .",
"Dieks made the following comment on his Eq.",
"(2) (writing $N$ for what I have called $N_T$ ).",
"Indeed, the dependence of the total entropy in Eq.",
"(2) on $N_1$ and $N_2$ is unrelated to how $N$ occurs in this formula (and to the choice of the zero of the total entropy)[11].",
"His comment confirms the validity of my derivation of the factors $1/N_1!$ and $1/N_2!$ in the entropies of subsystems 1 and 2, as well as the irrelevance of the value of the additive constant $X$ .",
"Dieks then calculates the entropy after the release of the constraint on the particle number and return to equilibrium.",
"He gets the result $X=k_B \\ln \\left( V_T^{N_T}/N_T!",
"\\right)$ .",
"Dieks claims that this was the way I had obtained a $-k_B \\ln N_T!$ dependence of the total entropy.",
"I did not fix the value of $X$ , so I did not derive an expression for the entropy after the release of constraints.",
"Actually, the form of the $k_B \\ln \\left( V_T^{N_T}/N_T!",
"\\right)$ term in the joint entropy does not come from choosing the constant $X$ to make $S_T = \\sum _{j=1}^M S_j $ , but rather from the simplicity of the example used.",
"If the properties of the subsystems are generalized, a different result is obtained."
],
[
"Two less simple subsystems",
"Consider the entropy, $S_j=k_B N_j\\left[\\frac{3}{2}\\ln \\left( \\frac{E_j - N_j a_j }{ N_j } \\right)+\\ln \\left( \\frac{V_j }{ N_j } \\right)+Y_j^{\\prime }\\right] ,$ where I have used Stirling's approximation.",
"The total entropy before allowing the systems to interact is $S_T = S_1+S_2$ .",
"The energy dependence is now given explicitly, and an energy shift per particle, $a_j$ , is given to each subsystem.",
"Assume that $a_1=0$ and $a_2>0$ .",
"Let subsystems 1 and 2 come into thermal contact and exchange energy and particles.",
"The temperature dependence of the energy in the $j$ -th subsystem is given by $E_j = \\frac{3}{2} k_B N_j T_j + N_j a_j ,$ so the condition of equilibrium with respect to energy exchange is $\\frac{ E^{\\prime }_1 }{ N_1 }=\\frac{ E^{\\prime }_2 }{ N_2} - a_2 ,$ where I have indicated the new values of the energies by $E^{\\prime }_1$ and $E^{\\prime }_2$ .",
"Now let the two subsystems exchange particles.",
"From the condition of equilibrium with respect to particle number, it is straightforward to derive $\\ln \\left( \\frac{ V_1 }{ N^{\\prime \\prime }_1 } \\right)=\\ln \\left( \\frac{ V_2 }{ N^{\\prime \\prime }_2 } \\right)-\\frac{3}{2}\\left[\\frac{1}{E^{\\prime \\prime }_2/N^{\\prime \\prime }_2 a_2 - 1}\\right] ,$ where I have indicated the new values of the energies and particle numbers by double primes, i.e: $E^{\\prime }_j$ and $N^{\\prime \\prime }_j$ .",
"Since $E^{\\prime \\prime }_1/N^{\\prime \\prime }_1 \\ne E^{\\prime \\prime }_2/N^{\\prime \\prime }_2$ and $V_1/N^{\\prime \\prime }_1 \\ne V_2/N^{\\prime \\prime }_2$ , the total entropy cannot be written as a function of $(E^{\\prime \\prime }_1+E^{\\prime \\prime }_2)$ , $(V_1+V_2)$ , and $(N^{\\prime \\prime }_1+N^{\\prime \\prime }_2)$ .",
"There is no term in $S^{\\prime \\prime }_T = S^{\\prime \\prime }_1(E^{\\prime \\prime }_1,V_1,N^{\\prime \\prime }_1) + S^{\\prime \\prime }_2(E^{\\prime \\prime }_2 ,V_2,N^{\\prime \\prime }_2)$ of the form $k_B \\ln \\left( V_T^{N_T}/N_T!",
"\\right)$ .",
"For the next example it will be sufficient to again consider ideal gases and ignore the energy dependence."
],
[
"Three simple subsystems",
"Dieks' analysis does not recognize that the thermodynamic variables in subsystems 1 and 2 remain $N_1$ and $N_2$ , even after the systems come to equilibrium.",
"They are not replaced by a single variable.",
"This can be seen most easily by considering $M \\ge 3$ subsystems.",
"To avoid confusion, denote the number of particles in subsystems 1 and 2 by $N_{1,2}=N_1+N_2$ , because it is no longer constant.",
"Now consider how subsystems 1 and 2 interact with a third subsystem.",
"Let subsystems 1 and 2 first come to equilibrium and then be separated again, denoting the new particle numbers by $N^{\\prime }_1$ and $N^{\\prime }_2$ .",
"Let subsystem 3 originally have a high number density, $N_3/V_3 > N^{\\prime }_1/V_1= N^{\\prime }_2/V_2$ .",
"Now let subsystem 2 exchange particles with system 3, so that $N_2$ increases ($N^{\\prime \\prime }_2>N^{\\prime }_2$ ).",
"Subsystems 2 and 3 come to a new equilibrium, for which $\\frac{N^{\\prime }_1}{V_1}<\\frac{N^{\\prime \\prime }_2}{V_2}=\\frac{N^{\\prime \\prime }_3}{V_3} .$ The entropy of subsystems 1 and 2 is (with Stirling's approximation), $S^{\\prime }_1+S^{\\prime \\prime }_2\\approx k_BN^{\\prime }_1\\ln \\left(\\frac{V_1}{ N^{\\prime }_1}\\right)+k_BN^{\\prime \\prime }_2\\ln \\left(\\frac{V_2}{ N^{\\prime \\prime }_2}\\right) .$ Since the number density is different in subsystems 1 and 2, it is clear that $S^{\\prime }_1+S^{\\prime \\prime }_2$ is not given by $k_B N_{1,2} \\ln \\left( V_{1,2} / N_{1,2} \\right)$ ."
],
[
"An arbitrary number of subsystems",
"When Dieks discusses the case of many systems, he writes that I require a “consistency” condition, that the entropy formula should be such that there will be no change in entropy when a partition is removed[11].",
"I do not require it, and it is not a consistency condition.",
"It is the condition that systems separated by a partition are in equilibrium, which is not generally true in the presence of a constraint.",
"To summarize, I have calculated the dependence of the entropy on the variables $\\lbrace E_j, V_j, N_j \\vert =1, \\dots ,M\\rbrace $ in the presence or absence of arbitrary constraints.",
"My definition enables the calculation of the equilibrium conditions and entropy changes.",
"The additive constant, $X$ , may be determined by convention."
],
[
"Peters' objection",
"A prominent question in the literature is whether entropy should be defined in one step or two.",
"The two-step approach can be described as hybrid because it starts with a definition of entropy, notes that the definition fails in some respect, and then corrects it to agree more closely with the thermodynamic properties of entropy.",
"The historical reason for this peculiar question lies in the effort to maintain a definition of entropy in the form of the logarithm of a volume in phase space by modifying it to correct the dependence on particle number[11], [12], [13], [28], [29], [30].",
"Since this process usually involves the inclusion of a negative term, $-k_B \\ln N!$ , the result is often called a “reduced entropy.” Peters has introduced an interesting hybrid definition of the entropy[13], [14].",
"In doing so, he explicitly rejected the derivation of entropy given in Section , although his only criticism turns out to be something we agree on.",
"We both recognized that macroscopic experiments do not identify individual particles, so we can never know which particles are in which system.",
"However, Peters claimed that my version was “imprecise” because it did not include the condition he denoted as being “harmonic,” defined as follows.",
"Systems for which all possible particle compositions are equiprobable will be called harmonic[13].",
"For comparison, I had written that, when a system of distinguishable particles is allowed to exchange particles with the rest of the world, we must include the permutations of all possible combinations of particles that might enter or leave the system[6].",
"It is clear that we have made essentially the same assumption.",
"Peters' takes a hybrid approach in that he chooses to define a form of the Shannon entropy, and then “reduces” it to arrive at the final form[31], [13].",
"$R_P&=&-k_B\\sum _{i=1}^M\\int d^{3N_i}p_i \\, \\int d^{3N_i}q_i \\nonumber \\\\&&\\times \\rho _i(p_i,q_i)\\ln \\left( \\rho _i(p_i,q_i) h^{3N_i} \\right) \\nonumber \\\\&&-k_B \\ln N!$ This form does have the correct $N$ -dependence, and for the correct reason.",
"However, $R_P$ fails to satisfy the second law of thermodynamics.",
"In Section 4.3.3.2 of Ref.",
"[13], Peters discusses an irreversible process initiated by the release of constraints to allow exchange of energy and particles between two subsystems.",
"He assumes that “both before and after the exchange” the two subsystems “are in microcanonical equilibrium.” The problem is that this assumption is contradicted by Liouville's theorem, which requires the total time derivative of the probability distribution in the phase space of the complete composite system to vanish.",
"This means $R_P$ does not increase during an irreversible process, so it does not satisfy the second law of thermodynamics.",
"Peters explicitly acknowledges the difficulty posed by Liouville's theorem in his Section 5.6.5, writing that, “the Liouville equation is entropy conserving and therefore cannot describe irreversible processes.” He does not comment on the contradiction between his Sections 4.3.3.2 and 5.6.5.",
"In contrast, the Liouville equation does not conserve the entropy as defined in this paper, and the Second Law is satisfied."
],
[
"Summary",
"I've argued for a definition of the thermodynamic entropy based on the probability distribution of the macroscopic variables in a composite system.",
"The entropy defined this way satisfies the postulates for thermodynamics[1], [2], [3].",
"I've addressed the objections by Dieks[11], [12] and Peters[13], [14] to this derivation of the entropy from statistical mechanics and shown that they are not valid.",
"Since the thermodynamic entropy is known to be unique apart from constants chosen by convention[4], any other valid definition of the entropy must be equivalent the one presented here."
],
[
"Acknowledgement",
"I would like to thank Roberta Klatzky for many helpful discussions.",
"This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors."
]
] | 1605.05690 |
[
[
"Analogs of the Shapiro Shapiro Conjecture in Positive Characteristic"
],
[
"Abstract Motivated by the Shapiro Shapiro conjecture, we consider the following: given a field $k$, under what conditions must a rational function with only $k$-rational ramification points be equivalent (after post-composition with a fractional linear transformation) to a rational function defined over $k$?",
"The main results of this paper answer this question when $k$ has characteristic 2 or 3.",
"We also show the insufficiency of several natural conditions in higher characteristic."
],
[
"Introduction",
"In [2], Eremenko and Gabrielov proved a special case of the Shapiro Shapiro conjecture.",
"Their result states that a (univariate) rational function defined over the complex numbers with only real ramification points is equivalent to a rational function defined over the real numbers.",
"Here and throughout this paper rational functions $f_1(x)$ and $f_2(x)$ are considered equivalent if there is a fractional linear transformation $\\sigma (x)$ such that $f_2=\\sigma \\circ f_1$ .",
"Motivated by this result, it is natural to ask the following: Question 1.1 Fix a field $k$ .",
"Under what conditions, on for example the degree or ramification indices, must a rational function defined over $\\overline{k}$ with only $k$ -rational ramification points be equivalent to a rational function defined over $k$ ?",
"In [3], Faber and Thompson provide a necessary and sufficient condition on a field $k$ of characteristic 0 so that every degree 3 rational function defined over $\\overline{k}$ with only $k$ -rational ramification points must be equivalent to a rational function defined over $k$ .",
"We note that the arguments in [3] also hold for fields $k$ of characteristic greater than 3, which we shall make use of in negativesect.",
"Instead of restricting to low degree rational functions, we consider mainquest over fields of positive characteristic.",
"The primary results of this paper can be summarized as follows (the Wronskian is defined in prelimsect below): Theorem 1.2 (mainthm, finitefieldsimple, char2cor, char3cor) Let $k$ be a field of characteristic 3.",
"Every simply ramified (i.e.",
"no ramification index is greater than two) rational function defined over $\\overline{k}$ with only $k$ -rational ramification points is equivalent to a rational function defined over $k$ .",
"Let $k$ be a finite field of characteristic greater than 3.",
"There exists a degree 3 simply ramified rational function defined over $\\overline{k}$ with only $k$ -rational ramification points which is not equivalent to a rational function defined over $k$ .",
"Let $k$ be a non-algebraically closed field of characteristic 2.",
"Suppose that $f(x)$ is a degree $d>1$ rational function defined over $\\overline{k}$ with only $k$ -rational ramification points.",
"Then there are infinitely many equivalence classes of degree $d$ rational functions with the same Wronskian as $f(x)$ which are not equivalent to a rational function defined over $k$ .",
"The same result holds if $k$ is a non-algebraically closed field of characteristic 3 when $f(x)$ is not simply ramified.",
"These results provide a complete answer to mainquest in characteristic 2 and 3.",
"In particular, summarythm asserts that the correct condition for mainquest in characteristic 3 is that the ramification points do not coalesce.",
"We briefly remark on the differences between previous results in characteristic 0 and the positive characteristic results of this paper.",
"In [2] it is shown that no additional conditions are necessary in mainquest when $k=\\mathbb {R}$ for a generic choice of distinct real ramification points.",
"Using the fact that the Wronskian (defined in prelimsect below) is a quasi-finite map, a limiting argument (see for example the discussion proceeding [4]) extends this result for any configuration of real ramification points.",
"However, the Wronskian is not quasi-finite in positive characteristic.",
"This explains why the simply ramified result in summarythm(i) does not imply the same result for the case when the ramification points coalesce (which would contradict summarythm(iii))."
],
[
"Definitions and Preliminaries",
"Throughout this paper we shall implicitly assume that all rational functions are separable.",
"Let $k$ be a field and $f(x)=g(x)/h(x)$ a degree $d$ rational function defined over $\\overline{k}$ (i.e.",
"$g(x),h(x)\\in \\overline{k}[x]$ ), where $g(x)$ and $h(x)$ are coprime polynomials.",
"We say that $f(x)$ is simply ramified if none of its ramification indices are greater than two.",
"In this paper we shall use a measure of ramification related to the ramification index, called the differential length.",
"The differential length of $f(x)$ at a closed point $P\\in \\mathbb {P}^1_{\\overline{k}}$ , denoted $l_P$ , is the length of the sheaf of relative differentials as an $\\mathcal {O}_{\\mathbb {P}^1_{\\overline{k}},P}$ module.",
"We introduce a convenient method to calculate the differential lengths below.",
"The Wronskian of $f(x)$ is the polynomial $Wr(f(x))=h(x)g^{\\prime }(x)-g(x)h^{\\prime }(x)$ .",
"After factoring over $\\overline{k}$ we have that $Wr(f(x))=\\alpha \\prod (x-c_i)^{l_i},$ where each $l_i$ is positive, $\\alpha \\in \\overline{k}$ , and each $c_i\\in \\overline{k}$ is distinct.",
"The $c_i$ , considered as points on $\\mathbb {P}^1_{\\overline{k}}$ , are precisely the affine ramification points of $f(x)$ .",
"Each $l_i$ is equal to the differential length of $f(x)$ at $c_i$ .",
"By the Riemann Hurwitz formula the differential length at $\\infty $ is equal to $2d-2-\\deg (Wr(f(x)))$ .",
"Note that if $f(x)$ is tamely ramified at $P$ then $l_P$ is equal to the ramification index minus one; if $f(x)$ is wildly ramified at $P$ then $l_P$ is strictly larger than that.",
"Given a positive integer $d$ , let $G_d^{sep}$ denote the space of separable two dimensional linear series on $\\mathbb {P}^1_k$ .",
"Any such linear series corresponds to an equivalence class of separable degree $d$ rational functions.",
"Since the Wronskian of an equivalence class of rational functions is defined up to multiplication by a constant, the Wronskian induces a morphism $Wr:G_d^{sep}\\rightarrow \\mathbb {P}^{2d-2}_k$ .",
"Here we consider $\\mathbb {P}^{2d-2}_k$ as the projectivization of the space of polynomials of degree at most $2d-2$ .",
"We consider $G_d^{sep}$ as a scheme defined over $k$ .",
"See [1] for a more detailed discussion.",
"Fix a positive integer $d$ and a finite tuple of positive integers $T=(l_1,...,l_n)$ such that $\\sum l_i=2d-2$ .",
"Following [1], let $X_T$ denote the locally closed subscheme of $G_d^{sep}$ corresponding to equivalence classes of degree $d$ rational functions ramified at unspecified points $P_1,...,P_n$ such that $l_{P_i}$ is equal to $l_i$ .",
"As mentioned in the introduction, in positive characteristic the map $Wr:G_d^{sep}\\rightarrow \\mathbb {P}^{2d-2}_k$ is not quasi-finite.",
"However, in certain circumstances the restriction of this map to one of the $X_T$ subschemes is quasi-finite, which will be an important ingredient in our results.",
"As our final preliminary, we find a unique representative for each equivalence class of rational functions which will be convenient in what follows.",
"Lemma 2.1 Let $k$ be a field.",
"Every rational function defined over $\\overline{k}$ is equivalent to a unique rational function $g(x)/h(x)$ , where $g(x)$ and $h(x)$ are monic, $\\deg (g(x))>\\deg (h(x))$ , and $g(x)$ contains no $\\deg (h(x))$ term.",
"This will be referred to as the standard form of a rational function.",
"Such a rational function is equivalent to a rational function defined over $k$ if and only if $g(x),h(x)\\in k[x]$ in its standard form.",
"The latter condition is equivalent to the corresponding point in $G^{sep}_d$ having residue field $k$ .",
"Let $f(x)$ be a degree $d$ rational function.",
"We begin with (i) and first show the existence of a standard form.",
"After post-composing $f(x)$ with a fractional linear transformation sending $f(\\infty )$ to $\\infty $ , we have a rational function of the form $g(x)/h(x)$ , where $\\deg (g(x))>\\deg (h(x))$ .",
"By dividing $g(x)$ and $h(x)$ by the leading coefficient of $h(x)$ we may assume that $h(x)$ is monic.",
"Post-composing with $x\\mapsto x/a$ , where $a$ is the leading coefficient of $g(x)$ , yields a form where $g(x)$ is also monic.",
"Finally, post-composing with $x\\mapsto x-b$ , where $b$ is the coefficient of $x^{\\deg (h(x))}$ in $g(x)$ , leads to the desired form.",
"To show uniqueness, it suffices to prove that distinct standard forms cannot be equivalent.",
"To this end, suppose that $f_1(x)=g_1(x)/h_1(x)$ and $f_2(x)=g_2(x)/h_2(x)$ are two standard forms such that $f_2=\\sigma \\circ f_1$ , where $\\sigma (x)$ is a fractional linear transformation.",
"Since $\\deg (g_i(x))>\\deg (h_i(x))$ and each $g_i(x)$ and $h_i(x)$ is monic, it must be the case that $\\sigma (x)=x+a$ , where $a\\in \\overline{k}$ .",
"Since $f_1(x)$ and $f_2(x)$ are the same degree and each fixes $\\infty $ , we have that $\\deg (g_1(x))=\\deg (g_2(x))$ .",
"This implies that $a=0$ , and hence that $\\sigma $ is the identity, since otherwise $g_2(x)$ would contain a $\\deg (h_2(x))$ term.",
"Therefore the two standard forms must be equal.",
"We now prove (ii).",
"By definition if $g(x),h(x)\\in k[x]$ , where $g(x)/h(x)$ is the standard form for $f(x)$ , then $f(x)$ is equivalent to a rational function defined over $k$ .",
"For the other direction, suppose that $f(x)$ is equivalent to $f_k(x)$ , a rational function defined over $k$ .",
"Applying the procedure from the first paragraph of this proof to produce the standard form of $f_k(x)$ , we see that all fractional linear transformations are defined over $k$ .",
"This implies that the standard form for $f_k(x)$ , which is also the standard form for $f(x)$ , is defined over $k$ .",
"Finally we show that having a standard form $g(x)/h(x)$ defined over $k$ is equivalent to the corresponding point in $G_d^{sep}$ having residue field $k$ .",
"Recall that $G_d^{sep}$ is an open subscheme of $Gr(2,Poly_k(d))$ , the Grassmannian of 2-planes in the space of polynomials of degree at most d. The equivalence class of $g(x)/h(x)$ is contained in the open subscheme of $Gr(2,Poly_k(d))$ consisting of equivalence classes of rational functions with a representative of the form ${\\widetilde{g}}(x)/{\\widetilde{h}}(x)$ , where ${\\widetilde{g}}(x)$ and ${\\widetilde{h}}(x)$ are monic, $\\deg ({\\widetilde{h}}(x))<\\deg ({\\widetilde{g}}(x))$ , and ${\\widetilde{g}}(x)$ contains no $\\deg (h(x))$ term.",
"This subscheme is isomorphic to $\\mathbb {A}^{2d-2}_k$ .",
"Under the natural isomorphism the unspecified coefficients of ${\\widetilde{g}}(x)$ and ${\\widetilde{h}}(x)$ serve as the coordinates of affine space, which establishes the desired equivalence."
],
[
"Sufficient Conditions in Characteristic 3",
"Roughly speaking, our strategy for proving mainthm will be to show that there is no residue field extension for the map $Wr:G_d^{sep}\\rightarrow \\mathbb {P}^{2d-2}_k$ over points corresponding to polynomials with no repeated roots and with all coefficients contained in $k$ .",
"Our initial argument will imply that each such point has a unique preimage, but we must then rule out the case of purely inseparable field extensions by appealing to the following: Lemma 3.1 Let $k$ be a field of characteristic $p>2$ and $d$ a positive integer.",
"The map $Wr:G_d^{sep}\\rightarrow \\mathbb {P}^{2d-2}_k$ is not generically purely inseparable (i.e.",
"the induced extension of function fields is not purely inseparable).",
"First note that by [5] there is a non-zero number of equivalence classes of degree $d$ rational functions simply ramified at a generic choice of points $P_1,...,P_{2d-2}\\in \\mathbb {P}^1_{\\overline{k}}$ .",
"Since a generic polynomial has no repeated roots, this implies that $Wr$ is dominant, and hence induces an extension of function fields.",
"We shall explicitly describe the map $Wr$ in coordinates.",
"Consider $c_1,...,c_{2d-2}$ as parameters on $\\mathbb {P}^1_{\\overline{k}}$ .",
"Let $g(x)/h(x)$ be the standard form of a degree $d$ rational function where $g(x)=\\sum a_i x^i$ and $h(x)=\\sum b_i x^i$ .",
"If the Wronskian of $g(x)/h(x)$ maps to $\\prod (x-c_i)$ (where if $c_i=\\infty $ we define $x-c_i$ to equal 1), it must be the case that $h(x)g^{\\prime }(x)-g(x)h^{\\prime }(x)=\\alpha \\prod (x-c_i)$ , where $\\alpha $ is the leading coefficient of the left hand side.",
"Expanding both sides and equating coefficients of $x^j$ for $0\\le j\\le 2d-1$ , we have that $q_j(a_i,b_i)=(-1)^j e_{2d-2-j}(c_i)$ , where $e_k(c_i)$ is the $k^{th}$ elementary symmetric polynomial in the $c_i$ and each $q_j(a_i,b_i)$ is a polynomial of total degree at most two.",
"The function field of the codomain of $Wr$ is the fixed field of $k(c_1,...,c_{2d-2})$ under the action of the symmetric group which permutes the $c_i$ .",
"It is well known that a $k$ -basis for this fixed field is the symmetric polynomials $e_1(c_i),...,e_{2d-2}(c_i)$ .",
"Therefore the field extension induced by $Wr$ is given by $e_k\\mapsto q_{2d-2-k}(a_i,b_i)$ .",
"Since the degree of each $q_j$ is at most two and there are no relations amongst the $a_i,b_i$ , this field extension is not purely inseparable.",
"Equipped with noinsep, we now prove our main result: Theorem 3.2 Let $k$ be a field of characteristic 3.",
"Every simply ramified rational function defined over $\\overline{k}$ with only $k$ -rational ramification points is equivalent to a rational function defined over $k$ .",
"Fix an integer $d>1$ .",
"By [5], there is exactly one equivalence class of degree $d$ rational functions simply ramified at a generic choice of points $P_1,...,P_{2d-2}\\in \\mathbb {P}^1_{\\overline{k}}$ .",
"Indeed, using the notation of [5], since the ramification is all simple we have that $e_1,...,e_{2d-2}=2$ .",
"The only possible $(2d-5)$ -tuple of $e_i^{\\prime }$ satisfying conditions (i) and (ii) in [5] begins with a 1 and alternates between 1 and 2, which implies that there is generically one such equivalence class, as claimed.",
"Switching back to the notation introduced in this paper, let $T$ be the $(2d-2)$ -tuple consisting of all ones.",
"Then $X_T$ is the space of equivalence classes of simply ramified degree $d$ rational functions.",
"By [1], the number of equivalence classes of such rational functions is finite.",
"In other words, $Wr:X_T\\rightarrow \\mathbb {P}^{2d-2}_k$ is quasi-finite.",
"By Zariski's Main Theorem, we have a factorization ${X_T [rr]^{Wr}[rd]_\\iota & & \\mathbb {P}_k^{2d-2} \\\\& X[ur]_\\psi }$ where $\\iota $ is an open immersion and $\\psi $ is finite.",
"Since $\\psi $ is finite and generically the fibers of $Wr$ contain a single point, this implies that $\\psi $ must either be degree 1 or purely inseparable.",
"Since the latter possibility is eliminated by noinsep, $\\psi $ must be degree 1.",
"Therefore there is no residue field extension over closed points for the map $Wr:X_T\\rightarrow \\mathbb {P}^{2d-2}_k$ .",
"Hence, any point corresponding to an equivalence class of rational functions with only $k$ -rational ramification points must have residue field $k$ , which implies the result by form(ii)."
],
[
"Insufficient Conditions",
"We now turn our attention to considering conditions under which we cannot guarantee a rational function with only $k$ -rational ramification points is equivalent to a rational function defined over $k$ .",
"Our first result establishes that mainthm does not hold in characteristic greater than 3. char2cor will establish an analogous result for characteristic 2.",
"Proposition 4.1 Let $k$ be a finite field of characteristic $p>3$ .",
"There exists a simply ramified degree 3 rational function defined over $\\overline{k}$ with only $k$ -rational ramification points which is not equivalent to a rational function defined over $k$ .",
"Rather than repeating the arguments, we note that the results in [3] stated in characteristic 0 in fact hold for any field $k$ of characteristic greater than 3.",
"In particular, using the notation from [3], since the resultant of $x^3+ux^2$ and $(2u+3)x-(u+2)$ is $2(u+2)^2(u+1)^2$ , the only values of $u$ for which $f_u(x)$ degenerates into a lower degree rational function remain $u=-2,-1$ when $\\operatorname{char}(k)>3$ .",
"By [3] every degree 3 rational function defined over $\\overline{k}$ ramified at $0,1,\\infty $ is equivalent to a unique rational function of the form $f_u(x)=\\frac{x^3+ux^2}{(2u+3)x-(u+2)},$ where $u\\in \\overline{k}$ .",
"By factoring the Wronskian, we see that the last ramification point of $f_u(x)$ is $\\varphi (u)=-(u^2+2u)/(2u+3)$ .",
"By [3], $f_u(x)$ is equivalent to a rational function defined over $k$ precisely when $u\\in k$ .",
"Therefore to conclude the proof it suffices to show that there exists a $u\\in \\overline{k}\\setminus k$ such that $\\phi (u)\\in k\\setminus \\lbrace 0,1\\rbrace $ .",
"Here we exclude 0 and 1 because we wish to produce a simply ramified rational function.",
"Note that $\\phi (-3),\\phi (-1)=1$ and $\\phi (0),\\phi (-2)=0$ .",
"Since $k$ is a finite field, this implies that $\\phi : k\\rightarrow k$ cannot be surjective.",
"Choose a $c\\in k\\setminus \\phi (k)$ .",
"Solving $\\phi (u)=c$ over $\\overline{k}$ yields the desired rational function.",
"Next we show that the situation is particularly bad when a ramification index is at least as large as the characteristic of the ground field (referred to as “low characteristic” in [5]).",
"Proposition 4.2 Let $k$ be a non-algebraically closed field of characteristic $p>0$ and $f(x)$ a rational function defined over $\\overline{k}$ with only $k$ -rational ramification points.",
"Suppose that $f(x)$ has a ramification point with ramification index at least $p$ .",
"Then there exist infinitely many equivalence classes of rational functions of the same degree and Wronskian as $f(x)$ which are not equivalent to a rational function defined over $k$ .",
"Let $d$ be the degree of $f(x)$ .",
"Since there exists a fractional linear transformation defined over $k$ sending $\\infty $ to the point with ramification index at least $p$ , we may without loss of generality assume that $\\infty $ is this point.",
"Let $g(x)/h(x)$ be the standard form for $f(x)$ .",
"Consider the rational functions $f_t(x)&=\\frac{g(x)}{h(x)}+t\\cdot x^p=\\frac{g(x)+t\\cdot x^p h(x)}{h(x)}$ with parameter $t\\in \\overline{k}$ .",
"By our assumptions, each $f_t$ is degree $d$ and its Wronskian is $h(x)g^{\\prime }(x)-g(x)h^{\\prime }(x)$ , independent of $t$ .",
"By post-composing $f_t$ with an arbitrary fractional linear transformation, one verifies that $f_t$ and $f_{t^{\\prime }}$ are equivalent only if $t=t^{\\prime }$ .",
"This implies that these rational functions correspond to a one dimensional subscheme of $G_d^{sep}$ on which $Wr$ is constant.",
"Any point on this subscheme with residue field unequal to $k$ corresponds to an equivalence class of rational functions, none of which are defined over $k$ by form(ii).",
"When the characteristic of $k$ is small, noshapiro gives us a great deal of information.",
"As our first case, since all ramified points have ramification index at least 2, it informally tells us that the analog of the Shapiro Shapiro conjecture does not hold at all in characteristic 2: Corollary 4.3 Let $k$ be a non-algebraically closed field of characteristic 2.",
"Suppose that $f(x)$ is a degree $d>1$ rational function defined over $\\overline{k}$ with only $k$ -rational ramification points.",
"Then there are infinitely many equivalence classes of degree $d$ rational functions with the same Wronskian as $f(x)$ which are not equivalent to a rational function defined over $k$ .",
"Additionally, noshapiro tells us that in characteristic 3 the analog of the Shapiro Shapiro conjecture does not hold when the ramification is not simple: Corollary 4.4 Let $k$ be a non-algebraically closed field of characteristic 3.",
"Suppose that $f(x)$ is a degree $d$ rational function defined over $\\overline{k}$ with only $k$ -rational ramification points.",
"If the ramification of $f(x)$ is not simple, then there are infinitely many equivalence classes of rational functions of the same degree and Wronskian as $f(x)$ which are not equivalent to a rational function defined over $k$ ."
]
] | 1605.06090 |
[
[
"A non-restricted counterexample to the first Kac-Weisfeiler conjecture"
],
[
"Abstract In 1971 Kac and Weisfeiler made two important conjectures regarding the representation theory of restricted Lie algebras over fields of positive characteristic.",
"The first of these predicts the maximal dimension of the simple modules, and can be stated without the hypothesis that the Lie algebra is restricted.",
"In this short article we construct the first example of a non-restricted Lie algebra for which the prediction of the first Kac--Weisfeiler conjecture fails.",
"Our method is to present pairs of Lie algebras which have isomorphic enveloping algebras but distinct indexes."
],
[
"Introduction",
"Let ${\\mathbb {k}}$ be an algebraically closed field of positive characteristic $p$ and $L$ be a finite dimensional Lie algebra over ${\\mathbb {k}}$ .",
"It is well known that all simple modules have finite dimension and that the dimensions are uniformly bounded above by some integer.",
"We denote by $M(L)$ the least upper bound of dimensions of simple $L$ -modules.",
"We remind the reader that the index of $L$ , denoted ${\\operatorname{ind}\\,}L$ , is the minimal dimension of a stabiliser of an element of the coadjoint representation.",
"The number $\\dim L-{\\operatorname{ind}\\,}L$ is easily seen to be even and the first Kac–Weisfeiler conjecture (KW1) predicts that the index of a restricted Lie algebra $L$ is involved in the representation theory in the following way: $M(L) = p^{\\frac{1}{2}(\\dim L - {\\operatorname{ind}\\,}L)}$ [5].",
"The conjecture is striking for both its simplicity and its generality, and has attracted much attention over the past 45 years.",
"Since the statement may be phrased without the hypothesis that $L$ is restricted, there has been some small hope that it may hold in general.",
"In this paper we shall show that for certain non-restricted Lie algebras (REF ) fails.",
"These are the very first examples of this kind in the literature.",
"For a given Lie algebra $L$ the problem of calculating ${\\operatorname{ind}\\,}L$ belongs to the realm of elementary linear algebra and the meat of the KW1 conjecture lies in computing $M(L)$ .",
"There is no procedure for determining this invariant in general, and practically nothing is known about representations of Lie algebras which are not restricted, which is undoubtedly why it has taken so long for (REF ) to be refuted for non-restricted algebras.",
"The most general result appears in [4] where it is shown that if a restricted Lie algebra $L$ admits a $\\chi \\in L^*$ such that $L_\\chi $ is a torus then KW1 holds for $L$ .",
"Over the past 10 years, various authors have been studying the isomorphism problem for enveloping algebras (see [2] for example).",
"In its most general form, the question is: can two non-isomorphic Lie algebras admit isomorphic enveloping algebras?",
"For finite dimensional Lie algebras over fields of characteristic zero there are no known examples of this pathalogical behaviour, however in characteristic $p$ such algebras are not hard to construct (we shall see new examples of this phenomenon in Proposition REF ).",
"Several weaker variants of the isomorphism problem have been considered, asking which properties are shared by Lie algebras $L$ and $L^{\\prime }$ such that $U(L) \\cong U(L^{\\prime })$ , for instance nilpotence, solvability, derived length.",
"The key observation of this article is that (REF ) implies a weak variant of the isomorphism problem: if (REF ) holds for all ${\\mathbb {k}}$ -Lie algebras and $U(L) \\cong U(L^{\\prime })$ then ${\\operatorname{ind}\\,}L = {\\operatorname{ind}\\,}L^{\\prime }$ .",
"This is simply because both $M(L)$ and $\\dim (L)$ depend only upon the isomorphism class of $U(L)$ ; in the language of [2] we would say that ${\\operatorname{ind}\\,}L$ is determined by $U(L)$ .",
"Our method is to disprove this corollary of (REF ) by exhibiting two Lie algebras with isomorphic enveloping algebras but distinct indexes.",
"For any set $X$ we use the notation $\\langle X \\rangle $ to denote the vector space spanned by $X$ .",
"We now describe a family of examples for which (REF ) fails.",
"Let $k \\ge 3$ and let $L$ be the Lie algebra $\\langle x_1, x_2, ... x_k, D_0, D \\rangle $ such that $D_0$ is central, $\\langle x_1,... x_{k}\\rangle $ is abelian, whilst $& & [D, x_i] = x_i \\text{ for } i=1,...,k-2,\\\\& & {[}D, x_{k-1}] = x_k,\\\\& & {[}D, x_k] = 0.$ In this article we shall prove that: Theorem We have $p^2 | M(L)$ and ${\\operatorname{ind}\\,}L = k$ so that $M(L) \\ne p^{\\frac{1}{2}(\\dim (L) - {\\operatorname{ind}\\,}(L))}$ .",
"Note that the example above is not restrictable, since ${\\operatorname{ad}}(D)^p \\notin {\\operatorname{ad}}(L)$ .",
"Question Does there exist a restricted Lie algebra for which KW1 fails?",
"Do there exist two restricted Lie algebras $L$ and $L^{\\prime }$ with $U(L) \\cong U(L^{\\prime })$ and ${\\operatorname{ind}\\,}L \\ne {\\operatorname{ind}\\,}L^{\\prime }$ ?",
"A positive answer to question (ii) would imply a positive answer to question (i), whilst a negative answer to (ii) would offer supporting evidence for the KW1 conjecture, as well as having independent value in the context of the isomorphism problem.",
"Acknowledgement.",
"I would like to thank Stephane Launois and Alexander Premet for useful comments on a first draft of this article.",
"I would also like to thank colleagues Giovanna Carnovale and Jay Taylor at the University of Padova for many interesting discussions.",
"The research leading to these results has received funding from the European Commission, Seventh Framework Programme, under Grant Agreement number 600376, as well as grants CPDA125818/12 and 60A01-4222/15 from the University of Padova."
],
[
"Lie algebras with isomorphic enveloping algebras",
"In this section we prove a basic result which allows us to construct families of Lie algebras which have isomorphic enveloping algebras.",
"For any Lie algebra $L$ we consider the restricted closure ${\\overline{L}}$ of ${\\operatorname{ad}}(L)$ inside ${\\operatorname{Der}}(L)$ , ie.",
"the smallest restricted subalgebra of ${\\operatorname{Der}}(L)$ containing ${\\operatorname{ad}}(L)$ .",
"Lemma 1 Every element of ${\\overline{L}}$ is of the form $\\sum _{i=0}^k {\\operatorname{ad}}(X_i)^{p^i}$ for some $k \\ge 0$ and elements $X_1,...,X_k \\in L$ .",
"We start by showing that for each $k \\ge 0$ the sum $\\sum _{i=0}^k {\\operatorname{ad}}(L)^{p^i}$ is a vector space.",
"The case $k = 0$ is obvious so we may proceed by induction.",
"Using the formulas derived in Chapter 2 of [1] we have ${\\operatorname{ad}}(X + Y)^{p^k} = {\\operatorname{ad}}(X)^{p^k} + {\\operatorname{ad}}(Y)^{p^k} \\mod {{\\operatorname{ad}}}(L)$ and so $\\sum _{i=0}^k({\\operatorname{ad}}(X_i)^{p^i} + {\\operatorname{ad}}(Y_i)^{p^i}) = \\sum _{i=0}^{k-1} ({\\operatorname{ad}}(X_i)^{p^i} + {\\operatorname{ad}}(Y_i)^{p^i}) + {\\operatorname{ad}}(X + Y)^{p^k} \\mod {{\\operatorname{ad}}}(L)$ and so by induction $\\sum _{i\\ge 0} {\\operatorname{ad}}(L)^{p^i}$ is a vector space.",
"If $X_1 = {\\operatorname{ad}}(X)$ and $Y_1 = {\\operatorname{ad}}(Y)$ then $[X_1^{p^k}, Y_1^{p^j}] = {\\operatorname{ad}}(X_1)^{p^k-1}({\\operatorname{ad}}(Y_1)^{p^j} X_1$ and so $\\sum _{i\\ge 0} {\\operatorname{ad}}(L)^{p^i}$ is closed under the bracket.",
"Using the same formulas mentioned in the first paragraph of the proof it is clear that $\\sum _{i\\ge 0} {\\operatorname{ad}}(L)^{p^i}$ is closed under taking $p$ th powers, and so it is a restricted algebra containing ${\\operatorname{ad}}(L)$ .",
"It is easy to see that it is the smallest such algebra.",
"For $D \\in {\\operatorname{Der}}(L)$ we write $L_D$ for the semidirect product $L \\rtimes {\\mathbb {k}}D$ .",
"Proposition 2 For every $D, D^{\\prime } \\in {\\overline{L}}$ we have $U(L_D) \\cong U(L_{D^{\\prime }})$ .",
"Let $D_0$ denote the zero derivation of $L$ .",
"We shall show that $U(L_D) \\cong U(L_{D_0})$ for every $D\\in {\\overline{L}}$ .",
"According to the previous lemma we can write $D = \\sum _{i=0}^k {\\operatorname{ad}}(X_i)^{p^i}$ for some $k \\ge 0$ and elements $X_1,...,X_k \\in L$ .",
"We define a linear map $\\phi & : & L_{D} \\longrightarrow U(L_{D_0});\\\\& & L \\overset{\\operatorname{Id}}{\\longmapsto } L;\\\\& & D \\longmapsto \\sum _{i=0}^k X_i^{p^i} + D_0.$ By construction $\\phi [X,Y] = \\phi (X) \\phi (Y) - \\phi (Y) \\phi (X)$ for all $X, Y \\in L_D$ .",
"Furthermore, every element of $L_{D_0} \\subseteq U(L_{D_0})$ lies in the algebra generated by the image and so, by the universal property of the enveloping algebra there is a surjective algebra homomorphism $\\Phi : U(L_{D}) \\twoheadrightarrow U(L_{D_0})$ .",
"To see that the map is injective we appeal to the graded algebra, as follows.",
"Suppose that $I = {\\operatorname{Ker}}\\Phi $ is a nonzero ideal of $U(L_D)$ .",
"Then $U(L_D) / I \\cong U(L_{D_0})$ and, in particular, their Gelfand–Kirillov dimensions coincide.",
"By [3] have $\\dim L = {\\operatorname{GKdim}\\,}U(L_{D_0}) = {\\operatorname{GKdim}\\,}(U(L_D)/I).$ The PBW filtration on $U(L_D)$ induces a filtration on $U(L_D)/I$ and, according to Proposition 7.6.13 of op.",
"cit.",
"we have $\\operatorname{gr}(U(L_D)/I) \\cong \\operatorname{gr}U(L_D) / \\operatorname{gr}I \\cong S(L_D) / \\operatorname{gr}I.$ Now Proposition 8.1.14 in op.",
"cit.",
"tells us that ${\\operatorname{GKdim}\\,}(U(L_D)/I) = {\\operatorname{GKdim}\\,}(S(L_D ) / \\operatorname{gr}I).$ Since $S(L_D) / \\operatorname{gr}I$ is a commutative affine algebra, Theorem 8.2.14(i) in that same book tells us that ${\\operatorname{GKdim}\\,}(S(L_D) / \\operatorname{gr}I)$ is equal to the Krull dimension of $S(L_D) / \\operatorname{gr}I$ , which is necessarily less than ${\\operatorname{GKdim}\\,}S(L_D) = \\dim L_D = \\dim L_{D_0},$ since $S(L_D) / \\operatorname{gr}I$ is a proper quotient.",
"This contradiction tells us that $I = 0$ as desired."
],
[
"Calculating indexes",
"We continue to let ${\\mathbb {k}}$ have characteristic $p > 0$ and pick $k \\ge 3$ .",
"Let ${\\mathcal {A}}= \\langle x_1,..., x_k\\rangle $ be an abelian Lie algebra and define a derivation of ${\\mathcal {A}}$ by $D(x_1) = x_i$ for all $i=1,...,{k-2}$ , $D(x_{k-1}) = x_k$ , $D(x_k) = 0$ .",
"We consider the semidirect product ${\\mathcal {A}}_D$ .",
"Now denote by $D_0$ the zero derivation of ${\\mathcal {A}}_D$ , and write $D^{\\prime }$ for the derivation $D^p$ of ${\\mathcal {A}}_D$ .",
"Define $L:= ({\\mathcal {A}}_D)_{D_0},\\\\L^{\\prime } := ({\\mathcal {A}}_D)_{D^{\\prime }}.$ According to Proposition REF we have $U(L) \\cong U(L^{\\prime })$ .",
"Lemma 3 We have $& & {\\operatorname{ind}\\,}L = k;\\\\& & {\\operatorname{ind}\\,}L^{\\prime } \\le k-2.$ Pick $\\chi \\in L^*$ and observe that $[L, L] = \\langle x_1 x_2,...,x_{k-2}, x_k\\rangle $ , which implies that $L_\\chi $ is completely determined by $(\\chi (x_1), \\chi (x_2) , ...,\\chi (x_{k-2}), \\chi (x_k)) \\in {\\mathbb {k}}^{k-1}$ .",
"Choosing scalars $a_i, b_i, n_j, m_j\\in {\\mathbb {k}}$ with $i =1,...,k$ and $j = 1,2$ determines two elements of $L$ : $X = \\sum _{i} a_ix_i + n_1 D + n_2 D_0;\\\\Y = \\sum _i b_i x_i + m_1 D + m_2 D_0.$ Observe that $\\chi [X, Y] = \\sum _{i=1}^{k-2}(n_1b_i - m_1 a_i)\\chi (x_i) + (n_1 b_{k-1} - m_1 a_{k-1})\\chi (x_k).\\medskip $ The assertion $X \\in L_\\chi $ is equivalent to saying that the right hand side of (REF ) vanishes for every choice of $b_i, m_j$ for $i=1,...,k$ and $j=1,2$ .",
"We shall use this observation to show that $\\dim L_\\chi \\ge k$ for all $\\chi \\in L^*$ .",
"If $\\chi (x_i) \\ne 0$ for some $i \\in \\lbrace 1,...,k-2\\rbrace $ then we may pick $Y$ by setting scalars $b_j = \\delta _{i,j}$ and $m_1 = m_2 = 0$ .",
"Now the vanishing of (REF ) ensures $n_1 = 0$ .",
"If $\\chi (x_k) \\ne 0$ then we may pick $b_j = \\delta _{j, k-1}$ and $m_1 = m_2 = 0$ to arrive at the conclusion $n_1 = 0$ similarly.",
"In either case, the assertion $X\\in L_\\chi $ is now equivalent to $m_1 (\\sum _{i=1}^{k-2} a_i \\chi (x_i)) + m_1 a_{k-1} \\chi (x_k) = 0$ .",
"This final condition on $L_\\chi $ is a single linear dependence between the scalars $a_1,...,a_k$ , and we conclude that $\\dim L_\\chi \\ge k$ for all $\\chi \\in L^*$ , and that this lower bound is attained whenever $(\\chi (x_1), \\chi (x_2) ..., \\chi (x_{k-2}), \\chi (x_k))\\ne (0,0,...,0).$ This proves ${\\operatorname{ind}\\,}L = k$ .",
"Next observe that $[L^{\\prime }, L^{\\prime }] = \\langle x_1, x_2,...,x_{k-2}, x_k\\rangle $ , and so $\\chi \\in L^{\\prime *}$ is determined by $(\\chi (x_1), \\chi (x_2) , ...,\\chi (x_{k-2}), \\chi (x_k)) \\in {\\mathbb {k}}^{k-1}$ .",
"We pick scalars as before so that $X = \\sum _{i} a_i x_i + n_1 D + n_2 D^{\\prime };\\\\Y = \\sum _i b_i x_i + m_1 D + m_2 D^{\\prime }.$ are arbitrary elements of $L^{\\prime }$ , and we have $\\chi [X, Y] = \\sum _{i=1}^{k-2}(n_1 b_i - m_1 a_i + n_2 b_i - m_2 a_i)\\chi (x_i) + ( n_1 b_{k-1} - m_1 a_{k-1})\\chi (x_k).$ Now $X \\in L^{\\prime }_\\chi $ is equivalent to the vanishing of the right hand side for every choice of $b_i, m_j$ .",
"It will suffice to exhibit $\\chi $ such that $\\dim L_\\chi ^{\\prime } \\le k-2$ .",
"To this end we take $\\chi (x_1) = \\chi (x_2) = \\cdots = \\chi (x_{k-2}) = \\chi (x_k) = 1$ .",
"Setting $b_j = \\delta _{j, k-1}$ and $m_1 = m_2 = 0$ we obtain $n_1 = 0$ , whilst taking $b_j = \\delta _{j,1}$ with $m_1 = m_2 =0$ subtends $n_2 = 0$ .",
"Now take $b_i = 0$ for all $i$ and $m_1 = 0$ , $m_2 = 1$ to get $\\sum _{i=1}^{k-2} a_i = 0$ .",
"Finally set $b_i = 0$ for all $i$ and $m_1 = 1$ , $m_2 = 0$ to get $a_k = 0$ .",
"This shows that $\\dim L_\\chi \\le k-2$ , and we are done."
],
[
"Proof of the theorem",
"We let $L$ and $L^{\\prime }$ be the Lie algebras discussed in Lemma REF .",
"Our goal is to show that (REF ) fails for $L$ .",
"Since the conjecture asserts that $M(L) = p^{\\frac{1}{2}(\\dim L - {\\operatorname{ind}\\,}L)} = p$ it will suffice to show that $p^2 | M(L)$ .",
"By Proposition REF we know that $U(L) \\cong U(L^{\\prime })$ which implies that $M(L) = M(L^{\\prime })$ .",
"By Lemma REF and [4] there is a non-negative integer $s$ such that $p^{\\frac{1}{2}(\\dim L^{\\prime } - {\\operatorname{ind}\\,}L^{\\prime })} = p^{2 + s} | M(L^{\\prime }).$ $\\hfill \\Box $"
]
] | 1605.06033 |
[
[
"Delay Bounds for Multiclass FIFO"
],
[
"Abstract FIFO is perhaps the simplest scheduling discipline.",
"For single-class FIFO, its delay guarantee performance has been extensively studied: The well-known results include a stochastic delay bound for $GI/GI/1$ by Kingman and a deterministic delay bound for $D/D/1$ by Cruz.",
"However, for multiclass FIFO, few such results are available.",
"To fill the gap, we prove delay bounds for multiclass FIFO in this work, considering both deterministic and stochastic cases.",
"Specifically, delay bounds are presented for multiclass D/D/1, GI/GI/1 and G/G/1.",
"In addition, examples are provided for several basic settings to demonstrate the obtained bounds in more explicit forms, which are also compared with simulation results."
],
[
"Introduction",
"Multiclass FIFO refers to the scheduling discipline where customers are served in the first-in-first-out (FIFO) manner and the services required by different classes may differ.",
"Compared to single-class FIFO where all customers typically have the same service requirements, multiclass FIFO is more general, providing a more natural way to model the system for scenarios where the service requirements of customers from different classes may differ.",
"One example is downlink-sharing in wireless networks, where a wireless base station, shared by multiple users, sends packets to them in the FIFO manner.",
"Since the characteristics of the wireless channel seen by these users may differ, the data rates to them may also be different.",
"Another example is input-queueing on a switch, where packets are FIFO-queued at the input port before being forwarded to the output ports that pick packets from the FIFO queue and serve at possibly different rates.",
"A third example is video conferencing, where the video part stream and the audio part stream have highly different characteristics.",
"However, the two streams are synchronized when they are generated.",
"In addition, inside the network, they may share the same FIFO queues, e.g.",
"in network interface cards and in switches.",
"Surprisingly, while there are a lot of results for single-class FIFO, few such results exist for multicalss FIFO.",
"The existing results for multiclass FIFO are mostly under the classic queueing theory (e.g.",
"[3]).",
"However, those available results are rather limited and their focus has been mainly on the queue stability condition.",
"In the context of input-queueing in packet-switched systems, multiclass FIFO has also been studied (e.g.",
"[4]).",
"However, in these studies, the focus has been on the throughput of the switch, assuming saturated traffic on each input port.",
"None of these studies has focused on the delay performance of multiclass FIFO.",
"In terms of delay bounds for FIFO, available results are almost all for single-class FIFO.",
"Among them are the two well-known delay bound results: one by Kingman [1] for the stochastic case $GI/GI/1$ , and one by Cruz [2] for the deterministic case $D/D/1$ .",
"For other or more general arrival and service processes, various delay bounds have also been derived, mainly in the context of network calculus (e.g.",
"[5][6][7]).",
"In particular, for some multiclass settings studied in this work, analytical bounds on their single-class counterparts can be found in the literature (e.g.",
"[8][9]).",
"With those results, one might expect that they could be readily or easily extended to multi-class FIFO.",
"Unfortunately, such an extension is surprisingly difficult and a direct extension may result in rather limited applicability of the obtained results.",
"This work is devoted, as an initial try, to bridging the gap.",
"The focus is on finding bounds (or approximations) for the tail distribution of delay or waiting time in multiclass FIFO.",
"The rest is organized as follows.",
"In Section , the difficulties in extending or applying single-class FIFO delay bound results to multiclass FIFO are first discussed, using the two well-known delay bounds as examples.",
"Then in Section , we prove delay bounds for multiclass FIFO, considering both deterministic and stochastic cases.",
"Specifically, delay analysis is performed and delay bounds are derived for $D/D/1$ , $GI/GI/1$ and $G/G/1$ , all under multiclass FIFO.",
"In Section , examples are further provided for several basic settings to demonstrate the obtained bounds in more explicit forms and compare the obtained bounds with simulation results."
],
[
"Difficulties",
"Below we use the Kingman's bound and the Cruz's bound as examples to discuss the difficulties or limitations in applying the single-class delay bounds to multiclass FIFO."
],
[
"Kingman's Bound",
"For a $GI/GI/1$ queue, the following delay bound on waiting time by Kingman [1] is well-known: $P\\lbrace W \\ge \\tau \\rbrace \\le e^{-\\vartheta \\tau }$ where $W$ denotes the steady-state waiting time in queue, $\\vartheta = \\sup \\lbrace \\theta >0: M_{Y(1)-X(1)}(\\theta ) < 1\\rbrace $ with $X(1)$ being the interarrival time and $Y(1)$ the service time of a customer, and $M_{Z}(\\theta )$ denotes the moment generating function of random variable $Z$ , i.e., $M_{Z}(\\theta ) \\equiv E[e^{\\theta Z}]$ .",
"The i.i.d.",
"condition on interarrival times and the i.i.d.",
"condition on service times imply that the bound (REF ) is mostly applicable only for single-class FIFO.",
"Extending it or directly applying it to multiclass FIFO can be difficult.",
"For example, for a multiclass FIFO system, even under that the two i.i.d.",
"conditions hold for each class, the applicability of (REF ) may still be limited due to three main reasons: For the aggregate of all classes, the two i.i.d.",
"conditions for the aggregate may not hold.",
"For example, for a two-class system with one class being $M/M$ and the other being $D/D$ , the i.i.d.",
"conditions, particularly the identical part, for the aggregate do not hold.",
"There are cases where, even though within each class, customers are independent, there is dependence between classes.",
"For example, in video conferencing, the video stream and the audio stream are synchronized when generated.",
"As a result, the independent part of the i.i.d.",
"condition, for the aggregate, may not be met.",
"For cases where the two i.i.d.",
"conditions for the aggregate also hold, it can still be challenging to find the probability distribution or characteristic functions of the interarrival times and the service times of the aggregate class of customers, which are needed by (REF ).",
"We remark that the above reasons also make other related single-class results (e.g.",
"[8][9]), which rely on similar i.i.d.",
"conditions, difficult to apply to multiclass FIFO."
],
[
"Cruz's Bound",
"For a $D/D/1$ system in communication networks, the following delay bound was initially shown by Cruz [2] $D \\le \\frac{\\sigma }{C}$ where $D$ denotes the system delay of any packet, $C$ (in bps) the service rate of the system, and $\\sigma $ (in bits) the traffic burstiness parameter.",
"The conditions of the Cruz's delay bound are $r \\le C$ and that the input traffic during any time interval $[s,s+t]$ , denoted as $A(s,s+t)$ , for all $s, t\\ge 0$ , is upper-constrained by $A(s, s+t) \\le r \\cdot t + \\sigma $ .",
"Unlike the Kingman's bound, the Cruz's bound may be readily used for multiclass FIFO.",
"To illustrate this, consider a FIFO queue with $N$ classes, where the traffic of each class $n$ is upper-constrained by $A_n(s,s+t) \\le r_n \\cdot t + \\sigma _n$ and the service rate of the class is $C_n$ with $r_n \\le C_n$ .",
"Without difficulty, (REF ) can be extended to this multiclass queue.",
"Specifically, for the aggregate, there holds $A(s, s+t) \\equiv \\sum _n A_n(s,s+t) \\le \\sum _n r_n \\cdot t + \\sum _n \\sigma _n$ .",
"Then, if there holds $\\sum _{n}r_n \\le \\min _{n} \\lbrace C_n\\rbrace ,$ the delay of any packet is upper-bounded by $D \\le \\frac{\\sum _n \\sigma _n}{\\min _{n} \\lbrace C_n\\rbrace }.$ Unfortunately, the condition $\\sum _{n}r_n \\le \\min _{n} \\lbrace C_n\\rbrace $ can be too restrictive, particularly when the service rates differ much.",
"As a result, the bound can be highly loose or even it cannot be used due to that the condition to use the bound is not met, as to be exemplified later in Section ."
],
[
"System Model and Notation",
"Consider a multiclass queueing system.",
"There is only one queue that is initially empty.",
"Customers are served in the FIFO manner.",
"If multiple customers arrive at the same time, the tie is broken arbitrarily.",
"The size of the queue is unlimited.",
"The serving part of the system is characterized by a work-conserving server.",
"There are $N(\\ge 1)$ classes of customers (e.g.",
"packets in a communication network).",
"Let $p_n^j$ denote the $j$ th customer of class $n$ , with $n \\in [1,N]$ and $j=1, 2, \\cdots $ .",
"Each customer $p_n^j$ is characterized by a traffic parameter $l_n^j$ that denotes the amount of traffic (in the number counted on a defined traffic unit, e.g.",
"bits in the communication network setting) carried by the customer.",
"To customers of class $n$ , the service rate (in traffic units per second, e.g.",
"bps - bits per second) of the server is constant, denoted by $C_n$ .",
"For each class $n$ , let $A_n(0,t) \\equiv A(t)$ denote the amount of traffic (in traffic units, e.g.",
"bits) that arrives within the time period $[0, t)$ , and $A_n(s,t) \\equiv A_n(t) - A_n(s)$ the traffic in $[s,t)$ .",
"For the aggregate traffic of all classes, $A(s,t) \\equiv \\sum _n A_n(s,t)$ and $A(t)$ are similarly defined.",
"Also for each class $n$ , we use $\\lambda _n$ to denote the average customer arrival rate and $\\mu _n$ the average customer service rate, and define $\\rho _n = \\frac{\\lambda _n}{\\mu _n}$ .",
"For any customer $p_n^j$ , let $a_n^j$ and $d_n^j$ respectively denote its arrival time and departure time.",
"By convention, we let $a_n^{0}= d_n^{0}= 0$ .",
"The delay in system of the customer is then $D_n^j = d_n^j-a_n^j$ , and the waiting time in queue is $W_n^j = D_n^j - l_n^j / C_n$ .",
"In addition, corresponding to the notation used in Section REF , we use $X_n(j)$ to denote the interarrival time between $p_n^{j-1}$ and $p_n^j$ , and $Y_n(j)$ the service time of $p_n^j$ .",
"By definition, $X_n(j) = a_n^{j} - a_n^{j-1}$ , for $j=1, 2, \\dots $ , and $Y_n(j) = l_n^j / C_n$ .",
"In this paper, we assume that for each class $n$ , the processes $X_n(j)$ and $Y_n(j)$ are both stationary.",
"Then by definition, we can also write $\\lambda _n = 1/E[X_n(1)]$ , $\\mu _n = 1/E[Y_n(1)]$ , and $\\rho _n = E[X_n(1)]^{-1}E[Y_n(1)]$ .",
"Like single-class FIFO, the dynamics of the multiclass FIFO system are also described by, for all $j = 1, 2, \\dots $ , $ d^{j} = \\max (a^j, d^{j-1}) + l^j / C^j$ where $j=1, 2, \\dots $ denotes the aggregate sequence of all customers ordered according to their arrival times, and $p^j, a^j, d^j, l^j$ and $C^j$ respectively denote the $j$ th customer in this ordered aggregate sequence, its arrival time, departure time, carried traffic amount and received service rate.",
"Similarly, we denote the delay of $p^j$ as $D^j = d^j-a^j$ , and its waiting time in queue as $W^j = D^j - l^j / C^j$ ."
],
[
"Delay Bound for Multiclass $D/D/1$",
"Suppose that the traffic of each class $n$ is constrained by $A_n(s, s+t) \\le r_n t + \\sigma _n$ for all $s, t \\ge 0$ .",
"For this multiclass $D/D/1$ queue, we have: Theorem 1 If $\\sum _{n}\\frac{r_n}{C_n} \\le 1$ , the delay of any customer $p^j$ is bounded by: $D^j \\le \\sum _{n}\\frac{\\sigma _n}{C_n}$ and the delay bound is tight.",
"For any $p^{j}$ , there exists some time $t^0$ that starts the busy period where it is in.",
"Note that such a busy period always exists, since in the extreme case, the period is only the service time period of $p^{j}$ and in this case, $t^0=a^{j}$ .",
"Since the system is work-conserving, it is busy with serving customers in $[t^0, d^{j}]$ .",
"So, $d^{j} = t^0 + \\sum _{n=1}^{N}Y_n(t^0, d^{j})$ , where $Y_n(t^0, d^{j})$ denotes the total service time of class $n$ customers that are served in $[t^0, d^{j}]$ .",
"Because of FIFO and that the system is empty at $t^0_{-}$ , $Y_n(t^0, d^{j})$ is hence limited by the amount of traffic that arrives in $[t^0, a^{j}]$ , i.e., $A(t^0, a^{j}_{+})$ , where $x_{-} \\equiv x - \\epsilon $ and $x_{+} \\equiv x + \\epsilon $ with $\\epsilon \\rightarrow 0$ .",
"Specifically, $Y_n(t^0, d^{j}) \\le \\frac{A_n(t^0, a^{j}_{+})}{C_n} $ , so we then have $d^{j} \\le t^0 + \\sum _{n=1}^{N}\\frac{A_n(t^0, a^{j}_{+})}{C_n}$ .",
"Under the condition $\\sum _{n=1}^{N}\\frac{r_n}{C_n} \\le 1$ , we then obtain: $D^{j} &\\le & \\sum _{n=1}^{N}\\frac{A_n(t^0, a^{j}_{+})}{C_n} + t^0-a^{j} \\\\&\\le & \\sum _{n=1}^{N}\\frac{r_n\\cdot (a^{j}-t^0) + \\sigma _n}{C_n} - (a^{j}- t^0) \\le \\sum _{n=1}^{N} \\frac{ \\sigma _{n}}{C_n} \\nonumber $ where the second last step is due to the traffic constraint and the last step is from $\\sum _{n}\\frac{r_n}{C_n} \\le 1$ and $a^{j} \\ge t^0$ .",
"Note that, for the system, consider that immidiately after time 0, every traffic class generates a burst with size $\\sigma _n$ .",
"In this case, the customer in these bursts, which receives service last, will experience delay $\\sum _{n=1}^{N} \\frac{ \\sigma _n}{C_n}$ that equals the delay bound.",
"So, the bound is tight."
],
[
"Delay Bounds for Multiclass $GI/GI/1$",
"To assist proving results for the ordinary multiclass $GI/GI/1$ system, we first consider a discrete time counterpart of the system, where time is indexed by $t=0, 1, 2, \\dots $ .",
"The length of unit time is $\\delta $ .",
"The discrete time system becomes the former by letting $\\delta \\rightarrow 0$ .",
"In the discrete time system, depending on the length of the unit time $\\delta $ , it could happen that multiple customers arrive at the same (discrete) time.",
"Because of this, in addition to waiting time, we introduce the concept of virtual waiting time.",
"The virtual waiting time at time $t$ is defined to be the time that a virtual customer, which arrives immediately before time $t$ , would experience: All arrivals at $t$ are excluded in the calculation of the virtual waiting time at $t$.",
"More specifically, for any $p^j$ , the corresponding virtual waiting time $V^j$ can be written as: $V^{j} \\equiv \\sup _{0 \\le s \\le a^{j}} \\left[ \\sum _{n=1}^{N}\\frac{A_n(s, a^{j})}{C_n} - (a^{j}-s) \\right].",
"$ Note that the definition of virtual waiting time also applies to continuous time systems.",
"If in a (continuous time or discrete time) system, there is at most one arrival at a time, then $V^{j}$ equals $W^{j}$ , i.e.",
"$W^{j} = V^{j}$ .",
"For the discrete-time counterpart of multiclass $GI/GI/1$ , we have the following result.",
"Its proof is in the Appendix.",
"Lemma 1 For the discrete time system, if there exists small $\\theta > 0$ such that $E[e^{\\theta (\\sum _{n=1}^{N}\\frac{A_n(1)}{C_n} - 1)}] \\le 1$ , then, for any $p^j$ and for all such $\\theta $ , the virtual waiting time at $a^j$ and the delay of the customer are respectively bounded by, $P \\lbrace V^{j} \\ge \\tau \\rbrace &\\le & M_{\\sum _{n}\\frac{A_n(1)}{C_n} - 1}(\\theta ) e^{-\\theta \\tau } \\\\P \\lbrace D^{j} \\ge \\tau \\rbrace &\\le & 1- {F}_{\\frac{A_1(1)}{C_1}} \\ast \\cdots \\ast {F}_{\\frac{A_N(1)}{C_N}} \\ast {F}_{{V}}(\\tau ) $ where $F_{X}$ denotes the CDF (or a lower bound on CDF) of $X$ , ${F}_{{V}}(\\tau ) \\equiv 1 - M_{\\sum _{n}\\frac{A_n(1)}{C_n} - 1}(\\theta ) e^{-\\theta \\tau }$ , and $\\ast $ denotes the convolution operation.",
"For Lemma REF , we highlight that, though in the term ${F}_{\\frac{A_1(1)}{C_1}} * \\cdots *{F}_{\\frac{A_N(1)}{C_N}}$ of (), the time index 1 is used, the term is actually contributed by concurrent arrivals of the considered $p^j$ , specifically by the total service time of all arrivals at time $a^j$ , as shown in the proof.",
"For the ordinary continuous time multiclass FIFO system, if there is at most one customer arrival at a time, the rest of this part disappears and the remaining is the contribution by the considered customer $p^j$ , which is the service time of $p^j$ .",
"Now by letting $\\delta \\rightarrow 0$ , the following result immediately follows from Lemma REF .",
"Theorem 2 For a multiclass $GI/GI/1$ system with no concurrent arrivals at any time, if there exists small $\\theta > 0$ such that $E[e^{\\theta (\\sum _{n=1}^{N}\\frac{A_n(1)}{C_n} - 1)}] \\le 1$ , then, the waiting time and delay of any customer $p^j$ are respectively bounded by, $P \\lbrace W^j \\ge \\tau \\rbrace &\\le & e^{-\\theta ^{*} \\tau } \\\\P \\lbrace D^j \\ge \\tau \\rbrace &\\le & 1- {F}_{Y(j)} * {F}_{{W}}(\\tau ) $ where ${F}_{Y(j)}$ is the CDF (or a lower bound on CDF) of the service time of the customer, ${F}_{{W}} = 1- e^{-\\theta ^{*} \\tau }$ and $\\theta ^{*} = \\sup \\lbrace \\theta >0: E[e^{\\theta (\\sum _{n}\\frac{A_n(1)}{C_n} - 1)}] \\le 1\\rbrace .$"
],
[
"Delay Bounds for Multiclass $G/G/1$",
"As discussed in Section REF , when there are multiple classes, even though the i.i.d.",
"conditions may hold for each class, such i.i.d.",
"conditions do not necessarily carry over to between classes.",
"As a consequence, Theorem REF may not be applicable.",
"To deal with this, we present the following bounds.",
"Theorem 3 Suppose the traffic of each class has generalized Stochastically Bounded Burstiness (gSBB) [10] [11], satisfying for some $R_n >0$ and $\\forall t >0$ , $P \\lbrace \\sup _{0 \\le s \\le t}[A_n(s, t) - R_n \\cdot (t-s)] > \\sigma \\rbrace \\le \\bar{F}_n(R_n, \\sigma )$ for all $\\sigma \\ge 0$ .",
"Then, for $\\forall (R_1, \\dots , R_N)$ , under the condition $\\sum _{n}\\frac{R_n}{C_n} \\le 1,$ the delay of any customer $p^j$ is bounded by (a.s.): $P \\lbrace D^j > \\tau \\rbrace \\le \\inf _{p_1 + \\cdots + p_N=1}\\sum _{n=1}^N \\bar{F}_n(R_n, p_n \\cdot C_n \\tau )$ and if the arrival processes of the $N$ classes are independent of each other, then the delay is bounded by (a.s.): $P \\lbrace D^j > \\tau \\rbrace \\le 1- {F}_1 * \\cdots *{F}_N(R_n, C_n\\tau )$ where ${F}_n(R_n, C_n \\tau ) \\equiv 1 - \\bar{F}_n(R_n, C_n \\tau )$ .",
"Following the proof of Theorem REF , we have obtained (REF ).",
"By applying $\\sum _{n=1}^{N}\\frac{R_n}{C_n} \\le 1$ to (REF ), we further obtain $D^{j}&\\le & \\sum _{n=1}^{N} \\frac{A_n(t^0, a^{j}_{+}) - R_n\\cdot (a^{j}-t^0)}{C_n}.",
"$ Note that in (REF ), $t^0$ is a random variable.",
"Taking all sample paths into consideration, with $A_n(t^0, a^{j}_{+}) - R_n\\cdot (a^{j}-t^0) \\le \\sup _{0 \\le s \\le a^{j}}[A_n(s, a^{j}_{+}) - R_n\\cdot (a^{j}-s)]$ , we get: $D^{j}&\\le & \\sum _{n=1}^{N} \\frac{\\sup _{0 \\le s \\le a^{j}}[A_n(s, a^{j}_{+}) - R_n\\cdot (a^{j}-s)]}{C_n}.",
"$ Since the traffic of each class has gSBB, with simple manipulation on the definition and applying $\\epsilon \\rightarrow 0$ , we have, $& & P \\lbrace \\frac{\\sup _{0 \\le s \\le a^{j}}[A_n(s, a^{j}_{+}) - R_n\\cdot (a^{j}_{+}-s)]+R_n \\epsilon }{C_n} > \\tau \\rbrace \\nonumber \\\\&\\le & \\bar{F}_n (R_n, C_n \\tau ).",
"\\nonumber $ The theorem then follows from probability theory results on sum of random variables.",
"We remark that it is easily verified that the deterministic traffic model is a special case of the gSBB model with $\\bar{F}(x) = 0$ for all $x \\ge \\sigma _n$ and $\\bar{F}(x) = 1$ otherwise.",
"In addition, a wide range of traffic processes have been proved to have gSBB [11].",
"To illustrate the obtained delay bounds, this section presents examples for some basic settings, whose single-class counterparts have been extensively studied in the literature, with more explicit expressions for the delay bounds.",
"In addition, the obtained bounds are compared with simulation results and discussed."
],
[
"Multiclass $D/D/1$",
"Consider a multiclass FIFO queue in a communication network.",
"Assume that there are two traffic classes, and for each class, packets have constant size $l_n$ , which arrive periodically with $X_n$ being the period length.",
"It is easily verified that for each class, its traffic arrival process satisfies $A_n(s,t) \\le r_n \\cdot (t-s) + \\sigma _n$ , with $r_n = l_n / X_n$ and $\\sigma _n = l_n$ .",
"Applying them to Theorem REF , a delay bound is found as: If $\\frac{r_1}{C_1}+\\frac{r_2}{C_2} \\le 1$ , the delay of any packet satisfies: $D \\le \\frac{l_1}{C_1} + \\frac{l_2}{C2}.$ To compare, recall the delay bound directly from single class $D/D/1$ network calculus analysis discussed in Section REF , which is, if $r_1+r_2 \\le \\min \\lbrace C_1, C_2\\rbrace $ , $D \\le \\frac{l_1+l_2}{\\min \\lbrace C_1,C2\\rbrace }.$ To illustrate the two bounds, Figure REF is presented, where two cases, Case 1 and Case 2, are considered.",
"For Case 1, $C_1 =20Mbps$ and $C_2 = 100Mbps$ ; $l_1=100$ bytes and $l_2=1250$ bytes; $X_1=0.1 ms$ and $X_2=1 ms$ .",
"For Case 2, the other settings are the same except that $C_1 = 10Mbps$ .",
"Figure: Case 2For both cases, $r_1 = 8Mbps$ and $r_2=10Mbps$ .",
"It is easily verified that while for both cases, the condition $\\frac{r_1}{C_1}+\\frac{r_2}{C_2} \\le 1$ is satisfied, the condition $r_1+r_2 \\le \\min \\lbrace C_1, C_2\\rbrace $ is only met for Case 1.",
"Figure REF shows that, for both cases, the bound by Theorem REF is not only able to bound the delays of all simulated packets but also tight, i.e., some packets can indeed experience delay equal to the bound.",
"However, (REF ) can be highly conservative as shown by Figure REF for Case 1, and may even not be applicable (N.A.)",
"due to that its required condition is not met as indicated by Figure REF for Case 2."
],
[
"Multiclass $GI/GI/1$",
"In this subsection, we give two examples for multiclass $GI/GI/1$ , which are multiclass $M/D/1$ and multiclass $M/M/1$ .",
"In both examples, customers of each class arrive according to a Poisson process with average interarrival time $X_n$ .",
"In addition, customers of the same class have the same expected service time $Y_n$ .",
"However, while for $M/M/1$ , the service time of each customer is exponentially distributed, it is constant for $M/D/1$ .",
"Similar to the $D/D/1$ example, for each of them, we try to give an expression for the tail of delay / waiting time in a closed-form format to help the use of the related bounds.",
"In particular, we have the following corollaries, for which we assume the stability condition is met, i.e.",
"$\\rho \\equiv \\sum _{n}\\frac{r_n}{C_n} < 1$ .",
"Their proofs are included in the Appendix.",
"Corollary 1 For multiclass $M/D/1$ , if all classes are independent, then for any customer $p_n^j$ , its waiting time satisfies: $P \\lbrace W_n^{j} > \\tau \\rbrace \\le e^{-\\theta ^* \\tau }$ with $\\theta ^{*} = \\sup \\lbrace \\theta >0: \\sum _n \\lambda _n (e^{\\theta Y_n}-1)-\\theta \\le 0\\rbrace $ an approximation of which is $\\theta ^* = 2(1-\\rho )\\tau /(\\sum _{n=1}^{N} X_n^{-1} Y_n^2).",
"$ Corollary 2 For multiclass $M/M/1$ , if all classes are independent, then for any customer $p_n^j$ , its waiting time satisfies $P \\lbrace W_n^{j} > \\tau \\rbrace \\le e^{-\\theta ^* \\tau }$ with $\\theta ^{*} = \\sup \\lbrace 0 < \\theta < \\min _n \\mu _n: \\sum _n\\frac{\\lambda _n}{\\mu _n - \\theta } \\le 1\\rbrace $ an approximation of which is $\\theta ^* = (1-\\rho )\\tau /(\\sum _{n=1}^{N} X_n^{-1} Y_n^2).",
"$ Figure: Case 4: M/M/1M/M/1We remark that, with the bound on waiting time, a bound on delay can be easily obtained, e.g.",
"for $M/D/1$ , $P \\lbrace D_n^{j} > Y_n + \\tau \\rbrace =P \\lbrace W_n^{j} > \\tau \\rbrace $ .",
"In addition, we remark that the single class version of the bounds in the above two corollaries resemble closely with the literature approximations for tail of delay / waiting time distribution in single class $M/G/1$ , e.g., (2.9), (2.124) and (2.143) in [12].",
"To illustrate the bounds, Figure REF is presented, where two cases, Case 3 and Case 4, are respectively considered.",
"For Case 3, the other settings are the same as for Case 2, except that packets of each class arrive according to an independent Poisson process, i.e.",
"Case 3 is multiclass $M/D/1$ .",
"For Case 4, the other settings are the same as for Case 3, except that the traffic of each packet results in an exponentially distributed service time, i.e.",
"Case 4 is multiclass $M/M/1$ .",
"In Figure REF , the delay CCDF simulation results of the 1st, the 10th and the 100th packet of Class 1, are included, in addition to the steady state delay CCDF and the analytical bound.",
"Figure REF shows that the delays of packets are stochastically increasing as the packet number goes higher and they converge to the steady state distribution.",
"This is as expected and it is a proven phenomenon for single-class FIFO (e.g.",
"[12]).",
"For this reason, in later figures, only steady-state delay or waiting time distribution will be focused.",
"In Figure REF , the curves are for Class 2, which include the simulated steady-state waiting time CCDF, the analytical bound based on (REF ) and the approximate analytical bound (REF ).",
"Figure REF shows that the analytical bounds are fairly tight and provide good approximations of the corresponding steady-state delay CCDF for both cases."
],
[
"Multiclass $G/G/1$",
"In this subsection, we consider two examples where, even though within each class, the interarrival times and the service times are still respectively i.i.d., the overall FIFO system no more has i.i.d.",
"interarrival times or i.i.d.",
"service times.",
"To ease expression, only two classes are considered.",
"Corresponding to the two examples are Case 5 and Case 6.",
"The settings of Case 5 are the same as Case 3 except that some dependenceThe same series of pseudo random numbers have been used in generating the interarrival times for both classes.",
"is introduced in the two Poisson arrival processes.",
"To denote this case, we use $M^*/D/1$ , where $^*$ indicates that some dependence exists among classes.",
"In Case 6, Class 1 has the same settings of Class 1 in Case 2, while Class 2 has the same settings of Class 2 in Case 4.",
"In other words, the system has two classes, where one is $D/D$ and the other is $M/M$ .",
"In Figure REF , we denote the system by $DM/DM/1$ .",
"As discussed in Section REF , in this $DM/DM/1$ system, customers do not identical service time distribution.",
"For the two cases, the following corollaries are obtained by directly applying Theorem REF with the characteristics of the corresponding processes.",
"The detailed proofs are omitted.",
"Corollary 3 For the $M^*/D/1$ example, we have for any customer $p_n^j$ , $P \\lbrace W_n^{j} > \\tau \\rbrace &\\lessapprox & N e^{-\\theta ^* \\tau / N}.",
"$ with $\\theta ^*$ as shown in (REF ).",
"Corollary 4 For the $DM/DM/1$ example, we have for any customer $p_1^j$ , $P \\lbrace W_1^{j} > \\tau \\rbrace \\le e^{- (\\mu _2 - \\frac{\\lambda _2}{1-\\rho _1}) \\tau }, $ and for any customer $p_2^j$ $P \\lbrace W_2^{j} - Y_1 > \\tau \\rbrace \\le e^{- (\\mu _2 - \\frac{\\lambda _2}{1-\\rho _1}) \\tau } .",
"$ Figure: Case 6:DM/DM/1DM/DM/1To illustrate the two bounds in Corollary REF and Corollary REF , Figure REF presents results for the two cases, both for Class 2.",
"Figure REF indicates that when there is dependence between the two classes, the analytical bound assuming independent classes is no more an upper-bound.",
"However, the general analytical bound (REF ) holds, even though there is a noticeable gap from the actual distribution.",
"Note that the general bound holds for any possible dependence structure between the classes.",
"By making use of the dependence information in the analysis, the analytical bound could be improved, but this is out of the scope of the present paper and we leave it for future investigation.",
"Figure REF indicates an interesting phenomenon, which is that the (steady state) waiting time distributions of the two classes are different.",
"Note that, here, the waiting time distribution has been intentionally used.",
"In Case 3 - Case 5, where both classes have Poisson arrivals, the waiting time distribution is the same for both classes.",
"However, in Case 6, while Class 2 still has Poisson arrivals, Class 1 has periodic arrivals.",
"This arrival process difference results in the waiting time distribution difference: The waiting time observed by a Poisson inspector is different from that by a periodic inspector.",
"Nevertheless, the bounds (REF ) and (REF ) are valid and are fairly good.",
"Though improvement might be further made, the bounds provide an initial step towards the analysis of similar problems."
],
[
"Proof of Lemma 1",
"Our starting point is (REF ).",
"From (REF ) and following the same argument as for (), the following inequality is readily obtained, which holds for all sample paths: $D^{j} &\\le & \\sup _{0 \\le s \\le a^{j}} \\left[ \\sum _{n=1}^{N}\\frac{A_n(s, a^{j}_+)}{C_n} - (a^{j}-s) \\right] \\\\ &=&V^{j} + \\sum _{n=1}^{N}\\frac{A_n(a^{j}, a^{j}_+)}{C_n}$ Define $Z(k) = e^{\\theta [\\sum _{n=1}^{N}\\frac{A_n(a^{j}-k, a^{j})}{C_n}-k]}$ , $k = 1, 2, \\dots , a^{j}$ , where $\\theta >0$ is a constant.",
"Then, under the condition $E[e^{\\theta (\\sum _{n=1}^{N}\\frac{A_n(1)}{C_n} - 1)}] \\le 1$ , it can be proved that $\\lbrace Z(k) \\rbrace $ forms a supermartingale.",
"We now have, $P \\lbrace V^{j} > \\tau \\rbrace &=& P \\lbrace e^{\\theta V^{j}} > e^{\\theta \\tau } \\rbrace \\nonumber \\\\&\\le & P \\lbrace e^{\\theta \\sup _{s \\le a^{j}} [ \\sum _{n=1}^{N}\\frac{A_n(s, a^{j})}{C_n} - (a^{j}-s)}] > e^{\\theta \\tau }\\rbrace \\nonumber \\\\&=& P \\lbrace \\sup _{1 \\le k \\le a^{j}} Z(k) > e^{\\theta \\tau }\\rbrace \\nonumber \\\\&\\le & E[Z(1)]e^{-\\theta \\tau } $ where the last step follows from the Doob's maximal inequality for supermartingale.",
"Since $E[Z(1)] = E[e^{\\theta (\\sum _{n=1}^{N}\\frac{A_n(1)}{C_n} - 1)}] \\equiv M_{\\frac{A_n(1)}{C_n} - 1}(\\theta )$ , the first part is proved.",
"For the second part, since $V^{j}$ and $\\frac{A_n(a^{j}, a^{j}+1)}{C_n}$ , $n=1, \\cdots , N$ , are independent, it follows from elementary probability theory results on sum of independent random variables and that $A_n(a^{j}, a^{j}_+) \\le A_n(a^{j}, a^{j}+1)=_{st} A_n(1)$ ."
],
[
"Proof of Corollary 1",
"Note that, $A_n(1)$ is a compound Poisson process with $A_n(1)=\\sum _{i=1}^{\\mathcal {N}_n(1)}l_n^i = \\mathcal {N}_n(1) \\times l_n$ , where $\\mathcal {N}_n(1)$ denotes the number of Class $n$ packets that arrive within a unit time.",
"In addition, since $C_n$ is constant, we can write $\\frac{A_n(1)}{C_n}= \\mathcal {N}_n(1) \\times \\frac{l_n}{C_n}$ , which is also a compound Poisson with MGF: $E[e^{\\theta \\frac{A_n(1)}{C_n}}] = e^{\\lambda _n (e^{\\theta l_n/C_n}-1)}.$ Then, $M_{\\sum _n\\frac{A_n(1)}{C_n}-1} = e^{\\sum _n \\lambda _n (e^{\\theta l_n/C_n}-1)-\\theta }$ , which implies that solving $M_{\\sum _n\\frac{A_n(1)}{C_n}-1} \\le 1$ to get $\\theta $ is equivalent to finding $\\theta $ from: $\\sum _n \\lambda _n (e^{\\theta l_n/C_n}-1) -\\theta \\le 0 $ which proves the first part.",
"With $e^{\\theta l_n/C_n} \\approx 1 + \\theta l_n/C_n + \\frac{1}{2}\\theta ^2 (l_n/C_n)^2$ from Taylor expansion and $\\theta > 0$ , (REF ) can be rewritten as $\\theta \\sum _n \\lambda _n Y_n + \\frac{\\theta ^2}{2} \\sum _n \\lambda _n Y_n^2-\\theta \\le 0.$ Since $\\theta >0$ , $\\rho =\\sum _n \\lambda _n Y_n$ and $\\lambda _n = X_n^{-1}$ , we then get $\\theta \\lessapprox 2(1-\\rho )/(\\sum _{n=1}^{N} X_n^{-1} Y_n^2).$ Taking $\\theta ^* =2(1-\\rho )/(\\sum _{n=1}^{N} X_n^{-1} Y_n^2)$ , the 2nd part is proved."
],
[
"Proof of Corollary 2",
"Note that, $A_n(1)$ is again a compound Poisson process with $A_n(1)=\\sum _{i=1}^{\\mathcal {N}_n(1)}Y_n(i)$ and similarly, $A_n(1)/C_n$ is also a compound Poisson.",
"Since each $Y_n(i)=l_n^i/C_n$ has exponential distribution, the MGF of $A_n(1)$ can be written as, with $0 < \\theta \\le \\min _n \\mu _n$ , $E[e^{\\theta A_n(1) / C_n}] = e^{\\frac{\\lambda _n}{\\mu _n - \\theta } \\theta }.$ Then, solving $M_{\\sum _n\\frac{A_n(1)}{C_n}-1} \\le 1$ to get $\\theta $ is equivalent to finding $\\theta $ from: $\\sum _n (\\frac{\\lambda _n}{\\mu _n - \\theta } \\theta )-\\theta \\le 0$ and with simple manipulation, it becomes $\\sum _n \\frac{\\lambda _n}{\\mu _n - \\theta } \\le 1.",
"$ which proves the first part.",
"While (REF ) looks neat, finding an explicit expression for $\\theta $ is not easy.",
"In the following, we adopt an approximation approach.",
"In particular, $\\frac{\\lambda _n}{\\mu _n - \\theta } = \\frac{\\rho _n}{1 - \\theta /\\mu _n} \\approx \\rho _n (1 + \\theta /\\mu _n)$ applying which to (REF ) gives $\\rho + \\theta \\sum _n \\frac{\\rho _n}{\\mu _n} \\lessapprox 1$ i.e., $\\theta \\lessapprox (1-\\rho )/(\\sum _{n=1}^{N} X_n^{-1} Y_n^2)$ since $\\mu _n = Y_n^{-1}$ and $\\lambda _n = X_n^{-1}$ .",
"Then taking $\\theta ^* =(1-\\rho )/(\\sum _{n=1}^{N} X_n^{-1} Y_n^2)$ , the 2nd part is proved."
],
[
"Proof of Corollary 3",
"Our starting point is ().",
"Without loss of generality, suppose $p^j$ is a customer of class $n$ .",
"Due to also that all customers of the same class have the same service time $Y_n$ and that at time $a^{j}_{+}$ , there is only one arrival that is $p^j$ and hence $W^j = D^j - Y_n$ , we now have $W^{j}&\\le & \\sum _{n=1}^{N} \\frac{\\sup _{0 \\le s \\le a^{j}}[A_n(s, a^{j}_{+}) - R_n\\cdot (a^{j}-s)]}{C_n} - Y_n \\nonumber \\\\&=& \\sum _{n=1}^{N} \\frac{\\sup _{0 \\le s \\le a^{j}}[A_n(s, a^{j}) - R_n\\cdot (a^{j}-s)]}{C_n} $ The right hand side of (REF ) has $N$ items.",
"Denote each as ${\\tilde{W}}_n^j = \\frac{\\sup _{0 \\le s \\le a^{j}}[A_n(s, a^{j}) - R_n\\cdot (a^{j}-s)]}{C_n}.",
"$ Following the proof of the first part of Theorem 3, we have the following inequality that holds without any assumption on the potential dependence condition among classes, $P\\lbrace W^j >\\tau \\rbrace \\le \\inf _{\\sum _n p_n = 1} \\sum _n P\\lbrace {\\tilde{W}}_n^j > p_n \\cdot \\tau \\rbrace $ We highlight that (REF ) has a form similar to (REF ).",
"Then following the same approach as for the proof of Lemma 1 and Theorem 2, we can get: $P\\lbrace {\\tilde{W}}_n^j \\ge \\tau \\rbrace \\le e^{-\\theta ^{*}_{\\omega _n} \\cdot \\tau }$ where $\\omega _n \\equiv \\frac{R_n}{C_n}$ , and $\\theta ^{*}_{\\omega _n} $ is the solution of $E[e^{\\theta (\\frac{A(1)}{C_n} - \\omega _n)}] = 1.$ For $M/D$ , using similar approximation as for Corollary 1, $\\theta ^{*}_{\\omega _n} =2(\\omega _n-\\rho _n) / (X_n^{-1} Y_n^2).",
"$ Finding the solution for $(\\omega _1, \\dots , \\omega _N)$ from $2(\\omega _1\\_-\\rho _1) / (X_1^{-1} Y_1^2) = \\cdots = 2(\\omega _N-\\rho _N) / (X_N^{-1} Y_N^2).$ under the conditions $\\omega _n \\le \\rho _n$ and $\\sum _n \\omega _n \\le 1$ , the resultant $\\theta ^{*}_{\\omega _n}$ becomes $\\theta ^{*}$ .",
"Finally, (REF ) is obtained by directly applying the resultant $P\\lbrace {\\tilde{W}}_n^j > \\tau \\rbrace $ to (REF )."
],
[
"Proof of Corollary 4",
"If $p^j$ belongs to Class 2, we can start also from ().",
"Following the same argument of (REF ), we get $W^{j} &\\le & \\sum _{n=1}^{N} \\frac{\\sup _{0 \\le s \\le a^{j}}[A_n(s, a^{j}) - R_n\\cdot (a^{j}-s)]}{C_n}.$ Note that for Class 1, its customers arrive at $Y_1, 2Y_1, 3Y_1, \\dots $ , so, for any time period $[s, t]$ , $A_1(s,t) \\le r_1 \\cdot (t-s) + l_1$ , applying which to (REF ), together with letting $R_1=r_1$ , gives: $W^{j}&\\le & \\frac{\\sup _{0 \\le s \\le a^{j}}[A_2(s, a^{j}) - R_2\\cdot (a^{j}-s)]}{C_2} + \\frac{l_1}{C_1}.",
"\\nonumber $ Letting $R_2 = (1-\\frac{R_1}{C_1})C_2 = (1-\\rho _1) C_2$ , we have $W^{j} -Y_1&\\le & \\sup _{0 \\le s \\le a^{j}}[\\frac{A_2(s, a^{j})}{C_2} - (1-\\rho _1)\\cdot (a^{j}-s)] $ Following the same approach, a bound on $W^{j} -Y_1$ can be found as $P\\lbrace W^{j} -Y_1 \\ge \\tau \\rbrace \\le e^{-\\theta ^* \\tau }$ where $\\theta ^*$ is the maximum $\\theta $ satisfying $E[e^{\\theta A_2(1)/C_2 - (1-\\rho _1)}] \\le 1.$ Since Class 2 is $M/M$ , applying it to $A_2(1)/C_2$ gives $\\frac{\\lambda _2}{\\mu _2 - \\theta } - (1-\\rho _1) \\le 0$ from which, we can further get $\\theta ^{*} = \\mu _2 - \\lambda _2 / (1-\\rho _1)$ .",
"However, if $p^j$ belongs to Class 1, we can start from (REF ) and apply $A_1(s,t) \\le r_1 \\cdot (t-s) + l_1$ to it directly.",
"What we then get is: $D^{j} &\\le & \\sum _{n=1}^{2}\\frac{A_n(t^0, a^{j}_{+})}{C_n} + t^0-a^{j} \\\\ &\\le & \\frac{r_1\\cdot (a^{j}-t^0) + l_1}{C_1} + \\frac{A_2(t^0, a^{j}_{+})}{C_2} - (a^{j}- t^0) \\nonumber \\\\&=& \\frac{A_2(t^0, a^{j}_{+})}{C_2} - (1-\\rho _1) (a^{j}- t^0) + \\frac{l_1}{C_1} \\nonumber \\\\&\\le & \\sup _{0 \\le s \\le a^{j}}[\\frac{A_2(s, a^{j})}{C_2} - (1-\\rho _1)\\cdot (a^{j}-s)] + \\frac{l_1}{C_1}.",
"\\nonumber $ Since $p^j$ belongs to Class 1, the above then gives $W^{j}&\\le & \\sup _{0 \\le s \\le a^{j}}[\\frac{A_2(s, a^{j})}{C_2} - (1-\\rho _1)\\cdot (a^{j}-s)] .$ Comparing (REF ) with (REF ), one can see the only difference is the $Y_1$ term on the left hand side of (REF ).",
"Following the same approach, the waiting time distribution bound for Class 1 is obtained."
]
] | 1605.05753 |
[
[
"Polynomial Carleson operators along monomial curves in the plane"
],
[
"Abstract We prove $L^p$ bounds for partial polynomial Carleson operators along monomial curves $(t,t^m)$ in the plane $\\mathbb{R}^2$ with a phase polynomial consisting of a single monomial.",
"These operators are \"partial\" in the sense that we consider linearizing stopping-time functions that depend on only one of the two ambient variables.",
"A motivation for studying these partial operators is the curious feature that, despite their apparent limitations, for certain combinations of curve and phase, $L^2$ bounds for partial operators along curves imply the full strength of the $L^2$ bound for a one-dimensional Carleson operator, and for a quadratic Carleson operator.",
"Our methods, which can at present only treat certain combinations of curves and phases, in some cases adapt a $TT^*$ method to treat phases involving fractional monomials, and in other cases use a known vector-valued variant of the Carleson-Hunt theorem."
],
[
"Historical background.",
"In 1966, Carleson [2] proved an $L^2$ bound for the Carleson operator $ f(x)\\longmapsto \\sup _{N\\in \\mathbb {R}} \\Big | p.v.\\int _\\mathbb {R}f(x-t) e^{iN t} \\frac{dt}{t}\\Big |.$ This provided the key step in proving almost everywhere convergence of Fourier series of $L^2$ functions and thereby resolved a conjecture of Luzin.",
"The $L^p$ boundedness of the Carleson operator for $1<p<\\infty $ was then shown by Hunt [11], and further proofs of Carleson's theorem were later given by Fefferman [7] and Lacey and Thiele [14].",
"E. M. Stein suggested the following generalization: fix a natural number $d$ and consider the operator given by $f(x)\\longmapsto \\sup _{P}\\left|\\int _{\\mathbb {R}^n} f(x-y) e^{iP(y)} K(y) dy\\right|,$ where $K$ is an appropriately chosen Calderón-Zygmund kernel and the supremum runs over all real-valued polynomials $P$ of degree at most $d$ in $n$ variables.",
"Stein asked whether this polynomial Carleson operator is bounded from $L^p$ to $L^p$ for $1<p<\\infty $ .",
"Stein and Wainger [22] used a $TT^*$ argument and certain oscillatory integral estimates of van der Corput type to obtain $L^p$ bounds for a variant of the operator (REF ), where the polynomial $P$ is restricted to the set of polynomials of degree at most $d$ that vanish to at least second order at the origin (so in particular, have no linear term; of course constant terms may be disregarded).",
"In dimension $n=1$ , a positive answer to Stein's full question was provided by Lie [12], [13], who developed a sophisticated time-frequency approach.",
"In higher dimensions $n>1$ , boundedness of the full polynomial Carleson operator remains an open problem.",
"Pierce and Yung [16] have introduced a new aspect to the study of polynomial Carleson operators, by considering an operator that also features Radon-type behavior in the sense of integration along an appropriate hypersurface.",
"More precisely, they considered the operator $f(x,y) \\longmapsto \\sup _{P}\\left|\\int _{\\mathbb {R}^n} f(x-t,y-|t|^2) e^{iP(t)} K(t) dt\\right|,$ acting on functions $f$ on $\\mathbb {R}^n\\times \\mathbb {R}$ where $n\\ge 2$ , $K$ is a Calderón-Zygmund kernel, and the supremum runs over a suitable vector subspace of the space of all real-valued polynomials $P$ of degree at most $d$ in $n$ variables.",
"In particular, this allowable subspace requires that the polynomials considered should omit linear as well as certain types of quadratic terms.",
"The key result of [16] then proves $L^p$ , $1<p<\\infty $ , bounds for this operator, via a method of proof based on square functions, $TT^*$ techniques in the spirit of Stein and Wainger [22], and certain refined van der Corput estimates.",
"Notably, the method of [16] does not work in the planar case $n=1$ , which is the main subject of the present paper.",
"Our goal here is to prove bounds for a new class of polynomial Carleson operators along curves in the plane, and to demonstrate the curious feature that even partial results for these new operators along curves are in some sense as strong as Carleson's original theorem (and its variants) in the purely one-dimensional setting."
],
[
"Statements of main results.",
"Let $m,d$ be positive integers and $f$ a Schwartz function on $\\mathbb {R}^2$ .",
"For $N\\in \\mathbb {R}$ let $ H_N f(x,y)=H^{m,d}_N f(x,y)=p.v.",
"\\int \\limits _{\\mathbb {R}}f(x-t, y-t^m) e^{iNt^d} \\frac{dt}{t}.$ The natural goal, in the spirit of Carleson operators, is to prove that for all $1<p<\\infty $ , $\\left\\Vert \\sup _{N \\in \\mathbb {R}} |H_N f| \\right\\Vert _{L^p(dxdy)} \\le C \\Vert f\\Vert _{L^p(dxdy)}.$ This would be analogous to the results of Stein and Wainger [22] in the Radon-type context of (REF ).",
"We recall the useful strategy of linearization via a linearizing stopping-time function: we define for an arbitrary measurable function $N(x,y): \\mathbb {R}^2 \\mapsto \\mathbb {R}$ the operator $f \\mapsto H_{N(x,y)}f(x,y)$ .",
"Then proving $\\Vert H_{N(x,y)}f(x,y)\\Vert _{L^p(dxdy)} \\le C \\Vert f\\Vert _{L^p(dxdy)}$ with a constant $C$ independent of the choice of the function $N$ is equivalent to proving (REF ).",
"To prove this inequality appears to be out of reach of our current methods.",
"Recalling instead that a special case of [22] already shows that for any integer $d>1$ the operator $f(x) \\longmapsto p.v.",
"\\int \\limits _{\\mathbb {R}}f(x-t) e^{iN(x) t^d} \\frac{dt}{t}$ is bounded on $L^p(\\mathbb {R})$ for $1 < p < \\infty $ , we are motivated to consider the case when we twist the operator (REF ) with an additional Radon transform, while preserving the dependence of the linearizing function $N$ on one variable only.",
"Thus for an arbitrary measurable function $N:\\mathbb {R}\\rightarrow \\mathbb {R}$ , we define our main operators of interest: $A_N^{m,d}f(x,y):=H^{m,d}_{N(x)} f(x,y)$ and $B_N^{m,d} f(x,y):=H^{m,d}_{N(y)} f(x,y).$ Before turning to our main results, we briefly note that certain special cases of these operators may be treated immediately: namely, for $d \\ge 1$ the operators $A^{1,d}_N$ and $B^{1,d}_N$ are bounded on $L^p(\\mathbb {R}^2)$ for $1 < p < \\infty $ .",
"Indeed even the operator $\\sup _{N \\in \\mathbb {R}} |H^{1,d}_N f(x,y)|$ is bounded on $L^p(\\mathbb {R}^2)$ for $1 < p < \\infty $ .",
"This follows immediately by integrating Carleson's theorem for (REF ) (in the case $d = 1$ ), or the result of Stein and Wainger [22] for (REF ) (in the case $d > 1$ ), along the straight lines of slope 1 in $\\mathbb {R}^2,$ using Fubini's thoerem.",
"The remaining cases, with $m > 1$ , are highly nontrivial.",
"We formulate our main results as two theorems, which despite superficial similarities have quite different flavors, due to the differing symmetry groups of the involved operators (see Section REF ).",
"Our first main result can be stated as follows.",
"Theorem 1.1 Let $N:\\mathbb {R}\\rightarrow \\mathbb {R}$ be a measurable function and $d,m> 1, d\\ne m$ integers.",
"Then for $1<p<\\infty $ , $\\Big \\Vert A_N^{m,d} f\\Big \\Vert _{p} &\\le C \\Vert f\\Vert _p, \\\\\\Big \\Vert B_N^{m,d} f\\Big \\Vert _{p} &\\le C \\Vert f\\Vert _p,$ with the constant $0<C<\\infty $ depending only on $d,m,p$ and not on $N,f$ .",
"Note that uniformity of (REF ) in $N$ is tantamount to the estimate $\\left\\Vert \\sup _{N\\in \\mathbb {R}} \\Vert H_N f(x,y)\\Vert _{L^p(dy)} \\right\\Vert _{L^p(dx)}\\le C \\Vert f\\Vert _p.$ Similarly, () corresponds to $\\left\\Vert \\sup _{N\\in \\mathbb {R}} \\Vert H_N f(x,y)\\Vert _{L^p(dx)} \\right\\Vert _{L^p(dy)}\\le C \\Vert f\\Vert _p.$ Our proof of Theorem REF proceeds via van der Corput estimates, and does not depend on Carleson's theorem; this is in contrast to our second result, which we state as follows.",
"Theorem 1.2 Let $N:\\mathbb {R}\\rightarrow \\mathbb {R}$ be a measurable function.",
"Then for $1<p<\\infty $ , $\\Big \\Vert A_N^{m,1} f\\Big \\Vert _{p} &\\le C \\Vert f\\Vert _p, \\quad \\text{for any integer $m \\ge 3$,}\\\\\\Big \\Vert B_N^{m,m} f\\Big \\Vert _{p} &\\le C \\Vert f\\Vert _p, \\quad \\text{for any integer $m \\ge 2,$}$ with the constant $C$ depending only on $m,p$ and not on $N,f$ .",
"A novel feature of our proof of Theorem REF is that we combine Carleson's theorem with $TT^*$ estimates in the spirit of Stein and Wainger.",
"One surprising feature of our proof, compared to the original work [22] is that these $TT^*$ estimates can handle certain cases of phase polynomials with a linear term (c.f.",
"estimates (REF )–(REF )).",
"Remark 1.3 One is led to ask what happens to the remaining nontrivial ($m >1$ ) cases that are not covered by Theorems REF and REF , namely $A_N^{2,1}$ , $A^{m,m}_N$ and $B^{m,1}_N$ where $m > 1$ is an integer.",
"The key again lies in the symmetries of these operators: they are different from the symmetries of the operators in Theorems REF and REF , and this points to why our current proofs do not apply in these situations.",
"Despite these difficulties, at least the $L^2$ bounds for all these problematic cases still follow from known Carleson theorems via partial Fourier transform and Plancherel's theorem; see Section REF .",
"The full $L^p$ bounds remain an open problem in these cases."
],
[
"Consequences of bounding partial Carleson operators",
"We now turn to the surprising feature that $L^2$ bounds for partial operators along curves imply $L^2$ bounds for Carleson-type operators acting on functions of one variable.",
"Here we summarize several deductions of this kind; proofs are given in Section .",
"First, $L^2$ bounds for certain operators $A^{m,1}$ and $B^{m,m}$ are in some sense equivalent to an $L^2$ bound for Carleson's operator.",
"More precisely, for any integer $m \\ge 1$ , the $L^2$ boundedness of $A^{m,1}_N$ implies the $L^2$ boundedness for the one-dimensional Carleson operator (REF ), by a Plancherel argument (see Section 6.1).",
"In the other direction, we use the boundedness of the maximal truncated Carleson operator (REF ) (itself dominated by the Carleson operator according to the inequality (REF )) to prove the $L^2$ boundedness of $A^{m,1}$ for $m \\ge 3$ in Theorem REF , while the $L^2$ bound for $A^{2,1}$ may be deduced from Carleson's theorem (see Section REF ).",
"Similarly, for any odd integer $m \\ge 1$ , the $L^2$ boundedness of $B^{m,m}_N$ implies an $L^2$ bound for the one-dimensional Carleson operator (see Section REF ), while in the other direction we use the maximal truncated Carleson operator to prove Theorem REF .",
"Of course, the most natural challenge in the setting of Carleson operators along curves in the plane is the quadratic Carleson operator along the parabola defined by ${C}^{\\text{par}} f(x,y)=\\sup _{N\\in \\mathbb {R}^2}\\left|H_{N}^{\\text{par}} f(x,y)\\right|,$ where for $f$ a Schwartz function on $\\mathbb {R}^2$ , $ H_{N}^{\\text{par}} f(x,y)=p.v.\\int _{\\mathbb {R}} f(x-t,y-t^2) e^{iN_1t+iN_2t^2} \\frac{dt}{t}.",
"$ This operator combines all the features that have proved troublesome in the study of (REF ) in [16]: apart from acting on functions in the plane, the phase consists entirely of the problematic linear and quadratic terms.",
"Assuming that $N_1,N_2:\\mathbb {R}\\rightarrow \\mathbb {R}$ are arbitrary measurable functions depending only on $x$ , observe that for $N_1=0$ this gives our operator $A^{2,2}$ (which our present arguments cannot treat) and for $N_2=0$ it gives our problematic operator $A^{2,1}$ (which again our present arguments cannot treat).",
"So we are quite far from knowing how to bound (REF ).",
"But in the spirit of studying partial versions of Carleson operators, we point out that even a partial estimate for $H_N^{\\text{par}}$ of the form $\\left\\Vert \\sup _{N\\in \\mathbb {R}^2} \\Vert H_N^{\\text{par}} f\\Vert _{L^2(dy)} \\right\\Vert _{L^2(dx)}\\le C\\Vert f\\Vert _2,$ would suffice to imply an analogue over $\\mathbb {R}$ of Lie's $L^2$ result on the quadratic Carleson operator [12]; see Section REF for details.",
"These considerations indicate the interest in pursuing the partial Carleson operators we consider."
],
[
"Symmetries of our operators",
"To make precise the differences between Theorems REF and REF , we now characterize symmetries of the operators $A_N^{m,d}$ and $B_N^{m,d}$ as $m$ and $d$ vary.",
"First there is an anisotropic dilation symmetry.",
"If we denote $D_\\lambda f(x,y)=f(\\lambda x,\\lambda ^m y)$ for $\\lambda >0$ , then $ D^{-1}_{\\lambda } H^{m,d}_N D_\\lambda = H^{m,d}_{\\lambda ^{-d} N}.",
"$ Second, due to the convolution structure, $H_N^{m,d}$ commutes with translations of the plane, for any $m,d$ .",
"Third, the operators in Theorem REF additionally have certain modulation symmetries.",
"Let $M_{\\xi ,\\zeta } f(x,y)=e^{ix\\xi +iy\\zeta } f(x,y)$ for $\\xi ,\\zeta \\in \\mathbb {R}$ .",
"Then if $d=1$ , we have $ M_{\\xi ,0}^{-1} A^{m,1}_N M_{\\xi ,0} = A^{m,1}_{N-\\xi }$ for all $\\xi \\in \\mathbb {R}$ .",
"Similarly if $d=m$ , we have $ M_{0,\\zeta }^{-1} B^{m,m}_N M_{0,\\zeta } = B^{m,m}_{N-\\zeta }$ for all $\\zeta \\in \\mathbb {R}$ .",
"Simultaneous translation and modulation invariance is a characteristic property of the Carleson operator.",
"Hence we are led to use Carleson's theorem in parts of the proof of Theorem REF .",
"Finally, we remark briefly that for $A^{2, 1}_N$ , the modulation symmetries are more involved.",
"The problem is that in addition to the modulation symmetry (REF ), it also has a certain quadratic modulation symmetry.",
"Let $Q_{b}f(x, y)=e^{ibx^2} f(x, y).$ Then $Q^{-1}_b M^{-1}_{0, b}A^{2,1}_N M_{0, b} Q_b=A^{2, 1}_{N-2bx}.$ Recall that for the operator $A^{2, 1}_N$ , the linearizing function $N$ depends on the variable $x$ .",
"Thus, by $N-2bx$ we mean the function $x\\mapsto N(x)-2bx$ , also only depending on $x$ .",
"Moreover, notice that in (REF ), the linear modulation acts on the $y$ variable, while the quadratic modulation acts on the $x$ variable.",
"Hence there is a certain “twist” in this modulation symmetry."
],
[
"Method of proof: Theorem ",
"We now sketch the proof of Theorem REF .",
"The strategy follows broadly that of Stein and Wainger, but the means of obtaining the key estimates is necessarily different.",
"More precisely, we proceed by splitting the integral defining $A_N^{m,d}$ or $B_N^{m,d}$ into two parts, according to the size of the phase $Nt^d$ : for $Nt^d$ sufficiently small, we compare the resulting operator to a maximal truncated Hilbert transform along a curve, and for $Nt^d$ large, we use $TT^*$ and van der Corput estimates to handle the operator that arises.",
"It is in the treatment of this latter operator where we must assume the stopping time depends on one variable only, so that we may perform a Fourier transform in the free variable, along which the linearizing function is constant.",
"This idea goes back to Coifman and El Kohen, who used it in the context of Hilbert transforms along vector fields (see the discussion in Bateman and Thiele [1]).",
"Another important ingredient is a certain refinement of Theorem 1 of [22].",
"The main novelty is that our core estimate, which we now record, allows us to consider phases with monomials of fractional exponents.",
"Lemma 2.1 Fix real numbers $\\alpha ,\\beta >0$ , $\\alpha \\ne \\beta $ , $\\alpha ,\\beta \\ne 1$ .",
"Let $\\psi $ be smooth and supported on $[1,2]$ .",
"For $\\lambda =(\\lambda _1, \\lambda _2)\\in \\mathbb {R}^2$ and $t > 0$ , let $\\Phi ^\\lambda (t)=e^{i\\lambda _1 t^{\\alpha } + i \\lambda _2 t^{\\beta }} \\psi (t)/t,$ and set $\\Phi ^\\lambda (-t)=0$ .",
"For $a >0$ , let $\\Phi _a^\\lambda (t)= a^{-1} \\Phi ^\\lambda (t/a).$ Let $|\\lambda | = |\\lambda _1| + |\\lambda _2|$ .",
"Then there exists $\\gamma _0>0$ such that for all $r \\ge 1$ and all $F \\in L^2(\\mathbb {R})$ , $\\left\\Vert \\sup _{a>0, |\\lambda |\\ge r} |F*\\Phi _a^\\lambda | \\right\\Vert _{L^2(dx)} \\lesssim r^{-\\gamma _0} \\Vert F\\Vert _{L^2(dx)}.$ Remark 2.2 We note that as a byproduct of the proof of Lemma REF , $\\gamma _0$ can be chosen to be independent of $\\alpha ,\\beta $ .",
"Remark 2.3 For $\\alpha ,\\beta \\in \\mathbb {N}$ this is merely a special case of Stein and Wainger's Theorem 1 in [22], but to prove Lemma REF in full generality requires estimates of a very different flavor.",
"See also work of the first author [10] for a similar result regarding a phase comprised of a single fractional monomial.",
"Fractional exponents appear naturally during the analysis of the operators $B^{m,d}$ via a change of variables $t^m\\rightarrow t$ (for instance, see (REF ) and (REF )).",
"(In addition, Theorems REF and REF could be somewhat generalized to non-integral $m,d$ , but we have chosen the integer setting for our main results, to avoid unnecessary complications.)",
"The key contrast of our proof of Lemma REF with the corresponding result in Stein and Wainger [22] appears in the proof of Lemma REF .",
"The strategy is to linearize the operator $F \\mapsto \\sup _{a>0, |\\lambda |\\ge r} |F*\\Phi _a^\\lambda |$ using stopping-times for $a,\\lambda $ , and to bound an oscillatory integral by showing that for all but a small exceptional region of the integral, the phase has a large derivative of some order.",
"Our proof enables us to make the exceptional region independent of the precise stopping-time $\\lambda $ , thus obviating the need for the small-set maximal functions that appear in [22]; at the cost of restricting our attention to phases with only two monomials, we are also able to handle fractional powers."
],
[
"Method of proof: Theorem ",
"Next, we sketch the proof of Theorem REF .",
"To analyze $A^{m,1}_N$ , where $m \\ge 3$ is an integer, we first decompose the operator as $A^{m,1}_N = \\sum _{k \\in \\mathbb {Z}} A^{m,1}_N \\circ P_k,$ where $P_k$ is a Littlewood-Paley projection onto frequency $\\sim 2^k$ in the $y$ -variable.",
"In view of the modulation invariance (REF ) in the $x$ -variable, this is the only viable Littlewood-Paley decomposition we can use for the operator $A^{m,1}$ ; a Littlewood-Paley decomposition in the $x$ -variable is doomed to fail.",
"We also note that the Littlewood-Paley projection in the $y$ -variable commutes with $A^{m,1}_N$ , since the stopping time $N$ in the operator $A^{m,1}_N$ depends only on $x$ but not on $y$ .",
"Now to analyze each Littlewood-Paley piece of $A^{m,1}_N$ , we decompose the integral $A^{m,1}_N \\circ P_k f(x,y) = \\int _{\\mathbb {R}} (P_k f)(x-t,y-t^{m}) e^{iN(x) t} \\frac{dt}{t}$ into two parts, where $t$ is small or large compared to the frequency $2^k$ .",
"For $t$ small, we compare the resulting integral to a maximally truncated Carleson operator in the $x$ -variable; this is natural in view of the remarks in Section REF .",
"The error will be given by a strong maximal function, since $P_k f$ is localized in frequency in the $y$ -variable.",
"For $t$ large, we need to use a van der Corput estimate: again we take advantage of the fact that the stopping time $N$ of $A^{m,1}_N$ depends only on $x$ , to take a partial Fourier transform in the $y$ -variable.",
"In order to reassemble the various Littlewood-Paley pieces, the main ingredient is a vector-valued estimate for the maximally truncated Carleson operator (Theorem REF ).",
"A similar strategy works for $B^{m,m}_N$ for $m \\ge 2$ an integer.",
"There is, however, an interesting distinction depending on whether $m$ is odd or even: when $m$ is odd, we need to use the maximally truncated Carleson operator in the $y$ -variable, whereas when $m$ is even, the component of the operator that would correspond to the maximally truncated Carleson operator magically vanishes.",
"(See equation (), and the discussion immediately thereafter.)",
"Roughly speaking, our proof of Theorem REF works because the linearizing function depends on the same variable in which the modulation invariance occurs, so the other variable is at our disposal to use Plancherel's theorem and localize in frequency via Littlewood-Paley decomposition.",
"Essential parts of this proof fail in the remaining cases $A^{2,1}$ , $A^{m,m}$ and $B^{m,1}$ , where $m>1$ .",
"For $A^{m,m}$ , the linearizing function varies with $x$ , so we would like to use Plancherel's theorem in $y$ and localize in the $y$ frequency.",
"However, the modulation invariance in (REF ) causes translation invariance in the $y$ frequency so that any attempt at doing a Littlewood-Paley decomposition is doomed from the start.",
"Similar behavior occurs for $B^{m, 1}$ ."
],
[
"Notation",
"The notation $A\\lesssim B$ always means $A\\le C\\cdot B$ with $0<C<\\infty $ depending only on $m,d$ and the function $\\psi $ chosen below (and within the proof of Lemma REF , on $\\alpha ,\\beta $ ).",
"Similarly, $A\\approx B$ means $C_1 A\\le B\\le C_2 A$ with $0<C_1\\le C_2<\\infty $ and the same dependence.",
"We use the Fourier transform $\\hat{f}(\\xi ) = \\int _{\\mathbb {R}} f(x) e^{-i \\xi x} dx$ with inverse $\\check{g} (x) =(2\\pi )^{-1} \\int _\\mathbb {R}g(\\xi ) e^{i x \\xi } d\\xi $ and Plancherel identity $\\Vert f\\Vert _2 = (2\\pi )^{-1/2} \\Vert \\hat{f} \\Vert _2$ ."
],
[
"Littlewood-Paley decomposition",
"Once and for all we fix a smooth function $\\psi :\\mathbb {R}\\rightarrow \\mathbb {R}$ supported on $\\lbrace t:1/2~\\le ~|t|~\\le ~2\\rbrace $ such that $0\\le \\psi (t)\\le 1$ and $\\sum _{k\\in \\mathbb {Z}} \\psi _k(t)=1$ for all $t\\ne 0$ , where $\\psi _k(t)=\\psi (2^{-k} t)$ .",
"Define the associated Littlewood-Paley projection of a function $F$ on $\\mathbb {R}$ by $F_k(w) = P_k F(w)=\\int \\limits _{\\mathbb {R}}F(u) \\check{\\psi }_k(w-u) du,$ where $\\check{\\psi }_k$ denotes the inverse Fourier transform of the function $\\psi _k$ .",
"The standard Littlewood-Paley estimates apply, in the form $ \\Vert F\\Vert _p \\lesssim \\Big \\Vert \\left(\\sum _{k} |P_k F|^2 \\right)^{1/2} \\Big \\Vert _{p} \\lesssim \\Vert F\\Vert _{p}.$ We will apply this in the $x$ -variable or $y$ -variable of $f(x,y)$ , depending which is free."
],
[
"Vector-valued inequalities",
"In this section we collect several vector-valued estimates that will play important roles in our work.",
"Define the maximally truncated Carleson operator by ${C}^* F(x) = \\sup _{N\\in \\mathbb {R},\\varepsilon >0} \\Big | p.v.\\int _{|t|\\le \\varepsilon } F(x-t) e^{iNt} \\frac{dt}{t}\\Big |.$ Note that this operator is usually studied with the inequality $|t|\\le \\varepsilon $ being reversed; we may of course reduce to that case by subtracting the Carleson operator from ${C}^*$ .",
"Theorem A For $1<p<\\infty ,$ $\\Big \\Vert \\Big (\\sum _{k\\in \\mathbb {Z}} |{C}^* F_k|^2\\Big )^{1/2}\\Big \\Vert _p \\lesssim \\Big \\Vert \\Big (\\sum _{k\\in \\mathbb {Z}} |F_k|^2\\Big )^{1/2}\\Big \\Vert _p,$ with a constant depending only on $p$ .",
"We assemble the necessary results to verify Theorem REF in a brief appendix (Section REF ).",
"Next, let ${M}$ be the maximal operator of Radon-type along the curve $(t,t^m)$ : ${M} f(x,y)=\\sup _{r>0} \\frac{1}{2r} \\int \\limits _{-r}^{r} |f(x-t,y-t^m)| dt.$ This is known to be a bounded operator of $L^p$ for $1<p\\le \\infty $ (e.g.",
"by a small modification of the proof in the case of the parabola $(t,t^2)$ , [19]).",
"We require a vector-valued inequality for $f_k := P_k f$ , with $P_k$ acting on either the $x$ -variable or $y$ -variable (to be specified later): Theorem B For $1<p<\\infty $ we have $\\Big \\Vert \\Big (\\sum _{k\\in \\mathbb {Z}} |{M} f_k|^2\\Big )^{1/2}\\Big \\Vert _p \\lesssim \\Big \\Vert \\Big (\\sum _{k\\in \\mathbb {Z}} |f_k|^2\\Big )^{1/2}\\Big \\Vert _p,$ with a constant depending only on $p$ .",
"This result is stated in [17], as a consequence obtainable from a more general theory.",
"For completeness, we offer a brief, self-contained proof for our special case in an appendix (Section REF ); we thank E. M. Stein for sharing with us this method of proof, which appears in a significantly more general form in the preprint [15].",
"Finally, we will need two one-variable Hardy-Littlewood maximal functions in the plane, denoted by $M_1$ and $M_2$ .",
"Indeed, they will act on the first and second variable respectively: $M_1 f(x,y)=\\sup _{r>0} \\frac{1}{2r} \\int _{-r}^r |f(x-u,y)| du \\\\M_2 f(x,y)=\\sup _{r>0} \\frac{1}{2r} \\int _{-r}^r |f(x,y-t)| dt .$ They are bounded on $L^p(\\mathbb {R}^2)$ for all $1 < p < \\infty $ , and satisfy the following vector-valued inequality, which follows easily by integrating a corresponding result of Fefferman and Stein: Theorem C For $1<p<\\infty $ and $i = 1,2$ , we have $\\Big \\Vert \\Big (\\sum _{k\\in \\mathbb {Z}} |M_i f_k|^2\\Big )^{1/2}\\Big \\Vert _p \\lesssim \\Big \\Vert \\Big (\\sum _{k\\in \\mathbb {Z}} |f_k|^2\\Big )^{1/2}\\Big \\Vert _p,$ with a constant depending only on $p$ .",
"See e.g.",
"[19] for further details."
],
[
"The asymmetric case: Theorem ",
"First we prove Theorem REF , assuming Lemma REF ; then in Section REF we prove the lemma.",
"For convenience, we define the auxiliary variable $z=z(x,y)$ to be understood as indicating either $z(x,y)=x$ or $z(x,y)=y$ , so that $N(z)$ can mean either $N(x)$ or $N(y)$ .",
"To simplify notations, we also define $Tf(x,y)=H^{m,d}_{N(z)} f(x,y),$ with $m,d$ satisfying the conditions of Theorem REF .",
"With $\\psi _\\ell $ as defined in Section REF , define for each $\\ell \\in \\mathbb {Z}$ $T_\\ell f(x,y)=\\int _{\\mathbb {R}} f(x-t,y-t^m) e^{iN(z)t^d} \\psi _\\ell (t) \\frac{dt}{t}.$ Let $n:\\mathbb {R}\\rightarrow \\mathbb {Z}$ be such that for all $z \\in \\mathbb {R}$ , $2^{-n(z)d}\\le |N(z)|< 2^{-(n(z)-1)d}.$ Then we decompose $T=T^{(1)}+T^{(2)}$ with $T^{(1)}f(x,y)=\\sum \\limits _{\\ell \\le n(z)} T_\\ell f(x,y)$ and $T^{(2)}f(x,y)=\\sum \\limits _{\\ell >0} T_{n(z)+\\ell } f(x,y).$ The motivation for this decomposition is that when $\\ell \\le n(z)$ , $\\psi _\\ell (t)$ localizes to $|t| \\le 2^{\\ell +1} \\le 2^{n(z)+1}$ and the exponential factor $e^{iN(z)t^d}$ is well approximated by 1.",
"Consequently we write $T^{(1)}f(x,y)$ as $\\sum _{\\ell \\le n(z)} \\int \\limits _{\\mathbb {R}}f(x-t,y-t^m) (e^{iN(z)t^d}-1) \\psi _\\ell (t) \\frac{dt}{t} + \\sum _{\\ell \\le n(z)}\\int \\limits _{\\mathbb {R}}f(x-t,y-t^m) \\psi _\\ell (t) \\frac{dt}{t}.$ We may estimate the absolute value of the first summand brutally by applying (REF ): $\\lesssim \\sum _{\\ell \\le n(z)} \\int \\limits _{\\mathbb {R}}|f(x-t,y-t^m)|\\cdot |N(z)t^{d-1} \\psi _\\ell (t)| dt\\lesssim \\frac{1}{2^{n(z)+2}} \\int \\limits _{-2^{n(z)+1}}^{2^{n(z)+1}} |f(x-t,y-t^m)| dt.$ The right hand side is bounded by ${M}f(x,y)$ , where ${M}$ denotes the maximal operator along $(t,t^m)$ defined in (REF ).",
"The second summand in (REF ) is bounded in absolute value by the maximal truncated Hilbert transform along the curve $(t,t^m)$ , defined by $\\mathcal {H}^* f(x,y)=\\sup _{\\varepsilon , R > 0}\\Big |\\int _{\\varepsilon < |t| < R} f(x-t,y-t^m) \\frac{dt}{t}\\Big |,$ plus an error term bounded by ${M}f(x,y)$ (which arises at the endpoint when passing from smooth bump functions to a sharp truncation).",
"Thus in total we have obtained the pointwise estimate $|T^{(1)}f|\\lesssim {M}f + \\mathcal {H}^* f.$ Since both $\\mathcal {H}^*$ , ${M}$ are known to be bounded in $L^p$ , $1<p<\\infty $ (for example, by slight modifications of Stein and Wainger's work for $(t,t^2)$ in [21]), we may conclude that $\\Vert T^{(1)} f\\Vert _p\\lesssim \\Vert f\\Vert _p$ for all $1<p<\\infty $ .",
"It remains to show the same for $T^{(2)}$ .",
"Let $S_\\ell f(x,y)=T_{n(z)+\\ell } f(x,y);$ we claim that it suffices to prove there exists some $\\gamma _0>0$ such that for all $\\ell > 0$ , $\\Vert S_\\ell f\\Vert _2\\lesssim 2^{-\\gamma _0 \\ell } \\Vert f\\Vert _2.$ Indeed, the triangle inequality implies the pointwise estimate $|S_\\ell f|\\lesssim {M}f$ , so that we immediately obtain $\\Vert S_\\ell f\\Vert _p\\lesssim \\Vert f\\Vert _p$ for all $1<p<\\infty $ ; by interpolation with (REF ) we then obtain for any $1<p<\\infty $ there exists some $\\gamma _p>0$ such that $\\Vert S_\\ell f\\Vert _p\\lesssim 2^{-\\gamma _p \\ell } \\Vert f\\Vert _p.$ Finally, summing over $\\ell \\ge 0$ gives $\\Vert T^{(2)}f\\Vert _p\\lesssim \\Vert f\\Vert _p.$ All that remains is to prove (REF ); we proceed by distinguishing two cases."
],
[
"The $B_N^{m,d}$ operators.",
"Here we consider the case $z(x,y)=y$ .",
"Applying Plancherel's theorem in the free $x$ -variable, we obtain $\\Vert S_{\\ell } f(x,y)\\Vert _{L^2(dx)} = (2\\pi )^{-1/2} \\Big \\Vert \\int \\limits _{\\mathbb {R}}g_\\xi (y-t^m) e^{iN(y)t^d - i\\xi t} \\psi _{n(y)+\\ell }(t) \\frac{dt}{t}\\Big \\Vert _{L^2(d\\xi )}, $ where $g_\\xi (y)=\\int _\\mathbb {R}e^{-i\\xi x} f(x,y) dx.$ Therefore to prove (REF ) it will suffice to prove a bound of the form $ \\Big \\Vert \\int \\limits _{\\mathbb {R}}F(y-t^m) e^{iN(y)t^d-i\\xi t} \\psi _{n(y)+\\ell }(t)\\frac{dt}{t} \\Big \\Vert _{L^2(dy)} \\lesssim 2^{-\\gamma _0 \\ell } \\Vert F\\Vert _2, $ uniformly in $\\xi \\in \\mathbb {R}$ , for all single variable functions $F$ .",
"Recall that the cutoff function $\\psi _{n(y)+\\ell }$ has supports both in the positive half line and in the negative half line.",
"Accordingly let us split the integration over $t$ into a positive and a negative part.",
"We consider the positive part; the negative component is treated in an entirely analogous way.",
"Changing variables $t^m \\mapsto t$ , we see that it suffices to show there exists some $\\gamma _0>0$ such that for all $\\ell >0$ and all $F \\in L^2(\\mathbb {R})$ , $\\Big \\Vert \\int \\limits _{0}^\\infty F(y-t) e^{iN(y)t^{d/m}-i\\xi t^{1/m}} \\psi _{n(y)+\\ell }(t^{1/m}) \\frac{dt}{t}\\Big \\Vert _{L^2(dy)}\\lesssim 2^{-\\gamma _0 \\ell } \\Vert F\\Vert _2,$ uniformly in $\\xi $ .",
"In fact (REF ) is an immediate consequence of the key Lemma REF , with $\\alpha =d/m$ , $\\beta =1/m$ .",
"To see this, we first re-write $N(y) = 2^{-n(y)d +r(y)d}$ with $0<r(y)<1$ for all $y$ .",
"Then for $a \\in \\mathbb {R}$ and $\\lambda \\in \\mathbb {R}^2$ , we define $\\Phi _a^\\lambda := a^{-1}\\Phi ^\\lambda (t/a)$ , where $\\Phi ^\\lambda (t) := e^{i \\lambda _1 t^{\\alpha } + i \\lambda _2 t^{\\beta }} \\psi (t^{1/m}) t^{-1} \\quad \\text{for $t > 0$},$ and $\\Phi ^\\lambda (t) = 0$ for $t \\le 0$ .",
"One then observes that the integral on the left hand side of (REF ) is equal to $F* \\Phi _{a}^\\lambda (y)$ , with parameters $ a=2^{(n(y) + \\ell )m}, \\qquad \\lambda _1 = 2^{\\ell d + r(y) d}, \\qquad \\lambda _2= -\\xi 2^{n(y)+\\ell }.$ Then (recalling $\\ell >0$ , $0<r(y)<1$ ), we have $ |\\lambda | = |\\lambda _1| + |\\lambda _2| \\ge 2^{\\ell d + r(y) d} \\ge 2^{\\ell d}, $ and we see from Lemma REF that for any fixed $\\ell >0$ , $ \\Big \\Vert \\int \\limits _{0}^\\infty F(y-t) e^{iN(y)t^{d/m}-i\\xi t^{1/m}} \\psi _{n(y)+\\ell }(t^{1/m}) \\frac{dt}{t}\\Big \\Vert _{L^2(dy)}\\lesssim \\Vert \\sup _{a>0, |\\lambda | \\ge 2^{\\ell d}} |F * \\Phi _a^\\lambda | \\Vert _2 \\lesssim 2^{-\\gamma _0 \\ell } \\Vert F\\Vert _2, $ as desired.",
"This proves (REF ) and hence (REF ) in this case."
],
[
"The $A_N^{m,d}$ operators.",
"Here we treat the case $z(x,y)=x$ .",
"Applying Plancherel's theorem in the free $y$ -variable, we obtain $\\Big \\Vert S_\\ell f(x,y)\\Big \\Vert _{L^2(dy)}=(2\\pi )^{-1/2}\\Big \\Vert \\int \\limits _{\\mathbb {R}} g_\\eta (x-t) e^{iN(x)t^d-i\\eta t^m} \\psi _{n(x)+\\ell }(t) \\frac{dt}{t}\\Big \\Vert _{L^2(d\\eta )}$ with $g_\\eta (x)=\\int _\\mathbb {R}e^{-i\\eta y} f(x,y) dy.$ By Plancherel's theorem it suffices to show that there exists $\\gamma _0>0$ such that for each $\\ell >0$ , $\\Big \\Vert \\int \\limits _{\\mathbb {R}} F(x-t) e^{iN(x)t^d-i\\eta t^m} \\psi _{n(x)+\\ell }(t) \\frac{dt}{t}\\Big \\Vert _{L^2(dx)}\\lesssim 2^{-\\gamma _0 \\ell } \\Vert F\\Vert _2$ uniformly in $\\eta $ .",
"Now it is clear that we may proceed similarly to (REF ), and deduce this bound from Lemma REF with $\\alpha =d$ , $\\beta =m$ ."
],
[
"Proof of Lemma ",
"In order to complete the proof of Theorem REF , it remains to prove Lemma REF .",
"Due to a minor technical issue we will assume the pair $\\lbrace \\alpha ,\\beta \\rbrace \\ne \\lbrace 2,3\\rbrace $ in the proof.",
"However, this case is of course already covered by Stein and Wainger's work [22].",
"In fact it suffices to prove there exists $\\gamma _0$ such that for all $r \\ge 1$ , $\\Vert \\sup _{a>0, \\; r\\le |\\lambda |\\le 2r} |F*\\Phi _a^\\lambda | \\Vert _2 \\lesssim r^{-\\gamma _0} \\Vert F\\Vert _2.$ For with this result in hand, we immediately obtain the desired result, $ \\Big \\Vert \\sup _{a>0, \\; |\\lambda |\\ge r} |F*\\Phi _a^\\lambda | \\Big \\Vert _2 \\le \\sum _{k=0}^\\infty \\Big \\Vert \\sup _{a>0,\\; 2^{k}r \\le |\\lambda |\\le 2^{k+1} r } |F*\\Phi _a^\\lambda | \\Big \\Vert _2 \\lesssim r^{-\\gamma _0} \\Vert F\\Vert _2.",
"$ We proceed by linearizing the supremum.",
"For measurable functions $a:\\mathbb {R}\\rightarrow (0,\\infty )$ , $\\lambda :\\mathbb {R}\\rightarrow \\mathbb {R}^2$ with $r\\le |\\lambda (u)|\\le 2r$ for all $u\\in \\mathbb {R}$ , we define an operator $\\Lambda :L^2(\\mathbb {R})\\rightarrow L^2(\\mathbb {R})$ by $ \\Lambda F(u)=F*\\Phi ^{\\lambda (u)}_{a(u)}(u)=\\int _\\mathbb {R}F(t) \\Phi _{a(u)}^{\\lambda (u)} (u-t) dt.",
"$ The bound (REF ) will follow from proving $\\Vert \\Lambda \\Vert _{2\\rightarrow 2}\\lesssim r^{-\\gamma _0}$ for some $\\gamma _0>0$ with the implicit constant independent of $a,\\lambda $ .",
"Since $\\Vert \\Lambda \\Vert _{2\\rightarrow 2}=\\Vert \\Lambda \\Lambda ^*\\Vert _{2\\rightarrow 2}^{1/2},$ we will in fact prove $\\Vert \\Lambda \\Lambda ^*\\Vert _{2\\rightarrow 2}\\lesssim r^{-2\\gamma _0}.$ We calculate $\\Lambda \\Lambda ^* F(u)&= \\int _\\mathbb {R}F(s) (\\Phi ^\\nu _{a_1}*\\tilde{\\Phi }^\\mu _{a_2}) (u-s) ds,$ with $\\tilde{\\Phi }(u):=\\overline{\\Phi (-u)}$ and $\\nu =\\lambda (u), \\mu =\\lambda (s), a_1=a(u), a_2=a(s)$ .",
"Note that by rescaling we may write $ (\\Phi ^\\nu _{a_1} * \\tilde{\\Phi }^\\mu _{a_2})(s)= a_2^{-1} (\\Phi ^\\nu _{a_1/a_2}*\\tilde{\\Phi }^\\mu _1)(a_2^{-1}s)=a_1^{-1} (\\Phi ^\\nu _{1}*\\tilde{\\Phi }^\\mu _{a_2/a_1})(a_1^{-1}s).$ Thus we will deduce (REF ) from applying the following bounds, which are the heart of the proof: Lemma 4.1 There exists $\\gamma _1>0$ such that for $0< h\\le 1$ , $r\\le |\\nu |, |\\lambda |\\le 2r$ we have $|(\\Phi ^\\nu _h * \\tilde{\\Phi }^\\mu _1)(s)| \\lesssim r^{-\\gamma _1} \\mathbf {1}_{\\lbrace |s|\\le 4\\rbrace }(s) + \\mathbf {1}_{\\lbrace |s|\\le r^{-\\gamma _1}\\rbrace }(s), \\\\|(\\Phi ^\\nu _1 * \\tilde{\\Phi }^\\mu _h)(s)| \\lesssim r^{-\\gamma _1} \\mathbf {1}_{\\lbrace |s|\\le 4\\rbrace }(s) + \\mathbf {1}_{\\lbrace |s|\\le r^{-\\gamma _1}\\rbrace }(s).$ Remark 4.2 Note that the exceptional sets in (REF ), () do not depend on $\\nu ,\\mu $ .",
"This is in contrast to [22].",
"As a consequence we do not require Stein and Wainger's small set maximal function [22].",
"We first proceed with the proof of Lemma REF , and then prove Lemma REF in Section REF .",
"Applying (REF ) and () appropriately (depending on whether $a_1 \\ge a_2$ or $a_1 \\le a_2$ ), we deduce $|(\\Phi ^\\nu _{a_1} * \\tilde{\\Phi }^\\mu _{a_2})(s)| \\lesssim r^{-\\gamma _1} \\sum _{k=1}^2 \\left(a_k^{-1} \\mathbf {1}_{\\lbrace |s|\\le 4 a_k\\rbrace }(s) + (a_k r^{-\\gamma _1})^{-1}\\mathbf {1}_{\\lbrace |s|\\le r^{-\\gamma _1} a_k\\rbrace }(s)\\right).$ Thus for any $G \\in L^2$ we may compute $|\\langle \\Lambda \\Lambda ^* F,G\\rangle | = \\Big | \\int _\\mathbb {R}\\int _\\mathbb {R}(\\Phi ^\\nu _{a_1} * \\tilde{\\Phi }^\\mu _{a_2})(u-s) F(s) \\overline{G(u)} ds du\\Big |\\\\ \\lesssim r^{-\\gamma _1} \\left( \\int _\\mathbb {R}MF(u) |G(u)| du + \\int _\\mathbb {R}|F(s)| MG(s) ds \\right),$ in which $M$ denotes the standard one-variable Hardy-Littlewood maximal function.",
"(Here the important point is that we have integrated first in whichever variable was independent of the stopping-time $a_k$ , for the two terms $k=1,2$ .)",
"Via the Cauchy-Schwarz inequality and the boundedness of $M$ on $L^2$ , we obtain $ |\\langle \\Lambda \\Lambda ^* F,G\\rangle |\\lesssim r^{-\\gamma _1} \\Vert F\\Vert _2 \\Vert G\\Vert _2.",
"$ This completes the proof of (REF ) with $\\gamma _0=\\gamma _1/2$ ."
],
[
"Proof of Lemma ",
"We will only prove (REF ), as () follows by symmetry.",
"By definition, $(\\Phi ^\\nu _h * \\tilde{\\Phi }^\\mu _1)(s)&= \\int _\\mathbb {R}e^{i\\nu _1 t^\\alpha +i\\nu _2 t^\\beta -i\\mu _1(ht-s)^\\alpha -i\\mu _2 (ht-s)^\\beta } \\frac{\\psi (t)}{t} \\frac{\\overline{\\psi (ht-s)}}{ht-s} dt.$ First notice that the support of $\\Phi ^\\nu _h*\\tilde{\\Phi }^\\mu _1 (s)$ is contained in $\\lbrace s:|s|\\le 4\\rbrace $ .",
"In order to apply van der Corput estimates, we need to analyze when the phase function $ Q(t,s)=\\nu _1 t^\\alpha + \\nu _2 t^\\beta - \\mu _1(ht-s)^\\alpha - \\mu _2(ht-s)^\\beta $ has a large derivative of some order.",
"Here we recall that $\\nu _i=\\lambda _i(u)$ are fixed with respect to $t,s$ (the relevant variables of integration in (REF )), and that $ r \\le |\\nu _1| + |\\nu _2| \\le 2 r$ .",
"On the other hand, $\\mu _i = \\lambda _i(s)$ depends on $s$ (in an unknown way), and thus our strategy is to make our argument independent of $\\mu _1,\\mu _2$ .",
"Case 1: Suppose that $0<h\\le h_0$ , where $0<h_0<1$ is to be determined later, depending on $r, \\alpha , \\beta $ ; this is the easier case.",
"Let $0<\\varepsilon _1<1$ be small and fixed.",
"Within the support of $\\psi (t)\\overline{\\psi (ht-s)}$ , we estimate $ |\\partial _t Q(t,s)| \\ge |\\alpha \\nu _1 t^{\\alpha -1} + \\beta \\nu _2 t^{\\beta -1} | - h C r, $ where $C$ is a positive constant only depending on the exponents $\\alpha ,\\beta $ .",
"Let us define the function $F(t)=\\alpha \\nu _1 t^{\\alpha -1}+\\beta \\nu _2 t^{\\beta -1},$ and its associated exceptional set $ E=\\lbrace t\\in [1/2,2]\\,:\\,|F(t)|\\le \\tau r^{1-\\varepsilon _1}\\rbrace ,$ where $\\tau $ is a positive constant that depends only on $\\alpha ,\\beta $ and is to be determined later.",
"Our strategy will be to choose $\\tau $ so that $|E|$ is small and then apply van der Corput's lemma outside of $E$ .",
"We will prove (at the end of the considerations for Case 1): Lemma 4.3 There exists a choice of $\\tau $ (depending only on $\\alpha ,\\beta $ ) such that $|E|\\lesssim r^{-\\varepsilon _2}$ with $\\varepsilon _2 = \\varepsilon _1/|\\beta -\\alpha |$ .",
"Assuming $\\tau $ is chosen as in the lemma, we now specify $h_0$ to be such that $h_0 C = \\frac{1}{2} \\tau r^{-\\varepsilon _1}.$ Then whenever $h \\le h_0$ , for all $t \\in [1/2,2] \\setminus E$ , $|\\partial _t Q(t,s)| \\gtrsim r^{1-\\varepsilon _1}.$ We now split the integral in (REF ) according to whether $t\\in [1/2,2]\\setminus E$ or $t\\in E$ .",
"We estimate the portion of the integral over $E$ trivially by the measure of $E$ , which is small $\\lesssim r^{-\\varepsilon _2}$ by Lemma REF .",
"We will estimate the portion of the integral over $[1/2,2] \\setminus E$ by applying van der Corput's lemma combined with the lower bound (REF ).",
"Here we encounter a delicate point: as stated in [19] van der Corput's lemma for a first derivative assumes monotonicity.",
"We circumvent this assumption as follows.",
"We first note that $E$ (and thus also $[1/2,2]\\backslash E$ ) is a finite union of intervals, with the number of intervals being bounded by a small absolute constant.",
"To see this note that the equation $ \\alpha \\nu _1 t^{\\alpha -1} + \\beta \\nu _2 t^{\\beta -1} \\pm \\tau r^{1-\\varepsilon _1} = 0$ has at most 3 solutions in $t>0$ (see for example [20]).",
"Thus we may apply the following slight variant of van der Corput's lemma (proved at the end of Case 1) to each such interval: Lemma 4.4 Suppose $\\phi $ is real-valued and smooth in $(a,b)$ and that both $|\\phi ^{\\prime } (x)| \\ge \\sigma _1 $ and $|\\phi ^{\\prime \\prime } (x) | \\le \\sigma _2$ for all $t \\in (a,b)$ .",
"Then $\\left| \\int _a^b e^{i \\lambda \\phi (t)}dt \\right| \\le (a-b) \\left( \\frac{\\sigma _2}{\\sigma _1^2}\\right) \\lambda ^{-1}.$ Here we note that for $s$ fixed, we have that $Q(t,s)$ is $C^\\infty $ with respect to $t$ for all $t$ in the support of $\\psi (t)\\overline{\\psi (ht-s)}$ ; in particular note that both $t, ht-s$ are bounded away from the origin.",
"We also verify trivially that for all such $t$ , $| \\partial _t^2 Q(t,s)| \\lesssim r,$ with a constant depending only on $\\alpha ,\\beta $ .",
"Hence applying Lemma REF with the bounds (REF ) and (REF ) to each of the finitely many finite-length intervals in $[1/2,2] \\setminus E$ , we obtain for each such portion of the integral a bound of size $\\lesssim r (r^{1-\\varepsilon _1})^{-2} = r^{-(1-2\\varepsilon _1)}$ .",
"In total, combining this with our trivial estimate for the portion of the integral over $E$ , we have proved $|(\\Phi _h^\\nu *\\tilde{\\Phi }^\\mu _1)(s)| \\lesssim r^{-(1-2\\varepsilon _1)} + r^{-\\varepsilon _2}\\lesssim r^{-\\varepsilon _3},$ for all $|s| \\le 4$ , for a suitable $\\varepsilon _3 >0$ , which suffices for (REF ) in this case.",
"All that remains is to verify Lemmas REF and REF .",
"We observe that if one of $|\\nu _1|,|\\nu _2|$ dominates the other then $|F(t)|$ is large, that is $|F(t)|\\gtrsim r$ .",
"More precisely, recall that $|\\nu _1| + |\\nu _2| \\approx r$ , and suppose that $ |\\nu _2|/|\\nu _1| \\le c_0$ for some small constant $c_0$ (so in particular $|\\nu _1| \\gtrsim r$ ).",
"Then $ |F(t)|= |\\nu _1| \\left| \\alpha t^{\\alpha -1} + \\beta \\frac{\\nu _2}{\\nu _1} t^{\\beta -1} \\right|;$ if $c_0$ is chosen sufficiently small (with respect to $\\alpha ,\\beta $ ) we may guarantee that for all $t \\in [1/2,2]$ , $ \\alpha t^{\\alpha -1} \\ge 2 \\left| \\beta \\frac{\\nu _2}{\\nu _1} t^{\\beta -1} \\right| $ and hence $ |F(t)| \\ge |\\nu _1| \\frac{\\alpha }{2} t^{\\alpha -1} \\ge c_1 r,$ say.",
"We may argue similarly to obtain $|F(t)| \\ge c_1^{\\prime }r$ if $|\\nu _1|/|\\nu _2| \\le c_0^{\\prime }$ where $c_0^{\\prime }$ depends only on $\\alpha ,\\beta $ .",
"By choosing $\\tau < \\min \\lbrace c_0,c_0^{\\prime }\\rbrace $ (hence depending only on $\\alpha ,\\beta $ ) we then see that if $E$ is to be non-empty, we must be in the regime where $c_0^{-1} \\le |\\nu _1|/|\\nu _2| \\le c_0^{\\prime }$ , that is, $|\\nu _1| \\approx |\\nu _2|$ .",
"In this case, we will deduce that (REF ) holds.",
"Suppose that $\\alpha < \\beta $ ; write $c:= \\nu _2/\\nu _1$ so that $|c| \\in [c_0^{-1},c_0^{\\prime }].$ Then $ F(t)=\\alpha \\nu _1 t^{\\alpha -1} (1+ c(\\beta /\\alpha ) t^{\\beta -\\alpha }),$ so that for all $t\\in E$ we must have $ r|1+c (\\beta /\\alpha )t^{\\beta -\\alpha }| \\le |F(t)| \\le \\tau r^{1-\\varepsilon _1},$ that is, $t$ must satisfy $ |1+c (\\beta /\\alpha )t^{\\beta -\\alpha }| \\le \\tau r^{-\\varepsilon _1}.$ The measure of such $t$ is $\\lesssim r^{-\\varepsilon _1/(\\beta -\\alpha )}$ , with an implicit constant dependent on $\\alpha ,\\beta $ .",
"For the case $\\alpha >\\beta $ we argue in an entirely analogous way.",
"This proves Lemma REF .",
"We recall the proof of the original van der Corput lemma in the case of a first derivative [19], which bounds the integral in question by $ \\lambda ^{-1} \\int _a^b \\left| \\frac{d}{dt} \\left( \\frac{1}{\\phi ^{\\prime }} \\right) \\right| dt = \\lambda ^{-1} \\int _a^b \\left| \\frac{\\phi ^{\\prime \\prime }(t)}{\\phi ^{\\prime }(t)^2} \\right| dt,$ where we have evaluated the derivative rather than invoking monotonicity of $\\phi ^{\\prime }$ to bring the absolute values outside the integral.",
"The inequality claimed in Lemma REF then clearly follows.",
"We have now concluded the proof of Lemma REF in Case 1.",
"Case 2.",
"In the remaining case, $h_0 \\le h \\le 1$ .",
"Fix any small $0<\\varepsilon _4<1$ ; if $|s|\\le r^{-\\varepsilon _4}$ we use the triangle inequality to bound (REF ) trivially by 1, which is sufficient for the second term in (REF ).",
"Thus from now on we assume that $|s|\\ge r^{-\\varepsilon _4}$ and work to obtain a small bound for the integral.",
"Note that as a vector, $\\left(\\begin{matrix} \\partial _t Q(t,s)\\\\\\partial _t^2 Q(t,s)\\\\\\partial _t^3 Q(t,s)\\\\\\partial _t^4 Q(t,s)\\end{matrix}\\right)= M_{t,s} \\left(\\begin{matrix}\\alpha \\nu _1 t^{\\alpha -1}\\\\-\\alpha \\mu _1 (ht-s)^{\\alpha -1}\\\\\\beta \\nu _2 t^{\\beta -1}\\\\-\\beta \\mu _2 (ht-s)^{\\beta -1}\\end{matrix}\\right),$ where $M_{t,s}$ is the $4 \\times 4$ matrix $M_{t,s} = \\left(\\begin{matrix}1 & h & 1 & h\\\\a_1 t^{-1} & a_1 h^2(ht-s)^{-1} & b_1 t^{-1} & b_1 h^2(ht-s)^{-1}\\\\a_2 t^{-2} & a_2 h^3(ht-s)^{-2} & b_2 t^{-2} & b_2 h^3(ht-s)^{-2}\\\\a_3 t^{-3} & a_3 h^4(ht-s)^{-3} & b_3 t^{-3} & b_3 h^4(ht-s)^{-3}\\end{matrix} \\right),$ and $a_1=\\alpha -1, \\quad a_2&=(\\alpha -1)(\\alpha -2), \\quad a_3=(\\alpha -1)(\\alpha -2)(\\alpha -3),\\\\b_1=\\beta -1, \\quad b_2&=(\\beta -1)(\\beta -2), \\quad b_3=(\\beta -1)(\\beta -2)(\\beta -3).$ If we can show that $|\\det M_{t,s}|$ is sufficiently large, that is $| \\det M_{t,s} | \\gtrsim r^{-\\kappa }$ for some $\\kappa >0$ , then we will apply the following lemma (whose proof we defer to the end of the section): Lemma 4.5 Let $A$ be an invertible $n\\times n$ matrix and $x\\in \\mathbb {R}^n$ .",
"Then $ |Ax| \\ge |\\det A| \\Vert A\\Vert ^{1-n} |x|, $ where $\\Vert A\\Vert $ denotes the matrix norm $\\sup _{|x|=1} |Ax|$ .",
"Note that $\\Vert M_{t,s} \\Vert \\lesssim 1$ (since we only consider $t$ in the support of $\\psi (t)\\overline{\\psi (ht-s)}$ , so that both $t, ht-s$ are bounded away from the origin).",
"If we have shown (REF ) for $t$ in a certain interval, then applying Lemma REF to (REF ), we see that throughout that interval, $\\left( \\sum _{k=1}^4 |\\partial _t^k Q(t,s)|^2 \\right)^{1/2} \\gtrsim r^{-\\kappa } |(\\nu _1,\\nu _2,\\mu _1,\\mu _2)^T| \\gtrsim r^{1-\\kappa }.$ Then applying the van der Corput lemma to that portion of the integral (REF ) shows that portion is bounded by $r^{-(1-\\kappa )/4}$ .",
"(Note: to be precise, if only the first order term $|\\partial _t Q(t,s)|$ dominates in (REF ), then we must apply the variant Lemma REF of the van der Corput lemma, using the trivial upper bound $|\\partial _t^2Q(t,s)| \\lesssim r$ , similar to our argument in Case 1.",
"This will result in a bound for the portion of the integral over that interval of size $\\lesssim r^{-(1-2\\kappa )}$ , which is sufficient.)",
"In fact, we will show that $|\\det M_{t,s}|$ is sufficiently large in this manner for all but a small exceptional set $E$ of $t$ , with measure $ \\lesssim r^{-\\kappa ^{\\prime }}$ for some small $\\kappa ^{\\prime }>0$ .",
"(As in our argument in Case 1, we will also note that this exceptional set is a union of a finite number of intervals, dependent only on $\\alpha ,\\beta $ , so that we may apply the above argument to each individual component of $[1/2,2]\\setminus E$ .)",
"Thus this strategy is sufficient to complete the proof of (REF ).",
"We require the following purely algebraic identity.",
"Lemma 4.6 Let $a_0,\\dots ,a_3,b_0,\\dots ,b_3,x,y$ be arbitrary real numbers.",
"Then $ \\left|\\begin{matrix}a_0 & a_0 & b_0 & b_0\\\\a_1 x & a_1 y & b_1 x & b_1 y\\\\a_2 x^2 & a_2 y^2 & b_2 x^2 & b_2 y^2\\\\a_3 x^3 & a_3 y^3 & b_3 x^3 & b_3 y^3\\end{matrix}\\right| = (c(x^2+y^2)+dxy)(x-y)^2 xy,$ where $c,d$ are given by $c = -\\left|\\begin{matrix}a_0 & b_0\\\\a_1 & b_1\\end{matrix}\\right|\\cdot \\left|\\begin{matrix}a_2 & b_2\\\\a_3 & b_3\\end{matrix}\\right|, \\qquad d = c+ \\left|\\begin{matrix}a_0 & b_0\\\\a_3 & b_3\\end{matrix}\\right|\\cdot \\left|\\begin{matrix}a_1 & b_1\\\\a_2 & b_2\\end{matrix}\\right|.$ Expand the determinant along the first row, combine the terms corresponding to the first and second columns, and those corresponding to the third and fourth columns, respectively.",
"Then again expand each of the two resulting $3\\times 3$ determinants along the first row.",
"Let $\\tilde{M}$ denote the matrix in Lemma REF .",
"Rescaling the individual rows and columns of $M_{t,s}$ appropriately to clear denominators, we see that $ \\det M_{t,s}= h^2 t^{-6}(ht-s)^{-6} \\det \\tilde{M}$ where within $\\tilde{M}$ we set $a_0=b_0=1,$ $x=ht-s$ and $y=ht$ .",
"Then we may apply Lemma REF to compute $\\det M_{t,s} =t^{-5} (ht-s)^{-5} s^2 h^3 S(t),$ with $ S(t)=h^2(2c+d)t^2 - h(2c+d)st + cs^2,$ in which $c,d$ are as in (REF ).",
"Note that with $a_0=b_0=1$ and the other $a_i,b_i$ as specified above, then $c\\ne 0$ is equivalent to $\\alpha \\ne \\beta $ , $\\alpha ,\\beta \\ne 1,2$ .",
"In order to now verify that $|\\det M_{t,s}|$ is sufficiently large, as in (REF ), we distinguish between two cases.",
"Case 2A.",
"Suppose first that $2c+d=0$ , so that $S(t)=cs^2$ .",
"We must then verify that $c \\ne 0$ .",
"Since $2c+d=0$ , clearly if $c=0$ then $d=0$ .",
"But recall from above that $c = 0$ implies that either $\\alpha =2$ or $\\beta =2$ (since the hypotheses of Lemma REF already ruled out $\\alpha =\\beta $ , $\\alpha =1$ or $\\beta =1$ ).",
"Recall also that we assume in this stage of the proof that the pair $(\\alpha ,\\beta )$ is not $(2,3)$ .",
"Suppose that $\\alpha =2$ .",
"Then we would have $d = (\\beta -1)^2(\\beta -2)^2(\\beta -3)(\\alpha -1),$ which is clearly non-zero (since $\\beta \\ne 3$ ).",
"Analogously we see that $\\beta =2$ leads to a contradiction.",
"Thus we may conclude that $c\\ne 0$ , and recalling $|s| \\ge r^{-\\varepsilon _4}$ from (REF ) and $h \\ge h_0 \\gtrsim r^{-\\varepsilon _1}$ from (REF ), we may compute immediately from (REF ) that $|\\det M_{t,s} |\\gtrsim r^{-4\\varepsilon _4-3\\varepsilon _1},$ holds for all $t \\in [1/2,2]$ .",
"This verifies (REF ) and allows us to apply the van der Corput lemma to bound the full integral (REF ) by $r^{-\\kappa }$ for some $\\kappa >0$ , completing the proof of (REF ) in this case.",
"Case 2B.",
"The final case we must consider is when $2c+d\\ne 0$ .",
"Fix any small $0<\\varepsilon _5<1$ and define $ E=\\lbrace t\\in [1/2,2]\\,:\\, |S(t)|\\le r^{-2\\varepsilon _5-2\\varepsilon _1}\\rbrace .$ Note first that $E$ is a union of at most two intervals, since $S$ is a quadratic polynomial.",
"Then for $t \\in [1/2,2] \\setminus E$ , (REF ) in combination with $|s| \\ge r^{-\\varepsilon _4},$ $h \\gtrsim r^{-\\varepsilon _1}$ implies $|\\det M_{t,s} | \\gtrsim r^{-2\\varepsilon _4-5\\varepsilon _1-2\\varepsilon _5},$ verifying (REF ) so that we may apply the van der Corput lemma to bound the portion of the integral over $[1/2,2] \\setminus E$ by $\\lesssim r^{-\\kappa }$ for some $\\kappa >0$ , which suffices for the first term in (REF ).",
"We will bound the portion of the integral over $E$ trivially, so all that remains is to verify that $E$ has small measure, for which we call upon the following lemma (see Christ [3]): Lemma 4.7 Let $I\\subset \\mathbb {R}$ be an interval, $k\\in \\mathbb {N}$ , $f\\in C^k(I)$ , and suppose that for some $\\sigma >0$ , $|f^{(k)}(x)|\\ge \\sigma $ for all $x\\in I$ .",
"Then there exists a constant $0<C<\\infty $ depending only on $k$ such that for every $\\rho >0$ , $|\\lbrace x\\in I\\,:\\, |f(x)|\\le \\rho \\rbrace |\\le C \\Big (\\frac{\\rho }{\\sigma }\\Big )^{1/k}.$ By the choice of $h_0$ in (REF ) we have $|S^{\\prime \\prime }(t)|\\gtrsim h^2 \\gtrsim h_0^2 \\gtrsim r^{-2\\varepsilon _1}.$ Thus by Lemma REF , we have $|E|\\lesssim r^{-\\varepsilon _5}$ , which suffices for the second term in (REF ).",
"All that remains to complete the proof of Lemma REF , and hence of the main Lemma REF , is to verify Lemma REF .",
"First we show $\\Vert A^{-1}\\Vert \\le \\Vert A\\Vert ^{n-1}/|\\det A |$ .",
"By homogeneity we can assume $\\Vert A\\Vert =1$ .",
"Then all the eigenvalues of $AA^*$ are between 0 and 1.",
"Let $\\lambda $ be the smallest eigenvalue of $AA^*$ .",
"Then $\\Vert A^{-1}\\Vert =\\lambda ^{-1/2}\\le \\det (AA^*)^{-1/2}=|\\det A|^{-1}.$ Therefore in general, $ |x|=|A^{-1}Ax|\\le \\Vert A^{-1}\\Vert \\cdot |Ax| \\le \\Vert A\\Vert ^{n-1} |\\det A |^{-1} |Ax|,$ as desired."
],
[
"The symmetric case: Theorem ",
"Here we prove Theorem REF .",
"We present the proof in detail only for $B^{m,m}$ ; thus in the following we write $T = B^{m,m}$ .",
"The proof for $A^{m,1}$ is, mutatis mutandi, analogous, and we merely sketch the necessary changes in Section REF .",
"Recalling the function $\\psi _k$ fixed in Section REF , we define the Littlewood-Paley projection in the free $x$ -variable by $f_k(x,y) = P_k f(x,y)=\\int \\limits _{\\mathbb {R}}f(u,y) \\check{\\psi }_k(x-u) du,$ where $\\check{\\psi }_k$ denotes the inverse Fourier transform of the function $\\psi _k$ .",
"In particular, note that $T P_k=P_k T$ ."
],
[
"Single annulus estimate.",
"We fix $k_0 \\in \\mathbb {Z}$ and split the operator as $T=T_{k_0}^{(1)}+T_{k_0}^{(2)}$ , where for any fixed $k,$ $T^{(1)}_k$ is defined as $ T_k^{(1)}f(x,y) := p.v.",
"\\int _{|t|\\le 2^{-k}} f(x-t,y-t^m) e^{iN(y)t^m} \\frac{dt}{t} $ and $T_k^{(2)} := T - T_{k}^{(1)}$ accordingly.",
"Our key estimate for $T^{(1)}_{k_0}$ is the pointwise bound: $|T^{(1)}_{k_0} P_{k_0} f(x,y)|\\lesssim {C}^*P_{k_0} f(x,y) + M_1 M_2 P_{k_0} f(x,y),$ in which the maximally truncated one-variable Carleson operator ${C}^*$ is defined as in (REF ); here our understanding is that ${C}^*$ acts only on the second variable.",
"Also, $M_1$ and $M_2$ refer to the Hardy-Littlewood maximal function in the first and second variable respectively, as defined in (REF ), ().",
"We will prove (REF ) by using the fact that under the single annulus assumption $f=P_{k_0} f,$ the function $f(x-t,y-t^m)$ is well approximated by $f(x,y-t^m)$ .",
"Precisely, we assume (REF ) and estimate $|T_{k_0}^{(1)}f(x,y)|\\le \\mathrm {\\bf I}+\\mathrm {\\bf II},$ where $\\mathrm {\\bf I}&=&\\int \\limits _{|t|\\le 2^{-k_0}} |f(x-t,y-t^m)-f(x,y-t^m)| \\frac{dt}{|t|}, \\\\\\mathrm {\\bf II}&=&\\Big | p.v.\\int \\limits _{|t|\\le 2^{-k_0}} f(x,y-t^m) e^{iN(y)t^m} \\frac{dt}{t}\\Big |.$ At this point there is a striking dichotomy in our treatment, depending on the parity of $m$ : if $m$ is even, the term $\\mathrm {\\bf II}$ vanishes identically due to the integrand being an odd function.",
"On the other hand, if $m$ is odd, we can change variables $t^m\\mapsto t$ (appropriately in the cases $t>0, t<0$ ) to see $\\mathrm {\\bf II}\\lesssim \\sup _{\\varepsilon >0}\\Big | p.v.",
"\\int \\limits _{|t|\\le \\varepsilon } f(x,y-t) e^{iN(y)t} \\frac{dt}{t}\\Big |\\le {C}^* f(x,y),$ in which the maximally truncated Carleson operator acts only on the second variable.",
"This contributes the first term in (REF ).",
"Next, we note that the first term $\\mathrm {\\bf I}$ can be estimated by a maximal function due to the single annulus assumption (REF ).",
"We write $f(x-t,y-t^m)-f(x,y-t^m)=\\int _{\\mathbb {R}} f(x-u,y-t^m)(\\check{\\psi }_{k_0}(u-t)-\\check{\\psi }_{k_0}(u)) du.$ By the rapid decay of the first derivative of $\\check{\\psi }$ we certainly have $|\\frac{d}{d\\xi }(\\psi _{k_0})\\check{\\;} | = |\\frac{d}{d\\xi } (2^{k_0} \\check{\\psi }(2^{k_0}\\xi ) )| \\le 2^{2k_0} (1+|2^{k_0}\\xi |)^{-2}.$ Now suppose that for some $j \\ge 0$ , $u$ is in the annulus $2^{-k_0+j-1}\\le |u|\\le 2^{-k_0+j},$ so that for $|t| \\le 2^{-k_0}$ we have both $2^{-k_0+j-2}\\le |u|, |u-t|\\le 2^{-k_0+j+1}.$ Thus applying the mean value theorem and the decay (REF ), for $u$ in the annulus (REF ) we have $ |\\check{\\psi }_{k_0}(u-t)-\\check{\\psi }_{k_0}(u)|\\lesssim |t| \\cdot 2^{2(k_0-j)},$ where the implicit constant depends only on the choice of $\\psi $ .",
"Therefore (REF ) can be estimated in absolute value by $ \\lesssim |t| 2^{2k_0} \\sum _{j=0}^\\infty 2^{-2j} \\int _{|u|\\le 2^{-k_0+j}} |f(x-u,y-t^m)| du.",
"$ This allows us to bound the term $\\mathrm {\\bf I}$ by $\\lesssim \\sum _{j=0}^\\infty 2^{-j} \\frac{1}{2^{-k_0} 2^{-k_0+j}} \\int _{|u|\\le 2^{-k_0+j}}\\int _{|t|\\le 2^{-k_0}} |f(x-u,y-t^m)| dt du .$ We may dominate this by the maximal functions $M_1$ and $M_2$ as follows.",
"Indeed, we focus temporarily on the inner integration in $t$ in (REF ): $& \\frac{1}{2^{-k_0}}\\int _{|t|\\le 2^{-k_0}} |f(x-u,y-t^m)| dt\\le & \\frac{C}{2^{-k_0}}\\int _{|s|\\le 2^{-k_0 m}} |f(x-u,y-s)| |s|^{\\frac{1}{m} -1} ds$ Since $|s|^{\\frac{1}{m} - 1} \\mathbf {1}_{|s|\\le 2^{-k_0 m}}$ is radially decreasing and integrable in $s$ , with integral equal to $C 2^{-k_0}$ , by [18], we can bound the above by $\\lesssim M_2f(x-u,y)$ .",
"Thus (REF ) is bounded by $M_1 M_2 f(x,y)$ .",
"This completes the proof of the inequality (REF ) for $T^{(1)}_{k_0}$ .",
"We now turn to estimating $T_{k_0}^{(2)}f$ , still under the single annulus assumption (REF ).",
"Let us define for any integer $\\ell $ $ T_{\\ell } f(x,y)=\\int _{\\mathbb {R}} f(x-t,y-t^m) e^{iN(y)t^m} \\psi _{\\ell }(t)\\frac{dt}{t}.",
"$ Then certainly $|T_{k_0}^{(2)}f|\\lesssim {M}f + \\sum _{\\ell =0}^\\infty |T_{-k_0+\\ell } f|;$ here we need merely observe that the maximal operator ${M}$ along $(t,t^m)$ (defined in (REF )) arises in (REF ) due to the transition to smooth cutoffs.",
"Next, we claim that (still under the assumption (REF )) there exists a constant $\\gamma >0$ such that for all $\\ell \\ge 0,$ $\\Vert T_{-k_0+\\ell } f\\Vert _2 \\lesssim 2^{-\\gamma \\ell } \\Vert f\\Vert _2.$ To prove (REF ) we proceed similarly to Section REF .",
"First we apply Plancherel's theorem in the free $x$ -variable, so that it is equivalent to prove that for $g_\\xi (y)=\\int _\\mathbb {R}e^{-i \\xi x} f(x,y) dx,$ $\\Big \\Vert \\int \\limits _{\\mathbb {R}} g_\\xi (y-t^m) e^{iN(y)t^m-i\\xi t} \\psi _{-k_0+\\ell }(t) \\frac{dt}{t}\\Big \\Vert _{L^2 (d\\xi , dy)}\\lesssim 2^{-\\gamma \\ell } \\Vert g_\\xi \\Vert _{L^2(d\\xi , dy)}.$ In particular, we note that due to the assumption (REF ), $g_\\xi $ is nonzero only in the frequency annulus $2^{k_0-1}\\le |\\xi |\\le 2^{k_0+1}$ .",
"We then split the integral into a positive and negative part, which are dealt with analogously.",
"We focus here on the positive portion of the integral; by a change of variables $t^m\\mapsto t$ the claim (REF ) is reduced to showing $\\Big \\Vert \\int \\limits _{0}^\\infty F(y-t) e^{iN(y)t-i\\xi t^{1/m}} \\psi _{-k_0+\\ell }(t^{1/m}) \\frac{dt}{t}\\Big \\Vert _{L^2(dy)}\\lesssim 2^{-\\gamma \\ell } \\Vert F\\Vert _{L^2(dy)}$ for all single variable functions $F$ , uniformly in $2^{k_0-1}\\le |\\xi |\\le 2^{k_0+1}$ .",
"As in the proof of Lemma REF we proceed by the $TT^*$ method.",
"For convenience we write $\\tilde{\\psi }_k(t)=\\psi _k(t^{1/m}) \\mathbf {1}_{(0,\\infty )}(t),$ and denote the operator on the left hand side of (REF ) by $\\tilde{T}$ .",
"Then $\\Vert \\tilde{T}\\Vert _{2\\rightarrow 2}=\\Vert \\tilde{T}\\tilde{T}^*\\Vert _{2\\rightarrow 2}^{1/2}$ , where $\\tilde{T}\\tilde{T}^* F(y)=\\int _\\mathbb {R}F(y-s) K_{N(y),N(y-s)}(s) ds$ and for any $\\lambda _1,\\lambda _2\\in \\mathbb {R}$ the kernel $K_{\\lambda _1,\\lambda _2}$ is given by $ K_{\\lambda _1,\\lambda _2}(s)=\\int _\\mathbb {R}e^{i\\lambda _1 t-i\\lambda _2 (t-s)-i\\xi (t^{1/m}- (t-s)^{1/m})} \\frac{\\tilde{\\psi }_{-k_0+\\ell }(t-s)}{t-s} \\frac{\\tilde{\\psi }_{-k_0+\\ell }(t)}{t} dt.",
"$ Via the substitution $t\\mapsto \\rho t$ with $\\rho =2^{m(-k_0+\\ell )}$ we obtain $ \\rho K_{\\lambda _1,\\lambda _2}(\\rho s)=\\int _\\mathbb {R}e^{i\\lambda _1\\rho t-i\\lambda _2\\rho (t-s)-i\\xi 2^{-k_0+\\ell }(t^{1/m}- (t-s)^{1/m})} \\frac{\\tilde{\\psi }_{0}(t-s)}{t-s} \\frac{\\tilde{\\psi }_{0}(t)}{t} dt.",
"$ We need to analyze the phase function $Q(t,s)=\\lambda _1\\rho t-\\lambda _2\\rho (t-s) + \\eta (t^{1/m}-(t-s)^{1/m}),$ where $\\eta =-\\xi 2^{-k_0+\\ell }$ , so in particular $2^{\\ell -1}\\le |\\eta |\\le 2^{\\ell +1}.$ On first sight this may not look promising, because the phase function includes linear terms which tend to cause trouble (compare Stein and Wainger [22]).",
"However, it turns out that we are allowed to take derivatives to isolate the non-linear term (recall $m \\ge 2$ ) because we know by (REF ) that its coefficient $\\eta $ is large.",
"Taking two derivatives with respect to $t,$ we obtain $ \\partial ^2_t Q(t,s)=c\\eta (t^{\\alpha }-(t-s)^{\\alpha }) $ with $\\alpha =\\frac{1}{m}-2$ and $c=\\frac{1}{m} (\\frac{1}{m}-1)$ .",
"Suppose that $|s|\\ge 2^{-\\ell /2}$ .",
"Then by the mean value theorem and (REF ), $|\\partial ^2_t Q(t,s)|\\gtrsim 2^{\\ell /2}$ throughout the region of $t$ and $t-s$ considered (depending only on the support of $\\tilde{\\psi }_0$ ).",
"In this case, an application of the second derivative test shows that $|\\rho K_{\\lambda _1,\\lambda _2}(\\rho s)|\\lesssim 2^{-\\ell /4}.$ On the other hand, if $|s|\\le 2^{-\\ell /2}$ we merely use the triangle inequality for the trivial estimate $|\\rho K_{\\lambda _1,\\lambda _2}(\\rho s)|\\lesssim 1.",
"$ Altogether we have proved, with $\\rho =2^{m(-k_0+\\ell )}$ , $|K_{\\lambda _1,\\lambda _2}(s)|\\lesssim 2^{-\\ell /4} \\rho ^{-1} \\mathbf {1}_{\\lbrace |s|\\le 4\\rho \\rbrace }(s) + \\rho ^{-1}\\mathbf {1}_{\\lbrace |s|\\le 2^{-\\ell /2}\\rho \\rbrace } (s), $ uniformly in $\\lambda _1,\\lambda _2$ .",
"Applying this in (REF ) allows us to deduce that $|\\tilde{T}\\tilde{T}^* F(y)| \\lesssim 2^{-\\ell /4} MF(y), $ where $MF$ denotes the standard one-variable Hardy-Littlewood maximal function.",
"An application of the $L^2$ estimate for $M$ now implies our claim (REF ) with $\\gamma =1/8$ ; by Plancherel we then finally obtain (REF )."
],
[
"Square function estimate",
"In this section we assemble the single annulus estimates of the previous section to derive the $L^p$ boundedness of our operator $T=B^{m,m}$ .",
"This application of the Littlewood-Paley theory is in the spirit of Bateman and Thiele [1].",
"In view of the relation $TP_k=P_kT$ and the standard Littlewood-Paley inequalities, we have $ \\Vert Tf\\Vert _p \\lesssim \\Big \\Vert \\left(\\sum _{k\\in \\mathbb {Z}} |TP_k f|^2\\right)^{1/2}\\Big \\Vert _p\\lesssim \\Big \\Vert \\left(\\sum _{k\\in \\mathbb {Z}} |T^{(1)}_k P_k f|^2\\right)^{1/2}\\Big \\Vert _p+\\Big \\Vert \\left(\\sum _{k\\in \\mathbb {Z}} |T^{(2)}_k P_k f|^2\\right)^{1/2}\\Big \\Vert _p$ In the term for $T^{(1)}_k$ on the right hand side we apply the estimate (REF ); then by applying the vector-valued estimates of Theorems REF and REF to the maximally truncated Carleson operator and the one-variable maximal function, we obtain $ \\Big \\Vert \\left(\\sum _{k\\in \\mathbb {Z}} |T^{(1)}_k P_k f|^2\\right)^{1/2}\\Big \\Vert _p\\lesssim \\Big \\Vert \\left(\\sum _{k\\in \\mathbb {Z}} |P_k f|^2\\right)^{1/2}\\Big \\Vert _p\\lesssim \\Vert f\\Vert _p.",
"$ For $T_k^{(2)}$ , we recall that by (REF ) $|T^{(2)}_k P_k f|\\lesssim {M}P_k f + \\sum _{\\ell =0}^\\infty |T_{-k+\\ell } P_k f|,$ so that by Minkowski's inequality for integrals, $ \\Big \\Vert \\left(\\sum _{k\\in \\mathbb {Z}} |T^{(2)}_k P_k f|^2\\right)^{1/2}\\Big \\Vert _p\\lesssim \\Big \\Vert \\left(\\sum _{k\\in \\mathbb {Z}} | {M} P_k f|^2\\right)^{1/2}\\Big \\Vert _p+\\sum _{\\ell =0}^\\infty \\Big \\Vert \\left(\\sum _{k\\in \\mathbb {Z}} |T_{-k+\\ell } P_k f|^2\\right)^{1/2}\\Big \\Vert _p .$ We may apply Theorem REF to obtain a bound $\\lesssim \\Vert f\\Vert _p$ for the first term; for the second term it would suffice to show that for each $1<p<\\infty $ there exists $\\gamma >0$ such that for every $\\ell \\ge 0,$ $\\Big \\Vert \\left(\\sum _{k\\in \\mathbb {Z}} |T_{-k+\\ell } P_k f|^2\\right)^{1/2}\\Big \\Vert _p\\lesssim 2^{-\\gamma \\ell }\\Big \\Vert \\left(\\sum _{k\\in \\mathbb {Z}} |P_k f|^2\\right)^{1/2}\\Big \\Vert _p\\lesssim 2^{-\\gamma \\ell } \\Vert f\\Vert _p.$ For $p=2$ this follows from (REF ).",
"But recall that we always have the simple estimate $|T_{\\ell } P_k f|\\lesssim {M} P_k f$ for all $k,\\ell \\in \\mathbb {Z}$ by the triangle inequality.",
"Therefore Theorem REF implies a bound without decay, namely $\\Big \\Vert \\left(\\sum _{k\\in \\mathbb {Z}} |T_{-k+\\ell } P_k f|^2\\right)^{1/2}\\Big \\Vert _r\\lesssim \\Vert f\\Vert _r,$ valid for all $1 < r < \\infty $ .",
"Now in general, (REF ) follows for all $1< p < \\infty $ by interpolating between the $L^2 \\rightarrow L^2(\\ell ^2)$ case of (REF ) and the $L^r \\rightarrow L^r(\\ell ^2)$ bound (REF ) for the vector-valued map $f \\mapsto \\lbrace T_{-k+\\ell } P_k f\\rbrace _{k \\in \\mathbb {Z}}$ .",
"The proof of Theorem REF is now complete."
],
[
"Remarks on the proof for $A^{m,1}$",
"As mentioned above, we will not repeat the proof explicitly for $A^{m,1}$ , but merely complement the sketch already provided in Section REF by pointing out two key modifications.",
"Of course one interchanges the roles of the $x$ and $y$ variables.",
"In addition: The cancellation miracle for even $m$ in the term $\\mathrm {\\bf II}$ of () does not occur for $A^{m,1}$ .",
"Instead one always needs to invoke Carleson's theorem in the form of Theorem REF , analogous to the computation already carried out for odd $m$ in (REF ).",
"In the treatment of $A^{m,1}$ , the restriction $m\\ne 2$ originates because the relevant phase function analogous to (REF ) is $Q(t,s) = \\lambda _1 \\rho t - \\lambda _2 \\rho (t-s) + \\eta (t^m - (t-s)^m).$ Visibly, when $m=2$ , the phase function $Q(t, s)$ is now linear in $t$ , so that its second derivative vanishes, and consequently we fail in this case to obtain a good bound for the kernel."
],
[
"$L^2$ consequences of partial Carleson bounds",
"As stated in Section REF , the $L^2$ boundedness of $A^{m,1}_N$ , for any fixed integer $m \\ge 1$ , implies the $L^2$ boundedness of Carleson's operator (REF ).",
"Similarly, the $L^2$ boundedness of $B^{m,m}_N$ , when $m \\ge 1$ is an odd integer, implies the $L^2$ boundedness of Carleson's operator.",
"(Of course, in the work of this paper, our logic is actually the other way round: in proving Theorem REF , we used Carleson's theorem as a black box.)",
"We will see how to carry out these deductions from a more general argument we now give in the context of the quadratic Carleson operator ${C}^{\\text{par}}$ along the parabola (defined in equation (REF )).",
"We prove that an inequality of the form (REF ) would imply the analogue over $\\mathbb {R}$ of Lie's result [12] on the one-variable quadratic Carleson operator ${C}_Q$ : Proposition 6.1 Assume the veracity of the estimate $\\Big \\Vert \\int \\limits _{\\epsilon \\le |t|\\le R} f(x-t,y-t^2) e^{iN_1(x)t+iN_2(x)t^2}\\frac{dt}{t}\\Big \\Vert _{L^2(dxdy)}\\le C\\Vert f\\Vert _2,$ for all Schwartz functions $f$ , where $N_1,N_2: \\mathbb {R}\\rightarrow \\mathbb {R}$ are measurable functions, $0<\\epsilon <R$ are real parameters, and the constant $C$ is independent of $f,N_1,N_2,\\epsilon ,R$ .",
"Then the operator $ f \\mapsto {C}_Qf(x) := \\sup _{N\\in \\mathbb {R}^2} \\Big |p.v.\\int _\\mathbb {R}f(x-t) e^{iN_1t+iN_2t^2} \\frac{dt}{t}\\Big |$ is bounded on $L^2(\\mathbb {R})$ .",
"Note that in our assumed bound (REF ), the linearizing functions $N_1,N_2$ are independent of $y$ , so this is a far weaker assumption than the conjectured $L^2$ bound for ${C}^{\\text{par}}$ in (REF ).",
"In the argument that we will now give for (REF ), if we replace the curve $(t,t^2)$ by $(t,t^m)$ and the phase by $N_1(x)t + N_2(x)t^m$ , and furthermore specify that $N_2$ is identically zero, we may deduce Carleson's original theorem from the partial bound for $A^{m,1}_N$ for any integer $m \\ge 1$ ; or, if we specify $N_1$ is identically zero, we may deduce Carleson's original theorem from the partial bound for $B^{m,m}_N$ for $m$ an odd integer.",
"(When $m$ is even, under the specification $N_1 \\equiv 0$ , the operator in (REF ) would vanish, due to the integrand being an odd function.)",
"In general, to prove Proposition REF , we use an elementary tensor $f(x,y)=h(x)g(y)$ , where $h,g$ are real Schwartz functions, in which case (REF ) implies $ \\Big \\Vert \\int \\limits _{\\epsilon \\le |t|\\le R} h(x-t)e^{iN_1(x)t+iN_2(x)t^2} g(y-t^2) \\frac{dt}{t}\\Big \\Vert _{L^2(dx dy)}\\le C\\Vert h\\Vert _2 \\Vert g\\Vert _2.",
"$ Applying Plancherel's theorem in the $y$ variable we obtain $\\Big \\Vert \\int \\limits _{\\epsilon \\le |t|\\le R} h(x-t)e^{iN_1(x)t+iN_2(x)t^2} \\widehat{g}(\\eta )e^{-i\\eta t^2} \\frac{dt}{t}\\Big \\Vert _{L^2(dx d\\eta )}\\le C\\Vert h\\Vert _2\\Vert g\\Vert _2.",
"$ Suppose for the time being that we have chosen $g$ such that we have an estimate of the form $ \\Big \\Vert \\int \\limits _{\\epsilon \\le |t|\\le R} h(x-t)e^{iN_1(x)t+iN_2(x)t^2} \\widehat{g}(\\eta )(e^{-i\\eta t^2}-1)\\frac{dt}{t}\\Big \\Vert _{L^2(dx d\\eta )}\\le C \\Vert h\\Vert _2 \\Vert g\\Vert _2.$ We would deduce from (REF ) and (REF ) that $ \\Big \\Vert \\int _{\\epsilon \\le |t|\\le R} h(x-t)e^{iN_1(x)t+iN_2(x)t^2} \\widehat{g}(\\eta ) \\frac{dt}{t}\\Big \\Vert _{L^2(dx d\\eta )}\\le C\\Vert h\\Vert _2\\Vert g\\Vert _2, $ so that by Plancherel and Fubini, $ \\Big \\Vert \\int _{\\epsilon \\le |t|\\le R} h(x-t)e^{iN_1(x)t+iN_2(x)t^2} \\frac{dt}{t}\\Big \\Vert _2\\le C\\Vert h\\Vert _2.",
"$ Via Fatou's lemma this gives the $L^2$ boundedness of the quadratic Carleson operator $h \\mapsto {C}_Q h$ , as claimed in Proposition REF .",
"To obtain the estimate (REF ), we choose $\\delta $ with $0<\\delta <1/R^2$ and specify that $g$ be a Schwartz function on $\\mathbb {R}$ such that $\\widehat{g}$ is supported on $[-\\delta ,\\delta ]$ and $\\Vert g\\Vert _2>0$ .",
"Then by Minkowski's inequality and Fubini, the left hand side of (REF ) is bounded by $\\Vert h\\Vert _2\\int \\limits _{\\epsilon \\le |t|\\le R} \\Vert \\widehat{g}(\\eta )(e^{-i\\eta t^2}-1)\\Vert _{L^2(d\\eta )} \\frac{dt}{|t|}.$ The mean value theorem, followed by Plancherel, shows that for $|t| \\le R$ , $\\int \\limits _{\\mathbb {R}}\\Big |\\widehat{g}(\\eta )(e^{-i\\eta t^2}-1)\\Big |^2 d\\eta \\le \\delta ^2 t^4 \\Vert g\\Vert _2^2.$ This implies that (REF ) is no greater than $\\Vert h\\Vert _2 \\Vert g\\Vert _2 \\cdot \\delta \\int \\limits _{\\epsilon \\le |t|\\le R} |t| dt\\le \\Vert h\\Vert _2\\Vert g\\Vert _2,$ which completes the proof of (REF ), and hence Proposition REF ."
],
[
"$L^2$ deductions for partial Carleson operators",
"Remark REF stated that $L^2$ bounds for $A_N^{2,1}$ , $A^{m,m}_N$ and $B^{m,1}_N$ (with $m>1$ ) follow from known Carleson theorems.",
"We briefly indicate these deductions, which are along the lines of arguments in Sections REF and REF .",
"By Plancherel's theorem in the free $y$ -variable, $ \\Vert A_N^{m,m} f \\Vert _{L^2 (dxdy)} = \\Big \\Vert \\int _{\\mathbb {R}} g_\\eta (x-t) e^{i (N(x) -\\eta ) t^m} \\frac{dt}{t} \\Big \\Vert _{L^2(dx d\\eta )}$ where $g_\\eta (x)=\\int _\\mathbb {R}e^{-i\\eta y} f(x,y) dy.$ Then an $L^2$ bound of the form $ \\Big \\Vert \\int _{\\mathbb {R}} g_\\eta (x-t) e^{i (N(x) -\\eta ) t^m} \\frac{dt}{t} \\Big \\Vert _{L^2(dx)} \\lesssim \\Vert g_\\eta \\Vert _{L^2(dx)},$ uniform in $\\eta $ , follows from Stein and Wainger [22] (since $m>1$ ), and this suffices.",
"In the next case, $ \\Vert A_N^{2,1} f \\Vert _{L^2 (dxdy)} = \\Big \\Vert \\int _{\\mathbb {R}} g_\\eta (x-t) e^{i N(x) t - i \\eta t^2} \\frac{dt}{t} \\Big \\Vert _{L^2(dx d\\eta )}.$ Observe that $ i\\eta t^2 = i\\eta (x-t)^2 - i\\eta x^2 + 2i\\eta x t. $ Define $Q_\\eta f(x)=e^{i\\eta x^2} f(x)$ and set $\\widetilde{N}(x)=N(x)-2\\eta x$ .",
"Then, $\\int _\\mathbb {R}g_\\eta (x-t) e^{iN(x)t-i\\eta t^2} \\frac{dt}{t}=e^{i\\eta x^2} \\int _\\mathbb {R}Q_{-\\eta } g_\\eta (x-t) e^{i\\widetilde{N}(x)t} \\frac{dt}{t}=Q_{\\eta } H_{\\widetilde{N}(x)} Q_{-\\eta } g_\\eta (x),$ where $H_N f(x)=\\int _\\mathbb {R}f(x-t) e^{iNt} \\frac{dt}{t}$ .",
"Since $Q_\\eta $ is an isometry in $L^2$ , our claim follows from the $L^2$ bound for the Carleson operator.",
"In the final case, by Plancherel's theorem in the free $x$ -variable, $ \\Vert B_N^{m,1} f \\Vert _{L^2 (dxdy)} = \\Big \\Vert \\int _{\\mathbb {R}} g_\\xi (x-t^m) e^{i (N(x) -\\eta ) t} \\frac{dt}{t} \\Big \\Vert _{L^2(d\\xi dy)}$ where $g_\\xi (x)=\\int _\\mathbb {R}e^{-i\\xi x} f(x,y) dx.$ Thus the required $L^2$ bound follows from sending $t \\mapsto t^{1/m}$ and applying Guo [10] to the resulting operator, which has one fractional monomial in the phase."
],
[
"Proof of Theorem ",
"We assemble results from the Grafakos texts [8], [9].",
"By [9], there is a positive constant $c>0$ such that for any $1\\le p<\\infty $ , for all $f \\in L^p(\\mathbb {R})$ we have the pointwise inequality ${C}^* f \\le c M f + M( {C} f),$ in which $M$ is the standard one-dimensional Hardy-Littlewood maximal function.",
"Since the vector-valued $L^p(\\ell ^2)$ inequality analogous to (REF ) is known to hold for the Hardy-Littlewood maximal function (see e.g.",
"[19]), the problem is then reduced to proving the analogue of (REF ) for the Carleson operator ${C}$ .",
"In fact, this is a special case of [9], which claims that for all $1<p,r<\\infty $ and all weights $w \\in A_p$ , $\\Big \\Vert \\left( \\sum _k | {C} f_k|^r \\right)^{1/r} \\Big \\Vert _{L^p(w)} \\le C_{p,r}(w) \\Big \\Vert \\left( \\sum _k | f_k|^r \\right)^{1/r} \\Big \\Vert _{L^p(w)}$ for all sequences of functions $f_k \\in L^p(w)$ .",
"This inequality may be verified, following Grafakos, by the method of extrapolation.",
"We need only note that [9] provides a weighted estimate $\\Vert {C}f\\Vert _{L^p(w)} \\le C(p,[w]_{A_p}) \\Vert f \\Vert _{L^p(w)},$ for every $1<p<\\infty $ and $w \\in A_p$ .",
"This is sufficient to prove (REF ) for all the stated values of $r,p$ by applying the vector-valued extrapolation result [8].",
"(Here we remark on the detail that Corollary 7.5.7, to which we appeal, requires that $C(p,[w]_{A_p})$ be an increasing function in $[w]_{A_p}$ .",
"We can insure this is the case if we have the statement, slightly stronger than (REF ), that for every $B>0$ there exists a constant $C_p(B)$ such that for all $w \\in A_p$ with $[w]_{A_p} \\le B$ we have $\\Vert {C}f \\Vert _{L^p(w)} \\le C_p(B) \\Vert f\\Vert _{L^p(w)}$ ; such a statement is verified by the explicit version of (REF ) given by Lerner and Di Plinio [4].)",
"Alternatively, once one has the pointwise inequality (REF ) and has consequently reduced matters to proving an $L^p(\\ell ^2)$ vector-valued inequality for ${C}$ , one can turn to the original result [17] in the $L^p(\\ell ^2)$ case, or the recent streamlined proof [6]."
],
[
"Proof of Theorem ",
"We recall that the scalar-valued $L^p$ -bound for ${M}$ was obtained by comparing it to a square function [19].",
"Indeed, let $\\chi (t)$ be a non-negative smooth function with compact support on the interval $[-2,2]$ , such that $\\chi (t) \\equiv 1$ on $[-1,1]$ .",
"For $k \\in \\mathbb {Z}$ , let $\\chi _k(t) = 2^{-k} \\chi (2^{-k}t)$ , $d\\mu _k(x,y) = \\delta _{y=x^m} \\chi _k(x)$ , and $A_k f(x,y) = f*d\\mu _k(x,y) = \\int _{\\mathbb {R}} f(x-t,y-t^m) \\chi _k(t) dt.$ Also let $\\phi (x,y)$ be a smooth function with compact support on the unit ball in $\\mathbb {R}^2$ , normalized such that $\\int _{\\mathbb {R}^2} \\phi (x,y) dxdy = \\int _{\\mathbb {R}} \\chi (t) dt.$ For $k \\in \\mathbb {Z}$ , let $\\phi _k(x,y) = 2^{-(m+1)k} \\phi (2^{-k}x,2^{-mk}y)$ , and $B_k f(x,y) = f*\\phi _k(x,y) = \\int _{\\mathbb {R}^2} f(x-u,y-v) \\phi _k(u,v) dudv.$ Then for non-negative functions $f$ , we have the pointwise inequality $ {M}f \\le \\sup _{k \\in \\mathbb {Z}} B_k f + Sf,$ where $S$ is the following square function: $ Sf := \\left( \\sum _{k \\in \\mathbb {Z}} |A_k f - B_k f|^2 \\right)^{1/2}.$ Now $\\sup _{k \\in \\mathbb {Z}} B_k f$ is bounded by the standard maximal function associated to non-isotropic `squares' of sizes $R \\times R^m$ on $\\mathbb {R}^2$ .",
"It is known that a vector-valued estimate holds for the maximal function associated to these non-isotropic squares; that is an analogue of Theorem REF .",
"Thus the inequality (REF ) of Theorem REF holds for $1 < p < \\infty $ if we have $\\sup _{k \\in \\mathbb {Z}} B_k$ in place of ${M}$ on the left-hand side.",
"Hence to prove the desired form of (REF ), all we need to do is to establish $ \\left\\Vert \\left( \\sum _{\\ell \\in \\mathbb {Z}} |S f_{\\ell }|^2 \\right)^{1/2} \\right\\Vert _{L^p} \\lesssim _p \\left\\Vert \\left( \\sum _{\\ell \\in \\mathbb {Z}} |f_{\\ell }|^2 \\right)^{1/2} \\right\\Vert _{L^p}$ where $S$ is defined by (REF ), and $1 < p < \\infty $ .",
"The following scalar-valued inequality for $1<p<\\infty $ is already known [19]: $ \\Vert Sf\\Vert _{L^p} \\lesssim _p \\Vert f\\Vert _{L^p}.$ But to deduce (REF ) we will instead use a related scalar-valued inequality for a signed operator.",
"For $\\epsilon _k$ a random sequence of signs $\\pm 1$ , define $Tf := \\sum _{k \\in \\mathbb {Z}} \\epsilon _k (A_k f - B_k f).$ It is known that $ \\Vert Tf\\Vert _{L^p} \\lesssim _p \\Vert f\\Vert _{L^p}$ for all $1 < p < \\infty $ , independent of the signs $\\epsilon _k$ .",
"At the end of this section, we briefly recall a proof of this, for which one uses crucially the non-vanishing of the curvature of the curve $(t,t^m)$ , but we first deduce (REF ) from (REF ).",
"To do so, note that since $T$ is linear, the Marcinkiewicz-Zygmund theorem implies that $\\Vert \\, |T f_{\\ell }|_{\\ell ^2} \\,\\Vert _{L^p} \\lesssim _p \\Vert \\, |f_{\\ell }|_{\\ell ^2} \\,\\Vert _{L^p}$ for $1 < p < \\infty $ , i.e.",
"$\\left\\Vert \\, \\left|\\sum _{k \\in \\mathbb {Z}} \\epsilon _k (A_k f_{\\ell }-B_k f_{\\ell }) \\right|_{\\ell ^2(d\\ell )} \\,\\right\\Vert _{L^p} \\lesssim _p \\Vert \\, |f_{\\ell }|_{\\ell ^2} \\,\\Vert _{L^p}.$ (We write $\\ell ^2(d\\ell )$ to emphasize that the $\\ell ^2$ norm is taken with respect to the variable $\\ell $ .)",
"Now we take the expectation, denoted $\\mathbb {E}$ , over all the possible choices of $\\epsilon _k$ ; by Khintchine's inequality, $\\left( \\sum _{k \\in \\mathbb {Z}} |A_k f_{\\ell }-B_k f_{\\ell }|^2 \\right)^{1/2} \\simeq \\mathbb {E} \\left|\\sum _{k \\in \\mathbb {Z}} \\epsilon _k (A_k f_{\\ell }-B_k f_{\\ell }) \\right| .$ Taking the $\\ell ^2(d\\ell )$ and then $L^p$ norms on both sides, we get $\\left\\Vert \\, \\left( \\sum _{k,\\ell \\in \\mathbb {Z}} |A_k f_{\\ell }-B_k f_{\\ell }|^2 \\right)^{1/2} \\,\\right\\Vert _{L^p}&\\simeq \\left\\Vert \\, \\left(\\mathbb {E} \\left|\\sum _{k \\in \\mathbb {Z}} \\epsilon _k (A_k f_{\\ell }-B_k f_{\\ell }) \\right|\\right)_{\\ell ^2(d\\ell )} \\,\\right\\Vert _{L^p} \\\\&\\le \\mathbb {E} \\left\\Vert \\, \\left|\\sum _{k \\in \\mathbb {Z}} \\epsilon _k (A_k f_{\\ell }-B_k f_{\\ell }) \\right|_{\\ell ^2(d\\ell )} \\,\\right\\Vert _{L^p} \\\\&\\lesssim _p \\mathbb {E} \\Vert \\, |f_{\\ell }|_{\\ell ^2} \\,\\Vert _{L^p} \\\\&= \\Vert \\, |f_{\\ell }|_{\\ell ^2} \\,\\Vert _{L^p}.$ (The first inequality is the Minkowski inequality.)",
"The left-hand side above is precisely $\\Vert \\,| Sf_{\\ell } |_{\\ell ^2} \\,\\Vert _{L^p}$ .",
"This proves (REF ), and hence (REF ) of Theorem REF , for $1 < p <\\infty $ .",
"There are at least two ways of proving (REF ).",
"One is by complex interpolation, along the lines of arguments in [21], which we will not discuss here.",
"Alternatively, we can deduce (REF ) from a result of Duoandikoetxea and Rubio de Francia [5] without using complex interpolation.",
"To do so, let $d\\sigma _k = \\epsilon _k ( d\\mu _k - \\phi _k dxdy ).$ Then $d\\sigma _k$ has total mass $\\Vert d\\sigma _k\\Vert \\lesssim 1$ , and its Fourier transform satisfies $|\\widehat{d\\sigma _k}(\\xi ,\\eta )| \\lesssim \\min \\lbrace 2^k \\Vert (\\xi ,\\eta )\\Vert ,(2^k \\Vert (\\xi ,\\eta )\\Vert )^{-1/m}\\rbrace .$ (Here we see the curvature of $(t,t^m)$ .)",
"Furthermore, the operator $\\sup _{k \\in \\mathbb {Z}} |f*|d\\sigma _k||$ is bounded by the maximal Radon transform along the curve $(t,t^m)$ plus the Hardy-Littlewood maximal operator adapted to certain non-isotropic balls in $\\mathbb {R}^2$ .",
"It follows that $\\sup _{k \\in \\mathbb {Z}} |f*|d\\sigma _k||$ is bounded on $L^q(\\mathbb {R}^2)$ for all $1 < q < \\infty $ .",
"Thus Theorem B of Duoandikoetxea and Rubio de Francia [5] applies, and shows that $Tf = \\sum _{k \\in \\mathbb {Z}} f * d\\sigma _k$ is bounded on $L^p$ for all $1 < p < \\infty $ .",
"This completes our proof of (REF )."
],
[
"Acknowledgements.",
"We would like to thank C. Thiele and E. M. Stein for many helpful comments and discussions.",
"Pierce is supported in part by NSF DMS-1402121.",
"Yung is supported in part by the Hong Kong Research Grant Council Early Career Grant CUHK24300915.",
"This collaboration was initiated at the Hausdorff Center for Mathematics and Oberwolfach, and continued at the joint AMS-EMS-SPM 2015 international meeting at Porto.",
"The authors thank all institutions involved for gracious and productive work environments.",
"Shaoming Guo, Indiana University Bloomington, 107 S Indiana Ave, Bloomington, IN 47405, USA email: [email protected] Lillian B.",
"Pierce, Duke University, Box 90320, 120 Science Drive, Durham NC 27708, USA email: [email protected] Joris Roos, University of Bonn, Mathematical Institute, Endenicher Allee 60, 53115 Bonn, Germany email: [email protected] Po-Lam Yung, The Chinese University of Hong Kong, Ma Liu Shui, Shatin, Hong Kong email: [email protected]"
]
] | 1605.05812 |
[
[
"Extreme Vortex States and the Growth of Enstrophy in 3D Incompressible\n Flows"
],
[
"Abstract In this investigation we study extreme vortex states defined as incompressible velocity fields with prescribed enstrophy $\\mathcal{E}_0$ which maximize the instantaneous rate of growth of enstrophy $d\\mathcal{E}/dt$.",
"We provide {an analytic} characterization of these extreme vortex states in the limit of vanishing enstrophy $\\mathcal{E}_0$ and, in particular, show that the Taylor-Green vortex is in fact a local maximizer of $d\\mathcal{E} / dt$ {in this limit}.",
"For finite values of enstrophy, the extreme vortex states are computed numerically by solving a constrained variational optimization problem using a suitable gradient method.",
"In combination with a continuation approach, this allows us to construct an entire family of maximizing vortex states parameterized by their enstrophy.",
"We also confirm the findings of the seminal study by Lu & Doering (2008) that these extreme vortex states saturate (up to a numerical prefactor) the fundamental bound $d\\mathcal{E} / dt < C \\, \\mathcal{E}^3$, for some constant $C > 0$.",
"The time evolution corresponding to these extreme vortex states leads to a larger growth of enstrophy than the growth achieved by any of the commonly used initial conditions with the same enstrophy $\\mathcal{E}_0$.",
"However, based on several different diagnostics, there is no evidence of any tendency towards singularity formation in finite time.",
"Finally, we discuss possible physical reasons why the initially large growth of enstrophy is not sustained for longer times."
],
[
"Introduction",
"The objective of this investigation is to study three-dimensional (3D) flows of viscous incompressible fluids which are constructed to exhibit extreme growth of enstrophy.",
"It is motivated by the question whether the solutions to the 3D incompressible Navier-Stokes system on unbounded or periodic domains corresponding to smooth initial data may develop a singularity in finite time [17].",
"By formation of a “singularity” we mean the situation when some norms of the solution corresponding to smooth initial data have become unbounded after a finite time.",
"This so-called “blow-up problem” is one of the key open questions in mathematical fluid mechanics and, in fact, its importance for mathematics in general has been recognized by the Clay Mathematics Institute as one of its “millennium problems” [21].",
"Questions concerning global-in-time existence of smooth solutions remain open also for a number of other flow models including the 3D Euler equations [25] and some of the “active scalar” equations [34].",
"While the blow-up problem is fundamentally a question in mathematical analysis, a lot of computational studies have been carried out since the mid-'90s in order to shed light on the hydrodynamic mechanisms which might lead to singularity formation in finite time.",
"Given that such flows evolving near the edge of regularity involve formation of very small flow structures, these computations typically require the use of state-of-the-art computational resources available at a given time.",
"The computational studies focused on the possibility of finite-time blow-up in the 3D Navier-Stokes and/or Euler system include [11], [51], [10], [30], [49], [13], [46], [45], [26], [25], [28], [48], [12], [47], all of which considered problems defined on domains periodic in all three dimensions.",
"Recent investigations by [20], [32], [24], [31] focused on the time evolution of vorticity moments and compared it with the predictions derived from analysis based on rigorous bounds.",
"We also mention the studies by [43] and [53], along with the references found therein, in which various complexified forms of the Euler equation were investigated.",
"The idea of this approach is that, since the solutions to complexified equations have singularities in the complex plane, singularity formation in the real-valued problem is manifested by the collapse of the complex-plane singularities onto the real axis.",
"Overall, the outcome of these investigations is rather inconclusive: while for the Navier-Stokes flows most of recent computations do not offer support for finite-time blow-up, the evidence appears split in the case of the Euler system.",
"In particular, the recent studies by [12] and [48] hinted at the possibility of singularity formation in finite time.",
"In this connection we also mention the recent investigations by [40], [41] in which blow-up was observed in axisymmetric Euler flows in a bounded (tubular) domain.",
"A common feature of all of the aforementioned investigations was that the initial data for the Navier-Stokes or Euler system was chosen in an ad-hoc manner, based on some heuristic arguments.",
"On the other hand, in the present study we pursue a fundamentally different approach, proposed originally by [38] and employed also by [5], [6], [7] for a range of related problems, in which the initial data leading to the most singular behaviour is sought systematically via solution of a suitable variational optimization problem.",
"We carefully analyze the time evolution induced by the extreme vortex states first identified by [38] and compare it to the time evolution corresponding to a number of other candidate initial conditions considered in the literature [11], [30], [49], [15], [48].",
"We demonstrate that the Taylor-Green vortex, studied in the context of the blow-up problem by [54], [11], [10], [15], is in fact a particular member of the family of extreme vortex states maximizing the instantaneous rate of enstrophy production in the limit of vanishing enstrophy.",
"In addition, based on these findings, we identify the set of initial data, parameterized by its energy and enstrophy, for which one can a priori guarantee global-in-time existence of smooth solutions.",
"This result therefore offers a physically appealing interpretation of an “abstract” mathematical theorem concerning global existence of classical solutions corresponding to “small” initial data [36].",
"We also emphasize that, in order to establish a direct link with the results of the mathematical analysis discussed below, in our investigation we follow a rather different strategy than in most of the studies referenced above.",
"While these earlier studies relied on data from a relatively small number of simulations performed at a high (at the given time) resolution, in the present investigation we explore a broad range of cases, each of which is however computed at a more moderate resolution (or, equivalently, Reynolds number).",
"With such an approach to the use of available computational resources, we are able to reveal trends resulting from the variation of parameters which otherwise would be hard to detect.",
"Systematic computations conducted in this way thus allow us to probe the sharpness of the mathematical analysis relevant to the problem.",
"The question of regularity of solution to the Navier-Stokes system is usually addressed using “energy” methods which rely on finding upper bounds (with respect to time) on certain quantities of interest, typically taken as suitable Sobolev norms of the solution.",
"A key intermediate step is obtaining bounds on the rate of growth of the quantity of interest, a problem which can be studied with ODE methods.",
"While for the Navier-Stokes system different norms of the velocity gradient or vorticity can be used to study the regularity of solutions, the use of enstrophy ${\\mathcal {E}}$ (see equation (REF ) below) is privileged by the well-known result of [22], where it was established that if the uniform bound $\\mathop {\\sup }_{0 \\le t \\le T} {\\mathcal {E}}({\\bf u}(t)) < \\infty $ holds, then the regularity of the solution ${\\bf u}(t)$ is guaranteed up to time $T$ (to be precise, the solution remains in a suitable Gevrey class).",
"From the computational point of view, the enstrophy ${\\mathcal {E}}(t) := {\\mathcal {E}}({\\bf u}(t))$ is thus a convenient indicator of the regularity of solutions, because in the light of (REF ), singularity formation must manifest itself by the enstrophy becoming infinite.",
"While characterization of the maximum possible finite-time growth of enstrophy in the 3D Navier-Stokes flows is the ultimate objective of this research program, analogous questions can also be posed in the context of more tractable problems involving the one-dimensional (1D) Burgers equation and the two-dimensional (2D) Navier-Stokes equation.",
"Although global-in-time existence of the classical (smooth) solutions is well known for both these problems [35], questions concerning the sharpness of the corresponding estimates for the instantaneous and finite-time growth of various quantities are relevant, because these estimates are obtained using essentially the same methods as employed to derive their 3D counterparts.",
"Since in 2D flows on unbounded or periodic domains the enstrophy may not increase ($d{\\mathcal {E}}/dt \\le 0$ ), the relevant quantity in this case is the palinstrophy ${\\mathcal {P}}(\\mathbf {u}) := \\frac{1}{2}\\int _\\Omega |\\nabla \\mathbf {\\omega }(\\mathbf {x},t) |^2 \\,d\\mathbf {x}$ , where $\\mathbf {\\omega }:=\\nabla \\times \\mathbf {u}$ is the vorticity (which reduces to a pseudo-scalar in 2D).",
"Different questions concerning sharpness of estimates addressed in our research program are summarized together with the results obtained to date in Table REF .",
"We remark that the best finite-time estimate for the 1D Burgers equation was found not to be sharp using the initial data obtained from both the instantaneous and the finite-time variational optimization problems [5].",
"On the other hand, in 2D the bounds on both the instantaneous and finite-time growth of palinstrophy were found to be sharp and, somewhat surprisingly, both estimates were realized by the same family of incompressible vector fields parameterized by energy ${\\mathcal {K}}$ and palinstrophy ${\\mathcal {P}}$ , obtained as the solution of an instantaneous optimization problem [6].",
"It is worth mentioning that while the estimate for the instantaneous rate of growth of palinstrophy $d{\\mathcal {P}}/dt \\le C{\\mathcal {K}}^{1/2}{\\mathcal {P}}^{3/2}/\\nu $ (see Table REF ) was found to be sharp with respect to variations in palinstrophy, the estimate is in fact not sharp with respect to the prefactor $C_{\\mathbf {u},\\nu } = {\\mathcal {K}}^{1/2}/\\nu $ [4], with the correct prefactor being of the form $\\widetilde{C}_{\\mathbf {u},\\nu } =\\sqrt{\\log \\left({\\mathcal {K}}^{1/2}/\\nu \\right)}$ .",
"We add that what distinguishes the 2D problem, in regard to both the instantaneous and finite-time bounds, is that the RHS of these bounds are expressed in terms of two quantities, namely, energy ${\\mathcal {K}}$ and enstrophy ${\\mathcal {E}}$ , in contrast to the enstrophy alone appearing in the 1D and 3D estimates.",
"As a result, the 2D instantaneous optimization problem had to be solved subject to two constraints.",
"In the present investigation we advance the research program summarized in Table REF by assessing to what extent the finite-time growth of enstrophy predicted by the analytic estimates (REF ) and (REF ) can be actually realized by flow evolution starting from different initial conditions, including the extreme vortex states found by [38] to saturate the instantaneous estimate (REF ).",
"The key finding is that, at least for the range of modest enstrophy values we considered, the growth of enstrophy corresponding to this initial data, which has the form of two colliding axisymmetric vortex rings, is rapidly depleted and there is no indication of singularity formation in finite time.",
"Thus, should finite-time singularity be possible in the Navier-Stokes system, it is unlikely to result from initial conditions instantaneously maximizing the rate of growth of enstrophy.",
"We also provide a comprehensive characterization of the extreme vortex states which realize estimate (REF ) together with the resulting flow evolutions.",
"The structure of the paper is as follows: in the next section we present analytic estimates on the instantaneous and finite-time growth of enstrophy in 3D flows.",
"In § we formulate the variational optimization problems which will be solved to find the vortex states with the largest rate of enstrophy production and in § we provide an asymptotic representation for these optimal states in the limit of vanishing enstrophy.",
"In § we present numerically computed extreme vortex states corresponding to intermediate and large enstrophy values, while in § we analyze the temporal evolution corresponding to different initial data in order to compare it with the predictions of estimates (REF ) and (REF ).",
"Our findings are discussed in §, whereas conclusions and outlook are deferred to §.",
"Table: Summary of selected estimates for the instantaneousrate of growth and the growth over finite time of enstrophy and palinstrophyin 1D Burgers, 2D and 3D Navier-Stokes systems.",
"The quantities 𝒦{\\mathcal {K}} andℰ{\\mathcal {E}} are defined in () and ()."
],
[
"Bounds on the Growth of Enstrophy in 3D Navier-Stokes Flows",
"We consider the incompressible Navier-Stokes system defined on the 3D unit cube $\\Omega = [0,1]^3$ with periodic boundary conditions $\\partial _t\\mathbf {u}+ \\mathbf {u}\\cdot \\nabla \\mathbf {u}+ \\nabla p - \\nu \\Delta \\mathbf {u}& = 0 & &\\qquad \\mbox{in} \\,\\,\\Omega \\times (0,T), \\\\\\nabla \\cdot \\mathbf {u}& = 0 & & \\qquad \\mbox{in} \\,\\,\\Omega \\times [0,T), \\\\\\mathbf {u}(\\mathbf {x},0) & = \\mathbf {u}_0(\\mathbf {x}), & &$ where the vector $\\mathbf {u}= [u_1, u_2, u_3]$ is the velocity field, $p$ is the pressure and $\\nu >0$ is the coefficient of kinematic viscosity (hereafter we will set $\\nu =0.01$ which is the same value as used in the seminal study by [38]).",
"The velocity gradient $\\nabla \\mathbf {u}$ is the tensor with components $[\\nabla \\mathbf {u}]_{ij} =\\partial _j u_i$ , $i,j=1,2,3$ .",
"The fluid density $\\rho $ is assumed to be constant and equal to unity ($\\rho =1$ ).",
"The relevant properties of solutions to system () can be studied using energy methods, with the energy ${\\mathcal {K}}(\\mathbf {u})$ and its rate of growth given by ${\\mathcal {K}}(\\mathbf {u}) & := & \\frac{1}{2}\\int _\\Omega |\\mathbf {u}(\\mathbf {x},t)|^2 \\,d\\mathbf {x}, \\\\\\frac{d{\\mathcal {K}}(\\mathbf {u})}{dt} & = & -\\nu \\int _\\Omega |\\nabla \\mathbf {u}|^2 \\, d\\mathbf {x}, $ where “$:=$ ” means “equal to by definition”.",
"The enstrophy ${\\mathcal {E}}(\\mathbf {u})$ and its rate of growth are given by ${\\mathcal {E}}(\\mathbf {u}) & := & \\frac{1}{2}\\int _\\Omega | \\nabla \\times \\mathbf {u}(\\mathbf {x},t) |^2 \\,d\\mathbf {x}, \\\\\\frac{d{\\mathcal {E}}(\\mathbf {u})}{dt} & = & -\\nu \\int _\\Omega |\\Delta \\mathbf {u}|^2\\,d\\mathbf {x}+\\int _{\\Omega } \\mathbf {u}\\cdot \\nabla \\mathbf {u}\\cdot \\Delta \\mathbf {u}\\, d\\mathbf {x}=: {\\mathcal {R}}(\\mathbf {u}).",
"$ For incompressible flows with periodic boundary conditions we also have the following identity [18] $\\int _{\\Omega } |\\nabla \\times \\mathbf {u}|^2\\,d\\mathbf {x}= \\int _{\\Omega } |\\nabla \\mathbf {u}|^2\\,d\\mathbf {x}.$ Hence, combining (REF )–(REF ), the energy and enstrophy satisfy the system of ordinary differential equations $\\frac{d{\\mathcal {K}}(\\mathbf {u})}{dt} & = -2\\nu {\\mathcal {E}}(\\mathbf {u}), \\\\\\frac{d{\\mathcal {E}}(\\mathbf {u})}{dt} & = {\\mathcal {R}}(\\mathbf {u}).",
"$ A standard approach at this point is to try to upper-bound $d{\\mathcal {E}}/ dt$ and using standard techniques of functional analysis it is possible to obtain the following well-known estimate in terms of ${\\mathcal {K}}$ and ${\\mathcal {E}}$ [17] $\\frac{d{\\mathcal {E}}}{dt} \\le -\\nu \\frac{{\\mathcal {E}}^2}{{\\mathcal {K}}} + \\frac{c}{\\nu ^3} {\\mathcal {E}}^3$ for $c$ an absolute constant.",
"A related estimate expressed entirely in terms of the enstrophy ${\\mathcal {E}}$ is given by $\\frac{d{\\mathcal {E}}}{dt} \\le \\frac{27}{8\\,\\pi ^4\\,\\nu ^3} {\\mathcal {E}}^3.$ By simply integrating the differential inequality in (REF ) with respect to time we obtain the finite-time bound ${\\mathcal {E}}(t) \\le \\frac{{\\mathcal {E}}(0)}{\\sqrt{1 - \\frac{27}{4\\,\\pi ^4\\,\\nu ^3}\\,{\\mathcal {E}}(0)^2\\, t}}$ which clearly becomes infinite at time $t_0 = 4\\,\\pi ^4\\,\\nu ^3 /[27\\,{\\mathcal {E}}(0)^2]$ .",
"Thus, based on estimate (REF ), it is not possible to establish the boundedness of the enstrophy ${\\mathcal {E}}(t)$ globally in time and hence the regularity of solutions.",
"Therefore, the question about the finite-time singularity formation can be recast in terms of whether or not estimate (REF ) can be saturated.",
"By this we mean the existence of initial data with enstrophy ${\\mathcal {E}}_0 := {\\mathcal {E}}(0)> 0$ such that the resulting time evolution realizes the largest growth of enstrophy ${\\mathcal {E}}(t)$ allowed by the right-hand side (RHS) of estimate (REF ).",
"A systematic search for such most singular initial data using variational optimization methods is the key theme of this study.",
"Although different notions of sharpness of an estimate can be defined, e.g., sharpness with respect to constants or exponents in the case of estimates in the form of power laws, the precise notion of sharpness considered in this study is the following Definition 2.1 Given a parameter $p\\in \\mathbb {R}$ and maps $f,g:\\mathbb {R}\\rightarrow \\mathbb {R}$ , the estimate $f(p) \\le g(p)$ is declared sharp in the limit $p \\rightarrow p_0\\in \\mathbb {R}$ if and only if $\\lim _{p \\rightarrow p_0} \\frac{f(p)}{g(p)} \\sim \\beta , \\quad \\beta \\in {\\mathbb {R}}.$ From this definition, the sharpness of estimates in the form $g(p) =C\\, p^{\\alpha }$ for some $C \\in {\\mathbb {R}}_+$ and $\\alpha \\in {\\mathbb {R}}$ can be addressed in the limit $p \\rightarrow \\infty $ by studying the adequacy of the exponent $\\alpha $ .",
"The question of sharpness of estimate (REF ) was addressed in the seminal study by [38], see also [37], who constructed a family of divergence-free velocity fields saturating this estimate.",
"More precisely, these vector fields were parameterized by their enstrophy and for sufficiently large values of ${\\mathcal {E}}$ the corresponding rate of growth $d{\\mathcal {E}}/dt$ was found to be proportional to ${\\mathcal {E}}^3$ .",
"Therefore, in agreement with definition REF , estimate (REF ) was declared sharp up to a numerical prefactor.",
"However, the sharpness of the instantaneous estimate alone does not allow us to conclude about the possibility of singularity formation, because for this situation to occur, a sufficiently large enstrophy growth rate would need to be sustained over a finite time window $[0,t_0)$ .",
"In fact, assuming the instantaneous rate of growth of enstrophy in the form $d{\\mathcal {E}}/ dt =C \\, {\\mathcal {E}}^{\\alpha }$ for some $C>0$ , any exponent $\\alpha > 2$ will produce blow-up of ${\\mathcal {E}}(t)$ in finite time if the rate of growth is sustained.",
"The fact that there is no blow-up for $\\alpha \\le 2$ follows from Grönwall's lemma and the fact that one factor of ${\\mathcal {E}}$ in (REF ) can be bounded in terms of the initial energy using () as follows $\\int _0^t {\\mathcal {E}}(s)\\, ds = \\frac{1}{2\\nu } \\left[ {\\mathcal {K}}(0) - {\\mathcal {K}}(t)\\right] \\le \\frac{1}{2\\nu } {\\mathcal {K}}(0).$ This relation also leads to an alternative form of the estimate for the finite-time growth of enstrophy, namely $\\frac{d{\\mathcal {E}}}{dt} & \\le \\frac{27}{8\\pi ^4\\nu ^3}{\\mathcal {E}}^3\\quad \\Longrightarrow \\nonumber \\\\\\int _{{\\mathcal {E}}(0)}^{{\\mathcal {E}}(t)}{\\mathcal {E}}^{-2}\\,d{\\mathcal {E}}& \\le \\frac{27}{8\\pi ^4\\nu ^3}\\int _0^t{\\mathcal {E}}(s)\\,ds \\quad \\Longrightarrow \\nonumber \\\\\\frac{1}{{\\mathcal {E}}(0)} - \\frac{1}{{\\mathcal {E}}(t)} & \\le \\frac{27}{(2\\pi \\nu )^4}\\left[{\\mathcal {K}}(0) - {\\mathcal {K}}(t) \\right] $ which is more convenient than (REF ) from the computational point of view and will be used in the present study.",
"We note, however, that since the RHS of this inequality cannot be expressed entirely in terms of properties of the initial data, this is not in fact an a priori estimate.",
"Estimate (REF ) also allows us to obtain a condition on the size of the initial data, given in terms of its energy ${\\mathcal {K}}(0)$ and enstrophy ${\\mathcal {E}}(0)$ , which guarantees that smooth solutions will exist globally in time, namely, $\\mathop {\\max }_{t \\ge 0} {\\mathcal {E}}(t) \\le \\frac{{\\mathcal {E}}(0)}{1 - \\frac{27}{(2\\pi \\nu )^4}{\\mathcal {K}}(0){\\mathcal {E}}(0)}$ from which it follows that ${\\mathcal {K}}(0){\\mathcal {E}}(0) < \\frac{(2\\pi \\nu )^4}{27}.$ Thus, flows with energy and enstrophy satisfying inequality (REF ) are guaranteed to be smooth for all time, in agreement with the regularity results available under the assumption of small initial data [36]."
],
[
"Instantaneously Optimal Growth of Enstrophy",
"Sharpness of instantaneous estimate (REF ), in the sense of definition REF , can be probed by constructing a family of “extreme vortex states” $\\widetilde{\\mathbf {u}}_{{\\mathcal {E}}_0}$ which, for each ${\\mathcal {E}}_0 > 0$ , have prescribed enstrophy ${\\mathcal {E}}(\\widetilde{\\mathbf {u}}_{{\\mathcal {E}}_0}) ={\\mathcal {E}}_0$ and produce the largest possible rate of growth of enstrophy ${\\mathcal {R}}(\\widetilde{\\mathbf {u}}_{{\\mathcal {E}}_0})$ .",
"Given the form of (), the fields $\\widetilde{\\mathbf {u}}_{{\\mathcal {E}}_0}$ can be expected to exhibit (at least piecewise) smooth dependence on ${\\mathcal {E}}_0$ and we will refer to the mapping ${\\mathcal {E}}_0\\longmapsto \\widetilde{\\mathbf {u}}_{{\\mathcal {E}}_0}$ as a “maximizing branch”.",
"Thus, information about the sharpness of estimate (REF ) can be deduced by analyzing the relation ${\\mathcal {E}}_0$ versus ${\\mathcal {R}}(\\widetilde{\\mathbf {u}}_{{\\mathcal {E}}_0})$ obtained for a possibly broad range of enstrophy values.",
"A maximizing branch is constructed by finding, for different values of ${\\mathcal {E}}_0$ , the extreme vortex states $\\widetilde{\\mathbf {u}}_{{\\mathcal {E}}_0}$ as solutions of a variational optimization problem defined below.",
"Hereafter, $H^2(\\Omega )$ will denote the Sobolev space of functions with square-integrable second derivatives endowed with the inner product [1] $\\forall \\,\\mathbf {z}_1, \\mathbf {z}_2 \\in H^2(\\Omega ) \\qquad \\Big \\langle \\mathbf {z}_1, \\mathbf {z}_2 \\Big \\rangle _{H^2(\\Omega )}= \\int _{\\Omega } \\mathbf {z}_1 \\cdot \\mathbf {z}_2+ \\ell _1^2 \\,\\nabla \\mathbf {z}_1 \\colon \\nabla \\mathbf {z}_2+ \\ell _2^4 \\,\\Delta \\mathbf {z}_1 \\cdot \\Delta \\mathbf {z}_2 \\, d\\mathbf {x}, $ where $\\ell _1,\\ell _2\\in {\\mathbb {R}}_+$ are parameters with the meaning of length scales (the reasons for introducing these parameters in the definition of the inner product will become clear below).",
"The inner product in the space $L_2(\\Omega )$ is obtained from (REF ) by setting $\\ell _1 = \\ell _2 = 0$ .",
"The notation $H^2_0(\\Omega )$ will refer to the Sobolev space $H^2(\\Omega )$ of functions with zero mean.",
"For every fixed value ${\\mathcal {E}}_0$ of enstrophy we will look for a divergence-free vector field $\\widetilde{\\mathbf {u}}_{{\\mathcal {E}}_0}$ maximizing the objective function ${\\mathcal {R}}\\; : \\; H^2_0(\\Omega ) \\rightarrow {\\mathbb {R}}$ defined in ().",
"We thus have the following Problem 3.1 Given ${\\mathcal {E}}_0\\in \\mathbb {R}_+$ and the objective functional ${\\mathcal {R}}$ from equation (), find $\\widetilde{\\mathbf {u}}_{{\\mathcal {E}}_0}& = & \\mathop {\\arg \\max }_{\\mathbf {u}\\in \\mathcal {S}_{{\\mathcal {E}}_0}} \\, {\\mathcal {R}}(\\mathbf {u}) \\\\\\mathcal {S}_{{\\mathcal {E}}_0} & = & \\left\\lbrace \\mathbf {u}\\in H_0^2(\\Omega )\\,\\colon \\,\\nabla \\cdot \\mathbf {u}= 0, \\; {\\mathcal {E}}(\\mathbf {u}) = {\\mathcal {E}}_0 \\right\\rbrace $ which will be solved for enstrophy ${\\mathcal {E}}_0$ spanning a broad range of values.",
"This approach was originally proposed and investigated by [38].",
"In the present study we extend and generalize these results by first showing how other fields considered in the context of the blow-up problem for both the Euler and Navier-Stokes system, namely the Taylor-Green vortex, also arise from variational problem REF .",
"We then thoroughly analyze the time evolution corresponding to our extreme vortex states and compare it with the predictions of the finite-time estimates (REF ) and (REF ).",
"As discussed at the end of this section, some important aspects of our approach to solving problem REF are also quite different from the method adopted by [38].",
"The smoothness requirement in the statement of problem REF ($\\mathbf {u}\\in H_0^2(\\Omega )$ ) follows from the definition of the objective functional ${\\mathcal {R}}$ in equation (), where both the viscous term $\\nu \\int _\\Omega |\\Delta \\mathbf {u}|^2\\,d\\mathbf {x}$ and the cubic term $\\int _{\\Omega }\\mathbf {u}\\cdot \\nabla \\mathbf {u}\\cdot \\Delta \\mathbf {u}\\, d\\mathbf {x}$ contain derivatives of order up to two.",
"The constraint manifold $\\mathcal {S}_{{\\mathcal {E}}_0}$ can be interpreted as an intersection of the manifold (a subspace) $\\mathcal {S}_{0}\\in H_0^2(\\Omega )$ of divergence-free fields and the manifold $\\mathcal {S}^{\\prime }_{{\\mathcal {E}}_0} \\in H_0^2(\\Omega )$ of fields with prescribed enstrophy ${\\mathcal {E}}_0$ .",
"The structure of these constraint manifolds is reflected in the definition of the corresponding projections $\\mathbb {P}_{\\mathcal {S}}:H_0^2\\rightarrow \\mathcal {S}$ (without a subscript, $\\mathcal {S}$ refers to a generic manifold) which is given for each of the two constraints as follows: (div-free)-constraint: the projection of a field $\\mathbf {u}$ onto the subspace of solenoidal fields $\\mathcal {S}_0$ is performed using the Helmholtz decomposition; accordingly, every zero-mean vector field $\\mathbf {u}\\in H_0^2(\\Omega )$ can be decomposed uniquely as $\\mathbf {u}= \\nabla \\phi + \\nabla \\times \\mathbf {A},$ where $\\phi $ and $\\mathbf {A}$ are scalar and vector potentials, respectively; it follows from the identity $\\nabla \\cdot (\\nabla \\times \\mathbf {A})\\equiv 0$ , valid for any sufficiently smooth vector field $\\mathbf {A}$ , that the projection $\\mathbb {P}_{\\mathcal {S}_0}(\\mathbf {u})$ is given simply by $\\nabla \\times \\mathbf {A}$ and is therefore calculated as $\\mathbb {P}_{\\mathcal {S}_0}(\\mathbf {u}) = \\mathbf {u}- \\nabla \\left[\\Delta ^{-1}(\\nabla \\cdot \\mathbf {u})\\right],$ where $\\Delta ^{-1}$ is the inverse Laplacian associated with the periodic boundary conditions; the operator $\\mathbb {P}_{\\mathcal {S}_0}$ is also known as the Leray-Helmholtz projector.",
"$({\\mathcal {E}}_0)$ -constraint: the projection onto the manifold $\\mathcal {S}^{\\prime }_{{\\mathcal {E}}_0}$ is calculated by the normalization $\\mathbb {P}_{\\mathcal {S}^{\\prime }_{{\\mathcal {E}}_0}}(\\mathbf {u}) = \\sqrt{\\frac{{\\mathcal {E}}_0}{{\\mathcal {E}}\\left(\\mathbf {u}\\right)}}\\,\\mathbf {u}.$ Thus, composing (REF ) with (REF ), the projection onto the manifold $\\mathcal {S}_{{\\mathcal {E}}_0}$ defined in problem REF is constructed as $\\mathbb {P}_{\\mathcal {S}_{{\\mathcal {E}}_0}}(\\mathbf {u}) = \\mathbb {P}_{\\mathcal {S}^{\\prime }_{{\\mathcal {E}}_0}}\\Big ( \\mathbb {P}_{\\mathcal {S}_0} (\\mathbf {u})\\Big ).$ This approach, which was already successfully employed by [5], [6], allows one to enforce the enstrophy constraint essentially with the machine precision.",
"For a given value of ${\\mathcal {E}}_0$ , the maximizer $\\widetilde{\\mathbf {u}}_{{\\mathcal {E}}_0}$ can be found as $\\widetilde{\\mathbf {u}}_{{\\mathcal {E}}_0}= \\lim _{n\\rightarrow \\infty } \\mathbf {u}_{{\\mathcal {E}}_0}^{(n)}$ using the following iterative procedure representing a discretization of a gradient flow projected on $\\mathcal {S}_{{\\mathcal {E}}_0}$ $\\begin{aligned}\\mathbf {u}_{{\\mathcal {E}}_0}^{(n+1)} & = \\mathbb {P}_{\\mathcal {S}_{{\\mathcal {E}}_0}}\\left(\\;\\mathbf {u}^{(n)}_{{\\mathcal {E}}_0} + \\tau _n \\nabla {\\mathcal {R}}\\left(\\mathbf {u}^{(n)}_{{\\mathcal {E}}_0}\\right)\\;\\right), \\\\\\mathbf {u}_{{\\mathcal {E}}_0}^{(1)} & = \\mathbf {u}^0,\\end{aligned}$ where $\\mathbf {u}^{(n)}_{{\\mathcal {E}}_0}$ is an approximation of the maximizer obtained at the $n$ -th iteration, $\\mathbf {u}^0$ is the initial guess and $\\tau _n$ is the length of the step in the direction of the gradient.",
"It is ensured that the maximizers $\\widetilde{\\mathbf {u}}_{{\\mathcal {E}}_0}$ obtained for different values of ${\\mathcal {E}}_0$ lie on the same maximizing branch by using the continuation approach, where the maximizer $\\widetilde{\\mathbf {u}}_{{\\mathcal {E}}_0}$ is employed as the initial guess $\\mathbf {u}^0$ to compute $\\widetilde{\\mathbf {u}}_{{\\mathcal {E}}_0+\\Delta {\\mathcal {E}}}$ at the next enstrophy level for some sufficiently small $\\Delta {\\mathcal {E}}> 0$ .",
"As will be demonstrated in §, in the limit ${\\mathcal {E}}_0 \\rightarrow 0$ optimization problem REF admits a discrete family of closed-form solutions and each of these vortex states is the limiting (initial) member $\\widetilde{\\mathbf {u}}_{0}$ of the corresponding maximizing branch.",
"As such, these limiting extreme vortex states are used as the initial guesses $\\mathbf {u}^0$ for the calculation of $\\widetilde{\\mathbf {u}}_{\\Delta {\\mathcal {E}}}$ , i.e., they serve as “seeds” for the calculation of an entire maximizing branch (as discussed in §, while there exist alternatives to the continuation approach, this technique in fact results in the fastest convergence of iterations (REF ) and also ensures that all computed extreme vortex states lie on a single branch).",
"The procedure outlined above is summarized as Algorithm , whereas all details are presented below.",
"[t!]",
"set ${\\mathcal {E}}_0 = 0$ set $\\widetilde{\\mathbf {u}}_{{\\mathcal {E}}_0}= \\widetilde{\\mathbf {u}}_{0}$ ———————— loop over increasing enstrophy values ${\\mathcal {E}}_0$ ———————— $\\mathbf {u}_{{\\mathcal {E}}_0}^{(0)} = \\widetilde{\\mathbf {u}}_{{\\mathcal {E}}_0}$ ${\\mathcal {E}}_0 = {\\mathcal {E}}_0 + \\Delta {\\mathcal {E}}$ $n = 0$ compute ${\\mathcal {R}}_0 = {\\mathcal {R}}\\left(\\mathbf {u}_{{\\mathcal {E}}_0}^{(0)}\\right)$ —————————— optimization iterations (REF ) —————————— compute the $L_2$ gradient $\\nabla ^{L_2}{\\mathcal {R}}\\left(\\mathbf {u}_{{\\mathcal {E}}_0}^{(n)}\\right)$ , see equation (REF ) compute the Sobolev gradient $\\nabla {\\mathcal {R}}\\left(\\mathbf {u}_{{\\mathcal {E}}_0}^{(n)}\\right)$ , see equation (REF ) compute the step size $\\tau _n$ , see equation (REF ) set $\\mathbf {u}_{{\\mathcal {E}}_0}^{(n+1)} = \\mathbb {P}_{\\mathcal {S}_{{\\mathcal {E}}_0}}\\left(\\;\\mathbf {u}_{{\\mathcal {E}}_0}^{(n)} + \\tau _n \\nabla {\\mathcal {R}}\\left(\\mathbf {u}_{{\\mathcal {E}}_0}^{(n)}\\right)\\;\\right)$ , see equations (REF )–(REF ) set ${\\mathcal {R}}_1 = {\\mathcal {R}}\\left(\\mathbf {u}_{{\\mathcal {E}}_0}^{(n+1)}\\right)$ compute the relative error $ = ({\\mathcal {R}}_1 - {\\mathcal {R}}_0)/{\\mathcal {R}}_0$ set ${\\mathcal {R}}_0 = {\\mathcal {R}}_1$ set $n=n+1$ relative error < $\\epsilon $ ${\\mathcal {E}}_0 > {\\mathcal {E}}_{\\text{max}}$ Computation of a maximizing branch using continuation approach.",
"Input: $\\widetilde{\\mathbf {u}}_{0}$ — limiting extreme vortex state (corresponding to ${\\mathcal {E}}_0 \\rightarrow 0$ , see Table REF ) ${\\mathcal {E}}_{\\text{max}}$ — maximum enstrophy $\\Delta {\\mathcal {E}}$ — (adjustable) enstrophy increment $\\epsilon $ — tolerance in the solution of optimization problem REF via iterations (REF ) $\\ell _1,\\ell _2$ — adjustable length scales defining inner product (REF ), see also (REF ) Output: branch of extreme vortex states $\\widetilde{\\mathbf {u}}_{{\\mathcal {E}}_0}$ , $0 \\le {\\mathcal {E}}_0 \\le {\\mathcal {E}}_{\\text{max}}$ A key step of Algorithm is the evaluation of the gradient $\\nabla {\\mathcal {R}}(\\mathbf {u})$ of the objective functional ${\\mathcal {R}}(\\mathbf {u})$ , cf.",
"(), representing its (infinite-dimensional) sensitivity to perturbations of the velocity field $\\mathbf {u}$ , and it is essential that the gradient be characterized by the required regularity, namely, $\\nabla {\\mathcal {R}}(\\mathbf {u}) \\in H^2(\\Omega )$ .",
"This is, in fact, guaranteed by the Riesz representation theorem [39] applicable because the Gâteaux differential ${\\mathcal {R}}^{\\prime }(\\mathbf {u};\\cdot ) : H_0^2(\\Omega ) \\rightarrow {\\mathbb {R}}$ , defined as ${\\mathcal {R}}^{\\prime }(\\mathbf {u};\\mathbf {u}^{\\prime }) := \\lim _{\\epsilon \\rightarrow 0}\\epsilon ^{-1}\\left[{\\mathcal {R}}(\\mathbf {u}+\\epsilon \\mathbf {u}^{\\prime }) - {\\mathcal {R}}(\\mathbf {u})\\right]$ for some perturbation $\\mathbf {u}^{\\prime } \\in H_0^2(\\Omega )$ , is a bounded linear functional on $H_0^2(\\Omega )$ .",
"The Gâteaux differential can be computed directly to give ${\\mathcal {R}}^{\\prime }(\\mathbf {u};\\mathbf {u}^{\\prime }) = \\int _{\\Omega }\\left[\\mathbf {u}^{\\prime }\\cdot \\nabla \\mathbf {u}\\cdot \\Delta \\mathbf {u}+\\mathbf {u}\\cdot \\nabla \\mathbf {u}^{\\prime }\\cdot \\Delta \\mathbf {u}+\\mathbf {u}\\cdot \\nabla \\mathbf {u}\\cdot \\Delta \\mathbf {u}^{\\prime } \\right]\\,d\\mathbf {x}-2\\nu \\int _{\\Omega }\\Delta ^2\\mathbf {u}\\cdot \\mathbf {u}^{\\prime }\\,d\\mathbf {x}$ from which, by the Riesz representation theorem, we obtain ${\\mathcal {R}}^{\\prime }(\\mathbf {u};\\mathbf {u}^{\\prime })= \\Big \\langle \\nabla {\\mathcal {R}}(\\mathbf {u}), \\mathbf {u}^{\\prime } \\Big \\rangle _{H^2(\\Omega )}= \\Big \\langle \\nabla ^{L_2}{\\mathcal {R}}(\\mathbf {u}), \\mathbf {u}^{\\prime } \\Big \\rangle _{L_2(\\Omega )}$ with the Riesz representers $\\nabla {\\mathcal {R}}(\\mathbf {u})$ and $\\nabla ^{L_2}{\\mathcal {R}}(\\mathbf {u})$ being the gradients computed with respect to the $H^2$ and $L_2$ topology, respectively, and the inner products defined in (REF ).",
"We remark that, while the $H^2$ gradient is used exclusively in the actual computations, cf.",
"(REF ), the $L_2$ gradient is computed first as an intermediate step.",
"Identifying the Gâteaux differential (REF ) with the $L_2$ inner product and performing integration by parts yields $\\nabla ^{L_2}{\\mathcal {R}}(\\mathbf {u}) = \\Delta \\left( \\mathbf {u}\\cdot \\nabla \\mathbf {u}\\right) + (\\nabla \\mathbf {u})^T\\Delta \\mathbf {u}-\\mathbf {u}\\cdot \\nabla (\\Delta \\mathbf {u}) - 2\\nu \\Delta ^2\\mathbf {u}.$ Similarly, identifying the Gâteaux differential (REF ) with the $H^2$ inner product (REF ), integrating by parts and using (REF ), we obtain the required $H^2$ gradient $\\nabla {\\mathcal {R}}$ as a solution of the following elliptic boundary-value problem $\\begin{aligned}&\\left[ \\operatorname{Id}\\, - \\,\\ell _1^2 \\,\\Delta + \\,\\ell _2^4 \\,\\Delta ^2 \\right] \\nabla {\\mathcal {R}}= \\nabla ^{L_2} {\\mathcal {R}}\\qquad \\text{in} \\ \\Omega , \\\\& \\text{Periodic Boundary Conditions}.\\end{aligned}$ The gradient fields $\\nabla ^{L_2}{\\mathcal {R}}(\\mathbf {u})$ and $\\nabla {\\mathcal {R}}(\\mathbf {u})$ can be interpreted as infinite-dimensional sensitivities of the objective function ${\\mathcal {R}}(\\mathbf {u})$ , cf.",
"(), with respect to perturbations of the field $\\mathbf {u}$ .",
"While these two gradients may point towards the same local maximizer, they represent distinct “directions”, since they are defined with respect to different topologies ($L_2$ vs. $H^2$ ).",
"As shown by [50], extraction of gradients in spaces of smoother functions such as $H^2(\\Omega )$ can be interpreted as low-pass filtering of the $L_2$ gradients with parameters $\\ell _1$ and $\\ell _2$ acting as cut-off length-scales and the choice of their numerical values will be discussed in §.",
"The step size $\\tau _n$ in algorithm (REF ) is computed as $\\tau _n = \\mathop {\\operatorname{argmax}}_{\\tau >0} \\left\\lbrace {\\mathcal {R}}\\left[\\mathbb {P}_{\\mathcal {S}_{{\\mathcal {E}}_0}}\\left( \\;\\mathbf {u}^{(n)} + \\tau \\,\\nabla {\\mathcal {R}}(\\mathbf {u}^{(n)}) \\;\\right)\\right] \\right\\rbrace $ which is done using a suitable derivative-free line-search algorithm [52].",
"Equation (REF ) can be interpreted as a modification of a standard line search method where the optimization is performed following an arc (a geodesic) lying on the constraint manifold $\\mathcal {S}_{{\\mathcal {E}}_0}$ , rather than a straight line.",
"This approach was already successfully employed to solve similar problems in [5], [6].",
"It ought to be emphasized here that the approach presented above in which the projections (REF )–(REF ) and gradients (REF )–(REF ) are obtained based on the infinite-dimensional (continuous) formulation to be discretized only at the final stage is fundamentally different from the method employed in the original study by [38] in which the optimization problem was solved in a fully discrete setting (the two approaches are referred to as “optimize-then-discretize” and “discretize-then-optimize”, respectively, cf. [27]).",
"A practical advantage of the continuous (“optimize-then-discretize”) formulation used in the present work is that the expressions representing the sensitivity of the objective functional ${\\mathcal {R}}$ , i.e.",
"the gradients $\\nabla ^{L_2}{\\mathcal {R}}$ and $\\nabla {\\mathcal {R}}$ , are independent of the specific discretization approach chosen to evaluate them.",
"This should be contrasted with the discrete (“discretize-then-optimize”) formulation, where a change of the discretization method would require rederivation of the gradient expressions.",
"In addition, the continuous formulation allows us to strictly enforce the regularity of maximizers required in problem REF .",
"Finally and perhaps most importantly, the continuous formulation of the maximization problem makes it possible to obtain elegant closed-form solutions of the problem in the limit ${\\mathcal {E}}_0 \\rightarrow 0$ , which is done in § below.",
"These analytical solutions will then be used in § to guide the computation of maximizing branches by numerically solving problem REF for a broad range of ${\\mathcal {E}}_0$ , as outlined in Algorithm ."
],
[
"Extreme Vortex States in the Limit ${\\mathcal {E}}_0 \\rightarrow 0$",
"It is possible to find analytic solutions to problem REF in the limit ${\\mathcal {E}}_0 \\rightarrow 0$ using perturbation methods.",
"To simplify the notation, in this section we will drop the subscript ${\\mathcal {E}}_0$ when referring to the optimal field.",
"The Euler-Lagrange system representing the first-order optimality conditions in optimization problem REF is given by [39] $\\mathcal {B}(\\widetilde{\\mathbf {u}},\\widetilde{\\mathbf {u}}) - 2\\nu \\Delta ^2\\widetilde{\\mathbf {u}}- \\lambda \\Delta \\widetilde{\\mathbf {u}}- \\nabla q & = 0 \\qquad \\mbox{in}\\,\\,\\Omega , \\\\\\nabla \\cdot \\widetilde{\\mathbf {u}}& = 0 \\qquad \\mbox{in}\\,\\,\\Omega , \\\\{\\mathcal {E}}(\\widetilde{\\mathbf {u}}) - {\\mathcal {E}}_0 & = 0, $ where $\\lambda \\in \\mathbb {R}$ and $q:\\Omega \\rightarrow \\mathbb {R}$ are the Lagrange multipliers associated with the constraints defining the manifold $\\mathcal {S}_{{\\mathcal {E}}_0}$ , and $\\mathcal {B}(\\mathbf {u},\\mathbf {v})$ , given by $\\mathcal {B}(\\mathbf {u},\\mathbf {v}) := \\Delta \\left( \\mathbf {u}\\cdot \\nabla \\mathbf {v}\\right) + (\\nabla \\mathbf {u})^T\\Delta \\mathbf {v}-\\mathbf {u}\\cdot \\nabla (\\Delta \\mathbf {v}),$ is the bilinear form from equation (REF ).",
"Using the formal series expansions with $\\alpha > 0$ $\\widetilde{\\mathbf {u}}& = \\mathbf {u}_0 + {\\mathcal {E}}_0^{\\alpha }\\mathbf {u}_1 + {\\mathcal {E}}_0^{2\\alpha }\\mathbf {u}_2 + \\ldots , \\\\\\lambda & = \\lambda _0 + {\\mathcal {E}}_0^{\\alpha }\\lambda _1 + {\\mathcal {E}}_0^{2\\alpha }\\lambda _2 + \\ldots , \\\\q & = q_0 + {\\mathcal {E}}_0^{\\alpha }q_1 + {\\mathcal {E}}_0^{2\\alpha }q_2 + \\ldots $ in () and collecting terms proportional to different powers of ${\\mathcal {E}}_0^{\\alpha }$ , it follows from (REF ) that, at every order $m=1,2,\\dots $ in ${\\mathcal {E}}_0^{\\alpha }$ , we have ${\\mathcal {E}}_0^{m\\alpha }: \\qquad \\sum _{j=0}^m \\mathcal {B}(\\mathbf {u}_j,\\mathbf {u}_{m-j}) - 2\\nu \\Delta ^2\\mathbf {u}_m -\\sum _{j=0}^m\\lambda _j\\Delta \\mathbf {u}_{m-j} - \\nabla q_m = 0 \\quad \\mbox{in}\\,\\,\\Omega .$ Similarly, equation () leads to $\\nabla \\cdot \\mathbf {u}_m = 0 \\quad \\mbox{in}\\,\\,\\Omega $ at every order $m$ in ${\\mathcal {E}}_0^{\\alpha }$ .",
"It then follows from equation () that ${\\mathcal {E}}(\\mathbf {u}) & = & {\\mathcal {E}}(\\mathbf {u}_0) -\\big \\langle \\mathbf {u}_0,\\Delta \\mathbf {u}_1\\big \\rangle _{L_2}{\\mathcal {E}}_0^{\\alpha } +\\left[ {\\mathcal {E}}(\\mathbf {u}_1) - \\big \\langle \\mathbf {u}_0,\\Delta \\mathbf {u}_2\\big \\rangle _{L_2}\\right]{\\mathcal {E}}_0^{2\\alpha } + \\ldots \\\\& = & {\\mathcal {E}}_0,$ which, for $\\alpha \\ne 0$ , forces ${\\mathcal {E}}(\\mathbf {u}_0) = 0$ .",
"Hence, $\\mathbf {u}_0\\equiv 0$ , $\\alpha = 1/2$ and ${\\mathcal {E}}(\\mathbf {u}_1) = 1$ .",
"The systems at orders ${\\mathcal {E}}_0^{1/2}$ and ${\\mathcal {E}}_0^1$ are given by: ${\\mathcal {E}}^{1/2}_0:\\quad \\qquad \\qquad \\qquad \\qquad \\qquad 2\\nu \\Delta ^2\\mathbf {u}_1 + \\lambda _0\\Delta \\mathbf {u}_1 +\\nabla q_1 & = 0 \\quad \\mbox{in}\\,\\,\\Omega , \\\\\\nabla \\cdot \\mathbf {u}_1 & = 0 \\quad \\mbox{in}\\,\\,\\Omega , \\\\{\\mathcal {E}}(\\mathbf {u}_1) & = 1, $ ${\\mathcal {E}}_0:\\qquad 2\\nu \\Delta ^2\\mathbf {u}_2 + \\lambda _0\\Delta \\mathbf {u}_2 + \\nabla q_2 - \\mathcal {B}(\\mathbf {u}_1,\\mathbf {u}_1) +\\lambda _1\\Delta \\mathbf {u}_1 & = 0 \\quad \\mbox{in}\\,\\,\\Omega , \\\\\\nabla \\cdot \\mathbf {u}_2 & = 0 \\quad \\mbox{in}\\,\\,\\Omega , \\\\\\langle \\Delta \\mathbf {u}_1, \\mathbf {u}_2 \\rangle _{L_2} & = 0,$ where the fact that $\\mathcal {B}(\\mathbf {u}_0,\\mathbf {u}_j) = 0$ for all $j$ has been used.",
"While continuing this process to larger values of $m$ may lead to some interesting insights, for the purpose of this investigation it is sufficient to truncate expansions () at the order $\\mbox{\\textit {O}}({\\mathcal {E}}_0)$ .",
"The corresponding approximation of the objective functional () then becomes ${\\mathcal {R}}(\\widetilde{\\mathbf {u}}) = - \\nu {\\mathcal {E}}_0\\int _{\\Omega } \\left| \\Delta \\mathbf {u}_1 \\right|^2 \\, d\\mathbf {x}+ \\mbox{\\textit {O}}({\\mathcal {E}}_0^{3/2}).$ It is worth noting that, in the light of relation (REF ), the maximum rate of growth of enstrophy in the limit of small ${\\mathcal {E}}_0$ is in fact negative, meaning that, for sufficiently small ${\\mathcal {E}}_0$ , the enstrophy itself is a decreasing function for all times.",
"This observation is consistent with the small-data regularity result discussed in Introduction.",
"As regards problem () defining the triplet $\\lbrace \\mathbf {u}_1, q_1,\\lambda _0\\rbrace $ , taking the divergence of equation (REF ) and using the condition $\\nabla \\cdot \\mathbf {u}_1=0$ leads to the Laplace equation $\\Delta q_1 =0$ in $\\Omega $ .",
"Since for zero-mean functions defined on $\\Omega $ , $\\operatorname{Ker}(\\Delta ) = \\lbrace 0 \\rbrace $ , it follows that $q_1\\equiv 0$ and equation (REF ) is reduced to the eigenvalue problem $2\\nu \\Delta \\mathbf {u}_1 + \\lambda _0\\mathbf {u}_1 = 0,$ with $\\mathbf {u}_1$ satisfying the incompressibility condition ().",
"Direct calculation using equation (REF ) and condition () leads to an asymptotic expression for the objective functional in the limit of small enstrophy ${\\mathcal {R}}(\\widetilde{\\mathbf {u}}) \\approx - \\lambda _0{\\mathcal {E}}_0.$ Solutions to the eigenvalue problem in equation (REF ) can be found using the Fourier expansion of $\\mathbf {u}_1$ given as (with hats denoting Fourier coefficients) $\\mathbf {u}_1(\\mathbf {x}) = \\sum _{\\mathbf {k}\\in \\mathcal {W}}\\widehat{\\mathbf {u}}_1(\\mathbf {k})\\textrm {e}^{2\\pi i\\mathbf {k}\\cdot \\mathbf {x}},$ where $\\mathcal {W}\\subseteq \\mathbb {Z}^3$ is a set of wavevectors $\\mathbf {k}$ for which $\\widehat{\\mathbf {u}}_1(\\mathbf {k}) \\ne 0$ .",
"The eigenvalue problem (REF ) then becomes $\\left[-2\\nu (2\\pi )^2|\\mathbf {k}|^2 + \\lambda _0\\right]\\widehat{\\mathbf {u}}_1(\\mathbf {k}) & = & 0 \\qquad \\forall \\,\\mathbf {k}\\in \\mathcal {W}, \\\\\\widehat{\\mathbf {u}}_1(\\mathbf {k})\\cdot \\mathbf {k}& = & 0 \\qquad \\forall \\,\\mathbf {k}\\in \\mathcal {W},$ with solutions obtained by choosing, for any $k\\in \\mathbb {Z}\\setminus \\lbrace 0\\rbrace $ , a set of wavevectors with the following structure $\\mathcal {W}_k = \\left\\lbrace \\mathbf {k}\\in \\mathbb {Z}^3\\colon |\\mathbf {k}|^2 = k \\right\\rbrace $ and $\\widehat{\\mathbf {u}}_1(\\mathbf {k})$ with an appropriate form satisfying the incompressibility condition $\\widehat{\\mathbf {u}}_1\\cdot \\mathbf {k}= 0$ .",
"For the solutions to equation (REF ) constructed in such manner it then follows that $\\lambda _0 = 2\\nu (2\\pi )^2|\\mathbf {k}|^2$ and the optimal asymptotic value of ${\\mathcal {R}}$ is given by ${\\mathcal {R}}(\\widetilde{\\mathbf {u}}) \\approx - 8\\pi ^2\\nu |\\mathbf {k}|^2{\\mathcal {E}}_0.$ Since the fields $\\mathbf {u}_1$ are real-valued, their Fourier modes must satisfy $\\widehat{\\mathbf {u}}_1(-\\mathbf {k}) =\\overline{\\widehat{\\mathbf {u}}_1(\\mathbf {k})}$ , where $\\overline{z}$ denotes the complex conjugate (C.C.)",
"of $z\\in \\mathbb {C}$ .",
"Depending on the choice of $\\mathcal {W}_k$ , a number of different solutions of () can be constructed and below we focus on the following three most relevant cases characterized by the largest values of ${\\mathcal {R}}(\\widetilde{\\mathbf {u}})$ : $\\mathcal {W}_1 = \\lbrace \\mathbf {k}_1, \\mathbf {k}_2, \\mathbf {k}_3, -\\mathbf {k}_1, -\\mathbf {k}_2,-\\mathbf {k}_3 \\rbrace $ , where $\\mathbf {k}_i = \\mathbf {e}_i$ , $i=1,2,3$ , is the $i^{\\textrm {th}}$ unit vector of the canonical basis of $\\mathbb {R}^3$ ; the most general solution can then be constructed as $\\mathbf {u}_1(\\mathbf {x}) = \\mathbf {A}\\textrm {e}^{2\\pi i\\mathbf {k}_1\\cdot \\mathbf {x}} +\\mathbf {B}\\textrm {e}^{2\\pi i\\mathbf {k}_2\\cdot \\mathbf {x}} +\\mathbf {C}\\textrm {e}^{2\\pi i\\mathbf {k}_3\\cdot \\mathbf {x}} + \\textrm {C.C.",
"}$ with the complex-valued constant vectors $\\mathbf {A} = [0,A_2,A_3]$ , $\\mathbf {B} = [B_1,0,B_3]$ and $\\mathbf {C} = [C_1,C_2,0]$ suitably chosen so that ${\\mathcal {E}}(\\mathbf {u}_1) = 1$ ; hereafter we will use the values $A_2 =A_3 = \\ldots = C_2 = 1/(48\\pi ^2)$ ; it follows that $|\\mathbf {k}|^2 =1$ $\\forall \\,\\,\\mathbf {k}\\in \\mathcal {W}_1$ , and the optimal asymptotic value of ${\\mathcal {R}}$ obtained from equation (REF ) is given by ${\\mathcal {R}}(\\widetilde{\\mathbf {u}}) \\approx - 8\\pi ^2\\nu {\\mathcal {E}}_0,$ $\\mathcal {W}_2 = \\mathcal {W}\\cup (-\\mathcal {W})$ , where $-\\mathcal {W}$ denotes the set whose elements are the negatives of the elements of set $\\mathcal {W}$ , for $\\mathcal {W}= \\lbrace \\mathbf {k}_1 + \\mathbf {k}_2, \\mathbf {k}_1 - \\mathbf {k}_2, \\mathbf {k}_1 + \\mathbf {k}_3, \\mathbf {k}_1 -\\mathbf {k}_3, \\mathbf {k}_2 + \\mathbf {k}_3, \\mathbf {k}_2 - \\mathbf {k}_3 \\rbrace $ ; the most general solution can be then constructed as $\\mathbf {u}_1(\\mathbf {x}) & = & \\mathbf {A}\\textrm {e}^{2\\pi i[1,1,0]\\cdot \\mathbf {x}} +\\mathbf {B}\\textrm {e}^{2\\pi i[1,-1,0]\\cdot \\mathbf {x}} +\\mathbf {C}\\textrm {e}^{2\\pi i[1,0,1]\\cdot \\mathbf {x}} + \\nonumber \\\\& & \\mathbf {D}\\textrm {e}^{2\\pi i[1,0,-1]\\cdot \\mathbf {x}} +\\mathbf {E}\\textrm {e}^{2\\pi i[0,1,1]\\cdot \\mathbf {x}} +\\mathbf {F}\\textrm {e}^{2\\pi i[0,1,-1]\\cdot \\mathbf {x}} + \\textrm {C.C.",
"}$ with the constants $\\mathbf {A},\\mathbf {B},\\ldots ,\\mathbf {F}\\in \\mathbb {C}^3$ suitably chosen so that $\\mathbf {A}\\cdot [1,1,0] = 0$ , $\\mathbf {B}\\cdot [1,-1,0]= 0 ,\\ldots ,\\mathbf {F}\\cdot [0,1,-1] = 0$ , which ensures that incompressibility condition () is satisfied, and that ${\\mathcal {E}}(\\mathbf {u}_1) = 1$ ; in this case, $|\\mathbf {k}|^2 =2$ , $\\forall \\,\\mathbf {k}\\in \\mathcal {W}_2$ , and the optimal asymptotic value of ${\\mathcal {R}}$ is ${\\mathcal {R}}(\\widetilde{\\mathbf {u}}) \\approx - 16\\pi ^2\\nu {\\mathcal {E}}_0,$ $\\mathcal {W}_3 = \\mathcal {W}\\cup (-\\mathcal {W})$ for $\\mathcal {W}= \\lbrace \\mathbf {k}_1+\\mathbf {k}_2+\\mathbf {k}_3,-\\mathbf {k}_1+\\mathbf {k}_2+\\mathbf {k}_3,\\mathbf {k}_1-\\mathbf {k}_2+\\mathbf {k}_3,\\mathbf {k}_1+\\mathbf {k}_2-\\mathbf {k}_3\\rbrace $ ; the most general solution can then be constructed as $\\mathbf {u}_1(\\mathbf {x}) & = & \\mathbf {A}\\textrm {e}^{2\\pi i[1,1,1]\\cdot \\mathbf {x}} +\\mathbf {B}\\textrm {e}^{2\\pi i[-1,1,1]\\cdot \\mathbf {x}} + \\nonumber \\\\& & \\mathbf {C}\\textrm {e}^{2\\pi i[1,-1,1]\\cdot \\mathbf {x}} +\\mathbf {D}\\textrm {e}^{2\\pi i[1,1,-1]\\cdot \\mathbf {x}} + \\textrm {C.C.",
"}$ with the constants $\\mathbf {A},\\mathbf {B},\\mathbf {C},\\mathbf {D}\\in \\mathbb {C}^3$ suitably chosen so that $\\mathbf {A}\\cdot [1,1,1] = 0$ , $\\mathbf {B}\\cdot [-1,1,1]= 0$ , $\\mathbf {C}\\cdot [1,-1,1] = 0$ and $\\mathbf {D}\\cdot [1,1,-1] = 0$ , which ensures that incompressibility condition () is satisfied, and that ${\\mathcal {E}}(\\mathbf {u}_1) = 1$ ; in this case, $|\\mathbf {k}|^2 =3$ , $\\forall \\,\\mathbf {k}\\in \\mathcal {W}_3$ , and the optimal asymptotic value of ${\\mathcal {R}}$ is ${\\mathcal {R}}(\\widetilde{\\mathbf {u}}) \\approx - 24\\pi ^2\\nu {\\mathcal {E}}_0.$ The three constructions of the extremal field $\\mathbf {u}_1$ given in (REF ), (REF ) and (REF ) are all defined up to arbitrary shifts in all three directions, reflections with respect to different planes and rotations by angles which are multiples of $\\pi / 2$ about the different axes.",
"As a result of this nonuniqueness, there is some freedom in choosing the constants $\\mathbf {A},\\ldots ,\\mathbf {F}$ .",
"Given that the optimal asymptotic value of ${\\mathcal {R}}$ depends exclusively on the wavevector magnitude $|\\mathbf {k}|$ , cf.",
"(REF ), any combination of constants $\\mathbf {A},\\ldots ,\\mathbf {F}$ will produce the same optimal rate of growth of enstrophy.",
"Thus, to fix attention, in our analysis we will set $\\mathbf {A}=\\mathbf {B}=\\mathbf {C}$ in case (i), $\\mathbf {A}=\\mathbf {B}=\\ldots =\\mathbf {F}$ in case (REF ) and $\\mathbf {A}=\\ldots =\\mathbf {D}$ in case (REF ).",
"With these choices, the contribution from each component of the field $\\mathbf {u}_1$ to the total enstrophy is the same.",
"The maximum (i.e., least negative) value of ${\\mathcal {R}}$ can be thus obtained by choosing the smallest possible $|\\mathbf {k}|^2$ .",
"This maximum is achieved in case (REF ) with the wavevectors $\\mathbf {k}_1 = [1,0,0]$ , $\\mathbf {k}_2 = [0,1,0]$ , $\\mathbf {k}_3 =[0,0,1]$ , and $-\\mathbf {k}_1$ , $-\\mathbf {k}_2$ and $-\\mathbf {k}_3$ , for which $|\\mathbf {k}|^2 = 1$ .",
"Because of this maximization property, this is the field we will focus on in our analysis in § and §.",
"The three fields constructed in (REF ), (REF ) and (REF ) are visualized in figure REF .",
"This analysis is performed using the level sets $\\Gamma _{s}(F)\\subset \\Omega $ defined as $\\Gamma _{s}(F) := \\lbrace \\mathbf {x}\\in \\Omega : F(\\mathbf {x}) = s \\rbrace ,$ for a suitable function $F:\\Omega \\rightarrow \\mathbb {R}$ .",
"In figures REF (a–c) we choose $F(\\mathbf {x}) =|\\nabla \\times \\mathbf {u}_1|(\\mathbf {x})$ with $s = 0.95||\\nabla \\times \\mathbf {u}_1||_{L_\\infty }$ .",
"To complement this information, in figures REF (d–f) we also plot the isosurfaces and cross-sectional distributions of the $x_1$ component of the field $\\mathbf {u}_1$ .",
"The fields shown in figure REF reveal interesting patterns involving well-defined “vortex cells”.",
"More specifically, we see that in case (REF ), given by equation (REF ) and shown in figures REF (a,d), the vortex cells are staggered with respect to the orientation of the cubic domain $\\Omega $ in all three planes, whereas in case (REF ), given by equation (REF ) and shown in figures REF (c,f), the vortex cells are aligned with the domain $\\Omega $ in all three planes.",
"On the other hand, in case (REF ), given by equation (REF ) and shown in figures REF (b,e), the vortex cells are staggered in one plane and aligned in another with the arrangement in the third plane resulting from the arrangement in the first two.",
"These geometric properties are also reflected in the $x_1$ -component of the field $\\mathbf {u}_1$ which is independent of $x_1$ in cases (REF ) and (REF ), but exhibits, respectively, a staggered and aligned arrangement of the cells in the $y-z$ plane in these two cases.",
"In case (REF ) the cells exhibit an aligned arrangement in all three planes.",
"The geometric properties of the extreme vortex states obtained in the limit ${\\mathcal {E}}_0 \\rightarrow 0$ are summarized in Table REF .",
"We remark that an analogous structure of the optimal fields, featuring aligned and staggered arrangements of vortex cells in the limiting case, was also discovered by [6] in their study of the maximum palinstrophy growth in 2D.",
"While due to a smaller spatial dimension only two optimal solutions were found in that study, the one characterized by the staggered arrangement also lead to a larger (less negative) rate of palinstrophy production.",
"Table: Summary of the properties of extreme vortex states 𝐮 1 \\mathbf {u}_1 obtained assolutions of optimization problem in the limit ℰ 0 →0{\\mathcal {E}}_0 \\rightarrow 0.Figure: Taylor-Green flowIt is also worth mentioning that the initial data for two well-known flows, namely, the Arnold-Beltrami-Childress (ABC) flow [42] and the Taylor-Green flow [54], are in fact particular instances of the optimal field $\\mathbf {u}_1$ corresponding to, respectively, cases (REF ) and (REF ).",
"Following the notation of [19], general ABC flows are characterized by the following velocity field $\\begin{array}{r@{\\,\\,}c@{\\,\\,}l}u_1(x_1,x_2,x_3) & = & A^{\\prime }\\sin (2\\pi x_3) + C^{\\prime }\\cos (2\\pi x_2), \\\\u_2(x_1,x_2,x_3) & = & B^{\\prime }\\sin (2\\pi x_1) + A^{\\prime }\\cos (2\\pi x_3), \\\\u_3(x_1,x_2,x_3) & = & C^{\\prime }\\sin (2\\pi x_2) + B^{\\prime }\\cos (2\\pi x_1),\\end{array}$ where $A^{\\prime }$ , $B^{\\prime }$ and $C^{\\prime }$ real constants.",
"The vector field in equation (REF ) can be obtained from equation (REF ) by choosing $\\mathbf {A} = (B^{\\prime }/2)[0,-i,1]$ , $\\mathbf {B} = (C^{\\prime }/2)[1,0,-i]$ and $\\mathbf {C} = (A^{\\prime }/2)[-i, 1,0]$ .",
"By analogy, we will refer to the state described by (REF ) as the “aligned ABC flow”.",
"Likewise, the well-known Taylor-Green vortex can be obtained as a particular instance of the field $\\mathbf {u}_1$ from equation (REF ) again using a suitable choice of the constants $\\mathbf {A},\\mathbf {B},\\mathbf {C},\\mathbf {D}$ .",
"Traditionally, the velocity field $\\mathbf {u}= [u_1,u_2,u_3]$ characterizing the Taylor-Green vortex is defined as [11] $\\begin{array}{r@{\\,\\,}c@{\\,\\,}l}u_1(x_1,x_2,x_3) & = & \\gamma _1\\sin (2\\pi x_1)\\cos (2\\pi x_2)\\cos (2\\pi x_3), \\\\u_2(x_1,x_2,x_3) & = & \\gamma _2\\cos (2\\pi x_1)\\sin (2\\pi x_2)\\cos (2\\pi x_3), \\\\u_3(x_1,x_2,x_3) & = & \\gamma _3\\cos (2\\pi x_1)\\cos (2\\pi x_2)\\sin (2\\pi x_3), \\\\0 & = & \\gamma _1+\\gamma _2+\\gamma _3,\\end{array}$ for $\\gamma _1,\\gamma _2,\\gamma _3\\in \\mathbb {R}$ .",
"For given values of $\\gamma _1$ ,$ \\gamma _2$ and $\\gamma _3$ in (REF ), the corresponding constants $\\mathbf {A},\\mathbf {B},\\mathbf {C},\\mathbf {D}$ in (REF ) can be found by separating them into their real and imaginary parts denoted, respectively, $\\mathbf {A}_{\\textrm {Re}},\\mathbf {B}_{\\textrm {Re}},\\mathbf {C}_{\\textrm {Re}},\\mathbf {D}_{\\textrm {Re}}$ and $\\mathbf {A}_{\\textrm {Im}},\\mathbf {B}_{\\textrm {Im}},\\mathbf {C}_{\\textrm {Im}},\\mathbf {D}_{\\textrm {Im}}$ .",
"Then, after choosing $\\mathbf {A}_{\\textrm {Re}}=\\mathbf {B}_{\\textrm {Re}}=\\mathbf {C}_{\\textrm {Re}}=\\mathbf {D}_{\\textrm {Re}}=\\mathbf {0} = [0,0,0],$ the imaginary parts can be determined by solving the following system of linear equations $2\\left[\\begin{array}{cccc}I_3 & -I_3 & -I_3 & -I_3 \\\\-I_3 & I_3 & -I_3 & -I_3 \\\\-I_3 & -I_3 & I_3 & -I_3 \\\\-I_3 & -I_3 & -I_3 & I_3\\end{array}\\right]\\left[\\begin{array}{c}\\mathbf {A}_{\\textrm {Im}} \\\\\\mathbf {B}_{\\textrm {Im}} \\\\\\mathbf {C}_{\\textrm {Im}} \\\\\\mathbf {D}_{\\textrm {Im}}\\end{array}\\right] =\\left[\\begin{array}{c}\\mathbf {0} \\\\\\gamma _1\\mathbf {e}_1 \\\\\\gamma _2\\mathbf {e}_2 \\\\\\gamma _3\\mathbf {e}_3\\end{array}\\right],$ where $I_3$ is the $3 \\times 3$ identity matrix.",
"The values of $\\mathbf {A}_{\\textrm {Im}},\\ldots ,\\mathbf {D}_{\\textrm {Im}}$ are thus given by $\\mathbf {A}_{\\textrm {Im}} = -\\frac{1}{8}\\left[ \\begin{array}{c}\\gamma _1 \\\\ \\gamma _2 \\\\ \\gamma _3 \\end{array} \\right],\\,\\mathbf {B}_{\\textrm {Im}} = -\\frac{1}{8}\\left[ \\begin{array}{c}-\\gamma _1 \\\\ \\gamma _2 \\\\ \\gamma _3 \\end{array} \\right],\\,\\mathbf {C}_{\\textrm {Im}} = -\\frac{1}{8}\\left[ \\begin{array}{c}\\gamma _1 \\\\ -\\gamma _2 \\\\ \\gamma _3 \\end{array} \\right],\\,\\mathbf {D}_{\\textrm {Im}} = -\\frac{1}{8}\\left[ \\begin{array}{c}\\gamma _1 \\\\ \\gamma _2 \\\\ -\\gamma _3 \\end{array} \\right].$ A typical choice of the parameters used in the numerical studies performed by [11] and [10] is $\\gamma _1 = - \\gamma _2= 1$ and $\\gamma _3 = 0$ .",
"We remark that the Taylor-Green vortex has been employed as the initial data in a number of studies aimed at triggering singular behaviour in both the Euler and Navier-Stokes systems [54], [11], [10], [12].",
"It is therefore interesting to note that it arises in the limit ${\\mathcal {E}}_0 \\rightarrow 0$ as one of the extreme vortex states in the variational formulation considered in the present study.",
"It should be emphasized, however, that out of the three optimal states identified above (see Table REF ), the Taylor-Green vortex is characterized by the smallest (i.e., the most negative) instantaneous rate of enstrophy production $d{\\mathcal {E}}/dt$ .",
"On the other hand, we are not aware of any prior studies involving ABC flows in the context of extreme behaviour and potential singularity formation.",
"The time evolution corresponding to these states and some other initial data will be analyzed in detail in §."
],
[
"Extreme Vortex States with Finite ${\\mathcal {E}}_0$",
"In this section we analyze the optimal vortex states $\\widetilde{\\mathbf {u}}_{{\\mathcal {E}}_0}$ obtained for finite values of the enstrophy in which we extend the results obtained in the seminal study by [38].",
"As was also the case in the analogous study in 2D [6], there is a distinct branch of extreme states $\\widetilde{\\mathbf {u}}_{{\\mathcal {E}}_0}$ parameterized by the enstrophy ${\\mathcal {E}}_0$ and corresponding to each of the three limiting states discussed in § (cf.",
"figure REF and Table REF ).",
"Each of these branches is computed using the continuation approach outlined in Algorithm .",
"As a key element of the gradient-based maximization technique (REF ), the gradient expressions (REF )–(REF ) are approximated pseudo-spectrally using standard dealiasing of the nonlinear terms and with resolutions varying from $128^3$ in the low-enstrophy cases to $512^3$ in the high-enstrophy cases, which necessitated a massively parallel implementation using the Message Passing Interface (MPI).",
"As regards the computation of the Sobolev $H^2$ gradients, cf.",
"(REF ), we set $\\ell _1 = 0$ , whereas the second parameter $\\ell _2$ was adjusted during the optimization iterations and was chosen so that $\\ell _2 \\in [\\ell _{\\min },\\ell _{\\max }]$ , where $\\ell _{\\min }$ is the length scale associated with the spatial resolution $N$ used for computations and $\\ell _{\\max }$ is the characteristic length scale of the domain $\\Omega $ , that is, $\\ell _{\\min } \\sim \\mbox{\\textit {O}}( 1/N) $ and $\\ell _{\\max } \\sim \\mbox{\\textit {O}}(1)$ .",
"We remark that, given the equivalence of the inner products (REF ) corresponding to different values of $\\ell _1$ and $\\ell _2$ (as long as $\\ell _2 \\ne 0$ ), these choices do not affect the maximizers found, but only how rapidly they are approached by iterations (REF ).",
"For further details concerning the computational approach we refer the reader to the dissertation by [2].",
"As was the case in the analogous 2D problem studied by [6], the largest instantaneous growth of enstrophy is produced by the states with vortex cells staggered in all planes, cf.",
"case (REF ) in Table REF .",
"Therefore, in our analysis we will focus exclusively on this branch of extreme vortex states which has been computed for ${\\mathcal {E}}_0 \\in [10^{-3},2\\times 10^2]$ .",
"The optimal instantaneous rate of growth of enstrophy ${\\mathcal {R}}_{{\\mathcal {E}}_0} ={\\mathcal {R}}(\\widetilde{\\mathbf {u}}_{{\\mathcal {E}}_0})$ and the energy of the optimal states ${\\mathcal {K}}(\\widetilde{\\mathbf {u}}_{{\\mathcal {E}}_0})$ are shown as functions of ${\\mathcal {E}}_0$ for small ${\\mathcal {E}}_0$ in figures REF (a) and REF (b), respectively.",
"As indicated by the asymptotic form of ${\\mathcal {R}}$ in (REF ) and the Poincaré limit ${\\mathcal {K}}_0={\\mathcal {E}}_0/(2\\pi )^2$ , both of which are marked in these figures, the behaviour of ${\\mathcal {R}}_{{\\mathcal {E}}_0}$ and ${\\mathcal {K}}(\\widetilde{\\mathbf {u}}_{{\\mathcal {E}}_0})$ as ${\\mathcal {E}}_0 \\rightarrow 0$ is correctly captured by the numerically computed optimal states.",
"In particular, we note that ${\\mathcal {R}}_{{\\mathcal {E}}_0}$ is negative for $0 \\le {\\mathcal {E}}_0 \\lessapprox 7$ and exhibits the same trend as predicted in (REF ) for ${\\mathcal {E}}_0 \\rightarrow 0$ .",
"For larger values of ${\\mathcal {E}}_0$ the rate of growth of enstrophy becomes positive.",
"Likewise, the asymptotic behaviour of the energy of the optimal fields does not come as a surprise since, as discussed in §, in the limit ${\\mathcal {E}}_0 \\rightarrow 0$ the maximizers of ${\\mathcal {R}}$ are eigenfunctions of the Laplacian, which also happen to saturate Poincaré's inequality.",
"The structure of the optimal vortex states $\\widetilde{\\mathbf {u}}_{{\\mathcal {E}}_0}$ is analyzed next.",
"They are visualized using (REF ) in which the vortex cores are identified as regions $\\Sigma := \\lbrace \\Gamma _{s}(Q): s\\ge 0 \\rbrace $ for $Q$ defined as [16] $ Q(\\mathbf {x}) := \\, \\frac{1}{2}\\left[ \\operatorname{tr}(\\mathbf {\\Omega }\\mathbf {\\Omega }^T) - \\operatorname{tr}(\\mathbf {S}\\mathbf {S}^T) \\right],$ where $\\mathbf {S}$ and $\\mathbf {\\Omega }$ are the symmetric and anti-symmetric parts of the velocity gradient tensor $\\nabla \\mathbf {u}$ , that is, $[\\mathbf {S}]_{ij} = \\frac{1}{2}(\\partial _j u_i + \\partial _iu_j)$ and $[\\mathbf {\\Omega }]_{ij} = \\frac{1}{2}(\\partial _j u_i -\\partial _i u_j )$ , $i,j=1,2,3$ .",
"The quantity $Q$ can be interpreted as the local balance between the strain rate and the vorticity magnitude.",
"The isosurfaces $\\Gamma _{0}(Q - 0.5||Q||_{L_\\infty })$ representing the optimal states $\\widetilde{\\mathbf {u}}_{{\\mathcal {E}}_0}$ with selected values of ${\\mathcal {E}}_0$ are shown in figures REF (c)-(e).",
"For the smallest values of ${\\mathcal {E}}_0$ , the optimal fields exhibit a cellular structure already observed in figure REF (a).",
"For increasing values of ${\\mathcal {E}}_0$ this cellular structure transforms into a vortex ring, as seen in figure REF (e).",
"The component of vorticity normal to the plane $P_x = \\lbrace \\mathbf {x}\\in \\Omega : \\mathbf {n}\\cdot (\\mathbf {x}-\\mathbf {x}_0) = 0\\rbrace $ for $\\mathbf {n} = [1,0,0]$ and $\\mathbf {x}_0 = [1/2,1/2,1/2]$ is shown in figures REF (f)-(h), where the transition from cellular structures to a localized vortex structure as enstrophy increases is evident.",
"The results corresponding to large values of ${\\mathcal {E}}_0$ are shown in figure REF with the maximum rate of growth of enstrophy ${\\mathcal {R}}_{{\\mathcal {E}}_0}$ plotted as a function of ${\\mathcal {E}}_0$ in figure REF (a).",
"We observe that, as ${\\mathcal {E}}_0$ increases, this relation approaches a power law of the form ${\\mathcal {R}}_{{\\mathcal {E}}_0} = C_1^{\\prime } \\,{\\mathcal {E}}_0^{\\alpha _1}$ .",
"In order to determine the prefactor $C_1^{\\prime }$ and the exponent $\\alpha _1$ we perform a local least-squares fit of the power law to the actual relation ${\\mathcal {R}}_{{\\mathcal {E}}_0}$ versus ${\\mathcal {E}}_0$ for increasing values of ${\\mathcal {E}}_0$ starting with ${\\mathcal {E}}_{0} = 20$ (this particular choice the starting value is justified below).",
"Then, the exponent $\\alpha _1$ is computed as the average of the exponents obtained from the local fits with their standard deviation providing the error bars, so that we obtain ${\\mathcal {R}}_{{\\mathcal {E}}_0} = C^{\\prime }_1{\\mathcal {E}}_0^{\\,\\alpha _1}, \\qquad C^{\\prime }_1 = 3.72 \\times 10^{-3} , \\ \\alpha _1 = 2.97 \\pm 0.02$ (the same approach is also used to determine the exponents in other power-law relations detected in this study).",
"We note that the exponent $\\alpha _1$ obtained in (REF ) is in fact very close to 3 which is the exponent in estimate (REF ).",
"For the value of the viscosity coefficient used in the computations ($\\nu =0.01$ ), the constant factor $C_1 = 27/(8\\pi ^4\\nu ^3)$ in estimate (REF ) has the value $C_1 \\approx 3.465 \\times 10^4$ which is approximately seven orders of magnitude larger than $C^{\\prime }_1$ given in (REF ).",
"To shed more light at the source of this discrepancy, the objective functional ${\\mathcal {R}}$ from equation () can be separated into a negative-definite viscous part ${\\mathcal {R}}_{\\nu }$ and a cubic part ${\\mathcal {R}}_{\\textrm {cub}}$ defined as ${\\mathcal {R}}_{\\nu }(\\mathbf {u}) & := -\\nu \\int _\\Omega |\\Delta \\mathbf {u}|^2\\,d\\mathbf {x}, \\\\{\\mathcal {R}}_{\\textrm {cub}}(\\mathbf {u}) & := \\int _{\\Omega } \\mathbf {u}\\cdot \\nabla \\mathbf {u}\\cdot \\Delta \\mathbf {u}\\, d\\mathbf {x}, $ so that ${\\mathcal {R}}(\\mathbf {u}) = {\\mathcal {R}}_{\\nu }(\\mathbf {u}) + {\\mathcal {R}}_{\\textrm {cub}}(\\mathbf {u})$ .",
"The values of ${\\mathcal {R}}_{\\textrm {cub}}(\\widetilde{\\mathbf {u}}_{{\\mathcal {E}}_0})$ are also plotted in figure REF (a) and it is observed that this quantity exhibits a power-law behaviour of the form ${\\mathcal {R}}_{\\textrm {cub}}(\\widetilde{\\mathbf {u}}_{{\\mathcal {E}}_0}) = C^{\\prime \\prime }_1{\\mathcal {E}}_0^{\\,\\alpha _2}, \\qquad C^{\\prime \\prime }_1 = 1.38\\times 10^{-2},\\ \\alpha _2 = 2.99 \\pm 0.05.$ While the value of $C^{\\prime \\prime }_1$ is slightly larger than the value of $C^{\\prime }_1$ in (REF ), it is still some six orders of magnitude smaller than the constant factor $C_1 = 27/(8\\pi ^4\\nu ^3)$ from estimate (REF ).",
"These differences notwithstanding, we may conclude that estimate (REF ) is sharp in the sense of definition REF .",
"The power laws from equations (REF ) and (REF ) are consistent with the results first presented by [37], [38], where the authors reported a power-law with exponent $\\alpha _{LD} = 2.99$ and a constant of proportionality $C_{LD} = 8.97\\times 10^{-4}$ .",
"The energy of the optimal fields ${\\mathcal {K}}(\\widetilde{\\mathbf {u}}_{{\\mathcal {E}}_0})$ for large values of ${\\mathcal {E}}_0$ is shown in figure REF (b) in which we observe that the energy stops to increase at about ${\\mathcal {E}}_0 \\approx 20$ .",
"This transition justifies using ${\\mathcal {E}}_0 = 20$ as the lower bound on the range of ${\\mathcal {E}}_0$ where the power laws are determined via least-square fits.",
"Figures REF (c)-(e) show the isosurfaces $\\Gamma _{0}(Q - 0.5||Q||_{L_\\infty })$ representing the optimal fields $\\widetilde{\\mathbf {u}}_{{\\mathcal {E}}_0}$ for selected large values of ${\\mathcal {E}}_0$ .",
"The formation of these localized vortex structures featuring two rings as ${\\mathcal {E}}_0$ increases is evident in these figures.",
"The formation process of localized vortex structures is also visible in figures REF (f)–(h), where the component of vorticity normal to the plane $P_{xz} = \\lbrace \\mathbf {x}\\in \\Omega : \\mathbf {n}\\cdot (\\mathbf {x}- \\mathbf {x}_0) = 0 \\rbrace $ for $\\mathbf {n} = [1,0,-1]$ and $\\mathbf {x}_0 =[1/2,1/2,1/2]$ is shown (we note that the planes used in figures REF (c)–(e) and REF (c)–(e) have different orientations).",
"Figure: ℰ 0 =10{\\mathcal {E}}_0 = 10Figure: ℰ 0 =100{\\mathcal {E}}_0 = 100Next we examine the variation of different diagnostics applied to the extreme states $\\widetilde{\\mathbf {u}}_{{\\mathcal {E}}_0}$ as enstrophy ${\\mathcal {E}}_0$ increases.",
"The maximum velocity $||\\widetilde{\\mathbf {u}}_{{\\mathcal {E}}_0}||_{L_\\infty }$ and maximum vorticity $||\\nabla \\times \\widetilde{\\mathbf {u}}_{{\\mathcal {E}}_0}||_{L_\\infty }$ of the optimal fields are shown, respectively, in figures REF (a) and REF (b) as functions of ${\\mathcal {E}}_0$ .",
"For each quantity, two distinct power laws are observed in the forms $||\\widetilde{\\mathbf {u}}_{{\\mathcal {E}}_0}||_{L_\\infty } & \\sim C_1{\\mathcal {E}}^{\\alpha _1}_0,& \\quad C_1 &= 0.263,& \\alpha _1 & = 0.5 \\pm 0.023,& \\quad &\\mbox{as }\\, {\\mathcal {E}}_0\\rightarrow 0, \\\\||\\widetilde{\\mathbf {u}}_{{\\mathcal {E}}_0}||_{L_\\infty } & \\sim C_2{\\mathcal {E}}^{\\alpha _2}_0,& \\quad C_2 &= 6.3\\times 10^{-2},& \\quad \\alpha _2 &= 1.04 \\pm 0.13,& \\qquad &\\mbox{as }\\, {\\mathcal {E}}_0\\rightarrow \\infty , $ and $||\\nabla \\times \\widetilde{\\mathbf {u}}_{{\\mathcal {E}}_0}||_{L_\\infty } & \\sim C_1{\\mathcal {E}}_0^{\\alpha _1},& \\quad C_1 &= 2.09,& \\alpha _1 & = 0.54 \\pm 0.03,& \\quad &\\mbox{as }\\, {\\mathcal {E}}_0\\rightarrow 0, \\\\||\\nabla \\times \\widetilde{\\mathbf {u}}_{{\\mathcal {E}}_0}||_{L_\\infty } & \\sim C_2{\\mathcal {E}}_0^{\\alpha _2},& \\quad C_2 &= 6.03\\times 10^{-2},& \\quad \\alpha _2 &= 1.99 \\pm 0.17,& \\quad &\\mbox{as }\\, {\\mathcal {E}}_0\\rightarrow \\infty .$ In order to quantify the variation of the relative size of the vortex structures, we will introduce two characteristic length scales.",
"The first one is based on the energy and enstrophy, and was defined by [18] as $\\Lambda := \\frac{1}{2\\pi }\\left[ \\frac{{\\mathcal {K}}(\\widetilde{\\mathbf {u}}_{{\\mathcal {E}}_0})}{{\\mathcal {E}}(\\widetilde{\\mathbf {u}}_{{\\mathcal {E}}_0})} \\right]^{1/2}.$ It is therefore equivalent to the Taylor microscale $\\lambda ^2= 15\\int _\\Omega |\\mathbf {u}|^2 d\\mathbf {x}/ \\int _\\Omega |\\mathbf {\\omega }|^2 d\\mathbf {x}$ used in turbulence research [16].",
"Another length scale, better suited to the ring-like vortex structures shown in figures REF (c)-(e), is the average radius $R_{\\Pi }$ of one of the vortex rings calculated as $R_{\\Pi } := \\frac{ \\int _\\Omega r(\\mathbf {x})\\chi _{\\Pi }(\\mathbf {x}) \\,d\\mathbf {x}}{ \\int _\\Omega \\chi _{\\Pi }(\\mathbf {x})d\\mathbf {x}},\\ \\text{where} \\ \\ r(\\mathbf {x}) = |\\mathbf {x}- \\overline{\\mathbf {x}}|,\\quad \\overline{\\mathbf {x}} = \\frac{\\int _{\\Omega } \\mathbf {x}\\chi _{\\Pi }(\\mathbf {x}) d\\mathbf {x}}{\\int _{\\Omega } \\chi _{\\Pi }(\\mathbf {x})d\\mathbf {x}},$ and $\\chi _{\\Pi }$ is the characteristic function of the set $\\Pi & = & \\lbrace \\Gamma _s( Q ) : s > 0.9|| Q ||_{L_\\infty }\\rbrace \\cap \\\\& & \\lbrace \\mathbf {x}\\in \\Omega : \\mathbf {n}\\cdot (\\mathbf {x}-\\mathbf {x}_0) > 0, \\ \\mathbf {n}= [1,1,1], \\ \\mathbf {x}_0 = [1/2,1/2,1/2] \\rbrace .$ In the above definition of the set $\\Pi $ , the intersection of the two regions is necessary to restrict the set $R_{\\Pi }$ to only one of the two ring structures visible in figures REF (c)–(e).",
"The quantity $\\overline{\\mathbf {x}}$ can be therefore interpreted as the geometric centre of one of the vortex rings.",
"The dependence of $\\Lambda $ and $R_\\Pi $ on ${\\mathcal {E}}_0$ is shown in figures REF (c,d) in which the following power laws can be observed $\\Lambda &\\sim \\mbox{\\textit {O}}(1)\\quad \\mbox{and}& \\quad R_\\Pi &\\sim \\mbox{\\textit {O}}(1)&&& \\quad &\\mbox{as } \\,{\\mathcal {E}}_0\\rightarrow 0, \\\\\\Lambda &\\sim C_1{\\mathcal {E}}_0^{\\alpha _1},& \\quad C_1 &= 10.96,& \\alpha _1 &= -0.886 \\pm 0.105,& \\qquad &\\mbox{as } \\, {\\mathcal {E}}_0\\rightarrow \\infty , \\\\R_\\Pi &\\sim C_2{\\mathcal {E}}_0^{\\alpha _2},& \\quad C_2 &= 2.692,& \\quad \\alpha _2 &= -1.01 \\pm 0.16,& \\quad &\\mbox{as } \\, {\\mathcal {E}}_0\\rightarrow \\infty .$ By comparing the error bars in the key power laws (REF ) and (REF ) with the error bars in power-law relations (REF ), (REF ), (REF ) and (), we observe that there is less uncertainty in the first case, indicating that the quantities $||\\widetilde{\\mathbf {u}}_{{\\mathcal {E}}_0}||_{L_\\infty }$ , $||\\nabla \\times \\widetilde{\\mathbf {u}}_{{\\mathcal {E}}_0}||_{L_\\infty }$ , $\\Lambda $ and $R_\\Pi $ tend to be more sensitive to approximation errors than ${\\mathcal {R}}_{{\\mathcal {E}}_0}(\\widetilde{\\mathbf {u}}_{{\\mathcal {E}}_0})$ .",
"Non-negligible error bars may also indicate that, due to modest enstrophy values attained in our computations, the ultimate asymptotic regime corresponding to ${\\mathcal {E}}_0 \\rightarrow \\infty $ has not been reached in some power laws.",
"A useful aspect of employing the average ring radius $R_{\\Pi }$ as the characteristic length scale is that its observed scaling with respect to ${\\mathcal {E}}_0$ can be used as an approximate indicator of the resolution $1/N$ required to numerically solve problem REF for large values of enstrophy.",
"From the scaling in relation (), it is evident that a two-fold increase in the value of ${\\mathcal {E}}_0$ will be accompanied by a similar reduction in $R_\\Pi $ , thus requiring an eight-fold increase in the resolution (a two-fold increase in each dimension).",
"This is one of the reasons why computation of extreme vortex states $\\widetilde{\\mathbf {u}}_{{\\mathcal {E}}_0}$ for large enstrophy values is a very challenging computational task.",
"In particular, this relation puts a limit on the largest value of ${\\mathcal {E}}_0$ for which problem REF can be in principle solved computationally at the present moment: a value of ${\\mathcal {E}}_0 =2000$ , a mere order of magnitude above the largest value of ${\\mathcal {E}}_0$ reported here, would require a resolution of $8192^3$ used by some of the largest Navier-Stokes simulations to date.",
"To summarize, as the enstrophy increases from ${\\mathcal {E}}_0 \\approx 0$ to ${\\mathcal {E}}_0 = \\mbox{\\textit {O}}(10^2)$ , the optimal vortex states change their structure from cellular to ring-like.",
"While with the exception of ${\\mathcal {R}}(\\widetilde{\\mathbf {u}}_{{\\mathcal {E}}_0})$ and ${\\mathcal {K}}(\\widetilde{\\mathbf {u}}_{{\\mathcal {E}}_0})$ , all of the diagnostic quantities behave in a monotonous manner, the corresponding power laws change at about ${\\mathcal {E}}_0\\approx 20$ , which approximately marks the transition from the cellular to the ring-like structure (cf.",
"figure REF (e) vs. REF (c)).",
"This is also the value of the enstrophy beyond which the energy ${\\mathcal {K}}(\\widetilde{\\mathbf {u}}_{{\\mathcal {E}}_0})$ starts to decrease (figure REF (b)).",
"This transition also coincides with a change of the symmetry properties of the extreme vortex states $\\widetilde{\\mathbf {u}}_{{\\mathcal {E}}_0}$ — while in the limit ${\\mathcal {E}}_0 \\rightarrow 0$ these fields feature reflection and discrete rotation symmetries (cf.",
"§), for $20 \\lessapprox {\\mathcal {E}}_0 \\rightarrow \\infty $ the optimal states are characterized by axial symmetry.",
"The asymptotic (as ${\\mathcal {E}}_0 \\rightarrow \\infty $ ) extreme vortex states on locally maximizing branches corresponding to the aligned ABC flow and the Taylor-Green vortex (cf.",
"Table REF ) are similar to the fields shown in figures REF (c)–(h), except for a different orientation of their symmetry axes with respect to the periodic domain $\\Omega $ (these results are not shown here for brevity).",
"The different power laws found here are compared to the corresponding results obtained in 2D in §.",
"It is also worth mentioning that, as shown by [3], all power laws discussed in this section, cf.",
"(REF ), (REF ), (REF ), (REF ) and (), can be deduced rigorously using arguments based on dimensional analysis under the assumption of axisymmetry for the optimal fields $\\widetilde{\\mathbf {u}}_{{\\mathcal {E}}_0}$ .",
"Figure: (a) Maximum velocity ||𝐮 ˜ ℰ 0 || L ∞ ||\\widetilde{\\mathbf {u}}_{{\\mathcal {E}}_0}||_{L_\\infty }, (b) maximum vorticity||∇×𝐮 ˜ ℰ 0 || L ∞ ||\\nabla \\times \\widetilde{\\mathbf {u}}_{{\\mathcal {E}}_0}||_{L_\\infty }, (c) characteristic length scaleΛ\\Lambda and (d) characteristic radius R Π R_{\\Pi } of the extremevortex states as functions of ℰ 0 {\\mathcal {E}}_0 (all marked with bluesolid lines).",
"In all cases two distinct behaviours, correspondingto ℰ 0 →0{\\mathcal {E}}_0 \\rightarrow 0 and ℰ 0 →∞{\\mathcal {E}}_0\\rightarrow \\infty , are evident with thecorresponding approximate power laws indicated withblack dashed lines.Figure: The phase space{𝒦,ℰ}\\lbrace {\\mathcal {K}},{\\mathcal {E}}\\rbrace .",
"The solid circles and triangles represent,respectively, the instantaneously optimal fields 𝐮 ˜ ℰ 0 \\widetilde{\\mathbf {u}}_{{\\mathcal {E}}_0}and 𝐮 ˜ 𝒦 0 ,ℰ 0 \\widetilde{\\mathbf {u}}_{{\\mathcal {K}}_0,{\\mathcal {E}}_0}, with the lines issuing from selected markersindicating the corresponding time-dependent trajectories (theoptimal states 𝐮 ˜ 𝒦 0 ,ℰ 0 \\widetilde{\\mathbf {u}}_{{\\mathcal {K}}_0,{\\mathcal {E}}_0} are discussed in §,cf.",
"problem ).",
"The two lines with a negativeslope represent condition () with the two differentconstants, whereas the line with a positive slope is the Poincarélimit 𝒦=(2π) 2 ℰ{\\mathcal {K}}= (2\\pi )^2 {\\mathcal {E}}.",
"The shaded areas represent regions of thephase space for which global regularity is not a priori guaranteedbased on estimate () combined with fits() and ().Finally, the findings of this section allow us to shed some light on the “small data” result (REF ) which provides the conditions on the size of the initial data $\\mathbf {u}_0$ , given in terms of its energy ${\\mathcal {K}}(0)$ and enstrophy ${\\mathcal {E}}(0)$ , in the Navier-Stokes system () guaranteeing that smooth solutions exist globally in time.",
"The power-law fits (REF ) and (REF ) allow us to sharpen condition (REF ) be replacing the constant on the RHS with either $2\\nu / C^{\\prime }_1$ or $2\\nu / C^{\\prime \\prime }_1$ , so that we obtain ${\\mathcal {K}}(0){\\mathcal {E}}(0) < \\left\\lbrace \\frac{2\\nu }{C^{\\prime }_1} \\ \\ \\text{or} \\ \\ \\frac{2\\nu }{C^{\\prime \\prime }_1} \\right\\rbrace .$ The region of the “phase space” $\\lbrace {\\mathcal {K}},{\\mathcal {E}}\\rbrace $ described by condition (REF ) is shown in white in figure REF .",
"The gray region represents the values of ${\\mathcal {K}}(0)$ and ${\\mathcal {E}}(0)$ for which long-time existence of smooth solutions cannot be a priori guaranteed (the two shades of gray correspond to the two constants which can be used in (REF )).",
"Solid circles represent the different extreme states found in this section, whereas the thin curves mark the time-dependent trajectories which will be analyzed in §.",
"We conclude from figure REF that the change of the properties of the optimal states $\\widetilde{\\mathbf {u}}_{{\\mathcal {E}}_0}$ discussed above occurs in fact for the values of enstrophy (${\\mathcal {E}}(0) \\approx 20$ ) for which the states $\\widetilde{\\mathbf {u}}_{{\\mathcal {E}}_0}$ are on the boundary of the region of guaranteed long-time regularity."
],
[
"Time Evolution of Extreme Vortex States",
"The goal of this section is to analyze the time evolution, governed by the Navier-Stokes system (), with extreme vortex states identified in § used as the initial data $\\mathbf {u}_0$ .",
"In particular, we are interested in the finite-time growth of enstrophy ${\\mathcal {E}}(t)$ and how it relates to estimates (REF ), (REF ) and (REF ).",
"We will compare these results with the growth of enstrophy obtained using other types of initial data which have also been studied in the context of the blow-up problem for both the Euler and Navier-Stokes systems, namely, the Taylor-Green vortex [54], [11], [10], [12], the Kida-Pelz flow [9], [49], [26], colliding Lamb-Chaplygin dipoles [48] and perturbed antiparallel vortex tubes [30], [32].",
"Precise characterization of these different initial conditions is provided in Table REF and, for the sake of completeness, the last three states are also visualized in figure REF .",
"We comment that, with the exception of the Taylor-Green vortex which was shown in § to be a local maximizer of problem REF in the limit ${\\mathcal {E}}_0 \\rightarrow 0$ , all these initial conditions were postulated based on rather ad-hoc physical arguments.",
"We also add that, in order to ensure a fair comparison, the different initial conditions listed in Table REF are rescaled to have the same enstrophy ${\\mathcal {E}}_0$ , which is different from the enstrophy values used in the original studies where these initial conditions were investigated [48], [32], [20], [47].",
"As regards our choices of the initial enstrophy ${\\mathcal {E}}_0$ , to illustrate different possible behaviours, we will consider initial data located in the two distinct regions of the phase space $\\lbrace {\\mathcal {K}},{\\mathcal {E}}\\rbrace $ shown in figure REF , corresponding to values of ${\\mathcal {K}}_0$ and ${\\mathcal {E}}_0$ for which global regularity may or may not be a priori guaranteed according to estimates (REF )–(REF ).",
"Table: Characterization of the different initial data used in timeevolution studies in §.System () is solved numerically with an approach combining a pseudo-spectral approximation of spatial derivatives with a third-order semi-implicit Runge-Kutta method [8] used to discretize the problem in time.",
"In the evaluation of the nonlinear term dealiasing was used based on the $2/3$ rule together with the Gaussian filtering proposed by [29].",
"Massively parallel implementation based on MPI and using the fftw routines [23] to perform Fourier transforms allowed us to use resolutions varying from $256^3$ to $1024^3$ in the low-enstrophy and high-enstrophy cases, respectively.",
"A number of different diagnostics were checked to ensure that all flows discussed below are well resolved.",
"We refer the reader to the dissertation by [2] for additional details and a validation of this approach.",
"Figure: Isosurfaces corresponding to Q(𝐱)=1 2||Q|| L ∞ Q(\\mathbf {x}) =\\tfrac{1}{2}||Q||_{L_\\infty } for different initial conditions, allnormalized to ℰ 0 =100{\\mathcal {E}}_0 = 100: (a) Kida-Pelz flow, (b) collidingLamb-Chaplygin dipoles and (c) perturbed antiparallel vortextubes.",
"Precise characterization of these different initialconditions is provided in Table .The time-dependent results will be shown with respect to a normalized time defined as $\\tau := U_c t/\\ell _c$ with $U_c := \\Vert \\widetilde{\\mathbf {u}}_{{\\mathcal {E}}_0}\\Vert _{L_2}$ and $\\ell _c = \\Lambda $ (cf.",
"equation (REF )) playing the roles of the characteristic velocity and length scale.",
"We begin by showing the time evolution of the enstrophy ${\\mathcal {E}}(\\tau )$ corresponding to the five different initial conditions listed in Table REF with ${\\mathcal {E}}_0 = 10$ and ${\\mathcal {E}}_0 = 100$ in figures REF (a) and REF (b), respectively (because of the faster time-scale, the time axis in the latter figure is scaled logarithmically).",
"We see that the maximizers $\\widetilde{\\mathbf {u}}_{{\\mathcal {E}}_0}$ of problem REF are the only initial data which triggers growth of enstrophy for these values of the initial enstrophy and, as expected, this growth is larger for ${\\mathcal {E}}_0 = 100$ than for ${\\mathcal {E}}_0 = 10$ .",
"The other initial condition which exhibits some tendency for growth when ${\\mathcal {E}}_0 = 100$ is the Taylor-Green vortex.",
"In all cases the enstrophy eventually decays to zero for large times.",
"Figure: ℰ 0 =100{\\mathcal {E}}_0 = 100Next we examine whether the flow evolutions starting from the instantaneous maximizers $\\widetilde{\\mathbf {u}}_{{\\mathcal {E}}_0}$ as the initial data saturate the finite-time estimate (REF ).",
"We do this by defining functions $f(\\tau ) & := \\frac{1}{{\\mathcal {E}}(0)} - \\frac{1}{{\\mathcal {E}}(\\tau )}\\qquad \\mbox{and} \\\\g(\\tau ) & := \\frac{C}{2\\nu }\\left[ {\\mathcal {K}}(0) - {\\mathcal {K}}(\\tau ) \\right] $ representing, respectively, the left- and right-hand side of the estimate and then plotting them with respect to the normalized time $\\tau $ , which is done in figures REF (a) and REF (b) for ${\\mathcal {E}}_0 = 10$ and ${\\mathcal {E}}_0 = 100$ , respectively.",
"The constant $C>0$ in the definition of $g(\\tau )$ is numerically computed from the power-law fit in (REF ).",
"It follows from estimate (REF ) that $f(\\tau ) \\le g(\\tau )$ pointwise in time.",
"The hypothetical extreme event of a finite-time blow-up can be represented graphically by an intersection of the graph of $f(\\tau )$ with the horizontal line $y = 1/{\\mathcal {E}}_0$ , which is also shown in figures REF (a)–(b).",
"The behaviour of $g(\\tau )$ , representing the upper bound in estimate (REF ), is quite distinct in figures REF (a) and REF (b) reflecting the fact that the initial data $\\widetilde{\\mathbf {u}}_{{\\mathcal {E}}_0}$ in the two cases comes from different regions of the phase diagram in figure REF .",
"In figure REF (a), corresponding to ${\\mathcal {E}}_0 = 10$ , the upper bound $g(\\tau )$ never reaches $1/{\\mathcal {E}}_0$ , in agreement with the fact that the finite-time blow-up is a priori ruled out in this case.",
"On the other hand, in figure REF (b), corresponding to ${\\mathcal {E}}_0 = 100$ , the upper bound $g(\\tau )$ does intersect $1/{\\mathcal {E}}_0$ implying that, in principle, finite-time blow-up might be possible in this case.",
"The sharpness of estimate (REF ) can be assessed by analyzing how closely the behaviour of $f(\\tau )$ matches that of $g(\\tau )$ .",
"In both figures REF (a) and REF (b) we observe that for a short period of time $f(\\tau )$ exhibits a very similar growth to the upper bound $g(\\tau )$ , but then this growth slows down and $f(\\tau )$ eventually starts to decrease short of ever approaching the limit $1/{\\mathcal {E}}_0$ .",
"Figure: ℰ 0 =100{\\mathcal {E}}_0 =100We further characterize the time evolution by showing the maximum enstrophy increase $\\delta {\\mathcal {E}}:= \\mathop {\\max }_{t \\ge 0} \\, \\lbrace {\\mathcal {E}}(t) -{\\mathcal {E}}(0) \\rbrace $ and the time when the maximum is achieved $T_{\\max } :=\\mathop {\\arg \\max }_{t \\ge 0}\\, {\\mathcal {E}}(t)$ as functions of ${\\mathcal {E}}_0$ in figures REF (a) and REF (b), respectively.",
"In both cases approximate power laws in the form $\\delta {\\mathcal {E}}\\sim {\\mathcal {E}}^{\\alpha _1}_0, \\quad \\alpha _1 = 0.95 \\pm 0.06 \\qquad \\mbox{and}\\qquad T_{\\max } \\sim {\\mathcal {E}}^{\\alpha _2}_0, \\quad \\alpha _2 = -2.03 \\pm 0.02$ are detected in the limit ${\\mathcal {E}}_0 \\rightarrow \\infty $ (as regards the second result, we remark that $T_{\\max }$ is not equivalent to the time until which the enstrophy grows at the sustained rate proportional to ${\\mathcal {E}}_0^3$ , cf.",
"figure REF ).",
"To complete presentation of the results, the dependence of the quantities $\\mathop {\\max }_{t \\ge 0} \\, \\left\\lbrace \\frac{1}{{\\mathcal {E}}_0} - \\frac{1}{{\\mathcal {E}}(t)}\\right\\rbrace \\qquad \\mbox{and}\\qquad [{\\mathcal {K}}(0) - {\\mathcal {K}}(T_{\\max })]$ on the initial enstrophy ${\\mathcal {E}}_0$ is shown in figures REF (c) and REF (d), respectively.",
"It is observed that both quantities approximately exhibit a power-law behaviour of the form ${\\mathcal {E}}^{-1}_0$ .",
"Discussion of these results in the context of the estimates recalled in § is presented in the next section.",
"Figure: Dependence on ℰ 0 {\\mathcal {E}}_0 of (a) the maximum enstrophy increase overfinite time δℰ\\delta {\\mathcal {E}}, (b) the time T max T_{\\max } when the enstrophymaximum is attained, (c) the maximum achieved by the LHS of estimate(), cf.",
"(), and (d) the energydissipation during [0,T max ][0,T_{\\max }]; all data corresponds to thetime evolution starting from the extreme vortex states𝐮 ˜ ℰ 0 \\widetilde{\\mathbf {u}}_{{\\mathcal {E}}_0}."
],
[
"Discussion",
"In this section we provide some comments about the results reported in §§, and .",
"First, we need to mention that our gradient-based approach to the solution of optimization problem REF can only yield local maximizers and, due to nonconvexity of the problem, it is not possible to guarantee a priori that the maximizers found are global.",
"To test for the possible presence of branches other than the ones found using the continuation approach described in §, cf.",
"Algorithm , we tried to find new maximizers by initializing the gradient iterations (REF ) with different initial guesses $\\mathbf {u}^0$ .",
"They were constructed as solenoidal vector fields with prescribed regularity and random structure, which was achieved by defining the Fourier coefficients of $\\mathbf {u}^0$ as $\\widehat{\\mathbf {u}}^0(\\mathbf {k}) = F(|\\mathbf {k}|)e^{i\\phi (\\mathbf {k})}$ with the amplitude $F(|\\mathbf {k}|) \\sim 1/|\\mathbf {k}|^2$ and the phases $\\phi (\\mathbf {k})$ chosen as random numbers uniformly distributed in $[0,2\\pi ]$ .",
"However, in all such tests conducted for ${\\mathcal {E}}_0 = O(1)$ the gradient optimization algorithm (REF ) would always converge to maximizers $\\widetilde{\\mathbf {u}}_{{\\mathcal {E}}_0}$ belonging to one of the branches discussed in § (modulo possible translations in the physical domain).",
"While far from settling this issue definitely, these observations lend some credence to the conjecture that the branch identified in § corresponds in fact to the global maximizers.",
"These states appear identical to the maximizers found by [38] and our search has also yielded two additional branches of locally maximizing fields, although we did not capture the lower branch reported by [38].",
"However, since that branch does not appear connected to any state in the limit ${\\mathcal {E}}_0 \\rightarrow 0$ , we speculate that it might be an artifact of the “discretize-then-optimize” formulation used by [38], in contrast to the “optimize-then-discretize” approach employed in our study which provides a more direct control over the analytic properties of the maximizers.",
"We add that the structure of the maximizing branches found here is in fact quite similar to what was discovered by [6] in an analogous problem in 2D.",
"Since the 2D problem is more tractable from the computational point of view, in that case we were able to undertake a much more thorough search for other maximizers which did not however yield any solutions not associated with the main branches.",
"The results reported in § and § clearly exhibit two distinct behaviours, depending on whether or not global-in-time regularity can be guaranteed a priori based on estimates (REF )–(REF ).",
"These differences are manifested, for example, in the power laws evident in figures REF and REF , as well as in the different behaviours of the RHS of estimate (REF ) with respect to time in figures REF (a) and REF (b).",
"However, for the initial data for which global-in-time regularity cannot be ensured a priori there is no evidence of sustained growth of enstrophy strong enough to signal formation of singularity in finite time.",
"Indeed, in figure REF (c) one sees that the quantity $\\mathop {\\max }_{t \\ge 0} \\, \\left\\lbrace 1/{\\mathcal {E}}_0 - 1/{\\mathcal {E}}(t)\\right\\rbrace $ behaves as $C_1 / {\\mathcal {E}}_0$ , where $C_1 < 1$ , when ${\\mathcal {E}}_0$ increases, revealing no tendency to approach $1/{\\mathcal {E}}_0$ which is a necessary precursor of a singular behaviour (cf.",
"discussion in §).",
"To further illustrate how the rate of growth of enstrophy achieved initially by the maximizers $\\widetilde{\\mathbf {u}}_{{\\mathcal {E}}_0}$ is depleted in time, in figure REF we show the flow evolution corresponding to $\\widetilde{\\mathbf {u}}_{{\\mathcal {E}}_0}$ with ${\\mathcal {E}}_0 = 100$ as a trajectory in the coordinates $\\lbrace {\\mathcal {E}}, d{\\mathcal {E}}/dt\\rbrace $ .",
"From the discussion in Introduction we know that in order for the singularity to occur in finite time, the enstrophy must grow at least at a sustained rate $d{\\mathcal {E}}/ dt \\sim {\\mathcal {E}}^\\alpha $ for some $\\alpha > 2$ .",
"In other words, a “blow-up trajectory” will be realized only if the trajectory of the flow, expressed in $\\lbrace {\\mathcal {E}},d{\\mathcal {E}}/dt \\rbrace $ coordinates, is contained in the region $\\mathcal {M} = \\lbrace ({\\mathcal {E}},d{\\mathcal {E}}/dt) \\; : \\; C_1{\\mathcal {E}}^2 < d{\\mathcal {E}}/dt \\le C_2{\\mathcal {E}}^3\\rbrace $ , for some positive constants $C_1$ and $C_2$ .",
"For the flow corresponding to the instantaneous optimizers $\\widetilde{\\mathbf {u}}_{{\\mathcal {E}}_0}$ , the initial direction of a trajectory in $\\lbrace {\\mathcal {E}},d{\\mathcal {E}}/dt \\rbrace $ coordinates is determined by the vector $\\mathbf {v}= \\left[1,\\left.\\tfrac{d{\\mathcal {R}}}{d{\\mathcal {E}}}\\right|_{{\\mathcal {E}}_0} \\right]$ and, for initial conditions $\\mathbf {u}_0$ satisfying ${\\mathcal {R}}(\\mathbf {u}_0) = C{\\mathcal {E}}^3(\\mathbf {u}_0)$ , it follows that $\\left.\\frac{d{\\mathcal {R}}}{d{\\mathcal {E}}}\\right|_{{\\mathcal {E}}_0} = 3C{\\mathcal {E}}_0^2.$ Since the optimal rate of growth is sustained only over a short interval of time, the trajectory of the flow in the $\\lbrace {\\mathcal {E}},d{\\mathcal {E}}/dt\\rbrace $ coordinates approaches the region $\\mathcal {M}$ only tangentially following the direction of the lower bound $C_1{\\mathcal {E}}^2$ , and remains outside $\\mathcal {M}$ for all subsequent times.",
"This behaviour is clearly seen in the inset of figure REF .",
"An interpretation of this behaviour can be proposed based on equation (REF ) from which it is clear that the evolution of the flow energy is closely related to the growth of enstrophy.",
"In particular, if the initial energy ${\\mathcal {K}}(0)$ is not sufficiently large, then its depletion due to the initial growth of enstrophy may render the flow incapable of sustaining this growth over a longer period of time.",
"This is in fact what seems to be happening in the present problem as evidenced by the data shown in figure REF .",
"We remark that, for a prescribed enstrophy ${\\mathcal {E}}(0)$ , the flow energy cannot be increased arbitrarily as it is upper-bounded by Poincaré's inequality ${\\mathcal {K}}(0) \\le (2\\pi )^2 {\\mathcal {E}}(0)$ .",
"This behaviour can also be understood in terms of the geometry of the extreme vortex states $\\widetilde{\\mathbf {u}}_{{\\mathcal {E}}_0}$ .",
"Figure REF shows a magnification of the pair of vortex rings corresponding to the optimal field $\\widetilde{\\mathbf {u}}_{{\\mathcal {E}}_0}$ with ${\\mathcal {E}}_0=100$ .",
"It is observed that the vorticity field $\\nabla \\times \\widetilde{\\mathbf {u}}_{{\\mathcal {E}}_0}$ inside the vortex core has an azimuthal component only which exhibits no variation in the azimuthal direction.",
"Thus, in the limit ${\\mathcal {E}}_0\\rightarrow \\infty $ the vortex ring shrinks with respect to the domain $\\Omega $ (cf.",
"figure REF (d)) and the field $\\widetilde{\\mathbf {u}}_{{\\mathcal {E}}_0}$ ultimately becomes axisymmetric (i.e., in this limit boundary effects vanish).",
"At the same time, it is known that the 3D Navier-Stokes problem on an unbounded domain and with axisymmetric initial data is globally well posed [33], a results which is a consequence of the celebrated theorem due to [14].",
"Figure: Trajectory of the flow corresponding to the initial condition𝐮 ˜ ℰ 0 \\widetilde{\\mathbf {u}}_{{\\mathcal {E}}_0} with ℰ 0 =100{\\mathcal {E}}_0 = 100 in the coordinates {ℰ,dℰ/dt}\\lbrace {\\mathcal {E}},d{\\mathcal {E}}/dt\\rbrace .For comparison, in the inset the thin line represents theborderline growth at the rate dℰ/dt∼ℰ 2 d{\\mathcal {E}}/dt \\sim {\\mathcal {E}}^2Figure: Vortex lines inside the region with the strongestvorticity in the extreme vortex state 𝐮 ˜ ℰ 0 \\widetilde{\\mathbf {u}}_{{\\mathcal {E}}_0} with ℰ 0 =100{\\mathcal {E}}_0 = 100.The colour coding of the vortex lines is for identification purposesonly.We close this section by comparing the different power laws characterizing the maximizers $\\widetilde{\\mathbf {u}}_{{\\mathcal {E}}_0}$ and the corresponding flow evolutions with the results obtained in analogous studies of extreme behaviour in 1D and 2D (see also Table REF ).",
"First, we note that the finite-time growth of enstrophy $\\delta {\\mathcal {E}}$ in 3D, cf.",
"figure REF (a), exhibits the same dependence on the enstrophy ${\\mathcal {E}}_0$ of the instantaneously optimal initial data as in 1D, i.e., is directly proportional to ${\\mathcal {E}}_0$ in both cases [5].",
"This is also analogous to the maximum growth of palinstrophy ${\\mathcal {P}}$ in 2D which was found by [6] to scale with the palinstrophy ${\\mathcal {P}}_0$ of the initial data, when the instantaneously optimal initial condition was computed subject to one constraint only (on ${\\mathcal {P}}_0$ ).",
"When the instantaneously optimal initial data was determined subject to two constraints, on ${\\mathcal {K}}_0$ and ${\\mathcal {P}}_0$ , then the maximum finite-time growth of palinstrophy was found to scale with ${\\mathcal {P}}_0^{3/2}$ [7].",
"On the other hand, the time $T_{\\max }$ when the maximum enstrophy is attained, cf.",
"figure REF (b), scales as ${\\mathcal {E}}_0^{-2}$ , which should be contrasted with the scalings ${\\mathcal {E}}_0^{-1/2}$ and ${\\mathcal {P}}_0^{-1/2}$ found in the 1D and 2D cases, respectively.",
"This implies that the time interval during which the extremal growth of enstrophy is sustained in 3D is shorter than the corresponding intervals in 1D and 2D."
],
[
"Conclusions and Outlook",
"By constructing the initial data to exhibit the most extreme behaviour allowed for by the mathematically rigorous estimates, this study offers a fundamentally different perspective on the problem of searching for potentially singular solutions from most earlier investigations.",
"Indeed, while the corresponding flow evolutions did not reveal any evidence for finite-time singularity formation, the initial data obtained by maximizing $d{\\mathcal {E}}/dt$ produced a significantly larger growth of enstrophy in finite time than any other candidate initial conditions (cf.",
"Table REF and figure REF ).",
"Admittedly, this observation is limited to the initial data with ${\\mathcal {E}}_0 \\le 100$ , which corresponds to Reynolds numbers $Re = \\sqrt{{\\mathcal {E}}_0\\, \\Lambda } / \\nu \\lessapprox 450$ lower than the Reynolds numbers achieved in other studies concerned with the extreme behaviour in the Navier-Stokes flows [48], [32], [20], [47].",
"Given that the definitions of the Reynolds numbers applicable to the various flow configurations considered in these studies were not equivalent, it is rather difficult to make a precise comparison in terms of specific numerical values, but it is clear that the largest Reynolds numbers attained in these investigations were at least an order of magnitude higher than used in the present study; for Euler flows such a comparison is obviously not possible at all.",
"However, from the mathematical point of view, based on estimates (REF )–(REF ), there is no clear indication that a very large initial enstrophy ${\\mathcal {E}}_0$ (or, equivalently, a high Reynolds number) should be a necessary condition for singularity formation in finite time.",
"In fact, blow-up cannot be a priori ruled out as soon as condition (REF ) is violated, which happens for all initial data lying on the gray region of the phase space in figure REF .",
"We remark that additional results were obtained (not reported in this paper) by studying the time evolution corresponding to the optimal initial data $\\widetilde{\\mathbf {u}}_{{\\mathcal {E}}_0}$ , but using smaller values of the viscosity coefficient $\\nu =10^{-3},10^{-4}$ , thereby artificially increasing the Reynolds number at the price of making the initial data suboptimal.",
"Although these attempts did increase the amplification of enstrophy as compared to what was observed in figures REF and REF , no signature of finite-time singularity formation could be detected either.",
"Our study confirmed the earlier findings of [38] about the sharpness of the instantaneous estimate (REF ).",
"We also demonstrated that the finite-time estimate (REF ) is saturated by the flow evolution corresponding to the optimal initial data $\\widetilde{\\mathbf {u}}_{{\\mathcal {E}}_0}$ , but only for short times, cf.",
"figure REF , which are not long enough to trigger a singular behaviour.",
"In § we speculated that a relatively small initial energy ${\\mathcal {K}}(0)$ , cf.",
"figure REF (b), might be the property of the extreme vortex states $\\widetilde{\\mathbf {u}}_{{\\mathcal {E}}_0}$ preventing the resulting flow evolutions from sustaining a significant growth of enstrophy over long times.",
"On the other hand, in Introduction we showed that estimate (REF ) need not be saturated for blow-up to occur in finite time and, in fact, sustained growth at the rate $d{\\mathcal {E}}/dt = C \\, {\\mathcal {E}}^\\alpha $ with any $\\alpha > 2$ will also produce singularity in finite time.",
"Thus, another strategy to construct initial data which could lead to a more sustained growth of enstrophy in finite time might be to increase its kinetic energy by allowing for a smaller instantaneous rate of growth (i.e., with an exponent $2 < \\alpha \\le 3$ instead of $\\alpha = 3$ ).",
"This can be achieved by prescribing an additional constraint in the formulation of the variational optimization problem, resulting in Problem 8.1 $\\widetilde{\\mathbf {u}}_{{\\mathcal {K}}_0,{\\mathcal {E}}_0}& = & \\mathop {\\arg \\max }_{\\mathbf {u}\\in \\mathcal {S}_{{\\mathcal {K}}_0,{\\mathcal {E}}_0}} \\, {\\mathcal {R}}(\\mathbf {u}) \\\\\\mathcal {S}_{{\\mathcal {K}}_0,{\\mathcal {E}}_0} & = & \\left\\lbrace \\mathbf {u}\\in H_0^2(\\Omega )\\,\\colon \\,\\nabla \\cdot \\mathbf {u}= 0, \\; {\\mathcal {K}}(\\mathbf {u}) = {\\mathcal {K}}_0, \\; {\\mathcal {E}}(\\mathbf {u}) = {\\mathcal {E}}_0 \\right\\rbrace .$ It differs from problem REF in that the maximizers are sought at the intersection of the original constraint manifold $\\mathcal {S}_{{\\mathcal {E}}_0}$ and the manifold defined by the condition ${\\mathcal {K}}(\\mathbf {u})= {\\mathcal {K}}_0$ , where ${\\mathcal {K}}_0 \\le (2\\pi )^2 {\\mathcal {E}}_0$ is the prescribed energy.",
"While computation of such maximizers is more complicated, robust techniques for the solution of optimization problems of this type have been developed and were successfully used in the 2D setting by [6].",
"Preliminary results obtained in the present setting by solving problem REF for ${\\mathcal {K}}_0 =1$ are indicated in figure REF , where we see that the flow evolutions starting from $\\widetilde{\\mathbf {u}}_{{\\mathcal {K}}_0,{\\mathcal {E}}_0}$ do not in fact produce a significant growth of enstrophy either.",
"An alternative, and arguably more flexible approach, is to formulate this problem in terms of multiobjective optimization [44] in which the objective function ${\\mathcal {R}}(\\mathbf {u})$ in problem REF would be replaced with ${\\mathcal {R}}_{\\eta }(\\mathbf {u}) := \\eta \\, {\\mathcal {R}}(\\mathbf {u}) + (1-\\eta )\\, {\\mathcal {K}}(\\mathbf {u}),$ where $\\eta \\in [0,1]$ .",
"Solution of such a multiobjective optimization problem has the form of a “Pareto front” parameterized by $\\eta $ .",
"Clearly, the limits $\\eta \\rightarrow 1$ and $\\eta \\rightarrow 0$ correspond, respectively, to the extreme vortex states already found in § and §, and to the Poincaré limit.",
"Another interesting possibility is to replace the energy ${\\mathcal {K}}(\\mathbf {u})$ with the helicity ${\\mathcal {H}}(\\mathbf {u}) :=\\int _{\\Omega } \\mathbf {u}\\cdot (\\nabla \\times \\mathbf {u})\\,d\\Omega $ in the multiobjective formulation (REF ), as this might allow one to obtain extreme vortex states with a more complicated topology (i.e., a certain degree of “knottedness”).",
"We note that all the extreme vortex states found in the present study were “unknotted”, i.e., were characterized by ${\\mathcal {H}}(\\widetilde{\\mathbf {u}}_{{\\mathcal {E}}_0}) = 0$ , as the vortex rings were in all cases disjoint (cf.",
"figure REF ).",
"Finally, another promising possibility to find initial data producing a larger growth of enstrophy is to solve a finite-time optimization problem of the type already studied by [5] in the context of the 1D Burgers equation, namely Problem 8.2 $\\tilde{\\mathbf {u}}_{0;{\\mathcal {E}}_0,T} = \\mathop {\\arg \\max }_{\\mathbf {u}_0\\in \\mathcal {S}_{{\\mathcal {E}}_0}} \\, {\\mathcal {E}}(T),$ where $T>0$ is the length of the time interval of interest and $\\mathbf {u}_0$ the initial data for the Navier-Stokes system ().",
"In contrast to problems REF and REF , solution of problem REF is more complicated as it involves flow evolution.",
"It represents therefore a formidable computational task for the 3D Navier-Stokes system.",
"However, it does appear within reach given the currently available computational resources and will be studied in the near future."
],
[
"Acknowledgements",
"The authors are indebted to Charles Doering for many enlightening discussions concerning the research problems studied in this work.",
"The authors are grateful to Nicholas Kevlahan for making his parallel Navier-Stokes solver available, which was used to obtain the results reported in §.",
"Anonymous referees provided many insightful comments which helped us improve this work.",
"This research was funded through an Early Researcher Award (ERA) and an NSERC Discovery Grant, whereas the computational time was made available by SHARCNET under its Dedicated Resource Allocation Program.",
"Diego Ayala was funded in part by NSF Award DMS-1515161 at the University of Michigan and by the Institute for Pure and Applied Mathematics at UCLA."
]
] | 1605.05742 |
[
[
"Two Quasi Orthogonal Space-Time Block Codes with Better Performance and\n Low Complexity Decoder"
],
[
"Abstract This paper presents two new space time block codes (STBCs) with quasi orthogonal structure for an open loop multi-input single-output (MISO) systems.",
"These two codes have been designed to transmit from three or four antennas at the transmitter and be given to one antenna at the receiver.",
"First, the proposed codes are introduced and their structures are investigated.",
"This is followed by the demonstration of how the decoder decodes half of transmitted symbols independent of the other half.",
"The last part of this paper discusses the simulation results, makes performance comparison against other popular approaches and concludes that the proposed solutions offer superiority."
],
[
"Introduction",
"Nowadays, on the one hand explosion of data traffic in wireless communication is real and undeniable [1], on the other hand, spectrum is the most valuable and scarce asset in communication.",
"In order to achieve a high data transmission rate in narrow band wireless communication with an acceptable performance, we need to combat the interference between received symbols at the receiver which is called fading phenomenon in Rayleigh channels.",
"One of the most advanced methods that is commonly deployed is space and time diversity which means the transmission of symbols not only from different places, but also at different times, requiring neither extra bandwidth nor more transmission power.",
"The codes which benefit from these types of diversity are called space time block codes (STBCs).",
"Today, many STBCs with various characteristics designed for open loop multi-input multi-output (MIMO) systems are in existence.",
"We can categorize some of codes into two groups, those with orthogonal and those with quasi orthogonal structures.",
"Although orthogonal space-time block codes (OSTBCs) decoder is much simpler than quasi-orthogonal space-time block codes (QOSTBCs) one, the rate of QOSTBCs is higher than that of OSTBCs for more than two transmitter antennas.",
"Let ponder on the STBCs which are presented in [2], [3] for quasi-static flat fading channels.",
"An OSTBC with very simple decoder and full rate is Alamouti code [2] and since of this feature it is a good choice for practical applications.",
"Relaxing the simple decoding capability of OSTBCs, not being exist full rate OSTBCs for more than 2 transmitter antennas is a drawback for these codes.",
"To overcome this shortcoming, QOSTBCs was introduced for the first time in [4] .",
"The presented code was for four transmitter antennas which can be approached to full rate at the expense of diversity.",
"Subsequently, both full rate and full diversity QOSTBC which was an enhancement of [4] appeared in [5], [6].",
"In both aforementioned studies, diversity has been improved from $2M$ to $4M$ , in which $M$ is the total number of antenna on the receiver side, by constellation rotation for some symbols.",
"The orthogonality space between the two groups of columns makes independent decoding of pairs of transmitted symbols possible.",
"Some codes with different structure from that of OSTBCs and QOSTBCs with dramatically better performance have also been reported.",
"For instance, the code proposed in [7] known as Golden Code, is full rate, full diversity and also non-vanishing constant minimum determinate with $2\\times 2$ transmission matrices, likewise Perfect Codes with $3 \\times 3, 4 \\times 4,$ and $6 \\times 6$ is studied in [8] .",
"The main advantage of these codes is that their rates are more than one, nonetheless this is at the expense of ML detection with high complexity and the requirement for more than two receiver antennas.",
"In this paper, we present two new STBCs, one for three and the other for four transmit antennas with quasi-orthogonal form.",
"For more clarification, consider a codeword in which some groups of its columns are orthogonal to the opposite groups' columns, while columns in the same group are not.",
"That was the reason that they have been called QOSTBCs.",
"Due to the space created between column, we can decode half of transmitting symbols without dependence the other half.",
"These two codes are both full rate and full diversity.",
"Note that full diversity can appear when some of the transmitted symbols are selected from a constellation and the others are selected from the rotation of that constellation.",
"This derivation has been proven in both [5] and [6].",
"In fact, the strategy which is used in this paper is to combat destructive fading coefficients.",
"In other words, we have combined symbols through scarifying of neither full rate nor full diversity.",
"The rest of the paper is organized as follows; In section II system model is introduced and in Section III structure of the codes and the decoders are investigated.",
"Simulation results are provided in Section IV.",
"Finally, Section V concludes the paper.",
"Notation: We used bold letter for matrices.",
"Superscripts $(.",
")^H, ||.||_F,$ and $(.",
")^*$ to indicate Hermitian, Frobenius norm and complex conjugation, respectively.",
"In addition, we use $\\mathbb {C}^{M\\times N}$ to represent the set of $M\\times N$ matrices over field of complex numbers.",
"Let consider a quasi-static flat fading channel with $M_T$ and $M_R$ antennas at the transmitter and the receiver, respectively.",
"Throughout this paper, we assume that there is no information about the channel state on the transmitter side, but the receiver has informed about channel state information well.",
"In addition they have been perfectly synchronized.",
"Suppose that the transmitter sends the codeword $\\textbf {C}\\in \\mathbb {C}^{T\\times M_T} $ over $T$ time slot, thus we can model the system as below: $\\mathbf {Y} = \\mathbf {CH}+\\mathbf {N},$ in which $\\mathbf {Y}\\in \\mathbb {C}^{T\\times M_R}$ is the received signal, $\\mathbf {H}\\in \\mathbb {C}^{M_T\\times M_R}$ is fading matrices and $\\mathbf {N}\\in \\mathbb {C}^{T\\times M_R}$ is the Additive White Gaussian Noise (AWGN).",
"For a system that has been designed for two blocks fading the following expression can be written as follows: $\\begin{bmatrix}\\mathbf {Y}_1 \\\\\\mathbf {Y}_2\\end{bmatrix}=\\begin{bmatrix}\\mathbf {C}_1 & 0 \\\\ 0 & \\mathbf {C}_2\\end{bmatrix}\\begin{bmatrix}\\mathbf {H}_1 \\\\ \\mathbf {H}_2\\end{bmatrix}+\\begin{bmatrix}\\mathbf {N}_1 \\\\ \\mathbf {N}_2\\end{bmatrix},$ where codeword can be defined as below: $\\mathbf {C}=\\text{diag}\\lbrace \\mathbf {C}_1,\\mathbf {C}_2\\rbrace \\in \\mathbb {C}^{2T\\times 2M_T}$"
],
[
"Decoder Model",
"For maximum-likelihood (ML) detection at the receiver, the decoder examines all possible answers for this equation and then decides on the minimum of the following equation: $\\hat{\\mathbf {C}}=\\arg \\min _{\\mathbf {C}^i}||\\mathbf {Y}-\\mathbf {C}^i\\mathbf {H}||^2_F.$"
],
[
"Four Transmitter and One Receiver Antennas",
"In this section, we present our two QOSTBCs and also its proposed decoder.",
"We begin with the QOSTBCs for $M_T=4$ and $M_R=1$ , and proceed with the analysis of ML decoder before returning to consider the case of $M_T=3$ and $M_R=1$ .",
"By using the Alamouti scheme [2] we can define ${G}$ as matrix generator, ${G}(x_1,x_2)\\triangleq \\begin{bmatrix}x_1 & x_2 \\\\ -x_2^* & x_1^*\\end{bmatrix}.$ Note that it is obvious that the two columns of this matrix are orthogonal to each other.",
"To construct $\\mathbf {C}_1$ and $\\mathbf {C}_2,$ we should write as follows: $\\mathbf {C}_1\\triangleq \\begin{bmatrix}{G}(S_1+jS_2) & {G}(\\tilde{S}_3+j\\tilde{S}_4) \\\\-({G}(\\tilde{S}_3+j\\tilde{S}_4))^* & ({G}(S_1+jS_2))^*\\end{bmatrix},$ and, $\\mathbf {C}_2\\triangleq \\begin{bmatrix}{G}(S_1-jS_2) & {G}(\\tilde{S}_3-j\\tilde{S}_4) \\\\-({G}(\\tilde{S}_3-j\\tilde{S}_4))^* & ({G}(S_1-jS_2))^*\\end{bmatrix},$ where $\\tilde{S}_k=S_k\\times e^{j\\theta _k}$ for $k = 3,4,7,8$ .",
"The aforementioned codewords are the outcome of a case where the careful combination of the two symbols results in the best possible performance.",
"In this case, as Eq.",
"(REF ) and Eq.",
"(REF ), are showing we combine $S_k,$ and $S_{k+4}$ , where $k = 1, 2, \\dots , 4$ , orthogonally.",
"After substituting (REF ) in both (REF ) and (REF ), we can represent $\\mathbf {C}_1$ and $\\mathbf {C}_2$ , as follows: $\\mathbf {C}_1=\\begin{bmatrix}S_1+jS_5 & S_2+jS_6 & \\tilde{S}_3+j\\tilde{S}_7&\\tilde{S}_4+j\\tilde{S}_8 \\\\-S_2^*+jS_6^* & S_1^*-jS^*_5 & -\\tilde{S}_4^*+j\\tilde{S}_8^*& \\tilde{S}_3^*-j\\tilde{S}^*_7 \\\\-\\tilde{S}_3^*+j\\tilde{S}_7^* & -\\tilde{S}_4^*+j\\tilde{S}^*_8 & S_1^*-jS_5^*& S_2^*-jS^*_6\\\\\\tilde{S}_4+j\\tilde{S}_8 & -\\tilde{S}_3-j\\tilde{S}_7 & -S_2-jS_6& S_1+jS_5\\end{bmatrix},$ $\\mathbf {C}_2=\\begin{bmatrix}S_1-jS_5 & S_2-jS_6 & \\tilde{S}_3-j\\tilde{S}_7&\\tilde{S}_4-j\\tilde{S}_8 \\\\-S_2^*-jS_6^* & S_1^*+jS^*_5 & -\\tilde{S}_4^*-j\\tilde{S}_8^*& \\tilde{S}_3^*+j\\tilde{S}^*_7 \\\\-\\tilde{S}_3^*-j\\tilde{S}_7^* & -\\tilde{S}_4^*-j\\tilde{S}^*_8 & S_1^*+jS_5^*& S_2^*+jS^*_6\\\\-\\tilde{S}_4-j\\tilde{S}_8 & -\\tilde{S}_3+j\\tilde{S}_7 & -S_2+jS_6& S_1-jS_5\\end{bmatrix}.$ The final codeword can be concluded using Eq.",
"(REF ).",
"When a system utilize this QOSTBC (the quasi-orthogonality will be provided) structure, it transmits $\\mathbf {C}_1$ in the first four time slots from its four transmitter antennas, on the other side, the receiver receives $\\mathbf {Y}_1,$ and buffer that, the second four timeslots is the time to send the next sub-codeword $\\mathbf {C}_2,$ and on the receiver side an approximation of $\\mathbf {C}_2$ ($\\mathbf {Y}_2$ ), will be received.",
"Note that the channel state while transmitting $\\mathbf {C}_1,$ should be different from while the transmitter is transmitting $\\mathbf {C}_2$ .",
"This recent scenario can implemented by using a reconfigurable antenna such as PIXEL antenna [9] on the receiver side.",
"PIXEL antennas are capable to provide up to 5 uncorrelated channel propagation states simultaneously, therefore even more diversity gain is achievable.",
"It can be stated, therefore, that by employing the proposed strategy the channel changes from quasi static to block fading and as a result the probability of destructive fading effect, will be mitigated near to zero.",
"Given that eight symbols are transmitted in eight time slots, the code rate is one and because it employs one antenna on the receiver side (MISO structure), it is a full rate code.",
"In order to achieve full diversity and to maximize the minimum of code gain distance (CGD), we have to select $S_k$ , in which $k= 1, 2, \\dots , 8$ , from rotated constellation.",
"Let $\\nu _k$ denotes the $k^{\\text{th}}$ column of codeword, and then we can write: $\\begin{aligned}<\\nu _1,\\nu _i>=0,i\\ne 1,4\\quad &,\\quad <\\nu _5,\\nu _i>=0, i\\ne 5,8, \\\\<\\nu _2,\\nu _i>=0, i\\ne 2,3\\quad &,\\quad <\\nu _6,\\nu _i>=0, i\\ne 6,7,\\end{aligned}$ in which $<\\nu _i,\\nu _j>=\\sum _{\\forall k} \\nu _{ki} \\nu ^*_{kj} $ , and $\\nu _{ki}$ is refer to $k^{\\text{th}}$ element from vector $\\nu _i$ .",
"The orthogonality between some columns made by ${G}$ can help to simplify our ML decoder.",
"As mentioned before, this type of coding creates block fading channels [10] which allows us to formulate an ML decoding equation as follows: $\\begin{aligned}\\hat{\\mathbf {C}}&=\\arg \\min _{\\mathbf {C}^i}\\sum _{L=1}^{2}||\\mathbf {Y}_L-\\mathbf {C}^i_L\\mathbf {H}_L||^2_F\\\\&=\\arg \\min _{\\mathbf {C}^i}\\sum _{L=1}^{2}\\text{Tr}\\lbrace (\\mathbf {C}^i_L \\mathbf {H}_L)^H\\mathbf {C}_L^i \\mathbf {H}_L\\rbrace \\\\&-2\\mathcal {R}e[\\text{Tr}\\lbrace (\\mathbf {H}^H_L)^H(\\mathbf {C}^i_L)^H\\mathbf {Y}_L\\rbrace ].\\end{aligned}$ On account of the quasi-orthogonal structure of the codeword, the above equation can be simplified into two independent parts.",
"Due to space limitation, we proposed these two detection formulas on top of the next page.",
"In Eq.",
"(REF ) and Eq.",
"(REF ), $y_i\\in \\mathbf {Y}$ and similarly, $h_i\\in \\mathbf {H}, \\forall i$ .",
"Since $F_1 (.)",
"$ is independent from $F_2 (.)",
"$ , entirely, we can state that the receiver is capable to decode $ (S_1, S_4, S_5, S_8) $ and $ (S_2, S_3, S_6, S_7) $ separately.",
"It means that the ML decoder is talented to minimize both Eq.",
"(REF ) and Eq.",
"(REF ) over all possible symbols.",
"It is therefore clear that the complexity of the ML decoder for such proposed codeword and the system is $\\mathcal {O}(m^4),$ instead of $\\mathcal {O}(m^8)$ .",
"All the reasons for this dramatic mitigation is creative symbols combination in the proposed codeword.",
"Figure: NO_CAPTION"
],
[
"Three Transmitter and One Receiver Antennas",
"By omitting the fourth column from both $\\mathbf {C}_1$ and $\\mathbf {C}_2$ in (REF ) and REF , respectively, a full rate QOSTBC can be recommended for $M_T=3$ and $M_R=1$ as it is illustrated in (REF ) and (REF ), $\\mathbf {C}^{\\prime }_1=\\begin{bmatrix}S_1+jS_5 & S_2+jS_6 & \\tilde{S}_3+j\\tilde{S}_7\\\\-S_2^*+jS_6^* & S_1^*-jS^*_5 & -\\tilde{S}_4^*+j\\tilde{S}_8^*\\\\-\\tilde{S}_3^*+j\\tilde{S}_7^* & -\\tilde{S}_4^*+j\\tilde{S}^*_8 & S_1^*-jS_5^*\\\\\\tilde{S}_4+j\\tilde{S}_8 & -\\tilde{S}_3-j\\tilde{S}_7 & -S_2-jS_6\\end{bmatrix},$ $\\mathbf {C}^{\\prime }_2=\\begin{bmatrix}S_1-jS_5 & S_2-jS_6 & \\tilde{S}_3-j\\tilde{S}_7\\\\-S_2^*-jS_6^* & S_1^*+jS^*_5 & -\\tilde{S}_4^*-j\\tilde{S}_8^*\\\\-\\tilde{S}_3^*-j\\tilde{S}_7^* & -\\tilde{S}_4^*-j\\tilde{S}^*_8 & S_1^*+jS_5^*\\\\-\\tilde{S}_4-j\\tilde{S}_8 & -\\tilde{S}_3+j\\tilde{S}_7 & -S_2+jS_6\\end{bmatrix}.$ The same approach applies when considering full diversity and maximizing the minimum of CGD.",
"After constructing codeword $\\mathbf {C}^{\\prime }$ based on Eq.",
"(REF ), it is easy to understand the following expressions, $\\begin{aligned}&<\\nu _1,\\nu _i>=0,i\\ne 1\\quad &&,\\quad <\\nu _5,\\nu _i>=0, i\\ne 4 \\\\&<\\nu _2,\\nu _i>=0, i\\ne 2,3\\quad &&,\\quad <\\nu _6,\\nu _i>=0, i\\ne 5,6\\end{aligned}$ Given (REF ), we have a quasi-orthogonal structure; Thus again we can break ML into two independent parts almost similar to Eq.",
"(REF ), and Eq.",
"(REF ).",
"Thus it is again possible to reduce complexity of the ML decoder from $\\mathcal {O}(m^8),$ to $\\mathcal {O}(m^4)$ .",
"Note that in order to minimize decoder complexity we can also use sphere-decoding algorithm based on [11] for both codewords $\\mathbf {C},$ and $\\mathbf {C}^{\\prime }$ .",
"In this case, two independent sphere-decoders can be used, one for detecting $(S_1, S_4, S_5, S_8),$ and the other to detect $(S_2, S_3, S_6, S_7)$ ."
],
[
"Simulation Results and Discussion",
"In this part of the paper, we scrutinized the performance of our proposed codes and a popular approach namely QOSTBC.",
"Simulations were executed in an open loop MISO system using two separated sphere-decoding algorithm for detection and then the results were compared against quasi-orthogonal codes in [5], [6], and [12].",
"Fig.",
"REF shows bit-error-rate (BER) versus signal to noise ratio (SNR) for the codeword $\\mathbf {C}$ , in which $M_T=4,$ and $M_R=1,$ using BPSK constellation.",
"Likewise, Fig.",
"REF depicts the results based on a 4QAM constellation and finally Fig.",
"REF represents the result for codeword $\\mathbf {C}^{\\prime },$ where $M_T = 3,$ and $M_R=1,$ using similar constellation to Fig.",
"2.",
"As simulations for CGD shows, the optimal rotation is $\\theta =\\frac{\\pi }{4},$ using aforementioned constellation.",
"With this amount of rotation we can reach the best performance for both two codewords.",
"Consider that this code scheme is flexible and can be implemented with higher order of QAM without any limitation.",
"By recent FPGA deployment, which provides higher speed for processing, if we ignore complexity of the decoder, the simulations were run under fair conditions.",
"Therefore, according to the simulations, the proposed codes outperform those reported in [6] and [12] by an amount near to 4 dB.",
"Figure: Bit error rate versus signal to noise ratio for BPSK constellation1 bit/sec/Hz (M T =4 M_T = 4).Figure: Bit error rate versus signal to noise ratio for 4QAM constellation2 bits/sec/Hz (M T =4 M_T = 4).Figure: Bit error rate versus signal to noise ratio for 4QAM constellation2 bit/sec/Hz (M T =3 M_T = 3)."
],
[
"Conclusion and Future Work",
"In this paper, we unveiled two full rates and full diversity quasi-orthogonal space-time block codes for open loop MISO systems employing three and four transmitter antennas and one antenna at the receiver.",
"We demonstrated that ML decoder can be separated into two independent functions to reduce complexity of the decoder.",
"By exploiting these codes, we changed property of the channel from quasi-static fading to block fading and discovered that this alteration improved the proposed codes' performance by up to 4dB in comparison with [6] and [12].",
"We can now pose this question: is it possible to reduce complexity of the decoder for these codewords?",
"Finding an answer to this question is an open field problem for future work."
]
] | 1605.05901 |
[
[
"Entangled Dynamics in Macroscopic Quantum Tunneling of Bose-Einstein\n Condensates"
],
[
"Abstract Tunneling of a quasibound state is a non-smooth process in the entangled many-body case.",
"Using time-evolving block decimation, we show that repulsive (attractive) interactions speed up (slow down) tunneling, which occurs in bursts.",
"While the escape time scales exponentially with small interactions, the maximization time of the von Neumann entanglement entropy between the remaining quasibound and escaped atoms scales quadratically.",
"Stronger interactions require higher order corrections.",
"Entanglement entropy is maximized when about half the atoms have escaped."
],
[
"Entangled Dynamics in Macroscopic Quantum Tunneling of Bose-Einstein Condensates Diego A. Alcala, Joseph A. Glick, and Lincoln D. Carr Department of Physics, Colorado School of Mines, Golden, CO 80401, USA Tunneling of a quasibound state is a non-smooth process in the entangled many-body case.",
"Using time-evolving block decimation, we show that repulsive (attractive) interactions speed up (slow down) tunneling, which occurs in bursts.",
"While the escape time scales exponentially with small interactions, the maximization time of the von Neumann entanglement entropy between the remaining quasibound and escaped atoms scales quadratically.",
"Stronger interactions require higher order corrections.",
"Entanglement entropy is maximized when about half the atoms have escaped.",
"Tunneling is one of the most pervasive concepts in quantum mechanics and is essential to contexts as diverse as $\\alpha $ -decay of nuclei [1], vacuum states in quantum cosmology [2] and chromodynamics [3], and photosynthesis [4].",
"Macroscopic quantum tunneling (MQT), the aggregate tunneling behavior of a quantum many-body wavefunction, has been demonstrated in many condensed matter systems [5], [6] and is one of the remarkable features of Bose-Einstein Condensates (BECs), ranging from Landau-Zener tunneling in tilted optical lattices [7] to the AC and DC Josephson effects in double wells [8], [9], as well as their quantum entangled generalizations [10].",
"The original vision of quantum tunneling was in fact the quantum escape or quasibound problem by Gurney and Condon in 1929 [1], and recently the first mean-field or semiclassical observation of quantum escape has been made in Toronto [11].",
"However, with the rise of entanglement as a key perspective on quantum many-body physics, the advent of powerful entangled dynamics matrix-product-state (MPS) methods [12], [13], and the possibility of observing the moment-to-moment time evolution of quasibound tunneling dynamics directly in the laboratory [14], [15], [16], [17], [11] it is the right time to revisit quantum escape.",
"In this Letter, we take advantage of the powerful new toolset for quantum many-body simulations [18], [13] to show that the many-body quantum tunneling problem differs in key respects from our expectations from semiclassical and other well-established approaches to tunneling.",
"Specifically, we use time-evolving block decimation (TEBD) to follow lowly entangled matrix product states [19], [12] for the quantum escape of a quasibound ultracold Bose gas initially confined behind a potential barrier.",
"Our use of a Bose-Hubbard Hamiltonian [20] can be viewed either as a discretization scheme or as an explicitly enforced optical lattice used to control the tunneling dynamics.",
"Unlike instanton and semiclassical approaches, we are able to follow the von Neumann entanglement entropy, number fluctuations, quantum depletion, and other quantum many-body aspects of time evolution of the many-body wavefunction.",
"Such measures clarify when semiclassical approaches are and are not applicable.",
"They also show that hiding in the semiclassical averaged picture are other many-body features with radically different scalings: the escape time $t_{\\mathrm {esc}}$ , i.e., the time at which the average number of remaining quasibound atoms falls to $1/e$ of its initial value, increases (decreases) as an exponential with attractive (repulsive) interactions for a limited range of interactions near zero.",
"We will show that in order to accurately describe the scaling of $t_{\\mathrm {esc}}$ and other many-body observables over many interaction strengths, one must include the effect of higher corrections.",
"Whether between discrete states in a double well [21], [8], in Landau-Zener [22] and orbital angular momentum contexts [23], for quantum escape [24], [25], or even in variational parameter space [26], MQT has up till now mainly been treated under semiclassical approximations such as the instanton approximation and JWKB, as well as the nonlinear Schrodinger equation (NLS).",
"The NLS approach already establishes non-smooth time-evolution of a quasibound state in the form of “blips” or bursts of condensate [27], although mean-field theory sometimes gives incorrect predictions in this regard; we demonstrate that the burst predictions are correct.",
"Beyond mean-field, semiclassical, and instanton approaches, two time-evolving many-body studies have been performed recently.",
"First, an explicit comparison between instanton and TEBD Bose-Hubbard based predictions has been performed for superfluid decay [28], [29], establishing explicit numerical limits on the instanton approach; this method is nearly identical to ours but treats discrete-to-discrete state or double-well type tunneling, in this case between two rotational states on a ring.",
"Second, the quantum escape problem has been studied with the first-quantization-based time-adaptive many-body method known as multi-configurational Hartree-Fock theory [30], [10]; this work treated quantum depletion but not von Neumann entropy and number fluctuations.",
"In contrast, our approach accesses a wide variety of quantum measures to elucidate the underlying many-body quantum features of quasibound escape dynamics, and shows the explicit convergence to mean-field type dynamics.",
"Consider a system of $N$ bosons at zero temperature in the canonical ensemble.",
"To simulate such a system, we can either invoke an explicit optical lattice of $L$ sites, deep enough for tight binding and single band approximations to be valid; or we can simply choose a discretization scheme.",
"Either way the Bose Hubbard Hamiltonian (BHH) is an appropriate model: $\\hat{H} = -J\\sum _{i=1}^{L-1}(\\hat{b}_{i+1}^\\dagger \\hat{b}_i+\\mathrm {h.c.})+\\sum _{i=1}^L [\\frac{U}{2}\\hat{n}_i(\\hat{n}_i-\\hat{1})+V^{\\mathrm {ext}}_i \\hat{n}_{i}].$ In Eq.",
"(REF ), $J$ is the energy of hopping and $U$ determines the on-site two-particle interactions.",
"An external rectangular potential barrier, of width $w$ and height $h$ , is given by $V_i^{\\mathrm {ext}}$ .",
"The field operator $\\hat{b}_{i}^\\dagger $ ($\\hat{b}_{i}$ ) creates (annihilates) a boson at the $i\\mathrm {th}$ site and $\\hat{n}_{i} \\equiv \\hat{b}_{i}^\\dagger \\hat{b}_{i}$ .",
"We will work in hopping units: energies are scaled to $J$ and time $t$ to $\\hbar /J$ .",
"We use open boundary conditions, as convenient for TEBD.",
"TEBD is a matrix product state numerical method that time evolves Eq.",
"(REF ) on a time-adaptive reduced Hilbert space, given that the system is lowly entangled.",
"TEBD is a superior method because it gives us access to quintessential many-body quantities like entanglement.",
"Instanton methods offer another approach towards calculating tunneling rates within a semiclassical approximation [31], but are rapidly rendered inaccurate for larger interaction strengths [32], whereas TEBD suffers from no such limitations.",
"To describe the system from a mean field perspective, the discrete NLS (DNLS) may either be obtained via discretization of the NLS or from a mean field approximation of the BHH.",
"In the latter case, one can propagate the field operator $\\hat{b}_{i}$ forward in time using the BHH in the Heisenberg picture: $i \\hbar \\partial {}_t \\hat{b}_{i}=[\\hat{b}_{i},\\hat{H}]$ .",
"Assuming the many-body state is a product of Glauber coherent states, $\\langle \\hat{b}_i^{\\dagger }\\hat{b}_i\\hat{b}_i\\rangle = \\psi _i^*\\psi _i\\psi _i$ , where $\\psi _i\\equiv \\langle \\hat{b}_{i} \\rangle $ , leads to the DNLS: $\\textstyle i\\hbar \\dot{\\psi _i}=-J(\\psi _{i+1}+\\psi _{i-1}) + g|\\psi _i|^2 \\psi _i+V^{\\mathrm {ext}}_{i} \\psi _i.$ In Eq.",
"(REF ), the condensate order parameter, $\\psi _i$ , is normalized to the number of atoms, $N = \\sum ^{L}_{i=1} |\\psi _i|^2$ .",
"Mean field simulations are performed using a fourth-order Runge-Kutta adaptation of Eq.",
"(REF ).",
"The BHH approaches the DNLS in the mean field limit $N\\rightarrow \\infty $ , $U \\rightarrow 0$ , $NU/J=\\mathrm {const.",
"}$ We emphasize that both the BHH and the DNLS are single band models, valid when the many-body wavefunction covers many sites and has variations larger than the lattice constant.",
"A true continuum limit is possible for $N J / L = \\mathrm {const.",
"}$ , $N/L \\rightarrow 0$ and $J \\rightarrow \\infty $ ; however, this would restrict us numerically to very small numbers of atoms [33] and prevent us from approaching the mean field limit of $NU=\\mathrm {const.",
"}$ , $N\\rightarrow \\infty $ , $U\\rightarrow 0$ ; it can also require different discretization schemes than the BHH depending on the interaction strength and regime of interest.",
"We therefore restrict ourselves to the semi-discrete regime appropriate to both the BHH and DNLS.",
"Figure: Initial Quasibound State.",
"The many-body wavefunction for N=20N=20 with NU/J=+0.15NU/J=+0.15 (blue shaded region, points show actual TEBD results for the density average 〈n ^ i 〉\\langle \\hat{n}_i \\rangle j) is first localized to the left behind the barrier (red line, red and pink shaded areas) via relaxation in imaginary time with a barrier of height hh and initial width w I w_{I}.",
"At t=0t=0 in real time propagation the barrier is reduced to width ww (solid red line, red shaded area) so the now quasibound Bose gas can commence macroscopic quantum tunneling.",
"The hard wall at the left and relatively small barrier area pushes the density tail to partially extend to the right.We initialize the many-body wavefunction via imaginary time relaxation to trap the atoms in a quasibound state behind the barrier as illustrated in Fig.",
"REF .",
"We set $V^{\\mathrm {ext}}$ to height $h=0.05$ and width $w_{I}$ , effectively reducing the system size.",
"At $t=0$ , in real time, the barrier is decreased to width $w$ , where $w$ is typically one to five sites, such that the atoms can escape on a time scale within reach of TEBD simulations.",
"We choose $L$ large enough so that reflections from the box boundary at the far right do not return to the barrier in simulation times of interest: $t_\\mathrm {reflect} \\gg t_{\\mathrm {esc}}$ .",
"Evolving in real time, we first make a coarse observation of the dynamics of MQT in Fig.",
"REF by plotting the average atom number in different regions for repulsive interactions, in order to determine $t_\\mathrm {esc}$ .",
"We find similar results for attractive interactions, but with larger $t_{\\mathrm {esc}}$ .",
"Figure: Many-Body Tunneling, and Calculation of Decay Time.",
"Barrier widths (a,d) w=1w=1, (b,e) w=3w=3 , and (c,f) w=5w=5.",
"Top row: average atom number per site.",
"Bottom row: number in well 〈n ^ well 〉\\langle \\hat{n}_{\\mathrm {well}} \\rangle (blue), number in barrier 〈n ^ bar 〉\\langle \\hat{n}_{\\mathrm {bar}} \\rangle (green), and escaped 〈n ^ esc 〉\\langle \\hat{n}_{\\mathrm {esc}} \\rangle (red) atoms; the 1/e1/e decay time all ±0.1\\pm 0.1.",
"All plots for NU/J=+0.30NU/J=+0.30 with N=20N=20.How do many-body predictions compare to mean field ones?",
"We define $t_{\\mathrm {esc}}^{\\mathrm {MF}}$ and $t_{\\mathrm {esc}}^{\\mathrm {MB}}$ as the mean field and many-body escape times, respectively.",
"For fixed $N U/J$ , $w$ , and $h$ , the DNLS gives the same result independent of $N$ and $U$ ; $t_{\\mathrm {esc}}^{\\mathrm {MB}} \\rightarrow t_{\\mathrm {esc}}^{\\mathrm {MF}}$ only in the large $N$ small $|U|$ mean field limit; and $w^2 h$ determines the barrier area.",
"Figure REF illustrates our exploration of this parameter space.",
"The dynamics of MQT predicted by the DNLS and BHH differ strongly when $N$ is small.",
"Generally, the DNLS predicts $t_{\\mathrm {esc}}^{\\mathrm {MB}}$ well when $N$ is sufficiently large.",
"For example, in Fig.",
"REF (c) for repulsive (attractive) interactions $NU/ J = +0.15$ ($NU/ J = -0.15$ ) and barrier width $w=5$ , the BHH predicts a decrease (increase) in $t_{\\mathrm {esc}}^{\\mathrm {MB}}$ , approaching a nearly constant value for $N \\gtrsim 20$ .",
"This same trend is apparent for various barrier areas, see Fig.",
"REF (a,b).",
"In Fig.",
"REF (d) we also show the quantum depletion $D$ or fragmentation, for $N U/J=\\pm 0.30$ , $w=5$ , $D \\equiv 1-(\\lambda _1) / (\\sum _{m=1}^{L}\\lambda _m)$ where $\\lbrace \\lambda _m\\rbrace $ are the eigenvalues of the single particle density matrix $\\langle \\hat{b}_i^{\\dagger }\\hat{b}_j\\rangle $ , and $\\lambda _1$ is the largest eigenvalue; larger $D$ corresponds to a more fragmented (less condensed) state.",
"The largest fragmentation for both attractive and repulsive interactions occurs for $N=2$ .",
"As $N$ increases, depletion decreases monotonically, with $N=20$ reaching $D \\approx 0.10$ ($D \\approx 0.04$ ) for attractive (repulsive) interactions.",
"This decreased fragmentation allows the DNLS to give accurate predictions for $t_{\\mathrm {esc}}^{\\mathrm {MB}}$ for larger $N$ .",
"Figure: Many Body (MB) vs.",
"Mean Field (MF) Escape Time Predictions.",
"Solid lines: Repulsive (REP).",
"Dashed lines: Attractive (ATT).",
"(a)-(b) Dependence of t esc MB t_{\\mathrm {esc}}^{\\mathrm {MB}} on barrier area and atom number for (a) NU/J=±0.15N U/J = \\pm 0.15 and (b) NU/J=±0.30NU/J=\\pm 0.30.",
"(c) t esc MB t_{\\mathrm {esc}}^{\\mathrm {MB}} plateaus towards t esc MF t_{\\mathrm {esc}}^{\\mathrm {MF}} for 10 to 80 atoms as shown for NU/J=±0.15NU/J = \\pm 0.15 and w=5w=5.",
"(d) Fragmentation for NU/J=±0.30NU/J=\\pm 0.30 and w=5w=5.",
"Curves are a guide to the eye, points represent actual data with error bars smaller than data point in all panels.",
"Panel (d) legend corresponds to (a),(b), and (d).Systematic error in TEBD [34] for $t_{\\mathrm {esc}}^{\\mathrm {MB}}$ results from the Schmidt truncation $(\\chi )$ , the truncation in the on-site Hilbert space dimension $(d)$ , and the time resolution at which we write out data $(\\delta t)$ .",
"The hardest many-body measures to converge, such as the block entropy, at $\\chi =35$ have an error $\\lesssim 10^{-3}$ for $N=70$ , and were checked up through $\\chi =55$ ; due to small $U$ and effective system size, much lower $\\chi $ is required than usual in TEBD.",
"For up to $N=10$ we have not truncated $d$ , but for larger $N$ up to 80, we truncated attractive (repulsive) to $d=20$ ($d=15$ ).",
"A lower truncation results in decreased $t_{\\mathrm {esc}}^{\\mathrm {MB}}$ , e.g.",
"by 10% for $d=5$ , $NU/J=-0.1$ , and $N=10$ , even though $\\mathrm {max}(\\langle \\hat{n}\\rangle ) < 1$ , since more weight is given to spread-out Fock states.",
"The attractive BHH requires much higher $d$ than the repulsive BHH, since $U<0$ increases number fluctuations in high density regions, i.e., behind the barrier at $t=0$ .",
"In both cases, in general we find on-site number fluctuations play a surprisingly strong role in tunneling processes compared to usual for TEBD.",
"The BHH also has a number of sources of systematic error, the most important of which is virtual fluctuations to the second band; however, since we compare single-band DNLS to single-band BHH this does not effect our comparison.",
"In general we expect fluctuations to higher bands will speed up tunneling; therefore out calculations may be taken as a lower bound for experiments.",
"Figure: Many-body Quantum Measures.",
"Solid lines: Repulsive.",
"Dashed lines: Attractive.",
"(a) Average number at the density peak shows bursts of atoms .",
"Early time attractive lines solid to make dynamics distinguishable.",
"(b) Time derivative of number fluctuations in the number of trapped atoms are smaller for repulsive interactions.",
"(c) Nearly universal curve for the entropy of entanglement vs. the average number of trapped atoms.",
"(d) Observables demonstrate very different scaling with interaction.",
"Points show actual data (error bars smaller than points), while lines are best fit curves.",
"All plots treat N=6N=6.",
"Panel (a) legend correspond to (a),(b), and (c).In Fig.",
"REF (a) we plot the average number at the peak of the many-body wavefunction.",
"There are points in time when the number density exhibits quadratic decay, and others during which it is nearly constant, similar to the density bursts found by Dekel et al.",
"[27]; thus their predictions are correct even in the many-body regime.",
"The first burst is nearly independent of $U$ .",
"The initial flat horizontal region and burst originate from the wave-function pushing away from the leftmost infinite boundary and interacting with the barrier.",
"For attractive interactions, the initial small increase in $\\hat{n}_{\\mathrm {peak}}$ is due to the atoms fluctuating towards the peak, attracted by the strong concentration of atoms.",
"For repulsive interactions this increase occurs because the wavefunction collides with the barrier after pushing away from the infinite wall, causing a slight swell in $\\hat{n}_{\\mathrm {peak}}$ .",
"All subsequent dynamics appear to be dependent on $U$ .",
"To characterize the quantum nature of MQT, in Fig.",
"REF (b) we plot the time derivative of fluctuations in the number of atoms behind the barrier $df_{l}/dt$ , where $f_l=(\\langle N^2_l \\rangle - \\langle N_l \\rangle ^2) / \\langle N_l \\rangle $ , $N_l$ is the number of atoms to the left of site $l$ , and $l$ is taken at the outer edge of the barrier.",
"Once MQT commences, the maximum value of $df_{l}/dt$ in time increases with decreasing $U$ because number densities just outside the barrier have more influence to “pull” additional atoms through the barrier for attractive interactions.",
"Repulsive interactions, in comparison, suppress tunneling, so $df_{l}/dt$ does not increase as much.",
"Of particular interest to MQT is the von Neumann block entropy characterizing entanglement between the remaining quasibound atoms and the escaped atoms, $S_l\\equiv -\\mathrm {Tr}(\\hat{\\rho }_l \\log \\hat{\\rho }_l)$ , where $\\hat{\\rho }_l$ is the reduced density matrix for the well plus barrier.",
"The key features of $S_l$ are illustrated in a nearly universal curve in Fig.",
"REF (c): on the lower right side tunneling has not yet commenced.",
"$S_l$ maximizes part way through the tunneling process in the center of the curve, at $N_l/N \\simeq 1/2$ ; and $S_l$ then decreases again to the left as the atoms finish tunneling out.",
"Define $t_{s}$ as the time at which $S_{l}$ is maximized and define $t_{f}$ as the time at which the slope of the number fluctuations ($df_l/dt$ ) is largest before $t_{f}$ .",
"We find $t_{s}$ , $t_{f}$ , and $t_\\mathrm {esc}^{\\mathrm {MB}}$ increase with decreasing $U$ , as shown in Fig.",
"REF (d).",
"As $NU/J$ decreases, we approach the self-trapping regime, where escape times become much longer than the lifetime of the system.",
"While $t_{\\mathrm {esc}}$ increases smoothly as $NU/J$ decreases, $df_{l}/dt$ is strongly influenced by change in $NU/J$ , with a noticeable increase near $NU/J \\approx -0.3$ , and a steady flattening-out as we approach self-trapping interaction strength.",
"A best fit line for $t_{f}$ covering all $NU/J$ requires an exponential of a second order polynomial, while an exponential fits well for $-0.3 < NU/J < 0.3$ , as also found in Ref.",
"[35] for bright solitons tunneling in a tilted optical lattice.",
"In the coarser measure $t_{\\mathrm {esc}}^{\\mathrm {MB}}$ , we find exponential scaling when $-0.4 < NU/J < 0.4$ .",
"In order to accurately capture the strong interaction regimes $N|U|/J \\gtrsim 0.4$ , we need a third order polynomial in the exponential, as shown in the fit in Fig.",
"REF (d).",
"We find that $t_{s}$ scales linearly only for $-0.1 < NU/J < 0.1$ , quadratically for $-0.4 < NU/J < 0.4$ , and requires a cubic polynomial fit to cover the entire interaction regime.",
"Results in Fig.",
"REF are for $N=6$ ; we found similar results for up to $N=20$ , although simulations are limited in the large $|U|$ regime.",
"Another experimental signature is the density-density correlations, $g^{(2)}_{ij}=\\langle \\hat{n}_i \\hat{n}_j \\rangle - \\langle \\hat{n}_i \\rangle \\langle \\hat{n}_j \\rangle $ , extractable from noise measurements [36], [37]; $g^{(2)}$ is zero in mean field theory.",
"As customary, we subtract off the large diagonal matrix elements of $g^{(2)}$ to view the underlying off-diagonal structure.",
"In Fig.",
"REF (a)-(c) we show $g^{(2)}$ for $N=40$ , $NU/J=-0.015$ , and $w=2$ , dividing up the system to observe correlations between the three physical regions: trapped, under the barrier, and escaped.",
"We initially observe near-zero correlations everywhere except near the many-body wavefunction peak.",
"At $t=62 \\approx t_{s}$ , $g^{(2)}$ shows many negatively-correlated regions ($g^{(2)}<0$ ) which are broken up by the potential barrier.",
"In Fig.",
"REF (d) we also show quantum depletion $D$ for $NU/J=\\pm 0.15$ with $N=2$ and $w=1,2,3,4,5$ .",
"$D$ increases with increasing $w$ .",
"In comparison to Fig.",
"REF (d) ($NU/J=\\pm 0.30$ ), $D$ doesn't become as large for Fig.",
"REF (d) ($NU/J=\\pm 0.15$ ) because of the smaller $N|U|/J$ value.",
"The growth in $D$ emphasizes the many-body nature of the escape process.",
"Figure: Time-dependence of Density-Density Correlations.",
"(a)-(c) g (2) g^{(2)} shows correlations between trapped and escaped atoms.",
"The barrier, indicated by dotted lines, breaks up negatively-correlated regions (red); shown are time slices at (a) t=0t=0, (b) t=62≈t s t=62 \\approx t_{s}, and (c) t=125≈t esc MB t=125 \\approx t_{\\mathrm {esc}}^{\\mathrm {MB}}.",
"(d) Quantum depletion grows rapidly for N=2N=2 with NU/J=±0.15NU/J=\\pm 0.15.",
"Solid lines: Repulsive.",
"Dashed lines: Attractive.",
"Curves are a guide to the eye, points represent actual data (error bars smaller than points).In conclusion, we have performed quantum many-body simulations of the macroscopic quantum tunneling of attractive and repulsive bosons using TEBD to time-evolve the Bose-Hubbard Hamiltonian, treating the original 1929 quasibound or quantum escape problem.",
"We found strong deviations from mean field predictions and provided quantitative boundaries by which one can judge the legitimacy of applying mean field theory to this problem.",
"Even a low average order moment like escape time was shown to deviate from simple exponential scaling for strong interactions.",
"Higher order quantum measures like entropy of entanglement between the quasibound and escaped atoms, and the slope of number fluctuations, reached a maximum at times which exhibited scaling behaviors with interactions ranging from polynomial to exponential to exponential of a polynomial, showing tunneling dynamics are far richer in the quantum many body picture.",
"We showed the many-body extension to the predictions of Dekel et al.",
"regarding number density bursts [27].",
"Finally, our study shows that many-body effects in macroscopic quantum tunneling can be experimentally observed via number fluctuations and density-density correlations as well as dependence of escape time on interactions.",
"We thank Veronica Ahufinger, Jen Glick, Mark Lusk, Kenji Maeda, Marie Mclain, Shreyas Potnis, Anna Sanpera, Aephraim Steinberg, Marc Valdez, Michael Wall, David Wood, and Xinxin Zhao for valuable discussions.",
"This work was supported by NSF and the AFOSR."
]
] | 1605.05768 |
[
[
"Giant Thermal Vibrations in the Framework Compounds Ba1-xSrxAl2O4"
],
[
"Abstract Synchrotron X-ray diffraction (XRD) experiments were performed on the network compounds Ba1-xSrxAl2O4 at temperatures between 15 and 800 K. The ferroelectric phase of the parent BaAl2O4 is largely suppressed by the Sr-substitution and disappears for $x\\geq0.1$.",
"Structural refinements reveal that the isotropic atomic displacement parameter ($B_{\\rm iso}$) in the bridging oxygen atom for $x\\geq0.05$ is largely independent of temperature and retains an anomalously large value in the adjacent paraelectric phase even at the lowest temperature.",
"The $B_{\\rm iso}$ systematically increases as $x$ increases, exhibiting an especially large value for $x\\geq0.5$.",
"According to previous electron diffraction experiments for Ba1-xSrxAl2O4 with $x\\geq0.1$, strong thermal diffuse scattering occurs at two reciprocal points relating to two distinct soft modes at the M- and K-points over a wide range of temperatures below 800 K [Y. Ishii et al., Sci.",
"Rep. 6, 19154 (2016)].",
"Although the latter mode disappears at approximately 200 K, the former does not condense, at least down to 100 K. On the contrary, it still survives at low-temperature.",
"The anomalously large $B_{\\rm iso}$ observed in this study is ascribed to these soft modes existing in a wide temperature range."
],
[
"Introduction",
"Since the discovery of the incipient perovskite-type quantum paraelectric oxides[1], [2], the quantum criticality in ferroelectrics has attracted substantial interest in condensed-matter physics.",
"On the border of ferroelectricity, the dielectric constant is enhanced in the vicinity of absolute zero.",
"This non-classical behavior was recently quantitatively explained[3], and the relevance of the ferroelectric quantum criticality to other condensed states, such as superconductivity in doped SrTiO$_3$[4], [5], has been reported.",
"The recently observed quantum critical behavior in other new ferroelectric compounds, $e.g.$ , the organic ferroelectric TSCaCl$_{2(1-x)}$ Br$_{2x}$[6] and TTF-QBr$_{4-n}$ I$_n$ complexes[7] and the multiferroic Ba$_2$ CoGe$_2$ O$_7$[8], has prompted fascinating studies on the quantum criticality in ferroelectrics.",
"Figure: (Color online) Crystal structures of BaAl 2 _2O 4 _4.",
"(a) The high-temperature phase (space group P6 3 22P6_322) and (b) the low-temperature phase (P6 3 P6_3).",
"(c) The split atom model for the P6 3 22P6_322 high-temperature phase.",
"In this structure model, the atomic positions of Ba/Sr and O2 are split into two sites along the cc-axis, and the O1 site is split into three sites around the threefold axis.BaAl$_2$ O$_4$ is a chiral improper ferroelectric[9] without an inversion center that crystallizes in a stuffed tridymite-type structure comprising a corner-sharing AlO$_4$ tetrahedral network with six-member cavities occupied by Ba ions.",
"This compound undergoes a structural phase transition from a high-temperature phase with a space group of $P6_322$ to a low-temperature phase with $P6_3$ at approximately 450 K[10], [11], [12].",
"This transition is accompanied by the significant tilting of the Al-O-Al bond angle along the $c$ -axis, giving rise to the enlargement of the cell volume to $2a\\times 2b\\times c$ .",
"The crystal structures of the high-temperature and low-temperature phases are shown in Figs.",
"REF (a) and (b), respectively.",
"This compound possesses low-energy phonon modes, which are associated with a tilting of the AlO$_4$ tetrahedra around the shared vertices without a large distortion in each AlO$_4$ block[13].",
"Such low-energy phonon modes have also been reported in SiO$_2$ modifications[14], [15], nepheline[16], and ZrW$_2$ O$_8$[17] and are often called rigid unit modes (RUMs).",
"RUMs can sometimes act as soft modes and cause structural phase transitions, as observed in quartz[18], tridymite[19], and nepheline[16].",
"RUMs have also been suggested as the origin of a negative thermal expansion, as in ZrW$_2$ O$_8$[17], [20].",
"Low-energy phonon modes, such as RUMs, can be observed as thermal diffuse scattering in electron and X-ray diffractions (XRD).",
"In electron diffraction experiments of BaAl$_2$ O$_4$ , a characteristic honeycomb-type diffuse scattering pattern has been reported over a wide range of temperatures below 800 K[21], [22].",
"A similar pattern was also observed in Ba$_{1-x}$ Sr$_x$ Al$_2$ O$_4$ with $x=0.4$[23].",
"In the structural refinements of the $x=0.4$ sample, significant disorder was noted in the two oxygen sites, O1($2d$ ) and O2($6g$ ), in the initial model of $P6_322$ symmetry, as shown in Fig REF (a).",
"In those analyses, a split atom model, as shown in Fig.",
"REF (c), was employed, in which the positions of Ba/Sr($2b$ ), O1($2d$ ), and O2($6g$ ) are off-center from their ideal positions and split into the less symmetric $4e$ , $6h$ and $12i$ sites, respectively.",
"This type of disorder has also been reported in Sr$_{0.864}$ Eu$_{0.136}$ Al$_2$ O$_4$[24] and BaGa$_2$ O$_4$[25].",
"We recently investigated the low-energy phonon modes in BaAl$_2$ O$_4$ in detail via synchrotron XRD using single crystals and first principles calculations[10].",
"According to the calculations, this compound possesses two unstable phonon modes at the M- and K-points with nearly the same energies.",
"Both of these unstable phonon modes gave rise to strong diffuse scattering intensities in a wide range of temperatures below 800 K. Interestingly, their intensities sharply increase towards $T_{\\rm C}$ , indicating that the two modes soften simultaneously.",
"Furthermore, the ordered phase with the $P6_3$ superstructure has been reported to be substantially suppressed by a small amount of Sr-substitution for Ba[22], [26].",
"According to our electron diffraction experiments on Ba$_{1-x}$ Sr$_x$ Al$_2$ O$_4$ with precisely controlled Sr concentrations, no superstructure is observed, at least down to 100 K, for $x\\ge 0.1$ .",
"In the temperature and compositional window of $T<200$ K and $x\\ge 0.1$ , although the soft mode at the K-point disappears, the soft mode at the M-point survives and shows further fluctuation as the temperature decreases[22].",
"These findings imply the presence of a new quantum critical state induced by the soft modes in Ba$_{1-x}$ Sr$_x$ Al$_2$ O$_4$ .",
"In the present study, we performed synchrotron powder XRD on Ba$_{1-x}$ Sr$_x$ Al$_2$ O$_4$ at 15–800 K, revealing an unusually large and temperature-independent thermal vibration at the bridging oxygens."
],
[
"Experimental",
"Polycrystalline samples of Ba$_{1-x}$ Sr$_x$ Al$_2$ O$_4$ ($x$ = 0, 0.02, 0.05, 0.06, 0.1, 0.3, and 0.5) were synthesized using a conventional solid-state reaction.",
"The sample preparation procedure is described elsewhere[22], [27].",
"The obtained samples were stored in a vacuum.",
"The synchrotron powder XRD patterns were obtained in the temperature range of 15–800 K at the BL02B2 beamline of SPring-8[28].",
"The samples to be measured were crushed into fine powder and filled into a fused quartz capillary with a diameter of 0.2 mm.",
"The diffraction intensities were recorded using an imaging plate and multiple microstrip solid-state detectors.",
"The incident X-ray beam was monochromatized to 25 keV using a Si (111) double-crystal.",
"The temperature was controlled with flowing helium and nitrogen gases.",
"The structure refinements were performed via the Rietveld method using the JANA2006 software package[29].",
"The split atom model was employed for the structural refinements of the high-temperature phase.",
"For the fitting of several profiles obtained at high temperatures, the Ba/Sr atom was placed on the $2b$ site of the $P6_322$ average structure rather than on the $4e$ site of the split atom model because the atomic displacement from the $2b$ site is so small that the $4e$ site cannot be distinguished from the $2b$ site at these temperatures.",
"The $P6_3$ structure model was used for the low-temperature phase below $T_{\\rm C}$ .",
"Several profiles just below $T_{\\rm C}$ were analyzed using the $P6_322$ split atom model because of the poor fitting results obtained using the $P6_3$ structure model.",
"The space groups and the atomic positions used for the refinements are summarized in Table S1."
],
[
"Results and Discussion",
"The obtained diffraction profiles for Ba$_{1-x}$ Sr$_x$ Al$_2$ O$_4$ with $x=0.02$ and 0.1 are shown in Figs.",
"REF (a) and (b), respectively, and the profiles for the other compositions are displayed in Fig.",
"S1.",
"The superlattice reflections of the low-temperature $P6_3$ phase can be clearly seen below 400 K for $x=0.02$ , as indicated by arrows in Fig.",
"REF (a).",
"These superlattice reflections were also observed for $x=0.05$ and 0.06, but they could not be observed for $x\\ge 0.1$ .",
"Thus, the structural phase transition does not occur down to 15 K for $x\\ge 0.1$ .",
"Figure: (Color online) Synchrotron powder XRD profiles obtained at 15–600 K for Ba 1-x _{1-x}Sr x _xAl 2 _2O 4 _4 with (a) x=0.02x = 0.02 and (b) 0.1.",
"All of the indices throughout the paper are based on the P6 3 22P6_322 parent phase.",
"The superlattice reflection grows below 400 K for x=0.02x=0.02, as marked by the arrows, indicating the structural phase transition from P6 3 22P6_322 to P6 3 P6_3.",
"No superlattice reflections can be seen for the x=0.1x=0.1 sample down to 15 K.Figure: (Color online) Phase diagram for Ba 1-x _{1-x}Sr x _xAl 2 _2O 4 _4.",
"T C T_{\\rm C} abruptly decreases as xx increases.Anomalously large and temperature-independent B iso B_{\\rm iso} values were observed in the region of x≥0.1x\\ge 0.1, which is labeled as a giant thermal vibration.T f T_{\\rm f} denotes the temperature at which the K-point mode disappears, as described in the text.$T_{\\rm C}$ was defined as the temperature at which the superlattice reflections appear.",
"A phase diagram for Ba$_{1-x}$ Sr$_x$ Al$_2$ O$_4$ is shown in Fig.",
"REF .",
"The $T_{\\rm C}$ values determined in this study were plotted together with the previously reported data[22], [10].",
"As shown in this figure, $T_{\\rm C}$ becomes largely suppressed as $x$ increases, in agreement with previous reports[26], [22].",
"Synchrotron XRD experiments using single crystals[10] revealed a continuous variation of the superlattice intensities at $T_{\\rm C}$ , indicating a second-order phase transition.",
"These findings indicate that Ba$_{1-x}$ Sr$_x$ Al$_2$ O$_4$ may plausibly show a new quantum critical state.",
"We performed structural refinements on the Ba$_{1-x}$ Sr$_x$ Al$_2$ O$_4$ sample using the Rietveld method.",
"Fig.",
"REF displays the powder diffraction pattern for $x=0.2$ at 15 K and the fitting results obtained using the split atom model.",
"The refinement using the initial model yielded a poor fitting result; the obtained $R$ -factors based on the weighted profile ($R_{\\rm WP}$ ) and the Bragg-integrated intensities ($R_{\\rm I}$ ) were 5.14 and 9.25 %, respectively.",
"In contrast, Fig.",
"REF shows that the split atom model accurately reproduces the experimental profile with high reliability ($R_{\\rm WP}$ = 4.11 % and $R_{\\rm I}$ = 3.20 %).",
"The refined structural parameters are listed in Table REF .",
"One important finding is that the isotropic atomic displacement parameter ($B_{\\rm iso}$ ) of the O1 site is extraordinarily large, which is rare in oxide insulators.",
"Because the $R$ -factors are sufficiently small, this large $B_{\\rm iso}$ should have an important physical meaning.",
"Such a large $B_{\\rm iso}$ in the O1 site has also been reported in previous works on Ba$_{0.6}$ Sr$_{0.4}$ Al$_2$ O$_4$[23], for which it was ascribed to the thermal diffuse scattering observed in the diffraction experiments, and Sr$_{0.864}$ Eu$_{0.136}$ Al$_2$ O$_4$[24] at room temperature.",
"Figure: (Color online) Synchrotron powder XRD patterns (triangles) of Ba 0.8 _{0.8}Sr 0.2 _{0.2}Al 2 _2O 4 _4 at 15 K. The solid red curve represents the result of the structure refinement using the split atom model.",
"The difference curve is shown in the lower part of the figure.",
"Vertical lines indicate the positions of possible Bragg peaks.The fitting result for the high-QQ region is depicted in the inset.The Ba/Sr atom is placed at the 4e4e site.Table: Results of the structure refinement for Ba 0.8 _{0.8}Sr 0.2 _{0.2}Al 2 _2O 4 _4 at 15 K using the split atom model.",
"The cation ratio of Ba:Sr is fixed to 0.8:0.2.Figure: (Color online) Temperature dependence of B iso B_{\\rm iso} of Ba/Sr, Al, the bridging oxygen (O br _{\\rm br}) and the basal oxygen (O bas _{\\rm bas}) atoms for (a) x=0x = 0, (b) 0.05, and (c) 0.5.",
"T C T_{\\rm C} is indicated by an arrow.",
"(d) B iso B_{\\rm iso} at 15 K plotted as a function of xx.",
"(e) The displacement of the Ba/Sr atom (-Δz-\\Delta z) from the ideal 2b2b site of z=0.25z=0.25.",
"The -Δz-\\Delta z values obtained using the P6 3 22P6_322 split atom model are indicated by the open symbols.Below 420 K, the Ba site for x=0x=0 is split into two sites of P6 3 P6_3, 2a2a and 6c6c, as indicated by the closed symbols.The temperature dependences of the obtained $B_{\\rm iso}$ for the $x$ = 0, 0.05, and 0.5 samples are shown in Figs.",
"REF (a), (b), and (c), respectively.",
"In these figures, the bridging oxygen, O$_{\\rm br}$ , and the basal oxygen, O$_{\\rm bas}$ , represent the O1 and O2 atoms in the $P6_322$ structure, respectively.",
"In the structural refinements for the $P6_3$ structure, the atoms were categorized into four groups: the Ba group of Ba1–2, the Al group of Al1–4, the basal oxygen group of O1–4, and the bridging oxygen group of O5–6.",
"The calculations were performed under the constraint that an equal $B_{\\rm iso}$ is used within each group.",
"As shown in Fig.",
"REF (a), the $B_{\\rm iso}$ of each atom for $x=0$ gradually decreases as the temperature decreases, and rapidly decreases at $T_{\\rm C}$ , except for the Ba atom, as marked by an arrow in Fig.",
"REF (a).",
"This is probably because of the structural phase transition accompanying the condensation of one of the soft modes and the resulting suppression of the thermal vibrations.",
"The $B_{\\rm iso}$ values of all atoms are small at 15 K. The sudden drops in the $B_{\\rm iso}$ at $T_{\\rm C}$ were also observed for the $x=0.02$ sample.",
"The temperature dependences of the $B_{\\rm iso}$ for all compositions are shown in Fig.",
"S2.",
"The full width at half maximum (FWHM) of the superlattice reflection has been reported to increase as $x$ increases[22], as observed in this study (Fig.",
"S3).",
"Thus, the long-range ordering of the low-temperature $P6_3$ structure is strongly suppressed by the Sr-substitution.",
"In Fig.",
"REF (b), no sudden drop in $B_{\\rm iso}$ can be observed for $x=0.05$ , although the superlattice reflections develop below 320 K in the $x = 0.05$ sample, as shown in Fig.",
"S1(c).",
"This is probably because of the suppression of the long-range ordering of the low-temperature structure in the $x=0.05$ sample.",
"In contrast, the observed $B_{\\rm iso}$ for $x=0.5$ is surprisingly large, as seen in Fig.",
"REF (c).",
"In addition, it is largely independent of the temperature, resulting in anomalously large values even at the lowest temperature.",
"Such a temperature-independent large $B_{\\rm iso}$ is also observed for $x=0.1$ and 0.3, as shown in Figs.",
"S2(e) and (f).",
"Other atoms also exhibit larger $B_{\\rm iso}$ over the whole temperature range than that of $x=0$ .",
"Because the $B_{\\rm iso}$ values for Ba/Sr, Al, and O$_{\\rm bas}$ are sufficiently small at 15 K, and the refinements yield the satisfactorily small $R$ -factors, as shown in Figs.",
"S4 and S5, the extraordinarily large $B_{\\rm iso}$ observed in O$_{\\rm br}$ cannot be attributed to the fitting error in the refinements.",
"Fig.",
"REF (d) represents the $B_{\\rm iso}$ for each atom at 15 K as a function of $x$ .",
"These values systematically increase as $x$ increses, particularly for the O$_{\\rm br}$ atom.",
"The $B_{\\rm iso}$ for the O$_{\\rm br}$ atoms are especially large for $x\\ge 0.06$ , indicating that the O$_{\\rm br}$ atom for $x\\ge 0.06$ exhibits an unusually large thermal vibration down to absolute zero.",
"Notably, this enhancement in $B_{\\rm iso}$ is observed on the border of the ferroelectric phase.",
"The Ba/Sr atom displaces only along the $c$ -axis in the split atom model.",
"The displacements of the Ba/Sr atoms ($-\\Delta z$ ) from the ideal $2b$ site of $z=0.25$ are plotted in Fig.",
"REF (e) as a function of temperature.",
"These gradually increase as the temperature decreases.",
"For $x=0$ , the Ba site splits into the $2a$ and $6c$ sites of the $P6_3$ low-temperature structure at $T_{\\rm C}$ , as indicated by the closed symbols in Fig.",
"REF (e), because of the structural phase transition.",
"Below $T_{\\rm C}$ , the $-\\Delta z$ of Ba at the $6c$ site increases as the temperature decreases.",
"For $x=0.1$ , 0.3, and 0.5, the $-\\Delta z$ increases as $x$ increases.",
"Notably, the $-\\Delta z$ for $x=0.5$ exceeds that for $x=0$ in the ferroelectric phase, although the $x=0.5$ sample does not exhibit a structural phase transition.",
"The temperature and compositional window of the anomalously large $B_{\\rm iso}$ is illustrated in Fig.",
"REF .",
"In general, the thermal vibration is large at high temperatures and becomes suppressed as temperature decreases, and thus $B_{\\rm iso}$ is expected to decrease as the temperature drops.",
"However, the $B_{\\rm iso}$ of the O$_{\\rm br}$ atom in this system with $x\\ge 0.1$ does not follow this general trend; instead, it exhibits fairly large values even at 15 K. According to our previous studies, this system possesses structural instabilities at the M- and K-points, leading to two energetically competing soft modes[10].",
"Mode analyses have revealed that the O$_{\\rm br}$ atom shows a remarkable in-plane vibration around the ideal $2d$ site.",
"In addition, strong thermal diffuse scattering because of these soft modes has been observed in the electron diffraction patterns for $x\\ge 0.1$ over a wide range of temperatures between 100 and 800 K[22].",
"Clearly, these soft modes are responsible for the unusually large $B_{\\rm iso}$ observed in this study.",
"The large enhancement in $B_{\\rm iso}$ by the Sr substitution indicates “a giant thermal vibration,” which might be a phonon-related quantum critical phenomenon.",
"For the $x\\ge 0.1$ sample, the M-point soft mode does not condense but survives down to at least 100 K, whereas the K-point mode weakens and disappears below 200 K[22].",
"The temperature at which the K-point mode disappears, $T_{\\rm f}$ , is indicated by a broken line in Fig.",
"REF .",
"Below this line, the M-point mode fluctuates with short-range correlations in nanoscale regions[10].",
"The anomalously large and temperature-independent $B_{\\rm iso}$ even at low temperatures can be attributed to this fluctuating M-point mode.",
"Large thermal displacements have also been reported in the metallic phase of VO$_2$[30], [31], which exhibits a drastic metal-insulator transition (MIT).",
"Several approaches, including thermal diffuse scattering just above the MIT[32], the symmetry analyses[33], and the phonon dispersion calculations[34], indicate the presence of a soft mode.",
"ZrW$_2$ O$_8$ is another example that shows a large $B_{\\rm iso}$ in its oxygen atoms, comprises a corner-sharing polyhedral network, and possesses RUMs.",
"However, the $B_{\\rm iso}$ values of these oxygen atoms have been reported to decrease as the temperature decreases[35].",
"No compound showing a temperature-independent and anomalously large thermal vibration even at low temperature has been reported to date.",
"To clarify the nature of the fluctuating state in Ba$_{1-x}$ Sr$_x$ Al$_2$ O$_4$ , measurements of the physical properties, such as the dielectric constant and specific heat, are now in progress."
],
[
"Conclusions",
"We performed synchrotron XRD experiments for Ba$_{1-x}$ Sr$_x$ Al$_2$ O$_4$ at 15–800 K and analyzed their crystal structures via the Rietveld method using the split atom model.",
"$T_{\\rm C}$ becomes substantially suppressed as $x$ increases, and the structural phase transition disappears for $x\\ge 0.1$ .",
"We observed an anomalously large and temperature-independent $B_{\\rm iso}$ in the bridging oxygen for the $x\\ge 0.1$ samples, with the value systematically increasing as $x$ increases.",
"These anomalously large $B_{\\rm iso}$ values can be attributed to the giant thermal vibration arising from the existence of soft modes over a wide temperature range.",
"This work was partially supported by a Grant-in-Aid for Scientific Research from the Ministry of Education, Culture, Sports, Science and Technology of Japan (MEXT) from the Japan Society for the Promotion of Science (JSPS).",
"The synchrotron radiation experiments were performed at BL02B2 of SPring-8 with the approval of the Japan Synchrotron Radiation Research Institute (JASRI) (Proposals No.",
"2015A2058, No.",
"2015A1510, and No.",
"2015B1488)."
]
] | 1605.05880 |
[
[
"Computing Small Certificates of Inconsistency of Quadratic Fewnomial\n Systems"
],
[
"Abstract B{\\'e}zout 's theorem states that dense generic systems of n multivariate quadratic equations in n variables have 2 n solutions over algebraically closed fields.",
"When only a small subset M of monomials appear in the equations (fewnomial systems), the number of solutions may decrease dramatically.",
"We focus in this work on subsets of quadratic monomials M such that generic systems with support M do not admit any solution at all.",
"For these systems, Hilbert's Nullstellensatz ensures the existence of algebraic certificates of inconsistency.",
"However, up to our knowledge all known bounds on the sizes of such certificates -including those which take into account the Newton polytopes of the polynomials- are exponential in n. Our main results show that if the inequality 2|M| -- 2n $\\le$ $\\sqrt$ 1 + 8{\\nu} -- 1 holds for a quadratic fewnomial system -- where {\\nu} is the matching number of a graph associated with M, and |M| is the cardinality of M -- then there exists generically a certificate of inconsistency of linear size (measured as the number of coefficients in the ground field K).",
"Moreover this certificate can be computed within a polynomial number of arithmetic operations.",
"Next, we evaluate how often this inequality holds, and we give evidence that the probability that the inequality is satisfied depends strongly on the number of squares.",
"More precisely, we show that if M is picked uniformly at random among the subsets of n + k + 1 quadratic monomials containing at least $\\Omega$(n 1/2+$\\epsilon$) squares, then the probability that the inequality holds tends to 1 as n grows.",
"Interestingly, this phenomenon is related with the matching number of random graphs in the Erd{\\\"o}s-Renyi model.",
"Finally, we provide experimental results showing that certificates in inconsistency can be computed for systems with more than 10000 variables and equations."
],
[
"Introduction",
"Context and problem statement.",
"Identifying classes of structured polynomial systems and designing dedicated algorithms to solve them is a central theme in computer algebra and in computational algebraic geometry, due to the wide range of applications where such systems appear.",
"We investigate quadratic systems involving a small number of monomials (quadratic fewnomial systems).",
"Let $\\mathbb {K}$ be a field, $\\overline{\\mathbb {K}}$ its algebraic closure and $\\mathbf {M}$ be a finite subset of monomials of degree at most two in a polynomial ring $\\mathbb {K}[X_1,\\ldots ,X_n]$ .",
"Suppose also that the constant 1 belongs to $\\mathbf {M}$ , and let $\\mathcal {L}_{\\mathbf {M}}$ be the $\\mathbb {K}$ -linear space of polynomials in $\\mathbb {K}[X_1,\\ldots , X_n]$ spanned by $\\mathbf {M}$ .",
"A classical question is to bound the number of solutions in $\\overline{\\mathbb {K}}^n$ of a system $f_1(X_1,\\ldots ,X_n)=\\dots =f_n(X_1,\\ldots , X_n)=0$ , where all polynomials lie in $\\mathcal {L}_{\\mathbf {M}}$ and have generic coefficients.",
"When the exponent vectors of the monomials in $\\mathbf {M}$ are the points with integer coordinates in a lattice polytope, Kushnirenko's theorem states that the number of toric solutions (i.e.",
"solutions whose all coordinates are nonzero) is bounded by the normalized volume of the polytope [24].",
"A variant of this theorem indicates that such generic polynomial systems do not admit any solution if the dimension of the $\\mathbb {Q}$ -linear space generated by the exponent vectors of the monomials in $\\mathbf {M}$ does not equal $n$ .",
"Solving a polynomial system and deciding if it has any solution in $\\overline{\\mathbb {K}}^n$ are two closely related questions.",
"One classical method to produce a certificate that a polynomial system does not have any solution is to provide an algebraic relation via Hilbert's Nullstellensatz: Problem 1 - Effective fewnomial Nullstellensatz.",
"Given a system $(f_1,\\ldots , f_m)\\in \\mathcal {L}_{\\mathbf {M}}^m$ such that $f_1(X)=\\dots =f_m(X)=0$ has no solution in $\\overline{\\mathbb {K}}^n$ , compute $h_1,\\ldots h_m\\in \\mathbb {K}[X_1,\\ldots , X_n]$ such that $\\sum _{i=1}^m f_i\\,h_i = 1.$ Bounding the sizes of the polynomials $h_i$ is a crucial question to estimate the complexity of this problem.",
"In this paper, the notion of size that we use is the number of coefficients in $\\mathbb {K}$ required to describe the polynomials $h_1,\\ldots , h_m$ .",
"On the other hand, the specification of “solving a polynomial system” depends on the context.",
"If the number of solutions in the algebraic closure $\\overline{\\mathbb {K}}$ is finite, one way to represent them symbolically is to provide a rational parametrization of their coordinates by the roots of a univariate polynomial.",
"For the sake of simplicity, we consider only the problem of computing a univariate polynomial whose roots parametrize the set of solutions: Problem 2 - Partial 0-dimensional fewnomial system solving.",
"Given a polynomial system $f_1=\\dots =f_m=0$ with support $\\mathbf {M}$ that have finitely-many solutions in $\\overline{\\mathbb {K}}^n$ and a monomial $\\mu \\in \\mathbf {M}$ , compute a univariate polynomial $P_\\mu \\in \\mathbb {K}[\\mu ]$ which vanishes at all the solutions of the system.",
"Hence, the roots of the univariate polynomial $P_\\mu $ contain the images of the solutions of the input multivariate system via the monomial map $(X_1,\\ldots , X_n)\\mapsto \\mu $ .",
"Related works.",
"Sparse elimination theory for solving systems with special monomial structures have been developed since the 80s [33].",
"Several lines of work have been initiated during this period.",
"When the exponent vectors of the monomials occurring in the polynomials of the system are the lattice points in a lattice polytope, connections with convex and toric geometry have been established and dedicated solving methods have been designed: homotopy continuation methods [18], [35], resultants [33], [6], Gröbner bases [34], [13], etc.",
"One important theme of these developments is to relate algebraic structures with combinatorial properties of convex bodies.",
"In particular, Kushnirenko and Bernshtein's theorems [24], [2] provide bounds on the number of isolated toric solutions in terms of the Newton polytopes of the input polynomials.",
"Another line of work have been initiated by Khovanskii in the 80s on fewnomial systems [19].",
"The main theme in this setting is to relate the algebraic and algorithmic complexity of several problems to the number of monomials occurring in the equations.",
"For instance, a classical and challenging question is to bound the number of real solutions in the positive orthant, see e.g.",
"[19], [30], [3], [4], [20], [21] for results on this topic.",
"Bounding the size of a certificate of inconsistency of a polynomial system via the Nullstellensatz is a classical problem.",
"Up to our knowledge, all known upper bounds on the size of such certificates are exponential in the number of variables $n$ ; moreover, examples by Masser and Philippon and by Lazard and Mora show that one cannot hope for better bounds in the worst case.",
"A classical bound is given by Kollar [22]: if the maximal degree of the input inconsistent system $f_1,\\ldots , f_m$ is $D$ , then there exist $h_1,\\ldots , h_m$ such that $\\sum _{i=1}^m f_i\\, h_i=1$ and the degrees of the $h_i$ are bounded by $n\\min (n,m)D^{\\min (n,m)}+\\min (n,m)D$ .",
"This bound is general and does not require any further assumption.",
"It was later improved to $\\deg (f_ih_i)\\le \\max (3,D)^n$ [14].",
"When there is no solution at infinity, the degrees of the polynomials $h_i$ are bounded by $(D-1)n$ [25], [5]: the number of coefficients in dense polynomials of this degree is still exponential in $n$ .",
"For general polynomial systems, it would be surprising that certificates of inconsistency with size polynomial in the input size exist, as this would imply $NP=coNP$ .",
"Estimates taking into account the bitsize of the coefficients that appear in the certificate in terms of the bitsize of the coefficients of the input system are provided by Arithmetic Nullstellensätze, see e.g.",
"[23] and references within.",
"Two milestones on the sparse effective Nullstellensatz are the bounds in [6] and [31]: these bounds provide certificates of size bounded by a quantity which depends on the Newton polytopes of the input polynomials.",
"However, their size is exponential in the size of the input, and these bounds do not take into account the sparsity of the support inside its Newton polytope.",
"One of the main difficulty to generalize these techniques to fewnomial quadratic systems is the fact that the proofs rely on algebraic properties (normality, Cohen-Macaulay algebras) that hold for semigroup algebras generated by lattice points in normal polytopes, but not for semigroup algebras generated by a scattered set of monomials.",
"Other models of sparse systems have also been investigated.",
"For instance, systems where each quadratic equation involves a small subset of variables have been investigated in [29] and [7].",
"Connections between combinatorial properties of graphs and polynomial systems is a classical topic which has been investigated from several viewpoints.",
"For instance, square-free monomial ideals have many combinatorial properties and can be seen as the edge ideals of graphs, see e.g.",
"[15], [17].",
"Connections between the regularity of the edge ideal of a graph and its matching number and co-chordal cover number are shown in [36].",
"Cohen-Macaulay criteria for such ideals are investigated in [16], [10].",
"Bounds on the size of certificate of inconsistency of polynomial systems are a important ingredient in algebraic proof complexity, see e.g.",
"the Nullstellensatz proof system [1] and related works [8].",
"Main results.",
"An open question is whether there exist certificates of inconsistency of polynomial size for general fewnomial systems involving $n+k+1$ monomials in $n$ variables.",
"The goal of this work is to investigate this question in the case of quadratic polynomials.",
"We present an explicit criterion which identifies subsets $\\mathbf {M}$ of monomials of degree at most 2 such that systems of $n$ equations in $n$ variables with support $\\mathbf {M}$ and generic coefficients do not have any solution, and such that there exists a sparse certificate $\\sum _{i=1}^n f_i h_i=1$ , where all polynomials $h_1,\\ldots , h_n$ lie in $\\mathcal {L}_{\\mathbf {M}}$ .",
"Therefore, the number of coefficients in $\\mathbb {K}$ required to represent the certificate is the same as that of the input system.",
"Moreover, when $\\mathbf {M}$ is such a subset, we propose a method which computes such $h_1,\\ldots , h_n$ within a polynomial number of arithmetic operations.",
"More precisely, we model the set $\\mathbf {M}$ by a graph $G$ on $n+1$ vertices, where each edge represents a nonconstant square-free monomial in $\\mathbf {M}$ .",
"The constant 1 and the squares in $\\mathbf {M}$ are distinguished with loops in the graph (the precise construction is described in Section ).",
"Let $\\nu (\\mathbf {M})$ denote the matching number (i.e.",
"the maximum cardinality of a matching) of the subgraph of vertices in $G$ with a loop.",
"Theorem 1.1 If $m\\ge \\vert \\mathbf {M}\\vert -\\frac{\\sqrt{1+8\\nu (\\mathbf {M})}-1}{2}$ , then a generic system $(f_1,\\ldots , f_m)\\in \\mathcal {L}_{\\mathbf {M}}^m$ has no solution in $\\overline{\\mathbb {K}}^n$ .",
"Moreover, there exists $(h_1,\\ldots , h_m)$ solving Problem s.t.",
"all $h_i$ lie in $\\mathcal {L}_{\\mathbf {M}}$ and they can be computed within $O\\left(m^\\omega \\binom{\\vert \\mathbf {M}\\vert +1}{2}^\\omega \\right)$ operations in $\\mathbb {K}$ , where $\\omega $ is a feasible exponent for matrix multiplication ($\\omega <2.37286$ with Le Gall's algorithm [26]).",
"We would like to emphasize that the inequality $m\\ge \\vert \\mathbf {M}\\vert -\\frac{\\sqrt{1+8\\nu (\\mathbf {M})}-1}{2}$ can be checked in polynomial time, since the matching number of a graph can be computed in polynomial time with Edmonds' algorithm [11].",
"Next, we relate how often the assumptions of Theorem REF hold with the number of squares in the support $\\mathbf {M}$ .",
"If the subset of square monomials and the subset of square-free monomials in $\\mathbf {M}$ are chosen at random, and the cardinality of $\\mathbf {M}$ is $n+k+1$ and the number of squares is larger than $\\Omega (n^{1/2+\\varepsilon })$ for some $\\varepsilon >0$ , then the assumptions of Theorem REF hold with large probability, leading to the following statement: Theorem 1.2 Let $k$ be a fixed integer, $a_n,b_n\\in \\mathbb {N}$ be such $a_n+b_n=n+k+1$ , and $\\mathbf {M}$ be a subset of monomials of degree at most 2 in $\\mathbb {K}[X_1,\\ldots , X_n]$ distributed uniformly at random among those that contain the constant 1, $a_n$ nonsquare monomials, and $b_n-1$ non-constant square monomials.",
"Assume further that $b_n=\\Omega (n^{1/2+\\varepsilon })$ , for $\\varepsilon >0$ .",
"Then the probability that the assumptions of Theorem REF with $m=n$ are satisfied for $\\mathbf {M}$ tends towards 1 as $n$ grows.",
"The cornerstones of the proof of this theorem rely on properties of random graphs in the Erdös-Renyi model.",
"Experiments suggest that this result is sharp: when there are at most $O(n^{1/2})$ square monomials in $\\mathbf {M}$ , we observe experimentally that the probability of having a certificate in $\\mathcal {L}_{\\mathbf {M}}^m$ seems to converge to a non-zero value smaller than 1 as $n$ grows.",
"This is also the case when the support is chosen uniformly at random (the expected number of squares is $O(1)$ ).",
"We propose a conjecture stating that the limit probability is nonzero in that case.",
"We also study a limit case: when $\\vert \\mathbf {M}\\vert =n+k+1$ and all the squares are in $\\mathbf {M}$ .",
"The generic number of solutions in this setting is given by the Bézout's theorem: it equals $2^n$ .",
"We shall see that with probability tending to 1, these solutions can be compactly represented as the orbits of $2^{2k+2}$ points under an action of $(\\mathbb {Z}/2\\mathbb {Z})^{n-2k-2}$ .",
"Computing these solutions amounts to solving a system of $2k+2$ equations in $2k+2$ variables: the time complexity of this task does not depend on $n$ .",
"A direct consequence is that computing a compact representation of the solutions of such systems require a number of operations in $\\mathbb {K}$ which is polynomial in $n$ , even though their number of solutions is exponential in $n$ .",
"This suggests that the number of solutions in the algebraic closure, which is often used to measure the complexity of solving polynomial systems, might in some cases greatly overestimate the complexity for fewnomial systems.",
"Another open issue is to extend this work to the non quadratic case.",
"Finally, we show experimental results obtained with our proof-of-concept implementation.",
"They show that certificates of inconsistency can be computed for quadratic fewnomial systems with more than 10000 variables and equations when there are sufficiently many squares in the monomial support.",
"Moreover, we also observe some unexpected behaviors which raise new questions about fewnomial systems.",
"For instance, as $n$ grows, there seems to be a phase transition in the probability of having a small certificate of inconsistency.",
"Moreover, in the case where there are few squares in the fewnomial system (this case is not covered by the theoretical analysis), there seems to be a non-zero probability that a fewnomial system has a small certificate of inconsistency.",
"These phenomenons remain to be explained.",
"Organization of the paper.",
"Section introduces notation and states preliminary results.",
"The core result of the paper is proved in Section , establishing a connection between the matching number and the existence of a small certificate of inconsistency.",
"Section is devoted to a probabilistic analysis of the matching number of some random graphs in the Erdös-Renyi model.",
"Section investigates some families of fewnomial systems where all squares appear in the equations.",
"Finally, we report experimental results in Section and state a conjecture for quadratic fewnomial systems involving few square monomials."
],
[
"Notation and preliminaries",
"Notation.",
"Throughout this paper, $\\mathbb {K}$ denotes a field of odd characteristic.",
"Its algebraic closure is denoted by $\\overline{\\mathbb {K}}$ .",
"If $X_1,\\ldots , X_n$ are variables, and $\\alpha \\in \\mathbb {N}^n$ , then the shorthand $X^\\alpha $ stands for the monomial $X_1^{\\alpha _1}\\dots X_n^{\\alpha _n}$ .",
"The symbol $\\mathbf {M}$ denotes a finite subset of monomials in $\\mathbb {K}[X_1,\\ldots , X_n]$ containing the constant 1.",
"For any $i\\in \\mathbb {N}$ , $\\mathbf {M}^i$ denotes the subset of all products of $i$ monomials in $\\mathbf {M}$ .",
"Its cardinality is denoted by $\\vert \\mathbf {M}^i\\vert $ .",
"By convention, $\\vert \\mathbf {M}^0\\vert =1.$ By slight abuse of notation, we call dimension of an ideal $I$ in a ring $R$ the Krull dimension of the quotient ring $R/I$ .",
"Complexity model.",
"Complexity bounds in this paper count the number of operations $\\lbrace +,-,\\times ,\\div \\rbrace $ in the field $\\mathbb {K}$ .",
"It is not our goal to take into account the bitsize of the coefficients in $\\mathbb {K}$ .",
"Hence, we count each arithmetic operation with unit cost.",
"We do not take into account operations on monomials.",
"The notion of size that we use for polynomial systems is the number of coefficients in $\\mathbb {K}$ required to represent them.",
"Note that if $\\mathbb {K}$ is a finite field, then the bitsizes of the elements in $\\mathbb {K}$ are bounded, and hence the bit complexity is the same as the arithmetic complexity.",
"Given partial functions $g, h$ from a set $I$ to $\\mathbb {N}$ , we use the following classical Landau notation: $f=O(g)$ means that $f/g$ is bounded above by a constant, $f=\\Omega (g)$ is equivalent to $g=O(f)$ , and $f=\\Theta (g)$ means that $f=O(g)$ and $g=O(f)$ .",
"Genericity.",
"Let $\\mathcal {L}_{\\mathbf {M}}$ denote the $\\mathbb {K}$ -linear space spanned by $\\mathbf {M}$ .",
"It has dimension $\\vert \\mathbf {M}\\vert $ .",
"We say that a property holds for a generic system $(f_1,\\ldots , f_m)\\in \\mathcal {L}_{\\mathbf {M}}^m$ if there exists a dense Zariski open subset $\\mathcal {O}$ of $\\mathcal {L}_{\\mathbf {M}}^m$ s.t.",
"this property holds for any system in $\\mathcal {O}$ .",
"Semigroup algebras.",
"The main algebraic structure that we consider are semigroup algebras (also called toric rings): if $\\mathbf {M}\\subset \\mathbb {K}[X_1,\\ldots , X_n]$ is a finite subset of monomials containing 1, we let $\\mathbb {K}[\\mathbf {M}]$ denote the subalgebra of $\\mathbb {K}[X_1,\\ldots , X_n]$ generated by $\\mathbf {M}$ .",
"We do not make any assumption on the Krull dimension of the ring $\\mathbb {K}[\\mathbf {M}]$ .",
"Semigroup algebras which are domains are the coordinate rings of affine toric varieties [9].",
"We refer to [28] for a more detailed presentation.",
"By slight abuse of notation, we call variety of a system $f_1,\\ldots , f_m\\in \\mathbb {K}[\\mathbf {M}]$ the variety in $\\overline{\\mathbb {K}}^n$ associated to the ideal $\\langle f_1,\\ldots ,f_n\\rangle \\subset \\mathbb {K}[X_1,\\ldots , X_n]$ .",
"The following proposition is a variant of the weak Nullstellensatz for the total coordinate ring of projective toric varieties (see e.g.",
"[9]).",
"Proposition 2.1 The variety associated with a system $f_1,\\ldots , f_m\\in \\mathbb {K}[\\mathbf {M}]$ is empty if and only if there exist $h_1,\\ldots , h_m\\in \\mathbb {K}[\\mathbf {M}]$ such that $\\sum _{i=1}^m f_i\\, h_i = 1.$ the ring $\\mathbb {K}[\\mathbf {M}]$ is isomorphic to $\\mathbb {K}[X]/I_M$ , where $I_M$ is a toric ideal generated by binomials $b_1,\\ldots , b_\\ell $ .",
"Let $\\widetilde{f_1},\\ldots ,\\widetilde{f_m}$ be the images of $f_1,\\ldots , f_m$ by the isomorphism.",
"Using the Nullstellensatz on the system $\\widetilde{f_1},\\ldots , \\widetilde{f_m}, b_1,\\ldots , b_\\ell $ in $\\mathbb {K}[X]$ and pulling it back to $\\mathbb {K}[\\mathbf {M}]$ proves the proposition.",
"Proposition REF indicates that we can look for polynomial relations in $\\mathbb {K}[\\mathbf {M}]$ instead of the whole algebra $\\mathbb {K}[X]$ .",
"Although narrowing the search for the certificate in $\\mathbb {K}[\\mathbf {M}]$ instead of $\\mathbb {K}[X]$ constrains the problem, we shall see that this approach enables us to find efficiently small certificates.",
"This leads to the following variant of Problem : Problem 3 - Effective fewnomial Nullstellensatz in $\\mathbb {K}[\\mathbf {M}]$ .",
"Given a system $f_1,\\ldots , f_m\\in \\mathbb {K}[\\mathbf {M}]$ and such that $f_1(X)=\\dots =f_m(X)=0$ has no solution in $\\overline{\\mathbb {K}}^n$ , compute $h_1,\\ldots ,h_m\\in \\mathbb {K}[\\mathbf {M}]$ such that $\\sum _{i=1}^m f_i\\,h_i = 1.$"
],
[
"Monomials and support graphs",
"In this section, we show a connection between graphs and properties of $\\mathbb {K}[\\mathbf {M}]$ .",
"In particular, we focus on quadratic relations between monomials in $\\mathbf {M}$ , i.e.",
"at $\\mathbb {K}$ -linear relations in the vector space spanned by $\\mathbf {M}^2$ .",
"We start by adding a new variable $X_0$ and by considering the homogenized support $\\mathbf {M}^h=\\lbrace X_0^{2-\\deg (\\mu )}\\mu \\rbrace _{\\mu \\in \\mathbf {M}}$ .",
"We associate with $\\mathbf {M}$ a simple labeled undirected graph $G$ on $S=\\lbrace 0,\\ldots , n\\rbrace $ whose edges are $E=\\lbrace (i,j) \\mid X_i X_j\\in \\mathbf {M}^h, i\\ne j\\rbrace $ .",
"There is a loop at a vertex $i$ iff $X_i^2\\in \\mathbf {M}^h$ .",
"Example 3.1 Let $\\mathbf {M}=\\lbrace 1,X_1^2,X_2^2, X_3^2, X_3, X_4, X_1 X_2, X_2 X_3, X_3X_4\\rbrace $ .",
"The following picture represents the graph $G$ ; squares in $\\mathbf {M}^h$ are indicated by a loop.",
"[scale=0.7] [vstyle=Normal] 2 [L=0]1 [L=1](1)2 [L=2](2)3 [L=3](3)4 [L=4](4)5 [dist=2cm,dir=SO,style=thick](1) [dist=2cm,dir=SO,style=thick](2) [dist=2cm,dir=EA,style=thick](3) [dist=2cm,dir=NO,style=thick](4) (2,3,4,5,4,1,5) Quadratic relations between elements of $\\mathbf {M}$ are of the form $\\mu _1\\, \\mu _2 = \\mu _3\\, \\mu _4$ for (not necessarily distinct) monomials $\\mu _1,\\mu _2,\\mu _3,\\mu _4\\in \\mathbf {M}$ .",
"For the quadratic supports $\\mathbf {M}$ that we consider in this paper, these quadratic relations come in three flavors that appear as subgraphs of $G$ and are described in Figure REF .",
"The next proposition shows how the cardinality of $\\mathbf {M}^2$ can be computed from the number of quadratic relations and the number of 4-cliques in $G$ .",
"We recall that a 4-clique is a subgraph on 4 vertices such that every pair of vertices is linked by an edge.",
"Figure: The three types of quadratic relationsProposition 3.2 The cardinality of $\\mathbf {M}^2$ equals $\\binom{\\vert \\mathbf {M}\\vert +1}{2}-\\lambda (G)+{\\rm clique}_4(G)$ , where $\\lambda (G)$ is the number of subgraphs of $G$ isomorphic to any of the three graphs in Figure REF and ${\\rm clique}_4(G)$ is the number of 4-cliques in $G$ .",
"Proof.",
"We can form $\\binom{\\vert \\mathbf {M}\\vert +1}{2}$ products of two (non-necessarily distinct) elements in $\\mathbf {M}$ .",
"However, some of these products are counted several times because of the quadratic relations between elements in $\\mathbf {M}$ .",
"This is corrected by the terms $-\\lambda (G)+{\\rm clique}_4(G)$ ; we detail below the possible cases: If $\\mu =X_iX_jX_kX_\\ell $ is a product of four distinct variables, then $\\mu $ can be obtained from $\\mathbf {M}$ by three different products, since $\\mu =(X_iX_j)(X_kX_\\ell )=(X_iX_k)(X_jX_\\ell )=(X_iX_\\ell )(X_jX_k)$ .",
"Depending on the number of pairs of such edges that lie in the graph, the monomial $\\mu $ is counted one, two or three times in $\\binom{\\vert \\mathbf {M}\\vert +1}{2}$ .",
"If there is only one way to obtain $\\mu $ (for example if $(X_iX_j)$ and $(X_kX_\\ell )$ are the only monomials in $\\mathbf {M}$ whose products are $\\mu $ ), then the subgraph associated with the vertices $\\lbrace X_i,X_j,X_k,X_\\ell \\rbrace $ is neither of type 1 nor a 4-clique.",
"Hence, $\\mu $ is counted only one time.",
"If there are two ways to obtain $\\mu $ , then the subgraph associated with the vertices $\\lbrace X_i,X_j,X_k,X_\\ell \\rbrace $ is of type 1 but not a 4-clique.",
"Hence, $\\mu $ is counted twice in $\\binom{\\vert \\mathbf {M}\\vert +1}{2}$ but this is corrected by the term $\\lambda (G)$ .",
"If all the three products are possible, then the subgraph associated with the vertices $\\lbrace X_i,X_j,X_k,X_\\ell \\rbrace $ contains three subgraphs of type 1, and is also a 4-clique.",
"Therefore $\\mu $ is counted 3 times in $\\binom{\\vert \\mathbf {M}\\vert +1}{2}$ , removed 3 times in $\\lambda (G)$ and counted once in ${\\rm clique}_4(G)$ .",
"If $\\mu =X_i^2X_jX_k$ is a monomial involving three distinct variables, then $\\mu $ is counted twice in $\\binom{\\vert \\mathbf {M}\\vert +1}{2}$ if and only the subgraph associated with the vertices $\\lbrace X_i,X_j,X_k\\rbrace $ is of type 2.",
"In this case one contribution is removed by the term $\\lambda (G)$ , hence $\\mu $ is counted one time.",
"Similarly, monomials $\\mu =X_i^2X_j^2$ are counted once or twice in the formula $\\binom{\\vert \\mathbf {M}\\vert +1}{2}$ : if it is counted twice (i.e.",
"when $X_iX_j, X_i^2, X_j^2\\in \\mathbf {M}$ ), then the subgraph associated with $\\lbrace X_i,X_j\\rbrace $ is of type 3.",
"$\\square $ Notation.",
"For a graph $G$ associated with a set of monomials $\\mathbf {M}$ , let $G^{\\prime }$ be the subgraph of squares (i.e.",
"the subgraph of vertices with a loop).",
"Definition 3.3 A matching (also called independent edge set) of $G^{\\prime }$ is a set of edges of $G^{\\prime }$ without common vertices.",
"We let $\\nu (\\mathbf {M})$ denote the matching number of $G^{\\prime }$ , i.e.",
"the maximum cardinality of a matching of $G^{\\prime }$ .",
"The matching number of a graph can be computed in polynomial time by Edmonds's algorithm [11].",
"We refer to [27] for more details on matching theory.",
"We state now the main result of this section, which connects the matching number of the graph $G^{\\prime }$ to the existence of a small certificate of inconsistency: Theorem 3.4 Let $(f_1,\\ldots , f_m)\\in \\mathcal {L}_{\\mathbf {M}}^m$ be a system with generic coefficients.",
"If $m\\ge \\vert \\mathbf {M}\\vert -\\frac{\\sqrt{1+8\\nu (\\mathbf {M})}-1}{2}$ , then there exist polynomials $h_1,\\ldots , h_m\\in \\mathcal {L}_{\\mathbf {M}}$ such that $\\sum _{i=1}^m f_i h_i =1.$ The proof of this theorem is postponed to the end of the section.",
"It is actually not surprising that systems satisfying the assumptions of Theorem REF do not have any solution, since the dimension of the $\\mathbb {Q}$ -vector space generated by the exponent vectors in $\\mathbf {M}$ is upper bounded by $\\vert \\mathbf {M}\\vert -\\nu (\\mathbf {M})$ : each edge $(i,j)$ in $G^{\\prime }$ means that $X_i^2, X_j^2, X_i X_j\\in \\mathbf {M}$ and the exponent vectors of these three monomials are linearly dependent over $\\mathbb {Q}$ .",
"The main point of Theorem REF is that, under the condition on $\\nu (\\mathbf {M})$ , the polynomials $(h_1,\\ldots , h_m)$ for the effective Nullstellensatz can be searched in $\\mathcal {L}_{\\mathbf {M}}$ .",
"This allows to obtain to get small certificates of inconsistency: Corollary 3.5 With the notation and under the assumptions of Theorem REF , there is an explicit algorithm which solves Problem within $O\\left(m\\vert \\mathbf {M}\\vert \\left(\\binom{\\vert \\mathbf {M}\\vert +1}{2}-\\lambda (G)+{\\rm clique}_4(G)\\right)^{\\omega -1}\\right)$ arithmetic operations, where $\\omega $ is a feasible exponent for matrix multiplication ($\\omega <2.37286$ with Le Gall's algorithm [26]).",
"This complexity is polynomial in the number of coefficients $m\\vert \\mathbf {M}\\vert $ of the input system.",
"According to Theorem REF , there exist polynomials $h_1,\\ldots , h_m$ with support $\\mathbf {M}$ s.t.",
"$\\sum _i h_i f_i=1$ , by decomposing the polynomials $h_i$ in the monomial basis there exists a relation $\\sum _i \\sum _{\\mu \\in \\mathbf {M}}\\alpha _{\\mu ,i}\\, \\mu f_i = 1$ where $\\alpha _{\\mu ,i}\\in \\mathbb {K}$ .",
"Let $V\\subset {\\sf Span}_\\mathbb {K}(\\mathbf {M}^2)$ be the linear space generated by the products $\\lbrace \\mu \\,f_i\\rbrace _{\\mu \\in \\mathbf {M},i\\in \\lbrace 1,\\ldots ,m\\rbrace }$ .",
"Consequently, computing the polynomials $h_i$ amounts to solving a linear system over $\\mathbb {K}$ with $m\\,\\vert \\mathbf {M}\\vert $ unknowns and $\\vert \\mathbf {M}^2\\vert $ equations.",
"Solving it requires $O(m\\vert \\mathbf {M}\\vert \\cdot \\vert \\mathbf {M}^2\\vert ^{\\omega -1})$ operations in $\\mathbb {K}$ [32].",
"Proposition REF concludes the proof.",
"The sequel of this section is devoted to this proof of Theorem REF .",
"The squareroot involved in the formula is a consequence of the following lemma, as the maximal value of $p$ for which $n\\ge \\binom{n-p+1}{2}$ .",
"Lemma 3.6 There exist linear forms $\\ell _1,\\ldots , \\ell _p\\in \\mathbb {K}[X_1,\\ldots , X_n]$ such that the ideal $I=\\langle X_1^2,\\ldots , X_n^2, \\ell _1(X_1,\\ldots , X_n),\\ldots ,\\ell _p(X_1,\\ldots , X_n)\\rangle $ contains all monomials of degree 2 iff $p\\ge n- \\frac{\\sqrt{1+8 n}-1}{2}$ .",
"The vector space of $(n-p)$ -variate quadratic forms has dimension $\\binom{n-p+1}{2}$ .",
"From the inequality $p\\ge n- \\frac{\\sqrt{1+8 n}-1}{2}$ , we obtain $n\\ge \\binom{n-p+1}{2}$ .",
"This inequality and the fact that any quadratic form can be written as a linear combination of squares of linear forms (since ${\\rm char}(\\mathbb {K})\\ne 2$ ), implies that there exist $\\ell ^{\\prime }_1,\\ldots , \\ell ^{\\prime }_n$ such that their squares ${\\ell ^{\\prime }_1}^2,\\ldots , {\\ell ^{\\prime }_n}^2$ generate the space of $(n-p)$ -variate quadratic forms.",
"Then the dimension of the linear space generated by $\\ell ^{\\prime }_1,\\ldots , \\ell ^{\\prime }_n$ is necessarily maximal and equals $n-p$ .",
"Up to permuting the indices, we assume also that $\\ell ^{\\prime }_1,\\ldots , \\ell ^{\\prime }_{n-p}$ are linearly independent.",
"Hence the ideal $I^{\\prime }=\\langle {\\ell ^{\\prime }_1}(X_1,\\ldots ,X_{n-p})^2,\\ldots ,{\\ell ^{\\prime }_n}(X_1,\\ldots ,X_{n-p})^2, X_{n-p+1},\\ldots , X_n\\rangle $ contains all monomials of degree 2.",
"We rewrite $I^{\\prime }$ as $I^{\\prime }=\\langle {\\ell ^{\\prime \\prime }_1}(X_1,\\ldots ,X_n)^2,\\ldots ,{\\ell ^{\\prime \\prime }_{n}}(X_1,\\ldots ,X_{n})^2, X_{n-p+1},\\ldots ,X_n\\rangle ,$ $\\quad \\quad {\\left\\lbrace \\begin{array}{ll}\\ell ^{\\prime \\prime }_i(X_1,\\ldots , X_n)=\\ell ^{\\prime }_i(X_1,\\ldots , X_{n-p})\\text{ if } 1\\le i\\le n-p\\\\\\ell ^{\\prime \\prime }_i(X_1,\\ldots , X_n)=X_i-\\ell ^{\\prime }_i(X_1,\\ldots , X_{n-p})\\text{ otherwise}.\\end{array}\\right.",
"}$ Note that the linear forms $\\ell ^{\\prime \\prime }_1,\\ldots ,\\ell ^{\\prime \\prime }_n$ are linearly independent by construction.",
"We consider the automorphism $\\theta $ of $\\mathbb {K}[X_1,\\ldots , X_n]$ defined by $\\theta (X_i)=\\ell ^{\\prime \\prime }_i(X_1,\\ldots , X_{n})$ , and we set $\\ell _i(X_1,\\ldots , X_n)=\\theta ^{-1}(X_{n-p+i})$ for $i\\in \\lbrace 1,\\ldots , p\\rbrace $ .",
"Therefore $I$ is the inverse image of $I^{\\prime }$ by $\\theta $ and hence contains all the monomials of degree 2.",
"It remains to prove the converse statement, i.e.",
"that $p< n- \\frac{\\sqrt{1+8 n}-1}{2}$ implies that there do not exist such linear forms $\\ell _1,\\ldots , \\ell _p$ .",
"This is achieved by a similar argument: if such linear forms existed, then there would exist a set of $n$ generators of the vector space of $(n-p)$ -variate quadratic forms.",
"This is not possible if $p< n- \\frac{\\sqrt{1+8 n}-1}{2}$ since this vector space has dimension $\\binom{n-p+1}{2}$ .",
"We can now prove the main theorem of this section: [Proof of Theorem REF ] We prove a homogeneous version of Theorem REF : let $f_1^{(h)},\\ldots , f_m^{(h)}\\in \\mathcal {L}_{\\mathbf {M}}^h\\subset \\mathbb {K}[X_0,\\ldots , X_n]$ be the homogenization of the generic system $f_1,\\ldots , f_m$ .",
"We shall show that any monomial in $(\\mathbf {M}^h)^2$ (see the definition of $\\mathbf {M}^h$ at the beginning of this section) belongs to the ideal $\\langle f_1^{(h)},\\ldots , f_m^{(h)} \\rangle \\subset \\mathbb {K}[\\mathbf {M}^h]$ .",
"This will imply that there exist $h_1^{(h)},\\ldots , h_m^{(h)}\\in \\mathbb {K}[\\mathbf {M}^h]$ such that $\\sum _{i=1}^m f_i^{(h)} h_i^{(h)}=X_0^4\\in (\\mathbf {M}^h)^2.$ Setting $X_0=1$ in this equation yields the desired relation.",
"First, we prove the existence of one system $f_1^{(h)},\\ldots , f_m^{(h)}$ such that all monomials of $(\\mathbf {M}^h)^2$ appear in the ideal $\\langle f_1^{(h)},\\ldots ,f_m^{(h)}\\rangle $ .",
"Throughout this proof, we let $A=\\lbrace \\lbrace a_1,b_1\\rbrace ,\\ldots ,\\lbrace a_{\\nu (\\mathbf {M})},b_{\\nu (\\mathbf {M})}\\rbrace \\rbrace \\subset \\lbrace 0,\\ldots , n\\rbrace ^2$ denote a matching of $G^{\\prime }$ of maximum cardinality.",
"We construct a system from $A$ whose polynomials are: all the monomials in $\\mathbf {M}^{h}$ of the form $X_i X_j$ with $i\\ne j$ ; all the monomials in $\\mathbf {M}^h$ of the form $X_i^2$ with $i$ not appearing in $A$ ; for each $i\\in \\lbrace 1,\\ldots ,\\nu (\\mathbf {M})\\rbrace $ , the polynomial $X_{a_i}^2-X_{b_i}^2$ ; the polynomials $\\ell _1(X_{a_1}^2,\\ldots ,X_{a_{\\nu (\\mathbf {M})}}^2),\\ldots ,\\ell _p(X_{a_1}^2,\\ldots ,X_{a_{\\nu (\\mathbf {M})}}^2)$ , where the $\\nu (\\mathbf {M})$ -variate linear forms $\\ell _1,\\ldots ,\\ell _p$ are obtained by replacing $n$ by $\\nu (\\mathbf {M})$ in Lemma REF .",
"This is a system of $\\vert \\mathbf {M}\\vert -\\left\\lfloor \\frac{\\sqrt{1+8\\nu (\\mathbf {M})}-1}{2}\\right\\rfloor $ polynomials, generating an ideal $I\\subset \\mathbb {K}[\\mathbf {M}^h]$ .",
"We claim that all monomials in $(\\mathbf {M}^h)^2$ are in the ideal of $\\mathbb {K}[\\mathbf {M}^h]$ generated by these polynomials: every monomial in $(\\mathbf {M}^h)^2$ involving at least 3 different variables belongs necessarily to the ideal generated by the monomials $X_i X_j$ with $i\\ne j$ ; the same holds for monomials of the form $X_i^3 X_j$ with $i\\ne j$ ; next, we look at monomials of the form $X_i^4$ .",
"If $i$ does not appear in $A$ , then by construction $X_i^2$ is in the ideal.",
"If $i$ is in $A$ , then there exists $j$ such that $i=a_j$ or $i=b_j$ .",
"Noticing that $X_{a_j}^4=X_{a_j}^2(X_{a_j}^2-X_{b_j}^2) - (X_{a_j} X_{b_j})^2$ or $X_{b_j}^4=X_{b_j}^2(X_{a_j}^2-X_{b_j}^2) - (X_{a_j} X_{b_j})^2$ shows that $X_i^4\\in I$ .",
"finally, we focus on monomials of the form $X_i^2 X_j^2$ .",
"If $i$ or $j$ do not appear in $A$ , then either $X_i^2$ or $X_j^2$ belongs to $I$ .",
"If both $i$ and $j$ appear in $A$ then Lemma REF tells us that $X_i^2 X_j^2$ belongs to the ideal generated by $\\langle X_{a_1}^4,\\ldots ,X_{a_{\\nu (\\mathbf {M})}}^4,\\ell _1(X_{a_1}^2,\\ldots ,X_{a_{\\nu (\\mathbf {M})}}^2),\\ldots ,\\ell _p(X_{a_1}^2,\\ldots ,X_{a_{\\nu (\\mathbf {M})}}^2),X_{a_1}^2-X_{b_1}^2,\\ldots , X_{a_{\\nu (\\mathbf {M})}}^2-X_{b_{\\nu (\\mathbf {M})}}^2\\rangle $ .",
"So far, we have proven that there exists at least one system such that Theorem REF is correct.",
"It remains to prove that this is true for a generic system.",
"To this end, we note that all monomials in $(\\mathbf {M}^h)^2$ belongs to $\\langle f_1^{(h)},\\ldots ,f_m^{(h)}\\rangle \\subset \\mathbb {K}[\\mathbf {M}^h]$ if and only if ${\\sf Span}_\\mathbb {K}(\\lbrace \\mu \\,f^{(h)}_i\\rbrace _{\\mu \\in \\mathbf {M}^h,i\\in \\lbrace 1,\\ldots , m\\rbrace })={\\sf Span}_\\mathbb {K}((\\mathbf {M}^h)^2)$ .",
"This is an open condition given by the non-vanishing of a product of minors of the matrix recording the coefficients of $\\lbrace \\mu \\,f_i^{(h)}\\rbrace _{\\mu \\in \\mathbf {M},i\\in \\lbrace 1,\\ldots , m\\rbrace }$ .",
"Consequently, there exists a Zariski open subset $\\mathcal {O}\\subset \\mathbb {K}[\\mathbf {M}^h]^m$ such that Theorem REF holds.",
"This open subset $\\mathcal {O}$ is non-empty by the construction above.",
"The proof is concluded by noticing that any non-empty open subset is dense in the Zariski topology."
],
[
"Random support graphs",
"In this section, we assume that the support $\\mathbf {M}$ is randomly generated and we estimate the probability that the assumptions of Theorem REF are satisfied.",
"Roughly speaking, the aim of this section is to show that the conditions of Theorem REF hold with large probability if $n$ is large enough and if there are sufficiently many squares in $\\mathbf {M}$ .",
"Let us consider the following variant of the Erdös-Rényi random graph model: for $n\\in \\mathbb {N}$ , we set two probabilities $p_n,q_n\\in [0,1]$ , and we consider a sequence of random supports $(\\mathbf {M}_n)_{n\\in \\mathbb {N}}$ where $\\mathbf {M}_n$ is a subset of quadratic monomials of $\\mathbb {K}[X_0,\\ldots , X_n]$ ; each square $X_i^2$ appears in $\\mathbf {M}_n$ independently with probability $q_n$ ; each monomial of the form $X_i X_j$ with $i\\ne j$ appears independently with probability $p_n$ .",
"The goal is to estimate in which cases the random variable $\\nu (\\mathbf {M}_n)$ grows sufficiently quickly so that the assumptions of Theorem REF are satisfied asymptotically with large probability.",
"In order to estimate $\\nu (\\mathbf {M}_n)$ , we first forget the meaning of the graph in terms of monomials and count the number of isolated edges in a random graph $G$ in this variant of the Erdös-Renyi model.",
"Proposition 4.1 Let $G$ be a random simple graph on $n+1$ vertices.",
"Each vertex has a loop with probability $q\\in [0,1]$ and an edge between any two vertices appear with probability $p\\in [0,1]$ .",
"Let $G^{\\prime }$ be the subgraph obtained by restricting $G$ to the vertices with a loop and $\\mathcal {E}$ be the random variable counting the number of isolated edges in $G^{\\prime }$ .",
"Then $\\mathcal {E}$ has expected value and variance $\\mathbb {E}(\\mathcal {E})=&\\displaystyle \\binom{n+1}{2}\\,q^2\\,p\\,(1-q(1-(1-p)^2))^{n-1}, \\\\\\operatorname{Var}(\\mathcal {E})=&\\displaystyle \\mathbb {E}(\\mathcal {E})-\\mathbb {E}(\\mathcal {E})^2+\\\\&6\\,\\binom{n+1}{4}\\,q^4\\,p^2\\,(1-p)^4(1-q(1-(1-p)^4))^{n-3}.$ For each possible edge $e$ between two vertices $i\\ne j$ , we denote by $X_e$ the random variable taking the value 1 if $e$ is an isolated edge of $G^{\\prime }$ , and 0 otherwise.",
"The probability that $i$ and $j$ appear as vertices with loops in $G$ is $q^2$ .",
"Hence, the probability that the edge $e$ lies in $G^{\\prime }$ is $q^2p$ .",
"Moreover, for a given vertex $k\\ne i,j$ , the probability that $k$ appears in $G^{\\prime }$ and at least one of the edges $(i,k)$ and $(j,k)$ belong to $G$ is $q(1-(1-p)^2)$ .",
"There are $n-1$ other vertices than $i$ and $j$ in $G$ , hence $X_e$ follows a Bernoulli law of parameter $q^2p(1-q(1-(1-p)^2))^{n-1}$ .",
"It follows that $\\mathbb {E}(\\mathcal {E}) = \\sum _e \\mathbb {E}(X_e)= \\binom{n+1}{2}q^2p(1-q(1-(1-p)^2))^{n-1} $ The computation of the variance can be done similarly.",
"We now apply the previous proposition in the case where $p$ and $q$ depends on $n$ , and analyze the convergence of $\\mathbb {E}(\\mathcal {E})$ and $\\operatorname{Var}(\\mathcal {E})$ as $n$ grows to infinity.",
"Corollary 4.2 Let $p_n=\\Theta (n^{-1})$ and $q_n=\\Theta (n^\\beta )$ .",
"With the notation of Proposition REF , if $-1/2<\\beta <0$ then $\\mathbb {E}(\\mathcal {E})=\\Theta \\left(n^{2\\beta +1}\\right)$ and $\\operatorname{Var}(\\mathcal {E})=\\Theta \\left(n^{2\\beta +1}\\right)$ .",
"First, note that $ \\log \\left[(1-q_n(1-(1-p_n)^2))^{n-1}\\right] =-2 n p_nq_n+O\\left(n^{-1}\\right) $ since $\\beta <0$ .",
"This shows that $\\mathbb {E}(\\mathcal {E}) = \\frac{q_n^2 p_nn^2}{2}e^{-2 n p_nq_n}(1+O(n^{-1}))$ .",
"The claim on the asymptotic behavior of $\\mathbb {E}(\\mathcal {E})$ follows from $e^{-2 n p_nq_n}=\\Theta (1)$ .",
"Next, let $\\lambda $ denote the last summand in Eq.",
"(REF ), namely $\\lambda =\\operatorname{Var}(\\mathcal {E})-\\mathbb {E}(\\mathcal {E})+\\mathbb {E}(\\mathcal {E})^2$ .",
"The asymptotic behavior of $\\lambda $ can be obtained by a similar analysis: $\\log \\left[(1-q_n(1-(1-p_n)^4))^{n-3}\\right] = -4 n p_nq_n+O\\left( n^{-1}\\right),$ hence $\\lambda =\\frac{q_n^4 p_n^2 n^4}{4}e^{-4 n p_nq_n} (1+O(n^{-1}))$ .",
"Notice that $\\mathbb {E}(\\mathcal {E})^2 = \\frac{q_n^4 p_n^2 n^4}{4} e^{-4 n p_nq_n}(1+O(n^{-1}))$ .",
"Consequently, $\\mathbb {E}(\\mathcal {E})^2-\\lambda =O(n^{4\\beta +1})$ , since $e^{-4 n p_nq_n}=\\Theta (1)$ .",
"Finally, putting all the estimates together, we obtain $\\operatorname{Var}(\\mathcal {E})=\\Theta (n^{2\\beta +1})+O(n^{4\\beta +1})=\\Theta (n^{2\\beta +1})$ since $\\beta <0$ .",
"Finally, we relate the distribution of $\\mathcal {E}$ with the probability that the assumptions of Theorem REF hold for fewnomial systems with $\\vert \\mathbf {M}\\vert =n+k+1$ .",
"If one wants that $\\mathbb {E}(\\vert \\mathbf {M}_n\\vert )=n+k+1$ for some fixed $k$ and that the expected number of squares is $(n+1)^{1/2+\\varepsilon }$ , then one has to choose $q_n=(n+1)^{-1/2+\\varepsilon }$ and $p_n=(n+k+1-(n+1)q_n)/\\binom{n+1}{2}$ .",
"The asymptotic expected behavior of the matching number in that case is described by the following statement: Lemma 4.3 Let $\\mathbf {M}_n$ be a sequence of random supports where each square monomial appears with probability $q_n$ , and each square-free monomial appears with probability $p_n$ .",
"If $p_n=\\Theta (n^{-1})$ and $q_n=\\Omega (n^{-1/2+\\varepsilon })$ , with $0<\\varepsilon <1/2$ , then for any $\\ell \\in \\mathbb {N}$ , $\\mathbf {P}\\left(\\nu (\\mathbf {M}_n)\\ge \\ell \\right)$ tends towards 1 as $n$ grows.",
"Chebyshev's inequality implies that $\\mathbf {P}(\\mathcal {E}\\le \\mathbb {E}(\\mathcal {E})/2)\\le & \\mathbf {P}(\\mid \\mathcal {E}-\\mathbb {E}(\\mathcal {E})\\mid \\ge \\mathbb {E}(\\mathcal {E})/2)\\le \\dfrac{4\\operatorname{Var}(\\mathcal {E})}{\\mathbb {E}(\\mathcal {E})^2}\\\\= &O\\left(n^{-2(-1/2+\\varepsilon )-1}\\right)= O\\left(n^{-2 \\varepsilon }\\right).$ Next, notice that $\\mathbb {E}(\\mathcal {E})/2=\\Theta \\left(n^{2\\varepsilon }\\right)$ by Corollary REF .",
"Also, note that $\\mathcal {E}\\le \\nu (\\mathbf {M}_n)$ , so that for $n$ sufficiently large, we have $\\displaystyle \\mathbf {P}\\left(\\nu (\\mathbf {M}_n)\\le n^\\varepsilon \\right)\\le \\displaystyle \\mathbf {P}\\left(\\mathcal {E}\\le n^{2\\varepsilon }\\right)= O\\left(n^{-2 \\varepsilon }\\right),$ which tends towards 0 as $n$ grows.",
"Next, we show that these estimates also hold for a different model of random monomial supports.",
"For $n\\in \\mathbb {N}$ and two integers $a,b\\in \\mathbb {N}$ , we consider the random sets $\\mathbf {U}_{n,a,b}$ of quadratic monomials in $\\mathbb {K}[X_0,\\ldots , X_n]$ distributed uniformly at random among those that contain $a$ non-squares and $b$ squares.",
"Theorem 4.4 Let $k$ be a fixed integer, $a_n,b_n\\in \\mathbb {N}$ be such $a_n+b_n=n+k+1$ , and $\\mathbf {U}_{n,a_n,b_n}$ be a subset of quadratic monomials in $\\mathbb {K}[X_0,\\ldots , X_n]$ distributed uniformly at random among those that contain $a_n$ non-square monomials and $b_n$ squares.",
"Assume further that $b_n=\\Omega (n^{1/2+\\varepsilon })$ , for $\\varepsilon >0$ .",
"Then the probability that the assumptions of Theorem REF with $m=n$ are satisfied for $\\mathbf {U}_{n,a_n,b_n}$ tends towards 1 as $n$ grows.",
"The proof of this theorem is technical and is similar to the classical techniques to prove properties of random graphs in the Erdös-Renyi models [12].",
"Details are provided in the appendix.",
"Theorem REF is a direct consequence of Theorem REF and is obtained by dehomogenization."
],
[
"Systems with all the squares",
"Next, we investigate the special case of fewnomial systems where all the squares $X_i^2$ belong to $\\mathbf {M}$ .",
"This corresponds to a limit case of Theorem REF : $\\varepsilon =1/2$ .",
"In this setting, the Newton polytopes of the polynomials are the same as those of dense quadratic polynomials, hence these systems have generically $2^n$ solutions in $\\overline{\\mathbb {K}}^n$ .",
"In the sequel of this section, $\\mathbf {M}$ is a set of monomials of degree at most 2 in $\\mathbb {K}[X_1,\\ldots ,X_n]$ , of cardinality $n+k+1$ , and which contains the constant 1 and all the squares $X_i^2$ .",
"We also assume that $n>2k$ .",
"We let $\\ell $ denote the number of variables $X_i$ which appear in a square-free monomial in $\\mathbf {M}$ .",
"Hence $\\ell \\le 2 k$ .",
"For a 0-dimensional system $(f_1,\\ldots , f_n)\\in \\mathcal {L}_{\\mathbf {M}}^n$ , we let $S$ denote the $n\\times (n-\\ell )$ matrix which contains the coefficients of the squares $X_i^2$ such that $X_i$ does not appear in a square-free monomial in $\\mathbf {M}$ .",
"Proposition 5.1 Let $(f_1,\\ldots , f_n)\\in \\mathcal {L}_{\\mathbf {M}}^n$ be a 0-dimensional system with support $\\mathbf {M}$ .",
"Then the system $f_1=\\dots =f_n=0$ has at most $2^n$ solutions in $\\overline{\\mathbb {K}}^n$ .",
"If the matrix $S$ has full rank, then the solutions are the orbits of at most $2^{\\ell }$ points under the action of $\\left(\\mathbb {Z}/2\\mathbb {Z}\\right)^{n-\\ell }$ given by $\\begin{array}{rccl}\\chi :&\\left(\\mathbb {Z}/2\\mathbb {Z}\\right)^{n-\\ell }\\times \\overline{\\mathbb {K}}^n&\\rightarrow &\\quad \\overline{\\mathbb {K}}^n\\\\&(\\mathbf {e}_i,(a_1,\\dots ,a_n))&\\mapsto & (a_1,\\dots ,-a_{i_j},\\ldots ,a_n)\\end{array},$ where the set $\\lbrace i_j\\rbrace $ is the set of indices such that $X_{i_j}$ does not appear in a square-free monomial in $\\mathbf {M}$ .",
"Up to a permutation of the indices, we can assume w.l.o.g.",
"that $X_1,\\ldots , X_{n-\\ell }$ are the variables that does not appear in a square-free monomial in $\\mathbf {M}$ .",
"Since the matrix $S$ is full-rank, we perform Gaussian elimination to remove the squares $X_{i_j}^2$ which do not belong to an edge of the graph.",
"This provides us with an equivalent system of the form $\\left\\lbrace \\begin{array}{l}X_1^2-g_1(X_{n-\\ell +1},\\ldots ,X_n) = 0\\\\\\quad \\quad \\quad \\vdots \\\\X_{n-\\ell }^2-g_{n-\\ell }(X_{n-\\ell +1},\\ldots ,X_n) = 0\\\\h_1(X_{n-\\ell +1},\\ldots ,X_n) = \\dots =h_\\ell (X_{n-\\ell +1},\\ldots ,X_n)=0.\\end{array}\\right.$ We end up with a system $(h_1,\\ldots , h_\\ell )$ of dense homogeneous polynomials in $\\ell $ variables.",
"Note that $\\ell $ is bounded by $2k$ , which does not depend on $n$ .",
"Consequently, this system can be solved within a constant number of operations as $n$ grows.",
"By Bézout theorem, this system has at most $2^{\\ell }$ solutions.",
"Finally, if $(a_{n-\\ell +1},\\ldots , a_n)$ is a solution of $h_1=\\dots =h_\\ell =0$ , then $(\\pm \\sqrt{g_1(a_{n-\\ell +1},\\ldots , a_n)},\\ldots ,\\linebreak \\pm \\sqrt{g_{n-\\ell }(a_{n-\\ell +1},\\ldots , a_n)},a_{n-\\ell +1},\\ldots ,a_n)$ is a solution of the input system.",
"Moreover, all solutions are of this form.",
"Therefore, even though the number of solutions of such systems depends exponentially on $n$ , they can be conveniently represented.",
"Moreover, we show next that computing this representation can be achieved within a number of operations in $\\mathbb {K}$ which is polynomial in $n$ : Corollary 5.2 Let $(f_1,\\ldots , f_n)\\in \\mathcal {L}_{\\mathbf {M}}^n$ be polynomials with support $\\mathbf {M}$ satisfying the above assumptions ($\\vert \\mathbf {M}\\vert =n+k+1$ , all squares are in $\\mathbf {M}$ , $S$ has full-rank) and $\\mu $ be a square-free monomial.",
"For fixed $k$ , Problem with input $(f_1,\\ldots ,f_n)$ and $\\mu $ can be solved within $O\\left(n^\\omega \\right)$ arithmetic operations as $n$ grows, where $\\omega $ is a feasible exponent for matrix multiplication.",
"With the same notations as in the proof of Proposition REF , and by noticing that $\\langle h_1,\\ldots ,h_\\ell \\rangle \\cap \\mathbb {K}[\\mu ]=\\langle f_1,\\ldots , f_n\\rangle \\cap \\mathbb {K}[\\mu ]$ , solving Problem with input $(h_1,\\ldots ,h_\\ell )$ and $\\mu $ yields a solution to Problem with input $(f_1,\\ldots , f_n)$ and $\\mu $ .",
"Solving Problem with input $h_1,\\ldots , h_\\ell $ can be achieved within a time complexity which does not depend on $n$ .",
"Consequently, the only complexity that depends on $n$ is the cost of computing the polynomials $h_1,\\ldots , h_\\ell $ .",
"This is done by linear algebra, within $O\\left(n^\\omega \\right)$ operations in $\\mathbb {K}$ ."
],
[
"Experimental results",
"In this section, we describe experimental results, validating the theoretical results and illustrating their practical relevance.",
"In particular, our prototype implementation of the algorithm in the proof of Corollary REF is able to compute Nullstellensatz' certificates of inconsistency for systems of 30000 equations and 30000 unknowns generated from the uniform model in Theorem REF .",
"This may be compared to the practical timings for solving the same problem with dense generic quadratic systems, where 20 unknowns is already a difficult challenge due to the exponential size of the certificates.",
"Experimental setting.",
"$\\mathbb {K}$ is the finite field ${\\rm GF}(65521)$ .",
"The experimental procedure depends on parameters $n,k$ and $\\beta $ : generate a random support $\\mathbf {M}$ of $n+k+1$ monomials of degree at most 2, containing 1 and $\\lfloor n^\\beta \\rfloor $ squares.",
"The subsets of square monomials and non-square monomials are respectively chosen uniformly at random; generate a random system of $n$ equations with support $\\mathbf {M}$ , where all the coefficients are chosen uniformly at random in $\\mathbb {K}$ ; return “success” if our implementation returns a relation $1=\\sum _{i=1}^n h_i f_i,$ with $h_i\\in \\mathcal {L}_{\\mathbf {M}}$ , else return “failure”.",
"By Theorem REF , for any choice of parameters $k\\in \\mathbb {N}$ and $0.5<\\beta <1$ , the probability that “success” is returned should tend towards 1 as $n$ grows.",
"First, we study the dependence of the asymptotic behavior on the choice of $\\beta $ .",
"To this end, we fix $k=1$ and we look at the experimental probability of success as $n$ grows.",
"Experimental results are reported in Figure REF .",
"The results are in accordance with Theorem REF : when $\\beta > 0.5$ , the probability that such systems have no solution and that there exists a Nullstellensatz certificate in $\\mathcal {L}_{\\mathbf {M}}$ seems to tend to 1 as $n$ grows.",
"We also observe that the convergence seems to depend strongly on $\\beta $ : when $\\beta $ becomes close to the limit value $0.5$ , the speed of convergence seems to decrease.",
"Next, we focus on the dependency on $k$ .",
"We fix $\\beta =0.9$ and let $n$ grow for different values of $k$ .",
"Experiments are reported in Figure REF .",
"Finally, we look at quadratic supports $\\mathbf {M}$ of cardinality $n+k+1$ generated uniformly at random without any constraint on the number of squares.",
"This case is not covered by the analysis of this paper and experiments show a different behavior: the probability of success of the algorithm does not seem to tend to 1 as $n$ grows, contrary to the case $\\beta >0.5$ .",
"However, this probability seems to converges to a nonzero value.",
"Conjecture 6.1 Let $k\\in \\mathbb {N}$ be a fixed integer.",
"For $n\\in \\mathbb {N}$ , let $\\mathbf {M}_n$ be a random subset of monomials in $n$ variables of degree at most 2, uniformly distributed among those of cardinality $n+k+1$ that contain 1.",
"Let $f_1,\\ldots , f_n\\in \\mathcal {L}_{\\mathbf {M}}$ be a system with support $\\mathbf {M}$ and generic coefficients.",
"Then the probability that there exist $h_1,\\ldots , h_n\\in \\mathcal {L}_{\\mathbf {M}}$ such that $\\sum _{i=1}^n f_i h_i = 1$ tends to a nonzero value as $n$ grows.",
"Finally, we report in Figure REF on experiments about the efficiency our prototype implementation for computing Nullstellensatz certificates.",
"The experiments were conducted on a Mac Retina 2.8Ghz Intel Core i7, and linear algebra computations were performed with Magma V2.20-3.",
"We see in these experiments that systems with several thousands of variables can be handled in a few seconds.",
"The algorithm works in two steps: first we reduce the quadratic system with linear algebra (the complexity of this step is independent of $\\beta $ and is represented by the dashed curve); then, the matrix in degree 4 (multiplying all the reduced polynomials by all the monomials in $\\mathbf {M}$ ) is constructed and reduced.",
"The time of this second step depends on $\\beta $ and is indicated by the plain curves.",
"Therefore, these graphs seem to indicate that the cost of computing certificates of inconsistency for these systems is approximately twice the time of computing the row echelon form of a dense $n\\times n$ matrix.",
"Figure: k=2k=2 fixed, nn grows, several value of β\\beta .",
"Every point is anaverage over 1000 tests.",
"The relative positions of the curves follow thevalues of β\\beta .Figure: β=0.9\\beta =0.9 fixed, nn grows, several value of kk.",
"Every point is anaverage over 10000 tests.Figure: Timings for the computation of the certificate ofinconsistency, k=2k=2."
],
[
"Proof of Theorem 4.4",
"Set $p_n=a_n/\\binom{n+1}{2} \\text{ and }q_n=b_n/(n+1),$ and let $\\mathbf {M}_n$ be the random support constructed as above with respect to the probabilities $p_n$ and $q_n$ .",
"We let $\\mathbf {MS}_n$ denote the subset of squares in $\\mathbf {M}_n$ and $\\mathbf {MNS}_n$ denote the subset of nonsquare monomials in $\\mathbf {M}_n$ .",
"Also, we set $\\ell =(k^2+3k+2)/2$ .",
"First, we notice that $\\mathbf {P}(\\nu (\\mathbf {M}_n)\\ge \\ell )$ equals $&\\displaystyle \\sum _{\\begin{array}{c}0\\le i\\le \\binom{n+1}{2}\\\\0\\le j\\le n\\end{array}}\\mathbf {P}(\\nu (\\mathbf {M}_n)\\ge \\ell \\mid \\vert \\mathbf {MS}\\vert =i\\text{ and}\\vert \\mathbf {MNS}\\vert =j)\\,c_{n,i,j} \\nonumber \\\\&=\\displaystyle \\sum _{\\begin{array}{c}0\\le i\\le \\binom{n+1}{2}\\\\0\\le j\\le n\\end{array}}\\mathbf {P}(\\nu (\\mathbf {U}_{n,i,j})\\ge \\ell )\\,c_{n,i,j},$ where $c_{n,i,j}=p_n^i(1-p_n)^{\\binom{n+1}{2}-i}q_n^j(1-q_n)^{n+1-j}\\binom{n+1}{j}\\binom{\\binom{n+1}{2}}{i}$ is the probability that $\\vert \\mathbf {MS}_n\\vert =i$ and $\\vert \\mathbf {MNS}_n\\vert =j$ .",
"Since the matching number is monotone with respect to the subgraph ordering, we obtain $i_1\\ge i_2 \\text{ and } j_1\\ge j_2 \\Longrightarrow \\mathbf {P}(\\nu (\\mathbf {U}_{n,i_1,j_1})\\ge \\ell )\\ge \\mathbf {P}(\\nu (\\mathbf {U}_{n,i_2,j_2})\\ge \\ell ).$ Consequently, Equation (REF ) implies $\\mathbf {P}(\\nu (\\mathbf {M}_n)\\ge \\ell )$ is bounded from above by $\\begin{array}{l}\\displaystyle \\mathbf {P}(\\nu (\\mathbf {U}_{n,a_n,b_n})\\ge \\ell )\\sum _{\\begin{array}{c}0\\le i\\le a_n\\\\0\\le j\\le b_n\\end{array}} c_{n,i,j}\\,\\,+\\\\\\sum _{\\begin{array}{c}0\\le i\\le \\binom{n+1}{2}\\\\b_n+1\\le j\\le n+1\\end{array}}\\mathbf {P}(\\nu (\\mathbf {U}_{n,i,j})\\ge \\ell )\\,c_{n,i,j}+\\\\\\sum _{\\begin{array}{c}a_n+1\\le i\\le \\binom{n+1}{2}\\\\0\\le j\\le b_n\\end{array}}\\mathbf {P}(\\nu (\\mathbf {U}_{n,i,j})\\ge \\ell )\\,c_{n,i,j}.\\end{array}$ Note that the first summand is bounded by $\\sum _{\\begin{array}{c}0\\le i\\le a_n\\\\0\\le j\\le b_n\\end{array}} c_{n,i,j}$ , the second summand is bounded by $\\sum _{\\begin{array}{c}0\\le i\\le \\binom{n+1}{2}\\\\b_n+1\\le j\\le n+1\\end{array}}c_{n,i,j}$ and the third one is bounded by $\\sum _{\\begin{array}{c}a_n+1\\le i\\le \\binom{n+1}{2}\\\\0\\le j\\le b_n\\end{array}}c_{n,i,j}$ .",
"Since the sum of these bounds equals 1, and since the left-hand side of the inequality tends to 1 as $n$ grows by Lemma REF , if $\\underset{n\\rightarrow \\infty }{\\liminf } \\sum _{\\begin{array}{c}0\\le i\\le a_n\\\\0\\le j\\le b_n\\end{array}} c_{n,i,j}>0$ , then $\\mathbf {P}(\\nu (\\mathbf {U}_{n,a_n,b_n})\\ge \\ell )$ must tend to 1 as $n$ grows.",
"We prove now that $\\underset{n\\rightarrow \\infty }{\\liminf }\\sum _{\\begin{array}{c}0\\le i\\le a_n\\\\0\\le j\\le b_n\\end{array}} c_{n,i,j}\\ge 1/4$ .",
"First, we rewrite $\\sum _{\\begin{array}{c}0\\le i\\le a_n\\\\0\\le j\\le b_n\\end{array}} c_{n,i,j}$ as $\\left(\\sum _{0\\le i\\le a_n} p_n^i(1-p_n)^{\\binom{n+1}{2}-i}\\binom{\\binom{n+1}{2}}{i}\\right)\\left(\\sum _{0\\le j\\le b_n} q_n^j(1-q_n)^{n+1-j}\\binom{n+1}{j}\\right).$ Notice that if $(\\mathcal {T}_n)$ is a sequence of random variables following a binomial distribution $B(n,s_n)$ (i.e.",
"the sum of $n$ Bernoulli independent variables of parameter $s_n$ ) such that $s_n\\underset{n\\rightarrow \\infty }{\\rightarrow }0$ and $n s_n\\underset{n\\rightarrow \\infty }{\\rightarrow }\\infty $ , then $(\\mathcal {T}_n-ns_n)/\\sqrt{n s_n}$ converges in distribution to the standard Gaussian distribution $\\mathcal {N}$ (this can be seen on the pointwise convergence of the moment generating function).",
"This implies $\\liminf _{n\\rightarrow \\infty }\\mathbf {P}(\\mathcal {T}_n\\le n s_n)\\ge \\mathbf {P}(\\mathcal {N}\\le 0)=1/2,$ where $\\mathcal {N}$ is a standard Gaussian distribution.",
"Then, we remark that by construction the first factor in Eq.",
"(REF ) equals $\\mathbf {P}(\\vert \\mathbf {MNS}_n\\vert \\le \\binom{n+1}{2}p_n)$ and $\\vert \\mathbf {MNS}_n\\vert $ follows a binomial distribution of parameters $(\\binom{n+1}{2},\\Omega (1/n))$ .",
"Therefore, we obtain $\\underset{n\\rightarrow \\infty }{\\liminf }\\left(\\sum _{0\\le i\\le a_n} p_n^i(1-p_n)^{\\binom{n+1}{2}-i}\\binom{\\binom{n+1}{2}}{i}\\right)\\ge 1/2.$ A similar argument shows the same lower bound for the second factor in Eq.",
"(REF ), finishing to prove that $\\underset{n\\rightarrow \\infty }{\\liminf }\\sum _{\\begin{array}{c}0\\le i\\le a_n\\\\0\\le j\\le b_n\\end{array}} c_{n,i,j}\\ge 1/4>0$ .",
"As explained above, this implies that $\\underset{n\\rightarrow \\infty }{\\lim }\\mathbf {P}(\\nu (\\mathbf {U}_{n,a_n,b_n})\\ge \\ell )=1$ for any $\\ell \\in \\mathbb {N}$ .",
"Finally, $\\begin{array}{r@{~}c@{~}l}\\mathbf {P}(\\nu (\\mathbf {U}_{n,a_n,b_n})\\ge \\ell )&=&\\mathbf {P}(\\nu (\\mathbf {U}_{n,a_n,b_n})\\ge (k^2+3k+2)/2)\\\\&=&\\displaystyle \\mathbf {P}\\left(\\frac{\\sqrt{1+8\\nu (\\mathbf {U}_{n,a_n,b_n})}-1}{2}\\ge k+1\\right)\\\\&\\underset{n\\rightarrow \\infty }{\\longrightarrow }&1.\\end{array}$ This proof is concluded by noticing that $k+1=\\vert \\mathbf {U}_{n,a_n,b_n}\\vert -n$ ."
]
] | 1605.05889 |
[
[
"Classification of differential symmetry breaking operators for\n differential forms"
],
[
"Abstract We give a complete classification of conformally covariant differential operators between the spaces of differential $i$-forms on the sphere $S^n$ and $j$-forms on the totally geodesic hypersphere $S^{n-1}$ by analyzing the restriction of principal series representations of the Lie group $O(n+1,1)$.",
"Further, we provide explicit formul\\ae{} for these matrix-valued operators in the flat coordinates and find factorization identities for them."
],
[
"Introduction",
"Suppose a Lie group $G$ acts conformally on a Riemannian manifold $(X,g)$ .",
"This means that there exists a positive-valued function $\\Omega \\in C^\\infty (G\\times X)$ (conformal factor) such that $L_h^*g_{h\\cdot x}=\\Omega (h,x)^2g_x \\quad \\mathrm {for\\; all}\\;h\\in G\\;\\mathrm {and}\\; x\\in X,$ where $L_h:X\\rightarrow X, x\\mapsto h\\cdot x$ denotes the action of $G$ on $X$ .",
"Since $\\Omega $ satisfies a cocycle condition, we can form a family of representations $\\varpi ^{(i)}_{u}$ for $u\\in and $ 0iX$ on the space $ Ei(X)$ of differential $ i$-forms on $ X$ by\\begin{equation}\\varpi ^{(i)}_{u}(h)\\alpha :=\\Omega (h^{-1},\\cdot )^uL_{h^{-1}}^*\\alpha \\quad (h\\in G).\\end{equation}The representation $ (i)u$ of the conformal group $ G$ on $ Ei(X)$will be simply denoted by$ Ei(X)u$.$ If $Y$ is a submanifold of $X$ , then we can also define a family of representations $\\varpi ^{(j)}_{v}$ on $\\mathcal {E}^j(Y)$ ($v\\in 0\\le j\\le \\dim Y$ ) of the subgroup $G^{\\prime }:=\\left\\lbrace h\\in G\\,:\\,h\\cdot Y=Y\\right\\rbrace ,$ which acts conformally on the Riemannian submanifold $(Y,g{\\;\\vert }_{ Y})$ .",
"We study differential operators $\\mathcal {D}:\\mathcal {E}^i(X)\\longrightarrow \\mathcal {E}^j(Y)$ that intertwine the two representations $\\varpi ^{(i)}_{u}\\vert _{G^{\\prime }}$ and $\\varpi ^{(j)}_{v}$ of $G^{\\prime }$ .",
"Here $\\varpi ^{(i)}_{u}\\vert _{G^{\\prime }}$ stands for the restriction of the $G$ -representation $\\varpi ^{(i)}_{u}$ to the subgroup $G^{\\prime }$ .",
"We say that such $\\mathcal {D}$ is a differential symmetry breaking operator, and denote by $\\mathrm {Diff}_{G^{\\prime }}(\\mathcal {E}^i(X)_{u},\\mathcal {E}^j(Y)_{v})$ the space of all differential symmetry breaking operators.",
"We address the following problems: Problem A Determine the dimension of the space $\\mathrm {Diff}_{G^{\\prime }}\\left(\\mathcal {E}^i(X)_{u}, \\mathcal {E}^j(Y)_{v}\\right)$ .",
"In particular, find a necessary and sufficient condition on a quadruple $(i,j,u,v)$ such that there exist nontrivial differential symmetry breaking operators.",
"Problem B Construct explicitly a basis of $\\mathrm {Diff}_{G^{\\prime }}\\left(\\mathcal {E}^i(X)_{u}, \\mathcal {E}^j(Y)_{v}\\right)$ .",
"In the case where $X=Y$ , $G=G^{\\prime }$ , and $i = j = 0$ , a classical prototype of such operators is a second order differential operator called the Yamabe operator $\\Delta +\\frac{n-2}{4(n-1)}\\kappa \\in \\mathrm {Diff}_{G}(\\mathcal {E}^0(X)_{\\frac{n}{2}-1},\\mathcal {E}^0(X)_{\\frac{n}{2}+1}),$ $\\Delta $ is the Laplace–Beltrami operator, where $n$ is the dimension of $X$ , and $\\kappa $ is the scalar curvature of $X$ .",
"Conformally covariant differential operators of higher order are also known: the Paneitz operator (fourth order) , which appears in four dimensional supergravity , or more generally, the so-called GJMS operators are such examples.",
"Analogous conformally covariant operators on forms ($i=j$ case) were studied by Branson .",
"On the other hand, the insight of representation theory of conformal groups is useful in studying Maxwell's equations, see , for instance.",
"Let us consider the more general case where $Y\\ne X$ and $G^{\\prime } \\ne G$ .",
"An obvious example of symmetry breaking operators is the restriction operator $\\mathrm {Rest}_Y$ which belongs to $\\mathrm {Diff}_{G^{\\prime }}\\left(\\mathcal {E}^i(X)_{u}, \\mathcal {E}^i(Y)_{u}\\right)$ for all $u\\in .",
"Another elementary exampleis $ RestYNY(X) DiffG'(Ei(X)u, Ei-1(Y)v)$if $ v = u+1$ where $ NY(X)$ denotesthe interior multiplication by the normal vector field to $ Y$when $ Y$ is of codimension one in $ X$.$ In the model space where $(X,Y)=(S^n,S^{n-1})$ , the pair $(G,G^{\\prime })$ of conformal groups amounts to $(O(n+1,1),O(n,1))$ modulo center, and Problems REF and REF have been recently solved for $i=j=0$ by Juhl , see also , and for different approaches by the F-method and the residue calculus, respectively.",
"Problems REF and REF for general $i$ and $j$ for the model space can be reduced to analogous problems for (nonspherical) principal series representations by the isomorphism () below.",
"In this note we shall give complete solutions to Problems REF and REF in those terms (see Theorems REF and ).",
"Notation: $\\mathbb {N}=\\lbrace 0,1,2,\\cdots \\rbrace $ , $\\mathbb {N}_+=\\lbrace 1,2,\\cdots \\rbrace $ ."
],
[
"Principal series representations of $G=O(n+1,1)$",
"We set up notations.",
"Let $P=MAN$ be a Langlands decomposition of a minimal parabolic subgroup of $G=O(n+1,1)$ .",
"For $0\\le i\\le n$ , $\\delta \\in \\mathbb {Z}/2\\mathbb {Z}$ , and $\\lambda \\in , we extend the outer tensor product representation $ i(n)(-1)$ of $ MA(O(n)O(1))R$ to $ P$ by letting $ N$ act trivially, and form a $ G$-equivariantvector bundle$ Vi,:=GP (i(n)(-1)) $over the real flag variety$ X=G/PSn$.",
"Then we define an unnormalized principalseries representations\\begin{equation}I(i,\\lambda )_\\delta :=\\mathrm {Ind}_P^G\\left(\\bigwedge ^i(n)\\otimes (-1)^\\delta \\otimes \\lambda \\right)\\end{equation}of $ G$ on the Fréchet space $ C(X,V,i)$ of smooth sections.$ In our parametrization, $I(i,n-2i)_\\delta $ and $I(i,i)_\\delta $ have the same infinitesimal character with the trivial one-dimensional representation of $G$ .",
"Then, for all $u\\in , we have a natural $ G$-isomorphism\\begin{equation}\\varpi ^{(i)}_u\\simeq I(i,u+i)_{i\\,\\mathrm {mod}\\,2}.\\end{equation}Similarly, for $ 0jn-1$, $ Z/2Z$ and $ , we define an unnormalized principal series representation $J(j,\\nu )_\\varepsilon :=\\mathrm {Ind}_{P^{\\prime }}^{G^{\\prime }}\\left(\\bigwedge ^j({n-1})\\otimes (-1)^\\varepsilon \\otimes \\nu \\right)$ of the subgroup $G^{\\prime }=O(n,1)$ on $C^\\infty (Y,\\mathcal {W}_{\\nu ,\\varepsilon }^j)$ , where $\\mathcal {W}_{\\nu ,\\varepsilon }^j:=G^{\\prime }\\times _{P^{\\prime }}\\left(\\bigwedge ^j({n-1})\\otimes (-1)^\\varepsilon \\otimes \\nu \\right)$ is a $G^{\\prime }$ -equivariant vector bundle over $Y=G^{\\prime }/P^{\\prime }\\simeq S^{n-1}$ ."
],
[
"Existence condition for differential symmetry breaking operators",
"A continuous $G^{\\prime }$ -intertwining operator $T:I(i,\\lambda )_\\delta \\longrightarrow J(j,\\nu )_\\varepsilon $ is said to be a symmetry breaking operator (SBO).",
"We say that $T$ is a differential operator if $T$ satisfies $\\mathrm {Supp}(Tf)\\subset \\mathrm {Supp} f$ for all $f\\in C^\\infty (X,\\mathcal {V}^i_{\\lambda ,\\delta })$ , and $\\mathrm {Diff}_{G^{\\prime }}\\left( I(i,\\lambda )_\\delta , J(j,\\nu )_\\varepsilon \\right)$ denotes the space of differential SBOs.",
"We give a complete solution to Problem REF for $(X,Y)=(S^n,S^{n-1})$ in terms of principal series representations: Theorem 3.1 Let $n\\ge 3$ .",
"Suppose $0\\le i\\le n$ , $0\\le j\\le n-1$ , $\\lambda ,\\nu \\in ,and $ ,Z/2Z$.Then the followingthree conditions on 6-tuple $ (i,j,,,,)$ are equivalent:\\begin{enumerate}\\item [\\emph {(i)}]\\mathrm {Diff}_{O(n,1)}(I(i,\\lambda )_\\delta ,J(j,\\nu )_{\\varepsilon })\\ne \\lbrace 0\\rbrace .\\item [\\emph {(ii)}]\\dim \\mathrm {Diff}_{O(n,1)}(I(i,\\lambda )_\\delta ,J(j,\\nu )_{\\varepsilon })=1.\\item [\\emph {(iii)}]The 6-tuple belongs to one of the following six cases:\\vspace{3.61371pt}\\begin{enumerate}\\item [] \\emph {Case 1}.",
"j = i,0\\le i \\le n-1, \\nu - \\lambda \\in \\mathbb {N},\\varepsilon -\\delta \\equiv \\nu -\\lambda \\; \\mathrm {mod}\\, 2.\\vspace{3.61371pt}\\end{enumerate}\\item [] \\emph {Case 2}.",
"j = i-1,1 \\le i \\le n, \\nu -\\lambda \\in \\mathbb {N},\\varepsilon -\\delta \\equiv \\nu - \\lambda \\; \\mathrm {mod}\\, 2.\\vspace{3.61371pt}\\end{enumerate}\\item [] \\emph {Case 3}.",
"$ j=i+1$,$ 1 i n-2$,$ (,) = (i,i+1)$,$ + 1 mod 2$.\\vspace{3.61371pt}$ Case 3$^{\\prime }$.",
"$(i,j) = (0,1)$ , $-\\lambda \\in \\mathbb {N}$ , $\\nu = 1$ , $\\varepsilon \\equiv \\delta + \\lambda +1 \\; \\mathrm {mod}\\, 2$ .",
"Case 4.",
"$j=i-2$ , $2\\le i \\le n-1$ , $(\\lambda ,\\nu ) = (n-i,n-i+1)$ , $\\varepsilon \\equiv \\delta + 1\\; \\mathrm {mod}\\, 2$ .",
"Case 4$^{\\prime }$.",
"$(i,j) = (n, n-2)$ , $-\\lambda \\in \\mathbb {N}$ , $\\nu =1$ , $\\varepsilon \\equiv \\delta + \\lambda +1\\; \\mathrm {mod}\\, 2$ .",
"We set $\\Xi :=\\lbrace (i,j,\\lambda ,\\nu )$ : the 6-tuple $(i,j,\\lambda ,\\nu ,\\delta ,\\varepsilon )$ satisfies one of the equivalent conditions of Theorem REF for some $\\delta ,\\varepsilon \\in \\mathbb {Z}/2\\mathbb {Z}\\rbrace $ .",
"Construction of differential symmetry breaking operators In this section, we describe an explicit generator of the space of differential SBOs if one of the equivalent conditions in Theorem REF is satisfied.",
"For this we use the flat picture of the principal series representations $I(i,\\lambda )_\\delta $ of $G$ which realizes the representation space $C^\\infty (X,\\mathcal {V}^i_{\\lambda ,\\delta })$ as a subspace of $C^\\infty (\\mathbb {R}^n,\\bigwedge ^i(n))$ by trivializing the bundle $\\mathcal {V}^i_{\\lambda ,\\delta }\\longrightarrow X$ on the open Bruhat cell $\\mathbb {R}^n\\hookrightarrow X,\\quad (x_1,\\cdots ,x_n)\\mapsto \\exp \\left(\\sum _{j=1}^nx_jN_j^-\\right)P.$ Here $\\lbrace N_1^-,\\cdots ,N_n^-\\rbrace $ is an orthonormal basis of the nilradical $\\mathfrak {n}_-(\\mathbb {R})$ of the opposite parabolic subalgebra with respect to an $M$ -invariant inner product.",
"Without loss of generality, we may and do assume that the open Bruhat cell $\\mathbb {R}^{n-1}\\hookrightarrow Y\\simeq G^{\\prime }/P^{\\prime }$ is given by putting $x_n=0$ .",
"Then the flat picture of the principal series representation $J(j,\\nu )_\\varepsilon $ of $G^{\\prime }$ is defined by realizing $C^\\infty (Y,\\mathcal {W}_{\\nu ,\\varepsilon }^j)$ as a subspace of $C^\\infty (\\mathbb {R}^{n-1},\\bigwedge ^j({n-1}))$ .",
"For the construction of explicit generators of matrix-valued SBOs, we begin with a scalar-valued differential operator.",
"For $\\alpha \\in and $ N$, we define a polynomial of two variables $ (s,t)$ by$$\\left( I_\\ell \\widetilde{C}^\\alpha _\\ell \\right)(s,t):=s^{\\frac{\\ell }{2}}\\widetilde{C}^\\alpha _\\ell \\left(\\frac{t}{\\sqrt{s}}\\right),$$where $ C(z)$ is the renormalized Gegenbauer polynomial given by$$\\widetilde{C}^\\alpha _\\ell (z):=\\frac{1}{\\Gamma \\left(\\alpha +\\left[\\frac{\\ell +1}{2}\\right]\\right)}\\sum _{k=0}^{\\left[\\frac{\\ell }{2}\\right]}(-1)^k\\frac{\\Gamma (\\ell -k+\\alpha )}{k!",
"(\\ell -2k)!",
"}(2z)^{\\ell -2k}.$$Then $ C(z)$ is a nonzero polynomial for all $ and $\\ell \\in \\mathbb {N}$ , and a (normalized) Juhl's conformally covariant operator $\\widetilde{_{\\lambda ,\\nu }: C^\\infty (\\mathbb {R}^n)\\longrightarrow C^\\infty (\\mathbb {R}^{n-1})is defined by\\widetilde{_{\\lambda ,\\nu }:=\\mathrm {Rest}_{x_n=0}\\circ \\left( I_\\ell \\widetilde{C}^{\\lambda -\\frac{n-1}{2}}_\\ell \\right)\\left(-\\Delta _{\\mathbb {R}^{n-1}},\\frac{\\partial }{\\partial x_n}\\right),for \\lambda ,\\nu \\in with \\ell :=\\nu -\\lambda \\in \\mathbb {N}.For instance,\\widetilde{_{\\lambda ,\\nu }=\\mathrm {Rest}_{x_n=0}\\circ {\\left\\lbrace \\begin{array}{ll}\\mathrm {id} & \\mathrm {if}\\; \\nu =\\lambda ,\\\\2\\frac{\\partial }{\\partial x_n} & \\mathrm {if}\\; \\nu =\\lambda +1,\\\\\\Delta _{\\mathbb {R}^{n-1}}+(2\\lambda -n+3)\\frac{\\partial ^2}{\\partial x_n^2}& \\mathrm {if}\\; \\nu =\\lambda +2.\\end{array}\\right.",
"}For (i,j,\\lambda ,\\nu )\\in \\Xi , we introduce a new family of matrix-valued differential operators\\widetilde{^{i,j}_{\\lambda ,\\nu }: C^\\infty (\\mathbb {R}^n,\\bigwedge ^i(n))\\longrightarrow C^\\infty (\\mathbb {R}^{n-1},\\bigwedge ^j({n-1})),by using the identifications \\mathcal {E}^i(\\mathbb {R}^n)\\simeq C^\\infty (\\mathbb {R}^n)\\otimes \\bigwedge ^i(n)and \\mathcal {E}^j(\\mathbb {R}^{n-1})\\simeq C^\\infty (\\mathbb {R}^{n-1})\\otimes \\bigwedge ^j({n-1}),as follows.",
"Let d^*_{\\mathbb {R}^n} be the codifferential,which is the formal adjoint of the differential d_{\\mathbb {R}^n},and \\iota _{\\frac{\\partial }{\\partial x_n}} the inner multiplication by the vector field\\frac{\\partial }{\\partial x_n}.",
"Both operators map \\mathcal {E}^i(\\mathbb {R}^n) to \\mathcal {E}^{i-1}(\\mathbb {R}^n).For \\alpha \\in and \\ell \\in \\mathbb {N}, let\\gamma (\\alpha ,\\ell ):=1 (\\ell is odd); =\\alpha +\\frac{\\ell }{2} ( \\ell is even).Then we set{\\begin{@align}{1}{-1}{i,i}_{\\lambda ,\\nu }&:=\\widetilde{_{\\lambda +1, \\nu -1} d_{\\mathbb {R}^n} d^*_{\\mathbb {R}^n}-\\gamma (\\lambda -\\frac{n}{2}, \\nu -\\lambda ) \\widetilde{_{\\lambda ,\\nu -1}d_{\\mathbb {R}^n}\\iota _{\\frac{\\partial }{\\partial x_n}}+ \\frac{1}{2}(\\nu -i)\\widetilde{_{\\lambda ,\\nu } \\nonumber \\qquad \\qquad \\qquad \\mathrm {for}\\; 0\\le i \\le n-1.\\\\{i,i-1}_{\\lambda ,\\nu }&:= - \\widetilde{\\mathcal {}}_{\\lambda +1, \\nu -1} d_{\\mathbb {R}^n}d^*_{\\mathbb {R}^n}\\iota _{\\frac{\\partial }{\\partial x_n}} -\\gamma (\\lambda -\\frac{n-1}{2}, \\nu -\\lambda )\\widetilde{_{\\lambda +1, \\nu }d^*_{\\mathbb {R}^n}+\\frac{1}{2}(\\lambda +i-n)\\widetilde{_{\\lambda ,\\nu } \\iota _{\\frac{\\partial }{\\partial x_n}}\\; \\mathrm {for}\\; 1\\le i \\le n.\\nonumber }}We note that there exist isolated parameters(\\lambda ,\\nu ) for which {i,i}_{\\lambda ,\\nu } =0or {i,i-1}_{\\lambda ,\\nu } =0.For instance, {0,0}_{\\lambda ,\\nu } = \\frac{1}{2}\\nu \\widetilde{_{\\lambda ,\\nu },and thus {0,0}_{\\lambda ,\\nu } = 0 if \\nu =0.To be precise, we have the following:\\begin{itemize}\\item []{i,i}_{\\lambda , \\nu } = 0if and only if \\lambda = \\nu =i or \\nu = i = 0;\\item []{i,i-1}_{\\lambda ,\\nu }=0 if and only if\\lambda =\\nu =n-i or \\nu = n-i = 0.\\end{itemize}}We renormalize these operators by\\begin{equation*}\\widetilde{^{i,i}_{\\lambda ,\\nu }:={\\left\\lbrace \\begin{array}{ll}\\mathrm {Rest}_{x_n=0} & \\text{if $\\lambda = \\nu $},\\\\\\widetilde{_{\\lambda ,\\nu } & \\text{if $i=0$},\\\\{i,i}_{\\lambda ,\\nu } & \\text{otherwise},}\\end{array}\\quad \\text{and}\\quad \\widetilde{^{i,i-1}_{\\lambda ,\\nu }:={\\left\\lbrace \\begin{array}{ll}\\mathrm {Rest}_{x_n=0} \\circ \\iota _{\\frac{\\partial }{\\partial x_n}} & \\text{if $\\lambda = \\nu $},\\\\\\widetilde{_{\\lambda ,\\nu } \\circ \\iota _{\\frac{\\partial }{\\partial x_n}} & \\text{if $i=n$},\\\\{i,i-1}_{\\lambda ,\\nu } & \\text{otherwise}.}\\end{array}\\right.",
"}Then \\widetilde{^{i,i}_{\\lambda ,\\nu } (0\\le i \\le n-1) and\\widetilde{^{i,i-1}_{\\lambda ,\\nu } (1 \\le i \\le n)are nonzero differential operators of order \\nu -\\lambda for any \\lambda , \\nu \\in with \\nu -\\lambda \\in \\mathbb {N}.",
"}The differential operators\\widetilde{^{i,i+1}_{\\lambda ,\\nu } and\\widetilde{^{i,i-2}_{\\lambda ,\\nu } aredefined only for special parameters (\\lambda ,\\nu ) as follows.",
"{\\begin{@align*}{1}{-1}\\widetilde{^{i,i+1}_{\\lambda ,i+1}&:={\\left\\lbrace \\begin{array}{ll}\\mathrm {Rest}_{x_n=0} \\circ d_{\\mathbb {R}^n}&\\mathrm {for}\\;1\\le i\\le n-2, \\lambda =i,\\\\d_{\\mathbb {R}^{n-1}} \\circ \\widetilde{_{\\lambda ,0}&\\mathrm {for}\\; i=0, \\lambda \\in -\\mathbb {N},}\\end{array}\\\\\\widetilde{^{i,i-2}_{\\lambda ,n-i+1}&:={\\left\\lbrace \\begin{array}{ll}\\mathrm {Rest}_{x_n=0}\\circ \\iota _{\\frac{\\partial }{\\partial x_n}} d^*_{\\mathbb {R}^n}&\\mathrm {for}\\; 2\\le i\\le n, \\lambda =n-i,\\\\-d^*_{\\mathbb {R}^{n-1}}\\circ {n,n-1}_{\\lambda ,0}&\\mathrm {for}\\; i=n, \\lambda \\in -\\mathbb {N}.\\end{array}\\right.",
"}}\\right.Then we give a complete solution to Problem \\ref {prob:2} forthe model space (X,Y)=(S^n, S^{n-1}) in terms of the flat picture ofprincipal series representations as follows:\\begin{thm}Suppose a 6-tuple (i,j,\\lambda ,\\nu ,\\delta ,\\varepsilon ) satisfies one of the equivalent conditions in Theorem \\ref {thm:A}.Then theoperators \\widetilde{^{i,j}_{\\lambda ,\\nu }: C^\\infty (\\mathbb {R}^n)\\otimes \\bigwedge ^i(n)\\longrightarrow C^\\infty (\\mathbb {R}^{n-1})\\otimes \\bigwedge ^j({n-1}) extend to differential SBOs I(i,\\lambda )_\\delta \\longrightarrow J(j,\\nu )_\\varepsilon , to be denoted by the same letters.",
"Conversely, anydifferential SBOfrom I(i,\\lambda )_\\delta to J(j,\\nu )_\\varepsilon is proportional to thefollowing differential operators:\\widetilde{^{i,i}_{\\lambda ,\\nu } in Case 1,\\widetilde{^{i,i-1}_{\\lambda ,\\nu } in Case 2,\\widetilde{^{i,i+1}_{i,i+1} in Case 3,\\widetilde{^{0,1}_{\\lambda ,1} in Case 3^{\\prime },\\widetilde{^{i,i-2}_{n-i,n-i+1} in Case 4, and\\widetilde{^{n,n-2}_{\\lambda ,1} in Case 4^{\\prime }.",
"}}\\section {Matrix-valued factorization identities}}Suppose thatT_X:I(i,\\lambda ^{\\prime })_\\delta \\rightarrow I(i,\\lambda )_\\delta or T_Y:J(j,\\nu )_\\varepsilon \\rightarrow J(j,\\nu ^{\\prime })_\\varepsilon are G- orG^{\\prime }-intertwining operators, respectively.",
"Thenthe composition T_Y\\circ D_{X\\rightarrow Y} or D_{X\\rightarrow Y}\\circ T_X of asymmetry breaking operator D_{X\\rightarrow Y}:I(i,\\lambda )_\\delta \\rightarrow J(j,\\nu )_\\varepsilon gives another symmetry breaking operator:{I(i,\\lambda )_\\delta [rr]^{D_{X\\rightarrow Y}}@{-->}[drr]&&J(j,\\nu )_\\varepsilon [d]_{T_Y}\\\\I(i,\\lambda ^{\\prime })_\\delta [u]^{T_X}@{-->}[urr]&&J(j,\\nu ^{\\prime })_\\varepsilon }}The multiplicity-free property (see Theorem \\ref {thm:A} (ii)) assures the existence of matrix-valued factorization identitiesfor differential SBOs, namely, D_{X\\rightarrow Y}\\circ T_X must be a scalar multiple of \\widetilde{_{\\lambda ^{\\prime },\\nu }^{i, j}, and T_Y\\circ D_{X\\rightarrow Y} must be a scalar multiple of \\widetilde{_{\\lambda ,\\nu ^{\\prime }}^{i, j}.We shall determine these constants explicitly when T_X or T_Y areBranson^{\\prime }s conformally covariant operators \\cite {Branson} defined below.Let 0\\le i\\le n. For \\ell \\in \\mathbb {N}_+, we set\\begin{equation}\\mathcal {T}_{2\\ell }^{(i)}:=((\\frac{n}{2}-i-\\ell )d_{\\mathbb {R}^{n}}d^*_{\\mathbb {R}^{n}}+(\\frac{n}{2}-i+\\ell )d^*_{\\mathbb {R}^{n}}d_{\\mathbb {R}^{n}})\\Delta ^{\\ell -1}_{\\mathbb {R}^n}=(-2\\ell \\,d_{\\mathbb {R}^{n}}d^*_{\\mathbb {R}^{n}}-(\\frac{n}{2}-i+\\ell )\\Delta _{\\mathbb {R}^n})\\Delta _{\\mathbb {R}^n}^{\\ell -1}\\nonumber .\\end{equation}}Then the differential operator\\mathcal {T}_{2\\ell }^{(i)}:\\mathcal {E}^i(\\mathbb {R}^n)\\longrightarrow \\mathcal {E}^i(\\mathbb {R}^n)induces a nonzero O(n+1,1)-intertwining operator, to be denoted by the same letter \\mathcal {T}_{2\\ell }^{(i)}, from I\\left(i,\\frac{n}{2}-\\ell \\right)_\\delta to I\\left(i,\\frac{n}{2}+\\ell \\right)_\\delta , for \\delta \\in \\mathbb {Z}/2\\mathbb {Z}.Similarly, we define a G^{\\prime }-intertwining operator\\mathcal {T^{\\prime }}_{2\\ell }^{(j)}:J\\left(j,\\frac{n-1}{2}-\\ell \\right)_\\varepsilon \\longrightarrow J\\left(j,\\frac{n-1}{2}+\\ell \\right)_\\varepsilon for 0\\le j\\le n-1 and \\varepsilon \\in \\mathbb {Z}/2\\mathbb {Z}as the lift of the differential operator \\mathcal {T^{\\prime }}_{2\\ell }^{(j)}:\\mathcal {E}^j(\\mathbb {R}^{n-1})\\longrightarrow \\mathcal {E}^j(\\mathbb {R}^{n-1}) which is given by\\begin{equation*}\\mathcal {T^{\\prime }}_{2\\ell }^{(j)}=((\\frac{ n-1}{2}-j-\\ell )d_{\\mathbb {R}^{n-1}}d_{\\mathbb {R}^{n-1}}^*+(\\frac{ n-1}{2}-j+\\ell )d_{\\mathbb {R}^{n-1}}^*d_{\\mathbb {R}^{n-1}})\\Delta ^{\\ell -1}_{\\mathbb {R}^{n-1}}.\\end{equation*}}Consider the following diagrams for j=i and j=i-1:\\begin{eqnarray*}{I\\left(i,\\frac{n}{2}-\\ell \\right)_\\delta [d]_{\\mathcal {T}_{2\\ell }^{(i)}}[drr]^{{\\widetilde{}^{i,j}_{\\frac{n}{2}-\\ell ,\\frac{n}{2}+a+\\ell }}&& \\\\I\\left(i,\\frac{n}{2}+\\ell \\right)_\\delta [rr]_{{\\widetilde{}^{i,j}_{\\frac{n}{2}+\\ell ,\\frac{n}{2}+a+\\ell }}&&J\\left(j,\\frac{n}{2}+a+\\ell \\right)_\\varepsilon ,}&&{I\\left(i,\\frac{n-1}{2}-a-\\ell \\right)_\\delta [rr]^{{\\widetilde{}^{i, j}_{\\frac{n-1}{2}-a-\\ell ,\\frac{n-1}{2}-\\ell }}[drr]_{{\\widetilde{}^{i, j}_{\\frac{n-1}{2}-a-\\ell ,\\frac{n-1}{2}+\\ell }}&&J\\left(j,\\frac{n-1}{2}-\\ell \\right)_\\varepsilon [d]^{{\\mathcal {T}^{\\prime }}^{(j)}_{2\\ell }}\\\\&&J\\left(j,\\frac{n-1}{2}+\\ell \\right)_\\varepsilon ,}}where parameters \\delta and \\varepsilon \\in \\mathbb {Z}/2\\mathbb {Z} are chosen according to Theorem \\ref {thm:A} (iii).In what follows, we putp_\\pm ={\\left\\lbrace \\begin{array}{ll}i\\pm \\ell -\\frac{n}{2}&\\mathrm {if}\\; a\\ne 0\\\\\\pm 2&\\mathrm {if}\\; a=0\\end{array}\\right.",
"},\\quad q={\\left\\lbrace \\begin{array}{ll}i+\\ell -\\frac{n-1}{2}&\\mathrm {if}\\;i\\ne 0, a\\ne 0\\\\-2&\\mathrm {if}\\; i\\ne 0, a=0\\\\-\\left(\\ell +\\frac{n-1}{2}\\right)&\\mathrm {if}\\; i=0\\end{array}\\right.",
"},\\quad r={\\left\\lbrace \\begin{array}{ll}i-\\ell -\\frac{n+1}{2}&\\mathrm {if}\\;i\\ne n, a\\ne 0\\\\2&\\mathrm {if}\\; i\\ne n, a=0\\\\-\\left(\\ell +\\frac{n+1}{2}\\right)&\\mathrm {if}\\; i=n\\end{array}\\right.",
"},\\begin{equation*}K_{\\ell ,a}:=\\prod _{k=1}^\\ell \\left(\\left[\\frac{a}{2}\\right]+k\\right).\\end{equation*}}Then the factorization identities for differential SBOs \\widetilde{^{i,j}_{\\lambda ,\\nu } for j\\in \\lbrace i-1,i\\rbrace andBranson^{\\prime }s conformally covariant operators \\mathcal {T}_{2\\ell }^{(i)} or \\mathcal {T^{\\prime }}_{2\\ell }^{(j)} are given as follows.\\begin{thm} Suppose 0\\le i\\le n-1, a\\in \\mathbb {N} and \\ell \\in \\mathbb {N}_+.Then\\begin{eqnarray*}&(1)&{\\widetilde{}^{i,i}_{\\frac{n}{2}+\\ell ,a+\\ell +\\frac{n}{2}}\\circ \\mathcal {T}_{2\\ell }^{(i)}=p_-K_{\\ell ,a}{\\widetilde{}^{i, i}_{\\frac{n}{2}-\\ell ,a+\\ell +\\frac{n}{2}}.\\\\&(2)& \\mathcal {T^{\\prime }}_{2\\ell }^{(i)}\\circ {\\widetilde{}^{i,i}_{\\frac{n-1}{2}-a-\\ell ,\\frac{n-1}{2}-\\ell }=qK_{\\ell ,a}{\\widetilde{}^{i,i}_{\\frac{n-1}{2}-a-\\ell ,\\frac{n-1}{2}+\\ell }.",
"}}}\\begin{thm}Suppose 1\\le i\\le n, a\\in \\mathbb {N} and \\ell \\in \\mathbb {N}_+.",
"Then\\begin{eqnarray*}&(1)&{\\widetilde{}^{i,i-1}_{\\frac{n}{2}+\\ell ,a+\\ell +\\frac{n}{2}}\\circ \\mathcal {T}_{2\\ell }^{(i)}=p_+K_{\\ell ,a}{\\widetilde{}^{i,i-1}_{\\frac{n}{2}-\\ell ,a+\\ell +\\frac{n}{2}}.\\\\&(2)&\\mathcal {T^{\\prime }}_{2\\ell }^{(i-1)}\\circ {\\widetilde{}^{i,i-1}_{\\frac{n-1}{2}-a-\\ell ,\\frac{n-1}{2}-\\ell }=rK_{\\ell ,a}{\\widetilde{}^{i,i-1}_{\\frac{n-1}{2}-a-\\ell ,\\frac{n-1}{2}+\\ell }.\\\\}}In the case where i=0, {\\widetilde{}^{i, i}_{\\lambda ,\\nu } isa scalar-valued operator, and the corresponding factorization identities in Theorem \\ref {thm:factor1} were studied in \\cite {Juhl,KS13,KOSS15}.",
"}The main results are proved by using the F-method \\cite {K14,KP1,KOSS15}.Details will appear elsewhere.",
"}\\begin{thebibliography}{99}\\end{thebibliography}\\bibitem {Branson} T.~P.",
"Branson,Conformally covariant equations on differential forms,\\emph {Comm.",
"Part.",
"Diff.",
"Eq.}",
"\\textbf {7}, (1982),\\href {http://dx.doi.org/10.1080/03605308208820228}{pp.",
"393--431}.",
"}\\bibitem {FT}E.~S.",
"Fradkin, A.~A.",
"Tseytlin,Asymptotic freedom in extended conformal supergravities,\\emph {Phys.",
"Lett.",
"B} \\textbf {110}, (1982)\\href {http://dx.doi.org/10.1016/0370-2693(82)91018-8}{pp.",
"117--122}.\\end{eqnarray*}\\bibitem {GJMS}C.~R.",
"Graham, R. Jenne, L.~J.",
"Mason, G.~A.~J.",
"Sparling,Conformally invariant powers of the Laplacian.",
"I. Existence.\\emph {J. London Math.",
"Soc.}",
"(2) \\textbf {46} (1992),\\href {http://dx.doi.org/10.1112/jlms/s2-46.3.557}{pp.",
"557--565}.\\end{thm}\\bibitem {Juhl}A. Juhl, \\emph { Families of conformally covariant differential operators, Q-curvature and holography.}",
"Progr.",
"Math.,\\href {http://link.springer.com/book/10.1007/978-3-7643-9900-9/page/1}{\\bf {275}}.",
"Birkhäuser, Basel, 2009.",
"}\\bibitem {K14} T.~Kobayashi,F-method for symmetry breaking operators,\\emph {Diff.",
"Geometry and its Appl.",
"}{\\textbf {33}}, (2014),\\href {http://dx.doi.org/10.1016/j.difgeo.2013.10.003}{pp.",
"272--289}.\\end{eqnarray*}\\bibitem {KP1}T.~Kobayashi, M.~Pevzner,Differential symmetry breaking operators.",
"I.",
"General theory andF-method,\\emph {Selecta.",
"Math.",
"(N.S.",
")}, \\textbf {22}, (2016),\\href {http://dx.doi.org/10.1007/s00029-015-0207-9}{pp.",
"801--845}.\\end{thm}\\bibitem {KP2}T.~Kobayashi, M.~Pevzner, Differential symmetry breaking operators.II.",
"Rankin--Cohen operators for symmetric pairs,\\emph {Selecta.",
"Math.",
"(N.S.",
")}, \\textbf {22}, (2016),\\href {http://dx.doi.org/10.1007/s00029-015-0208-8}{pp.",
"847--911}.",
"}\\bibitem {KS13}T.~Kobayashi, B.~Speh,Symmetry Breaking for Representations of Rank One Orthogonal Groups,\\href {http://dx.doi.org/10.1090/memo/1126}{Memoirs of American Mathematical Society, vol.",
"\\textbf {238}, 2015.}",
"118 pp.",
"{ISBN: 978-1-4704-1922-6}.",
"}\\bibitem {KOSS15}T.~Kobayashi, B.~Ørsted, P. Somberg, and V. Souček, Branching laws forVerma modules and applications in parabolic geometry.",
"I.",
"\\emph {Adv.",
"Math}., \\textbf {285},(2015),\\href {http://dx.doi.org/10.1016/j.aim.2015.08.020}{pp.",
"1796--1852}.",
"}\\bibitem {KW14}B.~Kostant, N. R.~Wallach, Action of the conformal group on steady state solutions to Maxwell^{\\prime }s equations and background radiation.",
"Symmetry: representation theory and its applications,\\href {http://dx.doi.org/10.1007/978-1-4939-1590-3_14}{pp.",
"385--418},\\emph {Progr.",
"Math.",
"},{\\textbf {257}}, Birkhäuser/Springer, New York, 2014.\\end{eqnarray*}\\bibitem {P08}S. Paneitz, A quartic conformally covariant differential operatorfor arbitrary pseudo-Riemannian manifolds,\\emph {SIGMA Symmetry Integrability Geom.",
"Methods Appl.",
"}\\href {http://www.emis.de/journals/SIGMA/2008/}{\\textbf {4} (2008)}, paper 036, 3 pp.",
"}}\\vspace{10.0pt}}\\footnotesize { Addresses:T. Kobayashi.",
"Kavli IPMU (WPI)and Graduate School of Mathematical Sciences,The University of Tokyo, 3-8-1 Komaba, Meguro, Tokyo, 153-8914 Japan;\\texttt {[email protected]}.\\vspace{5.0pt}}T. Kubo.",
"Graduate School of Mathematical Sciences, The University ofTokyo,3-8-1 Komaba, Meguro, Tokyo, 153-8914 Japan; \\texttt {{[email protected]}}.\\vspace{5.0pt}\\end{thm}}M. Pevzner.Laboratoire de Mathématiques de Reims, Universitéde Reims-Champagne-Ardenne, FR 3399 CNRS, F-51687, Reims, France; \\texttt {{[email protected].}}}\\end{@align*}}}}}}\\right.",
"}}\\end{equation*}}}}\\end{@align}}}}}}$"
],
[
"Construction of differential symmetry breaking operators",
"In this section, we describe an explicit generator of the space of differential SBOs if one of the equivalent conditions in Theorem REF is satisfied.",
"For this we use the flat picture of the principal series representations $I(i,\\lambda )_\\delta $ of $G$ which realizes the representation space $C^\\infty (X,\\mathcal {V}^i_{\\lambda ,\\delta })$ as a subspace of $C^\\infty (\\mathbb {R}^n,\\bigwedge ^i(n))$ by trivializing the bundle $\\mathcal {V}^i_{\\lambda ,\\delta }\\longrightarrow X$ on the open Bruhat cell $\\mathbb {R}^n\\hookrightarrow X,\\quad (x_1,\\cdots ,x_n)\\mapsto \\exp \\left(\\sum _{j=1}^nx_jN_j^-\\right)P.$ Here $\\lbrace N_1^-,\\cdots ,N_n^-\\rbrace $ is an orthonormal basis of the nilradical $\\mathfrak {n}_-(\\mathbb {R})$ of the opposite parabolic subalgebra with respect to an $M$ -invariant inner product.",
"Without loss of generality, we may and do assume that the open Bruhat cell $\\mathbb {R}^{n-1}\\hookrightarrow Y\\simeq G^{\\prime }/P^{\\prime }$ is given by putting $x_n=0$ .",
"Then the flat picture of the principal series representation $J(j,\\nu )_\\varepsilon $ of $G^{\\prime }$ is defined by realizing $C^\\infty (Y,\\mathcal {W}_{\\nu ,\\varepsilon }^j)$ as a subspace of $C^\\infty (\\mathbb {R}^{n-1},\\bigwedge ^j({n-1}))$ .",
"For the construction of explicit generators of matrix-valued SBOs, we begin with a scalar-valued differential operator.",
"For $\\alpha \\in and $ N$, we define a polynomial of two variables $ (s,t)$ by$$\\left( I_\\ell \\widetilde{C}^\\alpha _\\ell \\right)(s,t):=s^{\\frac{\\ell }{2}}\\widetilde{C}^\\alpha _\\ell \\left(\\frac{t}{\\sqrt{s}}\\right),$$where $ C(z)$ is the renormalized Gegenbauer polynomial given by$$\\widetilde{C}^\\alpha _\\ell (z):=\\frac{1}{\\Gamma \\left(\\alpha +\\left[\\frac{\\ell +1}{2}\\right]\\right)}\\sum _{k=0}^{\\left[\\frac{\\ell }{2}\\right]}(-1)^k\\frac{\\Gamma (\\ell -k+\\alpha )}{k!",
"(\\ell -2k)!",
"}(2z)^{\\ell -2k}.$$Then $ C(z)$ is a nonzero polynomial for all $ and $\\ell \\in \\mathbb {N}$ , and a (normalized) Juhl's conformally covariant operator $\\widetilde{_{\\lambda ,\\nu }: C^\\infty (\\mathbb {R}^n)\\longrightarrow C^\\infty (\\mathbb {R}^{n-1})is defined by\\widetilde{_{\\lambda ,\\nu }:=\\mathrm {Rest}_{x_n=0}\\circ \\left( I_\\ell \\widetilde{C}^{\\lambda -\\frac{n-1}{2}}_\\ell \\right)\\left(-\\Delta _{\\mathbb {R}^{n-1}},\\frac{\\partial }{\\partial x_n}\\right),for \\lambda ,\\nu \\in with \\ell :=\\nu -\\lambda \\in \\mathbb {N}.For instance,\\widetilde{_{\\lambda ,\\nu }=\\mathrm {Rest}_{x_n=0}\\circ {\\left\\lbrace \\begin{array}{ll}\\mathrm {id} & \\mathrm {if}\\; \\nu =\\lambda ,\\\\2\\frac{\\partial }{\\partial x_n} & \\mathrm {if}\\; \\nu =\\lambda +1,\\\\\\Delta _{\\mathbb {R}^{n-1}}+(2\\lambda -n+3)\\frac{\\partial ^2}{\\partial x_n^2}& \\mathrm {if}\\; \\nu =\\lambda +2.\\end{array}\\right.",
"}For (i,j,\\lambda ,\\nu )\\in \\Xi , we introduce a new family of matrix-valued differential operators\\widetilde{^{i,j}_{\\lambda ,\\nu }: C^\\infty (\\mathbb {R}^n,\\bigwedge ^i(n))\\longrightarrow C^\\infty (\\mathbb {R}^{n-1},\\bigwedge ^j({n-1})),by using the identifications \\mathcal {E}^i(\\mathbb {R}^n)\\simeq C^\\infty (\\mathbb {R}^n)\\otimes \\bigwedge ^i(n)and \\mathcal {E}^j(\\mathbb {R}^{n-1})\\simeq C^\\infty (\\mathbb {R}^{n-1})\\otimes \\bigwedge ^j({n-1}),as follows.",
"Let d^*_{\\mathbb {R}^n} be the codifferential,which is the formal adjoint of the differential d_{\\mathbb {R}^n},and \\iota _{\\frac{\\partial }{\\partial x_n}} the inner multiplication by the vector field\\frac{\\partial }{\\partial x_n}.",
"Both operators map \\mathcal {E}^i(\\mathbb {R}^n) to \\mathcal {E}^{i-1}(\\mathbb {R}^n).For \\alpha \\in and \\ell \\in \\mathbb {N}, let\\gamma (\\alpha ,\\ell ):=1 (\\ell is odd); =\\alpha +\\frac{\\ell }{2} ( \\ell is even).Then we set{\\begin{@align}{1}{-1}{i,i}_{\\lambda ,\\nu }&:=\\widetilde{_{\\lambda +1, \\nu -1} d_{\\mathbb {R}^n} d^*_{\\mathbb {R}^n}-\\gamma (\\lambda -\\frac{n}{2}, \\nu -\\lambda ) \\widetilde{_{\\lambda ,\\nu -1}d_{\\mathbb {R}^n}\\iota _{\\frac{\\partial }{\\partial x_n}}+ \\frac{1}{2}(\\nu -i)\\widetilde{_{\\lambda ,\\nu } \\nonumber \\qquad \\qquad \\qquad \\mathrm {for}\\; 0\\le i \\le n-1.\\\\{i,i-1}_{\\lambda ,\\nu }&:= - \\widetilde{\\mathcal {}}_{\\lambda +1, \\nu -1} d_{\\mathbb {R}^n}d^*_{\\mathbb {R}^n}\\iota _{\\frac{\\partial }{\\partial x_n}} -\\gamma (\\lambda -\\frac{n-1}{2}, \\nu -\\lambda )\\widetilde{_{\\lambda +1, \\nu }d^*_{\\mathbb {R}^n}+\\frac{1}{2}(\\lambda +i-n)\\widetilde{_{\\lambda ,\\nu } \\iota _{\\frac{\\partial }{\\partial x_n}}\\; \\mathrm {for}\\; 1\\le i \\le n.\\nonumber }}We note that there exist isolated parameters(\\lambda ,\\nu ) for which {i,i}_{\\lambda ,\\nu } =0or {i,i-1}_{\\lambda ,\\nu } =0.For instance, {0,0}_{\\lambda ,\\nu } = \\frac{1}{2}\\nu \\widetilde{_{\\lambda ,\\nu },and thus {0,0}_{\\lambda ,\\nu } = 0 if \\nu =0.To be precise, we have the following:\\begin{itemize}\\item []{i,i}_{\\lambda , \\nu } = 0if and only if \\lambda = \\nu =i or \\nu = i = 0;\\item []{i,i-1}_{\\lambda ,\\nu }=0 if and only if\\lambda =\\nu =n-i or \\nu = n-i = 0.\\end{itemize}}We renormalize these operators by\\begin{equation*}\\widetilde{^{i,i}_{\\lambda ,\\nu }:={\\left\\lbrace \\begin{array}{ll}\\mathrm {Rest}_{x_n=0} & \\text{if $\\lambda = \\nu $},\\\\\\widetilde{_{\\lambda ,\\nu } & \\text{if $i=0$},\\\\{i,i}_{\\lambda ,\\nu } & \\text{otherwise},}\\end{array}\\quad \\text{and}\\quad \\widetilde{^{i,i-1}_{\\lambda ,\\nu }:={\\left\\lbrace \\begin{array}{ll}\\mathrm {Rest}_{x_n=0} \\circ \\iota _{\\frac{\\partial }{\\partial x_n}} & \\text{if $\\lambda = \\nu $},\\\\\\widetilde{_{\\lambda ,\\nu } \\circ \\iota _{\\frac{\\partial }{\\partial x_n}} & \\text{if $i=n$},\\\\{i,i-1}_{\\lambda ,\\nu } & \\text{otherwise}.}\\end{array}\\right.",
"}Then \\widetilde{^{i,i}_{\\lambda ,\\nu } (0\\le i \\le n-1) and\\widetilde{^{i,i-1}_{\\lambda ,\\nu } (1 \\le i \\le n)are nonzero differential operators of order \\nu -\\lambda for any \\lambda , \\nu \\in with \\nu -\\lambda \\in \\mathbb {N}.",
"}The differential operators\\widetilde{^{i,i+1}_{\\lambda ,\\nu } and\\widetilde{^{i,i-2}_{\\lambda ,\\nu } aredefined only for special parameters (\\lambda ,\\nu ) as follows.",
"{\\begin{@align*}{1}{-1}\\widetilde{^{i,i+1}_{\\lambda ,i+1}&:={\\left\\lbrace \\begin{array}{ll}\\mathrm {Rest}_{x_n=0} \\circ d_{\\mathbb {R}^n}&\\mathrm {for}\\;1\\le i\\le n-2, \\lambda =i,\\\\d_{\\mathbb {R}^{n-1}} \\circ \\widetilde{_{\\lambda ,0}&\\mathrm {for}\\; i=0, \\lambda \\in -\\mathbb {N},}\\end{array}\\\\\\widetilde{^{i,i-2}_{\\lambda ,n-i+1}&:={\\left\\lbrace \\begin{array}{ll}\\mathrm {Rest}_{x_n=0}\\circ \\iota _{\\frac{\\partial }{\\partial x_n}} d^*_{\\mathbb {R}^n}&\\mathrm {for}\\; 2\\le i\\le n, \\lambda =n-i,\\\\-d^*_{\\mathbb {R}^{n-1}}\\circ {n,n-1}_{\\lambda ,0}&\\mathrm {for}\\; i=n, \\lambda \\in -\\mathbb {N}.\\end{array}\\right.",
"}}\\right.Then we give a complete solution to Problem \\ref {prob:2} forthe model space (X,Y)=(S^n, S^{n-1}) in terms of the flat picture ofprincipal series representations as follows:\\begin{thm}Suppose a 6-tuple (i,j,\\lambda ,\\nu ,\\delta ,\\varepsilon ) satisfies one of the equivalent conditions in Theorem \\ref {thm:A}.Then theoperators \\widetilde{^{i,j}_{\\lambda ,\\nu }: C^\\infty (\\mathbb {R}^n)\\otimes \\bigwedge ^i(n)\\longrightarrow C^\\infty (\\mathbb {R}^{n-1})\\otimes \\bigwedge ^j({n-1}) extend to differential SBOs I(i,\\lambda )_\\delta \\longrightarrow J(j,\\nu )_\\varepsilon , to be denoted by the same letters.",
"Conversely, anydifferential SBOfrom I(i,\\lambda )_\\delta to J(j,\\nu )_\\varepsilon is proportional to thefollowing differential operators:\\widetilde{^{i,i}_{\\lambda ,\\nu } in Case 1,\\widetilde{^{i,i-1}_{\\lambda ,\\nu } in Case 2,\\widetilde{^{i,i+1}_{i,i+1} in Case 3,\\widetilde{^{0,1}_{\\lambda ,1} in Case 3^{\\prime },\\widetilde{^{i,i-2}_{n-i,n-i+1} in Case 4, and\\widetilde{^{n,n-2}_{\\lambda ,1} in Case 4^{\\prime }.",
"}}\\section {Matrix-valued factorization identities}}Suppose thatT_X:I(i,\\lambda ^{\\prime })_\\delta \\rightarrow I(i,\\lambda )_\\delta or T_Y:J(j,\\nu )_\\varepsilon \\rightarrow J(j,\\nu ^{\\prime })_\\varepsilon are G- orG^{\\prime }-intertwining operators, respectively.",
"Thenthe composition T_Y\\circ D_{X\\rightarrow Y} or D_{X\\rightarrow Y}\\circ T_X of asymmetry breaking operator D_{X\\rightarrow Y}:I(i,\\lambda )_\\delta \\rightarrow J(j,\\nu )_\\varepsilon gives another symmetry breaking operator:{I(i,\\lambda )_\\delta [rr]^{D_{X\\rightarrow Y}}@{-->}[drr]&&J(j,\\nu )_\\varepsilon [d]_{T_Y}\\\\I(i,\\lambda ^{\\prime })_\\delta [u]^{T_X}@{-->}[urr]&&J(j,\\nu ^{\\prime })_\\varepsilon }}The multiplicity-free property (see Theorem \\ref {thm:A} (ii)) assures the existence of matrix-valued factorization identitiesfor differential SBOs, namely, D_{X\\rightarrow Y}\\circ T_X must be a scalar multiple of \\widetilde{_{\\lambda ^{\\prime },\\nu }^{i, j}, and T_Y\\circ D_{X\\rightarrow Y} must be a scalar multiple of \\widetilde{_{\\lambda ,\\nu ^{\\prime }}^{i, j}.We shall determine these constants explicitly when T_X or T_Y areBranson^{\\prime }s conformally covariant operators \\cite {Branson} defined below.Let 0\\le i\\le n. For \\ell \\in \\mathbb {N}_+, we set\\begin{equation}\\mathcal {T}_{2\\ell }^{(i)}:=((\\frac{n}{2}-i-\\ell )d_{\\mathbb {R}^{n}}d^*_{\\mathbb {R}^{n}}+(\\frac{n}{2}-i+\\ell )d^*_{\\mathbb {R}^{n}}d_{\\mathbb {R}^{n}})\\Delta ^{\\ell -1}_{\\mathbb {R}^n}=(-2\\ell \\,d_{\\mathbb {R}^{n}}d^*_{\\mathbb {R}^{n}}-(\\frac{n}{2}-i+\\ell )\\Delta _{\\mathbb {R}^n})\\Delta _{\\mathbb {R}^n}^{\\ell -1}\\nonumber .\\end{equation}}Then the differential operator\\mathcal {T}_{2\\ell }^{(i)}:\\mathcal {E}^i(\\mathbb {R}^n)\\longrightarrow \\mathcal {E}^i(\\mathbb {R}^n)induces a nonzero O(n+1,1)-intertwining operator, to be denoted by the same letter \\mathcal {T}_{2\\ell }^{(i)}, from I\\left(i,\\frac{n}{2}-\\ell \\right)_\\delta to I\\left(i,\\frac{n}{2}+\\ell \\right)_\\delta , for \\delta \\in \\mathbb {Z}/2\\mathbb {Z}.Similarly, we define a G^{\\prime }-intertwining operator\\mathcal {T^{\\prime }}_{2\\ell }^{(j)}:J\\left(j,\\frac{n-1}{2}-\\ell \\right)_\\varepsilon \\longrightarrow J\\left(j,\\frac{n-1}{2}+\\ell \\right)_\\varepsilon for 0\\le j\\le n-1 and \\varepsilon \\in \\mathbb {Z}/2\\mathbb {Z}as the lift of the differential operator \\mathcal {T^{\\prime }}_{2\\ell }^{(j)}:\\mathcal {E}^j(\\mathbb {R}^{n-1})\\longrightarrow \\mathcal {E}^j(\\mathbb {R}^{n-1}) which is given by\\begin{equation*}\\mathcal {T^{\\prime }}_{2\\ell }^{(j)}=((\\frac{ n-1}{2}-j-\\ell )d_{\\mathbb {R}^{n-1}}d_{\\mathbb {R}^{n-1}}^*+(\\frac{ n-1}{2}-j+\\ell )d_{\\mathbb {R}^{n-1}}^*d_{\\mathbb {R}^{n-1}})\\Delta ^{\\ell -1}_{\\mathbb {R}^{n-1}}.\\end{equation*}}Consider the following diagrams for j=i and j=i-1:\\begin{eqnarray*}{I\\left(i,\\frac{n}{2}-\\ell \\right)_\\delta [d]_{\\mathcal {T}_{2\\ell }^{(i)}}[drr]^{{\\widetilde{}^{i,j}_{\\frac{n}{2}-\\ell ,\\frac{n}{2}+a+\\ell }}&& \\\\I\\left(i,\\frac{n}{2}+\\ell \\right)_\\delta [rr]_{{\\widetilde{}^{i,j}_{\\frac{n}{2}+\\ell ,\\frac{n}{2}+a+\\ell }}&&J\\left(j,\\frac{n}{2}+a+\\ell \\right)_\\varepsilon ,}&&{I\\left(i,\\frac{n-1}{2}-a-\\ell \\right)_\\delta [rr]^{{\\widetilde{}^{i, j}_{\\frac{n-1}{2}-a-\\ell ,\\frac{n-1}{2}-\\ell }}[drr]_{{\\widetilde{}^{i, j}_{\\frac{n-1}{2}-a-\\ell ,\\frac{n-1}{2}+\\ell }}&&J\\left(j,\\frac{n-1}{2}-\\ell \\right)_\\varepsilon [d]^{{\\mathcal {T}^{\\prime }}^{(j)}_{2\\ell }}\\\\&&J\\left(j,\\frac{n-1}{2}+\\ell \\right)_\\varepsilon ,}}where parameters \\delta and \\varepsilon \\in \\mathbb {Z}/2\\mathbb {Z} are chosen according to Theorem \\ref {thm:A} (iii).In what follows, we putp_\\pm ={\\left\\lbrace \\begin{array}{ll}i\\pm \\ell -\\frac{n}{2}&\\mathrm {if}\\; a\\ne 0\\\\\\pm 2&\\mathrm {if}\\; a=0\\end{array}\\right.",
"},\\quad q={\\left\\lbrace \\begin{array}{ll}i+\\ell -\\frac{n-1}{2}&\\mathrm {if}\\;i\\ne 0, a\\ne 0\\\\-2&\\mathrm {if}\\; i\\ne 0, a=0\\\\-\\left(\\ell +\\frac{n-1}{2}\\right)&\\mathrm {if}\\; i=0\\end{array}\\right.",
"},\\quad r={\\left\\lbrace \\begin{array}{ll}i-\\ell -\\frac{n+1}{2}&\\mathrm {if}\\;i\\ne n, a\\ne 0\\\\2&\\mathrm {if}\\; i\\ne n, a=0\\\\-\\left(\\ell +\\frac{n+1}{2}\\right)&\\mathrm {if}\\; i=n\\end{array}\\right.",
"},\\begin{equation*}K_{\\ell ,a}:=\\prod _{k=1}^\\ell \\left(\\left[\\frac{a}{2}\\right]+k\\right).\\end{equation*}}Then the factorization identities for differential SBOs \\widetilde{^{i,j}_{\\lambda ,\\nu } for j\\in \\lbrace i-1,i\\rbrace andBranson^{\\prime }s conformally covariant operators \\mathcal {T}_{2\\ell }^{(i)} or \\mathcal {T^{\\prime }}_{2\\ell }^{(j)} are given as follows.\\begin{thm} Suppose 0\\le i\\le n-1, a\\in \\mathbb {N} and \\ell \\in \\mathbb {N}_+.Then\\begin{eqnarray*}&(1)&{\\widetilde{}^{i,i}_{\\frac{n}{2}+\\ell ,a+\\ell +\\frac{n}{2}}\\circ \\mathcal {T}_{2\\ell }^{(i)}=p_-K_{\\ell ,a}{\\widetilde{}^{i, i}_{\\frac{n}{2}-\\ell ,a+\\ell +\\frac{n}{2}}.\\\\&(2)& \\mathcal {T^{\\prime }}_{2\\ell }^{(i)}\\circ {\\widetilde{}^{i,i}_{\\frac{n-1}{2}-a-\\ell ,\\frac{n-1}{2}-\\ell }=qK_{\\ell ,a}{\\widetilde{}^{i,i}_{\\frac{n-1}{2}-a-\\ell ,\\frac{n-1}{2}+\\ell }.",
"}}}\\begin{thm}Suppose 1\\le i\\le n, a\\in \\mathbb {N} and \\ell \\in \\mathbb {N}_+.",
"Then\\begin{eqnarray*}&(1)&{\\widetilde{}^{i,i-1}_{\\frac{n}{2}+\\ell ,a+\\ell +\\frac{n}{2}}\\circ \\mathcal {T}_{2\\ell }^{(i)}=p_+K_{\\ell ,a}{\\widetilde{}^{i,i-1}_{\\frac{n}{2}-\\ell ,a+\\ell +\\frac{n}{2}}.\\\\&(2)&\\mathcal {T^{\\prime }}_{2\\ell }^{(i-1)}\\circ {\\widetilde{}^{i,i-1}_{\\frac{n-1}{2}-a-\\ell ,\\frac{n-1}{2}-\\ell }=rK_{\\ell ,a}{\\widetilde{}^{i,i-1}_{\\frac{n-1}{2}-a-\\ell ,\\frac{n-1}{2}+\\ell }.\\\\}}In the case where i=0, {\\widetilde{}^{i, i}_{\\lambda ,\\nu } isa scalar-valued operator, and the corresponding factorization identities in Theorem \\ref {thm:factor1} were studied in \\cite {Juhl,KS13,KOSS15}.",
"}The main results are proved by using the F-method \\cite {K14,KP1,KOSS15}.Details will appear elsewhere.",
"}\\begin{thebibliography}{99}\\end{thebibliography}\\bibitem {Branson} T.~P.",
"Branson,Conformally covariant equations on differential forms,\\emph {Comm.",
"Part.",
"Diff.",
"Eq.}",
"\\textbf {7}, (1982),\\href {http://dx.doi.org/10.1080/03605308208820228}{pp.",
"393--431}.",
"}\\bibitem {FT}E.~S.",
"Fradkin, A.~A.",
"Tseytlin,Asymptotic freedom in extended conformal supergravities,\\emph {Phys.",
"Lett.",
"B} \\textbf {110}, (1982)\\href {http://dx.doi.org/10.1016/0370-2693(82)91018-8}{pp.",
"117--122}.\\end{eqnarray*}\\bibitem {GJMS}C.~R.",
"Graham, R. Jenne, L.~J.",
"Mason, G.~A.~J.",
"Sparling,Conformally invariant powers of the Laplacian.",
"I. Existence.\\emph {J. London Math.",
"Soc.}",
"(2) \\textbf {46} (1992),\\href {http://dx.doi.org/10.1112/jlms/s2-46.3.557}{pp.",
"557--565}.\\end{thm}\\bibitem {Juhl}A. Juhl, \\emph { Families of conformally covariant differential operators, Q-curvature and holography.}",
"Progr.",
"Math.,\\href {http://link.springer.com/book/10.1007/978-3-7643-9900-9/page/1}{\\bf {275}}.",
"Birkhäuser, Basel, 2009.",
"}\\bibitem {K14} T.~Kobayashi,F-method for symmetry breaking operators,\\emph {Diff.",
"Geometry and its Appl.",
"}{\\textbf {33}}, (2014),\\href {http://dx.doi.org/10.1016/j.difgeo.2013.10.003}{pp.",
"272--289}.\\end{eqnarray*}\\bibitem {KP1}T.~Kobayashi, M.~Pevzner,Differential symmetry breaking operators.",
"I.",
"General theory andF-method,\\emph {Selecta.",
"Math.",
"(N.S.",
")}, \\textbf {22}, (2016),\\href {http://dx.doi.org/10.1007/s00029-015-0207-9}{pp.",
"801--845}.\\end{thm}\\bibitem {KP2}T.~Kobayashi, M.~Pevzner, Differential symmetry breaking operators.II.",
"Rankin--Cohen operators for symmetric pairs,\\emph {Selecta.",
"Math.",
"(N.S.",
")}, \\textbf {22}, (2016),\\href {http://dx.doi.org/10.1007/s00029-015-0208-8}{pp.",
"847--911}.",
"}\\bibitem {KS13}T.~Kobayashi, B.~Speh,Symmetry Breaking for Representations of Rank One Orthogonal Groups,\\href {http://dx.doi.org/10.1090/memo/1126}{Memoirs of American Mathematical Society, vol.",
"\\textbf {238}, 2015.}",
"118 pp.",
"{ISBN: 978-1-4704-1922-6}.",
"}\\bibitem {KOSS15}T.~Kobayashi, B.~Ørsted, P. Somberg, and V. Souček, Branching laws forVerma modules and applications in parabolic geometry.",
"I.",
"\\emph {Adv.",
"Math}., \\textbf {285},(2015),\\href {http://dx.doi.org/10.1016/j.aim.2015.08.020}{pp.",
"1796--1852}.",
"}\\bibitem {KW14}B.~Kostant, N. R.~Wallach, Action of the conformal group on steady state solutions to Maxwell^{\\prime }s equations and background radiation.",
"Symmetry: representation theory and its applications,\\href {http://dx.doi.org/10.1007/978-1-4939-1590-3_14}{pp.",
"385--418},\\emph {Progr.",
"Math.",
"},{\\textbf {257}}, Birkhäuser/Springer, New York, 2014.\\end{eqnarray*}\\bibitem {P08}S. Paneitz, A quartic conformally covariant differential operatorfor arbitrary pseudo-Riemannian manifolds,\\emph {SIGMA Symmetry Integrability Geom.",
"Methods Appl.",
"}\\href {http://www.emis.de/journals/SIGMA/2008/}{\\textbf {4} (2008)}, paper 036, 3 pp.",
"}}\\vspace{10.0pt}}\\footnotesize { Addresses:T. Kobayashi.",
"Kavli IPMU (WPI)and Graduate School of Mathematical Sciences,The University of Tokyo, 3-8-1 Komaba, Meguro, Tokyo, 153-8914 Japan;\\texttt {[email protected]}.\\vspace{5.0pt}}T. Kubo.",
"Graduate School of Mathematical Sciences, The University ofTokyo,3-8-1 Komaba, Meguro, Tokyo, 153-8914 Japan; \\texttt {{[email protected]}}.\\vspace{5.0pt}\\end{thm}}M. Pevzner.Laboratoire de Mathématiques de Reims, Universitéde Reims-Champagne-Ardenne, FR 3399 CNRS, F-51687, Reims, France; \\texttt {{[email protected].}}}\\end{@align*}}}}}}\\right.",
"}}\\end{equation*}}}}\\end{@align}}}}}}$"
]
] | 1605.05722 |
[
[
"Generalized Charged Nariai Solutions in Arbitrary Even Dimensions with\n Multiple Magnetic Charges"
],
[
"Abstract Higher-dimensional solutions for Einstein-Maxwell equations that generalize the charged Nariai spacetime are obtained.",
"The solutions presented here are made from the direct product of several 2-spaces of constant curvature.",
"These solutions turn out to have many magnetic charges, contrary to the usual higher-dimensional generalization of the Nariai spacetime, which has no magnetic charge at all.",
"These solutions are then used to generate black hole metrics.",
"Finally, it is analyzed how these generalized Nariai solutions are modified in more general theories of gravity."
],
[
"Introduction",
"The use of higher-dimensional spacetimes to explain our physical world has a long history in physics.",
"Indeed, even prior to the publication of the final form of general relativity, G. Nordström made use of a five-dimensional spacetime in order to unify gravity and electromagnetism into a single scheme [1].",
"Few years later, under the light of general relativity, T. Kaluza and O. Klein made this possibility clearer in influential works whose concepts are of relevance until now [2].",
"However, the idea that the universe can have more that four dimensions started to spread and be considered by a bigger community around the 70s, when string theory arose as a possibility of providing a quantum theory of gravity [3].",
"According to the current interpretation, the reason why we are not used to notice the six extra dimensions necessary in string theory is that they are very small and, therefore, can only be probed through highly energetic experiments.",
"Nevertheless, from the theoretical point of view, it is also possible to reconcile our impression that we live in a four-dimensional world with the existence of large extra dimensions [4], [5].",
"In this scenario, the interactions of the standard model are restricted to a four-dimensional brane while gravity permeates all dimensions.",
"Even infinity extra dimensions are not necessarily in contradiction with our daily experiences [6], [5].",
"Nevertheless, it is fair to say that astrophysical and earth-based high energy experiments have put huge constraints in the possibility of existing extra dimensions of sizes much greater than the Planck length [7], [8].",
"Nowadays, another important source of attention for higher-dimensional spacetimes is the AdS/CFT correspondence [9], which relates a gravitational theory in the bulk of an $n$ -dimensional spacetime with a conformal field theory in the $(n-1)$ -dimensional boundary.",
"Due to all these branches of physics that make use of higher-dimensional spacetimes, it is increasing the amount of research in such subject.",
"The Schwarzschild-de Sitter black hole is not in thermodynamical equilibrium, since the temperatures of the black hole horizon and cosmological horizon are not the same.",
"As the limit of equal temperatures is taken, the two horizons approach each other and the outcome of this limit process is the Nariai spacetime [10], [11], a solution first found in Ref.",
"[12].",
"The Nariai metric is a four-dimensional vacuum solution of Einstein's field equation in the presence of a positive cosmological constant that is the direct product of two spaces of constant curvature, namely $dS_2$ and $S^2$ .",
"Analogously, the so-called anti-Nariai, Bertotti-Robinson and Plabański-Hacyan solutions are other vacuum solutions formed by the direct product of two spaces of constant curvature [13], [14], [15].",
"For an account of impulsive gravitational waves in these spacetimes, see [15], [16], while the thermodynamics of the Nariai spacetime have been considered in Ref.",
"[17].",
"An electrically charged higher-dimensional generalization of the Nariai spacetime have been obtained in Ref.",
"[18].",
"This solution of Einstein-Maxwell equations is the direct product of the spaces $dS_2$ and $S^{n-2}$ , and can be obtained from Schwarzschild-Tangherlini black hole in the limit of equal temperatures of the horizons, as proved in [19].",
"Here, we will present a different higher-dimensional generalization of the Nariai solution that is formed from the direct product of $dS_2$ with several 2-spheres possessing different radii.",
"One interesting feature of the latter solution is that, besides having an electric charge, it also admits several magnetic charges, diversely from the solutions obtained in Refs.",
"[18], [19], which have no magnetic charge at all.",
"In addition, we will investigate whether these new generalized Nariai spacetimes are associated to black holes and if they have counterparts in theories of gravity that are more general than Einstein's theory.",
"The outline of the article is the following.",
"In Sec.",
", the new higher-dimensional generalizations of Nariai solutions are presented and some of their geometrical and physical properties are investigated.",
"Moreover, we also obtain higher-dimensional extensions of anti-Nariai, Bertotti-Robinson and Plabański-Hacyan solutions.",
"Then, in Sec.",
", we use the generalized Nariai metrics to obtain black hole solutions for Einstein-Maxwell equations.",
"In Sec.",
", we investigate metrics made from the product of 2-spaces of constant curvature that are solutions for more general gravitational theories coupled to an electromagnetic field through a minimal coupling.",
"Finally, in Sec.",
", we summarize the results and discuss some perspectives."
],
[
"Generalized Charged Nariai Solutions in Einstein's Theory",
"In this starting section we are interested in obtaining solutions for a gravitational system interacting, via minimal coupling, with an electromagnetic field and a cosmological constant $\\Lambda $ .",
"The system is described by the following action $\\mathcal {S} \\,=\\,\\int \\sqrt{-g} \\left[ \\mathcal {R} \\,-\\,(n-2) \\, \\Lambda \\,-\\,\\frac{1}{4}\\,\\mathcal {F}^{cd}\\mathcal {F}_{cd} \\right] \\,,$ where $\\mathcal {F}_{ab} = \\nabla _{[a}\\mathcal {A}_{b]}$ is the electromagnetic field, $\\mathcal {R}$ stands for the Ricci scalar and $n$ is the dimension of the spacetime.",
"The field equations of this system are: $\\mathcal {R}_{ab} - \\frac{1}{2}\\,\\mathcal {F}_a^{\\phantom{a}c}\\mathcal {F}_{bc} =\\frac{g_{ab}}{2} \\left[ \\mathcal {R} - (n-2) \\Lambda - \\frac{1}{4}\\,\\mathcal {F}^{cd}\\mathcal {F}_{cd} \\right] \\nonumber \\\\\\quad \\nonumber \\\\\\textrm {and} \\quad \\nabla ^a\\mathcal {F}_{ab} \\, = \\, 0 \\,.",
"$ In order to write the solutions that will be presented in the sequel, it is useful to adopt the notation $d\\Omega _j^2 = d\\theta _j^2 \\,+\\,\\sin ^2\\theta _j\\,d\\phi _j^2$ to represent the line element of the unit sphere $S^2$ .",
"In what follows, we will assume that the dimension $n$ is even and that the index $j$ ranges from 2 to $n/2$ .",
"The first result presented here is that a static solution for the equations of motion (REF ) is provided by the following fields: $ds^2 = R_1^2( - \\sin ^2x\\,dt^2 + dx^2 ) + \\sum _{j=2}^{n/2}\\,R_j^2\\,d\\Omega _j^2 \\,, \\\\\\mathcal {F} = q_1 R_1^2 \\sin x \\, dt\\wedge dx + \\sum _{j=2}^{n/2}\\,q_j R_j^2 \\sin \\theta _j \\,d\\phi _j\\wedge d\\theta _j , \\nonumber $ where $q_1$ is an electric charge and $q_j$ are magnetic charges.",
"The radii $R_1$ and $R_j$ are real constants that are related to the charges and the cosmological constants by the subsequent relations: $R_1 =& \\left[\\, \\Lambda \\,-\\,\\frac{1}{2} q_1^2 \\,+\\,\\frac{Q}{2(n-2)} \\,\\right]^{-1/2} \\;, \\\\R_j =& \\left[\\, \\Lambda \\,+\\,\\frac{1}{2} q_j^2 \\,+\\,\\frac{Q}{2(n-2)} \\,\\right]^{-1/2} \\;, $ where $Q \\, \\equiv \\, q_1^2 \\,-\\,\\sum _{j=2}^{n/2}\\, q_j^2 \\,.$ One can check that in order for the radii to be real, which assures the Lorentzian signature, the following constraint must be satisfied by the cosmological constant: $2\\,(n-2)\\,\\Lambda \\, \\ge \\, (n-3) \\, q_1^2 \\,+\\,\\sum _{j=2}^{n/2}\\, q_j^2 \\,.$ In particular, the cosmological constant must be positive.",
"If $n\\ne 4$ , the relations (REF ) and () can be inverted to write the charges $q_1$ and $q_j$ in terms of the radii, the final result being $q_1 =& \\sqrt{ \\,\\, \\frac{4\\,(n-2)}{(n-4)} \\Lambda \\,-\\,\\frac{4}{(n-4)R_0^2} \\,-\\,\\frac{2}{R_1^2} \\,} \\;, \\\\q_j =& \\sqrt{ -\\, \\frac{4\\,(n-2)}{(n-4)} \\Lambda \\,+\\,\\frac{4}{(n-4)R_0^2} \\,+\\,\\frac{2}{R_j^2} \\,} \\;, $ with $R_0$ being defined by $\\frac{1}{R_0^2} \\,\\equiv \\, \\frac{1}{R_1^2} \\,+\\,\\sum _{j=2}^{n/2}\\, \\frac{1}{R_j^2} \\,.$ Thus, instead of considering the electromagnetic charges as being arbitrary, we can, equivalently, suppose that the radii $R_1$ and $R_j$ are arbitrary, while the charges are determined in terms of these radii by Eqs.",
"(REF ) and ().",
"This is quite interesting from the point of view of compactification of extra dimensions, since we can set $R_1$ and $R_2$ to be much bigger than the other radii.",
"However, this freedom of choosing either $\\lbrace q_1,q_j\\rbrace $ or $\\lbrace R_1, R_j\\rbrace $ as being the independent parameters is not valid in the particular case $n=4$ , as can already be grasped from Eqs.",
"(REF ) and (), which diverge in such a case.",
"Indeed, by means of Eqs.",
"(REF ) and () one can see that if $n=4$ the radii $R_1$ and $R_2$ are not independent from each other, they must obey the constraint $\\frac{1}{R_1^2} \\,+\\,\\frac{1}{R_2^2} \\,=\\,2 \\, \\Lambda \\,.$ Thus, when $n=4$ the solution presented here reduces to the well-known charged Nariai solution [12].",
"Therefore, the metric (REF ) should be seen as a higher-dimensional generalization of the Nariai solution.",
"Just as the Nariai solution, the higher-dimensional version of the Nariai spacetime presented here has no singularities.",
"Indeed, its Riemann tensor and its Weyl tensor are both covariantly constant and, therefore, all invariant scalars that can be constructed from the curvature are constant.",
"For instance, the full contractions of the products of the Riemann tensor are given by $\\mathcal {R}^{a_1b_1}_{\\phantom{a_1b_1}a_2b_2} \\cdots \\mathcal {R}^{a_nb_n}_{\\phantom{a_nb_n}a_1b_1}= 2^n\\,\\left[ \\frac{1}{R_1^{2n}} + \\sum _{j=2}^{n/2}\\,\\frac{1}{R_j^{2n}} \\right] \\nonumber \\,.$ These generalized Nariai spacetimes are contained in the Kundt class of metrics [20], inasmuch as the null directions $\\ell = \\csc x \\,\\partial _t + \\partial _x \\quad \\textrm {and} \\quad n = \\csc x \\,\\partial _t - \\partial _x$ are geodesic, shear-free, twist-free and expansion-free.",
"They are also repeated principal null directions of the Weyl tensor [21], [22], $C_{ab[cd}\\,\\ell _{e]} \\ell ^b = 0 = C_{ab[cd}\\, n_{e]} n^b \\,,$ so that, according to the boost weight classification [23], [24], the algebraic type of the Weyl tensor is $D$ .",
"These properties are also in agreement with the four-dimensional Nariai solution.",
"Using the coordinates $\\tau =R_1\\,t$ and $r = R_1 \\cos x$ , the line element of the generalized Nariai spacetime is written as $ds^2 = -\\,h(r)\\,d\\tau ^2 + \\frac{1}{h(r)} dr^2 + \\sum _{j=2}^{n/2}\\,R_j^2\\,d\\Omega _j^2 \\,,$ where $h(r) = 1 - r^2/R_1^2$ .",
"In these coordinates, it is clear that the hyper-surfaces $r=\\pm R_1$ are closed null surfaces and comprise event horizons.",
"The entropies of these horizons are given by one quarter of their area, $S \\,=\\,\\frac{1}{4}\\, \\prod _{j=2}^{n/2}\\,4\\,\\pi \\,R_j^2 \\,.$ These null hyper-surfaces are Killing horizons associated to the Killing vector $\\partial _\\tau $ .",
"In order to calculate the temperature of such horizons, one must compute the surface acceleration of the of this Killing vector, $\\kappa = \\left.",
"\\sqrt{-\\,\\frac{1}{2}\\, \\nabla ^a \\xi ^b \\,\\nabla _a \\xi _b } \\, \\right|_{r=r_h} \\,,$ where $r_h$ is the value of the coordinate $r$ at the horizon and $\\xi ^a$ is the Killing vector field properly normalised.",
"A natural way to normalize $\\xi = \\lambda \\, \\partial _\\tau $ is to choose the multiplicative constant $\\lambda $ in such a way that $\\xi ^a\\xi _a = -1$ at the value of $r=r_{\\star }$ for which $\\xi $ is a geodesic vector field, just as happens at the asymptotic infinity of an asymptotically flat spacetime [11].",
"At $r=r_{\\star }$ the gravitational, electromagnetic and cosmological forces balance each other and the observer can stay still without acceleration.",
"Computing the Christoffel symbol $\\Gamma ^a_{\\tau \\tau }$ , we see that this value of $r$ is the one for which $h^{\\prime }(r_{\\star }) = 0$ , with $h^{\\prime }(r)$ standing for the derivative of $h(r)$ with respect to $r$ .",
"Therefore, the normalized Killing vector field and the temperature of the horizon are respectively given by $ \\xi = \\frac{1}{\\sqrt{h(r_{\\star })}}\\, \\partial _\\tau \\quad \\textrm {and} \\quad T \\,=\\,\\frac{\\kappa }{2\\pi } = \\frac{|h^{\\prime }(r_h)|}{4\\,\\pi \\,\\sqrt{h(r_{\\star })} }$ In the case of the metric considered here, we have $r_h= \\pm R_1$ and $r_{\\star } = 0$ , so that the temperature of the horizons of the generalized Nariai solutions considered here is $T = \\frac{1}{2\\,\\pi \\,R_1} \\,.$ This is the temperature measured by a still observer at $r=0$ .",
"Of course, other observers should measure different temperatures.",
"For a nice account on the thermodynamics of the four-dimensional Nariai spacetime, the reader is referred to Ref.",
"[17].",
"We have seen that, due to Eq.",
"(REF ), these generalized Nariai solutions exist only for positive cosmological constant.",
"Nonetheless, it is also possible to make analytical continuations on the coordinates and find the analogue of the latter solution in the case of negative cosmological constant, obtaining a generalized anti-Nariai solution.",
"Indeed, the line element (REF ) is the direct product of the de Sitter space $dS_2$ with $(\\frac{n}{2} -1)$ copies of the sphere $S^2$ , which are both two-dimensional spaces of constant positive curvatures.",
"Thus, in an analogous fashion, we can find a solution whose metric is the product of spaces of constant negative curvatures, namely the product of anti-de Sitter space $AdS_2$ with $(\\frac{n}{2} -1)$ copies of a hyperboloid $H^2$ , in which case the cosmological constant can be negative.",
"Actually, we can go one step further and consider the ansatz of a spacetime whose metric is the direct product of 2-spaces of arbitrary constant curvature, which provides a generalization of Bertotti-Robinson and Plebański-Hacyan four-dimensional solutions [13], [14].",
"In order to accomplish this goal it is useful to define the following function depending on a discrete parameter $\\epsilon $ : $S_{\\epsilon }(\\theta ) = \\frac{1}{\\sqrt{\\epsilon }}\\,\\sin (\\sqrt{\\epsilon } \\,\\theta ) =\\left\\lbrace \\begin{array}{ll}\\sin \\theta \\,,\\; &\\textrm {if \\epsilon = 1} \\\\\\;\\;\\theta \\,\\,,\\; &\\textrm {if \\epsilon = 0} \\\\\\sinh \\theta \\,,\\; &\\textrm {if \\epsilon = -1}\\end{array}\\right.",
".$ Then, the line element of a constant curvature two-dimensional space can be conveniently written as $d\\tilde{\\Omega }_{\\epsilon _j}^2 = d\\theta _j^2 \\,+\\,S_{\\epsilon _j}(\\theta _j)^2\\,d\\phi _j^2 \\,,$ with the case $\\epsilon _j = 1$ representing a sphere, the plane being represented by $\\epsilon _j = 0$ , while the hyperboloid corresponds to $\\epsilon _j = -1$ .",
"Then, we shall seek for solutions whose line elements are given by $ds^2 = R_1^2\\left[ - S_{\\epsilon _1}(x)^2\\,dt^2 + dx^2 \\right] + \\sum _{j=2}^{n/2}\\,R_j^2 \\, d\\tilde{\\Omega }_{\\epsilon _j}^2 \\,.$ One can check that metric (REF ) is a solution of the field equations (REF ) when $n\\ge 6$ provided that the electromagnetic field is given by $ \\mathcal {F} = q_1 \\, R_1^2 \\, S_{\\epsilon _1}(x) \\, dt\\wedge dx + \\sum _{j=2}^{n/2}\\,q_j\\, R_j^2\\, S_{\\epsilon _j}(\\theta _j) \\,d\\phi _j\\wedge d\\theta _j \\,, $ where the electric charge $q_1$ and the magnetic charges $q_j$ might be given by the following relations respectively $q_1 =& \\sqrt{ \\,\\, \\frac{4\\,(n-2)}{(n-4)} \\Lambda \\,-\\,\\frac{4}{(n-4)\\tilde{R}_0^2} \\,-\\,\\frac{2\\,\\epsilon _1}{R_1^2} \\,} \\;, \\\\q_j =& \\sqrt{ -\\,\\frac{4\\,(n-2)}{(n-4)} \\Lambda \\,+\\,\\frac{4}{(n-4)\\tilde{R}_0^2} \\,+\\,\\frac{2\\,\\epsilon _j}{R_j^2} \\,} \\;.",
"$ The constant $\\tilde{R}_0$ used in the latter expression is defined by $\\frac{1}{\\tilde{R}_0^2} \\,\\equiv \\, \\frac{\\epsilon _1}{R_1^2} \\,+\\,\\sum _{j=2}^{n/2}\\, \\frac{\\epsilon _j}{R_j^2} \\,.$ Now, let us define the following 1-forms constituting a local basis: $\\left.\\begin{array}{ll}e_1 \\,=\\,R_1\\, S_{\\epsilon _1}(x)\\, \\,dt \\;\\;&, \\quad \\tilde{e}_{1} = R_1\\, dx \\\\e_j = R_j\\, S_{\\epsilon _j}(\\theta _j) d\\phi _j \\;\\; &, \\quad \\tilde{e}_{j} = R_j\\, d\\theta _j \\,.\\end{array}\\right.$ Then, the vector fields associated to these 1-forms by means of the metric comprise a Lorentz frame.",
"In terms of such basis, the electromagnetic field is given by $ \\mathcal {F} = q_1 \\,e_1\\wedge \\tilde{e}_{1} \\,+\\,\\sum _{j=2}^{n/2}\\,q_j\\,e_j\\wedge \\tilde{e}_{j} \\,.",
"$ Since the components of $\\mathcal {F}$ in this Lorentz frame are constant, we interpret it as a uniform electromagnetic field throughout the spacetime.",
"From Eqs.",
"(REF ) and (), one can grasp that if either $\\epsilon _1$ or some $\\epsilon _j$ vanish then these equations cannot be inverted to write the radii $R_1$ and $R_j$ in terms of the charges $q_1$ and $q_j$ .",
"For example, let us say that $\\epsilon _1 = 0$ , then the expressions of $q_1$ and $q_j$ do not depend on $R_1$ , so that, generally, we cannot write $R_1$ as a function of the electromagnetic charges.",
"Nevertheless, if none of the 2-spaces that form the metric is flat, namely if $\\epsilon _1$ and $\\epsilon _j$ are all different from zero, then Eqs.",
"(REF ) and () can be inverted and solved for the radii $R_1$ and $R_j$ in terms of the charges $q_1$ and $q_j$ .",
"In the latter case, the electromagnetic charges can be arbitrarily assigned to the solution.",
"A different generalization of the charged Nariai solution to higher dimensions has also been found elsewhere [18], [19], but such spacetime is the direct product of just two spaces of constant curvature, namely $AdS_2\\times S^{n-2}$ .",
"On the other hand, the metric presented here is the direct product of several 2-spaces of constant curvature.",
"It is also worth pointing out that whereas the solution presented in Refs.",
"[18], [19] admits no magnetic charge at all, which is related to the fact that the second Betti number of the sphere $S^{n-2}$ is different from zero only if $n=4$ [26], the solution obtained here has several magnetic charges.",
"This interesting feature makes these metrics a rich arena for studying the physics of higher-dimensional spacetimes.",
"After the release of the pre-print of this article, Marcello Ortaggio kindly warned that the purely magnetic case of the solutions presented here, namely when $q_1=0$ , is contained in the broad class of solutions obtained in Ref.",
"[25]."
],
[
"Associated Black Hole Solutions ",
"It is well-known that the four-dimensional Nariai solution can be obtained from the Schwarzschild-de Sitter metric in the limit that the temperature of the black hole horizon coincides with the temperature of the cosmological horizon.",
"Indeed, the Nariai solution is the metric perceived by an observer between the two horizons as they coalesce into a single hyper-surface.",
"Since the coordinate range between the two horizons shrinks to zero as the temperatures of the horizons approach each other, a careful near horizon limit must be taken in order to obtain Nariai metric [10], [11].",
"Therefore, it is natural to think that a similar approach works in higher dimensions.",
"Indeed, this was the path taken in Ref.",
"[19] to obtain another higher-dimensional generalization of the Nariai solution.",
"In the latter reference, the limit of equal temperatures of the horizons of the Schwarzschild-Tangherlini solution [33] have been taken and the final result was a spacetime of the form $AdS_2\\times S^{n-2}$ .",
"Here, we already have a higher-dimensional generalization of the Nariai solution.",
"Thus, we could think the other way around: are the generalized Nariai metrics obtained in this article related to black hole solutions?",
"In order to answer this question, let us investigate whether it is possible to attain the metric (REF ) from a black hole solution with coalescing horizons.",
"Let us start with the following general static black hole ansatz that has the same topology of the spacetime presented in Eq.",
"(REF ): $ds^2 = -\\, f(r) \\, d\\tilde{t}^{\\,2} + \\frac{1}{f(r)}\\, dr^2 + \\sum _{j=2}^{n/2}\\,R_j(r)^2\\,d\\Omega _j^2 \\,.",
"$ Then, we take the limit in which two horizons have the same temperature while the horizons coalesce into a single surface $r=r_h$ .",
"The hallmark of a degenerate horizon is that $f(r)$ and $f^{\\prime }(r)$ both vanish at the horizon, so that near the horizon we can write $ f(r_h + \\rho ) \\,=\\,- \\, \\lambda \\, \\rho ^2 + O(\\rho ^3) \\,, $ where $\\lambda $ is some non-vanishing constant.",
"Therefore, using $\\rho = r - r_h$ as a coordinate while we take the near horizon limit, the general ansatz (REF ) becomes $ds^2 = \\lambda \\, \\rho ^2 \\, d\\tilde{t}^{\\,2} - \\frac{1}{\\lambda \\,\\rho ^2}\\, d\\rho ^2 + \\sum _{j=2}^{n/2}\\,R_j(r_h)^2\\,d\\Omega _j^2 \\,.",
"$ Then, defining $\\tilde{t}= \\lambda ^{-1}\\,e^{-t}\\, \\cot x$ , $\\rho = e^{t}\\,\\sin x$ , $R_j= R_j(r_h)$ and $R_1= \\lambda ^{-1/2}$ , we retrieve the generalized Nariai spacetime (REF ).",
"Therefore, we conclude that the generalized Nariai solution presented here induces the search for static black hole solutions of the form (REF ), which we do in the sequel.",
"Before proceeding, it is useful to introduce the following notation: $ \\mathcal {E}_{ab} \\,\\equiv \\, (\\mathcal {R}_{ab} - \\frac{1}{2}g_{ab}\\mathcal {R}) - \\mathcal {T}_{ab} \\,,$ where $\\mathcal {T}_{ab}$ is the energy-momentum tensor of the electromagnetic field.",
"Then, the condition $\\mathcal {E}^a_{\\;\\;b} \\,=\\,0$ is just Einstein's equation (REF ).",
"In spite of the simplicity of the line element (REF ), it turns out that it is quite hard to integrate Einstein-Maxwell equations for this metric in the general case.",
"Therefore, we shall make some simplifying assumptions.",
"First, inspired by Eq.",
"(REF ), let us adopt the following ansatz for the electromagnetic field $\\mathcal {F} = F_1(r) \\, d\\tilde{t}\\wedge dr + \\sum _{j=2}^{n/2}\\,F_j(r) \\sin \\theta _j \\,d\\phi _j\\wedge d\\theta _j \\,.$ Even with the latter assumption, it is difficult to integrate Einstein's equation without making restrictions over the functions $R_j(r)$ that appear in the metric (REF ).",
"Nevertheless, imposing equation $\\mathcal {E}^{\\tilde{t}}_{\\;\\;\\tilde{t}} - \\mathcal {E}^r_{\\;\\;r} = 0$ , we find that the following condition must hold: $\\sum _{j=2}^{n/2}\\, \\frac{R^{\\prime \\prime }_j(r)}{R_j(r)} \\,=\\,0 \\,.$ The simplest solution for the latter equation is attained when each term of the above sum vanishes, in which case the functions $R_j(r)$ have a linear dependence on $r$ : $R_j(r) \\,=\\,a_j\\,r \\,+\\,b_j \\,,$ where $a_j$ and $b_j$ are constants.",
"In particular, when the constants $a_j$ are all zero we retrieve the generalized Nariai metric (REF ).",
"Another case that is not so alluring happens when just one of the constants $a_j$ is non-vanishing, in which case we obtain the direct product of the Reissner-Nordström solution with $\\frac{n-4}{2}$ spheres of constant radii.",
"A particularly interesting case happens when we assume that $a_j$ are non-vanishing and the constants $b_j$ in Eq.",
"(REF ) are all equal.",
"In such a case, we can redefine the origin of the coordinate $r$ so that the constants $b_j$ vanish.",
"Then, after some calculations, the integration of the field equations leads to the following black hole solution: $ds^2 = -\\, f(r) \\, d\\tilde{t}^{\\,2} + \\frac{1}{f(r)}\\, dr^2 + r^2\\, \\sum _{j=2}^{n/2}\\,d\\Omega _j^2 \\,, \\nonumber \\\\\\mathcal {F} = \\sqrt{2(n-2)(n-3)}\\, \\frac{q_1}{r^{n-2}}\\,d\\tilde{t}\\wedge dr \\\\+ 2\\,q_2\\,\\sum _{j=2}^{n/2}\\,\\sin \\theta _j \\,d\\phi _j\\wedge d\\theta _j \\nonumber \\,,$ where the constants $q_1$ and $q_2$ are arbitrary parameters proportional to the electric and magnetic charges respectively, while the function $f(r)$ is given by $f(r) = \\frac{1}{n-3} - \\frac{2\\,\\mu }{r^{n-3}} + \\frac{q_1^2}{r^{2(n-3)}}-\\frac{q_2^2}{(n-5)r^2} - \\frac{\\Lambda \\,r^2}{n-1} \\,, $ with $\\mu $ being an arbitrary constant.",
"The latter parameter is proportional to the mass, i.e., it is proportional to the conserved charge associated to the Killing vector field $\\partial _{\\tilde{t}}$ .",
"One interesting feature of this solution is that it only has one magnetic charge $q_2$ , whereas in the generalized Nariai solution (REF ) a different charge can be assigned for each sphere factor in the line element.",
"As a consequence, all sphere factors in the black hole solution (REF ) must have the same radius.",
"Note that we started the integration process with arbitrary coefficients $a_j$ and, in principle, they were independent from each other.",
"Nonetheless, the field equations impose that all the constants $a_j$ must coincide, leading to the same radius and magnetic charge for each sphere factor.",
"This black hole solution has already been obtained in Ref.",
"[27], as a particular case of a broader solution in which the $\\left(n-2\\right)$ -dimensional spatial part of the metric is an Einstein manifold possessing an almost-Kähler structure.",
"Moreover, the cases $q_2=0$ (purely electric) and $q_1=0$ (purely magnetic) of this black hole solution can be attained in the limit of Einstein-Maxwell theory from the solutions found in Ref.",
"[26].",
"The spacetime presented in Eq.",
"(REF ) has a singularity at $r=0$ , which is indicated by the fact that the Kretschmann scalar diverges in the limit $r\\rightarrow 0$ .",
"Also, the null vector fields $\\ell = \\frac{1}{\\sqrt{f}} \\,\\partial _{\\tilde{t}} + \\sqrt{f} \\,\\partial _r \\quad \\textrm {and} \\quad n = \\frac{1}{\\sqrt{f}} \\,\\partial _{\\tilde{t}} - \\sqrt{f} \\,\\partial _r$ are repeated principal null directions of the Weyl tensor [21], [22], $C_{ab[cd}\\,\\ell _{e]} \\ell ^b = 0 = C_{ab[cd}\\, n_{e]} n^b \\,.$ Therefore, according to the boost weight classification [23], [24], the algebraic type of this spacetime is $D$ .",
"The latter classification is a generalization of the Petrov classification to arbitrary dimension.",
"These repeated principal null directions are geodesic, shear-free and twist-free, but have non-zero expansion.",
"Thus, the solution presented here is contained in the Robinson-Trautman class of spacetimes [28], [29].",
"In the special case $n=4$ , the solution (REF ) reduces to the well-known Reissner-Nordström solution in the presence of a cosmological constant, with $\\mu $ denoting the mass, $q_1$ is the electric charge and $q_2$ is the magnetic charge.",
"However, for $n>4$ the solution presented here differs, in several respects, from the charged version of the Schwarzschild-Tangherlini spacetime.",
"First, the topology of the spatial infinity and of the horizons are different in both cases.",
"For instance, while the topology of the horizon in Schwarzschild-Tangherlini spacetime is $\\mathbb {R}\\times S^{n-2}$ , in the solution presented here the topology of the horizon is the cartesian product of a real line with several 2-spheres.",
"Second, note that the additive constant term in the function $f(r)$ is different from 1 if $n\\ne 4$ , and this constant factor cannot be modified by a coordinate transformation.",
"Moreover, a higher-dimensional Schwarzschild metric possessing a magnetic charge does not exist, which is related to the fact that the second Betti number of the sphere $S^{n-2}$ is different from zero only if $n=4$ [26].",
"Regarding the magnetic charge $q_2$ , it is also worth stressing that the sign of in front of $q_2^2$ in Eq.",
"(REF ) becomes negative if $n\\ge 6$ , which implies the existence of an event horizon even in the case of vanishing mass parameter $\\mu $ .",
"Besides the Killing vector field $\\partial _{\\tilde{t}}$ , each sphere factor of the metric (REF ) provides three space-like Killing vector fields for the spacetime, so that the total number of Killing vector fields is $\\frac{3}{2}n-2$ .",
"On the other hand, the Schwarzschild-Tangherlini spacetime possesses $\\frac{1}{2}n^2-\\frac{3}{2}n+2$ Killing vectors.",
"Besides, for each sphere factor of the line element (REF ) there exists a Killing-Yano tensor given by $Y_j \\,=\\,r^3\\,\\sin \\theta _j\\,d\\theta _j\\wedge d\\phi _j \\,.$ Nevertheless, the conserved quantities along the geodesic motion associated to these Killing-Yano tensors are not independent from the ones generated by the Killing vector fields.",
"Finally, let us analyse some thermodynamic properties of the black hole solution (REF ).",
"In four dimensions, its is simple matter to interpret a magnetic charge and its associated potential, due to the electric/magnetic duality.",
"Nevertheless, in higher dimensions these two phenomena have different features, and the physical interpretation of the thermodynamic variable conjugated to a magnetic charge is trickier.",
"Hence, for simplicity, in what follows we will consider that the magnetic charge is zero, $q_2 =0$ .",
"Being $r_h$ the radius of the horizon, namely, the value of the larger root of $f(r)$ , the entropy of the horizon is just one quarter of its area: $S \\,=\\,\\frac{1}{4}\\, \\prod _{j=2}^{n/2}\\,4\\,\\pi \\,(r_h)^2 \\,.$ In its turn, the temperature is given by $T\\,=\\,\\kappa /(2\\pi )$ , where $\\kappa $ is the surface acceleration at the horizon, given by the expression (REF ) with the Killing vector being $\\xi = \\partial _{\\tilde{t}}$ .",
"Since the coordinate $r$ is not suitable at the horizon, we must use another coordinate system in order to do the calculation of the temperature properly.",
"For instance, using Eddington-Finkelstein coordinates we easily find $ T \\,=\\,\\frac{1}{2}\\, f^{\\prime }(r_h) \\,.",
"$ In the presence of electric charges, the electromagnetic field obeys $ d \\star \\mathcal {F} = 16\\pi \\star \\mathcal {J}$ , where $\\mathcal {J}$ is the 1-form representing the electromagnetic current.",
"Then, if $\\Omega $ is a space-like slice of the spacetime, the electric charge is given by: $Q & \\,=\\,\\int _{\\Omega } \\star \\mathcal {J} \\,=\\,\\frac{1}{16\\pi } \\int _{\\partial \\Omega } \\star \\mathcal {F} \\\\& \\,=\\,\\frac{(4\\pi )^{\\frac{n-2}{2}}}{16\\pi }\\,\\sqrt{2(n-2)(n-3)}\\,\\,q_1 \\,.$ Denoting the 1-form potential of the electromagnetic field by $\\mathcal {A}$ , the conjugated thermodynamic variable of the electric charge is the electric potential: $ \\Phi \\,=\\,\\xi ^a\\,\\mathcal {A}_a|_{r=r_h} \\,-\\,\\xi ^a\\,\\mathcal {A}_a|_{r=\\infty } \\,=\\,\\sqrt{\\frac{2(n-2)}{n-3}}\\,\\frac{q_1}{(r_h)^{n-3}} \\,.",
"$ Finally, the Komar mass of the spacetime is given by: $ M \\,=\\,\\frac{n-2}{8\\pi \\,(n-3)} \\, \\int dS_{ab}\\, \\nabla ^a\\,\\xi ^b \\,=\\,\\frac{n-2}{8\\pi }\\,(4\\pi )^{\\frac{n-2}{2}}\\, \\mu \\,, $ where a divergent piece arising from the cosmological constant has been subtracted, in accordance with the usual procedure [30].",
"Using these expressions, we can check that the first law of thermodynamics holds, namely $ dM \\,=\\,T\\,dS \\,+\\,\\Phi \\,dQ \\,.",
"$ Concerning the Smarr formula, one would expect that a relation of the type $M = \\alpha _1 T S \\,+\\,\\alpha _2 \\Phi Q$ would be valid, for some constants $\\alpha _1$ and $\\alpha _2$ .",
"Nevertheless, this is no true.",
"The reason is that the cosmological constant $\\Lambda $ should also be considered a thermodynamical variable, which is identified as the analogous of pressure [31], [32].",
"Indeed, the pressure is generally defined by $ P \\,=\\,-\\, \\frac{\\Lambda }{8\\pi }\\,.",
"$ In this formalism, the mass is identified with the enthalpy, rather than the energy, so that the variable thermodynamically conjugated to $P$ , the “volume”, is defined by $V \\,=\\,\\left( \\frac{\\partial M}{\\partial P} \\right)_{S,Q} \\,=\\,\\frac{n-2}{2(n-1)} \\, (4\\pi )^{\\frac{n-2}{2}}\\, (r_h)^{n-1} \\,.$ In the latter derivation, its has been used the relation $f(r_h)=0$ to write $M$ in terms of $S$ , $Q$ and $P$ .",
"With these definitions, one can check that the broader first law $ dM \\,=\\,T\\,dS \\,+\\,\\Phi \\,dQ \\,+\\,V \\, dP $ holds, as well as the Smarr formula $ M \\,=\\,\\frac{n-2}{n-3} \\, S\\, T \\,+\\,\\Phi \\, Q \\,-\\,\\frac{2}{n-3}\\, P\\,V \\,,$ which is in perfect accordance with the general results of Ref.",
"[32].",
"As a final remark on the thermodynamical aspects of the solution (REF ), note that the thermodynamical volume (REF ) is generally different from the geometric volume of the space inside the horizon, which is $ \\mathcal {V}_{\\textrm {Geom}} \\,=\\,\\int _0^{r_h} dr \\prod _{j=2}^{n/2}\\,r^2\\,d\\Omega _j^2\\,=\\,\\frac{2}{n-2} \\, V\\,.",
"$ Thus, the coincidence between the thermodynamical volume and the geometric volume occurs just in four dimensions, whereas in higher dimensions $\\mathcal {V}_{\\textrm {Geom}}$ is smaller than $V$ .",
"The discrepancy between these volumes should not raise any concern at all.",
"For instance, it is known that for rotating black holes this difference also occurs [31].",
"The nice thing of the solution presented here is that it is a static black hole in which the geometric volume and the thermodynamical volume do not agree, differently from the Schwarzschild-Tangherlini spacetime.",
"In order to obtain solution (REF ), we have assumed that the constants $b_j$ defined by Eq.",
"(REF ) are all equal.",
"Nevertheless, it is worth pointing out that there are also solutions for the general case in which the constants are distinct.",
"As an example, it will be shown a solution in six dimensions, $n=6$ .",
"Assuming that $a_2=a_3=a$ , where $a$ is non-zero, it follows that we can always redefine the origin of the coordinate $r$ in such a way that $b_3 = - b_2 = b$ , so that $R_2(r) \\,=\\,a\\,r \\,-\\,b \\quad \\textrm {and} \\quad R_3(r) \\,=\\,a\\,r \\,+\\,b \\,.$ In such a case, one can check that a solution is provided by the ansatz (REF ) along with the electromagnetic field (REF ), where the functions $F_1$ , $F_j$ and $f$ are fixed by the integration process.",
"Indeed, imposing Maxwell's equation (REF ) we find that $F_1(r) \\,=\\,\\sqrt{\\frac{8}{3}}\\, \\frac{q_1}{(b^2 \\,-\\,a^2 \\,r^2)^2} \\,,$ where $q_1$ is an integration constant that represents the electric charge.",
"Then, imposing relations $\\mathcal {E}^t_{\\;\\;t} + \\mathcal {E}^{\\theta _2}_{\\;\\;\\theta _2} = 0$ , $\\mathcal {E}^t_{\\;\\;t} + \\mathcal {E}^{\\theta _3}_{\\;\\;\\theta _3} = 0$ and $\\mathcal {E}^{\\theta _2}_{\\;\\;\\theta _2} + \\mathcal {E}^{\\theta _3}_{\\;\\;\\theta _3} = 0$ , which are immediate consequences of Einstein's equation, we end up with the following relations respectively $F_2(r) \\,=\\,& \\frac{b-a r}{b + a r} \\bigg [ -8 \\Lambda \\left(b^2-a^2 r^2\\right)^2 + 6 a^2 r^2 + 4 a br + 6 b^2 - 2 a^2 \\left(9 a^2 r^2+2 a b r-3 b^2\\right) f \\nonumber \\\\& \\phantom{1} \\quad \\quad \\quad + 2 a (b^2-a^2 r^2) (5 a r+b) f^{\\prime } -\\left(b^2-a^2 r^2\\right)^2 f^{\\prime \\prime } \\bigg ]^{1/2} \\,,\\nonumber \\\\\\nonumber \\\\F_3(r) \\,=\\,& \\frac{b+a r}{b - a r} \\bigg [ -8 \\Lambda \\left(b^2-a^2 r^2\\right)^2 + 6 a^2 r^2 - 4 a br + 6 b^2 + 2 a^2 \\left(-9 a^2 r^2+2 a b r+3 b^2\\right)f \\\\& \\phantom{1} \\quad \\quad \\quad + 2 a (b^2-a^2 r^2) (5 a r - b) f^{\\prime } -\\left(b^2-a^2 r^2\\right)^2 f^{\\prime \\prime } \\bigg ]^{1/2} \\,,\\nonumber \\\\\\nonumber \\\\f(r) \\,=\\,& \\frac{1}{15 \\left(b^2-a^2 r^2\\right)} \\,\\bigg \\lbrace C_1 -\\frac{1}{2} C_2 \\log \\left(\\frac{a r+b}{b-ar} \\right)+ \\frac{4 b^2}{a^2} \\left(4 b^2 \\Lambda -5\\right) \\log \\left(b^2-a^2r^2\\right)+\\frac{5 q_1^2}{4 a^2 b^4} \\log ^2\\left(\\frac{a r+b}{b-a r}\\right) \\nonumber \\\\& \\quad - \\frac{1}{a^2 b^4 \\left(b^2-a^2 r^2\\right)} \\bigg [ 14 a^2 b^8 \\Lambda r^2+a^2 b^6 r^2 \\left(5-17 a^2 \\Lambda r^2\\right)-5 b^2 q_1^2 + a^4 b^4 r^4 \\left(3 a^2 \\Lambda r^2-5\\right) \\bigg ] \\bigg \\rbrace \\nonumber $ with $C_1$ and $C_2$ being integration constants.",
"The remaining components of Einstein's equation are immediately satisfied once Eqs.",
"(REF ) and (REF ) are assumed to hold.",
"Computing the Ricci scalar and the Kretschmann scalar we see that they diverge at $r \\,=\\,\\pm b/a$ , indicating that this spacetime has two singularities.",
"Besides the latter solution, it has been checked that solutions also exist in six and eight dimensions for the general case in which the constants $a_j$ and $b_j$ are different from each other, contrasting with the assumption made in Eq.",
"(REF ).",
"Nevertheless, the expressions for the functions $F_j$ and $f$ turn out to be quite messy in such a broad case, so that they will not be presented here."
],
[
"Generalized Nariai Spacetimes in Higher Order Curvature Theories ",
"It is widely believed that Einstein's theory of gravitation is just the low-energy limit of a more complete (quantum) theory that is valid up to energies of the order of the Planck scale, as exemplified by String theory [34].",
"In this scheme, the Einstein-Hilbert Lagrangian density is supplemented by terms of higher order in the curvature.",
"Since these modified theories of gravitation have equations of motion that differ from Einstein's equation, it turns out that, generally, vacuum solutions of the latter are not vacuum solutions of the former theories.",
"Nevertheless, there are special metrics called universal that, apart from a possible rescalement by a constant multiplicative factor, are vacuum solutions in any gravitational theory arising from an action that is invariant under diffeomorphisms [35].",
"For instance, the maximally symmetric spacetimes, and the Kerr-Schild metrics that are in the Kundt class are examples of universal solutions [36].",
"These universal spacetimes are of great physical interest because they can be used as consistent vacuum states for the quantum theory of gravity, irrespective of its form.",
"Given the relevance of these issues, in this section we shall investigate the generalized charged Nariai solution in a much broader theory of gravity than general relativity, and we will show that the uncharged generalized (anti-)Nariai solution presented here is a universal metric.",
"The Einstein-Hilbert Lagrangian density is just a linear function of the Ricci scalar $\\mathcal {R}$ .",
"The simplest way to modify it is to consider that the gravitational Lagrangian density is a more general function of the Ricci scalar, $f(\\mathcal {R})$ , where $f$ could be chosen on phenomenological and experimental grounds.",
"So, the action of the system in such a case is given by $ \\mathcal {S} \\,=\\,\\int \\sqrt{-g} \\left[ \\, f(\\mathcal {R}) \\,+\\,\\mathcal {L}_{\\textrm {matter}}\\, \\right] \\,, $ where $\\mathcal {L}_{\\textrm {matter}}$ is the Lagrangian density of the matter content.",
"Performing the variation of this action with respect to the metric, we end up with the following field equation $f^{\\prime }(\\mathcal {R}) \\, \\mathcal {R}_{ab} - \\left[ \\nabla _a\\,\\nabla _b - g_{ab}\\,\\nabla _c\\nabla ^c \\right]f^{\\prime }(\\mathcal {R}) & \\nonumber \\\\- \\frac{1}{2}\\,f(\\mathcal {R})\\,g_{ab} & \\,=\\,\\mathcal {T}_{ab}\\,,$ with $\\mathcal {T}_{ab}$ denoting the energy-momentum tensor of the matter, which is obtained from the following relation: $ \\mathcal {T}_{ab} \\,=\\,\\frac{-1}{\\sqrt{-g}}\\, \\frac{ \\delta \\mathcal {S}_{\\textrm {matter}}}{\\delta g^{ab}} \\,.",
"$ It is worth pointing out that the field equation (REF ) generally involves derivatives of metric that are higher than second order.",
"Thus, it is natural to imagine that this feature would lead to a theory whose initial value problem is not well-posed.",
"However, it can be proved that $f(\\mathcal {R})$ theory is equivalent to Einstein's gravity coupled non-minimally with a scalar field [37], so that the Cauchy problem is well established, at least in four dimensions [38].",
"Another interesting way to systematically add higher order curvature terms to the Einstein-Hilbert Lagrangian is provided by the so-called Lovelock gravity [39], whose gravitational Lagrangian density is given by $\\mathcal {L}_{\\textrm {Lov.}}",
"\\,=\\,\\sum _{k=0}^{\\frac{n-2}{2}} \\, \\frac{\\alpha _k}{2^k} \\,\\delta ^{c_1d_1\\cdots c_k d_k}_{a_1b_1\\cdots a_k b_k}\\,\\mathcal {R}^{a_1b_1}_{\\phantom{a_1a_1}c_1d_1} \\cdots \\mathcal {R}^{a_{k}b_{k}}_{\\phantom{a_1a_2}c_{k}d_{k}} \\,,$ where $\\alpha _k$ are constants that should be fixed by phenomenology.",
"In $n$ dimensions, this sum could continue up to values $k\\ge n/2$ .",
"However, these extra terms would not contribute to the field equations, since they are purely topological.",
"The great feature that defines Lovelock gravity is that the above Lagrangian is the most general one that provides a field equation that is of second order on the derivatives of the metric [39].",
"The term $k=0$ in the above sum represents the cosmological constant part of the Lagrangian, while the term $k=1$ gives the Ricci scalar.",
"Thus, the term $k=2$ is the first “non-conventional” term and is called the Gauss-Bonnet (GB) Lagrangian density, whose expression is given by: $ \\mathcal {L}_{GB}\\,=\\,\\left( \\, \\mathcal {R}^2 - 4 \\,\\mathcal {R}^{ab}\\mathcal {R}_{ab} + \\mathcal {R}^{abcd}\\mathcal {R}_{abcd}\\, \\right) \\,.",
"$ In four dimensions, this term is purely topological and, hence, does not contribute to the gravitational field equation.",
"Nevertheless, in higher dimensions it changes Einstein's field equation.",
"It has recently been proved that Lovelock gravity can be seen as general relativity coupled with skew-symmetric auxiliary fields [40], but these form fields are non-dynamical [41].",
"In order to cover a great amount of gravitational theories, here we shall consider that the gravitational Lagrangian density is given by a sum of the term $f(\\mathcal {R})$ with the Gauss-Bonnet term.",
"Considering that the gravitational field is interacting with an electromagnetic field through a minimal coupling, we end up with the following final action $\\mathcal {S} \\,=\\,\\int \\sqrt{-g} \\left[ f(\\mathcal {R}) \\,+\\,\\alpha \\, \\mathcal {L}_{GB} \\,-\\,\\frac{1}{4}\\,\\mathcal {F}^{cd}\\mathcal {F}_{cd} \\right] \\,,$ where $f$ is an arbitrary function and $\\alpha $ is some arbitrary constant.",
"The field equations in such a case are given by $\\nabla ^a \\mathcal {F}_{ab}\\,=\\,0$ along with: $f^{\\prime }(\\mathcal {R}) \\, \\mathcal {R}_{ab} - \\left[ \\nabla _a\\,\\nabla _b - g_{ab}\\,\\nabla _c\\nabla ^c \\right]f^{\\prime }(\\mathcal {R})- \\frac{1}{2}\\,f(\\mathcal {R})\\,g_{ab} \\,+\\,\\alpha \\, \\mathcal {H}_{ab} \\,=\\,\\mathcal {T}_{ab}\\,,$ where the tensor $\\mathcal {H}_{ab}$ is defined by $\\mathcal {H}_{ab} \\,=\\,2 \\left( \\mathcal {R} \\mathcal {R}_{ab} - 2 \\mathcal {R}_{ac}\\mathcal {R}^{c}_{\\phantom{c}b} - 2\\mathcal {R}_{acbd}\\mathcal {R}^{cd} + \\mathcal {R}_{acde}\\mathcal {R}_{b}^{\\phantom{b}cde} \\right) \\,-\\,\\frac{1}{2}\\,g_{ab}\\, \\mathcal {L}_{GB}\\,,$ and the electromagnetic energy-momentum tensor is given by $\\mathcal {T}_{ab} \\,=\\,\\frac{1}{2}\\,\\mathcal {F}_a^{\\phantom{a}c}\\mathcal {F}_{bc} - \\frac{1}{8}\\,g_{ab}\\,\\mathcal {F}^{cd}\\mathcal {F}_{cd} \\,.$ We shall integrate these field equations starting with the following ansatz for the metric and for the electromagnetic field respectively: $ds^2 = R_1^2\\left[ - S_{\\epsilon _1}(x)^2\\,dt^2 + dx^2 \\right] + \\sum _{j=2}^{n/2}\\,R_j^2 \\, d\\tilde{\\Omega }_{\\epsilon _j}^2 \\;\\;, \\quad \\mathcal {F} = q_1 R_1^2 \\, S_{\\epsilon _1}(x) \\, dt\\wedge dx + \\sum _{j=2}^{n/2} q_j R_j^2\\, S_{\\epsilon _j}(\\theta _j)\\, d\\phi _j\\wedge d\\theta _j ,$ where $R_1$ and $R_j$ are constants, the functions $S_{\\epsilon }$ have been defined in Eq.",
"(REF ) and the line elements $d\\tilde{\\Omega }_{\\epsilon }^2$ are the ones presented in Eq.",
"(REF ).",
"Plugging such fields into the equation of motion for the electromagnetic field, $\\nabla ^a \\mathcal {F}_{ab}\\,=\\,0$ , and into the field equation for the gravitational field, Eq.",
"(REF ), lead us to the conclusion that these fields are a solution to such equations of motion provided that the electric charge $q_1$ and the magnetic charges $q_j$ are respectively given by $ q_1 = \\sqrt{- \\frac{4}{n-4} \\, f( {\\scriptstyle 2/\\tilde{R}_0^2})+ \\left[ \\frac{4\\, }{(n-4)\\tilde{R}_0^2 } - \\frac{2 \\epsilon _1}{R_1^2} \\right]\\, f^{\\prime }( {\\scriptstyle 2/\\tilde{R}_0^2}) +\\left( \\frac{1 }{\\tilde{R}_0^2 } - \\frac{\\epsilon _1}{R_1^2} \\right) \\frac{8\\,\\alpha \\,\\epsilon _1 }{R_1^2} } \\,, $ $ q_j = \\sqrt{ \\, \\frac{4}{n-4} \\, f( {\\scriptstyle 2/\\tilde{R}_0^2})- \\left[ \\frac{4\\, }{(n-4)\\tilde{R}_0^2 } - \\frac{2 \\epsilon _j}{R_j^2} \\right]\\, f^{\\prime }( {\\scriptstyle 2/\\tilde{R}_0^2}) -\\left( \\frac{1 }{\\tilde{R}_0^2 } - \\frac{\\epsilon _j}{R_j^2} \\right) \\frac{8\\,\\alpha \\,\\epsilon _j }{R_j^2} } \\,,$ where $\\tilde{R}_0$ was defined in Eq.",
"(REF ).",
"In particular, note that these expressions reduce to the ones presented in Eqs.",
"(REF ) and () when $\\alpha $ vanishes and $f(\\mathcal {R}) = \\mathcal {R} - (n-2)\\Lambda $ .",
"At this point, it is pertinent to mention that the entropy of horizons in theories of gravity with Lagrangian densities possessing higher order curvature terms is not just one quarter of the area [42], [43], so that Eq.",
"(REF ) is not valid in the context of this section.",
"Indeed, the total entropy will be the sum of a term arising from the Gauss-Bonnet part of the Lagrangian, which is essentially the integral of the Ricci scalar along the horizon [42], plus a term arising from the $f(R)$ part of the Lagrangian, which is essentially the integral of $f^{\\prime }(R)$ along the Horizon [44].",
"Since the expression for the entropy is not particularly illuminating, we will omit it here.",
"In the particular case of vanishing electromagnetic field, in which $q_1=0$ and $q_j=0$ , the gravitational field equation imposes that the radii $R_1$ and $R_j$ must be all equal to each other as well as the parameters $\\epsilon _1$ and $\\epsilon _j$ must all coincide.",
"In such a case, the only free parameters are $R_1$ and $\\epsilon _1$ , and the metric is given by $ds^2 = R_1^2\\bigg [ - S_{\\epsilon _1}&(x)^2\\, dt^2 + dx^2 \\nonumber \\\\& + \\sum _{j=2}^{n/2} \\left( d\\theta _j^2 + S_{\\epsilon _1}(\\theta _j)^2 d\\phi _j^2 \\right) \\bigg ] $ Irrespective of the parameters $R_1$ and $\\epsilon _1$ , it turns out that the above metric is such that the left hand side of the field equation (REF ) is equal to the metric times a constant multiplicative factor.",
"This means that, given some $R_1$ and $\\epsilon _1$ , we can always choose the value of the cosmological constant in such a way that the field equation is satisfied.",
"Actually, the same holds for any gravitational theory, not only for the ones covered by the action (REF ).",
"Indeed, since the spacetime (REF ) is the direct product of $n/2$ metrics of two-dimensional maximally symmetric spaces possessing the same Ricci scalar, it is a universal spacetime, as recently proved in Ref.",
"[45].",
"The proof goes as follows.",
"The Riemann tensor of this spacetime is the “sum” of the Riemann tensors associated to each these 2-spaces.",
"More precisely, the Riemann tensor is given by: $ \\mathcal {R}_{abcd} = \\mathcal {R}_{1\\,abcd} + \\sum _{j=2}^{n/2} \\mathcal {R}_{j\\,abcd} \\,, $ where $\\left.\\begin{array}{ll}\\mathcal {R}^{\\phantom{1}ab}_{1\\phantom{ab}cd} \\,=\\,- 4\\,\\epsilon _1 R_1^{-2} \\,\\, e_1^{\\;[a}\\tilde{e}_1^{\\;b]}e_{1\\,[c}\\tilde{e}_{1\\,d]}\\,, \\\\\\\\\\mathcal {R}^{\\phantom{j}ab}_{j\\phantom{ab}cd} \\,=\\,4\\,\\epsilon _1 R_1^{-2} \\,\\, e_j^{\\;[a}\\tilde{e}_j^{\\;b]}e_{j\\,[c}\\tilde{e}_{j\\,d]}\\,,\\end{array}\\right.$ where the Lorentz frame (REF ) has been used.",
"Thus, any symmetric tensor of rank two constructed from contractions of the curvature tensor is a sum of the analog contractions on the Riemann tensors of the 2-spaces.",
"For example, $ \\mathcal {R}_{acde}\\mathcal {R}_{b}^{\\phantom{b}cde} = \\mathcal {R}_{1\\,acde} \\mathcal {R}_{1\\,b}^{\\phantom{1\\,b}cde} +\\sum _{j=2}^{n/2} \\mathcal {R}_{j\\,acde} \\mathcal {R}_{j\\,b}^{\\phantom{j\\,b}cde}\\,.",
"$ But, if a space is maximally symmetric, any symmetric tensor of rank two built from contractions of its Riemann tensor must be proportional to its metric.",
"Thus, a rank two symmetric tensor constructed from contractions of the curvature of the whole spacetime (REF ) is a linear combination of the metrics of the 2-spaces.",
"But, since these 2-spaces all have the same Ricci scalar, the coefficients in front of each of the two-dimensional metrics are coincident, so that this linear combination is proportional to the metric of the whole spacetime (REF ).",
"This proves that the uncharged generalized (anti-)Nariai solution is a universal spacetime [35], [45]."
],
[
"Conclusions and Perspectives",
"In this article we have obtained higher-dimensional generalizations of the Nariai, anti-Nariai, Bertotti-Robinson and Plabański-Hacyan spacetimes that are made from the direct product of several 2-spaces of constant curvature.",
"In $n$ dimensions, these solutions admit $n/2$ electromagnetic charges, one being electric and the rest being magnetic.",
"These charges generate an uniform electromagnetic field throughout the spacetime.",
"Since in four dimensions the Nariai solution can be obtained from the Schwarzschild-de Sitter spacetime in the limit that the black hole horizon and the cosmological horizon have the same temperature, we have searched for higher-dimensional black holes whose limits of thermodynamical equilibrium converge to the generalized Nariai solutions presented here.",
"We have also seen that similar configurations of metric and electromagnetic fields provide solutions for more general theories of gravity, the only difference being the expressions for the electromagnetic charges.",
"One interesting feature of the generalized Nariai solutions and their associated black holes presented here is that they possess magnetic charge, while the usual higher-dimensional generalization of the Nariai metric [18], [19] and the Schwarzschild-Tangherlini black hole [33] are not magnetically charged.",
"Therefore, it would be interesting to study the physics of the solutions obtained here.",
"Particularly, one can seek for a rotating version of the black hole solution given by Eqs.",
"(REF ), (REF ).",
"Another nice feature of the solutions (REF ) is that the radii $R_j$ can be arbitrarily chosen.",
"In particular, one can set $R_3$ , $R_4$ , $\\cdots $ $R_{\\frac{n-2}{2}}$ much smaller than $R_2$ and study gravitational waves and other physical phenomena in such scenario.",
"The point being that in the latter case only four dimensions of the spacetime are accessible through low energy excitations, which is of relevance for theories and models that assume that our universe have tiny curled extra dimensions.",
"I would like to thank very much Andrés Anabalón for valuable suggestions in this work.",
"I also thank Marcello Ortaggio and Vojtěch Pravda for pointing out some important references."
]
] | 1605.05788 |
[
[
"Summing threshold logs in a parton shower"
],
[
"Abstract When parton distributions are falling steeply as the momentum fractions of the partons increases, there are effects that occur at each order in $\\alpha_s$ that combine to affect hard scattering cross sections and need to be summed.",
"We show how to accomplish this in a leading approximation in the context of a parton shower Monte Carlo event generator."
],
[
"Introduction",
"One often wants to calculate the cross section for a hard process in hadron-hadron collisions under the circumstance that one or both of the required momentum fractions $\\eta _\\mathrm {a}$ and $\\eta _\\mathrm {b}$ for the initial state partons is large.",
"Then the corresponding parton distribution functions $f_{a/A}(\\eta _\\mathrm {a},\\mu ^2)$ or $f_{b/B}(\\eta _\\mathrm {b},\\mu ^2)$ will be steeply falling as a function $\\eta _\\mathrm {a}$ and $\\eta _\\mathrm {b}$ .",
"In this case, the cross section calculated beyond the leading order is enhanced compared to the leading order cross section.",
"The effect is said to be due to “threshold logarithms.”For the emission of soft gluons from, say, parton “a”, we have an integration over the fraction $(1-z)$ of $\\eta _\\mathrm {a}$ that is taken by the gluons.",
"There is a parton factor $f_{a/A}(\\eta _\\mathrm {a}/z,\\mu ^2)$ and a singular function of $(1-z)$ .",
"The integration is limited by the falloff of $f_{a/A}(\\eta _\\mathrm {a}/z,\\mu ^2)$ as $(1-z)$ increases.",
"This integration produces logarithms of the effective upper bound on $(1-z)$ .",
"For details, see section .",
"The amount of enhancement is proportional to how fast $f_{a/A}(\\eta _\\mathrm {a},\\mu ^2)$ or $f_{b/B}(\\eta _\\mathrm {b},\\mu ^2)$ is falling.",
"The threshold enhancement grows as $\\eta _\\mathrm {a}$ and $\\eta _\\mathrm {b}$ increase, but $\\eta _\\mathrm {a}$ and $\\eta _\\mathrm {b}$ do not have to be close to 1 for the threshold logarithms to start to be significant.",
"One can count $\\eta _\\mathrm {a}> 0.1$ and $\\eta _\\mathrm {b}> 0.1$ as being part of the threshold region.",
"In the threshold region, it is useful to sum the enhanced contributions.",
"Already in 1987, Sterman [1] pointed out the issue and showed how to understand it and sum the threshold logarithms.",
"Since then, there has been a substantial development in the field, including both treatments following the traditional direct approach [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32] and treatments using soft-collinear effective theory [33], [34], [35], [36], [37], [38], [39], [40], [41], [42], [43], [44].",
"Comparisons of these approaches can be found in refs.",
"[45], [46], [47].",
"Even before the study of threshold logarithms began, Sjöstrand [48] and Marchesini and Webber [49] had introduced parton shower event generators that were important in the mid 1980s and whose direct descendants are still essential tools today [50], [51], [52], [53], [54], [55], [56], [57], [58], [59], [60], [61], [62], [63], [64], [65].",
"These programs are based on iterating parton splittings derived from perturbative quantum chromodynamics (QCD).",
"Because of this iterative structure based on the soft and collinear singularities of the theory, they sum many sorts of logarithms in a natural way.",
"For instance, parton shower event generators can sum logarithms of $k_\\mathrm {T}^2/M_\\mathrm {Z}^2$ , where $k_\\mathrm {T}$ is the transverse momentum of a Z-boson produced in the Drell-Yan process [66].",
"Also, the analysis of ref.",
"[4] connects the summation of threshold logarithms to the structure of a parton shower.",
"Current standard parton shower event generators do not sum threshold logarithms.",
"However, it is possible to rearrange them so that they do sum the leading threshold logarithms.",
"In this paper, we show how to do that.",
"Our analysis is based on our own parton shower event generator, Deductor [67], [68], [69], [70], [71], [72], [73], [74].",
"We make this definite choice because we need to be specific with respect to shower kinematics, the choice of shower ordering variable, splitting functions, and the treatment of color.",
"For this paper, we use a new version, 2.0.2, of Deductor [75].",
"Even though our analysis uses definitions specific to Deductor, we think that it will be apparent that the results could be adapted to other hardness based parton shower generators, such as Pythia [65] and Sherpa [61].",
"With somewhat more restructuring, it should be possible to adapt the results to the angle ordered parton shower generator Herwig [60].",
"We present some of the features of the theoretical framework in section .",
"Then, before we begin the analysis, we exhibit in section some of the main features of the result and of where this result comes from.",
"Inevitably, we leave a lot out.",
"The details are presented later in the paper.",
"Section discusses the role of parton distribution functions.",
"Section analyzes the Sudakov exponent conventionally used in parton shower algorithms.",
"Section presents a revised Sudakov exponent to be used if one wants to include threshold logarithms.",
"Sections and analyze the exponential that contains the threshold logarithms.",
"Section presents a comparison of the leading behavior of the threshold factor in this paper to the factor in previous treatments of threshold logarithms.",
"Section presents numerical results obtained with the Deductor parton shower.",
"There are three concluding sections.",
"In section , we summarize the main steps of the analysis that has been presented, leaving out the supporting arguments.",
"The reader may wish to read section first, then return to it after absorbing the supporting arguments.",
"In section , we describe some of the choices available in running Deductor.",
"In section , we say something about the advantages and disadvantages of putting threshold enhancements into a parton shower.",
"We have tried to keep the main presentation relatively short by relegating calculational details to three appendices."
],
[
"The ",
"Our analysis is based on the parton shower event generator, Deductor [67], [68], [69], [70], [71], [72], [73], [74], which uses specific choices with respect to shower kinematics, the shower ordering variable, the parton splitting functions, and the treatment of color.",
"In this section, we outline some of these choices that play a role in the analysis of this paper.",
"An exact treatment of leading threshold logarithms requires an exact treatment of color, which is available in the general formalism of Refs.",
"[67], [68], [69], [70], [71], [72], [73], [74].",
"The exact color treatment is not implemented in the code of Deductor.",
"Rather, we are able to use only an approximation, the leading-color-plus (LC+) approximation.",
"However, in this paper we develop the theory using exact color.",
"Then the LC+ approximation consists of simply dropping some terms.",
"In Deductor, we order splittings according to decreasing values of a hardness parameter $\\Lambda ^2$ .",
"The hardness parameter is based on virtuality.",
"For massless partons, the definition is $\\begin{split}\\Lambda ^2 ={}& \\frac{(\\hat{p}_l + \\hat{p}_{m+1})^2}{2 p_l\\cdot Q_0}\\ Q_0^2\\hspace{28.45274pt}{\\rm final\\ state},\\\\\\Lambda ^2 ={}& -\\frac{(\\hat{p}_\\mathrm {a}- \\hat{p}_{m+1})^2}{2 p_\\mathrm {a}\\cdot Q_0}\\ Q_0^2\\hspace{28.45274pt}{\\rm initial\\ state}.\\end{split}$ Here the mother parton in a final state splitting has momentum $p_l$ and the daughters have momenta $\\hat{p}_l$ and $\\hat{p}_{m+1}$ .",
"For an initial state splitting in hadron A, the mother parton has momentum $p_\\mathrm {a}$ , the new initial state parton has momentum $\\hat{p}_\\mathrm {a}$ and the final state parton created in the splitting has momentum $\\hat{p}_{m+1}$ .Here $p_\\mathrm {a}$ is the momentum of the mother parton in the sense of shower evolution that moves from hard interactions to softer interactions.",
"With initial state splittings, the shower development moves backwards in physical time.",
"For hadron B, we replace “a” $\\rightarrow $ “b.” We denote by $Q_0$ a fixed vector equal to the total momentum of all of the final state partons just after the hard scattering that initiates the shower.",
"We often use a dimensionless “shower time” variable given by $t = \\log (Q_0^2/\\Lambda ^2)\\;.$ Then $t$ increases as the shower develops.",
"One could use some other hardness parameter to order the shower.",
"For instance, various measures of the transverse momentum in a splitting are popular choices.",
"In this paper, we use $\\Lambda ^2$ because that is what we have implemented in Deductor.",
"In an initial state splitting, parton distribution functions enter the splitting functions.",
"Then we need a scale parameter for the parton distribution functions.",
"We use the virtuality of the splitting: $\\begin{split}\\mu _\\mathrm {a}^2(t) ={}& 2 p_\\mathrm {a}\\cdot Q_0 \\,e^{-t}\\;,\\\\\\mu _\\mathrm {b}^2(t) ={}& 2 p_\\mathrm {b}\\cdot Q_0 \\,e^{-t}\\;.\\end{split}$ Similarly, for a final state splitting of parton $l$ , the virtuality parameter is $\\mu _l^2(t) = 2 p_l\\cdot Q_0 \\,e^{-t}\\;.$ It proves convenient to use a dimensionless virtuality variable $y$ .",
"For an initial state splitting, $y = \\frac{\\mu _\\mathrm {a}^2(t)}{2 p_\\mathrm {a}\\cdot Q} = \\frac{\\mu _\\mathrm {a}^2(t)}{Q^2}\\;,$ where $Q = p_\\mathrm {a}+ p_\\mathrm {b}$ is the total momentum of the final state particles in the shower state before the splitting.",
"We collect these and other notations used throughout the paper in appendix .",
"Deductor 1.0 uses non-zero charm and bottom quark masses, both in the final and the initial states.",
"However, we have simplified the code in Deductor 2.0 [75] that we use here by setting the charm and bottom quark masses to zero in the parton kinematics (although charm and bottom quark thresholds in the evolution of parton distribution functions remain).",
"This is evidently an approximation, but this approximation does not matter for the physics investigated in this paper, the threshold logarithms at very large parton collision energies.",
"The general formalism includes parton spins, but the current implementation in Deductor simply eliminates spin by averaging over the mother parton spin before each splitting and summing over the daughter parton spins.",
"In this paper we keep the analysis as simple as we can by doing the same spin averaging, so that spin dynamics does not play a role."
],
[
"Some main features of the threshold enhancement",
"As in our previous work [67], [68], [69], [70], [71], [72], [73], [74], we find it useful to think of parton shower evolution as describing the evolution of a statistical state vector $\\big |{\\rho (t)}\\big )$ .",
"The shower time $t$ represents the hardness scale of the current state.",
"As $t$ increases, parton emissions get softer and softer.",
"At shower time $t$ , $\\big |{\\rho (t)}\\big )$ represents the distribution of partonic states in an ensemble of runs of the Monte Carlo style program.",
"Without color or spin, $\\big |{\\rho (t)}\\big )$ would simply represent a probability distribution in the number $m+2$ of partons ($m$ final state partons and two initial state partons), their flavors $f$ , and their momenta $p$ .",
"With the inclusion of color, $\\big |{\\rho (t)}\\big )$ is a little more complicated: it represents the probability distribution of parton flavors and momenta and the density matrix in the quantum color space.",
"As stated in the introduction, we ignore spin in this paper.",
"If spin were included, $\\big |{\\rho (t)}\\big )$ would represent the density matrix in the combined color and spin space.",
"In the notation we use here, a measurement of some property $F$ of the partonic system, like the number of jets with transverse momenta bigger than some given value, is represented by a vector $\\big ({F}\\big |$ .",
"The average cross section corresponding to measurement $F$ at time $t$ is $\\big ({F}\\big |{\\rho (t)}\\big )$ .",
"Of particular importance is the completely inclusive measurement, which we denote by $\\big ({1}\\big |$ .",
"Thus the total cross section represented by state $\\big |{\\rho (t)}\\big )$ is $\\big ({1}\\big |{\\rho (t)}\\big )$ .",
"To compute $\\big ({1}\\big |{\\rho (t)}\\big )$ , we sum over the number of partons and their flavors, integrate over the parton momenta, and take the trace of the density matrix in color space.",
"We represent the evolution of the shower by the equation $\\frac{d}{dt}\\,\\big |{\\rho (t)}\\big ) = [{\\cal H}_I(t) - {\\cal S}(t)]\\big |{\\rho (t)}\\big )\\;.$ Here ${\\cal H}_I(t)$ represents real parton emissions and increases the number of final state partons by one.",
"It contains splitting functions derived from QCD and a ratio of parton distribution functions.",
"The operator ${\\cal S}(t)$ leaves the number of partons, their momenta, and their flavors unchanged.",
"It is, however, an operator on the color space.",
"The operator ${\\cal S}(t)$ contains terms representing two effects.",
"It contains a term derived from one loop virtual graphs.",
"It also contains a term proportional to the derivative with respect to scale of the parton distribution functions.",
"This term is needed because parton distribution functions appear at the hard interaction and then (with a different scale) at each parton splitting.",
"Thus one needs a term in ${\\cal S}(t)$ that cancels the parton distributions at the old, harder, scale and inserts parton distributions at the new, softer, scale.",
"It is useful to solve the shower evolution equation (REF ) in the form $\\big |{\\rho (t)}\\big ) = {\\cal U}_{\\cal S}(t,t_0)\\big |{\\rho (t_0)}\\big )\\;,$ where ${\\cal U}_{\\cal S}(t,t_0) = {\\cal N}_{\\cal S}(t,t_0)+ \\int _{t_0}^t\\!d\\tau \\ {\\cal U}_{\\cal S}(t,\\tau ){\\cal H}_I(\\tau ){\\cal N}_{\\cal S}(\\tau ,t_0)\\;.$ Here ${\\cal N}_{\\cal S}(\\tau ,t_0)$ is the no-splitting operator, given by a time-ordered exponential of the integral of ${\\cal S}(\\tau ^{\\prime })$ , ${\\cal N}_{\\cal S}(\\tau ,t_0) = \\mathbb {T} \\exp \\left[-\\int _{t_0}^{\\tau } d\\tau ^{\\prime }\\ {\\cal S}(\\tau ^{\\prime })\\right]\\;.$ Thus, the number of partons, their flavors, and their momenta remain unchanged from time $t_0$ to some intermediate time $\\tau $ .",
"The probability for the state can be multiplied by a numerical factor or, in general, by a matrix in color space.",
"The no-splitting operator is usually called the Sudakov factor.",
"In eq.",
"(REF ), there is a splitting at the intermediate time $\\tau $ , as specified by the splitting operator ${\\cal H}_I(\\tau )$ .",
"After that, there is more evolution according to the full evolution operator ${\\cal U}_{\\cal S}(t,\\tau )$ .",
"The evolution equation eq.",
"(REF ) is what one gets in a rather direct way from first order QCD perturbation theory.",
"It is, however, not exactly what is computed in Deductor 1.0 or in other standard parton shower event generators.",
"The reason is that this evolution equation does not exactly conserve the total cross section: $\\frac{d}{dt}\\,\\big ({1}\\big |{\\rho (t)}\\big ) \\ne 0\\;.$ We can conserve the total cross section if we change the evolution equation to $\\frac{d}{dt}\\,\\big |{\\hat{\\rho }(t)}\\big ) = [{\\cal H}_I(t) - {\\cal V}(t)]\\big |{\\hat{\\rho }(t)}\\big )\\;,$ where $\\big ({1}\\big |[{\\cal H}_I(t) - {\\cal V}(t)] = 0\\;,$ so that $\\frac{d}{dt}\\,\\big ({1}\\big |{\\hat{\\rho }(t)}\\big ) = 0\\;.$ That is, we define ${\\cal V}(t)$ from ${\\cal H}_I(t)$ so that $\\big ({1}\\big |{\\cal V}(t) = \\big ({1}\\big |{\\cal H}_I(t)\\;.$ With this definition, the color trace of the revised Sudakov factor, ${\\cal N}_{{\\cal V}}(t_2,t_1) = \\mathbb {T} \\exp \\left[-\\int _{t_1}^{t_2} d\\tau \\ {\\cal V}(\\tau )\\right]\\;,$ represents the probability for the parton system not to split between shower time $t_1$ and shower time $t_2$ .",
"It is standard to use Sudakov factors that are exponentials of ${\\cal V}(t)$ defined by eq.",
"(REF ) to define a parton shower.",
"That is the choice in Deductor 1.0.",
"When one constructs a parton shower using ${\\cal V}(t)$ , the cross section associated with $\\big |{\\hat{\\rho }(t)}\\big )$ is what it was at the hard interaction, say $t = 0$ .",
"That is to say, the total cross section is the Born cross section.",
"What the parton shower does is to distribute the fixed cross section into cross sections for the different final states that the starting partons could evolve into.",
"The idea of the summation of threshold logarithms is that the total cross section is not just the Born cross section.",
"Rather it contains corrections from higher orders of perturbation theory.",
"To see this effect, we should use ${\\cal S}(t)$ instead of ${\\cal V}(t)$ .",
"When we do that, we have a Sudakov factor that we can write as ${\\cal N}_{\\cal S}(t_2,t_1) = \\mathbb {T} \\exp \\left[\\int _{t_1}^{t_2} d\\tau \\ (-{\\cal V}(\\tau )+ [{\\cal V}(\\tau ) - {\\cal S}(\\tau )])\\right]\\;.$ Compared to the standard Sudakov factor made using ${\\cal V}(\\tau )$ , there is then an extra term in the exponent, $[{\\cal V}(\\tau ) - {\\cal S}(\\tau )]$ .",
"The most important parts of this are those associated with initial state evolution, $[{\\cal V}_\\mathrm {a}(\\tau ) - {\\cal S}_\\mathrm {a}(\\tau )]$ and $[{\\cal V}_\\mathrm {b}(\\tau ) - {\\cal S}_\\mathrm {b}(\\tau )]$ .",
"There is a good deal of calculation needed to find $[{\\cal V}_\\mathrm {a}(\\tau ) - {\\cal S}_\\mathrm {a}(\\tau )]$ .",
"However, the most important term in the result is pretty simple to understand.",
"For a partonic state with $m+2$ partons with momenta $p$ , flavors $f$ , and colors $c^{\\prime },c$ , we find that the contribution is $\\begin{split}[{\\cal V}_{\\mathrm {a}}(t)&- {\\cal S}_\\mathrm {a}(t)]\\big |{\\lbrace p,f,c^{\\prime },c\\rbrace _{m}}\\big )\\\\={}&\\Bigg \\lbrace \\frac{\\alpha _{\\mathrm {s}}}{2\\pi }\\,\\int _0^{1/(1+y)}\\!dz\\,\\left(\\frac{f_{a/A}(\\eta _{\\mathrm {a}}/z,\\mu _\\mathrm {a}^2(t))}{zf_{a/A}(\\eta _{\\mathrm {a}},\\mu _\\mathrm {a}^2(t))}\\,P_{a a}(z)- \\frac{2C_a}{1-z}\\right)[1\\otimes 1]\\\\&-\\frac{\\alpha _{\\mathrm {s}}}{2\\pi }\\,\\int _0^{1}\\!dz\\left(\\frac{f_{a/A}(\\eta _{\\mathrm {a}}/z, \\mu _\\mathrm {a}^2(t))}{zf_{a/A}(\\eta _{\\mathrm {a}}, \\mu _\\mathrm {a}^2(t))}\\,P_{a a}\\!\\left(z\\right)-\\frac{2 C_a }{1-z}\\right)[1\\otimes 1]\\\\ & + \\cdots \\Bigg \\rbrace \\big |{\\lbrace p,f,c^{\\prime },c\\rbrace _{m}}\\big )\\;.\\end{split}$ The omitted terms here need not concern us at the moment.",
"The function $P_{a a}\\!\\left(z\\right)$ is the flavor conserving part of the evolution kernel for the parton distributions $f_{a/A}(\\eta _{\\mathrm {a}},\\mu ^2)$ .",
"The factors $[1\\otimes 1]$ represent unit operators on the color space.",
"The index $a$ represents the flavor of parton “a” in the state $\\big |{\\lbrace p,f,c^{\\prime },c\\rbrace _{m}}\\big )$ .",
"The factors $C_a$ are color factors, $C_\\mathrm {F}$ for quark or antiquark flavors, $C_\\mathrm {A}$ for gluons.",
"The parameter $y$ , defined in eq.",
"(REF ), is a dimensionless virtuality variable for a splitting at shower time $t$ .",
"For large $t$ , the splitting has small virtuality and $y \\ll 1$ .",
"The integration variable $z$ is the momentum fraction in the splitting: parton “a” carries momentum fraction $\\eta _\\mathrm {a}$ before the splitting and the new parton “a” carries momentum fraction $\\eta _\\mathrm {a}/z$ after the splitting.",
"The first term in eq.",
"(REF ) comes from ${\\cal V}_{\\mathrm {a}}(t)$ , which is given in eq.",
"(REF ) and involves parton distribution functions and the splitting functions that describe real emissions.",
"The second term comes from ${\\cal S}_\\mathrm {a}(t)$ , which is given in eq.",
"(REF ) and involves the evolution of the parton distribution functions together with contributions from virtual graphs.",
"The full ${\\cal V}_{\\mathrm {a}}(t) - {\\cal S}_\\mathrm {a}(t)$ is given in eq.",
"(REF ).",
"In the first term in eq.",
"(REF ), the upper limit $z < 1/(1+y)$ comes from the kinematics of parton splitting: for a finite virtuality, $(1-z)$ cannot be too small.",
"See eq.",
"(REF ).",
"In the second term, the upper limit $z < 1$ comes from the upper limit in the evolution equation for parton distribution functions.",
"We see that the two terms are almost identical.",
"They differ only in the upper limits of the $z$ -integrations.",
"Adding the two terms, we have $\\begin{split}[{\\cal V}_{\\mathrm {a}}(t)&- {\\cal S}_\\mathrm {a}(t)]\\big |{\\lbrace p,f,c^{\\prime },c\\rbrace _{m}}\\big )\\\\={}&\\Bigg \\lbrace -\\frac{\\alpha _{\\mathrm {s}}}{2\\pi }\\,\\int _{1/(1+y)}^1\\!dz\\,\\left(\\frac{f_{a/A}(\\eta _{\\mathrm {a}}/z,\\mu _\\mathrm {a}^2(t))}{z f_{a/A}(\\eta _{\\mathrm {a}},\\mu _\\mathrm {a}^2(t))}P_{a a}(z)- \\frac{2C_a}{1-z}\\right)[1\\otimes 1]\\\\ & + \\cdots \\Bigg \\rbrace \\big |{\\lbrace p,f,c^{\\prime },c\\rbrace _{m}}\\big )\\;.\\end{split}$ Only a small integration range, corresponding to soft gluon emission, remains.",
"In this range, we can use $P_{a a}(z) \\approx \\frac{2zC_a}{1-z}\\;.$ This gives $\\begin{split}[{\\cal V}_{\\mathrm {a}}(t)&- {\\cal S}_\\mathrm {a}(t)]\\big |{\\lbrace p,f,c^{\\prime },c\\rbrace _{m}}\\big )\\\\={}&\\Bigg \\lbrace \\frac{\\alpha _{\\mathrm {s}}}{2\\pi }\\,\\int _{1/(1+y)}^1\\!dz\\,\\left(1 -\\frac{f_{a/A}(\\eta _{\\mathrm {a}}/z,\\mu _\\mathrm {a}^2(t))}{f_{a/A}(\\eta _{\\mathrm {a}},\\mu _\\mathrm {a}^2(t))}\\right)\\frac{2C_a}{1-z}\\,[1\\otimes 1]\\\\ & + \\cdots \\Bigg \\rbrace \\big |{\\lbrace p,f,c^{\\prime },c\\rbrace _{m}}\\big )\\;.\\end{split}$ We integrate over a small range of $z$ near $z = 1$ .",
"There is a $1/(1-z)$ singularity, the “threshold singularity.” The ratio of parton distribution functions is 1 at $z = 1$ .",
"Thus the integral is not singular.",
"But the integral is large if $f_{a/A}(\\eta _{\\mathrm {a}},\\mu ^2)$ is a fast varying function of $\\eta _{\\mathrm {a}}$ .",
"When $\\eta _{\\mathrm {a}}$ is not small, say $\\eta _{\\mathrm {a}} > 0.1$ , $f_{a/A}(\\eta _{\\mathrm {a}},\\mu ^{2})$ is in fact a fast varying function of $\\eta _{\\mathrm {a}}$ .",
"For that reason, we retain this contribution.",
"The difference $[{\\cal V}_{\\mathrm {a}}(t) - {\\cal S}_\\mathrm {a}(t)]$ appears in the exponent of the Sudakov factor, so by keeping this contribution, we sum the effects that come from the threshold singularity.",
"This “shower” version of the summation of threshold logs looks rather different from the direct QCD or the SCET descriptions of the same physics.",
"We will discuss the connections in section .",
"With this introduction to provide hints about where we are going, we now proceed to a detailed examination of how the leading threshold logarithms can be summed within a parton shower event generator, at least after we apply the LC+ approximation [70] to simplify the color structure."
],
[
"The parton distribution functions",
"As was noted already in ref.",
"[1], the precise definition of parton distributions matters for the question of threshold logarithms.",
"In this paper, we are dealing with a parton shower algorithm.",
"As we will see in more detail in the following sections, it is important that the behavior of the parton distribution functions match the structure of the parton shower algorithm.",
"We discuss the needed definitions in this section."
],
[
"Parton evolution equations",
"Begin with the $\\overline{\\text{MS}}$ parton distribution functions with one loop evolution.",
"Call the scale variable $\\mu _\\perp ^2$ .",
"We have $\\begin{split}\\frac{d f_{a/A}^{\\overline{\\text{MS}}}(\\eta _\\mathrm {a},\\mu _\\perp ^2)}{d\\log \\mu _\\perp ^2}={}&\\int _0^1\\!dz \\sum _{\\hat{a}} \\frac{\\alpha _{\\mathrm {s}}(\\lambda _\\mathrm {R}\\mu _\\perp ^2)}{2\\pi }\\,\\theta (\\mu _\\perp ^2 > m_\\perp ^2(a))\\\\&\\times \\Bigg \\lbrace \\frac{1}{z}\\, P_{a\\hat{a}}(z)\\,f_{\\hat{a}/A}^{\\overline{\\text{MS}}}(\\eta _\\mathrm {a}/z,\\mu _\\perp ^2)- \\delta _{a, \\hat{a}}\\left(\\frac{2 C_a}{1-z} - \\gamma _a\\right)f_{a/A}^{\\overline{\\text{MS}}}(\\eta _\\mathrm {a},\\mu _\\perp ^2)\\Bigg \\rbrace \\;.\\end{split}$ In the case of a bottom or charm quark, evolution of $f_{a/A}(\\eta _\\mathrm {a},\\mu _\\perp ^2)$ occurs only for $\\mu _\\perp ^2 > m_\\perp ^2(a)$ , where $m_\\perp (a)$ is the quark mass.",
"Below this scale, $f_{a/A}(\\eta _\\mathrm {a},\\mu _\\perp ^2) = 0$ .This is to be understood as a one loop matching condition between five flavor $\\overline{\\text{MS}}$ renormalization and four flavor renormalization and then between four and three flavor renormalization.",
"For a light quark or gluon, we define $m_\\perp ^2(f) = m_\\perp ^2({\\rm start})$ to be a starting scale for shower evolution and for parton distribution function evolution.",
"In Deductor, we take $m_\\perp ^2({\\rm start})$ near $1 \\ \\mathrm {GeV}^2$ .",
"In eq.",
"(REF ), $P_{a\\hat{a}}(z)$ are the familiar DGLAP kernels and $C_a$ and $\\gamma _a$ are the standard flavor dependent constants recorded in eqs.",
"(REF ) and (REF ).",
"We take the argument of $\\alpha _{\\mathrm {s}}$ to be $\\lambda _\\mathrm {R}\\mu _\\perp ^2$ , where $\\lambda _\\mathrm {R}= \\exp \\left(- [ C_\\mathrm {A}(67 - 3\\pi ^2)- 10\\, n_\\mathrm {f}]/[3\\, (33 - 2\\,n_\\mathrm {f})]\\right) \\approx 0.4\\;.$ That is, we use the $\\overline{\\text{MS}}$ scale $\\mu _\\perp ^2$ for $\\alpha _{\\mathrm {s}}$ in the evolution equation and we insert an extra factor $\\lambda _\\mathrm {R}$ .",
"(In $\\lambda _\\mathrm {R}$ , the number $n_\\mathrm {f}$ of active flavors depends on $\\mu _\\perp ^2$ .)",
"The factor $\\lambda _\\mathrm {R}$ in the argument of $\\alpha _{\\mathrm {s}}$ is, strictly speaking, beyond the order of perturbation theory that we control in a leading order shower, but it is helpful in generating certain next-to-leading logarithms [51], [66].",
"To make the structure of the $z \\rightarrow 1$ singularities in this equation clearer, we write $P_{a\\hat{a}}(z) = \\delta _{a \\hat{a}}\\,\\frac{2 z C_a}{1-z}+ P_{a\\hat{a}}^{\\rm reg}(z)\\;,$ where $P_{a\\hat{a}}^{\\rm reg}(z)$ is nonsingular as $z \\rightarrow 1$ .",
"Then we can write $\\begin{split}&\\frac{d f_{a/A}^{\\overline{\\text{MS}}}(\\eta _\\mathrm {a},\\mu _\\perp ^2)}{d\\log \\mu _\\perp ^2}=\\\\& \\qquad f_{a/A}^{\\overline{\\text{MS}}}(\\eta _\\mathrm {a},\\mu _\\perp ^2)\\Bigg \\lbrace \\frac{\\alpha _{\\mathrm {s}}(\\lambda _\\mathrm {R}\\mu _\\perp ^2)}{2\\pi }\\,\\theta (\\mu _\\perp ^2 > m_\\perp ^2(a))\\,\\gamma _a +\\int _0^1\\!dz\\ \\frac{\\alpha _{\\mathrm {s}}(\\lambda _\\mathrm {R}\\mu _\\perp ^2)}{2\\pi }\\,\\theta (\\mu _\\perp ^2 > m_\\perp ^2(a))\\\\& \\qquad \\times \\Bigg (-\\frac{2 C_a}{1-z}\\left[1 -\\frac{f_{a/A}^{\\overline{\\text{MS}}}(\\eta _\\mathrm {a}/z,\\mu _\\perp ^2)}{f_{a/A}^{\\overline{\\text{MS}}}(\\eta _\\mathrm {a},\\mu _\\perp ^2)}\\right]+\\sum _{\\hat{a}} P_{a\\hat{a}}^{\\rm reg}(z)\\frac{f_{\\hat{a}/A}^{\\overline{\\text{MS}}}(\\eta _\\mathrm {a}/z,\\mu _\\perp ^2)}{z f_{a/A}^{\\overline{\\text{MS}}}(\\eta _\\mathrm {a},\\mu _\\perp ^2)}\\Bigg )\\Bigg \\rbrace \\;.\\end{split}$ Now we look at the parton evolution that matches shower evolution in Deductor, which uses the virtuality based ordering variable $\\Lambda ^2$ defined in eq.",
"(REF ).",
"It thus needs parton distribution functions $f_{a/A}(\\eta ,\\mu _\\Lambda ^2)$ , where $\\mu _\\Lambda ^2$ represents the virtuality in an initial state splitting, as in eq.",
"(REF ).",
"In the conventional $\\overline{\\text{MS}}$ parton distribution functions [76], the scale variable is the renormalization scale for the functions with $\\overline{\\text{MS}}$ renormalization.",
"In $\\overline{\\text{MS}}$ renormalization at one loop, we have a dimensionally regulated integration, $\\int _0^1\\!",
"dz\\ \\mu _\\perp ^{2\\epsilon }\\int \\frac{d^{2 - 2\\epsilon }\\mathbf {k}_\\perp }{(2\\pi )^{2 - 2\\epsilon }}\\cdots \\;,$ over the momentum fraction $z$ and the transverse momentum $\\mathbf {k}_\\perp $ of the splitting.",
"The integration is ultraviolet divergent at $\\epsilon = 0$ .",
"The $\\overline{\\text{MS}}$ prescription is to subtract the $1/\\epsilon $ pole along with certain conventional finite terms.",
"This gives us a finite result in which the scale of the squared transverse momentum $\\mathbf {k}_\\perp ^2$ is set by $\\mu _\\perp ^2$ .",
"This is not quite the same as setting the scale of the virtuality $|k^2|$ in the splitting.",
"The two variables are related,For a derivation, see eq (3.23) of ref.",
"[72] with all of the masses equal to zero.",
"Following the discussion in appendix , we adopt $(1-z) |k^2|$ as the definition of $\\mathbf {k}_\\perp ^2$ for an initial state splitting.",
"for small angle splittings, by $\\mathbf {k}_\\perp ^2 = (1-z) |k^2|$ .",
"Using $\\mathbf {k}_\\perp ^2 = (1-z)|k^2|$ , we see that the scale $\\mu _\\perp ^2$ of $\\mathbf {k}_\\perp ^2$ is related to the scale of $\\mu _\\Lambda ^2$ of $|k^2|$ by $\\mu _\\perp ^2 = (1-z)\\mu _\\Lambda ^2$ .",
"This works for $\\mu _\\perp ^2 > m_\\perp ^2(a)$ .",
"Below $\\mu _\\perp ^2 = m_\\perp ^2(a)$ , evolution for partons of flavor $a$ stops.",
"We want parton evolution and shower evolution to match.",
"Thus we define $\\mu _\\perp ^2(z,\\mu _\\Lambda ^2) = \\max [(1-z)\\mu _\\Lambda ^2, m_\\perp ^2(a)]\\;.$ With this definition of the scales, under under an infinitesimal change of $\\mu _\\Lambda ^2$ we have $\\delta \\log \\mu _\\perp ^2 ={\\left\\lbrace \\begin{array}{ll}\\delta \\log \\mu _\\Lambda ^2 & (1-z)\\mu _\\Lambda ^2 > m_\\perp ^2(a) \\\\0 & (1-z)\\mu _\\Lambda ^2 < m_\\perp ^2(a)\\end{array}\\right.",
"}\\;.$ Thus the one loop evolution equation for virtuality based parton distribution functions is almost the same as the evolution equation for $\\overline{\\text{MS}}$ parton distribution functions.",
"We need theta functions that turn off the evolution for flavor $a$ unless $(1-z) \\mu _\\Lambda ^2 > m_\\perp ^2(a)$ and we should use $(1-z) \\lambda _\\mathrm {R}\\mu _\\Lambda ^2$ in the argument of $\\alpha _{\\mathrm {s}}$ .",
"We simplify this a bit and use $\\begin{split}\\frac{d f_{a/A}(\\eta _\\mathrm {a},\\mu _\\Lambda ^2)}{d\\log \\mu _\\Lambda ^2}={}&f_{a/A}(\\eta _\\mathrm {a},\\mu _\\Lambda ^2)\\Bigg \\lbrace \\frac{\\alpha _{\\mathrm {s}}(\\lambda _\\mathrm {R}\\mu _\\Lambda ^2)}{2\\pi }\\,\\theta (\\mu _\\Lambda ^2 > m_\\perp ^2(a))\\,\\gamma _a\\\\&+\\int _0^1\\!dz\\ \\theta ((1-z)\\mu _\\Lambda ^2 > m_\\perp ^2(a))\\\\& \\qquad \\times \\Bigg (-\\frac{\\alpha _{\\mathrm {s}}((1-z)\\lambda _\\mathrm {R}\\mu _\\Lambda ^2)}{2\\pi }\\,\\frac{2 C_a}{1-z}\\left[1 -\\frac{f_{a/A}(\\eta _\\mathrm {a}/z,\\mu _\\Lambda ^2)}{f_{a/A}(\\eta _\\mathrm {a},\\mu _\\Lambda ^2)}\\right]\\\\&\\qquad \\qquad +\\frac{\\alpha _{\\mathrm {s}}(\\lambda _\\mathrm {R}\\mu _\\Lambda ^2)}{2\\pi }\\,\\sum _{\\hat{a}} P_{a\\hat{a}}^{\\rm reg}(z)\\frac{f_{\\hat{a}/A}(\\eta _\\mathrm {a}/z,\\mu _\\Lambda ^2)}{z f_{a/A}(\\eta _\\mathrm {a},\\mu _\\Lambda ^2)}\\Bigg )\\Bigg \\rbrace \\;.\\end{split}$ In the first term, there is no change from the $\\overline{\\text{MS}}$ evolution equation.",
"In the remaining two terms, there is an integration over $z$ and we identify $\\mu _\\perp ^2$ with $(1-z) \\mu _\\Lambda ^2$ in the theta function that provides an infrared cutoff on $\\mu _\\perp ^2$ .",
"In the second term, which has a $1/(1-z)$ singularity from soft gluon emission, we use $(1-z) \\lambda _\\mathrm {R}\\mu _\\Lambda ^2$ as the argument of $\\alpha _{\\mathrm {s}}$ , while in the third term, with no $1/(1-z)$ singularity, we simplify the evolution by omitting the factor $(1-z)$ in the argument of $\\alpha _{\\mathrm {s}}$ .",
"With these choices, parton evolution matches the conventions that we use in the shower splitting kernels in Deductor.",
"The parton distributions $f_{a/A}(\\eta _\\mathrm {a},\\mu _\\Lambda ^2)$ that we use are obtained by solving the evolution equation (REF ) using an $\\overline{\\text{MS}}$ parton distribution set as the initial condition at the starting scale $m_\\perp ^2({\\rm start})$ for the shower.",
"For the starting distributions, we use the CT14 NLO parton set [77]."
],
[
"Approximate analytic result",
"In section , we will present an analytical comparison of the results of this paper for the Drell-Yan cross section to standard analytical results in the leading log approximation.",
"For this purpose, we will need an approximate analytical relation between the parton distribution functions $f_{a/A}(\\eta _\\mathrm {a},\\mu _\\Lambda ^2)$ obtained by solving eq.",
"(REF ) and the $\\overline{\\text{MS}}$ parton distribution functions $f_{a/A}^{\\overline{\\text{MS}}}(\\eta _\\mathrm {a},\\mu _\\perp ^2)$ obtained by solving eq.",
"(REF ).",
"Our aim is to understand just the leading contributions to threshold behavior, so we do not specify the argument of $\\alpha _{\\mathrm {s}}$ and we make some approximations that correspond to including only the effect of soft gluon emissions.",
"We write eqs.",
"(REF ) and (REF ) as equations for the Mellin moments of the parton distributions, $\\tilde{f}(N) = \\int _0^1\\!\\frac{d\\eta }{\\eta }\\ \\eta ^{N} f(\\eta )\\;.$ We introduce a parameter $\\lambda $ with $\\lambda = 0$ corresponding to $\\overline{\\text{MS}}$ evolution, eq.",
"(REF ), and $\\lambda = 1$ corresponding to $\\Lambda ^2$ evolution, eq.",
"(REF ).",
"We denote by $\\mu _0^2$ the appropriate scale parameter, $\\mu _\\perp ^2$ for $\\lambda = 0$ or $\\mu _\\Lambda ^2$ for $\\lambda = 1$ .",
"Keeping only the leading singular terms in the kernel, we have $\\begin{split}\\frac{d \\tilde{f}_{a/A}(N,\\mu _0^2,\\lambda )}{d\\log \\mu _0^2}\\approx {}&\\int _0^1\\!dz\\ \\frac{\\alpha _{\\mathrm {s}}}{2\\pi }\\,\\theta ((1-z)^\\lambda \\mu _0^2 > m_\\perp ^2(a))\\\\&\\times \\frac{2 C_a}{1-z}\\left[z^N - 1\\right]\\tilde{f}_{a/A}(N,\\mu _0^2,\\lambda )\\;.\\end{split}$ The solution of this is $\\begin{split}\\tilde{f}_{a/A}(N,\\mu _0^2,\\lambda )\\approx {}&\\tilde{f}_{a/A}(N,m_\\perp ^2({\\rm start}),0)\\\\&\\times \\exp \\bigg \\lbrace \\int _{m_\\perp ^2({\\rm start})}^{\\mu _0^2}\\!\\frac{d\\bar{\\mu }^2}{\\bar{\\mu }^2}\\int _0^1\\!dz\\ \\frac{\\alpha _{\\mathrm {s}}}{2\\pi }\\,\\theta ((1-z)^\\lambda \\bar{\\mu }^2 > m_\\perp ^2(a))\\\\&\\qquad \\times \\frac{2 C_a}{1-z}\\left[z^N - 1\\right]\\bigg \\rbrace \\;.\\end{split}$ At the starting scale, $m_\\perp ^2({\\rm start})$ , we use the $\\overline{\\text{MS}}$ parton distributions ($\\lambda = 0$ ) as a boundary value.",
"Now we can regard $\\lambda $ as a continuous variable.",
"Then eq.",
"(REF ) gives us a differential equation for the $\\lambda $ dependence of $\\tilde{f}_{a/A}(N,\\mu _0^2,\\lambda )$ : $\\begin{split}\\frac{d\\tilde{f}_{a/A}(N,\\mu _0^2,\\lambda )}{d\\lambda }\\approx {}&\\tilde{f}_{a/A}(N,\\mu _0^2,\\lambda )\\\\&\\times \\int _0^1\\!dz\\ \\frac{\\alpha _{\\mathrm {s}}}{2\\pi }\\,\\theta ((1-z)^\\lambda \\mu _0^2 > m_\\perp ^2(a))\\log (1-z)\\frac{2 C_a}{1-z}\\left[z^N - 1\\right]\\;.\\end{split}$ When we take the inverse Mellin transform of this, we find $\\begin{split}\\frac{df_{a/A}(\\eta _\\mathrm {a},\\mu _0^2,\\lambda )}{d\\lambda }\\approx {}&-f_{a/A}(\\eta _\\mathrm {a},\\mu _0^2,\\lambda )\\int _0^1\\!dz\\ \\frac{\\alpha _{\\mathrm {s}}}{2\\pi }\\,\\theta ((1-z)^\\lambda \\mu _0^2 > m_\\perp ^2(a))\\log (1-z)\\\\&\\qquad \\times \\frac{2 C_a}{1-z}\\left[1-\\frac{f_{a/A}(\\eta _\\mathrm {a}/z, \\mu _0^2,\\lambda )}{f_{a/A}(\\eta _\\mathrm {a}, \\mu _0^2,\\lambda )}\\right]\\;.\\end{split}$ Notice the factor $R = \\frac{f_{a/A}(\\eta _\\mathrm {a}/z,\\mu _0^2,\\lambda )}{f_{a/A}(\\eta _\\mathrm {a},\\mu _0^2,\\lambda )}\\;.$ In the kinematic region of interest for threshold logarithms, $R$ is approximately independent of $\\lambda $ .",
"For instance, with the $\\lambda = 1$ parton distributions obtained by solving eq.",
"(REF ), we find for $a = \\mathrm {u}$ and $\\eta _a = 0.3$ , $\\mu = 2 \\ \\mathrm {TeV}$ , that $R \\approx z^{2.8}$ for $0.8 < z < 1$ .",
"For $\\lambda = 0$ , we solve the ordinary first order $\\overline{\\text{MS}}$ evolution equation.",
"Then $R \\approx z^{2.9}$ for $0.8< z < 1$ .",
"There is a difference, but it is small.",
"We can get an instructive analytic result if we make the approximation that the $R$ is independent of $\\lambda $ .",
"Then the solution of eq.",
"(REF ) is $f_{a/A}(\\eta _\\mathrm {a},\\mu _0^2) = Z_a(\\eta _\\mathrm {a},\\mu _0^2)\\,f^{\\overline{\\text{MS}}}_{a/A}(\\eta _\\mathrm {a},\\mu _0^2)\\;,$ where $f_{a/A}(\\eta _\\mathrm {a},\\mu _0^2) = f_{a/A}(\\eta _\\mathrm {a},\\mu _0^2,1)$ and $f^{\\overline{\\text{MS}}}_{a/A}(\\eta _\\mathrm {a},\\mu _0^2) = f_{a/A}(\\eta _\\mathrm {a},\\mu _0^2,0)$ and where $\\begin{split}Z_a(\\eta _\\mathrm {a},\\mu _0^2)={}&\\exp \\!\\Bigg ( -\\int _0^1\\!dz\\int _0^1\\!d\\lambda \\ \\frac{\\alpha _{\\mathrm {s}}}{2\\pi }\\,\\theta ((1-z)^\\lambda \\mu _0^2 > m_\\perp ^2(a))\\,\\log (1-z)\\\\& \\qquad \\times \\frac{2 C_a}{1-z} \\left\\lbrace 1 -\\frac{f_{a/A}(\\eta _\\mathrm {a}/z,\\mu _0^2)}{f_{a/A}(\\eta _\\mathrm {a},\\mu _0^2)}\\right\\rbrace \\Bigg )\\;.\\end{split}$ It will prove useful to change variables from $\\lambda $ to $\\mu _\\perp ^2 = (1-z)^\\lambda \\mu _0^2$ .",
"Then $\\log (1-z)\\, d \\lambda = d\\mu _\\perp ^2/\\mu _\\perp ^2$ and $\\lambda = 0$ corresponds to $\\mu _\\perp ^2 = \\mu _0^2$ while $\\lambda = 1$ corresponds to $\\mu _\\perp ^2 = (1-z) \\mu _0^2$ .",
"Thus (after also choosing a standard argument for $\\alpha _{\\mathrm {s}}$ ) $\\begin{split}Z_a(\\eta _\\mathrm {a},\\mu _0^2) ={}&\\exp \\!\\Bigg (\\int _0^1\\!dz\\int _{(1-z)\\,\\mu _0^2}^{\\mu _0^2} \\frac{d\\mu _\\perp ^2}{\\mu _\\perp ^2}\\,\\frac{\\alpha _{\\mathrm {s}}(\\lambda _\\mathrm {R}\\mu _\\perp ^2)}{2\\pi }\\,\\theta (\\mu _\\perp ^2 > m_\\perp ^2(a))\\\\& \\qquad \\times \\frac{2 C_a}{1-z} \\left\\lbrace 1 -\\frac{f_{a/A}(\\eta _\\mathrm {a}/z,\\mu _0^2)}{f_{a/A}(\\eta _\\mathrm {a},\\mu _0^2)}\\right\\rbrace \\Bigg )\\;.\\end{split}$ When does $Z_a(\\eta _\\mathrm {a},\\mu _0^2)$ differ significantly from 1?",
"There is a factor $\\alpha _{\\mathrm {s}}$ in the exponent, which generally makes the exponent small.",
"However, the factor multiplying $\\alpha _{\\mathrm {s}}$ can be large when the parton ratio $R$ is a fast varying function of $z$ .",
"We will use this result in section REF ."
],
[
"The probability preserving integrand ${\\cal V}(t)$",
"As outlined in section , we seek to calculate (with suitable approximations) the difference $({\\cal V}(\\tau ) - {\\cal S}(\\tau ))$ between the integrand of the Sudakov exponent ${\\cal V}(\\tau )$ that preserves probabilities as the shower evolves and the Sudakov integrand ${\\cal S}(\\tau )$ that is based on virtual diagrams and the evolution of the parton distribution functions.",
"We begin in this section with ${\\cal V}(\\tau )$ as defined in Deductor 1.0, but with all quark masses set to zero.",
"We sketch the needed calculations in appendix .",
"We will use the results in section .",
"The operator ${\\cal V}(\\tau )$ is used to define the Sudakov factor, ${\\cal N}_{{\\cal V}}(t_2,t_1) = \\mathbb {T} \\exp \\left[-\\int _{t_1}^{t_2}\\!",
"d\\tau \\ {\\cal V}(\\tau )\\right]\\;.$ The exponent $\\int _{t_1}^{t_2} d\\tau \\ {\\cal V}(\\tau )$ , when we take its trace in color, is the total probability to have a real parton splitting between shower times $t_1$ and $t_2$ .",
"Thus ${\\cal N}_{{\\cal V}}(t_2,t_1)$ is the probability not to have had a splitting.",
"The operator ${\\cal V}(\\tau )$ has, in general, a non-trivial color structure.",
"For that reason, the exponential function is ordered in shower time $\\tau $ , as indicated by the $\\mathbb {T}$ instruction.",
"In Deductor, we apply an approximation, the LC+ approximation, that makes ${\\cal V}(\\tau )$ diagonal in color.",
"Then the $\\mathbb {T}$ ordering instruction is not needed.",
"In this paper, we keep the full color structure for all operators, but then at the end we can apply the LC+ approximation.",
"For many of our formulas for ${\\cal V}(\\tau )$ , it is convenient to define energies and angles in a reference frame in which the total momentum $Q$ of all of the final state partons has the form $Q = (E_Q,\\vec{0})$ .",
"Then $\\vec{p}_l$ is the momentum of parton $l$ .",
"The angle $\\theta _{kl}$ between two parton momenta is defined in this same reference frame.",
"Thus $\\begin{split}E_Q^2 ={}& Q^2\\;,\\\\1 - \\cos \\theta _{kl} ={}& \\frac{p_l\\cdot p_k\\ Q^2}{p_l\\cdot Q\\ p_k\\cdot Q}\\;.\\end{split}$ It is convenient to define $a_l = \\frac{Q^2}{2\\,p_l\\cdot Q} = \\frac{E_Q}{2|\\vec{p}_l|}\\;.$ See appendix , where $C_{f_l}$ and $\\gamma _{f_l}$ are also defined.",
"In our analysis, we use a dimensionless virtuality variable $y$ given by eqs.",
"(REF ) and (REF ).",
"The variable $y$ is fixed by the shower time $\\tau $ according to eq.",
"(REF ).",
"The structure of ${\\cal V}(\\tau )$ is rather complicated, but we simplify it by taking the leading behavior as $y \\rightarrow 0$ .",
"That is, we use the leading behavior of the splitting functions in the limits of soft or collinear splittings.",
"The parton shower in Deductor has an infrared cutoff: no splittings with a transverse momentum smaller than $m_\\perp ({\\rm start}) \\sim 1 \\ \\mathrm {GeV}$ are generated.",
"In terms of $y$ and the momentum fraction $z$ in the splitting, the cutoff is $y (1-z) > m_\\perp ^2({\\rm start})/(2 p_\\mathrm {a}\\cdot Q)$ for an initial state splitting and $y z(1-z) > m_\\perp ^2({\\rm start})/(2 p_l\\cdot Q)$ for an final state splitting.",
"However, in calculating the leading $y \\ll 1$ behavior of ${\\cal V}(\\tau )$ , we assume that $y$ is not so small that we need to be concerned with this infrared cutoff.",
"In fact, we will find that very tiny values of $y$ are not relevant for the threshold effects that we investigate.",
"Thus we simply calculate ${\\cal V}(\\tau )$ with no infrared cutoff.",
"Some of the terms in ${\\cal V}(\\tau )$ depend on the angle $\\theta _{kl}$ between the splitting parton $l$ and a parton $k$ that forms part of a color dipole with parton $l$ .",
"This angle dependence arises because soft gluon emission from a color dipole depends on the angles among the partons.",
"The angle $\\theta _{kl}$ is small when partons $k$ and $l$ are the daughter partons of a previous splitting that was nearly collinear.",
"When this previous splitting was a final state splitting, ordering in $\\Lambda ^2$ for the new splitting of parton $l$ implies $a_l y < 1 - \\cos \\theta _{kl}$ .",
"When a previous splitting that produced partons $k$ and $l$ was an initial state splitting with a small momentum fraction $z_{kl}$ , one can also have $a_l y > 1 - \\cos \\theta _{kl}$ .",
"This happens in the case of multi-regge kinematics, as discussed in section 5.4 of ref. [72].",
"This is the opposite kinematic regime from that of threshold logarithms, so we ignore this possibility in this paper.",
"Thus we assume $a_l y \\ll 1 - \\cos \\theta _{kl}$ in evaluating ${\\cal V}(\\tau )$ .",
"The terms in ${\\cal V}(\\tau )$ contain operators like $[(\\mathbf {T}_l\\cdot \\mathbf {T}_k) \\otimes 1]$ that act on the color space.",
"This means that, in the ket part of the color density matrix, color generator $T^a$ acts on the color of parton $l$ , color generator $T^a$ acts on the color of parton $k$ , and we sum over the octet color index $a$ .",
"In this example, a unit operator acts on the bra part of the color density matrix.",
"In manipulating the color operators, we use the identity $\\sum _k \\mathbf {T}_k = 0$ and the identity $(\\mathbf {T}_l\\cdot \\mathbf {T}_l) = C_{f_l}$ , where $C_q = C_\\mathrm {F}$ for a quark or antiquark flavor $q$ and $C_\\mathrm {g}= C_\\mathrm {A}$ for a gluon.",
"At any stage in the shower, there are initial state partons “a” and “b” as well as final state partons $l$ with $l \\in \\lbrace 1,\\dots ,m\\rbrace $ .",
"There is a term in ${\\cal V}(\\tau )$ for each parton: ${\\cal V}(\\tau )={\\cal V}_\\mathrm {a}(\\tau )+ {\\cal V}_\\mathrm {b}(\\tau )+ \\sum _{l=1}^m {\\cal V}_l(\\tau )\\;.$ The limiting form for ${\\cal V}_l(\\tau )$ for a final state parton, from eq.",
"(REF ), is quite simple, $\\begin{split}{\\cal V}_l(\\tau )&\\big |{\\lbrace p,f,c^{\\prime },c\\rbrace _{m}}\\big )\\\\={}&\\frac{\\alpha _{\\mathrm {s}}}{2\\pi }\\bigg \\lbrace \\left(2 C_{f_l} \\log \\!\\left[\\frac{2|\\vec{p}_l|}{E_Q y}\\right]- \\gamma _{f_l}\\right)[1\\otimes 1]\\\\&\\qquad - \\sum _{k \\ne l}\\log \\!\\left[\\frac{1 - \\cos \\theta _{lk}}{2}\\right][(\\mathbf {T}_l\\cdot \\mathbf {T}_k) \\otimes 1+ 1\\otimes (\\mathbf {T}_l\\cdot \\mathbf {T}_k)]\\bigg \\rbrace \\\\ & \\times \\big |{\\lbrace p,f,c^{\\prime },c\\rbrace _{m}}\\big )\\;.\\end{split}$ The limiting form for ${\\cal V}_\\mathrm {a}(\\tau )$ for initial state parton “a” is not quite so simple because it involves parton distribution functions.",
"From eq.",
"(REF ), we have $\\begin{split}{\\cal V}_{\\mathrm {a}}(t)&\\big |{\\lbrace p,f,c^{\\prime },c\\rbrace _{m}}\\big )\\\\={}& \\bigg [\\frac{\\alpha _{\\mathrm {s}}}{2\\pi }\\,\\int _0^{1/(1+y)}\\!dz\\\\&\\times \\Bigg \\lbrace \\sum _{\\hat{a}}\\left(\\frac{f_{\\hat{a}/A}(\\eta _{\\mathrm {a}}/z,y Q^2)}{zf_{a/A}(\\eta _{\\mathrm {a}},y Q^2)}P_{a\\hat{a}}(z)- \\delta _{a\\hat{a}}\\frac{2C_a}{1-z}\\right)[1\\otimes 1]\\\\&\\quad +\\sum _{k\\ne \\mathrm {a},\\mathrm {b}}\\left[\\frac{f_{a/A}(\\eta _{\\mathrm {a}}/z,y Q^2)}{f_{a/A}(\\eta _{\\mathrm {a}},y Q^2)} - 1\\right]\\Delta _{\\mathrm {a}k}(z,y)\\big ([(\\mathbf {T}_\\mathrm {a}\\cdot \\mathbf {T}_k)\\otimes 1] + [1 \\otimes (\\mathbf {T}_\\mathrm {a}\\cdot \\mathbf {T}_k)]\\big )\\Bigg \\rbrace \\\\& -\\sum _{k\\ne \\mathrm {a},\\mathrm {b}}\\frac{\\alpha _{\\mathrm {s}}}{2\\pi }\\,\\log \\!\\left[\\frac{1 - \\cos \\theta _{\\mathrm {a}k}}{2}\\right]\\,\\big ([(\\mathbf {T}_\\mathrm {a}\\cdot \\mathbf {T}_k)\\otimes 1] + [1 \\otimes (\\mathbf {T}_\\mathrm {a}\\cdot \\mathbf {T}_k)]\\big )\\\\&+\\frac{\\alpha _{\\mathrm {s}}}{2\\pi }\\,2 C_a\\log \\left[\\frac{1}{y}\\right][1\\otimes 1]\\bigg ]\\\\&\\times \\big |{\\lbrace p,f,c^{\\prime },c\\rbrace _{m}}\\big )\\;.\\end{split}$ The first two terms involve ratios of parton distribution functions together with subtractions.",
"The ratios of parton distribution functions arise naturally in the Sudakov factors for initial state parton shower evolution from hard emissions to soft emissions [48].",
"Notice that there is an upper limit for the momentum fraction variable in the splitting: $z < 1/(1+y)$ .",
"With the definition of $z$ used in Deductor, $z \\rightarrow 1$ implies that the emitted parton has zero momentum, so that the virtuality variable $y$ must vanish.",
"Thus at finite $y$ there must be a limit on $z$ .",
"The precise limit depends on the splitting kinematics used in Deductor.",
"See eq.",
"(REF ).",
"In the second term, there is a non-trivial color factor and a function defined in eq.",
"(REF ), $\\begin{split}\\Delta _{\\mathrm {a}k}(z,y)=\\frac{1}{1-z}-\\frac{1}{\\sqrt{(1-z)^2 + y^2 z^2/\\psi _{\\mathrm {a}k}^2}}\\;,\\end{split}$ where $\\psi _{\\mathrm {a}k} = (1 - \\cos \\theta _{\\mathrm {a}k})/\\sqrt{8(1 + \\cos \\theta _{\\mathrm {a}k})}$ , as in eq.",
"(REF ).",
"For $(1-z) \\ll y/\\psi _{\\mathrm {a}k}$ , $\\Delta _{\\mathrm {a}k}$ is approximately $1/(1-z)$ .",
"However, for $(1-z) \\gg y/\\psi _{\\mathrm {a}k}$ , $\\Delta _{\\mathrm {a}k}$ tends to zero faster than $1/(1-z)$ .",
"Thus, when we integrate over $z$ , the important integration region is $y/(1+y) < (1-z) \\lesssim y/\\psi _{\\mathrm {a}k}$ provided that $\\psi _{\\mathrm {a}k} \\lesssim 1$ .",
"If $\\psi _{\\mathrm {a}k} \\gg 1$ , there is no important integration region.",
"In the final two terms, there is no dependence on parton distribution functions and the $z$ -integration has been performed.",
"One term depends on $\\theta _{\\mathrm {a}k}$ , while the other term, with a trivial color factor, depends only on $y$ ."
],
[
"The Sudakov integrand ${\\cal S}(t)$",
"In this section, we study the integrand ${\\cal S}(t)$ for the Sudakov exponent.",
"As in our discussion of ${\\cal V}(t)$ , we calculate without an infrared cutoff associated with the end of the shower at $k_\\perp ^2 = m_\\perp ^2({\\rm start}) \\sim 1 \\ \\mathrm {GeV}^2$ .",
"Our first task is to identify the parts of ${\\cal S}(t)$ that come from parton evolution and from virtual graphs."
],
[
"Decomposition of ${\\cal S}(t)$",
"The operator ${\\cal S}(t)$ affects the evolution of $\\big |{\\rho (t)}\\big )$ through eq.",
"(REF ), $\\frac{d}{dt}\\,\\big |{\\rho (t)}\\big ) = [{\\cal H}_I(t) - {\\cal S}(t)]\\big |{\\rho (t)}\\big )\\;.$ The color density matrix $\\big |{\\rho (t)}\\big )$ is defined to include the proper factors of parton distributions so that the color trace of $\\rho $ is a differential cross section [67].",
"It is related to the density matrix without parton distribution functions by $\\big |{\\rho (t)}\\big ) = {\\cal F}(t) \\big |{\\rho _{\\rm pert}(t)}\\big )\\;,$ where $\\begin{split}{\\cal F}(t)&\\big |{\\lbrace p,f,c^{\\prime },c\\rbrace _{m}}\\big )\\\\ &=\\frac{f_{a/A}(\\eta _{\\mathrm {a}},\\mu _\\mathrm {a}^2(t))f_{b/B}(\\eta _{\\mathrm {b}}, \\mu _\\mathrm {b}^2(t))}{4n_\\mathrm {c}(a) n_\\mathrm {c}(b)\\,4\\eta _{\\mathrm {a}}\\eta _{\\mathrm {b}}p_\\mathrm {A}\\!\\cdot \\!p_\\mathrm {B}}\\ \\big |{\\lbrace p,f,c^{\\prime },c\\rbrace _{m}}\\big )\\;.\\end{split}$ Here the virtuality scale for the parton distributions is a function of the shower time $t$ as defined in eq.",
"(REF ) or eq.",
"(REF ), (Cf.",
"section 2.3 of ref. [73].)",
"Then $\\big |{\\rho _{\\rm pert}(t)}\\big )$ obeys an evolution equation of the form $\\frac{d}{dt}\\big |{\\rho _{\\rm pert}(t)}\\big ) =[{\\cal H}^{\\rm pert}_I(t) - {\\cal S}^{\\rm pert}(t)]\\big |{\\rho _{\\rm pert}(t)}\\big )\\;.$ In eq.",
"(REF ), $\\begin{split}{\\cal H}^{\\rm pert}_I(t)={}&{\\cal F}(t)^{-1}{\\cal H}_I(t){\\cal F}(t)\\;.\\end{split}$ is the perturbative splitting function, with no factors representing parton distribution functions.",
"That is, the definition of ${\\cal H}_I(t)$ based on eq.",
"(8.26) of ref.",
"[67] contains explicit factors of ratios of parton distributions; to define ${\\cal H}^{\\rm pert}_I(t)$ , we remove these factors.",
"We then interpret $-{\\cal S}^{\\rm pert}(t)$ in eq.",
"(REF ), as giving the part of the evolution of $\\big |{\\rho _{\\rm pert}(t)}\\big )$ that does not involve parton splitting.",
"Thus we will calculate $-{\\cal S}^{\\rm pert}(t)$ using suitable approximations to virtual one loop Feynman graphs.",
"We do that in appendix .",
"Using eqs.",
"(REF ), (REF ), (REF ), and (REF ), we see that the operator ${\\cal S}(t)$ is related to ${\\cal S}^{\\rm pert}(t)$ by $\\begin{split}{\\cal S}(t)={}&{\\cal S}^{\\rm pert}(t)- {\\cal F}(t)^{-1}\\left[\\frac{d}{dt}\\,{\\cal F}(t)\\right]\\;.\\end{split}$ Acting on a statistical basis state, ${\\cal F}(t)^{-1}\\left[d{\\cal F}(t)/dt\\right]$ gives $\\begin{split}{\\cal F}(t)^{-1}\\left[\\frac{d}{dt}\\,{\\cal F}(t)\\right]&\\big |{\\lbrace p,f,c^{\\prime },c\\rbrace _{m}}\\big )\\\\ ={}&[\\lambda ^{\\cal F}_{\\mathrm {a}}(a,\\eta _\\mathrm {a},t)+ \\lambda ^{\\cal F}_{\\mathrm {b}}(b,\\eta _\\mathrm {b},t)]\\big |{\\lbrace p,f,c^{\\prime },c\\rbrace _{m}}\\big )\\;,\\end{split}$ where $\\lambda ^{\\cal F}_{\\mathrm {a}}(a,\\eta _\\mathrm {a},t)=\\frac{\\frac{d}{dt}\\,f_{a/A}(\\eta _\\mathrm {a},\\mu _\\mathrm {a}^2(t))}{f_{a/A}(\\eta _\\mathrm {a},\\mu _\\mathrm {a}^2(t))}$ with a corresponding expression for $\\lambda ^{\\cal F}_{\\mathrm {b}}$ .",
"Using the parton evolution equation, this is $\\begin{split}\\lambda ^{\\cal F}_{\\mathrm {a}}(a,\\eta _\\mathrm {a},t)={}&-\\sum _{\\hat{a}}\\int _0^{1}\\!dz\\ \\bigg \\lbrace \\frac{\\alpha _{\\mathrm {s}}}{2\\pi }\\,\\frac{1}{z}P_{a\\hat{a}}\\!\\left(z\\right)\\,\\frac{f_{\\hat{a}/A}(\\eta _{\\mathrm {a}}/z, \\mu _\\mathrm {a}^2(t))}{f_{a/A}(\\eta _{\\mathrm {a}}, \\mu _\\mathrm {a}^2(t))}\\\\&-\\delta _{a\\hat{a}}\\,\\frac{\\alpha _{\\mathrm {s}}}{2\\pi }\\left[\\frac{2 C_a}{1-z}-\\gamma _a\\right]\\bigg \\rbrace + {\\cal O}(\\alpha _{\\mathrm {s}}^2)\\;.\\end{split}$ To be more precise, we should use the evolution equation (REF ) appropriate for $\\Lambda ^2$ as the evolution variable.",
"For the moment, it suffices to use this simple form: we ignore the theta function that enforces an infrared cutoff $\\mu _\\perp ^2 > m_\\perp ^2(a)$ and we do not specify the argument of $\\alpha _{\\mathrm {s}}$ ."
],
[
"Approach to calculating ${\\cal S}^{\\rm pert}(t)$",
"We have argued that $-{\\cal S}^{\\rm pert}(t)$ should be calculated from virtual one loop Feynman diagrams.",
"There are some choices for how to do that.",
"First, we should choose a gauge.",
"Ultimately, we expect that Feynman gauge provides the most powerful method, especially if one wants a method that can be extended to higher orders of perturbation theory.",
"However, in Feynman gauge, factorization of softer interactions from harder interactions does not work graph by graph.",
"Gluons that are nearly collinear to external legs of a diagram or are very soft but that carry unphysical longitudinal polarization can couple to the interior of ultraviolet dominated subgraphs.",
"These attachments can be treated after a sum over graphs by the use of the gauge invariance of the theory.",
"However, if we use a physical gauge, only transversely polarized gluons can propagate over long distances.",
"Then we do not need to sum over graphs.",
"For this reason, in appendix , we use Coulomb gauge.",
"Second, virtual graphs do not come with a definition of the shower time $t$ , so to define $-{\\cal S}^{\\rm pert}(t)$ from virtual graphs, we need to provide a relation to $t$ .",
"Ultimately, we expect that the most powerful method is to consider ${\\cal G}(t) = - \\int _{t}^\\infty \\!d\\tau \\ {\\cal S}^{\\rm pert}(\\tau )\\;.$ Here we integrate over the shower time with a lower cutoff at $t$ .",
"This integral is infrared divergent and we understand it to be regulated with dimensional regularization.",
"Then we can obtain ${\\cal S}^{\\rm pert}(t)$ by differentiating with respect to $t$ .",
"This formulation gives us the possibility of incorporating the virtual graphs into a general theory that includes the real graphs and a systematic treatment of factorization and renormalization.",
"In this paper, we choose a method that is computationally a little involved, but relates the definition of the shower time in virtual graphs quite directly to that in the shower splitting functions.",
"We recognize that the splitting functions arise simply from cut Feynman diagrams for real parton emissions in an approximation in which one is close to the infrared singular soft or collinear limits.",
"There is a three dimensional integral over the momentum of the emitted parton, with a delta function that fixes this momentum as a function of the shower time.",
"For virtual diagrams, one can imitate the structure of the real emission diagrams.",
"We begin with an integration over four components of the loop momentum $q$ .",
"Working in a reference frame in which the total momentum $Q$ of the final state partons has only a time component, we can perform the integration over the energy $q^0$ in the loop.",
"This leaves us with an integration $\\int \\!d\\vec{q}$ over the three-momentum in the loop.",
"Then we have an integrand with a similar structure to that of the real emission diagrams, but with ${\\mathrm {i}}\\epsilon $ terms in the denominators modified.",
"For both real and virtual diagrams, we obtain integrands corresponding to the exchange of a soft gluon between partons $l$ and $k$ that form a color dipole.",
"We partition each of these contributions into two, one with a collinear singularity when the gluon momentum $q$ is collinear to the momentum of parton $l$ and one with a collinear singularity when $q$ is collinear to the momentum of parton $k$ .",
"In the splitting function corresponding to a real emission diagram, when a parton with momentum $\\vec{q}$ is emitted from a parton with velocity $\\vec{v}_l$ , there is an infrared singularity when $|\\vec{q}\\,| - \\vec{q} \\cdot \\vec{v}_l \\rightarrow 0$ .",
"This happens in the soft limit $|\\vec{q}\\,| \\rightarrow 0$ or in the collinear limit that $\\vec{q}$ becomes collinear with $\\vec{v}_l$ .",
"Note that $(|\\vec{q}\\,| - \\vec{q} \\cdot \\vec{v}_l)$ is proportional to the virtuality associated with the emission: $|\\vec{q}\\,| - \\vec{q} \\cdot \\vec{v}_l = 2 p_l\\cdot q\\ \\frac{\\sqrt{Q^2}}{2 p_l \\cdot Q}\\;.$ The shower time $t$ associated with this splitting is defined by $\\sqrt{Q^2}\\ \\frac{2 p_l\\cdot Q_0}{2 p_l\\cdot Q}\\,e^{-t} = |\\vec{q}\\,| - \\vec{q} \\cdot \\vec{v}_l\\;.$ Here $Q_0$ is the total momentum of the final state partons at the start of the shower, as in section .",
"In the virtual diagrams, we find that the infrared singularities are controlled by factors of the form $1/(|\\vec{q}\\,| - \\vec{q} \\cdot \\vec{v}_l)$ .",
"We simply insert a delta function that identifies this denominator with the shower time according to eq.",
"(REF ).",
"This leaves an integration over two dimensions, which we can arrange to have the form $\\int \\!dz\\,d\\phi $ to match the integrations in splitting functions.",
"With these manipulations, there is a term in ${\\cal S}^{\\rm pert}(t)$ for each parton in the shower at shower time $t$ : ${\\cal S}^{\\rm pert}(t)={\\cal S}_\\mathrm {a}^{\\rm pert}(t)+ {\\cal S}_\\mathrm {b}^{\\rm pert}(t)+ \\sum _{l=1}^m {\\cal S}_l^{\\rm pert}(t)\\;.$ In the following two subsections, we state our results for these terms.",
"In each case, the results apply in the limit that the dimensionless virtuality variable $y$ is small: $y \\ll 1$ and $y \\ll (1 - \\cos \\theta _{k l})$ ."
],
[
"Virtual contributions for final state partons",
"We sketch the calculation for ${\\cal S}_{l}^{\\rm pert}(t)$ for final state partons in appendix .",
"After combining the definitions (REF ), (REF ), and (REF ) with the results in eqs.",
"(REF ), (REF ), and (REF ), we obtain a simple result, $\\begin{split}{\\cal S}_{l}^{\\rm pert}(t)&\\big |{\\lbrace p,f,c^{\\prime },c\\rbrace _{m}}\\big )\\\\={}& \\bigg \\lbrace -\\frac{\\alpha _{\\mathrm {s}}}{2\\pi }\\,\\sum _{k \\ne l}\\log \\!\\left[\\frac{1 - \\cos \\theta _{k l}}{2}\\right]\\big ([(\\mathbf {T}_l\\cdot \\mathbf {T}_k)\\otimes 1]+ [1 \\otimes (\\mathbf {T}_l\\cdot \\mathbf {T}_k)]\\big )\\\\&\\ \\; + \\sum _{k \\ne \\mathrm {a},\\mathrm {b},l}\\frac{\\alpha _{\\mathrm {s}}}{2\\pi }\\,{\\mathrm {i}}\\pi \\,\\big ([(\\mathbf {T}_l\\cdot \\mathbf {T}_k)\\otimes 1]- [1 \\otimes (\\mathbf {T}_l\\cdot \\mathbf {T}_k)]\\big )\\\\&\\ \\; +\\frac{\\alpha _{\\mathrm {s}}}{2\\pi }\\,\\left(2 C_{f_l}\\log \\!\\left[\\frac{2 |\\vec{p}_l|}{E_Q y}\\right]-\\gamma _{f_l}\\right)[1\\otimes 1]\\bigg \\rbrace \\\\&\\times \\big |{\\lbrace p,f,c^{\\prime },c\\rbrace _{m}}\\big )\\;.\\end{split}$ The constants $C_f$ and $\\gamma _f$ are defined in appendix .",
"In assembling this result, we have used the color identities $T_l\\cdot T_l = C_{f_l}$ and $\\sum _k T_l\\cdot T_k = 0$ .",
"The most notable feature of eq.",
"(REF ) is that ${\\cal S}_{l}^{\\rm pert}(t)$ has contributions proportional to ${\\mathrm {i}}\\pi $ .",
"There is one such contribution for each final state parton $k$ that can form a color dipole with parton $l$ , but there is no contribution for $k = \\mathrm {a}$ or $k = \\mathrm {b}$ .",
"These contributions persist no matter how small $e^{-t}$ is."
],
[
"Virtual contributions for initial state partons",
"We sketch the calculation for ${\\cal S}_{l}^{\\rm pert}(t)$ for initial state partons in appendix .",
"After combining the definitions (REF ), (REF ), and (REF ) with the results in eqs.",
"(REF ), (REF ), and (REF ), we obtain $\\begin{split}{\\cal S}_{\\mathrm {a}}^{\\rm pert}(t)&\\big |{\\lbrace p,f,c^{\\prime },c\\rbrace _{m}}\\big )\\\\={}& \\bigg \\lbrace -\\frac{\\alpha _{\\mathrm {s}}}{2\\pi }\\,\\sum _{k \\ne \\mathrm {a},\\mathrm {b}}\\log \\!\\left[\\frac{1 - \\cos \\theta _{\\mathrm {a}k}}{2}\\right]\\big ([(\\mathbf {T}_\\mathrm {a}\\cdot \\mathbf {T}_k)\\otimes 1]+ [1 \\otimes (\\mathbf {T}_\\mathrm {a}\\cdot \\mathbf {T}_k)]\\big )\\\\&\\ \\; + \\frac{\\alpha _{\\mathrm {s}}}{2\\pi }\\,{\\mathrm {i}}\\pi \\,\\big ([(\\mathbf {T}_\\mathrm {a}\\cdot \\mathbf {T}_\\mathrm {b})\\otimes 1]- [1 \\otimes (\\mathbf {T}_\\mathrm {a}\\cdot \\mathbf {T}_\\mathrm {b})]\\big )\\\\&\\ \\; +\\frac{\\alpha _{\\mathrm {s}}}{2\\pi }\\,\\left(2 C_{a}\\log \\!\\left[\\frac{1}{y}\\right]-\\gamma _{a}\\right)[1\\otimes 1]\\bigg \\rbrace \\\\&\\times \\big |{\\lbrace p,f,c^{\\prime },c\\rbrace _{m}}\\big )\\;.\\end{split}$ In assembling this result, we have used the color identities $T_\\mathrm {a}\\cdot T_\\mathrm {a}= C_{a}$ and $\\sum _k T_\\mathrm {a}\\cdot T_k = 0$ and the kinematic identity $E_Q = 2 |\\vec{p}_\\mathrm {a}|$ .",
"As in the final state ${\\cal S}_{l}^{\\rm pert}(t)$ , there is an ${\\mathrm {i}}\\pi $ contribution, this time associated with the color dipole formed by parton “a” and parton “b.”"
],
[
"The complete ${\\cal S}(t)$",
"The complete operator ${\\cal S}(t)$ is a sum of pieces associated with the individual partons, ${\\cal S}(t)={\\cal S}_\\mathrm {a}(t)+ {\\cal S}_\\mathrm {b}(t)+ \\sum _{l=1}^m {\\cal S}_l(t)\\;.$ For the final state partons, we use simply ${\\cal S}_{l}^{\\rm pert}(t)$ from eq.",
"(REF ).",
"For initial state parton “a”, we use ${\\cal S}_{\\mathrm {a}}^{\\rm pert}(t)$ from eq.",
"(REF ) and add the contribution from parton evolution according to eq.",
"(REF ), (REF ) and (REF ).",
"For final state partons, this gives $\\begin{split}{\\cal S}_{l}(t)&\\big |{\\lbrace p,f,c^{\\prime },c\\rbrace _{m}}\\big )\\\\={}& \\bigg \\lbrace -\\frac{\\alpha _{\\mathrm {s}}}{2\\pi }\\,\\sum _{k \\ne l}\\log \\!\\left[\\frac{1 - \\cos \\theta _{k l}}{2}\\right]\\big ([(\\mathbf {T}_l\\cdot \\mathbf {T}_k)\\otimes 1]+ [1 \\otimes (\\mathbf {T}_l\\cdot \\mathbf {T}_k)]\\big )\\\\&\\ \\; + \\sum _{k \\ne \\mathrm {a},\\mathrm {b},l}\\frac{\\alpha _{\\mathrm {s}}}{2\\pi }\\,{\\mathrm {i}}\\pi \\,\\big ([(\\mathbf {T}_l\\cdot \\mathbf {T}_k)\\otimes 1]- [1 \\otimes (\\mathbf {T}_l\\cdot \\mathbf {T}_k)]\\big )\\\\&\\ \\; +\\frac{\\alpha _{\\mathrm {s}}}{2\\pi }\\,\\left(2 C_{f_l}\\log \\!\\left[\\frac{2 |\\vec{p}_l|}{E_Q y}\\right]-\\gamma _{f_l}\\right)[1\\otimes 1]\\bigg \\rbrace \\\\&\\times \\big |{\\lbrace p,f,c^{\\prime },c\\rbrace _{m}}\\big )\\;.\\end{split}$ For initial state partons, there is a small simplification because terms proportional to $\\gamma _a$ cancel, giving $\\begin{split}{\\cal S}_\\mathrm {a}(t)&\\big |{\\lbrace p,f,c^{\\prime },c\\rbrace _{m}}\\big )\\\\={}& \\bigg \\lbrace -\\frac{\\alpha _{\\mathrm {s}}}{2\\pi }\\,\\sum _{k \\ne \\mathrm {a},\\mathrm {b}}\\log \\!\\left[\\frac{1 - \\cos \\theta _{\\mathrm {a}k}}{2}\\right]\\big ([(\\mathbf {T}_\\mathrm {a}\\cdot \\mathbf {T}_k)\\otimes 1]+ [1 \\otimes (\\mathbf {T}_\\mathrm {a}\\cdot \\mathbf {T}_k)]\\big )\\\\&\\ \\; + \\frac{\\alpha _{\\mathrm {s}}}{2\\pi }\\,{\\mathrm {i}}\\pi \\,\\big ([(\\mathbf {T}_\\mathrm {a}\\cdot \\mathbf {T}_\\mathrm {b})\\otimes 1]- [1 \\otimes (\\mathbf {T}_\\mathrm {a}\\cdot \\mathbf {T}_\\mathrm {b})]\\big )\\\\&\\ \\; +\\frac{\\alpha _{\\mathrm {s}}}{2\\pi }\\,2 C_{a}\\log \\!\\left[\\frac{1}{y}\\right][1\\otimes 1]\\\\&\\ \\; +\\sum _{\\hat{a}}\\int _0^{1}\\!dz\\ \\frac{\\alpha _{\\mathrm {s}}}{2\\pi }\\bigg (\\frac{1}{z}P_{a\\hat{a}}\\!\\left(z\\right)\\,\\frac{f_{\\hat{a}/A}(\\eta _{\\mathrm {a}}/z, \\mu _\\mathrm {a}^2(t))}{f_{a/A}(\\eta _{\\mathrm {a}}, \\mu _\\mathrm {a}^2(t))}-\\delta _{a\\hat{a}}\\frac{2 C_a}{1-z}\\bigg )[1\\otimes 1]\\bigg \\rbrace \\\\&\\times \\big |{\\lbrace p,f,c^{\\prime },c\\rbrace _{m}}\\big )\\;.\\end{split}$ We will use these results in the following section."
],
[
"The cross section changing exponent",
"The operator ${\\cal N}_{{\\cal V}}(t_2,t_1) = \\mathbb {T} \\exp \\left[-\\int _{t_1}^{t_2} d\\tau \\ {\\cal V}(\\tau )\\right]\\;,$ inserted between parton splittings, will preserve the Born cross section throughout the shower.",
"This statement is exact in color as long as we use the exact ${\\cal V}(\\tau )$ and the exact splitting operator ${\\cal H}_\\mathrm {I}(t)$ .",
"In the simplest application, we use the leading color approximation or the LC+ approximation for both ${\\cal H}_\\mathrm {I}(t)$ and ${\\cal V}(\\tau )$ .",
"Then the Born cross section is also preserved in the shower.",
"In a leading order shower, it would be better to use ${\\cal N}_{{\\cal S}}(t_2,t_1) = \\mathbb {T} \\exp \\!\\left[-\\int _{t_1}^{t_2} d\\tau \\ {\\cal S}(\\tau )\\right]\\;,$ where ${\\cal S}(t)$ is obtained from virtual one loop graphs and parton evolution as explained above.",
"Then we have ${\\cal N}_{{\\cal S}}(t_2,t_1) = \\mathbb {T} \\exp \\!\\left[\\int _{t_1}^{t_2} d\\tau \\ [-{\\cal V}(\\tau ) + ({\\cal V}(\\tau ) - {\\cal S}(\\tau ))]\\right]\\;.$ The first term creates a cross section preserving shower, while the second term sums effects that change the cross section.",
"We thus need the cross section changing integrand $({\\cal V}(\\tau ) - {\\cal S}(\\tau ))$ and its integral over $\\tau $ .",
"We assemble this from our previous results."
],
[
"Final state partons",
"For the contribution from final state partons, we subtract eq.",
"(REF ) for ${\\cal S}_l(\\tau )$ from eq.",
"(REF ) for ${\\cal V}_l(\\tau )$ .",
"Almost everything cancels and we are left with $\\begin{split}({\\cal V}_{l}(t) - {\\cal S}_{l}(t))&\\big |{\\lbrace p,f,c^{\\prime },c\\rbrace _{m}}\\big )\\\\={}&\\bigg \\lbrace - \\sum _{k \\ne \\mathrm {a},\\mathrm {b},l}\\frac{\\alpha _{\\mathrm {s}}(\\mu ^2_l(t))}{2\\pi }\\,{\\mathrm {i}}\\pi \\,\\big ([(\\mathbf {T}_l\\cdot \\mathbf {T}_k)\\otimes 1]- [1 \\otimes (\\mathbf {T}_l\\cdot \\mathbf {T}_k)]\\big )\\bigg \\rbrace \\\\&\\times \\big |{\\lbrace p,f,c^{\\prime },c\\rbrace _{m}}\\big )\\;.\\end{split}$ That is, only the “${\\mathrm {i}}\\pi $ ” terms remain.",
"We have supplied the virtuality of the splitting as the argument of $\\alpha _{\\mathrm {s}}$ , using the definition (REF ).",
"Integrating over $t$ gives $\\begin{split}\\int _{t_1}^{t_2}\\!d\\tau \\ ({\\cal V}_{l}(t)& - {\\cal S}_{l}(t))\\big |{\\lbrace p,f,c^{\\prime },c\\rbrace _{m}}\\big )\\\\={}&\\bigg \\lbrace -\\int _{\\mu ^2_l(t_2)}^{\\mu ^2_l(t_1)} \\frac{d\\mu ^2}{\\mu ^2}\\sum _{k \\ne \\mathrm {a},\\mathrm {b},l}\\frac{\\alpha _{\\mathrm {s}}(\\mu ^2)}{2\\pi }\\,{\\mathrm {i}}\\pi \\,\\big ([(\\mathbf {T}_l\\cdot \\mathbf {T}_k)\\otimes 1]- [1 \\otimes (\\mathbf {T}_l\\cdot \\mathbf {T}_k)]\\big )\\bigg \\rbrace \\\\&\\times \\big |{\\lbrace p,f,c^{\\prime },c\\rbrace _{m}}\\big )\\;.\\end{split}$"
],
[
"Initial state partons",
"For the contribution from initial state parton “a,” we subtract eq.",
"(REF ) for ${\\cal S}_\\mathrm {a}(\\tau )$ from eq.",
"(REF ) for ${\\cal V}_\\mathrm {a}(\\tau )$ .",
"Quite a lot cancels and we are left with $\\begin{split}({\\cal V}_{\\mathrm {a}}(t) - {\\cal S}_{\\mathrm {a}}(t))&\\big |{\\lbrace p,f,c^{\\prime },c\\rbrace _{m}}\\big )\\\\={}& \\bigg [-\\int _{1/(1+y)}^1\\!dz\\ \\frac{\\alpha _{\\mathrm {s}}}{2\\pi }\\sum _{\\hat{a}}\\ \\left(\\frac{f_{\\hat{a}/A}(\\eta _{\\mathrm {a}}/z,\\mu _\\mathrm {a}^2(t))}{zf_{a/A}(\\eta _{\\mathrm {a}},\\mu _\\mathrm {a}^2(t))}P_{a\\hat{a}}(z)- \\delta _{a\\hat{a}}\\frac{2C_a}{1-z}\\right)[1\\otimes 1]\\\\& -\\int _0^{1/(1+y)}\\!dz\\ \\frac{\\alpha _{\\mathrm {s}}}{2\\pi }\\left[1 -\\frac{f_{a/A}(\\eta _{\\mathrm {a}}/z,\\mu _\\mathrm {a}^2(t))}{f_{a/A}(\\eta _{\\mathrm {a}},\\mu _\\mathrm {a}^2(t))}\\right]\\sum _{k\\ne \\mathrm {a},\\mathrm {b}}\\Delta _{\\mathrm {a}k}(z,y)\\\\&\\qquad \\quad \\times \\big ([(\\mathbf {T}_\\mathrm {a}\\cdot \\mathbf {T}_k)\\otimes 1] + [1 \\otimes (\\mathbf {T}_\\mathrm {a}\\cdot \\mathbf {T}_k)]\\big )\\\\&- \\frac{\\alpha _{\\mathrm {s}}}{2\\pi }\\,{\\mathrm {i}}\\pi \\,\\big ([(\\mathbf {T}_\\mathrm {a}\\cdot \\mathbf {T}_\\mathrm {b})\\otimes 1]- [1 \\otimes (\\mathbf {T}_\\mathrm {a}\\cdot \\mathbf {T}_\\mathrm {b})]\\big )\\bigg ]\\\\&\\times \\big |{\\lbrace p,f,c^{\\prime },c\\rbrace _{m}}\\big )\\;.\\end{split}$ There is an “${\\mathrm {i}}\\pi $ ” term that does not cancel.",
"There is a term with a non-trivial color factor that is proportional to the function $\\Delta _{\\mathrm {a}k}(z,y)$ defined in eq.",
"(REF ).",
"Finally, there is a term with a trivial color factor $[1 \\otimes 1]$ .",
"This term arises from using a contribution from ${\\cal V}_{\\mathrm {a}}(t)$ that is integrated over $0 < z < 1/(1+y)$ and subtracting a term with the same integrand from ${\\cal S}_{\\mathrm {a}}(t)$ that is integrated over $0 < z < 1$ .",
"In the difference, we integrate over $1/(1+y) < z < 1$ .",
"For small $y$ , this is a small range near $z = 1$ .",
"We can integrate this over a range of shower times, $t_1 < \\tau < t_2$ .",
"This gives the probability changing Sudakov exponent associated with going from a splitting at shower time $t_1$ to a later splitting at shower time $t_2$ .",
"We use $y = \\mu _\\mathrm {a}^2(\\tau )/Q^2$ , where $Q^2$ is the square of the total momentum of the final state particles at shower time $t_1$ .",
"Additionally, in the $[1 \\otimes 1]$ term, we decompose the splitting function $P_{a\\hat{a}}(z)$ into $P_{a\\hat{a}}(z) = \\delta _{a \\hat{a}}\\,{2 z C_a}/(1-z) + P_{a\\hat{a}}^{\\rm reg}(z)$ as in eq.",
"(REF ).",
"We keep the nonsingular $P_{a\\hat{a}}^{\\rm reg}(z)$ term because it can give a large contribution for certain flavor choices.",
"This gives $\\begin{split}\\int _{t_1}^{t_2}\\!d\\tau \\ ({\\cal V}_{\\mathrm {a}}(\\tau )& - {\\cal S}_{\\mathrm {a}}(\\tau ))\\big |{\\lbrace p,f,c^{\\prime },c\\rbrace _{m}}\\big )\\\\={}&\\int _{\\mu ^2_\\mathrm {a}(t_2)}^{\\mu ^2_\\mathrm {a}(t_1)} \\frac{d\\mu ^2}{\\mu ^2}\\ \\Bigg \\lbrace \\bigg [\\int _{1/(1+\\mu ^2/Q^2)}^1\\!dz\\ \\frac{\\alpha _{\\mathrm {s}}}{2\\pi }\\left[1- \\frac{f_{a/A}(\\eta _{\\mathrm {a}}/z,\\mu ^2)}{f_{a/A}(\\eta _{\\mathrm {a}},\\mu ^2)}\\right]\\frac{2C_a}{1-z}\\,[1\\otimes 1]\\\\&-\\int _{1/(1+\\mu ^2/Q^2)}^1\\!dz\\ \\frac{\\alpha _{\\mathrm {s}}}{2\\pi }\\sum _{\\hat{a}}\\,\\frac{f_{\\hat{a}/A}(\\eta _{\\mathrm {a}}/z,\\mu ^2)}{zf_{a/A}(\\eta _{\\mathrm {a}},\\mu ^2)}\\,P_{a\\hat{a}}^{\\rm reg}(z)\\,[1\\otimes 1]\\\\& -\\int _0^{1/(1+\\mu ^2/Q^2)}\\!dz\\ \\frac{\\alpha _{\\mathrm {s}}}{2\\pi }\\left[1 -\\frac{f_{a/A}(\\eta _{\\mathrm {a}}/z,\\mu ^2)}{f_{a/A}(\\eta _{\\mathrm {a}},\\mu ^2)}\\right]\\sum _{k\\ne \\mathrm {a},\\mathrm {b}}\\Delta _{\\mathrm {a}k}(z,\\mu ^2/Q^2)\\\\&\\qquad \\quad \\times \\big ([(\\mathbf {T}_\\mathrm {a}\\cdot \\mathbf {T}_k)\\otimes 1] + [1 \\otimes (\\mathbf {T}_\\mathrm {a}\\cdot \\mathbf {T}_k)]\\big )\\\\&- \\frac{\\alpha _{\\mathrm {s}}}{2\\pi }\\,{\\mathrm {i}}\\pi \\,\\big ([(\\mathbf {T}_\\mathrm {a}\\cdot \\mathbf {T}_\\mathrm {b})\\otimes 1]- [1 \\otimes (\\mathbf {T}_\\mathrm {a}\\cdot \\mathbf {T}_\\mathrm {b})]\\big )\\bigg ]\\Bigg \\rbrace \\\\&\\times \\big |{\\lbrace p,f,c^{\\prime },c\\rbrace _{m}}\\big )\\;.\\end{split}$ In the remainder of this section, we discuss the physics of the four terms in eq.",
"(REF ), revise the $\\Delta _{\\mathrm {a}k}$ term to better reflect the physics, implement an infrared cutoff, and specify the argument of $\\alpha _{\\mathrm {s}}$ in each term."
],
[
"Integration ranges",
"We first consider the effective ranges of $\\mu ^2$ and $z$ that appear in each term in eq.",
"(REF ).",
"All but the last term in eq.",
"(REF ) involve the ratio $y = \\mu ^2/Q^2$ for a potential splitting that might occur in hadron A between the previous splitting at shower time $t_1$ and the next shower time $t_2$ at which a splitting occurs.",
"Suppose that $\\tilde{y} = \\tilde{\\mu }^2/\\tilde{Q}^2$ is the corresponding hardness variable for a previous splitting in hadron A, perhaps the one at $t_1$ if it was in hadron A or perhaps an earlier splitting.",
"How is $\\mu ^2/Q^2$ related to $\\tilde{\\mu }^2/\\tilde{Q}^2$ ?",
"The relation between $\\mu ^2/Q^2$ and the shower time for initial state splittings is given in eq.",
"(REF ), $\\begin{split}\\frac{\\mu ^2}{Q^2}\\equiv y ={}& \\frac{2 p_\\mathrm {a}\\cdot Q_0}{2 p_\\mathrm {a}\\cdot p_\\mathrm {b}}\\,e^{-t}.\\end{split}$ As the shower progresses with increasing $t$ , the momentum $p_\\mathrm {b}$ can stay the same or increase.",
"Thus $\\mu ^2/Q^2$ decreases: $\\frac{\\mu ^2}{Q^2} < \\frac{\\tilde{\\mu }^2}{\\tilde{Q}^2}\\;.$ We now examine the terms in eq.",
"(REF ), beginning with the simplest term, the one proportional to ${\\mathrm {i}}\\pi $ .",
"This term can have important effects, but it does not change the inclusive probability $\\big ({1}\\big |{\\rho (t)}\\big )$ associated with the statistical state.",
"Its effect continues for all $\\mu ^2$ from the beginning of the shower to the end of the shower.",
"Consider next the second term, proportional to $P_{a\\hat{a}}^{\\rm reg}(z)$ .",
"The integrand here has no $z \\rightarrow 1$ singularity.",
"The integration over $z$ covers the range $0 < 1-z < \\mu ^2/Q^2/(1+\\mu ^2/Q^2)$ .",
"Thus this term contributes only near the start of the shower, when $\\mu ^2/Q^2 \\sim 1$ .",
"It does not contribute when $\\mu ^2/Q^2 \\ll 1$ .",
"Now consider the first term.",
"Here there is a factor $2 C_a/(1-z)$ , so we are sensitive to small $(1-z)$ .",
"We can understand the integration region, at least qualitatively, by approximatingWe will explore this approximation further in section .",
"the parton distribution functions as $f_{a/A}(\\eta _{\\mathrm {a}},\\mu ^2) \\propto \\eta _{\\mathrm {a}}^{-N}$ .",
"We consider the case that $N$ is large, which happens when $\\eta _\\mathrm {a}$ is bigger than about 0.1.",
"Here “$N$ is large” means something like $N \\sim 3$ in realistic cases.",
"With this approximation, we have $\\left[1- \\frac{f_{a/A}(\\eta _{\\mathrm {a}}/z,\\mu ^2)}{f_{a/A}(\\eta _{\\mathrm {a}},\\mu ^2)}\\right]\\sim [1 - z^N] \\sim [1 - \\exp (-N(1-z))]\\;.$ This factor in eq.",
"(REF ) is 1 for $(1-z) \\gg 1/N$ and tends to zero for $(1-z) \\ll 1/N$ .",
"Thus we approximate $\\left[1- \\frac{f_{a/A}(\\eta _{\\mathrm {a}}/z,\\mu ^2)}{f_{a/A}(\\eta _{\\mathrm {a}},\\mu ^2)}\\right]\\sim \\theta \\!\\left(\\frac{1}{N} < (1-z)\\right)\\;.$ We also approximate $1 - 1/(1+\\mu ^2/Q^2) \\sim \\mu ^2/Q^2$ .",
"Thus the integral in the first term is roughly $\\begin{split}\\int _{\\mu ^2(t_2)}^{\\mu ^2(t_1)}& \\frac{d\\mu ^2}{\\mu ^2}\\ \\int _{1/(1+\\mu ^2/Q^2)}^1\\!\\frac{dz}{1-z}\\ \\left[1- \\frac{f_{a/A}(\\eta _{\\mathrm {a}}/z,\\mu ^2)}{f_{a/A}(\\eta _{\\mathrm {a}},\\mu ^2)}\\right]\\\\\\sim {}&\\int _{\\mu ^2(t_2)}^{\\mu ^2(t_1)} \\frac{d\\mu ^2}{\\mu ^2}\\ \\theta \\!\\left(\\frac{1}{N} < \\frac{\\mu ^2}{Q^2}\\right)\\int _{0}^1\\!\\frac{dz}{1-z}\\ \\theta \\!\\left(\\frac{1}{N} < (1-z) < \\frac{\\mu ^2}{Q^2}\\right)\\;.\\end{split}$ Evidently, this is a rather crude approximation, but it is instructive.",
"The integral would be a small perturbative correction that we could simply ignore except that $N$ is large.",
"This gives a contribution to the exponent proportional to $\\log ^2 N$ .",
"However, the range of $\\mu ^2/Q^2$ does not extend down to infinitesimal values.",
"As soon as $\\mu ^2/Q^2 < 1/N$ , there is no more contribution.",
"That is, the threshold factor associated with this term in eq.",
"(REF ) comes from the first few steps in shower evolution.",
"The remaining term in eq.",
"(REF ) has more complicated structure.",
"It contains a sum over final state partons $k$ and factor $\\Delta _{\\mathrm {a}k}$ , defined in eq.",
"(REF ).",
"This factor depends on the angle $\\theta _{\\mathrm {a}k}$ the initial state parton “a” and parton $k$ .",
"The angle $\\theta _{\\mathrm {a}k}$ appears in the combination $\\psi _{\\mathrm {a}k} = (1 - \\cos \\theta _{\\mathrm {a}k})/\\sqrt{8(1 + \\cos \\theta _{\\mathrm {a}k})}$ .",
"From the definition, we see that $\\Delta _{\\mathrm {a}k} \\sim 1/(1-z)$ for small $(1-z)$ with $(1-z) \\ll (\\mu ^2/Q^2)/ \\psi _{\\mathrm {a}k}$ .",
"However, when $(\\mu ^2/Q^2)/\\psi _{\\mathrm {a}k} \\ll (1-z)$ , $\\Delta _{\\mathrm {a}k}$ is small compared to $1/(1-z)$ Thus we can roughly approximate $\\Delta _{\\mathrm {a}k}$ by $\\Delta _{\\mathrm {a}k} \\sim \\frac{1}{1-z}\\ \\theta \\!\\left((1-z) <\\frac{\\mu ^2}{2Q^2\\psi _{\\mathrm {a}k}}\\right)\\;.$ (The factor 2 here is rather arbitrary.)",
"For the parton factor in this term, we can use the rough approximation (REF ).",
"We can also approximate the upper limit of the $z$ -integration as $(1-z) < \\mu ^2/Q^2$ .",
"This gives us the integral $\\begin{split}\\int _{\\mu ^2(t_2)}^{\\mu ^2(t_1)}& \\frac{d\\mu ^2}{\\mu ^2}\\ \\int _0^{1/(1+\\mu ^2/Q^2)}\\!dz\\ \\left[1 -\\frac{f_{a/A}(\\eta _{\\mathrm {a}}/z,\\mu ^2)}{f_{a/A}(\\eta _{\\mathrm {a}},\\mu ^2)}\\right]\\Delta _{\\mathrm {a}k}(z,\\mu ^2/Q^2)\\\\\\sim {}&\\int _0^{1}\\!\\frac{dz}{1-z}\\ \\theta \\!\\left(\\frac{1}{N} < (1-z)\\right)\\int _{\\mu ^2(t_2)}^{\\mu ^2(t_1)} \\frac{d\\mu ^2}{\\mu ^2}\\ \\theta \\!\\left(2 \\psi _{\\mathrm {a}k} (1-z) < \\frac{\\mu ^2}{Q^2}< (1-z)\\right)\\;.\\end{split}$ We will consider the behavior of this integral both for $\\psi _{\\mathrm {a}k} \\sim 1$ and for $\\psi _{\\mathrm {a}k} \\ll 1$ .",
"Let us first consider the case that $\\psi _{\\mathrm {a}k} \\sim 1$ .",
"Then we integrate $(1-z)$ over a range that is large when $N$ is large, giving a $\\log N$ .",
"On the other hand, $\\mu ^2/Q^2$ is integrated over a finite range.",
"Thus we have only a single power of $\\log N$ .",
"We note also that $\\mu ^2/Q^2$ is never smaller than a number of order $1/N$ .",
"Now consider the case that $\\psi _{\\mathrm {a}k} \\ll 1$ , supposing that $\\mu ^2(t_2) \\rightarrow 0$ .",
"Then the lower bound $2 \\psi _{\\mathrm {a}k} (1-z)$ on $\\mu ^2/Q^2$ in eq.",
"(REF ) is very small and one may wonder if this leads to a large integral.",
"When $\\psi _{\\mathrm {a}k} \\ll 1$ , we have $2\\psi _{\\mathrm {a}k} \\approx \\theta ^2_{\\mathrm {a}k}/4\\;.$ Then parton $k$ makes a very small angle with respect to the beam axis.",
"This can happen with a large probability when, late in the shower, parton $k$ was emitted from the initial state parton from hadron A.",
"The most important case to consider is that the latest real emission before the shower time interval under consideration was the emission of parton $k$ .",
"Then the upper endpoint of the $\\mu ^2$ integration, $\\mu ^2 < \\mu ^2(t_1)$ corresponds to the bound (REF ), in which $\\tilde{\\mu }^2$ and $\\tilde{Q}^2$ refer to the splitting at which parton $k$ was created.",
"We have $\\tilde{\\mu }^2 = 2 p_\\mathrm {a}\\cdot p_k \\approx E_a E_k \\theta ^2_{\\mathrm {a}k}$ and $\\tilde{Q}^2 = 2 \\tilde{p}_\\mathrm {a}\\cdot p_\\mathrm {b}\\approx 4\\tilde{E}_\\mathrm {a}E_\\mathrm {b}$ .",
"Here $\\tilde{E}_\\mathrm {a}= \\tilde{z} E_\\mathrm {a}$ is the momentum of the mother parton of the previous splitting, while the energy of the of the emitted parton is approximately $E_k \\approx (1-\\tilde{z}) E_\\mathrm {a}$ .",
"Thus $\\frac{\\tilde{\\mu }^2}{\\tilde{Q}^2} \\approx \\frac{E_\\mathrm {a}[(1-\\tilde{z}) E_\\mathrm {a}] \\theta ^2_{\\mathrm {a}k}}{4\\tilde{z} E_\\mathrm {a}E_\\mathrm {b}}\\;.$ Since we define energies and angles in the $\\vec{Q} = 0$ frame, we have $E_\\mathrm {a}= E_\\mathrm {b}$ .",
"Thus $\\frac{\\tilde{\\mu }^2}{\\tilde{Q}^2} \\approx \\frac{(1-\\tilde{z}) \\theta ^2_{\\mathrm {a}k}}{4\\tilde{z} }\\;.$ The splitting function for the previous emission contained a ratio $f_{a/A}(\\tilde{\\eta }/\\tilde{z}, \\tilde{\\mu }^2)/ f_{a/A}(\\tilde{\\eta }, \\tilde{\\mu }^2)$ of parton distributions.",
"Applying the approximation (REF ) to this ratio, we have $(1 - \\tilde{z}) < 1/N$ .",
"Thus also, $\\tilde{z}$ is close to 1.",
"This gives $\\frac{\\tilde{\\mu }^2}{\\tilde{Q}^2} \\lesssim \\frac{\\theta ^2_{\\mathrm {a}k}}{4 N }\\;.$ Since $\\mu ^2/Q^2 < {\\tilde{\\mu }^2}/{\\tilde{Q}^2}$ , we have $\\frac{\\mu ^2}{Q^2} \\lesssim \\frac{\\theta ^2_{\\mathrm {a}k}}{4 N }\\;.$ In the integrand of eq.",
"(REF ), we have $(1-z)\\theta ^2_{\\mathrm {a}k}/4 < \\mu ^2/Q^2$ and $1/N < (1-z)$ .",
"This gives $\\frac{\\theta ^2_{\\mathrm {a}k}}{4 N }< \\frac{\\mu ^2}{Q^2}\\;.$ When we combine the upper bound (REF ) with the lower bound (REF ), we see that ${\\mu ^2}/{Q^2}$ can vary only over a small range around ${\\theta ^2_{\\mathrm {a}k}}/({4 N })$ .",
"Similarly, $(1-z)$ varies only over a small range around $1/N$ ."
],
[
"Revised $\\Delta _{\\mathrm {a}k}$ term",
"We conclude that there are no $\\log N$ factors associated with the integration in the $\\Delta _{\\mathrm {a}k}$ term eq.",
"(REF ) in the case that $\\psi _{\\mathrm {a}k} \\ll 1$ .",
"The integral does have a finite contribution proportional to $\\alpha _{\\mathrm {s}}$ with no $\\log N$ factors.",
"However, a first order parton shower is not adequate to calculate this contribution accurately: the definition of the parton shower splitting functions incorporates the strong ordering condition $\\mu ^2/Q^2 \\ll {\\tilde{\\mu }^2}/{\\tilde{Q}^2}$ and that condition is violated here.",
"Ordinarily, the inclusion of an inaccurately calculated small perturbative correction to the cross section would be of little consequence.",
"However, this correction can occur many times as the shower progresses, leading to a large, inaccurately calculated, correction.",
"Thus, we eliminate this contribution from the $\\Delta _{\\mathrm {a}k}$ term at small $\\psi _{\\mathrm {a}k}$ by multiplying this term by $\\theta (\\psi _{\\mathrm {a}k} > \\psi _{\\rm min})$ , where the default value of $\\psi _{\\rm min}$ is $10^{-2}$ ."
],
[
"Infrared cutoff",
"Recall now from section REF that the shower algorithm has an infrared cutoff that vetoes initial state splittings unless the transverse momentum in the splitting is above a minimum: $(1-z) \\mu ^2 > m_\\perp ^2(a)$ .",
"Here $m_\\perp (a)$ is the shower cutoff scale of order 1 GeV or the heavy quark mass in the case of a charm or bottom quark.",
"Eq.",
"(REF ) was derived with no infrared cutoff, but we can insert the cutoff by inserting a factor $\\theta ((1-z) \\mu ^2 > m_\\perp ^2(a))$ into the integrations over $z$ .",
"This cutoff has negligible effect as long as $\\mu ^2 \\gg m_\\perp (a) Q$ .",
"Since the whole integral is negligible unless $\\mu ^2 > Q^2/N$ , we see that inserting the infrared cutoff has negligible effect as long as $m_\\perp (a) < Q/N$ .",
"This is the case in situations of phenomenological interest, in which $Q$ is of order 1 TeV, or at least 100 GeV for Tevatron studies, and $m_\\perp (a)$ is at most 5 GeV.",
"Thus we make eq.",
"(REF ) consistent with the rest of the shower algorithm and with our treatment of parton evolution by inserting a factor $\\theta ((1-z) \\mu ^2 > m_\\perp ^2(a))$ into the $z$ -integrations in Eq.",
"(REF ).In a future publication, we hope to derive the real and virtual shower splitting functions in a more general framework in which the infrared cutoff is included from the start."
],
[
"Running coupling",
"Now we need to specify the argument of the running coupling $\\alpha _{\\mathrm {s}}$ .",
"The simplest choice would be $\\alpha _{\\mathrm {s}}(\\mu ^2)$ .",
"Instead, in the two soft-sensitive terms (proportional to $1/(1-z)$ and $\\Delta _{\\mathrm {a}k}$ ) we use $\\lambda _\\mathrm {R}\\mu _\\perp ^2 = \\lambda _\\mathrm {R}(1-z) \\mu ^2$ as the argument of $\\alpha _{\\mathrm {s}}$ .",
"Here $\\lambda _\\mathrm {R}$ [51] is the constant defined in eq.",
"(REF ).",
"The factors $(1-z)$ and $\\lambda _\\mathrm {R}$ in the argument of $\\alpha _{\\mathrm {s}}$ are, strictly speaking, beyond the order of perturbation theory that we control in a leading order shower, but it is helpful in generating next-to-leading logarithms for at least some inclusive observables [51], [66].",
"The use of $\\alpha _{\\mathrm {s}}(\\lambda _\\mathrm {R}(1-z) \\mu ^2)$ inside the $z$ integrations would create an artificial problem if we integrated down to $(1-z) = 0$ .",
"However, with the cutoff $(1-z) \\mu ^2 > m_\\perp ^2(a)$ to keep us out of the nonperturbative region, we do not encounter this problem.",
"For the remaining two terms in eq.",
"(REF ), we use simply $\\alpha _{\\mathrm {s}}(\\lambda _\\mathrm {R}\\mu ^2)$ ."
],
[
"Result",
"With these substitutions, we have $\\begin{split}\\int _{t_1}^{t_2}\\!d\\tau \\ &({\\cal V}_{\\mathrm {a}}(\\tau ) - {\\cal S}_{\\mathrm {a}}(\\tau ))\\big |{\\lbrace p,f,c^{\\prime },c\\rbrace _{m}}\\big )\\\\={}&\\int _{\\mu ^2_\\mathrm {a}(t_2)}^{\\mu ^2_\\mathrm {a}(t_1)} \\frac{d\\mu ^2}{\\mu ^2}\\ \\Bigg \\lbrace \\int _{1/(1 + \\mu ^2/Q^2)}^1\\!dz\\ \\frac{\\alpha _{\\mathrm {s}}(\\lambda _\\mathrm {R}(1-z) \\mu ^2)}{2\\pi }\\,\\theta ((1-z) \\mu ^2 > m_\\perp ^2(a))\\\\&\\quad \\times \\left[1 -\\frac{f_{a/A}(\\eta _{\\mathrm {a}}/z,\\mu ^2)}{f_{a/A}(\\eta _{\\mathrm {a}},\\mu ^2)}\\right]\\frac{2C_a}{1-z}\\ [1\\otimes 1]\\\\&-\\int _{1/(1 + \\mu ^2/Q^2)}^1\\!dz\\ \\frac{\\alpha _{\\mathrm {s}}(\\lambda _\\mathrm {R}\\mu ^2)}{2\\pi }\\,\\theta ((1-z) \\mu ^2 > m_\\perp ^2(a))\\\\&\\quad \\times \\sum _{\\hat{a}}\\frac{f_{\\hat{a}/A}(\\eta _{\\mathrm {a}}/z,\\mu _\\mathrm {a}^2(t))}{zf_{a/A}(\\eta _{\\mathrm {a}},\\mu _\\mathrm {a}^2(t))}\\,P_{a\\hat{a}}^{\\rm reg}(z)\\ [1\\otimes 1]\\\\& -\\int _0^{1/(1 + \\mu ^2/Q^2)}\\!dz\\ \\frac{\\alpha _{\\mathrm {s}}(\\lambda _\\mathrm {R}(1-z) \\mu ^2)}{2\\pi }\\,\\theta ((1-z) \\mu ^2 > m_\\perp ^2(a))\\\\&\\quad \\times \\left[1 -\\frac{f_{a/A}(\\eta _{\\mathrm {a}}/z,\\mu ^2)}{f_{a/A}(\\eta _{\\mathrm {a}},\\mu ^2)}\\right]\\sum _{k\\ne \\mathrm {a},\\mathrm {b}}\\Delta _{\\mathrm {a}k}(z,\\mu ^2/Q^2)\\,\\theta (\\psi _{\\mathrm {a}k} > \\psi _{\\rm min})\\\\&\\quad \\times \\big ([(\\mathbf {T}_\\mathrm {a}\\cdot \\mathbf {T}_k)\\otimes 1]+ [1 \\otimes (\\mathbf {T}_\\mathrm {a}\\cdot \\mathbf {T}_k)]\\big )\\\\&- \\frac{\\alpha _{\\mathrm {s}}(\\lambda _\\mathrm {R}\\mu ^2)}{2\\pi }\\,{\\mathrm {i}}\\pi \\,\\big ([(\\mathbf {T}_\\mathrm {a}\\cdot \\mathbf {T}_\\mathrm {b})\\otimes 1]- [1 \\otimes (\\mathbf {T}_\\mathrm {a}\\cdot \\mathbf {T}_\\mathrm {b})]\\big )\\Bigg \\rbrace \\\\&\\times \\big |{\\lbrace p,f,c^{\\prime },c\\rbrace _{m}}\\big )\\;.\\end{split}$ Some discussion of this result may be useful.",
"In the first term here, there is a singular factor $1/(1-z)$ .",
"As discussed above, the singularity is cancelled because the ratio of parton distribution functions approaches 1 as $(1-z) \\rightarrow 0$ .",
"This constant will be large if the derivative of $f_{a/A}(\\eta _{\\mathrm {a}},\\mu _\\mathrm {a}^2(t))$ with respect to $\\eta _\\mathrm {a}$ is large.",
"Then there is a “threshold enhancement” of the cross section.",
"Since the integration range in $z$ disappears when $\\mu ^2/Q^2 \\rightarrow 0$ , the important contribution comes from the region in which $\\mu ^2/Q^2$ is not too small.",
"In the second term, there is no singular factor $1/(1-z)$ .",
"In standard treatments, there is no “threshold log.” However, the ratio of parton distributions can be large, for instance for $a = \\mathrm {g}$ and $\\hat{a} = \\mathrm {u}$ .",
"For this reason, we retain this contribution.",
"The term in eq.",
"(REF ) proportional to $\\Delta _{\\mathrm {a}k}$ appears when there is a final state parton that can form a color dipole with the initial state parton.",
"This does not happen in the starting configuration of the Drell-Yan process, before any final state partons have been emitted.",
"However, the $\\Delta _{\\mathrm {a}k}$ term does appear in the starting parton configuration for jet production.",
"The $\\Delta _{\\mathrm {a}k}$ term also appears for any hard process once one or more final state partons have been emitted by initial state radiation.",
"The color factor for this term is non-trivial.",
"If we use the LC+ approximation instead of full color, the $\\Delta _{\\mathrm {a}k}$ term contributes when parton $k$ is color connected to parton “a.” For the reasons given above in this section, we turn this term off when $\\psi _{\\mathrm {a}k}$ is too small.",
"Here $\\Delta _{ak}(z,y)$ and $\\psi _{\\mathrm {a}k}$ were defined in eqs.",
"(REF ) and (REF ).",
"The last term in eq.",
"(REF ) is proportional to ${\\mathrm {i}}\\pi $ .",
"This term is not associated with $1/(1-z)$ singularities, but it is potentially important."
],
[
"The cross section changing exponent in the LC+ approximation",
"The color structure of ${\\cal V}(\\tau )$ and of $({\\cal V}(\\tau ) - {\\cal S}(\\tau ))$ is non-trivial.",
"However, the current version of Deductor uses the LC+ approximation.",
"In this approximation, the operators are diagonal in color and thus commute with each other.",
"Then ${\\cal N}_{{\\cal S}}(t_2,t_1) =K(t_2,t_1)\\,\\exp \\!\\left[-\\int _{t_1}^{t_2} d\\tau \\ {\\cal V}(\\tau )\\right]\\hspace{28.45274pt}{\\rm for\\ LC+}\\;,$ where $K(t_2,t_1) = \\exp \\!\\left[\\int _{t_1}^{t_2} d\\tau \\ ({\\cal V}(\\tau ) - {\\cal S}(\\tau ))\\right]\\hspace{28.45274pt}{\\rm for\\ LC+}\\;.$ Thus, within the LC+ approximation, we can generate the shower using ${\\cal V}(\\tau )$ and then, at each splitting, multiply by a numerical factor $K(t_2,t_1)$ .",
"The exponent in $K$ is a sum over partons $\\begin{split}K(t_2,&t_1) =\\\\& \\exp \\!\\left[\\int _{t_1}^{t_2} d\\tau \\left( ({\\cal V}_\\mathrm {a}(\\tau ) - {\\cal S}_\\mathrm {a}(\\tau ))+ ({\\cal V}_\\mathrm {b}(\\tau ) - {\\cal S}_\\mathrm {b}(\\tau ))+\\sum _{l = 1}^m({\\cal V}_l(\\tau ) - {\\cal S}_l(\\tau ))\\right)\\right]_{\\rm LC+}.\\end{split}$ The contributions for final state partons $l$ are given in eq.",
"(REF ).",
"The contribution for initial state parton “a” is given in eq.",
"(REF ) and for parton “b” we simply have to substitute $\\mathrm {a}\\leftrightarrow \\mathrm {b}$ .",
"With exact color, the operators in the exponent change the color state.",
"However, in the LC+ approximation these operators are color diagonal.",
"For a splitting of parton $l$ with helper parton $k$ , $[\\mathbf {T}_l\\cdot \\mathbf {T}_k \\otimes 1]$ vanishes unless $k$ is color connected to $l$ in the ket state and, if $k$ is color connected to $l$ , equals $C_\\mathrm {A}/2$ if parton $l$ is a gluon and equals $C_\\mathrm {F}$ if parton $l$ is a quark or antiquark.",
"For $[1 \\otimes \\mathbf {T}_l\\cdot \\mathbf {T}_k]$ , we have the same factors if $k$ is color connected to $l$ in the bra state.",
"This rule applies for $l$ and $k$ being either initial state or final state parton indices."
],
[
"Comparison to the standard summation",
"In this section, we consider the cross section $d\\sigma /(dQ^2 dY)$ to produce a muon pair with squared momentum $Q^2$ and rapidity $Y$ and compare our result to standard results.",
"To do that, we need two manipulations, which are interesting in their own right."
],
[
"The single power approximation for the Mellin transform",
"A parton distribution function can be expressed as an integral over its Mellin transform, $f_{a/A}(\\eta _\\mathrm {a},\\mu ^2)= \\frac{1}{2\\pi }\\int _{-\\infty }^\\infty \\!d\\omega \\ \\tilde{f}_{a/A}(N + {\\mathrm {i}}\\,\\omega ,\\mu ^2)\\, \\eta _\\mathrm {a}^{- (N + {\\mathrm {i}}\\,\\omega )}\\;.$ Here $\\tilde{f}$ is the Mellin transform of $f$ and is a function of the Mellin variable $n = N + {\\mathrm {i}}\\,\\omega $ .",
"The integration contour runs from $N - {\\mathrm {i}}\\, \\infty $ to $N + {\\mathrm {i}}\\, \\infty $ , where $N$ is chosen such that the contour runs to the right of any singularities of $\\tilde{f}_{a/A}(n,\\mu ^2)$ .",
"In the standard method, it is not the parton distribution function that appears in an exponent, but rather the Mellin moment variable $n$ .",
"However, there is a simple method that allows us to compare the results of this paper to the standard results.",
"We note that, with a reasonable model of the behavior of $f_{a/A}(\\eta _\\mathrm {a},\\mu ^2)$ , its Mellin transform $\\tilde{f}_{a/A}(n,\\mu ^2)$ has a saddle point at some point $n = N$ along the real axis.",
"If we choose to let the integration contour in eq.",
"(REF ) run through the saddle point, the integration will be dominatedThe large $N$ dependence of $\\tilde{f}_{a/A}(N,\\mu ^2)$ is determined by how fast $f_{a/A}(\\eta _\\mathrm {a},\\mu ^2)$ decreases as $\\eta _\\mathrm {a}\\rightarrow 1$ .",
"If we suppose that in this limit, $f_{a/A}(\\eta _\\mathrm {a},\\mu ^2) \\sim (1 - \\eta _\\mathrm {a})^{\\beta - 1}$ , then large $\\beta $ corresponds to a fast decrease.",
"Then one can easily show that the saddle point approximation is valid for $\\beta \\rightarrow \\infty $ , $\\eta _\\mathrm {a}\\rightarrow 1$ .",
"However, the saddle point approximation is not valid in the limit $\\eta _\\mathrm {a}\\rightarrow 1$ at fixed $\\beta $ , even though $N \\rightarrow \\infty $ in this limit.",
"by $n \\approx N$ .",
"(Cf.",
"ref.",
"[27] for a related use of the saddle point approximation.)",
"Now, when addressing threshold summation, we encounter $f_{a/A}(\\eta _\\mathrm {a}/z,\\mu ^2)$ , where $\\eta _\\mathrm {a}$ is the momentum fraction that appears in the Born cross section and we integrate over $z$ .",
"Thus what enters our calculation is $f_{a/A}(\\eta _\\mathrm {a}/z,\\mu ^2)=\\frac{1}{2\\pi }\\int _{-\\infty }^\\infty \\!d\\omega \\ \\tilde{f}_{a/A}(N + {\\mathrm {i}}\\,\\omega ,\\mu ^2)\\, (\\eta _\\mathrm {a}/z)^{- (N + {\\mathrm {i}}\\, \\omega )}\\;.$ The integration over $z$ is dominated by $z$ near 1.",
"Using the saddle point approximation, with the location $N$ of the saddle point determined for $z = 1$ , we have $f_{a/A}(\\eta _\\mathrm {a}/z,\\mu ^2)= \\tilde{f}_{a/A}(N,\\mu ^2)\\,(\\eta _\\mathrm {a}/z)^{- N}\\,I(N,\\mu ^2,z)\\;.$ Here $\\begin{split}I(N,\\mu ^2,z)={}&\\frac{1}{2\\pi }\\int _{-\\infty }^\\infty \\!d\\omega \\ \\exp \\left({\\mathrm {i}}\\,\\omega \\log z - \\frac{1}{2}\\omega ^2 B(N,\\mu ^2) + \\cdots \\right)\\;,\\end{split}$ where $B(N,\\mu ^2) = \\left[\\frac{d^2}{dn^2}\\,\\log (\\tilde{f}_{a/A}(n,\\mu ^2))\\right]_{n = N}\\;.$ Keeping the order $\\omega $ and $\\omega ^2$ terms indicated, we have $\\begin{split}I(N,\\mu ^2,z)\\approx {}&\\frac{1}{\\sqrt{2\\pi B(N,\\mu ^2)}}\\ \\exp \\left(-\\frac{\\log ^2(z)}{2 B(N,\\mu ^2)}\\right)\\;.\\end{split}$ We are interested in the behavior of $f_{a/A}(\\eta _\\mathrm {a}/z,\\mu ^2)$ for $(1-z) \\ll 1$ , so we neglect the $z$ dependence of $I(\\eta _\\mathrm {a},\\mu ^2,z)$ , which starts at order $(1-z)^2$ .",
"Then $f_{a/A}(\\eta _\\mathrm {a}/z,\\mu ^2)\\approx \\left[\\tilde{f}_{a/A}(N,\\mu ^2)I(N,\\mu ^2,1)\\right](\\eta _\\mathrm {a}/z)^{- N}\\;.$ This gives us what Sterman and Zeng [46] call the “single power approximation.” Sterman and Zeng argue that the single power approximation is numerically quite accurate in practice.",
"With the single power approximation, the ratios of parton distributions in eq.",
"(REF ) becomes $\\frac{f_{a/A}(\\eta _\\mathrm {a}/z,\\mu ^2)}{f_{a/A}(\\eta _\\mathrm {a},\\mu ^2)} = z^{N}\\;.$ Then the Mellin moment variable appears in the exponent of our expressions, so that we can compare to standard results that are written in this form."
],
[
"Rapidity dependence",
"We now consider the cross section to produce a muon pair with squared momentum $Q^2$ and rapidity $Y$ : $\\frac{d\\sigma }{dQ^2\\,dY}= \\sum _{a,b}\\int _{x_{\\mathrm {a}}}^1 d\\eta _\\mathrm {a}\\int _{x_{\\mathrm {b}}}^1 d\\eta _\\mathrm {b}\\ f_{a/A}^{\\overline{\\text{MS}}}(\\eta _\\mathrm {a},Q^2)\\,f_{b/B}^{\\overline{\\text{MS}}}(\\eta _\\mathrm {b},Q^2)\\,\\frac{d\\hat{\\sigma }(a,b)}{dQ^2\\,dY}\\;.$ Here the lower limits on $\\eta _\\mathrm {a}$ and $\\eta _b$ are the momentum fractions at the Born level: $\\begin{split}x_\\mathrm {a}={}& \\sqrt{Q^2/s}\\ e^Y\\;,\\\\x_\\mathrm {b}={}& \\sqrt{Q^2/s}\\ e^{-Y}\\;.\\end{split}$ We are interested in the threshold region for $Q^2$ and $Y$ , by which we mean that $x_\\mathrm {a}$ and $x_\\mathrm {b}$ are close to 1 and $f^{\\overline{\\text{MS}}}_{a/A}(\\eta _\\mathrm {a},\\mu ^2)$ and $f^{\\overline{\\text{MS}}}_{b/B}(\\eta _\\mathrm {b},\\mu ^2)$ are fast decreasing functions for $\\eta _\\mathrm {a}> x_\\mathrm {a}$ and $\\eta _\\mathrm {b}> x_\\mathrm {b}$ , respectively.",
"In the threshold region, the flavor structure simplifies.",
"Parton $a$ must be a quark and $b$ an antiquark, or vice versa.",
"The flavor structure is carried by a function $\\sigma _0$ that appears in the Born cross section: $\\frac{d\\sigma _{\\rm Born}}{dQ^2\\,dY}= \\sum _{a,b}f_{a/A}^{\\overline{\\text{MS}}}(x_\\mathrm {a},Q^2)\\,f_{b/B}^{\\overline{\\text{MS}}}(x_\\mathrm {b},Q^2)\\,\\sigma _0(a,b,Q^2,s)\\;.$ In the threshold region, we can write the parton level cross section in eq.",
"(REF ) in terms of a dimensionless and flavor independent coefficient function $C$ as $\\frac{d\\hat{\\sigma }(a,b)}{dQ^2\\,dY}\\approx \\frac{\\sigma _0(a,b,Q^2,s)}{\\eta _\\mathrm {a}\\eta _\\mathrm {b}}\\,C(\\alpha _{\\mathrm {s}}(Q^2),z,y)\\;,$ where $\\begin{split}z = {}& \\frac{Q^2/s}{\\eta _\\mathrm {a}\\eta _\\mathrm {b}}\\;,\\\\y ={}& Y - \\frac{1}{2} \\log (\\eta _\\mathrm {a}/\\eta _\\mathrm {b})\\;.\\end{split}$ This gives $\\begin{split}\\frac{d\\sigma }{dQ^2\\,dY}\\approx {}& \\sum _{a,b}\\frac{d\\sigma (a,b)}{dQ^2\\,dY}\\;,\\end{split}$ where $\\begin{split}\\frac{d\\sigma (a,b)}{dQ^2\\,dY}={}& \\sigma _0(a,b,Q^2,s)\\int _{x_{\\mathrm {a}}}^1 \\frac{d\\eta _\\mathrm {a}}{\\eta _\\mathrm {a}}\\int _{x_{\\mathrm {b}}}^1 \\frac{d\\eta _\\mathrm {b}}{\\eta _\\mathrm {b}}\\ f^{\\overline{\\text{MS}}}_{a/A}(\\eta _\\mathrm {a},Q^2)\\,f^{\\overline{\\text{MS}}}_{b/B}(\\eta _\\mathrm {a},Q^2)\\,C(\\alpha _{\\mathrm {s}}(\\mu ^2),z,y)\\;.\\end{split}$ Now we change integration variables to $z$ and $y$ defined in eq.",
"(REF ): $\\begin{split}\\frac{d\\sigma (a,b)}{dQ^2\\,dY}={}& \\sigma _0(a,b,Q^2,s)\\int _{0}^1 \\frac{dz}{z} \\int _{-\\frac{1}{2}\\log (1/z)}^{\\frac{1}{2}\\log (1/z)} dy\\\\&\\times f^{\\overline{\\text{MS}}}_{a/A}(x_\\mathrm {a}\\,e^{-y}/\\sqrt{z},Q^2)\\,f^{\\overline{\\text{MS}}}_{b/B}(x_\\mathrm {b}\\,e^{y}/\\sqrt{z},Q^2)\\ C(\\alpha _{\\mathrm {s}}(Q^2),z,y)\\;.\\end{split}$ The limits $-\\frac{1}{2}\\log (1/z) < y < \\frac{1}{2}\\log (1/z)$ come from the requirement that real emissions have positive components along the directions of $p_\\mathrm {a}$ and $p_\\mathrm {b}$ , so that $\\eta _\\mathrm {a}> x_\\mathrm {a}$ and $\\eta _\\mathrm {b}> x_\\mathrm {b}$ .",
"Separately, the arguments $x_\\mathrm {a}\\,e^{-y}/\\sqrt{z}$ and $x_\\mathrm {b}\\,e^{y}/\\sqrt{z}$ of the parton distribution functions must be less than 1 or else the parton distribution functions will vanish.",
"The requirements on the arguments of the parton distribution functions also implies that $z > Q^2/s$ .",
"We can now apply the single power approximation for the parton distributions, giving $\\begin{split}\\frac{d\\sigma (a,b)}{dQ^2\\,dY}={}& \\sigma _0(a,b,Q^2)\\,f^{\\overline{\\text{MS}}}_{a/A}(x_\\mathrm {a},Q^2)\\,f^{\\overline{\\text{MS}}}_{b/B}(x_\\mathrm {b},Q^2)\\int _{0}^1 \\frac{dz}{z}\\ z^{(N_\\mathrm {a}+ N_\\mathrm {b})/2}\\\\&\\times \\int _{-\\frac{1}{2}\\log (1/z)}^{\\frac{1}{2}\\log (1/z)} dy\\ e^{y(N_\\mathrm {a}- N_\\mathrm {b})}\\,C(\\alpha _{\\mathrm {s}}(Q^2),z,y)\\;.\\end{split}$ Here $N_\\mathrm {a}$ and $N_\\mathrm {b}$ are the saddle point Mellin powers for hadrons A and B, respectively.",
"We can simplify this for the purpose of comparing to standard results.",
"For a given choice of quark-antiquark flavors $a,b$ , the saddle point Mellin powers $N_\\mathrm {a}$ and $N_\\mathrm {b}$ depend on the rapidity $Y$ .",
"If we imagine replacing $Y \\rightarrow Y + \\delta Y$ while keeping the definitions of $x_\\mathrm {a}$ and $x_\\mathrm {b}$ unchanged, then the parton distribution function factor in eq.",
"(REF ) becomes $f_{a/A}(x_\\mathrm {a}e^{\\delta Y},\\mu ^2)\\,f_{b/B}(x_\\mathrm {b}e^{-\\delta Y},\\mu ^2)$ .",
"According to the single power approximation, this factor is $f^{\\overline{\\text{MS}}}_{a/A}(x_\\mathrm {a}e^{\\delta Y},\\mu ^2)\\,f^{\\overline{\\text{MS}}}_{b/B}(x_\\mathrm {b}e^{-\\delta Y},\\mu ^2)\\approx f^{\\overline{\\text{MS}}}_{a/A}(x_\\mathrm {a},\\mu ^2)\\,f^{\\overline{\\text{MS}}}_{b/B}(x_\\mathrm {b},\\mu ^2)\\,e^{(N_\\mathrm {b}- N_\\mathrm {a})\\delta Y}\\;.$ This implies that the parton distribution factor (and thus also the Born cross section) is maximum at that value of $Y$ such that $(N_\\mathrm {b}- N_\\mathrm {a}) = 0$ .",
"For our purpose of comparing to standard results, let us choose $Y$ such that the parton factor is close to its maximum.",
"Then $N_\\mathrm {b}\\approx N_\\mathrm {a}$ .",
"Now look at the integration in eq.",
"(REF ).",
"For the kinematic regime in which threshold summation is needed, $(N_\\mathrm {a}+ N_\\mathrm {b})/2$ is large.",
"Then in the $z$ -integration, $z$ near 1 dominates.",
"That means that the range of $y$ is small.",
"Since $(N_\\mathrm {b}- N_\\mathrm {a})$ is small (compared to $(N_\\mathrm {a}+ N_\\mathrm {b})/2$ ), the factor $\\exp ({y(N_\\mathrm {a}- N_\\mathrm {b})})$ can be approximated by 1, as argued in in refs.",
"[21], [22].",
"This gives $\\begin{split}\\frac{d\\sigma (a,b)}{dQ^2\\,dY}\\approx {}& \\sigma _0(a,b,Q^2)\\,f^{\\overline{\\text{MS}}}_{a/A}(x_\\mathrm {a},Q^2)\\,f^{\\overline{\\text{MS}}}_{b/B}(x_\\mathrm {b},Q^2)\\int _{0}^1 \\frac{dz}{z}\\ z^{N}\\,C(\\alpha _{\\mathrm {s}}(Q^2),z)\\;,\\end{split}$ where $N = (N_\\mathrm {a}+ N_\\mathrm {b})/2$ and $C(\\alpha _{\\mathrm {s}}(Q^2),z)=\\int _{-\\frac{1}{2}\\log (1/z)}^{\\frac{1}{2}\\log (1/z)} dy\\ C(\\alpha _{\\mathrm {s}}(Q^2),z,y)\\;.$ These manipulations have given us a function of one variable to work with instead of a function of two variables.",
"The function of one variable, $C(\\alpha _{\\mathrm {s}}(Q^2),z)$ is the function that appears in the rapidity-integrated cross section $d\\sigma /dQ^2$ and is well studied.",
"Eq.",
"(REF ) gives us $\\begin{split}\\frac{d\\sigma (a,b)}{dQ^2\\,dY}\\approx {}& \\sigma _0(a,b,Q^2)\\,f^{\\overline{\\text{MS}}}_{a/A}(x_\\mathrm {a},\\mu ^2)\\,f^{\\overline{\\text{MS}}}_{b/B}(x_\\mathrm {b},\\mu ^2)\\,\\widetilde{C}(\\alpha _{\\mathrm {s}}(Q^2),N)\\;,\\end{split}$ where $\\widetilde{C}$ is the Mellin transform of $C$ : $\\begin{split}\\widetilde{C}(\\alpha _{\\mathrm {s}}(\\mu ^2),N)=\\int _{0}^1 \\frac{dz}{z}\\ z^{N}\\,C(\\alpha _{\\mathrm {s}}(Q^2),z)\\;.\\end{split}$"
],
[
"The standard result",
"Now we need $\\widetilde{C}$ .",
"We use the standard result [1], [3] as given in eq.",
"(3.1) of ref.",
"[46] with factorization scale $\\mu _\\mathrm {f}^2 = Q^2$ .",
"We use the first term only in the cusp anomalous dimension, set the hard scattering function to 1, and omit the function $D$ , thus dropping terms that contribute non-leading logarithms of $N$ : $\\begin{split}\\widetilde{C}(& \\alpha _{\\mathrm {s}}(Q^2),N)\\\\={}&\\exp \\!\\Bigg (\\int _{0}^1\\!dz\\ \\frac{4 C_\\mathrm {F}}{1-z}\\left[1 - z^{N-1}\\right]\\int _{(1-z)^2 Q^2}^{Q^2} \\frac{d\\mu _\\perp ^2}{\\mu _\\perp ^2}\\,\\frac{\\alpha _{\\mathrm {s}}(\\mu _\\perp ^2)}{2\\pi }\\,\\theta (\\mu _\\perp ^2 > m_\\perp ^2)\\Bigg )\\;.\\end{split}$"
],
[
"The comparison",
"How does this compare to our results?",
"In the parton shower approach, there are Sudakov factors $K$ for the shower interval between the hard scattering that produces the muon pair and the first real parton splitting, then for the interval between the first real splitting and the second, and so forth.",
"The integrands in the exponent of $K$ decrease with decreasing splitting scale.",
"Therefore we assume for the purposes of making a comparison that the first splitting occurs at a sufficiently small scale that we can just set that scale to zero and ignore the Sudakov factors for further splittings.",
"This gives $\\begin{split}\\frac{d\\sigma (a,b)}{dQ^2\\,dY}\\approx {}& \\sigma _0(a,b,Q^2)\\,f_{a/A}(x_\\mathrm {a},Q^2)\\, f_{b/B}(x_\\mathrm {b},Q^2)\\,K_\\mathrm {a}K_\\mathrm {b}\\;.\\end{split}$ The factor $K_\\mathrm {a}$ comes from the exponential of ${\\cal V}_{\\mathrm {a}} - {\\cal S}_{\\mathrm {a}}$ for initial state radiation from parton “a.” The factor $K_\\mathrm {b}$ is the same expression applied to parton “b.” The parton distribution functions in eq.",
"(REF ) are those appropriate for a $\\Lambda ^2$ -ordered shower.",
"They are related to the $\\overline{\\text{MS}}$ parton distribution functions by factors $Z_\\mathrm {a}$ and $Z_\\mathrm {b}$ defined in eq.",
"(REF ), so that $\\begin{split}\\frac{d\\sigma (a,b)}{dQ^2\\,dY}\\approx {}& \\sigma _0(a,b,Q^2)\\,f^{\\overline{\\text{MS}}}_{a/A}(x_\\mathrm {a},Q^2)\\, f^{\\overline{\\text{MS}}}_{b/B}(x_\\mathrm {b},Q^2)\\,Z_\\mathrm {a}K_\\mathrm {a}Z_\\mathrm {b}K_\\mathrm {b}\\;.\\end{split}$ This matches the form of eq.",
"(REF ).",
"We need to check whether the factor $\\widetilde{C}$ given by eq.",
"(REF ) is the same as the leading approximation to $Z_\\mathrm {a}K_\\mathrm {a}Z_\\mathrm {b}K_\\mathrm {b}$ .",
"We worked out the leading approximation to $Z_\\mathrm {a}$ in eq.",
"(REF ): $\\begin{split}Z_a ={}&\\exp \\!\\Bigg (\\int _0^1\\!dz\\ \\frac{2 C_\\mathrm {F}}{1-z} \\left\\lbrace 1 -\\frac{f_{a/A}(\\eta _\\mathrm {a}/z,Q^2)}{f_{a/A}(\\eta _\\mathrm {a},Q^2)}\\right\\rbrace \\\\&\\qquad \\times \\int _{(1-z)\\,Q^2}^{Q^2} \\frac{d\\mu _\\perp ^2}{\\mu _\\perp ^2}\\,\\frac{\\alpha _{\\mathrm {s}}(\\lambda _\\mathrm {R}\\mu _\\perp ^2)}{2\\pi }\\,\\theta (\\mu _\\perp ^2 > m_\\perp ^2(a))\\Bigg )\\;.\\end{split}$ The factor $K_\\mathrm {a}$ comes from the exponential of ${\\cal V}_{\\mathrm {a}} - {\\cal S}_{\\mathrm {a}}$ for initial state radiation from parton “a,” integrated from an upper scale $\\mu _\\mathrm {a}^2(t_0) = Q^2$ , to a lower scale $\\mu _\\mathrm {a}^2(\\infty ) = 0$ .",
"We use eq.",
"(REF ) for $K_\\mathrm {a}$ .",
"The term in eq.",
"(REF ) proportional to $\\Delta _{\\mathrm {a}k}$ is absent from the first factor $K_\\mathrm {a}$ for the Drell-Yan process and the term proportional to $P_{a\\hat{a}}^{\\rm reg}(z)$ can be omitted because it is not large.",
"We also omit the ${\\mathrm {i}}\\,\\pi $ term.",
"Thus in eq.",
"(REF ) we include only the main term, proportional to $1/(1-z)$ , we take the argument of the parton distributions to by fixed at $Q^2$ , and we approximate the lower endpoint of the $z$ -integration as $1 - \\mu ^2/Q^2$ instead of $1/(1+ \\mu ^2/Q^2)$ .",
"This gives $\\begin{split}K_\\mathrm {a}={}&\\exp \\!\\Bigg (\\int _{0}^{Q^2} \\frac{d\\mu ^2}{\\mu ^2}\\ \\int _{1 - \\mu ^2/Q^2}^1\\!dz\\ \\frac{\\alpha _{\\mathrm {s}}(\\lambda _\\mathrm {R}(1-z) \\mu ^2)}{2\\pi }\\,\\theta ((1-z) \\mu ^2 > m_\\perp ^2(a))\\\\&\\qquad \\times \\left[1 -\\frac{f_{a/A}(\\eta _{\\mathrm {a}}/z,Q^2)}{f_{a/A}(\\eta _{\\mathrm {a}},Q^2)}\\right]\\frac{2C_a}{1-z}\\ \\Bigg )\\;.\\end{split}$ In eq.",
"(REF ) it is useful to interchange the order of integrations and change variables from $\\mu ^2$ to $\\mu _\\perp ^2 = (1-z) \\mu ^2$ .",
"This gives $\\begin{split}K_\\mathrm {a}={}&\\exp \\!\\Bigg (\\int _{0}^1\\!dz\\ \\frac{2C_\\mathrm {F}}{1-z}\\left[1 -\\frac{f_{a/A}(\\eta _{\\mathrm {a}}/z,Q^2)}{f_{a/A}(\\eta _{\\mathrm {a}},Q^2)}\\right]\\\\&\\qquad \\times \\int _{(1-z)^2 Q^2}^{(1-z)Q^2} \\frac{d\\mu _\\perp ^2}{\\mu _\\perp ^2}\\ \\frac{\\alpha _{\\mathrm {s}}(\\lambda _\\mathrm {R}\\mu _\\perp ^2)}{2\\pi }\\,\\theta (\\mu _\\perp ^2 > m_\\perp ^2(a))\\Bigg )\\;.\\end{split}$ In the product $Z_\\mathrm {a}K_\\mathrm {a}$ , the integrands combine in a nice way to give $\\begin{split}Z_\\mathrm {a}K_\\mathrm {a}={}&\\exp \\!\\Bigg (\\int _{0}^1\\!dz\\ \\frac{2C_\\mathrm {F}}{1-z}\\left[1 -\\frac{f_{a/A}(\\eta _{\\mathrm {a}}/z,Q^2)}{f_{a/A}(\\eta _{\\mathrm {a}},Q^2)}\\right]\\\\&\\qquad \\times \\int _{(1-z)^2 Q^2}^{Q^2} \\frac{d\\mu _\\perp ^2}{\\mu _\\perp ^2}\\ \\frac{\\alpha _{\\mathrm {s}}(\\lambda _\\mathrm {R}\\mu _\\perp ^2)}{2\\pi }\\,\\theta (\\mu _\\perp ^2 > m_\\perp ^2(a))\\Bigg )\\;.\\end{split}$ Although an analysis of other shower ordering schemes is beyond the scope of this paper, it is worth noting that if we had used $k_\\perp ^2$ ordering or angular ordering for the shower, then $Z_\\mathrm {a}$ and $K_\\mathrm {a}$ would be different from what we have with $\\Lambda ^2$ ordering, but $Z_\\mathrm {a}K_\\mathrm {a}$ would be the same.",
"Now, apply the single power approximation (REF ).",
"This gives $\\begin{split}Z_\\mathrm {a}K_\\mathrm {a}={}&\\exp \\!\\left(\\int _{0}^1\\!dz\\ \\frac{2C_\\mathrm {F}}{1-z}\\left(1 -z^{N}\\right)\\int _{(1-z)^2 Q^2}^{Q^2} \\frac{d\\mu _\\perp ^2}{\\mu _\\perp ^2}\\ \\frac{\\alpha _{\\mathrm {s}}(\\lambda _\\mathrm {R}\\mu _\\perp ^2)}{2\\pi }\\,\\theta (\\mu _\\perp ^2 > m_\\perp ^2)\\,\\right)\\;.\\end{split}$ We get the same factor for parton “b”, so $\\begin{split}Z_\\mathrm {a}K_\\mathrm {a}& Z_\\mathrm {b}K_\\mathrm {b}\\\\={}&\\exp \\!\\Bigg (\\int _{0}^1\\!dz\\ \\frac{4 C_\\mathrm {F}}{1-z}\\left[1 - z^{N}\\right]\\int _{(1-z)^2 Q^2}^{Q^2} \\frac{d\\mu _\\perp ^2}{\\mu _\\perp ^2}\\,\\frac{\\alpha _{\\mathrm {s}}(\\lambda _\\mathrm {R}\\mu _\\perp ^2)}{2\\pi }\\,\\theta (\\mu _\\perp ^2 > m_\\perp ^2)\\Bigg )\\;.\\end{split}$ This matches the standard result (REF ) with two small changes.",
"First, we have used a factor $\\lambda _\\mathrm {R}$ in the argument of $\\alpha _{\\mathrm {s}}$ .",
"This does not affect the leading logarithms.",
"Second, we have $\\left[1 - z^{N}\\right]$ instead of $\\left[1 - z^{N-1}\\right]$ .",
"These forms are equivalent for large $N$ .",
"We conclude that the form of threshold logarithm summation that arises naturally in a parton shower is equivalent to the traditional forms that one gets in a direct-QCD analysis [8], [9], [46] or in soft-collinear-effective-theory as in ref. [37].",
"The shower form is less precise in that it does not allow an analysis beyond the leading approximation.",
"On the other hand, it applies immediately to many processes with no further analysis."
],
[
"Numerical comparisons",
"In this section, we exhibit two numerical tests of the threshold summation presented in this paper.",
"In the first test, we look at the inclusive Drell-Yan cross section, $p + p \\rightarrow e^+ + e^- + X$ at 13 TeV.",
"In the second test we look at the one jet inclusive cross section in proton-proton collisions at 13 TeV.",
"We compare cross sections $d\\sigma $ , differential in whatever variables we choose, calculated with threshold summation to the corresponding cross sections without threshold summation.",
"The full cross section including threshold factors is $d\\sigma ({\\rm full})$ .",
"This includes all of the terms in eq.",
"(REF ) for $({\\cal V}_{\\mathrm {a}}(\\tau ) - {\\cal S}_{\\mathrm {a}}(\\tau ))$ except the ${\\mathrm {i}}\\pi $ term, with the color matrices calculated using the leading color approximation.We could use the LC+ approximation, but we find that this makes very little difference.",
"It also includes a factor $Z_a(\\eta _\\mathrm {a}, \\mu _\\mathrm {f}^2) Z_b(\\eta _\\mathrm {b}, \\mu _\\mathrm {f}^2)$ , as defined in eq.",
"(REF ).",
"This factor relates the $\\Lambda ^2$ -ordered parton distributions to the $\\overline{\\text{MS}}$ parton distributions.",
"Here $\\mu _\\mathrm {f}^2$ is the factorization scale, characteristic of the hard scattering.",
"For the jet cross section, $\\alpha _{\\mathrm {s}}$ at the hard interaction is evaluated at $\\mu _\\mathrm {r}= \\mu _\\mathrm {f}$ .",
"The parton shower then starts at scale $\\mu ^2(t) = \\mu _\\mathrm {f}^2$ as given in eqs.",
"(REF ) and (REF ).",
"We will sometimes find it of interest to exhibit a cross section $d\\sigma ({\\rm no\\ }\\Delta )$ in which we omit the term proportional to $\\Delta _{ak}$ in eq.",
"(REF ).",
"We can turn off all of the threshold effects to obtain a standard parton shower cross section $d\\sigma ({\\rm std.",
"})$ .",
"In all of these cross sections the default calculation in Deductor begins with a factor $f^{\\overline{\\text{MS}}}_{a/A}(\\eta _\\mathrm {a},\\mu _\\mathrm {f}^2) f^{\\overline{\\text{MS}}}_{b/B}(\\eta _\\mathrm {b},\\mu _\\mathrm {f}^2)$ , where these functions obey the first order evolution equation (REF ).",
"In the calculations presented in this section, we modify the initial parton factor to use NLO parton distributions by multiplying by a factor $R_{\\rm pdf} =\\frac{f^{\\overline{\\text{MS}},{\\rm NLO}}_{a/A}(\\eta _\\mathrm {a},\\mu _\\mathrm {f}^2)}{f^{\\overline{\\text{MS}}}_{a/A}(\\eta _\\mathrm {a},\\mu _\\mathrm {f}^2)}\\,\\frac{f^{\\overline{\\text{MS}},{\\rm NLO}}_{b/B}(\\eta _\\mathrm {b},\\mu _\\mathrm {f}^2)}{f^{\\overline{\\text{MS}}}_{b/B}(\\eta _\\mathrm {b},\\mu _\\mathrm {f}^2)}\\;.$ The NLO parton distribution functions used are from the central CT14 NLO fit [77].",
"The parton distributions that obey the first order evolution equation (REF ) are simply obtained by using eq.",
"(REF ) with the same starting distributions at the starting scale $\\mu _{\\rm start}^2$ .",
"For the Drell-Yan cross section, we find that $R_{\\rm pdf}$ is within about 5% of 1.",
"The Deductor (full) results depend on the parameter $\\psi _{\\rm min}$ introduced in section REF .",
"We use $\\psi _{\\rm min} = 0.01$ .",
"We have checked that varying $\\psi _{\\rm min}$ by a factor 2 or 1/2 affects the cross sections examined by $\\pm 2\\%$ or somewhat less, depending on the observable."
],
[
"Drell-Yan",
"In figure REF we look at the inclusive Drell-Yan cross section, $p + p \\rightarrow e^+ + e^- + X$ at $\\sqrt{s} = 13 \\ \\mathrm {TeV}$ , as a function of the mass $Q$ of the $e^+ e^-$ pair.",
"We take $\\mu _\\mathrm {f}= Q$ .",
"We show two curves from Deductor, one, $d\\sigma ({\\rm full})/dQ$ , with the threshold effects turned on, the other, $d\\sigma ({\\rm std.",
"})/dQ$ , with the threshold effects turned off.",
"Since the parton shower does not change $Q$ , $d\\sigma ({\\rm std.",
"})/dQ$ equals the leading order (LO) perturbative cross section.",
"For comparison, we show a perturbative next-to-leading order (NLO) calculation obtained from MCFM [78] with $\\mu _\\mathrm {f}= Q$ and CT14 NLO parton distributions.The MCFM results were adjusted to use the same running $\\alpha _{\\rm em}(Q)$ as Deductor.",
"We note that $d\\sigma /dQ$ decreases approximately exponentially as $Q$ increases in the threshold region $Q > 1 \\ \\mathrm {TeV}$ .",
"This reflects the fast decrease of the parton distribution functions as the momentum fraction increases.",
"All three computed cross sections display the same approximately exponential behavior.",
"However, the threshold correction has an effect that is large enough to notice even in this multi-decade semilog plot.",
"Figure: Ratios, KK, of Drell-Yan cross sections dσ/dQd\\sigma /dQ, as in figure , to the Born cross section dσ( std .",
")/dQ=dσ( LO )/dQd\\sigma ({\\rm std.",
"})/dQ = d\\sigma ({\\rm LO})/dQ calculated with a factorization scale μ f =Q\\mu _\\mathrm {f}= Q.",
"The solid green curve is K( no Δ)K({\\rm no\\ }\\Delta ) corresponding to dσ( no Δ)/dQd\\sigma ({\\rm no\\ }\\Delta )/dQ.",
"The solid red curve is K( full )K({\\rm full}).",
"In each case, we take μ f =Q\\mu _\\mathrm {f}= Q.",
"The dashed, black curve is K( NLO )K({\\rm NLO}) obtained from a perturbative calculation using MCFM with μ f =Q\\mu _\\mathrm {f}= Q.",
"The purple, dashed curve is the analytic result of Becher, Neubert, and Xu, comparable to the NNLO curve of figure 8 of ref.",
".There is a theoretical uncertainty associated with the parton shower calculation, which we can estimate by changing the factorization scale $\\mu _\\mathrm {f}$ at which the initial parton distributions are evaluated and at which shower evolution starts.",
"It is rather standard for the Drell-Yan cross section to choose the factorization scale to be $\\mu _\\mathrm {f}= Q$ .",
"However, the maximum value of the transverse momentum of the $e^-$ or $e^+$ is $Q/2$ , so, by analogy with jet production, for which $\\mu _\\mathrm {f}= P_\\mathrm {T}({\\rm jet})$ is a widely used choice, $\\mu _\\mathrm {f}= Q/2$ might seem a sensible choice here.",
"On the other hand, one could choose $\\mu _\\mathrm {f}= 2 Q$ .",
"In figure REF , we plot the ratios of $d\\sigma ({\\rm full})$ with these two scale choices to $d\\sigma ({\\rm full})$ with $\\mu _\\mathrm {f}= Q$ .",
"Based on this result, one might estimate a $\\pm 5\\%$ uncertainty.",
"The precision of the Deductor calculation could be improved by matching to a NLO calculation of the Drell-Yan cross section, but we have not done this.",
"It is difficult to see small effects in semilog plots like figure REF , so, in figure REF , we show ratios $K$ of cross sections to the Born cross section, $d\\sigma ({\\rm std.",
"})/dQ = d\\sigma ({\\rm LO})/dQ$ .",
"In each case shown, we use $\\mu _\\mathrm {f}= Q$ .",
"We show first, in red, $K({\\rm full})$ , corresponding to the cross section with the threshold correction, $d\\sigma ({\\rm full})/dQ$ .",
"We see that the threshold correction is quite substantial and increases with $Q$ .",
"We next show, in green, $K({\\rm no}\\ \\Delta )$ for the cross section including just the part of the threshold correction in which we omit the term proportional to $\\Delta _{ak}$ in eq.",
"(REF ).",
"The ratio $K_\\Delta \\equiv [d\\sigma ({\\rm full})/dQ]/[\\sigma ({\\rm no}\\ \\Delta )/dQ]$ is of some interest.",
"For the Drell-Yan process, the $\\Delta _{ak}$ term does not occur in the Sudakov factor between the hard interaction and the first real parton emission.",
"After the first emission, there is color flow transverse to the beam so that the pattern of virtual gluon exchange is changed and $\\Delta _{ak}$ can be non-zero.",
"Since $\\Delta _{ak}$ is itself proportional to $\\alpha _{\\mathrm {s}}$ , the perturbative expansion of $K_\\Delta - 1$ begins at order $\\alpha _{\\mathrm {s}}^2$ .",
"Thus we expect $K_\\Delta $ to be close to 1.",
"In fact, we find that $K_\\Delta \\approx 1.03$ for $Q > 1 \\ \\mathrm {TeV}$ .",
"The parameter $\\psi _{\\rm min}$ described in section REF controls the integration range over which $\\Delta _{ak}$ operates.",
"As noted earlier, the cross section is sensitive to a factor 2 or 1/2 change in $\\psi _{\\rm min}$ at a level of about $\\pm 2 \\%$ , so the numerical value of $K_\\Delta $ is not highly significant.",
"What is significant is that $K_\\Delta $ is indeed close to 1.",
"We display next, as a dashed black curve, the ratio $K({\\rm NLO})$ of the NLO cross section to the Born cross section, as given by MCFM [78].",
"We note that $K({\\rm NLO})$ is an increasing function of $Q$ , as we might have expected since it includes some of the effect of threshold logs.",
"We note also that the slope of the NLO curve remains rather constant as $Q$ increases, in contrast to $K({\\rm full})$ , which has an increasing slope as the threshold logs build up.",
"Finally, we show as a purple, dashed curve, $K$ obtained with the analytic threshold summation of Becher, Neubert, and Xu [37].",
"This curve is obtained by adapting the code for figure 8 of ref.",
"[37] to CT14 parton distributions and $\\sqrt{s} = 13\\ \\ \\mathrm {TeV}$ .",
"We regard the analytic B.N.X.",
"curve as more precise than the Deductor (full) curve since the analytic result contains a high order of approximation in the summation of logarithms, while Deductor (full) is based on only a leading order parton shower.",
"We note that, for $Q > 2 \\ \\mathrm {TeV}$ , the BNX curve agrees within about 3% with the Deductor result for $d\\sigma ({\\rm full})/dQ$ .",
"Figure: The normalized Drell-Yan transverse momentum distribution, dN/dQ T =(1/σ)dσ/dQ T dN/dQ_\\mathrm {T}= (1/\\sigma )\\, d\\sigma /dQ_\\mathrm {T} for the LHC at 13 TeV.",
"Here Q T Q_T is the transverse momentum of the e + e^+e - e^- pair, dσ/dQ T d\\sigma /dQ_\\mathrm {T} is dσ/(dQ T dQ)d\\sigma /(dQ_\\mathrm {T}\\,dQ) integrated over 2 TeV <Q<2.1 TeV 2 \\ \\mathrm {TeV}< Q < 2.1 \\ \\mathrm {TeV} and σ\\sigma is this cross section integrated over 0<Q T <100 GeV 0 < Q_\\mathrm {T}< 100 \\ \\mathrm {GeV}, so that the area under the curve is 1.",
"The red curve that is lowest at small Q T Q_\\mathrm {T} is the full result with threshold effects, dN( full )/dQ T dN({\\rm full})/dQ_\\mathrm {T}.",
"The blue curve that is very slightly higher at small Q T Q_\\mathrm {T} is the result with no threshold effects, dN( std .",
")/dQ T dN({\\rm std.",
"})/dQ_\\mathrm {T}.",
"In these calculations, we have chosen μ f =Q\\mu _\\mathrm {f}= Q.",
"We also show the corresponding result obtained using ResBos , as a dashed black curve.Figure: Ratios, KK, of the Drell-Yan cross section dσ/dQ T d\\sigma /dQ_\\mathrm {T} obtained by integrating dσ/(dQ T dQ)d\\sigma /(dQ_\\mathrm {T}\\,dQ) over 2 TeV <Q<2.1 TeV 2 \\ \\mathrm {TeV}< Q < 2.1 \\ \\mathrm {TeV} for the LHC at 13 TeV.",
"The numerator in the upper curve is the full result with threshold factors, dσ( full )/dQ T d\\sigma ({\\rm full})/dQ_\\mathrm {T}.",
"The numerator in the lower curve is the cross section dσ( no Δ)/dQ T d\\sigma ({\\rm no}\\ \\Delta )/dQ_\\mathrm {T} obtained by omitting the Δ\\Delta term in the threshold factor.",
"In each case the denominator is the cross section dσ( std .",
")/dQ T d\\sigma ({\\rm std.",
"})/dQ_\\mathrm {T} with no threshold factors.",
"The factorization scale in all of these cross sections is chosen to be μ f =Q\\mu _\\mathrm {f}= Q."
],
[
"Drell-Yan transverse momentum",
"In the previous subsection, we examined the $Q$ dependence of the Drell-Yan cross section $d\\sigma /dQ$ , looking for the effects of steeply falling parton distribution functions when $Q$ is large.",
"Parton shower event generators can also predict the distribution of the transverse momentum $Q_\\mathrm {T}$ of the $e^+ e^-$ pair in the region of small $Q_\\mathrm {T}/Q$ , where logarithms of $Q_\\mathrm {T}/Q$ need to be summed.",
"We have found [66] that a parton shower with virtuality based ordering, like Deductor, gives the same result at the next-to-leading-log level for logarithms of $Q_\\mathrm {T}/Q$ (without threshold logs) as the analytical treatment of ref. [79].",
"Now, with threshold effects included in a parton shower, we can examine both the logs of $Q_\\mathrm {T}/Q$ and the threshold effect at the same time, as in the analytical treatments of refs.",
"[18], [19], [43], [44].",
"We do not, however, have analytical knowledge of the level of accuracy of the parton shower treatment.",
"To study the $Q_\\mathrm {T}$ distribution at large $Q$ , we examine $\\frac{d\\sigma }{d Q_\\mathrm {T}}=\\int _{2.0 \\ \\mathrm {TeV}}^{2.1 \\ \\mathrm {TeV}}\\!dQ\\ \\frac{d\\sigma }{dQ\\,dQ_\\mathrm {T}}\\;.$ We we divide by the integral of this over the $Q_\\mathrm {T}$ range $0 < Q_\\mathrm {T}< 100 \\ \\mathrm {GeV}$ to produce a distribution $dN/dQ_\\mathrm {T}$ normalized to $\\int _0^{100 \\ \\mathrm {GeV}}\\!dQ_\\mathrm {T}\\ \\frac{dN}{dQ_\\mathrm {T}} = 1\\;.$ In figure REF , we show the result with the threshold correction, $dN({\\rm full})/dQ_\\mathrm {T}$ .",
"We choose the factorization scale to be $\\mu _\\mathrm {f}= Q$ .",
"For comparison, we show the result $dN({\\rm std.",
"})/dQ_\\mathrm {T}$ with the threshold correction omitted.",
"We see that the threshold correction has only a very small effect.",
"We also show the result $dN({\\rm ResBos})/dQ_\\mathrm {T}$ obtained with the analytical summation of logs of $Q_\\mathrm {T}/Q$ contained in ResBos [80], [81].",
"The ResBos result does not contain a summation of threshold logs and so should be compared to Deductor (std.).",
"The ResBos calculation contains smearing with non-perturbative functions that are fit to data.",
"This smearing has not been included in Deductor.",
"Thus it is not surprising that the Deductor distributions are somewhat narrower than the distribution from ResBos.",
"In figure REF , we examine directly $d\\sigma /dQ_\\mathrm {T}$ defined in eq.",
"(REF ) so that we can see the effect of the threshold factors on the normalization of the cross section.",
"We take $\\mu _\\mathrm {f}= Q$ .",
"We examine ratios $K$ obtained by dividing $d\\sigma /dQ_\\mathrm {T}$ by the cross section obtained with no threshold corrections, $d\\sigma ({\\rm std.",
"})/dQ_\\mathrm {T}$ .",
"We show two curves.",
"In the upper curve, the numerator of $K$ is the result with the full threshold correction, $d\\sigma ({\\rm full})/dQ_\\mathrm {T}$ .",
"We see that there is a substantial, about 25%, threshold enhancement.",
"This enhancement is weakly $Q_\\mathrm {T}$ dependent,There is a strong $Q_\\mathrm {T}$ dependence at about $Q_\\mathrm {T}= 1 \\ \\mathrm {GeV}$ .",
"This arises from the minimum $p_\\mathrm {T}$ allowed for emissions in the shower and is not really physical.",
"increasing from 23% to 26% over the range $0 < Q_\\mathrm {T}< 100 \\ \\mathrm {GeV}$ .",
"We examine where this $Q_\\mathrm {T}$ dependence comes from by plotting also the ratio $K$ obtained using $d\\sigma ({\\rm no}\\ \\Delta )/dQ_\\mathrm {T}$ in the numerator.",
"Without the $\\Delta $ contribution in the Sudakov exponent, the threshold enhancement is flat as a function of $Q_\\mathrm {T}$ .",
"Thus the small $Q_\\mathrm {T}$ dependence seen in $d\\sigma ({\\rm full})/dQ_\\mathrm {T}$ comes mainly from the $\\Delta $ term in the Sudakov exponent.",
"This is easy to understand.",
"The $\\Delta $ term appears only after we have an initial state emission.",
"Having an initial state emission gives a transverse momentum recoil to the $e^+ e^-$ pair, so larger $Q_\\mathrm {T}$ should have a positive correlation with a larger threshold factor from the $\\Delta $ term.",
"We can offer two observations.",
"First, the Deductor (std.)",
"curve for ${dN}/(dQ_\\mathrm {T})$ agrees nicely with the ResBos curve, considering that there should be differences from non-perturbative smearing.",
"Second, the effect of threshold logs reflected in the Deductor (full) curves in figures REF and REF is small and its sign appears to us to be quite sensible."
],
[
"Jets",
"We now examine the one jet inclusive cross section $d\\sigma /dP_\\mathrm {T}$ in proton-proton collisions at $\\sqrt{s} = 13 \\ \\mathrm {TeV}$ as a function of the jet transverse momentum, $P_\\mathrm {T}$ , integrated over the rapidity range $-2 < y < 2$ .",
"The jet is defined using the anti-$k_\\mathrm {T}$ algorithm [82] with $R = 0.4$ with the aid of FastJet [83].",
"Notice that the cross section $d\\sigma ({\\rm std.",
"})/dP_\\mathrm {T}$ obtained with a standard shower with no threshold corrections is not the same as the Born cross section $d\\sigma ({\\rm LO})/dP_\\mathrm {T}$ because partons generated in the shower from initial state splittings can become part of the jet, while partons generated as daughters of the starting final state partons can escape from the jet.",
"In figure REF , we display three versions of $d\\sigma /dP_\\mathrm {T}$ as functions of $P_\\mathrm {T}$ .",
"In each case, we take the renormalization and factorization scales and the starting scale of the shower to be $\\mu _\\mathrm {f}= \\mu _\\mathrm {r}= P_\\mathrm {T}$ .",
"The lower, blue curve is $d\\sigma ({\\rm std.",
"})/dP_\\mathrm {T}$ , obtained with the parton shower with threshold effects omitted.",
"The solid red curve is $d\\sigma ({\\rm full})/dP_\\mathrm {T}$ , obtained with threshold effects.",
"We see that the threshold effect is large enough that it is evident even in this semilog plot.",
"The black, dashed curve is the result of a purely perturbative next-to-leading order (NLO) calculation [84].",
"We note that the parton shower calculation including the threshold effect is quite close to the NLO result.",
"There is a fairly substantial theoretical uncertainty associated with the parton shower calculation.",
"To estimate this uncertainty, we examine the effect of changing the scale $\\mu _\\mathrm {f}= \\mu _\\mathrm {r}$ at which the initial parton distributions and strong coupling are evaluated and at which shower evolution starts.",
"In figure REF , we used $\\mu _\\mathrm {f}= \\mu _\\mathrm {r}= P_\\mathrm {T}$ .",
"However, the minimum value of the dijet mass in the Born process is $Q = 2 P_\\mathrm {T}$ .",
"Thus $\\mu _\\mathrm {f}= \\mu _\\mathrm {r}= 2 P_\\mathrm {T}$ might seem a sensible choice.",
"One the other hand, jet cross sections are sometimes evaluated with $\\mu _\\mathrm {f}= \\mu _\\mathrm {r}= P_\\mathrm {T}/2$ , so $P_\\mathrm {T}/2$ might seem a sensible choice.",
"In figure REF , we plot the ratios of $d\\sigma ({\\rm full})/dP_\\mathrm {T}$ with $\\mu _\\mathrm {f}= \\mu _\\mathrm {r}= 2 P_\\mathrm {T}$ and with $\\mu _\\mathrm {f}= \\mu _\\mathrm {r}= P_\\mathrm {T}/2$ to $d\\sigma ({\\rm full})/dP_\\mathrm {T}$ with $\\mu _\\mathrm {f}= \\mu _\\mathrm {r}= P_\\mathrm {T}$ .",
"Based on this result, we estimate a $\\pm 30\\%$ uncertainty in $d\\sigma ({\\rm full})/dP_\\mathrm {T}$ .",
"This uncertainty could be reduced by performing a showered calculation matched to the NLO calculation.",
"Figure: Scale dependence of the one jet inclusive cross section after showering and threshold effects.",
"We plot as functions of P T P_\\mathrm {T} the ratios [dσ( full ,λ)/dP T ]/[dσ( full ,1)/dP T ][d\\sigma ({\\rm full},\\lambda )/dP_\\mathrm {T}]/[d\\sigma ({\\rm full},1)/dP_\\mathrm {T}] with μ f =μ r =λP T \\mu _\\mathrm {f}= \\mu _\\mathrm {r}= \\lambda P_\\mathrm {T} in the numerator and μ f =μ r =P T \\mu _\\mathrm {f}= \\mu _\\mathrm {r}= P_\\mathrm {T} in the denominator.",
"From top to bottom, the three curves are for λ=0.5\\lambda = 0.5, λ=1\\lambda = 1 and λ=2\\lambda = 2.In figure REF , we turn to several calculations of $d\\sigma /dP_\\mathrm {T}$ presented as ratios $K$ to the perturbative Born cross section, $d\\sigma ({\\rm LO})/dP_\\mathrm {T}$ .",
"In this figure, all cross sections are evaluated at $\\mu _\\mathrm {f}= \\mu _\\mathrm {r}= P_\\mathrm {T}$ .",
"The lowest, blue curve is $K({\\rm std.",
"})$ , obtained using $d\\sigma ({\\rm std.",
"})/dP_\\mathrm {T}$ , in which there is a standard shower but the threshold effects are turned off.",
"We see that $d\\sigma ({\\rm std.",
"})/dP_\\mathrm {T}$ is only about 60% of the Born cross section.",
"Since the cross section is so steeply falling as a function of $P_\\mathrm {T}$ , just a small amount of $P_\\mathrm {T}$ leakage out of the jet because of showering makes the cross section substantially smaller.",
"We now include the threshold correction, plotting the ratio $K({\\rm full})$ .",
"This gives the red curve.",
"We see that the threshold effect is very large and multiplies $d\\sigma ({\\rm std.",
"})/dP_\\mathrm {T}$ by a factor between 1.3 and 2 for $P_\\mathrm {T}> 1 \\ \\mathrm {TeV}$ .",
"This produces a result $d\\sigma ({\\rm full})/dP_\\mathrm {T}$ that ranges from 90% to 120% of the Born cross section for $P_\\mathrm {T}> 1 \\ \\mathrm {TeV}$ .",
"We also show, as a solid green curve, the factor $K({\\rm no}\\ \\Delta )$ obtained by omitting the term proportional to $\\Delta _{\\mathrm {a}k}$ in the Sudakov exponent.",
"We see that, for the jet cross section, this term makes a contribution to the exponent that is not negligible.",
"We show also as a black, dashed curve, $K({\\rm NLO})$ corresponding to the perturbative NLO calculation from figure REF .",
"There are analytic summations of threshold logarithms [8], [10], [17], [23], [28].",
"We have used the computer programs of Kidonakis and Owens [17] and of de Florian, Hinderer, Mukherjee, Ringer and Vogelsang [28] to produce the cross sections $d\\sigma ({\\rm K.O.",
"})/dP_\\mathrm {T}$ and $d\\sigma ({\\rm FHMRV})/dP_\\mathrm {T}$ , respectively.",
"In these programs, the all-order threshold effect is expanded to order $\\alpha _{\\mathrm {s}}^2$ to produce the calculated cross sections.",
"In the Kidonakis-Owens formulation of threshold summation, there is no dependence on the algorithm used to define the jet.",
"The FHMRV calculation is more sophisticated and includes the dependence on the jet algorithm (for which we use the anti-$k_\\mathrm {T}$ algorithm with $R = 0.4$ ).",
"In figure REF , we plot $K({\\rm K.O.",
"})$ as the green, dashed curve and $K({\\rm FHMRV})$ as the purple, dashed curve.",
"We note that the Deductor (full) curve is roughly 30% below the analytic FHMRV curve for $P_\\mathrm {T}> 2 \\ \\mathrm {TeV}$ .",
"We also note that the scale variation test in figure REF suggests that the Deductor (full) curve in figure REF should be regarded as being uncertain to $\\pm 30 \\%$ .",
"In fact, we see in figure REF that changing the scales to $\\mu _\\mathrm {f}= \\mu _\\mathrm {r}= P_\\mathrm {T}/2$ makes the Deductor (full) cross section roughly 30% bigger.",
"Thus the level of agreement between the Deductor (full) and FHMRV curves seems not unreasonable.",
"It would, of course, be desirable to improve the precision of the shower cross section.",
"This can be achieved by matching the shower calculation to the perturbative NLO correction to the cross section, but we have not yet undertaken this task.",
"Figure: Illustrations of the one jet inclusive cross section as in figure .",
"We show the results of calculations of dσ/dP T d\\sigma /dP_\\mathrm {T} by plotting ratios KK of dσ/dP T d\\sigma /dP_\\mathrm {T} to the perturbative Born cross section dσ( LO )/dP T d\\sigma ({\\rm LO})/dP_\\mathrm {T}.",
"The scales in all cross sections are μ f =μ r =P T \\mu _\\mathrm {f}= \\mu _\\mathrm {r}= P_\\mathrm {T}.",
"Going from the lowest to the highest curves at P T =2 TeV P_\\mathrm {T}= 2 \\ \\mathrm {TeV}, the lowest curve is K( std .",
")K({\\rm std.})",
"using in the numerator the showered cross section obtained with threshold effects turned off.",
"The next is K( no Δ)K({\\rm no}\\ \\Delta ) obtained with the Δ ak \\Delta _{\\mathrm {a}k} contribution turned off.",
"The next curve is K( full )K({\\rm full}) using the showered cross section with threshold effects.",
"The next is K( LO )=1K({\\rm LO}) = 1.",
"The next curve is K( NLO )K({\\rm NLO}) using the perturbative NLO cross section.",
"The next highest curve is K(K.O.",
")K({\\rm K.O.})",
"using the cross section obtained using the Kidonakis-Owens threshold effects program .",
"The highest curve is K( FHMRV )K({\\rm FHMRV}) using the cross section calculated with the de Florian, Hinderer, Mukherjee, Ringer and Vogelsang algorithm .We can draw a further conclusion from these comparisons.",
"The parton shower has a hard job to perform, since it needs to include two large effects that act in opposite directions: loss of $P_\\mathrm {T}$ from the jet from showering and also the threshold enhancement.",
"It seems to us remarkable that the calculation works to within the expected uncertainty."
],
[
"Summary of the analysis",
"We have viewed parton shower evolution as the solution of equation (REF ), $\\frac{d}{dt}\\,\\big |{\\rho (t)}\\big ) = [{\\cal H}_I(t) - {\\cal S}(t)]\\big |{\\rho (t)}\\big )\\;,$ where $\\big |{\\rho (t)}\\big )$ represents the probability distribution of parton flavors and momenta and the density matrix in the quantum color and spin space in a statistical ensemble of event generation trials [67], [68], [69], [70], [71], [72], [73], [74].",
"In this paper, we have ignored spin but still consider a full treatment of quantum color, even though in an actual implementation as computer code one has to make some approximations with respect to color.",
"The shower time $t$ is the negative logarithm of the hardness scale $\\mu ^2$ considered.",
"The shower starts at a hard interaction and evolves to softer scales.",
"At shower time $t$ , interactions that are softer than $\\mu ^2$ are regarded as unresolvable, so that it is not meaningful to measure properties of the states described by $\\big |{\\rho (t)}\\big )$ at a finer scale.",
"In the evolution equation, ${\\cal H}_I(t)$ represents real parton splittings, while ${\\cal S}(t)$ leaves the number of partons, their momenta, and their flavors unchanged.",
"Both operators are order $\\alpha _{\\mathrm {s}}$ ; we do not examine higher order contributions.",
"The operator ${\\cal S}(t)$ gives us the Sudakov factor $\\exp (-\\int _{t_1}^{t_2}\\!dt\\, S(t))$ that appears between two parton splittings.",
"We have argued that ${\\cal S}(t)$ should consist of two parts, as in eq.",
"(REF ), $\\begin{split}{\\cal S}(t)={}&{\\cal S}^{\\rm pert}(t)- {\\cal F}(t)^{-1}\\left[\\frac{d}{dt}\\,{\\cal F}(t)\\right]\\;.\\end{split}$ Here ${\\cal F}(t)$ represents the parton distribution factor in $\\big |{\\rho (t)}\\big )$ , so that the second term in ${\\cal S}(t)$ gives the effect of changing the scale parameter in the parton distributions.",
"The first term, ${\\cal S}^{\\rm pert}(t)$ , accounts for order $\\alpha _{\\mathrm {s}}$ graphs that leave the number of partons unchanged.",
"That is, ${\\cal S}^{\\rm pert}(t)$ represents one loop virtual graphs.",
"We have (approximately) calculated ${\\cal S}^{\\rm pert}(t)$ in this paper.",
"In order to construct a parton shower based on eq.",
"(REF ), one can use a trick.",
"One can replace ${\\cal S}(t)$ by ${\\cal V}(t)$ , where ${\\cal V}(t)$ is constructed from the splitting operator ${\\cal H}_I(t)$ in such a way that the Born level cross section contained in $\\big |{\\rho (0)}\\big )$ is exactly conserved by the shower evolution.",
"In fact, parton shower algorithms are typically based on ${\\cal V}(t)$ instead of ${\\cal S}(t)$ .",
"The difference $[{\\cal V}(t) - {\\cal S}(t)]$ corrects $-{\\cal V}(t)$ .",
"If we approximate the color using the leading color (LC) approximation or the LC+ approximation [70], then the color matrices are diagonal and $\\exp (\\int _{t_1}^{t_2}\\!d t\\,[{\\cal V}(t) - {\\cal S}(t)])$ gives us a numerical weight factor that adjusts the cross section.",
"We found that ${\\cal S}^{\\rm pert}(t)$ contains factors $\\pm {\\mathrm {i}}\\pi $ times certain color matrices.",
"These terms are very well known.",
"(See, for example ref. [85].)",
"As noted in ref.",
"[70], these terms conserve the Born level cross section and could be included in ${\\cal V}(t)$ instead of ${\\cal S}(t)$ .",
"Although the ${\\mathrm {i}}\\pi $ terms are of considerable physical interest, they are of secondary interest in this paper and have not been included in our numerical results.",
"The integrand in $[{\\cal V}(t) - {\\cal S}(t)]$ contains ratios of parton distribution functions.",
"The most important term has the form $\\frac{\\alpha _{\\mathrm {s}}}{2\\pi }\\left[1 - \\frac{f_{a/A}(\\eta _\\mathrm {a}/z,\\mu ^2)}{f_{a/A}(\\eta _\\mathrm {a},\\mu ^2)}\\right]\\frac{1}{1-z}$ integrated over $z$ in a small range near $z = 1$ .",
"This term is large if $f_{a/A}(\\eta _\\mathrm {a}/z,\\mu ^2)$ falls steeply as $1-z$ increases.",
"Furthermore, $f_{a/A}(\\eta _\\mathrm {a}/z,\\mu ^2)$ does fall steeply as $1-z$ increases when $\\eta _\\mathrm {a}$ is large, say bigger than 0.1.",
"Thus a straightforward analysis of parton shower evolution leads us to the conclusion that there can be large corrections to the Born level cross section for a hard process.",
"These contributions are naturally summed in a parton shower algorithm that incorporates ${\\cal S}(t)$ because $[{\\cal V}(t) - {\\cal S}(t)]$ appears as part of the Sudakov exponent.",
"Of course, these contributions are not summed if the Sudakov factor is $\\exp [-\\int \\!dt\\,{\\cal V}(t)]$ , as is customary in parton showers (including ours, Deductor 1.0).",
"In this paper, we have presented results from Deductor 2.0 [75], in which $[{\\cal V}(t) - {\\cal S}(t)]$ is included within the LC+ approximation.",
"(In our numerical results, we used the LC approximation.)",
"The effects that arise from the term in eq.",
"(REF ) are clearly connected with the effects of what are usually called threshold logarithms, which have been extensively studied.",
"It thus is puzzling that the formulation in this paper contains ratios of parton distribution functions in an exponent, whereas standard threshold summation results never, to our knowledge, contain such factors.",
"How, then, can these formulations be connected?",
"The answer can be understood in the treatment of the DGLAP parton evolution equation in the formulation of a parton shower.",
"In a parton shower describing hadron-hadron collisions, at hardness scale $\\mu ^2$ , one needs to include “unresolvable” initial state interactions as parton distribution factors, $f_{a/A}(\\eta _\\mathrm {a},\\mu ^2)$ and $f_{b/B}(\\eta _\\mathrm {b},\\mu ^2)$ .",
"When we come to a softer scale, we need to cancel the previous parton distribution functions and supply new ones.",
"For this purpose, we can use $f_{a/A}(\\eta _\\mathrm {a},\\mu _2^2) = f_{a/A}(\\eta _\\mathrm {a},\\mu _1^2)\\,\\exp \\!\\left[-\\int _{\\mu _2^2}^{\\mu _1^2} \\!d\\mu ^2\\frac{d}{d\\mu ^2} \\log \\left(f_{a/A}(\\eta _\\mathrm {a},\\mu ^2)\\right)\\right]\\;.$ That is, making use of the first order evolution equation, $\\begin{split}f_{a/A}(\\eta _\\mathrm {a},\\mu _2^2) ={}& f_{a/A}(\\eta _\\mathrm {a},\\mu _1^2)\\,\\exp \\!\\Bigg [-\\int _{\\mu _2^2}^{\\mu _1^2} \\!\\frac{d\\mu ^2}{\\mu ^2}\\int \\!dz\\sum _{\\hat{a}}\\frac{\\alpha _{\\mathrm {s}}}{2 \\pi }\\\\&\\times \\bigg \\lbrace P_{a\\hat{a}}(z)\\,\\frac{f_{a/A}(\\eta _\\mathrm {a}/z,\\mu ^2)}{z f_{a/A}(\\eta _\\mathrm {a},\\mu ^2)}- \\delta _{a, \\hat{a}}\\left(\\frac{2 C_a}{1-z} - \\gamma _a\\right)\\bigg \\rbrace \\Bigg ]\\;.\\end{split}$ This is what appears in shower evolution algorithms [48].",
"On the other hand, if we use the Mellin transform of the parton distributions, eq.",
"(REF ), we have $\\tilde{f}_{a/A}(N,\\mu _2^2)= \\tilde{f}_{a^{\\prime }/A}(N,\\mu _1^2)\\exp \\!\\left[-\\int _{\\mu _2^2}^{\\mu _1^2} \\!\\frac{d\\mu ^2}{\\mu ^2}\\frac{\\alpha _{\\mathrm {s}}}{2 \\pi }\\,\\gamma (N)\\right]_{a a^{\\prime }}\\;.$ Here $\\gamma (N)$ it the matrix obtained by taking the Mellin transform of the evolution kernels.",
"There are no parton distribution functions in the exponent.",
"It is this formulation, or variations on it that do not work directly with the Mellin transformed evolution kernels, that typically appears in threshold summation calculations.",
"For general purposes, eq.",
"(REF ) is more powerful than eq.",
"(REF ) because one does not need to have already solved the evolution equation to use it.",
"However, eq.",
"(REF ) is not more, or less, accurate than eq.",
"(REF ).",
"In fact, we found in section that a standard threshold summation in the case of the Drell-Yan process matches what the analysis of this paper gives when we use a leading saddle point approximation to connect the two results.",
"We now comment on the role of parton distributions in the formalism that we have presented.",
"The $\\big |{\\rho (t)}\\big )$ in eq.",
"(REF ) contains a factor representing the parton distribution functions at the resolution scale corresponding to $t$ .",
"In eq.",
"(REF ), we defined an alternative statistical state $\\big |{\\rho _{\\rm pert}(t)}\\big )$ in which this parton distribution factor has been removed.",
"Then $\\big |{\\rho _{\\rm pert}(t)}\\big )$ obeys the evolution equation (REF ) in which the parton distribution functions do not appear.",
"At the end of the shower, we obtain the ordinary statistical state $\\big |{\\rho (t_\\mathrm {f})}\\big )$ by multiplying by parton distribution functions appropriate to a low $k_\\perp ^2$ scale, $m_\\perp ^2({\\rm start})$ , at which the shower turns off.",
"With this formulation, the whole hard scattering plus parton shower is a single perturbative process for which the resolution scale is $m_\\perp ^2({\\rm start})$ .",
"There are no parton distribution functions except at the low scale.",
"Now, the actual code works with $\\big |{\\rho (t)}\\big )$ and does use parton distribution functions at the hard scattering and at each shower stage.",
"However, this use of parton distribution functions is only a trick [48].",
"Actually, all of the parton distribution functions approximately cancel except for those at the final low scale.The cancellation is approximate because the operator $[{\\cal V}(t) - {\\cal S}(t)]$ is calculated approximately, using a limit in which successive splittings are strongly ordered.",
"In order to make this cancellation work, Deductor 2.0 matches parton evolution to shower evolution by using parton distribution functions $f_{a/A}(\\eta _\\mathrm {a},\\mu _\\Lambda ^2)$ that evolve from the starting scale according a modified leading order evolution equation (REF ).",
"It would, of course, be better to use an appropriate next-to-leading order evolution equation for the parton distribution functions, but we do not have a next-to-leading order shower algorithm.",
"Thus we are stuck with a parton shower summation of logarithms based on leading order perturbation theory."
],
[
"Choices in ",
"A user of Deductor 2.0 has some choices.",
"The default choice is the calculation described above.",
"With this choice, the parton shower sums threshold logarithms as described in this paper.",
"Another choice would be to eliminate the summation of threshold logarithms.",
"This is easy to do, as described in section .",
"There is a third possibility.",
"The user may want to retain the threshold logarithms but modify the way parton distribution functions appear in the calculated cross section.",
"The default result for a cross section has the form $d\\sigma = d\\hat{\\sigma }\\ Z_\\mathrm {a}Z_\\mathrm {b}\\ f^{\\overline{\\text{MS}}}_{a/A}(\\eta _\\mathrm {a},Q^2)f^{\\overline{\\text{MS}}}_{b/B}(\\eta _\\mathrm {b},Q^2)\\;,$ where $Q^2$ is the scale of the hard scattering and some of the summation of threshold logarithms is contained in the factor $Z_\\mathrm {a}Z_\\mathrm {b}=\\frac{f_{a/A}(\\eta _\\mathrm {a},Q^2)}{f^{\\overline{\\text{MS}}}_{a/A}(\\eta _\\mathrm {a},Q^2)}\\frac{f_{b/B}(\\eta _\\mathrm {b},Q^2)}{f^{\\overline{\\text{MS}}}_{b/B}(\\eta _\\mathrm {b},Q^2)}\\;.$ See section REF .",
"Here the parton distribution functions $f^{\\overline{\\text{MS}}}_{a/A}(\\eta _\\mathrm {a},Q^2)$ are obtained from the parton distribution functions at the scale $m_\\perp ^2({\\rm start})$ using the first order evolution equation (REF ).",
"Suppose that the user is only slightly interested in the details of the final state that a parton shower naturally specifies.",
"Rather, the user is most interested in the hard scattering that initiates the parton shower.",
"This user wants to have a calculation of the inclusive hard scattering cross section, including threshold corrections, that is as accurate as possible.",
"Such a user might not want to have a cross section based on parton distributions $f^{\\overline{\\text{MS}}}_{a/A}(\\eta _\\mathrm {a},Q^2)$ and $f^{\\overline{\\text{MS}}}_{b/B}(\\eta _\\mathrm {a},Q^2)$ at the hard scale, since these parton distributions have been obtained by lowest order evolution from $m_\\perp ^2({\\rm start})$ .",
"Instead, this user might prefer parton distributions $f^{\\overline{\\text{MS}}, {\\rm NLO}}_{a/A}(\\eta _\\mathrm {a},Q^2)$ and $f^{\\overline{\\text{MS}}, {\\rm NLO}}_{b/B}(\\eta _\\mathrm {a},Q^2)$ that have been obtained with next-to-leading order evolution.",
"That is easily arranged by multiplying the default $d\\sigma $ in eq.",
"(REF ) by a suitable weight factor: $d\\sigma ^{\\rm mod.}",
"= d\\sigma \\ \\frac{f^{\\overline{\\text{MS}}, {\\rm NLO}}_{a/A}(\\eta _\\mathrm {a},Q^2)}{f^{\\overline{\\text{MS}}}_{a/A}(\\eta _\\mathrm {a},Q^2)}\\,\\frac{f^{\\overline{\\text{MS}}, {\\rm NLO}}_{b/B}(\\eta _\\mathrm {b},Q^2)}{f^{\\overline{\\text{MS}}}_{b/B}(\\eta _\\mathrm {b},Q^2)}\\;.$ We have, in fact, done this in our numerical comparisons in section ."
],
[
"Outlook",
"We have presented a formulation of parton shower event generators in which the “threshold” enhancements of the cross section at large hardness scale $Q^2$ are included within the parton shower.",
"This has a disadvantage compared to analytical summations of threshold logs: as presented here, the calculation has not been systematically extended beyond the leading logarithm approximation, whereas many of the analytical results are for a much improved order of approximation.",
"However, the parton shower formulation has the advantage compared to analytical calculations that the same algorithm works for a wide variety of physical observables.",
"As long as the desired Born level hard process is included in the parton shower code, the user simply has to specify the observable that is to be measured at the end of the shower.",
"Compared to standard parton shower formulations that do not include threshold effects, the methods presented here have the advantage that they make the parton shower more accurate in a base level approximation in which matching to an NLO calculation has not been applied.",
"Every parton shower program is a little bit different.",
"We have presented the threshold algorithms as needed for our program, Deductor.",
"However, we believe that the methods presented here can be adapted with not much difficulty to other parton shower event generators.",
"This work was supported in part by the United States Department of Energy and by the Helmholtz Alliance “Physics at the Terascale.\"",
"This project was begun while the authors were at the Munich Institute for Astro- and Particle Physics program “Challenges, Innovations and Developments in Precision Calculations for the LHC.” It was completed while one of us (DS) was at the Kavli Institute for Theoretical Physics at the University of California, Santa Barbara, program “LHC Run II and the Precision Frontier” which was supported by the U. S. National Science Foundation under Grant No.",
"NSF PHY11-25915.",
"We thank the MIAPP and the KITP for providing stimulating research environments.",
"We thank Thomas Becher, Timothy Cohen, Hannes Jung, and George Sterman for helpful conversations.",
"We thank Frank Petriello for advice about the Drell-Yan cross section and its perturbative calculation.",
"We thank Jeff Owens for providing us with the Kidonakis-Owens code for threshold corrections to the one jet inclusive cross section.",
"We thank Werner Vogelsang for providing us with the de Florian, Hinderer, Mukherjee, Ringer, Vogalsang code for threshold corrections to the jet cross section."
],
[
"Notation and kinematics",
"In this appendix, we collect some notations used throughout the paper and put them in one place.",
"We only sketch the notation that we need.",
"Details can be found in refs.",
"[67], [68], [69], [70], [71], [72], [73], [74].",
"In the parton shower, the partons are described by a state vector $\\big |{\\rho (t)}\\big )$ that represents the probability distribution of parton flavors and momenta and the density matrix in the quantum color space.",
"(Recall that in this paper we effectively ignore quantum spin by summing over spins of the daughter partons after each splitting and averaging over the mother parton spin.)",
"In this paper, all of the partons except top quarks are massless.",
"We expand $\\big |{\\rho (t)}\\big )$ in basis states $\\big |{\\lbrace p,f,c^{\\prime },c\\rbrace _{m}}\\big )$ .",
"This basis state represents two initial state partons with labels “a” and “b” and $m$ final state partons with labels $l$ .",
"The partons have momenta $p$ , flavors $f$ , and colors $c^{\\prime },c$ .",
"The momentum fractions of the initial state partons are $\\eta _\\mathrm {a}$ and $\\eta _\\mathrm {b}$ .",
"In the context of a parton splitting, we generally use $p_l$ to denote the momentum of parton $l$ before the splitting and $\\hat{p}_l$ to denote the momentum of parton $l$ after the splitting.",
"When a final state parton labelled $l$ splits into partons $l$ and $m+1$ with momenta $\\hat{p}_l$ and $\\hat{p}_{m+1}$ , we characterize the splitting by its virtuality $(\\hat{p}_l + \\hat{p}_{m+1})^2$ .",
"Similarly, when the initial state parton with label “a” splits in backward evolution to a new initial state parton “a” and a new final state parton with label $m+1$ , we characterize the splitting by its spacelike virtuality $(\\hat{p}_\\mathrm {a}- \\hat{p}_{m+1})^2$ .",
"The shower time that we use is related to the virtuality of a splitting: $\\begin{split}e^{-t} ={}& \\frac{(\\hat{p}_l + \\hat{p}_{m+1})^2}{2 p_l\\cdot Q_0}\\hspace{42.67912pt}{\\rm final\\ state},\\\\e^{-t} ={}& -\\frac{(\\hat{p}_\\mathrm {a}- \\hat{p}_{m+1})^2}{2 p_\\mathrm {a}\\cdot Q_0}\\hspace{28.45274pt}{\\rm initial\\ state}.\\end{split}$ Here $Q_0$ is a fixed vector equal to the total momentum of all of the final state partons just after the hard scattering that initiates the shower.",
"With this notation, the virtuality in an initial state splitting is $\\begin{split}\\mu _\\mathrm {a}^2(t) ={}& 2 p_\\mathrm {a}\\cdot Q_0 \\,e^{-t}\\;.\\end{split}$ It is convenient to use a dimensionless virtuality variable $\\begin{split}y ={}& \\frac{(\\hat{p}_l + \\hat{p}_{m+1})^2}{2 p_l\\cdot Q}\\hspace{42.67912pt}{\\rm final\\ state},\\\\y ={}& -\\frac{(\\hat{p}_\\mathrm {a}- \\hat{p}_{m+1})^2}{2 p_\\mathrm {a}\\cdot Q}\\hspace{28.45274pt}{\\rm initial\\ state}.\\end{split}$ Here $Q = p_\\mathrm {a}+ p_\\mathrm {b}$ is the total momentum of the final state partons just before the splitting.",
"Then $y$ is related to shower time by $\\begin{split}y ={}& \\frac{2 p_l\\cdot Q_0}{2 p_l\\cdot Q}\\,e^{-t}\\hspace{28.45274pt}{\\rm final\\ state},\\\\y ={}& \\frac{2 p_\\mathrm {a}\\cdot Q_0}{2 p_\\mathrm {a}\\cdot Q}\\,e^{-t}\\hspace{28.45274pt}{\\rm initial\\ state}.\\end{split}$ Since $2 p_\\mathrm {a}\\cdot Q = Q^2$ , we have a convenient identity for $\\mu _\\mathrm {a}^2(t)$ : $\\mu _\\mathrm {a}^2(t) = y\\,Q^2\\;.$ We sometimes use a squared transverse momentum variable $\\mathbf {k}_\\perp ^2$ for a splitting of initial state parton “a.” With the exact kinematics and momentum mappings used in Deductor, the emitted parton has label $m+1$ and momentum $\\hat{p}_{m+1}$ .",
"The part of $\\hat{p}_{m+1}$ orthogonal to the momenta of both incoming partons after the splitting, $\\hat{p}_\\mathrm {a}$ and $\\hat{p}_\\mathrm {b}$ , is $\\hat{\\mathbf {p}}_{m+1}^\\perp $ , whose square is $(\\hat{\\mathbf {p}}_{m+1}^\\perp )^2 = y (1 - z - zy)\\, 2 p_\\mathrm {a}\\cdot Q\\;.$ Here $z$ is defined by $z = \\eta _\\mathrm {a}/\\hat{\\eta }_\\mathrm {a}$ , where $\\eta _\\mathrm {a}$ is the momentum fraction of parton “a” before the splitting (in backward evolution) and $\\hat{\\eta }_\\mathrm {a}$ is its momentum fraction after the splitting.",
"We note that the condition $(\\hat{\\mathbf {p}}_{m+1}^\\perp )^2 \\ge 0$ implies $z < 1/(1+y)\\;.$ The quantity $(\\hat{\\mathbf {p}}_{m+1}^\\perp )^2$ vanishes when the emitted parton is collinear to parton “a,” which corresponds to $y \\rightarrow 0$ with fixed $(1-z)$ .",
"However, it also vanishes when the emitted parton is collinear to parton “b,” which corresponds to $(1 - z - zy) \\rightarrow 0$ with fixed $y$ .",
"For our purposes, we prefer a variable that matches $(\\hat{\\mathbf {p}}_{m+1}^\\perp )^2$ in the collinear limit, but does not vanish in the anticollinear limit $(1 - z - zy) \\rightarrow 0$ .",
"We thus define a transverse momentum variable $\\mathbf {k}_\\perp ^2 = y (1 - z)\\, 2 p_\\mathrm {a}\\cdot Q\\;.$ This discussion of $\\mathbf {k}_\\perp ^2$ may be compared to that in section 2.3.1 of ref. [56].",
"We will need some standard flavor dependent constants.",
"The number of flavors is $N_\\mathrm {f}$ .",
"We have color factors $C_\\mathrm {A}= N_\\mathrm {c}$ (with $N_\\mathrm {c}= 3$ ) and $C_\\mathrm {F}= (N_\\mathrm {c}^2 - 1)/(2 N_\\mathrm {c})$ .",
"We use $f$ for a parton flavor, $f \\in \\lbrace \\mathrm {g}, \\mathrm {u}, \\bar{\\mathrm {u}}, \\mathrm {d}, \\dots \\rbrace $ .",
"We define $C_f$ by $\\begin{split}C_\\mathrm {g}={}& C_\\mathrm {A}\\;,\\\\C_q ={}& C_\\mathrm {F}\\qquad q\\in \\lbrace \\mathrm {u}, \\bar{\\mathrm {u}}, \\mathrm {d}, \\dots \\rbrace \\;.\\end{split}$ In the kernels for evolution of parton distributions, constants $\\gamma _f$ appear, with $\\begin{split}\\gamma _\\mathrm {g}={}& \\frac{11 C_\\mathrm {A}}{6} - \\frac{2 T_\\mathrm {R}N_\\mathrm {f}}{3}\\;,\\\\\\gamma _q ={}& \\frac{3 C_\\mathrm {F}}{2} \\qquad q\\in \\lbrace \\mathrm {u}, \\bar{\\mathrm {u}}, \\mathrm {d}, \\dots \\rbrace \\;,\\end{split}$ where $T_\\mathrm {R}= 1/2$ .",
"We will use three momenta $\\vec{p}_l$ in a reference frame in which the total momentum $Q$ of the final state partons before the splitting has zero space components.",
"(Thus $Q = p_\\mathrm {a}+ p_\\mathrm {b}$ .)",
"In the case of an on-shell virtual particle, $p_l^0 = |\\vec{p}_l|$ .",
"We then write $p_l = |\\vec{p}_l| v_l$ where $v_l = (1,\\vec{v}_l)\\;,$ with $\\vec{v}_l^{\\,2} = 1$ .",
"We use $E_Q$ for the energy component of $Q$ : $Q = (E_Q, \\vec{0}\\,)$ .",
"We often use the convenient shorthand $\\begin{split}a_l ={}& \\frac{Q^{2}}{2p_{l}\\!\\cdot \\!",
"Q} = \\frac{E_Q}{2 |\\vec{p}_l|}\\;.\\end{split}$ In the case of an initial state parton, this is $a_\\mathrm {a}= a_\\mathrm {b}= 1$ .",
"We also sometimes use the shorthand $\\psi _{kl} = \\frac{1 - \\cos \\theta _{kl}}{\\sqrt{8(1 + \\cos \\theta _{kl})}}\\;,$ where $\\theta _{kl}$ is the angle between a pair of partons with labels $k$ and $l$ ."
],
[
"Calculation of the probability conserving integrand ${\\cal V}(t)$",
"For each real splitting in a parton shower, we need a Sudakov factor associated with the evolution of the system from the time $t_1$ of the previous splitting and the time $t_2$ of the new splitting.",
"The standard form for this factor is $\\mathbb {T} \\exp [-\\int _{t_1}^{t_2}\\!",
"dt\\ {\\cal V}(t)]$ , where ${\\cal V}(t)$ is the probability per unit $dt$ to have a splitting at time $t$ .",
"Thus the Sudakov factor is the probability not to have had a splitting between $t_1$ and $t_2$ .",
"This is the structure of the Sudakov factor used in Deductor 1.0.",
"In this paper, we calculate a numerical factor $\\int _{t_1}^{t_2}\\!",
"dt\\ [{\\cal V}(t) - {\\cal S}(t)]$ , where ${\\cal S}(t)$ gives the effect of parton evolution and virtual graphs.",
"We are not able to calculate this factor exactly.",
"Rather, we calculate ${\\cal V}(t)$ and ${\\cal S}(t)$ for large t, or small $y \\propto e^{-t}$ as defined in eq.",
"(REF ).",
"In this appendix, we calculate ${\\cal V}(t)$ for small y. Fortunately, although ${\\cal V}(t)$ in Deductor is quite complicated, it has a simple structure for small $y$ .",
"Our aim in this appendix is to exhibit enough details of the calculation to enable the reader to reproduce it.",
"The operator ${\\cal V}(t)$ is a sum over contributions from the partons in existence after time $t_1$ , as in eq.",
"(REF ) [67]: ${\\cal V}(t)={\\cal V}_\\mathrm {a}(t)+ {\\cal V}_\\mathrm {b}(t)+ \\sum _{l=1}^m {\\cal V}_l(t)\\;.$ We will first examine the case of final state partons, with labels $l = 1, \\dots m$ .",
"Then we will turn to the initial state partons, with labels “a” and “b.”"
],
[
"Final state partons",
"We write ${\\cal V}_l(t)$ as [67] ${\\cal V}_l(t) =\\sum _k {\\cal V}_{l k}(t)\\;.$ The sum includes all parton labels $k = \\mathrm {a}, \\mathrm {b}, 1, \\dots , m$ .",
"When the operators ${\\cal V}_{l k}(t)$ act on a partonic state $\\big |{\\lbrace p,f,c^{\\prime },c\\rbrace _{m}}\\big )$ , they have the structure $\\begin{split}{\\cal V}_{l k}(t)\\big |{\\lbrace p,f,c^{\\prime },c\\rbrace _{m}}\\big )={}&\\overline{\\lambda }^{{\\cal V}}_{l k}(\\lbrace p,f\\rbrace _{m},t)\\,\\frac{1}{2}\\big ([(\\mathbf {T}_l\\cdot \\mathbf {T}_k)\\otimes 1] + [1 \\otimes (\\mathbf {T}_l\\cdot \\mathbf {T}_k)]\\big )\\\\ &\\times \\big |{\\lbrace p,f,c^{\\prime },c\\rbrace _{m}}\\big )\\;.\\end{split}$ As explained in section , the operators $[(\\mathbf {T}_l\\cdot \\mathbf {T}_k)\\otimes 1]$ act on the space of color density matrices for the parton state [67], [70], with basis elements $\\big |{\\lbrace c\\rbrace _m}\\big \\rangle \\big \\langle {\\lbrace c^{\\prime }\\rbrace _m}\\big |$ .",
"A color generator matrix $T^a$ acting on parton $l$ in the ket state multiplies a generator matrix $T_a$ acting on parton $k$ in the ket state.",
"The dot product indicates a sum over $a = 1,\\dots , N_\\mathrm {c}^2 - 1$ .",
"In $[1 \\otimes (\\mathbf {T}_l\\cdot \\mathbf {T}_k)]$ , we have the same construction with the color generators acting on the bra state.",
"From eq.",
"(5.28) of ref.",
"[70], we find that the functions $\\overline{\\lambda }^{{\\cal V}}_{l k}$ has the structure $\\begin{split}\\overline{\\lambda }^{{\\cal V}}_{l k}(\\lbrace p,& f\\rbrace _{m},t)\\\\={}&\\frac{1}{m!",
"}\\int \\!d\\lbrace \\hat{p},\\hat{f}\\rbrace _{m+1}\\delta (t - T(\\lbrace \\hat{p},\\hat{f}\\rbrace _{m+1}))\\,\\big ({\\lbrace \\hat{p},\\hat{f}\\rbrace _{m+1}}\\big |{\\cal P}_{l}\\big |{\\lbrace p,f\\rbrace _m}\\big )\\\\&\\times \\bigg [\\theta (k = l)\\,\\theta (\\hat{f}_{m+1} \\ne \\mathrm {g}))\\,\\frac{1}{2 C_\\mathrm {A}}\\,\\overline{w}_{ll}(\\lbrace \\hat{p},\\hat{f}\\rbrace _{m+1})\\\\&\\qquad +\\theta (k = l)\\,\\theta (\\hat{f}_{m+1} = \\mathrm {g}))[\\overline{w}_{ll}(\\lbrace \\hat{p},\\hat{f}\\rbrace _{m+1})- \\overline{w}_{ll}^{\\rm eikonal}(\\lbrace \\hat{p},\\hat{f}\\rbrace _{m+1})]\\\\&\\qquad -\\theta (k\\ne l)\\,\\,\\theta (\\hat{f}_{m+1} = \\mathrm {g}))\\,A^{\\prime }_{l k}(\\lbrace \\hat{p}\\rbrace _{m+1})\\overline{w}_{l k}^{\\rm dipole}(\\lbrace \\hat{p},\\hat{f}\\rbrace _{m+1})\\bigg ]\\;.\\end{split}$ Consider the first line on the right hand side of eq.",
"(REF ).",
"There is an integration over the variables that define the splitting of a mother parton with momentum $p_l$ into daughter partons with momenta $\\hat{p}_l$ and $\\hat{p}_{m+1}$ .",
"In this paper, we take all partons to be massless.",
"We will use the auxiliary variables $\\begin{split}a_l ={}& \\frac{Q^{2}}{2p_{l}\\!\\cdot \\!",
"Q}\\;,\\\\\\lambda (y) ={}&\\sqrt{(1 + y)^2 - 4 a_l y}\\;,\\\\h_\\pm (y) ={}& \\frac{1}{2}\\,[1 + y \\pm \\lambda (y)]\\;.\\end{split}$ Here $Q$ is the total momentum of all the final state partons in $\\big |{\\lbrace p,f,c^{\\prime },c\\rbrace _{m}}\\big )$ .",
"The dimensionless virtuality variable $y$ was defined in eqs.",
"(REF ) and (REF ): $y = \\frac{2 \\hat{p}_l\\cdot \\hat{p}_{m+1}}{2 p_l \\cdot Q}\\;.$ We define the momentum fraction in the splitting by $\\frac{\\hat{p}_{m+1}\\cdot \\tilde{n}_l}{\\hat{p}_{l}\\cdot \\tilde{n}_l}= \\frac{1-z}{z}\\;,$ where the auxiliary lightlike vector $\\tilde{n}_l$ is $\\tilde{n}_l = \\frac{1}{a_l}\\, Q - p_l\\;.$ We also define an azimuthal angle $\\phi $ of the splitting using the part $k_\\perp $ of $\\hat{p}_l$ that is orthogonal to $p_l$ and $\\tilde{n}_l$ .",
"The choice of flavors of the daughters can be specified by giving the flavor $\\hat{f}_{m+1}$ of daughter parton $m+1$ .",
"This gives us splitting variables $y,z,\\phi ,\\hat{f}_{m+1}$ .",
"We can write the daughter parton momenta in terms of $y,z,\\phi $ using $\\begin{split}\\hat{p}_l ={}&z\\,h_+(y)\\, p_l+ (1 - z) h_-(y)\\, \\tilde{n}_l+k_\\perp \\;,\\\\\\hat{p}_{m+1} ={}&(1-z)\\, h_+(y)\\, p_l+ z h_-(y)\\, \\tilde{n}_l- k_\\perp \\;.\\end{split}$ The magnitude of the transverse momentum $k_\\perp $ is given by $- \\frac{k_\\perp ^2}{2p_l\\cdot Q} =z (1-z)y\\;.$ Using eq.",
"(8.20) of ref.",
"[67], we find that integration over the splitting variables between shower times corresponding to $y$ values $y_1$ and $y_2$ is accomplished with $\\begin{split}\\frac{1}{m!",
"}\\int \\big [d\\lbrace \\hat{p},\\hat{f}&\\rbrace _{m+1}\\big ]\\big ({\\lbrace \\hat{p},\\hat{f}\\rbrace _{m+1}}\\big |{\\cal P}_{l}\\big |{\\lbrace p,f\\rbrace _m}\\big )\\cdots \\\\&= \\frac{p_{l}\\!\\cdot \\!",
"Q}{8\\pi ^2}\\,\\int _{y_2}^{y_1}\\!",
"dy\\ \\lambda (y)\\,\\int _{0}^{1}\\!",
"dz\\int _{-\\pi }^\\pi \\frac{d\\phi }{2\\pi }\\sum _{\\hat{f}_{m+1}}\\ \\cdots \\;.\\end{split}$ The delta function that specifies the shower time is, from eqs.",
"(REF ) and (REF ), $\\delta (t - T(\\lbrace \\hat{p},\\hat{f}\\rbrace _{m+1}))=\\delta \\!\\left(\\log y - \\log \\!\\left(\\frac{p_l\\cdot Q_0}{p_l\\cdot Q}e^{-t}\\right)\\right)\\;.$ Using eqs.",
"(REF ) and (REF ) in eq.",
"(REF ) gives $\\begin{split}\\overline{\\lambda }^{{\\cal V}}_{l k}(\\lbrace p,& f\\rbrace _{m},t)\\\\={}&\\frac{p_{l}\\!\\cdot \\!",
"Q}{8\\pi ^2}\\,y\\,\\lambda (y)\\,\\int _{0}^{1}\\!",
"dz\\int _{-\\pi }^\\pi \\frac{d\\phi }{2\\pi }\\ \\sum _{\\hat{f}_{m+1}}\\\\&\\times \\bigg [\\theta (k = l)\\,\\theta (\\hat{f}_{m+1} \\ne \\mathrm {g}))\\,\\frac{1}{2 C_\\mathrm {A}}\\,\\overline{w}_{ll}(\\lbrace \\hat{p},\\hat{f}\\rbrace _{m+1})\\\\&\\qquad +\\theta (k = l)\\,\\theta (\\hat{f}_{m+1} = \\mathrm {g}))[\\overline{w}_{ll}(\\lbrace \\hat{p},\\hat{f}\\rbrace _{m+1})- \\overline{w}_{ll}^{\\rm eikonal}(\\lbrace \\hat{p},\\hat{f}\\rbrace _{m+1})]\\\\&\\qquad -\\theta (k\\ne l)\\,\\,\\theta (\\hat{f}_{m+1} = \\mathrm {g}))\\,A^{\\prime }_{l k}(\\lbrace \\hat{p}\\rbrace _{m+1})\\overline{w}_{l k}^{\\rm dipole}(\\lbrace \\hat{p},\\hat{f}\\rbrace _{m+1})\\bigg ]\\;.\\end{split}$ The first line inside the square brackets here is for a gluon self-energy graph with a quark loop.",
"The color factor is $T_\\mathrm {R}= 1/2$ , but we follow the notation of eq.",
"(REF ), in which this term comes with a color operator $(\\mathbf {T}_l\\cdot \\mathbf {T}_l)\\otimes 1 = C_\\mathrm {A}1\\otimes 1$ or $1\\otimes (\\mathbf {T}_l\\cdot \\mathbf {T}_l)= C_\\mathrm {A}1\\otimes 1$ .",
"The factor $C_\\mathrm {A}$ does not really belong here, so we remove it by dividing $\\overline{w}_{ll}$ by $C_\\mathrm {A}$ .",
"The second line of eq.",
"(REF ) covers gluon emission in a cut self-energy graph, while the third line covers gluon exchange between two lines, $l$ and $k$ .",
"For $k = l$ and $\\hat{f}({m+1}) \\ne \\mathrm {g}$ , we have a $\\mathrm {g}\\rightarrow q + \\bar{q}$ splitting.",
"From ref.",
"[68], eq.",
"(A.1), we have $\\overline{w}_{ll}(\\lbrace \\hat{p},\\hat{f}\\rbrace _{m+1})= \\frac{8\\pi \\alpha _{\\mathrm {s}}}{(\\hat{p}_l + \\hat{p}_{m+1})^2}\\left(1 + \\frac{2 \\hat{p}_l\\cdot D(p_l,Q) \\cdot \\hat{p}_{m+1}}{(\\hat{p}_l + \\hat{p}_{m+1})^2}\\right)\\;.$ Here $D(p_l,Q)$ is the Coulomb gauge numerator function, $D(q)^{\\mu \\nu } =- g^{\\mu \\nu }-\\frac{q^\\mu \\tilde{q}^\\nu + \\tilde{q}^\\mu q^\\nu - q^\\mu q^\\nu }{|\\vec{q}\\,|^2}\\;,$ where $\\tilde{q} = (0,\\vec{q}\\,)$ .",
"Since here $q^2 = 0$ , this also equals $D(q)^{\\mu \\nu } =- g^{\\mu \\nu } + \\frac{q^\\mu Q^\\nu + Q^\\mu q^\\nu }{q\\cdot Q}- \\frac{Q^2\\ q^\\mu q^\\nu }{(q\\cdot Q)^2}\\;.$ With the help of eq.",
"(REF ), we find $\\hat{p}_l\\cdot D(p_l,Q) \\cdot \\hat{p}_{m+1}=-2 z (1-z) \\, y\\, p_l \\cdot Q\\;.$ This gives $\\overline{w}_{ll}(\\lbrace \\hat{p},\\hat{f}\\rbrace _{m+1})= \\frac{4\\pi \\alpha _{\\mathrm {s}}}{y\\,p_l\\cdot Q}\\left(1 - 2 z (1-z)\\right)\\;.$ The function $\\overline{w}_{ll}(\\lbrace \\hat{p},\\hat{f}\\rbrace _{m+1}) - \\overline{w}_{ll}^{\\rm eikonal}(\\lbrace \\hat{p},\\hat{f}\\rbrace _{m+1})$ is calculated in ref. [68].",
"For $\\hat{f}_l$ equal to a quark or antiquark flavor, it is given in eq.",
"(2.23) of ref.",
"[68] (for the quark mass equal to zero): $\\begin{split}\\overline{w}_{ll}(\\lbrace \\hat{p},\\hat{f}\\rbrace _{m+1})& - \\overline{w}_{ll}^{\\rm eikonal}(\\lbrace \\hat{p},\\hat{f}\\rbrace _{m+1})\\\\={}&\\frac{4\\pi \\alpha _{\\mathrm {s}}}{y\\, p_l\\cdot Q}\\bigg \\lbrace \\frac{(\\lambda (y) - 1 + y)^2 + 4 y}{4\\lambda (y)}\\,\\left[\\frac{2x}{1-x} - \\frac{2 a_l y}{(1-x)^2 (1 + y)^2}\\right]\\\\&\\qquad \\qquad + \\frac{1}{2}\\,(1-z)(1 + y + \\lambda (y))\\bigg \\rbrace \\;.\\end{split}$ Here $\\begin{split}x ={}& \\frac{\\lambda (y)}{1 + y}\\,z+ \\frac{2 a_l y}{(1+y)(1 + y + \\lambda (y))}\\;,\\\\1-x ={}& \\frac{\\lambda (y)}{1 + y}\\,(1-z)+ \\frac{2 a_l y}{(1+y)(1 + y + \\lambda (y))}\\;.\\end{split}$ Here we have changed conventions compared to ref.",
"[68] and exchanged $z \\leftrightarrow (1-z)$ and $x \\leftrightarrow (1-x)$ .",
"This expression is rather complicated, but we only need its $y \\rightarrow 0$ limit: $\\begin{split}\\overline{w}_{ll}(\\lbrace \\hat{p},\\hat{f}\\rbrace _{m+1})- \\overline{w}_{ll}^{\\rm eikonal}(\\lbrace \\hat{p},\\hat{f}\\rbrace _{m+1})\\sim {}&\\frac{4\\pi \\alpha _{\\mathrm {s}}}{y\\, p_l\\cdot Q}\\,(1-z)\\;.\\end{split}$ We also need $\\overline{w}_{ll}(\\lbrace \\hat{p},\\hat{f}\\rbrace _{m+1}) - \\overline{w}_{ll}^{\\rm eikonal}(\\lbrace \\hat{p},\\hat{f}\\rbrace _{m+1})$ for $\\mathrm {g}\\rightarrow \\mathrm {g}+ \\mathrm {g}$ splittings, $\\hat{f}_l = \\mathrm {g}$ .",
"It is given in eq.",
"(2.50) of ref.",
"[68]: $\\begin{split}\\overline{w}_{ll}(\\lbrace \\hat{p},\\hat{f}\\rbrace _{m+1})& - \\overline{w}_{ll}^{\\rm eikonal}(\\lbrace \\hat{p},\\hat{f}\\rbrace _{m+1})\\\\={}&\\frac{4\\pi \\alpha _{\\mathrm {s}}}{y\\, p_l\\cdot Q}\\ \\frac{z(1-z)}{2}\\left[1 + \\left(1 - \\frac{2 a_l y}{x(1-x) (1 + y)^2}\\right)^2\\right]\\;.\\end{split}$ We need only the $y\\rightarrow 0$ limit of this: $\\begin{split}\\overline{w}_{ll}(\\lbrace \\hat{p},\\hat{f}\\rbrace _{m+1})- \\overline{w}_{ll}^{\\rm eikonal}(\\lbrace \\hat{p},\\hat{f}\\rbrace _{m+1})\\sim {}&\\frac{4\\pi \\alpha _{\\mathrm {s}}}{y\\, p_l\\cdot Q}\\,z(1-z)\\;.\\end{split}$ Now we need the interference terms.",
"Using eq.",
"(5.3) of ref.",
"[70] for $\\overline{w}_{l k}$ and eq.",
"(7.12) of ref.",
"[69] for the partitioning function $A^{\\prime }_{l k}$ , we have $\\begin{split}A^{\\prime }_{l k}(\\lbrace \\hat{p}\\rbrace _{m+1})&\\, \\overline{w}_{l k}^{\\rm dipole}(\\lbrace \\hat{p},\\hat{f}\\rbrace _{m+1})\\\\={}& 4\\pi \\alpha _{\\mathrm {s}}\\ \\frac{2 \\hat{p}_k\\cdot \\hat{p}_l}{\\hat{p}_{m+1}\\cdot \\hat{p}_ l}\\ \\frac{\\hat{p}_l\\cdot Q}{\\hat{p}_{m+1}\\cdot \\hat{p}_k\\ \\hat{p}_l\\cdot Q+ \\hat{p}_{m+1}\\cdot \\hat{p}_l\\ \\hat{p}_k\\cdot Q}\\;.\\end{split}$ To evaluate this, we write the spectator momentum after the splitting, $\\hat{p}_k$ , as $\\hat{p}_k =A_k\\left[a_l e^{\\xi + \\omega (y)} p_l+ a_l e^{-\\xi - \\omega (y)}\\tilde{n}_l+ \\ell _{\\perp }\\right]\\;.$ where $\\xi $ is a boost angle that is related to the angle $\\theta _{lk}$ between $\\vec{p}_l$ and $\\vec{p}_k$ in the $\\vec{Q} = 0$ frame by $e^{2\\xi } = \\frac{1+\\cos \\theta _{lk}}{1-\\cos \\theta _{lk}}\\;.$ The parameter $\\omega (y)$ is an additional boost angle, $\\begin{split}e^{\\omega (y)} ={}&\\frac{a_l - h_+(y)}{a_l - h_+(0)}\\end{split}$ and in the case that $k = \\mathrm {a}$ or $k = \\mathrm {b}$ , $\\omega (y) = 0$ .",
"Since $\\omega (y)$ is of order $y$ and we are interested in the limit of small $y$ , we will replace $\\omega (y) \\rightarrow 0$ .",
"The normalization parameter $A_k$ drops out of our calculation.",
"The product in eq.",
"(REF ) is quite complicated in general.",
"However, it is reasonably simple in the limit of small $y$ .",
"We find $\\begin{split}A^{\\prime }_{l k}(\\lbrace \\hat{p}\\rbrace _{m+1})&\\, \\overline{w}_{l k}^{\\rm dipole}(\\lbrace \\hat{p},\\hat{f}\\rbrace _{m+1})\\\\\\approx {}&\\frac{4\\pi \\alpha _{\\mathrm {s}}}{y\\, p_l\\cdot Q}\\,\\frac{2 z }{(1-z)+ a_l y [2 e^{2\\xi } + 1]+ 2\\sqrt{a_l y e^{2\\xi }} \\sqrt{1-z} \\,\\cos \\phi }\\;.\\end{split}$ Here we have noted that the terms proportional to $y$ are negligible for small $y$ unless $(1-z)$ is small.",
"Therefore in the coefficients of $y$ and $\\sqrt{y(1-z)}$ , we have replaced $z$ by 1.",
"We can integrate this over $\\phi $ with the result $\\begin{split}\\int \\!\\frac{d\\phi }{2\\pi }\\ A^{\\prime }_{l k}(\\lbrace \\hat{p}\\rbrace _{m+1})&\\, \\overline{w}_{l k}^{\\rm dipole}(\\lbrace \\hat{p},\\hat{f}\\rbrace _{m+1})\\\\\\approx {}&\\frac{4\\pi \\alpha _{\\mathrm {s}}}{y\\, p_l\\cdot Q}\\,\\frac{2 z}{\\left[[(1-z) + a_l y]^2 + 4 a_l^2 y^2 e^{2\\xi }(1+e^{2\\xi })\\right]^{1/2}}\\;.\\end{split}$ We can then integrate this over $z$ and take the small $y$ limit of the result.",
"We get $\\begin{split}\\int _0^1\\!dz\\int \\!\\frac{d\\phi }{2\\pi }\\ A^{\\prime }_{l k}(\\lbrace \\hat{p}\\rbrace _{m+1})\\, \\overline{w}_{l k}^{\\rm dipole}(\\lbrace \\hat{p},\\hat{f}\\rbrace _{m+1})\\approx \\frac{8\\pi \\alpha _{\\mathrm {s}}}{y\\, p_l\\cdot Q}\\bigg \\lbrace \\log \\!\\left(\\frac{1 - \\cos \\theta _{kl}}{2 a_l y}\\right)- 1\\bigg \\rbrace \\;.\\end{split}$ We insert this result back into eq.",
"(REF ).",
"The integrals over $\\phi $ and $z$ of the other terms are simple, giving in the small $y$ limit, $\\begin{split}\\overline{\\lambda }^{{\\cal V}}_{l k}(\\lbrace p,f\\rbrace _{m},t)\\approx {}&\\frac{\\alpha _{\\mathrm {s}}}{2\\pi }\\,\\sum _{\\hat{f}_{m+1}}\\bigg [\\theta (k = l)\\,\\theta (\\hat{f}_{m+1} \\ne \\mathrm {g}))\\,\\frac{1}{3 C_\\mathrm {A}}\\\\&\\qquad +\\theta (k = l)\\,\\theta (\\hat{f}_{m+1} = \\mathrm {g}))\\theta (\\hat{f}_l \\ne \\mathrm {g}))\\ \\frac{1}{2}\\\\&\\qquad +\\theta (k = l)\\,\\theta (\\hat{f}_{m+1} = \\mathrm {g}))\\,\\theta (\\hat{f}_l = \\mathrm {g}))\\ \\frac{1}{6}\\\\&\\qquad -\\theta (k\\ne l)\\,\\,\\theta (\\hat{f}_{m+1} = \\mathrm {g}))\\,2\\bigg \\lbrace \\log \\!\\left(\\frac{1 - \\cos \\theta _{kl}}{2a_l y}\\right)- 1\\bigg \\rbrace \\bigg ]\\;.\\end{split}$ We can perform the sum over flavors $\\hat{f}_{m+1}$ .",
"The first term applies only if $f_l = \\mathrm {g}$ .",
"When $f_l = \\mathrm {g}$ , there are $N_\\mathrm {f}$ equal terms.",
"The third term applies if $f_l = \\mathrm {g}$ and there is one term.",
"The second term applies when $f_l \\ne g$ and there is one term.",
"The fourth term applies for any $f_l$ .",
"Thus $\\begin{split}\\overline{\\lambda }^{{\\cal V}}_{l k}(\\lbrace p,f\\rbrace _{m},t)\\approx {}&\\frac{\\alpha _{\\mathrm {s}}}{2\\pi }\\,\\bigg [\\theta (k = l)\\,\\theta (f_l = \\mathrm {g}))\\,\\frac{1}{C_\\mathrm {A}}\\left[\\frac{C_\\mathrm {A}}{6} + \\frac{N_\\mathrm {f}}{3}\\right]\\\\&\\qquad +\\theta (k = l)\\,\\theta (f_l \\ne \\mathrm {g})) \\ \\frac{1}{2}\\\\&\\qquad -\\theta (k\\ne l)\\,2\\bigg \\lbrace \\log \\!\\left(\\frac{1 - \\cos \\theta _{kl}}{2a_l y}\\right)- 1\\bigg \\rbrace \\bigg ]\\bigg ]\\;.\\end{split}$ Using the constants $\\gamma _f$ and $C_f$ from eqs.",
"(REF ) and eq.",
"(REF ), this is $\\begin{split}\\overline{\\lambda }^{{\\cal V}}_{l k}(\\lbrace p,f\\rbrace _{m},t)\\approx {}&\\frac{\\alpha _{\\mathrm {s}}}{2\\pi }\\,\\bigg [\\theta (k = l)\\,\\left[-\\frac{\\gamma _{f_l}}{C_{f_l}} + 2\\right]\\\\&\\qquad -\\theta (k\\ne l)\\,2\\bigg \\lbrace \\log \\!\\left(\\frac{1 - \\cos \\theta _{kl}}{2a_l y}\\right)- 1\\bigg \\rbrace \\bigg ]\\;.\\end{split}$ When we insert this result into eq.",
"(REF ), we have $\\begin{split}{\\cal V}_{l}(t)\\big |{\\lbrace p,f,c^{\\prime },c\\rbrace _{m}}\\big )\\hspace{-28.45274pt}{}&\\\\={}&\\frac{\\alpha _{\\mathrm {s}}}{2\\pi }\\bigg \\lbrace \\left[-\\frac{\\gamma _{f_l}}{C_{f_l}} + 2\\right]\\frac{1}{2}\\big ([(\\mathbf {T}_l\\cdot \\mathbf {T}_l)\\otimes 1] + [1 \\otimes (\\mathbf {T}_l\\cdot \\mathbf {T}_l)]\\big )\\\\&\\qquad -\\bigg [\\log \\!\\left(\\frac{1}{a_l y}\\right)- 1\\bigg ]\\sum _{k \\ne l}\\big ([(\\mathbf {T}_l\\cdot \\mathbf {T}_k)\\otimes 1] + [1 \\otimes (\\mathbf {T}_l\\cdot \\mathbf {T}_k)]\\big )\\\\&\\qquad -\\sum _{k \\ne l}\\log \\!\\left(\\frac{1 - \\cos \\theta _{kl}}{2}\\right)\\big ([(\\mathbf {T}_l\\cdot \\mathbf {T}_k)\\otimes 1] + [1 \\otimes (\\mathbf {T}_l\\cdot \\mathbf {T}_k)]\\big )\\bigg \\rbrace \\\\ &\\times \\big |{\\lbrace p,f,c^{\\prime },c\\rbrace _{m}}\\big )\\;.\\end{split}$ In the second line of the right hand side, we can use $\\sum _{k \\ne l} \\mathbf {T}_l\\cdot \\mathbf {T}_k = -\\mathbf {T}_l\\cdot \\mathbf {T}_l$ .",
"Then in lines 1 and 2 we can use $\\mathbf {T}_l\\cdot \\mathbf {T}_l = C_{f_l}$ .",
"This gives $\\begin{split}{\\cal V}_{l}(t)\\big |{\\lbrace p,f,c^{\\prime },c\\rbrace _{m}}\\big )\\hspace{-28.45274pt}{}&\\\\={}&\\frac{\\alpha _{\\mathrm {s}}}{2\\pi }\\bigg \\lbrace \\left[2 C_{f_l}\\log \\!\\left(\\frac{1}{a_l y}\\right)-\\gamma _{f_l} \\right](1\\otimes 1)\\\\&\\qquad -\\sum _{k \\ne l}\\log \\!\\left(\\frac{1 - \\cos \\theta _{kl}}{2}\\right)\\big ([(\\mathbf {T}_l\\cdot \\mathbf {T}_k)\\otimes 1] + [1 \\otimes (\\mathbf {T}_l\\cdot \\mathbf {T}_k)]\\big )\\bigg \\rbrace \\\\ &\\times \\big |{\\lbrace p,f,c^{\\prime },c\\rbrace _{m}}\\big )\\;.\\end{split}$ We use this result with $a_l = E_Q/(2 | \\vec{p}_l |)$ from eq.",
"(REF ) in eq.",
"(REF )."
],
[
"Initial state partons",
"We write ${\\cal V}_\\mathrm {a}(t)$ as [67] ${\\cal V}_\\mathrm {a}(t) =\\sum _k {\\cal V}_{\\mathrm {a}k}(t)\\;.$ The sum includes all parton labels $k = \\mathrm {a}, \\mathrm {b}, 1, \\dots , m$ .",
"When the operators ${\\cal V}_{\\mathrm {a}k}(t)$ act on a partonic state $\\big |{\\lbrace p,f,c^{\\prime },c\\rbrace _{m}}\\big )$ , they have the structure $\\begin{split}{\\cal V}_{\\mathrm {a}k}(t)\\big |{\\lbrace p,f,c^{\\prime },c\\rbrace _{m}}\\big )={}&\\overline{\\lambda }^{{\\cal V}}_{\\mathrm {a}k}(\\lbrace p,f\\rbrace _{m},t)\\,\\frac{1}{2}\\big ([(\\mathbf {T}_\\mathrm {a}\\cdot \\mathbf {T}_k)\\otimes 1] + [1 \\otimes (\\mathbf {T}_\\mathrm {a}\\cdot \\mathbf {T}_k)]\\big )\\\\ &\\times \\big |{\\lbrace p,f,c^{\\prime },c\\rbrace _{m}}\\big )\\;,\\end{split}$ as in eq.",
"(REF ).",
"We find from eq.",
"(5.28) of ref.",
"[70], $\\begin{split}\\overline{\\lambda }^{{\\cal V}}_{\\mathrm {a}k}(\\lbrace p,&f\\rbrace _{m},t)\\\\={}&\\frac{1}{m!",
"}\\int \\!d\\lbrace \\hat{p},\\hat{f}\\rbrace _{m+1}\\delta (t - T(\\lbrace \\hat{p},\\hat{f}\\rbrace _{m+1}))\\,\\big ({\\lbrace \\hat{p},\\hat{f}\\rbrace _{m+1}}\\big |{\\cal P}_{l}\\big |{\\lbrace p,f\\rbrace _m}\\big )\\\\&\\times \\frac{n_\\mathrm {c}(a) \\eta _{\\mathrm {a}}}{n_\\mathrm {c}(\\hat{a}) \\hat{\\eta }_{\\mathrm {a}}}\\,\\frac{f_{\\hat{a}/A}(\\hat{\\eta }_{\\mathrm {a}},\\mu _\\mathrm {a}^{2}(t))}{f_{a/A}(\\eta _{\\mathrm {a}},\\mu _\\mathrm {a}^{2}(t)))}\\\\&\\times \\bigg [\\theta (k = \\mathrm {a})\\,\\theta (\\hat{f}_{m+1} \\ne \\mathrm {g})\\,\\theta (a \\ne \\mathrm {g})\\overline{w}_{\\mathrm {a}\\mathrm {a}}(\\lbrace \\hat{p},\\hat{f}\\rbrace _{m+1})\\\\&\\qquad +\\theta (k = \\mathrm {a})\\,\\theta (\\hat{f}_{m+1} \\ne \\mathrm {g})\\,\\theta (a = \\mathrm {g})\\,\\frac{T_\\mathrm {R}}{C_\\mathrm {A}}\\,\\overline{w}_{\\mathrm {a}\\mathrm {a}}(\\lbrace \\hat{p},\\hat{f}\\rbrace _{m+1})\\\\&\\qquad +\\theta (k = \\mathrm {a})\\,\\theta (\\hat{f}_{m+1} = \\mathrm {g})\\,[\\overline{w}_{\\mathrm {a}\\mathrm {a}}(\\lbrace \\hat{p},\\hat{f}\\rbrace _{m+1})- \\overline{w}_{\\mathrm {a}\\mathrm {a}}^{\\rm eikonal}(\\lbrace \\hat{p},\\hat{f}\\rbrace _{m+1})]\\\\&\\qquad -\\theta (k\\ne \\mathrm {a})\\,\\theta (\\hat{f}_{m+1} = \\mathrm {g})\\,A^{\\prime }_{\\mathrm {a}k}(\\lbrace \\hat{p}\\rbrace _{m+1})\\overline{w}_{\\mathrm {a}k}^{\\rm dipole}(\\lbrace \\hat{p},\\hat{f}\\rbrace _{m+1})\\bigg ]\\;.\\end{split}$ In an initial state splitting, parton “a” with momentum $p_\\mathrm {a}$ becomes a new initial state parton with momentum $\\hat{p}_\\mathrm {a}$ and a final state parton with momentum $\\hat{p}_{m+1}$ .",
"The momentum of the other initial state parton is unchanged: $\\hat{p}_\\mathrm {b}= p_\\mathrm {b}$ .",
"The splitting kinematics is defined by $\\begin{split}\\hat{p}_\\mathrm {a}={}& \\frac{1}{z}\\,p_\\mathrm {a}\\;,\\\\\\hat{p}_\\mathrm {b}={}& p_\\mathrm {b}\\;,\\\\\\hat{p}_{m+1} ={}& \\left(\\frac{1-z}{z} - y\\right)\\,p_\\mathrm {a}+ z y\\, p_\\mathrm {b}+ \\sqrt{y(1-z-yz) Q^2}\\ n_\\perp \\;,\\end{split}$ where $n_\\perp \\cdot p_\\mathrm {a}= n_\\perp \\cdot p_\\mathrm {b}= 0$ and $n_\\perp ^2 = -1$ .",
"Note that $y = \\hat{p}_\\mathrm {a}\\cdot \\hat{p}_{m+1}/p_\\mathrm {a}\\cdot Q$ , where $Q = p_\\mathrm {a}+ p_\\mathrm {b}$ , as in eq.",
"(REF ).",
"This kinematics requires $y < (1-z)/z$ , or $z < \\frac{1}{1+y}\\;.$ Eq.",
"(REF ) contains a ratio of parton distributions, which are evaluated at momentum fractions defined by $p_\\mathrm {a}= \\eta _\\mathrm {a}\\, p_\\mathrm {A}$ and $\\hat{p}_\\mathrm {a}= \\hat{\\eta }_\\mathrm {a}\\, p_\\mathrm {A}$ .",
"We take the scales of the parton distributions to be the virtuality $\\mu ^2_\\mathrm {a}(t)$ defined in eq.",
"(REF ).",
"There are also factors $n_\\mathrm {c}(a)$ that count the number of colors (3 or 8) carried by partons of flavor $a$ .",
"Using eq.",
"(8.20) of ref.",
"[67] together with eqs.",
"(A.28) of ref.",
"[73], we find that integration over the splitting variables between shower times corresponding to $y$ values $y_1$ and $y_2$ is accomplished with $\\begin{split}\\frac{1}{m!",
"}\\int \\big [d\\lbrace \\hat{p},\\hat{f}&\\rbrace _{m+1}\\big ]\\big ({\\lbrace \\hat{p},\\hat{f}\\rbrace _{m+1}}\\big |{\\cal P}_{l}\\big |{\\lbrace p,f\\rbrace _m}\\big )\\cdots \\\\&= \\frac{Q^2}{16\\pi ^2}\\,\\int _{y_2}^{y_1}\\!",
"dy\\ \\int _{0}^{1}\\!\\frac{dz}{z}\\int _{-\\pi }^\\pi \\frac{d\\phi }{2\\pi }\\sum _{\\hat{a}}\\ \\cdots \\;.\\end{split}$ The delta function that specifies the shower time is, from eqs.",
"(REF ) and (REF ), $\\delta (t - T(\\lbrace \\hat{p},\\hat{f}\\rbrace _{m+1}))=\\delta \\!\\left(\\log y - \\log \\!\\left(\\frac{p_\\mathrm {a}\\cdot Q_0}{p_\\mathrm {a}\\cdot Q}e^{-t}\\right)\\right)\\;,$ so that $\\mu _\\mathrm {a}^2(t) = y Q^2$ .",
"Using eqs.",
"(REF ) and (REF ) in eq.",
"(REF ) gives $\\begin{split}\\overline{\\lambda }^{{\\cal V}}_{\\mathrm {a}k}(\\lbrace p,&f\\rbrace _{m},t)\\\\={}&\\frac{Q^2}{16 \\pi ^2}\\,\\int _0^1\\!\\frac{dz}{z}\\, \\int _0^{2\\pi }\\frac{d\\phi }{2\\pi }\\sum _{\\hat{a}}\\ y\\,\\theta (z < 1/(1+y))\\\\&\\times \\frac{n_\\mathrm {c}(a) z}{n_\\mathrm {c}(\\hat{a})}\\,\\frac{f_{\\hat{a}/A}(\\eta _{\\mathrm {a}}/z,y Q^2)}{f_{a/A}(\\eta _{\\mathrm {a}},y Q^2)}\\\\&\\times \\bigg [\\theta (k = \\mathrm {a})\\,\\theta (\\hat{a} \\ne a)\\,\\theta (a \\ne \\mathrm {g})\\,\\overline{w}_{\\mathrm {a}\\mathrm {a}}(\\lbrace \\hat{p},\\hat{f}\\rbrace _{m+1})\\\\&\\qquad +\\theta (k = \\mathrm {a})\\,\\theta (\\hat{a} \\ne a)\\,\\theta (a = \\mathrm {g})\\,\\frac{T_\\mathrm {R}}{C_\\mathrm {A}}\\,\\overline{w}_{\\mathrm {a}\\mathrm {a}}(\\lbrace \\hat{p},\\hat{f}\\rbrace _{m+1})\\\\&\\qquad +\\theta (k = \\mathrm {a})\\,\\theta (\\hat{a} = a)\\,[\\overline{w}_{\\mathrm {a}\\mathrm {a}}(\\lbrace \\hat{p},\\hat{f}\\rbrace _{m+1})- \\overline{w}_{\\mathrm {a}\\mathrm {a}}^{\\rm eikonal}(\\lbrace \\hat{p},\\hat{f}\\rbrace _{m+1})]\\\\&\\qquad -\\theta (k\\ne \\mathrm {a})\\,\\theta (\\hat{a} = a)\\,A^{\\prime }_{\\mathrm {a}k}(\\lbrace \\hat{p}\\rbrace _{m+1})\\overline{w}_{\\mathrm {a}k}^{\\rm dipole}(\\lbrace \\hat{p},\\hat{f}\\rbrace _{m+1})\\bigg ]\\;.\\end{split}$ There are four terms here.",
"The first three are for direct graphs.",
"The first is for $(\\hat{a},a,\\hat{f}_{m+1}) = (\\mathrm {g},q,\\bar{q})$ and $(\\hat{a},a,\\hat{f}_{m+1}) = (\\mathrm {g},\\bar{q}, q)$ .",
"The second is for $(\\hat{a},a,\\hat{f}_{m+1}) = (q,\\mathrm {g}, q)$ and $(\\hat{a},a,\\hat{f}_{m+1}) = (\\bar{q},\\mathrm {g}, \\bar{q})$ .",
"Here the color factor is $T_\\mathrm {R}= 1/2$ , but this term multiplies $\\mathbf {T}_\\mathrm {a}\\cdot \\mathbf {T}_\\mathrm {a}= C_A$ , so we need to divide by $C_\\mathrm {A}$ .",
"The third term is for $(\\hat{a},a,\\hat{f}_{m+1}) = (\\mathrm {g},\\mathrm {g},\\mathrm {g})$ , $(\\hat{a},a,\\hat{f}_{m+1}) = (\\bar{q},\\bar{q},\\mathrm {g})$ and $(\\hat{a},a,\\hat{f}_{m+1}) = (q,q,\\mathrm {g})$ .",
"Here there is a soft gluon singularity, which is subtracted.",
"The fourth term is for interference graphs, in which a gluon is exchanged between parton “a” and parton $k$ .",
"Now we need the splitting functions.",
"We take the limit $y \\ll 1$ .",
"From ref.",
"[73], eq.",
"(A.42), we have for $(\\hat{a},a,\\hat{f}_{m+1}) = (\\mathrm {g},q,\\bar{q})$ and $(\\hat{a},a,\\hat{f}_{m+1}) = (\\mathrm {g},\\bar{q}, q)$ , $\\overline{w}_{\\mathrm {a}\\mathrm {a}}(\\lbrace \\hat{p},\\hat{f}\\rbrace _{m+1})\\sim \\frac{8\\pi \\alpha _{\\mathrm {s}}}{Q^2}\\,\\frac{1}{yz}\\ \\frac{P_{a\\hat{a}}(z)}{T_\\mathrm {R}}\\;.$ From ref.",
"[73], eq.",
"(A.45), we have for $(\\hat{a},a,\\hat{f}_{m+1}) = (q,\\mathrm {g}, q)$ and $(\\hat{a},a,\\hat{f}_{m+1}) = (\\bar{q},\\mathrm {g}, \\bar{q})$ , $\\overline{w}_{\\mathrm {a}\\mathrm {a}}(\\lbrace \\hat{p},\\hat{f}\\rbrace _{m+1})\\sim \\frac{8\\pi \\alpha _{\\mathrm {s}}}{Q^2}\\,\\frac{1}{yz}\\ \\frac{P_{a\\hat{a}}(z)}{C_\\mathrm {F}}\\;.$ From ref.",
"[73], eqs.",
"(A.35) and (A.39), we have for $(\\hat{a},a,\\hat{f}_{m+1}) = (\\mathrm {g},\\mathrm {g},\\mathrm {g})$ , $(\\hat{a},a,\\hat{f}_{m+1}) = (\\bar{q},\\bar{q},\\mathrm {g})$ and $(\\hat{a},a,\\hat{f}_{m+1}) = (q,q,\\mathrm {g})$ , $\\overline{w}_{\\mathrm {a}\\mathrm {a}}(\\lbrace \\hat{p},\\hat{f}\\rbrace _{m+1})\\sim \\frac{8\\pi \\alpha _{\\mathrm {s}}}{Q^2}\\,\\frac{1}{yz}\\ \\left(\\frac{P_{aa}(z)}{C_a}- \\frac{2y}{(1-z)^2}\\right)\\;.$ The eikonal function is defined in eq.",
"(2.10) of ref.",
"[68] as $\\overline{w}_{\\mathrm {a}\\mathrm {a}}^{\\rm eikonal}(\\lbrace \\hat{p},\\hat{f}\\rbrace _{m+1})=4\\pi \\alpha _{\\mathrm {s}}\\frac{\\hat{p}_\\mathrm {a}\\cdot D(\\hat{p}_{m+1},\\hat{Q}) \\cdot \\hat{p}_\\mathrm {a}}{(\\hat{p}_{m+1} \\cdot \\hat{p}_\\mathrm {a})^2}\\;,$ where $\\hat{Q} = \\hat{p}_\\mathrm {a}+ \\hat{p}_\\mathrm {b}$ .",
"That is, $\\overline{w}_{\\mathrm {a}\\mathrm {a}}^{\\rm eikonal}(\\lbrace \\hat{p},\\hat{f}\\rbrace _{m+1})=4\\pi \\alpha _{\\mathrm {s}}\\left\\lbrace \\frac{2\\hat{p}_\\mathrm {a}\\cdot \\hat{Q}}{\\hat{p}_{m+1} \\cdot \\hat{p}_\\mathrm {a}\\ \\hat{p}_{m+1}\\cdot \\hat{Q}}-\\frac{\\hat{Q}^2}{(\\hat{p}_{m+1} \\cdot \\hat{Q})^2}\\right\\rbrace \\;.$ Using eq.",
"(REF ) gives $\\overline{w}_{\\mathrm {a}\\mathrm {a}}^{\\rm eikonal}(\\lbrace \\hat{p},\\hat{f}\\rbrace _{m+1})=\\frac{16\\pi \\alpha _{\\mathrm {s}}}{Q^2}\\frac{1}{yz}\\left\\lbrace \\frac{z}{(1-z)}-\\frac{yz^2}{(1-z)^2}\\right\\rbrace \\;.$ Thus $\\begin{split}\\overline{w}_{\\mathrm {a}\\mathrm {a}}(\\lbrace \\hat{p},\\hat{f}\\rbrace _{m+1})-{}&\\overline{w}_{\\mathrm {a}\\mathrm {a}}^{\\rm eikonal}(\\lbrace \\hat{p},\\hat{f}\\rbrace _{m+1})\\\\\\sim {}&\\frac{16\\pi \\alpha _{\\mathrm {s}}}{Q^2}\\,\\frac{1}{yz}\\ \\left(\\frac{P_{aa}(z)}{2C_a}-\\frac{z}{1-z}- \\frac{y (1+z)}{(1-z)}\\right)\\;.\\end{split}$ The term $z/(1-z)$ here removes the $1/(1-z)$ singularity from $P_{aa}(z)$ .",
"The third term, proportional to $y$ has a $1/(1-z)$ singularity.",
"This can give a $\\log (y)$ contribution to an integration over $z$ down to $(1-z) = y$ .",
"However, we can neglect $y \\log (y)$ .",
"Thus we can throw this term away.",
"This gives $\\begin{split}\\overline{w}_{\\mathrm {a}\\mathrm {a}}(\\lbrace \\hat{p},\\hat{f}\\rbrace _{m+1})-{}&\\overline{w}_{\\mathrm {a}\\mathrm {a}}^{\\rm eikonal}(\\lbrace \\hat{p},\\hat{f}\\rbrace _{m+1})\\sim \\frac{8\\pi \\alpha _{\\mathrm {s}}}{Q^2}\\,\\frac{1}{yz\\, C_a}\\ \\left(P_{aa}(z)-2C_a\\,\\frac{z}{1-z}\\right)\\;.\\end{split}$ Now we need the interference terms.",
"From ref.",
"[70], eq.",
"(5.3), we have $\\overline{w}_{\\mathrm {a}k}^{\\rm dipole}(\\lbrace \\hat{p},\\hat{f}\\rbrace _{m+1})= 4\\pi \\alpha _{\\mathrm {s}}\\ \\frac{2 \\hat{p}_k\\cdot \\hat{p}_\\mathrm {a}}{\\hat{p}_{m+1}\\cdot \\hat{p}_k\\ \\hat{p}_{m+1}\\cdot \\hat{p}_\\mathrm {a}}\\;.$ This multiplies $A^{\\prime }_{\\mathrm {a}k}$ .",
"We use eq.",
"(7.12) of ref.",
"[69]: $A^{\\prime }_{\\mathrm {a}k}(\\lbrace \\hat{p}\\rbrace _{m+1}) = \\frac{\\hat{p}_{m+1}\\cdot \\hat{p}_k\\ \\hat{p}_\\mathrm {a}\\cdot \\hat{Q}}{\\hat{p}_{m+1}\\cdot \\hat{p}_k\\ \\hat{p}_\\mathrm {a}\\cdot \\hat{Q}+ \\hat{p}_{m+1}\\cdot \\hat{p}_\\mathrm {a}\\ \\hat{p}_k\\cdot \\hat{Q}}\\;.$ The product is $\\begin{split}A^{\\prime }_{\\mathrm {a}k}(\\lbrace \\hat{p}\\rbrace _{m+1})\\, \\overline{w}_{\\mathrm {a}k}^{\\rm dipole}(\\lbrace \\hat{p},\\hat{f}\\rbrace _{m+1})={}&\\frac{16\\pi \\alpha _{\\mathrm {s}}\\ }{y\\,Q^2}\\ \\frac{\\hat{p}_k\\cdot \\hat{p}_\\mathrm {a}}{\\hat{p}_{m+1}\\cdot \\hat{p}_k+ yz\\, \\hat{p}_k\\cdot \\hat{Q}}\\;.\\end{split}$ To proceed further, we note that we need the function in eq.",
"(REF ) in the limit of small $y$ .",
"The momentum $\\hat{p}_k$ is related to the momentum $p_k$ by a Lorentz transformation that becomes the unit operator when $y \\rightarrow 0$ .",
"Thus we can neglect the difference between $\\hat{p}_k$ , which varies as we integrate over splitting variables $(y,z,\\phi )$ , and $p_k$ , which is fixed.",
"For this reason, we substitute $p_k$ for $\\hat{p}_k$ in eq.",
"(REF ).",
"Then we use $p_k = A_k\\big ((1+\\cos \\theta _{\\mathrm {a}k})\\,p_\\mathrm {a}+ (1-\\cos \\theta _{\\mathrm {a}k})\\,p_\\mathrm {b}+\\sqrt{Q^2}\\sin \\theta _{\\mathrm {a}k}\\ u_\\perp \\big )\\;.$ Here $A_k$ is a normalization factor that cancels in eq.",
"(REF ) and $\\theta _{\\mathrm {a}k}$ is the angle between the three-vector parts of $p_k$ and $p_\\mathrm {a}$ in the frame in which $Q$ has only a time component, while $u_\\perp $ is a vector transverse to $p_\\mathrm {a}$ and $p_\\mathrm {b}$ with $u_\\perp ^2 = -1$ .",
"Using the parameterizations in eqs.",
"(REF ) and (REF ), we find $\\begin{split}&\\frac{p_k\\cdot \\hat{p}_\\mathrm {a}}{\\hat{p}_{m+1}\\cdot p_k+ yz\\,p_k\\cdot \\hat{Q}}\\\\&\\ =\\frac{(1-\\cos \\theta _{\\mathrm {a}k})}{2(1 + \\cos \\theta _{\\mathrm {a}k})yz^2+(1 - \\cos \\theta _{\\mathrm {a}k})(1-z)-2z\\sin \\theta _{\\mathrm {a}k}\\cos \\phi \\sqrt{y(1-z-yz)}}\\;.\\end{split}$ We can perform the averaging of this over $\\phi $ exactly: $\\begin{split}\\int _0^{2\\pi }&\\frac{d\\phi }{2\\pi }\\,\\frac{p_k\\cdot \\hat{p}_\\mathrm {a}}{\\hat{p}_{m+1}\\cdot \\hat{p}_k+ yz\\,p_k\\cdot \\hat{Q}}\\\\&=\\frac{1 - \\cos \\theta _{\\mathrm {a}k}}{\\sqrt{(1 - \\cos \\theta _{\\mathrm {a}k})^2 (1-z)^2+ 4 y^2 z^3[z(1 + \\cos \\theta _{\\mathrm {a}k})^2 + \\sin ^2(\\theta _{\\mathrm {a}k})]}}\\;.\\end{split}$ We can write this in a suggestive form as $\\int _0^{2\\pi }\\frac{d\\phi }{2\\pi }\\,\\frac{\\hat{p}_k\\cdot \\hat{p}_\\mathrm {a}}{\\hat{p}_{m+1}\\cdot \\hat{p}_k+ yz\\,\\hat{p}_k\\cdot \\hat{Q}}=\\frac{1}{\\sqrt{(1-z)^2 + y^2 z^2 /\\Psi _{\\mathrm {a}k}(z)^2}}\\;,$ where $\\Psi _{\\mathrm {a}k}(z) = \\frac{1 - \\cos \\theta _{\\mathrm {a}k}}{\\sqrt{4 z[z(1 + \\cos \\theta _{\\mathrm {a}k})^2 + \\sin ^2(\\theta _{\\mathrm {a}k})]}}\\;.$ We are interested in this in the small $y$ limit, in which $y^2 z^2 /\\Psi _{\\mathrm {a}k}(z)^2 \\ll 1$ .",
"Then the second term in the denominator of eq.",
"(REF ) is non-negligible only when $z$ is close to 1.",
"Thus we can replace $\\Psi _{\\mathrm {a}k}(z)$ by $\\psi _{\\mathrm {a}k}$ defined in eq.",
"(REF ): $\\Psi _{\\mathrm {a}k}(1) = \\psi _{\\mathrm {a}k} = \\frac{1 - \\cos \\theta _{\\mathrm {a}k}}{\\sqrt{8 (1 + \\cos \\theta _{\\mathrm {a}k})}}\\;.$ Thus we use $\\int _0^{2\\pi }\\frac{d\\phi }{2\\pi }\\,\\frac{\\hat{p}_k\\cdot \\hat{p}_\\mathrm {a}}{\\hat{p}_{m+1}\\cdot \\hat{p}_k+ yz\\,\\hat{p}_k\\cdot \\hat{Q}}\\approx \\frac{1}{\\sqrt{(1-z)^2 + y^2 z^2 /\\psi _{\\mathrm {a}k}^2}}\\;.$ We may note that the angle $\\theta _{\\mathrm {a}k}$ is small when partons “a” and $k$ are the daughter partons of a previous initial state splitting that was nearly collinear.",
"It is allowed in Deductor to have an initial state splitting with small momentum fraction $z_{\\mathrm {a}k}$ , so that the new initial state parton “a”, which is the mother parton for the next splitting, has a much larger momentum fraction than the previous initial state parton.",
"In this case, the virtualities of the initial state partons in successive splittings are not strongly ordered, so that one can have $y > 1 - \\cos \\theta _{\\mathrm {a}k}$ .",
"This regime of multi-regge kinematics is discussed in section 5.4 of ref. [72].",
"This is the opposite kinematic regime from that of threshold logarithms, so we ignore this possibility in this paper.",
"However, we still use $y^2 z^2$ instead of just $y^2$ in the denominator of eq.",
"(REF ) in order to keep the result reasonably accurate even when $y \\gtrsim 1 - \\cos \\theta _{\\mathrm {a}k}$ .",
"We can now assemble our results: $\\begin{split}\\overline{\\lambda }^{{\\cal V}}_{\\mathrm {a}k}(\\lbrace p,f\\rbrace _{m},t)\\approx {}&\\frac{\\alpha _{\\mathrm {s}}}{2\\pi }\\,\\int _0^{1/(1+y)}\\!\\frac{dz}{z}\\,\\sum _{\\hat{a}}\\frac{f_{\\hat{a}/A}(\\eta _{\\mathrm {a}}/z,y Q^2)}{f_{a/A}(\\eta _{\\mathrm {a}},y Q^2)}\\\\&\\times \\Bigg [\\theta (k = \\mathrm {a})\\,\\theta (\\hat{a} \\ne a)\\,\\theta (a \\ne \\mathrm {g})\\frac{n_\\mathrm {c}(a)}{n_\\mathrm {c}(\\hat{a})}\\,\\frac{C_\\mathrm {F}}{T_\\mathrm {R}}\\frac{P_{a\\hat{a}}(z)}{C_\\mathrm {F}}\\\\&\\qquad +\\theta (k = \\mathrm {a})\\,\\theta (\\hat{a} \\ne a)\\,\\theta (a = \\mathrm {g})\\,\\frac{n_\\mathrm {c}(a)}{n_\\mathrm {c}(\\hat{a})}\\,\\frac{T_\\mathrm {R}}{C_\\mathrm {F}}\\,\\frac{P_{a\\hat{a}}(z)}{C_\\mathrm {A}}\\\\&\\qquad +\\theta (k = \\mathrm {a})\\,\\theta (\\hat{a} = a)\\,\\frac{1}{C_a}\\left(P_{aa}(z)-2C_a\\,\\frac{z}{1-z}\\right)\\\\&\\qquad -\\theta (k\\ne \\mathrm {a})\\,\\theta (\\hat{a} = a)\\,\\frac{2z}{\\sqrt{(1-z)^2 + y^2 z^2/\\psi _{\\mathrm {a}k}^2}}\\,\\Bigg ]\\;.\\end{split}$ In the first term, for $(\\hat{a},a,\\hat{f}_{m+1}) = (\\mathrm {g},q,\\bar{q})$ and $(\\hat{a},a,\\hat{f}_{m+1}) = (\\mathrm {g},\\bar{q}, q)$ , we have $n_\\mathrm {c}(a) = N_\\mathrm {c}$ and $n_\\mathrm {c}(\\hat{a}) = (N_\\mathrm {c}^2 - 1)$ .",
"Thus $[n_\\mathrm {c}(a)/n_\\mathrm {c}(\\hat{a})]\\times C_\\mathrm {F}/T_\\mathrm {R}= 1$ .",
"Also, $C_\\mathrm {F}= C_a$ .",
"In the second term, for for $(\\hat{a},a,\\hat{f}_{m+1}) = (q,\\mathrm {g}, q)$ and $(\\hat{a},a,\\hat{f}_{m+1}) = (\\bar{q},\\mathrm {g}, \\bar{q})$ , we have $n_\\mathrm {c}(a) = (N_\\mathrm {c}^2 - 1)$ and $n_\\mathrm {c}(\\hat{a}) = N_\\mathrm {c}$ .",
"Thus $[n_\\mathrm {c}(a)/n_\\mathrm {c}(\\hat{a})]\\times T_\\mathrm {R}/C_\\mathrm {F}= 1$ .",
"Also, $C_\\mathrm {A}= C_a$ .",
"After inserting $\\overline{\\lambda }^{{\\cal V}}_{\\mathrm {a}k}$ into eqs.",
"(REF ) and (REF ), this gives $\\begin{split}{\\cal V}_{\\mathrm {a}}(t)&\\big |{\\lbrace p,f,c^{\\prime },c\\rbrace _{m}}\\big )\\\\={}&\\frac{\\alpha _{\\mathrm {s}}}{2\\pi }\\,\\int _0^{1/(1+y)}\\!dz\\,\\sum _{\\hat{a}}\\ \\frac{f_{\\hat{a}/A}(\\eta _{\\mathrm {a}}/z, y Q^2)}{z f_{a/A}(\\eta _{\\mathrm {a}}, y Q^2)}\\\\&\\times \\Bigg \\lbrace \\frac{1}{C_a}\\left(P_{a\\hat{a}}(z)- \\delta _{a \\hat{a}} \\frac{2C_a\\,z}{1-z}\\right)\\frac{1}{2}\\big ([(\\mathbf {T}_\\mathrm {a}\\cdot \\mathbf {T}_\\mathrm {a})\\otimes 1]+ [1 \\otimes (\\mathbf {T}_\\mathrm {a}\\cdot \\mathbf {T}_\\mathrm {a})]\\big )\\\\&\\qquad -\\sum _{k\\ne \\mathrm {a}}\\,\\delta _{a \\hat{a}}\\,\\frac{2z}{\\sqrt{(1-z)^2 + y^2 z^2/\\psi _{\\mathrm {a}k}^2}}\\,\\frac{1}{2}\\big ([(\\mathbf {T}_\\mathrm {a}\\cdot \\mathbf {T}_k)\\otimes 1] + [1 \\otimes (\\mathbf {T}_\\mathrm {a}\\cdot \\mathbf {T}_k)]\\big )\\Bigg \\rbrace \\\\&\\times \\big |{\\lbrace p,f,c^{\\prime },c\\rbrace _{m}}\\big )\\;.\\end{split}$ In the first term in eq.",
"(REF ), we can replace $(\\mathbf {T}_\\mathrm {a}\\cdot \\mathbf {T}_\\mathrm {a})$ by $C_a$ .",
"We divide the second term into three terms by defining $\\Delta _{\\mathrm {a}k}(z,y)$ according to $\\begin{split}\\frac{1}{\\sqrt{(1-z)^2 + y^2 z^2/\\psi _{\\mathrm {a}k}^2}} ={}&\\frac{1}{1-z}- \\Delta _{\\mathrm {a}k}(z,y)\\;.\\end{split}$ This gives $\\begin{split}{\\cal V}_{\\mathrm {a}}(t)&\\big |{\\lbrace p,f,c^{\\prime },c\\rbrace _{m}}\\big )\\\\={}&\\frac{\\alpha _{\\mathrm {s}}}{2\\pi }\\,\\int _0^{1/(1+y)}\\!dz\\\\&\\times \\Bigg \\lbrace \\sum _{\\hat{a}}\\frac{f_{\\hat{a}/A}(\\eta _{\\mathrm {a}}/z,y Q^2)}{zf_{a/A}(\\eta _{\\mathrm {a}},y Q^2)}\\left(P_{a a}(z)- \\frac{2C_a\\,z}{1-z}\\right)[1\\otimes 1]\\\\&\\qquad -\\sum _{k\\ne \\mathrm {a}}\\,\\left[\\frac{f_{a/A}(\\eta _{\\mathrm {a}}/z,y Q^2)}{f_{a/A}(\\eta _{\\mathrm {a}},y Q^2)}-1\\right]\\frac{1}{1-z}\\,\\big ([(\\mathbf {T}_\\mathrm {a}\\cdot \\mathbf {T}_k)\\otimes 1] + [1 \\otimes (\\mathbf {T}_\\mathrm {a}\\cdot \\mathbf {T}_k)]\\big )\\\\&\\qquad +\\sum _{k\\ne \\mathrm {a}}\\left[\\frac{f_{a/A}(\\eta _{\\mathrm {a}}/z,y Q^2)}{f_{a/A}(\\eta _{\\mathrm {a}},y Q^2)}-1\\right]\\Delta _{\\mathrm {a}k}(z,y)\\,\\big ([(\\mathbf {T}_\\mathrm {a}\\cdot \\mathbf {T}_k)\\otimes 1] + [1 \\otimes (\\mathbf {T}_\\mathrm {a}\\cdot \\mathbf {T}_k)]\\big )\\\\&\\qquad -\\sum _{k\\ne \\mathrm {a}}\\,\\frac{1}{\\sqrt{(1-z)^2 + y^2 z^2/\\psi _{\\mathrm {a}k}^2}}\\,\\big ([(\\mathbf {T}_\\mathrm {a}\\cdot \\mathbf {T}_k)\\otimes 1] + [1 \\otimes (\\mathbf {T}_\\mathrm {a}\\cdot \\mathbf {T}_k)]\\big )\\Bigg \\rbrace \\\\&\\times \\big |{\\lbrace p,f,c^{\\prime },c\\rbrace _{m}}\\big )\\;.\\end{split}$ In the last term, we can perform the $z$ -integration approximately.",
"For $2 y/(1 - \\cos \\theta _{\\mathrm {a}k}) \\ll 1$ , one easily finds $\\begin{split}\\int _0^{1/(1+y)}&\\!dz\\frac{1}{\\sqrt{(1-z)^2 + y^2 z^2 /\\psi _{\\mathrm {a}k}^2}}\\approx \\log \\!\\left[\\frac{1 - \\cos \\theta _{\\mathrm {a}k}}{2y}\\right]\\;.\\end{split}$ We note that for $k = \\mathrm {b}$ , we have $\\cos \\theta _{\\mathrm {a}\\mathrm {b}} = -1$ and $1/\\psi _{\\mathrm {a}\\mathrm {b}}^2 = 0$ .",
"Then the integral is just $- \\log (y)$ .",
"This gives $\\begin{split}{\\cal V}_{\\mathrm {a}}&(t)\\big |{\\lbrace p,f,c^{\\prime },c\\rbrace _{m}}\\big )\\\\={}&\\bigg [\\frac{\\alpha _{\\mathrm {s}}}{2\\pi }\\,\\int _0^{1/(1+y)}\\!dz\\\\&\\times \\Bigg \\lbrace \\sum _{\\hat{a}}\\frac{f_{\\hat{a}/A}(\\eta _{\\mathrm {a}}/z,y Q^2)}{z f_{a/A}(\\eta _{\\mathrm {a}},y Q^2)}\\left(P_{a\\hat{a}}(z)- \\delta _{a \\hat{a}} \\frac{2C_a\\,z}{1-z}\\right)[1\\otimes 1]\\\\&\\quad -\\sum _{k\\ne \\mathrm {a}}\\,\\left[\\frac{f_{a/A}(\\eta _{\\mathrm {a}}/z,y Q^2)}{f_{a/A}(\\eta _{\\mathrm {a}},y Q^2)}- 1\\right]\\frac{1}{1-z}\\,\\big ([(\\mathbf {T}_\\mathrm {a}\\cdot \\mathbf {T}_k)\\otimes 1] + [1 \\otimes (\\mathbf {T}_\\mathrm {a}\\cdot \\mathbf {T}_k)]\\big )\\\\&\\quad +\\sum _{k\\ne \\mathrm {a},\\mathrm {b}}\\left[\\frac{f_{a/A}(\\eta _{\\mathrm {a}}/z,y Q^2)}{f_{a/A}(\\eta _{\\mathrm {a}},y Q^2)}-1\\right]\\Delta _{\\mathrm {a}k}(z,y)\\,\\big ([(\\mathbf {T}_\\mathrm {a}\\cdot \\mathbf {T}_k)\\otimes 1] + [1 \\otimes (\\mathbf {T}_\\mathrm {a}\\cdot \\mathbf {T}_k)]\\big )\\Bigg \\rbrace \\\\&\\quad -\\frac{\\alpha _{\\mathrm {s}}}{2\\pi }\\,\\sum _{k\\ne \\mathrm {a}}\\,\\left(\\log \\!\\left[\\frac{1 - \\cos \\theta _{\\mathrm {a}k}}{2}\\right]- \\log (y)\\right)\\big ([(\\mathbf {T}_\\mathrm {a}\\cdot \\mathbf {T}_k)\\otimes 1] + [1 \\otimes (\\mathbf {T}_\\mathrm {a}\\cdot \\mathbf {T}_k)]\\big )\\bigg ]\\\\&\\times \\big |{\\lbrace p,f,c^{\\prime },c\\rbrace _{m}}\\big )\\;.\\end{split}$ Now there are three $k \\ne \\mathrm {a}$ terms.",
"The first has the good feature that its only $k$ dependence is in the color factor, so that we can sum it over $k$ .",
"In the second term, we note that by its construction, $\\Delta _{\\mathrm {a}k}(z,y)$ has a $1/(1-z)$ singularity for $(1-z) \\ll y/\\psi _{\\mathrm {a}k}$ but is suppressed compared to $1/(1-z)$ when $(1-z) \\gg y/\\psi _{\\mathrm {a}k}$ .",
"Thus the second term has the good feature that, because of the structure of $\\Delta _{\\mathrm {a}k}(z,y)$ , the only important integration region for the $z$ -integration is $0 < (1-z) \\lesssim y/\\psi _{\\mathrm {a}k}$ .",
"We also note that $1/\\psi _{\\mathrm {a}k} = 0$ when $\\cos \\theta _{ak} = -1$ .",
"In this limit, $\\Delta _{\\mathrm {a}k}(z,y) = 0$ .",
"We have $\\cos \\theta _{ak} = -1$ for $k = \\mathrm {b}$ , so $\\Delta _{\\mathrm {a}\\mathrm {b}}(z,y) = 0$ .",
"This eliminates one term in our sum over $k$ .",
"The third term has the good feature that we have been able to integrate it, at least approximately.",
"In the second term in eq.",
"(REF ), we can sum over $k$ using $\\sum _{k \\ne \\mathrm {a}} (\\mathbf {T}_\\mathrm {a}\\cdot \\mathbf {T}_k) = -(\\mathbf {T}_\\mathrm {a}\\cdot \\mathbf {T}_\\mathrm {a}) \\rightarrow -C_a$ .",
"In the last term, we can separate the $\\log (1/y)$ contribution and perform the color sum in the same way.",
"This gives $\\begin{split}{\\cal V}_{\\mathrm {a}}(t)&\\big |{\\lbrace p,f,c^{\\prime },c\\rbrace _{m}}\\big )\\\\={}& \\bigg [\\frac{\\alpha _{\\mathrm {s}}}{2\\pi }\\,\\int _0^{1/(1+y)}\\!dz\\\\&\\times \\Bigg \\lbrace \\sum _{\\hat{a}}\\left(\\frac{f_{\\hat{a}/A}(\\eta _{\\mathrm {a}}/z,y Q^2)}{zf_{a/A}(\\eta _{\\mathrm {a}},y Q^2)}P_{a\\hat{a}}(z)- \\delta _{a\\hat{a}}\\frac{2C_a}{1-z}\\right)[1\\otimes 1]\\\\&\\quad +\\sum _{k\\ne \\mathrm {a},\\mathrm {b}}\\left[\\frac{f_{a/A}(\\eta _{\\mathrm {a}}/z,y Q^2)}{f_{a/A}(\\eta _{\\mathrm {a}},y Q^2)} - 1\\right]\\Delta _{\\mathrm {a}k}(z,y)\\big ([(\\mathbf {T}_\\mathrm {a}\\cdot \\mathbf {T}_k)\\otimes 1] + [1 \\otimes (\\mathbf {T}_\\mathrm {a}\\cdot \\mathbf {T}_k)]\\big )\\Bigg \\rbrace \\\\& -\\sum _{k\\ne \\mathrm {a},\\mathrm {b}}\\frac{\\alpha _{\\mathrm {s}}}{2\\pi }\\,\\log \\!\\left[\\frac{1 - \\cos \\theta _{\\mathrm {a}k}}{2}\\right]\\,\\big ([(\\mathbf {T}_\\mathrm {a}\\cdot \\mathbf {T}_k)\\otimes 1] + [1 \\otimes (\\mathbf {T}_\\mathrm {a}\\cdot \\mathbf {T}_k)]\\big )\\\\&+\\frac{\\alpha _{\\mathrm {s}}}{2\\pi }\\,2 C_a\\log \\left[\\frac{1}{y}\\right][1\\otimes 1]\\bigg ]\\\\&\\times \\big |{\\lbrace p,f,c^{\\prime },c\\rbrace _{m}}\\big )\\;.\\end{split}$ We will use this result in eq.",
"(REF )."
],
[
"Calculation of the virtual graphs",
"The Sudakov exponent ${\\cal S}$ that appears in eq.",
"(REF ) consists of two terms, as given in eq.",
"(REF ): a term ${\\cal S}^{\\rm pert}(t)$ that comes from virtual graphs and a term that accounts for the evolution of the parton distribution functions.",
"In this appendix, we outline the calculation of ${\\cal S}^{\\rm pert}(t)$ .",
"Consider first an operator ${\\cal S}^{\\rm pert}_{\\rm tot}$ that corresponds to the one loop virtual graphs that contribute to shower evolution.",
"(More precisely, because of the minus sign in eq.",
"(REF ), ${\\cal S}^{\\rm pert}_{\\rm tot}$ corresponds to the negative of the one loop virtual graphs.)",
"In these graphs, we integrate over a loop momentum $k$ .",
"There will be ultraviolet divergences that come from one loop corrections to QCD propagators and vertices.",
"We suppose that these are removed by renormalization.",
"There may be an additional ultraviolet divergence that arises from letting the scale of the loop momentum be much larger than the scale $Q^2$ of the hard interaction that initiates the shower.",
"This can happen if we make the approximation $Q^2 \\rightarrow \\infty $ inside the graph.",
"We should simply arrange to have a regulator in the integrand that eliminates such a divergence.",
"The operator ${\\cal S}^{\\rm pert}_{\\rm tot}$ will also have infrared divergences, corresponding to the integration regions in which $k \\rightarrow 0$ or $k$ becomes collinear with the momentum of one of the external lines of the graph.",
"We regulate the infrared divergences with dimensional regulation or some other method.",
"We will represent ${\\cal S}^{\\rm pert}_{\\rm tot}$ as an integral over a shower time $t$ , as we have done for real emission diagrams: ${\\cal S}^{\\rm pert}_{\\rm tot} = \\int \\!",
"dt\\ {\\cal S}^{\\rm pert}(t)\\;.$ The shower time $t$ corresponds to the negative of the logarithm of the hardness scale of the integrand, in analogy with the definition of $t$ in real emission graphs.",
"Thus the infrared divergences are associated with $t \\rightarrow \\infty $ .",
"This general idea does not, however, tell us exactly how to define $t$ .",
"One way to proceed would be to use dimensional regulation and subtract the infrared poles.",
"Then the result would depend on a parameter $\\mu ^2$ .",
"Then we could identify $e^{-t}$ with $\\mu ^2/Q^2$ and ${\\cal S}^{\\rm pert}$ with the derivative of the graphs with respect to $\\log (Q^2/\\mu ^2)$ .",
"However, in eq.",
"(REF ) below, we will make a more direct identification based on the physical meaning of our version of the shower time.",
"Using the notation analogous to that of eq.",
"(REF ) for real emission diagrams, we can write ${\\cal S}^{\\rm pert}(t)= \\sum _{l=\\mathrm {a}, \\mathrm {b}, 1,\\cdots , m} {\\cal S}_l^{\\rm pert}(t)\\;.$ The part associated with splitting of parton $l$ is ${\\cal S}_l^{\\rm pert}(t)$ .",
"As in eq.",
"(REF ) for real emission graphs, we write ${\\cal S}_l^{\\rm pert}(t)$ as ${\\cal S}_l^{\\rm pert}(t) = {\\cal S}_{ll}^{\\rm pert}(t) + \\sum _{k \\ne l} {\\cal S}_{l k}^{\\rm pert}(t)\\;.$ The first term describes self-energy interactions.",
"In the second term, a virtual gluon is exchanged between parton $l$ and parton $k$ .",
"The sum includes all parton labels $k = \\mathrm {a}, \\mathrm {b}, 1, \\dots , m$ except for $k = l$ .",
"The operators ${\\cal S}_{l k}(t)$ have the color structure $\\begin{split}{\\cal S}_{l k}^{\\rm pert}(t)&\\big |{\\lbrace p,f,c^{\\prime },c\\rbrace _{m}}\\big )\\\\={}&\\big \\lbrace S^{\\mathrm {L}}_{l k}(\\lbrace p,f\\rbrace _{m};t)[(\\mathbf {T}_l\\cdot \\mathbf {T}_k)\\otimes 1]+ S^{\\mathrm {R}}_{l k}(\\lbrace p,f\\rbrace _{m};t)[1 \\otimes (\\mathbf {T}_l\\cdot \\mathbf {T}_k)]\\big \\rbrace \\\\&\\times \\big |{\\lbrace p,f,c^{\\prime },c\\rbrace _{m}}\\big )\\;.\\end{split}$ As in appendix , $T^a_k$ inserts a color generator matrix $T^a$ on line $k$ .",
"The functions $S^{\\mathrm {R}}_{l k}(t)$ are just the complex conjugates of the functions $S^{\\mathrm {L}}_{l k}(t)$ , so we need only to define and analyze the functions $S^{\\mathrm {L}}_{l k}(t)$ .",
"We will begin with the case of interference diagrams: $k \\ne l$ .",
"We start with the case that both $l$ and $k$ represent final state partons.",
"Then we will look at the case in which both $l$ and $k$ represent initial state partons.",
"Finally we let one of the partons be in the initial state while the other is in the final state.",
"Once we have covered interference diagrams, we look at self-energy diagrams: $k = l$ ."
],
[
"Final state interference diagrams",
"Let's work with $S^{\\mathrm {L}}_{l k}(\\lbrace p,f\\rbrace _{m};t)$ in the case that $l$ and $k$ are (different) final state partons in the ket amplitude.",
"We look at the contribution to $S^{\\mathrm {L}}_{l k}(\\lbrace p,f\\rbrace _{m};t)$ from gluon exchange between two partons.",
"As we do throughout this paper, we take all partons to be massless.",
"The momenta of partons $l$ and $k$ are $p_l$ and $p_k$ .",
"The gluon carries momentum $q$ from line $k$ to line $l$ , so that, inside the loop, line $l$ carries momentum $p_l - q$ and line $k$ carries momentum $p_k + q$ .",
"In analyzing the virtual graphs, we follow as much as possible the treatment of the real graphs that we have used in refs.",
"[67], [68], [69].",
"In particular, this means that we calculate the virtual graphs in Coulomb gauge."
],
[
"Integral in Coulomb gauge",
"We start with the full interference graph.",
"We will want to identify the shower time, but we have not done that, so we start with the integral over $t$ of the functions that we ultimately want.",
"Also, when a gluon is exchanged between partons $l$ and $k$ , it is undefined which is the primary emitter and which is playing only a helping role.",
"That is, we want $S^{\\mathrm {L}}_{l k}$ to represent an emission from parton $l$ with parton $k$ as helper, while $S^{\\mathrm {L}}_{kl}$ will represent an emission from parton $k$ with parton $l$ as helper.",
"We have not yet defined how the total graph is partitioned into these two parts, so we begin with $S^{\\mathrm {L}}_{l k} + S^{\\mathrm {L}}_{kl}$ .",
"Thus we start with the definition (including the minus sign in eq.",
"(REF ), so that we write the negative of the usual Feynman diagram), $\\begin{split}\\int \\!dt\\ \\big [S^{\\mathrm {L}}_{l k}(\\lbrace p,f\\rbrace _{m};t) &+ S^{\\mathrm {L}}_{kl}(\\lbrace p,f\\rbrace _{m};t)\\big ]\\\\={}&{\\mathrm {i}}\\, \\frac{\\alpha _{\\mathrm {s}}}{(2\\pi )^{3}}\\,\\int \\!",
"d\\vec{q} \\int \\!",
"dE\\ \\frac{2 p_l\\cdot D(q)\\cdot p_k}{(q\\cdot p_l - {\\mathrm {i}}\\epsilon )(q\\cdot p_k + {\\mathrm {i}}\\epsilon )(q^2 + {\\mathrm {i}}\\epsilon )}\\,\\theta (|\\vec{q}\\,| < M)\\;.\\end{split}$ We work in Coulomb gauge in a reference frame in which $\\vec{Q} = 0$ .",
"The integration variables are defined by $q = (E,\\vec{q}\\,)$ in this frame.",
"The integral is both ultraviolet and infrared divergent.",
"For technical reasons have inserted an ultraviolet cutoff $|\\vec{q}\\,| < M$ , where we will take $M$ to be large.",
"The infrared divergence is associated with large positive values of the shower time $t$ .",
"We could imagine that the infrared divergences are regulated, but once we select a fixed value of $t$ , the regulation is not needed.",
"For this reason, we do not specify a regulation method.",
"In Coulomb gauge at a fixed shower time $t$ , only the soft integration region for $q$ is important.",
"For this reason, it is appropriate to use the eikonal approximation (as in Deductor).",
"We have applied the eikonal approximation in eq.",
"(REF ).",
"The numerator of the gluon propagator in Coulomb gauge is $D(q)^{\\mu \\nu } =- g^{\\mu \\nu }-\\frac{q^\\mu \\tilde{q}^\\nu + \\tilde{q}^\\mu q^\\nu - q^\\mu q^\\nu }{|\\vec{q}\\,|^2}\\;,$ where $\\tilde{q} = (0,\\vec{q}\\,)$ .",
"Now, the integrand in eq.",
"(REF ) is complicated because of the numerator $D(q)$ of the gluon propagator in Coulomb gauge.",
"However, we can simplify it by writing $\\begin{split}S^{\\mathrm {L}}_{l k}(\\lbrace p,f\\rbrace _{m};t)={}&S^{\\mathrm {L}}_{l k}(\\lbrace p,f\\rbrace _{m};t;{\\rm dipole}) +S^{\\mathrm {L}}_{ll}(\\lbrace p,f\\rbrace _{m};t;{\\rm eikonal})\\end{split}$ and $\\begin{split}S^{\\mathrm {L}}_{kl}(\\lbrace p,f\\rbrace _{m};t)={}&S^{\\mathrm {L}}_{kl}(\\lbrace p,f\\rbrace _{m};t;{\\rm dipole}) +S^{\\mathrm {L}}_{kk}(\\lbrace p,f\\rbrace _{m};t;{\\rm eikonal})\\;.\\end{split}$ Here $\\begin{split}\\int \\!dt\\ S^{\\mathrm {L}}_{ll}(\\lbrace p,f\\rbrace _{m};&t;{\\rm eikonal})\\\\={}&{\\mathrm {i}}\\, \\frac{\\alpha _{\\mathrm {s}}}{(2\\pi )^{3}}\\,\\int \\!",
"d\\vec{q} \\int \\!",
"dE\\ \\frac{p_l\\cdot D(q)\\cdot p_l}{(q\\cdot p_l - {\\mathrm {i}}\\epsilon )^2 (q^2 + {\\mathrm {i}}\\epsilon )}\\,\\theta (|\\vec{q}\\,| < M)\\;,\\end{split}$ with an analogous definition of $S^{\\mathrm {L}}_{kk}(\\lbrace p,f\\rbrace _{m};t;{\\rm eikonal})$ .",
"This is the eikonal approximation to the self-energy graph for parton $l$ in Coulomb gauge.",
"Recall that we associate $S^{\\mathrm {L}}_{l k}$ with emissions from parton $l$ and $S^{\\mathrm {L}}_{kl}$ with emissions from parton $k$ .",
"In keeping with that interpretation, we count $S^{\\mathrm {L}}_{ll}(\\lbrace p,f\\rbrace _{m};t;{\\rm eikonal})$ as contributing to $S^{\\mathrm {L}}_{l k}(\\lbrace p,f\\rbrace _{m};t)$ and we count $S^{\\mathrm {L}}_{kk}(\\lbrace p,f\\rbrace _{m};t;{\\rm eikonal})$ as contributing to $S^{\\mathrm {L}}_{kl}(\\lbrace p,f\\rbrace _{m};t)$ .",
"This defines $S^{\\mathrm {L}}_{l k}(\\lbrace p,f\\rbrace _{m};t;{\\rm dipole})$ and $S^{\\mathrm {L}}_{kl}(\\lbrace p,f\\rbrace _{m};t;{\\rm dipole})$ .",
"With simple algebra, we find $\\begin{split}\\int \\!dt\\, \\big [S^{\\mathrm {L}}_{l k}&(\\lbrace p,f\\rbrace _{m};t;{\\rm dipole})+ S^{\\mathrm {L}}_{kl}(\\lbrace p,f\\rbrace _{m};t;{\\rm dipole})\\big ]\\\\={}&{\\mathrm {i}}\\, \\frac{\\alpha _{\\mathrm {s}}}{(2\\pi )^{3}}\\,\\int \\!",
"d\\vec{q} \\int \\!",
"dE \\ \\frac{- P_{lk}\\cdot D(q)\\cdot P_{lk}}{(q\\cdot p_l - {\\mathrm {i}}\\epsilon )^2(q\\cdot p_k + {\\mathrm {i}}\\epsilon )^2(q^2 + {\\mathrm {i}}\\epsilon )}\\,\\theta (|\\vec{q}\\,| < M)\\;,\\end{split}$ where $P_{lk} = q\\cdot p_l\\, p_k - q\\cdot p_k\\, p_l\\;.$ Since $q\\cdot P_{lk} = 0$ , none of the $q$ dependent terms in $D(q)$ contribute and we are left with $- P_{lk}\\cdot D(q)\\cdot P_{lk} = P_{lk}^2= - 2 q\\cdot p_l\\, q\\cdot p_k\\, p_l\\cdot p_k\\;.$ Thus $\\begin{split}\\int \\!dt\\ \\big [S^{\\mathrm {L}}_{l k}&(\\lbrace p,f\\rbrace _{m};t;{\\rm dipole})+ S^{\\mathrm {L}}_{kl}(\\lbrace p,f\\rbrace _{m};t;{\\rm dipole})\\big ]\\\\={}&-{\\mathrm {i}}\\, \\frac{\\alpha _{\\mathrm {s}}}{(2\\pi )^{3}}\\,\\int \\!",
"d\\vec{q} \\int \\!",
"dE \\ \\frac{2\\,p_l\\cdot p_k}{(q\\cdot p_l - {\\mathrm {i}}\\epsilon )(q\\cdot p_k + {\\mathrm {i}}\\epsilon )(q^2 + {\\mathrm {i}}\\epsilon )}\\,\\theta (|\\vec{q}\\,| < M)\\;.\\end{split}$ That is, we get the familiar dipole formula for one gluon exchange in Feynman gauge."
],
[
"Dipole part",
"We now analyze the Feynman gauge eikonal integral in eq.",
"(REF ).",
"We write $\\frac{1}{q^2 + {\\mathrm {i}}\\epsilon } = \\frac{1}{2 |\\vec{q}|}\\left[\\frac{1}{E - |\\vec{q}| + {\\mathrm {i}}\\epsilon } - \\frac{1}{E + |\\vec{q}| - {\\mathrm {i}}\\epsilon }\\right]\\;.$ In the first term, the gluon propagates forward in time from parton $k$ to parton $l$ .",
"In the second term, the gluon propagates forward in time from parton $l$ to parton $k$ .",
"In the second term, we redefine $q \\rightarrow - q$ , so that the direction of $q$ is the direction of propagation forward in time.",
"Then $\\begin{split}\\int \\!dt\\ & \\big [S^{\\mathrm {L}}_{l k}(\\lbrace p,f\\rbrace _{m};t;{\\rm dipole})+ S^{\\mathrm {L}}_{kl}(\\lbrace p,f\\rbrace _{m};t;{\\rm dipole})\\big ]\\\\={}&-{\\mathrm {i}}\\, \\frac{\\alpha _{\\mathrm {s}}}{(2\\pi )^{3}}\\,2 p_l\\cdot p_k\\int \\!",
"\\frac{d\\vec{q} }{2 |\\vec{q}|}\\, \\theta (|\\vec{q}\\,| < M)\\int \\!",
"dE\\ \\\\&\\times \\frac{1}{E - |\\vec{q}| + {\\mathrm {i}}\\epsilon }\\left[\\frac{1}{(q\\cdot p_l - {\\mathrm {i}}\\epsilon )(q\\cdot p_k + {\\mathrm {i}}\\epsilon )}+\\frac{1}{(q\\cdot p_l + {\\mathrm {i}}\\epsilon )(q\\cdot p_k - {\\mathrm {i}}\\epsilon )}\\right]\\;.\\end{split}$ We let $p_l = |\\vec{p}_l|\\, v_l$ and $p_k = |\\vec{p}_k|\\, v_k$ , where $\\begin{split}v_l ={}& (1,\\vec{v}_l)\\;,\\\\v_k ={}& (1,\\vec{v}_k)\\;,\\end{split}$ with $\\vec{v}_\\mathrm {a}^{\\,2} = \\vec{v}_k^{\\,2} = 1$ .",
"Also, we define $Q = E_Q(1,\\vec{0})$ .",
"Then $\\begin{split}\\int \\!dt\\ &\\big [S^{\\mathrm {L}}_{l k}(\\lbrace p,f\\rbrace _{m};t;{\\rm dipole})+ S^{\\mathrm {L}}_{kl}(\\lbrace p,f\\rbrace _{m};t;{\\rm dipole})\\big ]\\\\={}&-{\\mathrm {i}}\\, \\frac{\\alpha _{\\mathrm {s}}}{(2\\pi )^{3}}\\,2 v_l\\cdot v_k\\int \\!",
"\\frac{d\\vec{q} }{2 |\\vec{q}|}\\, \\theta (|\\vec{q}\\,| < M)\\int \\!",
"dE\\ \\frac{1}{E - |\\vec{q}| + {\\mathrm {i}}\\epsilon }\\\\&\\times \\left[\\frac{1}{(E - \\vec{q}\\cdot \\vec{v}_l - {\\mathrm {i}}\\epsilon )(E - \\vec{q}\\cdot \\vec{v}_k + {\\mathrm {i}}\\epsilon )}+\\frac{1}{(E - \\vec{q}\\cdot \\vec{v}_l + {\\mathrm {i}}\\epsilon )(E - \\vec{q}\\cdot \\vec{v}_k - {\\mathrm {i}}\\epsilon )}\\right]\\;.\\end{split}$ We can immediately perform the $E$ -integration to give $\\begin{split}\\int \\!dt\\ &\\big [S^{\\mathrm {L}}_{l k}(\\lbrace p,f\\rbrace _{m};t;{\\rm dipole})+ S^{\\mathrm {L}}_{kl}(\\lbrace p,f\\rbrace _{m};t;{\\rm dipole})\\big ]\\\\={}&-\\frac{\\alpha _{\\mathrm {s}}}{(2\\pi )^{2}}\\,2 v_l\\cdot v_k\\int \\!",
"\\frac{d\\vec{q} }{2 |\\vec{q}|}\\, \\theta (|\\vec{q}\\,| < M)\\\\&\\times \\left[\\frac{1}{(|\\vec{q}| - \\vec{q}\\cdot \\vec{v}_l)(\\vec{q}\\cdot \\vec{v}_l - \\vec{q}\\cdot \\vec{v}_k + {\\mathrm {i}}\\epsilon )}-\\frac{1}{(|\\vec{q}| - \\vec{q}\\cdot \\vec{v}_k)(\\vec{q}\\cdot \\vec{v}_l - \\vec{q}\\cdot \\vec{v}_k - {\\mathrm {i}}\\epsilon )}\\right]\\;.\\end{split}$ Now, we can rewrite this using $\\frac{1}{x \\pm {\\mathrm {i}}\\epsilon } = \\frac{1}{[x]_\\mathrm {P}} \\mp {\\mathrm {i}}\\pi \\, \\delta (x)\\;.$ This gives $\\begin{split}\\int \\!dt\\ \\big [S^{\\mathrm {L}}_{l k} &(\\lbrace p,f\\rbrace _{m};t;{\\rm dipole})+ S^{\\mathrm {L}}_{kl}(\\lbrace p,f\\rbrace _{m};t;{\\rm dipole})\\big ]\\\\={}&-\\frac{\\alpha _{\\mathrm {s}}}{(2\\pi )^{2}}\\,2 v_l\\cdot v_k\\int \\!",
"\\frac{d\\vec{q} }{2 |\\vec{q}|}\\,\\theta (|\\vec{q}\\,| < M)\\\\&\\times \\bigg \\lbrace \\frac{1}{[\\vec{q}\\cdot \\vec{v}_l - \\vec{q}\\cdot \\vec{v}_k]_\\mathrm {P}}\\left[\\frac{1}{|\\vec{q}| - \\vec{q}\\cdot \\vec{v}_l}-\\frac{1}{|\\vec{q}| - \\vec{q}\\cdot \\vec{v}_k}\\right]\\\\& \\qquad - {\\mathrm {i}}\\pi \\,\\delta (\\vec{q}\\cdot \\vec{v}_k - \\vec{q}\\cdot \\vec{v}_l)\\left[\\frac{1}{|\\vec{q}| - \\vec{q}\\cdot \\vec{v}_l}+\\frac{1}{|\\vec{q}| - \\vec{q}\\cdot \\vec{v}_k}\\right]\\bigg \\rbrace \\;.\\end{split}$ That is $\\begin{split}\\int \\!dt\\ \\big [S^{\\mathrm {L}}_{l k}&(\\lbrace p,f\\rbrace _{m};t;{\\rm dipole})+ S^{\\mathrm {L}}_{kl}(\\lbrace p,f\\rbrace _{m};t;{\\rm dipole})\\big ]\\\\={}&-\\frac{\\alpha _{\\mathrm {s}}}{(2\\pi )^{2}}\\,2 v_l\\cdot v_k\\int \\!",
"\\frac{d\\vec{q} }{2 |\\vec{q}|}\\,\\theta (|\\vec{q}\\,| < M)\\bigg \\lbrace \\frac{1}{(|\\vec{q}| - \\vec{q}\\cdot \\vec{v}_l)(|\\vec{q}| - \\vec{q}\\cdot \\vec{v}_k)}\\\\&- {\\mathrm {i}}\\pi \\,\\delta (\\vec{q}\\cdot \\vec{v}_k - \\vec{q}\\cdot \\vec{v}_l)\\,\\left[\\frac{1}{|\\vec{q}| - \\vec{q}\\cdot \\vec{v}_l}+\\frac{1}{|\\vec{q}| - \\vec{q}\\cdot \\vec{v}_k}\\right]\\bigg \\rbrace \\;.\\end{split}$ Now consider the first term.",
"It has essentially the structure of the graphs for the emission of a real gluon with $q^2 = 0$ , although it is not quite the same as our real emission factors because it does not contain momentum mappings that allow an on-shell parton to split into two on-shell partons.",
"The integrand has poles at $|\\vec{q}| = \\vec{q}\\cdot \\vec{v}_l$ and at $|\\vec{q}| = \\vec{q}\\cdot \\vec{v}_k$ .",
"This reflects splittings both of parton $l$ and of parton $k$ .",
"In order to separate these splittings, we multiply the integrand by $1 = A^{\\prime }_{lk} + A^{\\prime }_{kl}$ , where $A^{\\prime }_{lk} = \\frac{|\\vec{q}| - \\vec{q}\\cdot \\vec{v}_k}{(|\\vec{q}| - \\vec{q}\\cdot \\vec{v}_k) + (|\\vec{q}| - \\vec{q}\\cdot \\vec{v}_l)}\\;.$ The term containing $A^{\\prime }_{lk}$ is associated with $S^{\\mathrm {L}}_{l k}(\\lbrace p,f\\rbrace _{m};t)$ , while the term containing $A^{\\prime }_{kl}$ is associated with $S^{\\mathrm {L}}_{kl}(\\lbrace p,f\\rbrace _{m};t)$ .",
"In the ${\\mathrm {i}}\\pi $ term, the two contributions are actually equal, but we can associate the first with $S^{\\mathrm {L}}_{l k}(\\lbrace p,f\\rbrace _{m};t)$ and the second with $S^{\\mathrm {L}}_{kl}(\\lbrace p,f\\rbrace _{m};t)$ .",
"This gives $\\begin{split}\\int \\!dt\\ S^{\\mathrm {L}}_{l k}(\\lbrace p,f\\rbrace _{m};t&;{\\rm dipole})\\\\={}&-\\frac{\\alpha _{\\mathrm {s}}}{(2\\pi )^{2}}\\,2 v_l\\cdot v_k\\int \\!",
"\\frac{d\\vec{q} }{2 |\\vec{q}|}\\,\\theta (|\\vec{q}\\,| < M)\\\\&\\times \\bigg \\lbrace \\frac{1}{(|\\vec{q}| - \\vec{q}\\cdot \\vec{v}_l)}\\,\\frac{1}{(|\\vec{q}| - \\vec{q}\\cdot \\vec{v}_k) + (|\\vec{q}| - \\vec{q}\\cdot \\vec{v}_l)}\\\\&\\qquad - {\\mathrm {i}}\\pi \\,\\delta (\\vec{q}\\cdot \\vec{v}_k - \\vec{q}\\cdot \\vec{v}_l)\\,\\frac{1}{|\\vec{q}| - \\vec{q}\\cdot \\vec{v}_l}\\bigg \\rbrace \\;.\\end{split}$ We can now identify the shower time with $\\begin{split}y = \\frac{1}{E_Q}\\ (|\\vec{q}| - \\vec{q}\\cdot \\vec{v}_l)\\;.\\end{split}$ This matches the definition (REF ) that we used for a real gluon emission.In the case of real gluon emission, we take $\\vec{q}$ and $\\vec{l}$ to be the parton momenta after the splitting.",
"The momentum of parton $l$ before the splitting is $\\vec{l} + \\vec{q}$ , but here we use just $\\vec{l}$ because $\\vec{q}$ is small.",
"We introduce this as a delta function, recognizing that $\\int \\!dt \\cdots $ is equivalent to $\\int \\!d\\log (y) \\cdots $ .",
"Thus $\\begin{split}S^{\\mathrm {L}}_{l k}(\\lbrace p,f\\rbrace _{m};&t;{\\rm dipole})\\\\={}&-\\frac{\\alpha _{\\mathrm {s}}}{(2\\pi )^{2}}\\,2 v_l\\cdot v_k\\int \\!",
"\\frac{d\\vec{q} }{2 |\\vec{q}|}\\,\\theta (|\\vec{q}\\,| < M)\\,\\delta \\!\\left[\\log (E_Q y) - \\log \\left(|\\vec{q}| - \\vec{q}\\cdot \\vec{v}_l\\right)\\right]\\\\&\\times \\bigg \\lbrace \\frac{1}{E_Q y}\\,\\frac{1}{(|\\vec{q}| - \\vec{q}\\cdot \\vec{v}_k) + E_Q y}- {\\mathrm {i}}\\pi \\,\\delta (\\vec{q}\\cdot \\vec{v}_k - \\vec{q}\\cdot \\vec{v}_l)\\,\\frac{1}{E_Q y}\\bigg \\rbrace \\;.\\end{split}$ We will want to apply different methods for the two integrations."
],
[
"Dipole real part",
"We examine first the real part of the integral (REF ).",
"We introduce transverse and longitudinal coordinates for $\\vec{q}$ : $\\vec{q} = (1 - z - 2 a_l y)\\vec{p}_l + \\vec{q}_\\perp \\;,$ where $\\vec{q}_\\perp \\cdot \\vec{p}_l = 0$ and where we have defined $a_l = \\frac{E_Q}{2 |\\vec{p}_l|}\\;,$ as in eq.",
"(REF ).",
"We denote by $\\phi $ the azimuthal angle of $\\vec{q}_\\perp $ relative to the $(\\vec{p}_k, \\vec{p}_l)$ plane.",
"We first need to find the value of $|\\vec{q}_\\perp |$ .",
"From eq.",
"(REF ), we have $|\\vec{q}| = E_Q y + (1 - z - 2 a_l y) |\\vec{p}_l|\\;.$ That is $|\\vec{q}| = (1 - z) |\\vec{p}_l|\\;.$ This implies that the cutoff $|\\vec{q}\\,| < M$ amounts to $(1-z) < M/|\\vec{p}_l|\\;.$ We have $|\\vec{q}|^2 = (1-z)^2 |\\vec{p}_l|^2\\;.$ But $|\\vec{q}|^2 = (1-z - 2 a_l y)^2 |\\vec{p}_l|^2 + \\vec{q}_\\perp ^{\\,2}$ .",
"Thus $\\begin{split}\\vec{q}_\\perp ^{\\,2} ={}& |\\vec{p}_l|^2 [ (1-z)^2 - (1-z - 2 a_l y)^2]\\\\={}& |\\vec{p}_l|^2 4 a_l y\\,[(1-z) - a_l y]\\;.\\end{split}$ Note that there is a minimum possible value of $(1-z)$ , corresponding to $q_\\perp ^2 = 0$ : $(1 - z) > a_l y\\;.$ We can write the integration over $\\vec{q}$ as $d\\vec{q} = \\pi |\\vec{p}_l|\\, dz\\ dq_\\perp ^2\\ \\frac{d\\phi }{2\\pi }\\;.$ To perform the $q_\\perp ^2$ integration against the delta function, we note that $\\int \\!",
"dq_\\perp ^2\\ \\delta \\!\\left[\\log (E_Q y) - \\log \\left((|\\vec{q}| - \\vec{v}_l \\cdot \\vec{q})\\right)\\right]= 2 |\\vec{q}| E_Q y\\;.$ This gives $\\begin{split}{\\rm Re}\\, S^{\\mathrm {L}}_{l k}(\\lbrace p,f\\rbrace _{m};t;{\\rm dipole})={}&-\\frac{\\alpha _{\\mathrm {s}}}{2\\pi }\\, v_l\\cdot v_k \\,\\int _{1 - M/|\\vec{p}_l|}^{1 - a_l y}\\!",
"dz\\ \\int \\!\\frac{d\\phi }{2\\pi }\\ \\frac{|\\vec{p}_l|}{(|\\vec{q}| - \\vec{q}\\cdot \\vec{v}_k) + E_Q y}\\;.\\end{split}$ We can integrate this over $\\phi $ : $\\begin{split}\\int \\!\\frac{d\\phi }{2\\pi }\\ \\frac{|\\vec{p}_l|}{(|\\vec{q}| - \\vec{q}\\cdot \\vec{v}_k) + E_Q y}={}&\\frac{1}{v_k\\cdot v_l}\\,\\frac{1}{\\sqrt{(1-z)^2 + a_l^2 y^2/\\psi _{kl}^2}}\\;,\\end{split}$ where $\\psi _{kl} = \\frac{1 - \\cos \\theta _{kl}}{\\sqrt{8 (1 + \\cos \\theta _{kl})}}\\;,$ as in eq.",
"(REF ).",
"We now need to integrate this over $z$ : $\\int _{1 - M/|\\vec{p}_l|}^{1 - a_l y}\\!\\frac{dz}{\\sqrt{(1-z)^2 + a_l^2 y^2/\\psi _{kl}^2}}= \\log \\left(\\frac{M/|\\vec{p}_l| + \\sqrt{M^2/|\\vec{p}_l|^2 + a_l^2 y^2/\\psi _{kl}^2}}{a_l y \\left(1 + \\sqrt{1 + 1/\\psi _{kl}^2}\\right)}\\right)\\;,$ Neglecting terms that vanish like a power of $|\\vec{p}_l|/M$ as $|\\vec{p}_l|/M \\rightarrow 0$ , this becomes $\\begin{split}\\int _{1 - M/|\\vec{p}_l|}^{1 - a_l y}\\!\\frac{dz}{\\sqrt{(1-z)^2 + a_l^2 y^2/\\psi _{kl}^2}}\\sim {}&\\log \\!\\left(\\frac{2}{a_l y \\left(1 + \\sqrt{1 + 1/\\psi _{kl}^2}\\right)}\\right)+ \\log (M/|\\vec{p}_l|)\\\\&={}\\log \\!\\left(\\frac{1 - \\cos \\theta _{kl}}{2 a_l y}\\right)+ \\log (M/|\\vec{p}_l|)\\;.\\end{split}$ This gives $\\begin{split}{\\rm Re}\\,S^{\\mathrm {L}}_{l k}(\\lbrace p,f\\rbrace _{m};t;{\\rm dipole})\\approx {}&-\\frac{\\alpha _{\\mathrm {s}}}{2\\pi }\\,\\left[\\log \\!\\left(\\frac{1 - \\cos \\theta _{kl}}{2 a_l y}\\right)+ \\log (M/|\\vec{p}_l|)\\right]\\;.\\end{split}$ The $\\log (M/|\\vec{p}_l|)$ term will cancel against an identical term in the integral that we subtracted and have to add back."
],
[
"Dipole imaginary part",
"Now we examine the imaginary part of the integral (REF ), which we rewrite slightly as $\\begin{split}{\\rm Im}\\,S^{\\mathrm {L}}_{l k}(\\lbrace p,f\\rbrace _{m}&;t;{\\rm dipole})\\\\ ={}&\\pi \\frac{\\alpha _{\\mathrm {s}}}{(2\\pi )^{2}}\\,\\frac{2 v_l\\cdot v_k}{E_Q y}\\int \\!",
"\\frac{d\\vec{q} }{2 |\\vec{q}|}\\,\\theta (|\\vec{q}\\,| < M)\\\\&\\times \\delta \\!\\left[\\log (2E_Q y) - \\log \\left(2|\\vec{q}| - \\vec{q}\\cdot (\\vec{v}_k + \\vec{v}_l)\\right)\\right]\\delta (\\vec{q}\\cdot (\\vec{v}_k - \\vec{v}_l))\\;.\\end{split}$ We can immediately take the limit $M \\rightarrow \\infty $ .",
"It will be useful to choose coordinates $(\\xi ,\\eta ,\\lambda )$ based on the orthogonal vectors $(\\vec{v}_k + \\vec{v}_l)$ and $(\\vec{v}_k - \\vec{v}_l)$ : $\\vec{q} = \\left(\\xi + E_Q y \\sqrt{\\frac{1 + \\cos \\theta _{kl}}{1 - \\cos \\theta _{kl}}}\\right)\\frac{\\vec{v}_k + \\vec{v}_l}{\\sin \\theta _{kl}}+ \\eta \\, \\vec{u}+ \\lambda \\,\\frac{\\vec{v}_k - \\vec{v}_l}{2(1 - \\cos \\theta _{kl})}\\;,$ where $\\vec{u}$ is a unit vector orthogonal to $\\vec{v}_k$ and $\\vec{v}_l$ .",
"We then have $d\\vec{q} = \\frac{1}{1 - \\cos \\theta _{kl}}\\, d\\xi \\,d\\eta \\,d\\lambda \\;.$ We can immediately perform the $\\lambda $ integration using $\\delta (\\vec{q}\\cdot (\\vec{v}_k - \\vec{v}_l)) = \\delta (\\lambda )\\;.$ For the other delta function, define $f(\\xi ,\\eta ) = 2|\\vec{q}| - \\vec{q}\\cdot (\\vec{v}_k + \\vec{v}_l)\\;.$ The delta function restricts $(\\xi ,\\eta )$ to the surface $f(\\xi ,\\eta ) = 2 E_Q y$ .",
"On this surface, $2|\\vec{q}| = \\vec{q}\\cdot (\\vec{v}_k + \\vec{v}_l) + 2E_Q y$ , or $4 \\vec{q}^{\\,2} = [\\vec{q}\\cdot (\\vec{v}_k + \\vec{v}_l) + 2 E_Q y]^2\\;.$ After setting $\\lambda = 0$ , we find that the surface is a circle in our chosen coordinates: $\\xi ^2 + \\eta ^2 = \\frac{2 E_Q^2 y^2}{1 - \\cos \\theta _{kl}}\\;.$ Consider what happens if $\\xi \\rightarrow \\xi + \\delta \\xi $ and $\\eta \\rightarrow \\eta + \\delta \\eta $ .",
"At the surface $f(\\xi ,\\eta ) = 2 E_Q y$ , this gives $\\begin{split}\\delta f(\\xi ,\\eta ) ={}&\\frac{2}{|\\vec{q}|}\\left\\lbrace \\xi \\delta \\xi + \\eta \\,\\delta \\eta \\right\\rbrace \\;.\\end{split}$ If we use polar coordinates $\\xi = R\\cos \\theta $ and $\\eta = R\\sin \\theta $ then the surface $f(\\xi ,\\eta ) = 2 E_Q y$ is at $R^2 = 2 E_Q^2 y^2/(1 - \\cos \\theta _{kl})$ and we have $\\begin{split}\\delta f(\\xi ,\\eta ) ={}&\\frac{2}{|\\vec{q}|}\\,R\\,\\delta R\\;.\\end{split}$ This gives $\\begin{split}\\int \\!",
"\\frac{d\\xi \\, d\\eta }{2|\\vec{q}|}\\ \\delta \\!\\left[\\log (2E_Q y) - \\log \\left(2|\\vec{q}| - \\vec{q}\\cdot (\\vec{v}_k + \\vec{v}_l)\\right)\\right]={}& 2\\pi \\,\\frac{E_Q y}{2}\\;.\\end{split}$ Inserting these results into eq.",
"(REF ) and using $1 - \\cos \\theta _{kl} = v_k\\cdot v_l$ , we have $\\begin{split}{\\rm Im}\\,S^{\\mathrm {L}}_{l k}(\\lbrace p,f\\rbrace _{m};t;{\\rm dipole}) ={}&\\pi \\,\\frac{\\alpha _{\\mathrm {s}}}{2\\pi }\\;.\\end{split}$"
],
[
"Dipole total",
"Adding eqs.",
"(REF ) and (REF ), we have $\\begin{split}S^{\\mathrm {L}}_{l k}(\\lbrace p,f\\rbrace _{m};t;{\\rm dipole})\\approx {}&\\frac{\\alpha _{\\mathrm {s}}}{2\\pi }\\,\\left[-\\log \\!\\left(\\frac{1 - \\cos \\theta _{kl}}{2 a_l y}\\right)- \\log (M/|\\vec{p}_l|)+ {\\mathrm {i}}\\pi \\right]\\;.\\end{split}$"
],
[
"Eikonal self-energy integral",
"In order to construct $S^{\\mathrm {L}}_{l k}(\\lbrace p,f\\rbrace _{m};t)$ in eq.",
"(REF ), we add $S^{\\mathrm {L}}_{ll}(\\lbrace p,f\\rbrace _{m};t;{\\rm eikonal})$ , defined in eq.",
"(REF ), to $S^{\\mathrm {L}}_{l k}(\\lbrace p,f\\rbrace _{m};t;{\\rm dipole})$ .",
"We thus need to calculate $S^{\\mathrm {L}}_{ll}(\\lbrace p,f\\rbrace _{m};t;{\\rm eikonal})$ in eq.",
"(REF ).",
"The product $p_l\\cdot D(q)\\cdot p_l$ appears.",
"This factor is $p_l\\cdot D(q)\\cdot p_l =\\frac{q\\cdot p_l\\, |\\vec{p}_l|}{|\\vec{q}|^2}\\ [E + \\vec{q}\\cdot \\vec{v}_l]\\;.$ This gives us $\\begin{split}\\int \\!dt\\ &S^{\\mathrm {L}}_{ll}(\\lbrace p,f\\rbrace _{m};t;{\\rm eikonal})\\\\={}&\\frac{{\\mathrm {i}}\\, \\alpha _{\\mathrm {s}}}{(2\\pi )^{3}}\\,\\int \\!",
"d\\vec{q}\\ \\frac{\\theta (|\\vec{q}\\,| < M)}{|\\vec{q}|^2}\\int \\!",
"dE\\ \\frac{E + \\vec{q}\\cdot \\vec{v}_l}{(E - \\vec{q}\\cdot \\vec{v}_l - {\\mathrm {i}}\\epsilon )(E - |\\vec{q}| + {\\mathrm {i}}\\epsilon )(E + |\\vec{q}| - {\\mathrm {i}}\\epsilon )}\\;.\\end{split}$ We can perform the $E$ -integration to get $\\begin{split}\\int \\!dt\\ S^{\\mathrm {L}}_{ll}(\\lbrace p,f\\rbrace _{m};t;{\\rm eikonal})={}&\\frac{\\alpha _{\\mathrm {s}}}{(2\\pi )^{2}}\\,\\int \\!",
"d\\vec{q}\\ \\frac{\\theta (|\\vec{q}\\,| < M)}{2|\\vec{q}|^3}\\ \\frac{|\\vec{q}| + \\vec{q}\\cdot \\vec{v}_l}{|\\vec{q}| - \\vec{q}\\cdot v_l}\\;.\\end{split}$ We recognize the denominator as defining the shower time, so $\\begin{split}S^{\\mathrm {L}}_{ll}(\\lbrace p,&f\\rbrace _{m};t;{\\rm eikonal})\\\\={}&\\frac{\\alpha _{\\mathrm {s}}}{(2\\pi )^{2}}\\,\\int \\!",
"d\\vec{q}\\ \\theta (|\\vec{q}\\,| < M)\\,\\delta \\!\\left[\\log (E_Q y) - \\log \\left(|\\vec{q}| - \\vec{q}\\cdot \\vec{v}_l\\right)\\right]\\frac{E_Q y + 2\\vec{q}\\cdot \\vec{v}_l}{2|\\vec{q}|^3\\,E_Q y}\\ \\;.\\end{split}$ Introducing variables $q_\\perp , z, \\phi $ with the aid of eqs.",
"(REF ), (REF ), (REF ), and (REF ), we have $\\begin{split}S^{\\mathrm {L}}_{ll}(\\lbrace p,f\\rbrace _{m};t;{\\rm eikonal})={}&\\frac{\\alpha _{\\mathrm {s}}}{2\\pi }\\,\\int _{1- M/|\\vec{p}_l|}^{1 - a_l y}\\!",
"dz\\ \\frac{(1-z) - a_l y}{(1-z)^2}\\ \\;.\\end{split}$ We can perform the integration to obtain $\\begin{split}S^{\\mathrm {L}}_{ll}(\\lbrace p,f\\rbrace _{m};t;{\\rm eikonal})={}&\\frac{\\alpha _{\\mathrm {s}}}{2\\pi }\\,\\left[\\log \\left(\\frac{M/|\\vec{p}_l|}{a_l y}\\right)- \\frac{M/|\\vec{p}_l| - a_l y}{M/|\\vec{p}_l|}\\right]\\;.\\end{split}$ We want the limit of this for large $M/|\\vec{p}_l|$ : $\\begin{split}S^{\\mathrm {L}}_{ll}(\\lbrace p,f\\rbrace _{m};t;{\\rm eikonal})={}&\\frac{\\alpha _{\\mathrm {s}}}{2\\pi }\\,\\left[\\log \\left(\\frac{1}{a_l y}\\right)- 1+\\log \\left(\\frac{M}{|\\vec{p}_l|}\\right)\\right]\\;.\\end{split}$"
],
[
"Total $l$ -{{formula:06dcf82a-59e2-4650-bc2f-6487acd12f5f}} interference graph",
"We put our contributions back together, inserting $G^{\\mathrm {L}}_{lk}(\\lbrace p,f\\rbrace _{m};t;{\\rm dipole})$ from eq.",
"(REF ) and $G^{\\mathrm {L}}_{ll}(\\lbrace p,f\\rbrace _{m};t;{\\rm eikonal})$ from eq.",
"(REF ) into eq.",
"(REF ): $\\begin{split}S^{\\mathrm {L}}_{l k}(\\lbrace p,f\\rbrace _{m};t)\\approx {}&\\frac{\\alpha _{\\mathrm {s}}}{2\\pi }\\,\\left[-\\log \\!\\left(\\frac{1 - \\cos \\theta _{kl}}{2}\\right)- 1+ {\\mathrm {i}}\\pi \\right]\\;.\\end{split}$ Here the cutoff dependent terms proportional to $\\log (M/|\\vec{p}_l|)$ have cancelled.",
"We will use this result in eq.",
"(REF )."
],
[
"Initial state interference diagram",
"Let's now look at the case that the active parton is one of the initial state partons, $l = \\mathrm {a}$ , and the helper parton $k$ is the other, $k = \\mathrm {b}$ .",
"Thus, we examine $S^{\\mathrm {L}}_{\\mathrm {a}\\mathrm {b}}(\\lbrace p,f\\rbrace _{m};t)$ , corresponding to gluon exchange between the two initial state partons.",
"The gluon carries momentum $q$ from line “a” to line “b,” so that, inside the loop, line “a” carries momentum $p_\\mathrm {a}- q$ and line “b” carries momentum $ p_\\mathrm {b}+ q$ .",
"We start with the eikonal approximation to the exchange in Coulomb gauge, $\\begin{split}\\int \\!dt\\ [S^{\\mathrm {L}}_{\\mathrm {a}\\mathrm {b}}&(\\lbrace p,f\\rbrace _{m};t)+ S^{\\mathrm {L}}_{\\mathrm {b}\\mathrm {a}}(\\lbrace p,f\\rbrace _{m};t)]\\\\={}&{\\mathrm {i}}\\, \\frac{\\alpha _{\\mathrm {s}}}{(2\\pi )^{3}}\\,\\int \\!",
"d\\vec{q} \\int \\!",
"dE\\ \\frac{2 p_\\mathrm {a}\\cdot D(q)\\cdot p_\\mathrm {b}}{(q\\cdot p_\\mathrm {a}- {\\mathrm {i}}\\epsilon )(q\\cdot p_\\mathrm {b}+ {\\mathrm {i}}\\epsilon )(q^2 + {\\mathrm {i}}\\epsilon )}\\,\\theta (|\\vec{q}\\,| < M)\\;.\\end{split}$ There is an ultraviolet cutoff $|\\vec{q}\\,| < M$ that we eventually remove.",
"This integral is exactly the same as we had in eq.",
"(REF ) for the final state case, with $p_l \\rightarrow p_\\mathrm {a}$ and $p_k \\rightarrow p_\\mathrm {b}$ .",
"We can apply the same treatment, partitioning the integral into two terms using the partitioning function (REF ) and identifying the shower time using eq.",
"(REF ).",
"Thus we can simply use the result in eq.",
"(REF ), noting that $\\cos \\theta _{kl} = -1$ : $\\begin{split}S^{\\mathrm {L}}_{\\mathrm {a}\\mathrm {b}}(\\lbrace p,f\\rbrace _{m};t)\\approx {}&\\frac{\\alpha _{\\mathrm {s}}}{2\\pi }\\,\\left[- 1+ {\\mathrm {i}}\\pi \\right]\\;.\\end{split}$ We will use this result in eq.",
"(REF )."
],
[
"Initial state - final state interference",
"We now examine the case of gluon exchange between one of the initial state partons, say $l = \\mathrm {a}$ , and a final state parton $k$ .",
"Thus, we examine $S^{\\mathrm {L}}_{\\mathrm {a}k}(\\lbrace p,f\\rbrace _{m};t)$ and $S^{\\mathrm {L}}_{k \\mathrm {a}}(\\lbrace p,f\\rbrace _{m};t)$ .",
"An exchanged gluon carries momentum $q$ from line “a” to line $k$ , so that, inside the loop, line “a” carries momentum $p_\\mathrm {a}- q$ and line $k$ carries momentum $p_k - q$ .",
"We start with the eikonal approximation to the exchange in Coulomb gauge, $\\begin{split}\\int \\!dt\\ [S^{\\mathrm {L}}_{\\mathrm {a}k}&(\\lbrace p,f\\rbrace _{m};t)+ S^{\\mathrm {L}}_{k \\mathrm {a}}(\\lbrace p,f\\rbrace _{m};t)]\\\\={}&{\\mathrm {i}}\\, \\frac{\\alpha _{\\mathrm {s}}}{(2\\pi )^{3}}\\,\\int \\!",
"d\\vec{q} \\int \\!",
"dE\\ \\frac{2 p_\\mathrm {a}\\cdot D(q)\\cdot p_k}{(q\\cdot p_\\mathrm {a}- {\\mathrm {i}}\\epsilon )(q\\cdot p_k - {\\mathrm {i}}\\epsilon )(q^2 + {\\mathrm {i}}\\epsilon )}\\,\\theta (|\\vec{q}\\,| < M)\\;.\\end{split}$ The integrand in eq.",
"(REF ) is complicated, but we can simplify it by writing $\\begin{split}S^{\\mathrm {L}}_{\\mathrm {a}k}(\\lbrace p,f\\rbrace _{m};t)={}&S^{\\mathrm {L}}_{\\mathrm {a}k}(\\lbrace p,f\\rbrace _{m};t;{\\rm dipole}) +S^{\\mathrm {L}}_{\\mathrm {a}\\mathrm {a}}(\\lbrace p,f\\rbrace _{m};t;{\\rm eikonal})\\;,\\\\S^{\\mathrm {L}}_{k\\mathrm {a}}(\\lbrace p,f\\rbrace _{m};t)={}&S^{\\mathrm {L}}_{k\\mathrm {a}}(\\lbrace p,f\\rbrace _{m};t;{\\rm dipole}) +S^{\\mathrm {L}}_{kk}(\\lbrace p,f\\rbrace _{m};t;{\\rm eikonal})\\;.\\end{split}$ Here $\\begin{split}\\int \\!dt\\ S^{\\mathrm {L}}_{\\mathrm {a}\\mathrm {a}}(\\lbrace p,f\\rbrace _{m};&t;{\\rm eikonal})\\\\={}&{\\mathrm {i}}\\, \\frac{\\alpha _{\\mathrm {s}}}{(2\\pi )^{3}}\\,\\int \\!",
"d\\vec{q} \\int \\!",
"dE\\ \\frac{p_\\mathrm {a}\\cdot D(q)\\cdot p_\\mathrm {a}}{(q\\cdot p_\\mathrm {a}- {\\mathrm {i}}\\epsilon )^2 (q^2 + {\\mathrm {i}}\\epsilon )}\\,\\theta (|\\vec{q}\\,| < M)\\;,\\end{split}$ with an analogous definition of $S^{\\mathrm {L}}_{kk}(\\lbrace p,f\\rbrace _{m};t;{\\rm eikonal})$ .",
"This defines $S^{\\mathrm {L}}_{\\mathrm {a}k}(\\lbrace p,f\\rbrace _{m};t;{\\rm dipole})$ and $S^{\\mathrm {L}}_{k\\mathrm {a}}(\\lbrace p,f\\rbrace _{m};t;{\\rm dipole})$ .",
"After some algebra, we find $\\begin{split}\\int \\!dt\\ \\big [S^{\\mathrm {L}}_{\\mathrm {a}k}&(\\lbrace p,f\\rbrace _{m};t;{\\rm dipole})+ S^{\\mathrm {L}}_{k\\mathrm {a}}(\\lbrace p,f\\rbrace _{m};t;{\\rm dipole})\\big ]\\\\={}&-{\\mathrm {i}}\\, \\frac{\\alpha _{\\mathrm {s}}}{(2\\pi )^{3}}\\,\\int \\!",
"d\\vec{q} \\int \\!",
"dE \\ \\frac{2\\,p_\\mathrm {a}\\cdot p_k}{(q\\cdot p_\\mathrm {a}- {\\mathrm {i}}\\epsilon )(q\\cdot p_k + {\\mathrm {i}}\\epsilon )(q^2 + {\\mathrm {i}}\\epsilon )}\\,\\theta (|\\vec{q}\\,| < M)\\;.\\end{split}$ As with eq.",
"(REF ) we write this using $E$ and $\\vec{q}$ : $\\begin{split}\\int \\!dt\\ \\bigg [S^{\\mathrm {L}}_{\\mathrm {a}k}(\\lbrace p,f\\rbrace _{m};&t;{\\rm dipole})+ S^{\\mathrm {L}}_{k \\mathrm {a}}(\\lbrace p,f\\rbrace _{m};t;{\\rm dipole})\\bigg ]\\\\={}&-{\\mathrm {i}}\\, \\frac{\\alpha _{\\mathrm {s}}}{(2\\pi )^{3}}\\,\\int \\!",
"d\\vec{q} \\int \\!",
"dE \\ \\frac{2\\,v_\\mathrm {a}\\cdot v_k}{(E - \\vec{q}\\cdot \\vec{v}_\\mathrm {a}- {\\mathrm {i}}\\epsilon )(E - \\vec{q}\\cdot \\vec{v}_k - {\\mathrm {i}}\\epsilon )}\\\\ & \\times \\frac{1}{2 |\\vec{q}|}\\left[\\frac{1}{E - |\\vec{q}| + {\\mathrm {i}}\\epsilon } - \\frac{1}{E + |\\vec{q}| - {\\mathrm {i}}\\epsilon }\\right]\\theta (|\\vec{q}\\,| < M)\\;.\\end{split}$ We can immediately perform the energy integration, noting that the term $1/(E + |\\vec{q}| - {\\mathrm {i}}\\epsilon )$ does not contribute: $\\begin{split}\\int \\!dt\\ \\bigg [S^{\\mathrm {L}}_{\\mathrm {a}k}(\\lbrace p,f\\rbrace _{m};&t;{\\rm dipole})+ S^{\\mathrm {L}}_{k \\mathrm {a}}(\\lbrace p,f\\rbrace _{m};t;{\\rm dipole})\\bigg ]\\\\={}&-\\frac{\\alpha _{\\mathrm {s}}}{(2\\pi )^{2}}\\,\\int \\!\\frac{d\\vec{q}}{2 |\\vec{q}|}\\ \\frac{2\\,v_\\mathrm {a}\\cdot v_k}{(|\\vec{q}| - \\vec{q}\\cdot \\vec{v}_\\mathrm {a})(|\\vec{q}| - \\vec{q}\\cdot \\vec{v}_k)}\\,\\theta (|\\vec{q}\\,| < M)\\;.\\end{split}$ Notice how this compares to the equivalent result for virtual gluon exchange between two final state partons, as in eq.",
"(REF ), or two initial state partons.",
"Here there is no imaginary part.",
"The integrand has poles at $|\\vec{q}| = \\vec{q}\\cdot \\vec{v}_\\mathrm {a}$ and at $|\\vec{q}| = \\vec{q}\\cdot \\vec{v}_k$ .",
"This reflects splittings both of parton “a” and of parton $k$ .",
"In order to separate these splittings, we multiply the integrand by $1 = A^{\\prime }_{\\mathrm {a}k} + A^{\\prime }_{k\\mathrm {a}}$ , where $A^{\\prime }_{\\mathrm {a}k} = \\frac{|\\vec{q}| - \\vec{q}\\cdot \\vec{v}_k}{(|\\vec{q}| - \\vec{q}\\cdot \\vec{v}_k) + (|\\vec{q}| - \\vec{q}\\cdot \\vec{v}_\\mathrm {a})}\\;.$ The term containing $A^{\\prime }_{\\mathrm {a}k}$ is associated with $G^{\\mathrm {L}}_{\\mathrm {a}k}(\\lbrace p,f\\rbrace _{m};t)$ , while the term containing $A^{\\prime }_{k\\mathrm {a}}$ is associated with $G^{\\mathrm {L}}_{k\\mathrm {a}}(\\lbrace p,f\\rbrace _{m};t)$ .",
"Thus we define $\\begin{split}\\int \\!dt\\ S^{\\mathrm {L}}_{\\mathrm {a}k}&(\\lbrace p,f\\rbrace _{m};t;{\\rm dipole})\\\\={}&-\\frac{\\alpha _{\\mathrm {s}}}{(2\\pi )^{2}}\\,\\int \\!\\frac{d\\vec{q}}{2 |\\vec{q}|}\\ \\frac{2\\,v_\\mathrm {a}\\cdot v_k}{(|\\vec{q}| - \\vec{q}\\cdot \\vec{v}_\\mathrm {a})[(|\\vec{q}| - \\vec{q}\\cdot \\vec{v}_k) + (|\\vec{q}| - \\vec{q}\\cdot \\vec{v}_\\mathrm {a})]}\\;\\theta (|\\vec{q}\\,| < M)\\;.\\end{split}$ We can now identify the shower time with $\\begin{split}y = \\frac{1}{E_Q}\\ (|\\vec{q}| - \\vec{q}\\cdot \\vec{v}_\\mathrm {a})\\end{split}$ as we did for a final state splitting.",
"We introduce this as a delta function, recognizing that $\\int \\!dt \\cdots $ is equivalent to $\\int \\!d\\log (y) \\cdots $ .",
"Thus $\\begin{split}S^{\\mathrm {L}}_{\\mathrm {a}k}(\\lbrace p,f\\rbrace _{m};&t;{\\rm dipole})\\\\={}&-\\frac{\\alpha _{\\mathrm {s}}}{(2\\pi )^{2}}\\,2v_\\mathrm {a}\\cdot v_k\\,\\int \\!",
"\\frac{d\\vec{q}}{2|\\vec{q}|}\\,\\theta (|\\vec{q}\\,| < M)\\,\\delta \\!\\left[\\log (E_Q y) - \\log \\left(|\\vec{q}| - \\vec{q}\\cdot \\vec{v}_\\mathrm {a}\\right)\\right]\\\\&\\times \\frac{1}{E_Q y\\,(|\\vec{q}| - \\vec{q}\\cdot \\vec{v}_k + E_Q y)}\\;\\;.\\end{split}$ This is precisely the real part of the integral in eq.",
"(REF ) with $l \\rightarrow \\mathrm {a}$ .",
"To calculate $S^{\\mathrm {L}}_{\\mathrm {a}k}(\\lbrace p,f\\rbrace _{m};t)$ using eq.",
"(REF ), we also need $S^{\\mathrm {L}}_{\\mathrm {a}\\mathrm {a}}(\\lbrace p,f\\rbrace _{m};t;{\\rm eikonal})$ , which is the same as $S^{\\mathrm {L}}_{ll}(\\lbrace p,f\\rbrace _{m};t;{\\rm eikonal})$ with $l \\rightarrow \\mathrm {a}$ .",
"Thus $S^{\\mathrm {L}}_{\\mathrm {a}k}(\\lbrace p,f\\rbrace _{m};t)$ is simply the real part of $S^{\\mathrm {L}}_{l k}(\\lbrace p,f\\rbrace _{m};t)$ , eq.",
"(REF ), with $l \\rightarrow \\mathrm {a}$ : $\\begin{split}S^{\\mathrm {L}}_{\\mathrm {a}k}(\\lbrace p,f\\rbrace _{m};t)\\approx {}&\\frac{\\alpha _{\\mathrm {s}}}{2\\pi }\\,\\left[-\\log \\!\\left(\\frac{1 - \\cos \\theta _{\\mathrm {a}k}}{2}\\right)- 1\\right]\\;.\\end{split}$ We will use this result in eq.",
"(REF ).",
"For $S^{\\mathrm {L}}_{k \\mathrm {a}}(\\lbrace p,f\\rbrace _{m};t)$ , essentially the same calculation gives the same result, $\\begin{split}S^{\\mathrm {L}}_{k \\mathrm {a}}(\\lbrace p,f\\rbrace _{m};t)\\approx {}&\\frac{\\alpha _{\\mathrm {s}}}{2\\pi }\\,\\left[-\\log \\!\\left(\\frac{1 - \\cos \\theta _{\\mathrm {a}k}}{2}\\right)- 1\\right]\\;.\\end{split}$ We will use this result in eq.",
"(REF )."
],
[
"Self-energy diagrams",
"In this section, we look at self-energy graphs.",
"As with the interference graphs, we use Coulomb gauge in the rest frame of the total momentum $Q$ of the final state partons.",
"Consider a gluon that enters the final state or comes from the initial state.",
"The gluon has momentum $p_l$ , with $l \\in \\lbrace 1,2,\\dots ,m\\rbrace $ for a final state gluon and with $l \\in \\lbrace \\mathrm {a},\\mathrm {b}\\rbrace $ for the initial state case.",
"We combine the self-energy subgraph $-{\\mathrm {i}}\\Pi (q)^{\\alpha \\beta }$ with the adjoining virtual propagator and the adjoining cut propagator.",
"Such a graph really represents a field strength renormalization for the gluon field.",
"We interpret the graph together with the propagators as $\\int \\!dt\\ S^{\\mathrm {L}}_{l l}(\\lbrace p,f\\rbrace _{m};t;{\\rm gluon})\\, D(p_l)^{\\mu \\nu } =-\\frac{1}{2}\\left[\\frac{1}{p_l^2}D(p_l)^\\mu _\\alpha \\,\\Pi (p_l)^{\\alpha \\beta }\\,D^\\nu _\\beta (p_l)\\right]_{p_l^2 = 0}\\;.$ The minus sign is from eq.",
"(REF ), so that we write the negative of the usual Feynman diagram.",
"We will also consider a quark that enters the final state or comes from the initial state.",
"The analogous definition is $\\int \\!dt\\ S^{\\mathrm {L}}_{l l}(\\lbrace p,f\\rbrace _{m};t;{\\rm quark})\\, { p}/$ /$p$ l = -12[ /$p$ lpl2 (pl) /$p$ l]pl2 = 0 .",
"We can use the results of ref. [86].",
"For the gluon case, these results include the contributions from a gluon loop, a quark loop, and a ghost loop.",
"For the quark case, we have a quark-gluon loop.",
"In the loop integral, we integrate over the energy going around the loop, giving a result in time-ordered perturbation theory, with on-shell partons and energy denominators.",
"We write the result using the three-momenta $\\vec{k}_\\pm $ of the two partons in the loop, with $\\vec{k}_+ + \\vec{k}_- = \\vec{p}_l$ .",
"Inside this integral, we need to identify the shower time or, equivalently, the dimensionless virtuality variable $y$ .",
"The definition (REF ) for a real splitting is $\\begin{split}y ={}& \\frac{(k_+ + k_-)^2}{2 p_l\\cdot Q}\\;,\\end{split}$ where $k_\\pm $ are the (on-shell) parton momenta after the splitting.",
"The virtuality $(k_+ + k_-)^2$ is $(|\\vec{k}_+| + |\\vec{k}_-|)^2 - (\\vec{k}_+ + \\vec{k}_-)^{2}$ .",
"This is normalized by dividing by $2 p_l\\cdot Q = 2 p_l^0 E_Q$ .",
"In a real emission, a small amount of momentum is taken from elsewhere in the event to put $p_l$ on shell.",
"For the virtual graph, we simply replace $p_l^0 \\rightarrow |\\vec{p}_l| = |\\vec{k}_+ + \\vec{k}_-|$ .",
"Thus we identify $\\begin{split}y ={}& \\frac{(|\\vec{k}_+| + |\\vec{k}_-|)^2 - \\vec{p}_l^{\\,2}}{2|\\vec{p}_l|E_Q}\\;.\\end{split}$ In the result from ref.",
"[86], the integral is written as an integration over $y$ , the azimuthal angle $\\phi $ of $\\vec{k}_+$ around the direction of $\\vec{p}_l$ , and and a variableIn the quark case, $k_-$ is the momentum of the quark line inside the loop, so $x$ is the momentum fraction of the gluon.",
"However, we present the results symmetrized over $x \\leftrightarrow (1-x)$ , so that it does not matter whether $x$ or $(1-x)$ is identified with the gluon in the loop.",
"Ref.",
"[86] explains how to remove the symmetrization, but we do not need to do that here.",
"$\\begin{split}x ={}& \\frac{|\\vec{k}_+| - |\\vec{k}_-| + |\\vec{p}_l|}{2|\\vec{p}_l|}\\;.\\end{split}$ The variable $\\bar{q}^2 = 2|\\vec{p}_l|E_Q\\,y$ appears instead of $y$ in ref. [86].",
"We include here the factor 1/2 in eqs.",
"(REF ) and (REF ).",
"Ref.",
"[86] gives directly the right hand side of these equations without the factor 1/2.",
"Also, we maintain the notation of eq.",
"(REF ) in which $S^{\\mathrm {L}}_{l l}$ multiplies a factor $[(\\mathbf {T}_l\\cdot \\mathbf {T}_l)\\otimes 1]$ .",
"For the gluon case, this is a factor $C_\\mathrm {A}[1\\otimes 1]$ and for the quark case, it is a factor $C_\\mathrm {F}[1\\otimes 1]$ .",
"For this reason, we remove a factor $C_\\mathrm {A}$ or $C_\\mathrm {F}$ from the results as given in ref. [86].",
"As in previous sections, we will use the convenient abbreviation $a_l = E_Q/(2|\\vec{p}_l|)$ from eq.",
"(REF ).",
"The variable $y$ is proportional to $e^{-t}$ , where $t$ is the shower time.",
"Thus integrating over $t$ is the same as integrating over $y$ , with $dt = d\\log y$ .",
"We are interested in $S^{\\mathrm {L}}_{l l}(\\lbrace p,f\\rbrace _{m};t)$ , the integrand of the $\\log y$ integration.",
"Integrations over $x$ and $\\phi $ will remain.",
"We now turn to the gluon and quark cases separately."
],
[
"Gluon self-energy",
"For the gluon self energy, we find from ref.",
"[86] $S^{\\mathrm {L}}_{l l}(\\lbrace p,f\\rbrace _{m};t;{\\rm gluon}) =\\int _0^1\\!dx \\int _{-\\pi }^\\pi \\!\\frac{d\\phi }{2\\pi }\\ \\frac{\\alpha _{\\mathrm {s}}}{8\\pi }\\,\\frac{1}{C_\\mathrm {A}}\\sum _{J=0}^2\\frac{A_{\\mathrm {T},J}}{[4 a_l y + 4 x (1-x)]^J}\\;.$ The coefficients $A_{\\mathrm {T},J}$ are $\\begin{split}A_{\\mathrm {T},0} ={}&- (2 C_\\mathrm {A}- N_\\mathrm {f})\\,\\frac{e^2 \\mu _\\mathrm {R}^2}{2 |\\vec{p}_l| E_Q y + e^2 \\mu _\\mathrm {R}^2}+ 2 C_\\mathrm {A}\\,x(1-x) \\,\\frac{e^{5/3} \\mu _\\mathrm {R}^2}{2 |\\vec{p}_l| E_Q y + e^{5/3} \\mu _\\mathrm {R}^2}\\\\&+ (4 C_\\mathrm {A}- 2 N_\\mathrm {f})\\,x(1-x) \\,\\frac{e^{8/3} \\mu _\\mathrm {R}^2}{2 |\\vec{p}_l| E_Q y + e^{8/3} \\mu _\\mathrm {R}^2}\\;,\\\\A_{\\mathrm {T},1} ={}& 2 C_\\mathrm {A}\\left[12 x(1-x) - 24 x^2 (1-x)^2\\right]\\;,\\\\A_{\\mathrm {T},2} ={}& 16 C_\\mathrm {A}x(1-x) \\left[2 - 8 x(1-x) + 8 x^2 (1-x)^2\\right]\\;.\\end{split}$ Here we have renormalized the graphs according to the $\\overline{\\text{MS}}$ prescription, but have subtracted a numerical function that gives the same result as subtracting a pole.",
"The parameter $\\mu _\\mathrm {R}^2$ is the $\\overline{\\text{MS}}$ renormalization scale.",
"We can perform the integrations.",
"The $\\phi $ -integral is trivial.",
"Performing the $x$ -integral gives $S^{\\mathrm {L}}_{l l}(\\lbrace p,f\\rbrace _{m};t;0;{\\rm gluon}) =\\frac{\\alpha _{\\mathrm {s}}}{8\\pi }\\,\\frac{1}{C_\\mathrm {A}}\\sum _{J=0}^2 I^{\\rm g}_J\\;,$ where $\\begin{split}I^{\\rm g}_0 ={}&- (2 C_\\mathrm {A}- N_\\mathrm {f})\\,\\frac{e^2 \\mu _\\mathrm {R}^2}{2 |\\vec{p}_l| E_Q y + e^2 \\mu _\\mathrm {R}^2}+ \\frac{1}{3} C_\\mathrm {A}\\,\\frac{e^{5/3} \\mu _\\mathrm {R}^2}{2 |\\vec{p}_l| E_Q y + e^{5/3} \\mu _\\mathrm {R}^2}\\\\&+ \\frac{1}{3}(2 C_\\mathrm {A}- N_\\mathrm {f})\\,\\frac{e^{8/3} \\mu _\\mathrm {R}^2}{2 |\\vec{p}_l| E_Q y + e^{8/3} \\mu _\\mathrm {R}^2}\\;,\\\\I^{\\rm g}_1 ={}& 2 C_\\mathrm {A}\\left[2 + 6 a_l y-\\frac{6 a_l y (1+ 2 a_l y)}{\\sqrt{1 + 4 a_l y}}\\,\\log \\!\\left(\\frac{(1 + \\sqrt{1 + 4 a_l y}\\,)^2}{4 a_l y}\\right)\\right]\\;,\\\\I^{\\rm g}_2 ={}& 4 C_\\mathrm {A}\\bigg [- \\frac{4(1 + 3 a_l y)(2 + 5 a_l y)}{3 ( 1 + 4 a_l y)}\\\\&\\quad +\\frac{(1 + 2 a_l y) (1 + 8 a_l y + 20 a_l^2 y^2)}{(1 + 4 a_l y)^{3/2}}\\,\\log \\!\\left(\\frac{(1 + \\sqrt{1 + 4 a_l y})^2}{4 a_l y}\\right)\\bigg ]\\;.\\end{split}$ We are interested in the small $y$ limit of this.",
"There is a constant term and a term proportional to $\\log (y)$ : $S^{\\mathrm {L}}_{l l}(\\lbrace p,f\\rbrace _{m};t;{\\rm gluon}) =\\frac{\\alpha _{\\mathrm {s}}}{2\\pi }\\,\\frac{1}{C_\\mathrm {A}}\\left(-\\frac{23 C_\\mathrm {A}}{12} + \\frac{N_\\mathrm {f}}{6}- C_\\mathrm {A}\\log \\left(\\frac{E_Q y}{2 |\\vec{p}_l|} \\right)\\right)\\;.$"
],
[
"Quark self-energy",
"For the quark self energy, we find from ref.",
"[86] $S^{\\mathrm {L}}_{l l}(\\lbrace p,f\\rbrace _{m};t;{\\rm quark}) =\\int _0^1\\!dx \\int _{-\\pi }^\\pi \\!\\frac{d\\phi }{2\\pi }\\ \\frac{\\alpha _{\\mathrm {s}}}{8\\pi }\\,\\frac{1}{C_\\mathrm {F}}\\sum _{J=0}^2\\frac{B_{\\mathrm {L},J}}{[4 a_l y + 4 x (1-x)]^J}\\;.$ The coefficients $B_{\\mathrm {T},J}$ are $\\begin{split}B_{\\mathrm {L},0} ={}&C_\\mathrm {F}\\left[- \\frac{e^3 \\mu _\\mathrm {R}^2}{2 |\\vec{p}_l| E_Q y + e^3 \\mu _\\mathrm {R}^2}+ 12 x(1-x) \\,\\frac{e^{5/3} \\mu _\\mathrm {R}^2}{2 |\\vec{p}_l| E_Q y + e^{5/3} \\mu _\\mathrm {R}^2}\\right]\\;,\\\\B_{\\mathrm {L},1} ={}& 2 C_\\mathrm {F}\\left[20 x(1-x) - 56 x^2 (1-x)^2\\right]\\;,\\\\B_{\\mathrm {L},2} ={}& 32 C_\\mathrm {F}x(1-x) \\left[1 - 6 x(1-x) + 8 x^2 (1-x)^2\\right]\\;.\\end{split}$ Here again $\\mu _\\mathrm {R}^2$ is the $\\overline{\\text{MS}}$ renormalization scale.",
"We can perform the integrations.",
"The $\\phi $ -integral is trivial.",
"Performing the $x$ -integral gives $S^{\\mathrm {L}}_{l l}(\\lbrace p,f\\rbrace _{m};t;{\\rm quark}) =\\frac{\\alpha _{\\mathrm {s}}}{8\\pi }\\,\\frac{1}{C_\\mathrm {F}}\\sum _{J=0}^2 I^{\\rm q}_J\\;,$ where $\\begin{split}I^{\\rm q}_0 ={}&C_\\mathrm {F}\\left[-\\frac{e^3 \\mu _\\mathrm {R}^2}{2 |\\vec{p}_l| E_Q y + e^3 \\mu _\\mathrm {R}^2}+ \\frac{2 e^{5/3} \\mu _\\mathrm {R}^2}{2 |\\vec{p}_l| E_Q y + e^{5/3} \\mu _\\mathrm {R}^2}\\right]\\;,\\\\I^{\\rm q}_1 ={}& C_\\mathrm {F}\\left[\\frac{16}{3} + 28 a_l y-\\frac{4 a_l y (5 + 14 a_l y)}{\\sqrt{1 + 4 a_l y}}\\,\\log \\!\\left(\\frac{(1 + \\sqrt{1 + 4 a_l y})^2}{4 a_l y}\\right)\\right]\\;,\\\\I^{\\rm q}_2 ={}& C_\\mathrm {F}\\bigg [- \\frac{40}{3} (1 + 3 a_l y)+\\frac{4(1 + 10 a_l y + 20 a_l^2 y^2)}{\\sqrt{1 + 4 a_l y}}\\,\\log \\left(\\frac{(1 + \\sqrt{1 + 4 a_l y})^2}{4 a_l y}\\right)\\bigg ]\\;.\\end{split}$ We are interested in the small $y$ limit of this.",
"There is a constant term and a term proportional to $\\log (y)$ : $S^{\\mathrm {L}}_{l l}(\\lbrace p,f\\rbrace _{m};t;{\\rm quark}) =\\frac{\\alpha _{\\mathrm {s}}}{2\\pi }\\,\\left(-\\frac{7}{4}- \\log \\left(\\frac{E_Q y}{2|\\vec{p}_l|} \\right)\\right)\\;.$"
],
[
"Self-energy in general",
"We can combine the results (REF ) and (REF ) using the notation from eqs.",
"(REF ) and (REF ): $S^{\\mathrm {L}}_{l l}(\\lbrace p,f\\rbrace _{m};t) =-\\frac{\\alpha _{\\mathrm {s}}}{2\\pi }\\,\\left(\\frac{\\gamma _{f_l}}{2 C_{f_l}}+\\log \\left(\\frac{E_Q y}{2|\\vec{p}_l|} \\right)+ 1\\right)\\;.$ We will use this result in eqs.",
"(REF ) and (REF )."
]
] | 1605.05845 |
[
[
"Parametric and Probabilistic Model Checking of Confidentiality in Data\n Dispersal Algorithms (Extended Version)"
],
[
"Abstract Recent developments in cloud storage architectures have originated new models of online storage as cooperative storage systems and interconnected clouds.",
"Such distributed environments involve many organizations, thus ensuring confidentiality becomes crucial: only legitimate clients should recover the information they distribute among storage nodes.",
"In this work we present a unified framework for verifying confidentiality of dispersal algorithms against probabilistic models of intruders.",
"Two models of intruders are given, corresponding to different types of attackers: one aiming at intercepting as many slices of information as possible, and the other aiming at attacking the storage providers in the network.",
"Both try to recover the original information, given the intercepted slices.",
"By using probabilistic model checking, we can measure the degree of confidentiality of the system exploring exhaustively all possible behaviors.",
"Our experiments suggest that dispersal algorithms ensure a high degree of confidentiality against the slice intruder, no matter the number of storage providers in the system.",
"On the contrary, they show a low level of confidentiality against the provider intruder in networks with few storage providers (e.g.",
"interconnected cloud storage solutions)."
],
[
"Motivation",
"[FS]check missing \"s\" for 3rd person all over the paper Recently, the spreading of online storage services (such as iCloud, Dropbox, Skydrive, etc.)",
"has seen a huge increase.",
"According to the classical paradigm, the service provider buys or rents a large number of servers in which authorized clients are able to store their data.",
"Recently, two other paradigms have emerged, viz.",
"cooperative storage services (CSS) and federated (storage) clouds (FC).",
"[FS]find citation In CSS, the storage capacity is provided directly by the clients themselves who, co-operating in the cloud, make their own storage facilities available to the others.",
"This approach offers some evident advantages: first of all the service provider only needs a small number of servers, acting as coordinators for the access to the service.",
"Secondly, increasing the number of users yields an increase of the storage capabilities.",
"In this context, data is stored by the users, each acting as a storage server.",
"In FC, organizations decide to agree on sharing their resources for solving common tasks.",
"In this scenario, data is read from (and written to) several storage providers, each managing its set of storage servers behinds its cloud.",
"A fundamental requirement of CSS and FC is confidentiality: only the legitimate client should be able to recover the original information.",
"A consolidated solution to achieve confidentiality in such contexts is based on data dispersal.",
"Dispersal algorithms provide a methodology for storing information in $n$ distinct pieces, or slices, (dispersed) across multiple locations, so that redundancy protects the information in the event of a location outage, but unauthorized access at any single location does not provide usable information.",
"Only the originator or a user which has access to, at least, $k$ out of the $n$ slices distributed among $m$ available servers (or providers) can properly assemble and recover the complete information, without the need of any pre-shared encryption key.",
"Instead, a client or attacker retrieving a number of slices lower than $k$ is not able to get any information.",
"This basic principle has been applied since the pioneer works by Shamir [1] and Rabin [2], and subsequently confirmed by McEliece and Sarwate [3] who disclosed the relationship with Reed-Solomon (RS) coding schemes.",
"Dispersal algorithms based on RS schemes have optimal performances, but they are subject to constraints due to the algebraic nature of the codes which practically limit the number of servers $m$ , i.e., the level of dispersion.",
"Since a large number of servers is convenient both from the client point of view (which can tolerate a greater number of failures) and from the security point of view (since the attacker needs to crack a larger number of nodes to steal the data), new solutions based on the Luby transform (LT) codes for the dispersal algorithm have been proposed [4].",
"These codes have no limit, in principle, on the value of $n$ .",
"On the other side, a characteristic feature of LT is that there are two thresholds $k_1$ and $k_2$ such that if an attacker has access to at least $k_1$ slices it has some probability of reconstructing the entire message, while if it has access to at least $k_2$ slices it has all the needed information to reconstruct the message.",
"For this reason, one can see the RS coding schemes as special cases of LT.[FS]check Both coding schemes, RS and LT, can be combined with an all-or-nothing-transform (AONT) [5] realising AONT-RS and AONT-LT [6] data dispersal algorithms.",
"[FS]check In this work we provide a unified formal framework to model check the mentioned dispersal algorithms against different kinds of attackers trying to intercept slices and reconstruct the original message.",
"We use the tool PRISM [7] to verify the degree of confidentiality of such algorithms.",
"Since the problem is intrinsically parametric, we also want to identify suitable conditions under which the verification outcome holds for any number of storage providers in the cloud.",
"To this aim, we repeatedly measured the probability of a confidentiality attack to understand how it varies w.r.t.",
"other parameters, and in particular when the number of slices $n$ increases.",
"Two different types of intruder are analyzed: the first one can eavesdrop the slices traveling to a server without interfering with the communication.",
"The second type of intruder can violate some providers and retrieve all the slices they store.",
"Both intruders are assumed to be passive and probabilistic, meaning that they can only read the exchanged information, and probabilities affect their capability of taking any action.",
"The organization of the paper is as follows: Section defines the modeling languages and the models used for our analysis; Section shows the formal analysis allowing us to measure the probability of an attack; Section comments the experimental data; Section compares our work with the existing literature, while Section summarizes our results."
],
[
"Modeling",
"Here, we formally describe a client process and two types of intruder.",
"The main client responsibility is sending a sequence of slices to several distinct storage servers, following the specific dispersal algorithm.",
"The set of slices constitutes the message.",
"A message includes an actual content, viz.",
"message body, and some extra information, viz.",
"message payload, containing, among other things, the replicated information allowing to reconstruct the message body even in presence of faulty storage nodes.",
"The main responsibility for both types of intruder is to intercept the traveling slices, and reconstruct the message body.",
"The key difference between them is that the first type intercepts every slice independently from the previous ones.",
"The second type of intruder, on the other side, attacks the storage providers and collect all the stored slices.",
"The actions of the intruders are probabilistic.",
"The following parameters affect the system behavior: $n$ is the number of slices that compose the message, $m$ is the number of storage providers (or servers) in the system, $c$ is the number of slices every server/provider can store, also called capacity, $k_1$ and $k_2$ are two thresholds such that if the attacker intercepts at least $k_1$ slices it has some probability of reconstructing the entire message body, while if it intercepts at least $k_2$ slices it has all the necessary information to reconstruct the message body.",
"Two series of probabilities are used: $a_i$ is the probability of intercepting a slice traveling to storage provider $i$ , for the first attacker, and it is the probability of attacking the storage provider $i$ , for the second attacker; $x_j$ is the probability of reconstructing the entire message body, given $j$ slices have been intercepted by the attacker.",
"The relation among parameters are the following: $k_1 \\le k_2 \\le n$ and $n \\le m \\cdot c$ .",
"Probabilities $a_i$ are defined for $i \\in [1,m]$ , while probabilities $x_j$ are defined for $j \\in [k_1,n]$ and is such that: $\\forall j \\in [k_1,k_2-1].\\ 0 < x_j < 1$ and $\\forall j \\in [k_2,n].\\ x_j = 1$ .",
"Also: $\\forall j \\in [k_1,n].\\ x_j \\le x_{j+1}$ ."
],
[
"The system",
"Markov Decision Processes, or MDP for short, is a formalism allowing the definition of systems with probabilistic and non-deterministic actions.",
"They are thus recognized as a good means to model randomized distributed systems: each process is described by its probabilistic transition function, and processes are interleaved by a non-deterministic scheduler [FS]here we take care of observation by Reviewer 3.",
"We briefly introduce MDPs using Baier and Katoen's notation [8].",
"Definition 1 (Markov Decision Process) Assume a finite set of atomic propositions $\\textit {AP}$ .",
"A Markov decision process is a tuple $\\mathcal {M}= (S, \\textit {Act}, {Pr}, \\iota , L)$ where: $S= \\lbrace s_1, s_2, \\ldots \\rbrace $ is a finite set of states, $\\textit {Act}= \\lbrace \\alpha _1, \\alpha _2, \\ldots \\rbrace $ is a finite set of actions, ${Pr}: S\\times \\textit {Act}\\times S\\rightarrow [0,1]$ is a probabilistic transition function such that $\\sum _{s^{\\prime } \\in S} {Pr}(s,\\alpha ,s^{\\prime }) \\in \\lbrace 0,1 \\rbrace $ , for all $s \\in S$ and $\\alpha \\in \\textit {Act}$ ; $\\iota : S\\rightarrow [0,1]$ is the initial distribution probability of states, such that $\\sum _{s \\in S} \\iota (s) = 1$ ; $L: S \\rightarrow 2^\\textit {AP}$ is a labeling function.",
"We write $(s,\\alpha ,p,s^{\\prime }) \\in {Pr}$ whenever ${Pr}(s,\\alpha ,s^{\\prime }) = p$ , for some suitable states $s,s^{\\prime }$ , action $\\alpha $ and probability $p > 0$ .",
"We call transition any such tuple.",
"If the MDP is in state $s$ , an action $\\alpha $ is enabled if $(s,\\alpha ,p,s^{\\prime }) \\in {Pr}$ , for some state $s^{\\prime }$ and probability $p$ .",
"If an action $\\alpha $ is enabled, than the probabilities among $\\alpha $ -transitions must form a probability distribution: $\\sum _{s^{\\prime } \\in S} {Pr}(s,\\alpha ,s^{\\prime }) = 1$ .",
"More than one action can be enabled in the same state $s$ , thus the sum of probabilities of all transitions leaving state $s$ sum up to the number of enabled actions.",
"Let us remark that while probabilities in MDP could be real values, for algorithmic purposes in this work we constrain them to be rational values.",
"In the following we make use of the uniform probability distribution $\\epsilon _m : [1,m] \\rightarrow [0,1]$ having the property: $\\forall i \\in [1,m].\\ \\epsilon _m(i) = \\frac{1}{m}$ .",
"We write states of MDPs as configurations of some given set of variables $V$ .",
"Given a state $s$ and a variable $v \\in V$ , we write $s.v$ to denote the value of the variable in that state.",
"Given two states $s$ and $s^{\\prime }$ and a set of variables $V$ , we write $s \\equiv _V s^{\\prime }$ meaning that the values of variables in $V$ are the same in both states: $\\forall v \\in V.\\ s.v = s^{\\prime }.v$ .",
"Given a labeling $L : S \\rightarrow 2^\\textit {AP}$ , we say that $L$ is invariant w.r.t.",
"$V$ iff $\\forall s,s^{\\prime } \\in S.\\ s \\equiv _V s^{\\prime } \\Rightarrow L(s) = L(s^{\\prime })$ .",
"Given MDPs $\\mathcal {M}_1$ and $\\mathcal {M}_2$ , we will denote with $\\mathcal {M}_1 \\parallel \\mathcal {M}_2$ the MDP resulting from their synchronous composition.",
"Let us list the graphical conventions used in this work to depict MDPs (see Fig.",
"REF and following): the circles represent the values of variables $\\textsf {pc}_c$ or $\\textsf {pc}_a$ ; transitions have labels of the form: $\\lbrace p \\rbrace [ \\alpha ] \\gamma $ where $p \\in [0,1]$ is the probability, $\\alpha $ is the action, and $\\gamma $ is a boolean formula mentioning the variables of the state.",
"For the sake of brevity, we use notation $\\exists i \\in [a,b].",
"\\lbrace p(i) \\rbrace [\\alpha ] \\gamma (i)$ as transition label, to denote a group of $(b-a+1)$ similar transitions, each obtained by replacing $i$ with one of the natural values in the interval $[a,b]$ .",
"Following the PRISM notation, the boolean formula on a transition can refer to variables in the source state by their name (e.g.",
"$\\textsf {ctr}_c$ ), and to variables in the target state by their primed name (e.g.",
"$\\textsf {ctr}^{\\prime }_c$ )."
],
[
"The client",
"[FS]fix bugs in the figure Let $\\textsc {Client}$ be the MDP encoding the client process described earlier (see Fig.",
"REF ).",
"It has the following local variables: $\\textsf {pc}_c$ : track the progress of the process, $\\textsf {s}_c$ : the identifier of the next recipient server/provider, $\\textsf {ctr}_c$ : the total number of sent slices, $\\textsf {ctr}_c^i$ : the number of slices sent to server/provider $i$ .",
"The MDP has a first block of $m$ transitions from $\\textsf {pc}_c = 0$ to $\\textsf {pc}_c = 1$ picking a storage server (or provider) to store the slice; each transition is subject to some probability $p_i$ .",
"Next, the client either sends the slice to the selected recipient, if the latter has not reached its capacity, or it tries again picking another one.",
"A total slice counter and a server/provider slice counter are increased whenever the slice is sent.",
"The loop terminates when all the slices are sent.",
"Let us remark that the sending transition is labeled with a special action $busy$ .",
"This is used when building the synchronous composition of $\\textsc {Client}$ with the MDP modeling the intruder, to synchronize the action of sending by the client and the action of intercepting by the intruder."
],
[
"The slice attacker",
"Let us name $\\textsc {SliceAtt}$ the MDP encoding the first type of intruder.",
"The reason for its name is that it tries to intercept every slice, independently from the previously intercepted ones.",
"The intruder is given in Fig.",
"REF .",
"It has two local variables, viz.",
"$\\textsf {pc}_a$ and $\\textsf {ctr}_a$ .",
"The former tracks the progress of the attack, while the latter counts the number of intercepted slices at any given moment.",
"The figure shows that the attack progresses linearly: it starts by intercepting the first $k_1$ slices, each with probability $a_i$ given that the slice is sent to server/provider $i$ .",
"Having intercepted less than $k_1$ slices, there is no possibility to reconstruct the message body.",
"After intercepting $k_1$ slices the next chain of states repeatedly alternate these steps: first it tries to reconstruct the message body with probability $x_j$ , given $j = \\textsf {ctr}_a$ ; if it fails it tries to intercept a new slice.",
"The state $\\textsf {pc}_a = \\textsf {done}$ denotes that the intruder reconstructed the message body."
],
[
"The provider attacker",
"Let $\\textsc {ProviderAtt}$ be the MDP encoding the second type of intruder.",
"The reason for its name is that it tries to obtain the credentials of the storage provider, and later it will intercept all the slices traveling towards that provider.",
"The intruder is depicted in Fig.",
"REF .",
"It has two local variables, viz.",
"$\\textsf {pc}_a$ and $\\textsf {ctr}_a$ .",
"Similarly to the previous intruder, the former variable models the progress of the attack, while the latter counts the number of intercepted slices.",
"The intruder has an initial chain of $m$ states where it tries to attack every provider tossing a coin with probability $a_i$ ; if the attack is successful it sets a flag $\\textsf {att}_a^i$ for provider $i$ .",
"State $\\textsf {pc}_a = m$ is reached when all attack attempts are decided (some succeeded and some failed).",
"In a loop the attacker synchronizes with the $busy$ action from the client that is sending a message, to intercept every slice sent to an attacked provider.",
"From state $(\\textsf {pc}_a = m, \\textsf {ctr}_a = j)$ there is a transition to some state with $\\textsf {pc} = \\textsf {done}$ labeled with probability $x_j$ , meaning that it has probability $x_j$ to reconstruct the content of the message, given $j$ intercepted slices.",
"Figure: The MDP ProviderAtt\\textsc {ProviderAtt}"
],
[
"Parametric formal verification",
"In Section we have seen that the problem at our hands is intrinsically parametric.",
"The model checking problem requires its input MDP to be finite, thus we must fix the system parameters.",
"On the other hand, this means that the outcome of our formal verification holds only for the specific configuration of the parameters themselves.",
"One of the common desiderata when doing parametric formal verification, is to prove universal properties, i.e.",
"we should check whether some property holds for any configuration of parameters.",
"In this work we are able to measure the confidentiality of dispersal algorithms for any number of storage providers in presence of a slice attacker, while in the case of the provider attacker the degree of confidentiality depends on the actual number of storage providers in the network.",
"Before showing the detailed formal analysis, we report the needed formal ingredients."
],
[
"Preliminaries",
"$\\textit {PCTL}^\\star $ is a temporal logic for describing qualitative and quantitative aspects of probabilistic systems.",
"The grammar of $\\textit {PCTL}^\\star $ formulae is the following: $\\begin{array}{ccl}\\Phi &::=& \\textit {true}\\ |\\ p \\ |\\ \\Phi \\wedge \\Phi \\ |\\ \\lnot \\Phi \\ |\\ \\mathbb {P}_J(\\varphi ) \\\\ \\varphi &::=& \\Phi \\ |\\ \\varphi \\wedge \\varphi \\ |\\ \\lnot \\varphi \\ |\\ X\\varphi \\ |\\ G\\varphi \\ |\\ F\\varphi \\\\\\end{array}$ where $p \\in \\textit {AP}$ and $J \\subseteq [0,1]$ is a rational interval.",
"Terms of $\\Phi $ are state formulae, while terms of $\\varphi $ are path formulae.",
"A thorough description of the logic satisfiability relation is beyond the aims of this paper, since the subject is covered by several textbooks (e.g.",
"see [8]).",
"Intuitively, formula $G\\varphi $ holds w.r.t.",
"some path iff every state visited in the path satisfies the sub-formula $\\varphi $ .",
"Formula $F\\varphi $ , instead, holds w.r.t.",
"some path iff some visited state satisfies sub-formula $\\varphi $ .",
"The state formula $\\mathbb {P}_{[a,b]}(\\varphi )$ holds w.r.t.",
"state $s$ iff the sub-formula $\\varphi $ holds in all paths starting from $s$ with some probability $p \\in [a,b]$ .",
"Given an MDP $\\mathcal {M}$ , let us write $\\mathcal {M}\\models \\Phi $ to express that all the initial states of $\\mathcal {M}$ satisfy the property $\\Phi $ .",
"Given a $\\textit {PCTL}^\\star $ path formula $\\varphi $ and an MDP $\\mathcal {M}$ , there exist polynomial time algorithms computing the minimum and maximum probabilities of $\\varphi $ w.r.t.",
"all the initial states of $\\mathcal {M}$ [8].",
"In the following we will write $\\mathcal {P}_{min}(\\varphi ,\\mathcal {M})$ and $\\mathcal {P}_{max}(\\varphi ,\\mathcal {M})$ to denote such computed probabilities.",
"Note that, in general, given any $\\textit {PCTL}^\\star $ formula $\\varphi $ and MDP $\\mathcal {M}$ , it is possible that $\\mathcal {P}_{min}(\\varphi , \\mathcal {M}) \\ne \\mathcal {P}_{max}(\\varphi , \\mathcal {M})$ .",
"This is a consequence of the sequence of non-deterministic choices that can be taken in the executions of $\\mathcal {M}$ , each leading to a (possibly) different probability outcome associated to $\\varphi $ .",
"This motivates the interest in discovering the minimum and maximum probabilities with which $\\varphi $ holds in $\\mathcal {M}$ .",
"From the definitions of $\\mathcal {M}\\models \\Phi $ , $\\mathcal {P}_{min}$ and $\\mathcal {P}_{max}$ , the following fact holds immediately.",
"Proposition 1 ([8]) Given any MDP $\\mathcal {M}$ , any $\\textit {PCTL}^\\star $ path formula $\\varphi $ , and any $0 \\le a \\le b \\le 1$ , then: $\\mathcal {M}\\models \\mathbb {P}_{[a,b]} \\varphi \\iff a \\le \\mathcal {P}_{min}(\\varphi ,\\mathcal {M}) \\wedge b \\ge \\mathcal {P}_{max}(\\varphi ,\\mathcal {M})$ Example 1 Assume two MDPs $\\mathcal {M}_1 := \\textsc {Client}\\parallel \\textsc {SliceAtt}$ and $\\mathcal {M}_2 := \\textsc {Client}\\parallel \\textsc {ProviderAtt}$ .",
"Assume a proposition $\\textsc {hacked} \\in \\textit {AP}$ and labeling $L_1$ of $\\mathcal {M}_1$ (resp.",
"$L_2$ of $\\mathcal {M}_2$ ) such that $\\textsc {hacked} \\in L_1(s)$ (resp.",
"$\\textsc {hacked} \\in L_2(s)$ ) iff $s.\\textsf {pc}_a = \\textsf {done}$ .",
"We can measure the likelihood of breaking the confidentiality requirement of $\\mathcal {M}_1$ (resp.",
"$\\mathcal {M}_2$ ) computing $\\mathcal {P}_{min}(F(\\textsc {hacked}), \\mathcal {M}_1)$ and $\\mathcal {P}_{max}(F(\\textsc {hacked}), \\mathcal {M}_1)$ (resp.",
"$\\mathcal {P}_{min}(F(\\textsc {hacked}), \\mathcal {M}_2)$ and $\\mathcal {P}_{max}(F(\\textsc {hacked}), \\mathcal {M}_2)$ ).",
"Due to Proposition REF , the probabilistic model checking problem may assume two different flavours: qualitative: take as input an MDP $\\mathcal {M}$ and a formula $\\Phi $ , and return $\\textit {true}$ iff $\\mathcal {M}\\models \\Phi $ ; quantitative: take as input an MDP $\\mathcal {M}$ and a path formula $\\varphi $ , and compute $\\mathcal {P}_{min}$ and $\\mathcal {P}_{max}$ .",
"Here we use the quantitative probabilistic model checking.",
"Given an MDP $\\mathcal {M}$ , one can show that two states $s$ and $s^{\\prime }$ are indistinguishable, from a probabilistic point of view, if (i) every step taken from $s$ is mimicked by some step taken from $s^{\\prime }$ , (ii) both steps end in equivalent states, and (iii) the viceversa is also true.",
"This is captured by the notion of probabilistic bisimulation.",
"Definition 2 (Probabilistic Bisimulation, [8]) Given an MDP $(S, \\textit {Act}, {Pr}, \\iota , L)$ , a probabilistic bisimulation is an equivalence relation $R \\subseteq S \\times S$ such that, for any $s,s^{\\prime } \\in S$ , $R(s,s^{\\prime })$ iff: $L(s) = L(s^{\\prime })$ , and ${Pr}(s,\\alpha ,X) = {Pr}(s^{\\prime },\\alpha ,X)$ , $\\forall \\alpha \\in \\textit {Act}, X \\in {S}{R}$ where ${S}{R}$ represents the quotient set of $S$ by $R$ .",
"Given two states $s,t$ , let us write $s \\approx _R t$ if $R(s,t)$ for some probabilistic bisimulation $R$ .",
"When $R$ is clear from the context, we may omit it.",
"The given definition of probabilistic bisimulation helps establishing that two states of the same MDP cannot be distinguished.",
"It is possible to use probabilistic bisimulation to check whether different MDPs are indistinguishable.",
"Given two MDPs, $\\mathcal {M}_1 = (S_{1}, \\textit {Act}_{1}, {Pr}_{1}, \\iota _{1}, L_{1})$ and $\\mathcal {M}_2 = (S_{2}, \\textit {Act}_{2}, {Pr}_{2}, \\iota _{2}, L_{2})$ , let $S := S_1 \\uplus S_2$ be the disjoint union of the state sets.",
"Let us write $\\mathcal {M}_1 \\approx _R \\mathcal {M}_2$ iff $\\textit {AP}_1 = \\textit {AP}_2$ and there exists a bisimulation relation $R \\subseteq S \\times S$ and $\\iota _1(X) = \\iota _2(X)$ for each $X \\in {S}{R}$ .",
"It is known that bisimilar MDPs satisfy the same $\\textit {PCTL}^\\star $ formulae.",
"Theorem 1 ([8]) Given two MDPs $\\mathcal {M}_1$ and $\\mathcal {M}_2$ such that $\\mathcal {M}_1 \\approx \\mathcal {M}_2$ , then $\\mathcal {M}_1 \\models \\Phi $ iff $\\mathcal {M}_2 \\models \\Phi $ , for any $\\Phi \\in \\textit {PCTL}^\\star $ .",
"Proposition REF and Theorem REF yield the following.",
"corollaryBisimMinMax Given two MDPs $\\mathcal {M}_1$ ,$\\mathcal {M}_2$ such that $\\mathcal {M}_1 \\approx \\mathcal {M}_2$ and any $\\textit {PCTL}^\\star $ path formula $\\varphi $ , then $\\mathcal {P}_{min}(\\varphi , \\mathcal {M}_1) = \\mathcal {P}_{min}(\\varphi , \\mathcal {M}_2)$ and $\\mathcal {P}_{max}(\\varphi , \\mathcal {M}_1) = \\mathcal {P}_{max}(\\varphi , \\mathcal {M}_2)$ ."
],
[
"Parameter abstraction for the slice attacker",
"Let us consider the MDP $\\mathcal {M}^{\\textit {slice}}= \\textsc {Client}\\parallel \\textsc {SliceAtt}$ .",
"In order to generalize our verification for any number of storage servers, we group the latter in channels, i.e.",
"collections of servers that are indistinguishable.",
"Formally we define a channel as a triple $(x, g, a)$ where $x \\in \\mathbb {N}_{> 0}$ is the number of servers in the channel, $g : [1,x] \\rightarrow [0,1]$ is a probability distribution and $a \\in [0,1]$ is a probability value.",
"Intuitively, $x$ is the number of storage servers belonging to the channel, $g$ is the probability distribution of picking any server in the channel when sending a slice, conditioned by the fact that the current channel has been chosen, and $a$ is the probability of attacking any server belonging to the channel.",
"It is easy to see that if a channel has size one, it can only be defined as follows: $(1, \\epsilon _1, a)$ , for some $a \\in [0,1]$ .",
"In this analysis we assume that any storage server can host any number of slices.",
"We also fix a subset of the model variables: $V = \\lbrace \\textsf {pc}_c, \\textsf {ctr}_c, \\textsf {pc}_a, \\textsf {ctr}_a \\rbrace $ .",
"Given an Markov Decision Process $\\mathcal {M}$ , let us write $\\mathcal {M}_f((x_1,g_1,a_1),\\ldots ,(x_k,g_k,a_k))$ denoting a copy of it where storage servers are grouped in the given $k$ channels.",
"Our claim is that if we check formulae that look only at variables in $V$ (and in particular that do not look at variables $\\textsf {s}_c$ and $\\textsf {ctr}_c^i$ , for $i \\in [1,m]$ ), then a system with one server per channel is indistinguishable from a system with an arbitrary number of servers per channel.",
"This means that the number of channels in the system defines an upper limit, or cutoff, to the size of the model to be verified.",
"theoremChannelCutoff Fix a positive number $k$ and any probability distribution $f : [0,k] \\rightarrow [0,1]$ .",
"For all $n_1,\\ldots ,n_k \\in \\mathbb {N}_{> 0}$ , distribution probabilities $g_j : [0,n_j] \\rightarrow [0,1]$ ($j \\in [1,k]$ ), probability values $a_1,\\ldots ,a_k \\in [0,1]$ , let $\\mathcal {M}_1 = \\mathcal {M}^{\\textit {slice}}_f((1,\\epsilon _1,a_1),\\ldots ,(1,\\epsilon _1,a_k))$ and $\\mathcal {M}_2 = \\mathcal {M}^{\\textit {slice}}_f((n_1,g_1,a_1),\\ldots ,(n_k,g_k,a_k))$ .",
"Let $L_1$ and $L_2$ be the respective labeling functions, and assume they are invariant with respect to $V$ .",
"Then: $\\mathcal {M}_1 \\approx \\mathcal {M}_2$ .",
"For the sake of readability, the detailed proofs can be found in the appendix of this paper.",
"Next corollary follows from Theorem REF and Theorem REF .",
"Corollary 1 Given $\\mathcal {M}_1$ and $\\mathcal {M}_2$ as of Theorem REF , then for all $\\Phi \\in \\textit {PCTL}^\\star $ : $\\mathcal {M}_1 \\models \\Phi \\iff \\mathcal {M}_2 \\models \\Phi $"
],
[
"Variable abstraction for the provider attacker",
"Here we focus on the scenario in which the intruder can break one or more providers, thus accessing all the slices they store.",
"Two applicative examples of this scenario are the cooperative storage systems, where each node joining the network can receive some of the slices, and the interconnected cloud, where a client may decide to split the message among several cloud storage providers.",
"In the latter case we ignore the fact the cloud storage provider may further distribute the received slices among its own servers, and treat each such provider as a single server.",
"The model checking problem: $\\mathcal {P}_{min}(\\varphi , \\textsc {Client}\\parallel \\textsc {ProviderAtt})$ , where $\\varphi $ encodes our confidentiality requirements, remains a problem parameterized by the actual numbers of storage providers (similarly for $\\mathcal {P}_{max}$ ).",
"On the other side, one can easily see that if every provider can host any number of slices (i.e.",
"$c \\ge n$ ) we can produce an indistinguishable model that is much smaller w.r.t.",
"the original one, by simply dropping the variables $W = \\lbrace \\textsf {ctr}_c^i : i \\in [1,m] \\rbrace $ from the local state of $\\textsc {Client}$ .",
"This reduces the problem to a feasible one, allowing us to experimentally measure the degree of confidentiality of the considered dispersal algorithm w.r.t.",
"the number of slices $n$ and the number of providers $m$ .",
"Call $V$ the set of remaining variables (i.e.",
"$V \\cap W = \\emptyset $ ).",
"Name $\\textsc {Client}^{\\prime }$ a copy of $\\textsc {Client}$ whose state does not contain variables in $W$ .",
"Intuitively, this means that $\\textsc {Client}^{\\prime }$ does not check whether a provider reached its capacity, but this is not a limitation since we assumed that every storage provider can host any number of slices.",
"theoremServerAbstraction Assume MDPs $\\mathcal {M}_1 = \\textsc {Client}\\parallel \\textsc {ProviderAtt}$ and $\\mathcal {M}_2 = \\textsc {Client}^{\\prime } \\parallel \\textsc {ProviderAtt}$ .",
"Let $L_1$ and $L_2$ be the respective labeling functions, and assume they are invariant with respect to $V$ .",
"Then: $\\mathcal {M}_1 \\approx \\mathcal {M}_2$ .",
"Corollary 2 Given $\\mathcal {M}_1$ and $\\mathcal {M}_2$ as of Theorem REF , then for all $\\Phi \\in \\textit {PCTL}^\\star $ : $\\mathcal {M}_1 \\models \\Phi \\iff \\mathcal {M}_2 \\models \\Phi $"
],
[
"Experiments",
"Here we show how to use model checking for measuring the likelihood of breaking the confidentiality requirement against systems of growing sizes.",
"In particular the parameter $n$ is increased among runs.",
"We remark that any cloud system is characterized by its own sets of parameters (e.g.",
"the probabilities of attacking the used providers may be part of their SLA).",
"In our experiments we choose parameters arbitrarily, mainly for showing the feasibility of the approach and underlining the weakness of data dispersal algorithms in some cloud environment , viz.",
"interconnected clouds.",
"[FS]This paragraph takes care of observation from Reviewer 1 The checked specifications are taken from Example REF .",
"Thanks to the parametric abstractions explained in Section , the verification outcomes using the intruder $\\textsc {SliceAtt}$ hold no matter the number of storage servers/providers in the system.",
"In the case of intruder $\\textsc {ProviderAtt}$ the results depend on the number of storage providers in the network.",
"The experiments were run on a machine Xeon Quad Core 2.3 Ghz with 8 GB RAM and Linux 2.6.32 64 bit.",
"The points in the graphs correspond to distinct instances of the model checking problem, all requiring from few seconds to 60 minutes to complete.",
"Figure: Compare AONT-RS vs. AONT-LT using ProviderAtt\\textsc {ProviderAtt}In the first set of experiments we compare two different types of dispersal algorithms, those based on RS transforms against those based on LT transforms.",
"For this analysis we fixed the number of channels/providers to 3 and the attacking probabilities to $a_1 = 0.1$ , $a_2 = 0.2$ , and $a_3 = 0.3$ .",
"In the RS case we assumed $k_1 = k_2 = 0.7 \\cdot n$ , while in the LT case we assumed $k_1 = 0.6 \\cdot n$ and $k_2 = 0.8 \\cdot n$ .",
"In the case of the LT transforms we assumed that the sequence of probabilities $x_j$ , for $j \\in [k_1,n]$ is defined as follows: let $\\delta = \\frac{1}{k_2 - k_1 + 1}$ , $x_j = (\\min (j,k_2) - k_1 + 1) \\cdot \\delta $ .",
"Intuitively, probabilities $x_j$ grow linearly in the interval $[k_1,k_2-1]$ and then stabilize at 1 for values greater than or equal to $k_2$ .",
"In Fig.",
"REF and REF we depict how the probability of breaking the confidentiality requirement varies w.r.t.",
"$n$ .",
"Let us remark that $\\mathcal {P}_{max}$ and $\\mathcal {P}_{min}$ versions of the formula coincide in every point of the series.",
"We also notice that the algorithm (under the given parameters) shows a high degree of confidentiality against $\\textsc {SliceAtt}$ , while it is sensibly less confidential against $\\textsc {ProviderAtt}$ .",
"Moreover, the confidentiality in the latter case, after some initial change, stabilizes and does not depend on the actual number of exchanged slices.",
"The latter observation is not surprising since the intruder mainly attacks providers, and few providers will receive many slices, thus giving the intruder a high probability of guessing the message body.",
"Figure: Compare provider probability of attack using ProviderAtt\\textsc {ProviderAtt}The second block of experimental data compares the effect of different attack probabilities, viz.",
"$a_1 = 0.1$ , $a_2 = 0.2$ , and $a_3 = 0.3$ (see Low) vs. $a_1 = 0.3$ , $a_2 = 0.4$ , and $a_3 = 0.5$ (see High).",
"[FS]check names in figures are consistent For this analysis we used only LT transforms and we fixed the number of channels/providers to 3.",
"As before, $k_1 = 0.6 \\cdot n$ and $k_2 = 0.8 \\cdot n$ and the series $x_j$ grows linearly as for the previous experiment.",
"Fig.",
"REF and REF summarize the model checking outcomes.",
"As would be expected, higher probabilities of intercepting slices give the data dispersal algorithm a very low level of confidentiality against intruder $\\textsc {SliceAtt}$ .",
"Again, the low number of providers causes a low degree of confidentiality against intruder $\\textsc {ProviderAtt}$ .",
"Figure: Compare different number of providers using ProviderAtt\\textsc {ProviderAtt}Finally, a third set of experiments compares the effects of different numbers of channels/providers in the system and the results are given in Fig.",
"REF and REF .",
"In both cases a LT transform was used, with $k_1 = 0.6 \\cdot n$ and $k_2 = 0.8 \\cdot n$ .",
"In one case we assumed 5 channels/providers in the network (see m5) and in the other 10 channels/providers (see m10).",
"The probabilities $a_i$ , for $i \\in [1,m]$ , are distributed uniformly in the interval $[0,0.25]$ in both cases.",
"$x_j$ is defined as for the previous experiment.",
"Once again the degree of confidentiality against $\\textsc {ProviderAtt}$ is considerably lower than that against $\\textsc {SliceAtt}$ .",
"We underline that the experiment with 10 providers could be verified only for a small number of slices ($n \\le 150$ ), before running out of memory.",
"Even though this fact represents a scalability issue of the presented methodology, the verification outcomes are still of practical interest since interconnected (or federated) cloud solutions usually employ a limited number of storage providers.",
"Confidentiality of data dispersal with a small number of storage nodes appears to be too weak against attacks directed to the storage provider."
],
[
"Related work",
"Formal verification of security requirements has a long history.",
"In this area, model checking plays a predominant role [9], [10], [11], [12].",
"The traditional approaches consist in model checking the security requirements of a system opposed to an adversary able to intercept, remove, modify the original messages as well as to inject new messages.",
"In this respect, the Dolev-Yao intruder model [13] is considered the most general model (the worst case) [14] as it assumes a non-deterministic attacker in full control of the communication channels.",
"Traditional model checking, though, is not suitable for verifying security of cloud systems: it can only verify whether a system can be attacked or not.",
"We assume, instead, that every component of a cloud system can be attacked with some degree of probability, and are more interested in measuring the likelihood of such attacks.",
"This motivated us to define custom probabilistic intruder models, in place of the Dolev-Yao intruder.",
"To the best of our knowledge, few authors used probabilistic model checking for measuring security of systems [15], [16], [17], [18].",
"[FS]this compares our work with traditional model checking, as required by Reviewer 3 It is well known that model checking techniques must face the state-explosion problem, that easily makes the verification of real-world protocols and systems unfeasible.",
"To overcome this limitation, one looks for abstraction techniques [19] that reduce the description of the system to a feasible state-space, still preserving the relevant properties.",
"Special forms of abstractions are required when the system state-space depends on given parameters and one wants to check whether some property holds for all values of such parameters [20], [21].",
"With regards to probabilistic models, several approaches use abstraction techniques.",
"Legay et al.",
"[22] and Nouri et al.",
"[23], for example, collect traces of real or simulated systems.",
"Next they sample them in order to build an MDP that abstracts the original system.",
"Such technique avoids to build a complete analysis of all the traces for large-scale (or even infinite) systems.",
"The larger is the sample, on the other side, the higher is the probability that the verification outcomes are correct.",
"Herd et al.",
"[24], instead, proposed a trace sampling technique combined with trace fragmentation, i.e.",
"only few fragments of a trace are considered.",
"Abate et al.",
"[25] proposed a method for transforming an MDP with an uncountable number of states into a Markov chain by means of a quotient-set based abstraction.",
"The paper proves that the produced Markov chain approximates a probabilistic bisimulation of the original MDP.",
"[FS]check Finally, let us remark that also the structure of the attacker may determine the feasibility of the verification of security properties.",
"In our work we employed a passive intruder model, and indeed several authors agree that this is enough when analyzing confidentiality requirements.",
"For example, Li and Pang [26], and Shmatikov [15] used passive intruders to verify anonymity of protocols, a special case of confidentiality.",
"The latter work also considers probabilistic attacks.",
"As far as we know, the use of a probabilistic passive attacker model for the analysis of data confidentiality is original."
],
[
"Conclusions",
"We presented a unified framework for the probabilistic model checking of a broad class of data dispersal algorithms in interconnected or cooperative cloud storage systems.",
"We verified confidentiality requirements of dispersal algorithms, checking the likelihood that an intruder has of intercepting slices of information and reconstruct the information.",
"In our framework we defined two types of probabilistic intruder, one tries to intercept each slice independently and the other attempts to attack the storage provider.",
"In the former case the attack surface is the set of slices while in the latter it is the set of providers.",
"The problem is inherently parametric, since the CPU time and memory required to complete the verification are highly affected by several parameters, e.g.",
"the number of slices used to split the information and the number of servers/providers.",
"By proving a probabilistic bisimulation property, we were able to generalize the results of the verification of confidentiality against the slice attacker to any number of servers in the network.",
"The key observation, codified in our channel abstraction, is that any group of servers sharing the same probability of being eavesdropped may form a channel and behave like a single server hosting all the slices.",
"The analysis of the confidentiality against the provider attacker suggests, on the contrary, that classic data dispersal algorithms may not be the best solution to ensure confidentiality in interconnected cloud environments, unless the number of storage providers is considerably high.",
"We should remark that the conducted experiments fix some parameters to specific values.",
"The conclusions thus are not fully generalizable w.r.t.",
"such parameters.",
"Among the modifiable parameters we remark that only $k_1$ and $k_2$ affect the state space and thus the complexity of the model checking problem.",
"We leave as future research the investigation of better abstractions suitable for the verification of the confidentiality of data dispersal algorithms against $\\textsc {ProviderAtt}$ on networks with many storage providers.",
"That would improve the scalability of our framework to handle the case of cooperative storage systems."
],
[
"Proofs of Theorems",
"* Assume MDPs $\\mathcal {M}_1$ and $\\mathcal {M}_2$ such that $\\mathcal {M}_1 \\approx \\mathcal {M}_2$ , and any path formula $\\varphi \\in \\textit {PCTL}^\\star $ .",
"Let us name $a = \\mathcal {P}_{min}(\\varphi , \\mathcal {M}_1)$ and $a^{\\prime } = \\mathcal {P}_{min}(\\varphi , \\mathcal {M}_2)$ .",
"Assume, by contradiction and w.l.o.g., that $a > a^{\\prime } \\ge 0$ .",
"It is immediate that some $a \\le b \\le 1$ must exist such that: $\\mathcal {M}_1 \\models \\mathbb {P}_{[a,b]} (\\varphi )$ , thus $\\mathcal {M}_2 \\models \\mathbb {P}_{[a,b]}(\\varphi )$ (by Theorem REF ).",
"As we observed earlier (see Proposition REF ), by definition of the model checking problem $\\textit {PCTL}^\\star $ , $\\mathcal {P}_{min}$ and $\\mathcal {P}_{max}$ (see e.g.",
"[8]), the following holds: $\\mathcal {M}\\models \\mathbb {P}_{[x,y]}(\\varphi )$ iff $x \\le \\mathcal {P}_{min}(\\varphi , \\mathcal {M})$ and $y \\ge \\mathcal {P}_{max}(\\varphi , \\mathcal {M})$ , for any MDP $\\mathcal {M}$ , $\\textit {PCTL}^\\star $ path formula $\\varphi $ and rationals $0 \\le x,y \\le 1$ .",
"In our case, and from $\\mathcal {M}_2 \\models \\mathbb {P}_{[a,b]}(\\varphi )$ , it follows that: $a \\le \\mathcal {P}_{min}(\\varphi , \\mathcal {M}_2) = a^{\\prime }$ which in turn contradicts our assumption $a > a^{\\prime }$ .",
"This proves that $\\mathcal {P}_{min}(\\varphi , \\mathcal {M}_1) = \\mathcal {P}_{min}(\\varphi , \\mathcal {M}_2)$ .",
"Using a symmetric argument, it is straightforward to prove that, given the corollary assumptions, $\\mathcal {P}_{max}(\\varphi , \\mathcal {M}_1) = \\mathcal {P}_{max}(\\varphi , \\mathcal {M}_2)$ .",
"* Let $n := \\sum _{i=1}^{k} n_i$ .",
"Let $H : [1,n] \\rightarrow [1,k]$ be a surjective mapping.",
"Intuitively, $H(j) = i$ iff $j$ belongs to the $i$ -th channel.",
"If for some $j,k$ , $H(j) = H(k)$ it means that $(p_j,q_j) = (p_k,q_k)$ , i.e.",
"they belong to the same channel, and we will write $j =_H k$ .",
"In this proof we call transition template a PRISM transition, possibly with indexed variables.",
"Let us call $\\Theta $ the set of the above transition templates.",
"Given any $\\theta \\in \\Theta $ and a state $s$ , we write $\\theta (s)$ to denote the actual MDP transition obtained by instantiating the variables in $\\theta $ with their values in state $s$ .",
"It is evident that for any $\\theta $ and $s$ there exists a unique target state $s^{\\prime }$ .",
"For example: let $\\theta $ be an indexed PRISM transition $[\\alpha ] \\textsf {foo}_i=1 \\rightarrow p : \\textsf {foo}^{\\prime }_i=2$ , and an MDP state $s$ such that $s.\\textsf {foo}_1=1$ and $s.\\textsf {foo}_2=1$ , then $\\theta (s)$ is the MDP transition: $(s,p,\\alpha ,s^{\\prime })$ such that $s^{\\prime }.\\textsf {foo}_1 = 2$ , and $s^{\\prime }.\\textsf {foo}_2 = 2$ .",
"[FS]check: a transition template may not be applicable to a state (e.g.",
"if the lhs does not match) Given a transition $\\tau = (s,\\alpha ,p,s^{\\prime }) \\in {Pr}$ we will denote with $\\textit {pre}(\\tau )$ the source state $s$ and with $\\textit {post}(\\tau )$ the target state $s^{\\prime }$ .",
"Name $S_1$ the set of states of $\\mathcal {M}_1$ , $S_2$ the set of states of $\\mathcal {M}_2$ , and $S$ their disjoint union.",
"Let us define the relation $R \\subseteq S\\times S$ as follows: $R(s_1,s_2)$ iff the following properties hold: $s_1 \\equiv _V s_2$ $s_1.\\textsf {s}_c =_H s_2.\\textsf {s}_c$ $\\forall i \\in [1,m].\\ s_1.\\textsf {ctr}_i^1 = \\sum _{j=1}^{n_i} s_2.\\textsf {ctr}_i^j$ In order to show that $R$ is a probabilistic bisimulation, let us first observe that $R$ is an equivalence relation (i.e.",
"it is transitive, reflexive and symmetric).",
"Secondly, let us underline that $L(s_1) = L(s_2)$ follows by our definition of $\\textit {AP}$ and the fact that $\\forall v \\in V.\\ s_1.v = s_2.v$ .",
"Finally, let us show that: ${Pr}(s_1,\\alpha ,X) = {Pr}(s_2,\\alpha ,X)$ , for all $X \\in {S}{R}$ and all $\\alpha \\in \\textit {Act}$ .",
"Let us consider two cases: either (C1) $X = [s_1]_R$ , or (C2) $X \\ne [s_1]_R$ .",
"Notice that by definition of $X$ and $R$ , $[s_1]_R = [s_2]_R$ .",
"Case (C1) is possible iff there is a transition $\\tau $ satisfying all the following conditions: $\\textit {pre}(\\tau ) \\equiv _V \\textit {post}(\\tau )$ i.e.",
"$\\tau $ does not change any variable $v \\in V$ (otherwise $s_1,s_2 \\notin X$ ); $\\forall i \\in [1,n].\\ \\textit {pre}(\\tau ) \\equiv _{\\lbrace \\textsf {ctr}_i \\rbrace } \\textit {post}(\\tau )$ , i.e.",
"$\\tau $ does not change any counter (otherwise the counters can only increase, thus their sum increase, and thus $s_1,s_2 \\notin X$ ); either $\\textit {pre}(\\tau ) = \\textit {post}(\\tau )$ or $\\textit {pre}(\\tau ) \\lnot \\equiv _{\\lbrace \\textsf {s}_c\\rbrace } \\textit {post}(\\tau )$ i.e.",
"it is either a self-loop or it changes variable $\\textsf {s}_c$ .",
"Inspecting all the transitions in the model, there is none that satisfies the above conditions, thus this case is impossible.",
"Let us split case (C2) in subcases.",
"Either (C2.1) $X$ differs from $[s_1]$ only for variables in $V$ , or (C2.2) $X$ differs from $[s_1]$ for variables in $V \\cup \\lbrace \\textsf {ctr}_i, \\textsf {s}_c \\rbrace $ .",
"Case (C2.1) is possible for the following transition templates: any transition of $\\textsc {SliceAtt}$ , or the $\\textsc {Client}$ transition template: $(\\textsf {pc}_c = 0, \\textsf {ctr}_c = \\textsf {n}) \\rightarrow (\\textsf {pc}_c^{\\prime }=3, \\textsf {s}_c=0)$ Let us observe the following facts, for any $\\theta \\in \\Theta $ : $\\exists q_1 \\in X, p \\in [0,1].\\ \\theta (s_1) = (s_1,p,\\alpha ,q_1) \\Rightarrow \\exists q_2 \\in X.\\ \\theta (s_2) = (s_2,p,\\alpha ,q_2)$ ; $\\exists q_2 \\in X, p \\in [0,1].\\ \\theta (s_2) = (s_2,p,\\alpha ,q_2) \\Rightarrow \\exists q_1 \\in X.\\ \\theta (s_1) = (s_1,p,\\alpha ,q_1)$ ; This means, intuitively, that whenever $\\tau $ is applicable to state $s_1$ and reaching some state $q_1 \\in X$ , it is also applicable to the equivalent state (by R) $s_2$ and reaches some state $q_2$ equivalent to $q_1$ .",
"This can be proven checking all transition templates in $\\Theta $ : their final statuses $q_1,q_2$ have the same values for variables in $V$ and they didn't change the values of variables $\\lbrace \\textsf {s}_c, \\textsf {ctr}_i \\rbrace $ w.r.t.",
"$s_1,s_2$ .",
"Since the correspondence preserves the value of probability $p$ , it follows that: $\\begin{array}{lll}{Pr}(s_1,\\alpha ,X) &= \\sum _{\\stackrel{q \\in X, \\theta \\in \\Theta ,}{\\theta (s_1) = (s_1,p, q)}} p \\\\&= \\sum _{\\stackrel{q^{\\prime } \\in X, \\theta ^{\\prime } \\in \\Theta ,}{\\theta ^{\\prime }(s_2) = (s_2,p,q^{\\prime })}} p &= {Pr}(s_2,\\alpha ,X)\\end{array}$ Case (C2.2): X differs from $[s_1]$ for variables in $V$ and for variables in $\\lbrace \\textsf {s}_c, \\textsf {ctr}_c^i \\rbrace $ .",
"The possible transition templates in this case are all $\\textsc {Client}$ templates: $[\\textsf {busy}] (\\textsf {pc}_c=1, \\textsf {s}_c=i, \\textsf {ctr}_c^i < c) \\rightarrow (\\textsf {pc}^{\\prime }_c=0, \\textsf {ctr}^{\\prime }_c=\\textsf {ctr}_c+1,\\textsf {s}^{\\prime }_c=0)$ $(\\textsf {pc}_c=0, \\textsf {ctr}_c < n) \\rightarrow p_i : (\\textsf {pc}^{\\prime }_c=1, \\textsf {s}_c=i)$ In the case of the first template again we can directly check that: $\\exists q_1 \\in X, p \\in [0,1].\\ \\theta (s_1) = (s_1,p,\\alpha ,q_1) \\Rightarrow \\exists q_2 \\in X.\\ \\theta (s_2) = (s_2,p,\\alpha ,q_2)$ ; $\\exists q_2 \\in X, p \\in [0,1].\\ \\theta (s_2) = (s_2,p,\\alpha ,q_2) \\Rightarrow \\exists q_1 \\in X.\\ \\theta (s_1) = (s_1,p,\\alpha ,q_1)$ ; Take all transitions induced by the first template: the final statuses satisfy the property $q_1 \\equiv _W s_1$ and $q_2 \\equiv _W s_2$ , where $W = V \\cup \\lbrace \\textsf {s}_c \\rbrace $ .",
"When variable $\\textsf {ctr}^j_c$ changes, for some $j$ , it increases by one.",
"In this case the same template $\\theta $ can be applied to state $s_2$ and variable $\\textsf {ctr}^h_c$ can increase its value by one (with same probability) for some other channel $h$ , such that $h =_H j$ .",
"In the case $\\theta $ is the second template, let us call $n = \\sum _{i=1}^{k}n_i$ , i.e.",
"$n$ represents the total number of servers belonging to some channel in the system.",
"Let us define the set $A_s = \\lbrace \\tau : \\tau \\in \\theta , \\exists q.\\ {Pr}(s,\\alpha ,q) > 0 \\rbrace $ , i.e.",
"$A_s$ contains all the transitions that are enabled in $s$ and are induced by template $\\theta $ .",
"Let us define the sets: $A_s^i = \\lbrace \\tau : \\tau \\in \\theta , \\tau = (s,p,\\alpha ,q), p > 0, \\left< \\textsf {ctr}^j_c < \\textsf {c} \\right> \\in \\textit {guard}(\\tau ), j =_H i \\rbrace $ .",
"We have that: $A_s = \\bigcup _{i=1}^n A_s^i$ Being in this case, $s_1$ and $s_2$ are the states where transition of template $\\theta $ is enabled and picks a channel to send the slice to[FS]formalise this statement.",
"We can observe that: $|A_{s_1}^i| = 1$ and $|A_{s_2}^i | = n_i$ , for any $i \\in [1,k]$ .",
"Intuitively: by definition, in the small system we have only one possible concrete transition that picks a server from channel $i$ , while in the big system we have $n_i$ concrete transitions each choosing a different server from channel $i$ .",
"Thus, we can write that: $A_{s_1}^i = \\lbrace (s_1,p,\\alpha ,q) \\rbrace $ , for some $p \\in [0,1]$ and $q \\in S_1$ , and $A_{s_2}^i = \\lbrace (s_1,p_1,\\alpha ,q_1), \\ldots , (s_1,p_{n_i},\\alpha ,q_{n_i}) \\rbrace $ , for some $p_1, \\ldots , p_{n_i} \\in [0,1]$ s.t.",
"$\\sum _{j=1}^{n_i} p_j = p$ , and $q_1, \\ldots , q_{n_i} \\in S_2$ .",
"Finally, we can show that, for all $i$ : $\\begin{array}{lll}{Pr}(s_1,\\alpha ,X) &= {Pr}(s_1,\\alpha ,q) = p \\\\&= \\sum _{j=1}^{n_i} {Pr}(s_2,\\alpha ,q_j) \\\\&= \\sum _{j=1}^{n_i} p_j &= {Pr}(s_2,\\alpha ,X)\\end{array}$ In fact: $\\sum _{j=1}^{n_i} p_j = \\sum _{j} g_i(j) \\cdot f(i) = f(i) \\cdot \\sum _j g_i(j) = f(i)$ , since $1 = \\sum _j g_i(j)$ , by our assumptions.",
"* First of all, let us observe that variables $\\textsf {ctr}_c^i$ is always compared with parameter $c$ , the storage capacity.",
"Since by assumption $c > n$ , for all $n \\in \\mathbb {N}_{> 0}$ , this guard can always be dropped from $\\textsc {Client}$ as it is a tautology.",
"What remains is a MDP that updates the variables $\\textsf {ctr}_c^i$ but never reads it.",
"The MDP $\\textsc {Client}^{\\prime }$ is identical to Client, except that it does not update the variables $\\textsf {ctr}_c^i$ .",
"It is immediate to see that every transition enabled in $\\textsc {Client}$ must be enabled also in $\\textsc {Client}^{\\prime }$ , and viceversa.",
"Name $S_1$ the set of states of $\\mathcal {M}_1$ , and $S_2$ the set of states of $\\mathcal {M}_2$ , and name $S= S_1 \\uplus S_2$ their disjoint union.",
"Let us define the relation $R \\subseteq S\\times S$ as follows: $R = \\lbrace (s_1,s_2) : s_1 \\equiv _V s_2 \\rbrace $ .",
"By our assumptions, $L_1$ and $L_2$ are invariant w.r.t.",
"variables in $V$ , meaning that $s_1 \\equiv _V s_2 \\Rightarrow L_1(s_1) = L_2(s_2)$ .",
"Combined with definition of $R$ we have that $R(s_1,s_2) \\Rightarrow L_1(s_1) = L_2(s_2)$ .",
"Since every transition enabled in $\\mathcal {M}_1$ is also enabled in $\\mathcal {M}_2$ , and since they have the same probability, the second requirement of a probabilistic bisimulation holds (see Definition REF )."
]
] | 1605.05930 |
[
[
"Coupling a thermal atomic vapor to an integrated ring resonator"
],
[
"Abstract Strongly interacting atom-cavity systems within a network with many nodes constitute a possible realization for a quantum internet which allows for quantum communication and computation on the same platform.",
"To implement such large-scale quantum networks, nanophotonic resonators are promising candidates because they can be scalably fabricated and interconnected with waveguides and optical fibers.",
"By integrating arrays of ring resonators into a vapor cell we show that thermal rubidium atoms above room temperature can be coupled to photonic cavities as building blocks for chip-scale hybrid circuits.",
"Although strong coupling is not yet achieved in this first realization, our approach provides a key step towards miniaturization and scalability of atom-cavity systems."
],
[
"Funding Information",
"We acknowledge support by the ERC under contract number 267100 and the Deutsche Forschungsgemeinschaft (DFG) with the project number LO1657/2.",
"R.R.",
"acknowledges funding from the Landesgraduiertenförderung Baden-Württemberg, H.K acknowledges support from the Carl-Zeiss-Foundation.",
"N.G.",
"acknowledges support by the Karlsruhe School of Optics and Photonics (KSOP)."
]
] | 1605.05961 |
[
[
"Spectrum Resource Management and Interference Mitigation for D2D\n Communications with Awareness of BER Constraint in mmWave 5G Underlay Network"
],
[
"Abstract The work presented in this paper deals with the issue of massive demands for higher capacity.",
"For that matter, we investigate the spectrum resource management in outdoor mmWave cell for the uplink of cellular and D2D communications.",
"Indeed, we provide a first insight how to optimize the system performance in terms of achievable throughput while realizing a compromise between the large number of admitted devices and the generated interference constraint.",
"We propose a mathematical formulation of the optimization objective which falls in the mixed integer-real optimization scheme.",
"To overcome its complexity, we apply a heuristic algorithm and test its efficiency through simulation results with a particular regard to the BER impact in the QoS."
],
[
"Introduction",
"The continual evolution of the technological era has advanced drastically making it so far difficult to gauge precisely where it is headed in the future.",
"Thus, key global fifth Generation (5G) players from around the world are working together to clear the vision and the strategic orientation of the standards.",
"Still, it is undoubted that nowadays communication converges to a new design where device-centric communication horn in human-centric communication as a direct result of the connected devices explosion.",
"The network function virtualization is an enabling technology that performs abstraction of physical resources by the means of slicing.",
"Combined with the software defined networking, it fosters the development and the diversification of services provided by telecommunication operators.",
"By introducing the massive MIMO, hundreds and thousands of antennas achieve coherent and highly precise transmission and reception leading to substantial gains in capacity and energy efficiency [1].",
"Moreover, revolutionary inventions have made possible the exploitation of the millimeter-wave bands ranging between 11 and 300 GHz.",
"It is considered as an effective solution in the struggle against the spectrum shortage.",
"This is because it managed to simplify the network design due to the wavelength shortness, improve the quality of wireless transmission as well as interference mitigation.",
"For that, extensive measures for channel modeling have been carried out for different frequencies to pave the way for new algorithms and protocols to provide multi-gigabit services.",
"To maximize the system throughput and ensure fairness among users, dynamic scheduling and congestion control scheme was proposed in [2] that exploits the benefits of the millimeter wave (mmWave) in 28 GHz band and boosts its performance by using the multihop relaying technology.",
"One of the main design concerns of realizing the separation of control and data plane in network architecture is the Heterogeneous Cloud Radio Access Network (H-CRAN).",
"As indicated by its name, the baseband processing and the network control are shifted to a centralized zone denoted base band unit (BBU) pool.",
"Environmental friendly, it has the capability to deal with very complex heterogenous structures and enablers in a low cost and effective way.",
"Another particularity of 5G system is their intense heterogeneity (heterogeneous networks) in terms of transmit powers, supported frequency bands, and of course the coexistence of cellular and peer-to-peer communications (i.e mobile-to-mobile (M2M) and device-to-device D2D communication for both underlay and overlay mode non orthogonal/orthogonal spectrum sharing with cellular users)[3].",
"Particularly, interdisciplinary research efforts have been carried to exploit the D2D communication advantage in spectrum reuse, network offloading, the massive access and the enhancement in user and cell throughput under the aggregated interference issue.",
"This subject has been tackled in different directions.",
"The majority of research studies prioritize cellular users over D2D users [4].",
"The strategy adopted in [5] for example guaranties a required rate for cellular users preserving hence a satisfactory quality of service (QoS).",
"This was done through global mechanisms of power control and resource allocation.",
"Even if the D2D communication needs less requirements compared to conventional users, it is necessary to protect them from the superposed interference and make them meet their target signal to interference plus noise ratio (SINR) as well as improve their data rates.",
"Therefore, interference mitigation schemes and different strategies of mode selection are used to ensure the symbiotic coexistence as proposed in [6], [7].",
"There is a great deal of concern in choosing between the centralized and decentralized approaches as surveyed in [8].",
"In the works where the centralized scheme is adopted, the base station (BS) fully manages the radio resources according to the channel state information and the traffic demand then decides of the scheduling.",
"Even if this choice alleviates the interference, it presents a main drawback that is the massive network signaling for control.",
"In the decentralized strategy, instead, the D2D equipments communicate and share resources with cellular users autonomously which decreases the overhead but generates less interference management [9], [10].",
"There have been many challenging works to ensure the critical trade-off between the massive connectivity and the interference issues.",
"This is enabled through two approaches.",
"The first allows the D2D users to reuse the spectrum resources from more than one cellular user terminal as investigated in [11], [12].",
"The second by allowing cellular users to share their resources by several D2D users as analyzed in [9], [13].",
"However, this approach is challenging and scarcely investigated especially in real scenarios where the number of D2D users is high.",
"So, the majority of works simplify it by allowing cellular users to share their spectral resources with at most two D2D users.",
"Such strategy is used to reduce the interference at the BS and to fulfill the QoS of cellular users.",
"An increasing interest in using D2D communications for underlay cellular network in mmWave bands has been experienced.",
"The work presented in [14] combines both H-CRAN and SDN technologies to provide an efficient scheduling for indoor environment.",
"Besides, it resorts to the cross layer design for a global control of physical (PHY), medium access control (MAC) and Network layers.",
"The work of [15] proposed the D2DMAC; a scheduling algorithm for both access and backhaul; in 60 Ghz mmWave band for small cells.",
"In this scheme, where a centralized control is adopted, the spatial reuse gain is achieved through the concurrent transmission and the priority of D2D users is ensured through the path selection.",
"For widespread adoption of D2D communication in underlay cellular network for the millimeter wave spectrum band range is undoubtable.",
"Nevertheless, the most foreseen applications are for the indoor and relatively rare are the works that investigate it in the outdoor.",
"Motivated by the above facts, the present paper deals with the spectrum reuse issue in mmWave 28GHz band for an outdoor scenario wherein the number of D2D pairs largely exceeds cellular users.",
"We develop a radio resource management scheme that aims to improve the network performance in terms of achievable data rate for both cellular and D2D users through spatial and multi-user gain combined with efficient interference policy.",
"A particular regard is dedicated to the role that plays the BER in the QoS transmission.",
"The paper is organized as follows.",
"Section II introduces the system description and assumption.",
"Section III deals with the mathematical formulation of the optimization problem.",
"In section IV, near-optimal solution algorithm for resource reuse is provided.",
"Section V unleashes the simulations results.",
"Finally, section V summarizes the achieved work.",
"The D2D communication enabled single cell environment, adopted for study in this paper, is presented in Fig.REF .",
"The signal coverage is ensured by a BS localized at the center that serves in the uplink mode $K$ cellular user terminals (UT) labelled $\\mathbf {S}=\\lbrace s_{1},s_{2},...,s_{K}\\rbrace $ .",
"These UTs share the same radio resources with $M$ device terminal (DT) pairs labelled $\\mathbf {M}=\\lbrace d_1, d_2,...,d_M\\rbrace $ on the underlay mode.",
"The DT transmitter is denoted by $d_{TX}$ and the receiver by $d_{RX}$ .",
"A binary indicator $\\rho _{d,s}$ is set to 1 when $UT_s$ shares its resource block (RB) with a $DT_d$ .",
"Note that DT pairs are allowed to reuse the RB of only one UT, contrarily to the UTs who can share their RBs to many DTs as long as their channel status allows them.",
"The received signal at the BS can be written as: $ \\begin{aligned}{y_B}&=\\sqrt{P_s}h_{s,B}x_s+n_{s,B}&+\\underset{d=1}{\\overset{M}{\\sum }} \\rho _{d,s}(\\sqrt{P_d}h_{d_{TX},B}x_d+n_{d,B})\\end{aligned}$ where $P_s$ and $P_d$ , $x_s$ and $x_d$ are the $UT_s$ and $DT_d$ transmit power and transmitted data.",
"$h_{X,Y}$ and $n_{X,Y}$ denote the channel $X-Y$ transfer function and the additive white gaussian noise (AWGN) power.",
"Likewise, we define the received signal at the $d_{RX}$ by: $ \\begin{aligned}y_{d_{RX}}&=\\sqrt{P_d}h_{d_{TX},d_{RX}}x_d+n_{d,d}\\\\&+ \\underset{s=1}{\\overset{K}{\\sum }} \\rho _{d,s}(\\sqrt{P_s}h_{s,d_{RX}}x_s+n_{s,d})\\\\&+\\underset{d^{^{\\prime }}\\in \\mathbb {M}\\setminus \\lbrace d\\rbrace }{\\overset{M}{\\sum }}\\underset{s=1}{\\overset{K}{\\sum }} \\rho _{d^{^{\\prime }},s}(\\sqrt{P_{{d^{^{\\prime }}}_{TX}}}h_{{d^{^{\\prime }}_{TX}},d_{RX}}x_{d^{^{\\prime }}}+n_{{d^{^{\\prime }}},d})\\end{aligned}$ where $d^{^{\\prime }}$ denotes another DT which reuses the RB of the same $UT_s$ as $DT_d$ ."
],
[
"Radio propagation model",
"The most studied bands in the millimeter wave are the 28, 38, 60, 71-76 and 81-86 GHz.",
"Results show that they provide ubiquitous throughput, high quality of wireless links, massive antenna deployment and clear network design.",
"However, these mmWave bands are extremely directive and usually subject to signal attenuation due to obstacles and atmospheric absorbtion.",
"Tab.REF gives an overview how these frequency bands are affected by rain attenuation and oxygen absorbtion.",
"The use of techniques such as beamforming and directional antennas has helped to address these challenges.",
"Table: The propagation characteristics of mmWave communications in different bandsIn this work, we consider 28 GHz mmWave band.",
"Realistic outdoor propagation conditions for 28 GHz mmWave are proposed by [17].",
"They were collected through large scale measurements which are carried out in New York city.",
"Famous for its very dense environment of users and obstacles, the line of sight is practically improbable in such locations.",
"It is characterized by angular signal copies with different delays [18].",
"Further, the path loss (PL) model separates the Line of sight (LOS) and the non line of sight (NLOS) components and associates to each of them the corresponding shadowing.",
"The PL is calculated as $PL=PL_{LOS}+PL_{NLOS}$ .",
"For a distance $d$ , each component is considered as $PL_{X}(d)[dB]=\\mu +10\\nu \\log 10[d(m)]+\\xi , \\ \\xi \\sim \\mathbf {N}(0,\\sigma ^2)$ with $\\mu $ as the PL coefficient, $\\nu $ as its exponent and $\\xi $ as its corresponding lognormal shadowing with mean 0 and variance $\\sigma ^2$ .",
"We insert probability to the lognormal path-loss and shadowing model.",
"By that, we favorite DTs to receive more LOS signals given their close TX-RX proximity.",
"The D2D link PL is calculated as in (REF ): $ \\begin{aligned}PL1=p1\\ PL_{LOS}+(1-p1)\\ PL_{NLOS}\\end{aligned}$ For the rest of links, i.e, (BS-D2D), (D2D-UT) or (UT-UT) and (UT-BS) the PL is expressed as follows: $ \\begin{aligned}PL2=p2\\ PL_{LOS}+(1-p2)\\ PL_{NLOS}\\end{aligned}$ Moreover, works in [19] indicate that the multipath fading is likely to be a Rician channel rather than Rayleigh."
],
[
"Achievable data rate",
"In this single cell network, it is assumed that the number of DTs largely exceeds the number of cellular users (UTs).",
"Moreover, the bandwidth is divided into $N$ RBs of bandwidth $B_{RB}$ where $(N \\ge K)$ .",
"Given the fact that we consider a fully loaded scenario, $V$ DTs are selected to be treated as cellular users who share their RBs with the remaining DTs in the case where $N>K$ .",
"Therefore, the set $\\mathbf {S}$ is extended to include the $V$ elements.",
"Besides, it is indexed $UT_s$ and referred to $DT_d$ such as $s=d^{*}$ .",
"It is worthy of noticing that a perfect channel state information is ensured at the base station and the inter-cell interference is well mitigated.",
"Owner of a RB in the set $\\mathbf {S}$ , each transmitter is allowed to share its spectrum resource with DTs if it satisfies the throughput-BER compromise that depends on the SINR.",
"It is denoted $\\gamma _s$ and can be written as: $ \\begin{aligned}\\gamma _s=\\frac{P_s (\\alpha _s H_{s,B} +\\beta _s H_{s,s}) }{ \\underset{d=1}{\\overset{M}{\\sum }} \\rho _{d,s} P_d(\\alpha _s H_{d_{TX},B} +\\beta _s H_{d_{TX},s} )+N_s}\\ \\\\\\ge {\\gamma _s}^{th},\\ s=1..N\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\end{aligned}$ In above, $\\alpha _s$ and $\\beta _s$ are opposite binary indicators which are used to differentiate between the UTs and DTs owners of RBs in $\\mathbf {S}$ .",
"$(\\alpha _s,\\beta _s)$ equals (1,0) when $s\\in [1,N-V]$ and $(\\alpha _s,\\beta _s)$ equals (0,1) otherwise.",
"$H$ is the channel gain, $N_s$ is the noise power and ${\\gamma _s}^{th}$ is the SINR threshold corresponding to $UT_s$ .",
"The DTs are also under SINR requirements defined by the throughput-BER compromise.",
"It is denoted $\\gamma _d$ that must be greater than a threshold ${\\gamma _d}^{th}$ and can be identified as: $ \\begin{aligned}\\frac{P_d H_{d_{TX},d_{RX}}}{ \\underset{s=1}{\\overset{N}{\\sum }} \\rho _{d,s}P_s H_{s,d_{RX}} +\\underset{s=1}{\\overset{N}{\\sum }} \\underset{d^{^{\\prime }}\\in \\mathbb {M}\\setminus \\lbrace d\\rbrace }{\\overset{M}{\\sum }} \\rho _{d^{^{\\prime }},s} P_{{d^{^{\\prime }}}_{TX}} H_{{d^{^{\\prime }}}_{TX},d_{RX}} +N_d}\\\\= \\gamma _d \\ge {\\gamma _d}^{th},\\ \\ \\ \\ d=1..M,\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\end{aligned}$ The data rates for $UT_s$ and $DT_d$ are given by $ \\begin{aligned}R_s= B_{RB}\\log _{2}(1+ cst_s \\gamma _s) \\ \\ \\ \\ \\ \\ \\ \\ \\ \\end{aligned}$ $ \\begin{aligned}R_d= \\underset{s=1}{\\overset{N}{\\sum }} \\rho _{d,s} B_{RB}\\log _{2}(1+ cst_d \\gamma _d)\\end{aligned}$ where, $cst_d = -1.5/ \\ln (5BER_d)$ and $cst_s = -1.5/ \\ln (5BER_s)$ .",
"Each of them identifies the QoS imposed by a minimum probability of error, i.e, Bit Error Rate (BER).",
"In this mmWave outdoor network design, the sum-rate of UTs and DTs is considered as the objective function to maximize.",
"The optimal solution for this resource sharing is to specify the admitted DTs per each UT (i.e a matrix $\\rho =[N;M]$ ), where both BER-aware SINR requirements and a large number of admitted D2D users are respected.",
"The resource sharing optimization problem can be formulated as follows: $ \\begin{aligned}R_{Tot}= \\underset{\\rho }{\\max }\\lbrace \\underset{s=1}{\\overset{N}{\\sum }} R_s + \\underset{d=1}{\\overset{M}{\\sum }} R_d\\rbrace \\ = \\underset{\\rho }{\\max }\\lbrace \\underset{s=1}{\\overset{N}{\\sum }} B_{RB}\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\\\\\log _{2}(1+cst_s \\times \\frac{P_s (\\alpha _s H_{s,B} +\\beta _s H_{s,s}) }{ \\underset{d=1}{\\overset{M}{\\sum }} \\rho _{d,s} P_d(\\alpha _s H_{d_{TX},B} +\\beta _s H_{d_{TX},s} )+N_s})\\ \\ \\\\+ \\underset{d=1}{\\overset{M}{\\sum }} \\underset{s=1}{\\overset{N}{\\sum }} \\rho _{d,s} B_{RB} \\log _{2}(1+cst_d \\times \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\\\\\frac{P_d H_{d_{TX},d_{RX}}}{ \\underset{s=1}{\\overset{N}{\\sum }} \\rho _{d,s}P_s H_{s,d_{RX}} + \\underset{s=1}{\\overset{N}{\\sum }} \\underset{d^{^{\\prime }}\\in \\mathbb {M}\\setminus \\lbrace d\\rbrace }{\\overset{M}{\\sum }} \\rho _{d^{^{\\prime }},s} P_{{d^{^{\\prime }}}_{TX}} H_{{d^{^{\\prime }}}_{TX},d_{RX}} +N_d} )\\rbrace \\end{aligned}$ Subject to : $ \\begin{aligned}\\frac{P_s (\\alpha _s H_{s,B} +\\beta _s H_{s,s}) }{ \\underset{d=1}{\\overset{M}{\\sum }} \\rho _{d,s} P_d(\\alpha _s H_{d_{TX},B} +\\beta _s H_{d_{TX},s} )+N_s}\\ \\ge {\\gamma _s}^{th},\\ s=1..N\\end{aligned}$ $ \\begin{aligned}\\frac{P_d H_{d_{TX},d_{RX}}}{ \\underset{s=1}{\\overset{N}{\\sum }} \\rho _{d,s}P_s H_{s,d_{RX}} +\\underset{s=1}{\\overset{N}{\\sum }} \\underset{d^{^{\\prime }}\\in \\mathbb {M}\\setminus \\lbrace d\\rbrace }{\\overset{M}{\\sum }} \\rho _{d^{^{\\prime }},s} P_{{d^{^{\\prime }}}_{TX}} H_{{d^{^{\\prime }}}_{TX},d_{RX}} +N_d}\\\\ \\ge {\\gamma _d}^{th},\\ \\ \\ \\ d=1..M,\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\end{aligned}$ $ \\begin{aligned}P_s \\le {P_s}^{max},\\ s\\in [1, N-V]\\ and \\\\P_s \\le {P_d}^{max},\\ s\\in [N-V+1,N]\\end{aligned}$ $ \\begin{aligned}P_d \\le {P_d}^{max}\\end{aligned}$ $ \\begin{aligned}\\underset{s=1}{\\overset{N}{\\sum }} \\rho _{d,s} \\in \\lbrace 0,1\\rbrace ,\\ d=1..M\\end{aligned}$ Here, the two first constraints are used to impose the SINR requirements for both UTs and DTs.",
"The third and the fourth are used to ensure a limit for the maximal transmit power.",
"The last constraint is adopted to guaranty that each DT can reuse one RB (that is equivalent to share the same RB of one UT).",
"Due to non-linear-integer formulation, the solution of this optimization problem is not straightforward and hard to find within short range time especially with large number of DTs.",
"Therefore, we resort to a low complexity algorithm that manages the spectrum resources between the different users."
],
[
"Spectrum Resource Management Scheme",
"The proposed algorithm for resource allocation is based on interference alleviation in the uplink.",
"For that purpose, we define the total interference received at $d_{RX}$ by $Interf_d= \\underset{s=1}{\\overset{N}{\\sum }} \\rho _{d,s}P_s H_{s,d_{RX}} +\\underset{s=1}{\\overset{N}{\\sum }} \\underset{d^{^{\\prime }}\\in \\mathbb {M}\\setminus \\lbrace d\\rbrace }{\\overset{M}{\\sum }} \\rho _{d^{^{\\prime }},s} P_{{d^{^{\\prime }}}_{TX}} H_{{d^{^{\\prime }}}_{TX},d_{RX}}$ .",
"To satisfy $\\gamma _d$ defined in (REF ), $Interf_d$ must not exceed the limit ${Interf_d}^{th}$ which is given as ${Interf_d}^{th}=\\frac{P_d H_{d_{TX},d_{RX}}}{{\\gamma _d}^{th}}-N_d$ .",
"Likewise, we define $InterfB_s$ as the sum of interferences received at BS created by the set of DTs which reuse the RB of $UT_s$ .",
"Besides, we recall that $s=d^{*}$ if $(s>K)$ .",
"So, the interference received at ${d^{*}}_{RX}$ of $UT_s$ pair is included in $InterfB_s$ .",
"Thus, it is calculated as $\\underset{d=1}{\\overset{M}{\\sum }} \\rho _{d,s} P_d(\\alpha _sH_{d_{TX},B}+\\beta _sH_{d_{TX},s}) $ .",
"To satisfy $\\gamma _s$ defined in (REF ), $InterfB_s$ must not exceed ${InterfB_s}^{th}$ that corresponds to $ {InterfB_s}^{th}=(\\alpha _s \\frac{P_s H_{s,B}}{ {\\gamma _s}^{th}}+\\beta _s \\frac{P_s H_{s,s}}{ {\\gamma _s}^{th}})+N_s$ .",
"In what follows, $\\mathbf {U}$ and $\\mathbf {M^{^{\\prime }}}$ are intermediate sets used to refer to not yet allocated RBs and untreated DTs.",
"The scheduling mechanism, starts by treating each $DT_d$ from $\\mathbf {M^{^{\\prime }}}$ apart.",
"It considers all the possible links between $DT_d$ and UTs achieving $InterfB_s<{InterfB_s}^{th}$ then calculates $R_s+R_d$ with the assumption that $d^{^{\\prime }}=\\emptyset , \\ \\forall d^{^{\\prime }}\\ne d$ .",
"Afterwards, it chooses $UT_s$ that corresponds to $\\mathrm {argmax}\\lbrace R_d + R_s \\rbrace $ and adds $DT_d$ to $\\Omega _s$ .",
"We refer to this step by cond1 in (II).",
"Once the $\\Omega $ are defined, the algorithm proceeds by checking the requirements in SINR for both DTs and UTs in order to manage the superposed interference for the users who reuse the same RBs (III).",
"The resulting $\\rho $ is the solution of the optimization problem (IV).",
"A summary of the resource allocation and interference alleviation process is given in Alg..",
"I: Initialize the network parameters: $\\mathbf {U}=\\mathbf {S},\\mathbf {M}^{^{\\prime }}=\\mathbf {M},$ Define the sets $\\Omega =\\lbrace \\Omega _1,\\Omega _2,...,\\Omega _N \\rbrace $ of possible assignment between UTs and DTs with $\\Omega _i = \\emptyset ,\\ i=1..N$ Define the sets $\\rho = \\lbrace \\rho _1, \\rho _2, ....,\\rho _N\\rbrace $ of the optimal assignment with $\\rho _i = \\emptyset ,\\ i=1..N$ II: Fill $\\Omega $ with the possible association between $\\mathbf {M}^{^{\\prime }}$ and $\\mathbf {U}$ according to cond1.",
"All elements in U are treated III: Treat each $\\mathbf {U}$ consecutively starting with the $UT_s$ who has the largest $\\Omega _s$ : (1): Eliminate the DTs from $\\Omega _s$ if their $Interf_d >{Interf_d}^{th}$ .",
"(2): Each DT in $\\Omega _s$ computes its own contribution in $InterfB_s$ .",
"(3): Exclude from $\\Omega _s$ the $DT_d$ with the greatest interference in the BS while $InterfB_s>{InterfB_s}^{th}$ .",
"IV: Update: (1): Mark $\\rho _s=\\Omega _s$ as the optimal assignment of $RB_s$ .",
"(2): Eliminate $UT_s$ from $\\mathbf {U}$ .",
"(4): Apply II for all removed DTs.",
"Spectrum Resource Management paradigm"
],
[
"Numerical results",
"In this section, simulation results are illustrated in order to assess the performance of our proposed algorithm.",
"Details of the simulation parameters are given in Tab.",
"REF Table: Simulation ParametersFigure: D2D throughput with fixed number of UTs and variable DTsFigure: System throughput with fixed number of UTs and variable DTsAs an alternative interpretation of the results shown by Fig.REF , the rate achieved by the D2D communication increases steadily when the number of DTs goes up for a fixed number of UTs.",
"This is reflected on the overall system performance, where the achievable rate of the system has exactly the same shape as given in Fig.REF .",
"For a maximum probability error equal to $10^{-1}$ the system rate rose by 0.08 Gbits/s from about 0.7 to 0.78 Gbits/s.",
"This is primarily the result of the spatial reuse and proximity gain which improves consequently the overall system throughput.",
"However, when the number of DTs becomes large, the case of $M=320$ , the achievable rate for both graphs drops.",
"This implies that eventually a very large number has a bad impact on the system performance that can be affected by the resulting interference.",
"Moreover regardless of the number of users, the rate for the two graphs has been decreasing as long as the QoS defined by the BER is relaxed.",
"Figure: D2D throughput with fixed number of DTs and variable UTsFigure: System throughput with fixed number of DTs and variable UTsFrom Fig.REF , the D2D achievable rate decreases as long as the number of cellular users rises for a given number of DTs.",
"This is a natural result, as the scheduling process of our proposed solution prioritizes the UTs since the RBs are firstly assigned to them and reused in the second order by DTs.",
"Results from Fig.REF show that even the system throughput decreases with the increase of UTs number.",
"This is due to the decrease in D2D data rate which affects the spatial reuse gain.",
"It is also noticed the impact of the QoS defined by the BER, as the achievable rates drop with greater bit error probabilities.",
"Figure: Satisfaction ratio with fixed number of UTs and variable DTsFigure: Satisfaction ratio with fixed number of DTs and variable UTsIn what follows, we introduce the D2D satisfaction ratio (SR) as a new metric to assess the performance of the proposed scheduling scheme.",
"The SR corresponds to the number of DTs whose data rate exceeds a minimum required level divided by their admitted number in the system.",
"It is observed from Fig.REF that the SR is about the same for $M=240$ till $M=300$ .",
"But when M exceeds 300, it decreases for all values of BER.",
"This is due to the superposed interference in the system between the pairs that reuse the same RB.",
"In Fig.REF , $UT=64$ and $UT=128$ correspond to the totality or the half of DTs who are assigned individually RBs (i.e treated as UTs).",
"Hence, their needs in spectrum resources are highly fulfilled compared to the other presented statistics.",
"And besides that, when the QoS defined by the BER is high, the SR is improved.",
"These results are in great agreement with those treating the achievable data rate discussed above.",
"They highlight how the spatial reuse and the multi-user reuse improve the network performance as long as the interference is well managed."
],
[
"Conclusion",
"Throughout this paper, we considered the spectrum resource management in outdoor mmWave cell for the uplink of conventional and D2D communication.",
"We aimed to optimize the system performance in terms of achievable throughput while achieving a compromise between the elevated number of admitted devices and the generated interference constraint.",
"We provided a mathematical formulation of the optimization problem which falls in the mixed integer-real optimization scheme.",
"To overcome its complexity, we proposed a heuristic algorithm and tested its efficiency through simulation results."
]
] | 1605.06020 |
[
[
"E6Tensors: A Mathematica Package for E6 Tensors"
],
[
"Abstract We present the Mathematica package E6Tensors, a tool for explicit tensor calculations in E6 gauge theories.",
"In addition to matrix expressions for the group generators of E6, it provides structure constants, various higher rank tensors and expressions for the representations 27, 78, 351 and 351'.",
"This paper comes along with a short manual including physically relevant examples.",
"I further give a complete list of gauge invariant, renormalisable terms for superpotentials and Lagrangians."
],
[
"Program Summary",
"Author: Thomas Deppisch Title: E6Tensors Licence: GNU LGPL Programming language / external routines: Wolfram Mathematica 8-10 Operating system: cross-platform Computer architecture: x86, x86_64 RAM: $>1$ GB RAM recommended Current version: 1.0.0 Web page: http://e6tensors.hepforge.org Contact: [email protected] — for bugs, possible improvements and questions"
],
[
"Introduction",
"The exceptional Lie group $E_6$ may be a suitable candidate for describing fundamental symmetries in particle physics [1].",
"In the discussion of $E_6$ , most authors rely on abstract group theoretic methods.",
"Besides, there is a paper by Kephart and Vaughn [2] that describes the generators of $E_6$ in terms of its maximal subgroup $SU(3)\\times SU(3)\\times SU(3)$ .",
"In practice, applying these methods may be cumbersome.",
"For writing down Lagrangians and superpotentials of such models it is useful to have explicit expressions for irreducible representations and invariants, preferably in a way suited for computer use.",
"To our knowledge, such a tool is still missing in the literature and the package E6Tensors tries to fill this gap.",
"Our package provides such explicit expressions for the represenations $\\mathbf {27}, \\mathbf {78}, \\mathbf {351},\\mathbf {351^{\\prime }}$ as well as structure constants and higher rank tensors.",
"E6Tensors enables the user to study Lagrangians and superpotentials including the aforementioned representations explicitly by components.",
"The outline of the paper is the following: In Section , we state the transformation law for the fundamental representation $\\mathbf {27}$ .",
"From that we construct the matrix expressions for the 78 group generators forming the adjoint representation $\\mathbf {78}$ .",
"Then it is possible to compute the (symmetric) structure constants and other properties of the matrix generators.",
"Since higher-order tensors can be built from the tensor products of (anti-)fundamental representations, we derive expressions for the irreducible representations $\\mathbf {351}$ and $\\mathbf {351^{\\prime }}$ in Section .",
"Section then provides a small manual for the Mathematica package including some remarks and examples.",
"A summary of possible gauge invariant terms in superpotentials and Lagrangians can be found in Appendix ."
],
[
"Matrix Expressions for the Group Generators",
"In [2], the authors give a prescription how the fundamental representation can be expressed as three $3\\times 3$ matrices $L,M,N$ and how the group generators act on them.",
"We briefly revise that prescription and point out that we performed the calculations described there using symbolic manipulation in Mathematica.",
"Throughout the paper we use Greek indices for the fundamental representation of $E_6$ $\\mu ,\\nu ,\\rho ,\\dots = 1,\\dots ,27.",
"$ For the adjoint representation we use Latin indices $k,l,m,\\dots $ $k,l,m,\\dots = 1,\\dots ,78.",
"$ For describing $E_6$ by its maximal subgroup $SU(3)_C\\times SU(3)_L \\times SU(3)_R$ we also use the $SU(3)$ indices $\\alpha ,\\beta ,\\gamma ,\\dots &= 1,2,3.\\\\a,b,c,\\dots &= 1,2,3.\\\\p,q,r,\\dots &= 1,2,3.\\\\A,B,C,\\dots &= 1,\\dots ,8.$"
],
[
"Transformation Law of the Fundamental Representation",
"With respect to its maximal subgroup $SU(3)_C\\times SU(3)_L\\times SU(3)_R$ , the fundamental representation of $E_6$ can be decomposed as $\\mathbf {27} \\rightarrow (\\mathbf {3},\\mathbf {\\bar{3}},\\mathbf {1}) + (\\mathbf {1},\\mathbf {3},\\mathbf {\\bar{3}}) + (\\mathbf {\\bar{3}},\\mathbf {1},\\mathbf {3}),$ and its adjoint representation can be decomposed as $\\mathbf {78} \\rightarrow (\\mathbf {8},\\mathbf {1},\\mathbf {1}) + (\\mathbf {1},\\mathbf {8},\\mathbf {1}) + (\\mathbf {1},\\mathbf {1},\\mathbf {8}) + (\\mathbf {3},\\mathbf {3},\\mathbf {3}) +(\\mathbf {\\bar{3}},\\mathbf {\\bar{3}},\\mathbf {\\bar{3}}).",
"$ First, the generators $T^A$ of the three SU(3) subgroups corresponding to the first three representations in eq.",
"(REF ) can be represented by the eight Gell-Mann matrices $\\lambda ^A/2$ and their action onto the three matrices $(L,M,N)$ according to eq.",
"(REF ) can be expressed by $ T^A_C\\ \\begin{pmatrix} L,& M,& N \\end{pmatrix} &= \\begin{pmatrix} \\tfrac{1}{2} \\lambda ^A L, & 0 ,& -\\tfrac{1}{2} N \\lambda ^A \\end{pmatrix}, \\nonumber \\\\T_L^A\\ \\begin{pmatrix} L,& M,& N \\end{pmatrix} &= \\begin{pmatrix} -\\tfrac{1}{2} L \\lambda ^A, & \\tfrac{1}{2} \\lambda ^A M, & 0 \\end{pmatrix}, \\\\T_R^A\\ \\begin{pmatrix} L,& M,& N \\end{pmatrix} &= \\begin{pmatrix}0, & -\\tfrac{1}{2} M \\lambda ^A, & \\tfrac{1}{2} \\lambda ^A N\\end{pmatrix}.",
"\\nonumber $ Second, there are the generators $T^{\\alpha ap}$ and $\\bar{T}^{\\alpha ap}$ that mediate shifts between the matrices $L,M,N$ $T_{\\alpha a p}\\ L_\\beta ^{\\ b} &= \\varepsilon _{\\alpha \\beta \\gamma }\\ \\delta _a^{\\ b}\\ N_p^{\\ \\gamma }, &\\bar{T}^{\\alpha a p} L_\\beta ^{\\ b} &= -\\varepsilon ^{abc}\\ \\delta _\\beta ^{\\ \\alpha }\\ M_c^{\\ p}, \\nonumber \\\\T_{\\alpha a p}\\ M_b^{\\ q} &= \\varepsilon _{abc}\\ \\delta _p^{\\ q}\\ L_\\alpha ^{\\ c}, &\\bar{T}^{\\alpha a p} M_b^{\\ q} &= -\\varepsilon ^{pqr}\\ \\delta _b^{\\ a}\\ N_r^{\\ \\alpha }, \\\\T_{\\alpha a p}\\ N_q^{\\ \\beta } &= \\varepsilon _{pqr}\\ \\delta _\\alpha ^{\\ \\beta }\\ M_\\alpha ^{\\ r}, &\\bar{T}^{\\alpha a p} N_q^{\\ \\beta } &= -\\varepsilon ^{\\alpha \\beta \\gamma }\\ \\delta _q^{\\ p}\\ L_\\gamma ^{\\ a}.",
"\\nonumber $ $\\delta _a^{b}$ is the Kronecker symbol and $\\varepsilon ^{abc}$ the Levi-Civita symbol with $\\varepsilon _{123} = \\varepsilon ^{123}=1$ .",
"With these set of generators an infinitesimal, unitary $E_6$ transformation reads $U(u,v,w,x,y) = \\mathbf {1} + \\mathrm {i}\\, u_A T^A_C + \\mathrm {i}\\, v_A T^A_L + \\mathrm {i}\\, w_A T^A_R + \\mathrm {i}\\, x_{\\alpha ap} T^{\\alpha ap} + \\mathrm {i}\\, y_{\\alpha ap} \\bar{T}^{\\alpha ap} + \\dots $ In total, $u_A,v_A,w_A,x_{\\alpha ap},y_{\\alpha ap}$ are 78 parameters."
],
[
"Explicit Matrix Expressions for the Group Generators",
"We now aim at writing the transformation in eq.",
"(REF ) with one set of 78 matrices $_k T$ and parameters $\\varepsilon ^k$ that act on a 27-dimensional vector $\\psi $ $U(\\varepsilon )\\ \\psi = \\left( \\mathbf {1}+\\mathrm {i}\\ \\varepsilon ^k\\, _k T + \\dots \\right) \\psi .$ For that purpose, we rearrange the transformation parameters and the matrices $L,M,N$ into column vectors in the following way: $\\psi &= (L_{11},L_{12},...,M_{11},...,N_{11},...,N_{33})^T \\nonumber \\\\\\varepsilon &= (u^1,u^2,...,v^1,...,w^1,...,x^{111},...,y^{111},...,y^{333})^T.$ By comparing coefficients in (REF ) and (REF ), the $27\\times 27$ matrices $_kT$ can be constructed.",
"To adjust the normalisation and obtain Hermiticity, we perform the following change of basis $_k\\tilde{T} &= \\frac{1}{2} \\left(\\, _k T +\\, _{k+27} T \\right), &24<k<52 \\nonumber \\\\_k\\tilde{T} &= \\frac{\\mathrm {i}}{2} \\left(\\, _k T -\\, _{k+27} T \\right), &51<k<78.$ The group generators are included in the Mathematica package as an $78\\times 27\\times 27$ dimensional array called E6gen.",
"They obey $\\operatorname{tr} \\left(\\, _k T \\, _l T \\right) &= 3\\ \\delta _{kl}\\\\\\sum \\limits _{k=1}^{78} \\, _kT \\, _kT &= \\tfrac{26}{3}\\ \\mathbf {1}_{27}.",
"$ This sets the Dynkin index and the quadratic Casimir invariant to $C(27) = 3 \\qquad \\mathrm {and} \\qquad C_2(27) = \\tfrac{26}{3},$ satisfying the well-known identity $C_2(R) &= \\frac{\\operatorname{dim}(G)}{\\operatorname{dim}(R)}\\ C(R),$ for a representation $R$ and the adjoint representation $G$ .",
"In addition to this consistency check, $C(27)$ and $C_2(27)$ match the same values Kephart and Vaughn state in their paper [2].",
"By construction, the generators are ordered in the following way $_1T \\dots \\, _8T &:\\qquad SU(3)_C,\\\\_{9}T \\dots \\, _{16} T &:\\qquad SU(3)_L,\\\\_{17}T \\dots \\, _{24} T &:\\qquad SU(3)_R.$ Therefore, the diagonal generators representing the Cartan subalgebra are $_3T,\\,_8 T,\\, _{11} T,\\, _{16} T,\\, _{19} T,\\, _{24} T.$ The generators $_{25}T$ to $_{78}T$ are the shifting operators defined in eq.",
"(REF ).",
"The structure constants $f_{klm}$ of a Lie Algebra are defined by $[\\, _kT,\\, _lT] = \\mathrm {i}f_{klm} \\, _m T.$ Applying the normalisation condition gives $f_{klm} = -\\dfrac{\\mathrm {i}}{C(27)} \\operatorname{tr} \\left( [\\, _k T,\\, _l T]\\ _m T \\right).$ In the Mathematica package they are encoded in the array E6f.",
"As a cross-check we calculated the normalisation to be $f_{kmn}f_{lmn} = 12\\ \\delta _{kl}$ which also matches the value in [2].",
"Since the structure constants are the generators of the adjoint representation, its quadratic Casimir and Dynkin index are $C(G)=C_2(G)=12.$ The symmetric structure constants $C_{klm}$ are defined by $ \\lbrace \\, _kT,\\, _lT\\rbrace = \\, _kT\\, _l T + \\, _l T \\, _k T = \\mathrm {i}\\ C_{klm} \\, _m T.$ This can be rewritten as $ C_{klm} = -\\dfrac{\\mathrm {i}}{C(27)} \\operatorname{tr} \\left( \\lbrace \\, _kT,\\, _l T\\rbrace \\ _mT \\right).$ Explicit computation then yields $C_{klm} = 0 \\quad \\forall \\ k,l,m =1,\\dots ,78.$ Hence, $E_6$ GUT models are in general free of chiral anomalies."
],
[
"Higher Rank Tensors",
"The transformation laws for fundamental 27, anti-fundamental $\\overline{\\mathbf {27}}$ and adjoint representation 78 are as follows $\\psi _\\mu &\\rightarrow \\psi _\\mu + \\mathrm {i}\\,\\varepsilon ^k\\ (_k T)_\\mu ^{\\,\\nu } \\psi _\\nu , \\\\\\bar{\\psi }^\\mu &\\rightarrow \\bar{\\psi }^\\mu - \\mathrm {i}\\,\\varepsilon ^k\\ (_k T)_\\nu ^{\\,\\mu } \\bar{\\psi }^\\nu , \\\\\\phi _l &\\rightarrow \\phi _l + \\varepsilon ^k\\ f_{klm} \\phi _m,$ with Greek indices running from 1 to 27 and Latin indices running from 1 to 78, cf.",
"eq.",
"(REF ) and eq.",
"(REF )."
],
[
"Higher Dimensional Representations",
"The representations $\\mathbf {351}$ , $\\mathbf {351^{\\prime }}$ and $\\mathbf {650}$ are included in the tensor products [3]The notation for $\\mathbf {351}$ and $\\mathbf {351^{\\prime }}$ differs in the literature.",
"In our notation, $\\mathbf {351}$ is symmetric, whereas $\\mathbf {351^{\\prime }}$ is anti-symmetric.",
"$\\overline{\\mathbf {27}} \\otimes \\overline{\\mathbf {27}} &= \\mathbf {27} \\oplus \\mathbf {351} \\oplus \\mathbf {351^{\\prime }},\\\\\\overline{\\mathbf {27}} \\otimes \\mathbf {27} &= \\mathbf {1} \\oplus \\mathbf {78} \\oplus \\mathbf {650}.$ Therefore, they can be represented as rank-two tensors.",
"Their transformation properties are implicitly given by (REF ) and ().",
"For $\\mathbf {351}$ and $\\mathbf {351^{\\prime }}$ , we labelled the entries for that tensors $\\chi _1,\\dots ,\\chi _{351}$ and choose them in a way that $\\bar{X}^{\\mu \\nu } X_{\\mu \\nu } = \\bar{\\chi }^1 \\chi _1 + \\dots + \\bar{\\chi }^{351} \\chi _{351}, \\qquad \\mu ,\\nu =1,\\dots 27,$ ensuring a canonical normalisation of the kinetic term.",
"A comment on this normalisation is given in Appendix .",
"$\\mathbf {351}$ can be represented by a second rank tensor $A_{\\mu \\nu }$ antisymmetric in its indices.",
"In the Mathematica package it is included as an $27\\times 27$ dimensional array called E6A.",
"$\\mathbf {351^{\\prime }}$ is also a rank two tensor $S_{\\mu \\nu }$ but symmetric in its indices and additionally satisfying $d^{\\mu \\nu \\lambda } S_{\\mu \\nu } = 0, \\qquad \\forall \\ \\lambda =1,\\dots ,27,$ with $d^{\\mu \\nu \\lambda }$ defined below in eq.",
"(REF ).",
"$\\mathbf {351^{\\prime }}$ is named E6S in the package.",
"650 has a fundamental and an anti-fundamental index with vanishing trace $\\psi _\\mu ^{\\ \\mu } =0$ and $_kT_\\nu ^{\\,\\mu }\\, \\psi _\\mu ^{\\,\\nu } = 0.$ It is not (yet) included in the package due to its memory usage."
],
[
"Invariants for the Fundamental Representation",
"The Kronecker symbol $\\delta ^\\mu _\\nu $ is the most simple way to define a quadratic invariant of a fundamental and an anti-fundamental repesentation $\\delta ^\\mu _\\nu \\, \\bar{\\psi }^\\nu \\psi _\\mu .$ A cubic invariant can be defined in the following way [2] $ d^{\\mu \\nu \\lambda }\\, \\psi _\\mu \\psi _\\nu \\psi _\\lambda = \\det {(L + M + N)} - \\operatorname{tr}(LMN).$ The entries of the tensor $d^{\\mu \\nu \\lambda }$ can be obtained by comparing the coefficients of the field components on each side of the equation.",
"It is provided as E6d in the Mathematica package.",
"Together with $d_{\\mu \\nu \\lambda }$ , defined by $d_{\\mu \\nu \\lambda }\\, \\bar{\\psi }^\\mu \\bar{\\psi }^\\nu \\bar{\\psi }^\\lambda = \\det {(L^\\dagger + M^\\dagger + N^\\dagger )} - \\operatorname{tr}(N^\\dagger M^\\dagger L^\\dagger ),$ it is normalised to $d_{\\mu \\nu \\lambda }d^{\\mu \\nu \\sigma } = 10\\, \\delta _\\lambda ^\\sigma .$ Additionally there are some compound tensors that are used to construct the invariants in Appendix .",
"A tensor with four indices is $D_{\\mu \\nu }^{\\sigma \\tau } = d_{\\mu \\nu \\lambda }d^{\\lambda \\sigma \\tau }.$ In the package it is called E6D.",
"There is also a compound tensor carrying five indices: $D_\\lambda ^{\\mu \\nu ,\\sigma \\tau } &= (d^{\\mu \\sigma \\xi } d^{\\nu \\tau \\eta }-d^{\\mu \\tau \\xi } d^{\\nu \\sigma \\eta })d_{\\xi \\eta \\lambda },\\\\D^\\lambda _{\\mu \\nu ,\\sigma \\tau } &= (d_{\\mu \\sigma \\xi } d_{\\nu \\tau \\eta }-d_{\\mu \\tau \\xi } d_{\\nu \\sigma \\eta })d^{\\xi \\eta \\lambda }.$ They are not included in the package to avoid excessive memory usage."
],
[
"Invariants for the Adjoint Representation",
"The normalisation condition for the generators can be used to define a quadratic invariant for the adjoint representation $\\phi $ .",
"$\\delta _{kl}\\ \\phi _k\\phi _l = \\tfrac{1}{3} \\operatorname{tr}(_kT_lT)\\phi _k\\phi _l$ With the structure constants one can form an invariant of three different adjoint representations $\\phi ,\\phi ^{\\prime },\\phi ^{\\prime \\prime }$ : $f_{klm}\\ \\phi _k\\phi ^{\\prime }_l\\phi ^{\\prime \\prime }_m.$ The completely symmetric tensor $\\chi ^5_{klmn} = \\delta _{kl}\\delta _{mn} +\\delta _{kl}\\delta _{mn} +\\delta _{kl}\\delta _{mn}$ in the package is called E6chi and can be used to form an quartic invariant $\\chi ^5_{klmn}\\phi ^{\\prime }_k\\phi ^{\\prime \\prime }_l\\phi ^{\\prime \\prime \\prime }_m\\phi ^{\\prime \\prime \\prime \\prime }_n$ ."
],
[
"Mixed Invariants",
"The generators $(_kT)_\\mu ^{\\,\\nu }$ form a tensor $_kT_\\mu ^{\\,\\nu }$ with an adjoint, a fundamental and an anti-fundamental index.",
"Further, there is a tensor that can be constructed from the anti-commutator and Kronecker symbols $_{kl}H_\\mu ^{\\,\\nu } = \\left\\lbrace T_k,T_l\\right\\rbrace _\\mu ^{\\,\\nu } - \\tfrac{2}{9} \\,\\delta _{kl} \\delta _\\mu ^\\nu ,$ called E6H in the package.",
"Contracting $_kT_\\mu ^{\\,\\nu }$ with $d_{\\nu \\rho \\lambda }$ gives the tensor $_kA_{\\mu \\nu ,\\lambda } = \\,_kT_\\mu ^{\\ \\sigma } d_{\\sigma \\nu \\lambda } - \\,_kT_\\nu ^{\\ \\sigma } d_{\\mu \\sigma \\lambda }$ which is antisymmetric w.r.t.",
"$\\mu \\leftrightarrow \\nu $ and the tensor $_kS_{\\mu \\nu ,\\lambda } = -\\,_kT_\\lambda ^{\\ \\sigma } d_{\\mu \\nu \\sigma }$ which is symmetric w.r.t.",
"$\\mu \\leftrightarrow \\nu $ .",
"They are called E6kA and E6kS, respectively."
],
[
"Download and Installation",
"E6Tensors can be downloaded from http://e6tensors.hepforge.org.",
"On that page, there are also some instructions on how to install it.",
"Currently, there are two versions at the download section: e6tensors_full-1.0.0.tar.gz and e6tensors_small-1.0.0.tar.gz.",
"e6tensors_small-1.0.0.tar.gz contains the following files: install.sh calls the command line version of MathematicaWe assume it to be called math.",
"Change that if it has another name on your system.",
"and runs create_E6Tensors.m.",
"This script uses E6gen.m an E6d.m to create the higher dimensional tensors and saves them as arrays to E6Tensors.m.",
"examples.nb shows some well-documented examples how E6Tensors.m can be used.",
"Note that E6Tensors.m has a size of roughly 150 MB.",
"Therefore make sure to provide enough RAM for loading it.",
"Running create_E6Tensors.m also may need some time.",
"On a quad-core i5 machine this took about half an hour working on four subkernels.",
"To use parallelisation, change LaunchKernels[1] in create_E6Tensors.m to the appropriate value.",
"For most users, we recommend to download e6tensors_full-1.0.0.tar.gz.",
"After extracting the tarball, it is ready to use and contains all files of e6tensors_small-1.0.0.tar.gz including E6Tensors.m.",
"You probably will not need all tensors in a single project.",
"In that case you can comment out the unnecessary tensors in create_E6Tensors.m and run it to get your own customised file E6Tensors.m."
],
[
"Structure of the Package",
"E6Tensors.m is a simple text file.",
"It contains the definitions of all tensors as nested lists.",
"In this way, it is very flexible to use: You can write your own functions and procedures that fit the problem you want to solve.",
"As an example, the Pauli matrices would look like {{{0,1},{1,0}},{{0,-I},{I,0}},{{1,0},{0,-1}}}.",
"The generators E6gen have exactly the same structure.",
"Hence, E6gen[[k,mu,nu]] is the element in the $\\mu ^{\\mathrm {th}}$ row and the $\\nu ^{\\mathrm {th}}$ column of the $k^{\\mathrm {th}}$ generator.",
"All tensors are listed in Table REF .",
"There, you can also find their symbolic name, the indices they carry and a short explanation.",
"For instance, E6gen has indices $78,\\overline{27},{27}$ refering to the gauge index $k=1,\\dots ,78$ , the row index $\\mu =1,\\dots ,27$ and the column index $\\nu =1,\\dots ,27$ .",
"The order follows the convention of the Part[] function in Mathematica.",
"Keep in mind that Mathematica does not make any difference between row and column vectors.",
"Table: Overview of Tensors in E6Tensors.m."
],
[
"Known Issues",
"It is not recommended to open E6Tensors.m via the graphical frontend of Mathematica.",
"To load it use Get[\"##/E6Tensors.m\"] instead, where ## refers to the correct path."
],
[
"Examples",
"In the download version, there is a notebook examples.nb hat contains some possible applications of the package.",
"It starts with loading E6gen.m and E6d.m which are sufficient for the first examples.",
"We identify the Standard Model generators among the $E_6$ generators: By construction, we can choose the gluons to be the first eight generators.",
"The generators of $SU(2)_L$ can then be defined as $T^{9}, T^{10}, T^{11},$ and hypercharge as $Y = \\sqrt{\\frac{3}{5}} \\left(-\\sqrt{\\frac{1}{3}} T^{16} - T^{19} - \\sqrt{\\frac{1}{3}} T^{24} \\right).$ There are two additional U(1) charges, which we can define by $Y^{\\prime } &= \\sqrt{\\frac{1}{40}} \\left( -2\\sqrt{3} T^{16} - T^{19} + 3 \\sqrt{3} T^{24}\\right),\\\\Y^{\\prime \\prime } &= \\sqrt{\\frac{1}{40}} \\left( -2\\sqrt{3} T^{16} +4 T^{19} -2 \\sqrt{3} T^{24}\\right).$ In this basis, there is a singlet in the fundamental representation for $Y^{\\prime }$ and $Y^{\\prime \\prime }$ each.",
"The generators for $SU(2)_R$ can be defined as $T^{17},T^{18},T^{19},$ and $B-L$ as $B-L = -\\sqrt{\\frac{1}{2}}\\left(T^{16} + T^{24}\\right) =\\sqrt{\\frac{5}{2}} Y + \\sqrt{\\frac{3}{2}}T^{19}.$ As a first check, we can write the fundamental representation as a list of field names and show their quantum numbers in a table.",
"We also check for the GUT normalisation of the $U(1)$ charges and the correct commutation relation for $SU(2)_L$ , $SU(2)_R$ and $Y$ .",
"In a next step, we use $d^{\\mu \\nu \\lambda }$ to write down the trilinear coupling in the superpotential $\\mathcal {W} = \\frac{\\lambda }{6}\\ d^{\\mu \\nu \\rho } \\psi _\\mu \\psi _\\nu \\psi _\\rho .$ For instruction, we once write out the explicit sum over all indices and once use the Dot[] operator.",
"In many cases, the latter one will be the faster way to do it.",
"The same holds for functions like TensorContract[].",
"For the next examples E6Tenors.m must be located in the same directory.",
"We first test the tensors for the higher dimensional representations $\\mathbf {351}$ and $\\mathbf {351^{\\prime }}$ , i.e.",
"their defining properties $d^{\\mu \\nu \\lambda } S_{\\mu \\nu } = 0, \\qquad S_{\\mu \\nu } = S_{\\nu \\mu },$ and $A_{\\mu \\nu } = - A_{\\nu \\mu }.$ The normalisation of the kinetic terms gives the wanted result.",
"The couplings to matter fields can be described by $\\mathcal {W}= \\bar{S}^{\\mu \\nu } \\psi _\\mu \\psi _\\nu $ and $\\mathcal {W}= \\bar{A}^{\\mu \\nu } \\psi _\\mu \\psi _\\nu $ .",
"Now, we can read off the fields that couple to down quarks.",
"Since $A^{\\mu \\nu }$ is anti-symmetric, it does not couple fields of the same representation (e.g.",
"the same flavour) to each other.",
"For a more advanced example, we discuss possible vacuum expectation values (VEVs) for $\\mathbf {351^{\\prime }}$ ($S^{\\mu \\nu }$ ) and $\\mathbf {351}$ ($A^{\\mu \\nu }$ ).",
"Since they are contained in the tensor product $\\mathbf {27}\\otimes \\mathbf {27}$ , we can write an infinitesimal $E_6$ transformation as $S^{\\mu \\nu } \\rightarrow \\left(\\delta ^\\mu _\\rho + \\mathrm {i}\\, \\alpha _k\\,_k T_\\rho ^{\\,\\mu } \\right)\\left(\\delta ^\\nu _\\sigma + \\mathrm {i}\\, \\alpha _k\\,_k T_\\sigma ^{\\,\\nu } \\right) S^{\\rho \\sigma }= S^{\\mu \\nu } + \\mathrm {i}\\, \\alpha _k \\left(_kT_\\nu ^{\\,\\sigma } S_{\\mu \\sigma }+ \\,_kT_\\mu ^{\\,\\rho } S_{\\rho \\nu } \\right).",
"$ $\\alpha _k$ is a set of parameters.",
"For a VEV $s^{\\mu \\nu }=\\langle S^{\\mu \\nu }\\rangle $ that transforms trivially under a set of generators $\\lbrace _kT\\rbrace $ , the last term in (REF ) must vanish.",
"Using the permutation symmetry of $S^{\\mu \\nu }$ , this can be written as a matrix equation $(_kT\\cdot s)^T + \\,_kT\\cdot s = 0.$ $A^{\\mu \\nu }$ is antisymmetric, therefore the condition reads $(_kT\\cdot a)^T - \\,_kT\\cdot a = 0.$ The conditions are implemented in example.nb.",
"As set of generators we use the gluons and $Q = I_L^3 + \\sqrt{\\frac{5}{3}} Y.$ This ensures that the vacuum does not carry electric charge or colour.",
"We then calculate the resulting mass terms that are generated by this VEV."
],
[
"Acknowledgements",
"The author likes to thank Martin Spinrath for helpful discussions and commments on this paper as well as Julia Gehrlein, Jakob Schwichtenberg and Georg Winner for testing the package and Ulrich Nierste for proofreading.",
"This work was supported by Studienstiftung des deutschen Volkes and the DFG-funded Research Training Group GRK 1694 - Elementarteilchenphysik bei höchster Energie und höchster Präzision."
],
[
"Renormalisable Potentials",
"For completeness, we write down all possible renormalisable and gauge invariant terms that may occur in superpotentials and Lagrangians.",
"They can also be found in [2]."
],
[
"Mass Terms",
"Possible mass terms for the various representations are: $\\overline{\\mathbf {27}}\\cdot \\mathbf {27} \\qquad &: \\qquad \\bar{\\psi }^\\mu \\psi _\\mu \\\\\\mathbf {78}^2 \\qquad &: \\qquad \\operatorname{tr}(_kT_lT) \\phi _k \\phi ^{\\prime }_l= 3\\, \\phi _k\\phi ^{\\prime }_k \\\\\\overline{\\mathbf {351}}\\cdot \\mathbf {351} \\qquad &:\\qquad \\bar{A}^{\\mu \\nu }A_{\\mu \\nu }\\\\\\overline{\\mathbf {351^{\\prime }}}\\cdot \\mathbf {351^{\\prime }} \\qquad &:\\qquad \\bar{S}^{\\mu \\nu }S_{\\mu \\nu }$ For the fundamental and adjoint representations, there are $\\mathbf {27}^2\\cdot \\overline{\\mathbf {27}}^2&:&\\bar{\\psi }^\\mu \\psi _\\mu \\bar{\\psi }^\\nu \\psi _\\nu \\\\&&D^{\\mu \\nu }_{\\sigma \\tau }\\bar{\\psi }^\\sigma \\bar{\\psi }^\\tau \\psi _\\mu \\psi _\\nu \\\\\\mathbf {78}^4&:&(\\phi _k\\phi _k)^2\\\\&& (\\phi _l\\phi _l)^2(\\phi ^{\\prime }_k\\phi ^{\\prime }_k)^2\\\\&& (\\phi _l\\phi ^{\\prime }_l)^2(\\phi _k\\phi _k^{\\prime })^2\\\\&& \\phi _k\\phi _l\\phi _m\\phi _n \\operatorname{tr}\\bigl ( \\lbrace _kT,_lT\\rbrace \\lbrace _mT,_nT\\rbrace \\bigr )\\\\&& \\chi ^5_{klmn}\\phi _k\\phi _l\\phi _m\\phi _n\\\\\\mathbf {27}\\cdot \\overline{\\mathbf {27}} \\cdot \\mathbf {78}^2 &:& \\bar{\\psi }^\\mu \\psi _\\mu \\phi _k \\phi _k\\\\&& (_{kl}H_\\mu ^{\\,\\nu }) \\bar{\\psi }^\\mu \\psi _\\nu \\phi _k\\phi _l$ Including $\\mathbf {351}$ and $\\mathbf {351^{\\prime }}$ gives $\\mathbf {351}^2\\overline{\\mathbf {351}}^2 &:& (A_{\\mu \\nu }\\bar{A}^{\\mu \\nu })^2\\\\&& A_{\\mu \\nu }\\bar{A}^{\\nu \\sigma }A_{\\sigma \\tau }\\bar{A}^{\\tau \\mu }\\\\&& d^{\\mu \\nu \\lambda }d_{\\xi \\eta \\lambda } A_{\\mu \\sigma }A_{\\nu \\tau }\\bar{A}^{\\xi \\sigma } \\bar{A}^{\\eta \\tau }\\\\&& d^{\\mu \\nu \\alpha }d^{\\sigma \\tau \\beta }d_{\\xi \\eta \\alpha }d_{\\lambda \\rho \\beta } A_{\\mu \\sigma } \\bar{A}_{\\nu \\tau } A^{\\xi \\lambda } \\bar{A}^{\\eta \\rho }\\\\&& d_{\\mu \\nu \\alpha } d^{\\sigma \\beta \\gamma } d_{\\xi \\eta \\beta } d_{\\lambda \\alpha \\gamma } A_{\\mu \\sigma } A_{\\nu \\tau } \\bar{A}_\\xi ^\\lambda \\bar{A}^{\\eta \\tau } \\\\&& d^{\\mu \\nu \\alpha } d^{\\sigma \\tau \\beta } d_{\\alpha \\beta \\gamma } d^{\\gamma \\xi \\chi } d_{\\xi \\eta \\zeta } d_{\\lambda \\rho \\chi } A_{\\mu \\sigma } A_{\\nu \\tau } \\bar{A}^{\\xi \\lambda } \\bar{A}^{\\eta \\rho }\\\\\\mathbf {351^{\\prime }}^2 \\overline{\\mathbf {351^{\\prime }}}^2 &:& (A_{\\mu \\nu }\\bar{A}^{\\mu \\nu })^2\\\\&& S_{\\mu \\nu }\\bar{S}^{\\nu \\sigma }S_{\\sigma \\tau }\\bar{S}^{\\tau \\mu }\\\\&& d^{\\mu \\nu \\lambda }d_{\\xi \\eta \\lambda } S_{\\mu \\sigma }S_{\\nu \\tau }\\bar{S}^{\\xi \\sigma } \\bar{S}^{\\eta \\tau }\\\\&& d^{\\mu \\nu \\alpha }d^{\\sigma \\tau \\beta }d_{\\xi \\eta \\alpha }d_{\\lambda \\rho \\beta } S_{\\mu \\sigma } \\bar{S}_{\\nu \\tau } S^{\\xi \\lambda } \\bar{S}^{\\eta \\rho }\\\\\\mathbf {351}\\cdot \\overline{\\mathbf {351}}\\cdot \\mathbf {78}^2 &:& \\bar{A}^{\\mu \\nu }A_{\\mu \\nu } \\phi _k\\phi _k\\\\&& (_{kl}H_\\mu ^{\\,\\nu }) \\bar{A}^{\\mu \\lambda }A_{\\nu \\lambda } \\phi _k\\phi _l\\\\&& (_{kl}H_\\mu ^{\\,\\nu }) d^{\\mu \\sigma \\alpha } d_{\\nu \\tau \\alpha } \\bar{A}^{\\tau \\lambda } A_{\\sigma \\lambda } \\phi _k\\phi _l\\\\&& (_kT_\\mu ^{\\,\\sigma }) d^{\\mu \\lambda \\alpha } (_lT_\\tau ^{\\,\\nu }) d_{\\nu \\rho \\alpha } \\bar{A}^{\\rho \\tau } A_{\\alpha \\lambda } \\phi _k \\phi _l\\\\\\mathbf {351^{\\prime }}\\cdot \\overline{\\mathbf {351^{\\prime }}}\\cdot \\mathbf {78}^2 &:& \\bar{S}^{\\mu \\nu }S_{\\mu \\nu } \\phi _k\\phi _k\\\\&& (_{kl}H_\\mu ^{\\,\\nu }) \\bar{S}^{\\mu \\lambda } S_{\\nu \\lambda } \\phi _k\\phi _l\\\\&& (_{kl}H_\\mu ^{\\,\\nu }) d^{\\mu \\sigma \\alpha } d_{\\nu \\tau \\alpha } \\bar{S}^{\\tau \\lambda } S_{\\sigma \\lambda } \\phi _k\\phi _l\\\\\\mathbf {27}\\cdot \\overline{\\mathbf {27}}\\cdot \\mathbf {351} \\cdot \\overline{\\mathbf {351}} &:& \\bar{\\psi }^\\mu \\psi _\\mu \\bar{A}^{\\sigma \\tau } A_{\\sigma \\tau }\\\\&& \\bar{\\psi }^\\mu \\psi _\\nu \\bar{A}^{\\nu \\tau }A_{\\mu \\tau }\\\\&& d_{\\mu \\nu \\lambda } d^{\\xi \\eta \\lambda } \\bar{\\psi }^\\mu \\psi _\\xi \\bar{A}^{\\nu \\tau } A_{\\eta \\tau }\\\\\\mathbf {27}\\cdot \\overline{\\mathbf {27}}\\cdot \\mathbf {351^{\\prime }} \\cdot \\overline{\\mathbf {351^{\\prime }}} &:& \\bar{\\psi }^\\mu \\psi _\\mu \\bar{S}^{\\sigma \\tau } S_{\\sigma \\tau }\\\\&& \\bar{\\psi }^\\mu \\psi _\\nu \\bar{S}^{\\nu \\tau }S_{\\mu \\tau }\\\\\\mathbf {27}^2 \\mathbf {351}^2 &:& d^{\\mu \\sigma \\xi }d^{\\nu \\tau \\eta } \\psi _\\mu \\psi _\\nu A_{\\sigma \\tau } A_{\\xi \\eta }\\\\&& d^{\\mu \\sigma \\alpha } d^{\\nu \\xi \\beta } d_{\\alpha \\beta \\gamma }d^{\\gamma \\tau \\eta } \\psi _\\mu \\psi _\\nu A_{\\sigma \\tau } A_{\\xi \\eta } \\\\\\mathbf {27}^2 \\mathbf {351^{\\prime }}^2 &:& d^{\\mu \\sigma \\xi }d^{\\nu \\tau \\eta } \\psi _\\mu \\psi _\\nu S_{\\sigma \\tau } S_{\\xi \\eta }\\\\\\mathbf {351}^3 \\mathbf {78} &:& (_kT_\\rho ^\\eta ) d^{\\mu \\sigma \\alpha } d^{\\nu \\tau \\beta } d_{\\alpha \\beta \\gamma } d^{\\gamma \\xi \\rho } A_{\\mu \\nu } A_{\\sigma \\tau } A_{\\xi \\eta } \\phi _k$"
],
[
"On the Normalisation of $\\mathbf {351^{\\prime }}$",
"The symmetric tensor $\\mathbf {351^{\\prime }}$ ($S_{\\mu \\nu }$ ) is defined by $S_{\\nu \\mu } = S_{\\mu \\nu } \\qquad \\mathrm {and}\\qquad d^{\\mu \\nu \\lambda }S_{\\mu \\nu }=0\\quad \\forall \\ \\lambda =1,\\dots ,27.$ The first condition is easy to construct: We label the off-diagonal entries $\\phi _1,\\dots ,\\phi _{351}$ and the diagonal ones $\\phi _{352},\\dots ,\\phi _{378}$ .",
"The second condition then eliminates 27 entries.",
"For $\\lambda =1$ , it reads $ \\phi _{122}+ \\phi _{207}+\\phi _{226}+\\phi _{244}-\\phi _{102}= 0.$ It is now tempting to solve e.g.",
"for $\\phi _{102}$ and subsitute that in $S_{\\mu \\nu }$ .",
"But then the kinetic term $\\partial ^\\alpha S^{\\mu \\nu }\\partial _\\alpha S_{\\mu \\nu }$ is not canonically normalised.$\\alpha $ is a space-time index in this case.",
"Another solution is to introduce new field names $\\psi _1,\\psi _2,\\psi _3,\\psi _4$ with $\\phi _{102} &= a (\\psi _1+\\psi _2+\\psi _3+\\psi _4)\\\\\\phi _{122} &= \\psi _1-b(\\psi _1+\\psi _2+\\psi _3+\\psi _4)\\\\\\phi _{207} &= \\psi _2-b(\\psi _1+\\psi _2+\\psi _3+\\psi _4)\\\\\\phi _{226} &= \\psi _3-b(\\psi _1+\\psi _2+\\psi _3+\\psi _4)\\\\\\phi _{244} &= \\psi _4-b(\\psi _1+\\psi _2+\\psi _3+\\psi _4)$ For $a=1-4b$ and $b=(5+\\sqrt{5})/20,$ the defining condition is fulfilled and the kinetic term for $S^{\\mu \\nu }$ takes the form $\\partial _\\alpha \\bar{S}^{\\mu \\nu } \\partial ^\\alpha S_{\\mu \\nu } = \\dots + \\partial _\\alpha \\bar{\\psi }_1 \\partial ^{\\alpha }\\psi _1+ \\partial _\\alpha \\bar{\\psi }_2 \\partial ^{\\alpha }\\psi _2+ \\partial _\\alpha \\bar{\\psi }_3 \\partial ^{\\alpha }\\psi _3+ \\partial _\\alpha \\bar{\\psi }_4 \\partial ^{\\alpha }\\psi _4+ \\dots $ The same procedure also works for all other values of $\\lambda $ .",
"It is important, that the component with the relative minus sign ($\\phi _{102}$ in eq.",
"(REF )) is replaced by the expression with $a$ in it.",
"This procedure is implemented in create_E6Tensors.m and used to construct E6S."
]
] | 1605.05920 |
[
[
"Resonance fluorescence of a laser cooled atom in a non-harmonic\n potential"
],
[
"Abstract We investigate a single laser driven atom trapped in a non-harmonic potential.",
"We present the performance of ground-state laser cooling and Doppler cooling and the signatures of the center-of-mass motion in the power spectrum of the scattered light.",
"In order to illustrate the results we provide two explicit examples for the confining potential: the infinite square well and the Morse potential."
],
[
"Introduction",
"In 1953, Dicke investigated the effect of photon recoil on the spectrum of light scattered by an atom [1]: When the atom emits a single photon, it experiences a recoil and thereby changes its motional state.",
"The required energy for this acceleration is taken the photon, which shows a shifted frequency after the scattering event.",
"In order to describe this effect most clearly, Dicke presented its calculation based on a simple confining potential, namely an infinite square well.",
"Trapping atoms in confining potentials has meanwhile become state of the art in experiments and exploiting the momentum recoil due to the spontaneous emission of single photons is routinely applied in laser cooling in today's laboratories[2], [3], [4], [5].",
"Laser cooling is achieved when the laser parameters are chosen such that photon scattering processes that diminish the atomic motional energy prevail over transitions that heat the motion [6], [7].",
"For trapped ions or ground-state cooled atoms, it is justified to approximate the trapping potential harmonically.",
"In the Lamb-Dicke limit [8] the atomic wavepacket is spatially confined on a scale much smaller than the laser wavelength.",
"Laser cooling theory then predicts that light scattering drives the atomic motion towards a thermal state [9].",
"In the spectrum of resonance fluorescence [10] two distinct peaks emerge, the motional sidebands, as the Stokes- and anti-Stokes components of the scattered light.",
"In this article we investigate the light scattering at an atom trapped in a non-harmonic potential in the Lamb-Dicke regime.",
"By this analysis we extend the theoretical tools for the description of laser cooling [11], [12] and the corresponding spectrum of resonance fluorescence [13], [14], [15] to the case of non-harmonic potentials.",
"Compared to the previous works, the details of the theoretical treatment for arbitrarily shaped potentials are presented and compared to the well-known harmonic case.",
"This extension becomes relevant, for example, for atoms cooled in optical lattices [16], [17], [18], where the harmonic approximation can be insufficient, especially at higher temperatures.",
"Similarly, in cavity cooling experiments [19], [20], [21], [22] the trapping potential's anharmonicity can manifest itself in the cooling dynamics and the spectrum of scattered light.",
"The work presented here focuses on the expansion of the theoretical description, guided by Dicke's original work based on a particle in a box.",
"The infinite square, but also the Morse potential, is used to exemplify the application of the extended theory.",
"Both potentials support an analytic solution of the eigenvalue problem given by the time independent Schrödinger equation for the center-of-mass motion, allowing for a clear and comprehensive treatment of laser cooling and the analysis of the scattered light.",
"The Morse potential describes the dynamics of the relative coordinate of diatomic molecules [23], [24] and therefore is of essential interest for the cooling [25], [26] and spectroscopy [27] of such systems.",
"The presented approach allows to identify details in the perturbative description in the Lamb-Dicke regime that are connected with the non-degeneracy of transition frequencies between the motional eigenstates.",
"For both potentials the resulting steady state of the center-of-mass motion cannot be written in terms of a thermal distribution and the motional sidebands consist of a series of peaks of finite width whose spectral position is connected to the transition frequencies between the relevant vibrational states.",
"The article is organized as follows: In Sec.",
"we introduce the system and present in Sec.",
"the elements of the theory of light scattering that are necessary to describe the signatures of the atomic motion in the spectrum of resonance fluorescence.",
"In order to discuss the results and to illustrate the deviations from harmonic trapping potentials, we present in Sec.",
"the laser cooling performance and the motional sidebands of the scattered light for the two previously mentioned potentials.",
"Finally, in Sec.",
"we draw the conclusions."
],
[
"System",
"We investigate the radiation scattered by a single laser-cooled two-level atom.",
"Along the $x$ -direction the atom's center of mass is tightly trapped in a non-harmonic potential $V(x)$ and we restrict ourselves to the one-dimensional problem.",
"The relevant electronic states are the ground state $\\vert g\\rangle $ and the excited state $\\vert e\\rangle $ which are energetically separated by the transition frequency $\\omega _0$ .",
"A running wave laser with wave number $k$ irradiates the atom under an angle $\\phi $ with respect to the motional axis and drives the dipole transition between the two levels with Rabi frequency $\\Omega $ .",
"The laser with frequency $\\omega _\\mathrm {L}$ is detuned from the atomic transition by $\\Delta =\\omega _\\mathrm {L}-\\omega _0$ .",
"The scattered photons are recorded by a narrow-band detector positioned at an angle $\\psi $ from the axis of motion.",
"The rate of spontaneous emission of the two-level system is given by $\\Gamma $ .",
"Figure REF depicts a sketch of the setup.",
"Figure: Two-level atom with excited state linewidth Γ\\Gamma .",
"The transition is detuned from an incidence laser with wave number kk by Δ\\Delta .",
"The center of mass is confined along the xx-direction in the non-harmonic potential V(x)V(x).",
"A detector is placed in the far field at an angle ψ\\psi while the laser illuminates the atom under the angle φ\\phi with respect to the motional axis.We assume that the coupling of the light field to the atomic motion is in a regime where the size of the center-of-mass wave packet is much smaller than the laser wavelength.",
"Formally this can be expressed by the necessary condition $\\eta =k\\xi \\ll 1,$ for the smallness parameter $\\eta $ , where $\\xi =\\sqrt{\\langle x^2\\rangle _0-\\langle x\\rangle _0^2}$ denotes the position uncertainty of the atomic ground-state wave function in the potential $V(x)$ .",
"The requirement on the atomic localization puts constraints on the occupation of higher excited motional states: Only a sufficiently low mean occupation number $\\bar{m}$ of energy eigenstates in the trapping potential is allowed for the treatment presented in this work to be valid.",
"We note that for harmonic trapping $\\eta $ corresponds to the Lamb-Dicke parameter [8], [11], [28].",
"The Hamiltonian of the system is composed of an internal part $H_{\\rm I}$ describing the electronic states, an external part $H_{\\rm E}$ accounting for the atomic center-of-mass motion and the coupling $W(x)$ of those two degrees of freedom by the mechanical effects of the laser light, viz.",
"$H=H_{{\\rm I}}+H_{{\\rm E}}+W(x).$ In the frame rotating with the laser frequency the internal Hamiltonian is given by $H_{{\\rm I}}=-\\hbar \\Delta \\sigma _+\\sigma _-,$ where $\\sigma _+=\\vert e\\rangle \\langle g\\vert $ and $\\sigma _-=\\vert g\\rangle \\langle e\\vert $ represent the atomic raising and lowering operators, respectively.",
"The external Hamiltonian reads $H_{{\\rm E}}=\\frac{p^2}{2M}+V(x)$ with the atomic mass $M$ and the momentum operator $p$ .",
"The coupling of the electronic and motional degrees of freedom due to the laser field takes on the form $W=\\hbar \\frac{\\Omega }{2}\\left(\\sigma _+ e^{ikx\\cos \\phi }+{\\rm H.c.}\\right).$ To complete the description, we take electronic relaxation processes into account using a master-equation formalism.",
"The time evolution of the system's density operator $\\varrho $ , covering the internal and external degrees of freedom, including spontaneous emission, is generated by the Liouville operator $\\mathcal {L}$ and obeys $\\frac{\\partial \\varrho }{\\partial t}=&\\mathcal {L}\\varrho \\nonumber \\\\=&\\frac{1}{i\\hbar }[H,\\varrho ]+\\frac{\\Gamma }{2}\\int _{-1}^1 du \\, w(u) \\mathcal {D}[\\sigma _-e^{-ikxu}]\\varrho $ with the abbreviation $\\mathcal {D}[X]\\varrho =2X\\varrho X^\\dagger -X^\\dagger X\\varrho -\\varrho X^\\dagger X$ for superoperators of Lindblad form.",
"The normalized radiation pattern $w(u)$ for the considered transition describes the probability of emitting a photon at an angle $\\psi =\\arccos u$ while the exponential accounts for momentum recoils due to spontaneously emitted photons projected on the axis of motion.",
"The specific form of the symmetric function $w(u)$ depends on the details of the electronic transition [11]."
],
[
"Theory of light scattering",
"Under stationary conditions the spectral signal at the detector is given by [14], [13] $\\mathcal {S}(\\omega ) = {\\rm Re}\\int _{0}^{\\infty } dt\\, e^{-i(\\omega -\\omega _{\\rm L}) t} \\langle D_+(t)D_-(0)\\rangle _{\\rm st},$ where in the far field the two mutually adjoint generalized atomic lowering and raising operators have the form $D_-(t)=&\\sigma _-(t)e^{-ikx\\cos \\psi },\\nonumber \\\\D_+(t)=&\\sigma _+(t)e^{ikx\\cos \\psi },$ respectively.",
"The exponential term in Eq.",
"(REF ) accounts for the recoil of the photon of wave number $k$ spontaneously emitted along the direction specified by $\\psi $ , projected on the axis of motion."
],
[
"Spectral decomposition and perturbative expansion of the Liouville operator",
"A convenient way to evaluate the power spectrum (REF ) is to employ the spectral decomposition of the Liouville operator defined in Eq.",
"(REF ), the so-called damping basis [29], [30].",
"This method has already been applied for the description of light scattering [13], [15], [31], [32] and laser cooling [11], [12] of harmonically trapped atoms in the Lamb-Dicke limit.",
"Formally the solution of the master equation (REF ) can be achieved by solving the eigenvalue equations of $\\mathcal {L}$ for the left and right eigenelements which read $\\mathcal {L}\\hat{\\varrho }_\\lambda &=\\lambda \\hat{\\varrho }_\\lambda ,\\\\\\check{\\varrho }_\\lambda ^\\dagger \\mathcal {L}&=\\lambda \\check{\\varrho }_\\lambda ^\\dagger .$ The eigenelements are orthogonal with respect to the scalar product ${\\rm Tr}\\lbrace \\check{\\varrho }_\\lambda ^\\dagger \\hat{\\varrho }_{\\lambda ^{\\prime }}\\rbrace =\\delta _{\\lambda ,\\lambda ^{\\prime }}$ and we assume they for a complete set, formally expressed in $\\sum _{\\lambda }\\hat{\\varrho }_\\lambda \\otimes \\check{\\varrho }_\\lambda =1,$ where the action of the projectors on an arbitrary operator $X$ is given by $(\\hat{\\varrho }_\\lambda \\otimes \\check{\\varrho }_\\lambda ) X={\\rm Tr}\\lbrace \\check{\\varrho }_\\lambda ^\\dagger X\\rbrace \\hat{\\varrho }_\\lambda $ .",
"A small value of $\\eta $ suggests an expansion of the Liouville operator $\\mathcal {L}$ with techniques described in [31], [15].",
"Up to second order in $\\eta $ we write the Liouville operator as $\\mathcal {L}=\\mathcal {L}_0+\\mathcal {L}_1+\\mathcal {L}_2$ , where the subscript indicates the order.",
"The individual terms read $\\mathcal {L}_0\\varrho &=\\mathcal {L}_{\\rm I}\\varrho +\\mathcal {L}_{\\rm E}\\varrho \\nonumber \\\\&=\\frac{1}{i\\hbar }[H_{{\\rm I}}+W_0,\\varrho ]+\\frac{\\Gamma }{2}\\mathcal {D}[\\sigma _-]\\varrho +\\frac{1}{i\\hbar }[H_{{\\rm E}},\\varrho ],\\\\\\mathcal {L}_1\\varrho &=\\frac{1}{i\\hbar }[W_1x,\\varrho ],\\\\\\mathcal {L}_2\\varrho &=\\frac{1}{i\\hbar }[W_2x^2,\\varrho ]+\\alpha \\frac{\\Gamma }{2}k^2\\sigma _-\\mathcal {D}[x]\\varrho \\sigma _+$ with the definition $\\alpha =\\int _{-1}^1 du \\, w(u)u^2$ (which evaluates to $2/5$ for the dipole pattern used here [11]).",
"The expansion $W_n=\\frac{1}{n!",
"}\\left.\\frac{\\partial ^nW}{\\partial x^n}\\right|_{x=0}$ of the interaction Hamiltonian up to second order is explicitly given by $W_0&=\\hbar \\frac{\\Omega }{2}\\left(\\sigma _++\\sigma _-\\right),\\\\W_1&=i\\hbar \\frac{\\Omega }{2} k\\cos \\phi \\left(\\sigma _+-\\sigma _-\\right),\\\\W_2&=-\\hbar \\frac{\\Omega }{4}k^2\\cos ^2\\phi \\left(\\sigma _++\\sigma _-\\right).$ We solve the eigenvalue equations (REF ) and () in zeroth order of $\\eta $ and then perform perturbation theory to obtain the eigenvalues and eigenelements in higher orders.",
"The zeroth order Liouville operator Eq.",
"(REF ) does not couple the internal and external degrees of freedom.",
"Hence, the eigenelements $\\hat{\\varrho }_\\lambda ^{(0)}&=\\hat{\\rho }_{\\lambda _{\\rm I}}\\hat{\\mu }_{\\lambda _{\\rm E}},\\\\\\check{\\varrho }^{\\dagger (0)}_\\lambda &=\\check{\\rho }_{\\lambda _{\\rm I}}^\\dagger \\check{\\mu }_{\\lambda _{\\rm E}}^\\dagger $ factorize, where $\\rho $ and $\\mu $ denote eigenelements of the internal and external degrees of freedom, respectively.",
"The eigenvalues are $\\lambda _0=\\lambda _{\\rm I}+\\lambda _{\\rm E}$ , with $\\lambda _{\\rm I}$ and $\\lambda _{\\rm E}$ denoting the internal and external eigenvalues of $\\mathcal {L}_{\\rm I}$ and $\\mathcal {L}_{\\rm E}$ , respectively.",
"Therefore, we only have to solve the eigenvalue equations of the internal and external motion separately, which read $\\mathcal {L}_{\\rm I}\\hat{\\rho }_{\\lambda _{\\rm I}}=\\lambda _{\\rm I}\\hat{\\rho }_{\\lambda _{\\rm I}},&\\quad \\check{\\rho }_{\\lambda _{\\rm I}}^{\\dagger }\\mathcal {L}_{\\rm I}=\\lambda _{\\rm I}\\check{\\rho }_{\\lambda _{\\rm I}}^{\\dagger },\\\\\\mathcal {L}_{\\rm E}\\hat{\\mu }_{\\lambda _{\\rm E}}=\\lambda _{\\rm E}\\hat{\\mu }_{\\lambda _{\\rm E}},&\\quad \\check{\\mu }_{\\lambda _{\\rm E}}^{\\dagger }\\mathcal {L}_{\\rm E}=\\lambda _{\\rm E}\\check{\\mu }_{\\lambda _{\\rm E}}^{\\dagger }.$ The eigenvalue equations for the internal Liouville operator can be readily solved using a matrix representation of the superoperator [33].",
"In App.",
"we give explicit expressions including the steady state $\\rho _{\\rm st}$ of the internal dynamics.",
"The external Liouvillian (REF ) does not include any non-unitary terms and its eigenelements $\\hat{\\mu }_{nm}&=\\vert n\\rangle \\langle m\\vert ,\\\\\\check{\\mu }_{nm}^\\dagger &=\\vert m\\rangle \\langle n\\vert $ can be constructed from the energy eigenstates $|n\\rangle $ satisfying $H_{\\rm E}\\vert n\\rangle =\\varepsilon _n\\vert n\\rangle .$ The corresponding external eigenvalues $\\lambda _{nm}=i\\omega _{nm}$ contain the transition frequencies $\\omega _{nm}=\\frac{\\varepsilon _m-\\varepsilon _n}{\\hbar }$ between the energy eigenstates $|m\\rangle $ and $|n\\rangle $ .",
"The perturbative corrections of interest for later calculations are the ones of first order, $\\check{\\varrho }_{\\lambda }^{\\dagger (1)}=&\\check{\\varrho }_{\\lambda }^{\\dagger (0)}\\mathcal {L}_1\\left(\\lambda _0-\\mathcal {L}_0\\right)^{-1}\\mathcal {Q}_\\lambda ,\\\\\\hat{\\varrho }_{\\lambda }^{(1)}=&\\left(\\lambda _0-\\mathcal {L}_0\\right)^{-1}\\mathcal {Q}_\\lambda \\mathcal {L}_1\\hat{\\varrho }_{\\lambda }^{(0)}.$ Here subscripts of the eigenvalues and superscripts of the eigenelements again label the corresponding order of $\\eta $ .",
"The projectors introduced in Eqs.",
"(REF ) and () are given by $\\mathcal {Q}_\\lambda =1-\\mathcal {P}_\\lambda $ and $\\mathcal {P}_\\lambda =\\hat{\\varrho }_{\\lambda }^{(0)}\\otimes \\check{\\varrho }_{\\lambda }^{\\dagger (0)}$ ."
],
[
"Resonance fluorescence",
"The time evolution of the operators in expression (REF ) for the spectrum of resonance fluorescence is determined by the Liouville operator $\\mathcal {L}$ and can be calculated using the quantum regression theorem [34], [35].",
"Together with the eigenvalue equations (REF ) and () of the Liouville operator and the completeness relation (REF ) the spectrum formula (REF ) can be cast into the form $\\mathcal {S}(\\omega ) = {\\rm Re} \\sum \\limits _{\\lambda } \\frac{w_{\\lambda }}{i(\\omega -\\omega _{\\rm L}) - \\lambda },$ i.e.",
"we can decompose the spectrum into contributions connected to the eigenvalues of the Liouville operator, weighted by $w_\\lambda ={\\rm Tr}\\lbrace D_+\\hat{\\varrho }_\\lambda \\rbrace {\\rm Tr}\\lbrace \\check{\\varrho }^\\dagger _\\lambda D_-\\varrho _{\\rm st}\\rbrace .$ Depending on the real and imaginary parts of $w_\\lambda $ the spectrum consists of a superposition of Lorentzians and Fano profiles.",
"We are mainly interested in the signatures of the atomic motion in the spectrum of the scattered light.",
"Therefore we only focus on contributions fulfilling the following criteria: (i) We only take the first non-vanishing correction, i.e.",
"the second order in $\\eta $ , of the spectrum into account.",
"(ii) We only consider eigenvalues with $\\lambda _{\\rm I}=0$ giving the motional sidebands of the elastic peak For the harmonic trapping potential the motional sidebands of the inelastic peaks were reported in Ref. [49]..",
"(iii) We do not report the contribution $\\lambda _{\\rm I}=0$ and $\\lambda _{\\rm E}=0$ resulting in a correction to the Rayleigh peak.",
"In order to evaluate the factors (REF ) we expand the generalized atomic lowering operators as $D_-=D_-^{(0)}+D_-^{(1)}+...$ with $D_-^{(0)}&=\\sigma _-,\\\\D_-^{(1)}&=-ik\\cos \\psi \\,\\sigma _-x$ and likewise the weight factors according to $w_\\lambda =w_\\lambda ^{(0)}+w_\\lambda ^{(1)}+...$ in orders of $\\eta $ .",
"In this expansion the zeroth order weight factors give rise to the Mollow-type spectrum of a laser driven two-level system [37].",
"It turns out that the first order does not contribute, while the second order takes on the form $w^{(2)}_\\lambda =\\sum _{\\begin{array}{c}\\alpha +\\beta +\\gamma \\\\+\\delta +\\epsilon =2\\end{array}}{\\rm Tr}\\big \\lbrace D_+^{(\\alpha )}\\hat{\\varrho }_\\lambda ^{(\\beta )}\\big \\rbrace {\\rm Tr}\\big \\lbrace \\check{\\varrho }_\\lambda ^{\\dagger (\\gamma )}D_-^{(\\delta )}\\varrho _{\\rm st}^{(\\epsilon )}\\big \\rbrace .$ The only contributions to the weight factors that fulfill the conditions (i)-(iii) read $w&_\\lambda ^{(2)}={\\rm Tr}\\big \\lbrace D_-^{(0)}\\hat{\\varrho }^{(1)}_\\lambda +D_-^{(1)}\\hat{\\varrho }^{(0)}_\\lambda \\big \\rbrace \\times \\nonumber \\\\&{\\rm Tr}\\Big \\lbrace \\big [\\check{\\varrho }^{\\dagger (1)}_\\lambda D_+^{(0)}+\\check{\\varrho }^{\\dagger (0)}_{\\lambda }D_+^{(1)}\\big ]\\varrho ^{(0)}_{\\rm st}+\\check{\\varrho }^{\\dagger (0)}_{\\lambda }D_+^{(0)}\\varrho ^{(1)}_{\\rm st}\\Big \\rbrace $ where in $\\varrho _{\\rm st}^{(0)}=\\rho _{\\rm st}\\mu _{\\rm st}$ the external steady state has the form $\\mu _{\\rm st}=\\sum _j p_j \\vert j\\rangle \\langle j\\vert $ in the energy eigenbasis (REF ) of the external Hamiltonian.",
"The populations $p_j=\\text{Tr}_{\\text{I}}\\langle j \\vert \\rho \\vert j \\rangle $ in the energy eigenstates $\\vert j\\rangle $ are determined by laser cooling as discussed in the next section.",
"An outline of the evaluation of the factors (REF ) is presented in App. .",
"The spectrum of resonance fluorescence can be brought into the form $\\mathcal {S}_{\\rm sb}(\\omega &) = {\\rm Re}\\sum _{n\\ne m}\\frac{\\vert \\langle n\\vert x\\vert m\\rangle \\vert ^2}{i(\\omega -\\omega _{\\rm L}-\\omega _{nm})-\\lambda _{nm}^{(1)}-\\lambda _{nm}^{(2)}}\\nonumber \\\\&\\times \\left[p_m |\\tilde{r}(\\omega _{nm})|^2+(p_n-p_m)\\tilde{r}(\\omega _{nm})q(\\omega _{nm})\\right]$ where $\\tilde{r}(\\omega )=r(\\omega )-[\\Delta +i\\Gamma /2]\\Omega k\\cos \\psi /2N$ and $N=\\Gamma ^2/4+\\Delta ^2+\\Omega ^2/2$ .",
"Furthermore, we defined the two functions [13] $r(\\omega ) & =\\frac{1}{\\hbar }\\int _0^{\\infty } dt \\, {\\rm e}^{-i\\omega t} \\, \\langle [\\sigma _+(t), W_1(0)] \\rangle _{{\\rm st}},\\\\q(\\omega ) & = \\frac{1}{\\hbar }\\int _{-\\infty }^{\\infty } dt \\, {\\rm e}^{-i\\omega t} \\, \\langle W_1(t) \\sigma _-(0) \\rangle _{{\\rm st}}.$ Explicit expressions for $r(\\omega )$ and $q(\\omega )$ are given in App. .",
"In the denominator of Eq.",
"(REF ) we used the perturbative expansion $\\lambda _{nm}=\\lambda _{nm}^{(0)}+\\lambda _{nm}^{(1)}+\\lambda _{nm}^{(2)}$ of the eigenvalues.",
"The higher order contributions of this expansion introduce a finite linewidth of the motional sidebands, since the denominator of Eq.",
"(REF ) is purely imaginary in zeroth order.",
"In the well studied case of a harmonic trapping potential the external eigenvalues $\\lambda _{nm}$ are degenerate due to the equidistant eigenenergies of the potential.",
"In that specific case, perturbation theory for degenerate eigenvalues has to be performed.",
"In the problem treated here the potential shows an appreciable anharmonicity such that generally only the eigenvalue $\\lambda _{\\rm E}=0$ is degenerate.",
"Hence, perturbation theory for non-degenerate eigenvalues has to be applied.",
"Such an approach is valid if the splitting of the eigenvalues due to the interaction with the laser is small compared to the energy differences of the vibrational levels [38].",
"Explicitly the condition $\\eta \\Omega \\ll {\\rm min}_{n,n^{\\prime }\\ne n}\\vert \\omega _{nn^{\\prime }}\\vert $ has to be fulfilled.",
"The first order correction to the eigenvalues is given by $ \\lambda _1={\\rm Tr}\\big \\lbrace \\check{\\varrho }^{\\dagger (0)}_\\lambda \\mathcal {L}_1\\hat{\\varrho }^{(0)}_\\lambda \\big \\rbrace $ which for $\\lambda _0=\\lambda _{nm}$ can be written as $\\lambda ^{(1)}_{nm}=i\\delta \\omega ^{(1)}_{nm}$ with $ \\delta \\omega ^{(1)}_{nm}=\\frac{\\Gamma \\Omega ^2 k\\cos \\phi }{4N}\\big (\\langle n\\vert x\\vert n\\rangle -\\langle m\\vert x\\vert m\\rangle \\big ).$ This constitutes a shift of the peak positions but does not add a finite width.",
"We further note that this shift vanishes in all even potentials due to the parity of the eigenstates.",
"The second order correction $ \\lambda _2={\\rm Tr}\\big \\lbrace \\check{\\varrho }^{\\dagger (0)}_\\lambda \\left[\\mathcal {L}_2+\\mathcal {L}_1(\\lambda _0-\\mathcal {L}_0)^{-1}\\mathcal {Q}_\\lambda \\mathcal {L}_1\\right]\\hat{\\varrho }^{(0)}_\\lambda \\big \\rbrace $ shows a non-vanishing real part.",
"Their evaluation is sketched in App.",
"where it is shown that for $\\lambda _0=\\lambda _{nm}$ one obtains $\\lambda ^{(2)}_{nm}=i\\delta \\omega ^{(2)}_{nm}-\\gamma _{nm}$ with the second order frequency shift $\\delta \\omega &^{(2)}_{nm}=\\frac{\\Delta \\Omega ^2k^2\\cos ^2\\phi }{4N} \\left(\\langle n\\vert x^2\\vert n\\rangle -\\langle m\\vert x^2\\vert m\\rangle \\right)\\nonumber \\\\&-{\\rm Im}\\sum _j\\left[s(\\omega _{jn})\\vert \\langle j\\vert x\\vert n\\rangle \\vert ^2+s(\\omega _{jm})\\vert \\langle j\\vert x\\vert m\\rangle \\vert ^2\\right]$ and the real part $\\gamma _{nm}=\\frac{1}{2}\\sum _j\\left(A_{jn}+A_{jm}\\right)-D\\langle n\\vert x\\vert n\\rangle \\langle m\\vert x\\vert m\\rangle $ describing finite width of the sidebands.",
"In this expression we introduced the diffusion coefficient $D=\\alpha \\Gamma k^2{\\rm Tr}\\lbrace \\sigma _+\\sigma _-\\rho _{\\rm st}\\rbrace =\\frac{\\Gamma \\Omega ^2k^2}{10N}$ and the transition rates $A_{nm} &=\\left[2\\,{\\rm Re}\\, s(\\omega _{nm})+D\\right] \\vert \\langle n\\vert x\\vert m\\rangle \\vert ^2$ with the fluctuation spectrum $s(\\omega )=\\frac{1}{\\hbar ^2}\\int _0^{\\infty }dt\\, e^{i\\omega t}\\langle W_1(t)W_1(0)\\rangle _{\\rm st}.$ Here, the two-time correlation function $s(\\omega )$ (for explicit expressions see App. )",
"is evaluated in the steady state $\\rho _{\\rm st}$ , viz.",
"$\\langle X\\rangle _{\\rm st}={\\rm Tr}\\lbrace X\\rho _{\\rm st}\\rbrace $ .",
"In the next section we will see that the coefficients $A_{nm}$ indeed describe the rate of population transfer between the energy eigenstate $\\vert m\\rangle $ and $\\vert n\\rangle $ due to the mechanical effects of light scattering.",
"Inspecting $\\gamma _{nm}$ defined in Eq.",
"(REF ) we find that the width of a sideband peak that originates in a transition from $\\vert m\\rangle $ to $\\vert n\\rangle $ involves a sum over the rates of transitions from $\\vert m\\rangle $ and $\\vert n\\rangle $ to all other states $|j\\rangle $ , i.e.",
"$A_{jm}$ and $A_{jn}$ .",
"We note that since ${\\rm Re}\\,s(\\omega )>0$ one also finds $\\gamma _{nm}>0$ ."
],
[
"Cooling of the atomic motion",
"In zeroth order of $\\eta $ the steady state $\\mu _{\\rm st}$ cannot be determined uniquely since the eigenvalue $\\lambda _{\\rm E}=0$ is infinitely degenerate.",
"This degeneracy is lifted in second order perturbation theory and the unique steady state of laser cooling can be found by adiabatic elimination of the internal degrees of freedom [39], [11], [28], [40].",
"This procedure applied to Eq.",
"(REF ) yields the equation $\\mathcal {P}_0\\left(\\mathcal {L}_1\\mathcal {L}_0^{-1}\\mathcal {Q}_0\\mathcal {L}_1-\\mathcal {L}_2\\right)\\mathcal {P}_0\\rho _{\\rm st}\\mu _{\\rm st}=0$ for the external steady state $\\mu _{\\rm st}$ .",
"Using the representation (REF ) of $\\mu _{\\rm st}$ and performing a partial trace ${\\rm Tr_{\\rm I}}\\lbrace \\cdot \\rbrace $ over the internal degrees of freedom in Eq.",
"(REF ) results in the a recursive equation for the probabilities $p_j$ .",
"One can rewrite this equation as $\\sum _mA_{nm}p_m-\\sum _mA_{mn}p_n=0$ with $A_{nm}$ defined in Eq.",
"(REF ), see App. .",
"This set of equations determines the steady state of the atomic motion.",
"In this rate equation the part of the coefficients $A_{nm}$ including $s(\\omega )$ reflects the rate of transitions induced by the laser field while the diffusive part connected with the diffusion coefficient $D$ stems from spontaneous emission."
],
[
"Comparison to harmonic trapping potential",
"We conclude this section by a comparison with the harmonic trapping potential $V(x)=M\\nu ^2x^2/2$ [8], [11], [28], [12].",
"The matrix elements of the position operator are only non-zero between neighboring energy states, $\\langle n\\vert x\\vert m\\rangle \\propto \\sqrt{m} \\delta _{n,m-1}+\\sqrt{m+1}\\delta _{n,m+1}$ , allowing only transitions between adjacent vibrational levels.",
"This is directly reflected in the transition rates $A_{nm}$ which are also non-zero only for $n=m\\pm 1$ , resulting in a recurrence relation Eq.",
"(REF ) that has the form of the detailed balance condition $nA_+p_{n-1}+(n+1)A_-p_{n+1}=[nA_-+(n+1)A_+]p_n$ with $A_\\pm =2\\,{\\rm Re}\\,s(\\mp \\nu )+D$ [12].",
"The normalized solution is a thermal distribution $p_n=\\bar{m}^n/(\\bar{m}+1)^{n+1}$ with the mean occupation number $\\bar{m}=A_+/(A_--A_+)$ .",
"The steady state (REF ) of the atomic motion can be cast in the canonical form $\\mu _{\\rm st}=\\exp (-\\beta H_{\\rm osc})/Z$ with the harmonic oscillator Hamiltonian $H_{\\rm osc}$ , the partition function $Z={\\rm Tr}\\lbrace \\exp (-\\beta H_{\\rm osc})\\rbrace $ and the inverse temperature $\\beta $ implicitly defined via $\\bar{m}=[\\exp (\\beta \\hbar \\nu )-1]^{-1}$ .",
"The harmonic potential is special in the sense that the effective dynamical equation (REF ) takes the shape of a master equation of a harmonic oscillator in contact with a thermal reservoir [12], resulting in a thermal distribution as a steady state.",
"In contrast, non-harmonic potentials do not lead to a recursive relation as Eq.",
"(REF ) and generally do not lead to a thermal external state.",
"According to Eq.",
"(REF ) the atomic motion causes the emergence of an infinite number of motional sidebands, each approximately centered around a possible transition between eigenstates of the external motion.",
"The property that only transitions between neighboring levels can occur together with the degeneracy of the transition frequencies results in a sideband spectrum Eq.",
"(REF ) consisting of two peaks centered around $\\pm \\nu $ .",
"The widths of both peaks can be obtained by employing perturbation theory for degenerate eigenvalues and is given by the cooling rate $\\gamma =A_--A_+$ , thereby giving a sideband spectrum of the form [13] $\\mathcal {S}_{\\rm sb}(\\omega ) =&{\\rm Re}\\frac{\\bar{m} |\\tilde{r}(\\nu )|^2+\\tilde{r}(\\nu )q(\\nu )}{i(\\omega -\\omega _{\\rm L}-\\tilde{\\nu })^2+\\gamma }\\nonumber \\\\&+{\\rm Re}\\frac{(\\bar{m}+1)|\\tilde{r}(-\\nu )|^2-\\tilde{r}(-\\nu )q(-\\nu )}{i(\\omega -\\omega _{\\rm L}+\\tilde{\\nu })^2+\\gamma }$ with a renormalized frequency $\\tilde{\\nu }$ .",
"We exemplify our results by means of two specific choices of the potential $V(x)$ , namely the infinite square well and the Morse potential.",
"For both potentials we will focus on two distinct parameter regimes of laser cooling: The regime of Doppler cooling ($\\omega _{nm}\\ll \\Gamma $ ) and the regime of resolved sideband cooling ($\\omega _{nm}\\gg \\Gamma $ ).",
"In the following we will assume the Rabi frequency to be small such that the atom is driven below saturation.",
"In this case the function $s(\\omega )$ in Eq.",
"(REF ) can be expanded as $s(\\omega )=\\frac{\\Omega ^2}{4}k^2\\cos ^2\\phi \\,\\frac{\\Gamma /2+i(\\omega +\\Delta )}{\\Gamma ^2/4+(\\omega +\\Delta )^2}+\\mathcal {O}(\\Omega ^4).$ In the spectrum formula (REF ) we find $|\\tilde{r}(\\omega )|^2=\\mathcal {O}(\\Omega ^2)$ while $\\tilde{r}(\\omega )q(\\omega )=\\mathcal {O}(\\Omega ^4)$ .",
"In the low intensity limit $\\Omega \\ll \\Gamma $ this implies that we can write the motional sidebands of the elastic peak in second order of $\\eta $ as $\\mathcal {S}_{\\rm sb}(\\omega ) = \\sum _{n\\ne m}\\frac{\\gamma _{nm}p_m\\vert \\langle n\\vert x\\vert m\\rangle \\vert ^2|\\tilde{r}(\\omega _{nm})|^2}{\\gamma _{nm}^2+(\\omega -\\omega _{\\rm L}-\\tilde{\\omega }_{nm})^2}$ where we introduced the renormalized frequencies $\\tilde{\\omega }_{n,m}=\\omega _{n,m}+\\delta \\omega _{nm}^{(1)}+\\delta \\omega _{nm}^{(2)}$ that slightly shift the sideband peaks from the natural transitions between vibrational states."
],
[
"Infinite square well",
"In Ref.",
"[1] a study of light scattered by an atom confined in a one-dimensional square well (see Fig.",
"REF ) of length $L$ has been performed.",
"Figure: Infinite square well potential with its first seven eigenenergies (dashed) and the corresponding probability densities |ϕ n (x)| 2 |\\varphi _n(x)|^2 (solid) with some transition frequencies of allowed transitions (in units of ν\\nu ).We revisit this problem using the theory developed in the previous sections.",
"As a first example we therefore consider the atom to be trapped in the symmetric potential $V(x)=\\left\\lbrace \\begin{array}{cl}\\infty , & \\ \\text{for} \\ |x|\\ge L/2\\\\0,& \\ \\text{for} \\ |x|< L/2.\\end{array}\\right.$ The eigensystem of $H_{\\rm E}$ is given by $\\varepsilon _n=&(n+1)^2\\hbar \\nu ,\\\\\\varphi _n(x)=&\\left\\lbrace \\begin{array}{cl}\\sqrt{\\frac{2}{L}}\\sin ((n+1)\\pi x/L), \\ \\text{for\\ odd} \\ n\\\\\\sqrt{\\frac{2}{L}}\\cos ((n+1)\\pi x/L), \\ \\text{for\\ even} \\ n.\\end{array}\\right.$ for $n=0,1,2,...$ with the frequency $\\nu =\\hbar \\pi ^2/2ML^2$ .",
"Figure REF shows the potential well with the vibrational eigenenergies in units of $\\hbar \\nu $ and the probability density of the associated wavefunctions $\\varphi _n(x)=\\langle x\\vert n\\rangle $ .",
"The transition frequency between neighboring eigenstates increases linearly with $n$ according to $\\omega _{n,n+1}=(2n+3)\\nu .$ With the help of the eigenstates (REF ) it is easy to calculate the matrix elements of the position operator which are needed to evaluate the coefficients $A_{nm}$ and the eigenvalue corrections Eqs.",
"(REF ) and (REF ).",
"They are given by $\\langle n\\vert x\\vert m\\rangle =-\\frac{8(n+1)(m+1)(-1)^{(n+m+1)/2}L}{\\pi ^2[(n+1)^2-(m+1)^2]^2}$ for $n,m$ of different parity and zero otherwise, being a consequence of the symmetry of the potential.",
"The matrix elements of $x^2$ are also readily evaluated and their diagonal elements read $\\langle n\\vert x^2\\vert n\\rangle =\\frac{L^2}{12}\\left[1-\\frac{6}{(n+1)^2\\pi ^2}\\right].$ For the ground state $n=0$ this yields a smallness parameter, as introduced in Eq.",
"(REF ), of $\\eta =\\frac{kL}{2\\pi }\\sqrt{\\frac{\\pi ^2-6}{3}}\\approx 0.18\\,kL.$ The parameter $kL$ was already mentioned in Ref.",
"[1] to quantify the influence of the atomic motion on the emitted radiation.",
"The above mentioned matrix elements can now be used to determine the steady state of the atomic motion.",
"Although the explicit shape of the rate equation (REF ) is easily obtained it is nevertheless necessary to solve the rate equation numerically due to the complexity of the coefficient matrix Eq.",
"(REF ).",
"In the Doppler regime the transition frequencies between the lower eigenstates are small compared to the linewidth of the atomic transition.",
"To fulfill this criterion we choose $\\nu =\\Gamma /30$ which corresponds to a lowest transition frequency of $\\omega _{0,1}=\\Gamma /10$ .",
"Figure REF (a) shows the dependence of the mean steady state occupation $\\bar{m}=\\Sigma _mp_mm$ on the laser detuning $\\Delta $ for $\\Omega =\\Gamma /5$ .",
"Figure: Doppler cooling of an atom trapped in an infinite square well potential for ν=Γ/30\\nu =\\Gamma /30 (ω 0,1 =Γ/10\\omega _{0,1}=\\Gamma /10), Ω=Γ/5\\Omega =\\Gamma /5 and η=0.1\\eta =0.1.",
"(a) Mean occupation m ¯\\bar{m} in dependence on the detuning Δ\\Delta .",
"The circle indicates optimal cooling for Δ≈-0.6Γ\\Delta \\approx -0.6\\Gamma resulting in m ¯≈1.24\\bar{m}\\approx 1.24.",
"The inset shows the logarithm of the steady state occupations p m p_m (m=0,...,10m=0,...,10) for optimal cooling.",
"In a thermal distribution the curve would be linear–clearly not the case here.",
"(b) Power spectrum Eq.",
"() of the motional sidebands for ψ=π/2\\psi =\\pi /2 consisting of Lorentzians centered around the transition frequencies ω n,n+1 =(2n+3)ν\\omega _{n,n+1}=(2n+3)\\nu .",
"Transitions of the kind n→n±ln\\rightarrow n\\pm l with l>1l>1 are not visible due to their small heights.Optimal Doppler cooling occurs at the detuning $\\Delta \\approx -0.59\\Gamma $ (indicated by a circle), where a steady state occupation of $\\bar{m}\\approx 1.24$ is reached.",
"The optimal detuning takes a higher value than in the harmonic trapping potential where best Doppler cooling is achieved for approximately $\\Delta =-\\Gamma /2$ with $\\bar{m}\\approx 3$ .",
"This shows that in the square well potential the atomic motion is Doppler cooled well below the final value for harmonically trapped atoms.",
"In the inset of Fig.",
"REF (a) we plot the negative logarithm of the populations $p_m$ for optimal Doppler cooling against the eigenenergies $\\varepsilon _m$ .",
"Since $-\\log (p_m)$ does not show a linear dependence on $\\varepsilon _m$ the steady state of the atomic motion cannot be written in the form of a thermal state, i.e.",
"$\\mu _{\\rm st}\\ne \\exp (-\\beta H_{\\rm E})/Z$ with $H_{\\rm E}=\\sum _n (n+1)^2\\hbar \\nu \\vert n\\rangle \\langle n\\vert $ .",
"In Fig.",
"REF (b) we show the motional sidebands detected under an angle $\\psi =\\pi /2$ calculated using Eq.",
"(REF ) for optimal cooling $\\Delta =-0.6\\Gamma $ and a smallness parameter of $\\eta =0.1$ .",
"The sideband spectrum consists of a series of Lorentzians centered around the transition frequencies between neighboring energy levels, viz.",
"$\\omega _{n,n+1}=(2n+3)\\nu $ such that they are separated by $2\\nu $ .",
"Their height decreases rapidly towards higher values of $n$ , because of the lower populations $p_n$ of higher motional states.",
"Transitions between non-adjacent levels are not visible due to the smallness of the position operator's matrix elements and vanishing values of $r(\\omega )$ ."
],
[
"Resolved sideband regime",
"In the regime of resolved sideband cooling the transition frequency between the lowest energy states is much larger than the atomic linewidth.",
"We choose $\\nu =10\\Gamma /3$ , which corresponds to a transition frequency $\\omega _{0,1}=10\\Gamma $ .",
"Figure REF (a) shows again the dependence of the mean occupation on the laser detuning for $\\Omega =\\Gamma /5$ .",
"Figure: (a) Cooling of an atom trapped in an infinite square well potential in the resolved sideband regime for ν=10Γ/3\\nu =10\\Gamma /3 (corresponding to ω 01 =10Γ\\omega _{01}=10\\Gamma ) and Ω=Γ/5\\Omega =\\Gamma /5.",
"The mean occupation m ¯\\bar{m} in dependence on the laser detuning Δ\\Delta takes its minimum m ¯≈0.086\\bar{m}\\approx 0.086 at Δ≈-3.35ν\\Delta \\approx -3.35\\nu (indicated by a circle), where sideband cooling of the 1→01\\rightarrow 0 transition occurs.",
"(b) Power spectrum of the motional sidebands for ψ=π/2\\psi =\\pi /2 and η=0.1\\eta =0.1.",
"The insets show magnifications of the first red and blue sideband which dominate the spectrum.Optimal cooling to a mean occupation $\\bar{m}\\approx 0.068$ is achieved when the laser is red-detuned from the atomic transition by roughly the first transition frequency $\\omega _{0,1}$ , i.e $\\Delta \\approx -3.35\\nu $ .",
"In contrast to Doppler cooling the cooling curve shows additional dips at the detunings $\\Delta /\\nu =-15,-21,-27,...$ , corresponding to frequencies where multiple transition between motional eigenstates coincide, e.g.",
"$\\omega _{3,0}=\\omega _{7,6}=-15\\nu $ or $\\omega _{4,1}=\\omega _{10,9}=-21\\nu $ .",
"Due to ground state cooling, the spectrum shown in Fig.",
"REF (b) is dominated by two peaks involving the lowest transition frequency $\\pm \\omega _{0,1}=\\pm 3\\nu $ .",
"The inset shows a magnification of the red and blue sideband where also the frequency shift due to the interaction with the laser becomes visible.",
"We also note that the half-width at half-maximum of both peaks is equal, due to the invariance of $\\gamma _{nm}$ under exchange of $n$ and $m$ , and that the asymmetry in the height vanishes when the spectrum is averaged over the detection angle $\\psi $ .",
"We turn to the discussion of the Morse potential [23], [24] sketched in Fig.",
"REF .",
"Figure: The asymmetric Morse potential and the eigenenergies (dashed) and corresponding probability densities |ϕ n (x)| 2 |\\varphi _n(x)|^2 of its bound states for a=7a=7 with some transition frequencies of possible transitions (in units of ν\\nu ).It has possible applications in the laser cooling of the vibrational degrees of freedom of di-atomic molecules [26], [41], [42].",
"Although the general results are well-known, we briefly summarize them for the sake of completeness.",
"The potential has the form $V(x)=U \\left(1-e^{-\\kappa x}\\right)^2$ with the depth $U$ and the characteristic length scale $1/\\kappa $ .",
"Its eigenenergies are given by [43] $\\varepsilon _n=\\left[\\left(n+\\frac{1}{2}\\right)-\\frac{1}{2a}\\left(n+\\frac{1}{2}\\right)^2\\right]\\hbar \\nu $ for $n=0,1,2,...$ with the frequency $\\nu =\\kappa \\sqrt{2U/M}$ and the dimensionless parameter $a=\\sqrt{2MU}/\\hbar \\kappa $ .",
"The corresponding energy eigenstates $\\vert n\\rangle $ in the position representation read [44] $\\varphi _n(x)=&\\langle x\\vert n\\rangle \\nonumber \\\\=&\\mathcal {N}_n \\zeta ^{a-n-1/2}e^{-\\zeta /2}L_n^{(2a-2n-1)}(\\zeta )$ with the abbreviation $\\zeta =2a\\exp (-\\kappa x)$ and the generalized Laguerre polynomials $L_n^{(\\alpha )}(z)$ and $\\mathcal {N}_n=[\\kappa (2a-2n-1)n!/\\Gamma (2a-n)]^{1/2}$ [43], [45].",
"The highest bound state in this potential has the index $n=\\left\\lfloor {a-1/2}\\right\\rfloor $ .",
"Figure REF shows the Morse potential and its eigenenergies in units of $\\hbar \\nu $ along with the probability density of the bound states for $a=7$ .",
"The transition frequency between adjacent energy eigenstates decreases according to $\\omega _{n,n+1}=\\frac{a-1-n}{a}\\nu $ The matrix elements of the position operator required for the evaluation of the transition rates $A_{nm}$ were reported in [45], [46], [47] and can be written in the form $&\\langle n\\vert x\\vert m\\rangle =\\frac{(-1)^{n+m}}{\\kappa (n-m)(2a-n-m-1)}\\nonumber \\\\&\\times \\left[\\frac{m!}{n!",
"}\\frac{\\Gamma (2a-m)}{\\Gamma (2a-n)}(2a-2n-1)(2a-2m-1)\\right]^{1/2}$ for $n<m$ , while for $n>m$ the two indices have to be interchanged on the right-hand side.",
"The moments of the position operator in the eigenstates of the Morse potential are obtained from the generating function [46] $\\langle n\\vert e^{sx}\\vert n\\rangle &=\\frac{(2a-2n-1)n!",
"}{\\Gamma (2a-n)}e^{s\\log (2a)/\\kappa }\\nonumber \\\\&\\times \\sum _{j,l=0}^n\\frac{(-1)^{j+l}}{j!l!",
"}\\binom{2a-2n-1}{n-j}\\binom{2a-2n-1}{n-l}\\nonumber \\\\&\\times \\Gamma (2a-2n-1+j+l-s/\\kappa )$ by differentiation with respect to $s$ .",
"For the ground state $n=0$ the first and second moments are $\\langle 0\\vert x\\vert 0\\rangle &=\\frac{1}{\\kappa }\\big [\\log (2a)-\\psi ^{(0)}(2a-1)\\big ],\\\\\\langle 0\\vert x^2\\vert 0\\rangle &=\\frac{1}{\\kappa ^2}\\big [\\psi ^{(1)}(2a-1)+\\langle 0\\vert x\\vert 0\\rangle ^2\\big ]$ with the polygamma functions $\\psi ^{(n)}(z)$ [48], resulting in the ground-state variance $\\xi ^2=\\psi ^{(1)}(2a-1)/\\kappa ^2$ and thereby a smallness parameter $\\eta =\\frac{k}{\\kappa }\\sqrt{\\psi ^{(1)}(2a-1)}.$"
],
[
"Doppler regime",
"In Fig.",
"REF we show the mean steady state occupation $\\bar{m}$ in dependence of $\\Delta $ for $a=30$ and $\\Omega =\\Gamma /5$ assuming that the transition frequency between the lowest two states is again given by $\\omega _{01}=\\Gamma /10$ which leads to $\\nu =a/(a-1) \\Gamma /10\\approx 0.1034\\Gamma $ .",
"Optimal cooling is achieved for $\\Delta \\approx 0.509\\Gamma $ with a mean occupation of $\\bar{m}\\approx 3.54$ .",
"Figure: Cooling of atoms trapped in a Morse potential in the Doppler regime for a=30a=30, ν=aΓ/10(a-1)\\nu =a\\Gamma /10(a-1) (corresponding to ω 01 =Γ/10\\omega _{01}=\\Gamma /10) and Ω=Γ/5\\Omega =\\Gamma /5.",
"(a) Mean occupation m ¯\\bar{m} in dependence of the laser detuning Δ\\Delta .",
"The circle indicates optimal cooling occurring at Δ≈-0.51Γ\\Delta \\approx -0.51\\Gamma resulting in m ¯=Σ m mp m ≈3.54\\bar{m}=\\Sigma _mmp_m\\approx 3.54.",
"The inset shows logarithm of the steady state occupations p m p_m (m=0,...,24m=0,...,24) for optimal cooling.",
"In a thermal distribution the resulting curves would be linear–only approximately the case for the lower energy levels.",
"(b) Power spectrum of the motional sidebands.",
"The thin lines correspond to the first ten transitions between neighboring states on both sidebands.",
"The smallness parameter is given by η=0.1\\eta =0.1 and the detector angle by ψ=π/2\\psi =\\pi /2.In the inset we plot $-\\log (p_m)$ against the eigenenergies $\\varepsilon _m$ .",
"It can be seen that only the populations of the lower energy levels can be approximated by a thermal distribution while for higher states this is clearly not the case.",
"In the calculation only the finite number $\\left\\lfloor {a-1/2}\\right\\rfloor +1$ of bound states of the potential, here 30 for $a=30$ , are taken into account while free solutions are disregarded.",
"This is a good approximation for states which are energetically well localized within the range of bound states.",
"In Fig.",
"REF (b) we show the motional sidebands of an atom trapped in a Morse potential ($a=30$ ) calculated using Eq.",
"(REF ).",
"We used the parameters of optimal cooling, i.e.",
"$\\Delta =-0.509\\Gamma $ , a detection angle $\\psi =\\pi /2$ and a smallness parameter $\\eta =0.1$ .",
"The decreasing transition frequency between neighboring states is reflected in the fact that in the sideband spectrum the modulus of the peak positions is smaller than the first transition frequency $\\omega _{01}$ .",
"The thin curves under the sideband spectrum are the main components, according to the decomposition Eq.",
"(REF ), corresponding to the transitions $n+1\\rightarrow n$ for $n=0,...,9$ on the blue sideband and $n-1\\rightarrow n$ for $n=1,...,10$ on the red sideband, which add up to the complete spectrum but overlap because of their finite width."
],
[
"Resolved sideband regime",
"In the resolved sideband case with $\\nu =10a/(a-1)\\Gamma $ , corresponding to $\\omega _{01}=10\\Gamma $ , optimal cooling occurs when the laser is red-detuned by the transition frequency $\\omega _{01}$ , leading to a mean occupation $\\bar{m}=0.0026$ .",
"This is only slightly higher than in the harmonic case where $\\bar{m}=0.0016$ is achieved (calculated using $\\bar{m}=A_+/(A_--A_+)$ , see Sec.",
"REF ).",
"The fact that in this case the cooling is more efficient than in the square well case is due to the comparatively small anharmonicity of the Morse potential in the lower energy levels.",
"Figure: Cooling of an atom trapped in a Morse potential in the resolved sideband regime for ν=10a/(a-1)Γ\\nu =10a/(a-1)\\Gamma (corresponding to ω 01 =10Γ\\omega _{01}=10\\Gamma ) and Ω=Γ/5\\Omega =\\Gamma /5.",
"The mean occupation m ¯=Σ m mp m \\bar{m}=\\Sigma _mmp_m in dependence on the laser detuning Δ\\Delta takes its minimum m ¯≈0.0026\\bar{m}\\approx 0.0026 at Δ≈-ω 01 \\Delta \\approx -\\omega _{01} (indicated by a circle), where sideband cooling of the 1→01\\rightarrow 0 transition occurs.",
"(b) Power spectrum of the motional sidebands for a smallness parameter η=0.1\\eta =0.1 and ψ=π/2\\psi =\\pi /2.Figure REF (a) shows the mean occupation in dependence of the detuning $\\Delta $ where again shallow dips are visible where multiple transitions frequencies coincide, for example $\\omega _{2,0}=\\omega _{12,9}=-1.9\\nu $ , $\\omega _{4,1}=\\omega _{19,13}=-2.7\\nu $ , and $\\omega _{3,0}=\\omega _{21,14}=-2.8\\nu $ .",
"In this regime the motional sideband spectrum, depicted in Fig.",
"REF (b), is also mainly given by the two peaks arising from the transitions between the lowest energy states.",
"The insets show a magnification of the red and blue sideband."
],
[
"Conclusion",
"Following up on Dicke's original work on light scattered by atoms confined to an infinite square well and subsequent works on harmonically trapped atoms we present the spectrum of resonance fluorescence of laser cooled atoms trapped in arbitrary potentials.",
"The treatment relies on a perturbative analysis of the power spectrum of the scattered light up to second order in the Lamb-Dicke parameter and is based on the solution of the corresponding master equation describing the laser cooling dynamics.",
"We applied the results to two exemplary potentials, the infinite square well and the Morse potential, and distinguished between different cooling regimes, namely the Doppler and resolved sideband cooling.",
"In contrast to the harmonic trapping potential the steady state of the atomic motion does not take the form of a thermal distribution.",
"The spectrum of the motional sidebands consists of a series of peaks centered around the transitions between eigenstates of the atomic center-of-mass motion.",
"From our treatment also follow the widths of the individual peaks which are determined by the transitions rates between the motional states due to the interaction with the laser.",
"In the case of separated motional sideband peaks, the temperature of the cooled atom can be extracted from the individual heights.",
"When the peaks overlap, the resulting asymmetric envelope of the spectral signals stemming from the atomic motion can be used to determine the temperature by using our theoretical results.",
"Asymmetric motional sidebands have been observed in recent atom-cavity experiments where instead of the resonance fluorescence the spectrum of the cavity output was measured.",
"In this case the atom is trapped in the sinusoidal optical-lattice potential of the cavity mode.",
"With the work presented here we lay the ground work for further investigation of such periodic potentials offering possible means for temperature extraction in cavity-cooling experiments."
],
[
"Acknowledgments",
"The authors gratefully acknowledge helpful discussions with Giovanna Morigi and financial support from the GradUS program of the Saarland University, the German Ministry of Education and Research (BMBF \"Q.com\") and the German Research Foundation (DFG) within the Project No.",
"BI1694/1-1."
],
[
"Matrix representation of the internal Liouville operator",
"In the basis $\\lbrace \\vert g\\rangle \\langle g\\vert ,\\vert e\\rangle \\langle g\\vert ,\\vert g\\rangle \\langle e\\vert ,\\vert e\\rangle \\langle e\\vert \\rbrace $ the internal Liouville operator in Eq.",
"(REF ) can be written in its matrix representation [33] $\\mathcal {L}_{\\rm I}=\\frac{1}{2}\\begin{bmatrix}0 & -i\\Omega & i\\Omega & 2\\Gamma \\\\-i\\Omega & 2i\\tilde{\\Delta } & 0 & i\\Omega \\\\i\\Omega & 0 & -2i\\tilde{\\Delta }^\\ast & -i\\Omega \\\\0 & i\\Omega & -i\\Omega & -2\\Gamma \\end{bmatrix}$ with the complex detuning $\\tilde{\\Delta }=\\Delta +i\\Gamma /2$ .",
"From this we can easily derive the actual form of the internal eigenelements, being the eigenvectors of the matrix (REF ).",
"For instance, the steady state $\\rho _{\\rm st}$ of the internal dynamics, fulfilling $\\mathcal {L}_{\\rm I}\\rho _{\\rm st}=0$ , is given by $\\rho _{\\rm st}=\\frac{1}{N} \\begin{pmatrix}|\\tilde{\\Delta }|^2+\\Omega ^2/4 & \\tilde{\\Delta }\\Omega /2\\\\\\tilde{\\Delta }^\\ast \\Omega /2 & \\Omega ^2/4\\\\\\end{pmatrix}$ in the basis $\\lbrace \\vert g\\rangle , \\vert e\\rangle \\rbrace $ , with the normalization constant $N=\\Gamma ^2/4+\\Delta ^2+\\Omega ^2/2$ defined in the main text."
],
[
"Weight factors",
"We will outline the main steps in the evaluation of the weight factors that contribute to the motional sidebands.",
"They can be calculated according to ${\\rm Tr}\\big \\lbrace D_-^{(0)}\\hat{\\varrho }^{(1)}_\\lambda +D_-^{(1)}\\hat{\\varrho }^{(0)}_\\lambda \\big \\rbrace =&{\\rm Tr}\\big \\lbrace \\sigma _+\\big [(\\lambda _{nm}-\\mathcal {L}_0)^{-1}\\mathcal {Q}_{nm}\\mathcal {L}_1+ik\\cos \\psi \\,x\\big ]\\rho _{\\rm st}\\hat{\\mu }_{nm}\\big \\rbrace \\nonumber \\\\=&-i\\left[ r(\\omega _{nm})-k\\cos \\psi {\\rm Tr}\\lbrace \\sigma _+\\rho _{\\rm st}\\rbrace \\right]{\\rm Tr}\\lbrace x\\hat{\\mu }_{nm}\\rbrace ,\\\\{\\rm Tr}\\big \\lbrace \\big (\\check{\\varrho }^{\\dagger (1)}_\\lambda D_+^{(0)}+\\check{\\varrho }^{\\dagger (0)}_{\\lambda }D_+^{(1)}\\big )\\varrho ^{(0)}_{\\rm st}\\big \\rbrace =&{\\rm Tr}\\big \\lbrace \\check{\\mu }_{nm}^\\dag \\big [\\mathcal {L}_1(\\lambda _{nm}-\\mathcal {L}_0)^{-1}\\mathcal {Q}_{nm}-ik\\cos \\psi \\,x\\big ]\\sigma _+\\rho _{\\rm st}\\hat{\\mu }_{nm}\\big \\rbrace \\nonumber \\\\=-i\\Big [\\frac{1}{\\hbar }{\\rm Tr}\\big \\lbrace W_1&(\\lambda _{nm}-\\mathcal {L}_{\\rm I})^{-1}\\mathcal {Q}_{nm}\\sigma _+\\rho _{\\rm st}\\big \\rbrace {\\rm Tr}\\lbrace \\check{\\mu }^\\dag _{nm}[x,\\mu _{\\rm st}]\\rbrace +k\\cos \\psi {\\rm Tr}\\lbrace \\sigma _+\\rho _{\\rm st}\\rbrace {\\rm Tr}\\lbrace \\check{\\mu }^\\dag _{nm}x\\rbrace \\Big ]\\\\{\\rm Tr}\\big \\lbrace \\check{\\varrho }^{\\dagger (0)}_{\\lambda }D_+^{(0)}\\varrho ^{(1)}_{\\rm st}\\big \\rbrace =&-{\\rm Tr}\\big \\lbrace \\check{\\mu }_{nm}^\\dag \\sigma _+\\mathcal {L}_0^{-1}\\mathcal {Q}_0\\mathcal {L}_1\\rho _{\\rm st}\\mu _{\\rm st}\\big \\rbrace \\nonumber \\\\=i\\Big [r^\\ast (\\omega _{nm})&{\\rm Tr}\\lbrace \\check{\\mu }^\\dag _{nm}x\\mu _{\\rm st}\\rbrace +\\frac{1}{\\hbar }{\\rm Tr}\\big \\lbrace \\sigma _+(\\lambda _{nm}+\\mathcal {L}_{\\rm I})^{-1}\\mathcal {Q}_{nm}\\rho _{\\rm st}W_1\\big \\rbrace {\\rm Tr}\\lbrace \\check{\\mu }^\\dag _{nm}[x,\\mu _{\\rm st}]\\rbrace \\Big ].$ where we have used the corrections Eq.",
"(REF ), Eq.",
"(), Eq.",
"(REF ) and Eq.",
"() as well as the explicit form of $\\mathcal {L}_1$ and the projector $\\mathcal {Q}_{nm}=1-\\mathcal {P}_{\\lambda _{nm}}$ .",
"Addition of these terms yields the result $ w^{(2)}_{\\lambda }&=\\vert \\langle n\\vert x\\vert m\\rangle \\vert ^2\\Big [p_m\\big | r(\\omega _{nm})-k\\cos \\psi {\\rm Tr}\\lbrace \\sigma _+\\rho _{\\rm st}\\rbrace \\big |^2 \\nonumber \\\\+&(p_n-p_m)\\big (r(\\omega _{nm})-k\\cos \\psi {\\rm Tr}\\lbrace \\sigma _+\\rho _{\\rm st}\\rbrace \\big )q(\\omega _{nm})\\Big ],$ where we already inserted the actual form of the motional steady state $\\mu _{\\rm st}=\\Sigma _j p_j \\vert j\\rangle \\langle j\\vert $ and the two definitions Eqs.",
"(REF ) and ().",
"By evaluating ${\\rm Tr}\\lbrace \\sigma _+\\rho _{\\rm st}\\rbrace =[\\Delta +i\\Gamma /2]\\Omega /2N$ , with help of the internal steady state (REF ), this corresponds to Eq.",
"(REF )."
],
[
"Explicit expressions for the functions $r(\\omega )$ , {{formula:3cfa16cb-b99d-462d-97bb-eba48c0ba682}} and {{formula:8075e0f4-68b2-4362-9291-c41487990e6e}}",
"In this section we give explicit expressions for the two functions $r(\\omega )$ and $q(\\omega )$ from Eqs.",
"(REF ) and () as well as $s(\\omega )$ from Eq.",
"(REF ).",
"Using the quantum regression theorem [35] $\\langle X(t)Y(0)\\rangle _{\\rm st}={\\rm Tr}\\lbrace X\\exp (\\mathcal {L}t)Y\\varrho _{\\rm st}\\rbrace $ and formal integration $\\int _0^\\infty {\\rm d}t\\, \\exp (-z+\\mathcal {L})t=(z-\\mathcal {L})^{-1}$ they can be rewritten as $r(\\omega )=&\\frac{1}{\\hbar }{\\rm Tr}\\big \\lbrace \\sigma _+(i\\omega -\\mathcal {L}_{\\rm I})^{-1}[W_1,\\rho _{\\rm st}]\\big \\rbrace =\\frac{\\Omega }{2}k\\cos \\phi \\frac{\\Gamma N \\tilde{\\Delta }+i[(\\tilde{\\Delta }+i\\Gamma )|\\tilde{\\Delta }|^2+\\Delta \\Omega ^2]\\omega -i|\\tilde{\\Delta }|^2\\omega ^2}{\\Gamma N^2+i[5\\Gamma ^2/4+\\Delta ^2+\\Omega ^2]N\\omega -2\\Gamma N\\omega ^2-iN\\omega ^3},\\\\q(\\omega )=&\\frac{1}{\\hbar }{\\rm Tr}\\big \\lbrace W_1(i\\omega -\\mathcal {L}_{\\rm I})^{-1}\\sigma _-\\rho _{\\rm st}\\big \\rbrace -\\frac{1}{\\hbar }{\\rm Tr}\\big \\lbrace \\sigma _-(i\\omega +\\mathcal {L}_{\\rm I})^{-1}\\rho _{\\rm st}W_1\\big \\rbrace \\nonumber \\\\=&\\frac{\\Omega ^3}{2}k\\cos \\phi \\frac{i\\Gamma ^3 N\\tilde{\\Delta }^\\ast -\\Gamma ^2\\tilde{\\Delta }^\\ast N\\omega +[\\Gamma ^2(B+\\Omega ^2)-i\\Gamma B\\tilde{\\Delta }^\\ast ]\\omega ^2+B\\tilde{\\Delta }^\\ast \\omega ^3+[B-i\\Gamma \\tilde{\\Delta }]\\omega ^4+\\tilde{\\Delta }^\\ast \\omega ^5+\\omega ^6}{2\\Gamma ^2N^3\\omega +[\\Gamma ^2(9\\Gamma ^2/8-3\\Delta ^2+\\Omega ^2)+2(\\Delta ^2+\\Omega ^2)^2]N\\omega ^3+4BN\\omega ^5},\\\\s(\\omega )=&-\\frac{1}{\\hbar ^2}{\\rm Tr}\\big \\lbrace W_1(i\\omega +\\mathcal {L}_{\\rm I})^{-1}W_1\\rho _{\\rm st}\\big \\rbrace \\nonumber \\\\=&\\frac{\\Omega ^2}{4}k^2\\cos ^2\\phi \\frac{\\Gamma ^3\\Omega ^2/(4N)+[\\Gamma (\\Delta -3i\\Gamma /2)+i\\Gamma ^2|\\tilde{\\Delta }|^2/N]\\omega +[i\\Delta |\\tilde{\\Delta }|^2/N-3\\Gamma /2-2i\\Delta ]\\omega ^2+i\\omega ^3}{-i\\Gamma N\\omega -[5\\Gamma ^2/4+\\Delta ^2+\\Omega ^2]\\omega ^2+2i\\Gamma \\omega ^3+\\omega ^4}.$ with $\\tilde{\\Delta }=\\Delta +i\\Gamma /2$ and the constant $B=3\\Gamma ^2/4-\\Delta ^2-\\Omega ^2$ .",
"In the evaluation of the trace we employed the explicit form of the matrices $(i\\omega \\mp \\mathcal {L}_{\\rm I})^{-1}$ which can be obtained by using the Liouville operator Eq.",
"(REF )."
],
[
"Second order eigenvalue corrections",
"We briefly go into the derivation of the perturbative corrections of the eigenvalues that lead to a non-vanishing real part which becomes immanent in the finite width of the sideband peaks.",
"Their form is given by $ \\lambda _2=&{\\rm Tr}\\big \\lbrace \\check{\\varrho }^{\\dagger (0)}_\\lambda \\big [\\mathcal {L}_2+\\mathcal {L}_1(\\lambda _0-\\mathcal {L}_0)^{-1}\\mathcal {Q}_\\lambda \\mathcal {L}_1\\big ]\\hat{\\varrho }^{(0)}_\\lambda \\big \\rbrace .$ We will treat the two terms separately, starting with the first one that involves $\\mathcal {L}_2$ .",
"For $\\lambda _0=\\lambda _{nm}$ this part yields $ {\\rm Tr}&\\lbrace \\check{\\mu }^\\dag _{nm}\\mathcal {L}_2\\rho _{\\rm st}\\hat{\\mu }_{nm}\\rbrace =\\frac{1}{i\\hbar }{\\rm Tr}\\lbrace W_1\\rho _{\\rm st}\\rbrace \\big [\\langle n\\vert x^2\\vert n\\rangle -\\langle m\\vert x^2\\vert m\\rangle \\big ]\\nonumber \\\\&+\\frac{D}{2}\\big [2\\langle n\\vert x\\vert n\\rangle \\langle m\\vert x\\vert m\\rangle -\\langle n\\vert x^2\\vert n\\rangle -\\langle m\\vert x^2\\vert m\\rangle \\big ]$ The second term in (REF ) can be brought in the form $ {\\rm Tr}&\\big \\lbrace \\check{\\mu }^\\dag _{nm}\\mathcal {L}_1(\\lambda _{nm}-\\mathcal {L}_0)^{-1}\\mathcal {Q}_\\lambda \\mathcal {L}_1\\rho _{\\rm st}\\hat{\\mu }_{nm}\\big \\rbrace =\\nonumber \\\\&-\\sum _j \\left[s^\\ast (\\omega _{jn})\\vert \\langle m\\vert x\\vert j\\rangle \\vert ^2+s(\\omega _{jm})\\vert \\langle n\\vert x\\vert j\\rangle \\vert ^2\\right].$ After recombining these two results using $\\langle n\\vert x^2\\vert n\\rangle =\\Sigma _j \\vert \\langle n\\vert x\\vert j\\rangle \\vert ^2$ we obtain $\\lambda _2^{nm}=&\\frac{i\\Delta \\Omega ^2k^2\\cos ^2\\phi }{4N} \\left(\\langle n\\vert x^2\\vert n\\rangle -\\langle m\\vert x^2\\vert m\\rangle \\right)\\nonumber \\\\&-\\frac{1}{2}\\sum _j\\left[2s(\\omega _{jn})+D\\right] \\vert \\langle j\\vert x\\vert n\\rangle \\vert ^2\\nonumber \\\\&-\\frac{1}{2}\\sum _j\\left[2s^\\ast (\\omega _{jm})+D\\right] \\vert \\langle j\\vert x\\vert m\\rangle \\vert ^2\\nonumber \\\\&+D\\langle n\\vert x\\vert n\\rangle \\langle m\\vert x\\vert m\\rangle .$ With the definition of the transition rates $A_{nm}$ given by Eq.",
"(REF ) this corresponds to Eqs.",
"(REF )- (REF )."
],
[
"Rate equation",
"We outline the main steps to transform the equation of the reduced dynamics in the subspace for $\\lambda =0$ , i.e.",
"Eq.",
"(REF ), into a linear system $\\mathcal {A}\\mathbf {p}=0$ with $\\mathbf {p}=(p_0,p_1,...)^T$ .",
"Since we aim at determining the steady state of the external dynamics it is necessary to trace out the internal degrees of freedom in (REF ) yielding $ {\\rm Tr}_{\\rm I}\\big \\lbrace \\mathcal {P}_0\\left(\\mathcal {L}_1\\mathcal {L}_0^{-1}\\mathcal {Q}_0\\mathcal {L}_1-\\mathcal {L}_2\\right)\\mathcal {P}_0\\rho _{\\rm st}\\mu _{\\rm st}\\big \\rbrace =0.$ The action of the projector $\\mathcal {P}_0$ is given by $\\mathcal {P}_0 \\rho \\mu =\\rho _{\\rm st}\\sum _n\\langle n\\vert \\mu \\vert n\\rangle \\vert n\\rangle \\!\\langle n\\vert $ with internal and external density operators $\\rho $ and $\\mu $ , respectively.",
"For the second term in Eq.",
"(REF ) we obtain $ \\langle n\\vert {\\rm Tr}_{\\rm I}\\lbrace \\mathcal {L}_2\\rho _{\\rm st}\\mu _{\\rm st}\\rbrace \\vert n\\rangle =D\\sum _{m} (p_m-p_n) \\vert \\langle n\\vert x\\vert m\\rangle \\vert ^2.$ The evaluation of the first term is somewhat more involved and reads $ \\langle n&\\vert {\\rm Tr}_{\\rm I}\\lbrace \\mathcal {L}_1\\mathcal {L}_0^{-1}\\mathcal {Q}_0\\mathcal {L}_1\\rho _{\\rm st}\\mu _{\\rm st}\\rbrace \\vert n\\rangle \\nonumber \\\\&=\\int _0^\\infty dt\\,\\langle n\\vert {\\rm Tr}_{\\rm I}\\lbrace \\mathcal {L}_1e^{\\mathcal {L}_0t}\\mathcal {Q}_0\\mathcal {L}_1\\rho _{\\rm st}\\mu _{\\rm st}\\rbrace \\vert n\\rangle \\nonumber \\\\&=2\\,{\\rm Re}\\sum _{m} \\big [p_ms(\\omega _{n,m})-p_ns(\\omega _{m,n})\\big ]\\vert \\langle n\\vert x\\vert m\\rangle \\vert ^2 $ with $s(\\omega )$ defined in Eq.",
"(REF ).",
"Here we inserted the actual form of the Liouville operators and separated the internal from the external expressions.",
"We can now recombine the two terms leading to $ \\sum _{m} A_{nm}p_m-\\sum _{m} A_{mn}p_n=0.$ This is the set of equations that determines $\\mathbf {p}$ and can be expressed in the matrix notation $\\mathcal {A}\\mathbf {p}=0$ with $ \\mathcal {A}=\\begin{bmatrix}-\\!\\sum \\limits _{m\\ne 0} A_{m0}& A_{01} & A_{02} & \\dots \\\\A_{10} & -\\!\\sum \\limits _{m\\ne 1} A_{m1} & A_{12} &\\\\A_{20} & A_{21} & -\\!\\sum \\limits _{m\\ne 2} A_{m2} &\\\\\\vdots & & &\\ddots \\end{bmatrix}.$"
]
] | 1605.05951 |
[
[
"Linked-cluster expansions for quantum magnets on the hypercubic lattice"
],
[
"Abstract For arbitrary space dimension $d$ we investigate the quantum phase transitions of two paradigmatic spin models defined on a hypercubic lattice, the coupled-dimer Heisenberg model and the transverse-field Ising model.",
"To this end high-order linked-cluster expansions for the ground-state energy and the one-particle gap are performed up to order 9 about the decoupled-dimer and high-field limits, respectively.",
"Extrapolations of the high-order series yield the location of the quantum phase transition and the correlation-length exponent $\\nu$ as a function of space dimension $d$.",
"The results are complemented by $1/d$ expansions to next-to-leading order of observables across the phase diagrams.",
"Remarkably, our analysis of the extrapolated linked-cluster expansion allows to extract the coefficients of the full $1/d$ expansion for the phase-boundary location in both models exactly in leading order and quantitatively for subleading corrections."
],
[
"Introdcution",
"Zero-temperature phase transitions[1], [2], [3] are a central topic in many domains of modern physics, in particular in correlated-electron systems.",
"Such a quantum phase transition (QPT), driven by varying a non-thermal parameter like magnetic field, pressure, or doping, implies a qualitative change of the system's ground state.",
"Near a continuous QPT one observes unconventional finite-temperature behavior which is universal, i.e., independent of microscopic details.",
"Quantum magnets are a perfect playground, both experimentally and theoretically, to investigate quantum criticality and the associated universal behavior.",
"Here, theoretical investigations can mainly be distinguished in two groups.",
"The first studies continuum-limit quantum field theories using, e.g., renormalization-group techniques – these are designed to access the long-wavelength behavior near criticality.",
"The second works with lattice models which are analyzed using approximate analytical or numerical techniques – such approaches capture microscopic aspects of the problem at hand, but are often unsuitable to describe critical behavior.",
"One powerful technique to study quantum lattice models and QPTs are linked-cluster expansions (LCEs) [4].",
"Here high-order series expansions are derived at zero temperature in the thermodynamic limit for relevant physical quantities of the microscopic model under consideration.",
"The obtained series can then be extrapolated, giving access to quantum critical points and critical exponents.",
"In practice, LCEs are done most efficiently via a full graph expansion, i.e.",
"the topologically distinct fluctuations are determined on finite graphs and are then embedded directly into the thermodynamic-limit system.",
"LCEs therefore consist of two independent steps: the explicit calculation on graphs and the embedding procedure.",
"The latter is essentially the combinatorial problem to find the number of possible embeddings of a graph on the infinite lattice.",
"LCEs are usually performed for a microscopic lattice Hamiltonian in fixed space dimension $d$ .",
"However, since it is only the embedding procedure and the selection of graphs which depend explicitly on $d$ , it is also possible to realize LCEs for arbitrary $d$ so that the high-order series of physical quantities contain $d$ a free parameter.",
"Such LCEs for general $d$ have been set up for classical Ising and Potts models on the hypercubic lattice in Refs.",
"Fisher90, Hellmund06.",
"Recently, we have extended such expansions to ground-state energies and one-particle dispersions for quantum lattice problems in Ref.",
"Joshi2015 where they were mainly used as cross-check of the $1/d$ expansion methodology developed there.",
"It is the purpose of this paper to study QPTs of quantum magnets on the hypercubic lattice for arbitrary $d$ using LCEs.",
"We have pushed the series expansions for general $d$ to considerably higher orders which allows us to investigate the QPT of both the coupled-dimer Heisenberg model (like in Ref.",
"Joshi2015) and the transverse-field Ising model.",
"Specifically, we calculate high-order LCEs for the ground-state energy and for the one-particle gap about the decoupled dimer (high-field) limit for both models.",
"Extrapolations of these quantities allow us to determine the location of the phase boundary as function of $d$ and to predict the behavior of the leading coefficients of the full $1/d$ -expansion.",
"Surprisingly, the first-order coefficient can be extracted exactly from the extrapolation.",
"We complement the LCE results by those from a systematic $1/d$ expansion to next-to-leading order in both the disordered and ordered phases: Such an expansion has been developed for coupled dimers in Refs.",
"Joshi2015, Joshi2015b and is extended here to the transverse-field Ising model.",
"The paper is organized as follows.",
"In Sect.",
", we introduce the two microscopic models studied in this work.",
"The technical aspects including the high-order series expansions for the ground-state energy and the one-particle gap as well as their extrapolation are contained in Sect.",
", while details of the $1/d$ expansions are given in Sect. .",
"The physical implications and findings are discussed in Sect.",
"and we conclude our work in Sect.",
"."
],
[
"Models",
"We consider two archetypical models on the hypercubic lattice with dimension $d$ : i) a coupled-dimer Heisenberg magnet (CDHM) of spins $1/2$ and ii) the transverse-field Ising model (TFIM).",
"In both cases an infinite-$d$ limit exists which features a non-trivial QPT."
],
[
"Coupled-dimer Heisenberg model",
"The CDHM is a model of dimers, i.e., pairs of spins 1/2, on the sites $i$ of a hypercubic lattice.",
"The Hamiltonian reads $\\mathcal {H}_{\\rm CDHM} =J \\sum _i \\vec{S}_{i1} \\cdot \\vec{S}_{i2} +\\sum _{\\langle ij\\rangle } (K^{11} \\vec{S}_{i1} \\cdot \\vec{S}_{j1} + K^{22} \\vec{S}_{i2} \\cdot \\vec{S}_{j2})\\,,$ where $\\sum _{\\langle ij\\rangle }$ denotes a summation over pairs of nearest-neighbor dimer sites $i,j$ , and $1,2$ refer to the individual spins on each dimer.",
"We restrict our attention to the symmetric case of $K^{11}=K^{22}\\equiv K$ .",
"For $d=1$ and $d=2$ the spin lattice of $\\mathcal {H}_{\\rm CDHM}$ in Eq.",
"(REF ) corresponds to the much-studied two-leg ladder and square-lattice bilayer magnets, respectively.",
"A non-trivial limit $d\\rightarrow \\infty $ is obtained if the inter-dimer coupling constant $K$ is scaled as $1/d$ in order to preserve a competition between the $K$ and $J$ terms in (REF ).",
"For $d\\ge 2$ and $K,J>0$ , the dimensionless parameter $q_{\\rm CDHM} = \\frac{Kd}{J}$ controls a QPT between a singlet paramagnet with gapped triplon excitations [9] at small $q$ and a gapless antiferromagnet with ordering wavevector $\\vec{Q}=(\\pi ,\\pi ,\\ldots )$ at large $q$ .",
"For $d=2$ this transition occurs at[10] $q_c= 0.793$ while $q_c= 0.620$ has been found recently for $d=3$ in Ref. Qin2015.",
"The upper critical dimension of the QPT, with O(3) order parameter, is $d=3$ .",
"Consequently, the critical exponents take mean-field values $\\alpha =0$ , $\\beta =1/2$ , $\\gamma =1$ , $\\delta =3$ , and $\\nu =1/2$ for all dimensions $d\\ge 3$ .",
"In contrast, one expects continuously varying anomalous exponents for $1<d<3$ .",
"For $d=3$ one finds multiplicative logarithmic corrections to mean-field behavior [12], [13], [11]."
],
[
"Transverse-field Ising model",
"The TFIM is described by the Hamiltonian $\\mathcal {H}_{\\rm TFIM} = -h\\sum _i \\sigma ^{x}_i -J \\sum _{\\langle i j\\rangle } \\sigma ^{z}_i\\sigma ^{z}_j\\,,$ where $\\sum _{\\langle ij\\rangle }$ denotes a summation over nearest-neighbor spins $i,j$ on the hypercubic lattice and $\\sigma ^\\alpha _i$ with $\\alpha \\in \\lbrace x,y,z\\rbrace $ are Pauli matrices representing spin-1/2 degrees of freedom.",
"In the following we assume a ferromagnetic Ising exchange $J>0$ , but our results are equally valid for the antiferromagnetic case, since both cases can be mapped onto each other via a sublattice rotation on the hypercubic lattice.",
"A non-trivial limit $d\\rightarrow \\infty $ is obtained again by scaling the Ising coupling constant $J$ with $1/d$ .",
"For $d\\ge 1$ , the dimensionless parameter $q_{\\rm TFIM} = \\frac{Jd}{h}$ controls the QPT between two gapped phases, the field-polarized phase at small $q$ and the ordered ferromagnetic state with broken $\\mathbb {Z}_2$ symmetry at large $q$ .",
"For $d=1$ , this model reduces to the well-known Ising chain in a transverse field, with the transition located at $q_c= 1$ .",
"For $d=2$ , the quantum critical point has been determined by series expansions and quantum Monte Carlo and is known to be[14], [15], [16], [17], [18] $q_c\\approx 0.657$.",
"For the simple cubic lattice ($d=3$ ) series expansions and quantum Monte Carlo finds[19], [17] $q_c\\approx 0.582$ .",
"The order-parameter symmetry of the TFIM is $\\mathbb {Z}_2$ .",
"The upper critical dimension is $d=3$ as well, implying mean-field exponents for $d\\ge 3$ , continuously varying exponents for $1\\le d<3$ (different from the ones of the CDHM), and multiplicative logarithmic corrections [20], [21], [22], [19] for $d=3$ ."
],
[
"Linked-cluster expansion for arbitrary $d$",
"In this section we provide the relevant technical aspects with respect to linked-cluster expansions for arbitrary dimension $d$ for the disordered phases of the CDHM and the TFIM on the hypercubic lattice.",
"In practice, we calculated high-order series expansions for the ground-state energy per dimer (spin) and for the one-particle gap up to order 9 in the relative strength of the inter-dimer (inter-site) coupling $k= K/J$ ($k= J/h$ ) for the CDHM (TFIM)."
],
[
"Method",
"We start by sketching the underlying method of the expansion; for details we refer the reader to Refs. gsu00,Knetter031,Coester2015.",
"The expansion's reference point corresponds to $k=0$ .",
"Here the ground state is given by a product state in both models.",
"For the CDHM there are singlets on the dimers, and elementary excitations are local triplets with excitation energy $\\Delta ^{\\rm CDHM}/J=1$ .",
"In contrast, the TFIM is in the fully polarized state with all spins pointing along $\\hat{x}$ , and the elementary excitations are local spin flips corresponding to a spin along $-\\hat{x}$ with excitation energy $\\Delta ^{\\rm TFIM}/2h=1$ .",
"After a global energy shift, we can rewrite both models in the form $\\mathcal {H}&=&\\mathcal {H}_0+k\\, \\hat{V} \\quad ,$ where $\\mathcal {H}_0$ has an equidistant spectrum bounded from below counting the number of excitations, namely triplets (spin flips) for the CDHM (TFIM).",
"The perturbing parts can then be written as $\\hat{V}=\\hat{T}_{-2}+\\hat{T}_0+\\hat{T}_{2} \\, ,$ where $\\hat{T}_m$ changes the total number of excitations by $m\\in \\lbrace \\pm 2, 0\\rbrace $.",
"Note that only terms with even $m$ appear in $\\hat{V}$ due to the exact reflection symmetry of both models leading to a conserved parity quantum number.",
"Each operator $\\hat{T}_m$ is a sum over local operators connecting two nearest-neighbor sites (either dimers or spins).",
"One can therefore write $\\hat{T}_m=\\sum _l \\hat{\\tau }_{m,l}\\,,$ with $\\hat{\\tau }_{m,l}$ effecting only the two sites connected by the link $l$ on the lattice.",
"The perturbative continuous unitary transformations (pCUTs) [23], [24], [25] map the original Hamiltonian to an effective quasiparticle conserving Hamiltonian of the form $\\hat{H}_\\text{eff}(k)=\\hat{H}_0+\\sum _{n=1}^{\\infty }k^n \\hspace*{-5.69054pt}\\sum _{{\\rm dim}(\\underline{m})=n \\atop \\,M(\\underline{m})=0} \\hspace*{-5.69054pt}C(\\underline{m})\\,\\hat{T}_{m_1}\\dots \\hat{T}_{m_n},$ where $n$ reflects the perturbative order.",
"The second sum is taken over all possible vectors $\\underline{m}\\!\\equiv \\!",
"(m_1, \\ldots , m_n)$ with $m_i\\in \\lbrace \\pm 2, 0\\rbrace $ and dimension ${\\rm dim}(\\underline{m})=n$ .",
"Each term of this sum is weighted by the rational coefficient $C(\\underline{m})\\in \\mathbb {Q}$ which has been calculated model-independently up to high orders.",
"[23] The additional restriction $M(\\underline{m})\\equiv \\sum m_i=0$ reflects the quasiparticle-conserving property of the effective Hamiltonian, i.e.",
", the resulting Hamiltonian is block-diagonal in the number of quasiparticles $[\\hat{H}_\\text{eff},\\hat{H}_0]=0$ .",
"Each quasiparticle block can then be investigated separately which represents a major simplification of the complicated many-body problem.",
"The operator products $\\hat{T}_{m_1}\\dots \\hat{T}_{m_n}$ appearing in order $n$ can be interpreted as virtual fluctuations of “length” $l\\le n$ leading to dressed quasiparticles.",
"According to the linked-cluster theorem, only linked fluctuations can have an overall contribution to the effective Hamiltonian $\\hat{H}_\\text{eff}$ .",
"Hence, the properties of interest can be calculated in the thermodynamic limit by applying the effective Hamiltonian on finite clusters.",
"Considering all linked fluctuations on the lattice (for arbitrary $d$ ), it becomes clear that the contribution of each fluctuation only depends on its topology.",
"The calculation separates therefore into two independent steps: (i) The model-dependent part of the calculation is performed on the finite set of topologically distinct graphs.",
"(ii) The model-independent part corresponds to the combinatorial problem of how many times the contribution on the graphs has to be embedded into the lattice in order to extract thermodynamic-limit properties.",
"For the calculation in $d$ dimensions, it is the second model-independent part which represents the real challenge.",
"As a consequence, we reach the same maximum order 9 for both models, both for the ground-state energy and the one-particle gap."
],
[
"Ground-state energy",
"In the following, we discuss the calculation of the ground-state energy per dimer (spin) $\\epsilon _0^{\\rm CDHM}$ ($\\epsilon _0^{\\rm TFIM}$ ) for the hypercubic CDHM (TFIM) for arbitrary $d$ .",
"The calculation is performed up to order $k^9$ , using pCUTs and a full graph decomposition.",
"The first step of the calculation is conventional and it is part of any LCE.",
"One determines the (reduced) contributions to the ground-state energy $\\epsilon _{0,n}$ , which is done in pCUTs by simply applying Eq.",
"(REF ) to the zero-particle state on the relevant graphs.",
"In order to avoid double counting of contributions, the reduced contribution $\\epsilon _{0,i}$ to $\\epsilon _0$ of each graph $\\mathcal {G}_i$ has to be calculated by subtracting the contributions of all subgraphs.",
"Up to perturbative order $n$ , only graphs up to $n$ links have to be considered due to the linked-cluster theorem.",
"Additionally, it is useful to check whether the graphs fit onto the lattice and whether each graph has a finite contribution in the order under consideration.",
"The latter depends on both, the model and the observable.",
"In the case of the ground-state energy one has a specific selection rule for both models that each link has to be touched twice by the perturbation as long as it is not part of a closed loop of links [7].",
"This property drastically reduces the total number of graphs which one has to treat.",
"The embedding factor $\\nu _i(d)$ for graph $\\mathcal {G}_i$ , being the number of possible embeddings of $\\mathcal {G}_i$ on the lattice, is a function of the spatial dimension $d$ .",
"The ground-state energy in the thermodynamic limit is then given by $\\epsilon _0 =\\sum _i \\nu _i(d)\\, \\epsilon _{0,i}\\,.$ The determination of the embedding factors $\\nu _i (d)$ for arbitrary $d$ is the most challenging part of the calculation.",
"Note that the embedding factors $\\nu _i (d)$ are exactly the same for both models; only the contributions $\\epsilon _{0,i}$ are model-dependent.",
"To determine the embedding factor in $d$ dimensions, we apply a scheme similar to the one presented in Ref. Hellmund06.",
"Each graph can be associated with a dimension defined by the number of dimensions that the graph can maximally occupy.",
"The $d$ -dependent embedding factor is then given by a polynomial in $d$ of degree $m_\\text{max}$ which is determined by the embedding factors of the graph in dimension $d=1$ , $d=2, \\dots , d=m_\\text{max}$ .",
"In order to determine the embedding factors it is necessary to divide the number of naive embeddings by the symmetry factor $S_i$ of $\\mathcal {G}_i$ .",
"Otherwise one overcounts contributions, since embeddings connected by a symmetry mapping of the graph represent exactly the same fluctuation on the lattice in the thermodynamic limit.",
"Following these principles, we find the following ground-state energy per dimer for the CDHM $\\frac{\\epsilon _0^{\\rm CDHM}}{J} &= -\\frac{3}{4}-\\frac{3}{8}d\\,{k}^{2}-\\frac{3}{16}d\\,{k}^{3}+\\Big (\\frac{21}{128}d-\\frac{9}{64}d^2\\Big ){k}^{4}\\\\&+\\Big (\\frac{57}{256}d-\\frac{3}{64}d^2\\Big ){k}^{5}+\\Big (-\\frac{2781}{1024}d + \\frac{273}{64}d^2 \\\\&-\\frac{357}{256}d^3 \\Big ){k}^{6} +\\Big (-\\frac{73293}{16384}d + \\frac{53205}{8192}d^2 \\\\&- \\frac{8499}{4096}d^3 \\Big ){k}^{7}+\\Big ( \\frac{1151577}{32768} d-\\frac{2270385}{32768} d^2\\\\&+\\frac{687885}{16384} d^3-\\frac{8313}{1024} d^4\\Big ) {k}^{8} +\\Big ( \\frac{80239263}{1048576} d\\\\&-\\frac{75882381}{524288} d^2+\\frac{21745125}{262144} d^3-\\frac{490731}{32768} d^4 \\Big ) {k}^{9}$ and the ground-state energy per spin for the TFIM $\\frac{\\epsilon _0^{\\rm TFIM}}{2h} &= -\\frac{1}{2} -\\frac{1}{8} {k}^2d+\\Big (\\frac{13}{128}d-\\frac{7}{64} d^2\\Big ) {k}^{4} \\\\& +\\Big ( -\\frac{367}{512}d+\\frac{311}{256}d^2-\\frac{1}{2}d^3 \\Big ) {k}^{6} \\\\& +\\Big (\\frac{7031}{512}d-\\frac{947263}{32768}d^2+\\frac{321187}{16384}d^3-\\frac{4535}{1024}d^4 \\Big ) {k}^{8}\\,.$ One can easily check that these results match the ones from literature for specific dimensions.",
"For $d=1$ , this formula reduces to the known results for the CDHM on a two-leg ladder [26] and for the TFIM on a chain [27], [4].",
"For $d=2$ , we reproduce the numerical results of the ground-state energy for the square-lattice Heisenberg bilayer [28] and the square lattice TFIM [14].",
"For $d=3$ , the series expansion for the TFIM is known [19] while for the CDHM the high-order series expansion was unknown."
],
[
"One-particle gap",
"For the one-particle dispersion $\\omega _{\\vec{k}}$ , the embedding procedure is more demanding, since each embedding is associated with different hopping elements in the thermodynamic limit.",
"However, if one is only interested in the one-particle gap which is located at $\\vec{k}=\\vec{Q}$ for the CDHM and at $\\vec{k}=0$ for the TFIM, it is possible to apply a similar scheme as for the ground-state energy: The hypercubic lattice is bipartite, i.e., it can be divided into two sublattices.",
"The hopping elements can then be classified into hopping elements on the same sublattice and hopping elements between both sublattices.",
"At $\\vec{k}=0$ , relevant for the TFIM, the contribution of all hopping elements is simply given by the embedding factor.",
"However, at $\\vec{k}=(\\pi ,\\pi ,\\ldots ,\\pi )$ relevant for the CDHM, only hopping elements on the same sublattice are given by the embedding factor, while the contribution of hopping elements between different sublattices must be modified with an additional sign due to the gap momentum.",
"Consequently, the one-particle gap $\\Delta $ of the CDHM is given by $\\Delta =\\sum _{i} \\nu _i (d) \\Delta ^{\\mathcal {G}_i}$ with $\\Delta ^{\\mathcal {G}_i}=\\sum _{l,m} \\sigma _{l,m} t^{(i)}_{l,m}$ where the double sum runs over all sites of the graph $\\mathcal {G}_i$ , $t^{(i)}_{l,m}$ corresponds to the (reduced) one-particle hopping element from site $l$ to site $m$ determined by the pCUT calculation, and $\\sigma _{l,m}=1$ ($\\sigma _{l,m}=-1$ ) if the number of links between sites $l$ and $m$ is even (odd).",
"The latter represents the additional sign to account for the momentum $\\vec{Q}$ of the one-particle gap in the case of the CDHM.",
"The resulting one-particle gaps for both models are given by $\\frac{\\Delta ^{\\rm CDHM}}{J} &= 1-d{k}+\\Big ( d-\\frac{1}{2}d^2\\Big ){k}^{2} +\\Big ( \\frac{1}{4}d+\\frac{1}{2}d^2 \\\\& -\\frac{1}{2}d^3\\Big ){k}^{3} +\\Big ( -\\frac{5}{4}d+\\frac{3}{4}d^2+d^3-\\frac{5}{8}d^4 \\Big ){k}^{4} \\\\& +\\Big ( \\frac{29}{128}d-\\frac{5}{2}d^2+\\frac{9}{8}d^3 +\\frac{7}{4}d^4-\\frac{7}{8}d^5\\Big ){k}^{5} \\\\& +\\Big ( \\frac{17169}{1024}d-\\frac{12585}{512}d^2+\\frac{1429}{256}d^3+\\frac{17}{64}d^4\\\\& +\\frac{25}{8}d^5-\\frac{21}{16}d^6\\Big ){k}^{6} +\\Big ( -\\frac{1845}{4096}d+\\frac{94275}{4096}d^2\\\\& -\\frac{29833}{1024}d^3+\\frac{4117}{1024}d^4-\\frac{213}{256}d^5+\\frac{91}{16}d^6 \\\\& -\\frac{33}{16}d^7 \\Big ){k}^{7}+\\Big ( -\\frac{31976937}{65536}d+\\frac{30652045}{32768}d^2\\\\& -\\frac{133669}{256}d^3+\\frac{580521}{8192}d^4+\\frac{553}{512}d^5-\\frac{4059}{1024}d^6\\\\& +\\frac{21}{2}d^7-\\frac{429}{128}d^8 \\Big ){k}^{8} +\\Big ( -\\frac{128426725}{524288}d\\\\& -\\frac{36029001}{262144}d^2+\\frac{110976899}{131072}d^3-\\frac{34605481}{65536}d^4\\\\& +\\frac{491585}{8192}d^5+\\frac{11615}{8192}d^6-\\frac{47333}{4096}d^7\\\\& +\\frac{627}{32}d^8-\\frac{715}{128}d^9\\Big ){k}^{9} \\,.$ for the CDHM and $\\frac{\\Delta ^{\\rm TFIM}}{2h} &= 1-dx+\\Big ( \\frac{1}{2}d -\\frac{1}{2}d^2 \\Big ) {k}^{2} +\\Big ( -\\frac{1}{4}d+\\frac{3}{4}d^2 \\\\& -\\frac{1}{2}d^3 \\Big ) {k}^{3} +\\Big ( -\\frac{5}{8}d+\\frac{1}{4}d^2+d^3-\\frac{5}{8}d^4 \\Big ) {k}^{4} \\\\& +\\Big ( \\frac{5}{4}d-\\frac{5}{2}d^2+\\frac{1}{2}d^3+\\frac{13}{8}d^4-\\frac{7}{8}d^5 \\Big ) {k}^{5} \\\\& +\\Big ( \\frac{3873}{512}d-\\frac{5577}{512}d^2+\\frac{273}{128}d^3-\\frac{39}{128}d^4+\\frac{45}{16}d^5 \\\\& -\\frac{21}{16}d^6 \\Big ) {k}^{6} +\\Big ( -\\frac{9635}{512}d+\\frac{95253}{2048}d^2-\\frac{72089}{2048}d^3 \\\\& +\\frac{3165}{512}d^4-\\frac{841}{512}d^5+\\frac{161}{32}d^6-\\frac{33}{16}d^7 \\Big ) {k}^{7} \\\\& +\\Big ( -\\frac{7102997}{32768}d+\\frac{13840105}{32768}d^2-\\frac{970563}{4096}d^3 \\\\& +\\frac{205211}{8192}d^4+\\frac{21305}{4096}d^5-\\frac{9695}{2048}d^6 +\\frac{147}{16}d^7 \\\\& -\\frac{429}{128}d^8 \\Big ) {k}^{8} +\\Big ( \\frac{67215987}{131072}d-\\frac{12296139}{8192}d^2 \\\\& +\\frac{205457057}{131072}d^3 -\\frac{5394851}{8192}d^4+\\frac{2336733}{32768}d^5 \\\\& +\\frac{134797}{16384}d^6-\\frac{96289}{8192}d^7+\\frac{1089}{64}d^8-\\frac{715}{128}d^9 \\Big ) {k}^{9}$ for the TFIM.",
"As for the ground-state energy, these results match with the ones known from literature[26], [4], [28], [14], [19] for specific values of $d$ ."
],
[
"Extrapolation",
"The obtained LCEs for the physical quantities have to be extrapolated in order to locate quantum critical points and to determine the associated critical exponents.",
"For a general review on series extrapolation we refer to Ref. Guttmann1989.",
"Here we give the relevant information on the specific extrapolation techniques we applied.",
"One expects the following critical behavior close to a QPT: $\\frac{\\partial ^2 e_0}{\\partial k^2} &\\propto & \\left( k-k_{\\rm c}\\right)^{-\\alpha } \\nonumber \\\\\\Delta &\\propto & \\left( k-k_{\\rm c}\\right)^{z\\nu }\\quad ,$ where $\\alpha $ , $z$ , and $\\nu $ are the specific-heat, dynamic, and correlation length exponents, respectively.",
"The models under consideration have $z=1$ .",
"Our series are all of the form $F(k)=\\sum _{n\\ge 0}^m a_n k^n=a_0+a_1k+a_2k^2+\\dots a_mk^m,$ with $k\\in \\mathbb {R}$ and $a_i \\in \\mathbb {R}$ .",
"If one has power-law behavior near a critical value $k_{\\rm c}$ like in Eqs.",
"(REF ), the true physical function $\\tilde{F}(k)$ close to $k_{\\rm c}$ is given by $\\tilde{F}(k)\\approx \\left(1-\\frac{k}{k_{\\rm c}}\\right)^{-\\theta } A(k),$ where $\\theta $ is the associated critical exponent.",
"If $A(k)$ is analytic at $k=k_{\\rm c}$ , we can write $\\tilde{F}(k)\\approx \\left(1-\\frac{k}{k_{\\rm c}}\\right)^{-\\theta }A|_{k=k_{\\rm c}}\\left(1+\\mathcal {O}\\left(1-\\frac{k}{k_{\\rm c}}\\right)\\right).$ Near the critical value $k_{\\rm c}$ , the logarithmic derivative is then given by $\\tilde{D}(k)&:=\\frac{\\text{d}}{\\text{d}k}\\ln {\\tilde{F}(k)}\\\\&\\approx \\frac{\\theta }{k_{\\rm c}-k}\\left\\lbrace 1+ \\mathcal {O}(k-k_{\\rm c})\\right\\rbrace \\nonumber .$ In the case of power-law behavior, the logarithmic derivative $\\tilde{D}(k)$ is therefore expected to exhibit a single pole.",
"The latter is the reason why so-called Dlog-Padé extrapolation is often used to extract critical points and critical exponents from high-order series expansions.",
"Dlog-Padé extrapolants of $F(k)$ are defined by $dP[L/M]_F(k)=\\exp \\left(\\int _{0}^kP[L/M]_{D}\\,\\,\\text{d}k^{\\prime }\\right)$ and represent physically grounded extrapolants in the case of a second-order phase transition.",
"Here $P[L/M]_{D}$ denotes a standard Padé extrapolation of the logarithmic derivative $P[L/M]_{D}:=\\frac{P_L(k)}{Q_M(k)}=\\frac{p_0+p_1k+\\dots + p_L k^L}{q_0+q_1k+\\dots q_M k^M}\\quad ,$ with $p_i\\in \\mathbb {R}$ and $q_i \\in \\mathbb {R}$ and $q_0=1$ .",
"Additionally, $L$ and $M$ have to be chosen so that $L+M-1\\le m$ .",
"Physical poles of $P[L/M]_{D}(k)$ then indicate critical values $k_{\\rm c}$ while the corresponding critical exponent of the pole $k_{\\rm c}$ can be deduced by $\\theta \\equiv \\left.\\frac{P_L(k)}{\\frac{\\text{d}}{\\text{d}k} Q_M(k)}\\right|_{k=k_{\\rm c}} .$ If the exact value (or a quantitative estimate from other approaches) of $k_{\\rm c}$ is known, one can obtain better estimates of the critical exponent by defining $\\theta ^*(k)&\\equiv (k_{\\rm c}-k)D(k)\\\\&\\approx \\theta +\\mathcal {O}(k-k_{\\rm c}),$ where $D(k)$ is given by Eq.",
"(REF ).",
"Then $P[L/M]_{\\theta ^*}\\big |_{k=k_{\\rm c}}=\\theta $ yields a (biased) estimate of the critical exponent.",
"At the upper critical dimension $d=3$ , both models display multiplicative corrections to Eqs.",
"(REF ) close to the quantum critical point so that one expects the following critical behavivour $\\bar{F}(k)\\approx \\left(1-\\frac{k}{k_{\\rm c}}\\right)^{-\\theta } \\left(\\ln \\left( 1-\\frac{k}{k_{\\rm c}}\\right)\\right)^{p} \\bar{A}(k),$ where $k_{\\rm c}$ ($\\theta $ ) are the associated critical point (exponent) as before while $p$ yields the power of multiplicative logarithmic corrections.",
"Clearly, the extraction of $p$ from a high-order series expansion is very demanding.",
"The only reasonable approach is to bias the extrapolation by fixing $k_{\\rm c}$ and $\\theta $ , e.g., the critical exponents $\\theta $ are given by the well-known mean-field values.",
"Assuming again that the function $\\bar{A}(k)$ is analytic close to $k_{\\rm c}$ , Eq.",
"(REF ) transforms into $\\bar{F}(k)&\\approx & \\left(1-\\frac{k}{k_{\\rm c}}\\right)^{-\\theta } \\left(\\ln \\left( 1-\\frac{k}{k_{\\rm c}}\\right)\\right)^{p}\\bar{A}|_{k=k_{\\rm c}}\\nonumber \\\\&& \\cdot \\left(1+\\mathcal {O}\\left(1-\\frac{k}{k_{\\rm c}}\\right)\\right).$ and the logarithmic derivative Eq.",
"(REF ) becomes $\\bar{D}(k)&\\approx \\frac{\\theta }{k_{\\rm c}-k} + \\frac{-p}{\\ln \\left(1-k/k_{\\rm c}\\right)\\left(k_{\\rm c}-k\\right)} + \\mathcal {O}\\left(k-k_{\\rm c}\\right)\\nonumber .$ One can then estimate the multiplicative logarithmic correction $p$ by defining $p^{*}(k)&\\equiv -\\ln \\left( 1-k/k_{\\rm c}\\right) \\left[ \\left( k_{\\rm c}-k\\right) D(k)-\\theta \\right]\\\\&\\approx p +\\mathcal {O}(k-k_{\\rm c}),$ and by performing Padé extrapolants of this function $P[L/M]_{p*}\\big |_{k=k_{\\rm c}}=p \\quad .$ Utilizing the limit of large spatial dimension $d$ , an analytic $1/d$ expansion for spin models with order–disorder QPT was developed in Refs.",
"Joshi2015 and Joshi2015b.",
"In the limit $d \\rightarrow \\infty $ , non-local fluctuations are suppressed, such that a suitable product-state wavefunction can be used as a reference state.",
"The $1/d$ expansion is obtained from a theory of interacting bosons which capture fluctuations on top of the product state; in this theory, factors of $1/d$ do not appear in the Hamiltonian but are generated via momentum summations.",
"For large $d$ , critical exponents take mean-field values which allows one to identify observables which are analytic even at the quantum critical point.",
"This paves the way to a fully analytic description across the QPT.",
"Refs.",
"Joshi2015 and Joshi2015b developed and applied this methodology to a model of coupled dimers on the hypercubic lattice.",
"In the present paper, we will make use of those results in Sect.",
"below.",
"In addition, we apply and extend the $1/d$ expansion method to the transverse-field Ising model via a suitable auxiliary-boson description.",
"In this section, $q\\equiv q_{\\rm TFIM}$ is the tuning parameter defined in Eq.",
"(REF )."
],
[
"Quantum paramagnetic phase",
"Let us first discuss $1/d$ expansion in the quantum paramagnetic phase of the TFIM, where a suitable reference state is given by $\\Psi = \\prod _{i} | 0 \\rangle _{i} \\,,~~~| 0 \\rangle _{i} = \\frac{| \\uparrow \\rangle _{i} + | \\downarrow \\rangle _{i}}{\\sqrt{2}},$ where $i$ denotes lattice site and $|0\\rangle _i$ is an eigenstate of $\\sigma ^{x}_{i}$ with an eigenvalue 1.",
"A local excitation is given by $| T \\rangle _{i} = \\frac{| \\uparrow \\rangle _{i} - | \\downarrow \\rangle _{i}}{\\sqrt{2}} \\,,$ which is an eigenstate of $\\sigma ^{x}_{i}$ with an eigenvalue $-1$ .",
"We can introduce an auxiliary boson to efficiently describe this process as follows: $T_{i}^{\\dagger } | 0 \\rangle _i = | T \\rangle _i \\,;~~~~~~ T_{i} | 0 \\rangle _{i} = 0 \\,,$ such that it satisfies the usual bosonic commutation relations.",
"The physical Hilbert space on each lattice site consists of only two states.",
"This implies a hard-core constraint for the $T_i$ bosons which we implement by introducing a projection operator[30] $P_i = 1 - T_{i}^{\\dagger } T_{i}^{\\phantom{\\dagger }}$ which is then used to express the spin operators in terms of $T_i$ bosons: $\\sigma _{i}^{x} &= 1 - 2 T_{i}^{\\dagger } T_{i}^{\\phantom{\\dagger }} \\,, \\\\\\sigma _{i}^{y} &= \\imath \\left( P_{i} T_{i} - T_{i}^{\\dagger } P_{i}^{\\phantom{\\dagger }} \\right) \\,, \\\\\\sigma _{i}^{z} &= P_{i} T_{i} + T_{i}^{\\dagger } P_{i}^{\\phantom{\\dagger }} \\,.$ Note that the usual spin commutation relations are satisfied within the physical Hilbert space.",
"Now, substituting Eqs.",
"(REF ) - () in the spin Hamiltonian (REF ) we obtain an interacting boson Hamiltonian (see Appendix ).",
"We use diagrammatic perturbation theory to treat the interacting boson piece in the Hamiltonian.",
"In the large-$d$ formalism, corrections to observables due to these terms are suppressed in powers of $1/d$ , thereby leading to a systematic $1/d$ expansion.",
"This follows from the fact that due to particular momentum-summation properties of the Fourier-transformed interaction, all self-energy diagrams can be expressed as a series in $1/d$ in the limit $d \\rightarrow \\infty $ .",
"The diagrammatic treatment in this case is similar to the one discussed in Ref.",
"Joshi2015.",
"We will therefore only quote the $1/d$ expansion of relevant observables here while relegating technical details and relevant expressions to Appendix .",
"Let us begin with the mode dispersion $\\Omega _{\\vec{k}}$ corresponding to the single-particle excitation.",
"Since the critical exponents $\\nu $ and $z$ have mean-field values $1/2$ and 1 respectively, the square of the excitation-energy gap is analytic at the critical point.",
"We therefore have an analytic $1/d$ expansion for the square of the mode dispersion as follows: $\\frac{\\Omega _{\\vec{k}}^2}{4 h^2} = 1 - 2 \\gamma _{\\vec{k}}q+ \\frac{q^2 }{4 d} (4 + 2 \\gamma _{\\vec{k}}q) \\,.$ Here, $\\gamma _{\\vec{k}}= \\frac{1}{d} \\sum _{i=1}^{d} \\cos k_{i}$ is the interaction structure factor such that $\\gamma _{\\vec{k}}\\in \\left[ -1, 1 \\right]$ .",
"The leading part i.e.",
"$\\mathcal {O}(1/d^{0})$ result in Eq.",
"(REF ) arises from the non-interacting boson piece (harmonic approximation) in the Hamiltonian, while contribution from the boson-interaction terms start at $\\mathcal {O}(1/d)$ .",
"Despite its appearance to this order, this is not an expansion in $q$ .",
"As discussed at length in Ref.",
"Joshi2015, one can as well convert Eq.",
"(REF ) into a $1/d$ expansion for $\\Omega _{\\vec{k}}$ , but this will not be well-defined at the QPT for $\\vec{k}=\\vec{Q}=0$ .",
"The minimum of the mode dispersion (REF ) is at $\\vec{Q}$ .",
"Thus the excitation energy gap is given by $\\Delta = \\Omega _{\\vec{Q}}$ .",
"Substituting $\\gamma _{\\vec{Q}} =1$ in Eq.",
"(REF ) we obtain the $1/d$ expansion for the energy gap $\\frac{\\Delta ^2}{4 h^2} = 1 - 2 q+ \\frac{q^2}{4 d} (4 + 2 q) \\,.$ As for the dispersion, a $1/d$ expansion for $\\Delta $ can be written away from the QPT.",
"In particular, in the small-$q$ limit the $1/d$ expansion of $\\Delta $ matches with the LCE result (REF ).",
"Since the single-particle excitation gap vanishes at the quantum critical point, we can use this criterion to obtain the $1/d$ expansion for the phase boundary to the ferromagnetic phase, $q_c= \\frac{1}{2} + \\frac{5}{32 d} +\\mathcal {O}\\left(\\frac{1}{d^2}\\right)\\,.$ Next, we consider the ground-state energy.",
"In this case, the harmonic approximation itself leads to an expression to order $1/d$ , while the diagrammatic contribution starts at $\\mathcal {O}(1/d^{2})$ .",
"After collecting all the contributions we obtain the following expression for the ground-state energy per site: $\\frac{E_{g}}{2 hN} = -\\frac{1}{2} - \\frac{q^2}{8 d} - \\frac{7 q^4}{64 d^2} +\\mathcal {O}\\left(\\frac{1}{d^3}\\right)\\,.$ This expression matches with the LCE result Eq.",
"(REF )."
],
[
"Ferromagnetic phase",
"We now discuss the symmetry-broken ferromagnetic phase, realized for $q> q_c$ .",
"Beyond the quantum critical point, the auxiliary boson introduced earlier is condensed.",
"As a consequence, a suitable reference state in the ordered phase is[31] $\\Psi _0 = \\prod _{i} |\\tilde{0} \\rangle _i \\,, ~~~| \\tilde{0} \\rangle _{i} = \\frac{| 0 \\rangle _i + \\lambda | T \\rangle _i}{\\sqrt{1 + \\lambda ^2}} \\,.$ In this case, $\\lambda $ is the condensation parameter which takes values between 0 and 1 (alternatively, $-1$ for the other $\\mathbb {Z}_2$ symmetry related choice of reference state) as a function of the tuning parameter $q$ .",
"Obviously, $\\Psi _0$ in the limit $\\lambda \\rightarrow 1$ is the fully polarized ferromagnet.",
"Apart from our reference state we have one more state in the physical Hilbert space, which is orthonormal to the above state and is given by $| \\tilde{T}\\rangle _{i} = \\frac{-\\lambda | 0 \\rangle _i + | T \\rangle _i}{\\sqrt{1 + \\lambda ^2}} \\,.$ This is a local spin-flip excitation on top of the reference state.",
"Again we introduce auxiliary bosons $\\tilde{T}$ such that $\\tilde{T}_{i}^{\\dagger } | \\bar{0} \\rangle _i = | \\tilde{T}\\rangle _i \\,;~~ \\tilde{T}_{i} | \\bar{0} \\rangle _{i} = 0 \\,,$ which obey the usual bosonic commutation relations.",
"As above, the hard-core constraint is implemented by introducing the projection operator $\\tilde{P}_{i} = 1 - \\tilde{T}_{i}^{\\dagger } \\tilde{T}_{i}^{\\phantom{\\dagger }} \\,.$ The spin operators, expressed using the rotated excitation operators $\\tilde{T}$ , read: $\\sigma _{i}^{x} &= \\frac{1}{1+ \\lambda ^2} \\left( (1 - \\lambda ^2)(1 - 2 \\tilde{T}_{i}^{\\dagger } \\tilde{T}_{i}^{\\phantom{\\dagger }}) - 2\\lambda (\\tilde{T}_{i}^{\\dagger } \\tilde{P}_{i}^{\\phantom{\\dagger }} + \\tilde{P}_{i} \\tilde{T}_{i}) \\right) \\,, \\\\\\sigma _{i}^{y} &= \\imath \\left( \\tilde{P}_{i} \\tilde{T}_{i} - \\tilde{T}_{i}^{\\dagger } \\tilde{P}_{i}^{\\phantom{\\dagger }} \\right) \\,, \\\\\\sigma _{i}^{z} &= \\frac{1}{1+ \\lambda ^2} \\left( 2\\lambda (1 - 2 \\tilde{T}_{i}^{\\dagger } \\tilde{T}_{i}^{\\phantom{\\dagger }}) + (1 - \\lambda ^2) (\\tilde{T}_{i}^{\\dagger } \\tilde{P}_{i}^{\\phantom{\\dagger }} + \\tilde{P}_{i} \\tilde{T}_{i}) \\right) \\,.$ Also for $\\lambda =0$ we recover the spin representation (REF )-() in the paramagnetic phase.",
"Parenthetically we note that local spin flips correspond to elementary excitations of the ordered phase only for $d>1$ whereas domain-wall excitations are relevant in $d=1$ .",
"Since we are interested in the large-$d$ limit, this is not an issue.",
"The calculation in the ordered phase is similar to that in the paramagnetic phase, with one important complication: The condensation parameter $\\lambda $ itself has a $1/d$ expansion, which is another source for $1/d$ corrections to other observables.",
"A detailed discussion regarding the corrections to $\\lambda $ and its influence can be found in Ref.",
"Joshi2015b.",
"We will now straightaway proceed to $1/d$ expansion of observables in this phase.",
"Technical details can be found in Appendix .",
"We begin by quoting the single-particle (i.e.",
"spin-wave) dispersion: $\\frac{\\tilde{\\Omega }_{\\vec{k}}^{2}}{4 h^{2}} &= 4 q^2 - \\gamma _{\\vec{k}}- \\frac{1}{d} \\frac{1}{128 q^{4} (12 q^{2} + \\gamma _{\\vec{k}})} \\big [ 2 \\gamma _{\\vec{k}}^3 \\nonumber \\\\&+ 1536 q^{6} (1+\\gamma _{\\vec{k}}) + 4 \\gamma _{\\vec{k}}q^{2} (1 - 4 \\gamma _{\\vec{k}}^2) \\nonumber \\\\&+ 16 q^4 (2 \\gamma _{\\vec{k}}^3 + 2 \\gamma _{\\vec{k}}^2 - 16 \\gamma _{\\vec{k}}- 15)\\big ] \\,.$ We obtain the excitation-energy gap $\\tilde{\\Delta } = \\tilde{\\Omega }_{\\vec{Q}}$ by substituting $\\gamma _{\\vec{Q}} = 1$ in the above expression: $\\frac{\\tilde{\\Delta }^{2}}{4 h^{2}} &= 4 q^2 - 1 - \\frac{1}{d} \\frac{1}{128 q^{4} (12 q^{2} + 1)} \\big [ 2 \\nonumber \\\\&- 12 q^{2} - 432 q^{4} + 3072 q^6 \\big ] +\\mathcal {O}\\left(\\frac{1}{d^2}\\right)\\,.$ Again, demanding that the gap vanishes at the quantum critical point we obtain the same phase boundary (REF ) as before.",
"Unlike the paramagnetic phase, the ground-state energy in this phase is evaluated only to order $1/d$ , because the next order would require corrections to $\\lambda $ to order $1/d^2$ which are beyond the scope of this work.",
"The $1/d$ expansion of the ground-state energy per site is then given by $\\frac{E_g}{2 h N} = -\\frac{4 q^2 + 1}{4 q} - \\frac{1}{d} \\frac{1}{256 q^3} +\\mathcal {O}\\left(\\frac{1}{d^2}\\right)\\,.$ Note that at the quantum critical point given by Eq.",
"(REF ), above expression matches the ground-state energy (REF ) calculated in the paramagnetic phase to order $1/d$ .",
"The ferromagnetic phase is characterized by the non-zero value of the order parameter, which is magnetization.",
"It is zero at the quantum critical point, while in the limit $h \\rightarrow 0$ it takes the value 1 (or $-1$ for the other choice of symmetry-broken state) corresponding to the fully polarized state.",
"Since the mean-field critical exponent $\\beta =1/2$ , we have an analytic $1/d$ expansion for the square of magnetization per site as follows: $M^2 = \\frac{4 q^2 - 1}{4 q^2} - \\frac{1}{d} \\frac{5}{128 q^4} +\\mathcal {O}\\left(\\frac{1}{d^2}\\right)\\,.$ Note that using the condition of vanishing magnetization at the quantum critical point we again get the phase boundary (REF )."
],
[
"Results for the quantum phase transition in arbitrary $d$",
"In the following we use the high-order LCE to estimate the location of the QPT – as the point where the gap of the symmetric (paramagnetic or field-polarized) phase closes – for both models and arbitrary $d$ .",
"Furthermore, we aim at extracting the correlation-length exponent as well as multiplicative logarithmic corrections at the upper critical dimension $d=3$ .",
"Finally, we use the extrapolation of the series expansions about the $d=\\infty $ limit to predict the leading coefficients of the full $1/d$ expansion."
],
[
"Coupled-dimer Heisenberg model",
"The most reliable way to locate the QPT is the use of Dlog-Padé extrapolation Eqs.",
"(REF ) and (REF ) for the one-triplon gap $\\Delta ^{\\rm CDHM}$ given in Eq.",
"(REF ).",
"To this end we set the inverse of dimension $1/d$ to fixed values in $ [0,0.5]$ and extract the critical point $k_{\\rm c}$ .",
"Representative Dlog-Padé extrapolants as well as known literature values for $d=2$ and $d=3$ are displayed in Fig.",
"REF .",
"Figure: Critical point versus inverse dimension 1/d1/d for the CDHM on the hypercubic lattice.",
"Red diamond (green triangle) corresponds to the critical value obtained from quantum Monte Carlo simulations in Ref.",
"sandvik06 (Ref.",
"Qin2015) for d=2d=2 (d=3d=3).",
"Green dashed lines represent the estimated full 1/d1/d expansion Eq.",
"() up to order 𝒪(n)\\mathcal {O}(n) with n∈{1,2,3}n\\in \\lbrace 1,2,3\\rbrace .One observes that the extrapolants vary smoothly with $1/d$ and agree well with the literature values for $d=2$ and $d=3$ as well as with the known $d=\\infty $ limit.",
"As one important example, where especially no series expansion for fixed dimensions is available, let us focus on $d=3$ .",
"If we average over all Dlog-Padé extrapolants $P[L/M]_{D}$ with $L+M=8$ and $L,M\\le 2$ , we obtain $q_c^{\\rm CDHM} =0.6206(5)$ which is in excellent agreement with $q_c^{\\rm CDHM} = 0.620$ from quantum Monte Carlo simulations [11].",
"Figure: Critical exponent ν\\nu versus inverse dimension 1/d1/d for the CDHM on the hypercubic lattice.",
"Dot-dashed line indicates the mean-field exponent ν=1/2\\nu =1/2 and the red diamond corresponds to the critical exponent obtained from quantum Monte Carlo simulations in Ref.",
"campostrini2002 for d=2d=2.Next we investigate the behavior of the critical exponent $\\nu $ of the one-triplon gap (recall $z=1$ ) close to the quantum critical line as a function of $1/d$ using Eq.",
"(REF ).",
"The corresponding results are given in Fig.",
"REF .",
"Above the upper critical dimension, $d\\ge 3$ , mean-field behavior with $\\nu =1/2$ is expected, whereas for $3>d\\ge 2$ one expects a continuously varying critical exponent.",
"In particular, $\\nu =0.7113(10)$ for $d=2$ [32].",
"Semi-quantitatively, this behavior is reproduced by the Dlog-Padé extrapolants.",
"However, for $d=2$ the extrapolated critical exponent overshoots the correct value $\\nu \\approx 0.71$ .",
"This difference can be attributed to the maximal order nine of our series expansion.",
"Indeed, if one performs high-order series expansions for fixed dimension $d=2$ , one is able to reach order 11 which yields a critical exponent much closer to $\\nu \\approx 0.71$ [28].",
"The other issue is the behavior close to the upper critical dimension $d=3$ .",
"By construction, the series expansion cannot be expected to give a constant value for $d\\ge 3$ ; it will rather yield a smooth behavior $\\nu (d)$ .",
"Nevertheless, it may be surprising that sizeable deviations from mean-field behavior are visible already for $d\\lesssim 6$ .",
"We attribute this behavior to a combination of small scaling dimension and large prefactor of the leading irrelevant perturbation for $d\\gtrsim 3$ which in turn spoils the estimate of the critical exponent from series expansions.",
"As described in subsection REF , it is also possible to estimate the multiplicative logarithmic corrections at $d=3$ using Eq.",
"(REF ).",
"To this end we fix the critical exponent to $1/2$ and the critical point to $q_c^{\\rm CDHM} =0.6206$ .",
"The corresponding values of $p^{\\rm CDHM}_{\\rm gap}$ as a function of perturbative order $L+M$ are displayed in Fig.",
"REF together with the exact value[33], [34] $(-5/22)$ derived from perturbative renormalization-group (RG) calculations.",
"Averaging over Dlog-Padé extrapolants with $L+M\\ge 7$ and $M-L\\le 2$ yields $p^{\\rm CDHM}_{\\rm gap}=-0.19(2)$ .",
"Let us stress that these value are quite sensitive to the critical value $q_c^{\\rm CDHM}$ entering the extrapolation.",
"If one uses the value $q_c^{\\rm CDHM} =0.620$ from quantum Monte Carlo simulations [11], the value for the multiplicative logarithmic correction changes to $p^{\\rm CDHM}_{\\rm gap}=-0.18(3)$.",
"Nevertheless, as can be seen from Fig.",
"REF , the extrapolation tend to decrease with increasing perturbative order so that our results are in full agreement with the exact value $(-5/22)$ .",
"Figure: Multiplicative logarithmic correction p gap CDHM p^{\\rm CDHM}_{\\rm gap} versus perturbative order L+ML+M used in the Dlog-Padeé extrapolation Eq.",
"() with q c CDHM =0.6206q_c^{\\rm CDHM} =0.6206 for the CDHM on the hypercubic lattice.",
"Extrapolants with constant c=L-Mc=L-M with c∈{-1,0,1,2}c\\in \\lbrace -1,0,1,2\\rbrace are shown with the same black symbols.",
"Dashed line indicates the exact exponent (-5/22)(-5/22)."
],
[
"Transverse-field Ising model",
"Let us turn to the TFIM on the hypercubic lattice.",
"For this model one expects a second-order QPT for all dimensions $d\\ge 1$ , separating the gapped polarized phase present at large fields from the gapped symmetry-broken phase for dominating Ising exchange.",
"We have again used DlogPadé extrapolation to estimate the location of the QPT as a function of $1/d$ .",
"DlogPadé extrapolants as well as the known results from literature are shown in Fig.",
"REF .",
"Figure: Critical point versus inverse dimension 1/d1/d for the TFIM on the hypercubic lattice.",
"Red diamond (green triangle) corresponds to the critical value obtained from series expansion (SE) in Ref.",
"He1990 (Ref.",
"Zheng1994) for d=2d=2 (d=3d=3) and quantum Monte Carlo (QMC) simulations from Ref.",
"dengqmc3d (the two techniques fully agree on the displayed scale and are therefore shown together as a single symbol).",
"Blue square corresponds to the exact solution for d=1d=1 .",
"Green dashed lines represent the estimated full 1/d1/d expansion Eq.",
"() up to order 𝒪(n)\\mathcal {O}(n) with n∈{1,2,3}n\\in \\lbrace 1,2,3\\rbrace .Overall, the series expansion yields convincing results capturing quantitatively the exactly known cases $1/d=1$ [27] and $1/d=0$ as well as the estimates of series expansions with higher maximal order for fixed dimensions $d=2$ and $d=3$ .",
"The associated critical exponent $\\nu $ ($z=1$ for the TFIM) as a function of $1/d$ is displayed in Fig.",
"REF .",
"In certain regimes in $1/d$ (the two shaded areas in Fig.",
"REF ) we found no consistent Dlog-Padé extrapolation, since there are two poles close to each other on the real axis in the denominator of Eq.",
"(REF ) which spoil the extrapolation scheme.",
"Similarly to the CDHM, one observes deviations from mean-field behavior for $3<d\\lesssim 6$ .",
"As above, we believe that these are caused by large subleading corrections with small scaling dimension.",
"For the TFIM we do not analyze the multiplicative logarithmic corrections the the gap behavior, since Ref.",
"Zheng1994 already performed such an analysis with the high-order series expansion of order 13 for $d=3$ .",
"This gives $p^{\\rm TFIM}_{\\rm gap}=-0.143(5)$ which is consistent with the exact value[20], [21], [22], [19] $(-1/6)$ from perturbative RG.",
"Figure: Critical exponent ν\\nu versus dimension for the TFIM on the hypercubic lattice.",
"Dot-dashed line indicates the mean-field exponent ν=1/2\\nu =1/2.",
"The red diamond corresponds to the critical exponent obtained from quantum Monte Carlo simulations in Ref.",
"Bloete1995 for d=2d=2.",
"The blue square illustrates the exact value for d=1d=1 .",
"In the shaded regions no consistent extrapolation has been achieved due to spurious poles in the Dlog-Padé extrapolation."
],
[
"Large-$d$ limit",
"Up to now, we have studied the one-particle gap for both models by fixing the dimension $d$ and using DlogPadé extrapolation to extract critical points and associated critical exponents.",
"In this part we use the extrapolation of the high-order series expansion in $k$ to get an estimate for the quantum critical line about the $1/d=0$ limit in powers of $1/d$ , i.e.",
"we are interested in determining the coefficients $c_n$ in $q_c=c_0+c_1\\,\\left(\\frac{1}{d}\\right)+ c_2\\,\\left( \\frac{1}{d} \\right)^2+ c_3\\,\\left( \\frac{1}{d}\\right)^3 \\ldots \\quad .$ In a first approach we used various DlogPadé extrapolants for small values of the inverse dimension $1/d=0,\\epsilon ,2\\epsilon ,\\ldots $ and we fitted the polynomial (REF ) to determine the coefficients $c_n$ .",
"We observed that the coefficients $c_n$ converged reliably under the variation of $\\epsilon $ and for different DlogPadé extrapolants, and our numerical estimates for $c_0$ and $c_1$ approach the exact value from the analytic $1/d$ expansion.",
"The latter findings motivate the second approach, where the $c_n$ are obtained as fractions from the DlogPadé extrapolation.",
"To this end we did not fix the value of the dimension in the DlogPadé extrapolation, but we keep it general when calculating the Padé of the logarithmic derivative Eq.",
"(REF ).",
"In a next step we determine the pole of the denominator, corresponding to the critical point, as a Taylor series in $1/d$ which exactly yields a polynomial of the form Eq.",
"(REF ).",
"And indeed, we find that the leading coefficients $c_0$ and $c_1$ do not depend on the specific DlogPadé extrapolant used and, more importantly, the values correspond exactly to the values from the analytical $1/d$ expansion.",
"In contrast, the higher orders $c_2$ and $c_3$ do depend on the specific extrapolant, but the obtained values are very close to each other numerically and consistent with the values from the first approach.",
"One then obtains the following expressions for the $1/d$ expansion up to order three $q_c^{\\rm CDHM}=\\frac{1}{2}+\\frac{3}{16}\\,\\left(\\frac{1}{d}\\right)+ 0.2311\\,\\left( \\frac{1}{d} \\right)^2+ 0.1233\\,\\left( \\frac{1}{d}\\right)^3 \\ldots $ for the CDHM and $q_c^{\\rm TFIM}=\\frac{1}{2}+\\frac{5}{32}\\,\\left(\\frac{1}{d}\\right)+ 0.1383\\,\\left( \\frac{1}{d} \\right)^2+ 0.0643\\,\\left( \\frac{1}{d}\\right)^3 \\ldots $ for the TFIM.",
"These results are also illustrated in the phase diagram of the CDHM (TFIM) in Fig.",
"REF (Fig.",
"REF ).",
"One observes that the full $1/d$ expansion for both models is well behaved in the sense that all deduced coefficients are positive and one therefore approaches the correct quantum critical line at finite dimensions steadily with increasing order in $1/d$ ."
],
[
"Conclusion",
"We used high-order series expansions for arbitrary dimensions $d$ to study the dimensional dependence of the quantum critical line for two paradigmatic models in quantum magnetism, namely the transverse-field Ising model and the coupled-dimer Heisenberg model on the hypercubic lattice.",
"In both cases we reached order 9 perturbation theory about the decoupled-dimer (high-field) limit for the ground-state energy per dimer (site) as well as for the one-particle gap for general dimension $d$ .",
"This is achieved by a full graph decomposition in linked graphs keeping in mind that the dimension $d$ enters only in the combinatorical embedding factor of the graphs.",
"We focused on analyzing for both models the behavior of the one-particle gap about the decoupled-dimer (high-field) limit.",
"In both cases the continuous critical line of the LCE is in quantitative agreement with the known results for fixed finite dimensions as well as with the leading orders of the $1/d$ expansion.",
"For the CDHM, the series expansion for fixed dimension $d=3$ were to the best of our knowledge unknown and the extrapolated critical point is found to be in quantitative agreement with QMC simulations [11].",
"Furthermore, we also extracted the multiplicative logarithmic correction for this case which is fully consistent with the value from perturbative renormalization group calculations.",
"Finally, we used the LCEs to predict the coefficients of the full $1/d$ expansion of the quantum critical line.",
"It is found that the series in $1/d$ up to order 3 has only positive coefficients for both models and therefore approach monotonously the correct values.",
"These subleading corrections are however not small.",
"In our opinion the most remarkable finding is that the Dlog-Padé extrapolation of the gap series yields exactly the leading coefficients $(1/d)^0$ and $(1/d)^1$ of the full $1/d$ expansion.",
"It would be interesting to study this issue in other models in future studies.",
"We thank B. Normand for helpful discussions concerning logarithmic corrections.",
"This work was in part supported by the Helmholtz Virtual Institute “New states of matter and their excitations” as well as from the Deutsche Forschungsgemeinschaft (DFG) via grants SCHM 2511/9-1 and SFB 1143."
],
[
"$1/d$ expansion details: Quantum paramagnetic phase",
"Inserting Eqs.",
"(REF ) - () in the spin Hamiltonian (REF ) we obtain the following interacting boson Hamiltonian $\\mathcal {H} = &-J \\sum _{\\langle ij \\rangle } \\left( P_{i}T_{i}P_{j}T_{j} + T_{i}^{\\dagger } P_{i}^{\\phantom{\\dagger }} P_{j}^{\\phantom{\\dagger }} T_{j}^{\\phantom{\\dagger }} + {\\rm h.c.} \\right) \\nonumber \\\\&- h \\sum _{i} \\left( 1 - 2 T_{i}^{\\dagger } T_{i}^{\\phantom{\\dagger }} \\right) \\,.$ Further, inserting the explicit expression for the projection operator (REF ) we can write down the various pieces $\\mathcal {H}_{n}$ , where $n$ denotes the number of $T$ operators.",
"In this case, we have only even-ordered $\\mathcal {H}_{n}$ .",
"It turns out that to order $1/d$ calculation we need terms only upto $\\mathcal {H}_{4}$ .",
"We therefore present the explicit expressions of the relevant terms below: $\\mathcal {H}_0 &= - N h \\,, ~~~ \\text{$N$ is the number of lattice sites} \\,,\\\\\\mathcal {H}_2 &= - J \\sum _{\\langle ij \\rangle } \\left( T_{i} T_{j} + T_{i}^{\\dagger } T_{j}^{\\phantom{\\dagger }} + {\\rm h.c.} \\right)+ 2 h \\sum _{i} T_{i}^{\\dagger } T_{i}^{\\phantom{\\dagger }} \\,, \\\\\\mathcal {H}_4 &= 2 J \\sum _{\\langle ij \\rangle } \\bigg [ T_{i}^{\\dagger } T_{i}^{\\phantom{\\dagger }} T_{i}^{\\phantom{\\dagger }} T_{j}^{\\phantom{\\dagger }} + T_{i}^{\\dagger } T_{i}^{\\dagger } T_{i}^{\\phantom{\\dagger }} T_{j}^{\\phantom{\\dagger }} + {\\rm h.c.} \\bigg ] \\,.$ Let us first solve the bilinear part (), which we call the harmonic approximation.",
"Utilizing the lattice translation symmetry, it is convenient to work in the Fourier space by introducing $T_{i} = \\frac{1}{\\sqrt{N}} \\sum _{\\vec{k}} T_{\\vec{k}} e^{-\\imath \\vec{k}\\cdot \\vec{r}_{i}} \\,.$ In terms of the Fourier transformed operator, $T_{\\vec{k}}$ , the bilinear part of the Hamiltonian is given by $\\mathcal {H}_2 = \\sum _{\\vec{k}} \\left[ A_{\\vec{k}} T_{\\vec{k}}^{\\dagger } T_{\\vec{k}}^{\\phantom{\\dagger }} + \\frac{B_{\\vec{k}}}{2} \\left( T_{\\vec{k}} T_{-\\vec{k}} + {\\rm h.c.}\\right) \\right]$ where $A_{\\vec{k}} = 2 h + B_{\\vec{k}}\\,,~~B_{\\vec{k}} = -\\gamma _{\\vec{k}}q\\,.$ The above Hamiltonian piece can be diagonalized by introducing bosonic Bogoliubov transformation, $T_{\\vec{k}} =u_{\\vec{k}}\\tau _{\\vec{k}}+v_{\\vec{k}}\\tau ^\\dagger _{-\\vec{k}} \\,.$ Here $u_{\\vec{k}}$ and $v_{\\vec{k}}$ are the Bogoliubov coefficients such that $u_{\\vec{k}}^{2} - v_{\\vec{k}}^{2} =1$ , and the $\\tau $ operators obey the usual bosonic commutation relations.",
"Within the harmonic approximation, the mode energy is given by $\\omega _{\\vec{k}}= \\sqrt{A_{\\vec{k}}^{2} - B_{\\vec{k}}^{2}} = 2 h \\sqrt{1 - 2 \\gamma _{\\vec{k}}q} \\,,$ and the Bogoliubov coefficients are given by $u_{\\vec{k}}^{2}, v_{\\vec{k}}^{2} = \\frac{1}{2} \\left( \\frac{A_{\\vec{k}}}{\\omega _{\\vec{k}}} \\pm 1 \\right) \\,; ~~~u_{\\vec{k}} v_{\\vec{k}} = - \\frac{B_{\\vec{k}}}{2 \\omega _{\\vec{k}}} \\,.$ Having solved the bilinear piece of the Hamiltonian, we shall treat it as the unperturbed part and take into account the interaction terms perturbatively.",
"To set-up the method, we must first normal-order our Hamiltonian with respect to the $\\tau $ operators.",
"Upon normal ordering, $\\mathcal {H}_4$ will generate additional bilinear terms which are expressed below: $\\mathcal {H}^{\\prime }_{2b} = \\sum _{\\vec{k}} \\bigg [ C_{\\vec{k}} \\tau _{\\vec{k}}^{\\dagger } \\tau _{\\vec{k}}^{\\phantom{\\dagger }} + \\frac{D_{\\vec{k}}}{2} (\\tau _{\\vec{k}} \\tau _{-\\vec{k}} + {\\rm h.c.})\\bigg ]$ where $C_{\\vec{k}} &= 4 qh \\big [ (u_{\\vec{k}}^{2} + v_{\\vec{k}}^{2}) (\\gamma _{\\vec{k}}R_{1} + 2 \\gamma _{\\vec{k}}R_{2} + 2R_{3}) \\nonumber \\\\&+ 2 u_{\\vec{k}} v_{\\vec{k}} (\\gamma _{\\vec{k}}R_{1} + 2 \\gamma _{\\vec{k}}R_{2} + R_{3}) \\big ] \\,, \\\\D_{\\vec{k}} &= 4 qh \\big [ (u_{\\vec{k}}^{2} + v_{\\vec{k}}^{2}) (\\gamma _{\\vec{k}}R_{1} + 2 \\gamma _{\\vec{k}}R_{2} + R_{3}) \\nonumber \\\\&+ 2 u_{\\vec{k}} v_{\\vec{k}} (\\gamma _{\\vec{k}}R_{1} + 2 \\gamma _{\\vec{k}}R_{2} + 2R_{3}) \\big ] \\,.$ Following are required expressions of R's to order $1/d$ in the large-$d$ limit: $R_1 &= \\frac{1}{N} \\sum _{\\vec{k}} u_{\\vec{k}} v_{\\vec{k}} ~~~= \\frac{q^2}{4 d} + \\mathcal {O}(d^{-2}) \\,, \\\\R_2 &= \\frac{1}{N} \\sum _{\\vec{k}} v^2_{\\vec{k}} ~~~~~~= \\frac{q^2}{8 d} + \\mathcal {O}(d^{-2}) \\,, \\\\R_3 &= \\frac{1}{N} \\sum _{\\vec{k}} \\gamma _{\\vec{k}}u_{\\vec{k}} v_{\\vec{k}} = \\frac{q}{4 d} + \\mathcal {O}(d^{-2}) \\,, \\\\R_4 &= \\frac{1}{N} \\sum _{\\vec{k}} \\gamma _{\\vec{k}} v^2_{\\vec{k}} ~~~\\,= \\mathcal {O}(d^{-2}) \\,.$ Thus the normal ordered bilinear piece is sum of the unperturbed part and the above contribution: $\\mathcal {H}^{\\prime }_{2} = \\mathcal {H}^{\\prime }_{2a} + \\mathcal {H}^{\\prime }_{2b}$ where, $\\mathcal {H}^{\\prime }_{2a} = \\sum _{\\vec{k}} \\omega _{\\vec{k}}\\tau _{\\vec{k}}^{\\dagger } \\tau _{\\vec{k}}^{\\phantom{\\dagger }}$ is the unperturbed piece.",
"Now we quote the normal ordered quartic term: $\\mathcal {H}^{\\prime }_4 &= \\frac{1}{N} \\sum _{1234}\\big [\\delta _{1+2+3+4} \\Gamma _{41}^{d}(\\tau ^\\dagger _{1}\\tau ^\\dagger _{2}\\tau ^\\dagger _{3}\\tau ^\\dagger _{4} +\\tau _{1}\\tau _{2}\\tau _{3}\\tau _{4}) \\nonumber \\\\&~~~~~~~~~~~+ \\delta _{1+2-3-4} (\\Gamma _{42}^{d}\\tau ^\\dagger _{1}\\tau ^\\dagger _{2}\\tau _{3}^{\\phantom{\\dagger }} \\tau _{4}^{\\phantom{\\dagger }}+\\Gamma _{43}^{d}\\tau ^\\dagger _{1}\\tau ^\\dagger _{2}\\tau _{3}^{\\phantom{\\dagger }} \\tau _{4}^{\\phantom{\\dagger }} )\\nonumber \\\\&~~~~~~~~~~~+\\delta _{1+2+3-4} \\Gamma _{44}^{d}(\\tau ^\\dagger _{1}\\tau ^\\dagger _{2}\\tau ^\\dagger _{3}\\tau _{4}^{\\phantom{\\dagger }} +\\tau ^\\dagger _{4} \\tau _{3}^{\\phantom{\\dagger }} \\tau _{2}^{\\phantom{\\dagger }} \\tau _{1}^{\\phantom{\\dagger }} )\\big ] \\,,$ with the relevant vertex functions given by $\\Gamma _{41}^{d}&= 2 qh \\big [ \\gamma _{4} u_{1} v_{2} v_{3} v_{4} + \\gamma _{4} v_{1} u_{2} u_{3} u_{4} \\nonumber \\\\&~~~~~~~~~~~+ (\\gamma _{1} + \\gamma _{4}) u_{1} u_{2} v_{3} v_{4} \\big ] \\,, \\\\\\Gamma _{44}^{d}&= 2 qh \\big [ (2 \\gamma _{3} + \\gamma _{4}) u_{1} v_{2} v_{3} u_{4} + (2 \\gamma _{3} + \\gamma _{4}) v_{1} u_{2} u_{3} v_{4} \\nonumber \\\\&~~~~~~~~~~~+ \\gamma _{3} v_{1} v_{2} v_{3} v_{4} + \\gamma _{3} u_{1} u_{2} u_{3} u_{4} \\nonumber \\\\&~~~~~~~~~~~+ (2 \\gamma _{1} + \\gamma _{3} + \\gamma _{4}) u_{1} u_{2} v_{3} u_{4} \\nonumber \\\\&~~~~~~~~~~~+ (\\gamma _{1} + 2 \\gamma _{3} + \\gamma _{4}) u_{1} v_{2} v_{3} v_{4} \\big ] \\,.$ The self-energy diagrams constructed using these quartic vertices along with the bilinear vertices coming from (REF ) are suppressed as $1/d$ .",
"After inserting the self-energy in the Dyson equation and identifying the pole of the Green's functions, we obtain the dispersion (REF ).",
"The diagrammatic expansion for the ground-state energy works along similar lines, and we refer to Ref.",
"Joshi2015 for more details."
],
[
"$1/d$ expansion details: Ferromagnetic phase",
"Using Eqs.",
"(REF ) - (), the Hamiltonian in the ordered phase is $\\mathcal {H} = &- \\frac{J}{ (1+\\lambda ^2)^2} \\sum _{\\langle ij \\rangle } \\bigg [ (1-\\lambda ^2)^{2} \\left( \\tilde{P}_{i} \\tilde{T}_{i} \\tilde{P}_{j} \\tilde{T}_{j} +\\tilde{T}_{i}^{\\dagger } \\tilde{P}_{i} \\tilde{P}_{j} \\tilde{T}_{j} + {\\rm h.c.} \\right)+ 4 \\lambda ^2 \\left( 1 - 2\\tilde{T}_{j}^{\\dagger } \\tilde{T}_{j} - 2\\tilde{T}_{i}^{\\dagger } \\tilde{T}_{i} + 4 \\tilde{T}_{i}^{\\dagger } \\tilde{T}_{i} \\tilde{T}_{j}^{\\dagger } \\tilde{T}_{j}\\right) \\nonumber \\\\&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~+ 2\\lambda (1-\\lambda ^2) \\left( \\tilde{T}_{i}^{\\dagger } \\tilde{P}_{i} -2 \\tilde{T}_{i}^{\\dagger } \\tilde{P}_{i} \\tilde{T}_{j}^{\\dagger } \\tilde{T}_{j} +\\tilde{T}_{j}^{\\dagger } \\tilde{P}_{j} -2 \\tilde{T}_{j}^{\\dagger } \\tilde{P}_{j} \\tilde{T}_{i}^{\\dagger } \\tilde{T}_{i} + {\\rm h.c.} \\right)\\bigg ] \\nonumber \\\\&- \\frac{h}{1+\\lambda ^2} \\sum _{i} \\bigg [ (1-\\lambda ^2)(1 - 2\\tilde{T}_{i}^{\\dagger }\\tilde{T}_{i}) - 2\\lambda (\\tilde{T}_{i}^{\\dagger } \\tilde{P}_{i} + {\\rm h.c.} ) \\bigg ] \\,.$ To order $1/d$ we need only terms up to fourth order in $\\tilde{T}$ operators.",
"Inserting the explicit expression of the projector operator (REF ), we can write the relevant pieces in the Hamiltonian as follows: $\\mathcal {H}_0 &= -\\frac{4 N qh \\lambda ^2}{(1+ \\lambda ^2)^2} - \\frac{Nh (1- \\lambda ^2)}{1+ \\lambda ^2} \\,, \\\\\\mathcal {H}_{1} &= \\left[ \\frac{2h\\lambda }{1+\\lambda ^2} -\\frac{4 h q\\lambda (1-\\lambda ^2)}{(1+\\lambda ^2)^2}\\right] \\sum _{i}\\left( \\tilde{T}_{i}^{\\dagger } + \\tilde{T}_{i} \\right) \\,, $ $\\mathcal {H}_{2} &= - \\frac{J}{(1+\\lambda ^2)^2} \\sum _{\\langle ij \\rangle } \\bigg [ (1-\\lambda ^2)^{2} \\left( \\tilde{T}_{i} \\tilde{T}_{j} +\\tilde{T}_{i}^{\\dagger } + {\\rm h.c.} \\right) - 8\\lambda ^{2} \\left( \\tilde{T}_{i}^{\\dagger } \\tilde{T}_{i} + \\tilde{T}_{j}^{\\dagger } \\tilde{T}_{j} \\right) \\bigg ] + \\frac{2 h (1-\\lambda ^2)}{1+\\lambda ^2} \\sum _{i} \\tilde{T}_{i}^{\\dagger } \\tilde{T}_{i} \\,, \\\\\\mathcal {H}_{3} &= \\frac{4 J \\lambda (1-\\lambda ^2)}{(1+\\lambda ^2)^2} \\sum _{\\langle ij \\rangle } \\bigg [ \\tilde{T}_{i}^{\\dagger } \\tilde{T}_{i}^{\\dagger } \\tilde{T}_{i} + 2 \\tilde{T}_{j}^{\\dagger } \\tilde{T}_{i}^{\\dagger } \\tilde{T}_{i} + {\\rm h.c.} \\bigg ] - \\frac{2 h \\lambda }{1+\\lambda ^2} \\sum _{i} \\bigg [ \\tilde{T}_{i}^{\\dagger } \\tilde{T}_{i}^{\\dagger } \\tilde{T}_{i} + {\\rm h.c.} \\bigg ] \\,, \\\\\\mathcal {H}_4 &= \\frac{2 J}{ (1+ \\lambda ^2)^2} \\sum _{\\langle ij \\rangle } \\bigg [ (1 - \\lambda ^2)^2\\left( \\tilde{T}_{i}^{\\dagger } \\tilde{T}_{i} \\tilde{T}_{i} \\tilde{T}_{j} + \\tilde{T}_{i}^{\\dagger } \\tilde{T}_{i}^{\\dagger } \\tilde{T}_{i} \\tilde{T}_{j} + {\\rm h.c.} \\right)- 8 \\lambda ^2 \\tilde{T}_{i}^{\\dagger } \\tilde{T}_{j}^{\\dagger } \\tilde{T}_{i} \\tilde{T}_{j} \\bigg ] \\,.$ Again note that for $\\lambda =0$ we recover the Hamiltonian in the disordered phase in terms of $T$ operators.",
"Vanishing of $\\mathcal {H}_{1}$ () gives the leading order $\\lambda $ , denoted by $\\lambda _0$ and is given in terms of $q$ as $\\lambda _{0} = \\sqrt{\\frac{2 q- 1}{2 q+ 1}} \\,.$ Using this we will separate the unperturbed piece from the bilinear Hamiltonian (REF ) in the Fourier space, which is given by $\\mathcal {H}_{2 \\vec{k}}^{(0)} = \\sum _{\\vec{k}} \\left[ \\tilde{A}_{\\vec{k}}^{(0)} \\tilde{T}_{\\vec{k}}^{\\dagger } \\tilde{T}_{\\vec{k}} + \\frac{\\tilde{B}_{\\vec{k}}^{(0)}}{2} \\left( \\tilde{T}_{\\vec{k}} \\tilde{T}_{-\\vec{k}} + {\\rm h.c.} \\right) \\right]$ with $\\tilde{A}_{\\vec{k}}^{(0)} = \\frac{h}{q} + \\frac{h(4 q^2 -1)}{q} + \\tilde{B}_{\\vec{k}}\\,,~~\\tilde{B}_{\\vec{k}}^{(0)} = - \\frac{\\gamma _{\\vec{k}}h}{2q} \\,.$ It is then diagonalized using bosonic Bogoliubov transformation $\\tilde{T}_{\\vec{k}} =\\tilde{u}_{\\vec{k}}\\tilde{\\tau }_{\\vec{k}}+\\tilde{v}_{\\vec{k}}\\tilde{\\tau }^\\dagger _{-\\vec{k}} \\,,$ with the Bogoliubov coefficients having a similar expression to eq.",
"(REF ) in terms of $\\tilde{A}_{\\vec{k}}^{(0)}$ , $\\tilde{B}_{\\vec{k}}^{(0)}$ , and $\\tilde{\\omega }_{\\vec{k}}$ , with $\\tilde{\\omega }_{\\vec{k}} = 2 h \\sqrt{4 q^2 - \\gamma _{\\vec{k}}} \\,.$ The perturbation strategy is then same as in the paramagnetic phase with complications arising from $1/d$ expansion of $\\lambda $ .",
"A detailed account on how to proceed with the diagrammatics in the presence of a $1/d$ expansion to $\\lambda $ can be found in Ref.",
"Joshi2015b.",
"Now we quote the expressions for normal-ordered Hamiltonian in the $\\tilde{\\tau }$ basis, relevant to order $1/d$ .",
"The bilinear piece has three contributions: (i) Unperturbed (diagonal) piece $\\mathcal {H}^{\\prime }_{2 a}$ , (ii) Bilinear terms, $\\mathcal {H}^{\\prime }_{2 b}$ , arising from normal ordering of quartic terms, and (iii) Order $1/d$ terms, $\\mathcal {H}^{\\prime }_{2 c}$ , arising from (REF ) due to $1/d$ corrections to $\\lambda $ .",
"So normal-ordered bilinear Hamiltonian is $\\mathcal {H}^{\\prime }_{2} = \\mathcal {H}^{\\prime }_{2 a} + \\mathcal {H}^{\\prime }_{2 b} + \\mathcal {H}^{\\prime }_{2 c}$ .",
"We now quote the explicit expressions as follows: $\\mathcal {H}^{\\prime }_{2 a} &= \\tilde{\\omega }_{\\vec{k}} \\tilde{\\tau }^{\\dagger } \\tilde{\\tau }\\,, \\\\\\mathcal {H}^{\\prime }_{2b} &= \\sum _{\\vec{k}} \\bigg [ C_{\\vec{k}} \\tau _{\\vec{k}}^{\\dagger } \\tau _{\\vec{k}}+ \\frac{D_{\\vec{k}}}{2} (\\tau _{\\vec{k}} \\tau _{-\\vec{k}} + {\\rm h.c.} )\\bigg ] \\,, \\\\\\mathcal {H}^{\\prime }_{2c} &= \\sum _{\\vec{k}} \\bigg [ \\left( (\\tilde{u}_{\\vec{k}}^2 + \\tilde{v}_{\\vec{k}}^2) \\tilde{A}_{\\vec{k}}^{(1)} + 2 \\tilde{u}_{\\vec{k}} \\tilde{v}_{\\vec{k}} \\tilde{B}_{\\vec{k}}^{(1)} \\right) \\tau _{\\vec{k}}^{\\dagger } \\tau _{\\vec{k}} \\nonumber \\\\&+ \\frac{ (\\tilde{u}_{\\vec{k}}^2 + \\tilde{v}_{\\vec{k}}^2) \\tilde{B}_{\\vec{k}}^{(1)} + 2 \\tilde{u}_{\\vec{k}} \\tilde{v}_{\\vec{k}} \\tilde{A}_{\\vec{k}}^{(1)} }{2} (\\tau _{\\vec{k}} \\tau _{-\\vec{k}} + {\\rm h.c.})\\bigg ] \\,,$ where $C_{\\vec{k}} &= 4 qh \\left(\\frac{1-\\lambda ^2}{1+\\lambda ^2}\\right)^{2} \\big [ (\\tilde{u}_{\\vec{k}}^{2} + \\tilde{v}_{\\vec{k}}^{2}) (\\gamma _{\\vec{k}}R_{1} + 2 \\gamma _{\\vec{k}}R_{2} + 2R_{3}) + 2 \\tilde{u}_{\\vec{k}} \\tilde{v}_{\\vec{k}} (\\gamma _{\\vec{k}}R_{1} + 2 \\gamma _{\\vec{k}}R_{2} + R_{3}) \\big ] \\nonumber \\\\&-\\frac{16 \\lambda ^{2} qh}{(1+\\lambda ^2)^2} \\big [ 2 (\\tilde{u}_{\\vec{k}}^{2} + \\tilde{v}_{\\vec{k}}^{2}) R_2 + 4 \\tilde{u}_{\\vec{k}} \\tilde{v}_{\\vec{k}} \\gamma _{\\vec{k}}R_3 \\big ] \\,, \\\\D_{\\vec{k}} &= 4 qh \\left(\\frac{1-\\lambda ^2}{1+\\lambda ^2}\\right)^{2} \\big [ (\\tilde{u}_{\\vec{k}}^{2} + \\tilde{v}_{\\vec{k}}^{2}) (\\gamma _{\\vec{k}}R_{1} + 2 \\gamma _{\\vec{k}}R_{2} + R_{3}) + 2 \\tilde{u}_{\\vec{k}} \\tilde{v}_{\\vec{k}} (\\gamma _{\\vec{k}}R_{1} + 2 \\gamma _{\\vec{k}}R_{2} + 2R_{3}) \\big ] \\nonumber \\\\&-\\frac{32 \\lambda ^{2} qh}{(1+\\lambda ^2)^2} \\big [ (\\tilde{u}_{\\vec{k}}^{2} + \\tilde{v}_{\\vec{k}}^{2}) \\gamma _{\\vec{k}}R_3 + 2 \\tilde{u}_{\\vec{k}} \\tilde{v}_{\\vec{k}} R_2 \\big ] \\,, \\\\\\tilde{A}_{\\vec{k}}^{(1)} &= -\\frac{2 h}{q} \\left( R_2 + R_3 \\right) \\left( 1 + \\gamma _{\\vec{k}}\\right) \\,, ~~~~~~\\tilde{B}_{\\vec{k}}^{(1)} = -\\frac{2 \\gamma _{\\vec{k}}h}{q} \\left( R_2 + R_3 \\right) \\,.$ The expressions for $R$ 's are as follows: $R_1 &= \\frac{1}{N} \\sum _{\\vec{k}} \\tilde{u}_{\\vec{k}} \\tilde{v}_{\\vec{k}} ~~~= \\frac{1}{256 q^4 d} + \\mathcal {O}(d^{-2}) \\,, \\\\R_2 &= \\frac{1}{N} \\sum _{\\vec{k}} \\tilde{v}^2_{\\vec{k}} ~~~~~~= \\frac{1}{512 q^4 d} + \\mathcal {O}(d^{-2}) \\,, \\\\R_3 &= \\frac{1}{N} \\sum _{\\vec{k}} \\gamma _{\\vec{k}}\\tilde{u}_{\\vec{k}} \\tilde{v}_{\\vec{k}} = \\frac{1}{32 q^2 d} + \\mathcal {O}(d^{-2}) \\,, \\\\R_4 &= \\frac{1}{N} \\sum _{\\vec{k}} \\gamma _{\\vec{k}} \\tilde{v}^2_{\\vec{k}} ~~~\\,= \\mathcal {O}(d^{-2}) \\,.$ Now, the normal-ordered cubic terms and relevant vertex functions are as follows: $\\mathcal {H}^{\\prime }_{3} &= \\frac{1}{\\sqrt{N}} \\sum _{123} \\big [ \\delta _{1+2+3} \\Gamma _{31}^{o}(\\tilde{\\tau }_{1}^{\\dagger } \\tilde{\\tau }_{2}^{\\dagger } \\tilde{\\tau }_{3}^{\\dagger } + \\tilde{\\tau }_{1} \\tilde{\\tau }_{2} \\tilde{\\tau }_{3}) \\nonumber \\\\&+ \\delta _{1+2-3} \\Gamma _{32}^{o}(\\tilde{\\tau }_{1}^{\\dagger } \\tilde{\\tau }_{2}^{\\dagger } \\tilde{\\tau }_{3} + \\tilde{\\tau }_{3}^{\\dagger } \\tilde{\\tau }_{2} \\tilde{\\tau }_{1}) \\big ] \\,,$ with $\\Gamma _{31}^{o}&= (2 J_1 \\gamma _{1} - h_1) (\\tilde{u}_1 \\tilde{u}_2 \\tilde{v}_3 + \\tilde{v}_1 \\tilde{v}_2 \\tilde{u}_3) \\,, \\\\\\Gamma _{32}^{o}&= 2 J_1 \\big [ \\gamma _{1} (\\tilde{u}_1 \\tilde{u}_2 \\tilde{u}_3 + \\tilde{v}_1 \\tilde{v}_2 \\tilde{v}_3 ) \\nonumber \\\\&+ (\\gamma _1 + \\gamma _3) (\\tilde{u}_1 \\tilde{v}_2 \\tilde{v}_3 + \\tilde{v}_1 \\tilde{u}_2 \\tilde{u}_3 ) \\big ] \\,,$ where $h_1 = \\frac{2 h \\lambda }{1 + \\lambda ^2} - J_1 \\,, ~~~ \\text{and} ~~~J_1 = \\frac{4 qh \\lambda (1-\\lambda ^2)}{(1+\\lambda ^2)^2} \\,.$ Normal ordering of cubic terms also lead to additional linear terms, which together with $\\mathcal {H}_{1}$ () must vanish.",
"This gives the $1/d$ expansion of $\\lambda $ as follows: $\\lambda ^2 = \\frac{2 q- 1}{2 q+ 1} - \\frac{1}{64 q^3 d} \\frac{1 + 16 q^2}{(2 q+1)^2} \\,.$ Again, using the condition that $\\lambda $ vanishes at the quantum critical point, we obtain the same phase boundary (REF ).",
"Lastly, we quote the normal-ordered quartic term and the relevant vertex functions: $\\mathcal {H}^{\\prime }_4 &= \\frac{1}{N} \\sum _{1234}\\big [\\delta _{1+2+3+4} \\Gamma _{41}^{o}(\\tilde{\\tau }^\\dagger _{1}\\tilde{\\tau }^\\dagger _{2}\\tilde{\\tau }^\\dagger _{3}\\tilde{\\tau }^\\dagger _{4} +\\tilde{\\tau }_{1}\\tilde{\\tau }_{2}\\tilde{\\tau }_{3}\\tilde{\\tau }_{4}) \\nonumber \\\\&~~~~~~~~~~~+ \\delta _{1+2-3-4} (\\Gamma _{42}^{o}\\tilde{\\tau }^\\dagger _{1}\\tilde{\\tau }^\\dagger _{2}\\tilde{\\tau }_{3}\\tilde{\\tau }_{4}+\\Gamma _{43}^{o}\\tilde{\\tau }^\\dagger _{1}\\tilde{\\tau }^\\dagger _{2}\\tilde{\\tau }_{3}\\tau _{4})\\nonumber \\\\&~~~~~~~~~~~+\\delta _{1+2+3-4} \\Gamma _{44}^{o}(\\tilde{\\tau }^\\dagger _{1}\\tilde{\\tau }^\\dagger _{2}\\tilde{\\tau }^\\dagger _{3}\\tilde{\\tau }_{4} +\\tilde{\\tau }^\\dagger _{4} \\tilde{\\tau }_{3}\\tilde{\\tau }_{2}\\tilde{\\tau }_{1})\\big ] \\,,$ with $\\Gamma _{41}^{o}&= \\left(\\frac{1-\\lambda ^2}{1+\\lambda ^2}\\right)^{2} 2 qh \\big [ \\gamma _{4} \\tilde{u}_{1} \\tilde{v}_{2} \\tilde{v}_{3} \\tilde{v}_{4} + \\gamma _{4} \\tilde{v}_{1} \\tilde{u}_{2} \\tilde{u}_{3} \\tilde{u}_{4} \\nonumber \\\\&~~~~~~~~~~~+ (\\gamma _{1} + \\gamma _{4}) \\tilde{u}_{1} \\tilde{u}_{2} \\tilde{v}_{3} \\tilde{v}_{4} \\big ] \\nonumber \\\\&-\\frac{16 \\lambda ^2 qh}{(1+\\lambda ^2)^2} \\tilde{u}_{1} \\tilde{u}_{2} \\tilde{v}_{3} \\tilde{v}_{4} \\gamma _{2-4} \\,, \\\\\\Gamma _{44}^{o}&= \\left(\\frac{1-\\lambda ^2}{1+\\lambda ^2}\\right)^{2} 2 qh \\big [ (2 \\gamma _{3} + \\gamma _{4}) \\tilde{u}_{1} \\tilde{v}_{2} \\tilde{v}_{3} \\tilde{u}_{4}\\nonumber \\\\&~~~~~~~~~~~+ (2 \\gamma _{3} + \\gamma _{4}) \\tilde{v}_{1} \\tilde{u}_{2} \\tilde{u}_{3} \\tilde{v}_{4} \\nonumber \\\\&~~~~~~~~~~~+ \\gamma _{3} \\tilde{v}_{1} \\tilde{v}_{2} \\tilde{v}_{3} \\tilde{v}_{4} + \\gamma _{3} \\tilde{u}_{1} \\tilde{u}_{2} \\tilde{u}_{3} \\tilde{u}_{4} \\nonumber \\\\&~~~~~~~~~~~+ (2 \\gamma _{1} + \\gamma _{3} + \\gamma _{4}) \\tilde{u}_{1} \\tilde{u}_{2} \\tilde{v}_{3} \\tilde{u}_{4} \\nonumber \\\\&~~~~~~~~~~~+ (\\gamma _{1} + 2 \\gamma _{3} + \\gamma _{4}) \\tilde{u}_{1} \\tilde{v}_{2} \\tilde{v}_{3} \\tilde{v}_{4} \\big ] \\nonumber \\\\&-\\frac{16 \\lambda ^2 qh}{(1+\\lambda ^2)^2} \\big [ (\\gamma _{2-4} + \\gamma _{2+3})\\tilde{u}_{1} \\tilde{u}_{2} \\tilde{v}_{3} \\tilde{u}_{4} \\nonumber \\\\&~~~~~~~~~~~~~~~~~~~~~~+ (\\gamma _{3-4} + \\gamma _{1+3}) \\tilde{u}_{1} \\tilde{v}_{2} \\tilde{v}_{3} \\tilde{v}_{4}\\big ] \\,.$ The diagrammatics to evaluate the dispersion and ground-state energy is same as in the paramagnetic phase, with the additional contribution from the cubic vertices."
]
] | 1605.05848 |
[
[
"Efficient Nonparametric Smoothness Estimation"
],
[
"Abstract Sobolev quantities (norms, inner products, and distances) of probability density functions are important in the theory of nonparametric statistics, but have rarely been used in practice, partly due to a lack of practical estimators.",
"They also include, as special cases, $L^2$ quantities which are used in many applications.",
"We propose and analyze a family of estimators for Sobolev quantities of unknown probability density functions.",
"We bound the bias and variance of our estimators over finite samples, finding that they are generally minimax rate-optimal.",
"Our estimators are significantly more computationally tractable than previous estimators, and exhibit a statistical/computational trade-off allowing them to adapt to computational constraints.",
"We also draw theoretical connections to recent work on fast two-sample testing.",
"Finally, we empirically validate our estimators on synthetic data."
],
[
"Introduction",
"$L^2$ quantities (i.e., inner products, norms, and distances) of continuous probability density functions are important information theoretic quantities with many applications in machine learning and signal processing.",
"For example, estimates of the $L^2$ norm as can be used for goodness-of-fit testing [10], image registration and texture classification [15], and parameter estimation in semi-parametric models [40].",
"$L^2$ inner products estimates can be used with linear or polynomial kernels to generalize kernel methods to inputs which are distributions rather than numerical vectors.",
"[31] Estimators of $L^2$ distance have been used for two-sample testing [1], [28], transduction learning [33], and machine learning on distributional inputs [30].",
"[32] gives further applications of $L^2$ quantities to adaptive information filtering, classification, and clustering.",
"$L^2$ quantities are a special case of less-well-known Sobolev quantities.",
"Sobolev norms measure global smoothness of a function in terms of integrals of squared derivatives.",
"For example, for a non-negative integer $s$ and a function $f : \\mathbb {R}\\rightarrow \\mathbb {R}$ with an $s^{th}$ derivative $f^{(s)}$ , the $s$ -order Sobolev norm $\\Vert \\cdot \\Vert _{H^s}$ is given by $\\Vert f\\Vert _{H^s} = \\int _\\mathbb {R}\\left( f^{(s)}(x) \\right)^2 \\, dx$ (when this quantity is finite).",
"See Section for more general definitions, and see [24] for an introduction to Sobolev spaces.",
"Estimation of general Sobolev norms has a long history in nonparametric statistics (e.g., [36], [16], [13], [3]) This line of work was motivated by the role of Sobolev norms in many semi- and non-parametric problems, including density estimation, density functional estimation, and regression, (see [39], Section 1.7.1) where they dictate the convergence rates of estimators.",
"Despite this, to our knowledge, these quantities have never been studied in real data, leaving an important gap between the theory and practice of nonparametric statistics.",
"We suggest this is in part due a lack of practical estimators for these quantities.",
"For example, the only one of the above estimators that is statistically minimax-optimal [3] is extremely difficult to compute in practice, requiring numerical integration over each of $O(n^2)$ different kernel density estimates, where $n$ denotes the sample size.",
"We know of no estimators previously proposed for Sobolev inner products and distances.",
"The main goal of this paper is to propose and analyze a family of computationally and statistically efficient estimators for Sobolev inner products, norms, and distances.",
"Our specific contributions are: We propose a family of nonparametric estimators for Sobolev norms, inner products, and distances (Section ).",
"We analyze the bias and variance of the estimators.",
"Assuming the underlying density functions have bounded support in $\\mathbb {R}^D$ and lie in a Sobolev class of sufficient smoothness parametrized by $s^{\\prime }$ , we show that the estimator for Sobolev quantities of order $s < s^{\\prime }$ converges to the true value at the “parametric” rate of $O(n^{-1})$ in mean squared error when $s^{\\prime } \\ge 2s + D/4$ , and at a slower rate of $O \\left( n^{\\frac{8(s - s^{\\prime })}{4s^{\\prime } + D}} \\right)$ otherwise.",
"(Section ).",
"We derive asymptotic distributions for our estimators, and we use these to derive tests for the general statistical problem of two-sample testing.",
"We also draw theoretical connections between our test and the recent work on nonparametric two-sample testing.",
"(Section ).",
"We validate our theoretical results on simulated data.",
"(Section ).",
"In terms of mean squared error, minimax lower bounds matching our convergence rates over Sobolev or Hölder smoothness classes have been shown by [20] for $s = 0$ (i.e., $L^2$ quantities), and [4] for Sobolev norms with integer $s$ .",
"We conjecture but do not prove that our estimator is minimax rate-optimal for all Sobolev quantities and $s \\in [0, \\infty )$ .",
"As described in Section , our estimators are computable in $O(n^{1 + \\varepsilon })$ time using only basic matrix operations, where $n$ is the sample size and $\\varepsilon \\in (0, 1)$ is a tunable parameter trading statistical and computational efficiency; the smallest value of $\\varepsilon $ at which the estimator continues to be minimax rate-optimal approaches 0 as we assume more smoothness of the true density."
],
[
"Problem setup and notation",
"Let $\\mathcal {X}= [-\\pi , \\pi ]^D$ and let $\\mu $ denote the Lebesgue measure on $\\mathcal {X}$ .",
"For $D$ -tuples $z \\in \\mathbb {Z}^D$ of integers, let $\\psi _z \\in L^2 = L^2(\\mathcal {X})$ We suppress dependence on $\\mathcal {X}$ ; all function spaces are over $\\mathcal {X}$ except as discussed in Section REF .",
"defined by $\\psi _z(x) = e^{-i \\langle z, x \\rangle }$ for all $x \\in \\mathcal {X}$ denote the $z^{th}$ element of the $L^2$ -orthonormal Fourier basis, and, for $f \\in L^2$ , let $\\widetilde{f}(z):= \\langle \\psi _z, f \\rangle _{L^2}= \\int _\\mathcal {X}\\psi _z(x) \\overline{f(x)} \\, d\\mu (x)$ denote the $z^{th}$ Fourier coefficient of $f$ .",
"Here, $\\langle \\cdot , \\cdot \\rangle $ denotes the dot product on $\\mathbb {R}^D$ .",
"For a complex number $c = a + bi$ , $\\overline{c} = a - bi$ denotes the complex conjugate of $c$ , and $|c| = \\sqrt{c\\overline{c}} = \\sqrt{a^2 + b^2}$ denotes the modulus of $c$ .",
"For any $s \\in [0, \\infty )$ , define the Sobolev space $H^s = H^s(\\mathcal {X}) \\subseteq L^2$ of order $s$ on $\\mathcal {X}$ by When $D > 1$ , $z^{2s} = \\prod _{j = 1}^D z_j^{2s}$ .",
"For $z < 0$ , $z^{2s}$ should be read as $(z^2)^s$ , so that $z^{2s} \\in \\mathbb {R}$ even when $2s \\notin \\mathbb {Z}$ .",
"In the $L^2$ case, we use the convention that $0^0 = 1$ .",
"$H^s= \\left\\lbrace f \\in L^2 :\\sum _{z \\in \\mathbb {Z}^D} z^{2s} \\left| \\widetilde{f}(z) \\right|^2 < \\infty \\right\\rbrace .$ Fix a known $s \\in [0, \\infty )$ and a unknown probability density functions $p, q \\in H^s$ , and suppose we have $n$ IID samples $X_1,...,X_n \\sim p$ and $Y_1,\\dots ,Y_n \\sim q$ from each of $p$ and $q$ .",
"We are interested in estimating the inner product $\\langle p, q \\rangle _{H^s}:= \\sum _{z \\in \\mathbb {Z}^D} z^{2s} \\widetilde{p}(z) \\overline{\\widetilde{q}(z)}\\quad \\mbox{ defined for all } \\quad p, q \\in H^s.$ Estimating the inner product gives an estimate for the (squared) induced norm and distance, since $\\Vert p\\Vert _{H^s}$ is pseudonorm on $H^s$ because it fails to distinguish functions identical almost everywhere up to additive constants; a combination of $\\Vert p\\Vert _{L^2}$ and $\\Vert p\\Vert _{H^s}$ is used when a proper norm is needed.",
"However, since probability densities integrate to 1, $\\Vert \\cdot - \\cdot \\Vert _{H^s}$ is a proper metric on the subset of (almost-everywhere equivalence classes of) probability density functions in $H^s$ , which is important for two-sample testing (see Section ).",
"For simplicity, we use the terms “norm”, “inner product”, and “distance” for the remainder of the paper.",
"$\\Vert p\\Vert _{H^s}^2:= \\sum _{z \\in \\mathbb {Z}^D} z^{2s} \\left| \\widetilde{p}(z) \\right|^2= \\langle p, p \\rangle _{H^s}\\quad \\mbox{ and } \\quad \\Vert p - q\\Vert _{H^s}^2= \\Vert p\\Vert _{H^s}^2 - 2 \\langle p, q \\rangle _{H^s} + \\Vert q\\Vert _{H^s}^2.$ Since our theoretical results assume the samples from $p$ and $q$ are independent, when estimating $\\Vert p\\Vert _{H^s}^2$ , we split the sample from $p$ in half to compute two independent estimates of $\\widetilde{p}$ , although this may not be optimal in practice.",
"For a more classical intuition, we note that, in the case $D = 1$ and $s \\in \\lbrace 0,1,2,\\dots \\rbrace $ , (via Parseval's identity and the identity $\\widetilde{f^{(s)}}(z) = (iz)^s \\widetilde{f}(z)$ ), that one can show the following: $H^s$ includes the subspace of $L^2$ functions with at least $s$ derivatives in $L^2$ and, if $f^{(s)}$ denotes the $s^{th}$ derivative of $f$ $\\Vert f\\Vert _{H^s}^2= 2\\pi \\int _\\mathcal {X}\\left( f^{(s)}(x) \\right)^2 \\, dx= 2\\pi \\left\\Vert f^{(s)} \\right\\Vert _{L^2}^2,\\quad \\forall f \\in H^s.$ In particular, when $s = 0$ , $H^s = L^2$ , $\\Vert \\cdot \\Vert _{H^s} = \\Vert \\cdot \\Vert _{L^2}$ , and $\\langle \\cdot , \\cdot \\rangle _{H^s} = \\langle \\cdot , \\cdot \\rangle _{L^2}$ .",
"As we describe in the supplement, equation (REF ) and our results generalizes trivially to weak derivatives, as well as to non-integer $s \\in [0, \\infty )$ via a notion of fractional derivative."
],
[
"Unbounded domains",
"A notable restriction above is that $p$ and $q$ are supported in $\\mathcal {X}:= [-\\pi , \\pi ]^D$ .",
"In fact, our estimators and tests are well-defined and valid for densities supported on arbitrary subsets of $\\mathbb {R}^D$ .",
"In this case, they act on the $2\\pi $ -periodic summation $p_{2\\pi } : [-\\pi , \\pi ]^D \\rightarrow [0, \\infty ]$ defined for $x \\in \\mathcal {X}$ by $p_{2\\pi }(x) := \\sum _{z \\in \\mathbb {Z}^D} p(x + 2 \\pi z)$ , which is itself a probability density function on $\\mathcal {X}$ .",
"For example, the estimator for $\\Vert p\\Vert _{H^s}$ will instead estimate $\\Vert p_{2\\pi }\\Vert _{H^s}$ , and the two-sample test for distributions $p$ and $q$ will attempt to distinguish $p_{2\\pi }$ from $q_{2\\pi }$ .",
"In most cases, this is not problematic; for example, for most realistic probability densities, $p$ and $p_{2\\pi }$ have similar orders of smoothness, and $p_{2\\pi } = q_{2\\pi }$ if and only if $p = q$ .",
"However, there are (meagre) sets of exceptions; for example, if $q$ is a translation of $p$ by exactly $2\\pi $ , then $p_{2\\pi } = q_{2\\pi }$ , and one can craft a highly discontinuous function $p$ such that $p_{2\\pi }$ is uniform on $\\mathcal {X}$ .",
"[43] These exceptions make it difficult to extend theoretical results to densities with arbitrary support, but in practice, they are fixed simply by randomly rescaling the data (similar to the approach of [5]).",
"If the densities have (known) bounded support, they can simply be shifted and scaled to be supported on $\\mathcal {X}$ ."
],
[
"Related work",
"There is a large body of work on estimating nonlinear functionals of probability densities, with various generalizations in terms of the class of functionals considered.",
"Table REF gives a subset of such work, for functionals related to Sobolev quantities.",
"As shown in Section , the functional form we consider is a strict generalization of $L^2$ norms, Sobolev norms, and $L^2$ inner products.",
"It overlaps with, but is neither a special case nor a generalization of the remaining functional forms in the table.",
"Table: Some related functional forms for which estimators for whichnonparametric estimators have been developed and analyzed.",
"p,p 1 ,...,p k p,p_1,...,p_k areunknown probability densities, from each of which we draw nn IID samples,ϕ\\varphi is a known real-valued measurable function, and kk is a non-negativeinteger.Nearly all of the above approaches compute an optimally smoothed kernel density estimate and then perform bias corrections based on Taylor series expansions of the functional of interest.",
"They typically consider distributions with densities that are $\\beta $ -Hölder continuous and satisfy periodicity assumptions of order $\\beta $ on the boundary of their support, for some constant $\\beta > 0$ (see, for example, Section 4 of [20] for details of these assumptions).",
"The Sobolev class we consider is a strict superset of this Hölder class, permitting, for example, certain “small” discontinuities.",
"In this regard, our results are slightly more general than most of these prior works.",
"Finally, there is much recent work on estimating entropies, divergences, and mutual informations, using methods based on kernel density estimates [37], [38], [27], [20], [19], [17] or $k$ -nearest neighbor statistics [23], [29], [26], [25].",
"In contrast, our estimators are more similar to orthogonal series density estimators, which are computationally attractive because they require no pairwise operations between samples.",
"However, they require quite different theoretical analysis; unlike prior work, our estimator is constructed and analyzed entirely in the frequency domain, and then related to the data domain via Parseval's identity.",
"We hope our analysis can be adapted to analyze new, computationally efficient information theoretic estimators."
],
[
"Motivation and construction of our estimator",
"For a non-negative integer parameter $Z_n$ (to be specified later), let $p_n:= \\sum _{\\Vert z\\Vert _\\infty \\le Z_n} \\widetilde{p}(z) \\psi _z\\quad \\mbox{ and } \\quad q_n:= \\sum _{\\Vert z\\Vert _\\infty \\le Z_n} \\widetilde{q}(z) \\psi _z\\quad \\mbox{ where } \\quad \\Vert z\\Vert _\\infty := \\max _{j \\in \\lbrace 1,\\dots ,D\\rbrace } z_j$ denote the $L^2$ projections of $p$ and $q$ , respectively, onto the linear subspace spanned by the $L^2$ -orthonormal family $\\mathcal {F}_n := \\lbrace \\psi _z : z \\in \\mathbb {Z}^D, |z| \\le Z_n\\rbrace $ .",
"Note that, since $\\widetilde{\\psi _z}(y) = 0$ whenever $y \\ne z$ , the Fourier basis has the special property that it is orthogonal in $\\langle \\cdot , \\cdot \\rangle _{H^s}$ as well.",
"Hence, since $p_n$ and $q_n$ lie in the span of $\\mathcal {F}_n$ while $p - p_n$ and $q - q_n$ lie in the span of $\\lbrace \\psi _z : z \\in \\mathbb {Z}\\rbrace \\backslash \\mathcal {F}_n$ , $\\langle p - p_n, q_n \\rangle _{H^s} = \\langle p_n, q - q_n \\rangle _{H^s} = 0$ .",
"Therefore, $\\langle p, q \\rangle _{H^s}& = \\langle p_n, q_n \\rangle _{H^s}+ \\langle p - p_n, q_n \\rangle _{H^s}+ \\langle p_n, q - q_n \\rangle _{H^s}+ \\langle p - p_n, q - q_n \\rangle _{H^s} \\\\& = \\langle p_n, q_n \\rangle _{H^s}+ \\langle p - p_n, q - q_n \\rangle _{H^s}.$ We propose an unbiased estimate of $S_n:= \\langle p_n, q_n \\rangle _{H^s}= \\sum _{\\Vert z\\Vert _\\infty \\le Z_n} z^{2s} \\widetilde{p}_n(z) \\overline{\\widetilde{q}_n(z)}$ .",
"Notice that Fourier coefficients of $p$ are the expectations $\\widetilde{p}(z) = \\mathop {\\mathbb {E}}_{X \\sim p} \\left[ \\psi _z(X) \\right]$ .",
"Thus, $\\hat{p}(z) := \\frac{1}{n} \\sum _{j = 1}^n \\psi _z(X_j)$ and $\\hat{q}(z) := \\frac{1}{n} \\sum _{j = 1}^n \\psi _z(Y_j)$ are independent unbiased estimates of $\\widetilde{p}$ and $\\widetilde{q}$ , respectively.",
"Since $S_n$ is bilinear in $\\widetilde{p}$ and $\\widetilde{q}$ , the plug-in estimator for $S_n$ is unbiased.",
"That is, our estimator for $\\langle p, q \\rangle _{H^s}$ is $\\hat{S}_n:= \\sum _{\\Vert z\\Vert _\\infty \\le Z_n} z^{2s} \\hat{p}(z) \\overline{\\hat{q}(z)}.$"
],
[
"Finite sample bounds",
"Here, we present our main theoretical results, bounding the bias, variance, and mean squared error of our estimator for finite $n$ .",
"By construction, our estimator satisfies $\\mathop {\\mathbb {E}}\\left[ \\hat{S}_n \\right]= \\sum _{\\Vert z\\Vert _\\infty \\le Z_n} z^{2s} \\mathop {\\mathbb {E}}\\left[ \\hat{p}(z) \\right]\\overline{\\mathop {\\mathbb {E}}\\left[ \\hat{q}(z) \\right]}= \\sum _{\\Vert z\\Vert _\\infty \\le Z_n} z^{2s} \\widetilde{p}_n(z) \\overline{\\widetilde{q}_n(z)}= S_n.$ Thus, via (REF ) and Cauchy-Schwarz, the bias of the estimator $\\hat{S}_n$ satisfies $\\left| \\mathop {\\mathbb {E}}\\left[ \\hat{S}_n \\right] - \\langle p, q \\rangle _{H^s} \\right|= \\left| \\langle p - p_n, q - q_n \\rangle _{H^s} \\right|\\le \\sqrt{\\left\\Vert p - p_n \\right\\Vert _{H^s}^2\\left\\Vert q - q_n \\right\\Vert _{H^s}^2}.$ $\\left\\Vert p - p_n \\right\\Vert _{H^s}$ is the error of approximating $p$ by an order-$Z_n$ trigonometric polynomial, a classic problem in approximation theory, for which Theorem 2.2 of [18] shows: $\\mbox{ if } p \\in H^{s^{\\prime }} \\mbox{ for some } s^{\\prime } > s,\\quad \\mbox{ then } \\quad \\left\\Vert p - p_n \\right\\Vert _{H^s}\\le \\Vert p\\Vert _{H^{s^{\\prime }}}Z_n^{s - s^{\\prime }}.$ In combination with (REF ), this implies the following bound on the bias of our estimator: Theorem 1 (Bias bound) If $p, q \\in H^{s^{\\prime }}$ for some $s^{\\prime } > s$ , then, for $C_B := \\Vert p\\Vert _{H^{s^{\\prime }}} \\Vert q\\Vert _{H^{s^{\\prime }}}$ , $\\left| \\mathop {\\mathbb {E}}\\left[ \\hat{S}_n \\right] - \\langle p, q \\rangle _{H^s} \\right|\\le C_B Z_n^{2(s - s^{\\prime })}$ Hence, the bias of $\\hat{S}_n$ decays polynomially in $Z_n$ , with a power depending on the “extra” $s^{\\prime } - s$ orders of smoothness available.",
"On the other hand, as we increase $Z_n$ , the number of frequencies at which we estimate $\\hat{p}$ increases, suggesting that the variance of the estimator will increase with $Z_n$ .",
"Indeed, this is expressed in the following bound on the variance of the estimator.",
"Theorem 2 (Variance bound) If $p, q \\in H^{s^{\\prime }}$ for some $s^{\\prime } \\ge s$ , then $\\mathbb {V}\\left[ \\hat{S}_n \\right]\\le 2C_1 \\frac{Z_n^{4s + D}}{n^2} + \\frac{C_2}{n},$ where $C_1$ and $C_2$ are the constants (in $n$ ) $C_1:= \\frac{2^D\\Gamma (4s + 1)}{\\Gamma (4s + D + 1)} \\Vert p\\Vert _{L^2} \\Vert q\\Vert _{L^2}$ and $C_2:= \\left( \\Vert p\\Vert _{H^s} + \\Vert q\\Vert _{H^s} \\right) \\Vert p\\Vert _{W^{2s,4}} \\Vert q\\Vert _{W^{2s,4}}+ \\Vert p\\Vert _{H^s}^4 \\Vert q\\Vert _{H^s}^4$ .",
"The proof of Theorem REF is perhaps the most significant theoretical contribution of this work.",
"Due to space constraints, the proof is given in the appendix.",
"Combining Theorems REF and REF gives a bound on the mean squared error (MSE) of $\\hat{S}_n$ via the usual decomposition into squared bias and variance: Corollary 3 (Mean squared error bound) If $p, q \\in H^{s^{\\prime }}$ for some $s^{\\prime } > s$ , then $\\mathop {\\mathbb {E}}\\left[ \\left( \\hat{S}_n - \\langle p, q \\rangle _{H^s} \\right)^2 \\right]\\le C_B^2 Z_n^{4(s - s^{\\prime })}+ 2C_1 \\frac{Z_n^{4s + D}}{n^2} + \\frac{C_2}{n}.$ If, furthermore, we choose $Z_n \\asymp n^{\\frac{2}{4s^{\\prime } + D}}$ (optimizing the rate in inequality REF ), then $\\mathop {\\mathbb {E}}\\left[ \\left( \\hat{S}_n - \\langle p, q \\rangle _{H^2} \\right)^2 \\right]\\asymp n^{\\max \\left\\lbrace \\frac{8(s - s^{\\prime })}{4s^{\\prime } + D}, -1 \\right\\rbrace }.$ Corollary REF recovers the phenomenon discovered by [3]: when $s^{\\prime } \\ge 2s + \\frac{D}{4}$ , the minimax optimal MSE decays at the “semi-parametric” $n^{-1}$ rate, whereas, when $s^{\\prime } \\in \\left( s, 2s + \\frac{D}{4} \\right)$ , the MSE decays at a slower rate.",
"Also, the estimator is $L^2$ -consistent if $Z_n \\rightarrow \\infty $ and $Z_n n^{-\\frac{2}{4s + D}} \\rightarrow 0$ as $n \\rightarrow \\infty $ .",
"This is useful in practice, since $s$ is known but $s^{\\prime }$ is not."
],
[
"Asymptotic distributions",
"In this section, we derive the asymptotic distributions of our estimator in two cases: (1) the inner product estimator and (2) the distance estimator in the case $p = q$ .",
"These results provide confidence intervals and two-sample tests without computationally intensive resampling.",
"While (1) is more general in that it can be used with (REF ) to bound the asymptotic distributions of the norm and distance estimators, (2) provides a more precise result leading to a more computationally and statistically efficient two-sample test.",
"Proofs are given in the supplementary material.",
"Theorem REF shows that our estimator has a normal asymptotic distribution, assuming $Z_n \\rightarrow \\infty $ slowly enough as $n \\rightarrow \\infty $ , and also gives a consistent estimator for its asymptotic variance.",
"From this, one can easily estimate asymptotic confidence intervals for inner products, and hence also for norms.",
"Theorem 4 (Asymptotic normality) Suppose that, for some $s^{\\prime } > 2s + \\frac{D}{4}$ , $p, q \\in H^{s^{\\prime }}$ , and suppose $Z_n n^{\\frac{1}{4(s - s^{\\prime })}} \\rightarrow \\infty $ and $Z_n n^{-\\frac{1}{4s + D}} \\rightarrow 0$ as $n \\rightarrow \\infty $ .",
"Then, $\\hat{S}_n$ is asymptotically normal with mean $\\langle p, q \\rangle _{H^s}$ .",
"In particular, for $j \\in \\lbrace 1, \\dots , n\\rbrace $ and $z \\in \\mathbb {Z}^D$ with $\\Vert z\\Vert _\\infty \\le Z_n$ , define $W_{j,z} := z^s e^{izX_j}$ and $V_{j,z} := z^s e^{izY_j}$ , so that $W_j$ and $V_j$ are column vectors in $\\mathbb {R}^{(2Z_n)^D}$ .",
"Let $\\overline{W} := \\frac{1}{n} \\sum _{j = 1}^n W_j,\\overline{V} := \\frac{1}{n} \\sum _{j = 1}^n V_j\\in \\mathbb {R}^{(2Z_n)^D},$ $\\Sigma _W := \\frac{1}{n} \\sum _{j = 1}^n (W_j - \\overline{W}) (W_j - \\overline{W})^T,\\quad \\mbox{ and } \\quad \\Sigma _V := \\frac{1}{n} \\sum _{j = 1}^n (V_j - \\overline{V}) (V_j - \\overline{V})^T\\in \\mathbb {R}^{(2Z_n)^D \\times (2Z_n)^D}$ denote the empirical means and covariances of $W$ and $V$ , respectively.",
"Then, for $\\hat{\\sigma }_n^2:= \\begin{bmatrix}\\overline{V} \\\\\\overline{W}\\end{bmatrix}^T\\begin{bmatrix}\\Sigma _W & 0 \\\\0 & \\Sigma _V\\end{bmatrix}\\begin{bmatrix}\\overline{V} \\\\\\overline{W}\\end{bmatrix},\\quad \\mbox{ we have } \\quad \\sqrt{n} \\left( \\frac{\\hat{S}_n - \\langle p, q \\rangle _{H^s}}{\\hat{\\sigma }_n} \\right)\\stackrel{D}{\\rightarrow } \\mathcal {N}(0, 1),$ where $\\stackrel{D}{\\rightarrow }$ denotes convergence in distribution.",
"Since distances can be written as a sum of three inner products (Eq.",
"(REF )), Theorem REF might suggest an asymptotic normal distribution for Sobolev distances.",
"However, extending asymptotic normality from inner products to their sum requires that the three estimates be independent, and hence that we split data between the three estimates.",
"This is inefficient in practice and somewhat unnatural, as we know, for example, that distances should be non-negative.",
"For the particular case $p = q$ (as in the null hypothesis of two-sample testing), the following theorem This result is closely related to Proposition 4 of [5].",
"However, in their situation, $s = 0$ and the set of test frequencies is fixed as $n \\rightarrow \\infty $ , whereas our set is increasing.",
"provides a more precise asymptotic ($\\chi ^2$ ) distribution of our Sobolev distance estimator, after an extra decorrelation step.",
"This gives, for example, a more powerful two-sample test statistic (see Section for details).",
"Theorem 5 (Asymptotic null distribution) Suppose that, for some $s^{\\prime } > 2s + \\frac{D}{4}$ , $p, q \\in H^{s^{\\prime }}$ , and suppose $Z_n n^{\\frac{1}{4(s - s^{\\prime })}} \\rightarrow \\infty $ and $Z_n n^{-\\frac{1}{4s + D}} \\rightarrow 0$ as $n \\rightarrow \\infty $ .",
"For $j \\in \\lbrace 1,\\dots ,n\\rbrace $ and $z \\in \\mathbb {Z}^D$ with $\\Vert z\\Vert _{\\infty } \\le Z_n$ , define $W_{j, z} := z^s \\left( e^{-izX_j} - e^{-izY_j} \\right)$ , so that $W_j$ is a column vector in $\\mathbb {R}^{(2Z_n)^D}$ .",
"Let $\\overline{W} := \\frac{1}{n} \\sum _{j = 1}^n W_j \\in \\mathbb {R}^{(2Z_n)^D}\\quad \\mbox{ and } \\quad \\Sigma := \\frac{1}{n} \\sum _{j = 1}^n \\left( W_j - \\overline{W} \\right)\\left( W_j - \\overline{W} \\right)^T\\in \\mathbb {R}^{(2Z_n)^D \\times (2Z_n)^D}$ denote the empirical mean and covariance of $W$ , and define $T := n \\overline{W}^T \\Sigma ^{-1}\\overline{W}$ .",
"Then, if $p = q$ , then $Q_{\\chi ^2((2Z_n)^D)}(T) \\stackrel{D}{\\rightarrow } \\operatorname{Uniform}([0, 1])\\quad \\mbox{ as } \\quad n \\rightarrow \\infty ,$ where $Q_{\\chi ^2(d)} : [0, \\infty ) \\rightarrow [0, 1]$ denotes the quantile function (inverse CDF) of the $\\chi ^2$ distribution $\\chi ^2(d)$ with $d$ degrees of freedom.",
"Let $\\hat{M}$ denote our estimator for $\\Vert p - q\\Vert _{H^s}$ (i.e., plugging $\\hat{S}_n$ into (REF )).",
"While Theorem REF immediately provides a valid two-sample test of desired level, it is not immediately clear how this relates to $\\hat{M}$ , nor is there any suggestion of why the test statistic ought to be a good (i.e., consistent) one.",
"Some intuition is as follows.",
"Notice that $\\hat{M} = \\overline{W}^T \\overline{W}$ .",
"Since, by the central limit theorem, $\\overline{W}$ has a normal asymptotic distribution, if the components of $\\overline{W}$ were uncorrelated (and $Z_n$ were fixed), we would expect $n \\hat{M}$ to have an asymptotic $\\chi ^2$ distribution with $(2Z_n)^D$ degrees of freedom.",
"However, because we use the same data to compute each component of $\\hat{M}$ , they are not typically uncorrelated, and so the asymptotic distribution of $\\hat{M}$ is difficult to derive.",
"This motivates the statistic $T = \\left( \\sqrt{\\Sigma _W^{-1}} \\overline{W} \\right)^T\\sqrt{\\Sigma _W^{-1}} \\overline{W}$ , since the components of $\\sqrt{\\Sigma _W^{-1}} \\overline{W}$ are (asymptotically) uncorrelated."
],
[
"Parameter selection and statistical/computational trade-off",
"Here, we give statistical and computational considerations for choosing the smoothing parameter $Z_n$ .",
"Statistical perspective: In practice, of course, we do not typically know $s^{\\prime }$ , so we cannot simply set $Z_n \\asymp n^{\\frac{2}{4s^{\\prime } + D}}$ , as suggested by the mean squared error bound (REF ).",
"Fortunately (at least for ease of parameter selection), when $s^{\\prime } \\ge 2s + \\frac{D}{4}$ , the dominant term of (REF ) is $C_2/n$ for $Z_n \\asymp n^{-\\frac{1}{4s + D}}$ .",
"Hence if we are willing to assume that the density has at least $2s + \\frac{D}{4}$ orders of smoothness (which may be a mild assumption in practice), then we achieve statistical optimality (in rate) by setting $Z_n \\asymp n^{-\\frac{1}{4s + D}}$ , which depends only on known parameters.",
"On the other hand, the estimator can continue to benefit from additional smoothness computationally.",
"Computational perspective One attractive property of the estimator discussed is its computational simplicity and efficiency with respect to $n$ , in low dimensions.",
"Most competing nonparametric estimators, such as kernel-based or nearest-neighbor methods, either take $O(n^2)$ time or rely on complex data structures such as $k$ -d trees or cover trees [34] for $O(2^D n\\log n)$ time performance.",
"Since computing the estimator takes $O(nZ_n^D)$ time and $O(Z_n^D)$ memory (that is, the cost of estimating each of $(2Z_n)^D$ Fourier coefficients by an average), a statistically optimal choice of $Z_n$ gives a runtime of $O \\left( n^{\\frac{4s^{\\prime } + 2D}{4s^{\\prime } + D}} \\right)$ .",
"Since the estimate requires only a vector outer product, exponentiation, and averaging, the constants involved are small and computations parallelize trivially over frequencies and data.",
"Under severe computational constraints, for very large data sets, or if $D$ is large relative to $s^{\\prime }$ , we can reduce $Z_n$ to trade off statistical for computational efficiency.",
"For example, if we want an estimator with runtime $O(n^{1 + \\theta })$ and space requirement $O(n^\\theta )$ for some $\\theta \\in \\left( 0, \\frac{2D}{4s^{\\prime } + D} \\right)$ , setting $Z_n \\asymp n^{\\theta /D}$ still gives a consistent estimator, with mean squared error of the order $O \\left( n^{\\max \\lbrace \\frac{4\\theta (s - s^{\\prime })}{D}, -1\\rbrace } \\right)$ .",
"Kernel- or nearest-neighbor-based methods, including nearly all of the methods described in Section , tend to require storing previously observed data, resulting in $O(n)$ space requirements.",
"In contrast, orthogonal basis estimation requires storing only $O(Z_n^D)$ estimated Fourier coefficients.",
"The estimated coefficients can be incrementally updated with each new data point, which may make the estimator or close approximations feasible in streaming settings."
],
[
"Experimental results",
"In this section, we use synthesized data to demonstrate the effectiveness of our methods.",
"A MATLAB implementation of our estimators, two-sample tests, and experiments is available at https://github.com/sss1/SobolevEstimation.",
"For all experiments, we use $10,100,1000,10000,100000$ samples for estimation.",
"We first test our estimators on 1D $L_2$ distances.",
"Figure REF shows estimated distance between $\\mathcal {N}\\left(0,1\\right)$ and $\\mathcal {N}\\left(1,1\\right)$ ; Figure REF shows estimated distance between $\\mathcal {N}\\left(0,1\\right)$ and $\\mathcal {N}\\left(0,4\\right)$ ; Figure REF shows estimated distance between Unif $\\left[0,1\\right]$ and Unif$\\left[0.5,1.5\\right]$ ; Figure REF shows estimated distance between $\\left[0,1\\right]$ and a triangular distribution whose density is highest at $x=0.5$ .",
"Error bars indicate asymptotic $95\\%$ confidence intervals based on Theorem REF .",
"These experiments suggest $10^5$ samples is sufficient to estimate $L_2$ distances with high confidence.",
"Note that we need fewer samples to estimate Sobolev quantities of Gaussians than, say, of uniform distributions, consistent with our theory, since (infinitely differentiable) Gaussians are smoothier than (discontinuous) uniform distributions.",
"Next, we test our estimators on $L_2$ distances of multivariate distributions.",
"Figure REF shows estimated distance between $\\mathcal {N}\\left(\\left[0,0,0\\right],\\mathbf {I}\\right)$ and $\\mathcal {N}\\left(\\left[1,1,1\\right],\\mathbf {I}\\right)$ ; Figure REF shows estimated distance between $\\mathcal {N}\\left(\\left[0,0,0\\right],\\mathbf {I}\\right)$ and $\\mathcal {N}\\left(\\left[0,0,0\\right],4\\mathbf {I}\\right)$ .",
"Again, these experiments show that our estimators can also handle multivariate distributions.",
"Lastly, we test our estimators for $H^s$ norms.",
"Figure REF shows estimated $H^0$ norm of $\\mathcal {N}\\left(0,1\\right)$ and Figure REF shows $H^1$ norm of $\\mathcal {N}\\left(0,1\\right)$ .",
"Notice that we need fewer samples to estimate $H_0$ than $H^1$ , which verifies our theory.",
"Figure: One uniform distribution and one triangular distribution.Figure: Estimation of H 1 H^1 norm of 𝒩0,1\\mathcal {N}\\left(0,1\\right)."
],
[
"Connections to two-sample testing",
"Here, we discuss the use of our estimator in two-sample testing.",
"There is a large literature on nonparametric two-sample testing, but we discuss only some recent approaches with theoretical connections to ours.",
"Let $\\hat{M}$ denote our estimate of the Sobolev distance, consisting of plugging $\\hat{S}$ into equation (REF ).",
"Since $\\Vert \\cdot - \\cdot \\Vert _{H^s}$ is a metric on the space of probability density functions in $H^s$ , computing $\\hat{M}$ leads naturally to a two-sample test on this space.",
"Theorem REF suggests an asymptotic test, which is computationally preferable to a permutation test.",
"In particular, for a desired Type I error rate $\\alpha \\in (0, 1)$ our test rejects the null hypothesis $p = q$ if and only if $Q_{\\chi ^2(2Z_n^D)}(T) < \\alpha $ .",
"When $s = 0$ , this approach is closely related to several two-sample tests in the literature based on comparing empirical characteristic functions (CFs).",
"Originally, these tests [14], [7] computed the same statistic $T$ with a fixed number of random $\\mathbb {R}^D$ -valued frequencies instead of deterministic $\\mathbb {Z}^D$ -valued frequencies.",
"This test runs in linear time, but is not generally consistent, since the two CFs need not differ almost everywhere.",
"Recently, [5] suggested using smoothed CFs, i.e., the convolution of the CF with a universal smoothing kernel $k$ .",
"This is computationally easy (due to the convolution theorem) and, when $p \\ne q$ , $(\\widetilde{p} * k)(z) \\ne (\\widetilde{q} * k)(z)$ for almost all $z \\in \\mathbb {R}^D$ , reducing the need for carefully choosing test frequencies.",
"Furthermore, this test is almost-surely consistent under very general alternatives.",
"However, it is not clear what sort of assumptions would allow finite sample analysis of the power of their test.",
"Indeed, the convergence as $n \\rightarrow \\infty $ can be arbitrarily slow, depending on the random test frequencies used.",
"Our analysis instead uses the assumption $p, q \\in H^{s^{\\prime }}$ to ensure that small, $\\mathbb {Z}^D$ -valued frequencies contain most of the power of $\\widetilde{p}$ .",
"Note that smooth CFs can be used in our test by replacing $\\hat{p}(z)$ with $\\frac{1}{n} \\sum _{j = 1}^n e^{-iz X_j} k(x)$ , where $k$ is the inverse Fourier transform of a characteristic kernel.",
"However, smoothing seems less desirable under Sobolev assumptions, as it spreads the power of the CF away from small $\\mathbb {Z}^D$ -valued frequencies where our test focuses.",
"These fixed-frequency approaches can be thought of as the extreme point $\\theta = 0$ of the computational/statistical trade-off described in section : they are computable in linear time and (with smoothing) are strongly consistent, but do not satisfy finite-sample bounds under general conditions.",
"At the other extreme ($\\theta = 1$ ) are MMD-based tests of [11], [12], which utilize the entire spectrum $\\widetilde{p}$ .",
"These tests are statistically powerful and have strong guarantees for densities in an RKHS, but have $O(n^2)$ computational complexity.",
"Fast MMD approximations have been proposed, including the Block MMD, [41] FastMMD, [42] and $\\sqrt{n}$ sub-sampled MMD, but these lack the statistical guarantees of MMD.",
"The computational/statistical trade-off discussed in Section can be thought of as an interpolation (controlled by $\\theta $ ) of these approaches, with runtime in the case $\\theta = 1$ approaching quadratic for large $D$ and small $s^{\\prime }$ ."
],
[
"Conclusions and future work",
"In this paper, we proposed nonparametric estimators for Sobolev inner products, norms and distances of probability densities, for which we derived finite-sample bounds and asymptotic distributions.",
"A natural follow-up question to our work is whether estimating smoothness of a density can guide the choice of smoothing parameters in nonparametric estimation.",
"For some problems, such as estimating functionals of a density, this may be especially useful, since no error metric is typically available for cross-validation.",
"Even when cross-validation is an option, as in density estimation or regression, estimating smoothness may be faster, or may suggest an appropriate range of parameter values."
],
[
"Proof of Variance Bound",
"Theorem 7 (Variance Bound) If $p, q \\in H^{s^{\\prime }}$ for some $s^{\\prime } > s$ , then $\\mathbb {V}\\left[ \\hat{S}_n \\right]\\le 2C_1 \\frac{Z_n^{4s + D}}{n^2} + \\frac{C_2}{n},$ where $C_1$ and $C_2$ are the constants (in $n$ ) $C_1 := \\frac{2^D\\Gamma (4s + 1)}{\\Gamma (4s + D + 1)} \\Vert p\\Vert _{L^2} \\Vert q\\Vert _{L^2}$ and $C_2:= \\left( \\Vert p\\Vert _{H^s} + \\Vert q\\Vert _{H^s} \\right) \\Vert p\\Vert _{W^{2s,4}} \\Vert q\\Vert _{W^{2s,4}}+ \\Vert p\\Vert _{H^s}^4 \\Vert q\\Vert _{H^s}^4$ .",
"We will use the Efron-Stein inequality [6] to bound the variance of $\\hat{S}_n$ .",
"To do this, suppose we were to draw $n$ additional IID samples $X_1^{\\prime },\\dots ,X_n^{\\prime } \\sim p$ , and define, for all $\\ell , j \\in \\lbrace 1,\\dots ,n\\rbrace $ , $X_j^{(\\ell )}= \\left\\lbrace \\begin{array}{ll}X_j^{\\prime } & \\mbox{ if } j = \\ell \\\\X_j & \\mbox{ else }\\end{array}\\right..$ Let $\\hat{S}_n^{(\\ell )}:= \\frac{1}{n^2}\\sum _{|z| \\le Z_n} z^{2s} \\sum _{j = 1}^n \\sum _{k = 1}^n\\psi _z(X_j^{(\\ell )}) \\overline{\\psi _z(Y_k)}$ denote our estimate when we replace $X_\\ell $ by $X_\\ell ^{\\prime }$ .",
"Noting the symmetry of $\\hat{S}_n$ in $p$ and $q$ , the Efron-Stein inequality tells us that $\\mathbb {V}\\left[ \\hat{S}_n \\right]\\le \\sum _{\\ell = 1}^n\\mathop {\\mathbb {E}}\\left[\\left|\\hat{S}_n - \\hat{S}_n^{(\\ell )}\\right|^2\\right],$ where the expectation above (and elsewhere in this section) is taken over all $3n$ samples $X_1,\\dots ,X_{2n},X_1^{\\prime },\\dots ,X_{2n}^{\\prime },Y_1,\\dots ,Y_n$ .",
"Expanding the difference in (REF ), note that any terms with $j \\ne \\ell $ cancel, so that It is useful here to note that $\\overline{\\psi _z(x)} = \\psi _{-z}(x)$ and that $\\psi _y\\psi _z = \\psi _{y + z}$ .",
"$\\hat{S}_n - \\hat{S}_n^{(\\ell )}& = \\frac{1}{n^2}\\sum _{|z| \\le Z_n} z^{2s}\\sum _{j = 1}^n \\sum _{k = 1}^n\\psi _z(X_j) \\overline{\\psi _z(Y_k)}- \\psi _z(X_j^{(\\ell )}) \\overline{\\psi _z(Y_k)} \\\\& = \\frac{1}{n^2}\\sum _{|z| \\le Z_n} z^{2s} (\\psi _z(X_\\ell ) - \\psi _z(X_\\ell ^{\\prime }))\\sum _{k = 1}^n \\psi _{-z}(Y_k),$ and so $& \\left| \\hat{S}_n - \\hat{S}_n^{(\\ell )} \\right|^2 \\\\& = \\frac{1}{n^4}\\sum _{|y|,|z| \\le Z_n}(yz)^{2s}(\\psi _y(X_\\ell ) - \\psi _y(X_\\ell ^{\\prime }))(\\psi _{-z}(X_\\ell ) - \\psi _{-z}(X_\\ell ^{\\prime }))\\left( \\sum _{k = 1}^n \\psi _{-y}(Y_k) \\right)\\left( \\sum _{k = 1}^n \\psi _z(Y_k) \\right).$ Since $X_\\ell $ and $X_\\ell ^{\\prime }$ are IID, $\\mathop {\\mathbb {E}}\\left[ (\\psi _y(X_\\ell ) - \\psi _y(X_\\ell ^{\\prime }))(\\psi _{-z}(X_\\ell ) - \\psi _{-z}(X_\\ell ^{\\prime })) \\right]& = 2 \\left( \\mathop {\\mathbb {E}}_{X \\sim p} \\left[ \\psi _{y - z}(X) \\right]- \\mathop {\\mathbb {E}}_{X \\sim p} \\left[ \\psi _y(X) \\right]\\mathop {\\mathbb {E}}_{X \\sim p} \\left[ \\psi _{-z}(X) \\right] \\right) \\\\& = 2 \\left( \\widetilde{p}(y - z) - \\widetilde{p}(y) \\widetilde{p}(-z) \\right),$ and, since $Y_1,\\dots ,Y_n$ are IID, $\\mathop {\\mathbb {E}}\\left[ \\left( \\sum _{k = 1}^n \\psi _{-y}(Y_k) \\right)\\left( \\sum _{k = 1}^n \\psi _z(Y_k) \\right) \\right]& = n \\mathop {\\mathbb {E}}_{Y \\sim q} \\left[ \\psi _{z - y}(Y) \\right]+ n(n - 1) \\mathop {\\mathbb {E}}_{Y \\sim q} \\left[ \\psi _{-y}(Y) \\right]\\mathop {\\mathbb {E}}_{Y \\sim q} \\left[ \\psi _z(Y) \\right] \\\\& = n \\widetilde{q}(z - y)+ n(n - 1) \\widetilde{q}(-y) \\widetilde{q}(z).$ In view of these two equalities, taking the expectation of (REF ) and using the fact that $X_\\ell $ and $X_\\ell ^{\\prime }$ are independent of $X_{n + 1},\\dots ,X_{2n}$ , (REF ) reduces: $& \\mathop {\\mathbb {E}}\\left[ \\left| \\hat{S}_n - \\hat{S}_n^{(\\ell )} \\right|^2 \\right]= \\frac{2}{n^3}\\sum _{|y|,|z| \\le Z_n}(yz)^{2s}\\left( \\widetilde{p}(y - z) - \\widetilde{p}(y) \\widetilde{p}(-z) \\right)\\left( \\widetilde{q}(z - y) + (n - 1) \\widetilde{q}(-y) \\widetilde{q}(z) \\right) \\\\& = \\frac{2}{n^3}\\sum _{|y|,|z| \\le Z_n}(yz)^{2s}\\left(\\widetilde{p}(y - z) \\widetilde{q}(z - y)- \\widetilde{p}(y) \\widetilde{p}(-z) \\widetilde{q}(z - y)\\right.",
"\\\\& \\hspace{99.58464pt}\\left.+ (n - 1) \\widetilde{p}(y - z) \\widetilde{q}(-y) \\widetilde{q}(z)- (n - 1) \\widetilde{p}(y) \\widetilde{p}(-z) \\widetilde{q}(-y) \\widetilde{q}(z)\\right).$ We now need to bound following terms in magnitude: $& \\sum _{|y|,|z| \\le Z_n} (yz)^{2s} \\widetilde{p}(y - z) \\widetilde{q}(z - y), \\quad \\\\& \\sum _{|y|,|z| \\le Z_n} (yz)^{2s} \\widetilde{p}(y - z) \\widetilde{q}(-y) \\widetilde{q}(z), \\\\\\mbox{ and } \\quad & \\sum _{|y|,|z| \\le Z_n} (yz)^{2s} \\widetilde{p}(y) \\widetilde{p}(-z) \\widetilde{q}(-y) \\widetilde{q}(z)$ (the second term in (REF ) is bounded identically to the third term).",
"To bound (REF ), we perform a change of variables, replacing $y$ by $k = y - z$ : $\\sum _{|y|,|z| \\le Z_n} (yz)^{2s} \\widetilde{p}(y - z) \\widetilde{q}(z - y)& = \\sum _{|k| \\le 2Z_n} \\widetilde{p}(k) \\widetilde{q}(-k)\\sum _{j = 1}^D \\sum _{z_j = \\max \\lbrace -Z_n, k_j - Z_n\\rbrace }^{\\min \\lbrace Z_n, k_j + Z_n\\rbrace } ((k - z)z)^{2s} \\\\& \\le \\frac{2^D\\Gamma (4s + 1)}{\\Gamma (4s + D + 1)} Z_n^{4s + D}\\sum _{|k| \\le 2Z_n} \\widetilde{p}(k) \\widetilde{q}(-k) \\\\& \\le C_1 Z_n^{4s + D},$ where $C_1$ is the constant (in $n$ and $Z_n$ ) $C_1:= \\frac{2^D\\Gamma (4s + 1)}{\\Gamma (4s + D + 1)} \\Vert p\\Vert _2 \\Vert q\\Vert _2.$ (REF ) and () follow from observing that $\\sum _{j = 1}^D \\sum _{z_j = \\max \\lbrace -Z_n, k_j - Z_n\\rbrace }^{\\min \\lbrace Z_n, k_j + Z_n\\rbrace }((k_j - z_j)z_j)^{2s}= (f * f)(k_j),$ where $f(z) := z^{2s} 1_{\\lbrace |z| \\le Z_n\\rbrace }, \\forall z \\in \\mathbb {Z}^D$ and $*$ denotes convolution (over $\\mathbb {Z}^D$ ).",
"This convolution is clearly maximized when $k = 0$ , in which case $(f * f)(k)= \\sum _{|z| \\le Z_n} z^{4s}\\le \\left( \\int _{B_\\infty (0, Z_n)} z^{4s} \\, dz\\right)= \\frac{2^D\\Gamma (4s + 1)}{\\Gamma (4s + D + 1)} Z_n^{4s + D},$ where we upper bounded the series by an integral over $B_\\infty (0, Z_n):= \\lbrace z \\in \\mathbb {R}^D : \\Vert z\\Vert _\\infty = \\max \\lbrace |z_1|,...,|z_D|\\rbrace \\le Z_n\\rbrace .$ () then follows via Cauchy-Schwarz.",
"Bounding () for general $s$ is more involved and requires rigorously defining more elaborate notions from the theory distributions, but the basic idea is as follows: $\\sum _{|y|,|z| \\le Z_n} (yz)^{2s} \\widetilde{p}(y - z) \\widetilde{q}(-y) \\widetilde{q}(z)& = \\sum _{|y| \\le Z_n}y^{2s} \\widetilde{q}(-y)\\sum _{|z| \\le Z_n} z^{2s} \\widetilde{p}(y - z) \\widetilde{q}(z) \\\\& = \\sum _{|y|,|z| \\le Z_n} y^{2s} \\widetilde{q}(-y) \\widetilde{\\left( p_n^{(s)} q_n^{(s)} \\right)}(y) \\\\& \\le \\sqrt{\\sum _{|y| \\le Z_n} y^{2s} \\left| \\widetilde{q}(y) \\right|^2\\sum _{|y| \\le Z_n} y^{2s}\\left( \\widetilde{\\left( p^{(s)} q^{(s)} \\right)}(y) \\right)^2} \\\\& = \\Vert q\\Vert _{H^s} \\Vert p_n^{(s)}q_n^{(s)}\\Vert _{H^s}\\le \\Vert q\\Vert _{H^s} \\Vert p_n\\Vert _{W^{2s, 4}} \\Vert q_n\\Vert _{W^{2s, 4}}.$ Here, $p_n^{(s)}$ and $q_n^{(s)}$ denote $s$ -order fractional derivatives of $p_n$ and $q_n$ , respectively, and $W^{2s,4}$ is a Sobolev space (with associated pseudonorm $\\Vert \\cdot \\Vert _{W^{2s,4}}$ ), which can be informally thought of as $W^{2s, 4} := \\left\\lbrace p \\in L^2 : \\left( p^{(s)} \\right)^2 \\in H^s \\right\\rbrace $ .",
"The equality between the first and second lines follows from Theorem REF , and both inequalities are simply applications of Cauchy-Schwarz.",
"For sake of intuition, it can be noted that the above steps are relatively elementary when $s = 0$ .",
"Now, it suffices to note that, by the Rellich-Kondrachov embedding theorem [35], [8], $W^{2s,4} \\subseteq H^{s^{\\prime }}$ , and hence $\\Vert p_n\\Vert _{W^{2s, 4}} \\le \\Vert p\\Vert _{W^{2s,4}} < \\infty $ , as long as $s^{\\prime } \\ge 2s + \\frac{D}{4}$ .",
"Bounding () is a simple application of Cauchy-Schwarz: $\\sum _{|y|,|z| \\le Z_n} (yz)^{2s} \\widetilde{p}(y) \\widetilde{p}(-z) \\widetilde{q}(-y) \\widetilde{q}(z)& = \\left( \\sum _{|y| \\le Z_n} y^{2s} \\widetilde{p}(y) \\widetilde{q}(-y) \\right)\\left( \\sum _{|z| \\le Z_n} z^{2s} \\widetilde{p}(-z) \\widetilde{q}(z) \\right) \\\\& \\le \\left( \\sum _{|z| \\le Z_n} z^{2s} \\left| \\widetilde{p}(z) \\right|^2 \\right)^2\\left( \\sum _{|z| \\le Z_n} z^{2s} \\left| \\widetilde{q}(z) \\right|^2 \\right)^2 \\\\& = \\Vert p\\Vert _{H^s}^4 \\Vert q\\Vert _{H^s}^4$ Plugging (), (REF ), and (REF ) into (REF ) gives $\\mathop {\\mathbb {E}}\\left[ \\left| \\hat{S}_n - \\hat{S}_n^{(\\ell )} \\right|^2 \\right]\\le 2C_1 \\frac{Z_n^{4s + D}}{n^3}+ \\frac{C_2}{n^2},$ where $C_2$ denotes the constant (in $n$ and $Z_n$ ) $C_2:= \\left( \\Vert p\\Vert _{H^s} + \\Vert q\\Vert _{H^s} \\right) \\Vert p\\Vert _{W^{2s,4}} \\Vert q\\Vert _{W^{2s,4}}+ \\Vert p\\Vert _{H^s}^4 \\Vert q\\Vert _{H^s}^4.$ Plugging this into the Efron-Stein inequality (REF ) gives, by symmetry of $\\hat{S}_n$ in $X_1,...,X_n$ , $\\mathbb {V}\\left[ \\hat{S}_n \\right]\\le 2C_1 \\frac{Z_n^{4s + D}}{n^2} + \\frac{C_2}{n}.$"
],
[
"Proofs of Asymptotic Distributions",
"Theorem 8 Suppose that, for some $s^{\\prime } > 2s + \\frac{D}{4}$ , $p, q \\in H^{s^{\\prime }}$ , and suppose $Z_n n^{\\frac{1}{4(s - s^{\\prime })}} \\rightarrow \\infty $ and $Z_n n^{-\\frac{1}{4s + D}} \\rightarrow 0$ as $n \\rightarrow \\infty $ .",
"Then, $\\hat{S}_n$ is asymptotically normal with mean $\\langle p, q \\rangle $ .",
"In particular, for $j \\in \\lbrace 1, \\dots , n\\rbrace $ , define the following quantities: $W_j:= \\begin{bmatrix}Z_n^s e^{i Z_n X_j} \\\\\\vdots \\\\e^{i X_j} \\\\e^{i X_j} \\\\\\vdots \\\\Z_n^s e^{-i Z_n X_j}\\end{bmatrix},\\quad V_j:= \\begin{bmatrix}Z_n^s e^{i Z_n Y_j} \\\\\\vdots \\\\e^{i Y_j} \\\\e^{i Y_j} \\\\\\vdots \\\\Z_n^s e^{-i Z_n Y_j}\\end{bmatrix},\\quad \\overline{W} := \\frac{1}{n} \\sum _{j = 1}^n W_j,\\quad \\overline{V} := \\frac{1}{n} \\sum _{j = 1}^n V_j\\in \\mathbb {R}^{2Z_n},$ $\\Sigma _W := \\frac{1}{n} \\sum _{j = 1}^n (W_j - \\overline{W}) (W_j - \\overline{W})^T,\\quad \\mbox{ and } \\quad \\Sigma _V := \\frac{1}{n} \\sum _{j = 1}^n (V_j - \\overline{V}) (V_j - \\overline{V})^T\\in \\mathbb {R}^{2Z_n \\times 2Z_n}.$ Then, for $\\hat{\\sigma }_n^2:= \\begin{bmatrix}\\overline{V} \\\\\\overline{W}\\end{bmatrix}^T\\begin{bmatrix}\\Sigma _W & 0 \\\\0 & \\Sigma _V\\end{bmatrix}\\begin{bmatrix}\\overline{V} \\\\\\overline{W}\\end{bmatrix},$ we have $\\sqrt{n} \\left( \\frac{\\hat{S}_n - \\langle p, q \\rangle _{H^s}}{\\hat{\\sigma }_n} \\right)\\stackrel{D}{\\rightarrow } \\mathcal {N}(0, 1).$ By the bias bound and the assumption $Z_n^{4(s - s^{\\prime })} n \\rightarrow \\infty $ , it suffices to show $\\sqrt{n} \\left( \\frac{\\hat{S}_n - \\mathop {\\mathbb {E}}\\left[ \\hat{S}_n \\right]}{\\sigma _n} \\right)\\stackrel{D}{\\rightarrow } \\mathcal {N}(0, 1).\\quad \\mbox{ as } \\quad n \\rightarrow \\infty .$ Let $\\widetilde{p}_{Z_n} :=\\begin{bmatrix}\\widetilde{p}(-Z_n) \\\\\\widetilde{p}(-Z_n + 1) \\\\\\vdots \\\\\\widetilde{p}(Z_n - 1) \\\\\\widetilde{p}(Z_n) \\\\\\end{bmatrix},\\quad \\hat{p}_{Z_n} :=\\begin{bmatrix}\\hat{p}(-Z_n) \\\\\\hat{p}(-Z_n + 1) \\\\\\vdots \\\\\\hat{p}(Z_n - 1) \\\\\\hat{p}(Z_n) \\\\\\end{bmatrix},$ $\\widetilde{q}_{Z_n} :=\\begin{bmatrix}\\widetilde{q}(-Z_n) \\\\\\widetilde{q}(-Z_n + 1) \\\\\\vdots \\\\\\widetilde{q}(Z_n - 1) \\\\\\widetilde{q}(Z_n) \\\\\\end{bmatrix},\\quad \\mbox{ and } \\quad \\hat{q} :=\\begin{bmatrix}\\hat{q}(-Z_n) \\\\\\hat{q}(-Z_n + 1) \\\\\\vdots \\\\\\hat{q}(Z_n - 1) \\\\\\hat{q}(Z_n) \\\\\\end{bmatrix}.$ Since $\\hat{p}_{Z_n}$ and $\\hat{q}_{Z_n}$ are empirical means of bounded random vectors with means $\\widetilde{p}_{Z_n}$ and $\\widetilde{q}_{Z_n}$ , respectively, by the central limit theorem, as $n \\rightarrow \\infty $ , $\\sqrt{n} \\left( \\hat{p}_{Z_n} - \\widetilde{p}_{Z_n} \\right) \\stackrel{D}{\\rightarrow }\\mathcal {N}(0, \\Sigma _p)\\quad \\mbox{ and } \\quad \\sqrt{n} \\left( \\hat{q}_{Z_n} - \\widetilde{q}_{Z_n} \\right) \\stackrel{D}{\\rightarrow }\\mathcal {N}(0, \\Sigma _q),$ where $\\left( \\Sigma _p \\right)_{w,z}:= \\operatornamewithlimits{Cov}_{X \\sim p} \\left( \\psi _w(X), \\psi _z(X) \\right)\\quad \\mbox{ and } \\quad \\left( \\Sigma _q \\right)_{w,z}:= \\operatornamewithlimits{Cov}_{X \\sim q} \\left( \\psi _w(X), \\psi _z(X) \\right).$ Define $h : \\mathbb {R}^{2Z_n + 1} \\times \\mathbb {R}^{2Z_n + 1} \\rightarrow \\mathbb {R}$ by $h(x,y) = \\sum _{z = -Z_n}^{Z_n} z^{2s} x_z y_{-z}$ , and note that $\\sigma _n^2:= \\left( \\nabla h \\left( \\widetilde{p}_{Z_n}, \\widetilde{q}_{Z_n} \\right) \\right)^{\\prime }\\begin{bmatrix}\\Sigma _p & 0 \\\\0 & \\Sigma _q\\end{bmatrix}\\left( \\nabla h \\left( \\widetilde{p}_{Z_n}, \\widetilde{q}_{Z_n} \\right) \\right).$ (REF ) follows by the delta method.",
"Theorem 9 Suppose that, for some $s^{\\prime } > 2s + \\frac{D}{4}$ , $p, q \\in H^{s^{\\prime }}$ , and suppose $Z_n n^{\\frac{1}{4(s - s^{\\prime })}} \\rightarrow \\infty $ and $Z_n n^{-\\frac{1}{4s + D}} \\rightarrow 0$ as $n \\rightarrow \\infty $ .",
"For $j \\in \\lbrace 1,\\dots ,n\\rbrace $ , define $W_j :=\\begin{bmatrix}Z_n^s \\left( e^{iZ_n X_j} - e^{iZ_n Y_j} \\right) \\\\\\vdots \\\\e^{iX_j} - e^{iY_j} \\\\e^{-iX_j} - e^{-iY_j} \\\\\\vdots \\\\Z_n^s \\left( e^{-iZ_n X_j} - e^{-iZ_n Y_j} \\right)\\end{bmatrix}\\in \\mathbb {R}^{2Z_n}.$ Let $\\overline{W} := \\frac{1}{n} \\sum _{j = 1}^n W_j\\quad \\mbox{ and } \\quad \\Sigma := \\frac{1}{n} \\sum _{j = 1}^n \\left( W_j - \\overline{W} \\right)\\left( W_j - \\overline{W} \\right)^T$ denote the empirical mean and covariance of $W$ , and define $T := n \\overline{W}^T \\Sigma ^{-1}\\overline{W}$ .",
"Then, if $p = q$ , then $Q_{\\chi ^2(2Z_n)}(T) \\stackrel{D}{\\rightarrow } \\operatorname{Uniform}([0, 1])\\quad \\mbox{ as } \\quad n \\rightarrow \\infty ,$ where $Q_{\\chi ^2(2Z_n)} : [0, \\infty ) \\rightarrow [0, 1]$ denotes the quantile function (inverse CDF) of the $\\chi ^2$ distribution $\\chi ^2(2Z_n)$ with $2Z_n$ degrees of freedom.",
"Since, as shown in the proof of the previous theorem, the distance estimate is a sum of squared asymptotically normal, zero-mean random variables, this is a standard result in multivariate statistics.",
"See, for example, Theorem 5.2.3 of [2]."
],
[
"Generalizations: Weak and Fractional Derivatives",
"As mentioned in the main text, our estimator and analysis can be generalized nicely to non-integer $s$ using an appropriate notion of fractional derivative.",
"For non-negative integers $s$ , let $\\delta ^{(s)}$ denote the measure underlying of the $s$ -order derivative operator at 0; that is, $\\delta ^{(s)}$ is the distribution such that $\\int _\\mathbb {R}f(x) \\delta ^{(s)}(x) \\, dx = f^{(s)}(0),$ for all test functions $f \\in H^s$ .",
"Then, for all $z \\in \\mathbb {R}$ , the Fourier transform of $\\delta ^{(s)}$ is $\\widetilde{\\delta }(z)= \\int _\\mathbb {R}e^{-izx} \\delta ^{(s)}(x) \\, dx= (-iz)^s.$ Thus, we can naturally generalize the derivative operator $\\delta ^{(s)}$ to general $s \\in [0, \\infty )$ as the inverse Fourier transform of the function $z \\mapsto (-iz)^s$ .",
"Generalization to differentiation at an arbitrary $y \\in \\mathbb {R}$ follows from translation properties of the Fourier transform, and, in multiple dimensions, for $s \\in \\mathbb {R}^D$ , we can consider the inverse Fourier transform of $z \\in \\mathbb {R}^D \\mapsto \\prod _{j = 1}^D (iz_j)^{s_j}$ .",
"With this definition in place, we can prove the following the Convolution Theorem, which equates a particular weighted convolution of Fourier transforms and a product of particular fractional derivatives.",
"Note that we will only need this result in the case that $f$ is a trigonometric polynomial (i.e., $\\widetilde{f}$ has finite support), because we apply it only to $p_n$ and $q_n$ .",
"Hence, the sum below has only finitely many non-zero terms and commutes freely with integrals.",
"Theorem 10 Suppose $p, q \\in L^2$ are trigonometric polynomials.",
"Then, $\\forall s \\in [0, \\infty )$ , and $y \\in \\mathbb {Z}^D$ , $\\sum _{z \\in \\mathbb {Z}^D} z^{2s} \\widetilde{p}(y - z) \\widetilde{q}(z)= \\widetilde{\\left( p^{(s)} q^{(s)} \\right)}(y).$ By linearity of the integral, $\\sum _{z \\in \\mathbb {Z}^D} z^{2s} \\widetilde{p}(y - z) \\widetilde{q}(z)& = \\sum _{z \\in \\mathbb {Z}^D} z^{2s}\\int _{\\mathbb {R}^D} p(x_1) e^{-i\\langle y - z, x_1 \\rangle } \\, dx_1\\int _{\\mathbb {R}^D} q(x_2) e^{-i\\langle z, x_2 \\rangle } \\, dx_2 \\\\& = \\int _{\\mathbb {R}^D} \\int _{\\mathbb {R}^D} p(x_1) q(x_2) e^{-i\\langle y, x_1 \\rangle }\\sum _{z \\in \\mathbb {Z}^D} z^{2s} e^{i\\langle z, x_1 - x_2 \\rangle } \\, dx_1 \\, dx_2 \\\\& = \\int _{\\mathbb {R}^D} \\int _{\\mathbb {R}^D} p(x_1) q(x_2) e^{-i\\langle y, x_1 \\rangle }\\delta ^{(s)}(x_1 - x_2) \\, dx_1 \\, dx_2 \\\\& = \\int _{\\mathbb {R}^D} p^{(s)}(x) q^{(s)}(x) e^{-i\\langle y, x \\rangle } \\, dx= \\widetilde{\\left( p^{(s)} q^{(s)} \\right)}(y).$"
],
[
"Acknowledgments",
"This material is based upon work supported by a National Science Foundation Graduate Research Fellowship to the first author under Grant No.",
"DGE-1252522."
]
] | 1605.05785 |
[
[
"MCMC for Imbalanced Categorical Data"
],
[
"Abstract Many modern applications collect highly imbalanced categorical data, with some categories relatively rare.",
"Bayesian hierarchical models combat data sparsity by borrowing information, while also quantifying uncertainty.",
"However, posterior computation presents a fundamental barrier to routine use; a single class of algorithms does not work well in all settings and practitioners waste time trying different types of MCMC approaches.",
"This article was motivated by an application to quantitative advertising in which we encountered extremely poor computational performance for common data augmentation MCMC algorithms but obtained excellent performance for adaptive Metropolis.",
"To obtain a deeper understanding of this behavior, we give strong theory results on computational complexity in an infinitely imbalanced asymptotic regime.",
"Our results show computational complexity of Metropolis is logarithmic in sample size, while data augmentation is polynomial in sample size.",
"The root cause of poor performance of data augmentation is a discrepancy between the rates at which the target density and MCMC step sizes concentrate.",
"In general, MCMC algorithms that have a similar discrepancy will fail in large samples - a result with substantial practical impact."
],
[
"Introduction",
"It has become common to collect very large data sets, but in many settings, there is limited information in the data about many of the unknowns of interest, particularly when data are sparse and imbalanced.",
"Bayesian approaches are useful for borrowing information and characterizing uncertainty in these settings, but a fundamental barrier to routine implementation is posterior computation.",
"In general, when we are faced with an applied problem involving Bayesian modeling of complex data, the most time consuming and challenging stage of the implementation is not the choice of the model or priors but the `design' of the MCMC algorithm for posterior computation.",
"MCMC design [10], [32] remains more of an art than a science [40], with expert Bayesian modelers using their substantial experience in choosing different types of algorithms, and combinations of algorithms, targeted to each new situation.",
"Although there are a variety of software packages for routine Bayesian computation in broad model classes – for example Stan [3] and R-INLA [38] – such packages often do not work well in large and complex settings.",
"Bayesian researchers continue to spend substantial time trying out many different types of algorithms before (hopefully) finding ones that work well in a particular setting.",
"The over-arching goal of this article is to take a step in the direction of improving fundamental understanding of the contexts in which a particular type of MCMC algorithm should work well or not, allowing one to limit the need for trial and error.",
"Given the sparsity of the relevant literature, we are necessarily quite modest in the scope of problems we focus on, but nonetheless obtain what we feel is a broadly useful result that should help practitioners to take more of a scientific approach to MCMC design.",
"To formally assess whether an algorithm `works well' we use the lens of computational complexity theory.",
"The goal of MCMC is to obtain samples from the posterior for use in constructing statistical estimators of posterior summaries of interest; we would like these estimators to have low mean squared error even if we have run our algorithms for a limited clock time.",
"In general, computational efficiency depends on time per iteration and the mixing rate of the Markov chain.",
"As the `problem size' increases, both of these factors tend to slow down; computational complexity theory describes the rate of this slow down.",
"If the rate is too high, then the MCMC algorithm may be practically useless in `big' problems.",
"Problem size is a general term but may correspond to the sample size, data dimensionality, parameter dimensionality or other aspects measuring the hardness of the problem.",
"Most of the existing literature studying the efficiency of MCMC algorithms has focused on showing the Markov chain mixes well in the sense of being geometrically or uniformly ergodic [22], [21], [34], [35]; for example, refer to [4] and [36], which show such conditions for two of the algorithms studied here.A parallel and interesting literature exists on optimal scaling, see e.g.",
"[33].",
"However, we find that such results tell a practitioner very little about whether the algorithm works well in a particular context or not; some simple examples where bounds of this type are very loose are given in [6].",
"Part of this is due to the fact that the constants in the bounds often depend critically on the problem size in a way that is inconsistent with empirical performance [31].",
"Therefore, to study computational compexity, it is necessary to obtain bounds that are sharp as a function of problem size, which is considerably more difficult; [14] is a prominent (successful) example.",
"See [2], [23], [41] for some precedents applying computational complexity theory to MCMC.",
"Most of these studies focus on a single model and MCMC algorithm and show either upper or lower bounds, whereas here we seek to compare different types of algorithms for the same model.",
"We are particularly interested in models for categorical data.",
"In such settings, it is routine to rely on data augmentation to simplify design of MCMC algorithms [39], [5].",
"An amazing variety of clever schemes have been introduced so that one can sample from simple conditional distributions for the parameters after introducing latent data [15], [9], [29], [1].",
"Key examples include the Gaussian data augmentation (DA) scheme for probit models of [1] and the Polya-gamma DA approach for logistic regression of [29].",
"We and many others routinely use these algorithms in all sorts of applied contexts, and they are often remarkably successful.",
"However, sometimes they fail dramatically for unknown reasons, producing very poor mixing.",
"In such cases, one can instead avoid introducing latent data, and use more generic Metropolis, adaptive Metropolis or Hamiltonian Monte Carlo (HMC) methods.",
"Two examples we have encountered include computational advertising (see Section ) and ecological modeling of biological communities (for example, [25]).",
"In these cases, we wasted months developing code, error checking and refining DA-MCMC algorithms before shifting focus to other types of approaches.",
"However, we noticed a commonality in these problematic applications for DA-MCMC: both involved large and very imbalanced data, with some events or species being rare.",
"We have found that this behavior occurs routinely, essentially regardless of the type and complexity of the statistical model, if the data are large and imbalanced.",
"To obtain insight into why this occurs with a goal of providing guidance to practitioners, we carefully study computational complexity of DA-MCMC and Metropolis algorithms under an infinitely imbalanced asymptotic regime introduced by [26] in studying estimation performance in logistic regression.",
"Although our theory is focused on a simple case, the technical details are very far from straightforward, and the results lead to substantial new insight, which has already led to new algorithms [8].",
"In particular, we find that the root case for the poor behavior of DA-MCMC is a discrepancy between the rates at which the target density and MCMC step sizes concentrate.",
"Such a discrepancy will lead to poor scaling of MCMC algorithms in other contexts as well, and to our knowledge we are the first to notice this.",
"This insight is possible because we consider a non-standard asymptotic framework that more accurately approximates the properties of the posterior in finite samples.",
"An important implication is to avoid DA in large imbalanced data contexts.",
"Section 2 motivates the problem and describes the practical behavior of various MCMC algorithms through our computational advertising application.",
"Section 3 contains our main theoretical results, while providing an intuition.",
"Section 4 shows that the predictions from our theory hold in broad imbalanced data applications, and Section 5 contains a discussion.",
"Proofs are included in an Appendix."
],
[
"Motivating application",
"This article was motivated by an application to quantitative advertising.",
"Advertisers seek to optimize the yield or click through rate for display advertisements.",
"There are thousands of websites serving ads – showing display ads for a fee – and advertisers must bid on these impressions – placements of an ad for their client's website in a particular location on a site serving ads – in auctions that take place in a fraction of a second when a user navigates to the serving site.",
"Advertisers develop models of the value of showing a particular advertisement to a user given features on the user, serving site, and the site being advertised.",
"An important component of these models is the estimated probability $p_i$ that a user visits serving site $i$ and the client site in the same browsing session.",
"The idea is that if visitors to site $i$ tend to be more interested in the client's products than visitors to most other serving sites even without being shown an ad, then a visitor to site $i$ will be more likely to click on an advertisement for the client's product(s).",
"Let $n_i$ be the total number of visitors to serving site $i$ in some study period, and $y_i$ be the number of users who also visited the client's site in the same browsing period, with $i=1,\\ldots ,N$ .",
"The $y_i$ tend to be small – in many cases, $y_i \\le 10$ – and the empirical probabilities $\\frac{y_i}{n_i}$ on the order of $10^{-3}$ to $10^{-5}$ .",
"A histogram of logit transformed $\\frac{y_i+1}{n_i}$ for the motivating dataset we obtained from MaxPoint Interactive is shown in Figure REF .",
"Also shown is a histogram of $\\log (y_i+1)$ .",
"The data are about 74% sparse, and of the nonzero observations of $y_i$ , the 25th percentile is 7 and the median is 13.",
"Figure: Histograms of co-browsing data as described in text.A natural Bayesian approach to obtaining low-risk estimates of the $p_i$ is to borrow information across the serving sites via a hierarchical model: yi ni, pi Binomial( ni, g-1(i) ), i iid No(0,2) 0 No(b,B), (), where $\\pi (\\sigma )$ is a half Cauchy prior on $\\sigma $ , recommended as a prior on variance components in hierarchical models by [11] and [28].",
"In this application, we use a moderately informative prior of $b=-12$ and $B = 36$ , consistent with Figure REF , though the results that follow were insensitive to the prior choice.",
"Hierarchical generalized linear models are commonly estimated using data augmentation Gibbs samplers of the form , y p(, y) , y No( (), () ).",
"where $p(\\omega \\mid \\theta ,y)$ is a Pólya-Gamma distribution when $g^{-1}$ is the inverse logit link, and truncated Gaussian when $g^{-1}$ is the inverse probit link.",
"Applying this approach to the MaxPoint data in the logistic case, the sampler had remarkably poor efficiency.",
"Running the MCMC sampler for 50,000 iterations, with 30,000 iterations discarded as burn-in, the right panel of Figure REF shows the empirical autocorrelation function up to lag 100 for the $N=59,317$ different $\\theta _i$ parameters.",
"Even at lag 100, the autocorrelations were high.",
"In contrast, as shown in the left panel, a simple adaptive Metropolis algorithm, having less computational time per iteration, had dramatically better mixing.",
"Figure: Boxplots of estimated autocorrelations for the θ i \\theta _i parameters.",
"Left panel: adaptive Metropolis.",
"Right panel: Polya-gamma data augmentation.",
"Outliers suppressed for readability.The much greater efficiency of Metropolis compared with data augmentation persists in the probit case and well beyond this particular setting – even when there is no hierarchical structure and we have only one site, so long as $y \\ll n$ .",
"Figure REF shows estimated autocorrelations for data augmentation and Metropolis with logit link for $y=1$ with increasing $n$ , i.e.",
"y n, Binomial( n,g-1()), No(0,B) with $B=100$ .",
"For data augmentation, the autocorrelations increase markedly with $n$ , while for Metropolis, the autocorrelations are insensitive to $n$ .",
"We have found this behavior to be unrelated to centering the prior on $\\theta $ at zero and the choice of $B$ .",
"In this case, Metropolis uses a Gaussian random walk proposal with unit variance.",
"At least part of the phenomenon observed in this application is a generic feature of data augmentation when the data are imbalanced.",
"The remainder of this paper aims to demonstrate theoretically why this occurs, showing that it is a general phenomenon for data augmentation samplers in the large sample, highly imbalanced data settings that dominate applications ranging from modeling human behavior in industry settings to ecology.",
"Our results are of direct interest to practitioners conducting applied Bayesian modeling in these settings, while producing important insights into factors controlling efficiency of augmentation-based MCMC algorithms in broad settings.",
"Figure: Plot of estimated autocorrelations in the intercept-only case as described in text."
],
[
"Theory results",
"In this section, we give a brief overview of the computational complexity of MCMC and its relationship to the spectral gap and conductance of the associated transition kernel.",
"We then give theoretical results that are consistent with the empirical performance of data augmentation and Metropolis for imbalanced data.",
"Our results are given for the intercept-only case with $y=1$ and increasing $n$ , as in Figure REF .",
"[26] referred to this as the infinitely imbalanced regime.",
"We focus on this simple case because it reflects the high level of imbalance that we observe in the application, the poor performance of DA in the hierarchical model persists in the intercept only case shown in Figure REF , and obtaining bounds that are sufficiently sharp to have relevance to the practical performance of these algorithms is highly non-trivial even in this simple setting.",
"We later show empirically that this behavior is found in a wide variety of data augmentation algorithms for binomial and multinomial likelihoods, including for regression applications, when the data are imbalanced."
],
[
"Computational complexity of MCMC",
"We assume some facility with concepts such as the integrated autocorrelation time, spectral gap, mixing time, and conductance of a Markov chain, as well as Gibbs sampling and the Metropolis-Hastings algorithm.",
"Please refer to the Supplement for a primer of these concepts.",
"Let $\\lbrace \\Theta _k\\rbrace $ be a Markov chain with transition kernel $\\mathcal {P}$ on a Polish state space $\\mathcal {T}$ having invariant measure $\\Pi $ .",
"In our application, $\\mathcal {T} = \\mathbb {R}^p$ .",
"The goal of MCMC is to approximate the expectation of functions $f : \\mathcal {T} \\rightarrow \\mathbb {R}$ under $\\Pi $ by Cesàro averages f = T f() (dy) 1T k=0T-1 f(k) fT, with $\\theta _0,\\theta _1,\\ldots ,\\theta _{T-1}$ a single realization of the Markov chain.",
"A common measure of performance is the MCMC mean squared error (f, fT) = E ( f - 1T k=0T-1 f(k) )2 = ( f - 1T k=0T-1 Pk f )2 + 1T2 j=0T-1 k=0T-1 cov(f(j),f(k)) = Bias2 + Variance, where $\\Theta _0 \\sim \\nu $ and the expectation is with respect to the law of $\\Theta _0,\\ldots ,\\Theta _{T-1}$ .",
"If one can compute $\\Delta $ as a function of $n$ and the computational complexity of one step from $\\mathcal {P}$ is known, then it is natural to measure overall computational complexity by multiplying these two factors (see e.g.",
"[17]).",
"For instance, if $\\Delta $ converges to $\\infty $ at the rate $n^a$ and one step from $\\mathcal {P}$ costs $n^b$ , then the overall computational complexity is $n^{a+b}$ up to constants.",
"The variance term in (REF ) motivates empirical analysis of the performance of the algorithm through estimates of the autocorrelations $\\rho _k$ at lag $k$ .",
"Another common empirical performance metric is the effective sample size $T_e$ , which is roughly proportional to $1/\\Delta (\\Pi f, \\widehat{f}_T)$ .",
"Informally, $T_e$ is the number of independent samples from $\\Pi f$ that would give variance equal to a path of length $T$ from $\\mathcal {P}$ with $\\Theta _0 \\sim \\Pi $ .",
"For interpretability, it is useful to compute $T_e/T$ or $T_e/t$ , where $t$ is total computation (wall clock) time.",
"The mean squared error $\\Delta $ can also be analyzed theoretically.",
"For reversible $\\mathcal {P}$ , one can obtain both asymptotic (in $T$ ) and finite $T$ bounds on $\\Delta $ in terms of the $L^2(\\Pi )$ spectral gap $\\delta (\\mathcal {P})$ of $\\mathcal {P}$ , with the leading terms order $1/(\\delta T)$ (see [18] for asymptotic bounds and [37], [27] for finite-time bounds, among others).",
"These bounds are generally given for the supremum over all $f \\in L^2(\\Pi )$ .",
"The asymptotic bound is sharp for worst case functions.",
"Transition kernels defined by Metropolis algorithms are in general reversible.",
"Although the transition kernels defined by data augmentation Gibbs samplers are not reversible, the marginal chain for $\\theta $ is reversible.",
"Our results for DA will pertain to the $\\theta $ -marginal chain.",
"We use two strategies to obtain bounds on $\\delta (\\mathcal {P})$ .",
"The first is to obtain lower bounds by a drift and minorization argument in the style of [35].",
"The second is to obtain upper bounds by upper bounding the conductance or Cheeger constant $\\kappa $ of $\\mathcal {P}$ , and then employing the inequality [19] 28 (P) .",
"The conductance also gives bounds on the mixing time of $\\mathcal {P}$ when $\\nu $ satisfies a “warm start” condition.",
"The resulting upper bounds on mixing times are approximately order $\\kappa ^{-2}$ (see the Supplement).",
"By obtaining bounds on $\\delta $ or $\\kappa $ and studying the rate at which these bounds converge to zero as $n \\rightarrow \\infty $ , we obtain estimates of the computational complexity of the algorithm."
],
[
"Main results",
"We now give bounds on $\\delta $ for a Metropolis algorithm as well as data augmentation algorithms for logit and probit.",
"The Metropolis result gives a lower bound on the spectral gap, showing that the computational complexity of the algorithm cannot be worse than $(\\log n)^3$ .",
"The results for the data augmentation algorithms give an upper bound on the spectral gap of order $n^{-1/2} (\\log n)^k$ , with $k=2.5 or 5.5$ , depending on the algorithm.",
"Since each step of the data augmentation sampler requires sampling $n$ auxiliary variables, this suggests the computational complexity is order $n^{3/2} (\\log n)^k$ .",
"In the results that follow, we use the notation $f(n) \\gtrsim g(n)$ (or $f(n) \\lesssim g(n)$ ) to mean there exists a constant $C$ and $n_0<\\infty $ such that for all $n>n_0$ , $f(n) > C g(n)$ (or $f(n) < C g(n)$ , respectively).",
"Theorem 3.1 Let $\\mathcal {P}$ be the transition kernel of a Metropolis algorithm for the model in () with $y=1$ and proposal kernel $q(\\theta ,\\cdot ) = \\text{Uniform}(\\theta -\\log n, \\theta +\\log n)$ .",
"Then n(P) (n)-3.",
"Since the Metropolis algorithm has cost per step that is independent of $n$ , this immediately implies that the computational complexity of the algorithm is at worst $(\\log n)^3$ via the upper bounds on $\\Delta (\\widehat{f}_T, \\Pi f)$ of order $1/(\\delta T)$ .",
"The proof proceeds by showing a Lyapunov function and a minorization condition, then applying [35].",
"The full proof is given in the Appendix, but we highlight an aspect of the argument that is to our knowledge unusual in a proof of this type and should be useful in proving drift conditions for Metropolis and Metropolis-Hastings algorithms generally.",
"When sampling from a target density $p(\\theta )$ , one always has the relationship p'() = ( dd p() ) p().",
"Suppose $z(\\theta ) > \\frac{d}{d\\theta } \\log p(\\theta )$ for all $\\theta $ .",
"Then $p^{\\prime }(\\theta ) \\le z(\\theta ) p(\\theta )$ and by Grönwall's inequality p() p(0) ( 0 z() d).",
"The usefulness of this strategy is that to control Metropolis acceptance probabilities, one needs a bound on the ratio $\\frac{p(\\theta )}{p(\\theta _0)}$ .",
"Often, it is easier to bound $\\frac{d}{d\\theta } \\log p(\\theta )$ than it is to bound $\\frac{p(\\theta )}{p(\\theta _0)}$ directly.",
"In a Bayesian model, $p(\\theta ) \\propto \\log (L(y \\mid \\theta ))+ \\log (\\pi (\\theta ))$ , the sum of the log prior and log likelihood.",
"In our setting, applying the mean value theorem to $\\frac{d}{d\\theta } \\log p(\\theta )$ gives us a bounding function $z(\\theta ) \\le \\frac{-\\theta }{B}$ , and we obtain exponentially decaying acceptance probabilities as we move away from the mode.",
"This allows us to show that $V(\\theta ) = \\exp (-|\\hat{\\theta }-\\theta |)$ is a Lyapunov function for $\\mathcal {P}$ , where $\\hat{\\theta }$ is the posterior mode.",
"We expect this strategy to be useful for constructing good Lyapunov functions for Metropolis and Metropolis-Hastings algorithms for unimodal targets.",
"In multiple dimensions, Grönwall-Bellman like inequalities for multivariate differential equations may allow extension of this approach.",
"The next theorem gives an upper bound for the spectral gap for data augmentation for logit or probit models.",
"Theorem 3.2 Let $\\mathcal {P}$ be the transition kernel of the Pólya-Gamma data augmentation sampler for the model in () with $y=1$ and $g^{-1}$ the inverse logit function.",
"Then the conductance of $\\mathcal {P}$ satisfies $ \\kappa _n(\\mathcal {P}) \\lesssim \\frac{ (\\log n)^{5.5}}{\\sqrt{n}}.$ Similarly, if $\\mathcal {P}$ is the transition kernel of the Albert and Chib data augmentation sampler for the model in () with $y=1$ and $g^{-1}$ the inverse probit link.",
"Then the conductance of $\\mathcal {P}$ satisfies n(P) (n)2.5n.",
"By (REF ), this immediately gives asymptotic (in $n$ ) upper bounds on the spectral gap $\\delta _n(\\mathcal {P})$ for both algorithms.",
"Discarding log factors, these bounds show that the spectral gap converges to zero in the square root of sample size or faster, explaining the results in Section 2 showing very poor mixing that got worse and worse with increasing $n$ .",
"To obtain a bound on computational complexity, one must factor in how the time per iteration increases with $n$ .",
"This suggests a computational complexity of order between $n^{3/2} (\\log n)^c$ and $n^{2} (\\log n)^c$ for both algorithms, with $c \\le 5.5$ .",
"These estimates are obtained by applying (REF ) to the bounds in Theorem REF , then multiplying by a factor of $n$ for the linear complexity of one step from either of these kernels.",
"In the Supplement, we provide empirical evidence that the computational complexity is approximately $n^{1.85}$ based on multiple very long runs of both algorithms."
],
[
"Intuition",
"The key cause of order $n^{-1/2}$ conductance for data augmentation for highly imbalanced data is a discrepancy between the width of the high-probability region of the posterior as a function of $n$ and the rate at which the step sizes for these algorithms converge to zero.",
"We give a rough characterization of this phenomenon for Pólya-Gamma; probit data augmentation is similar.",
"When $\\theta _t$ is close to the posterior mode, the mean and variance of $\\omega $ satisfy E[t t] n(n), var[t t] n((n))3, so by Chebyshev's inequality, $\\omega _{t+1}$ is in the interval $\\frac{n}{\\log (n)} \\pm \\sqrt{n} (\\log (n))^{-3/2}$ with high probability for large $n$ .",
"Also, E[t+1 t+1] -nt+1, var[t+1 t+1] -1, and $\\theta _{t+1} \\mid \\omega _{t+1}$ is conditionally Gaussian, so with probability converging to 1 exponentially fast, $\\theta _{t+1}$ is in the interval $-\\log (n) \\pm (\\log n)^{3/2} n^{-1/2}$ – a ball of radius $(\\log n )^{3/2} n^{-1/2}$ around the posterior mode, which is approximately $-\\log (n)$ .",
"This concentration of measure phenomenon means that step sizes in the bulk of the posterior are not much larger than $n^{-1/2}$ with high probability.",
"On the other hand, As $n \\rightarrow \\infty $ , the bulk of the posterior has width at least $(\\log n)^{-1}$ , so it is not contracting at the same rate as the step sizes.",
"The infinitely imbalanced regime is a non-standard asymptotic setup designed to reflect the extreme imbalance observed in many modern categorical data applications.",
"In classical statistical asymptotics, the samples size $n \\rightarrow \\infty $ for data of fixed dimension generated from a likelihood with a fixed true parameter value.",
"To mimic high dimensional data applications, it has become popular to study the regime where the dimension of the data and/or parameter $\\theta $ to diverge with $n$ ; the infinitely imbalanced setting instead fixes the number of successes $y$ to effectively drive the true parameter to zero as $n \\rightarrow \\infty $ .",
"This leads the posterior to concentrate around the mode at a rate no faster than $(\\log n)^{-1}$ , instead of the usual $n^{-1/2}$ rate, consistent with the empirical observation that substantial estimation uncertainty remains in highly imbalanced cases even when sample sizes are huge.",
"We emphasize that this setup was selected because the data in our motivating application were extremely imbalanced, and the theoretical results predict the observed performance, and not for theoretical convenience or to obtain any particular result.",
"This mismatch between typical step sizes of $\\mathcal {P}$ and the width of the posterior bulk results in the Markov chain becoming trapped too close to the mode, as illustrated by the graphic in Figure REF .",
"This reasoning applies to essentially any MCMC algorithm: if the width of the high probability region of the posterior and step sizes inside that region are of different order in $n$ , then the algorithm will converge slowly and exhibit very high autocorrelations in large samples.",
"Figure: Cartoon comparing the high posterior density region and typical move size.The success of Metropolis-Hastings in this case is easily explained.",
"The typical step sizes of the kernel can be tuned through the choice of $q$ .",
"Since the posterior is contracting at a rate no faster than $(\\log n)^{-1}$ , one sets $q$ to propose moves larger than $(\\log n)^{-1}$ .",
"In particular, the bound of $(\\log n)^{-3}$ is composed of a factor of $(\\log n)^{-1}$ from the return time bound, and a factor of $(\\log n)^{-2}$ from an application of two-step minorization using a set of width $\\log (n)$ outside of which the posterior is negligible and a set of width $(\\log n)^{-1}$ containing the mode.",
"The empirical analysis that follows shows little sensitivity of effective sample size in Metropolis-Hastings or Hamiltonian Monte Carlo algorithms to $n$ ."
],
[
"Empirical analysis of general imbalanced data applications",
"In this section, we aim to show through simulation studies that poor mixing of data augmentation samplers occurs in many imbalanced data settings, including binomial regression models.",
"In each case, alternative Metropolis algorithms perform much better.",
"In the supplement, we conduct additional simulation studies suggesting that data augmentation algorithms for multinomial logit and probit have similar behavior when data are imbalanced.",
"We conclude this section by returning to the original application."
],
[
"Binomial logit and probit",
"In the first set of examples, we consider the model in () with probit and logit link.",
"We set $y=1$ and vary $n$ between 10 and $10,000$ .",
"We perform computation using the Albert and Chib data augmentation algorithm for the probit link and the Pólya-Gamma data augmentation algorithm for the logit link, then estimate autocorrelations and effective sample sizes.",
"In each case we use a prior of $b=0, B=100$ .",
"For probit, we use the implementation in the bayesm package for R. For logit, we use the package BayesLogit.",
"For comparison, we implement random walk Metropolis with $q(\\theta ,\\cdot ) \\sim \\operatorname{No}(\\theta ,1)$ as proposal distribution for the model with logit link.",
"Table REF shows $T_e/t$ (rounded to the nearest integer), computed using the coda package for R. Effective samples per second is anemic for the data augmentation Gibbs samplers for large $n$ , but largely insensitive to $n$ for random walk Metropolis.",
"Table: T e /tT_e/t for data augmentation and Metropolis algorithmswith y=1y=1 and varying nnAlthough the theoretical results in Section consider the case where $y=1$ and $n$ is increasing, empirically we observe poor mixing whenever $y/n$ is small.",
"To demonstrate this, we perform another set of computational examples where $y$ and $n$ both vary in such a way that $y/n$ is constant.",
"Specifically, we consider $n=10,000$ , $n=20,000$ , and $n=50,000$ with $y=1, 2, 5$ .",
"Computation is performed for the two data augmentation Gibbs samplers as above, and effective sample sizes and autocorrelation functions estimated.",
"Fig.",
"REF shows estimated autocorrelations, which are similarly near 1 at lag 1 and decay slowly.",
"Table REF shows values of $T_e/t$ for the two algorithms.",
"Neither measure of computational efficiency shows a meaningful effect of increasing $y$ when $y/n$ remains constant.",
"Figure: Estimated autocorrelation functions for synthetic data examples that vary yy and nn with y/n=10 -4 y/n = 10^{-4} in each case.Table: T e /tT_e/t for data augmentation and Metropolis algorithmswith varying nn and yy with y/n=10 -4 y/n=10^{-4} in each case."
],
[
"Binomial regression",
"We now consider a binomial regression model of the form yi xi, Binomial(ni,g-1(xi )), i=1,...,N No(0,B).",
"We form the predictor matrix $X$ by putting $x_{i,1} = 1$ and sampling $x_{i,2:p} \\sim \\text{Uniform}(-1,1)$ .",
"We then simulate from (REF ) where $g^{-1}$ is the inverse logit link, with $\\beta _1 = \\alpha $ and $\\beta _{2:p} \\sim \\mathrm {No}(0,1)$ .",
"We set $n_i = 1,000$ for all $i$ , $N=1,000$ , and consider $p=20$ and $p=100$ .",
"We vary $\\alpha $ between $-5$ and $-10$ , giving a series of increasingly imbalanced data settings.",
"The means of $y_i$ are given in table REF .",
"Table: mean of y in simulation studyFor each simulation, we perform computation using Pólya-Gamma data augmentation with the BayesLogit package.",
"In this moderate dimension setting, construction of good Metropolis proposals can be challenging, so we use HMC implemented in Stan, which can be viewed as the use of simulated Hamiltonian dynamics to generate an efficient Metropolis proposal.",
"Results summarized by estimated values of $T_e/T$ and $T_e/t$ are shown in Figure REF .",
"In all cases, effective sample size and effective samples per second are orders of magnitude larger for HMC than for PG data augmentation.",
"Additionally, HMC shows little sensitivity to the level of imbalance, while the performance of data augmentation degrades noticeably as the level of imbalance increases.",
"Figure: T e /TT_e/T and T e /tT_e/t for general binomial regression examples for p=20,100p=20,100 for logistic regression computation by Pólya-Gamma data augmentation and Hamiltonian Monte Carlo.",
"Boxplots show distribution of the indicated quantity over the pp parameters of the model."
],
[
"Quantitative advertising reprise",
"We now give details on the adaptive Metropolis algorithm we employed with success in the quantitative advertising application, and provide some additional results.",
"Our alternative to data augmentation for the model in () with logit link has the update scheme update ({i} 0, ,y) for i =1,...,n using Metropolis update (0 , 1,...,n) using Gibbs update (0,1,...,n) using slice sampling.",
"The complete algorithm is given in the Appendix.",
"We detail the Metropolis update here, which we construct using a variation of the adaptive Metropolis algorithm of [13].",
"We construct a time-inhomogeneous proposal $\\theta _i^*$ for each component $\\theta _i$ of $\\theta $ using the proposal kernel qk(ik,*i) = ( (i*-ik)2/ik ), ik = sk j=0k-1 (ij-ik)2, where $\\phi (\\cdot )$ is the univariate standard Gaussian density and $\\bar{\\theta }_{ik}$ is the average of $\\theta _{i0},\\theta _{i1},\\ldots ,\\theta _{i(k-1)}$ , the first $k$ realizations of $\\theta _i$ .",
"We then make independent Metropolis acceptance decisions for each component $i$ ; since the $\\theta _i$ are all conditionally independent given $\\theta _0,\\sigma $ , these updates are made in parallel.",
"The original adaptive Metropolis update of [13] suggests making a joint proposal for $\\theta $ from multivariate Gaussian with covariance depending on the history of the chain of length $k$ .",
"In our case, $\\theta $ has dimension 59,317, so this approach is infeasible, but we find this simplified version works well.",
"We use $s=2.4$ in (REF ), the default recommended in [13].",
"Figure REF shows effective samples per second $T_e/t$ for Pólya-Gamma data augmentation and the algorithm in (REF ).",
"The histogram shows the distribution of $T_e/t$ for all 59,317 $\\theta _i$ parameters.",
"This shows definitively that the algorithm in (REF ) has lower computational complexity by orders of magnitude for this application.",
"Figure: T e /tT_e/t for data augmentation and the algorithm in ()"
],
[
"Discussion",
"For several decades, there has been substantial interest in easy to implement and reliable algorithms for posterior computation in generalized linear models.",
"Data augmentation Gibbs sampling, particularly for probit and logit links, has received much of this attention.",
"A series of data augmentation schemes [9], [15], [24], with [29] being the most popular of the recently developed algorithms, have steadily improved the accessibility of Gibbs sampling for logistic regression.",
"This is a specific case of the larger focus of Bayesian computation on Gibbs samplers, in algorithm development, routine use, and theoretical analysis.",
"The appeal of Gibbs samplers is largely due to their conceptual simplicity, minimal tuning, and widespread familiarity.",
"In addition, there is a common misconception among practitioners that Gibbs samplers are more efficient than alternative Metropolis-Hastings algorithms.",
"The literature studying theoretical efficiency of Gibbs samplers has largely focused on showing uniform or geometric ergodicity.",
"These bounds often say nothing about the computational complexity of the algorithm.",
"Here, we obtain upper and lower bounds on computational complexity that explain empirical performance.",
"Although we compare DA and Metropolis for a specific model and imbalanced data setting, the insight we obtain – that concentration of step sizes may occur at a different rate than concentration of the target – is relevant generally.",
"We showed step sizes for DA concentrating at rate $1/\\sqrt{n}$ but the target concentrating at the slower rate $1/\\log n$ .",
"Had we selected the “standard” asymptotic framework, the target would naturally have concentrated at the usual $1/\\sqrt{n}$ rate, implying that DA and Metropolis would have similar performance.",
"This result is uninformative for understanding the failure of DA in large imbalanced data settings, highlighting the importance of moving beyond standard asymptotics when studying algorithmic complexity of MCMC."
],
[
"Appendix",
"In the proofs, we use $\\mathcal {O}\\left( g(n) \\right)$ and $\\Omega (g(n))$ notation.",
"A function $f(n) = \\mathcal {O}\\left( g(n) \\right)$ indicates that there exist $C, n_0 < \\infty $ such that $n > n_0$ implies $f(n) < C g(n)$ .",
"Conversely, $f(n) = \\Omega (g(n))$ means that there exists $n_0, C$ such that $n > n_0$ implies $f(n) > C g(n)$ ."
],
[
"Introduction",
"In this section, we prove that a properly-tuned Metropolis-Hastings (M-H) algorithm converges quickly when targeting the distribution proportional to pn() = p() (1 + e)-n e e-22B, where $0 < B < \\infty $ is a constant and $n \\in \\mathbb {N}$ is a very large integer.",
"Our bound will be given for the following Markov chain: Definition A.1 (Metropolis-Hastings Kernel) For a fixed sequence $\\lbrace \\epsilon _{n} \\rbrace _{ n \\in \\mathbb {N}}$ of strictly positive real numbers, we define the kernel $\\mathcal {P}_{n}$ to be the Metropolis-Hastings kernel on $\\mathbb {R}$ with proposal kernel Ln(x,) = Unif([x - n, x+ n ]) and target distribution $p_{n}$ .",
"Throughout this section, we denote by n = (n) n(1,2) = ( 1, p(2)p(1) ) the step size and acceptance probability of the kernel $\\mathcal {P}_{n}$ , where $p(\\cdot )$ is the density of the target $\\Pi $ .",
"In the interest of using natural notation and avoiding decorations on frequently used symbols, throughout this proof we let $x$ and $y$ represent generic points in $\\mathbb {R}$ and $X$ represent random variables; we hope this does not cause any confusion with notation used for data in the main text."
],
[
"Preliminary Calculations",
"We define $\\widehat{\\theta } = \\mathrm {argmax}_{\\theta } p(\\theta )$ .",
"By straightforward calculus, $\\widehat{\\theta }$ satisfies B + n e1 + e = 1, and so = -(n) + O( (n ) ).",
"We note that $p(\\theta )$ has only one local maximum: Lemma A.1 (Unimodality of $p(\\theta )$ ) We have p'() >0, < p'() <0, > .",
"By direct calculation, p'() = (1 - n e1 + e - B ) p().",
"Define f() = (1 - n e1 + e - B ), so that p'() = f() p().",
"We then have that f'() = -1B - n e1 + e ( 1 - e1 + e ) - 1B < 0.",
"Since $f^{\\prime }(\\theta ) < 0$ for all $\\theta \\in \\mathbb {R}$ , it follows that $\\lbrace \\theta \\, : \\, f(\\theta ) = 0\\rbrace $ has at most one point.",
"Since $p(\\theta ) > 0$ for all $\\theta \\in \\mathbb {R}$ , we have by Equation (REF ) that { : p'() = 0} = { : f() = 0}.",
"Thus, $\\lbrace \\theta \\, : \\, p^{\\prime }(\\theta ) = 0\\rbrace $ has at most one point.",
"Since $p^{\\prime }(\\hat{\\theta }) = 0$ , the lemma follows immediately.",
"We bound the acceptance probability far from $\\hat{\\theta }_{n}$ : Lemma A.2 (Bound on Acceptance Probability) For $\\hat{\\theta } \\le x \\le y$ , n(x,y) e-(y-x)(x+y- 2 )2B e-x - B (y-x).",
"For $\\hat{\\theta } \\ge x \\ge y$ , n(x,y) e-(y-x)(x+y- 2 )2B e-x - B (y-x).",
"We prove Inequality (REF ) first.",
"Define $f$ as in Equation (REF ).",
"Recall from Equation (REF ) that $p^{\\prime }(\\theta ) = f(\\theta ) \\, p(\\theta ),$ and by direct calculation f'() = -1B - n e1 + e ( 1 - e1 + e ) - 1B < 0 f() = 0.",
"Thus, for all $z \\ge 0$ , $f(z + \\hat{\\theta }) \\le - \\frac{z}{B}$ .",
"Combining this with Equality (REF ), we have p'(z + ) - zB p(z + ) for all $z \\ge 0$ .",
"Let $q \\, : \\, \\mathbb {R}^{+} \\mapsto \\mathbb {R}^{+}$ be the solution to the ODE q(0) = p(x), q'(z) = -z+(x-)B q(z).",
"Note that $q(0) = p(x)$ and $q^{\\prime }(z) \\le p^{\\prime }(z + (x-\\hat{\\theta }))$ for all $z \\ge 0$ .",
"Thus, by Gronwall's inequality, p(y) q(y-x).",
"Solving the ODE that defines $q$ , we have q(z) = C e-z(2(x-) + z)2B for some $C \\in \\mathbb {R}$ .",
"Solving for $C$ , we have p(x) = q(0) = C, so q(z) = p(x) e-z(2(x-) + z)2B.",
"By Inequality (REF ), this implies p(y) q(y-x) = p(x) e-(y-x)(2 (x - ) + (y-x))2B = p(x) e-(y-x)(x+y- 2 )2B.",
"This completes the proof of Inequality (REF ).",
"The proof of Inequality (REF ) is essentially identical."
],
[
"Drift Bounds",
"We show that a Markov chain evolving according to $\\mathcal {P}_{n}$ will tend to drift towards $\\hat{\\theta }$ : Lemma A.3 Define $c_{n} \\equiv 1$ , $\\delta _{n} = \\sqrt{\\log (n)}$ , and $V_{n} \\, : \\, \\mathbb {R} \\mapsto \\mathbb {R}^{+}$ by Vn() = ecn | - n|.",
"Let $x \\in \\mathbb {R}^{+}$ and let $X \\sim \\mathcal {P}_{n}(x,\\cdot )$ .",
"Then for all $n > N_{0}$ sufficiently large, E[Vn(X)] 23 Vn(x) + e(n).",
"We prove this in three cases: $x > \\hat{\\theta }_{n} + \\max (\\epsilon _{n},\\delta _{n})$ , $x < \\hat{\\theta }_{n} - \\max (\\epsilon _{n},\\delta _{n})$ , and $\\hat{\\theta }_{n} - \\max (\\epsilon _{n},\\delta _{n}) \\le x \\le \\hat{\\theta }_{n} + \\max (\\epsilon _{n},\\delta _{n})$ .",
"We calculate: Case 1: $x > \\hat{\\theta }_{n} + \\max (\\epsilon _{n},\\delta _{n})$ .",
"In this case, 2 n E[Vn(X)] = x-nx Vn(y) n(x,y) dy + xx+n Vn(y) n(x,y) dy + Vn(x) x-nx+n ( 1 - n(x,y)) dy = x-nx Vn(y) dy + xx+n Vn(y) n(x,y) dy + Vn(x) xx+n ( 1 - n(x,y)) dy = 1cn (1 - e-cn n) Vn(x) + xx+n Vn(y) n(x,y) dy + Vn(x) xx+n ( 1 - n(x,y)) dy, where the inequality in the second line follows from Lemma REF .",
"Using Inequality (REF ), we continue by writing: 2 n E[Vn(X)] 1cn (1 - e-cn n) Vn(x) + xx+n Vn(y) n(x,y) dy + Vn(x) xx+n ( 1 - n(x,y)) dy, 1cn (1 - e-cn n) Vn(x) + xx+n e-nB (y-x) Vn(y) dy + Vn(x) xx+n ( 1 - e-nB (y-x)) dy = 1cn (1 - e-cn n) Vn(x) + xx+n e-nB (y-x) ecn(y-) dy + Vn(x) xx+n ( 1 - e-nB (y-x)) dy = 1cn (1 - e-cn n) Vn(x) + enBx e-cn xx+n e(cn-nB) y dy + Vn(x) (n - 1cn (1 - e-cn n)) = n Vn(x) + enBx e-cn 1cn - nB e(cn - nB)x (e(cn - nB) n - 1) = Vn(x) (n + 1cn - nB (e(cn - nB)n-1)) = Vn(x) n (1 + O(1(n))).",
"This completes the proof of Inequality (REF ) in the first case.",
"Case 2: $x < \\hat{\\theta }_{n} - \\max (\\epsilon _{n},\\delta _{n})$ .",
"The proof of this case is essentially identical to the proof of the first case, with the exception that Inequality (REF ) is used in place of Inequality (REF ), and this change is propogated through the remaining calculations.",
"The details are omitted.",
"Case 3: $\\hat{\\theta }_{n} - \\max (\\epsilon _{n},\\delta _{n}) \\le x \\le \\hat{\\theta }_{n} + \\max (\\epsilon _{n},\\delta _{n})$ .",
"In this case, we have Vn(X) ecn (n, n) = e(n)."
],
[
"Minorization Condition",
"Define the small set Cn = { R : Vn() 6 e(n) } = { R : | - | (n) + (6) }.",
"We have: Lemma A.4 (Minorization on Small Set) There exists some constants $c, N_{0} > 0$ and some sequence of probability measures $\\lbrace \\mu _{n} \\rbrace _{n \\in \\mathbb {N}}$ so that Pn2(x,) c(n)2 n() for all $n > N_{0}$ and all $x \\in \\mathcal {C}_{n}$ .",
"Let $X_{1} \\sim \\mathcal {P}_{n}(x,\\cdot )$ and then, conditional on $X_{1}$ , let $X_{2} \\sim \\mathcal {P}_{n}(X_{1},\\cdot )$ .",
"Define the event $\\mathcal {E}_{n} = \\lbrace |X_{1} - \\hat{\\theta }| < \\frac{1}{10} \\log (n) \\rbrace $ .",
"It is clear that there exists some uniform constant $c_{1} > 0$ so that P[En] c1.",
"By (REF ), there exists some constant $c_{2}$ so that Pn(y,) c2(n)2 Unif([n - c2(n),n + c2(n)]) for all $|y - \\hat{\\theta }| \\le \\frac{1}{10} \\log (n)$ .",
"The result now follows, with $\\mu _{n} = \\mathrm {Unif}([\\hat{\\theta }_{n} - \\frac{c_{2}}{\\log (n)},\\hat{\\theta }_{n} + \\frac{c_{2}}{\\log (n)}])$ , by combining Inequalities (REF ) and (REF ).",
"By Theorem 5 of [35], Lemmas REF and REF together imply that PnT(,) - () TV (1 - c1(n)2)Tc2 (n) + c3T(e|- | + c4 n), for some constants $c_{1},c_{2},c_{3},c_{4} > 0$ that do not depend on $n$ , where $0 <c_{3} < 1$ and $c_1, c_2$ are distinct from the constants $c_1, c_2$ in the proof of Lemma REF .",
"By Theorem 2 of [34], this completes our proof."
],
[
"Preparatory results",
"The following Corollary to Theorem REF will be used to obtain upper bounds on the conductance and spectral gap.",
"Observe that because the set that we consider in this Corollary is a product of the entire sample space for $\\omega $ with a subset of the sample space for $\\theta $ , verifying this Corollary immediately gives a bound on the conductance for the $\\theta $ -marginal chain, which is reversible.",
"Corollary B.1 Let $(\\theta _{t}, \\omega _t)$ be a data augmentation Markov chain on state space $\\Omega _{1} \\times \\Omega _{2} \\subset \\mathbb {R} \\times \\mathbb {R}^{n}$ .",
"Denote by $\\mathcal {P} = \\mathcal {P}_{1} \\mathcal {P}_{2}$ the transition kernel of this chain, where $\\mathcal {P}_{1}[(\\theta , \\omega ), \\Omega _{1} \\times \\lbrace \\omega \\rbrace ] = \\mathcal {P}_{2}[(\\theta ,\\omega ), \\lbrace \\theta \\rbrace \\times \\Omega _{2}] = 1$ for all $(\\theta , \\omega ) \\in \\Omega _{1} \\times \\Omega _{2}$ .",
"Denote by $\\Pi $ the stationary measure of $\\mathcal {P}$ , and denote by $\\Pi _{1}$ and $\\Pi _{2}$ the marginals of this stationary measure on $\\Omega _{1}$ and $\\Omega _{2}$ ; denote by $\\mu $ , $\\mu _{1}$ and $\\mu _{2}$ their densities.",
"Assume that there exists an interval $I = (a,b) \\subset \\Omega _{1}$ that satisfies 1(I) 1 - c* I 1() I 1() C* I, z 2 P[ (s+1-s)2 > r (s, s) = (,z) ] r-2 + for some $\\epsilon , \\zeta > 0$ , $0 \\le \\gamma < \\infty $ and $0 < c^* < C^* < \\infty $ , and for all $0 \\le r < (1-\\epsilon )/(4c^*)$ .",
"Since (REF ) is trivial for $r < 1$ , this is equivalent to (REF ) holding for $1 \\le r \\le (1-\\epsilon )/(4c^*)$ .",
"Assume that $\\zeta \\le \\frac{1-\\epsilon }{4C^*}$ .",
"Then $\\delta (\\mathcal {P}) \\le \\kappa (\\mathcal {P}) \\le \\frac{16 C^* \\zeta }{(1-\\epsilon )^{2}} + \\frac{2C^* \\gamma }{c^*(1-\\epsilon )}.$ Let $m = \\inf \\left\\lbrace x > a \\, : \\, \\int _{a}^{x} \\mu _{1}(y) dy \\ge \\frac{\\pi _{1}(I)}{2} \\right\\rbrace \\ge a + \\frac{1-\\epsilon }{2C^*}$ be the median of the restriction of $\\Pi _{1}$ to $I$ and let $S = (a,m] \\times \\Omega _{2}$ .",
"By inequality (REF ), 1-2c* m-a 1-2C*.",
"We now bound the conductance $\\kappa $ by showing an upper bound on $\\kappa (S)$ (S) = (x,y) S P((x,y),Sc) (x,y) dx dy (S) (1 - (S)) 4(1-)2 (x,y) S P((x,y),Sc) (x,y) dx dy 4(1-)2 am C* [ ( 1,2(x-a, m-x)2 ) + ] dx where in the last step we applied (REF ) with $r \\le \\max (x-a,m-x) \\le (1-\\epsilon )/(4c^*)$ on $[a,m]$ .",
"Continuing (S) 8 C*(1-)2 ( 0(1 + )dx + m-a2 ( 2x2 + ) dx ) = 8 C*(1-)2 ( + 2(-1 - 2m-a ) + m-a2 ) 16 C* (1-)2 + 8C* (1-)2 1 -4c* = 16 C* (1-)2 + 2C* c*(1-).",
"The result now follows immediately from an application of Theorem REF ."
],
[
"Verifying Corollary ",
"We briefly outline the strategy for showing the three conditions in Corollary REF .",
"To show (REF ), the existence of an interval $I(n)$ satisfying $\\pi _1(I(n)) \\ge 1-\\epsilon $ , we first find an interval $I(n)$ containing the posterior mode for large enough $n$ on which the posterior density ratio is bounded below by a constant.",
"Then, we find a second interval $I^{\\prime }(n)$ outside of which the posterior integrates to $o(1)$ and that satisfies $I(n) \\subset I^{\\prime }(n)$ .",
"By lower bounding the width of $I(n)$ and upper bounding the width of $I^{\\prime }(n)$ , we obtain a lower bound on $\\pi _1(I(n))$ and bounds on $c^*(n)$ and $C^*(n)$ , sequences of constants corresponding to $C^*$ and $c^*$ in (REF ).",
"To show (REF ), we study the dynamics of the chain on $I(n)$ and use concentration inequalities."
],
[
"Proof of (",
"We prove (REF ) with an application of Corollary REF .",
"(REF ) is proved in the Supplement.",
"The proof consists of verifying the three conditions given by inequalities (REF ), (REF ), and (REF ).",
"The proof proceeds in three parts: Showing an interval $I(n)$ on which the posterior density ratio is bounded by a constant and lower bounding its width; Showing an interval $I^{\\prime }(n) \\supset I(n)$ outside of which the posterior integrates to $o(1)$ and upper bounding its width; and, Showing a concentration result of the form (REF ) on an interval containing $I(n)$ .",
"Part (a) : showing the posterior is almost constant on an interval $I(n)$ containing the mode and lower bounding its width.",
"First, we provide bounds of the form (REF ) and (REF ).",
"Recall that the posterior density of $\\theta $ is p(| y = 1) = n(2 )1/2 B (1 + e)-n e e-22 B.",
"We begin by showing that $p(\\theta | y = 1)$ is near-constant on a small region around the mode $\\widehat{\\theta } \\equiv \\mathrm {argmax}_{\\theta } p(\\theta | y = 1$ ) of width $\\Omega ( (\\log n)^{-1})$ given by $I(n) = [\\widehat{\\theta }-(\\log (n))^{-1},\\widehat{\\theta }+(\\log (n))^{-1}]$ .",
"Recall from Inequality (REF ) = n = - (n) + O(((n))).",
"Therefore, there exists an $A< \\infty $ such that $\\widehat{\\theta } \\in [-\\log (n) - A\\log (\\log n),-\\log (n) + A\\log (\\log n)]$ for all $n>N_0$ , where $N_0$ depends only on $A$ .",
"Consider pairs $\\theta _{1}, \\theta _{2}$ that satisfy $|\\theta _{1} - \\theta _{2}| \\le (\\log n)^{-1}$ and also $|\\theta _{1} + \\log (n)|, \\, |\\theta _{2} + \\log (n) | \\le A \\log (\\log n )$ .",
"Define $\\zeta _{1}, \\zeta _{2}$ by $\\theta _{1} = - \\log (n) + \\zeta _{1}$ , $\\theta _{2} = - \\log (n) + \\zeta _{2}$ .",
"Then we calculate p(1 | y = 1)p(2 | y = 1) = e1-2 ( 1 + e21 + e1 )n e12B(22 - 12) = e1-2 ( 1 + 1n e21 + 1n e1 )n e12B(1-2)(2 (n) - 1 - 2) (e-2)(2e)-2A(e-2/B).",
"Since this holds for any pair of points satisfying $|\\theta _1-\\theta _2| \\le (\\log n)^{-1}$ inside the interval $-\\log (n) \\pm A\\log (\\log n)$ , and $\\widehat{\\theta }$ is inside this interval for $n > N_0$ , we conclude the posterior density ratio is bounded below by a constant on an interval $I(n)$ of width $\\Omega ((\\log n)^{-1})$ centered at $\\widehat{\\theta }$ for all $n > N_0$ .",
"Since the posterior density must integrate to 1, this shows that $\\mu _1(\\widehat{\\theta }) = \\mathcal {O}\\left( \\log n \\right)$ in (REF ), so we can take $C^*(n)=\\mathcal {O}\\left( \\log n \\right)$ in (REF ).",
"Part (b): showing the posterior is negligible outside an interval $I^{\\prime }(n) \\supset I(n)$ and upper bounding its width.",
"Next, we show that $p(\\theta | y = 1)$ is negligible outside of the interval $I^{\\prime }(n)=(-5 \\log (n), 3 \\log (n))$ .",
"This interval clearly contains $I(n)$ for all large $n$ , since $I(n)$ is an interval of width $\\mathcal {O}\\left( \\log (\\log n) \\right)$ containing $-\\log (n)$ .",
"If $\\theta = -\\log (n) + C \\log (n)$ for some $C \\ge 4$ , p(| y = 1) n(2 )1/2 B nC-1 (1 + nC-1)-n e- (C-1)2 ((n))22B 1(2 )1/2B nC - n(C-1) - (C-1)22B (n).",
"Thus, 3 (n) p(| y = 1) d C=4 (n)(2 )1/2B nC - n(C-1) - (C-1)22B (n) = o(1).",
"If $\\theta = -\\log (n) - C \\log (n)$ for some $C \\ge 4$ , then p(| y = 1) n(2 )1/2 B n-C-1 (1 + n-C-1)-n e- (C+1)2 ((n))22B 2(2 )1/2B n-C - (C+1)22B (n).",
"Thus, - -5 (n) p(| y = 1) d C=4 2 (n)(2 )1/2B n-C - (C+1)22B (n) = o(1).",
"Combining inequalities (REF ) and (REF ) gives {I'(n)}c p(| y = 1) d = o(1).",
"Therefore, since the posterior is negligible outside a region of width $\\mathcal {O}\\left( \\log n \\right)$ , and the density is unimodal and smooth, we can take $c=\\Omega ((\\log n)^{-1})$ in (REF ).",
"This also shows that $\\pi _1(I(n)) = \\Omega ((\\log n)^{-2})$ , so we can take $1-\\epsilon (n) = \\Omega ((\\log n)^{-2})$ in (REF ).",
"Part (c): Showing (REF ) on an interval containing the mode.",
"Fix a constant $0<C<1$ and consider the interval I*(n) = [-(n)(1+C),-(n)(1-C)].",
"This interval contains $\\widehat{\\theta } \\in -\\log (n) \\pm \\mathcal {O}\\left( \\log ( \\log n) \\right)$ for sufficiently large $n$ .",
"We will show (REF ) on $I^*(n)$ .",
"We can write values of $\\theta _t$ inside this interval as $\\theta _{t} = -\\log (n)(1 + a_{t})$ for $|a_{t}| \\le C$ .",
"Recall that we are considering the Pólya-Gamma sampler with an update rule consisting of sampling $\\omega _{t+1} \\mid \\theta _t, n$ and then sampling $\\theta _{t+1} \\mid \\omega _{t+1}, y, n$ .",
"We first obtain bounds on the conditional expectation and variance of $\\omega _{t+1} \\mid \\theta _t, n$ for $\\theta _t$ inside of $I^*(n)$ , which will be used to show concentration.",
"We have E( t+1 t, n) = n2 t tanh(t/2) = n -2 (n)(1 + at) 1 - e(n) (1 + at)1 + e(n) (1 + at) = n -2 (n)(1 + at) 1 - n1 + at1+n1+ at = n 2 (n)(1 + at)[ 1 - 2n-1-at(1- o(1)) ] and var(t+1 t, n) = n4 t3 (sinh(t) - t) sech2(t2 ) = -n4 (1 + at)3 (n)3 [ 1 - e2 (1 + at) (n)2 e(1 + at) (n) + (1 + at) (n)] [ 2 e12 (1 + at) (n) 1 + e(1+ at) (n) ]2 = n4 (1 + at)3 (n)3 [ 12 n1 + at(1 + o(1)) ] [ 4n1 + at( 1 + o(1)) ] = n2 (1 + at)3 (n)3( 1 + o(1)).",
"Define $\\zeta (n) = n^{1/2} (\\log n)^{-1.5}$ .",
"Combining (REF ) and (REF ), we have by Chebyshev's inequality that P( |t+1 - n2 (n) (1 + at) | > r n1/2(n)1.5 | t ) = O( r-2 ) for any $r > 0$ , $\\theta _t \\in I^*(n)$ .",
"Next, we bound $|\\theta _{t+1}-\\theta _t|$ for $\\theta _t \\in I^*(n)$ .",
"Recall t+1|t+1 No t+1-1 (y-n/2)t+1-1 , t+1-1 = (t+1 + B-1)-1.",
"Define $r_{t}$ by $\\omega _{t+1} = n(2 \\log (n) (1 + a_{t}))^{-1} + r_{t} n^{1/2} \\log (n)^{-1.5}$ .",
"In the following, we condition on $r_{t} (4 \\log (n))^{1/2}n^{-1/2} \\le 1/8$ and $4 B^{-1} \\log (n) n^{-1} \\le 1/8$ to obtain concentration results.",
"Clearly, the second condition holds for fixed $0<B < \\infty $ for all sufficiently large $n$ .",
"To show that the second condition holds for the relevant values of $r_t$ , recall that we need to show (REF ) only for $1 \\le r \\le (1-\\epsilon )/(4c^*)$ .",
"Since $1-\\epsilon (n) \\le 1$ and $c^*(n) = \\Omega ((\\log n)^{-1})$ , $r \\le (1-\\epsilon (n))/(4c^*(n))$ gives $r = \\mathcal {O}\\left( \\log (n) \\right)$ , so $r_{t} (4 \\log (n))^{1/2}n^{-1/2} = o(1)$ , as required.",
"Conditional on $r_{t} (4 \\log (n))^{1/2}n^{-1/2} \\le 1/8$ and $4 B^{-1} \\log (n) n^{-1} \\le 1/8$ , we have t+1-1 = [ n2 (n) (1 + at) + rt n1/2(n)1.5 + B-1 ]-1 = 2 (n) (1 + at)n [1 + B-1 2 (n) (1 + at)n + rt 2 (1 + at)(n n)1/2]-1 =2 (n) (1 + at)n [ 1 -O( B-1 2 (n) (1 + at)n + rt 2 (1 + at)(n n)1/2 ) ] =2 (n) (1 + at)n[ 1 - O( rt + 1(n n)1/2 ) ].",
"Thus, still conditional on $r_{t} (4 \\log n)^{1/2} n^{-1/2} \\le 1/8$ and $4 B^{-1} \\log (n) n^{-1} \\le 1/8$ , t+1|t+1 No t+1-1 (y-n/2) t+1-1 = No( (2-n) (n) (1 + at)n[1 + O( rt + 1(n n)1/2 ) ], (n) (1 + at)n[1 + O( rt + 1(n n)1/2 )] ) = No( - (n) (1 + at)[ 1 + O( rt + 1(n n)1/2 )], (n) (1 + at)n[ 1 + O( rt + 1(n n)1/2 )] ) = No t[1 + O( rt + 1(n n)1/2 )] (n) (1 + at)n[1 + O( rt + 1(n n)1/2 ) ].",
"Applying this bound along with Chebyshev's inequality to the second term, and applying inequality (REF ) to the first term, we conclude that P[ | t+1 - t | > 2r nn ] P[ |t+1 - n2 (n) (1 + at) | > r n1/2(n)1.5 ] + P[ | t+1 - t | > 2r (nn)1/2 | |t+1 - n2 (n) (1 + at) | r n1/2(n)1.5 ] = O( r-2 ) + O( (n n)1/2 ).",
"Thus, inequality (REF ) is satisfied for two sequences of constants $\\zeta = \\zeta (n)$ and $\\gamma = \\gamma (n)$ that satisfy (n) = O( (nn)1/2 ), (n) = O( (nn)1/2 ) on any sequence of sets $I^* = I^*(n)$ satisfying $I^*(n) \\subset (-\\log (n)(1 + C), -\\log (n)(1 - C))$ and fixed $0 < C < 1$ .",
"By inequalities (REF ) and (REF ), the Inequalities (REF ) and (REF ) are satisfied with $\\epsilon = \\epsilon (n)$ , $c = c(n)$ and $C = C(n)$ satisfying (1-(n))-1 = O( (n)2 ), c*(n) = ( (n)-1), C*(n) = O( n ) and a set $I(n) \\subset \\left(-\\log (n) - (\\log n)^{-1} - \\eta _{n}, -\\log (n) + (\\log n)^{-1} - \\eta _{n} \\right)$ , where by Inequality (REF ) we have $\\eta _{n} = \\mathcal {O}\\left( \\log (\\log n) \\right)$ .",
"Combining this with (REF ), Corollary REF completes the proof of Equation (REF ).",
"Finally, Equality (REF ) follows immediately from inequalities (REF ) and (REF ).",
"This completes the proof of the Theorem."
],
[
"Adaptive hybrid Metropolis algorithm",
"We give the full algorithm we use for the model in ().",
"Update $\\theta _i \\mid \\theta _0, \\sigma , y, n$ for $i=1,\\ldots ,N$ independently in parallel Metropolis steps using the adaptive proposal outlined in (REF ).",
"Update $\\theta _0 \\mid \\theta _1,\\ldots ,\\theta _N, \\sigma , y, n$ from No( s m, s), s = ( N2 + 1B )-1, m = i=1N i2 + bB.",
"Update $\\sigma \\mid \\theta _0,\\theta _1,\\ldots ,\\theta _N, y,n$ using slice sampling as in [30].",
"Specifically, put $\\eta = \\sigma ^{-2}$ , then sample u Uniform(0,1+1 ), then sample $\\eta $ from an exponential distribution with scale i=1N (i-0)22 truncated to the interval $\\left( 0,\\frac{1-u}{u} \\right)$ .",
"Now put $\\sigma ^2 = \\eta ^{-1}$ , giving a new sample of $\\sigma ^2$ ."
],
[
"Computational efficiency of MCMC",
"This section provides a more detailed introduction to Markov chain concepts relevant to our study of computational complexity and the results in Section .",
"Let $\\lbrace \\Theta _k\\rbrace $ be a Markov chain with transition kernel $\\mathcal {P}$ and target measure $\\Pi $ .",
"For $f : \\mathcal {T} \\rightarrow \\mathbb {R}$ , the expectation $\\Pi f$ is usually estimated by the time average fT = 1T k=0T-1 f(k).",
"For $\\Theta _1 \\sim \\mu $ , the mean squared error of $\\widehat{f}_T$ is (fT,f) E[ ( fT - f )2 ] = ( E [ fT ] - f )2 + E[ (fT - 1T k=0T-1 Pk-1 f )2].",
"The right side of () is analogous to a bias-variance decomposition, with var[ fT ] = E [ ( 1T k=0T-1 f(k)- 1T k=0T-1 Pk-1 f )2 ], bias(fT) = E [ fT ] - f. Under fairly general conditions, $\\Delta (\\widehat{f}_T,\\Pi f)$ decreases at the rate $T^{-1}$ , but the implied multiplicative constant may be enormous and is intimately related to the convergence properties of $\\mathcal {P}$ .",
"Further, this constant often increases with sample size, so that for large samples, huge MCMC path lengths are necessary to achieve acceptable error."
],
[
"Spectral gap, conductance, and approximation error",
"The $L^2(\\Pi )$ spectral gap (henceforth “spectral gap”) of a Markov operator $\\mathcal {P}$ is defined as Definition S1.1 (Spectral Gap) Let $\\mathcal {P}(\\theta ;\\cdot )$ be the transition kernel of a Markov chain with unique stationary distribution $\\Pi $ .",
"The spectrum of $\\mathcal {P}$ is S = { C {0} : (I - P)-1 is not a bounded linear operator on L2() }, where $L^2(\\Pi )$ is the space of $\\Pi $ -square integrable functions.",
"The spectral gap of $\\mathcal {P} $ is given by ( P) = 1 - { || : S, 1 } when the eigenvalue 1 has multiplicity 1, and $\\delta (\\mathcal {P}) = 0$ otherwise.",
"When $\\mathcal {P}$ is reversible, one can obtain both finite-time and asymptotic bounds on $\\Delta (\\widehat{f}_T,\\Pi f)$ in terms of $\\delta (\\mathcal {P})$ .",
"The following result from [37] gives a finite-time bound[37] actually gives a slightly sharper version of this bound, where in the first two terms the spectral gap is replaced by $1-\\sup \\lbrace \\lambda : \\lambda \\in S, \\lambda \\ne 1 \\rbrace $ , but for simplicity we give the bound in terms of spectral gap.",
"Theorem S1.2 Let $\\mathcal {P}$ be a reversible Markov kernel.",
"Suppose $f \\in L^p(\\Pi )$ for $p \\in (2,\\infty ]$ and $\\Vert \\frac{d\\nu }{d\\Pi } \\Vert _{p/(p-2) \\vee 2} < \\infty $ for $\\nu $ a probability measure on $\\Theta $ and $\\Vert \\cdot \\Vert _q$ the $L^q(\\Pi )$ norm.",
"Then 1f2 (fT,f) 2-T - 2 (1-T)T2 2 + fp2f2 64 pT2 (p-2) 2 d d -1 p/(p-2) 2, where $\\bar{\\delta } = 1-\\delta $ .",
"As a function of $T$ , (REF ) is order $T^{-1}$ , while as a function of $\\delta $ for finite $T$ , it is order $\\delta ^{-2}$ .",
"Taking $T \\rightarrow \\infty $ in (REF ), we obtain the well-known asymptotic bound 1f2 (fT,f) 2-T Thus, for large $T$ , $T \\Delta (\\widehat{f}_T, \\Pi f)$ behaves like $\\delta ^{-1}$ .",
"It is worth noting that the condition on the Radon-Nikodym derivative $\\Vert d\\nu /d\\Pi \\Vert _{p/(p-2) \\vee 2} < \\infty $ is similar to the condition $M < \\infty $ in Theorem REF , and the “warm start” distributions $\\nu $ that we construct for the kernels considered here satisfy $\\Vert d \\nu /d\\Pi \\Vert _2 < \\infty $ , so a bound of the form (REF ) holds for $p \\ge 4$ for the cases under study.",
"One can also obtain a general central limit theorem, which we review here because of its role in MCMC performance diagnostics.",
"The following result from [18] is valid for the Markov kernels considered here, all of which are geometrically ergodic and reversible.",
"A Markov chain evolving according to $\\mathcal {P}$ is geometrically ergodic if there exist constants $\\rho \\in (0,1), B < \\infty $ and a $\\Pi $ -almost everywhere finite measurable function $V : \\mathcal {T} \\rightarrow [1,\\infty )$ such that Pk(0;) - V BV(0) k, where for a probability measure $\\mu $ , $\\Vert \\mu \\Vert _V = \\sup _{f \\le V} |\\mu f|$ .",
"For a function $f \\in L^2(\\Pi )$ , define the quantity f2 = var[f(0)] + 2 k=1 cov[f(0),f(k)] for $\\Theta _0 \\sim \\Pi $ .",
"Then, if $\\mathcal {P}$ is reversible we have T T1/2( fT - f ) d= No(0,f2), for any initial distribution $\\mu $ on $\\Theta _0$ .",
"The asymptotic variance $\\sigma ^2_f$ is related to the spectral gap by the inequality (c.f.",
"page 479 of [12]) f2var(f) 1 + 2 k=1 k(f) 2(P)-1, which is identical to (REF ).",
"The quantity $2 \\sum _{k=1}^{\\infty } \\eta _k(f)$ is referred to as the integrated autocorrelation time.",
"Here, $\\eta _k(f)$ is the lag-$k$ autocorrelation $\\operatorname{cor}\\left[ f(\\Theta _0),f(\\Theta _k) \\right]$ with $\\lbrace \\Theta _k\\rbrace $ evolving according to $\\mathcal {P}$ .",
"A commonly used performance measure for MCMC is the effective sample size $T_e$ : Tevar(f) T2f, which measures how much the asymptotic variance is inflated by autocorrelation.",
"The bound in (REF ) is sharp for worst case functions when $\\mathcal {P}$ has no residual spectrum, which holds, for example, for reversible Markov operators on discrete state spaces.",
"So to a first approximation, the effective sample size is proportional to $\\delta (\\mathcal {P})$ .",
"Since $T_e$ is an asymptotic quantity, it is common to approximate it from finite-length sample paths using finite-time estimates of the spectral density at frequency zero.",
"Taken together, (REF ) and (REF ) indicate that $\\Delta (\\widehat{f}_T, \\Pi f)$ is proportional to $\\delta (\\mathcal {P})^{-2}$ for finite $T$ , and $\\delta (\\mathcal {P})^{-1}$ asymptotically (as $T \\rightarrow \\infty $ ).",
"We will generally refer to the asymptotic setting when discussing results.",
"The bias of finite-length Markov chains can be bounded in terms of the conductance of the associated $\\mathcal {P}$ , which also provides a double-sided bound on $\\delta $ .",
"Definition S1.3 (Conductance) For $\\Pi $ -measurable sets $S \\subset \\Theta $ with $0 < \\Pi (S) < 1$ , define $\\kappa (S) = \\frac{\\int _{\\theta \\in S} \\mathcal {P}(\\theta ,S^{c}) \\Pi (ds)}{\\Pi (S) (1 - \\Pi (S))}$ and the Cheeger constant or conductance $\\kappa = \\inf _{0 <\\Pi (S) < 1} \\kappa (S).$ The relationship between conductance and bias for Markov operators is quantified by the following Theorem from [20]: Theorem S1.4 (Warm Start Bound) Let $\\mathcal {P}(\\theta ;\\cdot )$ be the transition kernel of a Markov chain $\\lbrace \\Theta _j\\rbrace _{j \\in \\mathbb {N}}$ with invariant measure $\\Pi $ and conductance $\\kappa $ .",
"Then for all measurable sets $S \\subset \\mathcal {T}$ , $| \\operatorname{\\mathbb {P}}(\\Theta _{j+1} \\in S) - \\Pi (S)| \\le M^{1/2} \\left(1 - \\frac{\\kappa ^{2}}{2} \\right)^j, $ where $M = \\sup _{A \\subset \\Theta } \\frac{\\operatorname{\\mathbb {P}}(\\Theta _0 \\in A)}{\\Pi (A)}$ .",
"The total variation bound in (REF ) is enough to give useful bounds on the bias term in () for bounded $f$ .",
"When $\\kappa $ is near zero, Theorem REF implies that the number of steps required to attain bias $\\epsilon $ – the mixing time $\\tau _{\\epsilon }$ – scales approximately like $\\kappa ^{-2}$ .",
"For $j \\gg \\frac{\\log (\\frac{\\epsilon }{\\sqrt{M}})}{\\log (1 - \\frac{\\kappa ^{2}}{2})} \\approx \\kappa ^{-2} \\log \\big (\\frac{\\epsilon }{\\sqrt{M}} \\big )$ , we have S | P(t+1 S) - (S) | M (1 - 22)j M (-2 -2 (M) ) = M M = .",
"For reversible Markov operators $\\mathcal {P}$ , Theorem 2.1 of [19] relates conductance to the spectral gap: Theorem S1.5 The spectral gap $\\delta (\\mathcal {P})$ of a reversible Markov operator $\\mathcal {P}$ satisfies 28 (P) .",
"Clearly, one gives up a factor of either $\\delta $ or $\\kappa $ when transitioning between bounds on $\\kappa $ and bounds on $\\delta $ via (REF )."
],
[
"Results on mixing times",
"We give some additional results on mixing times for the data augmentation algorithms.",
"The following remark shows that Theorem REF applies to the Pólya-Gamma sampler.",
"It is established in the course of proving Theorem REF .",
"Corollary S2.1 (Warm start for Pólya-Gamma) Let $\\hat{\\theta }$ be the posterior mode for the model in () with $g^{-1}$ the inverse logit link.",
"Then n = Uniform( - 1(n), + 1(n) ) 2lsatisfies A R n(A)(A|y) (n)2 where $\\Pi $ is the posterior measure, and thus provides a `warm start' distribution for the Pólya-Gamma sampler.",
"Note that $\\hat{\\theta }$ is the unique solution to $\\frac{\\theta }{B} + n \\frac{e^\\theta }{1 + e^\\theta } = 1$ .",
"Combined with (REF ), Corollary REF therefore indicates that $\\epsilon $ -mixing times for the Pólya-Gamma sampler scale approximately like $\\kappa ^2_n(\\mathcal {P}) \\lesssim (\\log n)^{11} n^{-1}$ .",
"The following corollary shows that Theorem REF also applies to the Albert and Chib sampler.",
"It is also established in the proof of Theorem REF .",
"Corollary S2.2 (Warm start for Albert and Chib) Let $\\Phi $ be the standard Gaussian cumulative distribution function.",
"Then $\\mu _n = \\mathrm {Uniform}\\left(\\Phi ^{-1}\\left(\\frac{B+2}{2 (Bn+2)} \\right),\\Phi ^{-1}\\left( \\frac{2(B+2)}{Bn+2} \\right) \\right)$ satisfies $ \\sup _{A \\subset \\mathbb {R}} \\frac{\\mu _n(A)}{\\Pi (A|y)} \\lesssim \\log n$ and thus provides a `warm start' distribution for the Albert and Chib sampler.",
"Combined with (REF ), Corollary REF therefore indicates that $\\epsilon $ -mixing times for the Albert and Chib sampler scale approximately like $\\kappa ^2_n(\\mathcal {P}) \\lesssim (\\log n)^5 n^{-1}$"
],
[
"Empirical analysis of efficiency",
"In this section, we estimate empirically the efficiency of the two data augmentation Gibbs samplers via finite path estimates of $\\sigma ^2_f$ in (REF ).",
"These empirical estimates can be compared to our estimate based on the conductance that the asymptotic variance is approximately order $n^{1/2}$ to order $n$ for typical functions.",
"Let $\\sigma ^2_f(n)$ be the asymptotic variance when the sample size is $n$ , and assume $\\sigma ^2_f(n) \\approx C n^a$ for large $n$ , so that $\\log \\lbrace \\sigma ^2_f(n)\\rbrace = \\log (C) + a \\log (n)$ .",
"This suggests first estimating $\\sigma ^2_f(n)$ for different values of $n$ , and then estimating $a$ by regression of $\\log \\lbrace \\sigma ^2_f(n)\\rbrace $ on $\\log (n)$ .",
"We estimate $\\sigma ^2_f$ based on autocorrelations via a truncation of (REF ), f = 1 + 2 k=1K k, where $\\widehat{\\eta }_k$ is a point estimate of $\\eta _k$ .",
"It is important to choose $K \\gg \\lbrace \\delta _n(\\mathcal {P})\\rbrace ^{-1}$ .",
"The lower bounds derived in Section have $\\lbrace \\delta _n(\\mathcal {P})\\rbrace ^{-1} \\ge n^{1/2}$ up to a log factor and a universal constant, so we use $K = n$ to compute the sum in (REF ).",
"To further improve the estimates, we use multiple chains to compute $\\widehat{\\eta }_k$ and run all of the chains for $10^6$ iterations.",
"Figure REF shows $\\log (n)$ versus $\\log (\\widehat{\\rho }_f)$ for values of $n$ between 10 and 10,000 for the Pólya-Gamma and Albert and Chib sampler.",
"The relationships are linear, and the least squares estimate of the slope is $0.86$ for Pólya-Gamma and 0.84 for Albert and Chib.",
"This is in the range of $n^{1/2}$ to $n$ estimated from our upper bound on the conductance.",
"Figure: Plots of log(n)\\log (n) versus σ 2 (n)\\sigma ^2(n) for different values of nn.",
"The estimated values of aa are 0.860.86 and 0.840.84, respectively."
],
[
"Data augmentation algorithms for multinomial likelihoods",
"So far we have considered data augmentation algorithms for binomial likelihoods.",
"Similar algorithms exist for multinomial logit and probit models.",
"Specifically, let y Multinomial(n,), = g-1(), N(0,B), where $y$ is a length $d$ vector of nonnegative integers whose sum is $n$ , $\\pi = (\\pi _1,\\ldots ,\\pi _d)^{{ \\mathrm {\\scriptscriptstyle T} }}$ is a probability vector, and $g^{-1}(\\theta )$ is a multinomial logit or probit link function.",
"Posterior computation under (REF ) is commonly performed using data augmentation algorithms of the form in ().",
"[29] describe a Pólya-Gamma sampler for the multinomial logit, which is implemented in BayesLogit, while [16] propose a data augmentation Gibbs sampler for the multinomial probit, which is implemented in R package MNP.",
"We study a synthetic data example where $y$ is a $4 \\times 1$ count vector with entries adding to $n$ .",
"The first three entries of $y$ are always 1, the final entry is $n-3$ , and a series of values of $n$ between $n=10$ and $n=10,000$ are considered.",
"Estimated values of $T_e/T$ for the first three entries of $\\theta $ for both algorithms are shown in Table REF .",
"The results are similar to those for the binomial logit and probit, and are consistent across the different entries of $\\theta $ .",
"It is exceedingly common for contingency tables to have many cells with small or zero entries.",
"Our results suggest that data augmentation algorithms should be avoided in such settings.",
"Table: Estimated values of T eff /TT_{\\text{eff}}/T for the three entries of θ\\theta for multinomial logit and probit data augmentation for increasing values of nnwith data y=(1,1,1,n-3)y = (1,1,1,n-3).",
"Results are based on 5,000 samples gathered after discarding 5,000 samples as burn-in."
],
[
"Details of Data Augmentation samplers",
"We provide more detail on the two data augmentation samplers considered in the main article.",
"[29] introduce a data augmentation Gibbs sampler for posterior computation when $g^{-1}$ in () is the inverse logit link.",
"The sampler has update rule given by PG(n,) No (+B-1)-1 (+B-1)-1, where $\\alpha = y-n/2$ and $\\text{PG}(a,c)$ is the Pólya-Gamma distribution with parameters $a$ and $c$ .",
"The transition kernel $\\mathcal {P}(\\theta ;\\cdot )$ given by this update has $\\theta $ -marginal invariant measure the posterior $\\Pi (\\theta \\mid y)$ for the model in ().",
"A similar data augmentation scheme exists for the case where $g^{-1}$ is the inverse probit $\\Phi (\\cdot )$ .",
"Initially proposed by [1], the sampler has update rule = i=1y zi + i=1n-y ui, zi TN(,1;0,), ui TN(,1;-,0) No (n+B-1)-1 (n+B-1)-1, where $\\text{TN}(\\mu ,\\tau ^2;a,b)$ is the normal distribution with parameters $\\mu $ and $\\tau ^2$ truncated to the interval $(a,b)$ .",
"The transition kernel $\\mathcal {P}(\\theta ;\\cdot )$ for $\\theta $ defined by this update has $\\theta $ -marginal invariant distribution $\\Pi (\\theta \\mid y)$ for the model in () when $g^{-1} = \\Phi $ .",
"It is clear from () that the computational complexity per iteration scales linearly in $n$ for this algorithm.",
"Although a recent manuscript proposes some more efficient samplers, the samplers in [29] for $\\mathrm {PG}(n,\\theta )$ also scale linearly in $n$ ."
],
[
"Proof of (",
"First we give a lemma that is used in the main proof to bound $\\Phi ^{-1}(x)$ and $(\\Phi ^{-1}(x))^2$ .",
"Lemma S5.1 Let $\\Phi (\\cdot )$ be the standard normal distribution function and fix $x > 0$ .",
"Then, as $n \\rightarrow \\infty $ , -1(xn ) = -(2(n/x))1/2 ( 1 - (4 (n/x) )2(n/x) + O( 1( (n/x))1.5 ) )1/2 = -(2(n/x))1/2(1+o(1)), Furthermore, (-1(xn ))2 = 2 (nx ) - ( 2 ( nx )) + (2 ) + O( 1(n/x) ).",
"From equations 7.8.1 and 7.8.2 of [7], we have for $x\\ge 0$ 1x/2 + (x2/2 + 2)1/2 < ex2/2 x/2 e-t2 dt 1x/2 + (x2/2 + 4/)1/2 1x+ (x2 + 4)1/2 < ( /2)1/2 ex2/2 ( 1 - (x)) 1x+ (x2 + 8/)1/2 Thus, we can write (/2)1/2 ex2/2 {1-(x)} = 1x+(x2+h(x))1/2 for some function $h(x)$ that satisfies $8/\\pi \\le h(x) \\le 4$ for all $x \\ge 0$ , giving 1-(x) = (2/)1/2 e-x2/2 1x+(x2+h(x))1/2.",
"Writing $y = \\Phi (x)$ , so that $y>1/2$ from the original condition for the inequality, and inverting gives 1-y = (2/)1/2 e-x2/2 [x+(x2+h(x))1/2]-1 (1-y) = (2/)/2 -x2/2 -(x+(x2+h(x))1/2) x2 = -2(1-y) + (2/) -2(x+(x2+h(x))1/2).",
"We now claim that for any fixed $\\epsilon > 0$ and any sufficiently large $x > X(\\epsilon )$ , we have $[-(2 - \\epsilon )\\log (1-y)]^{1/2} < x < [-(2 + \\epsilon )\\log (1-y)]^{1/2}$ .",
"To see this, recall that by Inequality (), for any fixed $\\epsilon > 0$ , (2/)1/2 e-(1+) x2/2 1-(x) (2/)1/2 e-(1-) x2/2 for all sufficiently large $x$ .",
"Substituting this bound into () we obtain x2 = -2(1-y) + (2/) - (-2(1-y)) + O( 1-(1-y) ), for $1-y<1/2$ which gives x = [-2(1-y) + (2/) - (-2(1-y)) + O( 1-(1-y) )]1/2.",
"To get the result for arguments $1-y<1/2$ , we take the negative solution, giving x = - (-2(1-y))1/2 [ 1 + (-(4/) (1-y) )2(1-y) + O( 1[-(1-y)]1.5 ) ]1/2.",
"Also since $(1+o(1))^{1/2} = 1 + [(1+o(1))^{1/2}-1] = 1 + o(1)$ , x = - (-2(1-y))1/2 (1+o(1)), Setting $1-y = x/n$ for $x/n<1/2$ – the region where $\\Phi ^{-1}(x/n) < 0$ – we have (-1(x/n))2 = 2(n/x) + (2/) - (2(n/x)) + O( 1(n/x) ) -1(x/n) = -(2(n/x))1/2 ( 1 - ((4/) (n/x) )2(n/x) + O( 1[(n/x)]1.5 ) )1/2 = -(2(n/x))1/2(1+o(1)), completing the proof.",
"Proof of main result The main result is proved in four steps; the rationale for each step is outlined in §REF : Obtain bounds on quantities that will appear in steps (b) through (d); Find an interval $I^{\\prime }(n)$ outside of which the posterior is negligible, in the sense of integrating to $o(1)$ , and find an upper bound for its width; Find an interval $I(n) \\subset I^{\\prime }(n)$ containing the posterior mode $\\widehat{\\theta }$ on which the posterior density ratio is bounded below by a constant and show a lower bound on its width; and Show a concentration inequality for $|\\theta _t-\\theta _{t+1}|$ when $\\theta _t \\in I(n)$ .",
"Part (a) : obtaining additional bounds Recall that the Albert and Chib sampler has the update rule given by sampling $\\omega _{t+1} \\mid y, n, \\theta _t$ then sampling $\\theta _{t+1} \\mid \\omega _{t+1}$ from a Gaussian.",
"$\\omega _{t+1}$ is the sum of $n-y$ independent Gaussians truncated below by zero and $y$ independent Gaussians truncated above by zero; here we always have $y=1$ .",
"Then, the expectation and variance of $\\omega _{t+1}$ given $\\theta _t$ are E(t+1 t, n, y) = (n-1) [ t - (t)1-(t) ] + [ t + (t)(t) ] = nt - (n-1) (t)1-(t) + (t)(t) var(t+1 t, n, y) = vt = n + (n-1) [t (t)1-(t)-( (t)1-(t) )2 ] - t (t)(t) -(t)2(t)2.",
"We now compute the posterior mode $\\hat{\\theta }$ .",
"We begin by reparameterizing our problem by the one-to-one transformation $\\theta = \\Phi ^{-1}(x/n)$ .",
"We will compute $\\hat{x}$ , the posterior mode under this transformation, and then use this to compute the mode $\\hat{\\theta }$ on the original scale by the equation $\\hat{\\theta } = \\Phi ^{-1}(\\hat{x}/n)$ .",
"We will require an approximation to $\\phi _B(\\Phi ^{-1}(x/n))$ , where $\\phi _B$ is the density of $N(0,B)$ .",
"Using (REF ), B( -1( xn ) ) = 1(2 B)1/2 [-2 (n/x)2B + 12 B ( 2 ( nx ) ) -12B ( 2/) + o(1) ] = 1(2 B)1/2 ( xn )1/B ([2 (n/x)]1/2)1/B (/2)1/2B (o(1)).",
"The posterior density when $y=1$ is proportional to $p(\\theta | n,y) \\propto n \\Phi (\\theta ) (1-\\Phi (\\theta ))^{n-1} \\phi _B(\\theta ).$ Under our reparameterization, p(x n,y) x + (n-1) (1-xn ) -(-1(x/n))22B.",
"Differentiating to find the mode, x p(x n,y) = 1x - n-1n-x - (2 )1/2Bn ( -1(x/n)22 ) -1(x/n).",
"Using (REF ) and (REF ), we have x p(x n,y) = 1x - n-1n-x - (2)1/2Bn ( nx ) (2 (n/x))-1/2 (2/)1/2 (o(1)) [ -(2 (n/x))1/2 (1 + o(1)) ] = 1x - n-1n-x + 2Bx ( 1 + o(1) ), so in the limit as $n \\rightarrow \\infty $ the posterior mode is x = n(B+2 + o(1))Bn+2.",
"In particular, for large enough $n$ , $\\widehat{x}/n$ is in the interval $\\frac{\\widehat{x}}{n} \\in \\left[\\frac{B+2}{2(Bn+2)} ,\\frac{2(B+2)}{Bn+2}\\right].$ Part (b) : find an interval $I^{\\prime }(n)$ outside of which the posterior is negligible We now implement the first part of the approach to showing conditions (REF ) and (REF ).",
"As described in §REF , we show an interval $I^{\\prime }(n)$ outside of which the posterior is negligible, that is, integrates to $o(1)$ .",
"Fix $ C > 2$ and consider the interval $I^{\\prime }(n)=[\\Phi ^{-1}(n^{-C^2}),\\Phi ^{-1}(1-n^{-C^2})]$ .",
"First we bound the width of this interval and the size of the increments $|\\Phi ^{-1}(1-n^{-(C+1)^2}) - \\Phi ^{-1}(1-n^{-C^2})|$ .",
"Bounding the width from above is necessary for showing condition (REF ), and bounding the size of the increments is necessary to show that the posterior integrates to $o(1)$ outside this interval.",
"From (REF ): -1(1-n-C2) = --1(n-C2) = (2(nC2))1/2 (1+o(1)) = C (2 (n))1/2 (1+o(1)).",
"So then |-1(n-C2)--1(1-n-C2)| = 2 C (2 (n))1/2 (1+o(1)) and |-1(n-(C+1)2)--1(1-n-C2)| = (2 (n))1/2 (1+o(1)).",
"Now we bound the posterior density $\\Phi ^{-1}(1-n^{-C^2})$ , which will be used to bound the integral of the posterior on the complement of $I^{\\prime }(n)$ $p( \\theta \\mid y=1) &= n (1-n^{-C^2}) \\left( 1- (1-n^{-C^2}) \\right)^{n-1} \\phi _B(\\Phi ^{-1}(1-n^{-C^2})) \\\\&\\le n^{-C^2 n} n^{C^2+1} (2 \\pi B)^{-1/2}.$ We have with $p_1(\\theta ) = p(\\theta \\mid y=1)$ -1(1-n-C2) p1() d(2B)-1/2 C=2 (2 (n))1/2(1 + o(1)) nC2+1-C2n = o(1).",
"So the posterior measure of the part of $\\lbrace I^{\\prime }(n)\\rbrace ^c$ that contains values of $\\theta $ greater than those in $I^{\\prime }(n)$ is $o(1)$ .",
"Now we take the same approach to show this for the part of $\\lbrace I^{\\prime }(n)\\rbrace ^c$ consisting of values of $\\theta $ less than those in $I^{\\prime }(n)$ .",
"We have that $p(\\theta \\mid y=1)$ for $\\theta = \\Phi ^{-1}(n^{-C^2})$ satisfies p(y=1) = n (n-C2) (1-n-C2)n-1 (2 B)-1/2 n1-C2 e-n-(C2-1) (1-n-C2)-1 (2 B)-1/2 n1-C2 e-1 (4/3) (2 B)-1/2, when $n\\ge 2$ .",
"So then --1(n-C2) p0() d(4/3) (2 B)-1/2 C=2 (2 (n))1/2(1 + o(1)) n1-C2 = o(1).",
"We conclude {I'(n)}c p(y=1) d= o(1) for $I^{\\prime }(n)=[\\Phi ^{-1}(n^{-C^2}),\\Phi ^{-1}(1-n^{-C^2})]$ with $C > 2$ , so the posterior is negligible outside an interval of length $\\mathcal {O}\\left( \\sqrt{\\log (n)} \\right)$ based on ().",
"Part (c): Find an interval $I(n) \\subset I^{\\prime }(n)$ containing the mode on which the posterior is almost constant We now do the second step outlined in §REF to show (REF ) and (REF ).",
"We show an interval $I(n)$ containing the posterior mode on which the posterior is bounded below by a constant for all large $n$ .",
"Again fix a constant $2<C < \\infty $ .",
"We now show that for $n>N(C)$ sufficiently large, where the function $N(C)$ depends only on $C$ , the posterior is almost constant on the interval $I(n)=\\left[\\Phi ^{-1}\\left(\\frac{B+2}{C (Bn+2)} \\right),\\Phi ^{-1}\\left( \\frac{C(B+2)}{Bn+2} \\right) \\right].$ As shown in Equality (), this interval includes the posterior mode for all large enough $n$ .",
"This interval has width $\\Omega \\left( (\\log n)^{-1/2} \\right)$ .",
"To see this, put $q(n) = (B+2)^{-1}(Bn + 2)$ , then | I(n) | = [2 (C q(n)) ]1/2 [ 1- [ (4/) (C q(n)) ]2 (Cq(n)) + O( [(q(n))]-1.5 )]1/2 -[2 ( q(n)/C) ]1/2 [ 1- [ (4/) (q(n)/C) ]2 (q(n)/C) + O( [(q(n))]-1.5 ) ]1/2 f1(n,C)-f2(n,C), where $|I(n)|$ is the width of $I(n)$ .",
"Now multiply the right side by $f_1(n,C)+f_2(n,C)$ to get $&2 \\log (C q(n)) - \\log \\left[(4/\\pi ) \\log (C q(n)) \\right] + \\mathcal {O}\\left( [\\log (q(n))]^{-1/2} \\right) \\\\&-2 \\log (q(n)/C) + \\log \\left[ (4/\\pi ) \\log (q(n)/C) \\right] + \\mathcal {O}\\left( [\\log (q(n))]^{-1/2} \\right) \\\\&= 4\\log (C) - 2 \\log (\\log (C)) + o(1).$ Since f1(n,C) + f2(n,C) = O( (n)1/2 ) we get that | I(n) | = ( (n)-1/2 ).",
"Recall the posterior mode is $\\widehat{\\theta } = \\Phi ^{-1}((B+2+o(1))/(Bn+2))$ , which is contained in $I(n)$ for sufficiently large $n$ .",
"Set $\\theta _0 = \\Phi ^{-1}(C(B+2)/(Bn+2))$ .",
"We will bound the ratio of the posterior densities on the interval $I_1(n) = [\\widehat{\\theta },\\theta _0]$ , which is a subset of our interval $I(n)$ .",
"Repeatedly applying Lemma REF , we have p( y=1 )p( y=1 0 ) = n (B+2+o(1)Bn +2 ) (1-( B+2+o(1)Bn +2 ))n-1 B( B+2+o(1)Bn +2 )n (C(B+2)Bn +2 ) [1-( C(B+2)Bn +2 )]n-1 B( C(B+2)Bn +2 ) = n (B+2Bn +2 ) [1-( B+2Bn +2 )]n-1 B( B+2+o(1)Bn +2 )n (C(B+2)Bn +2 ) [1-( C(B+2)Bn +2 )]n-1 B( C(B+2)Bn +2 ) + o(1) = (B+2Bn +2 ) ( Bn+2Bn-B ) (1- B+2Bn +2 )n B( B+2Bn +2 )(C(B+2)Bn +2 ) ( Bn+2Bn-BC ) (1-C(B+2)Bn +2 )n B( C(B+2)Bn +2 ) + o(1) = (n-C) (1- B+2Bn +2 )n B( B+2Bn +2 )C (n-1) (1-C(B+2)Bn +1 )n B( C(B+2)Bn +2 ) + o(1) = ( 1/C + o(1) ) [e(B+2)(C-1)/(B+2/n) + o(1) ] B( B+2Bn +2 )B( C(B+2)Bn +2 ) + o(1) = ( 1/C + o(1) ) [e(B+2)(C-1)/(B+o(1)) + o(1) ] ( B+2+o(1)Bn+2 )1/B [2 ( Bn+2B+2+o(1) ) ]1/(2B) eo(1)( C(B+2)Bn+2 )1/B [2 ( Bn+2C(B+2) ) ]1/(2B) eo(1) +o(1) = ( 1/C + o(1) ) [e(B+2)(C-1)/(B+o(1)) + o(1) ] (1/C + o(1))1/B [( Bn+2B+2+o(1) )( Bn+2C(B+2) ) ]1/(2B) eo(1) + o(1) = ( 1/C + o(1) ) [e(B+2)(C-1)/(B+ o(1)) + o(1) ] (1/C + o(1) )1/B [(Bn+2)(Bn+2)-(C(B+2)) - o(1) ]1/(2B) eo(1) + o(1).",
"So then n p( y=1 )p( y=1 0 ) = (1/C )1+1/B e(B+2)(C-1)/B, so in particular, since the posterior is unimodal and $\\theta _0$ is the endpoint of $I_1(n)$ , there exists $N_0(C)<\\infty $ such that $n > N_0(C)$ implies 0 I1(n) p( y=1 )p( y=1 0 ) > (1/2) ( 1/C )1+1/B e(B+2)(C-1)/B.",
"Now define $I_2(n) = [\\theta _1,\\widehat{\\theta }]$ with $\\theta _1 = \\Phi ^{-1}((B+2)/[C(Bn+2)]) = \\Phi ^{-1}([(1/C)(B+2)]/(Bn+2))$ .",
"Then clearly, there exists $N_1(C)<\\infty $ such that $n > N_1(C)$ such that 0 I2(n) p( y=1 )p( y=1 1 ) > (1/2) C1+1/B e(B+2)(1/C-1)/B.",
"Put $N(C) = \\max (N_0(C),N_1(C))$ .",
"Then since $I(n) = I_1(n) \\cup I_2(n)$ , $n > N(C)$ implies 0 I(n) p( y=1 )p( y=1 1 ) > (1/2) ( (1/C)1+1/B e(B+2)(C-1)/B, C1+1/B e(B+2)(1/C-1)/B ).",
"Combining with (), this implies the posterior density is bounded below by a constant on an interval of width $\\Omega \\left( [\\log (n)]^{-1/2} \\right)$ .",
"Parts (b) and (c) together then give $c^*(n) = \\Omega ((\\log n)^{-1/2})$ , $C^*(n) = \\mathcal {O}\\left( (\\log n)^{1/2} \\right)$ , and $1-\\epsilon (n) = \\Omega ((\\log n)^{-1})$ .",
"Part (d) : show a concentration result for $|\\theta _{t+1} - \\theta _t|$ inside $I(n)$.",
"We now show a concentration inequality for $|\\theta _t-\\theta _{t+1}|$ for $\\theta _t$ inside the interval $I(n)$ .",
"Fix a constant $2<C<\\infty $ .",
"When we have $\\theta _t$ inside the interval $I(n) = \\left[\\Phi ^{-1}\\left(\\frac{B+2}{C (Bn+2)} \\right),\\Phi ^{-1}\\left( \\frac{C(B+2)}{Bn+2} \\right) \\right],$ we can write $\\theta _t = \\Phi ^{-1}\\left( \\frac{a_t (B+2)}{Bn +1} \\right)$ for $a_t \\in [C^{-1},C]$ , which by () contains the posterior mode for large enough $n$ .",
"The term $\\phi \\left(\\Phi ^{-1}\\left( \\frac{a_t(B+2)}{Bn+2} \\right)\\right)$ will appear often.",
"We have that ( -1 ( at(B+2)Bn+2 )) = O( [2 (Bn+2)]1/2Bn+2 ) by ().",
"The conditional mean of $\\theta _{t+1} \\mid \\omega _{t+1}$ will be approximately $\\omega _{t+1}/n$ for large $n$ , so we calculate the first two moments of $\\omega _{t+1}/n$ for use in the concentration argument that follows.",
"For $\\theta _t \\in I(n)$ as above we have E( t+1/n t, y, n ) = -1( at(B+2)Bn +1 ) - ( -1( at(B+2)Bn +1 ) ) (n-1n 11-(-1( at(B+2)Bn +1 )) - 1n(-1( at(B+2)Bn +1 ))) = -1( at(B+2)Bn +1 ) - ( -1( at(B+2)Bn +1 ) ) (n-1n Bn+2Bn+2-at(B+2) - Bn+2n at(B+2) ) = -1( at(B+2)Bn +2 ) - O( (2 (Bn+2))1/2Bn+2 ) O( 1 ) = -1( at(B+2)Bn +2 ) +O( n-1 (n)1/2 ), and var(t+1/n t, y, n) = 1n + n-1n2 [t (t)1-(t)-( (t)1-(t) )2 ] - tn2 (t)(t) -(t)2n2 (t)2 = 1n + t (t) (n-1n2 11-(t) - 1n2 1(t) ) - (t)2 ( n-1n2(1-(t))2 -1n2 (t)2 ) = 1n + t (t) (n-1n2 Bn+2Bn+2-at(B+2) - 1n2 Bn+2at(B+2) ) - (t)2 ( n-1 (Bn+2)2n2(Bn+2-at(B+2))2 -(Bn+2)2n2 (at(B+2))2 ) = 1n + t (t) O( 1 ) - (t)2 O( 1 ) = 1n + -1( at(B+2)Bn+2 ) O( (2 (Bn+2))1/2Bn+2 ) + O( 2 (Bn+2)(Bn+2)2 ) = 1n + ( (2 ( Bn+2at(B+2) ))1/2 + O( ( 2 ( Bn+2 ) )(2 ( Bn+2 ))1/2 ) ) O( (2 (Bn+2))1/2Bn+2 ) + O( 2 (Bn+2)(Bn+2)2 ) = 1n + O( 2 (Bn+2)Bn+2 ) + O( ( 2 (Bn+2))Bn+2 ) + O( 2 (Bn+2)(Bn+2)2 ) = O( nn ) Next, for $\\theta _t = \\Phi ^{-1}\\left\\lbrace \\frac{a_t(B+2)}{Bn+2} \\right\\rbrace $ – equivalently, $\\theta _t \\in I(n)$ – we want to show a uniform upper bound on $\\mathbb {P} \\left( |\\theta _t - \\theta _{t+1}| > r \\zeta \\right)$ .",
"Our strategy is to show a uniform lower bound on $\\mathbb {P} \\left( |\\theta _t - \\theta _{t+1}| < r \\zeta \\right)$ for $\\zeta > 0, r\\ge 1$ .",
"By the triangle inequality, |t - t+1| < |t - t+1n | + |t+1n - t+1 |.",
"It follows that, P ( |t - t+1| < r ) P ( |t - t+1n | < r 2, |t+1n - t+1 | < r 2 ) P ( |t - t+1n | < r 2 ) P ( |t+1/n - t+1 | < r 2 | |t - t+1/n | < r 2 ).",
"Since $\\theta _t = \\Phi ^{-1}\\left( \\frac{a_t(B+2)}{Bn+2} \\right)$ , the first term on the right side is P[ | -1( at(B+2)Bn+2 ) - t+1n | < r 2 ].",
"By () and (), there exists a constant $1<A<\\infty $ and an $N_0 < \\infty $ such that $n>N_0$ implies $\\mbox{var}(\\omega _{t+1}/n \\mid \\theta _t, y, n) < A^2 (\\log n) n^{-1}$ , and (n) = |E( t+1/n t, y, n ) - -1( at(B+2)Bn +2 ) | < 2 A (n)1/2 n-1/2 Putting $\\zeta = 8 A n^{-1/2} (\\log n)^{1/2}$ and recognizing that the distribution of $\\omega _{t+1} \\mid \\theta _t$ is sub-Gaussian, we have, applying () and () P[ | t+1n - -1( at(B+2)Bn +1 ) | > r 8 A (n)1/24n1/2 + (n) ] e-2 r2 P[ | t+1n - -1( at(B+2)Bn +1 ) | > r 8 A (n)1/22n1/2 ] e-2 r2 For the second term, recall t+1 t+1, n No( (n+B-1)-1 t+1, (n+B-1)-1) No( n(n+B-1) t+1n, (n+B-1)-1 ).",
"So then there exists $N_1<\\infty $ depending only on $A,B$ such that for all $n>N_1$ , the following holds using a Gaussian tail bound, P ( |t+1/n - t+1 | > r 8 A(n)1/24n1/2 + | t+1n-t+1n+B-1 | ) e-r2 (n) P ( |t+1/n - t+1 | > r 8 A(n)1/24 n1/2 + | (n+B-1)t+1-nt+1n(n+B-1) | ) e-r2 (n) P ( |t+1/n - t+1 | > r 8 A(n)1/24 n1/2 + | t+1Bn2 + n | ) e-r2 (n).",
"Conditional on $\\left| \\Phi ^{-1}\\left( \\frac{a_t(B+2)}{Bn+2} \\right) - \\frac{\\omega _{t+1}}{n} \\right| < \\frac{r 8 A (\\log n)^{1/2}}{2 n^{1/2}}$ we have | t+1n | < [2 ( Bn+2at(B+2) )]1/2 (1+o(1)) + r 8 A(n)1/22 n1/2, < [2 ( C(Bn+2)(B+2) )]1/2 (1+o(1)) + r 8 A(n)1/22 n1/2, where the second line followed since $a_t \\in [1/C,C]$ .",
"So there exists $C_0 < \\infty $ and a function $N_0(r)$ depending only on $r$ and $A<\\infty $ such that for every $r \\ge 1$ , $n>N_0(r)$ implies |t+1| < C0 n (n)1/2 | t+1Bn2 + n | < C0 (n)1/2Bn.",
"So then for any $r$ there exists $N_1(r) < \\infty $ depending on $r$ and $B$ such that $n>\\max (N_1(r),N_0(r))=N_{\\max }(r)$ implies | t+1Bn2 + n | < r 8 A(n)1/24 n1/2 Since $r 8 A (\\log n)^{1/2}(4 n^{1/2})^{-1}$ is increasing in $r$ and we have $r \\ge 1$ , choose $N_{\\max } = N_{\\max }(1)$ , so that $n>N_{\\max }$ implies () uniformly over $r$ .",
"Then for all $n>\\max (N_{\\max },N_1)$ , we have P ( |t+1/n - t+1 | > r 8 A (n)1/22 n1/2 | |t - t+1/n| < r 2 ) e-r2 (n) .",
"uniformly over $r \\ge 1$ .",
"Putting together () and () we have for all $n > \\max (N_{\\max }, N_1, N_0)$ , that $\\theta _t \\in I(n)$ implies P ( |t - t+1| < r 8 A(n)1/2n1/2 ) (1-e-r2 (n))(1-e-2r2) so P( | t+1 - t | > r 8 A (n)1/2n1/2 ) 1-(1-e-r2 (n))(1-e-2r2 ) e-r2 (n) + e-2r2 - e-{r2 (n)+2r2} e-r2 (n) (1 - e-2r2) + e-2r2.",
"For $r \\ge 1$ , the term $1 - e^{-2r^2}$ is bounded above by 1, and $e^{-2r^2} < r^{-2}$ .",
"So then P( | t+1 - t | > r 8 A (n)1/2n1/2 ) O( n-1 ) + r-2, uniformly over $r$ , since $e^{-r^2 \\log n} = \\mathcal {O}\\left( n^{-1} \\right)$ for $r \\ge 1$ .",
"Since the posterior is negligible outside a region of width $\\mathcal {O}\\left( (\\log n)^{1/2} \\right)$ by () and is almost constant on an interval of width $\\Omega \\left( (\\log n)^{-1/2} \\right)$ by () and (), we have $1-\\epsilon (n) = \\Omega \\left( (\\log n)^{-1} \\right)$ ; $\\zeta (n) = \\mathcal {O}\\left( n^{-1/2} (\\log n)^{1/2} \\right)$ from (), $\\gamma = \\mathcal {O}\\left( n^{-1} \\right)$ from () and $c^*(n) = \\Omega ( (\\log n)^{-1/2}), C^*(n) = \\mathcal {O}\\left( (\\log n)^{1/2} \\right)$ .",
"This gives (P) = O( (n)2.5 n-1/2 ) + O( n-1 (n)2 ) = O( (n)2.5 n-1/2 ) by Corollary REF .",
"Finally, we prove Inequality (REF ).",
"Combining inequalities () and () with Lemma REF , we have shown that the mode is contained within an interval of length $\\Omega ((\\log n)^{-1/2})$ for which the density is $\\Omega ((\\log n)^{-1/2})$ .",
"Combining inequality () with Lemma REF , we have shown that the posterior distribution is negligible outside of an interval of length $\\mathcal {O}\\left( (\\log n)^{1/2} \\right)$ .",
"Inequality (REF ) follows immediately."
]
] | 1605.05798 |
[
[
"Optimization via Separated Representations and the Canonical Tensor\n Decomposition"
],
[
"Abstract We introduce a new, quadratically convergent algorithm for finding maximum absolute value entries of tensors represented in the canonical format.",
"The computational complexity of the algorithm is linear in the dimension of the tensor.",
"We show how to use this algorithm to find global maxima of non-convex multivariate functions in separated form.",
"We demonstrate the performance of the new algorithms on several examples."
],
[
"Introduction",
"Finding global extrema of a multivariate function is an ubiquitous task with many approaches developed to address this problem (see e.g.",
"[19]).",
"Unfortunately, no existing optimization method can guarantee that the results of optimization are true global extrema unless restrictive assumptions are placed on the function.",
"Assumptions on smoothness of the function do not help since it is easy to construct an example of a function with numerous local extrema “hiding” the location of the true one.",
"While convexity assumptions are helpful for finding global maxima, in practical applications there are many non-convex functions.",
"For non-convex functions various randomized search strategies have been suggested and used but none can assure that the results are true global extrema (see, e.g., [25], [19]).",
"We propose a new approach to the problem of finding global extrema of a multivariate function under the assumption that the function has certain structure, namely, a separated representation with a reasonably small separation rank.",
"While this assumption limits the complexity of the function, i.e.",
"number of independent degrees of freedom in its representation, there is no restriction on its convexity.",
"Furthermore, a large number of functions that do not appear to have an obvious separated representation do in fact possess one, as was observed in [5], [6], [4].",
"In particular, separated representations have been used recently to address curse of dimensionality issues that occur when solving PDEs with random data in the context of uncertainty quantification (see, e.g., [11], [13], [14], [15], [17], [22], [23], [24]).",
"We present a surprisingly simple algorithm that uses canonical tensor decompositions (CTDs) for the optimization process.",
"A canonical decomposition of a tensor $\\mathbf {U}\\in \\mathbb {R}^{M_{1}\\times M_{2}\\times \\dots \\times M_{d}}$ is of the form, $\\mathbf {U}=U\\left(i_{1}\\dots i_{d}\\right)=\\sum _{l=1}^{r}s_{l}u_{i_{1}}^{l}u_{i_{2}}^{l}\\cdots u_{i_{d}}^{l},$ where $u_{i_{j}}^{l}$ , $j=1,\\dots ,d$ are entries of vectors in $\\mathbb {R}^{M_{d}}$ and $r$ is the separation rank.",
"Using the CTD format to find entries of a tensor with the maximum absolute value was first explored in [16] via an analogue of the matrix power method.",
"Unfortunately, the algorithm in [16] has the same weakness as the matrix power method: its convergence can be very slow, thus limiting its applicability.",
"Instead of using the power method, we introduce a quadratically convergent algorithm based on straightforward sequential squaring of tensor entries in order to find entries with the maximum absolute value.",
"For a tensor with even a moderate number of dimensions finding the maximum absolute value of the entries appears to be an intractable problem.",
"A brute-force search of a $d$ -directional tensor requires $\\mathcal {O}\\left(N^{d}\\right)$ operations, an impractical computational cost.",
"However, for tensors in CTD format the situation is different since the curse of dimensionality can be avoided.",
"Our approach to finding the entries with the maximum absolute value (abbreviated as “the maximum entry” where it does not cause confusion) is as follows: as in [16], we observe that replacing the maximum entries with 1's and the rest of the entries with zeros results in a tensor that has a low separation rank representation (at least in the case where the number of such extrema is reasonably small).",
"To construct such a tensor, we simply square the entries via the Hadamard product, normalize the result and repeat these steps a sufficient number of times.",
"The resulting algorithm is quadratically convergent as we raise the tensor entries to the power $2^{n}$ after $n$ steps.",
"Since the nominal separation rank of the tensor grows after each squaring, we use CTD rank reduction algorithms to keep the separation rank at a reasonable level.",
"The accuracy threshold used in rank reduction algorithms limits the overall algorithm to finding the global maximum within the introduced accuracy limitation.",
"However, in many practical applications, we can find the true extrema as this accuracy threshold is user-controlled.",
"These operations are performed with a cost that depends linearly on the dimension $d$ .",
"The paper is organized as follows.",
"In Section we briefly review separated representations of functions and demonstrate how they give rise to CTDs.",
"In Section we introduce the quadratically convergent algorithm for finding the maximum entry of a tensor in CTD format and discuss the selection of tensor norm for this algorithm.",
"In Section we use the new algorithm to find the maximum absolute value of continuous functions represented in separated form.",
"In Section we test both the CTD and separated representation optimization algorithms on numerical examples.",
"We provide our conclusions and a discussion of these algorithms in Section ."
],
[
"Separated representation of functions and the canonical tensor decomposition",
"Separated representations of multivariate functions and operators for computing in higher dimensions were introduced in [5], [6].",
"The separated representation of a function $u(x_{1},x_{2},\\dots ,x_{d})$ is a natural extension of separation of variables as we seek an approximation $u(x_{1},\\ldots ,x_{d})=\\sum _{l=1}^{r}s_{l}u_{1}^{(l)}(x_{1})\\cdots u_{d}^{(l)}(x_{d})+\\mathcal {O}\\left(\\epsilon \\right),$ where $s_{l}>0$ are referred to as $s$ -values.",
"In this approximation the functions $u_{j}^{(l)}(x_{j})$ , $j=1,\\dots ,d$ are not fixed in advance but are optimized in order to achieve the accuracy goal with (ideally) a minimal separation rank $r$ .",
"In (REF ) we set $x_{j}\\in \\mathbb {R}$ while noting that in general the variables $x_{j}$ may be complex-valued or low dimensional vectors.",
"Importantly, a separated representation is not a projection onto a subspace, but rather a nonlinear method to track a function in a high-dimensional space using a small number of parameters.",
"We note that the separation rank indicates just the nominal number of terms in the representation and is not necessarily minimal.",
"Any discretization of the univariate functions $u_{j}^{(l)}\\left(x_{j}\\right)$ in (REF ) with $u_{i_{j}}^{(l)}=u_{j}^{(l)}\\left(x_{i_{j}}\\right)$ , $i_{j}=1,\\dots ,M_{j}$ and $j=1,\\dots ,d$ , leads to a $d$ -dimensional tensor $\\mathbf {U}\\in \\mathbb {R}^{M_{1}\\times \\cdots \\times M_{d}}$ , a canonical tensor decomposition (CTD) of separation rank $r_{u}$ , $\\mathbf {U}=U\\left(i_{1},\\dots ,i_{d}\\right)=\\sum _{l=1}^{r_{u}}s_{l}^{u}\\prod _{j=1}^{d}u_{i_{j}}^{(l)},$ where the $s$ -values $s_{l}^{u}$ are chosen so that each vector $\\mathbf {u}_{j}^{(l)}=\\left\\lbrace u_{i_{j}}^{(l)}\\right\\rbrace _{i_{j}=1}^{M_{j}}$ has unit Euclidean norm $\\Vert \\mathbf {u}_{j}^{(l)}\\Vert _{2}=1$ for all $j,l$ .",
"The CTD has become one of the key tools in the emerging field of numerical multilinear algebra (see, e.g., the reviews [9], [26], [20]).",
"Given CTDs of two tensors $\\mathbf {U}$ and $\\mathbf {V}$ of separation ranks $r_{u}$ and $r_{v}$ , their inner product is defined as $\\left\\langle \\mathbf {U},\\mathbf {V}\\right\\rangle =\\sum _{l=1}^{r_{u}}\\sum _{l^{\\prime }=1}^{r_{v}}s_{l}^{u}s_{l^{\\prime }}^{v}\\prod _{j=1}^{d}\\left\\langle \\mathbf {u}_{j}^{(l)},\\mathbf {v}_{j}^{(l^{\\prime })}\\right\\rangle , $ where the inner product $\\left\\langle \\cdot ,\\cdot \\right\\rangle $ operating on vectors is the standard vector dot product.",
"The Frobenius norm is then defined as $\\left\\Vert \\mathbf {U}\\right\\Vert _{F}=\\sqrt{\\left\\langle \\mathbf {U},\\mathbf {U}\\right\\rangle }$ .",
"Central to our optimization algorithm is the Hadamard, or point-wise, product of two tensors represented in CTD format.",
"We define this product as, $\\mathbf {U}*\\mathbf {V}=\\left(U*V\\right)\\left(i_{1},\\dots ,i_{d}\\right)=\\sum _{l=1}^{r_{u}}\\sum _{l^{\\prime }=1}^{r_{v}}s_{l}^{u}s_{l^{\\prime }}^{v}\\prod _{j=1}^{d}\\left(u_{i_{j}}^{(l)}\\cdot v_{i_{j}}^{(l^{\\prime })}\\right).$ Common operations involving CTDs, e.g.",
"sum or multiplication, lead to CTDs with separation ranks that may be larger than necessary for a user-specified accuracy.",
"To keep computations with CTDs manageable, it is crucial to reduce the separation rank.",
"The separation rank reduction operation for CTDs, here referred to as $\\tau _{\\epsilon }$ , is defined as follows: given a tensor $\\mathbf {U}$ in CTD format with separation rank $r{}_{u}$ , $\\mathbf {U}=U\\left(i_{1},\\dots ,i_{d}\\right)=\\sum _{l=1}^{r_{u}}s_{l}^{u}\\prod _{j=1}^{d}u_{i_{j}}^{(l)},$ and user-supplied error $\\epsilon $ , find a representation $\\mathbf {V}=V\\left(i_{1},\\dots ,i_{d}\\right)=\\sum _{l^{\\prime }=1}^{r_{v}}s_{l^{\\prime }}^{v}\\prod _{j=1}^{d}v_{i_{j}}^{(l^{\\prime })},$ with lower separation rank, $r_{v}<r_{u}$ , such that $\\left\\Vert \\mathbf {U}-\\mathbf {V}\\right\\Vert <\\epsilon \\left\\Vert \\mathbf {U}\\right\\Vert $ .",
"The workhorse algorithm for the separation rank reduction problem is Alternating Least Squares (ALS) which was introduced originally for data fitting as the PARAFAC (PARAllel FACtor) [18] and the CANDECOMP [10] models.",
"ALS has been used extensively in data analysis of (mostly) three-way arrays (see e.g.",
"the reviews [9], [26], [20] and references therein).",
"For the experiments in this paper we use exclusively the randomized interpolative CTD tensor decomposition, CTD-ID, described in [8], as a faster alternative to ALS."
],
[
"Power method for finding the maximum entry of a tensor",
"The method for finding the maximum entry of a tensor in CTD format in [16] relies on a variation of the matrix power method adapted to tensors.",
"To see how finding entries with maximum absolute value can be cast as an eigenvalue problem, we define the low rank tensor $\\mathbf {Y}$ with 1s at the locations of such entries in $\\mathbf {U}$ , and set $\\lambda =\\max _{i_{1},\\dots ,i_{d}}\\left|U\\left(i_{1},i_{2},\\dots ,i_{d}\\right)\\right|$ .",
"The Hadamard product, $\\mathbf {U}*\\mathbf {Y}=\\lambda \\mathbf {Y}, $ reveals the underlying eigenvalue problem.",
"We describe the algorithm of [16] as Algorithm REF and use it for performance comparisons.",
"The resulting tensor $\\mathbf {Y}_{k}$ ideally has a low separation rank with zero entries except for a few non-zeros indicating the location of the extrema.",
"In the case of a single extremum the tensor $\\mathbf {Y}_{k}$ has separation rank 1."
],
[
"A quadratically convergent algorithm for finding maximum entries\nof tensors",
"In order to construct $\\mathbf {Y}$ from (REF ), we simply square all entries of the tensor, normalize the result, and repeat these steps a number of times.",
"Fortunately, the CTD format allows us to square tensor entries using the Hadamard product.",
"These operations increase the separation rank, so the squaring step is (usually) followed by the application of a rank reduction algorithm such as ALS, CTD-ID, or their combination.",
"If we identify the largest (by absolute value) entry as $a=\\left|F\\left(j_{1},j_{2},\\dots ,j_{d}\\right)\\right|$ and the next largest as $b$ , the ratio $b/a<1$ decreases rapidly, $\\left(\\frac{b}{a}\\right)^{2^{j}}\\le \\epsilon $ and the total number of iterations, $j$ , to reach $\\epsilon $ is logarithmic, $j\\le \\log _{2}\\left(\\frac{\\log \\left(\\epsilon \\right)}{\\log \\left(b\\right)-\\log \\left(a\\right)}\\right).$ Since this ratio is expected to be reasonably small for most entries, the tensor will rapidly become sparse after only a small number of iterations $j$ .",
"The quadratic rate of convergence of Algorithm REF is the key difference with the algorithm in [16] whose convergence rate is at best linear."
],
[
"Termination conditions for Algorithm ",
"We have explored three options to terminate the iteration in Algorithm REF .",
"First, the simplest, is to fix the number of iterations in advance provided some knowledge of the gap between the entry with the maximum absolute value and the rest of the entries is available.",
"The second option is to define $\\lambda _{k}=\\left\\langle \\mathbf {Y}_{k},\\mathbf {U}\\right\\rangle $ and terminate the iteration once the rate of its decrease becomes small, i.e., $\\left(\\lambda _{k}-\\lambda _{k-1}\\right)/\\lambda _{k-1}<\\delta $ , where $\\delta >0$ is a user-supplied parameter.",
"The third option is to observe the separation rank of $\\mathbf {Y}_{k}$ and terminate the iteration when it becomes smaller than a fixed, user-selected value.",
"In Section we indicate which termination condition is used in our numerical examples."
],
[
"Selection of tensor norm",
"Working with tensors, we have a rather limited selection of norms that are computable.",
"The usual norm chosen for work with tensors is the Frobenius norm which may not be appropriate in all situations where the goal is to find the maximum absolute value entries.",
"Unfortunately, the Frobenius norm is only weakly sensitive to changes of individual entries.",
"The alternative $s$ -norm (see [8]), i.e., the largest $s$ -value of the rank one separated approximation to the tensor in CTD format, is better in some situations.",
"In particular, it allows the user to lower the tolerance to below single precision (within a double precision environment).",
"The improved control over truncation accuracy makes this norm more sensitive to individual entries of the tensor.",
"We note that while it is straightforward to compute the $s$ -value of a rank one separated approximation via a convergent iteration, theoretically it is possible that this computed number is not the largest $s$ -value.",
"However, we have not encountered such a situation using the $s$ -norm in practical problems."
],
[
"Numerical demonstration of convergence",
"We construct an experiment using random tensors, with low separation rank ($r=4$ in what follows), in dimension $d=6$ with $M=32$ samples in each direction.",
"The input CTD is formed by adding a rank three CTD and a rank one CTD together.",
"The rank three CTD is formed using random factors ($\\mathbf {u}_{j}^{(l)}$ in (REF )) whose samples were drawn from a uniform distribution on $\\left[.9,1\\right]$ (the $s$ -values were set to 1).",
"The rank one CTD has all zeros except for a spike added in at a random location.",
"The size of this spike is chosen to make the maximum entry have a magnitude of $3.5$ .",
"For the experiments in this section we set the reduction tolerance of $\\tau _{\\epsilon }$ in Algorithm REF to $\\epsilon =10^{-6}$ and use the Frobenius norm.",
"We observe that as the iteration proceeds, the maximum entry of one of the CTD terms eventually becomes dominant relative to the rest.",
"The iteration continues until the rest of the terms are small enough to be eliminated by the separation rank reduction algorithm $\\tau _{\\epsilon }$ .",
"This is illustrated in Figure REF where we display the maximum entry from all rank one terms in the CTD for each iteration.",
"Before the first iteration (iteration 0 in Figure REF ), the gaps between the maxima of the rank one terms are not large.",
"As we iterate, the maximum entry of one of the rank one terms rapidly separates itself from the others and, by iteration 6, the reduction algorithm removes all terms except one.",
"This remaining term has an entry 1 corresponding to the location of the maximum entry of the original tensor and all other entries are 0.",
"Figure: Comparing the maximum absolutevalues of entries from rank one terms of a CTD.",
"After 6 iterationsAlgorithm converges to a rank 1tensor.Another interesting case occurs when a tensor has multiple maximum entry candidates, either close to or exactly the same in magnitude.",
"To explore this case, we construct an experiment similar to the previous one, namely, find the maximum absolute value of a CTD of dimension $d=6$ and $M=32$ samples in each direction.",
"This CTD is constructed with the same rank 3 random CTD as above, but this time we added in two spikes at random locations so that there are two maximal spikes of absolute value $3.5$ .",
"In cases such as this, Algorithm REF can be run for a predetermined number of iterations, after which the factors are examined and a subset of candidates can be identified and explicitly verified.",
"For example, in Figure REF , we observe that by iteration 6 there are only two candidates for the location of the maximum.",
"As a warning we note that allowing the code to continue to run many more iterations will leave only one term due to the finite reduction tolerance $\\epsilon $ of $\\tau _{\\epsilon }$ in Algorithm REF .",
"Indeed, in Figure REF we observe the maximum entries of the two rank one terms begin to split around iteration number 26, and by iteration 37 only one term remains.",
"This example illustrates a potential danger of insisting on a single term termination condition.",
"Figure: In this example there aretwo maxima of the same size.",
"The values at these locations convergeto 1/21/\\sqrt{2} and the rest of the entries converge to 0.",
"If allowedto run long enough, the value at one of the locations becomes dominantdue to the truncation error of the rank reduction algorithm.",
"The otherlocation is then eliminated by the algorithm and only one term remains."
],
[
"CTD-based global optimization of multivariate functions",
"We describe an approach using Algorithm REF for the global optimization of functions that admit low rank separated representations.",
"We consider two types of problems: first, where the objective function is already in a separated form, ready to yield a CTD appropriate for use with Algorithm REF .",
"The second, more general problem, is to construct a separated representation given data or an analytic expression of the function, from which a CTD can then be built.",
"In the first problem the objective function is already represented in separated form as in (REF ), and an interpolation scheme is associated with each direction.",
"Such problems have been subject to growing interest due to the use of CTDs for solving operator equations, e.g., deterministic or stochastic PDE/ODE systems (see, for example, [5], [6], [13], [24], [11], [12]).",
"In the second problem the goal is to optimize a multivariate function where no separated representation is readily available.",
"What typically is available is a data set of function values (scattered or on a grid).",
"In such cases our recourse is to first construct a separated representation of the underlying multivariate objective function and then re-sample it in order to obtain a CTD of the form (REF ).",
"The construction of a separated representation given a set of function values can be thought of as a regression problem, see [4], [15].",
"The algorithms in these papers use function samples to build a separated representation of form (REF ).",
"Following [4], an interpolation scheme is set up in each direction and the ALS algorithm is used to reduce the problem to regression in a single variable.",
"The univariate interpolation scheme may use a variety of possible bases, as well as nonlinear approximation techniques.",
"Once a separated representation is constructed, the CTD for finding the global maxima is then obtained to satisfy the local interpolation requirement stated below.",
"The resulting CTD is then used as input into Algorithm REF to produce an approximation of the global maxima.",
"If the objective function is given analytically, it is preferable to use analytic techniques to approximate the original function via a separated representation.",
"An example of this approach is included in Section REF below.",
"We note that sampling becomes an important issue when finding global maxima of a function via Algorithm REF .",
"The key sampling requirement is to ensure that the objective function can be interpolated at an arbitrary point from its sampled values up to a desired accuracy.",
"This implies that sampling rate must depend on the local behavior of the function, e.g., the sampling rate is greater near singularities.",
"For example, if a function has algebraic singularities, an efficient method to interpolate is to use wavelet bases.",
"In such a case we emphasize the importance of using interpolating scaling functions, i.e., scaling functions whose coefficients are function values or are simply related to those.",
"Since the global maxima are typically searched for in finite domains, the multiresolution basis should also work well in an interval (or a box in higher dimensions).",
"This leads us to suggest the use of Lagrange interpolating polynomials within an adaptive spatial subdivision framework.",
"We note that rescaled Lagrange interpolating polynomials form an orthonormal basis on an interval and the spatial subdivision is available as multiwavelet bases (see e.g., [1], [2]).",
"Finding the maximum entries of a CTD yields only an approximation of the maxima and their locations for the function.",
"However, due to the interpolation requirements stated above these are high-quality approximations so that, if needed, a local optimization algorithm can then be used to find the true global maxima.",
"Since in this case we use local optimization, it is preferable to oversample the CTD representation of a function, i.e.",
"to use large $M_{j}$ 's in (REF ).",
"This will not result in a prohibitive cost as the rank reduction algorithms, e.g.",
"ALS, that operate on CTDs scale linearly with respect to the number of samples in each direction [6].",
"However, the total number of samples needed to satisfy the local interpolation requirement may be large and, thus, additional steps to accelerate the reduction algorithm $\\tau _{\\epsilon }$ in Algorithm REF may be required.",
"One such technique consists of using the $QR$ factorization as explained in [8]."
],
[
"Comparison with power method algorithm from Section ",
"To test Algorithm REF , we construct tensors from random factors and find the entries with the largest absolute value.",
"For these examples we select dimension $d=8$ and set $M=32$ samples in each direction.",
"The test CTDs are constructed by adding rank four and rank one CTDs together.",
"The rank four CTDs are composed of random factors (vectors $\\mathbf {u}_{j}^{(l)}$ in (REF )) whose entries are drawn from the uniform distribution on $\\left[.9,1\\right]$ and the corresponding $s$ -values are set to 1.",
"The rank one CTDs contain all zeros except for a magnitude 4 spike added in at a random location.",
"The CTDs in these examples have a unique maximum entry, and we chose to terminate the algorithm when the output CTD reaches separation rank one.",
"We run tests for both Algorithm REF and Algorithm REF , using for the rank reduction operation $\\tau _{\\epsilon }$ the CTD-ID algorithm with accuracy threshold $\\epsilon =10^{-6}$ in the $s$ -norm.",
"Figure: Histograms illustrating the number ofiterations required for convergence to a rank one CTD by the powermethod optimization (hatched pattern) and the squaring method optimization(solid gray).A histogram displaying the required number of iterations to converge for 500 individual tests is shown in Figure REF .",
"While both algorithms find the correct maximum location in all tests, Algorithm REF outperforms Algorithm REF in terms of the number of iterations required.",
"While Algorithm REF requires much fewer iterations than the power method for the same problem, a lingering concern may be that the squaring process in Algorithm REF requires the reduction of CTDs with much larger separation ranks.",
"In Figure REF we show histograms of computation times.",
"We observe that despite reducing larger separation rank CTDs, the computation times using Algorithm REF are significantly smaller.",
"These computation times are for our MATLAB code running on individual cores of a Intel Xeon $2.40$ GHz processor.",
"Figure: Histograms illustrating the computationtimes, in seconds, required for convergence to a rank one CTD by thepower method optimization (hatched pattern) and the squaring methodoptimization (solid gray)."
],
[
"Optimization example",
"We reiterate that separated representations of general functions can be obtained numerically using only function evaluations, as described in [4], [15].",
"Once such a representation is obtained, an appropriate sampling yields a tensor in CTD format.",
"In some cases, as in this example, this can be accomplished analytically.",
"As an example of constructing separated representations for optimization, we consider Ackley's test function [3], commonly used for testing global optimization algorithms, $u\\left(\\mathbf {x}\\right)=a\\,e^{-b\\left(\\frac{1}{d}\\sum _{i=1}^{d}x_{i}^{2}\\right)^{1/2}}+e^{\\frac{1}{d}\\sum _{i=1}^{d}\\cos \\left(cx_{i}\\right)},$ where $d$ is the number of dimensions and $a$ , $b$ , and $c$ are parameters.",
"For our tests we set $d=10$ , $a=20$ , $b=0.2$ , and $c=2\\pi $ .",
"We choose this test function for two reasons: first, it has many local maxima, thus a local method without a good initial guess may not find the correct maximum.",
"Second, one of the terms is not in separated form, hence we must construct a separated approximation of the function first.",
"The true maximum of (REF ) occurs at $\\mathbf {x}=\\mathbf {0}$ with a value of $u\\left(\\mathbf {0}\\right)=a+e$ .",
"The first term in (REF ) is radially symmetric which we approximate with a linear combination of Gaussians using the approximation of $e^{-xy}$ from [7], $G_{e}\\left(x\\right)=\\frac{hb}{2\\sqrt{\\pi d}}\\sum _{j=0}^{R}\\exp \\left(-\\frac{b^{2}}{4d}e^{s_{j}}-x^{2}e^{s_{j}}+\\frac{1}{2}s_{j}\\right),$ where $s_{j}=s_{start}+jh$ , and $h,\\,\\, s_{start},$ and $R$ are parameters chosen such that given $\\epsilon ,\\delta >0$ , $\\left|G_{e}\\left(x\\right)-e^{-\\frac{b}{\\sqrt{d}}x}\\right|\\le \\epsilon $ for $0<\\delta <x<\\infty $ .",
"We arrive at the approximation of the first term in (REF ), $\\left|e^{ -b \\left(\\frac{1}{d}\\sum _{i=1}^{d}x_{i}^{2}\\right)^{1/2}}-\\sum _{j=0}^{R}w_{j}\\prod _{i=1}^{d}\\exp \\left(-x_{i}^{2}e^{s_{j}}\\right)\\right|\\le \\epsilon $ where $w_{j}=\\frac{hb}{2\\sqrt{\\pi d}}\\exp \\left(-\\frac{b^{2}}{4d}e^{s_{j}}+\\frac{1}{2}s_{j}\\right)$ .",
"To correctly sample (REF ), we first consider two terms of the function separately, the radially-symmetric exponential term, and the exponential cosine term which is already in separated form, $\\exp \\left(\\frac{1}{d}\\sum _{i=1}^{d}\\cos \\left(cx_{i}\\right)\\right)=\\prod _{i=1}^{d}\\exp \\left(\\frac{1}{d}\\cos \\left(cx_{i}\\right)\\right)$ .",
"These terms require different sampling strategies.",
"The sampling rate of the radially-symmetric exponential term near the origin should be significantly higher than away from it.",
"On the other hand, uniform sampling is sufficient for the exponential cosine term.",
"We briefly discuss the combined sampling of these terms to satisfy both requirements.",
"To sample the radially-symmetric exponential term in (REF ), we use the expansion by Gaussians (REF ) as a guide.",
"We choose 10 equally-spaced points centered at zero and sample the Gaussian $e^{-x^{2}}$ .",
"For each Gaussian in (REF ), we scale these points by the factor $e^{s_{j}/2}$ in order to consistently sample each Gaussian independent of its scale.",
"Since (REF ) includes a large number of rapidly decaying Gaussians, too many of the resulting samples will concentrate near the origin.",
"Therefore, we select a subset as follows: first, we keep all samples from the sharpest Gaussian, $x_{0l}$ , $l=1,\\dots ,10$ .",
"We then take the samples from second sharpest Gaussian and keep only those outside the range the previous samples, i.e., we keep $x_{1m}$ such that $\\left|x_{1m}\\right|>\\max _{l}\\left|x_{0l}\\right|$ , $m=1,\\dots ,10$ .",
"Repeating this process for all terms $j=0,1,\\dots ,R$ in expansion (REF ) yields samples for the radially-symmetric exponential term in (REF ).",
"We oversample the exponential cosine term using 16 samples per oscillation.",
"Notice that this sampling strategy is not sufficient for the radially-symmetric exponential term in (REF ) (and vice-versa).",
"To combine these samples, we first find the radius where the sampling rate of the radially-symmetric exponential term in (REF ) drops below the sampling rate chosen to sample the exponential cosine term.",
"We then remove all samples whose coordinates are greater in absolute value than the radius, and replace them with samples at the chosen rate of 16 samples per oscillation.",
"The resulting set of samples consists of $M=1080$ points for each direction $j=1,\\dots ,10$ .",
"Using expansion (REF ) leads to a 77 term CTD representation for the chosen accuracy $\\epsilon =10^{-8}$ and parameter $\\delta =3\\cdot 10^{-6}$ in (REF ).",
"An initial reduction of this CTD using the CTD-ID algorithm with accuracy threshold $\\epsilon =10^{-6}$ in the Frobenius norm produces a CTD with separation rank 11 representing the function $u\\left(\\mathbf {x}\\right)$ .",
"We use this CTD as an input into Algorithm REF , yielding a rank one solution after 24 iterations.",
"We set the reduction tolerance in Algorithm REF for this experiment to $\\epsilon =10^{-6}$ and find the maximum entry of the tensor located at a distance $2.13\\times 10^{-3}$ away from the function's true maximum location, the origin.",
"Using the output as the initial guess in a local, gradient-free, optimization algorithm (in our case a compass search, see e.g.",
"[21]) yields the maximum value with relative error of $1.03\\times 10^{-7}$ located at a distance $1.84\\times 10^{-6}$ away from the origin."
],
[
"Discussion and conclusions",
"Entries of a tensor in CTD format with the largest absolute value can be reliably identified by the new quadratically convergent algorithm, Algorithm REF .",
"This algorithm, in turn, can be used for global optimization of multivariate functions that admit separated representations.",
"Unlike algorithms based on random search strategies (see e.g.",
"[19]), our approach allows the user to estimate the potential error of the result.",
"The computational cost of Algorithm REF is dominated by the cost of reducing the separation ranks of intermediate CTDs, not the dimension of the optimization problem.",
"If ALS is used to reduce the separation rank and $d$ is the dimension, $r$ is the maximum separation rank (after reductions) during the iteration, and $M$ is the maximum number of components in each direction, then the computational cost of each rank reduction step can be estimated as $\\mathcal {O}\\left(r^{4}\\cdot M\\cdot d\\cdot N_{iter}\\right)$ , where $N_{iter}$ is the number of iterations required by the ALS algorithm to converge.",
"The computational cost of the CTD-ID algorithm is estimated as $\\mathcal {O}\\left(r^{3}\\cdot M\\cdot d\\right)$ and, if it is used instead of ALS, the reduction step is faster by a factor of $\\mathcal {O}\\left(r\\cdot N_{iter}\\right)$ [8].",
"We note that, while linear in dimension, Algorithm REF may require significant computational resources due to the cubic (or quartic for ALS) dependence on the separation rank $r$ .",
"Finally, we note that the applicability of our approach to global optimization extends beyond the use of separated representations of multivariate functions or tensors in CTD format.",
"In fact, any structural representation of a multivariate function that allows rapid computation of the Hadamard product of corresponding tensors and has an associated algorithm for reducing the complexity of the representation should work in a manner similar to Algorithm REF ."
]
] | 1605.05789 |
[
[
"Short-time height distribution in 1d KPZ equation: starting from a\n parabola"
],
[
"Abstract We study the probability distribution $\\mathcal{P}(H,t,L)$ of the surface height $h(x=0,t)=H$ in the Kardar-Parisi-Zhang (KPZ) equation in $1+1$ dimension when starting from a parabolic interface, $h(x,t=0)=x^2/L$.",
"The limits of $L\\to\\infty$ and $L\\to 0$ have been recently solved exactly for any $t>0$.",
"Here we address the early-time behavior of $\\mathcal{P}(H,t,L)$ for general $L$.",
"We employ the weak-noise theory - a variant of WKB approximation -- which yields the optimal history of the interface, conditioned on reaching the given height $H$ at the origin at time $t$.",
"We find that at small $H$ $\\mathcal{P}(H,t,L)$ is Gaussian, but its tails are non-Gaussian and highly asymmetric.",
"In the leading order and in a proper moving frame, the tails behave as $-\\ln \\mathcal{P}= f_{+}|H|^{5/2}/t^{1/2}$ and $f_{-}|H|^{3/2}/t^{1/2}$.",
"The factor $f_{+}(L,t)$ monotonically increases as a function of $L$, interpolating between time-independent values at $L=0$ and $L=\\infty$ that were previously known.",
"The factor $f_{-}$ is independent of $L$ and $t$, signalling universality of this tail for a whole class of deterministic initial conditions."
],
[
"Introduction",
"The Kardar-Parisi-Zhang (KPZ) equation [1] describes an important universality class of non-equilibrium interface growth [2], [3], [4], [5], [6].",
"In $1+1$ dimension the KPZ equation, $\\partial _{t}h=\\nu \\partial ^2_{x}h+(\\lambda /2)\\left(\\partial _{x}h\\right)^2+\\sqrt{D}\\,\\xi (x,t),$ governs the evolution of the interface height $h(x,t)$ driven by a Gaussian white noise $\\xi (x,t)$ with zero mean and $\\langle \\xi (x_{1},t_{1})\\xi (x_{2},t_{2})\\rangle = \\delta (x_{1}-x_{2})\\delta (t_{1}-t_{2})$ .",
"Without losing generality, we will assume that $\\lambda <0$ [7].",
"An extensive body of work was devoted to the long-time behavior of the KPZ interface [2], [3].",
"In $1+1$ dimension, the interface width grows at long times as $t^{1/3}$ , whereas the correlation length grows as $t^{2/3}$ , as confirmed in experiments [8].",
"The exponents $1/3$ and $2/3$ are hallmarks of the KPZ universality class.",
"In the recent years the focus of interest in the KPZ equation in $1+1$ dimension shifted toward the complete one-point probability distribution of height $H$ at a specified point in space and at a specified time [4], [5], [6].",
"Several groups derived exact representations of this distribution [that we will call ${\\mathcal {P}}(H,t,L)$ ] for an arbitrary time $t>0$ .",
"This remarkable progress has been achieved for three classes of initial conditions (and some of their combinations and variations): flat interface [9], sharp wedge [10], [11], [12], [13], [4], and stationary interface: a two-sided Brownian interface pinned at a point [14], [15].",
"In the long-time limit, and for typical fluctuations, ${\\mathcal {P}}(H,t)$ converges to the Gaussian orthogonal ensemble (GOE) Tracy-Widom distribution [16] for the flat interface, to the Gaussian unitary ensemble (GUE) Tracy-Widom distribution for the sharp wedge, and to the Baik-Rains distribution [17] for the stationary interface.",
"A series of ingenious experiments fully confirmed the long-time results [18].",
"Recently, Le Doussal et al used the exact results for the sharp-wedge initial condition to extract asymptotics corresponding to large deviations of $H$ at long [19] and short [20] times.",
"The long-time regime has traditionally attracted great interest [2], [3], [4], [5], [6], but the short-time regime is also interesting [21], [22], [23], [24].",
"Indeed, at short times one observes, for both flat and sharp-wedge initial conditions, crossover of the full one-point height statistics from the Edwards-Wilkinson universality class to the KPZ universality class as one moves away from the body of the distribution ${\\mathcal {P}}(H)$ to its strongly asymmetric tails [21], [23], [24], [25], [20].",
"In each of the exactly solved cases, ${\\mathcal {P}}(H,t)$ is given in terms of a generating function that involves a complicated determinant form.",
"Extracting useful asymptotics from these general results may require considerable effort.",
"It can be advantageous to use approximations which directly probe the desired asymptotic regimes.",
"This approach was taken in Refs.",
"[21], [22], [23], [25] which studied the short-time asymptotics of ${\\mathcal {P}}(H,t)$ when starting the process from a flat interface.",
"In these works the probability distribution ${\\mathcal {P}}(H,t)$ was evaluated by using the weak-noise theory (WNT) of Eq.",
"(REF ).",
"The WNT is a variant of WKB approximation.",
"It employs in a smart way the smallness of typical noise when studying large fluctuations.",
"The WNT originated from the Martin-Siggia-Rose path-integral formalism in physics [26] and the Freidlin-Wentzel large-deviation theory in mathematics [27].",
"The WNT is related to the optimal fluctuation method which goes back to Refs.",
"[28], [29], [30], see also Ref.",
"[31].",
"Similar approaches have been applied, under different names, to turbulence [32], [33], [34], stochastic reactions [35], [36], diffusive lattice gases [37], and non-equilibrium surface growth [38], [39], [40], [21], [22], [23], [41], [25] including the KPZ equation itself.",
"The WNT equations can be formulated as a classical Hamiltonian field theory.",
"After having solved the WNT equations, one can evaluate the action functional, which gives, up to a sub-leading prefactor, the probability to observe a specific large deviation.",
"The exactly soluble cases of the complete height statistics of the KPZ equation serve as excellent benchmarks for the WNT, which then can be applied to other initial conditions, to higher dimensions, and to other models, where exact solutions are unavailable.",
"Here we consider one such initial condition: a parabolic interface $h(x,t=0)=\\frac{x^2}{L}.$ The limit of $L\\rightarrow \\infty $ corresponds to the exactly soluble case of the flat interface.",
"As we explain in Section 2, the limit of $L\\rightarrow 0$ is intimately related to another exactly soluble case: of the sharp wedge interface.",
"Here we address the early-time behavior of $\\mathcal {P}(H,t,L)$ for arbitrary $L$ .",
"To this end, we determine the optimal (the most likely) history of the interface $h(x,t)$ conditioned on reaching the height $H$ at time $T$ .",
"We find that the tails of $\\mathcal {P}$ behave, in a proper moving frame [42], as $-\\ln \\mathcal {P}=f_+ H^{5/2}/T^{1/2}$ as $H\\rightarrow \\infty $ and $f_- |H|^{3/2}/T^{1/2}$ as $H\\rightarrow -\\infty $ .",
"The factor $f_{+}(L,T)$ increases with $L$ , interpolating between previously known, time-independent values at $L=0$ and $L=\\infty $ .",
"On the contrary, the factor $f_{-}$ is independent of $L$ and $T$ .",
"This indicates universality of this tail for a whole class of deterministic initial conditions, and we uncover the mechanism of, and the condition for, this universality.",
"Here is a plan of the remainder of this paper.",
"In Sec.",
"2 we formulate the problem, identify the scaling behavior of $\\mathcal {P}(H,L,T)$ and briefly discuss the connection between the problem with parabolic initial condition (REF ) and the problem with a sharp-wedge initial condition.",
"Our main results are presented in Sec.",
"3, where we employ the WNT are obtain leading-order analytical results for $-\\ln {\\mathcal {P}}(H,T,L)$ in three limiting cases: large positive $H$ , large negative $H$ and small $H$ .",
"Section 4 contains a summary and discussion of our results."
],
[
"Formulation of the problem",
"Without noise, the interface height is governed by the deterministic KPZ equation, $\\partial _{t}h=\\nu \\partial ^2_{x}h+(\\lambda /2)\\left(\\partial _{x}h\\right)^2.$ Its solution with the initial condition (REF ) is $h(x,t)=\\frac{x^2}{L-2\\lambda t}+\\frac{\\nu }{\\lambda } \\,\\ln \\frac{L}{L-2\\lambda t},$ so the average profile remains parabolic at all times.",
"For $\\lambda <0$ it is well-behaved at any $t>0$ .",
"Let us rescale $t$ by the given time $T$ (see below), $x$ by the diffusion length $\\sqrt{\\nu T}$ , and $h$ by the $\\nu /|\\lambda |$ .",
"Then Eq.",
"(REF ) becomes $\\partial _{t}h=\\partial ^2_{x}h-(1/2)\\left(\\partial _{x}h\\right)^2,$ while its solution (REF ) becomes $h_0(x,t)=\\frac{x^2}{L+2 t}+\\ln \\left(1+\\frac{2t}{L}\\right),$ where $L$ is rescaled by $|\\lambda |T$ .",
"When the rescaled $L$ is very small, the deterministic solution rapidly becomes $h_0(x,t)\\simeq \\frac{x^2}{2 t}+\\ln \\left(\\frac{2t}{L}\\right).$ A very similar deterministic profile appears in the problem of sharp wedge, when $h(x,t=0)=|x|/\\delta $ with $\\delta \\ll 1$ .",
"Here at $t\\gg \\delta ^2$ and $|x|\\ll t/\\delta $ a parabolic profile develops: $h_0(x,t)\\simeq \\frac{x^2}{2 t}+\\ln \\left(\\frac{t}{\\delta ^2}\\right).$ As one can see, the solutions (REF ) and (REF ) are identical up to notation.",
"Therefore, we will not distinguish in the following between the limit of $L\\rightarrow 0$ of the parabolic initial condition and the limit of $\\delta \\rightarrow 0$ of the wedge initial condition.",
"Now we return to the stochastic equation (REF ) and study the probability distribution $\\mathcal {P}(H,T,L)$ of observing (in a proper moving frame [42]) a given value $h(x=0,t=T) =H$ , considerably different from the prediction of the deterministic solution (REF ).",
"Upon the rescaling transformation introduced above, Eq.",
"(REF ) becomes $\\partial _{t}h=\\partial ^2_{x}h-(1/2) \\left(\\partial _{x}h\\right)^2+\\sqrt{\\epsilon } \\,\\xi (x,t),$ where $\\epsilon =\\frac{D\\lambda ^2 \\sqrt{T}}{\\nu ^{5/2}}$ is a dimensionless noise magnitude.",
"The rescaled initial condition coincides with Eq.",
"(REF ), with $L$ replaced by $\\tilde{L} = L/(|\\lambda | T)$ .",
"As one can see, $\\mathcal {P}(H,T,L)$ depends on three dimensionless parameters: $\\tilde{H}=|\\lambda | H /\\nu $ , $\\tilde{L}$ and $\\epsilon $ .",
"We will omit the tildes."
],
[
"Weak-noise theory",
"Formally, the WNT relies on the smallness of $\\epsilon $ .",
"In view of Eq.",
"(REF ), this makes the WNT especially suitable for short times.",
"A saddle-point evaluation of the path integral, corresponding to Eq.",
"(REF ), leads to a variational problem for the action [38], [21], [22], [23], [25].",
"As we show in the Appendix, the Euler-Lagrange equations can be presented as a pair of Hamilton equations for the optimal height history $h(x,t)$ and the canonically conjugate “momentum\" density field $\\rho (x,t)$ : $\\partial _{t} h &=& \\delta \\mathcal {H}/\\delta \\rho = \\partial _{x}^2 h -(1/2) \\left(\\partial _x h\\right)^2+\\rho , \\\\\\partial _{t}\\rho &=& - \\delta \\mathcal {H}/\\delta h = - \\partial _{x}^2 \\rho - \\partial _x \\left(\\rho \\partial _x h\\right) ,$ where $\\mathcal {H} = \\int dx \\,w$ is the Hamiltonian, and $w(x,t)= \\rho \\left[\\partial _x^2 h-(1/2) \\left(\\partial _x h\\right)^2+\\rho /2\\right]$ .",
"Note that $\\rho $ undergoes rescaling $|\\lambda | T \\rho /\\nu \\rightarrow \\rho $ .",
"The initial condition is Eq.",
"(REF ) with rescaled $L$ .",
"The behavior of $h(x,t)$ at large $|x|$ is governed by Eq.",
"(REF ), whereas $\\rho (|x| \\rightarrow \\infty )=0$ so that the action is bounded, see Eq.",
"(REF ) below.",
"Finally, the condition $h(x=0,t=1)=H$ translates into [21], [25] $\\rho (x,1)=\\Lambda \\,\\delta (x),$ where $\\Lambda $ is ultimately determined by the rescaled $H$ and $L$ .",
"Once the WNT problem is solved, we can evaluate $-\\ln \\mathcal {P}(H,T,L)&\\simeq & \\frac{1}{\\epsilon }\\,S\\left(\\frac{|\\lambda | H}{\\nu }, \\frac{L}{|\\lambda |T}\\right)\\nonumber \\\\&=& \\frac{\\nu ^{5/2}}{D\\lambda ^2\\sqrt{T}}\\,\\,S\\left(\\frac{|\\lambda | H}{\\nu }, \\frac{L}{|\\lambda |T}\\right),$ (in the physical units), where the rescaled action $S$ is $S = \\int _0^1 dt \\int dx\\,(\\rho \\partial _t h -w)= \\frac{1}{2}\\int _0^1 dt \\int dx\\,\\rho ^2 (x,t).",
"$ Now we consider three asymptotic limits where we can solve the problem analytically."
],
[
"Large positive heights",
"Here one can neglect the diffusion terms in Eqs.",
"(REF ) and () and obtain hydrodynamic equations $\\partial _t \\rho +\\partial _x (\\rho V)&=& 0, \\\\\\partial _t V +V \\partial _x V &=&\\partial _x \\rho , $ where $V(x,t) =\\partial _x h (x,t)$ .",
"These equations describe a non-stationary inviscid flow of a compressible gas with density $\\rho $ , velocity $V$ , and negative pressure $p(\\rho )=-\\rho ^2/2$ [25], [43].",
"The problem should be solved subject to the condition $V(x,t=0)=\\frac{2x}{L}$ and Eq.",
"(REF ).",
"Equations (REF ), () and (REF ) remain invariant under inviscid rescaling $x/\\Lambda ^{1/3} \\rightarrow x$ , $V/\\Lambda ^{1/3} \\rightarrow V$ , and $\\rho /\\Lambda ^{2/3} \\rightarrow \\rho $ .",
"In its turn, Eq.",
"(REF ) becomes $\\rho (x,t=1)=\\delta (x).$ Now Eq.",
"(REF ) yields $S=\\Lambda ^{5/3}\\, s(L),$ where, in the newly rescaled variables, $s(L) = \\frac{1}{2}\\int _0^1 dt \\int dx\\,\\rho ^2 (x,t).",
"$ What is the expected scaling behavior of $S$ entering Eq.",
"(REF )?",
"The rescaled height at $t=1$ is $h(x=0,t=1) \\equiv H_1(L) = H/\\Lambda ^{2/3}$ .",
"Therefore, $\\Lambda = (H/H_1)^{3/2}$ , and Eq.",
"(REF ) yields $S(H,L)=\\frac{s(L) H^{5/2}}{\\left[H_1(L)\\right]^{5/2}}$ leading, for any $L$ , to a $H^{5/2}$ tail.",
"What is left is to find $s(L)$ and $H_1(L)$ .",
"By virtue of the special boundary conditions (REF ) and (REF ), the solution of Eqs.",
"(REF ) and () with $\\rho >0$ has compact support and describes a uniform-strain flow: $V(x,t)=a(t)\\,x, \\quad |x|\\le \\ell (t), $ and (x,t) = r(t) [1-x2/2(t)], $|x|\\le \\ell (t)$ , 0, $|x|> \\ell (t)$ , where the functions $r(t)>0$ , $\\ell (t)\\ge 0$ and $a(t)$ are to be determined.",
"The “zero-pressure\" region of $|x|>\\ell (t)$ needs to be considered separately.",
"For the flat interface, $L\\rightarrow \\infty $ , this problem was solved previously in Ref.",
"[25], see also Ref.",
"[23].",
"In that case $a(t)$ starts from zero at $t=0$ and decreases monotonically, going to $-\\infty $ at $t\\rightarrow 1$ .",
"The solution describes an inflow of the gas, culminating in its collapse into the origin at $t=1$ .",
"For a finite $L$ one has $a(t=0)=2/L>0$ [see Eq.",
"(REF )] implying an outflow of the gas.",
"This outflow stops at some time $0<t_*<1$ , so that $a(t_*)=0$ , and then becomes an inflow, $a(t)<0$ at $t>t_*$ , until $a$ reaches $-\\infty $ , and the gas collapses into the origin, at $t=1$ .",
"Let us first consider the limit of $L\\rightarrow 0$ corresponding to the sharp-wedge initial condition.",
"Here $a(t)$ is equal to $-\\infty $ at $t=0$ , zero at $t=1/2$ , and $+\\infty $ at $t=1$ .",
"This outflow-inflow solution exhibits a remarkable symmetry in time around $t=1/2$ .",
"Here the “gas density\" $\\rho $ is equal to $\\delta (x)$ both at $t=0$ and at $t=1$ .",
"The mass conservation yields $\\ell (t) r(t)=3/4$ .",
"Using it, and plugging Eqs.",
"(REF ) and (REF ) into Eqs.",
"(REF ) and (), we obtain two coupled equations for $r(t)$ and $a(t)$ : $\\dot{r}=-ra$ and $\\dot{a}=-a^2-(32/9) r^3$ [25].",
"Their first integral can be written as $a=\\pm (8/3) r \\sqrt{r-r_*}$ , where $r_*\\equiv r(t=1/2)$ .",
"This yields $\\dot{r}=\\pm (8/3) r^2 \\sqrt{r-r_*},$ with the minus sign for $0<t<1/2$ and the plus sign for $1/2<t<1$ .",
"An implicit solution of Eq.",
"(REF ), obeying the conditions $r(t\\rightarrow 0)=r(t\\rightarrow 1) =\\infty $ (see Fig.",
"REF ), is $t=t_{\\pm }(r)= \\frac{1}{2}\\pm \\frac{3 \\sqrt{r-r_*}}{8 r r_*}\\pm \\frac{1}{\\pi }\\arctan \\left(\\sqrt{\\frac{r}{r_*}-1}\\,\\right),$ where $r_*=(3 \\pi /8)^{2/3}$ .",
"The minus signs correspond to $0< t\\le 1/2$ , the plus signs to $1/2\\le t< 1$ .",
"Figure: r=ρ(x=0,t)r=\\rho (x=0,t) as a function of time for H≫1H\\gg 1 and L→0L\\rightarrow 0 as determined by Eq.",
"().Now we can calculate $s$ : $\\!\\!\\!",
"s(L\\rightarrow 0) &=& \\!\\!",
"\\frac{1}{2}\\int _0^1 dt \\int _{-\\ell }^{\\ell } dx\\,r^2(t) \\left[1-x^2/\\ell (t)^2\\right]^2 \\nonumber \\\\\\!\\!\\!&=& \\!\\!\\frac{2}{5} \\int _0^1 dt \\,r(t) = \\frac{2}{5}\\int _0^{1/2} r(t) dt+ \\int _{1/2}^1 r(t) dt \\nonumber \\\\\\!\\!\\!&=&\\!\\!\\frac{2}{5} \\left(\\int _{\\infty }^{r_*} dr \\, r\\frac{dt_{-}}{dr}+\\int _{r_*}^{\\infty } dr \\, r\\frac{dt_{+}}{dr}\\right)\\nonumber \\\\\\!\\!\\!&=& \\!\\frac{(3 \\pi )^{2/3}}{5}.$ To determine $H_1$ , we can use Eq.",
"(REF ) at $x=0$ : $\\partial _{t}h(0,t)=\\partial _{x}^2 h(0,t) -\\frac{1}{2} \\left[\\partial _x h\\right(0,t)]^2+\\rho (0,t).$ As $\\partial _x h(0,t)=0$ (except at $t=0$ and $t=1$ ), and the diffusion is negligible, we obtain $\\partial _{t}h(0,t) \\simeq \\rho (0,t) = r(t),$ so $H_1= \\int _0^1 r(t) dt = \\frac{(3\\pi )^{2/3}}{2}.$ Now we plug $s$ and $H_1$ into Eq.",
"(REF ) and obtain the $H\\gg 1$ tail we are after.",
"In the physical units, $-\\ln \\mathcal {P}(H,T, L\\rightarrow 0) \\simeq \\frac{4\\sqrt{2 |\\lambda |}}{15 \\pi D}\\, \\frac{H^{5/2}}{T^{1/2}}.$ Equation (REF ) coincides with the asymptotic (5) of Ref.",
"[20], [44], extracted from the exact solution [10], [11], [12], [13], [4] at short times.",
"This leading-order asymptotic is controlled by the nonlinearity and independent of $\\nu $ .",
"It is twice as small as the corresponding result [23], [25] for $L\\rightarrow \\infty $ .",
"In the zero-pressure region $|x|>\\ell (t)$ the governing equation, $\\partial _t V + V\\partial _x V=0,$ describes the Hopf flow.",
"We will only consider $x>\\ell (t)$ ; the solution for $x<-\\ell (t)$ can be obtained from the symmetry $V(-x,t)=-V(x,t)$ .",
"The general solution of Eq.",
"(REF ) can be written as [45], [46] $x-Vt=F(V)\\,,$ where the arbitrary function $F(V)$ should be found from matching with the pressure-driven solution at $x=\\ell (t)$ .",
"The matching yields the equation $x-Vt=\\frac{3}{4r_*}-\\frac{V}{2}+\\frac{V}{\\pi }\\,\\arctan \\frac{V}{2\\sqrt{r_*}}$ which determines $V(x,t)$ in an implicit form.",
"Figure REF shows $V$ as a function of $x>0$ at different times.",
"Both the pressure-driven solution (REF ) and the Hopf solution (REF ) are shown.",
"Importantly, the Hopf solution complies with the large-$x$ asymptotic $V(x,t) \\simeq x/t$ , described by the inviscid limit of the deterministic solution (REF ) at $L\\rightarrow 0$ .",
"Notice the presence of the stagnation point at $r=r_*$ at $t\\ge 1/2$ .",
"We also calculated $h(x,t)$ in an implicit form, but we do not present these cumbersome formulas here.",
"Figure: The rescaled interface slope V(x,t)=∂ x h(x,t)V(x,t)=\\partial _x h(x,t), as described by the inviscid solution for H≫1H\\gg 1, is shown as a function of x>0x>0 at times t=0.1t=0.1 (a), 0.30.3 (b), 0.50.5 (c), 0.70.7 (d) and 0.90.9 (e) for L→0L\\rightarrow 0.",
"Both the “pressure-driven\" solution and the Hopf solution are shown.",
"A stagnation point V=0V=0 develops at r=r * r=r_* at t≥1/2t\\ge 1/2.",
"The interface height h(x,t)h(x,t) has a local minimum at this point at all times t≥1/2t\\ge 1/2.",
"The dashed line is the large-xx asymptotic V=x/tV=x/t at t=0.9t=0.9."
],
[
"$L>0$",
"In this case $a(t=0)=2/L$ , $a(t=1)=-\\infty $ , and $a(t=t_1)=0$ where $0<t_1<1$ is a priori unknown.",
"Let us denote $r(t_1)=r_1$ .",
"The first integral of the equations for $\\dot{a}$ and $\\dot{r}$ can be written as $a=\\pm (8/3) r \\sqrt{r-r_1}$ leading to $\\dot{r}=\\pm (8/3) r^2 \\sqrt{r-r_1}.$ An implicit solution of Eq.",
"(REF is $t=t_{\\pm }(r)= t_1\\pm \\frac{3}{8} \\left(\\frac{\\sqrt{r-r_1}}{r r_1}+\\frac{\\arctan \\sqrt{\\frac{r}{r_1}-1}}{r_1^{3/2}}\\right),$ where $r_0\\equiv r(t=0)$ is a priori unknown.",
"In Eqs.",
"(REF ) and (REF ) the minus signs correspond to $0<t<t_1$ , the plus signs to $t_1<t<1$ .",
"Let us evaluate the rescaled action: $\\!\\!\\!",
"s &=&\\!\\!\\frac{2}{5} \\int _0^1 dt \\,r(t) = \\frac{2}{5}\\int _0^{t_1} r(t) dt+ \\int _{t_1}^1 r(t) dt \\nonumber \\\\\\!\\!\\!&=&\\!\\!\\frac{2}{5} \\left(\\int _{r_0}^{r_1} dr \\, r\\frac{dt_{-}}{dr}+\\int _{r_1}^{\\infty } dr \\, r\\frac{dt_{+}}{dr}\\right) \\nonumber \\\\\\!\\!\\!&=&\\!\\!\\frac{3 \\left(\\pi +2 \\arccos \\sqrt{\\frac{r_1}{r_0}}\\right)}{20 \\sqrt{r_1}}.$ Also, $H_1 \\simeq \\int _0^1 r(t)\\,dt =\\frac{3 \\left(\\pi +2 \\arccos \\sqrt{\\frac{r_1}{r_0}}\\right)}{8 \\sqrt{r_1}}.$ The three unknown constants $r_0$ , $r_1$ and $t_1$ can be expressed via $L$ , the only parameter of the rescaled problem, with the help of three algebraic relations: $&&\\frac{8}{3} r_0 \\sqrt{r_0-r_1}= \\frac{2}{L}, \\\\&&t_1= \\frac{3}{8} \\left(\\frac{\\sqrt{r_0-r_1}}{r_0 r_1}+\\frac{\\arctan \\sqrt{\\frac{r_0}{r_1}-1}}{r_1^{3/2}}\\right), \\\\&& t_1+\\frac{3}{8}\\frac{\\pi }{2 r_1^{3/2}} =1.",
"$ The solution is unique and can be obtained in a parametric form.",
"Let us introduce the parameter $y=r_0/r_1$ that decreases monotonically from $\\infty $ to 1 as $L$ increases from 0 to $\\infty $ .",
"We can express $L, r_0, r_1$ and $t_1$ via $y$ as follows: $\\!\\!\\!\\!\\!\\!\\!\\!L&=&\\!\\!\\frac{4}{\\sqrt{y-1} \\left[\\pi y+2 \\sqrt{y-1}+2 y \\arctan \\left(\\sqrt{y-1}\\right)\\right]},\\\\\\!\\!\\!\\!\\!\\!\\!\\!r_0&=&\\!\\!\\frac{3^{2/3}}{4}y\\left(\\frac{\\pi }{2} +\\frac{\\sqrt{y-1}}{y}+\\arctan \\sqrt{y-1}\\right)^{2/3},\\\\\\!\\!\\!\\!\\!\\!\\!\\!r_1&=&\\!\\!",
"\\frac{3^{2/3}}{4}\\left(\\frac{\\pi }{2}+\\frac{\\sqrt{y-1}}{y}+\\arctan \\sqrt{y-1}\\right)^{2/3},\\\\\\!\\!\\!\\!\\!\\!\\!\\!t_1&=&\\!\\!2\\frac{\\frac{\\sqrt{y-1}}{y}+\\arctan \\sqrt{y-1}}{\\pi +2\\left(\\frac{\\sqrt{y-1}}{y}+\\arctan \\sqrt{y-1}\\right)}.$ Using these relations in conjunction with Eqs.",
"(REF ) and (REF ), and introducing $\\Phi (L)=s_1/H_1^{5/2}$ , we finally obtain, in physical units, $-\\ln \\mathcal {P}(H,T,L)\\simeq \\frac{4\\sqrt{2}\\,|\\lambda |^{1/2} H^{5/2}}{15 \\pi D T^{1/2}} \\Phi \\left(\\frac{L}{|\\lambda |T}\\right).$ Correspondingly, the factor $f_+(L,T)$ , mentioned in the Abstract and in the Introduction, is the following: $f_+=\\frac{4\\sqrt{2}\\,|\\lambda |^{1/2}}{15 \\pi D} \\Phi \\left(\\frac{L}{|\\lambda |T}\\right).$ A plot of the function $\\Phi =\\Phi (w)$ is shown in Fig.",
"REF .",
"Its small- and large-$w$ asymptotics are (w) 1+3 w1/324/3 2/3, $w\\ll 1$ , 2(1-42 w), $w\\gg 1$ , see Fig.",
"REF .",
"At $L\\rightarrow 0$ we obtain $\\Phi =1$ (the solid point on Fig.",
"REF ), in agreement with Eq.",
"(REF ) and Ref.",
"[20].",
"At $L\\rightarrow \\infty $ one has $\\Phi = 2$ (the horizontal dashed line) in agreement with Refs.",
"[23], [25].",
"Notice the non-analytic $w^{1/3}$ behavior of $f(w)$ at $w\\rightarrow 0$ .",
"Figure: Φ(w)\\Phi (w) from Eq.",
"() and its asymptotics, Eqs.",
"() and ().",
"The filled circle shows Φ(0)=1\\Phi (0)=1, the horizontal asymptotic shows Φ(∞)=2\\Phi (\\infty )=2.The Hopf flow regions $|x|>\\ell (t)$ for $L>0$ can be analyzed similarly to the case of $L\\rightarrow 0$ .",
"The Hopf-flow solution for $V(x,t)$ matches continuously with the pressure-driven solution at $|x|=\\ell (t)$ , complies with the deterministic behavior $V(x,t)=2x/(L+2t)$ at $|x|\\rightarrow \\infty $ , and exhibits, at $t\\ge t_1$ , two stagnation points $V=0$ at $x=\\pm r_1$ where $h(x,t)$ has a local minimum.",
"We do not show these cumbersome formulas here.",
"At very large negative $H$ , or $\\Lambda $ , the solution, at any $L$ , has the following character.",
"$\\rho $ is localized in a narrow boundary layer around $x=0$ and is almost independent of time except very close to $t=0$ and $t=1$ .",
"$V$ in the boundary layer is also almost independent of time.",
"There is also exterior, or bulk, region where $\\rho \\simeq 0$ , whereas $V(x,t)$ obeys the deterministic KPZ equation (REF )."
],
[
"The boundary layer",
"The stationary boundary-layer solution was previously found in the problem of flat interface [21], [25], see also Ref.",
"[38]: $\\rho _{\\text{bl}}(x) &=& -2 c\\,\\text{sech}^2 \\left( \\sqrt{c/2} \\,x \\right), \\\\V_{\\text{bl}}(x) &=& \\sqrt{2 c}\\,\\tanh \\left(\\sqrt{c/2} \\,x \\right), $ where $c=\\Lambda ^2/32$ .",
"The action in terms of $c$ or $\\Lambda $ is obtained immediately: $S =\\frac{1}{2} \\int _{-\\infty }^{\\infty } dx\\,\\rho _{\\text{bl}}^2(x) = \\frac{8\\sqrt{2}\\, c^{3/2}}{3}= -\\frac{\\Lambda ^3}{48},$ recall that $\\Lambda <0$ .",
"To express $c$ through $H$ , we need to rewrite the boundary-layer solution in terms of $h(x,t)$ [25], $h_{\\text{bl}}(x,t) = 2 \\ln \\cosh \\left( \\sqrt{c/2} \\,x \\right) -c t,$ and obtain $h_{\\text{bl}}(0,1) = -c =H$ which yields $c=-H$ and $\\Lambda = - 2^{5/2} |H|^{1/2}$ .",
"Using this result in Eqs.",
"(REF ) and (REF ), we obtain in the physical units $-\\ln \\mathcal {P} (H,T,L)\\simeq \\frac{8\\sqrt{2}\\,\\nu |H|^{3/2}}{3 D |\\lambda |^{1/2} T^{1/2}}.$ As one can see, the factor $f_- = \\frac{8\\sqrt{2}\\,\\nu }{3 D |\\lambda |^{1/2}}$ is independent of $L$ and $T$ .",
"It is not surprising, therefore, that the same expression (REF ) for the negative tail was previously obtained for $L\\rightarrow \\infty $ [21], [25] and $L\\rightarrow 0$ [20].",
"Interestingly, Eq.",
"(REF ) also coincides with the corresponding asymptotic of the GOE and GUE Tracy-Widom distributions, which describe the negative tail of ${\\mathcal {P}}(H,T)$ at long times, both for $L\\rightarrow \\infty $ [9], [21], [25] and for $L\\rightarrow 0$ [10], [11], [12], [13], [4]."
],
[
"The bulk region",
"Now we will show that the boundary-layer solution () can be properly matched with a deterministic bulk solution.",
"Not being interested in the structure of an additional narrow transition layer that emerges in the bulk solution (see below), we can neglect the diffusion term and, instead of Eq.",
"(REF ), deal with the inviscid equation $\\partial _t h +\\frac{1}{2}{} (\\partial _x h)^2=0,$ or the Hopf equation (REF ), where we allow for $V$ -shocks.",
"We will only consider $x>0$ : the solution for $x<0$ can obtained by a mirror reflection of $h(x,t)$ with respect to the origin.",
"The outer asymptotic of the boundary-layer solution (REF ) for $h(x,t)$ is $h_1(x,t) = \\sqrt{2c} \\,x - ct.$ Correspondingly, $V_1(x,t)=\\partial _x h_1(x,t)=\\sqrt{2c}=|\\Lambda |/4=\\text{const}$ .",
"Note that these asymptotics is independent of the diffusivity.",
"To satisfy the boundary conditions at $x \\rightarrow \\infty $ , $h_1(x,t)$ must be continuously matched with the inviscid limit of the deterministic solution (REF ), which holds at large distances, $h_2(x,t) \\simeq \\frac{x^2}{L+2t},$ and for which $V_2(x,t)\\simeq \\frac{2x}{L+2t}.$ At $L>0$ the equality $h_1(x,t)=h_2(x,t)$ is satisfied in two locations, $X_-(t)$ and $X_+(t)$ , where $X_{\\pm }(t) = \\sqrt{(c/2)\\left(L+t\\right)}\\,\\left(\\sqrt{L+t}\\pm \\sqrt{L}\\right) .$ While $h(x)$ is continuous in the matching points, $V(x)$ is generally not, so a shock appears.",
"$X_+(t)$ is inadmissible as a shock position, as it violates the condition $V_1[X(t),t] \\ge V_2[X(t),t]$ [46].",
"$X_-(t)$ does satisfy this condition, and so $V(x,t)$ exhibits a shock at this location.",
"The shock speed is equal to $V_{\\text{shock}}(t) = \\frac{dX_-}{dt} =\\sqrt{2 c}-\\sqrt{\\frac{cL}{2(L+2 t)}}.$ What happens in the limits of $L\\rightarrow \\infty $ and $L\\rightarrow 0$ ?",
"At $L\\rightarrow \\infty $ the deterministic solution at large distances is trivial: $h_2(x,t)=0$ .",
"Here the $V$ -shock is located at $X(t)=\\sqrt{c/2}\\,t$ and moves with a constant speed [25].",
"In the limit of $L\\rightarrow 0$ the two locations $X_-(t)$ and $X_+(t)$ merge.",
"In this special case $V(x)$ is continuous everywhere, and there is no shock.",
"There is only a discontinuity in the derivative $\\partial _x V$ at the moving point $X(t)= \\sqrt{2c}\\,t$ .",
"All the discontinuities, discussed here, are smoothed, and narrow transition layers appear, if one accounts for the diffusion."
],
[
"The variance",
"When $\\epsilon \\ll 1$ , low cumulants of ${\\mathcal {P}}$ can be calculated via a regular perturbation theory in $H$ , or in $\\Lambda $ , in the WNT framework [25], [47].",
"We set $\\!\\!\\!",
"h(x,t)&=& h_0(x,t)+\\Lambda h_1(x,t)+\\Lambda ^2 h_2(x,t) +\\dots ,\\\\\\!\\!\\!",
"\\rho (x,t) &=& \\Lambda \\rho _1(x,t)+\\Lambda ^2 \\rho _2(x,t) +\\dots .",
"$ where $h_0(x,t)$ is given by Eq.",
"(REF ).",
"Correspondingly, $S(\\Lambda )=\\Lambda ^2 S_1 +\\Lambda ^3 S_2 + \\dots $ .",
"Here we limit ourselves to the first order of this perturbation series which gives the distribution variance.",
"In the first order Eqs.",
"(REF ) and () yield $&& \\partial _t h_1 +\\partial _x h_0 \\,\\partial _x h_1 - \\partial _x^2 h_1 = \\rho _1, \\\\&& \\partial _t \\rho _1 +\\partial _x (\\partial _x h_0 \\,\\rho _1)+\\partial _x^2 \\rho _1= 0, $ or $&& \\partial _t h_1 +\\frac{2 x}{L+2t} \\,\\partial _x h_1 - \\partial _x^2 h_1=\\rho _1, \\\\&& \\partial _t \\rho _1 +\\partial _x \\left(\\frac{2 x}{L+2t} \\,\\rho _1\\right)+\\partial _x^2 \\rho _1=0.",
"$ In contrast to the flat case [25], [24], the KPZ nonlinearity kicks in already in the first order of the perturbation theory, so the variance of ${\\mathcal {P}}(H,T,L)$ is different from that for the Edwards-Wilkinson equation.",
"To solve Eqs.",
"(REF ) and (), we introduce new variables $z=\\frac{x}{L+2t},\\quad u(z,t) = (L+2t) \\rho _1.$ Equation () becomes $\\partial _t u+\\frac{\\partial _z^2 u}{(L+2t)^2}=0.$ Now we introduce new time, $\\tau =\\frac{t}{L(L+2t)},$ so that $t=\\tau L^2/(1-2 \\tau L)$ .",
"The new time grows monotonically on the interval $0\\le \\tau \\le \\tau _1$ , where $\\tau _1 = \\frac{1}{L(L+2)}$ corresponds to $t=1$ .",
"Equation (REF ) becomes the antidiffusion equation $\\partial _{\\tau } u +\\partial _{z}^2 u=0$ .",
"The boundary condition $\\rho _1(x,1)=\\delta (x)$ translates into $u(z,\\tau _1)=\\delta (z)$ , and the solution is $u(z,0\\le \\tau \\le \\tau _1)= \\frac{1}{\\sqrt{4 \\pi (\\tau _1-\\tau )}}\\,e^{-\\frac{z^2}{4(\\tau _1-\\tau )}},$ or $\\rho _1(x,t)=\\frac{e^{-\\frac{(L+2) x^2}{4 (1-t)(L+2 t)}}}{\\sqrt{\\frac{4\\pi \\,(1-t) (L+2t)}{L+2}}}$ As a result, $S_1(L) &=& \\frac{1}{2}\\int _0^1 dt \\int _{-\\infty }^{\\infty } dx\\,\\rho _1^2(x,t) \\nonumber \\\\&=& \\frac{\\sqrt{L+2} \\,\\arccos \\left(\\frac{L-2}{L+2}\\right)}{8 \\sqrt{\\pi }}, $ To express $\\Lambda $ via $H$ we need to solve Eq.",
"(REF ) for $h_1(x,t)$ with the initial condition $h_1(x,0)=0$ and the source term given by Eq.",
"(REF ).",
"It suffices to calculate $h_1(x=0,t=1)$ .",
"In the new variables Eq.",
"(REF ) becomes $\\partial _{\\tau } h = \\partial _z^2 h+\\frac{L u(z,\\tau )}{1-2\\tau L},$ with $u(z,\\tau )$ from Eq.",
"(REF ).",
"The solution can be obtained with the help of the Green's function of the diffusion equation.",
"As a result, $&& h_1(x=0, t=1)=h_1(z=0, \\tau =\\tau _1) \\nonumber \\\\&& =\\frac{1}{4\\pi }\\int _0^{\\tau _1} \\frac{L d\\tau }{(1-2L\\tau )(\\tau _1-\\tau )}\\,\\int _{-\\infty }^{\\infty } dz\\,e^{-\\frac{z^2}{2(\\tau _1-\\tau )}}\\nonumber \\\\&& =\\frac{\\sqrt{L+2}}{2\\sqrt{\\pi }}\\,\\arccos \\sqrt{\\frac{L}{L+2}}.$ Now we can express $\\Lambda $ through $H$ using the relation $\\Lambda \\,h_1(x=0,t=1) = H$ .",
"Finally, we obtain in the physical units $-\\ln {\\mathcal {P}}(H,T,L)\\simeq \\frac{\\nu ^{1/2}H^2}{D\\sqrt{T}}\\,\\phi \\left(\\frac{L}{|\\lambda |T}\\right),$ where $\\phi (w)=\\frac{\\sqrt{\\pi }\\,\\arccos \\left(\\frac{w-2}{w+2}\\right)}{2\\sqrt{w+2}\\,\\left(\\arccos \\sqrt{\\frac{w}{w+2}}\\right)^2}.$ The asymptotics of $\\phi (w)$ are the following: (w) 2(1+2w) , $w\\ll 1$ , 2(1-13w), $w\\gg 1$ , see Fig.",
"REF .",
"At $L\\rightarrow 0$ we obtain $\\phi =\\sqrt{2/\\pi }$ in agreement with Eq.",
"(6) of Ref.",
"[20], [44].",
"At $L\\rightarrow \\infty $ $\\phi =\\sqrt{\\pi /2}$ in agreement with Refs.",
"[24], [25].",
"Notice the non-analytic $w^{1/2}$ behavior of $\\Phi (w)$ at $w\\rightarrow 0$ .",
"Figure: φ(w)\\phi (w) from Eqs.",
"() and () and its asymptotics, Eqs.",
"() and ().",
"The filled circle shows φ(0)=2/π\\phi (0)=\\sqrt{2/\\pi }, the horizontal asymptotic shows φ(∞)=π/2\\phi (\\infty )=\\sqrt{\\pi /2}."
],
[
"Summary and Discussion",
"Let us briefly summarize our results for the probability distribution $\\mathcal {P}(H,t,L)$ of the surface height $h(x=0,t)=H$ in the KPZ equation in $1+1$ dimension when starting from a parabolic interface $h(x,t=0)=x^2/L$ .",
"At early times, $\\epsilon \\ll 1$ , the central part of the distribution is described by Eqs.",
"(REF ) and (REF ).",
"Although it is a Gaussian, it does not belong to the Edwards-Wilkinson universality class.",
"Indeed, the distribution variance explicitly depends on the nonlinearity coefficient $\\lambda $ and does not exhibit the customary $t^{1/4}$ scaling, see Eq.",
"(REF ).",
"The tails of $\\mathcal {P}(H,t,L)$ are described by Eqs.",
"(REF ) and (REF ): they are non-Gaussian and strongly asymmetric.",
"The asymmetry is manifested by very different optimal histories of the process conditioned on observing a large positive or negative value of $H$ at time $t$ .",
"As we observed, the positive tail (REF ) of ${\\mathcal {P}}(H,t,L)$ depends on $L$ monotonically, see Fig.",
"REF , interpolating between time-independent values at $L=0$ and $L=\\infty $ that were previously known.",
"On the contrary, the negative tail (REF ) of ${\\mathcal {P}}(H,t,L)$ is independent of $L$ , because it comes from the universal boundary-layer solution (REF ) and ().",
"We argue that exactly the same negative tail (REF ) should be observed for a whole class of deterministic initial conditions such that the boundary-layer solution () for $V(x,t)=\\partial _x h(x,t)$ can be matched with a (deterministic) bulk solution for $V(x,t)$ that satisfies correct boundary conditions at $|x|\\rightarrow \\infty $ .",
"An important role in this matching is played by $V$ -shocks (in the inviscid approximation) that, in general, develop inside the bulk region.",
"Are any of our early-time predictions, based on the WNT, expected to hold at long times?",
"(See Ref.",
"[25] for a similar discussion for the flat initial condition.)",
"At $\\epsilon \\gg 1$ , the WNT breaks down in the body of the height distribution, where the Gaussian distribution (REF ) and (REF ) must give way to a different distribution which reduces to the GUE Tracy-Widom statistics at $L\\rightarrow 0$ , and to the GOE Tracy-Widom statistics at $L\\rightarrow \\infty $ .",
"However, sufficiently far in the tails the action $S$ is very large.",
"Therefore, one can expect the WNT tails (REF ) and (REF ) to hold there.",
"Indeed, the universal $3/2$ tail agrees with the corresponding Tracy-Widom tail at $L\\rightarrow 0$ and $L\\rightarrow \\infty $ .",
"The $5/2$ tail is incompatible with the Tracy-Widom statistics.",
"We argue that it holds (see also Ref.",
"[25]) when it predicts a higher probability than the corresponding tail, $-\\ln \\mathcal {P}\\sim \\nu ^2 H^3/(|\\lambda |D^2 t)$ , of the Tracy-Widom distribution.",
"At fixed $t$ , and sufficiently far in the tail, $H\\gg D^2 |\\lambda |^3 t/\\nu ^4$ , this condition is satisfied.",
"It would be very interesting to test this prediction by extracting the $H\\gg D^2 |\\lambda |^3 t/\\nu ^4$ asymptotics of ${\\mathcal {P}}$ in the exactly soluble cases of $L\\rightarrow 0$ and $L\\rightarrow \\infty $ ."
],
[
"ACKNOWLEDGMENTS",
"A.K.",
"was supported by NSF grant DMR1306734.",
"B.M.",
"acknowledges financial support from the Israel Science Foundation (grant No.",
"807/16) and the United States-Israel Binational Science Foundation (BSF) (grant No.",
"2012145) and hospitality of the William I.",
"Fine Theoretical Physics Institute of the University of Minnesota.",
"For completeness, here we present a brief derivation of the WNT equations and boundary conditions.",
"Using Eq.",
"(1), we express the noise term as $\\sqrt{D}\\,\\xi (x,t)=\\partial _{t} h-\\nu \\partial _{x}^2 h-\\frac{\\lambda }{2} \\left(\\partial _{x} h\\right)^2.$ The corresponding Gaussian action is, therefore, $S=\\frac{1}{2}\\int _{0}^{T}dt\\int _{-\\infty }^{\\infty }dx \\left[\\partial _{t} h-\\nu \\partial _{x}^2 h-\\frac{\\lambda }{2} \\left(\\partial _{x} h\\right)^2\\right]^2.$ In the weak-noise limit the main contribution to the integral comes from the “optimal path\" $h(x,t)$ that minimizes $S$ .",
"The variation of $S$ $\\delta S&=& \\int _{0}^{T}dt\\int _{-L/2}^{L/2}dx\\left[\\partial _{t} h-\\nu \\partial _{x}^2 h-\\frac{\\lambda }{2} \\left(\\partial _{x} h\\right)^2\\right] \\nonumber \\\\&\\times & \\left(\\partial _t \\delta h -\\nu \\partial _x^2 \\delta h -\\lambda \\partial _x h \\,\\partial _x \\delta h\\right).$ By analogy with classical mechanics, one can introduce the “momentum density\" field $\\rho (x,t)=\\delta L/\\delta v$ , where $v\\equiv \\partial _t h$ , and $L\\lbrace h\\rbrace =\\frac{1}{2}\\int _{-\\infty }^{\\infty }dx \\left[\\partial _{t} h-\\nu \\partial _{x}^2 h-\\frac{\\lambda }{2} \\left(\\partial _{x} h\\right)^2\\right]^2$ is the Lagrangian.",
"In this way we obtain $\\partial _{t}h=\\nu \\partial _{x}^2 h +\\frac{\\lambda }{2} \\left(\\partial _x h\\right)^2+\\rho ,$ the first of the two Hamilton equations.",
"Rewriting the variation (REF ) as $\\delta S=\\int _{0}^{T}dt\\int _{-\\infty }^{\\infty }dx\\,\\rho \\,(\\partial _{t}\\delta h-\\nu \\partial _{x}^2\\delta h -\\lambda \\partial _x h \\,\\partial _x \\delta h),$ and integrating by parts, we arrive at the second Hamilton equation: $\\partial _{t}\\rho =-\\nu \\partial _{x}^2 \\rho +\\lambda \\partial _x \\left(\\rho \\partial _x h\\right).$ The boundary terms in $x$ , emerging in the integrations by parts, vanish because of the boundary conditions at $|x|\\rightarrow \\infty $ .",
"There also appear two boundary terms in time: at $t=0$ and $t=T$ .",
"The boundary term $\\int dx \\,\\rho (x,0) \\,\\delta h(x,0)$ vanishes because the height profile at $t=0$ is fixed by Eq.",
"(REF ).",
"The boundary term $\\int dx \\,\\rho (x,T) \\,\\delta h(x,T)$ must be also zero.",
"As we fixed $h(x=0,T)=H$ , we have $\\delta h(x=0,T) =0$ , but $\\rho (x=0,T)$ can be arbitrary.",
"On the contrary, $h(x\\ne 0,T)$ is not fixed, so we must have $\\rho (x\\ne 0,T)=0$ .",
"This leads to the boundary condition [21], [25] $\\rho (x,T)=\\Lambda \\,\\delta (x),$ where one introduces an unknown constant $\\Lambda $ which is ultimately set by the condition $h(x=0,T)=H$ .",
"Upon the rescaling $t/T\\rightarrow t$ , $x/\\sqrt{\\nu T} \\rightarrow x$ , $|\\lambda |h/\\nu \\rightarrow h$ and $|\\lambda |T\\rho /\\nu \\rightarrow \\rho $ , one arrives at Eqs.",
"(REF )-(REF ) of the main text, with rescaled $H$ and $\\Lambda $ , and Eq.",
"(REF ) with rescaled $L$ ."
]
] | 1605.06130 |
[
[
"Domain Theory: An Introduction"
],
[
"Abstract This monograph is an ongoing revision of \"Lectures On A Mathematical Theory of Computation\" by Dana Scott.",
"Scott's monograph uses a formulation of domains called neighborhood systems in which finite elements are selected subsets of a master set of objects called \"tokens\".",
"Since tokens have little intuitive significance, Scott has discarded neighborhood systems in favor of an equivalent formulation of domains called information systems.",
"Unfortunately, he has not rewritten his monograph to reflect this change.",
"We have rewritten Scott's monograph in terms of finitary bases instead of information systems.",
"A finitary basis is an information system that is closed under least upper bounds on finite consistent subsets.",
"This convention ensures that every finite answer is represented by a single basis object instead of a set of objects."
],
[
"Motivation",
"Computer programs perform computations by repeatedly applying primitive operations to data values.",
"The set of primitive operations and data values depends on the particular programming language.",
"Nearly all programming languages support a rich collection of data values including atomic objects, such as booleans, integers, characters, and floating point numbers, and composite objects, such as arrays, records, sequences, tuples, and infinite streams.",
"More advanced languages also support functions and procedures as data values.",
"To define the meaning of programs in a given language, we must first define the building blocks—the primitive data values and operations—from which computations in the language are constructed.",
"Domain theory is a comprehensive mathematical framework for defining the data values and primitive operations of a programming language.",
"A critical feature of domain theory (and expressive programming languages like Scheme, ML and Haskell) is the fact that program operations are also data values; in domain theory both operations and data values that the operations operate on are elements of computational domains.",
"In a language implementation, every data value and operation is represented by a finite configuration of symbols (e.g., a bitstring).",
"However, the choice of how data is represented should not affect the observable behavior of programs.",
"Otherwise, a programmer cannot reason about the behavior of programs independent of their implementations.",
"To achieve this goal, we must abandon finite representations for some data values.",
"The abstract meaning of a procedure, for example, is typically defined as a mathematical function from an infinite domain to an infinite codomain.",
"Although the graph of this function is recursively enumerable, it does not have an effective finite canonical representation; otherwise we could decide the equality of recursively enumerable sets by generating their canonical descriptions then comparing these descriptions.",
"Finite data values have canonical representations.",
"Data values that do not have finite canonical representations are called infinite data values.",
"Some common examples of infinite data values are functions over an infinite domain, infinite streams and infinite trees (sometimes also called “lazy\" lists and trees).",
"A data domain that contains infinite data values is called a higher order data domain.",
"To describe an infinite data value, we must use an infinite sequence of progressively better finite approximations, each obviously having a canonical representation.",
"We can interpret each finite approximation as a proposition asserting that a certain property is true of the approximated value.",
"By stating enough different properties (a countably infinite number of them in general), every infinite data value can be uniquely identified.",
"Higher order data domains can also contain ordinary finite data values.",
"As such, in a higher order data domain there are two separate kinds of finite values.",
"First, the finite elements used to approximate infinite values are legitimate data values themselves.",
"Even though these approximating elements are only “partially defined,” they can be produced as the final results of computations.",
"For example, a tree of the form $cons(\\alpha ,\\beta )$ , where $\\alpha $ and $\\beta $ are arbitrary data values, is a data value in its own right, because a computation yielding $cons(\\alpha ,\\beta )$ may subsequently diverge without producing any information about the values $\\alpha $ and $\\beta $ .",
"Second, higher order domains may contain “maximal” finite elements that do not approximate any other values.",
"These “maximal” values correspond to conventional finite data values.",
"For example, in the domain of potentially infinite binary trees of integers, the leaf carrying the integer 42 does not approximate any value other than itself.",
"In summary, a framework for defining computational data values and operations must accommodate infinite elements, partially-defined elements, and finite maximal elements.",
"In addition, the framework should support the construction of more complex values from simpler values, and it should support a notion of computation on these objects.",
"This monograph describes a framework—domain theory—satisfying all of these properties."
],
[
"Notation",
"The following notation is used throughout the monograph: Table: NO_CAPTION"
],
[
"Basic Definitions",
"To support the idea of describing data values by generating “better and better” approximations, we need to specify an (“information content\") ordering relation among the finite approximations to data values.",
"The following definitions describe the structure of the sets of finite approximations used to build domains; these sets of finite approximations are called finitary bases.",
"Definition 1.1: [Partial Order] A partial order B is a pair $\\langle B,\\sqsubseteq \\rangle $ consisting of $(i)$ a (non-empty) set $B$ , called the universe of B, and $(ii)$ a binary relation $\\sqsubseteq $ on the set $B$ , called the approximation ordering, that is reflexive: $\\forall x \\in B [x \\sqsubseteq x]$ , antisymmetric: $\\forall x,y \\in B [x \\sqsubseteq y] \\mbox{ and }[y \\sqsubseteq x] \\mbox{ implies } x = y$ , and transitive: $\\forall x,y,z \\in B [x \\sqsubseteq y] \\mbox{ and }[y \\sqsubseteq z] \\mbox{ implies } x \\sqsubseteq z$ .",
"Definition 1.2: [Upper Bounds, Lower Bounds, Consistency] Let $S$ be a subset of (the universe of) a partial order B.",
"An element $b \\in B$ is an upper bound of $S$ iff $\\forall s \\in S \\: s \\sqsubseteq b$ .",
"An element $b \\in B$ is a lower bound of $S$ iff $\\forall s \\in S \\: b \\sqsubseteq s$ .",
"$S$ is consistent (sometimes also called bounded) iff $S$ has an upper bound.",
"An upper bound $b$ of $S$ is the least upper bound of $S$ (denoted $\\bigsqcup S$ ) iff $b$ approximates all upper bounds of $S$ .",
"A lower bound $b$ of $S$ is the greatest lower bound of $S$ (denoted $\\mbox{\\Large $\\sqcap $}S$ ) iff every lower bound of $S$ approximates $b$ .",
"Remark 1.3: In domain theory upper bounds are much more important than lower bounds.",
"Definition 1.4: [Directed Set, Progressive Set, Chain] A subset $S$ of a partial order B is directed iff every finite subset of $S$ is consistent (i.e., has an upper bound) in $S$ .",
"A directed subset $S$ of ${\\bf B}$ is progressive iff $S$ does not contain a maximum element: $\\nexists b \\in S[\\forall s \\in S s\\sqsubseteq b]$ .",
"A directed subset $S$ of ${\\bf B}$ is a chain iff $S$ is totally ordered: $\\forall a,b \\in S a \\sqsubseteq b$ or $b \\sqsubseteq a$ .",
"Claim 1.5: The empty set is directed.",
"Definition 1.6: [Complete Partial Order] A complete partial order, abbreviated cpo (or sometimes dcpo), is a partial order $\\langle B,\\sqsubseteq \\rangle $ such that every directed subset has a least upper bound in $B$ .",
"Claim 1.7: A cpo has a least element.",
"Finitary Bases Definition 1.8: [Finitary Basis] A finitary basis B is a partial order $\\langle B,\\sqsubseteq \\rangle $ such that $B$ is countable and every finite consistent subset has a least upper bound in $B$ .",
"We call the elements of a finitary basis ${\\bf B}$ propositions since they can be interpreted as logical assertions about domain elements.",
"In propositional logic, the least upper bound of a set of propositions is the conjunction of all the propositions in the set.",
"Since the empty set, $\\emptyset $ , is a finite consistent subset of ${\\bf B}$ , it has a least upper bound, which is denoted $\\bot $ .",
"The $\\bot $ proposition is true for all domain elements; hence it does not give any “information” about an element.",
"Example 1.9: Let $B=\\lbrace \\bot ,0\\bot ,1\\bot , 00,01,10,11\\rbrace $ where $0\\bot $ describes strings that start with 0 and are indeterminate past that point; 00 describes the string consisting of two consecutive 0's.",
"The other propositions are defined similarly.",
"Let $\\sqsubseteq $ denote the implication (or, conversely, the approximation) relation between propositions.",
"Thus, $0\\bot \\sqsubseteq 00$ and $0\\bot \\sqsubseteq 01$ .",
"In pictorial form, the partial order $\\langle B,\\sqsubseteq \\rangle $ looks like: 000110110$\\bot $1$\\bot $$\\bot $$\\langle B,\\sqsubseteq \\rangle $ is clearly a partial order.",
"To show that $\\langle B,\\sqsubseteq \\rangle $ is a finitary basis, we must show that $B$ is countable and that all finite bounded (i.e., consistent) subsets of $B$ have least upper bounds.",
"Since $B$ is finite, it is obviously countable.",
"It is easy to confirm that every finite consistent subset has a least upper bound (by inspection).",
"In fact, the least upper bound of any consistent subset $S$ of $B$ is simply the greatest element of $S$ .This is a special property of this particular partial order.",
"It is not true of finitary bases in general.",
"Thus, $\\langle B,\\sqsubseteq \\rangle $ is a finitary basis.",
"$\\:\\:\\Box $ Example 1.10: Let $B=\\lbrace (n,m) \\: \\:\\vert \\:\\:n,m\\in {{N}}\\cup \\lbrace \\infty \\rbrace , n \\le m \\rbrace $ where the proposition $(n,m)$ represents an integer $x$ such that $n \\le x\\le m$ .",
"$\\bot $ in this example is the proposition $(0,\\infty )$ .",
"Let $\\sqsubseteq $ be defined as $(n,m) \\sqsubseteq (j,k) \\:\\:\\iff \\:\\: n \\le j ~{\\rm and}~ k \\le m$ For example, $(1,10) \\sqsubseteq (2,6)$ but $(2,6)$ and $(7,12)$ are incomparable, as are $(2,6)$ and $(4,8)$ .",
"It is easy to confirm that $B$ is countable and that $\\langle B,\\sqsubseteq \\rangle $ is a partial order.",
"A subset $S$ of $B$ is consistent if there is an integer for which the proposition in $S$ is true.",
"Thus, $(2,6)$ and $(4,8)$ are consistent since either 4, 5, or 6 could be represented by these propositions.",
"The least upper bound of these elements is $(4,6)$ .",
"In general, for a consistent subset $S = \\lbrace (n_i,m_i) \\: \\:\\vert \\:\\:i \\in I\\rbrace $ of $B$ , the least upper bound of $S$ is defined as $\\bigsqcup S = ({\\sf max}\\: \\lbrace n_i \\: \\:\\vert \\:\\:i \\in I\\rbrace , {\\sf min}\\:\\lbrace m_i \\: \\:\\vert \\:\\:i \\in I\\rbrace )\\;.$ Therefore, $\\langle B,\\sqsubseteq \\rangle $ is a finitary basis.",
"$\\:\\:\\Box $ Given a finitary basis ${\\bf B}$ , the corresponding domain ${\\cal D}_{\\bf B}$ (also called ${\\cal B}$ ) is constructed by forming all consistent subsets of ${\\bf B}$ that are “closed” under implication and finite conjunction (where $a \\sqsubseteq b$ corresponds to $b \\:\\Rightarrow \\:a$ and $a \\sqcup b$ then corresponds to $a \\wedge b$ ).",
"More precisely, a consistent subset $S \\subseteq {\\bf B}$ is an element of the corresponding domain ${\\cal D}_{\\bf B}$ iff $\\forall s \\in S \\;\\; \\forall b \\in {\\bf B} \\;\\; b \\sqsubseteq s \\:\\Rightarrow \\:b \\in S$ , and $\\forall r,s \\in S \\;\\; r \\sqcup s \\in S$ .",
"Corresponding to each basis element/proposition $p \\in B$ there is a unique element ${\\cal I}_p$ = ${\\lbrace {b \\in B \\: \\:\\vert \\:\\:b \\sqsubseteq p}\\rbrace }\\in {\\cal D}_{\\bf B}$ .",
"In addition, ${\\cal D}_{\\bf B}$ contains elements (“closed” subsets of $B$ ) corresponding to the “limits” of all progressive directed subsets of ${\\bf B}$ .",
"This construction “completes” the finitary basis ${\\bf B}$ by adding limit elements for all progressive directed subsets of $B$ .",
"In ${\\cal D}_{\\bf B}$ , every element $d$ is represented by the set of all the propositions in the finitary basis ${\\bf B}$ that describe (i.e., approximate) $d$ .",
"These sets are called ideals.",
"Definition 1.11: [Ideal] For finitary basis ${\\bf B}$ , a subset ${\\cal I}$ of $B$ is an ideal over ${\\bf B}$ iff ${\\cal I}$ is downward-closed (implication): $e\\in {{\\cal I}} \\:\\Rightarrow \\:(\\forall b\\in B \\; b\\sqsubseteq e \\:\\Rightarrow \\:b\\in {{\\cal I}})$ ${\\cal I}$ is closed under least upper bounds on finite subsets (finite conjunction): $\\forall r,s \\in I \\;\\; r \\sqcup s \\in {\\cal I}$ .Using induction, it is easy to prove that closure under lubs of pairs implies closure under lubs of finite sets, and trivially vice versa.",
"Domains Now, we construct domains as partially ordered sets of ideals.",
"Definition 1.12: [Constructed Domain] Let ${\\bf B}$ be a finitary basis.",
"The domain ${\\cal D}_{\\bf B}$ determined by ${\\bf B}$ is the partial order $\\langle D,\\sqsubseteq _D\\rangle $ where $D$ , the universe of ${\\cal D}_{\\bf B}$ , is the set of all ideals ${\\cal I}$ over B, and $\\sqsubseteq _D$ is the subset relation.",
"We will frequently write ${\\cal D}$ or ${\\cal B}$ instead of ${\\cal D}_{\\bf B}$ .",
"The proof of the following two claims are easy; they are left to the reader.",
"Claim 1.13: The least upper bound of two ideals ${\\cal I}_1$ and ${\\cal I}_2$ , closing over ${\\cal I}_1$ and ${\\cal I}_2$ , if it exists, is found by unioning ${\\cal I}_1$ and ${\\cal I}_2$ to form an ideal ${\\cal I}_1 \\cup {\\cal I}_2$ over ${\\bf B}$ .",
"Claim 1.14: The domain ${\\cal D}$ determined by a finitary basis ${\\bf B}$ is a complete partial order.",
"Each proposition $b$ in a finitary basis $B$ determines an ideal ${\\cal I}_b$ consisting of the set of propositions implied by $b$ .",
"An ideal of this form called a principal ideal of $B$ .",
"Definition 1.15: [Principal Ideals] For finitary basis ${\\bf B}=\\langle B,\\sqsubseteq \\rangle $ , the principal ideal determined by $b\\in B$ , is the ideal ${{\\cal I}_b}=\\lbrace b^{\\prime }\\in B \\: \\:\\vert \\:\\:b^{\\prime }\\sqsubseteq b\\rbrace \\;.$ We will use the notation ${\\cal I}_b$ to denote the principal ideal determined by an element $b$ throughout this monograph.",
"Since there is a natural one-to-one correspondence between the propositions of a finitary basis ${\\bf B}$ and the principal ideals over ${\\bf B}$ , the following theorem obviously holds.",
"Theorem 1.16: The principal ideals over a finitary basis ${\\bf B}$ form a finitary basis under the subset ordering.",
"Proof This is an instance of the universality of the subset relation over partial orders.$\\:\\:\\Box $ Within the domain ${\\cal D}$ determined by a finitary basis ${\\bf B}$ , the principal ideals are characterized by an important topological property called finiteness.",
"Definition 1.17: [Finite Elements] An element $e$ of a cpo ${\\cal D}=\\langle D,\\sqsubseteq \\rangle $ is finite iff for every directed subset $S$ of $D$ , $e =\\bigsqcup S \\:\\Rightarrow \\:e \\in S$ .This property is weaker in general than the corresponding property (called isolated or compact) that is widely used in topology.",
"In the context of cpos, the two properties are equivalent.",
"The set of finite elements in a cpo ${\\cal D}$ is denoted ${\\cal D}^0$ .",
"The proof of the following theorem is left to the reader.",
"Theorem 1.18: An element of the domain ${\\cal D}$ of ideals determined by a finitary basis ${\\bf B}$ is finite iff it is principal.",
"In ${\\cal D}$ , the principal ideal determined by the least proposition $\\bot $ is the set $\\lbrace \\bot \\rbrace $ .",
"This ideal is the least element in the domain (viewed as a cpo).",
"In contexts where there is no confusion, we will abuse notation and denote this ideal by the symbol $\\bot $ instead of ${\\cal I}_\\bot $ .",
"The next theorem identifies the relationship between an ideal and all the principal ideals that approximate it.",
"Theorem 1.19: Let ${\\cal D}$ be the domain determined by a finitary basis ${\\bf B}$ .",
"For any ${\\cal I}\\in {\\cal D}$ , ${\\cal I}=\\bigsqcup \\:\\lbrace {\\cal I}^{\\prime } \\in {\\cal D}^0 \\: \\:\\vert \\:\\:{\\cal I}^{\\prime } \\sqsubseteq {\\cal I}\\rbrace \\, .$ Proof See Exercise 9.",
"$\\:\\:\\Box $ The approximation ordering in a partial order allows us to differentiate partial elements from total elements.",
"Definition 1.20: [Partial and Total Elements] Let ${\\bf B}=\\langle B,\\sqsubseteq \\rangle $ be a partial order.",
"An element $b \\in B$ is partial iff there exists an element $b^{\\prime }\\in B$ such that $b\\ne b^{\\prime }$ and $b\\sqsubseteq b^{\\prime }$ (sometimes written $b \\sqsubset b^{\\prime }$ ).",
"An element $b\\in B$ is total iff for all $b^{\\prime }\\in B$ , $b\\sqsubseteq b^{\\prime }$ implies $b=b^{\\prime }\\, .$ Example 1.21: The domain determined by the finitary basis defined in Example REF consists only of elements for each proposition in the basis.",
"The four total elements are the principal ideals for the propositions $00,01,10$ , and 11.",
"In general, a finite basis determines a domain with this property (all ideals are principal ideals).",
"$\\:\\:\\Box $ Example 1.22: The domain determined by the basis defined in Example REF contains total elements for each of the natural numbers.",
"These elements are the principal ideals for propositions of the form $(n,n)$ .",
"In this case as well, there are no ideals formed that are not principal.",
"$\\:\\:\\Box $ Example 1.23: Let $\\Sigma = \\lbrace 0,1\\rbrace $ , and let $\\Sigma ^\\star $ be the set of all finite strings over $\\Sigma $ with $\\epsilon $ denoting the empty string.",
"$\\Sigma ^\\star $ forms a finitary basis under the prefix ordering on strings.",
"$\\epsilon $ is the least element in $\\Sigma ^\\star $ .",
"The domain ${\\cal S}$ determined by $\\Sigma ^\\star $ contains principal ideals for all the finite bitstrings.",
"In addition, ${\\cal S}$ contains nonprincipal ideals corresponding to all infinite bitstrings.",
"Given any infinite bitstring $s$ , the corresponding ideal ${\\cal I}_s$ is the set of all finite prefixes of $s$ .",
"In fact, these prefixes form a chain.Nonprincipal ideals in other domains are not necessarily chains.",
"Strings are a special case because finite elements can “grow” in only one direction.",
"In contrast, the ideals corresponding to infinite trees—other than vines—are not chains.",
"$\\:\\:\\Box $ If we view cpos abstractly, the names we associate with particular elements in the universe are unimportant.",
"Consequently, we introduce the notion of isomorphism: two domains are isomorphic iff they have exactly the same structure.",
"Definition 1.24: [Isomorphic Partial Orders] Two partial orders ${\\bf A}$ and ${\\bf B}$ are isomorphic, denoted ${\\bf A}\\approx {\\bf B}$ , iff there exists a one-to-one onto function $m : A \\rightarrow B$ that preserves the approximation ordering: $ \\forall a,b\\in A \\; a\\sqsubseteq _{\\bf A}b \\iff m(a)\\sqsubseteq _{\\bf B}m(b)\\,.",
"$ Theorem 1.25: Let ${\\cal D}$ be the domain determined by a finitary basis ${\\bf B}$ .",
"${\\cal D}^0$ forms a finitary basis ${\\bf B^{\\prime }}$ under the approximation ordering $\\sqsubseteq $ (restricted to ${\\cal D}^0$ ).",
"Moreover, the domain ${\\cal E}$ determined by the finitary basis ${\\bf B^{\\prime }}$ is isomorphic to ${\\cal D}$ .",
"Proof Since the finite elements of ${\\cal D}$ are precisely the principal ideals, it is easy to see that ${\\bf B^{\\prime }}$ is isomorphic to ${\\bf B}$ .",
"Hence, ${\\bf B^{\\prime }}$ is a finitary basis and ${\\cal E}$ is isomorphic to ${\\cal D}$ .",
"The isomorphism between ${\\cal D}$ and ${\\cal E}$ is given by the mapping $\\delta :{\\cal D}\\rightarrow {\\cal E}$ is defined by the equation $\\delta (d)=\\lbrace e\\in {\\cal D}^0 \\: \\:\\vert \\:\\:e\\sqsubseteq d\\rbrace \\, .$ $\\:\\:\\Box $ The preceding theorem justifies the following comprehensive definition for domains.",
"Definition 1.26: [Domain] A cpo ${\\cal D}=\\langle D,\\sqsubseteq \\rangle $ is a domain iff ${\\cal D}^0$ forms a finitary basis under the approximation ordering $\\sqsubseteq $ restricted to ${\\cal D}^0$ , and ${\\cal D}$ is isomorphic to the domain ${\\cal E}$ determined by ${\\cal D}^0$ .",
"In other words, a domain is a partial order that is isomorphic to a constructed domain.",
"To conclude this section, we state some closure properties on ${\\cal D}$ to provide more intuition about the approximation ordering.",
"Theorem 1.27: Let ${\\cal D}$ be the domain determined by a finitary basis ${\\bf B}$ .",
"For any subset $S$ of ${\\cal D}$ , the following properties hold: $\\bigcap S \\in {\\cal D}$ and $\\bigcap S = \\mbox{\\Large $\\sqcap $}S\\,.$ if $S$ is directed, then $\\bigcup S \\in {\\cal D}$ and $\\bigcup S = \\bigsqcup S\\,.$ Proof The conditions for ideals specified in Definition REF must be satisfied for these properties to hold.",
"The intersection case is trivial.",
"The union case requires the directedness restriction since ideals require closure under lubs.",
"$\\:\\:\\Box $ For the remainder of this monograph, we will ignore the distinction between principal ideals and the corresponding elements (i.e., propositions) of the finitary basis whenever it is convenient.",
"Exercises Exercise 1.28: Let $B=\\lbrace s_n\\: \\:\\vert \\:\\:s_n=\\lbrace m\\in {{N}}\\: \\:\\vert \\:\\:m\\ge n\\rbrace ,~ n\\in {{N}}\\rbrace $ What is the approximation ordering for $B$ ?",
"Verify that $B$ is a finitary basis.",
"What are the total elements of the domain determined by $B$ ?",
"Draw a partial picture demonstrating the approximation ordering in the basis.",
"Exercise 1.29: Example REF can be generalized to allow strings of any finite length.",
"Give the finitary basis for this general case.",
"What is the approximation ordering?",
"What does the domain look like?",
"What are the total elements in the domain?",
"Draw a partial picture of the approximation ordering.",
"Exercise 1.30: Let $B$ be all finite subsets of ${{N}}$ with the subset relation as the approximation relation.",
"Verify that this is a finitary basis.",
"What is the domain determined by $B$ ?",
"What are the total elements?",
"Draw a partial picture of the domain.",
"Exercise 1.31: Construct two non-isomorphic infinite domains in which all elements are finite but there is no infinite chain of elements (i.e., no sequence $\\langle x_n\\rangle ^\\infty _{n=0}$ with $x_n \\sqsubset x_{n+1}$ —i.e., $x_n \\sqsubseteq x_{n+1}$ but $x_n \\ne x_{n+1}$ — for all $n$ ).",
"Exercise 1.32: Let $B$ be the set of all non-empty open intervals on the real line with rational endpoints plus a “bottom” element.",
"What would a reasonable approximation ordering be?",
"Verify that $B$ is a finitary basis.",
"For any real number $r$ , show that $\\lbrace i\\in B \\: \\:\\vert \\:\\:r\\in i\\rbrace \\cup \\lbrace \\bot \\rbrace $ is an ideal element.",
"Is it a total element?",
"What are the total elements?",
"(Hint: When $r$ is rational consider all intervals with $r$ as a right-hand end point.)",
"Exercise 1.33: Let ${\\bf D}$ be a finitary basis for domain ${\\cal D}$ .",
"Define a new basis ${\\bf D}^{\\prime }=\\lbrace \\downarrow X\\: \\:\\vert \\:\\:X\\in {\\bf D}\\rbrace $ where $\\downarrow X=\\lbrace Y\\in {\\bf D}\\: \\:\\vert \\:\\:X\\sqsubseteq Y\\rbrace $ .",
"Show that ${\\bf D}^{\\prime }$ is a finitary basis and that ${\\bf D}$ and ${\\bf D^{\\prime }}$ are isomorphic.",
"Exercise 1.34: Let $\\langle B,\\sqsubseteq \\rangle $ be a finitary basis where $B=\\lbrace X_0,X_1,\\ldots ,X_n,\\ldots \\rbrace .$ Suppose that consistency of finite sequences of elements is decidable.",
"Let $Y_0&=&X_0\\\\Y_{n+1}&=&\\left\\lbrace \\begin{array}{ll}X_{n+1} &\\mbox{if $X_{n+1}$ is consistent with $Y_0,Y_1,\\ldots ,Y_n$}\\\\Y_n & \\mbox{otherwise}\\,.\\end{array}\\right.$ Show that $\\lbrace Y_0,\\ldots ,Y_n,\\ldots \\rbrace $ is a total element in the domain determined by $B$ .",
"(Hint: Show that $Y_0,\\ldots ,Y_{n-1}$ is consistent for all $n$ .)",
"Show that all ideals can be determined by such sequences.",
"Exercise 1.35: Devise a finitary basis ${\\bf B}$ with more than two elements such that every pair of elements in $B$ is consistent, but $B$ is not consistent.",
"Exercise 1.36: Prove Theorem REF .",
"Operations on Data Motivation Since program operations perform computations incrementally on data values that correspond to ideals (sets of approximations), operations must obey some critical topological constraints.",
"For any approximation $x^{\\prime }$ to the input value $x$ , a program operation $f$ must produce the output $f(x^{\\prime })$ .",
"Since program output cannot be withdrawn, every program operation $f$ is a monotonic function: $x_1 \\sqsubseteq x_2$ implies $f(x_1) \\sqsubseteq (x_2)$ .",
"We can describe this process in more detail by examining the structure of computations.",
"Recall that every value in a domain ${\\cal D}$ can be interpreted as a set of finite elements in ${\\cal D}$ that is closed under implication.",
"When an operation $f$ is applied to the input value $x$ , $f$ gathers information about $x$ by asking the program computing $x$ to generate a countable chain of finite elements $C$ where $\\bigsqcup \\: \\lbrace {\\cal I}_c \\: \\:\\vert \\:\\: c \\in C\\rbrace = x$ .",
"For the sake of simplicity, we can force the chain $C$ describing the input value $x$ to be infinite: if $C$ is finite, we can convert it to an equivalent infinite chain by repeating the last element.",
"Then we can view operation $f$ as a function on infinite streams that repeatedly “reads” the next element in an infinite chain $C$ and “writes” the next element in an infinite chain $C^{\\prime }$ where $\\bigsqcup \\: \\lbrace {\\cal I}_{c^{\\prime }} \\: \\:\\vert \\:\\: c^{\\prime } \\in C^{\\prime }\\rbrace = f(x)$ .",
"Any such function $f$ on ${\\cal D}$ is clearly monotonic.",
"In addition, $f$ obeys the stronger property that for any directed set $S$ , $f(\\bigsqcup S) = \\bigsqcup \\: \\lbrace f(s) \\: \\:\\vert \\:\\: s \\in S\\rbrace $ .",
"This property is called continuity.For the computational motivation behind continuity, the interested reader is referred to Stoy's detailed and highly-readable account in [5], in particular the derivation of Condition 6.39 on page 99 of [5], and the following discussion of its implications.",
"The formulation of computable operations as functions on streams of finite elements is concrete and intuitive, but it is not canonical.",
"Due to the elements of domains in general not being totally-ordered but partially-ordered, there are many different functions on streams of finite elements corresponding to the same continuous function $f$ over a domain ${\\cal D}$ .",
"For this reason, we will use a slightly different model of incremental computation as the basis for defining the form of operations on domains.",
"To produce a canonical representation for computable operations, we represent values as ideals rather than chains of finite elements.",
"In addition, we allow computations to be performed in parallel, producing finite answers incrementally in non-deterministic order.",
"It is important to emphasize that the result of every computation—an ideal ${\\cal I}$ —is still deterministic; only the order in which the elements of ${\\cal I}$ are enumerated is non-deterministic.",
"When an operation $f$ is applied to an input value $x$ , $f$ gathers information about $x$ by asking for the enumeration of the ideal of finite elements $I_x =\\lbrace d \\in {\\cal D}^0 \\: \\:\\vert \\:\\: d \\sqsubseteq x\\rbrace $ .",
"In response to each input approximation $d \\sqsubseteq x$ , $f$ enumerates the ideal $I_{f(d)} =\\lbrace e \\in {\\cal D}^0 \\: \\:\\vert \\:\\: e \\sqsubseteq f(d)\\rbrace $ .",
"Since $I_{f(d)}$ may be infinite, each enumeration must be an independent computation.",
"The operation $f$ merges all of these enumerations yielding an enumeration of the ideal $I_{f(x)} =\\lbrace e \\in {\\cal D}^0 \\: \\:\\vert \\:\\: e \\sqsubseteq f(x)\\rbrace $ .",
"A computable operation $f$ mapping domain ${\\cal A}$ , with basis ${\\bf A}$ , into domain ${\\cal B}$ , with basis ${\\bf B}$ , can be formalized as a consistent relation $F \\subseteq {\\bf A} \\times {\\bf B}$ (a subset of the Cartesian product of the two basis) such that the image $F(a) = {\\lbrace {b \\in {\\bf B} | a \\, F \\, b}\\rbrace }$ of any input element $a \\in {\\bf A}$ is an ideal, and $F$ is monotonic: $a \\sqsubseteq a^{\\prime } \\:\\Rightarrow \\:F(a) \\subseteq F(a^{\\prime })$ .",
"These closure properties ensure that the relation $F$ uniquely identifies a continuous function $f$ on $D$ .",
"Relations (i.e., subsets of ${\\bf A} \\times {\\bf B}$ ) satisfying these closure properties are called approximable mappings.",
"Approximable Mappings and Continuous Functions The following set of definitions restates the preceding descriptions in more rigorous form.",
"Definition 2.1: [Approximable Mapping] Let ${\\cal A}$ and ${\\cal B}$ be the domains determined by finitary bases ${\\bf A}$ and ${\\bf B}$ , respectively.",
"An approximable mapping $F \\subseteq {\\bf A} \\times {\\bf B}$ is a binary relation over ${\\bf A} \\times {\\bf B}$ such that $\\bot _A\\,F\\,\\bot _B$ If $a\\,F\\,b$ and $a\\,F\\,b^{\\prime }$ then $a\\,F\\,(b \\sqcup b^{\\prime })$ If $a\\,F\\,b$ and $b^{\\prime }\\sqsubseteq _B b$ , then $a\\,F\\,b^{\\prime }$ If $a\\,F\\,b$ and $a\\sqsubseteq _A a^{\\prime }$ , then $a^{\\prime }\\,F\\,b$ The partial order of approximable mappings $F \\subseteq {\\bf A} \\times {\\bf B}$ under the subset relation is denoted by the expression ${\\sf {Map}}({\\bf A},{\\bf B})$ .",
"Conditions 1, 2, and 3 force the image of an input ideal to be an ideal.",
"Condition 4 states that the function on ideals associated with $F$ is monotonic.",
"Definition 2.2: [Continuous Function] Let ${\\cal A}$ and ${\\cal B}$ be the domains determined by finitary bases ${\\bf A}$ and ${\\bf B}$ , respectively.",
"A function $f:{\\cal A} \\rightarrow {\\cal B}$ is continuous iff for any ideal ${\\cal I}$ in ${\\cal A}$ , $f({\\cal I}) = \\bigsqcup \\: \\lbrace f({\\cal I}_a) \\: \\:\\vert \\:\\: a \\in {\\cal I}\\rbrace $ .",
"The partial ordering $\\sqsubseteq _B$ from ${\\cal B}$ determines a partial ordering $\\sqsubseteq $ on continuous functions: $ f \\sqsubseteq g \\iff \\forall x \\in {\\cal A} \\; f(x) \\sqsubseteq _{{\\cal B}}g(x) \\,.$ The partial order consisting of the continuous functions from ${\\cal A}$ to ${\\cal B}$ under the pointwise ordering is denoted by the expression ${\\cal A} \\rightarrow _c {\\cal B}$ (or, sometimes, by the expression ${\\sf {Fun}}({\\cal A},{\\cal B})$ ).",
"It is easy to show that continuous functions satisfy a stronger condition than the definition given above.",
"Theorem 2.3: If a function $f:{\\cal A} \\rightarrow {\\cal B}$ is continuous, then for every directed subset $S$ of ${\\cal A}$ , $f(\\bigsqcup S) =\\bigsqcup \\: \\lbrace f({\\cal I}) \\: \\:\\vert \\:\\: {\\cal I}\\in S\\rbrace $ .",
"Proof By Theorem REF , $\\bigsqcup S$ is simply $\\bigcup S$ .",
"Since $f$ is continuous, $f(\\bigsqcup S) = \\bigsqcup \\: \\lbrace f({\\cal I}_a) \\: \\:\\vert \\:\\: \\exists {\\cal I}\\in S \\; a \\in {\\cal I}\\rbrace $ .",
"Similarly, for every ${\\cal I}\\in {\\cal A}$ , $f({\\cal I}) = \\bigsqcup \\: \\lbrace f({\\cal I}_a) \\: \\:\\vert \\:\\: a \\in {\\cal I}\\rbrace $ .",
"Hence, $\\bigsqcup \\: \\lbrace f({\\cal I}) \\: \\:\\vert \\:\\: {\\cal I}\\in S\\rbrace =\\bigsqcup \\: \\lbrace \\bigsqcup \\: \\lbrace f({\\cal I}_a) \\: \\:\\vert \\:\\: a \\in {\\cal I}\\rbrace \\: \\:\\vert \\:\\: {\\cal I}\\in S\\rbrace =\\bigsqcup \\: \\lbrace f({\\cal I}_a) \\: \\:\\vert \\:\\: \\exists {\\cal I}\\in S \\; a \\in {\\cal I}\\rbrace $ .",
"$\\:\\:\\Box $ Every approximable mapping $F$ over the finitary basis ${\\bf A} \\times {\\bf B}$ determines a continuous function $f: {\\cal A} \\rightarrow {\\cal B}$ .",
"Similarly, every continuous function $f: {\\cal A} \\rightarrow {\\cal B}$ determines an approximable mapping $F$ over the finitary basis ${\\bf A} \\times {\\bf B}$ .",
"Definition 2.4: [Image of Approximable Mapping] For approximable mapping $F \\subseteq {\\bf A} \\times {\\bf B}$ the image of $d\\in {\\cal A}$ under $F$ (denoted ${\\sf apply}(F,d)$ ) is the ideal $\\lbrace b\\in {\\bf B} \\: \\:\\vert \\:\\: \\exists a\\in {\\bf A},\\: a\\in d \\: \\wedge a\\,F\\,b\\rbrace \\,.$ The function $f:{\\cal A}\\rightarrow {\\cal B}$ determined by $F$ is defined by the equation: $f(d) = {\\sf apply}(F,d)\\,.$ Remark 2.5: It is easy to confirm that ${\\sf apply}(F,d)$ is an element of ${\\cal B}$ and that the function $f$ is continuous.",
"Given any ideal $d \\in {\\cal A}$ , ${\\sf apply}(F,d)$ is the subset of ${\\bf B}$ consisting of all the elements related by $F$ to finite elements in $d$ .",
"The set ${\\sf apply}(F,d)$ is an ideal in ${\\cal B}$ since $(i)$ the set $\\lbrace b \\in {\\bf B} \\: \\:\\vert \\:\\: a \\, F \\, b\\rbrace $ is downward-closed for all $a \\in {\\bf A}$ , and $(ii)$ $a \\, F \\, b \\wedge a \\, F \\, b^{\\prime }$ implies $a \\, F \\, (b \\sqcup b^{\\prime })$ .",
"The continuity of $f$ is an immediate consequence of the definition of $f$ and the definition of continuity.",
"The following theorem establishes that the partial order of approximable mappings over ${\\bf A} \\times {\\bf B}$ is isomorphic to the partial order of continuous functions in ${\\cal A}\\rightarrow _c{\\cal B}$ .",
"Theorem 2.6: Let ${\\bf A}$ and ${\\bf B}$ be finitary bases.",
"The partial order ${\\sf {Map}}({\\bf A},{\\bf B})$ consisting of the set of approximable mappings over ${\\bf A}$ and ${\\bf B}$ is isomorphic to the partial order ${\\cal A} \\rightarrow _c {\\cal B}$ of continuous functions mapping ${\\cal A}$ into ${\\cal B}$ .",
"The isomorphism is witnessed by the function ${{F}}: {\\sf {Map}}({\\bf A},{\\bf B}) \\rightarrow ({\\cal A} \\rightarrow _c {\\cal B})$ defined by $ {{F}}(F) = f$ where $f$ is the function defined by the equation $f(d) = {\\sf apply}(F,d)$ for all $d \\in {\\cal A}$ .",
"Proof The theorem is an immediate consequence of the following lemma.",
"$\\:\\:\\Box $ Lemma 2.7: For any approximable mappings $F,G \\subseteq {\\bf A} \\times {\\bf B}$ $\\forall a\\in {\\bf A}, b\\in {\\bf B} \\;\\;a\\,F\\,b \\iff {\\cal I}_b \\sqsubseteq {{F}}(F)({\\cal I}_a)$ .",
"$F\\subseteq G \\iff \\forall a\\in {\\bf A} \\;\\; {{F}}(F)(a) \\sqsubseteq {{F}}(G)(a)$ The function ${{F}}: {\\sf {Map}}({\\bf A},{\\bf B}) \\rightarrow ({\\cal A} \\rightarrow _c {\\cal B})$ is one-to-one and onto.",
"Proof (lemma) Part ($a$ ) is the immediate consequence of the definition of $f$ ($b \\in f({\\cal I}_a) \\iff a \\, F \\, b$ ) and the fact that $f({\\cal I}_a)$ is downward closed.",
"Part ($b$ ) follows directly from Part ($a$ ): $F \\subseteq G \\iff \\forall a \\in {\\cal A} \\;\\;\\lbrace b \\: \\:\\vert \\:\\: a \\, F \\, b\\rbrace \\subseteq \\lbrace b \\: \\:\\vert \\:\\:a \\, G \\, b\\rbrace $ .",
"But the latter holds iff $\\forall a\\in {\\bf A} \\;\\; (f(a) \\subseteq g(a) \\iff f(a) \\sqsubseteq g(a))$ .",
"Assume ${{F}}$ is not one-to-one.",
"Then there are distinct approximable mappings $F$ and $G$ such that ${{F}}(F) = {{F}}(G)$ .",
"Since ${{F}}(F) = {{F}}(G)$ , $\\forall a \\in {\\bf A}, b \\in {\\bf B} \\;\\;({\\cal I}_b \\sqsubseteq {{F}}(F)({\\cal I}_a) \\iff {\\cal I}_b \\sqsubseteq {{F}}(G)({\\cal I}_a))\\:.$ By Part 1 of the lemma, $\\forall a \\in {\\bf A}, b \\in {\\bf B} \\;\\;(a\\,F\\,b \\iff {\\cal I}_b \\sqsubseteq {{F}}(F)({\\cal I}_a) \\iff {\\cal I}_b \\sqsubseteq {{F}}(G)({\\cal I}_a) \\iff a\\,G\\,b)\\:.$ We can prove that ${{F}}$ is onto as follows.",
"Let $f$ be an arbitary continuous function in ${\\cal A}\\rightarrow {\\cal B}$ .",
"Define the relation $F \\subseteq {\\bf A} \\times {\\bf B}$ by the rule $a \\: F \\: b \\iff {\\cal I}_b \\sqsubseteq f({\\cal I}_a)\\;.$ It is easy to verify that $F$ is an approximable mapping.",
"By Part 1 of the lemma, $a \\: F \\: b \\iff {\\cal I}_b \\sqsubseteq {{F}}(F)({\\cal I}_a)\\;.$ Hence ${\\cal I}_b \\sqsubseteq f({\\cal I}_a) \\iff {\\cal I}_b \\sqsubseteq {{F}}(F)({\\cal I}_a)\\;,$ implying that $f$ and ${{F}}(F)$ agree on finite inputs.",
"Since $f$ and ${{F}}(F)$ are continuous, they are equal.",
"$\\:\\:\\Box $ The following examples show how approximable mappings and continuous functions are related.",
"Example 2.8: Let ${\\cal B}$ be the domain of infinite strings from the previous section and let ${\\cal T}$ be the truth value domain with two total elements, true and false, where $\\bot _{{\\cal T}}$ denotes that there is insufficient information to determine the outcome.",
"Let $p:{\\cal B}\\rightarrow {\\cal T}$ be the function defined by the equation: $p(x) =\\left\\lbrace \\begin{array}{ll}{\\tt true} & \\mbox{if $x = 0^n1y$ where $n$ is even}\\\\{\\tt false} & \\mbox{if $x = 0^n1y$ where $n$ is odd}\\\\\\bot _{{\\cal T}} & \\mbox{otherwise}\\end{array}\\right.$ The function $p$ determines whether or not there are an even number of 0's before the first 1 in the string.",
"If there is no 1 in the string, the result is $\\bot _{{\\cal T}}$ .",
"It is easy to show that $p$ is continuous.",
"The corresponding binary relation $P$ is defined by the rule: $a\\:P\\:b &\\iff & (b \\sqsubseteq _{{\\cal T}}\\bot _{{\\cal T}}) \\; \\vee \\;\\\\& & (0^{2n}1\\sqsubseteq _B a \\wedge b\\sqsubseteq _{{\\cal T}}{\\tt true}) \\; \\vee \\;\\\\& & (0^{2n+1}1\\sqsubseteq _B a \\wedge b\\sqsubseteq _{{\\cal T}}{\\tt false})$ The reader should verify that $P$ is an approximable mapping and that $p$ is the continuous function determined by $P$ .",
"$\\:\\:\\Box $ Example 2.9: Given the domain ${\\cal B}$ from the previous example, let $g:{\\cal B}\\rightarrow {\\cal B}$ be the function defined by the equation: $g(x) =\\left\\lbrace \\begin{array}{ll}0^{n+1}y & \\mbox{if $x = 0^n1^k0y$}\\\\\\bot _D & \\mbox{otherwise}\\end{array}\\right.$ The function $g$ eliminates the first substring of the form $1^k \\; (k > 0)$ from the input string $x$ .",
"If $x = 1^\\infty $ , the infinite string of ones, then $g(x) = \\bot _D$ .",
"Similarly, if $x = 0^n1^\\infty $ , then $g(x) = \\bot _D$ .",
"The reader should confirm that $g$ is continuous and determine the approximable mapping $G$ corresponding to $g$ .",
"$\\:\\:\\Box $ Approximable mappings and continuous functions can be composed and manipulated just like any other relations and functions.",
"In particular, the composition operators for approximable mappings and continuous functions behave as expected.",
"In fact, they form categories.",
"Categories of Approx.",
"Mappings and Cont.",
"Functions Approximable mappings and continuous functions form categories over finitary bases and domains, respectively.",
"Theorem 2.10: The approximable mappings form a category over finitary bases where the identity mapping for finitary basis $B$ , ${\\sf I}_B \\subseteq {\\bf B} \\times {\\bf B}$ , is defined for $a,b\\in {\\bf B}$ as $a\\:{\\sf I}_B \\:b \\iff b\\sqsubseteq a$ and the composition $G \\circ F \\subseteq {\\bf B_1} \\times {\\bf B_3}$ of approximable mappings $F \\subseteq {\\bf B_1}\\times {\\bf B_2}$ and $G \\subseteq {\\bf B_2}\\times {\\bf B_3}$ is defined for $a\\in {\\bf B_1}$ and $c\\in {\\bf B_3}$ by the rule: $a \\: (G\\circ F) \\:c \\iff \\exists b\\in {\\bf B_2} \\: a\\,F\\,b \\wedge b\\,G\\,c\\,.$ To show that this structure is a category, we must establish the following properties: the identity mappings are approximable mappings, the identity mappings composed with an approximable mapping defines the original mapping, the mappings formed by composition are approximable mappings, and composition of approximable mappings is associative.",
"Proof Let $F \\subseteq {\\bf B_1}\\times {\\bf B_2}$ and $G \\subseteq {\\bf B_2}\\times {\\bf B_3}$ be approximable mappings.",
"Let ${{\\sf I}}_1, {{\\sf I}}_2$ be identity mappings for ${\\bf B_1}$ and ${\\bf B_2}$ respectively.",
"The verification that the identity mappings satisfy the requirements for approximable mappings is straightforward and left to the reader.",
"To show $F\\circ {{\\sf I}}_1$ and ${{\\sf I}}_2\\circ F$ are approximable mappings, we prove the following equivalence: $F\\circ {\\sf I}_1 = {\\sf I}_2\\circ F = F$ For $a\\in {\\bf B_1}$ and $b\\in {\\bf B_2}$ , $a\\:(F\\circ {\\sf I}_1) \\:b \\iff \\exists c\\in {\\cal D}_1 \\;\\; (c\\sqsubseteq a\\wedge c\\,F\\,b)\\, .$ By the definition of approximable mappings, this property holds iff $a\\,F\\,b$ , implying that $F$ and $F\\circ {\\sf I}_1$ are the same relation.",
"The proof of the other half of the equivalence is similar.",
"We must show that the relation $G\\circ F$ is approximable given that the relations $F$ and $G$ are approximable.",
"To prove the first condition, we observe that $\\bot _1\\,F\\,\\bot _2$ and $\\bot _2\\,G\\,\\bot _3$ by assumption, implying that $\\bot _1\\:(G\\circ F)\\:\\bot _3$ .",
"Proving the second condition requires a bit more work.",
"If $a\\,(G\\circ F)\\,c$ and $a\\,(G\\circ F)\\,c^{\\prime }$ , then by the definition of composition, $a\\, F\\,b$ and $b\\,G\\,c$ for some $b$ and $a\\, F\\,b^{\\prime }$ and $b^{\\prime }\\,G\\,c^{\\prime }$ for some $b^{\\prime }$ .",
"Since $F$ and $G$ are approximable mappings, $a\\,F\\,(b\\sqcup b^{\\prime })$ and since $b^{\\prime }\\sqsubseteq (b\\sqcup b^{\\prime })$ , it must be true that $(b\\sqcup b^{\\prime })\\,G\\,c$ .",
"By an analogous argument, $(b\\sqcup b^{\\prime })\\,G\\,c^{\\prime }$ .",
"Therefore, $(b\\sqcup b^{\\prime })\\,G\\,(c\\sqcup c^{\\prime })$ since $G$ is an approximable mapping, implying that $a\\,(G\\circ F)\\,(c\\sqcup c^{\\prime })$ .",
"The final condition asserts that $G\\circ F$ is monotonic.",
"We can prove this as follows.",
"If $a\\sqsubseteq a^{\\prime }$ , $c^{\\prime }\\sqsubseteq c$ and $a\\,(G\\circ F)\\,c$ , then $a\\,F\\,b$ and $b\\,G\\,c$ for some $b$ .",
"So $a^{\\prime }\\,F\\,b$ and $b\\,G\\,c^{\\prime }$ and thus $a^{\\prime }\\,(G\\circ F)\\,c^{\\prime }$ .",
"Thus, $G\\circ F$ satisfies the conditions of an approximable mapping.",
"Associativity of composition implies that for approximable mapping $H$ with $F, G$ as above and $H:{\\cal D}_3\\rightarrow {\\cal D}_4$ , $H\\circ (G\\circ F) = (H\\circ G)\\circ F$ .",
"Assume $a\\,(H\\circ (G\\circ F))\\,z$ .",
"Then, $\\begin{array}{lcl}a\\,(H\\circ (G\\circ F))\\,z&\\iff &\\exists c\\in {\\cal D}_3 \\; a\\,(G\\circ F)\\,c\\wedge c\\,H\\,z\\\\&\\iff & \\exists c\\in {\\cal D}_3 \\, \\exists b\\in {\\cal D}_2 \\; a\\,F\\,b\\wedge b\\,G\\,c\\wedge c\\,H\\,z\\\\&\\iff & \\exists b\\in {\\cal D}_2 \\: \\exists c\\in {\\cal D}_3 \\; a\\,F\\,b\\wedge b\\,G\\,c\\wedge c\\:H\\,z\\\\&\\iff & \\exists b\\in {\\cal D}_2 \\: a\\,F\\,b\\wedge b\\,(H\\circ G)\\,z\\\\&\\iff & a\\,((H\\circ G)\\circ F)\\,z\\end{array}$ $\\:\\:\\Box $ Since finitary bases correspond to domains and approximable mappings correspond to continuous functions, we can restate the same theorem in terms of domains and continuous functions.",
"Corollary 2.11: The continuous functions form a category over domains determined by finitary bases; the identity function for domain ${\\cal B}$ , ${\\sf I}_B: {\\cal B} \\rightarrow {\\cal B}$ , is defined for by the equation ${\\sf I}_B(d) = d$ and the composition $g \\circ f \\in {\\cal B}_1 \\rightarrow {\\cal B}_3$ of continuous functions $f: {\\cal B}_1\\rightarrow {\\cal B}_2$ and $g:{\\cal B}_2\\rightarrow {\\cal B}_3$ is defined for $a\\in {\\cal B}_1$ by the equation $(g\\circ f)(a) = g(f(a))\\,.$ Proof The corollary is an immediate consequence of the preceding theorem and two facts: The partial order of finitary bases is isomorphic to the partial order of domains determined by finitary bases; the ideal completion mapping established the isomorphism.",
"The partial order of approximable mappings over ${\\bf A} \\times {\\bf B}$ is isomorphic to the partial order of continuous functions in ${\\cal A} \\rightarrow {\\cal B}$ .",
"$\\:\\:\\Box $ Based on Theorem REF and Corollary REF , we define $\\bf FB$ as the category having finitary bases as its objects and approximable mappings between finitary bases as its arrows, and we define $\\bf Dom$ as the category having domains as its objects and continuous functions over domains as its arrows.",
"Domain Isomorphisms Isomorphisms between domains are important.",
"We briefly state and prove one of their most important properties.",
"Theorem 2.12: Every isomorphism between domains is characterized by an approximable mapping between the finitary bases.",
"Additionally, finite elements are always mapped to finite elements.",
"Proof Let $f:{\\cal D}\\rightarrow {\\cal E}$ be a one-to-one and onto function that preserves the approximation ordering.",
"Using the earlier theorem characterizing approximable mappings and their associated functions, we can define the mapping as $a\\,F\\,b \\iff {\\cal I}_b\\sqsubseteq f({\\cal I}_a)$ where ${\\cal I}_a,{\\cal I}_b$ are the principal ideals for $a,b$ .",
"As shown in Exercise REF , monotone functions on finite elements always determine approximable mappings.",
"Thus, we need to show that the function described by this mapping, using the function image construction defined earlier, is indeed the original function $f$ .",
"To show this, the following equivalence must be established for $a\\in {\\cal D}$ : $f(a) = \\lbrace b\\in {\\bf E}\\: \\:\\vert \\:\\: \\exists a^{\\prime }\\in a \\:{\\cal I}_b\\sqsubseteq f({\\cal I}_{a^{\\prime }})\\rbrace $ The right-hand side of this equation, call it $e$ , is an ideal—for a proof of this, see Exercise REF .",
"Since $f$ is an onto function, there must be some $d\\in {\\cal D}$ such that $f(d)=e$ .",
"Since $a^{\\prime }\\in a$ , ${\\cal I}_{a^{\\prime }}\\sqsubseteq a$ holds.",
"Thus, $f({\\cal I}_{a^{\\prime }})\\sqsubseteq f(a)$ .",
"Since this holds for all $a^{\\prime }\\in a$ , $f(d)\\sqsubseteq f(a)$ .",
"Now, since $f$ is an order-preserving function, $d\\sqsubseteq a$ .",
"In addition, since $a^{\\prime }\\in a$ , $f({\\cal I}_{a^{\\prime }})\\sqsubseteq f(d)$ by the definition of $f(d)$ so ${\\cal I}_{a^{\\prime }}\\sqsubseteq d$ .",
"Thus, $a^{\\prime }\\in d$ and thus $a\\sqsubseteq d$ since $a^{\\prime }$ is an arbitrary element of $a$ .",
"Thus, $a=d$ and $f(a) = f(d)$ as desired.",
"To show that finite elements are mapped to finite elements, let ${\\cal I}_a\\in {\\cal D}$ for $a\\in {\\bf D}$ .",
"Since $f$ is one-to-one and onto, every $b\\in f({\\cal I}_a)$ has a unique ${\\cal I}_{b^{\\prime }}\\sqsubseteq {\\cal I}_a$ such that $f({\\cal I}_{b^{\\prime }})={\\cal I}_b$ .",
"This element is found using the inverse of $f$ , which must exist.",
"Now, let $z=\\bigsqcup \\: \\lbrace {\\cal I}_{b^{\\prime }}\\: \\:\\vert \\:\\: b\\in f({\\cal I}_a)\\rbrace $ Since $p\\sqsubseteq q$ implies ${\\cal I}_{p^{\\prime }}\\sqsubseteq {\\cal I}_{q^{\\prime }}$ , $z$ is also an ideal (see Exercise REF again).",
"Since ${\\cal I}_{b^{\\prime }}\\sqsubseteq {\\cal I}_a$ holds for each ${\\cal I}_{b^{\\prime }}$ , $z\\sqsubseteq {\\cal I}_a$ must also hold.",
"Also, since each ${\\cal I}_{b^{\\prime }}\\sqsubseteq z$ , ${\\cal I}_b=f({\\cal I}_{b^{\\prime }})\\sqsubseteq f(z)$ .",
"Therefore, $b\\in f(z)$ .",
"Since $b$ is an arbitrary element in $f({\\cal I}_a)$ , $f({\\cal I}_a)\\sqsubseteq f(z)$ must hold and thus ${\\cal I}_a\\sqsubseteq z$ .",
"Therefore, ${\\cal I}_a=z$ and $a\\in z$ .",
"But then $a\\in {\\cal I}_{c^{\\prime }}$ for some $c\\in f({\\cal I}_a)$ by the definition of $z$ .",
"Thus, ${\\cal I}_a\\sqsubseteq {\\cal I}_{c^{\\prime }}$ and $f({\\cal I}_a)\\sqsubseteq {\\cal I}_c$ .",
"Since $c$ was chosen such that ${\\cal I}_c\\sqsubseteq f({\\cal I}_a)$ , ${\\cal I}_{c^{\\prime }}\\sqsubseteq {\\cal I}_a$ and therefore ${\\cal I}_c=f({\\cal I}_a)$ and $f({\\cal I}_a)$ is finite.",
"The same argument holds for the inverse of $f$ ; therefore, the isomorphism preserves the finiteness of elements.",
"$\\:\\:\\Box $ Exercises Exercise 2.13: Show that the partial order of monotonic functions mapping ${\\cal D}^0$ to ${\\cal E}^0$ (using the pointwise ordering) is isomorphic to the partial order of approximable mappings $f:{\\bf D}\\times {\\bf E}$ .",
"Exercise 2.14: Prove that, if $F \\subseteq {\\bf D}\\times {\\bf E}$ is an approximable mapping, then the corresponding function $f:{\\cal D}\\rightarrow {\\cal E}$ satisfies the following equation: $f(x) = \\bigsqcup \\: \\lbrace e \\: \\:\\vert \\:\\: \\exists d \\in x \\; d F e\\rbrace $ for all $x\\in {\\cal D}$ .",
"Exercise 2.15: Prove the following claim: if $F,G \\subseteq {\\bf D}\\times {\\bf E}$ are approximable mappings, then there exists $H \\subseteq {\\bf D}\\times {\\bf E}$ such that $H = F \\cap G = F \\sqcap G$ .",
"Exercise 2.16: Let $\\langle I,\\le \\rangle $ be a non-empty partial order that is directed and let $\\langle D,\\sqsubseteq \\rangle $ be a finitary basis.",
"Suppose that $a:I\\rightarrow {\\cal D}$ is defined such that $i\\le j\\:\\Rightarrow \\:a(i)\\sqsubseteq a(j)$ for all $i,j\\in I$ .",
"Show that $\\bigcup \\lbrace a(i)\\: \\:\\vert \\:\\: i\\in I\\rbrace $ is an ideal in ${\\cal D}$ .",
"This says that the domain is closed under directed unions.",
"Prove also that for an approximable mapping $f:{\\bf D}\\rightarrow {\\bf E}$ , then for any directed union, $f(\\bigcup \\lbrace a(i)\\: \\:\\vert \\:\\: i\\in I\\rbrace ) = \\bigcup \\lbrace f(a(i))\\: \\:\\vert \\:\\: i\\in I\\rbrace $ This says that approximable mappings preserve directed unions.",
"If an elementwise function preserves directed unions, must it come from an approximable mapping?",
"(Hint: See Exercise REF ?",
"?).",
"Exercise 2.17: Let $\\langle I,\\le \\rangle $ be a directed partial order with $f_i:{\\bf D}\\rightarrow {\\bf E}$ as a family of approximable mappings indexed by $i\\in I$ .",
"And assume $i\\le j \\:\\Rightarrow \\:f_i(x)\\sqsubseteq f_j(x)$ for all $i,j\\in I$ and all $x\\in {\\cal D}$ .",
"Show that there is an approximable mapping $g:{\\bf D}\\rightarrow {\\bf E}$ where $g(x)=\\bigcup \\lbrace f_i(x)\\: \\:\\vert \\:\\: i\\in I\\rbrace $ for all $x\\in {\\cal D}$ .",
"Exercise 2.18: Let $f:{\\cal D}\\rightarrow {\\cal E}$ be an isomorphism between domains.",
"Let $\\phi :{\\bf D}\\rightarrow {\\bf E}$ be the one-to-one correspondence from Theorem REF where $f({\\cal I}_a)={\\cal I}_{\\phi (a)}$ for $a\\in {\\bf D}$ .",
"Show that the approximable mapping determined by $f$ is the relationship $\\phi (x)\\sqsubseteq b$ .",
"Show also that if $a$ and $a^{\\prime }$ are consistent in ${\\cal D}$ then $\\phi (a\\sqcup a^{\\prime }) = \\phi (a) \\sqcup \\phi (a^{\\prime })$ .",
"Show how this means that isomorphisms between domains correspond to isomorphisms between the bases for the domains.",
"Exercise 2.19: Show that the mapping defined in Example REF is approximable.",
"Is it uniquely determined by the following equations or are some missing?",
"$\\begin{array}{lcl}g(0x) &=& 0g(x)\\\\g(11x) &=& g(1x)\\\\g(10x) &=& 0x\\\\g(1) &=& \\bot \\end{array}$ Exercise 2.20: Define in words the effect of the approximable mapping $h:{\\bf B}\\rightarrow {\\bf B}$ using the bases defined in Example REF where $\\begin{array}{lcl}h(0x)&=&00h(x)\\\\h(1x)&=&10h(x)\\end{array}$ for all $x\\in {\\cal B}$ .",
"Is $h$ an isomorphism?",
"Does there exist a map $k:{\\bf B}\\rightarrow {\\bf B}$ such that $k\\circ h = {\\sf I}_B$ and is $k$ a one-to-one function?",
"Exercise 2.21: Generalize the definition of approximable mappings to define mappings $f:{\\bf D_1}\\times {\\bf D_2}\\rightarrow {\\bf D_3}$ of two variables.",
"(Hint: A mapping $f$ can be a ternary relation $f\\subseteq {\\bf D_1}\\times {\\bf D_2}\\times {\\bf D_3}$ where the relation among the basis elements is denoted $(a,b)\\,F\\,c$ .)",
"State a modified version of the theorem characterizing these mappings and their corresponding functions.",
"Exercise 2.22: Modify the construction of the domain ${\\cal B}$ from Example REF to construct a domain ${\\cal C}$ with both finite and infinite total elements (${\\cal B}\\subseteq {\\cal C}$ ).",
"Define an approximable map, $C$ , on this domain corresponding to the concatenation of two strings.",
"(Hint: Use 011 as an finite total element, 011$\\bot $ as the corresponding finite partial element.)",
"Recall that $\\epsilon $ , the empty sequence, is different from $\\bot $ , the undefined sequence.",
"Concatenation should be defined such that if $x$ is an infinite element from ${\\cal C}$ , then $\\forall y\\in {\\bf C} \\; (x,y)\\,C\\,x$ .",
"How does concatenation behave on partial elements on the left?",
"Exercise 2.23: Let ${\\bf A}$ and ${\\bf B}$ be arbitrary finitary bases.",
"Prove that the partial order of approximable mappings over ${\\bf A} \\times {\\bf B}$ is a domain.",
"(Hint: The finite elements are the closures of finite consistent relations.)",
"Prove that the partial order of continuous functions in ${\\cal A} \\rightarrow {\\cal B}$ is a domain.",
"Domain Constructors Now that the notion of a domain has been defined, we need to develop convenient methods for constructing specific domains.",
"The strategy that we will follow is to define the simplest domains (i.e., flat domains) directly (as initial or term algebras) and to construct more complex domains by applying domain constructors to simpler domains.",
"Since domains are completely determined by finitary bases, we will focus on how to construct composite finitary bases from simpler ones.",
"These constructions obviously determine corresponding constructions on domains.",
"The two most important constructions on finitary bases are (1) Cartesian products of finitary bases and (2) approximable mappings on finitary bases constructed using a function-space constructor.",
"Cartesian Products Definition 3.1: [Product Basis] Let ${\\bf D}$ and ${\\bf E}$ be the finitary bases generating domains ${\\cal D}$ and ${\\cal E}$ .",
"The product basis, ${\\bf D\\times E}$ , is the partial order consisting of the universe $D\\times E = \\lbrace [d,e]\\:\\vert \\:d\\in {\\bf D},e\\in {\\bf E}\\rbrace $ and the approximation ordering $[d,e] \\sqsubseteq [i,j] \\iff d\\sqsubseteq _D i~{\\rm and}~e\\sqsubseteq _E j.$ Theorem 3.2: The product basis of two finitary bases, as defined above, is a finitary basis.",
"Proof Let ${\\bf D}$ and ${\\bf E}$ be finitary bases and let ${\\bf D\\times E}$ be defined as above.",
"Since D and E are countable, the universe of ${\\bf D}\\times {\\bf E}$ must be countable.",
"It is easy to show that ${\\bf D}\\times {\\bf E}$ is a partial order.",
"By the construction, the bottom element of the product basis is $[\\bot _D,\\bot _E]$ .",
"For any finite bounded subset $R$ of ${\\bf D}\\times {\\bf E}$ where $R=\\lbrace [d_i,e_i]\\rbrace $ , the lub of $R$ is $[\\sqcup \\lbrace d_i\\rbrace ,\\sqcup \\lbrace e_i\\rbrace ]$ which must be defined since D and E are finitary bases and for $R$ to be bounded, each of the component sets must be bounded.",
"$\\:\\:\\Box $ It is straightforward to define projection mappings on product bases, corresponding to the standard projection functions defined on Cartesian products of sets.",
"Definition 3.3: [Projection Mappings] For a finitary basis ${\\bf D}\\times {\\bf E}$ , projection mappings ${\\sf {P}}_0 \\subseteq ({\\bf D}\\times {\\bf E}) \\times {\\bf D}$ and $P_1 \\subseteq ({\\bf D}\\times {\\bf E}) \\times {\\bf E}$ are the relations defined by the rules $[d,e] \\: P_0 \\:d^{\\prime } & \\iff & d^{\\prime } \\sqsubseteq _D d \\\\ [d,e] \\: P_1 \\:e^{\\prime } & \\iff & e^{\\prime } \\sqsubseteq _E e$ where $d$ ,$d^{\\prime }$ are arbitrary elements of ${\\bf D}$ and $e$ , $e^{\\prime }$ are arbitrary elements of ${\\bf E}$ .",
"Let ${\\bf A}$ , ${\\bf D}$ , and ${\\bf E}$ be finitary bases and let $F \\subseteq {\\bf A}\\times {\\bf D}$ and $G \\subseteq {\\bf A}\\times {\\bf E}$ be approximable mappings.",
"The paired mapping $\\langle F,G\\rangle \\subseteq {\\bf A}\\times ({\\bf D}\\times {\\bf E})$ is the relation defined by the rule $a\\:\\langle F,G\\rangle \\:[d,e] \\iff a\\:F\\:d\\wedge a\\:G\\:e$ for all $a\\in {\\bf A},d\\in {\\bf D}$ , and all $e\\in {\\bf E}$ .",
"It is an easy exercise to show that projection mappings and paired mappings are approximable mappings (as defined in the previous section).",
"Theorem 3.4: The mappings ${\\sf {P}}_0$ , ${\\sf {P}}_1$ , and $\\langle F,G\\rangle $ are approximable mappings if $F,G$ are.",
"In addition, ${\\sf {P}}_0\\circ \\langle F,G\\rangle = F$ and ${\\sf {P}}_1\\circ \\langle F,G\\rangle = G$ .",
"For $[d,e]\\in {\\bf D}\\times {\\bf E}$ and $d^{\\prime } \\in {\\bf D}$ , $[d,e] \\:{\\sf {P}}_0 \\: d^{\\prime } \\iff d^{\\prime } \\sqsubseteq d$ .",
"For $[d,e]\\in {\\bf D}\\times {\\bf E}$ and $e^{\\prime } \\in {\\bf E}$ , $[d,e] \\:{\\sf {P}}_1 \\: e^{\\prime } \\iff e^{\\prime } \\sqsubseteq d$ .",
"For approximable mapping $H \\subseteq {\\bf A}\\times ({\\bf D}\\times {\\bf E})$ , $H=\\langle ({\\sf {P}}_0\\circ H),({\\sf {P}}_1\\circ H)\\rangle $ .",
"For $a\\in {\\bf A}$ and $[d,e] \\in {\\bf D}\\times {\\bf E}$ , $[a, [d,e]] \\in \\langle F,G\\rangle \\iff [a,d] \\in F \\wedge [a,e] \\in G$ .",
"Proof The proof is left as an exercise to the reader.",
"$\\:\\:\\Box $ The projection mappings and paired mappings on finitary bases obviously correspond to continuous functions on the corresponding domains.",
"We will denote the continous functions corresponding to ${\\sf {P}}_0$ and ${\\sf {P}}_1$ by the symbols $p_1$ and $p_2$ .",
"Similarly, we will denote the function corresponding to the paired mapping $\\langle F, G\\rangle $ by the expression $\\langle f, g \\rangle $ .",
"It should be clear that the definition of projection mappings and paired mappings can be generalized to products of more than two domains.",
"This generalization enables us to treat a multi-ary continous function (or approximable mapping) as a special form of a unary continuous function (or approximable mapping) since multi-ary inputs can be treated as single objects in a product domain.",
"Moreover, it is easy to show that a relation $R \\subseteq ({\\bf A_1} \\times \\ldots \\times {\\bf A_n}) \\times {\\bf B}$ of arity $n+1$ (as in Exercise REF ) is an approximable mapping iff every restriction of $R$ to a single input argument (by fixing the other input arguments) is an approximable mapping.",
"Theorem 3.5: A relation $F \\subseteq ({\\bf A}\\times {\\bf B})\\times {\\bf C}$ is an approximable mapping iff for every $a\\in {\\bf A}$ and every $b\\in {\\bf B}$ , the derived relations $F_{a,*} & = & {\\lbrace { [y,z] \\; \\:\\vert \\:\\; [[a,y],z] \\in F}\\rbrace }\\\\F_{*,b} & = & {\\lbrace { [x,z] \\; \\:\\vert \\:\\; [[x,b],z] \\in F}\\rbrace }$ are approximable mappings.",
"Proof Before we prove the theorem, we need to introduce the class of constant relations ${\\sf {K}}_e \\subseteq {\\bf D}\\times {\\bf E}$ for arbitrary finitary bases ${\\bf D}$ and ${\\bf E}$ and show that they are approximable mappings.",
"Lemma 3.6: For each $e \\in {\\bf E}$ , let the “constant” relation ${\\sf {K}}_e \\subseteq {\\bf D} \\times {\\bf E}$ be defined by the equation ${\\sf {K}}_e = {\\lbrace {[d,e^{\\prime }] \\; \\:\\vert \\:\\; d \\in {\\bf D}, e^{\\prime } \\sqsubseteq e}\\rbrace }\\,.$ In other words, $d \\: {\\sf {K}}_e \\: e^{\\prime } \\; \\iff \\; e^{\\prime } \\sqsubseteq e \\, .$ For $e\\in {\\bf E}$ , the constant relation ${\\sf {K}}_e \\subseteq {\\bf D} \\times {\\bf E}$ is approximable.",
"Proof (lemma) The proof of this lemma is left to the reader.",
"$\\:\\:\\Box $ To prove the “if” direction of the theorem, we observe that we can construct the relations $F_{a,*}$ and $F_{*,b}$ for all $a \\in {\\bf A}$ and $b \\in {\\bf B}$ by composing and pairing primitive approximable mappings.",
"In particular, $F_{a,*}$ is the relation $F \\circ \\langle K_a, {\\sf I}_B \\rangle $ where ${\\sf I}_B$ denotes the identity relation on ${\\bf B}$ .",
"Similary, $F_{*,b}$ is the relation $F \\circ \\langle {\\sf I}_A, K_b\\rangle $ where ${\\sf I}_A$ denotes the identity relation on ${\\bf A}$ .",
"To prove the “only-if” direction, we assume that for all $a \\in {\\bf A}$ and $b \\in {\\bf B}$ , the relations $F_{a,*}$ and $F_{*,b}$ are approximable.",
"We must show that the four closure properties for approximable mappings hold for $F$ .",
"Since $F_{\\bot _A,*}$ is approximable, $[\\bot _B,\\bot _C] \\in F_{\\bot _A,*}$ , which implies that $[[\\bot _A,\\bot _B],\\bot _C] \\in F$ .",
"If $[[x,y],z] \\in F$ and $[[x,y],z^{\\prime }] \\in F$ , then $[y,z] \\in F_{x,*}$ and $[y,z^{\\prime }] \\in F_{x,*}$ .",
"Since $F_{x,*}$ is approximable, $[y,z \\sqcup z^{\\prime }] \\in F_{x,*}$ , implying $[[x,y],z \\sqcup z^{\\prime }] \\in F$ .",
"If $[[x,y],z] \\in F$ and $z^{\\prime } \\sqsubseteq z$ , then $[y,z] \\in F_{x,*}$ .",
"Since $F_{x,*}$ is approximable, $[y,z^{\\prime }] \\in F_{x,*}$ , implying $[[x,y],z^{\\prime }] \\in F$ .",
"If $[[x,y],z] \\in F$ and $[x,y] \\sqsubseteq [x^{\\prime },y^{\\prime }]$ , then $[y,z] \\in F_{x,*}$ , $x \\sqsubseteq x^{\\prime }$ , and $y \\sqsubseteq y^{\\prime }$ .",
"Since $F_{x,*}$ is approximable, $[y^{\\prime },z] \\in F_{x,*}$ , implying $[x,y^{\\prime }],z] \\in F$ , which is equivalent to $[x,z] \\in F_{*,y^{\\prime }}$ .",
"Since $F_{*,y^{\\prime }}$ is approximable, $[x^{\\prime },z] \\in F_{*,y^{\\prime }}$ , implying $[[x^{\\prime },y^{\\prime }],z] \\in F$ .",
"$\\:\\:\\Box $ The same result can be restated in terms of continous functions.",
"Theorem 3.7: A function of two arguments, $f:{\\cal A}\\times {\\cal B}\\rightarrow {\\cal C}$ is continuous iff for every $a\\in {\\cal A}$ and every $b\\in {\\cal B}$ , the unary functions $x\\mapsto f[a,x]~{\\rm and}~y\\mapsto f[y,b]$ are continuous.",
"Proof Immediate from the previous theorem and the fact that the domain of approximable mappings over $({\\bf A} \\times {\\bf B}) \\times {\\bf C}$ is isomorphic to the domain of continuous functions over $({\\cal A} \\times {\\cal B}) \\times {\\cal C}$ .",
"$\\:\\:\\Box $ Multiary Function Composition The composition of functions, as defined in Theorem REF , can be generalized to functions of several arguments.",
"But we need some new syntactic machinery to describe more general forms of function composition.",
"Definition 3.8: [Cartesian Types] Let $S$ be a set of symbols used to denote primitive types.",
"The set $S^*$ of Cartesian types over $S$ consists of the set of expressions denoting all finite non-empty Cartesian products over primitive types in $S$ : $S^* ::= S \\; \\:\\vert \\:\\; S \\times S \\; \\:\\vert \\:\\; \\ldots \\; .$ A signature $\\Sigma $ is a pair $\\langle S, O \\rangle $ consisting of a set $S$ of type names ${\\lbrace {s_1, \\ldots , s_m}\\rbrace }$ used to denote domains and a set $O={\\lbrace {o_i^{\\rho _i\\rightarrow \\sigma _i}\\; \\:\\vert \\:\\; 1 \\le i \\le m, \\; \\rho _i \\in S^*, \\; \\sigma _i \\in S}\\rbrace }$ of function symbols used to denote first-order functions over the domains $S$ .",
"Let $V={\\lbrace {v_i^{\\tau } \\; \\:\\vert \\:\\; \\tau \\in S,\\; i \\in {N}}\\rbrace }$ be a countably-infinite set of symbols (variables) distinct from the symbols in $\\Sigma $ .",
"The typed expressions over $\\Sigma $ (denoted ${\\cal Ex}(\\Sigma )$ ) is the set of “typed” terms determined by the following inductive definition: $v_i^{\\tau } \\in V$ is a term of type $\\tau $ , for $M_1^{\\tau _1}, \\dots , M_n^{\\tau _n} \\in {\\cal Ex}(\\Sigma )$ and $o^{(\\tau _1 \\times \\dots \\times \\tau _n) \\rightarrow \\tau _0} \\in O$ then $o^{(\\tau _1 \\times \\dots \\times \\tau _n) \\rightarrow \\tau _0}(M_1^{\\tau _1}, \\dots ,M_n^{\\tau _n})^{\\tau _0}$ is a term of type $\\tau _0$ .",
"We will restrict our attention to terms where every instance of a variable $v_i$ in a term has the same type $\\tau $ .",
"To simplify notation, we will drop the type superscripts from terms whenever they can be easily inferred from context.",
"Definition 3.9: [Finitary Algebra] A finitary algebra with signature $\\Sigma $ is a function ${\\bf A}$ mapping each primitive type $\\tau \\in S$ to a finitary basis ${{\\bf A}}{[\\!",
"[}\\tau {]\\!",
"]}$ , each operation type $\\tau ^1 \\times \\dots \\times \\tau ^n \\in S^*$ to the finitary basis ${{\\bf A}}{[\\!",
"[}\\tau ^1{]\\!]}",
"\\times \\dots \\times {{\\bf A}}{[\\!",
"[}\\tau ^n{]\\!",
"]}$ , each function symbol $o_i^{\\rho _i \\rightarrow \\sigma _i} \\in O$ to an approximable mapping ${{\\bf A}}{[\\![}o_i{]\\!]}",
"\\subseteq ({{\\bf A}}{[\\!",
"[}\\rho _i{]\\!]}",
"\\times {{\\bf A}}{[\\!",
"[}\\sigma _i{]\\!",
"]})$ .",
"(Recall that ${{\\bf A}}{[\\!",
"[}\\rho _i{]\\!",
"]}$ is a product basis.)",
"Definition 3.10: [Closed Term] A term $M \\in {\\cal Ex}(\\Sigma )$ is closed iff it contains no variables in $V$ .",
"The finitary algebra ${\\bf A}$ implicitly assigns a meaning to every closed term $M$ in ${\\cal Ex}(\\Sigma )$ .",
"This extension is inductively defined by the equation: ${{\\bf A}}{[\\!",
"[}o[M_1,\\dots ,M_n]{]\\!]}",
"= {{\\bf A}}{[\\![}o{]\\!",
"]}[{{\\bf A}}{[\\![}M_1{]\\!",
"]}, \\dots , {{\\bf A}}{[\\![}M_n{]\\!]}]",
"=$ ${\\lbrace { b_0 \\; \\:\\vert \\:\\; \\exists [b_1,\\dots ,b_n] \\in {{\\bf A}}{[\\!",
"[}\\rho _i{]\\!]}",
"\\;[b_1,\\dots ,b_n] {{\\bf A}}{[\\![}o{]\\!]}",
"b_0}\\rbrace } \\, .$ We can extend ${\\bf A}$ to terms $M$ with free variables by implicitly abstracting over all of the free variables in $M$ .",
"Definition 3.11: [Meaning of Terms] Let $M$ be a term in ${\\cal Ex}{(\\Sigma )}$ and let $l = x_1^{\\tau _1}, \\dots , x_n^{\\tau _n}$ be a list of distinct variables in $V$ containing all the free variables of $M$ .",
"Let ${\\bf A}$ be a finitary algebra with signature $\\Sigma $ and for each tuple $[d_1, \\dots , d_n] \\in {{\\bf A}}{[\\!",
"[}\\tau _1{]\\!]}",
"\\times \\dots \\times {{\\bf A}}{[\\!",
"[}\\tau _n{]\\!",
"]}$ , let ${\\bf A}_{\\lbrace {x_1 := d_1,\\dots , x_n := d_n}\\rbrace }$ denote the algebra ${\\bf A}$ extended by defining ${{\\bf A}}{[\\![}x_i{]\\!]}",
"= d_i$ for $1 \\le i \\le n$ .",
"The meaning of $M$ with respect to $l$ , denoted ${{\\bf A}}{[\\!",
"[}x_1^{\\tau _1}, \\dots , x_n^{\\tau _n} \\, \\mapsto \\,M{]\\!",
"]}$ , is the relation $F_M \\subseteq ({{\\bf A}}{[\\!",
"[}\\tau _1{]\\!]}",
"\\times {{\\bf A}}{[\\!",
"[}\\tau _n{]\\!]})",
"\\times {{\\bf A}}{[\\!",
"[}\\tau _0{]\\!",
"]}$ defined by the equation: $F_M[d_1, \\dots , d_n] = {{\\bf A}_{\\lbrace {x_1 := d_1,\\dots ,x_1 := d_n}\\rbrace }}{[\\![}M{]\\!",
"]}$ The relation denoted by ${{\\bf A}}{[\\!",
"[}x_1^{\\tau _1}, \\dots , x_n^{\\tau _n} \\, \\mapsto \\, M{]\\!",
"]}$ is often called a substitution.",
"The following theorem shows that the relation ${{\\bf A}}{[\\!",
"[}x_1^{\\tau _1}, \\dots , x_n^{\\tau _n} \\, \\mapsto \\, M{]\\!",
"]}$ is approximable.",
"Theorem 3.12: (Closure of continuous functions under substitution) Let $M$ be a term in ${\\cal Ex}{(\\Sigma )}$ and let $l = x_1^{\\tau _1}, \\dots , x_n^{\\tau _n}$ be a list of distinct variables in $V$ containing all the free variables of $M$ .",
"Let ${\\bf A}$ be a finitary algebra with signature $\\Sigma $ .",
"Then the relation $F_M$ denoted by the expression $x_1^{\\tau _1}, \\dots , x_n^{\\tau _n} \\mapsto M $ is approximable.",
"Proof The proof proceeds by induction on the structure of $M$ .",
"The base cases are easy.",
"If $M$ is a variable $x_i$ , the relation $F_M$ is simply the projection mapping ${\\sf {P}}_i$ .",
"If $M$ is a constant $c$ of type $\\tau $ , then $F_M$ is the contant relation ${\\sf {K}}_c$ of arity $n$ .",
"The induction step is also straightforward.",
"Let $M$ have the form $g[M_1^{\\sigma _1},\\ldots ,M_m^{\\sigma _m}]$ .",
"By the induction hypothesis, $x_1^{\\tau _1}, \\dots , x_n^{\\tau _n} \\mapsto M_i^{\\sigma _i} $ denotes an approximable mapping $F_{M_i} \\subseteq ({{\\bf A}}{[\\!",
"[}\\tau _1{]\\!]}",
"\\times {{\\bf A}}{[\\!",
"[}\\tau _n{]\\!]})",
"\\times {{\\bf A}}{[\\!",
"[}\\sigma _i{]\\!",
"]}$ .",
"But $F_M$ is simply the composition of the approximable mapping ${{\\bf A}}{[\\![}g{]\\!",
"]}$ with the mapping $\\langle F_{M_1}, \\dots F_{M_m} \\rangle $ .",
"Theorem REF tells us that the composition must be approximable.",
"$\\:\\:\\Box $ The preceding generalization of composition obviously carries over to continuous functions.",
"The details are left to the reader.",
"Function Spaces The next domain constructor, the function-space constructor, allows approximable mappings (or, equivalently, continuous functions) to be regarded as objects.",
"In this framework, standard operations on approximable mappings such as application and composition are approximable mappings too.",
"Indeed, the definitions of ideals and of approximable mappings are quite similar.",
"The space of approximable mappings is built by looking at the actions of mappings on finite sets, and then using progressively larger finite sets to construct the mappings in the limit.",
"To this end, the notion of a finite step mapping is required.",
"Definition 3.13: [Finite Step Mapping] Let ${\\bf A}$ and ${\\bf B}$ be finitary bases.",
"An approximable mapping $F \\subseteq {\\bf A} \\times {\\bf B}$ is a finite step mapping iff there exists a finite set $S\\subseteq {\\bf A}\\times {\\bf B}$ and $F$ is the least approximable mapping such that $S \\subseteq F$ .",
"It is easy to show that for every consistent finite set $S \\subseteq {\\bf A}\\times {\\bf B}$ , a least mapping $F$ always exists.",
"$F$ is simply the closure of $S$ under the four conditions that an approximable mapping must satisfy.",
"The least approximable mapping respecting the empty set is the relation ${\\lbrace {\\langle a, \\bot _B \\rangle \\; \\:\\vert \\:\\; a \\in {\\bf A}}\\rbrace }$ .",
"The space of approximable mappings is built from finite step mappings.",
"Definition 3.14: [Partial Order of Finite Step Mappings] For finitary bases ${\\bf A}$ and ${\\bf B}$ the mapping basis is the partial order ${\\bf A}\\Rightarrow {\\bf B}$ consisting of the universe of all finite step mappings, and the approximation ordering $F \\sqsubseteq G \\iff \\forall a\\in {\\bf A} \\; F(a) \\sqsubseteq _B G(a) \\, .$ The following theorem establishes that the constructor $\\Rightarrow $ maps finitary bases into finitary bases.",
"Theorem 3.15: Let ${\\bf A}$ and ${\\bf B}$ be finitary bases.",
"Then, the mapping basis ${\\bf A}\\Rightarrow {\\bf B}$ is a finitary basis.",
"Proof Since the elements are finite subsets of a countable set, the basis must be countable.",
"It is easy to confirm that ${\\bf A}\\Rightarrow {\\bf B}$ is a partial order; this task is left to the reader.",
"We must show that every finite consistent subset of ${\\bf A}\\Rightarrow {\\bf B}$ has a least upper bound in ${\\bf A}\\Rightarrow {\\bf B}$ .",
"Let $§$ be a finite consistent subset of the universe of ${\\bf A}\\Rightarrow {\\bf B}$ .",
"Each element of $§$ is a set of ordered pairs $\\langle a, b \\rangle $ that meets the approximable mapping closure conditions.",
"Since $§$ is consistent, it has an upper bound $§^{\\prime } \\in {\\bf A}\\Rightarrow {\\bf B}$ .",
"Let $U = \\bigcup §$ .",
"Clearly, $U \\subseteq §^{\\prime }$ .",
"But $U$ may not be approximable.",
"Let $S$ be the intersection of all relations in ${\\bf A}\\Rightarrow {\\bf B}$ above $§$ .",
"Clearly $U \\subseteq S$ , implying $S$ is a superset of every element of $§$ .",
"It is easy to verify that $S$ is approximable, because all the approximable mapping closure conditions are preserved by infinite intersections.",
"$\\:\\:\\Box $ Definition 3.16: [Function Domain] We will denote the domain of ideals determined by the finitary basis ${\\bf A}\\Rightarrow {\\bf B}$ by the expression ${\\cal A}\\Rightarrow {\\cal B}$ .",
"The justification for this notation will be explained shortly.",
"Since the partial order of approximable mappings is isomorphic to the partial order of continuous functions, the preceding definitions and theorems about approximable mappings can be restated in terms of continuous functions.",
"Definition 3.17: [Finite Step Function] Let ${\\cal A}$ and ${\\cal B}$ be the domains determined by the finitary bases ${\\bf A}$ and ${\\bf B}$ , respectively.",
"A continuous function $f$ in ${\\cal A}\\rightarrow _c{\\cal B}$ is finite iff there exists a finite step mapping $F \\subseteq {\\bf A} \\times {\\bf B}$ such that $f$ is the function determined by $F$ .",
"Definition 3.18: [Function Basis] For domains ${\\cal A}$ and ${\\cal B}$ , the function basis is the partial order $({\\cal A}\\rightarrow _c{\\cal B})^0$ consisting of a universe of all finite step functions, and the approximation order $f\\sqsubseteq g \\iff \\forall a\\in {\\cal A} \\; f(a) \\sqsubseteq _{\\cal B}g(a) \\, .$ Corollary 3.19: (to Theorem REF ) For domains ${\\cal A}$ and ${\\cal B}$ , the function basis $({\\cal A}\\rightarrow _c{\\cal B})^0$ is a finitary basis.",
"We can prove that the domain constructed by generating the ideals over ${\\bf A}\\Rightarrow {\\bf B}$ is isomorphic to the partial order ${\\sf {Map}}({\\bf A},{\\bf B})$ of approximable mappings defined in Section 2.",
"This result is not surprising; it merely demonstrates that ${\\sf {Map}}({\\bf A},{\\bf B})$ is a domain and that we have identified the finite elements correctly in defining ${\\bf A}\\Rightarrow {\\bf B}$ .",
"Theorem 3.20: The domain of ideals determined by ${\\bf A}\\Rightarrow {\\bf B}$ is isomorphic to the partial order of the approximable mappings ${\\sf {Map}}({\\bf A},{\\bf B})$ .",
"Hence, ${\\sf {Map}}({\\bf A},{\\bf B})$ is a domain.",
"Proof We must establish an isomorphism between the domain determined by ${\\bf A}\\Rightarrow {\\bf B}$ and the partial order of mappings from ${\\bf A}$ to ${\\bf B}$ .",
"Let $h: {\\cal A}\\Rightarrow {\\cal B} \\rightarrow {\\sf {Map}}({\\bf A},{\\bf B})$ be the function defined by rule $h \\: {\\cal {F}}= \\bigcup {\\lbrace {F \\in {\\cal {F}}}\\rbrace } \\,.$ It is easy to confirm that the relation on the right hand side of the preceding equation is an approximable mapping: if it violated any of the closure properties, so would a finite approximation in ${\\cal {F}}$ .",
"We must prove that the function $h$ is one-to-one and onto.",
"To prove the former, we note that each pair of distinct ideals has a witness $\\langle a, b \\rangle $ that belongs to a set in one ideal but not in any set in the other.",
"Hence, the images of the two ideals are distinct.",
"The function $h$ is onto because every approximable mapping is the image of the set of finite step maps that approximate it.",
"$\\:\\:\\Box $ The preceding theorem can be restated in terms of continuous functions.",
"Corollary 3.21: (to Theorem REF ) The domain of ideals determined by the finitary basis $({\\cal A}\\rightarrow _c{\\cal B})^0$ is isomorphic to the partial order of continuous functions ${\\cal A} \\rightarrow _c {\\cal B}$ .",
"Hence, ${\\cal A} \\rightarrow _c {\\cal B}$ is a domain.",
"Now that we have defined the approximable mapping and continous function domain constructions, we can show that operators on maps and functions introduced in Section 2 are continuous functions.",
"Theorem 3.22: Given finitary bases, ${\\bf A}$ and ${\\bf B}$ , there is an approximable mapping $Apply:(({\\bf A}\\Rightarrow {\\bf B}) \\times {\\bf A}) \\times {\\bf B}$ such that for all $F:{\\bf A}\\Rightarrow {\\bf B}$ and $a\\in {\\bf A}$ , $Apply[F,a] = F(a)\\,.$ Recall that for any approximable mapping $G \\subseteq {\\bf C} \\times {\\bf D}$ and any element $c \\in {\\bf C}$ $G(c) = {\\lbrace {d \\; \\:\\vert \\:\\; c \\: G \\: d}\\rbrace }\\,.$ Proof For $F\\in ({\\bf A}\\Rightarrow {\\bf B})$ , $a\\in {\\bf A}$ and $b\\in {\\bf B}$ , define the $Apply$ relation as follows: $[F,a]\\:Apply\\:b \\iff a\\:F\\:b\\,.$ It is easy to verify that $Apply$ is an approximable mapping; this step is left to the reader.",
"From the definition of $Apply$ , we deduce $Apply[F,a] = {\\lbrace {b \\; \\:\\vert \\:\\; [F,a]\\: Apply \\: b}\\rbrace } ={\\lbrace {b \\; \\:\\vert \\:\\; a \\: F \\: b}\\rbrace } = F(a)\\,.$ $\\:\\:\\Box $ This theorem can be restated in terms of continuous functions.",
"Corollary 3.23: Given domains, ${\\cal A}$ and ${\\cal B}$ , there is a continuous function $apply:(({\\cal A}\\rightarrow _c{\\cal B}) \\times {\\cal A}) \\rightarrow _c {\\cal B}$ such that for all $f:{\\cal A}\\rightarrow _c{\\cal B}$ and $a\\in {\\cal A}$ , $apply[f,a] = f(a)\\,.$ Proof (of corollary).",
"Let $apply: (({\\cal A}\\rightarrow _c {\\cal B}) \\times {\\cal A}) \\rightarrow _c {\\cal B}$ be the continuous function (on functions rather than relations!)",
"corresponding to $Apply$ .",
"From the definition of $apply$ and Theorem REF which relates approximable mappings on finitary bases to continous functions over the corresponding domains, we know that $apply[f,{\\cal I}_A]=\\lbrace b\\in {\\bf B}\\:\\vert \\:\\exists F^{\\prime }\\in ({\\bf A}\\Rightarrow {\\bf B}) \\ ; \\exists a\\in {\\cal I}_A \\wedge F^{\\prime } \\subseteq F \\wedge [F^{\\prime },a]\\:Apply\\:b\\rbrace $ where $F$ denotes the approximable mapping corresponding to $f$ .",
"Since $f$ is the continuous function corresponding to $F$ , $f({\\cal I}_A) = \\lbrace b\\in {\\bf B}\\:\\vert \\:\\exists a\\in {\\cal I}_A \\; a\\:F\\:b\\rbrace $ So, by the definition of the $Apply$ relation, $apply[f,{\\cal I}_A]\\subseteq f({\\cal I}_A)$ .",
"For every $b\\in f({\\cal I}_A)$ , there exists $a \\in {\\cal I}_A$ such that $a\\:F\\:b$ .",
"Let $F^{\\prime }$ be the least approximable mapping such that $a \\: F^{\\prime } b$ .",
"By definition, $F^{\\prime }$ is a finite step mapping.",
"Hence $b \\in apply[f,{\\cal I}_A]$ , implying $f({\\cal I}_A) \\subseteq apply[f,{\\cal I}_A]$ .",
"Therefore, $f({\\cal I}_A) = apply[f,{\\cal I}_A]$ for arbitrary ${\\cal I}_A$ .",
"$\\:\\:\\Box $ The preceding theorem and corollary demonstrate that approximable mappings and continuous functions can operate on other approximable mappings or continuous functions just like other data objects.",
"The next theorem shows that the currying operation is a continuous function.",
"Definition 3.24: [The Curry Operator] Let A, B, and C be finitary bases.",
"Given an approximable mapping $G$ in the basis $({\\bf A}\\times {\\bf B})\\Rightarrow {\\bf C}$ , $Curry_G:{\\bf A}\\Rightarrow ({\\bf B}\\Rightarrow {\\bf C})$ is the relation defined by the equation $Curry_G(a) = {\\lbrace {F \\in {\\bf B}\\Rightarrow {\\bf C} \\; \\:\\vert \\:\\;\\forall [b,c] \\in F \\; [a,b] \\: G \\: c}\\rbrace }$ for all $a \\in {\\bf A}$ .",
"Similarly, given any continuous function $g: ({\\cal A}\\times {\\cal B})\\rightarrow _c{\\cal C}$ , $curry_g:{\\cal A}\\rightarrow _c({\\cal B}\\rightarrow _c{\\cal C})$ is the function defined by the equation $curry_g({\\cal I}_A) = (y \\mapsto g[{\\cal I}_A,y])\\,.$ By Theorem 2.7?",
"?, $(y \\mapsto g[{\\cal I}_A,y])$ is a continous function.",
"Lemma 3.25: $Curry_G$ is an approximable mapping and $curry_g$ is the continuous function determined by $Curry_G$ .",
"Proof A straightforward exercise.$\\:\\:\\Box $ It is more convenient to discuss the currying operation in the context of continuous functions than approximable mappings.",
"Theorem 3.26: Let $g \\in ({\\cal A}\\times {\\cal B})\\rightarrow _c{\\cal C}$ and $h \\in ({\\cal A}\\rightarrow _c({\\cal B}\\rightarrow _c{\\cal C})$ .",
"The $curry$ operation satisfies the following two equations: $apply\\circ \\langle curry_g\\circ p_0,p_1\\rangle & = & g\\\\curry_{apply\\circ \\langle h\\circ p_0,p_1\\rangle } & = & h\\, .$ In addition, the function $curry:({\\cal A}\\times {\\cal B}\\rightarrow {\\cal C})\\rightarrow ({\\cal A}\\rightarrow _c({\\cal B}\\rightarrow _c{\\cal C}))$ defined by the equation $curry(g)({\\cal I}_A)({\\cal I}_B) = curry_g({\\cal I}_A)({\\cal I}_B)$ is continuous.",
"Proof Let $g$ be any continuous function in the domain $({\\cal A}\\times {\\cal B})\\rightarrow _c{\\cal C}$ .",
"Recall that $curry_g(a) = (y \\mapsto g[a,y])\\,.$ Using this definition and the definition of operations in the first equation, we can deduce $\\begin{array}{lcl}apply\\circ \\langle curry_g\\circ p_0,p_1\\rangle [a,b]& = & apply[\\langle curry_g \\circ p_0, p_1 \\rangle [a,b]] \\\\& = & apply[(curry_g \\circ p_0)[a,b], p_1[a,b]] \\\\& = & apply[curry_g p_0[a,b], b] \\\\& = & apply[curry_g \\: a, b] \\\\& = & curry_g \\: a \\: b \\\\& = & g[a,b]\\,.\\end{array}$ Hence, the first equation holds.",
"The second equation follows almost immediately from the first.",
"Define $g^{\\prime }:({\\cal A}\\times {\\cal B})\\rightarrow _c{\\cal C}$ by the equation $g^{\\prime }[a,b] = h \\; a \\; b\\,.$ The function $g^{\\prime }$ is defined so that $curry_{g^{\\prime }} = h$ .",
"This fact is easy to prove.",
"For $a \\in {\\cal A}$ : $curry_{g^{\\prime }}(a) & = & (y \\mapsto g^{\\prime }[a,y])\\\\& = & (y \\mapsto h (a)(y))\\\\& = & h(a) \\, .$ Since $h = curry_{g^{\\prime }}$ , the first equation implies that $apply\\circ \\langle h\\circ p_0,p_1\\rangle & = &apply\\circ \\langle curry_{g^{\\prime }}\\circ p_0,p_1\\rangle \\\\& = & g^{\\prime }\\,.$ Hence, $curry_{apply}\\circ \\langle h\\circ p_0,p_1\\rangle = curry_{g^{\\prime }} = h\\,.$ These two equations show that $({\\cal A}\\times {\\cal B})\\rightarrow _c{\\cal C}$ is isomorphic to $({\\cal A}\\rightarrow _c({\\cal B}\\rightarrow _c{\\cal C})$ under the $curry$ operation.",
"In addition, the definition of $curry$ shows that $curry(g) \\sqsubseteq curry(g^{\\prime }) \\iff g \\sqsubseteq g^{\\prime }\\,.$ Hence, $curry$ is an isomorphism.",
"Moreover, $curry$ must be continuous.",
"$\\:\\:\\Box $ The same theorem can be restated in terms of approximable mappings.",
"Corollary 3.27: The relation $Curry_G$ satisfies the following two equations: $Apply\\circ \\langle Curry_G \\circ {\\sf {P}}_0,{\\sf {P}}_1\\rangle & = & G\\\\Curry_{Apply\\circ \\langle G\\circ {\\sf {P}}_0,{\\sf {P}}_1\\rangle } & = & G\\, .$ In addition, the relation $Curry:({\\bf A}\\times {\\bf B})\\Rightarrow {\\bf C})\\Rightarrow ({\\bf A} \\Rightarrow ({\\bf B}\\Rightarrow {\\bf C}))$ defined by the equation $Curry(G) = {\\lbrace {[a,F] \\; \\:\\vert \\:\\; a \\in {\\bf A}, \\:F \\in ({\\bf B}\\Rightarrow {\\bf C}), \\; \\forall [b,c] \\in F \\:[a,b]\\: G \\: c }\\rbrace }$ is approximable.",
"Table: Domains and Finitary BasesTable REF summarizes the main elements of the correspondence between domains and finitary bases.",
"Whenever convenient, in the following sections we take liberty to confuse corresponding notions.",
"Context and notation should make clear which category is meant.",
"Exercises Exercise 3.28: We assume that there is a countable basis.",
"Thus, the basis elements could without loss of generality be defined in terms of $\\lbrace 0,1\\rbrace ^*$ .",
"Show that the product space ${\\bf A}\\times {\\bf B}$ could be defined as a finitary basis over $\\lbrace 0,1\\rbrace ^*$ such that ${\\bf A}\\times {\\bf B}=\\lbrace [0a,1b]\\:\\vert \\:a\\in {\\bf A},b\\in {\\bf B}\\rbrace $ Give the appropriate definition for the elements in the domain.",
"Also show that there exists an approximable mapping $diag:{\\bf D}\\rightarrow {\\bf D}\\times {\\bf D}$ where $diag(x)= [x,x]$ for all $x\\in {\\cal D}$ .",
"Exercise 3.29: Establish some standard isomorphisms: ${\\bf A}\\times {\\bf B}\\approx {\\bf B}\\times {\\bf A}$ ${\\bf A}\\times ({\\bf B}\\times {\\bf C})\\approx ({\\bf A}\\times {\\bf B})\\times {\\bf C}$ ${\\bf A}\\approx {\\bf A^{\\prime }},{\\bf B}\\approx {\\bf B^{\\prime }}\\:\\Rightarrow \\:{\\bf A}\\times {\\bf B}\\approx {\\bf A^{\\prime }}\\times {\\bf B^{\\prime }}$ for all finitary bases.",
"Exercise 3.30: Let $B\\subseteq \\lbrace 0,1\\rbrace ^*$ be a finitary basis.",
"Define $B^\\infty =\\bigcup \\limits _{n=0}^\\infty 1^n0B$ Thus, $B^\\infty $ contains infinitely many disjoint copies of $B$ .",
"Now let $D^\\infty $ be the least family of subsets over $\\lbrace 0,1\\rbrace ^*$ such that $B^\\infty \\in D^\\infty $ if $b\\in {\\bf B}$ and $d\\in D^\\infty $ , then $0X\\cup 1Y\\in D^\\infty $ .",
"Show that, with the superset relation as the approximation ordering, $D^\\infty $ is a finitary basis.",
"State any assumptions that must be made.",
"Show then that $D^\\infty \\approx D\\times D^\\infty $ .",
"Exercise 3.31: Using the product construction as a guide, generate a definition for the separated sum structure ${\\bf A}+{\\bf B}$ .",
"Show that there are mappings $in_A:{\\bf A}\\rightarrow {\\bf A}+{\\bf B}$ , $in_B:{\\bf B}\\rightarrow {\\bf A}+{\\bf B}$ , $out_A:{\\bf A}+{\\bf B}\\rightarrow {\\bf A}$ , and $out_B:{\\bf A}+{\\bf B}\\rightarrow {\\bf B}$ such that $out_A\\circ in_A = {\\sf I}_A$ where ${\\sf I}_A$ is the identity function on ${\\bf A}$ .",
"State any necessary assumptions to ensure this function equation is true.",
"Exercise 3.32: For approximable mappings $f:{\\bf A}\\rightarrow {\\bf A^{\\prime }}$ and $g:{\\bf B}\\rightarrow {\\bf B^{\\prime }}$ , show that there exist approximable mappings, $f\\times g:{\\bf A}\\times {\\bf B}\\rightarrow {\\bf A^{\\prime }}\\times {\\bf B^{\\prime }}$ and $f+g:{\\bf A}+{\\bf B}\\rightarrow {\\bf A^{\\prime }}+{\\bf B^{\\prime }}$ such that $(f\\times g)[a,b] = [f \\: a, g \\: b]$ and thus $f\\times g = \\langle f\\circ p_0,g\\circ p_1\\rangle $ Show also that $out_A\\circ (f+g)\\circ in_A = f$ and $out_B\\circ (f+g)\\circ in_B = g$ Is $f+g$ uniquely determined by the last two equations?",
"Exercise 3.33: Prove that the composition operator is an approximable mapping.",
"That is, show that $comp:({\\bf B}\\rightarrow {\\bf C})\\times ({\\bf A}\\rightarrow {\\bf B})\\rightarrow ({\\bf A}\\rightarrow {\\bf C})$ is an approximable mapping where for $f:{\\bf A}\\rightarrow {\\bf B}$ and $g:{\\bf B}\\rightarrow {\\bf C}$ , $comp[g,f] = g\\circ f$ .",
"Show this using the approach used in showing the result for $apply$ and $curry$ .",
"That is, define the relation and then build the function from $apply$ and $curry$ , using $\\circ $ and paired functions.",
"(Hint: Fill in mappings according to the following sequence of domains.)",
"$\\begin{array}{c}({\\bf A}\\rightarrow {\\bf B})\\times {\\bf A}\\rightarrow {\\bf B}\\\\({\\bf B}\\rightarrow {\\bf C})\\times (({\\bf A}\\rightarrow {\\bf B})\\times {\\bf A})\\rightarrow ({\\bf B}\\rightarrow {\\bf C})\\times {\\bf B}\\\\(({\\bf B}\\rightarrow {\\bf C})\\times ({\\bf A}\\rightarrow {\\bf B}))\\times {\\bf A}\\rightarrow ({\\bf B}\\rightarrow {\\bf C})\\times {\\bf B}\\\\(({\\bf B}\\rightarrow {\\bf C})\\times ({\\bf A}\\rightarrow {\\bf B}))\\times {\\bf A}\\rightarrow {\\bf C}\\\\({\\bf B}\\rightarrow {\\bf C})\\times ({\\bf A}\\rightarrow {\\bf B})\\rightarrow ({\\bf A}\\rightarrow {\\bf C}).\\end{array}$ This map shows only one possible solution.",
"Exercise 3.34: Show that for every domain ${\\cal D}$ there is an approximable mapping $cond:{\\bf T}\\times {\\bf D}\\times {\\bf D}\\rightarrow {\\bf D}$ called the conditional operator such that $cond[true,a,b]=a$ $cond[false,a,b]=b$ $cond[\\bot _T,a,b]=\\bot _D$ and ${\\bf T}=\\lbrace \\bot _T,true,false\\rbrace $ such that $\\bot _T\\sqsubseteq true$ , $\\bot _T\\sqsubseteq false$ , but $true$ and $false$ are incomparable.",
"(Hint: Define a $Cond$ relation).",
"Fixed Points and Recursion Fixed Points Functions can now be constructed by composing basic functions.",
"However, we wish to be able to define functions recursively as well.",
"The technique of recursive definition will also be useful for defining domains as we will see in Section .",
"Recursion can be thought of as (possibly infinite) iterated function composition.",
"The primary result for interpreting recursive definitions is the following Fixed Point Theorem.",
"Theorem 4.1: For any continuous function $f:{\\cal D}\\rightarrow {\\cal D}$ determined by an approximable mapping $F:{\\bf D}\\rightarrow {\\bf D}$ , there exists a least element $x\\in {\\cal D}$ such that $f(x) = x.$ Proof Let $f^n$ stand for the function $f$ composed with itself $n$ times, and similarly for $F^n$ .",
"Thus, for $\\begin{array}{lcl}f^0&=&I_{\\cal D}~\\\\f^{n+1}&=&f\\circ f^n\\\\F^0&=&{\\sf I}_D~{\\rm and}\\\\F^{n+1}&=&F\\circ F^n\\end{array}$ we define $x = \\lbrace d\\in {\\bf D}\\:\\vert \\:\\exists n\\in {{N}}.",
"\\bot \\:F^n\\:d\\rbrace .$ To show that $x\\in {\\cal D}$ , we must show it to be an ideal.",
"Map $F$ is an approximable mapping, so $\\bot \\in x$ since $\\bot \\:F\\:\\bot $ .",
"For $d\\in x$ and $d^{\\prime }\\sqsubseteq d$ , $d^{\\prime }\\in x$ must hold since, for $d\\in x$ , there must exist an $a\\in {\\bf D}$ such that $a\\:F\\:d$ .",
"But by the definition of an approximable mapping, $a\\:F\\:d^{\\prime }$ must hold as well so $d^{\\prime }\\in x$ .",
"Closure under lubs is direct since $F$ must include lubs to be approximable.",
"To see that $f(x)=x$ , or equivalently $x\\:F\\:x$ , note that for any $d\\in x$ , if $d\\:F\\:d^{\\prime }$ , then $d^{\\prime }\\in x$ .",
"Thus, $f(x)\\sqsubseteq x$ .",
"Now, $x$ is constructed to be the least element in ${\\cal D}$ with this property.",
"To see this is true, let $a\\in {\\cal D}$ such that $f(a)\\sqsubseteq a$ .",
"We want to show that $x\\sqsubseteq a$ .",
"Let $d\\in x$ be an arbitrary element.",
"Therefore, there exists an $n$ such that $\\bot \\: F^n\\:d$ and therefore $\\bot \\:F\\:d_1\\:F\\:d_2\\:\\ldots \\:F\\:d_{n-1}\\:F\\:d.$ Since $\\bot \\in a$ , $d_1\\in f(a)$ .",
"Thus, since $f(a)\\sqsubseteq a$ , $d_1\\in a$ .",
"Thus, $d_2\\in f(a)$ and therefore $d_2\\in a$ .",
"Using induction on $n$ , we can show that $d\\in f(a)$ .",
"Therefore, $d\\in a$ and thus $x\\sqsubseteq a$ .",
"Since $f$ is monotonic and $f(x)\\sqsubseteq x$ , $f(f(x))\\sqsubseteq f(x)$ .",
"Since $x$ is the least element with this property, $x\\sqsubseteq f(x)$ and thus $x=f(x)$ .",
"$\\:\\:\\Box $ Since the element $x$ above is the least element, it must be unique.",
"Thus we have defined a function mapping the domain ${\\cal D}\\rightarrow {\\cal D}$ into the domain ${\\cal D}$ .",
"The next step is to show that this mapping is approximable.",
"Theorem 4.2: For any domain ${\\cal D}$ , there is an approximable mapping $fix:({\\bf D}\\rightarrow {\\bf D})\\rightarrow {\\bf D}$ such that if $f:{\\bf D}\\rightarrow {\\bf D}$ is an approximable mapping, $fix(f) = f(fix(f))$ and for $x\\in {\\cal D}$ , $f(x)\\sqsubseteq x \\:\\Rightarrow \\:fix(f)\\sqsubseteq x$ This property implies that $fix$ is unique.",
"The function $fix$ is characterized by the equation $fix(f)=\\bigcup \\limits _{n=0}^\\infty f^n(\\bot )$ for all $f:{\\bf D}\\rightarrow {\\bf D}$ .",
"Proof The final equation can be simplified to $fix(f) = \\lbrace d\\in {\\bf D}\\:\\vert \\:\\exists n\\in {{N}}.\\bot \\:f^n\\:d\\rbrace $ which is the equation used in the previous theorem to define the fixed point.",
"Using the formula from Exercise REF on the above definition for $fix$ yields the following equation to be shown: $fix(f)=\\bigcup \\lbrace fix({\\cal I}_F)\\:\\vert \\:\\exists F\\in ({\\bf D}\\rightarrow {\\bf D}).F\\in f\\rbrace $ where ${\\cal I}_F$ denotes the ideal for $F$ in ${\\bf D}\\rightarrow {\\bf D}$ .",
"From its definition, $fix$ is monotonic since, if $f\\sqsubseteq g$ , then $fix(f)\\sqsubseteq fix(g)$ since $f^n\\sqsubseteq g^n$ .",
"Since $F\\in f$ , ${\\cal I}_F\\sqsubseteq f$ and since $fix$ is monotonic, $fix({\\cal I}_F)\\sqsubseteq fix(f)$ .",
"Let $x\\in fix(f)$ .",
"Thus, there is a finite sequence of elements such that $\\bot \\:f\\:x_1\\:f\\:\\ldots \\:f\\:x^{\\prime }\\:f\\:x$ .",
"Define $F$ as the basis element encompassing the step functions required for this sequence.",
"Clearly, $F\\in f$ .",
"In addition, this same sequence exists in $fix({\\cal I}_F)$ since we constructed $F$ to contain it, and thus, $x\\in fix({\\cal I}_F)$ and $fix(f)\\sqsubseteq fix({\\cal I}_F)$ .",
"The equality is therefore established.",
"The first equality is direct from the Fixed Point Theorem since the same definition is used.",
"Assume $f(x)\\sqsubseteq x$ for some $x\\in {\\cal D}$ .",
"Since $\\bot \\in x$ , $x\\ne \\emptyset $ .",
"Since $f$ is an approximable mapping, for $x^{\\prime }\\in x$ and $x^{\\prime }\\:f\\:y$ , $y\\in x$ must hold.",
"By induction, for any $\\bot \\:f\\:y$ , $y\\in x$ must hold.",
"Thus, $fix(f)\\sqsubseteq x$ .",
"To see that the operator is unique, define another operator $fax$ that satisfies the first two equations.",
"It can easily be shown that $\\begin{array}{lcll}fix(f)&\\sqsubseteq & fax(f)~{\\rm and}\\\\fax(f)&\\sqsubseteq &fix(f)\\end{array}$ Thus the two operators are the same.",
"$\\:\\:\\Box $ Recursive Definitions Recursion has played a part already in the definitions above.",
"Recall that $f^n$ was defined for all $n\\in {{N}}$ .",
"More complex examples of recursion are given below.",
"Example 4.3: Define a basis ${\\bf N}=\\langle N,\\sqsubseteq _N\\rangle $ where $N=\\lbrace \\lbrace n\\rbrace \\:\\vert \\:n\\in {{N}}\\rbrace \\cup \\lbrace {{N}}\\rbrace $ and the approximation ordering is the superset relation.",
"This generates a flat domain with $\\bot =\\lbrace \\lbrace {{N}}\\rbrace \\rbrace $ and the total elements being in a one-to-one correspondence with the natural numbers.",
"Using the construction outlined in Exercise REF , construct the basis $F=N^\\infty $ .",
"Its corresponding domain is the domain of partial functions over the natural numbers.",
"To see this, let $\\Phi $ be the set of all finite partial functions $\\varphi \\subseteq {{N}}\\times {{N}}$ .",
"Define $\\uparrow \\varphi =\\lbrace \\psi \\in \\Phi \\:\\vert \\:\\varphi \\subseteq \\psi \\rbrace $ Consider the finitary basis $\\langle F^{\\prime },\\sqsubseteq _F^{\\prime }\\rangle $ where $F^{\\prime }=\\lbrace \\uparrow \\varphi \\:\\vert \\:\\varphi \\in \\Phi \\rbrace $ and the approximation order is the superset relation.",
"The reader should satisfy himself that $F^{\\prime }$ and $F$ are isomorphic and that the elements are the partial functions.",
"The total elements are the total functions over the natural numbers.",
"The domains ${\\cal F}$ and $({\\cal N}\\rightarrow {\\cal N})$ are not isomorphic.",
"However, the following mapping $val:F\\times {\\bf N}\\rightarrow {\\bf N}$ can be defined as follows: $(\\uparrow \\varphi ,\\lbrace n\\rbrace )\\:val\\:\\lbrace m\\rbrace \\iff (n,m)\\in \\varphi $ and $(\\uparrow \\varphi ,{{N}})\\:val\\:{{N}}$ Define also as the ideal for $m\\in {\\cal N}$ , $\\hat{m} = \\lbrace \\lbrace m\\rbrace ,{{N}}\\rbrace $ It is easy to show then that for $\\pi \\in {\\cal F}$ and $n\\in {\\cal N}$ we have $\\begin{array}{lcll}val(\\pi ,\\hat{n})&=&\\hat{\\pi (n)}&{\\rm if}~\\pi (n)\\ne \\bot \\\\&=&\\bot &{\\rm otherwise}\\end{array}$ Thus, $curry(val):{\\bf F}\\rightarrow ({\\bf N}\\rightarrow {\\bf N})$ is a one-to-one function on elements.",
"(The problem is that (${\\bf N}\\rightarrow {\\bf N}$ ) has more elements than F does as the reader should verify for himself).",
"Now, what about mappings $f:{\\bf F}\\rightarrow {\\bf F}$ ?",
"Consider the function $\\begin{array}{lcll}f(\\pi )(n)&=&0&{\\rm if}~n=0\\\\&=&\\pi (n-1)+n-1&{\\rm for}~n>0\\end{array}$ If $\\pi $ is a total function, $f(\\pi )$ is a total function.",
"If $\\pi (k)$ is undefined, then $f(\\pi )(k+1)$ is undefined.",
"The function $f$ is approximable since it is completely determined by its actions on partial functions.",
"That is $f(\\pi )=\\bigcup \\lbrace f(\\varphi )\\:\\vert \\:\\exists \\varphi \\in \\Phi .\\varphi \\subseteq \\pi \\rbrace $ The Fixed Point Theorem defines a least fixed point for any approximable mapping.",
"Let $\\sigma =f(\\sigma )$ .",
"Now, $\\sigma (0)=0$ and $\\begin{array}{lcl}\\sigma (n+1)&=&f(\\sigma )(n+1)\\\\&=&\\sigma (n)+n\\end{array}$ By induction, $\\sigma (n)=\\sum \\limits _{i=0}^n i$ and therefore, $\\sigma $ is a total function.",
"Thus, $f$ has a unique fixed point.",
"Now, in looking at $({\\bf N}\\rightarrow {\\bf N})$ , we have $\\hat{0}\\in {\\cal N}$ (The symbols $n$ and $\\hat{n}$ will no longer be distinguished, but the usage should be clear from context.).",
"Now define the two mappings, $succ,pred:{\\bf N}\\rightarrow {\\bf N}$ as approximable mappings such that $\\begin{array}{lcl}n\\:succ\\: m& \\iff & \\exists p\\in {{N}}.n\\sqsubseteq p,m\\sqsubseteq p+1\\\\n\\:pred\\: m& \\iff & \\exists p+1\\in {{N}}.n\\sqsubseteq p+1,m\\sqsubseteq p\\end{array}$ In more familiar terms, the same functions are defined as $\\begin{array}{lcll}succ(n)&=&n+1\\\\pred(n)&=&n-1&{\\rm if}~n>0\\\\&=&\\bot &{\\rm if}~n=0\\end{array}$ The mapping $zero:{\\bf N}\\rightarrow {\\bf T}$ is also defined such that $\\begin{array}{lcll}zero(n)&=&true&{\\rm if}~n=0\\\\&=&false&{\\rm if}~n>0\\end{array}$ where ${\\cal T}$ is the domain of truth value defined in an earlier section.",
"The structured domain $\\langle N,0,succ,pred,zero\\rangle $ is called “The Domain of the Integers” in the present context.",
"The function element $\\sigma $ defined as the fixed point of the mapping $f$ can now be defined directly as a mapping $\\sigma :{\\bf N}\\rightarrow {\\bf N}$ as follows: $\\sigma (n)=cond(zero(n),0,\\sigma (pred(n))+pred(n))$ where the function $+$ must be suitably defined.",
"Recall that $cond$ was defined earlier as part of the structure of the domain ${\\cal T}$ .",
"This equation is called a functional equation; the next section will give another notation, the $\\lambda -calculus$ for writing such equations.",
"$\\:\\:\\Box $ Example 4.4: The domain ${\\cal B}$ defined in Example REF contained only infinite elements as total elements.",
"A related domain, ${\\cal C}$ defined in Exercise REF , can be regarded as a generalization on ${\\cal N}$ .",
"To demonstrate this, the structured domain corresponding to the domain of integers must be presented.",
"The total elements in ${\\cal C}$ are denoted $\\sigma $ while the partial elements are denoted $\\sigma \\bot $ for any $\\sigma \\in \\lbrace 0,1\\rbrace ^*$ .",
"The empty sequence $\\epsilon $ assumes the role of the number 0 in ${\\cal N}$ .",
"Two approximable mappings can serve as the successor function: $x\\mapsto 0x$ denoted $succ_0$ and $x\\mapsto 1x$ denoted $succ_1$ .",
"The predecessor function is filled by the $tail$ mapping defined as follows: $\\begin{array}{lcll}tail(0x)& =& x,\\\\tail(1x)& =& x&{\\rm and}\\\\tail(\\epsilon )& =& \\bot .\\end{array}$ The $zero$ predicate is defined using the $empty$ mapping defined as follows: $\\begin{array}{lcll}empty(0x)& =& false,\\\\empty(1x)& =& false&{\\rm and}\\\\empty(\\epsilon )& =& true.\\end{array}$ To distinguish the other types of elements in ${\\cal C}$ , the following mappings are also defined: $\\begin{array}{lcll}zero(0x)& =& true,\\\\zero(1x)& =& false&{\\rm and}\\\\zero(\\epsilon )& =& false.\\\\one(0x)& =& false,\\\\one(1x)& =& true&{\\rm and}\\\\one(\\epsilon )& =& false.\\end{array}$ The reader should verify the conditions for an approximable mapping are met by these functions.",
"An element of ${\\cal C}$ can be defined using a fixed point equation.",
"For example, the total element representing an infinite sequence of alternating zeroes and ones is defined by the fixed point of the equation $a=01a$ .",
"This same element is defined with the equation $a=0101a$ .",
"(Is the element defined as $b=010b$ the same as the previous two?)",
"Approximable mappings in ${\\cal C}\\rightarrow {\\cal C}$ can also be defined using equations.",
"For example, the mapping $\\begin{array}{lcll}d(\\epsilon ) &= &\\epsilon ,\\\\d(0x)&=&00d(x)&{\\rm and}\\\\d(1x)&=&11d(x)\\end{array}$ can be characterized with the functional equation $d(x)=cond(empty(x),\\epsilon ,cond(zero(x),succ_0(succ_0(d(tail(x)))),succ_1(succ_1(d(tail(x))))))$ The concatenation function of Exercise REF over ${\\cal C}\\times {\\cal C}\\rightarrow {\\cal C}$ can be defined with the functional equation $C(x,y)=cond(empty(x),y,cond(zero(x),succ_0(C(tail(x),y)),succ_1(C(tail(x),y))))$ The reader should verify that this definition is consistent with the properties required in the exercise.",
"These definitions all use recursion.",
"They rely on the object being defined for a base case ($\\epsilon $ for example) or on earlier values ($tail(x)$ for example).",
"These equations characterize the object being defined, but unless a theorem is proved to show that a solution to the equation exists, the definition is meaningless.",
"However, the Fixed Point Theorem for domains was established earlier in this section.",
"Thus, solutions exist to these equations provided that the variables in the equation range over domains and any other functions appearing in the equation are known to be continuous (that is approximable).",
"Peano's Axioms To illustrate one use of the Fixed Point Theorem as well as show the use of recursion in a more familiar setting, we will show that all second order models of Peano's axioms are isomorphic.",
"Recall that Definition 4.5: [Model for Peano's Axioms] A structured set $\\langle {{N}},0,succ\\rangle $ for $0\\in {{N}}$ and $succ:{{N}}\\times {{N}}$ is a model for Peano's axioms if all the following conditions are satisfied: $\\forall n\\in {{N}}.",
"0\\ne succ(n)$ $\\forall n,m \\in {{N}}.succ(n)=succ(m)\\:\\Rightarrow \\:n=m$ $\\forall x\\subseteq {{N}}.0\\in x\\wedge succ(x)\\subseteq x\\:\\Rightarrow \\:x={{N}}$ where $succ(x)=\\lbrace succ(n)\\:\\vert \\:n\\in x\\rbrace $ .",
"The final clause is usually referred to as the principle of mathematical induction.",
"Theorem 4.6: All second order models of Peano's axioms are isomorphic.",
"Proof Let $\\langle N,0,+\\rangle $ and $\\langle M,\\bullet ,\\#\\rangle $ be models for Peano's axioms.",
"Let $N\\times M$ be the cartesian product of the two sets and let ${\\cal P}(N\\times M)$ be the powerset of $N\\times M$ .",
"Recall from Exercise REF that the powerset can be viewed as a domain with the subset relation as the approximation order.",
"Define the following mapping: $u\\mapsto \\lbrace (0,\\bullet )\\rbrace \\cup \\lbrace (+(n),\\#(m))\\:\\vert \\:(n,m)\\in u\\rbrace $ The reader should verify that this mapping is approximable.",
"Since it is indeed approximable, a fixed point exists for the function.",
"Let $r$ be the least fixed point: $r=\\lbrace (0,\\bullet )\\rbrace \\cup \\lbrace (+(n),\\#(m))\\:\\vert \\:(n,m)\\in r\\rbrace $ But $r$ defines a binary relation which establishes the isomorphism.",
"To see that $r$ is an isomorphism, the one-to-one and onto features must be established.",
"By construction, $0\\:r\\:\\bullet $ and $n\\:r\\:m \\:\\Rightarrow \\:+(n)\\:r\\:\\#(m)$ .",
"Now, the sets $\\lbrace (0,\\bullet )\\rbrace $ and $\\lbrace (+(n),\\#(m))\\:\\vert \\:(n,m)\\in r\\rbrace $ are disjoint by the first axiom.",
"Therefore, 0 corresponds to only one element in $m$ .",
"Let $x\\subseteq N$ be the set of all elements of $N$ that correspond to only one element in $m$ .",
"Clearly, $0\\in x$ .",
"Now, for some $y\\in x$ let $z\\in M$ be the element in $M$ that $y$ uniquely corresponds to (that is $y\\:r\\:z$ ).",
"But this means that $+(y)\\:r\\#(z)$ by the construction of the relation.",
"If there exists $w\\in M$ such that $+(y)\\:r\\:w$ and since $(+(y),w)\\ne (0,\\bullet )$ , the fixed point equation implies that $(+(y)=+(n_0))$ and $(w=\\#(m_0))$ for some $(n_0,m_0)\\in r$ .",
"But then by the second axiom, $y=n_0$ and since $y\\in x$ , $z=m_0$ .",
"Thus, $\\#(z)$ is the unique element corresponding to $+(y)$ .",
"The third axiom can now be applied, and thus every element in $N$ corresponds to a unique element in $M$ .",
"The roles of $N$ and $M$ can be reversed in this proof.",
"Therefore, it can also be shown that every element of $M$ corresponds to a unique element in $N$ .",
"Thus, $r$ is a one-to-one and onto correspondence.",
"$\\:\\:\\Box $ Exercises Exercise 4.7: In Theorem REF , an equation was given to find the least fixed point of a function $f:{\\cal D}\\rightarrow {\\cal D}$ .",
"Suppose that for $a\\in {\\cal D}$ , $a\\sqsubseteq f(a)$ .",
"Will the fixed point $x=f(x)$ be such that $a\\sqsubseteq x$ ?",
"(Hint: How do we know that $\\bigcup \\limits _{n=0}^\\infty f^n(a)\\in {\\cal D}$ ?)",
"Exercise 4.8: Let $f:{\\cal D}\\rightarrow {\\cal D}$ and $S\\subseteq {\\cal D}$ satisfy $\\bot \\in S$ $x\\in S\\:\\Rightarrow \\:f(x)\\in S$ $[\\forall n .\\lbrace x_n\\rbrace \\subseteq S \\wedge x_n\\sqsubseteq x_{n+1}]\\:\\Rightarrow \\:\\bigcup \\limits _{n=0}^\\infty x_n \\in S$ Conclude that $fix(f)\\in S$ .",
"This is sometimes called the principle of fixed point induction.",
"Apply this method to the set $S=\\lbrace x\\in {\\cal D}\\:\\vert \\:a(x)= b(x)\\rbrace $ where $a,b:{\\cal D}\\rightarrow {\\cal D}$ are approximable, $a(\\bot )=b(\\bot )$ , and $f\\circ a=a\\circ f$ and $f\\circ b=b\\circ f$ .",
"Exercise 4.9: Show that there is an approximable operator $\\Psi :(({\\cal D}\\rightarrow {\\cal D})\\rightarrow {\\cal D})\\rightarrow (({\\cal D}\\rightarrow {\\cal D})\\rightarrow {\\cal D})$ such that for $\\Theta :({\\cal D}\\rightarrow {\\cal D})\\rightarrow {\\cal D}$ and $f:{\\cal D}\\rightarrow {\\cal D}$ , $\\Psi (\\Theta ) (f) = f(\\Theta (f))$ Prove also that $fix:({\\cal D}\\rightarrow {\\cal D})\\rightarrow {\\cal D}$ is the least fixed point of $\\Psi $ .",
"Exercise 4.10: Given a domain ${\\cal D}$ and an element $a\\in {\\cal D}$ , construct the domain ${\\cal D}_a$ where ${\\cal D}_a=\\lbrace x\\in {\\cal D}\\:\\vert \\:x\\sqsubseteq a\\rbrace $ Show that if $f:{\\cal D}\\rightarrow {\\cal D}$ is approximable, then $f$ can be restricted to another approximable map $f^{\\prime }:{\\cal D}_{fix(f)}\\rightarrow {\\cal D}_{fix(f)}$ where $\\forall x\\in {\\cal D}_{fix(f)}.f^{\\prime }(x)=f(x)$ How many fixed points does $f^{\\prime }$ have in ${\\cal D}_{fix(f)}$ ?",
"Exercise 4.11: The mapping ${\\bf fix}$ can be viewed as assigning a fixed point operator to any domain ${\\cal D}$ .",
"Show that ${\\bf fix}$ can be uniquely characterized by the following conditions on an assignment ${\\cal D}\\leadsto F_D$ : $F_D:({\\cal D}\\rightarrow {\\cal D})\\rightarrow {\\cal D}$ $F_D(f)=f(F_D(f))$ for all $f:{\\cal D}\\rightarrow {\\cal D}$ when $f_0:{\\cal D}_0\\rightarrow {\\cal D}_0$ and $f_1:{\\cal D}_1\\rightarrow {\\cal D}_1$ are given and $h:{\\cal D}_0\\rightarrow {\\cal D}_1$ is such that $h(\\bot )=\\bot $ and $h\\circ f_0=f_1\\circ h$ , then $h(F_{D_0}(f_0)) = F_{D_1}(f_1).$ Hint: Apply Exercise REF to show ${\\bf fix}$ satisfies the conditions.",
"For the other direction, apply Exercise REF .",
"Exercise 4.12: Must an approximable function have a maximum fixed point?",
"Give an example of an approximable function that has many fixed points.",
"Exercise 4.13: Must a monotone function $f:{\\cal P}(A)\\rightarrow {\\cal P}(A)$ have a maximum fixed point?",
"(Recall: ${\\cal P}(A)$ is the powerset of the set $A$ ).",
"Exercise 4.14: Verify the assertions made in the first example of this section.",
"Exercise 4.15: Verify the assertions made in the second example, in particular those in the discussion of “Peano's Axioms”.",
"Show that the predicate function $one:{\\cal C}\\rightarrow {\\cal T}$ could be defined using a fixed point equation from the other functions in the structure.",
"Exercise 4.16: Prove that $fix(f\\circ g)=f(fix(g\\circ f))$ for approximable functions $f,g:{\\cal D}\\rightarrow {\\cal D}$ .",
"Exercise 4.17: Show that the less-than-or-equal-to relation $l\\subseteq {{N}}\\times {{N}}$ is uniquely determined by $l=\\lbrace (n,n)\\:\\vert \\:n\\in {{N}}\\rbrace \\cup \\lbrace (n,succ(m)\\:\\vert \\:(n,m)\\in l\\rbrace $ for the structure called the “Domain of Integers”.",
"Exercise 4.18: Let $N^*$ be a structured set satisfying only the first two of the axioms referred to as “Peano's”.",
"Must there be a subset $S\\subseteq N^*$ such that all three axioms are satisfied?",
"(Hint: Use a least fixed point from ${\\cal P}(N^*)$ ).",
"Exercise 4.19: Let $f:{\\cal D}\\rightarrow {\\cal D}$ be an approximable map.",
"Let $a_n:{\\cal D}\\rightarrow {\\cal D}$ be a sequence of approximable maps such that $a_0(x)=\\bot $ for all $x\\in {\\cal D}$ $a_n\\sqsubseteq a_{n+1}$ for all $n\\in {{N}}$ $\\bigcup \\limits _{n=0}^\\infty a_n = {\\sf I}_D$ in ${\\cal D}\\rightarrow {\\cal D}$ $a_{n+1}\\circ f = a_{n+1}\\circ f\\circ a_n$ for all $n\\in {{N}}$ Show that $f$ has a unique fixed point.",
"(Hint: Show that if $x=f(x)$ then $a_n(x)\\sqsubseteq a_n(fix(f))$ for all $n\\in {{N}}$ .",
"Show this by induction on $n$ .)",
"Typed $\\lambda $ -Calculus As shown in the previous section, functions can be characterized by recursion equations which combine previously defined functions with the function being defined.",
"The expression of these functions is simplified in this section by introducing a notation for specifying a function without having to give the function a name.",
"The notation used is that of the typed $\\lambda $ -Calculus; a function is defined using a $\\lambda $ -abstraction.",
"Definition of Typed $\\lambda $ -Calculus An informal characterization of the $\\lambda $ -calculus suffices for this section; more formal descriptions are available elsewhere in the literature [1].",
"Thus, examples are used to introduce the notation.",
"An infinite number of variables, $x$ ,$y$ ,$z$ ,$\\ldots $ of various types are required.",
"While a variable has a certain type, type subscripts will not be used due to the notational complexity.",
"A distinction must also be made between type symbols and domains.",
"The domain ${\\cal A}\\times {\\cal B}$ does not uniquely determine the component domains ${\\cal A}$ and ${\\cal B}$ even though these domains are uniquely determined by the symbol for the domain.",
"The domain is the meaning that we attribute to the symbol.",
"In addition to variables, constants are also present.",
"For example, the symbol 0 is used to represent the zero element from the domain ${\\cal N}$ .",
"Another constant, present in each domain by virtue of Theorem REF , is $fix^{\\cal D}$ , the least fixed point operator for domain ${\\cal D}$ of type $({\\cal D}\\rightarrow {\\cal D})\\rightarrow {\\cal D}$ .",
"The constants and variables are the atomic (non-compound) terms.",
"Types can be associated with all atomic terms.",
"There are several constructions for compound terms.",
"First, given $\\tau ,\\ldots ,\\sigma $ , a list of terms, the ordered tuple $\\langle \\tau ,\\ldots ,\\sigma \\rangle $ is a compound term.",
"If the types of $\\tau ,\\ldots ,\\sigma $ are ${\\cal A},\\ldots ,{\\cal B}$ , the type of the tuple is ${\\cal A}\\times \\ldots \\times {\\cal B}$ since the tuple is to be an element of this domain.",
"The tuple notation for combining functions given earlier should be disregarded here.",
"The next construction is function application.",
"If the term $\\tau $ has type ${\\cal A}\\rightarrow {\\cal B}$ and the term $\\sigma $ has the type ${\\cal A}$ , then the compound term $\\tau (\\sigma )$ has type ${\\cal B}$ .",
"Function application denotes the value of a function at a given input.",
"The notation $\\tau (\\sigma _0,\\ldots ,\\sigma _n)$ abbreviates $\\tau (\\langle \\sigma _0,\\ldots ,\\sigma _n\\rangle )$ .",
"Functions applied to tuples allows us to represent applications of multi-variate functions.",
"The $\\lambda $ -abstraction is used to define functions.",
"Let $x_0,\\ldots ,x_n$ be a list of distinct variables of type ${\\cal D}_0,\\ldots ,{\\cal D}_n$ .",
"Let $\\tau $ be a term of some type ${\\cal D}_{n+1}$ .",
"$\\tau $ can be thought of as a function of $n+1$ variables with type $({\\cal D}_0\\times \\ldots \\times {\\cal D}_n)\\rightarrow {\\cal D}_{n+1}$ .",
"The name for this function is written $\\lambda x_0,\\ldots ,x_n.\\tau $ This expression denotes the entire function.",
"To look at some familiar functions in the new notation, consider $\\lambda x,y.x$ This notation is read “lambda ex wye (pause) ex”.",
"If the types of $x$ and $y$ are ${\\cal A}$ and ${\\cal B}$ respectively, the function has type $({\\cal A}\\times {\\cal B})\\rightarrow {\\cal A}$ .",
"This function is the first projection function $p_0$ .",
"This function and the second projection function can be defined by the following equations: $\\begin{array}{lcl}p_0&=&\\lambda x,y.x\\\\p_1&=&\\lambda x,y.y\\end{array}$ Recalling the function tuple notation introduced in an earlier section, the following equation holds: $\\langle f,g\\rangle =\\lambda w.\\langle f(w),g(w)\\rangle $ which defines a function of type ${\\cal D}_1\\rightarrow ({\\cal D}_2\\times {\\cal D}_3)$ .",
"Other familiar functions are defined by the following equations: $\\begin{array}{lcl}eval&=&\\lambda f,x.f(x)\\\\curry&=&=\\lambda g\\lambda x\\lambda y.g(x,y)\\end{array}$ The $curry$ example shows that this notation can be iterated.",
"A distinction is thus made between the terms $\\lambda x,y.x$ and $\\lambda x\\lambda y.x$ which have the types ${\\cal D}_0\\times {\\cal D}_1\\rightarrow {\\cal D}_0$ and ${\\cal D}_0\\rightarrow {\\cal D}_1\\rightarrow {\\cal D}_0$ respectively.",
"Thus, the following equation also holds: $curry(\\lambda x,y.\\tau )=\\lambda x\\lambda y.\\tau $ which relates the multi-variate form to the iterated or curried form.",
"Another true equation is $fix={\\bf fix}(\\lambda F\\lambda f.f(F(f)))$ where $fix$ has type $({\\cal D}\\rightarrow {\\cal D})\\rightarrow {\\cal D}$ and fix has type $(((({\\cal D}\\rightarrow {\\cal D})\\rightarrow {\\cal D})\\rightarrow (({\\cal D}\\rightarrow {\\cal D})\\rightarrow {\\cal D}))\\rightarrow (({\\cal D}\\rightarrow {\\cal D})\\rightarrow {\\cal D}))$ This is the content of Exercise REF .",
"This notation can now be used to define functions using recursion equations.",
"For example, the function $\\sigma $ in Example REF can be characterized by the following equation: $\\sigma =fix(\\lambda f\\lambda n.cond(zero(n),0,f(pred(n))+pred(n))$ which states that $\\sigma $ is the least recursively defined function $f$ whose value at $n$ is $cond(\\ldots )$ .",
"The variable $f$ occurs in the body of the $cond$ expression, but this is just the point of a recursive definition.",
"$f$ is defined in terms of its value on “smaller” input values.",
"The use of the fixed point operator makes the definition explicit by forcing there to be a unique solution to the equation.",
"In an abstraction $\\lambda x,y,z.\\tau $ , the variables $x$ ,$y$ , and $z$ are said to be bound in the term $\\tau $ .",
"Any other variables in $\\tau $ are said to be free variables in $\\tau $ unless they are bound elsewhere in $\\tau $ .",
"Bound variables are simply placeholders for values; the particular variable name chosen is irrelevant.",
"Thus, the equation $\\lambda x.\\tau =\\lambda y.\\tau [y/x]$ is true provided $y$ is not free in $\\tau $ .",
"The notation $\\tau [y/x]$ specifies the substitution of $y$ for $x$ everywhere $x$ occurs in $\\tau $ .",
"The notation $\\tau [\\sigma /x]$ for the substitution of the term $\\sigma $ for the variable $x$ is also legitimate.",
"Semantics of Typed $\\lambda $ -Calculus To show that the equations above with $\\lambda $ –terms are indeed meaningful, the following theorem relating $\\lambda $ –terms and approximable mappings must be proved.",
"Theorem 5.1: Every typed $\\lambda $ –term defines an approximable function of its free variables.",
"Proof Induction on the length of the term and its structure will be used in this proof.",
"Variables Direct since $x\\mapsto x$ is an approximable function.",
"Constants Direct since $x\\mapsto k$ is an approximable function for constant $k$ .",
"Tuples Let $\\tau =\\langle \\sigma _0,\\ldots ,\\sigma _n\\rangle $ .",
"Since the $\\sigma _i$ terms are less complex, they are approximable functions of their free variables by the induction hypothesis.",
"Using Theorem REF (generalized to the multi-variate case) then, $\\tau $ which takes tuples as values also defines an approximable function.",
"Application Let $\\tau =\\sigma _0(\\sigma _1)$ .",
"We assume that the types of the terms are appropriately matched.",
"The $\\sigma _i$ terms define approximable functions again by the induction hypothesis.",
"Recalling the earlier equations, the value of $\\tau $ is the same as the value of $eval(\\sigma _0,\\sigma _1)$ .",
"Since $eval$ is approximable, Theorem REF shows that the term defines an approximable function.",
"Abstraction Let $\\tau =\\lambda x.\\sigma $ .",
"By the induction hypothesis, $\\sigma $ defines a function of its free variables.",
"Let those free variables be of types ${\\cal D}_0,\\ldots ,{\\cal D}_n$ where ${\\cal D}_n$ is the type of $x$ .",
"Then $\\sigma $ defines an approximable function $g:{\\cal D}_0\\times \\ldots \\times {\\cal D}_n\\rightarrow {\\cal D}^{\\prime }$ where ${\\cal D}^{\\prime }$ is the type of $\\sigma $ .",
"Using Theorem REF , the function $curry(g):{\\cal D}_0\\times \\ldots \\times {\\cal D}_{n-1}\\rightarrow ({\\cal D}_n\\rightarrow {\\cal D}^{\\prime })$ yields an approximable function, but this is just the function defined by $\\tau $ .",
"The reader can generalize this proof for multiple bound variables in $\\tau $ .",
"$\\:\\:\\Box $ Given this, the equation $\\tau =\\sigma $ states that the two terms define the same approximable function of their free variables.",
"As an example, $\\lambda x.\\tau =\\lambda y.\\tau [y/x]$ provided $y$ is not free in $\\tau $ since the generation of the approximable function did not depend on the name $x$ but only on its location in $\\tau $ .",
"Other equations such as these are given in the exercises.",
"The most basic rule is shown below.",
"Theorem 5.2: For appropriately typed terms, the following equation is true: $(\\lambda x_0,\\ldots ,x_{1}.\\tau )(\\sigma _0,\\ldots ,\\sigma _{n-1})=\\tau [\\sigma _0/x_0,\\ldots ,\\sigma _{n-1}/x_{n-1}]$ Proof The proof is given for $n=1$ and proceeds again by induction on the length of the term and the structure of the term.",
"Variables This means $(\\lambda x.x)(\\sigma )=\\sigma $ must be true which it is.",
"Constants This requires $(\\lambda x.k)(\\sigma )= k$ must be true which it is for any constant $k$ .",
"Tuples Let $\\tau =\\langle \\tau _0,\\tau _1\\rangle $ .",
"This requires that $(\\lambda x.\\langle \\tau _0,\\tau _1\\rangle )(\\sigma ) =\\langle \\tau _0[\\sigma /x],\\tau _1[\\sigma /x]\\rangle $ must be true.",
"This equation holds since the left-hand side can be transformed using the following true equation: $(\\lambda x.\\langle \\tau _0,\\tau _1\\rangle )(\\sigma ) =\\langle (\\lambda x.\\tau _0)(\\sigma ),(\\lambda x.\\tau _1)(\\sigma )\\rangle $ Then the inductive hypothesis is applied to the $\\tau _i$ terms.",
"Applications Let $\\tau =\\tau _0(\\tau _1)$ .",
"Then, the result requires that the equation $(\\lambda x.\\tau _0(\\tau _1))(\\sigma ) =\\tau _0[\\sigma /x](\\tau _1[\\sigma /x])$ hold true.",
"To see that this is true, examine the approximable functions for the left-hand side of the equation.",
"$\\begin{array}{lcl}\\tau _0&\\mapsto &\\bar{V},x\\rightarrow t_0\\\\\\tau _1&\\mapsto &\\bar{V},x\\rightarrow t_1\\\\\\sigma &\\mapsto &\\bar{V}\\rightarrow s\\\\\\mbox{so}\\\\(\\lambda x.\\tau _0(\\tau _1))(\\sigma )&\\mapsto &\\bar{V}\\rightarrow [(x\\rightarrow t_0(t_1))(s)]\\\\&=&\\bar{V},x\\rightarrow [(x\\rightarrow t_0)(s)]([(x\\rightarrow t_1)(s)])\\end{array}$ From this last term, we use the induction hypothesis.",
"To see why the last step holds, start with the set representing the left-hand side and using the aprroximable mappings for the terms: $\\begin{array}{cl}&(\\lambda x.\\tau _0(\\tau _1))(\\sigma )\\\\\\mapsto &\\bar{V}\\rightarrow [(x\\rightarrow t_0(t_1))(s)]\\\\=&\\lbrace b\\:\\vert \\:\\exists a.a\\in s\\wedge a\\:[x\\rightarrow t_0(t_1)]\\: b\\rbrace \\\\=&\\lbrace b\\:\\vert \\:\\exists a. a\\in s\\wedge a\\:\\lbrace (x,u)\\:\\vert \\:v\\in x\\rightarrow t_1)\\wedge v\\:(x\\rightarrow t_0)\\: u\\rbrace \\: b\\rbrace \\\\=&\\lbrace b\\:\\vert \\:\\exists a.a\\in s\\wedge v\\in (x\\rightarrow t_1)(a)\\wedge v\\:(x\\rightarrow t_0)(a)\\:b\\rbrace \\\\=&\\lbrace b\\:\\vert \\:\\exists a,c.a\\in s\\wedge a\\:(x\\rightarrow t_1)\\: v\\wedge a\\:(x\\rightarrow t_0)\\:c\\wedge v\\:c\\:b\\rbrace \\\\=&\\lbrace b\\:\\vert \\:v\\in [(x\\rightarrow t_1)(s)]\\wedge c\\in (x\\rightarrow t_0)(s) \\wedge v\\:c\\:b\\rbrace \\\\=&\\lbrace b\\:\\vert \\:v\\in [(x\\rightarrow t_1)(s)]\\wedge v\\:[(x\\rightarrow t_0)(s)]\\:b\\rbrace \\\\=&[(x\\rightarrow t_0)(s)]([(x\\rightarrow t_1)(s)])\\end{array}$ Abstractions Let $\\tau =\\lambda y.\\tau _0$ .",
"The required equation is $(\\lambda x.\\lambda y.\\tau _0)(\\sigma )=\\lambda y.\\tau _0[\\sigma /x]$ provided that $y$ is not free in $\\sigma $ .",
"The following true equation applies here: $(\\lambda x.\\lambda y.\\tau )(\\sigma )=\\lambda y.",
"((\\lambda x.\\tau )(\\sigma ))$ To see that this equation holds, let $g$ be a function of $n+2$ free variables defined by $\\tau $ .",
"By Theorem REF , the term $\\lambda x.\\lambda y.\\tau $ defines the function $curry(curry(g))$ of $n$ variables.",
"Call this function $h$ .",
"Thus, $h(v)(\\sigma )(y) = g(v,\\sigma ,y)$ where $v$ is the list of the other free variables.",
"Using a combinator $inv$ which inverts the order of the last two arguments, $h(v)(\\sigma )(y)=curry(inv(g))(v,y)(\\sigma )$ But, $curry(inv(g))$ is the function defined by $\\lambda x.\\tau $ .",
"Thus, we have shown that $(\\lambda x.\\lambda y.\\tau )(\\sigma )(y)=(\\lambda x.\\tau )(\\sigma )$ is a true equation.",
"If $y$ is not free in $\\alpha $ and $\\alpha (y)=\\beta $ is true, then $\\alpha =\\lambda y.\\beta $ must also be true.",
"$\\:\\:\\Box $ If $\\tau ^{\\prime }$ is the term $\\lambda x,y.\\tau $ , then $\\tau ^{\\prime }(x,y)$ is the same as $\\tau $ .",
"This specifies that $x$ and $y$ are not free in $\\tau $ .",
"This notation is used in the proof of the following theorem.",
"Theorem 5.3: The least fixed point of $\\lambda x,y.\\langle \\tau (x,y),\\sigma (x,y)\\rangle $ is the pair with coordinates $fix(\\lambda x.\\tau (x,fix(\\lambda y.\\sigma (x,y))))$ and $fix(\\lambda y.\\sigma (fix(\\lambda x.\\tau (x,y)),y))$ .",
"Proof We are thus assuming that $x$ and $y$ are not free in $\\tau $ and $\\sigma $ .",
"The purpose here is to find the least solution to the pair of equations: $x=\\tau (x,y)~{\\rm and}~y=\\sigma (x,y)$ This generalizes the fixed point equation to two variables.",
"More variables could be included using the same method.",
"Let $y_*=fix(\\lambda y.\\sigma (fix(\\lambda x.\\tau (x,y)),y))$ and $x_*=fix(\\lambda x.\\tau (x,y))$ Then, $x_*=\\tau (x_*,y_*)$ and $\\begin{array}{lcl}y_*&=&\\sigma (fix(\\lambda x.\\tau (x,y_*),y_*))\\\\&=&\\sigma (x_*,y_*).\\end{array}$ This shows that the pair $\\langle x_*,y_*\\rangle $ is one fixed point.",
"Now, let $\\langle x_0,y_0\\rangle $ be the least solution.",
"(Why must a least solution exist?",
"Hint: Consider a suitable mapping of type $({\\cal D}_x\\times {\\cal D}_y)\\rightarrow ({\\cal D}_x\\times {\\cal D}_y)$ .)",
"Thus, we know that $x_0=\\tau (x_0,y_0)$ , $y_0=\\sigma (x_0,y_0)$ , and that $x_0\\sqsubseteq x_*$ and $y_0\\sqsubseteq y_*$ .",
"But this means that $\\tau (x_0,y_0)\\sqsubseteq x_0$ and thus $fix(\\lambda x.\\tau (x,y_0))\\sqsubseteq x_0$ and consequently $\\sigma (fix(\\lambda x.\\tau (x,y_0),y_0))\\sqsubseteq \\sigma (x_0,y_0)\\sqsubseteq y_0$ By the fixed point definition of $y_*$ , $y_*\\sqsubseteq y_0$ must hold as well so $y_0=y_*$ .",
"Thus, $x_*=fix(\\lambda x.\\tau (x,y*))=fix(\\lambda x.\\tau (x,y_0))\\sqsubseteq x_0.$ Thus, $x*=x_0$ must also hold.",
"A similar argument holds for $x_0$ .$\\:\\:\\Box $ The purpose of the above proof is to demonstrate the use of least fixed points in proofs.",
"The following are also true equations: $fix(\\lambda x.\\tau (x))=\\tau (fix(\\lambda x.\\tau (x)))$ and $\\tau (y)\\sqsubseteq y\\:\\Rightarrow \\:fix(\\lambda x.\\tau (x))\\sqsubseteq y$ if $x$ is not free in $\\tau $ .",
"These equations combined with the monotonicity of functions were the methods used in the proof above.",
"Another example is the proof of the following theorem.",
"Theorem 5.4: Let $x$ ,$y$ , and $\\tau (x,y)$ be of type ${\\cal D}$ and let $g:{\\cal D}\\rightarrow {\\cal D}$ be a function.",
"Then the equation $\\lambda x.fix(\\lambda y.\\tau (x,y))=fix(\\lambda g.\\lambda x.\\tau (x,g(x)))$ holds.",
"Proof Let $f$ be the function on the left-hand side.",
"Then, $f(x)=fix(\\lambda y.\\tau (x,y))=\\tau (x,f(x))$ holds using the equations stated above.",
"Therefore, $f=\\lambda x.\\tau (x,f(x))$ and thus $g_0=fix(\\lambda g.\\lambda x.\\tau (x,g(x)))\\sqsubseteq f. $ By the definition of $g_0$ we have $g_0(x)=\\tau (x,g_0(x))$ for any given $x$ .",
"By the definition of $f$ we find that $f(x)=fix(\\lambda y.\\tau (x,y))\\sqsubseteq g_0(x)$ must hold for all $x$ .",
"Thus $f\\sqsubseteq g_0$ and the equation is true.$\\:\\:\\Box $ This proof illustrates the use of inclusion and equations between functions.",
"The following principle was used: $(\\forall x.\\tau \\sqsubseteq \\sigma )\\:\\Rightarrow \\:\\lambda x.\\tau \\sqsubseteq \\lambda x.\\sigma $ This is a restatement of the first part of Theorem REF .",
"Combinators and Recursive Functions Below is a list of various combinators with their definitions in $\\lambda $ -notation.",
"The meanings of those combinators not previously mentioned should be clear.",
"$\\begin{array}{lcl}p_0&=&\\lambda x,y.x\\\\p_1&=&\\lambda x,y.y\\\\pair&=&\\lambda x.\\lambda y.\\langle x,y\\rangle \\\\n-tuple&=&\\lambda x_0\\lambda \\ldots \\lambda x_{n-1}.\\langle x_0,\\ldots ,x_{n-1}\\rangle \\\\diag&=&\\lambda x.\\langle x,x\\rangle \\\\funpair&=&\\lambda f.\\lambda g.\\lambda x.\\langle f(x),g(x)\\rangle \\\\proj^n_i&=&\\lambda x_0,\\ldots ,x_{n-1}.x_i\\\\inv^n_{i,j}&=&\\lambda x_0,\\ldots ,x_i,\\ldots ,x_j,\\ldots ,x_{n-1}.\\langle x0,\\ldots ,x_j,\\ldots ,x_i,\\ldots ,x_{n-1}\\rangle \\\\eval&=&\\lambda f,x.f(x)\\\\curry&=&\\lambda g.\\lambda x.\\lambda y.g(x,y)\\\\comp&=&\\lambda f,g.\\lambda x.g(f(x))\\\\const&=&\\lambda k.\\lambda x.k\\\\{\\bf fix}&=&\\lambda f.fix(\\lambda x.f(x))\\end{array}$ These combinators are actually schemes for combinators since no types have been specified and thus the equations are ambiguous.",
"Each scheme generates an infinite number of combinators for all the various types.",
"One interest in combinators is that they allow expressions without variables—if enough combinators are used.",
"This is useful at times but can be clumsy.",
"However, defining a combinator when the same combination of symbols repeatedly appears is also useful.",
"There are some familiar combinators that do not appear in the table.",
"Combinators such as $cond$ , $pred$ , and $succ$ cannot be defined in the pure $\\lambda $ -calculus but are instead specific to certain domains.",
"They are thus regarded as primitives.",
"A large number of other functions can be defined using these primitives and the $\\lambda $ -notation, as the following theorem shows.",
"Theorem 5.5: For every partial recursive function $h:{{N}}\\rightarrow {{N}}$ , there is a $\\lambda $ -term $\\tau $ of type ${\\cal N}\\rightarrow {\\cal N}$ such that the only constants occurring in $\\tau $ are $cond$ , $succ$ , $pred$ , $zero$ , and 0 and if $h(n)=m$ then $\\tau (n)=m$ .",
"If $h(n)$ is undefined, then $\\tau (n)=\\bot $ holds.",
"$\\tau (\\bot )=\\bot $ is also true.",
"Proof It is convenient in the proof to work with strict functions $f:{\\cal N}^k\\rightarrow {\\cal N}$ such that if any input is $\\bot $ , the result of the function is $\\bot $ .",
"The composition of strict functions is easily shown to be strict.",
"It is also easy to see that any partial function $g:{{N}}^k\\rightarrow {{N}}$ can be extended to a strict approximable function $\\bar{g}:{\\cal N}^k\\rightarrow {\\cal N}$ which yields the same values on inputs for which $g$ is defined.",
"Other input values yield $\\bot $ .",
"We want to show that $\\bar{g}$ is definable with a $\\lambda $ -expression.",
"First we must show that primitive recursive functions have $\\lambda $ -definitions.",
"Primitive recursive functions are formed from starting functions using composition and the scheme of primitive recursion.",
"The starting functions are the constant function for zero and the identity and projection functions.",
"These functions, however, must be strict so the term $\\lambda x,y.x$ is not sufficient for a projection function.",
"The following device reduces a function to its strict form.",
"Let $\\lambda x.cond(zero(x),x,x)$ be a function with $x$ of type ${\\cal N}$ .",
"This is the strict identity function.",
"The strict projection function attempted above can be defined as $\\lambda x,y.cond(zero(y),x,x)$ The three variable projection function can be defined as $\\lambda x,y,z.cond(zero(x),cond(zero(z),y,y),cond(zero(z),y,y))$ While not very elegant, this device does produce strict functions.",
"Strict functions are closed under substitution and composition.",
"Any substitution of a group of functions into another function can be defined with a $\\lambda $ -term if the functions themselves can be so defined.",
"Thus, we need to show that functions obtained by primitive recursion are definable.",
"Let $f:{\\cal N}\\rightarrow {\\cal N}$ , and $g:{\\cal N}^3\\rightarrow {\\cal N}$ be total functions with $\\bar{f}$ and $\\bar{g}$ being $\\lambda $ -definable.",
"We obtain the function $h:{\\cal N}^2\\rightarrow {\\cal N}$ by primitive recursion where $\\begin{array}{lcl}h(0,m)&=&f(m)\\\\h(n+1,m)&=&g(n,m,h(n,m))\\end{array}$ for all $n,m\\in {\\cal N}$ .",
"The $\\lambda $ -term for $\\bar{h}$ is $fix(\\lambda k.\\lambda x,y.cond(zero(x),\\bar{f}(y),\\bar{g}(pred(x),y,k(pred(x),y))))$ Note that the fixed point operator for the domain ${\\cal N}^2\\rightarrow {\\cal N}$ was used.",
"The variables $x$ and $y$ are of type ${\\cal N}$ .",
"The $cond$ function is used to encode the function requirements.",
"The fixed point function is easily seen to be strict and this function is $\\bar{h}$ .",
"Primitive recursive functions are now $\\lambda $ -definable.",
"To obtain partial (i.e., general) recursive functions, the $\\mu $ -scheme (the least number operator) is used.",
"Let $f(n,m)$ be a primitive recursive function.",
"Then, define $h$ , a partial function, as $h(m) =$ the least $n$ such that $f(n,m)=0$ .",
"This is written as $h(m)=\\mu n.f(n,m)=0$ .",
"Since $\\bar{f}$ is $\\lambda $ -definable as has just been shown, let $\\bar{g}=fix(\\lambda g.\\lambda x,y.cond(zero(\\bar{f}(x,y)),x,g(succ(x),y)))$ Then, the desired function $\\bar{h}$ is defined as $\\bar{h}=\\lambda y.\\bar{g}(0,y)$ .",
"It is easy to see that this is a strict function.",
"Note that, if $h(m)$ is defined, clearly $h(m)=\\bar{g}(0,m)$ is also defined.",
"If $h(m)$ is undefined, it is also true that $\\bar{g}(0,m)=\\bot $ due to the fixed point construction but it is less obvious.",
"This argument is left to the reader.$\\:\\:\\Box $ Theorem REF does not claim that all $\\lambda $ -terms define partial recursive functions although this is also true.",
"Further examples of recursion are found in the exercises.",
"Exercises Exercise 5.6: Find the definitions of $\\lambda x,y.\\tau ~{\\rm and}~\\sigma (x,y)$ which use only $\\lambda v$ with one variable and applications only to one argument at a time.",
"Note that use must be made of the combinators $p_0$ , $p_1$ , and $pair$ .",
"Generalize the result to functions of many variables.",
"Exercise 5.7: The table of combinators was meant to show how combinators could be defined in terms of $\\lambda $ -expressions.",
"Can the tables be turned to show that, with enough combinators available, every $\\lambda $ -expression can be defined by combining combinators using application as the only mode of combination?",
"Exercise 5.8: Suppose that $f,g:{\\cal D}\\rightarrow {\\cal D}$ are approximable and $f\\circ g=g\\circ f$ .",
"Show that $f$ and $g$ have a least common fixed point $x=f(x)=g(x)$ .",
"(Hint: See Exercise REF .)",
"If, in addition, $f(\\bot )=g(\\bot )$ , show that $fix(f)=fix(g)$ .",
"Will $fix(f)=fix(f^2)$ ?",
"What if the assumption is weakened to $f\\circ g=g^2\\circ f$ ?",
"Exercise 5.9: For any domain ${\\cal D}$ , ${\\cal D}^\\infty $ can be regarded as consisting of bottomless stacks of elements of ${\\cal D}$ .",
"Using this view, define the following combinators with their obvious meaning: $head:{\\cal D}^\\infty \\rightarrow {\\cal D}$ , $tail:{\\cal D}^\\infty \\rightarrow {\\cal D}^\\infty $ and $push:{\\cal D}\\times {\\cal D}^\\infty \\rightarrow {\\cal D}^\\infty $ .",
"Using the fixed point theorem, argue that there is a combinator $diag:{\\cal D}\\rightarrow {\\cal D}^\\infty $ where for all $x\\in {\\cal D}$ , $diag(x)=\\langle x\\rangle _{n=0}^\\infty $ .",
"(Hint: Try a recursive definition, such as $diag(x)=push(x,diag(x))$ but be sure to prove that all terms of $diag(x)$ are $x$ .)",
"Also introduce by an appropriate recursive definition a combinator $map:({\\cal D}\\rightarrow {\\cal D})^\\infty \\times {\\cal D}\\rightarrow {\\cal D}^\\infty $ where for elements of the proper type $map(\\langle f_n\\rangle _{n=0}^\\infty ,x)=\\langle f_n(x)\\rangle _{n=0}^\\infty $ Exercise 5.10: For any domain ${\\cal D}$ introduce, as a least fixed point, a combinator $while:({\\cal D}\\rightarrow {\\cal T})\\times ({\\cal D}\\rightarrow {\\cal D})\\rightarrow ({\\cal D}\\rightarrow {\\cal D})$ by the recursion equation $while(p,f)(x)=cond(p(x),while(p,f)(f(x)),x)$ Prove that $while(p,while(p,f))=while(p,f)$ Show how $while$ could be used to obtain the least number operator,$\\mu $ , mentioned in the proof of Theorem REF .",
"Generalize this idea to define a combinator $find:{\\cal D}^\\infty \\times ({\\cal D}\\rightarrow {\\cal T})\\rightarrow {\\cal D}$ which means “find the first term in the sequence (if any) which satisfies the given predicate”.",
"Exercise 5.11: Prove the existence of a one-one function $num:{{N}}\\times {{N}}\\leftrightarrow {{N}}$ such that $\\begin{array}{lcl}num(0,0)&=&0\\\\num(n,m+1)&=&num(n+1,m)\\\\num(n+1,0)&=&num(0,n)+1\\end{array}$ Draw a descriptive picture (an infinite matrix) for the function.",
"Find a closed form for the values if possible.",
"Use the function to prove the isomorphism between ${\\cal P}({{N}})$ ,${\\cal P}({{N}}\\times {{N}})$ , and ${\\cal P}({{N}})\\times {\\cal P}({{N}})$ .",
"Exercise 5.12: Show that there are approximable mappings $graph:({\\cal P}({{N}})\\rightarrow {\\cal P}({{N}}))\\rightarrow {\\cal P}({{N}})$ and $fun:{\\cal P}({{N}})\\rightarrow ({\\cal P}({{N}})\\rightarrow {\\cal P}({{N}}))$ where $fun\\circ graph = \\lambda f.f$ and $graph\\circ fun\\sqsubseteq \\lambda x.x$ .",
"(Hint: Using the notation $[n_0,\\ldots ,n_k]=num(n_0,[n_1,\\ldots ,n_k])$ , two such combinators can be given by the formulas $\\begin{array}{lcl}fun(u)(x)&=&\\lbrace m\\:\\vert \\:\\exists n_0,\\ldots ,n_{k-1}\\in x.",
"[n_0+1,\\ldots ,n_{k-1}+1,0,m]\\in u\\rbrace \\\\graph(f)&=&\\lbrace [n_0+1,\\ldots ,n_{k-1}+1,0,m]\\:\\vert \\:m\\in f(\\lbrace n_0,\\ldots ,n_{k-1}\\rbrace )\\rbrace \\end{array}$ where $k$ is a variable - meaning all finite sequences are to be considered.)",
"Introduction to Domain Equations As stressed in the introduction, the notion of computation with potentially infinite elements is an integral part of domain theory.",
"The previous sections have defined the notion of functions over domains, as well as a notation for expressing these functions.",
"In addition, the notion of computation through series of approximations has been addressed.",
"This computation is possible since the functions defined have been approximable and thus continuous.",
"This section addresses the construction of more complex domains with infinite elements.",
"The next section looks specifically at the notion of computability with respect to these infinite elements.",
"The last section looks at another approach to domain construction.",
"New domains have been constructed from existing ones using domain constructors such as the product construction ($\\times $ ), the function space construction ($\\rightarrow $ ) and the sum construction ($+$ ) of Exercise REF .",
"These constructors can be iterated similar to the way that function application was iterated to form recursive function definitions.",
"In this way, domains can be characterized using recursion equations, called domain equations.",
"Domain Equations A domain equation represents an isomorphism between the domain as a whole and the combination of domains that comprise it.",
"These recursive domains are frequently termed reflexive domains since, as in the following example, the domain contains a copy of itself in its structure.",
"Example 6.1: Consider the following domain equation: ${\\cal T}={\\cal A}+({\\cal T}\\times {\\cal T})$ where ${\\cal A}$ is a previously defined domain.",
"This domain can be thought of as containing atomic elements from ${\\cal A}$ or pairs of elements of ${\\cal T}$ .",
"What do the elements of this domain look like?",
"In particular, what are the finite elements of this domain?",
"How is the domain constructed?",
"What is an appropriate approximation ordering for the domain?",
"What do lubs in this domain look like?",
"What is the appropriate notion of consistency?",
"Does this domain even exist?",
"In other words, are we certain a solution to this domain equation exists?",
"And if a solution to the equation exists, is it a unique solution?",
"Each of these questions is examined below.",
"The domain equation tells us that an element of the domain is either an element from ${\\cal A}$ or is a pair of “smaller” elements from ${\\cal T}$ .",
"One method of constructing a sum domain is using pairs where some distinguished element denotes what type an element is.",
"Thus, for some $a\\in {\\cal A}$ , the pair $\\langle \\pi ,a\\rangle $ might represent the element in ${\\cal T}$ for the given element $a$ .",
"For some $s,t\\in {\\cal T}$ , the pair $\\langle \\langle s,t\\rangle ,\\pi \\rangle $ might then represent the element in ${\\cal T}$ for the pair $s,t$ .",
"Thus, $\\pi $ is the distinguished element, and the location of $\\pi $ in the pair specifies the type of the element.",
"The finite elements are either elements in ${\\cal T}$ representing the (finite) elements of ${\\cal A}$ or the pair elements from ${\\cal T}$ whose components are also finite elements in ${\\cal T}$ .",
"The question then arises about infinite elements.",
"Are there infinite elements in this domain?",
"Consider the following fixed point equation for some element for $a\\in {\\cal A}$ : $x=\\langle \\langle a,x\\rangle ,\\pi \\rangle .$ The fixed point of this equation is the infinite product of the element $a$ .",
"Does this element fit the definition for ${\\cal T}$ ?",
"From the informal description of the elements of ${\\cal T}$ given so far, $x$ does qualify as a member of ${\\cal T}$ .",
"Now that some intuition has been developed about this domain, a formal construction is required.",
"Let $\\langle {\\bf A},\\sqsubseteq _A\\rangle $ be the finitary basis used to generate the domain ${\\cal A}$ .",
"Let $\\pi $ be an object such that $\\pi \\notin {\\bf A}$ .",
"Define the bottom element of the finitary basis T as $\\Delta _T=\\langle \\pi ,\\pi \\rangle $ .",
"Next, all the elements of ${\\cal A}$ must be included so define an element in ${\\bf T}$ for each $a\\in {\\bf A}$ as $\\langle \\pi ,a\\rangle $ .",
"Finally, pair elements for all elements in ${\\bf T}$ must exist in ${\\bf T}$ to complete the construction.",
"The set ${\\bf T}$ can be defined inductively as the least set such that: $\\Delta _T\\in {\\bf T}$ , $\\langle \\pi ,a\\rangle \\in {\\bf T}$ whenever $a\\in {\\bf A}$ , $\\langle \\langle \\Delta _T,s\\rangle ,\\pi \\rangle \\in {\\bf T}$ whenever $s\\in {\\bf T}$ (necessary??",
"), $\\langle \\langle t,\\Delta _T\\rangle ,\\pi \\rangle \\in {\\bf T}$ whenever $t\\in {\\bf T}$ (necessary??",
"), and $\\langle \\langle t,s\\rangle ,\\pi \\rangle \\in {\\bf T}$ whenever $s, t\\in {\\bf T}$ .",
"The set can also be characterized by the following fixed point equation: ${\\bf T}=\\lbrace \\Delta _T\\rbrace \\cup \\lbrace \\langle \\pi ,a\\rangle \\:\\vert \\:a\\in {\\bf A}\\rbrace \\cup \\lbrace \\langle \\langle \\Delta _T,s\\rangle ,\\pi \\rangle \\:\\vert \\:s\\in {\\bf T}\\rbrace \\cup \\lbrace \\langle \\langle t,\\Delta _T\\rangle ,\\pi \\rangle \\:\\vert \\:t\\in {\\bf T}\\rbrace \\cup \\lbrace \\langle \\langle t,s\\rangle ,\\pi \\rangle \\:\\vert \\:s,t\\in {\\bf T}\\rbrace .$ A solution must exist for this equation by the fixed point theorem.",
"Now that the basis elements have been defined, we must show how to find lubs.",
"We will again use an inductive definition.",
"$\\langle \\pi ,\\pi \\rangle \\sqcup t=t$ for all $t\\in {\\bf T}$ For $a,b\\in {\\bf A}$ , $\\langle \\pi ,a\\rangle \\sqcup \\langle \\pi ,b\\rangle =\\langle \\pi ,a\\sqcup b\\rangle $ if $a\\sqcup b$ exists in ${\\bf A}$ $\\langle \\langle s,t\\rangle ,\\pi \\rangle \\sqcup \\langle \\langle s^{\\prime },t^{\\prime }\\rangle ,\\pi \\rangle =\\langle \\langle s\\sqcup s^{\\prime },t\\sqcup t^{\\prime }\\rangle ,\\pi \\rangle $ if $s\\sqcup s^{\\prime }$ and $t\\sqcup t^{\\prime }$ exist in ${\\bf T}$ .",
"The lub $\\langle \\pi ,a\\rangle \\sqcup \\langle \\langle s,t\\rangle ,\\pi \\rangle $ does not exist.",
"Next, the notion of consistency needs to be explored.",
"From the definition of lubs given above, the following sets are consistent: The empty set is consistent.",
"Everything is consistent with the bottom element.",
"A set of elements all from the basis A is consistent in T if the set of elements is consistent in A.",
"A set of product elements in T is consistent if the left component elements are consistent and the right component elements are consistent.",
"These conditions derive from the sum and product nature of the domain.",
"The approximation ordering in the basis has the following inductive definition: $\\Delta _T\\sqsubseteq _T s$ for all $s\\in {\\bf T}$ $y\\sqsubseteq _Tu\\sqcup \\Delta _T$ whenever $y\\sqsubseteq _Tu$ $\\langle \\pi ,a\\rangle \\sqsubseteq _T\\langle \\pi ,b\\rangle $ whenever $a\\sqsubseteq _Ab$ $\\langle \\langle s,t\\rangle ,\\pi \\rangle \\sqsubseteq _T\\langle \\langle u,v\\rangle ,\\pi \\rangle $ whenever $s\\sqsubseteq _Tu$ and $t\\sqsubseteq _Tv$ The next step is to verify that ${\\bf T}$ is indeed a finitary basis.",
"The basis is still countable.",
"The approximation is clearly a partial order.",
"The existence of lubs of finite bounded (i.e., consistent) subsets must be verified.",
"The definition of consistency gives us the requirements for a bounded subset.",
"Each of the conditions for consistency are examined inductively since the definitions are all inductive: The lub of the empty set is the bottom element $\\Delta _T$ .",
"The lub of a set containing the bottom element is the lub of the set without the bottom element which must exist by the induction hypothesis.",
"The lub of a set of elements all from the ${\\bf A}$ is the element in ${\\bf T}$ for the lub in ${\\bf A}$ .",
"This element must exist since ${\\bf A}$ is a finitary basis and all elements from ${\\bf A}$ have corresponding elements in ${\\bf T}$ .",
"The lub of a set of product elements is the pair of the lub of the left components and the lub of the right components.",
"These exist by the induction hypothesis.",
"Thus, a finitary basis has been created; the domain is formed as always from the basis.",
"The solution to the domain equation has been found since any element in the domain ${\\cal T}$ is either an element representing an element in ${\\cal A}$ or is the product of two other elements in ${\\cal T}$ .",
"Similarly, any element found on the left-hand side must also be in the domain ${\\cal T}$ by the construction.",
"Thus, the domain ${\\cal T}$ is identical to the domain ${\\cal A}+({\\cal T}\\times {\\cal T})$ .",
"To look at the question concerning the existence and uniqueness of the solution to this domain equation, recall the fixed point theorem.",
"This theorem states that a fixed point set exists for any approximable mapping over a domain.",
"Subdomains In Section , the concept of a universal domain is introduced.",
"A universal domain is a domain which contains all other domains as sub-domains.",
"These sub-domains are, roughly speaking, the image of approximable functions over the universal domain.",
"The domain equation for ${\\cal T}$ can be viewed as an approximable mapping over the universal domain.",
"As such, the fixed point theorem states that a least fixed point set for the function does exist and is unique.",
"Sub-domains are defined formally below.",
"Looking again at the informal discussion concerning the elements of the domain ${\\cal T}$ , the infinite element proposed does fit into the formal definition for elements of ${\\cal T}$ .",
"This element is an infinite tree with all left sub-trees containing only the element $a$ .",
"For this infinite element to be computable, it must be the lub of some ascending chain of finite approximations to it.",
"The element $x$ can, in fact, be defined by the following ascending sequence of finite trees: $\\begin{array}{lcl}x_0&=&\\bot \\\\x_{n+1}&=&\\langle \\langle a,x_n\\rangle ,\\pi \\rangle \\\\x&=&\\bigsqcup ^\\infty _{n=0}x_n\\end{array}$ Thus, using domain equations, a domain has been defined recursively.",
"This domain includes infinite as well as finite elements and allows computation on the infinite elements to proceed using the finite approximations, as with the more conventionally defined domains presented earlier.",
"The final topic of this section is the notion of a sub-domain.",
"Informally, a sub-domain is a structured part of a larger domain.",
"Earlier, a domain was described as a sub-domain of the universal domain.",
"Thus, the sub-domain starts with a subset of the elements of the larger domain while retaining the approximation ordering, consistency relation and lub relation, suitably restricted to the subset elements.",
"Definition 6.2: [Sub-Domain] A domain $\\langle {\\cal R},\\sqsubseteq _R\\rangle $ is a sub-domain of a domain $\\langle {\\cal D},\\sqsubseteq _D\\rangle $ , denoted ${\\cal R}\\lhd {\\cal D}$ iff ${\\cal R}\\subseteq {\\cal D}$ - The elements of ${\\cal R}$ are a subset of the elements of ${\\cal D}$ .",
"$\\bot _R=\\bot _D$ - The bottom elements are the same.",
"For $x,y\\in {\\cal R}$ , $x\\sqsubseteq _Ry\\iff x\\sqsubseteq _Dy$ - The approximation ordering for ${\\cal R}$ is the approximation ordering for ${\\cal D}$ restricted to elements in ${\\cal R}$ .",
"For $x,y,z\\in {\\cal R}$ , $x\\sqcup _Ry=z$ iff $x\\sqcup _Dy=z$ - The lub relation for ${\\cal R}$ is the lub relation for ${\\cal D}$ restricted to elements in ${\\cal R}$ .",
"${\\cal R}$ is a domain.",
"Equivalently, a sub-domain can be thought of as the image of an approximable function which approximates the identity function (also termed a projection).",
"The notion of a sub-domain is used in the final section in the discussions about the universal domain.",
"This mapping between the domains can be formalized as follows: Theorem 6.3: If ${\\cal D}\\lhd {\\cal E}$ , then there exists a projection pair of approximable mappings $i:{\\cal D}\\rightarrow {\\cal E}$ and $j:{\\cal E}\\rightarrow {\\cal D}$ where $j\\circ i={{\\sf I}}_{\\cal D}$ and $i\\circ j\\sqsubseteq {{\\sf I}}_{\\cal E}$ where $i$ and $j$ are determined by the following equations: $\\begin{array}{lcl}i(x)&=&\\lbrace y\\in {\\bf E}\\:\\vert \\:\\exists z\\in x.z\\sqsubseteq y\\rbrace \\\\j(y)&=&\\lbrace x\\in {\\bf D}\\:\\vert \\:x\\in y\\rbrace \\end{array}$ for all $x\\in {\\cal D}$ and $y\\in {\\cal E}$ .",
"The proof is left as an exercise.",
"By the definition of a sub-domain, it should be clear that ${\\cal D}_0\\lhd {\\cal E}\\wedge {\\cal D}_1\\lhd {\\cal E}\\:\\Rightarrow \\:({\\cal D}_0\\lhd {\\cal D}_1\\iff {\\cal D}_0\\subseteq {\\cal D}_1)$ Using this observation, the sub-domains of a domain can be ordered.",
"Indeed, the following theorem is a consequence of this ordering.",
"Theorem 6.4: For a given domain ${\\cal D}$ , the set of sub-domains $\\lbrace {\\cal D}_0\\:\\vert \\:{\\cal D}_0\\lhd {\\cal D}\\rbrace $ form a domain.",
"The proof proceeds using the inclusion relation defined as an approximation ordering and is left as an exercise.",
"Finally, a converse of Theorem REF can also be established: Theorem 6.5: For two domains ${\\cal D}$ and ${\\cal E}$ , if there exists a projection pair $i:{\\cal D}\\rightarrow {\\cal E}$ and $j:{\\cal E}\\rightarrow {\\cal D}$ with $j\\circ i={\\sf I}_{\\cal D}$ and $i\\circ j\\sqsubseteq {\\sf I}_{\\cal E}$ , then $\\exists {\\cal D}^{\\prime }\\lhd {\\cal E}$ where ${\\cal D}\\approx {\\cal D}^{\\prime }$ .",
"Proof We show that $i$ maps finite elements to finite elements and that ${\\cal D}^{\\prime }$ is the image of ${\\cal D}$ in ${\\cal E}$ .",
"For some $x\\in {\\bf D}$ with ${\\cal I}_x$ as the principal ideal of $x$ , we can write $i({\\cal I}_x)=\\sqcup \\lbrace {\\cal I}_y\\:\\vert \\:y\\in i({\\cal I}_x)\\rbrace $ Applying $j$ to both sides we get ${\\cal I}_x=j\\circ i({\\cal I}_x)=\\sqcup \\lbrace j({\\cal I}_y)\\:\\vert \\:y\\in i({\\cal I}_x)\\rbrace $ since $j\\circ i={\\sf I}_D$ and $j$ is continuous by assumption.",
"But, since $x\\in {\\cal I}_x$ , $x\\in j({\\cal I}_y)$ for some $y\\in i({\\cal I}_x)$ .",
"This means that ${\\cal I}_x\\subseteq j({\\cal I}_y)$ and thus $i({\\cal I}_x)\\subseteq i\\circ j({\\cal I}_y) \\subseteq {\\cal I}_y$ Since ${\\cal I}_y\\subseteq i({\\cal I}_x)$ must hold by the construction, $i({\\cal I}_x) ={\\cal I}_y$ .",
"This proves that finite elements are mapped to finite elements.",
"Next, consider the value for $i(\\bot _D)$ .",
"Since $\\bot _D\\sqsubseteq _Dj(\\bot _E)$ , $i(\\bot _D)\\sqsubseteq \\bot _E$ .",
"Thus $i(\\bot _D)=\\bot _E$ .",
"Thus, ${\\cal D}$ is isomorphic to the image of $i$ in ${\\cal E}$ .",
"We still must show that ${\\cal D}^{\\prime }$ is a domain.",
"Thus, we need to show that if a lub exists in ${\\cal E}$ for a finite subset in ${\\cal D}^{\\prime }$ , then the lub is also in ${\\cal D}^{\\prime }$ .",
"Let $y^{\\prime },z^{\\prime }\\in {\\bf D}^{\\prime }$ and $y^{\\prime }\\sqcup z^{\\prime }=x^{\\prime }\\in {\\bf E}$ .",
"Then, there exists $y,z\\in {\\bf D}$ such that $i({\\cal I}_y)={\\cal I}_{y^{\\prime }}$ and $i({\\cal I}_z)={\\cal I}_{z^{\\prime }}$ which implies that ${\\cal I}_y=j({\\cal I}_{y^{\\prime }})$ and ${\\cal I}_z=j({\\cal I}_{z^{\\prime }})$ .",
"Since ${\\cal I}_{y^{\\prime }}\\sqsubseteq {\\cal I}_{x^{\\prime }}$ and $j({\\cal I}_{y^{\\prime }})\\sqsubseteq j({\\cal I}_{x^{\\prime }})$ by monotonicity, $y\\in j({\\cal I}_{x^{\\prime }})$ must hold.",
"By the same reasoning, $z\\in j({\\cal I}_{x^{\\prime }})$ .",
"But then $x=y\\sqcup z\\in j({\\cal I}_{x^{\\prime }})$ must also hold and thus $y\\sqcup z \\in {\\cal D}$ since the element $j({\\cal I}_{x^{\\prime }})$ must be an ideal.",
"But, $\\begin{array}{lcl}{\\cal I}_y\\sqsubseteq {\\cal I}_x&\\:\\Rightarrow \\:& {\\cal I}_{y^{\\prime }}\\sqsubseteq i({\\cal I}_{x})\\\\{\\cal I}_z\\sqsubseteq {\\cal I}_x&\\:\\Rightarrow \\:& {\\cal I}_{z^{\\prime }}\\sqsubseteq i({\\cal I}_{x})\\end{array}$ This implies that $y^{\\prime }\\sqcup z^{\\prime }=x^{\\prime }\\in i({\\cal I}_x)$ .",
"We already know that $x\\in j({\\cal I}_{x^{\\prime }})$ so $i({\\cal I}_x)\\sqsubseteq {\\cal I}_{x^{\\prime }}$ .",
"Thus, $i({\\cal I}_x)={\\cal I}_{x^{\\prime }}$ and thus, $x^{\\prime }\\in {\\bf D^{\\prime }}$ .$\\:\\:\\Box $ Exercises Exercise 6.6: Show that there must exist domains satisfying $\\begin{array}{lcll}{\\cal A}&=&{\\cal A}+({\\cal A}\\times {\\cal B})&{\\rm and}\\\\{\\cal B}&=&{\\cal A}+{\\cal B}\\end{array}$ Decide what the elements will look like and define ${\\cal A}$ and ${\\cal B}$ using simultaneous fixed points.",
"Exercise 6.7: Prove Theorem REF Exercise 6.8: Prove Theorem REF Exercise 6.9: Show that if ${\\cal A}$ and ${\\cal B}$ are finite systems, that ${\\cal D}\\unlhd {\\cal E}\\unlhd {\\cal D}\\:\\Rightarrow \\:{\\cal D}\\approx {\\cal E}$ where ${\\cal D}\\approx {\\cal D}^{\\prime }$ and ${\\cal D}^{\\prime }\\lhd {\\cal E}$ is denoted ${\\cal D}^{\\prime }\\unlhd {\\cal E}$ .",
"Computability in Effectively Given Domains In the previous sections, we gave considerable emphasis to the notion of computation using increasingly accurate approximations of the input and output.",
"This section defines this notion of computability more formally.",
"In Section 5, we found that partial functions over the natural numbers were expressible in the $\\lambda $ -notation.",
"This relationship characterizes computation for a particular domain.",
"To describe computation over domains in general, a broader definition is required.",
"The way a domain is presented impacts the way computations are performed over it.",
"Indeed, the theorems of recursive function theory [6] rely in part on the normal presentation of the natural numbers.",
"A presentation for a domain is an enumeration of the elements of the domain.",
"The standard presentation of the natural numbers is simply the numbers in ascending order beginning with 0.",
"There are many permutations of the natural numbers, each of which can be considered a presentation.",
"Computation with these non-standard presentations may be impossible; that is a computable function on the standard presentation may be non-computable over a non-standard presentation.",
"Therefore, an effective presentation for a domain is defined as a presentation which makes the required information computable.",
"Effective Presentations Information about elements in a domain can be characterized completely by looking at the finite elements and their relationships.",
"Thus a presentation must enumerate the finite elements and allow the consistency and lub relationships on these elements to be computed to allow this style of computation.",
"The consistency relation and the lub relation depend on each other.",
"For example, if a set of elements is consistent, a lub must exist for the set.",
"Given that a set is consistent, the lub can be found in finite time by just enumerating the elements and checking to see if this element is the lub.",
"However, if the set is inconsistent, the enumeration will not reveal this fact.",
"Thus, the consistency relation must be assumed to be recursive in an effective presentation.",
"Exercise REF provides a description of presentations that should clarify the assumptions made.",
"Formally, a presentation is defined as follows: Definition 7.1: [Effective Presentation] The presentation of a finitary basis D is a function $\\pi :{{N}}\\rightarrow {\\bf D}$ such that $\\pi (0)=\\Delta _D$ and the range of $\\pi $ is the set of finite elements of D. The definition holds for a domain ${\\cal D}$ as well.",
"A presentation $\\pi $ is effective iff The consistency relation ($\\exists k.\\pi _i\\sqsubseteq \\pi _k\\wedge \\pi _j\\sqsubseteq \\pi _k$ ) for elements $\\pi _i$ and $\\pi _j$ is recursiveRecursive in this context means that the relation is decidable.",
"over $i$ and $j$ .",
"The lub relation ($\\pi _k=\\pi _i\\sqcup \\pi _j$ ) is recursive over $i$ , $j$ , and $k$ .",
"This definition supports our intuition about domains; we have stated that the important information about a domain is the set of finite elements, the ordering and consistency relationships between the elements and the lub relation.",
"Thus, an effective presentation provides, in a suitable (that is computable) form, the basic information about the structure and elements of a domain.",
"A presentation can also be viewed as an enumeration of the elements of the domain with the position of an element in the enumeration given by the index corresponding to the integer input for that element in the presentation function with the 0 element representing $\\bot $ .",
"This perspective is used in the majority of the proofs.",
"Computability Now that the presentation of a domain has been formalized, the notion of computability can be formally defined.",
"Thus, Definition 7.2: [Computable Mappings] Given two domains, ${\\cal D}$ and ${\\cal E}$ with effective presentations $\\pi _1$ and $\\pi _2$ respectively, an approximable mapping $f:{\\bf D}\\rightarrow {\\bf E}$ is computable iff the relation $x_n\\:f\\:y_m$ is recursively enumerable in $n$ and $m$ .",
"By considering the domain ${\\cal D}$ to be a single element domain, the above definition applies not only to computable functions but also to computable elements.",
"For $d\\in {\\cal D}$ where $d$ is the only element in the domain, the element $e=f(d)\\in {\\cal E}$ defines an element in ${\\cal E}$ .",
"The definition states that $e$ is a computable iff the set $\\lbrace m\\in {{N}}\\:\\vert \\:y_m\\sqsubseteq e\\rbrace $ is a recursively enumerable set of integers.",
"Clearly if the set of elements approximating another is finite, the set is recursive.",
"The notion of a recursively enumerable set simply requires that all elements approximating the element in question be listed eventually.",
"The computation then proceeds by accepting an enumeration representing the input element and enumerating the elements that approximate the desired output element.",
"Now that the notions of computability and effective presentations have been formalized, the methods of constructing domains and functions will be addressed.",
"The proof of the next theorem is trivial and is left to the reader.",
"Theorem 7.3: The identity map on an effectively given domain is computable.",
"The composition of computable mappings on effectively given domains are also computable.",
"The following corollary is a consequence of this theorem: Corollary 7.4: For computable function $f:{\\cal D}\\rightarrow {\\cal E}$ and a computable element $x\\in {\\cal D}$ , the element $f(x)\\in {\\cal E}$ is computable.",
"In addition, the standard domain constructors maintain effective presentations.",
"Theorem 7.5: For domains ${\\cal D}_0$ and ${\\cal D}_1$ with effective presentations, the domains ${\\cal D}_0+{\\cal D}_1~{\\rm and}~ {\\cal D}_0\\times {\\cal D}_1$ are also effectively given.",
"In addition, the projection functions are all computable.",
"Finally, if $f$ and $g$ are computable maps, then so are $f+g$ and $f\\times g$ .",
"Proof Let $\\lbrace X_i\\:\\vert \\:i\\in {{N}}\\rbrace $ be the enumeration of ${\\cal D}_0$ and $\\lbrace Y_i\\:\\vert \\:i\\in {{N}}\\rbrace $ be the enumeration of ${\\cal D}_1$ .",
"Another method of sum construction is to use two distinguishing elements in the first position to specify the element type.",
"Thus, a sum domain can be defined as follows: ${\\cal D}_0+{\\cal D}_1=\\lbrace (\\Delta _0,\\Delta _1)\\rbrace \\cup \\lbrace (0,x)\\:\\vert \\:x\\in {\\cal D}_0\\rbrace \\cup \\lbrace (1,y)\\:\\vert \\:y\\in {\\cal D}_1\\rbrace $ The enumeration can then be defined as follows for $n\\in {{N}}$ : $\\begin{array}{lcl}Z_0&=&(\\Delta _0,\\Delta _1)\\\\Z_{2n+1}&=&(0,X_n)\\\\Z_{2n+2}&=&(1,Y_n)\\end{array}$ The proof that $Z_i$ is an effective presentation is left as an exercise.",
"For the product construction, the domain appears as follows: ${\\cal D}_0\\times {\\cal D}_1=\\lbrace (x,y)\\:\\vert \\:x\\in {\\cal D}_0,y\\in {\\cal D}_1\\rbrace $ The enumeration can be defined in terms of the functions $p:{{N}}\\rightarrow {{N}}$ , $q:{{N}}\\rightarrow {{N}}$ , and $r:({{N}}\\times {{N}})\\rightarrow {{N}}$ where for $m$ , $n$ , $k\\in {{N}}$ : $\\begin{array}{lcl}p(r(n,m))&=&n\\\\q(r(n,m))&=&m\\\\r(p(k),q(k))&=&k\\end{array}$ Thus, $r$ is a one-to-one pairing function (see Exercise REF ) of which there are several.",
"The functions $p$ and $q$ extract the indices from the result of the pairing function.",
"The enumeration for the product domain is then defined as follows: $W_i = (X_{p(i)},Y_{q(i)})$ The proof that this is an effective presentation is also left as an exercise.",
"For the combinators, the relations will be defined in terms of the enumeration indices.",
"For example, $\\begin{array}{lcl}X_n\\:in_0\\:Z_m&\\iff & m=0~{\\rm or}\\\\&&\\exists k.m=2k+1\\wedge X_k\\sqsubseteq X_n\\\\W_k\\:proj_1\\:Y_m&\\iff & Y_m\\sqsubseteq Y_{q(k)}\\end{array}$ The reader should verify that these sets are recursively enumerable.",
"For this proof, recall that recursively enumerable sets are closed under conjunction, disjunction, substituting recursive functions, and applying an existential quantifier to the front of a recursive predicate.",
"The proof for the other combinators is left as an exercise.",
"$\\:\\:\\Box $ Product spaces formalize the notion of computable functions of several variables.",
"Note that the proof of Theorem REF shows that substitution of computable functions of severable variables into other computable functions are still computable.",
"The next step is to show that the function space constructor preserves effectiveness.",
"Theorem 7.6: For domains ${\\cal D}_0$ and ${\\cal D}_1$ with effective presentations, the domain ${\\cal D}_0\\rightarrow {\\cal D}_1$ also has an effective presentation.",
"The combinators $apply$ and $curry$ are computable if all input domains are effectively given.",
"The computable elements of the domain ${\\cal D}_0\\rightarrow {\\cal D}_1$ are the computable maps for ${\\bf D_0}\\rightarrow {\\bf D_1}$ .",
"Proof Let ${\\cal D}_0=\\lbrace X_i\\:\\vert \\:i\\in {{N}}\\rbrace $ and ${\\cal D}_1=\\lbrace Y_i\\:\\vert \\:i\\in {{N}}\\rbrace $ be the presentations for the domains.",
"The elements of ${\\bf D_0}\\rightarrow {\\bf D_1}$ are finite step functions which respect the mapping of some subset of ${\\bf D_0}\\times {\\bf D_1}$ .",
"Given the enumeration, each element can be associated with a set $\\lbrace (X_{n_i},Y_{m_i})\\:\\vert \\:\\exists q.",
"1\\le i\\le q\\rbrace $ Thus, there is a finite set of integers pairs that determine the element.",
"Given the definition of consistency from Theorem REF for elements in the function space domain and the decidability of consistency in ${\\cal D}_0$ and ${\\cal D}_1$ , consistency of any finite set of this form is decidable (tedious but decidable since all elements must be checked with all others, etc).",
"Since consistency is decidable, a systematic enumeration of pair sets which are consistent can be made; this enumeration is simply the enumeration of ${\\cal D}_0\\rightarrow {\\cal D}_1$ .",
"Finding the lub consists of making a finite series of tests to find the element that is the lub, which must exist since the set is consistent and we have closure on lubs of finite consistent subsets.",
"Finding the lub requires a finite series of checks in both ${\\cal D}_0$ and ${\\cal D}_1$ but these checks are decidable.",
"Thus, the lub relation is also decidable in ${\\cal D}_0\\rightarrow {\\cal D}_1$ .",
"This shows that ${\\cal D}_0\\rightarrow {\\cal D}_1$ is effectively given.",
"To show that $apply$ and $curry$ are computable, the mappings need to be examined.",
"The mapping defined for apply is $(F,a)\\:apply\\: b\\iff a\\:F\\:b$ The function $F$ is the lub of all the finite step functions that are consistent with it.",
"As such, $F$ can be viewed as the canonical representative of this set.",
"Since $F$ is a finite step function, this relation is decidable.",
"As such, the $apply$ relation is recursive and not just recursively enumerable and $apply$ is a computable function.",
"The reasoning for $curry$ is similar in that the relations are studied.",
"Given the increase in the number of domains, the construction is more tedious and is left for the exercises.",
"To see that the computable elements correspond to the computable maps, recall the relationship shown in Theorem REF between the maps and the elements in the function space.",
"Thus, we have $a\\:f\\:b \\iff b\\in f({\\cal I}_a)~{\\rm or}~{\\cal I}_b\\sqsubseteq f({\\cal I}_a)$ Since $f$ is a computable map, we know that the pairs in the map are recursively enumerable.",
"Using the previous techniques for deciding consistency of finite sets, the set of elements consistent with $f$ can be enumerated.",
"But this set is simply the ideal for $f$ in the function space.",
"The converse direction is trivial.",
"$\\:\\:\\Box $ The final combinator to be discussed, and perhaps the most important, is the fixed point combinator.",
"Theorem 7.7: For any effectively given domain, ${\\cal D}$ , the combinator $fix:({\\cal D}\\rightarrow {\\cal D})\\rightarrow {\\cal D}$ is computable.",
"Proof Let $\\lbrace X_n\\:\\vert \\:n\\in {{N}}\\rbrace $ be the presentation of the domain ${\\cal D}$ .",
"Recall that for $f\\in {\\cal D}\\rightarrow {\\cal D}$ , $f\\:fix\\:X\\iff \\exists k\\in {{N}}.\\Delta \\:f\\:X_1\\:f\\ldots f\\:X_k\\wedge X_k=X$ All of the checks in this finite sequence are decidable since ${\\cal D}$ is effectively given.",
"In addition, existential quantification of a decidable predicate gives a recursively enumerable predicate.",
"Thus, $fix$ is computable.",
"$\\:\\:\\Box $ Recap Now that this has been formalized, what has been accomplished?",
"The major consequence of the theorems to this point is that any expression over effectively given domains (that is effectively given types) combined with computable constants using the $\\lambda $ -notation and the fixed point combinator is a computable function of its free variables.",
"Such functions, applied to computable arguments, yield computable values.",
"These functions also have computable least fixed points.",
"All this gives us a mathematical programming language for defining computable operations.",
"Combining this language with the specification of types with domain equations gives a powerful language.",
"As an example, the effectiveness of the domain ${\\cal T}$ from Example REF is studied.",
"The complete proof is left as an exercise.",
"Example 7.8: Recall the domain ${\\cal T}$ from the previous section.",
"This domain is characterized by the domain equation ${\\cal T}={\\cal A}+({\\cal T}\\times {\\cal T})$ for some domain ${\\cal A}$ .",
"If ${\\cal A}$ is effectively given, we wish to show that ${\\cal T}$ is effectively given as well.",
"The elements are either atomic elements from ${\\cal A}$ or are pairs from ${\\cal T}$ .",
"Let $A=\\lbrace A_i\\:\\vert \\:i\\in {{N}}\\rbrace $ be the enumeration for ${\\cal A}$ .",
"An enumeration for ${\\cal T}$ can be defined as follows: $\\begin{array}{lcl}T_0&=&\\bot _T\\\\T_{2n+1}&=&3*A_n\\\\T_{2n+2}&=&3*T_{p(n)}+1\\cup 3*T_{q(n)}+2\\end{array}$ where for $A$ , a set of indices, $m*A+k=\\lbrace m*n+k\\:\\vert \\:n\\in A\\rbrace $ .",
"The functions $p$ and $q$ here are the inverses of the pairing function $r$ defined in Theorem REF .",
"These functions must be defined such that $p(n)\\le n$ and $q(n)\\le n$ so that the recursion is well defined by taking smaller indices.",
"The rest of the proof is left to the exercises.",
"Specifically, the claim that ${\\cal T}=\\lbrace T_i\\rbrace $ should be verified as well as the effectiveness of the enumeration.",
"These proofs rely either on the effectiveness of ${\\cal A}$ , on the effectiveness of elements in ${\\cal T}$ with smaller indices, or are trivial.",
"The final example uses the powerset construction.",
"We have repeatedly used the fact that a powerset is a domain.",
"Its effectiveness is now verified.",
"Example 7.9: Specifically, the powerset of the natural numbers, ${\\cal {P}({{N}})}$ is considered.",
"In this domain, all elements are consistent, and there is a top element, denoted $\\omega $ , which is the set of all natural numbers.",
"The ordering is the subset relation.",
"The lub of two subsets is the union of the two subsets, which is decidable.",
"To enumerate the finite subsets, the following enumeration is used: $E_n=\\lbrace k\\:\\vert \\:\\exists i,j.",
"i< 2^k\\wedge n=i+2^k+j*2^{k+1}\\rbrace $ This says that $k\\in E_n$ if the $k$ bit in the binary expansion of $n$ is a 1.",
"All finite subsets of ${{N}}$ are of the form $E_n$ for some $n$ .",
"Various combinators for ${\\cal P}({{N}})$ are presented in Exercise REF .",
"Exercises Exercise 7.10: Show that an effectively given domain can always be identified with a relation $INCL(n,m)$ on integers where the derived relations $\\begin{array}{lcl}CONS(n,m)&\\iff &\\exists k.INCL(k,n)\\wedge INCL(k,m)\\\\MEET(n,m,k)&\\iff &\\forall j.",
"[INCL(j,k)\\iff INCL(j,n)\\wedge INCL(j,m)]\\end{array}$ are recursively decidable and where the following axioms hold: $\\forall n.INCL(n,n)$ $\\forall n,m,k.",
"INCL(n,m)\\wedge INCL(m,k)\\:\\Rightarrow \\:INCL(n,k)$ $\\exists m.\\forall n. INCL(n,m)$ $\\forall n,m.",
"CONS(n,m)\\:\\Rightarrow \\:\\exists k.MEET(n,m,k)$ Exercise 7.11: Finish the proof of Theorem REF .",
"Exercise 7.12: Complete the proof of Theorem REF by defining $curry$ as a relation and showing it computable.",
"Is the set recursively enumerable or is it recursive?",
"Exercise 7.13: Two effectively given domains are effectively isomorphic iff $\\ldots $ Complete the statement of the theorem and prove it.",
"Exercise 7.14: Complete the proof about the powerset in Example REF .",
"Show that the combinators $fun$ and $graph$ from Exercise REF are computable.",
"Show the same for $\\lambda x,y.x\\cap y$ $\\lambda x,y.x\\cup y$ $\\lambda x,y.x+ y$ where for $x,y\\in {\\cal P}({{N}})$ , $x+y=\\lbrace n+m\\:\\vert \\:n\\in x, m\\in y\\rbrace $ What are the computable elements of ${\\cal P}({{N}})$ ?",
"Sub-Spaces of the Universal Domain To have a flexible method of solving domain equations and yielding effectively given domains as the solutions, the domains will be embedded in a universal domain which is “big” enough to hold all other domains as sub-domains.",
"This universal domain is shown to be effectively presented, and the mappings which define the sub-spaces are shown to be computable.",
"First, the correspondence between sub-spaces and mappings called retractions is investigated, leading us to the definition of mappings called projections.",
"It is then shown that these definitions can be written out using the $\\lambda $ -calculus notation, demonstrating the power of our mathematical programming language.",
"Retractions and Projections We start with the definition of retractions.",
"Definition 8.1: [Retractions] A retraction of a given domain ${\\cal E}$ is an approximable mapping $a:{\\bf E}\\rightarrow {\\bf E}$ such that $a\\circ a=a$ .",
"Thus, a retraction is the identity function on objects in the range of the retraction and maps other elements into range.",
"The next theorem relates these sets to sub-spaces.",
"Theorem 8.2: If ${\\cal D}\\lhd {\\cal E}$ and if $a:{\\bf E}\\rightarrow {\\bf E}$ is defined such that $X\\:a\\:Z \\iff \\exists Y\\in {\\cal D}.",
"Z\\sqsubseteq Y\\sqsubseteq X$ for all $X,Z\\in {\\bf E}$ , then $a$ is a retraction and ${\\cal D}$ is isomorphic to the fixed point set of $a$ , the set $\\lbrace y\\in {\\cal E}\\:\\vert \\:a(y)=y\\rbrace $ , ordered under inclusion.",
"Proof That $a$ is an approximable map is a direct consequence of the definition of sub-space (Definition REF ).",
"By Theorem REF , a projection pair, $i$ and $j$ , exist for ${\\cal D}$ and this tells us that $a=i\\circ j$ (also showing $a$ approximable since approximable mappings are closed under composition).",
"Theorem REF also tells us that $j\\circ i={\\sf I}_D$ .",
"To show that $a$ is a retraction, $a\\circ a=a$ must be established.",
"Thus, $a\\circ a = i\\circ j\\circ i\\circ j = i\\circ {\\sf I}_D\\circ j = i\\circ j =a$ holds, showing that $a$ is a retraction.",
"We now need to show the isomorphism to ${\\cal D}$ .",
"For $x\\in {\\cal D}$ , $i(x)\\in {\\cal E}$ and we can calculate: $a(i(x))=i\\circ j\\circ i(x) = i\\circ {\\sf I}_D(x) = i(x)$ Thus, $i(x)$ is in the fixed point set of $a$ .",
"For the other direction, let $a(y)=y$ .",
"Then $i(j(y)) = y$ holds.",
"But, $j(y)\\in {\\cal D}$ , so $i$ must map ${\\cal D}$ one-to-one and onto the fixed point set of $a$ .",
"Since $i$ and $j$ are approximable, they are certainly monotonic, and thus the map is an isomorphism with respect to set inclusion.",
"$\\:\\:\\Box $ Not all retractions are associated with a sub-domain relationship.",
"The retractions defined in the above theorem are all subsets as relations of the identity relation.",
"The retractions for sub-domains are characterized by the following definition: Definition 8.3: [Projections] A retraction $a:{\\cal E}\\rightarrow {\\cal E}$ is a projection if $a\\subseteq {\\sf I}_E$ as relations.",
"The retraction is finitary iff its fixed point set is isomorphic to some domain.",
"An example is in order.",
"Example 8.4: Consider a two element system, ${\\bf O}$ with objects $\\Delta $ and 0.",
"For any basis ${\\bf D}$ that is not trivial (has more than one element), ${\\bf O}$ comes from a retraction on ${\\bf D}$ .",
"Define a combinator $check:{\\bf D}\\rightarrow {\\bf O}$ by the relation $x\\:check\\: y \\iff y=\\Delta ~{\\rm or}~x\\ne \\Delta _D$ Thus, $check(x)=\\bot _O\\iff x=\\bot _D$ .",
"Another combinator can be defined, $fade:{\\bf O}\\times {\\bf D}\\rightarrow {\\bf D}$ such that for $t\\in {\\cal O}$ and $x\\in {\\cal D}$ $\\begin{array}{lcll}fade(t,x)&=&\\bot _D&{\\rm if}~t=\\bot _O\\\\&=&x&otherwise\\end{array}$ For $u\\in {\\cal D}$ and $u\\ne \\bot _D$ , the mapping $a$ is defined as $a(x)=fade(check(x),u)$ It can be seen that $a$ is a retraction, but not a projection in general, and the range of $a$ is isomorphic to ${\\bf O}$ .",
"These combinators can also be used to define the subset of functions in ${\\bf D}\\rightarrow {\\bf E}$ that are strict.",
"Define a combinator $strict:({\\bf D}\\rightarrow {\\bf E})\\rightarrow ({\\bf D}\\rightarrow {\\bf E})$ by the equation $strict(f)=\\lambda x.fade(check(x),f(x))$ with $fade$ defined as $fade:{\\bf O}\\times {\\bf E}\\rightarrow {\\bf E}$ .",
"The range of $strict$ is all the strict functions; $strict$ is a projection whose range is a domain.",
"The next theorem characterizes projections.",
"Theorem 8.5: For approximable mapping $a:{\\bf E}\\rightarrow {\\bf E}$ , the following are equivalent: $a$ is a finitary projection $a(x)=\\lbrace y\\in {\\bf E}\\:\\vert \\:\\exists x^{\\prime }\\in I_x.",
"x^{\\prime }\\:a\\:x^{\\prime }\\wedge y\\sqsubseteq x^{\\prime }\\rbrace $ for all $x\\in {\\bf E}$ .",
"Proof Assume that (2) holds.",
"We want to show that $a$ is a finitary projection.",
"By the closure properties on ideals, we know that for all $x\\in {\\cal E}$ , $x^{\\prime }\\in x\\wedge y\\sqsubseteq x^{\\prime }\\:\\Rightarrow \\:y\\in x$ Thus, $a(x)\\subseteq x$ must hold.",
"In addition, the following trivially holds: $x^{\\prime }\\in x\\wedge x^{\\prime }\\:a\\: x^{\\prime }\\:\\Rightarrow \\:x^{\\prime }\\in a(x)$ thus $a(x)\\subseteq a(a(x))$ holds for all $x\\in {\\cal E}$ .",
"This shows that $a$ is indeed a projection.",
"Let $D=\\lbrace x\\in {\\bf E}\\:\\vert \\:x\\:a\\:x\\rbrace $ .",
"It is easy to show that ${\\bf D}\\lhd {\\bf E}$ and that $a$ is determined from ${\\bf D}$ as required in Theorem REF .",
"Thus, the fixed point set of $a$ is isomorphic to a domain from the previous proofs.",
"Thus, (2)$\\:\\Rightarrow \\:$ (1).",
"For the converse, assume that $a$ is a finitary projection.",
"Let ${\\cal D}$ be isomorphic to the fixed point set of $a$ .",
"This means there is a projection pair $i$ and $j$ such that $j\\circ i={\\sf I}_D$ and $i\\circ j = a$ and $a\\subseteq {\\sf I}_E$ .",
"From Theorem REF then we have that ${\\cal D}\\approx {\\cal D}^{\\prime }$ and ${\\cal D}^{\\prime }\\lhd {\\cal E}$ .",
"We want to identify ${\\cal D}^{\\prime }$ as follows: ${\\cal D}^{\\prime }=\\lbrace x\\in {\\cal E}\\:\\vert \\:x\\:a\\:x\\rbrace $ From the proof of Theorem REF , the basis elements of ${\\bf D^{\\prime }}$ are the finite elements of ${\\bf D}$ .",
"Each of these elements is in the fixed point set of $a$ .",
"Thus, $x\\in {\\bf D^{\\prime }}\\:\\Rightarrow \\:a({{\\cal I}}_x) = {{\\cal I}}_x \\:\\Rightarrow \\:x\\:a\\:x$ Since $a$ is a projection, ${{\\cal I}}_x$ must also be a fixed point.",
"Since $i(j({{\\cal I}}_x)) = {{\\cal I}}_x$ implies that $j({{\\cal I}}_x)$ is a finite element of ${\\cal D}$ , $x\\in {\\cal D}^{\\prime }$ must hold.",
"Thus, the identification of ${\\cal D}^{\\prime }$ holds.",
"Finally, using $a=i\\circ j$ in the formula in Theorem REF , the formula in (2) is obtained, proving the converse.",
"$\\:\\:\\Box $ This characterization of projections provides a new and interesting combinator.",
"Theorem 8.6: For any domain ${\\cal E}$ , define $sub:({\\cal E}\\rightarrow {\\cal E})\\rightarrow ({\\cal E}\\rightarrow {\\cal E})$ using the relation $x\\: sub(f)\\: z \\iff \\exists y\\in {\\bf E}.y\\:f\\:y\\wedge y\\sqsubseteq x\\wedge z\\sqsubseteq y$ for all $x,z\\in {\\bf E}$ and all $f:{\\bf E}\\rightarrow {\\bf E}$ .",
"Then the range of $sub$ is exactly the set of finitary projections on ${\\cal E}$ .",
"In addition, $sub$ is a finitary projection on ${\\cal E}\\rightarrow {\\cal E}$ .",
"If ${\\cal E}$ is effectively given, then $sub$ is computable.",
"Proof Clearly, $sub(f)$ is approximable.",
"It is obvious from the definition that $f\\mapsto sub(f)$ preserves lubs and thus is approximable as well.",
"Thus, $y\\:f\\:y\\wedge y\\sqsubseteq x\\wedge z\\sqsubseteq y\\:\\Rightarrow \\:x\\:f\\:z$ obviously holds.",
"Thus, $sub(f)\\subseteq f$ holds.",
"Also $y\\:f\\:y\\:\\Rightarrow \\:y\\:sub(f)\\:y$ thus, $sub(f)\\subseteq sub(sub(f))$ holds as well.",
"Thus, $sub$ is a projection on ${\\cal E}\\rightarrow {\\cal E}$ .",
"The definition of the relation shows that it is computable when ${\\cal E}$ is effectively given.",
"Since $sub$ is a projection, its range is the same as its fixed point set.",
"If $sub(a)=a$ , it is easy to see that clause (2) of Theorem REF holds and conversely.",
"Thus, the range of $sub$ is the finitary projections.",
"To see that $sub$ is a finitary projection, we use Theorem REF and Theorem REF to say that the fixed point set of $sub$ is in a one-to-one inclusion preserving correspondence with the domain $\\lbrace D\\:\\vert \\:D\\lhd {\\cal E}\\rbrace $ .",
"$\\:\\:\\Box $ Universal Domain ${\\cal U}$ With these results and the universal domain to be defined next, the theory of sub-domains is translated into the $\\lambda $ -calculus notation using the $sub$ combinator.",
"The universal domain is defined by first defining a domain which has the desired structure but has a top element.",
"The top element is then removed to give the universal domain.",
"Definition 8.7: [Universal Domain] As in the section on domain equations, an inductive definition for a domain ${\\cal V}$ is given as follows: $\\Delta ,\\top \\in {\\bf V}$ $\\langle u,v\\rangle \\in {\\bf V}$ whenever $u,v\\in {\\bf V}$ Thus, we are starting with two objects, a bottom element and a top element, and making two flavors of copies of these objects.",
"Intuitively, we end up with finite binary trees with either the top or the bottom element as the leaves.",
"To simplify the definitions below, the pairs should be reduced such that: All occurrences of $\\langle \\Delta ,\\Delta \\rangle $ are replaced by $\\Delta $ and All occurrences of $\\langle \\top ,\\top \\rangle $ are replaced by $\\top $ .",
"These rewrite rules are easily shown to be finite Church-Rosser.The finitary basis should be defined as the equivalence classes induced by the reduction.",
"The presentation is simplified by considering only reduced trees.",
"As an example of the reduction the pair $\\langle \\langle \\langle \\top ,\\langle \\top ,\\top \\rangle \\rangle ,\\langle \\top ,\\Delta \\rangle \\rangle ,\\langle \\langle \\Delta ,\\Delta \\rangle ,\\langle \\top ,\\top \\rangle \\rangle \\rangle $ reduces to $\\langle \\langle \\top ,\\langle \\top ,\\Delta \\rangle \\rangle ,\\langle \\Delta ,\\top \\rangle \\rangle $ .",
"The approximation ordering is defined as follows: $\\Delta \\sqsubseteq v$ for all $v\\in {\\bf V}$ $v\\sqsubseteq \\top $ for all $v\\in {\\bf V}$ .",
"$\\langle u,v\\rangle \\sqsubseteq \\langle u^{\\prime },v^{\\prime }\\rangle $ iff $u\\sqsubseteq u^{\\prime }$ and $v\\sqsubseteq v^{\\prime }$ Since the top element is approximated by everything, all finite sets of trees are consistent.",
"The lub for a pair of trees is defined as follows: $u\\sqcup \\top =\\top $ for $u\\in {\\bf V}$ $\\top \\sqcup u=\\top $ for $u\\in {\\bf V}$ $u\\sqcup \\Delta =u$ for $u\\in {\\bf V}$ $\\Delta \\sqcup u=u$ for $u\\in {\\bf V}$ $\\langle u,v\\rangle \\sqcup \\langle u^{\\prime },v^{\\prime }\\rangle =\\langle u\\sqcup u^{\\prime },v\\sqcup v^{\\prime }\\rangle $ for $u,v\\in {\\bf V}$ The proof that this forms a finitary basis follows the same guidelines as the proofs in Section .",
"In addition, it should be clear that the presentation is effective.",
"To form the universal domain, the top element is simply removed.",
"Thus, the system ${\\bf U}={\\bf V}-\\lbrace \\top \\rbrace $ is the basis used to form the universal domain.",
"The proof that this is still a finitary basis with an effective presentation is also straightforward and left to the exercises.",
"Note that inconsistent sets can now exist since there is no top element.",
"A set is inconsistent iff its lub is $\\top $ .",
"We shall now prove the claims made for the universal domain.",
"Theorem 8.8: The domain ${\\cal U}$ is universal, in the sense that for every domain ${\\cal D}$ we have ${\\cal D}\\lhd {\\cal U}$ .",
"If ${\\cal D}$ is effectively given, then the projection pair for the embedding is computable.",
"In fact, there is a correspondence between the effectively presented domains and the computable finitary projections of ${\\cal U}$ .",
"Proof Recall that ${\\bf D}$ must be countable to be a finitary basis.",
"Thus, we can assume that the basis has an enumeration $D=\\lbrace X_n\\:\\vert \\:n\\in {{N}}\\rbrace $ where $X_0=\\Delta $ .",
"The effective and general cases are considered together in the proof; comments about computability are included for the effective case as required.",
"Thus, if ${\\cal D}$ is effectively given, the enumeration above is assumed to be computable.",
"To prove that the domain can be embedded in ${\\cal U}$ , the embedding will be shown.",
"To start, for each finite element $d_i$ in the basis, define two sets, $d_i^+$ and $d_i^-$ as follows: $\\begin{array}{lcl}d_i^+&=&\\lbrace d\\in {\\bf D}\\:\\vert \\:d_i\\sqsubseteq d\\rbrace \\\\d_i^-&=&D-d_i^+\\end{array}$ The $d_i^+$ set contains all the elements that $d_i$ approximates, while the $d_i^-$ set contains all the other elements, partitioning ${\\bf D}$ into two disjoint sets.",
"Sets for different elements can be intersected to form finer partitions of ${\\bf D}$ .",
"For $k>0$ , let $R\\in \\lbrace +,-\\rbrace ^k$ , let $R_i$ be the $ith$ symbol in the string $R$ , and define a region $D_R$ as $D_R=\\bigcap \\limits _{i=1}^k d_i^{R_i}$ where $k$ is the length of $R$ .",
"The set $\\lbrace D_{R}\\:\\vert \\:R\\in \\lbrace +,-\\rbrace ^k\\rbrace $ of regions partitions ${\\bf D}$ into $2^k$ disjoint sets.",
"Thus, for each element $e_i$ in the enumeration there is a corresponding partition of the basis given by the family of sets $\\lbrace D_{R}\\:\\vert \\:R\\in \\lbrace +,-\\rbrace ^i\\rbrace $ .",
"For strings $R,S\\in \\lbrace +,-\\rbrace ^*$ such that $R$ is a prefix of $S$ , denoted $R\\le S$ , $D_S\\subseteq D_R$ .",
"It is important to realize that the composition of these sets is dependent on the order in which the elements are enumerated.",
"Some of these regions are empty, but it is decidable if a given intersection is empty if ${\\cal D}$ is effectively presented.",
"It is also decidable if a given element is in a particular region.",
"Figure: Example Finite DomainTo see the function these regions are serving, consider the finite domain in Figure REF .This example is taken from Cartwright and Demers [2].",
"Consider the enumeration with $d_0=\\bot , d_1=b, d_2=c, d_3=a.$ The $d_i^+$ and $d_i^-$ sets are as follows: $\\begin{array}{lcl}d_1^+&=&\\lbrace a,b\\rbrace \\\\d_1^-&=&\\lbrace c,\\bot \\rbrace \\\\d_2^+&=&\\lbrace c\\rbrace \\\\d_2^-&=&\\lbrace a,b,\\bot \\rbrace \\\\d_3^+&=&\\lbrace a\\rbrace \\\\d_3^-&=&\\lbrace b,c,\\bot \\rbrace \\end{array}$ The regions are as follows: $\\begin{array}{lclclcl}D_+ &=&\\lbrace a,b\\rbrace &\\:\\:\\:\\:&D_{+++} &=&\\lbrace \\rbrace \\\\D_- &=&\\lbrace \\bot ,c\\rbrace &&D_{++-} &=&\\lbrace \\rbrace \\\\D_{++} &=&\\lbrace \\rbrace &&D_{+-+} &=&\\lbrace a\\rbrace \\\\D_{+-} &=&\\lbrace a,b\\rbrace &&D_{+--} &=&\\lbrace b\\rbrace \\\\D_{-+} &=&\\lbrace c\\rbrace &&D_{-++} &=&\\lbrace \\rbrace \\\\D_{--} &=&\\lbrace \\bot \\rbrace &&D_{-+-} &=&\\lbrace c\\rbrace \\\\&&&&D_{--+} &=&\\lbrace \\rbrace \\\\&&&&D_{---} &=&\\lbrace \\bot \\rbrace \\end{array}$ The regions generated by each successive element encode the relationships induced by the approximation ordering between the new element and all elements previously added.",
"The reader is encouraged to try this example with other enumerations of this basis and compare the results.",
"The embedding of the elements proceeds by building a tree based on the regions corresponding to the element.",
"The regions are used to find locations in the tree and to determine whether a $\\top $ or a $\\Delta $ element is placed in the location.",
"These trees preserve the relationships specified by the regions and thus, the tree embedding is isomorphic to the domain in question.",
"Once the tree is built, the reduction rules are applied until a non-reducible tree is reached.",
"This tree is the representative element in the universal domain, and the set of these trees form the sub-space.",
"The function to determine the location in the tree for a given domain, $Loc_D:\\lbrace +,-\\rbrace ^*\\rightarrow \\lbrace l,r\\rbrace ^*$ takes strings used to generate regions and outputs a path in a tree where $l$ stands for left sub-tree and $r$ stands for right sub-tree.",
"This path is computed using the following inductive definition: $\\begin{array}{lcll}Loc_D(\\epsilon )&=&\\epsilon .\\\\Loc_D(R+)&=&Loc_D(R)l&{\\rm if }~D_{R+}\\ne \\emptyset ~ {\\rm and }~D_{R-}\\ne \\emptyset .\\\\&=&Loc_D(R)&{\\rm otherwise}.\\\\Loc_D(R-)&=&Loc_D(R)r&{\\rm if }~D_{R+}\\ne \\emptyset ~{\\rm and }~D_{R-}\\ne \\emptyset .\\\\&=&Loc_D(R)&{\\rm otherwise}.\\end{array}$ The set of locations for each non-empty region is the set of paths to all leaves of some finite binary tree.",
"An induction argument is used to show the following properties of $Loc_D$ that ensure this: If $R\\le S$ for $R,S\\subseteq \\lbrace +,-\\rbrace ^*$ , then $Loc_D(R)\\le Loc_D(S)$ .",
"Let $S=\\lbrace Loc_D(R)\\:\\vert \\:R\\in \\lbrace +,-\\rbrace ^k\\wedge D_R\\ne \\emptyset \\rbrace $ for $k>0$ be a set of location paths for a given $k$ .",
"For any $p\\in \\lbrace l,r\\rbrace ^*$ there exists $q\\in S$ such that either $p\\le q$ or $q\\le p$ .",
"That is, every potential path is represented by some finite path.",
"Finally, for all $p,q\\in S$ if $p\\le q$ then $p=q$ .",
"This means that a unique leaf is associated with each location.",
"To find the tree for a given element $d_k$ in the enumeration, apply the following rules to each $R\\in \\lbrace +,-\\rbrace ^{k-1}$ .",
"If $D_{R-}\\ne \\emptyset $ then the leaf for path $Loc_D(R-)$ is labeled $\\top $ .",
"If $D_{R+}\\ne \\emptyset $ then the leaf for path $Loc_D(R+)$ is labeled $\\Delta $ .",
"These rules are used to assign a tree in ${\\bf U}$ , which is then reduced using the reduction rules, for each element in the enumeration of ${\\bf D}$ .",
"To see that the top element is never assigned by these rules, note that some region of the form $R+$ for every length $k$ must be non-empty since it must contain the element $e_k$ being embedded.",
"Returning to the example, the location function defines paths for these elements as follows: $\\begin{array}{lclclcl}Loc_D(+)&=&l&\\:\\:\\:\\:&Loc_D(+-+)&=&ll\\\\Loc_D(-)&=&r&&Loc_D(+--)&=&lr\\\\Loc_D(+-)&=&l&&Loc_D(-+-)&=&rl\\\\Loc_D(-+)&=&rl&&Loc_D(---)&=&rr\\\\Loc_D(--)&=&rr\\end{array}$ The trees generated for each of the elements are: $\\begin{array}{lcl}d_0&\\mapsto &\\Delta \\\\d_1&\\mapsto &\\langle \\Delta ,\\top \\rangle \\\\d_2&\\mapsto &\\langle \\top ,\\langle \\Delta ,\\top \\rangle \\rangle \\\\d_3&\\mapsto & \\langle \\langle \\Delta ,\\top \\rangle ,\\langle \\top ,\\top \\rangle \\rangle \\\\&\\mapsto & \\langle \\langle \\Delta ,\\top \\rangle ,\\top \\rangle \\end{array}$ To verify that the space generated is a valid sub-space, we must verify that the bottom element is mapped to $\\bot _U$ and that the consistency and lub relations are maintained.",
"The tree $\\Delta $ is clearly assigned to $X_0$ , the bottom element for the basis being embedded, since there are no strings of length $-1$ .",
"The embedding preserves inconsistency of elements by forcing the lub of the embedded elements to be $\\top $ .",
"The $D_{R-}$ regions represent the elements that the element being embedded does not approximate.",
"Note that the $D_{R-}$ sets cause the $\\top $ element to be added as the leaf.",
"Since the $D_R$ sets are built using the approximation ordering, it is straightforward to see that the approximation ordering is preserved by the embedding.",
"Lubs are also maintained by the embedding, although the reduction is required to see that this is the case.",
"It should be clear that, if the domain ${\\cal D}$ is effectively given, the sub-space can be computed since the embedding procedure uses the relationships given in the presentation.",
"Finally, suppose that $a$ is a computable, finitary projection on ${\\cal U}$ .",
"From the proof of Theorem REF , the domain of this projection is characterized by the set $\\lbrace y\\in {\\bf U}\\:\\vert \\:y\\:a\\:y\\rbrace $ If $a$ is computable, the set of pairs for $a$ is recursively enumerable.",
"Thus, the set above is also recursively enumerable since equality among basis elements is decidable.",
"Thus, the domain given by the projection must also be effectively given.",
"$\\:\\:\\Box $ Thus, the domain ${\\cal U}$ is an effectively presented universal domain in which all other domains can be embedded.",
"The sub-domains of ${\\cal U}$ include ${\\cal U}\\rightarrow {\\cal U}$ , ${\\cal U}\\times {\\cal U}$ , etc.",
"These domains must be sub-domains of ${\\cal U}$ since they are effectively presented based on our earlier theorems.",
"Domain Constructors in ${\\cal U}$ The next step is to see how to define the constructors commonly used.",
"Definition 8.9: [Domain Constructors] Let the computable projection pair, $i_+:{\\cal U}+{\\cal U}\\rightarrow {\\cal U}~{\\rm and}~j_+:{\\cal U}\\rightarrow {\\cal U}+{\\cal U}$ be fixed.",
"Fix suitable projection pairs $i_\\times ,j_\\times ,i_\\rightarrow $ , and $j_\\rightarrow $ as well.",
"Define $\\begin{array}{lcl}a+b&=&cond\\circ \\langle which,i_+\\circ in_0\\circ a\\circ out_0, i_+\\circ in_1\\circ b\\circ out_1\\rangle \\circ j_+\\\\a\\times b&=&i_x\\circ \\langle a\\circ proj_0,b\\circ proj_1\\rangle \\circ j_x\\\\a\\rightarrow b&=&i_\\rightarrow \\circ (\\lambda f.b\\circ f\\circ a)\\circ j_\\rightarrow \\end{array}$ for all $a,b:{\\cal U}\\rightarrow {\\cal U}$ .",
"From earlier theorems, we know that these combinators are all computable over an effectively presented domain.",
"The next theorem characterizes the effect these combinators have on projection functions.",
"Theorem 8.10: If $a,b:{\\cal U}\\rightarrow {\\cal U}$ are projections, then so are $a+b$ , $a\\times b$ , and $a\\rightarrow b$ .",
"If $a$ and $b$ are finitary, then so are the compound projections.",
"Proof Since $a$ and $b$ are retractions, $a=a\\circ a$ and $b=b\\circ b$ .",
"Then for $a\\times b$ using the definition of $\\times $ , $\\begin{array}{lcl}(a\\times b)\\circ (a\\times b)&=&i_x\\circ \\langle a\\circ proj_0,b\\circ proj_1\\rangle \\circ \\langle a\\circ proj_0,b\\circ proj_1\\rangle \\circ j_x\\\\&=&i_x\\circ \\langle a\\circ a\\circ proj_0,b\\circ b\\circ proj_1\\rangle \\circ j_x\\\\&=& a\\times b\\end{array}$ Thus, $a\\times b$ is a retraction.",
"The other cases follow similarly.",
"Since $a$ and $b$ are projections, $a,b\\subseteq {\\sf I}_U$ (denoted simply ${\\sf I}$ for the remainder of the proof).",
"Using the definition for $+$ along with the above relation and the definition of projection pairs, we can see that $a+b\\subseteq {\\sf I}+{\\sf I}=i_+\\circ j_+ \\subseteq {\\sf I}$ Thus, $a+b$ is a projection.",
"The other cases follow similarly.",
"To show that the projections are finitary, we must show that the fixed point sets are isomorphic to a domain.",
"Since $a$ and $b$ are assumed finitary, their fixed point sets are isomorphic to $\\begin{array}{lcl}D_a&=&\\lbrace x\\in {\\bf U}\\:\\vert \\:x\\:a\\: x\\rbrace \\\\D_b&=&\\lbrace y\\in {\\bf U}\\:\\vert \\:y\\:b\\: y\\rbrace \\end{array}$ We wish to show that ${\\cal D}_a\\rightarrow {\\cal D}_b\\approx {\\cal D}_{a\\rightarrow b}$ .",
"By the definition of the $\\rightarrow $ constructor, the fixed point set of $a\\rightarrow b$ over ${\\cal U}$ is the same as the fixed point set of $\\lambda f.b\\circ f\\circ a$ on ${\\cal U}\\rightarrow {\\cal U}$ .",
"(Hint: $i_\\rightarrow $ and $j_\\rightarrow $ set up the isomorphism.)",
"So, the fixed points for $f:{\\cal U}\\rightarrow {\\cal U}$ are of the form: $f=b\\circ f\\circ a$ We can think of $a$ as a function in ${\\cal U}\\rightarrow {\\cal D}_a$ and define the other half of the projection pair as $i_a:{\\cal D}_a\\rightarrow {\\cal U}$ where $i_a\\circ a = a$ and $a\\circ i_a=i_a$ .",
"Define a function $i_b$ for the projection pair for $b$ similarly.",
"For some $g:{\\cal D}_a\\rightarrow {\\cal D}_b$ let $f=i_b\\circ g\\circ a$ Substituting this definition for $f$ yields $b\\circ f\\circ a = b\\circ i_b\\circ g\\circ a\\circ a = i_b\\circ g \\circ a = f$ by the definition of $i_b$ and since $a$ is a retraction by assumption.",
"Conversely, for a function $f$ such that $i_b\\circ g\\circ a= f$ , let $g=b\\circ f\\circ i_a$ Substituting again, $i_b\\circ g\\circ a = i_b\\circ g\\circ f\\circ i_a\\circ a = b\\circ f\\circ a = f$ Thus, there is an order preserving isomorphism between $g:{\\cal D}_a\\rightarrow {\\cal D}_b$ and the functions $f=b\\circ f\\circ a$ .",
"The proofs of the isomorphisms for the other constructs are similar.",
"$\\:\\:\\Box $ Thus, the sub-domain relationship with the universal domain has been stated in terms of finitary projections over the universal domain.",
"In addition, all the domain constructors have been shown to be computable combinators on the domain of these finitary projections.",
"Recalling that all computable maps have computable fixed points, the standard fixed point method can be used to solve domain equations of all kinds if they can be defined on projections.",
"Returning to the $\\lambda $ -calculus for a moment, all objects in the $\\lambda $ -calculus are considered functions.",
"Since ${\\cal U}\\rightarrow {\\cal U}$ is a part of ${\\cal U}$ , every object in the $\\lambda $ -calculus is also an object of ${\\cal U}$ .",
"Transposing some of the familiar notation, where the old notation appears on the left, the new combinators are defined as follows: $\\begin{array}{lcl}which(z)=which(j_+(z))\\\\in_i(x)=i_+(in_i(x))~{\\rm where}~i=0,1\\\\out_i(x)=out_i(j_+(x))~{\\rm where}~i=0,1\\\\\\langle x,y\\rangle =i_x(\\langle x,y\\rangle )\\\\proj_i=proj_i(j_x(z))~{\\rm where}~i=0,1\\\\u(x) = j_\\rightarrow (u)(x)\\\\\\lambda x.\\tau =i_\\rightarrow (\\lambda x.\\tau )\\end{array}$ Thus, all functions, all constants, all combinators, and all constructs are elements of ${\\cal U}$ .",
"Indeed, everything computable is an element of ${\\cal U}$ .",
"Elements in ${\\cal U}$ play multiple roles by representing different objects under different projections.",
"While this notion may be difficult to get used to, there are many advantages, both notational and conceptual.",
"Exercises Exercise 8.11: A retraction $a:{\\cal D}\\rightarrow {\\cal D}$ is a closure operator iff ${\\sf I}_D\\subseteq a$ as relations.",
"On a domain like ${\\cal P}({{N}})$ , give some examples of closure operators.",
"(Hint: Close up the integers under addition.",
"Is this continuous on ${\\cal P}({{N}})$ ?)",
"Prove in general that for any closure $a:{\\cal D}\\rightarrow {\\cal D}$ , the fixed point set of $a$ is always a finitary domain.",
"(Hint: Show that the fixed point set is closed as required for a domain.)",
"What are the finite elements of the fixed point set?",
"Exercise 8.12: Give a direct proof that the domain $\\lbrace X\\:\\vert \\:X\\lhd {\\cal D}\\rbrace $ is effectively presented if ${\\cal D}$ is.",
"(Hint: The finite elements of the domain correspond exactly to the finite domains $X\\lhd {\\cal D}$ .)",
"In the case of ${\\cal D}={\\cal U}$ , show that the computable elements of the domain correspond exactly to the effectively presented domains (up to effective isomorphism).",
"Exercise 8.13: For finitary projections $a:{\\cal E}\\rightarrow {\\cal E}$ , write ${\\cal D}_a=\\lbrace x\\in {\\cal E}\\:\\vert \\:x\\:a\\:x\\rbrace $ Show that for any two such projections $a$ and $b$ , that $a\\subseteq b \\iff {\\cal D}_a\\lhd {\\cal D}_b$ Exercise 8.14: Find another universal domain that is not isomorphic to ${\\cal U}$ .",
"Exercise 8.15: Prove the remaining cases in Theorem REF .",
"Exercise 8.16: Suppose $S$ and $T$ are two binary constructors on domains that can be made into computable operators on projections over the universal domain.",
"Show that we can find a pair of effectively presented domains such that $D\\approx S(D,E)~{\\rm and}~E\\approx T(D,E).$ Exercise 8.17: Using the translations shown after the proof of Theorem REF , show how the whole typed-$\\lambda $ -calculus can be translated into ${\\cal U}$ .",
"(Hint: for $f:{\\cal D}_a\\rightarrow {\\cal B}$ , write $f=b\\circ f\\circ a$ for finitary projections $a$ and $b$ .",
"For $\\lambda x^{{\\cal D}_a}.\\sigma $ , write $\\lambda x.b(\\sigma ^{\\prime }[a(x)/x])$ where $\\sigma ^{\\prime }$ is the translation of $\\sigma $ into the untyped $\\lambda $ -calculus.",
"Be sure that the resulting term has the right type.)",
"Exercise 8.18: Show that the basis presented for the universal domain ${\\bf U}$ is indeed a finitary basis and that it has an effective presentation.",
"Exercise 8.19: Work out the embedding for the other enumerations for the example given in the proof of Theorem REF ."
],
[
"Motivation",
"Since program operations perform computations incrementally on data values that correspond to ideals (sets of approximations), operations must obey some critical topological constraints.",
"For any approximation $x^{\\prime }$ to the input value $x$ , a program operation $f$ must produce the output $f(x^{\\prime })$ .",
"Since program output cannot be withdrawn, every program operation $f$ is a monotonic function: $x_1 \\sqsubseteq x_2$ implies $f(x_1) \\sqsubseteq (x_2)$ .",
"We can describe this process in more detail by examining the structure of computations.",
"Recall that every value in a domain ${\\cal D}$ can be interpreted as a set of finite elements in ${\\cal D}$ that is closed under implication.",
"When an operation $f$ is applied to the input value $x$ , $f$ gathers information about $x$ by asking the program computing $x$ to generate a countable chain of finite elements $C$ where $\\bigsqcup \\: \\lbrace {\\cal I}_c \\: \\:\\vert \\:\\: c \\in C\\rbrace = x$ .",
"For the sake of simplicity, we can force the chain $C$ describing the input value $x$ to be infinite: if $C$ is finite, we can convert it to an equivalent infinite chain by repeating the last element.",
"Then we can view operation $f$ as a function on infinite streams that repeatedly “reads” the next element in an infinite chain $C$ and “writes” the next element in an infinite chain $C^{\\prime }$ where $\\bigsqcup \\: \\lbrace {\\cal I}_{c^{\\prime }} \\: \\:\\vert \\:\\: c^{\\prime } \\in C^{\\prime }\\rbrace = f(x)$ .",
"Any such function $f$ on ${\\cal D}$ is clearly monotonic.",
"In addition, $f$ obeys the stronger property that for any directed set $S$ , $f(\\bigsqcup S) = \\bigsqcup \\: \\lbrace f(s) \\: \\:\\vert \\:\\: s \\in S\\rbrace $ .",
"This property is called continuity.For the computational motivation behind continuity, the interested reader is referred to Stoy's detailed and highly-readable account in [5], in particular the derivation of Condition 6.39 on page 99 of [5], and the following discussion of its implications.",
"The formulation of computable operations as functions on streams of finite elements is concrete and intuitive, but it is not canonical.",
"Due to the elements of domains in general not being totally-ordered but partially-ordered, there are many different functions on streams of finite elements corresponding to the same continuous function $f$ over a domain ${\\cal D}$ .",
"For this reason, we will use a slightly different model of incremental computation as the basis for defining the form of operations on domains.",
"To produce a canonical representation for computable operations, we represent values as ideals rather than chains of finite elements.",
"In addition, we allow computations to be performed in parallel, producing finite answers incrementally in non-deterministic order.",
"It is important to emphasize that the result of every computation—an ideal ${\\cal I}$ —is still deterministic; only the order in which the elements of ${\\cal I}$ are enumerated is non-deterministic.",
"When an operation $f$ is applied to an input value $x$ , $f$ gathers information about $x$ by asking for the enumeration of the ideal of finite elements $I_x =\\lbrace d \\in {\\cal D}^0 \\: \\:\\vert \\:\\: d \\sqsubseteq x\\rbrace $ .",
"In response to each input approximation $d \\sqsubseteq x$ , $f$ enumerates the ideal $I_{f(d)} =\\lbrace e \\in {\\cal D}^0 \\: \\:\\vert \\:\\: e \\sqsubseteq f(d)\\rbrace $ .",
"Since $I_{f(d)}$ may be infinite, each enumeration must be an independent computation.",
"The operation $f$ merges all of these enumerations yielding an enumeration of the ideal $I_{f(x)} =\\lbrace e \\in {\\cal D}^0 \\: \\:\\vert \\:\\: e \\sqsubseteq f(x)\\rbrace $ .",
"A computable operation $f$ mapping domain ${\\cal A}$ , with basis ${\\bf A}$ , into domain ${\\cal B}$ , with basis ${\\bf B}$ , can be formalized as a consistent relation $F \\subseteq {\\bf A} \\times {\\bf B}$ (a subset of the Cartesian product of the two basis) such that the image $F(a) = {\\lbrace {b \\in {\\bf B} | a \\, F \\, b}\\rbrace }$ of any input element $a \\in {\\bf A}$ is an ideal, and $F$ is monotonic: $a \\sqsubseteq a^{\\prime } \\:\\Rightarrow \\:F(a) \\subseteq F(a^{\\prime })$ .",
"These closure properties ensure that the relation $F$ uniquely identifies a continuous function $f$ on $D$ .",
"Relations (i.e., subsets of ${\\bf A} \\times {\\bf B}$ ) satisfying these closure properties are called approximable mappings."
],
[
"Approximable Mappings and Continuous Functions",
"The following set of definitions restates the preceding descriptions in more rigorous form.",
"Definition 2.1: [Approximable Mapping] Let ${\\cal A}$ and ${\\cal B}$ be the domains determined by finitary bases ${\\bf A}$ and ${\\bf B}$ , respectively.",
"An approximable mapping $F \\subseteq {\\bf A} \\times {\\bf B}$ is a binary relation over ${\\bf A} \\times {\\bf B}$ such that $\\bot _A\\,F\\,\\bot _B$ If $a\\,F\\,b$ and $a\\,F\\,b^{\\prime }$ then $a\\,F\\,(b \\sqcup b^{\\prime })$ If $a\\,F\\,b$ and $b^{\\prime }\\sqsubseteq _B b$ , then $a\\,F\\,b^{\\prime }$ If $a\\,F\\,b$ and $a\\sqsubseteq _A a^{\\prime }$ , then $a^{\\prime }\\,F\\,b$ The partial order of approximable mappings $F \\subseteq {\\bf A} \\times {\\bf B}$ under the subset relation is denoted by the expression ${\\sf {Map}}({\\bf A},{\\bf B})$ .",
"Conditions 1, 2, and 3 force the image of an input ideal to be an ideal.",
"Condition 4 states that the function on ideals associated with $F$ is monotonic.",
"Definition 2.2: [Continuous Function] Let ${\\cal A}$ and ${\\cal B}$ be the domains determined by finitary bases ${\\bf A}$ and ${\\bf B}$ , respectively.",
"A function $f:{\\cal A} \\rightarrow {\\cal B}$ is continuous iff for any ideal ${\\cal I}$ in ${\\cal A}$ , $f({\\cal I}) = \\bigsqcup \\: \\lbrace f({\\cal I}_a) \\: \\:\\vert \\:\\: a \\in {\\cal I}\\rbrace $ .",
"The partial ordering $\\sqsubseteq _B$ from ${\\cal B}$ determines a partial ordering $\\sqsubseteq $ on continuous functions: $ f \\sqsubseteq g \\iff \\forall x \\in {\\cal A} \\; f(x) \\sqsubseteq _{{\\cal B}}g(x) \\,.$ The partial order consisting of the continuous functions from ${\\cal A}$ to ${\\cal B}$ under the pointwise ordering is denoted by the expression ${\\cal A} \\rightarrow _c {\\cal B}$ (or, sometimes, by the expression ${\\sf {Fun}}({\\cal A},{\\cal B})$ ).",
"It is easy to show that continuous functions satisfy a stronger condition than the definition given above.",
"Theorem 2.3: If a function $f:{\\cal A} \\rightarrow {\\cal B}$ is continuous, then for every directed subset $S$ of ${\\cal A}$ , $f(\\bigsqcup S) =\\bigsqcup \\: \\lbrace f({\\cal I}) \\: \\:\\vert \\:\\: {\\cal I}\\in S\\rbrace $ .",
"Proof By Theorem REF , $\\bigsqcup S$ is simply $\\bigcup S$ .",
"Since $f$ is continuous, $f(\\bigsqcup S) = \\bigsqcup \\: \\lbrace f({\\cal I}_a) \\: \\:\\vert \\:\\: \\exists {\\cal I}\\in S \\; a \\in {\\cal I}\\rbrace $ .",
"Similarly, for every ${\\cal I}\\in {\\cal A}$ , $f({\\cal I}) = \\bigsqcup \\: \\lbrace f({\\cal I}_a) \\: \\:\\vert \\:\\: a \\in {\\cal I}\\rbrace $ .",
"Hence, $\\bigsqcup \\: \\lbrace f({\\cal I}) \\: \\:\\vert \\:\\: {\\cal I}\\in S\\rbrace =\\bigsqcup \\: \\lbrace \\bigsqcup \\: \\lbrace f({\\cal I}_a) \\: \\:\\vert \\:\\: a \\in {\\cal I}\\rbrace \\: \\:\\vert \\:\\: {\\cal I}\\in S\\rbrace =\\bigsqcup \\: \\lbrace f({\\cal I}_a) \\: \\:\\vert \\:\\: \\exists {\\cal I}\\in S \\; a \\in {\\cal I}\\rbrace $ .",
"$\\:\\:\\Box $ Every approximable mapping $F$ over the finitary basis ${\\bf A} \\times {\\bf B}$ determines a continuous function $f: {\\cal A} \\rightarrow {\\cal B}$ .",
"Similarly, every continuous function $f: {\\cal A} \\rightarrow {\\cal B}$ determines an approximable mapping $F$ over the finitary basis ${\\bf A} \\times {\\bf B}$ .",
"Definition 2.4: [Image of Approximable Mapping] For approximable mapping $F \\subseteq {\\bf A} \\times {\\bf B}$ the image of $d\\in {\\cal A}$ under $F$ (denoted ${\\sf apply}(F,d)$ ) is the ideal $\\lbrace b\\in {\\bf B} \\: \\:\\vert \\:\\: \\exists a\\in {\\bf A},\\: a\\in d \\: \\wedge a\\,F\\,b\\rbrace \\,.$ The function $f:{\\cal A}\\rightarrow {\\cal B}$ determined by $F$ is defined by the equation: $f(d) = {\\sf apply}(F,d)\\,.$ Remark 2.5: It is easy to confirm that ${\\sf apply}(F,d)$ is an element of ${\\cal B}$ and that the function $f$ is continuous.",
"Given any ideal $d \\in {\\cal A}$ , ${\\sf apply}(F,d)$ is the subset of ${\\bf B}$ consisting of all the elements related by $F$ to finite elements in $d$ .",
"The set ${\\sf apply}(F,d)$ is an ideal in ${\\cal B}$ since $(i)$ the set $\\lbrace b \\in {\\bf B} \\: \\:\\vert \\:\\: a \\, F \\, b\\rbrace $ is downward-closed for all $a \\in {\\bf A}$ , and $(ii)$ $a \\, F \\, b \\wedge a \\, F \\, b^{\\prime }$ implies $a \\, F \\, (b \\sqcup b^{\\prime })$ .",
"The continuity of $f$ is an immediate consequence of the definition of $f$ and the definition of continuity.",
"The following theorem establishes that the partial order of approximable mappings over ${\\bf A} \\times {\\bf B}$ is isomorphic to the partial order of continuous functions in ${\\cal A}\\rightarrow _c{\\cal B}$ .",
"Theorem 2.6: Let ${\\bf A}$ and ${\\bf B}$ be finitary bases.",
"The partial order ${\\sf {Map}}({\\bf A},{\\bf B})$ consisting of the set of approximable mappings over ${\\bf A}$ and ${\\bf B}$ is isomorphic to the partial order ${\\cal A} \\rightarrow _c {\\cal B}$ of continuous functions mapping ${\\cal A}$ into ${\\cal B}$ .",
"The isomorphism is witnessed by the function ${{F}}: {\\sf {Map}}({\\bf A},{\\bf B}) \\rightarrow ({\\cal A} \\rightarrow _c {\\cal B})$ defined by $ {{F}}(F) = f$ where $f$ is the function defined by the equation $f(d) = {\\sf apply}(F,d)$ for all $d \\in {\\cal A}$ .",
"Proof The theorem is an immediate consequence of the following lemma.",
"$\\:\\:\\Box $ Lemma 2.7: For any approximable mappings $F,G \\subseteq {\\bf A} \\times {\\bf B}$ $\\forall a\\in {\\bf A}, b\\in {\\bf B} \\;\\;a\\,F\\,b \\iff {\\cal I}_b \\sqsubseteq {{F}}(F)({\\cal I}_a)$ .",
"$F\\subseteq G \\iff \\forall a\\in {\\bf A} \\;\\; {{F}}(F)(a) \\sqsubseteq {{F}}(G)(a)$ The function ${{F}}: {\\sf {Map}}({\\bf A},{\\bf B}) \\rightarrow ({\\cal A} \\rightarrow _c {\\cal B})$ is one-to-one and onto.",
"Proof (lemma) Part ($a$ ) is the immediate consequence of the definition of $f$ ($b \\in f({\\cal I}_a) \\iff a \\, F \\, b$ ) and the fact that $f({\\cal I}_a)$ is downward closed.",
"Part ($b$ ) follows directly from Part ($a$ ): $F \\subseteq G \\iff \\forall a \\in {\\cal A} \\;\\;\\lbrace b \\: \\:\\vert \\:\\: a \\, F \\, b\\rbrace \\subseteq \\lbrace b \\: \\:\\vert \\:\\:a \\, G \\, b\\rbrace $ .",
"But the latter holds iff $\\forall a\\in {\\bf A} \\;\\; (f(a) \\subseteq g(a) \\iff f(a) \\sqsubseteq g(a))$ .",
"Assume ${{F}}$ is not one-to-one.",
"Then there are distinct approximable mappings $F$ and $G$ such that ${{F}}(F) = {{F}}(G)$ .",
"Since ${{F}}(F) = {{F}}(G)$ , $\\forall a \\in {\\bf A}, b \\in {\\bf B} \\;\\;({\\cal I}_b \\sqsubseteq {{F}}(F)({\\cal I}_a) \\iff {\\cal I}_b \\sqsubseteq {{F}}(G)({\\cal I}_a))\\:.$ By Part 1 of the lemma, $\\forall a \\in {\\bf A}, b \\in {\\bf B} \\;\\;(a\\,F\\,b \\iff {\\cal I}_b \\sqsubseteq {{F}}(F)({\\cal I}_a) \\iff {\\cal I}_b \\sqsubseteq {{F}}(G)({\\cal I}_a) \\iff a\\,G\\,b)\\:.$ We can prove that ${{F}}$ is onto as follows.",
"Let $f$ be an arbitary continuous function in ${\\cal A}\\rightarrow {\\cal B}$ .",
"Define the relation $F \\subseteq {\\bf A} \\times {\\bf B}$ by the rule $a \\: F \\: b \\iff {\\cal I}_b \\sqsubseteq f({\\cal I}_a)\\;.$ It is easy to verify that $F$ is an approximable mapping.",
"By Part 1 of the lemma, $a \\: F \\: b \\iff {\\cal I}_b \\sqsubseteq {{F}}(F)({\\cal I}_a)\\;.$ Hence ${\\cal I}_b \\sqsubseteq f({\\cal I}_a) \\iff {\\cal I}_b \\sqsubseteq {{F}}(F)({\\cal I}_a)\\;,$ implying that $f$ and ${{F}}(F)$ agree on finite inputs.",
"Since $f$ and ${{F}}(F)$ are continuous, they are equal.",
"$\\:\\:\\Box $ The following examples show how approximable mappings and continuous functions are related.",
"Example 2.8: Let ${\\cal B}$ be the domain of infinite strings from the previous section and let ${\\cal T}$ be the truth value domain with two total elements, true and false, where $\\bot _{{\\cal T}}$ denotes that there is insufficient information to determine the outcome.",
"Let $p:{\\cal B}\\rightarrow {\\cal T}$ be the function defined by the equation: $p(x) =\\left\\lbrace \\begin{array}{ll}{\\tt true} & \\mbox{if $x = 0^n1y$ where $n$ is even}\\\\{\\tt false} & \\mbox{if $x = 0^n1y$ where $n$ is odd}\\\\\\bot _{{\\cal T}} & \\mbox{otherwise}\\end{array}\\right.$ The function $p$ determines whether or not there are an even number of 0's before the first 1 in the string.",
"If there is no 1 in the string, the result is $\\bot _{{\\cal T}}$ .",
"It is easy to show that $p$ is continuous.",
"The corresponding binary relation $P$ is defined by the rule: $a\\:P\\:b &\\iff & (b \\sqsubseteq _{{\\cal T}}\\bot _{{\\cal T}}) \\; \\vee \\;\\\\& & (0^{2n}1\\sqsubseteq _B a \\wedge b\\sqsubseteq _{{\\cal T}}{\\tt true}) \\; \\vee \\;\\\\& & (0^{2n+1}1\\sqsubseteq _B a \\wedge b\\sqsubseteq _{{\\cal T}}{\\tt false})$ The reader should verify that $P$ is an approximable mapping and that $p$ is the continuous function determined by $P$ .",
"$\\:\\:\\Box $ Example 2.9: Given the domain ${\\cal B}$ from the previous example, let $g:{\\cal B}\\rightarrow {\\cal B}$ be the function defined by the equation: $g(x) =\\left\\lbrace \\begin{array}{ll}0^{n+1}y & \\mbox{if $x = 0^n1^k0y$}\\\\\\bot _D & \\mbox{otherwise}\\end{array}\\right.$ The function $g$ eliminates the first substring of the form $1^k \\; (k > 0)$ from the input string $x$ .",
"If $x = 1^\\infty $ , the infinite string of ones, then $g(x) = \\bot _D$ .",
"Similarly, if $x = 0^n1^\\infty $ , then $g(x) = \\bot _D$ .",
"The reader should confirm that $g$ is continuous and determine the approximable mapping $G$ corresponding to $g$ .",
"$\\:\\:\\Box $ Approximable mappings and continuous functions can be composed and manipulated just like any other relations and functions.",
"In particular, the composition operators for approximable mappings and continuous functions behave as expected.",
"In fact, they form categories.",
"Categories of Approx.",
"Mappings and Cont.",
"Functions Approximable mappings and continuous functions form categories over finitary bases and domains, respectively.",
"Theorem 2.10: The approximable mappings form a category over finitary bases where the identity mapping for finitary basis $B$ , ${\\sf I}_B \\subseteq {\\bf B} \\times {\\bf B}$ , is defined for $a,b\\in {\\bf B}$ as $a\\:{\\sf I}_B \\:b \\iff b\\sqsubseteq a$ and the composition $G \\circ F \\subseteq {\\bf B_1} \\times {\\bf B_3}$ of approximable mappings $F \\subseteq {\\bf B_1}\\times {\\bf B_2}$ and $G \\subseteq {\\bf B_2}\\times {\\bf B_3}$ is defined for $a\\in {\\bf B_1}$ and $c\\in {\\bf B_3}$ by the rule: $a \\: (G\\circ F) \\:c \\iff \\exists b\\in {\\bf B_2} \\: a\\,F\\,b \\wedge b\\,G\\,c\\,.$ To show that this structure is a category, we must establish the following properties: the identity mappings are approximable mappings, the identity mappings composed with an approximable mapping defines the original mapping, the mappings formed by composition are approximable mappings, and composition of approximable mappings is associative.",
"Proof Let $F \\subseteq {\\bf B_1}\\times {\\bf B_2}$ and $G \\subseteq {\\bf B_2}\\times {\\bf B_3}$ be approximable mappings.",
"Let ${{\\sf I}}_1, {{\\sf I}}_2$ be identity mappings for ${\\bf B_1}$ and ${\\bf B_2}$ respectively.",
"The verification that the identity mappings satisfy the requirements for approximable mappings is straightforward and left to the reader.",
"To show $F\\circ {{\\sf I}}_1$ and ${{\\sf I}}_2\\circ F$ are approximable mappings, we prove the following equivalence: $F\\circ {\\sf I}_1 = {\\sf I}_2\\circ F = F$ For $a\\in {\\bf B_1}$ and $b\\in {\\bf B_2}$ , $a\\:(F\\circ {\\sf I}_1) \\:b \\iff \\exists c\\in {\\cal D}_1 \\;\\; (c\\sqsubseteq a\\wedge c\\,F\\,b)\\, .$ By the definition of approximable mappings, this property holds iff $a\\,F\\,b$ , implying that $F$ and $F\\circ {\\sf I}_1$ are the same relation.",
"The proof of the other half of the equivalence is similar.",
"We must show that the relation $G\\circ F$ is approximable given that the relations $F$ and $G$ are approximable.",
"To prove the first condition, we observe that $\\bot _1\\,F\\,\\bot _2$ and $\\bot _2\\,G\\,\\bot _3$ by assumption, implying that $\\bot _1\\:(G\\circ F)\\:\\bot _3$ .",
"Proving the second condition requires a bit more work.",
"If $a\\,(G\\circ F)\\,c$ and $a\\,(G\\circ F)\\,c^{\\prime }$ , then by the definition of composition, $a\\, F\\,b$ and $b\\,G\\,c$ for some $b$ and $a\\, F\\,b^{\\prime }$ and $b^{\\prime }\\,G\\,c^{\\prime }$ for some $b^{\\prime }$ .",
"Since $F$ and $G$ are approximable mappings, $a\\,F\\,(b\\sqcup b^{\\prime })$ and since $b^{\\prime }\\sqsubseteq (b\\sqcup b^{\\prime })$ , it must be true that $(b\\sqcup b^{\\prime })\\,G\\,c$ .",
"By an analogous argument, $(b\\sqcup b^{\\prime })\\,G\\,c^{\\prime }$ .",
"Therefore, $(b\\sqcup b^{\\prime })\\,G\\,(c\\sqcup c^{\\prime })$ since $G$ is an approximable mapping, implying that $a\\,(G\\circ F)\\,(c\\sqcup c^{\\prime })$ .",
"The final condition asserts that $G\\circ F$ is monotonic.",
"We can prove this as follows.",
"If $a\\sqsubseteq a^{\\prime }$ , $c^{\\prime }\\sqsubseteq c$ and $a\\,(G\\circ F)\\,c$ , then $a\\,F\\,b$ and $b\\,G\\,c$ for some $b$ .",
"So $a^{\\prime }\\,F\\,b$ and $b\\,G\\,c^{\\prime }$ and thus $a^{\\prime }\\,(G\\circ F)\\,c^{\\prime }$ .",
"Thus, $G\\circ F$ satisfies the conditions of an approximable mapping.",
"Associativity of composition implies that for approximable mapping $H$ with $F, G$ as above and $H:{\\cal D}_3\\rightarrow {\\cal D}_4$ , $H\\circ (G\\circ F) = (H\\circ G)\\circ F$ .",
"Assume $a\\,(H\\circ (G\\circ F))\\,z$ .",
"Then, $\\begin{array}{lcl}a\\,(H\\circ (G\\circ F))\\,z&\\iff &\\exists c\\in {\\cal D}_3 \\; a\\,(G\\circ F)\\,c\\wedge c\\,H\\,z\\\\&\\iff & \\exists c\\in {\\cal D}_3 \\, \\exists b\\in {\\cal D}_2 \\; a\\,F\\,b\\wedge b\\,G\\,c\\wedge c\\,H\\,z\\\\&\\iff & \\exists b\\in {\\cal D}_2 \\: \\exists c\\in {\\cal D}_3 \\; a\\,F\\,b\\wedge b\\,G\\,c\\wedge c\\:H\\,z\\\\&\\iff & \\exists b\\in {\\cal D}_2 \\: a\\,F\\,b\\wedge b\\,(H\\circ G)\\,z\\\\&\\iff & a\\,((H\\circ G)\\circ F)\\,z\\end{array}$ $\\:\\:\\Box $ Since finitary bases correspond to domains and approximable mappings correspond to continuous functions, we can restate the same theorem in terms of domains and continuous functions.",
"Corollary 2.11: The continuous functions form a category over domains determined by finitary bases; the identity function for domain ${\\cal B}$ , ${\\sf I}_B: {\\cal B} \\rightarrow {\\cal B}$ , is defined for by the equation ${\\sf I}_B(d) = d$ and the composition $g \\circ f \\in {\\cal B}_1 \\rightarrow {\\cal B}_3$ of continuous functions $f: {\\cal B}_1\\rightarrow {\\cal B}_2$ and $g:{\\cal B}_2\\rightarrow {\\cal B}_3$ is defined for $a\\in {\\cal B}_1$ by the equation $(g\\circ f)(a) = g(f(a))\\,.$ Proof The corollary is an immediate consequence of the preceding theorem and two facts: The partial order of finitary bases is isomorphic to the partial order of domains determined by finitary bases; the ideal completion mapping established the isomorphism.",
"The partial order of approximable mappings over ${\\bf A} \\times {\\bf B}$ is isomorphic to the partial order of continuous functions in ${\\cal A} \\rightarrow {\\cal B}$ .",
"$\\:\\:\\Box $ Based on Theorem REF and Corollary REF , we define $\\bf FB$ as the category having finitary bases as its objects and approximable mappings between finitary bases as its arrows, and we define $\\bf Dom$ as the category having domains as its objects and continuous functions over domains as its arrows.",
"Domain Isomorphisms Isomorphisms between domains are important.",
"We briefly state and prove one of their most important properties.",
"Theorem 2.12: Every isomorphism between domains is characterized by an approximable mapping between the finitary bases.",
"Additionally, finite elements are always mapped to finite elements.",
"Proof Let $f:{\\cal D}\\rightarrow {\\cal E}$ be a one-to-one and onto function that preserves the approximation ordering.",
"Using the earlier theorem characterizing approximable mappings and their associated functions, we can define the mapping as $a\\,F\\,b \\iff {\\cal I}_b\\sqsubseteq f({\\cal I}_a)$ where ${\\cal I}_a,{\\cal I}_b$ are the principal ideals for $a,b$ .",
"As shown in Exercise REF , monotone functions on finite elements always determine approximable mappings.",
"Thus, we need to show that the function described by this mapping, using the function image construction defined earlier, is indeed the original function $f$ .",
"To show this, the following equivalence must be established for $a\\in {\\cal D}$ : $f(a) = \\lbrace b\\in {\\bf E}\\: \\:\\vert \\:\\: \\exists a^{\\prime }\\in a \\:{\\cal I}_b\\sqsubseteq f({\\cal I}_{a^{\\prime }})\\rbrace $ The right-hand side of this equation, call it $e$ , is an ideal—for a proof of this, see Exercise REF .",
"Since $f$ is an onto function, there must be some $d\\in {\\cal D}$ such that $f(d)=e$ .",
"Since $a^{\\prime }\\in a$ , ${\\cal I}_{a^{\\prime }}\\sqsubseteq a$ holds.",
"Thus, $f({\\cal I}_{a^{\\prime }})\\sqsubseteq f(a)$ .",
"Since this holds for all $a^{\\prime }\\in a$ , $f(d)\\sqsubseteq f(a)$ .",
"Now, since $f$ is an order-preserving function, $d\\sqsubseteq a$ .",
"In addition, since $a^{\\prime }\\in a$ , $f({\\cal I}_{a^{\\prime }})\\sqsubseteq f(d)$ by the definition of $f(d)$ so ${\\cal I}_{a^{\\prime }}\\sqsubseteq d$ .",
"Thus, $a^{\\prime }\\in d$ and thus $a\\sqsubseteq d$ since $a^{\\prime }$ is an arbitrary element of $a$ .",
"Thus, $a=d$ and $f(a) = f(d)$ as desired.",
"To show that finite elements are mapped to finite elements, let ${\\cal I}_a\\in {\\cal D}$ for $a\\in {\\bf D}$ .",
"Since $f$ is one-to-one and onto, every $b\\in f({\\cal I}_a)$ has a unique ${\\cal I}_{b^{\\prime }}\\sqsubseteq {\\cal I}_a$ such that $f({\\cal I}_{b^{\\prime }})={\\cal I}_b$ .",
"This element is found using the inverse of $f$ , which must exist.",
"Now, let $z=\\bigsqcup \\: \\lbrace {\\cal I}_{b^{\\prime }}\\: \\:\\vert \\:\\: b\\in f({\\cal I}_a)\\rbrace $ Since $p\\sqsubseteq q$ implies ${\\cal I}_{p^{\\prime }}\\sqsubseteq {\\cal I}_{q^{\\prime }}$ , $z$ is also an ideal (see Exercise REF again).",
"Since ${\\cal I}_{b^{\\prime }}\\sqsubseteq {\\cal I}_a$ holds for each ${\\cal I}_{b^{\\prime }}$ , $z\\sqsubseteq {\\cal I}_a$ must also hold.",
"Also, since each ${\\cal I}_{b^{\\prime }}\\sqsubseteq z$ , ${\\cal I}_b=f({\\cal I}_{b^{\\prime }})\\sqsubseteq f(z)$ .",
"Therefore, $b\\in f(z)$ .",
"Since $b$ is an arbitrary element in $f({\\cal I}_a)$ , $f({\\cal I}_a)\\sqsubseteq f(z)$ must hold and thus ${\\cal I}_a\\sqsubseteq z$ .",
"Therefore, ${\\cal I}_a=z$ and $a\\in z$ .",
"But then $a\\in {\\cal I}_{c^{\\prime }}$ for some $c\\in f({\\cal I}_a)$ by the definition of $z$ .",
"Thus, ${\\cal I}_a\\sqsubseteq {\\cal I}_{c^{\\prime }}$ and $f({\\cal I}_a)\\sqsubseteq {\\cal I}_c$ .",
"Since $c$ was chosen such that ${\\cal I}_c\\sqsubseteq f({\\cal I}_a)$ , ${\\cal I}_{c^{\\prime }}\\sqsubseteq {\\cal I}_a$ and therefore ${\\cal I}_c=f({\\cal I}_a)$ and $f({\\cal I}_a)$ is finite.",
"The same argument holds for the inverse of $f$ ; therefore, the isomorphism preserves the finiteness of elements.",
"$\\:\\:\\Box $ Exercises Exercise 2.13: Show that the partial order of monotonic functions mapping ${\\cal D}^0$ to ${\\cal E}^0$ (using the pointwise ordering) is isomorphic to the partial order of approximable mappings $f:{\\bf D}\\times {\\bf E}$ .",
"Exercise 2.14: Prove that, if $F \\subseteq {\\bf D}\\times {\\bf E}$ is an approximable mapping, then the corresponding function $f:{\\cal D}\\rightarrow {\\cal E}$ satisfies the following equation: $f(x) = \\bigsqcup \\: \\lbrace e \\: \\:\\vert \\:\\: \\exists d \\in x \\; d F e\\rbrace $ for all $x\\in {\\cal D}$ .",
"Exercise 2.15: Prove the following claim: if $F,G \\subseteq {\\bf D}\\times {\\bf E}$ are approximable mappings, then there exists $H \\subseteq {\\bf D}\\times {\\bf E}$ such that $H = F \\cap G = F \\sqcap G$ .",
"Exercise 2.16: Let $\\langle I,\\le \\rangle $ be a non-empty partial order that is directed and let $\\langle D,\\sqsubseteq \\rangle $ be a finitary basis.",
"Suppose that $a:I\\rightarrow {\\cal D}$ is defined such that $i\\le j\\:\\Rightarrow \\:a(i)\\sqsubseteq a(j)$ for all $i,j\\in I$ .",
"Show that $\\bigcup \\lbrace a(i)\\: \\:\\vert \\:\\: i\\in I\\rbrace $ is an ideal in ${\\cal D}$ .",
"This says that the domain is closed under directed unions.",
"Prove also that for an approximable mapping $f:{\\bf D}\\rightarrow {\\bf E}$ , then for any directed union, $f(\\bigcup \\lbrace a(i)\\: \\:\\vert \\:\\: i\\in I\\rbrace ) = \\bigcup \\lbrace f(a(i))\\: \\:\\vert \\:\\: i\\in I\\rbrace $ This says that approximable mappings preserve directed unions.",
"If an elementwise function preserves directed unions, must it come from an approximable mapping?",
"(Hint: See Exercise REF ?",
"?).",
"Exercise 2.17: Let $\\langle I,\\le \\rangle $ be a directed partial order with $f_i:{\\bf D}\\rightarrow {\\bf E}$ as a family of approximable mappings indexed by $i\\in I$ .",
"And assume $i\\le j \\:\\Rightarrow \\:f_i(x)\\sqsubseteq f_j(x)$ for all $i,j\\in I$ and all $x\\in {\\cal D}$ .",
"Show that there is an approximable mapping $g:{\\bf D}\\rightarrow {\\bf E}$ where $g(x)=\\bigcup \\lbrace f_i(x)\\: \\:\\vert \\:\\: i\\in I\\rbrace $ for all $x\\in {\\cal D}$ .",
"Exercise 2.18: Let $f:{\\cal D}\\rightarrow {\\cal E}$ be an isomorphism between domains.",
"Let $\\phi :{\\bf D}\\rightarrow {\\bf E}$ be the one-to-one correspondence from Theorem REF where $f({\\cal I}_a)={\\cal I}_{\\phi (a)}$ for $a\\in {\\bf D}$ .",
"Show that the approximable mapping determined by $f$ is the relationship $\\phi (x)\\sqsubseteq b$ .",
"Show also that if $a$ and $a^{\\prime }$ are consistent in ${\\cal D}$ then $\\phi (a\\sqcup a^{\\prime }) = \\phi (a) \\sqcup \\phi (a^{\\prime })$ .",
"Show how this means that isomorphisms between domains correspond to isomorphisms between the bases for the domains.",
"Exercise 2.19: Show that the mapping defined in Example REF is approximable.",
"Is it uniquely determined by the following equations or are some missing?",
"$\\begin{array}{lcl}g(0x) &=& 0g(x)\\\\g(11x) &=& g(1x)\\\\g(10x) &=& 0x\\\\g(1) &=& \\bot \\end{array}$ Exercise 2.20: Define in words the effect of the approximable mapping $h:{\\bf B}\\rightarrow {\\bf B}$ using the bases defined in Example REF where $\\begin{array}{lcl}h(0x)&=&00h(x)\\\\h(1x)&=&10h(x)\\end{array}$ for all $x\\in {\\cal B}$ .",
"Is $h$ an isomorphism?",
"Does there exist a map $k:{\\bf B}\\rightarrow {\\bf B}$ such that $k\\circ h = {\\sf I}_B$ and is $k$ a one-to-one function?",
"Exercise 2.21: Generalize the definition of approximable mappings to define mappings $f:{\\bf D_1}\\times {\\bf D_2}\\rightarrow {\\bf D_3}$ of two variables.",
"(Hint: A mapping $f$ can be a ternary relation $f\\subseteq {\\bf D_1}\\times {\\bf D_2}\\times {\\bf D_3}$ where the relation among the basis elements is denoted $(a,b)\\,F\\,c$ .)",
"State a modified version of the theorem characterizing these mappings and their corresponding functions.",
"Exercise 2.22: Modify the construction of the domain ${\\cal B}$ from Example REF to construct a domain ${\\cal C}$ with both finite and infinite total elements (${\\cal B}\\subseteq {\\cal C}$ ).",
"Define an approximable map, $C$ , on this domain corresponding to the concatenation of two strings.",
"(Hint: Use 011 as an finite total element, 011$\\bot $ as the corresponding finite partial element.)",
"Recall that $\\epsilon $ , the empty sequence, is different from $\\bot $ , the undefined sequence.",
"Concatenation should be defined such that if $x$ is an infinite element from ${\\cal C}$ , then $\\forall y\\in {\\bf C} \\; (x,y)\\,C\\,x$ .",
"How does concatenation behave on partial elements on the left?",
"Exercise 2.23: Let ${\\bf A}$ and ${\\bf B}$ be arbitrary finitary bases.",
"Prove that the partial order of approximable mappings over ${\\bf A} \\times {\\bf B}$ is a domain.",
"(Hint: The finite elements are the closures of finite consistent relations.)",
"Prove that the partial order of continuous functions in ${\\cal A} \\rightarrow {\\cal B}$ is a domain.",
"Domain Constructors Now that the notion of a domain has been defined, we need to develop convenient methods for constructing specific domains.",
"The strategy that we will follow is to define the simplest domains (i.e., flat domains) directly (as initial or term algebras) and to construct more complex domains by applying domain constructors to simpler domains.",
"Since domains are completely determined by finitary bases, we will focus on how to construct composite finitary bases from simpler ones.",
"These constructions obviously determine corresponding constructions on domains.",
"The two most important constructions on finitary bases are (1) Cartesian products of finitary bases and (2) approximable mappings on finitary bases constructed using a function-space constructor.",
"Cartesian Products Definition 3.1: [Product Basis] Let ${\\bf D}$ and ${\\bf E}$ be the finitary bases generating domains ${\\cal D}$ and ${\\cal E}$ .",
"The product basis, ${\\bf D\\times E}$ , is the partial order consisting of the universe $D\\times E = \\lbrace [d,e]\\:\\vert \\:d\\in {\\bf D},e\\in {\\bf E}\\rbrace $ and the approximation ordering $[d,e] \\sqsubseteq [i,j] \\iff d\\sqsubseteq _D i~{\\rm and}~e\\sqsubseteq _E j.$ Theorem 3.2: The product basis of two finitary bases, as defined above, is a finitary basis.",
"Proof Let ${\\bf D}$ and ${\\bf E}$ be finitary bases and let ${\\bf D\\times E}$ be defined as above.",
"Since D and E are countable, the universe of ${\\bf D}\\times {\\bf E}$ must be countable.",
"It is easy to show that ${\\bf D}\\times {\\bf E}$ is a partial order.",
"By the construction, the bottom element of the product basis is $[\\bot _D,\\bot _E]$ .",
"For any finite bounded subset $R$ of ${\\bf D}\\times {\\bf E}$ where $R=\\lbrace [d_i,e_i]\\rbrace $ , the lub of $R$ is $[\\sqcup \\lbrace d_i\\rbrace ,\\sqcup \\lbrace e_i\\rbrace ]$ which must be defined since D and E are finitary bases and for $R$ to be bounded, each of the component sets must be bounded.",
"$\\:\\:\\Box $ It is straightforward to define projection mappings on product bases, corresponding to the standard projection functions defined on Cartesian products of sets.",
"Definition 3.3: [Projection Mappings] For a finitary basis ${\\bf D}\\times {\\bf E}$ , projection mappings ${\\sf {P}}_0 \\subseteq ({\\bf D}\\times {\\bf E}) \\times {\\bf D}$ and $P_1 \\subseteq ({\\bf D}\\times {\\bf E}) \\times {\\bf E}$ are the relations defined by the rules $[d,e] \\: P_0 \\:d^{\\prime } & \\iff & d^{\\prime } \\sqsubseteq _D d \\\\ [d,e] \\: P_1 \\:e^{\\prime } & \\iff & e^{\\prime } \\sqsubseteq _E e$ where $d$ ,$d^{\\prime }$ are arbitrary elements of ${\\bf D}$ and $e$ , $e^{\\prime }$ are arbitrary elements of ${\\bf E}$ .",
"Let ${\\bf A}$ , ${\\bf D}$ , and ${\\bf E}$ be finitary bases and let $F \\subseteq {\\bf A}\\times {\\bf D}$ and $G \\subseteq {\\bf A}\\times {\\bf E}$ be approximable mappings.",
"The paired mapping $\\langle F,G\\rangle \\subseteq {\\bf A}\\times ({\\bf D}\\times {\\bf E})$ is the relation defined by the rule $a\\:\\langle F,G\\rangle \\:[d,e] \\iff a\\:F\\:d\\wedge a\\:G\\:e$ for all $a\\in {\\bf A},d\\in {\\bf D}$ , and all $e\\in {\\bf E}$ .",
"It is an easy exercise to show that projection mappings and paired mappings are approximable mappings (as defined in the previous section).",
"Theorem 3.4: The mappings ${\\sf {P}}_0$ , ${\\sf {P}}_1$ , and $\\langle F,G\\rangle $ are approximable mappings if $F,G$ are.",
"In addition, ${\\sf {P}}_0\\circ \\langle F,G\\rangle = F$ and ${\\sf {P}}_1\\circ \\langle F,G\\rangle = G$ .",
"For $[d,e]\\in {\\bf D}\\times {\\bf E}$ and $d^{\\prime } \\in {\\bf D}$ , $[d,e] \\:{\\sf {P}}_0 \\: d^{\\prime } \\iff d^{\\prime } \\sqsubseteq d$ .",
"For $[d,e]\\in {\\bf D}\\times {\\bf E}$ and $e^{\\prime } \\in {\\bf E}$ , $[d,e] \\:{\\sf {P}}_1 \\: e^{\\prime } \\iff e^{\\prime } \\sqsubseteq d$ .",
"For approximable mapping $H \\subseteq {\\bf A}\\times ({\\bf D}\\times {\\bf E})$ , $H=\\langle ({\\sf {P}}_0\\circ H),({\\sf {P}}_1\\circ H)\\rangle $ .",
"For $a\\in {\\bf A}$ and $[d,e] \\in {\\bf D}\\times {\\bf E}$ , $[a, [d,e]] \\in \\langle F,G\\rangle \\iff [a,d] \\in F \\wedge [a,e] \\in G$ .",
"Proof The proof is left as an exercise to the reader.",
"$\\:\\:\\Box $ The projection mappings and paired mappings on finitary bases obviously correspond to continuous functions on the corresponding domains.",
"We will denote the continous functions corresponding to ${\\sf {P}}_0$ and ${\\sf {P}}_1$ by the symbols $p_1$ and $p_2$ .",
"Similarly, we will denote the function corresponding to the paired mapping $\\langle F, G\\rangle $ by the expression $\\langle f, g \\rangle $ .",
"It should be clear that the definition of projection mappings and paired mappings can be generalized to products of more than two domains.",
"This generalization enables us to treat a multi-ary continous function (or approximable mapping) as a special form of a unary continuous function (or approximable mapping) since multi-ary inputs can be treated as single objects in a product domain.",
"Moreover, it is easy to show that a relation $R \\subseteq ({\\bf A_1} \\times \\ldots \\times {\\bf A_n}) \\times {\\bf B}$ of arity $n+1$ (as in Exercise REF ) is an approximable mapping iff every restriction of $R$ to a single input argument (by fixing the other input arguments) is an approximable mapping.",
"Theorem 3.5: A relation $F \\subseteq ({\\bf A}\\times {\\bf B})\\times {\\bf C}$ is an approximable mapping iff for every $a\\in {\\bf A}$ and every $b\\in {\\bf B}$ , the derived relations $F_{a,*} & = & {\\lbrace { [y,z] \\; \\:\\vert \\:\\; [[a,y],z] \\in F}\\rbrace }\\\\F_{*,b} & = & {\\lbrace { [x,z] \\; \\:\\vert \\:\\; [[x,b],z] \\in F}\\rbrace }$ are approximable mappings.",
"Proof Before we prove the theorem, we need to introduce the class of constant relations ${\\sf {K}}_e \\subseteq {\\bf D}\\times {\\bf E}$ for arbitrary finitary bases ${\\bf D}$ and ${\\bf E}$ and show that they are approximable mappings.",
"Lemma 3.6: For each $e \\in {\\bf E}$ , let the “constant” relation ${\\sf {K}}_e \\subseteq {\\bf D} \\times {\\bf E}$ be defined by the equation ${\\sf {K}}_e = {\\lbrace {[d,e^{\\prime }] \\; \\:\\vert \\:\\; d \\in {\\bf D}, e^{\\prime } \\sqsubseteq e}\\rbrace }\\,.$ In other words, $d \\: {\\sf {K}}_e \\: e^{\\prime } \\; \\iff \\; e^{\\prime } \\sqsubseteq e \\, .$ For $e\\in {\\bf E}$ , the constant relation ${\\sf {K}}_e \\subseteq {\\bf D} \\times {\\bf E}$ is approximable.",
"Proof (lemma) The proof of this lemma is left to the reader.",
"$\\:\\:\\Box $ To prove the “if” direction of the theorem, we observe that we can construct the relations $F_{a,*}$ and $F_{*,b}$ for all $a \\in {\\bf A}$ and $b \\in {\\bf B}$ by composing and pairing primitive approximable mappings.",
"In particular, $F_{a,*}$ is the relation $F \\circ \\langle K_a, {\\sf I}_B \\rangle $ where ${\\sf I}_B$ denotes the identity relation on ${\\bf B}$ .",
"Similary, $F_{*,b}$ is the relation $F \\circ \\langle {\\sf I}_A, K_b\\rangle $ where ${\\sf I}_A$ denotes the identity relation on ${\\bf A}$ .",
"To prove the “only-if” direction, we assume that for all $a \\in {\\bf A}$ and $b \\in {\\bf B}$ , the relations $F_{a,*}$ and $F_{*,b}$ are approximable.",
"We must show that the four closure properties for approximable mappings hold for $F$ .",
"Since $F_{\\bot _A,*}$ is approximable, $[\\bot _B,\\bot _C] \\in F_{\\bot _A,*}$ , which implies that $[[\\bot _A,\\bot _B],\\bot _C] \\in F$ .",
"If $[[x,y],z] \\in F$ and $[[x,y],z^{\\prime }] \\in F$ , then $[y,z] \\in F_{x,*}$ and $[y,z^{\\prime }] \\in F_{x,*}$ .",
"Since $F_{x,*}$ is approximable, $[y,z \\sqcup z^{\\prime }] \\in F_{x,*}$ , implying $[[x,y],z \\sqcup z^{\\prime }] \\in F$ .",
"If $[[x,y],z] \\in F$ and $z^{\\prime } \\sqsubseteq z$ , then $[y,z] \\in F_{x,*}$ .",
"Since $F_{x,*}$ is approximable, $[y,z^{\\prime }] \\in F_{x,*}$ , implying $[[x,y],z^{\\prime }] \\in F$ .",
"If $[[x,y],z] \\in F$ and $[x,y] \\sqsubseteq [x^{\\prime },y^{\\prime }]$ , then $[y,z] \\in F_{x,*}$ , $x \\sqsubseteq x^{\\prime }$ , and $y \\sqsubseteq y^{\\prime }$ .",
"Since $F_{x,*}$ is approximable, $[y^{\\prime },z] \\in F_{x,*}$ , implying $[x,y^{\\prime }],z] \\in F$ , which is equivalent to $[x,z] \\in F_{*,y^{\\prime }}$ .",
"Since $F_{*,y^{\\prime }}$ is approximable, $[x^{\\prime },z] \\in F_{*,y^{\\prime }}$ , implying $[[x^{\\prime },y^{\\prime }],z] \\in F$ .",
"$\\:\\:\\Box $ The same result can be restated in terms of continous functions.",
"Theorem 3.7: A function of two arguments, $f:{\\cal A}\\times {\\cal B}\\rightarrow {\\cal C}$ is continuous iff for every $a\\in {\\cal A}$ and every $b\\in {\\cal B}$ , the unary functions $x\\mapsto f[a,x]~{\\rm and}~y\\mapsto f[y,b]$ are continuous.",
"Proof Immediate from the previous theorem and the fact that the domain of approximable mappings over $({\\bf A} \\times {\\bf B}) \\times {\\bf C}$ is isomorphic to the domain of continuous functions over $({\\cal A} \\times {\\cal B}) \\times {\\cal C}$ .",
"$\\:\\:\\Box $ Multiary Function Composition The composition of functions, as defined in Theorem REF , can be generalized to functions of several arguments.",
"But we need some new syntactic machinery to describe more general forms of function composition.",
"Definition 3.8: [Cartesian Types] Let $S$ be a set of symbols used to denote primitive types.",
"The set $S^*$ of Cartesian types over $S$ consists of the set of expressions denoting all finite non-empty Cartesian products over primitive types in $S$ : $S^* ::= S \\; \\:\\vert \\:\\; S \\times S \\; \\:\\vert \\:\\; \\ldots \\; .$ A signature $\\Sigma $ is a pair $\\langle S, O \\rangle $ consisting of a set $S$ of type names ${\\lbrace {s_1, \\ldots , s_m}\\rbrace }$ used to denote domains and a set $O={\\lbrace {o_i^{\\rho _i\\rightarrow \\sigma _i}\\; \\:\\vert \\:\\; 1 \\le i \\le m, \\; \\rho _i \\in S^*, \\; \\sigma _i \\in S}\\rbrace }$ of function symbols used to denote first-order functions over the domains $S$ .",
"Let $V={\\lbrace {v_i^{\\tau } \\; \\:\\vert \\:\\; \\tau \\in S,\\; i \\in {N}}\\rbrace }$ be a countably-infinite set of symbols (variables) distinct from the symbols in $\\Sigma $ .",
"The typed expressions over $\\Sigma $ (denoted ${\\cal Ex}(\\Sigma )$ ) is the set of “typed” terms determined by the following inductive definition: $v_i^{\\tau } \\in V$ is a term of type $\\tau $ , for $M_1^{\\tau _1}, \\dots , M_n^{\\tau _n} \\in {\\cal Ex}(\\Sigma )$ and $o^{(\\tau _1 \\times \\dots \\times \\tau _n) \\rightarrow \\tau _0} \\in O$ then $o^{(\\tau _1 \\times \\dots \\times \\tau _n) \\rightarrow \\tau _0}(M_1^{\\tau _1}, \\dots ,M_n^{\\tau _n})^{\\tau _0}$ is a term of type $\\tau _0$ .",
"We will restrict our attention to terms where every instance of a variable $v_i$ in a term has the same type $\\tau $ .",
"To simplify notation, we will drop the type superscripts from terms whenever they can be easily inferred from context.",
"Definition 3.9: [Finitary Algebra] A finitary algebra with signature $\\Sigma $ is a function ${\\bf A}$ mapping each primitive type $\\tau \\in S$ to a finitary basis ${{\\bf A}}{[\\!",
"[}\\tau {]\\!",
"]}$ , each operation type $\\tau ^1 \\times \\dots \\times \\tau ^n \\in S^*$ to the finitary basis ${{\\bf A}}{[\\!",
"[}\\tau ^1{]\\!]}",
"\\times \\dots \\times {{\\bf A}}{[\\!",
"[}\\tau ^n{]\\!",
"]}$ , each function symbol $o_i^{\\rho _i \\rightarrow \\sigma _i} \\in O$ to an approximable mapping ${{\\bf A}}{[\\![}o_i{]\\!]}",
"\\subseteq ({{\\bf A}}{[\\!",
"[}\\rho _i{]\\!]}",
"\\times {{\\bf A}}{[\\!",
"[}\\sigma _i{]\\!",
"]})$ .",
"(Recall that ${{\\bf A}}{[\\!",
"[}\\rho _i{]\\!",
"]}$ is a product basis.)",
"Definition 3.10: [Closed Term] A term $M \\in {\\cal Ex}(\\Sigma )$ is closed iff it contains no variables in $V$ .",
"The finitary algebra ${\\bf A}$ implicitly assigns a meaning to every closed term $M$ in ${\\cal Ex}(\\Sigma )$ .",
"This extension is inductively defined by the equation: ${{\\bf A}}{[\\!",
"[}o[M_1,\\dots ,M_n]{]\\!]}",
"= {{\\bf A}}{[\\![}o{]\\!",
"]}[{{\\bf A}}{[\\![}M_1{]\\!",
"]}, \\dots , {{\\bf A}}{[\\![}M_n{]\\!]}]",
"=$ ${\\lbrace { b_0 \\; \\:\\vert \\:\\; \\exists [b_1,\\dots ,b_n] \\in {{\\bf A}}{[\\!",
"[}\\rho _i{]\\!]}",
"\\;[b_1,\\dots ,b_n] {{\\bf A}}{[\\![}o{]\\!]}",
"b_0}\\rbrace } \\, .$ We can extend ${\\bf A}$ to terms $M$ with free variables by implicitly abstracting over all of the free variables in $M$ .",
"Definition 3.11: [Meaning of Terms] Let $M$ be a term in ${\\cal Ex}{(\\Sigma )}$ and let $l = x_1^{\\tau _1}, \\dots , x_n^{\\tau _n}$ be a list of distinct variables in $V$ containing all the free variables of $M$ .",
"Let ${\\bf A}$ be a finitary algebra with signature $\\Sigma $ and for each tuple $[d_1, \\dots , d_n] \\in {{\\bf A}}{[\\!",
"[}\\tau _1{]\\!]}",
"\\times \\dots \\times {{\\bf A}}{[\\!",
"[}\\tau _n{]\\!",
"]}$ , let ${\\bf A}_{\\lbrace {x_1 := d_1,\\dots , x_n := d_n}\\rbrace }$ denote the algebra ${\\bf A}$ extended by defining ${{\\bf A}}{[\\![}x_i{]\\!]}",
"= d_i$ for $1 \\le i \\le n$ .",
"The meaning of $M$ with respect to $l$ , denoted ${{\\bf A}}{[\\!",
"[}x_1^{\\tau _1}, \\dots , x_n^{\\tau _n} \\, \\mapsto \\,M{]\\!",
"]}$ , is the relation $F_M \\subseteq ({{\\bf A}}{[\\!",
"[}\\tau _1{]\\!]}",
"\\times {{\\bf A}}{[\\!",
"[}\\tau _n{]\\!]})",
"\\times {{\\bf A}}{[\\!",
"[}\\tau _0{]\\!",
"]}$ defined by the equation: $F_M[d_1, \\dots , d_n] = {{\\bf A}_{\\lbrace {x_1 := d_1,\\dots ,x_1 := d_n}\\rbrace }}{[\\![}M{]\\!",
"]}$ The relation denoted by ${{\\bf A}}{[\\!",
"[}x_1^{\\tau _1}, \\dots , x_n^{\\tau _n} \\, \\mapsto \\, M{]\\!",
"]}$ is often called a substitution.",
"The following theorem shows that the relation ${{\\bf A}}{[\\!",
"[}x_1^{\\tau _1}, \\dots , x_n^{\\tau _n} \\, \\mapsto \\, M{]\\!",
"]}$ is approximable.",
"Theorem 3.12: (Closure of continuous functions under substitution) Let $M$ be a term in ${\\cal Ex}{(\\Sigma )}$ and let $l = x_1^{\\tau _1}, \\dots , x_n^{\\tau _n}$ be a list of distinct variables in $V$ containing all the free variables of $M$ .",
"Let ${\\bf A}$ be a finitary algebra with signature $\\Sigma $ .",
"Then the relation $F_M$ denoted by the expression $x_1^{\\tau _1}, \\dots , x_n^{\\tau _n} \\mapsto M $ is approximable.",
"Proof The proof proceeds by induction on the structure of $M$ .",
"The base cases are easy.",
"If $M$ is a variable $x_i$ , the relation $F_M$ is simply the projection mapping ${\\sf {P}}_i$ .",
"If $M$ is a constant $c$ of type $\\tau $ , then $F_M$ is the contant relation ${\\sf {K}}_c$ of arity $n$ .",
"The induction step is also straightforward.",
"Let $M$ have the form $g[M_1^{\\sigma _1},\\ldots ,M_m^{\\sigma _m}]$ .",
"By the induction hypothesis, $x_1^{\\tau _1}, \\dots , x_n^{\\tau _n} \\mapsto M_i^{\\sigma _i} $ denotes an approximable mapping $F_{M_i} \\subseteq ({{\\bf A}}{[\\!",
"[}\\tau _1{]\\!]}",
"\\times {{\\bf A}}{[\\!",
"[}\\tau _n{]\\!]})",
"\\times {{\\bf A}}{[\\!",
"[}\\sigma _i{]\\!",
"]}$ .",
"But $F_M$ is simply the composition of the approximable mapping ${{\\bf A}}{[\\![}g{]\\!",
"]}$ with the mapping $\\langle F_{M_1}, \\dots F_{M_m} \\rangle $ .",
"Theorem REF tells us that the composition must be approximable.",
"$\\:\\:\\Box $ The preceding generalization of composition obviously carries over to continuous functions.",
"The details are left to the reader.",
"Function Spaces The next domain constructor, the function-space constructor, allows approximable mappings (or, equivalently, continuous functions) to be regarded as objects.",
"In this framework, standard operations on approximable mappings such as application and composition are approximable mappings too.",
"Indeed, the definitions of ideals and of approximable mappings are quite similar.",
"The space of approximable mappings is built by looking at the actions of mappings on finite sets, and then using progressively larger finite sets to construct the mappings in the limit.",
"To this end, the notion of a finite step mapping is required.",
"Definition 3.13: [Finite Step Mapping] Let ${\\bf A}$ and ${\\bf B}$ be finitary bases.",
"An approximable mapping $F \\subseteq {\\bf A} \\times {\\bf B}$ is a finite step mapping iff there exists a finite set $S\\subseteq {\\bf A}\\times {\\bf B}$ and $F$ is the least approximable mapping such that $S \\subseteq F$ .",
"It is easy to show that for every consistent finite set $S \\subseteq {\\bf A}\\times {\\bf B}$ , a least mapping $F$ always exists.",
"$F$ is simply the closure of $S$ under the four conditions that an approximable mapping must satisfy.",
"The least approximable mapping respecting the empty set is the relation ${\\lbrace {\\langle a, \\bot _B \\rangle \\; \\:\\vert \\:\\; a \\in {\\bf A}}\\rbrace }$ .",
"The space of approximable mappings is built from finite step mappings.",
"Definition 3.14: [Partial Order of Finite Step Mappings] For finitary bases ${\\bf A}$ and ${\\bf B}$ the mapping basis is the partial order ${\\bf A}\\Rightarrow {\\bf B}$ consisting of the universe of all finite step mappings, and the approximation ordering $F \\sqsubseteq G \\iff \\forall a\\in {\\bf A} \\; F(a) \\sqsubseteq _B G(a) \\, .$ The following theorem establishes that the constructor $\\Rightarrow $ maps finitary bases into finitary bases.",
"Theorem 3.15: Let ${\\bf A}$ and ${\\bf B}$ be finitary bases.",
"Then, the mapping basis ${\\bf A}\\Rightarrow {\\bf B}$ is a finitary basis.",
"Proof Since the elements are finite subsets of a countable set, the basis must be countable.",
"It is easy to confirm that ${\\bf A}\\Rightarrow {\\bf B}$ is a partial order; this task is left to the reader.",
"We must show that every finite consistent subset of ${\\bf A}\\Rightarrow {\\bf B}$ has a least upper bound in ${\\bf A}\\Rightarrow {\\bf B}$ .",
"Let $§$ be a finite consistent subset of the universe of ${\\bf A}\\Rightarrow {\\bf B}$ .",
"Each element of $§$ is a set of ordered pairs $\\langle a, b \\rangle $ that meets the approximable mapping closure conditions.",
"Since $§$ is consistent, it has an upper bound $§^{\\prime } \\in {\\bf A}\\Rightarrow {\\bf B}$ .",
"Let $U = \\bigcup §$ .",
"Clearly, $U \\subseteq §^{\\prime }$ .",
"But $U$ may not be approximable.",
"Let $S$ be the intersection of all relations in ${\\bf A}\\Rightarrow {\\bf B}$ above $§$ .",
"Clearly $U \\subseteq S$ , implying $S$ is a superset of every element of $§$ .",
"It is easy to verify that $S$ is approximable, because all the approximable mapping closure conditions are preserved by infinite intersections.",
"$\\:\\:\\Box $ Definition 3.16: [Function Domain] We will denote the domain of ideals determined by the finitary basis ${\\bf A}\\Rightarrow {\\bf B}$ by the expression ${\\cal A}\\Rightarrow {\\cal B}$ .",
"The justification for this notation will be explained shortly.",
"Since the partial order of approximable mappings is isomorphic to the partial order of continuous functions, the preceding definitions and theorems about approximable mappings can be restated in terms of continuous functions.",
"Definition 3.17: [Finite Step Function] Let ${\\cal A}$ and ${\\cal B}$ be the domains determined by the finitary bases ${\\bf A}$ and ${\\bf B}$ , respectively.",
"A continuous function $f$ in ${\\cal A}\\rightarrow _c{\\cal B}$ is finite iff there exists a finite step mapping $F \\subseteq {\\bf A} \\times {\\bf B}$ such that $f$ is the function determined by $F$ .",
"Definition 3.18: [Function Basis] For domains ${\\cal A}$ and ${\\cal B}$ , the function basis is the partial order $({\\cal A}\\rightarrow _c{\\cal B})^0$ consisting of a universe of all finite step functions, and the approximation order $f\\sqsubseteq g \\iff \\forall a\\in {\\cal A} \\; f(a) \\sqsubseteq _{\\cal B}g(a) \\, .$ Corollary 3.19: (to Theorem REF ) For domains ${\\cal A}$ and ${\\cal B}$ , the function basis $({\\cal A}\\rightarrow _c{\\cal B})^0$ is a finitary basis.",
"We can prove that the domain constructed by generating the ideals over ${\\bf A}\\Rightarrow {\\bf B}$ is isomorphic to the partial order ${\\sf {Map}}({\\bf A},{\\bf B})$ of approximable mappings defined in Section 2.",
"This result is not surprising; it merely demonstrates that ${\\sf {Map}}({\\bf A},{\\bf B})$ is a domain and that we have identified the finite elements correctly in defining ${\\bf A}\\Rightarrow {\\bf B}$ .",
"Theorem 3.20: The domain of ideals determined by ${\\bf A}\\Rightarrow {\\bf B}$ is isomorphic to the partial order of the approximable mappings ${\\sf {Map}}({\\bf A},{\\bf B})$ .",
"Hence, ${\\sf {Map}}({\\bf A},{\\bf B})$ is a domain.",
"Proof We must establish an isomorphism between the domain determined by ${\\bf A}\\Rightarrow {\\bf B}$ and the partial order of mappings from ${\\bf A}$ to ${\\bf B}$ .",
"Let $h: {\\cal A}\\Rightarrow {\\cal B} \\rightarrow {\\sf {Map}}({\\bf A},{\\bf B})$ be the function defined by rule $h \\: {\\cal {F}}= \\bigcup {\\lbrace {F \\in {\\cal {F}}}\\rbrace } \\,.$ It is easy to confirm that the relation on the right hand side of the preceding equation is an approximable mapping: if it violated any of the closure properties, so would a finite approximation in ${\\cal {F}}$ .",
"We must prove that the function $h$ is one-to-one and onto.",
"To prove the former, we note that each pair of distinct ideals has a witness $\\langle a, b \\rangle $ that belongs to a set in one ideal but not in any set in the other.",
"Hence, the images of the two ideals are distinct.",
"The function $h$ is onto because every approximable mapping is the image of the set of finite step maps that approximate it.",
"$\\:\\:\\Box $ The preceding theorem can be restated in terms of continuous functions.",
"Corollary 3.21: (to Theorem REF ) The domain of ideals determined by the finitary basis $({\\cal A}\\rightarrow _c{\\cal B})^0$ is isomorphic to the partial order of continuous functions ${\\cal A} \\rightarrow _c {\\cal B}$ .",
"Hence, ${\\cal A} \\rightarrow _c {\\cal B}$ is a domain.",
"Now that we have defined the approximable mapping and continous function domain constructions, we can show that operators on maps and functions introduced in Section 2 are continuous functions.",
"Theorem 3.22: Given finitary bases, ${\\bf A}$ and ${\\bf B}$ , there is an approximable mapping $Apply:(({\\bf A}\\Rightarrow {\\bf B}) \\times {\\bf A}) \\times {\\bf B}$ such that for all $F:{\\bf A}\\Rightarrow {\\bf B}$ and $a\\in {\\bf A}$ , $Apply[F,a] = F(a)\\,.$ Recall that for any approximable mapping $G \\subseteq {\\bf C} \\times {\\bf D}$ and any element $c \\in {\\bf C}$ $G(c) = {\\lbrace {d \\; \\:\\vert \\:\\; c \\: G \\: d}\\rbrace }\\,.$ Proof For $F\\in ({\\bf A}\\Rightarrow {\\bf B})$ , $a\\in {\\bf A}$ and $b\\in {\\bf B}$ , define the $Apply$ relation as follows: $[F,a]\\:Apply\\:b \\iff a\\:F\\:b\\,.$ It is easy to verify that $Apply$ is an approximable mapping; this step is left to the reader.",
"From the definition of $Apply$ , we deduce $Apply[F,a] = {\\lbrace {b \\; \\:\\vert \\:\\; [F,a]\\: Apply \\: b}\\rbrace } ={\\lbrace {b \\; \\:\\vert \\:\\; a \\: F \\: b}\\rbrace } = F(a)\\,.$ $\\:\\:\\Box $ This theorem can be restated in terms of continuous functions.",
"Corollary 3.23: Given domains, ${\\cal A}$ and ${\\cal B}$ , there is a continuous function $apply:(({\\cal A}\\rightarrow _c{\\cal B}) \\times {\\cal A}) \\rightarrow _c {\\cal B}$ such that for all $f:{\\cal A}\\rightarrow _c{\\cal B}$ and $a\\in {\\cal A}$ , $apply[f,a] = f(a)\\,.$ Proof (of corollary).",
"Let $apply: (({\\cal A}\\rightarrow _c {\\cal B}) \\times {\\cal A}) \\rightarrow _c {\\cal B}$ be the continuous function (on functions rather than relations!)",
"corresponding to $Apply$ .",
"From the definition of $apply$ and Theorem REF which relates approximable mappings on finitary bases to continous functions over the corresponding domains, we know that $apply[f,{\\cal I}_A]=\\lbrace b\\in {\\bf B}\\:\\vert \\:\\exists F^{\\prime }\\in ({\\bf A}\\Rightarrow {\\bf B}) \\ ; \\exists a\\in {\\cal I}_A \\wedge F^{\\prime } \\subseteq F \\wedge [F^{\\prime },a]\\:Apply\\:b\\rbrace $ where $F$ denotes the approximable mapping corresponding to $f$ .",
"Since $f$ is the continuous function corresponding to $F$ , $f({\\cal I}_A) = \\lbrace b\\in {\\bf B}\\:\\vert \\:\\exists a\\in {\\cal I}_A \\; a\\:F\\:b\\rbrace $ So, by the definition of the $Apply$ relation, $apply[f,{\\cal I}_A]\\subseteq f({\\cal I}_A)$ .",
"For every $b\\in f({\\cal I}_A)$ , there exists $a \\in {\\cal I}_A$ such that $a\\:F\\:b$ .",
"Let $F^{\\prime }$ be the least approximable mapping such that $a \\: F^{\\prime } b$ .",
"By definition, $F^{\\prime }$ is a finite step mapping.",
"Hence $b \\in apply[f,{\\cal I}_A]$ , implying $f({\\cal I}_A) \\subseteq apply[f,{\\cal I}_A]$ .",
"Therefore, $f({\\cal I}_A) = apply[f,{\\cal I}_A]$ for arbitrary ${\\cal I}_A$ .",
"$\\:\\:\\Box $ The preceding theorem and corollary demonstrate that approximable mappings and continuous functions can operate on other approximable mappings or continuous functions just like other data objects.",
"The next theorem shows that the currying operation is a continuous function.",
"Definition 3.24: [The Curry Operator] Let A, B, and C be finitary bases.",
"Given an approximable mapping $G$ in the basis $({\\bf A}\\times {\\bf B})\\Rightarrow {\\bf C}$ , $Curry_G:{\\bf A}\\Rightarrow ({\\bf B}\\Rightarrow {\\bf C})$ is the relation defined by the equation $Curry_G(a) = {\\lbrace {F \\in {\\bf B}\\Rightarrow {\\bf C} \\; \\:\\vert \\:\\;\\forall [b,c] \\in F \\; [a,b] \\: G \\: c}\\rbrace }$ for all $a \\in {\\bf A}$ .",
"Similarly, given any continuous function $g: ({\\cal A}\\times {\\cal B})\\rightarrow _c{\\cal C}$ , $curry_g:{\\cal A}\\rightarrow _c({\\cal B}\\rightarrow _c{\\cal C})$ is the function defined by the equation $curry_g({\\cal I}_A) = (y \\mapsto g[{\\cal I}_A,y])\\,.$ By Theorem 2.7?",
"?, $(y \\mapsto g[{\\cal I}_A,y])$ is a continous function.",
"Lemma 3.25: $Curry_G$ is an approximable mapping and $curry_g$ is the continuous function determined by $Curry_G$ .",
"Proof A straightforward exercise.$\\:\\:\\Box $ It is more convenient to discuss the currying operation in the context of continuous functions than approximable mappings.",
"Theorem 3.26: Let $g \\in ({\\cal A}\\times {\\cal B})\\rightarrow _c{\\cal C}$ and $h \\in ({\\cal A}\\rightarrow _c({\\cal B}\\rightarrow _c{\\cal C})$ .",
"The $curry$ operation satisfies the following two equations: $apply\\circ \\langle curry_g\\circ p_0,p_1\\rangle & = & g\\\\curry_{apply\\circ \\langle h\\circ p_0,p_1\\rangle } & = & h\\, .$ In addition, the function $curry:({\\cal A}\\times {\\cal B}\\rightarrow {\\cal C})\\rightarrow ({\\cal A}\\rightarrow _c({\\cal B}\\rightarrow _c{\\cal C}))$ defined by the equation $curry(g)({\\cal I}_A)({\\cal I}_B) = curry_g({\\cal I}_A)({\\cal I}_B)$ is continuous.",
"Proof Let $g$ be any continuous function in the domain $({\\cal A}\\times {\\cal B})\\rightarrow _c{\\cal C}$ .",
"Recall that $curry_g(a) = (y \\mapsto g[a,y])\\,.$ Using this definition and the definition of operations in the first equation, we can deduce $\\begin{array}{lcl}apply\\circ \\langle curry_g\\circ p_0,p_1\\rangle [a,b]& = & apply[\\langle curry_g \\circ p_0, p_1 \\rangle [a,b]] \\\\& = & apply[(curry_g \\circ p_0)[a,b], p_1[a,b]] \\\\& = & apply[curry_g p_0[a,b], b] \\\\& = & apply[curry_g \\: a, b] \\\\& = & curry_g \\: a \\: b \\\\& = & g[a,b]\\,.\\end{array}$ Hence, the first equation holds.",
"The second equation follows almost immediately from the first.",
"Define $g^{\\prime }:({\\cal A}\\times {\\cal B})\\rightarrow _c{\\cal C}$ by the equation $g^{\\prime }[a,b] = h \\; a \\; b\\,.$ The function $g^{\\prime }$ is defined so that $curry_{g^{\\prime }} = h$ .",
"This fact is easy to prove.",
"For $a \\in {\\cal A}$ : $curry_{g^{\\prime }}(a) & = & (y \\mapsto g^{\\prime }[a,y])\\\\& = & (y \\mapsto h (a)(y))\\\\& = & h(a) \\, .$ Since $h = curry_{g^{\\prime }}$ , the first equation implies that $apply\\circ \\langle h\\circ p_0,p_1\\rangle & = &apply\\circ \\langle curry_{g^{\\prime }}\\circ p_0,p_1\\rangle \\\\& = & g^{\\prime }\\,.$ Hence, $curry_{apply}\\circ \\langle h\\circ p_0,p_1\\rangle = curry_{g^{\\prime }} = h\\,.$ These two equations show that $({\\cal A}\\times {\\cal B})\\rightarrow _c{\\cal C}$ is isomorphic to $({\\cal A}\\rightarrow _c({\\cal B}\\rightarrow _c{\\cal C})$ under the $curry$ operation.",
"In addition, the definition of $curry$ shows that $curry(g) \\sqsubseteq curry(g^{\\prime }) \\iff g \\sqsubseteq g^{\\prime }\\,.$ Hence, $curry$ is an isomorphism.",
"Moreover, $curry$ must be continuous.",
"$\\:\\:\\Box $ The same theorem can be restated in terms of approximable mappings.",
"Corollary 3.27: The relation $Curry_G$ satisfies the following two equations: $Apply\\circ \\langle Curry_G \\circ {\\sf {P}}_0,{\\sf {P}}_1\\rangle & = & G\\\\Curry_{Apply\\circ \\langle G\\circ {\\sf {P}}_0,{\\sf {P}}_1\\rangle } & = & G\\, .$ In addition, the relation $Curry:({\\bf A}\\times {\\bf B})\\Rightarrow {\\bf C})\\Rightarrow ({\\bf A} \\Rightarrow ({\\bf B}\\Rightarrow {\\bf C}))$ defined by the equation $Curry(G) = {\\lbrace {[a,F] \\; \\:\\vert \\:\\; a \\in {\\bf A}, \\:F \\in ({\\bf B}\\Rightarrow {\\bf C}), \\; \\forall [b,c] \\in F \\:[a,b]\\: G \\: c }\\rbrace }$ is approximable.",
"Table: Domains and Finitary BasesTable REF summarizes the main elements of the correspondence between domains and finitary bases.",
"Whenever convenient, in the following sections we take liberty to confuse corresponding notions.",
"Context and notation should make clear which category is meant.",
"Exercises Exercise 3.28: We assume that there is a countable basis.",
"Thus, the basis elements could without loss of generality be defined in terms of $\\lbrace 0,1\\rbrace ^*$ .",
"Show that the product space ${\\bf A}\\times {\\bf B}$ could be defined as a finitary basis over $\\lbrace 0,1\\rbrace ^*$ such that ${\\bf A}\\times {\\bf B}=\\lbrace [0a,1b]\\:\\vert \\:a\\in {\\bf A},b\\in {\\bf B}\\rbrace $ Give the appropriate definition for the elements in the domain.",
"Also show that there exists an approximable mapping $diag:{\\bf D}\\rightarrow {\\bf D}\\times {\\bf D}$ where $diag(x)= [x,x]$ for all $x\\in {\\cal D}$ .",
"Exercise 3.29: Establish some standard isomorphisms: ${\\bf A}\\times {\\bf B}\\approx {\\bf B}\\times {\\bf A}$ ${\\bf A}\\times ({\\bf B}\\times {\\bf C})\\approx ({\\bf A}\\times {\\bf B})\\times {\\bf C}$ ${\\bf A}\\approx {\\bf A^{\\prime }},{\\bf B}\\approx {\\bf B^{\\prime }}\\:\\Rightarrow \\:{\\bf A}\\times {\\bf B}\\approx {\\bf A^{\\prime }}\\times {\\bf B^{\\prime }}$ for all finitary bases.",
"Exercise 3.30: Let $B\\subseteq \\lbrace 0,1\\rbrace ^*$ be a finitary basis.",
"Define $B^\\infty =\\bigcup \\limits _{n=0}^\\infty 1^n0B$ Thus, $B^\\infty $ contains infinitely many disjoint copies of $B$ .",
"Now let $D^\\infty $ be the least family of subsets over $\\lbrace 0,1\\rbrace ^*$ such that $B^\\infty \\in D^\\infty $ if $b\\in {\\bf B}$ and $d\\in D^\\infty $ , then $0X\\cup 1Y\\in D^\\infty $ .",
"Show that, with the superset relation as the approximation ordering, $D^\\infty $ is a finitary basis.",
"State any assumptions that must be made.",
"Show then that $D^\\infty \\approx D\\times D^\\infty $ .",
"Exercise 3.31: Using the product construction as a guide, generate a definition for the separated sum structure ${\\bf A}+{\\bf B}$ .",
"Show that there are mappings $in_A:{\\bf A}\\rightarrow {\\bf A}+{\\bf B}$ , $in_B:{\\bf B}\\rightarrow {\\bf A}+{\\bf B}$ , $out_A:{\\bf A}+{\\bf B}\\rightarrow {\\bf A}$ , and $out_B:{\\bf A}+{\\bf B}\\rightarrow {\\bf B}$ such that $out_A\\circ in_A = {\\sf I}_A$ where ${\\sf I}_A$ is the identity function on ${\\bf A}$ .",
"State any necessary assumptions to ensure this function equation is true.",
"Exercise 3.32: For approximable mappings $f:{\\bf A}\\rightarrow {\\bf A^{\\prime }}$ and $g:{\\bf B}\\rightarrow {\\bf B^{\\prime }}$ , show that there exist approximable mappings, $f\\times g:{\\bf A}\\times {\\bf B}\\rightarrow {\\bf A^{\\prime }}\\times {\\bf B^{\\prime }}$ and $f+g:{\\bf A}+{\\bf B}\\rightarrow {\\bf A^{\\prime }}+{\\bf B^{\\prime }}$ such that $(f\\times g)[a,b] = [f \\: a, g \\: b]$ and thus $f\\times g = \\langle f\\circ p_0,g\\circ p_1\\rangle $ Show also that $out_A\\circ (f+g)\\circ in_A = f$ and $out_B\\circ (f+g)\\circ in_B = g$ Is $f+g$ uniquely determined by the last two equations?",
"Exercise 3.33: Prove that the composition operator is an approximable mapping.",
"That is, show that $comp:({\\bf B}\\rightarrow {\\bf C})\\times ({\\bf A}\\rightarrow {\\bf B})\\rightarrow ({\\bf A}\\rightarrow {\\bf C})$ is an approximable mapping where for $f:{\\bf A}\\rightarrow {\\bf B}$ and $g:{\\bf B}\\rightarrow {\\bf C}$ , $comp[g,f] = g\\circ f$ .",
"Show this using the approach used in showing the result for $apply$ and $curry$ .",
"That is, define the relation and then build the function from $apply$ and $curry$ , using $\\circ $ and paired functions.",
"(Hint: Fill in mappings according to the following sequence of domains.)",
"$\\begin{array}{c}({\\bf A}\\rightarrow {\\bf B})\\times {\\bf A}\\rightarrow {\\bf B}\\\\({\\bf B}\\rightarrow {\\bf C})\\times (({\\bf A}\\rightarrow {\\bf B})\\times {\\bf A})\\rightarrow ({\\bf B}\\rightarrow {\\bf C})\\times {\\bf B}\\\\(({\\bf B}\\rightarrow {\\bf C})\\times ({\\bf A}\\rightarrow {\\bf B}))\\times {\\bf A}\\rightarrow ({\\bf B}\\rightarrow {\\bf C})\\times {\\bf B}\\\\(({\\bf B}\\rightarrow {\\bf C})\\times ({\\bf A}\\rightarrow {\\bf B}))\\times {\\bf A}\\rightarrow {\\bf C}\\\\({\\bf B}\\rightarrow {\\bf C})\\times ({\\bf A}\\rightarrow {\\bf B})\\rightarrow ({\\bf A}\\rightarrow {\\bf C}).\\end{array}$ This map shows only one possible solution.",
"Exercise 3.34: Show that for every domain ${\\cal D}$ there is an approximable mapping $cond:{\\bf T}\\times {\\bf D}\\times {\\bf D}\\rightarrow {\\bf D}$ called the conditional operator such that $cond[true,a,b]=a$ $cond[false,a,b]=b$ $cond[\\bot _T,a,b]=\\bot _D$ and ${\\bf T}=\\lbrace \\bot _T,true,false\\rbrace $ such that $\\bot _T\\sqsubseteq true$ , $\\bot _T\\sqsubseteq false$ , but $true$ and $false$ are incomparable.",
"(Hint: Define a $Cond$ relation).",
"Fixed Points and Recursion Fixed Points Functions can now be constructed by composing basic functions.",
"However, we wish to be able to define functions recursively as well.",
"The technique of recursive definition will also be useful for defining domains as we will see in Section .",
"Recursion can be thought of as (possibly infinite) iterated function composition.",
"The primary result for interpreting recursive definitions is the following Fixed Point Theorem.",
"Theorem 4.1: For any continuous function $f:{\\cal D}\\rightarrow {\\cal D}$ determined by an approximable mapping $F:{\\bf D}\\rightarrow {\\bf D}$ , there exists a least element $x\\in {\\cal D}$ such that $f(x) = x.$ Proof Let $f^n$ stand for the function $f$ composed with itself $n$ times, and similarly for $F^n$ .",
"Thus, for $\\begin{array}{lcl}f^0&=&I_{\\cal D}~\\\\f^{n+1}&=&f\\circ f^n\\\\F^0&=&{\\sf I}_D~{\\rm and}\\\\F^{n+1}&=&F\\circ F^n\\end{array}$ we define $x = \\lbrace d\\in {\\bf D}\\:\\vert \\:\\exists n\\in {{N}}.",
"\\bot \\:F^n\\:d\\rbrace .$ To show that $x\\in {\\cal D}$ , we must show it to be an ideal.",
"Map $F$ is an approximable mapping, so $\\bot \\in x$ since $\\bot \\:F\\:\\bot $ .",
"For $d\\in x$ and $d^{\\prime }\\sqsubseteq d$ , $d^{\\prime }\\in x$ must hold since, for $d\\in x$ , there must exist an $a\\in {\\bf D}$ such that $a\\:F\\:d$ .",
"But by the definition of an approximable mapping, $a\\:F\\:d^{\\prime }$ must hold as well so $d^{\\prime }\\in x$ .",
"Closure under lubs is direct since $F$ must include lubs to be approximable.",
"To see that $f(x)=x$ , or equivalently $x\\:F\\:x$ , note that for any $d\\in x$ , if $d\\:F\\:d^{\\prime }$ , then $d^{\\prime }\\in x$ .",
"Thus, $f(x)\\sqsubseteq x$ .",
"Now, $x$ is constructed to be the least element in ${\\cal D}$ with this property.",
"To see this is true, let $a\\in {\\cal D}$ such that $f(a)\\sqsubseteq a$ .",
"We want to show that $x\\sqsubseteq a$ .",
"Let $d\\in x$ be an arbitrary element.",
"Therefore, there exists an $n$ such that $\\bot \\: F^n\\:d$ and therefore $\\bot \\:F\\:d_1\\:F\\:d_2\\:\\ldots \\:F\\:d_{n-1}\\:F\\:d.$ Since $\\bot \\in a$ , $d_1\\in f(a)$ .",
"Thus, since $f(a)\\sqsubseteq a$ , $d_1\\in a$ .",
"Thus, $d_2\\in f(a)$ and therefore $d_2\\in a$ .",
"Using induction on $n$ , we can show that $d\\in f(a)$ .",
"Therefore, $d\\in a$ and thus $x\\sqsubseteq a$ .",
"Since $f$ is monotonic and $f(x)\\sqsubseteq x$ , $f(f(x))\\sqsubseteq f(x)$ .",
"Since $x$ is the least element with this property, $x\\sqsubseteq f(x)$ and thus $x=f(x)$ .",
"$\\:\\:\\Box $ Since the element $x$ above is the least element, it must be unique.",
"Thus we have defined a function mapping the domain ${\\cal D}\\rightarrow {\\cal D}$ into the domain ${\\cal D}$ .",
"The next step is to show that this mapping is approximable.",
"Theorem 4.2: For any domain ${\\cal D}$ , there is an approximable mapping $fix:({\\bf D}\\rightarrow {\\bf D})\\rightarrow {\\bf D}$ such that if $f:{\\bf D}\\rightarrow {\\bf D}$ is an approximable mapping, $fix(f) = f(fix(f))$ and for $x\\in {\\cal D}$ , $f(x)\\sqsubseteq x \\:\\Rightarrow \\:fix(f)\\sqsubseteq x$ This property implies that $fix$ is unique.",
"The function $fix$ is characterized by the equation $fix(f)=\\bigcup \\limits _{n=0}^\\infty f^n(\\bot )$ for all $f:{\\bf D}\\rightarrow {\\bf D}$ .",
"Proof The final equation can be simplified to $fix(f) = \\lbrace d\\in {\\bf D}\\:\\vert \\:\\exists n\\in {{N}}.\\bot \\:f^n\\:d\\rbrace $ which is the equation used in the previous theorem to define the fixed point.",
"Using the formula from Exercise REF on the above definition for $fix$ yields the following equation to be shown: $fix(f)=\\bigcup \\lbrace fix({\\cal I}_F)\\:\\vert \\:\\exists F\\in ({\\bf D}\\rightarrow {\\bf D}).F\\in f\\rbrace $ where ${\\cal I}_F$ denotes the ideal for $F$ in ${\\bf D}\\rightarrow {\\bf D}$ .",
"From its definition, $fix$ is monotonic since, if $f\\sqsubseteq g$ , then $fix(f)\\sqsubseteq fix(g)$ since $f^n\\sqsubseteq g^n$ .",
"Since $F\\in f$ , ${\\cal I}_F\\sqsubseteq f$ and since $fix$ is monotonic, $fix({\\cal I}_F)\\sqsubseteq fix(f)$ .",
"Let $x\\in fix(f)$ .",
"Thus, there is a finite sequence of elements such that $\\bot \\:f\\:x_1\\:f\\:\\ldots \\:f\\:x^{\\prime }\\:f\\:x$ .",
"Define $F$ as the basis element encompassing the step functions required for this sequence.",
"Clearly, $F\\in f$ .",
"In addition, this same sequence exists in $fix({\\cal I}_F)$ since we constructed $F$ to contain it, and thus, $x\\in fix({\\cal I}_F)$ and $fix(f)\\sqsubseteq fix({\\cal I}_F)$ .",
"The equality is therefore established.",
"The first equality is direct from the Fixed Point Theorem since the same definition is used.",
"Assume $f(x)\\sqsubseteq x$ for some $x\\in {\\cal D}$ .",
"Since $\\bot \\in x$ , $x\\ne \\emptyset $ .",
"Since $f$ is an approximable mapping, for $x^{\\prime }\\in x$ and $x^{\\prime }\\:f\\:y$ , $y\\in x$ must hold.",
"By induction, for any $\\bot \\:f\\:y$ , $y\\in x$ must hold.",
"Thus, $fix(f)\\sqsubseteq x$ .",
"To see that the operator is unique, define another operator $fax$ that satisfies the first two equations.",
"It can easily be shown that $\\begin{array}{lcll}fix(f)&\\sqsubseteq & fax(f)~{\\rm and}\\\\fax(f)&\\sqsubseteq &fix(f)\\end{array}$ Thus the two operators are the same.",
"$\\:\\:\\Box $ Recursive Definitions Recursion has played a part already in the definitions above.",
"Recall that $f^n$ was defined for all $n\\in {{N}}$ .",
"More complex examples of recursion are given below.",
"Example 4.3: Define a basis ${\\bf N}=\\langle N,\\sqsubseteq _N\\rangle $ where $N=\\lbrace \\lbrace n\\rbrace \\:\\vert \\:n\\in {{N}}\\rbrace \\cup \\lbrace {{N}}\\rbrace $ and the approximation ordering is the superset relation.",
"This generates a flat domain with $\\bot =\\lbrace \\lbrace {{N}}\\rbrace \\rbrace $ and the total elements being in a one-to-one correspondence with the natural numbers.",
"Using the construction outlined in Exercise REF , construct the basis $F=N^\\infty $ .",
"Its corresponding domain is the domain of partial functions over the natural numbers.",
"To see this, let $\\Phi $ be the set of all finite partial functions $\\varphi \\subseteq {{N}}\\times {{N}}$ .",
"Define $\\uparrow \\varphi =\\lbrace \\psi \\in \\Phi \\:\\vert \\:\\varphi \\subseteq \\psi \\rbrace $ Consider the finitary basis $\\langle F^{\\prime },\\sqsubseteq _F^{\\prime }\\rangle $ where $F^{\\prime }=\\lbrace \\uparrow \\varphi \\:\\vert \\:\\varphi \\in \\Phi \\rbrace $ and the approximation order is the superset relation.",
"The reader should satisfy himself that $F^{\\prime }$ and $F$ are isomorphic and that the elements are the partial functions.",
"The total elements are the total functions over the natural numbers.",
"The domains ${\\cal F}$ and $({\\cal N}\\rightarrow {\\cal N})$ are not isomorphic.",
"However, the following mapping $val:F\\times {\\bf N}\\rightarrow {\\bf N}$ can be defined as follows: $(\\uparrow \\varphi ,\\lbrace n\\rbrace )\\:val\\:\\lbrace m\\rbrace \\iff (n,m)\\in \\varphi $ and $(\\uparrow \\varphi ,{{N}})\\:val\\:{{N}}$ Define also as the ideal for $m\\in {\\cal N}$ , $\\hat{m} = \\lbrace \\lbrace m\\rbrace ,{{N}}\\rbrace $ It is easy to show then that for $\\pi \\in {\\cal F}$ and $n\\in {\\cal N}$ we have $\\begin{array}{lcll}val(\\pi ,\\hat{n})&=&\\hat{\\pi (n)}&{\\rm if}~\\pi (n)\\ne \\bot \\\\&=&\\bot &{\\rm otherwise}\\end{array}$ Thus, $curry(val):{\\bf F}\\rightarrow ({\\bf N}\\rightarrow {\\bf N})$ is a one-to-one function on elements.",
"(The problem is that (${\\bf N}\\rightarrow {\\bf N}$ ) has more elements than F does as the reader should verify for himself).",
"Now, what about mappings $f:{\\bf F}\\rightarrow {\\bf F}$ ?",
"Consider the function $\\begin{array}{lcll}f(\\pi )(n)&=&0&{\\rm if}~n=0\\\\&=&\\pi (n-1)+n-1&{\\rm for}~n>0\\end{array}$ If $\\pi $ is a total function, $f(\\pi )$ is a total function.",
"If $\\pi (k)$ is undefined, then $f(\\pi )(k+1)$ is undefined.",
"The function $f$ is approximable since it is completely determined by its actions on partial functions.",
"That is $f(\\pi )=\\bigcup \\lbrace f(\\varphi )\\:\\vert \\:\\exists \\varphi \\in \\Phi .\\varphi \\subseteq \\pi \\rbrace $ The Fixed Point Theorem defines a least fixed point for any approximable mapping.",
"Let $\\sigma =f(\\sigma )$ .",
"Now, $\\sigma (0)=0$ and $\\begin{array}{lcl}\\sigma (n+1)&=&f(\\sigma )(n+1)\\\\&=&\\sigma (n)+n\\end{array}$ By induction, $\\sigma (n)=\\sum \\limits _{i=0}^n i$ and therefore, $\\sigma $ is a total function.",
"Thus, $f$ has a unique fixed point.",
"Now, in looking at $({\\bf N}\\rightarrow {\\bf N})$ , we have $\\hat{0}\\in {\\cal N}$ (The symbols $n$ and $\\hat{n}$ will no longer be distinguished, but the usage should be clear from context.).",
"Now define the two mappings, $succ,pred:{\\bf N}\\rightarrow {\\bf N}$ as approximable mappings such that $\\begin{array}{lcl}n\\:succ\\: m& \\iff & \\exists p\\in {{N}}.n\\sqsubseteq p,m\\sqsubseteq p+1\\\\n\\:pred\\: m& \\iff & \\exists p+1\\in {{N}}.n\\sqsubseteq p+1,m\\sqsubseteq p\\end{array}$ In more familiar terms, the same functions are defined as $\\begin{array}{lcll}succ(n)&=&n+1\\\\pred(n)&=&n-1&{\\rm if}~n>0\\\\&=&\\bot &{\\rm if}~n=0\\end{array}$ The mapping $zero:{\\bf N}\\rightarrow {\\bf T}$ is also defined such that $\\begin{array}{lcll}zero(n)&=&true&{\\rm if}~n=0\\\\&=&false&{\\rm if}~n>0\\end{array}$ where ${\\cal T}$ is the domain of truth value defined in an earlier section.",
"The structured domain $\\langle N,0,succ,pred,zero\\rangle $ is called “The Domain of the Integers” in the present context.",
"The function element $\\sigma $ defined as the fixed point of the mapping $f$ can now be defined directly as a mapping $\\sigma :{\\bf N}\\rightarrow {\\bf N}$ as follows: $\\sigma (n)=cond(zero(n),0,\\sigma (pred(n))+pred(n))$ where the function $+$ must be suitably defined.",
"Recall that $cond$ was defined earlier as part of the structure of the domain ${\\cal T}$ .",
"This equation is called a functional equation; the next section will give another notation, the $\\lambda -calculus$ for writing such equations.",
"$\\:\\:\\Box $ Example 4.4: The domain ${\\cal B}$ defined in Example REF contained only infinite elements as total elements.",
"A related domain, ${\\cal C}$ defined in Exercise REF , can be regarded as a generalization on ${\\cal N}$ .",
"To demonstrate this, the structured domain corresponding to the domain of integers must be presented.",
"The total elements in ${\\cal C}$ are denoted $\\sigma $ while the partial elements are denoted $\\sigma \\bot $ for any $\\sigma \\in \\lbrace 0,1\\rbrace ^*$ .",
"The empty sequence $\\epsilon $ assumes the role of the number 0 in ${\\cal N}$ .",
"Two approximable mappings can serve as the successor function: $x\\mapsto 0x$ denoted $succ_0$ and $x\\mapsto 1x$ denoted $succ_1$ .",
"The predecessor function is filled by the $tail$ mapping defined as follows: $\\begin{array}{lcll}tail(0x)& =& x,\\\\tail(1x)& =& x&{\\rm and}\\\\tail(\\epsilon )& =& \\bot .\\end{array}$ The $zero$ predicate is defined using the $empty$ mapping defined as follows: $\\begin{array}{lcll}empty(0x)& =& false,\\\\empty(1x)& =& false&{\\rm and}\\\\empty(\\epsilon )& =& true.\\end{array}$ To distinguish the other types of elements in ${\\cal C}$ , the following mappings are also defined: $\\begin{array}{lcll}zero(0x)& =& true,\\\\zero(1x)& =& false&{\\rm and}\\\\zero(\\epsilon )& =& false.\\\\one(0x)& =& false,\\\\one(1x)& =& true&{\\rm and}\\\\one(\\epsilon )& =& false.\\end{array}$ The reader should verify the conditions for an approximable mapping are met by these functions.",
"An element of ${\\cal C}$ can be defined using a fixed point equation.",
"For example, the total element representing an infinite sequence of alternating zeroes and ones is defined by the fixed point of the equation $a=01a$ .",
"This same element is defined with the equation $a=0101a$ .",
"(Is the element defined as $b=010b$ the same as the previous two?)",
"Approximable mappings in ${\\cal C}\\rightarrow {\\cal C}$ can also be defined using equations.",
"For example, the mapping $\\begin{array}{lcll}d(\\epsilon ) &= &\\epsilon ,\\\\d(0x)&=&00d(x)&{\\rm and}\\\\d(1x)&=&11d(x)\\end{array}$ can be characterized with the functional equation $d(x)=cond(empty(x),\\epsilon ,cond(zero(x),succ_0(succ_0(d(tail(x)))),succ_1(succ_1(d(tail(x))))))$ The concatenation function of Exercise REF over ${\\cal C}\\times {\\cal C}\\rightarrow {\\cal C}$ can be defined with the functional equation $C(x,y)=cond(empty(x),y,cond(zero(x),succ_0(C(tail(x),y)),succ_1(C(tail(x),y))))$ The reader should verify that this definition is consistent with the properties required in the exercise.",
"These definitions all use recursion.",
"They rely on the object being defined for a base case ($\\epsilon $ for example) or on earlier values ($tail(x)$ for example).",
"These equations characterize the object being defined, but unless a theorem is proved to show that a solution to the equation exists, the definition is meaningless.",
"However, the Fixed Point Theorem for domains was established earlier in this section.",
"Thus, solutions exist to these equations provided that the variables in the equation range over domains and any other functions appearing in the equation are known to be continuous (that is approximable).",
"Peano's Axioms To illustrate one use of the Fixed Point Theorem as well as show the use of recursion in a more familiar setting, we will show that all second order models of Peano's axioms are isomorphic.",
"Recall that Definition 4.5: [Model for Peano's Axioms] A structured set $\\langle {{N}},0,succ\\rangle $ for $0\\in {{N}}$ and $succ:{{N}}\\times {{N}}$ is a model for Peano's axioms if all the following conditions are satisfied: $\\forall n\\in {{N}}.",
"0\\ne succ(n)$ $\\forall n,m \\in {{N}}.succ(n)=succ(m)\\:\\Rightarrow \\:n=m$ $\\forall x\\subseteq {{N}}.0\\in x\\wedge succ(x)\\subseteq x\\:\\Rightarrow \\:x={{N}}$ where $succ(x)=\\lbrace succ(n)\\:\\vert \\:n\\in x\\rbrace $ .",
"The final clause is usually referred to as the principle of mathematical induction.",
"Theorem 4.6: All second order models of Peano's axioms are isomorphic.",
"Proof Let $\\langle N,0,+\\rangle $ and $\\langle M,\\bullet ,\\#\\rangle $ be models for Peano's axioms.",
"Let $N\\times M$ be the cartesian product of the two sets and let ${\\cal P}(N\\times M)$ be the powerset of $N\\times M$ .",
"Recall from Exercise REF that the powerset can be viewed as a domain with the subset relation as the approximation order.",
"Define the following mapping: $u\\mapsto \\lbrace (0,\\bullet )\\rbrace \\cup \\lbrace (+(n),\\#(m))\\:\\vert \\:(n,m)\\in u\\rbrace $ The reader should verify that this mapping is approximable.",
"Since it is indeed approximable, a fixed point exists for the function.",
"Let $r$ be the least fixed point: $r=\\lbrace (0,\\bullet )\\rbrace \\cup \\lbrace (+(n),\\#(m))\\:\\vert \\:(n,m)\\in r\\rbrace $ But $r$ defines a binary relation which establishes the isomorphism.",
"To see that $r$ is an isomorphism, the one-to-one and onto features must be established.",
"By construction, $0\\:r\\:\\bullet $ and $n\\:r\\:m \\:\\Rightarrow \\:+(n)\\:r\\:\\#(m)$ .",
"Now, the sets $\\lbrace (0,\\bullet )\\rbrace $ and $\\lbrace (+(n),\\#(m))\\:\\vert \\:(n,m)\\in r\\rbrace $ are disjoint by the first axiom.",
"Therefore, 0 corresponds to only one element in $m$ .",
"Let $x\\subseteq N$ be the set of all elements of $N$ that correspond to only one element in $m$ .",
"Clearly, $0\\in x$ .",
"Now, for some $y\\in x$ let $z\\in M$ be the element in $M$ that $y$ uniquely corresponds to (that is $y\\:r\\:z$ ).",
"But this means that $+(y)\\:r\\#(z)$ by the construction of the relation.",
"If there exists $w\\in M$ such that $+(y)\\:r\\:w$ and since $(+(y),w)\\ne (0,\\bullet )$ , the fixed point equation implies that $(+(y)=+(n_0))$ and $(w=\\#(m_0))$ for some $(n_0,m_0)\\in r$ .",
"But then by the second axiom, $y=n_0$ and since $y\\in x$ , $z=m_0$ .",
"Thus, $\\#(z)$ is the unique element corresponding to $+(y)$ .",
"The third axiom can now be applied, and thus every element in $N$ corresponds to a unique element in $M$ .",
"The roles of $N$ and $M$ can be reversed in this proof.",
"Therefore, it can also be shown that every element of $M$ corresponds to a unique element in $N$ .",
"Thus, $r$ is a one-to-one and onto correspondence.",
"$\\:\\:\\Box $ Exercises Exercise 4.7: In Theorem REF , an equation was given to find the least fixed point of a function $f:{\\cal D}\\rightarrow {\\cal D}$ .",
"Suppose that for $a\\in {\\cal D}$ , $a\\sqsubseteq f(a)$ .",
"Will the fixed point $x=f(x)$ be such that $a\\sqsubseteq x$ ?",
"(Hint: How do we know that $\\bigcup \\limits _{n=0}^\\infty f^n(a)\\in {\\cal D}$ ?)",
"Exercise 4.8: Let $f:{\\cal D}\\rightarrow {\\cal D}$ and $S\\subseteq {\\cal D}$ satisfy $\\bot \\in S$ $x\\in S\\:\\Rightarrow \\:f(x)\\in S$ $[\\forall n .\\lbrace x_n\\rbrace \\subseteq S \\wedge x_n\\sqsubseteq x_{n+1}]\\:\\Rightarrow \\:\\bigcup \\limits _{n=0}^\\infty x_n \\in S$ Conclude that $fix(f)\\in S$ .",
"This is sometimes called the principle of fixed point induction.",
"Apply this method to the set $S=\\lbrace x\\in {\\cal D}\\:\\vert \\:a(x)= b(x)\\rbrace $ where $a,b:{\\cal D}\\rightarrow {\\cal D}$ are approximable, $a(\\bot )=b(\\bot )$ , and $f\\circ a=a\\circ f$ and $f\\circ b=b\\circ f$ .",
"Exercise 4.9: Show that there is an approximable operator $\\Psi :(({\\cal D}\\rightarrow {\\cal D})\\rightarrow {\\cal D})\\rightarrow (({\\cal D}\\rightarrow {\\cal D})\\rightarrow {\\cal D})$ such that for $\\Theta :({\\cal D}\\rightarrow {\\cal D})\\rightarrow {\\cal D}$ and $f:{\\cal D}\\rightarrow {\\cal D}$ , $\\Psi (\\Theta ) (f) = f(\\Theta (f))$ Prove also that $fix:({\\cal D}\\rightarrow {\\cal D})\\rightarrow {\\cal D}$ is the least fixed point of $\\Psi $ .",
"Exercise 4.10: Given a domain ${\\cal D}$ and an element $a\\in {\\cal D}$ , construct the domain ${\\cal D}_a$ where ${\\cal D}_a=\\lbrace x\\in {\\cal D}\\:\\vert \\:x\\sqsubseteq a\\rbrace $ Show that if $f:{\\cal D}\\rightarrow {\\cal D}$ is approximable, then $f$ can be restricted to another approximable map $f^{\\prime }:{\\cal D}_{fix(f)}\\rightarrow {\\cal D}_{fix(f)}$ where $\\forall x\\in {\\cal D}_{fix(f)}.f^{\\prime }(x)=f(x)$ How many fixed points does $f^{\\prime }$ have in ${\\cal D}_{fix(f)}$ ?",
"Exercise 4.11: The mapping ${\\bf fix}$ can be viewed as assigning a fixed point operator to any domain ${\\cal D}$ .",
"Show that ${\\bf fix}$ can be uniquely characterized by the following conditions on an assignment ${\\cal D}\\leadsto F_D$ : $F_D:({\\cal D}\\rightarrow {\\cal D})\\rightarrow {\\cal D}$ $F_D(f)=f(F_D(f))$ for all $f:{\\cal D}\\rightarrow {\\cal D}$ when $f_0:{\\cal D}_0\\rightarrow {\\cal D}_0$ and $f_1:{\\cal D}_1\\rightarrow {\\cal D}_1$ are given and $h:{\\cal D}_0\\rightarrow {\\cal D}_1$ is such that $h(\\bot )=\\bot $ and $h\\circ f_0=f_1\\circ h$ , then $h(F_{D_0}(f_0)) = F_{D_1}(f_1).$ Hint: Apply Exercise REF to show ${\\bf fix}$ satisfies the conditions.",
"For the other direction, apply Exercise REF .",
"Exercise 4.12: Must an approximable function have a maximum fixed point?",
"Give an example of an approximable function that has many fixed points.",
"Exercise 4.13: Must a monotone function $f:{\\cal P}(A)\\rightarrow {\\cal P}(A)$ have a maximum fixed point?",
"(Recall: ${\\cal P}(A)$ is the powerset of the set $A$ ).",
"Exercise 4.14: Verify the assertions made in the first example of this section.",
"Exercise 4.15: Verify the assertions made in the second example, in particular those in the discussion of “Peano's Axioms”.",
"Show that the predicate function $one:{\\cal C}\\rightarrow {\\cal T}$ could be defined using a fixed point equation from the other functions in the structure.",
"Exercise 4.16: Prove that $fix(f\\circ g)=f(fix(g\\circ f))$ for approximable functions $f,g:{\\cal D}\\rightarrow {\\cal D}$ .",
"Exercise 4.17: Show that the less-than-or-equal-to relation $l\\subseteq {{N}}\\times {{N}}$ is uniquely determined by $l=\\lbrace (n,n)\\:\\vert \\:n\\in {{N}}\\rbrace \\cup \\lbrace (n,succ(m)\\:\\vert \\:(n,m)\\in l\\rbrace $ for the structure called the “Domain of Integers”.",
"Exercise 4.18: Let $N^*$ be a structured set satisfying only the first two of the axioms referred to as “Peano's”.",
"Must there be a subset $S\\subseteq N^*$ such that all three axioms are satisfied?",
"(Hint: Use a least fixed point from ${\\cal P}(N^*)$ ).",
"Exercise 4.19: Let $f:{\\cal D}\\rightarrow {\\cal D}$ be an approximable map.",
"Let $a_n:{\\cal D}\\rightarrow {\\cal D}$ be a sequence of approximable maps such that $a_0(x)=\\bot $ for all $x\\in {\\cal D}$ $a_n\\sqsubseteq a_{n+1}$ for all $n\\in {{N}}$ $\\bigcup \\limits _{n=0}^\\infty a_n = {\\sf I}_D$ in ${\\cal D}\\rightarrow {\\cal D}$ $a_{n+1}\\circ f = a_{n+1}\\circ f\\circ a_n$ for all $n\\in {{N}}$ Show that $f$ has a unique fixed point.",
"(Hint: Show that if $x=f(x)$ then $a_n(x)\\sqsubseteq a_n(fix(f))$ for all $n\\in {{N}}$ .",
"Show this by induction on $n$ .)",
"Typed $\\lambda $ -Calculus As shown in the previous section, functions can be characterized by recursion equations which combine previously defined functions with the function being defined.",
"The expression of these functions is simplified in this section by introducing a notation for specifying a function without having to give the function a name.",
"The notation used is that of the typed $\\lambda $ -Calculus; a function is defined using a $\\lambda $ -abstraction.",
"Definition of Typed $\\lambda $ -Calculus An informal characterization of the $\\lambda $ -calculus suffices for this section; more formal descriptions are available elsewhere in the literature [1].",
"Thus, examples are used to introduce the notation.",
"An infinite number of variables, $x$ ,$y$ ,$z$ ,$\\ldots $ of various types are required.",
"While a variable has a certain type, type subscripts will not be used due to the notational complexity.",
"A distinction must also be made between type symbols and domains.",
"The domain ${\\cal A}\\times {\\cal B}$ does not uniquely determine the component domains ${\\cal A}$ and ${\\cal B}$ even though these domains are uniquely determined by the symbol for the domain.",
"The domain is the meaning that we attribute to the symbol.",
"In addition to variables, constants are also present.",
"For example, the symbol 0 is used to represent the zero element from the domain ${\\cal N}$ .",
"Another constant, present in each domain by virtue of Theorem REF , is $fix^{\\cal D}$ , the least fixed point operator for domain ${\\cal D}$ of type $({\\cal D}\\rightarrow {\\cal D})\\rightarrow {\\cal D}$ .",
"The constants and variables are the atomic (non-compound) terms.",
"Types can be associated with all atomic terms.",
"There are several constructions for compound terms.",
"First, given $\\tau ,\\ldots ,\\sigma $ , a list of terms, the ordered tuple $\\langle \\tau ,\\ldots ,\\sigma \\rangle $ is a compound term.",
"If the types of $\\tau ,\\ldots ,\\sigma $ are ${\\cal A},\\ldots ,{\\cal B}$ , the type of the tuple is ${\\cal A}\\times \\ldots \\times {\\cal B}$ since the tuple is to be an element of this domain.",
"The tuple notation for combining functions given earlier should be disregarded here.",
"The next construction is function application.",
"If the term $\\tau $ has type ${\\cal A}\\rightarrow {\\cal B}$ and the term $\\sigma $ has the type ${\\cal A}$ , then the compound term $\\tau (\\sigma )$ has type ${\\cal B}$ .",
"Function application denotes the value of a function at a given input.",
"The notation $\\tau (\\sigma _0,\\ldots ,\\sigma _n)$ abbreviates $\\tau (\\langle \\sigma _0,\\ldots ,\\sigma _n\\rangle )$ .",
"Functions applied to tuples allows us to represent applications of multi-variate functions.",
"The $\\lambda $ -abstraction is used to define functions.",
"Let $x_0,\\ldots ,x_n$ be a list of distinct variables of type ${\\cal D}_0,\\ldots ,{\\cal D}_n$ .",
"Let $\\tau $ be a term of some type ${\\cal D}_{n+1}$ .",
"$\\tau $ can be thought of as a function of $n+1$ variables with type $({\\cal D}_0\\times \\ldots \\times {\\cal D}_n)\\rightarrow {\\cal D}_{n+1}$ .",
"The name for this function is written $\\lambda x_0,\\ldots ,x_n.\\tau $ This expression denotes the entire function.",
"To look at some familiar functions in the new notation, consider $\\lambda x,y.x$ This notation is read “lambda ex wye (pause) ex”.",
"If the types of $x$ and $y$ are ${\\cal A}$ and ${\\cal B}$ respectively, the function has type $({\\cal A}\\times {\\cal B})\\rightarrow {\\cal A}$ .",
"This function is the first projection function $p_0$ .",
"This function and the second projection function can be defined by the following equations: $\\begin{array}{lcl}p_0&=&\\lambda x,y.x\\\\p_1&=&\\lambda x,y.y\\end{array}$ Recalling the function tuple notation introduced in an earlier section, the following equation holds: $\\langle f,g\\rangle =\\lambda w.\\langle f(w),g(w)\\rangle $ which defines a function of type ${\\cal D}_1\\rightarrow ({\\cal D}_2\\times {\\cal D}_3)$ .",
"Other familiar functions are defined by the following equations: $\\begin{array}{lcl}eval&=&\\lambda f,x.f(x)\\\\curry&=&=\\lambda g\\lambda x\\lambda y.g(x,y)\\end{array}$ The $curry$ example shows that this notation can be iterated.",
"A distinction is thus made between the terms $\\lambda x,y.x$ and $\\lambda x\\lambda y.x$ which have the types ${\\cal D}_0\\times {\\cal D}_1\\rightarrow {\\cal D}_0$ and ${\\cal D}_0\\rightarrow {\\cal D}_1\\rightarrow {\\cal D}_0$ respectively.",
"Thus, the following equation also holds: $curry(\\lambda x,y.\\tau )=\\lambda x\\lambda y.\\tau $ which relates the multi-variate form to the iterated or curried form.",
"Another true equation is $fix={\\bf fix}(\\lambda F\\lambda f.f(F(f)))$ where $fix$ has type $({\\cal D}\\rightarrow {\\cal D})\\rightarrow {\\cal D}$ and fix has type $(((({\\cal D}\\rightarrow {\\cal D})\\rightarrow {\\cal D})\\rightarrow (({\\cal D}\\rightarrow {\\cal D})\\rightarrow {\\cal D}))\\rightarrow (({\\cal D}\\rightarrow {\\cal D})\\rightarrow {\\cal D}))$ This is the content of Exercise REF .",
"This notation can now be used to define functions using recursion equations.",
"For example, the function $\\sigma $ in Example REF can be characterized by the following equation: $\\sigma =fix(\\lambda f\\lambda n.cond(zero(n),0,f(pred(n))+pred(n))$ which states that $\\sigma $ is the least recursively defined function $f$ whose value at $n$ is $cond(\\ldots )$ .",
"The variable $f$ occurs in the body of the $cond$ expression, but this is just the point of a recursive definition.",
"$f$ is defined in terms of its value on “smaller” input values.",
"The use of the fixed point operator makes the definition explicit by forcing there to be a unique solution to the equation.",
"In an abstraction $\\lambda x,y,z.\\tau $ , the variables $x$ ,$y$ , and $z$ are said to be bound in the term $\\tau $ .",
"Any other variables in $\\tau $ are said to be free variables in $\\tau $ unless they are bound elsewhere in $\\tau $ .",
"Bound variables are simply placeholders for values; the particular variable name chosen is irrelevant.",
"Thus, the equation $\\lambda x.\\tau =\\lambda y.\\tau [y/x]$ is true provided $y$ is not free in $\\tau $ .",
"The notation $\\tau [y/x]$ specifies the substitution of $y$ for $x$ everywhere $x$ occurs in $\\tau $ .",
"The notation $\\tau [\\sigma /x]$ for the substitution of the term $\\sigma $ for the variable $x$ is also legitimate.",
"Semantics of Typed $\\lambda $ -Calculus To show that the equations above with $\\lambda $ –terms are indeed meaningful, the following theorem relating $\\lambda $ –terms and approximable mappings must be proved.",
"Theorem 5.1: Every typed $\\lambda $ –term defines an approximable function of its free variables.",
"Proof Induction on the length of the term and its structure will be used in this proof.",
"Variables Direct since $x\\mapsto x$ is an approximable function.",
"Constants Direct since $x\\mapsto k$ is an approximable function for constant $k$ .",
"Tuples Let $\\tau =\\langle \\sigma _0,\\ldots ,\\sigma _n\\rangle $ .",
"Since the $\\sigma _i$ terms are less complex, they are approximable functions of their free variables by the induction hypothesis.",
"Using Theorem REF (generalized to the multi-variate case) then, $\\tau $ which takes tuples as values also defines an approximable function.",
"Application Let $\\tau =\\sigma _0(\\sigma _1)$ .",
"We assume that the types of the terms are appropriately matched.",
"The $\\sigma _i$ terms define approximable functions again by the induction hypothesis.",
"Recalling the earlier equations, the value of $\\tau $ is the same as the value of $eval(\\sigma _0,\\sigma _1)$ .",
"Since $eval$ is approximable, Theorem REF shows that the term defines an approximable function.",
"Abstraction Let $\\tau =\\lambda x.\\sigma $ .",
"By the induction hypothesis, $\\sigma $ defines a function of its free variables.",
"Let those free variables be of types ${\\cal D}_0,\\ldots ,{\\cal D}_n$ where ${\\cal D}_n$ is the type of $x$ .",
"Then $\\sigma $ defines an approximable function $g:{\\cal D}_0\\times \\ldots \\times {\\cal D}_n\\rightarrow {\\cal D}^{\\prime }$ where ${\\cal D}^{\\prime }$ is the type of $\\sigma $ .",
"Using Theorem REF , the function $curry(g):{\\cal D}_0\\times \\ldots \\times {\\cal D}_{n-1}\\rightarrow ({\\cal D}_n\\rightarrow {\\cal D}^{\\prime })$ yields an approximable function, but this is just the function defined by $\\tau $ .",
"The reader can generalize this proof for multiple bound variables in $\\tau $ .",
"$\\:\\:\\Box $ Given this, the equation $\\tau =\\sigma $ states that the two terms define the same approximable function of their free variables.",
"As an example, $\\lambda x.\\tau =\\lambda y.\\tau [y/x]$ provided $y$ is not free in $\\tau $ since the generation of the approximable function did not depend on the name $x$ but only on its location in $\\tau $ .",
"Other equations such as these are given in the exercises.",
"The most basic rule is shown below.",
"Theorem 5.2: For appropriately typed terms, the following equation is true: $(\\lambda x_0,\\ldots ,x_{1}.\\tau )(\\sigma _0,\\ldots ,\\sigma _{n-1})=\\tau [\\sigma _0/x_0,\\ldots ,\\sigma _{n-1}/x_{n-1}]$ Proof The proof is given for $n=1$ and proceeds again by induction on the length of the term and the structure of the term.",
"Variables This means $(\\lambda x.x)(\\sigma )=\\sigma $ must be true which it is.",
"Constants This requires $(\\lambda x.k)(\\sigma )= k$ must be true which it is for any constant $k$ .",
"Tuples Let $\\tau =\\langle \\tau _0,\\tau _1\\rangle $ .",
"This requires that $(\\lambda x.\\langle \\tau _0,\\tau _1\\rangle )(\\sigma ) =\\langle \\tau _0[\\sigma /x],\\tau _1[\\sigma /x]\\rangle $ must be true.",
"This equation holds since the left-hand side can be transformed using the following true equation: $(\\lambda x.\\langle \\tau _0,\\tau _1\\rangle )(\\sigma ) =\\langle (\\lambda x.\\tau _0)(\\sigma ),(\\lambda x.\\tau _1)(\\sigma )\\rangle $ Then the inductive hypothesis is applied to the $\\tau _i$ terms.",
"Applications Let $\\tau =\\tau _0(\\tau _1)$ .",
"Then, the result requires that the equation $(\\lambda x.\\tau _0(\\tau _1))(\\sigma ) =\\tau _0[\\sigma /x](\\tau _1[\\sigma /x])$ hold true.",
"To see that this is true, examine the approximable functions for the left-hand side of the equation.",
"$\\begin{array}{lcl}\\tau _0&\\mapsto &\\bar{V},x\\rightarrow t_0\\\\\\tau _1&\\mapsto &\\bar{V},x\\rightarrow t_1\\\\\\sigma &\\mapsto &\\bar{V}\\rightarrow s\\\\\\mbox{so}\\\\(\\lambda x.\\tau _0(\\tau _1))(\\sigma )&\\mapsto &\\bar{V}\\rightarrow [(x\\rightarrow t_0(t_1))(s)]\\\\&=&\\bar{V},x\\rightarrow [(x\\rightarrow t_0)(s)]([(x\\rightarrow t_1)(s)])\\end{array}$ From this last term, we use the induction hypothesis.",
"To see why the last step holds, start with the set representing the left-hand side and using the aprroximable mappings for the terms: $\\begin{array}{cl}&(\\lambda x.\\tau _0(\\tau _1))(\\sigma )\\\\\\mapsto &\\bar{V}\\rightarrow [(x\\rightarrow t_0(t_1))(s)]\\\\=&\\lbrace b\\:\\vert \\:\\exists a.a\\in s\\wedge a\\:[x\\rightarrow t_0(t_1)]\\: b\\rbrace \\\\=&\\lbrace b\\:\\vert \\:\\exists a. a\\in s\\wedge a\\:\\lbrace (x,u)\\:\\vert \\:v\\in x\\rightarrow t_1)\\wedge v\\:(x\\rightarrow t_0)\\: u\\rbrace \\: b\\rbrace \\\\=&\\lbrace b\\:\\vert \\:\\exists a.a\\in s\\wedge v\\in (x\\rightarrow t_1)(a)\\wedge v\\:(x\\rightarrow t_0)(a)\\:b\\rbrace \\\\=&\\lbrace b\\:\\vert \\:\\exists a,c.a\\in s\\wedge a\\:(x\\rightarrow t_1)\\: v\\wedge a\\:(x\\rightarrow t_0)\\:c\\wedge v\\:c\\:b\\rbrace \\\\=&\\lbrace b\\:\\vert \\:v\\in [(x\\rightarrow t_1)(s)]\\wedge c\\in (x\\rightarrow t_0)(s) \\wedge v\\:c\\:b\\rbrace \\\\=&\\lbrace b\\:\\vert \\:v\\in [(x\\rightarrow t_1)(s)]\\wedge v\\:[(x\\rightarrow t_0)(s)]\\:b\\rbrace \\\\=&[(x\\rightarrow t_0)(s)]([(x\\rightarrow t_1)(s)])\\end{array}$ Abstractions Let $\\tau =\\lambda y.\\tau _0$ .",
"The required equation is $(\\lambda x.\\lambda y.\\tau _0)(\\sigma )=\\lambda y.\\tau _0[\\sigma /x]$ provided that $y$ is not free in $\\sigma $ .",
"The following true equation applies here: $(\\lambda x.\\lambda y.\\tau )(\\sigma )=\\lambda y.",
"((\\lambda x.\\tau )(\\sigma ))$ To see that this equation holds, let $g$ be a function of $n+2$ free variables defined by $\\tau $ .",
"By Theorem REF , the term $\\lambda x.\\lambda y.\\tau $ defines the function $curry(curry(g))$ of $n$ variables.",
"Call this function $h$ .",
"Thus, $h(v)(\\sigma )(y) = g(v,\\sigma ,y)$ where $v$ is the list of the other free variables.",
"Using a combinator $inv$ which inverts the order of the last two arguments, $h(v)(\\sigma )(y)=curry(inv(g))(v,y)(\\sigma )$ But, $curry(inv(g))$ is the function defined by $\\lambda x.\\tau $ .",
"Thus, we have shown that $(\\lambda x.\\lambda y.\\tau )(\\sigma )(y)=(\\lambda x.\\tau )(\\sigma )$ is a true equation.",
"If $y$ is not free in $\\alpha $ and $\\alpha (y)=\\beta $ is true, then $\\alpha =\\lambda y.\\beta $ must also be true.",
"$\\:\\:\\Box $ If $\\tau ^{\\prime }$ is the term $\\lambda x,y.\\tau $ , then $\\tau ^{\\prime }(x,y)$ is the same as $\\tau $ .",
"This specifies that $x$ and $y$ are not free in $\\tau $ .",
"This notation is used in the proof of the following theorem.",
"Theorem 5.3: The least fixed point of $\\lambda x,y.\\langle \\tau (x,y),\\sigma (x,y)\\rangle $ is the pair with coordinates $fix(\\lambda x.\\tau (x,fix(\\lambda y.\\sigma (x,y))))$ and $fix(\\lambda y.\\sigma (fix(\\lambda x.\\tau (x,y)),y))$ .",
"Proof We are thus assuming that $x$ and $y$ are not free in $\\tau $ and $\\sigma $ .",
"The purpose here is to find the least solution to the pair of equations: $x=\\tau (x,y)~{\\rm and}~y=\\sigma (x,y)$ This generalizes the fixed point equation to two variables.",
"More variables could be included using the same method.",
"Let $y_*=fix(\\lambda y.\\sigma (fix(\\lambda x.\\tau (x,y)),y))$ and $x_*=fix(\\lambda x.\\tau (x,y))$ Then, $x_*=\\tau (x_*,y_*)$ and $\\begin{array}{lcl}y_*&=&\\sigma (fix(\\lambda x.\\tau (x,y_*),y_*))\\\\&=&\\sigma (x_*,y_*).\\end{array}$ This shows that the pair $\\langle x_*,y_*\\rangle $ is one fixed point.",
"Now, let $\\langle x_0,y_0\\rangle $ be the least solution.",
"(Why must a least solution exist?",
"Hint: Consider a suitable mapping of type $({\\cal D}_x\\times {\\cal D}_y)\\rightarrow ({\\cal D}_x\\times {\\cal D}_y)$ .)",
"Thus, we know that $x_0=\\tau (x_0,y_0)$ , $y_0=\\sigma (x_0,y_0)$ , and that $x_0\\sqsubseteq x_*$ and $y_0\\sqsubseteq y_*$ .",
"But this means that $\\tau (x_0,y_0)\\sqsubseteq x_0$ and thus $fix(\\lambda x.\\tau (x,y_0))\\sqsubseteq x_0$ and consequently $\\sigma (fix(\\lambda x.\\tau (x,y_0),y_0))\\sqsubseteq \\sigma (x_0,y_0)\\sqsubseteq y_0$ By the fixed point definition of $y_*$ , $y_*\\sqsubseteq y_0$ must hold as well so $y_0=y_*$ .",
"Thus, $x_*=fix(\\lambda x.\\tau (x,y*))=fix(\\lambda x.\\tau (x,y_0))\\sqsubseteq x_0.$ Thus, $x*=x_0$ must also hold.",
"A similar argument holds for $x_0$ .$\\:\\:\\Box $ The purpose of the above proof is to demonstrate the use of least fixed points in proofs.",
"The following are also true equations: $fix(\\lambda x.\\tau (x))=\\tau (fix(\\lambda x.\\tau (x)))$ and $\\tau (y)\\sqsubseteq y\\:\\Rightarrow \\:fix(\\lambda x.\\tau (x))\\sqsubseteq y$ if $x$ is not free in $\\tau $ .",
"These equations combined with the monotonicity of functions were the methods used in the proof above.",
"Another example is the proof of the following theorem.",
"Theorem 5.4: Let $x$ ,$y$ , and $\\tau (x,y)$ be of type ${\\cal D}$ and let $g:{\\cal D}\\rightarrow {\\cal D}$ be a function.",
"Then the equation $\\lambda x.fix(\\lambda y.\\tau (x,y))=fix(\\lambda g.\\lambda x.\\tau (x,g(x)))$ holds.",
"Proof Let $f$ be the function on the left-hand side.",
"Then, $f(x)=fix(\\lambda y.\\tau (x,y))=\\tau (x,f(x))$ holds using the equations stated above.",
"Therefore, $f=\\lambda x.\\tau (x,f(x))$ and thus $g_0=fix(\\lambda g.\\lambda x.\\tau (x,g(x)))\\sqsubseteq f. $ By the definition of $g_0$ we have $g_0(x)=\\tau (x,g_0(x))$ for any given $x$ .",
"By the definition of $f$ we find that $f(x)=fix(\\lambda y.\\tau (x,y))\\sqsubseteq g_0(x)$ must hold for all $x$ .",
"Thus $f\\sqsubseteq g_0$ and the equation is true.$\\:\\:\\Box $ This proof illustrates the use of inclusion and equations between functions.",
"The following principle was used: $(\\forall x.\\tau \\sqsubseteq \\sigma )\\:\\Rightarrow \\:\\lambda x.\\tau \\sqsubseteq \\lambda x.\\sigma $ This is a restatement of the first part of Theorem REF .",
"Combinators and Recursive Functions Below is a list of various combinators with their definitions in $\\lambda $ -notation.",
"The meanings of those combinators not previously mentioned should be clear.",
"$\\begin{array}{lcl}p_0&=&\\lambda x,y.x\\\\p_1&=&\\lambda x,y.y\\\\pair&=&\\lambda x.\\lambda y.\\langle x,y\\rangle \\\\n-tuple&=&\\lambda x_0\\lambda \\ldots \\lambda x_{n-1}.\\langle x_0,\\ldots ,x_{n-1}\\rangle \\\\diag&=&\\lambda x.\\langle x,x\\rangle \\\\funpair&=&\\lambda f.\\lambda g.\\lambda x.\\langle f(x),g(x)\\rangle \\\\proj^n_i&=&\\lambda x_0,\\ldots ,x_{n-1}.x_i\\\\inv^n_{i,j}&=&\\lambda x_0,\\ldots ,x_i,\\ldots ,x_j,\\ldots ,x_{n-1}.\\langle x0,\\ldots ,x_j,\\ldots ,x_i,\\ldots ,x_{n-1}\\rangle \\\\eval&=&\\lambda f,x.f(x)\\\\curry&=&\\lambda g.\\lambda x.\\lambda y.g(x,y)\\\\comp&=&\\lambda f,g.\\lambda x.g(f(x))\\\\const&=&\\lambda k.\\lambda x.k\\\\{\\bf fix}&=&\\lambda f.fix(\\lambda x.f(x))\\end{array}$ These combinators are actually schemes for combinators since no types have been specified and thus the equations are ambiguous.",
"Each scheme generates an infinite number of combinators for all the various types.",
"One interest in combinators is that they allow expressions without variables—if enough combinators are used.",
"This is useful at times but can be clumsy.",
"However, defining a combinator when the same combination of symbols repeatedly appears is also useful.",
"There are some familiar combinators that do not appear in the table.",
"Combinators such as $cond$ , $pred$ , and $succ$ cannot be defined in the pure $\\lambda $ -calculus but are instead specific to certain domains.",
"They are thus regarded as primitives.",
"A large number of other functions can be defined using these primitives and the $\\lambda $ -notation, as the following theorem shows.",
"Theorem 5.5: For every partial recursive function $h:{{N}}\\rightarrow {{N}}$ , there is a $\\lambda $ -term $\\tau $ of type ${\\cal N}\\rightarrow {\\cal N}$ such that the only constants occurring in $\\tau $ are $cond$ , $succ$ , $pred$ , $zero$ , and 0 and if $h(n)=m$ then $\\tau (n)=m$ .",
"If $h(n)$ is undefined, then $\\tau (n)=\\bot $ holds.",
"$\\tau (\\bot )=\\bot $ is also true.",
"Proof It is convenient in the proof to work with strict functions $f:{\\cal N}^k\\rightarrow {\\cal N}$ such that if any input is $\\bot $ , the result of the function is $\\bot $ .",
"The composition of strict functions is easily shown to be strict.",
"It is also easy to see that any partial function $g:{{N}}^k\\rightarrow {{N}}$ can be extended to a strict approximable function $\\bar{g}:{\\cal N}^k\\rightarrow {\\cal N}$ which yields the same values on inputs for which $g$ is defined.",
"Other input values yield $\\bot $ .",
"We want to show that $\\bar{g}$ is definable with a $\\lambda $ -expression.",
"First we must show that primitive recursive functions have $\\lambda $ -definitions.",
"Primitive recursive functions are formed from starting functions using composition and the scheme of primitive recursion.",
"The starting functions are the constant function for zero and the identity and projection functions.",
"These functions, however, must be strict so the term $\\lambda x,y.x$ is not sufficient for a projection function.",
"The following device reduces a function to its strict form.",
"Let $\\lambda x.cond(zero(x),x,x)$ be a function with $x$ of type ${\\cal N}$ .",
"This is the strict identity function.",
"The strict projection function attempted above can be defined as $\\lambda x,y.cond(zero(y),x,x)$ The three variable projection function can be defined as $\\lambda x,y,z.cond(zero(x),cond(zero(z),y,y),cond(zero(z),y,y))$ While not very elegant, this device does produce strict functions.",
"Strict functions are closed under substitution and composition.",
"Any substitution of a group of functions into another function can be defined with a $\\lambda $ -term if the functions themselves can be so defined.",
"Thus, we need to show that functions obtained by primitive recursion are definable.",
"Let $f:{\\cal N}\\rightarrow {\\cal N}$ , and $g:{\\cal N}^3\\rightarrow {\\cal N}$ be total functions with $\\bar{f}$ and $\\bar{g}$ being $\\lambda $ -definable.",
"We obtain the function $h:{\\cal N}^2\\rightarrow {\\cal N}$ by primitive recursion where $\\begin{array}{lcl}h(0,m)&=&f(m)\\\\h(n+1,m)&=&g(n,m,h(n,m))\\end{array}$ for all $n,m\\in {\\cal N}$ .",
"The $\\lambda $ -term for $\\bar{h}$ is $fix(\\lambda k.\\lambda x,y.cond(zero(x),\\bar{f}(y),\\bar{g}(pred(x),y,k(pred(x),y))))$ Note that the fixed point operator for the domain ${\\cal N}^2\\rightarrow {\\cal N}$ was used.",
"The variables $x$ and $y$ are of type ${\\cal N}$ .",
"The $cond$ function is used to encode the function requirements.",
"The fixed point function is easily seen to be strict and this function is $\\bar{h}$ .",
"Primitive recursive functions are now $\\lambda $ -definable.",
"To obtain partial (i.e., general) recursive functions, the $\\mu $ -scheme (the least number operator) is used.",
"Let $f(n,m)$ be a primitive recursive function.",
"Then, define $h$ , a partial function, as $h(m) =$ the least $n$ such that $f(n,m)=0$ .",
"This is written as $h(m)=\\mu n.f(n,m)=0$ .",
"Since $\\bar{f}$ is $\\lambda $ -definable as has just been shown, let $\\bar{g}=fix(\\lambda g.\\lambda x,y.cond(zero(\\bar{f}(x,y)),x,g(succ(x),y)))$ Then, the desired function $\\bar{h}$ is defined as $\\bar{h}=\\lambda y.\\bar{g}(0,y)$ .",
"It is easy to see that this is a strict function.",
"Note that, if $h(m)$ is defined, clearly $h(m)=\\bar{g}(0,m)$ is also defined.",
"If $h(m)$ is undefined, it is also true that $\\bar{g}(0,m)=\\bot $ due to the fixed point construction but it is less obvious.",
"This argument is left to the reader.$\\:\\:\\Box $ Theorem REF does not claim that all $\\lambda $ -terms define partial recursive functions although this is also true.",
"Further examples of recursion are found in the exercises.",
"Exercises Exercise 5.6: Find the definitions of $\\lambda x,y.\\tau ~{\\rm and}~\\sigma (x,y)$ which use only $\\lambda v$ with one variable and applications only to one argument at a time.",
"Note that use must be made of the combinators $p_0$ , $p_1$ , and $pair$ .",
"Generalize the result to functions of many variables.",
"Exercise 5.7: The table of combinators was meant to show how combinators could be defined in terms of $\\lambda $ -expressions.",
"Can the tables be turned to show that, with enough combinators available, every $\\lambda $ -expression can be defined by combining combinators using application as the only mode of combination?",
"Exercise 5.8: Suppose that $f,g:{\\cal D}\\rightarrow {\\cal D}$ are approximable and $f\\circ g=g\\circ f$ .",
"Show that $f$ and $g$ have a least common fixed point $x=f(x)=g(x)$ .",
"(Hint: See Exercise REF .)",
"If, in addition, $f(\\bot )=g(\\bot )$ , show that $fix(f)=fix(g)$ .",
"Will $fix(f)=fix(f^2)$ ?",
"What if the assumption is weakened to $f\\circ g=g^2\\circ f$ ?",
"Exercise 5.9: For any domain ${\\cal D}$ , ${\\cal D}^\\infty $ can be regarded as consisting of bottomless stacks of elements of ${\\cal D}$ .",
"Using this view, define the following combinators with their obvious meaning: $head:{\\cal D}^\\infty \\rightarrow {\\cal D}$ , $tail:{\\cal D}^\\infty \\rightarrow {\\cal D}^\\infty $ and $push:{\\cal D}\\times {\\cal D}^\\infty \\rightarrow {\\cal D}^\\infty $ .",
"Using the fixed point theorem, argue that there is a combinator $diag:{\\cal D}\\rightarrow {\\cal D}^\\infty $ where for all $x\\in {\\cal D}$ , $diag(x)=\\langle x\\rangle _{n=0}^\\infty $ .",
"(Hint: Try a recursive definition, such as $diag(x)=push(x,diag(x))$ but be sure to prove that all terms of $diag(x)$ are $x$ .)",
"Also introduce by an appropriate recursive definition a combinator $map:({\\cal D}\\rightarrow {\\cal D})^\\infty \\times {\\cal D}\\rightarrow {\\cal D}^\\infty $ where for elements of the proper type $map(\\langle f_n\\rangle _{n=0}^\\infty ,x)=\\langle f_n(x)\\rangle _{n=0}^\\infty $ Exercise 5.10: For any domain ${\\cal D}$ introduce, as a least fixed point, a combinator $while:({\\cal D}\\rightarrow {\\cal T})\\times ({\\cal D}\\rightarrow {\\cal D})\\rightarrow ({\\cal D}\\rightarrow {\\cal D})$ by the recursion equation $while(p,f)(x)=cond(p(x),while(p,f)(f(x)),x)$ Prove that $while(p,while(p,f))=while(p,f)$ Show how $while$ could be used to obtain the least number operator,$\\mu $ , mentioned in the proof of Theorem REF .",
"Generalize this idea to define a combinator $find:{\\cal D}^\\infty \\times ({\\cal D}\\rightarrow {\\cal T})\\rightarrow {\\cal D}$ which means “find the first term in the sequence (if any) which satisfies the given predicate”.",
"Exercise 5.11: Prove the existence of a one-one function $num:{{N}}\\times {{N}}\\leftrightarrow {{N}}$ such that $\\begin{array}{lcl}num(0,0)&=&0\\\\num(n,m+1)&=&num(n+1,m)\\\\num(n+1,0)&=&num(0,n)+1\\end{array}$ Draw a descriptive picture (an infinite matrix) for the function.",
"Find a closed form for the values if possible.",
"Use the function to prove the isomorphism between ${\\cal P}({{N}})$ ,${\\cal P}({{N}}\\times {{N}})$ , and ${\\cal P}({{N}})\\times {\\cal P}({{N}})$ .",
"Exercise 5.12: Show that there are approximable mappings $graph:({\\cal P}({{N}})\\rightarrow {\\cal P}({{N}}))\\rightarrow {\\cal P}({{N}})$ and $fun:{\\cal P}({{N}})\\rightarrow ({\\cal P}({{N}})\\rightarrow {\\cal P}({{N}}))$ where $fun\\circ graph = \\lambda f.f$ and $graph\\circ fun\\sqsubseteq \\lambda x.x$ .",
"(Hint: Using the notation $[n_0,\\ldots ,n_k]=num(n_0,[n_1,\\ldots ,n_k])$ , two such combinators can be given by the formulas $\\begin{array}{lcl}fun(u)(x)&=&\\lbrace m\\:\\vert \\:\\exists n_0,\\ldots ,n_{k-1}\\in x.",
"[n_0+1,\\ldots ,n_{k-1}+1,0,m]\\in u\\rbrace \\\\graph(f)&=&\\lbrace [n_0+1,\\ldots ,n_{k-1}+1,0,m]\\:\\vert \\:m\\in f(\\lbrace n_0,\\ldots ,n_{k-1}\\rbrace )\\rbrace \\end{array}$ where $k$ is a variable - meaning all finite sequences are to be considered.)",
"Introduction to Domain Equations As stressed in the introduction, the notion of computation with potentially infinite elements is an integral part of domain theory.",
"The previous sections have defined the notion of functions over domains, as well as a notation for expressing these functions.",
"In addition, the notion of computation through series of approximations has been addressed.",
"This computation is possible since the functions defined have been approximable and thus continuous.",
"This section addresses the construction of more complex domains with infinite elements.",
"The next section looks specifically at the notion of computability with respect to these infinite elements.",
"The last section looks at another approach to domain construction.",
"New domains have been constructed from existing ones using domain constructors such as the product construction ($\\times $ ), the function space construction ($\\rightarrow $ ) and the sum construction ($+$ ) of Exercise REF .",
"These constructors can be iterated similar to the way that function application was iterated to form recursive function definitions.",
"In this way, domains can be characterized using recursion equations, called domain equations.",
"Domain Equations A domain equation represents an isomorphism between the domain as a whole and the combination of domains that comprise it.",
"These recursive domains are frequently termed reflexive domains since, as in the following example, the domain contains a copy of itself in its structure.",
"Example 6.1: Consider the following domain equation: ${\\cal T}={\\cal A}+({\\cal T}\\times {\\cal T})$ where ${\\cal A}$ is a previously defined domain.",
"This domain can be thought of as containing atomic elements from ${\\cal A}$ or pairs of elements of ${\\cal T}$ .",
"What do the elements of this domain look like?",
"In particular, what are the finite elements of this domain?",
"How is the domain constructed?",
"What is an appropriate approximation ordering for the domain?",
"What do lubs in this domain look like?",
"What is the appropriate notion of consistency?",
"Does this domain even exist?",
"In other words, are we certain a solution to this domain equation exists?",
"And if a solution to the equation exists, is it a unique solution?",
"Each of these questions is examined below.",
"The domain equation tells us that an element of the domain is either an element from ${\\cal A}$ or is a pair of “smaller” elements from ${\\cal T}$ .",
"One method of constructing a sum domain is using pairs where some distinguished element denotes what type an element is.",
"Thus, for some $a\\in {\\cal A}$ , the pair $\\langle \\pi ,a\\rangle $ might represent the element in ${\\cal T}$ for the given element $a$ .",
"For some $s,t\\in {\\cal T}$ , the pair $\\langle \\langle s,t\\rangle ,\\pi \\rangle $ might then represent the element in ${\\cal T}$ for the pair $s,t$ .",
"Thus, $\\pi $ is the distinguished element, and the location of $\\pi $ in the pair specifies the type of the element.",
"The finite elements are either elements in ${\\cal T}$ representing the (finite) elements of ${\\cal A}$ or the pair elements from ${\\cal T}$ whose components are also finite elements in ${\\cal T}$ .",
"The question then arises about infinite elements.",
"Are there infinite elements in this domain?",
"Consider the following fixed point equation for some element for $a\\in {\\cal A}$ : $x=\\langle \\langle a,x\\rangle ,\\pi \\rangle .$ The fixed point of this equation is the infinite product of the element $a$ .",
"Does this element fit the definition for ${\\cal T}$ ?",
"From the informal description of the elements of ${\\cal T}$ given so far, $x$ does qualify as a member of ${\\cal T}$ .",
"Now that some intuition has been developed about this domain, a formal construction is required.",
"Let $\\langle {\\bf A},\\sqsubseteq _A\\rangle $ be the finitary basis used to generate the domain ${\\cal A}$ .",
"Let $\\pi $ be an object such that $\\pi \\notin {\\bf A}$ .",
"Define the bottom element of the finitary basis T as $\\Delta _T=\\langle \\pi ,\\pi \\rangle $ .",
"Next, all the elements of ${\\cal A}$ must be included so define an element in ${\\bf T}$ for each $a\\in {\\bf A}$ as $\\langle \\pi ,a\\rangle $ .",
"Finally, pair elements for all elements in ${\\bf T}$ must exist in ${\\bf T}$ to complete the construction.",
"The set ${\\bf T}$ can be defined inductively as the least set such that: $\\Delta _T\\in {\\bf T}$ , $\\langle \\pi ,a\\rangle \\in {\\bf T}$ whenever $a\\in {\\bf A}$ , $\\langle \\langle \\Delta _T,s\\rangle ,\\pi \\rangle \\in {\\bf T}$ whenever $s\\in {\\bf T}$ (necessary??",
"), $\\langle \\langle t,\\Delta _T\\rangle ,\\pi \\rangle \\in {\\bf T}$ whenever $t\\in {\\bf T}$ (necessary??",
"), and $\\langle \\langle t,s\\rangle ,\\pi \\rangle \\in {\\bf T}$ whenever $s, t\\in {\\bf T}$ .",
"The set can also be characterized by the following fixed point equation: ${\\bf T}=\\lbrace \\Delta _T\\rbrace \\cup \\lbrace \\langle \\pi ,a\\rangle \\:\\vert \\:a\\in {\\bf A}\\rbrace \\cup \\lbrace \\langle \\langle \\Delta _T,s\\rangle ,\\pi \\rangle \\:\\vert \\:s\\in {\\bf T}\\rbrace \\cup \\lbrace \\langle \\langle t,\\Delta _T\\rangle ,\\pi \\rangle \\:\\vert \\:t\\in {\\bf T}\\rbrace \\cup \\lbrace \\langle \\langle t,s\\rangle ,\\pi \\rangle \\:\\vert \\:s,t\\in {\\bf T}\\rbrace .$ A solution must exist for this equation by the fixed point theorem.",
"Now that the basis elements have been defined, we must show how to find lubs.",
"We will again use an inductive definition.",
"$\\langle \\pi ,\\pi \\rangle \\sqcup t=t$ for all $t\\in {\\bf T}$ For $a,b\\in {\\bf A}$ , $\\langle \\pi ,a\\rangle \\sqcup \\langle \\pi ,b\\rangle =\\langle \\pi ,a\\sqcup b\\rangle $ if $a\\sqcup b$ exists in ${\\bf A}$ $\\langle \\langle s,t\\rangle ,\\pi \\rangle \\sqcup \\langle \\langle s^{\\prime },t^{\\prime }\\rangle ,\\pi \\rangle =\\langle \\langle s\\sqcup s^{\\prime },t\\sqcup t^{\\prime }\\rangle ,\\pi \\rangle $ if $s\\sqcup s^{\\prime }$ and $t\\sqcup t^{\\prime }$ exist in ${\\bf T}$ .",
"The lub $\\langle \\pi ,a\\rangle \\sqcup \\langle \\langle s,t\\rangle ,\\pi \\rangle $ does not exist.",
"Next, the notion of consistency needs to be explored.",
"From the definition of lubs given above, the following sets are consistent: The empty set is consistent.",
"Everything is consistent with the bottom element.",
"A set of elements all from the basis A is consistent in T if the set of elements is consistent in A.",
"A set of product elements in T is consistent if the left component elements are consistent and the right component elements are consistent.",
"These conditions derive from the sum and product nature of the domain.",
"The approximation ordering in the basis has the following inductive definition: $\\Delta _T\\sqsubseteq _T s$ for all $s\\in {\\bf T}$ $y\\sqsubseteq _Tu\\sqcup \\Delta _T$ whenever $y\\sqsubseteq _Tu$ $\\langle \\pi ,a\\rangle \\sqsubseteq _T\\langle \\pi ,b\\rangle $ whenever $a\\sqsubseteq _Ab$ $\\langle \\langle s,t\\rangle ,\\pi \\rangle \\sqsubseteq _T\\langle \\langle u,v\\rangle ,\\pi \\rangle $ whenever $s\\sqsubseteq _Tu$ and $t\\sqsubseteq _Tv$ The next step is to verify that ${\\bf T}$ is indeed a finitary basis.",
"The basis is still countable.",
"The approximation is clearly a partial order.",
"The existence of lubs of finite bounded (i.e., consistent) subsets must be verified.",
"The definition of consistency gives us the requirements for a bounded subset.",
"Each of the conditions for consistency are examined inductively since the definitions are all inductive: The lub of the empty set is the bottom element $\\Delta _T$ .",
"The lub of a set containing the bottom element is the lub of the set without the bottom element which must exist by the induction hypothesis.",
"The lub of a set of elements all from the ${\\bf A}$ is the element in ${\\bf T}$ for the lub in ${\\bf A}$ .",
"This element must exist since ${\\bf A}$ is a finitary basis and all elements from ${\\bf A}$ have corresponding elements in ${\\bf T}$ .",
"The lub of a set of product elements is the pair of the lub of the left components and the lub of the right components.",
"These exist by the induction hypothesis.",
"Thus, a finitary basis has been created; the domain is formed as always from the basis.",
"The solution to the domain equation has been found since any element in the domain ${\\cal T}$ is either an element representing an element in ${\\cal A}$ or is the product of two other elements in ${\\cal T}$ .",
"Similarly, any element found on the left-hand side must also be in the domain ${\\cal T}$ by the construction.",
"Thus, the domain ${\\cal T}$ is identical to the domain ${\\cal A}+({\\cal T}\\times {\\cal T})$ .",
"To look at the question concerning the existence and uniqueness of the solution to this domain equation, recall the fixed point theorem.",
"This theorem states that a fixed point set exists for any approximable mapping over a domain.",
"Subdomains In Section , the concept of a universal domain is introduced.",
"A universal domain is a domain which contains all other domains as sub-domains.",
"These sub-domains are, roughly speaking, the image of approximable functions over the universal domain.",
"The domain equation for ${\\cal T}$ can be viewed as an approximable mapping over the universal domain.",
"As such, the fixed point theorem states that a least fixed point set for the function does exist and is unique.",
"Sub-domains are defined formally below.",
"Looking again at the informal discussion concerning the elements of the domain ${\\cal T}$ , the infinite element proposed does fit into the formal definition for elements of ${\\cal T}$ .",
"This element is an infinite tree with all left sub-trees containing only the element $a$ .",
"For this infinite element to be computable, it must be the lub of some ascending chain of finite approximations to it.",
"The element $x$ can, in fact, be defined by the following ascending sequence of finite trees: $\\begin{array}{lcl}x_0&=&\\bot \\\\x_{n+1}&=&\\langle \\langle a,x_n\\rangle ,\\pi \\rangle \\\\x&=&\\bigsqcup ^\\infty _{n=0}x_n\\end{array}$ Thus, using domain equations, a domain has been defined recursively.",
"This domain includes infinite as well as finite elements and allows computation on the infinite elements to proceed using the finite approximations, as with the more conventionally defined domains presented earlier.",
"The final topic of this section is the notion of a sub-domain.",
"Informally, a sub-domain is a structured part of a larger domain.",
"Earlier, a domain was described as a sub-domain of the universal domain.",
"Thus, the sub-domain starts with a subset of the elements of the larger domain while retaining the approximation ordering, consistency relation and lub relation, suitably restricted to the subset elements.",
"Definition 6.2: [Sub-Domain] A domain $\\langle {\\cal R},\\sqsubseteq _R\\rangle $ is a sub-domain of a domain $\\langle {\\cal D},\\sqsubseteq _D\\rangle $ , denoted ${\\cal R}\\lhd {\\cal D}$ iff ${\\cal R}\\subseteq {\\cal D}$ - The elements of ${\\cal R}$ are a subset of the elements of ${\\cal D}$ .",
"$\\bot _R=\\bot _D$ - The bottom elements are the same.",
"For $x,y\\in {\\cal R}$ , $x\\sqsubseteq _Ry\\iff x\\sqsubseteq _Dy$ - The approximation ordering for ${\\cal R}$ is the approximation ordering for ${\\cal D}$ restricted to elements in ${\\cal R}$ .",
"For $x,y,z\\in {\\cal R}$ , $x\\sqcup _Ry=z$ iff $x\\sqcup _Dy=z$ - The lub relation for ${\\cal R}$ is the lub relation for ${\\cal D}$ restricted to elements in ${\\cal R}$ .",
"${\\cal R}$ is a domain.",
"Equivalently, a sub-domain can be thought of as the image of an approximable function which approximates the identity function (also termed a projection).",
"The notion of a sub-domain is used in the final section in the discussions about the universal domain.",
"This mapping between the domains can be formalized as follows: Theorem 6.3: If ${\\cal D}\\lhd {\\cal E}$ , then there exists a projection pair of approximable mappings $i:{\\cal D}\\rightarrow {\\cal E}$ and $j:{\\cal E}\\rightarrow {\\cal D}$ where $j\\circ i={{\\sf I}}_{\\cal D}$ and $i\\circ j\\sqsubseteq {{\\sf I}}_{\\cal E}$ where $i$ and $j$ are determined by the following equations: $\\begin{array}{lcl}i(x)&=&\\lbrace y\\in {\\bf E}\\:\\vert \\:\\exists z\\in x.z\\sqsubseteq y\\rbrace \\\\j(y)&=&\\lbrace x\\in {\\bf D}\\:\\vert \\:x\\in y\\rbrace \\end{array}$ for all $x\\in {\\cal D}$ and $y\\in {\\cal E}$ .",
"The proof is left as an exercise.",
"By the definition of a sub-domain, it should be clear that ${\\cal D}_0\\lhd {\\cal E}\\wedge {\\cal D}_1\\lhd {\\cal E}\\:\\Rightarrow \\:({\\cal D}_0\\lhd {\\cal D}_1\\iff {\\cal D}_0\\subseteq {\\cal D}_1)$ Using this observation, the sub-domains of a domain can be ordered.",
"Indeed, the following theorem is a consequence of this ordering.",
"Theorem 6.4: For a given domain ${\\cal D}$ , the set of sub-domains $\\lbrace {\\cal D}_0\\:\\vert \\:{\\cal D}_0\\lhd {\\cal D}\\rbrace $ form a domain.",
"The proof proceeds using the inclusion relation defined as an approximation ordering and is left as an exercise.",
"Finally, a converse of Theorem REF can also be established: Theorem 6.5: For two domains ${\\cal D}$ and ${\\cal E}$ , if there exists a projection pair $i:{\\cal D}\\rightarrow {\\cal E}$ and $j:{\\cal E}\\rightarrow {\\cal D}$ with $j\\circ i={\\sf I}_{\\cal D}$ and $i\\circ j\\sqsubseteq {\\sf I}_{\\cal E}$ , then $\\exists {\\cal D}^{\\prime }\\lhd {\\cal E}$ where ${\\cal D}\\approx {\\cal D}^{\\prime }$ .",
"Proof We show that $i$ maps finite elements to finite elements and that ${\\cal D}^{\\prime }$ is the image of ${\\cal D}$ in ${\\cal E}$ .",
"For some $x\\in {\\bf D}$ with ${\\cal I}_x$ as the principal ideal of $x$ , we can write $i({\\cal I}_x)=\\sqcup \\lbrace {\\cal I}_y\\:\\vert \\:y\\in i({\\cal I}_x)\\rbrace $ Applying $j$ to both sides we get ${\\cal I}_x=j\\circ i({\\cal I}_x)=\\sqcup \\lbrace j({\\cal I}_y)\\:\\vert \\:y\\in i({\\cal I}_x)\\rbrace $ since $j\\circ i={\\sf I}_D$ and $j$ is continuous by assumption.",
"But, since $x\\in {\\cal I}_x$ , $x\\in j({\\cal I}_y)$ for some $y\\in i({\\cal I}_x)$ .",
"This means that ${\\cal I}_x\\subseteq j({\\cal I}_y)$ and thus $i({\\cal I}_x)\\subseteq i\\circ j({\\cal I}_y) \\subseteq {\\cal I}_y$ Since ${\\cal I}_y\\subseteq i({\\cal I}_x)$ must hold by the construction, $i({\\cal I}_x) ={\\cal I}_y$ .",
"This proves that finite elements are mapped to finite elements.",
"Next, consider the value for $i(\\bot _D)$ .",
"Since $\\bot _D\\sqsubseteq _Dj(\\bot _E)$ , $i(\\bot _D)\\sqsubseteq \\bot _E$ .",
"Thus $i(\\bot _D)=\\bot _E$ .",
"Thus, ${\\cal D}$ is isomorphic to the image of $i$ in ${\\cal E}$ .",
"We still must show that ${\\cal D}^{\\prime }$ is a domain.",
"Thus, we need to show that if a lub exists in ${\\cal E}$ for a finite subset in ${\\cal D}^{\\prime }$ , then the lub is also in ${\\cal D}^{\\prime }$ .",
"Let $y^{\\prime },z^{\\prime }\\in {\\bf D}^{\\prime }$ and $y^{\\prime }\\sqcup z^{\\prime }=x^{\\prime }\\in {\\bf E}$ .",
"Then, there exists $y,z\\in {\\bf D}$ such that $i({\\cal I}_y)={\\cal I}_{y^{\\prime }}$ and $i({\\cal I}_z)={\\cal I}_{z^{\\prime }}$ which implies that ${\\cal I}_y=j({\\cal I}_{y^{\\prime }})$ and ${\\cal I}_z=j({\\cal I}_{z^{\\prime }})$ .",
"Since ${\\cal I}_{y^{\\prime }}\\sqsubseteq {\\cal I}_{x^{\\prime }}$ and $j({\\cal I}_{y^{\\prime }})\\sqsubseteq j({\\cal I}_{x^{\\prime }})$ by monotonicity, $y\\in j({\\cal I}_{x^{\\prime }})$ must hold.",
"By the same reasoning, $z\\in j({\\cal I}_{x^{\\prime }})$ .",
"But then $x=y\\sqcup z\\in j({\\cal I}_{x^{\\prime }})$ must also hold and thus $y\\sqcup z \\in {\\cal D}$ since the element $j({\\cal I}_{x^{\\prime }})$ must be an ideal.",
"But, $\\begin{array}{lcl}{\\cal I}_y\\sqsubseteq {\\cal I}_x&\\:\\Rightarrow \\:& {\\cal I}_{y^{\\prime }}\\sqsubseteq i({\\cal I}_{x})\\\\{\\cal I}_z\\sqsubseteq {\\cal I}_x&\\:\\Rightarrow \\:& {\\cal I}_{z^{\\prime }}\\sqsubseteq i({\\cal I}_{x})\\end{array}$ This implies that $y^{\\prime }\\sqcup z^{\\prime }=x^{\\prime }\\in i({\\cal I}_x)$ .",
"We already know that $x\\in j({\\cal I}_{x^{\\prime }})$ so $i({\\cal I}_x)\\sqsubseteq {\\cal I}_{x^{\\prime }}$ .",
"Thus, $i({\\cal I}_x)={\\cal I}_{x^{\\prime }}$ and thus, $x^{\\prime }\\in {\\bf D^{\\prime }}$ .$\\:\\:\\Box $ Exercises Exercise 6.6: Show that there must exist domains satisfying $\\begin{array}{lcll}{\\cal A}&=&{\\cal A}+({\\cal A}\\times {\\cal B})&{\\rm and}\\\\{\\cal B}&=&{\\cal A}+{\\cal B}\\end{array}$ Decide what the elements will look like and define ${\\cal A}$ and ${\\cal B}$ using simultaneous fixed points.",
"Exercise 6.7: Prove Theorem REF Exercise 6.8: Prove Theorem REF Exercise 6.9: Show that if ${\\cal A}$ and ${\\cal B}$ are finite systems, that ${\\cal D}\\unlhd {\\cal E}\\unlhd {\\cal D}\\:\\Rightarrow \\:{\\cal D}\\approx {\\cal E}$ where ${\\cal D}\\approx {\\cal D}^{\\prime }$ and ${\\cal D}^{\\prime }\\lhd {\\cal E}$ is denoted ${\\cal D}^{\\prime }\\unlhd {\\cal E}$ .",
"Computability in Effectively Given Domains In the previous sections, we gave considerable emphasis to the notion of computation using increasingly accurate approximations of the input and output.",
"This section defines this notion of computability more formally.",
"In Section 5, we found that partial functions over the natural numbers were expressible in the $\\lambda $ -notation.",
"This relationship characterizes computation for a particular domain.",
"To describe computation over domains in general, a broader definition is required.",
"The way a domain is presented impacts the way computations are performed over it.",
"Indeed, the theorems of recursive function theory [6] rely in part on the normal presentation of the natural numbers.",
"A presentation for a domain is an enumeration of the elements of the domain.",
"The standard presentation of the natural numbers is simply the numbers in ascending order beginning with 0.",
"There are many permutations of the natural numbers, each of which can be considered a presentation.",
"Computation with these non-standard presentations may be impossible; that is a computable function on the standard presentation may be non-computable over a non-standard presentation.",
"Therefore, an effective presentation for a domain is defined as a presentation which makes the required information computable.",
"Effective Presentations Information about elements in a domain can be characterized completely by looking at the finite elements and their relationships.",
"Thus a presentation must enumerate the finite elements and allow the consistency and lub relationships on these elements to be computed to allow this style of computation.",
"The consistency relation and the lub relation depend on each other.",
"For example, if a set of elements is consistent, a lub must exist for the set.",
"Given that a set is consistent, the lub can be found in finite time by just enumerating the elements and checking to see if this element is the lub.",
"However, if the set is inconsistent, the enumeration will not reveal this fact.",
"Thus, the consistency relation must be assumed to be recursive in an effective presentation.",
"Exercise REF provides a description of presentations that should clarify the assumptions made.",
"Formally, a presentation is defined as follows: Definition 7.1: [Effective Presentation] The presentation of a finitary basis D is a function $\\pi :{{N}}\\rightarrow {\\bf D}$ such that $\\pi (0)=\\Delta _D$ and the range of $\\pi $ is the set of finite elements of D. The definition holds for a domain ${\\cal D}$ as well.",
"A presentation $\\pi $ is effective iff The consistency relation ($\\exists k.\\pi _i\\sqsubseteq \\pi _k\\wedge \\pi _j\\sqsubseteq \\pi _k$ ) for elements $\\pi _i$ and $\\pi _j$ is recursiveRecursive in this context means that the relation is decidable.",
"over $i$ and $j$ .",
"The lub relation ($\\pi _k=\\pi _i\\sqcup \\pi _j$ ) is recursive over $i$ , $j$ , and $k$ .",
"This definition supports our intuition about domains; we have stated that the important information about a domain is the set of finite elements, the ordering and consistency relationships between the elements and the lub relation.",
"Thus, an effective presentation provides, in a suitable (that is computable) form, the basic information about the structure and elements of a domain.",
"A presentation can also be viewed as an enumeration of the elements of the domain with the position of an element in the enumeration given by the index corresponding to the integer input for that element in the presentation function with the 0 element representing $\\bot $ .",
"This perspective is used in the majority of the proofs.",
"Computability Now that the presentation of a domain has been formalized, the notion of computability can be formally defined.",
"Thus, Definition 7.2: [Computable Mappings] Given two domains, ${\\cal D}$ and ${\\cal E}$ with effective presentations $\\pi _1$ and $\\pi _2$ respectively, an approximable mapping $f:{\\bf D}\\rightarrow {\\bf E}$ is computable iff the relation $x_n\\:f\\:y_m$ is recursively enumerable in $n$ and $m$ .",
"By considering the domain ${\\cal D}$ to be a single element domain, the above definition applies not only to computable functions but also to computable elements.",
"For $d\\in {\\cal D}$ where $d$ is the only element in the domain, the element $e=f(d)\\in {\\cal E}$ defines an element in ${\\cal E}$ .",
"The definition states that $e$ is a computable iff the set $\\lbrace m\\in {{N}}\\:\\vert \\:y_m\\sqsubseteq e\\rbrace $ is a recursively enumerable set of integers.",
"Clearly if the set of elements approximating another is finite, the set is recursive.",
"The notion of a recursively enumerable set simply requires that all elements approximating the element in question be listed eventually.",
"The computation then proceeds by accepting an enumeration representing the input element and enumerating the elements that approximate the desired output element.",
"Now that the notions of computability and effective presentations have been formalized, the methods of constructing domains and functions will be addressed.",
"The proof of the next theorem is trivial and is left to the reader.",
"Theorem 7.3: The identity map on an effectively given domain is computable.",
"The composition of computable mappings on effectively given domains are also computable.",
"The following corollary is a consequence of this theorem: Corollary 7.4: For computable function $f:{\\cal D}\\rightarrow {\\cal E}$ and a computable element $x\\in {\\cal D}$ , the element $f(x)\\in {\\cal E}$ is computable.",
"In addition, the standard domain constructors maintain effective presentations.",
"Theorem 7.5: For domains ${\\cal D}_0$ and ${\\cal D}_1$ with effective presentations, the domains ${\\cal D}_0+{\\cal D}_1~{\\rm and}~ {\\cal D}_0\\times {\\cal D}_1$ are also effectively given.",
"In addition, the projection functions are all computable.",
"Finally, if $f$ and $g$ are computable maps, then so are $f+g$ and $f\\times g$ .",
"Proof Let $\\lbrace X_i\\:\\vert \\:i\\in {{N}}\\rbrace $ be the enumeration of ${\\cal D}_0$ and $\\lbrace Y_i\\:\\vert \\:i\\in {{N}}\\rbrace $ be the enumeration of ${\\cal D}_1$ .",
"Another method of sum construction is to use two distinguishing elements in the first position to specify the element type.",
"Thus, a sum domain can be defined as follows: ${\\cal D}_0+{\\cal D}_1=\\lbrace (\\Delta _0,\\Delta _1)\\rbrace \\cup \\lbrace (0,x)\\:\\vert \\:x\\in {\\cal D}_0\\rbrace \\cup \\lbrace (1,y)\\:\\vert \\:y\\in {\\cal D}_1\\rbrace $ The enumeration can then be defined as follows for $n\\in {{N}}$ : $\\begin{array}{lcl}Z_0&=&(\\Delta _0,\\Delta _1)\\\\Z_{2n+1}&=&(0,X_n)\\\\Z_{2n+2}&=&(1,Y_n)\\end{array}$ The proof that $Z_i$ is an effective presentation is left as an exercise.",
"For the product construction, the domain appears as follows: ${\\cal D}_0\\times {\\cal D}_1=\\lbrace (x,y)\\:\\vert \\:x\\in {\\cal D}_0,y\\in {\\cal D}_1\\rbrace $ The enumeration can be defined in terms of the functions $p:{{N}}\\rightarrow {{N}}$ , $q:{{N}}\\rightarrow {{N}}$ , and $r:({{N}}\\times {{N}})\\rightarrow {{N}}$ where for $m$ , $n$ , $k\\in {{N}}$ : $\\begin{array}{lcl}p(r(n,m))&=&n\\\\q(r(n,m))&=&m\\\\r(p(k),q(k))&=&k\\end{array}$ Thus, $r$ is a one-to-one pairing function (see Exercise REF ) of which there are several.",
"The functions $p$ and $q$ extract the indices from the result of the pairing function.",
"The enumeration for the product domain is then defined as follows: $W_i = (X_{p(i)},Y_{q(i)})$ The proof that this is an effective presentation is also left as an exercise.",
"For the combinators, the relations will be defined in terms of the enumeration indices.",
"For example, $\\begin{array}{lcl}X_n\\:in_0\\:Z_m&\\iff & m=0~{\\rm or}\\\\&&\\exists k.m=2k+1\\wedge X_k\\sqsubseteq X_n\\\\W_k\\:proj_1\\:Y_m&\\iff & Y_m\\sqsubseteq Y_{q(k)}\\end{array}$ The reader should verify that these sets are recursively enumerable.",
"For this proof, recall that recursively enumerable sets are closed under conjunction, disjunction, substituting recursive functions, and applying an existential quantifier to the front of a recursive predicate.",
"The proof for the other combinators is left as an exercise.",
"$\\:\\:\\Box $ Product spaces formalize the notion of computable functions of several variables.",
"Note that the proof of Theorem REF shows that substitution of computable functions of severable variables into other computable functions are still computable.",
"The next step is to show that the function space constructor preserves effectiveness.",
"Theorem 7.6: For domains ${\\cal D}_0$ and ${\\cal D}_1$ with effective presentations, the domain ${\\cal D}_0\\rightarrow {\\cal D}_1$ also has an effective presentation.",
"The combinators $apply$ and $curry$ are computable if all input domains are effectively given.",
"The computable elements of the domain ${\\cal D}_0\\rightarrow {\\cal D}_1$ are the computable maps for ${\\bf D_0}\\rightarrow {\\bf D_1}$ .",
"Proof Let ${\\cal D}_0=\\lbrace X_i\\:\\vert \\:i\\in {{N}}\\rbrace $ and ${\\cal D}_1=\\lbrace Y_i\\:\\vert \\:i\\in {{N}}\\rbrace $ be the presentations for the domains.",
"The elements of ${\\bf D_0}\\rightarrow {\\bf D_1}$ are finite step functions which respect the mapping of some subset of ${\\bf D_0}\\times {\\bf D_1}$ .",
"Given the enumeration, each element can be associated with a set $\\lbrace (X_{n_i},Y_{m_i})\\:\\vert \\:\\exists q.",
"1\\le i\\le q\\rbrace $ Thus, there is a finite set of integers pairs that determine the element.",
"Given the definition of consistency from Theorem REF for elements in the function space domain and the decidability of consistency in ${\\cal D}_0$ and ${\\cal D}_1$ , consistency of any finite set of this form is decidable (tedious but decidable since all elements must be checked with all others, etc).",
"Since consistency is decidable, a systematic enumeration of pair sets which are consistent can be made; this enumeration is simply the enumeration of ${\\cal D}_0\\rightarrow {\\cal D}_1$ .",
"Finding the lub consists of making a finite series of tests to find the element that is the lub, which must exist since the set is consistent and we have closure on lubs of finite consistent subsets.",
"Finding the lub requires a finite series of checks in both ${\\cal D}_0$ and ${\\cal D}_1$ but these checks are decidable.",
"Thus, the lub relation is also decidable in ${\\cal D}_0\\rightarrow {\\cal D}_1$ .",
"This shows that ${\\cal D}_0\\rightarrow {\\cal D}_1$ is effectively given.",
"To show that $apply$ and $curry$ are computable, the mappings need to be examined.",
"The mapping defined for apply is $(F,a)\\:apply\\: b\\iff a\\:F\\:b$ The function $F$ is the lub of all the finite step functions that are consistent with it.",
"As such, $F$ can be viewed as the canonical representative of this set.",
"Since $F$ is a finite step function, this relation is decidable.",
"As such, the $apply$ relation is recursive and not just recursively enumerable and $apply$ is a computable function.",
"The reasoning for $curry$ is similar in that the relations are studied.",
"Given the increase in the number of domains, the construction is more tedious and is left for the exercises.",
"To see that the computable elements correspond to the computable maps, recall the relationship shown in Theorem REF between the maps and the elements in the function space.",
"Thus, we have $a\\:f\\:b \\iff b\\in f({\\cal I}_a)~{\\rm or}~{\\cal I}_b\\sqsubseteq f({\\cal I}_a)$ Since $f$ is a computable map, we know that the pairs in the map are recursively enumerable.",
"Using the previous techniques for deciding consistency of finite sets, the set of elements consistent with $f$ can be enumerated.",
"But this set is simply the ideal for $f$ in the function space.",
"The converse direction is trivial.",
"$\\:\\:\\Box $ The final combinator to be discussed, and perhaps the most important, is the fixed point combinator.",
"Theorem 7.7: For any effectively given domain, ${\\cal D}$ , the combinator $fix:({\\cal D}\\rightarrow {\\cal D})\\rightarrow {\\cal D}$ is computable.",
"Proof Let $\\lbrace X_n\\:\\vert \\:n\\in {{N}}\\rbrace $ be the presentation of the domain ${\\cal D}$ .",
"Recall that for $f\\in {\\cal D}\\rightarrow {\\cal D}$ , $f\\:fix\\:X\\iff \\exists k\\in {{N}}.\\Delta \\:f\\:X_1\\:f\\ldots f\\:X_k\\wedge X_k=X$ All of the checks in this finite sequence are decidable since ${\\cal D}$ is effectively given.",
"In addition, existential quantification of a decidable predicate gives a recursively enumerable predicate.",
"Thus, $fix$ is computable.",
"$\\:\\:\\Box $ Recap Now that this has been formalized, what has been accomplished?",
"The major consequence of the theorems to this point is that any expression over effectively given domains (that is effectively given types) combined with computable constants using the $\\lambda $ -notation and the fixed point combinator is a computable function of its free variables.",
"Such functions, applied to computable arguments, yield computable values.",
"These functions also have computable least fixed points.",
"All this gives us a mathematical programming language for defining computable operations.",
"Combining this language with the specification of types with domain equations gives a powerful language.",
"As an example, the effectiveness of the domain ${\\cal T}$ from Example REF is studied.",
"The complete proof is left as an exercise.",
"Example 7.8: Recall the domain ${\\cal T}$ from the previous section.",
"This domain is characterized by the domain equation ${\\cal T}={\\cal A}+({\\cal T}\\times {\\cal T})$ for some domain ${\\cal A}$ .",
"If ${\\cal A}$ is effectively given, we wish to show that ${\\cal T}$ is effectively given as well.",
"The elements are either atomic elements from ${\\cal A}$ or are pairs from ${\\cal T}$ .",
"Let $A=\\lbrace A_i\\:\\vert \\:i\\in {{N}}\\rbrace $ be the enumeration for ${\\cal A}$ .",
"An enumeration for ${\\cal T}$ can be defined as follows: $\\begin{array}{lcl}T_0&=&\\bot _T\\\\T_{2n+1}&=&3*A_n\\\\T_{2n+2}&=&3*T_{p(n)}+1\\cup 3*T_{q(n)}+2\\end{array}$ where for $A$ , a set of indices, $m*A+k=\\lbrace m*n+k\\:\\vert \\:n\\in A\\rbrace $ .",
"The functions $p$ and $q$ here are the inverses of the pairing function $r$ defined in Theorem REF .",
"These functions must be defined such that $p(n)\\le n$ and $q(n)\\le n$ so that the recursion is well defined by taking smaller indices.",
"The rest of the proof is left to the exercises.",
"Specifically, the claim that ${\\cal T}=\\lbrace T_i\\rbrace $ should be verified as well as the effectiveness of the enumeration.",
"These proofs rely either on the effectiveness of ${\\cal A}$ , on the effectiveness of elements in ${\\cal T}$ with smaller indices, or are trivial.",
"The final example uses the powerset construction.",
"We have repeatedly used the fact that a powerset is a domain.",
"Its effectiveness is now verified.",
"Example 7.9: Specifically, the powerset of the natural numbers, ${\\cal {P}({{N}})}$ is considered.",
"In this domain, all elements are consistent, and there is a top element, denoted $\\omega $ , which is the set of all natural numbers.",
"The ordering is the subset relation.",
"The lub of two subsets is the union of the two subsets, which is decidable.",
"To enumerate the finite subsets, the following enumeration is used: $E_n=\\lbrace k\\:\\vert \\:\\exists i,j.",
"i< 2^k\\wedge n=i+2^k+j*2^{k+1}\\rbrace $ This says that $k\\in E_n$ if the $k$ bit in the binary expansion of $n$ is a 1.",
"All finite subsets of ${{N}}$ are of the form $E_n$ for some $n$ .",
"Various combinators for ${\\cal P}({{N}})$ are presented in Exercise REF .",
"Exercises Exercise 7.10: Show that an effectively given domain can always be identified with a relation $INCL(n,m)$ on integers where the derived relations $\\begin{array}{lcl}CONS(n,m)&\\iff &\\exists k.INCL(k,n)\\wedge INCL(k,m)\\\\MEET(n,m,k)&\\iff &\\forall j.",
"[INCL(j,k)\\iff INCL(j,n)\\wedge INCL(j,m)]\\end{array}$ are recursively decidable and where the following axioms hold: $\\forall n.INCL(n,n)$ $\\forall n,m,k.",
"INCL(n,m)\\wedge INCL(m,k)\\:\\Rightarrow \\:INCL(n,k)$ $\\exists m.\\forall n. INCL(n,m)$ $\\forall n,m.",
"CONS(n,m)\\:\\Rightarrow \\:\\exists k.MEET(n,m,k)$ Exercise 7.11: Finish the proof of Theorem REF .",
"Exercise 7.12: Complete the proof of Theorem REF by defining $curry$ as a relation and showing it computable.",
"Is the set recursively enumerable or is it recursive?",
"Exercise 7.13: Two effectively given domains are effectively isomorphic iff $\\ldots $ Complete the statement of the theorem and prove it.",
"Exercise 7.14: Complete the proof about the powerset in Example REF .",
"Show that the combinators $fun$ and $graph$ from Exercise REF are computable.",
"Show the same for $\\lambda x,y.x\\cap y$ $\\lambda x,y.x\\cup y$ $\\lambda x,y.x+ y$ where for $x,y\\in {\\cal P}({{N}})$ , $x+y=\\lbrace n+m\\:\\vert \\:n\\in x, m\\in y\\rbrace $ What are the computable elements of ${\\cal P}({{N}})$ ?",
"Sub-Spaces of the Universal Domain To have a flexible method of solving domain equations and yielding effectively given domains as the solutions, the domains will be embedded in a universal domain which is “big” enough to hold all other domains as sub-domains.",
"This universal domain is shown to be effectively presented, and the mappings which define the sub-spaces are shown to be computable.",
"First, the correspondence between sub-spaces and mappings called retractions is investigated, leading us to the definition of mappings called projections.",
"It is then shown that these definitions can be written out using the $\\lambda $ -calculus notation, demonstrating the power of our mathematical programming language.",
"Retractions and Projections We start with the definition of retractions.",
"Definition 8.1: [Retractions] A retraction of a given domain ${\\cal E}$ is an approximable mapping $a:{\\bf E}\\rightarrow {\\bf E}$ such that $a\\circ a=a$ .",
"Thus, a retraction is the identity function on objects in the range of the retraction and maps other elements into range.",
"The next theorem relates these sets to sub-spaces.",
"Theorem 8.2: If ${\\cal D}\\lhd {\\cal E}$ and if $a:{\\bf E}\\rightarrow {\\bf E}$ is defined such that $X\\:a\\:Z \\iff \\exists Y\\in {\\cal D}.",
"Z\\sqsubseteq Y\\sqsubseteq X$ for all $X,Z\\in {\\bf E}$ , then $a$ is a retraction and ${\\cal D}$ is isomorphic to the fixed point set of $a$ , the set $\\lbrace y\\in {\\cal E}\\:\\vert \\:a(y)=y\\rbrace $ , ordered under inclusion.",
"Proof That $a$ is an approximable map is a direct consequence of the definition of sub-space (Definition REF ).",
"By Theorem REF , a projection pair, $i$ and $j$ , exist for ${\\cal D}$ and this tells us that $a=i\\circ j$ (also showing $a$ approximable since approximable mappings are closed under composition).",
"Theorem REF also tells us that $j\\circ i={\\sf I}_D$ .",
"To show that $a$ is a retraction, $a\\circ a=a$ must be established.",
"Thus, $a\\circ a = i\\circ j\\circ i\\circ j = i\\circ {\\sf I}_D\\circ j = i\\circ j =a$ holds, showing that $a$ is a retraction.",
"We now need to show the isomorphism to ${\\cal D}$ .",
"For $x\\in {\\cal D}$ , $i(x)\\in {\\cal E}$ and we can calculate: $a(i(x))=i\\circ j\\circ i(x) = i\\circ {\\sf I}_D(x) = i(x)$ Thus, $i(x)$ is in the fixed point set of $a$ .",
"For the other direction, let $a(y)=y$ .",
"Then $i(j(y)) = y$ holds.",
"But, $j(y)\\in {\\cal D}$ , so $i$ must map ${\\cal D}$ one-to-one and onto the fixed point set of $a$ .",
"Since $i$ and $j$ are approximable, they are certainly monotonic, and thus the map is an isomorphism with respect to set inclusion.",
"$\\:\\:\\Box $ Not all retractions are associated with a sub-domain relationship.",
"The retractions defined in the above theorem are all subsets as relations of the identity relation.",
"The retractions for sub-domains are characterized by the following definition: Definition 8.3: [Projections] A retraction $a:{\\cal E}\\rightarrow {\\cal E}$ is a projection if $a\\subseteq {\\sf I}_E$ as relations.",
"The retraction is finitary iff its fixed point set is isomorphic to some domain.",
"An example is in order.",
"Example 8.4: Consider a two element system, ${\\bf O}$ with objects $\\Delta $ and 0.",
"For any basis ${\\bf D}$ that is not trivial (has more than one element), ${\\bf O}$ comes from a retraction on ${\\bf D}$ .",
"Define a combinator $check:{\\bf D}\\rightarrow {\\bf O}$ by the relation $x\\:check\\: y \\iff y=\\Delta ~{\\rm or}~x\\ne \\Delta _D$ Thus, $check(x)=\\bot _O\\iff x=\\bot _D$ .",
"Another combinator can be defined, $fade:{\\bf O}\\times {\\bf D}\\rightarrow {\\bf D}$ such that for $t\\in {\\cal O}$ and $x\\in {\\cal D}$ $\\begin{array}{lcll}fade(t,x)&=&\\bot _D&{\\rm if}~t=\\bot _O\\\\&=&x&otherwise\\end{array}$ For $u\\in {\\cal D}$ and $u\\ne \\bot _D$ , the mapping $a$ is defined as $a(x)=fade(check(x),u)$ It can be seen that $a$ is a retraction, but not a projection in general, and the range of $a$ is isomorphic to ${\\bf O}$ .",
"These combinators can also be used to define the subset of functions in ${\\bf D}\\rightarrow {\\bf E}$ that are strict.",
"Define a combinator $strict:({\\bf D}\\rightarrow {\\bf E})\\rightarrow ({\\bf D}\\rightarrow {\\bf E})$ by the equation $strict(f)=\\lambda x.fade(check(x),f(x))$ with $fade$ defined as $fade:{\\bf O}\\times {\\bf E}\\rightarrow {\\bf E}$ .",
"The range of $strict$ is all the strict functions; $strict$ is a projection whose range is a domain.",
"The next theorem characterizes projections.",
"Theorem 8.5: For approximable mapping $a:{\\bf E}\\rightarrow {\\bf E}$ , the following are equivalent: $a$ is a finitary projection $a(x)=\\lbrace y\\in {\\bf E}\\:\\vert \\:\\exists x^{\\prime }\\in I_x.",
"x^{\\prime }\\:a\\:x^{\\prime }\\wedge y\\sqsubseteq x^{\\prime }\\rbrace $ for all $x\\in {\\bf E}$ .",
"Proof Assume that (2) holds.",
"We want to show that $a$ is a finitary projection.",
"By the closure properties on ideals, we know that for all $x\\in {\\cal E}$ , $x^{\\prime }\\in x\\wedge y\\sqsubseteq x^{\\prime }\\:\\Rightarrow \\:y\\in x$ Thus, $a(x)\\subseteq x$ must hold.",
"In addition, the following trivially holds: $x^{\\prime }\\in x\\wedge x^{\\prime }\\:a\\: x^{\\prime }\\:\\Rightarrow \\:x^{\\prime }\\in a(x)$ thus $a(x)\\subseteq a(a(x))$ holds for all $x\\in {\\cal E}$ .",
"This shows that $a$ is indeed a projection.",
"Let $D=\\lbrace x\\in {\\bf E}\\:\\vert \\:x\\:a\\:x\\rbrace $ .",
"It is easy to show that ${\\bf D}\\lhd {\\bf E}$ and that $a$ is determined from ${\\bf D}$ as required in Theorem REF .",
"Thus, the fixed point set of $a$ is isomorphic to a domain from the previous proofs.",
"Thus, (2)$\\:\\Rightarrow \\:$ (1).",
"For the converse, assume that $a$ is a finitary projection.",
"Let ${\\cal D}$ be isomorphic to the fixed point set of $a$ .",
"This means there is a projection pair $i$ and $j$ such that $j\\circ i={\\sf I}_D$ and $i\\circ j = a$ and $a\\subseteq {\\sf I}_E$ .",
"From Theorem REF then we have that ${\\cal D}\\approx {\\cal D}^{\\prime }$ and ${\\cal D}^{\\prime }\\lhd {\\cal E}$ .",
"We want to identify ${\\cal D}^{\\prime }$ as follows: ${\\cal D}^{\\prime }=\\lbrace x\\in {\\cal E}\\:\\vert \\:x\\:a\\:x\\rbrace $ From the proof of Theorem REF , the basis elements of ${\\bf D^{\\prime }}$ are the finite elements of ${\\bf D}$ .",
"Each of these elements is in the fixed point set of $a$ .",
"Thus, $x\\in {\\bf D^{\\prime }}\\:\\Rightarrow \\:a({{\\cal I}}_x) = {{\\cal I}}_x \\:\\Rightarrow \\:x\\:a\\:x$ Since $a$ is a projection, ${{\\cal I}}_x$ must also be a fixed point.",
"Since $i(j({{\\cal I}}_x)) = {{\\cal I}}_x$ implies that $j({{\\cal I}}_x)$ is a finite element of ${\\cal D}$ , $x\\in {\\cal D}^{\\prime }$ must hold.",
"Thus, the identification of ${\\cal D}^{\\prime }$ holds.",
"Finally, using $a=i\\circ j$ in the formula in Theorem REF , the formula in (2) is obtained, proving the converse.",
"$\\:\\:\\Box $ This characterization of projections provides a new and interesting combinator.",
"Theorem 8.6: For any domain ${\\cal E}$ , define $sub:({\\cal E}\\rightarrow {\\cal E})\\rightarrow ({\\cal E}\\rightarrow {\\cal E})$ using the relation $x\\: sub(f)\\: z \\iff \\exists y\\in {\\bf E}.y\\:f\\:y\\wedge y\\sqsubseteq x\\wedge z\\sqsubseteq y$ for all $x,z\\in {\\bf E}$ and all $f:{\\bf E}\\rightarrow {\\bf E}$ .",
"Then the range of $sub$ is exactly the set of finitary projections on ${\\cal E}$ .",
"In addition, $sub$ is a finitary projection on ${\\cal E}\\rightarrow {\\cal E}$ .",
"If ${\\cal E}$ is effectively given, then $sub$ is computable.",
"Proof Clearly, $sub(f)$ is approximable.",
"It is obvious from the definition that $f\\mapsto sub(f)$ preserves lubs and thus is approximable as well.",
"Thus, $y\\:f\\:y\\wedge y\\sqsubseteq x\\wedge z\\sqsubseteq y\\:\\Rightarrow \\:x\\:f\\:z$ obviously holds.",
"Thus, $sub(f)\\subseteq f$ holds.",
"Also $y\\:f\\:y\\:\\Rightarrow \\:y\\:sub(f)\\:y$ thus, $sub(f)\\subseteq sub(sub(f))$ holds as well.",
"Thus, $sub$ is a projection on ${\\cal E}\\rightarrow {\\cal E}$ .",
"The definition of the relation shows that it is computable when ${\\cal E}$ is effectively given.",
"Since $sub$ is a projection, its range is the same as its fixed point set.",
"If $sub(a)=a$ , it is easy to see that clause (2) of Theorem REF holds and conversely.",
"Thus, the range of $sub$ is the finitary projections.",
"To see that $sub$ is a finitary projection, we use Theorem REF and Theorem REF to say that the fixed point set of $sub$ is in a one-to-one inclusion preserving correspondence with the domain $\\lbrace D\\:\\vert \\:D\\lhd {\\cal E}\\rbrace $ .",
"$\\:\\:\\Box $ Universal Domain ${\\cal U}$ With these results and the universal domain to be defined next, the theory of sub-domains is translated into the $\\lambda $ -calculus notation using the $sub$ combinator.",
"The universal domain is defined by first defining a domain which has the desired structure but has a top element.",
"The top element is then removed to give the universal domain.",
"Definition 8.7: [Universal Domain] As in the section on domain equations, an inductive definition for a domain ${\\cal V}$ is given as follows: $\\Delta ,\\top \\in {\\bf V}$ $\\langle u,v\\rangle \\in {\\bf V}$ whenever $u,v\\in {\\bf V}$ Thus, we are starting with two objects, a bottom element and a top element, and making two flavors of copies of these objects.",
"Intuitively, we end up with finite binary trees with either the top or the bottom element as the leaves.",
"To simplify the definitions below, the pairs should be reduced such that: All occurrences of $\\langle \\Delta ,\\Delta \\rangle $ are replaced by $\\Delta $ and All occurrences of $\\langle \\top ,\\top \\rangle $ are replaced by $\\top $ .",
"These rewrite rules are easily shown to be finite Church-Rosser.The finitary basis should be defined as the equivalence classes induced by the reduction.",
"The presentation is simplified by considering only reduced trees.",
"As an example of the reduction the pair $\\langle \\langle \\langle \\top ,\\langle \\top ,\\top \\rangle \\rangle ,\\langle \\top ,\\Delta \\rangle \\rangle ,\\langle \\langle \\Delta ,\\Delta \\rangle ,\\langle \\top ,\\top \\rangle \\rangle \\rangle $ reduces to $\\langle \\langle \\top ,\\langle \\top ,\\Delta \\rangle \\rangle ,\\langle \\Delta ,\\top \\rangle \\rangle $ .",
"The approximation ordering is defined as follows: $\\Delta \\sqsubseteq v$ for all $v\\in {\\bf V}$ $v\\sqsubseteq \\top $ for all $v\\in {\\bf V}$ .",
"$\\langle u,v\\rangle \\sqsubseteq \\langle u^{\\prime },v^{\\prime }\\rangle $ iff $u\\sqsubseteq u^{\\prime }$ and $v\\sqsubseteq v^{\\prime }$ Since the top element is approximated by everything, all finite sets of trees are consistent.",
"The lub for a pair of trees is defined as follows: $u\\sqcup \\top =\\top $ for $u\\in {\\bf V}$ $\\top \\sqcup u=\\top $ for $u\\in {\\bf V}$ $u\\sqcup \\Delta =u$ for $u\\in {\\bf V}$ $\\Delta \\sqcup u=u$ for $u\\in {\\bf V}$ $\\langle u,v\\rangle \\sqcup \\langle u^{\\prime },v^{\\prime }\\rangle =\\langle u\\sqcup u^{\\prime },v\\sqcup v^{\\prime }\\rangle $ for $u,v\\in {\\bf V}$ The proof that this forms a finitary basis follows the same guidelines as the proofs in Section .",
"In addition, it should be clear that the presentation is effective.",
"To form the universal domain, the top element is simply removed.",
"Thus, the system ${\\bf U}={\\bf V}-\\lbrace \\top \\rbrace $ is the basis used to form the universal domain.",
"The proof that this is still a finitary basis with an effective presentation is also straightforward and left to the exercises.",
"Note that inconsistent sets can now exist since there is no top element.",
"A set is inconsistent iff its lub is $\\top $ .",
"We shall now prove the claims made for the universal domain.",
"Theorem 8.8: The domain ${\\cal U}$ is universal, in the sense that for every domain ${\\cal D}$ we have ${\\cal D}\\lhd {\\cal U}$ .",
"If ${\\cal D}$ is effectively given, then the projection pair for the embedding is computable.",
"In fact, there is a correspondence between the effectively presented domains and the computable finitary projections of ${\\cal U}$ .",
"Proof Recall that ${\\bf D}$ must be countable to be a finitary basis.",
"Thus, we can assume that the basis has an enumeration $D=\\lbrace X_n\\:\\vert \\:n\\in {{N}}\\rbrace $ where $X_0=\\Delta $ .",
"The effective and general cases are considered together in the proof; comments about computability are included for the effective case as required.",
"Thus, if ${\\cal D}$ is effectively given, the enumeration above is assumed to be computable.",
"To prove that the domain can be embedded in ${\\cal U}$ , the embedding will be shown.",
"To start, for each finite element $d_i$ in the basis, define two sets, $d_i^+$ and $d_i^-$ as follows: $\\begin{array}{lcl}d_i^+&=&\\lbrace d\\in {\\bf D}\\:\\vert \\:d_i\\sqsubseteq d\\rbrace \\\\d_i^-&=&D-d_i^+\\end{array}$ The $d_i^+$ set contains all the elements that $d_i$ approximates, while the $d_i^-$ set contains all the other elements, partitioning ${\\bf D}$ into two disjoint sets.",
"Sets for different elements can be intersected to form finer partitions of ${\\bf D}$ .",
"For $k>0$ , let $R\\in \\lbrace +,-\\rbrace ^k$ , let $R_i$ be the $ith$ symbol in the string $R$ , and define a region $D_R$ as $D_R=\\bigcap \\limits _{i=1}^k d_i^{R_i}$ where $k$ is the length of $R$ .",
"The set $\\lbrace D_{R}\\:\\vert \\:R\\in \\lbrace +,-\\rbrace ^k\\rbrace $ of regions partitions ${\\bf D}$ into $2^k$ disjoint sets.",
"Thus, for each element $e_i$ in the enumeration there is a corresponding partition of the basis given by the family of sets $\\lbrace D_{R}\\:\\vert \\:R\\in \\lbrace +,-\\rbrace ^i\\rbrace $ .",
"For strings $R,S\\in \\lbrace +,-\\rbrace ^*$ such that $R$ is a prefix of $S$ , denoted $R\\le S$ , $D_S\\subseteq D_R$ .",
"It is important to realize that the composition of these sets is dependent on the order in which the elements are enumerated.",
"Some of these regions are empty, but it is decidable if a given intersection is empty if ${\\cal D}$ is effectively presented.",
"It is also decidable if a given element is in a particular region.",
"Figure: Example Finite DomainTo see the function these regions are serving, consider the finite domain in Figure REF .This example is taken from Cartwright and Demers [2].",
"Consider the enumeration with $d_0=\\bot , d_1=b, d_2=c, d_3=a.$ The $d_i^+$ and $d_i^-$ sets are as follows: $\\begin{array}{lcl}d_1^+&=&\\lbrace a,b\\rbrace \\\\d_1^-&=&\\lbrace c,\\bot \\rbrace \\\\d_2^+&=&\\lbrace c\\rbrace \\\\d_2^-&=&\\lbrace a,b,\\bot \\rbrace \\\\d_3^+&=&\\lbrace a\\rbrace \\\\d_3^-&=&\\lbrace b,c,\\bot \\rbrace \\end{array}$ The regions are as follows: $\\begin{array}{lclclcl}D_+ &=&\\lbrace a,b\\rbrace &\\:\\:\\:\\:&D_{+++} &=&\\lbrace \\rbrace \\\\D_- &=&\\lbrace \\bot ,c\\rbrace &&D_{++-} &=&\\lbrace \\rbrace \\\\D_{++} &=&\\lbrace \\rbrace &&D_{+-+} &=&\\lbrace a\\rbrace \\\\D_{+-} &=&\\lbrace a,b\\rbrace &&D_{+--} &=&\\lbrace b\\rbrace \\\\D_{-+} &=&\\lbrace c\\rbrace &&D_{-++} &=&\\lbrace \\rbrace \\\\D_{--} &=&\\lbrace \\bot \\rbrace &&D_{-+-} &=&\\lbrace c\\rbrace \\\\&&&&D_{--+} &=&\\lbrace \\rbrace \\\\&&&&D_{---} &=&\\lbrace \\bot \\rbrace \\end{array}$ The regions generated by each successive element encode the relationships induced by the approximation ordering between the new element and all elements previously added.",
"The reader is encouraged to try this example with other enumerations of this basis and compare the results.",
"The embedding of the elements proceeds by building a tree based on the regions corresponding to the element.",
"The regions are used to find locations in the tree and to determine whether a $\\top $ or a $\\Delta $ element is placed in the location.",
"These trees preserve the relationships specified by the regions and thus, the tree embedding is isomorphic to the domain in question.",
"Once the tree is built, the reduction rules are applied until a non-reducible tree is reached.",
"This tree is the representative element in the universal domain, and the set of these trees form the sub-space.",
"The function to determine the location in the tree for a given domain, $Loc_D:\\lbrace +,-\\rbrace ^*\\rightarrow \\lbrace l,r\\rbrace ^*$ takes strings used to generate regions and outputs a path in a tree where $l$ stands for left sub-tree and $r$ stands for right sub-tree.",
"This path is computed using the following inductive definition: $\\begin{array}{lcll}Loc_D(\\epsilon )&=&\\epsilon .\\\\Loc_D(R+)&=&Loc_D(R)l&{\\rm if }~D_{R+}\\ne \\emptyset ~ {\\rm and }~D_{R-}\\ne \\emptyset .\\\\&=&Loc_D(R)&{\\rm otherwise}.\\\\Loc_D(R-)&=&Loc_D(R)r&{\\rm if }~D_{R+}\\ne \\emptyset ~{\\rm and }~D_{R-}\\ne \\emptyset .\\\\&=&Loc_D(R)&{\\rm otherwise}.\\end{array}$ The set of locations for each non-empty region is the set of paths to all leaves of some finite binary tree.",
"An induction argument is used to show the following properties of $Loc_D$ that ensure this: If $R\\le S$ for $R,S\\subseteq \\lbrace +,-\\rbrace ^*$ , then $Loc_D(R)\\le Loc_D(S)$ .",
"Let $S=\\lbrace Loc_D(R)\\:\\vert \\:R\\in \\lbrace +,-\\rbrace ^k\\wedge D_R\\ne \\emptyset \\rbrace $ for $k>0$ be a set of location paths for a given $k$ .",
"For any $p\\in \\lbrace l,r\\rbrace ^*$ there exists $q\\in S$ such that either $p\\le q$ or $q\\le p$ .",
"That is, every potential path is represented by some finite path.",
"Finally, for all $p,q\\in S$ if $p\\le q$ then $p=q$ .",
"This means that a unique leaf is associated with each location.",
"To find the tree for a given element $d_k$ in the enumeration, apply the following rules to each $R\\in \\lbrace +,-\\rbrace ^{k-1}$ .",
"If $D_{R-}\\ne \\emptyset $ then the leaf for path $Loc_D(R-)$ is labeled $\\top $ .",
"If $D_{R+}\\ne \\emptyset $ then the leaf for path $Loc_D(R+)$ is labeled $\\Delta $ .",
"These rules are used to assign a tree in ${\\bf U}$ , which is then reduced using the reduction rules, for each element in the enumeration of ${\\bf D}$ .",
"To see that the top element is never assigned by these rules, note that some region of the form $R+$ for every length $k$ must be non-empty since it must contain the element $e_k$ being embedded.",
"Returning to the example, the location function defines paths for these elements as follows: $\\begin{array}{lclclcl}Loc_D(+)&=&l&\\:\\:\\:\\:&Loc_D(+-+)&=&ll\\\\Loc_D(-)&=&r&&Loc_D(+--)&=&lr\\\\Loc_D(+-)&=&l&&Loc_D(-+-)&=&rl\\\\Loc_D(-+)&=&rl&&Loc_D(---)&=&rr\\\\Loc_D(--)&=&rr\\end{array}$ The trees generated for each of the elements are: $\\begin{array}{lcl}d_0&\\mapsto &\\Delta \\\\d_1&\\mapsto &\\langle \\Delta ,\\top \\rangle \\\\d_2&\\mapsto &\\langle \\top ,\\langle \\Delta ,\\top \\rangle \\rangle \\\\d_3&\\mapsto & \\langle \\langle \\Delta ,\\top \\rangle ,\\langle \\top ,\\top \\rangle \\rangle \\\\&\\mapsto & \\langle \\langle \\Delta ,\\top \\rangle ,\\top \\rangle \\end{array}$ To verify that the space generated is a valid sub-space, we must verify that the bottom element is mapped to $\\bot _U$ and that the consistency and lub relations are maintained.",
"The tree $\\Delta $ is clearly assigned to $X_0$ , the bottom element for the basis being embedded, since there are no strings of length $-1$ .",
"The embedding preserves inconsistency of elements by forcing the lub of the embedded elements to be $\\top $ .",
"The $D_{R-}$ regions represent the elements that the element being embedded does not approximate.",
"Note that the $D_{R-}$ sets cause the $\\top $ element to be added as the leaf.",
"Since the $D_R$ sets are built using the approximation ordering, it is straightforward to see that the approximation ordering is preserved by the embedding.",
"Lubs are also maintained by the embedding, although the reduction is required to see that this is the case.",
"It should be clear that, if the domain ${\\cal D}$ is effectively given, the sub-space can be computed since the embedding procedure uses the relationships given in the presentation.",
"Finally, suppose that $a$ is a computable, finitary projection on ${\\cal U}$ .",
"From the proof of Theorem REF , the domain of this projection is characterized by the set $\\lbrace y\\in {\\bf U}\\:\\vert \\:y\\:a\\:y\\rbrace $ If $a$ is computable, the set of pairs for $a$ is recursively enumerable.",
"Thus, the set above is also recursively enumerable since equality among basis elements is decidable.",
"Thus, the domain given by the projection must also be effectively given.",
"$\\:\\:\\Box $ Thus, the domain ${\\cal U}$ is an effectively presented universal domain in which all other domains can be embedded.",
"The sub-domains of ${\\cal U}$ include ${\\cal U}\\rightarrow {\\cal U}$ , ${\\cal U}\\times {\\cal U}$ , etc.",
"These domains must be sub-domains of ${\\cal U}$ since they are effectively presented based on our earlier theorems.",
"Domain Constructors in ${\\cal U}$ The next step is to see how to define the constructors commonly used.",
"Definition 8.9: [Domain Constructors] Let the computable projection pair, $i_+:{\\cal U}+{\\cal U}\\rightarrow {\\cal U}~{\\rm and}~j_+:{\\cal U}\\rightarrow {\\cal U}+{\\cal U}$ be fixed.",
"Fix suitable projection pairs $i_\\times ,j_\\times ,i_\\rightarrow $ , and $j_\\rightarrow $ as well.",
"Define $\\begin{array}{lcl}a+b&=&cond\\circ \\langle which,i_+\\circ in_0\\circ a\\circ out_0, i_+\\circ in_1\\circ b\\circ out_1\\rangle \\circ j_+\\\\a\\times b&=&i_x\\circ \\langle a\\circ proj_0,b\\circ proj_1\\rangle \\circ j_x\\\\a\\rightarrow b&=&i_\\rightarrow \\circ (\\lambda f.b\\circ f\\circ a)\\circ j_\\rightarrow \\end{array}$ for all $a,b:{\\cal U}\\rightarrow {\\cal U}$ .",
"From earlier theorems, we know that these combinators are all computable over an effectively presented domain.",
"The next theorem characterizes the effect these combinators have on projection functions.",
"Theorem 8.10: If $a,b:{\\cal U}\\rightarrow {\\cal U}$ are projections, then so are $a+b$ , $a\\times b$ , and $a\\rightarrow b$ .",
"If $a$ and $b$ are finitary, then so are the compound projections.",
"Proof Since $a$ and $b$ are retractions, $a=a\\circ a$ and $b=b\\circ b$ .",
"Then for $a\\times b$ using the definition of $\\times $ , $\\begin{array}{lcl}(a\\times b)\\circ (a\\times b)&=&i_x\\circ \\langle a\\circ proj_0,b\\circ proj_1\\rangle \\circ \\langle a\\circ proj_0,b\\circ proj_1\\rangle \\circ j_x\\\\&=&i_x\\circ \\langle a\\circ a\\circ proj_0,b\\circ b\\circ proj_1\\rangle \\circ j_x\\\\&=& a\\times b\\end{array}$ Thus, $a\\times b$ is a retraction.",
"The other cases follow similarly.",
"Since $a$ and $b$ are projections, $a,b\\subseteq {\\sf I}_U$ (denoted simply ${\\sf I}$ for the remainder of the proof).",
"Using the definition for $+$ along with the above relation and the definition of projection pairs, we can see that $a+b\\subseteq {\\sf I}+{\\sf I}=i_+\\circ j_+ \\subseteq {\\sf I}$ Thus, $a+b$ is a projection.",
"The other cases follow similarly.",
"To show that the projections are finitary, we must show that the fixed point sets are isomorphic to a domain.",
"Since $a$ and $b$ are assumed finitary, their fixed point sets are isomorphic to $\\begin{array}{lcl}D_a&=&\\lbrace x\\in {\\bf U}\\:\\vert \\:x\\:a\\: x\\rbrace \\\\D_b&=&\\lbrace y\\in {\\bf U}\\:\\vert \\:y\\:b\\: y\\rbrace \\end{array}$ We wish to show that ${\\cal D}_a\\rightarrow {\\cal D}_b\\approx {\\cal D}_{a\\rightarrow b}$ .",
"By the definition of the $\\rightarrow $ constructor, the fixed point set of $a\\rightarrow b$ over ${\\cal U}$ is the same as the fixed point set of $\\lambda f.b\\circ f\\circ a$ on ${\\cal U}\\rightarrow {\\cal U}$ .",
"(Hint: $i_\\rightarrow $ and $j_\\rightarrow $ set up the isomorphism.)",
"So, the fixed points for $f:{\\cal U}\\rightarrow {\\cal U}$ are of the form: $f=b\\circ f\\circ a$ We can think of $a$ as a function in ${\\cal U}\\rightarrow {\\cal D}_a$ and define the other half of the projection pair as $i_a:{\\cal D}_a\\rightarrow {\\cal U}$ where $i_a\\circ a = a$ and $a\\circ i_a=i_a$ .",
"Define a function $i_b$ for the projection pair for $b$ similarly.",
"For some $g:{\\cal D}_a\\rightarrow {\\cal D}_b$ let $f=i_b\\circ g\\circ a$ Substituting this definition for $f$ yields $b\\circ f\\circ a = b\\circ i_b\\circ g\\circ a\\circ a = i_b\\circ g \\circ a = f$ by the definition of $i_b$ and since $a$ is a retraction by assumption.",
"Conversely, for a function $f$ such that $i_b\\circ g\\circ a= f$ , let $g=b\\circ f\\circ i_a$ Substituting again, $i_b\\circ g\\circ a = i_b\\circ g\\circ f\\circ i_a\\circ a = b\\circ f\\circ a = f$ Thus, there is an order preserving isomorphism between $g:{\\cal D}_a\\rightarrow {\\cal D}_b$ and the functions $f=b\\circ f\\circ a$ .",
"The proofs of the isomorphisms for the other constructs are similar.",
"$\\:\\:\\Box $ Thus, the sub-domain relationship with the universal domain has been stated in terms of finitary projections over the universal domain.",
"In addition, all the domain constructors have been shown to be computable combinators on the domain of these finitary projections.",
"Recalling that all computable maps have computable fixed points, the standard fixed point method can be used to solve domain equations of all kinds if they can be defined on projections.",
"Returning to the $\\lambda $ -calculus for a moment, all objects in the $\\lambda $ -calculus are considered functions.",
"Since ${\\cal U}\\rightarrow {\\cal U}$ is a part of ${\\cal U}$ , every object in the $\\lambda $ -calculus is also an object of ${\\cal U}$ .",
"Transposing some of the familiar notation, where the old notation appears on the left, the new combinators are defined as follows: $\\begin{array}{lcl}which(z)=which(j_+(z))\\\\in_i(x)=i_+(in_i(x))~{\\rm where}~i=0,1\\\\out_i(x)=out_i(j_+(x))~{\\rm where}~i=0,1\\\\\\langle x,y\\rangle =i_x(\\langle x,y\\rangle )\\\\proj_i=proj_i(j_x(z))~{\\rm where}~i=0,1\\\\u(x) = j_\\rightarrow (u)(x)\\\\\\lambda x.\\tau =i_\\rightarrow (\\lambda x.\\tau )\\end{array}$ Thus, all functions, all constants, all combinators, and all constructs are elements of ${\\cal U}$ .",
"Indeed, everything computable is an element of ${\\cal U}$ .",
"Elements in ${\\cal U}$ play multiple roles by representing different objects under different projections.",
"While this notion may be difficult to get used to, there are many advantages, both notational and conceptual.",
"Exercises Exercise 8.11: A retraction $a:{\\cal D}\\rightarrow {\\cal D}$ is a closure operator iff ${\\sf I}_D\\subseteq a$ as relations.",
"On a domain like ${\\cal P}({{N}})$ , give some examples of closure operators.",
"(Hint: Close up the integers under addition.",
"Is this continuous on ${\\cal P}({{N}})$ ?)",
"Prove in general that for any closure $a:{\\cal D}\\rightarrow {\\cal D}$ , the fixed point set of $a$ is always a finitary domain.",
"(Hint: Show that the fixed point set is closed as required for a domain.)",
"What are the finite elements of the fixed point set?",
"Exercise 8.12: Give a direct proof that the domain $\\lbrace X\\:\\vert \\:X\\lhd {\\cal D}\\rbrace $ is effectively presented if ${\\cal D}$ is.",
"(Hint: The finite elements of the domain correspond exactly to the finite domains $X\\lhd {\\cal D}$ .)",
"In the case of ${\\cal D}={\\cal U}$ , show that the computable elements of the domain correspond exactly to the effectively presented domains (up to effective isomorphism).",
"Exercise 8.13: For finitary projections $a:{\\cal E}\\rightarrow {\\cal E}$ , write ${\\cal D}_a=\\lbrace x\\in {\\cal E}\\:\\vert \\:x\\:a\\:x\\rbrace $ Show that for any two such projections $a$ and $b$ , that $a\\subseteq b \\iff {\\cal D}_a\\lhd {\\cal D}_b$ Exercise 8.14: Find another universal domain that is not isomorphic to ${\\cal U}$ .",
"Exercise 8.15: Prove the remaining cases in Theorem REF .",
"Exercise 8.16: Suppose $S$ and $T$ are two binary constructors on domains that can be made into computable operators on projections over the universal domain.",
"Show that we can find a pair of effectively presented domains such that $D\\approx S(D,E)~{\\rm and}~E\\approx T(D,E).$ Exercise 8.17: Using the translations shown after the proof of Theorem REF , show how the whole typed-$\\lambda $ -calculus can be translated into ${\\cal U}$ .",
"(Hint: for $f:{\\cal D}_a\\rightarrow {\\cal B}$ , write $f=b\\circ f\\circ a$ for finitary projections $a$ and $b$ .",
"For $\\lambda x^{{\\cal D}_a}.\\sigma $ , write $\\lambda x.b(\\sigma ^{\\prime }[a(x)/x])$ where $\\sigma ^{\\prime }$ is the translation of $\\sigma $ into the untyped $\\lambda $ -calculus.",
"Be sure that the resulting term has the right type.)",
"Exercise 8.18: Show that the basis presented for the universal domain ${\\bf U}$ is indeed a finitary basis and that it has an effective presentation.",
"Exercise 8.19: Work out the embedding for the other enumerations for the example given in the proof of Theorem REF ."
],
[
"Domain Constructors",
"Now that the notion of a domain has been defined, we need to develop convenient methods for constructing specific domains.",
"The strategy that we will follow is to define the simplest domains (i.e., flat domains) directly (as initial or term algebras) and to construct more complex domains by applying domain constructors to simpler domains.",
"Since domains are completely determined by finitary bases, we will focus on how to construct composite finitary bases from simpler ones.",
"These constructions obviously determine corresponding constructions on domains.",
"The two most important constructions on finitary bases are (1) Cartesian products of finitary bases and (2) approximable mappings on finitary bases constructed using a function-space constructor."
],
[
"Cartesian Products",
"Definition 3.1: [Product Basis] Let ${\\bf D}$ and ${\\bf E}$ be the finitary bases generating domains ${\\cal D}$ and ${\\cal E}$ .",
"The product basis, ${\\bf D\\times E}$ , is the partial order consisting of the universe $D\\times E = \\lbrace [d,e]\\:\\vert \\:d\\in {\\bf D},e\\in {\\bf E}\\rbrace $ and the approximation ordering $[d,e] \\sqsubseteq [i,j] \\iff d\\sqsubseteq _D i~{\\rm and}~e\\sqsubseteq _E j.$ Theorem 3.2: The product basis of two finitary bases, as defined above, is a finitary basis.",
"Proof Let ${\\bf D}$ and ${\\bf E}$ be finitary bases and let ${\\bf D\\times E}$ be defined as above.",
"Since D and E are countable, the universe of ${\\bf D}\\times {\\bf E}$ must be countable.",
"It is easy to show that ${\\bf D}\\times {\\bf E}$ is a partial order.",
"By the construction, the bottom element of the product basis is $[\\bot _D,\\bot _E]$ .",
"For any finite bounded subset $R$ of ${\\bf D}\\times {\\bf E}$ where $R=\\lbrace [d_i,e_i]\\rbrace $ , the lub of $R$ is $[\\sqcup \\lbrace d_i\\rbrace ,\\sqcup \\lbrace e_i\\rbrace ]$ which must be defined since D and E are finitary bases and for $R$ to be bounded, each of the component sets must be bounded.",
"$\\:\\:\\Box $ It is straightforward to define projection mappings on product bases, corresponding to the standard projection functions defined on Cartesian products of sets.",
"Definition 3.3: [Projection Mappings] For a finitary basis ${\\bf D}\\times {\\bf E}$ , projection mappings ${\\sf {P}}_0 \\subseteq ({\\bf D}\\times {\\bf E}) \\times {\\bf D}$ and $P_1 \\subseteq ({\\bf D}\\times {\\bf E}) \\times {\\bf E}$ are the relations defined by the rules $[d,e] \\: P_0 \\:d^{\\prime } & \\iff & d^{\\prime } \\sqsubseteq _D d \\\\ [d,e] \\: P_1 \\:e^{\\prime } & \\iff & e^{\\prime } \\sqsubseteq _E e$ where $d$ ,$d^{\\prime }$ are arbitrary elements of ${\\bf D}$ and $e$ , $e^{\\prime }$ are arbitrary elements of ${\\bf E}$ .",
"Let ${\\bf A}$ , ${\\bf D}$ , and ${\\bf E}$ be finitary bases and let $F \\subseteq {\\bf A}\\times {\\bf D}$ and $G \\subseteq {\\bf A}\\times {\\bf E}$ be approximable mappings.",
"The paired mapping $\\langle F,G\\rangle \\subseteq {\\bf A}\\times ({\\bf D}\\times {\\bf E})$ is the relation defined by the rule $a\\:\\langle F,G\\rangle \\:[d,e] \\iff a\\:F\\:d\\wedge a\\:G\\:e$ for all $a\\in {\\bf A},d\\in {\\bf D}$ , and all $e\\in {\\bf E}$ .",
"It is an easy exercise to show that projection mappings and paired mappings are approximable mappings (as defined in the previous section).",
"Theorem 3.4: The mappings ${\\sf {P}}_0$ , ${\\sf {P}}_1$ , and $\\langle F,G\\rangle $ are approximable mappings if $F,G$ are.",
"In addition, ${\\sf {P}}_0\\circ \\langle F,G\\rangle = F$ and ${\\sf {P}}_1\\circ \\langle F,G\\rangle = G$ .",
"For $[d,e]\\in {\\bf D}\\times {\\bf E}$ and $d^{\\prime } \\in {\\bf D}$ , $[d,e] \\:{\\sf {P}}_0 \\: d^{\\prime } \\iff d^{\\prime } \\sqsubseteq d$ .",
"For $[d,e]\\in {\\bf D}\\times {\\bf E}$ and $e^{\\prime } \\in {\\bf E}$ , $[d,e] \\:{\\sf {P}}_1 \\: e^{\\prime } \\iff e^{\\prime } \\sqsubseteq d$ .",
"For approximable mapping $H \\subseteq {\\bf A}\\times ({\\bf D}\\times {\\bf E})$ , $H=\\langle ({\\sf {P}}_0\\circ H),({\\sf {P}}_1\\circ H)\\rangle $ .",
"For $a\\in {\\bf A}$ and $[d,e] \\in {\\bf D}\\times {\\bf E}$ , $[a, [d,e]] \\in \\langle F,G\\rangle \\iff [a,d] \\in F \\wedge [a,e] \\in G$ .",
"Proof The proof is left as an exercise to the reader.",
"$\\:\\:\\Box $ The projection mappings and paired mappings on finitary bases obviously correspond to continuous functions on the corresponding domains.",
"We will denote the continous functions corresponding to ${\\sf {P}}_0$ and ${\\sf {P}}_1$ by the symbols $p_1$ and $p_2$ .",
"Similarly, we will denote the function corresponding to the paired mapping $\\langle F, G\\rangle $ by the expression $\\langle f, g \\rangle $ .",
"It should be clear that the definition of projection mappings and paired mappings can be generalized to products of more than two domains.",
"This generalization enables us to treat a multi-ary continous function (or approximable mapping) as a special form of a unary continuous function (or approximable mapping) since multi-ary inputs can be treated as single objects in a product domain.",
"Moreover, it is easy to show that a relation $R \\subseteq ({\\bf A_1} \\times \\ldots \\times {\\bf A_n}) \\times {\\bf B}$ of arity $n+1$ (as in Exercise REF ) is an approximable mapping iff every restriction of $R$ to a single input argument (by fixing the other input arguments) is an approximable mapping.",
"Theorem 3.5: A relation $F \\subseteq ({\\bf A}\\times {\\bf B})\\times {\\bf C}$ is an approximable mapping iff for every $a\\in {\\bf A}$ and every $b\\in {\\bf B}$ , the derived relations $F_{a,*} & = & {\\lbrace { [y,z] \\; \\:\\vert \\:\\; [[a,y],z] \\in F}\\rbrace }\\\\F_{*,b} & = & {\\lbrace { [x,z] \\; \\:\\vert \\:\\; [[x,b],z] \\in F}\\rbrace }$ are approximable mappings.",
"Proof Before we prove the theorem, we need to introduce the class of constant relations ${\\sf {K}}_e \\subseteq {\\bf D}\\times {\\bf E}$ for arbitrary finitary bases ${\\bf D}$ and ${\\bf E}$ and show that they are approximable mappings.",
"Lemma 3.6: For each $e \\in {\\bf E}$ , let the “constant” relation ${\\sf {K}}_e \\subseteq {\\bf D} \\times {\\bf E}$ be defined by the equation ${\\sf {K}}_e = {\\lbrace {[d,e^{\\prime }] \\; \\:\\vert \\:\\; d \\in {\\bf D}, e^{\\prime } \\sqsubseteq e}\\rbrace }\\,.$ In other words, $d \\: {\\sf {K}}_e \\: e^{\\prime } \\; \\iff \\; e^{\\prime } \\sqsubseteq e \\, .$ For $e\\in {\\bf E}$ , the constant relation ${\\sf {K}}_e \\subseteq {\\bf D} \\times {\\bf E}$ is approximable.",
"Proof (lemma) The proof of this lemma is left to the reader.",
"$\\:\\:\\Box $ To prove the “if” direction of the theorem, we observe that we can construct the relations $F_{a,*}$ and $F_{*,b}$ for all $a \\in {\\bf A}$ and $b \\in {\\bf B}$ by composing and pairing primitive approximable mappings.",
"In particular, $F_{a,*}$ is the relation $F \\circ \\langle K_a, {\\sf I}_B \\rangle $ where ${\\sf I}_B$ denotes the identity relation on ${\\bf B}$ .",
"Similary, $F_{*,b}$ is the relation $F \\circ \\langle {\\sf I}_A, K_b\\rangle $ where ${\\sf I}_A$ denotes the identity relation on ${\\bf A}$ .",
"To prove the “only-if” direction, we assume that for all $a \\in {\\bf A}$ and $b \\in {\\bf B}$ , the relations $F_{a,*}$ and $F_{*,b}$ are approximable.",
"We must show that the four closure properties for approximable mappings hold for $F$ .",
"Since $F_{\\bot _A,*}$ is approximable, $[\\bot _B,\\bot _C] \\in F_{\\bot _A,*}$ , which implies that $[[\\bot _A,\\bot _B],\\bot _C] \\in F$ .",
"If $[[x,y],z] \\in F$ and $[[x,y],z^{\\prime }] \\in F$ , then $[y,z] \\in F_{x,*}$ and $[y,z^{\\prime }] \\in F_{x,*}$ .",
"Since $F_{x,*}$ is approximable, $[y,z \\sqcup z^{\\prime }] \\in F_{x,*}$ , implying $[[x,y],z \\sqcup z^{\\prime }] \\in F$ .",
"If $[[x,y],z] \\in F$ and $z^{\\prime } \\sqsubseteq z$ , then $[y,z] \\in F_{x,*}$ .",
"Since $F_{x,*}$ is approximable, $[y,z^{\\prime }] \\in F_{x,*}$ , implying $[[x,y],z^{\\prime }] \\in F$ .",
"If $[[x,y],z] \\in F$ and $[x,y] \\sqsubseteq [x^{\\prime },y^{\\prime }]$ , then $[y,z] \\in F_{x,*}$ , $x \\sqsubseteq x^{\\prime }$ , and $y \\sqsubseteq y^{\\prime }$ .",
"Since $F_{x,*}$ is approximable, $[y^{\\prime },z] \\in F_{x,*}$ , implying $[x,y^{\\prime }],z] \\in F$ , which is equivalent to $[x,z] \\in F_{*,y^{\\prime }}$ .",
"Since $F_{*,y^{\\prime }}$ is approximable, $[x^{\\prime },z] \\in F_{*,y^{\\prime }}$ , implying $[[x^{\\prime },y^{\\prime }],z] \\in F$ .",
"$\\:\\:\\Box $ The same result can be restated in terms of continous functions.",
"Theorem 3.7: A function of two arguments, $f:{\\cal A}\\times {\\cal B}\\rightarrow {\\cal C}$ is continuous iff for every $a\\in {\\cal A}$ and every $b\\in {\\cal B}$ , the unary functions $x\\mapsto f[a,x]~{\\rm and}~y\\mapsto f[y,b]$ are continuous.",
"Proof Immediate from the previous theorem and the fact that the domain of approximable mappings over $({\\bf A} \\times {\\bf B}) \\times {\\bf C}$ is isomorphic to the domain of continuous functions over $({\\cal A} \\times {\\cal B}) \\times {\\cal C}$ .",
"$\\:\\:\\Box $ Multiary Function Composition The composition of functions, as defined in Theorem REF , can be generalized to functions of several arguments.",
"But we need some new syntactic machinery to describe more general forms of function composition.",
"Definition 3.8: [Cartesian Types] Let $S$ be a set of symbols used to denote primitive types.",
"The set $S^*$ of Cartesian types over $S$ consists of the set of expressions denoting all finite non-empty Cartesian products over primitive types in $S$ : $S^* ::= S \\; \\:\\vert \\:\\; S \\times S \\; \\:\\vert \\:\\; \\ldots \\; .$ A signature $\\Sigma $ is a pair $\\langle S, O \\rangle $ consisting of a set $S$ of type names ${\\lbrace {s_1, \\ldots , s_m}\\rbrace }$ used to denote domains and a set $O={\\lbrace {o_i^{\\rho _i\\rightarrow \\sigma _i}\\; \\:\\vert \\:\\; 1 \\le i \\le m, \\; \\rho _i \\in S^*, \\; \\sigma _i \\in S}\\rbrace }$ of function symbols used to denote first-order functions over the domains $S$ .",
"Let $V={\\lbrace {v_i^{\\tau } \\; \\:\\vert \\:\\; \\tau \\in S,\\; i \\in {N}}\\rbrace }$ be a countably-infinite set of symbols (variables) distinct from the symbols in $\\Sigma $ .",
"The typed expressions over $\\Sigma $ (denoted ${\\cal Ex}(\\Sigma )$ ) is the set of “typed” terms determined by the following inductive definition: $v_i^{\\tau } \\in V$ is a term of type $\\tau $ , for $M_1^{\\tau _1}, \\dots , M_n^{\\tau _n} \\in {\\cal Ex}(\\Sigma )$ and $o^{(\\tau _1 \\times \\dots \\times \\tau _n) \\rightarrow \\tau _0} \\in O$ then $o^{(\\tau _1 \\times \\dots \\times \\tau _n) \\rightarrow \\tau _0}(M_1^{\\tau _1}, \\dots ,M_n^{\\tau _n})^{\\tau _0}$ is a term of type $\\tau _0$ .",
"We will restrict our attention to terms where every instance of a variable $v_i$ in a term has the same type $\\tau $ .",
"To simplify notation, we will drop the type superscripts from terms whenever they can be easily inferred from context.",
"Definition 3.9: [Finitary Algebra] A finitary algebra with signature $\\Sigma $ is a function ${\\bf A}$ mapping each primitive type $\\tau \\in S$ to a finitary basis ${{\\bf A}}{[\\!",
"[}\\tau {]\\!",
"]}$ , each operation type $\\tau ^1 \\times \\dots \\times \\tau ^n \\in S^*$ to the finitary basis ${{\\bf A}}{[\\!",
"[}\\tau ^1{]\\!]}",
"\\times \\dots \\times {{\\bf A}}{[\\!",
"[}\\tau ^n{]\\!",
"]}$ , each function symbol $o_i^{\\rho _i \\rightarrow \\sigma _i} \\in O$ to an approximable mapping ${{\\bf A}}{[\\![}o_i{]\\!]}",
"\\subseteq ({{\\bf A}}{[\\!",
"[}\\rho _i{]\\!]}",
"\\times {{\\bf A}}{[\\!",
"[}\\sigma _i{]\\!",
"]})$ .",
"(Recall that ${{\\bf A}}{[\\!",
"[}\\rho _i{]\\!",
"]}$ is a product basis.)",
"Definition 3.10: [Closed Term] A term $M \\in {\\cal Ex}(\\Sigma )$ is closed iff it contains no variables in $V$ .",
"The finitary algebra ${\\bf A}$ implicitly assigns a meaning to every closed term $M$ in ${\\cal Ex}(\\Sigma )$ .",
"This extension is inductively defined by the equation: ${{\\bf A}}{[\\!",
"[}o[M_1,\\dots ,M_n]{]\\!]}",
"= {{\\bf A}}{[\\![}o{]\\!",
"]}[{{\\bf A}}{[\\![}M_1{]\\!",
"]}, \\dots , {{\\bf A}}{[\\![}M_n{]\\!]}]",
"=$ ${\\lbrace { b_0 \\; \\:\\vert \\:\\; \\exists [b_1,\\dots ,b_n] \\in {{\\bf A}}{[\\!",
"[}\\rho _i{]\\!]}",
"\\;[b_1,\\dots ,b_n] {{\\bf A}}{[\\![}o{]\\!]}",
"b_0}\\rbrace } \\, .$ We can extend ${\\bf A}$ to terms $M$ with free variables by implicitly abstracting over all of the free variables in $M$ .",
"Definition 3.11: [Meaning of Terms] Let $M$ be a term in ${\\cal Ex}{(\\Sigma )}$ and let $l = x_1^{\\tau _1}, \\dots , x_n^{\\tau _n}$ be a list of distinct variables in $V$ containing all the free variables of $M$ .",
"Let ${\\bf A}$ be a finitary algebra with signature $\\Sigma $ and for each tuple $[d_1, \\dots , d_n] \\in {{\\bf A}}{[\\!",
"[}\\tau _1{]\\!]}",
"\\times \\dots \\times {{\\bf A}}{[\\!",
"[}\\tau _n{]\\!",
"]}$ , let ${\\bf A}_{\\lbrace {x_1 := d_1,\\dots , x_n := d_n}\\rbrace }$ denote the algebra ${\\bf A}$ extended by defining ${{\\bf A}}{[\\![}x_i{]\\!]}",
"= d_i$ for $1 \\le i \\le n$ .",
"The meaning of $M$ with respect to $l$ , denoted ${{\\bf A}}{[\\!",
"[}x_1^{\\tau _1}, \\dots , x_n^{\\tau _n} \\, \\mapsto \\,M{]\\!",
"]}$ , is the relation $F_M \\subseteq ({{\\bf A}}{[\\!",
"[}\\tau _1{]\\!]}",
"\\times {{\\bf A}}{[\\!",
"[}\\tau _n{]\\!]})",
"\\times {{\\bf A}}{[\\!",
"[}\\tau _0{]\\!",
"]}$ defined by the equation: $F_M[d_1, \\dots , d_n] = {{\\bf A}_{\\lbrace {x_1 := d_1,\\dots ,x_1 := d_n}\\rbrace }}{[\\![}M{]\\!",
"]}$ The relation denoted by ${{\\bf A}}{[\\!",
"[}x_1^{\\tau _1}, \\dots , x_n^{\\tau _n} \\, \\mapsto \\, M{]\\!",
"]}$ is often called a substitution.",
"The following theorem shows that the relation ${{\\bf A}}{[\\!",
"[}x_1^{\\tau _1}, \\dots , x_n^{\\tau _n} \\, \\mapsto \\, M{]\\!",
"]}$ is approximable.",
"Theorem 3.12: (Closure of continuous functions under substitution) Let $M$ be a term in ${\\cal Ex}{(\\Sigma )}$ and let $l = x_1^{\\tau _1}, \\dots , x_n^{\\tau _n}$ be a list of distinct variables in $V$ containing all the free variables of $M$ .",
"Let ${\\bf A}$ be a finitary algebra with signature $\\Sigma $ .",
"Then the relation $F_M$ denoted by the expression $x_1^{\\tau _1}, \\dots , x_n^{\\tau _n} \\mapsto M $ is approximable.",
"Proof The proof proceeds by induction on the structure of $M$ .",
"The base cases are easy.",
"If $M$ is a variable $x_i$ , the relation $F_M$ is simply the projection mapping ${\\sf {P}}_i$ .",
"If $M$ is a constant $c$ of type $\\tau $ , then $F_M$ is the contant relation ${\\sf {K}}_c$ of arity $n$ .",
"The induction step is also straightforward.",
"Let $M$ have the form $g[M_1^{\\sigma _1},\\ldots ,M_m^{\\sigma _m}]$ .",
"By the induction hypothesis, $x_1^{\\tau _1}, \\dots , x_n^{\\tau _n} \\mapsto M_i^{\\sigma _i} $ denotes an approximable mapping $F_{M_i} \\subseteq ({{\\bf A}}{[\\!",
"[}\\tau _1{]\\!]}",
"\\times {{\\bf A}}{[\\!",
"[}\\tau _n{]\\!]})",
"\\times {{\\bf A}}{[\\!",
"[}\\sigma _i{]\\!",
"]}$ .",
"But $F_M$ is simply the composition of the approximable mapping ${{\\bf A}}{[\\![}g{]\\!",
"]}$ with the mapping $\\langle F_{M_1}, \\dots F_{M_m} \\rangle $ .",
"Theorem REF tells us that the composition must be approximable.",
"$\\:\\:\\Box $ The preceding generalization of composition obviously carries over to continuous functions.",
"The details are left to the reader.",
"Function Spaces The next domain constructor, the function-space constructor, allows approximable mappings (or, equivalently, continuous functions) to be regarded as objects.",
"In this framework, standard operations on approximable mappings such as application and composition are approximable mappings too.",
"Indeed, the definitions of ideals and of approximable mappings are quite similar.",
"The space of approximable mappings is built by looking at the actions of mappings on finite sets, and then using progressively larger finite sets to construct the mappings in the limit.",
"To this end, the notion of a finite step mapping is required.",
"Definition 3.13: [Finite Step Mapping] Let ${\\bf A}$ and ${\\bf B}$ be finitary bases.",
"An approximable mapping $F \\subseteq {\\bf A} \\times {\\bf B}$ is a finite step mapping iff there exists a finite set $S\\subseteq {\\bf A}\\times {\\bf B}$ and $F$ is the least approximable mapping such that $S \\subseteq F$ .",
"It is easy to show that for every consistent finite set $S \\subseteq {\\bf A}\\times {\\bf B}$ , a least mapping $F$ always exists.",
"$F$ is simply the closure of $S$ under the four conditions that an approximable mapping must satisfy.",
"The least approximable mapping respecting the empty set is the relation ${\\lbrace {\\langle a, \\bot _B \\rangle \\; \\:\\vert \\:\\; a \\in {\\bf A}}\\rbrace }$ .",
"The space of approximable mappings is built from finite step mappings.",
"Definition 3.14: [Partial Order of Finite Step Mappings] For finitary bases ${\\bf A}$ and ${\\bf B}$ the mapping basis is the partial order ${\\bf A}\\Rightarrow {\\bf B}$ consisting of the universe of all finite step mappings, and the approximation ordering $F \\sqsubseteq G \\iff \\forall a\\in {\\bf A} \\; F(a) \\sqsubseteq _B G(a) \\, .$ The following theorem establishes that the constructor $\\Rightarrow $ maps finitary bases into finitary bases.",
"Theorem 3.15: Let ${\\bf A}$ and ${\\bf B}$ be finitary bases.",
"Then, the mapping basis ${\\bf A}\\Rightarrow {\\bf B}$ is a finitary basis.",
"Proof Since the elements are finite subsets of a countable set, the basis must be countable.",
"It is easy to confirm that ${\\bf A}\\Rightarrow {\\bf B}$ is a partial order; this task is left to the reader.",
"We must show that every finite consistent subset of ${\\bf A}\\Rightarrow {\\bf B}$ has a least upper bound in ${\\bf A}\\Rightarrow {\\bf B}$ .",
"Let $§$ be a finite consistent subset of the universe of ${\\bf A}\\Rightarrow {\\bf B}$ .",
"Each element of $§$ is a set of ordered pairs $\\langle a, b \\rangle $ that meets the approximable mapping closure conditions.",
"Since $§$ is consistent, it has an upper bound $§^{\\prime } \\in {\\bf A}\\Rightarrow {\\bf B}$ .",
"Let $U = \\bigcup §$ .",
"Clearly, $U \\subseteq §^{\\prime }$ .",
"But $U$ may not be approximable.",
"Let $S$ be the intersection of all relations in ${\\bf A}\\Rightarrow {\\bf B}$ above $§$ .",
"Clearly $U \\subseteq S$ , implying $S$ is a superset of every element of $§$ .",
"It is easy to verify that $S$ is approximable, because all the approximable mapping closure conditions are preserved by infinite intersections.",
"$\\:\\:\\Box $ Definition 3.16: [Function Domain] We will denote the domain of ideals determined by the finitary basis ${\\bf A}\\Rightarrow {\\bf B}$ by the expression ${\\cal A}\\Rightarrow {\\cal B}$ .",
"The justification for this notation will be explained shortly.",
"Since the partial order of approximable mappings is isomorphic to the partial order of continuous functions, the preceding definitions and theorems about approximable mappings can be restated in terms of continuous functions.",
"Definition 3.17: [Finite Step Function] Let ${\\cal A}$ and ${\\cal B}$ be the domains determined by the finitary bases ${\\bf A}$ and ${\\bf B}$ , respectively.",
"A continuous function $f$ in ${\\cal A}\\rightarrow _c{\\cal B}$ is finite iff there exists a finite step mapping $F \\subseteq {\\bf A} \\times {\\bf B}$ such that $f$ is the function determined by $F$ .",
"Definition 3.18: [Function Basis] For domains ${\\cal A}$ and ${\\cal B}$ , the function basis is the partial order $({\\cal A}\\rightarrow _c{\\cal B})^0$ consisting of a universe of all finite step functions, and the approximation order $f\\sqsubseteq g \\iff \\forall a\\in {\\cal A} \\; f(a) \\sqsubseteq _{\\cal B}g(a) \\, .$ Corollary 3.19: (to Theorem REF ) For domains ${\\cal A}$ and ${\\cal B}$ , the function basis $({\\cal A}\\rightarrow _c{\\cal B})^0$ is a finitary basis.",
"We can prove that the domain constructed by generating the ideals over ${\\bf A}\\Rightarrow {\\bf B}$ is isomorphic to the partial order ${\\sf {Map}}({\\bf A},{\\bf B})$ of approximable mappings defined in Section 2.",
"This result is not surprising; it merely demonstrates that ${\\sf {Map}}({\\bf A},{\\bf B})$ is a domain and that we have identified the finite elements correctly in defining ${\\bf A}\\Rightarrow {\\bf B}$ .",
"Theorem 3.20: The domain of ideals determined by ${\\bf A}\\Rightarrow {\\bf B}$ is isomorphic to the partial order of the approximable mappings ${\\sf {Map}}({\\bf A},{\\bf B})$ .",
"Hence, ${\\sf {Map}}({\\bf A},{\\bf B})$ is a domain.",
"Proof We must establish an isomorphism between the domain determined by ${\\bf A}\\Rightarrow {\\bf B}$ and the partial order of mappings from ${\\bf A}$ to ${\\bf B}$ .",
"Let $h: {\\cal A}\\Rightarrow {\\cal B} \\rightarrow {\\sf {Map}}({\\bf A},{\\bf B})$ be the function defined by rule $h \\: {\\cal {F}}= \\bigcup {\\lbrace {F \\in {\\cal {F}}}\\rbrace } \\,.$ It is easy to confirm that the relation on the right hand side of the preceding equation is an approximable mapping: if it violated any of the closure properties, so would a finite approximation in ${\\cal {F}}$ .",
"We must prove that the function $h$ is one-to-one and onto.",
"To prove the former, we note that each pair of distinct ideals has a witness $\\langle a, b \\rangle $ that belongs to a set in one ideal but not in any set in the other.",
"Hence, the images of the two ideals are distinct.",
"The function $h$ is onto because every approximable mapping is the image of the set of finite step maps that approximate it.",
"$\\:\\:\\Box $ The preceding theorem can be restated in terms of continuous functions.",
"Corollary 3.21: (to Theorem REF ) The domain of ideals determined by the finitary basis $({\\cal A}\\rightarrow _c{\\cal B})^0$ is isomorphic to the partial order of continuous functions ${\\cal A} \\rightarrow _c {\\cal B}$ .",
"Hence, ${\\cal A} \\rightarrow _c {\\cal B}$ is a domain.",
"Now that we have defined the approximable mapping and continous function domain constructions, we can show that operators on maps and functions introduced in Section 2 are continuous functions.",
"Theorem 3.22: Given finitary bases, ${\\bf A}$ and ${\\bf B}$ , there is an approximable mapping $Apply:(({\\bf A}\\Rightarrow {\\bf B}) \\times {\\bf A}) \\times {\\bf B}$ such that for all $F:{\\bf A}\\Rightarrow {\\bf B}$ and $a\\in {\\bf A}$ , $Apply[F,a] = F(a)\\,.$ Recall that for any approximable mapping $G \\subseteq {\\bf C} \\times {\\bf D}$ and any element $c \\in {\\bf C}$ $G(c) = {\\lbrace {d \\; \\:\\vert \\:\\; c \\: G \\: d}\\rbrace }\\,.$ Proof For $F\\in ({\\bf A}\\Rightarrow {\\bf B})$ , $a\\in {\\bf A}$ and $b\\in {\\bf B}$ , define the $Apply$ relation as follows: $[F,a]\\:Apply\\:b \\iff a\\:F\\:b\\,.$ It is easy to verify that $Apply$ is an approximable mapping; this step is left to the reader.",
"From the definition of $Apply$ , we deduce $Apply[F,a] = {\\lbrace {b \\; \\:\\vert \\:\\; [F,a]\\: Apply \\: b}\\rbrace } ={\\lbrace {b \\; \\:\\vert \\:\\; a \\: F \\: b}\\rbrace } = F(a)\\,.$ $\\:\\:\\Box $ This theorem can be restated in terms of continuous functions.",
"Corollary 3.23: Given domains, ${\\cal A}$ and ${\\cal B}$ , there is a continuous function $apply:(({\\cal A}\\rightarrow _c{\\cal B}) \\times {\\cal A}) \\rightarrow _c {\\cal B}$ such that for all $f:{\\cal A}\\rightarrow _c{\\cal B}$ and $a\\in {\\cal A}$ , $apply[f,a] = f(a)\\,.$ Proof (of corollary).",
"Let $apply: (({\\cal A}\\rightarrow _c {\\cal B}) \\times {\\cal A}) \\rightarrow _c {\\cal B}$ be the continuous function (on functions rather than relations!)",
"corresponding to $Apply$ .",
"From the definition of $apply$ and Theorem REF which relates approximable mappings on finitary bases to continous functions over the corresponding domains, we know that $apply[f,{\\cal I}_A]=\\lbrace b\\in {\\bf B}\\:\\vert \\:\\exists F^{\\prime }\\in ({\\bf A}\\Rightarrow {\\bf B}) \\ ; \\exists a\\in {\\cal I}_A \\wedge F^{\\prime } \\subseteq F \\wedge [F^{\\prime },a]\\:Apply\\:b\\rbrace $ where $F$ denotes the approximable mapping corresponding to $f$ .",
"Since $f$ is the continuous function corresponding to $F$ , $f({\\cal I}_A) = \\lbrace b\\in {\\bf B}\\:\\vert \\:\\exists a\\in {\\cal I}_A \\; a\\:F\\:b\\rbrace $ So, by the definition of the $Apply$ relation, $apply[f,{\\cal I}_A]\\subseteq f({\\cal I}_A)$ .",
"For every $b\\in f({\\cal I}_A)$ , there exists $a \\in {\\cal I}_A$ such that $a\\:F\\:b$ .",
"Let $F^{\\prime }$ be the least approximable mapping such that $a \\: F^{\\prime } b$ .",
"By definition, $F^{\\prime }$ is a finite step mapping.",
"Hence $b \\in apply[f,{\\cal I}_A]$ , implying $f({\\cal I}_A) \\subseteq apply[f,{\\cal I}_A]$ .",
"Therefore, $f({\\cal I}_A) = apply[f,{\\cal I}_A]$ for arbitrary ${\\cal I}_A$ .",
"$\\:\\:\\Box $ The preceding theorem and corollary demonstrate that approximable mappings and continuous functions can operate on other approximable mappings or continuous functions just like other data objects.",
"The next theorem shows that the currying operation is a continuous function.",
"Definition 3.24: [The Curry Operator] Let A, B, and C be finitary bases.",
"Given an approximable mapping $G$ in the basis $({\\bf A}\\times {\\bf B})\\Rightarrow {\\bf C}$ , $Curry_G:{\\bf A}\\Rightarrow ({\\bf B}\\Rightarrow {\\bf C})$ is the relation defined by the equation $Curry_G(a) = {\\lbrace {F \\in {\\bf B}\\Rightarrow {\\bf C} \\; \\:\\vert \\:\\;\\forall [b,c] \\in F \\; [a,b] \\: G \\: c}\\rbrace }$ for all $a \\in {\\bf A}$ .",
"Similarly, given any continuous function $g: ({\\cal A}\\times {\\cal B})\\rightarrow _c{\\cal C}$ , $curry_g:{\\cal A}\\rightarrow _c({\\cal B}\\rightarrow _c{\\cal C})$ is the function defined by the equation $curry_g({\\cal I}_A) = (y \\mapsto g[{\\cal I}_A,y])\\,.$ By Theorem 2.7?",
"?, $(y \\mapsto g[{\\cal I}_A,y])$ is a continous function.",
"Lemma 3.25: $Curry_G$ is an approximable mapping and $curry_g$ is the continuous function determined by $Curry_G$ .",
"Proof A straightforward exercise.$\\:\\:\\Box $ It is more convenient to discuss the currying operation in the context of continuous functions than approximable mappings.",
"Theorem 3.26: Let $g \\in ({\\cal A}\\times {\\cal B})\\rightarrow _c{\\cal C}$ and $h \\in ({\\cal A}\\rightarrow _c({\\cal B}\\rightarrow _c{\\cal C})$ .",
"The $curry$ operation satisfies the following two equations: $apply\\circ \\langle curry_g\\circ p_0,p_1\\rangle & = & g\\\\curry_{apply\\circ \\langle h\\circ p_0,p_1\\rangle } & = & h\\, .$ In addition, the function $curry:({\\cal A}\\times {\\cal B}\\rightarrow {\\cal C})\\rightarrow ({\\cal A}\\rightarrow _c({\\cal B}\\rightarrow _c{\\cal C}))$ defined by the equation $curry(g)({\\cal I}_A)({\\cal I}_B) = curry_g({\\cal I}_A)({\\cal I}_B)$ is continuous.",
"Proof Let $g$ be any continuous function in the domain $({\\cal A}\\times {\\cal B})\\rightarrow _c{\\cal C}$ .",
"Recall that $curry_g(a) = (y \\mapsto g[a,y])\\,.$ Using this definition and the definition of operations in the first equation, we can deduce $\\begin{array}{lcl}apply\\circ \\langle curry_g\\circ p_0,p_1\\rangle [a,b]& = & apply[\\langle curry_g \\circ p_0, p_1 \\rangle [a,b]] \\\\& = & apply[(curry_g \\circ p_0)[a,b], p_1[a,b]] \\\\& = & apply[curry_g p_0[a,b], b] \\\\& = & apply[curry_g \\: a, b] \\\\& = & curry_g \\: a \\: b \\\\& = & g[a,b]\\,.\\end{array}$ Hence, the first equation holds.",
"The second equation follows almost immediately from the first.",
"Define $g^{\\prime }:({\\cal A}\\times {\\cal B})\\rightarrow _c{\\cal C}$ by the equation $g^{\\prime }[a,b] = h \\; a \\; b\\,.$ The function $g^{\\prime }$ is defined so that $curry_{g^{\\prime }} = h$ .",
"This fact is easy to prove.",
"For $a \\in {\\cal A}$ : $curry_{g^{\\prime }}(a) & = & (y \\mapsto g^{\\prime }[a,y])\\\\& = & (y \\mapsto h (a)(y))\\\\& = & h(a) \\, .$ Since $h = curry_{g^{\\prime }}$ , the first equation implies that $apply\\circ \\langle h\\circ p_0,p_1\\rangle & = &apply\\circ \\langle curry_{g^{\\prime }}\\circ p_0,p_1\\rangle \\\\& = & g^{\\prime }\\,.$ Hence, $curry_{apply}\\circ \\langle h\\circ p_0,p_1\\rangle = curry_{g^{\\prime }} = h\\,.$ These two equations show that $({\\cal A}\\times {\\cal B})\\rightarrow _c{\\cal C}$ is isomorphic to $({\\cal A}\\rightarrow _c({\\cal B}\\rightarrow _c{\\cal C})$ under the $curry$ operation.",
"In addition, the definition of $curry$ shows that $curry(g) \\sqsubseteq curry(g^{\\prime }) \\iff g \\sqsubseteq g^{\\prime }\\,.$ Hence, $curry$ is an isomorphism.",
"Moreover, $curry$ must be continuous.",
"$\\:\\:\\Box $ The same theorem can be restated in terms of approximable mappings.",
"Corollary 3.27: The relation $Curry_G$ satisfies the following two equations: $Apply\\circ \\langle Curry_G \\circ {\\sf {P}}_0,{\\sf {P}}_1\\rangle & = & G\\\\Curry_{Apply\\circ \\langle G\\circ {\\sf {P}}_0,{\\sf {P}}_1\\rangle } & = & G\\, .$ In addition, the relation $Curry:({\\bf A}\\times {\\bf B})\\Rightarrow {\\bf C})\\Rightarrow ({\\bf A} \\Rightarrow ({\\bf B}\\Rightarrow {\\bf C}))$ defined by the equation $Curry(G) = {\\lbrace {[a,F] \\; \\:\\vert \\:\\; a \\in {\\bf A}, \\:F \\in ({\\bf B}\\Rightarrow {\\bf C}), \\; \\forall [b,c] \\in F \\:[a,b]\\: G \\: c }\\rbrace }$ is approximable.",
"Table: Domains and Finitary BasesTable REF summarizes the main elements of the correspondence between domains and finitary bases.",
"Whenever convenient, in the following sections we take liberty to confuse corresponding notions.",
"Context and notation should make clear which category is meant.",
"Exercises Exercise 3.28: We assume that there is a countable basis.",
"Thus, the basis elements could without loss of generality be defined in terms of $\\lbrace 0,1\\rbrace ^*$ .",
"Show that the product space ${\\bf A}\\times {\\bf B}$ could be defined as a finitary basis over $\\lbrace 0,1\\rbrace ^*$ such that ${\\bf A}\\times {\\bf B}=\\lbrace [0a,1b]\\:\\vert \\:a\\in {\\bf A},b\\in {\\bf B}\\rbrace $ Give the appropriate definition for the elements in the domain.",
"Also show that there exists an approximable mapping $diag:{\\bf D}\\rightarrow {\\bf D}\\times {\\bf D}$ where $diag(x)= [x,x]$ for all $x\\in {\\cal D}$ .",
"Exercise 3.29: Establish some standard isomorphisms: ${\\bf A}\\times {\\bf B}\\approx {\\bf B}\\times {\\bf A}$ ${\\bf A}\\times ({\\bf B}\\times {\\bf C})\\approx ({\\bf A}\\times {\\bf B})\\times {\\bf C}$ ${\\bf A}\\approx {\\bf A^{\\prime }},{\\bf B}\\approx {\\bf B^{\\prime }}\\:\\Rightarrow \\:{\\bf A}\\times {\\bf B}\\approx {\\bf A^{\\prime }}\\times {\\bf B^{\\prime }}$ for all finitary bases.",
"Exercise 3.30: Let $B\\subseteq \\lbrace 0,1\\rbrace ^*$ be a finitary basis.",
"Define $B^\\infty =\\bigcup \\limits _{n=0}^\\infty 1^n0B$ Thus, $B^\\infty $ contains infinitely many disjoint copies of $B$ .",
"Now let $D^\\infty $ be the least family of subsets over $\\lbrace 0,1\\rbrace ^*$ such that $B^\\infty \\in D^\\infty $ if $b\\in {\\bf B}$ and $d\\in D^\\infty $ , then $0X\\cup 1Y\\in D^\\infty $ .",
"Show that, with the superset relation as the approximation ordering, $D^\\infty $ is a finitary basis.",
"State any assumptions that must be made.",
"Show then that $D^\\infty \\approx D\\times D^\\infty $ .",
"Exercise 3.31: Using the product construction as a guide, generate a definition for the separated sum structure ${\\bf A}+{\\bf B}$ .",
"Show that there are mappings $in_A:{\\bf A}\\rightarrow {\\bf A}+{\\bf B}$ , $in_B:{\\bf B}\\rightarrow {\\bf A}+{\\bf B}$ , $out_A:{\\bf A}+{\\bf B}\\rightarrow {\\bf A}$ , and $out_B:{\\bf A}+{\\bf B}\\rightarrow {\\bf B}$ such that $out_A\\circ in_A = {\\sf I}_A$ where ${\\sf I}_A$ is the identity function on ${\\bf A}$ .",
"State any necessary assumptions to ensure this function equation is true.",
"Exercise 3.32: For approximable mappings $f:{\\bf A}\\rightarrow {\\bf A^{\\prime }}$ and $g:{\\bf B}\\rightarrow {\\bf B^{\\prime }}$ , show that there exist approximable mappings, $f\\times g:{\\bf A}\\times {\\bf B}\\rightarrow {\\bf A^{\\prime }}\\times {\\bf B^{\\prime }}$ and $f+g:{\\bf A}+{\\bf B}\\rightarrow {\\bf A^{\\prime }}+{\\bf B^{\\prime }}$ such that $(f\\times g)[a,b] = [f \\: a, g \\: b]$ and thus $f\\times g = \\langle f\\circ p_0,g\\circ p_1\\rangle $ Show also that $out_A\\circ (f+g)\\circ in_A = f$ and $out_B\\circ (f+g)\\circ in_B = g$ Is $f+g$ uniquely determined by the last two equations?",
"Exercise 3.33: Prove that the composition operator is an approximable mapping.",
"That is, show that $comp:({\\bf B}\\rightarrow {\\bf C})\\times ({\\bf A}\\rightarrow {\\bf B})\\rightarrow ({\\bf A}\\rightarrow {\\bf C})$ is an approximable mapping where for $f:{\\bf A}\\rightarrow {\\bf B}$ and $g:{\\bf B}\\rightarrow {\\bf C}$ , $comp[g,f] = g\\circ f$ .",
"Show this using the approach used in showing the result for $apply$ and $curry$ .",
"That is, define the relation and then build the function from $apply$ and $curry$ , using $\\circ $ and paired functions.",
"(Hint: Fill in mappings according to the following sequence of domains.)",
"$\\begin{array}{c}({\\bf A}\\rightarrow {\\bf B})\\times {\\bf A}\\rightarrow {\\bf B}\\\\({\\bf B}\\rightarrow {\\bf C})\\times (({\\bf A}\\rightarrow {\\bf B})\\times {\\bf A})\\rightarrow ({\\bf B}\\rightarrow {\\bf C})\\times {\\bf B}\\\\(({\\bf B}\\rightarrow {\\bf C})\\times ({\\bf A}\\rightarrow {\\bf B}))\\times {\\bf A}\\rightarrow ({\\bf B}\\rightarrow {\\bf C})\\times {\\bf B}\\\\(({\\bf B}\\rightarrow {\\bf C})\\times ({\\bf A}\\rightarrow {\\bf B}))\\times {\\bf A}\\rightarrow {\\bf C}\\\\({\\bf B}\\rightarrow {\\bf C})\\times ({\\bf A}\\rightarrow {\\bf B})\\rightarrow ({\\bf A}\\rightarrow {\\bf C}).\\end{array}$ This map shows only one possible solution.",
"Exercise 3.34: Show that for every domain ${\\cal D}$ there is an approximable mapping $cond:{\\bf T}\\times {\\bf D}\\times {\\bf D}\\rightarrow {\\bf D}$ called the conditional operator such that $cond[true,a,b]=a$ $cond[false,a,b]=b$ $cond[\\bot _T,a,b]=\\bot _D$ and ${\\bf T}=\\lbrace \\bot _T,true,false\\rbrace $ such that $\\bot _T\\sqsubseteq true$ , $\\bot _T\\sqsubseteq false$ , but $true$ and $false$ are incomparable.",
"(Hint: Define a $Cond$ relation).",
"Fixed Points and Recursion Fixed Points Functions can now be constructed by composing basic functions.",
"However, we wish to be able to define functions recursively as well.",
"The technique of recursive definition will also be useful for defining domains as we will see in Section .",
"Recursion can be thought of as (possibly infinite) iterated function composition.",
"The primary result for interpreting recursive definitions is the following Fixed Point Theorem.",
"Theorem 4.1: For any continuous function $f:{\\cal D}\\rightarrow {\\cal D}$ determined by an approximable mapping $F:{\\bf D}\\rightarrow {\\bf D}$ , there exists a least element $x\\in {\\cal D}$ such that $f(x) = x.$ Proof Let $f^n$ stand for the function $f$ composed with itself $n$ times, and similarly for $F^n$ .",
"Thus, for $\\begin{array}{lcl}f^0&=&I_{\\cal D}~\\\\f^{n+1}&=&f\\circ f^n\\\\F^0&=&{\\sf I}_D~{\\rm and}\\\\F^{n+1}&=&F\\circ F^n\\end{array}$ we define $x = \\lbrace d\\in {\\bf D}\\:\\vert \\:\\exists n\\in {{N}}.",
"\\bot \\:F^n\\:d\\rbrace .$ To show that $x\\in {\\cal D}$ , we must show it to be an ideal.",
"Map $F$ is an approximable mapping, so $\\bot \\in x$ since $\\bot \\:F\\:\\bot $ .",
"For $d\\in x$ and $d^{\\prime }\\sqsubseteq d$ , $d^{\\prime }\\in x$ must hold since, for $d\\in x$ , there must exist an $a\\in {\\bf D}$ such that $a\\:F\\:d$ .",
"But by the definition of an approximable mapping, $a\\:F\\:d^{\\prime }$ must hold as well so $d^{\\prime }\\in x$ .",
"Closure under lubs is direct since $F$ must include lubs to be approximable.",
"To see that $f(x)=x$ , or equivalently $x\\:F\\:x$ , note that for any $d\\in x$ , if $d\\:F\\:d^{\\prime }$ , then $d^{\\prime }\\in x$ .",
"Thus, $f(x)\\sqsubseteq x$ .",
"Now, $x$ is constructed to be the least element in ${\\cal D}$ with this property.",
"To see this is true, let $a\\in {\\cal D}$ such that $f(a)\\sqsubseteq a$ .",
"We want to show that $x\\sqsubseteq a$ .",
"Let $d\\in x$ be an arbitrary element.",
"Therefore, there exists an $n$ such that $\\bot \\: F^n\\:d$ and therefore $\\bot \\:F\\:d_1\\:F\\:d_2\\:\\ldots \\:F\\:d_{n-1}\\:F\\:d.$ Since $\\bot \\in a$ , $d_1\\in f(a)$ .",
"Thus, since $f(a)\\sqsubseteq a$ , $d_1\\in a$ .",
"Thus, $d_2\\in f(a)$ and therefore $d_2\\in a$ .",
"Using induction on $n$ , we can show that $d\\in f(a)$ .",
"Therefore, $d\\in a$ and thus $x\\sqsubseteq a$ .",
"Since $f$ is monotonic and $f(x)\\sqsubseteq x$ , $f(f(x))\\sqsubseteq f(x)$ .",
"Since $x$ is the least element with this property, $x\\sqsubseteq f(x)$ and thus $x=f(x)$ .",
"$\\:\\:\\Box $ Since the element $x$ above is the least element, it must be unique.",
"Thus we have defined a function mapping the domain ${\\cal D}\\rightarrow {\\cal D}$ into the domain ${\\cal D}$ .",
"The next step is to show that this mapping is approximable.",
"Theorem 4.2: For any domain ${\\cal D}$ , there is an approximable mapping $fix:({\\bf D}\\rightarrow {\\bf D})\\rightarrow {\\bf D}$ such that if $f:{\\bf D}\\rightarrow {\\bf D}$ is an approximable mapping, $fix(f) = f(fix(f))$ and for $x\\in {\\cal D}$ , $f(x)\\sqsubseteq x \\:\\Rightarrow \\:fix(f)\\sqsubseteq x$ This property implies that $fix$ is unique.",
"The function $fix$ is characterized by the equation $fix(f)=\\bigcup \\limits _{n=0}^\\infty f^n(\\bot )$ for all $f:{\\bf D}\\rightarrow {\\bf D}$ .",
"Proof The final equation can be simplified to $fix(f) = \\lbrace d\\in {\\bf D}\\:\\vert \\:\\exists n\\in {{N}}.\\bot \\:f^n\\:d\\rbrace $ which is the equation used in the previous theorem to define the fixed point.",
"Using the formula from Exercise REF on the above definition for $fix$ yields the following equation to be shown: $fix(f)=\\bigcup \\lbrace fix({\\cal I}_F)\\:\\vert \\:\\exists F\\in ({\\bf D}\\rightarrow {\\bf D}).F\\in f\\rbrace $ where ${\\cal I}_F$ denotes the ideal for $F$ in ${\\bf D}\\rightarrow {\\bf D}$ .",
"From its definition, $fix$ is monotonic since, if $f\\sqsubseteq g$ , then $fix(f)\\sqsubseteq fix(g)$ since $f^n\\sqsubseteq g^n$ .",
"Since $F\\in f$ , ${\\cal I}_F\\sqsubseteq f$ and since $fix$ is monotonic, $fix({\\cal I}_F)\\sqsubseteq fix(f)$ .",
"Let $x\\in fix(f)$ .",
"Thus, there is a finite sequence of elements such that $\\bot \\:f\\:x_1\\:f\\:\\ldots \\:f\\:x^{\\prime }\\:f\\:x$ .",
"Define $F$ as the basis element encompassing the step functions required for this sequence.",
"Clearly, $F\\in f$ .",
"In addition, this same sequence exists in $fix({\\cal I}_F)$ since we constructed $F$ to contain it, and thus, $x\\in fix({\\cal I}_F)$ and $fix(f)\\sqsubseteq fix({\\cal I}_F)$ .",
"The equality is therefore established.",
"The first equality is direct from the Fixed Point Theorem since the same definition is used.",
"Assume $f(x)\\sqsubseteq x$ for some $x\\in {\\cal D}$ .",
"Since $\\bot \\in x$ , $x\\ne \\emptyset $ .",
"Since $f$ is an approximable mapping, for $x^{\\prime }\\in x$ and $x^{\\prime }\\:f\\:y$ , $y\\in x$ must hold.",
"By induction, for any $\\bot \\:f\\:y$ , $y\\in x$ must hold.",
"Thus, $fix(f)\\sqsubseteq x$ .",
"To see that the operator is unique, define another operator $fax$ that satisfies the first two equations.",
"It can easily be shown that $\\begin{array}{lcll}fix(f)&\\sqsubseteq & fax(f)~{\\rm and}\\\\fax(f)&\\sqsubseteq &fix(f)\\end{array}$ Thus the two operators are the same.",
"$\\:\\:\\Box $ Recursive Definitions Recursion has played a part already in the definitions above.",
"Recall that $f^n$ was defined for all $n\\in {{N}}$ .",
"More complex examples of recursion are given below.",
"Example 4.3: Define a basis ${\\bf N}=\\langle N,\\sqsubseteq _N\\rangle $ where $N=\\lbrace \\lbrace n\\rbrace \\:\\vert \\:n\\in {{N}}\\rbrace \\cup \\lbrace {{N}}\\rbrace $ and the approximation ordering is the superset relation.",
"This generates a flat domain with $\\bot =\\lbrace \\lbrace {{N}}\\rbrace \\rbrace $ and the total elements being in a one-to-one correspondence with the natural numbers.",
"Using the construction outlined in Exercise REF , construct the basis $F=N^\\infty $ .",
"Its corresponding domain is the domain of partial functions over the natural numbers.",
"To see this, let $\\Phi $ be the set of all finite partial functions $\\varphi \\subseteq {{N}}\\times {{N}}$ .",
"Define $\\uparrow \\varphi =\\lbrace \\psi \\in \\Phi \\:\\vert \\:\\varphi \\subseteq \\psi \\rbrace $ Consider the finitary basis $\\langle F^{\\prime },\\sqsubseteq _F^{\\prime }\\rangle $ where $F^{\\prime }=\\lbrace \\uparrow \\varphi \\:\\vert \\:\\varphi \\in \\Phi \\rbrace $ and the approximation order is the superset relation.",
"The reader should satisfy himself that $F^{\\prime }$ and $F$ are isomorphic and that the elements are the partial functions.",
"The total elements are the total functions over the natural numbers.",
"The domains ${\\cal F}$ and $({\\cal N}\\rightarrow {\\cal N})$ are not isomorphic.",
"However, the following mapping $val:F\\times {\\bf N}\\rightarrow {\\bf N}$ can be defined as follows: $(\\uparrow \\varphi ,\\lbrace n\\rbrace )\\:val\\:\\lbrace m\\rbrace \\iff (n,m)\\in \\varphi $ and $(\\uparrow \\varphi ,{{N}})\\:val\\:{{N}}$ Define also as the ideal for $m\\in {\\cal N}$ , $\\hat{m} = \\lbrace \\lbrace m\\rbrace ,{{N}}\\rbrace $ It is easy to show then that for $\\pi \\in {\\cal F}$ and $n\\in {\\cal N}$ we have $\\begin{array}{lcll}val(\\pi ,\\hat{n})&=&\\hat{\\pi (n)}&{\\rm if}~\\pi (n)\\ne \\bot \\\\&=&\\bot &{\\rm otherwise}\\end{array}$ Thus, $curry(val):{\\bf F}\\rightarrow ({\\bf N}\\rightarrow {\\bf N})$ is a one-to-one function on elements.",
"(The problem is that (${\\bf N}\\rightarrow {\\bf N}$ ) has more elements than F does as the reader should verify for himself).",
"Now, what about mappings $f:{\\bf F}\\rightarrow {\\bf F}$ ?",
"Consider the function $\\begin{array}{lcll}f(\\pi )(n)&=&0&{\\rm if}~n=0\\\\&=&\\pi (n-1)+n-1&{\\rm for}~n>0\\end{array}$ If $\\pi $ is a total function, $f(\\pi )$ is a total function.",
"If $\\pi (k)$ is undefined, then $f(\\pi )(k+1)$ is undefined.",
"The function $f$ is approximable since it is completely determined by its actions on partial functions.",
"That is $f(\\pi )=\\bigcup \\lbrace f(\\varphi )\\:\\vert \\:\\exists \\varphi \\in \\Phi .\\varphi \\subseteq \\pi \\rbrace $ The Fixed Point Theorem defines a least fixed point for any approximable mapping.",
"Let $\\sigma =f(\\sigma )$ .",
"Now, $\\sigma (0)=0$ and $\\begin{array}{lcl}\\sigma (n+1)&=&f(\\sigma )(n+1)\\\\&=&\\sigma (n)+n\\end{array}$ By induction, $\\sigma (n)=\\sum \\limits _{i=0}^n i$ and therefore, $\\sigma $ is a total function.",
"Thus, $f$ has a unique fixed point.",
"Now, in looking at $({\\bf N}\\rightarrow {\\bf N})$ , we have $\\hat{0}\\in {\\cal N}$ (The symbols $n$ and $\\hat{n}$ will no longer be distinguished, but the usage should be clear from context.).",
"Now define the two mappings, $succ,pred:{\\bf N}\\rightarrow {\\bf N}$ as approximable mappings such that $\\begin{array}{lcl}n\\:succ\\: m& \\iff & \\exists p\\in {{N}}.n\\sqsubseteq p,m\\sqsubseteq p+1\\\\n\\:pred\\: m& \\iff & \\exists p+1\\in {{N}}.n\\sqsubseteq p+1,m\\sqsubseteq p\\end{array}$ In more familiar terms, the same functions are defined as $\\begin{array}{lcll}succ(n)&=&n+1\\\\pred(n)&=&n-1&{\\rm if}~n>0\\\\&=&\\bot &{\\rm if}~n=0\\end{array}$ The mapping $zero:{\\bf N}\\rightarrow {\\bf T}$ is also defined such that $\\begin{array}{lcll}zero(n)&=&true&{\\rm if}~n=0\\\\&=&false&{\\rm if}~n>0\\end{array}$ where ${\\cal T}$ is the domain of truth value defined in an earlier section.",
"The structured domain $\\langle N,0,succ,pred,zero\\rangle $ is called “The Domain of the Integers” in the present context.",
"The function element $\\sigma $ defined as the fixed point of the mapping $f$ can now be defined directly as a mapping $\\sigma :{\\bf N}\\rightarrow {\\bf N}$ as follows: $\\sigma (n)=cond(zero(n),0,\\sigma (pred(n))+pred(n))$ where the function $+$ must be suitably defined.",
"Recall that $cond$ was defined earlier as part of the structure of the domain ${\\cal T}$ .",
"This equation is called a functional equation; the next section will give another notation, the $\\lambda -calculus$ for writing such equations.",
"$\\:\\:\\Box $ Example 4.4: The domain ${\\cal B}$ defined in Example REF contained only infinite elements as total elements.",
"A related domain, ${\\cal C}$ defined in Exercise REF , can be regarded as a generalization on ${\\cal N}$ .",
"To demonstrate this, the structured domain corresponding to the domain of integers must be presented.",
"The total elements in ${\\cal C}$ are denoted $\\sigma $ while the partial elements are denoted $\\sigma \\bot $ for any $\\sigma \\in \\lbrace 0,1\\rbrace ^*$ .",
"The empty sequence $\\epsilon $ assumes the role of the number 0 in ${\\cal N}$ .",
"Two approximable mappings can serve as the successor function: $x\\mapsto 0x$ denoted $succ_0$ and $x\\mapsto 1x$ denoted $succ_1$ .",
"The predecessor function is filled by the $tail$ mapping defined as follows: $\\begin{array}{lcll}tail(0x)& =& x,\\\\tail(1x)& =& x&{\\rm and}\\\\tail(\\epsilon )& =& \\bot .\\end{array}$ The $zero$ predicate is defined using the $empty$ mapping defined as follows: $\\begin{array}{lcll}empty(0x)& =& false,\\\\empty(1x)& =& false&{\\rm and}\\\\empty(\\epsilon )& =& true.\\end{array}$ To distinguish the other types of elements in ${\\cal C}$ , the following mappings are also defined: $\\begin{array}{lcll}zero(0x)& =& true,\\\\zero(1x)& =& false&{\\rm and}\\\\zero(\\epsilon )& =& false.\\\\one(0x)& =& false,\\\\one(1x)& =& true&{\\rm and}\\\\one(\\epsilon )& =& false.\\end{array}$ The reader should verify the conditions for an approximable mapping are met by these functions.",
"An element of ${\\cal C}$ can be defined using a fixed point equation.",
"For example, the total element representing an infinite sequence of alternating zeroes and ones is defined by the fixed point of the equation $a=01a$ .",
"This same element is defined with the equation $a=0101a$ .",
"(Is the element defined as $b=010b$ the same as the previous two?)",
"Approximable mappings in ${\\cal C}\\rightarrow {\\cal C}$ can also be defined using equations.",
"For example, the mapping $\\begin{array}{lcll}d(\\epsilon ) &= &\\epsilon ,\\\\d(0x)&=&00d(x)&{\\rm and}\\\\d(1x)&=&11d(x)\\end{array}$ can be characterized with the functional equation $d(x)=cond(empty(x),\\epsilon ,cond(zero(x),succ_0(succ_0(d(tail(x)))),succ_1(succ_1(d(tail(x))))))$ The concatenation function of Exercise REF over ${\\cal C}\\times {\\cal C}\\rightarrow {\\cal C}$ can be defined with the functional equation $C(x,y)=cond(empty(x),y,cond(zero(x),succ_0(C(tail(x),y)),succ_1(C(tail(x),y))))$ The reader should verify that this definition is consistent with the properties required in the exercise.",
"These definitions all use recursion.",
"They rely on the object being defined for a base case ($\\epsilon $ for example) or on earlier values ($tail(x)$ for example).",
"These equations characterize the object being defined, but unless a theorem is proved to show that a solution to the equation exists, the definition is meaningless.",
"However, the Fixed Point Theorem for domains was established earlier in this section.",
"Thus, solutions exist to these equations provided that the variables in the equation range over domains and any other functions appearing in the equation are known to be continuous (that is approximable).",
"Peano's Axioms To illustrate one use of the Fixed Point Theorem as well as show the use of recursion in a more familiar setting, we will show that all second order models of Peano's axioms are isomorphic.",
"Recall that Definition 4.5: [Model for Peano's Axioms] A structured set $\\langle {{N}},0,succ\\rangle $ for $0\\in {{N}}$ and $succ:{{N}}\\times {{N}}$ is a model for Peano's axioms if all the following conditions are satisfied: $\\forall n\\in {{N}}.",
"0\\ne succ(n)$ $\\forall n,m \\in {{N}}.succ(n)=succ(m)\\:\\Rightarrow \\:n=m$ $\\forall x\\subseteq {{N}}.0\\in x\\wedge succ(x)\\subseteq x\\:\\Rightarrow \\:x={{N}}$ where $succ(x)=\\lbrace succ(n)\\:\\vert \\:n\\in x\\rbrace $ .",
"The final clause is usually referred to as the principle of mathematical induction.",
"Theorem 4.6: All second order models of Peano's axioms are isomorphic.",
"Proof Let $\\langle N,0,+\\rangle $ and $\\langle M,\\bullet ,\\#\\rangle $ be models for Peano's axioms.",
"Let $N\\times M$ be the cartesian product of the two sets and let ${\\cal P}(N\\times M)$ be the powerset of $N\\times M$ .",
"Recall from Exercise REF that the powerset can be viewed as a domain with the subset relation as the approximation order.",
"Define the following mapping: $u\\mapsto \\lbrace (0,\\bullet )\\rbrace \\cup \\lbrace (+(n),\\#(m))\\:\\vert \\:(n,m)\\in u\\rbrace $ The reader should verify that this mapping is approximable.",
"Since it is indeed approximable, a fixed point exists for the function.",
"Let $r$ be the least fixed point: $r=\\lbrace (0,\\bullet )\\rbrace \\cup \\lbrace (+(n),\\#(m))\\:\\vert \\:(n,m)\\in r\\rbrace $ But $r$ defines a binary relation which establishes the isomorphism.",
"To see that $r$ is an isomorphism, the one-to-one and onto features must be established.",
"By construction, $0\\:r\\:\\bullet $ and $n\\:r\\:m \\:\\Rightarrow \\:+(n)\\:r\\:\\#(m)$ .",
"Now, the sets $\\lbrace (0,\\bullet )\\rbrace $ and $\\lbrace (+(n),\\#(m))\\:\\vert \\:(n,m)\\in r\\rbrace $ are disjoint by the first axiom.",
"Therefore, 0 corresponds to only one element in $m$ .",
"Let $x\\subseteq N$ be the set of all elements of $N$ that correspond to only one element in $m$ .",
"Clearly, $0\\in x$ .",
"Now, for some $y\\in x$ let $z\\in M$ be the element in $M$ that $y$ uniquely corresponds to (that is $y\\:r\\:z$ ).",
"But this means that $+(y)\\:r\\#(z)$ by the construction of the relation.",
"If there exists $w\\in M$ such that $+(y)\\:r\\:w$ and since $(+(y),w)\\ne (0,\\bullet )$ , the fixed point equation implies that $(+(y)=+(n_0))$ and $(w=\\#(m_0))$ for some $(n_0,m_0)\\in r$ .",
"But then by the second axiom, $y=n_0$ and since $y\\in x$ , $z=m_0$ .",
"Thus, $\\#(z)$ is the unique element corresponding to $+(y)$ .",
"The third axiom can now be applied, and thus every element in $N$ corresponds to a unique element in $M$ .",
"The roles of $N$ and $M$ can be reversed in this proof.",
"Therefore, it can also be shown that every element of $M$ corresponds to a unique element in $N$ .",
"Thus, $r$ is a one-to-one and onto correspondence.",
"$\\:\\:\\Box $ Exercises Exercise 4.7: In Theorem REF , an equation was given to find the least fixed point of a function $f:{\\cal D}\\rightarrow {\\cal D}$ .",
"Suppose that for $a\\in {\\cal D}$ , $a\\sqsubseteq f(a)$ .",
"Will the fixed point $x=f(x)$ be such that $a\\sqsubseteq x$ ?",
"(Hint: How do we know that $\\bigcup \\limits _{n=0}^\\infty f^n(a)\\in {\\cal D}$ ?)",
"Exercise 4.8: Let $f:{\\cal D}\\rightarrow {\\cal D}$ and $S\\subseteq {\\cal D}$ satisfy $\\bot \\in S$ $x\\in S\\:\\Rightarrow \\:f(x)\\in S$ $[\\forall n .\\lbrace x_n\\rbrace \\subseteq S \\wedge x_n\\sqsubseteq x_{n+1}]\\:\\Rightarrow \\:\\bigcup \\limits _{n=0}^\\infty x_n \\in S$ Conclude that $fix(f)\\in S$ .",
"This is sometimes called the principle of fixed point induction.",
"Apply this method to the set $S=\\lbrace x\\in {\\cal D}\\:\\vert \\:a(x)= b(x)\\rbrace $ where $a,b:{\\cal D}\\rightarrow {\\cal D}$ are approximable, $a(\\bot )=b(\\bot )$ , and $f\\circ a=a\\circ f$ and $f\\circ b=b\\circ f$ .",
"Exercise 4.9: Show that there is an approximable operator $\\Psi :(({\\cal D}\\rightarrow {\\cal D})\\rightarrow {\\cal D})\\rightarrow (({\\cal D}\\rightarrow {\\cal D})\\rightarrow {\\cal D})$ such that for $\\Theta :({\\cal D}\\rightarrow {\\cal D})\\rightarrow {\\cal D}$ and $f:{\\cal D}\\rightarrow {\\cal D}$ , $\\Psi (\\Theta ) (f) = f(\\Theta (f))$ Prove also that $fix:({\\cal D}\\rightarrow {\\cal D})\\rightarrow {\\cal D}$ is the least fixed point of $\\Psi $ .",
"Exercise 4.10: Given a domain ${\\cal D}$ and an element $a\\in {\\cal D}$ , construct the domain ${\\cal D}_a$ where ${\\cal D}_a=\\lbrace x\\in {\\cal D}\\:\\vert \\:x\\sqsubseteq a\\rbrace $ Show that if $f:{\\cal D}\\rightarrow {\\cal D}$ is approximable, then $f$ can be restricted to another approximable map $f^{\\prime }:{\\cal D}_{fix(f)}\\rightarrow {\\cal D}_{fix(f)}$ where $\\forall x\\in {\\cal D}_{fix(f)}.f^{\\prime }(x)=f(x)$ How many fixed points does $f^{\\prime }$ have in ${\\cal D}_{fix(f)}$ ?",
"Exercise 4.11: The mapping ${\\bf fix}$ can be viewed as assigning a fixed point operator to any domain ${\\cal D}$ .",
"Show that ${\\bf fix}$ can be uniquely characterized by the following conditions on an assignment ${\\cal D}\\leadsto F_D$ : $F_D:({\\cal D}\\rightarrow {\\cal D})\\rightarrow {\\cal D}$ $F_D(f)=f(F_D(f))$ for all $f:{\\cal D}\\rightarrow {\\cal D}$ when $f_0:{\\cal D}_0\\rightarrow {\\cal D}_0$ and $f_1:{\\cal D}_1\\rightarrow {\\cal D}_1$ are given and $h:{\\cal D}_0\\rightarrow {\\cal D}_1$ is such that $h(\\bot )=\\bot $ and $h\\circ f_0=f_1\\circ h$ , then $h(F_{D_0}(f_0)) = F_{D_1}(f_1).$ Hint: Apply Exercise REF to show ${\\bf fix}$ satisfies the conditions.",
"For the other direction, apply Exercise REF .",
"Exercise 4.12: Must an approximable function have a maximum fixed point?",
"Give an example of an approximable function that has many fixed points.",
"Exercise 4.13: Must a monotone function $f:{\\cal P}(A)\\rightarrow {\\cal P}(A)$ have a maximum fixed point?",
"(Recall: ${\\cal P}(A)$ is the powerset of the set $A$ ).",
"Exercise 4.14: Verify the assertions made in the first example of this section.",
"Exercise 4.15: Verify the assertions made in the second example, in particular those in the discussion of “Peano's Axioms”.",
"Show that the predicate function $one:{\\cal C}\\rightarrow {\\cal T}$ could be defined using a fixed point equation from the other functions in the structure.",
"Exercise 4.16: Prove that $fix(f\\circ g)=f(fix(g\\circ f))$ for approximable functions $f,g:{\\cal D}\\rightarrow {\\cal D}$ .",
"Exercise 4.17: Show that the less-than-or-equal-to relation $l\\subseteq {{N}}\\times {{N}}$ is uniquely determined by $l=\\lbrace (n,n)\\:\\vert \\:n\\in {{N}}\\rbrace \\cup \\lbrace (n,succ(m)\\:\\vert \\:(n,m)\\in l\\rbrace $ for the structure called the “Domain of Integers”.",
"Exercise 4.18: Let $N^*$ be a structured set satisfying only the first two of the axioms referred to as “Peano's”.",
"Must there be a subset $S\\subseteq N^*$ such that all three axioms are satisfied?",
"(Hint: Use a least fixed point from ${\\cal P}(N^*)$ ).",
"Exercise 4.19: Let $f:{\\cal D}\\rightarrow {\\cal D}$ be an approximable map.",
"Let $a_n:{\\cal D}\\rightarrow {\\cal D}$ be a sequence of approximable maps such that $a_0(x)=\\bot $ for all $x\\in {\\cal D}$ $a_n\\sqsubseteq a_{n+1}$ for all $n\\in {{N}}$ $\\bigcup \\limits _{n=0}^\\infty a_n = {\\sf I}_D$ in ${\\cal D}\\rightarrow {\\cal D}$ $a_{n+1}\\circ f = a_{n+1}\\circ f\\circ a_n$ for all $n\\in {{N}}$ Show that $f$ has a unique fixed point.",
"(Hint: Show that if $x=f(x)$ then $a_n(x)\\sqsubseteq a_n(fix(f))$ for all $n\\in {{N}}$ .",
"Show this by induction on $n$ .)",
"Typed $\\lambda $ -Calculus As shown in the previous section, functions can be characterized by recursion equations which combine previously defined functions with the function being defined.",
"The expression of these functions is simplified in this section by introducing a notation for specifying a function without having to give the function a name.",
"The notation used is that of the typed $\\lambda $ -Calculus; a function is defined using a $\\lambda $ -abstraction.",
"Definition of Typed $\\lambda $ -Calculus An informal characterization of the $\\lambda $ -calculus suffices for this section; more formal descriptions are available elsewhere in the literature [1].",
"Thus, examples are used to introduce the notation.",
"An infinite number of variables, $x$ ,$y$ ,$z$ ,$\\ldots $ of various types are required.",
"While a variable has a certain type, type subscripts will not be used due to the notational complexity.",
"A distinction must also be made between type symbols and domains.",
"The domain ${\\cal A}\\times {\\cal B}$ does not uniquely determine the component domains ${\\cal A}$ and ${\\cal B}$ even though these domains are uniquely determined by the symbol for the domain.",
"The domain is the meaning that we attribute to the symbol.",
"In addition to variables, constants are also present.",
"For example, the symbol 0 is used to represent the zero element from the domain ${\\cal N}$ .",
"Another constant, present in each domain by virtue of Theorem REF , is $fix^{\\cal D}$ , the least fixed point operator for domain ${\\cal D}$ of type $({\\cal D}\\rightarrow {\\cal D})\\rightarrow {\\cal D}$ .",
"The constants and variables are the atomic (non-compound) terms.",
"Types can be associated with all atomic terms.",
"There are several constructions for compound terms.",
"First, given $\\tau ,\\ldots ,\\sigma $ , a list of terms, the ordered tuple $\\langle \\tau ,\\ldots ,\\sigma \\rangle $ is a compound term.",
"If the types of $\\tau ,\\ldots ,\\sigma $ are ${\\cal A},\\ldots ,{\\cal B}$ , the type of the tuple is ${\\cal A}\\times \\ldots \\times {\\cal B}$ since the tuple is to be an element of this domain.",
"The tuple notation for combining functions given earlier should be disregarded here.",
"The next construction is function application.",
"If the term $\\tau $ has type ${\\cal A}\\rightarrow {\\cal B}$ and the term $\\sigma $ has the type ${\\cal A}$ , then the compound term $\\tau (\\sigma )$ has type ${\\cal B}$ .",
"Function application denotes the value of a function at a given input.",
"The notation $\\tau (\\sigma _0,\\ldots ,\\sigma _n)$ abbreviates $\\tau (\\langle \\sigma _0,\\ldots ,\\sigma _n\\rangle )$ .",
"Functions applied to tuples allows us to represent applications of multi-variate functions.",
"The $\\lambda $ -abstraction is used to define functions.",
"Let $x_0,\\ldots ,x_n$ be a list of distinct variables of type ${\\cal D}_0,\\ldots ,{\\cal D}_n$ .",
"Let $\\tau $ be a term of some type ${\\cal D}_{n+1}$ .",
"$\\tau $ can be thought of as a function of $n+1$ variables with type $({\\cal D}_0\\times \\ldots \\times {\\cal D}_n)\\rightarrow {\\cal D}_{n+1}$ .",
"The name for this function is written $\\lambda x_0,\\ldots ,x_n.\\tau $ This expression denotes the entire function.",
"To look at some familiar functions in the new notation, consider $\\lambda x,y.x$ This notation is read “lambda ex wye (pause) ex”.",
"If the types of $x$ and $y$ are ${\\cal A}$ and ${\\cal B}$ respectively, the function has type $({\\cal A}\\times {\\cal B})\\rightarrow {\\cal A}$ .",
"This function is the first projection function $p_0$ .",
"This function and the second projection function can be defined by the following equations: $\\begin{array}{lcl}p_0&=&\\lambda x,y.x\\\\p_1&=&\\lambda x,y.y\\end{array}$ Recalling the function tuple notation introduced in an earlier section, the following equation holds: $\\langle f,g\\rangle =\\lambda w.\\langle f(w),g(w)\\rangle $ which defines a function of type ${\\cal D}_1\\rightarrow ({\\cal D}_2\\times {\\cal D}_3)$ .",
"Other familiar functions are defined by the following equations: $\\begin{array}{lcl}eval&=&\\lambda f,x.f(x)\\\\curry&=&=\\lambda g\\lambda x\\lambda y.g(x,y)\\end{array}$ The $curry$ example shows that this notation can be iterated.",
"A distinction is thus made between the terms $\\lambda x,y.x$ and $\\lambda x\\lambda y.x$ which have the types ${\\cal D}_0\\times {\\cal D}_1\\rightarrow {\\cal D}_0$ and ${\\cal D}_0\\rightarrow {\\cal D}_1\\rightarrow {\\cal D}_0$ respectively.",
"Thus, the following equation also holds: $curry(\\lambda x,y.\\tau )=\\lambda x\\lambda y.\\tau $ which relates the multi-variate form to the iterated or curried form.",
"Another true equation is $fix={\\bf fix}(\\lambda F\\lambda f.f(F(f)))$ where $fix$ has type $({\\cal D}\\rightarrow {\\cal D})\\rightarrow {\\cal D}$ and fix has type $(((({\\cal D}\\rightarrow {\\cal D})\\rightarrow {\\cal D})\\rightarrow (({\\cal D}\\rightarrow {\\cal D})\\rightarrow {\\cal D}))\\rightarrow (({\\cal D}\\rightarrow {\\cal D})\\rightarrow {\\cal D}))$ This is the content of Exercise REF .",
"This notation can now be used to define functions using recursion equations.",
"For example, the function $\\sigma $ in Example REF can be characterized by the following equation: $\\sigma =fix(\\lambda f\\lambda n.cond(zero(n),0,f(pred(n))+pred(n))$ which states that $\\sigma $ is the least recursively defined function $f$ whose value at $n$ is $cond(\\ldots )$ .",
"The variable $f$ occurs in the body of the $cond$ expression, but this is just the point of a recursive definition.",
"$f$ is defined in terms of its value on “smaller” input values.",
"The use of the fixed point operator makes the definition explicit by forcing there to be a unique solution to the equation.",
"In an abstraction $\\lambda x,y,z.\\tau $ , the variables $x$ ,$y$ , and $z$ are said to be bound in the term $\\tau $ .",
"Any other variables in $\\tau $ are said to be free variables in $\\tau $ unless they are bound elsewhere in $\\tau $ .",
"Bound variables are simply placeholders for values; the particular variable name chosen is irrelevant.",
"Thus, the equation $\\lambda x.\\tau =\\lambda y.\\tau [y/x]$ is true provided $y$ is not free in $\\tau $ .",
"The notation $\\tau [y/x]$ specifies the substitution of $y$ for $x$ everywhere $x$ occurs in $\\tau $ .",
"The notation $\\tau [\\sigma /x]$ for the substitution of the term $\\sigma $ for the variable $x$ is also legitimate.",
"Semantics of Typed $\\lambda $ -Calculus To show that the equations above with $\\lambda $ –terms are indeed meaningful, the following theorem relating $\\lambda $ –terms and approximable mappings must be proved.",
"Theorem 5.1: Every typed $\\lambda $ –term defines an approximable function of its free variables.",
"Proof Induction on the length of the term and its structure will be used in this proof.",
"Variables Direct since $x\\mapsto x$ is an approximable function.",
"Constants Direct since $x\\mapsto k$ is an approximable function for constant $k$ .",
"Tuples Let $\\tau =\\langle \\sigma _0,\\ldots ,\\sigma _n\\rangle $ .",
"Since the $\\sigma _i$ terms are less complex, they are approximable functions of their free variables by the induction hypothesis.",
"Using Theorem REF (generalized to the multi-variate case) then, $\\tau $ which takes tuples as values also defines an approximable function.",
"Application Let $\\tau =\\sigma _0(\\sigma _1)$ .",
"We assume that the types of the terms are appropriately matched.",
"The $\\sigma _i$ terms define approximable functions again by the induction hypothesis.",
"Recalling the earlier equations, the value of $\\tau $ is the same as the value of $eval(\\sigma _0,\\sigma _1)$ .",
"Since $eval$ is approximable, Theorem REF shows that the term defines an approximable function.",
"Abstraction Let $\\tau =\\lambda x.\\sigma $ .",
"By the induction hypothesis, $\\sigma $ defines a function of its free variables.",
"Let those free variables be of types ${\\cal D}_0,\\ldots ,{\\cal D}_n$ where ${\\cal D}_n$ is the type of $x$ .",
"Then $\\sigma $ defines an approximable function $g:{\\cal D}_0\\times \\ldots \\times {\\cal D}_n\\rightarrow {\\cal D}^{\\prime }$ where ${\\cal D}^{\\prime }$ is the type of $\\sigma $ .",
"Using Theorem REF , the function $curry(g):{\\cal D}_0\\times \\ldots \\times {\\cal D}_{n-1}\\rightarrow ({\\cal D}_n\\rightarrow {\\cal D}^{\\prime })$ yields an approximable function, but this is just the function defined by $\\tau $ .",
"The reader can generalize this proof for multiple bound variables in $\\tau $ .",
"$\\:\\:\\Box $ Given this, the equation $\\tau =\\sigma $ states that the two terms define the same approximable function of their free variables.",
"As an example, $\\lambda x.\\tau =\\lambda y.\\tau [y/x]$ provided $y$ is not free in $\\tau $ since the generation of the approximable function did not depend on the name $x$ but only on its location in $\\tau $ .",
"Other equations such as these are given in the exercises.",
"The most basic rule is shown below.",
"Theorem 5.2: For appropriately typed terms, the following equation is true: $(\\lambda x_0,\\ldots ,x_{1}.\\tau )(\\sigma _0,\\ldots ,\\sigma _{n-1})=\\tau [\\sigma _0/x_0,\\ldots ,\\sigma _{n-1}/x_{n-1}]$ Proof The proof is given for $n=1$ and proceeds again by induction on the length of the term and the structure of the term.",
"Variables This means $(\\lambda x.x)(\\sigma )=\\sigma $ must be true which it is.",
"Constants This requires $(\\lambda x.k)(\\sigma )= k$ must be true which it is for any constant $k$ .",
"Tuples Let $\\tau =\\langle \\tau _0,\\tau _1\\rangle $ .",
"This requires that $(\\lambda x.\\langle \\tau _0,\\tau _1\\rangle )(\\sigma ) =\\langle \\tau _0[\\sigma /x],\\tau _1[\\sigma /x]\\rangle $ must be true.",
"This equation holds since the left-hand side can be transformed using the following true equation: $(\\lambda x.\\langle \\tau _0,\\tau _1\\rangle )(\\sigma ) =\\langle (\\lambda x.\\tau _0)(\\sigma ),(\\lambda x.\\tau _1)(\\sigma )\\rangle $ Then the inductive hypothesis is applied to the $\\tau _i$ terms.",
"Applications Let $\\tau =\\tau _0(\\tau _1)$ .",
"Then, the result requires that the equation $(\\lambda x.\\tau _0(\\tau _1))(\\sigma ) =\\tau _0[\\sigma /x](\\tau _1[\\sigma /x])$ hold true.",
"To see that this is true, examine the approximable functions for the left-hand side of the equation.",
"$\\begin{array}{lcl}\\tau _0&\\mapsto &\\bar{V},x\\rightarrow t_0\\\\\\tau _1&\\mapsto &\\bar{V},x\\rightarrow t_1\\\\\\sigma &\\mapsto &\\bar{V}\\rightarrow s\\\\\\mbox{so}\\\\(\\lambda x.\\tau _0(\\tau _1))(\\sigma )&\\mapsto &\\bar{V}\\rightarrow [(x\\rightarrow t_0(t_1))(s)]\\\\&=&\\bar{V},x\\rightarrow [(x\\rightarrow t_0)(s)]([(x\\rightarrow t_1)(s)])\\end{array}$ From this last term, we use the induction hypothesis.",
"To see why the last step holds, start with the set representing the left-hand side and using the aprroximable mappings for the terms: $\\begin{array}{cl}&(\\lambda x.\\tau _0(\\tau _1))(\\sigma )\\\\\\mapsto &\\bar{V}\\rightarrow [(x\\rightarrow t_0(t_1))(s)]\\\\=&\\lbrace b\\:\\vert \\:\\exists a.a\\in s\\wedge a\\:[x\\rightarrow t_0(t_1)]\\: b\\rbrace \\\\=&\\lbrace b\\:\\vert \\:\\exists a. a\\in s\\wedge a\\:\\lbrace (x,u)\\:\\vert \\:v\\in x\\rightarrow t_1)\\wedge v\\:(x\\rightarrow t_0)\\: u\\rbrace \\: b\\rbrace \\\\=&\\lbrace b\\:\\vert \\:\\exists a.a\\in s\\wedge v\\in (x\\rightarrow t_1)(a)\\wedge v\\:(x\\rightarrow t_0)(a)\\:b\\rbrace \\\\=&\\lbrace b\\:\\vert \\:\\exists a,c.a\\in s\\wedge a\\:(x\\rightarrow t_1)\\: v\\wedge a\\:(x\\rightarrow t_0)\\:c\\wedge v\\:c\\:b\\rbrace \\\\=&\\lbrace b\\:\\vert \\:v\\in [(x\\rightarrow t_1)(s)]\\wedge c\\in (x\\rightarrow t_0)(s) \\wedge v\\:c\\:b\\rbrace \\\\=&\\lbrace b\\:\\vert \\:v\\in [(x\\rightarrow t_1)(s)]\\wedge v\\:[(x\\rightarrow t_0)(s)]\\:b\\rbrace \\\\=&[(x\\rightarrow t_0)(s)]([(x\\rightarrow t_1)(s)])\\end{array}$ Abstractions Let $\\tau =\\lambda y.\\tau _0$ .",
"The required equation is $(\\lambda x.\\lambda y.\\tau _0)(\\sigma )=\\lambda y.\\tau _0[\\sigma /x]$ provided that $y$ is not free in $\\sigma $ .",
"The following true equation applies here: $(\\lambda x.\\lambda y.\\tau )(\\sigma )=\\lambda y.",
"((\\lambda x.\\tau )(\\sigma ))$ To see that this equation holds, let $g$ be a function of $n+2$ free variables defined by $\\tau $ .",
"By Theorem REF , the term $\\lambda x.\\lambda y.\\tau $ defines the function $curry(curry(g))$ of $n$ variables.",
"Call this function $h$ .",
"Thus, $h(v)(\\sigma )(y) = g(v,\\sigma ,y)$ where $v$ is the list of the other free variables.",
"Using a combinator $inv$ which inverts the order of the last two arguments, $h(v)(\\sigma )(y)=curry(inv(g))(v,y)(\\sigma )$ But, $curry(inv(g))$ is the function defined by $\\lambda x.\\tau $ .",
"Thus, we have shown that $(\\lambda x.\\lambda y.\\tau )(\\sigma )(y)=(\\lambda x.\\tau )(\\sigma )$ is a true equation.",
"If $y$ is not free in $\\alpha $ and $\\alpha (y)=\\beta $ is true, then $\\alpha =\\lambda y.\\beta $ must also be true.",
"$\\:\\:\\Box $ If $\\tau ^{\\prime }$ is the term $\\lambda x,y.\\tau $ , then $\\tau ^{\\prime }(x,y)$ is the same as $\\tau $ .",
"This specifies that $x$ and $y$ are not free in $\\tau $ .",
"This notation is used in the proof of the following theorem.",
"Theorem 5.3: The least fixed point of $\\lambda x,y.\\langle \\tau (x,y),\\sigma (x,y)\\rangle $ is the pair with coordinates $fix(\\lambda x.\\tau (x,fix(\\lambda y.\\sigma (x,y))))$ and $fix(\\lambda y.\\sigma (fix(\\lambda x.\\tau (x,y)),y))$ .",
"Proof We are thus assuming that $x$ and $y$ are not free in $\\tau $ and $\\sigma $ .",
"The purpose here is to find the least solution to the pair of equations: $x=\\tau (x,y)~{\\rm and}~y=\\sigma (x,y)$ This generalizes the fixed point equation to two variables.",
"More variables could be included using the same method.",
"Let $y_*=fix(\\lambda y.\\sigma (fix(\\lambda x.\\tau (x,y)),y))$ and $x_*=fix(\\lambda x.\\tau (x,y))$ Then, $x_*=\\tau (x_*,y_*)$ and $\\begin{array}{lcl}y_*&=&\\sigma (fix(\\lambda x.\\tau (x,y_*),y_*))\\\\&=&\\sigma (x_*,y_*).\\end{array}$ This shows that the pair $\\langle x_*,y_*\\rangle $ is one fixed point.",
"Now, let $\\langle x_0,y_0\\rangle $ be the least solution.",
"(Why must a least solution exist?",
"Hint: Consider a suitable mapping of type $({\\cal D}_x\\times {\\cal D}_y)\\rightarrow ({\\cal D}_x\\times {\\cal D}_y)$ .)",
"Thus, we know that $x_0=\\tau (x_0,y_0)$ , $y_0=\\sigma (x_0,y_0)$ , and that $x_0\\sqsubseteq x_*$ and $y_0\\sqsubseteq y_*$ .",
"But this means that $\\tau (x_0,y_0)\\sqsubseteq x_0$ and thus $fix(\\lambda x.\\tau (x,y_0))\\sqsubseteq x_0$ and consequently $\\sigma (fix(\\lambda x.\\tau (x,y_0),y_0))\\sqsubseteq \\sigma (x_0,y_0)\\sqsubseteq y_0$ By the fixed point definition of $y_*$ , $y_*\\sqsubseteq y_0$ must hold as well so $y_0=y_*$ .",
"Thus, $x_*=fix(\\lambda x.\\tau (x,y*))=fix(\\lambda x.\\tau (x,y_0))\\sqsubseteq x_0.$ Thus, $x*=x_0$ must also hold.",
"A similar argument holds for $x_0$ .$\\:\\:\\Box $ The purpose of the above proof is to demonstrate the use of least fixed points in proofs.",
"The following are also true equations: $fix(\\lambda x.\\tau (x))=\\tau (fix(\\lambda x.\\tau (x)))$ and $\\tau (y)\\sqsubseteq y\\:\\Rightarrow \\:fix(\\lambda x.\\tau (x))\\sqsubseteq y$ if $x$ is not free in $\\tau $ .",
"These equations combined with the monotonicity of functions were the methods used in the proof above.",
"Another example is the proof of the following theorem.",
"Theorem 5.4: Let $x$ ,$y$ , and $\\tau (x,y)$ be of type ${\\cal D}$ and let $g:{\\cal D}\\rightarrow {\\cal D}$ be a function.",
"Then the equation $\\lambda x.fix(\\lambda y.\\tau (x,y))=fix(\\lambda g.\\lambda x.\\tau (x,g(x)))$ holds.",
"Proof Let $f$ be the function on the left-hand side.",
"Then, $f(x)=fix(\\lambda y.\\tau (x,y))=\\tau (x,f(x))$ holds using the equations stated above.",
"Therefore, $f=\\lambda x.\\tau (x,f(x))$ and thus $g_0=fix(\\lambda g.\\lambda x.\\tau (x,g(x)))\\sqsubseteq f. $ By the definition of $g_0$ we have $g_0(x)=\\tau (x,g_0(x))$ for any given $x$ .",
"By the definition of $f$ we find that $f(x)=fix(\\lambda y.\\tau (x,y))\\sqsubseteq g_0(x)$ must hold for all $x$ .",
"Thus $f\\sqsubseteq g_0$ and the equation is true.$\\:\\:\\Box $ This proof illustrates the use of inclusion and equations between functions.",
"The following principle was used: $(\\forall x.\\tau \\sqsubseteq \\sigma )\\:\\Rightarrow \\:\\lambda x.\\tau \\sqsubseteq \\lambda x.\\sigma $ This is a restatement of the first part of Theorem REF .",
"Combinators and Recursive Functions Below is a list of various combinators with their definitions in $\\lambda $ -notation.",
"The meanings of those combinators not previously mentioned should be clear.",
"$\\begin{array}{lcl}p_0&=&\\lambda x,y.x\\\\p_1&=&\\lambda x,y.y\\\\pair&=&\\lambda x.\\lambda y.\\langle x,y\\rangle \\\\n-tuple&=&\\lambda x_0\\lambda \\ldots \\lambda x_{n-1}.\\langle x_0,\\ldots ,x_{n-1}\\rangle \\\\diag&=&\\lambda x.\\langle x,x\\rangle \\\\funpair&=&\\lambda f.\\lambda g.\\lambda x.\\langle f(x),g(x)\\rangle \\\\proj^n_i&=&\\lambda x_0,\\ldots ,x_{n-1}.x_i\\\\inv^n_{i,j}&=&\\lambda x_0,\\ldots ,x_i,\\ldots ,x_j,\\ldots ,x_{n-1}.\\langle x0,\\ldots ,x_j,\\ldots ,x_i,\\ldots ,x_{n-1}\\rangle \\\\eval&=&\\lambda f,x.f(x)\\\\curry&=&\\lambda g.\\lambda x.\\lambda y.g(x,y)\\\\comp&=&\\lambda f,g.\\lambda x.g(f(x))\\\\const&=&\\lambda k.\\lambda x.k\\\\{\\bf fix}&=&\\lambda f.fix(\\lambda x.f(x))\\end{array}$ These combinators are actually schemes for combinators since no types have been specified and thus the equations are ambiguous.",
"Each scheme generates an infinite number of combinators for all the various types.",
"One interest in combinators is that they allow expressions without variables—if enough combinators are used.",
"This is useful at times but can be clumsy.",
"However, defining a combinator when the same combination of symbols repeatedly appears is also useful.",
"There are some familiar combinators that do not appear in the table.",
"Combinators such as $cond$ , $pred$ , and $succ$ cannot be defined in the pure $\\lambda $ -calculus but are instead specific to certain domains.",
"They are thus regarded as primitives.",
"A large number of other functions can be defined using these primitives and the $\\lambda $ -notation, as the following theorem shows.",
"Theorem 5.5: For every partial recursive function $h:{{N}}\\rightarrow {{N}}$ , there is a $\\lambda $ -term $\\tau $ of type ${\\cal N}\\rightarrow {\\cal N}$ such that the only constants occurring in $\\tau $ are $cond$ , $succ$ , $pred$ , $zero$ , and 0 and if $h(n)=m$ then $\\tau (n)=m$ .",
"If $h(n)$ is undefined, then $\\tau (n)=\\bot $ holds.",
"$\\tau (\\bot )=\\bot $ is also true.",
"Proof It is convenient in the proof to work with strict functions $f:{\\cal N}^k\\rightarrow {\\cal N}$ such that if any input is $\\bot $ , the result of the function is $\\bot $ .",
"The composition of strict functions is easily shown to be strict.",
"It is also easy to see that any partial function $g:{{N}}^k\\rightarrow {{N}}$ can be extended to a strict approximable function $\\bar{g}:{\\cal N}^k\\rightarrow {\\cal N}$ which yields the same values on inputs for which $g$ is defined.",
"Other input values yield $\\bot $ .",
"We want to show that $\\bar{g}$ is definable with a $\\lambda $ -expression.",
"First we must show that primitive recursive functions have $\\lambda $ -definitions.",
"Primitive recursive functions are formed from starting functions using composition and the scheme of primitive recursion.",
"The starting functions are the constant function for zero and the identity and projection functions.",
"These functions, however, must be strict so the term $\\lambda x,y.x$ is not sufficient for a projection function.",
"The following device reduces a function to its strict form.",
"Let $\\lambda x.cond(zero(x),x,x)$ be a function with $x$ of type ${\\cal N}$ .",
"This is the strict identity function.",
"The strict projection function attempted above can be defined as $\\lambda x,y.cond(zero(y),x,x)$ The three variable projection function can be defined as $\\lambda x,y,z.cond(zero(x),cond(zero(z),y,y),cond(zero(z),y,y))$ While not very elegant, this device does produce strict functions.",
"Strict functions are closed under substitution and composition.",
"Any substitution of a group of functions into another function can be defined with a $\\lambda $ -term if the functions themselves can be so defined.",
"Thus, we need to show that functions obtained by primitive recursion are definable.",
"Let $f:{\\cal N}\\rightarrow {\\cal N}$ , and $g:{\\cal N}^3\\rightarrow {\\cal N}$ be total functions with $\\bar{f}$ and $\\bar{g}$ being $\\lambda $ -definable.",
"We obtain the function $h:{\\cal N}^2\\rightarrow {\\cal N}$ by primitive recursion where $\\begin{array}{lcl}h(0,m)&=&f(m)\\\\h(n+1,m)&=&g(n,m,h(n,m))\\end{array}$ for all $n,m\\in {\\cal N}$ .",
"The $\\lambda $ -term for $\\bar{h}$ is $fix(\\lambda k.\\lambda x,y.cond(zero(x),\\bar{f}(y),\\bar{g}(pred(x),y,k(pred(x),y))))$ Note that the fixed point operator for the domain ${\\cal N}^2\\rightarrow {\\cal N}$ was used.",
"The variables $x$ and $y$ are of type ${\\cal N}$ .",
"The $cond$ function is used to encode the function requirements.",
"The fixed point function is easily seen to be strict and this function is $\\bar{h}$ .",
"Primitive recursive functions are now $\\lambda $ -definable.",
"To obtain partial (i.e., general) recursive functions, the $\\mu $ -scheme (the least number operator) is used.",
"Let $f(n,m)$ be a primitive recursive function.",
"Then, define $h$ , a partial function, as $h(m) =$ the least $n$ such that $f(n,m)=0$ .",
"This is written as $h(m)=\\mu n.f(n,m)=0$ .",
"Since $\\bar{f}$ is $\\lambda $ -definable as has just been shown, let $\\bar{g}=fix(\\lambda g.\\lambda x,y.cond(zero(\\bar{f}(x,y)),x,g(succ(x),y)))$ Then, the desired function $\\bar{h}$ is defined as $\\bar{h}=\\lambda y.\\bar{g}(0,y)$ .",
"It is easy to see that this is a strict function.",
"Note that, if $h(m)$ is defined, clearly $h(m)=\\bar{g}(0,m)$ is also defined.",
"If $h(m)$ is undefined, it is also true that $\\bar{g}(0,m)=\\bot $ due to the fixed point construction but it is less obvious.",
"This argument is left to the reader.$\\:\\:\\Box $ Theorem REF does not claim that all $\\lambda $ -terms define partial recursive functions although this is also true.",
"Further examples of recursion are found in the exercises.",
"Exercises Exercise 5.6: Find the definitions of $\\lambda x,y.\\tau ~{\\rm and}~\\sigma (x,y)$ which use only $\\lambda v$ with one variable and applications only to one argument at a time.",
"Note that use must be made of the combinators $p_0$ , $p_1$ , and $pair$ .",
"Generalize the result to functions of many variables.",
"Exercise 5.7: The table of combinators was meant to show how combinators could be defined in terms of $\\lambda $ -expressions.",
"Can the tables be turned to show that, with enough combinators available, every $\\lambda $ -expression can be defined by combining combinators using application as the only mode of combination?",
"Exercise 5.8: Suppose that $f,g:{\\cal D}\\rightarrow {\\cal D}$ are approximable and $f\\circ g=g\\circ f$ .",
"Show that $f$ and $g$ have a least common fixed point $x=f(x)=g(x)$ .",
"(Hint: See Exercise REF .)",
"If, in addition, $f(\\bot )=g(\\bot )$ , show that $fix(f)=fix(g)$ .",
"Will $fix(f)=fix(f^2)$ ?",
"What if the assumption is weakened to $f\\circ g=g^2\\circ f$ ?",
"Exercise 5.9: For any domain ${\\cal D}$ , ${\\cal D}^\\infty $ can be regarded as consisting of bottomless stacks of elements of ${\\cal D}$ .",
"Using this view, define the following combinators with their obvious meaning: $head:{\\cal D}^\\infty \\rightarrow {\\cal D}$ , $tail:{\\cal D}^\\infty \\rightarrow {\\cal D}^\\infty $ and $push:{\\cal D}\\times {\\cal D}^\\infty \\rightarrow {\\cal D}^\\infty $ .",
"Using the fixed point theorem, argue that there is a combinator $diag:{\\cal D}\\rightarrow {\\cal D}^\\infty $ where for all $x\\in {\\cal D}$ , $diag(x)=\\langle x\\rangle _{n=0}^\\infty $ .",
"(Hint: Try a recursive definition, such as $diag(x)=push(x,diag(x))$ but be sure to prove that all terms of $diag(x)$ are $x$ .)",
"Also introduce by an appropriate recursive definition a combinator $map:({\\cal D}\\rightarrow {\\cal D})^\\infty \\times {\\cal D}\\rightarrow {\\cal D}^\\infty $ where for elements of the proper type $map(\\langle f_n\\rangle _{n=0}^\\infty ,x)=\\langle f_n(x)\\rangle _{n=0}^\\infty $ Exercise 5.10: For any domain ${\\cal D}$ introduce, as a least fixed point, a combinator $while:({\\cal D}\\rightarrow {\\cal T})\\times ({\\cal D}\\rightarrow {\\cal D})\\rightarrow ({\\cal D}\\rightarrow {\\cal D})$ by the recursion equation $while(p,f)(x)=cond(p(x),while(p,f)(f(x)),x)$ Prove that $while(p,while(p,f))=while(p,f)$ Show how $while$ could be used to obtain the least number operator,$\\mu $ , mentioned in the proof of Theorem REF .",
"Generalize this idea to define a combinator $find:{\\cal D}^\\infty \\times ({\\cal D}\\rightarrow {\\cal T})\\rightarrow {\\cal D}$ which means “find the first term in the sequence (if any) which satisfies the given predicate”.",
"Exercise 5.11: Prove the existence of a one-one function $num:{{N}}\\times {{N}}\\leftrightarrow {{N}}$ such that $\\begin{array}{lcl}num(0,0)&=&0\\\\num(n,m+1)&=&num(n+1,m)\\\\num(n+1,0)&=&num(0,n)+1\\end{array}$ Draw a descriptive picture (an infinite matrix) for the function.",
"Find a closed form for the values if possible.",
"Use the function to prove the isomorphism between ${\\cal P}({{N}})$ ,${\\cal P}({{N}}\\times {{N}})$ , and ${\\cal P}({{N}})\\times {\\cal P}({{N}})$ .",
"Exercise 5.12: Show that there are approximable mappings $graph:({\\cal P}({{N}})\\rightarrow {\\cal P}({{N}}))\\rightarrow {\\cal P}({{N}})$ and $fun:{\\cal P}({{N}})\\rightarrow ({\\cal P}({{N}})\\rightarrow {\\cal P}({{N}}))$ where $fun\\circ graph = \\lambda f.f$ and $graph\\circ fun\\sqsubseteq \\lambda x.x$ .",
"(Hint: Using the notation $[n_0,\\ldots ,n_k]=num(n_0,[n_1,\\ldots ,n_k])$ , two such combinators can be given by the formulas $\\begin{array}{lcl}fun(u)(x)&=&\\lbrace m\\:\\vert \\:\\exists n_0,\\ldots ,n_{k-1}\\in x.",
"[n_0+1,\\ldots ,n_{k-1}+1,0,m]\\in u\\rbrace \\\\graph(f)&=&\\lbrace [n_0+1,\\ldots ,n_{k-1}+1,0,m]\\:\\vert \\:m\\in f(\\lbrace n_0,\\ldots ,n_{k-1}\\rbrace )\\rbrace \\end{array}$ where $k$ is a variable - meaning all finite sequences are to be considered.)",
"Introduction to Domain Equations As stressed in the introduction, the notion of computation with potentially infinite elements is an integral part of domain theory.",
"The previous sections have defined the notion of functions over domains, as well as a notation for expressing these functions.",
"In addition, the notion of computation through series of approximations has been addressed.",
"This computation is possible since the functions defined have been approximable and thus continuous.",
"This section addresses the construction of more complex domains with infinite elements.",
"The next section looks specifically at the notion of computability with respect to these infinite elements.",
"The last section looks at another approach to domain construction.",
"New domains have been constructed from existing ones using domain constructors such as the product construction ($\\times $ ), the function space construction ($\\rightarrow $ ) and the sum construction ($+$ ) of Exercise REF .",
"These constructors can be iterated similar to the way that function application was iterated to form recursive function definitions.",
"In this way, domains can be characterized using recursion equations, called domain equations.",
"Domain Equations A domain equation represents an isomorphism between the domain as a whole and the combination of domains that comprise it.",
"These recursive domains are frequently termed reflexive domains since, as in the following example, the domain contains a copy of itself in its structure.",
"Example 6.1: Consider the following domain equation: ${\\cal T}={\\cal A}+({\\cal T}\\times {\\cal T})$ where ${\\cal A}$ is a previously defined domain.",
"This domain can be thought of as containing atomic elements from ${\\cal A}$ or pairs of elements of ${\\cal T}$ .",
"What do the elements of this domain look like?",
"In particular, what are the finite elements of this domain?",
"How is the domain constructed?",
"What is an appropriate approximation ordering for the domain?",
"What do lubs in this domain look like?",
"What is the appropriate notion of consistency?",
"Does this domain even exist?",
"In other words, are we certain a solution to this domain equation exists?",
"And if a solution to the equation exists, is it a unique solution?",
"Each of these questions is examined below.",
"The domain equation tells us that an element of the domain is either an element from ${\\cal A}$ or is a pair of “smaller” elements from ${\\cal T}$ .",
"One method of constructing a sum domain is using pairs where some distinguished element denotes what type an element is.",
"Thus, for some $a\\in {\\cal A}$ , the pair $\\langle \\pi ,a\\rangle $ might represent the element in ${\\cal T}$ for the given element $a$ .",
"For some $s,t\\in {\\cal T}$ , the pair $\\langle \\langle s,t\\rangle ,\\pi \\rangle $ might then represent the element in ${\\cal T}$ for the pair $s,t$ .",
"Thus, $\\pi $ is the distinguished element, and the location of $\\pi $ in the pair specifies the type of the element.",
"The finite elements are either elements in ${\\cal T}$ representing the (finite) elements of ${\\cal A}$ or the pair elements from ${\\cal T}$ whose components are also finite elements in ${\\cal T}$ .",
"The question then arises about infinite elements.",
"Are there infinite elements in this domain?",
"Consider the following fixed point equation for some element for $a\\in {\\cal A}$ : $x=\\langle \\langle a,x\\rangle ,\\pi \\rangle .$ The fixed point of this equation is the infinite product of the element $a$ .",
"Does this element fit the definition for ${\\cal T}$ ?",
"From the informal description of the elements of ${\\cal T}$ given so far, $x$ does qualify as a member of ${\\cal T}$ .",
"Now that some intuition has been developed about this domain, a formal construction is required.",
"Let $\\langle {\\bf A},\\sqsubseteq _A\\rangle $ be the finitary basis used to generate the domain ${\\cal A}$ .",
"Let $\\pi $ be an object such that $\\pi \\notin {\\bf A}$ .",
"Define the bottom element of the finitary basis T as $\\Delta _T=\\langle \\pi ,\\pi \\rangle $ .",
"Next, all the elements of ${\\cal A}$ must be included so define an element in ${\\bf T}$ for each $a\\in {\\bf A}$ as $\\langle \\pi ,a\\rangle $ .",
"Finally, pair elements for all elements in ${\\bf T}$ must exist in ${\\bf T}$ to complete the construction.",
"The set ${\\bf T}$ can be defined inductively as the least set such that: $\\Delta _T\\in {\\bf T}$ , $\\langle \\pi ,a\\rangle \\in {\\bf T}$ whenever $a\\in {\\bf A}$ , $\\langle \\langle \\Delta _T,s\\rangle ,\\pi \\rangle \\in {\\bf T}$ whenever $s\\in {\\bf T}$ (necessary??",
"), $\\langle \\langle t,\\Delta _T\\rangle ,\\pi \\rangle \\in {\\bf T}$ whenever $t\\in {\\bf T}$ (necessary??",
"), and $\\langle \\langle t,s\\rangle ,\\pi \\rangle \\in {\\bf T}$ whenever $s, t\\in {\\bf T}$ .",
"The set can also be characterized by the following fixed point equation: ${\\bf T}=\\lbrace \\Delta _T\\rbrace \\cup \\lbrace \\langle \\pi ,a\\rangle \\:\\vert \\:a\\in {\\bf A}\\rbrace \\cup \\lbrace \\langle \\langle \\Delta _T,s\\rangle ,\\pi \\rangle \\:\\vert \\:s\\in {\\bf T}\\rbrace \\cup \\lbrace \\langle \\langle t,\\Delta _T\\rangle ,\\pi \\rangle \\:\\vert \\:t\\in {\\bf T}\\rbrace \\cup \\lbrace \\langle \\langle t,s\\rangle ,\\pi \\rangle \\:\\vert \\:s,t\\in {\\bf T}\\rbrace .$ A solution must exist for this equation by the fixed point theorem.",
"Now that the basis elements have been defined, we must show how to find lubs.",
"We will again use an inductive definition.",
"$\\langle \\pi ,\\pi \\rangle \\sqcup t=t$ for all $t\\in {\\bf T}$ For $a,b\\in {\\bf A}$ , $\\langle \\pi ,a\\rangle \\sqcup \\langle \\pi ,b\\rangle =\\langle \\pi ,a\\sqcup b\\rangle $ if $a\\sqcup b$ exists in ${\\bf A}$ $\\langle \\langle s,t\\rangle ,\\pi \\rangle \\sqcup \\langle \\langle s^{\\prime },t^{\\prime }\\rangle ,\\pi \\rangle =\\langle \\langle s\\sqcup s^{\\prime },t\\sqcup t^{\\prime }\\rangle ,\\pi \\rangle $ if $s\\sqcup s^{\\prime }$ and $t\\sqcup t^{\\prime }$ exist in ${\\bf T}$ .",
"The lub $\\langle \\pi ,a\\rangle \\sqcup \\langle \\langle s,t\\rangle ,\\pi \\rangle $ does not exist.",
"Next, the notion of consistency needs to be explored.",
"From the definition of lubs given above, the following sets are consistent: The empty set is consistent.",
"Everything is consistent with the bottom element.",
"A set of elements all from the basis A is consistent in T if the set of elements is consistent in A.",
"A set of product elements in T is consistent if the left component elements are consistent and the right component elements are consistent.",
"These conditions derive from the sum and product nature of the domain.",
"The approximation ordering in the basis has the following inductive definition: $\\Delta _T\\sqsubseteq _T s$ for all $s\\in {\\bf T}$ $y\\sqsubseteq _Tu\\sqcup \\Delta _T$ whenever $y\\sqsubseteq _Tu$ $\\langle \\pi ,a\\rangle \\sqsubseteq _T\\langle \\pi ,b\\rangle $ whenever $a\\sqsubseteq _Ab$ $\\langle \\langle s,t\\rangle ,\\pi \\rangle \\sqsubseteq _T\\langle \\langle u,v\\rangle ,\\pi \\rangle $ whenever $s\\sqsubseteq _Tu$ and $t\\sqsubseteq _Tv$ The next step is to verify that ${\\bf T}$ is indeed a finitary basis.",
"The basis is still countable.",
"The approximation is clearly a partial order.",
"The existence of lubs of finite bounded (i.e., consistent) subsets must be verified.",
"The definition of consistency gives us the requirements for a bounded subset.",
"Each of the conditions for consistency are examined inductively since the definitions are all inductive: The lub of the empty set is the bottom element $\\Delta _T$ .",
"The lub of a set containing the bottom element is the lub of the set without the bottom element which must exist by the induction hypothesis.",
"The lub of a set of elements all from the ${\\bf A}$ is the element in ${\\bf T}$ for the lub in ${\\bf A}$ .",
"This element must exist since ${\\bf A}$ is a finitary basis and all elements from ${\\bf A}$ have corresponding elements in ${\\bf T}$ .",
"The lub of a set of product elements is the pair of the lub of the left components and the lub of the right components.",
"These exist by the induction hypothesis.",
"Thus, a finitary basis has been created; the domain is formed as always from the basis.",
"The solution to the domain equation has been found since any element in the domain ${\\cal T}$ is either an element representing an element in ${\\cal A}$ or is the product of two other elements in ${\\cal T}$ .",
"Similarly, any element found on the left-hand side must also be in the domain ${\\cal T}$ by the construction.",
"Thus, the domain ${\\cal T}$ is identical to the domain ${\\cal A}+({\\cal T}\\times {\\cal T})$ .",
"To look at the question concerning the existence and uniqueness of the solution to this domain equation, recall the fixed point theorem.",
"This theorem states that a fixed point set exists for any approximable mapping over a domain.",
"Subdomains In Section , the concept of a universal domain is introduced.",
"A universal domain is a domain which contains all other domains as sub-domains.",
"These sub-domains are, roughly speaking, the image of approximable functions over the universal domain.",
"The domain equation for ${\\cal T}$ can be viewed as an approximable mapping over the universal domain.",
"As such, the fixed point theorem states that a least fixed point set for the function does exist and is unique.",
"Sub-domains are defined formally below.",
"Looking again at the informal discussion concerning the elements of the domain ${\\cal T}$ , the infinite element proposed does fit into the formal definition for elements of ${\\cal T}$ .",
"This element is an infinite tree with all left sub-trees containing only the element $a$ .",
"For this infinite element to be computable, it must be the lub of some ascending chain of finite approximations to it.",
"The element $x$ can, in fact, be defined by the following ascending sequence of finite trees: $\\begin{array}{lcl}x_0&=&\\bot \\\\x_{n+1}&=&\\langle \\langle a,x_n\\rangle ,\\pi \\rangle \\\\x&=&\\bigsqcup ^\\infty _{n=0}x_n\\end{array}$ Thus, using domain equations, a domain has been defined recursively.",
"This domain includes infinite as well as finite elements and allows computation on the infinite elements to proceed using the finite approximations, as with the more conventionally defined domains presented earlier.",
"The final topic of this section is the notion of a sub-domain.",
"Informally, a sub-domain is a structured part of a larger domain.",
"Earlier, a domain was described as a sub-domain of the universal domain.",
"Thus, the sub-domain starts with a subset of the elements of the larger domain while retaining the approximation ordering, consistency relation and lub relation, suitably restricted to the subset elements.",
"Definition 6.2: [Sub-Domain] A domain $\\langle {\\cal R},\\sqsubseteq _R\\rangle $ is a sub-domain of a domain $\\langle {\\cal D},\\sqsubseteq _D\\rangle $ , denoted ${\\cal R}\\lhd {\\cal D}$ iff ${\\cal R}\\subseteq {\\cal D}$ - The elements of ${\\cal R}$ are a subset of the elements of ${\\cal D}$ .",
"$\\bot _R=\\bot _D$ - The bottom elements are the same.",
"For $x,y\\in {\\cal R}$ , $x\\sqsubseteq _Ry\\iff x\\sqsubseteq _Dy$ - The approximation ordering for ${\\cal R}$ is the approximation ordering for ${\\cal D}$ restricted to elements in ${\\cal R}$ .",
"For $x,y,z\\in {\\cal R}$ , $x\\sqcup _Ry=z$ iff $x\\sqcup _Dy=z$ - The lub relation for ${\\cal R}$ is the lub relation for ${\\cal D}$ restricted to elements in ${\\cal R}$ .",
"${\\cal R}$ is a domain.",
"Equivalently, a sub-domain can be thought of as the image of an approximable function which approximates the identity function (also termed a projection).",
"The notion of a sub-domain is used in the final section in the discussions about the universal domain.",
"This mapping between the domains can be formalized as follows: Theorem 6.3: If ${\\cal D}\\lhd {\\cal E}$ , then there exists a projection pair of approximable mappings $i:{\\cal D}\\rightarrow {\\cal E}$ and $j:{\\cal E}\\rightarrow {\\cal D}$ where $j\\circ i={{\\sf I}}_{\\cal D}$ and $i\\circ j\\sqsubseteq {{\\sf I}}_{\\cal E}$ where $i$ and $j$ are determined by the following equations: $\\begin{array}{lcl}i(x)&=&\\lbrace y\\in {\\bf E}\\:\\vert \\:\\exists z\\in x.z\\sqsubseteq y\\rbrace \\\\j(y)&=&\\lbrace x\\in {\\bf D}\\:\\vert \\:x\\in y\\rbrace \\end{array}$ for all $x\\in {\\cal D}$ and $y\\in {\\cal E}$ .",
"The proof is left as an exercise.",
"By the definition of a sub-domain, it should be clear that ${\\cal D}_0\\lhd {\\cal E}\\wedge {\\cal D}_1\\lhd {\\cal E}\\:\\Rightarrow \\:({\\cal D}_0\\lhd {\\cal D}_1\\iff {\\cal D}_0\\subseteq {\\cal D}_1)$ Using this observation, the sub-domains of a domain can be ordered.",
"Indeed, the following theorem is a consequence of this ordering.",
"Theorem 6.4: For a given domain ${\\cal D}$ , the set of sub-domains $\\lbrace {\\cal D}_0\\:\\vert \\:{\\cal D}_0\\lhd {\\cal D}\\rbrace $ form a domain.",
"The proof proceeds using the inclusion relation defined as an approximation ordering and is left as an exercise.",
"Finally, a converse of Theorem REF can also be established: Theorem 6.5: For two domains ${\\cal D}$ and ${\\cal E}$ , if there exists a projection pair $i:{\\cal D}\\rightarrow {\\cal E}$ and $j:{\\cal E}\\rightarrow {\\cal D}$ with $j\\circ i={\\sf I}_{\\cal D}$ and $i\\circ j\\sqsubseteq {\\sf I}_{\\cal E}$ , then $\\exists {\\cal D}^{\\prime }\\lhd {\\cal E}$ where ${\\cal D}\\approx {\\cal D}^{\\prime }$ .",
"Proof We show that $i$ maps finite elements to finite elements and that ${\\cal D}^{\\prime }$ is the image of ${\\cal D}$ in ${\\cal E}$ .",
"For some $x\\in {\\bf D}$ with ${\\cal I}_x$ as the principal ideal of $x$ , we can write $i({\\cal I}_x)=\\sqcup \\lbrace {\\cal I}_y\\:\\vert \\:y\\in i({\\cal I}_x)\\rbrace $ Applying $j$ to both sides we get ${\\cal I}_x=j\\circ i({\\cal I}_x)=\\sqcup \\lbrace j({\\cal I}_y)\\:\\vert \\:y\\in i({\\cal I}_x)\\rbrace $ since $j\\circ i={\\sf I}_D$ and $j$ is continuous by assumption.",
"But, since $x\\in {\\cal I}_x$ , $x\\in j({\\cal I}_y)$ for some $y\\in i({\\cal I}_x)$ .",
"This means that ${\\cal I}_x\\subseteq j({\\cal I}_y)$ and thus $i({\\cal I}_x)\\subseteq i\\circ j({\\cal I}_y) \\subseteq {\\cal I}_y$ Since ${\\cal I}_y\\subseteq i({\\cal I}_x)$ must hold by the construction, $i({\\cal I}_x) ={\\cal I}_y$ .",
"This proves that finite elements are mapped to finite elements.",
"Next, consider the value for $i(\\bot _D)$ .",
"Since $\\bot _D\\sqsubseteq _Dj(\\bot _E)$ , $i(\\bot _D)\\sqsubseteq \\bot _E$ .",
"Thus $i(\\bot _D)=\\bot _E$ .",
"Thus, ${\\cal D}$ is isomorphic to the image of $i$ in ${\\cal E}$ .",
"We still must show that ${\\cal D}^{\\prime }$ is a domain.",
"Thus, we need to show that if a lub exists in ${\\cal E}$ for a finite subset in ${\\cal D}^{\\prime }$ , then the lub is also in ${\\cal D}^{\\prime }$ .",
"Let $y^{\\prime },z^{\\prime }\\in {\\bf D}^{\\prime }$ and $y^{\\prime }\\sqcup z^{\\prime }=x^{\\prime }\\in {\\bf E}$ .",
"Then, there exists $y,z\\in {\\bf D}$ such that $i({\\cal I}_y)={\\cal I}_{y^{\\prime }}$ and $i({\\cal I}_z)={\\cal I}_{z^{\\prime }}$ which implies that ${\\cal I}_y=j({\\cal I}_{y^{\\prime }})$ and ${\\cal I}_z=j({\\cal I}_{z^{\\prime }})$ .",
"Since ${\\cal I}_{y^{\\prime }}\\sqsubseteq {\\cal I}_{x^{\\prime }}$ and $j({\\cal I}_{y^{\\prime }})\\sqsubseteq j({\\cal I}_{x^{\\prime }})$ by monotonicity, $y\\in j({\\cal I}_{x^{\\prime }})$ must hold.",
"By the same reasoning, $z\\in j({\\cal I}_{x^{\\prime }})$ .",
"But then $x=y\\sqcup z\\in j({\\cal I}_{x^{\\prime }})$ must also hold and thus $y\\sqcup z \\in {\\cal D}$ since the element $j({\\cal I}_{x^{\\prime }})$ must be an ideal.",
"But, $\\begin{array}{lcl}{\\cal I}_y\\sqsubseteq {\\cal I}_x&\\:\\Rightarrow \\:& {\\cal I}_{y^{\\prime }}\\sqsubseteq i({\\cal I}_{x})\\\\{\\cal I}_z\\sqsubseteq {\\cal I}_x&\\:\\Rightarrow \\:& {\\cal I}_{z^{\\prime }}\\sqsubseteq i({\\cal I}_{x})\\end{array}$ This implies that $y^{\\prime }\\sqcup z^{\\prime }=x^{\\prime }\\in i({\\cal I}_x)$ .",
"We already know that $x\\in j({\\cal I}_{x^{\\prime }})$ so $i({\\cal I}_x)\\sqsubseteq {\\cal I}_{x^{\\prime }}$ .",
"Thus, $i({\\cal I}_x)={\\cal I}_{x^{\\prime }}$ and thus, $x^{\\prime }\\in {\\bf D^{\\prime }}$ .$\\:\\:\\Box $ Exercises Exercise 6.6: Show that there must exist domains satisfying $\\begin{array}{lcll}{\\cal A}&=&{\\cal A}+({\\cal A}\\times {\\cal B})&{\\rm and}\\\\{\\cal B}&=&{\\cal A}+{\\cal B}\\end{array}$ Decide what the elements will look like and define ${\\cal A}$ and ${\\cal B}$ using simultaneous fixed points.",
"Exercise 6.7: Prove Theorem REF Exercise 6.8: Prove Theorem REF Exercise 6.9: Show that if ${\\cal A}$ and ${\\cal B}$ are finite systems, that ${\\cal D}\\unlhd {\\cal E}\\unlhd {\\cal D}\\:\\Rightarrow \\:{\\cal D}\\approx {\\cal E}$ where ${\\cal D}\\approx {\\cal D}^{\\prime }$ and ${\\cal D}^{\\prime }\\lhd {\\cal E}$ is denoted ${\\cal D}^{\\prime }\\unlhd {\\cal E}$ .",
"Computability in Effectively Given Domains In the previous sections, we gave considerable emphasis to the notion of computation using increasingly accurate approximations of the input and output.",
"This section defines this notion of computability more formally.",
"In Section 5, we found that partial functions over the natural numbers were expressible in the $\\lambda $ -notation.",
"This relationship characterizes computation for a particular domain.",
"To describe computation over domains in general, a broader definition is required.",
"The way a domain is presented impacts the way computations are performed over it.",
"Indeed, the theorems of recursive function theory [6] rely in part on the normal presentation of the natural numbers.",
"A presentation for a domain is an enumeration of the elements of the domain.",
"The standard presentation of the natural numbers is simply the numbers in ascending order beginning with 0.",
"There are many permutations of the natural numbers, each of which can be considered a presentation.",
"Computation with these non-standard presentations may be impossible; that is a computable function on the standard presentation may be non-computable over a non-standard presentation.",
"Therefore, an effective presentation for a domain is defined as a presentation which makes the required information computable.",
"Effective Presentations Information about elements in a domain can be characterized completely by looking at the finite elements and their relationships.",
"Thus a presentation must enumerate the finite elements and allow the consistency and lub relationships on these elements to be computed to allow this style of computation.",
"The consistency relation and the lub relation depend on each other.",
"For example, if a set of elements is consistent, a lub must exist for the set.",
"Given that a set is consistent, the lub can be found in finite time by just enumerating the elements and checking to see if this element is the lub.",
"However, if the set is inconsistent, the enumeration will not reveal this fact.",
"Thus, the consistency relation must be assumed to be recursive in an effective presentation.",
"Exercise REF provides a description of presentations that should clarify the assumptions made.",
"Formally, a presentation is defined as follows: Definition 7.1: [Effective Presentation] The presentation of a finitary basis D is a function $\\pi :{{N}}\\rightarrow {\\bf D}$ such that $\\pi (0)=\\Delta _D$ and the range of $\\pi $ is the set of finite elements of D. The definition holds for a domain ${\\cal D}$ as well.",
"A presentation $\\pi $ is effective iff The consistency relation ($\\exists k.\\pi _i\\sqsubseteq \\pi _k\\wedge \\pi _j\\sqsubseteq \\pi _k$ ) for elements $\\pi _i$ and $\\pi _j$ is recursiveRecursive in this context means that the relation is decidable.",
"over $i$ and $j$ .",
"The lub relation ($\\pi _k=\\pi _i\\sqcup \\pi _j$ ) is recursive over $i$ , $j$ , and $k$ .",
"This definition supports our intuition about domains; we have stated that the important information about a domain is the set of finite elements, the ordering and consistency relationships between the elements and the lub relation.",
"Thus, an effective presentation provides, in a suitable (that is computable) form, the basic information about the structure and elements of a domain.",
"A presentation can also be viewed as an enumeration of the elements of the domain with the position of an element in the enumeration given by the index corresponding to the integer input for that element in the presentation function with the 0 element representing $\\bot $ .",
"This perspective is used in the majority of the proofs.",
"Computability Now that the presentation of a domain has been formalized, the notion of computability can be formally defined.",
"Thus, Definition 7.2: [Computable Mappings] Given two domains, ${\\cal D}$ and ${\\cal E}$ with effective presentations $\\pi _1$ and $\\pi _2$ respectively, an approximable mapping $f:{\\bf D}\\rightarrow {\\bf E}$ is computable iff the relation $x_n\\:f\\:y_m$ is recursively enumerable in $n$ and $m$ .",
"By considering the domain ${\\cal D}$ to be a single element domain, the above definition applies not only to computable functions but also to computable elements.",
"For $d\\in {\\cal D}$ where $d$ is the only element in the domain, the element $e=f(d)\\in {\\cal E}$ defines an element in ${\\cal E}$ .",
"The definition states that $e$ is a computable iff the set $\\lbrace m\\in {{N}}\\:\\vert \\:y_m\\sqsubseteq e\\rbrace $ is a recursively enumerable set of integers.",
"Clearly if the set of elements approximating another is finite, the set is recursive.",
"The notion of a recursively enumerable set simply requires that all elements approximating the element in question be listed eventually.",
"The computation then proceeds by accepting an enumeration representing the input element and enumerating the elements that approximate the desired output element.",
"Now that the notions of computability and effective presentations have been formalized, the methods of constructing domains and functions will be addressed.",
"The proof of the next theorem is trivial and is left to the reader.",
"Theorem 7.3: The identity map on an effectively given domain is computable.",
"The composition of computable mappings on effectively given domains are also computable.",
"The following corollary is a consequence of this theorem: Corollary 7.4: For computable function $f:{\\cal D}\\rightarrow {\\cal E}$ and a computable element $x\\in {\\cal D}$ , the element $f(x)\\in {\\cal E}$ is computable.",
"In addition, the standard domain constructors maintain effective presentations.",
"Theorem 7.5: For domains ${\\cal D}_0$ and ${\\cal D}_1$ with effective presentations, the domains ${\\cal D}_0+{\\cal D}_1~{\\rm and}~ {\\cal D}_0\\times {\\cal D}_1$ are also effectively given.",
"In addition, the projection functions are all computable.",
"Finally, if $f$ and $g$ are computable maps, then so are $f+g$ and $f\\times g$ .",
"Proof Let $\\lbrace X_i\\:\\vert \\:i\\in {{N}}\\rbrace $ be the enumeration of ${\\cal D}_0$ and $\\lbrace Y_i\\:\\vert \\:i\\in {{N}}\\rbrace $ be the enumeration of ${\\cal D}_1$ .",
"Another method of sum construction is to use two distinguishing elements in the first position to specify the element type.",
"Thus, a sum domain can be defined as follows: ${\\cal D}_0+{\\cal D}_1=\\lbrace (\\Delta _0,\\Delta _1)\\rbrace \\cup \\lbrace (0,x)\\:\\vert \\:x\\in {\\cal D}_0\\rbrace \\cup \\lbrace (1,y)\\:\\vert \\:y\\in {\\cal D}_1\\rbrace $ The enumeration can then be defined as follows for $n\\in {{N}}$ : $\\begin{array}{lcl}Z_0&=&(\\Delta _0,\\Delta _1)\\\\Z_{2n+1}&=&(0,X_n)\\\\Z_{2n+2}&=&(1,Y_n)\\end{array}$ The proof that $Z_i$ is an effective presentation is left as an exercise.",
"For the product construction, the domain appears as follows: ${\\cal D}_0\\times {\\cal D}_1=\\lbrace (x,y)\\:\\vert \\:x\\in {\\cal D}_0,y\\in {\\cal D}_1\\rbrace $ The enumeration can be defined in terms of the functions $p:{{N}}\\rightarrow {{N}}$ , $q:{{N}}\\rightarrow {{N}}$ , and $r:({{N}}\\times {{N}})\\rightarrow {{N}}$ where for $m$ , $n$ , $k\\in {{N}}$ : $\\begin{array}{lcl}p(r(n,m))&=&n\\\\q(r(n,m))&=&m\\\\r(p(k),q(k))&=&k\\end{array}$ Thus, $r$ is a one-to-one pairing function (see Exercise REF ) of which there are several.",
"The functions $p$ and $q$ extract the indices from the result of the pairing function.",
"The enumeration for the product domain is then defined as follows: $W_i = (X_{p(i)},Y_{q(i)})$ The proof that this is an effective presentation is also left as an exercise.",
"For the combinators, the relations will be defined in terms of the enumeration indices.",
"For example, $\\begin{array}{lcl}X_n\\:in_0\\:Z_m&\\iff & m=0~{\\rm or}\\\\&&\\exists k.m=2k+1\\wedge X_k\\sqsubseteq X_n\\\\W_k\\:proj_1\\:Y_m&\\iff & Y_m\\sqsubseteq Y_{q(k)}\\end{array}$ The reader should verify that these sets are recursively enumerable.",
"For this proof, recall that recursively enumerable sets are closed under conjunction, disjunction, substituting recursive functions, and applying an existential quantifier to the front of a recursive predicate.",
"The proof for the other combinators is left as an exercise.",
"$\\:\\:\\Box $ Product spaces formalize the notion of computable functions of several variables.",
"Note that the proof of Theorem REF shows that substitution of computable functions of severable variables into other computable functions are still computable.",
"The next step is to show that the function space constructor preserves effectiveness.",
"Theorem 7.6: For domains ${\\cal D}_0$ and ${\\cal D}_1$ with effective presentations, the domain ${\\cal D}_0\\rightarrow {\\cal D}_1$ also has an effective presentation.",
"The combinators $apply$ and $curry$ are computable if all input domains are effectively given.",
"The computable elements of the domain ${\\cal D}_0\\rightarrow {\\cal D}_1$ are the computable maps for ${\\bf D_0}\\rightarrow {\\bf D_1}$ .",
"Proof Let ${\\cal D}_0=\\lbrace X_i\\:\\vert \\:i\\in {{N}}\\rbrace $ and ${\\cal D}_1=\\lbrace Y_i\\:\\vert \\:i\\in {{N}}\\rbrace $ be the presentations for the domains.",
"The elements of ${\\bf D_0}\\rightarrow {\\bf D_1}$ are finite step functions which respect the mapping of some subset of ${\\bf D_0}\\times {\\bf D_1}$ .",
"Given the enumeration, each element can be associated with a set $\\lbrace (X_{n_i},Y_{m_i})\\:\\vert \\:\\exists q.",
"1\\le i\\le q\\rbrace $ Thus, there is a finite set of integers pairs that determine the element.",
"Given the definition of consistency from Theorem REF for elements in the function space domain and the decidability of consistency in ${\\cal D}_0$ and ${\\cal D}_1$ , consistency of any finite set of this form is decidable (tedious but decidable since all elements must be checked with all others, etc).",
"Since consistency is decidable, a systematic enumeration of pair sets which are consistent can be made; this enumeration is simply the enumeration of ${\\cal D}_0\\rightarrow {\\cal D}_1$ .",
"Finding the lub consists of making a finite series of tests to find the element that is the lub, which must exist since the set is consistent and we have closure on lubs of finite consistent subsets.",
"Finding the lub requires a finite series of checks in both ${\\cal D}_0$ and ${\\cal D}_1$ but these checks are decidable.",
"Thus, the lub relation is also decidable in ${\\cal D}_0\\rightarrow {\\cal D}_1$ .",
"This shows that ${\\cal D}_0\\rightarrow {\\cal D}_1$ is effectively given.",
"To show that $apply$ and $curry$ are computable, the mappings need to be examined.",
"The mapping defined for apply is $(F,a)\\:apply\\: b\\iff a\\:F\\:b$ The function $F$ is the lub of all the finite step functions that are consistent with it.",
"As such, $F$ can be viewed as the canonical representative of this set.",
"Since $F$ is a finite step function, this relation is decidable.",
"As such, the $apply$ relation is recursive and not just recursively enumerable and $apply$ is a computable function.",
"The reasoning for $curry$ is similar in that the relations are studied.",
"Given the increase in the number of domains, the construction is more tedious and is left for the exercises.",
"To see that the computable elements correspond to the computable maps, recall the relationship shown in Theorem REF between the maps and the elements in the function space.",
"Thus, we have $a\\:f\\:b \\iff b\\in f({\\cal I}_a)~{\\rm or}~{\\cal I}_b\\sqsubseteq f({\\cal I}_a)$ Since $f$ is a computable map, we know that the pairs in the map are recursively enumerable.",
"Using the previous techniques for deciding consistency of finite sets, the set of elements consistent with $f$ can be enumerated.",
"But this set is simply the ideal for $f$ in the function space.",
"The converse direction is trivial.",
"$\\:\\:\\Box $ The final combinator to be discussed, and perhaps the most important, is the fixed point combinator.",
"Theorem 7.7: For any effectively given domain, ${\\cal D}$ , the combinator $fix:({\\cal D}\\rightarrow {\\cal D})\\rightarrow {\\cal D}$ is computable.",
"Proof Let $\\lbrace X_n\\:\\vert \\:n\\in {{N}}\\rbrace $ be the presentation of the domain ${\\cal D}$ .",
"Recall that for $f\\in {\\cal D}\\rightarrow {\\cal D}$ , $f\\:fix\\:X\\iff \\exists k\\in {{N}}.\\Delta \\:f\\:X_1\\:f\\ldots f\\:X_k\\wedge X_k=X$ All of the checks in this finite sequence are decidable since ${\\cal D}$ is effectively given.",
"In addition, existential quantification of a decidable predicate gives a recursively enumerable predicate.",
"Thus, $fix$ is computable.",
"$\\:\\:\\Box $ Recap Now that this has been formalized, what has been accomplished?",
"The major consequence of the theorems to this point is that any expression over effectively given domains (that is effectively given types) combined with computable constants using the $\\lambda $ -notation and the fixed point combinator is a computable function of its free variables.",
"Such functions, applied to computable arguments, yield computable values.",
"These functions also have computable least fixed points.",
"All this gives us a mathematical programming language for defining computable operations.",
"Combining this language with the specification of types with domain equations gives a powerful language.",
"As an example, the effectiveness of the domain ${\\cal T}$ from Example REF is studied.",
"The complete proof is left as an exercise.",
"Example 7.8: Recall the domain ${\\cal T}$ from the previous section.",
"This domain is characterized by the domain equation ${\\cal T}={\\cal A}+({\\cal T}\\times {\\cal T})$ for some domain ${\\cal A}$ .",
"If ${\\cal A}$ is effectively given, we wish to show that ${\\cal T}$ is effectively given as well.",
"The elements are either atomic elements from ${\\cal A}$ or are pairs from ${\\cal T}$ .",
"Let $A=\\lbrace A_i\\:\\vert \\:i\\in {{N}}\\rbrace $ be the enumeration for ${\\cal A}$ .",
"An enumeration for ${\\cal T}$ can be defined as follows: $\\begin{array}{lcl}T_0&=&\\bot _T\\\\T_{2n+1}&=&3*A_n\\\\T_{2n+2}&=&3*T_{p(n)}+1\\cup 3*T_{q(n)}+2\\end{array}$ where for $A$ , a set of indices, $m*A+k=\\lbrace m*n+k\\:\\vert \\:n\\in A\\rbrace $ .",
"The functions $p$ and $q$ here are the inverses of the pairing function $r$ defined in Theorem REF .",
"These functions must be defined such that $p(n)\\le n$ and $q(n)\\le n$ so that the recursion is well defined by taking smaller indices.",
"The rest of the proof is left to the exercises.",
"Specifically, the claim that ${\\cal T}=\\lbrace T_i\\rbrace $ should be verified as well as the effectiveness of the enumeration.",
"These proofs rely either on the effectiveness of ${\\cal A}$ , on the effectiveness of elements in ${\\cal T}$ with smaller indices, or are trivial.",
"The final example uses the powerset construction.",
"We have repeatedly used the fact that a powerset is a domain.",
"Its effectiveness is now verified.",
"Example 7.9: Specifically, the powerset of the natural numbers, ${\\cal {P}({{N}})}$ is considered.",
"In this domain, all elements are consistent, and there is a top element, denoted $\\omega $ , which is the set of all natural numbers.",
"The ordering is the subset relation.",
"The lub of two subsets is the union of the two subsets, which is decidable.",
"To enumerate the finite subsets, the following enumeration is used: $E_n=\\lbrace k\\:\\vert \\:\\exists i,j.",
"i< 2^k\\wedge n=i+2^k+j*2^{k+1}\\rbrace $ This says that $k\\in E_n$ if the $k$ bit in the binary expansion of $n$ is a 1.",
"All finite subsets of ${{N}}$ are of the form $E_n$ for some $n$ .",
"Various combinators for ${\\cal P}({{N}})$ are presented in Exercise REF .",
"Exercises Exercise 7.10: Show that an effectively given domain can always be identified with a relation $INCL(n,m)$ on integers where the derived relations $\\begin{array}{lcl}CONS(n,m)&\\iff &\\exists k.INCL(k,n)\\wedge INCL(k,m)\\\\MEET(n,m,k)&\\iff &\\forall j.",
"[INCL(j,k)\\iff INCL(j,n)\\wedge INCL(j,m)]\\end{array}$ are recursively decidable and where the following axioms hold: $\\forall n.INCL(n,n)$ $\\forall n,m,k.",
"INCL(n,m)\\wedge INCL(m,k)\\:\\Rightarrow \\:INCL(n,k)$ $\\exists m.\\forall n. INCL(n,m)$ $\\forall n,m.",
"CONS(n,m)\\:\\Rightarrow \\:\\exists k.MEET(n,m,k)$ Exercise 7.11: Finish the proof of Theorem REF .",
"Exercise 7.12: Complete the proof of Theorem REF by defining $curry$ as a relation and showing it computable.",
"Is the set recursively enumerable or is it recursive?",
"Exercise 7.13: Two effectively given domains are effectively isomorphic iff $\\ldots $ Complete the statement of the theorem and prove it.",
"Exercise 7.14: Complete the proof about the powerset in Example REF .",
"Show that the combinators $fun$ and $graph$ from Exercise REF are computable.",
"Show the same for $\\lambda x,y.x\\cap y$ $\\lambda x,y.x\\cup y$ $\\lambda x,y.x+ y$ where for $x,y\\in {\\cal P}({{N}})$ , $x+y=\\lbrace n+m\\:\\vert \\:n\\in x, m\\in y\\rbrace $ What are the computable elements of ${\\cal P}({{N}})$ ?",
"Sub-Spaces of the Universal Domain To have a flexible method of solving domain equations and yielding effectively given domains as the solutions, the domains will be embedded in a universal domain which is “big” enough to hold all other domains as sub-domains.",
"This universal domain is shown to be effectively presented, and the mappings which define the sub-spaces are shown to be computable.",
"First, the correspondence between sub-spaces and mappings called retractions is investigated, leading us to the definition of mappings called projections.",
"It is then shown that these definitions can be written out using the $\\lambda $ -calculus notation, demonstrating the power of our mathematical programming language.",
"Retractions and Projections We start with the definition of retractions.",
"Definition 8.1: [Retractions] A retraction of a given domain ${\\cal E}$ is an approximable mapping $a:{\\bf E}\\rightarrow {\\bf E}$ such that $a\\circ a=a$ .",
"Thus, a retraction is the identity function on objects in the range of the retraction and maps other elements into range.",
"The next theorem relates these sets to sub-spaces.",
"Theorem 8.2: If ${\\cal D}\\lhd {\\cal E}$ and if $a:{\\bf E}\\rightarrow {\\bf E}$ is defined such that $X\\:a\\:Z \\iff \\exists Y\\in {\\cal D}.",
"Z\\sqsubseteq Y\\sqsubseteq X$ for all $X,Z\\in {\\bf E}$ , then $a$ is a retraction and ${\\cal D}$ is isomorphic to the fixed point set of $a$ , the set $\\lbrace y\\in {\\cal E}\\:\\vert \\:a(y)=y\\rbrace $ , ordered under inclusion.",
"Proof That $a$ is an approximable map is a direct consequence of the definition of sub-space (Definition REF ).",
"By Theorem REF , a projection pair, $i$ and $j$ , exist for ${\\cal D}$ and this tells us that $a=i\\circ j$ (also showing $a$ approximable since approximable mappings are closed under composition).",
"Theorem REF also tells us that $j\\circ i={\\sf I}_D$ .",
"To show that $a$ is a retraction, $a\\circ a=a$ must be established.",
"Thus, $a\\circ a = i\\circ j\\circ i\\circ j = i\\circ {\\sf I}_D\\circ j = i\\circ j =a$ holds, showing that $a$ is a retraction.",
"We now need to show the isomorphism to ${\\cal D}$ .",
"For $x\\in {\\cal D}$ , $i(x)\\in {\\cal E}$ and we can calculate: $a(i(x))=i\\circ j\\circ i(x) = i\\circ {\\sf I}_D(x) = i(x)$ Thus, $i(x)$ is in the fixed point set of $a$ .",
"For the other direction, let $a(y)=y$ .",
"Then $i(j(y)) = y$ holds.",
"But, $j(y)\\in {\\cal D}$ , so $i$ must map ${\\cal D}$ one-to-one and onto the fixed point set of $a$ .",
"Since $i$ and $j$ are approximable, they are certainly monotonic, and thus the map is an isomorphism with respect to set inclusion.",
"$\\:\\:\\Box $ Not all retractions are associated with a sub-domain relationship.",
"The retractions defined in the above theorem are all subsets as relations of the identity relation.",
"The retractions for sub-domains are characterized by the following definition: Definition 8.3: [Projections] A retraction $a:{\\cal E}\\rightarrow {\\cal E}$ is a projection if $a\\subseteq {\\sf I}_E$ as relations.",
"The retraction is finitary iff its fixed point set is isomorphic to some domain.",
"An example is in order.",
"Example 8.4: Consider a two element system, ${\\bf O}$ with objects $\\Delta $ and 0.",
"For any basis ${\\bf D}$ that is not trivial (has more than one element), ${\\bf O}$ comes from a retraction on ${\\bf D}$ .",
"Define a combinator $check:{\\bf D}\\rightarrow {\\bf O}$ by the relation $x\\:check\\: y \\iff y=\\Delta ~{\\rm or}~x\\ne \\Delta _D$ Thus, $check(x)=\\bot _O\\iff x=\\bot _D$ .",
"Another combinator can be defined, $fade:{\\bf O}\\times {\\bf D}\\rightarrow {\\bf D}$ such that for $t\\in {\\cal O}$ and $x\\in {\\cal D}$ $\\begin{array}{lcll}fade(t,x)&=&\\bot _D&{\\rm if}~t=\\bot _O\\\\&=&x&otherwise\\end{array}$ For $u\\in {\\cal D}$ and $u\\ne \\bot _D$ , the mapping $a$ is defined as $a(x)=fade(check(x),u)$ It can be seen that $a$ is a retraction, but not a projection in general, and the range of $a$ is isomorphic to ${\\bf O}$ .",
"These combinators can also be used to define the subset of functions in ${\\bf D}\\rightarrow {\\bf E}$ that are strict.",
"Define a combinator $strict:({\\bf D}\\rightarrow {\\bf E})\\rightarrow ({\\bf D}\\rightarrow {\\bf E})$ by the equation $strict(f)=\\lambda x.fade(check(x),f(x))$ with $fade$ defined as $fade:{\\bf O}\\times {\\bf E}\\rightarrow {\\bf E}$ .",
"The range of $strict$ is all the strict functions; $strict$ is a projection whose range is a domain.",
"The next theorem characterizes projections.",
"Theorem 8.5: For approximable mapping $a:{\\bf E}\\rightarrow {\\bf E}$ , the following are equivalent: $a$ is a finitary projection $a(x)=\\lbrace y\\in {\\bf E}\\:\\vert \\:\\exists x^{\\prime }\\in I_x.",
"x^{\\prime }\\:a\\:x^{\\prime }\\wedge y\\sqsubseteq x^{\\prime }\\rbrace $ for all $x\\in {\\bf E}$ .",
"Proof Assume that (2) holds.",
"We want to show that $a$ is a finitary projection.",
"By the closure properties on ideals, we know that for all $x\\in {\\cal E}$ , $x^{\\prime }\\in x\\wedge y\\sqsubseteq x^{\\prime }\\:\\Rightarrow \\:y\\in x$ Thus, $a(x)\\subseteq x$ must hold.",
"In addition, the following trivially holds: $x^{\\prime }\\in x\\wedge x^{\\prime }\\:a\\: x^{\\prime }\\:\\Rightarrow \\:x^{\\prime }\\in a(x)$ thus $a(x)\\subseteq a(a(x))$ holds for all $x\\in {\\cal E}$ .",
"This shows that $a$ is indeed a projection.",
"Let $D=\\lbrace x\\in {\\bf E}\\:\\vert \\:x\\:a\\:x\\rbrace $ .",
"It is easy to show that ${\\bf D}\\lhd {\\bf E}$ and that $a$ is determined from ${\\bf D}$ as required in Theorem REF .",
"Thus, the fixed point set of $a$ is isomorphic to a domain from the previous proofs.",
"Thus, (2)$\\:\\Rightarrow \\:$ (1).",
"For the converse, assume that $a$ is a finitary projection.",
"Let ${\\cal D}$ be isomorphic to the fixed point set of $a$ .",
"This means there is a projection pair $i$ and $j$ such that $j\\circ i={\\sf I}_D$ and $i\\circ j = a$ and $a\\subseteq {\\sf I}_E$ .",
"From Theorem REF then we have that ${\\cal D}\\approx {\\cal D}^{\\prime }$ and ${\\cal D}^{\\prime }\\lhd {\\cal E}$ .",
"We want to identify ${\\cal D}^{\\prime }$ as follows: ${\\cal D}^{\\prime }=\\lbrace x\\in {\\cal E}\\:\\vert \\:x\\:a\\:x\\rbrace $ From the proof of Theorem REF , the basis elements of ${\\bf D^{\\prime }}$ are the finite elements of ${\\bf D}$ .",
"Each of these elements is in the fixed point set of $a$ .",
"Thus, $x\\in {\\bf D^{\\prime }}\\:\\Rightarrow \\:a({{\\cal I}}_x) = {{\\cal I}}_x \\:\\Rightarrow \\:x\\:a\\:x$ Since $a$ is a projection, ${{\\cal I}}_x$ must also be a fixed point.",
"Since $i(j({{\\cal I}}_x)) = {{\\cal I}}_x$ implies that $j({{\\cal I}}_x)$ is a finite element of ${\\cal D}$ , $x\\in {\\cal D}^{\\prime }$ must hold.",
"Thus, the identification of ${\\cal D}^{\\prime }$ holds.",
"Finally, using $a=i\\circ j$ in the formula in Theorem REF , the formula in (2) is obtained, proving the converse.",
"$\\:\\:\\Box $ This characterization of projections provides a new and interesting combinator.",
"Theorem 8.6: For any domain ${\\cal E}$ , define $sub:({\\cal E}\\rightarrow {\\cal E})\\rightarrow ({\\cal E}\\rightarrow {\\cal E})$ using the relation $x\\: sub(f)\\: z \\iff \\exists y\\in {\\bf E}.y\\:f\\:y\\wedge y\\sqsubseteq x\\wedge z\\sqsubseteq y$ for all $x,z\\in {\\bf E}$ and all $f:{\\bf E}\\rightarrow {\\bf E}$ .",
"Then the range of $sub$ is exactly the set of finitary projections on ${\\cal E}$ .",
"In addition, $sub$ is a finitary projection on ${\\cal E}\\rightarrow {\\cal E}$ .",
"If ${\\cal E}$ is effectively given, then $sub$ is computable.",
"Proof Clearly, $sub(f)$ is approximable.",
"It is obvious from the definition that $f\\mapsto sub(f)$ preserves lubs and thus is approximable as well.",
"Thus, $y\\:f\\:y\\wedge y\\sqsubseteq x\\wedge z\\sqsubseteq y\\:\\Rightarrow \\:x\\:f\\:z$ obviously holds.",
"Thus, $sub(f)\\subseteq f$ holds.",
"Also $y\\:f\\:y\\:\\Rightarrow \\:y\\:sub(f)\\:y$ thus, $sub(f)\\subseteq sub(sub(f))$ holds as well.",
"Thus, $sub$ is a projection on ${\\cal E}\\rightarrow {\\cal E}$ .",
"The definition of the relation shows that it is computable when ${\\cal E}$ is effectively given.",
"Since $sub$ is a projection, its range is the same as its fixed point set.",
"If $sub(a)=a$ , it is easy to see that clause (2) of Theorem REF holds and conversely.",
"Thus, the range of $sub$ is the finitary projections.",
"To see that $sub$ is a finitary projection, we use Theorem REF and Theorem REF to say that the fixed point set of $sub$ is in a one-to-one inclusion preserving correspondence with the domain $\\lbrace D\\:\\vert \\:D\\lhd {\\cal E}\\rbrace $ .",
"$\\:\\:\\Box $ Universal Domain ${\\cal U}$ With these results and the universal domain to be defined next, the theory of sub-domains is translated into the $\\lambda $ -calculus notation using the $sub$ combinator.",
"The universal domain is defined by first defining a domain which has the desired structure but has a top element.",
"The top element is then removed to give the universal domain.",
"Definition 8.7: [Universal Domain] As in the section on domain equations, an inductive definition for a domain ${\\cal V}$ is given as follows: $\\Delta ,\\top \\in {\\bf V}$ $\\langle u,v\\rangle \\in {\\bf V}$ whenever $u,v\\in {\\bf V}$ Thus, we are starting with two objects, a bottom element and a top element, and making two flavors of copies of these objects.",
"Intuitively, we end up with finite binary trees with either the top or the bottom element as the leaves.",
"To simplify the definitions below, the pairs should be reduced such that: All occurrences of $\\langle \\Delta ,\\Delta \\rangle $ are replaced by $\\Delta $ and All occurrences of $\\langle \\top ,\\top \\rangle $ are replaced by $\\top $ .",
"These rewrite rules are easily shown to be finite Church-Rosser.The finitary basis should be defined as the equivalence classes induced by the reduction.",
"The presentation is simplified by considering only reduced trees.",
"As an example of the reduction the pair $\\langle \\langle \\langle \\top ,\\langle \\top ,\\top \\rangle \\rangle ,\\langle \\top ,\\Delta \\rangle \\rangle ,\\langle \\langle \\Delta ,\\Delta \\rangle ,\\langle \\top ,\\top \\rangle \\rangle \\rangle $ reduces to $\\langle \\langle \\top ,\\langle \\top ,\\Delta \\rangle \\rangle ,\\langle \\Delta ,\\top \\rangle \\rangle $ .",
"The approximation ordering is defined as follows: $\\Delta \\sqsubseteq v$ for all $v\\in {\\bf V}$ $v\\sqsubseteq \\top $ for all $v\\in {\\bf V}$ .",
"$\\langle u,v\\rangle \\sqsubseteq \\langle u^{\\prime },v^{\\prime }\\rangle $ iff $u\\sqsubseteq u^{\\prime }$ and $v\\sqsubseteq v^{\\prime }$ Since the top element is approximated by everything, all finite sets of trees are consistent.",
"The lub for a pair of trees is defined as follows: $u\\sqcup \\top =\\top $ for $u\\in {\\bf V}$ $\\top \\sqcup u=\\top $ for $u\\in {\\bf V}$ $u\\sqcup \\Delta =u$ for $u\\in {\\bf V}$ $\\Delta \\sqcup u=u$ for $u\\in {\\bf V}$ $\\langle u,v\\rangle \\sqcup \\langle u^{\\prime },v^{\\prime }\\rangle =\\langle u\\sqcup u^{\\prime },v\\sqcup v^{\\prime }\\rangle $ for $u,v\\in {\\bf V}$ The proof that this forms a finitary basis follows the same guidelines as the proofs in Section .",
"In addition, it should be clear that the presentation is effective.",
"To form the universal domain, the top element is simply removed.",
"Thus, the system ${\\bf U}={\\bf V}-\\lbrace \\top \\rbrace $ is the basis used to form the universal domain.",
"The proof that this is still a finitary basis with an effective presentation is also straightforward and left to the exercises.",
"Note that inconsistent sets can now exist since there is no top element.",
"A set is inconsistent iff its lub is $\\top $ .",
"We shall now prove the claims made for the universal domain.",
"Theorem 8.8: The domain ${\\cal U}$ is universal, in the sense that for every domain ${\\cal D}$ we have ${\\cal D}\\lhd {\\cal U}$ .",
"If ${\\cal D}$ is effectively given, then the projection pair for the embedding is computable.",
"In fact, there is a correspondence between the effectively presented domains and the computable finitary projections of ${\\cal U}$ .",
"Proof Recall that ${\\bf D}$ must be countable to be a finitary basis.",
"Thus, we can assume that the basis has an enumeration $D=\\lbrace X_n\\:\\vert \\:n\\in {{N}}\\rbrace $ where $X_0=\\Delta $ .",
"The effective and general cases are considered together in the proof; comments about computability are included for the effective case as required.",
"Thus, if ${\\cal D}$ is effectively given, the enumeration above is assumed to be computable.",
"To prove that the domain can be embedded in ${\\cal U}$ , the embedding will be shown.",
"To start, for each finite element $d_i$ in the basis, define two sets, $d_i^+$ and $d_i^-$ as follows: $\\begin{array}{lcl}d_i^+&=&\\lbrace d\\in {\\bf D}\\:\\vert \\:d_i\\sqsubseteq d\\rbrace \\\\d_i^-&=&D-d_i^+\\end{array}$ The $d_i^+$ set contains all the elements that $d_i$ approximates, while the $d_i^-$ set contains all the other elements, partitioning ${\\bf D}$ into two disjoint sets.",
"Sets for different elements can be intersected to form finer partitions of ${\\bf D}$ .",
"For $k>0$ , let $R\\in \\lbrace +,-\\rbrace ^k$ , let $R_i$ be the $ith$ symbol in the string $R$ , and define a region $D_R$ as $D_R=\\bigcap \\limits _{i=1}^k d_i^{R_i}$ where $k$ is the length of $R$ .",
"The set $\\lbrace D_{R}\\:\\vert \\:R\\in \\lbrace +,-\\rbrace ^k\\rbrace $ of regions partitions ${\\bf D}$ into $2^k$ disjoint sets.",
"Thus, for each element $e_i$ in the enumeration there is a corresponding partition of the basis given by the family of sets $\\lbrace D_{R}\\:\\vert \\:R\\in \\lbrace +,-\\rbrace ^i\\rbrace $ .",
"For strings $R,S\\in \\lbrace +,-\\rbrace ^*$ such that $R$ is a prefix of $S$ , denoted $R\\le S$ , $D_S\\subseteq D_R$ .",
"It is important to realize that the composition of these sets is dependent on the order in which the elements are enumerated.",
"Some of these regions are empty, but it is decidable if a given intersection is empty if ${\\cal D}$ is effectively presented.",
"It is also decidable if a given element is in a particular region.",
"Figure: Example Finite DomainTo see the function these regions are serving, consider the finite domain in Figure REF .This example is taken from Cartwright and Demers [2].",
"Consider the enumeration with $d_0=\\bot , d_1=b, d_2=c, d_3=a.$ The $d_i^+$ and $d_i^-$ sets are as follows: $\\begin{array}{lcl}d_1^+&=&\\lbrace a,b\\rbrace \\\\d_1^-&=&\\lbrace c,\\bot \\rbrace \\\\d_2^+&=&\\lbrace c\\rbrace \\\\d_2^-&=&\\lbrace a,b,\\bot \\rbrace \\\\d_3^+&=&\\lbrace a\\rbrace \\\\d_3^-&=&\\lbrace b,c,\\bot \\rbrace \\end{array}$ The regions are as follows: $\\begin{array}{lclclcl}D_+ &=&\\lbrace a,b\\rbrace &\\:\\:\\:\\:&D_{+++} &=&\\lbrace \\rbrace \\\\D_- &=&\\lbrace \\bot ,c\\rbrace &&D_{++-} &=&\\lbrace \\rbrace \\\\D_{++} &=&\\lbrace \\rbrace &&D_{+-+} &=&\\lbrace a\\rbrace \\\\D_{+-} &=&\\lbrace a,b\\rbrace &&D_{+--} &=&\\lbrace b\\rbrace \\\\D_{-+} &=&\\lbrace c\\rbrace &&D_{-++} &=&\\lbrace \\rbrace \\\\D_{--} &=&\\lbrace \\bot \\rbrace &&D_{-+-} &=&\\lbrace c\\rbrace \\\\&&&&D_{--+} &=&\\lbrace \\rbrace \\\\&&&&D_{---} &=&\\lbrace \\bot \\rbrace \\end{array}$ The regions generated by each successive element encode the relationships induced by the approximation ordering between the new element and all elements previously added.",
"The reader is encouraged to try this example with other enumerations of this basis and compare the results.",
"The embedding of the elements proceeds by building a tree based on the regions corresponding to the element.",
"The regions are used to find locations in the tree and to determine whether a $\\top $ or a $\\Delta $ element is placed in the location.",
"These trees preserve the relationships specified by the regions and thus, the tree embedding is isomorphic to the domain in question.",
"Once the tree is built, the reduction rules are applied until a non-reducible tree is reached.",
"This tree is the representative element in the universal domain, and the set of these trees form the sub-space.",
"The function to determine the location in the tree for a given domain, $Loc_D:\\lbrace +,-\\rbrace ^*\\rightarrow \\lbrace l,r\\rbrace ^*$ takes strings used to generate regions and outputs a path in a tree where $l$ stands for left sub-tree and $r$ stands for right sub-tree.",
"This path is computed using the following inductive definition: $\\begin{array}{lcll}Loc_D(\\epsilon )&=&\\epsilon .\\\\Loc_D(R+)&=&Loc_D(R)l&{\\rm if }~D_{R+}\\ne \\emptyset ~ {\\rm and }~D_{R-}\\ne \\emptyset .\\\\&=&Loc_D(R)&{\\rm otherwise}.\\\\Loc_D(R-)&=&Loc_D(R)r&{\\rm if }~D_{R+}\\ne \\emptyset ~{\\rm and }~D_{R-}\\ne \\emptyset .\\\\&=&Loc_D(R)&{\\rm otherwise}.\\end{array}$ The set of locations for each non-empty region is the set of paths to all leaves of some finite binary tree.",
"An induction argument is used to show the following properties of $Loc_D$ that ensure this: If $R\\le S$ for $R,S\\subseteq \\lbrace +,-\\rbrace ^*$ , then $Loc_D(R)\\le Loc_D(S)$ .",
"Let $S=\\lbrace Loc_D(R)\\:\\vert \\:R\\in \\lbrace +,-\\rbrace ^k\\wedge D_R\\ne \\emptyset \\rbrace $ for $k>0$ be a set of location paths for a given $k$ .",
"For any $p\\in \\lbrace l,r\\rbrace ^*$ there exists $q\\in S$ such that either $p\\le q$ or $q\\le p$ .",
"That is, every potential path is represented by some finite path.",
"Finally, for all $p,q\\in S$ if $p\\le q$ then $p=q$ .",
"This means that a unique leaf is associated with each location.",
"To find the tree for a given element $d_k$ in the enumeration, apply the following rules to each $R\\in \\lbrace +,-\\rbrace ^{k-1}$ .",
"If $D_{R-}\\ne \\emptyset $ then the leaf for path $Loc_D(R-)$ is labeled $\\top $ .",
"If $D_{R+}\\ne \\emptyset $ then the leaf for path $Loc_D(R+)$ is labeled $\\Delta $ .",
"These rules are used to assign a tree in ${\\bf U}$ , which is then reduced using the reduction rules, for each element in the enumeration of ${\\bf D}$ .",
"To see that the top element is never assigned by these rules, note that some region of the form $R+$ for every length $k$ must be non-empty since it must contain the element $e_k$ being embedded.",
"Returning to the example, the location function defines paths for these elements as follows: $\\begin{array}{lclclcl}Loc_D(+)&=&l&\\:\\:\\:\\:&Loc_D(+-+)&=&ll\\\\Loc_D(-)&=&r&&Loc_D(+--)&=&lr\\\\Loc_D(+-)&=&l&&Loc_D(-+-)&=&rl\\\\Loc_D(-+)&=&rl&&Loc_D(---)&=&rr\\\\Loc_D(--)&=&rr\\end{array}$ The trees generated for each of the elements are: $\\begin{array}{lcl}d_0&\\mapsto &\\Delta \\\\d_1&\\mapsto &\\langle \\Delta ,\\top \\rangle \\\\d_2&\\mapsto &\\langle \\top ,\\langle \\Delta ,\\top \\rangle \\rangle \\\\d_3&\\mapsto & \\langle \\langle \\Delta ,\\top \\rangle ,\\langle \\top ,\\top \\rangle \\rangle \\\\&\\mapsto & \\langle \\langle \\Delta ,\\top \\rangle ,\\top \\rangle \\end{array}$ To verify that the space generated is a valid sub-space, we must verify that the bottom element is mapped to $\\bot _U$ and that the consistency and lub relations are maintained.",
"The tree $\\Delta $ is clearly assigned to $X_0$ , the bottom element for the basis being embedded, since there are no strings of length $-1$ .",
"The embedding preserves inconsistency of elements by forcing the lub of the embedded elements to be $\\top $ .",
"The $D_{R-}$ regions represent the elements that the element being embedded does not approximate.",
"Note that the $D_{R-}$ sets cause the $\\top $ element to be added as the leaf.",
"Since the $D_R$ sets are built using the approximation ordering, it is straightforward to see that the approximation ordering is preserved by the embedding.",
"Lubs are also maintained by the embedding, although the reduction is required to see that this is the case.",
"It should be clear that, if the domain ${\\cal D}$ is effectively given, the sub-space can be computed since the embedding procedure uses the relationships given in the presentation.",
"Finally, suppose that $a$ is a computable, finitary projection on ${\\cal U}$ .",
"From the proof of Theorem REF , the domain of this projection is characterized by the set $\\lbrace y\\in {\\bf U}\\:\\vert \\:y\\:a\\:y\\rbrace $ If $a$ is computable, the set of pairs for $a$ is recursively enumerable.",
"Thus, the set above is also recursively enumerable since equality among basis elements is decidable.",
"Thus, the domain given by the projection must also be effectively given.",
"$\\:\\:\\Box $ Thus, the domain ${\\cal U}$ is an effectively presented universal domain in which all other domains can be embedded.",
"The sub-domains of ${\\cal U}$ include ${\\cal U}\\rightarrow {\\cal U}$ , ${\\cal U}\\times {\\cal U}$ , etc.",
"These domains must be sub-domains of ${\\cal U}$ since they are effectively presented based on our earlier theorems.",
"Domain Constructors in ${\\cal U}$ The next step is to see how to define the constructors commonly used.",
"Definition 8.9: [Domain Constructors] Let the computable projection pair, $i_+:{\\cal U}+{\\cal U}\\rightarrow {\\cal U}~{\\rm and}~j_+:{\\cal U}\\rightarrow {\\cal U}+{\\cal U}$ be fixed.",
"Fix suitable projection pairs $i_\\times ,j_\\times ,i_\\rightarrow $ , and $j_\\rightarrow $ as well.",
"Define $\\begin{array}{lcl}a+b&=&cond\\circ \\langle which,i_+\\circ in_0\\circ a\\circ out_0, i_+\\circ in_1\\circ b\\circ out_1\\rangle \\circ j_+\\\\a\\times b&=&i_x\\circ \\langle a\\circ proj_0,b\\circ proj_1\\rangle \\circ j_x\\\\a\\rightarrow b&=&i_\\rightarrow \\circ (\\lambda f.b\\circ f\\circ a)\\circ j_\\rightarrow \\end{array}$ for all $a,b:{\\cal U}\\rightarrow {\\cal U}$ .",
"From earlier theorems, we know that these combinators are all computable over an effectively presented domain.",
"The next theorem characterizes the effect these combinators have on projection functions.",
"Theorem 8.10: If $a,b:{\\cal U}\\rightarrow {\\cal U}$ are projections, then so are $a+b$ , $a\\times b$ , and $a\\rightarrow b$ .",
"If $a$ and $b$ are finitary, then so are the compound projections.",
"Proof Since $a$ and $b$ are retractions, $a=a\\circ a$ and $b=b\\circ b$ .",
"Then for $a\\times b$ using the definition of $\\times $ , $\\begin{array}{lcl}(a\\times b)\\circ (a\\times b)&=&i_x\\circ \\langle a\\circ proj_0,b\\circ proj_1\\rangle \\circ \\langle a\\circ proj_0,b\\circ proj_1\\rangle \\circ j_x\\\\&=&i_x\\circ \\langle a\\circ a\\circ proj_0,b\\circ b\\circ proj_1\\rangle \\circ j_x\\\\&=& a\\times b\\end{array}$ Thus, $a\\times b$ is a retraction.",
"The other cases follow similarly.",
"Since $a$ and $b$ are projections, $a,b\\subseteq {\\sf I}_U$ (denoted simply ${\\sf I}$ for the remainder of the proof).",
"Using the definition for $+$ along with the above relation and the definition of projection pairs, we can see that $a+b\\subseteq {\\sf I}+{\\sf I}=i_+\\circ j_+ \\subseteq {\\sf I}$ Thus, $a+b$ is a projection.",
"The other cases follow similarly.",
"To show that the projections are finitary, we must show that the fixed point sets are isomorphic to a domain.",
"Since $a$ and $b$ are assumed finitary, their fixed point sets are isomorphic to $\\begin{array}{lcl}D_a&=&\\lbrace x\\in {\\bf U}\\:\\vert \\:x\\:a\\: x\\rbrace \\\\D_b&=&\\lbrace y\\in {\\bf U}\\:\\vert \\:y\\:b\\: y\\rbrace \\end{array}$ We wish to show that ${\\cal D}_a\\rightarrow {\\cal D}_b\\approx {\\cal D}_{a\\rightarrow b}$ .",
"By the definition of the $\\rightarrow $ constructor, the fixed point set of $a\\rightarrow b$ over ${\\cal U}$ is the same as the fixed point set of $\\lambda f.b\\circ f\\circ a$ on ${\\cal U}\\rightarrow {\\cal U}$ .",
"(Hint: $i_\\rightarrow $ and $j_\\rightarrow $ set up the isomorphism.)",
"So, the fixed points for $f:{\\cal U}\\rightarrow {\\cal U}$ are of the form: $f=b\\circ f\\circ a$ We can think of $a$ as a function in ${\\cal U}\\rightarrow {\\cal D}_a$ and define the other half of the projection pair as $i_a:{\\cal D}_a\\rightarrow {\\cal U}$ where $i_a\\circ a = a$ and $a\\circ i_a=i_a$ .",
"Define a function $i_b$ for the projection pair for $b$ similarly.",
"For some $g:{\\cal D}_a\\rightarrow {\\cal D}_b$ let $f=i_b\\circ g\\circ a$ Substituting this definition for $f$ yields $b\\circ f\\circ a = b\\circ i_b\\circ g\\circ a\\circ a = i_b\\circ g \\circ a = f$ by the definition of $i_b$ and since $a$ is a retraction by assumption.",
"Conversely, for a function $f$ such that $i_b\\circ g\\circ a= f$ , let $g=b\\circ f\\circ i_a$ Substituting again, $i_b\\circ g\\circ a = i_b\\circ g\\circ f\\circ i_a\\circ a = b\\circ f\\circ a = f$ Thus, there is an order preserving isomorphism between $g:{\\cal D}_a\\rightarrow {\\cal D}_b$ and the functions $f=b\\circ f\\circ a$ .",
"The proofs of the isomorphisms for the other constructs are similar.",
"$\\:\\:\\Box $ Thus, the sub-domain relationship with the universal domain has been stated in terms of finitary projections over the universal domain.",
"In addition, all the domain constructors have been shown to be computable combinators on the domain of these finitary projections.",
"Recalling that all computable maps have computable fixed points, the standard fixed point method can be used to solve domain equations of all kinds if they can be defined on projections.",
"Returning to the $\\lambda $ -calculus for a moment, all objects in the $\\lambda $ -calculus are considered functions.",
"Since ${\\cal U}\\rightarrow {\\cal U}$ is a part of ${\\cal U}$ , every object in the $\\lambda $ -calculus is also an object of ${\\cal U}$ .",
"Transposing some of the familiar notation, where the old notation appears on the left, the new combinators are defined as follows: $\\begin{array}{lcl}which(z)=which(j_+(z))\\\\in_i(x)=i_+(in_i(x))~{\\rm where}~i=0,1\\\\out_i(x)=out_i(j_+(x))~{\\rm where}~i=0,1\\\\\\langle x,y\\rangle =i_x(\\langle x,y\\rangle )\\\\proj_i=proj_i(j_x(z))~{\\rm where}~i=0,1\\\\u(x) = j_\\rightarrow (u)(x)\\\\\\lambda x.\\tau =i_\\rightarrow (\\lambda x.\\tau )\\end{array}$ Thus, all functions, all constants, all combinators, and all constructs are elements of ${\\cal U}$ .",
"Indeed, everything computable is an element of ${\\cal U}$ .",
"Elements in ${\\cal U}$ play multiple roles by representing different objects under different projections.",
"While this notion may be difficult to get used to, there are many advantages, both notational and conceptual.",
"Exercises Exercise 8.11: A retraction $a:{\\cal D}\\rightarrow {\\cal D}$ is a closure operator iff ${\\sf I}_D\\subseteq a$ as relations.",
"On a domain like ${\\cal P}({{N}})$ , give some examples of closure operators.",
"(Hint: Close up the integers under addition.",
"Is this continuous on ${\\cal P}({{N}})$ ?)",
"Prove in general that for any closure $a:{\\cal D}\\rightarrow {\\cal D}$ , the fixed point set of $a$ is always a finitary domain.",
"(Hint: Show that the fixed point set is closed as required for a domain.)",
"What are the finite elements of the fixed point set?",
"Exercise 8.12: Give a direct proof that the domain $\\lbrace X\\:\\vert \\:X\\lhd {\\cal D}\\rbrace $ is effectively presented if ${\\cal D}$ is.",
"(Hint: The finite elements of the domain correspond exactly to the finite domains $X\\lhd {\\cal D}$ .)",
"In the case of ${\\cal D}={\\cal U}$ , show that the computable elements of the domain correspond exactly to the effectively presented domains (up to effective isomorphism).",
"Exercise 8.13: For finitary projections $a:{\\cal E}\\rightarrow {\\cal E}$ , write ${\\cal D}_a=\\lbrace x\\in {\\cal E}\\:\\vert \\:x\\:a\\:x\\rbrace $ Show that for any two such projections $a$ and $b$ , that $a\\subseteq b \\iff {\\cal D}_a\\lhd {\\cal D}_b$ Exercise 8.14: Find another universal domain that is not isomorphic to ${\\cal U}$ .",
"Exercise 8.15: Prove the remaining cases in Theorem REF .",
"Exercise 8.16: Suppose $S$ and $T$ are two binary constructors on domains that can be made into computable operators on projections over the universal domain.",
"Show that we can find a pair of effectively presented domains such that $D\\approx S(D,E)~{\\rm and}~E\\approx T(D,E).$ Exercise 8.17: Using the translations shown after the proof of Theorem REF , show how the whole typed-$\\lambda $ -calculus can be translated into ${\\cal U}$ .",
"(Hint: for $f:{\\cal D}_a\\rightarrow {\\cal B}$ , write $f=b\\circ f\\circ a$ for finitary projections $a$ and $b$ .",
"For $\\lambda x^{{\\cal D}_a}.\\sigma $ , write $\\lambda x.b(\\sigma ^{\\prime }[a(x)/x])$ where $\\sigma ^{\\prime }$ is the translation of $\\sigma $ into the untyped $\\lambda $ -calculus.",
"Be sure that the resulting term has the right type.)",
"Exercise 8.18: Show that the basis presented for the universal domain ${\\bf U}$ is indeed a finitary basis and that it has an effective presentation.",
"Exercise 8.19: Work out the embedding for the other enumerations for the example given in the proof of Theorem REF .",
"Functions can now be constructed by composing basic functions.",
"However, we wish to be able to define functions recursively as well.",
"The technique of recursive definition will also be useful for defining domains as we will see in Section .",
"Recursion can be thought of as (possibly infinite) iterated function composition.",
"The primary result for interpreting recursive definitions is the following Fixed Point Theorem.",
"Theorem 4.1: For any continuous function $f:{\\cal D}\\rightarrow {\\cal D}$ determined by an approximable mapping $F:{\\bf D}\\rightarrow {\\bf D}$ , there exists a least element $x\\in {\\cal D}$ such that $f(x) = x.$ Proof Let $f^n$ stand for the function $f$ composed with itself $n$ times, and similarly for $F^n$ .",
"Thus, for $\\begin{array}{lcl}f^0&=&I_{\\cal D}~\\\\f^{n+1}&=&f\\circ f^n\\\\F^0&=&{\\sf I}_D~{\\rm and}\\\\F^{n+1}&=&F\\circ F^n\\end{array}$ we define $x = \\lbrace d\\in {\\bf D}\\:\\vert \\:\\exists n\\in {{N}}.",
"\\bot \\:F^n\\:d\\rbrace .$ To show that $x\\in {\\cal D}$ , we must show it to be an ideal.",
"Map $F$ is an approximable mapping, so $\\bot \\in x$ since $\\bot \\:F\\:\\bot $ .",
"For $d\\in x$ and $d^{\\prime }\\sqsubseteq d$ , $d^{\\prime }\\in x$ must hold since, for $d\\in x$ , there must exist an $a\\in {\\bf D}$ such that $a\\:F\\:d$ .",
"But by the definition of an approximable mapping, $a\\:F\\:d^{\\prime }$ must hold as well so $d^{\\prime }\\in x$ .",
"Closure under lubs is direct since $F$ must include lubs to be approximable.",
"To see that $f(x)=x$ , or equivalently $x\\:F\\:x$ , note that for any $d\\in x$ , if $d\\:F\\:d^{\\prime }$ , then $d^{\\prime }\\in x$ .",
"Thus, $f(x)\\sqsubseteq x$ .",
"Now, $x$ is constructed to be the least element in ${\\cal D}$ with this property.",
"To see this is true, let $a\\in {\\cal D}$ such that $f(a)\\sqsubseteq a$ .",
"We want to show that $x\\sqsubseteq a$ .",
"Let $d\\in x$ be an arbitrary element.",
"Therefore, there exists an $n$ such that $\\bot \\: F^n\\:d$ and therefore $\\bot \\:F\\:d_1\\:F\\:d_2\\:\\ldots \\:F\\:d_{n-1}\\:F\\:d.$ Since $\\bot \\in a$ , $d_1\\in f(a)$ .",
"Thus, since $f(a)\\sqsubseteq a$ , $d_1\\in a$ .",
"Thus, $d_2\\in f(a)$ and therefore $d_2\\in a$ .",
"Using induction on $n$ , we can show that $d\\in f(a)$ .",
"Therefore, $d\\in a$ and thus $x\\sqsubseteq a$ .",
"Since $f$ is monotonic and $f(x)\\sqsubseteq x$ , $f(f(x))\\sqsubseteq f(x)$ .",
"Since $x$ is the least element with this property, $x\\sqsubseteq f(x)$ and thus $x=f(x)$ .",
"$\\:\\:\\Box $ Since the element $x$ above is the least element, it must be unique.",
"Thus we have defined a function mapping the domain ${\\cal D}\\rightarrow {\\cal D}$ into the domain ${\\cal D}$ .",
"The next step is to show that this mapping is approximable.",
"Theorem 4.2: For any domain ${\\cal D}$ , there is an approximable mapping $fix:({\\bf D}\\rightarrow {\\bf D})\\rightarrow {\\bf D}$ such that if $f:{\\bf D}\\rightarrow {\\bf D}$ is an approximable mapping, $fix(f) = f(fix(f))$ and for $x\\in {\\cal D}$ , $f(x)\\sqsubseteq x \\:\\Rightarrow \\:fix(f)\\sqsubseteq x$ This property implies that $fix$ is unique.",
"The function $fix$ is characterized by the equation $fix(f)=\\bigcup \\limits _{n=0}^\\infty f^n(\\bot )$ for all $f:{\\bf D}\\rightarrow {\\bf D}$ .",
"Proof The final equation can be simplified to $fix(f) = \\lbrace d\\in {\\bf D}\\:\\vert \\:\\exists n\\in {{N}}.\\bot \\:f^n\\:d\\rbrace $ which is the equation used in the previous theorem to define the fixed point.",
"Using the formula from Exercise REF on the above definition for $fix$ yields the following equation to be shown: $fix(f)=\\bigcup \\lbrace fix({\\cal I}_F)\\:\\vert \\:\\exists F\\in ({\\bf D}\\rightarrow {\\bf D}).F\\in f\\rbrace $ where ${\\cal I}_F$ denotes the ideal for $F$ in ${\\bf D}\\rightarrow {\\bf D}$ .",
"From its definition, $fix$ is monotonic since, if $f\\sqsubseteq g$ , then $fix(f)\\sqsubseteq fix(g)$ since $f^n\\sqsubseteq g^n$ .",
"Since $F\\in f$ , ${\\cal I}_F\\sqsubseteq f$ and since $fix$ is monotonic, $fix({\\cal I}_F)\\sqsubseteq fix(f)$ .",
"Let $x\\in fix(f)$ .",
"Thus, there is a finite sequence of elements such that $\\bot \\:f\\:x_1\\:f\\:\\ldots \\:f\\:x^{\\prime }\\:f\\:x$ .",
"Define $F$ as the basis element encompassing the step functions required for this sequence.",
"Clearly, $F\\in f$ .",
"In addition, this same sequence exists in $fix({\\cal I}_F)$ since we constructed $F$ to contain it, and thus, $x\\in fix({\\cal I}_F)$ and $fix(f)\\sqsubseteq fix({\\cal I}_F)$ .",
"The equality is therefore established.",
"The first equality is direct from the Fixed Point Theorem since the same definition is used.",
"Assume $f(x)\\sqsubseteq x$ for some $x\\in {\\cal D}$ .",
"Since $\\bot \\in x$ , $x\\ne \\emptyset $ .",
"Since $f$ is an approximable mapping, for $x^{\\prime }\\in x$ and $x^{\\prime }\\:f\\:y$ , $y\\in x$ must hold.",
"By induction, for any $\\bot \\:f\\:y$ , $y\\in x$ must hold.",
"Thus, $fix(f)\\sqsubseteq x$ .",
"To see that the operator is unique, define another operator $fax$ that satisfies the first two equations.",
"It can easily be shown that $\\begin{array}{lcll}fix(f)&\\sqsubseteq & fax(f)~{\\rm and}\\\\fax(f)&\\sqsubseteq &fix(f)\\end{array}$ Thus the two operators are the same.",
"$\\:\\:\\Box $ Recursive Definitions Recursion has played a part already in the definitions above.",
"Recall that $f^n$ was defined for all $n\\in {{N}}$ .",
"More complex examples of recursion are given below.",
"Example 4.3: Define a basis ${\\bf N}=\\langle N,\\sqsubseteq _N\\rangle $ where $N=\\lbrace \\lbrace n\\rbrace \\:\\vert \\:n\\in {{N}}\\rbrace \\cup \\lbrace {{N}}\\rbrace $ and the approximation ordering is the superset relation.",
"This generates a flat domain with $\\bot =\\lbrace \\lbrace {{N}}\\rbrace \\rbrace $ and the total elements being in a one-to-one correspondence with the natural numbers.",
"Using the construction outlined in Exercise REF , construct the basis $F=N^\\infty $ .",
"Its corresponding domain is the domain of partial functions over the natural numbers.",
"To see this, let $\\Phi $ be the set of all finite partial functions $\\varphi \\subseteq {{N}}\\times {{N}}$ .",
"Define $\\uparrow \\varphi =\\lbrace \\psi \\in \\Phi \\:\\vert \\:\\varphi \\subseteq \\psi \\rbrace $ Consider the finitary basis $\\langle F^{\\prime },\\sqsubseteq _F^{\\prime }\\rangle $ where $F^{\\prime }=\\lbrace \\uparrow \\varphi \\:\\vert \\:\\varphi \\in \\Phi \\rbrace $ and the approximation order is the superset relation.",
"The reader should satisfy himself that $F^{\\prime }$ and $F$ are isomorphic and that the elements are the partial functions.",
"The total elements are the total functions over the natural numbers.",
"The domains ${\\cal F}$ and $({\\cal N}\\rightarrow {\\cal N})$ are not isomorphic.",
"However, the following mapping $val:F\\times {\\bf N}\\rightarrow {\\bf N}$ can be defined as follows: $(\\uparrow \\varphi ,\\lbrace n\\rbrace )\\:val\\:\\lbrace m\\rbrace \\iff (n,m)\\in \\varphi $ and $(\\uparrow \\varphi ,{{N}})\\:val\\:{{N}}$ Define also as the ideal for $m\\in {\\cal N}$ , $\\hat{m} = \\lbrace \\lbrace m\\rbrace ,{{N}}\\rbrace $ It is easy to show then that for $\\pi \\in {\\cal F}$ and $n\\in {\\cal N}$ we have $\\begin{array}{lcll}val(\\pi ,\\hat{n})&=&\\hat{\\pi (n)}&{\\rm if}~\\pi (n)\\ne \\bot \\\\&=&\\bot &{\\rm otherwise}\\end{array}$ Thus, $curry(val):{\\bf F}\\rightarrow ({\\bf N}\\rightarrow {\\bf N})$ is a one-to-one function on elements.",
"(The problem is that (${\\bf N}\\rightarrow {\\bf N}$ ) has more elements than F does as the reader should verify for himself).",
"Now, what about mappings $f:{\\bf F}\\rightarrow {\\bf F}$ ?",
"Consider the function $\\begin{array}{lcll}f(\\pi )(n)&=&0&{\\rm if}~n=0\\\\&=&\\pi (n-1)+n-1&{\\rm for}~n>0\\end{array}$ If $\\pi $ is a total function, $f(\\pi )$ is a total function.",
"If $\\pi (k)$ is undefined, then $f(\\pi )(k+1)$ is undefined.",
"The function $f$ is approximable since it is completely determined by its actions on partial functions.",
"That is $f(\\pi )=\\bigcup \\lbrace f(\\varphi )\\:\\vert \\:\\exists \\varphi \\in \\Phi .\\varphi \\subseteq \\pi \\rbrace $ The Fixed Point Theorem defines a least fixed point for any approximable mapping.",
"Let $\\sigma =f(\\sigma )$ .",
"Now, $\\sigma (0)=0$ and $\\begin{array}{lcl}\\sigma (n+1)&=&f(\\sigma )(n+1)\\\\&=&\\sigma (n)+n\\end{array}$ By induction, $\\sigma (n)=\\sum \\limits _{i=0}^n i$ and therefore, $\\sigma $ is a total function.",
"Thus, $f$ has a unique fixed point.",
"Now, in looking at $({\\bf N}\\rightarrow {\\bf N})$ , we have $\\hat{0}\\in {\\cal N}$ (The symbols $n$ and $\\hat{n}$ will no longer be distinguished, but the usage should be clear from context.).",
"Now define the two mappings, $succ,pred:{\\bf N}\\rightarrow {\\bf N}$ as approximable mappings such that $\\begin{array}{lcl}n\\:succ\\: m& \\iff & \\exists p\\in {{N}}.n\\sqsubseteq p,m\\sqsubseteq p+1\\\\n\\:pred\\: m& \\iff & \\exists p+1\\in {{N}}.n\\sqsubseteq p+1,m\\sqsubseteq p\\end{array}$ In more familiar terms, the same functions are defined as $\\begin{array}{lcll}succ(n)&=&n+1\\\\pred(n)&=&n-1&{\\rm if}~n>0\\\\&=&\\bot &{\\rm if}~n=0\\end{array}$ The mapping $zero:{\\bf N}\\rightarrow {\\bf T}$ is also defined such that $\\begin{array}{lcll}zero(n)&=&true&{\\rm if}~n=0\\\\&=&false&{\\rm if}~n>0\\end{array}$ where ${\\cal T}$ is the domain of truth value defined in an earlier section.",
"The structured domain $\\langle N,0,succ,pred,zero\\rangle $ is called “The Domain of the Integers” in the present context.",
"The function element $\\sigma $ defined as the fixed point of the mapping $f$ can now be defined directly as a mapping $\\sigma :{\\bf N}\\rightarrow {\\bf N}$ as follows: $\\sigma (n)=cond(zero(n),0,\\sigma (pred(n))+pred(n))$ where the function $+$ must be suitably defined.",
"Recall that $cond$ was defined earlier as part of the structure of the domain ${\\cal T}$ .",
"This equation is called a functional equation; the next section will give another notation, the $\\lambda -calculus$ for writing such equations.",
"$\\:\\:\\Box $ Example 4.4: The domain ${\\cal B}$ defined in Example REF contained only infinite elements as total elements.",
"A related domain, ${\\cal C}$ defined in Exercise REF , can be regarded as a generalization on ${\\cal N}$ .",
"To demonstrate this, the structured domain corresponding to the domain of integers must be presented.",
"The total elements in ${\\cal C}$ are denoted $\\sigma $ while the partial elements are denoted $\\sigma \\bot $ for any $\\sigma \\in \\lbrace 0,1\\rbrace ^*$ .",
"The empty sequence $\\epsilon $ assumes the role of the number 0 in ${\\cal N}$ .",
"Two approximable mappings can serve as the successor function: $x\\mapsto 0x$ denoted $succ_0$ and $x\\mapsto 1x$ denoted $succ_1$ .",
"The predecessor function is filled by the $tail$ mapping defined as follows: $\\begin{array}{lcll}tail(0x)& =& x,\\\\tail(1x)& =& x&{\\rm and}\\\\tail(\\epsilon )& =& \\bot .\\end{array}$ The $zero$ predicate is defined using the $empty$ mapping defined as follows: $\\begin{array}{lcll}empty(0x)& =& false,\\\\empty(1x)& =& false&{\\rm and}\\\\empty(\\epsilon )& =& true.\\end{array}$ To distinguish the other types of elements in ${\\cal C}$ , the following mappings are also defined: $\\begin{array}{lcll}zero(0x)& =& true,\\\\zero(1x)& =& false&{\\rm and}\\\\zero(\\epsilon )& =& false.\\\\one(0x)& =& false,\\\\one(1x)& =& true&{\\rm and}\\\\one(\\epsilon )& =& false.\\end{array}$ The reader should verify the conditions for an approximable mapping are met by these functions.",
"An element of ${\\cal C}$ can be defined using a fixed point equation.",
"For example, the total element representing an infinite sequence of alternating zeroes and ones is defined by the fixed point of the equation $a=01a$ .",
"This same element is defined with the equation $a=0101a$ .",
"(Is the element defined as $b=010b$ the same as the previous two?)",
"Approximable mappings in ${\\cal C}\\rightarrow {\\cal C}$ can also be defined using equations.",
"For example, the mapping $\\begin{array}{lcll}d(\\epsilon ) &= &\\epsilon ,\\\\d(0x)&=&00d(x)&{\\rm and}\\\\d(1x)&=&11d(x)\\end{array}$ can be characterized with the functional equation $d(x)=cond(empty(x),\\epsilon ,cond(zero(x),succ_0(succ_0(d(tail(x)))),succ_1(succ_1(d(tail(x))))))$ The concatenation function of Exercise REF over ${\\cal C}\\times {\\cal C}\\rightarrow {\\cal C}$ can be defined with the functional equation $C(x,y)=cond(empty(x),y,cond(zero(x),succ_0(C(tail(x),y)),succ_1(C(tail(x),y))))$ The reader should verify that this definition is consistent with the properties required in the exercise.",
"These definitions all use recursion.",
"They rely on the object being defined for a base case ($\\epsilon $ for example) or on earlier values ($tail(x)$ for example).",
"These equations characterize the object being defined, but unless a theorem is proved to show that a solution to the equation exists, the definition is meaningless.",
"However, the Fixed Point Theorem for domains was established earlier in this section.",
"Thus, solutions exist to these equations provided that the variables in the equation range over domains and any other functions appearing in the equation are known to be continuous (that is approximable).",
"Peano's Axioms To illustrate one use of the Fixed Point Theorem as well as show the use of recursion in a more familiar setting, we will show that all second order models of Peano's axioms are isomorphic.",
"Recall that Definition 4.5: [Model for Peano's Axioms] A structured set $\\langle {{N}},0,succ\\rangle $ for $0\\in {{N}}$ and $succ:{{N}}\\times {{N}}$ is a model for Peano's axioms if all the following conditions are satisfied: $\\forall n\\in {{N}}.",
"0\\ne succ(n)$ $\\forall n,m \\in {{N}}.succ(n)=succ(m)\\:\\Rightarrow \\:n=m$ $\\forall x\\subseteq {{N}}.0\\in x\\wedge succ(x)\\subseteq x\\:\\Rightarrow \\:x={{N}}$ where $succ(x)=\\lbrace succ(n)\\:\\vert \\:n\\in x\\rbrace $ .",
"The final clause is usually referred to as the principle of mathematical induction.",
"Theorem 4.6: All second order models of Peano's axioms are isomorphic.",
"Proof Let $\\langle N,0,+\\rangle $ and $\\langle M,\\bullet ,\\#\\rangle $ be models for Peano's axioms.",
"Let $N\\times M$ be the cartesian product of the two sets and let ${\\cal P}(N\\times M)$ be the powerset of $N\\times M$ .",
"Recall from Exercise REF that the powerset can be viewed as a domain with the subset relation as the approximation order.",
"Define the following mapping: $u\\mapsto \\lbrace (0,\\bullet )\\rbrace \\cup \\lbrace (+(n),\\#(m))\\:\\vert \\:(n,m)\\in u\\rbrace $ The reader should verify that this mapping is approximable.",
"Since it is indeed approximable, a fixed point exists for the function.",
"Let $r$ be the least fixed point: $r=\\lbrace (0,\\bullet )\\rbrace \\cup \\lbrace (+(n),\\#(m))\\:\\vert \\:(n,m)\\in r\\rbrace $ But $r$ defines a binary relation which establishes the isomorphism.",
"To see that $r$ is an isomorphism, the one-to-one and onto features must be established.",
"By construction, $0\\:r\\:\\bullet $ and $n\\:r\\:m \\:\\Rightarrow \\:+(n)\\:r\\:\\#(m)$ .",
"Now, the sets $\\lbrace (0,\\bullet )\\rbrace $ and $\\lbrace (+(n),\\#(m))\\:\\vert \\:(n,m)\\in r\\rbrace $ are disjoint by the first axiom.",
"Therefore, 0 corresponds to only one element in $m$ .",
"Let $x\\subseteq N$ be the set of all elements of $N$ that correspond to only one element in $m$ .",
"Clearly, $0\\in x$ .",
"Now, for some $y\\in x$ let $z\\in M$ be the element in $M$ that $y$ uniquely corresponds to (that is $y\\:r\\:z$ ).",
"But this means that $+(y)\\:r\\#(z)$ by the construction of the relation.",
"If there exists $w\\in M$ such that $+(y)\\:r\\:w$ and since $(+(y),w)\\ne (0,\\bullet )$ , the fixed point equation implies that $(+(y)=+(n_0))$ and $(w=\\#(m_0))$ for some $(n_0,m_0)\\in r$ .",
"But then by the second axiom, $y=n_0$ and since $y\\in x$ , $z=m_0$ .",
"Thus, $\\#(z)$ is the unique element corresponding to $+(y)$ .",
"The third axiom can now be applied, and thus every element in $N$ corresponds to a unique element in $M$ .",
"The roles of $N$ and $M$ can be reversed in this proof.",
"Therefore, it can also be shown that every element of $M$ corresponds to a unique element in $N$ .",
"Thus, $r$ is a one-to-one and onto correspondence.",
"$\\:\\:\\Box $ Exercises Exercise 4.7: In Theorem REF , an equation was given to find the least fixed point of a function $f:{\\cal D}\\rightarrow {\\cal D}$ .",
"Suppose that for $a\\in {\\cal D}$ , $a\\sqsubseteq f(a)$ .",
"Will the fixed point $x=f(x)$ be such that $a\\sqsubseteq x$ ?",
"(Hint: How do we know that $\\bigcup \\limits _{n=0}^\\infty f^n(a)\\in {\\cal D}$ ?)",
"Exercise 4.8: Let $f:{\\cal D}\\rightarrow {\\cal D}$ and $S\\subseteq {\\cal D}$ satisfy $\\bot \\in S$ $x\\in S\\:\\Rightarrow \\:f(x)\\in S$ $[\\forall n .\\lbrace x_n\\rbrace \\subseteq S \\wedge x_n\\sqsubseteq x_{n+1}]\\:\\Rightarrow \\:\\bigcup \\limits _{n=0}^\\infty x_n \\in S$ Conclude that $fix(f)\\in S$ .",
"This is sometimes called the principle of fixed point induction.",
"Apply this method to the set $S=\\lbrace x\\in {\\cal D}\\:\\vert \\:a(x)= b(x)\\rbrace $ where $a,b:{\\cal D}\\rightarrow {\\cal D}$ are approximable, $a(\\bot )=b(\\bot )$ , and $f\\circ a=a\\circ f$ and $f\\circ b=b\\circ f$ .",
"Exercise 4.9: Show that there is an approximable operator $\\Psi :(({\\cal D}\\rightarrow {\\cal D})\\rightarrow {\\cal D})\\rightarrow (({\\cal D}\\rightarrow {\\cal D})\\rightarrow {\\cal D})$ such that for $\\Theta :({\\cal D}\\rightarrow {\\cal D})\\rightarrow {\\cal D}$ and $f:{\\cal D}\\rightarrow {\\cal D}$ , $\\Psi (\\Theta ) (f) = f(\\Theta (f))$ Prove also that $fix:({\\cal D}\\rightarrow {\\cal D})\\rightarrow {\\cal D}$ is the least fixed point of $\\Psi $ .",
"Exercise 4.10: Given a domain ${\\cal D}$ and an element $a\\in {\\cal D}$ , construct the domain ${\\cal D}_a$ where ${\\cal D}_a=\\lbrace x\\in {\\cal D}\\:\\vert \\:x\\sqsubseteq a\\rbrace $ Show that if $f:{\\cal D}\\rightarrow {\\cal D}$ is approximable, then $f$ can be restricted to another approximable map $f^{\\prime }:{\\cal D}_{fix(f)}\\rightarrow {\\cal D}_{fix(f)}$ where $\\forall x\\in {\\cal D}_{fix(f)}.f^{\\prime }(x)=f(x)$ How many fixed points does $f^{\\prime }$ have in ${\\cal D}_{fix(f)}$ ?",
"Exercise 4.11: The mapping ${\\bf fix}$ can be viewed as assigning a fixed point operator to any domain ${\\cal D}$ .",
"Show that ${\\bf fix}$ can be uniquely characterized by the following conditions on an assignment ${\\cal D}\\leadsto F_D$ : $F_D:({\\cal D}\\rightarrow {\\cal D})\\rightarrow {\\cal D}$ $F_D(f)=f(F_D(f))$ for all $f:{\\cal D}\\rightarrow {\\cal D}$ when $f_0:{\\cal D}_0\\rightarrow {\\cal D}_0$ and $f_1:{\\cal D}_1\\rightarrow {\\cal D}_1$ are given and $h:{\\cal D}_0\\rightarrow {\\cal D}_1$ is such that $h(\\bot )=\\bot $ and $h\\circ f_0=f_1\\circ h$ , then $h(F_{D_0}(f_0)) = F_{D_1}(f_1).$ Hint: Apply Exercise REF to show ${\\bf fix}$ satisfies the conditions.",
"For the other direction, apply Exercise REF .",
"Exercise 4.12: Must an approximable function have a maximum fixed point?",
"Give an example of an approximable function that has many fixed points.",
"Exercise 4.13: Must a monotone function $f:{\\cal P}(A)\\rightarrow {\\cal P}(A)$ have a maximum fixed point?",
"(Recall: ${\\cal P}(A)$ is the powerset of the set $A$ ).",
"Exercise 4.14: Verify the assertions made in the first example of this section.",
"Exercise 4.15: Verify the assertions made in the second example, in particular those in the discussion of “Peano's Axioms”.",
"Show that the predicate function $one:{\\cal C}\\rightarrow {\\cal T}$ could be defined using a fixed point equation from the other functions in the structure.",
"Exercise 4.16: Prove that $fix(f\\circ g)=f(fix(g\\circ f))$ for approximable functions $f,g:{\\cal D}\\rightarrow {\\cal D}$ .",
"Exercise 4.17: Show that the less-than-or-equal-to relation $l\\subseteq {{N}}\\times {{N}}$ is uniquely determined by $l=\\lbrace (n,n)\\:\\vert \\:n\\in {{N}}\\rbrace \\cup \\lbrace (n,succ(m)\\:\\vert \\:(n,m)\\in l\\rbrace $ for the structure called the “Domain of Integers”.",
"Exercise 4.18: Let $N^*$ be a structured set satisfying only the first two of the axioms referred to as “Peano's”.",
"Must there be a subset $S\\subseteq N^*$ such that all three axioms are satisfied?",
"(Hint: Use a least fixed point from ${\\cal P}(N^*)$ ).",
"Exercise 4.19: Let $f:{\\cal D}\\rightarrow {\\cal D}$ be an approximable map.",
"Let $a_n:{\\cal D}\\rightarrow {\\cal D}$ be a sequence of approximable maps such that $a_0(x)=\\bot $ for all $x\\in {\\cal D}$ $a_n\\sqsubseteq a_{n+1}$ for all $n\\in {{N}}$ $\\bigcup \\limits _{n=0}^\\infty a_n = {\\sf I}_D$ in ${\\cal D}\\rightarrow {\\cal D}$ $a_{n+1}\\circ f = a_{n+1}\\circ f\\circ a_n$ for all $n\\in {{N}}$ Show that $f$ has a unique fixed point.",
"(Hint: Show that if $x=f(x)$ then $a_n(x)\\sqsubseteq a_n(fix(f))$ for all $n\\in {{N}}$ .",
"Show this by induction on $n$ .)",
"Typed $\\lambda $ -Calculus As shown in the previous section, functions can be characterized by recursion equations which combine previously defined functions with the function being defined.",
"The expression of these functions is simplified in this section by introducing a notation for specifying a function without having to give the function a name.",
"The notation used is that of the typed $\\lambda $ -Calculus; a function is defined using a $\\lambda $ -abstraction.",
"Definition of Typed $\\lambda $ -Calculus An informal characterization of the $\\lambda $ -calculus suffices for this section; more formal descriptions are available elsewhere in the literature [1].",
"Thus, examples are used to introduce the notation.",
"An infinite number of variables, $x$ ,$y$ ,$z$ ,$\\ldots $ of various types are required.",
"While a variable has a certain type, type subscripts will not be used due to the notational complexity.",
"A distinction must also be made between type symbols and domains.",
"The domain ${\\cal A}\\times {\\cal B}$ does not uniquely determine the component domains ${\\cal A}$ and ${\\cal B}$ even though these domains are uniquely determined by the symbol for the domain.",
"The domain is the meaning that we attribute to the symbol.",
"In addition to variables, constants are also present.",
"For example, the symbol 0 is used to represent the zero element from the domain ${\\cal N}$ .",
"Another constant, present in each domain by virtue of Theorem REF , is $fix^{\\cal D}$ , the least fixed point operator for domain ${\\cal D}$ of type $({\\cal D}\\rightarrow {\\cal D})\\rightarrow {\\cal D}$ .",
"The constants and variables are the atomic (non-compound) terms.",
"Types can be associated with all atomic terms.",
"There are several constructions for compound terms.",
"First, given $\\tau ,\\ldots ,\\sigma $ , a list of terms, the ordered tuple $\\langle \\tau ,\\ldots ,\\sigma \\rangle $ is a compound term.",
"If the types of $\\tau ,\\ldots ,\\sigma $ are ${\\cal A},\\ldots ,{\\cal B}$ , the type of the tuple is ${\\cal A}\\times \\ldots \\times {\\cal B}$ since the tuple is to be an element of this domain.",
"The tuple notation for combining functions given earlier should be disregarded here.",
"The next construction is function application.",
"If the term $\\tau $ has type ${\\cal A}\\rightarrow {\\cal B}$ and the term $\\sigma $ has the type ${\\cal A}$ , then the compound term $\\tau (\\sigma )$ has type ${\\cal B}$ .",
"Function application denotes the value of a function at a given input.",
"The notation $\\tau (\\sigma _0,\\ldots ,\\sigma _n)$ abbreviates $\\tau (\\langle \\sigma _0,\\ldots ,\\sigma _n\\rangle )$ .",
"Functions applied to tuples allows us to represent applications of multi-variate functions.",
"The $\\lambda $ -abstraction is used to define functions.",
"Let $x_0,\\ldots ,x_n$ be a list of distinct variables of type ${\\cal D}_0,\\ldots ,{\\cal D}_n$ .",
"Let $\\tau $ be a term of some type ${\\cal D}_{n+1}$ .",
"$\\tau $ can be thought of as a function of $n+1$ variables with type $({\\cal D}_0\\times \\ldots \\times {\\cal D}_n)\\rightarrow {\\cal D}_{n+1}$ .",
"The name for this function is written $\\lambda x_0,\\ldots ,x_n.\\tau $ This expression denotes the entire function.",
"To look at some familiar functions in the new notation, consider $\\lambda x,y.x$ This notation is read “lambda ex wye (pause) ex”.",
"If the types of $x$ and $y$ are ${\\cal A}$ and ${\\cal B}$ respectively, the function has type $({\\cal A}\\times {\\cal B})\\rightarrow {\\cal A}$ .",
"This function is the first projection function $p_0$ .",
"This function and the second projection function can be defined by the following equations: $\\begin{array}{lcl}p_0&=&\\lambda x,y.x\\\\p_1&=&\\lambda x,y.y\\end{array}$ Recalling the function tuple notation introduced in an earlier section, the following equation holds: $\\langle f,g\\rangle =\\lambda w.\\langle f(w),g(w)\\rangle $ which defines a function of type ${\\cal D}_1\\rightarrow ({\\cal D}_2\\times {\\cal D}_3)$ .",
"Other familiar functions are defined by the following equations: $\\begin{array}{lcl}eval&=&\\lambda f,x.f(x)\\\\curry&=&=\\lambda g\\lambda x\\lambda y.g(x,y)\\end{array}$ The $curry$ example shows that this notation can be iterated.",
"A distinction is thus made between the terms $\\lambda x,y.x$ and $\\lambda x\\lambda y.x$ which have the types ${\\cal D}_0\\times {\\cal D}_1\\rightarrow {\\cal D}_0$ and ${\\cal D}_0\\rightarrow {\\cal D}_1\\rightarrow {\\cal D}_0$ respectively.",
"Thus, the following equation also holds: $curry(\\lambda x,y.\\tau )=\\lambda x\\lambda y.\\tau $ which relates the multi-variate form to the iterated or curried form.",
"Another true equation is $fix={\\bf fix}(\\lambda F\\lambda f.f(F(f)))$ where $fix$ has type $({\\cal D}\\rightarrow {\\cal D})\\rightarrow {\\cal D}$ and fix has type $(((({\\cal D}\\rightarrow {\\cal D})\\rightarrow {\\cal D})\\rightarrow (({\\cal D}\\rightarrow {\\cal D})\\rightarrow {\\cal D}))\\rightarrow (({\\cal D}\\rightarrow {\\cal D})\\rightarrow {\\cal D}))$ This is the content of Exercise REF .",
"This notation can now be used to define functions using recursion equations.",
"For example, the function $\\sigma $ in Example REF can be characterized by the following equation: $\\sigma =fix(\\lambda f\\lambda n.cond(zero(n),0,f(pred(n))+pred(n))$ which states that $\\sigma $ is the least recursively defined function $f$ whose value at $n$ is $cond(\\ldots )$ .",
"The variable $f$ occurs in the body of the $cond$ expression, but this is just the point of a recursive definition.",
"$f$ is defined in terms of its value on “smaller” input values.",
"The use of the fixed point operator makes the definition explicit by forcing there to be a unique solution to the equation.",
"In an abstraction $\\lambda x,y,z.\\tau $ , the variables $x$ ,$y$ , and $z$ are said to be bound in the term $\\tau $ .",
"Any other variables in $\\tau $ are said to be free variables in $\\tau $ unless they are bound elsewhere in $\\tau $ .",
"Bound variables are simply placeholders for values; the particular variable name chosen is irrelevant.",
"Thus, the equation $\\lambda x.\\tau =\\lambda y.\\tau [y/x]$ is true provided $y$ is not free in $\\tau $ .",
"The notation $\\tau [y/x]$ specifies the substitution of $y$ for $x$ everywhere $x$ occurs in $\\tau $ .",
"The notation $\\tau [\\sigma /x]$ for the substitution of the term $\\sigma $ for the variable $x$ is also legitimate.",
"Semantics of Typed $\\lambda $ -Calculus To show that the equations above with $\\lambda $ –terms are indeed meaningful, the following theorem relating $\\lambda $ –terms and approximable mappings must be proved.",
"Theorem 5.1: Every typed $\\lambda $ –term defines an approximable function of its free variables.",
"Proof Induction on the length of the term and its structure will be used in this proof.",
"Variables Direct since $x\\mapsto x$ is an approximable function.",
"Constants Direct since $x\\mapsto k$ is an approximable function for constant $k$ .",
"Tuples Let $\\tau =\\langle \\sigma _0,\\ldots ,\\sigma _n\\rangle $ .",
"Since the $\\sigma _i$ terms are less complex, they are approximable functions of their free variables by the induction hypothesis.",
"Using Theorem REF (generalized to the multi-variate case) then, $\\tau $ which takes tuples as values also defines an approximable function.",
"Application Let $\\tau =\\sigma _0(\\sigma _1)$ .",
"We assume that the types of the terms are appropriately matched.",
"The $\\sigma _i$ terms define approximable functions again by the induction hypothesis.",
"Recalling the earlier equations, the value of $\\tau $ is the same as the value of $eval(\\sigma _0,\\sigma _1)$ .",
"Since $eval$ is approximable, Theorem REF shows that the term defines an approximable function.",
"Abstraction Let $\\tau =\\lambda x.\\sigma $ .",
"By the induction hypothesis, $\\sigma $ defines a function of its free variables.",
"Let those free variables be of types ${\\cal D}_0,\\ldots ,{\\cal D}_n$ where ${\\cal D}_n$ is the type of $x$ .",
"Then $\\sigma $ defines an approximable function $g:{\\cal D}_0\\times \\ldots \\times {\\cal D}_n\\rightarrow {\\cal D}^{\\prime }$ where ${\\cal D}^{\\prime }$ is the type of $\\sigma $ .",
"Using Theorem REF , the function $curry(g):{\\cal D}_0\\times \\ldots \\times {\\cal D}_{n-1}\\rightarrow ({\\cal D}_n\\rightarrow {\\cal D}^{\\prime })$ yields an approximable function, but this is just the function defined by $\\tau $ .",
"The reader can generalize this proof for multiple bound variables in $\\tau $ .",
"$\\:\\:\\Box $ Given this, the equation $\\tau =\\sigma $ states that the two terms define the same approximable function of their free variables.",
"As an example, $\\lambda x.\\tau =\\lambda y.\\tau [y/x]$ provided $y$ is not free in $\\tau $ since the generation of the approximable function did not depend on the name $x$ but only on its location in $\\tau $ .",
"Other equations such as these are given in the exercises.",
"The most basic rule is shown below.",
"Theorem 5.2: For appropriately typed terms, the following equation is true: $(\\lambda x_0,\\ldots ,x_{1}.\\tau )(\\sigma _0,\\ldots ,\\sigma _{n-1})=\\tau [\\sigma _0/x_0,\\ldots ,\\sigma _{n-1}/x_{n-1}]$ Proof The proof is given for $n=1$ and proceeds again by induction on the length of the term and the structure of the term.",
"Variables This means $(\\lambda x.x)(\\sigma )=\\sigma $ must be true which it is.",
"Constants This requires $(\\lambda x.k)(\\sigma )= k$ must be true which it is for any constant $k$ .",
"Tuples Let $\\tau =\\langle \\tau _0,\\tau _1\\rangle $ .",
"This requires that $(\\lambda x.\\langle \\tau _0,\\tau _1\\rangle )(\\sigma ) =\\langle \\tau _0[\\sigma /x],\\tau _1[\\sigma /x]\\rangle $ must be true.",
"This equation holds since the left-hand side can be transformed using the following true equation: $(\\lambda x.\\langle \\tau _0,\\tau _1\\rangle )(\\sigma ) =\\langle (\\lambda x.\\tau _0)(\\sigma ),(\\lambda x.\\tau _1)(\\sigma )\\rangle $ Then the inductive hypothesis is applied to the $\\tau _i$ terms.",
"Applications Let $\\tau =\\tau _0(\\tau _1)$ .",
"Then, the result requires that the equation $(\\lambda x.\\tau _0(\\tau _1))(\\sigma ) =\\tau _0[\\sigma /x](\\tau _1[\\sigma /x])$ hold true.",
"To see that this is true, examine the approximable functions for the left-hand side of the equation.",
"$\\begin{array}{lcl}\\tau _0&\\mapsto &\\bar{V},x\\rightarrow t_0\\\\\\tau _1&\\mapsto &\\bar{V},x\\rightarrow t_1\\\\\\sigma &\\mapsto &\\bar{V}\\rightarrow s\\\\\\mbox{so}\\\\(\\lambda x.\\tau _0(\\tau _1))(\\sigma )&\\mapsto &\\bar{V}\\rightarrow [(x\\rightarrow t_0(t_1))(s)]\\\\&=&\\bar{V},x\\rightarrow [(x\\rightarrow t_0)(s)]([(x\\rightarrow t_1)(s)])\\end{array}$ From this last term, we use the induction hypothesis.",
"To see why the last step holds, start with the set representing the left-hand side and using the aprroximable mappings for the terms: $\\begin{array}{cl}&(\\lambda x.\\tau _0(\\tau _1))(\\sigma )\\\\\\mapsto &\\bar{V}\\rightarrow [(x\\rightarrow t_0(t_1))(s)]\\\\=&\\lbrace b\\:\\vert \\:\\exists a.a\\in s\\wedge a\\:[x\\rightarrow t_0(t_1)]\\: b\\rbrace \\\\=&\\lbrace b\\:\\vert \\:\\exists a. a\\in s\\wedge a\\:\\lbrace (x,u)\\:\\vert \\:v\\in x\\rightarrow t_1)\\wedge v\\:(x\\rightarrow t_0)\\: u\\rbrace \\: b\\rbrace \\\\=&\\lbrace b\\:\\vert \\:\\exists a.a\\in s\\wedge v\\in (x\\rightarrow t_1)(a)\\wedge v\\:(x\\rightarrow t_0)(a)\\:b\\rbrace \\\\=&\\lbrace b\\:\\vert \\:\\exists a,c.a\\in s\\wedge a\\:(x\\rightarrow t_1)\\: v\\wedge a\\:(x\\rightarrow t_0)\\:c\\wedge v\\:c\\:b\\rbrace \\\\=&\\lbrace b\\:\\vert \\:v\\in [(x\\rightarrow t_1)(s)]\\wedge c\\in (x\\rightarrow t_0)(s) \\wedge v\\:c\\:b\\rbrace \\\\=&\\lbrace b\\:\\vert \\:v\\in [(x\\rightarrow t_1)(s)]\\wedge v\\:[(x\\rightarrow t_0)(s)]\\:b\\rbrace \\\\=&[(x\\rightarrow t_0)(s)]([(x\\rightarrow t_1)(s)])\\end{array}$ Abstractions Let $\\tau =\\lambda y.\\tau _0$ .",
"The required equation is $(\\lambda x.\\lambda y.\\tau _0)(\\sigma )=\\lambda y.\\tau _0[\\sigma /x]$ provided that $y$ is not free in $\\sigma $ .",
"The following true equation applies here: $(\\lambda x.\\lambda y.\\tau )(\\sigma )=\\lambda y.",
"((\\lambda x.\\tau )(\\sigma ))$ To see that this equation holds, let $g$ be a function of $n+2$ free variables defined by $\\tau $ .",
"By Theorem REF , the term $\\lambda x.\\lambda y.\\tau $ defines the function $curry(curry(g))$ of $n$ variables.",
"Call this function $h$ .",
"Thus, $h(v)(\\sigma )(y) = g(v,\\sigma ,y)$ where $v$ is the list of the other free variables.",
"Using a combinator $inv$ which inverts the order of the last two arguments, $h(v)(\\sigma )(y)=curry(inv(g))(v,y)(\\sigma )$ But, $curry(inv(g))$ is the function defined by $\\lambda x.\\tau $ .",
"Thus, we have shown that $(\\lambda x.\\lambda y.\\tau )(\\sigma )(y)=(\\lambda x.\\tau )(\\sigma )$ is a true equation.",
"If $y$ is not free in $\\alpha $ and $\\alpha (y)=\\beta $ is true, then $\\alpha =\\lambda y.\\beta $ must also be true.",
"$\\:\\:\\Box $ If $\\tau ^{\\prime }$ is the term $\\lambda x,y.\\tau $ , then $\\tau ^{\\prime }(x,y)$ is the same as $\\tau $ .",
"This specifies that $x$ and $y$ are not free in $\\tau $ .",
"This notation is used in the proof of the following theorem.",
"Theorem 5.3: The least fixed point of $\\lambda x,y.\\langle \\tau (x,y),\\sigma (x,y)\\rangle $ is the pair with coordinates $fix(\\lambda x.\\tau (x,fix(\\lambda y.\\sigma (x,y))))$ and $fix(\\lambda y.\\sigma (fix(\\lambda x.\\tau (x,y)),y))$ .",
"Proof We are thus assuming that $x$ and $y$ are not free in $\\tau $ and $\\sigma $ .",
"The purpose here is to find the least solution to the pair of equations: $x=\\tau (x,y)~{\\rm and}~y=\\sigma (x,y)$ This generalizes the fixed point equation to two variables.",
"More variables could be included using the same method.",
"Let $y_*=fix(\\lambda y.\\sigma (fix(\\lambda x.\\tau (x,y)),y))$ and $x_*=fix(\\lambda x.\\tau (x,y))$ Then, $x_*=\\tau (x_*,y_*)$ and $\\begin{array}{lcl}y_*&=&\\sigma (fix(\\lambda x.\\tau (x,y_*),y_*))\\\\&=&\\sigma (x_*,y_*).\\end{array}$ This shows that the pair $\\langle x_*,y_*\\rangle $ is one fixed point.",
"Now, let $\\langle x_0,y_0\\rangle $ be the least solution.",
"(Why must a least solution exist?",
"Hint: Consider a suitable mapping of type $({\\cal D}_x\\times {\\cal D}_y)\\rightarrow ({\\cal D}_x\\times {\\cal D}_y)$ .)",
"Thus, we know that $x_0=\\tau (x_0,y_0)$ , $y_0=\\sigma (x_0,y_0)$ , and that $x_0\\sqsubseteq x_*$ and $y_0\\sqsubseteq y_*$ .",
"But this means that $\\tau (x_0,y_0)\\sqsubseteq x_0$ and thus $fix(\\lambda x.\\tau (x,y_0))\\sqsubseteq x_0$ and consequently $\\sigma (fix(\\lambda x.\\tau (x,y_0),y_0))\\sqsubseteq \\sigma (x_0,y_0)\\sqsubseteq y_0$ By the fixed point definition of $y_*$ , $y_*\\sqsubseteq y_0$ must hold as well so $y_0=y_*$ .",
"Thus, $x_*=fix(\\lambda x.\\tau (x,y*))=fix(\\lambda x.\\tau (x,y_0))\\sqsubseteq x_0.$ Thus, $x*=x_0$ must also hold.",
"A similar argument holds for $x_0$ .$\\:\\:\\Box $ The purpose of the above proof is to demonstrate the use of least fixed points in proofs.",
"The following are also true equations: $fix(\\lambda x.\\tau (x))=\\tau (fix(\\lambda x.\\tau (x)))$ and $\\tau (y)\\sqsubseteq y\\:\\Rightarrow \\:fix(\\lambda x.\\tau (x))\\sqsubseteq y$ if $x$ is not free in $\\tau $ .",
"These equations combined with the monotonicity of functions were the methods used in the proof above.",
"Another example is the proof of the following theorem.",
"Theorem 5.4: Let $x$ ,$y$ , and $\\tau (x,y)$ be of type ${\\cal D}$ and let $g:{\\cal D}\\rightarrow {\\cal D}$ be a function.",
"Then the equation $\\lambda x.fix(\\lambda y.\\tau (x,y))=fix(\\lambda g.\\lambda x.\\tau (x,g(x)))$ holds.",
"Proof Let $f$ be the function on the left-hand side.",
"Then, $f(x)=fix(\\lambda y.\\tau (x,y))=\\tau (x,f(x))$ holds using the equations stated above.",
"Therefore, $f=\\lambda x.\\tau (x,f(x))$ and thus $g_0=fix(\\lambda g.\\lambda x.\\tau (x,g(x)))\\sqsubseteq f. $ By the definition of $g_0$ we have $g_0(x)=\\tau (x,g_0(x))$ for any given $x$ .",
"By the definition of $f$ we find that $f(x)=fix(\\lambda y.\\tau (x,y))\\sqsubseteq g_0(x)$ must hold for all $x$ .",
"Thus $f\\sqsubseteq g_0$ and the equation is true.$\\:\\:\\Box $ This proof illustrates the use of inclusion and equations between functions.",
"The following principle was used: $(\\forall x.\\tau \\sqsubseteq \\sigma )\\:\\Rightarrow \\:\\lambda x.\\tau \\sqsubseteq \\lambda x.\\sigma $ This is a restatement of the first part of Theorem REF .",
"Combinators and Recursive Functions Below is a list of various combinators with their definitions in $\\lambda $ -notation.",
"The meanings of those combinators not previously mentioned should be clear.",
"$\\begin{array}{lcl}p_0&=&\\lambda x,y.x\\\\p_1&=&\\lambda x,y.y\\\\pair&=&\\lambda x.\\lambda y.\\langle x,y\\rangle \\\\n-tuple&=&\\lambda x_0\\lambda \\ldots \\lambda x_{n-1}.\\langle x_0,\\ldots ,x_{n-1}\\rangle \\\\diag&=&\\lambda x.\\langle x,x\\rangle \\\\funpair&=&\\lambda f.\\lambda g.\\lambda x.\\langle f(x),g(x)\\rangle \\\\proj^n_i&=&\\lambda x_0,\\ldots ,x_{n-1}.x_i\\\\inv^n_{i,j}&=&\\lambda x_0,\\ldots ,x_i,\\ldots ,x_j,\\ldots ,x_{n-1}.\\langle x0,\\ldots ,x_j,\\ldots ,x_i,\\ldots ,x_{n-1}\\rangle \\\\eval&=&\\lambda f,x.f(x)\\\\curry&=&\\lambda g.\\lambda x.\\lambda y.g(x,y)\\\\comp&=&\\lambda f,g.\\lambda x.g(f(x))\\\\const&=&\\lambda k.\\lambda x.k\\\\{\\bf fix}&=&\\lambda f.fix(\\lambda x.f(x))\\end{array}$ These combinators are actually schemes for combinators since no types have been specified and thus the equations are ambiguous.",
"Each scheme generates an infinite number of combinators for all the various types.",
"One interest in combinators is that they allow expressions without variables—if enough combinators are used.",
"This is useful at times but can be clumsy.",
"However, defining a combinator when the same combination of symbols repeatedly appears is also useful.",
"There are some familiar combinators that do not appear in the table.",
"Combinators such as $cond$ , $pred$ , and $succ$ cannot be defined in the pure $\\lambda $ -calculus but are instead specific to certain domains.",
"They are thus regarded as primitives.",
"A large number of other functions can be defined using these primitives and the $\\lambda $ -notation, as the following theorem shows.",
"Theorem 5.5: For every partial recursive function $h:{{N}}\\rightarrow {{N}}$ , there is a $\\lambda $ -term $\\tau $ of type ${\\cal N}\\rightarrow {\\cal N}$ such that the only constants occurring in $\\tau $ are $cond$ , $succ$ , $pred$ , $zero$ , and 0 and if $h(n)=m$ then $\\tau (n)=m$ .",
"If $h(n)$ is undefined, then $\\tau (n)=\\bot $ holds.",
"$\\tau (\\bot )=\\bot $ is also true.",
"Proof It is convenient in the proof to work with strict functions $f:{\\cal N}^k\\rightarrow {\\cal N}$ such that if any input is $\\bot $ , the result of the function is $\\bot $ .",
"The composition of strict functions is easily shown to be strict.",
"It is also easy to see that any partial function $g:{{N}}^k\\rightarrow {{N}}$ can be extended to a strict approximable function $\\bar{g}:{\\cal N}^k\\rightarrow {\\cal N}$ which yields the same values on inputs for which $g$ is defined.",
"Other input values yield $\\bot $ .",
"We want to show that $\\bar{g}$ is definable with a $\\lambda $ -expression.",
"First we must show that primitive recursive functions have $\\lambda $ -definitions.",
"Primitive recursive functions are formed from starting functions using composition and the scheme of primitive recursion.",
"The starting functions are the constant function for zero and the identity and projection functions.",
"These functions, however, must be strict so the term $\\lambda x,y.x$ is not sufficient for a projection function.",
"The following device reduces a function to its strict form.",
"Let $\\lambda x.cond(zero(x),x,x)$ be a function with $x$ of type ${\\cal N}$ .",
"This is the strict identity function.",
"The strict projection function attempted above can be defined as $\\lambda x,y.cond(zero(y),x,x)$ The three variable projection function can be defined as $\\lambda x,y,z.cond(zero(x),cond(zero(z),y,y),cond(zero(z),y,y))$ While not very elegant, this device does produce strict functions.",
"Strict functions are closed under substitution and composition.",
"Any substitution of a group of functions into another function can be defined with a $\\lambda $ -term if the functions themselves can be so defined.",
"Thus, we need to show that functions obtained by primitive recursion are definable.",
"Let $f:{\\cal N}\\rightarrow {\\cal N}$ , and $g:{\\cal N}^3\\rightarrow {\\cal N}$ be total functions with $\\bar{f}$ and $\\bar{g}$ being $\\lambda $ -definable.",
"We obtain the function $h:{\\cal N}^2\\rightarrow {\\cal N}$ by primitive recursion where $\\begin{array}{lcl}h(0,m)&=&f(m)\\\\h(n+1,m)&=&g(n,m,h(n,m))\\end{array}$ for all $n,m\\in {\\cal N}$ .",
"The $\\lambda $ -term for $\\bar{h}$ is $fix(\\lambda k.\\lambda x,y.cond(zero(x),\\bar{f}(y),\\bar{g}(pred(x),y,k(pred(x),y))))$ Note that the fixed point operator for the domain ${\\cal N}^2\\rightarrow {\\cal N}$ was used.",
"The variables $x$ and $y$ are of type ${\\cal N}$ .",
"The $cond$ function is used to encode the function requirements.",
"The fixed point function is easily seen to be strict and this function is $\\bar{h}$ .",
"Primitive recursive functions are now $\\lambda $ -definable.",
"To obtain partial (i.e., general) recursive functions, the $\\mu $ -scheme (the least number operator) is used.",
"Let $f(n,m)$ be a primitive recursive function.",
"Then, define $h$ , a partial function, as $h(m) =$ the least $n$ such that $f(n,m)=0$ .",
"This is written as $h(m)=\\mu n.f(n,m)=0$ .",
"Since $\\bar{f}$ is $\\lambda $ -definable as has just been shown, let $\\bar{g}=fix(\\lambda g.\\lambda x,y.cond(zero(\\bar{f}(x,y)),x,g(succ(x),y)))$ Then, the desired function $\\bar{h}$ is defined as $\\bar{h}=\\lambda y.\\bar{g}(0,y)$ .",
"It is easy to see that this is a strict function.",
"Note that, if $h(m)$ is defined, clearly $h(m)=\\bar{g}(0,m)$ is also defined.",
"If $h(m)$ is undefined, it is also true that $\\bar{g}(0,m)=\\bot $ due to the fixed point construction but it is less obvious.",
"This argument is left to the reader.$\\:\\:\\Box $ Theorem REF does not claim that all $\\lambda $ -terms define partial recursive functions although this is also true.",
"Further examples of recursion are found in the exercises.",
"Exercises Exercise 5.6: Find the definitions of $\\lambda x,y.\\tau ~{\\rm and}~\\sigma (x,y)$ which use only $\\lambda v$ with one variable and applications only to one argument at a time.",
"Note that use must be made of the combinators $p_0$ , $p_1$ , and $pair$ .",
"Generalize the result to functions of many variables.",
"Exercise 5.7: The table of combinators was meant to show how combinators could be defined in terms of $\\lambda $ -expressions.",
"Can the tables be turned to show that, with enough combinators available, every $\\lambda $ -expression can be defined by combining combinators using application as the only mode of combination?",
"Exercise 5.8: Suppose that $f,g:{\\cal D}\\rightarrow {\\cal D}$ are approximable and $f\\circ g=g\\circ f$ .",
"Show that $f$ and $g$ have a least common fixed point $x=f(x)=g(x)$ .",
"(Hint: See Exercise REF .)",
"If, in addition, $f(\\bot )=g(\\bot )$ , show that $fix(f)=fix(g)$ .",
"Will $fix(f)=fix(f^2)$ ?",
"What if the assumption is weakened to $f\\circ g=g^2\\circ f$ ?",
"Exercise 5.9: For any domain ${\\cal D}$ , ${\\cal D}^\\infty $ can be regarded as consisting of bottomless stacks of elements of ${\\cal D}$ .",
"Using this view, define the following combinators with their obvious meaning: $head:{\\cal D}^\\infty \\rightarrow {\\cal D}$ , $tail:{\\cal D}^\\infty \\rightarrow {\\cal D}^\\infty $ and $push:{\\cal D}\\times {\\cal D}^\\infty \\rightarrow {\\cal D}^\\infty $ .",
"Using the fixed point theorem, argue that there is a combinator $diag:{\\cal D}\\rightarrow {\\cal D}^\\infty $ where for all $x\\in {\\cal D}$ , $diag(x)=\\langle x\\rangle _{n=0}^\\infty $ .",
"(Hint: Try a recursive definition, such as $diag(x)=push(x,diag(x))$ but be sure to prove that all terms of $diag(x)$ are $x$ .)",
"Also introduce by an appropriate recursive definition a combinator $map:({\\cal D}\\rightarrow {\\cal D})^\\infty \\times {\\cal D}\\rightarrow {\\cal D}^\\infty $ where for elements of the proper type $map(\\langle f_n\\rangle _{n=0}^\\infty ,x)=\\langle f_n(x)\\rangle _{n=0}^\\infty $ Exercise 5.10: For any domain ${\\cal D}$ introduce, as a least fixed point, a combinator $while:({\\cal D}\\rightarrow {\\cal T})\\times ({\\cal D}\\rightarrow {\\cal D})\\rightarrow ({\\cal D}\\rightarrow {\\cal D})$ by the recursion equation $while(p,f)(x)=cond(p(x),while(p,f)(f(x)),x)$ Prove that $while(p,while(p,f))=while(p,f)$ Show how $while$ could be used to obtain the least number operator,$\\mu $ , mentioned in the proof of Theorem REF .",
"Generalize this idea to define a combinator $find:{\\cal D}^\\infty \\times ({\\cal D}\\rightarrow {\\cal T})\\rightarrow {\\cal D}$ which means “find the first term in the sequence (if any) which satisfies the given predicate”.",
"Exercise 5.11: Prove the existence of a one-one function $num:{{N}}\\times {{N}}\\leftrightarrow {{N}}$ such that $\\begin{array}{lcl}num(0,0)&=&0\\\\num(n,m+1)&=&num(n+1,m)\\\\num(n+1,0)&=&num(0,n)+1\\end{array}$ Draw a descriptive picture (an infinite matrix) for the function.",
"Find a closed form for the values if possible.",
"Use the function to prove the isomorphism between ${\\cal P}({{N}})$ ,${\\cal P}({{N}}\\times {{N}})$ , and ${\\cal P}({{N}})\\times {\\cal P}({{N}})$ .",
"Exercise 5.12: Show that there are approximable mappings $graph:({\\cal P}({{N}})\\rightarrow {\\cal P}({{N}}))\\rightarrow {\\cal P}({{N}})$ and $fun:{\\cal P}({{N}})\\rightarrow ({\\cal P}({{N}})\\rightarrow {\\cal P}({{N}}))$ where $fun\\circ graph = \\lambda f.f$ and $graph\\circ fun\\sqsubseteq \\lambda x.x$ .",
"(Hint: Using the notation $[n_0,\\ldots ,n_k]=num(n_0,[n_1,\\ldots ,n_k])$ , two such combinators can be given by the formulas $\\begin{array}{lcl}fun(u)(x)&=&\\lbrace m\\:\\vert \\:\\exists n_0,\\ldots ,n_{k-1}\\in x.",
"[n_0+1,\\ldots ,n_{k-1}+1,0,m]\\in u\\rbrace \\\\graph(f)&=&\\lbrace [n_0+1,\\ldots ,n_{k-1}+1,0,m]\\:\\vert \\:m\\in f(\\lbrace n_0,\\ldots ,n_{k-1}\\rbrace )\\rbrace \\end{array}$ where $k$ is a variable - meaning all finite sequences are to be considered.)",
"Introduction to Domain Equations As stressed in the introduction, the notion of computation with potentially infinite elements is an integral part of domain theory.",
"The previous sections have defined the notion of functions over domains, as well as a notation for expressing these functions.",
"In addition, the notion of computation through series of approximations has been addressed.",
"This computation is possible since the functions defined have been approximable and thus continuous.",
"This section addresses the construction of more complex domains with infinite elements.",
"The next section looks specifically at the notion of computability with respect to these infinite elements.",
"The last section looks at another approach to domain construction.",
"New domains have been constructed from existing ones using domain constructors such as the product construction ($\\times $ ), the function space construction ($\\rightarrow $ ) and the sum construction ($+$ ) of Exercise REF .",
"These constructors can be iterated similar to the way that function application was iterated to form recursive function definitions.",
"In this way, domains can be characterized using recursion equations, called domain equations.",
"Domain Equations A domain equation represents an isomorphism between the domain as a whole and the combination of domains that comprise it.",
"These recursive domains are frequently termed reflexive domains since, as in the following example, the domain contains a copy of itself in its structure.",
"Example 6.1: Consider the following domain equation: ${\\cal T}={\\cal A}+({\\cal T}\\times {\\cal T})$ where ${\\cal A}$ is a previously defined domain.",
"This domain can be thought of as containing atomic elements from ${\\cal A}$ or pairs of elements of ${\\cal T}$ .",
"What do the elements of this domain look like?",
"In particular, what are the finite elements of this domain?",
"How is the domain constructed?",
"What is an appropriate approximation ordering for the domain?",
"What do lubs in this domain look like?",
"What is the appropriate notion of consistency?",
"Does this domain even exist?",
"In other words, are we certain a solution to this domain equation exists?",
"And if a solution to the equation exists, is it a unique solution?",
"Each of these questions is examined below.",
"The domain equation tells us that an element of the domain is either an element from ${\\cal A}$ or is a pair of “smaller” elements from ${\\cal T}$ .",
"One method of constructing a sum domain is using pairs where some distinguished element denotes what type an element is.",
"Thus, for some $a\\in {\\cal A}$ , the pair $\\langle \\pi ,a\\rangle $ might represent the element in ${\\cal T}$ for the given element $a$ .",
"For some $s,t\\in {\\cal T}$ , the pair $\\langle \\langle s,t\\rangle ,\\pi \\rangle $ might then represent the element in ${\\cal T}$ for the pair $s,t$ .",
"Thus, $\\pi $ is the distinguished element, and the location of $\\pi $ in the pair specifies the type of the element.",
"The finite elements are either elements in ${\\cal T}$ representing the (finite) elements of ${\\cal A}$ or the pair elements from ${\\cal T}$ whose components are also finite elements in ${\\cal T}$ .",
"The question then arises about infinite elements.",
"Are there infinite elements in this domain?",
"Consider the following fixed point equation for some element for $a\\in {\\cal A}$ : $x=\\langle \\langle a,x\\rangle ,\\pi \\rangle .$ The fixed point of this equation is the infinite product of the element $a$ .",
"Does this element fit the definition for ${\\cal T}$ ?",
"From the informal description of the elements of ${\\cal T}$ given so far, $x$ does qualify as a member of ${\\cal T}$ .",
"Now that some intuition has been developed about this domain, a formal construction is required.",
"Let $\\langle {\\bf A},\\sqsubseteq _A\\rangle $ be the finitary basis used to generate the domain ${\\cal A}$ .",
"Let $\\pi $ be an object such that $\\pi \\notin {\\bf A}$ .",
"Define the bottom element of the finitary basis T as $\\Delta _T=\\langle \\pi ,\\pi \\rangle $ .",
"Next, all the elements of ${\\cal A}$ must be included so define an element in ${\\bf T}$ for each $a\\in {\\bf A}$ as $\\langle \\pi ,a\\rangle $ .",
"Finally, pair elements for all elements in ${\\bf T}$ must exist in ${\\bf T}$ to complete the construction.",
"The set ${\\bf T}$ can be defined inductively as the least set such that: $\\Delta _T\\in {\\bf T}$ , $\\langle \\pi ,a\\rangle \\in {\\bf T}$ whenever $a\\in {\\bf A}$ , $\\langle \\langle \\Delta _T,s\\rangle ,\\pi \\rangle \\in {\\bf T}$ whenever $s\\in {\\bf T}$ (necessary??",
"), $\\langle \\langle t,\\Delta _T\\rangle ,\\pi \\rangle \\in {\\bf T}$ whenever $t\\in {\\bf T}$ (necessary??",
"), and $\\langle \\langle t,s\\rangle ,\\pi \\rangle \\in {\\bf T}$ whenever $s, t\\in {\\bf T}$ .",
"The set can also be characterized by the following fixed point equation: ${\\bf T}=\\lbrace \\Delta _T\\rbrace \\cup \\lbrace \\langle \\pi ,a\\rangle \\:\\vert \\:a\\in {\\bf A}\\rbrace \\cup \\lbrace \\langle \\langle \\Delta _T,s\\rangle ,\\pi \\rangle \\:\\vert \\:s\\in {\\bf T}\\rbrace \\cup \\lbrace \\langle \\langle t,\\Delta _T\\rangle ,\\pi \\rangle \\:\\vert \\:t\\in {\\bf T}\\rbrace \\cup \\lbrace \\langle \\langle t,s\\rangle ,\\pi \\rangle \\:\\vert \\:s,t\\in {\\bf T}\\rbrace .$ A solution must exist for this equation by the fixed point theorem.",
"Now that the basis elements have been defined, we must show how to find lubs.",
"We will again use an inductive definition.",
"$\\langle \\pi ,\\pi \\rangle \\sqcup t=t$ for all $t\\in {\\bf T}$ For $a,b\\in {\\bf A}$ , $\\langle \\pi ,a\\rangle \\sqcup \\langle \\pi ,b\\rangle =\\langle \\pi ,a\\sqcup b\\rangle $ if $a\\sqcup b$ exists in ${\\bf A}$ $\\langle \\langle s,t\\rangle ,\\pi \\rangle \\sqcup \\langle \\langle s^{\\prime },t^{\\prime }\\rangle ,\\pi \\rangle =\\langle \\langle s\\sqcup s^{\\prime },t\\sqcup t^{\\prime }\\rangle ,\\pi \\rangle $ if $s\\sqcup s^{\\prime }$ and $t\\sqcup t^{\\prime }$ exist in ${\\bf T}$ .",
"The lub $\\langle \\pi ,a\\rangle \\sqcup \\langle \\langle s,t\\rangle ,\\pi \\rangle $ does not exist.",
"Next, the notion of consistency needs to be explored.",
"From the definition of lubs given above, the following sets are consistent: The empty set is consistent.",
"Everything is consistent with the bottom element.",
"A set of elements all from the basis A is consistent in T if the set of elements is consistent in A.",
"A set of product elements in T is consistent if the left component elements are consistent and the right component elements are consistent.",
"These conditions derive from the sum and product nature of the domain.",
"The approximation ordering in the basis has the following inductive definition: $\\Delta _T\\sqsubseteq _T s$ for all $s\\in {\\bf T}$ $y\\sqsubseteq _Tu\\sqcup \\Delta _T$ whenever $y\\sqsubseteq _Tu$ $\\langle \\pi ,a\\rangle \\sqsubseteq _T\\langle \\pi ,b\\rangle $ whenever $a\\sqsubseteq _Ab$ $\\langle \\langle s,t\\rangle ,\\pi \\rangle \\sqsubseteq _T\\langle \\langle u,v\\rangle ,\\pi \\rangle $ whenever $s\\sqsubseteq _Tu$ and $t\\sqsubseteq _Tv$ The next step is to verify that ${\\bf T}$ is indeed a finitary basis.",
"The basis is still countable.",
"The approximation is clearly a partial order.",
"The existence of lubs of finite bounded (i.e., consistent) subsets must be verified.",
"The definition of consistency gives us the requirements for a bounded subset.",
"Each of the conditions for consistency are examined inductively since the definitions are all inductive: The lub of the empty set is the bottom element $\\Delta _T$ .",
"The lub of a set containing the bottom element is the lub of the set without the bottom element which must exist by the induction hypothesis.",
"The lub of a set of elements all from the ${\\bf A}$ is the element in ${\\bf T}$ for the lub in ${\\bf A}$ .",
"This element must exist since ${\\bf A}$ is a finitary basis and all elements from ${\\bf A}$ have corresponding elements in ${\\bf T}$ .",
"The lub of a set of product elements is the pair of the lub of the left components and the lub of the right components.",
"These exist by the induction hypothesis.",
"Thus, a finitary basis has been created; the domain is formed as always from the basis.",
"The solution to the domain equation has been found since any element in the domain ${\\cal T}$ is either an element representing an element in ${\\cal A}$ or is the product of two other elements in ${\\cal T}$ .",
"Similarly, any element found on the left-hand side must also be in the domain ${\\cal T}$ by the construction.",
"Thus, the domain ${\\cal T}$ is identical to the domain ${\\cal A}+({\\cal T}\\times {\\cal T})$ .",
"To look at the question concerning the existence and uniqueness of the solution to this domain equation, recall the fixed point theorem.",
"This theorem states that a fixed point set exists for any approximable mapping over a domain.",
"Subdomains In Section , the concept of a universal domain is introduced.",
"A universal domain is a domain which contains all other domains as sub-domains.",
"These sub-domains are, roughly speaking, the image of approximable functions over the universal domain.",
"The domain equation for ${\\cal T}$ can be viewed as an approximable mapping over the universal domain.",
"As such, the fixed point theorem states that a least fixed point set for the function does exist and is unique.",
"Sub-domains are defined formally below.",
"Looking again at the informal discussion concerning the elements of the domain ${\\cal T}$ , the infinite element proposed does fit into the formal definition for elements of ${\\cal T}$ .",
"This element is an infinite tree with all left sub-trees containing only the element $a$ .",
"For this infinite element to be computable, it must be the lub of some ascending chain of finite approximations to it.",
"The element $x$ can, in fact, be defined by the following ascending sequence of finite trees: $\\begin{array}{lcl}x_0&=&\\bot \\\\x_{n+1}&=&\\langle \\langle a,x_n\\rangle ,\\pi \\rangle \\\\x&=&\\bigsqcup ^\\infty _{n=0}x_n\\end{array}$ Thus, using domain equations, a domain has been defined recursively.",
"This domain includes infinite as well as finite elements and allows computation on the infinite elements to proceed using the finite approximations, as with the more conventionally defined domains presented earlier.",
"The final topic of this section is the notion of a sub-domain.",
"Informally, a sub-domain is a structured part of a larger domain.",
"Earlier, a domain was described as a sub-domain of the universal domain.",
"Thus, the sub-domain starts with a subset of the elements of the larger domain while retaining the approximation ordering, consistency relation and lub relation, suitably restricted to the subset elements.",
"Definition 6.2: [Sub-Domain] A domain $\\langle {\\cal R},\\sqsubseteq _R\\rangle $ is a sub-domain of a domain $\\langle {\\cal D},\\sqsubseteq _D\\rangle $ , denoted ${\\cal R}\\lhd {\\cal D}$ iff ${\\cal R}\\subseteq {\\cal D}$ - The elements of ${\\cal R}$ are a subset of the elements of ${\\cal D}$ .",
"$\\bot _R=\\bot _D$ - The bottom elements are the same.",
"For $x,y\\in {\\cal R}$ , $x\\sqsubseteq _Ry\\iff x\\sqsubseteq _Dy$ - The approximation ordering for ${\\cal R}$ is the approximation ordering for ${\\cal D}$ restricted to elements in ${\\cal R}$ .",
"For $x,y,z\\in {\\cal R}$ , $x\\sqcup _Ry=z$ iff $x\\sqcup _Dy=z$ - The lub relation for ${\\cal R}$ is the lub relation for ${\\cal D}$ restricted to elements in ${\\cal R}$ .",
"${\\cal R}$ is a domain.",
"Equivalently, a sub-domain can be thought of as the image of an approximable function which approximates the identity function (also termed a projection).",
"The notion of a sub-domain is used in the final section in the discussions about the universal domain.",
"This mapping between the domains can be formalized as follows: Theorem 6.3: If ${\\cal D}\\lhd {\\cal E}$ , then there exists a projection pair of approximable mappings $i:{\\cal D}\\rightarrow {\\cal E}$ and $j:{\\cal E}\\rightarrow {\\cal D}$ where $j\\circ i={{\\sf I}}_{\\cal D}$ and $i\\circ j\\sqsubseteq {{\\sf I}}_{\\cal E}$ where $i$ and $j$ are determined by the following equations: $\\begin{array}{lcl}i(x)&=&\\lbrace y\\in {\\bf E}\\:\\vert \\:\\exists z\\in x.z\\sqsubseteq y\\rbrace \\\\j(y)&=&\\lbrace x\\in {\\bf D}\\:\\vert \\:x\\in y\\rbrace \\end{array}$ for all $x\\in {\\cal D}$ and $y\\in {\\cal E}$ .",
"The proof is left as an exercise.",
"By the definition of a sub-domain, it should be clear that ${\\cal D}_0\\lhd {\\cal E}\\wedge {\\cal D}_1\\lhd {\\cal E}\\:\\Rightarrow \\:({\\cal D}_0\\lhd {\\cal D}_1\\iff {\\cal D}_0\\subseteq {\\cal D}_1)$ Using this observation, the sub-domains of a domain can be ordered.",
"Indeed, the following theorem is a consequence of this ordering.",
"Theorem 6.4: For a given domain ${\\cal D}$ , the set of sub-domains $\\lbrace {\\cal D}_0\\:\\vert \\:{\\cal D}_0\\lhd {\\cal D}\\rbrace $ form a domain.",
"The proof proceeds using the inclusion relation defined as an approximation ordering and is left as an exercise.",
"Finally, a converse of Theorem REF can also be established: Theorem 6.5: For two domains ${\\cal D}$ and ${\\cal E}$ , if there exists a projection pair $i:{\\cal D}\\rightarrow {\\cal E}$ and $j:{\\cal E}\\rightarrow {\\cal D}$ with $j\\circ i={\\sf I}_{\\cal D}$ and $i\\circ j\\sqsubseteq {\\sf I}_{\\cal E}$ , then $\\exists {\\cal D}^{\\prime }\\lhd {\\cal E}$ where ${\\cal D}\\approx {\\cal D}^{\\prime }$ .",
"Proof We show that $i$ maps finite elements to finite elements and that ${\\cal D}^{\\prime }$ is the image of ${\\cal D}$ in ${\\cal E}$ .",
"For some $x\\in {\\bf D}$ with ${\\cal I}_x$ as the principal ideal of $x$ , we can write $i({\\cal I}_x)=\\sqcup \\lbrace {\\cal I}_y\\:\\vert \\:y\\in i({\\cal I}_x)\\rbrace $ Applying $j$ to both sides we get ${\\cal I}_x=j\\circ i({\\cal I}_x)=\\sqcup \\lbrace j({\\cal I}_y)\\:\\vert \\:y\\in i({\\cal I}_x)\\rbrace $ since $j\\circ i={\\sf I}_D$ and $j$ is continuous by assumption.",
"But, since $x\\in {\\cal I}_x$ , $x\\in j({\\cal I}_y)$ for some $y\\in i({\\cal I}_x)$ .",
"This means that ${\\cal I}_x\\subseteq j({\\cal I}_y)$ and thus $i({\\cal I}_x)\\subseteq i\\circ j({\\cal I}_y) \\subseteq {\\cal I}_y$ Since ${\\cal I}_y\\subseteq i({\\cal I}_x)$ must hold by the construction, $i({\\cal I}_x) ={\\cal I}_y$ .",
"This proves that finite elements are mapped to finite elements.",
"Next, consider the value for $i(\\bot _D)$ .",
"Since $\\bot _D\\sqsubseteq _Dj(\\bot _E)$ , $i(\\bot _D)\\sqsubseteq \\bot _E$ .",
"Thus $i(\\bot _D)=\\bot _E$ .",
"Thus, ${\\cal D}$ is isomorphic to the image of $i$ in ${\\cal E}$ .",
"We still must show that ${\\cal D}^{\\prime }$ is a domain.",
"Thus, we need to show that if a lub exists in ${\\cal E}$ for a finite subset in ${\\cal D}^{\\prime }$ , then the lub is also in ${\\cal D}^{\\prime }$ .",
"Let $y^{\\prime },z^{\\prime }\\in {\\bf D}^{\\prime }$ and $y^{\\prime }\\sqcup z^{\\prime }=x^{\\prime }\\in {\\bf E}$ .",
"Then, there exists $y,z\\in {\\bf D}$ such that $i({\\cal I}_y)={\\cal I}_{y^{\\prime }}$ and $i({\\cal I}_z)={\\cal I}_{z^{\\prime }}$ which implies that ${\\cal I}_y=j({\\cal I}_{y^{\\prime }})$ and ${\\cal I}_z=j({\\cal I}_{z^{\\prime }})$ .",
"Since ${\\cal I}_{y^{\\prime }}\\sqsubseteq {\\cal I}_{x^{\\prime }}$ and $j({\\cal I}_{y^{\\prime }})\\sqsubseteq j({\\cal I}_{x^{\\prime }})$ by monotonicity, $y\\in j({\\cal I}_{x^{\\prime }})$ must hold.",
"By the same reasoning, $z\\in j({\\cal I}_{x^{\\prime }})$ .",
"But then $x=y\\sqcup z\\in j({\\cal I}_{x^{\\prime }})$ must also hold and thus $y\\sqcup z \\in {\\cal D}$ since the element $j({\\cal I}_{x^{\\prime }})$ must be an ideal.",
"But, $\\begin{array}{lcl}{\\cal I}_y\\sqsubseteq {\\cal I}_x&\\:\\Rightarrow \\:& {\\cal I}_{y^{\\prime }}\\sqsubseteq i({\\cal I}_{x})\\\\{\\cal I}_z\\sqsubseteq {\\cal I}_x&\\:\\Rightarrow \\:& {\\cal I}_{z^{\\prime }}\\sqsubseteq i({\\cal I}_{x})\\end{array}$ This implies that $y^{\\prime }\\sqcup z^{\\prime }=x^{\\prime }\\in i({\\cal I}_x)$ .",
"We already know that $x\\in j({\\cal I}_{x^{\\prime }})$ so $i({\\cal I}_x)\\sqsubseteq {\\cal I}_{x^{\\prime }}$ .",
"Thus, $i({\\cal I}_x)={\\cal I}_{x^{\\prime }}$ and thus, $x^{\\prime }\\in {\\bf D^{\\prime }}$ .$\\:\\:\\Box $ Exercises Exercise 6.6: Show that there must exist domains satisfying $\\begin{array}{lcll}{\\cal A}&=&{\\cal A}+({\\cal A}\\times {\\cal B})&{\\rm and}\\\\{\\cal B}&=&{\\cal A}+{\\cal B}\\end{array}$ Decide what the elements will look like and define ${\\cal A}$ and ${\\cal B}$ using simultaneous fixed points.",
"Exercise 6.7: Prove Theorem REF Exercise 6.8: Prove Theorem REF Exercise 6.9: Show that if ${\\cal A}$ and ${\\cal B}$ are finite systems, that ${\\cal D}\\unlhd {\\cal E}\\unlhd {\\cal D}\\:\\Rightarrow \\:{\\cal D}\\approx {\\cal E}$ where ${\\cal D}\\approx {\\cal D}^{\\prime }$ and ${\\cal D}^{\\prime }\\lhd {\\cal E}$ is denoted ${\\cal D}^{\\prime }\\unlhd {\\cal E}$ .",
"Computability in Effectively Given Domains In the previous sections, we gave considerable emphasis to the notion of computation using increasingly accurate approximations of the input and output.",
"This section defines this notion of computability more formally.",
"In Section 5, we found that partial functions over the natural numbers were expressible in the $\\lambda $ -notation.",
"This relationship characterizes computation for a particular domain.",
"To describe computation over domains in general, a broader definition is required.",
"The way a domain is presented impacts the way computations are performed over it.",
"Indeed, the theorems of recursive function theory [6] rely in part on the normal presentation of the natural numbers.",
"A presentation for a domain is an enumeration of the elements of the domain.",
"The standard presentation of the natural numbers is simply the numbers in ascending order beginning with 0.",
"There are many permutations of the natural numbers, each of which can be considered a presentation.",
"Computation with these non-standard presentations may be impossible; that is a computable function on the standard presentation may be non-computable over a non-standard presentation.",
"Therefore, an effective presentation for a domain is defined as a presentation which makes the required information computable.",
"Effective Presentations Information about elements in a domain can be characterized completely by looking at the finite elements and their relationships.",
"Thus a presentation must enumerate the finite elements and allow the consistency and lub relationships on these elements to be computed to allow this style of computation.",
"The consistency relation and the lub relation depend on each other.",
"For example, if a set of elements is consistent, a lub must exist for the set.",
"Given that a set is consistent, the lub can be found in finite time by just enumerating the elements and checking to see if this element is the lub.",
"However, if the set is inconsistent, the enumeration will not reveal this fact.",
"Thus, the consistency relation must be assumed to be recursive in an effective presentation.",
"Exercise REF provides a description of presentations that should clarify the assumptions made.",
"Formally, a presentation is defined as follows: Definition 7.1: [Effective Presentation] The presentation of a finitary basis D is a function $\\pi :{{N}}\\rightarrow {\\bf D}$ such that $\\pi (0)=\\Delta _D$ and the range of $\\pi $ is the set of finite elements of D. The definition holds for a domain ${\\cal D}$ as well.",
"A presentation $\\pi $ is effective iff The consistency relation ($\\exists k.\\pi _i\\sqsubseteq \\pi _k\\wedge \\pi _j\\sqsubseteq \\pi _k$ ) for elements $\\pi _i$ and $\\pi _j$ is recursiveRecursive in this context means that the relation is decidable.",
"over $i$ and $j$ .",
"The lub relation ($\\pi _k=\\pi _i\\sqcup \\pi _j$ ) is recursive over $i$ , $j$ , and $k$ .",
"This definition supports our intuition about domains; we have stated that the important information about a domain is the set of finite elements, the ordering and consistency relationships between the elements and the lub relation.",
"Thus, an effective presentation provides, in a suitable (that is computable) form, the basic information about the structure and elements of a domain.",
"A presentation can also be viewed as an enumeration of the elements of the domain with the position of an element in the enumeration given by the index corresponding to the integer input for that element in the presentation function with the 0 element representing $\\bot $ .",
"This perspective is used in the majority of the proofs.",
"Computability Now that the presentation of a domain has been formalized, the notion of computability can be formally defined.",
"Thus, Definition 7.2: [Computable Mappings] Given two domains, ${\\cal D}$ and ${\\cal E}$ with effective presentations $\\pi _1$ and $\\pi _2$ respectively, an approximable mapping $f:{\\bf D}\\rightarrow {\\bf E}$ is computable iff the relation $x_n\\:f\\:y_m$ is recursively enumerable in $n$ and $m$ .",
"By considering the domain ${\\cal D}$ to be a single element domain, the above definition applies not only to computable functions but also to computable elements.",
"For $d\\in {\\cal D}$ where $d$ is the only element in the domain, the element $e=f(d)\\in {\\cal E}$ defines an element in ${\\cal E}$ .",
"The definition states that $e$ is a computable iff the set $\\lbrace m\\in {{N}}\\:\\vert \\:y_m\\sqsubseteq e\\rbrace $ is a recursively enumerable set of integers.",
"Clearly if the set of elements approximating another is finite, the set is recursive.",
"The notion of a recursively enumerable set simply requires that all elements approximating the element in question be listed eventually.",
"The computation then proceeds by accepting an enumeration representing the input element and enumerating the elements that approximate the desired output element.",
"Now that the notions of computability and effective presentations have been formalized, the methods of constructing domains and functions will be addressed.",
"The proof of the next theorem is trivial and is left to the reader.",
"Theorem 7.3: The identity map on an effectively given domain is computable.",
"The composition of computable mappings on effectively given domains are also computable.",
"The following corollary is a consequence of this theorem: Corollary 7.4: For computable function $f:{\\cal D}\\rightarrow {\\cal E}$ and a computable element $x\\in {\\cal D}$ , the element $f(x)\\in {\\cal E}$ is computable.",
"In addition, the standard domain constructors maintain effective presentations.",
"Theorem 7.5: For domains ${\\cal D}_0$ and ${\\cal D}_1$ with effective presentations, the domains ${\\cal D}_0+{\\cal D}_1~{\\rm and}~ {\\cal D}_0\\times {\\cal D}_1$ are also effectively given.",
"In addition, the projection functions are all computable.",
"Finally, if $f$ and $g$ are computable maps, then so are $f+g$ and $f\\times g$ .",
"Proof Let $\\lbrace X_i\\:\\vert \\:i\\in {{N}}\\rbrace $ be the enumeration of ${\\cal D}_0$ and $\\lbrace Y_i\\:\\vert \\:i\\in {{N}}\\rbrace $ be the enumeration of ${\\cal D}_1$ .",
"Another method of sum construction is to use two distinguishing elements in the first position to specify the element type.",
"Thus, a sum domain can be defined as follows: ${\\cal D}_0+{\\cal D}_1=\\lbrace (\\Delta _0,\\Delta _1)\\rbrace \\cup \\lbrace (0,x)\\:\\vert \\:x\\in {\\cal D}_0\\rbrace \\cup \\lbrace (1,y)\\:\\vert \\:y\\in {\\cal D}_1\\rbrace $ The enumeration can then be defined as follows for $n\\in {{N}}$ : $\\begin{array}{lcl}Z_0&=&(\\Delta _0,\\Delta _1)\\\\Z_{2n+1}&=&(0,X_n)\\\\Z_{2n+2}&=&(1,Y_n)\\end{array}$ The proof that $Z_i$ is an effective presentation is left as an exercise.",
"For the product construction, the domain appears as follows: ${\\cal D}_0\\times {\\cal D}_1=\\lbrace (x,y)\\:\\vert \\:x\\in {\\cal D}_0,y\\in {\\cal D}_1\\rbrace $ The enumeration can be defined in terms of the functions $p:{{N}}\\rightarrow {{N}}$ , $q:{{N}}\\rightarrow {{N}}$ , and $r:({{N}}\\times {{N}})\\rightarrow {{N}}$ where for $m$ , $n$ , $k\\in {{N}}$ : $\\begin{array}{lcl}p(r(n,m))&=&n\\\\q(r(n,m))&=&m\\\\r(p(k),q(k))&=&k\\end{array}$ Thus, $r$ is a one-to-one pairing function (see Exercise REF ) of which there are several.",
"The functions $p$ and $q$ extract the indices from the result of the pairing function.",
"The enumeration for the product domain is then defined as follows: $W_i = (X_{p(i)},Y_{q(i)})$ The proof that this is an effective presentation is also left as an exercise.",
"For the combinators, the relations will be defined in terms of the enumeration indices.",
"For example, $\\begin{array}{lcl}X_n\\:in_0\\:Z_m&\\iff & m=0~{\\rm or}\\\\&&\\exists k.m=2k+1\\wedge X_k\\sqsubseteq X_n\\\\W_k\\:proj_1\\:Y_m&\\iff & Y_m\\sqsubseteq Y_{q(k)}\\end{array}$ The reader should verify that these sets are recursively enumerable.",
"For this proof, recall that recursively enumerable sets are closed under conjunction, disjunction, substituting recursive functions, and applying an existential quantifier to the front of a recursive predicate.",
"The proof for the other combinators is left as an exercise.",
"$\\:\\:\\Box $ Product spaces formalize the notion of computable functions of several variables.",
"Note that the proof of Theorem REF shows that substitution of computable functions of severable variables into other computable functions are still computable.",
"The next step is to show that the function space constructor preserves effectiveness.",
"Theorem 7.6: For domains ${\\cal D}_0$ and ${\\cal D}_1$ with effective presentations, the domain ${\\cal D}_0\\rightarrow {\\cal D}_1$ also has an effective presentation.",
"The combinators $apply$ and $curry$ are computable if all input domains are effectively given.",
"The computable elements of the domain ${\\cal D}_0\\rightarrow {\\cal D}_1$ are the computable maps for ${\\bf D_0}\\rightarrow {\\bf D_1}$ .",
"Proof Let ${\\cal D}_0=\\lbrace X_i\\:\\vert \\:i\\in {{N}}\\rbrace $ and ${\\cal D}_1=\\lbrace Y_i\\:\\vert \\:i\\in {{N}}\\rbrace $ be the presentations for the domains.",
"The elements of ${\\bf D_0}\\rightarrow {\\bf D_1}$ are finite step functions which respect the mapping of some subset of ${\\bf D_0}\\times {\\bf D_1}$ .",
"Given the enumeration, each element can be associated with a set $\\lbrace (X_{n_i},Y_{m_i})\\:\\vert \\:\\exists q.",
"1\\le i\\le q\\rbrace $ Thus, there is a finite set of integers pairs that determine the element.",
"Given the definition of consistency from Theorem REF for elements in the function space domain and the decidability of consistency in ${\\cal D}_0$ and ${\\cal D}_1$ , consistency of any finite set of this form is decidable (tedious but decidable since all elements must be checked with all others, etc).",
"Since consistency is decidable, a systematic enumeration of pair sets which are consistent can be made; this enumeration is simply the enumeration of ${\\cal D}_0\\rightarrow {\\cal D}_1$ .",
"Finding the lub consists of making a finite series of tests to find the element that is the lub, which must exist since the set is consistent and we have closure on lubs of finite consistent subsets.",
"Finding the lub requires a finite series of checks in both ${\\cal D}_0$ and ${\\cal D}_1$ but these checks are decidable.",
"Thus, the lub relation is also decidable in ${\\cal D}_0\\rightarrow {\\cal D}_1$ .",
"This shows that ${\\cal D}_0\\rightarrow {\\cal D}_1$ is effectively given.",
"To show that $apply$ and $curry$ are computable, the mappings need to be examined.",
"The mapping defined for apply is $(F,a)\\:apply\\: b\\iff a\\:F\\:b$ The function $F$ is the lub of all the finite step functions that are consistent with it.",
"As such, $F$ can be viewed as the canonical representative of this set.",
"Since $F$ is a finite step function, this relation is decidable.",
"As such, the $apply$ relation is recursive and not just recursively enumerable and $apply$ is a computable function.",
"The reasoning for $curry$ is similar in that the relations are studied.",
"Given the increase in the number of domains, the construction is more tedious and is left for the exercises.",
"To see that the computable elements correspond to the computable maps, recall the relationship shown in Theorem REF between the maps and the elements in the function space.",
"Thus, we have $a\\:f\\:b \\iff b\\in f({\\cal I}_a)~{\\rm or}~{\\cal I}_b\\sqsubseteq f({\\cal I}_a)$ Since $f$ is a computable map, we know that the pairs in the map are recursively enumerable.",
"Using the previous techniques for deciding consistency of finite sets, the set of elements consistent with $f$ can be enumerated.",
"But this set is simply the ideal for $f$ in the function space.",
"The converse direction is trivial.",
"$\\:\\:\\Box $ The final combinator to be discussed, and perhaps the most important, is the fixed point combinator.",
"Theorem 7.7: For any effectively given domain, ${\\cal D}$ , the combinator $fix:({\\cal D}\\rightarrow {\\cal D})\\rightarrow {\\cal D}$ is computable.",
"Proof Let $\\lbrace X_n\\:\\vert \\:n\\in {{N}}\\rbrace $ be the presentation of the domain ${\\cal D}$ .",
"Recall that for $f\\in {\\cal D}\\rightarrow {\\cal D}$ , $f\\:fix\\:X\\iff \\exists k\\in {{N}}.\\Delta \\:f\\:X_1\\:f\\ldots f\\:X_k\\wedge X_k=X$ All of the checks in this finite sequence are decidable since ${\\cal D}$ is effectively given.",
"In addition, existential quantification of a decidable predicate gives a recursively enumerable predicate.",
"Thus, $fix$ is computable.",
"$\\:\\:\\Box $ Recap Now that this has been formalized, what has been accomplished?",
"The major consequence of the theorems to this point is that any expression over effectively given domains (that is effectively given types) combined with computable constants using the $\\lambda $ -notation and the fixed point combinator is a computable function of its free variables.",
"Such functions, applied to computable arguments, yield computable values.",
"These functions also have computable least fixed points.",
"All this gives us a mathematical programming language for defining computable operations.",
"Combining this language with the specification of types with domain equations gives a powerful language.",
"As an example, the effectiveness of the domain ${\\cal T}$ from Example REF is studied.",
"The complete proof is left as an exercise.",
"Example 7.8: Recall the domain ${\\cal T}$ from the previous section.",
"This domain is characterized by the domain equation ${\\cal T}={\\cal A}+({\\cal T}\\times {\\cal T})$ for some domain ${\\cal A}$ .",
"If ${\\cal A}$ is effectively given, we wish to show that ${\\cal T}$ is effectively given as well.",
"The elements are either atomic elements from ${\\cal A}$ or are pairs from ${\\cal T}$ .",
"Let $A=\\lbrace A_i\\:\\vert \\:i\\in {{N}}\\rbrace $ be the enumeration for ${\\cal A}$ .",
"An enumeration for ${\\cal T}$ can be defined as follows: $\\begin{array}{lcl}T_0&=&\\bot _T\\\\T_{2n+1}&=&3*A_n\\\\T_{2n+2}&=&3*T_{p(n)}+1\\cup 3*T_{q(n)}+2\\end{array}$ where for $A$ , a set of indices, $m*A+k=\\lbrace m*n+k\\:\\vert \\:n\\in A\\rbrace $ .",
"The functions $p$ and $q$ here are the inverses of the pairing function $r$ defined in Theorem REF .",
"These functions must be defined such that $p(n)\\le n$ and $q(n)\\le n$ so that the recursion is well defined by taking smaller indices.",
"The rest of the proof is left to the exercises.",
"Specifically, the claim that ${\\cal T}=\\lbrace T_i\\rbrace $ should be verified as well as the effectiveness of the enumeration.",
"These proofs rely either on the effectiveness of ${\\cal A}$ , on the effectiveness of elements in ${\\cal T}$ with smaller indices, or are trivial.",
"The final example uses the powerset construction.",
"We have repeatedly used the fact that a powerset is a domain.",
"Its effectiveness is now verified.",
"Example 7.9: Specifically, the powerset of the natural numbers, ${\\cal {P}({{N}})}$ is considered.",
"In this domain, all elements are consistent, and there is a top element, denoted $\\omega $ , which is the set of all natural numbers.",
"The ordering is the subset relation.",
"The lub of two subsets is the union of the two subsets, which is decidable.",
"To enumerate the finite subsets, the following enumeration is used: $E_n=\\lbrace k\\:\\vert \\:\\exists i,j.",
"i< 2^k\\wedge n=i+2^k+j*2^{k+1}\\rbrace $ This says that $k\\in E_n$ if the $k$ bit in the binary expansion of $n$ is a 1.",
"All finite subsets of ${{N}}$ are of the form $E_n$ for some $n$ .",
"Various combinators for ${\\cal P}({{N}})$ are presented in Exercise REF .",
"Exercises Exercise 7.10: Show that an effectively given domain can always be identified with a relation $INCL(n,m)$ on integers where the derived relations $\\begin{array}{lcl}CONS(n,m)&\\iff &\\exists k.INCL(k,n)\\wedge INCL(k,m)\\\\MEET(n,m,k)&\\iff &\\forall j.",
"[INCL(j,k)\\iff INCL(j,n)\\wedge INCL(j,m)]\\end{array}$ are recursively decidable and where the following axioms hold: $\\forall n.INCL(n,n)$ $\\forall n,m,k.",
"INCL(n,m)\\wedge INCL(m,k)\\:\\Rightarrow \\:INCL(n,k)$ $\\exists m.\\forall n. INCL(n,m)$ $\\forall n,m.",
"CONS(n,m)\\:\\Rightarrow \\:\\exists k.MEET(n,m,k)$ Exercise 7.11: Finish the proof of Theorem REF .",
"Exercise 7.12: Complete the proof of Theorem REF by defining $curry$ as a relation and showing it computable.",
"Is the set recursively enumerable or is it recursive?",
"Exercise 7.13: Two effectively given domains are effectively isomorphic iff $\\ldots $ Complete the statement of the theorem and prove it.",
"Exercise 7.14: Complete the proof about the powerset in Example REF .",
"Show that the combinators $fun$ and $graph$ from Exercise REF are computable.",
"Show the same for $\\lambda x,y.x\\cap y$ $\\lambda x,y.x\\cup y$ $\\lambda x,y.x+ y$ where for $x,y\\in {\\cal P}({{N}})$ , $x+y=\\lbrace n+m\\:\\vert \\:n\\in x, m\\in y\\rbrace $ What are the computable elements of ${\\cal P}({{N}})$ ?",
"Sub-Spaces of the Universal Domain To have a flexible method of solving domain equations and yielding effectively given domains as the solutions, the domains will be embedded in a universal domain which is “big” enough to hold all other domains as sub-domains.",
"This universal domain is shown to be effectively presented, and the mappings which define the sub-spaces are shown to be computable.",
"First, the correspondence between sub-spaces and mappings called retractions is investigated, leading us to the definition of mappings called projections.",
"It is then shown that these definitions can be written out using the $\\lambda $ -calculus notation, demonstrating the power of our mathematical programming language.",
"Retractions and Projections We start with the definition of retractions.",
"Definition 8.1: [Retractions] A retraction of a given domain ${\\cal E}$ is an approximable mapping $a:{\\bf E}\\rightarrow {\\bf E}$ such that $a\\circ a=a$ .",
"Thus, a retraction is the identity function on objects in the range of the retraction and maps other elements into range.",
"The next theorem relates these sets to sub-spaces.",
"Theorem 8.2: If ${\\cal D}\\lhd {\\cal E}$ and if $a:{\\bf E}\\rightarrow {\\bf E}$ is defined such that $X\\:a\\:Z \\iff \\exists Y\\in {\\cal D}.",
"Z\\sqsubseteq Y\\sqsubseteq X$ for all $X,Z\\in {\\bf E}$ , then $a$ is a retraction and ${\\cal D}$ is isomorphic to the fixed point set of $a$ , the set $\\lbrace y\\in {\\cal E}\\:\\vert \\:a(y)=y\\rbrace $ , ordered under inclusion.",
"Proof That $a$ is an approximable map is a direct consequence of the definition of sub-space (Definition REF ).",
"By Theorem REF , a projection pair, $i$ and $j$ , exist for ${\\cal D}$ and this tells us that $a=i\\circ j$ (also showing $a$ approximable since approximable mappings are closed under composition).",
"Theorem REF also tells us that $j\\circ i={\\sf I}_D$ .",
"To show that $a$ is a retraction, $a\\circ a=a$ must be established.",
"Thus, $a\\circ a = i\\circ j\\circ i\\circ j = i\\circ {\\sf I}_D\\circ j = i\\circ j =a$ holds, showing that $a$ is a retraction.",
"We now need to show the isomorphism to ${\\cal D}$ .",
"For $x\\in {\\cal D}$ , $i(x)\\in {\\cal E}$ and we can calculate: $a(i(x))=i\\circ j\\circ i(x) = i\\circ {\\sf I}_D(x) = i(x)$ Thus, $i(x)$ is in the fixed point set of $a$ .",
"For the other direction, let $a(y)=y$ .",
"Then $i(j(y)) = y$ holds.",
"But, $j(y)\\in {\\cal D}$ , so $i$ must map ${\\cal D}$ one-to-one and onto the fixed point set of $a$ .",
"Since $i$ and $j$ are approximable, they are certainly monotonic, and thus the map is an isomorphism with respect to set inclusion.",
"$\\:\\:\\Box $ Not all retractions are associated with a sub-domain relationship.",
"The retractions defined in the above theorem are all subsets as relations of the identity relation.",
"The retractions for sub-domains are characterized by the following definition: Definition 8.3: [Projections] A retraction $a:{\\cal E}\\rightarrow {\\cal E}$ is a projection if $a\\subseteq {\\sf I}_E$ as relations.",
"The retraction is finitary iff its fixed point set is isomorphic to some domain.",
"An example is in order.",
"Example 8.4: Consider a two element system, ${\\bf O}$ with objects $\\Delta $ and 0.",
"For any basis ${\\bf D}$ that is not trivial (has more than one element), ${\\bf O}$ comes from a retraction on ${\\bf D}$ .",
"Define a combinator $check:{\\bf D}\\rightarrow {\\bf O}$ by the relation $x\\:check\\: y \\iff y=\\Delta ~{\\rm or}~x\\ne \\Delta _D$ Thus, $check(x)=\\bot _O\\iff x=\\bot _D$ .",
"Another combinator can be defined, $fade:{\\bf O}\\times {\\bf D}\\rightarrow {\\bf D}$ such that for $t\\in {\\cal O}$ and $x\\in {\\cal D}$ $\\begin{array}{lcll}fade(t,x)&=&\\bot _D&{\\rm if}~t=\\bot _O\\\\&=&x&otherwise\\end{array}$ For $u\\in {\\cal D}$ and $u\\ne \\bot _D$ , the mapping $a$ is defined as $a(x)=fade(check(x),u)$ It can be seen that $a$ is a retraction, but not a projection in general, and the range of $a$ is isomorphic to ${\\bf O}$ .",
"These combinators can also be used to define the subset of functions in ${\\bf D}\\rightarrow {\\bf E}$ that are strict.",
"Define a combinator $strict:({\\bf D}\\rightarrow {\\bf E})\\rightarrow ({\\bf D}\\rightarrow {\\bf E})$ by the equation $strict(f)=\\lambda x.fade(check(x),f(x))$ with $fade$ defined as $fade:{\\bf O}\\times {\\bf E}\\rightarrow {\\bf E}$ .",
"The range of $strict$ is all the strict functions; $strict$ is a projection whose range is a domain.",
"The next theorem characterizes projections.",
"Theorem 8.5: For approximable mapping $a:{\\bf E}\\rightarrow {\\bf E}$ , the following are equivalent: $a$ is a finitary projection $a(x)=\\lbrace y\\in {\\bf E}\\:\\vert \\:\\exists x^{\\prime }\\in I_x.",
"x^{\\prime }\\:a\\:x^{\\prime }\\wedge y\\sqsubseteq x^{\\prime }\\rbrace $ for all $x\\in {\\bf E}$ .",
"Proof Assume that (2) holds.",
"We want to show that $a$ is a finitary projection.",
"By the closure properties on ideals, we know that for all $x\\in {\\cal E}$ , $x^{\\prime }\\in x\\wedge y\\sqsubseteq x^{\\prime }\\:\\Rightarrow \\:y\\in x$ Thus, $a(x)\\subseteq x$ must hold.",
"In addition, the following trivially holds: $x^{\\prime }\\in x\\wedge x^{\\prime }\\:a\\: x^{\\prime }\\:\\Rightarrow \\:x^{\\prime }\\in a(x)$ thus $a(x)\\subseteq a(a(x))$ holds for all $x\\in {\\cal E}$ .",
"This shows that $a$ is indeed a projection.",
"Let $D=\\lbrace x\\in {\\bf E}\\:\\vert \\:x\\:a\\:x\\rbrace $ .",
"It is easy to show that ${\\bf D}\\lhd {\\bf E}$ and that $a$ is determined from ${\\bf D}$ as required in Theorem REF .",
"Thus, the fixed point set of $a$ is isomorphic to a domain from the previous proofs.",
"Thus, (2)$\\:\\Rightarrow \\:$ (1).",
"For the converse, assume that $a$ is a finitary projection.",
"Let ${\\cal D}$ be isomorphic to the fixed point set of $a$ .",
"This means there is a projection pair $i$ and $j$ such that $j\\circ i={\\sf I}_D$ and $i\\circ j = a$ and $a\\subseteq {\\sf I}_E$ .",
"From Theorem REF then we have that ${\\cal D}\\approx {\\cal D}^{\\prime }$ and ${\\cal D}^{\\prime }\\lhd {\\cal E}$ .",
"We want to identify ${\\cal D}^{\\prime }$ as follows: ${\\cal D}^{\\prime }=\\lbrace x\\in {\\cal E}\\:\\vert \\:x\\:a\\:x\\rbrace $ From the proof of Theorem REF , the basis elements of ${\\bf D^{\\prime }}$ are the finite elements of ${\\bf D}$ .",
"Each of these elements is in the fixed point set of $a$ .",
"Thus, $x\\in {\\bf D^{\\prime }}\\:\\Rightarrow \\:a({{\\cal I}}_x) = {{\\cal I}}_x \\:\\Rightarrow \\:x\\:a\\:x$ Since $a$ is a projection, ${{\\cal I}}_x$ must also be a fixed point.",
"Since $i(j({{\\cal I}}_x)) = {{\\cal I}}_x$ implies that $j({{\\cal I}}_x)$ is a finite element of ${\\cal D}$ , $x\\in {\\cal D}^{\\prime }$ must hold.",
"Thus, the identification of ${\\cal D}^{\\prime }$ holds.",
"Finally, using $a=i\\circ j$ in the formula in Theorem REF , the formula in (2) is obtained, proving the converse.",
"$\\:\\:\\Box $ This characterization of projections provides a new and interesting combinator.",
"Theorem 8.6: For any domain ${\\cal E}$ , define $sub:({\\cal E}\\rightarrow {\\cal E})\\rightarrow ({\\cal E}\\rightarrow {\\cal E})$ using the relation $x\\: sub(f)\\: z \\iff \\exists y\\in {\\bf E}.y\\:f\\:y\\wedge y\\sqsubseteq x\\wedge z\\sqsubseteq y$ for all $x,z\\in {\\bf E}$ and all $f:{\\bf E}\\rightarrow {\\bf E}$ .",
"Then the range of $sub$ is exactly the set of finitary projections on ${\\cal E}$ .",
"In addition, $sub$ is a finitary projection on ${\\cal E}\\rightarrow {\\cal E}$ .",
"If ${\\cal E}$ is effectively given, then $sub$ is computable.",
"Proof Clearly, $sub(f)$ is approximable.",
"It is obvious from the definition that $f\\mapsto sub(f)$ preserves lubs and thus is approximable as well.",
"Thus, $y\\:f\\:y\\wedge y\\sqsubseteq x\\wedge z\\sqsubseteq y\\:\\Rightarrow \\:x\\:f\\:z$ obviously holds.",
"Thus, $sub(f)\\subseteq f$ holds.",
"Also $y\\:f\\:y\\:\\Rightarrow \\:y\\:sub(f)\\:y$ thus, $sub(f)\\subseteq sub(sub(f))$ holds as well.",
"Thus, $sub$ is a projection on ${\\cal E}\\rightarrow {\\cal E}$ .",
"The definition of the relation shows that it is computable when ${\\cal E}$ is effectively given.",
"Since $sub$ is a projection, its range is the same as its fixed point set.",
"If $sub(a)=a$ , it is easy to see that clause (2) of Theorem REF holds and conversely.",
"Thus, the range of $sub$ is the finitary projections.",
"To see that $sub$ is a finitary projection, we use Theorem REF and Theorem REF to say that the fixed point set of $sub$ is in a one-to-one inclusion preserving correspondence with the domain $\\lbrace D\\:\\vert \\:D\\lhd {\\cal E}\\rbrace $ .",
"$\\:\\:\\Box $ Universal Domain ${\\cal U}$ With these results and the universal domain to be defined next, the theory of sub-domains is translated into the $\\lambda $ -calculus notation using the $sub$ combinator.",
"The universal domain is defined by first defining a domain which has the desired structure but has a top element.",
"The top element is then removed to give the universal domain.",
"Definition 8.7: [Universal Domain] As in the section on domain equations, an inductive definition for a domain ${\\cal V}$ is given as follows: $\\Delta ,\\top \\in {\\bf V}$ $\\langle u,v\\rangle \\in {\\bf V}$ whenever $u,v\\in {\\bf V}$ Thus, we are starting with two objects, a bottom element and a top element, and making two flavors of copies of these objects.",
"Intuitively, we end up with finite binary trees with either the top or the bottom element as the leaves.",
"To simplify the definitions below, the pairs should be reduced such that: All occurrences of $\\langle \\Delta ,\\Delta \\rangle $ are replaced by $\\Delta $ and All occurrences of $\\langle \\top ,\\top \\rangle $ are replaced by $\\top $ .",
"These rewrite rules are easily shown to be finite Church-Rosser.The finitary basis should be defined as the equivalence classes induced by the reduction.",
"The presentation is simplified by considering only reduced trees.",
"As an example of the reduction the pair $\\langle \\langle \\langle \\top ,\\langle \\top ,\\top \\rangle \\rangle ,\\langle \\top ,\\Delta \\rangle \\rangle ,\\langle \\langle \\Delta ,\\Delta \\rangle ,\\langle \\top ,\\top \\rangle \\rangle \\rangle $ reduces to $\\langle \\langle \\top ,\\langle \\top ,\\Delta \\rangle \\rangle ,\\langle \\Delta ,\\top \\rangle \\rangle $ .",
"The approximation ordering is defined as follows: $\\Delta \\sqsubseteq v$ for all $v\\in {\\bf V}$ $v\\sqsubseteq \\top $ for all $v\\in {\\bf V}$ .",
"$\\langle u,v\\rangle \\sqsubseteq \\langle u^{\\prime },v^{\\prime }\\rangle $ iff $u\\sqsubseteq u^{\\prime }$ and $v\\sqsubseteq v^{\\prime }$ Since the top element is approximated by everything, all finite sets of trees are consistent.",
"The lub for a pair of trees is defined as follows: $u\\sqcup \\top =\\top $ for $u\\in {\\bf V}$ $\\top \\sqcup u=\\top $ for $u\\in {\\bf V}$ $u\\sqcup \\Delta =u$ for $u\\in {\\bf V}$ $\\Delta \\sqcup u=u$ for $u\\in {\\bf V}$ $\\langle u,v\\rangle \\sqcup \\langle u^{\\prime },v^{\\prime }\\rangle =\\langle u\\sqcup u^{\\prime },v\\sqcup v^{\\prime }\\rangle $ for $u,v\\in {\\bf V}$ The proof that this forms a finitary basis follows the same guidelines as the proofs in Section .",
"In addition, it should be clear that the presentation is effective.",
"To form the universal domain, the top element is simply removed.",
"Thus, the system ${\\bf U}={\\bf V}-\\lbrace \\top \\rbrace $ is the basis used to form the universal domain.",
"The proof that this is still a finitary basis with an effective presentation is also straightforward and left to the exercises.",
"Note that inconsistent sets can now exist since there is no top element.",
"A set is inconsistent iff its lub is $\\top $ .",
"We shall now prove the claims made for the universal domain.",
"Theorem 8.8: The domain ${\\cal U}$ is universal, in the sense that for every domain ${\\cal D}$ we have ${\\cal D}\\lhd {\\cal U}$ .",
"If ${\\cal D}$ is effectively given, then the projection pair for the embedding is computable.",
"In fact, there is a correspondence between the effectively presented domains and the computable finitary projections of ${\\cal U}$ .",
"Proof Recall that ${\\bf D}$ must be countable to be a finitary basis.",
"Thus, we can assume that the basis has an enumeration $D=\\lbrace X_n\\:\\vert \\:n\\in {{N}}\\rbrace $ where $X_0=\\Delta $ .",
"The effective and general cases are considered together in the proof; comments about computability are included for the effective case as required.",
"Thus, if ${\\cal D}$ is effectively given, the enumeration above is assumed to be computable.",
"To prove that the domain can be embedded in ${\\cal U}$ , the embedding will be shown.",
"To start, for each finite element $d_i$ in the basis, define two sets, $d_i^+$ and $d_i^-$ as follows: $\\begin{array}{lcl}d_i^+&=&\\lbrace d\\in {\\bf D}\\:\\vert \\:d_i\\sqsubseteq d\\rbrace \\\\d_i^-&=&D-d_i^+\\end{array}$ The $d_i^+$ set contains all the elements that $d_i$ approximates, while the $d_i^-$ set contains all the other elements, partitioning ${\\bf D}$ into two disjoint sets.",
"Sets for different elements can be intersected to form finer partitions of ${\\bf D}$ .",
"For $k>0$ , let $R\\in \\lbrace +,-\\rbrace ^k$ , let $R_i$ be the $ith$ symbol in the string $R$ , and define a region $D_R$ as $D_R=\\bigcap \\limits _{i=1}^k d_i^{R_i}$ where $k$ is the length of $R$ .",
"The set $\\lbrace D_{R}\\:\\vert \\:R\\in \\lbrace +,-\\rbrace ^k\\rbrace $ of regions partitions ${\\bf D}$ into $2^k$ disjoint sets.",
"Thus, for each element $e_i$ in the enumeration there is a corresponding partition of the basis given by the family of sets $\\lbrace D_{R}\\:\\vert \\:R\\in \\lbrace +,-\\rbrace ^i\\rbrace $ .",
"For strings $R,S\\in \\lbrace +,-\\rbrace ^*$ such that $R$ is a prefix of $S$ , denoted $R\\le S$ , $D_S\\subseteq D_R$ .",
"It is important to realize that the composition of these sets is dependent on the order in which the elements are enumerated.",
"Some of these regions are empty, but it is decidable if a given intersection is empty if ${\\cal D}$ is effectively presented.",
"It is also decidable if a given element is in a particular region.",
"Figure: Example Finite DomainTo see the function these regions are serving, consider the finite domain in Figure REF .This example is taken from Cartwright and Demers [2].",
"Consider the enumeration with $d_0=\\bot , d_1=b, d_2=c, d_3=a.$ The $d_i^+$ and $d_i^-$ sets are as follows: $\\begin{array}{lcl}d_1^+&=&\\lbrace a,b\\rbrace \\\\d_1^-&=&\\lbrace c,\\bot \\rbrace \\\\d_2^+&=&\\lbrace c\\rbrace \\\\d_2^-&=&\\lbrace a,b,\\bot \\rbrace \\\\d_3^+&=&\\lbrace a\\rbrace \\\\d_3^-&=&\\lbrace b,c,\\bot \\rbrace \\end{array}$ The regions are as follows: $\\begin{array}{lclclcl}D_+ &=&\\lbrace a,b\\rbrace &\\:\\:\\:\\:&D_{+++} &=&\\lbrace \\rbrace \\\\D_- &=&\\lbrace \\bot ,c\\rbrace &&D_{++-} &=&\\lbrace \\rbrace \\\\D_{++} &=&\\lbrace \\rbrace &&D_{+-+} &=&\\lbrace a\\rbrace \\\\D_{+-} &=&\\lbrace a,b\\rbrace &&D_{+--} &=&\\lbrace b\\rbrace \\\\D_{-+} &=&\\lbrace c\\rbrace &&D_{-++} &=&\\lbrace \\rbrace \\\\D_{--} &=&\\lbrace \\bot \\rbrace &&D_{-+-} &=&\\lbrace c\\rbrace \\\\&&&&D_{--+} &=&\\lbrace \\rbrace \\\\&&&&D_{---} &=&\\lbrace \\bot \\rbrace \\end{array}$ The regions generated by each successive element encode the relationships induced by the approximation ordering between the new element and all elements previously added.",
"The reader is encouraged to try this example with other enumerations of this basis and compare the results.",
"The embedding of the elements proceeds by building a tree based on the regions corresponding to the element.",
"The regions are used to find locations in the tree and to determine whether a $\\top $ or a $\\Delta $ element is placed in the location.",
"These trees preserve the relationships specified by the regions and thus, the tree embedding is isomorphic to the domain in question.",
"Once the tree is built, the reduction rules are applied until a non-reducible tree is reached.",
"This tree is the representative element in the universal domain, and the set of these trees form the sub-space.",
"The function to determine the location in the tree for a given domain, $Loc_D:\\lbrace +,-\\rbrace ^*\\rightarrow \\lbrace l,r\\rbrace ^*$ takes strings used to generate regions and outputs a path in a tree where $l$ stands for left sub-tree and $r$ stands for right sub-tree.",
"This path is computed using the following inductive definition: $\\begin{array}{lcll}Loc_D(\\epsilon )&=&\\epsilon .\\\\Loc_D(R+)&=&Loc_D(R)l&{\\rm if }~D_{R+}\\ne \\emptyset ~ {\\rm and }~D_{R-}\\ne \\emptyset .\\\\&=&Loc_D(R)&{\\rm otherwise}.\\\\Loc_D(R-)&=&Loc_D(R)r&{\\rm if }~D_{R+}\\ne \\emptyset ~{\\rm and }~D_{R-}\\ne \\emptyset .\\\\&=&Loc_D(R)&{\\rm otherwise}.\\end{array}$ The set of locations for each non-empty region is the set of paths to all leaves of some finite binary tree.",
"An induction argument is used to show the following properties of $Loc_D$ that ensure this: If $R\\le S$ for $R,S\\subseteq \\lbrace +,-\\rbrace ^*$ , then $Loc_D(R)\\le Loc_D(S)$ .",
"Let $S=\\lbrace Loc_D(R)\\:\\vert \\:R\\in \\lbrace +,-\\rbrace ^k\\wedge D_R\\ne \\emptyset \\rbrace $ for $k>0$ be a set of location paths for a given $k$ .",
"For any $p\\in \\lbrace l,r\\rbrace ^*$ there exists $q\\in S$ such that either $p\\le q$ or $q\\le p$ .",
"That is, every potential path is represented by some finite path.",
"Finally, for all $p,q\\in S$ if $p\\le q$ then $p=q$ .",
"This means that a unique leaf is associated with each location.",
"To find the tree for a given element $d_k$ in the enumeration, apply the following rules to each $R\\in \\lbrace +,-\\rbrace ^{k-1}$ .",
"If $D_{R-}\\ne \\emptyset $ then the leaf for path $Loc_D(R-)$ is labeled $\\top $ .",
"If $D_{R+}\\ne \\emptyset $ then the leaf for path $Loc_D(R+)$ is labeled $\\Delta $ .",
"These rules are used to assign a tree in ${\\bf U}$ , which is then reduced using the reduction rules, for each element in the enumeration of ${\\bf D}$ .",
"To see that the top element is never assigned by these rules, note that some region of the form $R+$ for every length $k$ must be non-empty since it must contain the element $e_k$ being embedded.",
"Returning to the example, the location function defines paths for these elements as follows: $\\begin{array}{lclclcl}Loc_D(+)&=&l&\\:\\:\\:\\:&Loc_D(+-+)&=&ll\\\\Loc_D(-)&=&r&&Loc_D(+--)&=&lr\\\\Loc_D(+-)&=&l&&Loc_D(-+-)&=&rl\\\\Loc_D(-+)&=&rl&&Loc_D(---)&=&rr\\\\Loc_D(--)&=&rr\\end{array}$ The trees generated for each of the elements are: $\\begin{array}{lcl}d_0&\\mapsto &\\Delta \\\\d_1&\\mapsto &\\langle \\Delta ,\\top \\rangle \\\\d_2&\\mapsto &\\langle \\top ,\\langle \\Delta ,\\top \\rangle \\rangle \\\\d_3&\\mapsto & \\langle \\langle \\Delta ,\\top \\rangle ,\\langle \\top ,\\top \\rangle \\rangle \\\\&\\mapsto & \\langle \\langle \\Delta ,\\top \\rangle ,\\top \\rangle \\end{array}$ To verify that the space generated is a valid sub-space, we must verify that the bottom element is mapped to $\\bot _U$ and that the consistency and lub relations are maintained.",
"The tree $\\Delta $ is clearly assigned to $X_0$ , the bottom element for the basis being embedded, since there are no strings of length $-1$ .",
"The embedding preserves inconsistency of elements by forcing the lub of the embedded elements to be $\\top $ .",
"The $D_{R-}$ regions represent the elements that the element being embedded does not approximate.",
"Note that the $D_{R-}$ sets cause the $\\top $ element to be added as the leaf.",
"Since the $D_R$ sets are built using the approximation ordering, it is straightforward to see that the approximation ordering is preserved by the embedding.",
"Lubs are also maintained by the embedding, although the reduction is required to see that this is the case.",
"It should be clear that, if the domain ${\\cal D}$ is effectively given, the sub-space can be computed since the embedding procedure uses the relationships given in the presentation.",
"Finally, suppose that $a$ is a computable, finitary projection on ${\\cal U}$ .",
"From the proof of Theorem REF , the domain of this projection is characterized by the set $\\lbrace y\\in {\\bf U}\\:\\vert \\:y\\:a\\:y\\rbrace $ If $a$ is computable, the set of pairs for $a$ is recursively enumerable.",
"Thus, the set above is also recursively enumerable since equality among basis elements is decidable.",
"Thus, the domain given by the projection must also be effectively given.",
"$\\:\\:\\Box $ Thus, the domain ${\\cal U}$ is an effectively presented universal domain in which all other domains can be embedded.",
"The sub-domains of ${\\cal U}$ include ${\\cal U}\\rightarrow {\\cal U}$ , ${\\cal U}\\times {\\cal U}$ , etc.",
"These domains must be sub-domains of ${\\cal U}$ since they are effectively presented based on our earlier theorems.",
"Domain Constructors in ${\\cal U}$ The next step is to see how to define the constructors commonly used.",
"Definition 8.9: [Domain Constructors] Let the computable projection pair, $i_+:{\\cal U}+{\\cal U}\\rightarrow {\\cal U}~{\\rm and}~j_+:{\\cal U}\\rightarrow {\\cal U}+{\\cal U}$ be fixed.",
"Fix suitable projection pairs $i_\\times ,j_\\times ,i_\\rightarrow $ , and $j_\\rightarrow $ as well.",
"Define $\\begin{array}{lcl}a+b&=&cond\\circ \\langle which,i_+\\circ in_0\\circ a\\circ out_0, i_+\\circ in_1\\circ b\\circ out_1\\rangle \\circ j_+\\\\a\\times b&=&i_x\\circ \\langle a\\circ proj_0,b\\circ proj_1\\rangle \\circ j_x\\\\a\\rightarrow b&=&i_\\rightarrow \\circ (\\lambda f.b\\circ f\\circ a)\\circ j_\\rightarrow \\end{array}$ for all $a,b:{\\cal U}\\rightarrow {\\cal U}$ .",
"From earlier theorems, we know that these combinators are all computable over an effectively presented domain.",
"The next theorem characterizes the effect these combinators have on projection functions.",
"Theorem 8.10: If $a,b:{\\cal U}\\rightarrow {\\cal U}$ are projections, then so are $a+b$ , $a\\times b$ , and $a\\rightarrow b$ .",
"If $a$ and $b$ are finitary, then so are the compound projections.",
"Proof Since $a$ and $b$ are retractions, $a=a\\circ a$ and $b=b\\circ b$ .",
"Then for $a\\times b$ using the definition of $\\times $ , $\\begin{array}{lcl}(a\\times b)\\circ (a\\times b)&=&i_x\\circ \\langle a\\circ proj_0,b\\circ proj_1\\rangle \\circ \\langle a\\circ proj_0,b\\circ proj_1\\rangle \\circ j_x\\\\&=&i_x\\circ \\langle a\\circ a\\circ proj_0,b\\circ b\\circ proj_1\\rangle \\circ j_x\\\\&=& a\\times b\\end{array}$ Thus, $a\\times b$ is a retraction.",
"The other cases follow similarly.",
"Since $a$ and $b$ are projections, $a,b\\subseteq {\\sf I}_U$ (denoted simply ${\\sf I}$ for the remainder of the proof).",
"Using the definition for $+$ along with the above relation and the definition of projection pairs, we can see that $a+b\\subseteq {\\sf I}+{\\sf I}=i_+\\circ j_+ \\subseteq {\\sf I}$ Thus, $a+b$ is a projection.",
"The other cases follow similarly.",
"To show that the projections are finitary, we must show that the fixed point sets are isomorphic to a domain.",
"Since $a$ and $b$ are assumed finitary, their fixed point sets are isomorphic to $\\begin{array}{lcl}D_a&=&\\lbrace x\\in {\\bf U}\\:\\vert \\:x\\:a\\: x\\rbrace \\\\D_b&=&\\lbrace y\\in {\\bf U}\\:\\vert \\:y\\:b\\: y\\rbrace \\end{array}$ We wish to show that ${\\cal D}_a\\rightarrow {\\cal D}_b\\approx {\\cal D}_{a\\rightarrow b}$ .",
"By the definition of the $\\rightarrow $ constructor, the fixed point set of $a\\rightarrow b$ over ${\\cal U}$ is the same as the fixed point set of $\\lambda f.b\\circ f\\circ a$ on ${\\cal U}\\rightarrow {\\cal U}$ .",
"(Hint: $i_\\rightarrow $ and $j_\\rightarrow $ set up the isomorphism.)",
"So, the fixed points for $f:{\\cal U}\\rightarrow {\\cal U}$ are of the form: $f=b\\circ f\\circ a$ We can think of $a$ as a function in ${\\cal U}\\rightarrow {\\cal D}_a$ and define the other half of the projection pair as $i_a:{\\cal D}_a\\rightarrow {\\cal U}$ where $i_a\\circ a = a$ and $a\\circ i_a=i_a$ .",
"Define a function $i_b$ for the projection pair for $b$ similarly.",
"For some $g:{\\cal D}_a\\rightarrow {\\cal D}_b$ let $f=i_b\\circ g\\circ a$ Substituting this definition for $f$ yields $b\\circ f\\circ a = b\\circ i_b\\circ g\\circ a\\circ a = i_b\\circ g \\circ a = f$ by the definition of $i_b$ and since $a$ is a retraction by assumption.",
"Conversely, for a function $f$ such that $i_b\\circ g\\circ a= f$ , let $g=b\\circ f\\circ i_a$ Substituting again, $i_b\\circ g\\circ a = i_b\\circ g\\circ f\\circ i_a\\circ a = b\\circ f\\circ a = f$ Thus, there is an order preserving isomorphism between $g:{\\cal D}_a\\rightarrow {\\cal D}_b$ and the functions $f=b\\circ f\\circ a$ .",
"The proofs of the isomorphisms for the other constructs are similar.",
"$\\:\\:\\Box $ Thus, the sub-domain relationship with the universal domain has been stated in terms of finitary projections over the universal domain.",
"In addition, all the domain constructors have been shown to be computable combinators on the domain of these finitary projections.",
"Recalling that all computable maps have computable fixed points, the standard fixed point method can be used to solve domain equations of all kinds if they can be defined on projections.",
"Returning to the $\\lambda $ -calculus for a moment, all objects in the $\\lambda $ -calculus are considered functions.",
"Since ${\\cal U}\\rightarrow {\\cal U}$ is a part of ${\\cal U}$ , every object in the $\\lambda $ -calculus is also an object of ${\\cal U}$ .",
"Transposing some of the familiar notation, where the old notation appears on the left, the new combinators are defined as follows: $\\begin{array}{lcl}which(z)=which(j_+(z))\\\\in_i(x)=i_+(in_i(x))~{\\rm where}~i=0,1\\\\out_i(x)=out_i(j_+(x))~{\\rm where}~i=0,1\\\\\\langle x,y\\rangle =i_x(\\langle x,y\\rangle )\\\\proj_i=proj_i(j_x(z))~{\\rm where}~i=0,1\\\\u(x) = j_\\rightarrow (u)(x)\\\\\\lambda x.\\tau =i_\\rightarrow (\\lambda x.\\tau )\\end{array}$ Thus, all functions, all constants, all combinators, and all constructs are elements of ${\\cal U}$ .",
"Indeed, everything computable is an element of ${\\cal U}$ .",
"Elements in ${\\cal U}$ play multiple roles by representing different objects under different projections.",
"While this notion may be difficult to get used to, there are many advantages, both notational and conceptual.",
"Exercises Exercise 8.11: A retraction $a:{\\cal D}\\rightarrow {\\cal D}$ is a closure operator iff ${\\sf I}_D\\subseteq a$ as relations.",
"On a domain like ${\\cal P}({{N}})$ , give some examples of closure operators.",
"(Hint: Close up the integers under addition.",
"Is this continuous on ${\\cal P}({{N}})$ ?)",
"Prove in general that for any closure $a:{\\cal D}\\rightarrow {\\cal D}$ , the fixed point set of $a$ is always a finitary domain.",
"(Hint: Show that the fixed point set is closed as required for a domain.)",
"What are the finite elements of the fixed point set?",
"Exercise 8.12: Give a direct proof that the domain $\\lbrace X\\:\\vert \\:X\\lhd {\\cal D}\\rbrace $ is effectively presented if ${\\cal D}$ is.",
"(Hint: The finite elements of the domain correspond exactly to the finite domains $X\\lhd {\\cal D}$ .)",
"In the case of ${\\cal D}={\\cal U}$ , show that the computable elements of the domain correspond exactly to the effectively presented domains (up to effective isomorphism).",
"Exercise 8.13: For finitary projections $a:{\\cal E}\\rightarrow {\\cal E}$ , write ${\\cal D}_a=\\lbrace x\\in {\\cal E}\\:\\vert \\:x\\:a\\:x\\rbrace $ Show that for any two such projections $a$ and $b$ , that $a\\subseteq b \\iff {\\cal D}_a\\lhd {\\cal D}_b$ Exercise 8.14: Find another universal domain that is not isomorphic to ${\\cal U}$ .",
"Exercise 8.15: Prove the remaining cases in Theorem REF .",
"Exercise 8.16: Suppose $S$ and $T$ are two binary constructors on domains that can be made into computable operators on projections over the universal domain.",
"Show that we can find a pair of effectively presented domains such that $D\\approx S(D,E)~{\\rm and}~E\\approx T(D,E).$ Exercise 8.17: Using the translations shown after the proof of Theorem REF , show how the whole typed-$\\lambda $ -calculus can be translated into ${\\cal U}$ .",
"(Hint: for $f:{\\cal D}_a\\rightarrow {\\cal B}$ , write $f=b\\circ f\\circ a$ for finitary projections $a$ and $b$ .",
"For $\\lambda x^{{\\cal D}_a}.\\sigma $ , write $\\lambda x.b(\\sigma ^{\\prime }[a(x)/x])$ where $\\sigma ^{\\prime }$ is the translation of $\\sigma $ into the untyped $\\lambda $ -calculus.",
"Be sure that the resulting term has the right type.)",
"Exercise 8.18: Show that the basis presented for the universal domain ${\\bf U}$ is indeed a finitary basis and that it has an effective presentation.",
"Exercise 8.19: Work out the embedding for the other enumerations for the example given in the proof of Theorem REF ."
],
[
"Typed $\\lambda $ -Calculus",
"As shown in the previous section, functions can be characterized by recursion equations which combine previously defined functions with the function being defined.",
"The expression of these functions is simplified in this section by introducing a notation for specifying a function without having to give the function a name.",
"The notation used is that of the typed $\\lambda $ -Calculus; a function is defined using a $\\lambda $ -abstraction."
],
[
"Definition of Typed $\\lambda $ -Calculus",
"An informal characterization of the $\\lambda $ -calculus suffices for this section; more formal descriptions are available elsewhere in the literature [1].",
"Thus, examples are used to introduce the notation.",
"An infinite number of variables, $x$ ,$y$ ,$z$ ,$\\ldots $ of various types are required.",
"While a variable has a certain type, type subscripts will not be used due to the notational complexity.",
"A distinction must also be made between type symbols and domains.",
"The domain ${\\cal A}\\times {\\cal B}$ does not uniquely determine the component domains ${\\cal A}$ and ${\\cal B}$ even though these domains are uniquely determined by the symbol for the domain.",
"The domain is the meaning that we attribute to the symbol.",
"In addition to variables, constants are also present.",
"For example, the symbol 0 is used to represent the zero element from the domain ${\\cal N}$ .",
"Another constant, present in each domain by virtue of Theorem REF , is $fix^{\\cal D}$ , the least fixed point operator for domain ${\\cal D}$ of type $({\\cal D}\\rightarrow {\\cal D})\\rightarrow {\\cal D}$ .",
"The constants and variables are the atomic (non-compound) terms.",
"Types can be associated with all atomic terms.",
"There are several constructions for compound terms.",
"First, given $\\tau ,\\ldots ,\\sigma $ , a list of terms, the ordered tuple $\\langle \\tau ,\\ldots ,\\sigma \\rangle $ is a compound term.",
"If the types of $\\tau ,\\ldots ,\\sigma $ are ${\\cal A},\\ldots ,{\\cal B}$ , the type of the tuple is ${\\cal A}\\times \\ldots \\times {\\cal B}$ since the tuple is to be an element of this domain.",
"The tuple notation for combining functions given earlier should be disregarded here.",
"The next construction is function application.",
"If the term $\\tau $ has type ${\\cal A}\\rightarrow {\\cal B}$ and the term $\\sigma $ has the type ${\\cal A}$ , then the compound term $\\tau (\\sigma )$ has type ${\\cal B}$ .",
"Function application denotes the value of a function at a given input.",
"The notation $\\tau (\\sigma _0,\\ldots ,\\sigma _n)$ abbreviates $\\tau (\\langle \\sigma _0,\\ldots ,\\sigma _n\\rangle )$ .",
"Functions applied to tuples allows us to represent applications of multi-variate functions.",
"The $\\lambda $ -abstraction is used to define functions.",
"Let $x_0,\\ldots ,x_n$ be a list of distinct variables of type ${\\cal D}_0,\\ldots ,{\\cal D}_n$ .",
"Let $\\tau $ be a term of some type ${\\cal D}_{n+1}$ .",
"$\\tau $ can be thought of as a function of $n+1$ variables with type $({\\cal D}_0\\times \\ldots \\times {\\cal D}_n)\\rightarrow {\\cal D}_{n+1}$ .",
"The name for this function is written $\\lambda x_0,\\ldots ,x_n.\\tau $ This expression denotes the entire function.",
"To look at some familiar functions in the new notation, consider $\\lambda x,y.x$ This notation is read “lambda ex wye (pause) ex”.",
"If the types of $x$ and $y$ are ${\\cal A}$ and ${\\cal B}$ respectively, the function has type $({\\cal A}\\times {\\cal B})\\rightarrow {\\cal A}$ .",
"This function is the first projection function $p_0$ .",
"This function and the second projection function can be defined by the following equations: $\\begin{array}{lcl}p_0&=&\\lambda x,y.x\\\\p_1&=&\\lambda x,y.y\\end{array}$ Recalling the function tuple notation introduced in an earlier section, the following equation holds: $\\langle f,g\\rangle =\\lambda w.\\langle f(w),g(w)\\rangle $ which defines a function of type ${\\cal D}_1\\rightarrow ({\\cal D}_2\\times {\\cal D}_3)$ .",
"Other familiar functions are defined by the following equations: $\\begin{array}{lcl}eval&=&\\lambda f,x.f(x)\\\\curry&=&=\\lambda g\\lambda x\\lambda y.g(x,y)\\end{array}$ The $curry$ example shows that this notation can be iterated.",
"A distinction is thus made between the terms $\\lambda x,y.x$ and $\\lambda x\\lambda y.x$ which have the types ${\\cal D}_0\\times {\\cal D}_1\\rightarrow {\\cal D}_0$ and ${\\cal D}_0\\rightarrow {\\cal D}_1\\rightarrow {\\cal D}_0$ respectively.",
"Thus, the following equation also holds: $curry(\\lambda x,y.\\tau )=\\lambda x\\lambda y.\\tau $ which relates the multi-variate form to the iterated or curried form.",
"Another true equation is $fix={\\bf fix}(\\lambda F\\lambda f.f(F(f)))$ where $fix$ has type $({\\cal D}\\rightarrow {\\cal D})\\rightarrow {\\cal D}$ and fix has type $(((({\\cal D}\\rightarrow {\\cal D})\\rightarrow {\\cal D})\\rightarrow (({\\cal D}\\rightarrow {\\cal D})\\rightarrow {\\cal D}))\\rightarrow (({\\cal D}\\rightarrow {\\cal D})\\rightarrow {\\cal D}))$ This is the content of Exercise REF .",
"This notation can now be used to define functions using recursion equations.",
"For example, the function $\\sigma $ in Example REF can be characterized by the following equation: $\\sigma =fix(\\lambda f\\lambda n.cond(zero(n),0,f(pred(n))+pred(n))$ which states that $\\sigma $ is the least recursively defined function $f$ whose value at $n$ is $cond(\\ldots )$ .",
"The variable $f$ occurs in the body of the $cond$ expression, but this is just the point of a recursive definition.",
"$f$ is defined in terms of its value on “smaller” input values.",
"The use of the fixed point operator makes the definition explicit by forcing there to be a unique solution to the equation.",
"In an abstraction $\\lambda x,y,z.\\tau $ , the variables $x$ ,$y$ , and $z$ are said to be bound in the term $\\tau $ .",
"Any other variables in $\\tau $ are said to be free variables in $\\tau $ unless they are bound elsewhere in $\\tau $ .",
"Bound variables are simply placeholders for values; the particular variable name chosen is irrelevant.",
"Thus, the equation $\\lambda x.\\tau =\\lambda y.\\tau [y/x]$ is true provided $y$ is not free in $\\tau $ .",
"The notation $\\tau [y/x]$ specifies the substitution of $y$ for $x$ everywhere $x$ occurs in $\\tau $ .",
"The notation $\\tau [\\sigma /x]$ for the substitution of the term $\\sigma $ for the variable $x$ is also legitimate."
],
[
"Semantics of Typed $\\lambda $ -Calculus",
"To show that the equations above with $\\lambda $ –terms are indeed meaningful, the following theorem relating $\\lambda $ –terms and approximable mappings must be proved.",
"Theorem 5.1: Every typed $\\lambda $ –term defines an approximable function of its free variables.",
"Proof Induction on the length of the term and its structure will be used in this proof.",
"Variables Direct since $x\\mapsto x$ is an approximable function.",
"Constants Direct since $x\\mapsto k$ is an approximable function for constant $k$ .",
"Tuples Let $\\tau =\\langle \\sigma _0,\\ldots ,\\sigma _n\\rangle $ .",
"Since the $\\sigma _i$ terms are less complex, they are approximable functions of their free variables by the induction hypothesis.",
"Using Theorem REF (generalized to the multi-variate case) then, $\\tau $ which takes tuples as values also defines an approximable function.",
"Application Let $\\tau =\\sigma _0(\\sigma _1)$ .",
"We assume that the types of the terms are appropriately matched.",
"The $\\sigma _i$ terms define approximable functions again by the induction hypothesis.",
"Recalling the earlier equations, the value of $\\tau $ is the same as the value of $eval(\\sigma _0,\\sigma _1)$ .",
"Since $eval$ is approximable, Theorem REF shows that the term defines an approximable function.",
"Abstraction Let $\\tau =\\lambda x.\\sigma $ .",
"By the induction hypothesis, $\\sigma $ defines a function of its free variables.",
"Let those free variables be of types ${\\cal D}_0,\\ldots ,{\\cal D}_n$ where ${\\cal D}_n$ is the type of $x$ .",
"Then $\\sigma $ defines an approximable function $g:{\\cal D}_0\\times \\ldots \\times {\\cal D}_n\\rightarrow {\\cal D}^{\\prime }$ where ${\\cal D}^{\\prime }$ is the type of $\\sigma $ .",
"Using Theorem REF , the function $curry(g):{\\cal D}_0\\times \\ldots \\times {\\cal D}_{n-1}\\rightarrow ({\\cal D}_n\\rightarrow {\\cal D}^{\\prime })$ yields an approximable function, but this is just the function defined by $\\tau $ .",
"The reader can generalize this proof for multiple bound variables in $\\tau $ .",
"$\\:\\:\\Box $ Given this, the equation $\\tau =\\sigma $ states that the two terms define the same approximable function of their free variables.",
"As an example, $\\lambda x.\\tau =\\lambda y.\\tau [y/x]$ provided $y$ is not free in $\\tau $ since the generation of the approximable function did not depend on the name $x$ but only on its location in $\\tau $ .",
"Other equations such as these are given in the exercises.",
"The most basic rule is shown below.",
"Theorem 5.2: For appropriately typed terms, the following equation is true: $(\\lambda x_0,\\ldots ,x_{1}.\\tau )(\\sigma _0,\\ldots ,\\sigma _{n-1})=\\tau [\\sigma _0/x_0,\\ldots ,\\sigma _{n-1}/x_{n-1}]$ Proof The proof is given for $n=1$ and proceeds again by induction on the length of the term and the structure of the term.",
"Variables This means $(\\lambda x.x)(\\sigma )=\\sigma $ must be true which it is.",
"Constants This requires $(\\lambda x.k)(\\sigma )= k$ must be true which it is for any constant $k$ .",
"Tuples Let $\\tau =\\langle \\tau _0,\\tau _1\\rangle $ .",
"This requires that $(\\lambda x.\\langle \\tau _0,\\tau _1\\rangle )(\\sigma ) =\\langle \\tau _0[\\sigma /x],\\tau _1[\\sigma /x]\\rangle $ must be true.",
"This equation holds since the left-hand side can be transformed using the following true equation: $(\\lambda x.\\langle \\tau _0,\\tau _1\\rangle )(\\sigma ) =\\langle (\\lambda x.\\tau _0)(\\sigma ),(\\lambda x.\\tau _1)(\\sigma )\\rangle $ Then the inductive hypothesis is applied to the $\\tau _i$ terms.",
"Applications Let $\\tau =\\tau _0(\\tau _1)$ .",
"Then, the result requires that the equation $(\\lambda x.\\tau _0(\\tau _1))(\\sigma ) =\\tau _0[\\sigma /x](\\tau _1[\\sigma /x])$ hold true.",
"To see that this is true, examine the approximable functions for the left-hand side of the equation.",
"$\\begin{array}{lcl}\\tau _0&\\mapsto &\\bar{V},x\\rightarrow t_0\\\\\\tau _1&\\mapsto &\\bar{V},x\\rightarrow t_1\\\\\\sigma &\\mapsto &\\bar{V}\\rightarrow s\\\\\\mbox{so}\\\\(\\lambda x.\\tau _0(\\tau _1))(\\sigma )&\\mapsto &\\bar{V}\\rightarrow [(x\\rightarrow t_0(t_1))(s)]\\\\&=&\\bar{V},x\\rightarrow [(x\\rightarrow t_0)(s)]([(x\\rightarrow t_1)(s)])\\end{array}$ From this last term, we use the induction hypothesis.",
"To see why the last step holds, start with the set representing the left-hand side and using the aprroximable mappings for the terms: $\\begin{array}{cl}&(\\lambda x.\\tau _0(\\tau _1))(\\sigma )\\\\\\mapsto &\\bar{V}\\rightarrow [(x\\rightarrow t_0(t_1))(s)]\\\\=&\\lbrace b\\:\\vert \\:\\exists a.a\\in s\\wedge a\\:[x\\rightarrow t_0(t_1)]\\: b\\rbrace \\\\=&\\lbrace b\\:\\vert \\:\\exists a. a\\in s\\wedge a\\:\\lbrace (x,u)\\:\\vert \\:v\\in x\\rightarrow t_1)\\wedge v\\:(x\\rightarrow t_0)\\: u\\rbrace \\: b\\rbrace \\\\=&\\lbrace b\\:\\vert \\:\\exists a.a\\in s\\wedge v\\in (x\\rightarrow t_1)(a)\\wedge v\\:(x\\rightarrow t_0)(a)\\:b\\rbrace \\\\=&\\lbrace b\\:\\vert \\:\\exists a,c.a\\in s\\wedge a\\:(x\\rightarrow t_1)\\: v\\wedge a\\:(x\\rightarrow t_0)\\:c\\wedge v\\:c\\:b\\rbrace \\\\=&\\lbrace b\\:\\vert \\:v\\in [(x\\rightarrow t_1)(s)]\\wedge c\\in (x\\rightarrow t_0)(s) \\wedge v\\:c\\:b\\rbrace \\\\=&\\lbrace b\\:\\vert \\:v\\in [(x\\rightarrow t_1)(s)]\\wedge v\\:[(x\\rightarrow t_0)(s)]\\:b\\rbrace \\\\=&[(x\\rightarrow t_0)(s)]([(x\\rightarrow t_1)(s)])\\end{array}$ Abstractions Let $\\tau =\\lambda y.\\tau _0$ .",
"The required equation is $(\\lambda x.\\lambda y.\\tau _0)(\\sigma )=\\lambda y.\\tau _0[\\sigma /x]$ provided that $y$ is not free in $\\sigma $ .",
"The following true equation applies here: $(\\lambda x.\\lambda y.\\tau )(\\sigma )=\\lambda y.",
"((\\lambda x.\\tau )(\\sigma ))$ To see that this equation holds, let $g$ be a function of $n+2$ free variables defined by $\\tau $ .",
"By Theorem REF , the term $\\lambda x.\\lambda y.\\tau $ defines the function $curry(curry(g))$ of $n$ variables.",
"Call this function $h$ .",
"Thus, $h(v)(\\sigma )(y) = g(v,\\sigma ,y)$ where $v$ is the list of the other free variables.",
"Using a combinator $inv$ which inverts the order of the last two arguments, $h(v)(\\sigma )(y)=curry(inv(g))(v,y)(\\sigma )$ But, $curry(inv(g))$ is the function defined by $\\lambda x.\\tau $ .",
"Thus, we have shown that $(\\lambda x.\\lambda y.\\tau )(\\sigma )(y)=(\\lambda x.\\tau )(\\sigma )$ is a true equation.",
"If $y$ is not free in $\\alpha $ and $\\alpha (y)=\\beta $ is true, then $\\alpha =\\lambda y.\\beta $ must also be true.",
"$\\:\\:\\Box $ If $\\tau ^{\\prime }$ is the term $\\lambda x,y.\\tau $ , then $\\tau ^{\\prime }(x,y)$ is the same as $\\tau $ .",
"This specifies that $x$ and $y$ are not free in $\\tau $ .",
"This notation is used in the proof of the following theorem.",
"Theorem 5.3: The least fixed point of $\\lambda x,y.\\langle \\tau (x,y),\\sigma (x,y)\\rangle $ is the pair with coordinates $fix(\\lambda x.\\tau (x,fix(\\lambda y.\\sigma (x,y))))$ and $fix(\\lambda y.\\sigma (fix(\\lambda x.\\tau (x,y)),y))$ .",
"Proof We are thus assuming that $x$ and $y$ are not free in $\\tau $ and $\\sigma $ .",
"The purpose here is to find the least solution to the pair of equations: $x=\\tau (x,y)~{\\rm and}~y=\\sigma (x,y)$ This generalizes the fixed point equation to two variables.",
"More variables could be included using the same method.",
"Let $y_*=fix(\\lambda y.\\sigma (fix(\\lambda x.\\tau (x,y)),y))$ and $x_*=fix(\\lambda x.\\tau (x,y))$ Then, $x_*=\\tau (x_*,y_*)$ and $\\begin{array}{lcl}y_*&=&\\sigma (fix(\\lambda x.\\tau (x,y_*),y_*))\\\\&=&\\sigma (x_*,y_*).\\end{array}$ This shows that the pair $\\langle x_*,y_*\\rangle $ is one fixed point.",
"Now, let $\\langle x_0,y_0\\rangle $ be the least solution.",
"(Why must a least solution exist?",
"Hint: Consider a suitable mapping of type $({\\cal D}_x\\times {\\cal D}_y)\\rightarrow ({\\cal D}_x\\times {\\cal D}_y)$ .)",
"Thus, we know that $x_0=\\tau (x_0,y_0)$ , $y_0=\\sigma (x_0,y_0)$ , and that $x_0\\sqsubseteq x_*$ and $y_0\\sqsubseteq y_*$ .",
"But this means that $\\tau (x_0,y_0)\\sqsubseteq x_0$ and thus $fix(\\lambda x.\\tau (x,y_0))\\sqsubseteq x_0$ and consequently $\\sigma (fix(\\lambda x.\\tau (x,y_0),y_0))\\sqsubseteq \\sigma (x_0,y_0)\\sqsubseteq y_0$ By the fixed point definition of $y_*$ , $y_*\\sqsubseteq y_0$ must hold as well so $y_0=y_*$ .",
"Thus, $x_*=fix(\\lambda x.\\tau (x,y*))=fix(\\lambda x.\\tau (x,y_0))\\sqsubseteq x_0.$ Thus, $x*=x_0$ must also hold.",
"A similar argument holds for $x_0$ .$\\:\\:\\Box $ The purpose of the above proof is to demonstrate the use of least fixed points in proofs.",
"The following are also true equations: $fix(\\lambda x.\\tau (x))=\\tau (fix(\\lambda x.\\tau (x)))$ and $\\tau (y)\\sqsubseteq y\\:\\Rightarrow \\:fix(\\lambda x.\\tau (x))\\sqsubseteq y$ if $x$ is not free in $\\tau $ .",
"These equations combined with the monotonicity of functions were the methods used in the proof above.",
"Another example is the proof of the following theorem.",
"Theorem 5.4: Let $x$ ,$y$ , and $\\tau (x,y)$ be of type ${\\cal D}$ and let $g:{\\cal D}\\rightarrow {\\cal D}$ be a function.",
"Then the equation $\\lambda x.fix(\\lambda y.\\tau (x,y))=fix(\\lambda g.\\lambda x.\\tau (x,g(x)))$ holds.",
"Proof Let $f$ be the function on the left-hand side.",
"Then, $f(x)=fix(\\lambda y.\\tau (x,y))=\\tau (x,f(x))$ holds using the equations stated above.",
"Therefore, $f=\\lambda x.\\tau (x,f(x))$ and thus $g_0=fix(\\lambda g.\\lambda x.\\tau (x,g(x)))\\sqsubseteq f. $ By the definition of $g_0$ we have $g_0(x)=\\tau (x,g_0(x))$ for any given $x$ .",
"By the definition of $f$ we find that $f(x)=fix(\\lambda y.\\tau (x,y))\\sqsubseteq g_0(x)$ must hold for all $x$ .",
"Thus $f\\sqsubseteq g_0$ and the equation is true.$\\:\\:\\Box $ This proof illustrates the use of inclusion and equations between functions.",
"The following principle was used: $(\\forall x.\\tau \\sqsubseteq \\sigma )\\:\\Rightarrow \\:\\lambda x.\\tau \\sqsubseteq \\lambda x.\\sigma $ This is a restatement of the first part of Theorem REF .",
"Combinators and Recursive Functions Below is a list of various combinators with their definitions in $\\lambda $ -notation.",
"The meanings of those combinators not previously mentioned should be clear.",
"$\\begin{array}{lcl}p_0&=&\\lambda x,y.x\\\\p_1&=&\\lambda x,y.y\\\\pair&=&\\lambda x.\\lambda y.\\langle x,y\\rangle \\\\n-tuple&=&\\lambda x_0\\lambda \\ldots \\lambda x_{n-1}.\\langle x_0,\\ldots ,x_{n-1}\\rangle \\\\diag&=&\\lambda x.\\langle x,x\\rangle \\\\funpair&=&\\lambda f.\\lambda g.\\lambda x.\\langle f(x),g(x)\\rangle \\\\proj^n_i&=&\\lambda x_0,\\ldots ,x_{n-1}.x_i\\\\inv^n_{i,j}&=&\\lambda x_0,\\ldots ,x_i,\\ldots ,x_j,\\ldots ,x_{n-1}.\\langle x0,\\ldots ,x_j,\\ldots ,x_i,\\ldots ,x_{n-1}\\rangle \\\\eval&=&\\lambda f,x.f(x)\\\\curry&=&\\lambda g.\\lambda x.\\lambda y.g(x,y)\\\\comp&=&\\lambda f,g.\\lambda x.g(f(x))\\\\const&=&\\lambda k.\\lambda x.k\\\\{\\bf fix}&=&\\lambda f.fix(\\lambda x.f(x))\\end{array}$ These combinators are actually schemes for combinators since no types have been specified and thus the equations are ambiguous.",
"Each scheme generates an infinite number of combinators for all the various types.",
"One interest in combinators is that they allow expressions without variables—if enough combinators are used.",
"This is useful at times but can be clumsy.",
"However, defining a combinator when the same combination of symbols repeatedly appears is also useful.",
"There are some familiar combinators that do not appear in the table.",
"Combinators such as $cond$ , $pred$ , and $succ$ cannot be defined in the pure $\\lambda $ -calculus but are instead specific to certain domains.",
"They are thus regarded as primitives.",
"A large number of other functions can be defined using these primitives and the $\\lambda $ -notation, as the following theorem shows.",
"Theorem 5.5: For every partial recursive function $h:{{N}}\\rightarrow {{N}}$ , there is a $\\lambda $ -term $\\tau $ of type ${\\cal N}\\rightarrow {\\cal N}$ such that the only constants occurring in $\\tau $ are $cond$ , $succ$ , $pred$ , $zero$ , and 0 and if $h(n)=m$ then $\\tau (n)=m$ .",
"If $h(n)$ is undefined, then $\\tau (n)=\\bot $ holds.",
"$\\tau (\\bot )=\\bot $ is also true.",
"Proof It is convenient in the proof to work with strict functions $f:{\\cal N}^k\\rightarrow {\\cal N}$ such that if any input is $\\bot $ , the result of the function is $\\bot $ .",
"The composition of strict functions is easily shown to be strict.",
"It is also easy to see that any partial function $g:{{N}}^k\\rightarrow {{N}}$ can be extended to a strict approximable function $\\bar{g}:{\\cal N}^k\\rightarrow {\\cal N}$ which yields the same values on inputs for which $g$ is defined.",
"Other input values yield $\\bot $ .",
"We want to show that $\\bar{g}$ is definable with a $\\lambda $ -expression.",
"First we must show that primitive recursive functions have $\\lambda $ -definitions.",
"Primitive recursive functions are formed from starting functions using composition and the scheme of primitive recursion.",
"The starting functions are the constant function for zero and the identity and projection functions.",
"These functions, however, must be strict so the term $\\lambda x,y.x$ is not sufficient for a projection function.",
"The following device reduces a function to its strict form.",
"Let $\\lambda x.cond(zero(x),x,x)$ be a function with $x$ of type ${\\cal N}$ .",
"This is the strict identity function.",
"The strict projection function attempted above can be defined as $\\lambda x,y.cond(zero(y),x,x)$ The three variable projection function can be defined as $\\lambda x,y,z.cond(zero(x),cond(zero(z),y,y),cond(zero(z),y,y))$ While not very elegant, this device does produce strict functions.",
"Strict functions are closed under substitution and composition.",
"Any substitution of a group of functions into another function can be defined with a $\\lambda $ -term if the functions themselves can be so defined.",
"Thus, we need to show that functions obtained by primitive recursion are definable.",
"Let $f:{\\cal N}\\rightarrow {\\cal N}$ , and $g:{\\cal N}^3\\rightarrow {\\cal N}$ be total functions with $\\bar{f}$ and $\\bar{g}$ being $\\lambda $ -definable.",
"We obtain the function $h:{\\cal N}^2\\rightarrow {\\cal N}$ by primitive recursion where $\\begin{array}{lcl}h(0,m)&=&f(m)\\\\h(n+1,m)&=&g(n,m,h(n,m))\\end{array}$ for all $n,m\\in {\\cal N}$ .",
"The $\\lambda $ -term for $\\bar{h}$ is $fix(\\lambda k.\\lambda x,y.cond(zero(x),\\bar{f}(y),\\bar{g}(pred(x),y,k(pred(x),y))))$ Note that the fixed point operator for the domain ${\\cal N}^2\\rightarrow {\\cal N}$ was used.",
"The variables $x$ and $y$ are of type ${\\cal N}$ .",
"The $cond$ function is used to encode the function requirements.",
"The fixed point function is easily seen to be strict and this function is $\\bar{h}$ .",
"Primitive recursive functions are now $\\lambda $ -definable.",
"To obtain partial (i.e., general) recursive functions, the $\\mu $ -scheme (the least number operator) is used.",
"Let $f(n,m)$ be a primitive recursive function.",
"Then, define $h$ , a partial function, as $h(m) =$ the least $n$ such that $f(n,m)=0$ .",
"This is written as $h(m)=\\mu n.f(n,m)=0$ .",
"Since $\\bar{f}$ is $\\lambda $ -definable as has just been shown, let $\\bar{g}=fix(\\lambda g.\\lambda x,y.cond(zero(\\bar{f}(x,y)),x,g(succ(x),y)))$ Then, the desired function $\\bar{h}$ is defined as $\\bar{h}=\\lambda y.\\bar{g}(0,y)$ .",
"It is easy to see that this is a strict function.",
"Note that, if $h(m)$ is defined, clearly $h(m)=\\bar{g}(0,m)$ is also defined.",
"If $h(m)$ is undefined, it is also true that $\\bar{g}(0,m)=\\bot $ due to the fixed point construction but it is less obvious.",
"This argument is left to the reader.$\\:\\:\\Box $ Theorem REF does not claim that all $\\lambda $ -terms define partial recursive functions although this is also true.",
"Further examples of recursion are found in the exercises.",
"Exercises Exercise 5.6: Find the definitions of $\\lambda x,y.\\tau ~{\\rm and}~\\sigma (x,y)$ which use only $\\lambda v$ with one variable and applications only to one argument at a time.",
"Note that use must be made of the combinators $p_0$ , $p_1$ , and $pair$ .",
"Generalize the result to functions of many variables.",
"Exercise 5.7: The table of combinators was meant to show how combinators could be defined in terms of $\\lambda $ -expressions.",
"Can the tables be turned to show that, with enough combinators available, every $\\lambda $ -expression can be defined by combining combinators using application as the only mode of combination?",
"Exercise 5.8: Suppose that $f,g:{\\cal D}\\rightarrow {\\cal D}$ are approximable and $f\\circ g=g\\circ f$ .",
"Show that $f$ and $g$ have a least common fixed point $x=f(x)=g(x)$ .",
"(Hint: See Exercise REF .)",
"If, in addition, $f(\\bot )=g(\\bot )$ , show that $fix(f)=fix(g)$ .",
"Will $fix(f)=fix(f^2)$ ?",
"What if the assumption is weakened to $f\\circ g=g^2\\circ f$ ?",
"Exercise 5.9: For any domain ${\\cal D}$ , ${\\cal D}^\\infty $ can be regarded as consisting of bottomless stacks of elements of ${\\cal D}$ .",
"Using this view, define the following combinators with their obvious meaning: $head:{\\cal D}^\\infty \\rightarrow {\\cal D}$ , $tail:{\\cal D}^\\infty \\rightarrow {\\cal D}^\\infty $ and $push:{\\cal D}\\times {\\cal D}^\\infty \\rightarrow {\\cal D}^\\infty $ .",
"Using the fixed point theorem, argue that there is a combinator $diag:{\\cal D}\\rightarrow {\\cal D}^\\infty $ where for all $x\\in {\\cal D}$ , $diag(x)=\\langle x\\rangle _{n=0}^\\infty $ .",
"(Hint: Try a recursive definition, such as $diag(x)=push(x,diag(x))$ but be sure to prove that all terms of $diag(x)$ are $x$ .)",
"Also introduce by an appropriate recursive definition a combinator $map:({\\cal D}\\rightarrow {\\cal D})^\\infty \\times {\\cal D}\\rightarrow {\\cal D}^\\infty $ where for elements of the proper type $map(\\langle f_n\\rangle _{n=0}^\\infty ,x)=\\langle f_n(x)\\rangle _{n=0}^\\infty $ Exercise 5.10: For any domain ${\\cal D}$ introduce, as a least fixed point, a combinator $while:({\\cal D}\\rightarrow {\\cal T})\\times ({\\cal D}\\rightarrow {\\cal D})\\rightarrow ({\\cal D}\\rightarrow {\\cal D})$ by the recursion equation $while(p,f)(x)=cond(p(x),while(p,f)(f(x)),x)$ Prove that $while(p,while(p,f))=while(p,f)$ Show how $while$ could be used to obtain the least number operator,$\\mu $ , mentioned in the proof of Theorem REF .",
"Generalize this idea to define a combinator $find:{\\cal D}^\\infty \\times ({\\cal D}\\rightarrow {\\cal T})\\rightarrow {\\cal D}$ which means “find the first term in the sequence (if any) which satisfies the given predicate”.",
"Exercise 5.11: Prove the existence of a one-one function $num:{{N}}\\times {{N}}\\leftrightarrow {{N}}$ such that $\\begin{array}{lcl}num(0,0)&=&0\\\\num(n,m+1)&=&num(n+1,m)\\\\num(n+1,0)&=&num(0,n)+1\\end{array}$ Draw a descriptive picture (an infinite matrix) for the function.",
"Find a closed form for the values if possible.",
"Use the function to prove the isomorphism between ${\\cal P}({{N}})$ ,${\\cal P}({{N}}\\times {{N}})$ , and ${\\cal P}({{N}})\\times {\\cal P}({{N}})$ .",
"Exercise 5.12: Show that there are approximable mappings $graph:({\\cal P}({{N}})\\rightarrow {\\cal P}({{N}}))\\rightarrow {\\cal P}({{N}})$ and $fun:{\\cal P}({{N}})\\rightarrow ({\\cal P}({{N}})\\rightarrow {\\cal P}({{N}}))$ where $fun\\circ graph = \\lambda f.f$ and $graph\\circ fun\\sqsubseteq \\lambda x.x$ .",
"(Hint: Using the notation $[n_0,\\ldots ,n_k]=num(n_0,[n_1,\\ldots ,n_k])$ , two such combinators can be given by the formulas $\\begin{array}{lcl}fun(u)(x)&=&\\lbrace m\\:\\vert \\:\\exists n_0,\\ldots ,n_{k-1}\\in x.",
"[n_0+1,\\ldots ,n_{k-1}+1,0,m]\\in u\\rbrace \\\\graph(f)&=&\\lbrace [n_0+1,\\ldots ,n_{k-1}+1,0,m]\\:\\vert \\:m\\in f(\\lbrace n_0,\\ldots ,n_{k-1}\\rbrace )\\rbrace \\end{array}$ where $k$ is a variable - meaning all finite sequences are to be considered.)",
"Introduction to Domain Equations As stressed in the introduction, the notion of computation with potentially infinite elements is an integral part of domain theory.",
"The previous sections have defined the notion of functions over domains, as well as a notation for expressing these functions.",
"In addition, the notion of computation through series of approximations has been addressed.",
"This computation is possible since the functions defined have been approximable and thus continuous.",
"This section addresses the construction of more complex domains with infinite elements.",
"The next section looks specifically at the notion of computability with respect to these infinite elements.",
"The last section looks at another approach to domain construction.",
"New domains have been constructed from existing ones using domain constructors such as the product construction ($\\times $ ), the function space construction ($\\rightarrow $ ) and the sum construction ($+$ ) of Exercise REF .",
"These constructors can be iterated similar to the way that function application was iterated to form recursive function definitions.",
"In this way, domains can be characterized using recursion equations, called domain equations.",
"Domain Equations A domain equation represents an isomorphism between the domain as a whole and the combination of domains that comprise it.",
"These recursive domains are frequently termed reflexive domains since, as in the following example, the domain contains a copy of itself in its structure.",
"Example 6.1: Consider the following domain equation: ${\\cal T}={\\cal A}+({\\cal T}\\times {\\cal T})$ where ${\\cal A}$ is a previously defined domain.",
"This domain can be thought of as containing atomic elements from ${\\cal A}$ or pairs of elements of ${\\cal T}$ .",
"What do the elements of this domain look like?",
"In particular, what are the finite elements of this domain?",
"How is the domain constructed?",
"What is an appropriate approximation ordering for the domain?",
"What do lubs in this domain look like?",
"What is the appropriate notion of consistency?",
"Does this domain even exist?",
"In other words, are we certain a solution to this domain equation exists?",
"And if a solution to the equation exists, is it a unique solution?",
"Each of these questions is examined below.",
"The domain equation tells us that an element of the domain is either an element from ${\\cal A}$ or is a pair of “smaller” elements from ${\\cal T}$ .",
"One method of constructing a sum domain is using pairs where some distinguished element denotes what type an element is.",
"Thus, for some $a\\in {\\cal A}$ , the pair $\\langle \\pi ,a\\rangle $ might represent the element in ${\\cal T}$ for the given element $a$ .",
"For some $s,t\\in {\\cal T}$ , the pair $\\langle \\langle s,t\\rangle ,\\pi \\rangle $ might then represent the element in ${\\cal T}$ for the pair $s,t$ .",
"Thus, $\\pi $ is the distinguished element, and the location of $\\pi $ in the pair specifies the type of the element.",
"The finite elements are either elements in ${\\cal T}$ representing the (finite) elements of ${\\cal A}$ or the pair elements from ${\\cal T}$ whose components are also finite elements in ${\\cal T}$ .",
"The question then arises about infinite elements.",
"Are there infinite elements in this domain?",
"Consider the following fixed point equation for some element for $a\\in {\\cal A}$ : $x=\\langle \\langle a,x\\rangle ,\\pi \\rangle .$ The fixed point of this equation is the infinite product of the element $a$ .",
"Does this element fit the definition for ${\\cal T}$ ?",
"From the informal description of the elements of ${\\cal T}$ given so far, $x$ does qualify as a member of ${\\cal T}$ .",
"Now that some intuition has been developed about this domain, a formal construction is required.",
"Let $\\langle {\\bf A},\\sqsubseteq _A\\rangle $ be the finitary basis used to generate the domain ${\\cal A}$ .",
"Let $\\pi $ be an object such that $\\pi \\notin {\\bf A}$ .",
"Define the bottom element of the finitary basis T as $\\Delta _T=\\langle \\pi ,\\pi \\rangle $ .",
"Next, all the elements of ${\\cal A}$ must be included so define an element in ${\\bf T}$ for each $a\\in {\\bf A}$ as $\\langle \\pi ,a\\rangle $ .",
"Finally, pair elements for all elements in ${\\bf T}$ must exist in ${\\bf T}$ to complete the construction.",
"The set ${\\bf T}$ can be defined inductively as the least set such that: $\\Delta _T\\in {\\bf T}$ , $\\langle \\pi ,a\\rangle \\in {\\bf T}$ whenever $a\\in {\\bf A}$ , $\\langle \\langle \\Delta _T,s\\rangle ,\\pi \\rangle \\in {\\bf T}$ whenever $s\\in {\\bf T}$ (necessary??",
"), $\\langle \\langle t,\\Delta _T\\rangle ,\\pi \\rangle \\in {\\bf T}$ whenever $t\\in {\\bf T}$ (necessary??",
"), and $\\langle \\langle t,s\\rangle ,\\pi \\rangle \\in {\\bf T}$ whenever $s, t\\in {\\bf T}$ .",
"The set can also be characterized by the following fixed point equation: ${\\bf T}=\\lbrace \\Delta _T\\rbrace \\cup \\lbrace \\langle \\pi ,a\\rangle \\:\\vert \\:a\\in {\\bf A}\\rbrace \\cup \\lbrace \\langle \\langle \\Delta _T,s\\rangle ,\\pi \\rangle \\:\\vert \\:s\\in {\\bf T}\\rbrace \\cup \\lbrace \\langle \\langle t,\\Delta _T\\rangle ,\\pi \\rangle \\:\\vert \\:t\\in {\\bf T}\\rbrace \\cup \\lbrace \\langle \\langle t,s\\rangle ,\\pi \\rangle \\:\\vert \\:s,t\\in {\\bf T}\\rbrace .$ A solution must exist for this equation by the fixed point theorem.",
"Now that the basis elements have been defined, we must show how to find lubs.",
"We will again use an inductive definition.",
"$\\langle \\pi ,\\pi \\rangle \\sqcup t=t$ for all $t\\in {\\bf T}$ For $a,b\\in {\\bf A}$ , $\\langle \\pi ,a\\rangle \\sqcup \\langle \\pi ,b\\rangle =\\langle \\pi ,a\\sqcup b\\rangle $ if $a\\sqcup b$ exists in ${\\bf A}$ $\\langle \\langle s,t\\rangle ,\\pi \\rangle \\sqcup \\langle \\langle s^{\\prime },t^{\\prime }\\rangle ,\\pi \\rangle =\\langle \\langle s\\sqcup s^{\\prime },t\\sqcup t^{\\prime }\\rangle ,\\pi \\rangle $ if $s\\sqcup s^{\\prime }$ and $t\\sqcup t^{\\prime }$ exist in ${\\bf T}$ .",
"The lub $\\langle \\pi ,a\\rangle \\sqcup \\langle \\langle s,t\\rangle ,\\pi \\rangle $ does not exist.",
"Next, the notion of consistency needs to be explored.",
"From the definition of lubs given above, the following sets are consistent: The empty set is consistent.",
"Everything is consistent with the bottom element.",
"A set of elements all from the basis A is consistent in T if the set of elements is consistent in A.",
"A set of product elements in T is consistent if the left component elements are consistent and the right component elements are consistent.",
"These conditions derive from the sum and product nature of the domain.",
"The approximation ordering in the basis has the following inductive definition: $\\Delta _T\\sqsubseteq _T s$ for all $s\\in {\\bf T}$ $y\\sqsubseteq _Tu\\sqcup \\Delta _T$ whenever $y\\sqsubseteq _Tu$ $\\langle \\pi ,a\\rangle \\sqsubseteq _T\\langle \\pi ,b\\rangle $ whenever $a\\sqsubseteq _Ab$ $\\langle \\langle s,t\\rangle ,\\pi \\rangle \\sqsubseteq _T\\langle \\langle u,v\\rangle ,\\pi \\rangle $ whenever $s\\sqsubseteq _Tu$ and $t\\sqsubseteq _Tv$ The next step is to verify that ${\\bf T}$ is indeed a finitary basis.",
"The basis is still countable.",
"The approximation is clearly a partial order.",
"The existence of lubs of finite bounded (i.e., consistent) subsets must be verified.",
"The definition of consistency gives us the requirements for a bounded subset.",
"Each of the conditions for consistency are examined inductively since the definitions are all inductive: The lub of the empty set is the bottom element $\\Delta _T$ .",
"The lub of a set containing the bottom element is the lub of the set without the bottom element which must exist by the induction hypothesis.",
"The lub of a set of elements all from the ${\\bf A}$ is the element in ${\\bf T}$ for the lub in ${\\bf A}$ .",
"This element must exist since ${\\bf A}$ is a finitary basis and all elements from ${\\bf A}$ have corresponding elements in ${\\bf T}$ .",
"The lub of a set of product elements is the pair of the lub of the left components and the lub of the right components.",
"These exist by the induction hypothesis.",
"Thus, a finitary basis has been created; the domain is formed as always from the basis.",
"The solution to the domain equation has been found since any element in the domain ${\\cal T}$ is either an element representing an element in ${\\cal A}$ or is the product of two other elements in ${\\cal T}$ .",
"Similarly, any element found on the left-hand side must also be in the domain ${\\cal T}$ by the construction.",
"Thus, the domain ${\\cal T}$ is identical to the domain ${\\cal A}+({\\cal T}\\times {\\cal T})$ .",
"To look at the question concerning the existence and uniqueness of the solution to this domain equation, recall the fixed point theorem.",
"This theorem states that a fixed point set exists for any approximable mapping over a domain.",
"Subdomains In Section , the concept of a universal domain is introduced.",
"A universal domain is a domain which contains all other domains as sub-domains.",
"These sub-domains are, roughly speaking, the image of approximable functions over the universal domain.",
"The domain equation for ${\\cal T}$ can be viewed as an approximable mapping over the universal domain.",
"As such, the fixed point theorem states that a least fixed point set for the function does exist and is unique.",
"Sub-domains are defined formally below.",
"Looking again at the informal discussion concerning the elements of the domain ${\\cal T}$ , the infinite element proposed does fit into the formal definition for elements of ${\\cal T}$ .",
"This element is an infinite tree with all left sub-trees containing only the element $a$ .",
"For this infinite element to be computable, it must be the lub of some ascending chain of finite approximations to it.",
"The element $x$ can, in fact, be defined by the following ascending sequence of finite trees: $\\begin{array}{lcl}x_0&=&\\bot \\\\x_{n+1}&=&\\langle \\langle a,x_n\\rangle ,\\pi \\rangle \\\\x&=&\\bigsqcup ^\\infty _{n=0}x_n\\end{array}$ Thus, using domain equations, a domain has been defined recursively.",
"This domain includes infinite as well as finite elements and allows computation on the infinite elements to proceed using the finite approximations, as with the more conventionally defined domains presented earlier.",
"The final topic of this section is the notion of a sub-domain.",
"Informally, a sub-domain is a structured part of a larger domain.",
"Earlier, a domain was described as a sub-domain of the universal domain.",
"Thus, the sub-domain starts with a subset of the elements of the larger domain while retaining the approximation ordering, consistency relation and lub relation, suitably restricted to the subset elements.",
"Definition 6.2: [Sub-Domain] A domain $\\langle {\\cal R},\\sqsubseteq _R\\rangle $ is a sub-domain of a domain $\\langle {\\cal D},\\sqsubseteq _D\\rangle $ , denoted ${\\cal R}\\lhd {\\cal D}$ iff ${\\cal R}\\subseteq {\\cal D}$ - The elements of ${\\cal R}$ are a subset of the elements of ${\\cal D}$ .",
"$\\bot _R=\\bot _D$ - The bottom elements are the same.",
"For $x,y\\in {\\cal R}$ , $x\\sqsubseteq _Ry\\iff x\\sqsubseteq _Dy$ - The approximation ordering for ${\\cal R}$ is the approximation ordering for ${\\cal D}$ restricted to elements in ${\\cal R}$ .",
"For $x,y,z\\in {\\cal R}$ , $x\\sqcup _Ry=z$ iff $x\\sqcup _Dy=z$ - The lub relation for ${\\cal R}$ is the lub relation for ${\\cal D}$ restricted to elements in ${\\cal R}$ .",
"${\\cal R}$ is a domain.",
"Equivalently, a sub-domain can be thought of as the image of an approximable function which approximates the identity function (also termed a projection).",
"The notion of a sub-domain is used in the final section in the discussions about the universal domain.",
"This mapping between the domains can be formalized as follows: Theorem 6.3: If ${\\cal D}\\lhd {\\cal E}$ , then there exists a projection pair of approximable mappings $i:{\\cal D}\\rightarrow {\\cal E}$ and $j:{\\cal E}\\rightarrow {\\cal D}$ where $j\\circ i={{\\sf I}}_{\\cal D}$ and $i\\circ j\\sqsubseteq {{\\sf I}}_{\\cal E}$ where $i$ and $j$ are determined by the following equations: $\\begin{array}{lcl}i(x)&=&\\lbrace y\\in {\\bf E}\\:\\vert \\:\\exists z\\in x.z\\sqsubseteq y\\rbrace \\\\j(y)&=&\\lbrace x\\in {\\bf D}\\:\\vert \\:x\\in y\\rbrace \\end{array}$ for all $x\\in {\\cal D}$ and $y\\in {\\cal E}$ .",
"The proof is left as an exercise.",
"By the definition of a sub-domain, it should be clear that ${\\cal D}_0\\lhd {\\cal E}\\wedge {\\cal D}_1\\lhd {\\cal E}\\:\\Rightarrow \\:({\\cal D}_0\\lhd {\\cal D}_1\\iff {\\cal D}_0\\subseteq {\\cal D}_1)$ Using this observation, the sub-domains of a domain can be ordered.",
"Indeed, the following theorem is a consequence of this ordering.",
"Theorem 6.4: For a given domain ${\\cal D}$ , the set of sub-domains $\\lbrace {\\cal D}_0\\:\\vert \\:{\\cal D}_0\\lhd {\\cal D}\\rbrace $ form a domain.",
"The proof proceeds using the inclusion relation defined as an approximation ordering and is left as an exercise.",
"Finally, a converse of Theorem REF can also be established: Theorem 6.5: For two domains ${\\cal D}$ and ${\\cal E}$ , if there exists a projection pair $i:{\\cal D}\\rightarrow {\\cal E}$ and $j:{\\cal E}\\rightarrow {\\cal D}$ with $j\\circ i={\\sf I}_{\\cal D}$ and $i\\circ j\\sqsubseteq {\\sf I}_{\\cal E}$ , then $\\exists {\\cal D}^{\\prime }\\lhd {\\cal E}$ where ${\\cal D}\\approx {\\cal D}^{\\prime }$ .",
"Proof We show that $i$ maps finite elements to finite elements and that ${\\cal D}^{\\prime }$ is the image of ${\\cal D}$ in ${\\cal E}$ .",
"For some $x\\in {\\bf D}$ with ${\\cal I}_x$ as the principal ideal of $x$ , we can write $i({\\cal I}_x)=\\sqcup \\lbrace {\\cal I}_y\\:\\vert \\:y\\in i({\\cal I}_x)\\rbrace $ Applying $j$ to both sides we get ${\\cal I}_x=j\\circ i({\\cal I}_x)=\\sqcup \\lbrace j({\\cal I}_y)\\:\\vert \\:y\\in i({\\cal I}_x)\\rbrace $ since $j\\circ i={\\sf I}_D$ and $j$ is continuous by assumption.",
"But, since $x\\in {\\cal I}_x$ , $x\\in j({\\cal I}_y)$ for some $y\\in i({\\cal I}_x)$ .",
"This means that ${\\cal I}_x\\subseteq j({\\cal I}_y)$ and thus $i({\\cal I}_x)\\subseteq i\\circ j({\\cal I}_y) \\subseteq {\\cal I}_y$ Since ${\\cal I}_y\\subseteq i({\\cal I}_x)$ must hold by the construction, $i({\\cal I}_x) ={\\cal I}_y$ .",
"This proves that finite elements are mapped to finite elements.",
"Next, consider the value for $i(\\bot _D)$ .",
"Since $\\bot _D\\sqsubseteq _Dj(\\bot _E)$ , $i(\\bot _D)\\sqsubseteq \\bot _E$ .",
"Thus $i(\\bot _D)=\\bot _E$ .",
"Thus, ${\\cal D}$ is isomorphic to the image of $i$ in ${\\cal E}$ .",
"We still must show that ${\\cal D}^{\\prime }$ is a domain.",
"Thus, we need to show that if a lub exists in ${\\cal E}$ for a finite subset in ${\\cal D}^{\\prime }$ , then the lub is also in ${\\cal D}^{\\prime }$ .",
"Let $y^{\\prime },z^{\\prime }\\in {\\bf D}^{\\prime }$ and $y^{\\prime }\\sqcup z^{\\prime }=x^{\\prime }\\in {\\bf E}$ .",
"Then, there exists $y,z\\in {\\bf D}$ such that $i({\\cal I}_y)={\\cal I}_{y^{\\prime }}$ and $i({\\cal I}_z)={\\cal I}_{z^{\\prime }}$ which implies that ${\\cal I}_y=j({\\cal I}_{y^{\\prime }})$ and ${\\cal I}_z=j({\\cal I}_{z^{\\prime }})$ .",
"Since ${\\cal I}_{y^{\\prime }}\\sqsubseteq {\\cal I}_{x^{\\prime }}$ and $j({\\cal I}_{y^{\\prime }})\\sqsubseteq j({\\cal I}_{x^{\\prime }})$ by monotonicity, $y\\in j({\\cal I}_{x^{\\prime }})$ must hold.",
"By the same reasoning, $z\\in j({\\cal I}_{x^{\\prime }})$ .",
"But then $x=y\\sqcup z\\in j({\\cal I}_{x^{\\prime }})$ must also hold and thus $y\\sqcup z \\in {\\cal D}$ since the element $j({\\cal I}_{x^{\\prime }})$ must be an ideal.",
"But, $\\begin{array}{lcl}{\\cal I}_y\\sqsubseteq {\\cal I}_x&\\:\\Rightarrow \\:& {\\cal I}_{y^{\\prime }}\\sqsubseteq i({\\cal I}_{x})\\\\{\\cal I}_z\\sqsubseteq {\\cal I}_x&\\:\\Rightarrow \\:& {\\cal I}_{z^{\\prime }}\\sqsubseteq i({\\cal I}_{x})\\end{array}$ This implies that $y^{\\prime }\\sqcup z^{\\prime }=x^{\\prime }\\in i({\\cal I}_x)$ .",
"We already know that $x\\in j({\\cal I}_{x^{\\prime }})$ so $i({\\cal I}_x)\\sqsubseteq {\\cal I}_{x^{\\prime }}$ .",
"Thus, $i({\\cal I}_x)={\\cal I}_{x^{\\prime }}$ and thus, $x^{\\prime }\\in {\\bf D^{\\prime }}$ .$\\:\\:\\Box $ Exercises Exercise 6.6: Show that there must exist domains satisfying $\\begin{array}{lcll}{\\cal A}&=&{\\cal A}+({\\cal A}\\times {\\cal B})&{\\rm and}\\\\{\\cal B}&=&{\\cal A}+{\\cal B}\\end{array}$ Decide what the elements will look like and define ${\\cal A}$ and ${\\cal B}$ using simultaneous fixed points.",
"Exercise 6.7: Prove Theorem REF Exercise 6.8: Prove Theorem REF Exercise 6.9: Show that if ${\\cal A}$ and ${\\cal B}$ are finite systems, that ${\\cal D}\\unlhd {\\cal E}\\unlhd {\\cal D}\\:\\Rightarrow \\:{\\cal D}\\approx {\\cal E}$ where ${\\cal D}\\approx {\\cal D}^{\\prime }$ and ${\\cal D}^{\\prime }\\lhd {\\cal E}$ is denoted ${\\cal D}^{\\prime }\\unlhd {\\cal E}$ .",
"Computability in Effectively Given Domains In the previous sections, we gave considerable emphasis to the notion of computation using increasingly accurate approximations of the input and output.",
"This section defines this notion of computability more formally.",
"In Section 5, we found that partial functions over the natural numbers were expressible in the $\\lambda $ -notation.",
"This relationship characterizes computation for a particular domain.",
"To describe computation over domains in general, a broader definition is required.",
"The way a domain is presented impacts the way computations are performed over it.",
"Indeed, the theorems of recursive function theory [6] rely in part on the normal presentation of the natural numbers.",
"A presentation for a domain is an enumeration of the elements of the domain.",
"The standard presentation of the natural numbers is simply the numbers in ascending order beginning with 0.",
"There are many permutations of the natural numbers, each of which can be considered a presentation.",
"Computation with these non-standard presentations may be impossible; that is a computable function on the standard presentation may be non-computable over a non-standard presentation.",
"Therefore, an effective presentation for a domain is defined as a presentation which makes the required information computable.",
"Effective Presentations Information about elements in a domain can be characterized completely by looking at the finite elements and their relationships.",
"Thus a presentation must enumerate the finite elements and allow the consistency and lub relationships on these elements to be computed to allow this style of computation.",
"The consistency relation and the lub relation depend on each other.",
"For example, if a set of elements is consistent, a lub must exist for the set.",
"Given that a set is consistent, the lub can be found in finite time by just enumerating the elements and checking to see if this element is the lub.",
"However, if the set is inconsistent, the enumeration will not reveal this fact.",
"Thus, the consistency relation must be assumed to be recursive in an effective presentation.",
"Exercise REF provides a description of presentations that should clarify the assumptions made.",
"Formally, a presentation is defined as follows: Definition 7.1: [Effective Presentation] The presentation of a finitary basis D is a function $\\pi :{{N}}\\rightarrow {\\bf D}$ such that $\\pi (0)=\\Delta _D$ and the range of $\\pi $ is the set of finite elements of D. The definition holds for a domain ${\\cal D}$ as well.",
"A presentation $\\pi $ is effective iff The consistency relation ($\\exists k.\\pi _i\\sqsubseteq \\pi _k\\wedge \\pi _j\\sqsubseteq \\pi _k$ ) for elements $\\pi _i$ and $\\pi _j$ is recursiveRecursive in this context means that the relation is decidable.",
"over $i$ and $j$ .",
"The lub relation ($\\pi _k=\\pi _i\\sqcup \\pi _j$ ) is recursive over $i$ , $j$ , and $k$ .",
"This definition supports our intuition about domains; we have stated that the important information about a domain is the set of finite elements, the ordering and consistency relationships between the elements and the lub relation.",
"Thus, an effective presentation provides, in a suitable (that is computable) form, the basic information about the structure and elements of a domain.",
"A presentation can also be viewed as an enumeration of the elements of the domain with the position of an element in the enumeration given by the index corresponding to the integer input for that element in the presentation function with the 0 element representing $\\bot $ .",
"This perspective is used in the majority of the proofs.",
"Computability Now that the presentation of a domain has been formalized, the notion of computability can be formally defined.",
"Thus, Definition 7.2: [Computable Mappings] Given two domains, ${\\cal D}$ and ${\\cal E}$ with effective presentations $\\pi _1$ and $\\pi _2$ respectively, an approximable mapping $f:{\\bf D}\\rightarrow {\\bf E}$ is computable iff the relation $x_n\\:f\\:y_m$ is recursively enumerable in $n$ and $m$ .",
"By considering the domain ${\\cal D}$ to be a single element domain, the above definition applies not only to computable functions but also to computable elements.",
"For $d\\in {\\cal D}$ where $d$ is the only element in the domain, the element $e=f(d)\\in {\\cal E}$ defines an element in ${\\cal E}$ .",
"The definition states that $e$ is a computable iff the set $\\lbrace m\\in {{N}}\\:\\vert \\:y_m\\sqsubseteq e\\rbrace $ is a recursively enumerable set of integers.",
"Clearly if the set of elements approximating another is finite, the set is recursive.",
"The notion of a recursively enumerable set simply requires that all elements approximating the element in question be listed eventually.",
"The computation then proceeds by accepting an enumeration representing the input element and enumerating the elements that approximate the desired output element.",
"Now that the notions of computability and effective presentations have been formalized, the methods of constructing domains and functions will be addressed.",
"The proof of the next theorem is trivial and is left to the reader.",
"Theorem 7.3: The identity map on an effectively given domain is computable.",
"The composition of computable mappings on effectively given domains are also computable.",
"The following corollary is a consequence of this theorem: Corollary 7.4: For computable function $f:{\\cal D}\\rightarrow {\\cal E}$ and a computable element $x\\in {\\cal D}$ , the element $f(x)\\in {\\cal E}$ is computable.",
"In addition, the standard domain constructors maintain effective presentations.",
"Theorem 7.5: For domains ${\\cal D}_0$ and ${\\cal D}_1$ with effective presentations, the domains ${\\cal D}_0+{\\cal D}_1~{\\rm and}~ {\\cal D}_0\\times {\\cal D}_1$ are also effectively given.",
"In addition, the projection functions are all computable.",
"Finally, if $f$ and $g$ are computable maps, then so are $f+g$ and $f\\times g$ .",
"Proof Let $\\lbrace X_i\\:\\vert \\:i\\in {{N}}\\rbrace $ be the enumeration of ${\\cal D}_0$ and $\\lbrace Y_i\\:\\vert \\:i\\in {{N}}\\rbrace $ be the enumeration of ${\\cal D}_1$ .",
"Another method of sum construction is to use two distinguishing elements in the first position to specify the element type.",
"Thus, a sum domain can be defined as follows: ${\\cal D}_0+{\\cal D}_1=\\lbrace (\\Delta _0,\\Delta _1)\\rbrace \\cup \\lbrace (0,x)\\:\\vert \\:x\\in {\\cal D}_0\\rbrace \\cup \\lbrace (1,y)\\:\\vert \\:y\\in {\\cal D}_1\\rbrace $ The enumeration can then be defined as follows for $n\\in {{N}}$ : $\\begin{array}{lcl}Z_0&=&(\\Delta _0,\\Delta _1)\\\\Z_{2n+1}&=&(0,X_n)\\\\Z_{2n+2}&=&(1,Y_n)\\end{array}$ The proof that $Z_i$ is an effective presentation is left as an exercise.",
"For the product construction, the domain appears as follows: ${\\cal D}_0\\times {\\cal D}_1=\\lbrace (x,y)\\:\\vert \\:x\\in {\\cal D}_0,y\\in {\\cal D}_1\\rbrace $ The enumeration can be defined in terms of the functions $p:{{N}}\\rightarrow {{N}}$ , $q:{{N}}\\rightarrow {{N}}$ , and $r:({{N}}\\times {{N}})\\rightarrow {{N}}$ where for $m$ , $n$ , $k\\in {{N}}$ : $\\begin{array}{lcl}p(r(n,m))&=&n\\\\q(r(n,m))&=&m\\\\r(p(k),q(k))&=&k\\end{array}$ Thus, $r$ is a one-to-one pairing function (see Exercise REF ) of which there are several.",
"The functions $p$ and $q$ extract the indices from the result of the pairing function.",
"The enumeration for the product domain is then defined as follows: $W_i = (X_{p(i)},Y_{q(i)})$ The proof that this is an effective presentation is also left as an exercise.",
"For the combinators, the relations will be defined in terms of the enumeration indices.",
"For example, $\\begin{array}{lcl}X_n\\:in_0\\:Z_m&\\iff & m=0~{\\rm or}\\\\&&\\exists k.m=2k+1\\wedge X_k\\sqsubseteq X_n\\\\W_k\\:proj_1\\:Y_m&\\iff & Y_m\\sqsubseteq Y_{q(k)}\\end{array}$ The reader should verify that these sets are recursively enumerable.",
"For this proof, recall that recursively enumerable sets are closed under conjunction, disjunction, substituting recursive functions, and applying an existential quantifier to the front of a recursive predicate.",
"The proof for the other combinators is left as an exercise.",
"$\\:\\:\\Box $ Product spaces formalize the notion of computable functions of several variables.",
"Note that the proof of Theorem REF shows that substitution of computable functions of severable variables into other computable functions are still computable.",
"The next step is to show that the function space constructor preserves effectiveness.",
"Theorem 7.6: For domains ${\\cal D}_0$ and ${\\cal D}_1$ with effective presentations, the domain ${\\cal D}_0\\rightarrow {\\cal D}_1$ also has an effective presentation.",
"The combinators $apply$ and $curry$ are computable if all input domains are effectively given.",
"The computable elements of the domain ${\\cal D}_0\\rightarrow {\\cal D}_1$ are the computable maps for ${\\bf D_0}\\rightarrow {\\bf D_1}$ .",
"Proof Let ${\\cal D}_0=\\lbrace X_i\\:\\vert \\:i\\in {{N}}\\rbrace $ and ${\\cal D}_1=\\lbrace Y_i\\:\\vert \\:i\\in {{N}}\\rbrace $ be the presentations for the domains.",
"The elements of ${\\bf D_0}\\rightarrow {\\bf D_1}$ are finite step functions which respect the mapping of some subset of ${\\bf D_0}\\times {\\bf D_1}$ .",
"Given the enumeration, each element can be associated with a set $\\lbrace (X_{n_i},Y_{m_i})\\:\\vert \\:\\exists q.",
"1\\le i\\le q\\rbrace $ Thus, there is a finite set of integers pairs that determine the element.",
"Given the definition of consistency from Theorem REF for elements in the function space domain and the decidability of consistency in ${\\cal D}_0$ and ${\\cal D}_1$ , consistency of any finite set of this form is decidable (tedious but decidable since all elements must be checked with all others, etc).",
"Since consistency is decidable, a systematic enumeration of pair sets which are consistent can be made; this enumeration is simply the enumeration of ${\\cal D}_0\\rightarrow {\\cal D}_1$ .",
"Finding the lub consists of making a finite series of tests to find the element that is the lub, which must exist since the set is consistent and we have closure on lubs of finite consistent subsets.",
"Finding the lub requires a finite series of checks in both ${\\cal D}_0$ and ${\\cal D}_1$ but these checks are decidable.",
"Thus, the lub relation is also decidable in ${\\cal D}_0\\rightarrow {\\cal D}_1$ .",
"This shows that ${\\cal D}_0\\rightarrow {\\cal D}_1$ is effectively given.",
"To show that $apply$ and $curry$ are computable, the mappings need to be examined.",
"The mapping defined for apply is $(F,a)\\:apply\\: b\\iff a\\:F\\:b$ The function $F$ is the lub of all the finite step functions that are consistent with it.",
"As such, $F$ can be viewed as the canonical representative of this set.",
"Since $F$ is a finite step function, this relation is decidable.",
"As such, the $apply$ relation is recursive and not just recursively enumerable and $apply$ is a computable function.",
"The reasoning for $curry$ is similar in that the relations are studied.",
"Given the increase in the number of domains, the construction is more tedious and is left for the exercises.",
"To see that the computable elements correspond to the computable maps, recall the relationship shown in Theorem REF between the maps and the elements in the function space.",
"Thus, we have $a\\:f\\:b \\iff b\\in f({\\cal I}_a)~{\\rm or}~{\\cal I}_b\\sqsubseteq f({\\cal I}_a)$ Since $f$ is a computable map, we know that the pairs in the map are recursively enumerable.",
"Using the previous techniques for deciding consistency of finite sets, the set of elements consistent with $f$ can be enumerated.",
"But this set is simply the ideal for $f$ in the function space.",
"The converse direction is trivial.",
"$\\:\\:\\Box $ The final combinator to be discussed, and perhaps the most important, is the fixed point combinator.",
"Theorem 7.7: For any effectively given domain, ${\\cal D}$ , the combinator $fix:({\\cal D}\\rightarrow {\\cal D})\\rightarrow {\\cal D}$ is computable.",
"Proof Let $\\lbrace X_n\\:\\vert \\:n\\in {{N}}\\rbrace $ be the presentation of the domain ${\\cal D}$ .",
"Recall that for $f\\in {\\cal D}\\rightarrow {\\cal D}$ , $f\\:fix\\:X\\iff \\exists k\\in {{N}}.\\Delta \\:f\\:X_1\\:f\\ldots f\\:X_k\\wedge X_k=X$ All of the checks in this finite sequence are decidable since ${\\cal D}$ is effectively given.",
"In addition, existential quantification of a decidable predicate gives a recursively enumerable predicate.",
"Thus, $fix$ is computable.",
"$\\:\\:\\Box $ Recap Now that this has been formalized, what has been accomplished?",
"The major consequence of the theorems to this point is that any expression over effectively given domains (that is effectively given types) combined with computable constants using the $\\lambda $ -notation and the fixed point combinator is a computable function of its free variables.",
"Such functions, applied to computable arguments, yield computable values.",
"These functions also have computable least fixed points.",
"All this gives us a mathematical programming language for defining computable operations.",
"Combining this language with the specification of types with domain equations gives a powerful language.",
"As an example, the effectiveness of the domain ${\\cal T}$ from Example REF is studied.",
"The complete proof is left as an exercise.",
"Example 7.8: Recall the domain ${\\cal T}$ from the previous section.",
"This domain is characterized by the domain equation ${\\cal T}={\\cal A}+({\\cal T}\\times {\\cal T})$ for some domain ${\\cal A}$ .",
"If ${\\cal A}$ is effectively given, we wish to show that ${\\cal T}$ is effectively given as well.",
"The elements are either atomic elements from ${\\cal A}$ or are pairs from ${\\cal T}$ .",
"Let $A=\\lbrace A_i\\:\\vert \\:i\\in {{N}}\\rbrace $ be the enumeration for ${\\cal A}$ .",
"An enumeration for ${\\cal T}$ can be defined as follows: $\\begin{array}{lcl}T_0&=&\\bot _T\\\\T_{2n+1}&=&3*A_n\\\\T_{2n+2}&=&3*T_{p(n)}+1\\cup 3*T_{q(n)}+2\\end{array}$ where for $A$ , a set of indices, $m*A+k=\\lbrace m*n+k\\:\\vert \\:n\\in A\\rbrace $ .",
"The functions $p$ and $q$ here are the inverses of the pairing function $r$ defined in Theorem REF .",
"These functions must be defined such that $p(n)\\le n$ and $q(n)\\le n$ so that the recursion is well defined by taking smaller indices.",
"The rest of the proof is left to the exercises.",
"Specifically, the claim that ${\\cal T}=\\lbrace T_i\\rbrace $ should be verified as well as the effectiveness of the enumeration.",
"These proofs rely either on the effectiveness of ${\\cal A}$ , on the effectiveness of elements in ${\\cal T}$ with smaller indices, or are trivial.",
"The final example uses the powerset construction.",
"We have repeatedly used the fact that a powerset is a domain.",
"Its effectiveness is now verified.",
"Example 7.9: Specifically, the powerset of the natural numbers, ${\\cal {P}({{N}})}$ is considered.",
"In this domain, all elements are consistent, and there is a top element, denoted $\\omega $ , which is the set of all natural numbers.",
"The ordering is the subset relation.",
"The lub of two subsets is the union of the two subsets, which is decidable.",
"To enumerate the finite subsets, the following enumeration is used: $E_n=\\lbrace k\\:\\vert \\:\\exists i,j.",
"i< 2^k\\wedge n=i+2^k+j*2^{k+1}\\rbrace $ This says that $k\\in E_n$ if the $k$ bit in the binary expansion of $n$ is a 1.",
"All finite subsets of ${{N}}$ are of the form $E_n$ for some $n$ .",
"Various combinators for ${\\cal P}({{N}})$ are presented in Exercise REF .",
"Exercises Exercise 7.10: Show that an effectively given domain can always be identified with a relation $INCL(n,m)$ on integers where the derived relations $\\begin{array}{lcl}CONS(n,m)&\\iff &\\exists k.INCL(k,n)\\wedge INCL(k,m)\\\\MEET(n,m,k)&\\iff &\\forall j.",
"[INCL(j,k)\\iff INCL(j,n)\\wedge INCL(j,m)]\\end{array}$ are recursively decidable and where the following axioms hold: $\\forall n.INCL(n,n)$ $\\forall n,m,k.",
"INCL(n,m)\\wedge INCL(m,k)\\:\\Rightarrow \\:INCL(n,k)$ $\\exists m.\\forall n. INCL(n,m)$ $\\forall n,m.",
"CONS(n,m)\\:\\Rightarrow \\:\\exists k.MEET(n,m,k)$ Exercise 7.11: Finish the proof of Theorem REF .",
"Exercise 7.12: Complete the proof of Theorem REF by defining $curry$ as a relation and showing it computable.",
"Is the set recursively enumerable or is it recursive?",
"Exercise 7.13: Two effectively given domains are effectively isomorphic iff $\\ldots $ Complete the statement of the theorem and prove it.",
"Exercise 7.14: Complete the proof about the powerset in Example REF .",
"Show that the combinators $fun$ and $graph$ from Exercise REF are computable.",
"Show the same for $\\lambda x,y.x\\cap y$ $\\lambda x,y.x\\cup y$ $\\lambda x,y.x+ y$ where for $x,y\\in {\\cal P}({{N}})$ , $x+y=\\lbrace n+m\\:\\vert \\:n\\in x, m\\in y\\rbrace $ What are the computable elements of ${\\cal P}({{N}})$ ?",
"Sub-Spaces of the Universal Domain To have a flexible method of solving domain equations and yielding effectively given domains as the solutions, the domains will be embedded in a universal domain which is “big” enough to hold all other domains as sub-domains.",
"This universal domain is shown to be effectively presented, and the mappings which define the sub-spaces are shown to be computable.",
"First, the correspondence between sub-spaces and mappings called retractions is investigated, leading us to the definition of mappings called projections.",
"It is then shown that these definitions can be written out using the $\\lambda $ -calculus notation, demonstrating the power of our mathematical programming language.",
"Retractions and Projections We start with the definition of retractions.",
"Definition 8.1: [Retractions] A retraction of a given domain ${\\cal E}$ is an approximable mapping $a:{\\bf E}\\rightarrow {\\bf E}$ such that $a\\circ a=a$ .",
"Thus, a retraction is the identity function on objects in the range of the retraction and maps other elements into range.",
"The next theorem relates these sets to sub-spaces.",
"Theorem 8.2: If ${\\cal D}\\lhd {\\cal E}$ and if $a:{\\bf E}\\rightarrow {\\bf E}$ is defined such that $X\\:a\\:Z \\iff \\exists Y\\in {\\cal D}.",
"Z\\sqsubseteq Y\\sqsubseteq X$ for all $X,Z\\in {\\bf E}$ , then $a$ is a retraction and ${\\cal D}$ is isomorphic to the fixed point set of $a$ , the set $\\lbrace y\\in {\\cal E}\\:\\vert \\:a(y)=y\\rbrace $ , ordered under inclusion.",
"Proof That $a$ is an approximable map is a direct consequence of the definition of sub-space (Definition REF ).",
"By Theorem REF , a projection pair, $i$ and $j$ , exist for ${\\cal D}$ and this tells us that $a=i\\circ j$ (also showing $a$ approximable since approximable mappings are closed under composition).",
"Theorem REF also tells us that $j\\circ i={\\sf I}_D$ .",
"To show that $a$ is a retraction, $a\\circ a=a$ must be established.",
"Thus, $a\\circ a = i\\circ j\\circ i\\circ j = i\\circ {\\sf I}_D\\circ j = i\\circ j =a$ holds, showing that $a$ is a retraction.",
"We now need to show the isomorphism to ${\\cal D}$ .",
"For $x\\in {\\cal D}$ , $i(x)\\in {\\cal E}$ and we can calculate: $a(i(x))=i\\circ j\\circ i(x) = i\\circ {\\sf I}_D(x) = i(x)$ Thus, $i(x)$ is in the fixed point set of $a$ .",
"For the other direction, let $a(y)=y$ .",
"Then $i(j(y)) = y$ holds.",
"But, $j(y)\\in {\\cal D}$ , so $i$ must map ${\\cal D}$ one-to-one and onto the fixed point set of $a$ .",
"Since $i$ and $j$ are approximable, they are certainly monotonic, and thus the map is an isomorphism with respect to set inclusion.",
"$\\:\\:\\Box $ Not all retractions are associated with a sub-domain relationship.",
"The retractions defined in the above theorem are all subsets as relations of the identity relation.",
"The retractions for sub-domains are characterized by the following definition: Definition 8.3: [Projections] A retraction $a:{\\cal E}\\rightarrow {\\cal E}$ is a projection if $a\\subseteq {\\sf I}_E$ as relations.",
"The retraction is finitary iff its fixed point set is isomorphic to some domain.",
"An example is in order.",
"Example 8.4: Consider a two element system, ${\\bf O}$ with objects $\\Delta $ and 0.",
"For any basis ${\\bf D}$ that is not trivial (has more than one element), ${\\bf O}$ comes from a retraction on ${\\bf D}$ .",
"Define a combinator $check:{\\bf D}\\rightarrow {\\bf O}$ by the relation $x\\:check\\: y \\iff y=\\Delta ~{\\rm or}~x\\ne \\Delta _D$ Thus, $check(x)=\\bot _O\\iff x=\\bot _D$ .",
"Another combinator can be defined, $fade:{\\bf O}\\times {\\bf D}\\rightarrow {\\bf D}$ such that for $t\\in {\\cal O}$ and $x\\in {\\cal D}$ $\\begin{array}{lcll}fade(t,x)&=&\\bot _D&{\\rm if}~t=\\bot _O\\\\&=&x&otherwise\\end{array}$ For $u\\in {\\cal D}$ and $u\\ne \\bot _D$ , the mapping $a$ is defined as $a(x)=fade(check(x),u)$ It can be seen that $a$ is a retraction, but not a projection in general, and the range of $a$ is isomorphic to ${\\bf O}$ .",
"These combinators can also be used to define the subset of functions in ${\\bf D}\\rightarrow {\\bf E}$ that are strict.",
"Define a combinator $strict:({\\bf D}\\rightarrow {\\bf E})\\rightarrow ({\\bf D}\\rightarrow {\\bf E})$ by the equation $strict(f)=\\lambda x.fade(check(x),f(x))$ with $fade$ defined as $fade:{\\bf O}\\times {\\bf E}\\rightarrow {\\bf E}$ .",
"The range of $strict$ is all the strict functions; $strict$ is a projection whose range is a domain.",
"The next theorem characterizes projections.",
"Theorem 8.5: For approximable mapping $a:{\\bf E}\\rightarrow {\\bf E}$ , the following are equivalent: $a$ is a finitary projection $a(x)=\\lbrace y\\in {\\bf E}\\:\\vert \\:\\exists x^{\\prime }\\in I_x.",
"x^{\\prime }\\:a\\:x^{\\prime }\\wedge y\\sqsubseteq x^{\\prime }\\rbrace $ for all $x\\in {\\bf E}$ .",
"Proof Assume that (2) holds.",
"We want to show that $a$ is a finitary projection.",
"By the closure properties on ideals, we know that for all $x\\in {\\cal E}$ , $x^{\\prime }\\in x\\wedge y\\sqsubseteq x^{\\prime }\\:\\Rightarrow \\:y\\in x$ Thus, $a(x)\\subseteq x$ must hold.",
"In addition, the following trivially holds: $x^{\\prime }\\in x\\wedge x^{\\prime }\\:a\\: x^{\\prime }\\:\\Rightarrow \\:x^{\\prime }\\in a(x)$ thus $a(x)\\subseteq a(a(x))$ holds for all $x\\in {\\cal E}$ .",
"This shows that $a$ is indeed a projection.",
"Let $D=\\lbrace x\\in {\\bf E}\\:\\vert \\:x\\:a\\:x\\rbrace $ .",
"It is easy to show that ${\\bf D}\\lhd {\\bf E}$ and that $a$ is determined from ${\\bf D}$ as required in Theorem REF .",
"Thus, the fixed point set of $a$ is isomorphic to a domain from the previous proofs.",
"Thus, (2)$\\:\\Rightarrow \\:$ (1).",
"For the converse, assume that $a$ is a finitary projection.",
"Let ${\\cal D}$ be isomorphic to the fixed point set of $a$ .",
"This means there is a projection pair $i$ and $j$ such that $j\\circ i={\\sf I}_D$ and $i\\circ j = a$ and $a\\subseteq {\\sf I}_E$ .",
"From Theorem REF then we have that ${\\cal D}\\approx {\\cal D}^{\\prime }$ and ${\\cal D}^{\\prime }\\lhd {\\cal E}$ .",
"We want to identify ${\\cal D}^{\\prime }$ as follows: ${\\cal D}^{\\prime }=\\lbrace x\\in {\\cal E}\\:\\vert \\:x\\:a\\:x\\rbrace $ From the proof of Theorem REF , the basis elements of ${\\bf D^{\\prime }}$ are the finite elements of ${\\bf D}$ .",
"Each of these elements is in the fixed point set of $a$ .",
"Thus, $x\\in {\\bf D^{\\prime }}\\:\\Rightarrow \\:a({{\\cal I}}_x) = {{\\cal I}}_x \\:\\Rightarrow \\:x\\:a\\:x$ Since $a$ is a projection, ${{\\cal I}}_x$ must also be a fixed point.",
"Since $i(j({{\\cal I}}_x)) = {{\\cal I}}_x$ implies that $j({{\\cal I}}_x)$ is a finite element of ${\\cal D}$ , $x\\in {\\cal D}^{\\prime }$ must hold.",
"Thus, the identification of ${\\cal D}^{\\prime }$ holds.",
"Finally, using $a=i\\circ j$ in the formula in Theorem REF , the formula in (2) is obtained, proving the converse.",
"$\\:\\:\\Box $ This characterization of projections provides a new and interesting combinator.",
"Theorem 8.6: For any domain ${\\cal E}$ , define $sub:({\\cal E}\\rightarrow {\\cal E})\\rightarrow ({\\cal E}\\rightarrow {\\cal E})$ using the relation $x\\: sub(f)\\: z \\iff \\exists y\\in {\\bf E}.y\\:f\\:y\\wedge y\\sqsubseteq x\\wedge z\\sqsubseteq y$ for all $x,z\\in {\\bf E}$ and all $f:{\\bf E}\\rightarrow {\\bf E}$ .",
"Then the range of $sub$ is exactly the set of finitary projections on ${\\cal E}$ .",
"In addition, $sub$ is a finitary projection on ${\\cal E}\\rightarrow {\\cal E}$ .",
"If ${\\cal E}$ is effectively given, then $sub$ is computable.",
"Proof Clearly, $sub(f)$ is approximable.",
"It is obvious from the definition that $f\\mapsto sub(f)$ preserves lubs and thus is approximable as well.",
"Thus, $y\\:f\\:y\\wedge y\\sqsubseteq x\\wedge z\\sqsubseteq y\\:\\Rightarrow \\:x\\:f\\:z$ obviously holds.",
"Thus, $sub(f)\\subseteq f$ holds.",
"Also $y\\:f\\:y\\:\\Rightarrow \\:y\\:sub(f)\\:y$ thus, $sub(f)\\subseteq sub(sub(f))$ holds as well.",
"Thus, $sub$ is a projection on ${\\cal E}\\rightarrow {\\cal E}$ .",
"The definition of the relation shows that it is computable when ${\\cal E}$ is effectively given.",
"Since $sub$ is a projection, its range is the same as its fixed point set.",
"If $sub(a)=a$ , it is easy to see that clause (2) of Theorem REF holds and conversely.",
"Thus, the range of $sub$ is the finitary projections.",
"To see that $sub$ is a finitary projection, we use Theorem REF and Theorem REF to say that the fixed point set of $sub$ is in a one-to-one inclusion preserving correspondence with the domain $\\lbrace D\\:\\vert \\:D\\lhd {\\cal E}\\rbrace $ .",
"$\\:\\:\\Box $ Universal Domain ${\\cal U}$ With these results and the universal domain to be defined next, the theory of sub-domains is translated into the $\\lambda $ -calculus notation using the $sub$ combinator.",
"The universal domain is defined by first defining a domain which has the desired structure but has a top element.",
"The top element is then removed to give the universal domain.",
"Definition 8.7: [Universal Domain] As in the section on domain equations, an inductive definition for a domain ${\\cal V}$ is given as follows: $\\Delta ,\\top \\in {\\bf V}$ $\\langle u,v\\rangle \\in {\\bf V}$ whenever $u,v\\in {\\bf V}$ Thus, we are starting with two objects, a bottom element and a top element, and making two flavors of copies of these objects.",
"Intuitively, we end up with finite binary trees with either the top or the bottom element as the leaves.",
"To simplify the definitions below, the pairs should be reduced such that: All occurrences of $\\langle \\Delta ,\\Delta \\rangle $ are replaced by $\\Delta $ and All occurrences of $\\langle \\top ,\\top \\rangle $ are replaced by $\\top $ .",
"These rewrite rules are easily shown to be finite Church-Rosser.The finitary basis should be defined as the equivalence classes induced by the reduction.",
"The presentation is simplified by considering only reduced trees.",
"As an example of the reduction the pair $\\langle \\langle \\langle \\top ,\\langle \\top ,\\top \\rangle \\rangle ,\\langle \\top ,\\Delta \\rangle \\rangle ,\\langle \\langle \\Delta ,\\Delta \\rangle ,\\langle \\top ,\\top \\rangle \\rangle \\rangle $ reduces to $\\langle \\langle \\top ,\\langle \\top ,\\Delta \\rangle \\rangle ,\\langle \\Delta ,\\top \\rangle \\rangle $ .",
"The approximation ordering is defined as follows: $\\Delta \\sqsubseteq v$ for all $v\\in {\\bf V}$ $v\\sqsubseteq \\top $ for all $v\\in {\\bf V}$ .",
"$\\langle u,v\\rangle \\sqsubseteq \\langle u^{\\prime },v^{\\prime }\\rangle $ iff $u\\sqsubseteq u^{\\prime }$ and $v\\sqsubseteq v^{\\prime }$ Since the top element is approximated by everything, all finite sets of trees are consistent.",
"The lub for a pair of trees is defined as follows: $u\\sqcup \\top =\\top $ for $u\\in {\\bf V}$ $\\top \\sqcup u=\\top $ for $u\\in {\\bf V}$ $u\\sqcup \\Delta =u$ for $u\\in {\\bf V}$ $\\Delta \\sqcup u=u$ for $u\\in {\\bf V}$ $\\langle u,v\\rangle \\sqcup \\langle u^{\\prime },v^{\\prime }\\rangle =\\langle u\\sqcup u^{\\prime },v\\sqcup v^{\\prime }\\rangle $ for $u,v\\in {\\bf V}$ The proof that this forms a finitary basis follows the same guidelines as the proofs in Section .",
"In addition, it should be clear that the presentation is effective.",
"To form the universal domain, the top element is simply removed.",
"Thus, the system ${\\bf U}={\\bf V}-\\lbrace \\top \\rbrace $ is the basis used to form the universal domain.",
"The proof that this is still a finitary basis with an effective presentation is also straightforward and left to the exercises.",
"Note that inconsistent sets can now exist since there is no top element.",
"A set is inconsistent iff its lub is $\\top $ .",
"We shall now prove the claims made for the universal domain.",
"Theorem 8.8: The domain ${\\cal U}$ is universal, in the sense that for every domain ${\\cal D}$ we have ${\\cal D}\\lhd {\\cal U}$ .",
"If ${\\cal D}$ is effectively given, then the projection pair for the embedding is computable.",
"In fact, there is a correspondence between the effectively presented domains and the computable finitary projections of ${\\cal U}$ .",
"Proof Recall that ${\\bf D}$ must be countable to be a finitary basis.",
"Thus, we can assume that the basis has an enumeration $D=\\lbrace X_n\\:\\vert \\:n\\in {{N}}\\rbrace $ where $X_0=\\Delta $ .",
"The effective and general cases are considered together in the proof; comments about computability are included for the effective case as required.",
"Thus, if ${\\cal D}$ is effectively given, the enumeration above is assumed to be computable.",
"To prove that the domain can be embedded in ${\\cal U}$ , the embedding will be shown.",
"To start, for each finite element $d_i$ in the basis, define two sets, $d_i^+$ and $d_i^-$ as follows: $\\begin{array}{lcl}d_i^+&=&\\lbrace d\\in {\\bf D}\\:\\vert \\:d_i\\sqsubseteq d\\rbrace \\\\d_i^-&=&D-d_i^+\\end{array}$ The $d_i^+$ set contains all the elements that $d_i$ approximates, while the $d_i^-$ set contains all the other elements, partitioning ${\\bf D}$ into two disjoint sets.",
"Sets for different elements can be intersected to form finer partitions of ${\\bf D}$ .",
"For $k>0$ , let $R\\in \\lbrace +,-\\rbrace ^k$ , let $R_i$ be the $ith$ symbol in the string $R$ , and define a region $D_R$ as $D_R=\\bigcap \\limits _{i=1}^k d_i^{R_i}$ where $k$ is the length of $R$ .",
"The set $\\lbrace D_{R}\\:\\vert \\:R\\in \\lbrace +,-\\rbrace ^k\\rbrace $ of regions partitions ${\\bf D}$ into $2^k$ disjoint sets.",
"Thus, for each element $e_i$ in the enumeration there is a corresponding partition of the basis given by the family of sets $\\lbrace D_{R}\\:\\vert \\:R\\in \\lbrace +,-\\rbrace ^i\\rbrace $ .",
"For strings $R,S\\in \\lbrace +,-\\rbrace ^*$ such that $R$ is a prefix of $S$ , denoted $R\\le S$ , $D_S\\subseteq D_R$ .",
"It is important to realize that the composition of these sets is dependent on the order in which the elements are enumerated.",
"Some of these regions are empty, but it is decidable if a given intersection is empty if ${\\cal D}$ is effectively presented.",
"It is also decidable if a given element is in a particular region.",
"Figure: Example Finite DomainTo see the function these regions are serving, consider the finite domain in Figure REF .This example is taken from Cartwright and Demers [2].",
"Consider the enumeration with $d_0=\\bot , d_1=b, d_2=c, d_3=a.$ The $d_i^+$ and $d_i^-$ sets are as follows: $\\begin{array}{lcl}d_1^+&=&\\lbrace a,b\\rbrace \\\\d_1^-&=&\\lbrace c,\\bot \\rbrace \\\\d_2^+&=&\\lbrace c\\rbrace \\\\d_2^-&=&\\lbrace a,b,\\bot \\rbrace \\\\d_3^+&=&\\lbrace a\\rbrace \\\\d_3^-&=&\\lbrace b,c,\\bot \\rbrace \\end{array}$ The regions are as follows: $\\begin{array}{lclclcl}D_+ &=&\\lbrace a,b\\rbrace &\\:\\:\\:\\:&D_{+++} &=&\\lbrace \\rbrace \\\\D_- &=&\\lbrace \\bot ,c\\rbrace &&D_{++-} &=&\\lbrace \\rbrace \\\\D_{++} &=&\\lbrace \\rbrace &&D_{+-+} &=&\\lbrace a\\rbrace \\\\D_{+-} &=&\\lbrace a,b\\rbrace &&D_{+--} &=&\\lbrace b\\rbrace \\\\D_{-+} &=&\\lbrace c\\rbrace &&D_{-++} &=&\\lbrace \\rbrace \\\\D_{--} &=&\\lbrace \\bot \\rbrace &&D_{-+-} &=&\\lbrace c\\rbrace \\\\&&&&D_{--+} &=&\\lbrace \\rbrace \\\\&&&&D_{---} &=&\\lbrace \\bot \\rbrace \\end{array}$ The regions generated by each successive element encode the relationships induced by the approximation ordering between the new element and all elements previously added.",
"The reader is encouraged to try this example with other enumerations of this basis and compare the results.",
"The embedding of the elements proceeds by building a tree based on the regions corresponding to the element.",
"The regions are used to find locations in the tree and to determine whether a $\\top $ or a $\\Delta $ element is placed in the location.",
"These trees preserve the relationships specified by the regions and thus, the tree embedding is isomorphic to the domain in question.",
"Once the tree is built, the reduction rules are applied until a non-reducible tree is reached.",
"This tree is the representative element in the universal domain, and the set of these trees form the sub-space.",
"The function to determine the location in the tree for a given domain, $Loc_D:\\lbrace +,-\\rbrace ^*\\rightarrow \\lbrace l,r\\rbrace ^*$ takes strings used to generate regions and outputs a path in a tree where $l$ stands for left sub-tree and $r$ stands for right sub-tree.",
"This path is computed using the following inductive definition: $\\begin{array}{lcll}Loc_D(\\epsilon )&=&\\epsilon .\\\\Loc_D(R+)&=&Loc_D(R)l&{\\rm if }~D_{R+}\\ne \\emptyset ~ {\\rm and }~D_{R-}\\ne \\emptyset .\\\\&=&Loc_D(R)&{\\rm otherwise}.\\\\Loc_D(R-)&=&Loc_D(R)r&{\\rm if }~D_{R+}\\ne \\emptyset ~{\\rm and }~D_{R-}\\ne \\emptyset .\\\\&=&Loc_D(R)&{\\rm otherwise}.\\end{array}$ The set of locations for each non-empty region is the set of paths to all leaves of some finite binary tree.",
"An induction argument is used to show the following properties of $Loc_D$ that ensure this: If $R\\le S$ for $R,S\\subseteq \\lbrace +,-\\rbrace ^*$ , then $Loc_D(R)\\le Loc_D(S)$ .",
"Let $S=\\lbrace Loc_D(R)\\:\\vert \\:R\\in \\lbrace +,-\\rbrace ^k\\wedge D_R\\ne \\emptyset \\rbrace $ for $k>0$ be a set of location paths for a given $k$ .",
"For any $p\\in \\lbrace l,r\\rbrace ^*$ there exists $q\\in S$ such that either $p\\le q$ or $q\\le p$ .",
"That is, every potential path is represented by some finite path.",
"Finally, for all $p,q\\in S$ if $p\\le q$ then $p=q$ .",
"This means that a unique leaf is associated with each location.",
"To find the tree for a given element $d_k$ in the enumeration, apply the following rules to each $R\\in \\lbrace +,-\\rbrace ^{k-1}$ .",
"If $D_{R-}\\ne \\emptyset $ then the leaf for path $Loc_D(R-)$ is labeled $\\top $ .",
"If $D_{R+}\\ne \\emptyset $ then the leaf for path $Loc_D(R+)$ is labeled $\\Delta $ .",
"These rules are used to assign a tree in ${\\bf U}$ , which is then reduced using the reduction rules, for each element in the enumeration of ${\\bf D}$ .",
"To see that the top element is never assigned by these rules, note that some region of the form $R+$ for every length $k$ must be non-empty since it must contain the element $e_k$ being embedded.",
"Returning to the example, the location function defines paths for these elements as follows: $\\begin{array}{lclclcl}Loc_D(+)&=&l&\\:\\:\\:\\:&Loc_D(+-+)&=&ll\\\\Loc_D(-)&=&r&&Loc_D(+--)&=&lr\\\\Loc_D(+-)&=&l&&Loc_D(-+-)&=&rl\\\\Loc_D(-+)&=&rl&&Loc_D(---)&=&rr\\\\Loc_D(--)&=&rr\\end{array}$ The trees generated for each of the elements are: $\\begin{array}{lcl}d_0&\\mapsto &\\Delta \\\\d_1&\\mapsto &\\langle \\Delta ,\\top \\rangle \\\\d_2&\\mapsto &\\langle \\top ,\\langle \\Delta ,\\top \\rangle \\rangle \\\\d_3&\\mapsto & \\langle \\langle \\Delta ,\\top \\rangle ,\\langle \\top ,\\top \\rangle \\rangle \\\\&\\mapsto & \\langle \\langle \\Delta ,\\top \\rangle ,\\top \\rangle \\end{array}$ To verify that the space generated is a valid sub-space, we must verify that the bottom element is mapped to $\\bot _U$ and that the consistency and lub relations are maintained.",
"The tree $\\Delta $ is clearly assigned to $X_0$ , the bottom element for the basis being embedded, since there are no strings of length $-1$ .",
"The embedding preserves inconsistency of elements by forcing the lub of the embedded elements to be $\\top $ .",
"The $D_{R-}$ regions represent the elements that the element being embedded does not approximate.",
"Note that the $D_{R-}$ sets cause the $\\top $ element to be added as the leaf.",
"Since the $D_R$ sets are built using the approximation ordering, it is straightforward to see that the approximation ordering is preserved by the embedding.",
"Lubs are also maintained by the embedding, although the reduction is required to see that this is the case.",
"It should be clear that, if the domain ${\\cal D}$ is effectively given, the sub-space can be computed since the embedding procedure uses the relationships given in the presentation.",
"Finally, suppose that $a$ is a computable, finitary projection on ${\\cal U}$ .",
"From the proof of Theorem REF , the domain of this projection is characterized by the set $\\lbrace y\\in {\\bf U}\\:\\vert \\:y\\:a\\:y\\rbrace $ If $a$ is computable, the set of pairs for $a$ is recursively enumerable.",
"Thus, the set above is also recursively enumerable since equality among basis elements is decidable.",
"Thus, the domain given by the projection must also be effectively given.",
"$\\:\\:\\Box $ Thus, the domain ${\\cal U}$ is an effectively presented universal domain in which all other domains can be embedded.",
"The sub-domains of ${\\cal U}$ include ${\\cal U}\\rightarrow {\\cal U}$ , ${\\cal U}\\times {\\cal U}$ , etc.",
"These domains must be sub-domains of ${\\cal U}$ since they are effectively presented based on our earlier theorems.",
"Domain Constructors in ${\\cal U}$ The next step is to see how to define the constructors commonly used.",
"Definition 8.9: [Domain Constructors] Let the computable projection pair, $i_+:{\\cal U}+{\\cal U}\\rightarrow {\\cal U}~{\\rm and}~j_+:{\\cal U}\\rightarrow {\\cal U}+{\\cal U}$ be fixed.",
"Fix suitable projection pairs $i_\\times ,j_\\times ,i_\\rightarrow $ , and $j_\\rightarrow $ as well.",
"Define $\\begin{array}{lcl}a+b&=&cond\\circ \\langle which,i_+\\circ in_0\\circ a\\circ out_0, i_+\\circ in_1\\circ b\\circ out_1\\rangle \\circ j_+\\\\a\\times b&=&i_x\\circ \\langle a\\circ proj_0,b\\circ proj_1\\rangle \\circ j_x\\\\a\\rightarrow b&=&i_\\rightarrow \\circ (\\lambda f.b\\circ f\\circ a)\\circ j_\\rightarrow \\end{array}$ for all $a,b:{\\cal U}\\rightarrow {\\cal U}$ .",
"From earlier theorems, we know that these combinators are all computable over an effectively presented domain.",
"The next theorem characterizes the effect these combinators have on projection functions.",
"Theorem 8.10: If $a,b:{\\cal U}\\rightarrow {\\cal U}$ are projections, then so are $a+b$ , $a\\times b$ , and $a\\rightarrow b$ .",
"If $a$ and $b$ are finitary, then so are the compound projections.",
"Proof Since $a$ and $b$ are retractions, $a=a\\circ a$ and $b=b\\circ b$ .",
"Then for $a\\times b$ using the definition of $\\times $ , $\\begin{array}{lcl}(a\\times b)\\circ (a\\times b)&=&i_x\\circ \\langle a\\circ proj_0,b\\circ proj_1\\rangle \\circ \\langle a\\circ proj_0,b\\circ proj_1\\rangle \\circ j_x\\\\&=&i_x\\circ \\langle a\\circ a\\circ proj_0,b\\circ b\\circ proj_1\\rangle \\circ j_x\\\\&=& a\\times b\\end{array}$ Thus, $a\\times b$ is a retraction.",
"The other cases follow similarly.",
"Since $a$ and $b$ are projections, $a,b\\subseteq {\\sf I}_U$ (denoted simply ${\\sf I}$ for the remainder of the proof).",
"Using the definition for $+$ along with the above relation and the definition of projection pairs, we can see that $a+b\\subseteq {\\sf I}+{\\sf I}=i_+\\circ j_+ \\subseteq {\\sf I}$ Thus, $a+b$ is a projection.",
"The other cases follow similarly.",
"To show that the projections are finitary, we must show that the fixed point sets are isomorphic to a domain.",
"Since $a$ and $b$ are assumed finitary, their fixed point sets are isomorphic to $\\begin{array}{lcl}D_a&=&\\lbrace x\\in {\\bf U}\\:\\vert \\:x\\:a\\: x\\rbrace \\\\D_b&=&\\lbrace y\\in {\\bf U}\\:\\vert \\:y\\:b\\: y\\rbrace \\end{array}$ We wish to show that ${\\cal D}_a\\rightarrow {\\cal D}_b\\approx {\\cal D}_{a\\rightarrow b}$ .",
"By the definition of the $\\rightarrow $ constructor, the fixed point set of $a\\rightarrow b$ over ${\\cal U}$ is the same as the fixed point set of $\\lambda f.b\\circ f\\circ a$ on ${\\cal U}\\rightarrow {\\cal U}$ .",
"(Hint: $i_\\rightarrow $ and $j_\\rightarrow $ set up the isomorphism.)",
"So, the fixed points for $f:{\\cal U}\\rightarrow {\\cal U}$ are of the form: $f=b\\circ f\\circ a$ We can think of $a$ as a function in ${\\cal U}\\rightarrow {\\cal D}_a$ and define the other half of the projection pair as $i_a:{\\cal D}_a\\rightarrow {\\cal U}$ where $i_a\\circ a = a$ and $a\\circ i_a=i_a$ .",
"Define a function $i_b$ for the projection pair for $b$ similarly.",
"For some $g:{\\cal D}_a\\rightarrow {\\cal D}_b$ let $f=i_b\\circ g\\circ a$ Substituting this definition for $f$ yields $b\\circ f\\circ a = b\\circ i_b\\circ g\\circ a\\circ a = i_b\\circ g \\circ a = f$ by the definition of $i_b$ and since $a$ is a retraction by assumption.",
"Conversely, for a function $f$ such that $i_b\\circ g\\circ a= f$ , let $g=b\\circ f\\circ i_a$ Substituting again, $i_b\\circ g\\circ a = i_b\\circ g\\circ f\\circ i_a\\circ a = b\\circ f\\circ a = f$ Thus, there is an order preserving isomorphism between $g:{\\cal D}_a\\rightarrow {\\cal D}_b$ and the functions $f=b\\circ f\\circ a$ .",
"The proofs of the isomorphisms for the other constructs are similar.",
"$\\:\\:\\Box $ Thus, the sub-domain relationship with the universal domain has been stated in terms of finitary projections over the universal domain.",
"In addition, all the domain constructors have been shown to be computable combinators on the domain of these finitary projections.",
"Recalling that all computable maps have computable fixed points, the standard fixed point method can be used to solve domain equations of all kinds if they can be defined on projections.",
"Returning to the $\\lambda $ -calculus for a moment, all objects in the $\\lambda $ -calculus are considered functions.",
"Since ${\\cal U}\\rightarrow {\\cal U}$ is a part of ${\\cal U}$ , every object in the $\\lambda $ -calculus is also an object of ${\\cal U}$ .",
"Transposing some of the familiar notation, where the old notation appears on the left, the new combinators are defined as follows: $\\begin{array}{lcl}which(z)=which(j_+(z))\\\\in_i(x)=i_+(in_i(x))~{\\rm where}~i=0,1\\\\out_i(x)=out_i(j_+(x))~{\\rm where}~i=0,1\\\\\\langle x,y\\rangle =i_x(\\langle x,y\\rangle )\\\\proj_i=proj_i(j_x(z))~{\\rm where}~i=0,1\\\\u(x) = j_\\rightarrow (u)(x)\\\\\\lambda x.\\tau =i_\\rightarrow (\\lambda x.\\tau )\\end{array}$ Thus, all functions, all constants, all combinators, and all constructs are elements of ${\\cal U}$ .",
"Indeed, everything computable is an element of ${\\cal U}$ .",
"Elements in ${\\cal U}$ play multiple roles by representing different objects under different projections.",
"While this notion may be difficult to get used to, there are many advantages, both notational and conceptual.",
"Exercises Exercise 8.11: A retraction $a:{\\cal D}\\rightarrow {\\cal D}$ is a closure operator iff ${\\sf I}_D\\subseteq a$ as relations.",
"On a domain like ${\\cal P}({{N}})$ , give some examples of closure operators.",
"(Hint: Close up the integers under addition.",
"Is this continuous on ${\\cal P}({{N}})$ ?)",
"Prove in general that for any closure $a:{\\cal D}\\rightarrow {\\cal D}$ , the fixed point set of $a$ is always a finitary domain.",
"(Hint: Show that the fixed point set is closed as required for a domain.)",
"What are the finite elements of the fixed point set?",
"Exercise 8.12: Give a direct proof that the domain $\\lbrace X\\:\\vert \\:X\\lhd {\\cal D}\\rbrace $ is effectively presented if ${\\cal D}$ is.",
"(Hint: The finite elements of the domain correspond exactly to the finite domains $X\\lhd {\\cal D}$ .)",
"In the case of ${\\cal D}={\\cal U}$ , show that the computable elements of the domain correspond exactly to the effectively presented domains (up to effective isomorphism).",
"Exercise 8.13: For finitary projections $a:{\\cal E}\\rightarrow {\\cal E}$ , write ${\\cal D}_a=\\lbrace x\\in {\\cal E}\\:\\vert \\:x\\:a\\:x\\rbrace $ Show that for any two such projections $a$ and $b$ , that $a\\subseteq b \\iff {\\cal D}_a\\lhd {\\cal D}_b$ Exercise 8.14: Find another universal domain that is not isomorphic to ${\\cal U}$ .",
"Exercise 8.15: Prove the remaining cases in Theorem REF .",
"Exercise 8.16: Suppose $S$ and $T$ are two binary constructors on domains that can be made into computable operators on projections over the universal domain.",
"Show that we can find a pair of effectively presented domains such that $D\\approx S(D,E)~{\\rm and}~E\\approx T(D,E).$ Exercise 8.17: Using the translations shown after the proof of Theorem REF , show how the whole typed-$\\lambda $ -calculus can be translated into ${\\cal U}$ .",
"(Hint: for $f:{\\cal D}_a\\rightarrow {\\cal B}$ , write $f=b\\circ f\\circ a$ for finitary projections $a$ and $b$ .",
"For $\\lambda x^{{\\cal D}_a}.\\sigma $ , write $\\lambda x.b(\\sigma ^{\\prime }[a(x)/x])$ where $\\sigma ^{\\prime }$ is the translation of $\\sigma $ into the untyped $\\lambda $ -calculus.",
"Be sure that the resulting term has the right type.)",
"Exercise 8.18: Show that the basis presented for the universal domain ${\\bf U}$ is indeed a finitary basis and that it has an effective presentation.",
"Exercise 8.19: Work out the embedding for the other enumerations for the example given in the proof of Theorem REF ."
],
[
"Introduction to Domain Equations",
"As stressed in the introduction, the notion of computation with potentially infinite elements is an integral part of domain theory.",
"The previous sections have defined the notion of functions over domains, as well as a notation for expressing these functions.",
"In addition, the notion of computation through series of approximations has been addressed.",
"This computation is possible since the functions defined have been approximable and thus continuous.",
"This section addresses the construction of more complex domains with infinite elements.",
"The next section looks specifically at the notion of computability with respect to these infinite elements.",
"The last section looks at another approach to domain construction.",
"New domains have been constructed from existing ones using domain constructors such as the product construction ($\\times $ ), the function space construction ($\\rightarrow $ ) and the sum construction ($+$ ) of Exercise REF .",
"These constructors can be iterated similar to the way that function application was iterated to form recursive function definitions.",
"In this way, domains can be characterized using recursion equations, called domain equations."
],
[
"Domain Equations",
"A domain equation represents an isomorphism between the domain as a whole and the combination of domains that comprise it.",
"These recursive domains are frequently termed reflexive domains since, as in the following example, the domain contains a copy of itself in its structure.",
"Example 6.1: Consider the following domain equation: ${\\cal T}={\\cal A}+({\\cal T}\\times {\\cal T})$ where ${\\cal A}$ is a previously defined domain.",
"This domain can be thought of as containing atomic elements from ${\\cal A}$ or pairs of elements of ${\\cal T}$ .",
"What do the elements of this domain look like?",
"In particular, what are the finite elements of this domain?",
"How is the domain constructed?",
"What is an appropriate approximation ordering for the domain?",
"What do lubs in this domain look like?",
"What is the appropriate notion of consistency?",
"Does this domain even exist?",
"In other words, are we certain a solution to this domain equation exists?",
"And if a solution to the equation exists, is it a unique solution?",
"Each of these questions is examined below.",
"The domain equation tells us that an element of the domain is either an element from ${\\cal A}$ or is a pair of “smaller” elements from ${\\cal T}$ .",
"One method of constructing a sum domain is using pairs where some distinguished element denotes what type an element is.",
"Thus, for some $a\\in {\\cal A}$ , the pair $\\langle \\pi ,a\\rangle $ might represent the element in ${\\cal T}$ for the given element $a$ .",
"For some $s,t\\in {\\cal T}$ , the pair $\\langle \\langle s,t\\rangle ,\\pi \\rangle $ might then represent the element in ${\\cal T}$ for the pair $s,t$ .",
"Thus, $\\pi $ is the distinguished element, and the location of $\\pi $ in the pair specifies the type of the element.",
"The finite elements are either elements in ${\\cal T}$ representing the (finite) elements of ${\\cal A}$ or the pair elements from ${\\cal T}$ whose components are also finite elements in ${\\cal T}$ .",
"The question then arises about infinite elements.",
"Are there infinite elements in this domain?",
"Consider the following fixed point equation for some element for $a\\in {\\cal A}$ : $x=\\langle \\langle a,x\\rangle ,\\pi \\rangle .$ The fixed point of this equation is the infinite product of the element $a$ .",
"Does this element fit the definition for ${\\cal T}$ ?",
"From the informal description of the elements of ${\\cal T}$ given so far, $x$ does qualify as a member of ${\\cal T}$ .",
"Now that some intuition has been developed about this domain, a formal construction is required.",
"Let $\\langle {\\bf A},\\sqsubseteq _A\\rangle $ be the finitary basis used to generate the domain ${\\cal A}$ .",
"Let $\\pi $ be an object such that $\\pi \\notin {\\bf A}$ .",
"Define the bottom element of the finitary basis T as $\\Delta _T=\\langle \\pi ,\\pi \\rangle $ .",
"Next, all the elements of ${\\cal A}$ must be included so define an element in ${\\bf T}$ for each $a\\in {\\bf A}$ as $\\langle \\pi ,a\\rangle $ .",
"Finally, pair elements for all elements in ${\\bf T}$ must exist in ${\\bf T}$ to complete the construction.",
"The set ${\\bf T}$ can be defined inductively as the least set such that: $\\Delta _T\\in {\\bf T}$ , $\\langle \\pi ,a\\rangle \\in {\\bf T}$ whenever $a\\in {\\bf A}$ , $\\langle \\langle \\Delta _T,s\\rangle ,\\pi \\rangle \\in {\\bf T}$ whenever $s\\in {\\bf T}$ (necessary??",
"), $\\langle \\langle t,\\Delta _T\\rangle ,\\pi \\rangle \\in {\\bf T}$ whenever $t\\in {\\bf T}$ (necessary??",
"), and $\\langle \\langle t,s\\rangle ,\\pi \\rangle \\in {\\bf T}$ whenever $s, t\\in {\\bf T}$ .",
"The set can also be characterized by the following fixed point equation: ${\\bf T}=\\lbrace \\Delta _T\\rbrace \\cup \\lbrace \\langle \\pi ,a\\rangle \\:\\vert \\:a\\in {\\bf A}\\rbrace \\cup \\lbrace \\langle \\langle \\Delta _T,s\\rangle ,\\pi \\rangle \\:\\vert \\:s\\in {\\bf T}\\rbrace \\cup \\lbrace \\langle \\langle t,\\Delta _T\\rangle ,\\pi \\rangle \\:\\vert \\:t\\in {\\bf T}\\rbrace \\cup \\lbrace \\langle \\langle t,s\\rangle ,\\pi \\rangle \\:\\vert \\:s,t\\in {\\bf T}\\rbrace .$ A solution must exist for this equation by the fixed point theorem.",
"Now that the basis elements have been defined, we must show how to find lubs.",
"We will again use an inductive definition.",
"$\\langle \\pi ,\\pi \\rangle \\sqcup t=t$ for all $t\\in {\\bf T}$ For $a,b\\in {\\bf A}$ , $\\langle \\pi ,a\\rangle \\sqcup \\langle \\pi ,b\\rangle =\\langle \\pi ,a\\sqcup b\\rangle $ if $a\\sqcup b$ exists in ${\\bf A}$ $\\langle \\langle s,t\\rangle ,\\pi \\rangle \\sqcup \\langle \\langle s^{\\prime },t^{\\prime }\\rangle ,\\pi \\rangle =\\langle \\langle s\\sqcup s^{\\prime },t\\sqcup t^{\\prime }\\rangle ,\\pi \\rangle $ if $s\\sqcup s^{\\prime }$ and $t\\sqcup t^{\\prime }$ exist in ${\\bf T}$ .",
"The lub $\\langle \\pi ,a\\rangle \\sqcup \\langle \\langle s,t\\rangle ,\\pi \\rangle $ does not exist.",
"Next, the notion of consistency needs to be explored.",
"From the definition of lubs given above, the following sets are consistent: The empty set is consistent.",
"Everything is consistent with the bottom element.",
"A set of elements all from the basis A is consistent in T if the set of elements is consistent in A.",
"A set of product elements in T is consistent if the left component elements are consistent and the right component elements are consistent.",
"These conditions derive from the sum and product nature of the domain.",
"The approximation ordering in the basis has the following inductive definition: $\\Delta _T\\sqsubseteq _T s$ for all $s\\in {\\bf T}$ $y\\sqsubseteq _Tu\\sqcup \\Delta _T$ whenever $y\\sqsubseteq _Tu$ $\\langle \\pi ,a\\rangle \\sqsubseteq _T\\langle \\pi ,b\\rangle $ whenever $a\\sqsubseteq _Ab$ $\\langle \\langle s,t\\rangle ,\\pi \\rangle \\sqsubseteq _T\\langle \\langle u,v\\rangle ,\\pi \\rangle $ whenever $s\\sqsubseteq _Tu$ and $t\\sqsubseteq _Tv$ The next step is to verify that ${\\bf T}$ is indeed a finitary basis.",
"The basis is still countable.",
"The approximation is clearly a partial order.",
"The existence of lubs of finite bounded (i.e., consistent) subsets must be verified.",
"The definition of consistency gives us the requirements for a bounded subset.",
"Each of the conditions for consistency are examined inductively since the definitions are all inductive: The lub of the empty set is the bottom element $\\Delta _T$ .",
"The lub of a set containing the bottom element is the lub of the set without the bottom element which must exist by the induction hypothesis.",
"The lub of a set of elements all from the ${\\bf A}$ is the element in ${\\bf T}$ for the lub in ${\\bf A}$ .",
"This element must exist since ${\\bf A}$ is a finitary basis and all elements from ${\\bf A}$ have corresponding elements in ${\\bf T}$ .",
"The lub of a set of product elements is the pair of the lub of the left components and the lub of the right components.",
"These exist by the induction hypothesis.",
"Thus, a finitary basis has been created; the domain is formed as always from the basis.",
"The solution to the domain equation has been found since any element in the domain ${\\cal T}$ is either an element representing an element in ${\\cal A}$ or is the product of two other elements in ${\\cal T}$ .",
"Similarly, any element found on the left-hand side must also be in the domain ${\\cal T}$ by the construction.",
"Thus, the domain ${\\cal T}$ is identical to the domain ${\\cal A}+({\\cal T}\\times {\\cal T})$ .",
"To look at the question concerning the existence and uniqueness of the solution to this domain equation, recall the fixed point theorem.",
"This theorem states that a fixed point set exists for any approximable mapping over a domain.",
"Subdomains In Section , the concept of a universal domain is introduced.",
"A universal domain is a domain which contains all other domains as sub-domains.",
"These sub-domains are, roughly speaking, the image of approximable functions over the universal domain.",
"The domain equation for ${\\cal T}$ can be viewed as an approximable mapping over the universal domain.",
"As such, the fixed point theorem states that a least fixed point set for the function does exist and is unique.",
"Sub-domains are defined formally below.",
"Looking again at the informal discussion concerning the elements of the domain ${\\cal T}$ , the infinite element proposed does fit into the formal definition for elements of ${\\cal T}$ .",
"This element is an infinite tree with all left sub-trees containing only the element $a$ .",
"For this infinite element to be computable, it must be the lub of some ascending chain of finite approximations to it.",
"The element $x$ can, in fact, be defined by the following ascending sequence of finite trees: $\\begin{array}{lcl}x_0&=&\\bot \\\\x_{n+1}&=&\\langle \\langle a,x_n\\rangle ,\\pi \\rangle \\\\x&=&\\bigsqcup ^\\infty _{n=0}x_n\\end{array}$ Thus, using domain equations, a domain has been defined recursively.",
"This domain includes infinite as well as finite elements and allows computation on the infinite elements to proceed using the finite approximations, as with the more conventionally defined domains presented earlier.",
"The final topic of this section is the notion of a sub-domain.",
"Informally, a sub-domain is a structured part of a larger domain.",
"Earlier, a domain was described as a sub-domain of the universal domain.",
"Thus, the sub-domain starts with a subset of the elements of the larger domain while retaining the approximation ordering, consistency relation and lub relation, suitably restricted to the subset elements.",
"Definition 6.2: [Sub-Domain] A domain $\\langle {\\cal R},\\sqsubseteq _R\\rangle $ is a sub-domain of a domain $\\langle {\\cal D},\\sqsubseteq _D\\rangle $ , denoted ${\\cal R}\\lhd {\\cal D}$ iff ${\\cal R}\\subseteq {\\cal D}$ - The elements of ${\\cal R}$ are a subset of the elements of ${\\cal D}$ .",
"$\\bot _R=\\bot _D$ - The bottom elements are the same.",
"For $x,y\\in {\\cal R}$ , $x\\sqsubseteq _Ry\\iff x\\sqsubseteq _Dy$ - The approximation ordering for ${\\cal R}$ is the approximation ordering for ${\\cal D}$ restricted to elements in ${\\cal R}$ .",
"For $x,y,z\\in {\\cal R}$ , $x\\sqcup _Ry=z$ iff $x\\sqcup _Dy=z$ - The lub relation for ${\\cal R}$ is the lub relation for ${\\cal D}$ restricted to elements in ${\\cal R}$ .",
"${\\cal R}$ is a domain.",
"Equivalently, a sub-domain can be thought of as the image of an approximable function which approximates the identity function (also termed a projection).",
"The notion of a sub-domain is used in the final section in the discussions about the universal domain.",
"This mapping between the domains can be formalized as follows: Theorem 6.3: If ${\\cal D}\\lhd {\\cal E}$ , then there exists a projection pair of approximable mappings $i:{\\cal D}\\rightarrow {\\cal E}$ and $j:{\\cal E}\\rightarrow {\\cal D}$ where $j\\circ i={{\\sf I}}_{\\cal D}$ and $i\\circ j\\sqsubseteq {{\\sf I}}_{\\cal E}$ where $i$ and $j$ are determined by the following equations: $\\begin{array}{lcl}i(x)&=&\\lbrace y\\in {\\bf E}\\:\\vert \\:\\exists z\\in x.z\\sqsubseteq y\\rbrace \\\\j(y)&=&\\lbrace x\\in {\\bf D}\\:\\vert \\:x\\in y\\rbrace \\end{array}$ for all $x\\in {\\cal D}$ and $y\\in {\\cal E}$ .",
"The proof is left as an exercise.",
"By the definition of a sub-domain, it should be clear that ${\\cal D}_0\\lhd {\\cal E}\\wedge {\\cal D}_1\\lhd {\\cal E}\\:\\Rightarrow \\:({\\cal D}_0\\lhd {\\cal D}_1\\iff {\\cal D}_0\\subseteq {\\cal D}_1)$ Using this observation, the sub-domains of a domain can be ordered.",
"Indeed, the following theorem is a consequence of this ordering.",
"Theorem 6.4: For a given domain ${\\cal D}$ , the set of sub-domains $\\lbrace {\\cal D}_0\\:\\vert \\:{\\cal D}_0\\lhd {\\cal D}\\rbrace $ form a domain.",
"The proof proceeds using the inclusion relation defined as an approximation ordering and is left as an exercise.",
"Finally, a converse of Theorem REF can also be established: Theorem 6.5: For two domains ${\\cal D}$ and ${\\cal E}$ , if there exists a projection pair $i:{\\cal D}\\rightarrow {\\cal E}$ and $j:{\\cal E}\\rightarrow {\\cal D}$ with $j\\circ i={\\sf I}_{\\cal D}$ and $i\\circ j\\sqsubseteq {\\sf I}_{\\cal E}$ , then $\\exists {\\cal D}^{\\prime }\\lhd {\\cal E}$ where ${\\cal D}\\approx {\\cal D}^{\\prime }$ .",
"Proof We show that $i$ maps finite elements to finite elements and that ${\\cal D}^{\\prime }$ is the image of ${\\cal D}$ in ${\\cal E}$ .",
"For some $x\\in {\\bf D}$ with ${\\cal I}_x$ as the principal ideal of $x$ , we can write $i({\\cal I}_x)=\\sqcup \\lbrace {\\cal I}_y\\:\\vert \\:y\\in i({\\cal I}_x)\\rbrace $ Applying $j$ to both sides we get ${\\cal I}_x=j\\circ i({\\cal I}_x)=\\sqcup \\lbrace j({\\cal I}_y)\\:\\vert \\:y\\in i({\\cal I}_x)\\rbrace $ since $j\\circ i={\\sf I}_D$ and $j$ is continuous by assumption.",
"But, since $x\\in {\\cal I}_x$ , $x\\in j({\\cal I}_y)$ for some $y\\in i({\\cal I}_x)$ .",
"This means that ${\\cal I}_x\\subseteq j({\\cal I}_y)$ and thus $i({\\cal I}_x)\\subseteq i\\circ j({\\cal I}_y) \\subseteq {\\cal I}_y$ Since ${\\cal I}_y\\subseteq i({\\cal I}_x)$ must hold by the construction, $i({\\cal I}_x) ={\\cal I}_y$ .",
"This proves that finite elements are mapped to finite elements.",
"Next, consider the value for $i(\\bot _D)$ .",
"Since $\\bot _D\\sqsubseteq _Dj(\\bot _E)$ , $i(\\bot _D)\\sqsubseteq \\bot _E$ .",
"Thus $i(\\bot _D)=\\bot _E$ .",
"Thus, ${\\cal D}$ is isomorphic to the image of $i$ in ${\\cal E}$ .",
"We still must show that ${\\cal D}^{\\prime }$ is a domain.",
"Thus, we need to show that if a lub exists in ${\\cal E}$ for a finite subset in ${\\cal D}^{\\prime }$ , then the lub is also in ${\\cal D}^{\\prime }$ .",
"Let $y^{\\prime },z^{\\prime }\\in {\\bf D}^{\\prime }$ and $y^{\\prime }\\sqcup z^{\\prime }=x^{\\prime }\\in {\\bf E}$ .",
"Then, there exists $y,z\\in {\\bf D}$ such that $i({\\cal I}_y)={\\cal I}_{y^{\\prime }}$ and $i({\\cal I}_z)={\\cal I}_{z^{\\prime }}$ which implies that ${\\cal I}_y=j({\\cal I}_{y^{\\prime }})$ and ${\\cal I}_z=j({\\cal I}_{z^{\\prime }})$ .",
"Since ${\\cal I}_{y^{\\prime }}\\sqsubseteq {\\cal I}_{x^{\\prime }}$ and $j({\\cal I}_{y^{\\prime }})\\sqsubseteq j({\\cal I}_{x^{\\prime }})$ by monotonicity, $y\\in j({\\cal I}_{x^{\\prime }})$ must hold.",
"By the same reasoning, $z\\in j({\\cal I}_{x^{\\prime }})$ .",
"But then $x=y\\sqcup z\\in j({\\cal I}_{x^{\\prime }})$ must also hold and thus $y\\sqcup z \\in {\\cal D}$ since the element $j({\\cal I}_{x^{\\prime }})$ must be an ideal.",
"But, $\\begin{array}{lcl}{\\cal I}_y\\sqsubseteq {\\cal I}_x&\\:\\Rightarrow \\:& {\\cal I}_{y^{\\prime }}\\sqsubseteq i({\\cal I}_{x})\\\\{\\cal I}_z\\sqsubseteq {\\cal I}_x&\\:\\Rightarrow \\:& {\\cal I}_{z^{\\prime }}\\sqsubseteq i({\\cal I}_{x})\\end{array}$ This implies that $y^{\\prime }\\sqcup z^{\\prime }=x^{\\prime }\\in i({\\cal I}_x)$ .",
"We already know that $x\\in j({\\cal I}_{x^{\\prime }})$ so $i({\\cal I}_x)\\sqsubseteq {\\cal I}_{x^{\\prime }}$ .",
"Thus, $i({\\cal I}_x)={\\cal I}_{x^{\\prime }}$ and thus, $x^{\\prime }\\in {\\bf D^{\\prime }}$ .$\\:\\:\\Box $ Exercises Exercise 6.6: Show that there must exist domains satisfying $\\begin{array}{lcll}{\\cal A}&=&{\\cal A}+({\\cal A}\\times {\\cal B})&{\\rm and}\\\\{\\cal B}&=&{\\cal A}+{\\cal B}\\end{array}$ Decide what the elements will look like and define ${\\cal A}$ and ${\\cal B}$ using simultaneous fixed points.",
"Exercise 6.7: Prove Theorem REF Exercise 6.8: Prove Theorem REF Exercise 6.9: Show that if ${\\cal A}$ and ${\\cal B}$ are finite systems, that ${\\cal D}\\unlhd {\\cal E}\\unlhd {\\cal D}\\:\\Rightarrow \\:{\\cal D}\\approx {\\cal E}$ where ${\\cal D}\\approx {\\cal D}^{\\prime }$ and ${\\cal D}^{\\prime }\\lhd {\\cal E}$ is denoted ${\\cal D}^{\\prime }\\unlhd {\\cal E}$ .",
"Computability in Effectively Given Domains In the previous sections, we gave considerable emphasis to the notion of computation using increasingly accurate approximations of the input and output.",
"This section defines this notion of computability more formally.",
"In Section 5, we found that partial functions over the natural numbers were expressible in the $\\lambda $ -notation.",
"This relationship characterizes computation for a particular domain.",
"To describe computation over domains in general, a broader definition is required.",
"The way a domain is presented impacts the way computations are performed over it.",
"Indeed, the theorems of recursive function theory [6] rely in part on the normal presentation of the natural numbers.",
"A presentation for a domain is an enumeration of the elements of the domain.",
"The standard presentation of the natural numbers is simply the numbers in ascending order beginning with 0.",
"There are many permutations of the natural numbers, each of which can be considered a presentation.",
"Computation with these non-standard presentations may be impossible; that is a computable function on the standard presentation may be non-computable over a non-standard presentation.",
"Therefore, an effective presentation for a domain is defined as a presentation which makes the required information computable.",
"Effective Presentations Information about elements in a domain can be characterized completely by looking at the finite elements and their relationships.",
"Thus a presentation must enumerate the finite elements and allow the consistency and lub relationships on these elements to be computed to allow this style of computation.",
"The consistency relation and the lub relation depend on each other.",
"For example, if a set of elements is consistent, a lub must exist for the set.",
"Given that a set is consistent, the lub can be found in finite time by just enumerating the elements and checking to see if this element is the lub.",
"However, if the set is inconsistent, the enumeration will not reveal this fact.",
"Thus, the consistency relation must be assumed to be recursive in an effective presentation.",
"Exercise REF provides a description of presentations that should clarify the assumptions made.",
"Formally, a presentation is defined as follows: Definition 7.1: [Effective Presentation] The presentation of a finitary basis D is a function $\\pi :{{N}}\\rightarrow {\\bf D}$ such that $\\pi (0)=\\Delta _D$ and the range of $\\pi $ is the set of finite elements of D. The definition holds for a domain ${\\cal D}$ as well.",
"A presentation $\\pi $ is effective iff The consistency relation ($\\exists k.\\pi _i\\sqsubseteq \\pi _k\\wedge \\pi _j\\sqsubseteq \\pi _k$ ) for elements $\\pi _i$ and $\\pi _j$ is recursiveRecursive in this context means that the relation is decidable.",
"over $i$ and $j$ .",
"The lub relation ($\\pi _k=\\pi _i\\sqcup \\pi _j$ ) is recursive over $i$ , $j$ , and $k$ .",
"This definition supports our intuition about domains; we have stated that the important information about a domain is the set of finite elements, the ordering and consistency relationships between the elements and the lub relation.",
"Thus, an effective presentation provides, in a suitable (that is computable) form, the basic information about the structure and elements of a domain.",
"A presentation can also be viewed as an enumeration of the elements of the domain with the position of an element in the enumeration given by the index corresponding to the integer input for that element in the presentation function with the 0 element representing $\\bot $ .",
"This perspective is used in the majority of the proofs.",
"Computability Now that the presentation of a domain has been formalized, the notion of computability can be formally defined.",
"Thus, Definition 7.2: [Computable Mappings] Given two domains, ${\\cal D}$ and ${\\cal E}$ with effective presentations $\\pi _1$ and $\\pi _2$ respectively, an approximable mapping $f:{\\bf D}\\rightarrow {\\bf E}$ is computable iff the relation $x_n\\:f\\:y_m$ is recursively enumerable in $n$ and $m$ .",
"By considering the domain ${\\cal D}$ to be a single element domain, the above definition applies not only to computable functions but also to computable elements.",
"For $d\\in {\\cal D}$ where $d$ is the only element in the domain, the element $e=f(d)\\in {\\cal E}$ defines an element in ${\\cal E}$ .",
"The definition states that $e$ is a computable iff the set $\\lbrace m\\in {{N}}\\:\\vert \\:y_m\\sqsubseteq e\\rbrace $ is a recursively enumerable set of integers.",
"Clearly if the set of elements approximating another is finite, the set is recursive.",
"The notion of a recursively enumerable set simply requires that all elements approximating the element in question be listed eventually.",
"The computation then proceeds by accepting an enumeration representing the input element and enumerating the elements that approximate the desired output element.",
"Now that the notions of computability and effective presentations have been formalized, the methods of constructing domains and functions will be addressed.",
"The proof of the next theorem is trivial and is left to the reader.",
"Theorem 7.3: The identity map on an effectively given domain is computable.",
"The composition of computable mappings on effectively given domains are also computable.",
"The following corollary is a consequence of this theorem: Corollary 7.4: For computable function $f:{\\cal D}\\rightarrow {\\cal E}$ and a computable element $x\\in {\\cal D}$ , the element $f(x)\\in {\\cal E}$ is computable.",
"In addition, the standard domain constructors maintain effective presentations.",
"Theorem 7.5: For domains ${\\cal D}_0$ and ${\\cal D}_1$ with effective presentations, the domains ${\\cal D}_0+{\\cal D}_1~{\\rm and}~ {\\cal D}_0\\times {\\cal D}_1$ are also effectively given.",
"In addition, the projection functions are all computable.",
"Finally, if $f$ and $g$ are computable maps, then so are $f+g$ and $f\\times g$ .",
"Proof Let $\\lbrace X_i\\:\\vert \\:i\\in {{N}}\\rbrace $ be the enumeration of ${\\cal D}_0$ and $\\lbrace Y_i\\:\\vert \\:i\\in {{N}}\\rbrace $ be the enumeration of ${\\cal D}_1$ .",
"Another method of sum construction is to use two distinguishing elements in the first position to specify the element type.",
"Thus, a sum domain can be defined as follows: ${\\cal D}_0+{\\cal D}_1=\\lbrace (\\Delta _0,\\Delta _1)\\rbrace \\cup \\lbrace (0,x)\\:\\vert \\:x\\in {\\cal D}_0\\rbrace \\cup \\lbrace (1,y)\\:\\vert \\:y\\in {\\cal D}_1\\rbrace $ The enumeration can then be defined as follows for $n\\in {{N}}$ : $\\begin{array}{lcl}Z_0&=&(\\Delta _0,\\Delta _1)\\\\Z_{2n+1}&=&(0,X_n)\\\\Z_{2n+2}&=&(1,Y_n)\\end{array}$ The proof that $Z_i$ is an effective presentation is left as an exercise.",
"For the product construction, the domain appears as follows: ${\\cal D}_0\\times {\\cal D}_1=\\lbrace (x,y)\\:\\vert \\:x\\in {\\cal D}_0,y\\in {\\cal D}_1\\rbrace $ The enumeration can be defined in terms of the functions $p:{{N}}\\rightarrow {{N}}$ , $q:{{N}}\\rightarrow {{N}}$ , and $r:({{N}}\\times {{N}})\\rightarrow {{N}}$ where for $m$ , $n$ , $k\\in {{N}}$ : $\\begin{array}{lcl}p(r(n,m))&=&n\\\\q(r(n,m))&=&m\\\\r(p(k),q(k))&=&k\\end{array}$ Thus, $r$ is a one-to-one pairing function (see Exercise REF ) of which there are several.",
"The functions $p$ and $q$ extract the indices from the result of the pairing function.",
"The enumeration for the product domain is then defined as follows: $W_i = (X_{p(i)},Y_{q(i)})$ The proof that this is an effective presentation is also left as an exercise.",
"For the combinators, the relations will be defined in terms of the enumeration indices.",
"For example, $\\begin{array}{lcl}X_n\\:in_0\\:Z_m&\\iff & m=0~{\\rm or}\\\\&&\\exists k.m=2k+1\\wedge X_k\\sqsubseteq X_n\\\\W_k\\:proj_1\\:Y_m&\\iff & Y_m\\sqsubseteq Y_{q(k)}\\end{array}$ The reader should verify that these sets are recursively enumerable.",
"For this proof, recall that recursively enumerable sets are closed under conjunction, disjunction, substituting recursive functions, and applying an existential quantifier to the front of a recursive predicate.",
"The proof for the other combinators is left as an exercise.",
"$\\:\\:\\Box $ Product spaces formalize the notion of computable functions of several variables.",
"Note that the proof of Theorem REF shows that substitution of computable functions of severable variables into other computable functions are still computable.",
"The next step is to show that the function space constructor preserves effectiveness.",
"Theorem 7.6: For domains ${\\cal D}_0$ and ${\\cal D}_1$ with effective presentations, the domain ${\\cal D}_0\\rightarrow {\\cal D}_1$ also has an effective presentation.",
"The combinators $apply$ and $curry$ are computable if all input domains are effectively given.",
"The computable elements of the domain ${\\cal D}_0\\rightarrow {\\cal D}_1$ are the computable maps for ${\\bf D_0}\\rightarrow {\\bf D_1}$ .",
"Proof Let ${\\cal D}_0=\\lbrace X_i\\:\\vert \\:i\\in {{N}}\\rbrace $ and ${\\cal D}_1=\\lbrace Y_i\\:\\vert \\:i\\in {{N}}\\rbrace $ be the presentations for the domains.",
"The elements of ${\\bf D_0}\\rightarrow {\\bf D_1}$ are finite step functions which respect the mapping of some subset of ${\\bf D_0}\\times {\\bf D_1}$ .",
"Given the enumeration, each element can be associated with a set $\\lbrace (X_{n_i},Y_{m_i})\\:\\vert \\:\\exists q.",
"1\\le i\\le q\\rbrace $ Thus, there is a finite set of integers pairs that determine the element.",
"Given the definition of consistency from Theorem REF for elements in the function space domain and the decidability of consistency in ${\\cal D}_0$ and ${\\cal D}_1$ , consistency of any finite set of this form is decidable (tedious but decidable since all elements must be checked with all others, etc).",
"Since consistency is decidable, a systematic enumeration of pair sets which are consistent can be made; this enumeration is simply the enumeration of ${\\cal D}_0\\rightarrow {\\cal D}_1$ .",
"Finding the lub consists of making a finite series of tests to find the element that is the lub, which must exist since the set is consistent and we have closure on lubs of finite consistent subsets.",
"Finding the lub requires a finite series of checks in both ${\\cal D}_0$ and ${\\cal D}_1$ but these checks are decidable.",
"Thus, the lub relation is also decidable in ${\\cal D}_0\\rightarrow {\\cal D}_1$ .",
"This shows that ${\\cal D}_0\\rightarrow {\\cal D}_1$ is effectively given.",
"To show that $apply$ and $curry$ are computable, the mappings need to be examined.",
"The mapping defined for apply is $(F,a)\\:apply\\: b\\iff a\\:F\\:b$ The function $F$ is the lub of all the finite step functions that are consistent with it.",
"As such, $F$ can be viewed as the canonical representative of this set.",
"Since $F$ is a finite step function, this relation is decidable.",
"As such, the $apply$ relation is recursive and not just recursively enumerable and $apply$ is a computable function.",
"The reasoning for $curry$ is similar in that the relations are studied.",
"Given the increase in the number of domains, the construction is more tedious and is left for the exercises.",
"To see that the computable elements correspond to the computable maps, recall the relationship shown in Theorem REF between the maps and the elements in the function space.",
"Thus, we have $a\\:f\\:b \\iff b\\in f({\\cal I}_a)~{\\rm or}~{\\cal I}_b\\sqsubseteq f({\\cal I}_a)$ Since $f$ is a computable map, we know that the pairs in the map are recursively enumerable.",
"Using the previous techniques for deciding consistency of finite sets, the set of elements consistent with $f$ can be enumerated.",
"But this set is simply the ideal for $f$ in the function space.",
"The converse direction is trivial.",
"$\\:\\:\\Box $ The final combinator to be discussed, and perhaps the most important, is the fixed point combinator.",
"Theorem 7.7: For any effectively given domain, ${\\cal D}$ , the combinator $fix:({\\cal D}\\rightarrow {\\cal D})\\rightarrow {\\cal D}$ is computable.",
"Proof Let $\\lbrace X_n\\:\\vert \\:n\\in {{N}}\\rbrace $ be the presentation of the domain ${\\cal D}$ .",
"Recall that for $f\\in {\\cal D}\\rightarrow {\\cal D}$ , $f\\:fix\\:X\\iff \\exists k\\in {{N}}.\\Delta \\:f\\:X_1\\:f\\ldots f\\:X_k\\wedge X_k=X$ All of the checks in this finite sequence are decidable since ${\\cal D}$ is effectively given.",
"In addition, existential quantification of a decidable predicate gives a recursively enumerable predicate.",
"Thus, $fix$ is computable.",
"$\\:\\:\\Box $ Recap Now that this has been formalized, what has been accomplished?",
"The major consequence of the theorems to this point is that any expression over effectively given domains (that is effectively given types) combined with computable constants using the $\\lambda $ -notation and the fixed point combinator is a computable function of its free variables.",
"Such functions, applied to computable arguments, yield computable values.",
"These functions also have computable least fixed points.",
"All this gives us a mathematical programming language for defining computable operations.",
"Combining this language with the specification of types with domain equations gives a powerful language.",
"As an example, the effectiveness of the domain ${\\cal T}$ from Example REF is studied.",
"The complete proof is left as an exercise.",
"Example 7.8: Recall the domain ${\\cal T}$ from the previous section.",
"This domain is characterized by the domain equation ${\\cal T}={\\cal A}+({\\cal T}\\times {\\cal T})$ for some domain ${\\cal A}$ .",
"If ${\\cal A}$ is effectively given, we wish to show that ${\\cal T}$ is effectively given as well.",
"The elements are either atomic elements from ${\\cal A}$ or are pairs from ${\\cal T}$ .",
"Let $A=\\lbrace A_i\\:\\vert \\:i\\in {{N}}\\rbrace $ be the enumeration for ${\\cal A}$ .",
"An enumeration for ${\\cal T}$ can be defined as follows: $\\begin{array}{lcl}T_0&=&\\bot _T\\\\T_{2n+1}&=&3*A_n\\\\T_{2n+2}&=&3*T_{p(n)}+1\\cup 3*T_{q(n)}+2\\end{array}$ where for $A$ , a set of indices, $m*A+k=\\lbrace m*n+k\\:\\vert \\:n\\in A\\rbrace $ .",
"The functions $p$ and $q$ here are the inverses of the pairing function $r$ defined in Theorem REF .",
"These functions must be defined such that $p(n)\\le n$ and $q(n)\\le n$ so that the recursion is well defined by taking smaller indices.",
"The rest of the proof is left to the exercises.",
"Specifically, the claim that ${\\cal T}=\\lbrace T_i\\rbrace $ should be verified as well as the effectiveness of the enumeration.",
"These proofs rely either on the effectiveness of ${\\cal A}$ , on the effectiveness of elements in ${\\cal T}$ with smaller indices, or are trivial.",
"The final example uses the powerset construction.",
"We have repeatedly used the fact that a powerset is a domain.",
"Its effectiveness is now verified.",
"Example 7.9: Specifically, the powerset of the natural numbers, ${\\cal {P}({{N}})}$ is considered.",
"In this domain, all elements are consistent, and there is a top element, denoted $\\omega $ , which is the set of all natural numbers.",
"The ordering is the subset relation.",
"The lub of two subsets is the union of the two subsets, which is decidable.",
"To enumerate the finite subsets, the following enumeration is used: $E_n=\\lbrace k\\:\\vert \\:\\exists i,j.",
"i< 2^k\\wedge n=i+2^k+j*2^{k+1}\\rbrace $ This says that $k\\in E_n$ if the $k$ bit in the binary expansion of $n$ is a 1.",
"All finite subsets of ${{N}}$ are of the form $E_n$ for some $n$ .",
"Various combinators for ${\\cal P}({{N}})$ are presented in Exercise REF .",
"Exercises Exercise 7.10: Show that an effectively given domain can always be identified with a relation $INCL(n,m)$ on integers where the derived relations $\\begin{array}{lcl}CONS(n,m)&\\iff &\\exists k.INCL(k,n)\\wedge INCL(k,m)\\\\MEET(n,m,k)&\\iff &\\forall j.",
"[INCL(j,k)\\iff INCL(j,n)\\wedge INCL(j,m)]\\end{array}$ are recursively decidable and where the following axioms hold: $\\forall n.INCL(n,n)$ $\\forall n,m,k.",
"INCL(n,m)\\wedge INCL(m,k)\\:\\Rightarrow \\:INCL(n,k)$ $\\exists m.\\forall n. INCL(n,m)$ $\\forall n,m.",
"CONS(n,m)\\:\\Rightarrow \\:\\exists k.MEET(n,m,k)$ Exercise 7.11: Finish the proof of Theorem REF .",
"Exercise 7.12: Complete the proof of Theorem REF by defining $curry$ as a relation and showing it computable.",
"Is the set recursively enumerable or is it recursive?",
"Exercise 7.13: Two effectively given domains are effectively isomorphic iff $\\ldots $ Complete the statement of the theorem and prove it.",
"Exercise 7.14: Complete the proof about the powerset in Example REF .",
"Show that the combinators $fun$ and $graph$ from Exercise REF are computable.",
"Show the same for $\\lambda x,y.x\\cap y$ $\\lambda x,y.x\\cup y$ $\\lambda x,y.x+ y$ where for $x,y\\in {\\cal P}({{N}})$ , $x+y=\\lbrace n+m\\:\\vert \\:n\\in x, m\\in y\\rbrace $ What are the computable elements of ${\\cal P}({{N}})$ ?",
"Sub-Spaces of the Universal Domain To have a flexible method of solving domain equations and yielding effectively given domains as the solutions, the domains will be embedded in a universal domain which is “big” enough to hold all other domains as sub-domains.",
"This universal domain is shown to be effectively presented, and the mappings which define the sub-spaces are shown to be computable.",
"First, the correspondence between sub-spaces and mappings called retractions is investigated, leading us to the definition of mappings called projections.",
"It is then shown that these definitions can be written out using the $\\lambda $ -calculus notation, demonstrating the power of our mathematical programming language.",
"Retractions and Projections We start with the definition of retractions.",
"Definition 8.1: [Retractions] A retraction of a given domain ${\\cal E}$ is an approximable mapping $a:{\\bf E}\\rightarrow {\\bf E}$ such that $a\\circ a=a$ .",
"Thus, a retraction is the identity function on objects in the range of the retraction and maps other elements into range.",
"The next theorem relates these sets to sub-spaces.",
"Theorem 8.2: If ${\\cal D}\\lhd {\\cal E}$ and if $a:{\\bf E}\\rightarrow {\\bf E}$ is defined such that $X\\:a\\:Z \\iff \\exists Y\\in {\\cal D}.",
"Z\\sqsubseteq Y\\sqsubseteq X$ for all $X,Z\\in {\\bf E}$ , then $a$ is a retraction and ${\\cal D}$ is isomorphic to the fixed point set of $a$ , the set $\\lbrace y\\in {\\cal E}\\:\\vert \\:a(y)=y\\rbrace $ , ordered under inclusion.",
"Proof That $a$ is an approximable map is a direct consequence of the definition of sub-space (Definition REF ).",
"By Theorem REF , a projection pair, $i$ and $j$ , exist for ${\\cal D}$ and this tells us that $a=i\\circ j$ (also showing $a$ approximable since approximable mappings are closed under composition).",
"Theorem REF also tells us that $j\\circ i={\\sf I}_D$ .",
"To show that $a$ is a retraction, $a\\circ a=a$ must be established.",
"Thus, $a\\circ a = i\\circ j\\circ i\\circ j = i\\circ {\\sf I}_D\\circ j = i\\circ j =a$ holds, showing that $a$ is a retraction.",
"We now need to show the isomorphism to ${\\cal D}$ .",
"For $x\\in {\\cal D}$ , $i(x)\\in {\\cal E}$ and we can calculate: $a(i(x))=i\\circ j\\circ i(x) = i\\circ {\\sf I}_D(x) = i(x)$ Thus, $i(x)$ is in the fixed point set of $a$ .",
"For the other direction, let $a(y)=y$ .",
"Then $i(j(y)) = y$ holds.",
"But, $j(y)\\in {\\cal D}$ , so $i$ must map ${\\cal D}$ one-to-one and onto the fixed point set of $a$ .",
"Since $i$ and $j$ are approximable, they are certainly monotonic, and thus the map is an isomorphism with respect to set inclusion.",
"$\\:\\:\\Box $ Not all retractions are associated with a sub-domain relationship.",
"The retractions defined in the above theorem are all subsets as relations of the identity relation.",
"The retractions for sub-domains are characterized by the following definition: Definition 8.3: [Projections] A retraction $a:{\\cal E}\\rightarrow {\\cal E}$ is a projection if $a\\subseteq {\\sf I}_E$ as relations.",
"The retraction is finitary iff its fixed point set is isomorphic to some domain.",
"An example is in order.",
"Example 8.4: Consider a two element system, ${\\bf O}$ with objects $\\Delta $ and 0.",
"For any basis ${\\bf D}$ that is not trivial (has more than one element), ${\\bf O}$ comes from a retraction on ${\\bf D}$ .",
"Define a combinator $check:{\\bf D}\\rightarrow {\\bf O}$ by the relation $x\\:check\\: y \\iff y=\\Delta ~{\\rm or}~x\\ne \\Delta _D$ Thus, $check(x)=\\bot _O\\iff x=\\bot _D$ .",
"Another combinator can be defined, $fade:{\\bf O}\\times {\\bf D}\\rightarrow {\\bf D}$ such that for $t\\in {\\cal O}$ and $x\\in {\\cal D}$ $\\begin{array}{lcll}fade(t,x)&=&\\bot _D&{\\rm if}~t=\\bot _O\\\\&=&x&otherwise\\end{array}$ For $u\\in {\\cal D}$ and $u\\ne \\bot _D$ , the mapping $a$ is defined as $a(x)=fade(check(x),u)$ It can be seen that $a$ is a retraction, but not a projection in general, and the range of $a$ is isomorphic to ${\\bf O}$ .",
"These combinators can also be used to define the subset of functions in ${\\bf D}\\rightarrow {\\bf E}$ that are strict.",
"Define a combinator $strict:({\\bf D}\\rightarrow {\\bf E})\\rightarrow ({\\bf D}\\rightarrow {\\bf E})$ by the equation $strict(f)=\\lambda x.fade(check(x),f(x))$ with $fade$ defined as $fade:{\\bf O}\\times {\\bf E}\\rightarrow {\\bf E}$ .",
"The range of $strict$ is all the strict functions; $strict$ is a projection whose range is a domain.",
"The next theorem characterizes projections.",
"Theorem 8.5: For approximable mapping $a:{\\bf E}\\rightarrow {\\bf E}$ , the following are equivalent: $a$ is a finitary projection $a(x)=\\lbrace y\\in {\\bf E}\\:\\vert \\:\\exists x^{\\prime }\\in I_x.",
"x^{\\prime }\\:a\\:x^{\\prime }\\wedge y\\sqsubseteq x^{\\prime }\\rbrace $ for all $x\\in {\\bf E}$ .",
"Proof Assume that (2) holds.",
"We want to show that $a$ is a finitary projection.",
"By the closure properties on ideals, we know that for all $x\\in {\\cal E}$ , $x^{\\prime }\\in x\\wedge y\\sqsubseteq x^{\\prime }\\:\\Rightarrow \\:y\\in x$ Thus, $a(x)\\subseteq x$ must hold.",
"In addition, the following trivially holds: $x^{\\prime }\\in x\\wedge x^{\\prime }\\:a\\: x^{\\prime }\\:\\Rightarrow \\:x^{\\prime }\\in a(x)$ thus $a(x)\\subseteq a(a(x))$ holds for all $x\\in {\\cal E}$ .",
"This shows that $a$ is indeed a projection.",
"Let $D=\\lbrace x\\in {\\bf E}\\:\\vert \\:x\\:a\\:x\\rbrace $ .",
"It is easy to show that ${\\bf D}\\lhd {\\bf E}$ and that $a$ is determined from ${\\bf D}$ as required in Theorem REF .",
"Thus, the fixed point set of $a$ is isomorphic to a domain from the previous proofs.",
"Thus, (2)$\\:\\Rightarrow \\:$ (1).",
"For the converse, assume that $a$ is a finitary projection.",
"Let ${\\cal D}$ be isomorphic to the fixed point set of $a$ .",
"This means there is a projection pair $i$ and $j$ such that $j\\circ i={\\sf I}_D$ and $i\\circ j = a$ and $a\\subseteq {\\sf I}_E$ .",
"From Theorem REF then we have that ${\\cal D}\\approx {\\cal D}^{\\prime }$ and ${\\cal D}^{\\prime }\\lhd {\\cal E}$ .",
"We want to identify ${\\cal D}^{\\prime }$ as follows: ${\\cal D}^{\\prime }=\\lbrace x\\in {\\cal E}\\:\\vert \\:x\\:a\\:x\\rbrace $ From the proof of Theorem REF , the basis elements of ${\\bf D^{\\prime }}$ are the finite elements of ${\\bf D}$ .",
"Each of these elements is in the fixed point set of $a$ .",
"Thus, $x\\in {\\bf D^{\\prime }}\\:\\Rightarrow \\:a({{\\cal I}}_x) = {{\\cal I}}_x \\:\\Rightarrow \\:x\\:a\\:x$ Since $a$ is a projection, ${{\\cal I}}_x$ must also be a fixed point.",
"Since $i(j({{\\cal I}}_x)) = {{\\cal I}}_x$ implies that $j({{\\cal I}}_x)$ is a finite element of ${\\cal D}$ , $x\\in {\\cal D}^{\\prime }$ must hold.",
"Thus, the identification of ${\\cal D}^{\\prime }$ holds.",
"Finally, using $a=i\\circ j$ in the formula in Theorem REF , the formula in (2) is obtained, proving the converse.",
"$\\:\\:\\Box $ This characterization of projections provides a new and interesting combinator.",
"Theorem 8.6: For any domain ${\\cal E}$ , define $sub:({\\cal E}\\rightarrow {\\cal E})\\rightarrow ({\\cal E}\\rightarrow {\\cal E})$ using the relation $x\\: sub(f)\\: z \\iff \\exists y\\in {\\bf E}.y\\:f\\:y\\wedge y\\sqsubseteq x\\wedge z\\sqsubseteq y$ for all $x,z\\in {\\bf E}$ and all $f:{\\bf E}\\rightarrow {\\bf E}$ .",
"Then the range of $sub$ is exactly the set of finitary projections on ${\\cal E}$ .",
"In addition, $sub$ is a finitary projection on ${\\cal E}\\rightarrow {\\cal E}$ .",
"If ${\\cal E}$ is effectively given, then $sub$ is computable.",
"Proof Clearly, $sub(f)$ is approximable.",
"It is obvious from the definition that $f\\mapsto sub(f)$ preserves lubs and thus is approximable as well.",
"Thus, $y\\:f\\:y\\wedge y\\sqsubseteq x\\wedge z\\sqsubseteq y\\:\\Rightarrow \\:x\\:f\\:z$ obviously holds.",
"Thus, $sub(f)\\subseteq f$ holds.",
"Also $y\\:f\\:y\\:\\Rightarrow \\:y\\:sub(f)\\:y$ thus, $sub(f)\\subseteq sub(sub(f))$ holds as well.",
"Thus, $sub$ is a projection on ${\\cal E}\\rightarrow {\\cal E}$ .",
"The definition of the relation shows that it is computable when ${\\cal E}$ is effectively given.",
"Since $sub$ is a projection, its range is the same as its fixed point set.",
"If $sub(a)=a$ , it is easy to see that clause (2) of Theorem REF holds and conversely.",
"Thus, the range of $sub$ is the finitary projections.",
"To see that $sub$ is a finitary projection, we use Theorem REF and Theorem REF to say that the fixed point set of $sub$ is in a one-to-one inclusion preserving correspondence with the domain $\\lbrace D\\:\\vert \\:D\\lhd {\\cal E}\\rbrace $ .",
"$\\:\\:\\Box $ Universal Domain ${\\cal U}$ With these results and the universal domain to be defined next, the theory of sub-domains is translated into the $\\lambda $ -calculus notation using the $sub$ combinator.",
"The universal domain is defined by first defining a domain which has the desired structure but has a top element.",
"The top element is then removed to give the universal domain.",
"Definition 8.7: [Universal Domain] As in the section on domain equations, an inductive definition for a domain ${\\cal V}$ is given as follows: $\\Delta ,\\top \\in {\\bf V}$ $\\langle u,v\\rangle \\in {\\bf V}$ whenever $u,v\\in {\\bf V}$ Thus, we are starting with two objects, a bottom element and a top element, and making two flavors of copies of these objects.",
"Intuitively, we end up with finite binary trees with either the top or the bottom element as the leaves.",
"To simplify the definitions below, the pairs should be reduced such that: All occurrences of $\\langle \\Delta ,\\Delta \\rangle $ are replaced by $\\Delta $ and All occurrences of $\\langle \\top ,\\top \\rangle $ are replaced by $\\top $ .",
"These rewrite rules are easily shown to be finite Church-Rosser.The finitary basis should be defined as the equivalence classes induced by the reduction.",
"The presentation is simplified by considering only reduced trees.",
"As an example of the reduction the pair $\\langle \\langle \\langle \\top ,\\langle \\top ,\\top \\rangle \\rangle ,\\langle \\top ,\\Delta \\rangle \\rangle ,\\langle \\langle \\Delta ,\\Delta \\rangle ,\\langle \\top ,\\top \\rangle \\rangle \\rangle $ reduces to $\\langle \\langle \\top ,\\langle \\top ,\\Delta \\rangle \\rangle ,\\langle \\Delta ,\\top \\rangle \\rangle $ .",
"The approximation ordering is defined as follows: $\\Delta \\sqsubseteq v$ for all $v\\in {\\bf V}$ $v\\sqsubseteq \\top $ for all $v\\in {\\bf V}$ .",
"$\\langle u,v\\rangle \\sqsubseteq \\langle u^{\\prime },v^{\\prime }\\rangle $ iff $u\\sqsubseteq u^{\\prime }$ and $v\\sqsubseteq v^{\\prime }$ Since the top element is approximated by everything, all finite sets of trees are consistent.",
"The lub for a pair of trees is defined as follows: $u\\sqcup \\top =\\top $ for $u\\in {\\bf V}$ $\\top \\sqcup u=\\top $ for $u\\in {\\bf V}$ $u\\sqcup \\Delta =u$ for $u\\in {\\bf V}$ $\\Delta \\sqcup u=u$ for $u\\in {\\bf V}$ $\\langle u,v\\rangle \\sqcup \\langle u^{\\prime },v^{\\prime }\\rangle =\\langle u\\sqcup u^{\\prime },v\\sqcup v^{\\prime }\\rangle $ for $u,v\\in {\\bf V}$ The proof that this forms a finitary basis follows the same guidelines as the proofs in Section .",
"In addition, it should be clear that the presentation is effective.",
"To form the universal domain, the top element is simply removed.",
"Thus, the system ${\\bf U}={\\bf V}-\\lbrace \\top \\rbrace $ is the basis used to form the universal domain.",
"The proof that this is still a finitary basis with an effective presentation is also straightforward and left to the exercises.",
"Note that inconsistent sets can now exist since there is no top element.",
"A set is inconsistent iff its lub is $\\top $ .",
"We shall now prove the claims made for the universal domain.",
"Theorem 8.8: The domain ${\\cal U}$ is universal, in the sense that for every domain ${\\cal D}$ we have ${\\cal D}\\lhd {\\cal U}$ .",
"If ${\\cal D}$ is effectively given, then the projection pair for the embedding is computable.",
"In fact, there is a correspondence between the effectively presented domains and the computable finitary projections of ${\\cal U}$ .",
"Proof Recall that ${\\bf D}$ must be countable to be a finitary basis.",
"Thus, we can assume that the basis has an enumeration $D=\\lbrace X_n\\:\\vert \\:n\\in {{N}}\\rbrace $ where $X_0=\\Delta $ .",
"The effective and general cases are considered together in the proof; comments about computability are included for the effective case as required.",
"Thus, if ${\\cal D}$ is effectively given, the enumeration above is assumed to be computable.",
"To prove that the domain can be embedded in ${\\cal U}$ , the embedding will be shown.",
"To start, for each finite element $d_i$ in the basis, define two sets, $d_i^+$ and $d_i^-$ as follows: $\\begin{array}{lcl}d_i^+&=&\\lbrace d\\in {\\bf D}\\:\\vert \\:d_i\\sqsubseteq d\\rbrace \\\\d_i^-&=&D-d_i^+\\end{array}$ The $d_i^+$ set contains all the elements that $d_i$ approximates, while the $d_i^-$ set contains all the other elements, partitioning ${\\bf D}$ into two disjoint sets.",
"Sets for different elements can be intersected to form finer partitions of ${\\bf D}$ .",
"For $k>0$ , let $R\\in \\lbrace +,-\\rbrace ^k$ , let $R_i$ be the $ith$ symbol in the string $R$ , and define a region $D_R$ as $D_R=\\bigcap \\limits _{i=1}^k d_i^{R_i}$ where $k$ is the length of $R$ .",
"The set $\\lbrace D_{R}\\:\\vert \\:R\\in \\lbrace +,-\\rbrace ^k\\rbrace $ of regions partitions ${\\bf D}$ into $2^k$ disjoint sets.",
"Thus, for each element $e_i$ in the enumeration there is a corresponding partition of the basis given by the family of sets $\\lbrace D_{R}\\:\\vert \\:R\\in \\lbrace +,-\\rbrace ^i\\rbrace $ .",
"For strings $R,S\\in \\lbrace +,-\\rbrace ^*$ such that $R$ is a prefix of $S$ , denoted $R\\le S$ , $D_S\\subseteq D_R$ .",
"It is important to realize that the composition of these sets is dependent on the order in which the elements are enumerated.",
"Some of these regions are empty, but it is decidable if a given intersection is empty if ${\\cal D}$ is effectively presented.",
"It is also decidable if a given element is in a particular region.",
"Figure: Example Finite DomainTo see the function these regions are serving, consider the finite domain in Figure REF .This example is taken from Cartwright and Demers [2].",
"Consider the enumeration with $d_0=\\bot , d_1=b, d_2=c, d_3=a.$ The $d_i^+$ and $d_i^-$ sets are as follows: $\\begin{array}{lcl}d_1^+&=&\\lbrace a,b\\rbrace \\\\d_1^-&=&\\lbrace c,\\bot \\rbrace \\\\d_2^+&=&\\lbrace c\\rbrace \\\\d_2^-&=&\\lbrace a,b,\\bot \\rbrace \\\\d_3^+&=&\\lbrace a\\rbrace \\\\d_3^-&=&\\lbrace b,c,\\bot \\rbrace \\end{array}$ The regions are as follows: $\\begin{array}{lclclcl}D_+ &=&\\lbrace a,b\\rbrace &\\:\\:\\:\\:&D_{+++} &=&\\lbrace \\rbrace \\\\D_- &=&\\lbrace \\bot ,c\\rbrace &&D_{++-} &=&\\lbrace \\rbrace \\\\D_{++} &=&\\lbrace \\rbrace &&D_{+-+} &=&\\lbrace a\\rbrace \\\\D_{+-} &=&\\lbrace a,b\\rbrace &&D_{+--} &=&\\lbrace b\\rbrace \\\\D_{-+} &=&\\lbrace c\\rbrace &&D_{-++} &=&\\lbrace \\rbrace \\\\D_{--} &=&\\lbrace \\bot \\rbrace &&D_{-+-} &=&\\lbrace c\\rbrace \\\\&&&&D_{--+} &=&\\lbrace \\rbrace \\\\&&&&D_{---} &=&\\lbrace \\bot \\rbrace \\end{array}$ The regions generated by each successive element encode the relationships induced by the approximation ordering between the new element and all elements previously added.",
"The reader is encouraged to try this example with other enumerations of this basis and compare the results.",
"The embedding of the elements proceeds by building a tree based on the regions corresponding to the element.",
"The regions are used to find locations in the tree and to determine whether a $\\top $ or a $\\Delta $ element is placed in the location.",
"These trees preserve the relationships specified by the regions and thus, the tree embedding is isomorphic to the domain in question.",
"Once the tree is built, the reduction rules are applied until a non-reducible tree is reached.",
"This tree is the representative element in the universal domain, and the set of these trees form the sub-space.",
"The function to determine the location in the tree for a given domain, $Loc_D:\\lbrace +,-\\rbrace ^*\\rightarrow \\lbrace l,r\\rbrace ^*$ takes strings used to generate regions and outputs a path in a tree where $l$ stands for left sub-tree and $r$ stands for right sub-tree.",
"This path is computed using the following inductive definition: $\\begin{array}{lcll}Loc_D(\\epsilon )&=&\\epsilon .\\\\Loc_D(R+)&=&Loc_D(R)l&{\\rm if }~D_{R+}\\ne \\emptyset ~ {\\rm and }~D_{R-}\\ne \\emptyset .\\\\&=&Loc_D(R)&{\\rm otherwise}.\\\\Loc_D(R-)&=&Loc_D(R)r&{\\rm if }~D_{R+}\\ne \\emptyset ~{\\rm and }~D_{R-}\\ne \\emptyset .\\\\&=&Loc_D(R)&{\\rm otherwise}.\\end{array}$ The set of locations for each non-empty region is the set of paths to all leaves of some finite binary tree.",
"An induction argument is used to show the following properties of $Loc_D$ that ensure this: If $R\\le S$ for $R,S\\subseteq \\lbrace +,-\\rbrace ^*$ , then $Loc_D(R)\\le Loc_D(S)$ .",
"Let $S=\\lbrace Loc_D(R)\\:\\vert \\:R\\in \\lbrace +,-\\rbrace ^k\\wedge D_R\\ne \\emptyset \\rbrace $ for $k>0$ be a set of location paths for a given $k$ .",
"For any $p\\in \\lbrace l,r\\rbrace ^*$ there exists $q\\in S$ such that either $p\\le q$ or $q\\le p$ .",
"That is, every potential path is represented by some finite path.",
"Finally, for all $p,q\\in S$ if $p\\le q$ then $p=q$ .",
"This means that a unique leaf is associated with each location.",
"To find the tree for a given element $d_k$ in the enumeration, apply the following rules to each $R\\in \\lbrace +,-\\rbrace ^{k-1}$ .",
"If $D_{R-}\\ne \\emptyset $ then the leaf for path $Loc_D(R-)$ is labeled $\\top $ .",
"If $D_{R+}\\ne \\emptyset $ then the leaf for path $Loc_D(R+)$ is labeled $\\Delta $ .",
"These rules are used to assign a tree in ${\\bf U}$ , which is then reduced using the reduction rules, for each element in the enumeration of ${\\bf D}$ .",
"To see that the top element is never assigned by these rules, note that some region of the form $R+$ for every length $k$ must be non-empty since it must contain the element $e_k$ being embedded.",
"Returning to the example, the location function defines paths for these elements as follows: $\\begin{array}{lclclcl}Loc_D(+)&=&l&\\:\\:\\:\\:&Loc_D(+-+)&=&ll\\\\Loc_D(-)&=&r&&Loc_D(+--)&=&lr\\\\Loc_D(+-)&=&l&&Loc_D(-+-)&=&rl\\\\Loc_D(-+)&=&rl&&Loc_D(---)&=&rr\\\\Loc_D(--)&=&rr\\end{array}$ The trees generated for each of the elements are: $\\begin{array}{lcl}d_0&\\mapsto &\\Delta \\\\d_1&\\mapsto &\\langle \\Delta ,\\top \\rangle \\\\d_2&\\mapsto &\\langle \\top ,\\langle \\Delta ,\\top \\rangle \\rangle \\\\d_3&\\mapsto & \\langle \\langle \\Delta ,\\top \\rangle ,\\langle \\top ,\\top \\rangle \\rangle \\\\&\\mapsto & \\langle \\langle \\Delta ,\\top \\rangle ,\\top \\rangle \\end{array}$ To verify that the space generated is a valid sub-space, we must verify that the bottom element is mapped to $\\bot _U$ and that the consistency and lub relations are maintained.",
"The tree $\\Delta $ is clearly assigned to $X_0$ , the bottom element for the basis being embedded, since there are no strings of length $-1$ .",
"The embedding preserves inconsistency of elements by forcing the lub of the embedded elements to be $\\top $ .",
"The $D_{R-}$ regions represent the elements that the element being embedded does not approximate.",
"Note that the $D_{R-}$ sets cause the $\\top $ element to be added as the leaf.",
"Since the $D_R$ sets are built using the approximation ordering, it is straightforward to see that the approximation ordering is preserved by the embedding.",
"Lubs are also maintained by the embedding, although the reduction is required to see that this is the case.",
"It should be clear that, if the domain ${\\cal D}$ is effectively given, the sub-space can be computed since the embedding procedure uses the relationships given in the presentation.",
"Finally, suppose that $a$ is a computable, finitary projection on ${\\cal U}$ .",
"From the proof of Theorem REF , the domain of this projection is characterized by the set $\\lbrace y\\in {\\bf U}\\:\\vert \\:y\\:a\\:y\\rbrace $ If $a$ is computable, the set of pairs for $a$ is recursively enumerable.",
"Thus, the set above is also recursively enumerable since equality among basis elements is decidable.",
"Thus, the domain given by the projection must also be effectively given.",
"$\\:\\:\\Box $ Thus, the domain ${\\cal U}$ is an effectively presented universal domain in which all other domains can be embedded.",
"The sub-domains of ${\\cal U}$ include ${\\cal U}\\rightarrow {\\cal U}$ , ${\\cal U}\\times {\\cal U}$ , etc.",
"These domains must be sub-domains of ${\\cal U}$ since they are effectively presented based on our earlier theorems.",
"Domain Constructors in ${\\cal U}$ The next step is to see how to define the constructors commonly used.",
"Definition 8.9: [Domain Constructors] Let the computable projection pair, $i_+:{\\cal U}+{\\cal U}\\rightarrow {\\cal U}~{\\rm and}~j_+:{\\cal U}\\rightarrow {\\cal U}+{\\cal U}$ be fixed.",
"Fix suitable projection pairs $i_\\times ,j_\\times ,i_\\rightarrow $ , and $j_\\rightarrow $ as well.",
"Define $\\begin{array}{lcl}a+b&=&cond\\circ \\langle which,i_+\\circ in_0\\circ a\\circ out_0, i_+\\circ in_1\\circ b\\circ out_1\\rangle \\circ j_+\\\\a\\times b&=&i_x\\circ \\langle a\\circ proj_0,b\\circ proj_1\\rangle \\circ j_x\\\\a\\rightarrow b&=&i_\\rightarrow \\circ (\\lambda f.b\\circ f\\circ a)\\circ j_\\rightarrow \\end{array}$ for all $a,b:{\\cal U}\\rightarrow {\\cal U}$ .",
"From earlier theorems, we know that these combinators are all computable over an effectively presented domain.",
"The next theorem characterizes the effect these combinators have on projection functions.",
"Theorem 8.10: If $a,b:{\\cal U}\\rightarrow {\\cal U}$ are projections, then so are $a+b$ , $a\\times b$ , and $a\\rightarrow b$ .",
"If $a$ and $b$ are finitary, then so are the compound projections.",
"Proof Since $a$ and $b$ are retractions, $a=a\\circ a$ and $b=b\\circ b$ .",
"Then for $a\\times b$ using the definition of $\\times $ , $\\begin{array}{lcl}(a\\times b)\\circ (a\\times b)&=&i_x\\circ \\langle a\\circ proj_0,b\\circ proj_1\\rangle \\circ \\langle a\\circ proj_0,b\\circ proj_1\\rangle \\circ j_x\\\\&=&i_x\\circ \\langle a\\circ a\\circ proj_0,b\\circ b\\circ proj_1\\rangle \\circ j_x\\\\&=& a\\times b\\end{array}$ Thus, $a\\times b$ is a retraction.",
"The other cases follow similarly.",
"Since $a$ and $b$ are projections, $a,b\\subseteq {\\sf I}_U$ (denoted simply ${\\sf I}$ for the remainder of the proof).",
"Using the definition for $+$ along with the above relation and the definition of projection pairs, we can see that $a+b\\subseteq {\\sf I}+{\\sf I}=i_+\\circ j_+ \\subseteq {\\sf I}$ Thus, $a+b$ is a projection.",
"The other cases follow similarly.",
"To show that the projections are finitary, we must show that the fixed point sets are isomorphic to a domain.",
"Since $a$ and $b$ are assumed finitary, their fixed point sets are isomorphic to $\\begin{array}{lcl}D_a&=&\\lbrace x\\in {\\bf U}\\:\\vert \\:x\\:a\\: x\\rbrace \\\\D_b&=&\\lbrace y\\in {\\bf U}\\:\\vert \\:y\\:b\\: y\\rbrace \\end{array}$ We wish to show that ${\\cal D}_a\\rightarrow {\\cal D}_b\\approx {\\cal D}_{a\\rightarrow b}$ .",
"By the definition of the $\\rightarrow $ constructor, the fixed point set of $a\\rightarrow b$ over ${\\cal U}$ is the same as the fixed point set of $\\lambda f.b\\circ f\\circ a$ on ${\\cal U}\\rightarrow {\\cal U}$ .",
"(Hint: $i_\\rightarrow $ and $j_\\rightarrow $ set up the isomorphism.)",
"So, the fixed points for $f:{\\cal U}\\rightarrow {\\cal U}$ are of the form: $f=b\\circ f\\circ a$ We can think of $a$ as a function in ${\\cal U}\\rightarrow {\\cal D}_a$ and define the other half of the projection pair as $i_a:{\\cal D}_a\\rightarrow {\\cal U}$ where $i_a\\circ a = a$ and $a\\circ i_a=i_a$ .",
"Define a function $i_b$ for the projection pair for $b$ similarly.",
"For some $g:{\\cal D}_a\\rightarrow {\\cal D}_b$ let $f=i_b\\circ g\\circ a$ Substituting this definition for $f$ yields $b\\circ f\\circ a = b\\circ i_b\\circ g\\circ a\\circ a = i_b\\circ g \\circ a = f$ by the definition of $i_b$ and since $a$ is a retraction by assumption.",
"Conversely, for a function $f$ such that $i_b\\circ g\\circ a= f$ , let $g=b\\circ f\\circ i_a$ Substituting again, $i_b\\circ g\\circ a = i_b\\circ g\\circ f\\circ i_a\\circ a = b\\circ f\\circ a = f$ Thus, there is an order preserving isomorphism between $g:{\\cal D}_a\\rightarrow {\\cal D}_b$ and the functions $f=b\\circ f\\circ a$ .",
"The proofs of the isomorphisms for the other constructs are similar.",
"$\\:\\:\\Box $ Thus, the sub-domain relationship with the universal domain has been stated in terms of finitary projections over the universal domain.",
"In addition, all the domain constructors have been shown to be computable combinators on the domain of these finitary projections.",
"Recalling that all computable maps have computable fixed points, the standard fixed point method can be used to solve domain equations of all kinds if they can be defined on projections.",
"Returning to the $\\lambda $ -calculus for a moment, all objects in the $\\lambda $ -calculus are considered functions.",
"Since ${\\cal U}\\rightarrow {\\cal U}$ is a part of ${\\cal U}$ , every object in the $\\lambda $ -calculus is also an object of ${\\cal U}$ .",
"Transposing some of the familiar notation, where the old notation appears on the left, the new combinators are defined as follows: $\\begin{array}{lcl}which(z)=which(j_+(z))\\\\in_i(x)=i_+(in_i(x))~{\\rm where}~i=0,1\\\\out_i(x)=out_i(j_+(x))~{\\rm where}~i=0,1\\\\\\langle x,y\\rangle =i_x(\\langle x,y\\rangle )\\\\proj_i=proj_i(j_x(z))~{\\rm where}~i=0,1\\\\u(x) = j_\\rightarrow (u)(x)\\\\\\lambda x.\\tau =i_\\rightarrow (\\lambda x.\\tau )\\end{array}$ Thus, all functions, all constants, all combinators, and all constructs are elements of ${\\cal U}$ .",
"Indeed, everything computable is an element of ${\\cal U}$ .",
"Elements in ${\\cal U}$ play multiple roles by representing different objects under different projections.",
"While this notion may be difficult to get used to, there are many advantages, both notational and conceptual.",
"Exercises Exercise 8.11: A retraction $a:{\\cal D}\\rightarrow {\\cal D}$ is a closure operator iff ${\\sf I}_D\\subseteq a$ as relations.",
"On a domain like ${\\cal P}({{N}})$ , give some examples of closure operators.",
"(Hint: Close up the integers under addition.",
"Is this continuous on ${\\cal P}({{N}})$ ?)",
"Prove in general that for any closure $a:{\\cal D}\\rightarrow {\\cal D}$ , the fixed point set of $a$ is always a finitary domain.",
"(Hint: Show that the fixed point set is closed as required for a domain.)",
"What are the finite elements of the fixed point set?",
"Exercise 8.12: Give a direct proof that the domain $\\lbrace X\\:\\vert \\:X\\lhd {\\cal D}\\rbrace $ is effectively presented if ${\\cal D}$ is.",
"(Hint: The finite elements of the domain correspond exactly to the finite domains $X\\lhd {\\cal D}$ .)",
"In the case of ${\\cal D}={\\cal U}$ , show that the computable elements of the domain correspond exactly to the effectively presented domains (up to effective isomorphism).",
"Exercise 8.13: For finitary projections $a:{\\cal E}\\rightarrow {\\cal E}$ , write ${\\cal D}_a=\\lbrace x\\in {\\cal E}\\:\\vert \\:x\\:a\\:x\\rbrace $ Show that for any two such projections $a$ and $b$ , that $a\\subseteq b \\iff {\\cal D}_a\\lhd {\\cal D}_b$ Exercise 8.14: Find another universal domain that is not isomorphic to ${\\cal U}$ .",
"Exercise 8.15: Prove the remaining cases in Theorem REF .",
"Exercise 8.16: Suppose $S$ and $T$ are two binary constructors on domains that can be made into computable operators on projections over the universal domain.",
"Show that we can find a pair of effectively presented domains such that $D\\approx S(D,E)~{\\rm and}~E\\approx T(D,E).$ Exercise 8.17: Using the translations shown after the proof of Theorem REF , show how the whole typed-$\\lambda $ -calculus can be translated into ${\\cal U}$ .",
"(Hint: for $f:{\\cal D}_a\\rightarrow {\\cal B}$ , write $f=b\\circ f\\circ a$ for finitary projections $a$ and $b$ .",
"For $\\lambda x^{{\\cal D}_a}.\\sigma $ , write $\\lambda x.b(\\sigma ^{\\prime }[a(x)/x])$ where $\\sigma ^{\\prime }$ is the translation of $\\sigma $ into the untyped $\\lambda $ -calculus.",
"Be sure that the resulting term has the right type.)",
"Exercise 8.18: Show that the basis presented for the universal domain ${\\bf U}$ is indeed a finitary basis and that it has an effective presentation.",
"Exercise 8.19: Work out the embedding for the other enumerations for the example given in the proof of Theorem REF ."
],
[
"Computability in Effectively Given Domains",
"In the previous sections, we gave considerable emphasis to the notion of computation using increasingly accurate approximations of the input and output.",
"This section defines this notion of computability more formally.",
"In Section 5, we found that partial functions over the natural numbers were expressible in the $\\lambda $ -notation.",
"This relationship characterizes computation for a particular domain.",
"To describe computation over domains in general, a broader definition is required.",
"The way a domain is presented impacts the way computations are performed over it.",
"Indeed, the theorems of recursive function theory [6] rely in part on the normal presentation of the natural numbers.",
"A presentation for a domain is an enumeration of the elements of the domain.",
"The standard presentation of the natural numbers is simply the numbers in ascending order beginning with 0.",
"There are many permutations of the natural numbers, each of which can be considered a presentation.",
"Computation with these non-standard presentations may be impossible; that is a computable function on the standard presentation may be non-computable over a non-standard presentation.",
"Therefore, an effective presentation for a domain is defined as a presentation which makes the required information computable."
],
[
"Effective Presentations",
"Information about elements in a domain can be characterized completely by looking at the finite elements and their relationships.",
"Thus a presentation must enumerate the finite elements and allow the consistency and lub relationships on these elements to be computed to allow this style of computation.",
"The consistency relation and the lub relation depend on each other.",
"For example, if a set of elements is consistent, a lub must exist for the set.",
"Given that a set is consistent, the lub can be found in finite time by just enumerating the elements and checking to see if this element is the lub.",
"However, if the set is inconsistent, the enumeration will not reveal this fact.",
"Thus, the consistency relation must be assumed to be recursive in an effective presentation.",
"Exercise REF provides a description of presentations that should clarify the assumptions made.",
"Formally, a presentation is defined as follows: Definition 7.1: [Effective Presentation] The presentation of a finitary basis D is a function $\\pi :{{N}}\\rightarrow {\\bf D}$ such that $\\pi (0)=\\Delta _D$ and the range of $\\pi $ is the set of finite elements of D. The definition holds for a domain ${\\cal D}$ as well.",
"A presentation $\\pi $ is effective iff The consistency relation ($\\exists k.\\pi _i\\sqsubseteq \\pi _k\\wedge \\pi _j\\sqsubseteq \\pi _k$ ) for elements $\\pi _i$ and $\\pi _j$ is recursiveRecursive in this context means that the relation is decidable.",
"over $i$ and $j$ .",
"The lub relation ($\\pi _k=\\pi _i\\sqcup \\pi _j$ ) is recursive over $i$ , $j$ , and $k$ .",
"This definition supports our intuition about domains; we have stated that the important information about a domain is the set of finite elements, the ordering and consistency relationships between the elements and the lub relation.",
"Thus, an effective presentation provides, in a suitable (that is computable) form, the basic information about the structure and elements of a domain.",
"A presentation can also be viewed as an enumeration of the elements of the domain with the position of an element in the enumeration given by the index corresponding to the integer input for that element in the presentation function with the 0 element representing $\\bot $ .",
"This perspective is used in the majority of the proofs.",
"Computability Now that the presentation of a domain has been formalized, the notion of computability can be formally defined.",
"Thus, Definition 7.2: [Computable Mappings] Given two domains, ${\\cal D}$ and ${\\cal E}$ with effective presentations $\\pi _1$ and $\\pi _2$ respectively, an approximable mapping $f:{\\bf D}\\rightarrow {\\bf E}$ is computable iff the relation $x_n\\:f\\:y_m$ is recursively enumerable in $n$ and $m$ .",
"By considering the domain ${\\cal D}$ to be a single element domain, the above definition applies not only to computable functions but also to computable elements.",
"For $d\\in {\\cal D}$ where $d$ is the only element in the domain, the element $e=f(d)\\in {\\cal E}$ defines an element in ${\\cal E}$ .",
"The definition states that $e$ is a computable iff the set $\\lbrace m\\in {{N}}\\:\\vert \\:y_m\\sqsubseteq e\\rbrace $ is a recursively enumerable set of integers.",
"Clearly if the set of elements approximating another is finite, the set is recursive.",
"The notion of a recursively enumerable set simply requires that all elements approximating the element in question be listed eventually.",
"The computation then proceeds by accepting an enumeration representing the input element and enumerating the elements that approximate the desired output element.",
"Now that the notions of computability and effective presentations have been formalized, the methods of constructing domains and functions will be addressed.",
"The proof of the next theorem is trivial and is left to the reader.",
"Theorem 7.3: The identity map on an effectively given domain is computable.",
"The composition of computable mappings on effectively given domains are also computable.",
"The following corollary is a consequence of this theorem: Corollary 7.4: For computable function $f:{\\cal D}\\rightarrow {\\cal E}$ and a computable element $x\\in {\\cal D}$ , the element $f(x)\\in {\\cal E}$ is computable.",
"In addition, the standard domain constructors maintain effective presentations.",
"Theorem 7.5: For domains ${\\cal D}_0$ and ${\\cal D}_1$ with effective presentations, the domains ${\\cal D}_0+{\\cal D}_1~{\\rm and}~ {\\cal D}_0\\times {\\cal D}_1$ are also effectively given.",
"In addition, the projection functions are all computable.",
"Finally, if $f$ and $g$ are computable maps, then so are $f+g$ and $f\\times g$ .",
"Proof Let $\\lbrace X_i\\:\\vert \\:i\\in {{N}}\\rbrace $ be the enumeration of ${\\cal D}_0$ and $\\lbrace Y_i\\:\\vert \\:i\\in {{N}}\\rbrace $ be the enumeration of ${\\cal D}_1$ .",
"Another method of sum construction is to use two distinguishing elements in the first position to specify the element type.",
"Thus, a sum domain can be defined as follows: ${\\cal D}_0+{\\cal D}_1=\\lbrace (\\Delta _0,\\Delta _1)\\rbrace \\cup \\lbrace (0,x)\\:\\vert \\:x\\in {\\cal D}_0\\rbrace \\cup \\lbrace (1,y)\\:\\vert \\:y\\in {\\cal D}_1\\rbrace $ The enumeration can then be defined as follows for $n\\in {{N}}$ : $\\begin{array}{lcl}Z_0&=&(\\Delta _0,\\Delta _1)\\\\Z_{2n+1}&=&(0,X_n)\\\\Z_{2n+2}&=&(1,Y_n)\\end{array}$ The proof that $Z_i$ is an effective presentation is left as an exercise.",
"For the product construction, the domain appears as follows: ${\\cal D}_0\\times {\\cal D}_1=\\lbrace (x,y)\\:\\vert \\:x\\in {\\cal D}_0,y\\in {\\cal D}_1\\rbrace $ The enumeration can be defined in terms of the functions $p:{{N}}\\rightarrow {{N}}$ , $q:{{N}}\\rightarrow {{N}}$ , and $r:({{N}}\\times {{N}})\\rightarrow {{N}}$ where for $m$ , $n$ , $k\\in {{N}}$ : $\\begin{array}{lcl}p(r(n,m))&=&n\\\\q(r(n,m))&=&m\\\\r(p(k),q(k))&=&k\\end{array}$ Thus, $r$ is a one-to-one pairing function (see Exercise REF ) of which there are several.",
"The functions $p$ and $q$ extract the indices from the result of the pairing function.",
"The enumeration for the product domain is then defined as follows: $W_i = (X_{p(i)},Y_{q(i)})$ The proof that this is an effective presentation is also left as an exercise.",
"For the combinators, the relations will be defined in terms of the enumeration indices.",
"For example, $\\begin{array}{lcl}X_n\\:in_0\\:Z_m&\\iff & m=0~{\\rm or}\\\\&&\\exists k.m=2k+1\\wedge X_k\\sqsubseteq X_n\\\\W_k\\:proj_1\\:Y_m&\\iff & Y_m\\sqsubseteq Y_{q(k)}\\end{array}$ The reader should verify that these sets are recursively enumerable.",
"For this proof, recall that recursively enumerable sets are closed under conjunction, disjunction, substituting recursive functions, and applying an existential quantifier to the front of a recursive predicate.",
"The proof for the other combinators is left as an exercise.",
"$\\:\\:\\Box $ Product spaces formalize the notion of computable functions of several variables.",
"Note that the proof of Theorem REF shows that substitution of computable functions of severable variables into other computable functions are still computable.",
"The next step is to show that the function space constructor preserves effectiveness.",
"Theorem 7.6: For domains ${\\cal D}_0$ and ${\\cal D}_1$ with effective presentations, the domain ${\\cal D}_0\\rightarrow {\\cal D}_1$ also has an effective presentation.",
"The combinators $apply$ and $curry$ are computable if all input domains are effectively given.",
"The computable elements of the domain ${\\cal D}_0\\rightarrow {\\cal D}_1$ are the computable maps for ${\\bf D_0}\\rightarrow {\\bf D_1}$ .",
"Proof Let ${\\cal D}_0=\\lbrace X_i\\:\\vert \\:i\\in {{N}}\\rbrace $ and ${\\cal D}_1=\\lbrace Y_i\\:\\vert \\:i\\in {{N}}\\rbrace $ be the presentations for the domains.",
"The elements of ${\\bf D_0}\\rightarrow {\\bf D_1}$ are finite step functions which respect the mapping of some subset of ${\\bf D_0}\\times {\\bf D_1}$ .",
"Given the enumeration, each element can be associated with a set $\\lbrace (X_{n_i},Y_{m_i})\\:\\vert \\:\\exists q.",
"1\\le i\\le q\\rbrace $ Thus, there is a finite set of integers pairs that determine the element.",
"Given the definition of consistency from Theorem REF for elements in the function space domain and the decidability of consistency in ${\\cal D}_0$ and ${\\cal D}_1$ , consistency of any finite set of this form is decidable (tedious but decidable since all elements must be checked with all others, etc).",
"Since consistency is decidable, a systematic enumeration of pair sets which are consistent can be made; this enumeration is simply the enumeration of ${\\cal D}_0\\rightarrow {\\cal D}_1$ .",
"Finding the lub consists of making a finite series of tests to find the element that is the lub, which must exist since the set is consistent and we have closure on lubs of finite consistent subsets.",
"Finding the lub requires a finite series of checks in both ${\\cal D}_0$ and ${\\cal D}_1$ but these checks are decidable.",
"Thus, the lub relation is also decidable in ${\\cal D}_0\\rightarrow {\\cal D}_1$ .",
"This shows that ${\\cal D}_0\\rightarrow {\\cal D}_1$ is effectively given.",
"To show that $apply$ and $curry$ are computable, the mappings need to be examined.",
"The mapping defined for apply is $(F,a)\\:apply\\: b\\iff a\\:F\\:b$ The function $F$ is the lub of all the finite step functions that are consistent with it.",
"As such, $F$ can be viewed as the canonical representative of this set.",
"Since $F$ is a finite step function, this relation is decidable.",
"As such, the $apply$ relation is recursive and not just recursively enumerable and $apply$ is a computable function.",
"The reasoning for $curry$ is similar in that the relations are studied.",
"Given the increase in the number of domains, the construction is more tedious and is left for the exercises.",
"To see that the computable elements correspond to the computable maps, recall the relationship shown in Theorem REF between the maps and the elements in the function space.",
"Thus, we have $a\\:f\\:b \\iff b\\in f({\\cal I}_a)~{\\rm or}~{\\cal I}_b\\sqsubseteq f({\\cal I}_a)$ Since $f$ is a computable map, we know that the pairs in the map are recursively enumerable.",
"Using the previous techniques for deciding consistency of finite sets, the set of elements consistent with $f$ can be enumerated.",
"But this set is simply the ideal for $f$ in the function space.",
"The converse direction is trivial.",
"$\\:\\:\\Box $ The final combinator to be discussed, and perhaps the most important, is the fixed point combinator.",
"Theorem 7.7: For any effectively given domain, ${\\cal D}$ , the combinator $fix:({\\cal D}\\rightarrow {\\cal D})\\rightarrow {\\cal D}$ is computable.",
"Proof Let $\\lbrace X_n\\:\\vert \\:n\\in {{N}}\\rbrace $ be the presentation of the domain ${\\cal D}$ .",
"Recall that for $f\\in {\\cal D}\\rightarrow {\\cal D}$ , $f\\:fix\\:X\\iff \\exists k\\in {{N}}.\\Delta \\:f\\:X_1\\:f\\ldots f\\:X_k\\wedge X_k=X$ All of the checks in this finite sequence are decidable since ${\\cal D}$ is effectively given.",
"In addition, existential quantification of a decidable predicate gives a recursively enumerable predicate.",
"Thus, $fix$ is computable.",
"$\\:\\:\\Box $ Recap Now that this has been formalized, what has been accomplished?",
"The major consequence of the theorems to this point is that any expression over effectively given domains (that is effectively given types) combined with computable constants using the $\\lambda $ -notation and the fixed point combinator is a computable function of its free variables.",
"Such functions, applied to computable arguments, yield computable values.",
"These functions also have computable least fixed points.",
"All this gives us a mathematical programming language for defining computable operations.",
"Combining this language with the specification of types with domain equations gives a powerful language.",
"As an example, the effectiveness of the domain ${\\cal T}$ from Example REF is studied.",
"The complete proof is left as an exercise.",
"Example 7.8: Recall the domain ${\\cal T}$ from the previous section.",
"This domain is characterized by the domain equation ${\\cal T}={\\cal A}+({\\cal T}\\times {\\cal T})$ for some domain ${\\cal A}$ .",
"If ${\\cal A}$ is effectively given, we wish to show that ${\\cal T}$ is effectively given as well.",
"The elements are either atomic elements from ${\\cal A}$ or are pairs from ${\\cal T}$ .",
"Let $A=\\lbrace A_i\\:\\vert \\:i\\in {{N}}\\rbrace $ be the enumeration for ${\\cal A}$ .",
"An enumeration for ${\\cal T}$ can be defined as follows: $\\begin{array}{lcl}T_0&=&\\bot _T\\\\T_{2n+1}&=&3*A_n\\\\T_{2n+2}&=&3*T_{p(n)}+1\\cup 3*T_{q(n)}+2\\end{array}$ where for $A$ , a set of indices, $m*A+k=\\lbrace m*n+k\\:\\vert \\:n\\in A\\rbrace $ .",
"The functions $p$ and $q$ here are the inverses of the pairing function $r$ defined in Theorem REF .",
"These functions must be defined such that $p(n)\\le n$ and $q(n)\\le n$ so that the recursion is well defined by taking smaller indices.",
"The rest of the proof is left to the exercises.",
"Specifically, the claim that ${\\cal T}=\\lbrace T_i\\rbrace $ should be verified as well as the effectiveness of the enumeration.",
"These proofs rely either on the effectiveness of ${\\cal A}$ , on the effectiveness of elements in ${\\cal T}$ with smaller indices, or are trivial.",
"The final example uses the powerset construction.",
"We have repeatedly used the fact that a powerset is a domain.",
"Its effectiveness is now verified.",
"Example 7.9: Specifically, the powerset of the natural numbers, ${\\cal {P}({{N}})}$ is considered.",
"In this domain, all elements are consistent, and there is a top element, denoted $\\omega $ , which is the set of all natural numbers.",
"The ordering is the subset relation.",
"The lub of two subsets is the union of the two subsets, which is decidable.",
"To enumerate the finite subsets, the following enumeration is used: $E_n=\\lbrace k\\:\\vert \\:\\exists i,j.",
"i< 2^k\\wedge n=i+2^k+j*2^{k+1}\\rbrace $ This says that $k\\in E_n$ if the $k$ bit in the binary expansion of $n$ is a 1.",
"All finite subsets of ${{N}}$ are of the form $E_n$ for some $n$ .",
"Various combinators for ${\\cal P}({{N}})$ are presented in Exercise REF .",
"Exercises Exercise 7.10: Show that an effectively given domain can always be identified with a relation $INCL(n,m)$ on integers where the derived relations $\\begin{array}{lcl}CONS(n,m)&\\iff &\\exists k.INCL(k,n)\\wedge INCL(k,m)\\\\MEET(n,m,k)&\\iff &\\forall j.",
"[INCL(j,k)\\iff INCL(j,n)\\wedge INCL(j,m)]\\end{array}$ are recursively decidable and where the following axioms hold: $\\forall n.INCL(n,n)$ $\\forall n,m,k.",
"INCL(n,m)\\wedge INCL(m,k)\\:\\Rightarrow \\:INCL(n,k)$ $\\exists m.\\forall n. INCL(n,m)$ $\\forall n,m.",
"CONS(n,m)\\:\\Rightarrow \\:\\exists k.MEET(n,m,k)$ Exercise 7.11: Finish the proof of Theorem REF .",
"Exercise 7.12: Complete the proof of Theorem REF by defining $curry$ as a relation and showing it computable.",
"Is the set recursively enumerable or is it recursive?",
"Exercise 7.13: Two effectively given domains are effectively isomorphic iff $\\ldots $ Complete the statement of the theorem and prove it.",
"Exercise 7.14: Complete the proof about the powerset in Example REF .",
"Show that the combinators $fun$ and $graph$ from Exercise REF are computable.",
"Show the same for $\\lambda x,y.x\\cap y$ $\\lambda x,y.x\\cup y$ $\\lambda x,y.x+ y$ where for $x,y\\in {\\cal P}({{N}})$ , $x+y=\\lbrace n+m\\:\\vert \\:n\\in x, m\\in y\\rbrace $ What are the computable elements of ${\\cal P}({{N}})$ ?",
"Sub-Spaces of the Universal Domain To have a flexible method of solving domain equations and yielding effectively given domains as the solutions, the domains will be embedded in a universal domain which is “big” enough to hold all other domains as sub-domains.",
"This universal domain is shown to be effectively presented, and the mappings which define the sub-spaces are shown to be computable.",
"First, the correspondence between sub-spaces and mappings called retractions is investigated, leading us to the definition of mappings called projections.",
"It is then shown that these definitions can be written out using the $\\lambda $ -calculus notation, demonstrating the power of our mathematical programming language.",
"Retractions and Projections We start with the definition of retractions.",
"Definition 8.1: [Retractions] A retraction of a given domain ${\\cal E}$ is an approximable mapping $a:{\\bf E}\\rightarrow {\\bf E}$ such that $a\\circ a=a$ .",
"Thus, a retraction is the identity function on objects in the range of the retraction and maps other elements into range.",
"The next theorem relates these sets to sub-spaces.",
"Theorem 8.2: If ${\\cal D}\\lhd {\\cal E}$ and if $a:{\\bf E}\\rightarrow {\\bf E}$ is defined such that $X\\:a\\:Z \\iff \\exists Y\\in {\\cal D}.",
"Z\\sqsubseteq Y\\sqsubseteq X$ for all $X,Z\\in {\\bf E}$ , then $a$ is a retraction and ${\\cal D}$ is isomorphic to the fixed point set of $a$ , the set $\\lbrace y\\in {\\cal E}\\:\\vert \\:a(y)=y\\rbrace $ , ordered under inclusion.",
"Proof That $a$ is an approximable map is a direct consequence of the definition of sub-space (Definition REF ).",
"By Theorem REF , a projection pair, $i$ and $j$ , exist for ${\\cal D}$ and this tells us that $a=i\\circ j$ (also showing $a$ approximable since approximable mappings are closed under composition).",
"Theorem REF also tells us that $j\\circ i={\\sf I}_D$ .",
"To show that $a$ is a retraction, $a\\circ a=a$ must be established.",
"Thus, $a\\circ a = i\\circ j\\circ i\\circ j = i\\circ {\\sf I}_D\\circ j = i\\circ j =a$ holds, showing that $a$ is a retraction.",
"We now need to show the isomorphism to ${\\cal D}$ .",
"For $x\\in {\\cal D}$ , $i(x)\\in {\\cal E}$ and we can calculate: $a(i(x))=i\\circ j\\circ i(x) = i\\circ {\\sf I}_D(x) = i(x)$ Thus, $i(x)$ is in the fixed point set of $a$ .",
"For the other direction, let $a(y)=y$ .",
"Then $i(j(y)) = y$ holds.",
"But, $j(y)\\in {\\cal D}$ , so $i$ must map ${\\cal D}$ one-to-one and onto the fixed point set of $a$ .",
"Since $i$ and $j$ are approximable, they are certainly monotonic, and thus the map is an isomorphism with respect to set inclusion.",
"$\\:\\:\\Box $ Not all retractions are associated with a sub-domain relationship.",
"The retractions defined in the above theorem are all subsets as relations of the identity relation.",
"The retractions for sub-domains are characterized by the following definition: Definition 8.3: [Projections] A retraction $a:{\\cal E}\\rightarrow {\\cal E}$ is a projection if $a\\subseteq {\\sf I}_E$ as relations.",
"The retraction is finitary iff its fixed point set is isomorphic to some domain.",
"An example is in order.",
"Example 8.4: Consider a two element system, ${\\bf O}$ with objects $\\Delta $ and 0.",
"For any basis ${\\bf D}$ that is not trivial (has more than one element), ${\\bf O}$ comes from a retraction on ${\\bf D}$ .",
"Define a combinator $check:{\\bf D}\\rightarrow {\\bf O}$ by the relation $x\\:check\\: y \\iff y=\\Delta ~{\\rm or}~x\\ne \\Delta _D$ Thus, $check(x)=\\bot _O\\iff x=\\bot _D$ .",
"Another combinator can be defined, $fade:{\\bf O}\\times {\\bf D}\\rightarrow {\\bf D}$ such that for $t\\in {\\cal O}$ and $x\\in {\\cal D}$ $\\begin{array}{lcll}fade(t,x)&=&\\bot _D&{\\rm if}~t=\\bot _O\\\\&=&x&otherwise\\end{array}$ For $u\\in {\\cal D}$ and $u\\ne \\bot _D$ , the mapping $a$ is defined as $a(x)=fade(check(x),u)$ It can be seen that $a$ is a retraction, but not a projection in general, and the range of $a$ is isomorphic to ${\\bf O}$ .",
"These combinators can also be used to define the subset of functions in ${\\bf D}\\rightarrow {\\bf E}$ that are strict.",
"Define a combinator $strict:({\\bf D}\\rightarrow {\\bf E})\\rightarrow ({\\bf D}\\rightarrow {\\bf E})$ by the equation $strict(f)=\\lambda x.fade(check(x),f(x))$ with $fade$ defined as $fade:{\\bf O}\\times {\\bf E}\\rightarrow {\\bf E}$ .",
"The range of $strict$ is all the strict functions; $strict$ is a projection whose range is a domain.",
"The next theorem characterizes projections.",
"Theorem 8.5: For approximable mapping $a:{\\bf E}\\rightarrow {\\bf E}$ , the following are equivalent: $a$ is a finitary projection $a(x)=\\lbrace y\\in {\\bf E}\\:\\vert \\:\\exists x^{\\prime }\\in I_x.",
"x^{\\prime }\\:a\\:x^{\\prime }\\wedge y\\sqsubseteq x^{\\prime }\\rbrace $ for all $x\\in {\\bf E}$ .",
"Proof Assume that (2) holds.",
"We want to show that $a$ is a finitary projection.",
"By the closure properties on ideals, we know that for all $x\\in {\\cal E}$ , $x^{\\prime }\\in x\\wedge y\\sqsubseteq x^{\\prime }\\:\\Rightarrow \\:y\\in x$ Thus, $a(x)\\subseteq x$ must hold.",
"In addition, the following trivially holds: $x^{\\prime }\\in x\\wedge x^{\\prime }\\:a\\: x^{\\prime }\\:\\Rightarrow \\:x^{\\prime }\\in a(x)$ thus $a(x)\\subseteq a(a(x))$ holds for all $x\\in {\\cal E}$ .",
"This shows that $a$ is indeed a projection.",
"Let $D=\\lbrace x\\in {\\bf E}\\:\\vert \\:x\\:a\\:x\\rbrace $ .",
"It is easy to show that ${\\bf D}\\lhd {\\bf E}$ and that $a$ is determined from ${\\bf D}$ as required in Theorem REF .",
"Thus, the fixed point set of $a$ is isomorphic to a domain from the previous proofs.",
"Thus, (2)$\\:\\Rightarrow \\:$ (1).",
"For the converse, assume that $a$ is a finitary projection.",
"Let ${\\cal D}$ be isomorphic to the fixed point set of $a$ .",
"This means there is a projection pair $i$ and $j$ such that $j\\circ i={\\sf I}_D$ and $i\\circ j = a$ and $a\\subseteq {\\sf I}_E$ .",
"From Theorem REF then we have that ${\\cal D}\\approx {\\cal D}^{\\prime }$ and ${\\cal D}^{\\prime }\\lhd {\\cal E}$ .",
"We want to identify ${\\cal D}^{\\prime }$ as follows: ${\\cal D}^{\\prime }=\\lbrace x\\in {\\cal E}\\:\\vert \\:x\\:a\\:x\\rbrace $ From the proof of Theorem REF , the basis elements of ${\\bf D^{\\prime }}$ are the finite elements of ${\\bf D}$ .",
"Each of these elements is in the fixed point set of $a$ .",
"Thus, $x\\in {\\bf D^{\\prime }}\\:\\Rightarrow \\:a({{\\cal I}}_x) = {{\\cal I}}_x \\:\\Rightarrow \\:x\\:a\\:x$ Since $a$ is a projection, ${{\\cal I}}_x$ must also be a fixed point.",
"Since $i(j({{\\cal I}}_x)) = {{\\cal I}}_x$ implies that $j({{\\cal I}}_x)$ is a finite element of ${\\cal D}$ , $x\\in {\\cal D}^{\\prime }$ must hold.",
"Thus, the identification of ${\\cal D}^{\\prime }$ holds.",
"Finally, using $a=i\\circ j$ in the formula in Theorem REF , the formula in (2) is obtained, proving the converse.",
"$\\:\\:\\Box $ This characterization of projections provides a new and interesting combinator.",
"Theorem 8.6: For any domain ${\\cal E}$ , define $sub:({\\cal E}\\rightarrow {\\cal E})\\rightarrow ({\\cal E}\\rightarrow {\\cal E})$ using the relation $x\\: sub(f)\\: z \\iff \\exists y\\in {\\bf E}.y\\:f\\:y\\wedge y\\sqsubseteq x\\wedge z\\sqsubseteq y$ for all $x,z\\in {\\bf E}$ and all $f:{\\bf E}\\rightarrow {\\bf E}$ .",
"Then the range of $sub$ is exactly the set of finitary projections on ${\\cal E}$ .",
"In addition, $sub$ is a finitary projection on ${\\cal E}\\rightarrow {\\cal E}$ .",
"If ${\\cal E}$ is effectively given, then $sub$ is computable.",
"Proof Clearly, $sub(f)$ is approximable.",
"It is obvious from the definition that $f\\mapsto sub(f)$ preserves lubs and thus is approximable as well.",
"Thus, $y\\:f\\:y\\wedge y\\sqsubseteq x\\wedge z\\sqsubseteq y\\:\\Rightarrow \\:x\\:f\\:z$ obviously holds.",
"Thus, $sub(f)\\subseteq f$ holds.",
"Also $y\\:f\\:y\\:\\Rightarrow \\:y\\:sub(f)\\:y$ thus, $sub(f)\\subseteq sub(sub(f))$ holds as well.",
"Thus, $sub$ is a projection on ${\\cal E}\\rightarrow {\\cal E}$ .",
"The definition of the relation shows that it is computable when ${\\cal E}$ is effectively given.",
"Since $sub$ is a projection, its range is the same as its fixed point set.",
"If $sub(a)=a$ , it is easy to see that clause (2) of Theorem REF holds and conversely.",
"Thus, the range of $sub$ is the finitary projections.",
"To see that $sub$ is a finitary projection, we use Theorem REF and Theorem REF to say that the fixed point set of $sub$ is in a one-to-one inclusion preserving correspondence with the domain $\\lbrace D\\:\\vert \\:D\\lhd {\\cal E}\\rbrace $ .",
"$\\:\\:\\Box $ Universal Domain ${\\cal U}$ With these results and the universal domain to be defined next, the theory of sub-domains is translated into the $\\lambda $ -calculus notation using the $sub$ combinator.",
"The universal domain is defined by first defining a domain which has the desired structure but has a top element.",
"The top element is then removed to give the universal domain.",
"Definition 8.7: [Universal Domain] As in the section on domain equations, an inductive definition for a domain ${\\cal V}$ is given as follows: $\\Delta ,\\top \\in {\\bf V}$ $\\langle u,v\\rangle \\in {\\bf V}$ whenever $u,v\\in {\\bf V}$ Thus, we are starting with two objects, a bottom element and a top element, and making two flavors of copies of these objects.",
"Intuitively, we end up with finite binary trees with either the top or the bottom element as the leaves.",
"To simplify the definitions below, the pairs should be reduced such that: All occurrences of $\\langle \\Delta ,\\Delta \\rangle $ are replaced by $\\Delta $ and All occurrences of $\\langle \\top ,\\top \\rangle $ are replaced by $\\top $ .",
"These rewrite rules are easily shown to be finite Church-Rosser.The finitary basis should be defined as the equivalence classes induced by the reduction.",
"The presentation is simplified by considering only reduced trees.",
"As an example of the reduction the pair $\\langle \\langle \\langle \\top ,\\langle \\top ,\\top \\rangle \\rangle ,\\langle \\top ,\\Delta \\rangle \\rangle ,\\langle \\langle \\Delta ,\\Delta \\rangle ,\\langle \\top ,\\top \\rangle \\rangle \\rangle $ reduces to $\\langle \\langle \\top ,\\langle \\top ,\\Delta \\rangle \\rangle ,\\langle \\Delta ,\\top \\rangle \\rangle $ .",
"The approximation ordering is defined as follows: $\\Delta \\sqsubseteq v$ for all $v\\in {\\bf V}$ $v\\sqsubseteq \\top $ for all $v\\in {\\bf V}$ .",
"$\\langle u,v\\rangle \\sqsubseteq \\langle u^{\\prime },v^{\\prime }\\rangle $ iff $u\\sqsubseteq u^{\\prime }$ and $v\\sqsubseteq v^{\\prime }$ Since the top element is approximated by everything, all finite sets of trees are consistent.",
"The lub for a pair of trees is defined as follows: $u\\sqcup \\top =\\top $ for $u\\in {\\bf V}$ $\\top \\sqcup u=\\top $ for $u\\in {\\bf V}$ $u\\sqcup \\Delta =u$ for $u\\in {\\bf V}$ $\\Delta \\sqcup u=u$ for $u\\in {\\bf V}$ $\\langle u,v\\rangle \\sqcup \\langle u^{\\prime },v^{\\prime }\\rangle =\\langle u\\sqcup u^{\\prime },v\\sqcup v^{\\prime }\\rangle $ for $u,v\\in {\\bf V}$ The proof that this forms a finitary basis follows the same guidelines as the proofs in Section .",
"In addition, it should be clear that the presentation is effective.",
"To form the universal domain, the top element is simply removed.",
"Thus, the system ${\\bf U}={\\bf V}-\\lbrace \\top \\rbrace $ is the basis used to form the universal domain.",
"The proof that this is still a finitary basis with an effective presentation is also straightforward and left to the exercises.",
"Note that inconsistent sets can now exist since there is no top element.",
"A set is inconsistent iff its lub is $\\top $ .",
"We shall now prove the claims made for the universal domain.",
"Theorem 8.8: The domain ${\\cal U}$ is universal, in the sense that for every domain ${\\cal D}$ we have ${\\cal D}\\lhd {\\cal U}$ .",
"If ${\\cal D}$ is effectively given, then the projection pair for the embedding is computable.",
"In fact, there is a correspondence between the effectively presented domains and the computable finitary projections of ${\\cal U}$ .",
"Proof Recall that ${\\bf D}$ must be countable to be a finitary basis.",
"Thus, we can assume that the basis has an enumeration $D=\\lbrace X_n\\:\\vert \\:n\\in {{N}}\\rbrace $ where $X_0=\\Delta $ .",
"The effective and general cases are considered together in the proof; comments about computability are included for the effective case as required.",
"Thus, if ${\\cal D}$ is effectively given, the enumeration above is assumed to be computable.",
"To prove that the domain can be embedded in ${\\cal U}$ , the embedding will be shown.",
"To start, for each finite element $d_i$ in the basis, define two sets, $d_i^+$ and $d_i^-$ as follows: $\\begin{array}{lcl}d_i^+&=&\\lbrace d\\in {\\bf D}\\:\\vert \\:d_i\\sqsubseteq d\\rbrace \\\\d_i^-&=&D-d_i^+\\end{array}$ The $d_i^+$ set contains all the elements that $d_i$ approximates, while the $d_i^-$ set contains all the other elements, partitioning ${\\bf D}$ into two disjoint sets.",
"Sets for different elements can be intersected to form finer partitions of ${\\bf D}$ .",
"For $k>0$ , let $R\\in \\lbrace +,-\\rbrace ^k$ , let $R_i$ be the $ith$ symbol in the string $R$ , and define a region $D_R$ as $D_R=\\bigcap \\limits _{i=1}^k d_i^{R_i}$ where $k$ is the length of $R$ .",
"The set $\\lbrace D_{R}\\:\\vert \\:R\\in \\lbrace +,-\\rbrace ^k\\rbrace $ of regions partitions ${\\bf D}$ into $2^k$ disjoint sets.",
"Thus, for each element $e_i$ in the enumeration there is a corresponding partition of the basis given by the family of sets $\\lbrace D_{R}\\:\\vert \\:R\\in \\lbrace +,-\\rbrace ^i\\rbrace $ .",
"For strings $R,S\\in \\lbrace +,-\\rbrace ^*$ such that $R$ is a prefix of $S$ , denoted $R\\le S$ , $D_S\\subseteq D_R$ .",
"It is important to realize that the composition of these sets is dependent on the order in which the elements are enumerated.",
"Some of these regions are empty, but it is decidable if a given intersection is empty if ${\\cal D}$ is effectively presented.",
"It is also decidable if a given element is in a particular region.",
"Figure: Example Finite DomainTo see the function these regions are serving, consider the finite domain in Figure REF .This example is taken from Cartwright and Demers [2].",
"Consider the enumeration with $d_0=\\bot , d_1=b, d_2=c, d_3=a.$ The $d_i^+$ and $d_i^-$ sets are as follows: $\\begin{array}{lcl}d_1^+&=&\\lbrace a,b\\rbrace \\\\d_1^-&=&\\lbrace c,\\bot \\rbrace \\\\d_2^+&=&\\lbrace c\\rbrace \\\\d_2^-&=&\\lbrace a,b,\\bot \\rbrace \\\\d_3^+&=&\\lbrace a\\rbrace \\\\d_3^-&=&\\lbrace b,c,\\bot \\rbrace \\end{array}$ The regions are as follows: $\\begin{array}{lclclcl}D_+ &=&\\lbrace a,b\\rbrace &\\:\\:\\:\\:&D_{+++} &=&\\lbrace \\rbrace \\\\D_- &=&\\lbrace \\bot ,c\\rbrace &&D_{++-} &=&\\lbrace \\rbrace \\\\D_{++} &=&\\lbrace \\rbrace &&D_{+-+} &=&\\lbrace a\\rbrace \\\\D_{+-} &=&\\lbrace a,b\\rbrace &&D_{+--} &=&\\lbrace b\\rbrace \\\\D_{-+} &=&\\lbrace c\\rbrace &&D_{-++} &=&\\lbrace \\rbrace \\\\D_{--} &=&\\lbrace \\bot \\rbrace &&D_{-+-} &=&\\lbrace c\\rbrace \\\\&&&&D_{--+} &=&\\lbrace \\rbrace \\\\&&&&D_{---} &=&\\lbrace \\bot \\rbrace \\end{array}$ The regions generated by each successive element encode the relationships induced by the approximation ordering between the new element and all elements previously added.",
"The reader is encouraged to try this example with other enumerations of this basis and compare the results.",
"The embedding of the elements proceeds by building a tree based on the regions corresponding to the element.",
"The regions are used to find locations in the tree and to determine whether a $\\top $ or a $\\Delta $ element is placed in the location.",
"These trees preserve the relationships specified by the regions and thus, the tree embedding is isomorphic to the domain in question.",
"Once the tree is built, the reduction rules are applied until a non-reducible tree is reached.",
"This tree is the representative element in the universal domain, and the set of these trees form the sub-space.",
"The function to determine the location in the tree for a given domain, $Loc_D:\\lbrace +,-\\rbrace ^*\\rightarrow \\lbrace l,r\\rbrace ^*$ takes strings used to generate regions and outputs a path in a tree where $l$ stands for left sub-tree and $r$ stands for right sub-tree.",
"This path is computed using the following inductive definition: $\\begin{array}{lcll}Loc_D(\\epsilon )&=&\\epsilon .\\\\Loc_D(R+)&=&Loc_D(R)l&{\\rm if }~D_{R+}\\ne \\emptyset ~ {\\rm and }~D_{R-}\\ne \\emptyset .\\\\&=&Loc_D(R)&{\\rm otherwise}.\\\\Loc_D(R-)&=&Loc_D(R)r&{\\rm if }~D_{R+}\\ne \\emptyset ~{\\rm and }~D_{R-}\\ne \\emptyset .\\\\&=&Loc_D(R)&{\\rm otherwise}.\\end{array}$ The set of locations for each non-empty region is the set of paths to all leaves of some finite binary tree.",
"An induction argument is used to show the following properties of $Loc_D$ that ensure this: If $R\\le S$ for $R,S\\subseteq \\lbrace +,-\\rbrace ^*$ , then $Loc_D(R)\\le Loc_D(S)$ .",
"Let $S=\\lbrace Loc_D(R)\\:\\vert \\:R\\in \\lbrace +,-\\rbrace ^k\\wedge D_R\\ne \\emptyset \\rbrace $ for $k>0$ be a set of location paths for a given $k$ .",
"For any $p\\in \\lbrace l,r\\rbrace ^*$ there exists $q\\in S$ such that either $p\\le q$ or $q\\le p$ .",
"That is, every potential path is represented by some finite path.",
"Finally, for all $p,q\\in S$ if $p\\le q$ then $p=q$ .",
"This means that a unique leaf is associated with each location.",
"To find the tree for a given element $d_k$ in the enumeration, apply the following rules to each $R\\in \\lbrace +,-\\rbrace ^{k-1}$ .",
"If $D_{R-}\\ne \\emptyset $ then the leaf for path $Loc_D(R-)$ is labeled $\\top $ .",
"If $D_{R+}\\ne \\emptyset $ then the leaf for path $Loc_D(R+)$ is labeled $\\Delta $ .",
"These rules are used to assign a tree in ${\\bf U}$ , which is then reduced using the reduction rules, for each element in the enumeration of ${\\bf D}$ .",
"To see that the top element is never assigned by these rules, note that some region of the form $R+$ for every length $k$ must be non-empty since it must contain the element $e_k$ being embedded.",
"Returning to the example, the location function defines paths for these elements as follows: $\\begin{array}{lclclcl}Loc_D(+)&=&l&\\:\\:\\:\\:&Loc_D(+-+)&=&ll\\\\Loc_D(-)&=&r&&Loc_D(+--)&=&lr\\\\Loc_D(+-)&=&l&&Loc_D(-+-)&=&rl\\\\Loc_D(-+)&=&rl&&Loc_D(---)&=&rr\\\\Loc_D(--)&=&rr\\end{array}$ The trees generated for each of the elements are: $\\begin{array}{lcl}d_0&\\mapsto &\\Delta \\\\d_1&\\mapsto &\\langle \\Delta ,\\top \\rangle \\\\d_2&\\mapsto &\\langle \\top ,\\langle \\Delta ,\\top \\rangle \\rangle \\\\d_3&\\mapsto & \\langle \\langle \\Delta ,\\top \\rangle ,\\langle \\top ,\\top \\rangle \\rangle \\\\&\\mapsto & \\langle \\langle \\Delta ,\\top \\rangle ,\\top \\rangle \\end{array}$ To verify that the space generated is a valid sub-space, we must verify that the bottom element is mapped to $\\bot _U$ and that the consistency and lub relations are maintained.",
"The tree $\\Delta $ is clearly assigned to $X_0$ , the bottom element for the basis being embedded, since there are no strings of length $-1$ .",
"The embedding preserves inconsistency of elements by forcing the lub of the embedded elements to be $\\top $ .",
"The $D_{R-}$ regions represent the elements that the element being embedded does not approximate.",
"Note that the $D_{R-}$ sets cause the $\\top $ element to be added as the leaf.",
"Since the $D_R$ sets are built using the approximation ordering, it is straightforward to see that the approximation ordering is preserved by the embedding.",
"Lubs are also maintained by the embedding, although the reduction is required to see that this is the case.",
"It should be clear that, if the domain ${\\cal D}$ is effectively given, the sub-space can be computed since the embedding procedure uses the relationships given in the presentation.",
"Finally, suppose that $a$ is a computable, finitary projection on ${\\cal U}$ .",
"From the proof of Theorem REF , the domain of this projection is characterized by the set $\\lbrace y\\in {\\bf U}\\:\\vert \\:y\\:a\\:y\\rbrace $ If $a$ is computable, the set of pairs for $a$ is recursively enumerable.",
"Thus, the set above is also recursively enumerable since equality among basis elements is decidable.",
"Thus, the domain given by the projection must also be effectively given.",
"$\\:\\:\\Box $ Thus, the domain ${\\cal U}$ is an effectively presented universal domain in which all other domains can be embedded.",
"The sub-domains of ${\\cal U}$ include ${\\cal U}\\rightarrow {\\cal U}$ , ${\\cal U}\\times {\\cal U}$ , etc.",
"These domains must be sub-domains of ${\\cal U}$ since they are effectively presented based on our earlier theorems.",
"Domain Constructors in ${\\cal U}$ The next step is to see how to define the constructors commonly used.",
"Definition 8.9: [Domain Constructors] Let the computable projection pair, $i_+:{\\cal U}+{\\cal U}\\rightarrow {\\cal U}~{\\rm and}~j_+:{\\cal U}\\rightarrow {\\cal U}+{\\cal U}$ be fixed.",
"Fix suitable projection pairs $i_\\times ,j_\\times ,i_\\rightarrow $ , and $j_\\rightarrow $ as well.",
"Define $\\begin{array}{lcl}a+b&=&cond\\circ \\langle which,i_+\\circ in_0\\circ a\\circ out_0, i_+\\circ in_1\\circ b\\circ out_1\\rangle \\circ j_+\\\\a\\times b&=&i_x\\circ \\langle a\\circ proj_0,b\\circ proj_1\\rangle \\circ j_x\\\\a\\rightarrow b&=&i_\\rightarrow \\circ (\\lambda f.b\\circ f\\circ a)\\circ j_\\rightarrow \\end{array}$ for all $a,b:{\\cal U}\\rightarrow {\\cal U}$ .",
"From earlier theorems, we know that these combinators are all computable over an effectively presented domain.",
"The next theorem characterizes the effect these combinators have on projection functions.",
"Theorem 8.10: If $a,b:{\\cal U}\\rightarrow {\\cal U}$ are projections, then so are $a+b$ , $a\\times b$ , and $a\\rightarrow b$ .",
"If $a$ and $b$ are finitary, then so are the compound projections.",
"Proof Since $a$ and $b$ are retractions, $a=a\\circ a$ and $b=b\\circ b$ .",
"Then for $a\\times b$ using the definition of $\\times $ , $\\begin{array}{lcl}(a\\times b)\\circ (a\\times b)&=&i_x\\circ \\langle a\\circ proj_0,b\\circ proj_1\\rangle \\circ \\langle a\\circ proj_0,b\\circ proj_1\\rangle \\circ j_x\\\\&=&i_x\\circ \\langle a\\circ a\\circ proj_0,b\\circ b\\circ proj_1\\rangle \\circ j_x\\\\&=& a\\times b\\end{array}$ Thus, $a\\times b$ is a retraction.",
"The other cases follow similarly.",
"Since $a$ and $b$ are projections, $a,b\\subseteq {\\sf I}_U$ (denoted simply ${\\sf I}$ for the remainder of the proof).",
"Using the definition for $+$ along with the above relation and the definition of projection pairs, we can see that $a+b\\subseteq {\\sf I}+{\\sf I}=i_+\\circ j_+ \\subseteq {\\sf I}$ Thus, $a+b$ is a projection.",
"The other cases follow similarly.",
"To show that the projections are finitary, we must show that the fixed point sets are isomorphic to a domain.",
"Since $a$ and $b$ are assumed finitary, their fixed point sets are isomorphic to $\\begin{array}{lcl}D_a&=&\\lbrace x\\in {\\bf U}\\:\\vert \\:x\\:a\\: x\\rbrace \\\\D_b&=&\\lbrace y\\in {\\bf U}\\:\\vert \\:y\\:b\\: y\\rbrace \\end{array}$ We wish to show that ${\\cal D}_a\\rightarrow {\\cal D}_b\\approx {\\cal D}_{a\\rightarrow b}$ .",
"By the definition of the $\\rightarrow $ constructor, the fixed point set of $a\\rightarrow b$ over ${\\cal U}$ is the same as the fixed point set of $\\lambda f.b\\circ f\\circ a$ on ${\\cal U}\\rightarrow {\\cal U}$ .",
"(Hint: $i_\\rightarrow $ and $j_\\rightarrow $ set up the isomorphism.)",
"So, the fixed points for $f:{\\cal U}\\rightarrow {\\cal U}$ are of the form: $f=b\\circ f\\circ a$ We can think of $a$ as a function in ${\\cal U}\\rightarrow {\\cal D}_a$ and define the other half of the projection pair as $i_a:{\\cal D}_a\\rightarrow {\\cal U}$ where $i_a\\circ a = a$ and $a\\circ i_a=i_a$ .",
"Define a function $i_b$ for the projection pair for $b$ similarly.",
"For some $g:{\\cal D}_a\\rightarrow {\\cal D}_b$ let $f=i_b\\circ g\\circ a$ Substituting this definition for $f$ yields $b\\circ f\\circ a = b\\circ i_b\\circ g\\circ a\\circ a = i_b\\circ g \\circ a = f$ by the definition of $i_b$ and since $a$ is a retraction by assumption.",
"Conversely, for a function $f$ such that $i_b\\circ g\\circ a= f$ , let $g=b\\circ f\\circ i_a$ Substituting again, $i_b\\circ g\\circ a = i_b\\circ g\\circ f\\circ i_a\\circ a = b\\circ f\\circ a = f$ Thus, there is an order preserving isomorphism between $g:{\\cal D}_a\\rightarrow {\\cal D}_b$ and the functions $f=b\\circ f\\circ a$ .",
"The proofs of the isomorphisms for the other constructs are similar.",
"$\\:\\:\\Box $ Thus, the sub-domain relationship with the universal domain has been stated in terms of finitary projections over the universal domain.",
"In addition, all the domain constructors have been shown to be computable combinators on the domain of these finitary projections.",
"Recalling that all computable maps have computable fixed points, the standard fixed point method can be used to solve domain equations of all kinds if they can be defined on projections.",
"Returning to the $\\lambda $ -calculus for a moment, all objects in the $\\lambda $ -calculus are considered functions.",
"Since ${\\cal U}\\rightarrow {\\cal U}$ is a part of ${\\cal U}$ , every object in the $\\lambda $ -calculus is also an object of ${\\cal U}$ .",
"Transposing some of the familiar notation, where the old notation appears on the left, the new combinators are defined as follows: $\\begin{array}{lcl}which(z)=which(j_+(z))\\\\in_i(x)=i_+(in_i(x))~{\\rm where}~i=0,1\\\\out_i(x)=out_i(j_+(x))~{\\rm where}~i=0,1\\\\\\langle x,y\\rangle =i_x(\\langle x,y\\rangle )\\\\proj_i=proj_i(j_x(z))~{\\rm where}~i=0,1\\\\u(x) = j_\\rightarrow (u)(x)\\\\\\lambda x.\\tau =i_\\rightarrow (\\lambda x.\\tau )\\end{array}$ Thus, all functions, all constants, all combinators, and all constructs are elements of ${\\cal U}$ .",
"Indeed, everything computable is an element of ${\\cal U}$ .",
"Elements in ${\\cal U}$ play multiple roles by representing different objects under different projections.",
"While this notion may be difficult to get used to, there are many advantages, both notational and conceptual.",
"Exercises Exercise 8.11: A retraction $a:{\\cal D}\\rightarrow {\\cal D}$ is a closure operator iff ${\\sf I}_D\\subseteq a$ as relations.",
"On a domain like ${\\cal P}({{N}})$ , give some examples of closure operators.",
"(Hint: Close up the integers under addition.",
"Is this continuous on ${\\cal P}({{N}})$ ?)",
"Prove in general that for any closure $a:{\\cal D}\\rightarrow {\\cal D}$ , the fixed point set of $a$ is always a finitary domain.",
"(Hint: Show that the fixed point set is closed as required for a domain.)",
"What are the finite elements of the fixed point set?",
"Exercise 8.12: Give a direct proof that the domain $\\lbrace X\\:\\vert \\:X\\lhd {\\cal D}\\rbrace $ is effectively presented if ${\\cal D}$ is.",
"(Hint: The finite elements of the domain correspond exactly to the finite domains $X\\lhd {\\cal D}$ .)",
"In the case of ${\\cal D}={\\cal U}$ , show that the computable elements of the domain correspond exactly to the effectively presented domains (up to effective isomorphism).",
"Exercise 8.13: For finitary projections $a:{\\cal E}\\rightarrow {\\cal E}$ , write ${\\cal D}_a=\\lbrace x\\in {\\cal E}\\:\\vert \\:x\\:a\\:x\\rbrace $ Show that for any two such projections $a$ and $b$ , that $a\\subseteq b \\iff {\\cal D}_a\\lhd {\\cal D}_b$ Exercise 8.14: Find another universal domain that is not isomorphic to ${\\cal U}$ .",
"Exercise 8.15: Prove the remaining cases in Theorem REF .",
"Exercise 8.16: Suppose $S$ and $T$ are two binary constructors on domains that can be made into computable operators on projections over the universal domain.",
"Show that we can find a pair of effectively presented domains such that $D\\approx S(D,E)~{\\rm and}~E\\approx T(D,E).$ Exercise 8.17: Using the translations shown after the proof of Theorem REF , show how the whole typed-$\\lambda $ -calculus can be translated into ${\\cal U}$ .",
"(Hint: for $f:{\\cal D}_a\\rightarrow {\\cal B}$ , write $f=b\\circ f\\circ a$ for finitary projections $a$ and $b$ .",
"For $\\lambda x^{{\\cal D}_a}.\\sigma $ , write $\\lambda x.b(\\sigma ^{\\prime }[a(x)/x])$ where $\\sigma ^{\\prime }$ is the translation of $\\sigma $ into the untyped $\\lambda $ -calculus.",
"Be sure that the resulting term has the right type.)",
"Exercise 8.18: Show that the basis presented for the universal domain ${\\bf U}$ is indeed a finitary basis and that it has an effective presentation.",
"Exercise 8.19: Work out the embedding for the other enumerations for the example given in the proof of Theorem REF ."
],
[
"Sub-Spaces of the Universal Domain",
"To have a flexible method of solving domain equations and yielding effectively given domains as the solutions, the domains will be embedded in a universal domain which is “big” enough to hold all other domains as sub-domains.",
"This universal domain is shown to be effectively presented, and the mappings which define the sub-spaces are shown to be computable.",
"First, the correspondence between sub-spaces and mappings called retractions is investigated, leading us to the definition of mappings called projections.",
"It is then shown that these definitions can be written out using the $\\lambda $ -calculus notation, demonstrating the power of our mathematical programming language."
],
[
"Retractions and Projections",
"We start with the definition of retractions.",
"Definition 8.1: [Retractions] A retraction of a given domain ${\\cal E}$ is an approximable mapping $a:{\\bf E}\\rightarrow {\\bf E}$ such that $a\\circ a=a$ .",
"Thus, a retraction is the identity function on objects in the range of the retraction and maps other elements into range.",
"The next theorem relates these sets to sub-spaces.",
"Theorem 8.2: If ${\\cal D}\\lhd {\\cal E}$ and if $a:{\\bf E}\\rightarrow {\\bf E}$ is defined such that $X\\:a\\:Z \\iff \\exists Y\\in {\\cal D}.",
"Z\\sqsubseteq Y\\sqsubseteq X$ for all $X,Z\\in {\\bf E}$ , then $a$ is a retraction and ${\\cal D}$ is isomorphic to the fixed point set of $a$ , the set $\\lbrace y\\in {\\cal E}\\:\\vert \\:a(y)=y\\rbrace $ , ordered under inclusion.",
"Proof That $a$ is an approximable map is a direct consequence of the definition of sub-space (Definition REF ).",
"By Theorem REF , a projection pair, $i$ and $j$ , exist for ${\\cal D}$ and this tells us that $a=i\\circ j$ (also showing $a$ approximable since approximable mappings are closed under composition).",
"Theorem REF also tells us that $j\\circ i={\\sf I}_D$ .",
"To show that $a$ is a retraction, $a\\circ a=a$ must be established.",
"Thus, $a\\circ a = i\\circ j\\circ i\\circ j = i\\circ {\\sf I}_D\\circ j = i\\circ j =a$ holds, showing that $a$ is a retraction.",
"We now need to show the isomorphism to ${\\cal D}$ .",
"For $x\\in {\\cal D}$ , $i(x)\\in {\\cal E}$ and we can calculate: $a(i(x))=i\\circ j\\circ i(x) = i\\circ {\\sf I}_D(x) = i(x)$ Thus, $i(x)$ is in the fixed point set of $a$ .",
"For the other direction, let $a(y)=y$ .",
"Then $i(j(y)) = y$ holds.",
"But, $j(y)\\in {\\cal D}$ , so $i$ must map ${\\cal D}$ one-to-one and onto the fixed point set of $a$ .",
"Since $i$ and $j$ are approximable, they are certainly monotonic, and thus the map is an isomorphism with respect to set inclusion.",
"$\\:\\:\\Box $ Not all retractions are associated with a sub-domain relationship.",
"The retractions defined in the above theorem are all subsets as relations of the identity relation.",
"The retractions for sub-domains are characterized by the following definition: Definition 8.3: [Projections] A retraction $a:{\\cal E}\\rightarrow {\\cal E}$ is a projection if $a\\subseteq {\\sf I}_E$ as relations.",
"The retraction is finitary iff its fixed point set is isomorphic to some domain.",
"An example is in order.",
"Example 8.4: Consider a two element system, ${\\bf O}$ with objects $\\Delta $ and 0.",
"For any basis ${\\bf D}$ that is not trivial (has more than one element), ${\\bf O}$ comes from a retraction on ${\\bf D}$ .",
"Define a combinator $check:{\\bf D}\\rightarrow {\\bf O}$ by the relation $x\\:check\\: y \\iff y=\\Delta ~{\\rm or}~x\\ne \\Delta _D$ Thus, $check(x)=\\bot _O\\iff x=\\bot _D$ .",
"Another combinator can be defined, $fade:{\\bf O}\\times {\\bf D}\\rightarrow {\\bf D}$ such that for $t\\in {\\cal O}$ and $x\\in {\\cal D}$ $\\begin{array}{lcll}fade(t,x)&=&\\bot _D&{\\rm if}~t=\\bot _O\\\\&=&x&otherwise\\end{array}$ For $u\\in {\\cal D}$ and $u\\ne \\bot _D$ , the mapping $a$ is defined as $a(x)=fade(check(x),u)$ It can be seen that $a$ is a retraction, but not a projection in general, and the range of $a$ is isomorphic to ${\\bf O}$ .",
"These combinators can also be used to define the subset of functions in ${\\bf D}\\rightarrow {\\bf E}$ that are strict.",
"Define a combinator $strict:({\\bf D}\\rightarrow {\\bf E})\\rightarrow ({\\bf D}\\rightarrow {\\bf E})$ by the equation $strict(f)=\\lambda x.fade(check(x),f(x))$ with $fade$ defined as $fade:{\\bf O}\\times {\\bf E}\\rightarrow {\\bf E}$ .",
"The range of $strict$ is all the strict functions; $strict$ is a projection whose range is a domain.",
"The next theorem characterizes projections.",
"Theorem 8.5: For approximable mapping $a:{\\bf E}\\rightarrow {\\bf E}$ , the following are equivalent: $a$ is a finitary projection $a(x)=\\lbrace y\\in {\\bf E}\\:\\vert \\:\\exists x^{\\prime }\\in I_x.",
"x^{\\prime }\\:a\\:x^{\\prime }\\wedge y\\sqsubseteq x^{\\prime }\\rbrace $ for all $x\\in {\\bf E}$ .",
"Proof Assume that (2) holds.",
"We want to show that $a$ is a finitary projection.",
"By the closure properties on ideals, we know that for all $x\\in {\\cal E}$ , $x^{\\prime }\\in x\\wedge y\\sqsubseteq x^{\\prime }\\:\\Rightarrow \\:y\\in x$ Thus, $a(x)\\subseteq x$ must hold.",
"In addition, the following trivially holds: $x^{\\prime }\\in x\\wedge x^{\\prime }\\:a\\: x^{\\prime }\\:\\Rightarrow \\:x^{\\prime }\\in a(x)$ thus $a(x)\\subseteq a(a(x))$ holds for all $x\\in {\\cal E}$ .",
"This shows that $a$ is indeed a projection.",
"Let $D=\\lbrace x\\in {\\bf E}\\:\\vert \\:x\\:a\\:x\\rbrace $ .",
"It is easy to show that ${\\bf D}\\lhd {\\bf E}$ and that $a$ is determined from ${\\bf D}$ as required in Theorem REF .",
"Thus, the fixed point set of $a$ is isomorphic to a domain from the previous proofs.",
"Thus, (2)$\\:\\Rightarrow \\:$ (1).",
"For the converse, assume that $a$ is a finitary projection.",
"Let ${\\cal D}$ be isomorphic to the fixed point set of $a$ .",
"This means there is a projection pair $i$ and $j$ such that $j\\circ i={\\sf I}_D$ and $i\\circ j = a$ and $a\\subseteq {\\sf I}_E$ .",
"From Theorem REF then we have that ${\\cal D}\\approx {\\cal D}^{\\prime }$ and ${\\cal D}^{\\prime }\\lhd {\\cal E}$ .",
"We want to identify ${\\cal D}^{\\prime }$ as follows: ${\\cal D}^{\\prime }=\\lbrace x\\in {\\cal E}\\:\\vert \\:x\\:a\\:x\\rbrace $ From the proof of Theorem REF , the basis elements of ${\\bf D^{\\prime }}$ are the finite elements of ${\\bf D}$ .",
"Each of these elements is in the fixed point set of $a$ .",
"Thus, $x\\in {\\bf D^{\\prime }}\\:\\Rightarrow \\:a({{\\cal I}}_x) = {{\\cal I}}_x \\:\\Rightarrow \\:x\\:a\\:x$ Since $a$ is a projection, ${{\\cal I}}_x$ must also be a fixed point.",
"Since $i(j({{\\cal I}}_x)) = {{\\cal I}}_x$ implies that $j({{\\cal I}}_x)$ is a finite element of ${\\cal D}$ , $x\\in {\\cal D}^{\\prime }$ must hold.",
"Thus, the identification of ${\\cal D}^{\\prime }$ holds.",
"Finally, using $a=i\\circ j$ in the formula in Theorem REF , the formula in (2) is obtained, proving the converse.",
"$\\:\\:\\Box $ This characterization of projections provides a new and interesting combinator.",
"Theorem 8.6: For any domain ${\\cal E}$ , define $sub:({\\cal E}\\rightarrow {\\cal E})\\rightarrow ({\\cal E}\\rightarrow {\\cal E})$ using the relation $x\\: sub(f)\\: z \\iff \\exists y\\in {\\bf E}.y\\:f\\:y\\wedge y\\sqsubseteq x\\wedge z\\sqsubseteq y$ for all $x,z\\in {\\bf E}$ and all $f:{\\bf E}\\rightarrow {\\bf E}$ .",
"Then the range of $sub$ is exactly the set of finitary projections on ${\\cal E}$ .",
"In addition, $sub$ is a finitary projection on ${\\cal E}\\rightarrow {\\cal E}$ .",
"If ${\\cal E}$ is effectively given, then $sub$ is computable.",
"Proof Clearly, $sub(f)$ is approximable.",
"It is obvious from the definition that $f\\mapsto sub(f)$ preserves lubs and thus is approximable as well.",
"Thus, $y\\:f\\:y\\wedge y\\sqsubseteq x\\wedge z\\sqsubseteq y\\:\\Rightarrow \\:x\\:f\\:z$ obviously holds.",
"Thus, $sub(f)\\subseteq f$ holds.",
"Also $y\\:f\\:y\\:\\Rightarrow \\:y\\:sub(f)\\:y$ thus, $sub(f)\\subseteq sub(sub(f))$ holds as well.",
"Thus, $sub$ is a projection on ${\\cal E}\\rightarrow {\\cal E}$ .",
"The definition of the relation shows that it is computable when ${\\cal E}$ is effectively given.",
"Since $sub$ is a projection, its range is the same as its fixed point set.",
"If $sub(a)=a$ , it is easy to see that clause (2) of Theorem REF holds and conversely.",
"Thus, the range of $sub$ is the finitary projections.",
"To see that $sub$ is a finitary projection, we use Theorem REF and Theorem REF to say that the fixed point set of $sub$ is in a one-to-one inclusion preserving correspondence with the domain $\\lbrace D\\:\\vert \\:D\\lhd {\\cal E}\\rbrace $ .",
"$\\:\\:\\Box $ Universal Domain ${\\cal U}$ With these results and the universal domain to be defined next, the theory of sub-domains is translated into the $\\lambda $ -calculus notation using the $sub$ combinator.",
"The universal domain is defined by first defining a domain which has the desired structure but has a top element.",
"The top element is then removed to give the universal domain.",
"Definition 8.7: [Universal Domain] As in the section on domain equations, an inductive definition for a domain ${\\cal V}$ is given as follows: $\\Delta ,\\top \\in {\\bf V}$ $\\langle u,v\\rangle \\in {\\bf V}$ whenever $u,v\\in {\\bf V}$ Thus, we are starting with two objects, a bottom element and a top element, and making two flavors of copies of these objects.",
"Intuitively, we end up with finite binary trees with either the top or the bottom element as the leaves.",
"To simplify the definitions below, the pairs should be reduced such that: All occurrences of $\\langle \\Delta ,\\Delta \\rangle $ are replaced by $\\Delta $ and All occurrences of $\\langle \\top ,\\top \\rangle $ are replaced by $\\top $ .",
"These rewrite rules are easily shown to be finite Church-Rosser.The finitary basis should be defined as the equivalence classes induced by the reduction.",
"The presentation is simplified by considering only reduced trees.",
"As an example of the reduction the pair $\\langle \\langle \\langle \\top ,\\langle \\top ,\\top \\rangle \\rangle ,\\langle \\top ,\\Delta \\rangle \\rangle ,\\langle \\langle \\Delta ,\\Delta \\rangle ,\\langle \\top ,\\top \\rangle \\rangle \\rangle $ reduces to $\\langle \\langle \\top ,\\langle \\top ,\\Delta \\rangle \\rangle ,\\langle \\Delta ,\\top \\rangle \\rangle $ .",
"The approximation ordering is defined as follows: $\\Delta \\sqsubseteq v$ for all $v\\in {\\bf V}$ $v\\sqsubseteq \\top $ for all $v\\in {\\bf V}$ .",
"$\\langle u,v\\rangle \\sqsubseteq \\langle u^{\\prime },v^{\\prime }\\rangle $ iff $u\\sqsubseteq u^{\\prime }$ and $v\\sqsubseteq v^{\\prime }$ Since the top element is approximated by everything, all finite sets of trees are consistent.",
"The lub for a pair of trees is defined as follows: $u\\sqcup \\top =\\top $ for $u\\in {\\bf V}$ $\\top \\sqcup u=\\top $ for $u\\in {\\bf V}$ $u\\sqcup \\Delta =u$ for $u\\in {\\bf V}$ $\\Delta \\sqcup u=u$ for $u\\in {\\bf V}$ $\\langle u,v\\rangle \\sqcup \\langle u^{\\prime },v^{\\prime }\\rangle =\\langle u\\sqcup u^{\\prime },v\\sqcup v^{\\prime }\\rangle $ for $u,v\\in {\\bf V}$ The proof that this forms a finitary basis follows the same guidelines as the proofs in Section .",
"In addition, it should be clear that the presentation is effective.",
"To form the universal domain, the top element is simply removed.",
"Thus, the system ${\\bf U}={\\bf V}-\\lbrace \\top \\rbrace $ is the basis used to form the universal domain.",
"The proof that this is still a finitary basis with an effective presentation is also straightforward and left to the exercises.",
"Note that inconsistent sets can now exist since there is no top element.",
"A set is inconsistent iff its lub is $\\top $ .",
"We shall now prove the claims made for the universal domain.",
"Theorem 8.8: The domain ${\\cal U}$ is universal, in the sense that for every domain ${\\cal D}$ we have ${\\cal D}\\lhd {\\cal U}$ .",
"If ${\\cal D}$ is effectively given, then the projection pair for the embedding is computable.",
"In fact, there is a correspondence between the effectively presented domains and the computable finitary projections of ${\\cal U}$ .",
"Proof Recall that ${\\bf D}$ must be countable to be a finitary basis.",
"Thus, we can assume that the basis has an enumeration $D=\\lbrace X_n\\:\\vert \\:n\\in {{N}}\\rbrace $ where $X_0=\\Delta $ .",
"The effective and general cases are considered together in the proof; comments about computability are included for the effective case as required.",
"Thus, if ${\\cal D}$ is effectively given, the enumeration above is assumed to be computable.",
"To prove that the domain can be embedded in ${\\cal U}$ , the embedding will be shown.",
"To start, for each finite element $d_i$ in the basis, define two sets, $d_i^+$ and $d_i^-$ as follows: $\\begin{array}{lcl}d_i^+&=&\\lbrace d\\in {\\bf D}\\:\\vert \\:d_i\\sqsubseteq d\\rbrace \\\\d_i^-&=&D-d_i^+\\end{array}$ The $d_i^+$ set contains all the elements that $d_i$ approximates, while the $d_i^-$ set contains all the other elements, partitioning ${\\bf D}$ into two disjoint sets.",
"Sets for different elements can be intersected to form finer partitions of ${\\bf D}$ .",
"For $k>0$ , let $R\\in \\lbrace +,-\\rbrace ^k$ , let $R_i$ be the $ith$ symbol in the string $R$ , and define a region $D_R$ as $D_R=\\bigcap \\limits _{i=1}^k d_i^{R_i}$ where $k$ is the length of $R$ .",
"The set $\\lbrace D_{R}\\:\\vert \\:R\\in \\lbrace +,-\\rbrace ^k\\rbrace $ of regions partitions ${\\bf D}$ into $2^k$ disjoint sets.",
"Thus, for each element $e_i$ in the enumeration there is a corresponding partition of the basis given by the family of sets $\\lbrace D_{R}\\:\\vert \\:R\\in \\lbrace +,-\\rbrace ^i\\rbrace $ .",
"For strings $R,S\\in \\lbrace +,-\\rbrace ^*$ such that $R$ is a prefix of $S$ , denoted $R\\le S$ , $D_S\\subseteq D_R$ .",
"It is important to realize that the composition of these sets is dependent on the order in which the elements are enumerated.",
"Some of these regions are empty, but it is decidable if a given intersection is empty if ${\\cal D}$ is effectively presented.",
"It is also decidable if a given element is in a particular region.",
"Figure: Example Finite DomainTo see the function these regions are serving, consider the finite domain in Figure REF .This example is taken from Cartwright and Demers [2].",
"Consider the enumeration with $d_0=\\bot , d_1=b, d_2=c, d_3=a.$ The $d_i^+$ and $d_i^-$ sets are as follows: $\\begin{array}{lcl}d_1^+&=&\\lbrace a,b\\rbrace \\\\d_1^-&=&\\lbrace c,\\bot \\rbrace \\\\d_2^+&=&\\lbrace c\\rbrace \\\\d_2^-&=&\\lbrace a,b,\\bot \\rbrace \\\\d_3^+&=&\\lbrace a\\rbrace \\\\d_3^-&=&\\lbrace b,c,\\bot \\rbrace \\end{array}$ The regions are as follows: $\\begin{array}{lclclcl}D_+ &=&\\lbrace a,b\\rbrace &\\:\\:\\:\\:&D_{+++} &=&\\lbrace \\rbrace \\\\D_- &=&\\lbrace \\bot ,c\\rbrace &&D_{++-} &=&\\lbrace \\rbrace \\\\D_{++} &=&\\lbrace \\rbrace &&D_{+-+} &=&\\lbrace a\\rbrace \\\\D_{+-} &=&\\lbrace a,b\\rbrace &&D_{+--} &=&\\lbrace b\\rbrace \\\\D_{-+} &=&\\lbrace c\\rbrace &&D_{-++} &=&\\lbrace \\rbrace \\\\D_{--} &=&\\lbrace \\bot \\rbrace &&D_{-+-} &=&\\lbrace c\\rbrace \\\\&&&&D_{--+} &=&\\lbrace \\rbrace \\\\&&&&D_{---} &=&\\lbrace \\bot \\rbrace \\end{array}$ The regions generated by each successive element encode the relationships induced by the approximation ordering between the new element and all elements previously added.",
"The reader is encouraged to try this example with other enumerations of this basis and compare the results.",
"The embedding of the elements proceeds by building a tree based on the regions corresponding to the element.",
"The regions are used to find locations in the tree and to determine whether a $\\top $ or a $\\Delta $ element is placed in the location.",
"These trees preserve the relationships specified by the regions and thus, the tree embedding is isomorphic to the domain in question.",
"Once the tree is built, the reduction rules are applied until a non-reducible tree is reached.",
"This tree is the representative element in the universal domain, and the set of these trees form the sub-space.",
"The function to determine the location in the tree for a given domain, $Loc_D:\\lbrace +,-\\rbrace ^*\\rightarrow \\lbrace l,r\\rbrace ^*$ takes strings used to generate regions and outputs a path in a tree where $l$ stands for left sub-tree and $r$ stands for right sub-tree.",
"This path is computed using the following inductive definition: $\\begin{array}{lcll}Loc_D(\\epsilon )&=&\\epsilon .\\\\Loc_D(R+)&=&Loc_D(R)l&{\\rm if }~D_{R+}\\ne \\emptyset ~ {\\rm and }~D_{R-}\\ne \\emptyset .\\\\&=&Loc_D(R)&{\\rm otherwise}.\\\\Loc_D(R-)&=&Loc_D(R)r&{\\rm if }~D_{R+}\\ne \\emptyset ~{\\rm and }~D_{R-}\\ne \\emptyset .\\\\&=&Loc_D(R)&{\\rm otherwise}.\\end{array}$ The set of locations for each non-empty region is the set of paths to all leaves of some finite binary tree.",
"An induction argument is used to show the following properties of $Loc_D$ that ensure this: If $R\\le S$ for $R,S\\subseteq \\lbrace +,-\\rbrace ^*$ , then $Loc_D(R)\\le Loc_D(S)$ .",
"Let $S=\\lbrace Loc_D(R)\\:\\vert \\:R\\in \\lbrace +,-\\rbrace ^k\\wedge D_R\\ne \\emptyset \\rbrace $ for $k>0$ be a set of location paths for a given $k$ .",
"For any $p\\in \\lbrace l,r\\rbrace ^*$ there exists $q\\in S$ such that either $p\\le q$ or $q\\le p$ .",
"That is, every potential path is represented by some finite path.",
"Finally, for all $p,q\\in S$ if $p\\le q$ then $p=q$ .",
"This means that a unique leaf is associated with each location.",
"To find the tree for a given element $d_k$ in the enumeration, apply the following rules to each $R\\in \\lbrace +,-\\rbrace ^{k-1}$ .",
"If $D_{R-}\\ne \\emptyset $ then the leaf for path $Loc_D(R-)$ is labeled $\\top $ .",
"If $D_{R+}\\ne \\emptyset $ then the leaf for path $Loc_D(R+)$ is labeled $\\Delta $ .",
"These rules are used to assign a tree in ${\\bf U}$ , which is then reduced using the reduction rules, for each element in the enumeration of ${\\bf D}$ .",
"To see that the top element is never assigned by these rules, note that some region of the form $R+$ for every length $k$ must be non-empty since it must contain the element $e_k$ being embedded.",
"Returning to the example, the location function defines paths for these elements as follows: $\\begin{array}{lclclcl}Loc_D(+)&=&l&\\:\\:\\:\\:&Loc_D(+-+)&=&ll\\\\Loc_D(-)&=&r&&Loc_D(+--)&=&lr\\\\Loc_D(+-)&=&l&&Loc_D(-+-)&=&rl\\\\Loc_D(-+)&=&rl&&Loc_D(---)&=&rr\\\\Loc_D(--)&=&rr\\end{array}$ The trees generated for each of the elements are: $\\begin{array}{lcl}d_0&\\mapsto &\\Delta \\\\d_1&\\mapsto &\\langle \\Delta ,\\top \\rangle \\\\d_2&\\mapsto &\\langle \\top ,\\langle \\Delta ,\\top \\rangle \\rangle \\\\d_3&\\mapsto & \\langle \\langle \\Delta ,\\top \\rangle ,\\langle \\top ,\\top \\rangle \\rangle \\\\&\\mapsto & \\langle \\langle \\Delta ,\\top \\rangle ,\\top \\rangle \\end{array}$ To verify that the space generated is a valid sub-space, we must verify that the bottom element is mapped to $\\bot _U$ and that the consistency and lub relations are maintained.",
"The tree $\\Delta $ is clearly assigned to $X_0$ , the bottom element for the basis being embedded, since there are no strings of length $-1$ .",
"The embedding preserves inconsistency of elements by forcing the lub of the embedded elements to be $\\top $ .",
"The $D_{R-}$ regions represent the elements that the element being embedded does not approximate.",
"Note that the $D_{R-}$ sets cause the $\\top $ element to be added as the leaf.",
"Since the $D_R$ sets are built using the approximation ordering, it is straightforward to see that the approximation ordering is preserved by the embedding.",
"Lubs are also maintained by the embedding, although the reduction is required to see that this is the case.",
"It should be clear that, if the domain ${\\cal D}$ is effectively given, the sub-space can be computed since the embedding procedure uses the relationships given in the presentation.",
"Finally, suppose that $a$ is a computable, finitary projection on ${\\cal U}$ .",
"From the proof of Theorem REF , the domain of this projection is characterized by the set $\\lbrace y\\in {\\bf U}\\:\\vert \\:y\\:a\\:y\\rbrace $ If $a$ is computable, the set of pairs for $a$ is recursively enumerable.",
"Thus, the set above is also recursively enumerable since equality among basis elements is decidable.",
"Thus, the domain given by the projection must also be effectively given.",
"$\\:\\:\\Box $ Thus, the domain ${\\cal U}$ is an effectively presented universal domain in which all other domains can be embedded.",
"The sub-domains of ${\\cal U}$ include ${\\cal U}\\rightarrow {\\cal U}$ , ${\\cal U}\\times {\\cal U}$ , etc.",
"These domains must be sub-domains of ${\\cal U}$ since they are effectively presented based on our earlier theorems.",
"Domain Constructors in ${\\cal U}$ The next step is to see how to define the constructors commonly used.",
"Definition 8.9: [Domain Constructors] Let the computable projection pair, $i_+:{\\cal U}+{\\cal U}\\rightarrow {\\cal U}~{\\rm and}~j_+:{\\cal U}\\rightarrow {\\cal U}+{\\cal U}$ be fixed.",
"Fix suitable projection pairs $i_\\times ,j_\\times ,i_\\rightarrow $ , and $j_\\rightarrow $ as well.",
"Define $\\begin{array}{lcl}a+b&=&cond\\circ \\langle which,i_+\\circ in_0\\circ a\\circ out_0, i_+\\circ in_1\\circ b\\circ out_1\\rangle \\circ j_+\\\\a\\times b&=&i_x\\circ \\langle a\\circ proj_0,b\\circ proj_1\\rangle \\circ j_x\\\\a\\rightarrow b&=&i_\\rightarrow \\circ (\\lambda f.b\\circ f\\circ a)\\circ j_\\rightarrow \\end{array}$ for all $a,b:{\\cal U}\\rightarrow {\\cal U}$ .",
"From earlier theorems, we know that these combinators are all computable over an effectively presented domain.",
"The next theorem characterizes the effect these combinators have on projection functions.",
"Theorem 8.10: If $a,b:{\\cal U}\\rightarrow {\\cal U}$ are projections, then so are $a+b$ , $a\\times b$ , and $a\\rightarrow b$ .",
"If $a$ and $b$ are finitary, then so are the compound projections.",
"Proof Since $a$ and $b$ are retractions, $a=a\\circ a$ and $b=b\\circ b$ .",
"Then for $a\\times b$ using the definition of $\\times $ , $\\begin{array}{lcl}(a\\times b)\\circ (a\\times b)&=&i_x\\circ \\langle a\\circ proj_0,b\\circ proj_1\\rangle \\circ \\langle a\\circ proj_0,b\\circ proj_1\\rangle \\circ j_x\\\\&=&i_x\\circ \\langle a\\circ a\\circ proj_0,b\\circ b\\circ proj_1\\rangle \\circ j_x\\\\&=& a\\times b\\end{array}$ Thus, $a\\times b$ is a retraction.",
"The other cases follow similarly.",
"Since $a$ and $b$ are projections, $a,b\\subseteq {\\sf I}_U$ (denoted simply ${\\sf I}$ for the remainder of the proof).",
"Using the definition for $+$ along with the above relation and the definition of projection pairs, we can see that $a+b\\subseteq {\\sf I}+{\\sf I}=i_+\\circ j_+ \\subseteq {\\sf I}$ Thus, $a+b$ is a projection.",
"The other cases follow similarly.",
"To show that the projections are finitary, we must show that the fixed point sets are isomorphic to a domain.",
"Since $a$ and $b$ are assumed finitary, their fixed point sets are isomorphic to $\\begin{array}{lcl}D_a&=&\\lbrace x\\in {\\bf U}\\:\\vert \\:x\\:a\\: x\\rbrace \\\\D_b&=&\\lbrace y\\in {\\bf U}\\:\\vert \\:y\\:b\\: y\\rbrace \\end{array}$ We wish to show that ${\\cal D}_a\\rightarrow {\\cal D}_b\\approx {\\cal D}_{a\\rightarrow b}$ .",
"By the definition of the $\\rightarrow $ constructor, the fixed point set of $a\\rightarrow b$ over ${\\cal U}$ is the same as the fixed point set of $\\lambda f.b\\circ f\\circ a$ on ${\\cal U}\\rightarrow {\\cal U}$ .",
"(Hint: $i_\\rightarrow $ and $j_\\rightarrow $ set up the isomorphism.)",
"So, the fixed points for $f:{\\cal U}\\rightarrow {\\cal U}$ are of the form: $f=b\\circ f\\circ a$ We can think of $a$ as a function in ${\\cal U}\\rightarrow {\\cal D}_a$ and define the other half of the projection pair as $i_a:{\\cal D}_a\\rightarrow {\\cal U}$ where $i_a\\circ a = a$ and $a\\circ i_a=i_a$ .",
"Define a function $i_b$ for the projection pair for $b$ similarly.",
"For some $g:{\\cal D}_a\\rightarrow {\\cal D}_b$ let $f=i_b\\circ g\\circ a$ Substituting this definition for $f$ yields $b\\circ f\\circ a = b\\circ i_b\\circ g\\circ a\\circ a = i_b\\circ g \\circ a = f$ by the definition of $i_b$ and since $a$ is a retraction by assumption.",
"Conversely, for a function $f$ such that $i_b\\circ g\\circ a= f$ , let $g=b\\circ f\\circ i_a$ Substituting again, $i_b\\circ g\\circ a = i_b\\circ g\\circ f\\circ i_a\\circ a = b\\circ f\\circ a = f$ Thus, there is an order preserving isomorphism between $g:{\\cal D}_a\\rightarrow {\\cal D}_b$ and the functions $f=b\\circ f\\circ a$ .",
"The proofs of the isomorphisms for the other constructs are similar.",
"$\\:\\:\\Box $ Thus, the sub-domain relationship with the universal domain has been stated in terms of finitary projections over the universal domain.",
"In addition, all the domain constructors have been shown to be computable combinators on the domain of these finitary projections.",
"Recalling that all computable maps have computable fixed points, the standard fixed point method can be used to solve domain equations of all kinds if they can be defined on projections.",
"Returning to the $\\lambda $ -calculus for a moment, all objects in the $\\lambda $ -calculus are considered functions.",
"Since ${\\cal U}\\rightarrow {\\cal U}$ is a part of ${\\cal U}$ , every object in the $\\lambda $ -calculus is also an object of ${\\cal U}$ .",
"Transposing some of the familiar notation, where the old notation appears on the left, the new combinators are defined as follows: $\\begin{array}{lcl}which(z)=which(j_+(z))\\\\in_i(x)=i_+(in_i(x))~{\\rm where}~i=0,1\\\\out_i(x)=out_i(j_+(x))~{\\rm where}~i=0,1\\\\\\langle x,y\\rangle =i_x(\\langle x,y\\rangle )\\\\proj_i=proj_i(j_x(z))~{\\rm where}~i=0,1\\\\u(x) = j_\\rightarrow (u)(x)\\\\\\lambda x.\\tau =i_\\rightarrow (\\lambda x.\\tau )\\end{array}$ Thus, all functions, all constants, all combinators, and all constructs are elements of ${\\cal U}$ .",
"Indeed, everything computable is an element of ${\\cal U}$ .",
"Elements in ${\\cal U}$ play multiple roles by representing different objects under different projections.",
"While this notion may be difficult to get used to, there are many advantages, both notational and conceptual.",
"Exercises Exercise 8.11: A retraction $a:{\\cal D}\\rightarrow {\\cal D}$ is a closure operator iff ${\\sf I}_D\\subseteq a$ as relations.",
"On a domain like ${\\cal P}({{N}})$ , give some examples of closure operators.",
"(Hint: Close up the integers under addition.",
"Is this continuous on ${\\cal P}({{N}})$ ?)",
"Prove in general that for any closure $a:{\\cal D}\\rightarrow {\\cal D}$ , the fixed point set of $a$ is always a finitary domain.",
"(Hint: Show that the fixed point set is closed as required for a domain.)",
"What are the finite elements of the fixed point set?",
"Exercise 8.12: Give a direct proof that the domain $\\lbrace X\\:\\vert \\:X\\lhd {\\cal D}\\rbrace $ is effectively presented if ${\\cal D}$ is.",
"(Hint: The finite elements of the domain correspond exactly to the finite domains $X\\lhd {\\cal D}$ .)",
"In the case of ${\\cal D}={\\cal U}$ , show that the computable elements of the domain correspond exactly to the effectively presented domains (up to effective isomorphism).",
"Exercise 8.13: For finitary projections $a:{\\cal E}\\rightarrow {\\cal E}$ , write ${\\cal D}_a=\\lbrace x\\in {\\cal E}\\:\\vert \\:x\\:a\\:x\\rbrace $ Show that for any two such projections $a$ and $b$ , that $a\\subseteq b \\iff {\\cal D}_a\\lhd {\\cal D}_b$ Exercise 8.14: Find another universal domain that is not isomorphic to ${\\cal U}$ .",
"Exercise 8.15: Prove the remaining cases in Theorem REF .",
"Exercise 8.16: Suppose $S$ and $T$ are two binary constructors on domains that can be made into computable operators on projections over the universal domain.",
"Show that we can find a pair of effectively presented domains such that $D\\approx S(D,E)~{\\rm and}~E\\approx T(D,E).$ Exercise 8.17: Using the translations shown after the proof of Theorem REF , show how the whole typed-$\\lambda $ -calculus can be translated into ${\\cal U}$ .",
"(Hint: for $f:{\\cal D}_a\\rightarrow {\\cal B}$ , write $f=b\\circ f\\circ a$ for finitary projections $a$ and $b$ .",
"For $\\lambda x^{{\\cal D}_a}.\\sigma $ , write $\\lambda x.b(\\sigma ^{\\prime }[a(x)/x])$ where $\\sigma ^{\\prime }$ is the translation of $\\sigma $ into the untyped $\\lambda $ -calculus.",
"Be sure that the resulting term has the right type.)",
"Exercise 8.18: Show that the basis presented for the universal domain ${\\bf U}$ is indeed a finitary basis and that it has an effective presentation.",
"Exercise 8.19: Work out the embedding for the other enumerations for the example given in the proof of Theorem REF ."
]
] | 1605.05858 |
[
[
"Automatic Selection of the Optimal Local Feature Detector"
],
[
"Abstract A large number of different feature detectors has been proposed so far.",
"Any existing approach presents strengths and weaknesses, which make a detector optimal only for a limited range of applications.",
"A tool capable of selecting the optimal feature detector in relation to the operating conditions is presented in this paper.",
"The input images are quickly analyzed to determine what type of image transformation is applied to them and at which amount.",
"Finally, the detector that is expected to obtain the highest repeatability under such conditions, is chosen to extract features from the input images.",
"The efficiency and the good accuracy in determining the optimal feature detector for any operating condition, make the proposed tool suitable to be utilized in real visual applications.",
"%A large number of different feature detectors has been proposed so far.",
"Any existing approach presents strengths and weaknesses, which make a detector optimal only for a limited range of applications.",
"A large number of different local feature detectors have been proposed in the last few years.",
"However, each feature detector has its own strengths ad weaknesses that limit its use to a specific range of applications.",
"In this paper is presented a tool capable of quickly analysing input images to determine which type and amount of transformation is applied to them and then selecting the optimal feature detector, which is expected to perform the best.",
"The results show that the performance and the fast execution time render the proposed tool suitable for real-world vision applications."
],
[
"Introduction",
"Local feature detection is an important and challenging task in most vision applications.",
"A large number of different approaches have been proposed so far [9].",
"All these techniques present various strengths and weaknesses, which make detectors' performance dependent on the application and, more generally, on the operating conditions, such as the transformation type and amount [8] [4].",
"To overcome this problem, an obvious solution is to run multiple feature detectors so that the shortcomings of one detector are countered by the strengths of the other detectors.",
"However, the computational demand of such an approach can be high and increases with the number of detectors employed.",
"An alternative solution consists of a tool capable of automatically selecting the optimal feature detector to cope with any operating conditions as suggested in [9].",
"To the best of our knowledge, such idea has received none or little attention so far [5].",
"This paper aims to bridge this gap by proposing a tool which can determine the transformation type ($T$ ) and amount ($A$ ) of input images and then select the detector that is expected to perform the best under those particular operating conditions.",
"The proposed approach requires to have a prior knowledge of how feature detectors perform under any of the considered operating conditions $(T, A)$ .",
"So, in order to design an effective selection stage (Fig.",
"REF ), the evaluation framework proposed in [7] is utilized to characterize the performance of a set of feature detectors under varying transformation types and amounts.",
"This performance characterization, as well as the results presented in this paper, are obtained with the image database available at [3].",
"This image database includes 539 scenes, which has been used for generating the datasets for three transformations, namely light reduction, JPEG compression and Gaussian blur.",
"Each dataset has a reference image and several target images, which are obtained by the application of the same transformation to a reference image with increasing amounts.",
"Considering that the JPEG and light reduction datasets include 13 target images and a blur dataset has 9 target images, the resulting number of operating conditions available in the image database [3] is 18865.",
"The rest of this paper is organized as follows.",
"The proposed selection tool is introduced in Section while its performance is discussed in Section .",
"Finally, Section , draws conclusions and discusses the future directions for the automatic selection of the optimal feature detector."
],
[
"The Automatic Selection Tool",
"The proposed system consists of four stages (Fig.",
"REF ).",
"The first stage extracts global features from the input images, then the second and the third stages determine the type ($T$ ) and the amount ($A$ ) of transformation respectively.",
"The last one selects the optimal detector, which is expected to obtain the highest repeatability.",
"The following subsections describe those four stages of the proposed system and provide more details about the selection criterion of the optimal feature detector."
],
[
"Global Feature Extraction",
"The first stage analyses the input pair of target and reference images and then builds a vector of three features: $F = [f_L, f_B, f_L]$ .",
"The component $f_L$ is the light reduction feature and is computed as the ratio between the mean of the image histogram of the target and the reference images: $f_L = h_t/h_r$ .",
"Hence, lower values of $f_L$ correspond to higher amount of light reduction.",
"The blur amount of an image is estimated with the perceptual blur metric proposed in [1].",
"The Gaussian Blur feature, $f_B$ , is computed as the ratio of the perceptual blur indices of the target and reference images respectively: $f_B = b_t/b_r$ .",
"A high value of $f_B$ corresponds to a relatively high level of blurring in the target image.",
"The JPEG feature $f_J$ is computed with the reference-less quality metric proposed in [10], which produces a quality index of an image by combining the blockiness and the zero-crossing rate of the image differential signal along vertical and horizontal lines.",
"Higher the compression rate of a JPEG image, lower is the value of $f_J$ ."
],
[
"Transformation Type Detection Stage",
"The transformation ($T$ ) is determined with a Support Vector Machine (SVM) classifier with a linear kernel function.",
"The SVM has been trained utilizing a portion of the datasets [3] of 339 scenes chosen randomly.",
"The related datasets for light changes, JPEG compression and Gaussian blur are employed to train the classifier.",
"This results in a training set of 11865 feature vectors (13 x 339 for JPEG compression and light reduction, and 9 x 339 for blurring).",
"The overall accuracy of the prediction is above 99%.",
"Almost all the classification errors occur between blurred and JPEG compressed images at the lowest amounts of transformation (10-20% of JPEG compression rate and 0.5-1.0$\\sigma $ for Gaussian blur)."
],
[
"Transformation Amount Detection Stage",
"The third stage is composed of a set of SVMs, each specifically trained to predict the amount $A$ of a single transformation type.",
"So, once $T$ is determined the corresponding SVM is activated to determine the transformation amount from the feature vector $F$ .",
"The overall accuracy for light reduction is close to 100% while the percentage of transformation amounts correctly classified by the JPEG and blur SVMs are just 75% and 73% respectively.",
"However, the results presented in Section , show the relatively low accuracy of the JPEG and blur classifiers do not significantly affect the overall performance of the automatic selection system."
],
[
"Selection of the Optimal Feature Detector",
"This stage is implemented as a set of rules, which associate each pair $(T, A)$ with the optimal feature detector $D$ to operate under such type and amount of transformation.",
"The evaluation framework from [7] is utilized to characterize the set of feature detectors available at runtime for selection.",
"Such characterization is carried out following the process described in [7] utilizing the training set (Section REF ) of 339 datasets per transformation.",
"First, the improved repeatability rate [2] is computed for each feature detector using the authors' original programs and the parameters values suggested by them.",
"The average of the repeatability rates is computed across all the scene images that are undergone to the same type and amount of transformation.",
"For example, the average repeatability of a detector at 20% of JPEG compression is obtained as the mean of the repeatability scored with the 339 JPEG images compressed at 20%.",
"Utilizing the outcomes of the performance characterization, the optimal feature detector for any operating condition is identified utilizing the highest average repeatability as a criterion.",
"The resulting set of associations, $(T,A) \\rightarrow D$ , is utilized by the proposed tool at runtime to select of the most suitable feature detector for any given input target image."
],
[
"Results and Discussion",
"This section presents the results of the comparison between the selection algorithm and several feature detectors working individually under varying uniform light reduction, Gaussian blur and JPEG compression.",
"The evaluation criteria are the accuracy, which is measured by means of the gap between the average repeatability of the best detector and the optimal detector selected by the tool, and the execution time.",
"The employed set of feature detectors represents a variety of different approaches [9] and includes the following: Edge-Based Region (EBR), Maximally Stable External Region (MSER), Intensity-Based Region (IBR), Salient Regions (SALIENT) Scale-invariant Feature Operator (SFOP), Speeded Up Robust Features (SURF).",
"The scenes utilized for the tests are the remaining 200 scenes at [3], which are not included in the training set (REF ).",
"Thus, 200 datasets each for light reduction, JPEG compression and blurring transformations have been utilized as a test set.",
"As it is done for characterization of detectors' performance, the repeatability data are obtained using the original authors' programs and with the recommended control parameter values suggested by them.",
"Fig.",
"REF shows a comparison of the average repeatability of the feature detectors working individually and the selection algorithm (red dotted line).",
"Under JPEG compression, the accuracy of the selection is very high with a negligible gap error.",
"Indeed, SURF performs the best under any transformation amount (Fig.",
"REF .c), so the accuracy of the selection depends only on the prediction of the transformation type, which is correct in more then 99% of the cases.",
"The automatic selection tool performs well also with light reduction as it can be appreciated from Fig.",
"REF .a where the red dotted line matches perfectly the SFOP's average curve up to 85% and the SALIENT's curve at 90% of light reduction (Fig.",
"REF .a).",
"To the contrary, under Guassian blur some selection errors occurs as shown in Fig.",
"REF .d, where the gap between the average repeatability of the best detector and the one chosen as optimal by the selection tools is plotted.",
"Between 1.5$\\sigma $ and 2.0$\\sigma $ (Fig.",
"REF .d) there is a dip of -1%.",
"In that range of blurring intensity the average curves of SURF and IBR intersect each other (Fig.",
"REF .b) and the wrong predictions of the transformation amount ($A$ ) causes some errors in the selection of the optimal feature detector.",
"Although, the probability that such classification error occurs is around 9%, the resulting gap error is just -1%.",
"This is due to the little difference between the average repeatability values of SURF and IBR, which are close to each other at 1.5$\\sigma $ (58.54% vs 55.78%) and at 2.0$\\sigma $ (43.6% vs 48.8%).",
"A complete run of the proposed tool, from image loading to the detector selection, requires a time comparable to the fastest of the feature detectors considered: MSER.",
"The hardware employed for the test is a laptop equipped with a i7-4710MQ CPU, 16Gb of RAM, and a SATA III SSD Hard drive and the test images have a resolution of 1080 x 717 pixels.",
"MSER and IBR are available as binary executables and have a running time of 150ms and 1.8 seconds respectively while the selection tool, which is a Matlab script, requires 170ms to load images and select a detector.",
"Hence, a system employing the proposed tool with those two feature detectors can extract features in 170 + 150 ms (when MSER is optimal) or 170 ms + 1.8 seconds (when IBR is optimal) while running both MSER and IBR with an image and select the best, would require always more than 1.9 seconds.",
"Thus, the proposed system is equally or more efficient than running more feature detectors with the same image, in addition, it scales really well with the number of feature detectors employed."
],
[
"Conclusions and Future Directions",
"The automatic tool for selecting the optimal feature detector proposed in this paper represents an attempt to achieve a fully adaptive feature detector system capable of coping with any operating condition.",
"The proposed approach is based on the knowledge of the behaviour of detectors under different operating conditions, which are the transformation type $T$ and the amount of such transformation, $A$ .",
"The next step towards a more robust automatic selection system is to consider the scene content as a part of the operating conditions as it is well known that a detector's performance depends also on that factor ([8], [6]).",
"Thus, modeling the scene content and designing a comprehensive evaluation framework that utilizes it together with the image transformation type and amount should, in our humble opinion, the direction to follow in order to achieve a robust selection tool."
]
] | 1605.06094 |
[
[
"Statistical solutions of hyperbolic conservation laws I: Foundations"
],
[
"Abstract We seek to define statistical solutions of hyperbolic systems of conservation laws as time-parametrized probability measures on $p$-integrable functions.",
"To do so, we prove the equivalence between probability measures on $L^p$ spaces and infinite families of \\textit{correlation measures}.",
"Each member of this family, termed a \\textit{correlation marginal}, is a Young measure on a finite-dimensional tensor product domain and provides information about multi-point correlations of the underlying integrable functions.",
"We also prove that any probability measure on a $L^p$ space is uniquely determined by certain moments (correlation functions) of the equivalent correlation measure.",
"We utilize this equivalence to define statistical solutions of multi-dimensional conservation laws in terms of an infinite set of equations, each evolving a moment of the correlation marginal.",
"These evolution equations can be interpreted as augmenting entropy measure-valued solutions, with additional information about the evolution of all possible multi-point correlation functions.",
"Our concept of statistical solutions can accommodate uncertain initial data as well as possibly non-atomic solutions even for atomic initial data.",
"For multi-dimensional scalar conservation laws we impose additional entropy conditions and prove that the resulting \\textit{entropy statistical solutions} exist, are unique and are stable with respect to the $1$-Wasserstein metric on probability measures on $L^1$."
],
[
"Introduction",
"Systems of conservation laws are nonlinear partial differential equations of the generic form $\\partial _t u + \\nabla _x\\cdot f(u) = 0$ $u(x,0) = \\bar{u}(x).$ Here, the unknown $u = u(x,t) : \\mathbb {R}^d\\times \\mathbb {R}_+ \\rightarrow \\mathbb {R}^N$ is the vector of conserved variables and $f = (f^1, \\dots , f^d) : \\mathbb {R}^N \\rightarrow \\mathbb {R}^{N\\times d}$ is the flux function.",
"We denote $\\mathbb {R}_+ := [0,\\infty )$ .",
"The system is termed hyperbolic if the flux Jacobian matrix has real eigenvalues [15].",
"Here and in the remainder, quantities with a bar (like $\\bar{u}$ ) denote prescribed initial data.",
"Hyperbolic systems of conservation laws arise in a wide variety of models in physics and engineering.",
"Prototypical examples include the compressible Euler equations of gas dynamics, the shallow water equations of oceanography, the magneto-hydrodynamics (MHD) equations of plasma physics and the equations of nonlinear elasticity [15].",
"It is well known that solutions of () can form discontinuities such as shock waves, even for smooth initial data $\\bar{u}$ .",
"Hence, solutions of systems of conservation laws () are sought in the sense of distributions.",
"These weak solutions are not necessarily unique.",
"They need to be augmented with additional admissibility criteria, often termed entropy conditions, to single out the physically relevant solution.",
"Entropy solutions are widely regarded as the appropriate solution paradigm for systems of conservation laws [15].",
"Global well-posedness (existence, uniqueness and continuous dependence on initial data) of entropy solutions of scalar conservation laws ($N = 1$ in ()), was established in the pioneering work of Kruzkhov [38].",
"For one-dimensional systems ($d=1$ , $N>1$ in ()), global existence, under the assumption of small initial total variation, was shown by Glimm in [32] and by Bianchini and Bressan in [6].",
"Uniqueness and stability of entropy solutions for one-dimensional systems has also been shown; see [8] and references therein.",
"Although existence results have been obtained for some very specific examples of multi-dimensional systems (see [4] and references therein), there are no global existence results for any generic class of multi-dimensional systems.",
"In fact, De Lellis, Székelyhidi et al.",
"have recently been able to construct infinitely many entropy solutions for prototypical multi-dimensional systems such as the Euler equations for polytropic gas dynamics (see [16], [17] and references therein).",
"Their construction involves a novel iterative procedure where oscillations at smaller and smaller scales are successively added to suitably constructed sub-solutions of ().",
"Given the lack of global existence and uniqueness results for entropy solutions of multi-dimensional systems of conservation laws, it is natural to seek alternative solution paradigms.",
"One option, advocated for instance in [3], is to augment entropy solutions with further admissibility criteria, such as the vanishing viscosity limit, in order to rule out “unphysical” solutions.",
"However, given the difficulties of obtaining existence results for the weaker concept of entropy solutions, it is unclear if such a narrowing of the solution concept would lead to any meaningful global existence results.",
"The other alternative is to extend the solution concept beyond entropy solutions (integrable functions) and seek possibly even weaker notions of solutions of (), together with suitable admissibility criteria to constrain these solutions and enforce uniqueness.",
"A recent paper [23] advocates such an approach.",
"Based on the extensive numerical simulations reported in [23] (see also [39]), the authors observe that approximate solutions of () can feature oscillations at smaller and smaller scales as mesh is refined.",
"Given this fact, they postulate that entropy measure-valued solutions may serve as an appropriate solution paradigm for systems of conservation laws in several space dimensions, particularly in characterizing limits of (numerical) approximations.",
"Measure-valued solutions, originally proposed by DiPerna in [20] (see also [21]), are space-time-parametrized probability measures, or Young measures, defined on the phase space $\\mathbb {R}^N$ of ().",
"In defining entropy measure-valued solutions, one requires consistency of certain functionals of this Young measure with the initial data, with the weak (distributional) form of (), and with a suitable (dissipative) form of the entropy conditions (see also [18]).",
"In recent papers [23], [24] (see also [33]), the authors were able to prove (global in time) existence of entropy measure-valued solutions for a very large class of systems of conservation laws, namely those endowed with a strictly convex entropy function, by showing convergence of numerical approximations of () based on a Monte Carlo algorithm.",
"Numerical experiments presented in these papers suggest that the measure-valued solution may be non-atomic, even when the initial data is atomic, i.e.",
"a Dirac Young measure concentrated on an integrable function.",
"The computed measure-valued solutions were observed to be stable with respect to the choice of numerical method and with respect to perturbations of initial data.",
"However, one can readily construct counter-examples to uniqueness of these entropy measure-valued solutions.",
"In particular, if the initial data is non-atomic then infinitely many entropy measure-valued solutions can be constructed, even for scalar conservation laws (see [41], [23]).",
"This lack of uniqueness, even for the scalar case, can be attributed to the fact that only certain functionals of the measure-valued solution (essentially the mean and the second moment) are required to be consistent with the initial data, the evolution equation () and the entropy conditions.",
"Since the the mean and the second moment uniquely specifies a measure only when the measure is atomic, one cannot expect uniqueness for generic (non-atomic) measure-valued solutions as considered in [23].",
"On the other hand, numerical experiments presented in [23] clearly suggest that one has to deal with non-atomic, “uncertain” measure-valued solutions of multi-dimensional systems of conservation laws, even when the initial data is atomic.",
"In a wide variety of applications, even the initial data can be non-atomic, carrying some uncertainty due to e.g.",
"measurement errors.",
"These measurements are inherently uncertain and can only be specified probabilistically, and this uncertainty inevitably propagates into the solution.",
"The modeling, analysis and numerical approximation of uncertain solutions, given uncertain inputs (such as the initial data), falls under the rubric of uncertainty quantification; see [7] and reference therein for an extensive discussion of the very large body of recent research activity on uncertainty quantification for systems of conservation laws.",
"Thus, in general, one has to deal with the possibility that physically relevant measure-valued solutions are non-atomic.",
"Given these considerations, we seek to find a solution framework that can deal with non-atomic measure-valued solutions of multi-dimensional systems of conservation laws, and can provide further constraints on these measure-valued solutions in order to enforce uniqueness and stability of the resulting solution concept.",
"A natural choice for such a solution framework is the notion of statistical solutions that was first proposed by Foiaş in [27], [28] (see also [29]) in the context of the incompressible Navier–Stokes equations of fluid dynamics.",
"As envisaged by Foiaş and co-workers, statistical solutions of the Navier–Stokes equations are time-parametrized probability measures on a given infinite-dimensional function space (divergence-free $L^2$ functions in the context of the Navier–Stokes equations).",
"This family of measures has to satisfy either a suitable infinite-dimensional Liouville equation that governs the time evolution of a class of functionals in a manner consistent with the Navier–Stokes dynamics, or equivalently, satisfy a Hopf equation, where the time-evolution of the characteristic functional of the probability measure (on $L^2$ ) is prescribed.",
"Both formulations result in evolution equations in infinite-dimensional spaces.",
"A detailed account of statistical solutions in the sense of Foiaş, and their relation to the description of turbulent incompressible flows, can be found in [29] and references therein.",
"However, it is far from straightforward to adapt the notion of statistical solutions to the context of systems of conservation laws.",
"There seems to be at least three main difficulties in this regard.",
"First, statistical solutions as defined in [27], [28], [29] are well suited to problems with viscosity, as they require some regularity of the underlying functions in order to define the infinite-dimensional Liouville or Hopf equations.",
"It is unclear how to extend them to inviscid problems such as systems of conservation laws where solutions are generally discontinuous.",
"Attempts to do so have been made in [9], [10], [5] (see also [35], [40]) for the special case of the one-dimensional inviscid Burgers equation.",
"The corresponding statistical solutions are probability measures on the space of distributions, and the infinite-dimensional Hopf equation is well-defined by using compactly supported infinitely differentiable test functions.",
"Although existence results for such statistical solutions of the inviscid Burgers equation have been obtained in the class of Levy processes with negative jumps, it is not possible to obtain uniqueness of these statistical solutions, even for the inviscid Burgers equation, in the class of probability measures on spaces as large as the space of distributions.",
"The second difficulty with statistical solutions in the sense of Foiaş, lies in the fact that the Liouville or Hopf equations are evolution equations on infinite-dimensional function spaces.",
"This makes the interpretation and computation of statistical solutions very hard for viscous problems, and the solution concept is not easily amenable to extension to inviscid PDEs such as systems of conservation laws.",
"Furthermore, probability measures on function spaces preclude a local (in space) description of the resulting solution, as it is unclear how to interpret statistical information at specific points (or collection of points) in space.",
"Finally, given our original motivation in constraining measure-valued solutions to recover uniqueness in the non-atomic case, the relationship between statistical solutions and measure-valued solutions is far from clear.",
"The only known results are presented in [11], [12] where a sequence of statistical solutions of the incompressible Navier–Stokes equations is shown to converge to a measure-valued solution of the incompressible Euler equations, as defined in [21], when the viscosity vanishes.",
"However, we are interested in investigating the more abstract question of the relationship between probability measures on function spaces (statistical solutions), and Young measures that represent one-point statistics (measure-valued solutions), with the aim of imposing further constraints on measure-valued solutions to enforce uniqueness.",
"With this background, the first aim of the current paper is to provide a novel representation of a probability measure on an infinite-dimensional function space (to be specific, $L^p$ space) in terms of an infinite hierarchy of Young measures called a correlation measure, defined on tensor products of the (finite-dimensional) spatial domain.",
"Each member of this hierarchy of measures, termed a correlation marginal, represents correlations (joint probabilities) in the values of the underlying functions at any finite collection of points.",
"Hence, this representation allows us to interpret probability measures on infinite-dimensional spaces as containing information about correlations across all possible finite collection of points in the spatial domain.",
"Consequently, we can “localize” any infinite-dimensional probability measure.",
"In particular, the first correlation marginal of this equivalent representation coincides with the classical notion of a Young measure.",
"Thus, a probability measure on an $L^p$ space augments a Young measure with multi-point correlations and provides significantly more information than the Young measure does.",
"We believe that this novel equivalence result could be of independent interest in stochastic analysis; see e.g.",
"[14].",
"Another consequence of the equivalence of probability measures on function spaces and hierarchies of finite-dimensional correlation marginals, is the fact that the probability measure can be uniquely determined by a family of moments of the corresponding correlation marginals.",
"Hence, the infinite-dimensional Liouville or Hopf equation for statistical solutions, as proposed in [29], can be replaced by an equivalent family of evolution equations (for moments) on finite-dimensional (tensor-product) domains.",
"The second aim of this paper is to utilize this novel representation to define a suitable notion of statistical solutions for systems of conservation laws ().",
"In particular, certain moments (correlation functions) of the (time-parametrized) correlation marginals are evolved in a manner consistent with the dynamics of the conservation law ().",
"Consequently, statistical solutions need to satisfy an infinite family of evolutionary PDEs, but each of these PDEs is defined on a finite-dimensional spatial domain.",
"The final aim of this paper is to study the well-posedness of the proposed notion of statistical solutions.",
"We will do so in the specific context of scalar conservation laws where we show existence of statistical solutions for a very large class of initial probability measures.",
"The harder issue of uniqueness of statistical solutions for scalar conservation laws is also addressed.",
"To this end, we propose a novel admissibility criterion that amounts to requiring stability of each admissible statistical solution with respect to a specific set of stationary statistical solutions, namely those probability measures supported on finite collections of constant functions.",
"Furthermore, we also show stability of the admissible statistical solution in the Wasserstein metric, with respect to probability measure-valued initial data: $W_1(\\mu _t,\\rho _t) \\leqslant W_1(\\bar{\\mu },\\bar{\\rho })$ .",
"Thus, a complete characterization — existence, uniqueness and stability — of statistical solutions for scalar conservation laws is provided.",
"The issues of existence and stability of admissible statistical solutions for the general case of systems of conservation laws will be presented in forthcoming papers in this series.",
"The rest of the paper is organized as follows.",
"In Section we prove the equivalence between probability measures on $L^p$ spaces and hierarchies of Young measures on finite-dimensional spaces.",
"Statistical solutions for systems of conservation laws are defined in Section and the well-posedness of statistical solutions for scalar conservation laws is presented in Section ."
],
[
"Probability measures on function spaces",
"The aim of this section is to establish the equivalence between probability measures on a function space, and families of measures describing the correlation of the values of underlying functions at different spatial points.",
"The function spaces that we have in mind are $L^p(D,U)$ for $1\\leqslant p <\\infty $ for some domain $D\\subset \\mathbb {R}^d$ and $U:=\\mathbb {R}^N$ (we will think of $D$ as physical space and $U$ as phase space).",
"For ease of notation we will denote ${F}:= L^p(D,U).$ Henceforth, we equip ${F}$ with its Borel $\\sigma $ -algebra ${B}({F})$ .",
"A short summary of the contents this section follows.",
"Given a probability measure $\\mu $ on ${F}= L^p(D,U)$ , we might be interested in local quantities such as the mean or the variance at a fixed point $x\\in D$ : $\\text{mean at }x = \\int _{F}u(x)\\,d\\mu (u), \\qquad \\text{variance at }x = \\int _{F}\\bigl (u(x) - \\text{mean})^2\\,d\\mu (u),$ or we might be interested in joint probability distributions at points $x, y\\in D$ : $\\text{probability that $u(x)\\in A$ \\textit {and} $u(y)\\in B$} = \\int _{F}\\mathbb {1}_A(u(x))\\mathbb {1}_B(u(y))\\,d\\mu (u).$ However, not only are the integrands in the above integrals non-measurable, they are ill-defined because point values $u(x)$ of a measurable function $u$ is not well-defined.",
"Thus, we would like an equivalent representation of $\\mu $ in terms of locally defined probability distributions $\\nu ^1_x$ or $\\nu ^2_{x,y}$ ; the above quantities could then be written as $\\int _U \\xi \\,d\\nu ^1_x(\\xi ), \\qquad \\int _U \\Bigl (\\xi -{\\textstyle \\int _U \\xi ^{\\prime }\\ d\\nu ^1_x(\\xi ^{\\prime })}\\Bigr )^2\\,d\\nu ^1_x(\\xi ), \\qquad \\int _{U^2}\\mathbb {1}_A(\\xi )\\mathbb {1}_B(\\zeta )\\,d\\nu ^2_{x,y}(\\xi ,\\zeta )= \\nu ^2_{x,y}(A\\times B),$ respectively.",
"As we will see, we will require all joint distributions across finitely many points in order to determine $\\mu $ uniquely.",
"This gives rise to an infinite hierarchy $\\nu = (\\nu ^1,\\nu ^2,\\dots )$ of maps $\\nu ^k$ from $D^k$ into ${P}(U^k)$ , the set of probability measures on $U^k$ .",
"Such a hierarchy is termed a correlation measure and each map $\\nu ^k$ a correlation marginal.",
"The complete definition of correlation measures is given in Section REF .",
"A similar construction is found in the Kolmogorov Extension Theorem (see e.g.",
"[44]).",
"However, this approach considers measures on the product space $U^D$ equipped with the cylinder $\\sigma $ -algebra, instead of measures on $L^p(D,U)$ equipped with its Borel $\\sigma $ -algebra.",
"In the former case, questions such as “is $u$ continuous” or “is $u$ Lebesgue integrable” are not measurable, thus disqualifying its use in our context."
],
[
"Preliminaries",
"We begin by recalling several definitions and results in functional analysis, measure theory and optimal transport theory.",
"If $\\xi ,\\zeta \\in U$ then $\\xi \\cdot \\zeta $ denotes their Euclidean inner product.",
"If $D$ is a Borel set then $D^k := \\underbrace{D\\times \\dots \\times D}_{k~{\\rm times}}$ and if $x=(x_1,\\dots ,x_k)\\in D^k$ then we denote $|x| = |x_1|+\\dots +|x_k|$ .",
"We denote the dual space of ${F}$ by ${F}^* = L^{p^{\\prime }}(D,U)$ (where $\\frac{1}{p}+\\frac{1}{p^{\\prime }}=1$ ), and the duality pairing between $\\varphi \\in {F}^*$ and $u\\in {F}$ by $\\bigl \\langle \\varphi ,\\, u\\bigr \\rangle = \\varphi (u) = \\int _D \\varphi (x)\\cdot u(x)\\,dx.$ For any normed space $X$ , we let $C_b(X)$ denote the space of bounded, continuous, real-valued functionals on $X$ , equipped with the supremum norm $\\Vert f\\Vert _{C_b(X)} = \\sup _{x\\in X}|f(x)|$ .",
"We let $C_c(X)$ be the set of $f\\in C_b(X)$ that have compact support, and we let $C_0(X)$ be the completion of $C_c(X)$ in the supremum norm.",
"The $k$ -dimensional Lebesgue measure of a Borel set $A\\subset \\mathbb {R}^k$ is denoted $|A|$ .",
"The average of a function $f$ over a set $A$ is denoted $\\mathop {}\\hspace{0.0pt}\\mathchoice{\\sbox 0{\\displaystyle \\int \\m@th }\\sbox 2{\\displaystyle \\int _{}\\m@th }\\sbox 4{\\displaystyle \\int _{\\hspace{0.00002pt}}^{\\hspace{0.00002pt}}\\m@th }\\sbox 0{\\scriptstyle A\\m@th }\\sbox 2{\\scriptstyle \\m@th }\\unknown.",
"{\\displaystyle \\hbox{t}o{\\hss \\textstyle {-}\\m@th \\hss }}}{}{}{}$ $\\textstyle \\hbox{t}o{\\hss \\scriptstyle {-}\\m@th \\hss }$$$$\\scriptstyle \\hbox{t}o{\\hss \\scriptscriptstyle {-}\\m@th \\hss }$$$$\\scriptscriptstyle \\hbox{t}o{\\hss \\scriptscriptstyle {-}\\m@th \\hss }$$$Af(x) dx = 1|A|A f(x) dx.",
"$$ The Borel $\\sigma $ -algebra on a Polish space $X$ (i.e., a complete, separable metric space) is denoted by ${B}(X)$ .",
"We let ${M}(X)$ denote the space of finite, signed Radon measures on $(X,\\, {B}(X))$ , and for $\\mu \\in {M}(X)$ and $f\\in L^1(X;\\mu )$ we write $\\bigl \\langle \\mu ,\\, f\\bigr \\rangle = \\int _X f(x)\\,d\\mu (x)$ .",
"The set ${P}(X)$ of probability measures on $X$ consist of those $\\mu \\in {M}(X)$ satisfying $\\mu \\geqslant 0$ and $\\mu (X)=1$ ."
],
[
"The Wasserstein distance",
"Let $X$ be a separable Banach space and let $\\mu , \\rho \\in {P}(X)$ have finite $p$ th moments, i.e.",
"$\\int _X |x|^p d\\mu (x) < \\infty $ and $\\int _X |x|^p d\\rho (x) < \\infty $ .",
"The $p$ -Wasserstein distance between $\\mu $ and $\\rho $ is defined as $W_p(\\mu ,\\rho ) = \\inf _{\\pi \\in \\Pi (\\mu ,\\rho )}\\int _{X^2} |x-y|^p\\,d\\pi (x,y);$ where the infimum is taken over the set $\\Pi (\\mu ,\\rho )\\subset {P}(X^2)$ of all transport plans from $\\mu $ to $\\rho $ , i.e.",
"those $\\pi \\in {P}(X^2)$ satisfying $\\int _{X^2} F(x)+G(y)\\,d\\pi (x,y) = \\int _X F(x)\\,d\\mu (x) + \\int _X G(y)\\,d\\rho (y) \\qquad \\forall \\ F,G\\in C_b(X)$ (see e.g.",
"[43]).",
"When $p=1$ we can write $W_1(\\mu ,\\rho ) = \\sup _{\\begin{array}{c}\\Psi \\in C_b(X) \\\\ \\Vert \\Psi \\Vert _{\\operatorname{Lip}}\\leqslant 1\\end{array}} \\int _X \\Psi (x)\\,d(\\mu -\\rho )(x),$ where the supremum is taken over all Lipschitz continuous functions with Lipschitz constant at most 1.",
"It is straightforward to show that there always exists an optimal transport plan $\\pi $ , i.e, one for which the infimum in (REF ) is attained [43].",
"The fact that (REF ) and (REF ) coincide when $p=1$ is a theorem in optimal transport theory often called the Kantorovich–Rubinstein theorem [43].",
"The Wasserstein distance is a complete metric on the set of probability measures with finite $p$ th moment, and metrizes the topology of weak convergence on this set [1]."
],
[
"Cylinder sets and -functions",
"Let $X$ be a normed vector space.",
"A function $\\Psi : X \\rightarrow \\mathbb {R}$ is a cylinder function if there exist functionals $\\varphi _1,\\dots ,\\varphi _n \\in X^*$ and a Borel measurable function $\\psi : \\mathbb {R}^n \\rightarrow \\mathbb {R}$ such that $\\Psi (u) = \\psi \\bigl (\\varphi _1(u), \\dots , \\varphi _n(u)\\bigr ) \\qquad \\forall \\ u\\in X.$ A set $E\\subset X$ is a cylinder set if the indicator function $u \\mapsto \\mathbb {1}_E(u)$ is a cylinder function, or equivalently, if $E$ is of the form $E = \\bigl \\lbrace u\\in X\\ :\\ \\bigl (\\varphi _1(u),\\dots ,\\varphi _n(u)\\bigr ) \\in F\\bigr \\rbrace $ for a Borel set $F\\subset \\mathbb {R}^n$ and $\\varphi _1,\\dots ,\\varphi _n\\in X^*$ .",
"We let $\\mathit {Cyl}(X)$ denote the collection of cylinder sets in $X$ .",
"Proposition 2.1 Let $X$ be a separable normed vector space.",
"Then: The $\\sigma $ -algebra generated by $\\mathit {Cyl}(X)$ is equal to ${B}(X)$ .",
"If $\\mu $ is a (signed) measure on $(X,{B}(X))$ such that $\\mu (A) = 0$ for all cylinder sets $A$ , then $\\mu \\equiv 0$ .",
"See the appendix."
],
[
"Correlation measures",
"A correlation measure is a collection $\\nu = (\\nu ^1, \\nu ^2, \\dots )$ of maps $\\nu ^k : D^k \\rightarrow {P}(U^k)$ satisfying the following properties: Weak* measurability: Each map $\\nu ^k : D^k \\rightarrow {P}(U^k)$ is weak*-measurable, in the sense that the map $x \\mapsto \\bigl \\langle \\nu ^k_x,\\, f\\bigr \\rangle $ from $x\\in D^k$ into $\\mathbb {R}$ is Borel measurable for all $f\\in C_0(U^k)$ and $k\\in \\mathbb {N}$ .",
"In other words, $\\nu ^k$ is a Young measure from $D^k$ to $U^k$ .",
"$L^p$ -boundedness: $\\nu $ is $L^p$ -bounded, in the sense that $\\int _{D} \\bigl \\langle \\nu ^1_x,\\, |\\xi |^p\\bigr \\rangle \\,dx < +\\infty .$ Symmetry: If $\\sigma $ is a permutation of $\\lbrace 1,\\dots ,k\\rbrace $ and $f\\in C_0(\\mathbb {R}^k)$ then $\\bigl \\langle \\nu ^k_{\\sigma (x)},\\, f(\\sigma (\\xi ))\\bigr \\rangle = \\bigl \\langle \\nu ^k_x,\\, f(\\xi )\\bigr \\rangle $ for a.e.",
"$x\\in D^k$ .",
"Here, we denote $\\sigma (x) = \\sigma (x_1,x_2,\\ldots , x_k) = (x_{\\sigma _1},x_{\\sigma _2},\\ldots ,x_{\\sigma _k})$ .",
"$\\sigma (\\xi )$ is denoted analogously.",
"Consistency: If $f\\in C_0(U^k)$ is of the form $f(\\xi _1,\\dots ,\\xi _k) = g(\\xi _1,\\dots ,\\xi _{k-1})$ for some $g\\in C_0(U^{k-1})$ , then $\\bigl \\langle \\nu ^k_{x_1,\\dots ,x_k},\\, f\\bigr \\rangle = \\bigl \\langle \\nu ^{k-1}_{x_1,\\dots ,x_{k-1}},\\, g\\bigr \\rangle $ for almost every $(x_1,\\dots ,x_k)\\in D^k$ .",
"Diagonal continuity (DC): If $B_r(x) := \\bigl \\lbrace y\\in D\\ :\\ |x-y|<r\\bigr \\rbrace $ then $\\lim _{r\\rightarrow 0}\\int _D\\mathop {}\\hspace{0.0pt}\\mathchoice{\\sbox 0{\\displaystyle \\int \\m@th }\\sbox 2{\\displaystyle \\int _{}\\m@th }\\sbox 4{\\displaystyle \\int _{\\hspace{0.00002pt}}^{\\hspace{0.00002pt}}\\m@th }\\sbox 0{\\scriptstyle B_r(x)\\m@th }\\sbox 2{\\scriptstyle \\m@th }\\unknown.",
"{\\displaystyle \\hbox{t}o{\\hss \\textstyle {-}\\m@th \\hss }}}{}{}{}$ $\\textstyle \\hbox{t}o{\\hss \\scriptstyle {-}\\m@th \\hss }$$$$\\scriptstyle \\hbox{t}o{\\hss \\scriptscriptstyle {-}\\m@th \\hss }$$$$\\scriptscriptstyle \\hbox{t}o{\\hss \\scriptscriptstyle {-}\\m@th \\hss }$$$Br(x)2x,y, |1-2|p dy dx = 0.",
"Each element $\\nu ^k$ is called a correlation marginal.",
"We let ${L}^{p} = {L}^{p}(D,U)$ denote the set of all correlation measures from $D$ to $U$ .",
"By combining the properties of symmetry and consistency, the expected value with respect to $\\nu ^k_x$ of a function depending on $l<k$ parameters $\\xi _{i_1}, \\dots , \\xi _{i_l}$ , can be written in terms of $\\nu ^l_{x_{i_1}, \\dots , x_{i_l}}$ .",
"Thus, the $k$ th correlation marginal $\\nu ^k$ contains all information about lower-order correlation marginals, but not vice-versa.",
"Hence, the family $\\nu = (\\nu ^k)_{k\\in \\mathbb {N}}$ constitutes a hierarchy.",
"Any function $u\\in L^p(D,U)$ gives rise to a correlation marginal $\\nu \\in {L}^p(D,U)$ by defining $\\nu ^k_x = \\delta _{u(x_1)}\\otimes \\cdots \\otimes \\delta _{u(x_k)}$ .",
"Correlation marginals of this form are called atomic.",
"It can be shown that the DC property is equivalent to $\\lim _{r\\rightarrow 0}\\int _D\\mathop {}\\hspace{0.0pt}\\mathchoice{\\sbox 0{\\displaystyle \\int \\m@th }\\sbox 2{\\displaystyle \\int _{}\\m@th }\\sbox 4{\\displaystyle \\int _{\\hspace{0.00002pt}}^{\\hspace{0.00002pt}}\\m@th }\\sbox 0{\\scriptstyle B_r(x)\\m@th }\\sbox 2{\\scriptstyle \\m@th }\\unknown.",
"{\\displaystyle \\hbox{t}o{\\hss \\textstyle {-}\\m@th \\hss }}}{}{}{}$ $\\textstyle \\hbox{t}o{\\hss \\scriptstyle {-}\\m@th \\hss }$$$$\\scriptstyle \\hbox{t}o{\\hss \\scriptscriptstyle {-}\\m@th \\hss }$$$$\\scriptscriptstyle \\hbox{t}o{\\hss \\scriptscriptstyle {-}\\m@th \\hss }$$$Br(x)2x,y, g(x,y,1,2) dy dx = D 1x,x, g(x,x,1,1) dx $for every $ H2$.",
"After possibly redefining $ 2$ on the zero-measure set $ {(x,y)D2 : x=y}$, this is equivalent to$$\\nu ^2_{x,x} = \\nu ^1_x \\qquad \\text{for a.e.\\ }x\\in D.$$In particular, $ 2x,x, 12 = 1x, 2$ -- i.e., the covariance between the value at the point $ x$ with itself is just the variance at $ x$.",
"Similarly, it can be shown that if $ Cb(Uk+1)$ is Lipschitz continuous then$$\\bigl \\langle \\nu ^{k+1}_{x_1,\\dots ,x_k,x_k},\\, \\psi (\\xi _1,\\dots ,\\xi _{k+1})\\bigr \\rangle = \\bigl \\langle \\nu ^{k}_{x_1,\\dots ,x_k},\\, \\psi (\\xi _1,\\dots ,\\xi _k,\\xi _{k})\\bigr \\rangle .$$$ We emphasize that diagonal continuity is an additional consistency requirement which is independent from consistency condition (iv) of Definition REF .",
"As an example of a “correlation measure” which is not diagonally continuous, let $\\nu ^1 : D \\rightarrow {P}(U)$ be any Young measure satisfying (REF ), and define $\\nu ^k_{x_1,\\dots ,x_k} := \\nu ^1_{x_1}\\otimes \\cdots \\otimes \\nu ^1_{x_k}$ for every $k\\in \\mathbb {N}$ .",
"Then $\\nu = (\\nu ^1, \\nu ^2, \\dots )$ satisfies properties (i)–(iv) of Definition REF , but is DC if and only if $\\nu ^1$ is atomic.",
"Indeed, by Jensen's inequality, $\\bigl \\langle \\nu ^2_{x,x},\\, \\xi _1\\xi _2\\bigr \\rangle = \\bigl \\langle \\nu ^1_x\\otimes \\nu ^1_x,\\, \\xi _1\\xi _2\\bigr \\rangle = \\bigl \\langle \\nu ^1_{x},\\, \\xi \\bigr \\rangle ^2 \\leqslant \\bigl \\langle \\nu ^1_{x},\\, \\xi ^2\\bigr \\rangle $ for a.e.",
"$x\\in D$ , with equality if and only if $\\nu ^1$ is atomic."
],
[
"The main theorem",
"Denote ${H}^k := L^1\\bigl (D^k,C_0(U^k)\\bigr )$ .",
"The proof of the following theorem, which is the main theorem of Section , will depend crucially on ${H}^k$ and its dual space; see Section REF .",
"Main Theorem 2.2 For every correlation measure $\\nu \\in {L}^p(D,U)$ there exists a unique probability measure $\\mu \\in {P}({F})$ satisfying $\\int _{F}\\Vert u\\Vert _{F}^{p}\\,d\\mu (u) < \\infty $ such that $\\int _{D^k}\\int _{U^k}g(x,\\xi )\\,d\\nu ^k_x(\\xi )dx = \\int _{F}\\int _{D^k}g(x,u(x))\\,dxd\\mu (u) \\qquad \\forall \\ g\\in {H}^k, \\quad \\forall \\ k\\in \\mathbb {N}$ (where $u(x)$ denotes the vector $(u(x_1), \\dots , u(x_k))$ ).",
"Conversely, for every probability measure $\\mu \\in {P}({F})$ with finite moment (REF ), there exists a unique correlation measure $\\nu \\in {L}^p(D,U)$ satisfying (REF ).",
"The relation (REF ) is also valid for any measurable $g:D\\times U\\rightarrow \\mathbb {R}$ such that $|g(x,\\xi )|\\leqslant C|\\xi |^p$ for a.e.",
"$x\\in D$ .",
"For a $g\\in {H}^k$ , define the functional $L_g : {F}\\rightarrow \\mathbb {R}$ by $L_g(u) := \\int _{D^k}g(x,u(x))\\,dx.$ Denoting $\\bigl \\langle \\nu ^k,\\, g\\bigr \\rangle := \\int _{D^k}\\int _{U^k}g(x,\\xi )\\,d\\nu ^k_x(\\xi )dx$ , we can write (REF ) as $\\bigl \\langle \\nu ^k,\\, g\\bigr \\rangle = \\bigl \\langle \\mu ,\\, L_g\\bigr \\rangle \\qquad \\forall \\ g\\in {H}^k, \\quad \\forall \\ k\\in \\mathbb {N}.\\qquad \\mathrm {(\\ref {eq:nukdef}^{\\prime })}$ To ensure that the terms appearing in (REF ) (or equivalently (REF )) are well-defined, we need to check that $\\nu ^k$ is a continuous linear functional on ${H}^k$ , and that $L_g : {F}\\rightarrow \\mathbb {R}$ is Borel measurable for every $g\\in {H}^k$ .",
"This is done in Theorem REF and Proposition REF , respectively.",
"The finite moment requirement (REF ) is the direct analogue of the $L^p$ bound (REF ).",
"Indeed, $\\int _{F}\\Vert u\\Vert _{F}^{p}\\,d\\mu (u) = \\int _{F}\\int _{D} |u(x)|^p\\,dxd\\mu (u) = \\int _{D}\\int _{U} |\\xi |^p\\,d\\nu ^1_x(\\xi ) dx.$"
],
[
"The spaces ${H}^k$ and {{formula:51f8f6ca-0531-4fdb-9818-3c8a5fae8feb}}",
"For any $k\\in \\mathbb {N}$ , denote ${H}^k := L^1\\bigl (D^k,C_0(U^k)\\bigr )$ , the space of measurable functions $g : x \\mapsto g(x)\\in C_0(U^k)$ such that $\\Vert g\\Vert _{{H}^k} = \\int _{D^k}\\bigl \\Vert g(x)\\bigr \\Vert _{C_0(U^k)}\\,dx < \\infty .$ (Here, $C_0(U^k)$ is equipped with its Borel $\\sigma $ -algebra.)",
"We will routinely write $g(x,\\xi )$ instead of $g(x)(\\xi )$ .",
"We let ${H}^{k*}:=L^\\infty _w(D^k,{M}(U^k))$ denote the space of weak* measurable maps $\\nu ^k : x \\mapsto \\nu ^k_x\\in {M}(U^k)$ such that $\\Vert \\nu ^k\\Vert _{{H}^{k*}} = \\operatornamewithlimits{ess\\ sup}_{x\\in D^k} \\Vert \\nu ^k_x\\Vert _{{M}(U^k)} < \\infty .$ (Recall that $\\nu ^k$ is weak* measurable if the map $x \\mapsto \\bigl \\langle \\nu ^k_x,\\, f\\bigr \\rangle $ from $D^k$ to $\\mathbb {R}$ is measurable for all $f\\in C_0(U^k)$ .)",
"Note that if $\\nu =(\\nu ^1,\\nu ^2,\\dots )$ is a correlation measure then each correlation marginal $\\nu ^k$ is an element of ${H}^{k*}$ , because $\\Vert \\nu ^k\\Vert _{{H}^{k*}} = 1$ .",
"The following result justifies the notation ${H}^{k*}$ .",
"Theorem 2.3 For any $k\\in \\mathbb {N}$ , the space ${H}^{k*}$ is isometrically isomorphic to the dual of ${H}^k$ through the pairing $\\bigl \\langle \\nu ^k,\\, g\\bigr \\rangle = \\int _{D^k} \\bigl \\langle \\nu ^k_x,\\, g(x,\\cdot )\\bigr \\rangle \\,dx, \\qquad g\\in {H}^k, \\ \\nu ^k\\in {H}^{k*}.$ See e.g.",
"[22] or [2].",
"Proposition 2.4 For any $g\\in {H}^k$ , the map $L_g:{F}\\rightarrow \\mathbb {R}$ defined by (REF ) is uniformly continuous and satisfies $\\Vert L_g\\Vert _{C_b({F})} \\leqslant \\Vert g\\Vert _{{H}^k}.$ Since $g\\in {H}^k=L^1(D^k,C_0(U^k))$ , there are simple functions $\\bar{g}_n(x) = \\sum _{i=1}^{n}\\mathbb {1}_{A_{n,i}}(x)\\bar{f}_{n,i}$ for functions $\\bar{f}_{n,i}\\in C_0(U^k)$ and sets $A_{n,i}\\subset D^k$ with positive and bounded Lebesgue measure, such that $\\bar{g}_n \\rightarrow g$ in ${H}^k$ .",
"Let $f_{n,i}$ be functions in $C_0(U^k)\\cap \\operatorname{Lip}(U^k)$ such that $\\Vert \\bar{f}_{n,i}-f_{n,i}\\Vert _{C_0(U^k)} \\leqslant \\frac{1}{|A_{n,i}|n^2}$ (constructed, for instance, by mollification of $\\bar{f}_{n,i}$ ), and define $g_n(x) := \\sum _{i=1}^{n}\\mathbb {1}_{A_{n,i}}(x)f_{n,i}$ .",
"If $u,v\\in {F}$ then $\\bigl |L_{g_n}(u)-L_{g_n}(v)\\bigr | &\\leqslant \\sum _{i=1}^{n}\\int _{A_{n,i}}\\bigl |f_{n,i}(u(x))-f_{n,i}(v(x))\\bigr |\\,dx \\\\&\\leqslant \\sum _{i=1}^{n}\\int _{A_{n,i}}\\Vert f_{n,i}\\Vert _{\\operatorname{Lip}(U^k)}\\bigl (|u(x_1)-v(x_1)|+ \\dots + |u(x_k)-v(x_k)|\\bigr )\\,dx \\\\&\\leqslant C_n\\Vert u-v\\Vert _{F}$ by Hölder's inequality, where $C_n>0$ depends on $|A_{n,i}|$ and $\\Vert f_{n,i}\\Vert _{\\operatorname{Lip}(U^k)}$ for $i=1,\\dots ,n$ .",
"Thus, $L_{g_n}$ is Lipschitz continuous.",
"Moreover, $|L_g(u)-L_{g_n}(u)| &\\leqslant \\int _{D^k}|g(x,u(x))-g_n(x,u(x))|\\,dx \\leqslant \\int _{D^k} \\Vert (g-g_n)(x)\\Vert _{C_0(U^k)}\\,dx \\\\&= \\Vert g-g_n\\Vert _{{H}^k} \\leqslant \\Vert g-\\bar{g}_n\\Vert _{{H}^k} + \\frac{1}{n} \\rightarrow 0 \\qquad \\text{as }n\\rightarrow \\infty ,$ and so $L_{g_n} \\rightarrow L_g$ uniformly on ${F}$ .",
"Since every uniform limit of Lipschitz continuous functions is uniformly continuous, we conclude that $L_g$ is uniformly continuous.",
"Finally, $|L_g(u)| \\leqslant \\int _{D^k}|g(x,u(x))|\\,dx \\leqslant \\int _{D^k}\\Vert g(x)\\Vert _{C_0(U^k)}\\,dx = \\Vert g\\Vert _{{H}^k} \\qquad \\forall \\ u\\in {F},$ which proves (REF )."
],
[
"Existence and uniqueness of $\\nu $",
"Theorem 2.5 Let $\\mu \\in {P}({F})$ satisfy (REF ).",
"Then (REF ) uniquely defines a correlation measure $\\nu \\in {L}^p$ .",
"We define each correlation marginal $\\nu ^k$ as an element of ${H}^{k*}$ through duality, and then show that it has the required properties.",
"The relation (REF ) uniquely defines $\\nu ^k$ as a linear functional on ${H}^k$ which is continuous since $|\\bigl \\langle \\nu ^k,\\, g\\bigr \\rangle | \\leqslant \\int _{F}\\int _{D^k}|g(x,u(x))|\\,dxd\\mu (u) \\leqslant \\int _{D^k}\\Vert g(x)\\Vert _{C_0(U^k)}\\,dx = \\Vert g\\Vert _{{H}^k}.$ Thus, $\\nu ^k$ is an element of the dual of ${H}^k$ , which by Theorem REF is ${H}^{k*} := L^\\infty _w(D^k,{M}(U^k))$ .",
"Hence, we can view $\\nu ^k$ as a weak* measurable map from $x\\in D^k$ to $\\nu ^k_x \\in {M}(U^k)$ .",
"We show next that $\\nu ^k_x \\in {P}(U^k)$ for Lebesgue-a.e.",
"$x\\in D^k$ .",
"For every $0\\leqslant f\\in C_0(U^k)$ and for every bounded Borel measurable $A\\subset D^k$ we have $\\bigl \\langle \\nu ^k,\\, \\mathbb {1}_A f\\bigr \\rangle = \\int _A \\bigl \\langle \\nu ^k_x,\\, f\\bigr \\rangle \\,dx = \\int _{F}\\int _A f(u(x_1),\\dots ,u(x_k))\\,dxd\\mu (u).$ But the right-hand side always lies between 0 and $|A|\\cdot \\Vert f\\Vert _{C_0}$ .",
"It follows from the arbitrariness of $A$ that $0\\leqslant \\bigl \\langle \\nu ^k_x,\\, f\\bigr \\rangle \\leqslant \\Vert f\\Vert _{C_0}$ for Lebesgue-a.e.",
"$x\\in D^k$ .",
"In particular, letting $f(\\xi ) \\equiv 1$ , we find that $\\Vert \\nu ^k_x\\Vert _{{M}} = 1$ for a.e.",
"$x\\in D$ , which proves the claim.",
"Next, we show that $\\nu =(\\nu ^1,\\nu ^2,\\dots )$ satisfies properties (ii)–(iv) of correlation measures (cf.",
"Definition REF ).",
"The properties of symmetry and consistency follow directly from (REF ), so it remains to show $L^p$ -boundedness.",
"By truncating the function $g:D\\times U\\rightarrow \\mathbb {R}$ defined by $g(x,\\xi ) = |\\xi _1|^p$ and applying Fatou's lemma and the dominated convergence theorem, we get that $\\int _{D} \\bigl \\langle \\nu ^1_x,\\, |\\xi |^p\\bigr \\rangle \\,dx &= \\bigl \\langle \\nu ^1,\\, |\\xi |^p\\bigr \\rangle = \\int _{{F}}\\int _{D} |u(x)|^p\\,dxd\\mu (u) = \\int _{{F}}\\Vert u\\Vert _{{F}}^{p}\\,d\\mu (u) < +\\infty .$ This proves (REF ).",
"Finally, we show that $\\nu $ is diagonally continuous (cf.",
"Definition REF (v)).",
"Indeed, $\\lim _{r\\rightarrow 0}\\int _D\\mathop {}\\hspace{0.0pt}\\mathchoice{\\sbox 0{\\displaystyle \\int \\m@th }\\sbox 2{\\displaystyle \\int _{}\\m@th }\\sbox 4{\\displaystyle \\int _{\\hspace{0.00002pt}}^{\\hspace{0.00002pt}}\\m@th }\\sbox 0{\\scriptstyle B_r(x)\\m@th }\\sbox 2{\\scriptstyle \\m@th }\\unknown.",
"{\\displaystyle \\hbox{t}o{\\hss \\textstyle {-}\\m@th \\hss }}}{}{}{}$ $\\textstyle \\hbox{t}o{\\hss \\scriptstyle {-}\\m@th \\hss }$$$$\\scriptstyle \\hbox{t}o{\\hss \\scriptscriptstyle {-}\\m@th \\hss }$$$$\\scriptscriptstyle \\hbox{t}o{\\hss \\scriptscriptstyle {-}\\m@th \\hss }$$$Br(x)2x,y, |1-2|p dy dx = r0FD$\\displaystyle \\hbox{t}o{\\hss \\textstyle {-}\\m@th \\hss }$$$$\\textstyle \\hbox{t}o{\\hss \\scriptstyle {-}\\m@th \\hss }$$$$\\scriptstyle \\hbox{t}o{\\hss \\scriptscriptstyle {-}\\m@th \\hss }$$$$\\scriptscriptstyle \\hbox{t}o{\\hss \\scriptscriptstyle {-}\\m@th \\hss }$$$Br(x)|u(x)-u(y)|p dy dx d(u) = FD |u(x)-u(x)|p dx d(u) = 0, the second equality following from Lebesgue's differentiation theorem and the dominated convergence theorem.",
"This completes the proof of existence of the correlation measure $\\nu $ .",
"We emphasize that uniqueness follows directly from the explicit definition of $\\nu ^k$ (for each $k$ ) from (REF )."
],
[
"Uniqueness of $\\mu $",
"Let now $\\nu \\in {L}^p(D,U)$ be a given correlation measure.",
"We begin by proving that there exists at most one probability measure $\\mu $ corresponding to $\\nu $ .",
"Theorem 2.6 If $\\mu ,\\tilde{\\mu }\\in {P}({F})$ both satisfy (REF ) and (REF ), then $\\mu = \\tilde{\\mu }$ .",
"By assumption we have $\\int _{F}\\int _{D^k}g(x,u(x))\\,dxd\\mu (u)= \\int _{F}\\int _{D^k}g(x,u(x))\\,dxd\\tilde{\\mu }(u)\\qquad \\forall \\ g\\in {H}^k \\quad \\forall \\ k\\in \\mathbb {N}.$ Fix a number $L>0$ .",
"By the dominated convergence theorem, Hölder's inequality and the $L^p$ -bound (REF ), this same equality holds for $g$ of the form $g(x,\\xi ) = \\varphi _1(x_1) \\cdots \\varphi _k(x_k)\\theta _L(x_1,\\xi _1) \\cdots \\theta _L(x_k,\\xi _k), \\qquad \\theta _L(x,\\xi ) ={\\left\\lbrace \\begin{array}{ll}\\xi & \\text{if } |\\xi | \\leqslant L \\text{ and } |x|\\leqslant L \\\\\\frac{\\xi }{|\\xi |}L & \\text{if } |\\xi |>L \\text{ and } |x|\\leqslant L \\\\0 & \\text{if } |x|> L\\end{array}\\right.",
"}$ where $\\varphi _1, \\dots , \\varphi _k\\in {F}^*$ .",
"Denoting $\\theta _L(u) = \\theta _L(\\cdot ,u(\\cdot ))$ for the sake of simplicity, we can write (REF ) with the above test function $g$ as $\\int _{F}\\langle \\varphi _1, \\theta _L(u) \\rangle \\cdots \\langle \\varphi _k, \\theta _L(u) \\rangle d\\mu (u)= \\int _{F}\\langle \\varphi _1, \\theta _L(u) \\rangle \\cdots \\langle \\varphi _k, \\theta _L(u) \\rangle d\\tilde{\\mu }(u).$ By repeating indices (i.e.",
"choosing some of the $\\varphi _i$ 's to be identical) and expanding integrals over the spatial domain, one can show that the above identity implies $\\int _{F}\\langle \\varphi _1, \\theta _L(u) \\rangle ^{\\alpha _1} \\cdots \\langle \\varphi _k, \\theta _L(u) \\rangle ^{\\alpha _k} d\\mu (u)= \\int _{F}\\langle \\varphi _1, \\theta _L(u) \\rangle ^{\\alpha _1} \\cdots \\langle \\varphi _k, \\theta _L(u) \\rangle ^{\\alpha _k} d\\tilde{\\mu }(u).$ for arbitrary $\\alpha _1, \\dots , \\alpha _k\\in \\mathbb {N}_0$ .",
"Define now $\\varphi :L^p(D) \\rightarrow \\mathbb {R}^k, \\qquad \\varphi (u) := \\Bigl (\\bigl \\langle \\varphi _1,\\, u\\bigr \\rangle , \\dots , \\bigl \\langle \\varphi _k,\\, u\\bigr \\rangle \\Bigr )$ and the truncation $\\varphi _L:L^p(D) \\rightarrow \\mathbb {R}^k, \\qquad \\varphi _L(u) := \\Bigl (\\bigl \\langle \\varphi _1,\\, \\theta _L(u)\\bigr \\rangle , \\dots , \\bigl \\langle \\varphi _k,\\, \\theta _L(u)\\bigr \\rangle \\Bigr )$ Since $|\\bigl \\langle \\varphi _i,\\, \\theta _L(u)\\bigr \\rangle | \\leqslant m_d^{d/p}L^{1+d/p}\\Vert \\varphi _i\\Vert _{{F}^*}$ for $i=1,\\dots ,k$ and with $m_d$ denoting the volume of the unit ball in $\\mathbb {R}^d$ , the map $\\varphi _L$ takes values only in the compact set $K_L := \\bigl [-cL^{1+d/p},cL^{1+d/p}\\bigr ]^k\\subset \\mathbb {R}^k$ , where $c = m_d^{d/p}\\max \\bigl (\\Vert \\varphi _1\\Vert _{{F}^*},\\dots ,\\Vert \\varphi _k\\Vert _{{F}^*}\\bigr )$ .",
"Let $\\psi \\in C^1_c(\\mathbb {R}^k)$ .",
"Then the restriction of $\\psi $ to $K_L$ can be approximated uniformly on $K_L$ by a sequence of polynomials $\\bigl (P_n\\bigr )_{n=1}^\\infty $ .",
"It follows that $P_n\\bigl (\\varphi _L(u)\\bigr ) \\rightarrow \\psi \\bigl (\\varphi _L(u)\\bigr ) \\qquad \\text{as } n\\rightarrow \\infty $ uniformly in $u$ .",
"On the other hand, equation (REF ) implies that for each polynomial $P_n$ , we have $\\int _{F}P_n\\bigl (\\varphi _L(u)\\bigr )\\,d\\mu (u) = \\int _{F}P_n\\bigl (\\varphi _L(u)\\bigr )\\,d\\tilde{\\mu }(u).$ From uniform convergence, we conclude that $\\int _{F}\\psi \\bigl (\\varphi _L(u)\\bigr )\\,d\\mu (u)= \\int _{F}\\psi \\bigl (\\varphi _L(u)\\bigr )\\,d\\tilde{\\mu }(u)$ for any $\\psi \\in C_c^1(\\mathbb {R}^k)$ .",
"Define now $\\Psi _L,\\Psi : {F}\\rightarrow \\mathbb {R}$ by $\\Psi _L(u) := \\psi \\bigl (\\varphi _L(u)\\bigr ), \\qquad \\Psi (u) := \\psi \\bigl (\\varphi (u)\\bigr ).$ Clearly, $|\\Psi _L(u)|, |\\Psi (u)| \\leqslant \\Vert \\psi \\Vert _{C_b(\\mathbb {R}^k)}$ and $\\lim _{L\\rightarrow \\infty } \\Psi _L(u) = \\Psi (u)$ for every $u\\in {F}$ , so by the dominated convergence theorem, $\\int _{F}\\Psi (u) d\\mu (u) = \\int _{F}\\Psi (u) d\\tilde{\\mu }(u)$ for any cylinder function $\\Psi (u) = \\psi \\bigl (\\bigl \\langle \\varphi _1,\\, u\\bigr \\rangle , \\dots , \\bigl \\langle \\varphi _k,\\, u\\bigr \\rangle \\bigr )$ with $\\psi \\in C_c^1(\\mathbb {R}^k)$ .",
"Given an open set $A \\subset \\mathbb {R}^k$ , we can find a sequence $\\psi _n \\in C_c^1(\\mathbb {R}^k)$ such that $0\\leqslant \\psi _n\\leqslant \\psi _{n+1}\\leqslant \\mathbb {1}_A$ for all $n\\in \\mathbb {N}$ , and $\\psi _n$ converges pointwise to the indicator function $\\mathbb {1}_A$ .",
"Again, by dominated convergence, we conclude that $\\int _{F}\\mathbb {1}_A\\bigl (\\bigl \\langle \\varphi _1,\\, u\\bigr \\rangle , \\dots , \\bigl \\langle \\varphi _k,\\, u\\bigr \\rangle \\bigr ) d\\mu (u)= \\int _{F}\\mathbb {1}_A\\bigl (\\bigl \\langle \\varphi _1,\\, u\\bigr \\rangle , \\dots , \\bigl \\langle \\varphi _k,\\, u\\bigr \\rangle \\bigr ) d\\tilde{\\mu }(u).$ By a standard argument, this equality also holds for any Borel measurable set $A\\subset \\mathbb {R}^k$ .",
"This means that $\\mu $ and $\\tilde{\\mu }$ agree on cylinder sets, so by Proposition REF , they must coincide."
],
[
"Existence of $\\mu $ for bounded {{formula:b7a832eb-d371-48b7-8e19-aaaa5b4c818c}}",
"To prove existence of a probability measure $\\mu $ corresponding to a given correlation measure $\\nu $ , we proceed in two steps, first proving the statement for bounded domains $D\\subset \\mathbb {R}^d$ , and then extending the result to arbitrary $D\\subset \\mathbb {R}^d$ .",
"We assume first that $D$ is bounded.",
"Our construction will consist of a piecewise constant approximation over successively finer partitions of $D$ .",
"A collection $\\mathcal {A}= \\lbrace A_1,\\dots ,A_N\\rbrace $ of subsets of $D$ is a partition of $D$ if $\\bigcup _{i=1}^N A_i = D,\\qquad A_i\\cap A_j = \\emptyset \\quad \\text{and}\\quad \\bigl |\\bar{A}_i\\cap \\bar{A}_j\\bigr | = 0 \\text{ for all } i\\ne j$ (where $\\bar{A}_i$ denotes the closure of $A_i$ ).",
"Another partition $\\widetilde{\\mathcal {A}}=\\big \\lbrace \\widetilde{A}_1,\\dots ,\\widetilde{A}_M\\big \\rbrace $ is a refinement of $\\mathcal {A}$ if for every $j=1,\\dots ,M$ , there is an $i\\in \\lbrace 1,\\dots ,N\\rbrace $ such that $\\widetilde{A}_j \\subset A_i$ .",
"Given a partition $\\mathcal {A}= \\lbrace A_1, \\dots , A_N\\rbrace $ of $D$ and a correlation measure $\\nu \\in {L}^p(D,U)$ , define the probability measure $\\rho _\\mathcal {A}\\in {P}(U^N)$ by $\\bigl \\langle \\rho _\\mathcal {A},\\, \\psi \\bigr \\rangle = \\mathop {}\\hspace{0.0pt}\\mathchoice{\\sbox 0{\\displaystyle \\int \\m@th }\\sbox 2{\\displaystyle \\int _{}\\m@th }\\sbox 4{\\displaystyle \\int _{\\hspace{0.00002pt}}^{\\hspace{0.00002pt}}\\m@th }\\sbox 0{\\scriptstyle A_1\\times \\cdots \\times A_N\\m@th }\\sbox 2{\\scriptstyle \\m@th }\\unknown.",
"{\\displaystyle \\hbox{t}o{\\hss \\textstyle {-}\\m@th \\hss }}}{}{}{}$ $\\textstyle \\hbox{t}o{\\hss \\scriptstyle {-}\\m@th \\hss }$$$$\\scriptstyle \\hbox{t}o{\\hss \\scriptscriptstyle {-}\\m@th \\hss }$$$$\\scriptscriptstyle \\hbox{t}o{\\hss \\scriptscriptstyle {-}\\m@th \\hss }$$$A1ANNx, dx, C0(UN).",
"$This is clearly a continuous, linear functional on $ 0(UN)$ with norm $ AM(UN) = C0A, C0 = 1$, and hence is a well-defined element of $ P(UN)$.",
"Next, define $ AP(F)$ by$$\\bigl \\langle \\mu _\\mathcal {A},\\, \\Psi \\bigr \\rangle = \\bigl \\langle \\rho _\\mathcal {A},\\, \\Psi \\Bigl ({\\textstyle \\sum _{i=1}^N\\xi _i\\mathbb {1}_{A_i}}\\Bigr )\\bigr \\rangle .$$Being the pushforward of $ A$ by the continuous function $ UNi=1Ni1AiF$, $ A$ is a well-defined element of $ P(F)$.",
"Finally, let $ ALp(D,U)$ be the unique correlation measure corresponding to $ A$, as constructed in Theorem \\ref {thm:nuexists}.",
"It is clear that $ A$ is the probability measure corresponding to $ A$, in the sense of Theorem \\ref {thm:main}.",
"Note that $ A$ and $ A$ are piecewise constant, in the sense that each correlation marginal $ A,xk$ is constant on sets of the form $ xAi1Aik$, and $ A$ is concentrated on functions $ u:DU$ of the form $ u(x)=i=1N i 1Ai(x)$.$ The correlation measure $\\nu _\\mathcal {A}\\in {L}^p(D,U)$ is called the projection of $\\nu $ onto $\\mathcal {A}$.",
"It is not difficult to see that $\\nu _\\mathcal {A}$ can be equivalently defined as $\\bigl \\langle \\nu ^k_{\\mathcal {A},x},\\, \\psi \\bigr \\rangle := \\sum _{\\alpha \\in [N]^k} \\mathbb {1}_{A_\\alpha }(x) \\mathop {}\\hspace{0.0pt}\\mathchoice{\\sbox 0{\\displaystyle \\int \\m@th }\\sbox 2{\\displaystyle \\int _{}\\m@th }\\sbox 4{\\displaystyle \\int _{\\hspace{0.00002pt}}^{\\hspace{0.00002pt}}\\m@th }\\sbox 0{\\scriptstyle A_1\\times \\cdots \\times A_N\\m@th }\\sbox 2{\\scriptstyle \\m@th }\\unknown.",
"{\\displaystyle \\hbox{t}o{\\hss \\textstyle {-}\\m@th \\hss }}}{}{}{}$ $\\textstyle \\hbox{t}o{\\hss \\scriptstyle {-}\\m@th \\hss }$$$$\\scriptstyle \\hbox{t}o{\\hss \\scriptscriptstyle {-}\\m@th \\hss }$$$$\\scriptscriptstyle \\hbox{t}o{\\hss \\scriptscriptstyle {-}\\m@th \\hss }$$$A1ANNy, () dy, xDk kN.",
"(Here, $[N]=\\lbrace 1,\\dots ,N\\rbrace $ , $A_\\alpha = A_{\\alpha _1}\\times \\cdots \\times A_{\\alpha _k}$ and $\\xi _\\alpha = (\\xi _{\\alpha _1}, \\dots ,\\xi _{\\alpha _k})$ .)",
"Given two partitions $\\mathcal {A}$ and $\\widetilde{\\mathcal {A}}$ of $D$ , where $\\widetilde{\\mathcal {A}}$ is a refinement of $\\mathcal {A}$ , the following lemma establishes an estimate for the distance between $\\mu _{\\mathcal {A}}$ and $\\mu _{\\widetilde{\\mathcal {A}}}$ .",
"Lemma 2.7 Let $\\nu \\in {L}^p(D,U)$ be given.",
"Let $\\mathcal {A}$ and $\\widetilde{\\mathcal {A}}$ be partitions of $D$ , where $\\widetilde{\\mathcal {A}}$ is a refinement of $\\mathcal {A}$ , and let $c,h>0$ be such that $|A_i| \\geqslant ch^d, \\qquad \\mathrm {diam}(A_i) \\leqslant h \\qquad \\forall \\ A_i \\in \\mathcal {A}.$ Let $\\mu _{\\mathcal {A}}, \\mu _{\\widetilde{\\mathcal {A}}}\\in {P}({F})$ be the probability measures corresponding to the projections of $\\nu $ onto $\\mathcal {A}$ and $\\widetilde{\\mathcal {A}}$ , respectively.",
"Then $W_1\\bigl (\\mu _\\mathcal {A},\\mu _{\\widetilde{\\mathcal {A}}}\\bigr ) \\leqslant C \\Biggl (\\int _{D}\\mathop {}\\hspace{0.0pt}\\mathchoice{\\sbox 0{\\displaystyle \\int \\m@th }\\sbox 2{\\displaystyle \\int _{}\\m@th }\\sbox 4{\\displaystyle \\int _{\\hspace{0.00002pt}}^{\\hspace{0.00002pt}}\\m@th }\\sbox 0{\\scriptstyle B_h(y)\\m@th }\\sbox 2{\\scriptstyle \\m@th }\\unknown.",
"{\\displaystyle \\hbox{t}o{\\hss \\textstyle {-}\\m@th \\hss }}}{}{}{}$ $\\textstyle \\hbox{t}o{\\hss \\scriptstyle {-}\\m@th \\hss }$$$$\\scriptstyle \\hbox{t}o{\\hss \\scriptscriptstyle {-}\\m@th \\hss }$$$$\\scriptscriptstyle \\hbox{t}o{\\hss \\scriptscriptstyle {-}\\m@th \\hss }$$$Bh(y)2x,y, |1-2|p dx dy)1/p, $where $ h(y):={xD : |x-y|<h}$ and $ C>0$ only depends on $ c$, $ p$ and $ d$ (the dimension of $ D$).$ Let $\\Psi : {F}\\rightarrow \\mathbb {R}$ be a Lipschitz function with $\\Vert \\Psi \\Vert _{\\operatorname{Lip}}=1$ .",
"Denote $\\mathcal {A}= \\big \\lbrace A_1, \\dots , A_N\\big \\rbrace , \\qquad \\widetilde{\\mathcal {A}} = \\big \\lbrace \\widetilde{A}_1, \\dots , \\widetilde{A}_M\\big \\rbrace .$ By definition, $\\int _{{F}} \\Psi (u) \\,d\\bigl (\\mu _{\\mathcal {A}} - \\mu _{\\widetilde{\\mathcal {A}}}\\bigr )&= \\mathop {}\\hspace{0.0pt}\\mathchoice{\\sbox 0{\\displaystyle \\int \\m@th }\\sbox 2{\\displaystyle \\int _{}\\m@th }\\sbox 4{\\displaystyle \\int _{\\hspace{0.00002pt}}^{\\hspace{0.00002pt}}\\m@th }\\sbox 0{\\scriptstyle A_1\\times \\cdots \\times A_N\\m@th }\\sbox 2{\\scriptstyle \\m@th }\\unknown.",
"{\\displaystyle \\hbox{t}o{\\hss \\textstyle {-}\\m@th \\hss }}}{}{}{}$ $\\textstyle \\hbox{t}o{\\hss \\scriptstyle {-}\\m@th \\hss }$$$$\\scriptstyle \\hbox{t}o{\\hss \\scriptscriptstyle {-}\\m@th \\hss }$$$$\\scriptscriptstyle \\hbox{t}o{\\hss \\scriptscriptstyle {-}\\m@th \\hss }$$$A1ANNx, (i=1N i 1Ai) dx - $\\displaystyle \\hbox{t}o{\\hss \\textstyle {-}\\m@th \\hss }$$$$\\textstyle \\hbox{t}o{\\hss \\scriptstyle {-}\\m@th \\hss }$$$$\\scriptstyle \\hbox{t}o{\\hss \\scriptscriptstyle {-}\\m@th \\hss }$$$$\\scriptscriptstyle \\hbox{t}o{\\hss \\scriptscriptstyle {-}\\m@th \\hss }$$$A1AMMy, (j=1M j 1Aj) dy = $\\displaystyle \\hbox{t}o{\\hss \\textstyle {-}\\m@th \\hss }$$$$\\textstyle \\hbox{t}o{\\hss \\scriptstyle {-}\\m@th \\hss }$$$$\\scriptstyle \\hbox{t}o{\\hss \\scriptscriptstyle {-}\\m@th \\hss }$$$$\\scriptscriptstyle \\hbox{t}o{\\hss \\scriptscriptstyle {-}\\m@th \\hss }$$$A1ANNx, (j=1M i(j) 1Ai(j)) dx - $\\displaystyle \\hbox{t}o{\\hss \\textstyle {-}\\m@th \\hss }$$$$\\textstyle \\hbox{t}o{\\hss \\scriptstyle {-}\\m@th \\hss }$$$$\\scriptstyle \\hbox{t}o{\\hss \\scriptscriptstyle {-}\\m@th \\hss }$$$$\\scriptscriptstyle \\hbox{t}o{\\hss \\scriptscriptstyle {-}\\m@th \\hss }$$$A1AMMy, (j=1M j 1Aj) dy where for any $j\\in \\lbrace 1, \\dots ,M\\rbrace $ , the index $i(j)$ is the unique integer in $\\lbrace 1,\\dots , N\\rbrace $ , such that $\\widetilde{A}_{j}\\subset A_{i(j)}$ , and $\\xi $ and $\\zeta $ are the integration variables with respect to $\\nu ^N_x$ and $\\nu ^M_x$ , respectively.",
"Denote $A = A_1\\times \\dots \\times A_N, \\qquad \\widetilde{A} = \\widetilde{A}_1 \\times \\dots \\times \\widetilde{A}_M.$ Then we can write $\\int _{{F}} \\Psi (u) \\, d\\bigl (\\mu _{\\mathcal {A}} - \\mu _{\\widetilde{\\mathcal {A}}}\\bigr )&= \\mathop {}\\hspace{0.0pt}\\mathchoice{\\sbox 0{\\displaystyle \\int \\m@th }\\sbox 2{\\displaystyle \\int _{}\\m@th }\\sbox 4{\\displaystyle \\int _{\\hspace{0.00002pt}}^{\\hspace{0.00002pt}}\\m@th }\\sbox 0{\\scriptstyle A\\m@th }\\sbox 2{\\scriptstyle \\m@th }\\unknown.",
"{\\displaystyle \\hbox{t}o{\\hss \\textstyle {-}\\m@th \\hss }}}{}{}{}$ $\\textstyle \\hbox{t}o{\\hss \\scriptstyle {-}\\m@th \\hss }$$$$\\scriptstyle \\hbox{t}o{\\hss \\scriptscriptstyle {-}\\m@th \\hss }$$$$\\scriptscriptstyle \\hbox{t}o{\\hss \\scriptscriptstyle {-}\\m@th \\hss }$$$ANx, (j=1M i(j) 1Aj) dx - $\\displaystyle \\hbox{t}o{\\hss \\textstyle {-}\\m@th \\hss }$$$$\\textstyle \\hbox{t}o{\\hss \\scriptstyle {-}\\m@th \\hss }$$$$\\scriptstyle \\hbox{t}o{\\hss \\scriptscriptstyle {-}\\m@th \\hss }$$$$\\scriptscriptstyle \\hbox{t}o{\\hss \\scriptscriptstyle {-}\\m@th \\hss }$$$AMy, (j=1M j 1Aj) dy (consistency of $\\nu $ ) = $\\displaystyle \\hbox{t}o{\\hss \\textstyle {-}\\m@th \\hss }$$$$\\textstyle \\hbox{t}o{\\hss \\scriptstyle {-}\\m@th \\hss }$$$$\\scriptstyle \\hbox{t}o{\\hss \\scriptscriptstyle {-}\\m@th \\hss }$$$$\\scriptscriptstyle \\hbox{t}o{\\hss \\scriptscriptstyle {-}\\m@th \\hss }$$$A$\\displaystyle \\hbox{t}o{\\hss \\textstyle {-}\\m@th \\hss }$$$$\\textstyle \\hbox{t}o{\\hss \\scriptstyle {-}\\m@th \\hss }$$$$\\scriptstyle \\hbox{t}o{\\hss \\scriptscriptstyle {-}\\m@th \\hss }$$$$\\scriptscriptstyle \\hbox{t}o{\\hss \\scriptscriptstyle {-}\\m@th \\hss }$$$AN+Mx,y, (j=1M i(j) 1Aj)-(j=1M j 1Aj) dx dy (Lipschitz continuity) $\\displaystyle \\hbox{t}o{\\hss \\textstyle {-}\\m@th \\hss }$$$$\\textstyle \\hbox{t}o{\\hss \\scriptstyle {-}\\m@th \\hss }$$$$\\scriptstyle \\hbox{t}o{\\hss \\scriptscriptstyle {-}\\m@th \\hss }$$$$\\scriptscriptstyle \\hbox{t}o{\\hss \\scriptscriptstyle {-}\\m@th \\hss }$$$A$\\displaystyle \\hbox{t}o{\\hss \\textstyle {-}\\m@th \\hss }$$$$\\textstyle \\hbox{t}o{\\hss \\scriptstyle {-}\\m@th \\hss }$$$$\\scriptstyle \\hbox{t}o{\\hss \\scriptscriptstyle {-}\\m@th \\hss }$$$$\\scriptscriptstyle \\hbox{t}o{\\hss \\scriptscriptstyle {-}\\m@th \\hss }$$$AN+Mx,y, j=1M i(j) 1Aj-j=1M j 1AjF dx dy = $\\displaystyle \\hbox{t}o{\\hss \\textstyle {-}\\m@th \\hss }$$$$\\textstyle \\hbox{t}o{\\hss \\scriptstyle {-}\\m@th \\hss }$$$$\\scriptstyle \\hbox{t}o{\\hss \\scriptscriptstyle {-}\\m@th \\hss }$$$$\\scriptscriptstyle \\hbox{t}o{\\hss \\scriptscriptstyle {-}\\m@th \\hss }$$$A$\\displaystyle \\hbox{t}o{\\hss \\textstyle {-}\\m@th \\hss }$$$$\\textstyle \\hbox{t}o{\\hss \\scriptstyle {-}\\m@th \\hss }$$$$\\scriptstyle \\hbox{t}o{\\hss \\scriptscriptstyle {-}\\m@th \\hss }$$$$\\scriptscriptstyle \\hbox{t}o{\\hss \\scriptscriptstyle {-}\\m@th \\hss }$$$AN+Mx,y, ( j=1M |Aj||i(j)-j|p )1/p dx dy (Jensen's inequality) $\\displaystyle \\hbox{t}o{\\hss \\textstyle {-}\\m@th \\hss }$$$$\\textstyle \\hbox{t}o{\\hss \\scriptstyle {-}\\m@th \\hss }$$$$\\scriptstyle \\hbox{t}o{\\hss \\scriptscriptstyle {-}\\m@th \\hss }$$$$\\scriptscriptstyle \\hbox{t}o{\\hss \\scriptscriptstyle {-}\\m@th \\hss }$$$A$\\displaystyle \\hbox{t}o{\\hss \\textstyle {-}\\m@th \\hss }$$$$\\textstyle \\hbox{t}o{\\hss \\scriptstyle {-}\\m@th \\hss }$$$$\\scriptstyle \\hbox{t}o{\\hss \\scriptscriptstyle {-}\\m@th \\hss }$$$$\\scriptscriptstyle \\hbox{t}o{\\hss \\scriptscriptstyle {-}\\m@th \\hss }$$$A(j=1M|Aj|N+Mx,y, |i(j)-j|p)1/p dx dy (Jensen's inequality) = ($\\displaystyle \\hbox{t}o{\\hss \\textstyle {-}\\m@th \\hss }$$$$\\textstyle \\hbox{t}o{\\hss \\scriptstyle {-}\\m@th \\hss }$$$$\\scriptstyle \\hbox{t}o{\\hss \\scriptscriptstyle {-}\\m@th \\hss }$$$$\\scriptscriptstyle \\hbox{t}o{\\hss \\scriptscriptstyle {-}\\m@th \\hss }$$$A$\\displaystyle \\hbox{t}o{\\hss \\textstyle {-}\\m@th \\hss }$$$$\\textstyle \\hbox{t}o{\\hss \\scriptstyle {-}\\m@th \\hss }$$$$\\scriptstyle \\hbox{t}o{\\hss \\scriptscriptstyle {-}\\m@th \\hss }$$$$\\scriptscriptstyle \\hbox{t}o{\\hss \\scriptscriptstyle {-}\\m@th \\hss }$$$Aj=1M|Aj|N+Mx,y, |i(j)-j|p dx dy)1/p (consistency of $\\nu $ ) = (j=1M|Aj|$\\displaystyle \\hbox{t}o{\\hss \\textstyle {-}\\m@th \\hss }$$$$\\textstyle \\hbox{t}o{\\hss \\scriptstyle {-}\\m@th \\hss }$$$$\\scriptstyle \\hbox{t}o{\\hss \\scriptscriptstyle {-}\\m@th \\hss }$$$$\\scriptscriptstyle \\hbox{t}o{\\hss \\scriptscriptstyle {-}\\m@th \\hss }$$$A$\\displaystyle \\hbox{t}o{\\hss \\textstyle {-}\\m@th \\hss }$$$$\\textstyle \\hbox{t}o{\\hss \\scriptstyle {-}\\m@th \\hss }$$$$\\scriptstyle \\hbox{t}o{\\hss \\scriptscriptstyle {-}\\m@th \\hss }$$$$\\scriptscriptstyle \\hbox{t}o{\\hss \\scriptscriptstyle {-}\\m@th \\hss }$$$A2xi(j),yj, |-|p dx dy)1/p = (j=1M|Aj|$\\displaystyle \\hbox{t}o{\\hss \\textstyle {-}\\m@th \\hss }$$$$\\textstyle \\hbox{t}o{\\hss \\scriptstyle {-}\\m@th \\hss }$$$$\\scriptstyle \\hbox{t}o{\\hss \\scriptscriptstyle {-}\\m@th \\hss }$$$$\\scriptscriptstyle \\hbox{t}o{\\hss \\scriptscriptstyle {-}\\m@th \\hss }$$$Aj$\\displaystyle \\hbox{t}o{\\hss \\textstyle {-}\\m@th \\hss }$$$$\\textstyle \\hbox{t}o{\\hss \\scriptstyle {-}\\m@th \\hss }$$$$\\scriptstyle \\hbox{t}o{\\hss \\scriptscriptstyle {-}\\m@th \\hss }$$$$\\scriptscriptstyle \\hbox{t}o{\\hss \\scriptscriptstyle {-}\\m@th \\hss }$$$Ai(j)2xi(j),yj, |-|p dxi(j) dyj)1/p.",
"Renaming variables $x_{i(j)}\\mapsto x$ and $y_j\\mapsto y$ in this summation, we obtain the estimate $\\int _{{F}} \\Psi (u) \\, d\\bigl (\\mu _{\\mathcal {A}} - \\mu _{\\widetilde{\\mathcal {A}}}\\bigr ) \\leqslant \\left(\\sum _{j=1}^M\\int _{\\widetilde{A}_j}\\mathop {}\\hspace{0.0pt}\\mathchoice{\\sbox 0{\\displaystyle \\int \\m@th }\\sbox 2{\\displaystyle \\int _{}\\m@th }\\sbox 4{\\displaystyle \\int _{\\hspace{0.00002pt}}^{\\hspace{0.00002pt}}\\m@th }\\sbox 0{\\scriptstyle A_{i(j)}\\m@th }\\sbox 2{\\scriptstyle \\m@th }\\unknown.",
"{\\displaystyle \\hbox{t}o{\\hss \\textstyle {-}\\m@th \\hss }}}{}{}{}\\right.",
"{\\sbox 0{\\textstyle \\int \\m@th }\\sbox 2{\\textstyle \\int _{}\\m@th }\\sbox 4{\\textstyle \\int _{\\hspace{0.00002pt}}^{\\hspace{0.00002pt}}\\m@th }\\sbox 0{\\scriptstyle A_{i(j)}\\m@th }\\sbox 2{\\scriptstyle \\m@th }\\unknown.",
"{\\textstyle \\hbox{t}o{\\hss \\scriptstyle {-}\\m@th \\hss }}}$ $\\scriptstyle \\hbox{t}o{\\hss \\scriptscriptstyle {-}\\m@th \\hss }$$$$\\scriptscriptstyle \\hbox{t}o{\\hss \\scriptscriptstyle {-}\\m@th \\hss }$$$Ai(j)2x,y, |-|p dx dy)1/p, $valid for any $ $-Lipschitz continuous $ : L1(D) R$.",
"Using (\\ref {eq:partitionassump}) we get the estimate{\\begin{@align*}{1}{-1}\\int _{{F}} \\Psi (u) \\, d\\bigl (\\mu _{\\mathcal {A}} - \\mu _{\\widetilde{\\mathcal {A}}}\\bigr )&\\leqslant \\left(\\sum _{j=1}^M\\frac{1}{|A_{i(j)}|}\\int _{\\widetilde{A}_j}\\int _{A_{i(j)}} \\bigl \\langle \\nu ^2_{x,y},\\, |\\xi -\\zeta |^p\\bigr \\rangle \\, dx\\, dy\\right)^{1/p} \\\\&\\leqslant \\left(\\sum _{j=1}^M\\frac{|B_h(y)|}{ch^d}\\int _{\\widetilde{A}_j}\\mathop {}\\hspace{0.0pt}\\mathchoice{\\sbox 0{\\displaystyle \\int \\m@th }\\sbox 2{\\displaystyle \\int _{}\\m@th }\\sbox 4{\\displaystyle \\int _{\\hspace{0.00002pt}}^{\\hspace{0.00002pt}}\\m@th }\\sbox 0{\\scriptstyle B_h(y)\\m@th }\\sbox 2{\\scriptstyle \\m@th }\\unknown.",
"{\\displaystyle \\hbox{t}o{\\hss \\textstyle {-}\\m@th \\hss }}}{}{}{}\\right.",
"{\\sbox 0{\\textstyle \\int \\m@th }\\sbox 2{\\textstyle \\int _{}\\m@th }\\sbox 4{\\textstyle \\int _{\\hspace{0.00002pt}}^{\\hspace{0.00002pt}}\\m@th }\\sbox 0{\\scriptstyle B_h(y)\\m@th }\\sbox 2{\\scriptstyle \\m@th }\\unknown.",
"{\\textstyle \\hbox{t}o{\\hss \\scriptstyle {-}\\m@th \\hss }}}\\end{@align*}{\\sbox 0{\\scriptstyle \\int \\m@th }\\sbox 2{\\scriptstyle \\int _{}\\m@th }\\sbox 4{\\scriptstyle \\int _{\\hspace{0.00002pt}}^{\\hspace{0.00002pt}}\\m@th }\\sbox 0{\\scriptscriptstyle B_h(y)\\m@th }\\sbox 2{\\scriptscriptstyle \\m@th }\\unknown.",
"{\\scriptstyle \\hbox{t}o{\\hss \\scriptscriptstyle {-}\\m@th \\hss }}}}{\\sbox 0{\\scriptscriptstyle \\int \\m@th }\\sbox 2{\\scriptscriptstyle \\int _{}\\m@th }\\sbox 4{\\scriptscriptstyle \\int _{\\hspace{0.00002pt}}^{\\hspace{0.00002pt}}\\m@th }\\sbox 0{\\scriptscriptstyle B_h(y)\\m@th }\\sbox 2{\\scriptscriptstyle \\m@th }\\unknown.",
"{\\scriptscriptstyle \\hbox{t}o{\\hss \\scriptscriptstyle {-}\\m@th \\hss }}}$ Bh(y)2x,y, |-|p dx dy)1/p (CD$\\displaystyle \\hbox{t}o{\\hss \\textstyle {-}\\m@th \\hss }$$$$\\textstyle \\hbox{t}o{\\hss \\scriptstyle {-}\\m@th \\hss }$$$$\\scriptstyle \\hbox{t}o{\\hss \\scriptscriptstyle {-}\\m@th \\hss }$$$$\\scriptscriptstyle \\hbox{t}o{\\hss \\scriptscriptstyle {-}\\m@th \\hss }$$$Bh(y)2x,y, |-|p dx dy)1/p where $C$ is given by the ratio of $c$ to the unit ball in $\\mathbb {R}^d$ .",
"Taking the supremum over all $\\Psi $ with $\\Vert \\Psi \\Vert _{\\operatorname{Lip}}\\leqslant 1$ on the left hand side and using the Kantorovich–Rubinstein definition (REF ) of $W_1$ yields the desired estimate.",
"With this bound in place we can complete the proof of existence of $\\mu $ .",
"Theorem 2.8 For any $\\nu \\in {L}^p(D,U)$ there exists a probability measure $\\mu \\in {P}({F})$ satisfying (REF ) and (REF ).",
"Let $(\\mathcal {A}_m)_{m\\in \\mathbb {N}}$ be a sequence of partitions of $D$ such that $\\mathcal {A}_{m+1}$ is a refinement of $\\mathcal {A}_m$ , there exists a constant $c>0$ and a sequence $h_m \\rightarrow 0$ , such that $|A|\\geqslant ch_m^d, \\qquad \\mathrm {diam}(A) \\leqslant h_m \\qquad \\forall \\ A\\in \\mathcal {A}_m, \\; \\forall \\ m\\in \\mathbb {N}.$ We show first that the sequence of probability measures $\\mu _{\\mathcal {A}_m}\\in {P}({F})$ converges weakly to some $\\mu \\in {P}({F})$ satisfying (REF ).",
"By Lemma REF , we have for any $m^{\\prime } > m$ $W_1\\bigl (\\mu _{\\mathcal {A}_m},\\mu _{\\mathcal {A}_{m^{\\prime }}}\\bigr ) \\leqslant C \\Biggl (\\int _{D}\\mathop {}\\hspace{0.0pt}\\mathchoice{\\sbox 0{\\displaystyle \\int \\m@th }\\sbox 2{\\displaystyle \\int _{}\\m@th }\\sbox 4{\\displaystyle \\int _{\\hspace{0.00002pt}}^{\\hspace{0.00002pt}}\\m@th }\\sbox 0{\\scriptstyle B_{h_m}(x)\\m@th }\\sbox 2{\\scriptstyle \\m@th }\\unknown.",
"{\\displaystyle \\hbox{t}o{\\hss \\textstyle {-}\\m@th \\hss }}}{}{}{}$ $\\textstyle \\hbox{t}o{\\hss \\scriptstyle {-}\\m@th \\hss }$$$$\\scriptstyle \\hbox{t}o{\\hss \\scriptscriptstyle {-}\\m@th \\hss }$$$$\\scriptscriptstyle \\hbox{t}o{\\hss \\scriptscriptstyle {-}\\m@th \\hss }$$$Bhm(x)2x,y, |-|p dy dx)1/p $where $ >0$ does not depend on $ m$.",
"By the DC property (\\ref {eq:dcproperty}), the right-hand side vanishes as $ m$.",
"It follows that $ m,m'W1(Am,Am') = 0$, so the sequence $ Am$ is Cauchy in the $ W1$ metric.",
"Since the $ W1$ metric turns $ P(F)$ into a complete metric space (see \\cite [Proposition 7.1.5]{AGS05}), we conclude that $ Am $ for some $ P(F)$.",
"Moreover, from the fact that $$ satisfies (\\ref {eq:corrlpbound}), it follows that $$ satisfies (\\ref {eq:lpbound}).$ We show next that the limit $\\mu $ satisfies (REF ).",
"Fix some $m\\in \\mathbb {N}$ and denote $\\mathcal {A}= \\mathcal {A}_m = \\lbrace A_1,\\dots ,A_N\\rbrace $ .",
"If $x\\in D^k$ then there is a unique index $\\alpha \\in [N]^k$ such that $x\\in A_\\alpha :=A_{\\alpha _1}\\times \\dots \\times A_{\\alpha _k}$ .",
"If $x$ is on the off-diagonal, i.e.",
"$\\alpha _i\\ne \\alpha _j$ for all $i\\ne j$ , then it follows from consistency that $\\bigl \\langle \\nu _{\\mathcal {A},x}^k,\\, \\psi \\bigr \\rangle = \\sum _{\\alpha \\in [N]^k} \\mathbb {1}_{A_\\alpha }(x) \\mathop {}\\hspace{0.0pt}\\mathchoice{\\sbox 0{\\displaystyle \\int \\m@th }\\sbox 2{\\displaystyle \\int _{}\\m@th }\\sbox 4{\\displaystyle \\int _{\\hspace{0.00002pt}}^{\\hspace{0.00002pt}}\\m@th }\\sbox 0{\\scriptstyle A_\\alpha \\m@th }\\sbox 2{\\scriptstyle \\m@th }\\unknown.",
"{\\displaystyle \\hbox{t}o{\\hss \\textstyle {-}\\m@th \\hss }}}{}{}{}$ $\\textstyle \\hbox{t}o{\\hss \\scriptstyle {-}\\m@th \\hss }$$$$\\scriptstyle \\hbox{t}o{\\hss \\scriptscriptstyle {-}\\m@th \\hss }$$$$\\scriptscriptstyle \\hbox{t}o{\\hss \\scriptscriptstyle {-}\\m@th \\hss }$$$Aky, dy $(compare with (\\ref {eq:nuprojdef})).",
"Hence, Lebesgue^{\\prime }s differentiation theorem implies that $ Am,xk, xk, $ as $ m0$ for almost every point $ xDk$ on the off-diagonal $ {xDk : xi xj for all ij}$.",
"But since the diagonal $ {xDk : xi = xj for some ij}$ has Lebesgue measure zero, we can conclude that$$\\operatornamewithlimits{w*-lim}_{m\\rightarrow \\infty } \\nu ^k_{\\mathcal {A}_m} = \\nu ^k \\qquad \\text{in } {H}^{k*} \\quad \\forall \\ k\\in \\mathbb {N},$$or in other words,\\begin{equation}\\lim _{m\\rightarrow \\infty } \\bigl \\langle \\nu ^k_{\\mathcal {A}_m},\\, g\\bigr \\rangle = \\bigl \\langle \\nu ^k,\\, g\\bigr \\rangle \\qquad \\forall \\ g\\in {H}^k \\quad \\forall \\ k\\in \\mathbb {N}.\\end{equation}$ We know that $\\mu _{\\mathcal {A}_m} \\rightharpoonup \\mu $ in ${P}({F})$ , that is, $\\lim _{m\\rightarrow \\infty }\\int _{F}\\Psi (u)\\,d\\mu _{\\mathcal {A}_m}(u) = \\int _{F}\\Psi (u)\\,d\\mu (u) \\qquad \\forall \\ \\Psi \\in C_b({F}).$ By Proposition REF , the functionals $L_g$ lie in $C_b({F})$ , so the above holds for $\\Psi =L_g$ for any $g\\in {H}^k$ .",
"Thus, for any $k\\in \\mathbb {N}$ and $g\\in {H}^k$ , we have $\\bigl \\langle \\mu ,\\, L_g\\bigr \\rangle = \\lim _{m\\rightarrow \\infty }\\bigl \\langle \\mu _{\\mathcal {A}_m},\\, L_g\\bigr \\rangle = \\lim _{m\\rightarrow \\infty }\\bigl \\langle \\nu ^k_{\\mathcal {A}_m},\\, g\\bigr \\rangle = \\bigl \\langle \\nu ^k,\\, g\\bigr \\rangle ,$ which is (REF )."
],
[
"Existence of $\\mu $ for unbounded {{formula:0eb66668-782e-4cae-823a-2a711c3c9a94}}",
"The next step is to prove existence of a probability measure $\\mu $ for a given correlation measure $\\nu $ on an arbitrary domain $D$ .",
"To this end, we first construct $\\mu $ on a bounded set $E\\subset D$ , and then pass to the limit $E\\uparrow D$ .",
"Lemma 2.9 Let $E\\subset D$ .",
"Let $r$ denote the restriction map $r: L^p(D,U) \\rightarrow L^p(E,U), \\qquad r(u) = u\\bigr |_E.$ If $\\mu \\in {P}\\bigl (L^p(D,U)\\bigr )$ has correlation measure $\\nu $ , then $r\\#\\mu \\in {P}\\bigl (L^p(E,U)\\bigr )$ has correlation measure $\\nu \\big |_E := \\Bigl (\\nu ^1\\big |_E,\\, \\nu ^2\\big |_{E^2},\\, \\nu ^3\\big |_{E^3},\\, \\dots \\Bigr ).$ Let $g\\in L^1(E^k,C_0(U^k))$ .",
"Then the function $x \\mapsto \\mathbb {1}_E(x)g(x,\\cdot )$ lies in ${H}^k=L^1(D^k,C_0(U^k))$ .",
"Hence, $\\int _{L^p(E,U)} \\int _{E^k} g(x,u(x)) \\, dx \\, d(r\\#\\mu )(u)&= \\int _{L^p(D,U)} \\int _{E^k} g(x,u|_E(x)) \\, dx \\, d\\mu (u)\\\\&= \\int _{L^p(D,U)} \\int _{D^k} \\mathbb {1}_{E^k}(x) g(x,u(x)) \\, dx \\, d\\mu (u) \\\\&= \\int _{D^k} \\bigl \\langle \\nu _x^k,\\, \\mathbb {1}_{E^k}(x)g(x,\\cdot )\\bigr \\rangle \\, dx \\\\&= \\int _{E^k} \\bigl \\langle \\bigl (\\nu ^k\\big |_{E^k}\\bigr )_x,\\, g(x,\\cdot )\\bigr \\rangle \\, dx.$ Thus, $\\nu |_E$ is the correlation measure associated with $r\\#\\mu $ .",
"Let now $\\nu \\in {L}^p(D,U)$ for an arbitrary measurable set $D\\subset \\mathbb {R}^d$ .",
"Given $L>0$ , let $D_L := D \\cap (-L,L)^d$ .",
"Let $\\tilde{\\mu }_L\\in {P}(L^p(D_L,U))$ be the unique probability measure associated with the restriction $\\nu |_{D_L}$ of $\\nu $ to $D_L$ , as constructed in Section REF .",
"Furthermore, let $\\mu _L\\in {P}(L^p(D,U))$ be the image of $\\tilde{\\mu }_L$ under the inclusion map obtained via extension by 0: $i_L: L^p(D_L,U) \\rightarrow L^p(D,U),\\qquad i_L(u) = u\\mathbb {1}_{D_L}.$ By Lemma REF , we expect the sequence $(\\tilde{\\mu }_L)_{L>0}$ to be related to the restriction of a probability measure $\\mu $ with correlation measure $\\nu $ .",
"In particular, we would then expect the sequence $\\mu _L$ to converge to a probability measure $\\mu $ as $L\\rightarrow \\infty $ .",
"The following theorem shows that this is indeed the case.",
"Theorem 2.10 The sequence $\\mu _L$ converges weakly as $L\\rightarrow \\infty $ to some $\\mu \\in {P}({F})$ satisfying (REF ) and (REF ).",
"Let $\\Psi \\in C_b({F})$ be an arbitrary 1-Lipschitz function.",
"Let $M < L,L^{\\prime }$ .",
"Then $\\int _{{F}} \\Psi (u) \\, d\\bigl (\\mu _L - \\mu _{L^{\\prime }}\\bigr )= \\int _{{F}} \\Psi (u)-\\Psi (\\mathbb {1}_{D_M}u) \\, d\\mu _L + \\int _{{F}} \\Psi (\\mathbb {1}_{D_M}u) \\, d\\bigl (\\mu _L - \\mu _{L^{\\prime }}\\bigr ) + \\int _{{F}}\\Psi (\\mathbb {1}_{D_M}u)-\\Psi (u) \\, d\\mu _{L^{\\prime }}$ The second term is zero as a consequence of Lemma REF .",
"For the first and third terms, we have the estimate $\\left|\\int _{{F}} \\Psi (u)-\\Psi (\\mathbb {1}_{D_M}u) \\, d\\mu _L \\right|&\\leqslant \\int _{{F}} \\Vert u-\\mathbb {1}_{D_M}u\\Vert _{L^p} \\, d\\mu _L \\\\&\\leqslant \\Bigg (\\int _{{F}} \\Vert \\mathbb {1}_{D_M^c}u\\Vert _{L^p}^p \\, d\\mu _L\\Bigg )^{1/p} \\\\&= \\Bigg (\\int _{D_L \\cap D_M^c} \\langle \\nu ^1_x, | \\xi |^p \\rangle \\, dx\\Bigg )^{1/p} \\\\&\\leqslant \\Bigg (\\int _{D\\setminus D_M} \\langle \\nu _x^1, | \\xi |^p \\rangle \\, dx\\Bigg )^{1/p}.$ It follows that $\\int _{{F}} \\Psi (u) \\, d\\bigl (\\mu _L - \\mu _{L^{\\prime }}\\bigr )\\leqslant 2\\left(\\int _{D\\setminus D_M} \\langle \\nu _x^1, | \\xi |^p \\rangle \\, dx\\right)^{1/p}$ Taking the supremum over all 1-Lipschitz $\\Psi \\in C_b({F})$ on the left, we obtain $W_1(\\mu _L,\\mu _{L^{\\prime }}) \\leqslant 2\\left(\\int _{D\\setminus D_M} \\langle \\nu _x^1, | \\xi |^p \\rangle \\, dx\\right)^{1/p}.$ By assumption, $\\int _{D} \\langle \\nu _x^1, | \\xi |^p \\rangle \\, dx$ is finite, so $\\int _{D\\setminus D_M} \\langle \\nu _x^1, | \\xi |^p \\rangle \\, dx$ goes to zero as $M \\rightarrow \\infty $ .",
"We conclude that $W_1(\\mu _L,\\mu _{L^{\\prime }})\\rightarrow 0$ as $L,L^{\\prime } \\rightarrow \\infty $ .",
"By completeness under the 1-Wasserstein distance, the sequence $\\mu _L$ converges to a limit $\\mu = \\operatornamewithlimits{w-lim}_{L \\rightarrow \\infty } \\mu _L$ .",
"We claim that the limit $\\mu $ has correlation measure $\\nu $ , in the sense of Theorem REF .",
"Indeed, we have $\\nu ^k\\bigl |_{D_L} \\overset{*}{\\rightharpoonup }\\nu ^k$ , $\\mu _L \\rightharpoonup \\mu $ and $\\bigl \\langle \\nu ^k\\bigl |_{D_L},\\, g\\bigr \\rangle = \\bigl \\langle \\mu _L,\\, L_g\\bigr \\rangle $ for all $g\\in {H}^k$ and $k\\in \\mathbb {N}$ .",
"It follows that $\\bigl \\langle \\nu ^k,\\, g\\bigr \\rangle =\\bigl \\langle \\mu ,\\, L_g\\bigr \\rangle $ ."
],
[
"Moments",
"We have now established the equivalence between probability measures $\\mu \\in {P}({F})$ satisfying $\\int _{F}\\Vert u\\Vert _{{F}}^p d\\mu <\\infty ,$ and so-called correlation measures $\\nu \\in {L}^p(D,U)$ .",
"In this section we introduce a third representation, that of moments.",
"The moments of a correlation measure $\\nu \\in {L}^p(D,U)$ are the functions $m^k : D^k\\rightarrow U^{\\otimes k}, \\qquad m^k(x) := \\int _{U^k} \\xi _1\\otimes \\cdots \\otimes \\xi _k\\,d\\nu ^k_x(\\xi ), \\qquad k\\in \\mathbb {N}.$ Here, $U^{\\otimes k}$ refers to the tensor product space $U\\otimes \\cdots \\otimes U$ (repeated $k$ times), and $\\xi _1\\otimes \\cdots \\otimes \\xi _k$ is a functional defined by its action on the dual space $\\bigl (U^{\\otimes k}\\bigr )^* = U^{\\otimes k}$ through $\\bigl (\\xi _1\\otimes \\cdots \\otimes \\xi _k\\bigr ):\\bigl (\\zeta _1\\otimes \\cdots \\otimes \\zeta _k\\bigr ) = (\\xi _1\\cdot \\zeta _1)\\cdots (\\xi _k\\cdot \\zeta _k).$ In the case $U=\\mathbb {R}$ , the moments can be written more simply as $m^k : D^k\\rightarrow \\mathbb {R}, \\qquad m^k(x) = \\int _{\\mathbb {R}^k} \\xi _1\\cdots \\xi _k\\,d\\nu ^k_x(\\xi ), \\qquad k\\in \\mathbb {N}.$ In either case, we will assume that $\\int _{D^k} \\int _{U^k} |\\xi _1|^p\\cdots |\\xi _k|^p\\,d\\nu ^k_x(\\xi ) dx < \\infty \\qquad \\forall \\ k\\in \\mathbb {N},$ or equivalently, $\\int _{F}\\Vert u\\Vert _{F}^{pk}\\,d\\mu (u) < \\infty \\qquad \\forall \\ k\\in \\mathbb {N}\\qquad \\mathrm {(\\ref {eq:nufinitemoments}^{\\prime })}$ (compare with (REF )).",
"This ensures that $m^k$ is a well-defined element of $L^p(D^k,\\, U^{\\otimes k})$ .",
"The following result uniquely characterizes a correlation measure in terms of the family of moments $(m^k)_{k\\in \\mathbb {N}}$ .",
"This result will be essential to the contents of the following sections.",
"Theorem 2.11 Let $\\nu \\in {L}^p(D,U)$ satisfy (REF ).",
"Then the moments (REF ) uniquely identify $\\nu $ , in the sense that if another correlation measure $\\tilde{\\nu }$ has the same moments $(m^k)_{k\\in \\mathbb {N}}$ , then $\\nu =\\tilde{\\nu }$ .",
"Denote by $\\mu ,\\tilde{\\mu }\\in {P}({F})$ the corresponding probability measures.",
"Recall that the characteristic functional of $\\mu $ is the functional $\\hat{\\mu } : {F}^*\\rightarrow \\mathbb {R}$ , $\\hat{\\mu }(\\varphi ) := \\int _{F}e^{i\\varphi (u)}\\,d\\mu (u), \\qquad \\varphi \\in {F}^*,$ and that $\\mu $ and $\\tilde{\\mu }$ coincide if and only if $\\hat{\\mu }=\\hat{\\tilde{\\mu }}$ (see [14]).",
"Using (REF ) we can interchange integration and summation in the following and obtain $\\hat{\\mu }(\\varphi ) &= \\int _{F}1+\\sum _{k=1}^\\infty \\frac{i^k}{k!",
"}\\varphi (u)^k\\,d\\mu (u) = 1 + \\sum _{k=1}^\\infty \\frac{i^k}{k!",
"}\\int _{F}\\left(\\int _D\\varphi (x)\\cdot u(x)\\,dx\\right)^kd\\mu (u) \\\\&= 1 + \\sum _{k=1}^\\infty \\frac{i^k}{k!",
"}\\int _{F}\\int _{D^k}\\bigl (\\varphi (x_1)\\cdot u(x_1)\\bigr )\\cdots \\bigl (\\varphi (x_k)\\cdot u(x_k)\\bigr )\\,dxd\\mu (u) \\\\&= 1 + \\sum _{k=1}^\\infty \\frac{i^k}{k!",
"}\\int _{F}\\int _{D^k}\\bigl (u(x_1)\\otimes \\cdots \\otimes u(x_k)\\bigr ) : \\bigl (\\varphi (x_1)\\otimes \\cdots \\otimes \\varphi (x_k)\\bigr )\\,dxd\\mu (u) \\\\&= 1 + \\sum _{k=1}^\\infty \\frac{i^k}{k!",
"}\\int _{D^k}\\int _{U^k}\\bigl (\\xi _1\\otimes \\cdots \\otimes \\xi _k\\bigr ):\\bigl (\\varphi (x_1)\\otimes \\cdots \\otimes \\varphi (x_k)\\bigr )\\,d\\nu ^k_xdx \\\\&= 1 + \\sum _{k=1}^\\infty \\frac{i^k}{k!",
"}\\int _{D^k}m^k(x):\\bigl (\\varphi (x_1)\\otimes \\cdots \\otimes \\varphi (x_k)\\bigr )\\,dx.$ Since the moments $m^k$ and $\\tilde{m}^k$ of $\\nu $ and $\\tilde{\\nu }$ coincide, we conclude that $\\mu =\\tilde{\\mu }$ ."
],
[
"Gaussian measures",
"As an example of the equivalence of probability measures on function spaces and correlation measures, we present here a (somewhat formal) computation which characterizes the correlation measure for Gaussian measures, a class of probability measures that is of great interest in stochastic analysis [14].",
"Although some of the following computations are quite standard in the literature on stochastic analysis, we include the details here for the sake of completeness.",
"We recall that a probability measure $\\rho \\in {P}(\\mathbb {R})$ is Gaussian if there is a number $\\sigma >0$ such that $\\bigl \\langle \\rho ,\\, f\\bigr \\rangle = \\frac{1}{\\sqrt{2\\pi \\sigma ^2}}\\int _\\mathbb {R}f(z) e^{-\\frac{z^2}{2\\sigma ^2}} dz$ for any $f\\in C_0(\\mathbb {R})$ .",
"(Note that we are implicitly assuming that $\\rho $ has mean zero, since the more general case of a nonzero mean can be easily obtained by translation.)",
"Given a Banach space $X$ , we say that a probability measure $\\mu \\in {P}(X)$ is Gaussian if $\\varphi \\#\\mu \\in {P}(\\mathbb {R})$ is Gaussian for every nonzero $\\varphi \\in X^*$ , that is, if for every $0\\ne \\varphi \\in X^*$ there is a number $\\sigma =\\sigma (\\varphi )>0$ such that $\\int _X f(\\varphi (u))\\,d\\mu (u) = \\frac{1}{\\sqrt{2\\pi \\sigma ^2}}\\int _\\mathbb {R}f(z)\\exp \\left(-\\frac{z^2}{2\\sigma ^2}\\right) dz \\qquad \\forall \\ f\\in C_0(\\mathbb {R}).$ We easily find that the variance $\\sigma (\\varphi )^2$ is given explicitly by $\\sigma (\\varphi )^2 = \\textrm {Var}(\\varphi \\#\\mu ) &= \\int _\\mathbb {R}y^2\\,d(\\varphi \\#\\mu )(y) = \\int _X \\varphi (u)^2\\,d\\mu (u) = \\bigl \\langle \\mu ,\\, \\varphi ^2\\bigr \\rangle .$ Choose now the Banach space $X={F}=L^p(D)$ .",
"For any $k\\in \\mathbb {N}$ and $0\\ne \\varphi \\in {F}^*$ , the expected value of the function $\\mathbb {R}\\ni z\\mapsto z^k$ with respect to $\\varphi \\#\\mu $ is $\\frac{1}{\\sqrt{2\\pi \\sigma (\\varphi )^2}}\\int _\\mathbb {R}z^k \\exp \\left(-\\frac{z^2}{2\\sigma (\\varphi )^2}\\right) dz &= \\bigl \\langle \\mu ,\\, \\varphi ^k\\bigr \\rangle = \\int _{F}\\int _{D^k} \\varphi (x_1)u(x_1)\\cdots \\varphi (x_k)u(x_k)\\,dxd\\mu (u) \\\\&= \\int _{D^k} \\int _{\\mathbb {R}^k} \\xi _1\\cdots \\xi _k \\varphi (x_1)\\cdots \\varphi (x_k)\\,d\\nu ^k_x(\\xi )dx \\\\&= \\int _{D^k} m^k(x)\\varphi (x_1)\\cdots \\varphi (x_k)\\,dx$ where $m^k(x):=\\int _{\\mathbb {R}^k}\\xi _1\\cdots \\xi _k\\,d\\nu ^k_x(\\xi )$ denotes the $k$ -th moment of $\\nu $ .",
"On the other hand, it is well-known that the $k$ -th moment $E[z^k]$ of a Gaussian distribution (with zero mean) is 0 when $k$ is odd, and $(k-1)!",
"!\\sigma ^k$ when $k$ is even, where $(k-1)!",
"!$ denotes the double factorial $(k-1)!!",
"= (k-1)(k-3)\\cdots 1 = \\frac{k!",
"}{(k/2)!2^{k/2}}$ .",
"Using the fact that $m^k(x_1,\\dots ,x_k)$ is symmetric in all arguments, we find that $m^k\\equiv 0$ when $k$ is odd.",
"When $k$ is even, i.e.",
"$k = 2l$ for some $l\\in \\mathbb {N}$ , we get $\\int _{D^{2l}} m^{2l}(x)\\varphi (x_1)\\cdots \\varphi (x_{2l})\\,dx &= \\frac{(2l)!",
"}{l!2^l}\\big (\\sigma (\\varphi )^2\\big )^l= \\frac{(2l)!",
"}{l!2^l}\\left(\\int _{D^{2}} m^2(x) \\varphi (x_1)\\varphi (x_2)\\,dx\\right)^l \\\\&= \\frac{(2l)!",
"}{l!2^l}\\int _{D^{2l}} m^2(x_1,x_2)\\cdots m^2(x_{2l-1},x_{2l}) \\varphi (x_1)\\cdots \\varphi (x_{2l})\\,dx.$ The above implies that the first integrand must be given by the symmetric part of the last integrand, i.e.",
"$m^{2l}(x) &= {\\rm Sym}\\left(\\frac{(2l)!",
"}{l!2^l}m^2\\otimes \\cdots \\otimes m^2\\right)(x) \\\\&= \\frac{1}{l!2^l}\\sum _{s\\in \\mathfrak {S}_{2l}}m^2\\bigl (x_{s(1)}, x_{s(2)}\\bigr )\\cdots m^2\\bigl (x_{s(2l-1)}, x_{s(2l)}\\bigr )$ where $\\mathfrak {S}_{k}$ is the symmetric group on $k$ symbols, consisting of all permutations of $\\lbrace 1, 2, \\dots , k\\rbrace $ (see e.g.",
"[13]).",
"Thus, all the moments—and thus all of $\\mu $ (or, equivalently, $\\nu $ )—is completely specified in terms of the second moment $m^2$ .",
"(This general rule is known as Isserlis' theorem [36]; see also [30].)",
"Finally, observe that $\\bigl \\langle \\nu ^1_x,\\, \\xi _1^n\\bigr \\rangle = m^n(x,\\dots ,x)= {\\left\\lbrace \\begin{array}{ll}0 & \\text{if $n$ is odd}, \\\\(n-1)!",
"!m^2(x,x)^{n/2} & \\text{if $n$ is even}\\end{array}\\right.",
"}$ (cf.",
"Remark REF (iii)).",
"Thus, for any $x\\in D$ , the probability measure $\\nu ^1_x$ is a Gaussian distribution with mean 0 and variance $m^2(x,x)$ .",
"More generally, for arbitrary $k$ we find that $\\nu ^k_{x_1,\\dots ,x_k}$ is a multivariate Gaussian distribution with mean $(0,\\dots ,0)$ and covariance $m^2(x_i,x_j)$ .",
"Thus, any function $m^2 : D^2 \\rightarrow \\mathbb {R}$ satisfying the properties of being a covariance function (see e.g.",
"[37]) corresponds to a unique Gaussian measure $\\mu \\in {P}(L^p(D))$ , and vice versa.",
"For instance, Brownian motion is obtained by letting $m^2(t,s)=\\min (t,s)$ for $t,s\\geqslant 0$ ."
],
[
"Statistical solutions",
"Equipped with the equivalence between probability measures on function spaces and correlation measures, we proceed in this section to define the concept of statistical solutions of multi-dimensional systems of conservation laws."
],
[
"Motivation and definition",
"To motivate the equations governing the time-evolution of statistical solutions, we consider a scalar, one-dimensional conservation law $\\partial _t u + \\partial _x f(u) = 0.$ This equation dictates the evolution of the quantity $u(x,t)$ over time.",
"For $x_1,x_2\\in \\mathbb {R}$ , consider the product $u(x_1,t)u(x_2,t)$ .",
"Assuming for the moment that $u$ is differentiable, we obtain $\\partial _t\\bigl [u(x_1,t)u(x_2,t)\\bigr ] &= \\bigl (\\partial _t u(x_1,t)\\bigr )u(x_2,t) + u(x_1,t)\\bigl (\\partial _t u(x_2,t)\\bigr ) \\\\&= -\\partial _{x_1} f(u(x_1,t))u(x_2,t) - \\partial _{x_2} u(x_1,t) f(u(x_2,t)),$ and for arbitrary $k\\in \\mathbb {N}$ , $\\partial _t \\bigl [u(x_1,t)\\cdots u(x_k,t)\\bigr ] + \\sum _{i=1}^k \\partial _{x_i}\\Bigl [u(x_1,t)\\cdots f(u(x_i,t)) \\cdots u(x_k,t)\\Bigr ] = 0.$ Since the above equation is in divergence form, it can be interpreted, in the sense of distributions, as $\\begin{split}\\int _{\\mathbb {R}_+}\\int _{\\mathbb {R}^k} \\partial _t \\varphi (x,t)\\, u(x_1,t)\\cdots u(x_k,t) + \\sum _{i=1}^k \\partial _{x_i}\\varphi (x,t)\\, u(x_1,t)\\cdots f(u(x_i,t)) \\cdots u(x_k,t)\\,dxdt \\\\+ \\int _{\\mathbb {R}^k}\\varphi (0,x)\\bar{u}(x_1)\\cdots \\bar{u}(x_k)\\,dx = 0\\end{split}$ for all $\\varphi \\in C_c^\\infty (\\mathbb {R}^k\\times \\mathbb {R}_+)$ .",
"For (multi-dimensional) systems, i.e.",
"when $u$ and $f(u)$ are vectors, we evolve the tensor product $u(x_1)\\otimes \\cdots \\otimes u(x_k)$ , and the resulting evolution equation (REF ) would read $\\partial _t \\bigl [u(x_1,t)\\otimes \\cdots \\otimes u(x_k,t)\\bigr ] + \\sum _{i=1}^k \\nabla _{x_i} \\cdot \\Bigl [u(x_1,t)\\otimes \\cdots \\otimes f(u(x_i,t))\\otimes \\cdots \\otimes u(x_k,t)\\Bigr ] = 0.$ Interpreting the above in the sense of distributions, we obtain $\\begin{split}\\int _{\\mathbb {R}_+}\\int _{\\mathbb {R}^k} \\partial _t \\varphi (x,t): \\bigl [u(x_1,t)\\otimes \\cdots \\otimes u(x_k,t)\\bigr ] + \\sum _{i=1}^k \\nabla _{x_i}\\cdot \\varphi (x,t) : \\Bigl [u(x_1,t)\\otimes \\cdots \\otimes f(u(x_i,t))\\otimes \\cdots \\otimes u(x_k,t)\\Bigr ]\\,dxdt \\\\+ \\int _{\\mathbb {R}^k}\\varphi (0,x):\\bigl [\\bar{u}(x_1)\\otimes \\cdots \\otimes \\bar{u}(x_k)\\bigr ]\\,dx = 0\\end{split}$ for all $\\varphi \\in C_c^\\infty \\Bigl (\\big (\\mathbb {R}^d\\big )^k\\times \\mathbb {R}_+,\\ \\big (\\mathbb {R}^N\\big )^{\\otimes k}\\Bigr )$ .",
"The above calculations can be made rigorous, as follows.",
"Lemma 3.1 If $u\\in L^1_{{\\rm loc}}(\\mathbb {R}^d\\times \\mathbb {R}_+,\\,\\mathbb {R}^N)$ is a weak solution of () then (REF ) holds for all $k\\in \\mathbb {N}$ .",
"For the sake of notational simplicity we present the proof only for the one-dimensional, scalar case ($d=N=1$ ).",
"The proof proceeds by induction.",
"Equation (REF ) with $k=1$ is precisely the definition of a weak solution, $\\int _{\\mathbb {R}_+}\\int _{\\mathbb {R}} \\partial _t \\psi u + \\partial _x\\psi u\\,dxdt + \\int _{\\mathbb {R}}\\psi (x,0)u(x)\\,dx = 0 \\qquad \\forall \\ \\psi \\in C_c^\\infty (\\mathbb {R}\\times \\mathbb {R}_+).$ Assume that (REF ) holds for some $k\\in \\mathbb {N}$ .",
"Let $\\omega _\\varepsilon :\\mathbb {R}\\rightarrow \\mathbb {R}$ be a symmetric mollifier with $\\operatorname{supp}\\omega _\\varepsilon \\subset [-\\varepsilon ,\\varepsilon ]$ , let $\\tilde{\\varphi }\\in C_c^\\infty (\\mathbb {R}^{k+1}\\times \\mathbb {R}_+)$ and define $\\varphi (x,t) := \\int _{\\mathbb {R}_+}\\int _\\mathbb {R}\\omega _\\varepsilon (t-s)\\tilde{\\varphi }(x,x_{k+1},s)u(x_{k+1},s)\\,dx_{k+1}ds$ for $x\\in \\mathbb {R}^k$ and any $0\\leqslant \\tilde{\\varphi }\\in C_c^\\infty (\\mathbb {R}^{k+1}\\times \\mathbb {R}_+)$ .",
"Then $\\varphi \\in C_c^\\infty (\\mathbb {R}^k\\times \\mathbb {R}_+)$ , and we have $\\partial _t\\varphi (x,t)&= \\int _{\\mathbb {R}_+}\\int _\\mathbb {R}\\omega _\\varepsilon ^{\\prime }(t-s)\\tilde{\\varphi }(x,x_{k+1},s)u(x_{k+1},s)\\,dx_{k+1}ds \\\\&= \\int _{\\mathbb {R}_+}\\int _\\mathbb {R}\\left[-\\partial _s\\Bigl (\\omega _\\varepsilon (t-s)\\tilde{\\varphi }(x,x_{k+1},s)\\Bigr ) + \\omega _\\varepsilon (t-s)\\partial _s\\tilde{\\varphi }(x,x_{k+1},s)\\right]u(x_{k+1},s)\\,dx_{k+1}ds \\\\&= \\int _{\\mathbb {R}_+}\\int _\\mathbb {R}\\omega _\\varepsilon (t-s)\\partial _{x_{k+1}}\\tilde{\\varphi }(x,x_{k+1},s)f(u(x_{k+1},s))\\,dx_{k+1}ds \\\\&\\quad + \\int _\\mathbb {R}\\omega _\\varepsilon (t)\\tilde{\\varphi }(x,x_{k+1},0) \\bar{u}(x_{k+1}) \\,dx_{k+1}\\\\&\\quad + \\int _{\\mathbb {R}_+}\\int _\\mathbb {R}\\omega _\\varepsilon (t-s)\\partial _s\\tilde{\\varphi }(x,x_{k+1},s)u(x_{k+1},s)\\,dx_{k+1}ds,$ the last equality following from (REF ).",
"Moreover, for $j=1,\\dots ,k$ we have $\\partial _{x_j}\\varphi (x,t) = \\int _{\\mathbb {R}_+}\\int _\\mathbb {R}\\omega _\\varepsilon (t-s)\\partial _{x_j}\\tilde{\\varphi }(x,x_{k+1},s)u(x_{k+1},s)\\,dx_{k+1}ds.$ Hence, inserting $\\varphi $ into (REF ) gives $0 &= \\int _{\\mathbb {R}_+}\\int _{\\mathbb {R}^k} u(x_1,t)\\cdots u(x_k,t)\\Biggl [\\int _{\\mathbb {R}_+}\\int _\\mathbb {R}\\omega _\\varepsilon (t-s)\\partial _{x_{k+1}}\\tilde{\\varphi }(x,x_{k+1},s)f(u(x_{k+1},s))\\,dx_{k+1}ds \\\\&\\quad + \\int _\\mathbb {R}\\omega _\\varepsilon (t)\\tilde{\\varphi }(x,x_{k+1},0) \\bar{u}(x_{k+1}) \\,dx_{k+1} + \\int _{\\mathbb {R}_+}\\int _\\mathbb {R}\\omega _\\varepsilon (t-s)\\partial _s\\tilde{\\varphi }(x,x_{k+1},s)u(x_{k+1},s)\\,dx_{k+1}ds\\Biggr ] \\\\&\\quad +\\sum _{j=1}^{k} u(x_1,t)\\cdots f(u(x_j,t))\\cdots u(x_k,t)\\int _{\\mathbb {R}_+}\\int _\\mathbb {R}\\omega _\\varepsilon (t-s)\\partial _{x_j}\\tilde{\\varphi }(x,x_{k+1},s)u(x_{k+1},s)\\,dx_{k+1}dsdxdt \\\\&\\quad + \\int _{\\mathbb {R}^k} \\bar{u}(x_1)\\cdots \\bar{u}(x_k)\\int _{\\mathbb {R}_+}\\int _\\mathbb {R}\\omega _\\varepsilon (-s)\\tilde{\\varphi }(x,x_{k+1},s)u(x_{k+1},s)\\,dx_{k+1}dsdx.$ In the limit $\\varepsilon \\rightarrow 0$ we get $0 &= \\int _{\\mathbb {R}_+}\\int _{\\mathbb {R}^k}\\int _\\mathbb {R}u(x_1,t)\\cdots u(x_k,t) \\partial _{x_{k+1}}\\tilde{\\varphi }(x,x_{k+1},t)f(u(x_{k+1},t))\\,dx_{k+1}dxdt \\\\&\\quad + \\frac{1}{2}\\int _{\\mathbb {R}^k}\\int _\\mathbb {R}\\bar{u}(x_1)\\cdots \\bar{u}(x_k) \\tilde{\\varphi }(x,x_{k+1},0) \\bar{u}(x_{k+1}) \\,dx_{k+1}dx \\\\&\\quad + \\int _{\\mathbb {R}_+}\\int _{\\mathbb {R}^k}\\int _\\mathbb {R}u(x_1,t)\\cdots u(x_k,t) \\partial _t\\tilde{\\varphi }(x,x_{k+1},t)u(x_{k+1},t)\\,dx_{k+1}dxdt \\\\&\\quad +\\int _{\\mathbb {R}_+}\\int _{\\mathbb {R}^k}\\int _\\mathbb {R}\\sum _{j=1}^{k} u(x_1,t)\\cdots f(u(x_j,t))\\cdots u(x_k,t) \\partial _{x_j}\\tilde{\\varphi }(x,x_{k+1},t)u(x_{k+1},t)\\,dx_{k+1}dxdt \\\\&\\quad + \\frac{1}{2}\\int _{\\mathbb {R}^k}\\int _\\mathbb {R}\\bar{u}(x_1)\\cdots \\bar{u}(x_k) \\tilde{\\varphi }(x,x_{k+1},0)\\bar{u}(x_{k+1})\\,dx_{k+1}dx$ (The factors $\\frac{1}{2}$ come from integrating $\\omega _\\varepsilon (-s)$ over $s\\in \\mathbb {R}_+$ and not $s\\in \\mathbb {R}$ .)",
"After reorganizing terms, we obtain (REF ) for $k+1$ .",
"Denoting the atomic correlation measure corresponding to $u(\\cdot ,t)$ by $\\nu _t = (\\nu ^{1}_t, \\nu ^{2}_t, \\dots )$ (cf.",
"Remark REF(ii)), we may write (REF ) equivalently as $\\partial _t \\bigl \\langle \\nu ^k_{t,x},\\, \\xi _1\\otimes \\cdots \\otimes \\xi _k\\bigr \\rangle + \\sum _{i=1}^k \\nabla _{x_i}\\cdot \\bigl \\langle \\nu ^k_{t,x},\\, \\xi _1\\otimes \\cdots \\otimes f(\\xi _i)\\otimes \\cdots \\otimes \\xi _k\\bigr \\rangle = 0$ for $x\\in \\mathbb {R}^k$ , $t>0$ and any $k\\in \\mathbb {N}$ .",
"Note that this expression makes sense even if $\\nu ^k_t$ is non-atomic.",
"We take this as the definition of a possibly non-atomic statistical solution.",
"In order for the terms appearing in (REF ) to be well-defined, we need to assume $\\int _{\\bar{D}^k} \\bigl \\langle \\nu ^k_{t,x},\\, |\\xi _1|\\cdots |\\xi _k|\\bigr \\rangle \\,dx < \\infty , \\qquad \\int _{\\bar{D}^k} \\bigl \\langle \\nu ^k_{t,x},\\, |\\xi _1|\\cdots |f(\\xi _i)|\\cdots |\\xi _k|\\bigr \\rangle \\,dx < \\infty \\qquad \\forall \\ k\\in \\mathbb {N}\\quad {\\rm and} \\quad i = 1,2,\\ldots ,k,$ for all compact subsets $\\bar{D} \\subset D$ .",
"We can write this in terms of the corresponding probability measure $\\mu _t\\in {P}(L^1)$ as $\\int _{L^1} \\Vert u\\Vert _{L^1(\\bar{D}^k)}\\,d\\mu _t(u) < \\infty , \\qquad \\int _{L^1(\\bar{D}^k)} \\Vert f\\circ u\\Vert _{L^1(\\bar{D}^k)}\\Vert u\\Vert _{L^1(\\bar{D}^k)}^k d\\mu _t(u) < \\infty \\qquad \\forall \\ k\\in \\mathbb {N},$ and for all compact subsets $\\bar{D} \\subset D$ .",
"Let $\\bar{\\mu }\\in {P}\\big (L^1\\big (\\mathbb {R}^d,\\mathbb {R}^N\\big )\\big )$ satisfy the decay rate (REF ).",
"A statistical solution of (REF ) with initial data $\\bar{\\mu }$ is a weak*-measurable mapping $t \\mapsto \\mu _t\\in {P}\\big (L^1\\big (\\mathbb {R}^d,\\mathbb {R}^N\\big )\\big )$ such that each $\\mu _t$ satisfies the decay rate (REF ), and such that the corresponding correlation measures $(\\nu ^k_t)_{k\\in \\mathbb {N}}$ satisfy (REF ) in the sense of distributions, i.e.",
"$\\int _{\\mathbb {R}_+}\\int _{(\\mathbb {R}^d)^k} \\bigl \\langle \\nu ^k_{t,x},\\, \\xi _1\\otimes \\cdots \\otimes \\xi _k\\bigr \\rangle :\\partial _t \\varphi + \\sum _{i=1}^k \\bigl \\langle \\nu ^k_{t,x},\\, \\xi _1\\otimes \\cdots \\otimes f(\\xi _i)\\otimes \\cdots \\otimes \\xi _k\\bigr \\rangle : \\nabla _{x_i}\\varphi \\,dxdt \\\\+ \\int _{(\\mathbb {R}^d)^k}\\bigl \\langle \\bar{\\nu }^k_x,\\, \\xi _1\\otimes \\cdots \\otimes \\xi _k\\bigr \\rangle \\varphi \\bigr |_{t=0}\\,dx = 0$ for every $\\varphi \\in C_c^\\infty \\Big (\\big (\\mathbb {R}^d\\big )^k\\times \\mathbb {R}_+,\\ \\big (\\mathbb {R}^N\\big )^{\\otimes k}\\Big )$ and for every $k\\in \\mathbb {N}$ .",
"We denote $\\bar{\\nu }$ to the correlation measure associated with initial probability measure $\\bar{\\mu }$ .",
"(A map $\\mu : t\\mapsto \\mu _t\\in {P}\\big (L^1\\big (\\mathbb {R}^d,\\mathbb {R}^N\\big )\\big )$ is weak*-measurable if the pairing $\\bigl \\langle \\mu _t,\\, G\\bigr \\rangle = \\int _{L^1}G(u)\\,d\\mu _t(u)$ with any $G\\in C_b\\big (L^1\\big (\\mathbb {R}^d,\\mathbb {R}^N\\big )\\big )$ is Lebesgue measurable in $t$ (see e.g.",
"[19]).)",
"Note carefully that the evolution equation (REF ) dictates the evolution of the moments $\\bigl \\langle \\nu ^k_{t,x},\\, \\xi _1\\otimes \\cdots \\otimes \\xi _k\\bigr \\rangle $ (see Section REF ).",
"Recall from Theorem REF that the moments of a correlation measure uniquely identify the correlation measure.",
"Thus, instead of determining the time evolution of functionals on infinite-dimensional function spaces as in the Liouville and Hopf equations of [29], we reduce the problem to the evolution of functions $\\bigl \\langle \\nu ^k_{t,x},\\, \\xi _1\\otimes \\cdots \\otimes \\xi _k\\bigr \\rangle $ defined on the finite-dimensional spaces $(x,t)\\in \\big (\\mathbb {R}^d\\big )^k\\times \\mathbb {R}_+$ .",
"Equation (REF ) for $k=1$ is simply the definition of $\\nu ^{1}$ being a measure-valued solution of (REF ), as introduced by DiPerna [20].",
"In light of the previous remark, we see that—except when the correlation measure is atomic—the evolution equation for measure-valued solutions (i.e., (REF ) with $k=1$ ) never uniquely determines the full correlation measure $\\nu _t$ (or equivalently, $\\mu _t$ ).",
"In other words, except in the case of an atomic statistical solution, the evolution equation for the $(k+1)$ th moment can contain strictly more information than the equation for the $k$ th moment.",
"Thus, statistical solutions are much more constrained than measure-valued solutions with additional information being provided by multi-point correlation measures.",
"This additional information provided by the correlation measures, opens the possibility of enforcing uniqueness of the statistical solutions, if necessary by augmenting them with further admissibility conditions.",
"If $\\bar{\\mu } = \\delta _{\\bar{u}}$ and $\\mu _t = \\delta _{u(t)}$ with $\\bar{u}, u(t) \\in L^1\\big (\\mathbb {R}^d,\\mathbb {R}^N\\big )$ for a.e.",
"$t>0$ , then Definition REF reduces to the classical definition of a weak solution of (REF )."
],
[
"Statistical solutions for scalar conservation laws",
"In Section we defined statistical solutions for multi-dimensional systems of conservation laws.",
"In this section we investigate the well-posedness of statistical solutions of (multi-dimensional) scalar conservation laws.",
"To this end, we can utilize the well-posedness of the deterministic problem () to show existence of a statistical solution for a multi-dimensional scalar conservation law."
],
[
"The canonical statistical solution",
"Recall that for scalar conservation laws, the Cauchy problem () is well-posed for any $\\bar{u}\\in {U}:= L^1\\cap L^\\infty (\\mathbb {R}^d\\times \\mathbb {R}_+)$ , and the entropy solution $u(t) = S_t\\bar{u}$ lies in ${U}$ for all $t>0$ [38].",
"Here, $S_t : {U}\\rightarrow {U}$ denotes the entropy solution semi-group.",
"Denote ${F}:= L^1(\\mathbb {R}^d)$ .",
"Given initial data $\\bar{\\mu }\\in {P}({F})$ with $\\operatorname{supp}\\mu \\subset {U}$ , we define the canonical statistical solution by $\\mu _t := S_t\\#\\bar{\\mu }, \\qquad t\\geqslant 0,$ where the pushforward operator $\\#$ applies $S_t$ to each element of the support of $\\bar{\\mu }$ : $\\int _{F}G(u)\\,d\\left(S_t\\#\\bar{\\mu }\\right)(u) = \\int _{F}G(S_tu)\\,d\\bar{\\mu }(u), \\qquad G\\in C_b({F}).$ Thus, the canonical statistical solution is concentrated on the entropy solutions of every initial data in the support of $\\bar{\\mu }$ , and each entropy solution is given the same weight as $\\bar{\\mu }$ gives to the corresponding initial data.",
"The semi-group $S_t$ is a continuous map, so it is easy to see that the canonical statistical solution is a weak*-measurable map from $t\\in \\mathbb {R}_+$ to ${P}({F})$ .",
"Moreover, it is in fact a statistical solution: For every $k\\in \\mathbb {N}$ and $\\varphi \\in C_c(\\mathbb {R}^k\\times \\mathbb {R}_+)$ , we have $&\\int _{\\mathbb {R}_+}\\int _{(\\mathbb {R}^d)^k} \\partial _t \\varphi \\, \\bigl \\langle \\nu ^k_{t,x},\\, \\xi _1\\cdots \\xi _k\\bigr \\rangle + \\sum _{i=1}^k \\nabla _{x_i}\\varphi \\,: \\bigl \\langle \\nu ^k_{t,x},\\, \\xi _1\\cdots f(\\xi _i)\\cdots \\xi _k\\bigr \\rangle \\,dxdt + \\int _{(\\mathbb {R}^d)^k}\\varphi \\bigr |_{t=0}\\,\\bigl \\langle \\bar{\\nu }^k_x,\\, \\xi _1\\cdots \\xi _k\\bigr \\rangle \\,dx \\\\=&\\ \\int _{\\mathbb {R}_+}\\int _{F}\\int _{(\\mathbb {R}^d)^k} \\partial _t \\varphi \\, u(x_1)\\cdots u(x_k) + \\sum _{i=1}^k \\nabla _{x_i}\\varphi : \\Bigl [u(x_1)\\cdots f\\bigl (u(x_i)\\bigr ) \\cdots u(x_k)\\Bigr ]\\,dxd\\mu _t(u)dt \\\\&+ \\int _{F}\\int _{(\\mathbb {R}^d)^k}\\varphi (0,x)\\bar{u}(x_1)\\cdots \\bar{u}(x_k)\\,dxd\\bar{\\mu }(\\bar{u}) \\qquad \\qquad ({\\rm by ~(\\ref {eq:nukdef})}) \\\\=&\\ \\int _{F}\\Biggl [\\int _{\\mathbb {R}_+}\\int _{(\\mathbb {R}^d)^k} \\partial _t \\varphi \\, S_t\\bar{u}(x_1)\\cdots S_t\\bar{u}(x_k) + \\sum _{i=1}^k \\nabla _{x_i} \\varphi : \\Bigl [S_t\\bar{u}(x_1)\\cdots f\\bigl (S_t\\bar{u}(x_i)\\bigr ) \\cdots S_t\\bar{u}(x_k)\\Bigr ]\\,dxdt \\\\&+ \\int _{(\\mathbb {R}^d)^k}\\varphi (0,x)\\bar{u}(x_1)\\cdots \\bar{u}(x_k)\\,dx\\Biggr ]d\\bar{\\mu }(\\bar{u}) \\\\=&\\ 0$ by Lemma REF , since $S_t \\bar{u}$ is a weak solution of () for every $\\bar{u}\\in {U}$ .",
"It is also quite easy to see that the canonical statistical solution is stable with respect to the initial data.",
"We measure this stability in the 1-Wasserstein metric $W_1$ on ${F}$ (cf.",
"Definition REF ).",
"Let $\\bar{\\mu }, \\bar{\\rho } \\in {P}({F})$ be given initial data and let $\\bar{\\pi }\\in \\Pi (\\bar{\\mu },\\bar{\\rho })$ be an optimal transport plan from $\\bar{\\mu }$ to $\\bar{\\rho }$ .",
"For each $t\\geqslant 0$ we define $\\pi _t := (S_t,S_t)\\#\\bar{\\pi }$ , which lies in $\\Pi (\\mu _t,\\rho _t)$ (where $\\mu _t,\\rho _t$ are the corresponding canonical statistical solutions).",
"We find that $W_1(\\mu _t,\\rho _t) &\\leqslant \\int _{{F}^2} \\Vert u-v\\Vert _{{F}}\\,d\\pi _t(u,v) = \\int _{{F}^2} \\Vert S_t\\bar{u}-S_t\\bar{v}\\Vert _{{F}}\\,d\\bar{\\pi }(\\bar{u},\\bar{v}) \\\\&\\leqslant \\int _{{F}^2} \\Vert \\bar{u}-\\bar{v}\\Vert _{{F}}\\,d\\bar{\\pi }(\\bar{u},\\bar{v}) = W_1(\\bar{\\mu },\\bar{\\rho }),$ where the first inequality comes from picking a particular plan $\\pi _t\\in \\Pi (\\mu _t,\\rho _t)$ in (REF ), and the second inequality follows from the $L^1$ contraction property of $S_t$ .",
"We summarize these observations as follows.",
"Theorem 4.1 Let $\\bar{\\mu }\\in {P}({F})$ be a probability measure on ${F}$ satisfying (REF ), and define the canonical statistical solution $\\mu _t := S_t\\#\\bar{\\mu }$ for each $t\\in \\mathbb {R}_+$ .",
"Then $t\\mapsto \\mu _t$ is a statistical solution of (REF ) with data $\\bar{\\mu }$ , and if $\\rho _t$ is another canonical statistical solution with initial data $\\bar{\\rho }\\in {P}({F})$ then $W_1(\\mu _t,\\rho _t) \\leqslant W_1(\\bar{\\mu },\\bar{\\rho }).$"
],
[
"Well-posedness of statistical solutions",
"As shown in Section REF , there always exists a statistical solution for scalar conservation laws, and this solution is stable with respect to initial data.",
"This does not imply, however, that the canonical solution is unique, in the same way that there might exist several weak solutions for the deterministic equation ().",
"As in the deterministic setting, entropy conditions must be imposed in order to single out a unique solution.",
"Recall that the (Kruzkov) entropy condition for () is $\\partial _t |u-c| + \\nabla _x \\cdot q(u,c) \\leqslant 0 \\qquad \\text{in } \\mathcal {D}^{\\prime }(\\mathbb {R}^d\\times \\mathbb {R}_+)$ for all constants $c\\in \\mathbb {R}$ , where $q(u,c) := {\\rm sgn}(u-c)(f(u)-f(c))$ .",
"Although not usually phrased as such, the Kruzkov entropy condition imposes stability with respect to a certain family of stationary (steady-state) solutions, namely the constant solutions.",
"The key to proving uniqueness of statistical solutions lies in finding the right family of stationary (time-invariant) solutions.",
"A natural first attempt follows from integrating (REF ) over the phase-space variable, which yields $\\partial _t \\bigl \\langle \\nu ^1,\\, |\\xi -c|\\bigr \\rangle + \\nabla _x \\cdot \\bigl \\langle \\nu ^1,\\, q(\\xi ,c)\\bigr \\rangle \\leqslant 0 \\qquad \\text{in } \\mathcal {D}^{\\prime }(\\mathbb {R}^d\\times \\mathbb {R}_+).$ This is the entropy condition enforced by DiPerna in the context of measure-valued solutions [20].",
"By a standard doubling-of-variables argument (see [20] and [23]), this leads to the stability estimate $\\int _{\\mathbb {R}^d} \\bigl \\langle \\nu ^1_{t,x},\\, \\big |\\xi -v(x,t)\\big |\\bigr \\rangle \\,dx \\leqslant \\int _{\\mathbb {R}^d} \\bigl \\langle \\bar{\\nu }^1_x,\\, \\big |\\xi -\\bar{v}(x)\\big |\\bigr \\rangle \\,dx$ for any entropy solution $v$ .",
"Thus, if $\\bar{\\nu }^1_x = \\delta _{\\bar{u}(x)}$ then also $\\nu ^1_{t,x} = \\delta _{u(x,t)}$ —in other words, (REF ) provides stability with respect to entropy solutions $u(x,t)$ , realized as atomic entropy measure-valued solutions.",
"Note, however, that if $\\bar{\\nu }$ is non-atomic then the right-hand side of (REF ) is $O(1)$ .",
"Hence, (REF ) only imposes stability with respect to atomic statistical solutions.",
"We propose instead the following: Entropy condition: The physically meaningful statistical solution must be stable not just with respect to single constant functions, but to any finite convex combination of constant functions.",
"Since constant functions do not lie in $L^1(\\mathbb {R}^d)$ , we need to introduce the following auxiliary lemma, which characterizes the set of transport plans, $\\Pi (\\mu ,\\rho )$ , when $\\rho $ is a convex combination of Dirac measures.",
"Lemma 4.2 Let $\\mu ,\\rho \\in {P}({F})$ such that $\\rho $ is of the form $\\rho = \\sum _{i=1}^M \\alpha _i \\delta _{u_i}$ for coefficients $\\alpha _i\\geqslant 0$ , $\\sum _i\\alpha _i=1$ and functions $u_1,\\dots ,u_M\\in {F}$ .",
"Then a measure $\\pi $ lies in $\\Pi (\\mu ,\\rho )$ if and only if there are $\\mu _1,\\dots ,\\mu _M\\in {P}({F})$ such that $\\pi = \\sum _{i=1}^M \\alpha _i\\mu _i\\otimes \\delta _{u_i}$ (and, in particular, $\\sum _{i=1}^M \\alpha _i\\mu _i = \\mu $ ).",
"Necessity is immediate.",
"For sufficiency, let $\\pi \\in \\Pi (\\mu ,\\rho )$ and define $\\mu _i(A) := \\frac{\\pi (A\\times \\lbrace u_i\\rbrace )}{\\alpha _i}$ .",
"Without loss of generality, we may assume that $\\alpha _i>0$ and that $u_1,\\dots ,u_M$ are distinct.",
"Since $\\pi ({F}\\times \\lbrace u_i\\rbrace ) = \\rho (\\lbrace u_i\\rbrace ) = \\alpha _i$ we have $\\mu _i\\in {P}({F})$ for each $i$ .",
"Moreover, $\\pi (A\\times \\lbrace u_i\\rbrace ) = \\alpha _i\\mu _i(A) = \\alpha _i(\\mu _i\\otimes \\delta _{u_i})(A\\times \\lbrace u_i\\rbrace )$ for each $i$ , so (REF ) follows.",
"Based on this simple observation we conclude that whenever $\\rho $ is $M$ -atomic with weights $\\alpha _i$ , there is a one-to-one correspondence between transport plans $\\pi \\in \\Pi (\\mu ,\\rho )$ and elements of the set $\\Lambda (\\alpha ,\\mu ) := \\Bigl \\lbrace (\\mu _1,\\dots ,\\mu _M)\\ :\\ \\textstyle \\sum _{i=1}^M \\alpha _i\\mu _i = \\mu \\Bigr \\rbrace \\qquad \\text{for } \\alpha =(\\alpha _1,\\dots ,\\alpha _M),\\ \\alpha _i\\geqslant 0,\\ \\sum _{i=1}^M \\alpha _i=1.$ This set is never empty since $(\\mu ,\\dots ,\\mu ) \\in \\Lambda (\\alpha ,\\mu )$ for any $\\alpha $ .",
"Note that the set $\\Lambda (\\alpha ,\\mu )$ depends on the target measure $\\rho $ only through the weights $\\alpha _1,\\dots ,\\alpha _M$ .",
"A statistical solution $\\mu _t$ is termed an entropy statistical solution if for every choice of coefficients $\\alpha _i>0$ with $\\sum _{i=1}^M \\alpha _i=1$ and for every $(\\bar{\\mu }_1,\\dots ,\\bar{\\mu }_M)\\in \\Lambda (\\alpha ,\\bar{\\mu })$ , there exists a map $t\\mapsto (\\mu _{1,t},\\dots ,\\mu _{M,t})\\in \\Lambda (\\alpha ,\\mu _t)$ such that $\\mu _{i,0}=\\bar{\\mu }_i$ and $\\begin{split}\\sum _{i=1}^M\\alpha _i\\left[\\int _{\\mathbb {R}_+}\\int _{{F}}\\int _{\\mathbb {R}^d} \\big |u(x)-c_i\\big |\\partial _t\\varphi + q\\big (u(x),c_i\\big )\\cdot \\nabla _x\\varphi \\,dxd\\mu _{i,t}(u)dt +\\int _{{F}}\\int _{\\mathbb {R}^d}\\big |\\bar{u}(x)-c_i\\big |\\varphi \\Bigr |_{t=0}\\,dxd\\bar{\\mu }_i(\\bar{u})\\right] \\geqslant 0\\end{split}$ for all $0\\leqslant \\varphi \\in C_c^\\infty (\\mathbb {R}^d\\times \\mathbb {R}_+)$ and for all constants $c_1,\\dots ,c_M\\in \\mathbb {R}$ .",
"(Here, $q(u,c)$ is the Kruzkov entropy flux function.)",
"Lemma 4.3 The canonical statistical solution is an entropy statistical solution.",
"Select $(\\bar{\\mu }_1,\\dots ,\\bar{\\mu }_M)\\in \\Lambda (\\alpha ,\\bar{\\mu })$ for an arbitrary weight $\\alpha $ and define $\\mu _{i,t} := S_t\\#\\bar{\\mu }_i$ .",
"Then $(\\mu _{1,t},\\dots ,\\mu _{M,t})\\in \\Lambda (\\alpha ,\\mu _t)$ , and $&\\sum _{i=1}^M\\alpha _i\\left[\\int _{\\mathbb {R}_+}\\int _{{F}}\\int _{\\mathbb {R}^d} \\bigl |u(x)-c_i\\bigr |\\partial _t\\varphi + q\\big (u(x),c_i\\big )\\cdot \\nabla _x \\varphi \\,dxd\\mu _{i,t}(u)dt + \\int _{{F}}\\int _{\\mathbb {R}^d}\\big |\\bar{u}(x)-c_i\\big |\\varphi \\Bigr |_{t=0}\\,dxd\\bar{\\mu }_i(\\bar{u})\\right] \\\\=&\\ \\sum _{i=1}^M\\alpha _i\\int _{F}\\left[\\int _{\\mathbb {R}_+}\\int _{\\mathbb {R}^d} \\big |S_t\\bar{u}(x)-c_i\\big |\\partial _t\\varphi + q\\big (S_t\\bar{u}(x),c_i\\big )\\cdot \\nabla _x \\varphi \\,dxdt + \\int _{\\mathbb {R}^d}\\big |\\bar{u}(x)-c_i\\big |\\varphi \\Bigr |_{t=0}\\,dx\\right]d\\bar{\\mu }_i(\\bar{u}) \\\\\\geqslant &\\ 0,$ since the map $(x,t)\\mapsto S_t\\bar{u}(x)$ is an entropy solution of the deterministic problem.",
"Note that Lemma 4.4 Let $\\mu _t$ be an arbitrary entropy statistical solution with initial data $\\bar{\\mu }\\in {P}({F})$ satisfying $\\operatorname{supp}\\bar{\\mu }\\subset {U}$ .",
"Fix $\\alpha _1,\\dots ,\\alpha _M>0$ with $\\sum _{i=1}^{M}\\alpha _i=1$ .",
"Let $v_1,\\dots ,v_M : \\mathbb {R}_+\\rightarrow {U}$ be entropy solutions of (REF ) with initial data $\\bar{v}_1,\\dots ,\\bar{v}_M\\in {U}$ , respectively, and define $\\bar{\\rho }:=\\sum _{i=1}^M\\alpha _i\\delta _{v_i}, \\qquad \\rho _t:=\\sum _{i=1}^M\\alpha _i\\delta _{v_i(t)} \\qquad \\forall \\ t\\in \\mathbb {R}_+.$ Then $W_1(\\rho _t,\\mu _t) \\leqslant W_1(\\bar{\\rho },\\bar{\\mu }) \\qquad \\forall \\ t>0.$ Let $(\\bar{\\mu }_i)_{i=1}^M\\in \\Lambda (\\alpha ,\\bar{\\mu })$ be an optimal transport plan from $\\bar{\\mu }$ to $\\bar{\\rho }$ .",
"The entropy condition for $\\mu _t$ gives the existence of a map $t\\mapsto \\big (\\mu _{i,t}\\big )_{i=1}^n$ such that $\\sum _{i=1}^M\\alpha _i\\left[\\int _{\\mathbb {R}_+}\\int _{{F}}\\int _{\\mathbb {R}^d} \\big |u(x)-c_i\\big |\\partial _t\\varphi + q\\big (u(x),c_i\\big )\\cdot \\nabla _x \\ \\varphi \\,dxd\\mu _{i,t}(u)dt + \\int _{{F}}\\int _{\\mathbb {R}^d}\\big |\\bar{u}(x)-c_i\\big |\\varphi \\Bigr |_{t=0}\\,dxd\\bar{\\mu }_i(\\bar{u})\\right] \\geqslant 0$ for any choice of $\\varphi \\in C_c^\\infty (\\mathbb {R}^d \\times \\mathbb {R}_+)$ and $c_i\\in \\mathbb {R}$ .",
"Let $\\varphi = \\varphi (x,y,t,s)\\in C_c^\\infty ((\\mathbb {R}^d)^2\\times \\mathbb {R}_+^2)$ .",
"Set $c_i = v_i(y,s)$ for some point $(y,s)$ and integrate over $y\\in \\mathbb {R}$ and $s\\in \\mathbb {R}_+$ : $\\begin{split}\\int _{\\mathbb {R}_+}\\int _{\\mathbb {R}^d}\\sum _{i=1}^M\\alpha _i\\Biggl [\\int _{\\mathbb {R}_+}\\int _{{F}}\\int _{\\mathbb {R}^d} \\big |u(x)-v_i(y,s)\\big |\\partial _t\\varphi + q\\big (u(x),v_i(y,s)\\big )\\cdot \\nabla _x \\varphi \\,dxd\\mu _{i,t}(u)dt \\\\+ \\int _{{F}}\\int _{\\mathbb {R}^d}\\big |\\bar{u}(x)-v_i(y,s)\\big |\\varphi \\Bigr |_{t=0}\\,dxd\\bar{\\mu }_i(\\bar{u})\\Biggr ]dyds \\geqslant 0.\\end{split}$ (The expression in the brackets is measurable with respect to $(y,s)$ since (REF ) is continuous with respect to $c_i$ .)",
"Next, since each $v_i$ is an entropy solution, we have for all $\\xi \\in \\mathbb {R}$ and $0\\leqslant \\varphi \\in C_c^\\infty (\\mathbb {R}^d\\times \\mathbb {R}_+)$ $\\int _{\\mathbb {R}_+}\\int _{\\mathbb {R}^d} \\big |\\xi -v_i(y,s)\\big |\\partial _s\\varphi + q\\big (\\xi ,v_i(y,s)\\big )\\cdot \\nabla _y\\varphi \\,dyds + \\int _{\\mathbb {R}^d} \\big |\\xi -\\bar{v}_i(y)\\big |\\varphi \\Bigr |_{s=0}\\,dy \\geqslant 0.$ Set $\\xi =u(x)$ for some $u\\in {F}$ and $x\\in \\mathbb {R}$ .",
"Integrate the above over $x\\in \\mathbb {R}$ and over $u\\in {F}$ with respect to $\\mu _{i,t}$ for some $t\\in \\mathbb {R}_+$ .",
"Integrate over $t\\in \\mathbb {R}_+$ , multiply by $\\alpha _i$ and sum over $i=1,\\dots ,M$ : $\\begin{split}\\sum _{i=1}^{M}\\alpha _i\\int _{\\mathbb {R}_+}\\int _{F}\\int _{\\mathbb {R}^d} \\Biggl [\\int _{\\mathbb {R}_+}\\int _{\\mathbb {R}^d} \\big |u(x)-v_i(y,s)\\big |\\partial _s\\varphi + q\\big (u(x),v_i(y,s)\\big )\\cdot \\nabla _y\\varphi \\,dyds \\\\+ \\int _{\\mathbb {R}^d} \\big |u(x)-\\bar{v}_i(y)\\big |\\varphi \\Bigr |_{s=0}\\,dy \\Biggr ]dxd\\mu _{i,t}(u)dt \\geqslant 0.\\end{split}$ Applying Fubini's theorem to this and equation (REF ) and adding the two, we obtain $\\sum _{i=1}^M\\alpha _i \\Biggl [\\int _{\\mathbb {R}_+}\\int _{\\mathbb {R}_+}\\int _{F}\\int _{\\mathbb {R}^d}\\int _{\\mathbb {R}^d} \\big |u(x)-v_i(y,s)\\big |(\\partial _t+\\partial _s)\\varphi + q\\big (u(x),v_i(y,s)\\big )\\cdot (\\nabla _x+\\nabla _y)\\varphi \\,dxdyd\\mu _{i,t}(u)dtds \\\\+ \\int _{\\mathbb {R}_+}\\int _{F}\\int _{\\mathbb {R}^d}\\int _{\\mathbb {R}^d} \\big |u(x)-\\bar{v}_i(y)\\big |\\varphi \\Bigr |_{s=0}\\,dxdyd\\mu _{i,t}(u)dt + \\int _{\\mathbb {R}_+}\\int _{{F}}\\int _{\\mathbb {R}^d}\\int _{\\mathbb {R}^d} \\big |\\bar{u}(x)-v_i(y,s)\\big |\\varphi \\Bigr |_{t=0}\\,dxdyd\\bar{\\mu }_i(\\bar{u})ds\\Biggr ] \\geqslant 0.$ Now set $\\varphi (x,y,t,s) := \\psi \\Bigl (\\frac{x+y}{2}, \\frac{t+s}{2}\\Bigr )\\omega _\\varepsilon (x-y)\\omega _{\\varepsilon ^{\\prime }}(t-s)$ for some nonnegative $\\psi \\in C_c^\\infty (\\mathbb {R}^d\\times \\mathbb {R}_+)$ and a mollifier $\\omega _\\varepsilon $ .",
"Using the dominated convergence theorem on the integrals over ${F}$ , we find that as $\\varepsilon \\rightarrow 0$ , the above converges to $\\sum _{i=1}^M\\alpha _i \\Biggl [\\int _{\\mathbb {R}_+}\\int _{\\mathbb {R}_+}\\int _{F}\\int _{\\mathbb {R}^d} \\big |u(x)-v_i(x,s)\\big |(\\partial _t+\\partial _s)\\tilde{\\varphi } + 2q\\big (u(x),v_i(x,s)\\big )\\cdot \\nabla _x\\tilde{\\varphi }\\,dxd\\mu _{i,t}(u)dtds \\\\+ \\int _{\\mathbb {R}_+}\\int _{F}\\int _{\\mathbb {R}^d} \\big |u(x)-\\bar{v}_i(x)\\big |\\tilde{\\varphi }\\Bigr |_{s=0}\\,dxd\\mu _{i,t}(u)dt + \\int _{\\mathbb {R}_+}\\int _{{F}}\\int _{\\mathbb {R}^d} \\big |\\bar{u}(x)-v_i(x,s)\\big |\\tilde{\\varphi }\\Bigr |_{t=0}\\,dxd\\bar{\\mu }_i(\\bar{u})ds\\Biggr ] \\geqslant 0,$ where $\\tilde{\\varphi }(x,t,s) := \\psi \\Bigl (x,\\frac{t+s}{2}\\Bigr )\\omega _{\\varepsilon ^{\\prime }}(t-s)$ .",
"Finally, letting $\\varepsilon ^{\\prime }\\rightarrow 0$ we get $\\sum _{i=1}^M\\alpha _i \\Biggl [\\int _{\\mathbb {R}_+}\\int _{F}\\int _{\\mathbb {R}^d} \\big |u(x)-v_i(x,t)\\big |\\partial _t\\psi + q\\big (u(x),v_i(x,t)\\big )\\cdot \\nabla _x\\psi \\,dxd\\mu _{i,t}(u)dt \\\\+ \\int _{F}\\int _{\\mathbb {R}^d} \\big |\\bar{u}(x)-\\bar{v}_i(x)\\big |\\psi \\Bigr |_{t=0}\\,dxd\\bar{\\mu }_i(\\bar{u})\\Biggr ] \\geqslant 0.$ We now set $\\psi (x,\\tau ) := \\mathbb {1}_{[0,t]}(\\tau )$ for some $t\\in \\mathbb {R}_+$ to get $\\sum _{i=1}^M\\alpha _i\\Biggl [-\\int _{F}\\big \\Vert u-v_i(t)\\big \\Vert _{{F}}\\,d\\mu _{i,t}(u) + \\int _{F}\\big \\Vert \\bar{u}-\\bar{v}_i\\big \\Vert _{F}\\,d\\bar{\\mu }_i(\\bar{u})\\Biggr ] \\geqslant 0.$ Using the fact that $(\\bar{\\mu }_i)$ is an optimal transport plan from $\\bar{\\mu }$ to $\\bar{\\rho }$ , we end up with (REF ).",
"To complete our proof of well-posedness of statistical solutions we need the following well-known result, whose proof is included in the appendix for the sake of completeness.",
"Lemma 4.5 Let $X$ be a Polish space equipped with its Borel $\\sigma $ -algebra.",
"Then the convex hull of Dirac measures on $X$ is dense in ${P}(X)$ with respect to the topology of weak convergence.",
"In other words, for every $\\mu \\in {P}(X)$ , there is a sequence $\\rho _n\\in {P}(X)$ of convex combinations of Dirac measures such that $\\rho _n\\rightharpoonup \\mu $ as $n\\rightarrow \\infty $ .",
"Theorem 4.6 Let $\\bar{\\mu }\\in {P}({F})$ with $\\operatorname{supp}\\bar{\\mu }\\subset {U}:=L^1\\cap L^\\infty (\\mathbb {R}^d)$ .",
"Then the entropy statistical solution with initial data $\\bar{\\mu }$ is unique and coincides with the canonical statistical solution.",
"Any two entropy statistical solutions $\\mu _t$ , $\\rho _t$ satisfy $W_1(\\mu _t,\\rho _t) \\leqslant W_1(\\bar{\\mu },\\bar{\\rho }).$ Let $\\mu _t$ be an entropy statistical solution with initial data $\\bar{\\mu }$ .",
"By Lemma REF , the convex hull of Dirac measures is dense in ${P}({F})$ , so we can find a sequence $\\bar{\\mu }_n\\in {P}({F})$ ($n\\in \\mathbb {N}$ ) of convex combinations of Dirac measures such that $\\bar{\\mu }_n \\rightharpoonup \\bar{\\mu }$ in ${P}({F})$ as $n\\rightarrow \\infty $ .",
"Let $\\mu _{n,t} := S_t\\#\\bar{\\mu }_n$ be the corresponding canonical statistical solutions, and note that also $\\mu _{n,t} \\rightharpoonup S_t\\#\\bar{\\mu }$ as $n\\rightarrow \\infty $ .",
"From Lemma REF we find that $W_1(\\mu _t,\\mu _{n,t}) \\leqslant W_1(\\bar{\\mu },\\bar{\\mu }_n) \\rightarrow 0 \\qquad \\text{as } n\\rightarrow \\infty .$ Thus, $\\mu _t = \\operatornamewithlimits{w-lim}_{n\\rightarrow \\infty }\\mu _{n,t} = S_t\\#\\bar{\\mu }$ , whence $\\mu _t$ is the canonical statistical solution."
],
[
"Discussion",
"Given the lack of global in time existence results, and the recent non-uniqueness results of [16], [17], the acceptance of entropy solutions as the standard solution paradigm for multi-dimensional systems of conservation laws is being increasingly questioned.",
"Based on extensive numerical results, recent papers such as [23] have advocated entropy measure-valued solutions (MVS), as defined by DiPerna [20], as an appropriate solution paradigm for systems of conservation laws.",
"However, entropy MVS are not necessarily unique, even for scalar conservation laws, if the MVS is non-atomic.",
"Since numerical results of [23] strongly hint at the possibility of non-atomic MVS even when the initial data is a atomic, it is natural to seek additional constraints on entropy MVS to enforce uniqueness.",
"Given this background, and the need for developing a solution concept that can accommodate uncertain initial data (and corresponding uncertain solutions) that arise frequently in the area of uncertainty quantification (UQ), we seek to adapt the notion of statistical solutions, originally developed in [27], [28] for the incompressible Navier–Stokes equations, to systems of conservation laws.",
"Statistical solutions are time-parametrized probability measures on some (infinite-dimensional) function space.",
"Infinite-dimensional Liouville or Hopf equations track the evolution of the time-parametrized measure.",
"However, the extension of statistical solutions as defined in [27], [28], [29], to systems of conservation laws, is highly non-trivial as the “natural” function spaces for the dynamics of conservation laws consists merely of integrable functions, and may lack the regularity required to define the Liouville or Hopf equations.",
"Although one can work with probability measures on distributions in the specific case of the inviscid Burgers equation (as suggested in [9], [10], [5]), it is very difficult to enforce uniqueness on such a large space of measures.",
"Another disadvantage of probability measures on functions is that they do not readily provide any local (statistical) information at specific (collections of) points in the spatial domain.",
"We define statistical solutions for systems of conservation laws in a different manner.",
"To this end, we prove a novel equivalence theorem between probability measures on $L^p$ spaces ($1 \\leqslant p < \\infty $ ) and a family (hierarchy) of Young measures, the so-called correlation measures, on finite-dimensional tensor product spatial domains.",
"For all $k \\in \\mathbb {N}$ , the $k$ -th member of this hierarchy, the so-called $k$ -point correlation marginal, is a Young measure that provides information on correlations of the underlying functions at $k$ distinct points in the spatial domain.",
"In particular, the first correlation marginal is classical one-point Young measure.",
"Thus, a probability measure on an $L^p$ space can be realized as an Young measure, augmented with multi-point correlations on the spatial domain.",
"This representation enables us to localize probability measures on function spaces and view them as a collection of all possible multi-point correlation marginals.",
"We also show that moments of the correlation marginals uniquely determine the corresponding probability measure on the infinite-dimensional function space.",
"We believe that this representation of probability measures will be of independent interest in stochastic analysis, particularly stochastic partial differential equations [14], in uncertainty quantification of evolutionary PDEs [31] and in Bayesian inversion and data assimilation for time-dependent PDEs [42].",
"In particular, the use of statistical solutions will provide a framework for uncertainty quantification that does not depend on any particular parametrization of the solution in terms of random fields, as is customary in UQ [31].",
"In this paper, we use the equivalence between probability measures on $L^p$ and families of correlation measures to define statistical solutions of systems of conservation laws.",
"In particular, we utilize the fact that moments of correlation measures uniquely determine the underlying probability measure, to evolve these moments in a manner consistent with the dynamics of the system (REF ).",
"Thus, a statistical solution has to satisfy an (infinite) family of nonlinear PDEs, but each of these PDEs is defined on a finite-dimensional (tensor-product) spatial domain.",
"This should be contrasted with the infinite-dimensional Liouville or Hopf equations that the statistical solutions of [27], [28], [29] need to satisfy.",
"Moreover, our notion of statistical solutions restricts the class of probability measures to those on $L^p$ spaces, rather than on distributions (as in [10]) and makes it more amenable to analysis, particularly from the point of view of uniqueness.",
"At the same time, our notion of statistical solutions augment the standard concept of measure-valued solutions, with additional information in the form of multi-point correlations, and paves the way for constraining the solutions sufficiently to guarantee uniqueness.",
"We investigate the well-posedness of the proposed concept of statistical solutions in the specific context of multi-dimensional scalar conservation laws in this paper.",
"We show existence by proving that the push forward of the initial probability measure on $L^1 \\cap L^{\\infty }$ by the Kruzkhov entropy solution semi-group is a statistical solution, and we term this solution the canonical statistical solution.",
"We propose a novel admissibility criteria, based on stability with respect to a suitable stationary statistical solution, namely probability measures supported on finite collections of constant functions.",
"These entropy statistical solutions are a generalization of the standard Kruzkhov entropy solutions for scalar conservation laws.",
"We show that the canonical statistical solution is the unique entropy statistical solution.",
"Furthermore, we show that it is contractive with respect to the 1-Wasserstein metric on probability measures on $L^1$ .",
"Thus, entropy statistical solutions for multi-dimensional scalar conservation laws are shown to be well-posed and are thus completely characterized.",
"This article is the first in a series of papers investigating statistical solutions of multi-dimensional systems of conservation laws.",
"We lay out the measure theoretic basis, define statistical solutions for systems and show well-posedness in the scalar case.",
"Forthcoming papers in the series will deal with numerical approximation of entropy statistical solutions of scalar conservation laws [25] and global existence of statistical solutions for a large class of multi-dimensional systems of conservation laws by showing convergence of a Monte Carlo based numerical approximation algorithm [26].",
"Admissibility criteria that single out physically relevant statistical solutions are the topic of current and future work."
],
[
"Acknowledgments",
"U.S.F.",
"was supported in part by the grant Waves and Nonlinear Phenomena (WaNP) from the Research Council of Norway.",
"S.M.",
"was supported in part by ERC STG.",
"N 306279, SPARCCLE.",
"The authors thanks Kjetil O. Lye and Franziska Weber (SAM, ETH) for their helpful comments."
],
[
"Appendix",
"For completeness we provide the proof of Proposition REF .",
"The proof relies on the following two lemmas.",
"Lemma A.1 $\\mathit {Cyl}(X)$ is a ring.1A collection of sets ${X}\\subset 2^X$ is a ring if $\\emptyset \\in {X}$ and if both $A\\cup B$ and $A\\setminus B$ lie in ${X}$ whenever $A,B\\in {X}$ .",
"Clearly, $\\emptyset \\in \\mathit {Cyl}(X)$ , and if $A_1, A_2\\in \\mathit {Cyl}(X)$ are of the form $A_i = \\left\\lbrace u\\in X\\ :\\ \\bigl (\\varphi _1^i, \\dots , \\varphi _{n_i}^i\\bigr )(u) \\in F_i\\right\\rbrace , \\qquad i=1,2$ then both $A_1\\cup A_2 = \\left\\lbrace u\\in X\\ :\\ \\bigl (\\varphi _1^1, \\dots , \\varphi _{n_1}^1, \\varphi _1^2, \\dots , \\varphi _{n_2}^2\\bigr )(u) \\in \\left(F_1\\times \\mathbb {R}^{n_2}\\right) \\cup \\left(\\mathbb {R}^{n_1}\\times F_2\\right)\\right\\rbrace $ and $A_1\\setminus A_2 = \\left\\lbrace u\\in X\\ :\\ \\bigl (\\varphi _1^1, \\dots , \\varphi _{n_1}^1, \\varphi _1^2, \\dots , \\varphi _{n_2}^2\\bigr )(u) \\in F_1\\times \\bigl (F_2\\bigr )^c\\right\\rbrace $ are cylinder sets.",
"Lemma A.2 If $X$ is a separable normed vector space then there exists a countable family $\\lbrace \\varphi _n\\rbrace _{n\\in \\mathbb {N}} \\subset X^*$ such that $\\Vert u\\Vert _X=\\sup _{n\\in \\mathbb {N}}\\,\\varphi _n(u) \\quad \\text{ for every $u\\in X$}.$ Let $\\lbrace u_n\\rbrace _{n\\in \\mathbb {N}} \\subset X$ be a countable dense subset of the unit sphere $\\partial B_1(0) \\subset X$ .",
"For each $n\\in \\mathbb {N}$ , let $\\varphi _n\\in X^*$ satisfy $\\varphi _n(u_n) = 1$ and $\\Vert \\varphi _n\\Vert _{X^*} = 1$ .",
"If $u\\in \\partial B_1(0)$ is arbitrary and $\\varepsilon >0$ , find an $u_n$ such that $\\Vert u-u_n\\Vert _X < \\varepsilon $ .",
"Then $1 \\geqslant \\varphi _n(u) = \\varphi _n(u_n) - \\varphi _n(u_n-u) \\geqslant 1-\\varepsilon ,$ so $\\Vert u\\Vert _X=1$ can be approximated from below by $\\varphi _n(u)$ .",
"Equation (REF ) follows.",
"[Proof of Proposition REF ] Let $\\lbrace \\varphi _n\\rbrace _{n\\in \\mathbb {N}}$ be as in Lemma REF .",
"For a $u_0\\in X$ and $r>0$ , the open ball of radius $r$ with centre $u_0$ can be written $B_r(u_0) &= \\Bigl \\lbrace u\\in X\\ :\\ \\varphi _n(u-u_0) < r\\ \\forall \\ n\\in \\mathbb {N}\\Bigr \\rbrace \\\\&= \\bigcap _{n\\in \\mathbb {N}} \\Bigl \\lbrace u\\in X\\ :\\ \\varphi _n(u) \\in \\bigl (-\\infty ,\\ \\varphi _n(u_0)+r\\bigr )\\Bigr \\rbrace ,$ which is a countable intersection of cylinder sets.",
"It follows that $\\sigma (\\mathit {Cyl}(X))$ , the $\\sigma $ -algebra generated by $\\mathit {Cyl}(X)$ , contains the $\\sigma $ -algebra generated by the open balls in $X$ , which is precisely ${B}(X)$ .",
"But every cylinder set is a Borel set; hence the two $\\sigma $ -algebras coincide, and (i) follows.",
"By Lemma REF , $\\mathit {Cyl}(X)$ is a ring which, by (i), generates ${B}(X)$ .",
"Assertion (ii) then follows from the fact that (signed) measures vanishing on a ring, vanish on the $\\sigma $ -algebra generated by the ring.",
"[Proof of Lemma REF ] Recall that the topology of weak convergence on ${P}(X)$ for a Polish metric space $X$ is the coarsest topology for which the map $\\mu \\mapsto \\int \\varphi \\,d\\mu $ is continuous for every $\\varphi \\in C_b(X)$ [37].",
"Thus, the topology of weak convergence is generated by the open sets $U_{\\varphi ,\\mu ,\\varepsilon } := \\left\\lbrace \\rho \\in {P}(X)\\ :\\ \\Bigl |\\int \\varphi \\, d\\mu - \\int \\varphi \\,d\\rho \\Bigr | < \\varepsilon \\right\\rbrace $ for $\\mu \\in {P}(X)$ , $\\varepsilon >0$ and $\\varphi \\in C_b(X)$ .",
"It suffices to show that every nonempty open set $U_{\\varphi ,\\mu ,\\varepsilon }$ contains a measure which is a convex combination of Dirac measures.",
"Let $\\bar{\\varphi }(x) = \\sum _{i=1}^n a_i\\mathbb {1}_{A_i}(x)$ be a simple function such that $\\sup _{x\\in X}|\\varphi (x)-\\bar{\\varphi }(x)| < \\varepsilon /2$ .",
"Fix $x_i\\in A_i$ and define $\\rho := \\sum _{i=1}^n \\mu (A_i)\\delta _{x_i}$ .",
"Since $|\\varphi (x_i)-\\varphi (x)|<\\varepsilon $ for every $x\\in A_i$ , we find that $\\bigg |\\int _X \\varphi \\,d\\rho - \\int _X \\varphi \\,d\\mu \\bigg | = \\biggl |\\sum _{i=1}^n\\int _{A_i}\\varphi (x_i)- \\varphi (x)\\,d\\mu \\biggr | \\leqslant \\sum _{i=1}^n\\int _{A_i}|\\varphi (x_i)-\\varphi (x)|\\,d\\mu < \\varepsilon .$ Hence, $\\rho \\in U_{\\varphi ,\\mu ,\\varepsilon }$ ."
],
[
"Statistical solutions",
"Equipped with the equivalence between probability measures on function spaces and correlation measures, we proceed in this section to define the concept of statistical solutions of multi-dimensional systems of conservation laws."
],
[
"Motivation and definition",
"To motivate the equations governing the time-evolution of statistical solutions, we consider a scalar, one-dimensional conservation law $\\partial _t u + \\partial _x f(u) = 0.$ This equation dictates the evolution of the quantity $u(x,t)$ over time.",
"For $x_1,x_2\\in \\mathbb {R}$ , consider the product $u(x_1,t)u(x_2,t)$ .",
"Assuming for the moment that $u$ is differentiable, we obtain $\\partial _t\\bigl [u(x_1,t)u(x_2,t)\\bigr ] &= \\bigl (\\partial _t u(x_1,t)\\bigr )u(x_2,t) + u(x_1,t)\\bigl (\\partial _t u(x_2,t)\\bigr ) \\\\&= -\\partial _{x_1} f(u(x_1,t))u(x_2,t) - \\partial _{x_2} u(x_1,t) f(u(x_2,t)),$ and for arbitrary $k\\in \\mathbb {N}$ , $\\partial _t \\bigl [u(x_1,t)\\cdots u(x_k,t)\\bigr ] + \\sum _{i=1}^k \\partial _{x_i}\\Bigl [u(x_1,t)\\cdots f(u(x_i,t)) \\cdots u(x_k,t)\\Bigr ] = 0.$ Since the above equation is in divergence form, it can be interpreted, in the sense of distributions, as $\\begin{split}\\int _{\\mathbb {R}_+}\\int _{\\mathbb {R}^k} \\partial _t \\varphi (x,t)\\, u(x_1,t)\\cdots u(x_k,t) + \\sum _{i=1}^k \\partial _{x_i}\\varphi (x,t)\\, u(x_1,t)\\cdots f(u(x_i,t)) \\cdots u(x_k,t)\\,dxdt \\\\+ \\int _{\\mathbb {R}^k}\\varphi (0,x)\\bar{u}(x_1)\\cdots \\bar{u}(x_k)\\,dx = 0\\end{split}$ for all $\\varphi \\in C_c^\\infty (\\mathbb {R}^k\\times \\mathbb {R}_+)$ .",
"For (multi-dimensional) systems, i.e.",
"when $u$ and $f(u)$ are vectors, we evolve the tensor product $u(x_1)\\otimes \\cdots \\otimes u(x_k)$ , and the resulting evolution equation (REF ) would read $\\partial _t \\bigl [u(x_1,t)\\otimes \\cdots \\otimes u(x_k,t)\\bigr ] + \\sum _{i=1}^k \\nabla _{x_i} \\cdot \\Bigl [u(x_1,t)\\otimes \\cdots \\otimes f(u(x_i,t))\\otimes \\cdots \\otimes u(x_k,t)\\Bigr ] = 0.$ Interpreting the above in the sense of distributions, we obtain $\\begin{split}\\int _{\\mathbb {R}_+}\\int _{\\mathbb {R}^k} \\partial _t \\varphi (x,t): \\bigl [u(x_1,t)\\otimes \\cdots \\otimes u(x_k,t)\\bigr ] + \\sum _{i=1}^k \\nabla _{x_i}\\cdot \\varphi (x,t) : \\Bigl [u(x_1,t)\\otimes \\cdots \\otimes f(u(x_i,t))\\otimes \\cdots \\otimes u(x_k,t)\\Bigr ]\\,dxdt \\\\+ \\int _{\\mathbb {R}^k}\\varphi (0,x):\\bigl [\\bar{u}(x_1)\\otimes \\cdots \\otimes \\bar{u}(x_k)\\bigr ]\\,dx = 0\\end{split}$ for all $\\varphi \\in C_c^\\infty \\Bigl (\\big (\\mathbb {R}^d\\big )^k\\times \\mathbb {R}_+,\\ \\big (\\mathbb {R}^N\\big )^{\\otimes k}\\Bigr )$ .",
"The above calculations can be made rigorous, as follows.",
"Lemma 3.1 If $u\\in L^1_{{\\rm loc}}(\\mathbb {R}^d\\times \\mathbb {R}_+,\\,\\mathbb {R}^N)$ is a weak solution of () then (REF ) holds for all $k\\in \\mathbb {N}$ .",
"For the sake of notational simplicity we present the proof only for the one-dimensional, scalar case ($d=N=1$ ).",
"The proof proceeds by induction.",
"Equation (REF ) with $k=1$ is precisely the definition of a weak solution, $\\int _{\\mathbb {R}_+}\\int _{\\mathbb {R}} \\partial _t \\psi u + \\partial _x\\psi u\\,dxdt + \\int _{\\mathbb {R}}\\psi (x,0)u(x)\\,dx = 0 \\qquad \\forall \\ \\psi \\in C_c^\\infty (\\mathbb {R}\\times \\mathbb {R}_+).$ Assume that (REF ) holds for some $k\\in \\mathbb {N}$ .",
"Let $\\omega _\\varepsilon :\\mathbb {R}\\rightarrow \\mathbb {R}$ be a symmetric mollifier with $\\operatorname{supp}\\omega _\\varepsilon \\subset [-\\varepsilon ,\\varepsilon ]$ , let $\\tilde{\\varphi }\\in C_c^\\infty (\\mathbb {R}^{k+1}\\times \\mathbb {R}_+)$ and define $\\varphi (x,t) := \\int _{\\mathbb {R}_+}\\int _\\mathbb {R}\\omega _\\varepsilon (t-s)\\tilde{\\varphi }(x,x_{k+1},s)u(x_{k+1},s)\\,dx_{k+1}ds$ for $x\\in \\mathbb {R}^k$ and any $0\\leqslant \\tilde{\\varphi }\\in C_c^\\infty (\\mathbb {R}^{k+1}\\times \\mathbb {R}_+)$ .",
"Then $\\varphi \\in C_c^\\infty (\\mathbb {R}^k\\times \\mathbb {R}_+)$ , and we have $\\partial _t\\varphi (x,t)&= \\int _{\\mathbb {R}_+}\\int _\\mathbb {R}\\omega _\\varepsilon ^{\\prime }(t-s)\\tilde{\\varphi }(x,x_{k+1},s)u(x_{k+1},s)\\,dx_{k+1}ds \\\\&= \\int _{\\mathbb {R}_+}\\int _\\mathbb {R}\\left[-\\partial _s\\Bigl (\\omega _\\varepsilon (t-s)\\tilde{\\varphi }(x,x_{k+1},s)\\Bigr ) + \\omega _\\varepsilon (t-s)\\partial _s\\tilde{\\varphi }(x,x_{k+1},s)\\right]u(x_{k+1},s)\\,dx_{k+1}ds \\\\&= \\int _{\\mathbb {R}_+}\\int _\\mathbb {R}\\omega _\\varepsilon (t-s)\\partial _{x_{k+1}}\\tilde{\\varphi }(x,x_{k+1},s)f(u(x_{k+1},s))\\,dx_{k+1}ds \\\\&\\quad + \\int _\\mathbb {R}\\omega _\\varepsilon (t)\\tilde{\\varphi }(x,x_{k+1},0) \\bar{u}(x_{k+1}) \\,dx_{k+1}\\\\&\\quad + \\int _{\\mathbb {R}_+}\\int _\\mathbb {R}\\omega _\\varepsilon (t-s)\\partial _s\\tilde{\\varphi }(x,x_{k+1},s)u(x_{k+1},s)\\,dx_{k+1}ds,$ the last equality following from (REF ).",
"Moreover, for $j=1,\\dots ,k$ we have $\\partial _{x_j}\\varphi (x,t) = \\int _{\\mathbb {R}_+}\\int _\\mathbb {R}\\omega _\\varepsilon (t-s)\\partial _{x_j}\\tilde{\\varphi }(x,x_{k+1},s)u(x_{k+1},s)\\,dx_{k+1}ds.$ Hence, inserting $\\varphi $ into (REF ) gives $0 &= \\int _{\\mathbb {R}_+}\\int _{\\mathbb {R}^k} u(x_1,t)\\cdots u(x_k,t)\\Biggl [\\int _{\\mathbb {R}_+}\\int _\\mathbb {R}\\omega _\\varepsilon (t-s)\\partial _{x_{k+1}}\\tilde{\\varphi }(x,x_{k+1},s)f(u(x_{k+1},s))\\,dx_{k+1}ds \\\\&\\quad + \\int _\\mathbb {R}\\omega _\\varepsilon (t)\\tilde{\\varphi }(x,x_{k+1},0) \\bar{u}(x_{k+1}) \\,dx_{k+1} + \\int _{\\mathbb {R}_+}\\int _\\mathbb {R}\\omega _\\varepsilon (t-s)\\partial _s\\tilde{\\varphi }(x,x_{k+1},s)u(x_{k+1},s)\\,dx_{k+1}ds\\Biggr ] \\\\&\\quad +\\sum _{j=1}^{k} u(x_1,t)\\cdots f(u(x_j,t))\\cdots u(x_k,t)\\int _{\\mathbb {R}_+}\\int _\\mathbb {R}\\omega _\\varepsilon (t-s)\\partial _{x_j}\\tilde{\\varphi }(x,x_{k+1},s)u(x_{k+1},s)\\,dx_{k+1}dsdxdt \\\\&\\quad + \\int _{\\mathbb {R}^k} \\bar{u}(x_1)\\cdots \\bar{u}(x_k)\\int _{\\mathbb {R}_+}\\int _\\mathbb {R}\\omega _\\varepsilon (-s)\\tilde{\\varphi }(x,x_{k+1},s)u(x_{k+1},s)\\,dx_{k+1}dsdx.$ In the limit $\\varepsilon \\rightarrow 0$ we get $0 &= \\int _{\\mathbb {R}_+}\\int _{\\mathbb {R}^k}\\int _\\mathbb {R}u(x_1,t)\\cdots u(x_k,t) \\partial _{x_{k+1}}\\tilde{\\varphi }(x,x_{k+1},t)f(u(x_{k+1},t))\\,dx_{k+1}dxdt \\\\&\\quad + \\frac{1}{2}\\int _{\\mathbb {R}^k}\\int _\\mathbb {R}\\bar{u}(x_1)\\cdots \\bar{u}(x_k) \\tilde{\\varphi }(x,x_{k+1},0) \\bar{u}(x_{k+1}) \\,dx_{k+1}dx \\\\&\\quad + \\int _{\\mathbb {R}_+}\\int _{\\mathbb {R}^k}\\int _\\mathbb {R}u(x_1,t)\\cdots u(x_k,t) \\partial _t\\tilde{\\varphi }(x,x_{k+1},t)u(x_{k+1},t)\\,dx_{k+1}dxdt \\\\&\\quad +\\int _{\\mathbb {R}_+}\\int _{\\mathbb {R}^k}\\int _\\mathbb {R}\\sum _{j=1}^{k} u(x_1,t)\\cdots f(u(x_j,t))\\cdots u(x_k,t) \\partial _{x_j}\\tilde{\\varphi }(x,x_{k+1},t)u(x_{k+1},t)\\,dx_{k+1}dxdt \\\\&\\quad + \\frac{1}{2}\\int _{\\mathbb {R}^k}\\int _\\mathbb {R}\\bar{u}(x_1)\\cdots \\bar{u}(x_k) \\tilde{\\varphi }(x,x_{k+1},0)\\bar{u}(x_{k+1})\\,dx_{k+1}dx$ (The factors $\\frac{1}{2}$ come from integrating $\\omega _\\varepsilon (-s)$ over $s\\in \\mathbb {R}_+$ and not $s\\in \\mathbb {R}$ .)",
"After reorganizing terms, we obtain (REF ) for $k+1$ .",
"Denoting the atomic correlation measure corresponding to $u(\\cdot ,t)$ by $\\nu _t = (\\nu ^{1}_t, \\nu ^{2}_t, \\dots )$ (cf.",
"Remark REF(ii)), we may write (REF ) equivalently as $\\partial _t \\bigl \\langle \\nu ^k_{t,x},\\, \\xi _1\\otimes \\cdots \\otimes \\xi _k\\bigr \\rangle + \\sum _{i=1}^k \\nabla _{x_i}\\cdot \\bigl \\langle \\nu ^k_{t,x},\\, \\xi _1\\otimes \\cdots \\otimes f(\\xi _i)\\otimes \\cdots \\otimes \\xi _k\\bigr \\rangle = 0$ for $x\\in \\mathbb {R}^k$ , $t>0$ and any $k\\in \\mathbb {N}$ .",
"Note that this expression makes sense even if $\\nu ^k_t$ is non-atomic.",
"We take this as the definition of a possibly non-atomic statistical solution.",
"In order for the terms appearing in (REF ) to be well-defined, we need to assume $\\int _{\\bar{D}^k} \\bigl \\langle \\nu ^k_{t,x},\\, |\\xi _1|\\cdots |\\xi _k|\\bigr \\rangle \\,dx < \\infty , \\qquad \\int _{\\bar{D}^k} \\bigl \\langle \\nu ^k_{t,x},\\, |\\xi _1|\\cdots |f(\\xi _i)|\\cdots |\\xi _k|\\bigr \\rangle \\,dx < \\infty \\qquad \\forall \\ k\\in \\mathbb {N}\\quad {\\rm and} \\quad i = 1,2,\\ldots ,k,$ for all compact subsets $\\bar{D} \\subset D$ .",
"We can write this in terms of the corresponding probability measure $\\mu _t\\in {P}(L^1)$ as $\\int _{L^1} \\Vert u\\Vert _{L^1(\\bar{D}^k)}\\,d\\mu _t(u) < \\infty , \\qquad \\int _{L^1(\\bar{D}^k)} \\Vert f\\circ u\\Vert _{L^1(\\bar{D}^k)}\\Vert u\\Vert _{L^1(\\bar{D}^k)}^k d\\mu _t(u) < \\infty \\qquad \\forall \\ k\\in \\mathbb {N},$ and for all compact subsets $\\bar{D} \\subset D$ .",
"Let $\\bar{\\mu }\\in {P}\\big (L^1\\big (\\mathbb {R}^d,\\mathbb {R}^N\\big )\\big )$ satisfy the decay rate (REF ).",
"A statistical solution of (REF ) with initial data $\\bar{\\mu }$ is a weak*-measurable mapping $t \\mapsto \\mu _t\\in {P}\\big (L^1\\big (\\mathbb {R}^d,\\mathbb {R}^N\\big )\\big )$ such that each $\\mu _t$ satisfies the decay rate (REF ), and such that the corresponding correlation measures $(\\nu ^k_t)_{k\\in \\mathbb {N}}$ satisfy (REF ) in the sense of distributions, i.e.",
"$\\int _{\\mathbb {R}_+}\\int _{(\\mathbb {R}^d)^k} \\bigl \\langle \\nu ^k_{t,x},\\, \\xi _1\\otimes \\cdots \\otimes \\xi _k\\bigr \\rangle :\\partial _t \\varphi + \\sum _{i=1}^k \\bigl \\langle \\nu ^k_{t,x},\\, \\xi _1\\otimes \\cdots \\otimes f(\\xi _i)\\otimes \\cdots \\otimes \\xi _k\\bigr \\rangle : \\nabla _{x_i}\\varphi \\,dxdt \\\\+ \\int _{(\\mathbb {R}^d)^k}\\bigl \\langle \\bar{\\nu }^k_x,\\, \\xi _1\\otimes \\cdots \\otimes \\xi _k\\bigr \\rangle \\varphi \\bigr |_{t=0}\\,dx = 0$ for every $\\varphi \\in C_c^\\infty \\Big (\\big (\\mathbb {R}^d\\big )^k\\times \\mathbb {R}_+,\\ \\big (\\mathbb {R}^N\\big )^{\\otimes k}\\Big )$ and for every $k\\in \\mathbb {N}$ .",
"We denote $\\bar{\\nu }$ to the correlation measure associated with initial probability measure $\\bar{\\mu }$ .",
"(A map $\\mu : t\\mapsto \\mu _t\\in {P}\\big (L^1\\big (\\mathbb {R}^d,\\mathbb {R}^N\\big )\\big )$ is weak*-measurable if the pairing $\\bigl \\langle \\mu _t,\\, G\\bigr \\rangle = \\int _{L^1}G(u)\\,d\\mu _t(u)$ with any $G\\in C_b\\big (L^1\\big (\\mathbb {R}^d,\\mathbb {R}^N\\big )\\big )$ is Lebesgue measurable in $t$ (see e.g.",
"[19]).)",
"Note carefully that the evolution equation (REF ) dictates the evolution of the moments $\\bigl \\langle \\nu ^k_{t,x},\\, \\xi _1\\otimes \\cdots \\otimes \\xi _k\\bigr \\rangle $ (see Section REF ).",
"Recall from Theorem REF that the moments of a correlation measure uniquely identify the correlation measure.",
"Thus, instead of determining the time evolution of functionals on infinite-dimensional function spaces as in the Liouville and Hopf equations of [29], we reduce the problem to the evolution of functions $\\bigl \\langle \\nu ^k_{t,x},\\, \\xi _1\\otimes \\cdots \\otimes \\xi _k\\bigr \\rangle $ defined on the finite-dimensional spaces $(x,t)\\in \\big (\\mathbb {R}^d\\big )^k\\times \\mathbb {R}_+$ .",
"Equation (REF ) for $k=1$ is simply the definition of $\\nu ^{1}$ being a measure-valued solution of (REF ), as introduced by DiPerna [20].",
"In light of the previous remark, we see that—except when the correlation measure is atomic—the evolution equation for measure-valued solutions (i.e., (REF ) with $k=1$ ) never uniquely determines the full correlation measure $\\nu _t$ (or equivalently, $\\mu _t$ ).",
"In other words, except in the case of an atomic statistical solution, the evolution equation for the $(k+1)$ th moment can contain strictly more information than the equation for the $k$ th moment.",
"Thus, statistical solutions are much more constrained than measure-valued solutions with additional information being provided by multi-point correlation measures.",
"This additional information provided by the correlation measures, opens the possibility of enforcing uniqueness of the statistical solutions, if necessary by augmenting them with further admissibility conditions.",
"If $\\bar{\\mu } = \\delta _{\\bar{u}}$ and $\\mu _t = \\delta _{u(t)}$ with $\\bar{u}, u(t) \\in L^1\\big (\\mathbb {R}^d,\\mathbb {R}^N\\big )$ for a.e.",
"$t>0$ , then Definition REF reduces to the classical definition of a weak solution of (REF )."
],
[
"Statistical solutions for scalar conservation laws",
"In Section we defined statistical solutions for multi-dimensional systems of conservation laws.",
"In this section we investigate the well-posedness of statistical solutions of (multi-dimensional) scalar conservation laws.",
"To this end, we can utilize the well-posedness of the deterministic problem () to show existence of a statistical solution for a multi-dimensional scalar conservation law."
],
[
"The canonical statistical solution",
"Recall that for scalar conservation laws, the Cauchy problem () is well-posed for any $\\bar{u}\\in {U}:= L^1\\cap L^\\infty (\\mathbb {R}^d\\times \\mathbb {R}_+)$ , and the entropy solution $u(t) = S_t\\bar{u}$ lies in ${U}$ for all $t>0$ [38].",
"Here, $S_t : {U}\\rightarrow {U}$ denotes the entropy solution semi-group.",
"Denote ${F}:= L^1(\\mathbb {R}^d)$ .",
"Given initial data $\\bar{\\mu }\\in {P}({F})$ with $\\operatorname{supp}\\mu \\subset {U}$ , we define the canonical statistical solution by $\\mu _t := S_t\\#\\bar{\\mu }, \\qquad t\\geqslant 0,$ where the pushforward operator $\\#$ applies $S_t$ to each element of the support of $\\bar{\\mu }$ : $\\int _{F}G(u)\\,d\\left(S_t\\#\\bar{\\mu }\\right)(u) = \\int _{F}G(S_tu)\\,d\\bar{\\mu }(u), \\qquad G\\in C_b({F}).$ Thus, the canonical statistical solution is concentrated on the entropy solutions of every initial data in the support of $\\bar{\\mu }$ , and each entropy solution is given the same weight as $\\bar{\\mu }$ gives to the corresponding initial data.",
"The semi-group $S_t$ is a continuous map, so it is easy to see that the canonical statistical solution is a weak*-measurable map from $t\\in \\mathbb {R}_+$ to ${P}({F})$ .",
"Moreover, it is in fact a statistical solution: For every $k\\in \\mathbb {N}$ and $\\varphi \\in C_c(\\mathbb {R}^k\\times \\mathbb {R}_+)$ , we have $&\\int _{\\mathbb {R}_+}\\int _{(\\mathbb {R}^d)^k} \\partial _t \\varphi \\, \\bigl \\langle \\nu ^k_{t,x},\\, \\xi _1\\cdots \\xi _k\\bigr \\rangle + \\sum _{i=1}^k \\nabla _{x_i}\\varphi \\,: \\bigl \\langle \\nu ^k_{t,x},\\, \\xi _1\\cdots f(\\xi _i)\\cdots \\xi _k\\bigr \\rangle \\,dxdt + \\int _{(\\mathbb {R}^d)^k}\\varphi \\bigr |_{t=0}\\,\\bigl \\langle \\bar{\\nu }^k_x,\\, \\xi _1\\cdots \\xi _k\\bigr \\rangle \\,dx \\\\=&\\ \\int _{\\mathbb {R}_+}\\int _{F}\\int _{(\\mathbb {R}^d)^k} \\partial _t \\varphi \\, u(x_1)\\cdots u(x_k) + \\sum _{i=1}^k \\nabla _{x_i}\\varphi : \\Bigl [u(x_1)\\cdots f\\bigl (u(x_i)\\bigr ) \\cdots u(x_k)\\Bigr ]\\,dxd\\mu _t(u)dt \\\\&+ \\int _{F}\\int _{(\\mathbb {R}^d)^k}\\varphi (0,x)\\bar{u}(x_1)\\cdots \\bar{u}(x_k)\\,dxd\\bar{\\mu }(\\bar{u}) \\qquad \\qquad ({\\rm by ~(\\ref {eq:nukdef})}) \\\\=&\\ \\int _{F}\\Biggl [\\int _{\\mathbb {R}_+}\\int _{(\\mathbb {R}^d)^k} \\partial _t \\varphi \\, S_t\\bar{u}(x_1)\\cdots S_t\\bar{u}(x_k) + \\sum _{i=1}^k \\nabla _{x_i} \\varphi : \\Bigl [S_t\\bar{u}(x_1)\\cdots f\\bigl (S_t\\bar{u}(x_i)\\bigr ) \\cdots S_t\\bar{u}(x_k)\\Bigr ]\\,dxdt \\\\&+ \\int _{(\\mathbb {R}^d)^k}\\varphi (0,x)\\bar{u}(x_1)\\cdots \\bar{u}(x_k)\\,dx\\Biggr ]d\\bar{\\mu }(\\bar{u}) \\\\=&\\ 0$ by Lemma REF , since $S_t \\bar{u}$ is a weak solution of () for every $\\bar{u}\\in {U}$ .",
"It is also quite easy to see that the canonical statistical solution is stable with respect to the initial data.",
"We measure this stability in the 1-Wasserstein metric $W_1$ on ${F}$ (cf.",
"Definition REF ).",
"Let $\\bar{\\mu }, \\bar{\\rho } \\in {P}({F})$ be given initial data and let $\\bar{\\pi }\\in \\Pi (\\bar{\\mu },\\bar{\\rho })$ be an optimal transport plan from $\\bar{\\mu }$ to $\\bar{\\rho }$ .",
"For each $t\\geqslant 0$ we define $\\pi _t := (S_t,S_t)\\#\\bar{\\pi }$ , which lies in $\\Pi (\\mu _t,\\rho _t)$ (where $\\mu _t,\\rho _t$ are the corresponding canonical statistical solutions).",
"We find that $W_1(\\mu _t,\\rho _t) &\\leqslant \\int _{{F}^2} \\Vert u-v\\Vert _{{F}}\\,d\\pi _t(u,v) = \\int _{{F}^2} \\Vert S_t\\bar{u}-S_t\\bar{v}\\Vert _{{F}}\\,d\\bar{\\pi }(\\bar{u},\\bar{v}) \\\\&\\leqslant \\int _{{F}^2} \\Vert \\bar{u}-\\bar{v}\\Vert _{{F}}\\,d\\bar{\\pi }(\\bar{u},\\bar{v}) = W_1(\\bar{\\mu },\\bar{\\rho }),$ where the first inequality comes from picking a particular plan $\\pi _t\\in \\Pi (\\mu _t,\\rho _t)$ in (REF ), and the second inequality follows from the $L^1$ contraction property of $S_t$ .",
"We summarize these observations as follows.",
"Theorem 4.1 Let $\\bar{\\mu }\\in {P}({F})$ be a probability measure on ${F}$ satisfying (REF ), and define the canonical statistical solution $\\mu _t := S_t\\#\\bar{\\mu }$ for each $t\\in \\mathbb {R}_+$ .",
"Then $t\\mapsto \\mu _t$ is a statistical solution of (REF ) with data $\\bar{\\mu }$ , and if $\\rho _t$ is another canonical statistical solution with initial data $\\bar{\\rho }\\in {P}({F})$ then $W_1(\\mu _t,\\rho _t) \\leqslant W_1(\\bar{\\mu },\\bar{\\rho }).$"
],
[
"Well-posedness of statistical solutions",
"As shown in Section REF , there always exists a statistical solution for scalar conservation laws, and this solution is stable with respect to initial data.",
"This does not imply, however, that the canonical solution is unique, in the same way that there might exist several weak solutions for the deterministic equation ().",
"As in the deterministic setting, entropy conditions must be imposed in order to single out a unique solution.",
"Recall that the (Kruzkov) entropy condition for () is $\\partial _t |u-c| + \\nabla _x \\cdot q(u,c) \\leqslant 0 \\qquad \\text{in } \\mathcal {D}^{\\prime }(\\mathbb {R}^d\\times \\mathbb {R}_+)$ for all constants $c\\in \\mathbb {R}$ , where $q(u,c) := {\\rm sgn}(u-c)(f(u)-f(c))$ .",
"Although not usually phrased as such, the Kruzkov entropy condition imposes stability with respect to a certain family of stationary (steady-state) solutions, namely the constant solutions.",
"The key to proving uniqueness of statistical solutions lies in finding the right family of stationary (time-invariant) solutions.",
"A natural first attempt follows from integrating (REF ) over the phase-space variable, which yields $\\partial _t \\bigl \\langle \\nu ^1,\\, |\\xi -c|\\bigr \\rangle + \\nabla _x \\cdot \\bigl \\langle \\nu ^1,\\, q(\\xi ,c)\\bigr \\rangle \\leqslant 0 \\qquad \\text{in } \\mathcal {D}^{\\prime }(\\mathbb {R}^d\\times \\mathbb {R}_+).$ This is the entropy condition enforced by DiPerna in the context of measure-valued solutions [20].",
"By a standard doubling-of-variables argument (see [20] and [23]), this leads to the stability estimate $\\int _{\\mathbb {R}^d} \\bigl \\langle \\nu ^1_{t,x},\\, \\big |\\xi -v(x,t)\\big |\\bigr \\rangle \\,dx \\leqslant \\int _{\\mathbb {R}^d} \\bigl \\langle \\bar{\\nu }^1_x,\\, \\big |\\xi -\\bar{v}(x)\\big |\\bigr \\rangle \\,dx$ for any entropy solution $v$ .",
"Thus, if $\\bar{\\nu }^1_x = \\delta _{\\bar{u}(x)}$ then also $\\nu ^1_{t,x} = \\delta _{u(x,t)}$ —in other words, (REF ) provides stability with respect to entropy solutions $u(x,t)$ , realized as atomic entropy measure-valued solutions.",
"Note, however, that if $\\bar{\\nu }$ is non-atomic then the right-hand side of (REF ) is $O(1)$ .",
"Hence, (REF ) only imposes stability with respect to atomic statistical solutions.",
"We propose instead the following: Entropy condition: The physically meaningful statistical solution must be stable not just with respect to single constant functions, but to any finite convex combination of constant functions.",
"Since constant functions do not lie in $L^1(\\mathbb {R}^d)$ , we need to introduce the following auxiliary lemma, which characterizes the set of transport plans, $\\Pi (\\mu ,\\rho )$ , when $\\rho $ is a convex combination of Dirac measures.",
"Lemma 4.2 Let $\\mu ,\\rho \\in {P}({F})$ such that $\\rho $ is of the form $\\rho = \\sum _{i=1}^M \\alpha _i \\delta _{u_i}$ for coefficients $\\alpha _i\\geqslant 0$ , $\\sum _i\\alpha _i=1$ and functions $u_1,\\dots ,u_M\\in {F}$ .",
"Then a measure $\\pi $ lies in $\\Pi (\\mu ,\\rho )$ if and only if there are $\\mu _1,\\dots ,\\mu _M\\in {P}({F})$ such that $\\pi = \\sum _{i=1}^M \\alpha _i\\mu _i\\otimes \\delta _{u_i}$ (and, in particular, $\\sum _{i=1}^M \\alpha _i\\mu _i = \\mu $ ).",
"Necessity is immediate.",
"For sufficiency, let $\\pi \\in \\Pi (\\mu ,\\rho )$ and define $\\mu _i(A) := \\frac{\\pi (A\\times \\lbrace u_i\\rbrace )}{\\alpha _i}$ .",
"Without loss of generality, we may assume that $\\alpha _i>0$ and that $u_1,\\dots ,u_M$ are distinct.",
"Since $\\pi ({F}\\times \\lbrace u_i\\rbrace ) = \\rho (\\lbrace u_i\\rbrace ) = \\alpha _i$ we have $\\mu _i\\in {P}({F})$ for each $i$ .",
"Moreover, $\\pi (A\\times \\lbrace u_i\\rbrace ) = \\alpha _i\\mu _i(A) = \\alpha _i(\\mu _i\\otimes \\delta _{u_i})(A\\times \\lbrace u_i\\rbrace )$ for each $i$ , so (REF ) follows.",
"Based on this simple observation we conclude that whenever $\\rho $ is $M$ -atomic with weights $\\alpha _i$ , there is a one-to-one correspondence between transport plans $\\pi \\in \\Pi (\\mu ,\\rho )$ and elements of the set $\\Lambda (\\alpha ,\\mu ) := \\Bigl \\lbrace (\\mu _1,\\dots ,\\mu _M)\\ :\\ \\textstyle \\sum _{i=1}^M \\alpha _i\\mu _i = \\mu \\Bigr \\rbrace \\qquad \\text{for } \\alpha =(\\alpha _1,\\dots ,\\alpha _M),\\ \\alpha _i\\geqslant 0,\\ \\sum _{i=1}^M \\alpha _i=1.$ This set is never empty since $(\\mu ,\\dots ,\\mu ) \\in \\Lambda (\\alpha ,\\mu )$ for any $\\alpha $ .",
"Note that the set $\\Lambda (\\alpha ,\\mu )$ depends on the target measure $\\rho $ only through the weights $\\alpha _1,\\dots ,\\alpha _M$ .",
"A statistical solution $\\mu _t$ is termed an entropy statistical solution if for every choice of coefficients $\\alpha _i>0$ with $\\sum _{i=1}^M \\alpha _i=1$ and for every $(\\bar{\\mu }_1,\\dots ,\\bar{\\mu }_M)\\in \\Lambda (\\alpha ,\\bar{\\mu })$ , there exists a map $t\\mapsto (\\mu _{1,t},\\dots ,\\mu _{M,t})\\in \\Lambda (\\alpha ,\\mu _t)$ such that $\\mu _{i,0}=\\bar{\\mu }_i$ and $\\begin{split}\\sum _{i=1}^M\\alpha _i\\left[\\int _{\\mathbb {R}_+}\\int _{{F}}\\int _{\\mathbb {R}^d} \\big |u(x)-c_i\\big |\\partial _t\\varphi + q\\big (u(x),c_i\\big )\\cdot \\nabla _x\\varphi \\,dxd\\mu _{i,t}(u)dt +\\int _{{F}}\\int _{\\mathbb {R}^d}\\big |\\bar{u}(x)-c_i\\big |\\varphi \\Bigr |_{t=0}\\,dxd\\bar{\\mu }_i(\\bar{u})\\right] \\geqslant 0\\end{split}$ for all $0\\leqslant \\varphi \\in C_c^\\infty (\\mathbb {R}^d\\times \\mathbb {R}_+)$ and for all constants $c_1,\\dots ,c_M\\in \\mathbb {R}$ .",
"(Here, $q(u,c)$ is the Kruzkov entropy flux function.)",
"Lemma 4.3 The canonical statistical solution is an entropy statistical solution.",
"Select $(\\bar{\\mu }_1,\\dots ,\\bar{\\mu }_M)\\in \\Lambda (\\alpha ,\\bar{\\mu })$ for an arbitrary weight $\\alpha $ and define $\\mu _{i,t} := S_t\\#\\bar{\\mu }_i$ .",
"Then $(\\mu _{1,t},\\dots ,\\mu _{M,t})\\in \\Lambda (\\alpha ,\\mu _t)$ , and $&\\sum _{i=1}^M\\alpha _i\\left[\\int _{\\mathbb {R}_+}\\int _{{F}}\\int _{\\mathbb {R}^d} \\bigl |u(x)-c_i\\bigr |\\partial _t\\varphi + q\\big (u(x),c_i\\big )\\cdot \\nabla _x \\varphi \\,dxd\\mu _{i,t}(u)dt + \\int _{{F}}\\int _{\\mathbb {R}^d}\\big |\\bar{u}(x)-c_i\\big |\\varphi \\Bigr |_{t=0}\\,dxd\\bar{\\mu }_i(\\bar{u})\\right] \\\\=&\\ \\sum _{i=1}^M\\alpha _i\\int _{F}\\left[\\int _{\\mathbb {R}_+}\\int _{\\mathbb {R}^d} \\big |S_t\\bar{u}(x)-c_i\\big |\\partial _t\\varphi + q\\big (S_t\\bar{u}(x),c_i\\big )\\cdot \\nabla _x \\varphi \\,dxdt + \\int _{\\mathbb {R}^d}\\big |\\bar{u}(x)-c_i\\big |\\varphi \\Bigr |_{t=0}\\,dx\\right]d\\bar{\\mu }_i(\\bar{u}) \\\\\\geqslant &\\ 0,$ since the map $(x,t)\\mapsto S_t\\bar{u}(x)$ is an entropy solution of the deterministic problem.",
"Note that Lemma 4.4 Let $\\mu _t$ be an arbitrary entropy statistical solution with initial data $\\bar{\\mu }\\in {P}({F})$ satisfying $\\operatorname{supp}\\bar{\\mu }\\subset {U}$ .",
"Fix $\\alpha _1,\\dots ,\\alpha _M>0$ with $\\sum _{i=1}^{M}\\alpha _i=1$ .",
"Let $v_1,\\dots ,v_M : \\mathbb {R}_+\\rightarrow {U}$ be entropy solutions of (REF ) with initial data $\\bar{v}_1,\\dots ,\\bar{v}_M\\in {U}$ , respectively, and define $\\bar{\\rho }:=\\sum _{i=1}^M\\alpha _i\\delta _{v_i}, \\qquad \\rho _t:=\\sum _{i=1}^M\\alpha _i\\delta _{v_i(t)} \\qquad \\forall \\ t\\in \\mathbb {R}_+.$ Then $W_1(\\rho _t,\\mu _t) \\leqslant W_1(\\bar{\\rho },\\bar{\\mu }) \\qquad \\forall \\ t>0.$ Let $(\\bar{\\mu }_i)_{i=1}^M\\in \\Lambda (\\alpha ,\\bar{\\mu })$ be an optimal transport plan from $\\bar{\\mu }$ to $\\bar{\\rho }$ .",
"The entropy condition for $\\mu _t$ gives the existence of a map $t\\mapsto \\big (\\mu _{i,t}\\big )_{i=1}^n$ such that $\\sum _{i=1}^M\\alpha _i\\left[\\int _{\\mathbb {R}_+}\\int _{{F}}\\int _{\\mathbb {R}^d} \\big |u(x)-c_i\\big |\\partial _t\\varphi + q\\big (u(x),c_i\\big )\\cdot \\nabla _x \\ \\varphi \\,dxd\\mu _{i,t}(u)dt + \\int _{{F}}\\int _{\\mathbb {R}^d}\\big |\\bar{u}(x)-c_i\\big |\\varphi \\Bigr |_{t=0}\\,dxd\\bar{\\mu }_i(\\bar{u})\\right] \\geqslant 0$ for any choice of $\\varphi \\in C_c^\\infty (\\mathbb {R}^d \\times \\mathbb {R}_+)$ and $c_i\\in \\mathbb {R}$ .",
"Let $\\varphi = \\varphi (x,y,t,s)\\in C_c^\\infty ((\\mathbb {R}^d)^2\\times \\mathbb {R}_+^2)$ .",
"Set $c_i = v_i(y,s)$ for some point $(y,s)$ and integrate over $y\\in \\mathbb {R}$ and $s\\in \\mathbb {R}_+$ : $\\begin{split}\\int _{\\mathbb {R}_+}\\int _{\\mathbb {R}^d}\\sum _{i=1}^M\\alpha _i\\Biggl [\\int _{\\mathbb {R}_+}\\int _{{F}}\\int _{\\mathbb {R}^d} \\big |u(x)-v_i(y,s)\\big |\\partial _t\\varphi + q\\big (u(x),v_i(y,s)\\big )\\cdot \\nabla _x \\varphi \\,dxd\\mu _{i,t}(u)dt \\\\+ \\int _{{F}}\\int _{\\mathbb {R}^d}\\big |\\bar{u}(x)-v_i(y,s)\\big |\\varphi \\Bigr |_{t=0}\\,dxd\\bar{\\mu }_i(\\bar{u})\\Biggr ]dyds \\geqslant 0.\\end{split}$ (The expression in the brackets is measurable with respect to $(y,s)$ since (REF ) is continuous with respect to $c_i$ .)",
"Next, since each $v_i$ is an entropy solution, we have for all $\\xi \\in \\mathbb {R}$ and $0\\leqslant \\varphi \\in C_c^\\infty (\\mathbb {R}^d\\times \\mathbb {R}_+)$ $\\int _{\\mathbb {R}_+}\\int _{\\mathbb {R}^d} \\big |\\xi -v_i(y,s)\\big |\\partial _s\\varphi + q\\big (\\xi ,v_i(y,s)\\big )\\cdot \\nabla _y\\varphi \\,dyds + \\int _{\\mathbb {R}^d} \\big |\\xi -\\bar{v}_i(y)\\big |\\varphi \\Bigr |_{s=0}\\,dy \\geqslant 0.$ Set $\\xi =u(x)$ for some $u\\in {F}$ and $x\\in \\mathbb {R}$ .",
"Integrate the above over $x\\in \\mathbb {R}$ and over $u\\in {F}$ with respect to $\\mu _{i,t}$ for some $t\\in \\mathbb {R}_+$ .",
"Integrate over $t\\in \\mathbb {R}_+$ , multiply by $\\alpha _i$ and sum over $i=1,\\dots ,M$ : $\\begin{split}\\sum _{i=1}^{M}\\alpha _i\\int _{\\mathbb {R}_+}\\int _{F}\\int _{\\mathbb {R}^d} \\Biggl [\\int _{\\mathbb {R}_+}\\int _{\\mathbb {R}^d} \\big |u(x)-v_i(y,s)\\big |\\partial _s\\varphi + q\\big (u(x),v_i(y,s)\\big )\\cdot \\nabla _y\\varphi \\,dyds \\\\+ \\int _{\\mathbb {R}^d} \\big |u(x)-\\bar{v}_i(y)\\big |\\varphi \\Bigr |_{s=0}\\,dy \\Biggr ]dxd\\mu _{i,t}(u)dt \\geqslant 0.\\end{split}$ Applying Fubini's theorem to this and equation (REF ) and adding the two, we obtain $\\sum _{i=1}^M\\alpha _i \\Biggl [\\int _{\\mathbb {R}_+}\\int _{\\mathbb {R}_+}\\int _{F}\\int _{\\mathbb {R}^d}\\int _{\\mathbb {R}^d} \\big |u(x)-v_i(y,s)\\big |(\\partial _t+\\partial _s)\\varphi + q\\big (u(x),v_i(y,s)\\big )\\cdot (\\nabla _x+\\nabla _y)\\varphi \\,dxdyd\\mu _{i,t}(u)dtds \\\\+ \\int _{\\mathbb {R}_+}\\int _{F}\\int _{\\mathbb {R}^d}\\int _{\\mathbb {R}^d} \\big |u(x)-\\bar{v}_i(y)\\big |\\varphi \\Bigr |_{s=0}\\,dxdyd\\mu _{i,t}(u)dt + \\int _{\\mathbb {R}_+}\\int _{{F}}\\int _{\\mathbb {R}^d}\\int _{\\mathbb {R}^d} \\big |\\bar{u}(x)-v_i(y,s)\\big |\\varphi \\Bigr |_{t=0}\\,dxdyd\\bar{\\mu }_i(\\bar{u})ds\\Biggr ] \\geqslant 0.$ Now set $\\varphi (x,y,t,s) := \\psi \\Bigl (\\frac{x+y}{2}, \\frac{t+s}{2}\\Bigr )\\omega _\\varepsilon (x-y)\\omega _{\\varepsilon ^{\\prime }}(t-s)$ for some nonnegative $\\psi \\in C_c^\\infty (\\mathbb {R}^d\\times \\mathbb {R}_+)$ and a mollifier $\\omega _\\varepsilon $ .",
"Using the dominated convergence theorem on the integrals over ${F}$ , we find that as $\\varepsilon \\rightarrow 0$ , the above converges to $\\sum _{i=1}^M\\alpha _i \\Biggl [\\int _{\\mathbb {R}_+}\\int _{\\mathbb {R}_+}\\int _{F}\\int _{\\mathbb {R}^d} \\big |u(x)-v_i(x,s)\\big |(\\partial _t+\\partial _s)\\tilde{\\varphi } + 2q\\big (u(x),v_i(x,s)\\big )\\cdot \\nabla _x\\tilde{\\varphi }\\,dxd\\mu _{i,t}(u)dtds \\\\+ \\int _{\\mathbb {R}_+}\\int _{F}\\int _{\\mathbb {R}^d} \\big |u(x)-\\bar{v}_i(x)\\big |\\tilde{\\varphi }\\Bigr |_{s=0}\\,dxd\\mu _{i,t}(u)dt + \\int _{\\mathbb {R}_+}\\int _{{F}}\\int _{\\mathbb {R}^d} \\big |\\bar{u}(x)-v_i(x,s)\\big |\\tilde{\\varphi }\\Bigr |_{t=0}\\,dxd\\bar{\\mu }_i(\\bar{u})ds\\Biggr ] \\geqslant 0,$ where $\\tilde{\\varphi }(x,t,s) := \\psi \\Bigl (x,\\frac{t+s}{2}\\Bigr )\\omega _{\\varepsilon ^{\\prime }}(t-s)$ .",
"Finally, letting $\\varepsilon ^{\\prime }\\rightarrow 0$ we get $\\sum _{i=1}^M\\alpha _i \\Biggl [\\int _{\\mathbb {R}_+}\\int _{F}\\int _{\\mathbb {R}^d} \\big |u(x)-v_i(x,t)\\big |\\partial _t\\psi + q\\big (u(x),v_i(x,t)\\big )\\cdot \\nabla _x\\psi \\,dxd\\mu _{i,t}(u)dt \\\\+ \\int _{F}\\int _{\\mathbb {R}^d} \\big |\\bar{u}(x)-\\bar{v}_i(x)\\big |\\psi \\Bigr |_{t=0}\\,dxd\\bar{\\mu }_i(\\bar{u})\\Biggr ] \\geqslant 0.$ We now set $\\psi (x,\\tau ) := \\mathbb {1}_{[0,t]}(\\tau )$ for some $t\\in \\mathbb {R}_+$ to get $\\sum _{i=1}^M\\alpha _i\\Biggl [-\\int _{F}\\big \\Vert u-v_i(t)\\big \\Vert _{{F}}\\,d\\mu _{i,t}(u) + \\int _{F}\\big \\Vert \\bar{u}-\\bar{v}_i\\big \\Vert _{F}\\,d\\bar{\\mu }_i(\\bar{u})\\Biggr ] \\geqslant 0.$ Using the fact that $(\\bar{\\mu }_i)$ is an optimal transport plan from $\\bar{\\mu }$ to $\\bar{\\rho }$ , we end up with (REF ).",
"To complete our proof of well-posedness of statistical solutions we need the following well-known result, whose proof is included in the appendix for the sake of completeness.",
"Lemma 4.5 Let $X$ be a Polish space equipped with its Borel $\\sigma $ -algebra.",
"Then the convex hull of Dirac measures on $X$ is dense in ${P}(X)$ with respect to the topology of weak convergence.",
"In other words, for every $\\mu \\in {P}(X)$ , there is a sequence $\\rho _n\\in {P}(X)$ of convex combinations of Dirac measures such that $\\rho _n\\rightharpoonup \\mu $ as $n\\rightarrow \\infty $ .",
"Theorem 4.6 Let $\\bar{\\mu }\\in {P}({F})$ with $\\operatorname{supp}\\bar{\\mu }\\subset {U}:=L^1\\cap L^\\infty (\\mathbb {R}^d)$ .",
"Then the entropy statistical solution with initial data $\\bar{\\mu }$ is unique and coincides with the canonical statistical solution.",
"Any two entropy statistical solutions $\\mu _t$ , $\\rho _t$ satisfy $W_1(\\mu _t,\\rho _t) \\leqslant W_1(\\bar{\\mu },\\bar{\\rho }).$ Let $\\mu _t$ be an entropy statistical solution with initial data $\\bar{\\mu }$ .",
"By Lemma REF , the convex hull of Dirac measures is dense in ${P}({F})$ , so we can find a sequence $\\bar{\\mu }_n\\in {P}({F})$ ($n\\in \\mathbb {N}$ ) of convex combinations of Dirac measures such that $\\bar{\\mu }_n \\rightharpoonup \\bar{\\mu }$ in ${P}({F})$ as $n\\rightarrow \\infty $ .",
"Let $\\mu _{n,t} := S_t\\#\\bar{\\mu }_n$ be the corresponding canonical statistical solutions, and note that also $\\mu _{n,t} \\rightharpoonup S_t\\#\\bar{\\mu }$ as $n\\rightarrow \\infty $ .",
"From Lemma REF we find that $W_1(\\mu _t,\\mu _{n,t}) \\leqslant W_1(\\bar{\\mu },\\bar{\\mu }_n) \\rightarrow 0 \\qquad \\text{as } n\\rightarrow \\infty .$ Thus, $\\mu _t = \\operatornamewithlimits{w-lim}_{n\\rightarrow \\infty }\\mu _{n,t} = S_t\\#\\bar{\\mu }$ , whence $\\mu _t$ is the canonical statistical solution."
],
[
"Discussion",
"Given the lack of global in time existence results, and the recent non-uniqueness results of [16], [17], the acceptance of entropy solutions as the standard solution paradigm for multi-dimensional systems of conservation laws is being increasingly questioned.",
"Based on extensive numerical results, recent papers such as [23] have advocated entropy measure-valued solutions (MVS), as defined by DiPerna [20], as an appropriate solution paradigm for systems of conservation laws.",
"However, entropy MVS are not necessarily unique, even for scalar conservation laws, if the MVS is non-atomic.",
"Since numerical results of [23] strongly hint at the possibility of non-atomic MVS even when the initial data is a atomic, it is natural to seek additional constraints on entropy MVS to enforce uniqueness.",
"Given this background, and the need for developing a solution concept that can accommodate uncertain initial data (and corresponding uncertain solutions) that arise frequently in the area of uncertainty quantification (UQ), we seek to adapt the notion of statistical solutions, originally developed in [27], [28] for the incompressible Navier–Stokes equations, to systems of conservation laws.",
"Statistical solutions are time-parametrized probability measures on some (infinite-dimensional) function space.",
"Infinite-dimensional Liouville or Hopf equations track the evolution of the time-parametrized measure.",
"However, the extension of statistical solutions as defined in [27], [28], [29], to systems of conservation laws, is highly non-trivial as the “natural” function spaces for the dynamics of conservation laws consists merely of integrable functions, and may lack the regularity required to define the Liouville or Hopf equations.",
"Although one can work with probability measures on distributions in the specific case of the inviscid Burgers equation (as suggested in [9], [10], [5]), it is very difficult to enforce uniqueness on such a large space of measures.",
"Another disadvantage of probability measures on functions is that they do not readily provide any local (statistical) information at specific (collections of) points in the spatial domain.",
"We define statistical solutions for systems of conservation laws in a different manner.",
"To this end, we prove a novel equivalence theorem between probability measures on $L^p$ spaces ($1 \\leqslant p < \\infty $ ) and a family (hierarchy) of Young measures, the so-called correlation measures, on finite-dimensional tensor product spatial domains.",
"For all $k \\in \\mathbb {N}$ , the $k$ -th member of this hierarchy, the so-called $k$ -point correlation marginal, is a Young measure that provides information on correlations of the underlying functions at $k$ distinct points in the spatial domain.",
"In particular, the first correlation marginal is classical one-point Young measure.",
"Thus, a probability measure on an $L^p$ space can be realized as an Young measure, augmented with multi-point correlations on the spatial domain.",
"This representation enables us to localize probability measures on function spaces and view them as a collection of all possible multi-point correlation marginals.",
"We also show that moments of the correlation marginals uniquely determine the corresponding probability measure on the infinite-dimensional function space.",
"We believe that this representation of probability measures will be of independent interest in stochastic analysis, particularly stochastic partial differential equations [14], in uncertainty quantification of evolutionary PDEs [31] and in Bayesian inversion and data assimilation for time-dependent PDEs [42].",
"In particular, the use of statistical solutions will provide a framework for uncertainty quantification that does not depend on any particular parametrization of the solution in terms of random fields, as is customary in UQ [31].",
"In this paper, we use the equivalence between probability measures on $L^p$ and families of correlation measures to define statistical solutions of systems of conservation laws.",
"In particular, we utilize the fact that moments of correlation measures uniquely determine the underlying probability measure, to evolve these moments in a manner consistent with the dynamics of the system (REF ).",
"Thus, a statistical solution has to satisfy an (infinite) family of nonlinear PDEs, but each of these PDEs is defined on a finite-dimensional (tensor-product) spatial domain.",
"This should be contrasted with the infinite-dimensional Liouville or Hopf equations that the statistical solutions of [27], [28], [29] need to satisfy.",
"Moreover, our notion of statistical solutions restricts the class of probability measures to those on $L^p$ spaces, rather than on distributions (as in [10]) and makes it more amenable to analysis, particularly from the point of view of uniqueness.",
"At the same time, our notion of statistical solutions augment the standard concept of measure-valued solutions, with additional information in the form of multi-point correlations, and paves the way for constraining the solutions sufficiently to guarantee uniqueness.",
"We investigate the well-posedness of the proposed concept of statistical solutions in the specific context of multi-dimensional scalar conservation laws in this paper.",
"We show existence by proving that the push forward of the initial probability measure on $L^1 \\cap L^{\\infty }$ by the Kruzkhov entropy solution semi-group is a statistical solution, and we term this solution the canonical statistical solution.",
"We propose a novel admissibility criteria, based on stability with respect to a suitable stationary statistical solution, namely probability measures supported on finite collections of constant functions.",
"These entropy statistical solutions are a generalization of the standard Kruzkhov entropy solutions for scalar conservation laws.",
"We show that the canonical statistical solution is the unique entropy statistical solution.",
"Furthermore, we show that it is contractive with respect to the 1-Wasserstein metric on probability measures on $L^1$ .",
"Thus, entropy statistical solutions for multi-dimensional scalar conservation laws are shown to be well-posed and are thus completely characterized.",
"This article is the first in a series of papers investigating statistical solutions of multi-dimensional systems of conservation laws.",
"We lay out the measure theoretic basis, define statistical solutions for systems and show well-posedness in the scalar case.",
"Forthcoming papers in the series will deal with numerical approximation of entropy statistical solutions of scalar conservation laws [25] and global existence of statistical solutions for a large class of multi-dimensional systems of conservation laws by showing convergence of a Monte Carlo based numerical approximation algorithm [26].",
"Admissibility criteria that single out physically relevant statistical solutions are the topic of current and future work."
],
[
"Acknowledgments",
"U.S.F.",
"was supported in part by the grant Waves and Nonlinear Phenomena (WaNP) from the Research Council of Norway.",
"S.M.",
"was supported in part by ERC STG.",
"N 306279, SPARCCLE.",
"The authors thanks Kjetil O. Lye and Franziska Weber (SAM, ETH) for their helpful comments."
],
[
"Appendix",
"For completeness we provide the proof of Proposition REF .",
"The proof relies on the following two lemmas.",
"Lemma A.1 $\\mathit {Cyl}(X)$ is a ring.1A collection of sets ${X}\\subset 2^X$ is a ring if $\\emptyset \\in {X}$ and if both $A\\cup B$ and $A\\setminus B$ lie in ${X}$ whenever $A,B\\in {X}$ .",
"Clearly, $\\emptyset \\in \\mathit {Cyl}(X)$ , and if $A_1, A_2\\in \\mathit {Cyl}(X)$ are of the form $A_i = \\left\\lbrace u\\in X\\ :\\ \\bigl (\\varphi _1^i, \\dots , \\varphi _{n_i}^i\\bigr )(u) \\in F_i\\right\\rbrace , \\qquad i=1,2$ then both $A_1\\cup A_2 = \\left\\lbrace u\\in X\\ :\\ \\bigl (\\varphi _1^1, \\dots , \\varphi _{n_1}^1, \\varphi _1^2, \\dots , \\varphi _{n_2}^2\\bigr )(u) \\in \\left(F_1\\times \\mathbb {R}^{n_2}\\right) \\cup \\left(\\mathbb {R}^{n_1}\\times F_2\\right)\\right\\rbrace $ and $A_1\\setminus A_2 = \\left\\lbrace u\\in X\\ :\\ \\bigl (\\varphi _1^1, \\dots , \\varphi _{n_1}^1, \\varphi _1^2, \\dots , \\varphi _{n_2}^2\\bigr )(u) \\in F_1\\times \\bigl (F_2\\bigr )^c\\right\\rbrace $ are cylinder sets.",
"Lemma A.2 If $X$ is a separable normed vector space then there exists a countable family $\\lbrace \\varphi _n\\rbrace _{n\\in \\mathbb {N}} \\subset X^*$ such that $\\Vert u\\Vert _X=\\sup _{n\\in \\mathbb {N}}\\,\\varphi _n(u) \\quad \\text{ for every $u\\in X$}.$ Let $\\lbrace u_n\\rbrace _{n\\in \\mathbb {N}} \\subset X$ be a countable dense subset of the unit sphere $\\partial B_1(0) \\subset X$ .",
"For each $n\\in \\mathbb {N}$ , let $\\varphi _n\\in X^*$ satisfy $\\varphi _n(u_n) = 1$ and $\\Vert \\varphi _n\\Vert _{X^*} = 1$ .",
"If $u\\in \\partial B_1(0)$ is arbitrary and $\\varepsilon >0$ , find an $u_n$ such that $\\Vert u-u_n\\Vert _X < \\varepsilon $ .",
"Then $1 \\geqslant \\varphi _n(u) = \\varphi _n(u_n) - \\varphi _n(u_n-u) \\geqslant 1-\\varepsilon ,$ so $\\Vert u\\Vert _X=1$ can be approximated from below by $\\varphi _n(u)$ .",
"Equation (REF ) follows.",
"[Proof of Proposition REF ] Let $\\lbrace \\varphi _n\\rbrace _{n\\in \\mathbb {N}}$ be as in Lemma REF .",
"For a $u_0\\in X$ and $r>0$ , the open ball of radius $r$ with centre $u_0$ can be written $B_r(u_0) &= \\Bigl \\lbrace u\\in X\\ :\\ \\varphi _n(u-u_0) < r\\ \\forall \\ n\\in \\mathbb {N}\\Bigr \\rbrace \\\\&= \\bigcap _{n\\in \\mathbb {N}} \\Bigl \\lbrace u\\in X\\ :\\ \\varphi _n(u) \\in \\bigl (-\\infty ,\\ \\varphi _n(u_0)+r\\bigr )\\Bigr \\rbrace ,$ which is a countable intersection of cylinder sets.",
"It follows that $\\sigma (\\mathit {Cyl}(X))$ , the $\\sigma $ -algebra generated by $\\mathit {Cyl}(X)$ , contains the $\\sigma $ -algebra generated by the open balls in $X$ , which is precisely ${B}(X)$ .",
"But every cylinder set is a Borel set; hence the two $\\sigma $ -algebras coincide, and (i) follows.",
"By Lemma REF , $\\mathit {Cyl}(X)$ is a ring which, by (i), generates ${B}(X)$ .",
"Assertion (ii) then follows from the fact that (signed) measures vanishing on a ring, vanish on the $\\sigma $ -algebra generated by the ring.",
"[Proof of Lemma REF ] Recall that the topology of weak convergence on ${P}(X)$ for a Polish metric space $X$ is the coarsest topology for which the map $\\mu \\mapsto \\int \\varphi \\,d\\mu $ is continuous for every $\\varphi \\in C_b(X)$ [37].",
"Thus, the topology of weak convergence is generated by the open sets $U_{\\varphi ,\\mu ,\\varepsilon } := \\left\\lbrace \\rho \\in {P}(X)\\ :\\ \\Bigl |\\int \\varphi \\, d\\mu - \\int \\varphi \\,d\\rho \\Bigr | < \\varepsilon \\right\\rbrace $ for $\\mu \\in {P}(X)$ , $\\varepsilon >0$ and $\\varphi \\in C_b(X)$ .",
"It suffices to show that every nonempty open set $U_{\\varphi ,\\mu ,\\varepsilon }$ contains a measure which is a convex combination of Dirac measures.",
"Let $\\bar{\\varphi }(x) = \\sum _{i=1}^n a_i\\mathbb {1}_{A_i}(x)$ be a simple function such that $\\sup _{x\\in X}|\\varphi (x)-\\bar{\\varphi }(x)| < \\varepsilon /2$ .",
"Fix $x_i\\in A_i$ and define $\\rho := \\sum _{i=1}^n \\mu (A_i)\\delta _{x_i}$ .",
"Since $|\\varphi (x_i)-\\varphi (x)|<\\varepsilon $ for every $x\\in A_i$ , we find that $\\bigg |\\int _X \\varphi \\,d\\rho - \\int _X \\varphi \\,d\\mu \\bigg | = \\biggl |\\sum _{i=1}^n\\int _{A_i}\\varphi (x_i)- \\varphi (x)\\,d\\mu \\biggr | \\leqslant \\sum _{i=1}^n\\int _{A_i}|\\varphi (x_i)-\\varphi (x)|\\,d\\mu < \\varepsilon .$ Hence, $\\rho \\in U_{\\varphi ,\\mu ,\\varepsilon }$ ."
]
] | 1605.05960 |
[
[
"Kostka-Shoji polynomials and Lusztig's convolution diagram"
],
[
"Abstract We propose an $r$-variable version of Kostka-Shoji polynomials $K^-_{\\lambda\\mu}$ for $r$-multipartitions $\\lambda,\\mu$.",
"Our version has positive integral coefficients and encodes the graded multiplicities in the space of global sections of a line bundle over Lusztig's iterated convolution diagram for the cyclic quiver $\\tilde{A}_{r-1}$."
],
[
"Introduction",
"Let $G$ be a reductive complex algebraic group.",
"According to G. Lusztig [9], the IC-stalks of $G[[z]]$ -orbit closures in the affine Grassmannian $\\operatorname{Gr}_G$ are encoded by the Kostka polynomials associated to the Langlands dual group $G^\\vee $ .",
"According to R. Brylinski [4], the same Kostka polynomials encode the graded multiplicities in the global sections of line bundles on the cotangent bundle of the flag variety of $G^\\vee $ .",
"According to [2], [5], the IC-stalks of $GL_N[[z]]$ -orbit closures in the mirabolic affine Grassmannian of $GL_N$ are encoded by the Kostka-Shoji polynomials [16].",
"We note that the same Kostka-Shoji polynomials encode the graded multiplicities in the global sections of line bundles on a certain vector bundle over the square of the flag variety of $GL_N$ .",
"This vector bundle is nothing but Lusztig's iterated convolution diagram for the cyclic $\\tilde{A}_1$ quiver [11].",
"The higher cohomology vanishing of the above line bundles follows from the Frobenius splitting of this convolution diagram which in turn follows from the fact that the convolution diagram is related to a Bott-Samelson-Demazure-Hansen (BSDH for short) variety of affine type $A$ [10].",
"The dilation action of ${\\mathbb {G}}_m$ on Lusztig's convolution diagram extends to an action of ${\\mathbb {G}}_m\\times {\\mathbb {G}}_m$ which gives rise to a 2-variable version of Kostka-Shoji polynomials $K_{{\\lambda }{\\mu }}(t_1,t_2)$ such that $K_{{\\lambda }{\\mu }}(t,t)=K_{{\\lambda }{\\mu }}(t)$ (the classical Kostka-Shoji polynomial).",
"Note that the realization of Kostka-Shoji polynomials via the IC-stalks on mirabolic affine Grassmannian cannot give rise to a 2-variable version since these stalks are pure Tate [2], [18], [5].",
"The (multi)graded multiplicities in the global sections of line bundles on Lusztig's iterated convolution diagram for the cyclic $\\tilde{A}_{r-1}$ quiver are encoded conjecturally by an $r$ -variable version of Kostka-Shoji polynomials $K^-_{{\\lambda }{\\mu }}(t)$ [16] for $r$ -multipartitions ${\\lambda },{\\mu }$ .",
"The higher cohomology vanishing is proved by the same argument as above.",
"It would be interesting to find out if Lusztig's convolution diagrams for more general quivers are Frobenius split.",
"We are grateful to P. Achar, R. Bezrukavnikov, A. Braverman, B. Feigin, V. Ginzburg, S. Kato, A. Kuznetsov, L. Rybnikov, V. Serganova, T. Shoji, R. Travkin, M. Vikulina, L. Yanushevich for the helpful discussions.",
"The study has been funded by the Russian Academic Excellence Project `5-100'.",
"A. I. was supported in part by Dobrushin stipend and grant RFBR 15-01-09242."
],
[
"Dominance order on multipartitions",
"We denote by ${\\mathcal {P}}^r_N\\subset {\\mathbb {Z}}^{rN}$ the set of generalized $r$ -multipartitions ${\\lambda }=(\\lambda ^{(1)},\\ldots ,\\lambda ^{(r)})$ such that for any $s=1,\\ldots ,r$ the corresponding $\\lambda ^{(s)}=(\\lambda ^{(s)}_1\\ge \\lambda ^{(s)}_2\\ge \\ldots \\ge \\lambda ^{(s)}_N)$ is a weakly decreasing sequence of integers of length $N$ .",
"We order the entries of ${\\lambda }$ lexicographically as follows: $\\lambda ^{(1)}_1,\\lambda ^{(2)}_1,\\ldots ,\\lambda ^{(r)}_1,\\lambda ^{(1)}_2,\\lambda ^{(2)}_2,\\ldots ,\\lambda ^{(r)}_2,\\ldots ,\\lambda ^{(r-1)}_N,\\lambda ^{(r)}_N.$ For any $n=1,\\ldots ,rN$ we denote by $\\Sigma _n({\\lambda })$ the sum of the first $n$ entries in the above order.",
"We say that ${\\lambda }\\ge {\\mu }$ in the dominance order if $\\Sigma _n({\\lambda })\\ge \\Sigma _n({\\mu })$ for any $n=1,\\ldots ,rN-1$ , and $\\Sigma _{rN}({\\lambda })=\\Sigma _{rN}({\\mu })$ .",
"If ${\\lambda }\\ge {\\mu }$ , then $\\alpha :={\\lambda }-{\\mu }\\in {\\mathbb {N}}^{rN-1}$ is the vector with coordinates $(\\Sigma _n({\\lambda })-\\Sigma _n({\\mu }))_{n=1,\\ldots ,rN-1}$ ."
],
[
"Partition function",
"Let $\\delta _n,\\ 1\\le n\\le rN-1$ , be the base of ${\\mathbb {N}}^{rN-1}$ .",
"For $1\\le m<n\\le rN$ we set $\\alpha _{mn}:=\\sum _{l=m}^{n-1}\\delta _l$ .",
"We define a finite subset $R^+_r\\subset {\\mathbb {N}}^{rN-1}$ of positive pseudoroots as follows: $R^+_r:=\\lbrace \\alpha _{mn}\\rbrace _{n-m=1\\pmod {r}}$ .",
"Given $\\alpha \\in {\\mathbb {N}}^{rN-1}$ we define a polynomial $L_r^\\alpha (t)$ (Lusztig's partition function) as follows: $L_r^\\alpha (t):=\\sum p_dt^d$ where $p_d$ is the number of (unordered) partitions of $\\alpha $ into a sum of $d$ positive pseudoroots.",
"We extend $L_r(t)$ from ${\\mathbb {N}}^{rN-1}$ to ${\\mathbb {Z}}^{rN-1}$ by zero.",
"We also introduce a multivariable version of $L_r^\\alpha (t_1,\\ldots ,t_r)$ where the variables are numbered by ${\\mathbb {Z}}/r{\\mathbb {Z}}=\\lbrace 1,\\ldots ,r\\rbrace $ .",
"Namely, $L_r^\\alpha (t_1,\\ldots ,t_r)=\\sum p_{\\underline{d}}\\prod _{s\\in {\\mathbb {Z}}/r{\\mathbb {Z}}}t_s^{d_s}$ where $\\underline{d}=(d_1,\\ldots ,d_r)\\in {\\mathbb {N}}^{{\\mathbb {Z}}/r{\\mathbb {Z}}}$ , and $p_{\\underline{d}}$ is the number of unorderd partitions of $\\alpha $ into a sum of positive pseudoroots having $d_s$ summands $\\alpha _{mn}$ with $m=d_s\\pmod {r}$ for any $s\\in {\\mathbb {Z}}/r{\\mathbb {Z}}$ .",
"Clearly, the restriction of $L_r^\\alpha (t_1,\\ldots ,t_r)$ to the diagonal $t_1=\\ldots =t_r=t$ coincides with $L_r^\\alpha (t)$ .",
"We extend $L_r(t_1,\\ldots ,t_r)$ from ${\\mathbb {N}}^{rN-1}$ to ${\\mathbb {Z}}^{rN-1}$ by zero."
],
[
"Lusztig-Kato formula",
"We set $\\rho =(N,N-1,\\ldots ,2,1)$ , and ${\\rho }=(\\rho ,\\ldots ,\\rho )\\in {\\mathcal {P}}^r_N$ .",
"Given ${\\lambda },{\\mu }\\in {\\mathcal {P}}^r_N$ we define $K_{{\\lambda }{\\mu }}(t):=\\sum _{{\\sigma }\\in {\\mathfrak {S}}^r_N}(-1)^{\\sigma }L_r^{{\\sigma }({\\lambda }+{\\rho })-{\\rho }-{\\mu }}(t)$ , the sum over the product of $r$ copies of the symmetric group ${\\mathfrak {S}}_N$ acting on $({\\mathbb {Z}}^N)^r$ by permutations of entries of each composition.",
"We also introduce a multivariable version $K_{{\\lambda }{\\mu }}(t_1,\\ldots ,t_r):=\\sum _{{\\sigma }\\in {\\mathfrak {S}}^r_N}(-1)^{\\sigma }L_r^{{\\sigma }({\\lambda }+{\\rho })-{\\rho }-{\\mu }}(t_1,\\ldots ,t_r).$ Clearly, $K_{{\\lambda }{\\mu }}(t,\\ldots ,t)=K_{{\\lambda }{\\mu }}(t)$ .",
"Recall the Kostka-Shoji polynomials $K^\\pm _{{\\lambda }{\\mu }}(t)$ [16].",
"In case $r=1,\\ K^+_{\\lambda \\mu }(t)=K^-_{\\lambda \\mu }(t)$ is the classical Kostka polynomial, and it was proved by I. G. Macdonald [12] that $K^+_{\\lambda \\mu }(t)=K^-_{\\lambda \\mu }(t)=K_{\\lambda \\mu }(t)$ for $\\lambda \\ge \\mu $ .A similar identity for arbitrary finite root systems was conjectured by G. Lusztig [9] and proved by S.-I. Kato [7].",
"In case $r=2$ , the identity $K^+_{{\\lambda }{\\mu }}(t)=K^-_{{\\lambda }{\\mu }}(t)=K_{{\\lambda }{\\mu }}(t)$ for ${\\lambda }\\ge {\\mu }$ was proved by T. Shoji [16].",
"The following generalization of these identities for arbitrary $r$ is supported by the calculations by L. Yanushevich for multipartitions of total size $\\le 7$ , using P. Achar's code [1].",
"Conjecture 2.4 For multipartitions ${\\lambda }\\ge {\\mu }\\in {\\mathcal {P}}^r_N$ we have $K^-_{{\\lambda }{\\mu }}(t)=K_{{\\lambda }{\\mu }}(t)$ ."
],
[
"A vector bundle over a flag variety",
"We consider the following ordered base of an $rN$ -dimensional vector space ${\\mathbb {C}}^{rN}\\colon v_1^{(1)},\\ldots ,v_1^{(r)},v_2^{(1)},\\ldots ,v_2^{(r)},\\ldots ,v_N^{(1)},\\ldots ,v_N^{(r)}$ .",
"Sometimes, for $1\\le s\\le r,\\ 1\\le j\\le N$ , we denote $v_j^{(s)}$ by $v_{r(j-1)+s}$ .",
"It gives rise to an embedding $GL_N^r\\hookrightarrow GL_{rN}$ ($s$ -th copy of $GL_N$ acts in the summand spanned by $v_1^{(s)},\\ldots ,v_N^{(s)}$ ), and also to an embedding of the Borel upper triangular subgroups $B_N^r\\hookrightarrow B_{rN}$ .",
"In the adjoint representation of $GL_{rN}$ restricted to $B_N^r$ we consider a subrepresentation ${\\mathfrak {n}}_r$ (of $B_N^r$ ) spanned by the elementary matrices $E_{mn},\\ 1\\le m<n\\le rN$ such that $n-m=1\\pmod {r}$ .",
"It gives rise to a $GL_N^r$ -equivariant vector bundle ${\\mathcal {T}}^*_r{\\mathcal {B}}_N^r=GL_N^r\\times ^{B_N^r}{\\mathfrak {n}}_r$ over the flag variety ${\\mathcal {B}}_N^r$ of $GL_N^r$ .",
"Note that when $r=1$ , the vector bundle ${\\mathcal {T}}^*_1{\\mathcal {B}}_N$ over the flag variety ${\\mathcal {B}}_N$ is nothing but the cotangent bundle.",
"Let $x_1,\\ldots ,x_{rN}$ stand for the characters of the diagonal Cartan torus $T_{rN}$ of $GL_{rN}$ corresponding to the diagonal matrix entries.",
"Sometimes, for $1\\le s\\le r,\\ 1\\le j\\le N$ , we denote $x_{r(j-1)+s}$ by $x_j^{(s)}$ .",
"For $1\\le m<n\\le rN$ we set $x^{\\alpha _{mn}}=x_m^{-1}x_n$ .",
"This is the weight of the elementary matrix $E_{nm}$ .",
"This rule extends to a homomorphism ${\\mathbb {N}}^{rN-1}\\rightarrow X^*(T_{rN}),\\ \\alpha \\rightarrow x^\\alpha $ .",
"The symmetric algebra ${\\mathop {\\operatorname{Sym}}}^\\bullet {\\mathfrak {n}}_r^\\vee $ is graded, and its character is a formal series in $x_1,\\ldots ,x_{rN},t$ .",
"In fact, ${\\mathop {\\operatorname{Sym}}}^\\bullet {\\mathfrak {n}}_r^\\vee $ has a finer grading by ${\\mathbb {N}}^{{\\mathbb {Z}}/r{\\mathbb {Z}}}$ arising from a ${\\mathbb {Z}}/r{\\mathbb {Z}}$ -grading of ${\\mathfrak {n}}_r^\\vee \\colon \\deg E_{nm}:=m\\pmod {r}$ .",
"Hence the character of ${\\mathop {\\operatorname{Sym}}}^\\bullet {\\mathfrak {n}}_r^\\vee $ is a formal series $\\chi $ in $x_1,\\ldots ,x_{rN},t_1,\\ldots ,t_r$ .",
"Lemma 3.2 $\\chi =\\sum _{\\alpha \\in {\\mathbb {N}}^{rN-1}}L^\\alpha (t_1,\\ldots ,t_r)x^\\alpha $ .",
"Clear.",
"Given a multipartition ${\\mu }\\in {\\mathcal {P}}^r_N$ , we consider the corresponding $GL_N^r$ -equivariant line bundle ${\\mathcal {O}}({\\mu })$ on ${\\mathcal {B}}^r_N$ : the action of $B_N^r$ on its fiber at the point $B_N^r\\in {\\mathcal {B}}_N^r$ is via the character $\\prod (x_j^{(s)})^{-\\mu _j^{(s)}}$ .",
"Its global sections $\\Gamma ({\\mathcal {B}}^r_N,{\\mathcal {O}}({\\mu }))$ is an irreducible $GL_N^r$ -module $V^{\\mu }$ with lowest weight $-{\\mu }$ .",
"The character of $V^{\\mu }$ will be denoted $\\chi ^{\\mu }\\in {\\mathbb {Z}}[x_1^{\\pm 1},\\ldots ,x_{rN}^{\\pm 1}]$ .",
"The pullback of ${\\mathcal {O}}({\\mu })$ to ${\\mathcal {T}}_r^*{\\mathcal {B}}^r_N$ will be also denoted ${\\mathcal {O}}({\\mu })$ when no confusion is likely.",
"We consider the equivariant Euler characteristic $\\chi ({\\mathcal {T}}_r^*{\\mathcal {B}}^r_N,{\\mathcal {O}}({\\mu }))=\\chi ({\\mathcal {B}}^r_N,{\\mathop {\\operatorname{Sym}}}^\\bullet {\\mathcal {T}}_r{\\mathcal {B}}^r_N\\otimes {\\mathcal {O}}({\\mu }))$ where ${\\mathcal {T}}_r{\\mathcal {B}}^r_N=GL_N^r\\times ^{B_N^r}{\\mathfrak {n}}_r^\\vee $ stands for the vector bundle over ${\\mathcal {B}}^r_N$ dual to ${\\mathcal {T}}_r^*{\\mathcal {B}}^r_N$ .",
"The ${\\mathbb {N}}^{{\\mathbb {Z}}/r{\\mathbb {Z}}}$ -grading of ${\\mathop {\\operatorname{Sym}}}^\\bullet {\\mathfrak {n}}_r^\\vee $ gives rise to a ${\\mathbb {N}}^{{\\mathbb {Z}}/r{\\mathbb {Z}}}$ -grading of ${\\mathop {\\operatorname{Sym}}}^\\bullet {\\mathcal {T}}_r{\\mathcal {B}}^r_N\\otimes {\\mathcal {O}}({\\mu })$ , and hence $\\chi ({\\mathcal {B}}^r_N,{\\mathop {\\operatorname{Sym}}}^\\bullet {\\mathcal {T}}_r{\\mathcal {B}}^r_N\\otimes {\\mathcal {O}}({\\mu }))$ is a formal series in $x_1,\\ldots ,x_{rN},t_1,\\ldots ,t_r$ .",
"Corollary 3.3 $\\chi ({\\mathcal {B}}^r_N,{\\mathop {\\operatorname{Sym}}}^\\bullet {\\mathcal {T}}_r{\\mathcal {B}}^r_N\\otimes {\\mathcal {O}}({\\mu }))=\\sum _{{\\lambda }\\ge {\\mu }}K_{{\\lambda }{\\mu }}(t_1,\\ldots ,t_r)\\chi ^{\\lambda }$ .",
"Same as the proof of [4]."
],
[
"Convolution diagram",
"Recall the notations of [11].",
"We consider the type $\\tilde{A}_{r-1}$ cyclic quiver $Q$ with the set ${\\mathbb {Z}}/r{\\mathbb {Z}}$ of vertices, and with arrows $s\\rightarrow s-1,\\ s\\in {\\mathbb {Z}}/r{\\mathbb {Z}}$ .",
"Let ${\\mathbf {V}}$ be a ${\\mathbb {Z}}/r{\\mathbb {Z}}$ -graded vector space such that $\\dim {\\mathbf {V}}_s=N$ for any $s\\in {\\mathbb {Z}}/r{\\mathbb {Z}}$ .",
"Let ${\\mathbf {i}}$ be a length $rN$ periodic sequence $(r,r-1,\\ldots ,2,1,r,r-1,\\ldots ,2,1,\\ldots ,r,\\ldots ,1)$ of vertices, and let ${\\mathbf {a}}$ be a length $rN$ sequence $(1,1,\\ldots ,1)$ of positive integers.",
"Then the variety ${\\mathcal {F}}_{{\\mathbf {i}},{\\mathbf {a}}}$ of all flags of type $({\\mathbf {i}},{\\mathbf {a}})$ in ${\\mathbf {V}}$ is nothing but ${\\mathcal {B}}^r_N$ .",
"Moreover, the iterated convolution diagram $\\widetilde{{\\mathcal {F}}}_{{\\mathbf {i}},{\\mathbf {a}}}$ of [11] is nothing but ${\\mathcal {T}}_r^*{\\mathcal {B}}^r_N$ .",
"In effect, we identify ${\\mathbf {V}}_s$ with a vector subspace of ${\\mathbb {C}}^{rN}$ spanned by $v_1^{(s)},\\ldots ,v_N^{(s)}$ (notations of Section REF ).",
"Then the fiber of the natural $GL({\\mathbf {V}})=GL_N^r$ -equivariant projection $\\widetilde{{\\mathcal {F}}}_{{\\mathbf {i}},{\\mathbf {a}}}\\rightarrow {\\mathcal {F}}_{{\\mathbf {i}},{\\mathbf {a}}}$ over the flag ${\\mathbf {V}}={\\mathbb {C}}v_1^{(1)}\\oplus \\ldots \\oplus {\\mathbb {C}}v_N^{(r)}\\supset {\\mathbb {C}}v_1^{(1)}\\oplus \\ldots \\oplus v_N^{(r-1)}\\supset \\ldots \\supset {\\mathbb {C}}v_1^{(1)}\\oplus \\ldots \\oplus {\\mathbb {C}}v_1^{(r)}\\oplus {\\mathbb {C}}v_2^{(1)}\\supset {\\mathbb {C}}v_1^{(1)}\\oplus \\ldots \\oplus {\\mathbb {C}}v_1^{(r)}\\supset \\ldots \\supset {\\mathbb {C}}v_1^{(1)}\\oplus {\\mathbb {C}}v_1^{(2)}\\supset {\\mathbb {C}}v_1^{(1)}\\supset 0$ is nothing but ${\\mathfrak {n}}_r$ ."
],
[
"Frobenius splitting of ${\\mathcal {T}}_r^*{\\mathcal {B}}^r_N$",
"In this section we replace ${\\mathbb {C}}$ by an algebraic closure ${\\mathsf {k}}$ of the finite field ${\\mathbb {F}}_p$ of characteristic $p$ .",
"The present section is devoted to the proof of the following Theorem 4.1 ${\\mathcal {T}}_r^*{\\mathcal {B}}^r_N$ is Frobenius split.",
"Our proof is a variation of the one in [14]."
],
[
"The canonical bundle of ${\\mathcal {T}}_r^*{\\mathcal {B}}^r_N$",
"We have a subgroup $SL_N^r\\subset GL_N^r$ (product of $r$ copies of $SL_N$ ).",
"Lemma 4.3 The canonical line bundle $\\omega $ of ${\\mathcal {T}}_r^*{\\mathcal {B}}^r_N$ is $SL_N^r$ -equivariantly trivial.",
"The product $w_1$ of $T_{rN}$ -weights in ${\\mathfrak {n}}_r$ is $\\prod _{s=2}^r\\prod _{1\\le k\\le l\\le N}x_{r(k-1)+s-1}x^{-1}_{r(l-1)+s}\\cdot \\prod _{1\\le k<l\\le N}x_{rk}x^{-1}_{r(l-1)+1}=\\prod _{s=1}^r\\prod _{k=1}^N x_{r(k-1)+s}^{N+1-2k}\\cdot \\prod _{k=1}^Nx_{rk}x^{-1}_{r(k-1)+1}.$ The product $w_2$ of $T_{rN}$ -weights in the tangent space of ${\\mathcal {B}}^r_N$ at $B^r_N\\in {\\mathcal {B}}^r_N$ is $\\prod _{s=1}^r\\prod _{N\\ge k>l\\ge 1}x_{r(k-1)+s}x^{-1}_{r(l-1)+s}=\\prod _{s=1}^r\\prod _{k=1}^N x_{r(k-1)+s}^{2k-N-1}.$ The $T_{rN}$ -weight in the fiber of the canonical bundle at the point $B^r_N\\in {\\mathcal {B}}^r_N\\subset {\\mathcal {T}}_r^*{\\mathcal {B}}^r_N$ is $w=w_1^{-1}w_2^{-1}=\\prod _{k=1}^Nx^{-1}_{rk}x_{r(k-1)+1}$ .",
"When we restrict $w$ to the maximal torus of $SL_N^r$ we get the trivial weight, hence the canonical line bundle $\\omega $ is $SL_N^r$ -equivariantly trivial."
],
[
"A splitting section",
"According to [3], in order to prove the Frobenius splitting of ${\\mathcal {T}}_r^*{\\mathcal {B}}^r_N$ it suffices to construct a section $\\phi \\in \\Gamma ({\\mathcal {T}}_r^*{\\mathcal {B}}^r_N,\\omega ^{1-p})$ in whose expansion with respect to some local coordinates $z_1,\\ldots ,z_d$ the monomial $z_1^{p-1}\\cdots z_d^{p-1}$ occurs with coefficient 1.",
"Since $\\omega $ is $SL_N^r$ -equivariantly trivial, it has an $SL_N^r$ -invariant nowhere vanishing section $\\varpi $ , and we will look for the desired section $\\phi $ in the form $\\phi =f\\varpi ^{1-p}$ for some $f\\in {\\mathsf {k}}[{\\mathcal {T}}_r^*{\\mathcal {B}}^r_N]$ .",
"To this end recall the decomposition ${\\mathsf {k}}^{rN}=\\bigoplus _{1\\le s\\le r}{\\mathbf {V}}_s$ of Section REF .",
"Accordingly, we will write down the matrices $A\\in {\\mathfrak {gl}}_{rN}$ in the block form $\\begin{pmatrix}A_{11} & A_{12} & \\ldots & A_{1r} \\\\A_{21} & A_{22} & \\ldots & A_{1n} \\\\\\vdots & \\vdots & \\ddots & \\vdots \\\\A_{r1} & A_{r2} & \\ldots & A _{rr}\\end{pmatrix}$ (for $1\\le s,u\\le r$ the corresponding block $A_{su}$ is an $N\\times N$ -matrix).",
"The subgroup $B^r_N\\subset GL^r_N\\subset GL_{rN}$ is formed by all the matrices with upper-triangular diagonal blocks and vanishing non-diagonal blocks.",
"The subspace ${\\mathfrak {n}}_r\\subset {\\mathfrak {gl}}_{rN}$ is formed by all the matrices with strictly upper triangular block $A_{r1}$ , nonstrictly upper triangular blocks $A_{s,s+1},\\ 1\\le s\\le r-1$ , and all the other blocks vanishing.",
"Hence ${\\mathcal {T}}_r^*{\\mathcal {B}}^r_N=GL^r_N\\times ^{B^r_N}{\\mathfrak {n}}_r$ is the quotient of $GL^r_N\\times {\\mathfrak {n}}_r=\\lbrace (g_1,\\ldots ,g_r;A_{12},A_{23},\\ldots ,A_{r-1,r},A_{r1})\\rbrace $ modulo the action of $B^r_N=\\lbrace (b_1,\\ldots ,b_r)\\rbrace $ given by $(b_1,\\ldots ,b_r)\\cdot (g_1,\\ldots ,g_r;A_{12},A_{23},\\ldots ,A_{r-1,r},A_{r1})=(g_1b_1^{-1},\\ldots ,g_rb_r^{-1};b_1A_{12}b_2^{-1}\\ldots ,b_{r-1}A_{r-1,r}b_r^{-1},b_rA_{r1}b_1^{-1})$ .",
"We define $ f(g_1,\\ldots ,g_r;A_{12},A_{23},\\ldots ,A_{r-1,r},A_{r1}):=\\prod _{s=1}^{r-1}\\prod _{j=1}^N\\Delta _j(g_sA_{s,s+1}g_{s+1}^{-1})\\cdot \\prod _{j=1}^{N-1}\\Delta _j(g_rA_{r1}g_1^{-1})$ where $\\Delta _j$ stands for the principal $j\\times j$ -minor in the upper left corner.",
"Proposition 4.5 The section $\\phi =f^{p-1}\\varpi ^{1-p}\\in \\Gamma ({\\mathcal {T}}_r^*{\\mathcal {B}}^r_N,\\omega ^{1-p})$ splits ${\\mathcal {T}}_r^*{\\mathcal {B}}^r_N$ .",
"The proof is given in Section REF after a preparation in Section REF ."
],
[
"Residues",
"We recall the following construction [14].",
"Given a smooth divisor $Z$ in a smooth variety $Y$ and a global section of the anticanonical class $\\eta \\in \\Gamma (Y,\\omega _Y^{-1})$ we construct the residue ${\\operatorname{res}}\\eta \\in \\Gamma (Z,\\omega _Z^{-1})$ as follows.",
"We start with an open subvariety $U\\subset Z$ such that the normal bundle ${\\mathcal {N}}_{Z/Y}|_U$ restricted to $U$ is trivial.",
"We choose a nowhere vanishing section $\\sigma \\in \\Gamma (U,{\\mathcal {N}}_{Z/Y}|_U)$ .",
"Then ${\\operatorname{res}}\\eta |_U$ is defined by the requirement $\\langle {\\operatorname{res}}\\eta ,\\zeta \\rangle =\\langle \\eta ,\\zeta \\frac{d\\sigma }{\\sigma }\\rangle $ for any $\\zeta \\in \\Gamma (U,\\omega _U)$ where $\\langle ,\\rangle $ is the pairing between the anticanonical and canonical bundles.",
"One can check that ${\\operatorname{res}}\\eta |_U$ is independent of the choice of $\\sigma $ , and for $U^{\\prime }\\subset U$ we have $({\\operatorname{res}}\\eta |_U)|_{U^{\\prime }}={\\operatorname{res}}\\eta |_{U^{\\prime }}$ , so the local sections ${\\operatorname{res}}\\eta |_U$ glue to the desired ${\\operatorname{res}}\\eta $ .",
"If we have a chain of smooth divisors $Y\\supset Z_1\\supset \\ldots \\supset Z_n$ we can iterate the above construction to obtain ${\\operatorname{res}}\\colon \\Gamma (Y,\\omega _Y^{-1})\\rightarrow \\Gamma (Z_n,\\omega _{Z_n}^{-1})$ .",
"Lemma 4.7 There is a chain of smooth divisors ${\\mathcal {T}}_r^*{\\mathcal {B}}^r_N=Y\\supset Z_1\\supset \\ldots \\supset Z_n={\\mathcal {B}}^r_N$ such that $({\\operatorname{res}}f\\varpi ^{-1})^{p-1}\\in \\Gamma ({\\mathcal {B}}^r_N,\\omega _{{\\mathcal {B}}^r_N}^{-1})$ gives rise to a Frobenius splitting of ${\\mathcal {B}}^r_N$ compatible with the splitting $\\varphi \\colon \\operatorname{Fr}_*{\\mathcal {O}}_{{\\mathcal {B}}^r_N}\\rightarrow {\\mathcal {O}}_{{\\mathcal {B}}^r_N}$ arising from $f^{p-1}\\varpi ^{1-p}$ .",
"It suffices to argue generically on ${\\mathcal {B}}^r_N$ .",
"Let $X_N\\subset {\\mathcal {B}}_N$ be an open Bruhat cell: the open orbit of the strictly lower triangular subgroup $U^-_N\\subset GL_N$ .",
"We consider an open cell $X:=X_N^r\\times {\\mathfrak {n}}_r\\subset {\\mathcal {T}}_r^*{\\mathcal {B}}^r_N$ ; we have $X\\cap {\\mathcal {B}}^r_N=X^r_N$ .",
"We will calculate residues on $X$ , so in the definition (REF ) of the function $f$ we will assume that $g_s$ are strictly lower triangular for any $1\\le s\\le r$ .",
"But for $g\\in U^-_N$ and $A\\in {\\mathfrak {gl}}_N$ we have $\\Delta _j(gA)=\\Delta _j(A)$ for any $1\\le j\\le N$ .",
"Hence $\\Delta _j(g_sA_{s,s+1}g_{s+1}^{-1})=\\Delta _j(A_{s,s+1}g_{s+1}^{-1})=\\Delta _j(g_{s+1}A_{s,s+1}g_{s+1}^{-1})$ .",
"Therefore, we can identify $X\\subset {\\mathcal {T}}_r^*{\\mathcal {B}}^r_N$ with $\\hat{X}:=(U^-_N\\times {\\mathfrak {b}}_N)^{r-1}\\times (U^-_N\\times {\\mathfrak {u}}_N)\\subset (GL_N\\times ^{B_N}{\\mathfrak {b}}_N)^{r-1}\\times (GL_N\\times ^{B_N}{\\mathfrak {u}}_N)$ (where ${\\mathfrak {b}}_N\\supset {\\mathfrak {u}}_N$ is the Lie algebra of $B_N$ and its nilpotent radical) so that $f\\varpi ^{-1}|_X=\\hat{f}\\hat{\\varpi }{}^{-1}|_{\\hat{X}}$ .",
"Here $\\hat{\\varpi }$ is an $SL_N^r$ -invariant nowhere vanishing volume form on $\\hat{X}$ , and $\\hat{f}(g_1,A_1,\\ldots ,g_r,A_r):=\\prod _{s=1}^r\\prod _{j=1}^N\\Delta _j(g_sA_sg_s^{-1}).$ According to [14], the chains of divisors required in the lemma exist for each factor $\\hat{X}_s$ of $\\hat{X}$ (that is, $GL_N\\times ^{B_N}{\\mathfrak {b}}_N\\supset {\\mathcal {B}}_N$ or $GL_N\\times ^{B_N}{\\mathfrak {u}}_N\\supset {\\mathcal {B}}_N$ ) equipped with the function $\\hat{f}_s:=\\prod _{j=1}^N\\Delta _j(g_sA_sg_s^{-1})$ and section $\\hat{\\varpi }_s$ , and the required compatibilities hold.",
"Hence the desired compatibility holds for their external product."
],
[
"Proof of Proposition ",
"The section $f^{p-1}\\varpi ^{1-p}\\in \\Gamma ({\\mathcal {T}}_r^*{\\mathcal {B}}^r_N,\\omega ^{1-p})$ gives rise to a morphism $\\varphi \\colon \\operatorname{Fr}_*{\\mathcal {O}}_{{\\mathcal {T}}_r^*{\\mathcal {B}}^r_N}\\rightarrow {\\mathcal {O}}_{{\\mathcal {T}}_r^*{\\mathcal {B}}^r_N}$ , and we have to check that the composition ${\\mathcal {O}}_{{\\mathcal {T}}_r^*{\\mathcal {B}}^r_N}\\rightarrow \\operatorname{Fr}_*{\\mathcal {O}}_{{\\mathcal {T}}_r^*{\\mathcal {B}}^r_N}\\rightarrow {\\mathcal {O}}_{{\\mathcal {T}}_r^*{\\mathcal {B}}^r_N}$ is the identity morphism.",
"It suffices to check $\\varphi (1)=1$ .",
"We know from Lemma REF that $\\varphi (1)|_{{\\mathcal {B}}^r_N}=1$ .",
"We consider a one-parametric central subgroup ${\\mathbb {G}}_m\\hookrightarrow GL^r_N$ whose $s$ -th component is $t\\mapsto t^s$ for $1\\le s\\le r$ .",
"Then $f\\varpi ^{-1}$ is a ${\\mathbb {G}}_m$ -eigensection with a nontrivial character.",
"Hence the zero divisor of $\\varphi (1)$ is ${\\mathbb {G}}_m$ -invariant.",
"Since $({\\mathcal {T}}_r^*{\\mathcal {B}}^r_N)^{{\\mathbb {G}}_m}={\\mathcal {B}}^r_N$ and $\\varphi (1)|_{{\\mathcal {B}}^r_N}=1$ , the function $\\varphi (1)$ has an empty zero divisor.",
"Since the fibers of the projection ${\\mathcal {T}}_r^*{\\mathcal {B}}^r_N\\rightarrow {\\mathcal {B}}^r_N$ are vector spaces, and a nowhere vanishing function on a vector space is constant, we conclude $\\varphi (1)=1$ .",
"$\\Box $"
],
[
"Frobenius splitting of Lusztig's convolution diagrams",
"The aim of this Section is a proof of the following generalization of Theorem REF : Theorem 5.1 Let ${\\mathbf {i}}$ be an arbitrary length $\\ell $ sequence of vertices of the cyclic quiver $Q$ .",
"Let ${\\mathbf {a}}$ be an arbitrary length $\\ell $ sequence of positive integers.",
"Then the iterated convolution diagram $\\widetilde{{\\mathcal {F}}}_{{\\mathbf {i}},{\\mathbf {a}}}$ is Frobenius split.",
"Our proof follows [10] covering $\\widetilde{{\\mathcal {F}}}_{{\\mathbf {i}},{\\mathbf {a}}}$ by an open subvariety of an affine type $A$ BSDH resolution, and then applying Frobenius splitting for BSDH resolutions [13]."
],
[
"Recollections of {{cite:c9a8aac281e5ad0d57d227d194ca39070417c511}}, {{cite:119f41dbc8c27ccb144ff6ed3802a9ccf233dbb6}}",
"We fix ${\\mathbf {d}}=(d_1,\\ldots ,d_r)\\in {\\mathbb {N}}^{{\\mathbb {Z}}/r{\\mathbb {Z}}}$ , and consider a ${\\mathbb {Z}}/r{\\mathbb {Z}}$ -graded vector space ${\\mathbf {V}}=\\bigoplus _{s\\in {\\mathbb {Z}}/r{\\mathbb {Z}}}{\\mathbf {V}}_s$ such that $\\dim {\\mathbf {V}}_s=d_s$ .",
"Given a length $\\ell $ sequence ${\\mathbf {i}}=(s_1,\\ldots ,s_\\ell )\\in ({\\mathbb {Z}}/r{\\mathbb {Z}})^\\ell $ and a length $\\ell $ sequence ${\\mathbf {a}}=(a_1,\\ldots ,a_\\ell )\\in {\\mathbb {N}}^\\ell $ such that $\\sum _{n\\colon s_n=s}a_n=d_s$ for any $s\\in {\\mathbb {Z}}/r{\\mathbb {Z}}$ , we consider the iterated convolution diagram $\\widetilde{{\\mathcal {F}}}_{{\\mathbf {i}},{\\mathbf {a}}}=\\lbrace ({\\mathbf {V}}^\\bullet ,f)\\rbrace $ .",
"Here ${\\mathbf {V}}^\\bullet $ is a ${\\mathbb {Z}}/r{\\mathbb {Z}}$ -graded flag in ${\\mathbf {V}}\\colon {\\mathbf {V}}={\\mathbf {V}}^0\\supset {\\mathbf {V}}^1\\supset \\ldots \\supset {\\mathbf {V}}^\\ell =0$ such that ${\\mathbf {V}}^{n-1}/{\\mathbf {V}}^n$ is an $a_n$ -dimensional vector space supported at the vertex $s_n$ for any $n=1,\\ldots ,\\ell $ , and $f=(f_s\\colon {\\mathbf {V}}_s\\rightarrow {\\mathbf {V}}_{s-1})_{s\\in {\\mathbb {Z}}/r{\\mathbb {Z}}}$ is a $Q$ -module structure on ${\\mathbf {V}}$ such that $f{\\mathbf {V}}^{n-1}\\subset {\\mathbf {V}}^n$ for any $1\\le n\\le \\ell $ .",
"The convolution diagram $\\widetilde{{\\mathcal {F}}}_{{\\mathbf {i}},{\\mathbf {a}}}$ is smooth, being a vector bundle over a flag variety of $GL({\\mathbf {V}})=\\prod _{s\\in {\\mathbb {Z}}/r{\\mathbb {Z}}}GL({\\mathbf {V}}_s)$ .",
"We have a projection $\\pi \\colon \\widetilde{{\\mathcal {F}}}_{{\\mathbf {i}},{\\mathbf {a}}}\\rightarrow {\\mathbf {E}}_{\\mathbf {V}},\\ ({\\mathbf {V}}^\\bullet ,f)\\mapsto f$ to the vector space ${\\mathbf {E}}_{\\mathbf {V}}$ of $Q$ -modules with underlying space ${\\mathbf {V}}$ .",
"The morphism $\\pi $ is proper, and its image is the closure $\\overline{{\\mathbb {O}}}_{{\\mathbf {i}},{\\mathbf {a}}}$ of a nilpotent $GL({\\mathbf {V}})$ -orbit in ${\\mathbf {E}}_{\\mathbf {V}}$ .",
"The union of all nilpotent $GL({\\mathbf {V}})$ -orbits in ${\\mathbf {E}}_{\\mathbf {V}}$ is a closed subvariety ${\\mathbf {E}}^{\\operatorname{nil}}_{\\mathbf {V}}\\subset {\\mathbf {E}}_{\\mathbf {V}}$ (possibly reducible).",
"Let $d:=d_1+\\ldots +d_r$ .",
"Let $F$ be a $d$ -dimensional vector space over the Laurent series field ${\\mathsf {k}}((\\epsilon ))$ .",
"We fix a flag of lattices $\\ldots \\supset L_{-1}\\supset L_0\\supset L_1\\supset \\ldots $ in $F$ such that $L_{s+r}=\\epsilon L_s,\\ L_s\\supset L_{s+1}$ and $\\dim (L_s/L_{s+1})=d_{s\\pmod {r}}$ for any $s\\in {\\mathbb {Z}}$ .",
"We consider a type $\\tilde{A}_{d-1}$ affine Schubert variety $Z$ formed by all the flags of lattices $\\ldots \\supset M_{-1}\\supset M_0\\supset M_1\\supset \\ldots $ in $F$ such that $M_{s+r}=\\epsilon M_s,\\ M_s\\supset M_{s+1}$ and $M_s\\subset L_s,\\ \\dim (L_s/M_s)=d_{s\\pmod {r}}$ for any $s\\in {\\mathbb {Z}}$ .",
"In [10], G. Lusztig constructs an open dense embedding $\\varphi \\colon {\\mathbf {E}}^{\\operatorname{nil}}_{\\mathbf {V}}\\hookrightarrow Z$ (the image ${\\overset{\\circ }{Z}}=\\varphi ({\\mathbf {E}}^{\\operatorname{nil}}_{\\mathbf {V}})\\subset Z$ is specified by certain transversality conditions) such that for any nilpotent $GL({\\mathbf {V}})$ -orbit closure $\\overline{{\\mathbb {O}}}_{{\\mathbf {i}},{\\mathbf {a}}}\\subset {\\mathbf {E}}^{\\operatorname{nil}}_{\\mathbf {V}}$ its image $\\varphi (\\overline{{\\mathbb {O}}}_{{\\mathbf {i}},{\\mathbf {a}}})$ is the intersection of ${\\overset{\\circ }{Z}}$ with a Schubert subvariety $Z_{{\\mathbf {i}},{\\mathbf {a}}}\\subset Z$ .",
"The construction of [10] yields an isomorphism $\\widetilde{\\mathcal {F}}_{{\\mathbf {i}},{\\mathbf {a}}}\\simeq {\\overset{\\circ }{Z}}\\times _Z\\widetilde{Z}{}_{{\\mathbf {i}},{\\mathbf {a}}}$ for a BSDH type resolution $\\widetilde{Z}{}_{{\\mathbf {i}},{\\mathbf {a}}}\\rightarrow Z_{{\\mathbf {i}},{\\mathbf {a}}}$ formed by all the collections $(M^n_s)^{0\\le n\\le \\ell }_{s\\in {\\mathbb {Z}}}$ of double flags of lattices such that (a) $M^0_s=L_s$ ; (b) $M^n_s\\supset M^n_{s+1}$ and $M^n_{s+r}=\\epsilon M^n_s$ ; (c) $M^{n-1}_s=M^n_s$ unless $s=s_n\\pmod {r}$ ; (d) if $s=s_n\\pmod {r}$ , then $M^{n-1}_s\\supset M^n_s$ , and $\\dim M^{n-1}_s/M^n_s=a_n$ ."
],
[
"BSDH resolution",
"According to [3], in order to construct a Frobenius splitting of $\\widetilde{Z}{}_{{\\mathbf {i}},{\\mathbf {a}}}$ (and hence of its open subvariety $\\widetilde{\\mathcal {F}}_{{\\mathbf {i}},{\\mathbf {a}}}$ ) it suffices to construct a proper dominant morphism $\\varpi \\colon \\widehat{Z}{}_{{\\mathbf {i}},{\\mathbf {a}}}\\rightarrow \\widetilde{Z}{}_{{\\mathbf {i}},{\\mathbf {a}}}$ with connected fibers from a Frobenius split variety $\\widehat{Z}{}_{{\\mathbf {i}},{\\mathbf {a}}}$ .",
"We will construct $\\varpi \\colon \\widehat{Z}{}_{{\\mathbf {i}},{\\mathbf {a}}}\\rightarrow \\widetilde{Z}{}_{{\\mathbf {i}},{\\mathbf {a}}}$ in two steps.",
"First we define $\\widehat{Z}{}^{\\prime }_{{\\mathbf {i}},{\\mathbf {a}}}:=\\widetilde{Z}{}_{{\\mathbf {i}},{\\mathbf {a}}}\\times _Z{\\mathcal {F}\\ell }_Z$ where ${\\mathcal {F}\\ell }_Z$ is formed by all the complete flags of lattices $\\ldots \\supset K_{-1}\\supset K_0\\supset K_1\\supset \\ldots $ (so that $K_{u+d}=\\epsilon K_u,\\ K_u\\supset K_{u+1}$ and $\\dim K_u/K_{u+1}=1$ for any $u\\in {\\mathbb {Z}}$ ) such that for $s\\ge 0,\\ L_s\\supset K_{d_0+d_1+\\ldots +d_s}$ and $\\dim L_s/K_{d_0+d_1+\\ldots +d_s}=d_s$ , while for $s<0,\\ L_s\\supset K_{d_0-d_{-1}-\\ldots -d_s}$ and $\\dim L_s/K_{d_0-d_{-1}-\\ldots -d_s}=d_s$ .",
"The evident projection ${\\mathcal {F}\\ell }_Z\\rightarrow Z$ sends $K_\\bullet $ to $M_\\bullet $ where for $s\\ge 0,\\ M_s=K_{d_0+d_1+\\ldots +d_s}$ , while for $s<0,\\ M_s=K_{d_0-d_{-1}-\\ldots -d_s}$ .",
"This projection is a fibration with a fiber isomorphic to a (finite) flag variety of type $A$ .",
"Let us choose a base point $K^0_\\bullet \\in {\\mathcal {F}\\ell }_Z$ such that $K^0_0=L_0$ , and for $s>0,\\ K^0_{d_1+\\ldots +d_s}=L_s$ , and for $s<0,\\ K^0_{-d_{-1}-\\ldots -d_s}=L_s$ .",
"Then the connected component ${\\mathcal {F}\\ell }$ of the ind-variety of complete flags of lattices containing $K^0_\\bullet $ is identified with the affine flag variety of $SL_d$ .",
"The simple reflections of its affine Weyl group are numbered by ${\\mathbb {Z}}/d{\\mathbb {Z}}$ , and any finite sequence $\\underline{u}=(u(1),u(2),\\ldots ,u(k)),\\ u(j)\\in {\\mathbb {Z}}/d{\\mathbb {Z}}$ , gives rise to a BSDH variety $D_{\\underline{u}}\\rightarrow {\\mathcal {F}\\ell }$ projecting to ${\\mathcal {F}\\ell }$ with connected fibers.",
"We consider a concatenated sequence $\\underline{u}=(\\underline{u}_\\ell ,\\ldots ,\\underline{u}_1)$ where for $1\\le n\\le \\ell \\ \\underline{u}_n$ is a sequence of integers in the interval $\\left[d_1+\\ldots +d_{s_n}+1+\\sum _{m<n\\colon s_m=s_n}a_m,\\ d_1+\\ldots +d_{s_n+1}-1+\\sum _{m<n\\colon s_m=s_n+1}a_m\\right]$ giving a reduced expression of the longest element of the (finite) parabolic Weyl subgroup generated by the simple reflections numbered by (the residues modulo $d$ of) the integers in the above interval.",
"Then there is a dominant projection $D_{\\underline{u}}\\rightarrow \\widehat{Z}{}^{\\prime }_{{\\mathbf {i}},{\\mathbf {a}}}$ with connected fibers.",
"Finally, $D_{\\underline{u}}$ is Frobenius split according to [13].",
"Theorem REF is proved.",
"$\\Box $"
],
[
"Cohomology vanishing",
"We say that a generalized multipartition ${\\mu }=(\\mu ^{(1)},\\ldots ,\\mu ^{(r)})\\in {\\mathcal {P}}^r_N$ is regular if for any $s=1,\\ldots ,r$ we have $\\mu ^{(s)}_1>\\mu ^{(s)}_2>\\ldots >\\mu ^{(s)}_N$ .",
"In this case the line bundle ${\\mathcal {O}}({\\mu })$ on ${\\mathcal {T}}_r^*{\\mathcal {B}}^r_N$ is very ample, and we deduce from Theorem REF and [3] the following Corollary 6.1 For a regular multipartition ${\\mu }\\in {\\mathcal {P}}^r_N$ , we have the higher cohomology vanishing $H^{>0}({\\mathcal {T}}_r^*{\\mathcal {B}}^r_N,{\\mathcal {O}}({\\mu }))=0$ .",
"$\\Box $ Similarly to [3] we put forth the following Conjecture 6.2 For a multipartition ${\\mu }\\in {\\mathcal {P}}^r_N$ , we have the higher cohomology vanishing $H^{>0}({\\mathcal {T}}_r^*{\\mathcal {B}}^r_N,{\\mathcal {O}}({\\mu }))=0$ .",
"From Corollary REF and Conjecture REF we deduce Corollary 6.3 For any multipartition ${\\mu }$ we have $[\\Gamma ({\\mathcal {B}}^r_N,{\\mathop {\\operatorname{Sym}}}^\\bullet {\\mathcal {T}}_r{\\mathcal {B}}^r_N\\otimes {\\mathcal {O}}({\\mu }))]=\\sum _{{\\lambda }\\ge {\\mu }}K_{{\\lambda }{\\mu }}(t_1,\\ldots ,t_r)\\chi ^{\\lambda }$ .",
"Hence for any multipartitions ${\\lambda }\\ge {\\mu }$ we have $K_{{\\lambda }{\\mu }}(t_1,\\ldots ,t_r)\\in {\\mathbb {N}}[t_1,\\ldots ,t_r]$ .",
"$\\Box $"
],
[
"Added in proof",
"Conjecture REF is proved in [17].",
"Conjecture REF follows from [15] (see [6]).",
"We are grateful to Wen-Wei Li and Yue Hu for bringing [15] to our attention.",
"M.F.",
": National Research University Higher School of Economics, Russian Federation, Department of Mathematics, 6 Usacheva st, Moscow 119048; Institute for Information Transmission Problems of RAS; [email protected] A.I.",
": National Research University Higher School of Economics, Russian Federation, Department of Mathematics, 6 Usacheva st, Moscow 119048; [email protected]"
]
] | 1605.05806 |
[
[
"Empowering line intensity mapping to study early galaxies"
],
[
"Abstract Line intensity mapping is a superb tool to study the collective radiation from early galaxies.",
"However, the method is hampered by the presence of strong foregrounds, mostly produced by low-redshift interloping lines.",
"We present here a general method to overcome this problem which is robust against foreground residual noise and based on the cross-correlation function $\\psi_{\\alpha L}(r)$ between diffuse line emission and Ly$\\alpha$ emitters (LAE).",
"We compute the diffuse line (Ly$\\alpha$ is used as an example) emission from galaxies in a $(800{\\rm Mpc})^3$ box at $z = 5.7$ and $6.6$.",
"We divide the box in slices and populate them with $14000(5500)$ LAEs at $z = 5.7(6.6)$, considering duty cycles from $10^{-3}$ to $1$.",
"Both the LAE number density and slice volume are consistent with the expected outcome of the Subaru HSC survey.",
"We add gaussian random noise with variance $\\sigma_{\\rm N}$ up to 100 times the variance of the Ly$\\alpha$ emission, $\\sigma_\\alpha$, to simulate foregrounds and compute $\\psi_{\\alpha L}(r)$.",
"We find that the signal-to-noise of the observed $\\psi_{\\alpha L}(r)$ does not change significantly if $\\sigma_{\\rm N} \\le 10 \\sigma_\\alpha$ and show that in these conditions the mean line intensity, $I_{Ly\\alpha}$, can be precisely recovered independently of the LAE duty cycle.",
"Even if $\\sigma_{\\rm N} = 100 \\sigma_\\alpha$, $I_\\alpha$ can be constrained within a factor $2$.",
"The method works equally well for any other line (e.g.",
"HI 21 cm, [CII], HeII) used for the intensity mapping experiment."
],
[
"Introduction",
"The Epoch of Reionization (EoR, redshift $z$ > $$ 5.5$) is a key phase of cosmic evolution for a number of reasons: (i) it is the last major cosmic phase transition \\cite {2001PhR...349..125B}; (ii) it affected a vast majority of the baryons in the universe; (iii) it has been at the edge of our observational capabilities for many years \\cite {2014ApJ...793..115B, 2014ApJ...786..108O, 2015ApJ...804L..30O, 2010ApJ...723..869O, 2008ApJS..176..301O, 2015MNRAS.451..400M}.",
"Most importantly, it is intimately connected with the formation of the first galaxies and to the very beginning of the universe as we know it.$ Unfortunately its observational investigation has proven extremely challenging.",
"EoR galaxies are extremely faint and therefore hard to detect.",
"Moreover the most popular theoretical scenario foresees that reionization is powered by a population of low mass, numerous galaxies [26] whose cumulative emission, dominates over more massive outliers.",
"These “typical” high redshift galaxies are individually out of reach of even the most powerful observatories of the next decades (such as JWSThttp://www.jwst.nasa.gov ,TMThttp://www.tmt.org or ELThttps://www.eso.org/sci/facilities/eelt/).",
"These difficulties make the development of new observational strategies targeting the cumulative emission of galaxies [15], [6], rather than individual detection of sources, quite compelling.",
"Intensity mapping (IM) is one implementation of this approach and targets the 3D fluctuations in the large scale cumulative line emission [25], [32], [33].",
"In principle, it can be applied independently of the particular line chosen, and several alternatives have been proposed in recent years (e. g. HI 21cm [11], CO [16], [25], [3], [CII] [13], [27], [37], H$_2$ [12], HeII [31] and Ly$\\alpha $ [24], [28], [5].",
"Figure: Left: diffuse Lyα\\alpha emission in a slice at z=6.6z=6.6.",
"Right: LAEs position in the same slice at z=6.6z = 6.6 with f d =0.1f_d = 0.1.However, although promising, the feasibility of an IM experiment has not yet been fully demonstrated.",
"The major challenge is that foregrounds dominate over line emission by several orders of magnitude.",
"Although cleaning algorithms have been developed for, e.g.",
"21 cm radiation [35], [4], [36], it is not obvious that they are effective for other lines.",
"Moreover some lines (such as Ly$\\alpha $ and [CII]) present additional difficulties, such as line confusion (Comaschi et al.",
"2016, [14]).",
"It is therefore quite possible that the first generation of intensity mappers will not be able to clean the signal up to a level sufficient to extract the targeted line power spectrum (PS).",
"Thus it is crucial to study and develop alternative strategies, given that the first instruments are starting to be proposed [10], [7], [8].",
"[9] recently showed that cross-correlating IM with point sources (specifically, QSOs in their case) can be very effective, even when only low quality data is available.",
"This approach has two important advantages: (i) the nature and redshift of the point sources are generally well known; (ii) one can observe both the cross-correlation and the auto-correlation of the point sources and use the latter to better constrain the diffuse line emission.",
"Here we propose to combine Ly$\\alpha $ IM with Ly$\\alpha $ emitters (LAEs) data.",
"This technique appears timely and it is motivated by the expectation that the next generation of LAEs surveys (such as the Subaruhttp://www.naoj.org/Projects/HSC/surveyplan.html ones) should find thousands of LAEs in the late EoR ($z = 5.7$ and $6.6$ ).",
"Beyond the low redshift work by [9], this approach was previously investigated only for the 21 cm line at $z = 6.6$ and $7.3$ .",
"[34] used N-body and radiative transfer simulations to compute the 21 cm-galaxy cross-power spectrum.",
"They conclude that the Subaru Hyper Suprime Cam (HSC) and the LOw-Frequency Array for Radio astronomy (LOFAR) [30] observations should show an anti-correlation on scales $k \\approx 0.1 h {\\rm Mpc}^{-1}$ .",
"[29] used seminumerical simulations to compute the 21cm-LAE cross-PS and cross-correlation function, studying different EoR scenarios and LAE duty cycles.",
"In this work we study the properties of the cross-correlation between diffuse Ly$\\alpha $ emission and LAEs with seminumerical simulations [18].",
"We use the model from [5] to populate with diffuse emission two $(800 {\\rm Mpc})^3$ boxes at $z = 5.7$ and $6.6$ .",
"Next we stochastically generate the LAE population in several redshift slices.",
"Each slice is comparable to a Subaru HSC observation both in terms of the number of LAEs it contains, and for its angular extension.",
"In order to test the robustness of such observation to foreground residuals, we add a gaussian random noise with a variance up to 100 times the diffuse emission one and then try to recover the initial line intensity by comparing the measured cross-correlation function with the LAE auto-correlation one.",
"We remark here that the present results depend only weakly on the specific line chosen, as the noise amplitude is scaled with the diffuse line variance.",
"The paper is organized as follows: in Sec.",
"we present our approach; Sec.",
"contains the results.",
"Finally, Sec.",
"draws the conclusions.",
"We assume a flat $\\Lambda $ CDM cosmology compatible with the latest Planck results ($h =0.67$ , $\\Omega _m = 0.32$ , $\\Omega _b = 0.049$ , $\\Omega _\\Lambda = 1-\\Omega _m$ , $n = 0.97$ , $\\sigma _8 = 0.83$ , [23])."
],
[
"Method",
"We aim at deriving the cross-correlation between diffuse line emission and LAEs by producing mock observations with the code DexMhttp://homepage.sns.it/mesinger/Download.html [18].",
"We generate two large (800 cMpc on a side) boxes at $z = 5.7$ and $z=6.6$ corresponding to $30.4(27.9){\\rm deg}^2$ at $z=5.7(6.6)$ , and resolve DM halos with minimum mass $M_{\\rm min}(z = 5.7[6.6]) = 4.1[3.4]\\times 10^{8}M_\\odot $ .",
"The Subaru HSC survey has a redshift precision $\\Delta z = 0.1$ , corresponding to $\\Delta l_z = 45.7(37.9)$ cMpc at $z = 5.7(6.6)$ .",
"Therefore we divide each box in slices of thickness $\\Delta l_z$ as above, obtaining 17 (21) slices at $z = 5.7$ ($z = 6.6$ ).",
"In order to model diffuse line emission, we need to associate a line luminosity to each DM halo.",
"We used the model by [5] (hereafter CF16).",
"Although such method can be applied to any emission line, it is natural to consider IM in the Ly$\\alpha $ line as LAE surveys are in the same band and share similar technical solutions (for example, a comparable angular resolution).",
"Note, however, that the main conclusions of this work do not depend on the specific line used, for reasons discussed in Sec.",
".",
"We compute the Ly$\\alpha $ intensity map by dividing each slice in pixels of size 1 Mpc (corresponding to $\\Delta \\theta = 24.8(23.8)$ arcsec at $z = 5.7(6.6)$ ), considering only the Ly$\\alpha $ emission from the ISM of galaxies.",
"According to CF16 the IGM emission dominates the mean Ly$\\alpha $ intensity; however, as we are here interested in fluctuations and IGM emission is very smooth on scales $k > 0.01 h{\\rm Mpc}^{-1}$ , we neglect such process.",
"Fig.",
"REF (left) shows the Ly$\\alpha $ emission map of a representative slice at $z=6.6$ .",
"The mean Ly$\\alpha $ intensity is $I_ {Ly\\alpha } = 3.51(1.99)\\times 10^{-2} {\\rm nW~}{\\rm m}^{-2}{\\rm sr}^{-1}$ at $z = 5.7(6.6)$ , with a variance $\\sigma _\\alpha = 5.23(3.43)\\times 10^{-2} {\\rm nW~}{\\rm m}^{-2}{\\rm sr}^{-1}$ .",
"Next we populate the slices with LAEs.",
"Since the LAE duty cycle ($f_d$ , the fraction of galaxies that are classified as LAEs at a given time) is not well constrained, we considered five different values, $f_d =(1, 0.2, 0.1, 0.01, 0.001)$ .",
"We generate the LAE list stochastically among the galaxies in a given slice, fixing the total number of LAEs as $N_{\\rm LAE} = 14000(5500)$ at $z=5.7(6.6)$ .",
"The value of $N_{\\rm LAE}$ is consistent with the expected performance of the Subaru HSC survey (M. Ouchi, private communication).",
"We rank halos by mass and, starting from the most massive, we decide if a halo is a LAE randomly with probability $f_d$ ; we stop when we reach $N_{\\rm LAE}$ extractions.",
"Fig.",
"REF (right) shows the LAE distribution in the same slice of Fig.",
"REF (left), with $f_d = 0.1$ .",
"It is evident that both the diffuse Ly$\\alpha $ emission and the LAE distribution trace the large scale DM distribution.",
"We do not include LAE emission in the diffuse Ly$\\alpha $ emission.",
"This is a good approximation because the bulk of diffuse Ly$\\alpha $ radiation is dominated by LBG emission (e. g. [9]).",
"CF16 is consistent with this hypothesis and it predicts the Ly$\\alpha $ diffuse intensity independently of the observed LAE LF.",
"Moreover, the LAE flux is likely to be will be lost anyway because pixels containing a LAE might be removed from the intensity map to better recover the signal from unresolved sources.",
"Finally, we compute the two-point correlation function between Ly$\\alpha $ intensity and LAE with the same approach followed by [9] $\\psi _{\\alpha L}(r, \\Delta r) = \\frac{1}{N_c} \\sum _{i=1}^{N_c} \\Delta I_ {Ly\\alpha } ({\\bf x}_i);$ $N_c$ is the number of LAE-pixels pairs with distance between $r$ and $r+\\Delta r$ ; $\\Delta I_ {Ly\\alpha } ({\\bf x}_i)$ is the fluctuation of the Ly$\\alpha $ intensity in the $i$ -th pair pixel.",
"For comparison we computed also the two-point LAE auto-correlation function, i.e.",
"$\\psi _{\\rm LL}(r, \\Delta r) = \\frac{N_{c, l} - \\langle N_{c, l} \\rangle }{\\langle N_{c, l} \\rangle },$ where $N_{c, l}$ is the number of LAE pairs with distance between $r$ and $r + \\Delta r$ and $\\langle N_{c, l} \\rangle $ in the expected value for a random LAE distribution."
],
[
"Results",
"Fig.",
"REF shows $\\psi _{\\alpha L}(r)$ for $z = 5.7$ and $6.6$ (computed considering bins with $\\Delta r = 1.5 r$ ); the variance is the one among different slices.",
"For all the five $f_d$ considered the amplitude of the correlation signal is sufficient to derive a model-dependent estimate of the diffuse line intensity.",
"Figure: S/N of the cross-correlation function in Fig.",
".",
"In the top panel no noise is included.",
"In the lower panels a gaussian random noise is added to the Lyα\\alpha fluctuations; the variance of the noise is σ N =(0,1,10,100)σ α \\sigma _{\\rm N} = (0, 1, 10, 100)\\sigma _\\alpha , from top to bottom.",
"Different colors refer to different duty cycle values, f d f_d, as in Fig.",
".A difficulty arises, though.",
"A realistic intensity map will contain several sources of foregrounds that need to be removed before it is cross-correlated with LAEs.",
"Because of this, a key step is to assess the robustness of the predicted cross-correlation signal to residual noise.",
"To this aim we superimpose a gaussian random noise with variance $\\sigma _N = n \\sigma _\\alpha $ , $n = (0, 1, 10, 100)$ to the Ly$\\alpha $ intensity map, and compute again $\\psi _{\\alpha L}$ .",
"Fig.",
"REF shows the dependence of the signal-to-noise (S/N) ratio on the noise.",
"One can see that the Ly$\\alpha $ -LAE cross-correlation is very solid and can lead to model dependent estimates of the diffuse line intensity even with a residual noise with a variance 2 orders of magnitude larger than the signal.",
"We point out that this result is almost independent of the particular line chosen for the diffuse emission.",
"In fact, we have assumed a random noise variance purely proportional to the variance of the diffuse emission, without specifying at any level the actual nature of the noise.",
"The only influence of the line choice on the results is the relative distribution of the emitted line luminosity on galaxy mass.",
"This relation can influence both the bias and the shot-noise of line emission.",
"However we do not expect that it could significantly affect the noise resilience and S/N of this cross-correlation.",
"To estimate the diffuse line intensity we proceed as follows.",
"We assume that both $\\psi _{\\alpha L}(r)$ and $\\psi _{LL}(r)$ are proportional to the dark matter two-point correlation function, $\\psi _{\\rm DM}(r)$ : $\\psi _{\\alpha L}(r) = \\langle b \\rangle _ {Ly\\alpha } I_ {Ly\\alpha } \\langle b \\rangle _{\\rm LAE} \\psi _{\\rm DM}(r)\\\\\\psi _{LL}(r) = \\langle b \\rangle _{\\rm LAE}^2 \\psi _{\\rm DM}(r)$ where $\\langle b \\rangle _ {Ly\\alpha } $ is the Ly$\\alpha $ luminosity weighted mean bias, and $\\langle b \\rangle _{\\rm LAE}$ is the mean LAE bias.",
"$\\psi _{LL}(r)$ is computed using the estimator in eq.",
"(REF ).",
"Therefore $\\frac{\\psi _{\\alpha L}(r)}{\\psi _{LL}(r)} = \\frac{\\langle b \\rangle _ {Ly\\alpha } }{\\langle b \\rangle _{\\rm LAE}} I_ {Ly\\alpha } .$ We use eq.",
"(REF ) to estimate $I_ {Ly\\alpha } $ recovered from our mock observations, and compute $\\langle b \\rangle _ {Ly\\alpha } $ and $\\langle b \\rangle _{\\rm LAE}$ from the CF16 modelThe mass function in CF16 is slightly different from the one in the DexM code.",
"We neglect this small inconsistency..",
"Figure: Mean Lyα\\alpha intensity recovered with eq.",
"().",
"Dark(Light) blue points are for z=5.7(6.6)z=5.7(6.6), as a function of the LAE duty cycle f d f_d.",
"At each f d f_d four points show the results for different σ N \\sigma _{\\rm N}: σ N =(0,1,10,100)σ α \\sigma _{\\rm N} = (0, 1, 10, 100)\\sigma _\\alpha from left to right.",
"The horizontal lines and the shaded region show the mean Lyα\\alpha intensity in the mock slices and its 1-σ\\sigma variance.The results (shown in Fig.",
"REF ) are in surprisingly good agreement with the actual $I_ {Ly\\alpha } $ values from the mock slices.",
"Fig.",
"REF shows that the Ly$\\alpha $ -LAE cross-correlation can be successfully used to determine the diffuse line intensity level and that this estimate is solid to foregrounds with a variance $$ < $$ 10$ times larger than the signal.",
"Even if the variance is as large as $ 100 $ our method can recover at least the order of magnitude of $ I Ly $.\\section {Summary and Conclusions}We have studied the cross-correlation $ L(r)$ between diffuse Ly$$~line emission and the Subaru HSC Ly$$~emitters and its robustness to foregrounds.",
"We computed the diffuse Ly$$~emission from dark matter halos in a $ (800Mpc)3$ box at $ z = 5.7$ and $ 6.6$.",
"We divided the box in slices and populate them stochastically with $ 14000(5500)$ LAEs at $ z = 5.7(6.6)$, considering duty cycles from $ 10-3$ to $ 1$; both the LAE number density and the size of the slices are consistent with the expected outcome of the Subaru HSC survey.",
"We added gaussian random noise with variance $ N$ up to 100 times the variance of the Ly$$~emission $$ to simulate foregrounds and compute $ L(r)$.$ We found that the signal-to-noise of the observed $\\psi _{\\alpha L}(r)$ did not change significantly if $\\sigma _{\\rm N} \\le 10 \\sigma _\\alpha $ and we have showed that in these conditions the mean line intensity $I_ {Ly\\alpha } $ can be recovered independently of the LAE duty cycle.",
"Even if $\\sigma _{\\rm N} = 100 \\sigma _\\alpha $ , $I_\\alpha $ could be constrained within a factor 2.",
"We point out that these results depend only very weakly on the line chosen to perform the intensity mapping experiment.",
"These results are very promising because they show that even with a dirty IM observation it is possible to recover the diffuse line intensity by cross-correlating with point sources, such as LAE.",
"Since removing the residual foregrounds (mostly coming from interloping lines) in future IM surveys will not be an easy task, relying on a solid cross-correlation is crucial to extract information from IM experiments.",
"In addition, the cross-correlation with point sources can provide an independent test of the quality of the recovered line signal after foreground removal.",
"These results represent only a first step towards a more comprehensive feasibility study of the proposed technique.",
"Nevertheless they clearly demonstrate the potentiality of the approach.",
"Future work should be devoted to simulate mock observations with realistic foreground residuals including interlopers, and more sophisticated LAE models.",
"The estimate of the diffuse line intensity might also benefit by a more advanced treatment of all the contributing processes.",
"These improvements are certainly encouraged by the success of the simple model presented here."
]
] | 1605.06124 |
[
[
"Magnetic Domain Wall Engineering in a Nanoscale Permalloy Junction"
],
[
"Abstract Nanoscale magnetic junction provides a useful approach to act as the building block for magnetoresistive random access memories (MRAM), where one of the key issues is to control the magnetic domain configuration.",
"Here, we study the domain structure and the magnetic switching in the Permalloy (Fe20Ni80) nanoscale magnetic junctions with different thicknesses by using micromagnetic simulations.",
"It is found that both the 90-degree and 45-degree domain walls can be formed between the junctions and the wire arms depending on the thickness of the device.",
"The magnetic switching fields show distinct thickness dependencies with a broad peak varying from 7 nm to 22 nm depending on the junction sizes, and the large magnetic switching fields favor the stability of the MRAM operation."
],
[
"Supplementary Material",
"See supplementary material for magnetization distribution in device at different layers and side views of the magnetization distribution in device.",
"This work was supported by State Key Program for Basic Research of China (Grant No.",
"2014CB921101, 2016YFA0300803 ), NSFC (Grants No.",
"61427812, 11574137), Jiangsu NSF (BK20140054), Jiangsu Shuangchuang Team Program, Shenzhen Fundamental Research Fund under Grant No.",
"JCYJ20160331164412545 and the UK EPSRC (EP/G010064/1)."
]
] | 1605.05998 |
[
[
"A Multi-Batch L-BFGS Method for Machine Learning"
],
[
"Abstract The question of how to parallelize the stochastic gradient descent (SGD) method has received much attention in the literature.",
"In this paper, we focus instead on batch methods that use a sizeable fraction of the training set at each iteration to facilitate parallelism, and that employ second-order information.",
"In order to improve the learning process, we follow a multi-batch approach in which the batch changes at each iteration.",
"This can cause difficulties because L-BFGS employs gradient differences to update the Hessian approximations, and when these gradients are computed using different data points the process can be unstable.",
"This paper shows how to perform stable quasi-Newton updating in the multi-batch setting, illustrates the behavior of the algorithm in a distributed computing platform, and studies its convergence properties for both the convex and nonconvex cases."
],
[
"Introduction",
"It is common in machine learning to encounter optimization problems involving millions of parameters and very large datasets.",
"To deal with the computational demands imposed by such applications, high performance implementations of stochastic gradient and batch quasi-Newton methods have been developed [1], [11], [9].",
"In this paper we study a batch approach based on the L-BFGS method [20] that strives to reach the right balance between efficient learning and productive parallelism.",
"In supervised learning, one seeks to minimize empirical risk, $ {F(w) := \\frac{1}{n} \\sum _{i=1}^{n}f(w;x^{i},y^{i}) \\stackrel{\\rm def}{=}\\frac{1}{n} \\sum _{i=1}^{n}f_i(w)},$ where $ (x^i, y^i)_{i=1}^n$ denote the training examples and $f(\\cdot ;x,y) : \\mathbb {R}^d \\rightarrow \\mathbb {R}$ is the composition of a prediction function (parametrized by $w$ ) and a loss function.",
"The training problem consists of finding an optimal choice of the parameters $w \\in \\mathbb {R}^d$ with respect to $F$ , i.e., $ \\min _{w\\in \\mathbb {R}^d}F(w) =\\frac{1}{n} \\sum _{i=1}^{n}f_i(w).$ At present, the preferred optimization method is the stochastic gradient descent (SGD) method [23], [5], and its variants [14], [24], [12], which are implemented either in an asynchronous manner (e.g.",
"when using a parameter server in a distributed setting) or following a synchronous mini-batch approach that exploits parallelism in the gradient evaluation [2], [22], [13].",
"A drawback of the asynchronous approach is that it cannot use large batches, as this would cause updates to become too dense and compromise the stability and scalability of the method [16], [22].",
"As a result, the algorithm spends more time in communication as compared to computation.",
"On the other hand, using a synchronous mini-batch approach one can achieve a near-linear decrease in the number of SGD iterations as the mini-batch size is increased, up to a certain point after which the increase in computation is not offset by the faster convergence [26].",
"An alternative to SGD is a batch method, such as L-BFGS, which is able to reach high training accuracy and allows one to perform more computation per node, so as to achieve a better balance with communication costs [27].",
"Batch methods are, however, not as efficient learning algorithms as SGD in a sequential setting [6].",
"To benefit from the strength of both methods some high performance systems employ SGD at the start and later switch to a batch method [1].",
"Multi-Batch Method.",
"In this paper, we follow a different approach consisting of a single method that selects a sizeable subset (batch) of the training data to compute a step, and changes this batch at each iteration to improve the learning abilities of the method.",
"We call this a multi-batch approach to differentiate it from the mini-batch approach used in conjunction with SGD, which employs a very small subset of the training data.",
"When using large batches it is natural to employ a quasi-Newton method, as incorporating second-order information imposes little computational overhead and improves the stability and speed of the method.",
"We focus here on the L-BFGS method, which employs gradient information to update an estimate of the Hessian and computes a step in $O(d)$ flops, where $d$ is the number of variables.",
"The multi-batch approach can, however, cause difficulties to L-BFGS because this method employs gradient differences to update Hessian approximations.",
"When the gradients used in these differences are based on different data points, the updating procedure can be unstable.",
"Similar difficulties arise in a parallel implementation of the standard L-BFGS method, if some of the computational nodes devoted to the evaluation of the function and gradient are unable to return results on time — as this again amounts to using different data points to evaluate the function and gradient at the beginning and the end of the iteration.",
"The goal of this paper is to show that stable quasi-Newton updating can be achieved in both settings without incurring extra computational cost, or special synchronization.",
"The key is to perform quasi-Newton updating based on the overlap between consecutive batches.",
"The only restriction is that this overlap should not be too small, something that can be achieved in most situations.",
"Contributions.",
"We describe a novel implementation of the batch L-BFGS method that is robust in the absence of sample consistency; i.e., when different samples are used to evaluate the objective function and its gradient at consecutive iterations.",
"The numerical experiments show that the method proposed in this paper — which we call the multi-batch L-BFGS method — achieves a good balance between computation and communication costs.",
"We also analyze the convergence properties of the new method (using a fixed step length strategy) on both convex and nonconvex problems."
],
[
"The Multi-Batch Quasi-Newton Method",
"In a pure batch approach, one applies a gradient based method, such as L-BFGS [20], to the deterministic optimization problem (REF ).",
"When the number $n$ of training examples is large, it is natural to parallelize the evaluation of $F$ and $\\nabla F$ by assigning the computation of the component functions $f_i$ to different processors.",
"If this is done on a distributed platform, it is possible for some of the computational nodes to be slower than the rest.",
"In this case, the contribution of the slow (or unresponsive) computational nodes could be ignored given the stochastic nature of the objective function.",
"This leads, however, to an inconsistency in the objective function and gradient at the beginning and at the end of the iteration, which can be detrimental to quasi-Newton methods.",
"Thus, we seek to find a fault-tolerant variant of the batch L-BFGS method that is capable of dealing with slow or unresponsive computational nodes.",
"A similar challenge arises in a multi-batch implementation of the L-BFGS method in which the entire training set $ T= \\lbrace (x^i, y^i)_{i=1}^n\\rbrace $ is not employed at every iteration, but rather, a subset of the data is used to compute the gradient.",
"Specifically, we consider a method in which the dataset is randomly divided into a number of batches — say 10, 50, or 100 — and the minimization is performed with respect to a different batch at every iteration.",
"At the $k$ -th iteration, the algorithm chooses a batch $S_k \\subset \\lbrace 1, \\ldots , n\\rbrace $ , computes $ {F}^{S_{k}}(w_k)=\\frac{1}{\\left|S_{k}\\right|} \\sum _{i\\in S_{k}}f_i\\left(w_{k}\\right), \\qquad \\nabla {F}^{S_{k}}(w_k) = {g}_{k}^{S_{k}} = \\frac{1}{\\left|S_{k}\\right|} \\sum _{i\\in S_{k}}\\nabla f_i\\left(w_{k}\\right) ,$ and takes a step along the direction $- H_k g_k^{S_k}$ , where $H_k$ is an approximation to $\\nabla ^2 F(w_k)^{-1}$ .",
"Allowing the sample $S_k$ to change freely at every iteration gives this approach flexibility of implementation and is beneficial to the learning process, as we show in Section .",
"(We refer to $S_k$ as the sample of training points, even though $S_k$ only indexes those points.)",
"The case of unresponsive computational nodes and the multi-batch method are similar.",
"The main difference is that node failures create unpredictable changes to the samples $S_k$ , whereas a multi-batch method has control over sample generation.",
"In either case, the algorithm employs a stochastic approximation to the gradient and can no longer be considered deterministic.",
"We must, however, distinguish our setting from that of the classical SGD method, which employs small mini-batches and noisy gradient approximations.",
"Our algorithm operates with much larger batches so that distributing the function evaluation is beneficial and the compute time of $g_k^{S_k}$ is not overwhelmed by communication costs.",
"This gives rise to gradients with relatively small variance and justifies the use of a second-order method such as L-BFGS.",
"Robust Quasi-Newton Updating.",
"The difficulties created by the use of a different sample $S_k$ at each iteration can be circumvented if consecutive samples $S_{k}$ and $S_{k+1}$ overlap, so that $ O_k= S_{k} \\cap S_{k+1} \\ne \\emptyset .$ One can then perform stable quasi-Newton updating by computing gradient differences based on this overlap, i.e., by defining $ y_{k+1}=g_{k+1}^{O_{k}}-g_{k}^{O_{k}}, \\qquad s_{k+1} = w_{k+1}-w_k,$ in the notation given in (REF ).",
"The correction pair $(y_k, s_k)$ can then be used in the BFGS update.",
"When the overlap set $O_k$ is not too small, $y_k$ is a useful approximation of the curvature of the objective function $F$ along the most recent displacement, and will lead to a productive quasi-Newton step.",
"This observation is based on an important property of Newton-like methods, namely that there is much more freedom in choosing a Hessian approximation than in computing the gradient [7], [3].",
"Thus, a smaller sample $O_k$ can be employed for updating the inverse Hessian approximation $H_k$ than for computing the batch gradient $g_k^{S_k}$ in the search direction $- H_k g_k^{S_k}$ .",
"In summary, by ensuring that unresponsive nodes do not constitute the vast majority of all working nodes in a fault-tolerant parallel implementation, or by exerting a small degree of control over the creation of the samples $S_k$ in the multi-batch method, one can design a robust method that naturally builds upon the fundamental properties of BFGS updating.",
"We should mention in passing that a commonly used strategy for ensuring stability of quasi-Newton updating in machine learning is to enforce gradient consistency [25], i.e., to use the same sample $S_k$ to compute gradient evaluations at the beginning and the end of the iteration.",
"Another popular remedy is to use the same batch $S_k$ for multiple iterations [19], alleviating the gradient inconsistency problem at the price of slower convergence.",
"In this paper, we assume that achieving such sample consistency is not possible (in the fault-tolerant case) or desirable (in a multi-batch framework), and wish to design a new variant of L-BFGS that imposes minimal restrictions in the sample changes."
],
[
"Specification of the Method",
"At the $k$ -th iteration, the multi-batch BFGS algorithm chooses a set $S_k \\subset \\lbrace 1, \\ldots , n\\rbrace $ and computes a new iterate $ w_{k+1}=w_k-\\alpha _{k}H_{k} g_{k}^{S_{k}} ,$ where $\\alpha _{k}$ is the step length, $g_k^{S_k}$ is the batch gradient (REF ) and $H_{k}$ is the inverse BFGS Hessian matrix approximation that is updated at every iteration by means of the formula $ H_{k+1}=V_{k}^{T}H_{k}V_{k}+\\rho _{k}s_{k}s_{k}^{T}, \\qquad \\rho _{k}=\\tfrac{1}{y_{k}^{T}s_{k}}, \\qquad V_{k}=I-\\rho _{k}y_{k}s_{k}^{T} .$ To compute the correction vectors $(s_k, y_k)$ , we determine the overlap set $O_k = S_{k} \\cap S_{k+1}$ consisting of the samples that are common at the $k$ -th and $k+1$ -st iterations.",
"We define $ {F}^{O_{k}}(w_k)=\\frac{1}{\\left|O_{k}\\right|} \\sum _{i\\in O_{k}}f_i\\left(w_k\\right),\\qquad \\nabla F^{O_k}(w_k)={g}_{k}^{O_{k}}=\\frac{1}{\\left|O_{k}\\right|} \\sum _{i\\in O_{k}}\\nabla f_i\\left(w_k\\right),$ and compute the correction vectors as in (REF ).",
"In this paper we assume that $\\alpha _k$ is constant.",
"In the limited memory version, the matrix $H_k$ is defined at each iteration as the result of applying $m$ BFGS updates to a multiple of the identity matrix, using a set of $m$ correction pairs $\\lbrace s_i, y_i\\rbrace $ kept in storage.",
"The memory parameter $m$ is typically in the range 2 to 20.",
"When computing the matrix-vector product in (REF ) it is not necessary to form that matrix $H_k$ since one can obtain this product via the two-loop recursion [20], using the $m$ most recent correction pairs $\\lbrace s_i, y_i\\rbrace $ .",
"After the step has been computed, the oldest pair $(s_j, y_j)$ is discarded and the new curvature pair is stored.",
"A pseudo-code of the proposed method is given below, and depends on several parameters.",
"The parameter $r$ denotes the fraction of samples in the dataset used to define the gradient, i.e., $r = \\frac{\\left| S\\right|}{n}$ .",
"The parameter $o$ denotes the length of overlap between consecutive samples, and is defined as a fraction of the number of samples in a given batch $S$ , i.e., $o = \\frac{\\left| O\\right|}{\\left| S\\right|}$ .",
"Multi-Batch L-BFGS Input: $w_{0}$ (initial iterate), $ T= \\lbrace (x^i, y^i)$ , for $i=1, \\ldots , n\\rbrace $ (training set), $m$ (memory parameter), $r$ (batch, fraction of $n$ ), $o$ (overlap, fraction of batch), $k\\leftarrow 0$ (iteration counter).",
"[1] Create initial batch $S_{0}$ As shown in Firgure REF $k=0,1,2,...$ Calculate the search direction $p_{k}=-H_{k}g_{k}^{S_{k}}$ Using L-BFGS formula Choose the step length $\\alpha _{k} >0$ Compute $w_{k+1}=w_k+\\alpha _{k}p_{k}$ Create the next batch $S_{k+1}$ Compute the curvature pairs $s_{k+1}=w_{k+1}-w_k$ and $y_{k+1}=g_{k+1}^{O_{k}}-g_{k}^{O_{k}}$ Replace the oldest pair $(s_i, y_i)$ by $s_{k+1}, y_{k+1}$"
],
[
"Sample Generation",
"We now discuss how the sample $S_{k+1}$ is created at each iteration (Line 8 in Algorithm REF ).",
"Distributed Computing with Faults.",
"Consider a distributed implementation in which slave nodes read the current iterate $w_k$ from the master node, compute a local gradient on a subset of the dataset, and send it back to the master node for aggregation in the calculation (REF ).",
"Given a time (computational) budget, it is possible for some nodes to fail to return a result.",
"The schematic in Figure REF a illustrates the gradient calculation across two iterations, $k$ and $k+1$ , in the presence of faults.",
"Here $\\mathcal {B}_i$ , $i=1,...,B$ denote the batches of data that each slave node $i$ receives (where $T = \\cup _i \\mathcal {B}_i$ ), and $\\tilde{\\nabla }f(w)$ is the gradient calculation using all nodes that responded within the preallocated time.",
"Figure: Sample and Overlap formation.Let $\\mathcal {J}_k\\subset \\lbrace 1,2,...,B\\rbrace $ and $\\mathcal {J}_{k+1}\\subset \\lbrace 1,2,...,B\\rbrace $ be the set of indices of all nodes that returned a gradient at the $k$ -th and $k+1$ -st iterations, respectively.",
"Using this notation $S_k = \\cup _{j\\in \\mathcal {J}_k} \\mathcal {B}_j$ and $S_{k+1} = \\cup _{j\\in \\mathcal {J}_{k+1}} \\mathcal {B}_j$ , and we define $O_k = \\cup _{j \\in \\mathcal {J}_k\\cap \\mathcal {J}_{k+1}} \\mathcal {B}_j$ .",
"The simplest implementation in this setting preallocates the data on each compute node, requiring minimal data communication, i.e., only one data transfer.",
"In this case the samples $S_k$ will be independent if node failures occur randomly.",
"On the other hand, if the same set of nodes fail, then sample creation will be biased, which is harmful both in theory and practice.",
"One way to ensure independent sampling is to shuffle and redistribute the data to all nodes after a certain number of iterations.",
"Multi-batch Sampling.",
"We propose two strategies for the multi-batch setting.",
"Figure REF b illustrates the sample creation process in the first strategy.",
"The dataset is shuffled and batches are generated by collecting subsets of the training set, in order.",
"Every set (except $S_0$ ) is of the form $S_k= \\lbrace O_{k-1}, N_k, O_k\\rbrace $ , where $O_{k-1}$ and $O_k$ are the overlapping samples with batches $S_{k-1}$ and $S_{k+1}$ respectively, and $N_k$ are the samples that are unique to batch $S_k$ .",
"After each pass through the dataset, the samples are reshuffled, and the procedure described above is repeated.",
"In our implementation samples are drawn without replacement, guaranteeing that after every pass (epoch) all samples are used.",
"This strategy has the advantage that it requires no extra computation in the evaluation of $g_k^{O_k}$ and $g_{k+1}^{O_k}$ , but the samples $\\lbrace S_k\\rbrace $ are not independent.",
"The second sampling strategy is simpler and requires less control.",
"At every iteration $k$ , a batch $S_k$ is created by randomly selecting $\\left| S_k \\right|$ elements from $\\lbrace 1,\\ldots n\\rbrace $ .",
"The overlapping set $O_k$ is then formed by randomly selecting $\\left| O_k \\right|$ elements from $S_k$ (subsampling).",
"This strategy is slightly more expensive since $g_{k+1}^{O_k}$ requires extra computation, but if the overlap is small this cost is not significant."
],
[
"Convergence Analysis",
"In this section, we analyze the convergence properties of the multi-batch L-BFGS method (Algorithm REF ) when applied to the minimization of strongly convex and nonconvex objective functions, using a fixed step length strategy.",
"We assume that the goal is to minimize the empirical risk $F$ given in (REF ), but note that a similar analysis could be used to study the minimization of the expected risk."
],
[
"Strongly Convex case",
"Due to the stochastic nature of the multi-batch approach, every iteration of Algorithm REF employs a gradient that contains errors that do not converge to zero.",
"Therefore, by using a fixed step length strategy one cannot establish convergence to the optimal solution $w^{\\star }$ , but only convergence to a neighborhood of $w^{\\star }$ [18].",
"Nevertheless, this result is of interest as it reflects the common practice of using a fixed step length and decreasing it only if the desired testing error has not been achieved.",
"It also illustrates the tradeoffs that arise between the size of the batch and the step length.",
"In our analysis, we make the following assumptions about the objective function and the algorithm.",
"Assumptions A.",
"$F$ is twice continuously differentiable.",
"There exist positive constants $\\hat{\\lambda }$ and $\\hat{\\Lambda }$ such that $\\hat{\\lambda } I \\preceq \\nabla ^2F^O(w) \\preceq \\hat{\\Lambda } I$ for all $w \\in \\mathbb {R}^d$ and all sets $O \\subset \\lbrace 1,2,\\ldots ,n\\rbrace $ .",
"There is a constant $\\gamma $ such that $\\mathbb {E}_{S}\\left[ \\Vert \\nabla F^{S}(w) \\Vert \\right]^2 \\le \\gamma ^2$ for all $w \\in \\mathbb {R}^d$ and all sets $S\\subset \\lbrace 1,2,\\ldots ,n\\rbrace $ .",
"The samples $S$ are drawn independently and $\\nabla F^{S}(w)$ is an unbiased estimator of the true gradient $\\nabla F(w)$ for all $w \\in \\mathbb {R}^d$ , i.e., $\\mathbb {E}_{S}[ \\nabla F^{S}(w)] = \\nabla F(w).$ Note that Assumption $A.2$ implies that the entire Hessian $\\nabla ^2F(w)$ also satisfies $\\lambda I \\preceq \\nabla ^2F(w) \\preceq \\Lambda I, \\quad \\forall w \\in \\mathbb {R}^d,$ for some constants $ \\lambda , \\Lambda >0$ .",
"Assuming that every sub-sampled function $F^O(w)$ is strongly convex is not unreasonable as a regularization term is commonly added in practice when that is not the case.",
"We begin by showing that the inverse Hessian approximations $H_k$ generated by the multi-batch L-BFGS method have eigenvalues that are uniformly bounded above and away from zero.",
"The proof technique used is an adaptation of that in [8].",
"Lemma 3.1 If Assumptions A.1-A.2 above hold, there exist constants $0<\\mu _1\\le \\mu _2$ such that the Hessian approximations $\\lbrace H_k\\rbrace $ generated by Algorithm REF satisfy $ \\mu _1 I \\preceq H_k \\preceq \\mu _2 I,\\qquad \\text{for } k=0,1,2,\\dots $ Utilizing Lemma REF , we show that the multi-batch L-BFGS method with a constant step length converges to a neighborhood of the optimal solution.",
"Theorem 3.2 Suppose that Assumptions A.1-A.4 hold and let $F^{\\star } = F(w^{\\star })$ , where $w^{\\star }$ is the minimizer of $F$ .",
"Let $\\lbrace w_k\\rbrace $ be the iterates generated by Algorithm REF with $\\alpha _k = \\alpha \\in (0,\\frac{1}{2\\mu _1 \\lambda })$ , starting from $w_0$ .",
"Then for all $k\\ge 0$ , $\\mathbb {E} [ F(w_k) - F^{\\star } ] & \\le ( 1-2\\alpha \\mu _1 \\lambda )^k [ F(w_0) - F^{\\star } ] + [ 1-(1-\\alpha \\mu _1 \\lambda )^k ]\\frac{\\alpha \\mu _2^2 \\gamma ^2 \\Lambda }{4 \\mu _1 \\lambda } \\\\ &\\xrightarrow{} \\frac{\\alpha \\mu _2^2 \\gamma ^2 \\Lambda }{4 \\mu _1 \\lambda }.",
"$ The bound provided by this theorem has two components: (i) a term decaying linearly to zero, and (ii) a term identifying the neighborhood of convergence.",
"Note that a larger step length yields a more favorable constant in the linearly decaying term, at the cost of an increase in the size of the neighborhood of convergence.",
"We will consider again these tradeoffs in Section , where we also note that larger batches increase the opportunities for parallelism and improve the limiting accuracy in the solution, but slow down the learning abilities of the algorithm.",
"One can establish convergence of the multi-batch L-BFGS method to the optimal solution $w^\\star $ by employing a sequence of step lengths $\\lbrace \\alpha _k \\rbrace $ that converge to zero according to the schedule proposed by Robbins and Monro [23].",
"However, that provides only a sublinear rate of convergence, which is of little interest in our context where large batches are employed and some type of linear convergence is expected.",
"In this light, Theorem REF is more relevant to practice."
],
[
"Nonconvex case",
"The BFGS method is known to fail on noconvex problems [17], [10].",
"Even for L-BFGS, which makes only a finite number of updates at each iteration, one cannot guarantee that the Hessian approximations have eigenvalues that are uniformly bounded above and away from zero.",
"To establish convergence of the BFGS method in the nonconvex case cautious updating procedures have been proposed [15].",
"Here we employ a cautious strategy that is well suited to our particular algorithm; we skip the update, i.e., set $H_{k+1} = H_k$ , if the curvature condition $ y_k^Ts_k \\ge {\\epsilon } \\Vert s_k \\Vert ^2$ is not satisfied, where $\\epsilon >0$ is a predetermined constant.",
"Using said mechanism we show that the eigenvalues of the Hessian matrix approximations generated by the multi-batch L-BFGS method are bounded above and away from zero (Lemma REF ).",
"The analysis presented in this section is based on the following assumptions.",
"Assumptions B.",
"$F$ is twice continuously differentiable.",
"The gradients of $F$ are $\\Lambda $ -Lipschitz continuous, and the gradients of $F^{O}$ are $\\Lambda _{O}$ -Lipschitz continuous for all $w \\in \\mathbb {R}^d$ and all sets $O \\subset \\lbrace 1,2,\\ldots ,n\\rbrace $ .",
"The function $ F(w)$ is bounded below by a scalar $\\widehat{F}$ .",
"There exist constants $\\gamma \\ge 0$ and $\\eta >0$ such that $\\mathbb {E}_{S}\\left[ \\Vert \\nabla F^{S}(w) \\Vert \\right]^2 \\le \\gamma ^2 + \\eta \\Vert \\nabla F(w)\\Vert ^2$ for all $w \\in \\mathbb {R}^d$ and all sets $S\\subset \\lbrace 1,2,\\ldots ,n\\rbrace $ .",
"The samples $S$ are drawn independently and $\\nabla F^{S}(w)$ is an unbiased estimator of the true gradient $\\nabla F(w)$ for all $w \\in \\mathbb {R}^d$ , i.e., $\\mathbb {E} [ \\nabla F^{S}(w) ] = \\nabla F(w).$ Lemma 3.3 Suppose that Assumptions B.1-B.2 hold and let $\\epsilon >0$ be given.",
"Let $\\lbrace H_k \\rbrace $ be the Hessian approximations generated by Algorithm REF , with the modification that $H_{k+1} = H_k$ whenever (REF ) is not satisfied.",
"Then, there exist constants $0<\\mu _1\\le \\mu _2$ such that $ \\mu _1 I \\preceq H_k \\preceq \\mu _2 I,\\qquad \\text{for } k=0,1,2,\\dots $ We can now follow the analysis in [4] to establish the following result about the behavior of the gradient norm for the multi-batch L-BFGS method with a cautious update strategy.",
"Theorem 3.4 Suppose that Assumptions B.1-B.5 above hold, and let $\\epsilon >0$ be given.",
"Let $\\lbrace w_k\\rbrace $ be the iterates generated by Algorithm REF , with $\\alpha _k = \\alpha \\in (0,\\frac{\\mu _1}{\\mu _2^2\\eta \\Lambda } )$ , starting from $w_0$ , and with the modification that $H_{k+1} = H_k$ whenever (REF ) is not satisfied.",
"Then, $ \\mathbb {E} \\Big [\\frac{1}{L}\\sum _{k=0}^{L-1} \\Vert \\nabla F(w_k) \\Vert ^2 \\Big ] & \\le \\frac{\\alpha \\mu _2^2 \\gamma ^2 \\Lambda }{ \\mu _1 } + \\frac{2[ F(w_0) - \\widehat{F} ]}{\\alpha \\mu _1 L }\\\\& \\xrightarrow{}\\frac{\\alpha \\mu _2^2 \\gamma ^2 \\Lambda }{ \\mu _1 }.$ This result bounds the average norm of the gradient of $F$ after the first $L-1$ iterations, and shows that the iterates spend increasingly more time in regions where the objective function has a small gradient."
],
[
"Numerical Results ",
"In this Section, we present numerical results that evaluate the proposed robust multi-batch L-BFGS scheme (Algorithm REF ) on logistic regression problems.",
"Figure REF shows the performance on the webspam datasetLIBSVM: https://www.csie.ntu.edu.tw/~cjlin/libsvmtools/datasets/binary.html., where we compare it against three methods: (i) multi-batch L-BFGS without enforcing sample consistency (L-BFGS), where gradient differences are computed using different samples, i.e., $y_k = g_{k+1}^{S_{k+1}}-g_{k}^{S_{k}}$ ; (ii) multi-batch gradient descent (Gradient Descent), which is obtained by setting $H_k = I$ in Algorithm REF ; and, (iii) serial SGD, where at every iteration one sample is used to compute the gradient.",
"We run each method with 10 different random seeds, and, where applicable, report results for different batch ($r$ ) and overlap ($o$ ) sizes.",
"The proposed method is more stable than the standard L-BFGS method; this is especially noticeable when $r$ is small.",
"On the other hand, serial SGD achieves similar accuracy as the robust L-BFGS method and at a similar rate (e.g., $r=1\\%$ ), at the cost of $n$ communications per epochs versus $\\frac{1}{r(1-o)}$ communications per epoch.",
"Figure REF also indicates that the robust L-BFGS method is not too sensitive to the size of overlap.",
"Similar behavior was observed on other datasets, in regimes where $r\\cdot o$ was not too small; see Appendix REF .",
"We mention in passing that the L-BFGS step was computed using the a vector-free implementation proposed in [9].",
"Figure: webspam dataset.",
"Comparison of Robust L-BFGS, L-BFGS (multi-batch L-BFGS without enforcing sample consistency), Gradient Descent (multi-batch Gradient method) and SGD for various batch (rr) and overlap (oo) sizes.",
"Solid lines show average performance, and dashed lines show worst and best performance, over 10 runs (per algorithm).",
"K=16K=16 MPI processes.We also explore the performance of the robust multi-batch L-BFGS method in the presence of node failures (faults), and compare it to the multi-batch variant that does not enforce sample consistency (L-BFGS).",
"Figure REF illustrates the performance of the methods on the webspam dataset, for various probabilities of node failures $p \\in \\lbrace 0.1, 0.3, 0.5 \\rbrace $ , and suggests that the robust L-BFGS variant is more stable; see Appendix REF for further results.",
"Figure: webspam dataset.",
"Comparison of Robust L-BFGS and L-BFGS (multi-batch L-BFGS without enforcing sample consistency), for various node failure probabilities pp.",
"Solid lines show average performance, and dashed lines show worst and best performance, over 10 runs (per algorithm).",
"K=16K=16 MPI processes.Lastly, we study the strong and weak scaling properties of the robust L-BFGS method on artificial data (Figure REF ).",
"We measure the time needed to compute a gradient (Gradient) and the associated communication (Gradient+C), as well as, the time needed to compute the L-BFGS direction (L-BFGS) and the associated communication (L-BFGS+C), for various batch sizes ($r$ ).",
"The figure on the left shows strong scaling of multi-batch LBFGS on a $d=10^4$ dimensional problem with $n=10^7$ samples.",
"The size of input data is 24GB, and we vary the number of MPI processes, $K \\in \\lbrace 1,2,\\dots , 128\\rbrace $ .",
"The time it takes to compute the gradient decreases with $K$ , however, for small values of $r$ , the communication time exceeds the compute time.",
"The figure on the right shows weak scaling on a problem of similar size, but with varying sparsity.",
"Each sample has $10\\cdot K$ non-zero elements, thus for any $K$ the size of local problem is roughly $1.5$ GB (for $K=128$ size of data 192GB).",
"We observe almost constant time for the gradient computation while the cost of computing the L-BFGS direction decreases with $K$ ; however, if communication is considered, the overall time needed to compute the L-BFGS direction increases slightly.",
"For more details see Appendix .",
"Figure: Strong and weak scaling of multi-batch L-BFGS method."
],
[
"Conclusion",
"This paper describes a novel variant of the L-BFGS method that is robust and efficient in two settings.",
"The first occurs in the presence of node failures in a distributed computing implementation; the second arises when one wishes to employ a different batch at each iteration in order to accelerate learning.",
"The proposed method avoids the pitfalls of using inconsistent gradient differences by performing quasi-Newton updating based on the overlap between consecutive samples.",
"Numerical results show that the method is efficient in practice, and a convergence analysis illustrates its theoretical properties."
],
[
"Acknowledgements",
"The first two authors were supported by the Office of Naval Research award N000141410313, the Department of Energy grant DE-FG02-87ER25047 and the National Science Foundation grant DMS-1620022.",
"Martin Takáč was supported by National Science Foundation grant CCF-1618717."
],
[
"Assumptions",
"We first restate the Assumptions that we use in the Convergence Analysis section (Section ).",
"Assumption $A$ and $B$ are used in the strongly convex and nonconvex cases, respectively."
],
[
"Assumptions A",
" $F$ is twice continuously differentiable.",
"There exist positive constants $\\hat{\\lambda }$ and $\\hat{\\Lambda }$ such that $ \\hat{\\lambda } I \\preceq \\nabla ^2F^O(w) \\preceq \\hat{\\Lambda } I,$ for all $w \\in \\mathbb {R}^d$ and all sets $O \\subset \\lbrace 1,2,\\ldots ,n\\rbrace $ .",
"There is a constant $\\gamma $ such that $ \\mathbb {E}_{S}\\left[ \\Vert \\nabla F^{S}(w) \\Vert \\right]^2 \\le \\gamma ^2,$ for all $w \\in \\mathbb {R}^d$ and all batches $S\\subset \\lbrace 1,2,\\ldots ,n\\rbrace $ .",
"The samples $S$ are drawn independently and $\\nabla F^{S}(w)$ is an unbiased estimator of the true gradient $\\nabla F(w)$ for all $w \\in \\mathbb {R}^d$ , i.e., $ \\mathbb {E}\\Big [ \\nabla F^{S}(w)\\Big ] = \\nabla F(w).$ Note that Assumption $A.2$ implies that the entire Hessian $\\nabla ^2F(w)$ also satisfies $ \\lambda I \\preceq \\nabla ^2F(w) \\preceq \\Lambda I, \\forall w \\in \\mathbb {R}^d,$ for some constants $ \\lambda , \\Lambda >0$ ."
],
[
"Assumptions B",
" $F$ is twice continuously differentiable.",
"The gradients of $F$ are $\\Lambda $ -Lipschitz continuous and the gradients of $F^{O}$ are $\\Lambda _{O}$ -Lipschitz continuous for all $w \\in \\mathbb {R}^d$ and all sets $O \\subset \\lbrace 1,2,\\ldots ,n\\rbrace $ .",
"The function $ F(w)$ is bounded below by a scalar $\\widehat{F}$ .",
"There exist constants $\\gamma \\ge 0$ and $\\eta >0$ such that $ \\mathbb {E}_{S}\\left[ \\Vert \\nabla F^{S}(w) \\Vert \\right]^2 \\le \\gamma ^2 + \\eta \\Vert \\nabla F(w)\\Vert ^2,$ for all $w \\in \\mathbb {R}^d$ and all batches $S\\subset \\lbrace 1,2,\\ldots ,n\\rbrace $ .",
"The samples $S$ are drawn independently and $\\nabla F^{S}(w)$ is an unbiased estimator of the true gradient $\\nabla F(w)$ for all $w \\in \\mathbb {R}^d$ , i.e., $ \\mathbb {E}\\Big [ \\nabla F^{S}(w)\\Big ] = \\nabla F(w).$"
],
[
"Proof of Lemma 3.1 (Strongly Convex Case)",
"The following Lemma shows that the eigenvalues of the matrices generated by the multi-batch L-BFGS method are bounded above and away from zero if $F$ is strongly convex.",
"Lemma 3.1 If Assumptions A.1-A.2 above hold, there exist constants $0<\\mu _1\\le \\mu _2$ such that the Hessian approximations $\\lbrace H_k\\rbrace $ generated by the multi-batch L-BFGS method (Algorithm 1) satisfy $ \\mu _1 I \\preceq H_k \\preceq \\mu _2 I,\\qquad \\text{for } k=0,1,2,\\dots $ Instead of analyzing the inverse Hessian approximation $H_k$ , we study the direct Hessian approximation $B_k = H_k^{-1}$ .",
"In this case, the limited memory quasi-Newton updating formula is given as follows Set $B_k^{(0)}=\\frac{y_k^Ty_k}{s_k^Ty_k}I$ and $\\tilde{m} = \\min \\lbrace k,m\\rbrace $ ; where $m$ is the memory in L-BFGS.",
"For $i=0,...,\\tilde{m}-1$ set $j=k-\\tilde{m}+1+i$ and compute $ B_k^{(i+1)}=B_k^{(i)}-\\frac{B_k^{(i)}s_js_j^TB_k^{(i)}}{s_j^TB_k^{(i)}s_j} + \\frac{y_jy_j^T}{y_j^Ts_j}.$ Set $B_{k+1} = B_k^{(\\tilde{m})}.$ The curvature pairs $s_k$ and $y_k$ are updated via the following formulae $ y_{k+1}=g_{k+1}^{O_{k}}-g_{k}^{O_{k}}, \\qquad s_k = w_{k+1}-w_k.$ A consequence of Assumption $A.2$ is that the eigenvalues of any sub-sampled Hessian ($\\left| O \\right|$ samples) are bounded above and away from zero.",
"Utilizing this fact, the convexity of component functions and the definitions (REF ), we have $ y_k^Ts_k \\ge \\frac{1}{\\hat{\\Lambda }}\\Vert y_k\\Vert ^2 \\quad & \\Rightarrow \\quad \\frac{\\Vert y_k\\Vert ^2 }{y_k^Ts_k} \\le \\hat{\\Lambda }.$ On the other hand, strong convexity of the sub-sampled functions, the consequence of Assumption $A.2$ and definitions (REF ), provide a lower bound, $ y_k^Ts_k \\le \\frac{1}{\\hat{\\lambda }}\\Vert y_k\\Vert ^2 \\quad & \\Rightarrow \\quad \\frac{\\Vert y_k\\Vert ^2 }{y_k^Ts_k} \\ge \\hat{\\lambda }.$ Combining the upper and lower bounds (REF ) and (REF ) $ \\hat{\\lambda } \\le \\frac{\\Vert y_k\\Vert ^2}{y_k^Ts_k} \\le \\hat{\\Lambda }.$ The above proves that the eigenvalues of the matrices $B_k^{(0)}=\\frac{y_k^Ty_k}{s_k^Ty_k}I$ at the start of the L-BFGS update cycles are bounded above and away from zero, for all $k$ .",
"We now use a Trace-Determinant argument to show that the eigenvalues of $B_k$ are bounded above and away from zero.",
"Let $Tr(B)$ and $\\det (B)$ denote the trace and determinant of matrix $B$ , respectively, and set $j_i = k-\\tilde{m}+i$ .",
"The trace of the matrix $B_{k+1}$ can be expressed as, $ Tr(B_{k+1}) &= Tr(B_k^{(0)}) - Tr\\sum _{i=1}^{\\tilde{m}}\\big (\\frac{B_k^{(i)}s_{j_i}s_{j_i}^TB_k^{(i)}}{s_{j_i}^TB_k^{(i)}s_{j_i}}\\big ) + Tr\\sum _{i=1}^{\\tilde{m}} \\frac{y_{j_i}y_{j_i}^T}{y_{j_i}^Ts_{j_i}}\\nonumber \\\\&\\le Tr(B_k^{(0)}) + \\sum _{i=1}^{\\tilde{m}} \\frac{\\Vert y_{j_i}\\Vert ^2}{y_{j_i}^Ts_{j_i}}\\nonumber \\\\&\\le Tr(B_k^{(0)}) + \\tilde{m}\\hat{\\Lambda } \\nonumber \\\\& \\le C_1,$ for some positive constant $C_1$ , where the inequalities above are due to (REF ), and the fact that the eigenvalues of the initial L-BFGS matrix $B_k^{(0)}$ are bounded above and away from zero.",
"Using a result due to Powell [21], the determinant of the matrix $B_{k+1}$ generated by the multi-batch L-BFGS method can be expressed as, $ \\det (B_{k+1}) &= \\det (B_{k}^{(0)}) \\prod _{i=1}^{\\tilde{m}} \\frac{y_{j_i}^Ts_{j_i}}{s_{j_i}^TB_{k}^{(i-1)}s_{j_i}} \\nonumber \\\\& = \\det (B_{k}^{(0)}) \\prod _{i=1}^{\\tilde{m}} \\frac{y_{j_i}^Ts_{j_i}}{s_{j_i}^Ts_{j_i}} \\frac{s_{j_i}^Ts_{j_i}}{s_{j_i}^TB_{k}^{(i-1)}s_{j_i}}\\nonumber \\\\& \\ge \\det (B_{k}^{(0)}) \\Big ( \\frac{\\hat{\\lambda }}{C_1} \\Big )^{\\tilde{m}} \\nonumber \\\\& \\ge C_2,$ for some positive constant $C_2$ , where the above inequalities are due to the fact that the largest eigenvalue of $B_{k}^{(i)}$ is less than $C_1$ and Assumption $A.2$ .",
"The trace (REF ) and determinant (REF ) inequalities derived above imply that largest eigenvalues of all matrices $B_k$ are bounded above, uniformly, and that the smallest eigenvalues of all matrices $B_k$ are bounded away from zero, uniformly."
],
[
"Proof of Theorem 3.2 (Strongly Convex Case)",
"Utilizing the result from Lemma REF , we now prove a linear convergence result to a neighborhood of the optimal solution, for the case where Assumptions $A$ hold.",
"Theorem 3.2 Suppose that Assumptions A.1-A.4 above hold, and let $F^{\\star } = F(w^{\\star })$ , where $w^{\\star }$ is the minimizer of $F$ .",
"Let $\\lbrace w_k\\rbrace $ be the iterates generated by the multi-batch L-BFGS method (Algorithm 1) with $\\alpha _k = \\alpha \\in (0,\\frac{1}{2\\mu _1 \\lambda }),$ starting from $w_0$ .",
"Then for all $k\\ge 0$ , $ \\mathbb {E} [ F(w_k) - F^{\\star } ] &\\le ( 1-2\\alpha \\mu _1 \\lambda )^k [ F(w_0) - F^{\\star } ] + [ 1-(1-\\alpha \\mu _1 \\lambda )^k ]\\frac{\\alpha \\mu _2^2 \\gamma ^2 \\Lambda }{4 \\mu _1 \\lambda } \\\\&\\xrightarrow{} \\frac{\\alpha \\mu _2^2 \\gamma ^2 \\Lambda }{4 \\mu _1 \\lambda }.$ We have that $ F(w_{k+1}) & = F(w_k -\\alpha H_k \\nabla F^{S_k}(w_k)) \\nonumber \\\\& \\le F(w_k) + \\nabla F(w_k)^T (-\\alpha H_k \\nabla F^{S_k}(w_k)) + \\frac{\\Lambda }{2}\\Vert \\alpha H_k \\nabla F^{S_k}(w_k)\\Vert ^2 \\nonumber \\\\& \\le F(w_k) - \\alpha \\nabla F(w_k)^T H_k \\nabla F^{S_k}(w_k) + \\frac{\\alpha ^2 \\mu _2^2 \\Lambda }{2} \\Vert \\nabla F^{S_k}(w_k)\\Vert ^2,$ where the first inequality arises due to (REF ), and the second inequality arises as a consequence of Lemma REF .",
"Taking the expectation (over $S_k$ ) of equation (REF ) $ \\mathbb {E}_{S_k}[ F(w_{k+1})] & \\le F(w_k) - \\alpha \\nabla F(w_k)^T H_k \\nabla F(w_k) + \\frac{\\alpha ^2 \\mu _2^2 \\Lambda }{2} \\mathbb {E}_{S_k} \\Big [ \\Vert \\nabla F^{S_k}(w_k)\\Vert \\Big ]^2 \\nonumber \\\\& \\le F(w_k) - \\alpha \\mu _1 \\Vert \\nabla F(w_k) \\Vert ^2 + \\frac{\\alpha ^2 \\mu _2^2 \\gamma ^2 \\Lambda }{2},$ where in the first inequality we make use of Assumption $A.5$ , and the second inequality arises due to Lemma REF and Assumption $A.4$ .",
"Since $F$ is $\\lambda $ -strongly convex, we can use the following relationship between the norm of the gradient squared, and the distance of the $k$ -th iterate from the optimal solution.",
"$ 2\\lambda [F(w_k) - F^{\\star }] \\le \\Vert \\nabla F(w_k)\\Vert ^2.$ Using the above with (REF ), $ \\mathbb {E}_{S_k} [F(w_{k+1})] & \\le F(w_k) - \\alpha \\mu _1 \\Vert \\nabla F(w_k) \\Vert ^2 + \\frac{\\alpha ^2 \\mu _2^2 \\gamma ^2 \\Lambda }{2} \\nonumber \\\\& \\le F(w_k) - 2\\alpha \\mu _1 \\lambda [F(w_k) - F^{\\star }] + \\frac{\\alpha ^2 \\mu _2^2 \\gamma ^2 \\Lambda }{2}.$ Let $ \\phi _k = \\mathbb {E}[F(w_k) - F^{\\star }],$ where the expectation is over all batches $S_0,S_1,...,S_{k-1}$ and all history starting with $w_0$ .",
"Thus (REF ) can be expressed as, $ \\phi _{k+1} \\le (1 - 2\\alpha \\mu _1 \\lambda ) \\phi _k + \\frac{\\alpha ^2 \\mu _2^2 \\gamma ^2 \\Lambda }{2},$ from which we deduce that in order to reduce the value with respect to the previous function value, the step length needs to be in the range $\\alpha \\in \\Big (0, \\frac{1}{2\\mu _1 \\lambda }\\Big ).$ Subtracting $\\frac{\\alpha \\mu _2^2\\gamma ^2\\Lambda }{4\\mu _1 \\lambda }$ from either side of (REF ) yields $ \\phi _{k+1} - \\frac{\\alpha \\mu _2^2\\gamma ^2\\Lambda }{4\\mu _1 \\lambda } & \\le (1 - 2\\alpha \\mu _1 \\lambda ) \\phi _k + \\frac{\\alpha ^2 \\mu _2^2 \\gamma ^2 \\Lambda }{2} - \\frac{\\alpha \\mu _2^2\\gamma ^2\\Lambda }{4\\mu _1 \\lambda } \\nonumber \\\\& = (1 - 2\\alpha \\mu _1 \\lambda ) \\Big [ \\phi _k - \\frac{\\alpha \\mu _2^2\\gamma ^2\\Lambda }{4\\mu _1 \\lambda } \\Big ].$ Recursive application of (REF ) yields $ \\phi _{k} - \\frac{\\alpha \\mu _2^2\\gamma ^2\\Lambda }{4\\mu _1 \\lambda }& \\le (1 - 2\\alpha \\mu _1 \\lambda )^k \\Big [ \\phi _0 - \\frac{\\alpha \\mu _2^2\\gamma ^2\\Lambda }{4\\mu _1 \\lambda } \\Big ],$ and thus, $ \\phi _{k} \\le (1 - 2\\alpha \\mu _1 \\lambda )^k \\phi _0 + \\Big [ 1-(1-\\alpha \\mu _1 \\lambda )^k\\Big ]\\frac{\\alpha \\mu _2^2 \\gamma ^2 \\Lambda }{4 \\mu _1 \\lambda }.$ Finally using the definition of $\\phi _k$ (REF ) with the above expression yields the desired result, $\\mathbb {E}\\Big [ F(w_k) - F^{\\star }\\Big ] \\le \\Big ( 1-2\\alpha \\mu _1 \\lambda \\Big )^k \\Big [ F(w_0) - F^{\\star } \\Big ] + \\Big [ 1-(1-\\alpha \\mu _1 \\lambda )^k\\Big ]\\frac{\\alpha \\mu _2^2 \\gamma ^2 \\Lambda }{4 \\mu _1 \\lambda }.",
"$"
],
[
"Proof of Lemma 3.3 (Nonconvex Case)",
"The following Lemma shows that the eigenvalues of the matrices generated by the multi-batch L-BFGS method are bounded above and away from zero (nonconvex case).",
"Lemma 3.3 Suppose that Assumptions B.1-B.2 hold and let $\\epsilon >0$ be given.",
"Let $\\lbrace H_k \\rbrace $ be the Hessian approximations generated by the multi-batch L-BFGS method (Algorithm REF ), with the modification that the Hessian approximation $H_k$ update is performed only when $ y_k^Ts_k \\ge {\\epsilon } \\Vert s_k \\Vert ^2,$ else $H_{k+1} = H_k$ .",
"Then, there exist constants $0<\\mu _1\\le \\mu _2$ such that $ \\mu _1 I \\preceq H_k \\preceq \\mu _2 I,\\qquad \\text{for } k=0,1,2,\\dots $ Similar to the proof of Lemma REF , we study the direct Hessian approximation $B_k = H_k^{-1}$ .",
"The curvature pairs $s_k$ and $y_k$ are updated via the following formulae $ y_{k+1}=g_{k+1}^{O_{k}}-g_{k}^{O_{k}}, \\qquad s_k = w_{k+1}-w_k.$ The skipping mechanism (REF ) provides both an upper and lower bound on the quantity $\\frac{\\Vert y_k\\Vert ^2 }{y_k^Ts_k}$ , which in turn ensures that the initial L-BFGS Hessian approximation is bounded above and away from zero.",
"The lower bound is attained by repeated application of Cauchy's inequality to condition (REF ).",
"We have from (REF ) that $\\epsilon \\Vert s_k \\Vert ^2 &\\le y_k^Ts_k \\le \\Vert y_k \\Vert \\Vert s_k \\Vert ,$ and therefore $\\Vert s_k \\Vert \\le \\frac{1}{\\epsilon } \\Vert y_k \\Vert .$ It follows that $s_k^Ty_k \\le \\Vert s_k \\Vert \\Vert y_k \\Vert \\le \\frac{1}{\\epsilon } \\Vert y_k \\Vert ^2$ and hence $ \\frac{\\Vert y_k \\Vert ^2}{s_k^Ty_k} \\ge \\epsilon .$ The upper bound is attained by the Lipschitz continuity of sample gradients, $y_k^Ts_k & \\ge \\epsilon \\Vert s_k \\Vert ^2\\\\&\\ge \\epsilon \\frac{ \\Vert y_k \\Vert ^2}{\\Lambda _{O_k}^2},$ Re-arranging the above expression yields the desired upper bound, $ \\frac{\\Vert y_k \\Vert ^2}{s_k^Ty_k} \\le \\frac{\\Lambda _{O_k}^2}{\\epsilon }.$ Combining (REF ) and (REF ), $\\epsilon \\le \\frac{\\Vert y_k\\Vert ^2 }{y_k^Ts_k} \\le \\frac{\\Lambda _{O_k}^2}{\\epsilon }.$ The above proves that the eigenvalues of the matrices $B_k^{(0)}=\\frac{y_k^Ty_k}{s_k^Ty_k}I$ at the start of the L-BFGS update cycles are bounded above and away from zero, for all $k$ .",
"The rest of the proof follows the same trace-determinant argument as in the proof of Lemma REF , the only difference being that the last inequality in REF comes as a result of the cautious update strategy."
],
[
"Proof of Theorem 3.4 (Nonconvex Case)",
"Utilizing the result from Lemma REF , we can now establish the following result about the behavior of the gradient norm for the multi-batch L-BFGS method with a cautious update strategy.",
"Theorem 3.4 Suppose that Assumptions B.1-B.5 above hold.",
"Let $\\lbrace w_k\\rbrace $ be the iterates generated by the multi-batch L-BFGS method (Algorithm REF ) with $\\alpha _k = \\alpha \\in (0,\\frac{\\mu _1}{\\mu _2^2\\eta \\Lambda } ),$ where $w_0$ is the starting point.",
"Also, suppose that if $ y_k^Ts_k < {\\epsilon } \\Vert s_k \\Vert ^2,$ for some ${\\epsilon }>0$ , the inverse L-BFGS Hessian approximation is skipped, $H_{k+1}=H_k$ .",
"Then, for all $k\\ge 0$ , $ \\mathbb {E} \\Big [\\frac{1}{L}\\sum _{k=0}^{L-1} \\Vert \\nabla F(w_k) \\Vert ^2 \\Big ] & \\le \\frac{\\alpha \\mu _2^2 \\gamma ^2 \\Lambda }{ \\mu _1 } + \\frac{2[ F(w_0) - \\widehat{F}]}{\\alpha \\mu _1 L }\\\\& \\xrightarrow{}\\frac{\\alpha \\mu _2^2 \\gamma ^2 \\Lambda }{ \\mu _1 }.$ Starting with (REF ), $\\mathbb {E}_{S_k}[ F(w_{k+1})] &\\le F(w_k) - \\alpha \\mu _1 \\Vert \\nabla F(w_k) \\Vert ^2 + \\frac{\\alpha ^2 \\mu _2^2 \\Lambda }{2} \\mathbb {E}_{S_k} \\Big [ \\Vert \\nabla F^{S_k}(w_k)\\Vert \\Big ]^2 \\nonumber \\\\& \\le F(w_k) - \\alpha \\mu _1 \\Vert \\nabla F(w_k) \\Vert ^2 + \\frac{\\alpha ^2 \\mu _2^2 \\Lambda }{2} (\\gamma ^2 + \\eta \\Vert \\nabla F(w)\\Vert ^2)\\\\& = F(w_k) - \\alpha \\big (\\mu _1 - \\frac{\\alpha \\mu _2^2 \\eta \\Lambda }{2}\\big ) \\Vert \\nabla F(w_k) \\Vert ^2 + \\frac{\\alpha ^2 \\mu _2^2 \\gamma ^2\\Lambda }{2}\\\\& \\le F(w_k) - \\frac{\\alpha \\mu _1}{2} \\Vert \\nabla F(w_k) \\Vert ^2 + \\frac{\\alpha ^2 \\mu _2^2 \\gamma ^2\\Lambda }{2},$ where the second inequality holds due to Assumption $B.4$ , and the fourth inequality is obtained by using the upper bound on the step length.",
"Taking an expectation over all batches $S_0,S_1,...,S_{k-1}$ and all history starting with $w_0$ yields $ \\phi _{k+1}-\\phi _k \\le - \\frac{\\alpha \\mu _1}{2} \\mathbb {E}\\Vert \\nabla F(w_k) \\Vert ^2 + \\frac{\\alpha ^2 \\mu _2^2 \\gamma ^2 \\Lambda }{2},$ where $\\phi _k = \\mathbb {E}[F(w_k)] $ .",
"Summing (REF ) over the first $L-1$ iterations $ \\sum _{k=0}^{L-1} [\\phi _{k+1}-\\phi _k] &\\le - \\frac{\\alpha \\mu _1}{2} \\sum _{k=0}^{L-1} \\mathbb {E}\\Vert \\nabla F(w_k) \\Vert ^2 + \\sum _{k=0}^{L-1} \\frac{\\alpha ^2 \\mu _2^2 \\gamma ^2 \\Lambda }{2} \\nonumber \\\\&= - \\frac{\\alpha \\mu _1}{2} \\mathbb {E} \\Big [\\sum _{k=0}^{L-1} \\Vert \\nabla F(w_k) \\Vert ^2 \\Big ] + \\frac{\\alpha ^2 \\mu _2^2 \\gamma ^2 \\Lambda L}{2}.$ The left-hand-side of the above inequality is a telescoping sum $\\sum _{k=0}^{L-1} [\\phi _{k+1}-\\phi _k] &= \\phi _{L}-\\phi _0 \\nonumber \\\\&= \\mathbb {E}[F(w_{L})] -F(w_0) \\nonumber \\\\& \\ge \\widehat{F} -F(w_0).$ Substituting the above expression into (REF ) and re-arranging terms $\\mathbb {E} \\Big [\\sum _{k=0}^{L-1} \\Vert \\nabla F(w_k) \\Vert ^2 \\Big ] \\le \\frac{\\alpha \\mu _2^2 \\gamma ^2 \\Lambda L}{ \\mu _1 } + \\frac{2[ F(w_0) - \\widehat{F}]}{\\alpha \\mu _1 }.$ Dividing the above equation by $L$ completes the proof."
],
[
"Extended Numerical Experiments - Real Datasets",
"In this Section, we present further numerical results, on the datasets listed in Table REF , in both the multi-batch and fault-tolerant settings.",
"Note, that some of the datasets are too small, and thus, there is no reason to run them on a distributed platform; however, we include them as they are part of the standard benchmarking datasets.",
"Notation.",
"Let $n$ denote the number of training samples in a given dataset, $d$ the dimension of the parameter vector $w$ , and $K$ the number of MPI processes used.",
"The parameter $r$ denotes the fraction of samples in the dataset used to define the gradient, i.e., $r = \\frac{\\left| S\\right|}{n} $ .",
"The parameter $o$ denotes the length of overlap between consecutive samples, and is defined as a fraction of the number of samples in a given batch $S$ , i.e., $o = \\frac{\\left| O\\right|}{\\left| S\\right|}$ .",
"Table: Datasets together with basic statistics.",
"All datasets are available at https://www.csie.ntu.edu.tw/~cjlin/libsvmtools/datasets/binary.html.We focus on logistic regression classification; the objective function is given by $\\min _{w \\in \\mathbb {R}^d} F(w) = \\frac{1}{n}\\sum _{i=1}^{n}\\log (1+e^{-y^i(w^Tx^i)})+ \\frac{\\sigma }{2} \\Vert w\\Vert ^2,$ where $ (x^i, y^i)_{i=1}^n$ denote the training examples and $\\sigma = \\frac{1}{n}$ is the regularization parameter."
],
[
"Multi-batch L-BFGS Implementation",
"For the experiments in this section (Figures REF -REF ), we run four methods: (Robust L-BFGS) robust multi-batch L-BFGS (Algorithm REF ), (L-BFGS) multi-batch L-BFGS without enforcing sample consistency; gradient differences are computed using different samples, i.e., $y_k = g_{k+1}^{S_{k+1}}-g_{k}^{S_{k}}$ , (Gradient Descent) multi-batch gradient descent; obtained by setting $H_k = I$ in Algorithm REF , (SGD) serial SGD; at every iteration one sample is used to compute the gradient.",
"In Figures REF -REF we show the evolution of $\\Vert \\nabla F(w)\\Vert $ for different step lengths $\\alpha $ , and for various batch ($\\left| S\\right| = r\\cdot n $ ) and overlap ($\\left| O\\right| = o \\cdot \\left| S\\right| $ ) sizes.",
"Each Figure (REF -REF ) consists of 10 plots that illustrate the performance of the methods with the following parameters: Top 3 plots: $\\alpha =1$ , $o=20\\%$ and $r=1\\%,5\\%,10\\%$ Middle 3 plots: $\\alpha =0.1$ , $o=20\\%$ and $r=1\\%,5\\%,10\\%$ Bottom 4 plots: $\\alpha =1$ , $r=1\\%$ and $o=5\\%,10\\%,20\\%,30\\%$ As is expected for quasi-Newton methods, robust L-BFGS performs best with a step-size $\\alpha =1$ , for the most part.",
"Figure: ijcnn1 dataset.",
"Comparison of Robust L-BFGS, L-BFGS (multi-batch L-BFGS without enforcing sample consistency), Gradient Descent (multi-batch Gradient method) and SGD.",
"Top part:we used α∈{1,0.1}\\alpha \\in \\lbrace 1, 0.1\\rbrace ,r∈{1%,5%,10%}r\\in \\lbrace 1\\%, 5\\%, 10\\%\\rbrace and o=20%o=20\\%.Bottom part: we used α=1\\alpha =1, r=1%r=1\\% ando∈{5%,10%,20%,30%}o\\in \\lbrace 5\\%, 10\\%, 20\\%, 30\\%\\rbrace .",
"Solid lines show average performance, and dashed lines show worst and best performance, over 10 runs (per algorithm).",
"K=4K=4 MPI processes.Figure: cov dataset.",
"Comparison of Robust L-BFGS, L-BFGS (multi-batch L-BFGS without enforcing sample consistency), Gradient Descent (multi-batch Gradient method) and SGD.",
"Top part:we used α∈{1,0.1}\\alpha \\in \\lbrace 1, 0.1\\rbrace ,r∈{1%,5%,10%}r\\in \\lbrace 1\\%, 5\\%, 10\\%\\rbrace and o=20%o=20\\%.Bottom part: we used α=1\\alpha =1, r=1%r=1\\% ando∈{5%,10%,20%,30%}o\\in \\lbrace 5\\%, 10\\%, 20\\%, 30\\%\\rbrace .",
"Solid lines show average performance, and dashed lines show worst and best performance, over 10 runs (per algorithm).",
"K=4K=4 MPI processes.Figure: news20 dataset.",
"Comparison of Robust L-BFGS, L-BFGS (multi-batch L-BFGS without enforcing sample consistency), Gradient Descent (multi-batch Gradient method) and SGD.",
"Top part:we used α∈{1,0.1}\\alpha \\in \\lbrace 1, 0.1\\rbrace ,r∈{1%,5%,10%}r\\in \\lbrace 1\\%, 5\\%, 10\\%\\rbrace and o=20%o=20\\%.Bottom part: we used α=1\\alpha =1, r=1%r=1\\% ando∈{5%,10%,20%,30%}o\\in \\lbrace 5\\%, 10\\%, 20\\%, 30\\%\\rbrace .",
"Solid lines show average performance, and dashed lines show worst and best performance, over 10 runs (per algorithm).",
"K=4K=4 MPI processes.Figure: rcvtest dataset.",
"Comparison of Robust L-BFGS, L-BFGS (multi-batch L-BFGS without enforcing sample consistency), Gradient Descent (multi-batch Gradient method) and SGD.",
"Top part:we used α∈{1,0.1}\\alpha \\in \\lbrace 1, 0.1\\rbrace ,r∈{1%,5%,10%}r\\in \\lbrace 1\\%, 5\\%, 10\\%\\rbrace and o=20%o=20\\%.Bottom part: we used α=1\\alpha =1, r=1%r=1\\% ando∈{5%,10%,20%,30%}o\\in \\lbrace 5\\%, 10\\%, 20\\%, 30\\%\\rbrace .",
"Solid lines show average performance, and dashed lines show worst and best performance, over 10 runs (per algorithm).",
"K=16K=16 MPI processes.Figure: url dataset.",
"Comparison of Robust L-BFGS, L-BFGS (multi-batch L-BFGS without enforcing sample consistency), Gradient Descent (multi-batch Gradient method) and SGD.",
"Top part:we used α∈{1,0.1}\\alpha \\in \\lbrace 1, 0.1\\rbrace ,r∈{1%,5%,10%}r\\in \\lbrace 1\\%, 5\\%, 10\\%\\rbrace and o=20%o=20\\%.Bottom part: we used α=1\\alpha =1, r=1%r=1\\% ando∈{5%,10%,20%,30%}o\\in \\lbrace 5\\%, 10\\%, 20\\%, 30\\%\\rbrace .",
"Solid lines show average performance, and dashed lines show worst and best performance, over 10 runs (per algorithm).",
"K=16K=16 MPI processes.Figure: kdda dataset.",
"Comparison of Robust L-BFGS, L-BFGS (multi-batch L-BFGS without enforcing sample consistency), Gradient Descent (multi-batch Gradient method) and SGD.",
"Top part:we used α∈{1,0.1}\\alpha \\in \\lbrace 1, 0.1\\rbrace ,r∈{1%,5%,10%}r\\in \\lbrace 1\\%, 5\\%, 10\\%\\rbrace and o=20%o=20\\%.Bottom part: we used α=1\\alpha =1, r=1%r=1\\% ando∈{5%,10%,20%,30%}o\\in \\lbrace 5\\%, 10\\%, 20\\%, 30\\%\\rbrace .",
"Solid lines show average performance, and dashed lines show worst and best performance, over 10 runs (per algorithm).",
"K=16K=16 MPI processes.Figure: kddb dataset.",
"Comparison of Robust L-BFGS, L-BFGS (multi-batch L-BFGS without enforcing sample consistency), Gradient Descent (multi-batch Gradient method) and SGD.",
"Top part:we used α∈{1,0.1}\\alpha \\in \\lbrace 1, 0.1\\rbrace ,r∈{1%,5%,10%}r\\in \\lbrace 1\\%, 5\\%, 10\\%\\rbrace and o=20%o=20\\%.Bottom part: we used α=1\\alpha =1, r=1%r=1\\% ando∈{5%,10%,20%,30%}o\\in \\lbrace 5\\%, 10\\%, 20\\%, 30\\%\\rbrace .",
"Solid lines show average performance, and dashed lines show worst and best performance, over 10 runs (per algorithm).",
"K=16K=16 MPI processes.Figure: webspam dataset.",
"Comparison of Robust L-BFGS, L-BFGS (multi-batch L-BFGS without enforcing sample consistency), Gradient Descent (multi-batch Gradient method) and SGD.",
"Top part:we used α∈{1,0.1}\\alpha \\in \\lbrace 1, 0.1\\rbrace ,r∈{1%,5%,10%}r\\in \\lbrace 1\\%, 5\\%, 10\\%\\rbrace and o=20%o=20\\%.Bottom part: we used α=1\\alpha =1, r=1%r=1\\% ando∈{5%,10%,20%,30%}o\\in \\lbrace 5\\%, 10\\%, 20\\%, 30\\%\\rbrace .",
"Solid lines show average performance, and dashed lines show worst and best performance, over 10 runs (per algorithm).",
"K=16K=16 MPI processes.Figure: splice-cite dataset.",
"Comparison of Robust L-BFGS, L-BFGS (multi-batch L-BFGS without enforcing sample consistency), Gradient Descent (multi-batch Gradient method) and SGD.",
"Top part:we used α∈{1,0.1}\\alpha \\in \\lbrace 1, 0.1\\rbrace ,r∈{1%,5%,10%}r\\in \\lbrace 1\\%, 5\\%, 10\\%\\rbrace and o=20%o=20\\%.Bottom part: we used α=1\\alpha =1, r=1%r=1\\% ando∈{5%,10%,20%,30%}o\\in \\lbrace 5\\%, 10\\%, 20\\%, 30\\%\\rbrace .",
"Solid lines show average performance, and dashed lines show worst and best performance, over 10 runs (per algorithm).",
"K=16K=16 MPI processes.",
"(No Serial SGD experiments due to memory limitations of our cluster.)"
],
[
"Fault-tolerant L-BFGS Implementation",
"If we run a distributed algorithm, for example on a shared computer cluster, then we may experience delays.",
"Such delays can be caused by other processes running on the same compute node, node failures and for other reasons.",
"As a result, given a computational (time) budget, these delays may cause nodes to fail to return a value.",
"To illustrate this behavior, and to motivate the robust fault-tolerant L-BFGS method, we run a simple benchmark MPI code on two different environments: Amazon EC2 – Amazon EC2 is a cloud system provided by Amazon.",
"It is expected that if load balancing is done properly, the execution time will have small noise; however, the network and communication can still be an issue.",
"(4 MPI processes) Shared Cluster – In our shared cluster, multiple jobs run on each node, with some jobs being more demanding than others.",
"Even though each node has 16 cores, the amount of resources each job can utilize changes over time.",
"In terms of communication, we have a GigaBit network.",
"(11 MPI processes, running on 11 nodes) We run a simple code on the cloud/cluster, with MPI communication.",
"We generate two matrices $A,B \\in R^{n \\times n}$ , then synchronize all MPI processes and compute $C=A\\cdot B$ using the GSL C-BLAS library.",
"The time is measured and recorded as computational time.",
"After the matrix product is computed, the result is sent to the master/root node using asynchronous communication, and the time required is recorded.",
"The process is repeated 3000 times.",
"Figure: Distribution of Computation and Communication Time for Amazon EC2 and Shared Cluster.Figures show worst and best time, average time and 10% and 90% quantiles.Amazon Cloud EC: In the experiment: 4 MPI processes; Shared Cluster: 11 MPI processes.The results of the experiment described above are captured in Figure REF .",
"As expected, on the Amazon EC2 cloud, the matrix-matrix multiplication takes roughly the same time for all replications and the noise in communication is relatively small.",
"In this example the cost of communication is negligible when compared to the cost of computation.",
"On our shared cluster, one cannot guarantee that all resources are exclusively used for a specific process, and thus, the computation and communication time is considerably more stochastic and unbalanced.",
"For some cases the difference between the minimum and maximum computation (communication) time varies by an order of magnitude or more.",
"Hence, on such a platform a fault-tolerant algorithm that only uses information from nodes that return an update within a preallocated budget is a natural choice.",
"In Figures REF -REF we show a comparison of the proposed robust multi-batch L-BFGS method and the multi-batch L-BFGS method that does not enforce sample consistency (L-BFGS).",
"In these experiments, $p$ denotes the probability that a single node (MPI process) will not return a gradient evaluated on local data within a given time budget.",
"We illustrate the performance of the methods for $\\alpha =0.1$ and $p\\in \\lbrace 0.1, 0.2, 0.3, 0.4, 0.5\\rbrace $ .",
"We observe that the robust implementation is not affected much by the failure probability $p$ .",
"Figure: rcvtest dataset.",
"Comparison of Robust L-BFGS and L-BFGS in the presence of faults.We used α=0.1\\alpha =0.1 and p∈{0.1,0.2,0.3,0.4,0.5}p\\in \\lbrace 0.1, 0.2, 0.3, 0.4, 0.5\\rbrace .",
"Solid lines show average performance, and dashed lines show worst and best performance, over 10 runs (per algorithm).",
"K=16K=16 MPI processes.Figure: webspam dataset.",
"Comparison of Robust L-BFGS and L-BFGS in the presence of faults.We used α=0.1\\alpha =0.1 and p∈{0.1,0.2,0.3,0.4,0.5}p\\in \\lbrace 0.1, 0.2, 0.3, 0.4, 0.5\\rbrace .",
"Solid lines show average performance, and dashed lines show worst and best performance, over 10 runs (per algorithm).",
"K=16K=16 MPI processes.Figure: kdda dataset.",
"Comparison of Robust L-BFGS and L-BFGS in the presence of faults.We used α=0.1\\alpha =0.1 and p∈{0.1,0.2,0.3,0.4,0.5}p\\in \\lbrace 0.1, 0.2, 0.3, 0.4, 0.5\\rbrace .",
"Solid lines show average performance, and dashed lines show worst and best performance, over 10 runs (per algorithm).",
"K=16K=16 MPI processes.Figure: kddb dataset.",
"Comparison of Robust L-BFGS and L-BFGS in the presence of faults.We used α=0.1\\alpha =0.1 and p∈{0.1,0.2,0.3,0.4,0.5}p\\in \\lbrace 0.1, 0.2, 0.3, 0.4, 0.5\\rbrace .",
"Solid lines show average performance, and dashed lines show worst and best performance, over 10 runs (per algorithm).",
"K=16K=16 MPI processes.Figure: url dataset.",
"Comparison of Robust L-BFGS and L-BFGS in the presence of faults.We used α=0.1\\alpha =0.1 and p∈{0.1,0.2,0.3,0.4,0.5}p\\in \\lbrace 0.1, 0.2, 0.3, 0.4, 0.5\\rbrace .",
"Solid lines show average performance, and dashed lines show worst and best performance, over 10 runs (per algorithm).",
"K=16K=16 MPI processes."
],
[
"Scaling of Robust Multi-Batch L-BFGS Implementation",
"In this Section, we study the strong and weak scaling properties of the robust multi-batch L-BFGS method on an artificial dataset.",
"For various values of $r$ and $K$ , we measure the time needed to compute a gradient (Gradient) and the time needed to compute and communicate the gradient (Gradient+C), as well as, the time needed to compute the L-BFGS direction (L-BFGS) and the associated communication overhead (L-BFGS+C)."
],
[
"Strong Scaling",
"Figure REF depicts the strong scaling properties of our proposed algorithm.",
"We generate a dataset with $n=10^7$ samples and $d=10^4$ dimensions, where each sample has 160 randomly chosen non-zero elements (dataset size 24GB).",
"We run our code for different values of $r$ (different batch sizes $S_k$ ), with $K= 1, 2, \\dots , 128$ number of MPI processes.",
"One can observe that the compute time for the gradient and the L-BFGS direction decreases as $K$ is increased.",
"However, when communication time is considered, the combined cost increases slightly as $K$ is increased.",
"Notice that for large $K$ , even when $r=10\\%$ (i.e., $10\\%$ of all samples processed in one iteration, $\\sim $ 18MB of data), the amount of local work is not sufficient to overcome the communication cost.",
"Figure: Strong scaling of robust multi-batch L-BFGS on a problem with artificial data;n=10 7 n=10^7 and d=10 4 d=10^4.",
"Each sample has 160 non-zero elements.",
"+C+C indicates that we include communication time to the gradient computation and L-BFGS update computation."
],
[
"Weak Scaling - Fixed Problem Dimension, Increasing Data Size",
"In order to illustrate the weak scaling properties of the algorithm, we generate a data-matrix $X \\in R^{10^7 \\times 10^4}$ , and run it on a shared cluster with $K=1,2,4,8,\\dots ,128$ MPI processes.",
"For a given number of MPI processes ($K$ ), each sample contains $10\\cdot K$ non-zero elements.",
"Effectively, the dimension of the problem is fixed, but sparsity of the data is decreased as more MPI processes are used.",
"The size of the input data is 1.5 $\\cdot K$ GB (i.e., 1.5GB per MPI process).",
"The compute time for the gradient is almost constant, this is because the amount of work per MPI process (rank) is almost identical; see Figure REF .",
"On the other hand, because we are using a Vector-Free L-BFGS implementation [9] for computing the L-BFGS direction, the amount of time needed for each node to compute the L-BFGS direction is decreasing as $K$ is increased.",
"However, increasing $K$ does lead to larger communication overhead, which can be observed in Figure REF .",
"For $K=128$ (192GB of data) and $r=10\\%$ , almost 20GB of data are processed per iteration in less than 0.1 seconds, which implies that one epoch would take around 1 second.",
"Figure: Weak scaling of robust multi-batch L-BFGS on a problem with artificial data;n=10 7 n=10^7 and d=10 4 d=10^4.",
"Each sample has 10·K10\\cdot K non-zero elements.",
"+C+C indicates that we also include communication time to the gradient computation and L-BFGS update computation."
],
[
"Increasing Problem Dimension, Fixed Data Size and $K$",
"In this experiment, we investigate the effect of a change in the dimension $d$ of the problem on the performance of the algorithm.",
"We fix the size of data ($29GB$ ) and the number of MPI processes ($K=8$ ).",
"We generate data with $n=10^7$ samples, where each sample has 200 non-zero elements.",
"Figure REF shows that increasing the dimension $d$ has a mild effect on the computation time of the gradient, while the effect on the time needed to compute the L-BFGS direction is more apparent.",
"However, if communication time is taken into consideration, the time required for the gradient computation and the L-BFGS direction computation increase as $d$ is increased.",
"Figure: Scaling of robust multi-batch L-BFGS on a problem with artificial data; n=10 7 n=10^7 samples, with increasing dd and K=8K=8 MPI processes.",
"Each sample had 200 non-zero elements.",
"+C+C indicates that we also include communication time to the gradient computation and L-BFGS update computation."
]
] | 1605.06049 |
[
[
"The fate of non-polynomial interactions in scalar field theory"
],
[
"Abstract We present an exact RG (renormalization group) analysis of $O(N)$-invariant scalar field theory about the Gaussian fixed point.",
"We prove a series of statements that taken together show that the non-polynomial eigen-perturbations found in the LPA (local potential approximation) at the linearised level, do not lead to new interactions, \\textit{i.e.}",
"enlarge the universality class, neither in the LPA or treated exactly.",
"Non-perturbatively, their RG flow does not emanate from the fixed point.",
"For the equivalent Wilsonian effective action they can be re-expressed in terms of the usual couplings to polynomial interactions, which can furthermore be tuned to be as small as desired for all finite RG time.",
"For the infrared cutoff Legendre effective action, this can also be done for the infrared evolution.",
"We explain why this is nevertheless consistent with the fact that the large field behaviour is fixed by these perturbations."
],
[
"Introduction",
"The Wilsonian RG (renormalization group), in particular when adapted to the continuum (where it was christened the “exact RG” by Wilson [1]), is an important framework for understanding quantum field theory outside the realm where perturbation theory is a secure guide (for introductions and reviews see refs.",
"[2], [3], [4], [5], [6], [7], [8], [9]).",
"Without the security of perturbation theory, it is clearly important if possible to make rigorous conclusions about such non-perturbative RG properties.",
"A persuasive example is provided by the investigations into asymptotic safety in quantum gravity [10].",
"Following ref.",
"[11], this has been a major area where these ideas have been applied (for reviews and introductions see [12], [13], [14], [15], [16]).",
"However this is also an area where there is little guidance from current experimental observation or other techniques, and therefore one must place particular reliance on a rigorous understanding of the mathematical structure that the exact RG exposes, in so far as this is possible.",
"This is especially so with recent work on “functional truncations” [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36].",
"In particular an analysis of what is legitimate for relevant eigen-perturbations in functional truncations [23], [24], [37], [36], thus determining what is the set of renormalised couplings in the continuum theory (see e.g.",
"[1], [2]), is clearly crucial.",
"But this has to begin by building on a thorough understanding of what is legitimate in a well-understood situation such as that of scalar field theory.",
"Moreover, recent work has shown that, despite the complications induced by diffeomorphism invariance and background independence [18], [20], [37], the overall scalar factor part of the metric (at least within the so-called conformal truncation) has RG properties that are just those of a scalar field theory – but crucially with wrong sign kinetic term [28], [36].",
"Furthermore this one sign profoundly alters the RG properties (unless the field is continued to the imaginary axis) [36].",
"Clearly then it is important to begin by being sure of the facts for standard scalar field theory (i.e.",
"with right sign kinetic term).",
"Indeed the RG properties of (standard) scalar field theory could be regarded as long settled, and in particular the classification of relevant perturbations in functional truncations [38], [39], [2], see also [40], [41], [42].",
"However, even for perturbations around the Gaussian fixed point, a debate has continued [38], [39], [43], [44], [45], [46], [47], [48], [49], [50], [51], [52], [53], [54], [55], [56], [57] over the significance of certain non-polynomial scaling solutions to the linearised LPA (Local Potential Approximation) flow equations (i.e.",
"eigen-perturbations) starting with Halpern and Huang's observations [58], [59], [60].",
"Although these ideas were criticised early on [38], [39], [43], researchers have continued to pursue these ideas [44], [45], [46], [47], [48], [49], [50], [51], [52], [53], [54], [55], [56], [57], no doubt inspired in large part by the hope that scalar fields might in certain circumstances enjoy asymptotically free interactions.",
"This is of course of particular relevance in four dimensions for the Higgs sector of the Standard Model, but if true would have profound consequences also for constructing theories beyond the Standard Model and indeed for continuous phase transitions in condensed matter and statistical physics.",
"In this paper we re-analyse the RG properties of standard scalar field theory about the Gaussian fixed point, in particular concentrating on such non-polynomial solutions for the linearised perturbations.",
"We prove a series of statements that taken together show that these solutions cannot be regarded as providing extra relevant directions.",
"In this way we aim to consolidate as rigorously as possible what is known about scalar field theory around the Gaussian fixed point, and to settle conclusively the debate over Halpern-Huang directions.",
"Although these non-polynomial solutions were found in the LPA, and we will proceed by demonstrating that they fail to provide new relevant directions within this approximation scheme, these results extend to the derivative expansion and also to the exact equations, as we outline at the end of the paper.",
"To be concrete we concentrate on the case of most interest for Higgs physics, namely $O(N)$ invariant $N$ -component scalar field theory around the Gaussian fixed point in four Euclidean space-time dimensions (where for the Higgs field, $N=4$ and the usual Wick rotation has been performed).",
"All our arguments can however be straightforwardly adapted to other dimensions $d>2$ and to other fixed points.In $d=2$ dimensions the situation is very different since the engineering dimension of the scalar field vanishes, leading to bounded (oscillatory) solutions for the LPA [61].",
"In particular in less than four dimensions, there are non-perturbative Wilson-Fisher fixed points [62], [1].",
"For these fixed points, linearised solutions divide into quantised eigen-perturbations (that is scaling fields with quantised scaling dimension as is true for the polynomial solutions about the Gaussian fixed point) and non-quantised eigen-perturbations (with continuous scaling dimension as also true for non-polynomial solutions about the Gaussian fixed point) [38], [39], [2].",
"Since we concentrate on the Gaussian fixed point, in this paper we will use the terms (non)quantised and (non)polynomial interchangeably.",
"In broad outline, the arguments are as follows.",
"The key observation is that solutions of the linear partial differential equation that result from linearisation of the flow equation around a fixed point, a priori must remain uniformly small.",
"That is, given an $\\epsilon $ multiplying the linearised solution, there must exist a $\\delta $ such that the solution is smallerFor the Gaussian fixed point `smaller' here does just mean “less than\" in magnitude.",
"In general `smaller' is determined by ratios involving the behaviour of the fixed point itself.",
"than this for all values of the field.",
"In other words it is not enough that such a perturbation is small for some given value of the field: linearisation requires that it must be small for all values of the field.",
"Such perturbations that grow sufficiently fast for large field violate this condition, thus ruling out the procedure used for finding them in the first place.",
"For the quantised solutions, the mistakenly-deduced RG evolution can nevertheless be (sufficiently) recovered, while for the non-quantised solutions they cannot.",
"Instead the large field behaviour undergoes mean-field RG evolution, which is to say in fact that in physical (unscaled) variables nothing happens: the large field behaviour remains fixed by the initial perturbation.In fact the same is true for the quantised perturbations but for these the large field behaviour can also be interpreted as the expected RG evolution of the corresponding coupling.",
"Despite the fact that such a linearisation procedure has been adopted since the earliest days [63], [64], [4] (the so-called scaling fields), and envisioned as taking a tangent to the flow in function space, it is not a justified procedure for finding solutions that really flow into or out of the fixed point, without the uniformity condition or further analysis.",
"The non-quantised solutions could therefore logically be viewed as no more special than any other finite perturbation added to the fixed point.",
"Nevertheless given the significance attached to the evolution of scaling fields at finite field, we will concentrate on their RG behaviour.",
"We recall and then further develop detailed analysis of their renormalization group evolution, both in the UV (ultraviolet) and especially we further develop the analysis in the IR (infrared).",
"In refs.",
"[38], [39], [2] we already demonstrated that beyond a certain RG time, the IR evolution of the non-quantised perturbations lives inside the Hilbert spacei.e.",
"Banach space with inner product, not to be confused with the quantum mechanical state space.",
"spanned by the quantised perturbations and therefore can be expressed in terms of the IR evolution of their couplings.",
"We will make further progress by analysing these couplings both for the flow of the (IR cutoff) Legendre effective action [65], [66], [67] and the flow of the Wilsonian effective action [1], [68], and in particular providing bounds on these couplings depending on $\\epsilon $ .",
"In this way we are able to show that, although an infinite number of (irrelevant) quantised couplings are involved, they can be made as small as desired (for all finite RG time $t$ ) by taking $\\epsilon \\rightarrow 0$ .",
"The RG evolution at finite field is thus expected to be the normal one for the universality class determined by quantised relevant couplings.",
"We explain why this is nevertheless consistent with the large field behaviour being still that determined by the initial perturbation.",
"Let us also mention that an elegant approach to studying the behaviour of the non-polynomial eigenperturbations was performed in ref.",
"[48].",
"There the subtleties of the large $N$ limit for such perturbations, were negotiated, allowing large $N$ methods to be used to solve for the full trajectory towards the infrared ($t\\rightarrow \\infty $ ).",
"Apart from the “fine-tuning” and “naturalness” properties found in that case, which are likely to be an artefact of $N=\\infty $ [48], the results are consistent with our conclusion that the RG evolution is determined at finite field by the universality class described by the quantised relevant couplings, and at large field by the original perturbation.",
"Since the $t\\rightarrow \\infty $ limit is also addressed in detail, the study is complementary to ours.",
"On the other hand, finite $N$ , the UV evolution, and the realisation that the IR evolution can be expressed in terms of the quantised couplings, were missing, and indeed do not look straightforward to access from the (somewhat subtle) $N=\\infty $ limit.",
"In the Conclusions, sec.",
", we list the statements that we prove and thus this final section also provides a more detailed overview of the paper."
],
[
"Universality of linearised LPA about the Gaussian fixed point",
"Halpern and Huang noticed that, within the (LPA) Local Potential Approximation [69], [70], [71], [72], [73], [74], [75], [76], [77] linearised perturbations about the Gaussian fixed point not only have solutions that are polynomial in the scalar field $\\phi $ , which can be identified with the operators usually considered in perturbation theory, but also solutions that are non-polynomial in the field with a continuous scaling dimension [58], [60].Refs.",
"[58], [60] also proposed a line of non-perturbative fixed points following from such equations but this matter was conclusively settled in ref.",
"[38] and will thus not be further discussed.",
"It is a particular set of these latter perturbations that Halpern and Huang identified as being asymptotically free.",
"They used the Wegner-Houghton equation [78].",
"But the same equation can also be reached by taking the sharp cutoff limit of any exact renormalisation group equation [67], [75].",
"In fact after appropriate rescalings, the linearised perturbations satisfy the same equation whatever cutoff profile is used providing it actually regulates.",
"This is actually clear from refs.",
"[74], [44] for the Wilson/Polchinski version, and for the Legendre effective (average) action version from ref.",
"[48] after finite rescalings of potential and field.",
"The reason for this can be traced to the fact that at the linearised level, the equations are only summing up tadpole terms, whose value can be rescaled by changes of variables [49] (see also [36]).",
"Indeed in this way, it is clear that the linearised flow equation (REF ) for the potential, that we are about to derive within LPA, may in fact be derived exactly, i.e.",
"without approximation, whatever cutoff profile is used.",
"We reserve further comments on this for the end of the paper.",
"Thus we can easily avoid the limitations of the sharp cutoff that concerned the authors in refs.",
"[58], [59], [60].",
"Also we see that the non-polynomial perturbations with continuous scaling dimension are thus universal in this sense.",
"Here let us demonstrate this universality by example.",
"At the same time we will set up the equations and notation we will need for the later discussion.",
"We specialise to the most interesting case of four space-time dimensions (but as usual work in Wick-rotated Euclidean space).",
"We work with an $O(N)$ invariant effective potential $V(z,t)$ , where $z=\\phi ^a\\phi _a/2$ with $\\phi ^a$ an $N$ -component real scalar field, and $t=\\ln (\\mu /\\Lambda )$ is renormalisation group `time'.",
"$\\Lambda $ is the running effective cutoff scale (often called $k$ in the functional renormalisation group literature) and $\\mu $ is some arbitrary fixed energy scale.",
"We have written all quantities in dimensionless terms using the running cutoff scale, thus in terms of physical quantities $V=V_{phys}/\\Lambda ^4\\qquad {\\rm and}\\qquad z=z_{phys}/\\Lambda ^2\\,.$ (The field's anomalous dimension is neglected in this version of the local potential approximation [69], [1], [72].",
"See ref.",
"[79] for discussions on this.)",
"For sharp cutoff one then has [71] ${\\dot{V}} + 2z V^{\\prime } - 4 V = (N-1) \\ln (1+V^{\\prime }) + \\ln (1+V^{\\prime } + 2 zV^{\\prime \\prime })\\,,$ where prime is differentiation with respect to $z$ and dot is differentiation with respect to $t$ .",
"If instead one uses Litim's optimised cutoff [80], [81], eqn.",
"(REF ), in the flow for the Legendre effective action (effective average action) [65], [66], [67], then one finds [82] ${\\dot{V}} + 2z V^{\\prime } - 4V = -\\frac{N-1}{1+V^{\\prime }} -\\frac{1}{1+V^{\\prime }+2zV^{\\prime \\prime }}+N\\,.$ The last term comes from a constant shift in the vacuum energy $V\\mapsto V+N/4$ , which has no physical consequences but streamlines the arguments later.",
"And if instead we use Polchinski's equation [68], equivalent at this level to Wilson's [1] by a change of variables [67], [83], [42], then for a continuous class of cutoff profiles one finds after appropriate rescalings the same flow equation [74]: ${\\dot{U}} + 2yU^{\\prime } - 4U = -2y\\!\\left(U^{\\prime }\\right)^2 +N U^{\\prime } + 2y U^{\\prime \\prime }\\,.$ where for later purposes we rename the potential and fields as $U(y,t)$ , where $y=\\Phi ^a\\Phi _a/2$ .",
"The Gaussian fixed point corresponds to the solution $V=0$ ($U=0$ ) in all three cases (REF –REF ).",
"Linearising about the fixed point and separating variables, we find for such infinitesimally small perturbations $V(z,t) = \\epsilon \\, w(z) \\, {\\rm e}^{\\lambda t} \\qquad {\\rm and}\\qquad U(y,t) = \\epsilon \\, w(y) \\, {\\rm e}^{\\lambda t}\\,,$ where $\\epsilon $ is considered `strictly infinitesimal', in the sense that we will for the moment accept without question the validity of the linearisation step required to form these eigen-perturbations $w$ or “scaling fields”.",
"The scaling dimension $\\lambda $ of the associated infinitesimal coupling $g=\\epsilon \\, {\\rm e}^{\\lambda t}\\,,$ implies a physical coupling $g_{phys} = \\epsilon \\, \\mu ^\\lambda $ of mass-dimension $\\lambda $ , so $g = g_{phys}/\\Lambda ^\\lambda $ .",
"As advertised, one then finds the same renormalisation group eigenvalue equation in all three cases: $\\lambda w + 2 z w^{\\prime } - 4 w = N w^{\\prime } + 2z w^{\\prime \\prime }$ (up to renaming $z$ to $y$ in the third case).",
"This equation is Kummer's equation and the solutions divide into two sets [84], [85], [86] depending on whether the solutions have quantised eigenvalues $\\lambda $ or not."
],
[
"Perturbations and the continuum limit",
"Before we develop the consequences of these solutions let us recall that, to get a bona fide field theory we need to take the continuum limit which means that we need to find a solution, called the Renormalised Trajectory, that in terms of dimensionless variables, shoots out from the fixed pointin our case the Gaussian fixed point at $t=-\\infty $ (the ultraviolet end corresponding to $\\Lambda =\\infty $ ) along a relevant (or marginally relevant) direction, and survives all the way down to the infrared end at $t=+\\infty $ ($\\Lambda =0$ ), see e.g.",
"[1], [2].",
"The exact renormalisation group may be viewed as performing a partial functional integral down to the infrared cutoff scale $\\Lambda $ (see e.g.",
"[66], [67], [2]).",
"We therefore need sensible IR behaviour at the least in order to be able to complete the integration.",
"On the other hand, the UV behaviour is crucially important for a continuum limit.",
"We require that the perturbation shoots out from the fixed point, or equivalently that the perturbations vanish into the fixed point as $t\\rightarrow -\\infty $ , so that interactions in this sense `switch off' in the high energy limit, allowing the overall ultraviolet regulator to be removed.",
"It is for this reason that the authors of ref.",
"[58], [59], [60] focus on this crucial property.",
"They refer to any interactions that satisfy this as “asymptotically free”.",
"We will however continue to use the established term “relevant” (reserving “asymptotically free” to mean marginally relevant - as is true of the QCD coupling for example, a property which will in fact play no rôle in the present discussions)."
],
[
"Quantised and non-quantised solutions of the linearised flow equation",
"Returning to Kummer's equation (REF ), we recall that the quantised perturbations are polynomials of rank $n$ , the generalised Laguerre polynomials: $\\lambda _{n} = 4-2n\\,,\\qquad w = w_n(z) = (-2)^n L^{\\frac{N}{2}-1}_n(z)\\,,\\qquad n=0,1,2,\\cdots \\,.$ They thus form a complete set of orthogonal functions under the generalised Laguerre weight [84], [85]: $\\int _0^\\infty \\!\\!dz\\, z^{\\frac{N}{2}-1}\\,{\\rm e}^{-z}\\, w_n(z)\\, w_m(z) = 2^{2n}\\frac{\\Gamma (n+N/2)}{\\Gamma (n+1)}\\, \\delta _{n,m}\\,,$ allowing any potential $V(z,t)$ to be expanded in terms of them as $V(z,t) = \\sum _{n=0}^\\infty g_{2n}(t)\\, w_n(z)\\,,$ providing the potential is square-integrable with the generalised Laguerre weight, i.e.",
"providing $\\int _0^\\infty \\!\\!dz\\, z^{\\frac{N}{2}-1}\\,{\\rm e}^{-z}\\, V^2$ converges.",
"The couplings $g_{2n}(t)$ are defined by $g_{2n} = \\frac{\\Gamma (n+1)}{2^{2n}\\Gamma (n+N/2)} \\int _0^\\infty \\!\\!dz\\, z^{\\frac{N}{2}-1}\\,{\\rm e}^{-z}\\, w_n(z)\\, V(z,t)\\,,$ providing these integrals converge.",
"Convergence of the series in (REF ) is in the almost-always sense, i.e.",
"mathematically this forms a Hilbert space structure, and convergence is with respect to the norm defined by the generalised Laguerre weight, i.e.",
"$\\int _0^\\infty \\!\\!dz\\, z^{\\frac{N}{2}-1}\\,{\\rm e}^{-z}\\, \\left(V(z,t) - \\sum _{n=0}^{n_{\\rm max}} g_{2n}(t)\\, w_n(z) \\right)^2 \\rightarrow 0 \\qquad {\\rm as} \\qquad n_{\\rm max}\\rightarrow \\infty \\,.$ With the $(-2)^n$ factor in (REF ), the highest power in $ w_n(z)$ is canonically normalised to $(\\phi ^a\\phi _a)^n/n!\\,.$ The lower powers of $\\phi ^a\\phi _a$ in $ w_n(z)$ are generated by successive tadpole corrections.",
"The $(\\phi ^a\\phi _a)^n/n!$ are of course nothing but the usual perturbative interactions with the expected scaling dimensions $\\lambda _n=4-2n$ for the associated couplings $g_{2n}$ at the linearised level.",
"Thus up to an additive constant, $ w_1(z)$ is the mass term associated to $g_2= m^2/\\Lambda ^2$ .",
"It has scaling dimension $\\lambda _1=2$ , and physical `coupling' $m^2$ .",
"Neglecting vacuum energy ($ w_0=1$ ), this is the sole relevant direction in the $O(N)$ symmetric potential.",
"$ w_2(z)$ starts with the $\\phi ^4$ interaction, and also carries additive constant and mass-term corrections.",
"Its associated coupling, $g_4$ , is marginally irrelevant (from second order perturbation theory it vanishes as $t\\rightarrow +\\infty $ ).",
"All higher powers, i.e.",
"$ w_n(z)$ , $n\\ge 3$ , give couplings $g_{2n}$ that are irrelevant, i.e.",
"at the linearised level they also vanish as $t\\rightarrow +\\infty $ .",
"Since from this linear analysis we see that the only relevant direction is the mass term, we recover the usual picture of triviality for scalar field theory in four dimensions which is that the continuum limit is just free massive scalar field theory.",
"The $\\Lambda \\rightarrow 0$ limit is sometimes known as the high temperature fixed point [73], [2], [4] corresponding (in units of $\\Lambda $ ) to an infinite mass (or zero correlation length).",
"The non-quantised solutions to (REF ), corresponding to any real $\\lambda \\ne 4-2n$ ($n=0,1,2,\\cdots $ ) are given by Kummer M functions[84], [85], [86]:For $\\lambda =4-2n$ the M functions are just the generalised Laguerre polynomial solutions given above.",
"$w = w^\\lambda (z)= s_\\lambda \\, {\\rm M}\\!\\!\\left(\\frac{\\lambda -4}{2},\\frac{N}{2},z\\right)\\,.$ Here $s_\\lambda $ is the sign of $(\\lambda -4)(\\lambda -2)$ .",
"We include it in the normalisation so that $ w^\\lambda (z)$ is positive for large $z$ , thus providing a perturbation to the potential that is bounded below for a positive associated coupling.",
"The M function has a Taylor series expansion around $z=0$ , and is normalised to 1 at $z=0$ .",
"In other words, $ w^\\lambda (0)=s_\\lambda $ .",
"Asympotically [86], $w^\\lambda (z) = C\\, {\\rm e}^z\\, z^{-p} \\left[ 1+ O(1/z) \\right]\\,,$ where $p = \\frac{N}{2} + 2 -\\frac{\\lambda }{2} \\qquad {\\rm and}\\qquad C = \\frac{\\Gamma \\!\\left({N/2}\\right)}{ \\left|\\Gamma \\!\\left( {\\lambda /2-2}\\right)\\right|}\\,.$ Since these non-polynomial solutions solve the linear equation (REF ) for real $\\lambda \\ne 4-2n$ , we already know of course that they cannot be expanded using (REF ).",
"Substituting (REF ) into (REF ), we readily confirm that (for sufficiently large $n$ ) these integrals do not converge.",
"These non-quantised solutions therefore live outside the Hilbert space spanned by the quantised solutions (REF ).",
"However we will see in sec.",
"that once correctly solved for, the IR ($t$ increasing) evolution drives them back immediately into this space.",
"Since for $\\lambda >0$ , the $\\exp (\\lambda t)$ term in (REF ) clearly vanishes as $t\\rightarrow -\\infty $ , Halpern and Huang [58], [59], [60] conclude that these interactions are “asymptotically free” (i.e.",
"relevant) if $\\lambda >0$ .They also note that $ w^\\lambda (z)$ has a minimum away from the origin if $\\lambda <2$ , thus potentially yielding symmetry breaking potentials, of direct applicability for $N=4$ to the Higgs sector of the Standard Model.",
"Obvious as this conclusion seems to be at first sight, it is not true, as one of us has already emphasised, albeit within the tight constraints of Physical Review Letter's comment section [38].",
"Our comments were based on the results already found for non-perturbative fixed points in refs.",
"[87], [61], [72], and were further expanded and developed for any fixed point in ref.",
"[39].",
"We first review these conclusions, specialised to the current context, attempting to make them both clearer and more precise."
],
[
"Ultraviolet properties: (non)existence of a continuum limit",
"Before we do so, let us underline the point that the existence of a continuum of non-quantised eigen-perturbations, is common to all fixed points found using these techniques, not just the Gaussian fixed point considered here.",
"In less than four dimensions, the flow equations (REF –REF ) have non-perturbative Wilson-Fisher fixed points [62], [1] which famously are known to describe the universal properties of a wide variety of statistical physics and condensed matter systems.",
"Perturbing around these fixed points using (REF ), it is obvious that the equations (REF –REF ) yield linear second order ordinary differential equations and therefore it is a mathematical theorem that there are solutions to these equations for any $\\lambda $ , i.e.",
"that there exist a spectrum of solutions with non-quantised values of $\\lambda $ .Being second order, but with a fixed singularity at $z=0$ , there is a one-parameter set of solutions for each $\\lambda $ , and this parameter can therefore be identified with the normalisation which we are free to choose since the equation is linear.",
"Therefore up to normalisation, there is a unique solution for every $\\lambda $ .",
"See e.g.",
"refs.",
"[23], [37], [36] for recent applications of these arguments in very different exact RG contexts.",
"If one wants to maintain that the non-quantised perturbations (REF ) actually do correspond to a continuous infinity of physical couplings $g_\\lambda $ in the particle physics of a scalar field, then one also has to find a convincing rationale to exclude the equivalent perturbations in all the so far experimentally realisable continuous phase transitions described by scalar field theory (including in simulations) since a continuous spectrum of couplings $g_\\lambda $ would evidently destroy the observed universalitynamely the dependence in this case on only one parameter: the overall scale, which can be identified with the correlation length.",
"of these phase transitions.",
"So what is the logical weakness in the arguments above in sec.",
"REF ?",
"The linearised solutions (REF ) were derived by assuming that $V\\ll 1$ ($U\\ll 1$ ).",
"But for any fixed $\\epsilon $ this condition is violated for sufficiently large $z$ , since the solutions (REF ) grow like $z^n$ , and solutions (REF ) grow as $\\exp z$ , as in (REF ).",
"Mathematically the issue is the difference between uniform and point-wise convergence: it is not possible to find an $\\epsilon $ such that the perturbation is uniformly bounded by a $\\delta >0$ of our choice.",
"Indeed we can always find an $\\epsilon $ small enough that linearisation, and thus separation of variables (REF ), is valid over some domain of $z$ and $t$ , but there is no value of $\\epsilon $ no matter how small, that ensures that linearisation of the flow equations and subsequent separation of variables is a valid procedure over the full range $0\\le z<\\infty $ (for any $t$ ).",
"One might be tempted to put aside this problem for the present, for example cutting off the potential at some very large $z=\\phi ^a\\phi _a/2$ (setting $V$ to be small or zero beyond this point), or simply assuming implicitly that $z$ can be assumed somehow sufficiently small, as is effectively done in any case in perturbation theory or more generally for scaling fields.",
"However as noted in refs.",
"[38], [59], [39] the large $z$ dependence is crucial to the arguments [58], [59], [60] in favour of (REF ) being a new interaction, since it is only the large $z$ behaviour (REF ) that prevents the integrals converging in (REF ) and thus forbids $ w^\\lambda (z)$ being expanded in terms of the quantised perturbations as in (REF ).",
"We emphasise that we have thus established that precisely in the regime where these non-polynomial solutions are distinguished from all those spanned by the quantised perturbations, it is incorrect to use the linearised solution (REF ) to deduce their $t$ dependence.",
"Further steps are therefore necessary before we can draw any conclusions as to the relevance of these non-quantised perturbations, and indeed rigorously, further steps are also necessary to classify the quantised perturbations.apart from of course the vacuum energy $w_0$ which has no field dependence Since the flow equations are first order differential equations in $t$ , we can if we wish specify the $z$ dependence of $V(z,t)$ at some `initial point' $t=0$ .",
"(Since $t=\\ln (\\mu /\\Lambda )$ and $\\mu $ is arbitrary, this is in fact some arbitrary point on the flow.)",
"The $t$ evolution can then in principle be solved for uniquely by the given flow equation.",
"We have seen that the evolution (REF ) is incorrect for the non-polynomial interactions (REF ).",
"Now we ask what the correct $t$ evolution is for these interactions.",
"In order for such a question to make any sense, one must set the potential to be (REF ) at some initial $t$ , which without loss of generality we can take to be $t=0$ .",
"(Later, in particular at the end of the paper, we will discuss the question of the $t$ evolution of general potentials that lie outside the space spanned by the quantised interactions.)",
"Therefore we will proceed by setting: $V(z,0)=\\epsilon \\, v(z)\\,,$ agreeing with (REF ) at $t=0$ .",
"Clearly however we now need a separate analysis to (REF ) and (REF ) for the $t$ dependence when $z$ is large enough that we have $V(z,0)\\gtrsim 1$ .",
"Such an analysis will therefore have to be non-perturbative.",
"Obviously similar comments apply to $U(y,t)$ .",
"However in this regime, the analysis for the effective average potential (Legendre effective potential), including (REF ) and also (REF ), will differ from that for the general cutoff profile Polchinski's equation (REF ).",
"We will exploit this later on.",
"For now, we concentrate on the behaviour of $V(z,t)$ .",
"Consider first the quantised perturbations.",
"The linearised mass term perturbation is given by $V(z,t)= \\epsilon \\, w_1(z) \\exp 2t$ , where $ w_1(z)=2z-N$ .",
"Setting $V(z,0)=\\epsilon \\, w_1(z)$ , since $ w_1(z)$ is linear inhomogeneous in $z$ , we see that the right hand side of the flow equations (REF ,REF ) are actually independent of $z$ .",
"Since the left hand side is linear anyway, the linearised solution for the $z$ piece is therefore correct whatever its magnitude, so non-perturbatively $V(z,t) = 2\\epsilon \\, z\\exp 2t +c(t)$ where $c(t)$ is to be solved for.",
"The non-linear terms on the right hand side of the flow equation are then known explicitly thus allowing the constant term $c(t)$ to be solved for exactly.",
"$c(t)$ therefore is not given by its linearised solution $-N\\epsilon \\, \\exp 2t$ .",
"But since there is no $z$ dependence in the non-linear terms, the $z$ dependence of the linearised solution is after all valid for all values.",
"Furthermore, $c(t)=-N\\epsilon \\, \\exp 2t$ is a good approximation for all $t\\ll t_c$ where $t_c=-\\frac{1}{2}\\ln (N\\epsilon )$ is the large positive RG time such that $-N\\epsilon \\, \\exp 2t\\sim 1$ .",
"We therefore conclude that the mass term $V(z,t)\\rightarrow 0$ as $t\\rightarrow -\\infty $ and thus indeed has the status of a relevant direction as the linearised analysis suggested.",
"Note that in physical variables (REF ), the mass term perturbation is $V_{phys}(z_{phys},t) =2\\epsilon \\,\\mu ^2 z_{phys} - N\\epsilon \\,\\mu ^4\\,{\\rm e}^{-2t}\\, =2\\epsilon \\,\\mu ^2 z_{phys} - N\\epsilon \\,\\mu ^2\\Lambda ^2\\,,$ for $t\\ll t_c$ .",
"Therefore in physical variables the field dependent piece is actually independent of $t$ (i.e.",
"has mean field evolution) while the constant term actually diverges as $t\\rightarrow -\\infty $ .",
"As expected, we see that it is important that we use the scaled variables appropriate to the Gaussian fixed point to determine whether or not $V(z,t)$ falls into the Gaussian fixed point ($V(z,t)\\rightarrow 0$ ) as $t\\rightarrow -\\infty $ .",
"For the quantised perturbations that are (marginally) irrelevant at the linearised level ($n\\ge 2$ ), we are already certain that they do not have the correct behaviour to fall into the Gaussian fixed point since even when $V\\ll 1$ , the $t$ dependence is such that the perturbation grows as $t$ becomes increasingly negative.Further comments on quantised (ir)relevant perturbations about both the Gaussian fixed point and general fixed points can be found in refs.",
"[38], [39], [2].",
"See also sec.",
"REF .",
"Now we answer the crucial question of whether the non-quantised perturbations (REF ) qualify as relevant perturbations when $\\lambda >0$ , as suggested by the linearised analysis.",
"Setting $V(z,0) = \\epsilon \\, w^\\lambda (z)\\,,$ we cannot now write down an analytic form for the exact solution $V(z,t)$ .",
"However the right hand side of the flow equations (REF ,REF ) cannot contribute to the $t$ dependence of the asymptotic behaviour (REF ).",
"In fact from (REF ), for large $z$ the right hand side of (REF ) can contribute at most a term of $O(z)$ and the right hand side of (REF ) is actually exponentially suppressed.",
"Therefore the $t$ dependence of the asymptotic expansion is found by requiring that the left hand side of the flow equations vanish.",
"Since the left hand side of the flow equations actually came about by the dimensional assignments in (REF ) it follows that in physical units the potential for large $z$ then does not evolve at all (i.e.",
"again has the so-called mean field evolution): $V_{phys}(z_{phys}, t) = V_{phys}(z_{phys},0) = \\epsilon \\, w_{phys}^\\lambda (z_{phys})\\,$ (using (REF )), where we have set $w_{phys}^\\lambda (z_{phys}) = \\mu ^4 w^\\lambda (z_{phys}/\\mu ^2)\\,,$ using (REF ) and (REF ), while in scaled variables appropriate to determining its evolution around the Gaussian fixed point, this impliesas of course also follows directly from the vanishing of the left hand side of eqns.",
"(REF ,REF ) $V(z,t) = \\,{\\rm e}^{4t}\\,V(z\\,{\\rm e}^{-2t}\\,,0)= \\epsilon \\,\\,{\\rm e}^{4t}\\, w^\\lambda (z\\,{\\rm e}^{-2t}\\,)\\,,$ i.e.",
"asymptotically $V(z,t) = \\epsilon \\,C\\, \\frac{\\,{\\rm e}^{(8+N-\\lambda )t}\\,}{ z^p}\\, \\exp {\\left(z\\,{\\rm e}^{-2t}\\,\\right)}\\left[ 1+ O\\left(\\frac{\\,{\\rm e}^{2t}\\,}{ z}\\right)\\right]\\,.$ We see that far from falling in to the Gaussian fixed point as $t\\rightarrow -\\infty $ , in the large $z$ regime the perturbation actually diverges rapidly as $t\\rightarrow -\\infty $ , dominated by the $\\exp \\!",
"{\\left(z\\,{\\rm e}^{-2t}\\,\\right)}$ term.",
"In the large field regime, the ultraviolet behaviour of the non-quantised perturbations therefore fail to behave correctly as relevant perturbations about the Gaussian fixed point: they do not generate a Renormalised Trajectory emanating from the Gaussian fixed point and thus cannot be used to form a continuum limit governed by the Gaussian fixed point.",
"We have seen that (REF ) can be deduced from neglecting the right hand sides of the flow equations.",
"This neglect is justified providing $V(z,t)\\gg 1$ .",
"Despite its simplicity, this argument is therefore inherently non-perturbative.",
"In fact as mentioned above, for large $z$ the right hand side of (REF ) can contribute at most a term of $O(z)$ and the right hand side of (REF ) is actually exponentially suppressed.",
"It is important to recognise that contributions from the right hand side of the flow equation are therefore infinitely suppressed in the large $z$ regime, in the sense that they make no contribution to the $t$ dependence of the asymptotic expansion (REF ), even if we had carried the multiplicative series corrections in $1/z$ in (REF ) to infinite order.",
"A closely related remark is to note that the $t$ evolution given in (REF ) can thus be used at large $z$ to derive the $t$ evolution for $V^{(n)}(z,t)$ , the potential differentiated with respect to $z$ to any finite order $n$ .",
"We emphasise again that the large field properties are crucial in this discussion since it is only these that keep $V(z,t)$ outside the space of the quantised interactions, i.e.",
"prevent it from being expanded in terms of the quantised perturbations as in (REF ).",
"At the same time we thus see that if we replace (REF ) by an initial perturbation that is a general linear combination of the non-quantised interactions: $V(z,0) = \\epsilon \\!\\int \\!\\!",
"d\\lambda \\, \\rho (\\lambda )\\, w^\\lambda (z)\\,,$ for some sufficiently well-behaved density factor $\\rho (\\lambda )$ , then either this perturbation can already be re-expressed in terms of the polynomial interactions as in (REF ), and thus will have RG evolution determined by them, or it lies outside this Hilbert space in which case by (REF ) its grows at large $z$ at least as fast as $z^{-N/4}\\exp (z/2)$ .",
"In this latter case we see again, by trivially adapting the argument below (REF ), that in the large $z$ regime the perturbation will diverge away from the Gaussian fixed point as $t\\rightarrow -\\infty $ ."
],
[
"Couplings in the Legendre effection action",
"Now we put aside the ultraviolet problems and turn to explore the infrared behaviour i.e.",
"the $t>0$ domain.",
"We can always choose some large enough $z$ that (REF ) still applies, i.e.",
"such that neglect of the right hand side of the flow equations continues to be justified.",
"To be precise, let $z=z_{asy}(t)$ be such that $V(z,t)\\gg 1$ according to (REF ).",
"Inverting, we find $z_{asy} \\sim \\,{\\rm e}^{2t}\\,\\ln (1/\\epsilon )+A$ for some large constant $A$ .See sec.",
"REF for the higher order terms.",
"Then for all $t<t_f$ , where $t_f$ is any fixed RG time, and for all $z>z_{asy}(t_f)$ , the solution $V(z,t)$ is well approximated by (REF ).",
"Using (REF ) and approximating $w_n(z)\\sim 2^n z^n/n!$ by its leading power, we thus see that for large $z$ the integrand in (REF ) has $z$ dependence $z^{n+\\frac{\\lambda }{2}-3} \\,{\\rm e}^{-az}\\,\\,,\\quad {\\rm where}\\quad a=1-\\,{\\rm e}^{-2t}\\,>0\\,,$ and thus the integrals defining $g_{2n}(t)$ converge.",
"Therefore as soon as $t>0$ (no matter how small), the couplings in the expansion (REF ) are well defined [38].",
"Furthermore, the expression (REF ) has a maximum at $z=(n-3+\\lambda /2)/(1-\\,{\\rm e}^{-2t}\\,)$ , which is an $O(1)$ value for any $t$ larger than infinitesimal, after which the exponential decay takes over.",
"Therefore large $z$ values actually make a negligible contribution to (REF ), and the integral is dominated by the bounded region $0<z\\lesssim O(1)$ .",
"Recalling the overall $\\epsilon $ multiplier, we have thus established that for all finite $t>0$ , the couplings $g_{2n}(t)$ are $O(\\epsilon )$ .",
"Finally, the series converges to the exact solution $V(z,t)$ , as in (REF ), once the integral in (REF ) converges.",
"Using (REF ), we see that this happens as soon as $1-2\\,{\\rm e}^{-2t}\\,>0$ .",
"Therefore we have proved that the series (REF ) converges, and thus $V(z,t)$ falls back into the Hilbert space spanned by the quantised interactions $w_n(z)$ , for all $t>{\\ln 2}/{2}$ .",
"It is then reasonable to expect that the infrared ($t\\rightarrow \\infty $ ) fate of the non-polynomial directions is just governed by the perturbative evolution of these quantised perturbations, which as we have reviewed, leads in the infrared to the high temperature fixed point with all interactions decayed away leaving only a diverging $w_1(z)\\,{\\rm e}^{2t}\\,$ mass term.",
"From (REF ) we see that for $n<2-\\lambda /2$ , the integrals $g_{2n}(t)$ converge and thus are of $O(\\epsilon )$ , even for $t=0$ .",
"For the potentially-symmetry-breaking potentials with $\\lambda <2$ (see footnote REF ) this corresponds to $O(\\epsilon )$ values for the running mass $g_2(t)$ and vacuum energy $g_0(t)$ for all finite $t\\ge 0$ .",
"Since we cannot have $n=2-\\lambda /2$ (see footnote REF ) the remaining possibility for $n$ is $n>2-\\lambda /2$ .",
"In this case for small positive $t$ the integrals are dominated by the behaviour at very large $z$ since by (REF ) $a\\approx 2t$ for small positive $t$ , so that they only just converge.",
"We can therefore compute the leading $t$ dependence of the coefficients $g_{2n}$ from the large $z$ behaviour.",
"Thus for $n>2-\\lambda /2$ and $0<t\\ll 1$In the first line, we drop the $O(1)$ integral over $0<z<z_{asy}$ and insert the leading term from $ w_n(z)$ .",
"Thus we ignore additive $O(\\epsilon )$ and multiplicative $O(t)$ corrections respectively.",
"In the second line we drop multiplicative corrections of $O(tz_{asy})$ coming from the lower limit.",
"$g_{2n} &\\approx & \\frac{\\epsilon }{2^n} \\,\\frac{C}{ \\Gamma (n+N/2)}\\, \\int _{z_{asy}}^\\infty \\!\\!\\!dz\\ z^{n+\\lambda /2-3}\\,\\,{\\rm e}^{-2tz}\\, \\nonumber \\\\&\\approx &\\ \\epsilon \\, 2^{\\lambda /2-2} \\frac{\\Gamma \\!\\left({N/2}\\right)}{ \\Gamma (n+N/2)} \\frac{\\Gamma \\!\\left( {n+\\lambda /2-2}\\right)}{ \\left|\\Gamma \\!\\left( {\\lambda /2-2}\\right)\\right|} \\left( \\frac{1}{4t}\\right)^{n+\\lambda /2-2}\\,.$ Therefore although $V(z,t)$ does have an expansion in the quantised perturbations when $t>0$ , and such that the corresponding couplings are infinitesimal for finite $t>0$ , all the above couplings $g_{2n}(t)$ , and in particular all the irrelevant couplings, diverge as $t\\rightarrow 0^+$ .",
"In this sense we see that the $t=0$ point is in fact already infinitely far from the Gaussian fixed point.",
"However, as $t$ increases to positive values, these couplings rapidly shrink as expected for the irrelevant couplings and as we have seen, for $t>\\ln 2/2$ the series (REF ) is then convergent.",
"Indeed from (REF ), for sufficiently small $\\epsilon $ , all couplings $g_{2n}$ of increasing irrelevancy up to some maximum $n<n_{max}(\\epsilon )$ already shrink to the regime $g_{2n}(t)\\ll 1$ before $t$ violates the bound $t\\ll 1$ .",
"We might expect to have to take $\\epsilon \\rightarrow 0$ to form a continuum limit, despite the poor behaviour in the ultraviolet.",
"If so, since $n_{max}\\propto \\ln (1/\\epsilon )$ increases without bound, this means that eventually all couplings shrink to infinitesimal with increasing $t$ already for $t\\ll 1$ ."
],
[
"Consistency with large field behaviour",
"We have seen that if the potential is set equal to an infinitesimal amount of non-polynomial direction at RG time $t=0$ , cf.",
"(REF ), then on IR evolution it immediately can be re-expressed in terms of quantised couplings which furthermore are $O(\\epsilon )$ , for all finite $t>0$ , and such that from $t>\\ln 2/2$ the potential is fully inside this Hilbert space in the sense that the series then converges to the potential.",
"This in turn gives us confidence that the universality class is just the usual one controlled by small initial values of these couplings (namely in four dimensions non-interacting scalar field theory).",
"At first sight this picture seems deeply at variance with the fact that the large field behaviour is fixed to be mean-field, viz.",
"(REF ), meaning that in physical variables the large field dependence (REF ) remains that of the original non-polynomial perturbation (REF ) and does not actually depend on $t$ at all.",
"Actually, these two pictures are consistent with each other as can be seen by the following observations.",
"We have already seen from eqn.",
"(REF ) that the RG properties are only manifest in scaled variables.",
"This is why we do not see in physical variables that the non-polynomial potential falls back into the Hilbert space spanned by the quantised directions.",
"Instead in physical variables the Hilbert space is growing to accommodate the non-polynomial interaction.",
"This follows from (REF ) which maps the explicit exponential in the measure in (REF ) to $\\exp (-z_{phys}/\\Lambda ^2)$ , and which thus overcomes the $\\exp (z_{phys}/\\mu ^2)$ present in $w_{phys}^\\lambda $ , as soon as $\\Lambda <\\mu $ .",
"(The latter exponential is evident from (REF ) and (REF ).)",
"Working again with scaled variables, we note that since the quantised couplings are $O(\\epsilon )$ , we expect that for finite $t$ they behave independently with the linearised $t$ dependence of form given in (REF ).",
"Then from (REF ), (REF ) and (REF ), we have for large $z$ , $g_{2n}(t)\\, w_n(z) \\sim g_{2n}(0)\\, {\\rm e}^{(4-2n) t} (2z)^n/n!",
"\\sim g_{2n}(0)\\, {\\rm e}^{4t} \\,w_n\\!\\left(z\\,{\\rm e}^{-2t}\\,\\right)\\,,$ where $g_{2n}(0)\\sim \\epsilon $ is the $t=0$ coefficient in (REF ).",
"We see that for the quantised interactions the $t$ -dependence of their renormalised coupling can at large $z$ be instead attributed to mean-field evolution as in (REF ).",
"(This is just the RG argument for accepting these as renormalised couplings [38], [39], [2], run in reverse.)",
"It is then natural to expect that the full sum (REF ) also satisfies mean-field evolution for large $z$ .",
"Notice that this result is general and holds for any infinitesimal perturbation that can be expanded as in (REF ) i.e.",
"such that the integrals in (REF ) converge.",
"We see that although the couplings $g_{2n}(t)$ are (marginally) irrelevant for all $n>1$ , meaning that their influence should die away as $t$ increases (and completely as $t\\rightarrow +\\infty $ ), this universality property is true only for finite field.",
"For large $z_{phys}$ none of the quantised interactions die away and thus they preserve the large $z_{phys}$ non-universal form of the perturbation, since mean-field evolution is nothing more than the statement that the potential is frozen in form in physical variables, cf.",
"(REF )."
],
[
"Couplings in the Wilsonian effective action",
"To further address these issues, we now use the fact that the flows (REF ) and (REF ) are equivalent under an exact duality [42].",
"Using this map we can turn the exact solution of the flow (REF ) for the LPA effective average potential $V(z,t)$ with boundary condition (REF ) (and optimised cutoff), into an exact solution $U(y,t)$ of the LPA Polchinski flow equation (REF ) (with general cutoff profile).",
"This $U(y,t)$ representation, which effectively just builds back in the one-particle reducible contributions missing from the one-particle irreducible $V(z,t)$ [42], [88], has improved properties.",
"This potential will also have an expansion in the quantised perturbations $U(y,t) = \\sum _{n=0}^\\infty h_{2n}(t)\\, w_n(y)\\,,$ but the couplings $h_{2n}(t)$ in this case converge and remain $O(\\epsilon )$ for all finite $t$ including for $t\\le 0$ , and furthermore the series converges to $U(y,t)$ for all finite $t$ .",
"In appendix we review the theory behind this duality.",
"The starting point for our discussion is the generalised Legendre transform relation [42]: $U(y,t) = V(z,t) +\\frac{1}{2}(\\phi -\\Phi )^2\\,.$ We have already seen that the Gaussian fixed point for the Polchinski flow (REF ) is $U=0$ , and that infinitesimally small perturbations around this, namely (REF ), give the same Kummer's equation (REF ) and thus the same quantised solutions $w= w_n(y)$ as in (REF ).",
"As already emphasised in sec.",
"REF , they also have the same non-quantised linearised solutions $w= w^\\lambda (y)$ as in (REF ).",
"However we are interested here in studying the non-perturbative flow where the potentials are $O(1)$ or larger, generated by the large $z$ dependence of the boundary condition (REF ).",
"Therefore we take as initial condition $U(y,0)$ as determined from $U(y,0) = \\epsilon \\, w^\\lambda (z) +\\frac{1}{2}(\\phi -\\Phi )^2\\,,$ consistently with (REF ) and (REF ).",
"The exact solution $U(y,t)$ of the flow equation (REF ) with this boundary condition, is then given by (REF ).",
"Before analysing the results of setting the boundary condition (REF ), we review the non-linear evolution of the mass perturbation $U(y,0)=\\epsilon \\,w_1(y)$ using the Wilson/Polchinski flow equation.",
"Its evolution follows from the analysis above (REF ) and the exact map (REF ), but we can just as easily solve for it directly using (REF ).",
"This time we neglect the uninteresting vacuum energy term, so the solution takes the form $U(y,t)= 2y h_2(t)$ .",
"Substituting in (REF ) we find ${\\dot{h}}_2 = 2h_2-2h_2^2$ , implying $h_2(t) = {\\epsilon \\,\\,{\\rm e}^{2t}\\,\\over 1+2\\epsilon \\left({\\rm e}^{2t}-1\\right)}\\,.$ We see that the non-linear term on the right hand side of (REF ) moderates the growth of this relevant coupling in the infrared and instead of diverging like the corresponding coupling $g_2(t)=\\epsilon \\,\\,{\\rm e}^{2t}\\,$ of the effective average action, as analysed above (REF ), it obtains a limiting value $h_2(t)\\rightarrow 1/2$ as $t\\rightarrow \\infty $ .",
"Thus $U(y,t)=U_*(y)=y$ is in fact a fixed point of the flow, known in the literature as the high temperature fixed point.",
"It is this `compactification' of the flow of the relevant coupling which will make it easier to see what is going on with the infrared evolution of the non-polynomial directions.",
"From (REF ) we see that [42]: $\\Phi ^a - \\phi ^a = \\Phi ^a U^{\\prime } = \\phi ^a V^{\\prime }\\,,$ Since it follows that $\\phi ^a$ and $\\Phi ^a$ point in the same direction we deduce [41], [42] $\\sqrt{y}-\\sqrt{z} = \\sqrt{y}\\, U^{\\prime } = \\sqrt{z}\\, V^{\\prime }$ and hence $\\sqrt{z\\over y} = 1-U^{\\prime } = {1\\over 1 + V^{\\prime }}\\,.$ Thus $y = z\\, (1+V^{\\prime })^2\\,.$ This provides us with the map from $z$ to $y$ .",
"Setting $V(z,0)=\\epsilon \\, w^\\lambda (z)$ as in (REF ), we have that for finite $z$ , i.e.",
"$z\\sim O(1)$ , the linearised evolution is valid for all finite $t$ : $V(z,t)=\\epsilon \\, w^\\lambda (z)\\,{\\rm e}^{\\lambda t}\\,$ .",
"Thus the $V^{\\prime }$ term in this regime remains $O(\\epsilon )$ and can be neglected in (REF ).",
"Hence we see that $y=z$ for all $O(1)$ values and for all finite RG time.Evidently $y(z)$ is then monotonic increasing, justifying the use of the Legendre transform, despite the fact (see footnote REF ) that $w^\\lambda (z)$ is initially a decreasing function and has a non-trivial minimum for $\\lambda <2$ .",
"When $z\\rightarrow \\infty $ , (REF ) applies, and thus we see from the latter two equations in (REF ) that $U^{\\prime }\\rightarrow 1$ as $y\\rightarrow \\infty $ .",
"This implies that $U(y,t)$ cannot grow faster than $y$ , as $y\\rightarrow \\infty $ , for all finite RG time, and thus in fact for all RG time.",
"In fact we see from (REF ) that whenever $V^{\\prime }\\rightarrow \\infty $ as $z\\rightarrow \\infty $ , that $U^{\\prime }\\rightarrow 1$ as $y\\rightarrow \\infty $ .",
"This is true for any $V$ growing faster than $z$ and thus also includes all polynomial interactions (REF ) with $n>1$ .",
"We see therefore that the large field behaviour is generally under much better control in the Wilsonian picture.",
"On the other hand, let us remark that the solution we are analysing is very different from the solution of the Wilson/Polchinski flow equation that one would obtain from the boundary condition $U(y,0)=\\epsilon \\, w^\\lambda (y)$ (or by setting this to be $\\epsilon \\,w_n(y)$ with $n>1$ ).",
"Indeed in the large $y$ region where $U(y,t)\\gg 1$ , the right hand side of (REF ) cannot be neglected and thus mean-field evolution like in (REF ) does not take place.",
"In particular the classical part itself (the first term on the right hand side of (REF )) continues to contribute to the evolution of $U(y,t)$ .",
"Furthermore, we see from (REF ) that $z$ must vanish at the point $y_0$ where $U^{\\prime }(y_0,t)=1$ .",
"Therefore $z(y)$ is no longer monotonic increasing in these cases, causing the Legendre transform relation to break down.",
"Following (REF ), we can compute the couplings in the expansion (REF ) from $h_{2n}(t) ={\\Gamma (n+1)\\over 2^{2n}\\Gamma (n+N/2)} \\int _0^\\infty \\!\\!dy\\, y^{{N\\over 2}-1}\\,{\\rm e}^{-y}\\, w_n(y)\\, U(y,t)\\,.$ Since $U(y,t)$ grows no faster than $y$ for large $y$ , these integrals always converge.",
"Furthermore, following (REF ),the series representation (REF ) will always converge back to $U(y,t)$ .",
"In other words we have proven that the dual potential $U(y,t)$ of the non-perturbative evolution of the non-polynomial directions (REF ) can always (viz.",
"for all $t$ ) be expanded in terms the quantised perturbations such that the series (REF ) converges to right limit, and such that the couplings $h_{2n}(t)$ remain finite for all finite $t$ .",
"We have shown that for large $y$ , $U(y,t)= y$ up to corrections that grow slower than $y$ .",
"We see therefore that the large field behaviour is governed by the high temperature fixed point (REF ) for all RG time including the initial $t=0$ .",
"Now we compute the leading correction.",
"Again using the fact that $\\phi ^a$ and $\\Phi ^a$ point in the same direction, (REF ) implies $U(y,t) = (\\sqrt{z}-\\sqrt{y})^2 + V(z,t) &=&(\\sqrt{z}-\\sqrt{y})^2+ \\int \\!\\!",
"dz\\left( \\sqrt{y\\over z} -1\\right) \\\\&=& y-2\\sqrt{zy} + \\int \\!\\!",
"dy\\, {dz\\over dy}\\, \\sqrt{y\\over z}\\,,$ on using (REF ).",
"The second line is in a useful form for substituting a series expansion for $z(y,t)$ .",
"Let us mention however that the first line gives, by integration by parts, the most compact expression: $U(y,t)&=& (\\sqrt{z}-\\sqrt{y})^2-z+2\\sqrt{zy} -\\int \\!\\!",
"{dz\\over \\sqrt{y}} {dy\\over dz} \\sqrt{z} \\nonumber \\\\&=& y - \\int \\!\\!dy\\,\\sqrt{z\\over y}\\,.$ Inverting (REF ) using (REF ), since the $(V^{\\prime })^2$ term dominates, one finds $z= \\,{\\rm e}^{2t}\\, \\ln (\\sqrt{y}/\\epsilon )+\\cdots $ .",
"Therefore for both $y$ and $z$ of order of magnitude $\\,{\\rm e}^{2t}\\,\\ln 1/\\epsilon $ or larger, one can show that $z(y,t) = \\,{\\rm e}^{2t}\\,\\left\\lbrace \\ln {\\sqrt{y}\\over \\epsilon } +\\left(p-{1\\over 2}\\right)\\ln \\ln {\\sqrt{y}\\over \\epsilon } -3t-\\ln C+O\\left({\\ln \\ln {(\\sqrt{y}/\\epsilon )}\\over \\ln (\\sqrt{y}/\\epsilon )}\\right)\\right\\rbrace \\,.$ Thus we find that $U(y,t) = y -2\\,{\\rm e}^{t}\\,\\sqrt{y\\,\\ln {(\\sqrt{y}/\\epsilon )}}+O\\left(\\sqrt{y}\\,{\\ln \\ln {(\\sqrt{y}/\\epsilon )}\\over \\sqrt{\\ln (\\sqrt{y}/\\epsilon )}}\\right)\\,.$ Despite appearances the subleading term remains of the same size as the leading term as $t$ increases, since the formula is only valid for $y\\gtrsim \\,{\\rm e}^{2t}\\,\\ln 1/\\epsilon $ .",
"Carefully discarding the terms that are the same size or smaller than the neglected corrections $O(\\cdots )$ , one can verify directly that (REF ) solves the Wilson/Polchinski LPA flow equation (REF ).",
"Finally, we compute estimates for the couplings $h_{2n}(t)$ in the expansion (REF ).",
"From (REF ), using the leading power for $w_n(z)$ , we see they are bounded by an integral expression of form $h_{2n}(t) \\lesssim {1\\over 2^{n}\\Gamma (n+N/2)} \\int _0^\\infty \\!\\!\\!dy\\,\\, y^{{N\\over 2}+n-1}\\,{\\rm e}^{-y}\\, U(y,t)\\,.$ Using (REF ), and (REF ) with (REF ), we can write this as an integral over $z$ : $h_{2n}(t) \\lesssim \\\\{1\\over 2^{n}\\Gamma (n+N/2)} \\int _0^\\infty \\!\\!\\!dz\\,\\, z^{{N\\over 2}+n-1}\\, \\left(1+V^{\\prime }+2zV^{\\prime \\prime }\\right)(V+z{V^{\\prime }}^2)\\left(1+V^{\\prime }\\right)^{N+2n-1}\\,{\\rm e}^{-z\\left(1+V^{\\prime }\\right)^2}\\,.$ The integral is cut-off sharply for $z>z_1(t)$ where $z=z_1(t)$ satisfies $z{V^{\\prime }}^2=1$ : as a consequence of (REF ), $z\\left(1+V^{\\prime }\\right)^2$ grows exponentially as $z$ increases beyond this point.",
"In other words, the last term in the integral vanishes as the negative exponential of an exponential for $z>z_1(t)$ .",
"Since (REF ) in fact solves $z{V^{\\prime }}^2=y$ , we already know that $z_1(t)=z(1,t)$ , where the latter is defined in (REF ).",
"Figure: The exact integrand in () for h 6 (t)h_6(t) is plotted (together with the pre-multiplier) for the case t=0t=0, as follows from () and ().",
"We have set N=4N=4, n=3n=3 and ϵ=10 -12 \\epsilon =10^{-12}.",
"In red is shown the result for λ=1\\lambda =1 (a symmetry-breaking potential) and in blue is plotted the integrand/10/10 for the case λ=3\\lambda =3 (a symmetry preserving potential).",
"(The small minimum in the latter case appears because the potential starts out negative.)",
"The dotted line curves show the equivalent integrands for g 6 (t)g_{6}(t) as it appears in ().",
"The integrands only deviate once zz approaches z 1 z_1.",
"Shown in orange and violet are the values z=z 1 (0)z=z_1(0) for the cases λ=1,3\\lambda =1,3 respectively.",
"The figure thus verifies that the values for z 1 z_1 closely approximate the effective cutoff points for the h 6 h_{6} integrands.For $z\\sim O(1)$ , the integrand is the same as in (REF ), more precisely collapses to the approximation $z^{{N\\over 2}+n-1} V(z,t) \\,{\\rm e}^{-z}\\,\\,,$ since in this case $V\\sim O(\\epsilon )$ allowing one to drop all the `correction' terms in (REF ).",
"The balance of terms changes as $z$ approaches $z_1(t)$ .",
"From (REF ), we see that $z_1(t)=z(1,t)$ diverges as $z_1\\sim \\,{\\rm e}^{2t}\\,\\ln {1\\over \\epsilon }$ .",
"Thus $V^{\\prime } = 1/\\sqrt{z_1}\\sim \\,{\\rm e}^{-t}\\,/\\sqrt{\\ln 1/\\epsilon }$ and thus from (REF ), $V\\sim \\,{\\rm e}^{t}\\,/\\sqrt{\\ln 1/\\epsilon }$ and $V^{\\prime \\prime }\\sim \\,{\\rm e}^{-3t}\\,/\\sqrt{\\ln 1/\\epsilon }$ .",
"Applying these estimates to (REF ), we see that in this regime the first two brackets in the integrand are now dominated by $2zV^{\\prime \\prime }$ and $zV^{\\prime 2}$ respectively, but it is the $\\,{\\rm e}^{-z}\\,\\sim \\epsilon ^{\\exp 2t}$ that makes the most significant contribution.",
"The situation is illustrated in fig.",
"REF .",
"Putting all these observations together we see that for finite $t>0$ only the $z\\sim O(1)$ regime contributes at leading order and thus the leading estimates for the couplings in the two pictures agree $h_{2n}(t)=g_{2n}(t)\\sim O(\\epsilon )$ .",
"For finite $t<0$ , the $z\\sim z_1$ regime dominates and thus one finds $h_{2n}(t)\\sim \\epsilon ^{\\exp 2t}$ to leading order.",
"Both of these estimates receive multiplicative corrections of form $(\\ln 1/\\epsilon )^q$ for some finite power $q$ whose precise value would require a more in-depth analysis.",
"However already we confirm that $U(y,t)$ can be expanded as a convergent series in the quantised perturbations, as in (REF ), for all $t$ , and such that the couplings $h_{2n}(t)$ are infinitesimal for infinitesimal $\\epsilon $ and all finite $t$ ."
],
[
"Conclusions",
"In sec.",
"we saw that for four-dimensional $O(N)$ scalar field theory, the LPA of different versions of the exact RG flow equation, for any sensible cutoff profile, give the same ordinary differential equation (REF ) for linearised perturbations about the Gaussian fixed point.",
"This is Kummer's equation and its solutions are therefore universal in this sense.",
"Solutions of this equation divide into polynomial perturbations (REF ) with quantised RG eigenvalue, which are generated by the usual perturbative interactions, and non-polynomial perturbations (REF ) with continuous (non-quantised) RG eigenvalue.",
"The quantised solutions form a Hilbert space structure, allowing any potential $V(z,t)$ to be expanded in terms of them as in (REF ), providing the integral (REF ) converges.",
"The only property that prevents the non-quantised perturbations $V(z,0) = \\epsilon \\,w^\\lambda (z)$ from being expanded in terms of the quantised perturbations is their large $z$ asymptotic behaviour (REF ).",
"But as we saw in sec.",
", it is also precisely this property that prevents $V(z,t) = \\epsilon \\, w^\\lambda (z) \\, {\\rm e}^{\\lambda t}$ from being the correct $t$ evolution for large $z$ .",
"Instead, mean-field evolution takes over, forcing, even for $\\lambda >0$ , the perturbation to diverge away from the Gaussian fixed point as $t\\rightarrow -\\infty $ .",
"In this way we proved that in the large field regime, the ultraviolet behaviour of the non-quantised perturbations fail to behave correctly as relevant perturbations.",
"They thus cannot generate a Renormalised Trajectory emanating from the Gaussian fixed point and therefore cannot be used to form a continuum limit governed by this fixed point.",
"We also saw that any linear combination of these non-quantised perturbations which is still not expandable in terms of the quantised perturbations, likewise fails to behave correctly as $t\\rightarrow -\\infty $ .",
"Despite the simplicity of the argument, this $V(z,t)\\gg 1$ analysis is inherently non-perturbative.",
"The same asymptotic mean-field behaviour can be established for all $z$ larger than some $z_{asy}(t)$ in the IR ($t>0$ ) domain.",
"In this way in sec.",
"REF , we proved that as soon as $t>0$ (no matter how small), the couplings $g_{2n}(t)$ are well defined, while for $t>\\ln 2/2$ the potential $V(z,t)$ falls fully back into the Hilbert space spanned by the quantised interactions $w_n(z)$ , and can be expanded as a convergent series as in (REF ) [39].",
"Furthermore we proved that for all finite $t>0$ , the corresponding couplings $g_{2n}(t)$ are $O(\\epsilon )$ .",
"For $n<2-\\lambda /2$ the couplings are $O(\\epsilon )$ even at $t=0$ , while for $n>2-\\lambda /2$ , they diverge as a power of $t$ as $t\\rightarrow 0^+$ .",
"In this sense we see that the $t=0$ point is in fact already infinitely far from the Gaussian fixed point.",
"With all the above properties in mind, we see why at finite $\\epsilon $ , the non-quantised solutions should more properly be viewed as no more special than any other finite perturbation added to the fixed point.",
"However from eqn.",
"(REF ), we also establish that in the limit as $\\epsilon \\rightarrow 0$ , all couplings shrink to infinitesimal already at arbitrarily small positive $t$ .",
"We therefore conclude that, at least for sufficiently small $\\epsilon $ , the infrared ($t\\rightarrow +\\infty $ ) fate of the non-polynomial directions is just governed by the perturbative evolution of these quantised perturbations, which (in four space-time dimensions with thus marginally irrelevant $g_4$ ) leads in the infrared to the high temperature fixed point with all interactions decayed away leaving only a diverging $w_1(z)\\,{\\rm e}^{2t}\\,$ mass term.",
"In sec.",
"REF we saw why this universality conclusion is nevertheless consistent with the fact that the large field dependence remains determined by the initial perturbation, cf.",
"(REF ), and indeed in physical variables fixed to be that in eqn.",
"(REF ).",
"We first noted that in physical variables it is the Hilbert space that grows to accommodate this interaction.",
"And then we noted that the universality properties in any case only hold for finite field.",
"The self-similar evolution of the quantised couplings can instead be attributed at large field to mean-field (equivalently classical) evolution, or in physical variables no evolution at all, cf.",
"(REF ).",
"Therefore for large $z_{phys}$ none of the quantised interactions die away and thus they preserve the large $z_{phys}$ non-universal form of the perturbation.",
"Finally in sec.",
"REF we used the generalised Legendre transform relation (REF ) to study the RG evolution of the corresponding exact solution $U(y,t)$ to the Wilson/Polchinski LPA flow equation (REF ).",
"Here the high temperature fixed point is given by a genuine fixed point of the flow, and this `compactification' leads to better control of the corresponding quantised couplings $h_{2n}(t)$ in the expansion (REF ).",
"Indeed we saw that the Legendre transform relation remains valid and the large field behaviour is governed by the high temperature fixed point (REF ), for all finite RG time, both positive and negative.",
"We also proved that the expansion (REF ) is well defined and convergent always, i.e.",
"for all $t$ .",
"Finally we found the leading dependence for the couplings in the limit $\\epsilon \\rightarrow 0$ , showing that they vanish in this limit for all finite $t$ .",
"In detail, $h_{2n}(t)\\sim \\epsilon ^{\\exp 2t}$ for finite $t<0$ , while the leading behaviour $h_{2n}(t)\\sim \\epsilon $ agrees with that for $g_{2n}(t)$ for finite $t>0$ .",
"As already remarked in the Introduction, and further touched on in sec.",
", although for concreteness we developed the arguments for the Gaussian fixed point and four space-time dimensions, it is clear that the arguments can be straightforwardly adapted to other space-time dimensions $d>2$ and also to the non-quantised eigen-perturbations about the Wilson-Fisher fixed points [38], [39], [2].",
"Finally, the main arguments can be extended to apply to the solutions of the flow equations derived at higher orders in the derivative expansion, and indeed for the exact flow equations, as we will now show.",
"From the above works, the $O(\\partial ^2)$ linearised flow equations about the Gaussian fixed point can be found, for both the potential and kinetic term interactions.",
"In fact these can be read off essentially directly from ref.",
"[36] (together with expressions for the quantised polynomial solutions for both potential and kinetic terms).",
"The linearised equation (REF ) for the potential itself in fact holds exactly, i.e.",
"without LPA or any other approximation, as we will now confirm.",
"The terms on the right hand side arise from a tadpole integral [49], see also [44], [36], and as we will see, are independent of cutoff profile after rescaling variables.",
"The interactions satisfy the exact flow equation (REF ).",
"(From the split (REF ) or (REF ), it is clear that $\\Gamma $ here stands for the interactions.)",
"Linearisation therefore just requires to make this $\\Gamma $ arbitrarily small.",
"Therefore the exact flow equation for a perturbation about the Gaussian fixed point becomes (in unscaled variables) ${\\partial \\over \\partial \\Lambda } {\\Gamma }[\\phi ] = -{1\\over 2} {\\rm tr} \\left[{\\delta ^2{\\Gamma }\\over \\delta \\phi ^a\\delta \\phi ^a}{\\partial \\Delta _{UV}\\over \\partial \\Lambda }\\right]\\,,$ where tr is a space-time trace and we have used (REF ) to make explicit the connection to a UV regulated tadpole diagram.",
"If the perturbation corresponds just to a potential piece $V(z,t)$ , then it is straightforward to compute the second order $\\phi ^a$ derivative and the above evaluates to $\\dot{V} + 2z V^{\\prime } -4 V = a\\, (N V^{\\prime } +2z V^{\\prime \\prime })\\,,$ where we used (REF ) and (REF ), rescaled using (REF ), and expressed the tadpole integral as a pure (but non-universal) number: $a =\\frac{1}{2\\Lambda } \\frac{\\partial }{\\partial \\Lambda }\\int \\!\\!",
"\\frac{d^4q}{(2\\pi )^4} \\frac{C_{UV}(q,\\Lambda )}{q^2}\\,.$ This number can however now be absorbed by rescaling $z\\mapsto a z$ .",
"After separation of variables (REF ) we thus rederive (REF ), this time without any approximation.",
"It follows immediately that the non-polynomial scaling solutions (REF ), hold also for linearisation at the exact level.",
"Of course none of this implies that they should lead to new relevant directions any more than they did within the LPA.",
"On the contrary, the properties we derived, all follow from the fact that the RG evolution of (standard) scalar field theory at large field is given by mean field.",
"This conclusion surely holds for the exact equations.",
"Indeed it is clear from the fact that the Hessian appears as an inverse in the exact flow equations (REF ) or (REF ), that if the Legendre effective action diverges for large field, then the right hand side of the flow equation can be neglected in this limit and thus mean-field evolution (i.e.",
"no evolution in physical variables) has to take over.",
"The main conclusions then follow also at the exact level, in particular the non-polynomial potential perturbations diverge away from the Gaussian fixed point as $t\\rightarrow -\\infty $ , while for $t>\\ln 2/2$ they can be expanded as a convergent series in the quantised polynomial interactions.",
"In fact since all of our conclusions stem from the mean field evolution (REF ) at large field, they depend very little on the fact that $w^\\lambda (z)$ was a solution to Kummer's equation (REF ).",
"The same conclusions would therefore follow for any initial perturbation $V(z,0)=\\epsilon \\, v(z)$ , with large $z$ asymptotics given by (REF ), even if it did not satisfy (REF ) for some $\\lambda $ .",
"Straightforward generalisations would therefore establish similar conclusions for any perturbation that grows as an exponential $z=\\phi ^2/2$ .",
"With some adaptation it ought to be possible to make the corresponding universality statements for initial perturbations whose growth for large field is so severe that under RG evolution they remain forever outside the space spanned by the quantised perturbations, for example $\\sim \\exp (\\exp \\phi ^2)$ ."
],
[
"Acknowledgments",
"IHB and TRM acknowledge support through an STFC studentship and Consolidated Grant ST/J000396/1 respectively.",
"We thank Alfio Bonanno for sharing with us his insights gained from numerical investigations of the RG flow of non-polynomial interactions."
],
[
"Duality relation for effective potentials",
"In this appendix we work in physical (i.e.",
"unscaled) variables although unlike in eqn.",
"(REF ) we will not label them explicitly as such.",
"Let us recall that the flow equation for the effective average action [66] ${\\partial \\over \\partial \\Lambda } {\\tilde{\\Gamma }}[\\phi ] = {1\\over 2} {\\rm tr} \\left[ {\\cal R}\\delta _{ab} + {\\delta ^2{\\tilde{\\Gamma }}\\over \\delta \\phi ^a\\delta \\phi ^b}\\right]^{-1} {\\partial {\\cal R}\\over \\partial \\Lambda }$ can trivially be alternatively written as [67], [2] ${\\partial \\over \\partial \\Lambda } {\\Gamma }[\\phi ] = -{1\\over 2} {\\rm tr} \\left[ \\delta _{ab} + \\Delta _{IR}\\cdot {\\delta ^2{\\Gamma }\\over \\delta \\phi ^a\\delta \\phi ^b}\\right]^{-1} \\Delta _{IR}^{-1} {\\partial \\Delta _{IR}\\over \\partial \\Lambda }\\,,$ where we have split off from the effective average action a canonical kinetic term ${\\tilde{\\Gamma }}[\\phi ] = \\Gamma [\\phi ] + {1\\over 2}\\int (\\partial _\\mu \\phi ^a)^2\\,,$ so that the total infrared regulated Legendre effective action is now given by $\\Gamma ^{tot} = {1\\over 2}\\phi ^a\\cdot \\Delta _{IR}^{-1}\\cdot \\phi _a+\\Gamma [\\phi ]\\,,$ where the propagator in momentum space is regulated by a multiplicative infrared regulating function $C_{IR}$ : $\\Delta _{IR} = {C_{IR}(q,\\Lambda )/ q^2} \\quad \\hbox{such that}\\quad q^2 C_{IR}^{-1} = q^2 +{\\cal R}\\,,$ and $\\Gamma [\\phi ]$ describes all the interactions (including mass terms).",
"Introducing an effective ultraviolet cutoff $C_{UV}$ via the sum rule [67], [2]: $C_{UV}+C_{IR}=1\\,,$ there is then an exact duality with Wilson/Polchinski's exact renormalisation group [1], [68]: ${\\partial S \\over \\partial \\Lambda }={1\\over 2}\\,{\\delta S\\over \\delta \\Phi ^a}\\cdot {\\partial \\Delta _{UV}\\over \\partial \\Lambda }\\cdot {\\delta S\\over \\delta \\Phi _a} -{1\\over 2}\\, {\\rm tr}\\,{\\partial \\Delta _{UV}\\over \\partial \\Lambda }\\cdot {\\delta ^2S\\over \\delta \\Phi ^a\\delta \\Phi _a}\\,,$ where $S[\\Phi ]$ is the interaction part of the Wilsonian effective action: $S^{tot}= \\frac{1}{2}\\Phi ^a.\\Delta _{UV}^{-1}.\\Phi _a +S[\\Phi ]\\,,$ and $\\Delta _{UV}(q,\\Lambda )=C_{UV}(q,\\Lambda )/q^2$ is the ultraviolet regularised propagator.",
"The flow equations (REF ) and (REF ) are therefore in fact two equivalent realizations of the same exact RG [67].",
"This exact duality is implemented via the generalised Legendre transform relation [67]: $S[\\Phi ]=\\Gamma [\\phi ]+\\frac{1}{2}(\\phi ^a-\\Phi ^a)\\cdot \\Delta _{IR}^{-1}\\cdot (\\phi _a-\\Phi _a)$ (see also ref.",
"[88]).",
"As shown in ref.",
"[42], remarkably this exact duality survives the Local Potential Approximation (LPA) if we employ the optimised cutoff [80], [81]: ${\\cal R} = (\\Lambda ^2-q^2)\\theta (\\Lambda ^2-q^2)\\,.$ See also refs.",
"[82], [89] for explorations of this.",
"From (REF ) we then have that $\\Delta _{IR} &=&1/\\Lambda ^2\\qquad {\\rm for}\\qquad q<\\Lambda \\,,\\nonumber \\\\\\Delta _{IR}&=&1/q^2 \\qquad \\, {\\rm for}\\qquad q>\\Lambda \\,,$ and thus from (REF ), $\\Delta _{UV} &=&{1 -q^2/\\Lambda ^2\\over q^2}\\qquad {\\rm for}\\qquad q<\\Lambda ,\\nonumber \\\\\\Delta _{UV}&=&0 \\qquad \\qquad \\quad \\ \\ \\, {\\rm for}\\qquad q>\\Lambda \\,.$ We see that the effective ultraviolet cutoff does indeed regulate as required inheriting its linear features and continuity from the properties of the optimised cutoff (REF ) [88].",
"At the LPA level effectively the fields are space-time independent.",
"In this case the relation between interactions (REF ), becomes an exact relation between the effective potentials: $U(\\Phi ) = V(\\phi ) +{1\\over 2}\\Lambda ^2 (\\phi -\\Phi )^2\\,,$ which in scaled variables is eqn.",
"(REF )."
]
] | 1605.06075 |
[
[
"The scaling of boson sampling experiments"
],
[
"Abstract Boson sampling is the problem of generating a quantum bit stream whose average is the permanent of a $n\\times n$ matrix.",
"The bitstream is created as the output of a prototype quantum computing device with $n$ input photons.",
"It is a fundamental challenge to verify boson sampling, and the question of how output count rates scale with matrix size $n$ is crucial.",
"Here we apply results from random matrix theory to establish scaling laws for average count rates in boson sampling experiments with arbitrary inputs and losses.",
"The results show that, even with losses included, verification of nonclassical behaviour at large $n$ values is indeed possible."
],
[
"3 The scaling of boson sampling experiments P. D. Drummond, B. Opanchuk, L. Rosales-Zárate, M. D. Reid Centre for Quantum and Optical Science, Swinburne University of Technology, Hawthorn, Victoria 3122, Australia P. J. Forrester Department of Mathematics and Statistics, ARC Centre of Excellence for Mathematical & Statistical Frontiers, The University of Melbourne, Victoria 3010, Australia.",
"Boson sampling is the problem of generating a quantum bit stream whose average is the permanent of a $n\\times n$ matrix.",
"The bitstream is created as the output of a prototype quantum computing device with $n$ input photons.",
"It is a fundamental challenge to verify boson sampling, and the question of how output count rates scale with matrix size $n$ is crucial.",
"Here we apply results from random matrix theory to establish scaling laws for average count rates in boson sampling experiments with arbitrary inputs and losses.",
"The results show that, even with losses included, verification of nonclassical behaviour at large $n$ values is indeed possible.",
"Much recent attention has been given to the application of multichannel linear photonic networks to solving computational tasks thought to be inaccessible to any classical computer.",
"Such devices are the prototypes of quantum computers [1], [2] and novel metrology devices [3], [4].",
"Exponentially hard problems that are not soluble with digital classical technology have many potential applications [5].",
"In particular, “BosonSampling” [6], [7] is the hard problem of how to generate a bitstream of photon-counts with the distribution of a unitary or Gaussian permanent.",
"This result is created as the output from a single photon input to each of $n$ distinct channels.",
"This is conjectured to be exponentially hard at large $n$ , while being relatively straightforward to implement physically.",
"It is widely appreciated that to verify the solution is correct is an important and significant challenge [8], [9], [10], [11], [12], [13], [14], [15].",
"The task is to measure the coincidence rates of counting $n$ photons in $n$ output channels and to confirm that they correspond to the modulus squared of an $n\\times n$ sub-permanent of a unitary matrix.",
"The matrices are under experimental control [16], [17], [18], [19], and an average over a random ensemble of them is necessary to verify the experiments.",
"A number of known strategies exist.",
"Yet, since permanents are intrinsically exponentially hard to compute [20], it is nontrivial to verify the output is correct [7], [8] at large $n$ values.",
"This computational issue makes it difficult to estimate how count rates scale unless averaged over all unitaries.",
"Understanding scaling is essential because count rates decline exponentially fast as $n$ increases, requiring a strategy that overcomes this.",
"In this Letter, we solve the average scaling problem for arbitrary inputs and losses.",
"We obtain the maximum scaling of improvements that are possible with recent channel grouping strategies [9].",
"To achieve this, we combine the generalized P-representation with methods from random matrix theory to obtain averages over unitary transformations.",
"This allows to describe realistic photonic network experiments, with arbitrary inputs, outputs, and losses.",
"Such losses in boson sampling have recently been investigated elsewhere [21], [22].",
"The scaling improvement with channel grouping depends on the channel occupation ratio, $k=m/n$ , reaching over a hundred orders of magnitude at $k=6$ , $n=100$ .",
"This is well beyond the capability of any classical, exact computation of matrix permanents.",
"Our results show that boson-sampling verification with such large $n$ values is possible provided high efficiency detectors are available.",
"Scaling issues like this arising in random matrix theory are widespread [23], as averages over unitaries are fundamental to quantum physics.",
"We note an unexpected analogy with the statistics of a classical device for generating random counts.",
"On averaging over all unitaries, the probability of $n$ single-photon counts in $n$ preselected channels — a quantum Galton's board [24] — is identical to a classical Galton's board.",
"The only difference is that there are now $n-1$ additional virtual channels, which describe multi-photon events in an output mode.",
"These extra channels can be thought of as non-classical communication channels.",
"Such channel capacity improvements are known from quantum communication theory [25], and are closely related to Arkhipov and Kuperberg's “birthday paradox” for bosons [26].",
"We start with a result [27], [28] from quantum optics: any bosonic correlation function is obtainable from the normally-ordered quantum characteristic function, $\\chi \\left({\\xi }\\right)=\\left\\langle :e^{{\\xi }\\cdot \\hat{{a}}^{\\dagger }-{\\xi }^{*}\\cdot \\hat{{a}}}:\\right\\rangle .$ Here, $\\left\\langle \\hat{O}\\right\\rangle \\equiv Tr\\left[\\hat{\\rho }\\hat{O}\\right]$ is a quantum average, which we calculate using a generalized P-representation [29].",
"This approach extends the Glauber P-function [30], giving a distribution $P\\left({\\alpha },{\\beta }\\right)$ over are two $m$ -component complex vectors, which exists for any $m-$ mode bosonic state $\\hat{\\rho }$ .",
"The quantum characteristic is then obtained from [31] $\\chi \\left({\\xi }\\right)=\\int P\\left({\\alpha },{\\beta }\\right)\\chi \\left({\\xi }|{\\alpha },{\\beta }\\right)d\\mu \\left({\\alpha },{\\beta }\\right)\\,,$ where $d\\mu \\left({\\alpha },{\\beta }\\right)$ is the integration measure, and $\\chi \\left({\\xi }|{\\alpha },{\\beta }\\right)\\equiv \\exp \\left({\\xi }\\cdot {\\beta }-{\\xi }^{*}\\cdot {\\alpha }\\right)$ is the conditional characteristic function for ${\\xi }$ given a particular quantum phase-space trajectory ${\\alpha },{\\beta }$ .",
"Transmission through a linear network changes the input density matrix $\\hat{\\rho }^{(\\mathrm {in})}$ to an output density matrix $\\hat{\\rho }^{(\\mathrm {out})}$ .",
"An amplitude transmission matrix $T$ transforms the coherent amplitudes [32], so that ${\\alpha }^{(\\mathrm {out})},\\,{\\beta }^{(\\mathrm {out})}=T{\\alpha },\\,T^{*}{\\beta }$ .",
"The output characteristic $\\chi ^{(\\mathrm {out})}$ now depends on the input phase-space amplitude ${\\alpha },{\\beta }$ in an intuitively understandable way: $\\chi ^{(\\mathrm {out})}\\left({\\xi }|{\\alpha },{\\beta }\\right)=e^{{\\xi }\\cdot {T}^{*}{\\beta }-{\\xi }^{*}\\cdot {T}{\\alpha }}\\,.$ To calculate the average scaling behavior, we consider the case of $T=\\sqrt{t}U$ , in which the unitary mode transformation $U$ of the photonic network is combined with an absorptive transmission coefficient $t$ , representing losses and detector inefficiencies.",
"We compute the average output correlations over all possible unitaries, indicated by $\\left\\langle \\right\\rangle _{U}$ , from random matrix theory [33].",
"This allows one to evaluate averages of exponentials of the unitary matrices in the conditional characteristic function, $\\chi ^{(out)}$ of Eq.",
"(REF ).",
"The result is an averaged conditional characteristic $\\left\\langle \\chi ^{(\\mathrm {out})}\\left({\\xi }|{\\alpha },{\\beta }\\right)\\right\\rangle _{U}=\\left(m-1\\right)!\\sum _{j=0}^{\\infty }\\frac{\\left[-t\\left|{\\xi }\\right|^{2}{\\beta }\\cdot {\\alpha }\\right]^{j}}{j!\\left(m-1+j\\right)!",
"}\\,.$ Inserting this unitary average in Eq.",
"(REF ) gives an exact solution for any averaged observable in the photonic network, with arbitrary inputs and losses.",
"The output photon statistics depend on ${\\beta }\\cdot {\\alpha }$ , which is the phase-space equivalent of the total input photon number $\\hat{N}$ .",
"As a result, the characteristic function after unitary and quantum averaging is: $\\left\\langle \\chi ^{(\\mathrm {out})}\\left({\\xi }\\right)\\right\\rangle _{U}=\\left(m-1\\right)!\\sum _{j=0}^{\\infty }\\frac{\\left(-t\\left|{\\xi }\\right|^{2}\\right)^{j}\\left\\langle :\\hat{N}^{j}:\\right\\rangle }{j!\\left(m-1+j\\right)!",
"}\\,.$ The effect of unitary averaging is such that all output channel and phase information is lost, since the characteristic function now only depends on $\\left|{\\xi }\\right|^{2}$ .",
"This also shows that all output averages are obtained solely from the normally ordered input photon number moments, $\\left\\langle :\\hat{N}^{j}:\\right\\rangle $ , regardless of which input channels are used.",
"As a common example, for an input $n$ -photon number state $\\left\\langle :\\hat{N}^{j}:\\right\\rangle =n!/(n-j)!$ , so the sum in Eq.",
"(REF ) vanishes for $j>n$ .",
"This is a consequence of photon-number conservation and the purely absorptive loss reservoirs.",
"Only the photon number observables are non-vanishing after unitary phase-averaging.",
"These are most readily obtained from taking derivatives of the photon-number generating function [34], $G\\left({\\gamma }\\right)\\equiv Tr\\left(\\hat{\\rho }\\prod _{i}\\left(1-\\gamma _{i}\\right)^{\\hat{n}_{i}}\\right)\\,.$ Using the relationship between photon-number generator and characteristic function [35], we find that: $G\\left({\\gamma }\\right)\\equiv \\left(m-1\\right)!\\sum _{j=0}^{\\infty }\\frac{\\left(-t\\right)^{j}\\left\\langle :\\hat{N}^{j}:\\right\\rangle }{\\left(m-1+j\\right)!",
"}\\sum _{\\sum \\mathbf {j}=j}\\gamma _{1}^{j_{i}}\\ldots \\gamma _{m}^{j_{m}}$ This result is completely general for a photonic network with arbitrary inputs, outputs and losses.",
"A case of special interest is $P_{n|m}$ , the probability of observing 1 photon in each of $n$ channels, given an $n$ -photon input and an $m$ -mode network.",
"This is found on taking $n$ first derivatives of $G\\left({\\gamma }\\right)$ , so that: $P_{n|m}=\\frac{t^{n}\\left(m-1\\right)!n!}{\\left(m-1+n\\right)!",
"}=t^{n}\\left[C_{n}^{m+n-1}\\right]^{-1}\\,$ We now wish to relate these results to the permanent of the transmission matrix $T$ .",
"The permanent is a sum over all permutations $\\sigma $ of the matrix indices of the product of $n$ terms, in which neither row nor column indices are repeated.",
"It is an exponentially hard object to compute, and is one of the fundamental quantities addressed in boson sampling theory and in linear optical networks [36], [37].",
"For a pure, unitary state evolution, the photon counting probability is the permanent of a sub-matrix, $\\left\\langle \\left|\\mathrm {perm}(U_{n|m})\\right|^{2}\\right\\rangle $ , where $U_{n|m}$ is any $n\\times n$ sub-matrix of $U$ [38].",
"More generally, we replace $U_{n|m}\\rightarrow T_{n|m}$ , so as to include losses.",
"The permanent of a sub-matrix of $T$ is obtainable [39] from the permanental polynomial, which has similarities with moment generating function.",
"This is given by: $p(x)=\\mathrm {perm}(xI-T)\\equiv \\sum _{n=0}^{m}b_{n}x^{m-n}\\,.$ Next, we consider how to compute the unitary average of products of the permanents, using$\\left\\langle .\\right\\rangle _{U}$ to indicate averages over the circular unitary ensemble, with a Haar measure.",
"This is achieved through an elegant result in random matrix theory [33].",
"The unitary average of permanental polynomials in Eq.",
"(REF ) is: $\\left\\langle p(x)p(y)^{*}\\right\\rangle _{U} & = & m!",
"(m\\text{—}1)!\\sum _{j=0}^{m}\\frac{t^{j}\\left(xy^{*}\\right)^{m-j}}{\\left(m-j\\right)!(m-1+j)!",
"}\\,.$ If $\\omega ^{(n)}=(\\omega _{1},\\ldots \\omega _{n})$ where $\\omega _{1}<\\omega _{2}\\ldots <\\omega _{n}$ , we can define an $n\\times n$ sub-matrix $T_{{\\omega }}\\equiv T_{\\omega _{i}\\omega _{j}}$ .",
"The coefficients of this polynomial are simply the sums over the permanents of all possible distinct sub-matrices: $b_{n}=(-1)^{n}\\sum _{{\\omega }^{(n)}}\\mathrm {perm}(T_{{\\omega }^{(n)}})\\,.$ The number of sub-matrices in the sum has a multiplicity given by the binomial coefficient $C_{n}^{m}=m!/\\left(n!",
"(m-n)!\\right)$ , corresponding to the different ways to choose the distinct indices $\\omega _{j}$ .",
"These indices have a straightforward physical interpretation: they are the channel numbers of the input or output modes of the photonic device.",
"Expanding the product average, and noting that products of different sub-matrices vanish under unitary ensemble averaging, we consider the sub-permanents with $j=n$ .",
"The average sum over all possible sub-permanents of this size is a ratio of two binomial coefficients, which we define as $R_{n|m}$ : $\\sum _{{\\omega }^{(n)}}\\left\\langle \\left|\\mathrm {perm}(T_{{\\omega }^{(n)}})\\right|^{2}\\right\\rangle _{U}=R_{n|m}=\\frac{t^{n}m!(m\\text{—}1)!}{(m-n)!(m+n-1)!",
"}\\,.$ All the averages are the same for every sub-matrix.",
"Therefore, we can replace $T_{{\\omega }^{(n)}}$ by any particular sub-matrix $T_{n|m}$ , and make use of the sub-matrix multiplicity.",
"The final result is the same as in Eq.",
"(REF ), except expressed using permanents, so that $P_{n|m}=\\left\\langle \\left|\\mathrm {perm}(T_{n|m})\\right|^{2}\\right\\rangle _{U}$ .",
"In the lossless limit of $t=1$ , this result agrees with the “bosonic birthday paradox” of Arkhipov and Kuperberg [26], derived using different techniques.",
"We turn next to some limiting cases for large $n$ , where $\\log P_{n|m}\\approx n\\epsilon $ for a scaling exponent $\\epsilon $ .",
"If the matrix is the entire transmission matrix, then $n=m$ .",
"The scaling exponent is $\\epsilon =\\log \\left(t/4\\right)$ , and $\\log P_{n|m}\\,\\mathop {\\sim }\\limits _{n\\rightarrow \\infty }\\,\\,n\\epsilon +\\frac{1}{2}\\log \\left[4\\pi n\\right]\\,.$ This result generalizes one of Fyodorov [33].",
"Next, take $n\\ll m$ , so that $k\\gg 1$ .",
"Standard methods for approximating a binomial coefficient in this limit give the scaling exponent $\\epsilon =\\log \\left(t/\\left(k+1/2\\right)\\right)-1$ , where $k=m/n$ , so that: $\\log P_{n|m}\\,\\,\\mathop {\\sim }\\limits _{n\\rightarrow \\infty }\\,\\,n\\epsilon +\\frac{1}{2}\\log \\left[2\\pi n\\right]\\,.$ This is consistent with the fact that for large $k$ , unitary sub-matrices reduce to matrices with complex Gaussian random entries [40], [7].",
"For the general case, the scaling exponent is $\\epsilon =\\log t+k\\log k-(1+k)\\log (1+k)$ , and the asymptotic result is: $\\log P_{n|m}\\,\\mathop {\\sim }\\limits _{n\\rightarrow \\infty }\\,\\,n\\epsilon +\\frac{1}{2}\\log \\left[2\\pi n(1+1/k)\\right]$ The exact result is plotted in Figure REF , with different values of $k=m/n$ .",
"The power law is so close that it cannot be told apart from the exact result on this scale.",
"For all numerical results, we choose $t=1$ , since results in more realistic cases with losses are readily obtained by adding $\\log t$ to the scaling exponents.",
"At large $k$ values, one obtains the Gaussian limit of Eq.",
"(REF ).",
"This gives an increasingly negative exponent, with exponentially small count-rates.",
"Figure: Average sub-unitary permanent squared P n|m P_{n|m} with t=1t=1 for k=1,2,4,6k=1,2,4,6,with k=1k=1 at the top and k=6k=6 at the bottom.Next, we show the result of an actual average over a finite, random ensemble with $S=40000$ unitary samples.",
"In the graphs, for $k=m/n=2$ and $n\\le 25$ we give the relative errorWes in the asymptotic approximation and a numerical average, compared to the exact solution.",
"To estimate statistical error-bars, we take an ensemble $S$ , and divide it into $\\sqrt{S}$ sub-ensembles, giving sub-ensemble means which are approximately Gaussian from the central limit theorem.",
"These are averaged, and the error in the mean $\\sigma _{m}$ is obtained using standard techniques.",
"The error-bars are given in the plots as $\\pm \\sigma _{m}$ .",
"The results agree with the exact equation with a relative error comparable to the sampling error-bars.",
"For the plotted ratio of $k=2$ , the errors are around $\\pm 1\\%$ for over twenty orders of magnitude range of values, and we see that the relative sampling error over unitaries is independent of matrix size for $n\\ge 10$ .",
"Figure: Relative sampling errors in the sub-unitary permanent squared P n|m P_{n|m}with t=1t=1 for k=2k=2.",
"Numerical results for 40,00040,000 random unitaries(solid lines with error bars) are compared to exact results (dashedlines) and the asymptotic power law form (dotted line) for up to n=25n=25.While the scaling is better than the Gaussian limit of $k\\gg 1$ , it is still a problem for boson sampling verification.",
"Even in the unlikely case of perfect efficiency, the average permanent for a photon number of $n=24$ is $\\sim 10^{-18}$ with a sub-matrix ratio of $k=2$ .",
"This is the probability of a coincidence count, so one needs $10^{18}$ samples to obtain one count for a typical unitary.",
"At a repetition rate of $10^{12}\\,\\mathrm {Hz}$ , which is the maximum one can reasonably expect from the technology, one would require $10^{6}\\,\\mathrm {s}$ of measurement time for each count.",
"The reason for this is simple: many-body complexity.",
"There are too many quantum states possible.",
"Monitoring the coincidence channels for one many-body state takes too long, even though it is these counts that are of interest.",
"This property, although making verification hard, is the most interesting feature of these experiments.",
"They give a uniquely controllable access to a laboratory system in which one can unravel the complexity of a many-body system, in order to examine each state, or arbitrary combinations of the quantum states.",
"Accordingly, suppose we consider what happens when one groups multiple output channels together, by using logic gate operations on the detector circuits, as in a recent, pioneering experiment [9].",
"A large number of randomized sets of channels can be combined, to obtain a unique distinguishing signature for each unitary.",
"This strategy has important advantages over previous proposals.",
"It increases count-rates by exponentially large factors, and may allow a test of the unitary output bitstream for matrices with permanents larger than $n=40$ , beyond the classical computation limits.",
"Yet, it is not restricted to any particular unitary, reducing the chance that the device may only work in special cases.",
"One can also use the strategy to efficiently test for null counts, in edge cases like Fourier matrices [14].",
"Our scaling laws predict the upper bound of the count-rate gain that can achieved through sub-matrix multiplicity.",
"The upper bound from channel grouping is given by $R_{n|m}$ , in Eq.",
"(REF ).",
"This has an exponent of $\\lambda =\\log t+2k\\log k-(k-1)\\log (k-1)-(k+1)\\log (k+1)$ , so that the scaling is: $\\log R_{n|m}\\,\\,\\mathop {\\sim }\\limits _{n\\rightarrow \\infty }\\,\\,n\\lambda +\\frac{1}{2}\\log \\left[\\frac{k+1}{k-1}\\right]$ For the Gaussian limit of $n,k\\gg 1$ , one finds that $\\lambda \\rightarrow \\log t-1/k$ .",
"Unlike the single coincidence case, the grouped channel count rate is maximized for large $k$ , rather than minimized as before.",
"The corresponding upper-bound result is plotted in Figure REF , for different values of $k=m/n$ , again taking $t=1$ for simplicity.",
"At at $k=6$ , and $n=100$ , there is now a dramatic increase of more than 100 orders of magnitude in total count-rate.",
"The improvement is greatest for large $k$ values, which are the cases of most interest.",
"High $n$ verification still requires total efficiencies of above $90\\%$ , which is possible as the technology improves.",
"Figure: Upper bound on count-rates R n m R_{n}^{m} for nn photons occurring,without bunching, in any nn output channels, for k=2,...6k=2,\\ldots 6,with k=6k=6 at the top.It is an intriguing result, mathematically and physically, that the quantum Galton's board has a close relationship with the binomial coefficients normally found in classical combinatorics.",
"How can we interpret this result?",
"Suppose that we replaced the photonic network, equivalent to a unitary transformation, by an updated Galton's board device, which simply switched the photons from the $n$ input channels to the $n$ output channels in a random way.",
"This would not involve interference, and would have a similar behavior to a mechanical board, apart from an increased number of inputs.",
"Under these conditions, the average probability of all counts occurring in a preselected set of $n$ output channels is an inverse binomial $\\left[C_{n}^{m}\\right]^{-1}$ .",
"We now see a truly remarkable result.",
"Apart from losses, the quantum Galton's board has, on averaging over all unitaries, identical output coincidence probabilities to a classical Galton's board with a number of channels given by $\\tilde{m}=m+n-1$ .",
"In other words, the fact that photons can bunch — one channel may carry up to $n$ photons, after all — has a similar effect on the output statistics as if the device were classical, but with $n-1$ additional channels available.",
"Needless to say, these virtual channels do not exist.",
"They represent, on average, the additional output possibilities available owing to the fact that several bosons can occupy the same mode, and hence occur in the same channel.",
"This is the large-scale consequence of the famous Hong-Ou-Mandel effect in quantum optics [41], [42].",
"It is this additional, virtual channel capacity that allows more quantum information to be transmitted in a quantum photonic network than is feasible if each channel was used separately, with one bit per channel.",
"Such extra capacity is a fundamental and important property of quantum photonic networks [25].",
"In summary, the unitary average of moduli of sub-permanents has remarkable properties.",
"Each permanent itself has no simple closed-form expression, and one might imagine that taking an average over all possible unitaries would only make things harder.",
"Yet the average over the unitary ensemble is just as simple as the closed form expression applicable to a classical Galton's board.",
"As well as improved measurements, verification for an individual large unitary requires improved computational methods, as standard methods take exponentially long times.",
"These results will be treated elsewhere.",
"In addition to understanding the scaling laws for verifying boson sampling, our results may suggest how random matrix theory can be applied to other, large-scale quantum technologies.",
"We wish to thank the Australian Research Council for their funding support."
]
] | 1605.05796 |
[
[
"Towards the fundamental spectrum of the Quantum Yang-Mills Theory"
],
[
"Abstract In this work we focus on the quantum Einstein-Yang-Mills sector quantised by the methods of Loop Quantum Gravity (LQG).",
"We point out the improved UV behaviour of the coupled system as compared to pure quantum Yang-Mills theory on a fixed, classical background spacetime as was considered in a seminal work by Kogut and Susskind.",
"Furthermore, we develop a calculational scheme by which the fundamental spectrum of the quantum Yang-Mills Hamiltonian can be computed in principle and by which one can make contact to the Wilsonian renormalization group, possibly purely within the Hamiltonian framework.",
"Finally, we comment on the relation of the fundamental spectrum to that of pure Yang-Mills theory on a (flat) classical spacetime."
],
[
"Introduction",
"The Hamiltonian approach to pure quantum Yang-Mills theory on Minkowski space was much developed by Kogut and Susskind [1].",
"These authors regularised the classical expression for the Yang-Mills Hamiltonian on a regular spatial lattice of cubic topology embedded in $\\mathbb {R}^3$ , which comes with a lattice length parameter $\\epsilon $ as measured by the spatial Euclidean background metric induced by the Minkowski metric on spatial hypersurfaces of Minkowski space.",
"The quantum Hamiltonian was written in terms of non-abelian fluxes through the faces of the cubic cell complex dual to the lattice for the electric degrees of freedom and in terms of non-abelian holonomies along the plaquette loops of the lattice.",
"Furthermore, those authors assumed a representation of holonomies and fluxes on a Hilbert space of square integrable functions of the magnetic loop functions just introduced, where the natural Haar measure on the compact gauge group is used in order to define the Hilbert space measure.",
"While well defined at finite $\\epsilon $ , the necessary continuum limit $\\epsilon \\rightarrow 0$ is problematic in this approach: Namely, the regularized Hamiltonian involves an inverse power of $\\epsilon $ and thus blows up at fixed Yang-Mills coupling.",
"This leads to the conclusion that the Yang-Mills coupling entering the Hamiltonian is to be considered a bare coupling that must be renormalized suitably in the continuum limit.",
"Since the renormalization is, arguably, easier to study in the path integral formulation, the Hamiltonian approach to quantum Yang-Mills theory was basically dropped and research focused on the functional integral approach, whose underlying mathematical framework is the constructive Euclidean program [2], [3], [4], [5], [6], [7].",
"Starting from the Euclidean action, not the Hamiltonian, hence involves an additional integral and thus in 4 spacetime dimensions does not involve $\\epsilon $ explicitly.",
"The well established and very active research field of Lattice Quantum chromodynamics (LQCD) is the practical implementation of that program and has produced many spectacular results, see e.g.",
"[8], [9], yet the existence of pure quantum Yang-Mills theory has not been proven.",
"In fact, the Clay Mathematical Institute http://www.claymath.org/millenium-problems/yang-mills-and-mass-gap has devoted one of its millennium prizes to this research topic.",
"To circumvent these problems this paper does not deal with the Euclidean formulation at all.",
"Futhermore, we will leave the realm of QFT on curved spacetime [10], [11], [12], [13] completely and pass to quantum gravity, because we wish to examine here the old idea that quantum gravity itself resolves the UV divergences of QFT.",
"We do this in the Hamiltonian approach to quantum gravity, one incarnation of which is Loop Quantum Gravity (LQG) [14], [15], [16].",
"This approach is ideally suited to Yang-Mills theory, because the gravitational field, in its canonical formulation, can be viewed as a Yang-Mills theory for the gauge group SU(2) with a very complicated interaction.",
"Thus the quantisation methods developed for Yang-Mills fields, in fact pioneered by Kogut and Susskind, can also be applied to the gravitational degrees of freedom, as has been done in [17].",
"Indeed, a rigorous Hilbert space representation can be found for the so called holonomy flux algebra, in fact for any compact gauge group and any spacetime dimension, which consists of holonomies along one dimensional paths and non-Abelian fluxes through two dimensional surfaces (in 3+1 spacetime dimensions).",
"This is in fact very similar to the Kogut-Susskind program, but the difference is that in LQG there is no fixed lattice and dual cell complex, there is also no lattice regulator $\\epsilon $ at all.",
"Rather, one considers all paths and all surfaces in one big Hilbert space, that is to say, one considers all graphs and dual cell complexes.",
"LQG is therefore a continuum theory without a lattice cut-off.",
"We will see that in the corresponding quantum operator the factor $1/\\epsilon $ of the Kogut-Susskind Hamiltonian is replaced by the $1/\\ell _P$ where $\\ell _P$ is the Planck length.",
"At that level therefore, there is no problem in taking the continuum limit.",
"However, renormalisation group ideas are still important as we see later on.",
"Just in order to avoid possible confusion from the outset, we mention here that LQG comes in two versions.",
"In the first version one solves the constraints of the theory, which arise due to the spacetime diffeomorphism invariance of Einstein's theory, in the quantum theory [18], [19].",
"In the second version one solves those constraints classically by gauge fixing the freedom to choose coordinates in terms of scalar matter fields (see e.g.",
"[20], [21], [22], [23]).",
"These two approaches are technically and conceptually very different, because in the first version the primary task is to solve the quantum constraints and to supply a Hilbert space structure on the resulting space of (distributional) solutions and it is a non-trivial task to find appropriate gauge invariant observables acting on it.",
"There is no Hamiltonian in this first approach, because time translations are regarded as gauge transformations.",
"In the second approach these tasks are already implemented classically.",
"Furthermore, the classical construction automatically supplies a Hamiltonian that generates time evolution.",
"In this paper we will therefore follow the second route, specifically the choice of scalar matter considered in [24], [28] as this brings us maximally close to the situation of pure Yang-Mills theory on Minkowski space.",
"The LQG Hilbert space, which was originally designed for the first approach, is necessarily non separable.",
"This comes about because one considers the huge algebra of all fluxes and all holonomies, which in turn are needed if one wishes to implement the (spatial) diffeomorphism invariance of the theory in a (cyclic) representation of the holonomy - flux algebra [29], [30].",
"On the other hand, classically, far fewer functions on the phase space would suffice in order to separate all of its points, that is to say, much fewer paths and surfaces would suffice.",
"In [25], [26], [27], [28] the observation was made, that - since in the second approach one has fixed the (spatial) diffeomorphism invariance of the theory - one may indeed restrict to a much smaller algebra.",
"For instance, if the topology of spacetime is that of $\\mathbb {R}^4$ then it suffices to consider rectangular paths and surfaces along the coordinate axes and planes, respectively.",
"A further reduction of the number of degrees of freedom is obtained by passing to an abstract infinite graph and dual cell complex respectively, which have no information about their embedding into $\\mathbb {R}^3$ .",
"The quantum theory is then formulated in terms of these abstract elementary holonomy and flux operators.",
"The embedding scale reappears in the semiclassical limit in terms of coherent states [31] for the gravitational degrees of freedom and can be chosen as small as one wishes.",
"In this paper we therefore consider the approach of [28] to Einstein-Yang-Mills theory on the differential manifold $\\mathbb {R}^4$ in the gauge fixed version of LQGThat is, the coordinate freedom is fixed but not the Yang-Mills like gauge freedom.",
"with scalar matter content and focus on the Yang-Mills contribution to the Hamiltonian, which then in the classical theory simply reads: $ H=\\frac{1}{2Q^2} \\;\\int _{\\mathbb {R}^3}\\; d^3x\\; \\frac{q_{ab}}{\\sqrt{\\det (q)}}[{\\rm Tr}(\\underline{E}^a \\underline{E}^b)+{\\rm Tr}(\\underline{B}^a \\underline{B}^b)]$ Here $\\underline{E},\\underline{B}$ denote the electric and magnetic Yang-Mills field, $Q$ is the Yang-Mills coupling constant and $q_{ab}$ is the induced spatial metric on the Cauchy surface $\\mathbb {R}^3$ .",
"The spatial indices are $a,b,c, ..=1,2,3$ and the traces are taken in the adjoint representation of the Lie algebra $\\mathfrak {g}$ of the Yang-Mills gauge group $G$ , e.g.",
"su(N) for $G=SU(N)$ .",
"The architecture of this paper is as follows: In section we will briefly review the quantisation of (REF ), more details can be found in [17], [28].",
"We also review the essentials of [1] and compare these two theories.",
"Section reviews useful facts about the representation theory of SU(3) (QCD gauge group) needed in sections and , while analogous knowledge for SU(2) (gravitational gauge group) are shifted to the appendix.",
"In section we compute basic building blocks necessary in order to compute the background spectrum of (REF ) with fixed Minkowski background metric, that is $q_{ab} = \\delta _{ab}$ , on a lattice of size $\\epsilon $ , i.e.",
"we treat the Kogut & Susskind situation.",
"In section we do the same, but with $q_{ab}$ being a quantum operator on the LQG Hilbert space.",
"The calculational steps performed here are the preparation for computing the fundamental spectrum of $H$ on the tensor product Hilbert space corresponding to both geometry and matter degrees of freedom.",
"In section we summarize our findings and elucidate the necessary steps for our future research."
],
[
"Review of Einstein-Yang-Mills-Theory",
"In this chapter we recap elements of the classical and quantum Einstein-Yang-Mills theories.",
"In the first section we review the classical canonical formulation and in the second we formulate the quantum theory using the techniques of Loop Quantum Gravity (LQG).",
"We also review the derivation of the Kogut-Susskind lattice Hamiltonian on Minkowski space.",
"Notice that our quantisation makes use of the presence of additional scalar matter fields that do not explicitly appear in the Hamiltonian since they serve to fix the general coordinate freedom and therefore are “Higgsed away”.",
"See [20] for all the details."
],
[
"Classical Einstein-Yang-Mills-Theory",
"The Yang-Mills action for a unitary gauge group $G$ in general relativity is: $S_{YM} = -\\frac{1}{4Q^2}\\underset{M}{\\int } d^4x \\sqrt{\\left| det(g) \\right|} g^{\\mu \\nu } g^{\\rho \\sigma } \\underline{F}^I_{\\mu \\rho }\\underline{F}^I_{\\nu \\sigma }$ where $\\underline{F}$ is the curvature of the $G$ -connection $\\underline{A}$ and $Q$ is the coupling constant and $g^{\\mu \\nu }$ is the metric on the manifold $M$ .",
"The aim of this chapter is to cast this action into canonical form.",
"This is done using the ADM-formalism, the details of which can be found in [32].",
"The idea is to assume that $M$ may be splitted as $M=\\mathbb {R}\\times S$ .",
"This foliation into space-like hypersurfaces allows the replacement of the ten components of the spacetime metric by the six components of the induced Riemann metric $q_{ab}$ of $S$ and the three components of the shift vector $N_a$ and the lapse function $N$ .",
"Also the co-triad field $e^i_a$ is transformed to the densitized triad $E^a_i = \\frac{1}{2}\\epsilon ^{abc}\\epsilon _{ijk}e^j_be^k_c = \\sqrt{det(q)} e^a_i$ which serves as the canonical pair on the gravitational phase space together with the extrinsic curvature: $K_{ab}=sgn\\left(det(e^j_c)\\right)K^i_ae^i_b$ REF and REF , together with the connection $A^i_a = \\Gamma ^i_a+K^i_a$ form the Asthekar-Barbero-Variables [33], [34], [35], [36], where $\\Gamma ^i_a$ is the spin-connection of $e^i_a$ .",
"In conjunction with the canonical pair from Yang-Mills-theory $\\left(\\underline{A}^i_a,\\frac{1}{Q^2}\\underline{E}^a_i\\right)$ , where the first is the above mentioned $G$ -connection and the second the associated electric field, one is set up to start working on $SU(2)\\times G$ .",
"Due to the gauge fixing dynamically induced by additional matter fields, lapse and shift get frozen to $N=1, \\; N^a=0$ respectively.",
"After performing the Legendre transformation, one finds [14] $ S_{YM} = \\frac{1}{Q^2}\\underset{\\mathbb {R}}{\\int }dt\\underset{S}{\\int }d^3x\\left(\\underline{\\dot{A}}_a^I\\underline{E}^a_I-\\left(-\\underline{A}^I_t\\underline{D}_a\\underline{E}^a_I+N^a\\underline{F}^I_{ab}\\underline{E}^b_I+\\frac{q_{ab}}{2\\sqrt{det(q)}}\\left(\\underline{E}^a_I\\underline{E}^b_I+\\underline{B}^a_I\\underline{B}^b_I\\right)\\right)\\right)$ where $\\underline{B}^a_I=\\epsilon ^{abc}\\underline{F}^I_{bc}$ and $\\underline{D}_a$ acts like the Levi-Civita connection on tensor indices.",
"The contributions to the spatial diffeomorphism constraint and the Hamiltonian can be directly read off: the Hamiltonian is $ H_{YM} = \\frac{q_{ab}}{2Q^2\\sqrt{det(q)}}\\left(\\underline{E}^a_I\\underline{E}^b_I+\\underline{B}^a_I\\underline{B}^b_I\\right)$"
],
[
"Quantum Einstein-Yang-Mills-Theory",
"In this chapter one will construct a Hamiltonian for a Quantum Einstein-Yang-Mills-Theory.",
"As already stated, the methods of quantisation (REF ) will be those of Loop Quantum Gravity.",
"We will present the construction separately for the Einstein-Term and the Yang-Mills-Term.",
"Finally we show how the classical Kogut-Susskind-Hamiltonian emerges from the theory in the limit of a flat spacetime.",
"Let us stress again, that we are working in the framework of deparametrised models: a suitable gauge fixing leads to a reduced phase spacetime that (when quantised via the methods of LQG) provides a model where all the constraints are solved, all operators are spacetime diffeomorphism invariant and physical states respectively.",
"In this formulation there is no Hamiltonian constraint, but a Hamiltonian operator [20], [26], [28], [37].",
"Also the idea of Algebraic Quantum Gravity is used, where we work solely on abstract graphs, which do not care about their embedding.",
"Instead only the nodes and their connection among themselves are of interest.",
"In our case the graphs are of cubic topology (i.e.",
"a general vertex will have six edges adjacent to it) which is very alike to the situation in Lattice Gauge Theory.",
"In this manner we follow the proposal of [25], meaning that physics now happens on such a given graph leaving it invariant, a feature in which AQG differs from the first route of LQG, where there is no Hamiltonian but an infinite number of constraints which must commute with each other on the kernel of the diffeomorphism constraint.",
"The only known way to achieve this without anomalies in this sense is to let the Hamiltonian constraint act by adding new edges.",
"By contrast, with only one Hamiltonian, there is no anomaly to worry about anymore and the quantization of the Hamiltonian can be done in the way that is customary in lattice gauge theory.",
"With every edge $e$ one associates an element $A(e)$ of SU(2) for the gravitational sector and an element $\\underline{A}(e)$ of the Yang-Mills gauge group $G$ , as well as elements $E(e)$ , $\\underline{E}(e)$ , respectively, for the corresponding Lie algebra.",
"Hence in both cases there are the following algebraic relations, with $Q$ being the coupling constant and $f_{jkl}$ the structure constant of SU(2) or $G$ respectively: $ \\left[A(e),A(e^{\\prime })\\right]=0$ $ \\left[E_j(e),A(e^{\\prime })\\right] = i\\hbar Q^2\\delta _{e,e^{\\prime }}\\tau _j/2 A(e)$ $ \\left[E_j(e), E_k(e^{\\prime })\\right] = i \\hbar Q^2\\delta _{e,e^{\\prime }}f_{jkl}E_l(e^{\\prime })$ A nice representation of this algebra is the Infinite Tensor Product Hilbert space $\\mathcal {H}=\\underset{e}{\\bigotimes } \\mathcal {H}_e$ , where on every edge $\\mathcal {H}_e=L^2\\left(G,d\\mu _H(G)\\right)\\otimes L^2\\left(SU(2),d\\mu _H(SU(2)\\right)$ [28].",
"Here $A(e)$ is a unitary matrix valued operator and $E(e)$ an essential self-adjoint derivation operator.",
"So e.g.",
"the action of $E(e)$ on a function $f_e$ on $e$ is: $ E_j(e)f_e (h)=i\\hbar Q^2 \\frac{d}{ds}\\left(f_e\\left(e^{s\\tau _j/2}h\\right)\\right)_{s=0}$ where $\\tau _j$ are the generators of the corresponding Lie algebra.",
"This choice gives a parallel to the concept of LQG.",
"And although there is no strict derivation of an algebraic Hamiltonian, it appears sensible to take the quantum version of the operators derived in the LQG framework and use them in AQG.",
"The derivation of those in LQG was first performed in [18], [19] for the gravitational sector and in [17] for the Yang-Mills sector).",
"Considering all this, the gravitational Hamiltonian is set to: $\\hat{H}_{Einstein}(v)= \\hat{S}_E^{(1/2)}(v)-2(1+\\gamma ^2)\\hat{T}(v)$ with $\\hat{S}_E^{(r)}(v)=\\frac{1}{N_v}\\underset{e_1\\cap e_2\\cap e_3 =v}{\\sum }\\frac{\\epsilon (e_1,e_2,e_3)}{\\left|L(v,e_1,e_2\\right|}\\underset{\\beta \\in L(v,e_1,e_2)}{\\sum }tr\\left(\\left(\\hat{A}(\\beta )-\\hat{A}(\\beta )^{-1}\\right)A(e_3)\\left[A(e_3)^{-1},\\hat{V}_v^r\\right]\\right)$ $\\hat{T}(v)=\\frac{1}{N_v}\\underset{e_1\\cap e_2\\cap e_3 =v}{\\sum }\\epsilon (e_1,e_2,e_3) tr\\left(\\hat{A}(e_1)\\left[\\hat{A}(e_1)^{-1},\\hat{K}\\right]\\hat{A}(e_2)\\left[\\hat{A}(e_2)^{-1},\\hat{K}\\right]\\hat{A}(e_3)\\left[\\hat{A}(e_3)^{-1},\\sqrt{\\hat{V}}\\right]\\right)$ where $\\hat{K}=\\left[\\hat{S}^{(1)}_E,\\hat{V}\\right]$ and $\\hat{S}^{(1)}_E=\\underset{v}{\\sum }\\hat{S}_E^{(1)}$ , $N_v$ is the number of unordered triples of mutually distinct edges incident at $v$ and $L(v,e,e^{\\prime })$ the set of minimal loops.",
"These are all loops, which start at $v$ along $e$ and end at $v$ along $(e^{\\prime })^{-1}$ and are minimal in the sense that there are no other loops with the same restrictions and fewer edges traversed.",
"In our case, where one is restricted to the once and for all fixed cubic graph, the elementary loops are the plaquettes, consisting of four edges.",
"$\\hat{V}$ is the algebraic quantum Volume Operator: $\\hat{V}=\\underset{N\\rightarrow \\infty }{\\textrm {lim}}\\underset{I=1}{\\overset{N}{\\sum }}\\sqrt{\\left|\\frac{1}{3!",
"}\\epsilon \\left(a,b,c\\right)\\hat{E}_{i}\\left(S_{I}^{a}\\right)\\hat{E_{j}}\\left(S_{I}^{b}\\right)\\hat{E}_{k}\\left(S_{I}^{c}\\right)\\epsilon ^{ijk}\\right|}$ where the skew function $\\epsilon $ is chosen such that it matches that of the embedding dependent Ashtekar-Lewandowski-Volume operator of LQG [38] when the algebraic graph is embedded in a generic way (see [25] for further details).",
"One can show that its spectrum has to be discrete and further analysis has been performed in greater detail in [39].",
"Consequently, the action of the Hamiltonian on an algebraic graph or others is quite involved and the solution of eigenstates cannot be computed analytically, however it is numerically [40] and semiclassically [27] under good control.",
"Some calculations have been done for the LQG Hamiltonian-constraint, which maybe could transfer directly to the algebraic version.",
"For further reading see e.g.",
"[41], [42].",
"For the Yang-Mills Hamiltonian one sets: $\\hat{H}_{YM}(v)=\\frac{1}{2Q^2}\\left(\\hat{H}_E(v)+\\hat{H}_B(v)\\right)$ with $\\hat{H}_E(v)=\\frac{1}{P_v}\\underset{e_1\\cap e_2=v}{\\sum }tr\\left(\\hat{A}(e_1)\\left[\\hat{A}(e_1)^{-1},\\sqrt{\\hat{V}}\\right]\\hat{A}(e_2)\\left[\\hat{A}(e_2)^{-1},\\sqrt{\\hat{V}}\\right]\\right)\\underline{\\hat{E}}_J(e_1)\\underline{\\hat{E}}_J(e_2)$ $\\hat{H}_B(v)=\\frac{1}{T_v^2}\\underset{e_1\\cap e_2\\cap e_3=v}{\\sum }\\underset{e_4\\cap e_5\\cap e_6=v}{\\sum }\\frac{\\epsilon (e_1,e_2,e_3)}{\\left|L(v,e_2,e_3)\\right|}\\frac{\\epsilon (e_4,e_5,e_6)}{\\left|L(v,e_5,e_6)\\right|}\\underset{\\beta \\in L(v,e_2,e_3)}{\\sum }\\underset{\\beta ^{\\prime }\\in L(v,e_5,e_6}{\\sum }\\times $ $\\times tr\\left(\\hat{\\tau }_j\\hat{A}(e_1)\\left[\\hat{A}(e_1)^{-1},\\sqrt{\\hat{V}}\\right]\\right)tr\\left(\\hat{\\tau }_j\\hat{A}(e_4)\\left[\\hat{A}(e_4)^{-1},\\sqrt{\\hat{V}}\\right]\\right)tr\\left(\\underline{\\hat{\\tau }}_J\\underline{\\hat{A}}(\\beta )\\right)tr\\left(\\underline{\\hat{\\tau }}_J\\underline{\\hat{A}}(\\beta ^{\\prime })\\right)$ where $P_v$ is the number of all pairs of edges incident at $v$ , $T_v$ is the number of all non-trivial triples of edges incident at $v$ and the $\\epsilon $ -term is that of the Volume-operator.",
"Note that as in the Kogut-Susskind case, while the Hamiltonian expressed in terms of lattice variables has the correct continuum limit when the lattice embedding becomes sufficiently fine, it is but one of infinitely many possible discretisations that have this property.",
"For instance one could consider discretisations that also have next to next neighbor interaction terms.",
"For the moment one should also notice that the gravitational Gauss constraint as well as the Yang-Mills Gauss constraint have their algebraic quantum versions as well.",
"Going over to the invariant subspace where these Gauss constraints are solved, leads (as in LQG) to the fact that one needs to introduce intertwiners $\\pi $ of both gauge groups respectively on every vertex.",
"The obtained subspace $\\mathcal {H}^\\mathcal {G}_{kin}$ is commonly referred to in the literature as the space of spin-network functions $T_{\\gamma ,{j_e},{\\pi _v}} \\left[A,\\underline{A}\\right] = \\underset{v\\subset \\gamma }{\\bigotimes } \\hspace{5.69054pt}\\underline{\\pi }_v\\otimes \\pi _v \\hspace{5.69054pt}\\underset{e\\subset \\gamma }{\\bigotimes }\\hspace{5.69054pt} h^{j_e}(e)\\otimes \\underline{h}^{\\underline{j}_e}(e)$ where $h^{j_e}(e)=h^{j_e}(e)\\left(A_e\\right)$ corresponds to the irreducible representation of label $j_e$ of the holonomy of $SU(2)$ and $\\underline{h}^{\\underline{j}_e}(e)$ respectively of the Yang-Mills gauge group $G$ .",
"For more information on these see section .",
"To compute the spectrum of the Hamiltonian one would have to compute its matrix elements and their calculation shall be done in chapter .",
"In the following the gauge group for the Gravitational spin-networks is of course $SU(2)$ and for the Yang-Mills gauge group we pick the case of QCD, i.e.",
"$SU(3)$ .",
"This section finishes with a last remark on the Kogut-Susskind-Hamiltonian.",
"While there are a lot of ways to derive it from the Wilson action (see e.g.",
"[1], [9], having this Yang-Mills-Hamiltonian of Quantum Gravity at hand gives an easy derivation of the Kogut-Susskind, which should be seen as the classical limit of the theory.",
"Hence we will replace the general metric with the flat Euclidean one and only quantise the Yang-Mills-Field.",
"After embedding the graph in Minkowski space with a sufficiently small lattice length $\\epsilon $ , one arrives, still with only nearest neighbor interactions (as in the case of the Wilson action), indeed at a version of the Kogut-Susskind Hamiltonian: $\\hat{H}_{KS}=\\frac{1}{2Q^2\\epsilon }\\left(\\underset{e\\in \\gamma }{\\sum }\\hat{E}_J(e)\\hat{E}_J(e)+\\underset{\\beta ,\\beta ^{\\prime }\\in \\gamma }{\\sum }tr\\left(\\tau _j\\hat{A}(\\beta )\\right)tr\\left(\\tau _j\\hat{A}(\\beta ^{\\prime })\\right)\\right)$ This is not the form generally found in the literature (e.g.",
"[1]), because for the derivation of the LQG version of (REF ) a different approximation scheme for the curvature of the $G$ -connection $F_{ab}$ is used.",
"The approximation used in [18], [19] is $\\text{Im}\\left(A(\\beta )\\right) \\approx \\epsilon ^2F_{ab}^j\\tau _j+\\mathcal {O}\\left(\\epsilon ^4\\right)$ , while the other one - which is in case of a flat background metric equivalent - is $\\text{Re}\\left(A(\\beta )\\right) \\approx d_n +\\epsilon ^4 F_{ab}^iF_i^{ab} + \\mathcal {O}\\left(\\epsilon ^6\\right)$ .",
"Kogut and Susskind used the latter one, however in the case of a non-trivial background it is not applicable.",
"In any case this second approximation leads to the addition of a constant, the dimension of the group matrices $d_n$ , which is treated in LQCD as a simple energy shift.",
"Going along this road one obtains: $\\hat{H}_{KS,\\text{lit}}=\\frac{1}{2Q^2\\epsilon }\\left(\\underset{e\\in \\gamma }{\\sum }\\hat{E}_J(e)\\hat{E}_J(e)+\\underset{\\beta \\in \\gamma }{\\sum }tr\\left(\\hat{A}(\\beta )\\right)+tr\\left(\\hat{A}(\\beta )^{\\dagger } \\right) - 2d_n\\right)$"
],
[
"Representation Theory and Graphical Calculus of $SU(3)$",
"Loop Quantum Gravity and Lattice Gauge theory both very heavily depend on the representation theory of the corresponding gauge group.",
"($SU(2)$ for the gravitational sector and for the purpose of this article we restrict ourselve to the $SU(3)$ for the Yang-Mills field).",
"Brink and Satchler have introduced a formalism called graphical calculus [43] for $SU(2)$ , which simplifies the manipulations one wants to perform on the coupled representations of the spin-network by suppressing many of indices from the irreducible representations and makes the coupling of different links more obvious.",
"There has also been a proposal for a graphical calculus in [44] for any Lie Group but this works only in its defining representation, while for our purpose we want to combine different irreducible representations.",
"The methods we will use throughout this paper regarding the computations of the gravitational degrees of freedom have been introduced in [41].",
"With this framework it has been accomplished to evaluate the matrix elements of the Euclidian Part of the Hamiltonian constraint from [18], [19] and the matrix elements of its Lorentzian Part in [42].",
"The matrix elements for the Euclidian and Lorentzian part have been found analytically modulo the matrix elements of the volume operator, which must be determined non analytically.",
"To make this paper self-contained we provide a list of the most important identities of this $SU(2)$ -related calculus to the appendix.",
"In this chapter we aim at the construction of a similar calculus for the gauge group of $SU(3)$ .",
"For this purpose we revisit the representation theory of $SU(3)$ in the following section.",
"The familiar reader may jump forward to REF ."
],
[
"Representation Theory of $SU(3)$",
"In this chapter, we recall some general properties of the finite dimensional representations of the unitary, compact and semi-simple Lie-Group $SU(3)$ and we will construct its Clebsch-Gordan-Coefficients.",
"We start by choosing a suitable basis for the Lie algebra $su(3)$ as in [45].",
"This Lie algebra has a real form and we may pick a basis $\\left\\lbrace A_{i,k}\\right\\rbrace $ (where $i,k=1,2,3$ ), with the following commutation relations: $\\left[ A_{i,k},A_{j,l}\\right] = \\delta _{k,j} A_{i,l} - \\delta _{i,l} A_{j,k}$ These are subject to the restriction $A_{11}+A_{22}+A_{33}=0$ and $A_{i,k}^{+}=A_{k,i}$ , where the adjoint is taken in the respective representation.",
"We will now consider representations of these commutation- and $\\ast $ -relations considered as an abstract Lie algebra.",
"Out of this set one can construct two (so-called) weight operators: $H_{1}=A_{11}-A_{22}$ $H_{2}=A_{22}-A_{33}$ Now given a finite dimensional representation $\\left(D,V\\right)$ over the vectorspace $V$ of $su(3)$ or equivalently $SU(3)$ (since any representation of $SU(3)$ corresponds to a unique one of $su(3)$ and vice versa, due to $SU(3)$ being simply connected), one can simultaneously diagonalize $D(H_{1})$ and $D(H_{2})$ as $\\left[H_{1},H_{2}\\right]=0$ .",
"A pair $j=(a,b)\\in \\mathbb {C}$ is called a weight for $D$ if there exists a $v\\text{$\\ne $}0$ in $V$ such that $D(H_{1})v=av$ $D(H_{2})v=bv$ Additionally $j$ is called highest weight, if for all weights $j^{\\prime }$ of $D$ and $\\mu ,\\nu \\ge 0$ holds $j-j^{\\prime }=\\mu \\alpha _{1}+\\nu \\alpha _{2}$ where the $\\alpha _{i}$ are roots (a non-zero pair $(\\alpha _{i,1},\\alpha _{i,2})\\in \\mathbb {C}^{2}$ , such that $\\left[H_{j},Z_i\\right]=\\alpha _{i,j}Z_i$ with a non-zero $Z_i\\in SU(3)$ ).",
"In the following the irreducible representation of highest weight $j$ is denoted by $D^{(j)}$ .",
"According to the Theorem of the highest weight [46] the following is true for an irreducible representation $D$ of $SU(3)$ $D$ is the direct sum of weight spaces $D$ has a unique highest weight $j=\\left(a,b\\right)$ with $a,b\\in \\mathbb {N}^{+}$ $D$ and $D^{\\prime }$ are equivalent $\\Leftrightarrow $ $j=j^{\\prime }$ From this we may can also deduce the following: The dimension of the irreducible representation with highest weight $j=(a,b)$ is $d_{j}=\\frac{1}{2}\\cdot (a+1)(b+1)(a+b+2)$ A proof for this formula can be found e.g.",
"in [47].",
"We work with finite dimensional representations of $SU(3)$ , which is thus completely reducible [49].",
"Consequently, the Tensor product of these representations can be rewritten as the sum of irreducible representations: $ D^{(j_{1})}\\otimes D^{(j_{2})}=\\underset{j}{\\sum }\\mu _{j}D^{(j)}$ Let the vector-spaces on which these act be called $V_{j}$ and choose orthonormal bases in these spaces.",
"Then a basis for $V_{j_{1}}\\otimes V_{j_{2}}$ is $\\left\\lbrace e_{m_{1}}^{j_{1}}\\otimes e_{m_{2}}^{j_{2}}\\right\\rbrace $ and equivalently for $V_{j}$ $\\left\\lbrace e_{m}^{j,s}\\right\\rbrace $ , where $j$ labels the weight and $s=1,...,\\mu _{j}$ is used to distinguish the multiplicities.",
"These bases can be connected by a unitary matrix: $ e_{m}^{j,s}=\\underset{m_{1},m_{2}}{\\sum }\\left\\langle e_{m_{1}}^{j_{1}},e_{m_{2}}^{j_{2}}\\mid e_{m}^{j,s}\\right\\rangle e_{m_{1}}^{j_{1}}\\otimes e_{m_{2}}^{j_{2}}$ where the entries of the matrix are called the Clebsch-Gordan-Coefficients of the Tensor product.",
"As they are elements of a unitary matrix, the following orthogonality relations hold: $ \\underset{m_{1},m_{2}}{\\sum }\\bigl \\langle e_{m}^{j,s}\\mid e_{m_{1}}^{j_{1}},e_{m_{2}}^{j_{2}}\\bigr \\rangle \\bigl \\langle e_{m_{1}}^{j_{1}},e_{m_{2}}^{j_{2}}\\mid e_{m^{\\prime }}^{j^{\\prime },s^{\\prime }}\\bigr \\rangle =\\delta _{j,j^{\\prime }}\\delta _{s,s^{\\prime }}\\delta _{m,m^{\\prime }}$ $ \\underset{j,s,m}{\\sum }\\left\\langle e_{m_{1}}^{j_{1}},e_{m_{2}}^{j_{2}}\\mid e_{m}^{j,s}\\right\\rangle \\bigl \\langle e_{m}^{j,s}\\mid e_{m_{1}^{\\prime }}^{j_{1}},e_{m_{2}^{\\prime }}^{j_{2}}\\bigr \\rangle =\\delta _{m_{1},m_{1}^{\\prime }}\\delta _{m_{2},m_{2}^{\\prime }}$ To construct these Clebsch-Gordan-Coefficients explicitly, we follow the formalism developed by Pluha$\\check{\\mathrm {r}}$ et al.",
"in [50], [51].",
"It is useful to introduce additional linear combinations of the $A_{i,j}$ .",
"In addition to $H_1$ and $H_2$ one introduces the following operators: The two Casimir operators $F_2 = \\frac{3}{2}\\underset{i,k}{\\sum }A_{i,j}A_{j,i}$ $F_3= 9 \\underset{i,j,k}{\\sum }A_{i,j}A_{j,k}$ which, in the $D^{(j)}$ -representation, have the eigenvalues $f_{2}=\\left(a+b+3\\right)\\left(a+b\\right)-ab$ $f_{3}=\\left(a-b\\right)\\left(2a+b+3\\right)\\left(a+2b+3\\right)$ Also let us look at two sub-algebras, one isomorphic to $su(2)$ : $I_{z}=\\frac{1}{2}\\left(A_{11}-A_{22}\\right)\\text{, {\\color {white}.}",
"}I_{+}=A_{12}\\text{ and }I_{-}=A_{21}$ There exist two eigenvalues for the group $SU(2)$ , which we call isospin $i$ (from the total angular momentum operator $I^2$ ) and isospin projection $i_z$ (from the operator $I_z$ ).",
"Also there is a different sub-algebra isomorphic to $su(2)$ : $\\Lambda _{z}=A_{11}-A_{33}\\text{, {\\color {white}.}",
"}\\Lambda _{+}=\\sqrt{2}\\left(A_{12}-A_{23}\\right)\\text{ and }\\Lambda _{-}=\\sqrt{2}\\left(A_{21}+A_{32}\\right)$ the eigenvalues of which are labeled $\\lambda _0, \\lambda _{0,z}$ .",
"Both sub-algebras contain a linear combination of the weight operators.",
"Thus their quantum numbers $i,\\lambda _0$ can at most be $i_{0}=\\frac{1}{2}a$ and $\\lambda _{0}=a+b$ , respectively [50].",
"The eighth independent operator shall be: $Y=\\frac{1}{3}\\left(A_{11}+A_{22}-2A_{33}\\right)$ called the Hypercharge-Operator, whose Eigenvalues $y$ can be maximally $y_{0}=\\frac{1}{3}\\left(a+2b\\right)$ .",
"This operator comes from particle physics where it unifies isospin and flavor into a single charge.",
"$Y$ is just a linear combination of the $I_z$ and $\\Lambda _z$ and thus the group, spanned from the latter operators, is, in principle, redundant.",
"Hypercharge and isospin projection are weight components for SU(3).",
"Now one has to find how many quantum numbers are needed in general to describe a state in the vectorspace V of a irreducible highest weight representation $D^{(j)}$ .",
"With $su(n)$ being a complex, semisimple Lie algebra one can do a splitting in the cartan sub-algebra $\\mathfrak {h}$ , which is the maximal sub-lie-algebra of all abelian sub-algebras, consisting of semisimple elements.",
"Thus $su(n)=\\mathfrak {h}\\oplus \\mathfrak {g}_+ \\oplus \\mathfrak {g}_-$ where $\\mathfrak {g}_\\pm $ are the sub-algebras to the corresponding to positive/negative roots with respect to a choice of simple positive roots.",
"While $\\mathfrak {h}$ has dimension $n-1$ , $\\mathfrak {g}_\\pm $ have dimension $\\frac{n(n-1)}{2}$ .",
"Every irreducible highest-weight representation is cyclic, i.e.",
"there exists a non-trivial vector $v\\in V$ , which is a weight vector for $j$ , with $D(\\mathfrak {g}_+)v=0$ and the smallest subspace containing $v$ is all of $V$ .",
"The cyclic highest-weight representation depends on $r$ quantum numbers, where $r$ is the rank of the Lie algebra.",
"These quantum numbers correspond to the highest weight vector eigenvalues of the Cartan sub-algebra generators.",
"Moreover the “occupation numbers” are given by the generators of $\\mathfrak {g}_-$ , which are thus $\\frac{n(n-1)}{2}$ many.",
"So now for $n=3$ one may see, that an additional quantum number next to the two weights $i_z$ and $y$ from the Cartan generators $I_z$ and $Y$ is needed.",
"As the Casimir of the $su(2)$ -subgroup $I^2$ commutates with both, it is convenient to use it.",
"Moreover, for a general rank $r$ semisimple Lie algebra the highest weight labels (here $a,b$ ) are in one-to-one correspondence with the eigenvalues of the $r$ algebraically independent Casimirs of rank $2,..,r+1$ (here $F_2,F_3$ ), hence $F_2,F_3,I_z,Y,I^2$ provides a maximally commuting set of self-adjoint operators characterising the irreducible representation completely.",
"Now one labels the basis states of $D^{(j)}$ with hypercharge $y$ , isospin $i$ and isospin projection $i_{z}$ as $\\left|\\left(a,b\\right),\\left(y,i,i_{z}\\right)\\right\\rangle \\equiv \\left|j,m\\right\\rangle $ .",
"To reduce the product $D^{(j_{1})}\\otimes D^{(j_{2})}$ one has to deal with the multiplicity factors.",
"These contribute non-trivially here (in contrast to $SU(2)$ ), as can be seen very easily by looking at the corresponding sets of commutating operators.",
"While there should be 10 commutating operators in the representation of $D^{(j_1)}\\otimes D^{(j_2)}$ , namely $\\left(F_2,F_3,I_z,Y,I^2\\right)^{(1)},\\left(F_2,F_3,I_z,Y,I^2\\right)^{(2)}$ , after looking at the decomposition into irreducible representations there seem to be only 9 commutating ones: $\\left(F_2,F_3,I_z,Y,I^2,F_2^{(1)},F_3^{(1)},F_2^{(2)},F_3^{(2)}\\right)$ .",
"This strange occurrence is solved by introducing an additional operator $S$ , which is a Casimir operator for the Lie algebra generated by $D^{(j_1)}(X)\\otimes 1_{D^{(j_2)}}+ 1_{D^{(j_1)}}\\otimes D^{(j_2)}(X),X\\in su(3)$ , and the s-classified reduced states, which are solutions to the eigenvalue problem $ S\\left(\\left\\lbrace A\\right\\rbrace _{1},\\left\\lbrace A\\right\\rbrace _{2}\\right)\\left|(j_1,j_2),j,m,s\\right\\rangle =s\\left|(j_1,j_2),j,m,s\\right\\rangle $ where we define $S\\left(\\left\\lbrace A\\right\\rbrace _{1},\\left\\lbrace A\\right\\rbrace _{2}\\right)=27\\underset{i,j,k}{\\sum }\\left(A_{i,j;1}A_{j,k;2}A_{k,i;2}-A_{i,j;2}A_{j,k;1}A_{k,i;1}\\right)-2F_{3;2}+2F_{3;1}$ This operator is seen to fulfill some symmetry relations when acting on $D_{j_1}\\otimes D_{j_2}\\otimes D_{j_3}$ $S\\left(\\left\\lbrace A\\right\\rbrace _{1},\\left\\lbrace A\\right\\rbrace _{2}\\right)=-S\\left(\\left\\lbrace A\\right\\rbrace _{2},\\left\\lbrace A\\right\\rbrace _{1}\\right)=-S\\left(\\left\\lbrace A\\right\\rbrace _{1},\\left\\lbrace A\\right\\rbrace _{3}\\right)=-S\\left(\\bar{\\left\\lbrace A\\right\\rbrace }_{1},\\bar{\\left\\lbrace A\\right\\rbrace }_{2}\\right)$ where $D^{(j_{3})}$ stands for the coupled representation and the $\\bar{A}_{ij}:=-A_{ij}$ define the generators of the conjugate (i.e.",
"contragredient) representation.",
"Finally these states have a phase ambiguity which can be resolved by setting: $\\left\\langle j_{1},j_{2}\\lambda _{0;2},\\lambda _{0,z;2}\\mid j_{1},j_{2},j_{3},s\\right\\rangle >0$ It should be noted, however, that the $s$ are in general neither integral nor rational.",
"Pluha$\\mathrm {\\check{r}}$ et al.",
"[50] have proposed a computational algorithm, where for a given set of highest weights the matrix $S\\left(\\left\\lbrace A\\right\\rbrace _{1},\\left\\lbrace A\\right\\rbrace _{2}\\right)$ is finite dimensional.",
"With the last two equations it can be shown, that the Clebsch-Gordan-Coefficients $\\langle j_{1},m_{1},j_{2},m_{2}\\mid (j_1,j_2),j_{3},m_{3},s\\rangle $ , which couple the two representations $j_1,j_2$ to the resulting third $j_3$ , while $m_1+m_2=m_3$ , fulfil the following symmetry relations [50]: $\\begin{aligned}\\langle j_{1},m_{1},j_{2},m_{2}\\mid (j_1,j_2),\\bar{j}_{3},\\bar{m}_{3},s\\rangle & =\\langle j_{2},m_{2},j_{1},m_{1}\\mid (j_1,j_2),\\bar{j}_{3},\\bar{m}_{3},\\bar{s}\\rangle \\left(-\\right)^{j_{1}+j_{2}+j_{3}}\\\\& =\\langle (j_{1},m_{1},j_{3},m_{3}\\mid (j_1,j_2),\\bar{j}_{2},\\bar{m}_{2},\\bar{s}\\rangle \\left(-\\right)^{j_{1}+m_{1}}\\sqrt{d_{j_2}/\\mathrm {d}_{j_3}} \\\\& =\\langle \\bar{j}_{1},\\bar{m}_{1},\\bar{j}_{2},\\bar{m}_{2}\\mid (j_1,j_2),j_{3},m_{3},\\bar{s}\\rangle \\left(-\\right)^{j_{1}+j_{2}+j_{3}}\\\\\\end{aligned}$ with $d_j=dim((a,b))$ the dimension of the space on which the irreducible representation corresponding to highest weight $(a,b)$ lives.",
"Also the following abbreviations have been introduced: $\\bar{j}=\\left(b,a\\right)\\text{, {\\color {white}.}",
"}\\bar{m}=\\left(-y,i,-i_{z}\\right)\\text{ and }\\bar{s}=-s$ $\\left(-\\right)^{j}=\\left(-1\\right)^{a+b}\\text{ and }\\left(-\\right)^{m}=\\left(-1\\right)^{\\frac{3}{2}y+i_{z}}$"
],
[
"Graphical Calculus of $SU(3)$",
"We will now develop a method to simplify computations involving the gauge group $SU(3)$ .",
"To the best of our knowledge, the graphical calculus developed here for $SU(3)$ , while building on the one developed for $SU(2)$ , is novel.",
"We start by defining the so called s-classified 3j-Wigner-Symbol, an object, which represents the symmetry relations of the Clebsch-Gordan-Coeffecients in an easy way: [51] $\\left(\\begin{array}{cccc}j_{1} & j_{2} & j_{3} & s\\\\m_{1} & m_{2} & m_{3}\\end{array}\\right)=\\langle j_{1},m_{1},j_{2},m_{2}\\mid (j_1,j_2),\\bar{j}_{3},\\bar{m}_{3},s\\rangle \\frac{\\left(-\\right)^{\\bar{j}_{3}+\\bar{m}_{3}}}{\\sqrt{d_{\\bar{j}_3}}}$ The symmetry relations from the last chapter (REF ) become: $\\begin{aligned}\\left(\\begin{array}{cccc}j_{1} & j_{2} & j_{3} & s\\\\m_{1} & m_{2} & m_{3}\\end{array}\\right) & =\\left(\\begin{array}{cccc}j_{2} & j_{\\text{1}} & j_{3} & s\\\\m_{2} & m_{1} & m_{3}\\end{array}\\right)\\left(-\\right)^{j_{1}+j_{2}+j_{3}}\\\\& =\\left(\\begin{array}{cccc}j_{1} & j_{3} & j_{2} & s\\\\m_{1} & m_{3} & m_{2}\\end{array}\\right)\\left(-\\right)^{j_{1}+j_{2}+j_{3}}\\\\& =\\left(\\begin{array}{cccc}\\bar{j}_{1} & \\bar{j}_{2} & \\bar{j}_{3} & \\bar{s}\\\\\\bar{m}_{1} & \\bar{m}_{2} & \\bar{m}_{3}\\end{array}\\right)\\left(-\\right)^{j_{1}+j_{2}+j_{3}}\\\\\\end{aligned}$ From this,it is apparent, that the s-classified 3j-symbols are invariant under even permutations and pick up a sign of $(-)^{j_1+j_2+j_3}$ for odd permutations.",
"The usefulness of this Symbol lies in the fact, that any coupling of N representations can be expressed via 3j-symbols.",
"The aim now is to construct a graphical representation that allows one to represent multiple 3j-symbols and their distinct coupling (e.g.",
"the s-classified 6j-symbols).",
"We choose our notation such that it closely resembles the established calculus of [43].",
"The graphical representation of the s-classified Wigner 3j-Symbol is a node, where the three representations are joined in, which are represented as lines $\\left(\\begin{array}{cccc}j_{1} & j_{2} & j_{3} & s\\\\m_{1} & m_{2} & m_{3}\\end{array}\\right)=\\begin{array}{c}\\includegraphics [scale=0.8]{thesis-2_5pic01-node}\\end{array}=\\begin{array}{c}\\includegraphics [scale=0.8]{thesis-2_5pic02-reversed_node}\\end{array}$ here the $+$ sign means that the elements of the 3j are ordered in an anti-clockwise orientation.",
"Equivalently a $-$ sign indicates a clockwise orientation.",
"E.g.",
"a symmetry relation for the 3j is: $\\begin{array}{c}\\includegraphics [scale=0.8]{thesis-2_5pic01-node}\\end{array}=\\left(-\\right)^{j_{1}+j_{2}+j_{3}}\\begin{array}{c}\\includegraphics [scale=0.8]{thesis-2_5pic03-negative_node}\\end{array}$ Additionally arrows will be introduced on the lines to indicate the “metric tensor”.",
"A line with no arrows means $\\begin{array}{c}\\includegraphics {thesis-2_5pic05-link}\\end{array}=\\delta _{j_{1},j_{2}}\\delta _{m_{1},m_{2}}$ while a line with an arrow denotes the 1j-symbol: $\\begin{array}{c}\\includegraphics {3-30}\\end{array}=\\delta _{j_{1},\\bar{j_{2}}}\\left(\\begin{array}{c}j_{1}\\\\m_{1},m_{2}\\end{array}\\right)=\\delta _{\\bar{j}_{1},j_{2}}\\delta _{\\bar{m}_1,m_{2}}\\left(-\\right)^{j_{1}+m_{1}}$ In the following we suppress the magnetic quantum numbers in the pictures.",
"Having multiple arrows on one line, one can realize that (as well as for other orientations of the two arrows) $\\begin{array}{c}\\includegraphics {thesis-2_5pic28-two_arrows}\\end{array}=\\begin{array}{c}\\includegraphics {thesis-2_5pic29-no_arrow}\\end{array}$ Given all of this we may calculate further: A contraction of 1j and 3j is: $\\begin{array}{c}\\includegraphics [scale=0.8]{thesis-2_5pic10-node+arrow}\\end{array}=\\underset{m}{\\sum }\\left(\\begin{array}{cccc}j_{1} & j_{2} & j & s\\\\m_{1} & m_{2} & m\\end{array}\\right)\\left(\\begin{array}{c}j_{3}\\\\m_{3},m\\end{array}\\right)\\delta _{j_{3},\\bar{j}}$ $=\\underset{m}{\\sum }\\left(\\begin{array}{cccc}j_{1} & j_{2} & j & s\\\\m_{1} & m_{2} & m\\end{array}\\right)\\delta _{m,\\bar{m}_{3}}\\delta _{j_{3},\\bar{j}}\\left(-\\right)^{j_{3}+m_{3}}$ $=\\left(\\begin{array}{cccc}j_{1} & j_{2} & \\bar{j}_{3} & s\\\\m_{1} & m_{2} & \\bar{m_3}\\end{array}\\right)\\left(-\\right)^{j_{3}+m_{3}}\\delta _{j_{3},\\bar{j}}$ Similarly we can write: $\\begin{array}{c}\\includegraphics [scale=0.8]{thesis-2_5pic11-node+arrow_detailed}\\end{array}=\\underset{m_{1}^{\\prime },m_{2}^{\\prime },m_{3}^{\\prime }}{\\sum }\\left(\\begin{array}{cccc}\\bar{j}_{1} & \\bar{j}_{2} & \\bar{j}_{3} & s\\\\m_{1}^{\\prime } & m_{2}^{\\prime } & m_{3}^{\\prime }\\end{array}\\right)\\left(\\begin{array}{c}j_{1}\\\\m_{1},m_{1}^{\\prime }\\end{array}\\right)\\left(\\begin{array}{c}j_{2}\\\\m_{2},m_{2}^{\\prime }\\end{array}\\right)\\left(\\begin{array}{c}j_{3}\\\\m_{3},m_{3}^{\\prime }\\end{array}\\right)$ $=\\underset{m_{1}^{\\prime },m_{2}^{\\prime },m_{3}^{\\prime }}{\\sum }\\left(\\begin{array}{cccc}\\bar{j}_{1} & \\bar{j}_{2} & \\bar{j}_{3} & s\\\\m_{1}^{\\prime } & m_{2}^{\\prime } & m_{3}^{\\prime }\\end{array}\\right)\\delta _{m_{1}^{\\prime },\\bar{m}_{1}}\\delta _{m_{2}^{\\prime },\\bar{m}_{2}}\\delta _{m_{3}^{\\prime },\\bar{m}_{3}}\\left(-\\right)^{\\underset{i}{\\sum }j_{i}+m_{i}}$ $=\\left(\\begin{array}{cccc}\\bar{j}_{1} & \\bar{j}_{2} & \\bar{j}_{3} & s\\\\\\bar{m}_{1} & \\bar{m}_{2} & \\bar{m}_{3}\\end{array}\\right)\\left(-\\right)^{j_{1}+j_{2}+j_{3}}\\left(-\\right)^{m_{1}+m_{2}+m_{3}}$ $=\\left(\\begin{array}{cccc}j_{1} & j_{2} & j_{3} & \\bar{s}\\\\m_{1} & m_{2} & m_{3}\\end{array}\\right)=\\begin{array}{c}\\includegraphics [scale=0.8]{thesis-2_5pic12-node}\\end{array}$ where we have used, that $(-)^{m_1+m_2+m_3}=0$ .",
"In the following one uses the abbreviation: $\\begin{array}{c}\\includegraphics {3-34}\\end{array}=\\begin{array}{c}\\includegraphics {thesis-2_5pic04-link+arrow}\\end{array}$ and thus only writes one index to each line from now on.",
"For lines without arrow it indicates the highest weights of its irreducible representation, and if the line has an arrow it indicates the highest weight of the representation where the arrow points towards.",
"Also the arrows can be changed by dualising the j.",
"$\\begin{array}{c}\\includegraphics {3-34}\\end{array}=\\begin{array}{c}\\includegraphics {3-35}\\end{array}$ In order to represent more complex structures, lines can be joined as long as they carry the same highest weight.",
"Note that the lines also carry a distinct group element.",
"Joining them means that the magnetic quantum numbers are set to equal and summed over.",
"In the following these numbers are omitted in the graphs as already stated.",
"With this definition one is, for example, able to represent the s-classified 6j-symbol, an object defined in the following way (similar to [51]): $\\left\\lbrace \\begin{array}{cccc}j_{1} & j_{2} & j_{3}\\\\j_{4} & j_{5} & j_{6}\\\\s_{1} & s_{2} & s_{3} & s_{4}\\end{array}\\right\\rbrace =\\underset{\\left\\lbrace m\\right\\rbrace }{\\sum }\\left(-\\right)^{\\underset{i}{\\sum }j_{i}+m_{i}}\\left(\\begin{array}{cccc}j_{1} & j_{2} & j_{3} & s_{1}\\\\m_{1} & m_{2} & m_{3}\\end{array}\\right)\\left(\\begin{array}{cccc}\\bar{j}_{1} & j_{5} & \\bar{j}_{6} & s_{2}\\\\\\bar{m}_{1} & m_{5} & \\bar{m}_{6}\\end{array}\\right)\\cdot $ $\\cdot \\left(\\begin{array}{cccc}\\bar{j}_{4} & \\bar{j}_{2} & j_{6} & s_{3}\\\\\\bar{m}_{4} & \\bar{m}_{2} & m_{6}\\end{array}\\right)\\left(\\begin{array}{cccc}j_{4} & \\bar{j}_{5} & \\bar{j}_{3} & s_{4}\\\\m_{4} & \\bar{m}_{5} & \\bar{m}_{3}\\end{array}\\right)=\\begin{array}{c}\\includegraphics {3-36}\\end{array}$ This object has a lot of symmetries at hand, so e.g., it holds $\\left\\lbrace \\begin{array}{cccc}j_{1} & j_{2} & j_{3}\\\\j_{4} & j_{5} & j_{6}\\\\s_{1} & s_{2} & s_{3} & s_{4}\\end{array}\\right\\rbrace =\\left\\lbrace \\begin{array}{cccc}\\bar{j}_{2} & \\bar{j}_{1} & \\bar{j}_{3}\\\\j_{5} & j_{4} & j_{6}\\\\s_{1} & s_{3} & s_{2} & s_{4}\\end{array}\\right\\rbrace =\\left\\lbrace \\begin{array}{cccc}\\bar{j}_{1} & \\bar{j}_{3} & \\bar{j}_{2}\\\\j_{4} & j_{6} & j_{5}\\\\s_{1} & s_{2} & s_{4} & s_{3}\\end{array}\\right\\rbrace =$ $=\\left\\lbrace \\begin{array}{cccc}j_{4} & \\bar{j}_{5} & \\bar{j}_{3}\\\\j_{1} &\\bar{j}_{2} & \\bar{j}_{6}\\\\s_{4} & s_{3} & s_{2} & s_{1}\\end{array}\\right\\rbrace =\\left\\lbrace \\begin{array}{cccc}\\bar{j}_{1} & \\bar{j}_{2} & \\bar{j}_{3}\\\\\\bar{j}_{4} & \\bar{j}_{5} & \\bar{j}_{6}\\\\\\bar{s}_{1} & \\bar{s}_{2} & \\bar{s}_{3} & \\bar{s}_{4}\\end{array}\\right\\rbrace $ Also, for such a closed diagram (meaning that no open links remain) the object infers the invariance of the change of + $\\leftrightarrow $ -, since every link obviously meets exactly two nodes, and $(-)^{2j}=1$ , because - recalling the theorem of the highest weight - $j=(a,b)$ with $a,b\\in \\mathbb {N}$ .",
"Important relations in the theory of group representations are the two orthogonality relations (REF ) and (REF ) Their form follows from the very definition of the 3j-symbols and the fact that they are real: $\\underset{m_1,m_2}{\\sum } \\left( \\begin{array}{cccc}j_{1} & j_2 & j_3 & s\\\\m_1 & m_2 & m_3\\end{array}\\right)\\left( \\begin{array}{cccc}j_{1} & j_2 & j^{\\prime }_3 & s^{\\prime }\\\\m_1 & m_2 & m^{\\prime }_3\\end{array}\\right)=\\frac{1}{d_{\\bar{j_3}}} \\delta _{j_3,j^{\\prime }_3}\\delta _{m_3,m^{\\prime }_3}\\delta _{s,s^{\\prime }}$ $\\underset{j_3,m_3,s}{\\sum } \\left( \\begin{array}{cccc}j_{1} & j_2 & j_3 & s\\\\m_1 & m_2 & m_3\\end{array}\\right)\\left( \\begin{array}{cccc}j_{1} & j_2 & j_3 & s\\\\m_1^{\\prime } & m_2^{\\prime } & m_3\\end{array}\\right)=\\frac{1}{d_{\\bar{j}_3}} \\delta _{m_1,m_1^{\\prime }}\\delta _{m_2,m_2^{\\prime }}$ Graphically these orthogonality relations can be encoded as: $ \\begin{array}{c}\\includegraphics [scale=0.8]{thesis-2_5pic06-first_orthog_rel}\\end{array}=\\frac{\\delta _{s,s^{\\prime }}}{d_{j_3}}\\begin{array}{c}\\includegraphics [scale=0.8]{thesis-2_5pic07-link}\\end{array}$ $ \\underset{s,j_3}{\\sum } d_{j_3}\\begin{array}{c}\\includegraphics [scale=0.8]{thesis-2_5pic08-sec_orthog_rel}\\end{array}=\\begin{array}{c}\\includegraphics [scale=0.8]{thesis-2_5pic09-parallel_links}\\end{array}$ It should be noted at this point that the sum over $s$ goes over all the solutions from (REF ) and is highly dependent on the coupled weights $j_1$ , $j_2$ and $j_3$ .",
"While $j_3$ itself has to be chosen such that the three representations together form a triad (as for $SU(2)$ ) [52], [48], [49], i.e.",
"if $j_3$ is inside the set $\\Pi _{j_1}+j_2$ , with $\\Pi _{j_1}$ denoting the set of all weights of the corresponding representation with heighest weight $j_1$ .",
"One can immediately see that the expression of the second orthogonality with arrows on the links is stated as: $\\underset{\\bar{j}_3s}{\\sum } d_{j_3}\\begin{array}{c}\\includegraphics [scale=0.8]{3-39--5}\\end{array}=\\begin{array}{c}\\includegraphics [scale=0.8]{thesis-2_5pic31-parallel_links+arrows}\\end{array}$ It is now obvious that transforming the algebraic expression of a graph alters its distinct representation, such that there also must exist some rules for transforming the graphs directly.",
"We have already seen that e.g.",
"the arrows can be changed in their direction, by going from weight $j=(a,b)$ to $\\bar{j}=(b,a)$ .",
"Also: a line with two arrows is equivalent to a line with no arrows.",
"Furthermore at a node one can add and remove arrows of the same direction on each line at the same time, while only changing the node internal index $s\\rightarrow \\bar{s}$ .",
"Since one has for any general Lie group [49], that $\\underset{m_1^{\\prime }m_2^{\\prime }}{\\sum }\\langle e^{j_3,s}_{m_3} \\mid e^{j_1}_{m_1} e^{j_2}_{m_2}\\rangle D^{(j_1)}_{m_1^{\\prime }m_1}(g)D^{(j_2)}_{m_2^{\\prime }m_2}(g)=\\underset{m_3^{\\prime }}{\\sum }\\langle e^{j_3,s}_{m_3^{\\prime }} \\mid e^{j_1}_{m_1}e^{j_2}_{m_2} \\rangle D^{(j_3)}_{m_3^{\\prime }m_3}(g)$ this translates as a transformation rule for our graphical calculus: $\\begin{array}{c}\\includegraphics {3-40-1}\\end{array} =\\begin{array}{c}\\includegraphics {3-40-2}\\end{array}$ We now look at further rules, which change the lines and their coupling itself.",
"For this purpose we define objects equivalent to the $SU(2)$ $jm$ -coefficients from [53], which are blocks of connected nodes with an arrow on each line, whose explicit internal structure is of no importance.",
"They have $n$ external lines with label $j_1 ... j_n$ .",
"Their graphical representation is: $F_n\\left( \\begin{array}{ccc}j_1 & ... & j_n \\\\m_1 & ... & m_n\\end{array}\\right)=\\begin{array}{c}\\includegraphics {thesis-2_3_2pic19-block}\\end{array}$ Using the orthogonality relations from above, a lot of manipulation on these external lines can be done.",
"First one has to notice that a block with only one external line, i.e.",
"$F_1\\left(\\begin{array}{c}j\\\\m\\end{array}\\right)$ , is equivalent to a scalar times a Clebsch-Gordan-Coefficient with two labels equal to zero and hence zero itself, if not $j=m=0$ : $F_1\\left(\\begin{array}{c}j\\\\m\\end{array}\\right)=F_1\\left(\\begin{array}{c}0\\\\0\\end{array}\\right)\\delta _{j,0}\\delta _{m,0}=\\left(\\begin{array}{ccc}0 & 0 & j\\\\0 & 0 & m\\end{array}\\right) \\mathrm {const.",
"}$ This and the second orthogonality relation (REF ) on an $F_2$ coefficient leads to: $\\begin{array}{c}\\includegraphics {thesis-2_5pic13-2block}\\end{array}=\\underset{j,s}{\\sum }\\begin{array}{c}\\includegraphics {3-43-1}\\end{array}=\\delta _{j_1,j_2}\\begin{array}{c}\\includegraphics {3-43-2}\\end{array}$ since the one connection link vanishes and the node reduces to a 1j-symbol and thus the sum over $s$ reduces to a $\\delta _{s,2f_3(j_1)}$ .",
"With a similar calculation and using (REF ) we arrive at: $\\begin{array}{c}\\includegraphics {thesis-2_5pic15-3block}\\end{array}=\\underset{s}{\\sum }\\begin{array}{c}\\includegraphics {thesis-2_5pic16-3block_coupled}\\end{array}$ With this at hand, all the tools of a graphical calculus necessary to simplify calculations involving the gauge group $SU(3)$ are provided.",
"Before we dive into the computations of the matrix elements of the Quantum Yang-Mills Hamiltonian, we provide a final example: The following structure will be encountered numerous times in the remainder of this article: $\\begin{array}{c}\\includegraphics {3-45-1}\\end{array}=\\underset{s}{\\sum }\\begin{array}{c}\\includegraphics {3-45-2}\\end{array} \\cdot \\begin{array}{c}\\includegraphics {3-45-3}\\end{array}$ $=\\underset{s}{\\sum } \\left\\lbrace \\begin{array}{cccc}j_1 & j_2 & j_3 \\\\j_4 & j_5 & j_6 \\\\s & s_2 & s_3 & s_4\\end{array}\\right\\rbrace \\begin{array}{c}\\includegraphics {3-45-3}\\end{array}$"
],
[
"Einstein-Yang-Mills-Theory in the Kogut-Susskind-Case",
"In this chapter we present the results, when applying the developed methods in case of the background spectrum of the Kogut-Susskind-Hamiltonian in flat space.",
"In this work we will not focus on any analytical solvable problem, e.g.",
"the one-plaquette-graph, whose eigenstates are given in terms of Mathieu-functions [54] in case of $U(1)$ or $SU(2)$ Gauge Theory[55], [56], [57].",
"Instead we concentrate on the physically interesting case of multiple-plaquette problems, which so far could be tackled using numerical investigations.",
"A lot of work has been done on this, see e.g.",
"[58], [59], [60], [61], [62], [63] and many more.",
"The most promising approach up to today is still to calculate the matrix elements and continue afterwards with numerical simulations.",
"For this reason this chapter will present the exact calculation of said matrix elements for further - yet to be done - computations.",
"The calculation is done in the notation of spin-networks, since this basis has certain advantages: e.g.",
"the first Term, consisting of the Casimir Operators, diagonalzies here and gives the corresponding quadric Casimir $C_2(j)^2$ of the group [59].",
"Furthermore (hence in the Kogut-Susskind-formalism one deals exclusively with it) a 3-dimensional spatial cubic lattice shall be considered.",
"Thus at each vertex 6 links meet and the first question to answer is, how to choose the intertwiner at this node, which couples all six $j$ 's to a resulting seventh which vanishes.",
"There are multiple ways to do this and choosing one corresponds to the choice of a basis.",
"Here we take the pairs of parallel edges (say e.g.",
"in $\\bar{e}_1$ -direction) and couple these to a resulting third (e.g.",
"$\\pi _1$ ).",
"At the end we couple all the three new representations $\\pi _1,\\pi _2, \\pi _3$ to a vanishing fourth.",
"This is independent of the gauge group and afterwards one single node looks as follows: $\\left|\\nu \\left(\\left\\lbrace \\pi \\right\\rbrace _{\\bar{k}};\\left\\lbrace j\\right\\rbrace _{\\bar{k}};\\left\\lbrace s\\right\\rbrace _{\\bar{k}}\\right)\\right\\rangle =\\begin{array}{c}\\includegraphics [scale=1]{thesis-3_4_0pic01-arbitrary_node}\\end{array}$ $=\\left|\\nu \\left(\\pi _{1,\\bar{k}},\\pi _{2,\\bar{k}},\\pi _{3,\\bar{k}};j_{1,\\bar{k}},j_{2,\\bar{k}},j_{3,\\bar{k}},j_{1,\\bar{k}-\\bar{e}_{1}},j_{1,\\bar{k}-\\bar{e}_{2}},j_{1,\\bar{k}-\\bar{e}_{3}};s_{0,\\bar{k}},s_{1.\\bar{k}}s_{2,\\bar{k}},s_{3.\\bar{k}}\\right)\\right\\rangle $ For $SU(2)$ of course all the $s$ would vanish and thus be omitted.",
"Out notation is chosen such that every edge is associated with its direction $\\bar{e}_{i}$ and one point on the lattice $\\bar{k}$ .",
"In total we write for the corresponding group element $\\hat{A}_{i,\\bar{k}}$ .",
"The group elements themselves however will not be written explicitly.",
"If one recalls formula (REF ) one sees, that when multiplies two representations of the same group element (as is done, when acting with the plaquette part of the Kogut-Susskind-Hamiltonian) one can shift it to the coupled representation.",
"In this manner, one sees easily that one always ends up with the same lattice one started with (regarding the group elements), only its distinct irreducible representation will have changed.",
"Since this concept translates to all the following calculations, all the corresponding group elements will obviously be omitted in the graphs.",
"Also, the lines, which are dashed in the picture, are those that are infinitesimally small (like those of $\\pi _{i,\\bar{k}}$ ), due to existing only at the vertex itself (and of course not carrying a group element).",
"To fix the orientation, we choose $\\forall i\\in \\left\\lbrace 1,2,3\\right\\rbrace ,\\forall \\bar{k}\\in \\mathbb {Z}^{3}$ $\\begin{aligned}\\begin{array}{c}\\includegraphics [scale=0.4]{thesis-3_4_0pic02-node}\\end{array}\\mathrm { and }\\begin{array}{c}\\includegraphics [scale=0.8]{thesis-3_4_0pic03-edge}\\end{array}\\end{aligned}$ Let $\\Psi $ be an arbitrary state of the lattice.",
"As was already stated the Electric Term is diagonal, so we see immediately that $2Q^2 \\epsilon \\hat{H}_{KS,lit} \\left| \\Psi \\right\\rangle = \\underset{i,\\bar{k}}{\\sum } C_2(j_{i,\\bar{k}})^2 + \\underset{\\beta }{\\sum } tr\\left(\\hat{A}(\\beta )\\right) + tr\\left(\\hat{A}(\\beta )\\right)^\\dagger \\left| \\Psi \\right\\rangle $ meaning we can restrict ourselves to the evaluation of the trace over all plaquettes.",
"Even more: using that $\\hat{A}+\\hat{A}^\\dagger = 2 Re(\\hat{A})$ we focus only on the $tr(\\hat{A}(\\beta )$ .",
"Given the set $\\left\\lbrace k,m,n\\right\\rbrace $ as an even permutation of $\\left\\lbrace 1,2,3\\right\\rbrace $ , one can look w.l.o.g at the plaquette in $\\left(m,n\\right)$ -direction containing amongst others the vertex $\\bar{k}$ .",
"In this notation the second term of the Hamiltonian is written: $\\frac{1}{Q^2}\\underset{\\bar{k}}{\\sum }\\overset{3}{\\underset{k=1}{\\sum }}tr\\left(\\hat{A}_{m,\\bar{k}}\\hat{A}_{n,\\bar{k}+\\bar{e}_{m}}\\hat{A}_{m,\\bar{k}+\\bar{e_{n}}}^{-1}\\hat{A}_{n,\\bar{k}}^{-1}\\right)$ We first present the application of the graphical calculus to evaluate the matrix elements of (REF ) in case of the gauge group $SU(2)$ and later on state the corresponding results in case of the $SU(3)$ gauge group.",
"We note in passing that the Kogut.Susskind computation of the magnetic term performed here would be the same (for $SU(2)$ ) as the Euclidian piece of the gravitational contribution to the Hamiltonian, which also has not been done in the non graph changing setting before, although it was done for its semi classically valid $U(1)^3$ approximation [26].",
"The action of the trace on a general graph $\\left|\\psi _{\\bar{j},\\bar{\\pi }}\\right\\rangle $ is written as $\\begin{array}{c}\\includegraphics [scale=0.8]{thesis-3_4_1pic04-plaquette}\\end{array}$ $=\\underset{\\begin{array}{c}j_{n,\\bar{k}}^{\\prime }j_{m,\\bar{k}}^{\\prime }\\\\j{}_{n,\\bar{k}+\\bar{e}_{m}}^{\\prime },j_{m,\\bar{k}+\\bar{e}_{n}}^{\\prime }\\end{array}}{\\sum }\\underset{\\begin{array}{c}\\pi _{n,\\bar{k}}^{\\prime },\\pi _{m,\\bar{k}}^{\\prime },\\pi _{n,\\bar{k}+\\bar{e}_{m}}^{\\prime },\\pi _{n,\\bar{k}+\\bar{e}_{m}}^{\\prime }\\\\\\pi _{n,\\bar{k}+\\bar{e}_{n}}^{\\prime },\\pi _{n,\\bar{k}+\\bar{e}_{n}}^{\\prime },\\pi _{n,\\bar{k}+\\bar{e}_{m}+\\bar{e}_{n}}^{\\prime },\\pi _{n,\\bar{k}+\\bar{e}_{m}+\\bar{e}_{n}}^{\\prime }\\end{array}}{\\sum }$ $\\cdot \\left(-\\right)^{2\\left(j_{m,\\bar{k}}^{\\prime }+j_{n,\\bar{k}}+j_{m,\\bar{k}+\\bar{e}_{n}}+j_{n,\\bar{k}+\\bar{e}_{m}}^{\\prime }+\\pi _{m,\\bar{k}}^{\\prime }+\\pi _{n,\\bar{k}}+\\pi _{m,\\bar{k}+\\bar{e}_{m}}+\\pi _{n,\\bar{k}+\\bar{e}_{m}}^{\\prime }+\\pi _{m,\\bar{k}+\\bar{e}_{n}}+\\pi _{n,\\bar{k}+\\bar{e}_{n}}^{\\prime }+\\pi _{m,\\bar{k}+\\bar{e}_{m}+\\bar{e}_{n}}^{\\prime }+\\pi _{n,\\bar{k}+\\bar{e}_{m}+\\bar{e}_{n}}\\right)}\\cdot $ $\\cdot \\begin{array}{c}\\includegraphics [scale=0.8]{thesis-3_4_1pic05-plaquettecoupled}\\end{array}$ Now all the 6j-symbols have to be recoupled.",
"One starts with the bottom left one in the figures (which is the easiest one with the 6j being exactly in the form as in equation (REF ) and then one brings the orientation of the node back to normal order and continues clockwise.",
"Finally if we define $\\mathfrak {P}_{SU(2)}\\left(\\left\\lbrace \\pi \\right\\rbrace _{\\bar{k}},\\left\\lbrace \\pi \\right\\rbrace _{\\bar{k}+\\bar{e}_{m}},\\left\\lbrace \\pi \\right\\rbrace _{\\bar{k}+\\bar{e}_{n}},\\left\\lbrace \\pi \\right\\rbrace _{\\bar{k}+\\bar{e}_{m}+\\bar{e}_{n}};\\left\\lbrace j\\right\\rbrace ;\\pi _{n,\\bar{k}}^{\\prime },\\pi _{m,\\bar{k}}^{\\prime },\\pi _{m,\\bar{k}+\\bar{e}_{m}}^{\\prime },...;j_{n,\\bar{k}}^{\\prime },j_{m,\\bar{k}}^{\\prime },...\\right)\\equiv $ $\\underset{i=0,1}{\\prod }d_{j^{\\prime }_{m,\\bar{k}+i\\bar{e}_n}}d_{j^{\\prime }_{n,\\bar{k}+i\\bar{e}_m}}\\underset{j=0,1}{\\prod }d_{\\pi ^{\\prime }_{m,\\bar{k}+i\\bar{e}_m+j\\bar{e}_n}}d_{\\pi ^{\\prime }_{n,\\bar{k}+i\\bar{e}_m+j\\bar{e}_n}}\\cdot $ $\\left(-\\right)^{2\\left(j_{m,\\bar{k}}^{\\prime }+j_{n,\\bar{k}}+j_{m,\\bar{k}+\\bar{e}_{n}}+j_{n,\\bar{k}+\\bar{e}_{m}}^{\\prime }+\\pi _{k,\\bar{k}}+\\pi _{k,\\bar{k}+\\bar{e}_{n}}+\\pi _{k,\\bar{k}+\\bar{e}_{m}}+\\pi _{k,\\bar{k}+\\bar{e}_{m}+\\bar{e}_{n}}\\right)+j_{n,\\bar{k}}^{\\prime }+j_{n,\\bar{k}-\\bar{e}_{n}}+\\pi _{n,\\bar{k}}^{\\prime }}\\left\\lbrace \\begin{array}{ccc}j_{n,\\bar{k}-\\bar{e}_{n}} & j_{n,\\bar{k}} & \\pi _{n,\\bar{k}}\\\\1 & \\pi _{n,\\bar{k}}^{\\prime } & j_{n,\\bar{k}}^{\\prime }\\end{array}\\right\\rbrace \\cdot $ $\\cdot \\left(-\\right)^{2\\pi _{n,\\bar{k}}}\\left\\lbrace \\begin{array}{ccc}\\pi _{k,\\bar{k}} & \\pi _{n,\\bar{k}} & \\pi _{m,\\bar{k}}\\\\1 & \\pi _{m,\\bar{k}}^{\\prime } & \\pi _{n,\\bar{k}}^{\\prime }\\end{array}\\right\\rbrace \\left(-\\right)^{\\pi _{k,\\bar{k}}+\\pi _{m,\\bar{k}}^{\\prime }+\\pi _{n,\\bar{k}}^{\\prime }}\\cdot \\left(-\\right)^{j_{m,\\bar{k}}+j_{m,\\bar{k}-\\bar{e}_{m}}+\\pi _{m,\\bar{k}}+2j_{m,\\bar{k}-\\bar{e}_{m}}}\\left\\lbrace \\begin{array}{ccc}j_{m,\\bar{k}-\\bar{e}_{m}} & \\pi _{m,\\bar{k}} & j_{m,\\bar{k}}\\\\1 & j_{m,\\bar{k}}^{\\prime } & \\pi _{m,\\bar{k}}^{\\prime }\\end{array}\\right\\rbrace \\cdot $ $\\left(-\\right)^{2\\pi _{m,\\bar{k}+\\bar{e}_{m}}+j_{m,\\bar{k}}^{\\prime }+j_{m,\\bar{k}+\\bar{e}_{m}}+\\pi _{m,\\bar{k}+\\bar{e}_{m}}^{\\prime }}\\left\\lbrace \\begin{array}{ccc}j_{m,\\bar{k}}^{\\prime } & 1 & j_{m,\\bar{k}}\\\\\\pi _{m,\\bar{k}+\\bar{e}_{m}} & j_{m,\\bar{k}+\\bar{e}_{m}} & \\pi _{m,\\bar{k}+\\bar{e}_m}^{\\prime }\\end{array}\\right\\rbrace \\cdot \\left\\lbrace \\begin{array}{ccc}\\pi _{m,\\bar{k}+\\bar{e}_{m}}^{\\prime } & 1 & \\pi _{m,\\bar{k}+\\bar{e}_{m}}\\\\\\pi _{n,\\bar{k}+\\bar{e}_{m}} & \\pi _{k,\\bar{k}+\\bar{e}_{m}} & \\pi _{n,\\bar{k}+\\bar{e}_{m}}^{\\prime }\\end{array}\\right\\rbrace \\cdot $ $\\left(-\\right)^{2\\pi _{n,\\bar{k}+\\bar{e}_{m}}}\\left(-\\right)^{\\pi _{n,\\bar{k}+\\bar{e}_{m}}^{\\prime }+\\pi _{k,\\bar{k}+\\bar{e}_{m}}+\\pi _{m,\\bar{k}+\\bar{e}_{m}}^{\\prime }}\\cdot \\left\\lbrace \\begin{array}{ccc}\\pi _{n,\\bar{k}+\\bar{e}_{m}}^{\\prime } & 1 & \\pi _{n,\\bar{k}+\\bar{e}_{m}}\\\\j_{n,\\bar{k}+\\bar{e}_{m}} & j_{n,\\bar{k}+\\bar{e}_{m}-\\bar{e}_{n}} & j_{n,\\bar{k}+\\bar{e}_{m}}^{\\prime }\\end{array}\\right\\rbrace \\left(-\\right)^{\\pi _{n,\\bar{k}+\\bar{e}_{m}}^{\\prime }+j_{n,\\bar{k}+\\bar{e}_{m}-\\bar{e}_{n}}+j_{n,\\bar{k}+\\bar{e}_{m}}^{\\prime }}\\cdot $ $\\left(-\\right)^{j_{n,\\bar{k}+\\bar{e}_{m}}+j_{n,\\bar{k}+\\bar{e}_{m}+\\bar{e}_{n}}+\\pi _{n,\\bar{k}+\\bar{e}_{m}+\\bar{e}_{n}}+2j_{n,\\bar{k}+\\bar{e}_{m}}}\\left\\lbrace \\begin{array}{ccc}j_{n,\\bar{k}+\\bar{e}_{m}}^{\\prime } & 1 & j_{n,\\bar{k}+\\bar{e}_{m}}\\\\\\pi _{n,\\bar{k}+\\bar{e}_{m}+\\bar{e}_{n}} & j_{n,\\bar{k}+\\bar{e}_{m}+\\bar{e}_{n}} & \\pi _{n,\\bar{k}+\\bar{e}_{m}+\\bar{e}_{n}}^{\\prime }\\end{array}\\right\\rbrace \\cdot \\left(-\\right)^{2\\pi _{n,\\bar{k}+\\bar{e}_{m}+\\bar{e}_{n}}}$ $\\left\\lbrace \\begin{array}{ccc}\\pi _{n,\\bar{k}+\\bar{e}_{m}+\\bar{e}_{n}}^{\\prime } & 1 & \\pi _{n,\\bar{k}+\\bar{e}_{m}+\\bar{e}_{n}}\\\\\\pi _{m,\\bar{k}+\\bar{e}_{m}+\\bar{e}_{n}} & \\pi _{n,\\bar{k}+\\bar{e}_{m}+\\bar{e}_{n}} & \\pi _{m,\\bar{k}+\\bar{e}_{m}+\\bar{e}_{n}}^{\\prime }\\end{array}\\right\\rbrace \\left(-\\right)^{\\pi _{n,\\bar{k}+\\bar{e}_{m}+\\bar{e}_{n}}^{\\prime }+\\pi _{m,\\bar{k}+\\bar{e}_{m}+\\bar{e}_{n}}^{\\prime }+\\pi _{k,\\bar{k}+\\bar{e}_{m}+\\bar{e}_{n}}}\\cdot \\left(-\\right)^{2\\pi _{m,\\bar{k}+\\bar{e}_{m}+\\bar{e}_{n}}}$ $\\left\\lbrace \\begin{array}{ccc}\\pi _{m,\\bar{k}+\\bar{e}_{m}+\\bar{e}_{n}}^{\\prime } & 1 & \\pi _{m,\\bar{k}+\\bar{e}_{m}+\\bar{e}_{n}}\\\\j_{m,\\bar{k}+\\bar{e}_{n}} & j_{m,\\bar{k}+\\bar{e}_{m}+\\bar{e}_{n}} & j_{m,\\bar{k}+\\bar{e}_{n}}^{\\prime }\\end{array}\\right\\rbrace \\left(-\\right)^{j_{m,\\bar{k}+\\bar{e}_{n}}^{\\prime }+\\pi _{m,\\bar{k}+\\bar{e}_{m}+\\bar{e}_{n}}^{\\prime }+j_{m,\\bar{k}+\\bar{e}_{m}+\\bar{e}_{n}}}\\cdot \\left(-\\right)^{j_{m,\\bar{k}+\\bar{e}_{n}}+j_{m,\\bar{k}-\\bar{e}_{m}+\\bar{e}_{n}}+\\pi _{m,\\bar{k}+\\bar{e}_{n}}}$ $\\left(-\\right)^{2j_{m,\\bar{k}+\\bar{e}_{n}}+2\\pi _{m,\\bar{k}+\\bar{e}_{n}}}\\cdot \\left\\lbrace \\begin{array}{ccc}\\pi _{m,\\bar{k}+\\bar{e}_{n}}^{\\prime } & \\pi _{m,\\bar{k}+\\bar{e}_{n}} & 1\\\\j_{m,\\bar{k}+\\bar{e}_{n}} & j_{m,\\bar{k}+\\bar{e}_{n}}^{\\prime } & j_{m,\\bar{k}-\\bar{e}_{m}+\\bar{e}_{n}}\\end{array}\\right\\rbrace \\cdot \\left(-\\right)^{2\\pi _{n,\\bar{k}+\\bar{e}_{n}}}\\left\\lbrace \\begin{array}{ccc}\\pi _{n,\\bar{k}+\\bar{e}_{n}}^{\\prime } & \\pi _{n,\\bar{k}+\\bar{e}_{n}} & 1\\\\\\pi _{m,\\bar{k}+\\bar{e}_{n}} & \\pi _{m,\\bar{k}+\\bar{e}_{n}}^{\\prime } & \\pi _{k,\\bar{k}+\\bar{e}_{n}}\\end{array}\\right\\rbrace $ $\\left(-\\right)^{\\pi _{k,\\bar{k}+\\bar{e}_{n}}+\\pi _{m,\\bar{k}+\\bar{e}_{n}}^{\\prime }+\\pi _{n,\\bar{k}+\\bar{e}_{n}}^{\\prime }}\\cdot \\left(-\\right)^{j_{n,\\bar{k}}+j_{n,\\bar{k}+\\bar{e}_{n}}+\\pi _{n,\\bar{k}+\\bar{e}_{n}}+2j_{n,\\bar{k}}}\\left\\lbrace \\begin{array}{ccc}j_{n,\\bar{k}}^{\\prime } & j_{n,\\bar{k}} & 1\\\\\\pi _{n,\\bar{k}+\\bar{e}_{n}} & \\pi _{n,\\bar{k}+\\bar{e}_{n}}^{\\prime } & j_{n,\\bar{k}+\\bar{e}_{n}}\\end{array}\\right\\rbrace $ we can write the complete Matrix element for the gauge group $SU(2)$ : $\\left\\langle \\psi _{\\bar{j}^{\\prime },\\bar{\\pi }^{\\prime }}\\mid \\hat{H}_{YM}\\mid \\psi _{\\bar{j},\\bar{\\pi }}\\right\\rangle =\\frac{1}{2Q^2}\\underset{\\bar{k}}{\\sum }j_{\\bar{k}}\\left(j_{\\bar{k}}+1\\right)+\\frac{1}{Q^2}\\underset{\\bar{k}}{\\sum }\\underset{m<n}{\\sum }$ $ \\mathfrak {P}_{SU(2)}\\left(\\left\\lbrace \\pi \\right\\rbrace _{\\bar{k}},\\left\\lbrace \\pi \\right\\rbrace _{\\bar{k}+\\bar{e}_{m}},\\left\\lbrace \\pi \\right\\rbrace _{\\bar{k}+\\bar{e}_{n}},\\left\\lbrace \\pi \\right\\rbrace _{\\bar{k}+\\bar{e}_{m}+\\bar{e}_{n}};\\left\\lbrace j\\right\\rbrace ;\\pi _{n,\\bar{k}}^{\\prime },\\pi _{m,\\bar{k}}^{\\prime },\\pi _{m,\\bar{k}+\\bar{e}_{m}}^{\\prime },...;j_{n,\\bar{k}}^{\\prime },j_{m,\\bar{k}}^{\\prime },...\\right)$ A similar calculation with the beforehand established calculus for $SU(3)$ gives us the new plaquette term with $\\mathcal {S}_{int}:=\\left\\lbrace s_{j_m,\\bar{k}}, s_{j_n,\\bar{k}}, s_{j_n,\\bar{k}+\\bar{e}_m}, s_{j_m,\\bar{k}+\\bar{e}_n},s_{\\pi _m,\\bar{k}}, s_{\\pi _n,\\bar{k}}, s_{\\pi _n,\\bar{k}+\\bar{e}_m+\\bar{e}_n}\\right.$ , $s_{\\pi _m,\\bar{k}+\\bar{e}_m+\\bar{e}_n}, s_{\\pi _m,\\bar{k}+\\bar{e}_m}$ , $\\left.",
"s_{\\pi _n,\\bar{k}+\\bar{e}_n}, s_{\\pi _n,\\bar{k}+\\bar{e}_m}, s_{\\pi _m, \\bar{k}+\\bar{e}_n} \\right\\rbrace $ which denotes the internal set of multiplicities over which we have to sum this time (in contrast note the absence of an additional sign factor here) $\\underset{\\mathcal {S}_{int}}{\\sum }\\mathfrak {P}_{SU(3)}\\left(\\left\\lbrace \\pi ,s\\right\\rbrace _{\\bar{k}},\\left\\lbrace \\pi ,s\\right\\rbrace _{\\bar{k}+\\bar{e}_{m}},\\left\\lbrace \\pi ,s\\right\\rbrace _{\\bar{k}+\\bar{e}_{n}},\\left\\lbrace \\pi ,s\\right\\rbrace _{\\bar{k}+\\bar{e}_{m}+\\bar{e}_{n}};\\left\\lbrace j\\right\\rbrace ;\\pi _{n,\\bar{k}}^{\\prime },\\pi _{m,\\bar{k}}^{\\prime },\\pi _{m,\\bar{k}+\\bar{e}_{m}}^{\\prime },...; \\right.$ $\\left.s_{n,\\bar{k}}^{\\prime },s_{0,\\bar{k}}^{\\prime },s_{m,\\bar{k}}^{\\prime },...;j_{n,\\bar{k}}^{\\prime },j_{m,\\bar{k}}^{\\prime },...\\right)\\equiv \\underset{\\mathcal {S}_{int}}{\\sum }d_{j_{m,\\bar{k}}}d_{j_{n,\\bar{k}}}d_{j_{m,\\bar{k}+\\bar{e}_n}}d_{j_{n,\\bar{k}+\\bar{e}_m}}\\left( \\underset{i,j=0,1}{\\prod }d_{\\pi _{m,\\bar{k}+i\\bar{e}_n+j\\bar{e}_m}} d_{\\pi _{n,\\bar{k}+i\\bar{e}_n+j\\bar{e}_m}}\\right)\\cdot $ $\\left\\lbrace \\begin{array}{cccc}j_{n,\\bar{k}-\\bar{e}_n} & \\bar{j}_{n,\\bar{k}}^{\\prime } & \\pi _{n,\\bar{k}}^{\\prime } \\\\1 &\\pi _{n,\\bar{k}}& j_{n,\\bar{k}} \\\\s_{n,\\bar{k}}^{\\prime } & s_{n,\\bar{k}} & \\bar{s}_{j_n,\\bar{k}} & s_{\\pi _n,\\bar{k}}\\end{array}\\right\\rbrace \\left\\lbrace \\begin{array}{cccc}\\bar{\\pi }_{k,\\bar{k}} & \\bar{\\pi }_{n,\\bar{k}}^{\\prime } & \\bar{\\pi }_{m,\\bar{k}}^{\\prime } \\\\1 & \\bar{\\pi }_{m,\\bar{k}} & \\pi _{n,\\bar{k}} \\\\s_{0,\\bar{k}}^{\\prime } & s_{0,\\bar{k}} & \\bar{s}_{\\pi _m,\\bar{k}} & \\bar{s}_{\\pi _n,\\bar{k}}\\end{array}\\right\\rbrace \\left\\lbrace \\begin{array}{cccc}j_{m,\\bar{k}-\\bar{e}_m} & \\pi _{m,\\bar{k}}^{\\prime } & \\bar{j}^{\\prime }_{m,\\bar{k}} \\\\1 & \\bar{j}_{m,\\bar{k}} & \\bar{\\pi }_{m,\\bar{k}} \\\\s_{m,\\bar{k}}^{\\prime } & s_{m,\\bar{k}} & s_{\\pi _m,\\bar{k}} & s_{j_m,\\bar{k}}\\end{array}\\right\\rbrace $ $\\left\\lbrace \\begin{array}{cccc}j_{m,\\bar{k}}^{\\prime } & \\pi _{m,\\bar{k}+\\bar{e}_m}^{\\prime } & \\bar{j}_{m,\\bar{k}+\\bar{e}_m} \\\\\\pi _{m,\\bar{k}+\\bar{e}_m} & \\bar{j}_{m,\\bar{k}} & 1 \\\\s^{\\prime }_{m,\\bar{k}+\\bar{e}_m} & s_{j_m,\\bar{k}} & s_{\\pi _m,\\bar{k}+\\bar{e}_m} & s_{m,\\bar{k}+\\bar{e}_m}\\end{array}\\right\\rbrace \\left\\lbrace \\begin{array}{cccc}\\bar{\\pi }^{\\prime }_{m,\\bar{k}+\\bar{e}_m} & \\bar{\\pi }^{\\prime }_{n,\\bar{k}+\\bar{e}_m} & \\bar{\\pi }_{k,\\bar{k}+\\bar{e}_m} \\\\\\bar{\\pi }_{n,\\bar{k}+\\bar{e}_m} & \\pi _{m,\\bar{k}+\\bar{e}_m} & 1 \\\\s^{\\prime }_{0,\\bar{k}+\\bar{e}_m} & \\bar{s}_{\\pi _m,\\bar{k}+\\bar{e}_m}& \\bar{s}_{\\pi _n,\\bar{k}+\\bar{e}_m} & s_{0,\\bar{k}+\\bar{e}_m}\\end{array}\\right\\rbrace $ $\\left\\lbrace \\begin{array}{cccc}\\pi ^{\\prime }_{n,\\bar{k}+\\bar{e}_m} & \\bar{j}^{\\prime }_{n,\\bar{k}+\\bar{e}_m} & j_{n,\\bar{k}+\\bar{e}_m-\\bar{e}_n} \\\\\\bar{j}_{n,\\bar{k}+\\bar{e}_m} & \\bar{\\pi }_{n,\\bar{k}+\\bar{e}_m} & 1 \\\\s^{\\prime }_{n,\\bar{k}+\\bar{e}_m} & s_{\\pi _{n},\\bar{k}+\\bar{e}_m} & \\bar{s}_{j_n,\\bar{k}+\\bar{e}_n} & s_{n,\\bar{k}+\\bar{e}_m}\\end{array}\\right\\rbrace \\left\\lbrace \\begin{array}{cccc}\\pi ^{\\prime }_{m,\\bar{k}+\\bar{e}_n} & j_{m,\\bar{k}+\\bar{e}_n-\\bar{e}_m} & \\bar{j}^{\\prime }_{m,\\bar{k}+\\bar{e}_n} \\\\j_{m,\\bar{k}+\\bar{e}_n} & 1 & \\pi ^{\\prime }_{m,\\bar{k}+\\bar{e}_n} \\\\s^{\\prime }_{m,\\bar{k}+\\bar{e}_n} & s_{\\pi _m,\\bar{k}+\\bar{e}_n} & s_{m,\\bar{k}+\\bar{e}_n} & \\bar{s}_{j_m,\\bar{k}+\\bar{e}_n}\\end{array}\\right\\rbrace $ $\\left\\lbrace \\begin{array}{cccc}\\bar{\\pi }^{\\prime }_{n,\\bar{k}+\\bar{e}_n} & \\bar{\\pi }_{k,\\bar{k}+ \\bar{e}_n} & \\bar{\\pi }^{\\prime }_{m,\\bar{k}+\\bar{e}_n} \\\\\\pi _{m,\\bar{k}+\\bar{e}_n} & 1 & \\bar{\\pi }_{n,\\bar{k}+\\bar{e}_n} \\\\s_{0,\\bar{k}+\\bar{e}_n}^{\\prime } & \\bar{s}_{\\pi _m,\\bar{k}+\\bar{e}_n} & s_{0,\\bar{k}+\\bar{e}_n} & \\bar{s}_{\\pi _m,\\bar{k}+\\bar{e}_n}\\end{array}\\right\\rbrace \\left\\lbrace \\begin{array}{cccc}j^{\\prime }_{n,\\bar{k}+\\bar{e}_m} & \\pi ^{\\prime }_{n,\\bar{k}+\\bar{e}_m+\\bar{e}_n} & \\bar{j}_{n,\\bar{k}+\\bar{e}_m+\\bar{e}_n} \\\\\\pi _{n,\\bar{k}+\\bar{e}_m+\\bar{e}_n} & \\bar{j}_{n,\\bar{k}+\\bar{e}_m} & 1 \\\\s^{\\prime }_{n,\\bar{k}+\\bar{e}_m+\\bar{e}_n} & s_{j_m,\\bar{k}+\\bar{e}_m} & s_{\\pi _n,\\bar{k}+\\bar{e}_m+\\bar{e}_n} & s_{n,\\bar{k}+\\bar{e}_m+\\bar{e}_n}\\end{array}\\right\\rbrace $ $\\left\\lbrace \\begin{array}{cccc}j^{\\prime }_{n,k} & \\bar{j}_{n,\\bar{k}+\\bar{e}_n} & \\pi ^{\\prime }_{n,\\bar{k}+\\bar{e}_n} \\\\\\pi _{n,\\bar{k}+\\bar{e}_n} & 1 & j_{n,\\bar{k}} \\\\s^{\\prime }_{n,\\bar{k}+\\bar{e}_n} & s_{j_n,\\bar{k}} & s_{n,\\bar{k}+\\bar{e}_n} & s_{\\pi _n,\\bar{k}+\\bar{e}_n}\\end{array}\\right\\rbrace \\left\\lbrace \\begin{array}{cccc}\\bar{\\pi }^{\\prime }_{n,\\bar{k}+\\bar{e}_m+\\bar{e}_n} & \\bar{\\pi }^{\\prime }_{m,\\bar{k}+\\bar{e}_m+\\bar{e}_n} & \\bar{\\pi }_{k,\\bar{k}+\\bar{e}_m+\\bar{e}_n} \\\\\\bar{\\pi }_{m,\\bar{k}+\\bar{e}_m+\\bar{e}_n} & \\pi _{n,\\bar{k}+\\bar{e}_m+\\bar{e}_n} & 1 \\\\s^{\\prime }_{0,\\bar{k}+\\bar{e}_m+\\bar{e}_n} & \\bar{s}_{\\pi _m,\\bar{k}+\\bar{e}_m+\\bar{e}_n} & \\bar{s}_{\\pi _m,\\bar{k}+\\bar{e}_m+\\bar{e}_n} & s_{0,\\bar{k}+\\bar{e}_m+\\bar{e}_n}\\end{array}\\right\\rbrace $ $\\left\\lbrace \\begin{array}{cccc}\\pi _{m,\\bar{k}+\\bar{e}_m+\\bar{e}_n} & j^{\\prime }_{m,\\bar{k}+\\bar{e}_n} & \\bar{j}_{m,\\bar{k}+\\bar{e}_m+\\bar{e}_n} \\\\j_{m,\\bar{k}+e_n} & \\bar{\\pi }_{m,\\bar{k}+\\bar{e}_m+\\bar{e}_n} & 1 \\\\s^{\\prime }_{m,\\bar{k}+\\bar{e}_m+\\bar{e}_n} & s_{\\pi _m,\\bar{k}+\\bar{e}_m+\\bar{e}_n} & s_{j_m,\\bar{k}+\\bar{e}_n} & s_{m,\\bar{k}+\\bar{e}_m+\\bar{e}_n}\\end{array}\\right\\rbrace $ So the complete matrix-element is the same as in (REF ) with this sum over the new plaquette term and the new Casimir.",
"Note, that in the action of the Kogut-Susskind-Hamiltonian the group elements of the plaquette are in the defining representation.",
"However the same calculation could be done for an arbitrary $m$ -representation.",
"Since this will be used later, there have been no simplifications in the above expressions, such that one can easily replace $1 \\rightarrow m$ and denote the new plaquette term as $\\mathfrak {P}\\left(\\ldots \\mid m\\right)$ to distinguish it from the QCD case."
],
[
"Einstein-Yang-Mills-Theory in Quantum Gravity",
"To compute the matrix elements of the full Quantum Gravity Yang-Mills-Hamiltonian, we adopt the same notation as in , and denote the gravity-quantum-numbers with $j_i$ and the Yang-Mills-quantum-numbers with $\\underline{j}_i$ , whose gauge group will be set to $SU(3)$ for the remainder of this paper.",
"The basis functions $\\Psi $ on our cubic graph are labelled by $\\left| \\Psi \\left(\\lbrace j\\rbrace \\right) \\underline{\\Psi }\\left(\\lbrace \\underline{j}\\rbrace ,\\lbrace \\underline{\\pi }\\rbrace ;\\lbrace \\underline{s}\\rbrace \\right)\\right\\rangle = \\underset{\\bar{k}\\in \\mathbb {Z}^3}{\\sum } \\left| \\nu \\left(\\lbrace \\pi \\rbrace _{\\bar{k}},\\lbrace j\\vphantom{\\underline{j}}\\rbrace _{\\bar{k}}\\right) \\right\\rangle \\otimes \\left| \\nu \\left(\\lbrace \\underline{\\pi }\\rbrace _{\\bar{k}},\\lbrace \\underline{j}\\rbrace _{\\bar{k}}\\right) \\right\\rangle $ Due to the fact, that the result is quite lengthy and splits up into a lot of sub-cases, we split up this section.",
"The Quantum Gravity Yang-Mills-Hamiltonian $\\hat{H}_{YM}(v)=\\frac{1}{2Q^2}\\left(\\hat{H}_E(v)+\\hat{H}_B(v)\\right)$ consists out of two big parts.",
"The first being the Electric Term and the second one being the Magnetic Term.",
"For both one can look separately at the gravitational degrees of freedom and at the Yang-Mills-degrees of freedom, i.e.",
"the Electric Fluxes and the plaquette part respectively.",
"Each of these four parts is calculated in its corresponding sub-chapter below."
],
[
"Gravity-Part of the Electric Term",
"The Gravity-Part of the YM-Hamiltonian is $tr\\left(\\hat{A}_{j}\\left[\\hat{A}_{j}^{-1},\\sqrt{\\hat{V}}\\right]\\hat{A}_{m}\\left[\\hat{A}_{m}^{-1},\\sqrt{\\hat{V}}\\right]\\right)$ Due to the commutators one gets four different parts.",
"The first one is just the definition of the elements of the action of the Volume: $\\hat{V}\\left|\\nu \\left(\\left\\lbrace \\pi \\right\\rbrace _{\\bar{k}},\\left\\lbrace j\\right\\rbrace _{\\bar{k}}\\right)\\right\\rangle \\equiv \\underset{\\left\\lbrace \\pi \\right\\rbrace _{\\bar{k}}^{2}}{\\tilde{\\sum }}V_{\\bar{k}}\\left(\\left\\lbrace \\pi \\right\\rbrace _{\\bar{k}},\\left\\lbrace \\pi \\right\\rbrace _{\\bar{k}}^{2};\\left\\lbrace j_{\\bar{k}}\\right\\rbrace \\right)\\left|\\nu \\left(\\left\\lbrace \\pi \\right\\rbrace _{\\bar{k}}^{2},\\left\\lbrace j\\right\\rbrace _{\\bar{k}}\\right)\\right\\rangle $ The label $\\bar{k}$ is purely of interest for the valency of the vertex (with $\\bar{k}\\in \\mathbb {Z}^3$ there are six edges meeting at the node).",
"Moreover one realizes that the Volume operator only changes the intertwiners, not the graph itself.",
"We have also introduced the weighted sum: $\\underset{j}{\\tilde{\\sum }}=\\underset{j}{\\sum }d_j$ .",
"For the second one the action of the Volume on a non-gauge invariant node is needed.",
"The notation here ($\\sqrt{V}_{\\bar{k}+\\bar{e}_j}$ ) means that on the edge in $j$ -direction a non gauge-invariant edge in $m$ representation is glued.",
"The additional representation $j_j$ that changes to $j_j^2$ , where one needs to sum over, is also displayed after the first semicolon: $\\sqrt{V}_{\\bar{k}+\\bar{e}_j}\\left(\\ldots ;j_j,j_j^2;\\ldots \\mid m\\right)$ .",
"If $j=1,2,3$ only half of the edges are calculated.",
"For the remaining ones, carrying the representation $\\left(j_{j,\\bar{k}-\\bar{e}_{j}}\\right)$ , the calculation broadly remains exactly the same when replacing $j_{j,\\bar{k}-\\bar{e}_{j}}\\Leftrightarrow j_{j,\\bar{k}}$ .",
"However one wants to work on a vertex where all edges are outgoing to maximize the degree of symmetry, which explains the (temporary) additional sign in the second line of the computation.",
"Moreover one also has to switch the orientation of the vertex itself, since the “+”-sign would elsewhere become “-”.",
"To combine both cases in one in the following, we will introduce the parameter $p_{j}\\in \\left\\lbrace \\bar{0},\\bar{e}_{j}\\right\\rbrace $ , which distinguishes the cases, using $j_{j,\\bar{k}-\\bar{p}_{j}}$ and $j_{j,\\bar{k}-\\bar{e}_{j}+\\bar{p}_{j}}$ .",
"So for one we will get a sign of ${\\left|\\bar{p}_{j}\\right|\\left(\\pi _{j,\\bar{k}}+j_{j,\\bar{k}}+j_{j,\\bar{k}+\\bar{e}_{j}}\\right)}$ to ensure that the sign at the vertex is always “+”.",
"With all of this the action for the second part is (where one also uses the $SU(2)$ version of the orthogonality relation (REF ) in the last line after having coupled the last holonomy to the graph): $tr\\left(\\sqrt{\\hat{V}}\\hat{A}_{m,\\bar{p}_{m}}\\sqrt{\\hat{V}}\\hat{A}_{m,\\bar{p}_{m}}^{-1}\\right)\\left|\\nu \\left(\\left\\lbrace \\pi \\right\\rbrace _{\\bar{k}};\\left\\lbrace j\\right\\rbrace _{\\bar{k}}\\right)\\right\\rangle =$ $=tr\\left(\\sqrt{\\hat{V}}\\hat{A}_{m,\\bar{p}_{m}}\\sqrt{\\hat{V}}\\right)\\left(-\\right)^{2\\left(j_{1,\\bar{k}-\\bar{e}_{1}}+j_{2,\\bar{k}-\\bar{e}_{2}}+j_{3,\\bar{k}-\\bar{e}_{3}}\\right)}\\left(-\\right)^{\\left|\\bar{p}_{m}\\right|\\left(\\pi _{m,\\bar{k}}+j_{m,\\bar{k}}+j_{m,\\bar{k}+\\bar{e}_{m}}\\right)}$ $\\underset{j_{m,\\bar{k}-\\bar{p}_{m}}^{2}}{\\tilde{\\sum }}\\left(-\\right)^{2j_{m,\\bar{k}-\\bar{p}_{m}}^{2}}\\Biggl |\\begin{array}{c}\\includegraphics [scale=0.5]{thesis-4_4_1pic03-edge+hol3}\\end{array}\\Biggr \\rangle =$ $=tr\\left(\\sqrt{\\hat{V}}\\hat{A}_{m,\\bar{p}_{m}}\\right)\\sqrt{V}_{\\bar{k}+\\bar{e}_{m}-2\\bar{p}_{m}}\\left(\\left\\lbrace \\pi \\right\\rbrace _{\\bar{k}},\\left\\lbrace \\pi \\right\\rbrace _{\\bar{k}}^{2};j_{m,\\bar{k}-\\bar{p}_{m}},j_{m,\\bar{k}-\\bar{p}_{m}}^{3};\\ldots j_{m,\\bar{k}-\\bar{p}_{m}}^{2}\\ldots \\mid m\\right)$ $\\left(-\\right)^{2j_{m,\\bar{k}-\\bar{p}_{m}}^{2}}\\Biggl |\\begin{array}{c}\\includegraphics [scale=0.5]{thesis-4_4_1pic19-edge+hol5}\\end{array}\\Biggr \\rangle =$ $=\\underset{j_{m,\\bar{k}+\\bar{p_{m}}}^{2},\\left\\lbrace \\pi \\right\\rbrace _{\\bar{k}}^{3},\\left\\lbrace \\pi \\right\\rbrace _{\\bar{k}}^{2}}{\\tilde{\\sum }}\\sqrt{V}_{\\bar{k}+\\bar{e}_{m}-2\\bar{p}_{m}}\\left(\\left\\lbrace \\pi \\right\\rbrace _{\\bar{k}},\\left\\lbrace \\pi \\right\\rbrace _{\\bar{k}}^{2};j_{m,\\bar{k}-\\bar{p}_{m}},j_{m,\\bar{k}-\\bar{p}_{m}};\\ldots j_{m,\\bar{k}-\\bar{p}_{m}}^{2}\\ldots \\right)\\cdot $ $\\cdot \\sqrt{V}_{\\bar{k}}\\left(\\left\\lbrace \\pi \\right\\rbrace _{\\bar{k}}^{2},\\left\\lbrace \\pi \\right\\rbrace _{\\bar{k}}^{3};\\left\\lbrace j\\right\\rbrace _{\\bar{k}}\\right)\\cdot \\left(-\\right)^{\\left|\\bar{p}_{m}\\right|\\left(\\pi _{m,\\bar{k}}^{3}+\\pi _{m,\\bar{k}}\\right)}\\left|\\nu \\left(\\left\\lbrace \\pi \\right\\rbrace _{\\bar{k}}^{3};\\left\\lbrace j\\right\\rbrace _{\\bar{k}}\\right)\\right\\rangle $ And correspondingly the third part is: $tr\\left(\\hat{A}_{j,\\bar{p}_{j}}\\sqrt{\\hat{V}}\\hat{A}_{j,\\bar{p}_{j}}^{-1}\\sqrt{\\hat{V}}\\right)\\left|\\nu \\left(\\left\\lbrace \\pi \\right\\rbrace _{\\bar{k}};\\left\\lbrace j\\right\\rbrace _{\\bar{k}}\\right)\\right\\rangle =$ $=\\underset{\\left\\lbrace \\pi \\right\\rbrace _{\\bar{k}}^{2},\\left\\lbrace \\pi \\right\\rbrace _{\\bar{k}}^{3},j_{j,\\bar{k}-\\bar{p}_{j}}^{2}}{\\tilde{\\sum }}\\left(-\\right)^{\\left|\\bar{p}_{j}\\right|\\left(\\pi _{j,\\bar{k}}-\\pi _{j,\\bar{k}}^{3}\\right)}\\sqrt{V_{\\bar{k}}}\\left(\\left\\lbrace \\pi \\right\\rbrace _{\\bar{k}},\\left\\lbrace \\pi \\right\\rbrace _{\\bar{k}}^{2};\\left\\lbrace j\\right\\rbrace _{\\bar{k}}\\right)\\cdot $ $\\cdot \\sqrt{V}_{\\bar{k}+\\bar{e}_{j}-2\\bar{p}_{j}}\\left(\\left\\lbrace \\pi \\right\\rbrace _{\\bar{k}}^{2},\\left\\lbrace \\pi \\right\\rbrace _{\\bar{k}}^{3};j_{j,\\bar{k}-\\bar{p}_{j}},j_{j,\\bar{k}-\\bar{p}_{j}};\\ldots j_{j,\\bar{k}-\\bar{p}_{j}}^{2}\\ldots \\mid m\\right)\\left|\\nu \\left(\\left\\lbrace \\pi \\right\\rbrace ^3_{\\bar{k}};\\left\\lbrace j\\right\\rbrace _{\\bar{k}}\\right)\\right\\rangle $ The fourth and last part of REF needs some more detailed treatment, since we deal now, with two holonomies that are glued to the graph, and that may go in different directions.",
"The term of interest is $\\hat{A}_j \\sqrt{\\hat{V}}\\hat{A}_j^-1\\hat{A}_m \\sqrt{\\hat{V}}\\hat{A}_m^-1$ , where $j,m$ denote the different directions of the glued edges.",
"Summing over all possible combinations of choosing two (possibly the same) edges emanating from one vertex $\\bar{k}$ we have 36 combinations, from which many due to symmetry reasons give the same result.",
"In total we have thus only to distinguish three case: Both holonomies may i) lie on the same edge $\\left(j_{m,\\bar{k}}=j_{j,\\bar{k}}\\right)$ ii) lie on parallel edges $\\left(j_{m,\\bar{k}+\\bar{e}_{m}}=j_{j,\\bar{k}}\\right)$ iii) go in different directions For i) it is obvious that the holonomies in the middle cancel, leaving us with a rather simple expression: $tr\\left(\\hat{A}_{j,\\bar{p}_{j}}\\hat{V}\\hat{A}_{j,\\bar{p}_j}^{-1}\\right)\\left|\\nu \\left(\\left\\lbrace \\pi \\right\\rbrace _{\\bar{k}},\\left\\lbrace j\\right\\rbrace _{\\bar{k}}\\right)\\right\\rangle =$ $=\\underset{j_{j,\\bar{k}-\\bar{p}_{j}}^{2}\\left\\lbrace \\pi \\right\\rbrace _{\\bar{k}}^{2}}{\\tilde{\\sum }}\\left(-\\right)^{\\left|\\bar{p}_{j}\\right|\\left(\\pi _{j,\\text{$\\bar{k}$}}+\\pi _{j,\\bar{k}}^{2}\\right)}V_{\\bar{k}+\\bar{e}_{j}-2\\bar{p}_{j}}\\left(\\left\\lbrace \\pi \\right\\rbrace _{\\bar{k}},\\left\\lbrace \\pi \\right\\rbrace _{\\bar{k}}^{2};j_{j,\\bar{k}-\\bar{p}_{j}},j_{j,\\bar{k}-\\bar{p}_{j}};\\ldots j_{j,\\bar{k}-\\bar{p}_{j}}^{2}\\ldots \\mid m\\right)\\left|\\nu \\left(\\left\\lbrace \\pi \\right\\rbrace _{\\bar{k}}^{2},\\left\\lbrace j\\right\\rbrace _{\\bar{k}}\\right)\\right\\rangle $ The second part of course incorporates now a change from one link to the other and back to close the trace of the holonomies at the end.",
"As one can easily see, the structures appearing again look similar to equation REF from the appendix and thus represent 6j-symbols.",
"Note, moreover, that the open edges in the $m$ -representation in the third line denotes the open ends of the holonomy.",
"One is attached infinitesimal close to the vertex, hence the action of the volume elements also changes the link between these two, and the other open end (on the $j_{j,\\bar{k}-\\bar{p}_j}$ -edge) is attached after the group element, which we have suppressed and trivially shifted to the $j^2_{j,\\bar{k}-\\bar{p}_j}$ -edge.",
"$tr\\left(\\hat{A}_{j,p_j}\\sqrt{\\hat{V}}\\hat{A}_{j,p_j}^{-1}\\hat{A}_{m,p_m}\\sqrt{\\hat{V}}\\hat{A}_{m,p_m}^{-1}\\right)\\left|\\nu (\\left\\lbrace \\pi \\right\\rbrace _{\\bar{k}},\\left\\lbrace j\\right\\rbrace _{\\bar{k}})\\right\\rangle =\\left(-\\right)^{2\\overset{3}{\\underset{i=1}{\\sum }}j_{i,\\bar{k}-\\bar{e}_{i}}+\\left|\\bar{p}_{j}\\right|\\left(j_{j,\\bar{k}-\\bar{e}_{j}}+j_{j,\\bar{k}}+\\pi _{j,\\bar{k}}\\right)}\\cdot $ $\\cdot \\underset{\\begin{array}{c}j_{j,\\bar{k}-\\bar{e}_{j}+\\bar{p}_{j}}^{2},\\left\\lbrace \\pi \\right\\rbrace _{\\bar{k}}^{2},\\\\j_{j,\\bar{k}-\\bar{e}_{j}+\\bar{p}_{j}}^{3},\\\\j_{j,\\bar{k}-\\bar{p}_{J}}^{2},j_{j,\\bar{k}-\\bar{e}_{j}+\\bar{p}_{j}}^{4}\\\\\\end{array}}{\\tilde{\\sum }}\\left(-\\right)^{2j_{j,\\bar{k}-\\bar{e}_{j}+\\bar{p}_{j}}^{2}}\\sqrt{V}_{\\bar{k}-\\bar{e}_{j}+2\\bar{p}_{j}}\\left(\\left\\lbrace \\pi \\right\\rbrace _{\\bar{k}},\\left\\lbrace \\pi \\right\\rbrace _{\\bar{k}}^{2};j_{j,\\bar{k}-\\bar{e}_{j}+\\bar{p}_{j}},j_{j,\\bar{k}-\\bar{e}_{j}+\\bar{p}_{j}}^{3};\\ldots j_{j,\\bar{k}-\\bar{e}_{j}+\\bar{p}_{j}}^{2}\\ldots \\mid m\\right)\\cdot $ $\\cdot \\left(-\\right)^{2j_{j,\\bar{k}-\\bar{p}_{j}}^{2}+2j_{j,\\bar{k}-\\bar{e}_{j}+\\bar{p}_{j}}^{3}}tr\\left(\\hat{A}_{j,\\bar{p}_{j}}\\sqrt{\\hat{V}}\\right)\\Biggl |\\begin{array}{c}\\includegraphics [scale=0.7]{thesis-4_4_1pic08-edge+holcoupled}\\end{array}\\Biggr \\rangle =$ $=\\underset{\\begin{array}{cc}j_{j,\\bar{k}-\\bar{e}_{j}+\\bar{p}_{j}}^{2\\ldots 5}, & j_{j,\\bar{k}-\\bar{p}_{J}}^{2},\\\\\\left\\lbrace \\pi \\right\\rbrace _{\\bar{k}}^{2}, & \\left\\lbrace \\pi \\right\\rbrace _{\\bar{k}}^{3},\\end{array}}{\\tilde{\\sum }}\\sqrt{V}_{\\bar{k}-\\bar{e}_{j}+2\\bar{p}_{j}}\\left(\\left\\lbrace \\pi \\right\\rbrace _{\\bar{k}},\\left\\lbrace \\pi \\right\\rbrace _{\\bar{k}}^{2};j_{j,\\bar{k}-\\bar{e}_{j}+\\bar{p}_{j}},j_{j,\\bar{k}-\\bar{e}_{j}+\\bar{p}_{j}}^{3};\\ldots j_{j,\\bar{k}-\\bar{e}_{j}+\\bar{p}_{j}}^{2}\\ldots \\mid m\\right)$ $\\sqrt{V}_{\\bar{k}-\\bar{e}_{j}+2\\bar{p}_{j}}\\left(\\left\\lbrace \\pi \\right\\rbrace _{\\bar{k}}^{2},\\left\\lbrace \\pi \\right\\rbrace _{\\bar{k}}^{3};j_{j,\\bar{k}-\\bar{e}_{j}+\\bar{p}_{j}}^{4},j_{j,\\bar{k}-\\bar{e}_{j}+\\bar{p}_{j}}^{5};\\ldots j_{j,\\bar{k}-\\bar{p}_{j}}^{2}\\ldots \\mid m\\right)$ $\\left(-\\right)^{\\left|\\bar{p}_{j}\\right|\\left(\\pi _{j,\\bar{k}}+\\pi _{j,\\bar{k}}^{3}\\right)}(-)^{\\pi ^2_{\\bar{k}}-\\pi ^3_{\\bar{k}}}(-)^{2j^5_{j,\\bar{k}-\\bar{e}_j}+\\bar{p}_j}\\left\\lbrace \\begin{array}{ccc}j_{j,\\bar{k}-\\bar{e}_j+\\bar{p}_j} & j^2_{j,\\bar{k}-\\bar{e}_j+\\bar{p}_j} & m \\\\j^3_{j,\\bar{k}-\\bar{e}_j+\\bar{p}_j} & j^4_{j,\\bar{k}-\\bar{e}_j+\\bar{p}_j} & m\\end{array}\\right\\rbrace $ $\\left\\lbrace \\begin{array}{ccc}j^4_{j,\\bar{k}-\\bar{e}_j+\\bar{p}_j} & j^3_{j,\\bar{k}-\\bar{e}_j+\\bar{p}_j} & m \\\\j_{j,\\bar{k}-\\bar{p}_j} & j^2_{j,\\bar{k}-\\bar{p}_j} & \\pi ^2_{j\\bar{k}}\\end{array}\\right\\rbrace \\left\\lbrace \\begin{array}{ccc}j_{j,\\bar{k}-\\bar{e}_j+\\bar{p}_j} & j^5_{j,\\bar{k}-\\bar{e}_j+\\bar{p}_j} & m \\\\j^2_{j,\\bar{k}-\\bar{p}_j} & j_{j,\\bar{k}-\\bar{p}_j} & \\pi ^3_{j,\\bar{k}}\\end{array}\\right\\rbrace \\left|\\nu \\left(\\lbrace \\pi \\rbrace ^3_{\\bar{k}},\\lbrace j\\rbrace _{\\bar{k}}\\right)\\right\\rangle $ Note that the additional sign of $\\pi ^2_{\\bar{k}}-\\pi ^3_{\\bar{k}}$ stems from the fact, that one has to reorient the vertices in between to act with the second Volume-operator in the way it was defined on a node with given orientation.",
"For iii) things get again more complicated.",
"We have to switch from one edge to another edge, which does not lie in the same direction.",
"Explicitly, we are interested in the action of the holonomy $\\hat{A}^{-1}_{j,\\bar{p}_j}\\hat{A}_{m,\\bar{p}_m}$ on a vertex, which we will find useful to write in the following form, where $\\sigma $ gives us the sign of the permutation of $m,j,q$ : $\\left|\\nu \\left(\\left\\lbrace \\pi \\right\\rbrace _{\\bar{k}},\\left\\lbrace j\\right\\rbrace _{\\bar{k}}\\right)\\right\\rangle =\\left(-\\right)^{2\\underset{i=1}{\\overset{3}{\\sum }}j_{i,\\bar{k}-\\bar{e}_{i}}}\\left(-\\right)^{\\left|\\bar{p}_{j}\\right|\\left(\\pi _{j,\\bar{k}}+j_{j,\\bar{k}}+j_{j,\\bar{k}-\\bar{e}_{j}}\\right)}\\left(-\\right)^{\\left(1-\\left|\\bar{p}_{m}\\right|\\right)\\left(\\pi _{m,\\bar{k}}+j_{m,\\bar{k}}+j_{m,\\bar{k}-\\bar{e}_{m}}\\right)}$ $\\left(-\\right)^{\\sigma \\left(m,j,q\\right)\\left(\\pi _{j,\\bar{k}}+\\pi _{m,\\bar{k}}+\\pi _{q,\\bar{k}}\\right)}\\Biggl |\\begin{array}{c}\\includegraphics [scale=0.8]{thesis-4_4_1pic11-node}\\end{array}\\Biggr \\rangle $ Once our Hamiltonian acts on the state, we see that traversing the node results in a couple of 6j-symbols (four when going from $j_{m,\\bar{k}-\\bar{p}_m}$ to $j_{j,\\bar{k}-\\bar{p}_j}$ and three when going back.",
"Remember that in between we have to bring the signs back into an orientation such that $\\hat{V}$ can act and after its action we have to restore the given orientation, such that one can close the holonomies.",
"In total one ends up with a fairly complicated expression: $tr\\left(\\hat{A}_{j,\\bar{p}_{j}}\\sqrt{\\hat{V}}\\right)\\underset{\\begin{array}{c}j_{m,\\bar{k}-\\bar{p}_{m}}^{2},j_{m,\\bar{k}-\\bar{p}_{m}}^{3}\\\\\\left\\lbrace \\pi \\right\\rbrace _{\\bar{k}}^{2}\\end{array}}{\\tilde{\\sum }}\\left(-\\right)^{2j_{m,\\bar{k}-\\bar{p}_{m}}^{2}}\\left(-\\right)^{2\\underset{i=1}{\\overset{3}{\\sum }}j_{i,\\bar{k}-\\bar{e}_{i}}+\\left|\\bar{p}_{m}\\right|\\left(j_{m,\\bar{k}}+j_{m,\\bar{k}-\\bar{e}_{m}}+\\pi _{m,\\bar{k}}\\right)}\\left(-\\right)^{\\sigma \\left(m,j,q\\right)\\left(\\pi _{j,\\bar{k}}^{2}+\\pi _{m,\\bar{k}}^{2}+\\pi _{q,\\bar{k}}^{2}\\right)}$ $\\left(-\\right)^{\\left|\\bar{p}_{j}\\right|\\left(j_{j,\\bar{k}}+j_{j,\\bar{k}-\\bar{e}_{j}}+\\pi _{j,\\bar{k}}\\right)}\\sqrt{V}_{\\bar{k}+\\bar{e}_{m}-2\\bar{p}_{m}}\\left(\\left\\lbrace \\pi \\right\\rbrace _{\\bar{k}},\\left\\lbrace \\pi \\right\\rbrace _{\\bar{k}}^{2};j_{m,\\bar{k}-\\bar{p}_{m}},j_{m,\\bar{k}-\\bar{p}_{m}}^{3};\\ldots j_{m,\\bar{k}-\\bar{p}_{m}}^{2}\\ldots \\mid m\\right)$ $\\cdot \\underset{\\begin{array}{c}j_{m,\\bar{k}-\\bar{p}_{m}}^{4},j_{j,\\bar{k}-\\bar{p}_{j}}^{2}\\\\\\left\\lbrace \\pi \\right\\rbrace _{\\bar{k}}^{2},\\pi _{m,\\bar{k}}^{3},\\pi _{j,\\bar{k}}^{3}\\end{array}}{\\tilde{\\sum }}\\left(-\\right)^{2j_{m,\\bar{k}-\\bar{p}_{m}}^{3}+2\\pi _{m,\\bar{k}}^{2}+2\\pi _{j,\\bar{k}}^{3}+2j_{j,\\bar{k}-\\bar{p}_{j}}^{2}}\\Biggl |\\begin{array}{c}\\includegraphics [scale=0.8]{thesis-4_4_1pic15-nodecoupled}\\end{array}\\Biggr \\rangle =$ $=tr\\left(\\hat{A}_{j,\\bar{p}_{j}}\\right)\\underset{\\begin{array}{c}j_{m,\\bar{k}-\\bar{p}_{m}}^{2\\ldots 4},j_{j,\\bar{k}-\\bar{p}_{j}}^{2}\\\\\\left\\lbrace \\pi \\right\\rbrace _{\\bar{k}}^{2},\\pi _{m,\\bar{k}}^{3},\\pi _{j,\\bar{k}}^{3}\\\\\\left\\lbrace \\pi \\right\\rbrace _{\\bar{k}}^{4},j_{m,\\bar{k}-\\bar{p}_{m}}^{5}\\end{array}}{\\tilde{\\sum }}\\left(-\\right)^{2\\underset{i=1}{\\overset{3}{\\sum }}j_{i,\\bar{k}-\\bar{e}_{i}}}\\left(-\\right)^{\\sigma \\left(m,j,q\\right)\\left(\\pi _{j,\\bar{k}}^{2}+\\pi _{m,\\bar{k}}^{2}+2\\pi _{q,\\bar{k}}^{2}+\\pi _{j,\\bar{k}}^{3}+\\pi _{m,\\bar{k}}^{3}+\\pi _{j,\\bar{k}}^{4}+\\pi _{m,\\bar{k}}^{4}+\\pi _{q,\\bar{k}}^{4}\\right)}$ $\\left(-\\right)^{\\left|\\bar{p}_{m}\\right|\\left(j_{m,\\bar{k}}+j_{m,\\bar{k}-\\bar{e}_{m}}+\\pi _{m,\\bar{k}}+\\pi _{m,\\bar{k}}^{3}+j_{m,\\bar{k}-\\bar{p}_{m}}^{4}+2j_{m,\\bar{k}-\\bar{e}_{m}+\\bar{p}_{m}}+j_{m,\\bar{k}-\\bar{p}_{m}}^{5}+\\pi _{m,\\bar{k}}^{4}\\right)}$ $\\left(-\\right)^{\\left|\\bar{p}_{j}\\right|\\left(j_{j,\\bar{k}}+j_{j,\\bar{k}-\\bar{e}_{j}}+\\pi _{j,\\bar{k}}+\\pi _{j,\\bar{k}}^{3}+2j_{j,\\bar{k}-\\bar{p}_{j}}^{2}+2j_{j,\\bar{k}-\\bar{e}_{j}+\\bar{p}_{j}}+\\pi _{j,\\bar{k}}^{4}\\right)}(-)^{m+j_{m,\\bar{k}-\\bar{p}_m}+j_{m,\\bar{k}-\\bar{e}_m+\\bar{p}_m}+j^2_{j,\\bar{k}-\\bar{p}_j}+j_{j,\\bar{k}-\\bar{e}_j+\\bar{p}_j}}$ $(-)^{\\pi ^3_{m,\\bar{k}}+\\pi ^3_{j,\\bar{k}}+\\pi ^2_{m,\\bar{k}}+\\pi ^2_{j,\\bar{k}}+\\pi ^2_{q,\\bar{k}}}\\sqrt{V}_{\\bar{k}+\\bar{e}_{m}-2\\bar{p}_{m}}\\left(\\left\\lbrace \\pi \\right\\rbrace _{\\bar{k}},\\left\\lbrace \\pi \\right\\rbrace _{\\bar{k}}^{2};j_{m,\\bar{k}-\\bar{p}_{m}},j_{m,\\bar{k}-\\bar{p}_{m}}^{3};\\ldots j_{m,\\bar{k}-\\bar{p}_{m}}^{2}\\ldots \\mid m\\right)$ $\\sqrt{V}_{\\bar{k}+\\bar{e}_{m}-2\\bar{p}_{m}}\\left(\\pi _{m,\\bar{k}}^{3},\\pi _{j,\\bar{k}}^{3},\\pi _{q,\\bar{k}}^{2},\\left\\lbrace \\pi \\right\\rbrace _{\\bar{k}}^{4};j_{m,\\bar{k}-\\bar{p}_{m}}^{4},j_{m,\\bar{k}-\\bar{p}_{m}}^{5};\\ldots j_{j,\\bar{k}-\\bar{p}_{m}}^{2}\\ldots \\mid m\\right)$ $\\left\\lbrace \\begin{array}{ccc}\\pi _{j,\\bar{k}}^{3} & \\pi _{j,\\bar{k}}^{2} & m\\\\j_{j,\\bar{k}-\\bar{p}_{j}} & j_{j,\\bar{k}-\\bar{p}_{j}}^{2} & j_{j,\\bar{k}+\\bar{e}_{j}-\\bar{p}_{j}}\\end{array}\\right\\rbrace \\left\\lbrace \\begin{array}{ccc}\\pi _{q,\\bar{k}}^{2} & \\pi _{j,\\bar{k}}^{2} & \\pi _{m,\\bar{k}}^{2}\\\\m & \\pi _{m,\\bar{k}}^{3} & \\pi _{j,\\bar{k}}^{3}\\end{array}\\right\\rbrace \\left\\lbrace \\begin{array}{ccc}m & j_{m,\\bar{k}-\\bar{p}_{m}}^{3} & j_{m,\\bar{k}-\\bar{p}_{m}}\\\\m & j_{m,\\bar{k}-\\bar{p}_{m}} & j_{m,\\bar{k}-\\bar{p}_{m}}^{4}\\end{array}\\right\\rbrace $ $\\left\\lbrace \\begin{array}{ccc}j_{m,\\bar{k}-\\bar{e}_{m}+\\bar{p}_{m}} & \\pi _{m,\\bar{k}}^{2} & j_{m,\\bar{k}-\\bar{p}_{m}}^{3}\\\\m & j_{m,\\bar{k}-\\bar{p}_{m}}^{4} & \\pi _{m,\\bar{k}}^{3}\\end{array}\\right\\rbrace \\Biggl |\\begin{array}{c}\\includegraphics [scale=0.7]{thesis-4_4_1pic17-noderecoupled2}\\end{array}\\Biggr \\rangle =$ $=\\underset{\\begin{array}{c}j_{m,\\bar{k}-\\bar{p}_{m}}^{2\\ldots 4},j_{j,\\bar{k}-\\bar{p}_{j}}^{2}\\\\\\left\\lbrace \\pi \\right\\rbrace _{\\bar{k}}^{2},\\pi _{m,\\bar{k}}^{3},\\pi _{j,\\bar{k}}^{3}, \\left\\lbrace \\pi \\right\\rbrace _{\\bar{k}}^{4}\\\\j_{m,\\bar{k}-\\bar{p}_{m}}^{5}, \\pi ^5_{m,\\bar{k}},\\pi ^5_{j,\\bar{k}}\\end{array}}{\\tilde{\\sum }}\\left(-\\right)^{\\sigma \\left(m,j,q\\right)\\left(\\pi _{j,\\bar{k}}^{2}+\\pi _{m,\\bar{k}}^{2}+2\\pi _{q,\\bar{k}}^{2}+\\pi _{j,\\bar{k}}^{3}+\\pi _{m,\\bar{k}}^{3}+\\pi _{j,\\bar{k}}^{4}+\\pi _{m,\\bar{k}}^{4}+2\\pi _{q,\\bar{k}}^{4}+\\pi _{m,\\bar{k}}^{5}+\\pi _{j,\\bar{k}}^{5}\\right)}$ $\\left(-\\right)^{\\left|\\bar{p}_{m}\\right|\\left(2j_{m,\\bar{k}-\\bar{p}_{m}}+j_{m,\\bar{k}-\\bar{p}_{m}}^{4}+j_{m,\\bar{k}-\\bar{p}_{m}}^{5}+\\pi _{m,\\bar{k}}+\\pi _{m,\\bar{k}}^{3}+\\pi _{m,\\bar{k}}^{4}+\\pi _{m,\\bar{k}}^{5}\\right)}\\left(-\\right)^{\\left|\\bar{p_{j}}\\right|\\left(\\pi _{j,\\bar{k}}+\\pi _{j,\\bar{k}}^{3}+\\pi _{j,\\bar{k}}^{4}+\\pi _{j,\\bar{k}}^{5}+2m\\right)}$ $(-)^{2\\pi ^4_{q,\\bar{k}}+\\pi ^5_{j,\\bar{k}}+\\pi ^4_{m,\\bar{k}}+\\pi ^4_m,\\bar{k}+\\pi ^4_{j,\\bar{k}}+\\pi ^4_{q,\\bar{k}}+\\pi ^3_{m,\\bar{k}}+\\pi ^3_{j,\\bar{k}}+\\pi ^2_{m,\\bar{k}}+\\pi ^2_{m,\\bar{k}}+\\pi ^2_{j,\\bar{k}}+\\pi ^2_{q,\\bar{k}}}$ $\\sqrt{V}_{\\bar{k}+\\bar{e}_{m}-2\\bar{p}_{m}}\\left(\\left\\lbrace \\pi \\right\\rbrace _{\\bar{k}},\\left\\lbrace \\pi \\right\\rbrace _{\\bar{k}}^{2};j_{m,\\bar{k}-\\bar{p}_{m}},j_{m,\\bar{k}-\\bar{p}_{m}}^{3};\\ldots j_{m,\\bar{k}-\\bar{p}_{m}}^{2}\\ldots \\mid m\\right)$ $\\sqrt{V}_{\\bar{k}+\\bar{e}_{m}-2\\bar{p}_{m}}\\left(\\pi _{m,\\bar{k}}^{3},\\pi _{j,\\bar{k}}^{3},\\pi _{q,\\bar{k}}^{2},\\left\\lbrace \\pi \\right\\rbrace _{\\bar{k}}^{4};j_{m,\\bar{k}-\\bar{p}_{m}}^{4},j_{m,\\bar{k}-\\bar{p}_{m}}^{5};\\ldots j_{j,\\bar{k}-\\bar{p}_{m}}^{2}\\ldots \\mid m\\right)$ $\\left\\lbrace \\begin{array}{ccc}\\pi _{j,\\bar{k}}^{3} & \\pi _{j,\\bar{k}}^{2} & m\\\\j_{j,\\bar{k}-\\bar{p}_{j}} & j_{j,\\bar{k}-\\bar{p}_{j}}^{2} & j_{j,\\bar{k}+\\bar{e}_{j}-\\bar{p}_{j}}\\end{array}\\right\\rbrace \\left\\lbrace \\begin{array}{ccc}\\pi _{q,\\bar{k}}^{2} & \\pi _{j,\\bar{k}}^{2} & \\pi _{m,\\bar{k}}^{2}\\\\m & \\pi _{m,\\bar{k}}^{3} & \\pi _{j,\\bar{k}}^{3}\\end{array}\\right\\rbrace \\left\\lbrace \\begin{array}{ccc}m & j_{m,\\bar{k}-\\bar{p}_{m}}^{3} & j_{m,\\bar{k}-\\bar{p}_{m}}\\\\m & j_{m,\\bar{k}-\\bar{p}_{m}} & j_{m,\\bar{k}-\\bar{p}_{m}}^{4}\\end{array}\\right\\rbrace $ $\\left\\lbrace \\begin{array}{ccc}j_{m,\\bar{k}-\\bar{e}_{m}+\\bar{p}_{m}} & \\pi _{m,\\bar{k}}^{2} & j_{m,\\bar{k}-\\bar{p}_{m}}^{3}\\\\m & j_{m,\\bar{k}-\\bar{p}_{m}}^{4} & \\pi _{m,\\bar{k}}^{3}\\end{array}\\right\\rbrace \\left\\lbrace \\begin{array}{ccc}j_{m,\\bar{k}-\\bar{e}_{m}+\\bar{p}_{m}} & \\pi _{m,\\bar{k}}^{4} & j_{m,\\bar{k}-\\bar{p}_{m}}^{5}\\\\m & j_{m,\\bar{k}-\\bar{p}_{m}} & \\pi _{m,\\bar{k}}^{5}\\end{array}\\right\\rbrace \\left\\lbrace \\begin{array}{ccc}\\pi _{m,\\bar{k}}^{5} & \\pi _{m,\\bar{k}}^{4} & m\\\\\\pi _{j,\\bar{k}}^{4} & \\pi _{j,\\bar{k}}^{5} & \\pi _{q,\\bar{k}}^{4}\\end{array}\\right\\rbrace $ $\\left\\lbrace \\begin{array}{ccc}\\pi _{j,\\bar{k}}^{5} & \\pi _{j,\\bar{k}}^{4} & m\\\\j_{j,\\bar{k}-\\bar{p}_{j}}^{2} & j_{j,\\bar{k}-\\bar{p}_{j}} & j_{j,\\bar{k}-\\bar{e}_{j}+\\bar{p}_{j}}\\end{array}\\right\\rbrace \\left|\\nu \\left(\\pi _{m,\\bar{k}}^{5},\\pi _{j,\\bar{k}}^{5},\\pi _{q,\\bar{k}}^{4};\\left\\lbrace j\\right\\rbrace _{\\bar{k}}\\right)\\right\\rangle $"
],
[
"Gluon electric Fluxes of the Electric Term",
"The Electric Part of the Hamiltonian is $\\underline{\\hat{E}}_I(e_1)\\underline{\\hat{E}}_I(e_2)$ where $e_1$ and $e_2$ correspond again to all possible tuples of edges incident at a vertex $v$ .",
"The Electric Fluxes $\\underline{\\hat{E}}_{I}(j)$ themselves are the grasping operators, whose action on a group element has been defined in (REF ).",
"The operator adds a generator of the Lie-Algebra, which can be viewed as a new intertwiner on the holonomy in the defining (i.e.",
"$\\underline{j}=1$ ) representation.",
"Hence the action is determined up to a normalization factor, which depends on the gauge group and possibly also on the multiplicity-factor corresponding to the chosen intertwiner.",
"However it is easy to check, that when choosing an arbitrary $\\underline{s}_I$ multiplicity everywhere, the normalization does not depend on it and becomes $N^{(j)}=\\sqrt{C_2(j)d_j}$ (the computation for this is, in principle, the same as in [64]).",
"Writing everything down in our graphical calculus: $\\underline{\\hat{E}}(j)\\begin{array}{c}\\includegraphics [scale=1]{thesis-4_3_1pic14-edge}\\end{array}=i\\sqrt{C_{2}\\left(j\\right)d_{j}}\\begin{array}{c}\\includegraphics [scale=1]{thesis-4_3_1pic15-edge+tau}\\end{array}$ With this at hand we turn again to the three cases i)-iii) from REF : However, due to the nature of the SU(3) gauge group, one cannot obtain a node with all edges outgoing by simply multiplying it with a sign factor.",
"Instead, one now has to take care of the fact, that the switched edges carry the dual representation.",
"So one works in the following with an oriented graph, denoted the following way: $\\left|\\nu _{orient}\\left(\\underline{j}_{1,\\bar{k}},\\underline{j}_{2,\\bar{k}},\\underline{j}_{3,\\bar{k}},\\bar{\\underline{j}}_{1,\\bar{k}-\\bar{e}_{1}}\\underline{\\bar{j}}_{2,\\bar{k}-\\bar{e}_{2}}\\underline{\\bar{j}}_{3,\\bar{k}-\\bar{e}_{3}},\\ldots \\right)\\right\\rangle =\\left|\\nu \\left(\\underline{j}_{1,\\bar{k}},\\underline{j}_{2,\\bar{k}},\\underline{j}_{3,\\bar{k}},\\underline{j}_{1,\\bar{k}-\\bar{e}_{1}}\\underline{j}_{2,\\bar{k}-\\bar{e}_{2}}\\underline{j}_{3,\\bar{k}-\\bar{e}_{3}},\\ldots \\right)\\right\\rangle $ The first case i) $\\left(\\underline{j}_{j,\\bar{k}}=\\underline{j}_{m,\\bar{k}}\\right)$ means, that both grasping operators act on the same edge, hence we get twice the square root of the corresponding quadric Casimir and using the orthogonality relation (REF ) one calculates: $\\underline{\\hat{E}}(\\underline{j}_j)^I\\underline{\\hat{E}}_(\\underline{j}_j)^{I}\\left|\\nu _{orient}\\left(\\left\\lbrace \\underline{\\pi }\\right\\rbrace _{\\bar{k}};\\left\\lbrace \\underline{j}\\right\\rbrace _{\\bar{k}};\\left\\lbrace \\underline{s}\\right\\rbrace _{\\bar{k}}\\right)\\right\\rangle =-C_{2}\\left(\\underline{j}_{j,\\bar{k}-\\bar{p}_{j}}\\right)\\Biggl |\\begin{array}{c}\\includegraphics [scale=0.6]{thesis-4_4_2pic01-edge+hol}\\end{array}\\Biggr \\rangle $ $=C_{2}\\left(\\underline{j}_{j,\\bar{k}-\\bar{p}_{j}}\\right)\\left|\\nu _{orient}\\left(\\left\\lbrace \\underline{\\pi }\\right\\rbrace _{\\bar{k}};\\left\\lbrace \\underline{j}\\right\\rbrace _{\\bar{k}};\\left\\lbrace \\underline{s}\\right\\rbrace _{\\bar{k}}\\right)\\right\\rangle $ The second case ii), where the edges in question lie in parallel direction $\\left(\\underline{j}_{j,\\bar{k}}=\\underline{j}_{m,\\bar{k}-\\bar{e}_{m}}\\right)$ uses again the extraction of the s-classified 3j-symbol and thus one gets: $\\underline{\\hat{E}}(\\underline{j}_{j,\\bar{k}-\\bar{e}_{j}+\\bar{p}_{j}})^{I}\\underline{\\hat{E}}(\\underline{j}_{j,\\bar{k}-\\bar{p}_{j}})^{I}\\left|\\nu _{orient}\\left(\\left\\lbrace \\underline{\\pi }\\right\\rbrace _{\\bar{k}};\\left\\lbrace \\underline{j}\\right\\rbrace _{\\bar{k}};\\left\\lbrace \\underline{s}\\right\\rbrace _{\\bar{k}}\\right)\\right\\rangle =\\sqrt{C_{2}\\left(\\underline{j}_{j,\\bar{k}-\\bar{p}_{j}}\\right)C_{2}\\left(\\underline{j}_{j,\\bar{k}-\\bar{e}_{j}+\\bar{p}_{j}}\\right)} \\cdot $ $\\cdot \\underset{\\underline{s}_{j,\\bar{k}}^{\\prime }}{\\sum }\\left\\lbrace \\begin{array}{cccc}\\underline{\\bar{j}}_{j,\\bar{k}-\\bar{e}_i+\\bar{p}_j} & \\underline{\\pi }_{j,\\bar{k}} & \\underline{\\bar{j}}_{j,\\bar{k}-\\bar{p}_j} \\\\\\underline{j}_{j,\\bar{k}-\\bar{p}_j} & 1 & \\underline{\\bar{j}}_{j,\\bar{k}-\\bar{e}_j+\\bar{p}_j} \\\\\\underline{s}^{\\prime }_{j,\\bar{k}} & \\underline{s}_I & \\underline{s}_{j,\\bar{k}} & \\underline{s}_I\\end{array}\\right\\rbrace \\left|\\nu _{orient}\\left(\\left\\lbrace \\underline{\\pi }\\right\\rbrace _{\\bar{k}},\\left\\lbrace \\underline{j}\\right\\rbrace _{\\bar{k}},\\ldots \\underline{s}_{j,\\bar{k}}^{\\prime }\\ldots \\right)\\right\\rangle $ Lastly we look at iii), where both holonomies go in different directions.",
"With the same strategy as before, we see: $\\underline{\\hat{E}}_{j}^{I}\\underline{\\hat{E}}_{m}^{I}\\left|\\nu _{out}\\left(\\left\\lbrace \\underline{\\pi }\\right\\rbrace _{\\bar{k}};\\left\\lbrace \\underline{j}\\right\\rbrace _{\\bar{k}};\\left\\lbrace \\underline{s}\\right\\rbrace _{\\bar{k}}\\right)\\right\\rangle =\\underset{\\begin{array}{c}\\underline{\\bar{\\pi }}_{m,\\bar{k}}^{2}\\underline{\\bar{\\pi }}_{j,\\bar{k}}^{2}\\\\\\underline{s}_{\\pi _{j,\\bar{k}}}\\underline{s}_{\\pi _{m,\\bar{k}}}\\end{array}}{\\tilde{\\sum }}\\left(-\\right)^{\\left(1-\\left|\\bar{p}_{j}\\right|\\right)\\left(\\underline{\\pi }_{j,\\bar{k}}+\\underline{j}_{j,\\bar{k}}+\\underline{j}_{j,\\bar{k}-\\bar{e}_{j}}\\right)+\\left|\\bar{p}_{m}\\right|\\left(\\underline{\\pi }_{m,\\bar{k}}+\\underline{j}_{m,\\bar{k}}+\\underline{j}_{m,\\bar{k}-\\bar{e}_{m}}\\right)}\\cdot $ $\\left(-\\right)^{\\sigma \\left(m,j,q\\right)\\left(\\underline{\\pi }_{j,\\bar{k}}+\\underline{\\pi }_{m,\\bar{k}}+\\underline{\\pi }_{q,\\bar{k}}\\right)}\\sqrt{C_{2}\\left(\\underline{j}_{m,\\bar{k}-\\bar{p}_{m}}\\right)C_{2}\\left(\\underline{j}_{j,\\bar{k}-\\bar{p}_{j}}\\right)}\\Biggl |\\begin{array}{c}\\includegraphics [scale=0.7]{thesis-4_4_2pic07-nodecoupled}\\end{array}\\Biggr \\rangle $ $=\\underset{\\begin{array}{c}\\underline{\\pi }_{m,\\bar{k}}^{2}\\underline{\\pi }_{j,\\bar{k}}^{2}\\\\\\underline{s}_{\\pi _{j,\\bar{k}}}\\underline{s}_{\\pi _{m,\\bar{k}}}\\\\\\underline{s}_{0,\\bar{k}}^{\\prime }\\underline{s}_{m,\\bar{k}}^{\\prime }\\underline{s}_{j,\\bar{k}}^{\\prime }\\end{array}}{\\sum }\\left(-\\right)^{\\sigma \\left(m,j,q\\right)\\left(\\underline{\\pi }_{m,\\bar{k}}^{2}+\\underline{\\pi }_{j,\\bar{k}}^{2}+\\underline{\\pi }_{m,\\bar{k}}+\\underline{\\pi }_{j,\\bar{k}}\\right)+\\left(1-\\left|\\bar{p}_{j}\\right|\\right)\\left(\\underline{\\pi }_{j,\\bar{k}}+\\underline{\\pi }_{j,\\bar{k}}^{2}\\right)+\\left|\\bar{p}_{m}\\right|\\left(\\underline{\\pi }_{m,\\bar{k}}+\\underline{\\pi }_{m,\\bar{k}}^{2}\\right)}(-)^{\\pi ^3_{j,\\bar{k}}+\\pi _{m,\\bar{k}}+1}$ $\\sqrt{C_{2}\\left(\\underline{j}_{m,\\bar{k}-\\bar{p}_{m}}\\right)C_{2}\\left(\\underline{j}_{j,\\bar{k}-\\bar{p}_{j}}\\right)}\\left\\lbrace \\begin{array}{cccc}\\underline{\\bar{j}_{m,\\bar{k}-\\bar{e}_m+\\bar{p}_m}} & \\underline{\\pi }^2_{m,\\bar{k}} & \\underline{\\bar{j}}_{m,\\bar{k}-\\bar{p}_m} \\\\1 & \\underline{\\bar{j}}_{m,\\bar{k}-\\bar{p}_m} & \\underline{\\bar{\\pi }}_{m,\\bar{k}} \\\\\\underline{s}^{\\prime }_{m,\\bar{k}} & s_{m,\\bar{k}} & \\underline{s}_{\\pi _m,\\bar{k}} & \\underline{s}_I\\end{array}\\right\\rbrace \\left\\lbrace \\begin{array}{cccc}\\underline{\\bar{\\pi }}^2_{m,\\bar{k}} & \\underline{\\bar{\\pi }}_{q,\\bar{k}} & \\underline{\\bar{\\pi }}^2_{j,\\bar{k}} \\\\\\underline{\\pi }_{j,\\bar{k}} & 1 & \\underline{\\pi }_{m,\\bar{k}} \\\\\\underline{s}^{\\prime }_{0,\\bar{k}} & \\underline{s}_{\\pi _m,\\bar{k}} & s_{0,\\bar{k}} & s_{\\pi _j,\\bar{k}}\\end{array}\\right\\rbrace $ $\\left\\lbrace \\begin{array}{cccc}\\underline{\\pi }^2_{j,\\bar{k}} & \\underline{\\bar{j}}_{j,\\bar{k}-\\bar{e}_j+\\bar{p}_j} & \\underline{\\bar{j}}_{\\bar{k}-\\bar{p}_j} \\\\\\underline{j}_{j,\\bar{k}-\\bar{p}_j} & 1 & \\underline{\\pi }_{j,\\bar{k}} \\\\\\underline{s}^{\\prime }_{j,\\bar{k}} & \\underline{s}_{\\pi _j,\\bar{k}} & s_{j,\\bar{k}} & \\underline{s}_I\\end{array}\\right\\rbrace \\left|\\nu _{orient}\\left(\\lbrace \\underline{\\pi }\\rbrace _{\\bar{k}};\\lbrace \\underline{j}\\rbrace ;\\underline{s}^{\\prime }_{0,\\bar{k}},\\underline{s}^{\\prime }_{j,\\bar{k}},\\underline{s}^{\\prime }_{m,\\bar{k}},\\underline{s}_{q,\\bar{k}}\\right)\\right\\rangle $"
],
[
"Gravity-Part of the Magnetic Term",
"The Gravity-Part of the Magnetic Term is $tr\\left(\\hat{\\tau }_{i}\\hat{A}_{l}\\left[\\hat{A}_{l}^{-1},\\sqrt{\\hat{V}}\\right]\\right)tr\\left(\\hat{\\tau }_{i}\\hat{A}_{p}\\left[\\hat{A}_{p}^{-1},\\sqrt{\\hat{V}}\\right]\\right)$ Since there are again two commutators we have, in principle, four different terms to look at.",
"However three of them vanish trivially.",
"For example look at the expression, where the $\\hat{A}_{p}$ cancel: $tr\\left(\\hat{\\tau }_{i}\\hat{A}_{l}\\left[\\hat{A}_{l}^{-1},\\sqrt{\\hat{V}}\\right]\\right)tr\\left(\\hat{\\tau }_{i}\\sqrt{\\hat{V}}\\right)\\left|\\nu _{out}\\left(\\left\\lbrace \\pi \\right\\rbrace _{\\bar{k}};\\left\\lbrace j\\right\\rbrace _{\\bar{k}}\\right)\\right\\rangle =$ $tr\\left(\\hat{\\tau }_{i}\\hat{A}_{l}\\left[\\hat{A}_{l}^{-1},\\sqrt{\\hat{V}}\\right]\\right)tr\\left(\\hat{\\tau }_{i}\\right)\\underset{\\left\\lbrace \\pi \\right\\rbrace _{\\bar{k}}^{2}}{\\tilde{\\sum }}\\sqrt{V}_{\\bar{k}}\\left(\\left\\lbrace \\pi \\right\\rbrace _{\\bar{k}},\\left\\lbrace \\pi \\right\\rbrace _{\\bar{k}}^{2};\\left\\lbrace j\\right\\rbrace _{\\bar{k}}\\right)\\left|\\nu _{out}\\left(\\left\\lbrace \\pi \\right\\rbrace _{\\bar{k}}^{2};\\left\\lbrace j\\right\\rbrace _{\\bar{k}}\\right)\\right\\rangle =0$ since $tr\\left(\\hat{\\tau }_{i}\\right)=0$ for $\\tau _{i}\\in SU(2)$ .",
"The same argument is of course also true, in case of the $A_l$ canceling.",
"Thus only the term with both Volume operators nested remains.",
"Again we distinguish on which edges the holonomies lie (cases i)-iii) from section REF ).",
"Since one has seen that the orientation of the arrows of the edges does not change the result, we will suppress this temporary sign from now on and just assume the vertex has been brought in a form such that all links are outgoing.",
"If i) $\\left(j_{p,\\bar{k}}=j_{l,\\bar{k}}\\right)$ then one gets from the first trace an 6j-symbol and the inserted $\\hat{\\tau }_i$ acts like adding an intertwiner in the defining representation, which hence remains open, after closing the first trace.",
"To close the second one, one uses again (REF ) twice.",
"In total one obtains: $tr\\left(\\hat{\\tau }_{i}\\hat{A}_{p}\\sqrt{\\hat{V}}\\hat{A}_{p}^{-1}\\right)tr\\left(\\hat{\\tau }_{i}\\hat{A}_{p}\\sqrt{\\hat{V}}\\hat{A}_{p}^{-1}\\right)\\left|\\nu \\left(\\left\\lbrace \\pi \\right\\rbrace _{\\bar{k}};\\left\\lbrace j\\right\\rbrace _{\\bar{k}}\\right)\\right\\rangle =tr\\left(\\hat{\\tau }_{i}\\hat{A}_{p}\\sqrt{\\hat{V}}\\hat{A}_{p}^{-1}\\right)\\cdot $ $\\underset{\\begin{array}{c}\\left\\lbrace \\pi \\right\\rbrace _{\\bar{k}}^{2}j_{p,\\bar{k}-\\bar{p}_{p}}^{2}\\\\j_{p,\\bar{k}-\\bar{p}_{p}}^{3}\\end{array}}{\\tilde{\\sum }}\\sqrt{V}_{\\bar{k}-\\bar{e}_{p}+2\\bar{p}_{p}}\\left(\\left\\lbrace \\pi \\right\\rbrace _{\\bar{k}},\\left\\lbrace \\pi \\right\\rbrace _{\\bar{k}}^{2};j_{p,\\bar{k}-\\bar{p}_{p}},j_{p,\\bar{k}-\\bar{p}_{p}}^{3};\\ldots j_{p,\\bar{k}-\\bar{p}_{p}}^{2}\\dots \\right)\\left\\lbrace \\begin{array}{ccc}j_{p,\\bar{k}-\\bar{p}_{p}}^{3} & j_{p,\\bar{k}-\\bar{p}_{p}}^{2} & m\\\\m & 1 & j_{p,\\bar{k}-\\bar{p}_{p}}\\end{array}\\right\\rbrace $ $\\left(-\\right)^{\\left|\\bar{p_{p}}\\right|\\left(\\pi _{p,\\bar{k}}^{2}+j_{p,\\bar{k}-\\bar{p}_{p}}^{3}+j_{p,\\bar{k}-\\bar{e}_{p}+\\bar{p}_{p}}\\right)+2m+j_{p,\\bar{k}-\\bar{p}_{p}}^{2}+j_{p,\\bar{k}-\\bar{p}_{p}}+m+1}\\Biggl |\\begin{array}{c}\\includegraphics [scale=0.8]{thesis-4_4_3pic04-edge+hol3}\\end{array}\\Biggr \\rangle $ $=\\underset{\\begin{array}{c}\\left\\lbrace \\pi \\right\\rbrace _{\\bar{k}}^{2}j_{p,\\bar{k}-\\bar{p}_{p},m_{1}}^{2}\\\\\\left\\lbrace \\pi \\right\\rbrace _{\\bar{k}}^{3}j_{p,\\bar{k}-\\bar{p}_{p}}^{3}\\\\j_{p,\\bar{k}-\\bar{p}_{p}}^{4}\\end{array}}{\\tilde{\\sum }}\\sqrt{V}_{\\bar{k}-\\bar{e}_{p}+2\\bar{p}_{p}}\\left(\\left\\lbrace \\pi \\right\\rbrace _{\\bar{k}},\\left\\lbrace \\pi \\right\\rbrace _{\\bar{k}}^{2};j_{p,\\bar{k}-\\bar{p}_{p}},j_{p,\\bar{k}-\\bar{p}_{p}}^{3};\\ldots j_{p,\\bar{k}-\\bar{p}_{p}}^{2}\\dots \\mid m\\right)$ $\\sqrt{V}_{\\bar{k}-\\bar{e}_{p}+2\\bar{p}_{p}}\\left(\\left\\lbrace \\pi \\right\\rbrace _{\\bar{k}}^{2},\\left\\lbrace \\pi \\right\\rbrace _{\\bar{k}}^{3};j_{p,\\bar{k}-\\bar{p}_{p}}^{3},j_{p,\\bar{k}-\\bar{p}_{p}};\\ldots j_{p,\\bar{k}-\\bar{p}_{p}}^{4}\\dots \\mid m\\right)(-)^{2m+1+j^2_{p,\\bar{k}-\\bar{p}_p}+j^4_{p,\\bar{k}-\\bar{p}_p}}$ $\\left\\lbrace \\begin{array}{ccc}j_{p,\\bar{k}-\\bar{p}_{p}}^{3} & m & j_{p,\\bar{k}-\\bar{p}_{p}}^{2}\\\\m & j_{p,\\bar{k}-\\bar{p}_{p}}^{2} & 1\\end{array}\\right\\rbrace \\left\\lbrace \\begin{array}{ccc}j_{p,\\bar{k}-\\bar{p}_{p}}^{3} & 1 & j_{p,\\bar{k}-\\bar{p}_p} \\\\m & j^4_{,\\bar{k}-\\bar{p}_p} & m\\end{array}\\right\\rbrace \\left|\\nu \\left(\\left\\lbrace \\pi \\right\\rbrace _{\\bar{k}}^{3};\\left\\lbrace j\\right\\rbrace _{\\bar{k}}\\right)\\right\\rangle $ With the same methods as established before, we get for case ii), meaning both links go in parallel direction $\\left(j_{p,\\bar{k}}=j_{l,\\bar{k}-\\bar{e}_{l}}\\right)$ , that: $tr\\left(\\hat{\\tau }_{i}\\hat{A}_{p,\\bar{p}_{p}}\\sqrt{\\hat{V}}\\hat{A}_{p,\\bar{p}_{p}}^{-1}\\right)tr\\left(\\hat{\\tau }_{i}\\hat{A}_{p,\\bar{p}_{p}-\\bar{e}_{p}}\\sqrt{\\hat{V}}\\hat{A}_{p,\\bar{p}_{p}-\\bar{e}_{p}}^{-1}\\right)\\left|\\nu \\left(\\left\\lbrace \\pi \\right\\rbrace _{\\bar{k}};\\left\\lbrace j\\right\\rbrace _{\\bar{k}}\\right)\\right\\rangle =\\underset{\\begin{array}{c}\\left\\lbrace \\pi \\right\\rbrace _{\\bar{k}}^{2}\\left\\lbrace \\pi \\right\\rbrace _{\\bar{k}}^{3}\\\\j_{p,\\bar{k}-\\bar{e}_{p}+\\bar{p}_{p}}^{2}j_{p,\\bar{k}-\\bar{e}_{p}+\\bar{p}_{p}}^{3}\\\\j_{p,\\bar{k}-\\bar{p}_{p}}^{2}j_{p,\\bar{k}-\\bar{p}_{p}}^{3}\\end{array}}{\\tilde{\\sum }}$ $\\left(-\\right)^{\\left|\\bar{p}_{p}\\right|\\left(2\\pi _{p,\\bar{k}}^{3}+j_{p,\\bar{k}-\\bar{p}_{p}}^{3}+j_{p,\\bar{k}-\\bar{p}_{p}}^{5}+j_{p,\\bar{k}-\\bar{e}_{p}+\\bar{p}}+j_{p,\\bar{k}-\\bar{e_{p}+}\\bar{p}}^{3}\\right)}(-)^{j^2_{p,\\bar{k}-\\bar{p}_p}+j_{p,\\bar{k}-\\bar{e}_p+\\bar{p}_p}+j^2_{p,\\bar{k}-\\bar{e}_p+\\bar{p}_p}+j^3_{p,\\bar{k}-\\bar{e}_p+\\bar{p}_p}+\\pi ^2_{p,\\bar{k}}+1}$ $\\left\\lbrace \\begin{array}{ccc}j_{p,\\bar{k}-\\bar{p}_{p}}^{3} & j_{p,\\bar{k}-\\bar{p}_{p}} & 1\\\\m & m & j_{p,\\bar{k}-\\bar{p}_{p}}^{2}\\end{array}\\right\\rbrace \\left\\lbrace \\begin{array}{ccc}j_{p,\\bar{k}-\\bar{e}_{p}+\\bar{p}_{p}} & j_{p,\\bar{k}-\\bar{e}_{p}+\\bar{p}_{p}}^{3} & 1\\\\j_{p,\\bar{k}-\\bar{p}_{p}} & j_{p,\\bar{k}-\\bar{p}_{p}}^{3} & \\pi _{p,\\bar{k}}^{2}\\end{array}\\right\\rbrace \\left\\lbrace \\begin{array}{ccc}j_{p,\\bar{k}-\\bar{e}_p+\\bar{p}_p} & j^2_{p,\\bar{k}-\\bar{e}_p+\\bar{p}_p} & m \\\\m & 1 & j^3_{p,\\bar{k}-\\bar{e}_p+\\bar{p}_p}\\end{array}\\right\\rbrace $ $\\sqrt{V}_{\\bar{k}-2\\bar{p}_{p}}\\left(\\left\\lbrace \\pi \\right\\rbrace _{\\bar{k}},\\left\\lbrace \\pi \\right\\rbrace _{\\bar{k}}^{2};j_{p,\\bar{k}-\\bar{e}_{p}+\\bar{p}_{p}},j_{p,\\bar{k}-\\bar{e}_{p}+\\bar{p}_{p}}^{3};\\ldots j_{p,\\bar{k}-\\bar{e}_{p}+\\bar{p}_{p}}^{2}\\ldots \\right)$ $\\sqrt{V}_{\\bar{k}+\\bar{e}_{p}-2\\bar{p}_{p}}\\left(\\left\\lbrace \\pi \\right\\rbrace _{\\bar{k}}^{2},\\left\\lbrace \\pi \\right\\rbrace _{\\bar{k}}^{3};j_{p,\\bar{k}-\\bar{p}_{p}}^{3},j_{p,\\bar{k}-\\bar{p}_{p}}^{4};\\ldots j_{p,\\bar{k}-\\bar{p}_{p}}^{2}\\ldots \\right)\\left|\\nu \\left(\\lbrace \\pi \\rbrace ^3_{\\bar{k}},\\lbrace j\\rbrace _{\\bar{k}}\\right)\\right\\rangle $ And finally with more suppressed calculation, it follows iii) (both holonomies go into different directions): $tr\\left(\\hat{\\tau }_{i}\\hat{A}_{l}\\sqrt{\\hat{V}}\\hat{A}_{l}^{-1}\\right)tr\\left(\\hat{\\tau }_{i}\\hat{A}_{p}\\sqrt{\\hat{V}}\\hat{A}_{p}^{-1}\\right)\\left|\\nu \\left(\\left\\lbrace \\pi \\right\\rbrace _{\\bar{k}};\\left\\lbrace j\\right\\rbrace _{\\bar{k}}\\right)\\right\\rangle =$ $=\\underset{\\begin{array}{c}\\left\\lbrace \\pi \\right\\rbrace _{\\bar{k}}^{2}\\pi _{p,\\bar{k}}^{3}\\pi _{l,\\bar{k}}^{3}\\left\\lbrace \\pi \\right\\rbrace _{\\bar{k}}^{4}\\\\j_{p,\\bar{k}-\\bar{p}_{p}}^{2}j_{p,\\bar{k}-\\bar{p}_{p}}^{3}j_{l,\\bar{k}-\\bar{p}_{l}}^{2}j_{l,\\bar{k}-\\bar{p}_{l}}^{3}\\end{array}}{\\tilde{\\sum }}\\sqrt{V}_{\\bar{k}+\\bar{e}_{p}-2\\bar{p}_{p}}\\left(\\left\\lbrace \\pi \\right\\rbrace _{\\bar{k}},\\left\\lbrace \\pi \\right\\rbrace _{\\bar{k}}^{2};j_{p,\\bar{k}-\\bar{p}_{p}},j_{p,\\bar{k}-\\bar{p}_{p}}^{3};\\ldots j_{p,\\bar{k}-\\bar{p}_{p}}^{2}\\ldots \\right)$ $\\sqrt{V}_{\\bar{k}-\\bar{e}_{l}-2\\bar{p}_{l}}\\left(\\pi _{l,\\bar{k}}^{3},\\pi _{p,\\bar{k}}^{3},\\pi _{q,\\bar{k}}^{2},\\left\\lbrace \\pi \\right\\rbrace _{\\bar{k}}^{4};j_{l,\\bar{k}-\\bar{p}_{l}}^{3},j_{l,\\bar{k}-\\bar{p}_{l}};\\ldots j_{l,\\bar{k}-\\bar{p}_{l}}^{2}\\ldots \\right)\\left(-\\right)^{\\sigma \\left(p,l,q\\right)\\left(\\pi _{p,\\bar{k}}^{2}+\\pi _{l,\\bar{k}}^{2}+2\\pi _{q,\\bar{k}}^{2}+\\pi _{p,\\bar{k}}^{3}+\\pi _{l,\\bar{k}}^{3}\\right)}$ $(-)^{\\left|\\bar{p}_{p}\\right|\\left(\\pi ^2_{p,\\bar{k}}+j^2_{p,\\bar{k}-\\bar{p}_p}+2j_{p,\\bar{k}-\\bar{e}_p+\\bar{p}_p}+j_{l,\\bar{k}-\\bar{p}_l}+\\pi ^3_{l,\\bar{k}}\\right)+\\left(1-\\left|\\bar{p}_l\\right|\\right)\\left(\\pi ^2_{l,\\bar{k}}+j_{l,\\bar{k}-\\bar{p}_l}+2j_{l,\\bar{k}-\\bar{e}_l+\\bar{p}_l}+\\pi ^3_{l,\\bar{k}}+j^3_{l,\\bar{k}-\\bar{p}_l}+\\pi ^4_{l,\\bar{k}}+j_{l,\\bar{k}-\\bar{p}_l}+j_{l,\\bar{k}-\\bar{e}_l+\\bar{p}_l}\\right)}$ $(-)^{2m+m+j_{p,\\bar{k}-\\bar{p}_p}+j^2_{p,\\bar{k}-\\bar{p}_p}+j^3_{p,\\bar{k}-\\bar{p}_p}+j_{p,\\bar{k}-\\bar{e}_p+\\bar{p}_p}-j_{l,\\bar{k}-\\bar{e}_l+\\bar{p}_l}+j^3_{l,\\bar{k}-\\bar{p}_l}+\\pi ^2_{p,\\bar{k}}+\\pi ^3_{p,\\bar{k}}+\\pi ^2_{q,\\bar{k}}+\\pi ^2_{l,\\bar{k}}+\\pi ^3_{l,\\bar{k}}}$ $\\left\\lbrace \\begin{array}{ccc}1 & m & m\\\\j_{p,\\bar{k}-\\bar{p}_{p}}^{2} & j_{p,\\bar{k}-\\bar{p}_{p}} & j_{p,\\bar{k}-\\bar{p}_{p}}^{3}\\end{array}\\right\\rbrace \\left\\lbrace \\begin{array}{ccc}j_{p,\\bar{k}-\\bar{e}_{p}+\\bar{p}_{p}} & \\pi _{p,\\bar{k}}^{2} & j_{p,\\bar{k}-\\bar{p}_{p}}^{3}\\\\1 & j_{p,\\bar{k}-\\bar{p}_{p}} & \\pi _{p,\\bar{k}}^{3}\\end{array}\\right\\rbrace $ $\\left\\lbrace \\begin{array}{ccc}\\pi _{p,\\bar{k}}^{3} & \\pi _{p,\\bar{k}}^{2} & 1\\\\\\pi _{l,\\bar{k}}^{2} & \\pi _{l,\\bar{k}}^{3} & \\pi _{q,\\bar{k}}^{2}\\end{array}\\right\\rbrace \\left\\lbrace \\begin{array}{ccc}\\pi _{l,\\bar{k}}^{3} & \\pi _{l,\\bar{k}}^{2} & 1\\\\j_{l,\\bar{k}-\\bar{p}_{l}} & j_{l,\\bar{k}-\\bar{p}_{l}}^{3} & j_{l,\\bar{k}-\\bar{e}_{l}+\\bar{p}_{l}}\\end{array}\\right\\rbrace \\left\\lbrace \\begin{array}{ccc}m_{1} & 1 & m\\\\j_{l,\\bar{k}-\\bar{p}_{l}} & j_{l,\\bar{k}-\\bar{p}_{l}}^{2} & j_{l,\\bar{k}-\\bar{p}_{l}}^{3}\\end{array}\\right\\rbrace \\left|\\nu \\left(\\left\\lbrace \\pi \\right\\rbrace _{\\bar{k}}^{4},\\left\\lbrace j\\right\\rbrace _{\\bar{k}}\\right)\\right\\rangle $"
],
[
"Gluons plaquette of the Magnetic Term",
"The plaquette part is given by $tr\\left(\\underline{\\hat{\\tau }}_I\\underline{\\hat{A}}_{jk}\\right)tr\\left(\\underline{\\hat{\\tau }}\\underline{\\hat{A}}_{mn}\\right)$ which again acts only on the magnetic graph.",
"Each of these two plaquettes, which we add, looks very similar in its structure to chapter .",
"Using this resemblance and inserting again the corresponding plaquette terms $\\mathfrak {P}_{SU(3)}$ will simplify the task at hand.",
"Again one has to distinguish different cases, i.e.",
"determined by the possible combinations of $j,k,m$ and $n$ .",
"The most simple one is $j=m$ and $k=n$ : $tr\\left(\\underline{\\hat{\\tau }}_{I}\\underline{\\hat{A}}_{mn}\\right)tr\\left(\\underline{\\hat{\\tau }}_{I}\\underline{\\hat{A}}_{mn}\\right)\\left|\\nu _{orient}\\left(\\left\\lbrace \\underline{\\pi }\\right\\rbrace _{\\bar{k}};\\left\\lbrace \\underline{j}\\right\\rbrace \\right)\\right\\rangle =\\left(-\\right)^{\\sigma \\left(n,m,p\\right)\\left(\\underline{\\pi }_{m,\\bar{k}}+\\underline{\\pi }_{n,\\bar{k}}+\\underline{\\pi }_{p,\\bar{k}}\\right)}$ $\\left(-\\right)^{\\left|\\bar{p}_{n}\\right|\\left(\\underline{\\pi }_{n,\\bar{k}}+\\underline{j}_{n,\\bar{k}-\\bar{p}_{n}}+\\underline{j}_{n,\\bar{k}-\\bar{e}_{n}+\\bar{p}_{n}}\\right)+\\left(1-\\left|\\bar{p}_{m}\\right|\\right)\\left(\\underline{\\pi }_{m,\\bar{k}}+\\underline{j}_{m,\\bar{k}}+\\underline{j}_{m,\\bar{k}-\\bar{e}_{m}}\\right)}\\Biggl |\\begin{array}{c}\\includegraphics [scale=0.7]{thesis-4_4_4pic00-node+twoplaquettes}\\end{array}\\Biggr \\rangle $ $=\\left(-\\right)^{\\sigma \\left(n,m,p\\right)\\left(\\ldots \\right)+\\left|\\bar{p}_{n}\\right|\\left(\\ldots \\right)+\\left(1-\\left|\\bar{p}_{m}\\right|\\right)\\left(\\dots \\right)}$ $\\underset{\\underline{m}_{1},\\underline{s}}{\\sum }\\left(-\\right)^{1}\\left\\lbrace \\begin{array}{cccc}\\underline{m} & \\underline{m} & \\underline{\\bar{m}}_1\\\\1 & \\underline{m} & \\underline{\\bar{m}} \\\\\\underline{\\bar{s}} & \\underline{s}_I & \\underline{s}_I & \\underline{s}\\end{array}\\right\\rbrace \\Biggl |\\begin{array}{c}\\includegraphics [scale=0.7]{thesis-4_4_4pic01-node+plaquette}\\end{array}\\Biggr \\rangle $ Now we have exactly the same plaquette we inserted in the Kogut Susskind case.",
"To extract exactly the same term again we have to bring the graph in an ordered form, which means we have to take care of the fact, that the Loop also touches four other nodes.",
"In contrast to the Kogut-Susskind case these signs of the intertwiners now only depend on the chosen permutation of $n,m,p$ which means that we get a somewhat more complicated sign factor in front: $=\\underset{\\underline{\\bar{m}}_{1},\\mathcal {S}}{\\sum }\\left(-\\right)^{1+\\underline{m}_{1}}\\left\\lbrace \\begin{array}{cccc}\\underline{m} & \\underline{m} & \\underline{\\bar{m}}_1\\\\1 & \\underline{m} & \\underline{\\bar{m}} \\\\\\underline{\\bar{s}} & \\underline{s}_I & \\underline{s}_I & \\underline{s}\\end{array}\\right\\rbrace \\cdot $ $(-)^{\\sigma (n,m,p)\\left(\\underset{i,j=0,1}{\\sum }\\pi _{m,\\bar{k}+i\\bar{e}_m+j\\bar{e}_n}+\\pi _{n,\\bar{k}+i\\bar{e}_m+j\\bar{e}_n}+2\\pi _{p,\\bar{k}+i\\bar{e}_m+j\\bar{e}_n}+\\pi ^2_{m,\\bar{k}+i\\bar{e}_m+j\\bar{e}_n}+\\pi ^2_{n,\\bar{k}+i\\bar{e}_m+j\\bar{e}_n}\\right)}$ $(-)^{|\\bar{p}_n|\\left(\\underset{i,j=0,1}{\\sum }\\pi _{n,\\bar{k}+i\\bar{e}_n+j\\bar{e}_m}+j_{n,\\bar{k}-\\bar{p}_n+i\\bar{e}_m}+2j_{n,\\bar{k}-\\bar{e}_n+\\bar{p}_n+i\\bar{e}_m+2j\\bar{e}_n}+j^2_{n,\\bar{k}-\\bar{p}_n+i\\bar{e}_m}+\\pi ^2_{n,\\bar{k}+i\\bar{e}_m+j\\bar{e}_n}\\right)}$ $(-)^{(1-|\\bar{p}_m|)\\left(\\underset{i,j=0,1}{\\sum }\\pi _{m,\\bar{k}+i\\bar{e}_m+j\\bar{e}_m}+j_{m,\\bar{k}-\\bar{p}_m+i\\bar{e}_n}+2j_{m,\\bar{k}-\\bar{e}_m+\\bar{p}_m+i\\bar{e}_n+2j\\bar{e}_m}+j^2_{m,\\bar{k}-\\bar{p}_m+i\\bar{e}_n}+\\pi ^2_{m,\\bar{k}+\\bar{e}_i+\\bar{e}_j}\\right)}$ $\\mathfrak {P}_{SU(3)}\\left(\\left\\lbrace \\underline{\\pi }\\right\\rbrace _{\\bar{k}}\\ldots ;\\left\\lbrace \\underline{j}\\right\\rbrace _{\\bar{k}};\\left\\lbrace \\underline{s}\\right\\rbrace _{\\bar{k}}\\ldots ;\\underline{\\pi }_{n,\\bar{k}}^{2},\\ldots ;\\underline{j}_{n,\\bar{k}}^{2}\\ldots ;\\underline{s}_{0,\\bar{k}}^{2}\\ldots \\mid \\bar{m}\\right)$ $\\left|\\nu _{orient}\\left(\\underline{\\pi }_{p,\\bar{k}},\\underline{\\pi }_{m,\\bar{k}}^{2},\\underline{\\pi }_{n,\\bar{k}}^{2};\\underline{j}_{n,\\bar{k}}^{2},\\underline{j}_{m,\\bar{k}}^{2},\\ldots ;\\underline{s}_{0,\\bar{k}}^{2},\\underline{s}_{m,\\bar{k}}^{2},\\underline{s}_{n,\\bar{k}}^{2},\\underline{s}_{p,\\bar{k}}\\right)\\right\\rangle $ where $\\mathcal {S}$ is the set of all new appearing labels in the state, which are the ones one has to sum over.",
"There are now four different cases, one has to look at, left: i) $j=m$ $\\left(p_{j}=p_{m}\\right)$ and $k=n$ $\\left(p_{k}\\ne p_{n}\\right)$ ii) $j=m$ $\\left(p_{j}=p_{m}\\right)$ and $k\\ne n$ iii) $j=m$ $\\left(p_{j}\\ne p_{m}\\right)$ and $k=n$ $\\left(p_{k}\\ne p_{n}\\right)$ iv) $j=m$ $\\left(p_{j}\\ne p_{m}\\right)$ and $k\\ne n$ Everything else is (up to a relabelling or switching the orientation of the loop) one of theses cases.",
"We could draw them as seen in Figure 5.2.",
"Figure: Case iv)Each loop can be recoupled with the previous techniques, giving a $\\mathfrak {P}_{SU(3)}\\left(\\ldots \\right)$ -term up to one 6j each, which is due to the coupled $\\hat{\\tau }_j$ .",
"Instead one will get a 12j-symbol, which is defined in the following way: $\\left\\lbrace \\begin{array}{ccccc}j_{1} & j_{2} & j_{3} & j_{4}\\\\l_{1} & l_{2} & l_{3} & l_{4}\\\\k_{1} & k_{2} & k_{3} & k_{4}\\\\& s_{1} & s_{2} & s_{3} & s_{4}\\\\& s_{5} & s_{6} & s_{7} & s_{8}\\end{array}\\right\\rbrace =\\begin{array}{c}\\includegraphics [scale=0.7]{thesis-4_3_2pic10-Def12j}\\end{array}$ For instance it can be used to recouple the following object: $\\Biggl |\\begin{array}{c}\\includegraphics [scale=0.8]{thesis-4_4_4pic13-coupled_hols}\\end{array}\\Biggr \\rangle $ $=\\underset{\\underline{s}_{n,\\bar{k}}^{3}}{\\sum }\\begin{array}{c}\\includegraphics [scale=0.7]{thesis-4_4_4pic06-12j}\\end{array}\\Biggl |\\begin{array}{c}\\includegraphics [scale=0.8]{thesis-4_4_4pic07-graph}\\end{array}\\Biggr \\rangle \\ $ And this is exactly the non-trivial operation for case i).",
"So using it, we obtain: $tr\\left(\\underline{\\hat{\\tau }}_{I}\\underline{\\hat{A}}_{m,n,p_{m},1-p_{n}}\\right)tr\\left(\\underline{\\hat{\\tau }}_{I}\\underline{\\hat{A}}_{m,n}\\right)\\left|\\nu _{orient}\\left(\\left\\lbrace \\underline{\\pi }\\right\\rbrace _{\\bar{k}};\\left\\lbrace \\underline{j}\\right\\rbrace _{\\bar{k}};\\left\\lbrace \\underline{s}\\right\\rbrace _{\\bar{k}}\\right)\\right\\rangle =$ $\\underset{\\mathcal {S}}{\\sum } (-)^{(...)}(-)^{\\underline{\\pi }^3_{n,\\bar{k}}+\\underline{\\pi }^2_{n,\\bar{k}}+\\underline{m}+j_{n,\\bar{k}-\\bar{p}_n}+\\underline{j}^2_{n,\\bar{k}-\\bar{p}_n}+\\underline{\\pi }^3_{n,\\bar{k}}+\\underline{\\pi }^2_{n,\\bar{k}}+1}\\left\\lbrace \\begin{array}{ccccc}\\underline{\\bar{j}}_{n,\\bar{k}-\\bar{e}_{n}+\\bar{p}_{n}} & \\underline{\\bar{\\pi }}_{n,\\bar{k}} & \\underline{\\bar{\\pi }}_{n,\\bar{k}}^{2} & \\underline{\\bar{\\pi }}_{n,\\bar{k}}^{3}\\\\\\underline{j}_{n,\\bar{k}-\\bar{p}_{n}} & \\underline{\\bar{m}} & \\underline{\\bar{m}} & \\underline{j}_{n,\\bar{k}-\\bar{e}_{n}+\\bar{p}_{n}}^{2}\\\\\\underline{\\bar{j}}_{n,\\bar{k}-\\bar{p}_{n}}^{2} & \\underline{m} & 1 & \\underline{\\bar{m}}\\\\& \\underline{s}_{j_{n},\\bar{k}-\\bar{e}_{n}+\\bar{p}_{n}} & \\underline{s}_{n,\\bar{k}} & \\underline{\\bar{s}}_{\\pi _{n},\\bar{k}}^{2} & \\underline{\\bar{s}}_{\\pi _{n},\\bar{k}}^{3}\\\\& \\underline{s}_{n,\\bar{k}}^{3} & \\underline{s}_{j_{n},\\bar{k}-\\bar{p}_{n}} & \\underline{s}_{I} & \\underline{s}_{I}\\end{array}\\right\\rbrace $ $\\underset{\\underline{s}^2_{n,\\bar{k}}}{\\sum }\\left\\lbrace \\begin{array}{cccc}\\underline{\\bar{j}}_{n,\\bar{k}-\\bar{e}_n}+\\bar{p}_n & \\underline{\\pi }^2_{n,\\bar{k}} & \\underline{\\bar{j}}^2_{n,\\bar{k}-\\bar{p}_n} \\\\\\underline{\\bar{m}} & \\underline{\\bar{j}}_{n,\\bar{k}-\\bar{p}_n} & \\underline{\\bar{\\pi }}_{n,\\bar{k}} \\\\\\underline{s}^2_{n,\\bar{k}} & \\underline{s}_{n,\\bar{k}} & \\underline{s}^2_{\\pi _n,\\bar{k}} & \\underline{s}_{j_n,\\bar{k}-\\bar{p}_n}\\end{array}\\right\\rbrace ^{-1}\\left\\lbrace \\begin{array}{cccc}\\underline{j}^2_{n,\\bar{k}-\\bar{e}_n+\\bar{p}_n} & \\underline{\\pi }^3_{n,\\bar{k}} & \\underline{j}^2_{n,\\bar{k}-\\bar{p}_n} \\\\\\underline{\\pi }^2_{n,\\bar{k}} & \\underline{\\bar{j}}_{n,\\bar{k}-\\bar{e}_n+\\bar{p}_n} & \\underline{m} \\\\\\underline{s}^3_{n,\\bar{k}} & \\underline{s}_{j_n,\\bar{k}-\\bar{e}_n+\\bar{p}_n} & \\underline{s}^3_{\\pi _n,\\bar{k}} & \\underline{s}^2_{n,\\bar{k}}\\end{array}\\right\\rbrace ^{-1}$ $\\mathfrak {P}_{SU(3)}\\left(\\underline{\\pi }_{n,\\bar{k}},\\underline{\\pi }_{p,\\bar{k}},\\underline{\\pi }_{m,\\bar{k}},\\left\\lbrace \\underline{\\pi }\\right\\rbrace _{\\bar{k}+\\bar{e}_{n}},\\ldots ;\\left\\lbrace \\underline{j}\\right\\rbrace ;\\underline{s}_{0,\\bar{k}},\\underline{s}_{m,\\bar{k}},\\underline{s}_{n,\\bar{k}},\\underline{s}_{p,\\bar{k}},\\left\\lbrace \\underline{s}\\right\\rbrace _{\\bar{k}+\\bar{e}_{n}},\\ldots ;\\right.$ $\\left.\\underline{\\pi }_{n,\\bar{k}}^{2},\\underline{\\pi }_{m,\\bar{k}}^{2},\\underline{\\pi }_{n,\\bar{k}+\\bar{e}_{n}}^{2},\\ldots ;\\underline{j}_{n,\\bar{k}-\\bar{p}_{n}}^{2},\\underline{j}_{m,\\bar{k}-\\bar{p}_{m}}^{2}\\ldots ;\\underline{s}_{0,\\bar{k}}^{2},\\underline{s}_{m,\\bar{k}}^{2},\\underline{s}_{n,\\bar{k}}^{2},\\ldots \\mid \\underline{m}\\right)$ $\\mathfrak {P}_{SU(3)}\\left(\\underline{\\pi }_{n,\\bar{k}}^{2},\\underline{\\pi }_{m,\\bar{k}}^{2},\\underline{\\pi }_{p,\\bar{k}}\\ldots ;\\underline{j}_{n,\\bar{k}-\\bar{p}_{n}}^{2},\\underline{j}_{m,\\bar{k}-\\bar{p}_{m}}^{2},\\ldots ;\\underline{s}_{0,\\bar{k}}^{2},\\underline{s}_{m,\\bar{k}},\\underline{s}_{n,\\bar{k}}^{2}\\ldots ;\\right.$ $\\left.\\underline{\\pi }_{n,\\bar{k}}^{3},\\underline{\\pi }_{m,\\bar{k}}^{3},\\ldots ;\\underline{j}_{n,\\bar{k}-\\bar{e}_{n}+\\bar{p}_{n}}^{2},\\underline{j}_{m,\\bar{k}-\\bar{p}_{m}}^{3};\\underline{s}_{0,\\bar{k}}^{3},\\underline{s}_{m,\\bar{k}}^{3},\\underline{s}_{n,.\\bar{k}}^{3},\\ldots \\mid \\underline{m}\\right)$ $\\left|\\nu _{orient}\\left(\\underline{\\pi }_{n,\\bar{k}}^{3},\\underline{\\pi }_{m,\\bar{k}}^{3}\\underline{\\pi }_{p,\\bar{k}},\\ldots ;\\underline{j}_{n,\\bar{k}}^{2},\\underline{j}_{n,\\bar{k}-\\bar{e}_{n}}^{2},\\underline{j}_{m,\\bar{k}-\\bar{p}_{m}}^{3},\\underline{j}_{m,\\bar{k}-\\bar{e}_{m}+\\bar{p}_{m}},\\ldots ;\\underline{s}_{0,\\bar{k}}^{3},\\underline{s}_{n,\\bar{k}}^{3},\\underline{s}_{m,\\bar{k}}^{3}\\underline{s}_{p,\\bar{k}}\\right)\\right\\rangle $ The additional sign $(-)^{(...)}$ contains again the resulting sign, which stems from the permutation of $m,n,p$ and the choices of $\\bar{p}_n,\\bar{p}_m$ .",
"Since its construction is the same as before we refrain from writing it down explicitly.",
"The inverse $s$ -classified 6j-symbols are chosen in such a way that they cancel the corresponding elements in both $\\mathcal {P}_{SU(3)}$ expressions.",
"For case ii) we get: $tr\\left(\\underline{\\hat{\\tau }}_{I}\\underline{\\hat{h}}_{m,p}\\right)tr\\left(\\underline{\\hat{\\tau }}_{I}\\underline{\\hat{h}}_{m,n}\\right)\\left|\\nu _{orient}\\left(\\left\\lbrace \\underline{\\pi }\\right\\rbrace _{\\bar{k}};\\left\\lbrace \\underline{j}\\right\\rbrace _{\\bar{k}};\\left\\lbrace \\underline{s}\\right\\rbrace _{\\bar{k}}\\right)\\right\\rangle =$ $=\\underset{\\mathcal {S}}{\\sum }\\left(-\\right)^{\\left(\\ldots \\right)}\\left\\lbrace \\begin{array}{ccccc}\\underline{\\pi }_{p,\\bar{k}} & \\underline{\\pi }_{m,\\bar{k}} & \\underline{\\pi }_{m,\\bar{k}}^{2} & \\underline{\\pi }_{m,\\bar{k}}^{3}\\\\\\underline{\\pi }_{n,\\bar{k}} & \\underline{\\bar{m}} & \\underline{\\bar{m}} & \\underline{\\bar{\\pi }}_{p,\\bar{k}}^{2}\\\\\\underline{\\bar{\\pi }}_{n,\\bar{k}}^{2} & \\underline{m} & 1 & \\underline{\\bar{m}}\\\\& \\underline{s}^2_{\\pi _{p},\\bar{k}} & \\underline{s}_{0,\\bar{k}} & \\underline{\\bar{s}}^2_{\\pi _{m},\\bar{k}} & \\underline{\\bar{s}}^3_{\\pi _{m},\\bar{k}}\\\\& \\underline{s}^3_{0,\\bar{k}} & \\underline{s}^2_{\\pi _{n},\\bar{k}} & \\underline{s}_{I} & \\underline{s}_{I}\\end{array}\\right\\rbrace \\left(-\\right)^{\\underline{\\pi }^2_{m,\\bar{k}}+\\underline{\\pi }_{m,\\bar{k}}^{3}+1}$ $\\left\\lbrace \\begin{array}{cccc}\\underline{\\bar{\\pi }}_{p,\\bar{k}} & \\underline{\\bar{\\pi }}^2_{m,\\bar{k}} & \\underline{\\bar{\\pi }}^2_{n,\\bar{k}} \\\\\\underline{\\bar{m}} & \\underline{\\bar{\\pi }}_{n,\\bar{k}} & \\underline{\\pi }_{m,\\bar{k}}\\\\\\underline{s}^2_{0,\\bar{k}} & \\underline{s}_{0,\\bar{k}} & \\underline{s}^2_{\\pi _m,\\bar{k}} & \\underline{s}^2_{\\pi _n,\\bar{k}}\\end{array}\\right\\rbrace \\left\\lbrace \\begin{array}{cccc}\\underline{\\bar{\\pi }}^2_{p,\\bar{k}} & \\underline{\\bar{\\pi }}^3_{m,\\bar{k}} & \\underline{\\bar{\\pi }}^2_{n,\\bar{k}}\\\\\\underline{\\bar{\\pi }}^2_{m,\\bar{k}} & \\underline{\\pi }_{p,\\bar{k}} & m\\\\\\underline{s}^3_{0,\\bar{k}} & \\underline{s}^2_{\\pi _p,\\bar{k}} & \\underline{s}^3_{\\pi _m,\\bar{k}} & \\underline{s}^2_{0,\\bar{k}}\\end{array}\\right\\rbrace $ $\\mathfrak {P}_{SU(3)}\\left(\\underline{\\pi }_{n,\\bar{k}},\\underline{\\pi }_{p,\\bar{k}},\\underline{\\pi }_{m,\\bar{k}},\\left\\lbrace \\underline{\\pi }\\right\\rbrace _{\\bar{k}+\\bar{e}_{n}},\\ldots ;\\left\\lbrace \\underline{j}\\right\\rbrace ;\\underline{s}_{n,\\bar{k}},\\underline{s}_{m,\\bar{k}},\\underline{s}_{0,\\bar{k}},\\left\\lbrace \\underline{s}\\right\\rbrace _{\\bar{k}+\\bar{e}_{n}},\\ldots ;\\right.$ $\\left.\\underline{\\pi }_{n,\\bar{k}},\\underline{\\pi }_{m,\\bar{k}}^{4},\\underline{\\pi }_{n,\\bar{k}+\\bar{e}_{n}}^{2},\\ldots ;\\underline{j}_{n,\\bar{k}}^{2}\\ldots ;\\underline{s}_{0,\\bar{k}}^{2},\\underline{s}_{n,\\bar{k}}^{2},\\underline{s}_{m,\\bar{k}}^{2},\\ldots \\mid \\bar{m}\\right)$ $\\mathfrak {P}_{SU(3)}\\left(\\underline{\\pi }_{n,\\bar{k}}^{2},\\underline{\\pi }_{p,\\bar{k}},\\underline{\\pi }_{m,\\bar{k}}^{2},\\left\\lbrace \\underline{\\pi }\\right\\rbrace _{\\bar{k}+\\bar{e}_{p}},\\ldots ;\\ldots \\underline{j}_{p,\\bar{k}},\\underline{j}_{m,\\bar{k}}^{2},\\ldots ;\\underline{s}_{m,\\bar{k}}^{2},\\underline{s}_{p,\\bar{k}},\\underline{s}_{0,\\bar{k}}^{2},\\left\\lbrace \\underline{s}\\right\\rbrace _{\\bar{k}+\\bar{e}_{n}},\\ldots ;\\right.$ $\\left.\\underline{\\pi }_{n,\\bar{k}}^{3},\\underline{\\pi }_{p,\\bar{k}}^{2},\\ldots ;\\underline{j}_{p,\\bar{k}}^{2},\\underline{j}_{m,\\bar{k}}^{3}\\ldots ;\\underline{s}_{0,\\bar{k}}^{3},\\underline{s}_{m,\\bar{k}}^{3},\\underline{s}_{p,\\bar{k}}^{2},\\ldots \\mid \\bar{m}\\right)$ $\\left|\\nu _{out}\\left(\\underline{\\pi }_{n,\\bar{k}}^{2},\\underline{\\pi }_{m,\\bar{k}}^{3},\\underline{\\pi }_{p,\\bar{k}}^{2}\\ldots ;\\underline{j}_{n,\\bar{k}}^{2},\\underline{j}_{m,\\bar{k}}^{3},\\underline{j}_{p,\\bar{k}}^{2},\\ldots ;\\underline{s}_{0,\\bar{k}}^{3},\\underline{s}_{m,\\bar{k}}^{3},\\underline{s}_{n,\\bar{k}}^{2},\\underline{s}_{p,\\bar{k}}^{2}\\right)\\right\\rangle $ For iii) one gets almost the same as for i): $tr\\left(\\underline{\\hat{\\tau }}_{I}\\hat{A}_{m,n,1-\\bar{p}_{m},1-\\bar{p}_{n}}\\right)tr\\left(\\underline{\\hat{\\tau }}_{I}\\underline{\\hat{A}}_{m,n}\\right)\\left|\\nu _{orient}\\left(\\left\\lbrace \\underline{\\pi }\\right\\rbrace _{\\bar{k}},\\left\\lbrace \\underline{j}\\right\\rbrace ;\\left\\lbrace \\underline{s}\\right\\rbrace _{\\bar{k}}\\right)\\right\\rangle =$ $\\underset{\\mathcal {S}}{\\sum } (-)^{(...)}(-)^{\\underline{\\pi }^3_{n,\\bar{k}}+\\underline{\\pi }^2_{n,\\bar{k}}+\\underline{m}+j_{n,\\bar{k}-\\bar{p}_n}+\\underline{j}^2_{n,\\bar{k}-\\bar{p}_n}+\\underline{\\pi }^3_{n,\\bar{k}}+\\underline{\\pi }^2_{n,\\bar{k}}+1}\\left\\lbrace \\begin{array}{ccccc}\\underline{\\bar{j}}_{n,\\bar{k}-\\bar{e}_{n}+\\bar{p}_{n}} & \\underline{\\bar{\\pi }}_{n,\\bar{k}} & \\underline{\\bar{\\pi }}_{n,\\bar{k}}^{2} & \\underline{\\bar{\\pi }}_{n,\\bar{k}}^{3}\\\\\\underline{j}_{n,\\bar{k}-\\bar{p}_{n}} & \\underline{\\bar{m}} & \\underline{\\bar{m}} & \\underline{j}_{n,\\bar{k}-\\bar{e}_{n}+\\bar{p}_{n}}^{2}\\\\\\underline{\\bar{j}}_{n,\\bar{k}-\\bar{p}_{n}}^{2} & \\underline{m} & 1 & \\underline{\\bar{m}}\\\\& \\underline{s}_{j_{n},\\bar{k}-\\bar{e}_{n}+\\bar{p}_{n}} & \\underline{s}_{n,\\bar{k}} & \\underline{\\bar{s}}_{\\pi _{n},\\bar{k}}^{2} & \\underline{\\bar{s}}_{\\pi _{n},\\bar{k}}^{3}\\\\& \\underline{s}_{n,\\bar{k}}^{3} & \\underline{s}_{j_{n},\\bar{k}-\\bar{p}_{n}} & \\underline{s}_{I} & \\underline{s}_{I}\\end{array}\\right\\rbrace $ $\\underset{\\underline{s}^2_{n,\\bar{k}}}{\\sum }\\left\\lbrace \\begin{array}{cccc}\\underline{\\bar{j}}_{n,\\bar{k}-\\bar{e}_n}+\\bar{p}_n & \\underline{\\pi }^2_{n,\\bar{k}} & \\underline{\\bar{j}}^2_{n,\\bar{k}-\\bar{p}_n} \\\\\\underline{\\bar{m}} & \\underline{\\bar{j}}_{n,\\bar{k}-\\bar{p}_n} & \\underline{\\bar{\\pi }}_{n,\\bar{k}} \\\\\\underline{s}^2_{n,\\bar{k}} & \\underline{s}_{n,\\bar{k}} & \\underline{s}^2_{\\pi _n,\\bar{k}} & \\underline{s}_{j_n,\\bar{k}-\\bar{p}_n}\\end{array}\\right\\rbrace ^{-1}\\left\\lbrace \\begin{array}{cccc}\\underline{j}^2_{n,\\bar{k}-\\bar{e}_n+\\bar{p}_n} & \\underline{\\pi }^3_{n,\\bar{k}} & \\underline{j}^2_{n,\\bar{k}-\\bar{p}_n} \\\\\\underline{\\pi }^2_{n,\\bar{k}} & \\underline{\\bar{j}}_{n,\\bar{k}-\\bar{e}_n+\\bar{p}_n} & \\underline{m} \\\\\\underline{s}^3_{n,\\bar{k}} & \\underline{s}_{j_n,\\bar{k}-\\bar{e}_n+\\bar{p}_n} & \\underline{s}^3_{\\pi _n,\\bar{k}} & \\underline{s}^2_{n,\\bar{k}}\\end{array}\\right\\rbrace ^{-1}$ $\\mathfrak {P}_{SU(3)}\\left(\\underline{\\pi }_{n,\\bar{k}},\\underline{\\pi }_{p,\\bar{k}},\\underline{\\pi }_{m,\\bar{k}},\\left\\lbrace \\underline{\\pi }\\right\\rbrace _{\\bar{k}+\\bar{e}_{n}},\\ldots ;\\left\\lbrace \\underline{j}\\right\\rbrace ;\\underline{s}_{m,\\bar{k}},\\underline{s}_{0,\\bar{k}},\\underline{s}_{n,\\bar{k}},\\left\\lbrace \\underline{s}\\right\\rbrace _{\\bar{k}+\\bar{e}_{n}}\\ldots ;\\right.$ $\\left.\\underline{\\pi }_{n,\\bar{k}}^2,\\underline{\\pi }_{m,\\bar{k}}^{2},\\underline{\\pi }_{n,\\bar{k}+\\bar{e}_{n}}^{2},\\ldots ;\\underline{j}_{n,\\bar{k}-\\bar{p}_{n}}^{2},\\ldots ;\\underline{s}_{0,\\bar{k}}^{2},\\underline{s}_{m,\\bar{k}}^{2},\\underline{s}_{n,\\bar{k}}^{2},\\ldots \\mid \\bar{m}\\right)$ $\\mathfrak {P}_{SU(3)}\\left(\\underline{\\pi }_{n,\\bar{k}}^2,\\underline{\\pi }_{p,\\bar{k}},\\underline{\\pi }_{m,\\bar{k}}^{2},\\left\\lbrace \\underline{\\pi }\\right\\rbrace _{\\bar{k}+\\bar{e}_{n}}^{2},\\ldots ;\\ldots \\underline{j}_{m,\\bar{k}-\\bar{p}_{m}}^{2},\\underline{j}_{n,\\bar{k}-\\bar{p}_{n}}^{2},\\ldots ;\\underline{s}_{m,\\bar{k}}^{2},\\underline{s}_{0,\\bar{k}}^{2},\\underline{s}_{n,\\bar{k}}^{2},\\left\\lbrace \\underline{s}\\right\\rbrace _{\\bar{k}+\\bar{e}_{n}}^{2}\\ldots ;\\right.$ $\\left.\\underline{\\pi }_{n,\\bar{k}}^{3},\\underline{\\pi }_{m,\\bar{k}}^{3},\\ldots ;\\underline{j}_{n,\\bar{k}-\\bar{e}_{n}+\\bar{p}_{n}}^{2},\\underline{j}_{m,\\bar{k}-\\bar{e}_{m}+\\bar{p}_{m}}^{3},\\ldots ;\\underline{s}_{0,\\bar{k}}^{3},\\underline{s}_{m,\\bar{k}}^{3},\\underline{s}_{n,\\bar{k}}^{3},\\ldots \\mid \\bar{m}\\right)$ $\\left|\\nu _{orient}\\left(\\underline{\\pi }_{n,\\bar{k}}^{3},\\underline{\\pi }_{m,\\bar{k}}^{3}\\underline{\\pi }_{p,\\bar{k}},\\ldots ;\\underline{j}_{n,\\bar{k}}^{2},\\underline{j}_{n,\\bar{k}-\\bar{e}_{n}}^{2},\\underline{j}_{m,\\bar{k}-\\bar{p}_{m}}^{3},\\underline{j}^2_{m,\\bar{k}-\\bar{e}_{m}+\\bar{p}_{m}},\\ldots ;\\underline{s}_{0,\\bar{k}}^{3},\\underline{s}_{n,\\bar{k}}^{3},\\underline{s}_{m,\\bar{k}}^{3}\\underline{s}_{p,\\bar{k}}\\right)\\right\\rangle $ For iv) finally (compare to (ii)): $tr\\left(\\underline{\\hat{\\tau }}_{I}\\hat{A}_{m,p,1-\\bar{p}_{m},\\bar{p}_{p}}\\right)tr\\left(\\underline{\\hat{\\tau }}_{I}\\underline{\\hat{A}}_{m,n}\\right)\\left|\\nu _{orient}\\left(\\left\\lbrace \\underline{\\pi }\\right\\rbrace _{\\bar{k}},\\left\\lbrace \\underline{j}\\right\\rbrace ;\\left\\lbrace \\underline{s}\\right\\rbrace _{\\bar{k}}\\right)\\right\\rangle =$ $=\\underset{\\mathcal {S}}{\\sum }\\left(-\\right)^{\\left(\\ldots \\right)}\\left\\lbrace \\begin{array}{ccccc}\\underline{\\pi }_{p,\\bar{k}} & \\underline{\\pi }_{m,\\bar{k}} & \\underline{\\pi }_{m,\\bar{k}}^{2} & \\underline{\\pi }_{m,\\bar{k}}^{3}\\\\\\underline{\\pi }_{n,\\bar{k}} & \\underline{\\bar{m}} & \\underline{\\bar{m}} & \\underline{\\bar{\\pi }}_{p,\\bar{k}}^{2}\\\\\\underline{\\bar{\\pi }}_{n,\\bar{k}}^{2} & \\underline{m} & 1 & \\underline{\\bar{m}}\\\\& \\underline{s}^2_{\\pi _{p},\\bar{k}} & \\underline{s}_{0,\\bar{k}} & \\underline{\\bar{s}}^2_{\\pi _{m},\\bar{k}} & \\underline{\\bar{s}}^3_{\\pi _{m},\\bar{k}}\\\\& \\underline{s}^3_{0,\\bar{k}} & \\underline{s}^2_{\\pi _{n},\\bar{k}} & \\underline{s}_{I} & \\underline{s}_{I}\\end{array}\\right\\rbrace \\left(-\\right)^{\\underline{\\pi }^2_{m,\\bar{k}}+\\underline{\\pi }_{m,\\bar{k}}^{3}+1}$ $\\left\\lbrace \\begin{array}{cccc}\\underline{\\bar{\\pi }}_{p,\\bar{k}} & \\underline{\\bar{\\pi }}^2_{m,\\bar{k}} & \\underline{\\bar{\\pi }}^2_{n,\\bar{k}} \\\\\\underline{\\bar{m}} & \\underline{\\bar{\\pi }}_{n,\\bar{k}} & \\underline{\\pi }_{m,\\bar{k}}\\\\\\underline{s}^2_{0,\\bar{k}} & \\underline{s}_{0,\\bar{k}} & \\underline{s}^2_{\\pi _m,\\bar{k}} & \\underline{s}^2_{\\pi _n,\\bar{k}}\\end{array}\\right\\rbrace \\left\\lbrace \\begin{array}{cccc}\\underline{\\bar{\\pi }}^2_{p,\\bar{k}} & \\underline{\\bar{\\pi }}^3_{m,\\bar{k}} & \\underline{\\bar{\\pi }}^2_{n,\\bar{k}}\\\\\\underline{\\bar{\\pi }}^2_{m,\\bar{k}} & \\underline{\\pi }_{p,\\bar{k}} & m\\\\\\underline{s}^3_{0,\\bar{k}} & \\underline{s}^2_{\\pi _p,\\bar{k}} & \\underline{s}^3_{\\pi _m,\\bar{k}} & \\underline{s}^2_{0,\\bar{k}}\\end{array}\\right\\rbrace $ $\\mathfrak {P}_{SU(3)}\\left(\\underline{\\pi }_{n,\\bar{k}},\\underline{\\pi }_{p,\\bar{k}},\\underline{\\pi }_{m,\\bar{k}},\\left\\lbrace \\underline{\\pi }\\right\\rbrace _{\\bar{k}+\\bar{e}_{n}},\\ldots ;\\left\\lbrace \\underline{j}\\right\\rbrace ;\\underline{s}_{n,\\bar{k}},\\underline{s}_{m,\\bar{k}},\\underline{s}_{0,\\bar{k}},\\left\\lbrace \\underline{s}\\right\\rbrace _{\\bar{k}+\\bar{e}_{n}},\\ldots ;\\right.$ $\\left.\\underline{\\pi }_{n,\\bar{k}},\\underline{\\pi }_{m,\\bar{k}}^{4},\\underline{\\pi }_{n,\\bar{k}+\\bar{e}_{n}}^{2},\\ldots ;\\underline{j}_{n,\\bar{k}}^{2}\\ldots ;\\underline{s}_{0,\\bar{k}}^{2},\\underline{s}_{n,\\bar{k}}^{2},\\underline{s}_{m,\\bar{k}}^{2},\\ldots \\mid \\bar{m}\\right)$ $\\mathfrak {P}_{SU(3)}\\left(\\underline{\\pi }_{n,\\bar{k}}^{2},\\underline{\\pi }_{p,\\bar{k}},\\underline{\\pi }_{m,\\bar{k}}^{2},\\left\\lbrace \\underline{\\pi }\\right\\rbrace _{\\bar{k}+\\bar{e}_{p}-2\\bar{p}_{p}}\\ldots ;\\ldots \\underline{j}_{p,\\bar{k}},\\underline{j}_{m,\\bar{k}}^{2},\\ldots ;\\underline{s}_{m,\\bar{k}}^{2},\\underline{s}_{p,\\bar{k}},\\underline{s}_{0,\\bar{k}},\\left\\lbrace \\underline{s}\\right\\rbrace _{\\bar{k}+\\bar{e}_{p}-2\\bar{p}_{p}},\\ldots ;\\right.$ $\\left.\\underline{\\pi }_{m,\\bar{k}}^{3},\\underline{\\pi }_{p,\\bar{k}}^{2},\\ldots ;\\underline{j}_{p,\\bar{k}}^{2},\\underline{j}_{m,\\bar{k}-\\bar{e}_{m}+\\bar{p}_{m}}^{3},\\ldots ;\\underline{s}_{0,\\bar{k}}^{3},\\underline{s}_{m,\\bar{k}}^{3},\\underline{s}_{p,\\bar{k}}^{2},\\ldots \\mid \\bar{m}\\right)$ $\\left|\\nu _{out}\\left(\\underline{\\pi }_{n,\\bar{k}}^{2},\\underline{\\pi }_{m,\\bar{k}}^{3},\\underline{\\pi }_{p,\\bar{k}}^{2}\\ldots ;\\underline{j}_{n,\\bar{k}}^{2},\\underline{j}_{m,\\bar{k}}^{3},\\underline{j}_{p,\\bar{k}}^{2},\\underline{j}_{m,\\bar{k}-\\bar{e}_{m}}^{2}\\ldots ;\\underline{s}_{0,\\bar{k}}^{3},\\underline{s}_{m,\\bar{k}}^{3},\\underline{s}_{n,\\bar{k}}^{2},\\underline{s}_{p,\\bar{k}}^{2}\\right)\\right\\rangle $"
],
[
"Conclusion",
"In this paper we have taken the first steps towards the computation of the fundamental QCD spectrum within the LQG approach to quantum gravity.",
"More precisely, we have computed the matrix elements of the Yang-Mills contribution to the Hamiltonian analytically in closed form as far as the gluon field is concerned, while for the gravitational degrees of freedom a fully analytical analysis is not possible due to the necessity of computing the spectrum of the volume operator, which is known to be possible only numerically.",
"Obviously, more analytical and numerical work is necessary to determine the spectrum with sufficient precision.",
"However the focus of this paper was not so much on the actual computation of the spectrum, but rather to prepare the necessary analytical tools.",
"The other message that we wanted to communicate is that the Hamiltonian that we considered in this paper needs to be improved by methods comming from renormalisation theory.",
"For this reason, we refrain from investigating more closely the spectrum of the Hamiltonian considered here from [25], but one should rather analyse the improved Hamiltonian.",
"We hope that, once one has found a Hamiltonian description of renormalisation, its fixed point Hamiltonian can be used, as this Hamiltonian has minimal if not vanishing discretisation errors.",
"Once this point has been understood, we can address the important question of how the picture of the running of the Yang-Mills coupling on a gravitational background is changed in the context of the quantum gravity coupled system.",
"Namely it transpires that the background dependent Hamiltonian depends on a cut-off while the background independent one does not.",
"Thus, the mechanism for the running of the coupling is very different for these two theories.",
"We reserve this analysis for future research."
],
[
"Acknowledgements",
"This project was supported in part by an Emerging Fields Initiative granted by the Friedrich-Alexander University.",
"KL thanks the German National Merit Foundation for financial support."
],
[
"Brief review on the $3j$ 's and {{formula:8641f1b0-3a2b-4a6c-a1ea-ffd3d34c4eaa}} 's for {{formula:814323b2-7721-4723-a0fa-fd3e316ffb50}}",
"For self-containedness some important properties of $nj$ -Symbols for the group $SU(2)$ are listed here.",
"Introductions to recoupling theory can be found in various textbooks on quantum mechanics and quantum angular momentum, e.g.",
"[43].",
"For an extensive list of properties of $nj$ -symbols see e.g.",
"[65] 3j-Symbols Relation to Clebsh-Gordan coefficients: $\\langle a,\\alpha ; b,\\beta | c,\\gamma \\rangle =(-)^{b-a+\\gamma }\\sqrt{2c+1}\\begin{pmatrix} a&b&c\\\\\\alpha &\\beta &-\\gamma \\end{pmatrix}$ where $|b,\\beta ;a,\\alpha \\rangle = |b,\\beta \\rangle \\otimes |a,\\alpha \\rangle $ Compatibility criteria If one (or several) of the following rules is violated, then $\\begin{pmatrix} a&b&c\\\\\\alpha &\\beta &\\gamma \\end{pmatrix}$ is vanishing: $a,b,c\\in \\frac{1}{2}\\mathbb {N}$ , $a\\pm \\alpha \\in \\mathbb {N}$ , $-a\\le \\alpha \\le a$ , $\\cdots $ $\\alpha +\\beta +\\gamma =0$ $a+b+c\\in \\mathbb {N}$ , $|a-b|\\le c\\le a+b$ (triangle inequality) Symmetries $\\begin{pmatrix} a&b&c\\\\\\alpha &\\beta &\\gamma \\end{pmatrix}=(-)^{a+b+c}\\begin{pmatrix} a&b&c\\\\-\\alpha &-\\beta &-\\gamma \\end{pmatrix}=(-)^{a+b+c}\\begin{pmatrix} b&a&c\\\\\\beta &\\alpha &\\gamma \\end{pmatrix}=\\begin{pmatrix} b&c&a\\\\\\beta &\\gamma &\\alpha \\end{pmatrix}$ 6j-Symbols Definition in terms of $3j$ 's $&\\left\\lbrace \\begin{array}{ccc} j_1&j_2&j_3\\\\j_4&j_5&j_6\\end{array}\\right\\rbrace =\\sum _{\\mu _1,\\cdots ,\\mu _6} (-)^{\\sum \\limits _{i=1}^6 (j_i-\\mu _i)}\\\\&\\begin{pmatrix} j_1&j_2&j_3\\\\\\mu _1&\\mu _2&-\\mu _3\\end{pmatrix}\\begin{pmatrix} j_1&j_5&j_6\\\\-\\mu _1&\\mu _5&\\mu _6\\end{pmatrix}\\begin{pmatrix} j_4&j_5&j_3\\\\\\mu _4&-\\mu _5&\\mu _3\\end{pmatrix}\\begin{pmatrix} j_4&j_2&j_6\\\\-\\mu _4&-\\mu _2&-\\mu _6\\end{pmatrix}$ Symmetries $\\left\\lbrace \\begin{array}{ccc} a&b&c\\\\d&e&f\\end{array}\\right\\rbrace =\\left\\lbrace \\begin{array}{ccc} b&a&c\\\\e&d&f\\end{array}\\right\\rbrace =\\left\\lbrace \\begin{array}{ccc} b&c&a\\\\e&f&d\\end{array}\\right\\rbrace =$ $\\left\\lbrace \\begin{array}{ccc} d&e&c\\\\a&b&f\\end{array}\\right\\rbrace =\\left\\lbrace \\begin{array}{ccc} d&b&f\\\\a&e&c\\end{array}\\right\\rbrace =\\left\\lbrace \\begin{array}{ccc} a&e&f\\\\d&b&c\\end{array}\\right\\rbrace $ Compatibility $\\left\\lbrace \\begin{array}{ccc} a&b&c\\\\d&e&f\\end{array}\\right\\rbrace =0$ unless the triangle inequalities hold for $\\lbrace a,b,c\\rbrace , \\lbrace a,e,f\\rbrace ,\\lbrace d,b,f\\rbrace $ and $\\lbrace d,e,c\\rbrace $ Orthogonality $\\sum _x d_x \\left\\lbrace \\begin{array}{ccc} a&b&x\\\\d&e&c\\end{array}\\right\\rbrace \\left\\lbrace \\begin{array}{ccc} a&b&x\\\\d&e&c^{\\prime }\\end{array}\\right\\rbrace =\\delta _{c,c^{\\prime }}\\frac{1}{d_c}$ if the compatibility requirements are fulfilled.",
"Graphical Calculus of $SU(2)$ The definitions of the basic objects in this graphical calculus are the same as in [41], [42] and thus reduce to the same labeling as has been done for the $SU(3)$ case.",
"Some of the rules for changing the graphs however have altered, e.g.",
"since the magnetic numbers are now in $\\frac{1}{2}\\mathbb {N}$ an arrow may change its direction by adding a sign factor of $(-)^{2a}$ $\\begin{array}{c}\\includegraphics {thesis-2_3_2pic14-arrow}\\end{array}=\\left(-1\\right)^{2a}\\begin{array}{c}\\includegraphics {thesis-2_3_2pic15-negative_arrow}\\end{array}$ This changes some of the more compley recoupling schemes (for a full list see [43]).",
"E.g.",
"the extraction of a 6j-symbol $\\begin{array}{c}\\includegraphics {thesis-2_3_2pic27-node+6j}\\end{array}=\\begin{array}{c}\\includegraphics {thesis-2_3_2pic28-6j}\\end{array}\\cdot \\begin{array}{c}\\includegraphics {thesis-2_3_2pic29-node}\\end{array}=$ $ \\left\\lbrace \\begin{array}{ccc}a & f & e\\\\d & c & b\\end{array}\\right\\rbrace \\cdot \\begin{array}{c}\\includegraphics {thesis-2_3_2pic30-node}\\end{array}$"
]
] | 1605.05975 |
[
[
"Electric-field tunable spin diode FMR in patterned PMN-PT/NiFe\n structures"
],
[
"Abstract Dynamic properties of NiFe thin films on PMN-PT piezoelectric substrate are investigated using the spin-diode method.",
"Ferromagnetic resonance (FMR) spectra of microstrips with varying width are measured as a function of magnetic field and frequency.",
"The FMR frequency is shown to depend on the electric field applied across the substrate, which induces strain in the NiFe layer.",
"Electric field tunability of up to 100 MHz per 1 kV/cm is achieved.",
"An analytical model based on total energy minimization and the LLG equation, with magnetostriction effect taken into account, is developed to explain the measured dynamics.",
"Based on this model, conditions for strong electric-field tunable spin diode FMR in patterned NiFe/PMN-PT structures are derived."
],
[
"Acknowledgement",
"This work is supported by the Polish National Science Center grant Harmonia-DEC-2012/04/M/ST7/00799.",
"S.v.D.",
"acknowledges financial support from the European Research Council (ERC-2012-StG 307502-E-CONTROL).",
"S.Z acknowledges the Dean's grant 15.11.230.198."
]
] | 1605.06019 |
[
[
"Many new variable stars discovered in the core of the globular cluster\n NGC 6715 (M54) with EMCCD observations"
],
[
"Abstract We show the benefits of using Electron-Multiplying CCDs and the shift-and-add technique as a tool to minimise the effects of the atmospheric turbulence such as blending between stars in crowded fields and to avoid saturated stars in the fields observed.",
"We intend to complete, or improve, the census of the variable star population in globular cluster NGC~6715.",
"Our aim is to obtain high-precision time-series photometry of the very crowded central region of this stellar system via the collection of better angular resolution images than has been previously achieved with conventional CCDs on ground-based telescopes.",
"Observations were carried out using the Danish 1.54-m Telescope at the ESO La Silla observatory in Chile.",
"The telescope is equipped with an Electron-Multiplying CCD that allowed to obtain short-exposure-time images (ten images per second) that were stacked using the shift-and-add technique to produce the normal-exposure-time images (minutes).",
"The high precision photometry was performed via difference image analysis employing the DanDIA pipeline.",
"We attempted automatic detection of variable stars in the field.",
"We statistically analysed the light curves of 1405 stars in the crowded central region of NGC~6715 to automatically identify the variable stars present in this cluster.",
"We found light curves for 17 previously known variable stars near the edges of our reference image (16 RR Lyrae and 1 semi-regular) and we discovered 67 new variables (30 RR Lyrae, 21 long-period irregular, 3 semi-regular, 1 W Virginis, 1 eclipsing binary, and 11 unclassified).",
"Photometric measurements for these stars are available in electronic form through the Strasbourg Astronomical Data Centre."
],
[
"Introduction",
"Galactic globular clusters are interesting stellar systems in Astronomy as they are fossils of the early Galaxy formation and evolution.",
"This makes them excellent laboratories in a wide range of topics from Stellar Evolution to Cosmology, from observations to theory.",
"NGC 6715 (M54) was discovered on July 24, 1778 by Charles Messierhttp://messier.seds.org/m/m054.html.",
"The cluster is in the Sagittarius Dwarf Spheroidal galaxy at a distance of 26.5 kpc from our Sun and 18.9 kpc from our Galactic centre.",
"It has a metallicity [Fe/H]=$-$ 1.49 dex and a distance modulus (m$-$ M)$_V$ =17.58 mag.",
"The magnitude of its horizontal branch is V$_{\\mathrm {HB}}$ =18.16 mag [23].",
"The stellar population and morphology of NGC 6715 have attracted the attention of several studies.",
"Although NGC 6715’s position is aligned with the central region of the Sagittarius Dwarf Spheroidal galaxy, recent studies have shown that this cluster is not the nucleus of the mentioned galaxy [29], [32], [37], [5].",
"[10] found that the metallicity spread in this cluster is intermediate between smaller-normal Galactic globular clusters and metallicity values associated with dwarf galaxies.",
"It has a very peculiar colour-magnitude diagram with multiple main sequences, turnoff points and an extended blue horizontal branch (HB) [47], [36].",
"[43] found that this cluster hosts a blue hook stellar population in its blue HB.",
"It is thought that NGC 6715 might host an intermediate mass black hole (IMBH) in its centre [24], [64] as well.",
"[41] found that globular clusters can be classified into two groups based on the mean periods and the number ratios of their RR0 and RR1 stars.",
"These are the Oosterhoff I (OoI) and Oosterhoff II (OoII) groups, and this is known as the Oosterhoff dichotomy.",
"It was also found that the metallicity of the clusters plays an important role in this classification [27] and that it could be related with the formation history of the Galactic halo [30], [11], [12], [53], [56].",
"The first Oosterhoff classification for NGC 6715 was done by [29] by comparing the amplitudes and periods of 67 RR Lyrae in this cluster with the period-amplitude relation obtained for M3 (OoI) and M9 (OoII).",
"NGC 6715 follows the relation defined by M3 and this implies that it is an OoI cluster.",
"They also found that the mean periods of the RR Lyrae also agree with the OoI classification.",
"Similarly, [55] in their study of variable stars in NGC 6715 discovered a new set of 80 RR Lyrae, and they used 95 RR0 and 33 RR1 (after excluding non-members or problematic stars) to find that this cluster shares some properties of both OoI and OoII clusters.",
"For instance, the average period of its RR0 and RR1 stars was found to be intermediate between the values for the OoI and OoII classifications.",
"We note that [12] also classified this cluster as an intermediate Oosterhoff type based on its position in the $<P_{\\mathrm {RR0}}>$ -metallicity diagram.",
"Several time-series photometric studies focused on the variable star population [44], [45], [29], [55], [31], but due to the high concentration of stars in its core, many variables have previously been missed due to blending.",
"NGC 6715 is very massive and it is among the densest known globular clusters [42].",
"Due to this, NGC 6715 also took our attention as an excellent candidate for our study of globular clusters using Electron-Multiplying CCDs (EMCCDs) and the shift-and-add technique [50], [49], [20].",
"Section summarises the observations, the reduction, and photometric techniques employed.",
"Section explains the calibration applied to the instrumental magnitudes.",
"Section contains the technique used in the detection and extraction of variable stars.",
"Sections and show the methodology used to classify variables and the color-magnitude diagrams employed, respectively.",
"Section presents our results and Section discusses the Oosterhoff classification of this cluster.",
"Our conclusions are presented in ."
],
[
"Data, reduction, and photometry",
"EMCCDs, also known as low light level charge-coupled devices (L3CCD) [54], [25], are conventional CCDs that have an extended readout register where the signal is amplified by impact ionisation before they are read out.",
"Hence the readout noise is negligible when compared to the signal and very high frame-rates become feasible (10-100 frames/s).",
"This provides an opportunity to compensate for the blurring effect of the turbulence in the atmosphere.",
"By shifting and adding the individual frames appropriately, it is possible to construct much higher resolution images than is possible using conventional CCD imaging from the ground.",
"Furthermore, the dynamic range of the stacked images is greatly increased and the saturation of bright stars is therefore not an issue except for the very brightest stars in the sky.",
"However, the main drawback with this technique is that the process whereby the signal is amplified also increases the photon noise component in the images by a factor of $\\sqrt{2}$ when compared to conventinal CCD images.",
"One should also be aware that EMCCD imaging data need to be calibrated in a different way to conventional CCD data.",
"Several studies where EMCCDs have been used can be found in the literature e.g.",
"high-resolution imaging of exoplanet host stars in the search for unseen companions [58], [13], [7], [59], [18] photometric and astrometric measurements of a pair of very close brown dwarfs [33], and time-series photometry of crowded globular cluster cores aiming to complete the census of variable stars [49], [20].",
"The data presented in this paper are the result of EMCCD observations performed over three consecutive years: 2013, 2014 and 2015 in April to September each year.",
"The 1.54 m Danish telescope at the ESO Observatory in La Silla, Chile was used with an Andor Technology iXon+897 EMCCD camera, which has a 512 $\\times $ 512 array of 16 $\\mu $ m pixels, giving a pixel scale of $\\sim 0^{\\prime \\prime }$ .09 per pixel and a total field of view of $\\sim 45\\times 45$ arcsec$^{2}$ .",
"For this research the EMCCD camera was set to work at a frame-rate of 10 Hz (ten frames per second) and an EM gain of $300e^-$ /photon.",
"The camera was placed behind a dichroic mirror which works as a long-pass filter.",
"Taking the mirror and the sensitivity of the camera into consideration, it is possible to cover the wavelength range from 650 nm to 1050 nm.",
"This is roughly a combination of SDSS $\\mathrm {i}^{\\prime }+\\mathrm {z}^{\\prime }$ filters [6].",
"More details about the instrument can be found in [51].",
"The total exposure time employed for a single observation was 10 minutes, which means that each observation is the result of shifting-and-adding 6000 exposures.",
"The resulting PSF FWHM in the reference image employed in the photometric reductions (see below) was 0.44$^{\\prime \\prime }$ .",
"In Figure REF , histograms of the number of observations per night during each year are shown.",
"Data in the left-hand panel correspond to 2013, data in the middle panel to 2014 and data in the right-hand panel to 2015.",
"We aimed to always take two observations per night, although it was not always possible due to weather conditions or time slots needed for other projects as is the case of the monitoring of microlensing events carried out by the MiNDSTEp consortium.",
"Figure: Histograms with the number of observations per night for globularcluster NGC 6715.",
"Left: 2013.",
"Middle 2014.",
"Right: 2015.Bias, flat-field and tip-tilt corrections, were performed using the procedures and algorithms described in [22].",
"In particular, the tip-tilt correction allows high-resolution stacked images to be created as described in detail in [49], [20].",
"Briefly, the method uses the Fourier cross correlation theorem where, for each bias- and flat-corrected exposure $I_k(i,j)$ within an observation, we calculate the cross correlation image between $I_k(i,j)$ and an average of 100 randomly chosen exposures.",
"The peak in the cross correlation image gives the appropriate shift to correct the tip-tilt error.",
"We then stack the shifted exposures according to batches grouped by image quality (a measure of image sharpness), to create ten-layer image cubes.",
"The best quality layers from each cube (i.e.",
"the sharpest layers) are extracted from all of the available observations to create a high-resolution reference image for subsequent difference image analysis (DIA).",
"Finally, each ten-layer cube is stacked to create a single image corresponding to each observation.",
"To extract the photometry in each of the stacked images we used the DanDIADanDIA is built from the DanIDL library of IDL routines available at http://www.danidl.co.uk pipeline [8], [9], which is based on difference image analysis [2], [1].",
"The pipeline works by aligning all images to the reference image, solving for a set of convolution kernels modelled as discrete pixel arrays, and subtracting the convolved reference image in each case to create a set of difference images.",
"Stars are detected on the reference image and their reference fluxes $f_{\\mathrm {ref}}$ (ADU/s) are measured using PSF photometry.",
"Difference fluxes $f_{\\mathrm {diff}}(t)$ (ADU/s) for each star detected in the reference image are measured in each of the difference images by optimally scaling the PSF model for the star to the difference image.",
"The light curve for each star in instrumental magnitudes $m_{\\mathrm {ins}}$ are built as shown in Eqn.",
"(REF ) $m_{\\mathrm {ins}}(t)=17.5-2.5\\log (f_{\\mathrm {tot}}(t)),$ where $f_{\\mathrm {tot}}(t)$ is the total flux in ADU/s defined as $f_{\\mathrm {tot}}(t)=f_{\\mathrm {ref}}+\\frac{f_{\\mathrm {diff}}(t)}{p(t)}.$ The quantity $p(t)$ is the photometric scale factor used to scale the reference frame to each image as part of the kernel model explained in [8].",
"An electronic table with photometric measurements and fluxes for all the variable stars presented in this work is available through the CDShttp://cds.u-strasbg.fr/ database with the format illustrated in Table REF .",
"Table: Time-series I-band photometry for all known and new variables in the field of view covered in globular cluster NGC 6715.",
"The standard M std M_{\\mathrm {std}} and instrumental m ins m_{\\mathrm {ins}} magnitudes are listed in Columns 4 and 5, respectively, corresponding to the variable star, filter, and epoch of mid-exposure listed in Columns 1-3, respectively.",
"The uncertainty on m ins m_{\\mathrm {ins}} is listed in Column 6, which also corresponds to the uncertainty on M std M_{\\mathrm {std}}.",
"For completeness, we also list the quantities f ref f_{\\mathrm {ref}}, f diff f_{\\mathrm {diff}} and pp from Eqn.",
"in Columns 7, 9 and 11, along with the uncertainties σ ref \\sigma _{\\mathrm {ref}} and σ diff \\sigma _{\\mathrm {diff}} in Columns 8 and 10.",
"This is an extract from the full table, which is available with the electronic version of the article at the CDS."
],
[
"Astrometry and finding chart",
"To create a reference image with the astrometric information for each star in the field covered in NGC 6715, we used the celestial coordinates available in the ACS Globular Cluster Surveyhttp://www.astro.ufl.edu/~ata/public_hstgc/ [3] which were uploaded for the field of the cluster through GAIA (Graphical Astronomy and Image Analysis Tool; [17]).",
"An (x,y) shift to match their respective stars in our reference image was applied.",
"Stars lying outside the field of view and those without a clear match were removed, and the (x, y) shift was refined by minimising the squared coordinate residuals.",
"A total of 305 stars over the entire field was used to guarantee that the astrometric solution applied to the reference image considered enough stars.",
"The radial Root Mean Square (RMS) scatter obtained in the residuals was $\\sim $ 0$^{\\prime \\prime }$ .028 ($\\sim $ 0.3 pixels).",
"This astrometrically calibrated reference image was used to produce a finding chart for NGC 6715 on which we marked the positions and identifications of all variable stars studied in this work (Fig.",
"REF ).",
"Finally, a table with the equatorial J2000 celestial coordinates of all variables is given in Table ."
],
[
"Photometric calibration",
"The photometric transformation of instrumental magnitudes to the standard system was accomplished using information available in the ACS Globular Cluster Survey, which provides calibrated magnitudes for selected stars in the fields of 50 globular clusters extracted from images taken with the Hubble Space Telescope (HST) instruments ACS and WFPC.",
"By matching the positions of the stars in the field of the HST images with those in our reference image, we obtained the photometric transformation shown in Figure REF .",
"The I magnitude obtained from the ACS [48] is plotted versus the instrumental $\\mathrm {i}^{\\prime }+\\mathrm {z}^{\\prime }$ magnitude obtained in this study.",
"The red line is a linear fit with slope unity yielding the zero point labelled in the title where $N$ is the number of stars used in the fit and $R$ is the correlation coefficient obtained.",
"Due to the substantial differences between the $\\mathrm {i}^{\\prime }+\\mathrm {z}^{\\prime }$ and standard I wavebands, there are non-linear colour terms in the transformation that we have not accounted for.",
"However, we have opted for an approximate absolute photometric calibration since variable star discovery and classification do not require a precise calibration.",
"Furthermore, our non-standard $\\mathrm {i}^{\\prime }+\\mathrm {z}^{\\prime }$ waveband precludes the possibility of using our RR Lyrae light curves for physical parameter estimation.",
"Figure: Standard I magnitude taken from the HST observations as a function of the instrumental i ' +z ' \\mathrm {i}^{\\prime }+\\mathrm {z}^{\\prime } magnitude.",
"The red line is the fit that best matches the data and it is described by the equation in the title.",
"The correlation coefficient is 0.999."
],
[
"Variable star searches",
"Light curves for a total of 1405 stars were obtained with the DanDIA pipeline in the field covered by the reference image.",
"To detect and extract the variable stars from all the non-variables, three automatic (or semi-automatic) techniques were employed.",
"They are described in Sections REF , REF , and REF ."
],
[
"Root mean square",
"A diagram of root mean square (RMS) magnitude deviation against mean I magnitude (see top Fig.",
"REF ) was constructed for the cluster.",
"In this diagram, we measure not only the photometric scatter for each star, but also the intrinsic variation of the variable stars over time, which gives them a higher RMS than the non-variables.",
"The classification is indicated by the colour as detailed in Table REF .",
"To select candidate variable stars, we fit a polynomial to the rms values as a function of magnitude and flag all stars with an rms greater than two times the model value."
],
[
"$S_B$ statistic",
"A detailed discussion can be found about the benefits of using the $S_B$ statistic to detect variable stars [19] and RR Lyrae with Blazhko effect [4].",
"The $S_B$ statistic is defined as $S_B=\\left(\\frac{1}{NM}\\right)\\sum _{i=1}^M\\left(\\frac{r_{i,1}}{\\sigma _{i,1}}+\\frac{r_{i,2}}{\\sigma _{i,2}}+...+\\frac{r_{i,k_i}}{\\sigma _{i,k_i}}\\right)^2,$ where $N$ is the number of data points for a given light curve and $M$ is the number of groups formed of time-consecutive residuals of the same sign from a constant-brightness light curve model (e. g. mean or median).",
"The residuals $r_{i,1}$ to $r_{i,k_i}$ correspond to the $i$ th group of $k_i$ time-consecutive residuals of the same sign with corresponding uncertainties $\\sigma _{i,1}$ to $\\sigma _{i,k_i}$ .",
"The $S_B$ statistic is larger in value for light curves with long runs of consecutive data points above or below the mean, which is the case for variable stars with periods longer than the typical photometric cadence.",
"A plot of $S_B$ versus mean I magnitude is given for NGC 6715 (see bottom Fig.",
"REF ), variable stars are plotted in colour.",
"To select candidate variable stars, the same technique employed in Section REF was used, with the threshold also set at two times the model $S_B$ values."
],
[
"Stacked difference image",
"Based on the results obtained using the DanDIA pipeline, a stacked difference image was built for NGC 6715 with the aim of detecting the difference fluxes that correspond to variable stars in the field of the reference image.",
"The stacked image is the result of summing the absolute values of the difference images divided by the respective pixel uncertainty $S_{ij}=\\sum _k\\frac{|D_{kij}|}{\\sigma _{kij}},$ where $S_{ij}$ is the stacked image, $D_{kij}$ is the $k$ th difference image, $\\sigma _{kij}$ is the pixel uncertainty associated with each image $k$ and the indexes $i$ and $j$ correspond to pixel positions.",
"All of the variable star candidates obtained by using the RMS and $S_B$ diagrams explained in Sections REF and REF were inspected visually in the stacked image and by blinking the difference images to confirm or refute their variability."
],
[
"Variable star classification",
"To define the type of variation in each of the variable stars found, several steps were done.",
"First, we used their position in the colour-magnitude diagram (CMD; Fig.",
"REF ) as a reference for their evolutionary stage as most types of variable stars are placed in very well defined zones in the CMD.",
"Second, we implemented a period search for each of the light curves by using the string method [28] and by minimising the $\\chi ^2$ in a Fourier analysis fit.",
"Periods found and light curve shapes were also taken into account.",
"Finally, to classify the variable stars, we used the conventions defined in the General Catalogue of Variable Stars [46].",
"In Table REF , the classification, corresponding symbols, and colours used in the plots throughout the paper are shown."
],
[
"Colour magnitude diagram",
"As our sample has data available for only one filter, we decided to build the colour-magnitude diagram (CMD; see Fig.",
"REF ) by using the information available from the HST images at the ACS Globular Cluster Survey.",
"The data used correspond to the V and I photometry obtained in [48].",
"The CMD was useful in classifying the variable stars, especially those with poorly defined light curves such as long period variables and semi-regular variables, as well as corroborating cluster membership."
],
[
"NGC 6715 / C1851-305 / Messier 54",
"The details of all variable stars in our FoV that are discussed in this section are listed in Table , and all light curves are plotted in Figure REF .",
"Table: Convention used in the variable star classification of this work based on the definitions of the General Catalogue of Variable Stars .Figure: Root mean square (RMS) magnitude deviation (top) and S B S_B statistic (bottom) versus the mean II magnitude for the 1402 stars detected in the field of view of the reference image for NGC 6715.",
"Coloured points follow the convention adopted in Table to identify the types of variables found in the field of this globular cluster.Figure: Colour magnitude diagram of the globular cluster NGC 6715 built with V and I magnitudes available in the ACS globular cluster survey extracted from HST images."
],
[
"Known variables",
"This globular cluster has of the order of 200 known variable stars listed in the Catalogue of Variable Stars in Galactic Globular Clusters [14].",
"Most of them are of the RR Lyrae type but a few are long-period irregular, semi-regular, W Virginis, eclipsing binaries, and SX Phoenicis.",
"To date four studies report variable star discoveries in this globular cluster: V1-V28 from [44], V29-V82 from [45], V83-V117 from [29], and V118-V211 from [55].",
"In the field of view covered by our reference image there are only five known variable stars (V112, V160, V173, V181, V192).",
"All of them lie towards the edges of the image as can be seen in Figure REF .",
"The star V112 was previously classified as long-period irregular.",
"However, we were able to find a period of $\\sim $ 100 days in the variability of this star and due to this we have reclassified it as a semi-regular variable.",
"For V160, we were not able to produce a good phased light curve using the published period of 0.6194848 d. The discovery observations by [55] cover a time baseline of only 6 days.",
"With our time baseline of more than two years, our derived periods are much more precise and the period found is in agreement with that found by the Optical Gravitational Lensing Experiment [60].",
"For V160, we list the OGLE period of 0.62813716 d in Table .",
"For V173, we improved the period estimate over that from [55].",
"For V181, we find a very different period with respect to the one estimated by [55].",
"The new period of 0.877072 d makes this RR Lyrae the one with the longest period in the cluster.",
"The phased light curve in Figure REF is somewhat noisy because this variable is highly blended with a brighter star.",
"The case of V192 is particularly interesting because the star was classified as RR1 with a period of 0.3986799 d. However, at the astrometric position reported for this star we found a RR Lyrae type RR0 with a period of 0.600373 d. This is also in agreement with the period and classification found by OGLE (see below) which we list in Table .",
"Again, the phased light curve of this variable is noisy because of blending with a brighter star.",
"It is clear then that the [55] periods and RR Lyrae classifications are not robust based on relatively few observations.",
"We will discuss the consequences of this later on.",
"Recently, [38] announced 50 RR Lyrae candidates based on observations taken with the HST.",
"However, the data obtained consist of 12 epochs covering only 8 hours, which made the study unsuitable for a period search and certainly some RR Lyrae stars will have been missed with such a short time baseline.",
"No light curves were presented in their paper.",
"Of these 50 candidate variable stars, 17 lie outside of our field of view (VC2, VC11-VC18, VC22, VC24, VC38, VC44, VC45, VC47-VC49).",
"As pointed out in the Catalogue of Variable Stars in Galactic Globular Clusters [14], 11 of these candidates are previously known variables, thus VC2=V127, VC11=V162, VC12=V163, VC13=V95, VC14=V164, VC15=V142, VC17=V129, VC18=V179, VC44=V46, VC45=V148, and VC47=V76 (see also Appendix ).",
"For 8 of their candidates in our field of view, we could not detect variations in our difference images at their coordinates (VC3, VC6, VC19, VC21, VC29, VC42, VC43, VC50), and we therefore cannot confirm their variability.",
"We plot their positions in Figure REF with a green squareThe [38] coordinates differ by RA$\\sim $ 0.3$^{\\prime \\prime }$ and Dec$\\sim $ 0.8$^{\\prime \\prime }$ from our coordinates, and we have corrected for this in Figure REF.",
"Three of their candidates in our field of view, VC28, VC34 and VC46, are the known variables V181, V160 and V192, respectively.",
"We confirm the variable nature of the remaining 22 candidates in our field of view, and we have assigned them “V” numbers as part of our study (see Section REF ).",
"Table lists their VC identification in Column 2.",
"We classify 18 of them as RR Lyrae stars, one as an eclipsing binary, and are unable to classify VC27, VC32, and VC33.",
"We plot their positions in Figure REF with a square symbol.",
"The study by [34] using the VISTA survey also covered the cluster and presents candidate variables based on typically $\\sim $ 12-13 epochs spread over $\\sim $ 100-200 days, which was insufficient to derive periods for many of them.",
"Of the short period candidate variables listed in their Table 1, 18 are inside the field of view covered by our reference image.",
"None of the bright long-period variables listed in their Table 2 and faint candidates listed in their Table 3 are inside the field covered in our study.",
"Positions of these stars inside our field of view are plotted in Figure REF with a red circle.",
"It is worth noting that all of these candidates are located more toward the edges of the reference image.",
"We detect variability in only two of the [34] candidates within our field of view (SPVSgr18550405-3028580 and SPVSgr18550386-3028593, which we assign V identifications as V229 and V263, respectively -see Section REF ).",
"Similarly, the field of this cluster was also covered by OGLE, particularly with their OGLE-IV survey [61].",
"We found in this survey 15 of the variable stars studied in our work of which two are previously known variables (V160 and V192) and 13 are new discoveries by OGLE (we assigned the following V identifications: V227-V229, V233, V236, V237, V240, V244, V246, V250-V253).",
"The OGLE light curves for these stars have typically $\\sim $ 150 epochs covering a baseline of $\\sim $ 2.5 years and OGLE derived precise periods for them.",
"With our data we were also able to find the same periods and type of classification assigned by OGLE.",
"The positions of these stars are plotted in Figure REF with a blue colour.",
"Again, it is worth noticing that all of these variables are located more toward the edges of the reference image.",
"In the particular case of V229, V244, and V246, the pipeline was not able to detect these stars in the reference image but their variation is clear in the difference images.",
"Their differential fluxes against phase are plotted in Figure REF .",
"Finally, epochs, periods, mean magnitudes, amplitudes, and classifications for these three stars were taken from the OGLE database, although the number of data points listed in Table correspond to our light curves.",
"In Appendix (Table REF ), we provide the cross identifications for the previously known RR Lyrae stars in NGC 6715 between the CVSGGC [14], the variable star candidates from [38], and the OGLE RR Lyrae stars [61].",
"Figure: Finding chart for the globular cluster NGC 6715.",
"The image usedcorresponds to the reference image constructed during the reduction.",
"All knownvariables and new discoveries are labelled with their V numbers.",
"Known variables, or new variables discovered in this work, are plotted with black symbols.",
"Variables discovered by the OGLE survey are plotted with blue symbols.",
"Those variables that were candidate variables from are plotted using squares.",
"Otherwise symbols are circles.",
"Green squares and red circles, both without labels, are candidate variables from and respectively for which we do not detect variability in our survey.",
"Image size is ∼40×40\\sim 40\\times 40arcsec 2 ^2."
],
[
"New variables",
"After employing the methods described in Section , we were able to extract 67 new variable stars in the core of NGC 6715 of which 30 are RR Lyrae, 1 is a W Virginis star (CWA), 21 are long-period irregular, 3 are semi regular, 1 is an eclipsing binary, and 11 remain without classification."
],
[
"RR Lyrae",
"V213-V226, V230-V232, V234-V235, V238-V239, V241-V243, V245, V247-V249, V254-V255: These 30 newly discovered variable stars are clear RR Lyrae variables.",
"Their positions in the CMD, light curve shapes, periods, and amplitudes corroborate their variability type.",
"We found that 17 are pulsating in the fundamental mode (RR0); 8 are pulsating in the first overtone (RR1); 1 is a double-mode pulsator (RR01) and 4 RR Lyrae remain with an uncetain subtype (3 RR0?",
"and 1 RR1?).",
"As shown in Table , their periods range from $\\sim $ 0.28 d to $\\sim $ 0.76 d with amplitudes between 0.06 and 1.69 mag.",
"The periodogram analysis for V221 showed two predominant frequencies typical of double mode RR Lyrae stars, one equivalent to the fundamental period $P_0=$ 0.459608 d and one equivalent to the first overtone period $P_1=$ 0.343828 d giving a period ratio $P_1/P_0=$ 0.748 which falls in the expected ratio range of $\\sim 0.725$ to $\\sim 0.748$ for this type of pulsating RR Lyrae stars [40], [39], [15].",
"Further data for a more detailed analysis of this star will be useful to corroborate its pulsational properties."
],
[
"W Virginis",
"V256: Particularly interesting is the case of this star as its variation (and position in the CMD) do not follow the pattern found for the other variable stars studied and classified in this work.",
"We found a very well phased light curve with a period of $\\sim $ 14.771 d and an amplitude of 0.71 mag.",
"This is the only bright variable star on the blue side of the colour-magnitude diagram far away from the red giant branch.",
"The properties found in the variation of this star and its position in the CMD match very well with the W Virginis type of variable star described in [46], particularly with the subtype CWA which have periods longer than 8 days [62].",
"Although these types of stars have not been commonly found in globular clusters in contrast to RR Lyrae stars, they are not entirely uncommon.",
"In the statistics of variable stars in Galactic globular clusters reported by [14] it is possible to notice that 60 variable stars are Cepheids, which include Population II Cepheids, anomalous Cepheids, and RV Tauri stars.",
"V256 is the first CWA star discovered in this cluster."
],
[
"Long-period irregular",
"V260-V280: These 21 stars are located at the top of the red giant branch as shown with blue squares in Figure REF .",
"Their amplitudes range from 0.05 to 0.46 mag.",
"We found no clear periods for these stars.",
"Due to this and also their position in the colour-magnitude diagram we classified them as long-period irregular.",
"Light curves for all these variables are found in Figure REF ."
],
[
"Semi regular",
"V257-V259: Based on the position of these 3 stars in the colour-magnitude diagram (see Fig.",
"REF ), their periods, and the shape of their light curves, we have classified them as semi-regular.",
"Light curves for these variables may be found in Figure REF .",
"Their amplitudes range from $\\sim $ 0.04 to $\\sim $ 0.45 mag and their periods span between $\\sim $ 20 and $\\sim $ 150 d."
],
[
"Eclipsing Binary",
"V212: The light curve variations for this star are very similar to those presented in eclipsing binary systems.",
"We found that the amplitude of the deeper eclipse is of the order of $\\sim $ 0.8 mag and the amplitude of the secondary eclipse is $\\sim $ 0.5 mag.",
"The phased light curve shown in Figure REF represented a period of $P=$ 0.202144 d."
],
[
"Other variable stars",
"V281-V285: These 5 stars are clear variable stars.",
"They show clear variability by blinking the difference images.",
"They have amplitudes between 0.45 and 0.94 mag.",
"Several attempts to determine periods for these stars were done without success.",
"Due to this their light curves in Figure REF are plotted against HJD.",
"We note that the variable source V281 is ony $\\sim $ 0.24 arcsec from the photometric centre of the cluster as measured by [21].",
"V286: Towards the beginning of the 2013 data, we found that the flux of this star was increasing to a maximum at $\\sim $ 2456464.9081 d of about $\\sim $ 39400 ADU/s on the flux scale of the reference image, corresponding to a peak magnitude of $\\sim $ 15.04 mag.",
"After that, its flux strongly decreased during the rest of the observational campaign.",
"During 2014 and 2015, we found that the object seems to be at baseline and it is not detected in the original images.",
"The centre of NGC 6715 is $\\sim $ 4.45 arcseconds from this source.",
"Also, at a distance of 3.73 arcseconds, there is a X-ray source studied by [64] using data from Chandra and Hubble Space Telescope.",
"Given the relatively small astrometric uncertainties in these positions, V286 is not associated with either.",
"The nature and classification of this variable will need further studies, and it will remain without classification in this work.",
"V287-V291: These 5 stars were not detected by the pipeline in the reference image.",
"Unfortunately, there are no light curves available in the OGLE database for these stars, but their variations are clear in the difference images.",
"In Figure REF their differential fluxes against HJD are plotted.",
"As we were not able to produce phased light curves for them, they will remain without classification until future studies are done."
],
[
"Oosterhoff dichotomy",
"In this work we discovered 30 new RR Lyrae, and OGLE also discovered 17 more (of which 13 are inside our FoV).",
"After removing the stars without a secure classification, there are 33 new RR0 and 9 new RR1 stars, which represent a significant increment in the known RR Lyrae population.",
"The vast majority of these new RR Lyrae stars are cluster members since we have studied the core of NGC 6715.",
"Hence it is pertinent to recalculate the mean periods and number ratios of the RR Lyrae stars to see if it modifies the current conclusions about the Oosterhoff type of NGC 6715.",
"However, we note that NGC 6715 lies projected against the Sagittarius Dwarf Spheroidal galaxy and behind the Galactic bulge.",
"Therefore, the sample of RR Lyrae stars in the field of the cluster is a mixture of cluster members, Bulge stars, and Sgr dSph stars.",
"Towards the centre of NGC 6715, the cluster member RR Lyraes dominate, but limiting our sample to the cluster core also reduces the number of RR Lyraes that can be used to calculate the mean periods and number ratios.",
"Hence, in Figure REF , we have plotted the total number of RR Lyrae stars, the mean period of the RR0 stars, the mean period of the RR1 stars, and the number ratio of the RR1 to all RR Lyrae stars (nRR1/(nRR0 + nRR1)), all as a function of distance from the cluster centre [21].",
"To convert angles on the sky into distances in parsecs we used the distance from the Sun to NGC 6715 of 26500 pc [23].",
"The inner and outer red vertical lines are the tidal radii estimated by [35] based on [26] and [63] models, respectively.",
"The horizontal black lines correspond to the OoI (solid) and OoII (dashed) types given by [52] in his Table 3.2.",
"We only used RR Lyrae stars with certain classifications (i.e.",
"published phased light curves with reliable period estimates).",
"Figure REF clearly shows that within the range of the two estimates of the tidal radii, where the RR Lyrae stars from the cluster still dominate, the values of $<P_{\\mathrm {RR0}}>$ , $<P_{\\mathrm {RR1}}>$ and nRR1/(nRR0 + nRR1) are intermediate between those expected for OoI and OoII clusters.",
"We obtain $<P_{\\mathrm {RR0}}>$ =0.613529 d, $<P_{\\mathrm {RR1}}>$ =0.334107 d, and nRR1/(nRR0 + nRR1)= 0.24050633 when calculated for the 79 RR Lyraes within the outer estimate of the tidal radius (i.e.",
"380 pc).",
"The contaminating RR Lyrae populations have ($<P_{\\mathrm {RR0}}>$ , $<P_{\\mathrm {RR1}}>$ )=(0.556 d, 0.310 d) and ($<P_{\\mathrm {RR0}}>$ ,$<P_{\\mathrm {RR1}}>$ )=(0.574 d, 0.322 d) for the Bulge and Sgr dSph, respectively [57], [16], placing them in the OoI region in the $<P_{\\mathrm {RR0}}>$ -metallicity diagram of [12].",
"This explains why the values of $<P_{\\mathrm {RR0}}>$ and nRR1/(nRR0 + nRR1) tend towards the OoI values beyond the outer estimate of the tidal radius.",
"The value of $<P_{\\mathrm {RR1}}>$ is hardly influenced though because there are only 6 RR1 stars beyond the outer estimate of the tidal radius.",
"Hence the new RR Lyrae discoveries in this paper have served to confirm that NGC 6715 is of intermediate Oosterhoff type.",
"Figure: The number of RR Lyraes, mean periods, and number ratios as a function of the distance from the cluster centre.",
"The red lines correspond to the tidal radii calculated by ."
],
[
"Conclusions",
"The globular cluster NGC 6715 turns out to be a very interesting stellar system, for which images with the highest angular resolution have ever been obtained so far with ground-based telescopes.",
"The use of the EMCCD and shift-and-add technique was demonstrated to be an excellent procedure to minimise the effect of the atmospheric turbulence which is one of the main constraints when doing ground-based observations.",
"Thanks to this and the use of difference image analysis it was possible to obtain high-precision time series photometry in the core of this cluster down to $I\\sim $ 18.3 mag.",
"A total of 1405 stars in the field covered by the reference image were statistically studied for variable star detection.",
"We presented light curves for 17 previously known variables that were found toward the edges of the reference image (16 RR Lyrae and 1 SR).",
"We also discovered 67 new variable stars, which consist of 30 RR Lyrae, 21 long-period irregular, 3 semi-regular, 1 W Virginis, 1 eclipsing binary, and 11 unclassified stars.",
"We estimated periods and ephemerides for all variable stars in the field of our reference image.",
"Our new RR Lyrae star discoveries help confirm that NGC 6715 is of intermediate Oosterhoff type.",
"Finally, our photometric measurements for all variable stars studied in this work are available in electronic form through the Strasbourg Astronomical Data Centre.",
"Our thanks go to Christine Clement for clarifying the known variable star content in NGC 6715 and the numbering systems of the variable stars while we were working on these clusters.",
"This support to the astronomical community is very much appreciated.",
"The Danish 1.54m telescope is operated based on a grant from the Danish Natural Science Foundation (FNU).",
"This publication was made possible by NPRP grant # X-019-1-006 from the Qatar National Research Fund (a member of Qatar Foundation).",
"The statements made herein are solely the responsibility of the authors.",
"KH acknowledges support from STFC grant ST/M001296/1.",
"GD acknowledges Regione Campania for support from POR-FSE Campania 2014-2020.",
"D.F.E.",
"is funded by the UK Science and Technology Facilities Council.",
"TH is supported by a Sapere Aude Starting Grant from the Danish Council for Independent Research.",
"Research at Centre for Star and Planet Formation is funded by the Danish National Research Foundation.",
"TCH acknowledges support from the Korea Research Council of Fundamental Science & Technology (KRCF) via the KRCF Young Scientist Research Fellowship.",
"Programme and for financial support from KASI travel grant number 2013-9-400-00, 2014-1-400-06 & 2015-1-850-04.",
"NP acknowledges funding by the Gemini-Conicyt Fund, allocated to project No.",
"32120036 and by the Portuguese FCT - Foundation for Science and Technology and the European Social Fund (ref: SFRH/BGCT/113686/2015).",
"CITEUC is funded by National Funds through FCT - Foundation for Science and Technology (project: UID/Multi/00611/2013) and FEDER - European Regional Development Fund through COMPETE 2020 -Operational Programme Competitiveness and Internationalisation (project: POCI-01-0145-FEDER-006922).",
"OW and J. Surdej acknowledge support from the Communauté française de Belgique - Actions de recherche concertées - Académie Wallonie-Europe.",
"This work has made extensive use of the ADS and SIMBAD services, for which we are thankful.",
"Var Other RA Dec Epoch $P$ $I_{median}$ $A_{\\mathrm {i}^{\\prime }+\\mathrm {z}^{\\prime }}$ $N$ Type id id J2000 J2000 HJD d mag mag (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) 9cContinued from previous page Var Other RA Dec Epoch $P$ $I_{median}$ $A_{\\mathrm {i}^{\\prime }+\\mathrm {z}^{\\prime }}$ $N$ Type id id J2000 J2000 HJD d mag mag (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) 9c$^{a}$ Epochs, periods, mean magnitudes, amplitudes, and classifications taken from OGLE database 9c$^{b}$ =SPVSgr18550405-3028580; $^{c}$ =SPVSgr18550386-3028593; $^{d}$ =Peak magnitude 9cContinued on next page 9c$^{a}$ Epochs, periods, mean magnitudes, amplitudes, and classifications taken from OGLE database 9c$^{b}$ =SPVSgr18550405-3028580; $^{c}$ =SPVSgr18550386-3028593; $^{d}$ =Peak magnitude 9cConcluded Ephemerides and main characteristics of the variable stars in the field of globular cluster NGC 6715.",
"Column 1 is the id assigned to the variable star, Column 2 is a previously known id assigned to the stars (5 digit numbers correspond to OGLE identifications of the form OGLE-BLG-RRLYR-NNNNN), Columns 3 and 4 correspond to the right ascension and declination (J2000), Column 5 is the epoch used, Column 6 is the period measured in this work unless the variable is an OGLE star in which case we use their period, Column 7 is median magnitude, Column 8 is the peak-to-peak amplitude in the light curve, Column 9 is the number of epochs and Column 10 is the classification of the variable.",
"The numbers in parentheses indicate the uncertainty on the last decimal place of the period.",
"cccccccccc V112 - 18:55:02.010 -30:28:35.13 2457190.7419 100(1) 13.63 0.34 44 SR V160 VC34; 37595 18:55:03.954 -30:28:29.83 2457189.7725 0.62813716(382) 17.65 0.75 40 RR0 V173 - 18:55:02.473 -30:29:00.04 2457206.6555 0.360284(152) 17.69 0.30 44 RR1 V181 VC28 18:55:02.395 -30:28:34.20 2457145.9107 0.877072(898) 17.60 0.87 42 RR0 V192 VC46; 37568 18:55:01.973 -30:29:02.27 2456784.9179 0.60035438(1060) 17.63 0.60 44 RR0 V212 VC35 18:55:03.777 -30:29:03.51 2457214.8470 0.202144(48) 17.59 0.79 42 E V213 VC40 18:55:02.126 -30:28:39.63 2456771.8623 0.286053(96) 17.95 0.64 44 RR1 V214 VC39 18:55:02.964 -30:28:35.37 2456765.9179 0.305386(109) 16.97 0.45 44 RR1 V215 VC25 18:55:02.701 -30:28:52.71 2456789.8857 0.307160(110) 17.45 0.34 44 RR1 V216 VC9 18:55:04.011 -30:28:37.73 2457292.6392 0.331073(128) 16.49 0.20 44 RR1 V217 - 18:55:02.070 -30:29:03.41 2457205.7553 0.331556(128) 17.76 0.35 44 RR1 V218 - 18:55:03.556 -30:28:56.77 2456891.5200 0.348450(142) 16.92 0.20 44 RR1 V219 - 18:55:03.233 -30:28:59.48 2457183.7770 0.382637(171) 17.21 0.24 44 RR1 V220 VC4 18:55:03.197 -30:28:38.54 2456765.9102 0.388893(177) 16.86 0.27 44 RR1 V221 VC8 18:55:04.353 -30:28:51.16 2456846.9179 0.459608(247) 17.39 0.71 44 RR01 V222 - 18:55:02.504 -30:28:46.40 2456765.9179 0.465653(253) 17.70 0.70 44 RR0?",
"V223 VC10 18:55:02.425 -30:28:45.20 2456823.8857 0.471901(260) 17.63 0.81 44 RR0 V224 VC20 18:55:03.549 -30:28:45.09 2457292.6392 0.482336(272) 16.16 0.27 44 RR0?",
"V225 VC37 18:55:03.862 -30:28:43.86 2456464.9081 0.483029(272) 17.28 0.35 44 RR1?",
"V226 - 18:55:03.948 -30:28:42.65 2456784.9179 0.497941(289) 16.39 0.36 44 RR0?",
"V227 37582 18:55:03.050 -30:28:35.81 2457190.7991 0.50805795(495) 17.66 0.90 44 RR0 V228 37575 18:55:02.757 -30:28:50.88 2456534.6709 0.52522562(844) 16.87 0.41 44 RR0 V229$^{a}$ 37597; $^{b}$ 18:55:04.033 -30:28:58.20 2456788.9321 0.52679780(281) 16.38 0.28 41 RR0 V230 - 18:55:03.637 -30:28:55.39 2456465.8700 0.532617(331) 16.04 0.29 44 RR0 V231 - 18:55:02.894 -30:28:58.30 2457189.8090 0.534194(333) 17.80 0.83 44 RR0 V232 VC41 18:55:03.581 -30:28:47.60 2456543.5260 0.556536(362) 16.35 0.29 44 RR0 V233 VC30; 37591 18:55:03.735 -30:28:40.24 2456846.9179 0.55752742(766) 17.41 0.93 44 RR0 V234 VC7 18:55:02.255 -30:28:37.13 2457206.6400 0.559765(366) 17.72 0.60 44 RR0 V235 - 18:55:03.411 -30:29:05.75 2457292.6392 0.566711(375) 17.41 0.69 43 RR0 V236 VC36; 37585 18:55:03.149 -30:28:34.86 2457152.8500 0.56847125(715) 16.53 0.28 44 RR0 V237 VC31; 37570 18:55:02.112 -30:28:44.40 2456435.8882 0.57985290(735) 17.57 0.65 44 RR0 V238 - 18:55:03.242 -30:28:42.94 2456846.9179 0.584918(399) 17.63 1.69 44 RR0 V239 - 18:55:02.848 -30:29:06.91 2457189.8090 0.596179(432) 17.57 0.69 39 RR0 V240 37576 18:55:02.725 -30:28:30.46 2456909.5275 0.59629933(890) 17.06 0.54 43 RR0 V241 VC1 18:55:04.190 -30:28:39.15 2456896.5087 0.602588(424) 17.79 0.65 44 RR0 V242 - 18:55:02.756 -30:29:00.15 2456891.4700 0.604762(427) 16.23 0.16 44 RR0 V243 - 18:55:02.587 -30:28:38.62 2456770.8691 0.610170(435) 16.97 0.36 44 RR0 V244$^{a}$ 37581 18:55:03.031 -30:29:01.15 2456891.4945 0.61517949(819) 16.90 0.27 44 RR0 V245 VC26 18:55:02.849 -30:28:57.81 2456771.8623 0.626797(459) 17.68 0.47 44 RR0 V246$^{a}$ 37573 18:55:02.616 -30:28:40.91 2457207.8691 0.62879467(894) 16.11 0.06 42 RR0 V247 - 18:55:02.826 -30:28:43.51 2456846.9179 0.650946(495) 16.86 0.42 44 RR0 V248 - 18:55:03.795 -30:28:54.42 2457258.7147 0.668967(522) 17.05 0.46 44 RR0 V249 - 18:55:03.226 -30:28:43.99 2457191.7238 0.673703(530) 15.91 0.21 44 RR0 V250 VC23; 37590 18:55:03.705 -30:29:00.36 2456543.5260 0.68064951(790) 17.54 0.58 44 RR0 V251 37579 18:55:02.876 -30:28:30.48 2457189.7725 0.68928616(716) 17.54 0.88 43 RR0 V252 37593 18:55:03.792 -30:29:07.83 2457189.8090 0.72866844(702) 17.28 0.50 17 RR0 V253 37586 18:55:03.268 -30:28:28.69 2457206.6555 0.74391157(1160) 17.32 0.29 17 RR0 V254 - 18:55:02.808 -30:28:49.69 2456436.8969 0.747628(652) 17.06 0.71 44 RR0 V255 VC5 18:55:03.123 -30:28:37.99 2456765.8300 0.760088(674) 16.48 0.19 44 RR0 V256 - 18:55:03.730 -30:28:44.94 2456784.9179 14.771(25) 14.72 0.71 44 CWA V257 - 18:55:02.996 -30:28:43.18 2456823.8857 20.747(50) 14.04 0.09 44 SR V258 - 18:55:03.732 -30:28:48.34 2457224.5000 37.980(168) 13.16 0.04 44 SR V259 - 18:55:04.216 -30:28:49.91 2456464.9081 154(3) 13.30 0.45 44 SR V260 - 18:55:03.170 -30:28:54.44 – – 12.76 0.06 44 L V261 - 18:55:03.399 -30:28:44.99 – – 13.78 0.07 44 L V262 - 18:55:01.876 -30:28:43.79 – – 13.99 0.07 40 L V263 $^{c}$ 18:55:03.877 -30:28:59.31 – – 14.71 0.07 44 L V264 - 18:55:03.753 -30:28:49.36 – – 13.87 0.08 44 L V265 - 18:55:02.987 -30:28:51.96 – – 14.00 0.08 44 L V266 - 18:55:03.458 -30:28:46.47 – – 14.04 0.08 44 L V267 - 18:55:03.463 -30:28:45.05 – – 13.66 0.09 44 L V268 - 18:55:02.998 -30:28:44.30 – – 13.79 0.09 44 L V269 - 18:55:03.712 -30:28:43.08 – – 13.62 0.10 44 L V270 - 18:55:02.603 -30:28:48.92 – – 13.70 0.11 44 L V271 - 18:55:04.004 -30:28:37.05 – – 13.97 0.12 44 L V272 - 18:55:03.065 -30:28:47.99 – – 14.25 0.05 44 L V273 - 18:55:03.708 -30:28:35.30 – – 13.40 0.15 44 L V274 - 18:55:03.833 -30:29:03.89 – – 13.47 0.08 44 L V275 - 18:55:02.346 -30:28:43.40 – – 13.67 0.16 44 L V276 - 18:55:03.659 -30:28:50.52 – – 14.06 0.16 44 L V277 - 18:55:03.309 -30:28:49.79 – – 13.71 0.19 44 L V278 - 18:55:04.070 -30:28:49.79 – – 13.13 0.33 44 L V279 - 18:55:04.045 -30:28:41.41 – – 13.42 0.34 44 L V280 - 18:55:03.692 -30:28:52.01 – – 13.25 0.46 44 L V281 - 18:55:03.324 -30:28:47.26 – – 16.71 0.45 44 NC V282 - 18:55:03.010 -30:28:35.82 – – 17.52 0.74 44 NC V283 - 18:55:03.747 -30:28:53.88 – – 16.97 0.76 44 NC V284 VC32 18:55:03.367 -30:28:37.97 – – 17.14 0.94 44 NC V285 VC33 18:55:02.686 -30:28:32.48 – – 17.52 0.55 44 NC V286 - 18:55:03.536 -30:28:51.06 – – 15.04$^{d}$ – 43 NC V287 - 18:55:03.373 -30:28:35.86 – – – – 44 NC V288 - 18:55:03.412 -30:28:52.76 – – – – 44 NC V289 - 18:55:02.904 -30:28:39.89 – – – – 42 NC V290 - 18:55:03.475 -30:29:06.23 – – – – 39 NC V291 VC27 18:55:03.258 -30:28:41.09 – – – – 42 NC Figure: Light curves of the known and new variables discovered in globular cluster NGC 6715.",
"Red, blue, and yellow triangles correspond to the data obtained during the years 2013, 2014, and 2015, respectively.",
"For V229, V244, V246, V286-V291, we plot the quantity f diff (t)/p(t)f_{\\mathrm {diff}}(t)/p(t) since a reference flux is not available.Figure: Light curves of the known and new variables discovered in globular cluster NGC 6715.",
"Red, blue, and yellow triangles correspond to the data obtained during the years 2013, 2014, and 2015, respectively.",
"For V229, V244, V246, V286-V291, we plot the quantity f diff (t)/p(t)f_{\\mathrm {diff}}(t)/p(t) since a reference flux is not available (cont.",
").Figure: Light curves of the known and new variables discovered in globular cluster NGC 6715.",
"Red, blue, and yellow triangles correspond to the data obtained during the years 2013, 2014, and 2015, respectively.",
"For V229, V244, V246, V286-V291, we plot the quantity f diff (t)/p(t)f_{\\mathrm {diff}}(t)/p(t) since a reference flux is not available (cont.",
").Figure: Light curves of the known and new variables discovered in globular cluster NGC 6715.",
"Red, blue, and yellow triangles correspond to the data obtained during the years 2013, 2014, and 2015, respectively.",
"For V229, V244, V246, V286-V291, we plot the quantity f diff (t)/p(t)f_{\\mathrm {diff}}(t)/p(t) since a reference flux is not available (cont.",
")."
]
] | 1605.06141 |
[
[
"The tensor hierarchy of 8-dimensional field theories"
],
[
"Abstract We construct the tensor hierarchy of generic, bosonic, 8-dimensional field theories.",
"We first study the form of the most general 8-dimensional bosonic theory with Abelian gauge symmetries only and no massive deformations.",
"This study determines the tensors that occur in the Chern-Simons terms of the (electric and magnetic) field strengths and the action for the electric fields, which we determine.",
"Having constructed the most general Abelian theory we study the most general gaugings of its global symmetries and the possible massive deformations using the embedding tensor formalism, constructing the complete tensor hierarchy using the Bianchi identities.",
"We find the explicit form of all the field strengths of the gauged theory up to the 6-forms.",
"Finally, we find the equations of motion comparing the Noether identities with the identities satisfied by the Bianchi identities themselves.",
"We find that some equations of motion are not simply the Bianchi identities of the dual fields, but combinations of them."
],
[
"Introduction",
"Over the last years, a great effort has been made to explore the most general field theories.",
"This exploration has been motivated by two main reasons.",
"First of all there is the need to search for viable candidates to describe the fundamental interactions known to us (specially gravity) and the universe at the cosmological scale, solving the theoretical problems encountered by the theories available today.",
"The second reason is the desire to map the space of possible theories and the different relations and dualities existing between them.",
"In the String Theory context, the landscape of $\\mathcal {N}=1,d=4$ vacua has focused most of the attention, but more general compactifications have also been studied.",
"At the level of the effective field theories the exploration has been carried out within the space of supergravity theories.",
"Most ungauged supergravity theories (excluding those of higher order in curvature) and some of the gauged ones have been constructed in the past century [1], but the space of possible gaugings and massive deformations (related to fluxes, symmetry enhancements etc.",
"in String theory) has started to be studied in a systematic way more recently with the introduction of the embedding-tensor formalism in Refs.",
"[2], [3], [4].",
"The formalism was developed in the context of the study of the gauging of $\\mathcal {N}=8,d=4$ supergravity in Refs.",
"[5], [6], but it has later been used in theories with less supersymmetry in different dimensions.See, for instance, Chapter 2 in Ref.",
"[15], which contains a pedagogical introduction to the formalism and references.",
"The embedding-tensor formalism comes with a bonus: the tensor hierarchy [25], [6], [26], [23], [27].",
"Using electric and magnetic vector fields in $d=4$ dimensions as gauge fields requires the introduction of 2-form-potentials in the theory, which would be dual to the scalars.",
"In $d=6$ dimensions certain gaugings require the introduction of magnetic 2-form and 3-form potentials [28].",
"But the addition of higher-rank potentials does not stop there: as a general rule, the construction of gauge-invariant field strengths for the new $p$ -form fields requires the introduction of $(p+1)$ -form fields with Stückelberg couplings.",
"This leads to a tensor hierarchy that includes all the electric and magnetic fields of the theory and opens up the systematic construction of gauged theories: construct the hierarchy using gauge invariance as a principle expressed through the Bianchi identities and find the equations of motion by using the duality relations between electric and magnetic fields of ranks $p$ and $d-p-2$ .",
"This approach has been used in Refs.",
"[23], [18] to construct the most general 4-, 5- and 6-dimensional field theoriesNot only supergravities, since use the embedding-tensor formalism is not restricted to supergravity theories.",
"with gauge invariance with at most two derivatives.",
"In this paper we want to consider the 8-dimensional case and construct the most general 8-dimensional field theory with gauge invariance and of second order in derivatives in the action: tensor hierarchy, Bianchi identities, field strengths, duality relations and action.The tensor hierarchy of maximal 8-dimensional supergravity has been constructed in Ref.",
"[17] in the context if exceptional field theory.",
"Our main motivation for considering this problem is to simplify and systematize the construction of a one-parameter family of inequivalent gaugings with the same SO$(3)$ group of maximal 8-dimensional supergravity, whose existence was conjectured in Ref.",
"[8]:By inequivalent here we mean theories which have different interactions, including, in particular, different scalar potentials.",
"A more restrictive definition of inequivalent theories (a more general concept of equivalence of theories) is often used in the literature (in Ref.",
"[22], for instance): theories related by a field redefinition (including non-local field redefinitions such as electric-magnetic dualities) are not considered to be inequivalent.",
"With this definition, the theories in the family we are talking about would not be considered to be inequivalent.",
"using Scherk-Schwarz's generalized dimensional reduction [13] Salam and Sezgin obtained from 11-dimensional reduction an 8-dimensional SO$(3)$ -gauged maximal supergravity in which the 3 Kaluza-Klein vectors played the role of gauge fields [14].Other, more general, gaugings can be obtained via Scherk-Schwarz reduction [11], [12], but it is always the Kaluza-Klein vectors that play the role of gauge fields.",
"The ungauged theory, though, has a second triplet of vector fields coming from the reduction of the 11-dimensional 3-form that can also be used as gauge fields and an SL$(2,\\mathbb {R})$ global symmetry that relates these two triplets of vectors, suggesting one could use as gauge fields any linear combination of these triplets.",
"The gauged theory in which the second tripet of vectors (those coming from the reduction of the 11-dimensional 3-form) played the role of gauge fields was obtained in Ref.",
"[8] by dimensional reduction of a non-covariant deformation of 11-dimensional supergravity proposed in Ref.",
"[7], [9].Many gauged supergravities whose 11-dimensional origin is unkonown or, in more modern parlance, they contain non-geometrical fluxes (like Roman's 10-dimensional massive supergravity or alternative, inequivalent gaugings of other theories) can be obtained systematically from this non-covariant deformation of 11-dimensional supergravity [10], which seems to encode many of these non-geometrical fluxes.",
"This theory has different Chern-Simons terms and a different scalar potential and provides an early example of inequivalent gauging with the same gauge group of a given supergravity theory.",
"However, for the reasons explained above, the existence of a full 1-parameter family of inequivalent SO$(3)$ gaugings is expected and it would be interesting to construct it and compare it with the 1-parameter family of inequivalentInequivalent in the more restrictive sense explained above.",
"SO$(8)$ gaugings of $\\mathcal {N}=8,d=4$ supergravity obtained in Ref.",
"[22] and consider the possible higher-dimensional origin of the new parameter.",
"The construction of that 1-parameter family interpolating between Salam-Sezgin's theory and that of Ref.",
"[8] is a complicated problem that will be addressed in a forthcoming publication [29].",
"In this paper we want to consider the general deformations (gaugings and massive transformations) of generic 8-dimensional theories.",
"This result paves the way for the constraction of the 1-parameter family of gauged $\\mathcal {N}=2,d=8$ theories which is our ultimate goal.",
"However, it is an interesting problem by itself whose solution will provide us with the most general theories with gauge symmetry in 8 dimensions up to two derivatives.",
"The construction of the most general 8-dimensional theory with gauge symmetry and at most two derivatives, and this paper, are organized as follows: first, in Section , we study the structure and symmetries (including electric-magnetic dualities of the 3-form potentials) of generic (up to second order in derivatives) 8-dimensional theories with Abelian gauge symmetry and no Chern-Simons terms.",
"In Section , we consider Abelian, massless deformations of those theories, which consist, essentially, in the introduction via some constant “$d$ -tensors” of Chern-Simons terms in the field strengths and action.",
"The new intereactions are required to preserve the Abelian gauge symmetries and, formally, the symplectic structure of the electric-magnetic duality transformations of the 3-form potentials.",
"We determine explicitly the form of all the electric and magnetic field strengths up to the 7-form field strengths, and give the gauge-invariant action in terms of the electric potentials.",
"This will be our starting point for the next stage.",
"In Section we consider the most general gauging and massive deformations (Stückelberg couplings) of the Abelian theory constructed in the previous section using the embedding-tensor formalism.",
"We proceed as in Refs.",
"[18], [15], finding Bianchi identities for field strengths from the identities satisfied by the Bianchi identities of the lower-rank field strengths and, then, solving them.",
"We have found the Bianchi identities satisfied by all the field strengths and we have managed to find the explicit form of the field strengths up to the 6-form.",
"In this approach, the “$d$ -tensors” that define the Chern-Simons terms will be treated in a different way as in Ref.",
"[18]: they will not be treated as deformations of the theory to be gauged, but as part of its definition.",
"Therefore, we will not associate to them any dual 7-form potentials.",
"In Section we study the construction of an action for the theory.",
"The equations of motion are related to the Bianchi identities by the duality relations between electric and magnetic field strengths, but, at least in this case, they are not directly equal to them.",
"In general they can be combinations of the Bianchi identities.",
"To find the right combinations we derive the Noether identities that the off-shell equations of motion of these theories should satisfy as a consistency condition that follows from gauge invariance.",
"Then, we compare those Noether identities with the identities satisfied by the Bianchi identities.",
"Once the equations of motion have been determined in this way, we proceed to the construction of the action, which we achieve up to terms that only contain 1-forms and their derivatives, whose form is too complicated.",
"Section contains our conclusions and the main formulae (field strengths, Bianchi identities etc.)",
"of the ungauged and gauged theories are collected in the appendices to simplify their use."
],
[
"Ungauged $d=8$ theories",
"In this section we are going to consider the construction of generic (bosonic) $d=8$ theories coupled to gravity containing terms of second order or lower in derivatives of any given fieldThe Chern–Simons (CS) terms may have terms with more than two derivatives, but they do not act on the same field..",
"The field content of a generic $d=8$ theory are the metric $g_{\\mu \\nu }$ , scalar fields $\\phi ^{x}$ , 1-form fields $A^{I}=A^{I}{}_{\\mu }dx^{\\mu }$ , 2-form fields $B_{m}=\\tfrac{1}{2}B_{m\\, \\mu \\nu }dx^{\\mu }\\wedge dx^{\\nu }$ and 3-form fields $C^{a}= \\tfrac{1}{3!",
"}C^{a}{}_{\\mu \\nu \\rho }dx^{\\mu }\\wedge dx^{\\nu }\\wedge dx^{\\rho }$ .",
"For the moment, we place no restrictions on the range of the indices labeling these fields nor on the symmetry groups that may act on them leaving the theory invariant.",
"We are going to start by the simplest theory one can construct with these fields to later gauge it and deform it in different ways.",
"The simplest field strengths one can construct for these fields are their exterior derivatives: $F^{I}\\equiv dA^{I}\\, ,\\hspace{28.45274pt}H_{m}\\equiv dB_{m}\\, ,\\hspace{28.45274pt}G^{a} \\equiv dC^{a}\\, .$ They are invariant under the gauge transformations $\\delta _{\\sigma }A^{I} = d\\sigma ^{I}\\, ,\\hspace{28.45274pt}\\delta _{\\sigma }B_{m} = d\\sigma _{m}\\, ,\\hspace{28.45274pt}\\delta _{\\sigma }C^{a} = d\\sigma ^{a}\\, ,$ where the local parameters $\\sigma ^{I},\\sigma _{m},\\sigma ^{a}$ are, respectively, 0-, 1-, and 2-forms.",
"The most general gauge-invariant action which one can write for these fields is $\\begin{array}{rcl}S& = &{\\displaystyle \\int }\\left\\lbrace -\\star \\mathbb {1} R+\\tfrac{1}{2} \\mathcal {G}_{xy}d\\phi ^{x}\\wedge \\star d\\phi ^{y}+\\frac{1}{2}\\mathcal {M}_{IJ}F^{I}\\wedge \\star F^{J}+\\frac{1}{2}\\mathcal {M}^{mn} H_{m}\\wedge \\star H_{n}\\right.\\\\& & \\\\& &\\left.-\\frac{1}{2}\\Im \\mathfrak {m}\\mathcal {N}_{ab} G^{a} \\wedge \\star G^{b}-\\frac{1}{2}\\Re \\mathfrak {e}\\mathcal {N}_{ab} G^{a} \\wedge G^{b}\\right\\rbrace \\, ,\\\\\\end{array}$ where the kinetic matrices $\\mathcal {G}_{xy},\\mathcal {M}_{IJ},\\mathcal {M}^{mn},\\Im \\mathfrak {m}\\mathcal {N}_{ab}$ as well as the matrix $\\Re \\mathfrak {e}\\mathcal {N}_{ab}$ are scalar-dependentIf $\\Re \\mathfrak {e}\\mathcal {N}_{ab}$ is constant, then the last term is a total derivative.. One could add CS terms to this action, but this possibility will arise naturally in what follows.",
"The equations of motion of the 3-forms $C^{a}$ can be written in the formThe equation of motion of a $p$ -form field, $\\delta S/\\delta \\omega ^{(p)}$ , is an $(8-p)-form$ defined by $\\delta S=+\\frac{\\delta S}{\\delta \\phi ^{x}}\\wedge \\delta \\phi ^{x}+\\frac{\\delta S}{\\delta A_{I}}\\wedge \\delta A^{I}+\\frac{\\delta S}{\\delta B^{m}}\\wedge \\delta B^{m}+\\frac{\\delta S}{\\delta C^{a}}\\wedge \\delta C^{a}\\, .$ With our conventions, when acting on $p$ -forms, $\\star ^{2}=(-1)^{p-1}$ .",
"$\\frac{\\delta S}{\\delta C^{a}}=-d\\frac{\\delta S}{\\delta G^{a}}=0\\, ,\\hspace{28.45274pt}\\frac{\\delta S}{\\delta G^{a}}=R_{a}\\equiv -\\Re \\mathfrak {e}\\mathcal {N}_{ab} G^{b}-\\Im \\mathfrak {m}\\mathcal {N}_{ab} \\star G^{b}\\, .$ These equations can be solved locally by introducing a set of dual 3-forms $C_{a}$ implicitly defined through their field strengths $G_{a}$ $R_{a}=G_{a}\\equiv dC_{a}\\, .$ It is convenient to construct vectors containing the fundamental and dual 3-forms: $(C^{i})\\equiv \\left(\\begin{array}{c}C^{a} \\\\ C_{a} \\\\\\end{array}\\right)\\, ,\\hspace{28.45274pt}G^{i}\\equiv dC^{i}\\, ,$ so that the equations of motion and the Bianchi identities for the fundamental field strengths take the simple form $dG^{i} = 0\\, .$ In other words: we have traded an equation of motion by a Bianchi identity and a duality relation.",
"In what follows we will do the same for all the fields in the action so that, in the end, we will have only a set of Bianchi identities and a set of duality relations between magnetic and electric fields.",
"The vector of field strengths $G^{i}$ satisfies the following linear, twisted, self-duality constraint $\\star G^{i} = \\Omega ^{ij}\\mathcal {W}_{jk}G^{k}\\, ,$ where $( \\Omega _{ij})=(\\Omega ^{ij})\\equiv \\left(\\begin{array}{cc}0 & \\mathbb {1} \\\\-\\mathbb {1} & 0 \\\\\\end{array}\\right)\\, ,$ is the symplectic metric and $(\\mathcal {W}_{ij}(\\mathcal {N}))\\equiv -\\left(\\begin{array}{cc}I_{ab} +R_{ac}I^{cd}R_{db}\\,\\,\\,\\,&R_{ac}I^{cb}\\\\& \\\\I^{ac}R_{cb}&I^{ab}\\\\\\end{array}\\right)\\, ,\\hspace{28.45274pt}\\Omega \\mathcal {W} \\Omega ^{T} = \\mathcal {W}^{-1}\\, ,$ is a symplectic symmetric matrixBasically the same that occurs in $d=4$ theories, $\\mathcal {M}(\\mathcal {N})$ see e.g. Ref.[21].",
"We use a slightly different convention for the sake of convenience and $\\mathcal {M}(\\mathcal {N})=\\mathcal {M}(-\\mathcal {N})$ due to the unconventional sign on the definition of $G_{a}$ ..",
"The equations (REF ) are formally invariant under arbitrary $\\mathrm {GL}(2n_{3},\\mathbb {R})$ transformations ($n_{3}$ being the number of fundamental 3-forms) but, just as it happens for 1-forms in $d=4$ , the self-duality constraint Eq.",
"(REF ) is only preserved by $\\mathrm {Sp}(2n_{3},\\mathbb {R})$ .",
"As usual, the only $\\mathrm {Sp}(2n_{3},\\mathbb {R})$ transformations which are true symmetries of the equations of motion are those associated to the transformations of the scalars which are isometries of $\\mathcal {G}_{xy}$ and which also induce linear transformations of the other kinetic matrices.",
"We will discuss this point in more detail later on.",
"The dualization of the other fields does not lead to any further restrictions.",
"In what follows we are going to generalize the simple Abelian theory that we have constructed by deforming it, adding new couplings.",
"We will use two guiding principles: preservation of gauge symmetry (even if it needs to be deformed as well) and preservation of the formal symplectic invariance that we have just discussed."
],
[
"Abelian, massless deformations",
"The deformations that we are going to consider in this section consist, essentially, in the introduction of CS terms in the field strengths and in the action.",
"Stückelberg coupling will be considered later.",
"Only the 3- and 4-form field strengths admit these massless Abelian deformations.",
"It is convenient to start by considering this simple modification of $G^{a}$ :We will often suppress the wedge product symbols $\\wedge $ in order to simplify the expressions that involve differential forms.",
"$G^{a} = dC^{a} +d^{a}{}_{I}{}^{m}F^{I}B_{m}\\, ,$ where $d^{a}{}_{I}{}^{m}$ is a constant tensor.",
"The gauge transformations need to be deformed accordingly: $\\delta _{\\sigma }A^{I} = d\\sigma ^{I}\\, ,\\hspace{28.45274pt}\\delta _{\\sigma }B_{m} = d\\sigma _{m}\\, ,\\hspace{28.45274pt}\\delta _{\\sigma }C^{a} = d\\sigma ^{a}-d^{a}{}_{I}{}^{m}F^{I}\\sigma _{m}\\, .$ The action Eq.",
"(REF ) remains gauge-invariant but the formal symplectic invariance is broken: if we do not modify the action, the dual 4-form field strengths are just $G_{a}=dC_{a}$ and $\\mathrm {Sp}(2n_{3},\\mathbb {R})$ cannot rotate these into $G^{a}$ in Eq.",
"(REF ).",
"Furthermore, the 1-form and 2-form equations of motion do not have a symplectic-invariant form.",
"This problem can be solved by adding a CS term to the action: $S_{CS}=\\int \\lbrace -d_{aI}{}^{m}dC^{a}F^{I}B_{m}\\rbrace \\, ,$ that modifies the equations of motion of the 3-forms $-d\\frac{\\delta S}{\\delta dC^{a}}=0\\, ,\\hspace{28.45274pt}\\frac{\\delta S}{\\delta dC^{a}}=R_{a}-d_{aI}{}^{m}F^{I}B_{m}\\, .$ The local solution is now $dC_{a}\\equiv R_{a}-d_{aI}{}^{m}F^{I}B_{m}\\, ,$ and, since $R_{a}$ is gauge-invariant, the dual, gauge-invariant, field strength must be defined by $R_{a}=dC_{a} +d_{aI}{}^{m}F^{I}B_{m}\\equiv G_{a}\\, .$ Again, $(C^{i})=\\left({\\begin{matrix}C^{a}\\\\ C_{a} \\\\\\end{matrix}}\\right)$ transforms linearly as a symplectic vector if $(d^{i}{}_{I}{}^{m})\\equiv \\left({\\begin{matrix}d^{a}{}_{I}{}^{m} \\\\ d_{aI}{}^{m} \\\\\\end{matrix}}\\right)$ also does.",
"Then, we can define the symplectic vector of 4-form field strengths $G^{i} = dC^{i} +d^{i}{}_{I}{}^{m}F^{I}B_{m}\\, ,$ invariant under the deformed gauge transformations $\\delta _{\\sigma }A^{I} = d\\sigma ^{I}\\, ,\\hspace{28.45274pt}\\delta _{\\sigma }B_{m} = d\\sigma _{m}\\, ,\\hspace{28.45274pt}\\delta _{\\sigma }C^{i} = d\\sigma ^{i}-d^{i}{}_{I}{}^{m}F^{I}\\sigma _{m}\\, .$ However, the deformed gauge transformations do not leave invariant the CS term Eq.",
"(REF ).",
"The only solutionWe have not found any other.",
"is to add another term of the formWe use the compact notation $A^{IJ\\ldots }=A^{I}A^{J} \\cdots $ , $F^{IJ\\ldots }=F^{I} F^{J} \\cdots $ , $B_{mn\\ldots }=B_{m}B_{n}\\cdots $ etc., where we have suppressed the wedge product symbols.",
"$S_{CS}=\\int \\lbrace -d_{aI}{}^{m}dC^{a}F^{I}B_{m}-\\tfrac{1}{2}d_{aI}{}^{m}d^{a}{}_{J}{}^{m}F^{IJ}B_{mn}\\rbrace \\, ,$ provided the following constraint holds: $d_{a(I}{}^{[m}d^{a}{}_{J)}{}^{n]}=0\\, ,\\,\\,\\,\\,\\,\\mbox{so}\\,\\,\\,\\,\\,d_{i(I}{}^{(m}d^{i}{}_{J)}{}^{n)}=0\\, .$ Observe that we are just using formal symplectic invariance: the symplectic vector $d^{i}{}_{I}{}^{m}$ is transformed into a different one.",
"Thus, in general, one gets $\\mathrm {Sp}(2n_{3},\\mathbb {R})$ multiplets of theories, except when $d^{i}{}_{I}{}^{m}$ is a symplectic invariant tensor,The only symplectic-invariant vector is 0. which requires, at least, one of the indices $I$ or $m$ to be a symplectic index.",
"In most cases the part of the symmetry group of the theory acting on the 3-forms, while embedded in $\\mathrm {Sp}(2n_{3},\\mathbb {R})$ , will be a much smaller group and, then, full symplectic invariance of $d^{i}{}_{I}{}^{m}$ may not be required.",
"As a nice check of the formal symplectic invariance of the deformed theory, we can check this invariance on the dual field strengths of the remaining fieldsWe leave aside the scalars for the moment., which is tantamount to checking the invariance of the equations of motion of the fundamental fields.",
"Using the duality relation $R_{a}=G_{a}$ the equations of motion of the 1-forms can be written in the form $\\frac{\\delta S}{\\delta A^{I}}=-d\\left\\lbrace \\mathcal {M}_{IJ}\\star F^{J}-d_{iI}{}^{m}G^{i}B_{m}-\\tfrac{1}{2} d_{iI}{}^{m}d^{i}{}_{J}{}^{n}F^{J}B_{mn}\\right\\rbrace =0\\, ,$ and can be solved by identifying all the terms inside the brackets with $d\\tilde{A}_{I}$ , where $\\tilde{A}_{I}$ is a set of 5-forms.",
"Taking into account gauge invariance, the 6-form field strengths $\\tilde{F}_{I}$ have the following definition, duality relation and Bianchi identities: $\\tilde{F}_{I}& \\equiv &d\\tilde{A}_{I}+d_{iI}{}^{m}G^{i}B_{m}+\\tfrac{1}{2} d_{iI}{}^{m}d^{i}{}_{J}{}^{n}F^{J}B_{mn}\\, ,\\\\& & \\nonumber \\\\\\tilde{F}_{I}& = &\\mathcal {M}_{IJ}\\star F^{J}\\, ,\\\\& & \\nonumber \\\\d\\tilde{F}_{I}& = &d_{iI}{}^{m}G^{i}H_{m}\\, ,$ and the equations of motion are of the 1-forms given by the Bianchi identities of the dual 6-form field strengths up to duality relations: $\\frac{\\delta S}{\\delta A^{I}}=-\\left\\lbrace d\\tilde{F}_{I}-d_{iI}{}^{m}G^{i}H_{m}\\right\\rbrace \\, .$ Using the duality relation $R_{a}=G_{a}$ and following the same steps for the 2-forms , we find $\\tilde{H}^{m}& = &d\\tilde{B}^{m} +d^{i}{}_{I}{}^{m}F^{I}C_{i}\\, ,\\\\& & \\nonumber \\\\\\tilde{H}^{m}& = &\\mathcal {M}^{mn}\\star H_{n}\\, ,\\\\& & \\nonumber \\\\d\\tilde{H}^{m}& = &-d_{iI}{}^{m}G^{i}F^{I}\\, ,$ and the equations of motion of the 2-forms are given by the Bianchi identities of the dual 5-form field strengths up to duality relations: $\\frac{\\delta S}{\\delta B_{m}}=-\\left\\lbrace d\\tilde{H}^{m}+d_{iI}{}^{m}G^{i}F^{I}\\right\\rbrace \\, .$ This completes the first Abelian deformation.",
"The second non-trivial deformation of $G^{a}$ that one could consider is the addition of a CS 4-form term $\\sim d^{a}{}_{IJK}A^{I}F^{J}A^{K}$ .",
"The gauge transformation of this term is not a total derivative and we cannot make $G^{a}$ gauge-invariant by deforming the gauge transformation rule of $C^{a}$ only: we must also deform that of $B_{m}$ , which, in its turn, induces a deformation of $H_{m}$ by addition of a CS 3-form term.",
"Since the deformation of $H_{m}$ is essentially unique, it is more convenient to start from this side and redefine $H_{m}= dB_{m} - d_{mIJ}F^{I}A^{J}\\, ,$ where $d_{mIJ}=d_{mJI}$The antisymmetric part is a total derivative that can be absorbed into a redefinition of $B_{m}$ .",
"which is invariant under the gauge transformations $\\delta _{\\sigma }A^{I} = d\\sigma ^{I}\\, ,\\hspace{28.45274pt}\\delta _{\\sigma }B_{m} = d\\sigma _{m} + d_{mIJ}F^{I}\\sigma ^{J}\\, ,$ and satisfies the Bianchi identities $dH_{m}=-d_{mIJ}F^{IJ}\\, .$ Under these gauge transformations and a generic $\\delta _{\\sigma }C^{a}$ $\\delta _{\\sigma }G^{a}=d\\left(\\delta _{\\sigma }C^{a} + d^{a}{}_{I}{}^{m}F^{I}\\sigma _{m} \\right)+d^{a}{}_{I}{}^{m}d_{mJK} F^{IJ}\\sigma ^{K}\\, .$ Adding a CS 4-form term to $G^{a}$ $G^{a}=dC^{a} +d^{a}{}_{I}{}^{m}F^{I}B_{m}-\\alpha d^{a}{}_{I}{}^{m}d_{mJK} A^{I}F^{J}A^{K}\\, .$ we find $\\begin{array}{rcl}\\delta _{\\sigma }G^{a}& = &d\\left[\\delta _{\\sigma }C^{a}+d^{a}{}_{I}{}^{m}F^{I}\\sigma _{m}-\\alpha d^{a}{}_{I}{}^{m}d_{mJK}(\\sigma ^{I}F^{J}A^{K} - A^{I}F^{J}\\sigma ^{K})\\right]\\\\& & \\\\& &+d^{a}{}_{I}{}^{m}d_{mJK}\\left[\\alpha \\sigma ^{I} F^{JK} +(1-\\alpha ) F^{IJ}\\sigma ^{K}\\right]\\, .\\end{array}$ The last term can be made to vanish by simply requiring $\\alpha d^{a}{}_{I}{}^{m}d_{mJK} = (\\alpha -1)d^{a}{}_{(J|}{}^{m}d_{m|K)I}\\, .$ Symmetrizing both sides of this equation w.r.t.",
"$IJK$ we conclude that $d^{a}{}_{(I|}{}^{m}d_{m|JK)}=0\\, ,$ and going back to the original (unsymmetrized) equation this implies that $\\alpha =1/3$ .",
"We arrive to the field strength, gauge transformation and Bianchi identities $G^{a}& = &dC^{a} +d^{a}{}_{I}{}^{m}F^{I}B_{m}-\\tfrac{1}{3} d^{a}{}_{I}{}^{m}d_{mJK} A^{I}F^{J}A^{K}\\, ,\\\\& & \\nonumber \\\\\\delta _{\\sigma }C^{a}& = &d\\sigma ^{a}-d^{a}{}_{I}{}^{m}F^{I}\\sigma _{m}+\\tfrac{1}{3} d^{a}{}_{I}{}^{m}d_{mJK}(\\sigma ^{I}F^{J}A^{K} - A^{I}F^{J}\\sigma ^{K})\\, ,\\\\& & \\nonumber \\\\dG^{a}& = &d^{a}{}_{I}{}^{m}F^{I}H_{m}\\, .$ If these deformations are going to preserve formal symplectic invariance, we expect that these results extend to the dual 3-forms and 4-forms field strengths, that is: $G^{i}& = &dC^{i} +d^{i}{}_{I}{}^{m}F^{I}B_{m}-\\tfrac{1}{3} d^{i}{}_{I}{}^{m}d_{mJK} A^{I}F^{J}A^{K}\\, ,\\\\& & \\nonumber \\\\\\delta _{\\sigma }C^{i}& = &d\\sigma ^{i}-d^{i}{}_{I}{}^{m}F^{I}\\sigma _{m}+\\tfrac{1}{3} d^{i}{}_{I}{}^{m}d_{mJK}(\\sigma ^{I}F^{J}A^{K} - A^{I}F^{J}\\sigma ^{K})\\, ,\\\\& & \\nonumber \\\\dG^{i}& = &d^{i}{}_{I}{}^{m}F^{I}H_{m}\\, ,$ while the identity $d^{i}{}_{(I|}{}^{m}d_{m|JK)}=0\\, .$ This requires the introduction of new CS terms in the action.",
"If we define the CS terms in the 4-form field strengths by $\\Delta G^{i}$ ($G^{i}=dC^{i}+\\Delta G^{i}$ ), then we expect the following terms to be present: $S_{CS}=-\\int \\lbrace dC^{a}\\Delta G_{a} +\\tfrac{1}{2}\\Delta G^{a} \\Delta G_{a}\\rbrace \\, .$ Instead of checking in detail the gauge-invariance of these terms, it is more convenient to take the formal exterior derivative and check whether it is entirely given in terms of the gauge-invariant field strengths found above.",
"if it is not, it should fail only by a total derivative which we can compensate by adding the corresponding terms to the action.",
"We have found that one has to relate $d^{i}{}_{(I|}{}^{m}d_{i|J)}{}^{n}$ to the tensor $d_{mIJ}$ .",
"The relation can be established by introducing a new tensor $d^{mnp}= -d^{nmp}$ and is given by $d^{i}{}_{(I|}{}^{m}d_{i|J)}{}^{n}=-2d^{mnp}d_{pIJ}\\, .$ Observe that $d^{i}{}_{[I|}{}^{m}d_{i|J]}{}^{n}$ does not necessarily vanish.",
"Using the above relation we find a result of the expected formWe use repeatedly the identity $2d^{i}{}_{I}{}^{m}d_{iJ}{}^{n}F^{I}A^{J}\\Delta H_{n}=-6 d^{mnp}\\Delta H_{n} \\Delta H_{p}+d\\lbrace \\tfrac{1}{2}d^{i}{}_{I}{}^{m}d_{iJ}{}^{n} A^{IJ} \\Delta H_{n} \\rbrace \\, .$ $\\begin{array}{rcl}d\\lbrace dC^{a}\\Delta G_{a} +\\tfrac{1}{2}\\Delta G^{a} \\Delta G_{a}\\rbrace & = &d_{aI}{}^{m}G^{a}F^{I}H_{m} -\\tfrac{1}{3}d^{mnp}H_{mnp}+d\\left\\lbrace - \\tfrac{1}{6}d^{mnp}B_{m}dB_{n}dB_{p}\\right.\\\\& & \\\\& &\\left.+\\tfrac{1}{2}d^{mnp}B_{m}H_{np}+\\tfrac{1}{24}d^{i}{}_{I}{}^{m}d_{iJ}{}^{n} A^{IJ}\\Delta H_{m}dB_{n}\\right\\rbrace \\, ,\\end{array}$ from which it follows that the gauge-invariant CS term in the action is given, up to total derivatives, by $\\begin{array}{rcl}S_{CS}& = &{\\displaystyle \\int }\\left\\lbrace -dC^{a}\\Delta G_{a} -\\tfrac{1}{2}\\Delta G^{a} \\Delta G_{a}-\\tfrac{1}{6}d^{mnp}B_{m}dB_{n}dB_{p}+\\tfrac{1}{2}d^{mnp}B_{m}H_{np}\\right.\\\\& & \\\\& &\\left.+\\tfrac{1}{24}d^{i}{}_{I}{}^{m}d_{iJ}{}^{n} A^{IJ}\\Delta H_{m}dB_{n}\\right\\rbrace \\, .\\end{array}$ Observe that only the completely antisymmetric part of $d^{mnp}$ enters the action, even though we have only assumed it to be antisymmetric in the first two indices.",
"We will henceforth assume that $d^{mnp}$ is completely antisymmetric.",
"Now, as a final check of the consistency of our results, we can compute the dual field strengths $\\tilde{H}^{m}$ and $\\tilde{F}_{I}$ , which should be formally symplectic invariant if the theory is, and their Bianchi identities, which should be given entirely in terms of other field strengths if the theory is indeed gauge invariant.",
"We find $\\tilde{H}^{m}& = &d\\tilde{B}^{m}+d^{i}{}_{I}{}^{m} C_{i}F^{I}+d^{mnp}B_{n}(H_{p}+\\Delta H_{p})+\\tfrac{1}{12}d^{i}{}_{I}{}^{m}d_{iJ}{}^{n} A^{IJ}\\Delta H_{n}\\, ,\\\\& & \\nonumber \\\\d\\tilde{H}^{m}& = &d^{i}{}_{I}{}^{m}G_{i}F^{I} + d^{mnp}H_{np}\\, ,\\\\& & \\nonumber \\\\\\tilde{F}_{I}& = &d\\tilde{A}_{I} +2d_{mIJ}A^{J}(\\tilde{H}_{m}-\\tfrac{1}{2}\\Delta \\tilde{H}_{m})-\\left(d^{i}{}_{I}{}^{m}B_{m} -\\tfrac{1}{3}d^{i}{}_{J}{}^{m}d_{mIK}A^{JK}\\right)(G_{i}-\\tfrac{1}{2}\\Delta G_{i})\\nonumber \\\\& & \\nonumber \\\\& &-\\tfrac{1}{3}\\left(d^{i}{}_{I}{}^{m}d_{mJK} -d^{i}{}_{K}{}^{m}d_{mIJ}\\right)F^{J}A^{K}C_{i}-d^{mnp}d_{mIJ}A^{J}B_{n}H_{p}\\nonumber \\\\& & \\nonumber \\\\& &+\\tfrac{1}{24}\\left(d^{i}{}_{K}{}^{m}d_{iL}{}^{n}d_{mIJ}+2d^{i}{}_{[I|}{}^{m}d_{i|K]}{}^{n}d_{mJL}\\right)F^{J}A^{KL}B_{n}+\\tfrac{1}{24} d^{i}{}_{J}{}^{m} d_{iK}^{n} d_{mIL}A^{JKL}dB_{n}\\nonumber \\\\& & \\nonumber \\\\& &-\\tfrac{1}{180} d^{i}{}_{L}{}^{n}d_{iQ}{}^{m}d_{mIJ} d_{nPK} A^{JKLQ}F^{P}\\, ,\\\\& & \\nonumber \\\\d\\tilde{F}_{I}& = &2d_{mIJ}F^{J}\\tilde{H}^{m} +d_{iI}{}^{m}G^{i}H_{m}\\, .$ The duality relations are the same as in the undeformed case.",
"As a further check of this construction, taking the exterior derivative of the Bianchi identities of all the field strengths one finds consistent results upon use of the properties of the deformation tensors $d^{i}{}_{I}{}^{m},d_{mIJ},d^{mnp}$ .",
"We will not compute the gauge transformations of the higher-rank form fields since they will not be necessary in what follows."
],
[
"The 6-form potentials and their 7-form field strengths",
"On general grounds (see [19] and references therein) the 6-form potentials are expected to be the duals of the scalars.",
"However, maintaining the manifest invariances of the theory in the dualization procedure requires the introduction of as many 6-forms $D_{A}$ as generators of global transformations $\\delta _{A}$ leaving the equations of motion (not just the action) invariant.",
"Hence, the index $A$ labels the adjoint representation of the duality group.",
"The 7-form field strengths $K_{A}$ are the Hodge duals of the piece $j^{(\\sigma )}_{A}(\\phi )$ of the Noether–Gaillard–Zumino (NGZ) conserved 1-form currents $j_{A}=j^{(\\sigma )}_{A}(\\phi )+\\Delta j_{A}$ associated to those symmetries (or, better, dualities) [20] which only depend on the scalar fieldsThis is the contribution of the $\\sigma $ -model to the Noether current.",
"The symmetries of the equations of motion are necessarily symmetries of the $\\sigma $ -model, i.e.",
"isometries of the $\\sigma $ -model metric $\\mathcal {G}_{xy}(\\phi )$ generated by Killing vectors $k_{A}{}^{x}$ .",
"The indices $A,B,C$ label the symmetries of the theory and, therefore, run over the adjoint representation of the Lie algebra of that symmetry group G. The contribution of the $\\sigma $ -model to the NGZ 1-form is $j^{(\\sigma )}_{A} = k_{A}{}^{x}\\mathcal {G}_{xy}d\\phi ^{y}$ .",
"$K_{A}\\equiv -\\star j^{(\\sigma )}_{A}\\, ,$ and their Bianchi identities follow from the conservation law for those currents $d K_{A} =d \\star j^{(\\sigma )}_{A}= d\\star (j^{NGZ}_{A}- \\Delta j_{A})=-d\\star \\Delta j_{A}\\, ,$ where we have used the conservation of the NGZ current.",
"The simplest procedure to compute $\\Delta j_{A}$ is to contract the equations of motion of the scalars with the Killing vectors $k_{A}{}^{x}(\\phi )$ of the $\\sigma $ -model metric $\\mathcal {G}_{xy}(\\phi )$ , which is given by $\\begin{array}{rcl}{\\displaystyle \\frac{\\delta S}{\\delta \\phi ^{x}}}& = &-d(\\star \\mathcal {G}_{xy}d\\phi ^{y}) +\\tfrac{1}{2} \\partial _{x}\\mathcal {G}_{yz}d\\phi ^{y}\\wedge \\star d\\phi ^{z}\\\\& & \\\\& &+\\tfrac{1}{2}\\partial _{x}\\mathcal {M}_{IJ}F^{I}\\wedge \\star F^{J}+\\tfrac{1}{2}\\partial _{x}\\mathcal {M}^{mn}H_{m}\\wedge \\star H_{n}+G^{a}\\partial _{x} G_{a}\\, .\\end{array}$ Using the Killing equation, we get $k_{A}{}^{x}\\frac{\\delta S}{\\delta \\phi ^{x}}=-d\\star j_{A}^{(\\sigma )}+\\tfrac{1}{2}k_{A}{}^{x}\\partial _{x}\\mathcal {M}_{IJ}F^{I}\\wedge \\star F^{J}+\\tfrac{1}{2}k_{A}{}^{x}\\partial _{x}\\mathcal {M}^{mn}H_{m}\\wedge \\star H_{n}+G^{a}k_{A}{}^{x}\\partial _{x} G_{a}\\, .$ We must now use the fact that the isometry generated by $k_{A}$ will only be a symmetry of the equations of motion ifThe transformation rule for the period matrix is unconventional because our definition of the lower component of the symplectic vector of 4-form field strengths, $G_{a}=R_{a}$ is unconventional (the sign is the oposite to the conventional one).",
"$\\begin{array}{rcl}k_{A}{}^{x}\\partial _{x}\\mathcal {M}_{IJ}& = &-2T_{A}{}^{K}{}_{(I}\\mathcal {M}_{J)K}\\, ,\\\\& & \\\\k_{A}{}^{x}\\partial _{x}\\mathcal {M}^{mn}& = &2T_{A}{}^{(m}{}_{p}\\mathcal {M}^{n)p}\\, ,\\\\& & \\\\k_{A}{}^{x}\\partial _{x} \\mathcal {N}_{ab}& = &-T_{A\\, ab}-\\mathcal {N}_{ac}T_{A}{}^{c}{}_{b}+T_{A\\, a}{}^{c}\\mathcal {N}_{cb}+\\mathcal {N}_{ac}T_{A}{}^{cd}\\mathcal {N}_{db}\\, ,\\end{array}$ where the matrices $T_{A}{}^{I}{}_{J},T_{A}{}^{m}{}_{n}$ and $\\left( T_{A}{}^{i}{}_{j} \\right)\\equiv \\left(\\begin{array}{cc}T_{A}{}^{a}{}_{b} & T_{A}{}^{ab} \\\\T_{A\\, ab} & T_{A\\, a}{}^{b} \\\\\\end{array}\\right)\\, ,$ are generators of the symmetry group G in the representation in which the 1-forms, 2-forms and 3-forms transform $[T_{A},T_{B}] = f_{AB}{}^{C}T_{C}\\, ,\\hspace{28.45274pt}[k_{A},k_{B}] = -f_{AB}{}^{C}k_{C}\\, .$ As we have discussed, this implies that the matrices $T_{A}{}^{i}{}_{j}$ are generators of the symplectic group $T_{A}{}^{i}{}_{[j}\\Omega _{k]i} = 0\\, .$ Upon use of the duality relations between field strengths, we find that $-k_{A}{}^{x}\\frac{\\delta S}{\\delta \\phi ^{x}}=d\\star j_{A}^{(\\sigma )} +T_{A}{}^{J}{}_{I}F^{I}\\tilde{F}_{J}+T_{A}{}^{m}{}_{n}\\tilde{H}^{n}H_{m}-\\tfrac{1}{2}T_{Aij}G^{ij}=0\\, ,$ on shell.",
"The exterior derivative of the whole expression vanishes due to the Bianchi identities of the field strengths and to the invariance of the deformation tensors $d_{mIJ},d^{i}{}_{I}{}^{m}$ and $d^{mnp}$ under the $\\delta _{A}$ transformations: $\\begin{array}{rcl}\\delta _{A}d_{mIJ}& = &-T_{A}{}^{n}{}_{m} d_{nIJ}-2T_{A}{}^{K}{}_{(I|} d_{n|J)K}=0\\, ,\\\\& & \\\\\\delta _{A}d^{i}{}_{I}{}^{m}& = &T_{A}{}^{i}{}_{j} d^{j}{}_{I}{}^{m}-T_{A}{}^{J}{}_{I} d^{i}{}_{J}{}^{m}+T_{A}{}^{m}{}_{n} d^{i}{}_{I}{}^{n}=0\\, ,\\\\& & \\\\\\delta _{A} d^{mnp}& = &3T_{A}{}^{[m|}_{q}d^{q|np]}=0\\, .\\end{array}$ This means that we can rewrite that equation locally as the conservation of the NGZ current $d\\star j_{A}^{NGZ} =0\\, ,\\hspace{28.45274pt}j_{A}^{NGZ} \\equiv j^{(\\sigma )}_{A}+\\Delta j_{A}\\, ,$ where $\\Delta j_{A}$ is a very long and complicated expression whose explicit form will not be useful for us.",
"A local solution is provided by $\\star [j^{(\\sigma )}_{A}+\\Delta j_{A}] = -d D_{A}$ for the 6-form potential $D_{A}$ and we get the definition of the 7-form field strength $\\star j^{(\\sigma )}_{A}= -dD_{A} +\\star \\Delta j_{A} \\equiv K_{A}\\, .$ Its Bianchi identity is given by $dK_{A}=-d\\star j^{(\\sigma )}_{A}=T_{A}{}^{I}{}_{J}F^{J}\\tilde{F}_{I}+T_{A}{}^{m}{}_{n}\\tilde{H}^{n}H_{m}+\\tfrac{1}{2}T_{Aij}G^{ij}\\, .$ In the kind of theories that we are considering here there is no reason to include potentials of rank higher than 6, unless we introduce a scalar potential depending on new coupling constants: one can then introduce 7-form potentials dual to those coupling constants.",
"Since the introduction of these parameters would be purely ad hoc, we will postpone the study of this duality to the next section in which we will be able to use in the definition of the scalar potential the embedding tensor and the massive deformation parameters, which have well-defined properties.",
"One can also generalize the theory by adding a scalar potential.",
"This addition is associated to the introduction of new deformation parameters.",
"In gauged supergravity, which is the main case of interest, these deformation parameters are the components of the embedding tensor and the scalar potential arises in the gauging procedure, associated to the fermion shifts in the fermion's supersymmetry transformations.",
"Thus, it is natural to deal with the scalar potential in the next section too.",
"The results obtained in this and the previous Section are summarized in Appendix ."
],
[
"Non-Abelian and massive deformations: the tensor hierarchy",
"The next step in the construction of the most general $d=8$ field theory is the gauging of the global symmetries of the theory.",
"The most general possibilities can be explored using the embedding tensor formalismIn this section we will follow Ref.",
"[15], where the essential references on the embedding tensor formalism can be found.",
"We will also use the same notation.",
"and in this section we are going to set it up for the Abelian theories we have just found.Observer that, in general, the theories that we are considering are just the bosonic sector of a theory that also contains fermions and whose symmetry group may include symmetries that only act on them.",
"The total symmetry group would, then, be larger and the embedding tensor should take this fact into account.",
"For the sake of convenience we are going to reproduce some of the formulae obtained above.",
"The starting point is the assumption that the equations of motion of the theory are invariant under a global symmetry group with infinitesimal generators $\\lbrace T_{A}\\rbrace $ satisfying the algebra $[T_{A},T_{B}] = f_{AB}{}^{C}T_{C}\\, .$ The group acts linearly on all the forms of rank $\\ge 1$ , including the 3-forms if the electric and magnetic 3-forms $C^{a}$ and $C_{a}$ are combined into a single symplectic vector of 3-forms $(C^{i})=\\left({\\begin{matrix}C^{a}\\\\ C_{a} \\\\\\end{matrix}}\\right) $ as explained above and codify electric-magnetic transformations involving the scalars.",
"The matrices that represent the generators are denoted by $\\lbrace T_{A}{}^{I}{}_{J}\\rbrace ,\\lbrace T_{A}{}^{m}{}_{n}\\rbrace ,\\lbrace T_{A}{}^{i}{}_{j}\\rbrace $ and the adjoint generators are $T_{A}{}^{B}{}_{C}= f_{AC}{}^{B}$ .",
"The matrices $T_{A}{}^{i}{}_{j}$ are generators of the symplectic group $T_{A}{}^{i}{}_{[j}\\Omega _{k]i} = 0\\, ,\\hspace{28.45274pt}( \\Omega _{ij})=(\\Omega ^{ij})\\equiv \\left(\\begin{array}{cc}0 & \\mathbb {1} \\\\-\\mathbb {1} & 0 \\\\\\end{array}\\right)\\, .$ We have $\\begin{array}{rclrclrcl}\\delta _{\\alpha } A^{I}& = &\\alpha ^{A}T_{A}{}^{I}{}_{J}A^{J}\\, ,\\hspace{28.45274pt}&\\delta _{\\alpha } B_{m}& = &-\\alpha ^{A}T_{A}{}^{n}{}_{m}B_{n}\\, ,\\hspace{28.45274pt}&\\delta _{\\alpha } C^{i}& = &\\alpha ^{A}T_{A}{}^{i}{}_{j}C^{j}\\, ,\\\\& & & & & & & & \\\\\\delta _{\\alpha } \\tilde{A}_{I}& = &-\\alpha ^{A}T_{A}{}^{J}{}_{I}\\tilde{A}_{J}\\, ,\\hspace{28.45274pt}&\\delta _{\\alpha } \\tilde{B}^{m}& = &\\alpha ^{A}T_{A}{}^{m}{}_{n}\\tilde{B}^{n}\\, ,\\hspace{28.45274pt}& & & \\\\\\end{array}$ (the dual potentials transform in the dual covariant-contravariant representation).",
"The kinetic matrices $\\mathcal {M}_{IJ},\\mathcal {M}^{mn},\\mathcal {W}_{ij}(\\mathcal {N})$ also transform linearly: if $\\delta _{\\alpha }\\equiv \\alpha ^{A}\\delta _{A}$ $\\delta _{A}\\mathcal {M}_{IJ}=-2T_{A}{}^{K}{}_{(I}\\mathcal {M}_{J)K}\\, ,\\hspace{14.22636pt}\\delta _{A}\\mathcal {M}^{mn}=2T_{A}{}^{(m}{}_{p}\\mathcal {M}^{n)p}\\, ,\\hspace{14.22636pt}\\delta _{A}\\mathcal {W}_{ij}=-2T_{A}{}^{k}{}_{(i}\\mathcal {W}_{j)k}\\, ,$ but the period matrix undergoes fractional-linear transformations which, infinitesimally, take the form $\\delta _{A} \\mathcal {N}_{ab}=-T_{A\\, ab}-\\mathcal {N}_{ac}T_{A}{}^{c}{}_{b}+T_{A\\, a}{}^{c}\\mathcal {N}_{cb}+\\mathcal {N}_{ac}T_{A}{}^{cd}\\mathcal {N}_{db}\\, ,\\hspace{28.45274pt}\\left( T_{A}{}^{i}{}_{j} \\right)=\\left(\\begin{array}{cc}T_{A}{}^{a}{}_{b} & T_{A}{}^{ab} \\\\T_{A\\, ab} & T_{A\\, a}{}^{b} \\\\\\end{array}\\right)\\, .$ The $k$ -form field strengths will transform in the same representation as the corresponding $(k-1)$ -form potential, but only if the $d$ -tensors $d_{mIJ},d^{i}{}_{I}{}^{m}$ and $d^{mnp}$ are invariant under the global symmetry group, i.e.",
"if they must satisfy $\\begin{array}{rcl}\\delta _{A}d_{mIJ}& = &-T_{A}{}^{n}{}_{m} d_{nIJ}-2T_{A}{}^{K}{}_{(I|} d_{n|J)K}=0\\, ,\\\\& & \\\\\\delta _{A}d^{i}{}_{I}{}^{m}& = &T_{A}{}^{i}{}_{j} d^{j}{}_{I}{}^{m}-T_{A}{}^{J}{}_{I} d^{i}{}_{J}{}^{m}+T_{A}{}^{m}{}_{n} d^{i}{}_{I}{}^{n}=0\\, ,\\\\& & \\\\\\delta _{A} d^{mnp}& = &3T_{A}{}^{[m|}_{q}d^{q|np]}=0\\, .\\end{array}$ The theories we have constructed are invariant under Abelian gauge transformations with 0-, 1- and 2-form parameters $\\sigma ^{I},\\sigma _{m},\\sigma ^{i}$ : $\\delta _{\\sigma }A^{I} \\sim d\\sigma ^{I}\\, ,\\hspace{28.45274pt}\\delta _{\\sigma }B_{m} \\sim d\\sigma _{m}\\, ,\\hspace{28.45274pt}\\delta _{\\sigma }C^{i} \\sim d\\sigma ^{i}\\, .$ In order to gauge the global symmetries, we promote the global parameters $\\alpha ^{A}$ to local ones $\\alpha ^{A}(x)$ and we identify them with some combinations of the gauge parameters of the 1-forms $\\sigma ^{I}$ via the embedding tensor $\\vartheta _{I}{}^{A}$ as follows: $\\alpha ^{A}\\equiv \\sigma ^{I}\\vartheta _{I}{}^{A}\\, .$ Using this redefinition in the transformation of the kinetic matrices $\\mathcal {M}_{IJ},\\mathcal {M}^{mn},\\mathcal {W}_{ij}$ one immediately finds their gauge transformations: $\\delta _{\\sigma }\\mathcal {M}_{IJ}=-2\\sigma ^{L} X_{L}{}^{K}{}_{(I}\\mathcal {M}_{J)K}\\, ,\\hspace{14.22636pt}\\delta _{\\sigma }\\mathcal {M}^{mn}=2\\sigma ^{I}T_{I}{}^{(m}{}_{p}\\mathcal {M}^{n)p}\\, ,\\hspace{14.22636pt}\\delta _{\\sigma }\\mathcal {W}_{ij}=-2\\sigma ^{I}X_{I}{}^{k}{}_{(i}\\mathcal {W}_{j)k}\\, ,$ where we have defined the matrices $X_{I}{}^{J}{}_{K} \\equiv \\vartheta _{I}{}^{A} T_{A}{}^{J}{}_{K}\\, ,\\hspace{14.22636pt}X_{I}{}^{m}{}_{n} \\equiv \\vartheta _{I}{}^{A} T_{A}{}^{m}{}_{n}\\, .\\hspace{14.22636pt}X_{I}{}^{i}{}_{j} \\equiv \\vartheta _{I}{}^{A} T_{A}{}^{i}{}_{j}\\, .$ The gauge fields for these symmetries are given by $A^{A}\\equiv A^{I}\\vartheta _{I}{}^{A}\\, .$ With them we can construct gauge-covariant derivatives, which we will then use to derive Bianchi identities.",
"Is is convenient to start by constructing the covariant derivatives of the kinetic matrices $\\mathcal {M}_{IJ},\\mathcal {M}^{mn},\\mathcal {W}_{ij}(\\mathcal {N})$ which transform linearly.",
"According to the general rule, the covariant derivative of a field $\\Phi $ transforming as $\\delta _{A}\\Phi $ is given by $\\mathcal {D}\\Phi \\equiv d\\Phi -A^{A}\\delta _{A}\\Phi \\, .$ Then, with the above definition of gauge fields $\\mathcal {D}\\mathcal {M}^{mn}& = &d\\mathcal {M}^{mn}-2A^{I}X_{I}{}^{(m}{}_{p}\\mathcal {M}^{n)p}\\, ,\\\\& & \\nonumber \\\\\\mathcal {D}\\mathcal {M}_{IJ}& = &d\\mathcal {M}_{IJ}+2A^{L}X_{L}{}^{K}{}_{(I}\\mathcal {M}_{J)K}\\, ,\\\\& & \\nonumber \\\\\\mathcal {D}\\mathcal {W}_{ij}& = &d\\mathcal {W}_{ij}+2A^{I}X_{I}{}^{k}{}_{(i}\\mathcal {W}_{j)k}\\, .$ These derivatives transform covariantly under gauge transformations $\\delta _{\\sigma }= \\sigma ^{I}\\vartheta _{I}{}^{A}\\delta _{A}$ provided that the embedding tensor is gauge-invariant $\\delta _{\\sigma }\\vartheta _{I}{}^{A}=0\\, ,$ and provided that the 1-forms transform as $\\delta _{\\sigma }A^{I} = \\mathcal {D} \\sigma ^{I} +\\Delta _{\\sigma } A^{I}\\, ,\\,\\,\\,\\,\\,\\,\\mbox{where}\\,\\,\\,\\,\\,\\,\\left\\lbrace \\begin{array}{rcl}\\Delta _{\\sigma } A^{I} \\vartheta _{I}{}^{A}& = & 0\\, ,\\\\& & \\\\\\mathcal {D} \\sigma ^{I}& = &d \\sigma ^{I}- A^{J} X_{J}{}^{I}{}_{K}\\sigma ^{K}\\, ,\\end{array}\\right.$ The condition Eq.",
"(REF ) leads to the so-called quadratic constraint $\\vartheta _{J}{}^{B}\\left[T_{B}{}^{K}{}_{I}\\vartheta _{K}{}^{A} -f_{BC}{}^{A}\\vartheta _{I}{}^{C}\\right]=0\\, .$ To determine $\\Delta _{\\sigma } A^{I}$ we have to construct the gauge-covariant 2-form field strengths $F^{I}$ ."
],
[
"2-form field strengths",
"The simplest way to find the 2-form field strengths $F^{I}$ is through the Ricci identities.",
"A straightforward calculation using the quadratic constraint Eq.",
"(REF ) leads to $\\mathcal {D}\\mathcal {D}\\mathcal {M}_{mn} =-F^{I}\\vartheta _{I}{}^{A}\\delta _{A}\\mathcal {M}_{mn}\\, ,$ and analogous equations for $\\mathcal {M}_{IJ}$ and $\\mathcal {W}_{ij}(\\mathcal {N})$ , with $F^{I}=dA^{I} -\\tfrac{1}{2}X_{J}{}^{I}{}_{K}A^{JK} +\\Delta F^{I}\\, ,\\,\\,\\,\\,\\,\\,\\mbox{where}\\,\\,\\,\\,\\,\\,\\Delta F^{I}\\vartheta _{I}{}^{A}=0\\, .$ Under gauge transformations, $\\delta _{\\sigma }F^{I}=\\sigma ^{J}X_{J}{}^{I}{}_{K}F^{K}\\, ,$ provided that $\\delta _{\\sigma }\\Delta F^{I}=-\\mathcal {D}\\Delta _{\\sigma } A^{I}+2X_{(J}{}^{I}{}_{K)}\\left(F^{J} \\sigma ^{K}-\\tfrac{1}{2}A^{J}\\delta _{\\sigma } A^{K} \\right)\\, .$ Given the field content of the theory, the natural candidate to $\\Delta F^{I}$ and $\\Delta _{\\sigma } A^{I}$ are $\\Delta F^{I} = Z^{I\\, m}B_{m}\\, ,\\hspace{28.45274pt}\\Delta _{\\sigma } A^{I} = -Z^{I\\, m}\\sigma _{m}\\, ,$ where the new tensor $Z^{I\\, m}$ is gauge-invariant and orthogonal to the embedding tensor: $\\delta _{\\sigma } Z^{I\\, m}& = &0\\, ,\\\\& & \\nonumber \\\\Z^{I\\, m}\\vartheta _{I}{}^{A}& = &0\\, .$ Then, the consistency of Eq.",
"(REF ) with the above choice requires $\\vartheta _{(J|}{}^{A}T_{A}{}^{I}{}_{|K)} = Z^{I\\, m}d_{mJK}\\, ,$ for some tensor $d_{mJK}= d_{mKJ}$ which will turn out to coincide with the tensor we introduced as an Abelian deformation in the previous sections.",
"Since we have assumed $\\vartheta _{I}{}^{A}$ and $d_{mJK}$ to be gauge-invariant, $Z^{I\\, m}$ is automatically gauge-invariant and we have one constraint less.",
"We conclude thatOn general grounds, we expect a term of the form $-\\sigma ^{I}X_{I}{}^{n}{}_{m}B_{n}$ in the gauge transformation rule of $B_{m}$ .",
"This term is indeed present, but in a disguised form.",
"$F^{I}& = &dA^{I} -\\tfrac{1}{2}X_{J}{}^{I}{}_{K}A^{JK} +Z^{I\\, m}B_{m}\\, ,\\\\& & \\nonumber \\\\\\delta _{\\sigma }F^{I}& = &\\sigma ^{J}X_{J}{}^{I}{}_{K}F^{K}\\, ,\\\\& & \\nonumber \\\\\\delta _{\\sigma }A^{I}& = &\\mathcal {D} \\sigma ^{I} -Z^{I\\, m}\\sigma _{m}\\, ,\\\\& & \\nonumber \\\\\\delta _{\\sigma }B_{m}& = &\\mathcal {D}\\sigma _{m}+2d_{mJK}\\left(F^{J} \\sigma ^{K}-\\tfrac{1}{2}A^{J}\\delta _{\\sigma }A^{K} \\right)+\\Delta _{\\sigma } B_{m}\\, ,\\,\\,\\,\\mbox{with}\\,\\,\\,Z^{I\\, m}\\Delta _{\\sigma } B_{m} =0\\, .\\hspace{28.45274pt}~$ In the ungauged limit $\\vartheta _{I}{}^{A}=Z^{I\\, m}=0$ we get the Abelian gauge transformations of the 2-form Eq.",
"(REF ) if we identify the above 1-form $\\sigma _{m}$ (relabeled $\\sigma _{\\mathrm {g}\\, m}$ ) with $\\sigma _{m}-d_{mIJ}A^{I}\\sigma ^{J}$ , $\\sigma _{\\mathrm {g}\\, m}=\\sigma _{m}-d_{mIJ}A^{I}\\sigma ^{J}\\, ,$ confirming the identification of the $d$ -tensor.",
"Using that variable makes the non-Abelian gauge transformations more complicated and, therefore, we will stick to the above $\\sigma _{m}$ ."
],
[
"3-form field strengths",
"Again, the shortest way to find $\\Delta B_{m}$ and the gauge-covariant 3-form field strength $H_{m}$ is through the Bianchi identities.",
"Taking the covariant derivative of the 2-form field strength, and using the generalized Jacobi identity $X_{[I|}{}^{K}{}_{M}X_{|J}{}^{M}{}_{L]}=\\tfrac{2}{3} Z^{K\\, m}d_{mM[I}X_{J}{}^{M}{}_{L]}\\, ,$ we get $\\mathcal {D} F^{I} = Z^{I\\, m}H_{m}\\, ,$ where $H_{m}=\\mathcal {D}B_{m} -d_{mIJ}dA^{I}A^{J} +\\tfrac{1}{3}d_{mMI}X_{J}{}^{M}{}_{K}A^{IJK}+\\Delta H_{m}\\, ,\\,\\,\\,\\,\\,\\,\\mbox{with}\\,\\,\\,\\,\\,\\,Z^{I\\, m}\\Delta H_{m} =0\\, .$ In the ungauged limit $\\vartheta _{I}{}^{A}=Z^{I\\, m}=0$ we recover the Abelian 3-form field strength in Eq.",
"(REF ).",
"On the other hand, by construction, the above field strength is gauge-covariant up to terms annihilated by $Z^{I\\, m}$ under Eqs.",
"(REF ) and ().",
"To show this explicitly, we will need further identities between the tensors of the theory that are more easily discovered by computing first the 4-form field strengths."
],
[
"4-form field strengths",
"From this moment, following Ref.",
"[18], we will determine the general form of the field strengths using the Bianchi identities and their consistency relations.",
"This procedure yields gauge-covariant field strengths and one can later find explicitly the gauge transformations of the fields that produce that result.",
"Thus, we take the covariant derivative of both sides of Eq.",
"(REF ), use the Ricci identity Eq.",
"(REF ) for the l.h.s.",
"and the explicit form of $H_{m}$ in Eq.",
"(REF ) for the r.h.s., and we find the Bianchi identity for $H_{m}$ to be $\\mathcal {D}H_{m}=-d_{mIJ}F^{IJ} +\\Delta \\mathcal {D}H_{m}\\, ,\\,\\,\\,\\,\\,\\,\\mbox{where}\\,\\,\\,\\,\\,\\,Z^{I\\, m}\\Delta \\mathcal {D}H_{m} =0\\, .$ $\\Delta \\mathcal {D}H_{m}$ has to be gauge-invariant and scalar-independent and the only possibility is a 4-form combination of field strengths.",
"$F^{I}\\wedge F^{J}$ has already been used and we must use $G^{i}$ , whose explicit form will be determined by consistency.",
"We need to introduce a new gauge-invariant tensor $Z_{im}$ orthogonal to $Z^{I\\, m}$ $Z^{I\\, m}Z_{jm}=0\\, ,$ and, then, we arrive to the Bianchi identity $\\mathcal {D}H_{m}=-d_{mIJ}F^{IJ} +Z_{im}G^{i}\\, .$ A direct calculation of $\\mathcal {D}H_{m}$ using the explicit expression of $H_{m}$ in Eq.",
"(REF ) with $\\Delta H_{m}= Z_{im}C^{i}$ can only give a consistent result if we introduce a tensor $d^{i}{}_{I}{}^{m}$ such that $X_{I}{}^{m}{}_{n} +2d_{nIJ}Z^{J\\, m} = Z_{in}d^{i}{}_{I}{}^{m}\\, .$ The tensor $d^{i}{}_{I}{}^{m}$ coincides with the one we introduced as an Abelian deformation.",
"Also, observe that this relation makes the condition of gauge invariance of $Z_{in}$ redundant.",
"We get $G^{i}=\\mathcal {D}C^{i}+d^{i}{}_{I}{}^{n}\\left[F^{I}B_{n} -\\tfrac{1}{2}Z^{I\\, p}B_{n}B_{p}-\\tfrac{1}{3}d_{nJK}dA^{J}A^{IK}+\\tfrac{1}{12}d_{nMJ}X_{K}{}^{M}{}_{L} A^{IJKL}\\right]+\\Delta G^{i}\\, ,$ with $Z_{im}\\Delta G^{i}=0\\, .$ These 4-form field strengths reduce exactly to the Abelian ones in Eq.",
"(REF ).",
"Now we are ready to check explicitly using the identity/constraint Eq.",
"(REF ) that $H_{m}$ in Eq.",
"(REF ) with $\\Delta H_{m} = Z_{im} C^{i}$ is gauge covariant up to terms proportional to $Z_{im}$ , which are automatically annihilated by $Z^{I\\, m}$ .",
"We find that $\\begin{array}{rcl}\\delta _{\\sigma } C^{i}& = &\\mathcal {D}\\sigma ^{i}-d^{i}{}_{I}{}^{n}\\left[\\sigma ^{I}H_{n}+F^{I}\\sigma _{n}+\\delta _{\\sigma }A^{I} B_{n}-\\tfrac{1}{3} d_{nJK}\\delta _{\\sigma }A^{J}A^{IK}\\right]+\\Delta _{\\sigma } C^{i}\\, ,\\\\& & \\\\& &\\mbox{with}\\,\\,\\,\\,\\,Z_{im}\\Delta _{\\sigma } C^{i} =0\\, ,\\\\& & \\\\\\Delta _{\\sigma } B_{m}& = &-Z_{im}\\sigma ^{i}\\, .\\end{array}$ These gauge transformations reduce to the Abelian ones Eq.",
"() upon use of the property of the $d$ -tensors Eq.",
"(REF ) and the identifications Eq.",
"(REF ) and $\\sigma _{\\mathrm {g}}^{i}=\\sigma ^{i} +d^{i}{}_{I}{}^{n}(B_{n}\\sigma ^{I}-\\tfrac{1}{3} d_{nJK}A^{IJ}\\sigma ^{K})\\, .$"
],
[
"5-form field strengths",
"Taking, once again, the covariant derivatives of both sides of the Bianchi identity for $H_{m}$ , Eq.",
"(REF ), and using the Bianchi identity for $F^{I}$ , Eq.",
"(REF ) and the newly introduced tensor $d^{i}{}_{I}{}^{m}$ , we find that $\\mathcal {D}G^{i} = d^{i}{}_{I}{}^{m} F^{I}H_{m} -Z^{i}{}_{m}\\tilde{H}^{m}\\, ,$ where $Z^{i}{}_{m}$ is a new gauge-invariant tensor orthogonal to $Z_{im}$ $Z^{i}{}_{m}Z_{in}=0\\, ,$ and where the sign of that term has been chosen so as to get the same signs as in the ungauged case.",
"In principle these two tensors could be completely unrelated (except for the constraints).",
"However, since, in the physical theory, $G^{i}$ is self-dual and $\\tilde{H}^{m}$ is the electric-magnetic dual of $H_{m}$ , it is natural to expect that the same tensors appear in both field strengths.",
"Thus, we are going to assume that $Z^{i}{}_{m}$ has been obtained from $Z_{jm}$ by raising the index with the symplectic metric tensor $\\Omega ^{ij}$ , that is $Z^{i}{}_{m} \\equiv \\Omega ^{ji}Z_{jm}\\, .$ Then, there is no new constraint associated to its gauge invariance and, we just have the constraint Eq.",
"(REF ) analogous to a constraint satisfied by the embedding tensor in 4-dimensional field theories."
],
[
"6-form field strengths",
"Taking the covariant derivative of both sides of the Bianchi identity for $G^{i}$ , Eq.",
"(REF ) and using the Bianchi identities for the field strengths of lower rank, we find that we need to introduce three new tensors $d_{iI}{}^{m}, d^{mnp}, d^{m}{}_{IJK}$ and demand that $d^{i}{}_{I}{}^{m}Z_{jm} +X_{I}{}^{i}{}_{j}& = &-Z^{i}{}_{m}d_{jI}{}^{m}\\, ,\\\\& & \\nonumber \\\\d^{i}{}_{I}{}^{[m|}Z^{I\\, |n]}& = &-Z^{i}{}_{p}d^{pmn}\\, ,\\\\& & \\nonumber \\\\d^{i}{}_{(I|}{}^{m}d_{m|JK)}& = &-Z^{i}{}_{p} d^{p}{}_{IJK}\\, .$ Lowering the $i$ indices in the first equation with $\\epsilon _{ik}$ and taking into account that $X_{I[kj]}=0$ , we conclude that it is natural to identify $d_{iI}{}^{m}=\\Omega _{ij}d^{j}{}_{I}{}^{m}\\, ,$ and rewrite the constraint as $X_{Iij} = -2 Z_{(i|m}d_{|j)I}{}^{m}\\, .$ Using these constraints and the same reasoning as in the previous cases we find the next Bianchi identity and we can also solve itActually, it is easier to find $\\tilde{H}^{M}$ from the previous Bianchi identity Eq.",
"(REF ) taking the covariant derivative of the 4-form field strengths $G^{i}$ in Eq.",
"(REF ) with $\\Delta G^{i}=-Z^{i}{}_{m}\\tilde{B}^{m}$ .",
"$\\mathcal {D}\\tilde{H}^{m}& = &-d_{iI}{}^{m}G^{i}F^{I} +d^{mnp}H_{np}+d^{m}{}_{IJK}F^{IJK}+Z^{I\\, m}\\tilde{F}_{I}\\, ,\\\\& & \\nonumber \\\\\\tilde{H}^{m}& = &\\mathcal {D}\\tilde{B}^{m} -d_{iI}{}^{m}F^{I}C^{i}+2d^{mnp}B_{n}\\left(H_{p}-Z_{ip}C^{i}-\\tfrac{1}{2}\\mathcal {D}B_{p}\\right)\\nonumber \\\\& & \\nonumber \\\\& &+d^{m}{}_{IJK}dA^{I}dA^{J}A^{K}\\nonumber \\\\& & \\nonumber \\\\& &+\\left(\\tfrac{1}{12}d_{iJ}{}^{m}d^{i}{}_{K}{}^{n}d_{nIL}-\\tfrac{3}{4} d^{m}{}_{IJM} X_{K}{}^{M}{}_{L}\\right)dA^{I} A^{JKL}\\nonumber \\\\& & \\nonumber \\\\& &+\\left(\\tfrac{3}{20} d^{m}{}_{NPM}X_{I}{}^{N}{}_{J}-\\tfrac{1}{60} d_{iM}{}^{m}d^{i}_{I}{}^{n}d_{nPJ}\\right)X_{K}{}^{P}{}_{L} A^{IJKLM}\\nonumber \\\\& & \\nonumber \\\\& &+Z^{Im}\\tilde{A}_{I}\\, ,$"
],
[
"7-form field strengths",
"Provided that we impose the additional constraintThis constraint reduces to Eq.",
"(REF ) in the ungauged, massless limit.",
"$d^{i}{}_{(I}{}^{m}d_{iJ)}{}^{n} +2 d^{mnp}d_{pIJ} +3d^{m}{}_{IJK} Z^{K\\, n}=+3d^{n}{}_{IJK} Z^{K\\, m}\\, ,$ the covariant derivative of the Bianchi identity Eq.",
"(REF ) leads to the Bianchi identity for the 6-form field strengths $\\mathcal {D}\\tilde{F}_{I}& = &2d_{mIJ} F^{J}\\tilde{H}^{m} +d_{iI}{}^{m}G^{i}H_{m}-3 d^{m}{}_{IJK}F^{JK}H_{m} -\\vartheta _{I}{}^{A}K_{A}\\, ,\\\\& & \\nonumber \\\\\\tilde{F}_{I}& = &\\mathcal {D}\\tilde{A}_{I}+2d_{mIJ}\\tilde{B}^{m}F^{J}+d_{iI}{}^{m}C^{i}(H_{m}-\\tfrac{1}{2}Z_{jm}C^{j})\\nonumber \\\\& & \\nonumber \\\\& &-3d^{m}{}_{IJK}B_{m}(F^{J}-\\tfrac{1}{2}Z^{Jn}B_{n})(F^{K}-\\tfrac{1}{2}Z^{Kp}B_{p})-\\tfrac{1}{4}d^{m}{}_{IJK}Z^{Jn}Z^{Kp}B_{mnp}\\nonumber \\\\& & \\nonumber \\\\& &+\\tfrac{1}{2}d_{iI}{}^{m}d^{i}{}_{J}{}^{n}(F^{J}-\\tfrac{2}{3}Z^{Jp}B_{p})B_{mn}-d_{iI}{}^{m}B_{m}\\Box G^{i} +\\cdots $ where we are denoting by $\\Box G^{i}$ the part of $G^{i}$ that only contains 1-forms $A^{I}$ and their derivatives $dA^{I}$ ."
],
[
"8-form field strengths",
"Taking the covariant derivative of Eq.",
"(REF ) and using several of the constraints imposed above, we find that $\\vartheta _{I}{}^{A}\\mathcal {D}K_{A}=X_{I}{}^{K}{}_{J}F^{J}\\tilde{F}_{K}+X_{I}{}^{m}{}_{n}\\tilde{H}^{n}H_{m}+\\tfrac{1}{2}X_{Iij}G^{ij}+5d_{m(IJ}d^{m}{}_{KLM)}F^{JKLM}\\, .$ According to the general arguments in Ref.",
"[19] the last term must vanish.",
"It cannot arise in the Bianchi identity of the dual Noether-Gaillard-Zumino current associated to the global symmetries of the theory.",
"Thus, we impose $d_{m(IJ}d^{m}{}_{KLM)} = 0\\, ,$ and, from the definition of the $X$ tensors, we get $\\mathcal {D}K_{A}=T_{A}{}^{K}{}_{J}F^{J}\\tilde{F}_{K}+T_{A}{}^{m}{}_{n}\\tilde{H}^{n}H_{m}-\\tfrac{1}{2}T_{A\\, ij}G^{ij}+Y_{A}{}^{\\sharp }L_{\\sharp }\\, ,$ where $Y_{A}{}^{\\sharp }$ is a tensor orthogonal to the embedding tensor $\\vartheta _{I}{}^{A}Y_{A}{}^{\\sharp } =0\\, ,$ and where the index $\\sharp $ runs over all the deformation tensors introduced so far, that we are going to denote collectively by $c^{\\sharp }$ .",
"As argued in Ref.",
"[18], the natural candidates for the $Y_{A}{}^{\\sharp }$ tensors are the variations of the deformation tensors $c^{\\sharp }$ under the global symmetries of the theory $Y_{A}{}^{\\sharp } = \\delta _{A}c^{\\sharp }\\, ,$ where $A$ runs over the whole Lie algebra of the global symmetry group, because all the deformation tensors are required to be gauge invariant $\\vartheta _{I}{}^{A}\\delta _{A} c^{\\sharp } =\\vartheta _{I}{}^{A}Y_{A}{}^{\\sharp }\\equiv \\mathcal {Q}_{I}{}^{\\sharp }= 0\\, ,$ where we have defined the constraints $\\mathcal {Q}_{A}{}^{\\sharp }$ .",
"At this point there are two possibilities: We can consider that all the independent tensorsThe tensors $d^{mnp},d^{m}{}_{IJK}$ are related to these and their gauge invariance is not an independent condition.",
"$\\lbrace \\vartheta _{I}{}^{A},Z^{Im},Z_{im},-d_{mIJ},d^{i}{}_{I}{}^{m}\\rbrace $ are deformations of the original theory introduced at the same time as the gauging of the global symmetries of the original symmetry is carried out.",
"In this case they only have to be invariant under the global symmetries that have been gauged and not the stronger condition $\\delta _{A} c^{\\sharp } =0\\, ,$ for any of them.",
"We can consider only the tensors $\\lbrace \\vartheta _{I}{}^{A},Z^{Im},Z_{im}\\rbrace $ are deformations of the original theory, whose definition includes the tensors $\\lbrace -d_{mIJ},d^{i}{}_{I}{}^{m}\\rbrace $ .",
"In this case, the latter must be invariant under the whole global symmetry group by hypothesis.",
"The corresponding $Y_{A}{}^{\\sharp }$ tensors are assumed to vanish identically, before they are contracted with the embedding tensor.",
"This is the point of view that we have adopted here and it implies that there are only three sets of 8-form field strengths $\\lbrace L_{\\sharp }\\rbrace =\\lbrace L_{A}{}^{I},L_{Im},L^{im}\\rbrace $ and only three corresponding sets of 7-form potentials $\\lbrace E_{\\sharp }\\rbrace =\\lbrace E_{A}{}^{I},E_{Im},E^{im}\\rbrace $ which are dual to the deformation tensors $\\lbrace \\vartheta _{I}{}^{A},Z^{Im},Z_{im}\\rbrace $ .",
"In an action in which these tensors are generalized to spacetime-dependent fields, these dual potentials appear as Lagrange multipliers enforcing their constancy [24], [23].",
"We, thus, have to consider three constraints associated to gauge invariance $\\mathcal {Q}_{IJ}{}^{A} & \\equiv & \\vartheta _{I}{}^{B}Y_{BJ}{}^{A}\\, ,\\hspace{28.45274pt}Y_{BJ}{}^{A} \\equiv \\delta _{B}\\vartheta _{J}{}^{A}=-T_{B}{}^{K}{}_{J}\\vartheta _{K}{}^{A}+T_{B}{}^{A}{}_{C}\\vartheta _{J}{}^{C}\\, ,\\\\& & \\nonumber \\\\\\mathcal {Q}_{I}{}^{Jm} & \\equiv & \\vartheta _{I}{}^{B}Y_{B}{}^{Jm}\\, ,\\hspace{28.45274pt}Y_{B}{}^{Jm} \\equiv \\delta _{B}Z^{Jm}=T_{B}{}^{J}{}_{K}Z^{Km}+T_{B}{}^{m}{}_{n}Z^{Jn}\\, ,\\\\& & \\nonumber \\\\\\mathcal {Q}_{Iim} & \\equiv & \\vartheta _{I}{}^{B}Y_{Bim}\\, ,\\hspace{28.45274pt}Y_{Bim} \\equiv \\delta _{B}Z_{im}=-T_{B}{}^{j}{}_{i}Z_{jm}-T_{B}{}^{n}{}_{m}Z_{in}\\, ,$ and two constraints associated to global invariance $\\mathcal {Q}_{AmIJ}& \\equiv &Y_{AmIJ}=-\\delta _{A}d_{mIJ}=T_{A}{}^{n}{}_{m}d_{nIJ}+2T_{A}^{K}{}_{(I}d_{|m|J)K}\\, ,\\\\& & \\nonumber \\\\\\mathcal {Q}_{A}{}^{i}{}_{I}{}^{m}& \\equiv &Y_{A}{}^{i}{}_{I}{}^{m}=\\delta _{A}d^{i}{}_{I}{}^{m}=T_{A}{}^{i}{}_{j}d^{j}{}_{I}{}^{m}-T_{A}{}^{J}{}_{I}d^{i}{}_{J}{}^{m}+T_{A}{}^{m}{}_{n}d^{i}{}_{I}{}^{n}\\, ,$ and the final form of the Bianchi identity for the 7-form field strengths is $\\mathcal {D}K_{A}=T_{A}{}^{K}{}_{J}F^{J}\\tilde{F}_{K}+T_{A}{}^{m}{}_{n}\\tilde{H}^{n}H_{m}-\\tfrac{1}{2}T_{A\\, ij}G^{ij}+Y_{AI}{}^{B}L_{B}{}^{I}+ Y_{A}{}^{Im}L_{Im} + Y_{Aim}L^{im}\\, .$ The occurrence of these $Y_{A}{}^{\\sharp }$ has to be confirmed by taking again the covariant derivative of this Bianchi identity."
],
[
"9-form field strengths",
"Taking the covariant derivative of both sides of the Bianchi identity Eq.",
"(REF ) we arrive toBy direct computation we have not found any constraint or $Y_{A}{}^{\\sharp }$ tensor associated to either $d^{m}{}_{IJK}$ or $d^{mnp}$ .",
"$\\begin{array}{rcl}Y_{AI}{}^{B}\\left[\\mathcal {D}L_{B}{}^{I} +F^{I}K_{B}\\right]+Y_{A}{}^{Im}\\left[ \\mathcal {D}L_{Im} +\\tilde{F}_{I}H_{m}\\right]+Y_{Aim}\\left[\\mathcal {D}L^{im}+G^{i}\\tilde{H}^{m}\\right]& & \\\\& & \\\\+\\mathcal {Q}_{AmIJ} \\tilde{H}^{m}F^{IJ}+\\mathcal {Q}_{A}{}^{i}{}_{I}{}^{m}G_{i}F^{I}H_{m}& = &0\\, .\\end{array}$ Since we have assumedObserve that, the alternative assumption is equally valid and can be made to work by including the 8-form field strengths $L^{mIJ},L_{i}{}^{I}{}_{m}$ .",
"$\\mathcal {Q}_{AmIJ}=\\mathcal {Q}_{A}{}^{i}{}_{I}{}^{m}=0$ , we arrive to the Bianchi identities $\\mathcal {D}L_{B}{}^{I}& = & -F^{I}K_{B} -W_{B}{}^{I\\beta }M_{\\beta }\\, ,\\\\& & \\nonumber \\\\\\mathcal {D}L_{Im} & = & -\\tilde{F}_{I}H_{m} -W_{Im}{}^{\\beta }M_{\\beta }\\, ,\\\\& & \\nonumber \\\\\\mathcal {D}L^{im} & = & -G^{i}\\tilde{H}^{m} -W^{im\\beta }M_{\\beta }\\, ,$ where the $W_{\\sharp }{}^{\\beta }$ tensors are invariant tensors annihilated by the $Y_{A}{}^{\\sharp }$ ones $Y_{A}{}^{\\sharp }W_{\\sharp }{}^{\\beta } =0\\, .$ As shown in Ref.",
"[18] these tensors are nothing but the derivatives of all the constraints satisfied by the deformation tensors (labeled by $\\beta $ ) with respect to the deformation tensors themselves.",
"This means that there are as many 9-form field strengths $M_{\\beta }$ and corresponding 8-form potentials $N_{\\beta }$ as constraints $\\mathcal {Q}^{\\beta }=0$ .",
"In a general action the top-form potentials $N_{\\beta }$ would occur as the Lagrange multipliers enforcing the constraints $\\mathcal {Q}^{\\beta }=0$ .",
"As usual, this can be confirmed by acting yet again with the covariant derivative on the above three Bianchi identities.",
"Let us first list all the constraints we have met: First of all we have the gauge-invariance constraints $\\mathcal {Q}_{IJ}{}^{A}\\, ,\\,\\,\\, \\mathcal {Q}_{I}{}^{Jm}\\, ,\\,\\,\\,\\mathcal {Q}_{Iim}\\, ,$ defined in Eqs.",
"(REF )-().",
"Secondly, we have the global-invariance constraints $\\mathcal {Q}_{AmIJ}\\, ,\\,\\,\\, \\mathcal {Q}_{A}{}^{i}{}_{I}{}^{m}\\, ,$ defined in Eqs.",
"(REF ) and (()).",
"Thirdly we have the orthogonality constraints between the three deformation tensors $\\mathcal {Q}^{mA} & \\equiv & -Z^{Im}\\vartheta _{I}{}^{A}\\, ,\\\\& & \\nonumber \\\\\\mathcal {Q}_{i}{}^{I} & \\equiv & Z_{im}Z^{Im}\\, ,\\\\& & \\nonumber \\\\\\mathcal {Q}_{mn} & \\equiv & Z_{im}Z^{i}{}_{n}\\, .$ Next, we have the constraints relating the gauge transformations to the $d$ -tensors $\\mathcal {Q}_{I}{}^{J}{}_{K} & \\equiv & X_{(I}{}^{J}{}_{K)}-Z^{Jm}d_{mIK}\\, ,\\\\& & \\nonumber \\\\\\mathcal {Q}_{I}{}^{m}{}_{n} & \\equiv & X_{I}{}^{m}{}_{n} +2d_{nIJ}Z^{Jm}+Z_{in}d^{i}{}_{I}{}^{m}\\, ,\\\\& & \\nonumber \\\\\\mathcal {Q}_{Iij} & \\equiv & -X_{Iij} -2Z_{(i|m}d_{|j)I}{}^{m}\\, ,$ Finally, we have the constraints that related the $d$ -tensors amongst them via the massive deformations $Z$ $\\mathcal {Q}^{imn} & \\equiv & d^{i}{}_{I}{}^{[m|}Z^{I|n]}+Z^{i}{}_{p}d^{pmn}\\, ,\\\\& & \\nonumber \\\\\\mathcal {Q}_{IJ}{}^{mn} & \\equiv &\\tfrac{1}{2}d^{i}{}_{(I|}{}^{m}d_{i|J)}{}^{n}+d^{mnp}d_{pIJ} +3 d^{[m|}{}_{IJK}Z^{K|n]}\\, ,\\\\& & \\nonumber \\\\\\mathcal {Q}_{iIJK} & \\equiv & Z_{im}d^{m}{}_{IJK} -d_{i(I|}{}^{m}d_{m|JK)}\\, .$ From Eq.",
"(REF ) we get $\\frac{\\partial \\mathcal {Q}_{IJ}{}^{A}}{\\partial \\vartheta _{K}{}^{B}}\\left[\\mathcal {D}M^{IJ}{}_{A}+F^{I}L_{A}{}^{J} \\right]+\\frac{\\partial \\mathcal {Q}^{mA}}{\\partial \\vartheta _{K}{}^{B}}\\left[\\mathcal {D}M_{mA}+ H_{m}K_{A}\\right]& & \\nonumber \\\\& & \\nonumber \\\\+\\frac{\\partial \\mathcal {Q}_{I}{}^{J}{}_{K}}{\\partial \\vartheta _{K}{}^{B}}\\left[\\mathcal {D}M^{I}{}_{J}{}^{K}+ F^{IK}\\tilde{F}_{J}\\right]+\\frac{\\partial \\mathcal {Q}_{I}{}^{m}{}_{n}}{\\partial \\vartheta _{K}{}^{B}}\\left[\\mathcal {D}M^{I}{}_{m}{}^{n}+ F^{I}\\tilde{H}^{m}H_{n}\\right]& & \\nonumber \\\\& & \\nonumber \\\\+\\frac{\\partial \\mathcal {Q}_{Iij}}{\\partial \\vartheta _{K}{}^{B}}\\left[\\mathcal {D}M^{Iij}+ F^{I}G^{ij}\\right]& = & 0\\, .$ From Eqs.",
"() and () we get very similar equations which guarantee the consistency of the whole construction of the tensor hierarchy that we have carried out in this section."
],
[
"Gauge-invariant action for the 1-, 2- and 3-forms",
"The Bianchi identities of the full tensor hierarchy give rise to the equations of motion of the electric fields of the theory upon use of the duality relations (on-duality-shell).",
"For field strengths of the 6-,5-, 4-forms they are given by $K_{A}=-\\star j_{A}^{(\\sigma )}\\, ,\\hspace{28.45274pt}\\tilde{F}_{I}=\\mathcal {M}_{IJ}\\star F^{J}\\, ,\\hspace{28.45274pt}\\tilde{H}^{m}=\\star \\mathcal {M}^{mn}H_{n}\\, .$ For the field strengths of the magnetic 3-forms they are given by $G_{a} = R_{a}\\, ,$ where $R_{a}$ has been defined in Eq.",
"(REF ).",
"Finally, the field strength of the 7-forms is, according to Refs.",
"[23], [18], dual to the derivatives of the gauge-invariant scalar potential with respect to the deformation parameters, denoted collectively by $c^{\\sharp }$ $L_{\\sharp } = \\star \\frac{\\partial V}{\\partial c^{\\sharp }}\\, .$ This identity follows from the scalar equation of motion in presence of a scalar potential together with the condition $k_{A}{}^{x}\\frac{\\partial V}{\\partial \\phi ^{x}} =Y_{A}{}^{\\sharp }\\frac{\\partial V}{\\partial c^{\\sharp }}\\, ,$ which implies, after multiplication by the embedding tensor $\\vartheta _{I}{}^{A}$ , the gauge-invariance of the scalar potential.",
"In general, the equations of motion are combinations of different Bianchi identities on-duality-shell.",
"In order to determine the combinations that correspond to the equations of motion we have to examine which combinations of Bianchi identities satisfy the Noether identities associated to the gauge invariances of the theory.",
"To start with, we need to introduce some notation for the Bianchi identities.",
"This has been done in Appendix REF .",
"These Bianchi identities are related by a hierarchy of identities that are obtained by taking the covariant derivative of those with lower rank, as we have shown.",
"These identities of Bianchi identities are collected in Appendix REF .",
"Now, let us assume that a standard gauge-invariant action for the 0-forms $\\mathcal {M}$ (or $\\phi ^{x}$ ), 1-forms $A^{I}$ , 2-forms $B_{m}$ and electric 3-forms $C^{a}$ exists.",
"This means that the Bianchi identities $\\mathcal {B}(\\mathcal {Q}^{\\beta }),\\mathcal {B}(c^{\\sharp })$ and $\\mathcal {B}(\\mathcal {D}\\mathcal {M}),\\mathcal {B}(F^{I}),\\mathcal {B}(H_{m}),\\mathcal {B}(G_{a})$ are satisfied, at least up to duality relations.",
"The kinetic terms of the electric fields are written in terms of the gauge-invariant field strengths and this implies that the magnetic fields $C_{a},\\tilde{B}^{m}$ must necessarily occur in the action, albeit not as dynamical fields: their equations of motion will be trivial on-duality-shell.",
"Under these assumptions, the identities satisfied by the non-trivial Bianchi identities (i.e.",
"those involving the magnetic field strengths) take the simplified formWe have also ignored the identities whose rank, as differential forms, is higher than eight.",
"$\\mathcal {D}\\mathcal {B}(H_{m}) -Z^{a}{}_{m}G_{a}& = & 0\\, ,\\\\& & \\nonumber \\\\\\mathcal {D}\\mathcal {B}(G_{a})-Z_{am}\\mathcal {B}(\\tilde{H}^{m})\\ & = & 0\\, ,\\\\& & \\nonumber \\\\\\mathcal {D}\\mathcal {B}(\\tilde{H}^{m})+d^{a}{}_{I}{}^{m}\\mathcal {B}(G_{a})F^{I}+Z^{Im}\\mathcal {B}(\\tilde{F}_{I})& = & 0\\, ,\\\\& & \\nonumber \\\\\\mathcal {D}\\mathcal {B}(\\tilde{F}_{I})+2d_{mIJ}\\mathcal {B}(\\tilde{H}^{m})F^{J}-d^{a}{}_{I}{}^{m}\\mathcal {B}(G_{a})H_{m}+\\vartheta _{I}{}^{A}\\mathcal {B}(K_{A})& = & 0\\, .$ If such an action exists, its invariance with respect to gauge transformations with parameters $\\sigma ^{m},\\sigma ^{i},\\sigma _{m},\\sigma ^{I}$ will imply that the equations of motion satisfy, off-shell, associated Noether identities.",
"Up to the field equations of $\\tilde{B}^{m}$ and $C_{a}$ which are assumed to be satisfied up to dualities, they take the form $\\mathcal {D}\\frac{\\delta S}{\\delta \\tilde{B}^{m}} -Z^{a}{}_{m}\\frac{\\delta S}{\\delta C_{a}}& = &0\\, ,\\\\& & \\nonumber \\\\\\mathcal {D}\\frac{\\delta S}{\\delta C^{a}}-Z_{am}\\frac{\\delta S}{\\delta B_{m}}& = &0\\, ,\\\\& & \\nonumber \\\\\\mathcal {D}\\frac{\\delta S}{\\delta B_{m}}+Z^{Im}\\left[\\frac{\\delta S}{\\delta A^{I}}-d_{nIJ}A^{J}\\frac{\\delta S}{\\delta B_{n}}-\\left(d^{a}{}_{I}{}^{n}B_{n}-\\tfrac{1}{3}d^{a}{}_{J}{}^{n}d_{nIK}A^{JK}\\right)\\frac{\\delta S}{\\delta C^{a}}\\right]& & \\nonumber \\\\& & \\nonumber \\\\+d^{a}{}_{I}{}^{m}F^{I}\\frac{\\delta S}{\\delta C^{a}}& = &0\\, ,\\\\& & \\nonumber \\\\\\mathcal {D}\\left[\\frac{\\delta S}{\\delta A^{I}}-d_{nIJ}A^{J}\\frac{\\delta S}{\\delta B_{n}}-\\left(d^{a}{}_{I}{}^{n}B_{n}-\\tfrac{1}{3}d^{a}{}_{J}{}^{n}d_{nIK}A^{JK}\\right)\\frac{\\delta S}{\\delta C^{a}}\\right]& & \\nonumber \\\\& & \\nonumber \\\\+2d_{mIJ}F^{J}\\frac{\\delta S}{\\delta B_{m}}+d^{a}{}_{I}{}^{m}H_{m}\\frac{\\delta S}{\\delta C^{a}}+\\vartheta _{I}{}^{A}k_{A}{}^{x}\\frac{\\delta S}{\\delta \\phi ^{x}}& = &0\\, .$ Comparing directly with the above identities satisfied by the Bianchi identities, we conclude that, up to dualities, the equations of motion of the electric fields are related to the Bianchi identities of the magnetic field strengths by $k_{A}{}^{x}\\frac{\\delta S}{\\delta \\phi ^{x}}& = &\\mathcal {B}(K_{A})\\, ,\\\\& & \\nonumber \\\\\\frac{\\delta S}{\\delta A^{I}}& = &\\mathcal {B}(\\tilde{F}_{I})+\\left(d^{a}{}_{I}{}^{m}B_{m}-\\tfrac{1}{3}d^{a}{}_{J}{}^{m}d_{mIK}A^{JK}\\right)\\mathcal {B}(G_{a})+d_{mIJ}A^{J}\\mathcal {B}(\\tilde{H}^{m})\\, ,\\\\& & \\nonumber \\\\\\frac{\\delta S}{\\delta B_{m}}& = &\\mathcal {B}(\\tilde{H}^{m})\\, ,\\\\& & \\nonumber \\\\\\frac{\\delta S}{\\delta C^{a}}& = &\\mathcal {B}(G_{a})\\, .$ This identification determines completely the field theory.",
"For instance, the equation of motion for the electric 3-forms $C^{a}$ must be $\\frac{\\delta S}{\\delta C^{a}}=\\mathcal {D}\\left(\\Im \\mathfrak {m}\\mathcal {N}_{ab} \\star G^{b}+\\Re \\mathfrak {e}\\mathcal {N}_{ab} G^{b}\\right)+d_{aI}{}^{m} F^{I}H_{m} -Z_{am}\\mathcal {M}^{mn}\\star H_{n}\\, ,$ etc.",
"Can we write an action gauge-invariant action for the electric fields $\\phi ^{x},A^{I},B_{m}$ and $C^{a}$ from which these equations of motion follow, up to duality relations?",
"We can follow the step-by-step procedure used in Ref.",
"[18] for the 5- and 6-dimensional cases.",
"This procedure consists in considering first an action $S^{(0)}$ containing only the gauge-invariant kinetic terms for the all the electric potentials $\\phi ^{x},A^{I},B_{m},C^{a}$ and start adding the necessary Chern-Simons terms $S^{(1)},S^{(2)},\\ldots $ to obtain the equations of motion of all the potentials occurring in $S^{(0)}$ in order of decreasing rank: $\\tilde{B}^{m},C^{a},C_{a},B_{m},A^{I}$ .",
"At the first step it will be necessary to introduce terms $S^{(1)}$ containing $\\tilde{B}^{m}$ but no new terms containing this potential will be introduced in the following steps.",
"At the second step we will introduce terms $S^{(2)}$ containing $C^{a}$ (but no $\\tilde{B}^{m}$ ) and in the following steps we will not introduce any more terms containing it and so on and so forth.",
"We will not carry this procedure to the end because in eight dimensions the number of Chern-Simons terms involving just 2- and 1-form potentials is huge and its structure is very complicated.",
"Nevertheless we are going to check that everything works as expected for the potentials of highest rank $\\tilde{B}^{m},C^{a},C_{a}$ and we are going to find that only under certain conditions the action we are looking for exists Our starting point is, therefore, the action $\\begin{array}{rcl}S^{(0)}& = &{\\displaystyle \\int }\\left\\lbrace -\\star \\mathbb {1} R+\\tfrac{1}{2} \\mathcal {G}_{xy}\\mathcal {D}\\phi ^{x}\\wedge \\star \\mathcal {D}\\phi ^{y}+\\frac{1}{2}\\mathcal {M}_{IJ}F^{I}\\wedge \\star F^{J}+\\frac{1}{2}\\mathcal {M}^{mn} H_{m}\\wedge \\star H_{n}\\right.\\\\& & \\\\& &\\left.+\\frac{1}{2} G^{a} \\wedge R_{a}-\\star \\mathbb {1} V(\\phi )\\right\\rbrace \\, ,\\\\\\end{array}$ where we have added a scalar potential $V(\\phi )$ .",
"This action gives $\\frac{\\delta S^{(0)}}{\\delta \\tilde{B}^{m}}=-Z^{a}{}_{m}R_{a}\\, .$ This equations should be trivial on-duality-shell and, therefore, we must add to the action $S^{(0)}$ $S^{(1)}=\\int Z^{a}{}_{m}(G_{a}+\\tfrac{1}{2}Z_{an}\\tilde{B}^{n})\\tilde{B}^{m}\\, ,$ so that $\\frac{\\delta (S^{(0)}+S^{(1)})}{\\delta \\tilde{B}^{m}}=-Z^{a}{}_{m}(R_{a}-G_{a})\\, .$ The equation for $C^{a}$ that follows from $S^{(0)}+S^{(1)}$ is $\\frac{\\delta (S^{(0)}+S^{(1)})}{\\delta C^{a}}=-\\mathcal {D}R_{a}-Z_{am}\\mathcal {M}^{mn}\\star H_{n}\\, ,$ and, comparing with Eq.",
"(REF ), we see that the term $+d_{aI}{}^{m}F^{I}H_{m}$ is missing and we must add a term of the form $S^{(2)}=\\int d_{aI}{}^{m} F^{I}\\left(H_{m}-\\tfrac{1}{2}Z_{bm}C^{b} \\right) C^{a}\\, ,$ Observe that $\\tilde{B}^{m}$ does not appear in this term and its equation of motion is, therefore, not modified by it.",
"However in this term or in any other similar term the only part of $d_{aI}{}^{m}Z_{bm}$ that can occur is the antisymmetric one $d_{[a|I}{}^{m}Z_{|b]m}$ while in the term $+d_{aI}{}^{m}F^{I}H_{m}$ both the antisymmetric and the symmetric parts occur.",
"The only way in which we can get that term in the equations of motion is by requiring $d_{(a|I}{}^{m}Z_{|b)m} = -\\tfrac{1}{2} X_{I\\, ab}=0\\, .$ Under this assumption, which will also prove crucial to obtain the equations of motion of other fields, the equation of motion of $C^{a}$ is Eq.",
"(REF ), as we wanted.",
"The equation of the magnetic potential $C_{a}$ , which should be trivial on-duality-shell which follows from the action we have constructed is $\\frac{\\delta (S^{(0)}+S^{(1)}+S^{(2)})}{\\delta C^{a}}=Z^{a}{}_{m}\\left[\\mathcal {M}^{mn}\\star H_{n} -\\mathcal {D}\\tilde{B}^{m} + d_{bI}{}^{m}F^{I}C^{b}\\right]\\, .$ The last two terms belong to the field strength $\\tilde{H}^{m}$ and we need to add $S^{(3)}=\\int \\left\\lbrace -\\tfrac{1}{2}d^{b}{}_{I}{}^{m}Z^{a}{}_{m}F^{I}C_{b}C_{a}-\\left[2d^{mnp}B_{n}(H_{p}-2Z_{ip}C^{i}-\\tfrac{1}{2}\\mathcal {D}B_{p})+\\Box \\tilde{H}^{m}\\right]Z^{a}{}_{m}C_{a}\\right\\rbrace \\, ,$ where $\\Box \\tilde{H}^{m}$ is the part of the field strength $\\tilde{H}^{m}$ that only contains 1-form potentials and their exterior derivatives.",
"Observe that neither $\\tilde{B}^{m}$ nor $C^{a}$ appear in this term and, therefore, their equations of motion are not modified.",
"Observe also that we are facing here the same problem we faced in getting the equation of motion of $C^{a}$ : only $d^{[b|}{}_{I}{}^{m}Z^{|a]}{}_{m}$ can enter the action while the equation of motion contains also the symmetric part.",
"The solution to this problem is the same: we demand $d^{(a|}{}_{I}{}^{m}Z^{|b)}{}_{m} = -\\tfrac{1}{2} X_{I}{}^{ab}=0\\, .$ Using Eqs.",
"(REF ) and (REF ) The equation of motion of $B_{m}$ that follows from the action $S^{(0)}+\\cdots +S^{(3)}$ can be put in the form $\\begin{array}{rcl}{\\displaystyle \\frac{\\delta (S^{(0)}+\\cdots +S^{(3)})}{\\delta B_{m}}}& = &-\\left[\\mathcal {D}(\\mathcal {M}^{mn}\\star H_{n}) +d_{aI}{}^{m}F^{I}G^{a}-d^{a}{}_{I}{}^{m}F^{I}R_{a}\\right.\\\\& & \\\\& &\\left.-d^{mnp}H_{np}-Z^{Im}\\mathcal {M}_{IJ}\\star F^{J}\\right]-d^{mnp}B_{n}Z^{a}{}_{p}(R_{a}-G_{a})\\\\& & \\\\& &-d^{mnp}Z^{a}{}_{p}d_{aI}{}^{q} B_{n}B_{q}(F^{I}-\\tfrac{1}{2} Z^{Ir}B_{r})\\\\& & \\\\& &+d_{aI}{}^{m}d^{a}{}_{J}{}^{n}F^{I}(F^{I}-\\tfrac{1}{2} Z^{Ip}B_{p})B_{n}\\\\& & \\\\& &+d_{aI}{}^{m}F^{I}\\Box G^{a}+d^{a}{}_{I}{}^{[m|} Z^{I|n]} B_{n}\\Box G_{a}\\\\& & \\\\& &-d^{mnp}(H_{n}-Z_{in}C^{i})(H_{p}-Z_{jp}C^{j})\\, .\\end{array}$ The expression in brackets in the r.h.s.",
"is identical to $\\mathcal {B}(\\tilde{H}^{m})$ up to dualities and up to the term $d^{m}{}_{IJK}F^{IJK}$ .",
"The next term vanishes on-duality-shell and the remaining terms should be eliminated.",
"Observe that in the terms that need to be eliminated and introduced neither $\\tilde{B}^{m}$ nor $C^{i}$ occur (they only depend on $B_{m},A^{I}$ and their derivatives) and, therefore, their equations of motion will not be modified."
],
[
"Conclusions",
"Following the same procedure as in Refs.",
"[18], [15], in this paper we have constructed the most general 8-dimensional theory with gauge symnmetries and with at most two derivatives: field strengths (up to 6-forms), all the Bianchi identities and duality relations satisfied by all the field strengths (up to the 9-formsThese identities are, of course, just formal, but they encode the gauge transformations of the 8-form potentials.",
"), and the equations of motion of the fundamental fields.",
"We have shown that they are characterized by a small number of invariant tensors ($d$ -tensor, embedding tensor $\\vartheta $ and massive deformations $Z$ ) that satisfy certain constraints that relate them among themselves and to the structure constants and generators of the global symmetry group, which has to act on the $n_{3}$ 3-form potentials of the theory as a subgroup of Sp$(2n_{3},\\mathbb {R})$ .",
"We have found that the Bianchi identities satisfied by the 7-form field strengths (dual to the generalized Noether-Gaillard-Zumino current) have the general form predicted in Ref.",
"[19], although in this case it is very difficult to find the explicit form of the 7-form field strengths.",
"We have constructed an action from which one can derive all the equations of motion except for those of the 1-form potentials because identifying the terms that only contain 1-forms becomes extremely complicated and time-consuming.",
"This general result can be applied to any 8-dimensional theory with a given field content, $d$ -tensors defining Chern-Simons interactions and global symmetry group, such as maximal $d=8$ supergravity.",
"In a forthcoming publication we will solve the constraints satisfied by the deformation tensors ($d$ -tensor, embedding tensor $\\vartheta $ and massive deformations $Z$ ) searching for a 1-parameter family of different SO$(3)$ gaugings of that theory."
],
[
"Acknowledgments",
"This work has been supported in part by the Spanish Ministry of Science and Education grant FPA2012-35043-C02-01, the Centro de Excelencia Severo Ochoa Program grant SEV-2012-0249, and the Spanish Consolider-Ingenio 2010 program CPAN CSD2007-00042.",
"The work of OLA was further supported by a scholarship of the Ecuadorian Secretary of Science, Technology and Innovation.",
"TO wishes to thank M.M.",
"Fernández for her permanent support."
],
[
"Bianchi identities",
"The Bianchi identities satisfied by the field strengths of the gauged theory are $\\mathcal {B}(\\cdot )=0$ where $\\mathcal {B}(L_{A}{}^{I})& = &-\\left[\\mathcal {D}L_{A}{}^{I} +F^{I}K_{A} +W^{I}{}_{A}{}^{\\beta }M_{\\beta }\\right]\\, ,\\\\& & \\nonumber \\\\\\mathcal {B}(L_{Im})& = &-\\left[\\mathcal {D}L_{Im} +\\tilde{F}_{I}H_{m} +W_{Im}{}^{\\beta }M_{\\beta }\\right]\\, ,\\\\& & \\nonumber \\\\\\mathcal {B}(L^{im})& = &-\\left[\\mathcal {D}L^{im} +G^{i}\\tilde{H}^{m} +W^{im\\, \\beta }M_{\\beta }\\right]\\, ,\\\\& & \\nonumber \\\\\\mathcal {B}(K_{A})& = &\\mathcal {D}K_{A}-T_{A}{}^{I}{}_{J}F^{J}\\tilde{F}_{I} -T_{A}{}^{m}{}_{n}\\tilde{H}^{n}H_{m}+\\tfrac{1}{2}T_{A\\, ij}G^{ij} -Y_{A}{}^{\\sharp }L_{\\sharp }\\, ,\\\\& & \\nonumber \\\\\\mathcal {B}(\\tilde{F}_{I})& = &-\\left[\\mathcal {D}\\tilde{F}_{I}-2d_{mIJ} F^{J}\\tilde{H}^{m}-d_{iI}{}^{m}G^{i}H_{m}+2d^{m}{}_{IJK}F^{JK}H_{m}+\\vartheta _{I}{}^{A}K_{A}\\right]\\, ,\\\\& & \\nonumber \\\\\\mathcal {B}(\\tilde{H}^{m})& = &-\\left[\\mathcal {D} \\tilde{H}^{m}+d_{iI}{}^{m}F^{I}G^{i} -d^{mnp}H_{np} -d^{m}{}_{IJK}F^{IJK}-Z^{Im}\\tilde{F}_{I}\\right]\\, ,\\\\& & \\nonumber \\\\\\mathcal {B}(G_{i})& = &-\\left[\\mathcal {D}G_{i}-d_{iI}{}^{m} F^{I}H_{m} +Z_{im}\\tilde{H}^{m}\\right]\\, ,\\\\& & \\nonumber \\\\\\mathcal {B}(H_{m})& = &-\\left[\\mathcal {D}H_{m}+d_{mIJ}F^{IJ}-Z_{im}G^{i}\\right]\\, ,\\\\& & \\nonumber \\\\\\mathcal {B}(F^{I})& = &-\\left[\\mathcal {D}F^{I} - Z^{Im}H_{m}\\right]\\, ,\\\\& & \\nonumber \\\\\\mathcal {B}(\\mathcal {D}\\mathcal {M})& = &-\\left[\\mathcal {D}\\mathcal {D}\\mathcal {M}+F^{I}\\vartheta _{I}{}^{A}\\delta _{A}\\mathcal {M}\\right]\\, ,\\\\& & \\nonumber \\\\\\mathcal {B}(c^{\\sharp })& = &\\mathcal {D}c^{\\sharp }\\, ,\\\\& & \\nonumber \\\\\\mathcal {B}(\\mathcal {Q}^{\\beta })& = &\\mathcal {Q}^{\\beta }\\, .$ Here $\\sharp $ labels the deformation parameters and $\\beta $ the constraints, as discussed in Sections REF and REF ."
]
] | 1605.05882 |
[
[
"Accretion disk signatures in Type I X-ray Bursts: prospects for future\n missions"
],
[
"Abstract Type I X-ray bursts and superbursts from accreting neutron stars illuminate the accretion disk and produce a reflection signal that evolves as the burst fades.",
"Examining the evolution of reflection features in the spectra will give insight into the burst-disk interaction, a potentially powerful probe of accretion disk physics.",
"At present, reflection has been observed during only two bursts of exceptional duration.",
"We investigate the detectability of reflection signatures with four of the latest well-studied X-ray observatory concepts: Hitomi, NICER, Athena, and LOFT.",
"Burst spectra are modeled for different values for the flux, temperature, and the disk ionization parameter, which are representative for most known bursts and sources.",
"The effective area and through-put of a Hitomi-like telescope are insufficient for characterizing burst reflection features.",
"NICER and Athena will detect reflection signatures in Type I bursts with peak fluxes $\\ge 10^{-7.5}$ erg cm$^{-2}$ s$^{-1}$, and also effectively constrain the reflection parameters for bright bursts with fluxes of $10^{-7}$ erg cm$^{-2}$ s$^{-1}$ in exposures of several seconds.",
"Thus, these observatories will provide crucial new insight into the interaction of accretion flows and X-ray bursts.",
"For sources with low line-of-sight absorption, the wide band-pass of these instruments allows for the detection of soft X-ray reflection features, which are sensitive to the disk metallicity and density.",
"The large collecting area that is part of the LOFT design would revolutionize the field by tracing the evolution of the accretion geometry in detail throughout short bursts."
],
[
"Introduction",
"In low-mass X-ray binaries (LMXBs) that host a neutron star, the donor star may fill its Roche lobe, and transfer material via an accretion disk to the neutron star.",
"Hydrogen and helium accumulated on the neutron star surface can undergo runaway thermonuclear burning, powering a brief ($10-100\\,\\mathrm {s}$ ) Type I X-ray burst [26], [9], [75], [53], [48].",
"Recurring on typical timescales of hours to days, these bursts are the most frequent thermonuclear flashes in nature [51], [63], [24].",
"A bright X-ray burst briefly outshines all other X-ray emitting regions in the system.",
"The X-ray spectrum from the neutron star during the burst is close to a blackbody ([68], [72]; see also, e.g., [67]), and this spectrum may be reprocessed (or, reflected) by and scattered off the surrounding disk and the companion star.",
"UV and optical reprocessing is thought to originate predominantly from the outer regions of the disk and the companion star [32], [58], whereas X-ray reflection occurs mostly off the inner disk ([6], [43]; see also [15]) which is struck by a particularly strong ionizing radiation field.",
"As a result, an Fe K$\\alpha $ emission line may be visible in the spectrum near $6.4\\,\\mathrm {keV}$ as well as an Fe absorption edge at slightly higher energies, the properties of which depend on the ionization state of the inner disk.",
"Further spectral features produced by X-ray reflection include a multitude of lines and a bremsstrahlung continuum below $\\sim 1\\,\\mathrm {keV}$ [3].",
"When originating from the inner disk, the shape of these spectral features is modified by rotational Doppler broadening and gravitational redshifting [18].",
"The strength of these effects depends on the distance from the neutron star, and, therefore, the location of the reflection site can be measured.",
"In addition, the magnitude of the reflection signal encodes information about the disk geometry [6], [30].",
"Therefore, the reflection features in a burst spectrum can potentially reveal a treasure of information on the properties of the accretion environment, similar to how reflection spectroscopy has been invaluable for the study of accretion onto black holes in Active Galactic Nuclei (AGN) and compact binaries [19], [55].",
"The short duration of X-ray bursts and the fast evolution of their spectral properties mean that spectra can only be collected in short time intervals.",
"Around the peak of the burst this is at most a few seconds.",
"The quality of the spectra is, therefore, rather limited.",
"In normal bursts, reflection features have never been clearly detected, and burst reflection is not distinguished from the directly observed thermal emission from the neutron star [49], [22].",
"The highest quality burst spectra were obtained with the Proportional Counter Array [38] on the Rossi X-Ray Timing Explorer [10] for two so-called “superbursts” with durations of several hours [64], [65].",
"The long duration allowed for more detailed spectral analysis that revealed reflection features [6], [43], [44], and in one case an evolving persistent component [42].",
"The reflection features showed that the inner disk is highly ionized by the burst, and may temporarily be disrupted.",
"These two observations show that X-ray reflection during bursts is a powerful tool for investigating the behavior of accretion disks under sudden strong irradiation.",
"[4] considered inflow, outflow, and thermodynamic processes during the interaction of an X-ray burst with the surrounding accretion disk, and found that nearly all processes may be relevant.",
"The variety of possible physical effects means that further theoretical studies as well as improved observational constraints are needed to better understand the impact of an X-ray burst on its surroundings.",
"Aside from a reflection component, the X-ray spectrum additionally includes components for thermal emission from the accretion disk, Comptonized emission from the accretion disk corona, as well as thermal emission from a boundary or spreading layer where freshly accreted material reaches the neutron star surface [37], [60].",
"These components are also present before the burst as “persistent” emission.",
"Typically, their spectral properties are measured at that time, and one assumes them to remain unchanged during the burst.",
"All of the burst spectrum in excess of the combined persistent parts is assumed to be blackbody emission from the neutron star [46].",
"However, there is now evidence that the persistent emission is being altered by the burst, as its normalization may increase [76], [35], [42], [77], [16], possibly due to Poynting-Robertson drag [73], and its spectrum appears to soften [52], [12], [11], [39], [42].",
"Therefore, when studying burst reflection, care must be taken to distinguish reflection from an evolving persistent spectral component.",
"In this paper we investigate the capabilities of different instrumentation for detecting burst reflection.",
"We simulate a wide range of burst observations with four X-ray observatories: NICER, Hitomi, Athena, and LOFT (Section ).",
"Due to their large collecting areas, NICER, Athena, and LOFT will be able to detect reflection features in bright bursts, even if their duration is short (Section ).",
"We discuss how burst reflection can be applied to study the accretion environment of neutron stars in LMXBs, and how it impacts neutron star science such as constraints on the dense matter equation of state (Section ).",
"We conclude that X-ray burst reflection will become an important method to study the properties and behavior of accretion disks in LMXBs.",
"(Section ).",
"The detection of X-ray burst reflection is highly dependent on the properties of the instrumentation, such as the energy band, the effective area, and the spectral resolution.",
"We study the detectability of reflection using the expected response of four X-ray observatories that are either recently launched, being constructed, or planned for a launch opportunity further in the future.",
"The Neutron Star Interior Composition Explorer [25] is planned for launch in 2016, and will be placed on the International Space Station.",
"Its X-ray Timing Instrument (XTI) consists of 56 units, each with an X-ray concentrator optic and a silicon strip detector.",
"It is not an imaging instrument, but it collects all photons from the $30\\ \\mathrm {arcmin}^2$ field of view.",
"Its detectors are sensitive in the $0.2-10\\ \\mathrm {keV}$ energy band, and its effective area is approximately $2000\\ \\mathrm {cm^2}$ at $1\\ \\mathrm {keV}$ .",
"By using of a large number of detectors, NICER can handle high photon count rates.",
"Athena [8] is a proposed large mission for the European Space Agency to be launched in the late 2020s.",
"It will combine a collecting area larger than NICER with imaging capabilities and high spectral resolution.",
"We consider the imaging CCD detector: the Wide Field Imager [54].",
"It is sensitive in the $0.3-12\\ \\mathrm {keV}$ band, and has a $\\sim 17,000\\ \\mathrm {cm^2}$ effective area around $1\\ \\mathrm {keV}$ .",
"Beyond Athena, we may speculate about a future observatory with an even larger detector area.",
"The Large Observatory For X-ray Timing [20] was a mission concept proposed to ESA.",
"Although it was not selected, parts of its design may be incorporated in other missions.",
"With an effective area of $8.5\\,\\mathrm {m^{2}}$ combined with high throughput, it promised unprecedented opportunities for studying short transient events such as Type I X-ray bursts [36].",
"We use the M4 configuration of its Large Area Detector (LAD).",
"Contrary to the other three instruments that we consider, the LAD is not sensitive below $2\\ \\mathrm {keV}$ : its energy band is $2-80\\ \\mathrm {keV}$ .",
"Hitomi [69] was launched in February 2016 and operated until the end of March.",
"As it had significantly different capabilities than the other instruments considered here, it is valuable to include it in the present study.",
"Hitomi hosted several instruments, and we consider the Soft X-ray Imager [29].",
"SXI was a CCD imaging detector sensitive in the $0.4-12\\ \\mathrm {keV}$ energy band, and at $1\\ \\mathrm {keV}$ the effective area was $\\sim 590\\ \\mathrm {cm^2}$ .",
"Compared to NICER's detectors, the imaging capability allows for a lower background, but it reduces the maximum photon count rate that the instrument can process.",
"In Section REF we will discuss the impact of this on the ability to detect reflection.",
"Table: Properties of the simulated instruments, as well as the employed response and background files.Figure: The effective area as a function of energy, EE, for the instruments considered in this paper including RXTE/PCA (Sect.).",
"The vertical line indicates the location of the Fe Kα\\alpha line near 6.46.4 keV.",
"LOFT has a larger area than RXTE (5 PCUs) at this energy.",
"However, the effective areas of Hitomi, NICER, and Athena extend to much lower energies than RXTE, allowing for additional reflection features to be detected.We simulate burst reflection spectra for all missions using the response matrix files (RMFs), ancillary response files (ARFs), and background files that have been made available by the instrument teams.",
"Table REF lists which exact files are employed, and Figure REF compares the effective areas.",
"The figure also includes the area of RXTE/PCA (assuming all 5 Proportional Counter Units [PCUs] are operating), the only instrument to have successfully detected burst reflection.",
"A more detailed comparison to RXTE is made in Sect.",
"REF ."
],
[
"Spectral Model",
"Three distinct emission components can contribute to the X-ray spectrum during a burst: the persistent emission from the underlying accretion disk, the thermal blackbody emission from the NS surface, and the reflection of that blackbody from the disk surface.",
"Additionally, a thin boundary or spreading layer may be present, which during a burst may cover a substantial part of the neutron star [37], [60].",
"We do not explicitly include a spreading layer, but our setup is equivalent to complete coverage by this layer.",
"We aim for our spectral model to be representative of a wide range of bursts, covering the majority of the known bursting sources.",
"We employ XSPEC version 12.8.2 [2] to create a spectral model that includes all three components, along with interstellar absorption (Figure REF ).",
"Using XSPEC terminology, the model reads as phabs*(constant*(diskbb + compTT) + rdblur*cflux*atable(bbrefl_1xsolar_0-5r.fits)) with the parameter values shown in Table REF .",
"Below we describe the components in detail.",
"We employ a blackbody component with temperature $kT$ as well as reflection of this blackbody off a photoionized accretion disk [3].",
"The reflection spectrum is sensitive to the ionization state of the illuminated layer and, as this is a constant density model, can be parameterized using the ionization parameter $\\xi =4\\pi F_{x}/n_{\\mathrm {H}}$ , where $F_{x}$ is the irradiating flux at the surface of the reflecting slab and $n_{\\mathrm {H}}$ is its density.",
"Both $kT$ and $\\xi $ evolve during a burst [6], [43]; therefore, we consider a range of values for these parameters in order to determine how they may affect the detectability of the reflection signal.",
"The reflection fraction, $R$ , is defined as the flux ratio of the reflection and the blackbody parts.",
"We use $R=0.5$ , which is approximately the value predicted for a flat accretion disk viewed under a small inclination [30], and which is a typical value found in the superburst analyses [6], [43].",
"The composition of the accretion disk is assumed to be solar, consistent with most bursters.",
"A subset of bursters are so-called Ultra-Compact X-ray Binaries (UCXBs), where the accretion composition is thought to be hydrogen-deficient [33].",
"However, because the metal content for UCXBs is likely similar to solar, and metals produce the prominent reflection features such as the Fe K$\\alpha $ line, our conclusions on the detectability of reflection will also be applicable to these sources.",
"A cflux component operating on the combined blackbody and reflection model sets the unabsorbed bolometric flux of the burst, and allows for a range of burst fluxes to be simulated.",
"The reflection signal may arise from the inner regions of the accretion disk and will be sculpted by relativistic effects [18], [6].",
"Thus, the reflection model is convolved with the rdblur relativistic blurring kernel that accounts for the effects of the emission being released in a Schwarzschild metric.",
"As we are simply interested in the detectability of the reflection component, the rdblur parameters are kept fixed throughout the experiment at values indicative of reflection from the inner disk, as seen from 4U 1820-30 [6], assuming a typical inclination angle of the disk with respect to the line of sight of $30^\\circ $ .",
"Although the location of the reflection signal may evolve throughout a burst, these changes will likely be undetectable for most of the considered instruments unless they arise from a superburst.",
"The persistent emission of LMXBs depends on their spectral state [28].",
"It is thought that in the soft state, the inner disk extends closest to the neutron star, which maximizes the reflection signal.",
"Furthermore, the soft state is associated with higher persistent flux and a higher burst frequency.",
"Therefore, we use a soft persistent spectrum in our model, but note that reflection will be even better detectable in the hard state when the persistent flux is lower (for equal reflection fraction).",
"As a typical example of a soft spectrum, we employ the spectral shape measured in broad-band BeppoSAX observations of the transient source 4U 1608-522 during outburst [45].",
"This includes thermal emission from the disk (diskbb) plus a Comptonized component (compTT).",
"When in outburst, 4U 1608-522 exhibits frequent bright bursts with peak fluxes of $F\\simeq 10^{-7}$ erg cm$^{-2}$ s$^{-1}$ , making it a prime candidate for detecting burst reflection.",
"Our results will not be strongly dependent on the details of the persistent spectrum, because we test how well reflection can be distinguished from the persistent components by fitting for a multiplicative factor for the latter.",
"The multiplicative factor, denoted $f$ , in front of the persistent emission (constant in XSPEC) may also represent the possibility of a change in the persistent emission during the burst.",
"Such behavior has been inferred for a variety of Type I bursts [76], [77] and measured in the 2001 superburst from 4U 1636-536 [42].",
"The exact interpretation of this increase in persistent emission is not understood, and it may be related to an increase of accretion onto to the neutron star or to changes in the disk corona due to the influence of the burst.",
"This parameter is set to unity for a burst flux of $F=10^{-7}$ erg cm$^{-2}$ s$^{-1}$ (typical peak flux of a bright burst), and is scaled appropriately as the flux is changed so that it remains the same fraction of the total flux.",
"Photoelectric interstellar absorption is taken into account using the phabs model with cross-sections from [7] and abundances from [1].",
"Most X-ray bursting sources are near the Galactic Center, where absorption is relatively strong ($N_\\mathrm {H}\\sim 10^{22}\\ \\mathrm {cm^{-2}}$ ).",
"For consistency, we use the absorption column of $N_\\mathrm {H}=0.891\\times 10^{22}\\ \\mathrm {cm^{-2}}$ measured for 4U 1608-522 during the same observation from which our persistent spectrum originates.",
"Additionally, in Section REF we investigate the case of a smaller absorption column.",
"Table: Parameters of the simulated Type I X-ray burst spectra."
],
[
"Spectral Simulations",
"Using the spectral model (Table REF ) and the responses of the instruments considered here (Table REF ) we create a set of simulated spectra.",
"The instrumental background is included as well as statistical fluctuations following a Poisson distribution.",
"Neighboring spectral bins with fewer than 20 counts are grouped to ensure $\\chi ^2$ statistics are applicable.",
"In order to cover the wide variety of known bursts and sources, we investigate a broad range of values for the burst flux $F$ , the burst temperature $kT$ , and the ionization parameter of the reflector $\\log \\xi $ .",
"Therefore, for each instrument, the spectral model is simulated over a three-dimensional space defined by these three parameters.",
"The range of values for temperature, $1.5\\le kT \\le 3.0$ , is resolved in steps of 0.5, and covers the peak of the hottest bursts as well as the tails of weaker bursts [24].",
"The strength of the reflection features are a strong function of $\\xi $ and $kT$ [3].",
"Therefore, we study a range of values for the ionization parameter, $1.5\\le \\log \\xi \\le 3.5$ , again resolved in steps of 0.5, which includes the values observed during superbursts [6], [43].",
"The statistical quality of the spectra depends on the observed number of counts, which is proportional to the product of the flux and the exposure time.",
"For simplicity, the exposure time of each simulation is maintained at one second.",
"This is approximately the time resolution required for time-resolved spectral analysis of the shortest bursts.",
"For longer bursts, especially during the tail when the spectral parameters evolve more slowly, longer exposures can in principle be taken.",
"Results for longer exposure times can be found by appropriately scaling the burst flux, such that the same number of counts is obtained.",
"For the bolometric burst flux we use the values $-8.0\\le \\log F \\le -4.5$ , with steps of $0.5$ .",
"This range includes $\\log F \\approx -7.0$ for the typical brightest known bursts [24] as well as an order of magnitude lower fluxes.",
"Furthermore, for the two mentioned superburst observations spectra were collected over 64 s intervals [6], [47].",
"This is equivalent to looking for 1 second at a burst that is 64 times brighter, and for the brightest bursts this extends the flux range to $\\log (64\\times 10^{-7}) = -5.2$ .",
"Spectra should not be accumulated over a longer duration because the spectrum changes as the temperature of the neutron star changes significantly, but to be inclusive, we take one extra step of 0.5, and include a flux of $\\log F=-4.5$ .",
"These choices of parameter values span the full relevant range of photospheric temperatures observed from all known bursting sources, as well as the full range of expected ionization states of the accretion disk.",
"Moreover, $F$ not only represents the intrinsic burst flux, but can be scaled to match different exposure times and source distances."
],
[
"Detectability of Reflection",
"To quantify the detectability of burst reflection, we investigate whether the reflection parameters can be retrieved from the simulated data, or whether a blackbody alone adequately describes the data.",
"Each simulated spectrum is fitted twice for each instrument: once with the same reflection model used to generate the simulation (with $f$ , $kT$ , $R$ , $\\log F$ and $\\log \\xi $ as fit parameters) and once with a standard blackbody model.",
"The latter model replaces the reflection component with a simple blackbody, and now $f$ , $kT$ , and the blackbody normalization are the fit parameters.",
"All other parameters are kept fixed.",
"The reduced $\\chi ^2$ , $\\chi _\\nu ^2$ , of each fit is recorded along with the best-fit values of the parameters, including their $1\\sigma $ uncertainties.",
"A quick overview of the results is shown in Table REF , where the $\\chi _\\nu ^2$ values (averaged over all $\\log \\xi $ and $\\log F$ values) are listed for each $kT$ and for each instrument (a comparison to RXTE follows in Section REF ).",
"Unsurprisingly, the reflection model is able to provide a good description of the simulated spectra with little scatter in the $\\chi _\\nu ^2$ values.",
"However, the blackbody model — which is commonly used in burst analyses — may produce very poor fits to the simulated observations from NICER, Athena, and LOFT.",
"The $\\chi _\\nu ^2$ values also appear to become progressively worse as $kT$ increases, as the peak of the blackbody spectrum moves to the edge of the bandpass.",
"A Hitomi-like mission, on the other hand, will have difficulty to distinguish the reflection features in the burst spectra because of its smaller effective area.",
"Table: The average χ ν 2 \\chi _\\nu ^2 of fits to reflection spectra.The standard deviations of the average $\\chi _\\nu ^2$ shown in the lower half of Table REF are similar in magnitude to the average $\\chi _\\nu ^2$ values themselves and increase with $kT$ .",
"Therefore, there is clearly a wide range of fit results obtained within the parameter space defined by $(kT, \\log \\xi , \\log F)$ .",
"A more detailed view of the variety of $\\chi ^2_\\nu $ obtained when applying the blackbody model to the simulated data is presented in Figure REF , where contours of $\\chi _\\nu ^2$ are shown as functions of $\\log F$ and $\\log \\xi $ at $kT=3$ keV for each instrument considered (the results are qualitatively similar at other blackbody temperatures).",
"Figure: Contours of goodness of fit, χ ν 2 \\chi _\\nu ^2, from fitting the blackbody model to simulated burst reflection spectra for four instruments.",
"Shown are simulations for a range of values for the ionization parameter, logξ\\log \\xi , and the unabsorbed bolometric burst flux, FF.",
"A blackbody temperature of kT=3kT=3 keV is used (the results at other temperatures are qualitatively similar).",
"The simulations have an exposure time of 1s1\\ \\mathrm {s}, and a high FF is equivalent to a longer exposure at correspondingly lower FF.",
"The contour levels are χ ν 2 \\chi _\\nu ^2=1.0 (solid), 1.5 (dashed), 2.0 (dotted), and 3.0 (dashed-dotted).Figure: Residuals of blackbody fits totwo NICER reflection spectra (in units of the 1σ1\\sigma uncertainty of the data points).",
"The spectra are simulated for kT=3 keV kT=3\\,\\mathrm {keV},logξ=3\\log \\xi =3, and logF=-7\\log F=-7.",
"We show a 1s1\\,\\mathrm {s} (top) anda 10s10\\,\\mathrm {s} exposure (bottom).",
"The iron line and the soft excess are clearlyvisible.The effective area of a Hitomi-like telescope is relatively small, such that reflection signatures in Type I X-ray bursts are unlikely to be strongly detected by this mission unless it was a burst of historic brightness or duration (see also Section REF ).",
"In contrast, the NICER reflection spectra significantly deviate from a blackbody for $\\log F -6.0$ .",
"The large effective area of Athena could enable the detection of reflection signatures at even lower flux levels.",
"However, it may not be able to handle the large photon counts that an X-ray burst provides (Section REF ).",
"The LOFT design includes an even larger collecting area, and the blackbody model is inadequate at all but the lowest considered flux values.",
"NICER and LOFT will, therefore, be best suited to detect reflection.",
"Figure REF illustrates the deviations of a NICER reflection spectrum from a blackbody.",
"Clearly visible are the Fe K$\\alpha $ line near $6.4$ keV as well as a soft excess below $\\sim 2$ keV.",
"The contours of Figure REF show for all instruments that reflection signatures are more easily detected for $\\log \\xi \\approx 2.5$ .",
"This is the result of two features in the X-ray reflection spectrum [3].",
"First, the equivalent width of the Fe K$\\alpha $ line is strongest at this ionization parameter because it is dominated by recombination onto He-like Fe, and the other metals in the irradiated slab (e.g., C, N and O) are highly ionized, leading to less absorption at energies around 6 keV and a more prominent Fe line.",
"In addition, the gas at these ionization parameters is full of hot electrons and ionized metals that together produce a significant bremsstrahlung-dominated soft excess at lower energies.",
"As all the instruments considered here with the exception of LOFT have a band-pass that extends substantially below 1 keV (Table REF , Figure REF ), this soft excess provides a significant deviation from the blackbody shape and allows the reflection signal to be more easily detected.",
"At larger values of $\\log \\xi $ , the Fe K$\\alpha $ line becomes weaker due to the increased ionization and the soft excess becomes stronger, but the overall detectability of the reflection signal is reduced.",
"For $\\log \\xi 2.5$ , both the soft excess and the Fe K$\\alpha $ equivalent width are smaller, and it becomes significantly more challenging for the instruments to detect the reflection signal in the burst spectra.",
"Figure REF indicates the region of parameter space where the reflection features are strong enough that a blackbody model cannot provide a good fit to the simulated spectrum.",
"However, the reflection signal remains in the data even if a blackbody model is a good statistical fit to the spectrum.",
"An example of this can be seen in the top panel of Figure REF , where we show the residuals of the blackbody fit to a 1 s NICER observation for a burst with $\\log F=-7$ .",
"According to the contours of Figure REF , the blackbody model is an acceptable fit to these data, yet the residuals clearly show the effects of the reflection spectrum with excesses at $\\approx 6$ keV and at lower energy.",
"In this case, a fit with the reflection model would likely be a significant improvement over the blackbody model.",
"Therefore, it is interesting to examine our fits results in a new way and determine the region of parameter space where the reflection model is a significant improvement over the blackbody model.",
"Figure: Contours of the F-test probability, illustrating the significance of the improvement in χ 2 \\chi ^2 when a reflection model is fit to the simulated spectra compared to the simple blackbody model.",
"The results for a blackbody temperature of kT=3kT=3 keV are shown (the results are qualitatively similar at other temperatures).",
"The contour levels of significance are 90% (solid), 95% (dashed), 99% (dotted), and 99.99% (dashed-dotted).To this end, we compute the F-test probability for each spectrum to determine the significance of any improvement the reflection model makes over the blackbody model.",
"Contours at different significance levels are shown in Figure REF for the $kT=3$ keV models.",
"These contours indicate which bursts are better described by the reflection model, whereas Figure REF shows where the blackbody model cannot adequately describe the spectra (a more stringent requirement).",
"The reflection model is a significant improvement over the blackbody model for $\\log F -8$ (LOFT, Athena), $-7.5$ (NICER), $-7$ (Hitomi).",
"These fluxes are substantially smaller than the limits needed for the blackbody model to provide a poor description of the data (Figure REF ), and therefore implies that reflection features will be detectable down to these flux limits, but constraints on the reflection parameters may be relatively poor."
],
[
"Uncertainties in the Measured Parameters",
"Many spectral analyses of Type I X-ray bursts omit the possibility of a reflection component in the model.",
"It is therefore interesting to consider if this omission has an effect on the uncertainty in the spectral parameters.",
"We plot in Figure REF the relative uncertainty in two crucial X-ray burst parameters ($kT$ and $f$ ) as a function of flux for simulated NICER bursts.",
"The specific reflection model used for the figure has $kT=3$ keV and $\\log \\xi =3$ , although the results are qualitatively similar for other parameters and other observatories.",
"The relative error on the parameters is defined using the $1\\sigma $ uncertainty from the spectral fit compared to the true value of the parameter.",
"We compare the errors for fits with the reflection model and with the model that omits reflection.",
"Figure: (Top) Relative error in measurements of kTkT from a Type I X-ray burst with a reflection component as a function of burst flux logF\\log F. The simulated X-ray spectra assume observations by NICER, kT=3kT=3 keV and a disk ionization parameter of logξ=3\\log \\xi =3.",
"Data are fit with models that includes disk reflection (solid line) and one that includes emission from only the blackbody and persistent components (dashed line).",
"The dotted line plots the slope expected for a Poisson distribution to help guide the eye.",
"Deviations from this slope at small logF\\log F are due to statistical fluctuations.",
"Both models constrain kTkT to similar precision.",
"(Bottom) As above, but now plotting the error in determining ff, the scale factor of the persistent emission.",
"The extra component in the reflection model leads to additional degeneracies in determining the normalization of the spectra and thus gives a larger error in ff.For both $f$ and $kT$ the measured error decreases with flux as expected for a Poisson distribution.",
"For the case of $kT$ , it is independent of the spectral model used to fit the simulated data.Deviations from the behavior expected for a Poisson distribution are seen in Figure REF , in particular at small $\\log F$ .",
"At these small fluxes, the statistically random noise added to the simulation is most significant.",
"Thus, these deviations are simply due to the single realization of a spectrum with a significant contribution from Poisson noise.",
"Thus, the measured relative error of $kT$ appears to be robust to neglecting reflection in the spectral fitting.",
"In contrast, the uncertainty in $f$ , which measures the relative strength of the persistent emission, is higher when fit with a model that includes reflection, even though this model was used to generate the simulated data.",
"This is because the reflection model has three parameters that can be adjusted to set the overall normalization of the spectrum ($f$ , $\\log F$ , and the reflection fraction $R$ ), whereas the blackbody model only has $f$ and $\\log F$ .",
"This extra degree of freedom in the reflection model leads to a larger uncertainty in the normalization of the relatively weak persistent emission.",
"Next, we consider the relative uncertainties in the reflection parameters: the reflection fraction $R$ and the ionization parameter $\\log \\xi $ , as a function of flux for a burst with $kT=3$ keV and $\\log \\xi =3$ as observed by all four observatories (Figure REF ).",
"Figure: (Top) Relative error in measurements of the reflection fraction RR from bursts observed by all four facilities as a function of burst flux logF\\log F. The simulated X-ray spectra assume kT=3kT=3 keV and a disk ionization parameter of logξ=3\\log \\xi =3.",
"The solid line near the bottom of the panel shows the slope for Poisson noise.",
"(Bottom) As above, but now showing the error in logξ\\log \\xi , the ionization parameter of the reflecting disk.As with the burst parameters, the errors on the parameters increase to lower fluxes due to Poisson noise, but for NICER and Hitomi the relative errors of both parameters increase sharply at $\\log F \\le -8$ where the reflection signal is no longer significantly detectable (Figure REF ).",
"NICER can measure $\\log \\xi $ with a relative error $<0.1$ for fluxes $\\log F -7$ .",
"Athena and LOFT measure $\\log \\xi $ with this relative error for all our modeled bursts, and can even constrain $\\log \\xi $ with an error $< 0.01$ for $\\log F -6$ (Athena) and $\\log F -7$ (LOFT), respectively.",
"As $R$ is related to the covering factor of the irradiated accretion disk, measurements of $R$ will provide information on the disk geometry.",
"Constraining $R$ is a more difficult challenge and requires a stronger detection.",
"Therefore, a Hitomi-like telescope would require $\\log F -5$ to be able to measure $R$ with a relative error $<0.1$ .",
"However, NICER can achieve that precision for $\\log F -6$ which is not much larger than the limit for Athena ($\\log F -6.5$ ), whereas LOFT could provide such precision for $\\log F -7$ ."
],
[
"Comparison to ",
"The most detailed burst observations to date have been performed with RXTE/PCA [23].",
"RXTE spectra of bright bursts with the full array of 5 PCUs exhibit a hint of the Fe K$\\alpha $ line [24], and reflection was detected for two superbursts [6], [43].",
"For comparison with the other instruments, and to check the reliability of our method, we also create simulations for RXTE/PCA.",
"This instrument consisted of 5 collimated PCUs that are sensitive in the $2-60$ keV band and have a combined collecting area of $6500\\ \\mathrm {cm^2}$ [38].",
"Although only a subset of the 5 PCUs was active during the majority of observations, we employ the response of the full array, which represents the highest quality burst observations.",
"The response is generated using the tool pcarsp for the top layer and event mode data with 64 channels, and the background is estimated with pcabackest, where the gain settings from Epoch 5C and background conditions from March 2011 were used as template.",
"First, to confirm that our method is obtaining the correct uncertainties in the reflection parameters, we include the uncertainties in $\\log \\xi $ and $R$ for the RXTE simulations in Fig.",
"REF .",
"We find that the reflection uncertainties are consistent with those measured at the peak of the superbursts near $\\log F\\simeq -6.0$ for 4U 1636-536 and $\\log F\\simeq -5.6$ for 4U 1820-30 (corrected for the different exposure times and the number of active PCUs).",
"Thus, we are confident that the plotted uncertainties in the reflection parameters will be an accurate guide for these future instruments.",
"Since RXTE has been the principle instrument for burst analyses for over a decade, it is useful to show how neglecting the presence of reflection will impact the derived $kT$ and $f$ for RXTE observations (Figure REF ).",
"Figure: Similar to Figure for RXTE/PCA.",
"For the other instruments the curves are consistent with Poisson noise, especially for large flux values (e.g., Figure ).",
"RXTE, however, exhibits a shallower decline at high flux.At low flux values, the uncertainties in $kT$ and $f$ are consistent with being scattered around a trend expected for a Poisson distribution.",
"This is similar to our results for NICER, where at high flux the scatter is reduced and the trend is purely described by Poisson noise (Figure REF ).",
"For RXTE, however, the trend appears somewhat less steep than a Poisson trend, and it is similar for fits with the blackbody and the reflection models.",
"The same behavior is present for the reflection parameters (Figure REF ).",
"Although at certain flux values RXTE's uncertainties are similar to NICER's, the scatter is larger.",
"At the largest considered flux, where the statistical quality of the spectra is highest, RXTE exhibits the largest relative uncertainties of all considered observatories: a factor 5 larger for $\\log \\xi $ and a factor 3 larger for $R$ compared to NICER.",
"Interestingly, the relative uncertainty in $kT$ is systematically larger by a factor $\\sim 3$ for the reflection fits compared to the blackbody fits.",
"Given the PCA's band-pass and modest spectral resolution, the broadening of the continuum by reflection may introduce a degeneracy between $kT$ and the reflection parameters.",
"As $kT$ determines the shape of the main spectral component, this also affects the other spectral parameters, including $f$ .",
"Furthermore, the main reflection feature observable by the PCA, the Fe K$\\alpha $ line, is sampled by just a few energy bins.",
"Measurement of the reflection parameters is highly sensitive to statistical noise in those bins, and this may explain why the RXTE simulations exhibit a relatively large scatter in the uncertainties of the reflection parameters.",
"The other instruments sample the reflection signal with a larger number of spectral bins, often including features below 2 keV, resulting in a much better behaved measurement of the reflection parameters.",
"However, the RXTE simulations employ the full array of 5 PCUs, whereas in practice not all were enabled during a particular observation.",
"A typical number of 2 active PCUs reduces the burst flux by $\\Delta \\log F\\simeq -0.4$ .",
"Also, most RXTE burst observations have been analyzed at a shorter time resolution of $0.25\\ \\mathrm {s}$ [24], which is equivalent to a further shift in flux of $\\Delta \\log F\\simeq -0.6$ .",
"Therefore, in practice most RXTE burst analyses are equivalent to the lowest considered flux values, where reflection is not detectable and the behavior of the uncertainty in $kT$ is closest to that of a Poisson distribution.",
"We conclude that neglecting the possibility of reflection in RXTE fits of bursts did not greatly impact the measured temperatures, with the deviations being at most $8\\%$ in our simulations."
],
[
"Constraining the Reflection Geometry with ",
"Contrary to the other considered missions, the band-pass of LOFT precludes it from detecting the reflection signatures below $2\\ \\mathrm {keV}$ .",
"However, its large collecting area enables such a detailed view of the Fe K$\\alpha $ line that LOFT would provide the most precise measurements of the spectral parameters (Figure REF ).",
"We illustrate the detectability of the iron line and edge by showing the residuals of blackbody fits to two reflection spectra with $kT=3\\,\\mathrm {keV}$ and $\\log \\xi =3$ (Figure REF ).",
"Because of the high signal-to-noise ratio, additional parameters can be constrained.",
"We repeat the fits of the reflection model to the simulated LOFT spectra, leaving the inner radius of the reflection site and the inclination angle of the disk free.",
"The relative errors of the two additional free parameters roughly follow a trend as a function of flux that is expected for a Poisson distribution (Figure REF ).",
"For a few second exposure at $\\log F=-7$ , the inner radius of the accretion disk can be crudely constrained to several tens of percent, whereas the disk's inclination can be constrained within a few percent.",
"Figure: Relative error in measurements of the inner radius of the reflection location, r in r_\\mathrm {in}, and the inclination angle of the disk as a function of the flux, FF, for LOFT.",
"Shown are simulations with kT=3 keV kT=3\\,\\mathrm {keV} andlogξ=3\\log \\xi =3.",
"The dot-dashed line illustrates the slope of the trend of a Poisson distribution."
],
[
"Sources with Low Absorption",
"We have employed a relatively high absorption column that is typical for sources in the Galactic plane [45], and therefore the predicted soft features are largely erased by interstellar absorption (Figure REF ).",
"A select few known bursting sources have a smaller absorption column of $N_\\mathrm {H} \\approx 0.1\\times 10^{22}\\ \\mathrm {cm^{-2}}$ .",
"This allows for the detection of emission features in the spectrum below $\\sim 1\\ \\mathrm {keV}$ (Figure REF ), where burst reflection dominates the soft part of the spectrum.",
"For Hitomi, NICER, and Athena, which cover the soft band, we redo the simulations using $N_\\mathrm {H}=0.12\\times 10^{22}\\ \\mathrm {cm^{-2}}$ .",
"This is the absorption column measured for SAX J1808.4-3658 [74].",
"The rest of the spectral model is the same as before.",
"For all three instruments the uncertainties in the reflection parameters are reduced for a given flux compared to the simulation with a larger $N_\\mathrm {H}$ value (Figure REF ).",
"For example, NICER reaches a precision for $\\log \\xi $ of $\\sim 2\\%$ in $1\\ \\mathrm {s}$ and measurements of the reflection fraction are improved as well.",
"This is due to the large number of soft photons below 1 keV that constitute the most prominent part of the reflection signal in this case (Figure REF ).",
"Furthermore, the scatter around the Poisson trend at low flux is reduced with respect to the simulations with higher absorption (Figure REF )."
],
[
"Limitations on Burst Observations due to Pile up",
"A common issue for X-ray detectors is “pile up.” If multiple photons are detected by the same or neighboring pixels within one read-out cycle, they may be recorded as a single photon with an energy that is the sum of the photon energies.",
"This reduces the observed count rates and distorts spectra.",
"For Hitomi SXI in the fastest read-out mode, a burst spectrum with $kT=3.0$ keV and $\\log \\xi =3.0$ is expected to be piled up at a level of 10% for fluxes in excess of $\\gtrsim 2\\times 10^{-9}\\ \\mathrm {erg\\ s^{-1}cm^{-2}}$ [70], which covers the entire flux range that we considered in our simulations.",
"Possibly, this issue could have been mitigated by a faster read-out mode, similar to the “burst” mode of XMM-Newton's EPIC pn instrument.",
"Such a mode was not planned for Hitomi [70].",
"Operation of Athena WFI will include a small window mode that facilitates high count rates of up to $7.8\\times 10^3\\ \\mathrm {c\\ s^{-1}}$ [54].",
"X-ray bursts may, however, reach this limit already for a flux of $\\sim 2\\times 10^{-8}\\ \\mathrm {erg\\ s^{-1}cm^{-2}}$ .",
"This flux is close to the minimum flux required for detecting reflection.",
"For bursts with fluxes below this limit, an integration time of several seconds may be sufficient to collect spectra with detectable reflection features.",
"Conversely, for brighter bursts, the usual strategy of dealing with pile up may be applied: the center of the point spread function where pile up is present, is excluded from the spectra.",
"This reduces the statistical quality of the spectra.",
"However, similar to the weaker bursts, reflection features may still be detected.",
"Moreover, as the launch date of Athena is still far in the future, technical solutions could be found to increase the count rate limit, such as a fast read-out mode.",
"The detectors on board NICER are of a different type, and were specifically designed to handle very high count rates.",
"For $\\log F=-7.0$ , $kT=3.0\\ \\mathrm {keV}$ , and $\\log \\xi =3.0$ , our spectral model predicts a count rate of $7.2 \\times 10^3\\ \\mathrm {c\\ s^{-1}}$ .",
"The signal will be divided among 56 detectors, allowing NICER to easily handle such large rates.",
"Similarly, dead time (the time a detector is inactive when an event is being processed) is negligible for typical bright bursts.",
"Observations with NICER can, therefore, be used to the full extent to detect burst reflection.",
"The same is true for LOFT, which likewise was designed to handle high throughput."
],
[
"Detectability of Burst Reflection",
"Detecting the presence of accretion disk reflection in X-ray burst spectra is challenging because of the short duration of the bursts and the mixture of multiple emission components all contributing to the observed source.",
"We showed that the detectability of reflection features depends only mildly on the blackbody temperature $kT$ , with lower $kT$ being easier to detect as more of the emission falls in the band-passes of most of the modeled instruments.",
"In contrast, the reflection features are significantly easier to detect for ionization parameters $\\log \\xi \\approx 2.5-3$ (Figures REF and REF ) because of the large equivalent width of the Fe K$\\alpha $ line and a strong soft excess.",
"The larger values of the ionization parameter will likely occur near the onset of any burst when the flux on the disk is largest [6], [43].",
"Thus, searches for reflection features in burst spectra will have the best chance for success with observations that are as close to the start of the burst as possible.",
"We investigated the detection of burst reflection with four instruments.",
"For bursts with a flux of $\\log F -7.5$ , NICER will significantly detect the presence of disk reflection in burst spectra, and will provide tight constraints on reflection parameters for 1 s exposures at $\\log F -6$ .",
"Our simulations all employ exposure times of 1 s, but the results are applicable to longer exposures for proportionally lower fluxes (such that the total number of counts is the same).",
"Therefore, for spectra originating from a bright burst of sufficient duration, NICER will be able to also track how the features change in time in response to the burst.",
"Unfortunately, the relatively low effective area of a Hitomi-like telescope means that although it will be able to statistically detect the reflection signal for bursts with $\\log F -7$ , it will be unable to constrain reflection parameters for any realistic burst or superburst.",
"Aside from the instrument that we consider (the SXI), Hitomi also hosted the Soft X-ray Spectrometer (SXS).",
"The SXI and SXS had overlapping band-passes and similar limitations on the photon rate that can be processed.",
"Including the SXS would have doubled the collecting area, and this reduces the flux requirements from our simulations by a factor 2 ($\\Delta \\log F \\simeq 0.3$ ).",
"This is insufficient to substantially improve the detectability of burst reflection with Hitomi.",
"In the next decade, Athena will provide an order-of-magnitude improvement over NICER in detecting reflection features from bursts.",
"With Athena it may be possible to directly constrain reflection features in a 1 s exposure of bright bursts, opening up the possibility of tracing the evolution of the reflection parameters during a burst in detail.",
"An important issue is for an observing mode to be devised for the WFI to handle the large count rates expected from the brightest bursts in order to avoid pile up.",
"Similar to Hitomi, Athena will also include a spectrometer that has a similar energy band as the WFI, and that can be employed to increase the collected number of photons by approximately a factor 2.",
"A LOFT-like mission with a $\\sim 8.5$ m$^{2}$ collecting area would revolutionize this field and be able to perform accretion disk tomography using the Fe K$\\alpha $ line from bright Type I X-ray bursts.",
"This would provide detailed information on the time evolution of the accretion environment of neutron stars under the influence of strong X-ray irradiation."
],
[
"The Soft X-Ray Band and the Influence of Absorption",
"The broad band-passes of the new and upcoming instruments present new opportunities to study the disk reflection signal.",
"Models of the reflection features predict a soft excess in the reflection spectrum due to both a bremsstrahlung continuum and recombination lines [3].",
"Unfortunately, for most known bursting sources the interstellar absorption is large, and the signal in the soft band is substantially reduced.",
"Still, the remaining signal is sufficient to enable the measurement of the reflection parameters during short exposures with NICER and Athena.",
"Conversely, RXTE does not cover the soft band, and only detects the Fe K$\\alpha $ line and absorption edge.",
"During short exposures, we find that RXTE's reflection detection is strongly impacted by statistical noise in the few spectral bins that cover these features, which explains why reflection was only observed clearly in longer exposures for two superbursts.",
"Therefore, coverage of the soft band is crucial for NICER and Athena to measure the reflection properties, even in the presence of strong absorption.",
"The recombination lines below 1 keV carry information on the disk composition and density [3].",
"For a few sources the interstellar absorption is low enough to observe these features in detail: only $2.6\\%$ of known bursts originate from sources with $N_\\mathrm {H}\\le 0.2\\times 10^{22}\\ \\mathrm {cm^{-2}}$ [13], [24].",
"An example of a source with a small absorption column is SAX J1808.4-3658 with $N_\\mathrm {H}=0.12\\times 10^{22}\\ \\mathrm {cm^{-2}}$ [74].",
"A Chandra observation of a burst from this source (with contemporaneous RXTE/PCA coverage) exhibits a soft excess with respect to a blackbody model, which is well fit with a model of reflection off a highly ionized accretion disk [35].",
"However, the quality of the spectra was insufficient to distinguish burst reflection from, e.g., increased persistent emission.",
"We find that observations with future instruments that are sensitive to the soft X-ray band will provide the strongest constraints on the reflection parameters.",
"The recently launched ASTROSAT [61] has an array of Large Area Xenon Proportional Counters (LAXPC) similar to the PCA, as well as a Soft X-ray Telescope (SXT).",
"The effective area at $6.4$ keV of LAXPC is similar to that of RXTE/PCA.",
"The SXT extends ASTROSAT's coverage to lower energies.",
"Its effective area at 1 keV is, however, only $120\\ \\mathrm {cm^2}$ , which is smaller than Chandra.",
"We, therefore, expect that ASTROSAT will only observe reflection in long bursts, similar to RXTE."
],
[
"Applications of Burst Reflection",
"In the same way as X-ray reflection has proved to be invaluable for studying the accretion processes around black holes, we expect burst reflection to provide invaluable insight into accretion physics in the vicinity of neutron stars.",
"Here we discuss several important applications that will be enabled by observations of burst reflection."
],
[
"Mass and Radius Measurements of Neutron Stars",
"In recent years X-ray bursts have been employed to measure the masses and radii of neutron stars in order to constrain the equation of state of dense matter [71], [56], [62], [27], [59].",
"X-ray bursts provide the rare opportunity to determine both quantities simultaneously [50].",
"A crucial issue is the accurate interpretation of burst spectra, as deviations from a blackbody introduce systematic uncertainties.",
"For example, free-free and Compton scattering in the neutron star atmosphere introduce subtle changes to the spectral shape.",
"Detailed spectral models have been created for neutron star atmospheres during X-ray bursts [67], and have been successfully fit to certain burst spectra [66].",
"For other observations, however, the atmosphere models do not reproduce the expected behavior [40], especially in the high-flux soft persistent state.",
"In this spectral state, the accretion disk is thought to extend closest to the neutron star [17], such that the reflection signal is maximal.",
"The reflection spectrum is also a reprocessed blackbody, which deviates more strongly from a blackbody (Figure REF ).",
"Therefore, a substantial contribution of reflection to the burst spectrum may explain those observations that are not well described by burst atmosphere models.",
"By detecting burst reflection, its contribution to the spectrum can be quantified.",
"This provides guidance for mass-radius measurements to select those bursts that are least “contaminated.” Moreover, if the reflection parameters can be sufficiently well determined, the properties of the burst atmosphere and the reflection components may be constrained simultaneously.",
"This will also be important for studies that use the shape of burst light curves to constrain nuclear reactions among short-lived proton-rich isotopes [21], [14], [57]."
],
[
"Evolution of the Accretion Environment",
"For X-ray reflection of coronal emission in AGN and compact binaries, the reflection signal is observed to evolve over time [5], [41].",
"These observations simultaneously probe changes in the corona and in the disk, making it challenging to uncover the evolution of the individual regions.",
"For burst reflection the situation is less complicated, as the neutron star's geometry remains largely constant, with the possible exception of a brief well discernible period of radius expansion.",
"Furthermore, the time evolution of the burst emission from the neutron star can be successfully modeled in great detail [31], [34].",
"Therefore, the evolution of the reflection parameters during X-ray bursts predominantly probes changes in the accretion disk.",
"X-ray bursts provide a repeating experiment to investigate the response of accretion disks to sudden strong irradiation.",
"During the two superbursts that at present provided the only clear detections of burst reflection, the accretion disks were found to be strongly ionized at the onset of the events, and the ionization parameter was reduced during the tail of the bursts [6], [43].",
"For the 1999 superburst from 4U 1820-30 the reflection signal was sufficiently strong to trace the location of reflection: it initially was located further from the neutron star before returning to smaller radii in the tail, suggesting that this superburst initially disrupted the inner disk.",
"In this paper we find that with future instruments, this type of analysis can be performed also for short bursts, which are detected at a thousand times higher rate than superbursts.",
"The reflection fraction, $R$ , is an important parameter for tracing changes in the geometry of the disk.",
"A flat disk produces a value of $R=0.5$ [49], [22].",
"For the two mentioned superbursts, larger values may have been present [6], [43], [44], which could indicate that the inner disk was puffed up due to the burst irradiation [30].",
"X-ray heating of the disk may, however, not be the only important process during bursts.",
"The generation of winds or the inflow of material by Poynting-Robertson drag could play a role as well [4].",
"Further theoretical studies are required to investigate this in more detail, and we expect that burst observations with future instruments will provide constraints for such models."
],
[
"Conclusions",
"Reflection spectroscopy during Type I X-ray bursts holds the promise to open a new avenue for studying accretion physics in the vicinity of neutron stars, by revealing the response of accretion disks to sudden strong irradiation.",
"Furthermore, burst reflection can act as a quantitative measure of which bursts are most suitable for mass-radius determination.",
"Previously, reflection has only been detected during two superbursts.",
"We investigate the detectability of burst reflection using a large set of simulated spectra that are representative of most known bursts.",
"Taking into account the instrumental energy response, effective area, and ability to handle large count rates, we find that future X-ray observatories will be able to detect reflection during the frequent short Type I bursts.",
"Considering all these factors leads us to conclude that NICER provides an excellent opportunity to study the interaction between X-ray bursts and the surrounding accretion disk that will not be significantly surpassed until the launch of Athena.",
"Further in the future, an observatory with a large collecting area similar to the LOFT design would enable studying the accretion processes in unprecedented detail.",
"The authors thank R.E.",
"Rutledge for encouraging to write this paper and T.E.",
"Strohmayer for helpful comments.",
"LK is supported by NASA under award number NNG06EO90A.",
"LK thanks the International Space Science Institute in Bern, Switzerland for hosting an International Team on X-ray bursts."
]
] | 1605.06113 |
[
[
"Some Mathematical Aspects of Price Optimisation"
],
[
"Abstract Calculation of an optimal tariff is a principal challenge for pricing actuaries.",
"In this contribution we are concerned with the renewal insurance business discussing various mathematical aspects of calculation of an optimal renewal tariff.",
"Our motivation comes from two important actuarial tasks, namely a) construction of an optimal renewal tariff subject to business and technical constraints, and b) determination of an optimal allocation of certain premium loadings.",
"We consider both continuous and discrete optimisation and then present several algorithmic sub-optimal solutions.",
"Additionally, we explore some simulation techniques.",
"Several illustrative examples show both the complexity and the importance of the optimisation approach."
],
[
"Introduction",
"Commonly, insurance contracts are priced based on a tariff, here referred to as the market tariff.",
"In mathematical terms such a tariff is a function say $f: \\mathbb {R}^d \\rightarrow [m,M]$ where $m,M$ are the minimal and the maximal premiums.",
"For instance, a motor third party liability (MTPL) market tariff of key insurance market players in Switzerland has $d> 15$ .",
"Typically, the function $f$ is neither linear nor a product of simple functions.",
"In non-life insurance, many insurance companies use different $f$ for new business and renewal business.",
"There are statistical and marketing reasons behind this practice.",
"In this paper we are primarily concerned with non-life renewal business.",
"Yet, some findings are of importance for general pricing of insurance and other non-insurance products.",
"We shall discuss two important actuarial tasks and present various mathematical aspects of relevance for pricing actuaries.",
"Practical actuarial task T1: Given that a portfolio of $N$ policyholders is priced under a given market tariff $f$ , determine an optimal market tariff $f^*$ that will be applied in the next portfolio renewal.",
"Typically, actuarial textbooks are concerned with the calculation of the pure premium, which is determined by applying different statistical and actuarial methods to historical portfolio data, see e.g., [1], [2], [3], [4].",
"The tariff that determines the pure premium of a given insurance contract will be here referred to as the pure risk tariff.",
"In mathematical terms this is a function say $g: \\mathbb {R}^{d_1} \\rightarrow [m_1, M_1]$ with $d_1 \\ge 1$ .",
"In the actuarial practice, pure premiums are loaded, for instance for large claims, provisions, direct expenses and other costs (overheads, profit, etc.).",
"Actuarial mathematics explains various approaches to load premiums; in practice very commonly a linear loading is applied.",
"We shall refer to the function that is utilised for the calculation of the premium of an insurance coverage based on the costs related to that coverage as actuarial tariff; write $g_A:\\mathbb {R}^{d_2} \\rightarrow [m_2, M_2]$ for that function.",
"Practical actuarial task T2: Given a pure risk tariff $g$ , construct an optimal actuarial tariff $g_A$ that includes various premium loadings.",
"Since by definition there is no unique optimal actuarial tariff, the calculations leading to it can be performed depending on the resources of pricing and implementation team.",
"To this end, let us briefly mention an instance which motivates T2: Suppose for simplicity that the portfolio in question consists of two groups of policyholders A and B.",
"In group A there are $n_A$ policyholders and in group $B$ there are already $n_B$ policyholders.",
"All the contracts are to be renewed at the next 1st January.",
"The pricing actuary calculates the actuarial tariff which shows that for group A, the yearly premium to be paid from each policyholder is 2'000 CHF and for group B, 500 CHF.",
"For this portfolio, overhead expenses (not directly allocated to an insurance policy) are calculated (estimated) to be X CHF for the next insurance period (one year in this case).",
"The amount $X$ can be distributed to $N=n_A+n_B$ policyholders in different ways, for instance each policyholder will have to pay $X/(n_A+n_B)$ of those expenses.",
"Another alternative approach could be to calculate it as a fix percentage of the pure premiums.",
"The principal challenge for pricing actuaries is that the policyholders already are in the portfolio and know their current premiums.",
"At renewal (abbreviated as $@ \\mathcal {R} \\ $ in the following) given that the risk does not change, if the new offered premium is different from the current one, the policyholder can cancel the contract.",
"Another reason for cancelling the policy could also be the competition in the insurance market.",
"Consequently, the solution of both T1 and T2 needs to take into account the probability of cancellation of the policies at the point of renewal.",
"Both T1 and T2 are in general very difficult to solve.",
"A simpler problem in renewal pricing is the following: Practical actuarial task T3: Modify for any $i\\le N$ the premium $P_i$ of the $i$ th policyholder $@ \\mathcal {R} \\ $ by a fixed percentage, say $\\delta _i$ with $\\delta _i \\in \\Delta _i=\\lbrace 0\\%, 5\\%\\rbrace $ so that the new set of premiums $P_i^*= P_i+ \\tau _i, \\quad 1 \\le i\\le N \\text{ with } \\tau _i= P_i\\delta _i$ are optimal.",
"Moreover, determine the new market tariff $f^*$ which yields $P_i^*$ 's.",
"Indeed, the actuarial task described in T3 is very common in actuarial practice, if the actual performance of the portfolio is not as expected, and a premium increase is to be applied at the next renewal.",
"There are several difficulties related to the solutions of tasks T1-T3.",
"In practice the market tariff is very complex for key insurance coverages such as motor or household insurance.",
"A typical $f$ used in practice is as follows (consider only two arguments for simplicity) $ f(x,y)= \\min \\Bigl (M_0, \\max ( e^{a x+ bx},m_0 + m_1 x+ m_2 y) \\Bigr ).",
"$ Even if we know the $P_i^*$ 's that solve T3, when the structure of $f$ (and also of $f^*$ ) is fixed say as in (REF ), then the existence of an optimal $f^*$ that gives exactly $P_i^*$ 's is in general not guaranteed.",
"Note that due to technical reasons, the actuaries can change the coefficients that determine $f$ , say $a,b$ and so on, but the structure of the tariff, i.e., the form of $f$ in (REF ) is in general fixed when preparing a new renewal tariff due to huge implementation costs.",
"The main goal of this contribution is to discuss various mathematical aspects that lead to optimal solutions of the actuarial tasks T1-T3.",
"Further we analyse eventual implementations of our optimisation problems for renewal business.",
"Optimisation problems related to new business are much more involved and will therefore be treated in a forthcoming contribution.",
"To this end, we observe that in the last 10 years many insurance companies in Europe have already used price optimisation techniques (mostly through consultancy companies).",
"So far in the literature, there is no precise mathematical description of the optimisation problems solved in such applications.",
"Very recent contributions focus on the issues of price optimisation, mainly from the ethical and regulation points of view, see [5], [6].",
"It is important to note that optimality issues in insurance and reinsurance business, not directly related to the problems treated in this contribution, have been discussed in various context, see [7], [8], [9], [10], [11], [12], [13] and the references therein.",
"Brief organisation of the rest of the paper: Section 2 describes the different optimisation settings from the insurer's point of view.",
"In Section 3, we provide partial solutions for problem T1.",
"Section 4 describes the different algorithms used to solve the optimisation problems followed by some insurance applications to the motor line of business presented in Section 5."
],
[
"Theoretical Settings",
"For simplicity, and without loss of generality, we shall assume that the renewal time is fixed for all $i=1, \\ldots , N$ policyholders already insured in the portfolio with the $i$ th policyholder paying $P_i$ for the current insurance period.",
"Each policyholder can be insured for different insurance periods.",
"Without loss of generality, we shall suppose that at renewal each insurance contract has the option to be renewed for say one year, with a renewal premium of $P_i+ \\tau _i $ .",
"Suppose that the cancellation probability for the $i$ th contract is a function of $P_i$ .",
"At renewal, by changing the premium, the cancellation probability will depend on the premium change, say $\\tau _i$ and the initial premium $P_i$ .",
"Therefore we shall assume that this probability is given by $ \\pi _i(P_i+ \\tau _i)= \\Psi _i(P_i, \\delta _i), \\text{ with } \\tau _i=P_i\\delta _i, $ where $\\Psi _i$ is a strictly positive monotone function depending eventually on $i$ .",
"This is a common assumption in logistic regression, where $\\Psi _i$ is the inverse of the logit function (called also expit), or $\\Psi _i$ is a univariate distribution function.",
"In order to consider the cancellation probabilities in the tariff and premium optimisation tasks, the actuary needs to know/determine $\\pi _i(P_i + \\tau _i)$ for any $\\delta _i \\in \\Delta _i$ , where $\\Delta _i$ is the range of possible changes of premium with $0 \\in \\Delta _i$ .",
"Estimation of $\\pi _i$ 's is difficult, and can be handled for instance using logistic regression, see Section REF below for more details.",
"In practice, depending on the market position and the strategy of the company, different objective functions can be used for the determination of an optimal actuarial tariff or market tariff.",
"We discuss below two important objective functions: O1) Maximise the future expected premium volume $@ \\mathcal {R}$: In our model, the current premium volume for the portfolio in question is $V=\\sum _{i=1}^N P_i$ , whereas the premium volume in case of complete renewal is $\\sum _{i=1}^N P_i^*= \\sum _{i=1}^N (P_i+ \\tau _i).$ Since not all policies might renew, let us denote by $ N_{@ \\mathcal {R}}$ the number of policies which will be renewed.",
"Since we can treat each contract as an independent risk, then $ N_{@ \\mathcal {R}}= \\sum _{i=1}^N I_i,$ with $I_1 , \\ldots ,I_N$ independent Bernoulli random variables with $\\mathbb {P} \\left\\lbrace I_i=1 \\right\\rbrace = \\pi _i(P_i+ \\tau _i), \\quad 1 \\le i\\le N.$ Clearly, the expected percentage of the portfolio to renew is given by $ \\theta (\\tau _1 , \\ldots ,\\tau _N)= \\frac{\\mathbb {E}\\left\\lbrace N_{@ \\mathcal {R}} \\right\\rbrace }{N}= \\sum _{i=1}^N \\frac{\\mathbb {E}\\left\\lbrace I_i \\right\\rbrace }{N} =\\frac{1}{N} \\sum _{i=1}^N \\pi _i(P_i+ \\tau _i).",
"$ The premium volume $@ \\mathcal {R} \\ $ (which is random) will be denoted by $ \\mathcal {V}_{@\\mathcal {R}} $ .",
"It is simply given by $ \\mathcal {V}_{@\\mathcal {R}} :=\\sum _{i=1}^N I_i (P_i+ \\tau _i).$ Consequently, the objective function is given by (set below $\\tau =(\\tau _1 , \\ldots ,\\tau _N)$ ) $ q_{vol} (\\tau ):= \\mathbb {E}\\left\\lbrace \\mathcal {V}_{@\\mathcal {R}} \\right\\rbrace = \\sum _{i=1}^N (P_i+ \\tau _i) \\mathbb {E}\\left\\lbrace I_i \\right\\rbrace =\\sum _{i=1}^N (P_i+ \\tau _i) \\pi _i(P_i+ \\tau _i).",
"$ Note that $P_1 , \\ldots ,P_N$ are known, therefore the optimisation will be performed with respect to $\\tau _i$ 's only.",
"O1') Minimise the variance of $ \\mathcal {V}_{@\\mathcal {R}} $: If the variance of $ \\mathcal {V}_{@\\mathcal {R}} $ is large, the whole renewal process can be ruined.",
"Therefore along O1 the minimisation of the variance of $ \\mathcal {V}_{@\\mathcal {R}} $ is important.",
"In this model we have $ q_{var}(\\tau ):=Var( \\mathcal {V}_{@\\mathcal {R}} )&=& \\sum _{i=1}^N(P_i+ \\tau _i) \\pi _i(P_i+ \\tau _i)[1- \\pi _i(P_i+ \\tau _i)] .",
"$ O2) Maximise the expected premium difference $@ \\mathcal {R}$: The premium difference for each policyholder in our notation is $\\tau _i$ and thus at renewal we have $\\sum _{i=1}^N I_i \\tau _i$ .",
"The expectation of this random variable is simply $ q_{dif}(\\tau )= \\mathbb {E}\\left\\lbrace \\sum _{i=1}^N I_i \\tau _i \\right\\rbrace = \\sum _{i=1}^N \\tau _i \\pi _i(P_i+ \\tau _i).",
"$ It is not difficult to formulate other objective functions, for instance related to the classical ruin probability, Parisian ruin, or future solvency and market position of the insurance company.",
"Moreover, the objective functions can be formulated over multiple insurance periods.",
"Due to the nature of insurance business, there are several constraints that should be taken into account, see for instance [14] and the references therein.",
"Typically, the most important business contraints relate to the strategy of the company and the concrete insurance market.",
"We formulate few important constraints below: C1) Expected retention level $@ \\mathcal {R} \\ $ should be bounded from below: Although the profit and the volume of premiums at renewal are important, all insurance companies are interested in keeping most of the policyholders in their portfolio.",
"Therefore there is commonly a lower bound on the expected retention level $\\ell \\in [0.7, 1]$ at renewal.",
"For instance $\\ell = 90\\%$ means that the expected percentage of customers that will not renew their contrat should not exceed 10%.",
"In mathematical terms, this is formulated as $ \\theta _{rlevel}(\\tau )= \\frac{\\mathbb {E}\\left\\lbrace N^* \\right\\rbrace }{N} \\ge \\ell .",
"$ C2) A simple constraint is to require that the renewal premiums $P_i^*$ 's are not too different from the \"old\" ones, i.e., $ \\frac{\\tau _i }{P_i} \\in [a,b], \\quad \\tau _i \\in [A,B], \\quad 1 \\le i\\le N $ for instance $a=-5\\%$ and $b=10\\%$ and $A= -50, B=300$ .",
"Several other constraints including those related to reputational risk, decrease of provision level for tied-agents, and loss of loyal customers can be formulated similarly and will therefore not be treated in detail."
],
[
"Practical Settings",
"In insurance practice the cost of optimisation itself (actuarial and other resources) needs to be also taken into account.",
"Additionally, since the total volume of premium at renewal is large, an optimal renewal tariff is of interest if it produces a significant improvement to the current tariff.",
"Therefore, for practical implementations, we need to redefine the objective functions.",
"For a given positive constant $c$ , say $c=1^{\\prime }000$ , we redefine (REF ) as $ q_{vol}^c (\\tau _1 , \\ldots ,\\tau _N):= c \\Bigl \\lfloor \\mathbb {E}\\left\\lbrace \\mathcal {V}_{@\\mathcal {R}} \\right\\rbrace /c \\Bigl \\rfloor =c \\Bigl \\lfloor \\sum _{i=1}^N (P_i+ \\tau _i) \\pi _i(P_i+ \\tau _i)/c \\Bigl \\rfloor , $ where $\\lfloor x \\rfloor $ denotes the largest integer smaller than $x$ .",
"Similarly, we redefine (REF ) as $ c \\Bigl \\lfloor Var( \\mathcal {V}_{@\\mathcal {R}} )/ c \\Bigl \\rfloor &=& c \\Bigl \\lfloor \\sum _{i=1}^N(P_i+ \\tau _i) \\pi _i(P_i+ \\tau _i)[1- \\pi _i(P_i+ \\tau _i)] / c \\Bigl \\rfloor .",
"$ Finally, (REF ) can be written as $ q^c_{dif}(\\tau _1 , \\ldots ,\\tau _N)= c \\Bigl \\lfloor \\sum _{i=1}^N \\tau _i \\pi _i(P_i+ \\tau _i)/ c \\Bigl \\rfloor .",
"$ For implementation purposes and due to business constrains, $\\tau _i$ 's can be assumed to be certain given numbers.",
"Therefore a modification of (REF ) can be as follows $ \\delta _i:= \\frac{\\tau _i }{P_i} \\in [a,b] \\cap (c_1^{-1} \\mathbb {Z}) , \\quad \\tau _i \\in [A,B] \\cap (c_1 \\mathbb {Z}), \\quad 1 \\le i\\le N, $ where $c_1>0$ , for instance $c_1=100$ .",
"Such modifications of both objective functions and constraints show that for practical implementation, there is no unique optimal solution of the optimisation problem of interest.",
"Remarks 2.1 i) In this contribution we are not directly concerned with distributional channels.",
"For example, if two policies say the $i$ th and the $k$ th ones are renewed through different distributional channels, then perhaps different contraints are to be applied to each of those policies.",
"Additionally, the cancellation probabilities could be different, even in the case where both policyholders have the same risk profile.",
"Therefore, in order to allow for different distributional channels, we only need to adjust the constraints and assume an appropriate cancellation pattern.",
"ii) From the practical point of view, $\\Psi _i$ 's are estimated by using for instance logistic regression.",
"At random, customers are offered higher/lower premiums than their $P_i$ 's at renewal, i.e., $\\tau _i$ 's are chosen randomly with respect to some prescribed distribution function.",
"An application of the logistic regression to the data obtained (renewal/non renewal) explains the cancellation (or renewal) probability in terms of risk factors as well as other predictors (social status, etc.)",
"In an insurance market dominated by tied-agents this approach is quite difficult to apply.",
"iii) Different policyholders can renew for different periods.",
"This case is included in our assumptions above.",
"iv) Most tariffs, like say a MTPL one, consist of hundreds of coefficients (typically more than 400).",
"Due to a dominating product-structure, advanced tariffs contains many individual cells, say 200'000 in average.",
"However, most of these tariff cells are empty.",
"For instance, it is quite rare that a Ferrari is insured for a TPL risk by a 90 years old lady, living in a very small village.",
"With this in mind, typically, the relevant number $N$ in practical optimisation problems does not exceed 50'000.",
"Our algorithms and simulation methods work fairly well for such $N$ ."
],
[
"Solutions for ",
"The chief difficulty when dealing with the actuarial tasks T1-T3 lies on the complexity of $\\Psi _i$ 's since these functions are: a) in general not known, b) difficult to estimate if past data are partially available, c) even when these functions are known, the constraints C1-C2 and the objective functions O1,O1',O2 are in general not convex.",
"We discuss next a partial solution for T1: Problem T1a: Given $P_1 , \\ldots ,P_N$ , determine $\\tau ^*=(\\tau _1^* , \\ldots ,\\tau _N^*)$ such that $ q_{vol}(\\tau ^*) \\text{ is maximal}, \\quad q_{var}(\\tau ^*) \\text{ is minimal} $ under the constraints $ \\theta _{rlevel}(\\tau ) \\ge \\ell , \\quad \\delta _i= \\frac{\\tau _i}{P_i} \\in [a,b], \\quad 1\\le i \\le N. $ Problem T1b: Determine $f^*$ from $P_1^* , \\ldots ,P_N^*$ .",
"The solution (an approximate one) of T1b can be easily derived.",
"Given $P_1^* , \\ldots ,P_N^*$ , and since the structure of the market tariff $f$ is known, then $f^* $ can be determined (approximately), by running a non-linear regression analysis.",
"Therefore, below we focus on T1a.",
"Consequently, the main question that we shall discuss here is how to determine optimal premiums $P_i^*$ 's at renewal.",
"In insurance practice, the functions $\\Psi _i,i\\le n$ can be assumed to be piece-wise linear and non-decreasing.",
"This assumption is indeed reasonable, since for very small $\\tau _i$ or $\\delta _i$ , the policyholder will not be aware of premium changes.",
"The latter assumption can be violated if for instance at renewal the competition modifies also their new business premiums.",
"For simplicity, these cases will be excluded in our analysis, and thus we assume that the decision for accepting the renewal offer is not influenced by the competitors.",
"We list below some tractable choices for $\\Psi _i$ 's: Ma) Suppose that for given known constants $\\pi _i, a_i,b_i$ $ \\Psi _i(P_i, \\delta _i)= \\pi _i (1+ a_i \\delta _i+ b_i \\delta _i^2), \\quad 1 \\le i\\le N. $ In practice, $\\pi _i, a_i,b_i$ need to be estimated.",
"Clearly, the case that $b_i^{\\prime }$ s are equal to 0 is quite simple and tractable.",
"Note in passing that an extension of the above model is by allowing $a_i$ and $b_i$ to differ depending on the sign of $\\delta _i$ .",
"Mb) One choice motivated by the logistic regression model commonly used for estimation of cancellation probabilities is the expit function, i.e., $ \\Psi _i(P_i, \\delta _i)= \\frac{1}{1+ c_i^{-1} e^{- T_i \\delta _i }}, \\quad 1 \\le i \\le N, $ where $c_i,T_i$ 's are known constants (to be estimated in applications).",
"We note that Model Ma) can be seen as an approximation of Model Mb).",
"Mc) A simple specification is when $\\Psi $ is given only for few values of $\\delta _i$ 's as follows.",
"n illustrative data of a policyholder $i$ are presented in REF Table: NO_CAPTION tableRenewal probabilities as a function of premiums for the $i$ th policyholder.",
"The Model Ma) is simple and tractable and it can be seen as an approximate model of a more complex one.",
"Moreover, it leads to some crucial simplification of the objective functions in question.",
"Toy Model 1: We suppose that $P_1=P_2= \\cdots = P_N$ .",
"This leads to a simplification of the objective function.",
"Toy Model 2: Assume that $\\Psi _i(P,\\delta )= \\Psi (P,\\delta )$ for any $i\\le N$ , i.e., we assume that all policyholders have the same behavior with respect to cancellation probability."
],
[
"O1) Maximise the future expected premium volume $@ \\mathcal {R}$",
"In this section, we denote by $t=0$ the present time and by $t=1$ the time at renewal.",
"We consider the case where the insurer would like to maximise the future expected premium volume while simultaneously keeping a minimum number of policyholders in his portfolio at $t=1$ .",
"Therefore, we denote by $\\ell $ the retention level.",
"$ N \\ell $ is just the minimum number of policyholders that the insurer would like to keep in his portfolio $@ \\mathcal {R}$ .",
"In this respect, the optimisation problem can be formulated as follows $\\begin{aligned}&\\underset{\\delta }{ \\text{max }} \\sum _ {i=1}^{N} P _i (1 + \\delta _{i}) \\Psi _i(P_i,\\delta _i),\\\\&\\text{subject to } \\frac{1}{N}\\sum _ {i=1}^{N} \\Psi _i(P_i,\\delta _i)\\geqslant \\ell .",
"\\\\\\end{aligned}$"
],
[
"Probability of renewal $\\Psi _i$ as in ",
"We consider the case where the probability of renewal $\\Psi _i$ is of the form $\\Psi _i:= \\Psi _i(P_i,\\delta _i)= \\pi _i (1+a_i \\delta _i+ b_i \\delta _i^2)$ as defined in Ma).",
"Setting $b_i=0$ , we have $ \\Psi _i:=\\Psi _i(P_i,\\delta _i)= \\pi _i (1+a_i \\delta _i),$ where the condition $\\Psi _i \\in (0,1)$ should hold for all policyholders $i\\le N$ .",
"Actually, it is satisfied when $a_i \\in (1-\\frac{1}{\\pi _i},\\frac{1}{\\pi _i}-1)$ for all $i \\le N$ .",
"Since $b_i=0$ , then (REF ) is just a quadratic programming (QP) problem subject to linear constraints and can be rewritten as follows $\\begin{aligned}&\\underset{\\delta }{\\text{max }} \\sum _ {i=1}^{N} P _i \\pi _i (1 + (1+a_i)\\delta _{i}+ a_i\\delta _i^2),\\\\&\\text{subject to } \\frac{1}{N}\\sum _ {i=1}^{N} \\pi _i(1+a_i \\delta _i) \\geqslant \\ell .\\\\\\end{aligned}$ (REF ) has a maximum if and only if its objective function is concave.",
"However, this is satisfied when $a_i < 0$ .",
"Thus, we assume that $a_i \\in (1-\\frac{1}{\\pi _i},0)$ for any $i \\le N$ .",
"In order to solve (REF ), we use the quadratic programming method summarised in Appendix A. Hereafter we shall assume that $b_i \\ne 0$ implying that $\\Psi _i$ is of the form $\\Psi _i:= \\Psi _i(P_i,\\delta _i)= \\pi _i (1+a_i \\delta _i+ b_i \\delta _i^2).$ We have that $\\Psi _i \\in (0,1)$ holds if and only if $a_i$ and $b_i$ satisfy the following conditions ${\\left\\lbrace \\begin{array}{ll}a_i \\in \\Bigl (\\text{max }(1-\\frac{1}{\\pi _i},-1-b_i), \\text{min }(1+b_i,\\frac{1}{\\pi _i}-1)\\Bigr ),\\\\b_i \\in (-1,0).\\end{array}\\right.",
"}$ Moreover, (REF ) can be rewritten as $\\begin{aligned}& \\underset{\\delta }{\\text{max }} \\sum _ {i=1}^{N} P _i \\pi _i (1 + (1+a_i)\\delta _{i}+ (a_i+b_i)\\delta _i^2+b_i \\delta _i^3),\\\\&\\text{subject to }\\frac{1}{N}\\sum _ {i=1}^{N} \\pi _i(1+a_i \\delta _i+b_i \\delta _i^2) \\geqslant \\ell .\\\\\\end{aligned}$ Clearly, (REF ) is a non-linear optimisation problem subject to non-linear constraints.",
"The most popular method discussed in the literature for solving this type of optimisation problem is the Sequential Quadratic Programming method (SQP), see [15], [16], [17].",
"It is an iterative method that generates a sequence of quadratic programs to be solved at each iterate.",
"Typically, at a given iterate $x_k$ , (REF ) is modelled by a QP subproblem subject to linear constraints and the solution to the latter is used as a search direction to construct a new iterate $x_{k+1}$ .",
"The optimisation problem at hand in this case is given by $\\begin{aligned}&\\underset{\\delta }{\\text{min }} f(\\delta )= - \\sum _ {i=1}^{N} P _i \\pi _i (1 + (1+a_i)\\delta _{i}+ (a_i+b_i)\\delta _i^2+b_i \\delta _i^3),\\\\&\\text{subject to } g(\\delta )= - \\sum _ {i=1}^{N} \\pi _i(1+a_i \\delta _i+b_i \\delta _i^2)+ N \\ell \\le 0,\\end{aligned}$ where $f$ and $g$ are continuous and twice differentiable.",
"See Appendix B for the main steps required to solve (REF ).",
"Probability of renewal $\\Psi _i$ as in Mb) We consider the case Mb) where the renewal probability is determined by the logistic regression model and is given by $ \\Psi _i:=\\Psi _i(P_i, \\delta _i)= \\frac{1}{1+ c_i^{-1} e^{- T_i \\delta _i }}, \\quad 1 \\le i \\le N. $ $c_i$ is a constant that depends on the probability of renewal before premium change, $\\pi _i$ , and is given by $c_i=\\frac{\\pi _i}{1-\\pi _i},$ and $T_i < 0$ is a constant (to be estimated in applications) that measures the elasticity of the policyholder relative to premium change.",
"The greater $|T_i|$ , the more elastic the policyholder is to premium change.",
"In this regard, the optimisation problem can be formulated as follows $\\begin{aligned}& \\underset{\\delta }{\\text{max }} \\sum _ {i=1}^{N} P _i (1 + \\delta _{i})/(1+e^{-T_i \\delta _i}/c_i),\\\\&\\text{subject to } \\frac{1}{N}\\sum _ {i=1}^{N} 1/(1+e^{-T_i \\delta _i}/{c_i}) \\geqslant \\ell .\\end{aligned}$ (REF ) is a non-linear optimisation problem subject to non-linear constraints.",
"Therefore, in order to find the optimal solution $\\delta $ , we use the SQP algorithm described in Appendix B.",
"Remarks 4.1 It should be noted that Mb) can be approximated by Ma) when the range of $\\delta _i$ is close to 0.",
"In this case, $ \\Psi _i(P_i,\\delta _i)=\\frac{c_i}{1+c_i}\\Bigl (1+\\frac{c_i T_i}{1+c_i} \\delta _i-\\frac{T_i^2 (c_i-1)}{2(1+c_i)^2} \\delta _i^2\\Bigr ),$ where $\\begin{aligned}& \\pi _i=\\frac{c_i}{1+c_i} ,& a_i=\\frac{c_i T_i}{1+c_i} \\text{, }& \\text{ } b_i=-\\frac{T_i^2 (c_i-1)}{2(1+c_i)^2}.\\end{aligned}$ Probability of renewal $\\Psi _i$ as in Mc) Finally, we consider the case Mc) where the optimal solutions $\\delta _i$ take its values from a discrete set.",
"Also, the probabilities of renewal $\\Psi _i$ at time 1 are fixed for each insured $i$ based on $\\delta _i$ for $i \\le N$ , as defined in Table REF .",
"In this section, we deal with a Mixed Discrete Non-Linear Programming (MDNLP) optimisation problem.",
"In this regard, we consider the discrete set $\\mathbf {D}= \\lbrace -20\\%,-15\\%,-10\\%,-5\\%,0\\%,5\\%,10\\%,15\\%,20\\%\\rbrace $ which corresponds to the optimal values that $\\delta _i$ can take.",
"Thus, (REF ) can be reformulated as follows $\\begin{aligned}&\\underset{\\delta }{ \\text{min }} f(\\delta )=- \\sum _ {i=1}^{N} P _i (1 + \\delta _{i,j}) \\Psi _{i,j}(P_i,\\delta _{i,j}),\\\\&\\text{subject to } g(\\delta )=- \\sum _ {i=1}^{N} \\Psi _{i,j}(P_i,\\delta _{i,j}) + N \\ell \\leqslant 0 \\text{ for } j=1,\\ldots ,9, \\\\& \\text{and }\\delta \\in \\mathbf {D}^N.\\end{aligned}$ In general, this type of optimisation problem is very difficult to solve due to the fact that the discrete space is non-convex.",
"Several methods were discussed in the literature for the resolution of (REF ), see [18].",
"The contribution [19] proposed a new method for solving the MDNLP optimisation problem subject to non-linear constraints.",
"It consists in approximating the original non-linear model by a sequence of mixed discrete linear problems evaluated at each point iterate $\\delta _k$ .",
"Also, a new method for solving a MDNLP was introduced by using a penalty function, see the recent contribution [20], [21] for more details.",
"The algorithm for solving this type of optimisation problem is described in Appendix C. Maximisation of the retention level $@ \\mathcal {R}$ We consider the case where the insurer would like to keep the maximum number of policyholders in the portfolio $@ \\mathcal {R}$ .",
"Therefore, the optimisation problem of interest consists in finding the optimal retention level $@ \\mathcal {R} \\ $ whilst increasing the expected premium volume by an amount say $C$ in the portfolio at time 1.",
"Hence, the optimisation problem can be formulated as such $\\begin{aligned}&\\underset{\\delta }{ \\text{max }} \\frac{1}{N} \\sum _ {i=1}^{N} \\Psi _i(P_i,\\delta _i),\\\\&\\text{subject to } \\mathbb {E}(P^{*}) \\geqslant \\mathbb {E}(P)+ C ,\\\\\\end{aligned}$ where $\\mathbb {E}(P^*)=\\sum _ {i=1}^{N} P_i(1+\\delta _i)\\Psi _i(P_i,\\delta _i)$ is the expected premium volume $@ \\mathcal {R}$ , $\\mathbb {E}(P)=\\sum _ {i=1}^{N}P_i \\pi _i$ is the expected premium volume at time 0, and $C$ is a fixed amount which can be expressed as a percentage of the expected premium volume at time 0; it represents a loading to increase the premium volume at renewal.",
"In order to solve (REF ), we use the SQP algorithm described in Appendix B.",
"Insurance Applications In this section, we consider a dataset that describes the production of the motor line of business of an insurance portfolio.",
"We assume that the premiums are exponentially increasing.",
"Also, the probability of renewal at time 0, $\\pi _i$ for $i=1,\\ldots ,N$ , are known and estimated by the insurance company for each category of policyholders based on historical data.",
"Given that the behavior of the policyholders is unknown at the time of renewal, the probability of renewal at time 1, $\\Psi _i$ , depends on $\\pi _i$ and $\\delta _i$ for $i=1,\\ldots ,N$ .",
"If $\\delta _i$ is positive, then $\\Psi _i$ decreases whereas if $\\delta _i$ is negative, it is more likely that the policyholder will renew his insurance policy at time 1, thus generating a greater $\\Psi _i$ .",
"In the following paragraphs, we are going to present some results relative to the optimisation problems at hand described in the last section.",
"Optimisation problem Ma) Maximise the expected premium volume $@ \\mathcal {R}$ .",
"We consider, first, the optimisation problem defined in (REF ).",
"In this case, the probability of renewal $\\Psi _i$ is defined in Ma) and set $b_i=0$ for $i=1,\\ldots ,N$ .",
"Given that $a_i < 0 \\text{ for } i\\le N $ , the probability of renewal $\\Psi _i$ increases when $\\delta _i$ is negative and decreases when $\\delta _i$ is positive, thus describing perfectly the behavior of the policyholders that are subject to a decrease, respectively increase, in their premiums $@ \\mathcal {R}$ .",
"The table below describes some statistics on the data for $10^{\\prime }000$ policyholders.",
"Table: NO_CAPTION tableProduction statistics for the motor business.",
"We consider that the insurance company would like to keep 85% of its policyholders in its portfolio $@ \\mathcal {R}$ .",
"By solving (REF ) in Matlab with the function quadprog, we obtain the optimal $\\delta $ for each policyholder.",
"Next, we consider two scenarios: Scenario 1 The insurer would like to keep 75% of the policyholders in his portfolio $@ \\mathcal {R}$ , Scenario 2 The insurer would like to keep 85% of the policyholders in his portfolio $@ \\mathcal {R}$ .",
"Table REF below summarises the optimal results when solving (REF ) and examines the effect of both scenarios on the expected premium volume and the expected number of policyholders in the portfolio $@ \\mathcal {R}$ .",
"Table: NO_CAPTION tableScenarios testing.",
"Scenario 1 The optimal $\\delta $ for both bounds corresponds approximately to the maximum value (upper bound) of the interval.",
"This is mainly due to the fact that the insurer would like to keep only 75% of the portfolio $@ \\mathcal {R}$ .",
"Therefore, his main goal is to maximise the expected premium volume at time 1.",
"Scenario 2 For a retention level of 85%, Table REF shows an increase in the expected premium volume which is less important than the one observed in Scenario 1.",
"However, the expected number of policyholders in the portfolio $@ \\mathcal {R} \\ $ is higher and is approximately the same as at t=0.",
"Hereafter, we shall consider a retention level of 85%.",
"Usually, in practice, the size of a motor insurance portfolio exceeds $10^{\\prime }000$ policyholders.",
"However, solving the optimisation problems for $\\delta $ using the described algorithms when $N$ is large requires a lot of time and heavy computation and may be costly for the insurance company.",
"Thus, an idea to overcome this problem is to split the original portfolio into sub-portfolios and compute the optimal $\\delta $ for the sub-portfolios.",
"One criteria that can be taken into account for the split is the amount of premium in our case.",
"However, in practice, insurance companies have a more detailed data, thus more information on each policyholders, so the criterion that are of interest for the split are the age of the policyholders, the car brand, car value ...Table REF and Table REF below describe the results when splitting the original portfolio into 3 and 4 sub-portfolios respectively.",
"Table: NO_CAPTION tableSplit into 3 sub-portfolios.",
"Table: NO_CAPTION tableSplit into 4 sub-portfolios.",
"In Table REF and REF , we consider that the insurer would like to keep 85% of the policyholders in each sub-portfolios, thus a total of 85% of the original portfolio.",
"However, in practice, the constraints on the retention level $@ \\mathcal {R} \\ $ are specific to each sub-portfolio and this depending on the insurer's decision whether he would like to keep the policies with large premium amounts or small premium amounts in his portfolio $@ \\mathcal {R}$ .",
"In this regard, the insurance company sets the constraints on the expected number of policies for each sub-portfolios so that the constraint of the overall portfolio is approximately equal to 85%.",
"The error from the split into 3, respectively 4 sub-portfolios is relatively small and is of -0.13%, respectively -0.23% for the expected premium volume $@ \\mathcal {R}$ .",
"It should be noted that the error margin increases as the number of splits increases.",
"Remarks 5.1 In the following sections, we limit the size of the insurance portfolio to 1'000 policyholders as the algorithms used thereafter to solve the optimisation problems are based on an iterative process and requires a lot of computation and time.",
"Hence, an idea to solve the optimisation problem for an insurance portfolio of size $n$ with $n\\ge 1^{\\prime }000$ is to split the original portfolio into sub-portfolios and compute the optimal results for the sub-portfolios, as discussed in the previous Section.",
"Maximise the premium volume and minimise the variance of the premium volume Similarly to the asset allocation optimisation problem in finance introduced by Markowitz [22], the insurer performs a trade-off between the maximum aggregate expected premiums and the minimum variance of the total earned premiums; see also [23] for a different optimality criteria.",
"We show next in Table REF the optimal results for the different constraints on the retention level and the possible range of premium changes.",
"Table: NO_CAPTION tableVolume and variance objective optimal results.",
"It can be seen that the optimal variance $@ \\mathcal {R} \\ $ increases with the range of the possible premium changes $\\delta $ .",
"For instance when the insurer would like to keep $75\\%$ of the policyholders, the variance $@ \\mathcal {R} \\ $ increases from $109.76$ for $\\delta \\in (-10\\%, 20\\% ) $ to $113.08$ for $\\delta \\in (-20\\%, 30\\% ) $ , respectively.",
"Furthermore, the increase in variance $@ \\mathcal {R} \\ $ is associated with an increase of the expected volume $@ \\mathcal {R}$ .",
"This means that the riskier the portfolio the more the insurance company earns premiums.",
"Maximise the retention level $@ \\mathcal {R}$ .",
"We consider here that the insurer would like to maximise his retention level whilst increasing the expected premium volume $@ \\mathcal {R} \\ $ by a certain amount $C$ needed to cover, for instance, the operating costs and other expenses of the insurance company.",
"$C$ can be expressed as a loading on the expected premium volume at time 0.",
"In practice, the amount $C$ needed to cover the expenses of the company is set by the insurers.",
"As stated previously, $C$ can be expressed as a percentage of the expected premium volume at time 0.",
"Therefore, we consider three different loadings: 9%, 10% and 11% thus adding an amount of $85^{\\prime }000$ , respectively $95^{\\prime }000$ and $105^{\\prime }000$ to the expected premium volume at time 0.",
"Also, we consider two ranges for $\\delta $ , $\\delta \\in (-10\\%,-20\\%)$ and $\\delta \\in (-20\\%,-30\\%)$ .",
"Table: NO_CAPTION tableScenario testing - Retention Table REF shows that when $C$ increases, the expected number of policyholders $@ \\mathcal {R} \\ $ decreases whereas the average optimal $\\delta $ increases.",
"Optimisation problem Mb) We consider the optimisation problem defined in (REF ) where the probability of renewal $\\Psi _i$ is defined in Mb).",
"As discussed in Section REF , $T_i$ describes the behavior of the policyholders subject to premium change.",
"For instance, let's consider a policyholder whose probability of renewal without premium change $\\pi _i$ is 0.95.",
"In this Section, we will only consider the case where the insurer would like to maximise the expected premium volume $@ \\mathcal {R}$ .",
"The constraint on the retention level is assumed to be of 85%.",
"As stated in Section REF , insurers are more likely to increase the premiums of policyholders with small premium amounts and decrease the premiums of policyholders with large premium amounts.",
"At the time of renewal, the insurer sets the constraints on the expected number of policyholders that he would like to keep in the portfolio.",
"His decision is based on the maximum premium volume that he expects to have $@ \\mathcal {R}$ .",
"Typically, when the retention level is low, the expected premium volume $@ \\mathcal {R} \\ $ is greater compared to the case when the retention level is high.",
"Therefore, we consider two scenarios: Scenario 1 The retention level is of 75%, Scenario 2 The retention level is of 85%.",
"The table below summarises the optimal results when solving (REF ) for the different constraints.",
"Table: NO_CAPTION tableScenarios testing.",
"Scenario 1 Table REF shows that all policyholders are subject to an increase in their premiums and the average optimal $\\delta $ for the whole portfolio corresponds to the maximum change in premium for both bounds of $\\delta $ .",
"Scenario 2 As seen in Table REF , the expected number of policyholders $@ \\mathcal {R} \\ $ is approximately the same as the one before premium change.",
"However, the growth in expected premium volume is lower than in Scenario 1 due to the fact that the average optimal $\\delta $ for both bounds is lower.",
"Remarks 5.2 It should be noted that the probability of renewal defined in Mb) can be approximated by the probability of renewal defined in Ma) for $\\delta $ relatively small (refer to Remark REF ).",
"Therefore, let's consider $\\delta \\in (-5\\%,5\\%)$ and a retention level $\\ell = 85\\%$ $@ \\mathcal {R}$ .",
"The table below describes the optimal results when using the logit model Mb) and the polynomial model defined in Ma).",
"Table: NO_CAPTION tableComparison between Ma) and Mb).",
"Table REF shows that for a small range of $\\delta $ , the difference between the exact results obtained from Mb) and the approximate results obtained from Ma) is relatively small and is of around 1% for the expected premium volume $@ \\mathcal {R} \\ $ and is of 0% for the expected number of policyholders $@ \\mathcal {R}$ .",
"Thus, the approximate values tend to the real ones when the range of $\\delta $ tends to 0.",
"Toy Models In this Section, we consider two toy models.",
"The first model consists in setting the same premium amounts among all policyholders.",
"Whereas in the second model, we assume that the policyholders have the same probability of renewal at time 0 irrespective of their premium amounts.",
"For both models, we compute the optimal results relative to the following scenarios: Scenario 1 Maximise the expected premium volume $@ \\mathcal {R}$ , Scenario 2 Maximise the expected premium volume and minimise the corresponding variance $@ \\mathcal {R}$ , Scenario 3 Maximise the retention level $@ \\mathcal {R}$ .",
"Toy Model 1 The premiums are constant among all policyholders.",
"We consider that $P_i=P= 200$ for all $i\\le N$ .",
"Table: NO_CAPTION tableToy Model 1.",
"Toy Model 2 The probabilities $\\pi _i$ are constant at 0.9 for all policyholders.",
"Table: NO_CAPTION tableToy Model 2.",
"These results are of interest when splitting the portfolios into sub-portfolios based on the premium amounts or the probability of renewal at time 0 of each policyholder.",
"Optimisation problem Mc) and Simulation studies In this Section, we consider the case where the renewal probabilities $\\Psi _i$ are fixed for each insured $i$ , as defined in Table REF .",
"To solve the optimisation problem (REF ), we use the MDNLP method described in Appendix C. The table below summarises the optimal results for a portfolio of 100'000 policyholders with respect to different constraints on the retention level at renewal.",
"Table: NO_CAPTION tableScenario testing-Discrete optimisation Table REF shows that when the retention level increases, the expected number of policies increases whereas the expected premium volume $@ \\mathcal {R} \\ $ decreases.",
"In fact, the average optimal $\\delta $ decreases gradually from 15% for a retention level of 85% to -6% for a retention level of 97.5%.",
"Also, it can be seen that for a retention level of 95% the optimisation has a negligible effect on the expected number of policies and premium volume $@ \\mathcal {R} \\ $ as the average optimal $\\delta $ is approximately null.",
"Hence, no optimisation is needed in this case.",
"In addition to the MDNLP approach, we have implemented a simulation technique which consists in simulating the premium change $\\delta $ for each policyholder as described in the following pseudo algorithm: Step 1: Based on a chosen prior distribution for $\\delta $ , sample the premium change for each policyholder, Step 2: Repeat Step 1 until the constraint on the retention level is satisfied, Step 3: Repeat Step 2 $m$ times, Step 4: Among the $m$ simulations take the simulated $\\delta $ which gives out the maximum expected profit.",
"Next, we present the optimal results obtained through $1^{\\prime }000$ simulations for the same portfolio.",
"We shall consider three different assumptions on the prior distribution of $\\delta $ , namely: Case 1: Simulation based on the Uniform distribution In this simulation approach, we assume that the prior distribution of $\\delta $ is uniformly distributed.",
"As highlighted in Table REF - REF , the parameters of the uniform distribution and the possible values of the premium change are chosen so that the constraint on the retention level is fulfilled.",
"We present in Table REF the simulation results.",
"Table: NO_CAPTION tableScenario testing- simulation approach: Uniform distribution.",
"Case 2: Simulation based on practical experience In this case we assume a prior distribution of $\\delta $ which is based on historical premium change of each policyholder.",
"Table: NO_CAPTION tableScenario testing- simulation approach: practical experience Case 3: Simulation based on the results of the MDNLP We use the distribution of the optimal $\\delta $ obtained from the MDNLP algorithm as a prior distribution.",
"Table REF below summarises the optimal results.",
"Table: NO_CAPTION tableScenario testing- simulation approach.",
"It can be seen that the simulation approaches yield approximately to the same results as the MDNLP algorithm presented in Table REF .",
"Appendix A: Constrained quadratic programming We present next the steps for the quadratic programming method utilised in our paper.",
"Step 1: (REF ) can be reformulated as follows $\\begin{aligned}& \\underset{\\delta }{ \\text{min }} f(\\delta )= \\frac{1}{2} \\delta ^{\\top } Q \\delta + c^\\top \\delta ,\\\\& \\text{subject to } g(\\delta ) = A^\\top \\delta \\le b ,\\end{aligned}$ where $\\delta =(\\delta _1,\\ldots ,\\delta _N)^\\top $ and $c$ is a vector describing the coefficient of the linear terms of $f$ given by $c= (-\\pi _1 P_1(1+a_1),\\ldots ,-\\pi _N P_N(1+a_N))^\\top .$ Here $Q$ is a diagonal and positive definite matrix describing the coefficients of the quadratic terms of $f$ determined by $Q= \\begin{pmatrix}-2\\pi _1P_ {1}a_1 &0&0 &\\ldots \\ & 0 \\\\0& -2\\pi _2 P_ {2}a_2&0 &\\ldots &0\\\\0&\\ldots &-2\\pi _iP_ {i}a_i &\\ldots &0\\\\0&0&0 &\\ldots &-2\\pi _NP_ {N}a_N\\\\\\end{pmatrix}.$ Since (REF ) has only one constraint, $A$ is a vector related to the linear coefficients of $g$ given by $A= - ( \\pi _ {1}a_1 ,\\pi _ {2}a_2,\\ldots ,\\pi _Na_N)^\\top ,$ and finally, $b= N \\ell - \\sum _{i=1}^{N} \\pi _i$ .",
"It should be noted that the constant term of the objective function $f$ is not accounted for in the resolution of (REF ).",
"Step 2: Let $\\mathcal {L}(\\delta ,{\\lambda })=f(\\delta )+{\\lambda } g(\\delta )$ be the Lagrangian function of (REF ) where $\\lambda $ is the Lagrangian multiplier.",
"Given that $Q$ is a positive definite matrix, the well-known Karush-Kuhn-Tucker (KKT) conditions (see for details [20][page 342]) defined below are sufficient for a global minimum of (REF ) if they are satisfied for a given vector $(\\delta ^*,{\\lambda }^*)$ $ \\left\\lbrace \\begin{array}{lcl}\\nabla \\mathcal {L}(\\delta ^*,\\lambda ^*)&=& 0,\\\\{{\\lambda }^{*}} g(\\delta ^*) &=& 0, \\\\g(\\delta ^*) & \\le & 0, \\\\\\lambda ^* &\\ge & 0.\\end{array}\\right.$ Remarks 6.1 In the settings of Problem T1a where the insurer aims at maximising the premium volume and minimising the corresponding variance, the optimisation problem can be expressed as follows $\\begin{aligned}&\\underset{\\delta }{\\text{max }} \\sum _ {i=1}^{N} P _i \\pi _i (1 + (1+a_i)\\delta _{i}+ a_i\\delta _i^2), \\\\&\\underset{\\delta }{\\text{min }} \\sum _ {i=1}^{N} P _i \\pi _i (1 + (1+a_i)\\delta _{i}+ a_i\\delta _i^2) (1- \\pi _i (1+a_i\\delta _{i})),\\quad \\\\& \\text{subject to } \\frac{1}{N} \\sum _ {i=1}^{N} \\pi _i(1+a_i \\delta _i) \\geqslant \\ell .\\\\\\end{aligned}$ Appendix B: Solution of (REF ) Step 1: Let $\\mathcal {L}(\\delta ,\\lambda )=f(\\delta )+\\lambda g(\\delta )$ be the Lagrangian function of (REF ) where $\\lambda \\in \\mathbb {R}$ is the Lagrangian multiplier and ($\\delta _0$ ,$\\lambda _0$ ) an initial estimate of the solution.",
"It should be noted that the SQP is not a feasible point method.",
"This means that neither the initial point nor the subsequent iterate ought to satisfy the constraints of the optimisation problem.",
"Step 2: In order to find the next point iterate $(\\delta _1,\\lambda _1)$ , the SQP determines a step vector $s=(s_\\delta ,s_\\lambda )$ solution of the QP sub-problem evaluated at ($\\delta _0$ ,$\\lambda _0$ ) and defined below $\\begin{aligned}&\\underset{s}{\\text{min}}&&\\frac{1}{2}s^\\top H s +\\nabla f(\\delta _0)^\\top \\mathbf {s},\\\\&\\text{subject to }&& \\nabla g(\\delta _0)^\\top s+ g(\\delta _0)\\le 0,\\\\\\end{aligned}$ where $H$ is an approximation of the Hessian matrix of $\\mathcal {L}$ , $\\nabla f$ and $\\nabla g$ are the gradient of the objective and the constraint functions respectively.",
"The Hessian matrix $H$ is updated at each iteration by the BFGS quazi Newton formula.",
"The SQP method maintains the sparsity of the approximation of the Hessian matrix and its positive definetness, a necessary condition for a unique solution.",
"Step 3: In order to ensure the convergence of the SQP method to a global solution, the latter uses a merit function $\\phi $ whose reduction implies progress towards a solution.",
"Thus, a step length, denoted by $\\alpha \\in (0,1)$ , is chosen in order to guarantee the reduction of $\\phi $ after each iteration such that $\\phi (\\delta _k+\\alpha s_k) \\le \\phi (\\delta _k),$ with $ \\phi (x)= f(x)+r g(x) \\text{ and } r>|\\lambda |.$ Step 4: The new point iterate is given by $(\\delta _1,\\lambda _1)=(\\delta _0+\\alpha s_{\\delta }, \\lambda _0+\\alpha s_{\\lambda })$ .",
"If $(\\delta _1,\\lambda _1)$ satisfies the KKT conditions (REF ), the SQP converges at that point.",
"If not, set $k=k+1$ and go back to Step 2.",
"Remarks 7.1 It should be noted that the KKT conditions defined in (REF ) are known as the first order optimality conditions, see e.g., [20].",
"Hence, if, for a given vector $(\\delta ^*,\\lambda ^*)$ , the KKT conditions are satisfied, then $(\\delta ^*,\\lambda ^*)$ is a local minimum of (REF ).",
"Appendix C: MDNLP optimisation problem (REF ) Step 1: Given that $\\Psi _i$ is discrete and depends on the values of $\\delta _i$ , we assume that $\\Psi _i$ can be written as a function of $\\delta _i$ as follows $ \\Psi _i(\\delta _i)= -0.9775 \\delta _i^2-0.4287 \\delta _i +0.9534 \\text{ for } \\delta _i \\in \\mathbf {D}$ (REF ) is then treated as a continuous optimisation problem and the optimal solution is found by using one of the methods described previously.",
"We denote by $\\delta ^*$ the continuous optimal solution.",
"Step 2: Let $\\delta _0$ be the rounded up vector of $\\delta ^*$ to the nearby discrete values of the set $\\mathbf {D}$ .",
"$\\delta _0$ is considered to be the initial point iterate.",
"If $\\delta _0$ is not a feasible point of (REF ), then (REF ) is approximated by a mixed discrete linear optimisation problem at $\\delta _0$ and is given by $\\begin{aligned}&\\underset{\\delta }{ \\text{min }} \\nabla f(\\delta _0)^\\top (\\delta -\\delta _0),\\\\&\\text{subject to } g(\\delta _0)+\\nabla g(\\delta _0)^\\top (\\delta -\\delta _0)\\leqslant 0,\\\\& \\text{ and }\\delta \\in \\mathbf {D}^N.\\end{aligned}$ Step 3: (REF ) is solved by using a linear programming method and the branch and bound method, see [24] for more details.",
"We denote by $\\delta _k$ the new point iterate.",
"If $\\delta _k$ is feasible and $||\\delta _k-\\delta _{k-1}|| < \\epsilon $ with $\\epsilon >0$ small, then the iteration is stopped.",
"Else $k=k+1$ and go back to Step 2.",
"Remarks 8.1 If, for a certain point iterate $\\delta $ , the constraint of (REF ) is satisfied and $\\delta \\in \\mathbf {D}^N$ then $\\delta $ is a feasible solution of the optimisation problem.",
"In general, it is very hard to find the global minimum of a MDNLP optimisation problem due to the fact that there are multiple local minimums.",
"Therefore, $\\delta ^*$ is said to be a global minimum if $\\delta ^*$ is feasible and $f(\\delta ^*)\\le f(\\delta )$ for all feasible $\\delta $ .",
"Appendix D: Prior distribution for simulation Simulation based on the Uniform distribution (simulation Case 1) The tables below describe the range of $\\delta $ with their respective distribution based on the different retention levels.",
"Table: NO_CAPTION tablePossible range of $\\delta $ and prior distribution uniformly distributed.",
"Table: NO_CAPTION tablePossible range of $\\delta $ and prior distribution uniformly distributed."
],
[
"Insurance Applications",
"In this section, we consider a dataset that describes the production of the motor line of business of an insurance portfolio.",
"We assume that the premiums are exponentially increasing.",
"Also, the probability of renewal at time 0, $\\pi _i$ for $i=1,\\ldots ,N$ , are known and estimated by the insurance company for each category of policyholders based on historical data.",
"Given that the behavior of the policyholders is unknown at the time of renewal, the probability of renewal at time 1, $\\Psi _i$ , depends on $\\pi _i$ and $\\delta _i$ for $i=1,\\ldots ,N$ .",
"If $\\delta _i$ is positive, then $\\Psi _i$ decreases whereas if $\\delta _i$ is negative, it is more likely that the policyholder will renew his insurance policy at time 1, thus generating a greater $\\Psi _i$ .",
"In the following paragraphs, we are going to present some results relative to the optimisation problems at hand described in the last section."
],
[
"Maximise the expected premium volume $@ \\mathcal {R}$ .",
"We consider, first, the optimisation problem defined in (REF ).",
"In this case, the probability of renewal $\\Psi _i$ is defined in Ma) and set $b_i=0$ for $i=1,\\ldots ,N$ .",
"Given that $a_i < 0 \\text{ for } i\\le N $ , the probability of renewal $\\Psi _i$ increases when $\\delta _i$ is negative and decreases when $\\delta _i$ is positive, thus describing perfectly the behavior of the policyholders that are subject to a decrease, respectively increase, in their premiums $@ \\mathcal {R}$ .",
"The table below describes some statistics on the data for $10^{\\prime }000$ policyholders.",
"Table: NO_CAPTION tableProduction statistics for the motor business.",
"We consider that the insurance company would like to keep 85% of its policyholders in its portfolio $@ \\mathcal {R}$ .",
"By solving (REF ) in Matlab with the function quadprog, we obtain the optimal $\\delta $ for each policyholder.",
"Next, we consider two scenarios: Scenario 1 The insurer would like to keep 75% of the policyholders in his portfolio $@ \\mathcal {R}$ , Scenario 2 The insurer would like to keep 85% of the policyholders in his portfolio $@ \\mathcal {R}$ .",
"Table REF below summarises the optimal results when solving (REF ) and examines the effect of both scenarios on the expected premium volume and the expected number of policyholders in the portfolio $@ \\mathcal {R}$ .",
"Table: NO_CAPTION tableScenarios testing.",
"Scenario 1 The optimal $\\delta $ for both bounds corresponds approximately to the maximum value (upper bound) of the interval.",
"This is mainly due to the fact that the insurer would like to keep only 75% of the portfolio $@ \\mathcal {R}$ .",
"Therefore, his main goal is to maximise the expected premium volume at time 1.",
"Scenario 2 For a retention level of 85%, Table REF shows an increase in the expected premium volume which is less important than the one observed in Scenario 1.",
"However, the expected number of policyholders in the portfolio $@ \\mathcal {R} \\ $ is higher and is approximately the same as at t=0.",
"Hereafter, we shall consider a retention level of 85%.",
"Usually, in practice, the size of a motor insurance portfolio exceeds $10^{\\prime }000$ policyholders.",
"However, solving the optimisation problems for $\\delta $ using the described algorithms when $N$ is large requires a lot of time and heavy computation and may be costly for the insurance company.",
"Thus, an idea to overcome this problem is to split the original portfolio into sub-portfolios and compute the optimal $\\delta $ for the sub-portfolios.",
"One criteria that can be taken into account for the split is the amount of premium in our case.",
"However, in practice, insurance companies have a more detailed data, thus more information on each policyholders, so the criterion that are of interest for the split are the age of the policyholders, the car brand, car value ...Table REF and Table REF below describe the results when splitting the original portfolio into 3 and 4 sub-portfolios respectively.",
"Table: NO_CAPTION tableSplit into 3 sub-portfolios.",
"Table: NO_CAPTION tableSplit into 4 sub-portfolios.",
"In Table REF and REF , we consider that the insurer would like to keep 85% of the policyholders in each sub-portfolios, thus a total of 85% of the original portfolio.",
"However, in practice, the constraints on the retention level $@ \\mathcal {R} \\ $ are specific to each sub-portfolio and this depending on the insurer's decision whether he would like to keep the policies with large premium amounts or small premium amounts in his portfolio $@ \\mathcal {R}$ .",
"In this regard, the insurance company sets the constraints on the expected number of policies for each sub-portfolios so that the constraint of the overall portfolio is approximately equal to 85%.",
"The error from the split into 3, respectively 4 sub-portfolios is relatively small and is of -0.13%, respectively -0.23% for the expected premium volume $@ \\mathcal {R}$ .",
"It should be noted that the error margin increases as the number of splits increases.",
"Remarks 5.1 In the following sections, we limit the size of the insurance portfolio to 1'000 policyholders as the algorithms used thereafter to solve the optimisation problems are based on an iterative process and requires a lot of computation and time.",
"Hence, an idea to solve the optimisation problem for an insurance portfolio of size $n$ with $n\\ge 1^{\\prime }000$ is to split the original portfolio into sub-portfolios and compute the optimal results for the sub-portfolios, as discussed in the previous Section."
],
[
"Maximise the premium volume and minimise the variance of the premium volume",
"Similarly to the asset allocation optimisation problem in finance introduced by Markowitz [22], the insurer performs a trade-off between the maximum aggregate expected premiums and the minimum variance of the total earned premiums; see also [23] for a different optimality criteria.",
"We show next in Table REF the optimal results for the different constraints on the retention level and the possible range of premium changes.",
"Table: NO_CAPTION tableVolume and variance objective optimal results.",
"It can be seen that the optimal variance $@ \\mathcal {R} \\ $ increases with the range of the possible premium changes $\\delta $ .",
"For instance when the insurer would like to keep $75\\%$ of the policyholders, the variance $@ \\mathcal {R} \\ $ increases from $109.76$ for $\\delta \\in (-10\\%, 20\\% ) $ to $113.08$ for $\\delta \\in (-20\\%, 30\\% ) $ , respectively.",
"Furthermore, the increase in variance $@ \\mathcal {R} \\ $ is associated with an increase of the expected volume $@ \\mathcal {R}$ .",
"This means that the riskier the portfolio the more the insurance company earns premiums."
],
[
"Maximise the retention level $@ \\mathcal {R}$ .",
"We consider here that the insurer would like to maximise his retention level whilst increasing the expected premium volume $@ \\mathcal {R} \\ $ by a certain amount $C$ needed to cover, for instance, the operating costs and other expenses of the insurance company.",
"$C$ can be expressed as a loading on the expected premium volume at time 0.",
"In practice, the amount $C$ needed to cover the expenses of the company is set by the insurers.",
"As stated previously, $C$ can be expressed as a percentage of the expected premium volume at time 0.",
"Therefore, we consider three different loadings: 9%, 10% and 11% thus adding an amount of $85^{\\prime }000$ , respectively $95^{\\prime }000$ and $105^{\\prime }000$ to the expected premium volume at time 0.",
"Also, we consider two ranges for $\\delta $ , $\\delta \\in (-10\\%,-20\\%)$ and $\\delta \\in (-20\\%,-30\\%)$ .",
"Table: NO_CAPTION tableScenario testing - Retention Table REF shows that when $C$ increases, the expected number of policyholders $@ \\mathcal {R} \\ $ decreases whereas the average optimal $\\delta $ increases."
],
[
"Optimisation problem Mb)",
"We consider the optimisation problem defined in (REF ) where the probability of renewal $\\Psi _i$ is defined in Mb).",
"As discussed in Section REF , $T_i$ describes the behavior of the policyholders subject to premium change.",
"For instance, let's consider a policyholder whose probability of renewal without premium change $\\pi _i$ is 0.95.",
"In this Section, we will only consider the case where the insurer would like to maximise the expected premium volume $@ \\mathcal {R}$ .",
"The constraint on the retention level is assumed to be of 85%.",
"As stated in Section REF , insurers are more likely to increase the premiums of policyholders with small premium amounts and decrease the premiums of policyholders with large premium amounts.",
"At the time of renewal, the insurer sets the constraints on the expected number of policyholders that he would like to keep in the portfolio.",
"His decision is based on the maximum premium volume that he expects to have $@ \\mathcal {R}$ .",
"Typically, when the retention level is low, the expected premium volume $@ \\mathcal {R} \\ $ is greater compared to the case when the retention level is high.",
"Therefore, we consider two scenarios: Scenario 1 The retention level is of 75%, Scenario 2 The retention level is of 85%.",
"The table below summarises the optimal results when solving (REF ) for the different constraints.",
"Table: NO_CAPTION tableScenarios testing.",
"Scenario 1 Table REF shows that all policyholders are subject to an increase in their premiums and the average optimal $\\delta $ for the whole portfolio corresponds to the maximum change in premium for both bounds of $\\delta $ .",
"Scenario 2 As seen in Table REF , the expected number of policyholders $@ \\mathcal {R} \\ $ is approximately the same as the one before premium change.",
"However, the growth in expected premium volume is lower than in Scenario 1 due to the fact that the average optimal $\\delta $ for both bounds is lower.",
"Remarks 5.2 It should be noted that the probability of renewal defined in Mb) can be approximated by the probability of renewal defined in Ma) for $\\delta $ relatively small (refer to Remark REF ).",
"Therefore, let's consider $\\delta \\in (-5\\%,5\\%)$ and a retention level $\\ell = 85\\%$ $@ \\mathcal {R}$ .",
"The table below describes the optimal results when using the logit model Mb) and the polynomial model defined in Ma).",
"Table: NO_CAPTION tableComparison between Ma) and Mb).",
"Table REF shows that for a small range of $\\delta $ , the difference between the exact results obtained from Mb) and the approximate results obtained from Ma) is relatively small and is of around 1% for the expected premium volume $@ \\mathcal {R} \\ $ and is of 0% for the expected number of policyholders $@ \\mathcal {R}$ .",
"Thus, the approximate values tend to the real ones when the range of $\\delta $ tends to 0."
],
[
"Toy Models",
"In this Section, we consider two toy models.",
"The first model consists in setting the same premium amounts among all policyholders.",
"Whereas in the second model, we assume that the policyholders have the same probability of renewal at time 0 irrespective of their premium amounts.",
"For both models, we compute the optimal results relative to the following scenarios: Scenario 1 Maximise the expected premium volume $@ \\mathcal {R}$ , Scenario 2 Maximise the expected premium volume and minimise the corresponding variance $@ \\mathcal {R}$ , Scenario 3 Maximise the retention level $@ \\mathcal {R}$ .",
"Toy Model 1 The premiums are constant among all policyholders.",
"We consider that $P_i=P= 200$ for all $i\\le N$ .",
"Table: NO_CAPTION tableToy Model 1.",
"Toy Model 2 The probabilities $\\pi _i$ are constant at 0.9 for all policyholders.",
"Table: NO_CAPTION tableToy Model 2.",
"These results are of interest when splitting the portfolios into sub-portfolios based on the premium amounts or the probability of renewal at time 0 of each policyholder."
],
[
"Optimisation problem Mc) and Simulation studies",
"In this Section, we consider the case where the renewal probabilities $\\Psi _i$ are fixed for each insured $i$ , as defined in Table REF .",
"To solve the optimisation problem (REF ), we use the MDNLP method described in Appendix C. The table below summarises the optimal results for a portfolio of 100'000 policyholders with respect to different constraints on the retention level at renewal.",
"Table: NO_CAPTION tableScenario testing-Discrete optimisation Table REF shows that when the retention level increases, the expected number of policies increases whereas the expected premium volume $@ \\mathcal {R} \\ $ decreases.",
"In fact, the average optimal $\\delta $ decreases gradually from 15% for a retention level of 85% to -6% for a retention level of 97.5%.",
"Also, it can be seen that for a retention level of 95% the optimisation has a negligible effect on the expected number of policies and premium volume $@ \\mathcal {R} \\ $ as the average optimal $\\delta $ is approximately null.",
"Hence, no optimisation is needed in this case.",
"In addition to the MDNLP approach, we have implemented a simulation technique which consists in simulating the premium change $\\delta $ for each policyholder as described in the following pseudo algorithm: Step 1: Based on a chosen prior distribution for $\\delta $ , sample the premium change for each policyholder, Step 2: Repeat Step 1 until the constraint on the retention level is satisfied, Step 3: Repeat Step 2 $m$ times, Step 4: Among the $m$ simulations take the simulated $\\delta $ which gives out the maximum expected profit.",
"Next, we present the optimal results obtained through $1^{\\prime }000$ simulations for the same portfolio.",
"We shall consider three different assumptions on the prior distribution of $\\delta $ , namely: Case 1: Simulation based on the Uniform distribution In this simulation approach, we assume that the prior distribution of $\\delta $ is uniformly distributed.",
"As highlighted in Table REF - REF , the parameters of the uniform distribution and the possible values of the premium change are chosen so that the constraint on the retention level is fulfilled.",
"We present in Table REF the simulation results.",
"Table: NO_CAPTION tableScenario testing- simulation approach: Uniform distribution.",
"Case 2: Simulation based on practical experience In this case we assume a prior distribution of $\\delta $ which is based on historical premium change of each policyholder.",
"Table: NO_CAPTION tableScenario testing- simulation approach: practical experience Case 3: Simulation based on the results of the MDNLP We use the distribution of the optimal $\\delta $ obtained from the MDNLP algorithm as a prior distribution.",
"Table REF below summarises the optimal results.",
"Table: NO_CAPTION tableScenario testing- simulation approach.",
"It can be seen that the simulation approaches yield approximately to the same results as the MDNLP algorithm presented in Table REF ."
],
[
"Appendix A: Constrained quadratic programming",
"We present next the steps for the quadratic programming method utilised in our paper.",
"Step 1: (REF ) can be reformulated as follows $\\begin{aligned}& \\underset{\\delta }{ \\text{min }} f(\\delta )= \\frac{1}{2} \\delta ^{\\top } Q \\delta + c^\\top \\delta ,\\\\& \\text{subject to } g(\\delta ) = A^\\top \\delta \\le b ,\\end{aligned}$ where $\\delta =(\\delta _1,\\ldots ,\\delta _N)^\\top $ and $c$ is a vector describing the coefficient of the linear terms of $f$ given by $c= (-\\pi _1 P_1(1+a_1),\\ldots ,-\\pi _N P_N(1+a_N))^\\top .$ Here $Q$ is a diagonal and positive definite matrix describing the coefficients of the quadratic terms of $f$ determined by $Q= \\begin{pmatrix}-2\\pi _1P_ {1}a_1 &0&0 &\\ldots \\ & 0 \\\\0& -2\\pi _2 P_ {2}a_2&0 &\\ldots &0\\\\0&\\ldots &-2\\pi _iP_ {i}a_i &\\ldots &0\\\\0&0&0 &\\ldots &-2\\pi _NP_ {N}a_N\\\\\\end{pmatrix}.$ Since (REF ) has only one constraint, $A$ is a vector related to the linear coefficients of $g$ given by $A= - ( \\pi _ {1}a_1 ,\\pi _ {2}a_2,\\ldots ,\\pi _Na_N)^\\top ,$ and finally, $b= N \\ell - \\sum _{i=1}^{N} \\pi _i$ .",
"It should be noted that the constant term of the objective function $f$ is not accounted for in the resolution of (REF ).",
"Step 2: Let $\\mathcal {L}(\\delta ,{\\lambda })=f(\\delta )+{\\lambda } g(\\delta )$ be the Lagrangian function of (REF ) where $\\lambda $ is the Lagrangian multiplier.",
"Given that $Q$ is a positive definite matrix, the well-known Karush-Kuhn-Tucker (KKT) conditions (see for details [20][page 342]) defined below are sufficient for a global minimum of (REF ) if they are satisfied for a given vector $(\\delta ^*,{\\lambda }^*)$ $ \\left\\lbrace \\begin{array}{lcl}\\nabla \\mathcal {L}(\\delta ^*,\\lambda ^*)&=& 0,\\\\{{\\lambda }^{*}} g(\\delta ^*) &=& 0, \\\\g(\\delta ^*) & \\le & 0, \\\\\\lambda ^* &\\ge & 0.\\end{array}\\right.$ Remarks 6.1 In the settings of Problem T1a where the insurer aims at maximising the premium volume and minimising the corresponding variance, the optimisation problem can be expressed as follows $\\begin{aligned}&\\underset{\\delta }{\\text{max }} \\sum _ {i=1}^{N} P _i \\pi _i (1 + (1+a_i)\\delta _{i}+ a_i\\delta _i^2), \\\\&\\underset{\\delta }{\\text{min }} \\sum _ {i=1}^{N} P _i \\pi _i (1 + (1+a_i)\\delta _{i}+ a_i\\delta _i^2) (1- \\pi _i (1+a_i\\delta _{i})),\\quad \\\\& \\text{subject to } \\frac{1}{N} \\sum _ {i=1}^{N} \\pi _i(1+a_i \\delta _i) \\geqslant \\ell .\\\\\\end{aligned}$"
],
[
"Appendix B: Solution of (",
"Step 1: Let $\\mathcal {L}(\\delta ,\\lambda )=f(\\delta )+\\lambda g(\\delta )$ be the Lagrangian function of (REF ) where $\\lambda \\in \\mathbb {R}$ is the Lagrangian multiplier and ($\\delta _0$ ,$\\lambda _0$ ) an initial estimate of the solution.",
"It should be noted that the SQP is not a feasible point method.",
"This means that neither the initial point nor the subsequent iterate ought to satisfy the constraints of the optimisation problem.",
"Step 2: In order to find the next point iterate $(\\delta _1,\\lambda _1)$ , the SQP determines a step vector $s=(s_\\delta ,s_\\lambda )$ solution of the QP sub-problem evaluated at ($\\delta _0$ ,$\\lambda _0$ ) and defined below $\\begin{aligned}&\\underset{s}{\\text{min}}&&\\frac{1}{2}s^\\top H s +\\nabla f(\\delta _0)^\\top \\mathbf {s},\\\\&\\text{subject to }&& \\nabla g(\\delta _0)^\\top s+ g(\\delta _0)\\le 0,\\\\\\end{aligned}$ where $H$ is an approximation of the Hessian matrix of $\\mathcal {L}$ , $\\nabla f$ and $\\nabla g$ are the gradient of the objective and the constraint functions respectively.",
"The Hessian matrix $H$ is updated at each iteration by the BFGS quazi Newton formula.",
"The SQP method maintains the sparsity of the approximation of the Hessian matrix and its positive definetness, a necessary condition for a unique solution.",
"Step 3: In order to ensure the convergence of the SQP method to a global solution, the latter uses a merit function $\\phi $ whose reduction implies progress towards a solution.",
"Thus, a step length, denoted by $\\alpha \\in (0,1)$ , is chosen in order to guarantee the reduction of $\\phi $ after each iteration such that $\\phi (\\delta _k+\\alpha s_k) \\le \\phi (\\delta _k),$ with $ \\phi (x)= f(x)+r g(x) \\text{ and } r>|\\lambda |.$ Step 4: The new point iterate is given by $(\\delta _1,\\lambda _1)=(\\delta _0+\\alpha s_{\\delta }, \\lambda _0+\\alpha s_{\\lambda })$ .",
"If $(\\delta _1,\\lambda _1)$ satisfies the KKT conditions (REF ), the SQP converges at that point.",
"If not, set $k=k+1$ and go back to Step 2.",
"Remarks 7.1 It should be noted that the KKT conditions defined in (REF ) are known as the first order optimality conditions, see e.g., [20].",
"Hence, if, for a given vector $(\\delta ^*,\\lambda ^*)$ , the KKT conditions are satisfied, then $(\\delta ^*,\\lambda ^*)$ is a local minimum of (REF )."
],
[
"Appendix C: MDNLP optimisation problem (",
"Step 1: Given that $\\Psi _i$ is discrete and depends on the values of $\\delta _i$ , we assume that $\\Psi _i$ can be written as a function of $\\delta _i$ as follows $ \\Psi _i(\\delta _i)= -0.9775 \\delta _i^2-0.4287 \\delta _i +0.9534 \\text{ for } \\delta _i \\in \\mathbf {D}$ (REF ) is then treated as a continuous optimisation problem and the optimal solution is found by using one of the methods described previously.",
"We denote by $\\delta ^*$ the continuous optimal solution.",
"Step 2: Let $\\delta _0$ be the rounded up vector of $\\delta ^*$ to the nearby discrete values of the set $\\mathbf {D}$ .",
"$\\delta _0$ is considered to be the initial point iterate.",
"If $\\delta _0$ is not a feasible point of (REF ), then (REF ) is approximated by a mixed discrete linear optimisation problem at $\\delta _0$ and is given by $\\begin{aligned}&\\underset{\\delta }{ \\text{min }} \\nabla f(\\delta _0)^\\top (\\delta -\\delta _0),\\\\&\\text{subject to } g(\\delta _0)+\\nabla g(\\delta _0)^\\top (\\delta -\\delta _0)\\leqslant 0,\\\\& \\text{ and }\\delta \\in \\mathbf {D}^N.\\end{aligned}$ Step 3: (REF ) is solved by using a linear programming method and the branch and bound method, see [24] for more details.",
"We denote by $\\delta _k$ the new point iterate.",
"If $\\delta _k$ is feasible and $||\\delta _k-\\delta _{k-1}|| < \\epsilon $ with $\\epsilon >0$ small, then the iteration is stopped.",
"Else $k=k+1$ and go back to Step 2.",
"Remarks 8.1 If, for a certain point iterate $\\delta $ , the constraint of (REF ) is satisfied and $\\delta \\in \\mathbf {D}^N$ then $\\delta $ is a feasible solution of the optimisation problem.",
"In general, it is very hard to find the global minimum of a MDNLP optimisation problem due to the fact that there are multiple local minimums.",
"Therefore, $\\delta ^*$ is said to be a global minimum if $\\delta ^*$ is feasible and $f(\\delta ^*)\\le f(\\delta )$ for all feasible $\\delta $ ."
],
[
"Simulation based on the Uniform distribution (simulation Case 1)",
"The tables below describe the range of $\\delta $ with their respective distribution based on the different retention levels.",
"Table: NO_CAPTION tablePossible range of $\\delta $ and prior distribution uniformly distributed.",
"Table: NO_CAPTION tablePossible range of $\\delta $ and prior distribution uniformly distributed."
]
] | 1605.05814 |
[
[
"Numerical Solution of the Steady-State Navier-Stokes Equations using\n Empirical Interpolation Methods"
],
[
"Abstract Reduced-order modeling is an efficient approach for solving parameterized discrete partial differential equations when the solution is needed at many parameter values.",
"An offline step approximates the solution space and an online step utilizes this approximation, the reduced basis, to solve a smaller reduced problem at significantly lower cost, producing an accurate estimate of the solution.",
"For nonlinear problems, however, standard methods do not achieve the desired cost savings.",
"Empirical interpolation methods represent a modification of this methodology used for cases of nonlinear operators or nonaffine parameter dependence.",
"These methods identify points in the discretization necessary for representing the nonlinear component of the reduced model accurately, and they incur online computational costs that are independent of the spatial dimension $N$.",
"We will show that empirical interpolation methods can be used to significantly reduce the costs of solving parameterized versions of the Navier-Stokes equations, and that iterative solution methods can be used in place of direct methods to further reduce the costs of solving the algebraic systems arising from reduced-order models."
],
[
"Introduction",
"Methods of reduced-order modeling are designed to obtain the numerical solution of parameterized partial differential equations (PDEs) efficiently.",
"In settings where solutions of parameterized PDEs are required for many parameters, such as uncertainty quantification, design optimization, and sensitivity analysis, the cost of obtaining high-fidelity solutions at each parameter may be prohibitive.",
"In this scenario, reduced-order models can often be used to keep computation costs low by projecting the model onto a space of smaller dimension with minimal loss of accuracy.",
"We begin with a brief statement of the reduced basis method for constructing a reduced-order model.",
"Consider an algebraic system of equations $G(u) = 0$ where $u : = u(\\xi )$ is an unknown vector of dimension $N$ , and $\\xi $ is vector of $m$ input parameters.",
"We are interested in the case where this system arises from the discretization of a PDE and $N$ is large, as would be the case for a high-fidelity discretization.",
"We will refer to this system as the full model.",
"We would like to compute solutions for many parameters $\\xi $ .",
"Reduced basis methods compute a (relatively) small number of full model solutions, $u(\\xi _1)$ , ...$u(\\xi _k)$ , known as snapshots, and then for other parameters, $\\xi \\ne \\xi _j$ , construct approximations of $u(\\xi )$ in the space spanned by $\\lbrace u(\\xi _j) \\rbrace _{j=1}^k$ .",
"In the offline-online paradigm, the offline step, which may be expensive, computes the snapshots using traditional (PDE) solvers.",
"The offline step builds a basis of the low-dimensional vector space spanned by the snapshots.",
"The online step, which is intended to be inexpensive (because $k$ is small), uses a projected version of the original problem (determined, for example, by a Galerkin projection) in the $k$ -dimensional space.",
"The projected problem, known as the reduced model, has a solution $\\tilde{u}(\\xi )$ which is an approximation of the solution $u(\\xi )$ .",
"A straightforward implementation of the reduced-basis method is only possible for linear problems that have affine dependence on the parameters.",
"Such problems have the form $G(u) = 0 $ where $G(u) = A(\\xi )u - b = \\left(\\sum _{i=1}^{l} \\varphi _i(\\xi ) A_i \\right) u - b$ and $\\lbrace A_i\\rbrace _{i=1}^l$ are parameter-independent matrices.",
"Let $Q$ be a matrix of dimensions $N \\times k$ whose columns span the space spanned by the snapshots.",
"For example, $Q$ can be taken to be an orthogonal matrix obtained using the Gram-Schmidt process applied to $[u(\\xi _1),..., u(\\xi _k)]$ ; construction of $Q$ is part of the offline step.",
"With this decomposition, the reduced model obtained from a Galerkin condition is $G^r(\\hat{u}) = 0$ where $G^r(\\hat{u}) = Q^T A(\\xi ) Q \\hat{u} - Q^T b= \\left(\\sum _{i=1}^l \\varphi _i(\\xi ) (Q^T A_i Q)\\right)\\hat{u}- Q^T b \\; .$ Computation of the matrices $\\lbrace Q^T A_iQ\\rbrace $ can be included as part of the offline step.",
"With this preliminary computation, the online step requires only the summation of the terms in equation (REF ), an $O(lk^2)$ operation, and then the solution of the system of order $k$ .",
"Clearly, the cost of this online computation is independent of $N$ , the dimension of the full model.",
"However, when this approach is applied to a nonlinear problem, the reduced model is not independent of the dimension of the full model.",
"Consider a problem with a nonlinear component $F(u( \\xi ))$ , so the full model is $G(u(\\xi )) = Au(\\xi ) + F(u(\\xi )) - b = 0 \\; .$ The reduced model obtained from the Galerkin projection is $G^r(\\hat{u}(\\xi )) = Q^T A Q \\hat{u}(\\xi ) + Q^T F(Q \\hat{u}(\\xi )) - Q^T b = 0$ Although the reduced operator $Q^T F(Q\\hat{u}( \\xi ))$ is a mapping from $\\mathbb {R}^{k} \\rightarrow \\mathbb {R}^{k}$ , any nonlinear solution algorithm (e.g.",
"Picard iteration) requires the evaluation of the operator $F(Q\\hat{u}(\\xi ))$ as well as the multiplication by $Q^T$ .",
"Both computations have costs that depend on $N$ , the dimension of the full model.",
"Empirical interpolation methods [2], [7], [13], [15] use interpolation to reduce the cost of the online construction for nonlinear operators or nonaffine parameter dependence.",
"The premise of these methods is to interpolate the operator using a subset of indices from the model.",
"The interpolation depends on an empirically derived basis that can also be constructed as part of an offline procedure.",
"This ensures that $F(Q\\hat{u}(\\xi ))$ is evaluated only at a relatively small number of indices.",
"These values are used in conjunction with a separate basis constructed to approximate the nonlinear operator.",
"The efficiency of this approach also depends on the fact that for all $i$ , $F_i(u(\\xi ))$ depends on a relatively small, $O(1)$ , number of components of $u$ .",
"Computing the solution of the reduced model for a nonlinear operator requires a nonlinear iteration based on a linearization strategy, which requires the solution of a reduced linear system at each step.",
"Thus, each iteration has two primary costs, the computation of the Jacobian corresponding to $Q^T F(u(\\xi ))$ , and the solution of the linear system at each step of the nonlinear iteration.",
"Empirical interpolation addresses the first cost, by using an approximation of $Q^T F (u(\\xi ))$ .",
"To address the second cost, one option is to use direct methods to solve the reduced linear systems.",
"In [9], however, we have seen that iterative methods are effective for solving reduced models of linear operators of a certain size.",
"In this paper, we extend this approach, using preconditioners that are precomputed in the offline stage, to nonlinear problems solved using empirical interpolation.",
"We explore this approach using a Picard iteration for the linearization strategy.",
"We will demonstrate the efficiency of combining empirical interpolation with an iterative linear solver by computing solutions of the steady-state incompressible Navier-Stokes equations with random viscosity coefficient: $\\begin{array}{rclcc}- \\nabla \\cdot \\nu (\\cdot ,\\xi )\\nabla {\\vec{u}}(\\cdot ,\\xi ) + {\\vec{u}}(\\cdot ,\\xi ) \\cdot \\nabla {\\vec{u}}(\\cdot ,\\xi ) + \\nabla p(\\cdot ,\\xi ) &=& f & \\text{ in } & D \\times \\Gamma \\\\ \\nonumber \\nabla \\cdot {\\vec{u}}(\\cdot ,\\xi ) &=& 0 & \\text{ in } & D \\times \\Gamma \\\\{\\vec{u}}(\\cdot ,\\xi ) & =& b & \\text{ on }& \\partial D \\times \\Gamma \\; , \\nonumber \\end{array}$ where ${\\vec{u}}(\\cdot , \\xi )$ is the flow velocity, $p(\\cdot ,\\xi )$ is the scalar pressure, and $b$ determines the Dirichlet boundary conditions, and the viscosity coefficient satisfies $\\nu (\\cdot ,\\xi ) > 0$ .",
"Models of this type have been used to model the viscosity in multiphase flows [14], [17], [24].",
"The boundary data $b$ and forcing function $f$ could also be parameter-dependent, although we will not consider such examples here.",
"An outline of this paper is as follows.",
"In Section , we outline the details of the empirical interpolation strategy that we use, the so-called discrete empirical interpolation method (DEIM) [7].",
"(See [19] for discussion and comparison of different variants of this idea.)",
"In Section we introduce the steady-state Navier-Stokes equations with an uncertain viscosity coefficient and describe the full, reduced, and DEIM models for this problem.",
"We present numerical results in Section including a comparison of snapshot selection methods for DEIM and a discussion of accuracy of the DEIM.",
"In addition, we discuss a generalization of this approach known as a gappy-POD method [6], [11].",
"Finally, in Section , we discuss the use of iterative methods for solving the reduced linear systems that arise from DEIM.",
"This includes a presentation of new preconditioning techniques for use in this setting and discussion of their effectiveness."
],
[
"The discrete empirical interpolation method",
"The discrete empirical interpolation method utilizes an approximation $\\bar{F}(u)$ of a nonlinear function $F(u)$ [7].",
"The keys to efficiency in this algorithm are that only a small number of indices of $F$ are used in each component each component of the nonlinear function depends only on a few indices of the input variable.",
"The key to the accuracy of this method is to select the indices of the discrete PDE that are most important to produce an accurate representation of the nonlinear component of the solution projected on the reduced space.",
"The efficiency requirement is clearly satisfied when the nonlinear function is a PDE discretized using the finite element method [1].",
"Given a PDE that depends on a set of parameters $\\xi = [\\xi _1, ..., \\xi _m]^T$ with solution $u(\\cdot ,\\xi )$ , the full model is the discretized equation such that $G(u(\\cdot ,\\xi )) = 0$ .",
"The offline step for the traditional approach to the reduced basis method takes full solutions at several parameters and constructs a matrix $Q$ of rank $k$ such that $\\text{range}(Q) = \\text{span}\\lbrace u(\\cdot ,\\xi ^{(1)}), ..., u(\\cdot ,\\xi ^{(k)})\\rbrace $ , where the solutions $u(\\cdot ,\\xi ^{(1)}), ..., u(\\cdot ,\\xi ^{(k)})$ are known as snapshots.",
"The online step approximates the true solution $u$ with an approximation $\\tilde{u}\\approx Q \\hat{u}$ .",
"Using the Galerkin projection, the reduced model, $G^r$ , is $G^r(\\hat{u}) = Q^T G(Q \\hat{u}) \\; .",
"$ and the Jacobian of $G^r(\\hat{u})$ is $J_{G^r}(\\hat{u}) \\ = \\ \\frac{\\partial G^{r}(\\hat{u})}{\\partial \\hat{u}} \\ = \\ Q^T \\frac{\\partial }{\\partial \\hat{u}} G(\\tilde{u}(\\hat{u}))\\ = \\ Q^T \\frac{\\partial G }{\\partial \\tilde{u}}\\frac{\\partial \\tilde{u}}{\\partial \\hat{u}} \\ = \\ Q^T J_G(\\tilde{u}) Q = Q^T J_G(Q\\hat{u}) Q \\; .",
"$ As observed above, when the operator is linear and affinely dependent on the parameters, the online costs, (of forming and solving the reduced system (REF )) are independent of $N$ .",
"This is not true for nonlinear or nonaffine operators.",
"Consider Newton's method for the reduced model with a nonlinear operator: $\\hat{u}_{n+1} = \\hat{u}_{n} - J_{G^r}(\\hat{u}_n)^{-1} G^r(\\hat{u}_n)$ Each iteration in equation (REF ) requires the construction of $J_{G^r}(\\hat{u}_n)= Q^T J_G(Q\\hat{u}_n)Q$ .",
"The construction of the matrix $J_G(Q\\hat{u}_n)$ as well as the multiplications by $Q$ and $Q^T$ have costs that depend on $N$ .",
"In the DEIM approach, the snapshots $u(\\xi ^{(1)}), ..., u(\\xi ^{(k)})$ are obtained from the full model and the reduced basis is constructed to span these snapshots.",
"In addition, DEIM requires a separate basis to represent the nonlinear component of the solution.",
"This basis is constructed using a matrix of snapshots of function values $S = [F(u(\\xi ^{(1)})), F(u(\\xi ^{(2)})), ..., F(u(\\xi ^{(s)}))]$ .",
"Then, using methods similar to finding the reduced basis, a basis is chosen to approximately span the space spanned by these snapshots.",
"One approach for doing this is to use a proper orthogonal decomposition (POD) of the snapshot matrix $S$ $S = \\bar{V} \\Sigma W^T$ where the singular values in $\\Sigma $ are sorted in order of decreasing magnitude and $\\bar{V}$ and $W$ are orthogonal.",
"DEIM will use columns of $\\bar{V}$ to approximate $F(\\tilde{u}(\\xi ))$ .",
"It may happen that $n_{deim}<k$ columns of $\\bar{V}$ are used here, giving a submatrix $V$ of $\\bar{V}$ .",
"We will discuss the method used to select $n_{deim}$ in Section .",
"Given the nonlinear basis from $V$ , the DEIM selects indices of $F$ so the interpolated nonlinear component on the range of $V$ in some sense represents a good approximation to the complete set of values of $F(u(\\xi ))$ .",
"In particular, the approximation of the nonlinear operator is $\\bar{F} (u(\\xi )) = V(P^T V)^{-1}P^T F(u(\\xi ))$ where $P^T$ extracts entries of $F(u)$ corresponding to the interpolation points from the spatial grid.An implementation does not literally construct the matrix $P$ ; instead, an index list is used to extract the required entries of $V$ .",
"This approximation satisfies $P^T \\bar{F} = P^T F$ .",
"To construct $P$ , a greedy procedure is used to minimize the error compared with the full representation of $F(u)$ [7].",
"For each column of $V$ , $v_i$ , the DEIM algorithm selects the row index for which the difference between the column $v_i$ and the approximation of $v_i$ obtained using the DEIM model with nonlinear basis and the first $i-1$ columns of V is maximal, that is, the index of the maximal entry of $r = v_i - \\hat{V}(P^T \\hat{V})^{-1} P^T v_i$ , where $\\hat{V}$ denotes the first $i-1$ columns of $V$ .",
"We present this in Algorithm .",
"DEIM [7] Input: $V = [v_1, ..., v_{n_{deim}}]$ , an $N \\times n_{deim}$ matrix with columns made up of the left singular vectors from the POD of the nonlinear snapshot matrix $S$ .",
"Output: $P$ , extracts the indices used for the interpolation.",
"[1] $\\rho = \\text{argmax}(|v_1|)$ , the index of the maximal entry of $|v_1|$ $\\widehat{V} = [v_1]$ , $P = [e_{\\rho }]$ $i =2:n_{deim}$ Solve $(P^T \\widehat{V})c = P^T v_i$ for $c$ $r = v_i - \\widehat{V}c$ $\\rho = \\text{argmax}(|r|)$ $\\widehat{V} = [\\widehat{V}, v_i]$ , $P = [P,e_{\\rho }] $ Incorporating this approximation into the reduced model, equation (REF ), yields $\\bar{F}^r(\\hat{u}) = Q^T \\bar{F}(\\tilde{u}) = Q^T V ( P^T V) ^{-1} P^T F(Q\\hat{u}) \\; .$ The construction of nonlinear basis matrix $V$ and the interpolation points are part of the offline computation.",
"Since $L^T = Q^T V(P^T V)^{-1}$ is parameter independent, it too can be computed offline.",
"Therefore, the online computations required are to compute $P^T J_F(u)$ and assemble $L^T (P^T J_F(u))Q$ .",
"For $P^TJ_F(u)$ , we need only to compute the components of $J_F(u)$ that are nonzero at the interpolation points.",
"This is where the assumption that each component of $F(u)$ (and thus $J_F(u)$ ) depends on only a few entries of $u$ is utilized.",
"With a finite element discretization, a component $F_i(u)$ depends on the components $u_j$ for which the intersection of the support of the basis functions have measure that is nonzero.",
"See [1] for additional discussion of this point.",
"The elements that must be tracked in the DEIM computations are referred to as the sample mesh.",
"When the sample mesh is small, the computational cost of assembling $L^T (P^T J_F(u)) Q$ scales not with $N$ but with the number of interpolation points.",
"Therefore, DEIM will decrease the online cost associated with assembling the nonlinear component of the solution.",
"For the Navier-Stokes equations, the nonlinear component is a function of the velocity.",
"We will discretize the velocity space using biquadratic ($Q_2$ ) elements.",
"In this case, an entry in $F_i(u)$ depends on at most nine entries of $u$ .",
"Thus this nonlinearity is amenable to using DEIM.",
"An existing finite element routine can be used for the assembly of the required entries of the Jacobian using the sample mesh, a subset of the original mesh, as the input.",
"The accuracy of this approximation is determined primarily by the quality of the nonlinear basis $V$ .",
"This can be seen by considering the error bound $|| F - \\bar{F} ||_2 \\le || (P^T V )^{-1} ||_2 ||(I - VV^T)F||_2$ which is derived and discussed in more detail in [7].",
"There it is shown that the greedy selection of indices in Algorithm limits the growth of $||(P^T V)^{-1}||_2$ as the dimension of $V$ grows.",
"The second term $||(I - VV^T)F||_2$ is the quantity that is determined by the quality of $V$ .",
"Note that if $V$ is taken from the truncated POD of $S$ , the matrix of nonlinear snapshots, then $||(I-VV^T)S||_F^2$ is minimized [1] where $||\\cdot ||_F$ is the Frobenius norm ($||X||_F^2 = \\sum _i \\sum _j |x_{ij}|^2$ ).",
"So the accuracy of the DEIM approximation depends on two factors.",
"The first is the number, $n_{deim}$ , of singular vectors kept in the POD.",
"The truncated matrix $V \\Sigma _{deim}W_{deim}^T$ is the optimal rank-$n_{deim}$ approximation of $S$ , but a higher rank approximation will improve accuracy of the DEIM model.",
"In fact the error $||(I - VV^T)F||_2$ approaches 0 in the limit as $n_{deim}$ approaches $N$ .",
"The second factor is the quality of the nonlinear snapshots in $S$ .",
"The nonlinear component should be sampled well enough to capture the variations of the nonlinear component throughout the solution space.",
"A comparison of methods for selecting the snapshot set is given in Section REF .",
"Given the full model defined in equation (REF ), let $J_G(u) = A + J_F(u)$ denote the Jacobian matrix.",
"The Jacobian of the reduced model equation (REF ) is then $J_{G^r}(\\hat{u}) = Q^T J_G(Q\\hat{u}) Q = Q^T A Q + Q^T J_F(Q\\hat{u}) Q, $ and the Jacobian $\\bar{F}^r(\\hat{u})$ of (REF ) is $J_{\\bar{F}^r}(\\hat{u}) \\ = \\ Q^T J_{\\bar{F}}(Q\\hat{u}) Q \\ = \\ Q^T V (P^T V)^{-1} P^T J_F(u) Q \\; .",
"$"
],
[
"Steady-state Navier-Stokes equations",
"A discrete formulation of the steady-state Navier-Stokes equations (REF ) is to find ${\\vec{u}}_h \\in X_E^h$ and $p_h \\in M^h$ such that $\\begin{array}{rclc}(\\nu (\\cdot ,\\xi ) \\nabla {\\vec{u}}_h, \\nabla {\\vec{v}}_h) + ({\\vec{u}}_h \\cdot \\nabla {\\vec{u}}_h,{\\vec{v}}_h) - (p_h,\\nabla \\cdot {\\vec{v}}_h) &=& (f,{\\vec{v}}_h) &\\forall {\\vec{v}}_h \\in X_0^h \\\\ \\nonumber (\\nabla \\cdot {\\vec{u}}_h,q_h) &=& 0 &\\forall q_h \\in M^h\\end{array}$ where $X_E^h$ and $M^h$ are finite-dimensional subspaces of the Sobolev spaces $H_0^1=\\lbrace \\vec{v}\\in H^1|\\vec{v} = 0 \\text{ on } \\partial D\\rbrace $ and $L_2(D)$ ; see [10], [12] for details.",
"We will use div-stable $Q_2$ -$P_{-1}$ finite element (biquadratic velocities, piecewise constant discontinuous pressure).",
"Let $\\lbrace \\phi _1, ..., \\phi _{n_u}\\rbrace $ represent a basis of $Q_2$ and $\\lbrace \\psi _1, ..., \\psi _{n_p}\\rbrace $ represent a basis of $P_{-1}$ .",
"We define the following vectors and matrices, where $u$ and $p$ here represent the vectors of coefficients that determine ${\\vec{u}}_h$ and $p_h$ , respectively: $\\textbf {z}= \\begin{bmatrix} u \\\\ p \\end{bmatrix},$ $[A(\\xi )]_{ij} = \\int \\nu (\\xi ) \\nabla \\phi _i : \\nabla \\phi _j \\,, \\; \\; \\; \\; \\;[B]_{ij} = - \\int \\psi _i ( \\nabla \\cdot \\phi _j )\\,, \\; \\; \\; \\; \\;[N(u)]_{ij} = \\int ({\\vec{u}}_h \\cdot \\nabla \\phi _j) \\cdot \\phi _i\\,,$ $[\\textbf {f}]_i = (f, \\phi _i), \\; \\; \\; \\; \\;[g({\\vec{u}})]_i = - (\\nabla \\cdot {\\vec{u}}_h, \\psi _i) , \\; \\; \\; \\; \\;\\textbf {b}(\\xi ) = \\begin{bmatrix} \\textbf {f}- A(\\xi )u_{bc} \\\\ g(u_{bc}) \\end{bmatrix}\\,,$ where $u_{bc}$ is the vector of coefficients of the discrete velocity field ${\\vec{u}}_{bc}$ that interpolates the Dirichlet boundary data $b(\\cdot ,\\xi )$ and is zero everywhere on the interior of the mesh.",
"We denote the velocity solution on the interior of the mesh, ${\\vec{u}}_{in}$ , so that ${\\vec{u}}= {\\vec{u}}_{bc} + {\\vec{u}}_{in}$ and ${\\vec{u}}_{in}$ satisfies homogenous Dirichlet boundary conditions.",
"The reduced basis is constructed using snapshots of ${\\vec{u}}_{in}$ so the approximation of the velocity solution generated by the reduced model is of the form $\\tilde{u}= u_{bc} + Q_u \\hat{u}$ where the columns of $Q_u$ correspond to a basis spanning the space of velocity snapshots with homogeneous Dirichlet boundary conditions."
],
[
"Full model",
"With this notation, the full discrete model for the Navier-Stokes problem with parameter $\\xi $ is to find $\\textbf {z}(\\xi )$ such that $G(\\textbf {z}(\\xi )) = 0$ where $G(\\textbf {z}(\\xi )) = \\begin{bmatrix}A(\\xi ) & B^T \\\\B & 0 \\\\\\end{bmatrix}\\begin{bmatrix}u \\\\ p\\end{bmatrix}+\\begin{bmatrix}N(u) & 0 \\\\0 & 0 \\\\\\end{bmatrix}\\begin{bmatrix}u \\\\ p\\end{bmatrix}-\\begin{bmatrix}\\textbf {f}\\\\ 0\\end{bmatrix}\\; .$ We utilize a Picard iteration to solve the full model, monitoring the norm of the nonlinear residual $G(\\textbf {z}_n(\\xi ))$ for convergence.",
"The nonlinear Picard iteration to solve this model is described in Algorithm REF .",
"Picard iteration for solving the discrete steady-state Navier-Stokes equations [1] The nonlinear iteration is initialized with the solution to a Stokes problem $\\begin{bmatrix} A(\\xi ) & B^T \\\\B & 0 \\\\\\end{bmatrix} \\begin{bmatrix} {\\vec{u}}_{in,0} \\\\ p_0 \\end{bmatrix}= \\textbf {b}(\\xi ) \\; .$ Incorporate the boundary conditions ${\\vec{u}}_{0} = {\\vec{u}}_{bc} + {\\vec{u}}_{in,0} \\; .",
"$ Solve $\\left( \\begin{bmatrix} A(\\xi ) & B^T \\\\B & 0 \\\\\\end{bmatrix} + \\begin{bmatrix} N({\\vec{u}}_n) & 0 \\\\0 & 0 \\\\\\end{bmatrix} \\right) \\begin{bmatrix} \\delta {\\vec{u}}\\\\ \\delta p \\end{bmatrix}= -G(\\textbf {z}_n) \\; .$ Update the solutions ${\\vec{u}}_{n+1} &= {\\vec{u}}_{n} + \\delta {\\vec{u}}\\\\ \\nonumber p_{n+1} &= p_{n} + \\delta p \\; .$ Exit if $||G(\\textbf {z}_{n+1})||_2 < \\delta \\left| \\left| \\textbf {b}(\\xi ) \\right| \\right|_2 \\; , $ otherwise return to step 2.",
"Picard iteration for solving the reduced steady-state Navier-Stokes equations [1] Initialize the Picard iteration by solving the reduced Stokes problem $ Q^T \\begin{bmatrix} A(\\xi ) & B^T \\\\B & 0 \\\\\\end{bmatrix} Q \\begin{bmatrix} \\hat{u}_{0} \\\\ \\hat{p}_0 \\end{bmatrix}= Q^T \\textbf {b}(\\xi ) \\; .",
"$ Solve the reduced problem for the Picard iteration $\\left(Q^T \\begin{bmatrix} A(\\xi ) & B^T \\\\B & 0 \\\\ \\end{bmatrix} Q+ Q^T \\begin{bmatrix} N(\\tilde{u}_n) & 0 \\\\ 0 & 0 \\\\ \\end{bmatrix}Q \\right)\\begin{bmatrix} \\delta \\hat{u}\\\\ \\delta \\hat{p}\\end{bmatrix}= -G^r(\\tilde{\\textbf {z}}_n) \\; .$ Note that the when the dependence on the parameters is affine, the first term in the left hand side can be computed primarily offline as in equation (REF ).",
"Update the reduced solutions $\\hat{u}_{n+1} &= \\hat{u}_{n} + \\delta \\hat{u}\\\\\\hat{p}_{n+1} &= \\hat{p}_{n} + \\delta \\hat{p}\\; .",
"\\\\$ Update the approximation to the full solution $\\tilde{u}_{n+1} &= {\\vec{u}}_{bc} + Q_u \\hat{u}_{n+1} \\\\\\tilde{p}_{n+1} &= Q_p \\hat{p}_{n+1} \\; .",
"\\\\$ Compute $N(\\tilde{u}_{n+1})$ .",
"Compute $G^r(\\tilde{\\textbf {z}}_{n+1})$ .",
"Exit if $||G^r(\\tilde{\\textbf {z}}_{n+1})||_2 < \\delta \\left| \\left| Q^T \\textbf {b}(\\xi ) \\right| \\right|_2 \\; , $ otherwise return to step 2.",
"Reduced model Next, we present a reduced model that does not make use of the DEIM strategy.",
"This is not meant to be practical strategy, but it is presented as a comparison to illustrate how the additional approximation used for DEIM affects both the accuracy of solutions obtained using DEIM and the speed with which they are obtained.",
"Offline we compute a reduced basis $Q = \\begin{bmatrix} Q_u & 0 \\\\ 0 & Q_p \\end{bmatrix} \\; ,$ where $Q_u$ represents the reduced basis of the velocity space and $Q_p$ the reduced basis for the pressure space.",
"We defer the details of this offline construction to Section REF .",
"For a given $Q$ , the Galerkin reduced model is $G^r(\\textbf {z}) \\ = \\ Q^T G(\\textbf {z})\\ = \\ \\left( Q^T \\begin{bmatrix}A(\\xi ) & B^T \\\\B & 0 \\\\\\end{bmatrix}Q \\right)\\begin{bmatrix}\\hat{u}\\\\ \\hat{p}\\end{bmatrix}+\\left(Q^T\\begin{bmatrix}N(\\tilde{u}) & 0 \\\\0 & 0 \\\\\\end{bmatrix}Q \\right)\\begin{bmatrix}\\hat{u}\\\\ \\hat{p}\\end{bmatrix}-Q^T\\begin{bmatrix}\\textbf {f}\\\\ 0\\end{bmatrix}\\; .$ Using the nonlinear Picard iteration, the reduced model is described in Algorithm REF .",
"After the convergence of the Picard iteration determined by $G^r(\\tilde{\\textbf {z}}_{n+1})$ , we compute the “full” residual $G(\\tilde{\\textbf {z}}_n)$ .",
"Note that this residual is computed only once; because the cost of computing it is $O(N)$ , it is not monitored during the course of the iteration.",
"The full residual indicates how well the reduced model approximates the full solution, so it is used to measure the quality of the reduced model via the error indicator $\\eta _{\\xi } = ||G(\\tilde{\\textbf {z}}_n(\\xi ))||_2/\\left|\\left| \\textbf {b}(\\xi ) \\right|\\right|_2 \\; .$ DEIM model The DEIM model has the structure of the reduced model but with the nonlinear component $F$ replaced by the approximation $\\bar{F}$ .",
"First in the offline step, we compute $V$ , $P$ , and $L^T = Q_u^T V ( P^T V) ^{-1}$ .",
"The DEIM model is $G^{deim}(\\textbf {z}) =\\left( Q^T \\begin{bmatrix}A(\\xi ) & B^T \\\\B & 0 \\\\\\end{bmatrix}Q \\right)\\begin{bmatrix}\\hat{u}\\\\ \\hat{p}\\end{bmatrix}+\\begin{bmatrix}L^TP^T N(\\tilde{u})Q_u & 0 \\\\0 & 0 \\\\\\end{bmatrix}\\begin{bmatrix}\\hat{u}\\\\ \\hat{p}\\end{bmatrix}- Q^T\\begin{bmatrix}\\textbf {f}\\\\ 0\\end{bmatrix}\\; .$ The computations are shown in Algorithm REF .",
"Recall from the earlier discussion of DEIM that $P^T N(u)$ is not computed by forming the matrix $N(u)$ .",
"Instead, only the components of $N(u)$ corresponding to the indices required by $P^T$ are constructed, using the elements of the discretization mesh that contribute to those indices.",
"Note that error indicator $\\eta _{\\xi }$ in equation (REF ) depends on $G(\\tilde{\\textbf {z}})$ .",
"This quantity contains $N(\\tilde{u}_n)$ and not $P^T N(\\tilde{u}_n)$ , so that computing it requires assembly of $N(\\tilde{u}_n)$ on the entire mesh.",
"As for the reduced model, to avoid this expense, this computation is performed only after convergence of the nonlinear iteration.",
"DEIM model for the steady-state Navier-Stokes equations [1] Initialize the Picard iteration by solving the reduced Stokes problem $Q^T \\begin{bmatrix} A(\\xi ) & B^T \\\\B & 0 \\\\\\end{bmatrix} Q \\begin{bmatrix} \\hat{u}_{0} \\\\ \\hat{p}_0 \\end{bmatrix}= Q^T \\textbf {b}(\\xi ) \\; .$ Solve the reduced problem for the Picard iteration $\\left(Q^T \\begin{bmatrix} A(\\xi ) & B^T \\\\B & 0 \\\\\\end{bmatrix} Q + \\begin{bmatrix} L^T (P^T N(\\tilde{u}_n))Q_u & 0 \\\\0 & 0 \\\\\\end{bmatrix} \\right) \\begin{bmatrix} \\delta \\hat{u}\\\\ \\delta \\hat{p}\\end{bmatrix}= -G^{deim}(\\tilde{\\textbf {z}}_n) \\; .$ Note that the term on the left can be computed cheaply as in equation (REF ) so we only need to update the upper left corner of the matrix as the Picard iteration proceeds.",
"Update the reduced solutions $\\hat{u}_{n+1} &= \\hat{u}_{n} + \\delta \\hat{u}\\\\\\hat{p}_{n+1} &= \\hat{p}_{n} + \\delta \\hat{p}\\; .",
"\\\\$ Compute $P^TN(\\tilde{u}_{n+1})= P^TN(Q\\hat{u}_{n+1})$ at required indices.",
"Compute $G^{deim}(\\tilde{\\textbf {z}}_{n+1})$ .",
"Exit if $||G^{deim}(\\tilde{\\textbf {z}}_{n+1})||_2 < \\delta \\left| \\left|Q^T \\textbf {b}(\\xi ) \\right| \\right|_2 \\; , $ otherwise, return to step 2.",
"Inf-sup condition We turn now to the construction of the reduced basis $Q = \\begin{bmatrix} Q_u & 0 \\\\ 0 & Q_p \\end{bmatrix} \\; .$ Given $k_s$ snapshots of the full model, a natural choice is to have the following spaces generated by these snapshots, $\\begin{array}{rcl}\\text{span}(Q_u) &=& \\text{span} \\lbrace {\\vec{u}}_{in}(\\xi ^{(1)}), ...,{\\vec{u}}_{in}(\\xi ^{(k_s)}) \\rbrace \\\\\\text{span}(Q_p) &=& \\text{span} \\lbrace p(\\xi ^{(1)}), ...,p(\\xi ^{(k_s)}) \\rbrace \\; .",
"\\\\\\end{array}$ However, this choice of basis does not satisfy an inf-sup condition $\\gamma _R := \\min _{0\\ne q_R \\in \\text{span}(Q_p)} \\max _{0\\ne {\\vec{v}}_R \\in \\text{span}(Q_u)}\\frac{(q_R,\\nabla \\cdot {\\vec{v}}_R)}{|{\\vec{v}}_R|_1 ||q_R||_0} \\ge \\gamma ^* > 0$ where $\\gamma ^*$ is independent of $Q_u$ and $Q_p$ .",
"To address this issue, we follow the enrichment procedure of [20], [21].",
"For $i = 1, ..., k_s$ , let ${\\vec{r}}_h(\\cdot ,\\xi ^{(i)})$ be the solution to the Poisson problem $(\\nabla {\\vec{r}}_h(\\cdot ,\\xi ^{(i)}),\\nabla {\\vec{v}}_h) = (p_h(\\cdot ,\\xi ^{(i)}),\\nabla \\cdot {\\vec{v}}_h) \\; \\; \\forall {\\vec{v}}_h \\in X_0^h \\; ,$ and let $Q_u$ of equation (REF ) be augmented by the corresponding discrete solutions $\\lbrace {\\vec{r}}(\\xi ^{(i)})\\rbrace $ , giving the enriched space $\\text{span}(Q_u) = \\text{span} \\lbrace {\\vec{u}}_{in}(\\xi ^{(1)}), ...,{\\vec{u}}_{in}(\\xi ^{(k_s)}),{\\vec{r}}(\\xi ^{(1)}), ..., {\\vec{r}}(\\xi ^{(k_s)} ) \\rbrace \\; .",
"$ These enriching functions satisfy ${\\vec{r}}_h(\\cdot ,\\xi ^{(i)}) = \\arg \\sup _{{\\vec{v}}_h \\in X_0^h} \\frac{(p_h(\\cdot ,\\xi ^{(i)}),\\nabla \\cdot {\\vec{v}}_h)}{|{\\vec{v}}_h|_1} \\; ,$ and thus $\\gamma _R$ defined for the enriched velocity space, $\\text{span}(Q_u)$ , together with $\\text{span}(Q_p)$ , satisfies the inf-sup condition $\\gamma _R \\ge \\gamma _h := \\min _{0\\ne q_h \\in M_h} \\max _{0\\ne {\\vec{v}}_h \\in X_0^h} \\frac{(q_h,\\nabla \\cdot {\\vec{v}}_h)}{|{\\vec{v}}_h|_1 ||q_h||_0} \\; .$ Experiments We consider the steady-state Navier-Stokes equations (REF ) for driven cavity flow posed on a square domain $D = (-1,1) \\times (-1,1)$ .",
"The lid, the top boundary ($y = 1$ ), has velocity profile $u_x = 1 - x^4, \\; \\; u_y = 0 \\; , $ and no-slip conditions ${\\vec{u}}= (0,0)^T$ hold on other boundaries.",
"The source term is $f \\equiv 0$ .",
"The discretization is done on a div-stable $n \\times n$ $Q_2$ -$P_{-1}$ (biquadratic velocities, discontinuous piecewise constant pressures) element grid, giving a discrete velocity space of order $(n+1)^2$ and pressure space of order $3(n/2)^2$ .",
"To define the uncertain viscosity, we divide the domain $D$ into $m = n_d \\times n_d$ subdomains as seen in Figure REF , and the viscosity is taken to be constant and random on each subdomain, $\\nu (\\xi ) = \\xi _i$ .",
"The random parameter vector, $\\xi = [\\xi _1, ..., \\xi _m]^T \\in \\Gamma $ , is comprised of uniform random variables such that $\\xi _i \\in \\Gamma _i = [0.01,1]$ for each $i$ .",
"Therefore, the local subdomain-dependent Reynolds number, $\\mathcal {R}= 2/\\nu $ , will vary between 2 and 200 for this problem.",
"Constant Reynolds numbers in this range give rise to stable steady solutions [10].",
"Figure: Flow domain with piecewise random coefficients for viscosity.The implementation uses IFISS [23] to generate the finite element matrices for the full model.",
"The matrices are then imported into Python and the full, reduced, and DEIM models are constructed and solved using a Python implementation run on an Intel 2.7 GHz i5 processor and 8 GB of RAM.",
"The full model is solved with the method described in equation (REF ) using sparse direct methods implemented in the UMFPACK suite [8] for the system solves in equations (REF ) and (REF ).",
"For this benchmark problem (with enclosed flow), these linear systems are singular [4].",
"This issue is addressed by augmenting the matrix, for example that of (REF ), as $\\begin{bmatrix} A(\\xi ) & B^T & 0 \\\\B & 0 & 0 \\\\0 & \\bar{p} & 0 \\\\\\end{bmatrix} \\begin{bmatrix} {\\vec{u}}_{in} \\\\ p \\\\ z \\end{bmatrix}= \\begin{bmatrix} {2}{*}{\\textbf {b}(\\xi )} \\\\\\\\0 \\end{bmatrix} \\; ,$ where $\\bar{p}$ is a vector corresponding to the element areas of the pressure elements.",
"This removes the singularity by adding a constraint via a Lagrange multiplier so that the average pressure of the solution is zero [22].",
"The same constraint is added to the systems in equation (REF ).",
"Construction of $Q$ and $V$ Construction of reduced basis $Q$ via random sampling, construction of nonlinear basis $V$ Cost: $n_{trial}$ reduced problems and $k$ full problems.",
"[1] Compute the reduced basis and the nonlinear snapshots Solve the full problem $G(\\textbf {z}(\\xi ^{(0)}))=0$ to tolerance $\\delta $ for $\\textbf {z}_n(\\xi ^{(0)})$ .",
"Compute the enriched velocity, $\\vec{r}(\\xi ^{(0)})$ .",
"Initialize $Q_u = [{\\vec{u}}_{in,n}(\\xi ^{(0)}), \\vec{r}(\\xi ^{(0)})]$ and $Q_p = [p_n(\\xi ^{(0)})]$ .",
"Save the nonlinear component of the solution $S = [N({\\vec{u}}_n){\\vec{u}}_n]$ .",
"$k_s=0$ $i =1:n_{trial}$ Randomly select $\\xi ^{(i)}$ .",
"Solve reduced model $G^r(\\tilde{\\textbf {z}}(\\xi ^{(i)}))= 0$ to tolerance $\\delta $ and compute the residual indicator $\\eta _{\\xi ^{(i)}}$ .",
"$\\eta _{\\xi ^{(i)}} > \\tau $ $k_s = k_s + 1$ Solve full model $G(\\textbf {z}(\\xi ^{(i)})) = 0$ .",
"Compute the enriched velocity, $\\vec{r}(\\xi ^{(i)})$ .",
"Add ${\\vec{u}}_{in,n}(\\xi ^{(i)})$ and $\\vec{r}(\\xi ^{(i)})$ to $Q_u$ and $p_n(\\xi ^{(i)})$ to $Q_p$ using modified Gram-Schmidt.",
"Add the nonlinear component to the matrix of nonlinear snapshots, $S = [S, N({\\vec{u}}_n){\\vec{u}}_n]$ .",
"The nonlinear snapshot matrix is $S = [F(\\textbf {z}(\\xi ^{(1)})), ..., F(\\textbf {z}(\\xi ^{(k_s)}))]$ .",
"Compute the POD of the nonlinear snapshot matrix: $S = \\bar{V}\\begin{bmatrix}\\sigma _{1} && \\\\ &\\ddots &\\\\ &&\\sigma _{k_s}\\end{bmatrix}W^T .",
"$ Choose $n_{deim}=rank(S)$ .",
"Define $V = \\bar{V}[:,1:n_{deim}]$ .",
"Compute $P$ using Algorithm with input $V$ .",
"We now describe the methodology we used to compute the reduced bases, $Q_u$ and $Q_p$ , and the nonlinear basis $V$ .",
"The description of the construction of $Q_u$ and $Q_p$ is presented in Algorithm REF .",
"The reduced bases $Q_u$ and $Q_p$ are constructed using random sampling of $n_{trial}$ samples of $\\Gamma $ , denoted $\\Gamma _{trial}$ .",
"The bases are constructed so that all samples $\\xi \\in \\Gamma _{trial}$ have a residual indicator, $\\eta _{\\xi }$ , less than a tolerance, $\\tau $ .",
"The procedure begins with single snapshot $\\textbf {z}(\\xi ^{(0)})$ where $\\xi ^{(0)} = E(\\xi )$ .",
"The bases are initialized using this snapshot, such that $Q_u= [{\\vec{u}}_{in,n}(\\xi ^{(0)}), {\\vec{r}}(\\xi ^{(0)})]$ and $Q_p = [p_n(\\xi ^{(0)})]$ .",
"Then for each sample of $\\Gamma _{trial}$ , the reduced-order model is solved with the current bases $Q_u$ and $Q_p$ .",
"The quality of the reduced solution produced by this reduced-order model can be evaluated using the error indicator $\\eta _{\\xi }$ of (REF ).",
"If $\\eta _{\\xi }$ is smaller than the tolerance $\\tau $ , the computation proceeds to the next sample.",
"If the error indicator exceeds the tolerance, the full model is solved, and then the new snapshots, $u_{in,n}(\\xi )$ and $p_n(\\xi )$ , and the enriched velocity ${\\vec{r}}(\\xi )$ , are used to augment $Q_u$ and $Q_p$ .",
"The experiments use $\\tau = 10^{-4}$ and $n_{trial} = 2000$ parameters to produce bases $Q_u$ and $Q_p$ .",
"An alternative to this strategy of random sampling is greedy sampling, which produces a basis of quasi-optimal dimension [3], [5], in the sense that the the maximum error differs from the Kolmogorov n-width by a constant factor.",
"Our experience [9] is that the performance of the random sampling strategy used here is comparable to that of a greedy strategy.",
"In particular, for several linear benchmark problems, to achieve comparable accuracy, we found that the size of the reduced basis generated by sampling was never more than 10% larger than that produced by a greedy algorithm and in many cases the basis sizes were identical.",
"The computational cost (in CPU time) of the random sampling strategy is significantly lower.",
"(See a discussion of this point in Section REF .)",
"Since our concern in this study is online strategies for reducing the cost of the reduced model, we use the random sampling strategy for the offline computation and remark that the online solution strategies considered here can be used for a reduced basis obtained using any method.",
"We turn now to the methodology for determining the nonlinear basis $V$ , which is the truncated form of $\\bar{V}$ defined in equation (REF ).",
"It was shown in [7] that the choice of the nonlinear basis, $V$ in equation (REF ), is important for the accuracy of the DEIM model.",
"DEIM uses a POD approach for constructing the nonlinear basis.",
"This POD has a two inputs, $S$ , the matrix of nonlinear snapshots, and $n_{deim}$ , the number of vectors retained after truncation.",
"We will compare three strategies for sampling $S$ specified in line REF of Algorithm REF .",
"Full($n_{trial}$ ).",
"This method is most similar to the method used to generate the nonlinear basis in [7].",
"The matrix of nonlinear snapshots, $S$ , is computed from the full solution at every random sample (i.e.",
"$\\lbrace N({\\vec{u}}(\\xi ^{(i)})) {\\vec{u}}(\\xi ^{(i)})\\rbrace _{i=1}^{n_{trial}}$ ).",
"This computation is part of the offline step but its cost, for solving solving $n_{trial}$ full problems, can be quite high.",
"Full($k_s$ ).",
"This is the sampling strategy included in Algorithm REF .",
"It saves the nonlinear component only when the full model is solved for augmenting the reduced basis, $Q$ .",
"Therefore the snapshot set $S$ contains $k_s$ snapshots.",
"Mixed($n_{trial})$ .",
"The final approach aims to mimic the Full($n_{trial}$ ) method with less offline work.",
"This method generates a nonlinear snapshot for each of the $n_{trial}$ random samples using full solutions when they are available (from full solution used for augmenting the reduced basis) and reduced solutions when they are not.",
"As the reduced basis is constructed, when the solution to the full problem is not needed for the reduced basis (i.e.",
"when $\\eta _{\\xi ^{(i)}} < \\tau $ ), use the reduced solution $\\tilde{u}(\\xi ^{(i)})$ to generate the nonlinear snapshot $N(\\tilde{u}(\\xi ^{(i)})) \\tilde{u}(\\xi ^{(i)})$ , where $\\tilde{u}(\\xi ^{(i)}) = {\\vec{u}}_{bc} + Q_u \\hat{u}(\\xi ^{(i)})$ and $Q_u$ is the basis at this value of $i$ .",
"Thus $S$ contains $n_{trial}$ snapshots, but it is constructed using only $k_s$ full model solves.",
"Figure REF compares the performance of the three methods for generating $S$ when Algorithm REF is used to generate $Q_u$ and $Q_p$ .",
"For each $S$ , we take the SVD and truncate with a varying number of vectors, $n_{deim}$ , and plot the average of the residuals of the DEIM solution for 100 randomly generated samples of $\\xi $ .",
"The average residual for the reduced model without DEIM is also shown.",
"It can be seen that as $n_{deim}$ increases, the residuals of the DEIM models approach the residual obtained without using DEIM.",
"It is also evident that the three methods perform similarly.",
"Thus, the Mixed($n_{trial}$ ) approach provides accurate nonlinear snapshots with fewer full solutions than the Full($n_{trial}$ ) method.",
"The Full($k_s$ ) method is essentially as effective as the others and requires both fewer full-system solves than Full($n_{trial}$ ) and the SVD of a smaller matrix than Mixed($n_{trial}$ ); we used Full($k_s$ ) for the remainder of this study.",
"For this method, $n_{deim}\\le k_s$ (in fact, in all cases $n_{deim}=k_s$ ), which is why its results do not fully extend across the horizontal axis in Figure REF ; if the (slightly higher) accuracy exhibited by the other methods is needed, it could be obtained using the Mixed($n_{trial}$ ) method at relatively little extra cost.",
"Figure: A comparison of methods to generate nonlinear snapshots for the DEIM method.",
"DEIM residualvs.",
"n deim n_{deim} averaged for n s =100n_s = 100 samples, n=32n = 32, m=4m = 4, τ=10 -4 \\tau = 10^{-4}, k=306k =306 ,n deim n_{deim} varies.",
"Q u Q_u, Q p Q_p are generated using the Algorithm .",
"Online component - DEIM model versus reduced model In Section , we presented analytic bounds for how accurately the DEIM approximates the nonlinear component of the model.",
"To examine how the approximation affects the accuracy of the reduced model, we will compare the error indicators of DEIM with those obtained using the reduced model without DEIM.",
"We perform the following offline and online computations: Offline: Use Algorithm REF with input $\\tau = 10^{-4}$ and $n_{trial} = 2000$ to generate the reduced bases $Q_u$ , $Q_p$ , $V$ and the indices in $P$ .",
"Online: Solve the problem using the full model, reduced model without DEIM, and the reduced model with DEIM for $n_s=10$ sample parameter sets.",
"Table REF presents the results of these tests.",
"For each benchmark problem, there are four entries.",
"Three of the entries, the first, third and fourth, show the time required to meet the stopping criterion in each of Algorithms 2 (full system), 3 (reduced system) and 4 (reduced system with DEIM) with tolerance $\\delta =10^{-8}$ , together with the final relative residual of the full nonlinear system.",
"The reduced models use a reduced residual for the stopping test; as we have observed, this is done for DEIM to make the cost of the iteration depend on the size of the reduced model rather than $N$ , the size of the full model.",
"(Although the reduced model without DEIM has costs that depend on $N$ , we used the same stopping criterion in order to assess the difference between the two reduced models.)",
"The other entries of Table REF , the second for each example, are the time and residual data for the full system solve with a milder tolerance, $\\delta = 10^{-4}$ ; with this choice, the relative residual is comparable in size to that obtained for the reduced models.",
"For each test problem, the smallest computational time is in boldface.",
"The times presented for the reduced and DEIM models are the CPU time spent in the online computation for the nonlinear iteration only and do not include assembly time or the time to compute the full nonlinear residual after the iteration for the reduced model has converged.",
"The results demonstrate the tradeoff between accuracy and time for the three models.",
"For example, in the case of $m = 16$ and $n = 65$ , the spatial dimension and parameter dimension are large enough that the DEIM model is fastest, and this is generally true for higher-resolution models.",
"If smaller residuals are needed, the accuracy of the DEIM model can be improved by either increasing $n_{deim}$ or improving the accuracy of the reduced model.",
"Increasing $n_{deim}$ has negligible effect on online costs (since $L^T$ is computed offline), but the benefits of doing this are limited.",
"As can be seen from Figure 4.2, the accuracy of the DEIM solution is limited by the accuracy of the reduced solution; once that accuracy is obtained, increasing $n_{deim}$ will produce no additional improvement.",
"The accuracy of the reduced model can be improved using a stricter tolerance $\\tau $ during the offline computation.",
"Since choosing a stricter tolerance for the reduced model causes the size of the reduced basis k to increase, the cost of the reduced and DEIM models will also increase.",
"This means that the benefits of more highly accurate DEIM computations will be obtained only for higher-resolution models.",
"In Table REF , the results for $n = 32$ and $m \\ge 25$ are not shown.",
"For these problems, the number of snapshots required to construct the reduced model, $k_s$ exceeds the size of the pressure space $n_p = 3(n/2)^2 = 768$ .",
"This means that the number of snapshots required for the accuracy of the velocity is higher than the number of degrees of freedom in the full discretized pressure space.",
"Therefore, the spatial discretization is not fine enough for reduced-order modeling to be necessary.",
"Remark.",
"Although offline computations are typically viewed as being inconsequential, we also note that for some of the larger examples tested, where $n = 128$ , the costs of the offline construction are significant.",
"First, the full solutions require over 2 minutes of CPU time.",
"For $m = 36$ , 1013 full solutions and 2000 reduced solutions were required.",
"The cost of each full solve is 132 seconds and the costs of the reduced solves are as high as 98.1 seconds.",
"The computation time for the assembly of the reduced models are not presented in Table REF and are also high.",
"Since $Q$ is changing during the offline stage, the assembly process cannot be made independent of $N$ .",
"For these experiments, the offline computation for $m=49$ took approximately five days, in contrast to 25 seconds for the DEIM solution on the finest grid.",
"Table: Solution time and accuracy for Full, Reduced and DEIM models.Figure REF illustrates the tradeoff between accuracy and time for the DEIM.",
"The top plot compares the error indicators for an average $n_s = 10$ parameters and the bottom plot shows the CPU time for the two methods.",
"While the cost of the DEIM does increase with the number of vectors $n_{deim}$ , we reach similar accuracy as for the reduced model at a much lower cost.",
"It can also be seen that the cost of increasing $n_{deim}$ is small since the maximum considered here ($n_{deim}=96$ ) is significantly smaller than $N = 1089$ .",
"Figure: Top: Error indicator for DEIM model versus n deim n_{deim}.",
"Bottom: CPU time to solve using DEIM direct versus n deim n_{deim}.",
"For n=32n = 32, m=4m = 4, τ=10 -4 \\tau = 10^{-4}, k=306k = 306, averaged over n s =10n_s = 10 samples.Gappy POD Another way to increase the accuracy of the reduced model is to increase the number of interpolation points in the approximation, while keeping the number of basis vectors fixed.",
"This alternative to DEIM for selecting the indices is the so-called gappy POD method [11].",
"This method allows the number or rows selected by $P^T$ to exceed the number of columns of $V$ .",
"The approximation of the function using gappy POD looks similar to DEIM, but the inverse of $P^T V$ is replaced with the Moore-Penrose pseudoinverse $(P^T V)^{\\dagger }$ [6] $\\hat{F}(u) = V (P^T V)^{\\dagger } P^T F(u) \\; .$ To use this operator, we compute $P^T F(u)$ and solve the least squares problem $\\alpha = \\arg \\min _{\\hat{\\alpha }} ||P^T V\\hat{\\alpha } - P^T F(u) ||_2 \\; , $ which leads to the approximation $\\hat{F}(u) = V \\alpha $ .",
"Like $(P^T V)^{-1}$ , the pseudoinverse can be precomputed, in this case using the SVD $P^TV = U\\Sigma W^T$ , giving $(P^T V)^{\\dagger }=W\\Sigma ^{\\dagger }U^T $ where $\\Sigma ^{\\dagger }$ is [16] $\\begin{bmatrix}1/\\sigma _{1} && \\\\ &\\ddots && 0\\\\ &&1/\\sigma _{n_{deim}} & \\end{bmatrix} \\; .$ With this approximation, the index selection method is described in Algorithm REF .",
"Given $V$ computed as in Algorithm REF , we require a method to determine the selection of the row indices that will lead to an accurate representation of the nonlinear component.",
"The approach in [6] is an extension of the greedy algorithm used for DEIM (Algorithm ).",
"It takes as an input the number of grid points and the basis vectors.",
"It simply chooses additional indices per basis vector where the indices correspond to the maximum index of the difference of the basis vector and its projection via the gappy POD model.",
"Recall that in DEIM, the index associated with vector $v_i$ that maximizes $|v_i - V(P^TV)^{-1}P^T v_i|$ (called $\\rho $ in Algorithm 1) is found, and $P$ is augmented by $e_{\\rho }$ .",
"In this extension of the algorithm, at each step $i$ , this type of construction is done $n_g/n_v$ times for $v_i$ , where an index $\\arg \\max (|v_i - V(P^TV)^{\\dagger }P^T v_i|)$ is found and then P and the projection of $v_i$ are updated each time a new such index is chosen.",
"Index selection using gappy POD [6] Input: $n_{g}$ number of indices to choose, $V = [v_1, ..., v_{n_{v}}]$ , an $N \\times n_{v}$ matrix with columns made up of the left singular vectors from the POD of the nonlinear snapshot matrix $S$ .",
"Output: $P$ , extracts the indices used for the interpolation.",
"[1] $n_b = 1$ , $n_{it} = \\min (n_v,n_{g})$ $n_{c,\\min } = \\left\\lfloor {\\frac{n_v}{n_{it}}}\\right\\rfloor $ $n_{a,\\min } = \\left\\lfloor {\\frac{n_{g}}{n_v}}\\right\\rfloor $ $i =1, ..., n_{it}$ $n_c = n_{c,\\min }$ , $n_a = n_{a,\\min }$ $i <=(n_v\\mod {n}_{it}) $ $n_c = n_c + 1$ $i <=(n_{g}\\mod {n}_v)$ $n_a = n_a + 1$$i == 1$ $r = \\sum _{q = 1}^{n_c} v_{q}^2$ $j = 1, ..., n_a$ $\\rho _j = \\text{argmax}(r)$ , $r[\\rho _j] = 0$ $P = [e_{\\rho _1}, ..., e_{\\rho _{n_a}}]$ , $\\widehat{V} = [v_1, ..., v_{n_c}]$ $q = 1, ..., n_c$ $\\alpha = \\min _{\\hat{\\alpha }} ||P^T \\widehat{V}\\hat{\\alpha } - P^T v_{n_b+q}||_2$ $R_q = v_{n_b+q} - \\widehat{V}\\alpha $ $r = \\sum _{q = 1}^{n_c} R_{q}^2$ $j=1, .., n_a$ $\\rho _j = \\text{argmax}(r)$ , $P = [P, e_{\\rho _j}]$ $q =1, ..., n_c$ $\\alpha = \\min _{\\hat{\\alpha }} ||P^T \\widehat{V}\\hat{\\alpha } - P^T v_{n_b+q}||_2$ $R_q = v_{n_b+q} - \\widehat{V}\\alpha $ $r = \\sum _{q = 1}^{n_c} R_{q}^2$ $\\widehat{V} = [\\widehat{V}, v_{n_b+1}, ..., v_{n_b+n_c}]$ , $n_b = n_b + n_c$ To compare the accuracy of this method with DEIM, we use Algorithm REF to compute DEIM and modify line REF to use Algorithm REF with $n_g= 2n_{deim}$ for a range of values of $n_{deim}$ , so that there are two indices selected for each $v_i$ in the gappy algorithm.",
"We use both methods to approximate the nonlinear component and solve the resulting models.",
"Figure REF shows the error indicators as functions of $n_{deim}$ for both methods.",
"It is evident that for smaller number of basis vectors the gappy POD provides additional accuracy.",
"However, for larger number of basis vectors, the additional accuracy provided by gappy POD is small.",
"Thus, the gappy POD method can be used to improve the accuracy when the number of basis vectors is limited.",
"Since $S$ is generated using the Full($k_s$ ) described in Section REF , no additional accuracy is gained for the DEIM method when $n_{deim} > 102$ .",
"Figure: Average error indicator as a function of basis vectors for reduced, DEIM, and gappy POD methods.",
"For n=32n = 32, m=4m = 4, τ=10 -4 \\tau = 10^{-4}, k=306k = 306, n deim n_{deim} varies, and n g =2n deim n_{g} = 2n_{deim}.",
"Averaged over n s =100n_s = 100 samples.",
"Iterative methods We have seen that the DEIM generates reduced-order models that produce solutions essentially as accurate as the reduced solution.",
"In addition, Table REF and Figure REF illustrate that as expected, the DEIM model significantly decreases the online time spent constructing the nonlinear component of the reduced model.",
"Since $Q_u^T N({\\vec{u}}) Q_u$ has been replaced by a cheap approximation, $L^T P^T N({\\vec{u}})Q_u$ , the remaining cost of the nonlinear iteration in the DEIM is that of the linear system solve in line REF of Algorithm REF .",
"The cost of this computation depends on the rank of the reduced basis $k$ , which is $3k_s$ in our setting, where $k_s$ is the number of snapshots used to construct the reduced basis.",
"These quantities depend on properties of the problem such as the number of parameters in the model or the desired level of accuracy in the reduced model.",
"In contrast, the cost of solving the full model is independent of the number of parameters, and it could be as small as $O(N)$ if multigrid methods can be utilized.",
"Thus, it may happen that $k$ is much less than $N$ , but the $O(k^3)$ cost of solving the DEIM model is larger than the cost of solving the full model.",
"This is the case, for example, for $n=64$ , $m = 49$ in Table REF , where the CPU time to solve the full model is half that of solving the DEIM model using direct methods.",
"An alternative is to use iterative methods to solve the reduced model.",
"Their cost is $O(k^2p)$ where $p$ is the number of iterations required for convergence, so that there are values of $k$ where, if $p$ is small enough, iterative methods will be preferable to direct methods.",
"We have seen examples of this for linear problems in [9].",
"In this section, we discuss the use of iterative methods based on preconditioned Krylov subspace methods to improve the efficiency of the DEIM model.",
"For iterative methods to be efficient, effective preconditioners are needed.",
"In the offline-online paradigm, it is also desirable to make the construction of the preconditioner independent of parameters, so that this construction can be part of the offline step.",
"Thus, we will develop preconditioners that depend only on the mean value $\\xi ^{(0)}$ from the parameter set, and refer to such techniques as “offline” preconditioners.",
"To see the impact of this choice, we will also compare the performance of these approaches with “online” versions of them, where the preconditioning operator for a model with parameter $\\xi $ is built using that parameter; this approach is not meant to be used in practice since its online cost will depend on $N$ , but it provides insight concerning a lower bound on the iteration count that can be achieved using an offline preconditioner.",
"Versions of the offline approach have been used with stochastic Galerkin methods in [18].",
"We consider two preconditioners of the DEIM model: The Stokes preconditioner is the matrix used for the reduced Stokes solve in equation (REF ), $M_r(\\xi ) = Q^T \\begin{bmatrix}A(\\xi ) & B^T \\\\B & 0 \\\\\\end{bmatrix}Q \\; .$ The Navier-Stokes preconditioner uses the converged solution of the full model ${\\vec{u}}_n$ as the input for $P^TN({\\vec{u}}_n)$ and uses the operator from the DEIM model from equation (REF ), $M_r(\\xi ) = Q^T \\begin{bmatrix}A(\\xi ) & B^T \\\\B & 0 \\\\\\end{bmatrix}Q + \\begin{bmatrix}L^T P^T N({\\vec{u}}_n) Q_u & 0 \\\\0 & 0 \\\\\\end{bmatrix} \\; .$ In both cases, the preconditioned coefficient matrix is $\\mathcal {A}(\\xi ) M_r^{-1}$ where $\\mathcal {A}(\\xi )$ is the coefficient matrix in (REF ).",
"According to the comments above, we consider two versions of each of these, the offline version, $M_r(\\xi _0)$ , and the online version, $M_r(\\xi )$ .",
"Clearly, for the first step of the reduced nonlinear iteration, a Stokes solve, only one linear iteration will be required using the online Stokes preconditioner.",
"For these experiments, we solve the steady-state Navier-Stokes equations for the driven cavity flow problem using the full model, reduced model, and the DEIM model.",
"For the DEIM model, the linear systems are solved using both direct and iterative methods.",
"The iterative methods use the preconditioned bicgstab method [25].",
"The offline construction is described in Algorithm REF ; we use $\\tau = 10^{-4}$ and $n_{trial} = 2000$ .",
"The algorithm chooses $k_s$ snapshots and produces $Q_u$ of rank $2k_s$ and $Q_p$ of rank $k_s$ , yielding reduced models of rank $k = 3k_s$ .",
"The online experiments are run for $n_s = 10$ random parameters.",
"The average number of iterations required for the convergence of the linear systems is presented in Table REF and the average time for the entire nonlinear solve of each model is presented in Table REF .",
"The nonlinear solve time includes the time to compute $N$ or $P^T N$ , but not the time for assembly of the linear component of the model nor the computation time for $\\eta _{\\xi }$ , the norm of the full residual (REF ) for the approximate solution found by the reduced and DEIM models.",
"The iterative methods presented in this table use offline preconditioners.",
"The lowest online CPU time is in bold.",
"The nonlinear iterations are run to tolerance $\\delta = 10^{-8}$ and the bicgstab method for a reduced system with coefficient matrix, $Q^T \\mathcal {A}Q$ , stops when the solution $x^{(i)}$ satisfies $ \\frac{||r - Q^T \\mathcal {A}Qx^{(i)}||}{||r||} < 10^{-9} \\; $ where $r$ is $Q^T \\begin{bmatrix}\\mathbf {f}\\\\ 0 \\end{bmatrix}$ .",
"Table: Average iteration counts of preconditioned bicgstab for solvingequation (),over n s =10n_s = 10 parameter samples.Table: Average time for the entire nonlinear solve, over n s =10n_s = 10 parameter samples.Table: CPU time to construct the preconditionerTable REF illustrates that the performance of the offline preconditioners using the mean parameter compared to the versions that use the exact parameter, with the offline preconditioners requiring more iterations, as expected.",
"In Table REF we compare these offline parameters with the direct DEIM method and determine that for large enough $m$ (number of parameters) and $k$ (size of the reduced basis), the iterative methods are faster than direct methods.",
"For example, for $m=9$ , the direct methods are slightly faster whereas for $m =16$ the iterative methods are faster for all values of $n$ .",
"We also note that the fastest DEIM method is faster than the full model for all cases.",
"Returning to the motivating example of $n=64$ and $m = 49$ , the DEIM iterative method is faster than the full model, whereas the DEIM direct method performs twice as slowly as the full model.",
"Thus, utilizing iterative methods increases the range of $k$ where reduced-order modeling is practical.",
"Table REF presents the (offline) cost of constructing the preconditioners.",
"Since the Navier-Stokes preconditioner uses the full solution, the cost of constructing this preconditioner scales with the costs of the full solution.",
"The cost of constructing the Stokes preconditioner is significantly smaller, because it does not require solving the full nonlinear problem at the mean parameter.",
"It performs similarly to the exact preconditioner in the online computations.",
"Thus, the exact Stokes preconditioner is an efficient option for both offline and online components of this problem.",
"Remark.",
"A variant of the offline preconditioning methods discussed here is a blended approach, in which a small number of preconditioners corresponding to a small set of parameters is constructed offline.",
"Then for online solution of a problem with parameter $\\xi $ , the preconditioner derived from the parameter closest to $\\xi $ can be applied.",
"We found that this approach did not improve performance of the preconditioners considered here.",
"Conclusion We have shown that the discrete interpolation method is effective for solving the steady-state Navier-Stokes equations.",
"This approach produces a reduced-order model that is essentially as accurate as a naive implementation of a reduced basis method without incurring online costs of order $N$ .",
"In cases where the dimension of the reduced basis is large, performance of the DEIM is improved through the use of preconditioned iterative methods to solve the linear systems arising at each nonlinear Picard iteration.",
"This is achieved using the mean parameter to construct preconditioners.",
"These preconditioners are effective for preconditioning the reduced model in the entire parameter space."
],
[
"Experiments",
"We consider the steady-state Navier-Stokes equations (REF ) for driven cavity flow posed on a square domain $D = (-1,1) \\times (-1,1)$ .",
"The lid, the top boundary ($y = 1$ ), has velocity profile $u_x = 1 - x^4, \\; \\; u_y = 0 \\; , $ and no-slip conditions ${\\vec{u}}= (0,0)^T$ hold on other boundaries.",
"The source term is $f \\equiv 0$ .",
"The discretization is done on a div-stable $n \\times n$ $Q_2$ -$P_{-1}$ (biquadratic velocities, discontinuous piecewise constant pressures) element grid, giving a discrete velocity space of order $(n+1)^2$ and pressure space of order $3(n/2)^2$ .",
"To define the uncertain viscosity, we divide the domain $D$ into $m = n_d \\times n_d$ subdomains as seen in Figure REF , and the viscosity is taken to be constant and random on each subdomain, $\\nu (\\xi ) = \\xi _i$ .",
"The random parameter vector, $\\xi = [\\xi _1, ..., \\xi _m]^T \\in \\Gamma $ , is comprised of uniform random variables such that $\\xi _i \\in \\Gamma _i = [0.01,1]$ for each $i$ .",
"Therefore, the local subdomain-dependent Reynolds number, $\\mathcal {R}= 2/\\nu $ , will vary between 2 and 200 for this problem.",
"Constant Reynolds numbers in this range give rise to stable steady solutions [10].",
"Figure: Flow domain with piecewise random coefficients for viscosity.The implementation uses IFISS [23] to generate the finite element matrices for the full model.",
"The matrices are then imported into Python and the full, reduced, and DEIM models are constructed and solved using a Python implementation run on an Intel 2.7 GHz i5 processor and 8 GB of RAM.",
"The full model is solved with the method described in equation (REF ) using sparse direct methods implemented in the UMFPACK suite [8] for the system solves in equations (REF ) and (REF ).",
"For this benchmark problem (with enclosed flow), these linear systems are singular [4].",
"This issue is addressed by augmenting the matrix, for example that of (REF ), as $\\begin{bmatrix} A(\\xi ) & B^T & 0 \\\\B & 0 & 0 \\\\0 & \\bar{p} & 0 \\\\\\end{bmatrix} \\begin{bmatrix} {\\vec{u}}_{in} \\\\ p \\\\ z \\end{bmatrix}= \\begin{bmatrix} {2}{*}{\\textbf {b}(\\xi )} \\\\\\\\0 \\end{bmatrix} \\; ,$ where $\\bar{p}$ is a vector corresponding to the element areas of the pressure elements.",
"This removes the singularity by adding a constraint via a Lagrange multiplier so that the average pressure of the solution is zero [22].",
"The same constraint is added to the systems in equation (REF )."
],
[
"Construction of $Q$ and {{formula:ee4db7dc-0eee-4da6-9b51-c5029b48ecbe}}",
"Construction of reduced basis $Q$ via random sampling, construction of nonlinear basis $V$ Cost: $n_{trial}$ reduced problems and $k$ full problems.",
"[1] Compute the reduced basis and the nonlinear snapshots Solve the full problem $G(\\textbf {z}(\\xi ^{(0)}))=0$ to tolerance $\\delta $ for $\\textbf {z}_n(\\xi ^{(0)})$ .",
"Compute the enriched velocity, $\\vec{r}(\\xi ^{(0)})$ .",
"Initialize $Q_u = [{\\vec{u}}_{in,n}(\\xi ^{(0)}), \\vec{r}(\\xi ^{(0)})]$ and $Q_p = [p_n(\\xi ^{(0)})]$ .",
"Save the nonlinear component of the solution $S = [N({\\vec{u}}_n){\\vec{u}}_n]$ .",
"$k_s=0$ $i =1:n_{trial}$ Randomly select $\\xi ^{(i)}$ .",
"Solve reduced model $G^r(\\tilde{\\textbf {z}}(\\xi ^{(i)}))= 0$ to tolerance $\\delta $ and compute the residual indicator $\\eta _{\\xi ^{(i)}}$ .",
"$\\eta _{\\xi ^{(i)}} > \\tau $ $k_s = k_s + 1$ Solve full model $G(\\textbf {z}(\\xi ^{(i)})) = 0$ .",
"Compute the enriched velocity, $\\vec{r}(\\xi ^{(i)})$ .",
"Add ${\\vec{u}}_{in,n}(\\xi ^{(i)})$ and $\\vec{r}(\\xi ^{(i)})$ to $Q_u$ and $p_n(\\xi ^{(i)})$ to $Q_p$ using modified Gram-Schmidt.",
"Add the nonlinear component to the matrix of nonlinear snapshots, $S = [S, N({\\vec{u}}_n){\\vec{u}}_n]$ .",
"The nonlinear snapshot matrix is $S = [F(\\textbf {z}(\\xi ^{(1)})), ..., F(\\textbf {z}(\\xi ^{(k_s)}))]$ .",
"Compute the POD of the nonlinear snapshot matrix: $S = \\bar{V}\\begin{bmatrix}\\sigma _{1} && \\\\ &\\ddots &\\\\ &&\\sigma _{k_s}\\end{bmatrix}W^T .",
"$ Choose $n_{deim}=rank(S)$ .",
"Define $V = \\bar{V}[:,1:n_{deim}]$ .",
"Compute $P$ using Algorithm with input $V$ .",
"We now describe the methodology we used to compute the reduced bases, $Q_u$ and $Q_p$ , and the nonlinear basis $V$ .",
"The description of the construction of $Q_u$ and $Q_p$ is presented in Algorithm REF .",
"The reduced bases $Q_u$ and $Q_p$ are constructed using random sampling of $n_{trial}$ samples of $\\Gamma $ , denoted $\\Gamma _{trial}$ .",
"The bases are constructed so that all samples $\\xi \\in \\Gamma _{trial}$ have a residual indicator, $\\eta _{\\xi }$ , less than a tolerance, $\\tau $ .",
"The procedure begins with single snapshot $\\textbf {z}(\\xi ^{(0)})$ where $\\xi ^{(0)} = E(\\xi )$ .",
"The bases are initialized using this snapshot, such that $Q_u= [{\\vec{u}}_{in,n}(\\xi ^{(0)}), {\\vec{r}}(\\xi ^{(0)})]$ and $Q_p = [p_n(\\xi ^{(0)})]$ .",
"Then for each sample of $\\Gamma _{trial}$ , the reduced-order model is solved with the current bases $Q_u$ and $Q_p$ .",
"The quality of the reduced solution produced by this reduced-order model can be evaluated using the error indicator $\\eta _{\\xi }$ of (REF ).",
"If $\\eta _{\\xi }$ is smaller than the tolerance $\\tau $ , the computation proceeds to the next sample.",
"If the error indicator exceeds the tolerance, the full model is solved, and then the new snapshots, $u_{in,n}(\\xi )$ and $p_n(\\xi )$ , and the enriched velocity ${\\vec{r}}(\\xi )$ , are used to augment $Q_u$ and $Q_p$ .",
"The experiments use $\\tau = 10^{-4}$ and $n_{trial} = 2000$ parameters to produce bases $Q_u$ and $Q_p$ .",
"An alternative to this strategy of random sampling is greedy sampling, which produces a basis of quasi-optimal dimension [3], [5], in the sense that the the maximum error differs from the Kolmogorov n-width by a constant factor.",
"Our experience [9] is that the performance of the random sampling strategy used here is comparable to that of a greedy strategy.",
"In particular, for several linear benchmark problems, to achieve comparable accuracy, we found that the size of the reduced basis generated by sampling was never more than 10% larger than that produced by a greedy algorithm and in many cases the basis sizes were identical.",
"The computational cost (in CPU time) of the random sampling strategy is significantly lower.",
"(See a discussion of this point in Section REF .)",
"Since our concern in this study is online strategies for reducing the cost of the reduced model, we use the random sampling strategy for the offline computation and remark that the online solution strategies considered here can be used for a reduced basis obtained using any method.",
"We turn now to the methodology for determining the nonlinear basis $V$ , which is the truncated form of $\\bar{V}$ defined in equation (REF ).",
"It was shown in [7] that the choice of the nonlinear basis, $V$ in equation (REF ), is important for the accuracy of the DEIM model.",
"DEIM uses a POD approach for constructing the nonlinear basis.",
"This POD has a two inputs, $S$ , the matrix of nonlinear snapshots, and $n_{deim}$ , the number of vectors retained after truncation.",
"We will compare three strategies for sampling $S$ specified in line REF of Algorithm REF .",
"Full($n_{trial}$ ).",
"This method is most similar to the method used to generate the nonlinear basis in [7].",
"The matrix of nonlinear snapshots, $S$ , is computed from the full solution at every random sample (i.e.",
"$\\lbrace N({\\vec{u}}(\\xi ^{(i)})) {\\vec{u}}(\\xi ^{(i)})\\rbrace _{i=1}^{n_{trial}}$ ).",
"This computation is part of the offline step but its cost, for solving solving $n_{trial}$ full problems, can be quite high.",
"Full($k_s$ ).",
"This is the sampling strategy included in Algorithm REF .",
"It saves the nonlinear component only when the full model is solved for augmenting the reduced basis, $Q$ .",
"Therefore the snapshot set $S$ contains $k_s$ snapshots.",
"Mixed($n_{trial})$ .",
"The final approach aims to mimic the Full($n_{trial}$ ) method with less offline work.",
"This method generates a nonlinear snapshot for each of the $n_{trial}$ random samples using full solutions when they are available (from full solution used for augmenting the reduced basis) and reduced solutions when they are not.",
"As the reduced basis is constructed, when the solution to the full problem is not needed for the reduced basis (i.e.",
"when $\\eta _{\\xi ^{(i)}} < \\tau $ ), use the reduced solution $\\tilde{u}(\\xi ^{(i)})$ to generate the nonlinear snapshot $N(\\tilde{u}(\\xi ^{(i)})) \\tilde{u}(\\xi ^{(i)})$ , where $\\tilde{u}(\\xi ^{(i)}) = {\\vec{u}}_{bc} + Q_u \\hat{u}(\\xi ^{(i)})$ and $Q_u$ is the basis at this value of $i$ .",
"Thus $S$ contains $n_{trial}$ snapshots, but it is constructed using only $k_s$ full model solves.",
"Figure REF compares the performance of the three methods for generating $S$ when Algorithm REF is used to generate $Q_u$ and $Q_p$ .",
"For each $S$ , we take the SVD and truncate with a varying number of vectors, $n_{deim}$ , and plot the average of the residuals of the DEIM solution for 100 randomly generated samples of $\\xi $ .",
"The average residual for the reduced model without DEIM is also shown.",
"It can be seen that as $n_{deim}$ increases, the residuals of the DEIM models approach the residual obtained without using DEIM.",
"It is also evident that the three methods perform similarly.",
"Thus, the Mixed($n_{trial}$ ) approach provides accurate nonlinear snapshots with fewer full solutions than the Full($n_{trial}$ ) method.",
"The Full($k_s$ ) method is essentially as effective as the others and requires both fewer full-system solves than Full($n_{trial}$ ) and the SVD of a smaller matrix than Mixed($n_{trial}$ ); we used Full($k_s$ ) for the remainder of this study.",
"For this method, $n_{deim}\\le k_s$ (in fact, in all cases $n_{deim}=k_s$ ), which is why its results do not fully extend across the horizontal axis in Figure REF ; if the (slightly higher) accuracy exhibited by the other methods is needed, it could be obtained using the Mixed($n_{trial}$ ) method at relatively little extra cost.",
"Figure: A comparison of methods to generate nonlinear snapshots for the DEIM method.",
"DEIM residualvs.",
"n deim n_{deim} averaged for n s =100n_s = 100 samples, n=32n = 32, m=4m = 4, τ=10 -4 \\tau = 10^{-4}, k=306k =306 ,n deim n_{deim} varies.",
"Q u Q_u, Q p Q_p are generated using the Algorithm ."
],
[
"Online component - DEIM model versus reduced model",
"In Section , we presented analytic bounds for how accurately the DEIM approximates the nonlinear component of the model.",
"To examine how the approximation affects the accuracy of the reduced model, we will compare the error indicators of DEIM with those obtained using the reduced model without DEIM.",
"We perform the following offline and online computations: Offline: Use Algorithm REF with input $\\tau = 10^{-4}$ and $n_{trial} = 2000$ to generate the reduced bases $Q_u$ , $Q_p$ , $V$ and the indices in $P$ .",
"Online: Solve the problem using the full model, reduced model without DEIM, and the reduced model with DEIM for $n_s=10$ sample parameter sets.",
"Table REF presents the results of these tests.",
"For each benchmark problem, there are four entries.",
"Three of the entries, the first, third and fourth, show the time required to meet the stopping criterion in each of Algorithms 2 (full system), 3 (reduced system) and 4 (reduced system with DEIM) with tolerance $\\delta =10^{-8}$ , together with the final relative residual of the full nonlinear system.",
"The reduced models use a reduced residual for the stopping test; as we have observed, this is done for DEIM to make the cost of the iteration depend on the size of the reduced model rather than $N$ , the size of the full model.",
"(Although the reduced model without DEIM has costs that depend on $N$ , we used the same stopping criterion in order to assess the difference between the two reduced models.)",
"The other entries of Table REF , the second for each example, are the time and residual data for the full system solve with a milder tolerance, $\\delta = 10^{-4}$ ; with this choice, the relative residual is comparable in size to that obtained for the reduced models.",
"For each test problem, the smallest computational time is in boldface.",
"The times presented for the reduced and DEIM models are the CPU time spent in the online computation for the nonlinear iteration only and do not include assembly time or the time to compute the full nonlinear residual after the iteration for the reduced model has converged.",
"The results demonstrate the tradeoff between accuracy and time for the three models.",
"For example, in the case of $m = 16$ and $n = 65$ , the spatial dimension and parameter dimension are large enough that the DEIM model is fastest, and this is generally true for higher-resolution models.",
"If smaller residuals are needed, the accuracy of the DEIM model can be improved by either increasing $n_{deim}$ or improving the accuracy of the reduced model.",
"Increasing $n_{deim}$ has negligible effect on online costs (since $L^T$ is computed offline), but the benefits of doing this are limited.",
"As can be seen from Figure 4.2, the accuracy of the DEIM solution is limited by the accuracy of the reduced solution; once that accuracy is obtained, increasing $n_{deim}$ will produce no additional improvement.",
"The accuracy of the reduced model can be improved using a stricter tolerance $\\tau $ during the offline computation.",
"Since choosing a stricter tolerance for the reduced model causes the size of the reduced basis k to increase, the cost of the reduced and DEIM models will also increase.",
"This means that the benefits of more highly accurate DEIM computations will be obtained only for higher-resolution models.",
"In Table REF , the results for $n = 32$ and $m \\ge 25$ are not shown.",
"For these problems, the number of snapshots required to construct the reduced model, $k_s$ exceeds the size of the pressure space $n_p = 3(n/2)^2 = 768$ .",
"This means that the number of snapshots required for the accuracy of the velocity is higher than the number of degrees of freedom in the full discretized pressure space.",
"Therefore, the spatial discretization is not fine enough for reduced-order modeling to be necessary.",
"Remark.",
"Although offline computations are typically viewed as being inconsequential, we also note that for some of the larger examples tested, where $n = 128$ , the costs of the offline construction are significant.",
"First, the full solutions require over 2 minutes of CPU time.",
"For $m = 36$ , 1013 full solutions and 2000 reduced solutions were required.",
"The cost of each full solve is 132 seconds and the costs of the reduced solves are as high as 98.1 seconds.",
"The computation time for the assembly of the reduced models are not presented in Table REF and are also high.",
"Since $Q$ is changing during the offline stage, the assembly process cannot be made independent of $N$ .",
"For these experiments, the offline computation for $m=49$ took approximately five days, in contrast to 25 seconds for the DEIM solution on the finest grid.",
"Table: Solution time and accuracy for Full, Reduced and DEIM models.Figure REF illustrates the tradeoff between accuracy and time for the DEIM.",
"The top plot compares the error indicators for an average $n_s = 10$ parameters and the bottom plot shows the CPU time for the two methods.",
"While the cost of the DEIM does increase with the number of vectors $n_{deim}$ , we reach similar accuracy as for the reduced model at a much lower cost.",
"It can also be seen that the cost of increasing $n_{deim}$ is small since the maximum considered here ($n_{deim}=96$ ) is significantly smaller than $N = 1089$ .",
"Figure: Top: Error indicator for DEIM model versus n deim n_{deim}.",
"Bottom: CPU time to solve using DEIM direct versus n deim n_{deim}.",
"For n=32n = 32, m=4m = 4, τ=10 -4 \\tau = 10^{-4}, k=306k = 306, averaged over n s =10n_s = 10 samples."
],
[
"Gappy POD",
"Another way to increase the accuracy of the reduced model is to increase the number of interpolation points in the approximation, while keeping the number of basis vectors fixed.",
"This alternative to DEIM for selecting the indices is the so-called gappy POD method [11].",
"This method allows the number or rows selected by $P^T$ to exceed the number of columns of $V$ .",
"The approximation of the function using gappy POD looks similar to DEIM, but the inverse of $P^T V$ is replaced with the Moore-Penrose pseudoinverse $(P^T V)^{\\dagger }$ [6] $\\hat{F}(u) = V (P^T V)^{\\dagger } P^T F(u) \\; .$ To use this operator, we compute $P^T F(u)$ and solve the least squares problem $\\alpha = \\arg \\min _{\\hat{\\alpha }} ||P^T V\\hat{\\alpha } - P^T F(u) ||_2 \\; , $ which leads to the approximation $\\hat{F}(u) = V \\alpha $ .",
"Like $(P^T V)^{-1}$ , the pseudoinverse can be precomputed, in this case using the SVD $P^TV = U\\Sigma W^T$ , giving $(P^T V)^{\\dagger }=W\\Sigma ^{\\dagger }U^T $ where $\\Sigma ^{\\dagger }$ is [16] $\\begin{bmatrix}1/\\sigma _{1} && \\\\ &\\ddots && 0\\\\ &&1/\\sigma _{n_{deim}} & \\end{bmatrix} \\; .$ With this approximation, the index selection method is described in Algorithm REF .",
"Given $V$ computed as in Algorithm REF , we require a method to determine the selection of the row indices that will lead to an accurate representation of the nonlinear component.",
"The approach in [6] is an extension of the greedy algorithm used for DEIM (Algorithm ).",
"It takes as an input the number of grid points and the basis vectors.",
"It simply chooses additional indices per basis vector where the indices correspond to the maximum index of the difference of the basis vector and its projection via the gappy POD model.",
"Recall that in DEIM, the index associated with vector $v_i$ that maximizes $|v_i - V(P^TV)^{-1}P^T v_i|$ (called $\\rho $ in Algorithm 1) is found, and $P$ is augmented by $e_{\\rho }$ .",
"In this extension of the algorithm, at each step $i$ , this type of construction is done $n_g/n_v$ times for $v_i$ , where an index $\\arg \\max (|v_i - V(P^TV)^{\\dagger }P^T v_i|)$ is found and then P and the projection of $v_i$ are updated each time a new such index is chosen.",
"Index selection using gappy POD [6] Input: $n_{g}$ number of indices to choose, $V = [v_1, ..., v_{n_{v}}]$ , an $N \\times n_{v}$ matrix with columns made up of the left singular vectors from the POD of the nonlinear snapshot matrix $S$ .",
"Output: $P$ , extracts the indices used for the interpolation.",
"[1] $n_b = 1$ , $n_{it} = \\min (n_v,n_{g})$ $n_{c,\\min } = \\left\\lfloor {\\frac{n_v}{n_{it}}}\\right\\rfloor $ $n_{a,\\min } = \\left\\lfloor {\\frac{n_{g}}{n_v}}\\right\\rfloor $ $i =1, ..., n_{it}$ $n_c = n_{c,\\min }$ , $n_a = n_{a,\\min }$ $i <=(n_v\\mod {n}_{it}) $ $n_c = n_c + 1$ $i <=(n_{g}\\mod {n}_v)$ $n_a = n_a + 1$$i == 1$ $r = \\sum _{q = 1}^{n_c} v_{q}^2$ $j = 1, ..., n_a$ $\\rho _j = \\text{argmax}(r)$ , $r[\\rho _j] = 0$ $P = [e_{\\rho _1}, ..., e_{\\rho _{n_a}}]$ , $\\widehat{V} = [v_1, ..., v_{n_c}]$ $q = 1, ..., n_c$ $\\alpha = \\min _{\\hat{\\alpha }} ||P^T \\widehat{V}\\hat{\\alpha } - P^T v_{n_b+q}||_2$ $R_q = v_{n_b+q} - \\widehat{V}\\alpha $ $r = \\sum _{q = 1}^{n_c} R_{q}^2$ $j=1, .., n_a$ $\\rho _j = \\text{argmax}(r)$ , $P = [P, e_{\\rho _j}]$ $q =1, ..., n_c$ $\\alpha = \\min _{\\hat{\\alpha }} ||P^T \\widehat{V}\\hat{\\alpha } - P^T v_{n_b+q}||_2$ $R_q = v_{n_b+q} - \\widehat{V}\\alpha $ $r = \\sum _{q = 1}^{n_c} R_{q}^2$ $\\widehat{V} = [\\widehat{V}, v_{n_b+1}, ..., v_{n_b+n_c}]$ , $n_b = n_b + n_c$ To compare the accuracy of this method with DEIM, we use Algorithm REF to compute DEIM and modify line REF to use Algorithm REF with $n_g= 2n_{deim}$ for a range of values of $n_{deim}$ , so that there are two indices selected for each $v_i$ in the gappy algorithm.",
"We use both methods to approximate the nonlinear component and solve the resulting models.",
"Figure REF shows the error indicators as functions of $n_{deim}$ for both methods.",
"It is evident that for smaller number of basis vectors the gappy POD provides additional accuracy.",
"However, for larger number of basis vectors, the additional accuracy provided by gappy POD is small.",
"Thus, the gappy POD method can be used to improve the accuracy when the number of basis vectors is limited.",
"Since $S$ is generated using the Full($k_s$ ) described in Section REF , no additional accuracy is gained for the DEIM method when $n_{deim} > 102$ .",
"Figure: Average error indicator as a function of basis vectors for reduced, DEIM, and gappy POD methods.",
"For n=32n = 32, m=4m = 4, τ=10 -4 \\tau = 10^{-4}, k=306k = 306, n deim n_{deim} varies, and n g =2n deim n_{g} = 2n_{deim}.",
"Averaged over n s =100n_s = 100 samples."
],
[
"Iterative methods",
"We have seen that the DEIM generates reduced-order models that produce solutions essentially as accurate as the reduced solution.",
"In addition, Table REF and Figure REF illustrate that as expected, the DEIM model significantly decreases the online time spent constructing the nonlinear component of the reduced model.",
"Since $Q_u^T N({\\vec{u}}) Q_u$ has been replaced by a cheap approximation, $L^T P^T N({\\vec{u}})Q_u$ , the remaining cost of the nonlinear iteration in the DEIM is that of the linear system solve in line REF of Algorithm REF .",
"The cost of this computation depends on the rank of the reduced basis $k$ , which is $3k_s$ in our setting, where $k_s$ is the number of snapshots used to construct the reduced basis.",
"These quantities depend on properties of the problem such as the number of parameters in the model or the desired level of accuracy in the reduced model.",
"In contrast, the cost of solving the full model is independent of the number of parameters, and it could be as small as $O(N)$ if multigrid methods can be utilized.",
"Thus, it may happen that $k$ is much less than $N$ , but the $O(k^3)$ cost of solving the DEIM model is larger than the cost of solving the full model.",
"This is the case, for example, for $n=64$ , $m = 49$ in Table REF , where the CPU time to solve the full model is half that of solving the DEIM model using direct methods.",
"An alternative is to use iterative methods to solve the reduced model.",
"Their cost is $O(k^2p)$ where $p$ is the number of iterations required for convergence, so that there are values of $k$ where, if $p$ is small enough, iterative methods will be preferable to direct methods.",
"We have seen examples of this for linear problems in [9].",
"In this section, we discuss the use of iterative methods based on preconditioned Krylov subspace methods to improve the efficiency of the DEIM model.",
"For iterative methods to be efficient, effective preconditioners are needed.",
"In the offline-online paradigm, it is also desirable to make the construction of the preconditioner independent of parameters, so that this construction can be part of the offline step.",
"Thus, we will develop preconditioners that depend only on the mean value $\\xi ^{(0)}$ from the parameter set, and refer to such techniques as “offline” preconditioners.",
"To see the impact of this choice, we will also compare the performance of these approaches with “online” versions of them, where the preconditioning operator for a model with parameter $\\xi $ is built using that parameter; this approach is not meant to be used in practice since its online cost will depend on $N$ , but it provides insight concerning a lower bound on the iteration count that can be achieved using an offline preconditioner.",
"Versions of the offline approach have been used with stochastic Galerkin methods in [18].",
"We consider two preconditioners of the DEIM model: The Stokes preconditioner is the matrix used for the reduced Stokes solve in equation (REF ), $M_r(\\xi ) = Q^T \\begin{bmatrix}A(\\xi ) & B^T \\\\B & 0 \\\\\\end{bmatrix}Q \\; .$ The Navier-Stokes preconditioner uses the converged solution of the full model ${\\vec{u}}_n$ as the input for $P^TN({\\vec{u}}_n)$ and uses the operator from the DEIM model from equation (REF ), $M_r(\\xi ) = Q^T \\begin{bmatrix}A(\\xi ) & B^T \\\\B & 0 \\\\\\end{bmatrix}Q + \\begin{bmatrix}L^T P^T N({\\vec{u}}_n) Q_u & 0 \\\\0 & 0 \\\\\\end{bmatrix} \\; .$ In both cases, the preconditioned coefficient matrix is $\\mathcal {A}(\\xi ) M_r^{-1}$ where $\\mathcal {A}(\\xi )$ is the coefficient matrix in (REF ).",
"According to the comments above, we consider two versions of each of these, the offline version, $M_r(\\xi _0)$ , and the online version, $M_r(\\xi )$ .",
"Clearly, for the first step of the reduced nonlinear iteration, a Stokes solve, only one linear iteration will be required using the online Stokes preconditioner.",
"For these experiments, we solve the steady-state Navier-Stokes equations for the driven cavity flow problem using the full model, reduced model, and the DEIM model.",
"For the DEIM model, the linear systems are solved using both direct and iterative methods.",
"The iterative methods use the preconditioned bicgstab method [25].",
"The offline construction is described in Algorithm REF ; we use $\\tau = 10^{-4}$ and $n_{trial} = 2000$ .",
"The algorithm chooses $k_s$ snapshots and produces $Q_u$ of rank $2k_s$ and $Q_p$ of rank $k_s$ , yielding reduced models of rank $k = 3k_s$ .",
"The online experiments are run for $n_s = 10$ random parameters.",
"The average number of iterations required for the convergence of the linear systems is presented in Table REF and the average time for the entire nonlinear solve of each model is presented in Table REF .",
"The nonlinear solve time includes the time to compute $N$ or $P^T N$ , but not the time for assembly of the linear component of the model nor the computation time for $\\eta _{\\xi }$ , the norm of the full residual (REF ) for the approximate solution found by the reduced and DEIM models.",
"The iterative methods presented in this table use offline preconditioners.",
"The lowest online CPU time is in bold.",
"The nonlinear iterations are run to tolerance $\\delta = 10^{-8}$ and the bicgstab method for a reduced system with coefficient matrix, $Q^T \\mathcal {A}Q$ , stops when the solution $x^{(i)}$ satisfies $ \\frac{||r - Q^T \\mathcal {A}Qx^{(i)}||}{||r||} < 10^{-9} \\; $ where $r$ is $Q^T \\begin{bmatrix}\\mathbf {f}\\\\ 0 \\end{bmatrix}$ .",
"Table: Average iteration counts of preconditioned bicgstab for solvingequation (),over n s =10n_s = 10 parameter samples.Table: Average time for the entire nonlinear solve, over n s =10n_s = 10 parameter samples.Table: CPU time to construct the preconditionerTable REF illustrates that the performance of the offline preconditioners using the mean parameter compared to the versions that use the exact parameter, with the offline preconditioners requiring more iterations, as expected.",
"In Table REF we compare these offline parameters with the direct DEIM method and determine that for large enough $m$ (number of parameters) and $k$ (size of the reduced basis), the iterative methods are faster than direct methods.",
"For example, for $m=9$ , the direct methods are slightly faster whereas for $m =16$ the iterative methods are faster for all values of $n$ .",
"We also note that the fastest DEIM method is faster than the full model for all cases.",
"Returning to the motivating example of $n=64$ and $m = 49$ , the DEIM iterative method is faster than the full model, whereas the DEIM direct method performs twice as slowly as the full model.",
"Thus, utilizing iterative methods increases the range of $k$ where reduced-order modeling is practical.",
"Table REF presents the (offline) cost of constructing the preconditioners.",
"Since the Navier-Stokes preconditioner uses the full solution, the cost of constructing this preconditioner scales with the costs of the full solution.",
"The cost of constructing the Stokes preconditioner is significantly smaller, because it does not require solving the full nonlinear problem at the mean parameter.",
"It performs similarly to the exact preconditioner in the online computations.",
"Thus, the exact Stokes preconditioner is an efficient option for both offline and online components of this problem.",
"Remark.",
"A variant of the offline preconditioning methods discussed here is a blended approach, in which a small number of preconditioners corresponding to a small set of parameters is constructed offline.",
"Then for online solution of a problem with parameter $\\xi $ , the preconditioner derived from the parameter closest to $\\xi $ can be applied.",
"We found that this approach did not improve performance of the preconditioners considered here."
],
[
"Conclusion",
"We have shown that the discrete interpolation method is effective for solving the steady-state Navier-Stokes equations.",
"This approach produces a reduced-order model that is essentially as accurate as a naive implementation of a reduced basis method without incurring online costs of order $N$ .",
"In cases where the dimension of the reduced basis is large, performance of the DEIM is improved through the use of preconditioned iterative methods to solve the linear systems arising at each nonlinear Picard iteration.",
"This is achieved using the mean parameter to construct preconditioners.",
"These preconditioners are effective for preconditioning the reduced model in the entire parameter space."
]
] | 1605.06138 |
[
[
"On centers of blocks with one simple module"
],
[
"Abstract Let G be a finite group, and let B be a non-nilpotent block of G with respect to an algebraically closed field of characteristic 2.",
"Suppose that B has an elementary abelian defect group of order 16 and only one simple module.",
"The main result of this paper describes the algebra structure of the center of B.",
"This is motivated by a similar analysis of a certain 3-block of defect 2 in [Kessar, 2012]."
],
[
"Introduction",
"This paper is concerned with the algebra structure of the center of a $p$ -block $B$ of a finite group $G$ .",
"In order to make precise statements let $(K,\\mathcal {O},F)$ be a $p$ -modular system where $\\mathcal {O}$ is a complete discrete valuation ring of characteristic 0, $K$ is the field of fractions of $\\mathcal {O}$ , and $F=\\mathcal {O}/J(\\mathcal {O})=\\mathcal {O}/(\\pi )$ is an algebraically closed field of characteristic $p$ .",
"As usual, we assume that $K$ is a splitting field for $G$ .",
"A well-known result by Broué-Puig [8] asserts that if $B$ is nilpotent, then the number of irreducible Brauer characters in $B$ equals $l(B)=1$ .",
"Since the algebra structure of nilpotent blocks is well understood by work of Puig [26], it is natural to study non-nilpotent blocks with only one irreducible Brauer character.",
"These blocks are necessarily non-principal (see [24]) and maybe the first example was given by Kiyota [17].",
"Here, $p=3$ and $B$ has elementary abelian defect group of order 9.",
"More generally, a theorem by Puig-Watanabe [28] states that if the defect group of $B$ is abelian, then $B$ has a Brauer correspondent with more than one simple module.",
"Ten years later, Benson-Green [2] and others [13], [16] have developed a general theory of these blocks by making use of quantum complete intersections.",
"Applying this machinery, Kessar [15] was able to describe the algebra structure of Kiyota's example explicitly.",
"Her arguments were simplified recently in [21].",
"We also mention two more recent papers dealing with these blocks.",
"Malle-Navarro-Späth [23] have shown that the unique irreducible Brauer character in $B$ is the restriction of an ordinary irreducible character.",
"Finally, Benson-Kessar-Linckelmann [3] studied Hochschild cohomology in order to obtain results on blocks of defect 2 with only one irreducible Brauer character.",
"In the present paper we deal with the second smallest example in terms of defect groups.",
"Here, $p=2$ and $B$ has elementary abelian defect group $D$ of order 16.",
"In [22] the numerical invariants of $B$ have been determined.",
"In particular, it is known that the number of irreducible ordinary characters (of height 0) of $B$ is $k(B)=k_0(B)=8$ .",
"Moreover, the inertial quotient $I(B)$ of $B$ is elementary abelian of order 9.",
"Examples for $B$ are given by the non-principal blocks of $G=\\texttt {SmallGroup}(432,526)\\cong D\\rtimes 3^{1+2}_+$ where $3^{1+2}_+$ denotes the extraspecial group of order 27 and exponent 3.",
"Since the algebra structure of $B$ seems too difficult to describe at the moment, we are content with studying the center $\\operatorname{Z}(B)$ as an algebra over $F$ .",
"As a consequence of Broué's Abelian Defect Group Conjecture, the isomorphism type of $\\operatorname{Z}(B)$ should be independent of $G$ .",
"In fact, our main theorem is the following.",
"Theorem 1.1 Let $B$ be a non-nilpotent 2-block with elementary abelian defect group of order 16 and only one irreducible Brauer character.",
"Then $\\operatorname{Z}(B)\\cong F[X,Y,Z_1,\\ldots ,Z_4]/\\langle X^2+1,\\,Y^2+1,\\,(X+1)Z_i,\\,(Y+1)Z_i,\\,Z_iZ_j\\rangle .$ In particular, $\\operatorname{Z}(B)$ has Loewy length 3.",
"The paper is organized as follows.",
"In the second section we consider the generalized decomposition matrix $Q$ of $B$ .",
"Up to certain choices there are essentially three different possibilities for $Q$ .",
"A result by Puig [27] (cf.",
"[9]) describes the isomorphism type of $\\operatorname{Z}(B)$ (regarded over $\\mathcal {O}$ ) in terms of $Q$ .",
"In this way we prove that there are at most two isomorphism types for $\\operatorname{Z}(B)$ .",
"In the two subsequent sections we apply ring-theoretical arguments to the basic algebra of $B$ in order to exclude one possibility for $\\operatorname{Z}(B)$ .",
"Finally, we give some concluding remarks in the last section.",
"Our notation is standard and can be found in [24], [29]."
],
[
"The generalized decomposition matrix",
"From now on we will always assume that $B$ is given as in main with defect group $D$ .",
"Since a Sylow 3-subgroup of $\\operatorname{Aut}(D)\\cong \\operatorname{GL}(4,2)\\cong A_8$ has order 9, the action of $I(B)$ on $D$ is essentially unique.",
"In particular, the $I(B)$ -conjugacy classes of $D$ have lengths 1, 3, 3 and 9.",
"Let $\\mathcal {R}=\\lbrace 1,x,y,xy\\rbrace $ be a set of representatives for these classes.",
"For $u\\in \\mathcal {R}$ we fix a $B$ -subsection $(u,b_u)$ .",
"Recall that $b_u$ is a Brauer correspondent of $B$ in $G(u)$ with defect group $D$ .",
"Moreover, the inertial quotient of $b_u$ is given by $I(b_u)\\cong {I(B)}(u)$ .",
"Since $D$ has exponent 2, the generalized decomposition numbers $d^u_{\\chi \\varphi }$ for $\\chi \\in \\operatorname{Irr}(B)$ and $\\varphi \\in \\operatorname{IBr}(b_u)$ are (rational) integers.",
"We set $Q_u\\mathrel {\\mathop }=(d^u_{\\chi \\varphi }\\mathrel {\\mathop }\\chi \\in \\operatorname{Irr}(B),\\varphi \\in \\operatorname{IBr}(b_u))$ for $u\\in \\mathcal {R}$ .",
"Then $C_u\\mathrel {\\mathop }=Q_u^\\text{T}Q_u$ is the Cartan matrix of $b_u$ where $Q_u^\\text{T}$ denotes the transpose of $Q_u$ .",
"On the other hand, the orthogonality relation implies $Q_u^\\text{T}Q_v=0\\in \\mathbb {Z}^{l(b_u)\\times l(b_v)}$ for $u\\ne v\\in \\mathcal {R}$ .",
"A basic set for $b_u$ is a basis for the $\\mathbb {Z}$ -module of class functions on the 2-regular elements of $G(u)$ spanned by $\\operatorname{IBr}(b_u)$ .",
"If we change the underlying basic set, the matrix $Q_u$ transforms into $Q_uS$ where $S\\in \\operatorname{GL}(l(b_u),\\mathbb {Z})$ .",
"Similarly, $C_u$ becomes $S^\\text{T}C_uS$ .",
"By [27] the isomorphism type of $\\operatorname{Z}(B)$ does not depend on the chosen basic sets.",
"Following Brauer [4], we define the contribution matrix of $b_u$ by $M^u\\mathrel {\\mathop }=(m^u_{\\chi \\psi })_{\\chi ,\\psi \\in \\operatorname{Irr}(B)}\\mathrel {\\mathop }=Q_uC_u^{-1}Q_u^\\text{T}\\in \\mathbb {Q}^{8\\times 8}.$ Observe that $M^u$ does not depend on the choice of the basic set, but on the order of $\\operatorname{Irr}(B)$ .",
"Since the largest elementary divisor of $C_u$ equals 16, it follows that $16M^u\\in \\mathbb {Z}^{8\\times 8}$ .",
"Moreover, all entries of $16M^u$ are odd, because all irreducible characters of $B$ have height 0 (see [29]).",
"We may assume that $l(b_x)=l(b_y)=3$ and $l(b_{xy})=1$ .",
"Then the Cartan matrices of $b_x$ and $b_y$ are given by $C_x=C_y=4\\begin{pmatrix}2&1&1\\\\1&2&1\\\\1&1&2\\end{pmatrix}$ up to basic sets (see e. g. [30]).",
"It is well-known that the entries of $Q_1$ are positive.",
"Since $C_1=C_{xy}=(16)$ , we may choose the order of $\\operatorname{Irr}(B)$ such that $Q_1=(3,1,1,1,1,1,1,1)^\\text{T}.$ Now we do some computations with the $*$ -construction introduced in [7].",
"Observe that the following generalized characters of $D$ are $I(B)$ -stable: Table: NO_CAPTIONSince $\\sum _{u\\in \\mathcal {R}}{\\lambda _i(u)m_{\\chi \\psi }^u}=(\\chi \\mathbin {\\ast }\\lambda _i,\\psi )_G\\in \\mathbb {Z}&&(\\chi ,\\psi \\in \\operatorname{Irr}(B))$ for $i=1,2,3$ , we obtain the following relations between the contribution matrices: $16M^1+16M^x\\equiv 16M^1+16M^y\\equiv 16M^x+16M^{xy}\\equiv 0_8\\pmod {4}.$ For the trivial character $\\lambda $ we obtain $\\sum _{u\\in \\mathcal {R}}{M^u}=1_8$ .",
"Therefore, $d_{11}^{xy}=\\pm 1$ .",
"After changing the basic set for $b_{xy}$ (i. e. multiplying $\\varphi \\in \\operatorname{IBr}(b_{xy})$ by a sign), we may assume that $d_{11}^{xy}=1$ .",
"Now (REF ) implies $Q_{xy}=(1,3,-1,-1,-1,-1,-1,-1)^\\text{T}$ for a suitable order of $\\operatorname{Irr}(B)$ .",
"Observe that the orthogonality relation is satisfied.",
"The matrices $Q_x$ and $Q_y$ are (integral) solutions of the matrix equation $X^\\text{T}X=C_x.$ We solve (REF ) by using an algorithm of Plesken [25].",
"In the first step we compute all possible rows $r=(r_1,r_2,r_3)\\in \\mathbb {Z}^3$ of $X$ .",
"These rows satisfy $rC_x^{-1}r^\\text{T}\\le 1$ where $C_x^{-1}=\\frac{1}{16}(-1+4\\delta _{ij})$ .",
"Since in our case the numbers $rC_x^{-1}r^\\text{T}$ are contributions, we get the additional constraint $16rC_x^{-1}r^\\text{T}\\equiv 1\\pmod {2}$ .",
"It follows that $r_1^2+r_2^2+r_3^2+(r_1-r_2)^2+(r_1-r_3)^2+(r_2-r_3)^2\\le 15.$ Thus, up to permutations of $r_i$ and signs we have the following solutions for $r$ : $(1,0,0),(1,1,1),(0,1,2),(1,1,-1),(1,2,2).$ Observe that the first two solutions give a contribution of $3/16$ while the other three solutions give $11/16$ .",
"By [25], the matrix $X$ contains five rows contributing $3/16$ and three rows contributing $11/16$ in the sense above.",
"If we change the basic set of $b_x$ according to the transformation matrix $S\\mathrel {\\mathop }=\\begin{pmatrix}1&.&.\\\\.&1&.\\\\-1&-1&-1\\end{pmatrix},$ then $C_x$ does not change (in fact, $C_x$ is the Gram matrix of the $A_3$ lattice and its automorphism group is $S_4\\times C_2$ ).",
"Doing so, we may assume that the first row of $X$ is $(2,2,1)$ .",
"Now we need to discuss the possibilities for the other rows where we will ignore their signs.",
"We may assume that the second and third row also contribute $11/16$ .",
"It is easy to see that the rows $(1,2,2)$ , $(2,1,2)$ , $(2,2,1)$ , $(1,2,0)$ , $(2,1,0)$ and $(1,1,-1)$ are excluded.",
"Now suppose that the second row is $(2,0,1)$ .",
"Then we may certainly assume that the third row is $(0,1,2)$ or $(0,2,1)$ .",
"In both cases the remaining rows are essentially determined (up to signs and order) as $(I)\\mathrel {\\mathop }\\begin{pmatrix}2&2&1\\\\2&.&1\\\\.&2&1\\\\.&.&1\\\\.&.&1\\\\.&.&1\\\\.&.&1\\\\.&.&1\\end{pmatrix},&&(II)\\mathrel {\\mathop }\\begin{pmatrix}2&2&1\\\\2&.&1\\\\.&1&2\\\\.&1&.\\\\.&1&.\\\\.&1&.\\\\.&.&1\\\\.&.&1\\end{pmatrix}.$ Suppose next that the second row is $(0,1,2)$ .",
"If the third row is $(2,0,1)$ , then we end up in case (II) (interchange the second and third row).",
"Hence, the third row must be $(1,-1,1)$ .",
"Again the remaining rows are essentially determined.",
"In order to avoid negative entries, we give a slightly different representative $(III)\\mathrel {\\mathop }\\begin{pmatrix}2&1&.\\\\.&2&1\\\\1&.&2\\\\1&1&1\\\\1&1&1\\\\1&.&.\\\\.&1&.\\\\.&.&1\\end{pmatrix}.$ Finally, suppose that the second row is $(1,-1,1)$ .",
"Observe that the third row cannot be $(1,0,2)$ .",
"If it is $(0,1,2)$ , then we are in case (III).",
"Therefore, we may assume that the third row is $(-1,1,1)$ .",
"Here a transformation similar to the matrix $S$ above gives case (II).",
"Summarizing we have seen that by ignoring the order and signs of the rows, there exists a matrix $S\\in \\operatorname{GL}(3,\\mathbb {Z})$ such that $XS$ is exactly one of the possibilities (I), (II) or (III).",
"The fact that these solutions are essentially different can be seen by computing the elementary divisors which are $(1,2,2)$ , $(1,1,2)$ and $(1,1,1)$ respectively.",
"In the following we will refer to (I), (II) or (III) whenever $Q_x$ belongs to (I), (II) or (III) respectively.",
"Then the corresponding contribution matrices (multiplied by 16) are given as follows $\\begin{pmatrix}11&5&5&-1&-1&-1&-1&-1\\\\5&11&-5&1&1&1&1&1\\\\5&-5&11&1&1&1&1&1\\\\-1&1&1&3&3&3&3&3\\\\-1&1&1&3&3&3&3&3\\\\-1&1&1&3&3&3&3&3\\\\-1&1&1&3&3&3&3&3\\\\-1&1&1&3&3&3&3&3\\end{pmatrix},\\begin{pmatrix}11&5&1&3&3&3&-1&-1\\\\5&11&-1&-3&-3&-3&1&1\\\\1&-1&11&1&1&1&5&5\\\\3&-3&1&3&3&3&-1&-1\\\\3&-3&1&3&3&3&-1&-1\\\\3&-3&1&3&3&3&-1&-1\\\\-1&1&5&-1&-1&-1&3&3\\\\-1&1&5&-1&-1&-1&3&3\\end{pmatrix},\\\\\\begin{pmatrix}11&-1&-1&3&3&5&1&-3\\\\-1&11&-1&3&3&-3&5&1\\\\-1&-1&11&3&3&1&-3&5\\\\3&3&3&3&3&1&1&1\\\\3&3&3&3&3&1&1&1\\\\5&-3&1&1&1&3&-1&-1\\\\1&5&-3&1&1&-1&3&-1\\\\-3&1&5&1&1&-1&-1&3\\end{pmatrix}.$ Note that the order of the rows does not correspond to the order of $\\operatorname{Irr}(B)$ chosen above.",
"Suppose that case (I) occurs.",
"Then, using (REF ), we may choose a basic set for $b_x$ and the order of the last six characters of $\\operatorname{Irr}(B)$ such that $Q_x=\\begin{pmatrix}.&.&1\\\\.&.&-1\\\\-2&-2&-1\\\\2&.&1\\\\.&2&1\\\\.&.&-1\\\\.&.&-1\\\\.&.&-1\\end{pmatrix}.$ Since $M^1+M^x+M^y+M^{xy}=1_8$ , we obtain $16M^y=\\begin{pmatrix}3&-3&-3&-3&-3&1&1&1\\\\-3&3&3&3&3&-1&-1&-1\\\\-3&3&3&3&3&-1&-1&-1\\\\-3&3&3&3&3&-1&-1&-1\\\\-3&3&3&3&3&-1&-1&-1\\\\1&-1&-1&-1&-1&11&-5&-5\\\\1&-1&-1&-1&-1&-5&11&-5\\\\1&-1&-1&-1&-1&-5&-5&11\\end{pmatrix}.$ Thus, also $Q_y$ corresponds to the first solution above.",
"After choosing an order of the last three characters in $\\operatorname{Irr}(B)$ , we get $Q_y=\\begin{pmatrix}.&.&-1\\\\.&.&1\\\\.&.&1\\\\.&.&1\\\\.&.&1\\\\2&2&1\\\\-2&.&-1\\\\.&-2&-1\\end{pmatrix}.$ Hence, the generalized decomposition matrix of $B$ in case (I) is given by: $(I)\\mathrel {\\mathop }\\begin{pmatrix}3&1&.&.&1&.&.&-1\\\\1&3&.&.&-1&.&.&1\\\\1&-1&-2&-2&-1&.&.&1\\\\1&-1&2&.&1&.&.&1\\\\1&-1&.&2&1&.&.&1\\\\1&-1&.&.&-1&2&2&1\\\\1&-1&.&.&-1&-2&.&-1\\\\1&-1&.&.&-1&.&-2&-1\\end{pmatrix}.$ Now we consider case (II).",
"Here, at first sight it is not clear if the first row of $Q_x$ is $(0,0,1)$ or $(0,1,0)$ .",
"Suppose that it is $(0,0,1)$ .",
"Then we may assume that $16m_{13}^x=5$ .",
"This gives $16(m_{13}^1+m_{13}^x+m_{13}^{xy})=7$ .",
"However, $16m_{13}^y$ can never be $-7$ .",
"Therefore, we may assume that the first row of $Q_x$ is $(0,1,0)$ .",
"Now it is straight forward to obtain the generalized decomposition matrix of $B$ as $(II)\\mathrel {\\mathop }\\begin{pmatrix}3&1&.&-1&.&.&-1&.\\\\1&3&.&1&.&.&1&.\\\\1&-1&2&2&1&.&.&1\\\\1&-1&-2&.&-1&.&.&1\\\\1&-1&.&-1&-2&.&1&.\\\\1&-1&.&1&.&.&-1&-2\\\\1&-1&.&.&1&-2&.&-1\\\\1&-1&.&.&1&2&2&1\\end{pmatrix}.$ Similarly, in case (III) we compute $(III)\\mathrel {\\mathop }\\begin{pmatrix}3&1&-1&-1&-1&1&1&1\\\\1&3&1&1&1&-1&-1&-1\\\\1&-1&2&1&.&.&1&.\\\\1&-1&.&2&1&.&.&1\\\\1&-1&1&.&2&1&.&.\\\\1&-1&-1&.&.&-1&.&-2\\\\1&-1&.&-1&.&-2&-1&.\\\\1&-1&.&.&-1&.&-2&-1\\end{pmatrix}.$ Now let $Q=(q_{ij})$ be the transpose of one of these three generalized decomposition matrices.",
"Let $e$ be the block idempotent of $B$ in $\\mathcal {O}G$ .",
"Then [27] gives an isomorphism $\\operatorname{Z}(\\mathcal {O}Ge)\\cong D_8(K)\\cap Q^{-1}\\mathcal {O}^{8\\times 8}Q=D_8(\\mathcal {O})\\cap Q^{-1}\\mathcal {O}^{8\\times 8}Q=\\mathrel {\\mathop }Z$ where $D_8(K)$ (respectively $D_8(\\mathcal {O})$ ) is the ring of $8\\times 8$ diagonal matrices over $K$ (respectively $\\mathcal {O}$ ).",
"For a matrix $A=(a_{ij})\\in \\mathcal {O}^{8\\times 8}$ the condition $Q^{-1}AQ\\in D_8(K)$ transforms into a homogeneous linear system in $a_{ij}$ with $8^2-8=56$ equations of the form $\\sum _{i,j=1}^8{q^{\\prime }_{ri}q_{js}a_{ij}}=0&&(r\\ne s).$ After multiplying with a common denominator, we may assume that the coefficients of this system are (rational) integers.",
"(Even if $Q$ were not rational, one could get an integral coefficient matrix by using the Galois action of a suitable cyclotomic field.)",
"Using the Smith normal form, it is easy to construct an $\\mathcal {O}$ -basis $\\beta _1,\\ldots ,\\beta _8$ of $Z$ consisting of integral matrices (this can be done conveniently in GAP [11]).",
"For instance, in case (I) such a basis is given by $(I)\\mathrel {\\mathop }\\begin{pmatrix}1&-1&-1&.&.&-3&.&-4\\\\1&7&3&.&.&9&.&12\\\\1&3&-1&-8&.&9&.&12\\\\1&3&7&.&8&9&.&12\\\\1&3&7&8&8&9&.&12\\\\1&3&3&.&.&13&8&12\\\\1&-5&-5&.&-8&-11&.&-12\\\\1&-5&-5&.&-8&-11&-8&-12\\end{pmatrix}$ where each column is the diagonal of a basis vector.",
"The canonical ring epimorphism $\\operatorname{Z}(\\mathcal {O}G)\\rightarrow \\operatorname{Z}(FG)$ sending class sums to class sums restricts to an epimorphism $\\operatorname{Z}(\\mathcal {O}Ge)=\\operatorname{Z}(\\mathcal {O}G)e\\rightarrow \\operatorname{Z}(B)$ with kernel $\\operatorname{Z}(\\mathcal {O}G)\\pi \\cap \\operatorname{Z}(\\mathcal {O}Ge)=\\operatorname{Z}(\\mathcal {O}Ge)\\pi $ .",
"This gives an isomorphism of $F$ -algebras $\\operatorname{Z}(B)\\cong \\operatorname{Z}(\\mathcal {O}Ge)/\\operatorname{Z}(\\mathcal {O}Ge)\\pi \\cong Z/\\pi Z.$ Obviously, the elements $\\beta _i+\\pi Z$ form an $F$ -basis of $Z/\\pi Z$ .",
"Thus, in order to obtain a presentation for $\\operatorname{Z}(B)$ it suffices to reduce the structure constants coming from $\\beta _i$ modulo 2.",
"An even nicer presentation can be achieved by replacing the generators with some $\\mathbb {F}_2$ -linear combinations.",
"Eventually, this proves the following result.",
"Proposition We have $\\operatorname{Z}(B)\\cong {\\left\\lbrace \\begin{array}{ll}F[X,Y,Z_1,\\ldots ,Z_4]/\\langle X^2+1,\\,Y^2+1,\\,(X+1)Z_i,\\,(Y+1)Z_i,\\,Z_iZ_j\\rangle &\\text{case (I) or (II)},\\\\F[X,Z_1,\\ldots ,Z_6]/\\langle X^2+1,\\,XZ_{2i}+Z_{2i-1},\\,Z_iZ_j\\rangle &\\text{case (III)}.\\end{array}\\right.",
"}$ These two algebras are non-isomorphic, since $\\dim _FJ(\\operatorname{Z}(B))^2$ differs.",
"In the following two sections we will see that the second alternative in prop2 does not occur."
],
[
"Tools from ring theory",
"In this section we will gather some well known facts about local symmetric $F$ -algebras and applications thereof to our block $B$ .",
"We start with some basic lemmas: Lemma ([15]) Let $A$ be a local symmetric $F$ -algebra.",
"Then the following hold: $\\operatorname{dim}_F \\operatorname{soc}(A)=1$ .",
"$\\operatorname{soc}(A)\\subseteq \\operatorname{soc}(\\operatorname{Z}(A))$ .",
"$\\operatorname{soc}(A)\\cap [A,A]=0$ .",
"$\\operatorname{dim}_F A=\\operatorname{dim}_F \\operatorname{Z}(A)+\\operatorname{dim}_F [A,A]$ .",
"$\\operatorname{Z}(A)$ is local and $J(A)\\cap \\operatorname{Z}(A) =J(\\operatorname{Z}(A))$ .",
"If $n$ is the least natural number such that $J^{n+1}(A)=0$ , then $J^n(A)=\\operatorname{soc}(A)$ .",
"Lemma ([19]) Let $A$ be an $F$ -algebra, let $I$ be a two-sided ideal in $A$ and let $n\\in \\mathbb {N}$ .",
"Suppose $I^n=F\\lbrace x_{i1}\\dots x_{in} \\mid i=1,\\dots ,d\\rbrace +I^{n+1}$ with elements $x_{ij}\\in I$ .",
"Then we have $I^{n+1}=F\\lbrace x_{j1}x_{i1}\\dots x_{in} \\mid i,j=1,\\dots ,d\\rbrace +I^{n+2},$ and also $I^{n+1}=F\\lbrace x_{i1}\\dots x_{in}x_{jn} \\mid i,j=1,\\dots ,d\\rbrace +I^{n+2} .$ The proof of the last statement of this lemma goes exactly as in [19].",
"We just have to do everything from the opposing side.",
"Lemma ([19]) Let $A$ be a local symmetric $F$ -algebra and let $n\\in \\mathbb {N}$ with $\\operatorname{dim}_F(J^n(A)\\slash J^{n+1}(A))=1$ .",
"Then $J^{n-1}(A)\\subseteq \\operatorname{Z}(A)$ .",
"Finally we have the following.",
"Lemma Let $A$ be a local symmetric $F$ -algebra.",
"Then $[A,A]\\subseteq J^2(A)$ .",
"This is an easy consequence since $[A,A]=[F 1+J(A),F 1+J(A)]=[J(A),J(A)]\\subseteq J^2(A)$ .",
"We recall the definition of the Külshammer spaces from [18].",
"Let $A$ be a finite dimensional $F$ -algebra and $n\\in \\mathbb {N}_0$ .",
"Then we define $T_n(A)\\mathrel {\\mathop }=\\lbrace a\\in A \\mid a^{2^n}\\in [A,A]\\rbrace $ and $T(A)\\mathrel {\\mathop }=\\lbrace a\\in A \\mid a^{2^n}\\in [A,A]\\text{ for some } n\\in \\mathbb {N} \\rbrace .$ It is well known (see [12]) that $T(A)=J(A)+[A,A]$ , and that there is a chain of inclusions $[A,A]=T_0(A)\\subseteq T_1(A)\\subseteq T_2(A)\\subseteq \\ldots \\subseteq T(A)$ .",
"From this and [18] we can deduce the following.",
"Lemma We have $T(B)=T_1(B)$ .",
"In particular, $a^2\\in [B,B]$ for every $a\\in J(B)$ .",
"There is a remarkable property of group algebras and their blocks considering the rate of growth of a minimal projective resolution of any of their finite dimensional modules.",
"Let $A$ be a finite dimensional $F$ -algebra and $M$ a finite dimensional $A$ -module.",
"Furthermore let $\\dots \\longrightarrow P_2\\longrightarrow P_1\\longrightarrow P_0\\longrightarrow M\\longrightarrow 0$ be a minimal projective resolution of $M$ .",
"If there is a smallest integer $c\\in \\mathbb {N}_0$ such that for some positive number $\\lambda $ we have $\\operatorname{dim}_FP_n\\le \\lambda n^{c-1}$ for every sufficiently large $n$ , then we say that $M$ has complexity $c$ .",
"If there is no such number, then we say that $M$ has infinite complexity.",
"Using [1] we get the following.",
"Lemma The maximal complexity of any indecomposable finite dimensional $B$ -module equals 4.",
"We will conclude this section with a proposition which gives us a sufficient condition for a finite dimensional $F$ -algebra $A$ to have a module with infinite complexity.",
"Although it might seem quite special at first, this condition will be crucial in the next section.",
"Proposition Let $A$ be a local $F$ -algebra and let $x,z\\in J(A)$ be such that $\\lbrace x+J^2(A),z+J^2(A)\\rbrace $ is $F$ -linearly independent in $J(A)\\slash J^2(A)$ and such that $xz=zx=z^2=0$ holds.",
"Furthermore, we denote by $(f_i)_{i=-1}^{\\infty }$ the shifted Fibonacci sequence given by $f_{-1}=1=f_0$ , and $f_{i}=f_{i-1}+f_{i-2}$ for $i\\in \\mathbb {N}$ .",
"Then there are a minimal projective resolution $\\ldots \\longrightarrow P_2\\stackrel{\\varphi _2}{\\longrightarrow }P_1\\stackrel{\\varphi _1}{\\longrightarrow }P_0\\stackrel{\\varphi _0}{\\longrightarrow }F\\longrightarrow 0$ of the trivial $A$ -module $F\\cong A\\slash J(A)$ and, for $i\\in \\mathbb {N}_0$ , an $A$ -basis $\\lbrace b_{i,1},\\dots ,b_{i,n_i}\\rbrace $ of $P_i$ with the following properties: $n_0=1=f_0$ and $zb_{0,1},xb_{0,1}\\in K_0\\mathrel {\\mathop }=\\operatorname{Ker}(\\varphi _0)$ .",
"For $i\\in \\mathbb {N}$ we have $n_i\\ge f_i$ and $zb_{i,1},\\dots ,zb_{i,f_i},xb_{i1},\\dots ,xb_{i,f_{i-1}}\\in K_i\\mathrel {\\mathop }=\\operatorname{Ker}(\\varphi _i)$ .",
"In particular, the $A$ -module $F$ has infinite complexity.",
"The first claim is clear, since $P_0=A$ and $\\operatorname{Ker}(\\varphi _0)=J(A)$ , so that we can choose $b_{0,1}=1$ .",
"Let us now assume that for some $i\\in \\mathbb {N}_0$ we have already constructed $P_0,\\dots ,P_i$ and $\\varphi _0,\\dots ,\\varphi _i$ with the properties from above.",
"We will show that the claim also holds true for $i+1$ .",
"First we notice that from $\\varphi _i\\mathrel {\\mathop }\\,P_i\\rightarrow K_{i-1}$ being a projective cover we get $K_i=\\operatorname{Ker}(\\varphi _i)\\subseteq J(A)P_i$ and, therefore, $J(A)K_i\\subseteq J^2(A)P_i$ .",
"Since $\\lbrace b_{i,1},\\dots ,b_{i,f_i}\\rbrace $ is $A$ -linearly independent in $P_i$ , we see that $\\lbrace zb_{i,1}+J^2(A)P_i,\\dots ,zb_{i,f_i}+J^2(A)P_i, xb_{i,1}+J^2(A)P_i,\\dots ,xb_{i,f_{i-1}}+J^2(A)P_i\\rbrace $ is an $F$ -linearly independent set in $J(A)P_i\\slash J^2(A)P_i$ .",
"Hence, the set $\\lbrace zb_{i,1}+J(A)K_i,\\dots ,zb_{i,f_i}+J(A)K_i, xb_{i,1}+J(A)K_i,\\dots ,xb_{i,f_{i-1}}+J(A)K_i\\rbrace $ is $F$ -linearly independent in $K_i\\slash J(A)K_i$ .",
"Therefore, there is a projective cover $\\varphi _{i+1}\\mathrel {\\mathop }P_{i+1}\\rightarrow K_i$ together with an $A$ -basis $\\lbrace b_{i+1,1},\\dots ,b_{i+1,n_{i+1}}\\rbrace $ of $P_{i+1}$ with the properties $n_{i+1}\\ge f_i+f_{i-1}=f_{i+1}$ and $\\varphi _{i+1}(b_{i+1,j})=zb_{i,j}$ for $j=1,\\dots ,f_i$ , and $\\varphi _{i+1}(b_{i+1,f_i+j})=xb_{i,j}$ for $j=1,\\dots ,f_{i-1}$ .",
"Since $zx=z^2=0$ , we have $\\varphi _{i+1}(zb_{i+1,j})=z\\varphi _{i+1}(b_{i+1,j})=0$ for $j\\in \\lbrace 1,\\dots ,f_{i+1}\\rbrace $ and since $xz=0$ , we have $\\varphi _{i+1}(xb_{i+1,j})=x\\varphi _{i+1}(b_{i+1,j})=0$ for $j\\in \\lbrace 1,\\dots ,f_{i}\\rbrace $ .",
"We thus have constructed a projective cover $\\varphi _{i+1}\\mathrel {\\mathop }P_{i+1}\\rightarrow K_i$ with the claimed properties.",
"From the exponential growth of the Fibonacci sequence and the shown properties of a minimal projective resolution of the $A$ -module $F$ and the fact that $A$ was assumed to be a local algebra, we deduce that $\\operatorname{dim}_FP_i\\ge f_i\\operatorname{dim}_FA$ , so that $F$ has, in fact, infinite complexity.",
"We mention that another version of the proposition which is due to J.F.",
"Carlson can be found in the upcoming paper [21].",
"In that version it is proved that the trivial $A$ -module has infinite complexity provided $x,y,z\\in J(A)$ with $\\lbrace x+J^2(A),y+J^2(A),z+J^2(A)\\rbrace $ is $F$ -linearly independent in $J(A)\\slash J^2(A)$ and $xz=zx=yz=zy=0$ .",
"We will need this statement in our paper too."
],
[
"Determining the isomorphism type of the center",
"Let $A$ be the basic algebra of $B$ over $F$ .",
"Since $A$ and $B$ are Morita equivalent, we can deduce a number of properties which are shared by these algebras.",
"Lemma $\\operatorname{dim}_FA=16$ , $\\operatorname{dim}_F\\operatorname{Z}(A)=8$ and $\\operatorname{dim}_F[A,A]=8$ .",
"$A$ is a local symmetric $F$ -algebra.",
"$\\operatorname{Z}(A)\\cong \\operatorname{Z}(B)$ .",
"For every $a\\in J(A)$ we have $a^2\\in [A,A]$ .",
"Every indecomposable $A$ -module $M$ has finite complexity.",
"Part (iii) is well-known.",
"From the introduction we already know that $\\dim _F\\operatorname{Z}(A)=\\dim _F\\operatorname{Z}(B)=k(B)=8$ .",
"Moreover, the dimension of $A$ equals the order of a defect group of $B$ (see [19]).",
"This proves the first part of (i).",
"Since $B$ has exactly one irreducible Brauer character, we infer that $B$ , and therefore $A$ , has just one isomorphism class of simple modules.",
"Together with the property of $A$ of being a basic $F$ -algebra this yields $A\\slash J(A)\\cong F$ , so that $A$ is a local $F$ -algebra.",
"It is a well known fact that blocks of finite groups are symmetric algebras and that symmetry is a Morita invariant.",
"Thus, also $A$ is a symmetric $F$ -algebra which shows (ii).",
"The third part of (i), and (iv) follow at once by combining the results in [12], lemma:lem1(iv) and lemma:lem5.",
"Finally, since Morita equivalences preserve projectivity and also projective covers, (v) follows easily from lemma:lem6.",
"From now on we will assume that $\\operatorname{Z}(B)\\cong \\operatorname{Z}(A)\\cong F[X,Z_1,\\dots ,Z_6]/\\langle X^2+1,XZ_{2i}+Z_{2i-1},Z_iZ_j\\rangle $ (see prop2).",
"We are seeking a contradiction.",
"To avoid initial confusion about signs it is to be noted explicitly that we calculate over a field of characteristic 2.",
"We introduce a new $F$ -basis for $\\operatorname{Z}(A)$ by setting: $W_0&\\mathrel {\\mathop }=1,&W_1&\\mathrel {\\mathop }=X+1,&W_2&\\mathrel {\\mathop }=Z_1,&W_3&\\mathrel {\\mathop }=Z_3,\\\\W_4&\\mathrel {\\mathop }=Z_5,&W_5&\\mathrel {\\mathop }=Z_1+Z_2,&W_6&\\mathrel {\\mathop }=Z_3+Z_4,&W_7&\\mathrel {\\mathop }=Z_5+Z_6.$ The structure constants with respect to $W_i$ are given as follows.",
"Table: NO_CAPTIONBy abuse of notation we will identify $\\operatorname{Z}(A)$ with $F\\lbrace W_0,\\dots ,W_7\\rbrace $ .",
"For every $z\\in J(\\operatorname{Z}(A))=F\\lbrace W_1,\\dots ,W_7\\rbrace $ we have $z^2=0$ since $\\operatorname{char}(F)=2$ .",
"From lemma:lem7(ii) we know that $A$ is a symmetric $F$ -algebra.",
"Let $s\\mathrel {\\mathop }\\, A\\rightarrow F$ be a symmetrizing form for $A$ .",
"Hence, $s$ is $F$ -linear, for every $a,b\\in A$ we have $s(ab)=s(ba)$ .",
"Moreover, the kernel $\\operatorname{Ker}(s)$ includes no non-zero (one-sided) ideal of $A$ .",
"For a subspace $U\\subseteq A$ we define the set $U^{\\perp }\\mathrel {\\mathop }=\\lbrace a\\in A \\mid s(aU)=0\\rbrace .$ It is well known that we always have $\\operatorname{dim}_FA=\\operatorname{dim}_FU+\\operatorname{dim}_F(U^{\\perp })$ and $U^{\\perp \\perp }=U$ .",
"In particular, the identities $\\operatorname{Z}(A)^{\\perp }=[A,A]$ and $\\operatorname{soc}(A)^{\\perp }=J(A)$ are known to hold.",
"Defining $\\operatorname{soc}^2(A)\\mathrel {\\mathop }=\\lbrace a\\in A \\mid aJ^2(A)=0 \\rbrace $ we easily see $\\operatorname{soc}^2(A)&=\\lbrace a\\in A \\mid a J^2(A)=0 \\rbrace =\\lbrace a\\in A \\mid s(aJ^2(A))=0 \\rbrace =\\lbrace a\\in A \\mid s(J^2(A) a)=0 \\rbrace \\\\&=\\lbrace a\\in A \\mid J^2(A) a=0 \\rbrace =(J^2(A))^{\\perp }.$ In particular, $\\operatorname{soc}^2(A)$ is a two-sided ideal in $A$ .",
"We will now collect some basic facts about the $F$ -algebra $A$ .",
"Lemma $J(\\operatorname{Z}(A))=F\\lbrace W_1,\\dots ,W_7\\rbrace $ and $\\operatorname{soc}(\\operatorname{Z}(A))=J^2(\\operatorname{Z}(A))=F\\lbrace W_5,W_6,W_7\\rbrace $ .",
"In particular, $\\operatorname{dim}_FJ(\\operatorname{Z}(A))=7$ and $\\operatorname{dim}_F\\operatorname{soc}(\\operatorname{Z}(A))=3$ .",
"$\\operatorname{soc}(\\operatorname{Z}(A))^{\\perp }=[A,A]+J(\\operatorname{Z}(A))\\cdot A=J(\\operatorname{Z}(A))+J^2(A)$ and this is an ideal in $A$ .",
"In particular, $\\operatorname{soc}(\\operatorname{Z}(A))$ is an ideal in $A$ .",
"$J(A)\\cdot \\operatorname{soc}(\\operatorname{Z}(A))=\\operatorname{soc}(A)$ .",
"$\\operatorname{dim}_F((J(\\operatorname{Z}(A))+J^2(A))\\slash J^2(A))\\le 2$ .",
"For any $a\\in \\operatorname{soc}^2(A)$ and $b\\in J(A)$ we have $ab,ba\\in \\operatorname{soc}(A)$ and $ab=ba$ .",
"This can be read off immediately from the multiplication table of $\\operatorname{Z}(A)$ .",
"For an element $z\\in \\operatorname{Z}(A)$ we have $z\\in \\operatorname{soc}(\\operatorname{Z}(A))\\Leftrightarrow z J(\\operatorname{Z}(A))=0\\Leftrightarrow s((z J(\\operatorname{Z}(A)))\\cdot A)=0\\Leftrightarrow z\\in (J(\\operatorname{Z}(A))\\cdot A)^{\\perp }.$ Hence, $\\operatorname{soc}(\\operatorname{Z}(A))=\\operatorname{Z}(A)\\cap (J(\\operatorname{Z}(A))\\cdot A)^{\\perp }$ and therefore, by going over to the orthogonal spaces, $\\operatorname{soc}(\\operatorname{Z}(A))^{\\perp }=[A,A]+ J(\\operatorname{Z}(A))\\cdot A.$ This shows the first equality in (ii).",
"From this and (i) we also get $\\operatorname{dim}_F([A,A]+ J(\\operatorname{Z}(A))\\cdot A)=13$ .",
"Now since $A$ is a local symmetric $F$ -algebra we have $[A,A]\\subseteq J^2(A)$ by lemma:lem4 and from $A=F 1\\oplus J(A)$ and lemma:lem1(v) we get $J(\\operatorname{Z}(A))\\cdot A\\subseteq J(\\operatorname{Z}(A))+J^2(A)$ .",
"Hence, we obtain $[A,A]+J(\\operatorname{Z}(A))\\cdot A \\subseteq J(\\operatorname{Z}(A))+J^2(A).$ If we had $[A,A]+J(\\operatorname{Z}(A))\\cdot A \\ne J(\\operatorname{Z}(A))+J^2(A)$ , it would follow that $\\operatorname{dim}_F(J(\\operatorname{Z}(A))+J^2(A))\\ge 14$ , so that $\\operatorname{dim}_F(J(A)\\slash (J(\\operatorname{Z}(A))+J^2(A)))\\le 1$ .",
"But then we could find subsets $\\mathcal {B}_1\\subseteq J(A)$ and $\\mathcal {B}_2\\subseteq J(\\operatorname{Z}(A))$ with $\\vert \\mathcal {B}_1\\vert \\le 1$ such that $\\lbrace 1\\rbrace \\cup \\mathcal {B}_1 \\cup \\mathcal {B}_2$ generated $A$ as an algebra.",
"Since $\\vert \\mathcal {B}_1\\vert \\le 1$ , however, all the generators would commute with each other and so $A$ would be a commutative algebra, a contradiction.",
"Hence, $[A,A]+J(\\operatorname{Z}(A))\\cdot A = J(\\operatorname{Z}(A))+J^2(A)$ and we have shown the second equality.",
"Finally we note that, since $A$ is a local algebra, every subspace of $J(A)$ containing $J^2(A)$ automatically is an ideal in $A$ .",
"Using this fact on $J(\\operatorname{Z}(A))+J^2(A)$ we see that $\\operatorname{soc}(\\operatorname{Z}(A))^{\\perp }$ , and therefore also $\\operatorname{soc}(\\operatorname{Z}(A))$ , is an ideal in $A$ .",
"From (ii) we have $\\operatorname{soc}(\\operatorname{Z}(A))^{\\perp }=J(\\operatorname{Z}(A))+J^2(A)$ , so that $s(J^2(A)\\cdot \\operatorname{soc}(\\operatorname{Z}(A)))=0$ .",
"Since $J^2(A)\\cdot \\operatorname{soc}(\\operatorname{Z}(A))$ is an ideal in $A$ and $s$ is non-degenerate, we get $J^2(A)\\cdot \\operatorname{soc}(\\operatorname{Z}(A))=0$ .",
"But this implies $J(A)\\cdot \\operatorname{soc}(\\operatorname{Z}(A))\\subseteq J(A)^{\\perp }=\\operatorname{soc}(A)$ .",
"If we even had $J(A)\\cdot \\operatorname{soc}(\\operatorname{Z}(A))=0$ , then $\\operatorname{soc}(\\operatorname{Z}(A))\\subseteq J(A)^{\\perp }=\\operatorname{soc}(A)$ , a contradiction.",
"Hence, the claim follows.",
"Let us assume to the contrary that $\\operatorname{dim}_F((J(\\operatorname{Z}(A))+J^2(A))\\slash J^2(A))\\ge 3$ .",
"Then we can find elements $z_1,z_2,z_3\\in J(\\operatorname{Z}(A))$ such that the set $\\lbrace z_1+J^2(A),z_2+J^2(A),z_3+J^2(A)\\rbrace $ is $F$ -linearly independent in $J(A)\\slash J^2(A)$ .",
"We write $z_i=\\alpha _iW_1+b_i$ with $\\alpha _i\\in F$ and $b_i\\in F\\lbrace W_2,\\dots ,W_7\\rbrace $ for $i=1,2,3$ .",
"We can assume that $\\alpha _1=\\alpha _2=0$ .",
"For if $\\alpha _1\\ne 0$ or $\\alpha _2\\ne 0$ we may say for instance that $\\alpha _1\\ne 0$ (after possibly swapping $z_1$ and $z_2$ ).",
"By defining $z^{\\prime }_1\\mathrel {\\mathop }=z_2-\\frac{\\alpha _2}{\\alpha _1}z_1$ , $z^{\\prime }_2\\mathrel {\\mathop }=z_3-\\frac{\\alpha _3}{\\alpha _1}z_1$ and $z^{\\prime }_3\\mathrel {\\mathop }=z_1$ we obtain elements $z^{\\prime }_1,z^{\\prime }_2,z^{\\prime }_3\\in J(\\operatorname{Z}(A))$ such that $\\lbrace z^{\\prime }_1+J^2(A),z^{\\prime }_2+J^2(A),z^{\\prime }_3+J^2(A)\\rbrace $ is again $F$ -linearly independent in $J(A)\\slash J^2(A)$ and such that $z^{\\prime }_1,z^{\\prime }_2\\in F\\lbrace W_2,\\dots ,W_7\\rbrace $ .",
"After renaming $z^{\\prime }_i$ into $z_i$ for $i=1,2,3$ we get $\\alpha _1=\\alpha _2=0$ as claimed.",
"But from this we get $z_1z_2=z_2z_1=z_2^2=0$ (see the multiplication table for $\\operatorname{Z}(A)$ ) which implies that the simple $A$ -module $F$ has infinite complexity by propo:propo1.",
"This, however, contradicts lemma:lem7(v).",
"Hence, (iv) holds true.",
"Let $a\\in \\operatorname{soc}^2(A)$ and $b\\in J(A)$ .",
"Moreover, as before, we fix a symmetrizing form $s\\mathrel {\\mathop }\\, A\\rightarrow F$ for the symmetric $F$ -algebra $A$ .",
"Then $s(abJ(A))\\subseteq s(aJ^2(A))=0$ by definition of $\\operatorname{soc}^2(A)$ , and thus $ab\\in (J(A))^{\\perp }=\\operatorname{soc}(A)$ .",
"Similarly, from $s(J(A)ba)=s(aJ(A)b)\\subseteq s(aJ^2(A))=0$ we get $ba\\in \\operatorname{soc}(A)$ .",
"This shows the first claim.",
"In order to show the second, we note that by lemma:lem1(i) and lemma:lem7(ii) we can find an element $c\\in A$ with $\\operatorname{soc}(A)=Fc$ .",
"Since $s$ is non-degenerate we deduce $s(c)=\\mathrel {\\mathop }\\gamma \\in F\\backslash \\lbrace 0\\rbrace $ .",
"We have already shown that $ab,ba\\in \\operatorname{soc}(A)=Fc$ .",
"Hence, there are $\\alpha ,\\beta \\in F$ with $ab=\\alpha c$ and $ba=\\beta c$ .",
"Using that $s$ is symmetric and $F$ -linear, we obtain $0=s(ab-ba)=s((\\alpha -\\beta )c)=(\\alpha -\\beta )\\gamma $ .",
"This implies $\\alpha = \\beta $ , since $\\gamma \\ne 0$ , and therefore $ab=\\alpha c=\\beta c=ba$ which shows the second claim and finishes the proof.",
"Corollary One of the following three cases occurs: There are $x,y\\in J(A)$ with $xy\\ne yx$ and $A=F 1\\oplus F x\\oplus F y\\oplus J^2(A)$ .",
"In particular, $\\operatorname{dim}_FJ^2(A)=13$ .",
"There are $x,y\\in J(A)$ and $z\\in J(\\operatorname{Z}(A))$ with $xy\\ne yx$ and $A=F 1\\oplus F x\\oplus F y\\oplus F z\\oplus J^2(A)$ .",
"In particular, $\\operatorname{dim}_FJ^2(A)=12$ .",
"There are $x,y\\in J(A)$ and $z_1,z_2\\in J(\\operatorname{Z}(A))$ with $xy\\ne yx$ , $z_1z_2\\ne 0$ , and $A=F 1\\oplus F x\\oplus F y\\oplus F z_1\\oplus F z_2\\oplus J^2(A)$ .",
"In particular, $\\operatorname{dim}_FJ^2(A)=11$ .",
"By lemma:lem8(iv) we have $\\operatorname{dim}_F((J(\\operatorname{Z}(A))+J^2(A))\\slash J^2(A))\\le 2$ .",
"Now if $\\operatorname{dim}_F((J(\\operatorname{Z}(A))+J^2(A))\\slash J^2(A))=0$ , then $J(\\operatorname{Z}(A))+J^2(A)=J^2(A)$ and by lemma:lem8(i,ii) we obtain $\\operatorname{dim}_F(J(\\operatorname{Z}(A))+J^2(A))=13$ .",
"Therefore, since $A$ is local and $\\operatorname{dim}_FA=16$ , there are $x,y\\in J(A)$ such that $A=F 1\\oplus F x\\oplus F y\\oplus (J(\\operatorname{Z}(A))+J^2(A))=F 1\\oplus F x\\oplus F y\\oplus J^2(A)$ .",
"By a similar argument as used in the proof of lemma:lem8(ii) we must have $xy\\ne yx$ since $A$ is non-commutative.",
"This gives case (I).",
"If $\\operatorname{dim}_F((J(\\operatorname{Z}(A))+J^2(A))\\slash J^2(A))=1$ , then $J(\\operatorname{Z}(A))+J^2(A)=F z\\oplus J^2(A)$ for some $z\\in J(\\operatorname{Z}(A))$ and, again, by lemma:lem8(i,ii) we obtain $\\operatorname{dim}_F(J(\\operatorname{Z}(A))+J^2(A))=13$ .",
"Now there are $x,y\\in J(A)$ such that $A=F 1\\oplus F x\\oplus F y\\oplus (J(\\operatorname{Z}(A))+J^2(A))=F 1\\oplus F x\\oplus F y\\oplus F z\\oplus J^2(A)$ .",
"Since $z\\in J(\\operatorname{Z}(A))$ we have $xz=zx$ and $yz=zy$ , so that we must have $xy\\ne yx$ since $A$ is non-commutative.",
"This gives case (II).",
"Finally, if $\\operatorname{dim}_F((J(\\operatorname{Z}(A))+J^2(A))\\slash J^2(A))=2$ , then $J(\\operatorname{Z}(A))+J^2(A)=F z_1\\oplus F z_2\\oplus J^2(A)$ for some $z_1,z_2\\in J(\\operatorname{Z}(A))$ .",
"For the same reason as before there are $x,y\\in J(A)$ with $A=F 1\\oplus F x\\oplus F y\\oplus (J(\\operatorname{Z}(A))+J^2(A))=F 1\\oplus F x\\oplus F y\\oplus F z_1\\oplus F z_2\\oplus J^2(A)$ and $xy\\ne yx$ .",
"Because of propo:propo1 and lemma:lem7(v) we must have $z_1z_2\\ne 0$ since $z_1^2=z_2^2=0$ (see the multiplication table for $\\operatorname{Z}(A)$ ).",
"This gives case (III).",
"The aim for the remainder of this section is to show that none of the cases (I), (II) or (III) of coroll:cor1 can actually occur.",
"This will give the desired contradiction.",
"Before we start to exclude the three cases one by one, we need two more crucial lemmas.",
"Lemma We have $\\operatorname{dim}_F(([A,A]+J^3(A))\\slash J^3(A))=1$ .",
"Moreover, there is an $a\\in J(A)$ with $a^2\\notin J^3(A)$ .",
"In particular, $a\\notin \\operatorname{Z}(A)$ .",
"In all the cases from coroll:cor1 we have $A=F 1\\oplus F x\\oplus F y\\oplus (J(\\operatorname{Z}(A))+J^2(A))$ with $xy\\ne yx$ .",
"Therefore, we get $[A,A]=[F x+F y+\\operatorname{Z}(A)+J^2(A),F x+F y+\\operatorname{Z}(A)+J^2(A)]\\subseteq F[x,y]+J^3(A).$ Hence, the coset of $[x,y]= xy+yx$ in $J^3(A)$ spans $([A,A]+J^3(A))\\slash J^3(A)$ over $F$ and so $\\operatorname{dim}_F(([A,A]+J^3(A))\\slash J^3(A))\\le 1$ .",
"If we show that there is an $a\\in J(A)$ with $a\\notin J^3(A)$ , we will get $\\operatorname{dim}_F(([A,A]+J^3(A))\\slash J^3(A))\\ge 1$ using lemma:lem7(iv), so that all the remaining claims will follow at once from this (note that for every $w\\in J(\\operatorname{Z}(A))$ we have $w^2=0$ , so that $w^2\\in J^3(A)$ ).",
"In order to show that there is such an $a$ , we will now assume to the contrary that $a^2\\in J^3(A)$ for every $a\\in J(A)$ .",
"For arbitrary $a,b\\in J(A)$ this implies that $[a,b]=ab+ba=(a+b)^2+a^2+b^2\\in J^3(A)$ , so that $ab+J^3(A)=ba+J^3(A)$ holds true for every $a,b\\in J(A)$ .",
"We will now separately deduce a contradiction for every case.",
"Let $A$ be as in case (I) from coroll:cor1.",
"Then $J(A)=F \\lbrace x,y\\rbrace + J^2(A)$ .",
"Using lemma:lem2 and our assumption we get $J^2(A)=F \\lbrace x^2,xy,yx,y^2\\rbrace + J^3(A)=F \\lbrace xy\\rbrace + J^3(A)$ since $x^2,y^2,[x,y]\\in J^3(A)$ .",
"Again by lemma:lem2 we get $J^3(A)=F\\lbrace x^2y\\rbrace +J^4(A)=J^4(A)$ since $x^2\\in J^3(A)$ and so $x^2y\\in J^4(A)$ .",
"Therefore, $J^3(A)=0$ by Nakayama's Lemma.",
"But then $A=F\\lbrace 1,x,y,xy\\rbrace $ and hence $\\operatorname{dim}_FA\\le 4$ which contradicts $\\operatorname{dim}_FA=16$ .",
"Next let $A$ be as in case (II).",
"Then $J(A)=F \\lbrace x,y,z\\rbrace + J^2(A)$ and $z\\in J(\\operatorname{Z}(A))$ .",
"Using the same facts as before we successively obtain $J^2(A)&=F \\lbrace x^2,xy,xz,yx,y^2,yz,zx,zy,z^2\\rbrace +J^3(A)=F \\lbrace xy,xz,yz\\rbrace +J^3(A),\\\\J^3(A)&=F \\lbrace x^2y,x^2z,xyz,yxy,yxz,y^2z\\rbrace +J^4(A)=F \\lbrace xyz\\rbrace +J^4(A),\\\\J^4(A)&=F \\lbrace x^2yz\\rbrace +J^5(A)=J^5(A).$ Again by Nakayama's Lemma we have $J^4(A)=0$ and $A=F \\lbrace 1,x,y,z,xy,xz,yz,xyz\\rbrace $ .",
"This yields the contradiction $\\operatorname{dim} _FA\\le 8$ .",
"Finally let $A$ be as in case (III).",
"Then $J(A)=F \\lbrace x,y,z_1,z_2\\rbrace + J^2(A)$ with $z_1,z_2\\in J(\\operatorname{Z}(A))$ .",
"As before: $J^2(A)&=F \\lbrace x^2, xy, xz_1, xz_2, yx, y^2, yz_1, yz_2, z_1x, z_1y, z_1^2, z_1z_2, z_2x, z_2y, z_2z_1, z_2^2\\rbrace +J^3(A)\\\\&=F \\lbrace xy, xz_1, xz_2, yz_1, yz_2, z_1z_2\\rbrace +J^3(A),\\\\J^3(A)&=F \\lbrace x^2y, x^2z_1, x^2z_2, xyz_1, xyz_2, xz_1z_2, yxy, yxz_1, yxz_2, y^2z_1, y^2z_2, yz_1z_2, \\dots \\\\&\\quad \\,\\dots , z_1xy, z_1xz_1, z_1xz_2, z_1yz_1, z_1yz_2, z_1^2z_2\\rbrace +J^4(A)\\\\&=F \\lbrace xyz_1, xyz_2, xz_1z_2, yz_1z_2\\rbrace +J^4(A),\\\\J^4(A)&=F \\lbrace x^2yz_1, x^2yz_2, x^2z_1z_2, xyz_1z_2, yxyz_1, yxyz_2, yxz_1z_2, y^2z_1z_2\\rbrace +J^5(A)\\\\&=F \\lbrace xyz_1z_2\\rbrace +J^5(A)=J^5(A).$ The last equality is a consequence of lemma:lem8(i,iii).",
"For, we have $xyz_1z_2\\in J^2(A)\\cdot J^2(\\operatorname{Z}(A))=J^2(A)\\cdot \\operatorname{soc}(\\operatorname{Z}(A))=J(A)\\cdot \\operatorname{soc}(A)=0.$ Now $J^4(A)=0$ by Nakayama and $A=F\\lbrace 1, x, y, z_1, z_2, xy, xz_1, xz_2, yz_1, yz_2, z_1z_2, xyz_1, xyz_2, xz_1z_2, yz_1z_2\\rbrace ,$ so that $\\operatorname{dim}_FA\\le 15$ , a contradiction.",
"This completes the proof.",
"Lemma With the notation of coroll:cor1 we may assume the following: $x^2\\notin J^3(A)$ , There is an $\\alpha \\in F\\backslash \\lbrace 0\\rbrace $ such that $xy\\equiv yx +\\alpha x^2 \\pmod {J^3(A)}$ , $y^2\\in J^3(A)$ .",
"Moreover, with the $\\alpha $ from the second item above we have for any $m\\in \\mathbb {N}$ : $x^{m+1}\\equiv \\frac{1}{\\alpha }[x,x^{m-1}y]\\pmod {J^{m+2}(A)}$ , $x^{2m}y\\equiv \\frac{1}{\\alpha }[y,x^{2m-1}y]\\pmod {J^{2m+2}(A)}$ , $x^{4m-1}y\\equiv \\frac{1}{\\alpha }(x^{2m-1}y)^2 \\pmod {J^{4m+1}(A)}$ , $x^{m+1}w\\equiv \\frac{1}{\\alpha }[x^{m-1}y,xw]\\pmod {J^{m+3}(A)}$ , where the last item is to be omitted in case (I), $w=z$ in case (II), and $w\\in \\lbrace z_1,z_2\\rbrace $ in case (III).",
"In particular: $x^n\\in [A,A]+J^{n+1}(A)$ for $n\\ge 2$ , $x^{n-1}y \\in [A,A]+J^{n+1}(A)$ for $n\\ge 3$ being odd or $n\\ge 4$ being divisible by 4, $x^{n-1}z \\in [A,A]+J^{n+1}(A)$ for $n\\ge 3$ in case (II), $x^{n-1}z_1,x^{n-1}z_2 \\in [A,A]+J^{n+1}(A)$ for $n\\ge 3$ in case (III).",
"By lemma:lem9 we can find an $a\\in J(A)$ with $a^2\\notin J^3(A)$ .",
"From this we deduce $a\\notin J^2(A)$ .",
"Since the square of any element from $J(\\operatorname{Z}(A))+J^2(A)$ is in $J^3(A)$ , we get $a\\notin J(\\operatorname{Z}(A))+J^2(A)$ .",
"Hence, $a+(J(\\operatorname{Z}(A))+J^2(A))\\ne 0$ in $J(A)\\slash (J(\\operatorname{Z}(A))+J^2(A))$ and we may therefore assume without loss of generality that $x=a$ (after possibly swapping $x$ and $y$ ).",
"This shows the first item.",
"Again, by lemma:lem9, we have $\\operatorname{dim}_F(([A,A]+J^3(A))\\slash J^3(A))=1$ .",
"Since in any of the cases (I), (II) and (III) we have $[A,A]=[Fx + Fy + \\operatorname{Z}(A) + J^2(A),Fx + Fy + \\operatorname{Z}(A) + J^2(A)]\\subseteq F [x,y]+J^3(A)$ and, by the first item and lemma:lem7(iv), we have $[A,A]\\subseteq Fx^2 + J^3(A)$ , we conclude that $\\lbrace [x,y]+J^3(A)\\rbrace $ and $\\lbrace x^2+J^3(A)\\rbrace $ are two $F$ -bases for $([A,A]+J^3(A))\\slash J^3(A)$ .",
"Hence, there is an $\\alpha \\in F\\backslash \\lbrace 0\\rbrace $ such that $xy+yx=[x,y]\\equiv \\alpha x^2 \\pmod {J^3(A)}$ .",
"From this the second item follows at once.",
"Now by lemma:lem7(iv) we have $y\\in [A,A]$ , so that there is a $\\beta \\in F$ with $y^2\\equiv \\beta x^2 \\pmod {J^3(A)}$ .",
"Let $\\zeta \\in F$ be a zero of the polynomial $p(X)=X^2+\\alpha X+\\beta $ .",
"Replacing $y$ by $y^{\\prime }\\mathrel {\\mathop }=y+\\zeta x$ we obtain $A=F 1\\oplus F x\\oplus F y^{\\prime }\\oplus J^2(A)$ and $[x,y^{\\prime }]&=[x,y+\\zeta x]=[x,y],\\\\(y^{\\prime })^2&=(y+\\zeta x)^2=y^2+\\zeta (xy+yx) +\\zeta ^2x^2\\\\&\\equiv (\\zeta ^2+\\alpha \\zeta +\\beta )x^2\\equiv 0 \\pmod {J^3(A)}.$ Renaming $y^{\\prime }$ into $y$ we obtain the third item.",
"Now we just have to show the four desired congruences and from those the other claims follow at once together with lemma:lem7(iv).",
"Let $m\\in \\mathbb {N}$ .",
"Then we have $\\frac{1}{\\alpha }[x,x^{m-1}y]=\\frac{1}{\\alpha }(x^{m}y+x^{m-1}yx)\\equiv \\frac{1}{\\alpha }(2\\cdot x^{m}y+\\alpha x^{m+1})\\equiv x^{m+1} \\pmod {J^{m+2}(A)}$ by applying $xy\\equiv yx +\\alpha x^2 \\pmod {J^3(A)}$ once.",
"Moreover we obtain $\\frac{1}{\\alpha }[y,x^{2m-1}y]=\\frac{1}{\\alpha }(yx^{2m-1}y+x^{2m-1}y^2)\\equiv \\frac{1}{\\alpha }(2\\cdot x^{2m-1}y^2+(2m-1)\\cdot \\alpha x^{2m}y)\\equiv x^{2m}y \\pmod {J^{2m+2}(A)}$ by repeatedly ($2m-1$ times to be more exact) applying $xy\\equiv yx +\\alpha x^2 \\pmod {J^3(A)}$ .",
"Doing the same thing we also get $\\frac{1}{\\alpha }(x^{2m-1}y)^2= \\frac{1}{\\alpha }(x^{2m-1}yx^{2m-1}y)\\equiv \\frac{1}{\\alpha }(x^{4m-2}y^2+(2m-1)\\cdot \\alpha x^{4m-1}y)\\equiv x^{4m-1}y \\pmod {J^{4m+1}(A)}$ keeping in mind that $y^2\\in J^3(A)$ .",
"Finally by the same arguments and using $w\\in \\operatorname{Z}(A)$ we get $\\frac{1}{\\alpha }[x^{m-1}y,xw]=\\frac{1}{\\alpha }(x^{m-1}yxw+x^myw)\\equiv \\frac{1}{\\alpha }(2\\cdot x^myw+\\alpha x^{m+1}w)\\equiv x^{m+1}w \\pmod {J^{m+3}(A)}$ which finishes the proof.",
"In the following we will always assume that $A$ fulfills all the properties stated in lemma:lem10 and we will use them without further mentioning.",
"We have everything we need in order to show that none of the cases (I), (II) or (III) from coroll:cor1 can occur for the $F$ -algebra $A$ under consideration.",
"Proposition The case (I) of coroll:cor1 cannot occur.",
"In case (I) the algebra $A$ has the decomposition $A=F 1\\oplus F x\\oplus F y\\oplus J^2(A)$ .",
"Using lemma:lem2 and $J(A)=F \\lbrace x,y\\rbrace +J^2(A)$ we get $J^2(A)=F \\lbrace x^2,xy,yx,y^2\\rbrace +J^3(A)=F \\lbrace x^2,xy\\rbrace +J^3(A)$ .",
"From here we get $J^n(A)=F \\lbrace x^n,x^{n-1}y\\rbrace +J^{n+1}(A)$ for every integer $n\\ge 2$ by inductively applying lemma:lem2.",
"Therefore, we get $\\operatorname{dim}_F(J^n(A)\\slash J^{n+1}(A))\\le 2$ for every $n\\in \\mathbb {N}$ .",
"Also by lemma:lem2 we see that if $\\operatorname{dim}_F(J^m(A)\\slash J^{m+1}(A))=1$ for some $m\\in \\mathbb {N}$ , then $\\operatorname{dim}_F(J^n(A)\\slash J^{n+1}(A))\\le 1$ for every $n\\ge m$ .",
"Since there is always such an $m$ by lemma:lem1(vi) and since $\\operatorname{dim}_FJ(\\operatorname{Z}(A))=7$ , we obtain the following three possibilities, denoted by (I.1), (I.2) and (I.3), for the dimensions of the Loewy layers of $A$ by keeping in mind lemma:lem3: Table: NO_CAPTIONIn case (I.1) we have $\\operatorname{soc}(A)=F\\lbrace x^8,x^7y\\rbrace $ .",
"On the other hand, since $J^9(A)=0$ , lemma:lem10 yields $x^8,x^7y\\in [A,A]$ .",
"Hence, $\\operatorname{soc}(A)\\cap [A,A]\\ne 0$ , a contradiction.",
"In case (I.2) we have $\\operatorname{soc}(A)=F\\lbrace x^9,x^8y\\rbrace $ .",
"Again, by lemma:lem10 and $J^{10}(A)=0$ , we have $x^9,x^8y\\in [A,A]$ and hence a contradiction.",
"Finally in case (I.3) we have $J^{11}(A)=0$ and $J^{10}(A)=F\\lbrace x^{10},x^9y\\rbrace \\ne 0$ , so that $x^9\\notin J^{10}(A)$ .",
"Hence, $J^9(A)=F\\lbrace x^9\\rbrace +J^{10}(A)$ and $\\operatorname{soc}(A)=J^{10}(A)=F\\lbrace x^{10}\\rbrace $ .",
"But on the other hand $x^{10}\\in [A,A]$ , since $J^{11}(A)=0$ , and therefore $\\operatorname{soc}(A)\\cap [A,A]\\ne 0$ , again a contradiction.",
"This shows that neither of the cases (I.1), (I.2) or (I.3) can occur and so the proposition is proven.",
"Proposition The case (II) of coroll:cor1 cannot occur.",
"In case (II) the algebra $A$ decomposes into $A=F 1\\oplus F x\\oplus F y\\oplus F z\\oplus J^2(A)$ .",
"Using this and lemma:lem2 and $z^2=0$ we easily see $J(A)&=F\\lbrace x,y,z\\rbrace +J^2(A),\\\\J^2(A)&=F\\lbrace x^2, xy, xz, yz\\rbrace +J^3(A),\\\\J^3(A)&=F\\lbrace x^3, x^2y, x^2z, xyz\\rbrace +J^4(A),\\multicolumn{2}{l}{\\text{ and inductively}}\\\\J^n(A)&=F\\lbrace x^n, x^{n-1}y, x^{n-1}z, x^{n-2}yz\\rbrace +J^{n+1}(A)$ for any integer $n\\ge 3$ .",
"Now we will distinguish between the different cases that can occur for $\\operatorname{dim}_F(J^2(A)\\slash J^3(A))$ .",
"We note that $2\\le \\operatorname{dim}_F(J^2(A)\\slash J^3(A))\\le 4$ .",
"The upper bound is clear by the preceding discussion, and if $\\operatorname{dim}_F(J^2(A)\\slash J^3(A))= 1$ , then $J^2(A)\\subseteq \\operatorname{Z}(A)$ by lemma:lem3 which is a contradiction to $\\operatorname{dim}_F\\operatorname{Z}(A)=8$ .",
"The case $\\operatorname{dim}_F(J^2(A)\\slash J^3(A))= 0$ leads to $J^2(A)=0$ by Nakayama's Lemma and this is clearly false.",
"Case (II.1): $\\operatorname{dim}_F(J^2(A)\\slash J^3(A))=2$ .",
"Since $x^2\\notin J^3(A)$ we proceed by distinguishing three subcases for an $F$ -basis of $J^2(A)\\slash J^3(A)$ .",
"More specifically there is always a basis of $J^2(A)\\slash J^3(A)$ given by $\\lbrace x^2+J^3(A),d+J^3(A)\\rbrace $ for some $d\\in \\lbrace xy, xz, yz\\rbrace $ .",
"(1): $J^2(A)=F \\lbrace x^2,xy\\rbrace +J^3(A)$ .",
"We inductively obtain $J^n(A)=F \\lbrace x^n,x^{n-1}y\\rbrace +J^{n+1}(A)$ for every $n\\ge 2$ .",
"With the same arguments as in the proof of propo:propo2 we see that there are the following two possibilities for the dimensions of the Loewy layers of $A$ : Table: NO_CAPTIONIn case (II.1.1) we have $\\operatorname{soc}(A)=F\\lbrace x^8,x^7y\\rbrace $ and $x^8,x^7y\\in [A,A]$ , a contradiction.",
"Similarly, in case (II.1.2) we have $\\operatorname{soc}(A)=F\\lbrace x^9,x^8y\\rbrace $ and $x^9,x^8y\\in [A,A]$ , again a contradiction.",
"$(2)$ : $J^2(A)=F \\lbrace x^2,xz\\rbrace +J^3(A)$ .",
"We can assume that $xy\\in F\\lbrace x^2\\rbrace +J^3(A)$ since otherwise we are in the first subcase.",
"Let $xy\\equiv \\gamma x^2 \\pmod {J^3(A)}$ .",
"Using this we obtain $x^3\\equiv \\frac{1}{\\alpha }[x,xy]\\equiv \\frac{\\gamma }{\\alpha }[x,x^2]=0 \\pmod {J^4(A)}.$ This, however, implies $J^3(A)=F\\lbrace x^3,x^2z\\rbrace +J^4(A)=F\\lbrace x^2z\\rbrace +J^4(A)$ and $J^4(A)=F\\lbrace x^3z\\rbrace +J^5(A)=J^5(A)$ .",
"Hence, $J^4(A)=0$ by Nakayama's Lemma and therefore $\\operatorname{dim}_FA\\le 1+3+2+1=7$ , a contradiction.",
"$(3)$ : $J^2(A)=F \\lbrace x^2,yz\\rbrace +J^3(A)=F \\lbrace x^2,zy\\rbrace +J^3(A)$ .",
"We may assume that $xy,xz\\in F\\lbrace x^2\\rbrace +J^3(A)$ since otherwise we are in one of the previous two subcases.",
"Using this we obtain $J^3(A)=F\\lbrace x^3, zxy, x^2z, z^2y\\rbrace +J^4(A)=F\\lbrace x^3, xyz, x^2z\\rbrace +J^4(A)=F\\lbrace x^3\\rbrace +J^4(A)$ .",
"Hence, $J^2(A)\\subseteq \\operatorname{Z}(A)$ by lemma:lem3, and so $\\operatorname{dim}_F\\operatorname{Z}(A)\\ge \\operatorname{dim}_FJ^2(A)=12$ , a contradiction.",
"We have thus shown that $\\operatorname{dim}_F(J^2(A)\\slash J^3(A))\\ne 2$ .",
"Case (II.2): $\\operatorname{dim}_F(J^2(A)\\slash J^3(A))=3$ .",
"Again, since $x^2\\notin J^3(A)$ , there is always an $F$ -basis of $J^2(A)\\slash J^3(A)$ of the form $\\lbrace x^2+J^3(A), d_1+J^3(A), d_2+J^3(A)\\rbrace $ for some $d_1,d_2\\in \\lbrace xy,xz,yz\\rbrace $ .",
"Hence, we can proceed by distinguishing three subcases for a basis of $J^2(A)\\slash J^3(A)$ .",
"$(1)$ : $J^2(A)=F\\lbrace x^2,xy,xz\\rbrace +J^3(A)$ .",
"We have $J^n(A)=F\\lbrace x^n,x^{n-1}y,x^{n-1}z\\rbrace +J^{n-1}(A)$ for every $n\\ge 2$ .",
"We obtain the following possibilities for the dimensions of the Loewy layers of $A$ : Table: NO_CAPTIONIn cases (II.2.2), (II.2.3) and (II.2.5) we have $\\operatorname{soc}(A)=F\\lbrace x^7,x^6y,x^6z\\rbrace $ , but $x^7,x^6y,x^6z\\in [A,A]$ , a contradiction.",
"Similarly, in cases (II.2.4) and (II.2.6) we have $\\operatorname{soc}(A)=F\\lbrace x^8,x^7y,x^7z\\rbrace $ and $x^8,x^7y,x^7z\\in [A,A]$ , again a contradiction.",
"Finally let us consider case (II.2.1).",
"By lemma:lem10 we obtain that $\\operatorname{dim}_F((([A,A]\\cap J^2(A))+J^{3}(A))\\slash J^{3}(A))=1$ , since this space is spanned by $\\lbrace x^2+J^3(A)\\rbrace $ .",
"Moreover $\\operatorname{dim}_F((([A,A]\\cap J^3(A))+J^{4}(A))\\slash J^{4}(A))=3$ , since this space is spanned by $\\lbrace x^3+J^4(A),x^2y+J^4(A),x^2z+J^4(A)\\rbrace $ .",
"Analogously $\\operatorname{dim}_F((([A,A]\\cap J^4(A))+J^{5}(A))\\slash J^{5}(A))=3$ and $\\operatorname{dim}_F((([A,A]\\cap J^5(A))+J^{6}(A))\\slash J^{6}(A))=2$ .",
"Using the canonical isomorphism $(([A,A]\\cap J^n(A))+J^{n+1}(A))\\slash J^{n+1}(A)\\cong ([A,A]\\cap J^n(A))\\slash ([A,A]\\cap J^{n+1}(A))$ for $n\\in \\mathbb {N}$ we obtain $8=\\operatorname{dim}_F[A,A]\\ge \\sum \\limits _{n=2}^{5}\\operatorname{dim}_F(([A,A]\\cap J^n(A))\\slash ([A,A]\\cap J^{n+1}(A)))=1+3+3+2=9,$ a contradiction.",
"$(2)$ : $J^2(A)=F\\lbrace x^2,xy,yz\\rbrace +J^3(A)=F\\lbrace x^2,xy,zy\\rbrace +J^3(A)$ .",
"Here we can assume $xz\\in F\\lbrace x^2,xy\\rbrace +J^3(A)$ since otherwise we are in the subcase $J^2(A)=F\\lbrace x^2,xy,xz\\rbrace +J^3(A)$ again.",
"We obtain $J^3(A)=F\\lbrace x^3, x^2y, xzy, zx^2, zxy, zxz\\rbrace +J^4(A)=F\\lbrace x^3,x^2y\\rbrace +J^4(A).$ Hence, we get the following two possibilities for the dimensions of the Loewy layers of $A$ : Table: NO_CAPTIONSince $x^7,x^6y\\in [A,A]+J^8(A)$ and $x^8,x^7y\\in [A,A]+J^9(A)$ , similar arguments as used before show that both cases lead to a contradiction.",
"$(3)$ : $J^2(A)=F\\lbrace x^2,xz,yz\\rbrace +J^3(A)$ .",
"Here we can assume $xy\\in F x^2+J^3(A)$ , since we are in one of the previous two subcases otherwise.",
"Hence, $J^3(A)=F\\lbrace x^3, xzx, yzx, x^2z, xz^2, yz^2\\rbrace +J^4(A)=F\\lbrace x^3, x^2z\\rbrace +J^4(A).$ Inductively we get $J^n(A)=F\\lbrace x^n,x^{n-1}z\\rbrace +J^{n+1}(A)$ for $n\\ge 3$ .",
"But together with lemma:lem10 this implies $J^3(A)\\subseteq [A,A]$ which is a contradiction.",
"This shows that $\\operatorname{dim}_F(J^2(A)\\slash J^3(A))\\ne 3$ .",
"Case (II.3): $\\operatorname{dim}_F(J^2(A)\\slash J^3(A))=4$ .",
"In this case we have $J^2(A)=F\\lbrace x^2,xy,xz,yz\\rbrace +J^3(A)$ and the cosets of the elements in $\\lbrace x^2,xy,xz,yx\\rbrace $ in $J^3(A)$ form an $F$ -basis of $J^2(A)\\slash J^3(A)$ .",
"Inductively we get $J^n(A)=F\\lbrace x^n,x^{n-1}y,x^{n-1}z,x^{n-2}yz\\rbrace +J^{n+1}(A)$ for $n\\ge 2$ (cf.",
"the beginning of this proof).",
"Arguing as before we see that there are the following possible cases for the dimensions of the Loewy layers of $A$ : Table: NO_CAPTIONIn case (II.3.1) we have $\\operatorname{soc}(A)=J^5(A)=F\\lbrace x^5, x^4y, x^4z, x^3yz\\rbrace $ .",
"By lemma:lem10 we obtain $x^5, x^4y, x^4z\\in [A,A]$ .",
"Since $x^3y\\in [A,A]+J^5(A)$ and $\\operatorname{Z}(A)\\cdot [A,A]\\subseteq [A,A]$ , we also get $x^3yz\\in [A,A]$ .",
"But this contradicts $\\operatorname{soc}(A)\\cap [A,A]=0$ .",
"In cases (II.3.2) and (II.3.3) we have $\\operatorname{soc}(A)=J^6(A)=F\\lbrace x^6, x^5y, x^5z, x^4yz\\rbrace $ and $J^4(A)\\subseteq \\operatorname{Z}(A)$ by lemma:lem3.",
"Therefore, $x^3y\\in \\operatorname{Z}(A)$ .",
"Now we have $x^6,x^5z\\in [A,A]$ and, using $\\operatorname{Z}(A)\\cdot [A,A]\\subseteq [A,A]$ again, we also obtain $x^5y,x^4yz\\in [A,A]$ since $x^2\\in [A,A]$ , $x^4z\\in [A,A]+ J^6(A)$ , and $x^3y, z\\in \\operatorname{Z}(A)$ .",
"Therefore, $\\operatorname{soc}(A)\\cap [A,A]\\ne 0$ , a contradiction.",
"In cases (II.3.5) and (II.3.6) we have $\\operatorname{soc}(A)=J^7(A)=F\\lbrace x^7, x^6y, x^6z, x^5yz\\rbrace $ and $J^5(A)\\subseteq \\operatorname{Z}(A)$ .",
"Since $x^2\\in [A,A]$ and $x^3yz\\in \\operatorname{Z}(A)$ we get $x^5yz\\in [A,A]$ .",
"Moreover $x^7, x^6y, x^6z\\in [A,A]$ and therefore $\\operatorname{soc}(A)\\subseteq [A,A]$ , a contradiction.",
"In case (II.3.7) we have $\\operatorname{soc}(A)=F\\lbrace x^8, x^7y, x^7z, x^6yz\\rbrace $ .",
"As before we obtain a contradiction using $x^8, x^7y, x^7z\\in [A,A]$ , $x^6y\\in [A,A]+J^8(A)$ , and $z\\in \\operatorname{Z}(A)$ .",
"There remains case (II.3.4) and excluding this one requires some additional arguments.",
"We have $J^7(A)=0$ , $\\operatorname{soc}(A)=J^6(A)=F\\lbrace x^6, x^5y, x^5z, x^4yz\\rbrace $ , and $J^5(A)\\subseteq \\operatorname{Z}(A)$ .",
"Since $x^6, x^5z\\in [A,A]$ , $x^4y\\in [A,A]+J^6(A)$ , and $z\\in \\operatorname{Z}(A)$ , we obtain $x^6,x^5z,x^4yz\\in \\operatorname{soc}(A)\\cap [A,A]=0$ .",
"Hence, $x^6=x^5z=x^4yz=0$ and $\\operatorname{soc}(A)=F\\lbrace x^5y\\rbrace $ .",
"This also yields $x^3\\notin J^4(A)$ and $x^4\\notin J^5(A)$ .",
"If $x^2y\\in F\\lbrace x^3\\rbrace +J^4(A)$ , then $x^5y\\in F\\lbrace x^6\\rbrace +J^7(A)=0$ , a contradiction.",
"Hence, $\\lbrace x^3+J^4(A), x^2y+J^4(A)\\rbrace $ is $F$ -linearly independent in $J^3(A)\\slash J^4(A)$ .",
"With similar arguments one gets that $\\lbrace x^4+J^5(A), x^3y+J^5(A)\\rbrace $ is $F$ -linearly independent in $J^4(A)\\slash J^5(A)$ .",
"Therefore, there is a pair $(\\lambda _1,\\lambda _2)\\in F^2\\backslash \\lbrace (0,0)\\rbrace $ such that $\\lbrace 1, x, y, z, x^2, xy, xz, yz, x^3, x^2y, \\lambda _1 x^2z+\\lambda _2 xyz, x^4, x^3y, x^5, x^4y, x^5y\\rbrace $ is an $F$ -basis of $A$ .",
"We will proceed by showing in two steps that $x^2z$ must be zero.",
"The first step will be to show $x^2z\\in J^4(A)$ .",
"In order to do this, assume that $x^2z\\notin J^4(A)$ and define the subspace $T\\mathrel {\\mathop }=F\\lbrace x^2,xy,xz,yz,x^3,x^2y,x^3y\\rbrace $ of $J^2(A)$ .",
"We will show that $T\\cap \\operatorname{Z}(A) =0$ .",
"This will imply the inequality $12=\\operatorname{dim}_FJ^2(A)\\ge \\operatorname{dim}_FT+\\operatorname{dim}_F(\\operatorname{Z}(A)\\cap J^2(A))=7+6=13$ which is certainly false, so that $x^2z$ must be in $J^4(A)$ .",
"Let $w=\\delta _1 x^2+\\delta _2 xy+\\delta _3 xz+\\delta _4 yz+\\delta _5 x^3+\\delta _6 x^2y+\\delta _7 x^3y\\in T\\cap \\operatorname{Z}(A)$ with $\\delta _i\\in F$ for $i=1,\\dots ,7$ be arbitrary.",
"We have to show $w=0$ .",
"Considering $x^6=x^5z=x^4yz=0$ , $J^7(A)=0$ , $w\\in \\operatorname{Z}(A)$ , and $x^4=(x^2)^2\\in [A,A]$ , we obtain $\\delta _2 x^5y=x^4w\\in \\operatorname{soc}(A)\\cap [A,A]=0$ , so that $\\delta _2=0$ and $w=\\delta _1 x^2+\\delta _3 xz+\\delta _4 yz+\\delta _5 x^3+\\delta _6 x^2y+\\delta _7 x^3y$ .",
"Using $w\\in \\operatorname{Z}(A)$ again, we obtain $0=xw+wx\\equiv \\delta _1 (x^3+x^3)+\\delta _3 (x^2z+x^2z) +\\delta _4 (xy+yx)z\\equiv \\delta _4(\\alpha x^2z) \\pmod {J^4(A)}$ and $0=yw+wy\\equiv \\delta _1 (yx^2+x^2y)+\\delta _3 (yx+xy)z +\\delta _4 (y^2z+y^2z)z\\equiv \\delta _3(\\alpha x^2z) \\pmod {J^4(A)}.$ Thus, $\\delta _3=\\delta _4=0$ since $\\alpha \\ne 0$ and we assumed $x^2z\\notin J^4(A)$ .",
"Hence, $w=\\delta _1 x^2+\\delta _5 x^3+\\delta _6 x^2y+\\delta _7 x^3y$ , and using $x^3y\\in [A,A]+J^5(A)$ and $w\\in \\operatorname{Z}(A)$ we get $\\delta _1 x^5y=wx^3y\\in \\operatorname{soc}(A)\\cap [A,A]=0$ .",
"Therefore, $\\delta _1=0$ and $w=\\delta _5 x^3+\\delta _6 x^2y+\\delta _7 x^3y$ .",
"Using $x^3\\in [A,A]+J^4(A)$ and $x^6=0$ we get $\\delta _6 x^5y=x^3w\\in \\operatorname{soc}(A)\\cap [A,A]=0$ , so that $\\delta _6=0$ and $w=\\delta _5 x^3+\\delta _7 x^3y$ .",
"With $x^2y\\in [A,A]+J^4(A)$ we conclude $\\delta _5 x^5y=wx^2y=0$ and hence $w=\\delta _7 x^3y$ .",
"Now, again, $\\delta _7 x^5y=x^2w=0$ , so that $w=0$ .",
"Therefore, we have shown $T\\cap \\operatorname{Z}(A)=0$ and by the argument above we obtain a contradiction.",
"We have thus shown that $x^2z$ can be assumed to be in $J^4(A)$ .",
"This also implies that the following elements form an $F$ -basis of $A$ : $1, x, y, z, x^2, xy, xz, yz, x^3, x^2y, xyz, x^4, x^3y, x^5, x^4y, x^5y.$ In the second step we will show $x^2z=0$ .",
"Since $x^2z\\in J^4(A)$ , there are $\\varepsilon _1,\\dots ,\\varepsilon _5\\in F$ such that $x^2z=\\varepsilon _1 x^4+ \\varepsilon _2 x^3y+ \\varepsilon _3 x^5+ \\varepsilon _4 x^4y+ \\varepsilon _5 x^5y.$ Since $J^4(A)=F\\lbrace x^4, x^3y, x^5, x^4y, x^5y\\rbrace $ , we observe that from $x^3, x^2y\\in [A,A]+J^4(A)$ , and $x^4, x^3y\\in [A,A]+J^5(A)$ , and $x^5, x^4y\\in [A,A]+J^6(A)$ it follows that $x^3, x^2y, x^3y\\in [A,A]+J^6(A)$ .",
"Now as before, $x^4\\in [A,A]$ and therefore $\\varepsilon _2 x^5y=x^2\\cdot x^2z=x^4z\\in \\operatorname{soc}(A)\\cap [A,A]=0$ , so that $\\varepsilon _2=0$ and $x^2z=\\varepsilon _1 x^4+ \\varepsilon _3 x^5+ \\varepsilon _4 x^4y+ \\varepsilon _5 x^5y$ .",
"Since $x^3y\\in [A,A]+J^6(A)$ , we get $\\varepsilon _1 x^5y=x^2z\\cdot xy=(x^3y)z\\in \\operatorname{soc}(A)\\cap [A,A]=0$ , so that $\\varepsilon _1=0$ and $x^2z=\\varepsilon _3 x^5+ \\varepsilon _4 x^4y+ \\varepsilon _5 x^5y$ .",
"Using $x^3\\in [A,A]+J^6(A)$ next, we obtain $\\varepsilon _4 x^5y=x\\cdot x^2z=x^3z\\in \\operatorname{soc}(A)\\cap [A,A]=0$ , so that $\\varepsilon _4=0$ and $x^2z=\\varepsilon _3 x^5+ \\varepsilon _5 x^5y$ .",
"Similarly, we get $\\varepsilon _3 x^5y=x^2z\\cdot y=(x^2y)z=0$ , so that $\\varepsilon _3 =0$ and $x^2z=\\varepsilon _5 x^5y$ .",
"But now $x^2z\\in [A,A]$ and $x^5y\\in \\operatorname{soc}(A)$ imply that $x^2z=\\varepsilon _5 x^5y=0$ .",
"Hence, we have shown $x^2z=0$ as claimed.",
"Since the three elements $x,y,z$ generate $A$ as an $F$ -algebra, and since $z\\in \\operatorname{Z}(A)$ , we observe that the center $\\operatorname{Z}(A)$ consists exactly of all elements $w\\in A$ which commute with both $x$ and $y$ .",
"Using this and the fact $x^2z=0$ one can easily show that in our case the elements $ 1,z,xz,yz,xyz,x^3y,x^5,x^4y,x^5y$ are central in $A$ .",
"Since they are also $F$ -linearly independent and $\\operatorname{dim}_F\\operatorname{Z}(A)=8$ , we obtain $\\operatorname{Z}(A)=F\\lbrace 1,z,xz,yz,xyz,x^5,x^4y,x^5y\\rbrace $ and, in particular, $J(\\operatorname{Z}(A))=F\\lbrace z,xz,yz,xyz,x^5,x^4y,x^5y\\rbrace $ .",
"But then for any $w_1,w_2\\in J(\\operatorname{Z}(A))$ we get $w_1\\cdot w_2=0$ and this contradicts lemma:lem7(iii) and the subsequent multiplication table for $\\operatorname{Z}(A)$ .",
"This finishes the proof.",
"Proposition The case (III) of coroll:cor1 cannot occur.",
"In this final case (III) the algebra $A$ has a decomposition $A=F 1\\oplus F x\\oplus F y\\oplus F z_1\\oplus F z_2\\oplus J^2(A)$ with $z_1,z_2\\in J(\\operatorname{Z}(A))$ and $\\operatorname{dim}_FJ^2(A)=11$ .",
"In the following we will frequently make use of $J(A)\\cdot J^2(\\operatorname{Z}(A))=\\operatorname{soc}(A)$ (see lemma:lem8(i,iii)) without further mentioning it.",
"From $J(A)=F\\lbrace x,y,z_1,z_2\\rbrace +J^2(A)$ we get $J^2(A)=F\\lbrace x^2, xy, xz_1, xz_2, yz_1, yz_2, z_1z_2\\rbrace +J^3(A)$ and this implies $J^3(A)\\ne \\operatorname{soc}(A)$ because of dimension reasons.",
"Hence, $J(A)\\cdot J^2(\\operatorname{Z}(A))=\\operatorname{soc}(A)\\subseteq J^4(A)$ and so $J^3(A)=F\\lbrace x^3, x^2y, x^2z_1, x^2z_2, xyz_1, xyz_2\\rbrace +J^4(A)$ .",
"Now since $[A,A]$ does not contain any non-zero ideal of $A$ and since $J^3(A)\\ne 0$ , we get $[A,A]\\ne [A,A]+J^3(A)$ and therefore $\\operatorname{dim}_F(A\\slash J^3(A))=\\operatorname{dim}_F(A\\slash ([A,A]+J^3(A)))+\\operatorname{dim}_F(([A,A]+J^3(A))\\slash J^3(A))\\le 7+1=8.$ Hence, $\\operatorname{dim}_FJ^3(A)\\ge 8$ and together with $\\operatorname{dim}_FJ^2(A)=11$ and $J(A)\\lnot \\subseteq \\operatorname{Z}(A)$ we get $\\operatorname{dim}_FJ^3(A)\\in \\lbrace 8,9\\rbrace $ by lemma:lem3.",
"We thus have to distinguish two cases.",
"Case (III.1): $\\operatorname{dim}_F(J^2(A)\\slash J^3(A))=2$ .",
"We will again consider several subcases corresponding to possible choices of an $F$ -basis of $J^2(A)\\slash J^3(A)$ .",
"Since $x^2\\notin J^3(A)$ we can fix $x+J^3(A)$ as a basis element of $J^2(A)\\slash J^3(A)$ .",
"Then there always is an $F$ -basis $\\lbrace x^2+J^3(A), d+J^3(A)\\rbrace $ of $J^2(A)\\slash J^3(A)$ for some $d\\in \\lbrace xy,xz_1,xz_2,yz_1,yz_2,z_1z_2\\rbrace $ .",
"$(1)$ : $J^2(A)=F\\lbrace x^2,xy\\rbrace +J^3(A)$ .",
"As in the proof of propo:propo3 we get the following possibilities for the dimensions of the Loewy layers of $A$ : Table: NO_CAPTIONCase (III.1.1) cannot occur since $\\operatorname{soc}(A)=F\\lbrace x^7,x^6y\\rbrace \\subseteq [A,A]$ and case (III.1.2) cannot occur since $\\operatorname{soc}(A)=F\\lbrace x^8,x^7y\\rbrace \\subseteq [A,A]$ .",
"$(2)$ : $J^2(A)=F\\lbrace x^2,xz_i\\rbrace +J^3(A)$ for some $i\\in \\lbrace 1,2\\rbrace $ .",
"We may assume that $xy\\in F\\lbrace x^2\\rbrace +J^3(A)$ since otherwise we are in the situation we have just considered.",
"Let $xy\\equiv \\gamma x^2 \\pmod {J^3(A)}$ .",
"Then we get $x^3\\equiv \\frac{1}{\\alpha }[x,xy]\\equiv \\frac{\\gamma }{\\alpha }[x,x^2]=0 \\pmod {J^4(A)}$ which implies $J^3(A)=F\\lbrace x^3,x^2z_i\\rbrace +J^4(A)$ and $J^4(A)=F\\lbrace x^3z_i\\rbrace +J^5(A)=J^5(A)$ , so that $J^4(A)=0$ by Nakayama's Lemma, a contradiction.",
"$(3)$ : $J^2(A)=F\\lbrace x^2,yz_i\\rbrace +J^3(A)=F\\lbrace x^2,z_iy\\rbrace +J^3(A)$ for some $i\\in \\lbrace 1,2\\rbrace $ .",
"We may, as before, assume that $xy,xz_1,z_2\\in F\\lbrace x^2\\rbrace +J^3(A)$ .",
"We obtain $J^3(A)=F\\lbrace x^3,xyz_i,x^2z_i,yz_i^2\\rbrace +J^4(A)=F\\lbrace x^3\\rbrace +J^4(A).$ This implies $J^2(A)\\subseteq \\operatorname{Z}(A)$ by lemma:lem3, a contradiction.",
"$(4)$ : $J^2(A)=F\\lbrace x^2,z_1z_2\\rbrace +J^3(A)$ .",
"Here we may assume that $xy,xz_1,xz_2,yz_1,yz_2\\in F\\lbrace x^2\\rbrace +J^3(A)$ .",
"As before we obtain a contradiction by $J^3(A)=F\\lbrace x^3,xz_1z_2,x^2z_1,z_1^2z_2\\rbrace +J^4(A)=F\\lbrace x^3\\rbrace +J^4(A).$ This completes case (III.1).",
"Case (III.2): $\\operatorname{dim}_F(J^2(A)\\slash J^3(A))=3$ .",
"We will again distinguish the different cases for a possible basis of $J^2(A)\\slash J^3(A)$ .",
"Since $x^2\\notin J^3(A)$ , we may fix $x^2+J^3(A)$ as a basis element of $J^2(A)\\slash J^3(A)$ and we have to look through the possibilities for the remaining two basis elements.",
"Since those two elements can be chosen from $\\lbrace xy,xz_1,xz_2,yz_1,yz_2,z_1z_2\\rbrace $ , there are essentially 8 different cases.",
"$(1)$ : $J^2(A)=F\\lbrace x^2,xy,xz_i\\rbrace +J^3(A)$ for some $i\\in \\lbrace 1,2\\rbrace $ .",
"We get the following four possibilities for the dimensions of the Loewy layers of $A$ : Table: NO_CAPTIONCases (III.2.3) and (III.2.4) can be excluded immediately since there we have $\\operatorname{soc}(A)=F\\lbrace x^7,x^6y,x^6z_i\\rbrace \\subseteq [A,A]$ , a contradiction.",
"Similarly, in case (III.2.1) we have $x^6,x^5z_i\\in [A,A]$ and by lemma:lem3 we have $x^3y\\in \\operatorname{Z}(A)$ .",
"Since $x^2\\in [A,A]$ we conclude $x^5y=x^2\\cdot x^3y\\in [A,A]$ and thus $\\operatorname{soc}(A)\\cap [A,A]\\ne 0$ , a contradiction.",
"Case (III.2.2) needs some more calculation to exclude.",
"After interchanging $z_1$ and $z_2$ if necessary, we may assume that $i=1$ .",
"Since $x^6,x^5z_1\\in \\operatorname{soc}(A)\\cap [A,A]=0$ we have $x^6=x^5z_1=0$ and $\\operatorname{soc}(A)=J^6(A)=F\\lbrace x^5y\\rbrace $ .",
"In particular, we have $x^4\\notin J^5(A)$ and $x^5\\notin J^6(A)$ .",
"As in the last subcase of the proof of propo:propo3 we get $J^4(A)=F\\lbrace x^4,x^3y\\rbrace +J^5(A)$ and $J^5(A)=F\\lbrace x^5,x^4y\\rbrace +J^6(A)$ from this.",
"Hence, the following elements form an $F$ -basis of $A$ : $1,x,y,z_1,z_2,x^2,xy,xz_1,x^3,x^2y,x^2z_1,x^4,x^3y,x^5,x^4y,x^5y.$ Now we define the subspace $T\\mathrel {\\mathop }=F\\lbrace 1,x,y,z_1,x^2,xy,xz_1,x^3,x^2y,x^4,x^3y\\rbrace $ of $A$ .",
"We will first show that $T\\cap \\operatorname{soc}^2(A)=0$ .",
"In order to do so we remark that $J(A)\\cdot \\operatorname{soc}^2(A)\\subseteq \\operatorname{soc}(A)$ by lemma:lem8(v).",
"In particular, $x\\operatorname{soc}^2(A)\\subseteq \\operatorname{soc}(A)$ .",
"But now $xT=F\\lbrace x,x^2,xy,xz_1,x^3,x^2y,x^2z_1,x^4,x^3y,x^5,x^4y\\rbrace $ and therefore $xT\\cap \\operatorname{soc}(A)=0$ .",
"This implies $T\\cap \\operatorname{soc}^2(A)=0$ .",
"Since $\\operatorname{dim}_FA=16$ , $\\operatorname{dim}_FT=11$ , $\\operatorname{dim}_F\\operatorname{soc}^2(A)=\\operatorname{dim}_F((J^2(A))^{\\perp })=5$ , and $T\\cap \\operatorname{soc}^2(A)=0$ , we obtain $A=T\\oplus \\operatorname{soc}^2(A)$ .",
"Decomposing $z_2\\in A$ into its direct summands, we find $\\lambda _0,\\lambda _1,\\lambda _2,\\lambda _3\\in F$ and $u\\in T\\cap J^2(A)$ and $v\\in \\operatorname{soc}^2(A)$ such that $z_2=\\lambda _0 1+\\lambda _1 x+\\lambda _2 y+\\lambda _3 z_1+u+v$ or, equivalently, $v=z_2+\\lambda _0 1+\\lambda _1 x+\\lambda _2 y+\\lambda _3 z_1+u\\in \\operatorname{soc}^2(A).$ Furthermore, $\\lambda _0$ must vanish since otherwise $vJ^2(A)\\subseteq J^2(A)\\backslash J^3(A)$ which would contradict $v\\in \\operatorname{soc}^2(A)$ .",
"Hence, $v=z_2+\\lambda _1 x+\\lambda _2 y+\\lambda _3 z_1+u\\in \\operatorname{soc}^2(A)$ .",
"Replacing $z_2$ by $v$ we obtain a new set of elements $\\lbrace x,y,z_1,v\\rbrace $ in $J(A)$ such that $\\lbrace x+J^2(A),y+J^2(A),z_1+J^2(A),v+J^2(A)\\rbrace $ is an $F$ -basis of $J(A)\\slash J^2(A)$ .",
"Since $v\\in \\operatorname{soc}^2(A)$ , we also have $xv,vx,yv,vy,z_1v,vz_1\\in \\operatorname{soc}(A)$ and $xv=vx,yv=vy,z_1v=vz_1$ by lemma:lem8(v).",
"Now if $z_1v=0$ , then $z_1v=vz_1=z_1^2=0$ and so, by propo:propo1, the $A$ -module $F$ would have infinite complexity in contradiction to lemma:lem7(v).",
"Hence, we may assume that $z_1v\\ne 0$ .",
"Since $\\operatorname{dim}_F\\operatorname{soc}(A)=1$ , $xv,yv,z_1v\\in \\operatorname{soc}(A)$ , and $z_1v\\ne 0$ , there are $\\gamma _1,\\gamma _2\\in F$ such that $(x-\\gamma _1z_1)v=0=(y-\\gamma _2z_1)v$ .",
"Replacing $x$ by $x^{\\prime }\\mathrel {\\mathop }=x-\\gamma _1z_1$ and $y$ by $y^{\\prime }\\mathrel {\\mathop }=y-\\gamma _2z_1$ , we obtain a set of elements $\\lbrace x^{\\prime },y^{\\prime },z_1,v\\rbrace $ in $J(A)$ such that $\\lbrace x^{\\prime }+J^2(A),y^{\\prime }+J^2(A),z_1+J^2(A),v+J^2(A)\\rbrace $ is an $F$ -basis in $J(A)\\slash J^2(A)$ with $x^{\\prime }v=0=y^{\\prime }v$ .",
"By lemma:lem8(v) we also have $x^{\\prime }v=vx^{\\prime }=y^{\\prime }v=vy^{\\prime }=0$ and this implies that the $A$ -module $F$ has infinite complexity by the remark after propo:propo1 in contradiction to lemma:lem7(v).",
"This finishes the first subcase.",
"$(2)$ : $J^2(A)=F\\lbrace x^2,xy,yz_i\\rbrace +J^3(A)=F\\lbrace x^2,xy,z_iy\\rbrace +J^3(A)$ for some $i\\in \\lbrace 1,2\\rbrace $ .",
"We may assume that $xz_1,xz_2\\in F\\lbrace x^2,xy\\rbrace +J^3(A)$ for otherwise we are in the first subcase again.",
"Using this we get $J^3(A)=F\\lbrace x^3,x^2y,xz_iy,x^2z_i,yz_i^2\\rbrace +J^4(A)=F\\lbrace x^3,x^2y\\rbrace +J^4(A).$ Now there is only one possibility for the dimensions of the Loewy layers of $A$ (see the table above in case (III.2.4)), namely $\\operatorname{dim}_F(J^3(A)\\slash J^4(A))=\\operatorname{dim}_F(J^4(A)\\slash J^5(A))=\\operatorname{dim}_F(J^5(A)\\slash J^6(A))=2$ and $\\operatorname{dim}_F(J^6(A)\\slash J^7(A))=\\operatorname{dim}_FJ^7(A)=1$ .",
"But this implies $\\operatorname{soc}(A)=F\\lbrace x^7,x^6y\\rbrace \\subseteq [A,A]$ , a contradiction.",
"$(3)$ : $J^2(A)=F\\lbrace x^2,xy,z_1z_2\\rbrace +J^3(A)$ .",
"We may assume that $xz_1,xz_2,yz_1,yz_2\\in F\\lbrace x^2,xy\\rbrace +J^3(A)$ for otherwise we are in one of the previous subcases.",
"Consequently, $J^3(A)=F\\lbrace x^3,x^2y,xz_1z_2,x^2z_1,xyz_1,z_1^2z_2\\rbrace +J^4(A)=F\\lbrace x^3,x^2y\\rbrace +J^4(A).$ From here on we get a contradiction exactly as in the previous subcase.",
"$(4)$ : $J^2(A)=F\\lbrace x^2,xz_1,xz_2\\rbrace +J^3(A)$ .",
"We may assume that $xy\\in F\\lbrace x^2\\rbrace +J^3(A)$ for otherwise we are in one of the subcases considered before.",
"We inductively obtain $J^n(A)=F\\lbrace x^n,x^{n-1}z_1,x^{n-1}z_2\\rbrace +J^{n+1}(A)$ for every $n\\ge 2$ .",
"But since $x^n,x^{n-1}z_1,x^{n-1}z_2\\in [A,A]+J^{n+1}(A)$ for any $n\\ge 3$ , we get the contradiction $\\operatorname{soc}(A)\\cap [A,A]\\ne 0$ .",
"$(5)$ : $J^2(A)=F\\lbrace x^2,xz_i,yz_j\\rbrace +J^3(A)=F\\lbrace x^2,xz_i,z_jy\\rbrace +J^3(A)$ for some $i,j\\in \\lbrace 1,2\\rbrace $ .",
"We may assume that $xy\\in F\\lbrace {x^2}\\rbrace +J^3(A)$ and $xz_1,xz_2\\in F\\lbrace x^2,xz_i\\rbrace +J^3(A)$ .",
"Then we get $J^3(A)=F\\lbrace x^3,x^2z_i,xyz_j,x^2z_j,xz_iz_j,yz_j^2\\rbrace +J^4(A)=F\\lbrace x^3,x^2z_i\\rbrace +J^4(A).$ Hence, $J^n(A)=F\\lbrace x^n,x^{n-1}z_i\\rbrace +J^{n+1}(A)$ for any $n\\ge 3$ .",
"But since $x^n,x^{n-1}z_i\\in [A,A]+J^{n+1}(A)$ for any $n\\ge 3$ , this yields, as before, the contradiction $\\operatorname{soc}(A)\\cap [A,A]\\ne 0$ .",
"$(6)$ : $J^2(A)=F\\lbrace x^2,xz_i,z_1z_2\\rbrace +J^3(A)$ for some $i\\in \\lbrace 1,2\\rbrace $ .",
"We may assume that $xy\\in F\\lbrace x^2\\rbrace +J^3(A)$ and $xz_1$ , $xz_2$ , $yz_1$ , $yz_2\\in F\\lbrace x^2,xz_i\\rbrace +J^3(A)$ .",
"Hence, $J^3(A)=F\\lbrace x^3,x^2z_i,xz_1z_2,xz_1^2,z_1^2z_2\\rbrace +J^4(A)=F\\lbrace x^3,x^2z_i\\rbrace +J^4(A)$ which leads to the same contradiction as the subcase before.",
"$(7)$ : $J^2(A)=F\\lbrace x^2,yz_1,yz_2\\rbrace +J^3(A)=F\\lbrace x^2,z_1y,z_2y\\rbrace +J^3(A)$ .",
"We may assume that $xy,xz_1,xz_2\\in F\\lbrace x^2\\rbrace +J^3(A)$ .",
"Hence, $J^3(A)=F\\lbrace x^3,xyz_1,xyz_2,x^2z_1,yz_1^2,yz_1z_2,x^2z_2,yz_2^2\\rbrace +J^4(A)=F\\lbrace x^3\\rbrace +J^4(A).$ Therefore, $J^2(A)\\subseteq \\operatorname{Z}(A)$ by lemma:lem3, a contradiction.",
"$(8)$ : $J^2(A)=F\\lbrace x^2,yz_i,z_1z_2\\rbrace +J^3(A)=F\\lbrace x^2,z_iy,z_1z_2\\rbrace +J^3(A)$ for some $i\\in \\lbrace 1,2\\rbrace $ .",
"We may assume that $xy$ , $xz_1$ , $xz_2\\in F\\lbrace x^2\\rbrace +J^3(A)$ and $yz_1,yz_2\\in F\\lbrace x^2,yz_i\\rbrace +J^3(A)$ .",
"Then $J^3(A)=F\\lbrace x^3,xyz_i,xz_1z_2,x^2z_i,yz_i^2,z_1z_2z_i\\rbrace +J^4(A)=F\\lbrace x^3\\rbrace +J^4(A)$ which is a contradiction just as before.",
"This finishes the proof of this proposition."
],
[
"Concluding remarks",
"Coming back to the analysis of the generalized decomposition matrix $Q$ in Section 2, we now know that only the possibilities (I) and (II) can occur for $Q$ .",
"In the example $G=D\\rtimes 3^{1+2}_+$ mentioned in the introduction, one can show that case (I) occurs (for both of the two non-principal blocks of $G$ ).",
"Thus, by Külshammer [20], case (I) occurs whenever $D$ is normal (see also [29]).",
"In view of [27], one might think that the generalized decomposition matrices $Q_I$ and $Q_{II}$ in case (I) and (II) respectively are linked via $PQ_IS=Q_{II}$ where $P,S\\in \\operatorname{GL}(8,\\mathbb {Z})$ and $P$ is a signed permutation matrix (this is more general than changing basic sets).",
"However, this is not the case.",
"In fact, we conjecture that case (II) never occurs for $Q$ .",
"By [14], $Q$ determines the perfect isometry class of $B$ .",
"Now we consider isotypies.",
"Since the block $b_{xy}$ is nilpotent, all its ordinary decomposition numbers equal 1.",
"Let $Q_{b_x}\\in \\mathbb {Z}^{16\\times 3}$ be the ordinary decomposition matrix of $b_x$ with respect to the basic set in Section 2.",
"As usual, the trace of the contribution matrix $Q_{b_x}C_x^{-1}Q_{b_x}^\\text{T}$ equals $l(b_x)=3$ (see [25]).",
"Hence, its diagonal entries are all $3/16$ and the rows of $Q_{b_x}$ have the form $(1,0,0),(0,1,0),(0,0,1),(1,1,1)$ (see (REF )).",
"It follows that $Q_{b_x}=\\begin{pmatrix}1&1&1&1&1&1&1&1&1&.&.&.&.&.&.&.&.\\\\1&1&1&1&1&.&.&.&.&1&1&1&1&.&.&.&.\\\\1&1&1&1&1&.&.&.&.&.&.&.&.&1&1&1&1\\\\\\end{pmatrix}^\\text{T}$ for a suitable order of $\\operatorname{Irr}(b_x)$ .",
"Exactly the same arguments apply for $b_y$ .",
"This shows that also the isotypy class of $B$ is uniquely determined by $Q$ (see [6], [5]).",
"Note that Usami [31] showed that there is only one such class provided $I(B)\\cong C_3\\times C_3$ and $p\\ne 2$ .",
"Generalizing our result, we note that the isomorphism type of $\\operatorname{Z}(B)$ is uniquely determined by local data whenever $B$ has elementary abelian defect group of order 16 (not necessarily $l(B)=1$ ).",
"In fact, one can use the methods from the second section to construct the generalized decomposition matrix in the remaining cases (this has been done to some extend in [30]).",
"In particular, the character-theoretic version of Broué's Conjecture can be verified unless $l(B)=1$ .",
"We omit the details.",
"Even more, $\\operatorname{Z}(B)$ can be computed whenever $B$ is any 2-block of defect at most 4.",
"To see this one has to construct the generalized decomposition matrix for the non-abelian defect groups (see [29]).",
"Again, we do not go into the details.",
"We remark that it is also possible the determine the isomorphism type of $\\operatorname{Z}(B)$ as an algebra over $\\mathcal {O}$ .",
"In fact, we may compute its structure constants as in Section 2 (these are integral).",
"Charles Eaton has communicated privately that he determined the Morita equivalence class of $B$ by relying heavily on the classification of the finite simple groups (his methods are described in [10] where he handles the elementary abelian defect group of order 8).",
"We believe that the methods of the present paper are of independent interest."
],
[
"Acknowledgment",
"The second author is supported by the German Research Foundation (project SA 2864/1-1) and the Daimler and Benz Foundation (project 32-08/13).",
"Parts of this work will contribute the first author's PhD thesis which he currently writes under the supervision of Burkhard Külshammer."
]
] | 1605.05953 |
[
[
"Uniform parameterization of subanalytic sets and diophantine\n applications"
],
[
"Abstract We prove new parameterization theorems for sets definable in the structure $\\mathbb{R}_{an}$ (i.e.",
"for globally subanalytic sets) which are uniform for definable families of such sets.",
"We treat both $C^r$-parameterization and (mild) analytic parameterization.",
"In the former case we establish a polynomial (in $r$) bound (depending only on the given family) for the number of parameterizing functions.",
"However, since uniformity is impossible in the latter case (as was shown by Yomdin via a very simple family of algebraic sets), we introduce a new notion, analytic quasi-parameterization (where many-valued complex analytic functions are used), which allows us to recover a uniform result.",
"We then give some diophantine applications motivated by the question as to whether the $H^{o(1)}$ bound in the Pila-Wilkie counting theorem can be improved, at least for certain reducts of $\\mathbb{R}_{an}$.",
"Both parameterization results are shown to give uniform $(\\log H)^{O(1)}$ bounds for the number of rational points of height at most $H$ on $\\mathbb{R}_{an}$-definable Pfaffian surfaces.",
"The quasi-parameterization technique produces the sharper result, but the uniform $C^r$-parametrization theorem has the advantage of also applying to $\\mathbb{R}_{an}^{pow}$-definable families."
],
[
"Introduction",
"The aim of this section is to give an informal account of the results appearing in this paper.",
"Precise definitions and statements are given in the next section.",
"So, we are concerned with parameterizations of bounded definable subsets of real euclidean space.",
"The definability here is with respect to some fixed (and, for the moment, arbitrary) o-minimal expansion of the real field.",
"By a parameterization of such a set $X \\subseteq \\mathbb {R}^n$ , we mean a finite collection of definable maps from $(0, 1)^m$ to $\\mathbb {R}^n$ , where $m:= \\text{dim}(X)$ , whose ranges cover $X$ .",
"The fact that parameterizations always exist is an easy consequence of the cell decomposition theorem, but the aim is to construct them with certain differentiability conditions imposed on the parameterizing functions together with bounds on their derivatives.",
"The first result in this generality was obtained in [24] (by adapting methods of Yomdin [32] and Gromov [13] who dealt with the semi-algebraic case), where it was shown that for each positive integer $r$ there exists a parametrization consisting of $C^r$ functions all of whose derivatives (up to order $r$ ) are bounded by 1.",
"Further, the parameterizing functions may be found uniformly.",
"This means that if $\\mathcal {X} = \\lbrace X_t : t \\in T \\rbrace $ is a definable family of $m$ -dimensional subsets of $(0, 1)^n$ (say), i.e.",
"the relation “$t \\in T \\hspace{5.69054pt} \\text{and} \\hspace{5.69054pt} x \\in X_t$ ” is definable in both $x$ and $t$ , then there exists a positive integer $N_r$ such that for each $t \\in T$ , at most $N_r$ functions are required to parameterize $X_t$ and each such function is definable in $t$ .",
"(The bound $N_r$ does, of course, also depend on the family $\\mathcal {X}$ , but we usually suppress this in the notation.",
"The point is that it is independent of $t$ .)",
"Unfortunately, the methods of [24] do not give an explicit bound for $N_r$ and it is the first aim of this paper to do so in the case that the ambient o-minimal structure is the restricted analytic field $\\mathbb {R}_{an}$ (where the bounded definable sets are precisely the bounded subanalytic sets), or a suitable reduct of it.",
"We prove, in this case, that $N_r$ may be taken to be a polynomial in $r$ (which depends only on the given family $\\mathcal {X}$ ).",
"While we have only diophantine applications in mind here, this result already gives a complete answer to an open question, raized by Yomdin, coming from the study of entropy and dynamical systems (see e.g.",
"[32], [31], [13], [3]).",
"In fact, even in the case that the ambient structure is just the ordered field of real numbers (which is certainly a suitable reduct of $\\mathbb {R}_{an}$ to which our result applies), the polynomial bound appears to be new and, indeed, gives a partial answer to a question raized in [3] (just below Remark 3.8).",
"Our uniform $C^r$ -parameterization theorem also holds for the expansion of $\\mathbb {R}_{an}$ by all power functions (i.e.",
"the structure usually denoted $\\mathbb {R}_{an}^{pow}$ ) and suitable reducts (to be clarified in section 2) of it.",
"Next we consider mild parameterizations.",
"Here it is more convenient to consider parameterizing functions with domain $(-1, 1)^m$ (where $m$ is the dimension of the set being parameterized) and we demand that they are $C^{\\infty }$ and we put a bound on all the derivatives.",
"We shall only be concerned with functions that satisfy a so called 0-mild condition, namely that there exists an $R>1$ such that for each positive integer $d$ , all their $d$ 'th derivatives have a bound of order $R^{-d} \\cdot d!$ (which in fact forces the functions to be real analytic).",
"It was shown in [15] that any reduct of $\\mathbb {R}_{an}$ has the 0-mild parameterization property: every definable subset of $(-1, 1)^n$ has a parameterization by a finite set of 0-mild functions.",
"However, this result cannot be made uniform.",
"For Yomdin showed in [33] (see also [34]) that the number of 0-mild functions required to parameterize the set $\\lbrace \\langle x_1 , x_2 \\rangle \\in (-1, 1)^2 : x_1 \\cdot x_2 = t \\rbrace $ necessarily tends to infinity as $t \\rightarrow 0$ .",
"Our second parametrization result recovers uniformity in the 0-mild setting but at the expense of, firstly, covering larger sets than, but ones having the same dimension as, the sets in the given family and secondly, covering not by ranges of 0-mild maps but by solutions to (a definable family of) Weierstrass polynomials with 0-mild functions as coefficients.",
"In [24] the parameterization theorem is applied to show that any definable subset of $(0, 1)^n$ (the ambient o-minimal structure being, once again, arbitrary) either contains an infinite semi-algebraic subset or else, for all $H \\ge 1$ , contains at most $H^{o(1)}$ rational points whose coordinates have denominators bounded by $H$ .",
"(For the purposes of this introduction we refer to such points as $H$ -bounded rational points.)",
"Although this result is best possible in general, and is so even for one dimensional subsets of $(0, 1)^2$ definable in the structure $\\mathbb {R}_{an}$ , it has been conjectured that the $H^{o(1)}$ bound may be improved to $(\\log H)^{O(1)}$ for certain reducts of $\\mathbb {R}_{an}$ (specifically, for sets definable from restricted Pfaffian functions), and it is our final aim in this paper is to take a small step towards such a conjecture.",
"We first observe that the point counting theorem from [24] quoted above follows (by induction on dimension) from the following uniform result (the main lemma of [24] on page 610).",
"Namely, if $m<n$ and $\\mathcal {X} = \\lbrace X_t : t \\in T \\rbrace $ is a definable family of $m$ -dimensional subsets of $(0, 1)^n$ , and $\\epsilon > 0$ , then there exists a positive integer $d= d(\\epsilon , n)$ such that for each $t \\in T$ and for all $H \\ge 1$ , all the $H$ -bounded rational points of $X_t$ are contained in the union of at most $O(H^{\\epsilon })$ algebraic hypersurfaces of degree at most $d$ , where the implied constant depends only on $\\mathcal {X}$ and $\\epsilon $ .",
"Now, for the structure $\\mathbb {R}_{an}^{pow}$ (or any of its suitable reducts), our uniform $C^r$ -parameterization theorem allows us to improve the bound here on the number of hypersurfaces to $O((\\log H)^{O(1)})$ (for $H > e$ , with the implied constants depending only on the family $\\mathcal {X}$ ) but, unfortunately, their degrees have this order of magnitude too.",
"Actually, the bound on the degrees is completely explicit, namely $[(\\log H)^{m/(n-m)}]$ , but as this tends to infinity with $H$ , the inductive argument used in [24] (where the degree d only depended on $\\epsilon $ and $n$ ) breaks down at this point.",
"Our 0-mild (quasi-) parameterization theorem does give a better result for (suitable reducts of) the structure $\\mathbb {R}_{an}$ in that the number of hypersurfaces is bounded by a constant (depending only on $\\mathcal {X}$ ), but the bound for their degrees is the same as above and so, once again, the induction breaks down.",
"We can, however, tease out a uniform result for rational points on one and two dimensional sets definable from restricted Pfaffian functions, but for the general conjecture a completely new uniform parameterization theorem that applies to the intersection of a definable set of constant complexity with an algebraic hypersurfaces of nonconstant degree is badly needed.",
"Note.",
"As this paper was being finalised, the arXiv preprints [1], [2] appeared.",
"There is some similarity in the methods used there and the complex analytic approach here.",
"There seems to be no inclusion in either direction in the parameterisation results nor in the diophantine applications; the diophantine result of [2] deals with sets of arbitrary dimension but in a smaller structure."
],
[
"Precise statements",
"2.1 $C^r$ -parameterizations The largest expansion of the real field (that is, the expansion with the most definable sets) to which our uniform $C^r$ -parameterization theorem applies is the structure $\\mathbb {R}_{an}^{pow}$ , i.e.",
"the expansion by all restricted analytic functions and all power functions $(0, \\infty ) \\rightarrow (0, \\infty ): x \\mapsto x^s$ (for $s \\in \\mathbb {R}$ ).",
"However, when it comes to applications there is considerable advantage to be gained from working in suitable reducts of $\\mathbb {R}_{an}^{pow}$ for which more effective topological and geometric information is available for the definable sets.",
"(For example, for sets definable from restricted Pfaffian functions one has, through the work of Khovanskii ([17]) and Gabrielov and Vorobjov ([11]), good bounds (in terms of natural data) on the number of their connected components.)",
"And of course, it is important for the inductive arguments involved in such applications that the parameterizing functions are definable in the same reduct as is the set (or family of sets) being parameterized.",
"It turns out that for our proof here to go through, the property required of the ambient o-minimal structure is that it should be a reduct of $\\mathbb {R}_{an}^{pow}$ in which a suitable version of the Weierstrass Preparation Theorem holds for definable functions.",
"Now, a large class of such reducts has been identified and extensively studied, by D. J. Miller in his Ph.D. thesis ([20], which was inspired by the subanalytic case obtained by Lion and Rolin in [19] and by Parusinski in [22]).",
"These are based on a language for functions in a Weierstrass system together with a certain class of power functions.",
"There is no need for us to go into precise definitions here-we will quote the relevant results from [20] when needed.",
"Suffice it to say that examples include the real ordered field itself, $\\mathbb {R}_{an}$ , $\\mathbb {R}_{an}^{pow}$ or, indeed, the expansion of $\\mathbb {R}_{an}$ by any collection of power functions that is closed under multiplication, inverse and composition (i.e.",
"such that the exponents form a subfield of $\\mathbb {R}$ ).",
"Many more examples appear in the literature (see [8], [7] and [20]).",
"We shall assume, in the precise statement of the theorem below and throughout section 4, that all notions of definability are with respect to some such fixed reduct of $\\mathbb {R}_{an}^{pow}$ : 2.1.1 Convention We fix a reduct of $\\mathbb {R}_{an}^{pow}$ based on a Weierstrass system $\\mathcal {F}$ and a subfield $K$ of its field of exponents as described in Definition 2.1 of [20].",
"As there, we denote its language by $\\mathcal {L}^{K}_{\\mathcal {F}}$ .",
"Note that, for the smallest possible choice of $\\mathcal {F}$ and $K=\\mathbb {Q}$ , the $\\mathcal {L}^{K}_{\\mathcal {F}}$ -definable sets are precisely the semi-algebraic sets.",
"2.1.2 Definition.",
"Let $r$ a non-negative integer or $+\\infty $ .",
"The $C^r$ -norm $\\Vert f \\Vert _{C^r}$ of a $C^r$ -function $f : U \\subset \\mathbb {R}^m \\rightarrow \\mathbb {R}$ , with $U$ open, is defined (in $\\mathbb {R}\\cup \\lbrace + \\infty \\rbrace $ ) by $\\sup _{x \\in U} \\sup _{\\underset{\\alpha \\in \\mathbb {N}^m}{|\\alpha | \\le r}} |f^{(\\alpha )} (x)|,$ where we have used the standard multi-index notation, namely $f^{(\\alpha )}$ stands for $\\partial ^{\\alpha } f / \\partial x^{\\alpha }$ ($= f$ for $\\alpha = 0$ ), and $|\\alpha |$ denotes $\\sum _{i=1}^{m} \\alpha _i$ .",
"By the $C^r$ -norm of a $C^r$ -map $f : U \\subset \\mathbb {R}^m \\rightarrow \\mathbb {R}^s$ , with $U$ open, we mean the maximum of the $C^r$ -norms of the component functions of $f$ .",
"2.1.3 Theorem (The uniform $C^r$ -parameterization theorem.)",
"Let $n, k$ be positive integers and $m$ be a nonnegative integer with $m \\le n$ .",
"Let $\\mathcal {X} = \\lbrace X_t : t \\in T \\rbrace $ be an $\\mathcal {L}^{K}_{\\mathcal {F}}$ -definable family of $m$ -dimensional subsets of $(0, 1)^n$ , where $T$ is some $\\mathcal {L}^{K}_{\\mathcal {F}}$ -definable subset of $\\mathbb {R}^k$ .",
"Then there exists a polynomial $D$ , depending only on the family $\\mathcal {X}$ , such that for each positive integer $r$ , and for each $t \\in T$ , there exist analytic maps $\\phi _{r, i, t} : (0, 1)^m \\rightarrow X_t$ for $i = 1, \\ldots , D(r)$ , whose $C^r$ -norms are bounded by 1 and whose ranges cover $X_t$ .",
"Moreover, for each $i$ and $r$ , $\\lbrace \\phi _{r,i,t} : t \\in T \\rbrace $ is an $\\mathcal {L}^{K}_{\\mathcal {F}}$ -definable family of functions.",
"The proof of 2.1.3 is given in section 4.",
"2.2 Quasi-parametrization For the main result of this section we require our ambient o-minimal structure to be a reduct of $\\mathbb {R}_{an}$ : we do not know whether theorem 2.2.3 below (or some version of it) holds if power functions with irrational exponents are admitted.",
"We shall be working with complex valued definable functions of several complex variables where the definability here is via the usual identification of $ with $$\\mathbb {R}$ 2$.",
"Naturally enough we will requirethere to be enough {\\it definable} holomorphic functions:$ 2.2.1 Convention We fix a reduct of $\\mathbb {R}_{an}$ with the following property.",
"If $f : U \\subset \\mathbb {R}^m \\rightarrow \\mathbb {R}$ , with $U$ open, is a definable, real analytic function, then for each $a \\in U$ there exists an open $V \\subset m$ with $a \\in V \\cap \\mathbb {R}^m \\subset U$ and a definable holomorphic function $\\tilde{f} : V \\rightarrow such that for all $ b V $\\mathbb {R}$ m$, $ f(b) = f(b)$.",
"For the remainder of this subsection and throughout section 5(unless otherwise stated) definability will be with respect to this structure.$ The main examples are the real field and $\\mathbb {R}_{an}$ itself.",
"Others may be constructed as follows.",
"Let $\\mathcal {F}$ be a collection of restricted (real) analytic functions closed under partial differentiation and under the operation implicit in 2.2.1 (i.e.",
"under taking the real and imaginary part functions of the local complex extensions).",
"Then the expansion of the real field by $\\mathcal {F}$ will be a reduct of $\\mathbb {R}_{an}$ satisfying 2.2.1.",
"This follows fairly easily from the theorem of Gabrielov ([10]) asserting that such a reduct is model complete.",
"(For a local description of the complex holomorphic functions that are definable in such a structure (at least, in a neighbourhood of a generic point) see [30].)",
"For $R>0$ we denote by $\\Delta (R)$ the open disc in $ of radius $ R$ and centred at the origin.$ 2.2.2 Definition Let $R>0$ , $K>0$ and let $m$ be a positive integer.",
"Then a definable family $\\Lambda = \\lbrace F_t : t \\in T \\rbrace $ , where $T$ is a definable subset of $\\mathbb {R}^k$ for some $k$ , is called an $(R, m, K)$ -family if for each $t \\in T$ , the function $F_t : \\Delta (R)^m \\rightarrow is holomorphicand for all $ z (R)m$ we have $ |Ft(z)| K$.$ We shall develop a considerable amount of theory for such families in section 5.",
"To mention just one result, which is perhaps of independent interest, we will show that if $R>1$ and, for each $t \\in T$ , $F_t(z) = \\sum a_{\\alpha }^{(t)} \\cdot z_1^{\\alpha _{1}}, \\cdots ,z_m^{\\alpha _{m}}$ is the Taylor expansion of $F_t$ around $0 \\in m$ (where the summation is over all $m$ -tuples $\\alpha = \\langle \\alpha _1, \\ldots , \\alpha _m \\rangle \\in \\mathbb {N}^m$ ), then there exists $M = M(\\Lambda ) \\in \\mathbb {N}$ such that $|a_{\\alpha }^{(t)}|$ achieves its maximum value for some $\\alpha $ with $|\\alpha | \\le M$ .",
"(See 2.1.2 for the multi-index notation.)",
"The fact that $M$ is independent of $t$ here is crucial for all the uniformity results that follow and leads to the following 2.2.3 Theorem (The quasi-parameterization theorem.)",
"Let $n$ and $m$ be non-negative integers with $m<n$ and let $\\mathcal {X} =\\lbrace X_s : s \\in S \\rbrace $ be a definable family of subsets of $[-1, 1]^n$ , each of dimension at most $m$ , where $S$ is a definable subset of $\\mathbb {R}^k$ for some $k$ .",
"Then there exists $R>1$ , $K > 0$ , a positive integer $d$ and an $(R, m+1, K)$ -family $\\Lambda = \\lbrace F_t : t \\in T \\rbrace $ such that each $F_t$ is a monic polynomial of degree $d$ in its first variable and for all $s \\in S$ , there exists $t \\in T$ such that $X_s \\subseteq \\lbrace x = \\langle x_1 , \\ldots , x_n \\rangle \\in [-1, 1]^n :\\exists w \\in [-1, 1]^m \\bigwedge _{i=1}^{n} F_t (x_i , w) =0 \\rbrace .$ The proof of 2.2.3 is given in section 5.",
"2.3 Diophantine applications The above parameterisation results may be applied to obtain results about the distribution of rational points on definable sets.",
"The height of a rational number $q=a/b$ where $a,b\\in \\mathbb {Z}$ with $b>0$ and ${\\rm gcd}(a,b)=1$ is defined to be $H(q)=\\max (|a|, b)$ and the height of a tuple $q=(q_1,\\ldots , q_n)\\in \\mathbb {Q}^n$ is $H(q)=\\max (H(q_i), i=1,\\ldots ,n)$ .",
"For $X\\subset \\mathbb {R}^n$ we set $X(\\mathbb {Q}, H)=\\lbrace x\\in X\\cap \\mathbb {Q}^n: H(x)\\le H\\rbrace $ and define the counting function $N(X,H)=\\#X(\\mathbb {Q}, H).$ In the following, ${\\mathcal {X}}\\subset T\\times [0,1]^n$ is a family of sets $X_t\\subset [0,1]^n, t\\in T$ , which is definable in a suitable o-minimal structure (specified in each result).",
"We assume that each fibre $X_t$ has dimension $m<n$ .",
"Note that “hypersurfaces” as used below may be reducible, that “degree” means total degree, and that $[x]$ denotes the integer part of a real number $x$ : $[x]\\in \\mathbb {Z}$ and $[x]\\le x< x+1$ .",
"2.3.1.",
"Theorem.",
"Let ${\\mathcal {X}}\\subset T\\times (0,1)^n$ be a family of sets $X_t, t\\in T$ , definable in $\\mathbb {R}_{an}^{pow}$ .",
"Then there exist positive constants $C_1=C_1({\\mathcal {X}}), c_1=c_1({\\mathcal {X}})$ such that, for $H\\ge e$ and $t\\in T$ , $X_t(\\mathbb {Q},H)$ is contained in the union of at most $C_1(\\log H)^{c_1}$ real algebraic hypersurfaces of degree at most $[(\\log H)^{m/(n-m)}].$ For a definable family in the smaller structure $\\mathbb {R}_{an}$ we get a more precise result.",
"2.3.2.",
"Theorem.",
"Let ${\\mathcal {X}}\\subset T\\times [-1,1]^n$ be a family of sets $X_t, t\\in T$ , definable in $\\mathbb {R}_{an}$ .",
"Then there exists a positive constant $C_2=C_2({\\mathcal {X}})$ such that, if $H\\ge e$ and $t\\in T$ then $X_t(\\mathbb {Q}, H)$ is contained in the union of at most $C_2$ real algebraic hypersurfaces of degree at most $[(\\log H)^{m/(n-m)}].$ If these results could be iterated on the intersections for we would be able to prove a poly-log bound for rational points.",
"However, as the degrees of the hypersurfaces increase with $H$ , even a second iteration would require a result for such non-definable families.",
"However, for certain families of pfaffian sets of dimension 2 (see the basic definitions below) we can carry this out using estimates due to Gabrielov and Vorbjov [11].",
"They have the right form of dependencies to give a suitable result for the curves arising when the surface is intersected with algebraic hypersurfaces of growing degree.",
"This idea has been used in several previous papers [4], [15], [16], [26], [27].",
"2.3.3.",
"Definition.",
"A pfaffian chain of order $r\\ge 0$ and degree $\\alpha \\ge 1$ in an open domain $G\\subset \\mathbb {R}^n$ is a sequence of analytic functions $f_1,\\ldots , f_r$ on $G$ satisfying differential equations $df_j=\\sum _{i=1}^ng_{ij}(x, f_1(x),\\ldots , xf_j(x))dx_i$ for $1\\le j\\le r$ , where $g_{ij}\\in \\mathbb {R}[x_1,\\ldots , x_n, y_1,\\ldots , y_j]$ are polynomials of degree not exceeding $\\alpha $ .",
"A function $f=P(x_1,\\ldots , x_n, f_1,\\ldots , f_r)$ where $P$ is a polynomial in $n+r$ variables with real coefficients of degree not exceeding $\\beta \\ge 1$ is called a pfaffian function of order $r$ and degree $(\\alpha , \\beta )$ .",
"A pfaffian set will mean the set of common zeros of some finite set pfaffian functions.",
"2.3.4.",
"Definition.",
"By a pfaffian surface we will mean the graph in $\\mathbb {R}^3$ of pfaffian functions of two variables with a common pfaffian chain of order and degree $(r, \\alpha )$ , defined on a “simple” domain $G$ in the sense of [11].",
"Namely, a domain of the form $\\mathbb {R}^2, (-1,1)^2, (0, \\infty )^2$ or $\\lbrace (u,v): u^2+v^2<1\\rbrace $ .",
"We take the complexity of the surface to be the triple $(r, \\alpha , \\beta )$ , where $\\beta $ is the maximum of the degrees of the coordinate functions defining the surface.",
"2.3.5.",
"Definition.",
"Let $X\\subset \\mathbb {R}^n$ .",
"The algebraic part of $X$ , denoted $X^{\\rm alg}$ is the union of all connected positive dimensional semi-algebraic subsets of $X$ .",
"The complement $X-X^{\\rm alg}$ is called the transcendental part of $X$ and denoted $X^{\\rm trans}$ .",
"By combining 2.1.3 with the methods of [26], [27] we get a uniform result for a family of pfaffian surfaces definable in $\\mathbb {R}_{an}^{pow}$ .",
"For an individual surface definable in the structure $\\mathbb {R}_{\\rm resPfaff}$ such a bound is due to Jones-Thomas [16].",
"Note that a definable surface in $\\mathbb {R}_{\\rm pfaff}$ is more general than a pfaffian surface.",
"Perhaps a combination of the methods could give uniformity for $\\mathbb {R}_{an}$ -definable families of restricted-pfaffian-definable sets of dimension 2.",
"2.3.6.",
"Proposition.",
"Let $r$ be a non-negative integer and $\\alpha , \\beta $ positive integers.",
"Let ${\\mathcal {X}}\\subset T\\times (0,1)^3$ be a family of surfaces $X_t, t\\in T$ , definable in $\\mathbb {R}_{an}^{pow}$ such that each fibre $X_t$ is the intersection of $(0,1)^3$ with a pfaffian surface with complexity (at most) $(r, \\alpha , \\beta )$ .",
"Then there exists $C_3({\\mathcal {X}}), c_3({\\mathcal {X}})$ such that, for $H\\ge e$ and $t\\in T$ , $N(X_t^{\\rm trans}, H)\\le C_3(\\log H)^{c_3}.$ When the family is definable in $\\mathbb {R}_{an}$ we can prove a more precise uniform result in which the exponent depends only on the complexity of the pfaffian surfaces.",
"2.3.7.",
"Proposition.",
"Let $r$ be a non-negative integer and $\\alpha , \\beta $ positive integers.",
"Let ${\\mathcal {X}}\\subset T\\times [-1,1]^3$ be a family of surfaces $X_t, t\\in T$ , definable in $\\mathbb {R}_{an}$ such that each fibre $X_t$ is the intersection of $[-1,1]^3$ with a pfaffian surface with complexity (at most) $(r, \\alpha , \\beta )$ .",
"Then there exists $C_4({\\mathcal {X}})$ depending only on ${\\mathcal {X}}$ and $c_4(r, \\alpha , \\beta ))$ such that, for $H\\ge e$ and $t\\in T$ , $N(X_t^{\\rm trans}, H)\\le C_4(\\log H)^{c_4}.$ The proofs of Theorems 2.3.1 and 2.3.2 and Propositions 2.3.6 and 2.3.7, assuming the parameterisation results, are given in Section 3."
],
[
"Some preliminaries for 2.3.1 and 2.3.2",
"For a positive integer $k$ and non-negative integer $\\delta $ we let $\\Lambda _k(\\delta )=\\lbrace \\mu =(\\mu _1,\\ldots ,\\mu _k)\\in \\mathbb {N}^k:|\\mu |=\\mu _1+\\ldots +\\mu _k=\\delta \\rbrace ,$ $\\Delta _k(\\delta )=\\lbrace \\mu =(\\mu _1,\\ldots ,\\mu _k)\\in \\mathbb {N}^k:|\\mu |=\\mu _1+\\ldots +\\mu _k\\le \\delta \\rbrace ,$ $L_k(\\delta )=\\#\\Lambda _k(\\delta ),\\quad D_k(\\delta )=\\#\\Delta _k(\\delta ),$ where $\\mathbb {N}$ is the set of nonnegative integers.",
"Let $X=X_t$ be a fibre of our definable family.",
"We will adapt the methods of [27], in which we explore $X(\\mathbb {Q}, H)$ with hypersurfaces of degree $d=[(\\log H)^{m/(n-m)}].$ This leads us to consider $D_n(d)\\times D_n(d)$ determinants $\\Delta $ whose entries are the monomials of degree $d$ (indexed by $\\Delta _n(d)$ ) evaluated at $D_n(d)$ points of $X$ .",
"These points lie on some algebraic hypersurface of degree $d$ if and only if $\\Delta =0$ .",
"Given some suitable parameterisation of $X$ by functions of $k$ variables, we estimate the above determinant by a Taylor expansion of the monomial functions to a suitable order $b$ (remainder term order $b+1$ ).",
"The order of the Taylor expansion will match the size of the matrix, and so we define $b(m,n,d)$ as the unique integer $b$ with $D_k(b)\\le D_n(d) < D_k(b+1).$ It is an elementary computation, carried out in [27], that $b=b(m,n,d)=\\left(\\frac{m!}{n!",
"}\\right)^{1/m}d^{n/m}(1+o(1))$ where the $o(1)$ means, here and below, as $d\\rightarrow \\infty $ with $m,n$ fixed.",
"In particular, $b(m,n,d)+1\\le 2\\left(\\frac{m!",
"d^n}{n!",
"}\\right)^{1/m}$ provided $d\\ge d_0(m,n)$ and hence provided $H\\ge H_0(m,n)$ .",
"The fact that $b$ is rather larger than $d$ is crucial to the estimates."
],
[
"Proof of 2.3.1",
"In this and subsequent subsections, $C,c,\\ldots $ will denote constants depending on ${\\mathcal {X}}$ , while $E$ denotes a constant depending only on $m,n$ , and in both cases they may differ at each occurrence.",
"Let $X=X_t$ be a fibre of ${\\mathcal {X}}$ .",
"We assume for now that $H\\ge H_0(m,n)$ for some $H_0(m,n)$ to be specified in the course of the proof.",
"According to Theorem 2.1.3, we can parameterise $X$ by functions $\\phi : (0,1)^m\\rightarrow (0,1)^n$ such that all partial derivatives of all component functions up to degree $b+1$ are bounded in absolute value by 1, and we can cover $X$ using at most $Cb^{c}$ such functions, where $C, c$ depend on ${\\mathcal {X}}$ .",
"Let us fix one such function $\\phi =(\\phi _1,\\ldots , \\phi _n)$ , where $\\phi _i: (0,1)^m\\rightarrow (0,1)$ .",
"From now on we deal only with $\\phi $ , whose properties of relevance depend only on $m,n$ .",
"Thus, from now on, all constants will depend only on $m,n$ .",
"We consider a $D_n(d)\\times D_n(d)$ determinant of the form $\\Delta =\\det \\left((x^{(\\nu )})^\\mu \\right)$ with $\\nu =1,\\ldots , D_n(d)$ indexing rows and $\\mu \\in \\Delta _d(n)$ indexing columns, where $x^{(\\nu )}=(x^{(\\nu )}_1,\\ldots ,x^{(\\nu )}_n)\\in X(\\mathbb {Q}, H)$ are points in the image of $\\phi $ , say $x^{(\\nu )}=\\phi (z^{(\\nu )})$ where $z^{(\\nu )}\\in (0,1)^m$ , later to be taken to be in a small disc in $(0,1)^m$ , and $x^\\mu =\\prod x_i^{\\mu _i}$ .",
"As each $x^{(\\nu )}$ is a rational number with denominator $\\le H$ , we find that there is a positive integer $K$ such that $K\\Delta \\in \\mathbb {Z}$ and $K\\le H^{ndD_n(d)}.$ If we write $\\Phi _\\mu =\\prod _{i=1}^n \\phi _i^{\\mu _i}$ for the corresponding monomial function on the $\\phi _i$ then we have $\\Delta =\\det \\left(\\Phi _\\mu (z^{(\\nu )})\\right).$ We now assume that the $z^{(\\nu )}$ all lie in a small disc of radius $r$ centred at some $z^{(0)}$ and expand the $\\Phi _\\mu $ in Taylor polynomials to order $b$ (with remainder term of order $b+1$ ).",
"For $\\alpha \\in \\Delta _k(b), \\beta \\in \\Lambda _k(b+1)$ we write $Q_{\\nu , \\mu }^\\alpha =\\frac{\\partial ^\\alpha \\Phi _\\mu (z^{(0)})}{\\alpha !",
"}\\left(z^{(\\nu )}-z^{(0)}\\right)^\\alpha ,\\quad Q_{\\nu , \\mu }^\\beta =\\frac{\\partial ^\\beta \\Phi _\\mu (z^{(0)})}{\\beta !",
"}\\left(\\zeta ^{(\\nu )}_\\mu -z^{(0)}\\right)^\\beta $ with a suitable intermediate point $\\zeta ^{(\\nu )}_\\mu $ (on the line joining $z^{(0)}$ to $z^{(\\nu )}$ ), and with $\\alpha !=\\prod _{j=1}^k\\alpha _j!$ , so that the Taylor polynomial is $\\Phi _\\mu (z^{(\\nu )})=\\sum _{\\alpha \\in \\Delta _k(b)}Q_{\\nu , \\mu }^\\alpha +\\sum _{\\beta \\in \\Lambda _m(b+1)}Q_{\\nu , \\mu }^\\beta $ and we have $\\Delta =\\det \\left(\\sum _{\\alpha \\in \\Delta _m(b+1)}Q_{\\nu , \\mu }^\\alpha \\right).$ We expand the determinant as in [25] (see also [28], eqn (2), p48), using column linearity to get $\\Delta =\\sum _{\\tau }\\Delta _\\tau ,\\quad \\Delta _\\tau =\\det \\left(Q_{\\nu , \\mu }^{\\tau (\\mu )}\\right).$ with the summation over $\\lbrace {\\tau : \\Delta _n(d)\\rightarrow \\Delta _m(b+1)}\\rbrace $ .",
"Now if, for some $k\\in \\lbrace 1,\\ldots , b\\rbrace $ , we have $\\#\\tau ^{-1}(\\Lambda _m(k))> L_m(k)$ then $\\Delta _\\tau =0$ as the corresponding columns are dependent (the space of homogeneous forms in $(z^{(\\nu )}-z^{(0)})$ of degree $k$ has rank $L_m(k)$ ).",
"Thus all surviving terms have a high number of factors of the form $(z^{(\\nu )}-z^{(0)})$ and/or $(\\zeta ^{(\\nu )}_\\mu -z^{(0)})$ .",
"We quantify this.",
"The function $\\Phi _\\mu $ is a product of $|\\mu |\\le d$ functions $\\phi _i$ , which have suitably bounded derivatives.",
"Let us consider more generally a function $\\Theta =\\prod _{i=1}^\\delta \\theta _i$ where $\\theta _i$ have $|\\theta _i^{(\\alpha )}(z)|\\le 1$ for all $|\\alpha |\\le b+1$ .",
"Then, for $\\alpha $ with $|\\alpha |\\le b+1$ , we have $\\Theta ^{(\\alpha )}=\\sum _{\\alpha ^{(i)}}{\\rm Ch}(\\alpha ^{(1)},\\ldots ,\\alpha ^{(\\delta )})\\prod _{i=1}^\\delta \\theta ^{(\\alpha ^{(i)})}$ with the summation over $\\alpha ^{(1)}+\\ldots +\\alpha ^{(\\delta )}=\\alpha $ where ${\\rm Ch}(\\alpha ^{(1)},\\ldots ,\\alpha ^{(\\delta )})=\\prod _{j=1}^m\\left(\\frac{\\alpha _j!",
"}{\\alpha ^{(1)}_j!\\ldots \\alpha ^{(\\delta )}_j!",
"}\\right).$ Since $|\\theta _i^{(\\alpha )}(z)|\\le 1$ for all $|\\alpha |\\le b+1$ we have $|\\Theta ^{(\\alpha )}(z)|\\le \\sum _{\\alpha ^{(i)}_1}\\left(\\frac{\\alpha _1!",
"}{\\alpha ^{(1)}_1!\\ldots \\alpha ^{(\\delta )}_1!",
"}\\right)\\times \\ldots \\times \\sum _{\\alpha ^{(i)}_m}\\left(\\frac{\\alpha _m!",
"}{\\alpha ^{(1)}_m!\\ldots \\alpha ^{(\\delta )}_m!",
"}\\right)$ each summation subject to $\\sum _{i} \\alpha ^{(i)}_j=\\alpha _j$ hence $|\\Theta ^{(\\alpha )}(z)|\\le \\delta ^{\\alpha _1}\\cdots \\delta ^{\\alpha _m}=\\delta ^{|\\alpha |}.$ Therefore $|Q_{\\nu , \\mu }^\\alpha |\\le \\frac{(|\\mu |r)^{|\\alpha |}}{\\alpha !",
"}\\le e^{m|\\mu |}r^{|\\alpha |} \\le e^{md}r^{|\\alpha |},$ and for a $\\tau $ which avoids the condition above (under which $\\Delta _\\tau =0$ ) all terms in its expansion are bounded above in size by $e^{mdD_n(d)}r^B$ where $B=B(m,n,d)=\\sum _{k=0}^bL_m(k)k+\\left(D_n(d)-\\sum _{\\kappa =0}^bL_m(\\kappa )\\right)(b+1).$ Note that $\\left(D_n(d)-\\sum _{\\kappa =0}^bL_m(\\kappa )\\right)\\ge 0$ by our choice of $b$ .",
"We have the asymptotic expression (see [27]) $B=B(m,n,d)=\\frac{1}{(m+1)!(m-1)!}\\left(\\frac{m!}{n!",
"}\\right)^{(m+1)/m}d^{n+n/m} (1+o(1)).$ The number of terms from all the $\\Delta _\\tau $ is $D_m(b+1)^{D_n(d)}D_n(d)!$ and we conclude that $|\\Delta |\\le D_m(b+1)^{D_n(d)}D_n(d)!e^{mdD_n(d)}r^B.$ Thus we have an integer $K\\Delta $ with $K|\\Delta |\\le H^{ndD_n(d)}D_m(b+1)^{D_n(d)}D_n(d)!e^{mdD_n(d)}r^B$ and if $K|\\Delta |\\ge 1$ then so is its $B$ th root.",
"Now with our choice of $d$ we find that $\\frac{dnD_n(d)}{B}=E\\frac{d^{n+1}}{d^{n+n/m}}(1+o(1))=Ed^{-(n-m)/m}(1+o(1))$ thus $H^{ndD_n(d)/B}\\le E(1+o(1))$ is bounded (as $d\\rightarrow \\infty $ ).",
"The remaining terms $\\left(D_m(b+1)^{D_n(d)}D_n(d)!e^{mdD_n(d)}\\right)^{1/B}$ are also bounded as $d\\rightarrow \\infty $ (see [27]) and so $\\left(K|\\Delta |\\right)^{1/B}\\le Er$ where $E$ is a constant depending only on $n,m$ , provided $H\\ge H_0(n,m)$ is sufficiently large.",
"If $rE<1$ then all points of $X(\\mathbb {Q}, H)$ parameterised by $\\phi $ from this disk lie on one algebraic hypersurface of degree $d$ , because the rank of the rectangular matrix formed by evaluating all monomials of degree $\\le d$ at all such points is less than $D_n(d)$ .",
"Since $(0,1)^m$ may be covered by some $E^{\\prime }$ such discs, and there are $C(b+1)^c\\le C^{\\prime }(\\log H)^{c^{\\prime }}$ maps $\\phi $ which cover $X$ , the required conclusion follows for $H\\ge H_0(m,n)$ .",
"However for $H\\le H_0$ the number of points is bounded depending only on $H_0, m, n$ .",
"${\\vbox {\\hrule height .4pt\\hbox{\\vrule width .4pt height 4pt\\hspace{4.0pt}\\vrule width .4pt}\\hrule height .4pt}}$$$ Note that, in [27] it is tacitly assumed that the mildness parameter $A$ satisfies $A\\ge 1$ ."
],
[
"Setup for 2.3.2",
"This and the subsequent two subsections are devoted to the proof of Proposition 2.3.2.",
"After some preliminary results, the proof itself is in 3.5.",
"We have a family of sets $X_t\\subset [0,1]^n$ , of dimension $m$ , definable in $\\mathbb {R}_{an}$ , and hence admitting a quasi-parameterisation of the following form.",
"There exist a positive integer $N$ , overconvergent analytic functions $h_{i,j}: [-1,1]^{m+\\eta }\\rightarrow \\mathbb {R}, \\quad i=1,\\ldots , n,\\quad j=0,\\ldots , N-1,$ converging on a disc of radius $r_0>1$ , and functions $u_j: T\\rightarrow [-1,1]^\\eta $ (which need not be definable) such that, setting $u(t)=(u_1(t),\\ldots , u_\\eta (t))$ , for all $t\\in T$ and $x=(x_1,\\ldots , x_n)\\in X_t$ there exists $w=(w_1,\\ldots , w_m)\\in [-1,1]^m$ such that $x_i^N=h_{i,N-1}(w_i, u(t))x_i^{N-1}+ \\ldots +h_{i, 0}(w_i, u(t)).$ In fact, we can assume that the $h_{i,j}$ are independent of $i$ , but don't need this.",
"We keep the previous convention regarding constants."
],
[
"Preliminary estimates",
"For each $i$ , setting $x=x_i$ and $h_j=h_{i,j}$ and suppressing the subscript $i$ and the arguments of the $h_j$ , we have a relation $x^N=h_{N-1}x^{N-1}+ \\ldots + h_{0}.$ By means of this relation, all powers $x^\\nu , \\nu \\in \\mathbb {N}$ may be expressed as suitable linear combinations of $1,x,\\ldots , x^{N-1}$ , namely $x^\\nu = \\sum _{j=0}^{N-1} q_{\\nu , j}x^j$ with coefficients $q_{\\nu , j}\\in \\mathbb {Z}[h_0,\\ldots , h_{N-1}]$ .",
"In particular $q_{N,j}=h_j$ .",
"We need an estimate for the degree and integer coefficients of the $q_{\\nu , j}$ .",
"Let $H$ be the $N\\times N$ matrix of analytic functions $\\left(\\begin{array}{cccccc}0&1&0&0&\\ldots &0\\\\0&0&1&0&\\ldots &0\\\\\\vdots &\\vdots &\\vdots &\\vdots &\\ddots &\\vdots \\\\h_0&h_1&h_2&h_3&\\ldots &h_{N-1}\\end{array}\\right)$ Then $H$ acts as a linear transformation on the vector space with (ordered) basis $\\lbrace 1, x, \\ldots , x^{N-1}\\rbrace $ and the $q_{\\nu , j}$ are the entries of the column vector $H^\\nu \\left(\\begin{array}{c}1\\\\0\\\\\\vdots \\\\0\\end{array}\\right)$ Inductively, the entry $H^\\nu _{ij}$ is in $\\mathbb {Z}[h_1,\\ldots , h_N]$ of degree $\\max (0, j-n+\\nu )$ and the sum of at most $\\max (n^{\\nu -1+j-n},1)$ pure (i.e.",
"with coefficient 1) monomials.",
"We have proved the following.",
"3.4.1.",
"Lemma.",
"For all $\\nu \\in \\mathbb {N}$ and $j=0,\\ldots , N-1$ , $q_{\\nu , j}$ is a sum of at most $2^\\nu $ monomials of degree at most $\\nu $ in the $h_j$ .",
"${\\vbox {\\hrule height .4pt\\hbox{\\vrule width .4pt height 4pt\\hspace{4.0pt}\\vrule width .4pt}\\hrule height .4pt}}$$$ The above is for one variable.",
"We now return to the multivariate setting with $x=(x_1,\\ldots , x_n)$ .",
"For $\\nu \\in \\mathbb {N}$ we then have $x_i^\\nu =\\sum _{j=0}^{N-1}q_{i, \\nu , j} x^j$ where $q_{i, \\nu , j}$ is the previously labelled $q_{\\nu , j}$ for the relevant $h_j=h_{i,j}$ .",
"If $\\nu \\in \\mathbb {N}^n$ we have $x^\\nu =\\prod _{i=1}^nx_i^{\\nu _i}=\\prod _{i=1}^n\\sum _{j=0}^{N-1}q_{i, \\nu _i ,j}x^j=\\sum _{\\lambda \\in M}q_{\\nu ,\\lambda }x^\\lambda $ where $q_{\\nu ,\\lambda }=\\prod _{i=1}^nq_{i, \\nu _i, \\lambda _i}$ .",
"We now want to bound the derivatives (in the $w$ variables) of the $q_{\\mu , \\lambda }$ .",
"For $\\alpha =(\\alpha _1,\\ldots , \\alpha _m)\\in \\mathbb {N}^m$ we set $\\overline{\\alpha }=\\max (\\alpha _i)$ .",
"3.4.2.",
"Lemma.",
"For suitable constants $C, R$ and all $\\alpha \\in \\mathbb {N}^m$ we have $\\frac{|q_{\\nu , \\lambda }^{(\\alpha )}(w, u)|}{\\alpha !",
"}\\le 2^{|\\nu |} (\\overline{\\alpha }+1)^{(|\\nu |-1)m}C^{|\\nu |} R^{|\\alpha |}.$ Proof.",
"For derivatives (in the $w$ -variables) of the $h_{i,j}$ we have a bound of the form $\\frac{|h_{i,j}^{(\\alpha )}(w, u)|}{\\alpha !}",
"\\le \\,C\\,R^{|\\alpha |},$ where $\\alpha \\in \\mathbb {N}^m$ , valid for every $i,j$ , by Cauchy's theorem, since they are analytic on some disc of radius $r_0>1$ .",
"We have that $q_{i,\\nu _i, j}$ is a sum of at most $2^{\\nu _i}$ monomials in the $h_{i,j}$ , each of degree at most $\\nu _i$ .",
"Therefore $q_{\\nu , \\lambda }$ is a sum of at most $2^{|\\nu |}$ monomials each of degree $|\\nu |$ in the $h_{i,j}$ .",
"Consider one such monomial $g=\\prod _{h=1}^\\ell \\phi _h$ where each $\\phi _h\\in \\lbrace h_{i,j},: i=1,\\ldots , n, j=0,\\ldots , N-1\\rbrace $ and $\\ell \\le |\\nu |$ .",
"As before $\\partial ^\\alpha g =\\partial ^\\alpha \\phi _1\\ldots \\phi _\\ell =\\sum _{\\alpha ^{(1)}+\\ldots + \\alpha ^{(\\ell )}=\\alpha }{\\rm Ch}(\\alpha ^{(1)},\\ldots ,\\alpha ^{(\\ell )})\\prod _{h=1}^\\ell \\partial ^{\\alpha ^{(h)}}\\phi _h.$ Thus $\\frac{\\partial ^\\alpha g}{\\alpha !",
"}\\le \\sum _{\\alpha ^{(1)}+\\ldots + \\alpha ^{(\\ell )}=\\alpha }\\prod _{h=1}^\\ell \\frac{\\partial ^{\\alpha ^{(h)}}\\phi _i}{\\alpha ^{(h)}!",
"}\\le C^{|\\nu |} R^{|\\alpha |}\\sum _{\\alpha ^{(1)}+\\ldots + \\alpha ^{(\\ell )}=\\alpha }1.$ The number of summands is at most $(\\overline{\\alpha }+1)^{(|\\nu |-1)m}$ .",
"${\\vbox {\\hrule height .4pt\\hbox{\\vrule width .4pt height 4pt\\hspace{4.0pt}\\vrule width .4pt}\\hrule height .4pt}}$$$ 3.4.3.",
"Corollary.",
"For $\\alpha \\in \\mathbb {N}^m$ we have $\\frac{|q_{\\nu , \\lambda }^{(\\alpha )}(w, u)|}{\\alpha !",
"}\\le 2^{|\\nu |} (|\\alpha |+1)^{|\\nu |m}C^{|\\nu |} R^{|\\alpha |}.\\ {\\vbox {\\hrule height .4pt\\hbox{\\vrule width .4pt height 4pt\\hspace{4.0pt}\\vrule width .4pt}\\hrule height .4pt}}$ $$"
],
[
"Proof of Proposition 2.3.2",
"We let $X=X_t$ be a fibre of ${\\mathcal {X}}$ .",
"We will explore $X(\\mathbb {Q}, H)$ with real algebraic hypersurfaces of degree $d=[(\\log H)^{m/(n-m)}],$ and again consider $D_n(d)\\times D_n(d)$ determinats $\\Delta =\\det \\left((x^{(\\nu )})^\\mu \\right)=0$ where $\\mu \\in \\Delta _n(d)$ indexes monomials and $x^{(\\nu )}, \\nu =1,\\ldots , D_n(d)$ are points of $X$ .",
"For each $x^{(\\nu )}$ there is some $z^{(\\nu )}$ such that the quasiparmeterisation conditions hold, i.e.",
"“$x^{(\\nu )}$ is parameterised by the point $z^{(\\nu )}$ ”.",
"Later we will assume that all the $z^{(\\nu )}$ are in the disc of radius $r$ entered at some $z^{(0)}$ .",
"By our assumptions, there is a positive integer $K\\le H^{ndD_n(d)}$ such that $K\\Delta \\in \\mathbb {Z}.$ Now let $M=\\lbrace \\lambda \\in \\mathbb {N}^n: \\lambda _i<N, i=1,\\ldots , n\\rbrace $ .",
"Then if $x\\in X$ we have $x^\\mu =\\sum _{\\lambda \\in M} x^\\lambda q_{\\mu ,\\lambda }(w, u(t))$ for some $z\\in [-1,1]^m$ where $q_{\\mu , \\lambda }=\\prod _{i=1}^n q_{\\mu _i, \\lambda _i}.$ There is a unique $b=b(m,n,d, N)$ such that $N^nD_m(b)\\le D_n(d)\\le N^nD_m(b+1).$ Since $m<n$ there are fewer monomials in $m$ variables than in $n$ variables, and so if $d$ suitably large in terms of $N, n$ then $b$ is somewhat larger than $d$ .",
"Set $B(m,n,d, N)=\\sum _{\\beta =0}^b N^nL_m(\\beta )\\beta +\\Big (D_n(d)-{N^n}\\,\\sum _{\\beta =0}^b L_m(\\beta )\\Big )(b+1).$ Since $b$ is somewhat larger than $d$ , we will have that $B$ is somewhat larger than $ndD_n(d)$ (as $d\\rightarrow \\infty $ ), as will be crucial.",
"Now we assume that all the $z^{(\\nu )}$ are all in the disc of radius $r$ entered at some $z^{(0)}$ .",
"We expand each $q=q_{\\mu , \\lambda }$ in a Taylor polynomial with remainder term of order $b+1$ .",
"For $\\alpha \\in \\Delta _k(b), \\beta \\in \\Lambda _k(b+1)$ we write $Q_{\\nu , \\mu }^{\\lambda , \\alpha }=\\frac{\\partial ^\\alpha q_{\\mu , \\lambda }}{\\alpha !",
"}\\left(z^{(0)}\\right)\\left(z^{(\\nu )}-z^{(0)}\\right)^\\alpha , \\quad Q_{\\nu , \\mu }^{\\lambda , \\beta }=\\frac{\\partial ^\\beta q_{\\mu , \\lambda }}{\\beta !",
"}\\left(z^{(0)}\\right)\\left(\\zeta ^{(\\nu )}_{\\lambda , \\beta }-z^{(0)}\\right)^\\beta $ for some suitable intermediate point $\\zeta ^{(\\nu )}_{\\lambda , \\beta }$ .",
"Then we have $\\Delta =\\det \\Big (\\sum _{(\\lambda ,\\beta )}Q_{\\mu , \\nu }^{\\lambda ,\\alpha }\\Big ),$ with the summation over $(\\lambda ,\\alpha )\\in \\lbrace 0,\\ldots , N-1\\rbrace ^n\\times \\Delta _k(b+1)$ .",
"We expand the determinant as previously in terms of maps $\\tau : \\Delta _n(d)\\rightarrow \\lbrace 0,\\ldots , N-1\\rbrace ^n\\times \\Delta _k(b+1)$ giving $\\Delta =\\sum _{\\tau } \\Delta _\\tau ,\\quad \\Delta _\\tau =\\det \\Big (Q^{\\tau (\\mu )}_{\\mu , \\nu }\\Big ).$ with the summation over $\\tau $ as above.",
"Now if for some $\\lambda \\in M, k$ with $0\\le k \\le b$ we have $\\#\\tau ^{-1}\\big (\\lbrace \\lambda \\rbrace \\times \\Lambda _m(k)\\big )> L_m(k)$ then $\\Delta _\\tau =0$ because the corresponding columns are dependent (the factors $(x^{(\\nu )})^\\lambda $ are constant on the rows in those columns).",
"Since there are $N^n$ possibilities for $\\lambda $ , we have that the total number of columns from which an expansion term of degree $k$ may be drawn for a surviving term is $N^nL_m(k)$ .",
"We now assume that $rR<1$ .",
"Then every surviving term is estimated by $\\big [\\big (n(b+1)^mC\\big )^d\\big ]^{D_n(d)}\\, (rR)^{B^{\\prime }}$ for some $B^{\\prime }\\ge B=B(m,n,d, N)$ , and since $rR<1$ every term is estimated by the above with $B^{\\prime }=B$ .",
"The total number of terms, assuming no cancellation, is $D_n(d)!\\left(N^n\\,D_m(b+1)\\right)^{D_n(d)}.$ Thus we have an integer $K\\Delta $ with $K|\\Delta | \\le H^{ndD_n(d)}\\, D_n(d)!\\left(N^nD_m(b+1)\\right)^{D_n(d)}\\,\\big [\\big (n(b+1)^mC\\big )^d\\,\\big ]^D\\, (rR)^{B}.$ And if $K|\\Delta |\\ge 1$ then so is its $B$ th root.",
"Now we have (see [27]) $L_m(d)=\\left(\\begin{array}{c}m-1+d\\\\ m-1\\end{array}\\right)=\\frac{d^{m-1}}{(m-1)!",
"}(1+o(1))$ where here and below $o(1)$ means as $d\\rightarrow \\infty $ for fixed $m, n, N$ .",
"Thus likewise $D_m(d)=L_{m+1}(d)=\\frac{d^{m}}{m!",
"}(1+o(1)).$ We find that $b(n,m,N,d)= \\left(\\frac{m!d^n}{n!",
"N^n}\\right)^{1/m}(1+o(1))$ for $d\\rightarrow \\infty $ with $n,m,N$ fixed.",
"Thus (by replacing the sum $\\sum _{\\delta =0}^bL_m(\\delta )\\delta $ by an integral) $B(m,n,N,d)=E(m,n,N)d^{n(m+1)/m}(1+o(1))$ where $E$ is a suitable combinatorial expression.",
"With our choice of $d$ we have as before that $H^{ndD_n(d)/B}\\le E_1$ is bounded.",
"We also have $\\Big (D_n(d)!",
"(N^nD_m(b+1))^{D_n(d)}\\Big )^{1/B}=1+o(1)$ as $d\\rightarrow \\infty $ , so is bounded by some $E$ .",
"Finally, we have that $\\frac{dD}{B}=\\frac{E}{d^{n/m-1}}(1+o(1))\\le \\frac{E}{d^{n/m-1}},$ while $\\frac{bD}{B}=E(1+o(1))\\le E.$ Therefore $|K\\Delta |^{1/B}\\le E\\,C\\,Rr$ and all the points of $X(\\mathbb {Q}, H)$ in the image of the disc lie on one hypersurface of degree at most $d$ provided $r<(C\\, E\\, R)^{-1}.$ The box $(0,1)^m$ may be covered by $C_2=\\left(c(n)C_2E_5R+1\\right)^m$ such discs, where $c(n)$ is the maximum side of a cube inscribed in a unit $n$ -sphere.",
"This gives the desired conclusion for $H\\ge H_0(m,n,N)$ and for smaller $H$ it follows as the number of such rational points is bounded.",
"${\\vbox {\\hrule height .4pt\\hbox{\\vrule width .4pt height 4pt\\hspace{4.0pt}\\vrule width .4pt}\\hrule height .4pt}}$$$"
],
[
"Proof of 2.3.3 and 2.3.4",
"By Theorem 2.1.3 in the case of 2.3.5, and Theorem 2.3.3 in the case of 2.3.6, we find that $X_t(\\mathbb {Q}, H)$ is contained in the intersection of $X_t$ with at most $c_5({\\mathcal {X}})$ hypersurfaces of degree $d=[(\\log H)^{m/(n-m)}]$ .",
"If any such intersection has dimension2, then the pfaffian functions parameterising the surface $X_t$ identically satisfy some algebraic relation.",
"Then the surface $X_t$ is algebraic, and $X_t^{\\rm trans}$ is empty.",
"Thus we may assume that all the intersections have dimension at most 1.",
"We will treat these following the method in [27] by dividing the intersections into graphs of functions with suitable properties, and estimating the rational points on any such graphs which are not semi-algebraic using the Gabrielov-Vorobjov estimates.",
"Suppose that the fibre $X_t$ is the intersection of $[-1,1]^3$ with the pfaffian surface defined by the pfaffian functions $x, y, z: G\\rightarrow \\mathbb {R}$ of complexity (at most) $(r, \\alpha , \\beta )$ .",
"Write $(p,q)$ for the variables in $G$ .",
"Suppose that the polynomial $F(x,y,z)$ of degree at most $d$ defines the hypersurface $V=V_F$ .",
"The intersection $X_t\\cap V$ is the image of the one-dimensional subset $W\\subset G$ defined by $\\phi (p,q)=F(x(p,q), y(p,q), z(p,q))=0.$ It is thus the zero-set of a pfaffian function of complexity $(r, \\alpha , d\\beta )$ .",
"The singular set $W_s\\subset W$ is defined by $\\phi =\\phi _p=\\phi _q=0$ , the zero-set pfaffian functions of complexity $(r, \\alpha , \\alpha +d\\beta -1)$ (see [11]).",
"At a point of $W-W_s$ , $W$ is locally the graph of a real-analytic function parameterised by $p$ if $\\phi _p\\ne 0$ , or $q$ if $\\phi _q\\ne 0$ .",
"Proceeding as in [27], we decompose $V_F$ into “good” curves, and points.",
"Here a “good” curve is a connected subset whose projection into each coordinate plane of $\\mathbb {R}^3$ is a “good” graph with respect to one or other of the axes; namely, the graph of a function $\\psi $ which is real analytic on an interval, has slope of absolute value 1 at every point, and such that the derivative of $\\psi $ of each order $1,\\ldots , [\\log H]$ is either non-vanishing in the interior of the interval or identically zero.",
"Using the topological estimates of Gabrielov-Vorobjov [11], Zell [35], and estimates for the complexities of the various pfaffian functions involved as in [27], one shows that $V_F$ decomposes into a union of at most $C_6(r, \\alpha , \\beta ) d^{C_7(r, \\alpha , \\beta )}$ points and “good” curves $Z$ .",
"If such a “good” curve $Z$ is semi-algebraic, then so are its projections to each coordinate plane, and also conversely.",
"On a non-algebraic plane “good” graph $Y$ , one has $N(Y, H)\\le C_8(r,\\alpha , \\beta ) (d\\log H)^{C_9(r, \\alpha , \\beta )}$ as in [27], using [26] and estimates for pfaffian complexity.",
"${\\vbox {\\hrule height .4pt\\hbox{\\vrule width .4pt height 4pt\\hspace{4.0pt}\\vrule width .4pt}\\hrule height .4pt}}$$$"
],
[
"Proof of the $C^r$ -parameterization theorem",
"In this section we prove Theorem 2.1.3.",
"We will use a combination of the preparation result of [20] with a preparation result of [21] which enables us to have centers of preparation with bounded first derivatives, see Theorem REF below.",
"Another ingredient for us is the result in [23] on cell decomposition with Lipschitz continuous cell walls, up to transformations coming just from coordinate permutations, see Theorem REF below.",
"Pawłucki's results in [23] refine Kurdyka's [18] and Valette's [29] in the sense that only coordinate permutations are needed, and no more general orthogonal transformations.",
"In section REF we give some results about derivatives of compositions, related to mild functions, Gevrey functions, Faá di Bruno's formula, and a notion which we call weakly mild functions.",
"Let us first fix some terminology.",
"For any language ${\\mathcal {L}}$ on $\\mathbb {R}$ and any subfield $K$ of $\\mathbb {R}$ , let us denote by ${\\mathcal {L}}^K$ the expansion of ${\\mathcal {L}}$ by the power maps $x \\mapsto {\\left\\lbrace \\begin{array}{ll} x^r, \\mbox{ if } x> 0,\\\\ 0 \\mbox{ otherwise, }\\end{array}\\right.",
"}$ for $r\\in K$ .",
"Let ${\\mathcal {L}}_{an}$ be the subanalytic language on $\\mathbb {R}$ .",
"We will use some reducts of ${\\mathcal {L}}_{an}^K$ , following [20].",
"Let $\\mathcal {F}$ be a Weierstrass system and let ${\\mathcal {L}}_\\mathcal {F}$ be the corresponding language as in Definition 2.1 of [20].",
"(The language ${\\mathcal {L}}_\\mathcal {F}$ is a reduct of ${\\mathcal {L}}_{an}$ .)",
"By the field of exponents of $\\mathcal {F}$ is meant the set of real $r$ such that $(0,1)\\rightarrow \\mathbb {R}:x\\mapsto (1+x)^r$ is ${\\mathcal {L}}_\\mathcal {F}$ -definable; this set is moreover a field, see Remark 2.3.5 of [20].",
"Let $K$ be a subfield of the field of exponents of $\\mathcal {F}$ .",
"From Section REF up to the end of Section we will work with the ${\\mathcal {L}}_\\mathcal {F}^K$ -structure on $\\mathbb {R}$ ."
],
[
"Compositions",
"We refine the notion of mild functions from [27] (which resemble Gevrey functions, [12]).",
"Definition 4.1.1 Let $A>0$ and $C\\ge 0$ be real, and let $r>0$ be either an integer or $+\\infty $ .",
"A function $f:U\\subset (0,1)^m\\rightarrow (0,1)$ with $U$ open is called $(A,C)$ -mild up to order $r$ if it is $C^r$ and for all $\\alpha \\in \\mathbb {N}^m$ with $|\\alpha |\\le r$ and all $x\\in U$ one has $| f^{(\\alpha )}(x) | \\le \\alpha !",
"(A|\\alpha |^C)^{|\\alpha |}.$ Call a map $f:U\\subset (0,1)^m\\rightarrow (0,1)^n$ $(A,C)$ -mild up to order $r$ if all of its component functions are.",
"We introduce a weaker notion, namely that of weakly mild functions.",
"Definition 4.1.2 Let $A>0$ and $C\\ge 0$ be real, and $r>0$ be either an integer or $+\\infty $ .",
"A function $f:U\\subset (0,1)^m\\rightarrow (0,1)$ with $U$ open is called weakly $(A,C)$ -mild up to order $r$ if it is $C^r$ and for all $\\alpha \\in \\mathbb {N}^m$ with $|\\alpha |\\le r$ and all $x\\in U$ one has $| f^{(\\alpha )}(x) | \\le \\frac{ \\alpha !",
"(A|\\alpha |^C)^{|\\alpha |} }{x^\\alpha } ,$ where $x^\\alpha $ stands for $\\prod _{j=1}^m x_j^{\\alpha _j}$ .",
"Call a map $f:U\\subset (0,1)^m\\rightarrow (0,1)^n$ weakly $(A,C)$ -mild up to order $r$ if all of its component functions are.",
"By the theory of Gevrey functions [12], it is known that a composition of mild functions is mild.",
"Here we study some related results about compositions, with proofs based on Faá di Bruno's formula.",
"(We do no effort to control the bounds beyond what we need.)",
"The next lemma is obvious by the chain rule for derivation.",
"Lemma 4.1.3 Let $r>0$ be an integer and let $f:U\\subset (0,1)^m\\rightarrow (0,1)$ be $(A,C)$ -mild up to order $r$ .",
"Then the function $V\\rightarrow (0,1):x\\mapsto f(x/Ar^{C+1})$ has $C^r$ -norm bounded by 1, where $V\\subset (0,1)^m$ is the open consisting of $x$ such that $x/Ar^{C+1} := (x_1/Ar^{C+1},\\ldots , x_m/Ar^{C+1})$ lies in $U$ .",
"We give a basic corollary of a multi-variable version of Faá di Bruno's formula from e.g. [5].",
"Lemma 4.1.4 Let $m\\ge 1$ and $d\\ge 1$ be integers.",
"Then there exist $A>0$ and $C\\ge 0$ , depending only on $m$ and $d$ , such that the following holds.",
"Consider a composition $h=f\\circ g$ , with $g:U\\subset \\mathbb {R}^d\\rightarrow V\\subset \\mathbb {R}^m$ , $f:V\\rightarrow \\mathbb {R}$ and $U$ and $V$ open.",
"Let $\\nu \\in \\mathbb {N}^d$ be a nonzero multi-index, write $|\\nu | = n$ and suppose that $f$ and $g$ are $C^n$ .",
"Then $h^{(\\nu )}$ is the sum of no more than $(An^{C})^{n}$ terms of the form (in multi-index notation) $f^{(\\lambda )} \\prod _{j=1}^s \\big ( g^{(\\ell _j)} \\big )^{k_j},$ where $1\\le s \\le n$ , $\\lambda ,k_j\\in \\mathbb {N}^m$ , $\\ell _j\\in \\mathbb {N}^d$ , $0< |\\lambda |\\le n$ , $0<|k_j|\\le n$ , $0<|\\ell _j|\\le n$ , and where $\\sum _{j=1}^s k_j = \\lambda ,\\ \\mbox{ and } \\sum _{j=1}^s |k_j|\\ell _j = \\nu ,$ where $0^0$ is set equal to 1, and where $h^{(\\nu )}$ and $g^{(\\ell _j)}$ are evaluated at $u\\in U$ and $f^{(\\lambda )}$ at $g(u)$ .",
"The lemma follows from Theorem 2.1 of [5], by using $\\frac{\\nu !",
"}{\\prod _{j=1}^s k_j!",
"(\\ell _j!",
")^{|k_j|}}\\le \\nu !",
"\\le n^n,$ and by letting the $s$ , $\\lambda _i$ , $k_{ji}$ and $\\ell _{jr}$ for $i=1,\\ldots ,m$ , $j=1,\\ldots ,n$ , and $r=1,\\ldots ,d$ run independently from 0 to $n$ when estimating the number of terms in (2.1) of Theorem 2.1 of [5].",
"We give some results about compositions.",
"Proposition 4.1.5 Let $A>0$ and $C\\ge 0$ be real numbers and let $m>0$ be an integer.",
"Let $f:V\\rightarrow (0,1)$ be a function on some open $V\\subset (0,1)^m$ .",
"Assume for $\\beta \\in \\mathbb {N}^m$ with $|\\beta |\\le 1$ , that $f^{(\\beta )}$ is weakly $(A,C)$ -mild up to order $+\\infty $ .",
"Then, there is $(A^{\\prime },C^{\\prime })$ , depending only on $m$ , $A$ and $C$ , such that for any integers $r>0$ and $L_i\\ge r$ , the composition $h=f\\circ g$ of $f$ with $g :x\\mapsto x^L := (x_1^{L_1},\\ldots ,x_m^{L_m})$ on the open $U\\subset (0,1)^m$ consisting of $x$ with $x^L\\in V$ , is $(L_{i_0}A^{\\prime },C^{\\prime })$ -mild up to order $r$ , with $i_0$ such that $L_{i_0}=\\max _i L_i$ .",
"By Lemma REF with $d=m$ , and up to enlarging $A$ and $C$ if necessary, we only have to estimate a single term of the form (REF ) for $\\nu $ with $|\\nu |\\le r$ .",
"Fix $\\nu $ with $1\\le |\\nu |\\le r$ and write $|\\nu |=n$ .",
"If $n=1$ the statement follows from the conditions on $f^{(\\beta )}$ for $\\beta $ with $|\\beta |\\le 1$ .",
"So let us suppose $n>1$ .",
"Fix $s$ with $1\\le s\\le n$ and $\\lambda \\in \\mathbb {N}^m$ with $|\\lambda | \\le n$ .",
"Choose $\\lambda ^{\\prime }$ and $\\beta $ in $\\mathbb {N}^m$ with $\\lambda ^{\\prime } + \\beta = \\lambda $ and $|\\beta |=1$ .",
"(Near the end we will optimize the choice of $\\beta $ .)",
"By the weak $(A,C)$ -mildness of $f^{(\\beta )}$ , we have $|f^{(\\lambda )}(x^L)| = |( f^{(\\beta )})^{(\\lambda ^{\\prime })} (x^L) | \\le \\frac{ \\lambda ^{\\prime }!",
"(A|\\lambda ^{\\prime }|^C)^{|\\lambda ^{\\prime }|} }{x^{L\\lambda ^{\\prime }}},$ where $L\\lambda ^{\\prime } = (L_i\\lambda ^{\\prime }_i)_i.$ We may and do suppose that $i_0=1$ for convenience of notation, so that $L_1=\\max _i L_i$ .",
"We have $|\\prod _{j=1}^s \\big ( g^{(\\ell _j)}(x) \\big )^{k_j}| \\le |\\prod _{j=1}^s L_1^{|\\ell _j|\\cdot |k_j|} x^{(L-\\ell _j)k_j} |$ where $k_j$ and $\\ell _j$ are as in (REF ) and where $(L-\\ell _j)k_j = ((L_i-\\ell _{ji})k_{ji} )_i.$ By (REF ) it follows that $\\frac{\\prod _{j=1}^s L_1^{|\\ell _j|\\cdot |k_j|} x^{(L-\\ell _j)k_j} }{x^{L\\lambda ^{\\prime }}} & = & L_1^{\\sum _{j=1}^s |\\ell _j|\\cdot |k_j|} x^{-L\\lambda ^{\\prime } + \\sum _{j=1}^s(L-\\ell _j)k_j}\\\\& \\le & L_1^{n} x^{L\\beta - \\sum _{j=1}^s \\ell _jk_j},$ where $L\\beta =(L_i\\beta _i)_i$ and similarly $\\ell _jk_j=(\\ell _{ji}k_{ji})_i$ .",
"Since this last inequality holds for any choice of $\\beta $ with $|\\beta |=1$ (and the corresponding $\\lambda ^{\\prime }$ ), given $x$ we can choose $\\beta $ with $\\beta _{i_1}=1$ when $x_{i_1}=\\min _i x_i$ .",
"Now we are done since $|\\sum _{j=1}^s \\ell _jk_j |\\le n$ by (REF ) and $|L\\beta |=L_{i_1} \\ge r \\ge n$ .",
"Proposition 4.1.6 Let $A>0,A^{\\prime }>0,C\\ge 0,C^{\\prime }\\ge 0$ be real numbers, let $m>0$ and $d>0$ be integers, and let $r$ be either a positive integer or $+\\infty $ .",
"Let $f:V\\rightarrow \\mathbb {R}$ and $g:U\\rightarrow V$ be $C^r$ functions on some open sets $U\\subset (0,1)^d$ and $V\\subset (0,1)^m$ .",
"Assume that $f$ is $(A,C)$ -mild up to order $r$ and that $g$ is weakly $(A^{\\prime },C^{\\prime })$ -mild up to order $r$ .",
"Then, there is $(A^{\\prime \\prime },C^{\\prime \\prime })$ , depending only on $m,d$ and $A,A^{\\prime },C,C^{\\prime }$ , such that the composition $h = f\\circ g$ is weakly $(A^{\\prime \\prime },C^{\\prime \\prime })$ -mild up to order $r$ .",
"As above, we only have to estimate a single term of the form (REF ) for $\\nu $ with $|\\nu |\\le r$ , by Lemma REF .",
"Fix $\\nu $ with $1\\le |\\nu |\\le r$ and write $|\\nu |=n$ .",
"Fix $s$ with $1\\le s\\le n$ and $\\lambda \\in \\mathbb {N}^m$ with $|\\lambda | \\le n$ .",
"By the $(A,C)$ -mildness of $f$ up to order $r$ we have $|f^{(\\lambda )}(x)| \\le \\lambda !",
"(A|\\alpha |^C)^{|\\lambda |}.$ By the weak $(A^{\\prime },C^{\\prime })$ -mildness of $g$ up to order $r$ we have $|\\prod _{j=1}^s \\big ( g^{(\\ell _j)}(x) \\big )^{k_j}| \\le | \\prod _{j=1}^s \\frac{ \\ell _j!",
"(A^{\\prime } |\\ell _j|^{C^{\\prime }} )^{|\\ell _j|\\cdot |k_j|} }{ x^{\\ell _j |k_j|} } |$ where $k_j$ and $\\ell _j$ are as in (REF ).",
"Now by (REF ), and assuming $A^{\\prime }\\ge 1$ , $|\\prod _{j=1}^s \\big ( g^{(\\ell _j)}(x) \\big )^{k_j}| \\le | \\frac{ \\nu !",
"(A^{\\prime } n^{C^{\\prime }} )^{\\sum _j |\\ell _j|\\cdot |k_j|} }{ x^{\\sum _j \\ell _j |k_j|} } |\\le | \\frac{ \\nu !",
"(A^{\\prime } n^{C^{\\prime }} )^{n} }{ x^{\\nu } } |.$ Putting these together we find bounds as desired.",
"We give one concrete instance of Proposition REF with polynomial control on the constants.",
"Lemma 4.1.7 Let $\\varepsilon >0$ , $A>0$ and $C\\ge 0$ be real numbers and let $m>0$ be an integer.",
"Let $f:U\\rightarrow (0,1)$ be weakly $(A,C)$ -mild up to order $+\\infty $ on some open $U$ of $(0,1)^m$ .",
"Assume furthermore that $f(x)>\\varepsilon $ on $U$ .",
"Then there are polynomials $A_1(\\ell )$ and $C_1(\\ell )$ in one variable $\\ell $ , depending only on $\\varepsilon $ , $A$ , $C$ and $m$ such that, for any integer $\\ell >0$ , the function sending $x$ in $U$ to $\\@root \\ell \\of {f(x)}$ is weakly $(A_1(\\ell ),C_1(\\ell ))$ -mild up to order $+\\infty $ .",
"If moreover $\\partial f/\\partial x_i$ is also weakly $(A,C)$ -mild up to order $+\\infty $ , for some $i$ , then $A_1(\\ell )$ , $C_1(\\ell )$ can be taken so that, for any integer $\\ell >0$ , $\\varepsilon \\cdot \\frac{\\partial \\@root \\ell \\of {f}}{\\partial x_i}$ is also weakly $(A_1(\\ell ),C_1(\\ell ))$ -mild up to order $+\\infty $ .",
"The lemma follows easily from the chain rule for derivation.",
"We apply our results on compositions to a specific kind of functions, called a-b-m functions, and defined as follows.",
"Definition 4.1.8 Call a function $U\\subset (0,1)^m\\rightarrow \\mathbb {R}$ monomial if it is of the form $x\\mapsto x^\\mu := \\prod _{i=1}^m x_i^{\\mu _{i}}$ for some $\\mu _{i}$ in $\\mathbb {R}$ .",
"Call a function $U \\rightarrow \\mathbb {R}$ bounded-monomial if it is monomial and its range is a bounded set in $\\mathbb {R}$ .",
"Call a map $U \\rightarrow \\mathbb {R}^n$ monomial, resp.",
"bounded-monomial, if all of its component functions are.",
"Call a function $U\\rightarrow \\mathbb {R}$ analytic-bounded-monomial, abbreviated by a-b-m, if it is of the form $x \\mapsto F( b(x) )$ for some bounded-monomial map $b:U\\subset (0,1)^m\\rightarrow \\mathbb {R}^n$ for some $n\\ge 0$ and for some function $F:b(U)\\rightarrow \\mathbb {R}$ which is analytic on some open neighborhood of the closure of $b(U)$ in $\\mathbb {R}^n$ .",
"Say that a function $f:U\\subset (0,1)^m\\rightarrow \\mathbb {R}$ is 0-prepared if there is an a-b-m function $u:U\\rightarrow \\mathbb {R}$ which takes values in $(1/S,S)$ for some $S>1$ and a bounded-monomial function $x\\mapsto x^\\mu $ on $U$ such that $f(x) = x^\\mu u(x).$ Call a map $U\\subset (0,1)^m \\rightarrow \\mathbb {R}^n$ a-b-m, resp.",
"0-prepared, if all of its component functions are.",
"In our terminology, a 0-prepared function is automatically a-b-m and has bounded range.",
"Lemma 4.1.9 Let $f:U\\subset (0,1)^m\\rightarrow (0,1)$ be an a-b-m function.",
"Then the function $f$ is weakly $(A,C)$ -mild up to order $+\\infty $ for some $A>0$ and some $C\\ge 0$ .",
"Any function $F:V\\rightarrow (0,1)$ on an open $V$ in $(0,1)^n$ such that $F$ is analytic on some open neighborhood of the closure of $V$ in $\\mathbb {R}^n$ is $(A_0,C_0)$ -mild up to order $+\\infty $ for some $A_0>0$ and some $C_0\\ge 0$ , see e.g. [12].",
"Also, for any real $S\\ge 1$ and any bounded-monomial function $b:U\\subset (0,1)^m\\rightarrow (0,S) : x\\mapsto x^\\mu $ with $U$ open in $(0,1)^m$ , the function $b/S$ is weakly $(A_1,C_1)$ -mild up to order $+\\infty $ for some $A_1>0$ and $C_1\\ge 0$ .",
"One may for example take $A_1=C_1=\\max _i |\\mu _i|$ .",
"The lemma now follows from Proposition REF and the definition of a-b-m functions."
],
[
"Cell decomposition with Lipschitz continuous cell walls",
"We use the definition of cells for an o-minimal structure $R$ as in [9], Chapter 3.",
"Recall that this definition uses an ordering of the coordinates.",
"The results in this section will be later used for $R=\\mathbb {R}$ with ${\\mathcal {L}}_\\mathcal {F}^K$ -structure.",
"Definition 4.2.1 (Cell walls) Let $R$ be an o-minimal structure as in [9].",
"Let $C\\subset R^n$ be a definable cell, and write $C = \\lbrace x\\in R^n\\mid \\wedge _{i=1}^n\\, \\alpha _i(x_{<i}) \\mathrel {\\square }_{i1} x_i \\mathrel {\\square }_{i2} \\beta _i(x_{<i}) \\rbrace $ for continuous definable functions $\\alpha _i$ and $\\beta _i$ with $\\alpha _i < \\beta _i$ , $x_{<i} = (x_1,\\ldots ,x_{i-1})$ , and with $\\mathrel {\\square }_{i1}$ either $=$ , $<$ , or no condition, and with $\\mathrel {\\square }_{i2}$ either $<$ or no condition.",
"If $\\mathrel {\\square }_{i1}$ is $=$ or $<$ then we call $\\alpha _i$ a wall of $C$ .",
"Likewise, if $\\mathrel {\\square }_{i2}$ is $<$ , then we also call $\\beta _i$ a wall of $C$ , where we use the convention that $\\mathrel {\\square }_{i2}$ is no condition if $\\mathrel {\\square }_{i1}$ is equality.",
"We give a family version of Theorem 3$_n$ of [23].",
"In the following result, some of the coordinates will be reordered in order to make cell walls Lipschitz continuous.",
"Theorem 4.2.2 ([23], family version) Let $R$ be a real closed field equipped with an o-minimal structure.",
"Let $T\\subset R^{k}$ and $G\\subset T\\times R^n$ be definable.",
"Then there exist a real constant $M_n\\ge 1$ depending only on $n$ , and a finite partition of $G$ $G = S_1\\cup \\ldots \\cup S_\\ell ,$ such that for every $i$ there exists a reordering of the coordinates on $R^n$ such that $S_{i}$ is a definable cell (w.r.t.",
"to the new ordering of the coordinates on $R^n$ ) whose walls $\\alpha $ are $C^1$ and satisfy $| \\frac{\\partial \\alpha }{\\partial x_j } (t,x) |\\le M_n$ for $j=1,\\ldots ,n$ .",
"Here, we write $(t,x)$ for coordinates $t$ running over $R^k$ and $x$ over $R^n$ .",
"By Theorem 3$_n$ of [23], the theorem holds when $T$ is a singleton, and up to an extra part in the partition which is a definable subset $\\Sigma $ of $G$ of dimension less than $n$ .",
"To deal with the part $\\Sigma $ is easy and classical by induction on $k+n$ .",
"(See the proof of Step 1 in Section REF for a similar such argument.)",
"The family version follows from logical compactness."
],
[
"Preparation with a Lipschitz continuous center",
"We combine the Main Theorem of [20] with Proposition 5.2 of [21], to get preparation of ${\\mathcal {L}}_\\mathcal {F}^K$ -definable functions with centers which are moreover Lipschitz continuous.",
"Let us first recall some definitions, with notation from Section 2.1.",
"Let $C\\subset \\mathbb {R}^n$ be an ${\\mathcal {L}}_\\mathcal {F}^K$ -definable cell.",
"Write $\\Pi _{<n}:\\mathbb {R}^n\\rightarrow \\mathbb {R}^{n-1}$ for the projection sending $x$ in $\\mathbb {R}^n$ to $x_{<n}=(x_1,\\ldots ,x_{n-1})$ .",
"Write $B$ for $\\Pi _{<n}(C)$ .",
"Let $\\theta :B\\rightarrow \\mathbb {R}$ be a continuous, ${\\mathcal {L}}_\\mathcal {F}^K$ -definable function whose graph is disjoint from $C$ .",
"Definition 4.3.1 (Units with center) Let $B,C,\\theta ,x_{<n}$ be as above.",
"An ${\\mathcal {L}}_\\mathcal {F}^K$ -definable function $u:C\\rightarrow \\mathbb {R}$ is called an ${\\mathcal {L}}_\\mathcal {F}^K$ -unit with center $\\theta $ if $u$ can be written in the form $u(x) = F (\\varphi (x))$ , where $\\varphi :C\\rightarrow \\mathbb {R}^s$ sends $x$ to $( a_1(x_{<n})|x_n - \\theta (x_{<n})|^{r_1},\\ldots , a_s(x_{<n})|x_n - \\theta (x_{<n})|^{r_s} )$ for some integer $s\\ge 0$ , some ${\\mathcal {L}}_\\mathcal {F}^K$ -definable functions $a_1,\\ldots ,a_s$ on $B$ , and some $r_i$ in $K$ (possibly zero), such that moreover $\\varphi (C)$ is a bounded subset of $\\mathbb {R}^s$ , and such that $F$ is an ${\\mathcal {L}}_\\mathcal {F}^K$ -definable, analytic, non-vanishing function on an open neighborhood of the closure of $\\varphi (C)$ in $\\mathbb {R}^{s}$ .",
"We call the set of the functions $\\theta ,a_1,\\ldots ,a_s$ a set of accessory functions of $u$ .",
"Note that there is no unique set of accessory functions of an ${\\mathcal {L}}_\\mathcal {F}^K$ -unit $u$ with unit $\\theta $ , but rather, a set of accessory functions comes by writing $u$ as a composition $F(\\varphi )$ as in Definition REF .",
"Definition 4.3.2 (Preparation with center) Let $B,C,\\theta , x_{<n}$ be as above.",
"An ${\\mathcal {L}}_\\mathcal {F}^K$ -definable function $f:C\\rightarrow \\mathbb {R}$ is called ${\\mathcal {L}}_\\mathcal {F}^K$ -prepared with center $\\theta $ if $f$ can be written as $f(x) = g(x_{<n}) |x_n - \\theta (x_{<n})|^{r}u(x)$ where $g:B\\rightarrow \\mathbb {R}$ is ${\\mathcal {L}}_\\mathcal {F}^K$ -definable, $u$ is an ${\\mathcal {L}}_\\mathcal {F}^K$ -unit with center $\\theta $ , and $r$ is in $K$ .",
"Say that an ${\\mathcal {L}}_\\mathcal {F}^K$ -definable map $f:C\\rightarrow \\mathbb {R}^\\ell $ for $\\ell \\ge 1$ is ${\\mathcal {L}}_\\mathcal {F}^K$ -prepared with center $\\theta $ if all its component functions are.",
"We call the collection consisting of $g$ and a set of accessory functions of $u$ , a set of accessory functions of $f$ .",
"In the following result, some of the coordinates will be reordered, with reordering depending on the piece in a finite partition, in order to ensure Lipschitz continuity for the occurring centers.",
"We state and will need a family version.",
"Theorem 4.3.3 Let $T\\subset \\mathbb {R}^k$ , ${\\mathcal {X}}\\subset T\\times \\mathbb {R}^n$ , and $f:{\\mathcal {X}}\\rightarrow \\mathbb {R}^s$ be ${\\mathcal {L}}_\\mathcal {F}^K$ -definable for some $s\\ge 0$ .",
"Then there exist $M\\ge 1$ and a finite partition partition of ${\\mathcal {X}}$ into ${\\mathcal {L}}_\\mathcal {F}^K$ -definable parts $S_i$ for $i=1,\\ldots ,\\ell $ for some $\\ell \\ge 0$ and an ${\\mathcal {L}}_\\mathcal {F}^K$ -definable part $\\Sigma $ which is nowhere dense in $\\mathbb {R}^{k+n}$ and such that, for each $i$ there exists a reordering of the coordinates on $\\mathbb {R}^n$ such that the following holds.",
"The set $S_i$ is an open cell (w.r.t.",
"the reordered coordinates), and, with $\\Pi _{<n}:\\mathbb {R}^{k+n}\\rightarrow \\mathbb {R}^{k+n-1}$ the projection sending $(t,x)$ to $(t,x_1,\\ldots ,x_{n-1})$ (still in the reordered coordinates), there is an ${\\mathcal {L}}_\\mathcal {F}^K$ -definable $C^1$ function $\\theta :\\Pi _{<n}(S_i)\\rightarrow \\mathbb {R}$ whose graph is disjoint from $S_i$ and which satisfies $|\\partial \\theta /\\partial x_i | \\le M$ for each $i=1,\\ldots ,n-1$ , and such that the restriction $f_{|S_i}$ is ${\\mathcal {L}}_\\mathcal {F}^K$ -prepared with center $\\theta $ .",
"The proof of Theorem REF is similar to the proof of Proposition 5.2 of [21], where one uses the Main Theorem of [20] instead of Theorem 3.5 of [21], and Theorem REF instead of Proposition 3 of [29].",
"We give the details for the convenience of the reader.",
"Write $\\Pi _{<n}$ for the projection $\\mathbb {R}^{k+n}\\rightarrow \\mathbb {R}^{k+n-1}$ sending $(t,x)$ to $(t,x_1,\\ldots ,x_{n-1})$ for $t\\in \\mathbb {R}^k$ and $x\\in \\mathbb {R}^n$ .",
"Apply the Main Theorem of [20] (in combination with the more classical Lemma 4.4 of [20] when $s>1$ ), to find a finite partition of ${\\mathcal {X}}$ into ${\\mathcal {L}}_\\mathcal {F}^K$ -definable cells such that, for each occurring cell $S$ , the restriction $f_{|S}$ is ${\\mathcal {L}}_\\mathcal {F}^K$ -prepared with center $\\theta $ for some ${\\mathcal {L}}_\\mathcal {F}^K$ -definable $C^1$ function $\\theta $ on $\\Pi _{<n}(S)$ .",
"(By a cell decomposition we mean a finite partition into ${\\mathcal {L}}_\\mathcal {F}^K$ -definable cells.)",
"Note that the variable $x_n$ plays a special role in this application of the Main Theorem of [20].",
"Next we do the same with each of the variables $x_i$ in this special role (which amounts to reordering $(x_1,\\ldots ,x_n)$ as $(x_1,\\ldots ,x_{i-1},x_{i},\\ldots ,x_{n},x_i)$ ), as follows.",
"Write, for each $i=1,\\ldots ,n$ , $\\Pi ^{i}$ for the coordinate projection $\\mathbb {R}^{k+n}\\rightarrow \\mathbb {R}^{k+n-1}$ forgetting the variable $x_i$ , where $(t,x_1,\\ldots ,x_n)$ run over $\\mathbb {R}^{k+n}$ .",
"Now, for $i$ subsequently equal to $n-1$ , $n-2$ , and up to $i=1$ , refine the obtained cell decomposition of ${\\mathcal {X}}$ by applying the Main Theorem of [20] to the restrictions of $f$ to the occurring parts, with the variable $x_i$ in the special role (so that in particular, the obtained centers will be functions defined on subsets of $\\Pi ^{i}({\\mathcal {X}})$ ).",
"After having applied the Main Theorem of [20] in total $n$ times, we obtain a finite partition of ${\\mathcal {X}}$ into ${\\mathcal {L}}_\\mathcal {F}^K$ -definable parts.",
"It is enough to prove the theorem for the restriction of $f$ to each part occurring in this partition, and hence, we may suppose that ${\\mathcal {X}}$ is equal to a single part, that is, the partition consists of the part $cX$ only.",
"Now apply Theorem REF to find a finite partition of ${\\mathcal {X}}$ into ${\\mathcal {L}}_\\mathcal {F}^K$ -definable parts $S_i$ and a part $\\Sigma $ , and for each part $S_i$ an ordering of the coordinates.",
"Similarly as above, we may assume that ${\\mathcal {X}}=S_i$ , and, that we have changed our notation so that we have the natural ordering $(x_1,\\ldots ,x_n)$ on the coordinates on $\\mathbb {R}^n$ (the ordering of the coordinates $(t_1,\\ldots ,t_k)$ on $\\mathbb {R}^k$ has never been changed).",
"Recapitulating, we may assume that ${\\mathcal {X}}$ is an open cell with extra information on the cell walls coming from our application of Theorem REF , and that $f$ is ${\\mathcal {L}}_\\mathcal {F}^K$ -prepared with center $\\theta $ , with $\\theta $ an ${\\mathcal {L}}_\\mathcal {F}^K$ -definable $C^1$ function.",
"We still have to ensure (REF ) for some $M$ , which we will now do.",
"Let $\\alpha (t,x_{<n})$ be a cell wall of ${\\mathcal {X}}$ bounding the variable $x_n$ .",
"Note that $|\\partial \\alpha /\\partial x_i| <M_n$ for each $i=1,\\ldots ,n-1$ , with $M_n$ as given by Theorem REF .",
"By symmetry (or, up to changing the sign of $x_n$ ), we may suppose that $\\alpha (t,x_{<n}) < x_n$ on ${\\mathcal {X}}$ , and, that $\\theta \\le \\alpha $ on $\\Pi _{<n}({\\mathcal {X}})$ .",
"Up to an extra finite further partition (and neglecting a lower dimensional piece), we may assume that, on ${\\mathcal {X}}$ , either $x_n > 2\\alpha - \\theta $ , or, $x_n < 2\\alpha - \\theta $ .",
"We will ensure (REF ) with $M=M_n$ , by showing that $f$ is ${\\mathcal {L}}_\\mathcal {F}^K$ -prepared with center $\\alpha $ which satisfies (REF ).",
"To this end, it suffices to show that the function ${\\mathcal {X}}\\rightarrow \\mathbb {R}:(t,x)\\mapsto x_n-\\theta (t,x_{<n})$ is ${\\mathcal {L}}_\\mathcal {F}^K$ -prepared with center $\\alpha $ .",
"Let us for the rest of the proof abbreviate $\\alpha (t,x_{<n})$ and $\\theta (t,x_{<n})$ by $\\alpha $ , resp.",
"by $\\theta $ .",
"In the case that $x_n > 2\\alpha - \\theta $ on ${\\mathcal {X}}$ , one has $x_n - \\theta = (x_n- \\alpha )(1+ \\frac{\\alpha -\\theta }{x_n-\\alpha }),$ and $(1+ \\frac{\\alpha -\\theta }{x_n-\\alpha })$ is clearly an ${\\mathcal {L}}_\\mathcal {F}^K$ -unit with center $\\alpha $ , which finishes this case.",
"In the case that $x_n < 2\\alpha - \\theta $ , one has $x_n - \\theta = (\\alpha - \\theta )(1+ \\frac{x_n - \\alpha }{\\alpha - \\theta }),$ and $(1+ \\frac{x_n - \\alpha }{\\alpha - \\theta })$ is an ${\\mathcal {L}}_\\mathcal {F}^K$ -unit with center $\\alpha $ , which finishes the second and final case and ends the proof of the theorem."
],
[
"Suppose we want to reparameterize ${\\mathcal {X}}$ as in Theorem 2.1.3, in the case that ${\\mathcal {X}}$ is the graph of an ${\\mathcal {L}}_\\mathcal {F}^K$ -definable function $f:U\\subset (0,1)^m\\rightarrow \\mathbb {R}^{n-m}$ , where we omit the parameters space $T$ for simplicity.",
"Of course $f$ is not automatically 0-prepared.",
"Nevertheless, after applying Theorem REF recursively $m$ times, and after translating in each coordinate by the center, we can reduce to a situation which is 0-prepared.",
"A great deal of effort will lie in controlling the first derivatives along this process: we will work towards the property that $f$ as well as the occurring centers of preparation have bounded $C^1$ -norm.",
"Having this, a triangular translation by the center in each coordinate becomes not only 0-prepared but still has bounded $C^1$ -norm.",
"In fact, we will even want each first partial derivative of $f$ to become a-b-m after some basic transformations, and, we have similar aims for the walls of the occurring cells and their first partial derivatives.",
"To achieve all these requirements jointly we will use induction on $m$ and an interplay of Theorems REF and REF .",
"Once we have achieved this we can finish off the proof of Theorem 2.1.3 by the results of Section REF , as follows.",
"By Lemma REF , a-b-m functions are automatically weakly mild.",
"One finishes by applying Proposition REF to get $C^r$ -parameterizations in a simple and controlled way, for any integer $r>0$ .",
"We proceed in four steps, each proved separately and implying Theorem 2.1.3.",
"In this step we show a reduction to the assumptions of Case 1, namely, we show that it is enough to prove Theorem 2.1.3 under the extra conditions of Case 1.",
"This reduction is classical.",
"We give the details for the convenience of the reader.",
"Case 1.",
"Each $X_t$ is the graph of $f_t$ for some ${\\mathcal {L}}_\\mathcal {F}^K$ -definable family of functions $f_t:U_t\\subset (0,1)^m\\rightarrow (0,1)^{n-m}$ with $C^1$ -norm bounded by 1 and with an ${\\mathcal {L}}_\\mathcal {F}^K$ -definable family $U_t$ such that ${\\mathcal {U}}=\\lbrace (t,x)\\mid t\\in T,\\ x\\in U_t\\rbrace $ is an open cell in $(0,1)^{k+m}$ .",
"It is harmless to assume that $T$ is a subset of $(0,1)^{k}$ instead of a subset of $\\mathbb {R}^k$ .",
"Up to a finite partition of ${\\mathcal {X}}=\\lbrace (t,x)\\mid t\\in T,\\ x\\in X_t\\rbrace $ into cells and up to reordering the coordinates we may suppose that ${\\mathcal {X}}$ is the graph of $f$ for some ${\\mathcal {L}}_\\mathcal {F}^K$ -definable function $f:{\\mathcal {U}}\\subset (0,1)^{k+m}\\rightarrow (0,1)^{n-m}$ .",
"By induction on $k+m$ we may suppose that ${\\mathcal {U}}$ is an open cell in $(0,1)^{k+m}$ .",
"By further partitioning we may suppose that $|f_t^{(\\beta )}|$ is maximal on $U_t$ for some $\\beta $ with $|\\beta |=1$ among all first partial order derivatives of $f_t$ , and that either $|f_t^{(\\beta )}|>1$ or $|f_t^{(\\beta )}|\\le 1$ holds on each $X_t$ .",
"If $|f_t^{(\\beta )}|\\le 1$ then one is done.",
"In the case that $|f_t^{(\\beta )}|>1$ , we may suppose that the function $g:x_i\\mapsto f_t(x_1,\\ldots , x_{i-1},x_i,x_{i+1},\\ldots ,x_m)$ is injective for each $t$ and each $(x_1,\\ldots , x_{i-1},x_{i+1},\\ldots ,x_m)$ , where $i$ is such that $\\beta _i=1$ .",
"Now inverting the role of $x_i$ and $g$ we are done by the chain rule for derivation.",
"This finishes the reduction of Theorem 2.1.3 to the extra conditions of Case 1 and thus Step 1 is done.",
"The above Case 1 yields an ${\\mathcal {L}}_\\mathcal {F}^K$ -definable family of functions $f_t$ .",
"From now on we will write $f:{\\mathcal {U}}\\rightarrow \\mathbb {R}^{n-m}$ for the map sending $(t,x)$ with $x\\in U_t$ and $t\\in T$ to $f_t(x)$ and with ${\\mathcal {U}}$ consisting of $(t,x)$ with $t\\in T$ and $x\\in U_t$ .",
"We will also continue to write ${\\mathcal {X}}$ for the ${\\mathcal {L}}_\\mathcal {F}^K$ -definable family of the sets $X_t$ with parameter $t\\in T$ , and which are the graphs of the $f_t$ .",
"In this step we prove Theorem 2.1.3 under the special assumptions summarized as Case 2.",
"We mainly use results from Section REF in this step.",
"Case 2.",
"The set ${\\mathcal {X}}$ is the graph of $f:{\\mathcal {U}}\\rightarrow (0,1)^{n-m}$ where $f$ is 0-prepared and ${\\mathcal {U}}\\subset T\\times (0,1)^{m} \\subset (0,1)^{k+m}$ is a cell whose walls are 0-prepared.",
"Furthermore, for each wall $\\alpha $ of ${\\mathcal {U}}$ and each $i=1,\\ldots ,m$ , the maps $\\partial f/\\partial x_i$ and $\\partial \\alpha /\\partial x_i$ are a-b-m. (In particular, for each $t$ , the map $f_t$ and the walls of $U_t$ have bounded $C^1$ -norm.)",
"Lemmas REF , REF and Proposition REF do the job for this step, as follows.",
"Let $r>0$ be any integer.",
"For each $t$ , let $U_t^{\\prime }$ be the subset of $(0,1)^m$ such that the map $\\varphi _t:x\\mapsto (x_1^{r^m},x_2^{r^{m-1}},\\ldots ,x_m^r)$ is a bijection from $U_t^{\\prime }$ to $U_t$ .",
"Note that $U_t^{\\prime }$ is an open cell in $(0,1)^m$ .",
"Claim 1.",
"There are polynomials $A(r)$ and $C(r)$ such that for each $t$ and any integer $r$ , the function $f_t\\circ \\varphi _t$ and all the walls of the cell $U_t^{\\prime }$ are $(A(r),C(r))$ -mild up to order $r$ .",
"For each $t$ , the function $f_t\\circ \\varphi _t$ satisfies the conditions from the claim by Lemma REF , Proposition REF and since a 0-prepared function is a-b-m. Let us now concentrate on a cell wall of $U_t^{\\prime }$ .",
"To name some of the cell boundaries, let us suppose that $x_m$ runs between $\\alpha _t(x_{<m})$ and $\\beta _t(x_{<m})$ for $x$ in $U_t$ , where $x_{<m} = (x_1,\\ldots ,x_{m-1})$ .",
"Let us further suppose, for $x\\in U_t^{\\prime }$ , that $x_m$ runs between $a_t(x_{<m})$ and $b_t(x_{<m})$ in $U_t^{\\prime }$ .",
"By induction on $m$ and the similar role of $a_t$ and $b_t$ it is enough to prove the claim for $a_t$ .",
"By the assumptions of Case 2, there is $S>1$ , an a-b-m function $\\alpha _0$ taking values in $(1/S,S)$ and a bounded-monomial function $x_{<m}^\\mu t^\\nu $ on ${\\mathcal {U}}$ such that, for $x\\in U_t$ , $\\alpha _t(x_{<m}) = x_{<m}^\\mu t^\\nu \\alpha _0(t,x_{<m}).$ But then we can write, for $x\\in U_t^{\\prime }$ , $a_t(x_{<m}) &=& \\@root r \\of { (\\varphi _t(x))_{<m}^\\mu t^\\nu \\alpha _0(t, (\\varphi _t(x))_{<m} ) }\\\\& = & (x_1^{r^{m-1}},x_2^{r^{m-1}},\\ldots ,x_{m-1}^r)^\\mu \\@root r \\of { t^\\nu \\alpha _0(t,x_1^{r^m}, \\ldots ,x_{m-1}^{2r})}.$ When $\\mu =0$ and $\\nu =0$ , the claim follows from Proposition REF and Lemma REF .",
"The case for general $\\mu $ and $\\nu $ is a similar exercise as the proof of Lemma REF .",
"This proves Claim 1.",
"Now one makes all the $f_t$ and all the walls of $U_t$ of $C^r$ -norm bounded by 1 by parameterizing according to Lemma REF , say with domain $U^{\\prime \\prime }_t$ .",
"Note that $U_t^{\\prime \\prime }$ is an open cell whose walls have $C^r$ -norm bounded by 1.",
"One finishes this case by mapping $(0,1)^m$ onto $U^{\\prime \\prime }_t$ in the obvious way.",
"This finishes the proof of Theorem 2.1.3 under the special assumption of Case 2.",
"Step 2 is done.",
"The rest of the proof will consist of a reduction from Case 1 to Case 2, using induction on $k+m$ .",
"In this step we want to reduce (starting from Case 1) to the following Case 3.",
"Namely, we show that it is enough to prove Theorem 2.1.3 under the extra conditions of Case 3.",
"Case 3.",
"On top of the assumptions of Case 1, the map $f$ as well as the maps $\\partial f/\\partial x_i$ for $i=1,\\ldots ,m$ and the coordinate function ${\\mathcal {U}}\\rightarrow \\mathbb {R}:(t,x)\\mapsto x_m$ are ${\\mathcal {L}}_\\mathcal {F}^K$ -prepared with center $\\theta $ .",
"Moreover, one has $|\\partial \\theta /\\partial x_i|\\le 1$ for each $i=1,\\ldots ,m$ .",
"Let $f$ and ${\\mathcal {U}}$ be as given by Case 1.",
"Now, up to a reordering of the coordinates $x_1,\\ldots ,x_m$ and up to a finite partition into ${\\mathcal {L}}_\\mathcal {F}^K$ -definable parts, we can reduce to the conditions of Case 3 by Theorem REF , and, obviously, by a further finite partitioning and scaling by $M$ to make the $C^1$ -norm of the maps $\\theta _t$ bounded by 1 instead of by the constant $M$ provided by Theorem REF .",
"This finishes the proof of Step 3.",
"In this step we show that it is enough to prove Theorem 2.1.3 under the extra conditions of Case 2.",
"By Step 2 itself, this will complete the proof of Theorem 2.1.3.",
"Let us start from the assumptions of Case 3.",
"We use a recursive procedure starting from Case 3, which goes roughly by recursively translating with the center and applying Theorem REF to sets of accessory functions in less variables.",
"To give names to some cell boundaries, let us suppose that $x_m$ runs between $\\alpha _t(x_{<m})$ and $\\beta _t(x_{<m})$ for $x$ in $U_t$ , where $x_{<m} = (x_1,\\ldots ,x_{m-1})$ .",
"Let ${\\mathcal {S}}$ be the finite set of functions consisting of $\\alpha $ , $\\beta $ , (the functions in) a set of accessory functions for the ${\\mathcal {L}}_\\mathcal {F}^K$ -prepared maps $f$ , a set of accessory functions for the maps $\\partial f/\\partial x_i$ for $i=1,\\ldots ,m$ , and a set of accessory functions for the coordinate function ${\\mathcal {U}}\\rightarrow \\mathbb {R}:(t,x)\\mapsto x_m$ (each of these maps being prepared with center $\\theta $ by Case 3).",
"Up to a finite partition, a translation by a constant in the $x_m$ -variable and a basic transformation $x_m\\mapsto -x_m+1$ if needed, we may suppose that $0\\le \\theta (t,x_{<m})< x_m$ on ${\\mathcal {U}}$ .",
"Now transform via the map $(t,x) \\mapsto (t,x_{<m},x_m + \\theta (t,x_{<m})).$ Then, up to identifying our notation for $f$ and ${\\mathcal {X}}$ with the new situation, we are in a situation satisfying the conditions of Case 1 where moreover $f$ is ${\\mathcal {L}}_\\mathcal {F}^K$ -prepared with center 0, and where the $\\partial f/\\partial x_i$ are a finite sum of bounded maps which are ${\\mathcal {L}}_\\mathcal {F}^K$ -prepared with center 0.",
"(Here, several case assumptions of Case 3 are used, in particular, it is used that $|\\partial \\theta /\\partial x_i|\\le 1$ on ${\\mathcal {U}}$ , for $i=1,\\ldots ,m$ , and, that the coordinate functions $(t,x)\\mapsto x_m$ on ${\\mathcal {U}}$ was also ${\\mathcal {L}}_\\mathcal {F}^K$ -prepared with center $\\theta $ .)",
"Let $\\Pi _{<m}:\\mathbb {R}^{k+m}\\rightarrow \\mathbb {R}^{k+m-1}$ be the projection sending $(t,x)$ to $(t,x_{<m})=(t,x_1,\\ldots ,x_{m-1})$ .",
"It is harmless to assume that each function in ${\\mathcal {S}}$ either takes values in $(0,1)$ , or, is identically zero.",
"Now use induction on $k+m$ , to obtain the reduction to Case 2 from the set-up of the Theorem where ${\\mathcal {X}}$ is the graph $\\Pi _{<m}({\\mathcal {U}})\\rightarrow (0,1)^s$ for some $s\\ge 0$ of the map whose component functions are the functions from ${\\mathcal {S}}$ which are not identically zero.",
"By nesting this information in a classical way one reduces to a situation where $f$ and the walls of ${\\mathcal {U}}$ are 0-prepared, and, by the chain rule for derivation, that the maps $\\partial f/\\partial x_i$ and $\\partial \\alpha /\\partial x_i$ are finite sums of 0-prepared functions for each $i=1,\\ldots ,m$ and each wall $\\alpha $ of ${\\mathcal {U}}$ .",
"Since finite sums of 0-prepared functions are clearly a-b-m, this finishes the proof of Step 4 and thus of Theorem 2.1.3.",
"To conclude this section, note that although the $\\phi _{r,i,t}$ of Theorem 2.1.3 are analytic for each $r,i,t$ , we do not have that they can be extended to an analytic map on an open neighborhood of the closed box $[0,1]^m$ ."
],
[
"The proof of the quasi-parameterization theorem",
"In this section we prove theorem 2.2.3.",
"First of all, however, we require some general results about definable families of holomorphic functions.",
"The definability here is with respect to an arbitrary polynomially bounded o-minimal expansion of the real field, which we now fix.",
"Let us recall the following definition from subsection 2.2 from which we also recall that $\\Delta (R)$ denotes the (open) disc in $ of radius $ R$ and centred at the origin.$ 5.1 Definition A definable family $\\Lambda = \\lbrace F_t : t \\in T \\rbrace $ is called an $(R, m, K)$ -family, where $R$ , $K$ are positive real numbers and $m$ is a positive integer, if for each $t \\in T$ , $F_t : \\Delta (R)^m \\rightarrow is holomorphic and for all $ z (R)m$,$ |Ft (z)| K$.$ Let us first observe that for $\\Lambda $ such a family it follows from the Cauchy inequalities (for all general results from the theory of functions of several complex variables we refer the reader to the first chapter of [14]) that we have the following bounds on the Taylor coefficients of each $F_t$ at $0 \\in m$ : 5.1.1 for all $\\alpha \\in \\mathbb {N}^m$ and all $t \\in T$ , $\\frac{|F^{(\\alpha )}_t (0)|}{\\alpha !",
"}\\le \\frac{K}{R^{|\\alpha |}}$ .",
"In particular, if $R>1$ then $\\frac{|F^{(\\alpha )}_t (0)|}{\\alpha !",
"}\\rightarrow 0$ as $|\\alpha | \\rightarrow \\infty $ and so for each $t \\in T$ there exists some $M_t \\in \\mathbb {N}$ such that 5.1.2 for all $\\alpha \\in \\mathbb {N}^m$ , $\\frac{|F^{(\\alpha )}_t (0)|}{\\alpha !",
"}\\le $ max$\\lbrace \\frac{|F^{(\\alpha )}_t (0)|}{\\alpha !",
"}: \\alpha \\in \\mathbb {N}^m , |\\alpha | \\le M_t \\rbrace $ .",
"The crucial uniformity result, from which the quasi-parameterization theorem will follow, is that $M_t$ may be chosen to be independent of $t$ .",
"This in turn will follow from the maximum modulus theorem and the following simple 5.2 Lemma Let $1<r<R$ , $0< \\lambda \\le and let $ { t : t T }$ be a definable family of functions from $ (0, R)$ to $ (0, R)$.Then there exists $ (0, )$ such that for all $ t T$, there exists $ yt (r, R)$ such that$ t (yt - ) 12 t (yt )$.$ Suppose not.",
"Then there exists a function $\\eta : (0, \\lambda ) \\rightarrow T$ , which by the principle of definable choice we may take to be definable, such that 5.2.1 for all $x \\in (0, \\lambda )$ and all $y \\in (r, R)$ , $\\theta _{\\eta (x)} (y-x) < _{\\eta (x)} (y)$ .",
"Pick some $\\gamma \\in (r, R)$ and consider the definable function $x \\mapsto \\theta _{\\eta (x)} (\\gamma )$ for $x \\in (0, \\lambda )$ .",
"It follows from polynomial boundedness that there exists a positive integer $N$ and $\\nu \\in (0, \\lambda )$ such that 5.2.2 for all $x \\in (0, \\nu )$ , $\\theta _{\\eta (x)} (\\gamma ) > x^{N}$ .",
"Now let $k$ be a positive integer and set $x_0 := \\frac{R-\\gamma }{2k}$ , so that $x_0 \\in (0, \\nu )$ for large enough $k$ .",
"By applying 5.2.1 successively with $x=x_0$ and $y= \\gamma + x_0, \\ldots , \\gamma + kx_0$ we see that $0 < \\theta _{\\eta (x_0)} (\\gamma ) < _{\\eta (x_0)} (\\gamma + x_0) < \\cdots < (^k \\theta _{\\eta (x_0 )}(\\gamma + kx_0 ) < (^k R$ .",
"So by 5.2.2 we obtain $(\\frac{R-\\gamma }{2k})^N = x_0^N < \\theta _{\\eta (x_0 )}(\\gamma ) < (^k R$ , which is the required contradiction if $k$ is sufficiently large.",
"5.3 Theorem Let $\\Lambda = \\lbrace F_t : t \\in T \\rbrace $ be an $(R, m, K)$ -family with $R>1$ .",
"Then there exists $M = M(\\Lambda ) \\in \\mathbb {N}$ such that for all $t \\in T$ and all $\\alpha \\in \\mathbb {N}^m$ , $\\frac{|F^{(\\alpha )}_t (0)|}{\\alpha !",
"}\\le \\text{max} \\lbrace \\frac{|F^{(\\alpha )}_t (0)|}{\\alpha !",
"}: \\alpha \\in \\mathbb {N}^m, |\\alpha | \\le M \\rbrace .$ Since the conclusion is trivially true for those $t \\in T$ such that $F_t \\equiv 0$ (no matter how $M$ is chosen) we may assume that no $F_t$ is identically zero.",
"For $t \\in T$ define $\\theta _t : (0, R) \\rightarrow (0, R)$ by 5.3.1 $\\theta _t (y) := \\frac{R}{K} \\text{sup} \\lbrace |F_t (z)| : z \\in \\Delta (y)^m \\rbrace .$ Now let $r := R^{\\frac{3}{4}}$ , $\\lambda = \\text{min}\\lbrace R^{\\frac{3}{4}} - R^{\\frac{2}{3}} \\rbrace $ and apply 5.2 to obtain $\\epsilon \\in (0, \\lambda )$ such that 5.3.2 for all $t \\in T$ , there exists $y_t \\in (r, R)$ such that $\\theta _t (y_t - \\epsilon ) \\ge _t (y_t)$ .",
"Now choose $D \\in \\mathbb {N}$ so that 5.3.3 $(1-\\frac{\\epsilon }{R} )^D < \\frac{1}{4}$ , and 5.3.4 $5(D+1)^m \\le 2R^{\\frac{D}{3}}$ .",
"We show that $M:= 2D$ satisfies the required conclusion.",
"So fix some arbitrary $t \\in T$ and let $B_t := \\text{max}\\lbrace \\frac{|F^{(\\alpha )}_t (0)|}{\\alpha !",
"}: |\\alpha | \\le D \\rbrace $ .",
"(The $\\alpha $ 's range over $\\mathbb {N}^m$ for the rest of this proof.)",
"It is clearly sufficient to show that (*) for all $\\alpha $ with $|\\alpha | \\ge 2D$ we have $\\frac{|F^{(\\alpha )}_t (0)|}{\\alpha !",
"}\\le B_t$ .",
"To this end we consider the truncated Taylor expansion of $F_t$ , namely $P_t (z) := \\sum _{|\\alpha | \\le D} \\frac{F_t^{(\\alpha )} (0)}{\\alpha !}",
"z^{\\alpha }$ .",
"Clearly we have 5.3.5 for all $z \\in \\Delta (R)^m$ , $|P_t (z)| \\le (D+1)^m B_t R^D$ .",
"Now choose $w = \\langle w_1 , \\ldots , w_m \\rangle \\in \\Delta (R)^m$ with $|w_i| = y_t - \\epsilon $ and $|F_t(w)| = \\frac{K}{R} \\theta _t (y_t - \\epsilon )$ (which is possible by 5.3.1 and the maximum modulus theorem), and let $\\eta _i := \\frac{w_i}{y_t - \\epsilon }$ so that $|\\eta _i| = 1$ for $i = 1, \\ldots , m$ .",
"Consider the function $H_t : \\Delta (R) \\rightarrow given by$$H_t (u) := \\frac{F_t(u\\eta ) - P_t (u\\eta )}{u^{D+1}}.$$This is clearly a well-defined analytic function and by the maximum modulus theorem there exists $ ut (R)$ with $ |ut| = yt$such that $ |Ht (yt - )| |Ht (ut )|$.",
"Thus$$|F_t((y_t-\\epsilon )\\eta ) - P_t((y_t-\\epsilon )\\eta )| \\le (\\frac{y_t-\\epsilon }{y_t})^{D+1} |F_t((u_t\\eta ) - P_t((u_t\\eta )|.$$However, by 5.3.3, $ (yt-yt)D+1 (1-R )D+1 < 14$ so, upon recallingthat $ w = (yt - )$ and using 5.3.5, we see that$ 5.3.6 $|F_t(w)| \\le \\frac{1}{4}|F_t(u_t \\eta )| + \\frac{5}{4}(D+1)^m B_t R^D$ .",
"But $|F_t (w)| = \\frac{K}{R} \\theta _t (y_t - \\epsilon ) \\ge \\frac{K}{2R} \\theta _t (y_t)$ by 5.3.2, and since $|u_t\\eta | = y_t$ we obtain from this and 5.3.1 that $|F_t (w)| \\ge F_t (u_t\\eta )|$ .",
"Putting this into 5.3.6 we obtain 5.3.7 $|F_t(w)| \\le \\frac{5}{2}(D+1)^m B_t R^D$ .",
"Now, with a view to proving (*), let $|\\alpha | \\ge 2D$ .",
"Then by applying the Cauchy inequalities in the polydisk $\\Delta (y_t - \\epsilon )^m$ and using 5.3.1 and 5.3.7 we obtain $\\frac{|F^{(\\alpha )}_t (0)|}{\\alpha !",
"}\\le \\frac{K}{R} \\theta _t (y_t - \\epsilon ) \\cdot (y_t - \\epsilon )^{-|\\alpha |}= |F_t (w)|(y_t - \\epsilon )^{-|\\alpha |} \\le \\frac{5}{2}(D+1)^m B_t R^D(y_t - \\epsilon )^{-|\\alpha |}.$ But $y_t - \\epsilon \\ge r-\\lambda \\ge R^{\\frac{2}{3}}$ (by the definitions of $r$ and $\\lambda $ ), so by 5.3.4 $\\frac{|F^{(\\alpha )}_t (0)|}{\\alpha !",
"}\\le \\frac{5}{2}(D+1)^mB_tR^D \\cdot (R^{\\frac{2}{3}})^{-2D} = \\frac{5}{2}(D+1)^m R^{\\frac{-D}{3}} B_t \\le B_t$ as required.",
"It follows immediately from 5.3 that the function $\\kappa _{\\Lambda } : T \\rightarrow \\mathbb {R}$ given by 5.3.8 $\\kappa _{\\Lambda }(t) := \\text{max} \\lbrace \\frac{|F^{(\\alpha )}_t (0)|}{\\alpha !",
"}: \\alpha \\in \\mathbb {N}^m \\rbrace $ is (well-defined and) definable.",
"It also determines the topology on $\\Lambda $ in the following sense.",
"5.4 Theorem Let $\\Lambda $ be an $(R, m, K)$ -family with $R>1$ as above and let $r$ be a real number satisfying $0<r<R$ .",
"Then there exists a positive real number $B_{\\Lambda } (r)$ such that for all $t \\in T$ and all $z \\in \\Delta (r)^m$ we have $|F_t(z)| \\le B_{\\Lambda }(r) \\cdot \\kappa _{\\Lambda } (t)$ .",
"Choose a real number $r_0$ such that $\\text{max} \\lbrace 1,r \\rbrace <r_0<R$ and for each $t \\in T$ define $G_t : \\Delta (\\frac{R}{r_0})^m \\rightarrow by $ Gt(z) := Ft(r0z)$.",
"Then $ * := {Gt : t T }$ is an $ (Rr0 , m, K)$-family and since $ Rr0 > 1$, wemay apply 5.3 to it and obtain some $ M(*) $\\mathbb {N}$$ such that$ *(t) = max { |Gt()(0)|!",
": $\\mathbb {N}$ m , || M(* }.$Fix $ t T$.",
"Then since $ Gt()(0) = r0||Ft()(0)$ (for all $$\\mathbb {N}$ m$) it follows thatfor all $ z (r)m$ we have that $ |Ft(z)| = |$\\mathbb {N}$ m r0-|| Gt()(0)!",
"z| $$ * (t) (r0r0 - r)m (t) r0M(*) (r0r0 - r)m$, whichgives the required result upon setting $ B(r) := r0M(*) (r0r0 - r)m$.$ The topology we are referring to here is determined by the metrics $\\delta _r$ ($0<r<R$ ) where, for any two bounded holomorphic functions $F, G : \\Delta (R)^m \\rightarrow , we define $ r (F, G) := sup {|F(z) - G(z)| : z (r)m }$.",
"It turns outthat if $$ is any $ (R, m, K)$-family and if $ 0< r, r' <R$, then the metric spaces $ (, r )$ and $ (, r' )$are quasi-isometric via the identity function on $$.",
"(Note that this is certainly not true in general for families of$ K$-bounded holomorphic functions on $ (R)m$, e.g.",
"consider, for $ R=2, m=K=1$, the family $ {0 } { (z2)q : q $\\mathbb {N}$ }$.)",
"In fact, as we now explain, theyare quasi-isometric to a bounded subset of $ N$ for some sufficiently large $ N$ (depending only on $$) endowed withthe metric induced by the usual sup-metric on $ N$: $ w1 , ..., wN := max{ |wi| : 1 i N }$.$ 5.5 Definition We say that an $(R, m, K)$ -family $\\Lambda = \\lbrace F_t : t \\in T \\rbrace $ is well-indexed if, for some $N \\in \\mathbb {N}$ , $T$ is a bounded subset of $N$ and for each $r$ with $0<r<R$ , there exist positive real numbers $c_r$ , $C_r$ such that for all $t, t^{\\prime } \\in T$ we have $c_r \\delta _r (F_t , F_{t^{\\prime }} ) \\le \\Vert t-t^{\\prime } \\Vert \\le C_r \\delta _r (F_t , F_{t^{\\prime }} )$ .",
"That is, the map $t \\mapsto F_t$ is a quasi-isometry from the metric space $\\langle T, \\Vert \\cdot \\Vert \\rangle $ to the metric space $\\langle \\Lambda , \\delta _r \\rangle $ .",
"5.6 Theorem Every $(R, m, K)$ -family with $R>1$ can be well-indexed.",
"Let $\\Lambda = \\lbrace F_t : t \\in T \\rbrace $ be an $(R, m, K)$ -family with $R>1$ .",
"What we are required to prove is that there exists a well-indexed $(R, m, K)$ -family, $\\Lambda ^{\\prime } = \\lbrace G_t : t \\in T^{*} \\rbrace $ say, such that $\\Lambda = \\Lambda ^{\\prime }$ as sets.",
"Consider the $(R, m, 2K)$ -family $\\Omega := \\lbrace F_t - F_{t^{\\prime }}: \\langle t, t^{\\prime } \\rangle \\in T^2 \\rbrace $ and let $M(\\Omega ) \\in \\mathbb {N}$ be as given by 5.3 (with $\\Omega $ in place of $\\Lambda $ ).",
"We take our $N = N(\\Lambda )$ to be the cardinality of the set $\\lbrace \\alpha \\in \\mathbb {N}^m : |\\alpha | \\le M(\\Omega ) \\rbrace $ .",
"Define the map $\\omega : T \\rightarrow N$ by $\\omega (t) := \\langle \\frac{F_t^{(\\alpha )} (0)}{\\alpha !}",
": \\alpha \\in \\mathbb {N}^m, |\\alpha | \\le M(\\Omega ) \\rangle $ and set $T^{*} := \\omega [T]$ .",
"If $t, t^{\\prime } \\in T$ and $\\omega (t) =\\omega (t^{\\prime })$ then $\\frac{F_t^{(\\alpha )} (0)}{\\alpha !}",
"= \\frac{F_{t^{\\prime }}^{(\\alpha )} (0)}{\\alpha !",
"}$ holds for all $\\alpha \\le M(\\Omega )$ and hence, by 5.3 (and the linearity of the derivatives), it holds for all $\\alpha \\in \\mathbb {N}^m$ .",
"Thus $F_t = F_{t^{\\prime }}$ .",
"This does not necessarily imply that $t=t^{\\prime }$ but, by the principle of definable choice, we may choose a definable right inverse $\\omega ^{-1}:T^{*} \\rightarrow T$ of $\\omega $ and setting $G_t = F_{\\omega ^{-1}(t)}$ (for $t \\in T^{*}$ ), we have that, as a set, $\\Lambda = \\lbrace G_{t} : t \\in T^{*} \\rbrace $ .",
"We complete the proof by showing that the $(R,m,K)$ -family $\\lbrace G_{t} : t \\in T^{*} \\rbrace $ is well-indexed.",
"Firstly, by 5.1.1 we have that $T^{*} \\subseteq \\overline{\\Delta (K)}^N$ , so $T^{*}$ is a bounded subset of $N$ .",
"For the quasi-isometric inequalities consider some $r \\in (0, R)$ .",
"We may take $C_r = \\text{max} \\lbrace 1, r^{-M(\\Omega )} \\rbrace $ .",
"Indeed, suppose $t, t^{\\prime } \\in T^{*}$ .",
"Let $s = \\omega ^{-1}(t)$ and $s^{\\prime } = \\omega ^{-1}(t^{\\prime })$ .",
"Then by the Cauchy inequalities applied to the function $(F_{s} - F_{s^{\\prime }})$ restricted to the disk $\\Delta (r)^m$ , we have that for all $\\alpha \\in \\mathbb {N}^m$ , $\\frac{|(F_{s} - F_{s^{\\prime }})^{\\alpha }(0)|}{\\alpha !",
"}\\le \\frac{\\delta _r(F_s , F_{s^{\\prime }})}{r^{|\\alpha |}}$ .",
"In particular, $\\Vert t-t^{\\prime } \\Vert = \\Vert \\omega (s) - \\omega (s^{\\prime }) \\Vert \\le C_r \\delta _r (F_s , F_{s^{\\prime }}) =C_r \\delta _r (G_ {t}, G_{t^{\\prime }})$ .",
"Finally, we take $c_r$ to be $B_{\\Omega }(r)^{-1}$ , where $B_{\\Omega }(r)$ is as in 5.4 (with $\\Omega $ in place of $\\Lambda $ ).",
"Then, with $t,t^{\\prime },s,s^{\\prime }$ as above, and $z \\in \\Delta (r)^m$ we have by 5.3 and 5.4, $|(F_s - F_{s^{\\prime }})(z)| \\le B_{\\Omega }(r)\\cdot \\kappa _{\\Omega }(\\langle s, s^{\\prime } \\rangle ) = B_{\\Omega }(r) \\cdot \\Vert \\omega (s) - \\omega (s^{\\prime }) \\Vert =B_{\\Omega }(r) \\cdot \\Vert t - t^{\\prime } \\Vert $ .",
"So $\\Vert t - t^{\\prime } \\Vert \\ge c_r \\delta _r (G_ {t}, G_{t^{\\prime }})$ , as required.",
"This result suggests a natural way of compactifying definable $(R, m, K)$ -families.",
"For let $\\Lambda = \\lbrace F_t : t \\in T \\rbrace $ be such a family with $R>1$ and assume, as now we may, that it is well-indexed (with $T$ a bounded subset of $N$ , say).",
"We wish to extend $\\Lambda $ to a family $\\overline{\\Lambda }$ well-indexed by the closure $\\overline{T}$ of $T$ in $N$ .",
"So for $t \\in \\overline{T}$ choose a Cauchy sequence $\\langle t^{(i)} : i \\in \\mathbb {N}\\rangle $ in $T$ converging to $t$ (in the space $\\langle N , \\Vert \\cdot \\Vert \\rangle $ ).",
"Then by the quasi-isometric property of the indexing it follows that $\\langle F_{t^{(i)}} : i \\in \\mathbb {N}\\rangle $ is a Cauchy sequence in $\\langle \\Lambda , \\delta _r \\rangle $ for every $r \\in (0, R)$ .",
"So by Weierstrass' theorem on uniformly convergent sequences, there exists a holomorphic function $F_t : \\Delta (R)^m \\rightarrow such that, for each $ r (0, R)$, $ r (Ft(i), Ft) 0$ as $ i $.",
"It is easy tocheck that $ Ft$ depends only on $ t$ (and not on the particular choice of Cauchy sequence) and that our notation is consistent if $ t$happens to lie in $ T$.",
"We have the following\\vspace{5.69054pt}$ 5.7 Theorem The collection $\\overline{\\Lambda } := \\lbrace F_t : t \\in \\overline{T} \\rbrace $ as defined above is a well-indexed $(R, m, K)$ -family.",
"Everything follows from elementary facts on convergence (and we may take the same constants $c_r$ , $C_r$ for the quasi-isometric inequalities) apart from the definability of $\\overline{\\Lambda }$ .",
"To see that this holds too, let $\\text{graph}(\\Lambda ) := \\lbrace \\langle t, z, w \\rangle \\in {N+m+1} : t \\in T, z \\in \\Delta (R)^m , F_t (z) = w \\rbrace .$ Then $\\text{graph}(\\Lambda )$ is a definable subset of ${N+m+1}$ (this being the definition of what it means for $\\Lambda $ to be a definable family).",
"We complete the proof by showing that (*) for all $\\langle t, z, w \\rangle \\in {N+m+1}$ , $t \\in \\overline{T}, z \\in \\Delta (R)^m$ and $F_t (z) = w$ if and only if $z \\in \\Delta (R)^m$ and $\\langle t, z, w \\rangle \\in \\overline{\\text{graph}(\\Lambda )}$ .",
"So let $\\langle t, z, w \\rangle \\in {N+m+1}$ .",
"Suppose first that $t \\in \\overline{T}, z \\in \\Delta (R)^m$ and $F_t (z) = w$ .",
"Choose a sequence $\\langle t^{(i)} : i \\in \\mathbb {N}\\rangle $ in $T$ converging to $t$ .",
"Choose $r$ so that $|z|<r<R$ .",
"Then by the construction of $F_t$ we have that $\\delta _r (F_{t^{(i)}}, F_t) \\rightarrow 0$ as $i \\rightarrow \\infty $ .",
"In particular, $|F_{t^{(i)}}(z) - F_t(z)| \\rightarrow 0$ as $i \\rightarrow \\infty $ , i.e.",
"$F_{t^{(i)}}(z) \\rightarrow w$ as $i \\rightarrow \\infty $ .",
"So $\\langle t^{(i)} , z , F_{t^{(i)}}(z) \\rangle \\rightarrow \\langle t, z, w \\rangle $ as $i \\rightarrow \\infty $ .",
"Since $\\langle t^{(i)} , z , F_{t^{(i)}}(z) \\rangle \\in \\text{graph}(\\Lambda )$ for each $i \\in \\mathbb {N}$ , it follows that $\\langle t, z, w \\rangle \\in \\overline{\\text{graph}(\\Lambda )}$ as required.",
"For the converse, suppose that $z \\in \\Delta (R)$ and that $\\langle t, z, w \\rangle \\in \\overline{\\text{graph}(\\Lambda )}$ .",
"Then certainly $t \\in \\overline{T}$ and we must show that $F_t(z) = w$ , thereby completing the proof of (*).",
"Let $\\langle \\langle t^{(i)}, z^{(i)} , w_i \\rangle : i \\in \\mathbb {N}\\rangle $ be a sequence in $\\text{graph}(\\Lambda )$ converging to $\\langle t, z, w \\rangle $ .",
"Then $z^{(i)} \\rightarrow z$ as $i \\rightarrow \\infty $ and since $z \\in \\Delta (R)$ , we may choose $r<R$ so that $z \\in \\Delta (r)^m$ and $z^{(i)} \\in \\Delta (r)^m$ for each $i \\in \\mathbb {N}$ .",
"Since $t^{(i)} \\rightarrow t$ as $i \\rightarrow \\infty $ , it follows from the construction of $F_t$ that $\\delta _r (F_{t^{(i)}}, F_t) \\rightarrow 0$ as $i \\rightarrow \\infty $ .",
"In particular, $|F_{t^{(i)}}(z^{(i)}) - F_t(z^{(i)})| \\rightarrow 0$ as $i \\rightarrow \\infty $ .",
"But by the definition of $\\text{graph}(\\Lambda )$ , $F_{t^{(i)}}(z^{(i)}) = w_i$ for all $i \\in \\mathbb {N}$ and hence $|w_i - F_t(z^{(i)})| \\rightarrow 0$ as $i \\rightarrow \\infty $ .",
"However, $F_t(z^{(i)}) \\rightarrow F_t(z)$ as $i \\rightarrow \\infty $ (because $F_t$ is certainly continuous on $\\Delta (r)^m$ ) and hence $w_i \\rightarrow F_t(z)$ as $i \\rightarrow \\infty $ .",
"Since $w_i \\rightarrow w$ as $i \\rightarrow \\infty $ it now follows that $w = F_t(z)$ as required.",
"Having shown how to compactify $(R,m,K)$ -families, we now projectivize them.",
"5.8 Theorem Let $\\Lambda = \\lbrace F_t : t \\in T \\rbrace $ be an $(R,m,K)$ -family with $R>1$ .",
"Assume that for no $t \\in T$ does $F_t$ vanish identically.",
"Let $R_0$ satisfy $1<R_0 < R$ .",
"Then there exists a positive real number $K_0$ and an $(R_0, m, K_0 )$ -family $\\Lambda ^{\\dag } = \\lbrace G_t : t\\in T^{\\dag } \\rbrace $ such that 5.8.1 $\\Lambda ^{\\dag }$ is well-indexed and $T^{\\dag }$ is closed in its ambient space $N$ ; 5.8.2 for every $t \\in T$ , there exists $A_t > 0$ and $t^{\\dag } \\in T^{\\dag }$ such that $G_{t^{\\dag }} = A_t \\cdot F_t \\upharpoonright \\Delta (R_0 )^m$ ; 5.8.3 the (real) dimension of $T^{\\dag }$ is at most that of $T$ ; 5.8.4 for no $t \\in T^{\\dag }$ is $G_t$ identically zero.",
"We consider the $(R_0 , m , K_0)$ -family $\\lbrace \\frac{F_t}{\\kappa _{\\Lambda }(t)} \\upharpoonright \\Delta (R_0)^m : t \\in T \\rbrace $ (cf.",
"5.3.6), where $K_0 = B_{\\Lambda } (R_0)$ (cf.",
"5.4).",
"Using 5.6, let $\\Lambda ^{*} = \\lbrace G_t : t \\in T^{*} \\rbrace $ be a well-indexing of it.",
"Clearly $\\text{dim}(T^{*}) \\le \\text{dim}(T)$ .",
"We set $T^{\\dag } := \\overline{T^{*}}$ and $\\Lambda ^{\\dag } := \\overline{\\Lambda ^{*}}$ as in 5.7.",
"Then 5.8.1-3 are clear.",
"For 5.8.4, let us first note that if $t \\in T^{*}$ then for some $s \\in T$ , $G_t = \\frac{F_s}{\\kappa _{\\Lambda }(s)} \\upharpoonright \\Delta (R_0)^m$ and hence there exists $\\alpha \\in \\mathbb {N}^m$ with $|\\alpha | \\le M(\\Lambda )$ such that $\\frac{|G_t^{(\\alpha )}(0)|}{\\alpha !}",
"= 1$ (see 5.5).",
"Now let $t^{\\dag } \\in \\overline{T^{*}}$ .",
"We must show that $G_{t^{\\dag }}$ does not vanish identically.",
"For this, choose $t \\in T^{*}$ such that $\\Vert t-t^{\\dag }\\Vert < \\frac{c_1}{2}$ so that $\\delta _1 (G_t ,G_{t^{\\dag }} ) < \\frac{1}{2}$ (by 5.5 with $r=1$ ).",
"It now follows from the Cauchy inequalities applied to the function $G_t - G_{t^{\\dag }}$ restricted to the unit polydisk $\\Delta (1)^m$ , that for all $\\alpha \\in \\mathbb {N}^m$ we have $\\frac{|G_t^{(\\alpha )}(0)-G_{t^{\\dag }}^{(\\alpha )}(0)|}{\\alpha !}",
"< \\frac{1}{2}$ .",
"So choosing $\\alpha $ with $\\frac{|G_t^{(\\alpha )}(0)|}{\\alpha !}",
"= 1$ as above, we see that $\\frac{|G_{t^{\\dag }}^{(\\alpha )}(0)|}{\\alpha !}",
"> \\frac{1}{2}$ .",
"In particular, $G_{t^{\\dag }}$ does not vanish identically.",
"Remark The hypothesis that $R_0 < R$ is necessary here: the reader may easily verify that for the $(2,1,1)$ -family $\\Lambda = \\lbrace g_t : t \\in [0, \\frac{1}{2}) \\rbrace $ where $g_t(z) = \\frac{1-2t}{1-tz}$ , and for each given $K_0 > 0$ , there is no $(2, 1, K_0)$ -family $\\Lambda ^{\\dag }$ satisfying 5.8.1-4.",
"We are almost ready for the proof of the quasi-parameterization theorem (2.2.3).",
"This will proceed by induction on the dimension of the given family $\\lbrace X_t : t \\in T \\rbrace $ , i.e.",
"the (minimum, real) dimension of the indexing set $T$ .",
"The inductive step will involve a use of the Weierstrass Preparation Theorem (or, rather, a modification of the argument used in the complex analytic proof of the Weierstrass Preparation Theorem) and, as usual, one first has to make a transformation so that the function being prepared is regular in one of its variables.",
"Further, in our case the transformation will have to work uniformly for all members of a certain definable family of functions and for all values of the other variables.",
"Unfortunately, the usual linear change of variables does not have this property.",
"Instead we use a variation of the transformation used by Denef and van den Dries in their proof of quantifier elimination for the structure $\\mathbb {R}_{an}$ (see [6]).",
"The result we require is contained in the following 5.9 Theorem Let $\\Lambda = \\lbrace F_t : t \\in T \\rbrace $ be an $(R,m,K)$ -family with $R>1$ (and, for non-triviality, with $m \\ge 2$ ) such that for no $t \\in T$ does $F_t$ vanish identically.",
"Let $R^{\\prime }$ and $R^{\\prime \\prime }$ be real numbers satisfying $1<R^{\\prime \\prime }<R^{\\prime }<R$ .",
"Then there exist positive integers $D_1,\\ldots ,D_{m-1}$ and a positive real number $\\eta $ such that the bijection $\\theta : {m} \\rightarrow {m} : z = \\langle z_1,\\ldots ,z_m \\rangle \\mapsto \\langle z_1 + \\eta z_m^{D_1},\\ldots ,z_{m-1} + \\eta z_m^{D_{m-1}}, z_m \\rangle $ satisfies 5.9.1 $\\theta [ \\overline{\\Delta (R^{\\prime })}^m ] \\subseteq \\Delta (R)^m$ , 5.9.2 $\\theta ^{-1}[\\overline{\\Delta (1)}^m] \\subseteq \\Delta (R^{\\prime \\prime })^m$ , and 5.9.3 for each $t \\in T$ and $z^{\\prime } \\in \\Delta (R^{\\prime })^{m-1}$ the function $z_m \\mapsto F_t \\circ \\theta (z^{\\prime },z_m)$ (for $z_m \\in \\Delta (R^{\\prime })$ ) does not vanish identically in $z_m$ .",
"This will follow from the following general 5.10 Lemma Let $m \\ge 1$ and suppose that $\\mathcal {X} = \\lbrace X_t : t \\in T \\rbrace $ is a definable family of subsets of $\\mathbb {R}^m$ such that for all $t \\in T$ , $\\text{dim}(X_t) < m$ .",
"Then there exist positive integers $D_1,\\ldots ,D_{m-1}$ such that for all $t \\in T$ , all $\\eta > 0$ and all $w_1,\\ldots , w_{m-1} \\in \\mathbb {R}$ , there exists $\\epsilon = \\epsilon (t, \\eta , w_1,\\ldots , w_{m-1} ) >0$ such that $X_t \\cap \\lbrace \\langle w_1 + \\eta x^{D_1},\\ldots ,w_{m-1} + \\eta x^{D_{m-1}}, x \\rangle \\in \\mathbb {R}^m : 0<x< \\epsilon \\rbrace = \\emptyset .$ Induction on $m$ .",
"For $m=1$ , each $X_t$ is (uniformly) finite.",
"So obviously we can find, for each $t \\in T$ , an $\\epsilon = \\epsilon (t) > 0$ such that $X_t \\cap (0, \\epsilon ) = \\emptyset $ , which is the required conclusion in this case.",
"Now assume that the lemma holds for some $m \\ge 1$ and that $\\lbrace X_t : t \\in T \\rbrace $ is a definable family of subsets of $\\mathbb {R}^{m+1}$ each having dimension at most $m$ .",
"For $t \\in T$ define $S_t := \\lbrace s \\in \\mathbb {R}^m : \\lbrace y \\in \\mathbb {R}: \\langle y, s \\rangle \\in X_t \\rbrace \\hspace{2.84526pt} \\text{is infinite} \\rbrace $ .",
"Then $\\lbrace S_t : t \\in T \\rbrace $ is a definable family of subsets of $\\mathbb {R}^m$ and clearly $\\text{dim}(S_t )<m$ for each $t \\in T$ .",
"So we may apply the inductive hypothesis to this family and obtain (with a small shift in notation) positive integers $D_2 , \\ldots ,D_m$ such that for all $t \\in T$ , all $\\eta >0$ and all $w_2,\\ldots ,w_m \\in \\mathbb {R}$ , there exists $\\epsilon = \\epsilon (t, \\eta , w_2,\\ldots , w_{m} ) >0$ such that for all $x \\in (0, \\epsilon )$ , we have that $\\langle w_2 + \\eta x^{D_2},\\ldots ,w_{m} + \\eta x^{D_{m}}, x \\rangle \\notin S_t$ , i.e.",
"there are at most finitely many $y \\in \\mathbb {R}$ such that $\\langle y, w_2 + \\eta x^{D_2},\\ldots ,w_{m} + \\eta x^{D_{m}}, x \\rangle \\in X_t$ .",
"Now, by the principle of definable choice, there exists a definable function $H: T \\times (0, \\infty ) \\times \\mathbb {R}^m \\times \\mathbb {R}\\rightarrow (0, 1]$ such that for all $t \\in T$ , all $\\eta \\in \\mathbb {R}$ , all $w \\in \\mathbb {R}^m$ and all $x \\in \\mathbb {R}$ , its value $H(t, \\eta , w, x)$ is some $y \\in (0,1)$ such that for no $u \\in (0,y)$ do we have $\\langle w_1 + \\eta u, w_{2} + \\eta x^{D_{2}},\\ldots , w_{m} + \\eta x^{D_{m}}, x \\rangle \\in X_t$ , if such a $y$ exists (and is, say, 1 otherwise).",
"Notice that by the discussion above, such a $y$ does indeed exist whenever $x \\in (0, \\epsilon (t, \\eta , w_2,\\ldots ,w_m))$ .",
"We now apply polynomial boundedness to obtain a positive integer $D_1$ such that for all $t \\in T$ , all $\\eta > 0$ and all $w = \\langle w_1 ,\\ldots , w_m \\rangle \\in \\mathbb {R}^m$ , there exists $\\gamma = \\gamma (t, \\eta , w) > 0$ , which we may assume is strictly less than $\\epsilon (t, \\eta , w_2,\\ldots ,w_m)$ , such that for all $x \\in (0 , \\gamma )$ , we have $H(t, \\eta , w, x) > x^{D_1}$ .",
"So if $x \\in (0, \\gamma )$ , then $x \\in (0, \\epsilon )$ and hence for all $u \\in (0, H(t, \\eta , w, x))$ we have that $\\langle w_1 + \\eta u, w_{2} + \\eta x^{D_{2}}, \\ldots , w_{m} + \\eta x^{D_{m}}, x \\rangle \\notin X_t$ .",
"But $x^{D_1} \\in (0, H(t, \\eta , w, x) )$ so $\\langle w_1 + \\eta x^{D_1}, w_{2} + \\eta x^{D_{2}}, \\ldots , w_{m} + \\eta x^{D_{m}}, x \\rangle \\notin X_t$ .",
"Since this holds for arbitrary $x \\in (0, \\gamma )$ , we are done (upon taking $\\epsilon (t, \\eta , w_1,\\ldots ,w_m) := \\gamma (t, \\eta , w)$ ).",
"Proof of 5.9 For $t \\in T$ and $u \\in (-R, R)^m$ , let $X_{\\langle t, u \\rangle } := \\lbrace w \\in (-R^{\\prime }, R^{\\prime })^m : w+ u\\text{i} \\in \\Delta (R)^m \\hspace{5.69054pt}\\text{and} \\hspace{5.69054pt} F_t(w+ u\\text{i}) = 0 \\rbrace $ .",
"Then $\\text{dim}(X_{\\langle t, u \\rangle }) < m$ because if $U$ is some non-empty, open subset of $(-R^{\\prime }, R^{\\prime })^m$ such that $w+ u\\text{i} \\in \\Delta (R)^m \\hspace{5.69054pt}\\text{and} \\hspace{5.69054pt} F_t(w+ u\\text{i}) = 0$ for all $w \\in U$ , then $F_t$ , being holomorphic, would vanish identically on $\\Delta (R)^m$ which is contrary to hypothesis.",
"So we may apply 5.10 to the family $\\mathcal {X} := \\lbrace X_{\\langle t, u \\rangle )} : \\langle t, u \\rangle \\in T \\times (-R, R)^m \\rbrace $ and obtain positive integers $D_1, \\ldots ,D_{m-1}$ with the property stated in the conclusion of 5.10.",
"Now choose $\\eta $ so small that the resulting map $\\theta $ satisfies 5.9.1 and 5.9.2.",
"(The inverse of $\\theta $ is given by $\\theta ^{-1} (z_1, \\ldots ,z_m) = \\langle z_1 - \\eta z_m^{D_1}, \\ldots , z_{m-1} - \\eta z_{m}^{D_{m-1}}, z_m \\rangle $ .)",
"To verify 5.9.3, let $t \\in T$ and let $z^{\\prime } = \\langle z_1,\\ldots ,z_{m-1} \\rangle \\in \\Delta (R^{\\prime })^{m-1}$ .",
"If $F_t \\circ \\theta (z^{\\prime } , z_m ) = 0$ for all $z_m \\in \\Delta (R^{\\prime })$ then, in particular, $\\langle z_1 + \\eta x^{D_1},\\ldots ,z_{m-1} + \\eta x^{D_{m-1}}, x \\rangle \\in \\Delta (R^{\\prime })^m$ and $F_t (z_1 + \\eta x^{D_1},\\ldots ,z_{m-1} + \\eta x^{D_{m-1}}, x ) = 0$ for all sufficiently small positive $x \\in \\mathbb {R}$ .",
"However, if the real and imaginary parts of $z_i$ are, respectively, $a_i$ and $b_i$ (for $i = 1,\\ldots ,m-1$ ), this implies that $\\langle a_1 + \\eta x^{D_1},\\ldots ,a_{m-1} + \\eta x^{D_{m-1}}, x \\rangle \\in X_{\\langle t, b \\rangle }$ for all sufficiently small $x>0$ , where $b:= \\langle b_1,\\ldots ,b_{m-1} , 0 \\rangle $ .",
"But this clearly contradicts the conclusion of 5.10.",
"We now come to the proof of the quasi-parameterization theorem (2.2.3), so definability is now, and henceforth, with respect to a structure as described in 2.2.1.",
"Recall that we are given a definable family $\\mathcal {X} = \\lbrace X_s : s \\in S \\rbrace $ of subsets of $[-1, 1 ]^n$ each of dimension at most $m$ , where $m<n$ .",
"We assume that the indexing set $S$ has been chosen of minimal dimension and we denote this dimension by $\\text{dim}(\\mathcal {X})$ .",
"We are required to find some $R>1, K>0$ , a positive integer $d$ , and an $(R, m+1, K)$ -family $\\Lambda ^{*}$ , each element of which is a monic polynomial of degree at most $d$ in its first variable, such that (*) for all $s \\in S$ , there exists $F \\in \\Lambda ^{*}$ such that $X_s \\subseteq \\lbrace x = \\langle x_1,\\ldots ,x_n \\rangle \\in [-1, 1]^n : \\exists w \\in [-1, 1]^m \\bigwedge _{i=1}^n F(x_i ,w ) =0 \\rbrace .$ Let us first consider the case $\\text{dim}(\\mathcal {X})=0$ , i.e.",
"the case that $\\mathcal {X}$ is finite.",
"In fact, it is sufficient to consider the case that $\\mathcal {X}$ consists of a single set, $X$ say, where $X \\subseteq [-1,1]^n$ and $\\text{dim}(X) \\le m<n$ .",
"5.11 Remark Indeed, it is obvious that, in general, if the conclusion of the quasi-parameterization theorem holds for the families $\\lbrace X_s : s \\in S_1 \\rbrace $ and $\\lbrace X_s : s \\in S_2 \\rbrace $ , then it also holds for the family $\\lbrace X_s : s \\in S_1 \\cup S_2 \\rbrace $ .",
"Since we are now assuming that our ambient o-minimal structure is a reduct of $\\mathbb {R}_{an}$ , we may apply the 0-mild parameterization theorem (Proposition 1.5 of [15]) which tells us that (after routine translation and scaling) there exists a finite set $\\lbrace \\Phi _j : 1 \\le j \\le l \\rbrace $ of definable, real analytic maps $\\Phi _j = \\langle \\phi _{j,1},\\ldots ,\\phi _{j,n} \\rangle :(-3,3)^m \\rightarrow \\mathbb {R}^n$ (say) whose images on $[-1,1]^m$ cover $X$ , and are such that $\\frac{|\\phi ^{(\\alpha )}_{j,i} (0)|}{\\alpha !}",
"\\le c\\cdot 3^{-|\\alpha |}$ for some constant $c$ , and all $\\alpha \\in \\mathbb {N}^m$ , $j = 1,\\ldots ,l$ and $i=1,\\ldots ,n$ .",
"If we now invoke our other assumption on the ambient 0-minimal structure, then (again, after translation and scaling at the expense of increasing the number of parameterizing functions) there is no harm in assuming that each function $\\phi _{j.i}$ has a definable, complex extension (for which we use the same notation) to the polydisk $\\Delta (2)^m$ .",
"We now set $F(z_1,z_2,\\ldots ,z_{m+1}) := \\prod _{j=1}^{l}\\prod _{i=1}^{n}(z_1 - \\phi _{j,i}(z_2,\\ldots ,z_{m+1}))$ for $z_1,z_2,\\ldots ,z_{m+1} \\in \\Delta (2)$ .",
"Then $F$ is a monic polynomial of degree $d=ln$ in its first variable.",
"Further, $\\lbrace F \\rbrace $ is, for some $K>0$ , a $(2,m+1,K)$ -family which clearly has the required property (*).",
"We now proceed by induction on $\\text{dim}(\\mathcal {X})$ .",
"So consider some $k,m,n \\in \\mathbb {N}$ with $k = \\text{dim}(\\mathcal {X})\\ge 1$ and $m<n$ , and a definable family $\\mathcal {X} = \\lbrace X_s : s \\in S \\rbrace $ of subsets of $[-1,1]^n$ with $\\text{dim}(X_t) \\le m$ for each $s \\in S$ , and assume that the theorem holds for families of dimension $< k$ (for arbitrary $m,n$ ).",
"Now it is easy to show that we may represent $\\mathcal {X}$ in the form $\\lbrace X_u : u \\in [-1, 1 ]^k \\rbrace $ .",
"In order to apply the inductive hypothesis we define the family $\\mathcal {Y} := \\lbrace Y_{u^{\\prime }} : u^{\\prime } \\in [-1,1]^{k-1} \\rbrace $ of subsets of $[-1,1]^{n+1}$ where, for each $u^{\\prime } \\in [-1,1]^{k-1}$ , 5.12 $Y_{u^{\\prime }} := \\lbrace \\langle x, u_{k} \\rangle \\in [-1,1]^{n+1} : x \\in X_u \\rbrace .$ (In the course of this proof we shall use the convention that if $v$ is a tuple whose length, $p$ say, is clear from the context, then $v = \\langle v_1,\\ldots ,v_p \\rangle $ , and $v^{\\prime } = \\langle v_1,\\ldots ,v_{p-1} \\rangle $ .",
"Also, by convention, $[-1, 1]^0 := \\lbrace 0\\rbrace $ .)",
"Clearly $\\text{dim}(\\mathcal {Y}) < k$ and, for each $u^{\\prime } \\in [-1,1]^{k-1}$ , $\\text{dim}(Y_{u^{\\prime }}) \\le m+1 < n+1$ , so we may indeed apply our inductive hypothesis to $\\mathcal {Y}$ and obtain some $R>1$ , $K>0$ , an $(R,m+2,K)$ -family $\\Lambda = \\lbrace H_t : t \\in T \\rbrace $ , and a positive integer $d$ such that 5.13 each $H_t$ is a monic polynomial of degree at most $d$ in its first variable, and 5.14 for each $u^{\\prime } \\in [-1,1]^{k-1}$ there exists $t = t(u^{\\prime }) \\in T$ such that $Y_{u^{\\prime }} \\subseteq \\lbrace \\langle x, x_{n+1} \\rangle \\in [-1, 1]^{n+1} :\\exists w \\in [-1, 1]^{m+1} (\\bigwedge _{i=1}^{n+1} H_t (x_i ,w ) =0 \\rbrace .$ In order to prepare the functions in $\\Lambda $ as discussed above, we must first remove those $t$ from $T$ such that for some $z_1$ the function $H_t(z_1, \\cdot )$ vanishes identically (in its last $m+1$ variables).",
"To do this, we first note that, by the principle of definable choice, the correspondence $u^{\\prime } \\mapsto t(u^{\\prime })$ (for $u^{\\prime } \\in [-1,1]^{k-1}$ ) may be taken to be a definable function and so the set $E := \\lbrace u \\in [-1, 1]^{k}: H_{t(u^{\\prime })} (u_{k}, {\\bf 0}) = 0 \\rbrace $ is definable (where ${\\bf 0}$ is the origin of $\\mathbb {R}^{m+1}$ ).",
"We have $\\text{dim}(E) < k$ because if $E$ contained a non-empty open subset of $[-1,1]^{k}$ , then we could find some $u^{\\prime } \\in [-1,1]^{k-1}$ such that $H_{t(u^{\\prime })} (u_{k}, {\\bf 0}) = 0$ for all $u_{k}$ lying in some non-empty open interval, which is impossible as $H_{t(u^{\\prime })} (\\cdot , {\\bf 0})$ is a monic polynomial.",
"Thus, by another use of the inductive hypothesis, the family $\\lbrace X_u : u \\in E \\rbrace $ satisfies the conclusion of the quasi-parameterization theorem and so, by 5.11, it is sufficient to consider the family $\\lbrace X_u : u \\in [-1,1]^k \\setminus E \\rbrace $ .",
"For this we define, for each $u \\in [-1,1]^k \\setminus E$ , the function $H^{*}_u : \\Delta (R)^{m+1} \\rightarrow by$ 5.15 $H^{*}_u (z) := H_{t(u^{\\prime })} (u_k , z)$ , so that for all $u \\in [-1,1]^k \\setminus E $ , $H^{*}_u({\\bf 0}) \\ne 0$ .",
"Now set 5.16 $\\Lambda _0 := \\lbrace H_u^{*} :u \\in [-1, 1]^k \\setminus E \\rbrace $ .",
"Then $\\Lambda _0$ is an $(R, m+1, K)$ -family which does not contain the zero function.",
"So we may apply 5.8 to it with, say, $R_0 = \\frac{1+R}{2}$ (and $m+1$ in place of $m$ ) and obtain, for some $K_0> 0$ , an $(R_0, m+1, K_0)$ -family $\\Lambda _0^{\\dag } = \\lbrace G_t : t \\in T_0^{\\dag } \\rbrace $ having properties 5.8.1-4.",
"I claim that 5.17 for all $u \\in [-1,1]^k \\setminus E$ , there exists $t^{\\dag } \\in T_0^{\\dag }$ such that $X_u \\subseteq \\lbrace x \\in [-1,1]^n : \\exists w \\in [-1,1]^{m+1} ( \\bigwedge _{i=1}^n H_{t(u^{\\prime })} (x_i , w)=0 \\wedge G_{t^{\\dag }}(w)=0) \\rbrace $ .",
"Indeed, let $u \\in [-1,1]^k \\setminus E$ .",
"By 5.8.2, there is some $t^{\\dag } \\in T_0^{\\dag }$ and $A>0$ such that for all $z \\in \\Delta (R_0)^{m+1}$ , 5.17.1 $G_{t^{\\dag }}(z) = A \\cdot H^{*}_u (z)$ .",
"Now let $x \\in X_u$ .",
"Then, by 5.12, $\\langle x, u_k \\rangle \\in Y_{u^{\\prime }}$ .",
"Hence, by 5.14, we may choose $w \\in [-1,1]^{m+1}$ such that $H_{t(u^{\\prime })}(x_i ,w) =0$ for $i = 1, \\ldots , n$ and $H_{t(u^{\\prime })}(u_k ,w) = 0$ .",
"Since $[-1,1]^{m+1} \\subseteq \\Delta (R_0)^{m+1}$ , 5.17 now follows from 5.17.1 and 5.15.",
"In order to complete the proof we must reduce the range of the $w$ -variable in 5.17 from $[-1,1]^{m+1}$ to $[-1,1]^{m}$ .",
"The idea is simple: we use the relation $G_{t^{\\dag }}(w)=0$ to express $w_{m+1}$ as a function of $w_1, \\ldots ,w_m$ , and then substitute this function for $w_{m+1}$ in the first conjunct appearing in 5.17.",
"Of course, there are some technical difficulties to be overcome.",
"Firstly, we must ensure that $G_{t^{\\dag }}(w)$ really does depend on $w_{m+1}$ and this is achieved by the transformation described in 5.9.",
"Secondly, the argument only works locally.",
"However, the compactness of $T_0^{\\dag }$ will guarantee that this is sufficient.",
"And finally, the functional dependence of $w_{m+1}$ on $w_1,\\ldots ,w_m$ will, in general, be a many-valued one.",
"This is precisely why we only obtain quasi-parameterization rather than parameterization.",
"So, to carry out the first step, we apply 5.9 to the $(R_0,m+1,K_0)$ -family $\\Lambda _0^{\\dag } = \\lbrace G_t : t \\in T_0^{\\dag } \\rbrace $ (which is permissible as it satisfies 5.8.4) with $R^{\\prime } = 1+\\frac{2(R_0 -1)}{3}$ and $R^{\\prime \\prime } = 1+\\frac{(R_0 -1)}{3}$ (and $m+1$ in place of $m$ ).",
"Let $\\theta : {m+1} \\rightarrow {m+1}$ be as in 5.9 and, for each $t \\in T_0^{\\dag }$ set 5.18 $\\tilde{G_t} := G_t \\circ \\theta \\upharpoonright \\Delta (R^{\\prime })^{m+1}$ .",
"Then $\\lbrace \\tilde{G_t} : t \\in T_0^{\\dag } \\rbrace $ is an $(R^{\\prime },m+1, K_0)$ -family.",
"Further, since the family $\\Lambda _0^{\\dag }$ is well indexed (5.8.1), it immediately follows (from 5.5 and the Cauchy inequalities) that for each $\\alpha \\in \\mathbb {N}^{m+1}$ , the function $G^{(\\alpha )}_t(z)$ is continuous in both $t$ and $z$ , for $\\langle t, z \\rangle \\in T_0^{\\dag } \\times \\Delta (R_0)^{m+1}$ .",
"Since $\\theta $ is holomorphic throughout ${m+1}$ we obtain 5.19 for each $\\alpha \\in \\mathbb {N}^{m+1}$ , the function $\\tilde{G}^{(\\alpha )}_t(z)$ is continuous in both $t$ and $z$ for $\\langle t, z \\rangle \\in T_0^{\\dag } \\times \\Delta (R^{\\prime })^{m+1}$ .",
"Also, it follows from 5.9.3 that 5.20 for all $t \\in T_0^{\\dag }$ and all $z^{\\prime } \\in \\Delta (R^{\\prime })^m$ , the function $\\tilde{G_t} (z^{\\prime }, \\cdot )$ does not vanish identically on $\\Delta (R^{\\prime })$ .",
"Having modified the functions $G_t$ , we must now adjust the functions $H_{t(u^{\\prime })}$ in order to preserve 5.17.",
"Accordingly, we define, for each $u \\in [-1,1]^{k} \\setminus E$ , the function $\\tilde{H}_{u} : \\Delta (R^{\\prime })^{m+2} \\rightarrow (which, in fact, onlydepends on $ u'$) by$ 5.21 $\\tilde{H}_{u} (z_1,z_2,\\ldots ,z_{m+2}) := H_{t(u^{\\prime })} (z_1, \\theta (z_2,\\ldots ,z_{m+2})).$ Then $\\lbrace \\tilde{H}_{u} : u \\in [-1,1]^k \\setminus E \\rbrace $ is an $(R^{\\prime }, m+2, K)$ -family and, as we show below, the following version of 5.17 holds.",
"5.22 for all $u \\in [-1,1]^k \\setminus E$ , there exists $t^{\\dag } \\in T_0^{\\dag }$ such that $X_u \\subseteq \\lbrace x \\in [-1,1]^n : \\exists v \\in (-R^{\\prime \\prime },R^{\\prime \\prime })^{m+1} ( \\bigwedge _{i=1}^n \\tilde{H}_{u} (x_i , v)=0 \\wedge \\tilde{G}_{t^{\\dag }}(v)=0) \\rbrace $ .",
"Indeed, let $u = \\langle u^{\\prime }, u_{k} \\rangle \\in [-1,1]^k \\setminus E$ and choose $t^{\\dag } \\in T_0^{\\dag }$ as in 5.17.",
"Suppose $x \\in X_u$ and (by 5.17) choose $w \\in [-1,1]^{m+1}$ such that $\\bigwedge _{i=1}^n H_{t(u^{\\prime })} (x_i , w)=0 \\wedge G_{t^{\\dag }}(w)=0$ .",
"Let $v = \\theta ^{-1}(w)$ .",
"Then by 5.9.2, $v \\in \\Delta (R^{\\prime \\prime })^{m+1}$ .",
"But all the coordinates of $v$ are real, so $v \\in (-R^{\\prime \\prime },R^{\\prime \\prime })^{m+1}$ .",
"Also, for $i=1,\\ldots ,n$ we have, by 5.21, that $\\tilde{H}_{u} (x_i , v) = H_{t(u^{\\prime })} (x_i , \\theta (v)) = H_{t(u^{\\prime })} (x_i , w) = 0$ .",
"Similarly, by 5.18, $\\tilde{G}_{t^{\\dag }} (v) = G_{t^{\\dag }} (w) =0$ and 5.22 follows.",
"Let us also record here the fact that in view of 5.13, and since the transformation 5.21 does not affect the variable $z_1$ , we have 5.23 for each $u \\in [-1,1]^k \\setminus E$ , the function $\\tilde{H}_u$ is a monic polynomial of degree at most $d$ in its first variable.",
"We now carry out the local argument, as sketched above, that expresses $z_{m+1}$ as a many-valued function of $z^{\\prime } = \\langle z_1,\\ldots ,z_m \\rangle $ via the relation $\\tilde{G}_{t^{\\dag }}(z^{\\prime }, z_{m+1}) = 0$ .",
"Firstly, fix some $R_1$ with $R^{\\prime \\prime } < R_1 < R^{\\prime }$ and for each $r$ with $R^{\\prime \\prime }<r<R_1$ let $C_r$ be the circle in $ with centre $ 0$ andradius $ r$.",
"Consider the set$ 5.24 $V_r := \\lbrace \\langle t, z^{\\prime } \\rangle \\in T_0^{\\dag } \\times \\overline{\\Delta (R_1)}^m : \\hspace{2.84526pt} \\text{for all} \\hspace{5.69054pt}z_{m+1} \\in C_r , \\hspace{5.69054pt} \\tilde{G}_t(z^{\\prime }, z_{m+1}) \\ne 0 \\rbrace $ .",
"It follows from 5.19 that $V_r$ is an open subset of $T_0^{\\dag } \\times \\overline{\\Delta (R_1)}^m$ (for the $\\Vert \\cdot \\Vert $ -metric inherited from ${N+m}$ , where $N$ is as in 5.8.1).",
"Further, it follows easily from 5.20 that the collection $\\lbrace V_r : R^{\\prime \\prime }<r<R_1 \\rbrace $ covers the compact space $T_0^{\\dag } \\times \\overline{\\Delta (R_1)}^m$ .",
"Now, by the Lebesgue Covering Lemma, there exists $\\epsilon > 0$ , a positive integer $M$ , and points $t^{(1)},\\ldots ,t^{(M)} \\in T_0^{\\dag }$ , $a^{(1)},\\ldots ,a^{(M)} \\in [-R^{\\prime \\prime }, R^{\\prime \\prime }]^m$ such that 5.25 the collection $\\lbrace t^{(h)} + \\Delta (\\epsilon )^N : h=1,\\ldots , M \\rbrace $ covers $T_0^{\\dag }$ , 5.26 each set $a^{(j)} + \\Delta (2\\epsilon )^m$ is contained in $\\Delta (R_1)^m$ and the collection $\\lbrace a^{(j)} + (-\\epsilon , \\epsilon )^m : j=1,\\ldots , M \\rbrace $ covers $[-R^{\\prime \\prime },R^{\\prime \\prime }]^m$ , and 5.27 for each $h, j = 1, \\ldots , M$ , there exists $r_{h,j} \\in (R^{\\prime \\prime }, R_1)$ such that $(\\langle t^{(h)}, a^{(j)} \\rangle + \\overline{\\Delta (2 \\epsilon )}^{N+m}) \\cap (T_0^{\\dag } \\times \\Delta (R_1)^m) \\subseteq V_{r_{h,j}}.$ Fix, for the moment, $h, j \\in \\lbrace 1,\\ldots ,M \\rbrace $ .",
"Then for each $t \\in T_0^{\\dag } \\cap (t^{(h)} + \\Delta (2\\epsilon )^N)$ and each $z^{\\prime } \\in a^{(j)} + \\Delta (2\\epsilon )^m$ it follows from 5.26, 5.27 and 5.24 that the contour integral $\\frac{1}{2\\pi \\text{i}} \\int _{C_{r_{h,j}}} \\frac{\\partial \\tilde{G}_t }{\\partial z_{m+1}}(z^{\\prime } , z_{m+1}) \\cdot (\\tilde{G}_t (z^{\\prime } , z_{m+1}))^{-1} \\hspace{2.84526pt} dz_{m+1}$ is well defined.",
"It counts the number of zeros (with multiplicity) of the function $\\tilde{G}_t (z^{\\prime } , \\cdot )$ lying within the circle $C_{r_{h,j}}$ .",
"Further, by 5.19, 5.24 and 5.27, the integral is a continuous function of $\\langle t, z^{\\prime } \\rangle $ in the stated domain, and so is constant there.",
"Let its value be $q_{h,j}$ and let $Z(t,z^{\\prime }) = \\langle \\rho _{1} (t,z^{\\prime }),\\ldots ,\\rho _{q_{h,j}}(t,z^{\\prime }) \\rangle $ be a listing of the zeros of $\\tilde{G}_t (z^{\\prime } , \\cdot )$ lying within the circle $C_{r_{h,j}}$ (each one counted according to its multiplicity).",
"Now, for $t \\in T^{\\dag }_0 \\cap (t^{(h)} + \\Delta (2\\epsilon )^N)$ , $\\langle z_2,\\ldots ,z_{m+1} \\rangle \\in a^{(j)} + \\Delta (2\\epsilon )^m$ , $u \\in [-1,1]^k \\setminus E$ , and each $l=1,\\ldots ,q_{h,j}$ , we have, by 5.26 and the fact that $r_{h, j} < R_1$ , that $z_2,\\ldots ,z_{m+1}, \\rho _{l} (t,z_2,\\ldots ,z_{m+1}) \\in \\Delta (R_1)$ and hence that the function $L_{t,u}^{h,j} : (a^{(j)} + \\Delta (2\\epsilon )^m) \\rightarrow $ given by 5.28 $L_{t,u}^{h,j}( z_1,z_2,\\ldots ,z_{m+1}) := \\prod _{l=1}^{q_{h,j}} \\tilde{H}_u (z_1, z_2,\\ldots ,z_{m+1}, \\rho _l (t,z_2,\\ldots ,z_{m+1}))$ is well defined and, is a monic polynomial of degree at most $d\\cdot q_{h,j}$ in $z_1$ (by 5.23).",
"Now, since $L_{t,u}^{h,j}( z_1,z_2,\\ldots ,z_{m+1})$ is symmetric in the $\\rho _l (t,z_2,\\ldots ,z_{m+1})$ (i.e.",
"it does not depend on our particular ordering of the list $Z(t, z_2,\\ldots ,z_{m+1})$ ), it follows easily that it is a definable function of all the variables $t,u,z_1,\\ldots ,z_{m+1}$ (restricted to the stated domain) and, as a standard argument shows, it is holomorphic in $z_1,z_2,\\ldots ,z_{m+1}$ .",
"We now scale and translate the function $L_{t,u}^{h,j}$ by setting 5.29 $P_{t,u}^{h,j}( z_1,z_2,\\ldots ,z_{m+1}) := L_{t,u}^{h,j}( z_1,a_1^{(j)} + \\epsilon z_2,\\ldots , a_{m}^{(j)} + \\epsilon z_{m+1})$ so that each $P_{t,u}^{h,j}$ maps $\\Delta (2)^m$ to $, is a monic polynomial of degree at most $ dqh,j$in $ z1$, and is bounded by $ Kqh,j$.",
"(Notice that this holds true, by our convention concerning the monic polynomialof degree $ 0$, even if $ qh,j = 0$.",
")$ We now combine the functions $P_{t,u}^{h,j}$ as $h$ and $j$ vary over $\\lbrace 1,\\ldots ,M \\rbrace $ .",
"Firstly, for $h \\in \\lbrace 1,\\ldots ,M \\rbrace $ , $u \\in [-1,1]^k \\setminus E$ and $t \\in T^{\\dag }_0 \\cap (t^{(h)} + \\Delta (2\\epsilon )^N)$ define 5.30 $P_{t,u}^{h} := \\prod _{j=1}^{M} P_{t,u}^{h,j}$ so that each $P_{t,u}^{h}$ maps $\\Delta (2)^m$ to $, is a monic polynomial of degree at most $ dh := j=1M dqh,j$in $ z1$, and is bounded by $ (K+1)Mqh$,where $ qh := max { qh,j : j = 1,...,M }$.$ Finally, we set 5.31 $\\Lambda ^{*} := \\bigcup _{h=1}^{M} \\lbrace P^h_{t,u} : u \\in [-1,1]^k , t \\in T_0^{\\dag } \\cap (t^{(h)} + \\Delta (\\epsilon )^N \\rbrace $ .",
"Then $\\Lambda ^{*}$ is a $(2, m+1, (K+1)^{Mq})$ -family, where $q:= \\text{max} \\lbrace q_h : h= 1,\\ldots ,M \\rbrace $ , each element of which is a monic polynomial of degree at most $\\text{max} \\lbrace d_h : h=1,\\ldots ,M \\rbrace $ in its first variable.",
"We now verify (*) (stated just before 5.11) which will complete the proof.",
"In fact, it just remains to show that if $u \\in [-1,1]^k \\setminus E$ , then there exists $F \\in \\Lambda ^{*}$ such that $X_u \\subseteq \\lbrace x \\in [-1, 1]^n : \\exists w \\in [-1, 1]^m \\bigwedge _{i=1}^n F(x_i ,w ) =0 \\rbrace .$ So let such a $u$ be given.",
"Choose $t^{\\dag } \\in T_0^{\\dag }$ as in 5.22.",
"By 5.25 we may choose $h \\in \\lbrace 1,\\ldots , M \\rbrace $ such that $t^{\\dag } \\in t^{(h)} + \\Delta (\\epsilon )^N$ .",
"We let our $F$ be the function $P_{t^{\\dag },u}^h$ (see 5.30), which of course lies in $\\Lambda ^{*}$ (see 5.31).",
"Now pick any $x = \\langle x_1,\\ldots ,x_n \\rangle \\in X_u$ .",
"By 5.22 we may pick $v = \\langle v^{\\prime } , v_{m+1} \\rangle \\in (-R^{\\prime \\prime },R^{\\prime \\prime })^{m+1}$ such that $\\bigwedge _{i=1}^n \\tilde{H}_{u} (x_i , v)=0 \\wedge \\tilde{G}_{t^{\\dag }}(v)=0$ .",
"By 5.26, there exists $j \\in \\lbrace 1,\\ldots ,M \\rbrace $ such that $v^{\\prime } \\in a^{(j)} + (-\\epsilon ,\\epsilon )^m$ .",
"Now, since $v_{m+1}$ lies within the circle $C_{r_{h,j}}$ and is a zero of the function $\\tilde{G}_{t^{\\dag }}(v^{\\prime }, \\cdot )=0$ , it follows that $q_{h,j} > 0$ and that for some $l=1,\\ldots , q_{i,j}$ , we have $v_{m+1} = \\rho _l (t^{\\dag }, v^{\\prime })$ .",
"Thus $\\bigwedge _{i=1}^n \\tilde{H}_{u} (x_i , v^{\\prime }, \\rho _l (t^{\\dag }, v^{\\prime } ))=0$ , and hence $\\bigwedge _{i=1}^n L_{t^{\\dag },u}^{h,j} (x_i , v^{\\prime }) =0$ (see 5.28).",
"We now choose $w \\in [-1,1]^m$ such that $v^{\\prime }= a^{(j)} + \\epsilon w$ .",
"Then $\\bigwedge _{i=1}^n P_{t^{\\dag },u}^{h,j} (x_i , w) =0$ (see 5.29).",
"It follows that $\\bigwedge _{i=1}^n P_{t^{\\dag },u}^{h} (x_i , w) =0$ (see 5.30), i.e.",
"$\\bigwedge _{i=1}^n F(x_i , w) =0$ , and we are done."
]
] | 1605.05916 |
[
[
"Interfacial magnetic anisotropy from a 3-dimensional Rashba substrate"
],
[
"Abstract We study the magnetic anisotropy which arises at the interface between a thin film ferromagnet and a 3-d Rashba material.",
"The 3-d Rashba material is characterized by the spin-orbit strength $\\alpha$ and the direction of broken bulk inversion symmetry $\\hat n$.",
"We find an in-plane uniaxial anisotropy in the $\\hat{z}\\times\\hat{n}$ direction, where $\\hat z$ is the interface normal.",
"For realistic values of $\\alpha$, the uniaxial anisotropy is of a similar order of magnitude as the bulk magnetocrystalline anisotropy.",
"Evaluating the uniaxial anisotropy for a simplified model in 1-d shows that for small band filling, the in-plane easy axis anisotropy scales as $\\alpha^4$ and results from a twisted exchange interaction between the spins in the 3-d Rashba material and the ferromagnet.",
"For a ferroelectric 3-d Rashba material, $\\hat n$ can be controlled with an electric field, and we propose that the interfacial magnetic anisotropy could provide a mechanism for electrical control of the magnetic orientation."
],
[
"Introduction",
"Interfacial magnetic anisotropy plays a key role in thin film ferromagnetism.",
"For ultra thin magnetic layers (less than 1 nm thickness), the reduced symmetry at the interface and orbital hybridization between the ferromagnet and substrate can lead to perpendicular magnetic anisotropy [1], [2], [3].",
"Perpendicular magnetization in magnetic multilayers can enable current-induced magnetic switching at lower current densities [4], [5].",
"Interfacial magnetic anisotropy is also at the heart of several schemes of electric-field based magnetic switching.",
"In this case an externally applied field can modify the electronic properties of the interface, changing the magnetic anisotropy and leading to efficient switching of the magnetic layer [6], [7], [8], [9].",
"The combination of symmetry breaking at the interface and the materials' spin-orbit coupling generally leads to an effective Rashba-like interaction acting on the orbitals at the interface [10], [11].",
"The interfacial magnetic anisotropy can be studied in terms of a minimal model containing both ferromagnetism and Rashba spin-orbit coupling [12].",
"The interfacial magnetic anisotropy direction is a structural property of the sample leading to easy- or hard-axis out-of-plane anisotropy, and isotropic in-plane anisotropy.",
"There has been recent interest in materials with strong spin-orbit coupling which lack structure inversion symmetry in the bulk.",
"These are known as 3-d Rashba materials, and examples include BiTeI [13] and GeTe [14].",
"In BiTeI, the structure inversion asymmetry results from the asymmetric stacking of Bi, Te, and I layers, and photoemission studies reveal an exceptionally large Rashba parameter $\\alpha $ [13].",
"In GeTe, a polar distortion of the rhombohedral unit cell leads to inversion asymmetry and ferroelectricity [15], [16], [17].",
"Both materials are semiconductors in which the valence and conduction bands are described by an effective Rashba model with symmetry-breaking direction $\\hat{n}$ , which is determined by the crystal structure.",
"There is interest in finding other ferroelectric materials with strong spin-orbit coupling, motivated by the desire to control the direction of $\\hat{n}$ with an applied electric field [18], [19], [20], [21], [22], [17].",
"In this work, we study the influence of a 3-d (nonmagnetic) Rashba material on the magnetic anisotropy of an adjacent ferromagnetic layer.",
"The interface between these materials breaks the symmetry along the $\\hat{z}$ -direction, and the addition of another symmetry breaking direction $\\hat{n}$ enriches the magnetic anisotropy energy landscape.",
"For a general direction of $\\hat{n}$ , we find a complex dependence of the system energy on magnetic orientation.",
"In our model system, we find the out-of-plane anisotropy is much smaller than the demagnetization energy.",
"However an in-plane component of $\\hat{n}$ leads to a uniaxial in-plane magnetic anisotropy which can be on the order of (or larger than) the magnetocrystalline anisotropy of bulk ferromagnetic materials.",
"Control of $\\hat{n}$ (for example via an electric field in a 3-d Rashba ferroelectric) can therefore modify the magnetization orientation, opening up new routes to magnetic control.",
"Figure: (a) shows the system geometry of a thin film ferromagnet adjacent to a nonmagnetic material described by a 3-d Rashba model.",
"(b) shows the unit cell of the model system."
],
[
"Numerical evaluation of 2-d model",
"We first consider a bilayer system as shown in Fig.",
"REF (a), with unit cell as shown in Fig.",
"REF (b).",
"We take 4 layers of the Rashba material and 2 layers of the ferromagnetic material with stacking along the $z$ -direction, and assume a periodic square lattice in the $x-y$ plane with lattice constant $a$ .",
"The Hamiltonian of the system is given by $H=H_{\\rm TB}+H_{\\rm F}+H_{\\rm R}+H_{\\rm F-R}$ , where: $H_{\\rm TB} &=& t \\sum _{\\langle ij\\rangle } c_i^\\dagger c_j \\\\H_{\\rm F} &=&\\frac{\\Delta }{2} \\sum _{i\\gamma \\nu } c_{i\\gamma }^\\dagger c_{i\\nu } \\left(\\hat{M} \\cdot \\vec{\\sigma }_{\\gamma \\nu }\\right) + H_{\\rm TB} , \\\\H_{\\rm R} &=&i \\frac{\\alpha }{2a}\\sum _{\\langle ij\\rangle \\gamma \\nu } c_{i\\gamma }^\\dagger c_{j\\nu } \\left[\\left(\\hat{r}_{ij} \\times \\vec{\\sigma }_{\\gamma \\nu }\\right)\\cdot \\hat{n}\\right] + H_{\\rm TB} , \\\\H_{\\rm F-R} &=& H_{\\rm TB} + \\frac{\\alpha }{2a}\\left( i c_{F\\gamma }^\\dagger c_{R\\nu } \\sigma ^x_{\\gamma \\nu } + {\\rm h.c.}\\right) .$ $H_{\\rm TB}$ describes nearest-neighbor hopping with amplitude $t$ .",
"$H_{\\rm F}$ is the on-site spin-dependent exchange interaction in the ferromagnet.",
"Its magnitude is $\\Delta $ and is directed along the magnetization orientation $\\hat{M}$ .",
"$H_{\\rm R}$ describes the Rashba layer: spin-orbit coupling and the broken symmetry direction $\\hat{n}$ lead to spin-dependent hopping between sites $i$ and $j$ which is aligned along the $\\hat{r}_{ij}\\times {\\hat{n}}$ direction, where ${\\hat{r}}_{ij}$ is the direction of the vector connecting sites $i$ and $j$ .",
"$\\alpha $ is the Rashba parameter (with units of energy$\\times $ length).",
"$H_{F-R}$ is the coupling between ferromagnetic and Rashba layers - it includes both spin-independent hopping and spin-dependent hopping.",
"In Eq.",
", the $F$ ($R$ ) subscript in the creation and annihilation operators labels the interfacial ferromagnet (Rashba) layer.",
"We find the model results are similar if $H_{\\rm F-R}$ includes only spin-independent hopping.",
"The Fermi energy $E_F$ is determined by the electron density $\\rho $ and temperature $T$ according to: $\\rho &=& \\frac{1}{\\left(2\\pi \\right)^2} \\int d{\\bf k}~ f\\left(\\frac{ E_{\\bf k}-E_F}{k_{\\rm B}T}\\right)$ where $k_{\\rm B}$ is the Boltzmann constant, and $f\\left(x\\right)$ is the Fermi distribution function: $f\\left(x\\right) = \\left(1+e^{x}\\right)^{-1}$ .",
"The integral is taken over the two-dimensional Brillouin zone.",
"For a given electron density $\\rho $ , Eq.",
"REF determines the Fermi energy (which generally depends on $\\hat{M}$ ).",
"The total electronic energy is then given by: $E(\\hat{M}) &=& \\frac{1}{\\left(2\\pi \\right)^2} \\int d{\\bf k}~ E_{\\bf k} f\\left(\\frac{ E_{\\bf k}-E_F}{k_{\\rm B}T}\\right)$ The default parameters we use are $\\Delta =t,~\\alpha =0.4\\times ta$ .",
"For $t=1~{\\rm eV}$ and $a=0.4~{\\rm nm}$ , this corresponds to a Rashba parameter of $0.16~{\\rm eV\\cdot nm}$ (compared to a value $\\alpha =0.38~{\\rm eV\\cdot nm}$ for BiTeI [13]).",
"We let $T=0.1~{\\rm K}$ and use a minimum of $1200^2~{\\bf k}$ -points to evaluate the integrals in Eqs.",
"REF -REF .",
"Fig.",
"REF (a) shows the total energy versus magnetic orientation for $\\hat{n}=\\hat{y}$ .",
"We observe an out-of-plane magnetic anisotropy, however its magnitude is much smaller than the demagnetization energy, which is typically on the order of $1000~{\\rm \\mu J/m^2}$ .",
"In the rest of the paper, we assume that the demagnetization energy leads to easy-plane anisotropy of the ferromagnet, so that the most relevant features of the Rashba substrate-induced anisotropy energy are confined to the $x-y$ plane.",
"The energy versus easy-plane magnetic orientation (parameterized by the azimuthal angle $\\phi $ ) is shown in Fig.",
"REF (b).",
"The anisotropy is uniaxial and favors orientation in the $\\pm \\hat{y}$ -directions.",
"This is in contrast to the bulk magnetocrystalline anisotropy of cubic transition metal ferromagnets, which has in-plane biaxial anisotropy.",
"As a point of reference for the magnitude of the calculated substrate-induced uniaxial anisotropy, we compare it to the magnetocrystalline anisotropy $E_{{\\rm MC}}$ for 2-monolayer thick film of Fe, Ni, and Co, for which $E_{\\rm MC}$ =$\\left(34,~3.5,~318\\right)~{\\rm \\mu J/m^2}$ , respectively [23].",
"Fig.",
"REF (b) shows that for the material parameters in our model, the Rashba-induced uniaxial anisotropy is larger than the magnetocrystalline anisotropy of Ni.",
"We also note that permalloy, commonly used as a thin film ferromagnet, has a vanishing magnetocrystalline anisotropy [24].",
"Figure: (a) The system energy as a function of the magnetic orientation for n ^=y ^\\hat{n}=\\hat{y}, (c) shows the same for n ^=y ^+z ^/2\\hat{n}=\\left(\\hat{y} + \\hat{z}\\right)/\\sqrt{2}.",
"(b) shows the system energy versus magnetic orientation within the x-yx-y plane, parameterized by the azimuthal angle φ\\phi for n ^=y ^\\hat{n}=\\hat{y} (d) shows the same for n ^=y ^+z ^/2\\hat{n}=\\left(\\hat{y} + \\hat{z}\\right)/\\sqrt{2}.Next we consider Rashba layer with both in-plane and out-of-plane components of $\\hat{n}$ : $\\hat{n} = \\left(\\hat{y} + \\hat{z}\\right)/\\sqrt{2}$ .",
"This is motivated in part by the fact that the symmetry is strongly broken along the $\\hat{z}$ -direction by the interface.",
"Our interest is in the influence of an out-of-plane component $\\hat{n}$ when there is also an in-plane component of $\\hat{n}$ .",
"The resulting $E(\\hat{M})$ shown in Fig.",
"REF (c).",
"As before, there is an in-plane uniaxial anisotropy, as shown in Fig.",
"REF (d), with a larger energy barrier as the previous $\\hat{n}=\\hat{y}$ case.",
"Note that the $\\hat{y}$ direction is now a hard axis.",
"In general we find that for an in-plane component of $\\hat{n}$ , the $\\hat{n}\\times \\hat{z}$ direction can be either a hard or an easy axis, depending on details of the electronic structure.",
"We define the uniaxial in-plane anisotropy energy $E_A$ as the difference in energy for $\\hat{M}=\\hat{y}$ and $\\hat{M}=\\hat{x}$ .",
"Fig.",
"REF (a) shows $E_A$ as the electron density $\\rho $ is varied, and indicates that sign of the anisotropy can change depending on the value of $\\rho $ .",
"Fig.",
"REF (b) shows the dependence of $E_A$ on $\\alpha $ for two values of the electron density $\\rho $ .",
"We find that the uniaxial anisotropy energy varies as a power of $\\alpha $ which depends on $\\rho $ (or equivalently on the band filling).",
"The origin of this dependence is discussed in more detail in the analytic model we develop next.",
"Figure: (a) E A E_A versus dimensionless electron density ρa 2 \\rho a^2.",
"(b) E A E_A versus α\\alpha for two values of ρ\\rho (ρ 1 a 2 =0.02,ρ 2 a 2 =2.1\\rho _1 a^2=0.02,\\rho _2 a^2=2.1), along with fitting to powers of α\\alpha .",
"For small ρ\\rho , the α 4 \\alpha ^4 dependence can be understood in terms of perturbation theory."
],
[
"Analytical treatment of 1-d model",
"To gain some insight into the physical origin of the uniaxial in-plane anisotropy, we consider a simplified system of a 1-d chain of atoms extending in the $x$ -direction with 2 sites in the unit cell, as shown in Fig.",
"REF (a), and take $\\hat{n}=\\hat{y}$ .",
"The Hamiltonian is the same as Eqs.",
"REF -.",
"We compute $E(k_x)$ in a perturbation expansion of the spin-orbit parameter $\\alpha $ , then determine the total energy as a function of $\\alpha $ and $\\hat{M}$ .",
"The lowest order term in $\\alpha $ is of the form $E(k_x)\\propto \\alpha k_x M_z$ .",
"This can be understood as a simple magnetic exchange interaction between the F and R sites: spin-orbit coupling leads to effective magnetic field on R along the $z$ -direction.",
"This effective magnetic field is proportional to $\\alpha k_x$ and is exchange coupled to the effective magnetic field on the F site (see Fig.",
"REF (a)).",
"The linear-in-$k_x$ term in $E(k_x)$ shifts the energy bands downward by an amount proportional to the square of the coefficient multiplying $k_x$ .",
"The total energy decreases by the same factor.",
"This finally results in a magnetic anisotropy energy which is proportional to $\\alpha ^2 M_z^2$ , describing an out-of-plane anisotropy.",
"As discussed earlier, we assume the ferromagnet is easy-plane and are therefore interested in the magnetic anisotropy within the plane.",
"In-plane anisotropy in $E(k)$ appears only at 2nd order in $\\alpha $ .",
"Taking $M_z=0$ , we find $E(k)$ takes a simple form in the limits that $\\alpha \\ll \\Delta \\ll 2t$ and $ka\\ll 1$ .",
"We present the result for the lowest energy band: $E\\left(k_x\\right) &=& A k_x^2 + B\\alpha ^2 k_x M_y + C $ Where $A,~B,~C$ depend on $\\Delta $ and $t$ , whose precise form is not essential for this discussion [25].",
"Fig.",
"REF (b) shows the numerically computed dispersion for two orientations of the ferromagnet.",
"For $\\hat{M}=\\hat{x}$ , the energy bands are purely quadratic in $k_x$ , while for $\\hat{M}=\\hat{y}$ , the energy bands acquire a linear-in-$k_x$ component, which is of opposite sign for the two lowest energy bands.",
"In the case where only the lowest band is occupied, the total energy again depends on the square of the linear-in-$k_x$ coefficient, resulting in an in-plane anisotropy energy which is proportional to $\\alpha ^4 M_y^2$ .",
"Figure: (a) cartoon of the unit cell of a 1-d model which extends along the xx-direction.",
"The top site is ferromagnetic and the bottom site includes Rashba spin-orbit coupling.",
"The arrow along the +z ^+\\hat{z}-direction on the R site indicates the direction of the Rashba effective magnetic field (we've assumed k x >0k_x>0 in the figure).",
"The double arrow along the ±x\\pm x-direction depicts the spin direction of the spin-dependent hopping between F and R sites.",
"(b) The energy dispersion for the lower two bands for two magnetic orientations (Δ=1,α=0.5\\Delta =1,~\\alpha =0.5 for this plot).We can understand the physical origin of the dependence of the energy on $M_y$ : the spin-dependent hopping between F and R sites induces a twisted exchange interaction between the spins on these sites, which favors a noncollinear configuration in which both spins lie in the $y-z$ plane [26].",
"The effective magnetic field on the R-site is always along the $z$ -direction, so the exchange energy therefore differs when the ferromagnet is aligned in the $x$ versus $y$ direction.",
"The twisted exchange interaction energy contains a factor proportional to the effective magnetic field on R (which varies as $\\alpha k_x$ ), and a factor proportional to the spin-dependent hopping (which is linear in $\\alpha $ ) - so the energy varies as finally as $\\alpha ^2 k_x M_y$ .",
"We find that the lowest energy band of the 2-d system of the previous section also contains a linear-in-$k_x$ term which varies as $\\alpha ^2 k_x M_y$ , indicating that the physical picture developed for the 1-d system also applies for the 2-d system.",
"In the case where multiple bands are occupied, we're unable to find a closed form solution for the anisotropy energy, and find that it can vary with a power of $\\alpha $ that depends on the electron density (and corresponding Fermi level).",
"This is shown in Fig.",
"REF (b) where the uniaxial anisotropy varies as $\\alpha ^3$ for multiply filled bands (we've observed several different power-law scalings with $\\alpha $ for different system parameters).",
"Nevertheless the case of a singly occupied band is sufficient to illustrate the physical mechanism underlying the in-plane magnetic anisotropy."
],
[
"Conclusion",
"We've examined the influence of a 3-d Rashba material on the magnetic properties of an adjacent ferromagnetic layer.",
"A uniaxial magnetic anisotropy is developed within the plane of the ferromagnetic layer with easy-axis direction determined by $\\hat{n}$ .",
"Depending on material parameters, the easy-axis can be parallel or perpendicular to $\\hat{n}$ .",
"For large but realistic values of the bulk Rashba parameter of the substrate, the magnitude of the anisotropy indicates that the effect should be observable.",
"For materials in which the direction of the bulk symmetry breaking is tunable - for example in a ferroelectric 3-d Rashba material - the interfacial magnetic anisotropy offers a novel route to controlling the magnetic orientation.",
"The magnitude of the Rashba-induced anisotropy is much less than the demagnetization energy, so its influence is confined to fixing the in-plane component of $\\hat{M}$ .",
"This control would nevertheless be useful in a bilayer geometry in which electrical current flows in plane.",
"In this case the anisotropic magnetoresistance effect yields a resistance which varies as $(\\hat{J}\\cdot \\hat{M})^2$ [27], which can be utilized to read out the orientation of $\\hat{M}$ .",
"We also note recent works have utilized the reduced crystal symmetry of the substrate to achieve novel directions of current-induced spin-orbit torques [28].",
"This indicates that the symmetry of the substrate can influence the nonequilibrium properties of the magnetization dynamics, in addition to modifying the equilibrium magnetic properties, as studied in this work."
],
[
"Acknowledgment",
"J. Li acknowledges support under the Cooperative Research Agreement between the University of Maryland and the National Institute of Standards and Technology Center for Nanoscale Science and Technology, Award 70NANB10H193, through the University of Maryland."
]
] | 1605.05739 |
[
[
"Cosmological Simulations of Dwarf Galaxies with Cosmic Ray Feedback"
],
[
"Abstract We perform zoom-in cosmological simulations of a suite of dwarf galaxies, examining the impact of cosmic-rays generated by supernovae, including the effect of diffusion.",
"We first look at the effect of varying the uncertain cosmic ray parameters by repeatedly simulating a single galaxy.",
"Then we fix the comic ray model and simulate five dwarf systems with virial masses range from 8-30 $\\times 10^{10}$ Msun.",
"We find that including cosmic ray feedback (with diffusion) consistently leads to disk dominated systems with relatively flat rotation curves and constant star formation rates.",
"In contrast, our purely thermal feedback case results in a hot stellar system and bursty star formation.",
"The CR simulations very well match the observed baryonic Tully-Fisher relation, but have a lower gas fraction than in real systems.",
"We also find that the dark matter cores of the CR feedback galaxies are cuspy, while the purely thermal feedback case results in a substantial core."
],
[
"Introduction",
"Isolated dwarf galaxies are commonly observed in the local Universe, and provide an interesting way to test models of galaxy formation and evolution.",
"Below a stellar mass limit of about $10^{10}$ M$_\\odot $ (or rotation velocity of about 100 km/s), such systems are largely star-forming systems except when they become satellites of larger halos [18], [5].",
"Naively, one might expect such systems to be easy to understand, as they have small or non-existent bulges and are free from uncertainties related to AGN feedback.",
"However, detailed comparisons with cosmological models have encountered two kinds of problems.",
"The first problem has to do with the baryonic content of the halos – simply put, galaxy formation must become increasingly less efficient as the halo mass decreases, for standard $\\Lambda $ CDM to match observations.",
"It has long been recognized that this is implied by the different slopes of the low-mass end of the dark matter halo mass function and the faint end of the luminosity function [73].",
"More recent work has sharpened this mismatch with both semi-analytic models [24], [2], [37] and simulations [64], [20], [57].",
"A closely related problem is the \"missing\" dwarf satellites around Milky-Way mass halos [40], [63].",
"A second issue has to do with the inner structure of dwarf galaxies – in particular the relation between the halo mass and the rotational velocity near the core.",
"Dark matter only simulations predict quite cuspy profiles with relatively high dark matter densities in the center, while observations generally find cored profiles [39], [54], [10], [70].",
"A related version of this problem has recently been highlighted among the dwarf satellites as the \"too big to fail\" problem, which highlights the paucity of satellites with rotation velocities in the 30-70 km/s range [4].",
"Solutions to both of these problems in an astrophysical context require efficient feedback mechanisms that can blow out a large amount of gas, suppress star formation, bring down the baryonic fraction, and possibly reduce the central dark matter density of lower mass dwarfs.",
"It has long been clear that the low potential depth of dwarf systems makes them particularly susceptible to stellar feedback [12]; however, modeling feedback from supernovae is numerically challenging [42].",
"A variety of numerical feedback mechanisms have been tested in different papers.",
"For example, by suppressing cooling immediately after a star formation event to counteract the artificially enhanced radiative losses [21], [23], simulated galaxies produced less massive bulges.",
"Supernovae feedback which injects momentum directly in to the surrounding gas has also been shown to drive mass-loaded outflows [45].",
"This has been combined with other physical processes – for example, [56] combined supernovae feedback with photoionization from ultraviolet background, and produced dwarf galaxies with high mass-to-light ratios.",
"More recent simulations have made progresses in reproducing the observed properties using subgrid models with strong feedback [13], [28], [58].",
"For example, [62] showed that they could reproduce the star formation main sequence relation at $z = 0$ .",
"Zoom-in simulations afford more resolution, and have shown much recent promise in reproducing the stellar mass content of dwarf satellites [20], [22], [28], [43].",
"[6] found that energetic supernovae feedback and subsequent tidal stripping has a significant effect on reducing the dense core of Milky Way mass galaxies Here we explore the impact of a specific physical feedback mechanism on dwarf galaxies, namely dynamical cosmic ray (CR) feedback.",
"CRs are high-energy, quickly diffusing particles accelerated by shock waves.",
"They have a wide range of energies, with the largest contribution to the energy density coming from those particles with energies around a few GeV.",
"The energy density of CRs is comparable to the magnetic energy and turbulent energy in galaxies [36].",
"In this paper, we present our cosmological simulations of dwarf galaxies with CR feedback.",
"We use the Enzo code, with an implementation of CR feedback by [51].",
"This adopts a relatively simple two-fluid model that provides an extra pressure term and also allows for diffusion of the CRs [14].",
"In [51] and [52], we applied this model to the cosmological simulation of a Milky Way mass galaxy.",
"Here we examine the effect of CR feedback and diffusion on dwarf galaxies with halo masses in the range of 8 to $30 \\times 10^{10}$ M$_\\odot $ .",
"In section , we describe the simulation model, the feedback scheme, initial conditions, and the selection of halos for further simulations.",
"In section REF , we choose the best set of parameters by comparing the simulation result of a single system with varying CR physics.",
"Then we apply this model to a series of dwarf galaxies, with halo mass spanning from 8 to $30 \\times 10^{10}$ M$_\\odot $ in section REF , and we evaluate the results by comparing to observed scaling relations.",
"Finally, in Section , we discuss our results.",
"Figure: Edge-on views of the same halo modeled with different CR physics.",
"From left to right, the panels show a simulation with no CRs, followed by runs with CR diffusion coefficients of 0, 3×10 27 3\\times 10^{27}, 1×10 28 1\\times 10^{28}, 3×10 28 3\\times 10^{28} cm 2 ^2s -1 ^{-1}.",
"From top to bottom, the panels show the projected (density-weighted) gas density, stellar density, and CR energy density.",
"Each panel represents a 30 kpc ×30\\times 30 kpc area."
],
[
"Method",
"We begin with a brief introduction to our hydrodynamics method, and then describe the two-fluid model used to model the CRs, and finally outline the cosmological initial conditions."
],
[
"Numerical code: ",
"Our simulations use the open-source code Enzo, which is a three-dimensional, Cartesian, grid-based hydrodynamics code that includes adaptive mesh refinement [7], [46], [30], [8].",
"We include a non-equilibrium chemical model following H, H$^+$ , He, He$^+$ , He$^{++}$ , and include radiative cooling from all the above chemical species as well as metal cooling computed in a lookup table as described in [61].",
"We include a metagalactic radiative background as computed in [25].",
"The star formation recipe of Enzo is based on the criteria presented in [9] and described in detail in [8].",
"Briefly: when a parcel of dense gas is identified to be Jeans unstable and collapsing, some of its gas may be converted into a star particle with some efficiency.",
"For a more detailed discussion of the star formation (and thermal-only) feedback scheme used in this paper, see [30] – in this work we adopt an efficiency of star formation per free fall time of 1% and an energetic feedback efficiency of $e_{\\rm SN} = 3 \\times 10^{-6}$ of the stellar rest mass energy."
],
[
"Cosmic Ray Feedback Scheme",
"We use a model for CR feedback and diffusion as implemented, tested and described in [51].",
"The model uses a two-fluid method [14], [15], [32].",
"It treats the high-energy particles as an ideal gas with $\\gamma = 4/3$ .",
"Anisotropic CR pressure and the dynamical impact of magnetic fields are assumed to be subdominant.",
"The fluid equations for the thermal gas stay as usual, except that the momentum of the gas is affected by the pressure of both the gas and CR components.",
"In addition, we optionally include isotropic diffusion of the cosmic rays.",
"Here we simply highlight the additional equation that describes the evolution of the CR energy density $\\epsilon _\\mathrm {CR}$ : $\\begin{split}\\partial _t \\epsilon _\\mathrm {CR} + \\nabla \\cdot (\\epsilon _\\mathrm {CR} \\textbf {\\textit {u}}) =& - P_\\mathrm {CR} (\\nabla \\cdot \\textbf {\\textit {u}}) \\\\& + \\nabla \\cdot (\\kappa _\\mathrm {CR} \\nabla \\epsilon _\\mathrm {CR}) + \\Gamma _\\mathrm {CR}\\end{split}$ where $\\textbf {\\textit {u}}$ is the fluid velocity, P$_\\mathrm {CR}$ is the pressure from CR, $\\kappa _\\mathrm {CR}$ is the CR isotropic diffusion coefficient, and $\\Gamma _\\mathrm {CR}$ is the source term for CR.",
"Except for the isotropic diffusion, the CRs are assumed to be tightly coupled to the gas through a tangled magnetic field (which is not explicitly simulated).",
"While this model is approximate, simulations including magnetic fields and/or anisotropic diffusion have also been shown to drive winds with similar mass-loading factors [27], [19].",
"We also do not include any non-adiabatic losses in the CRs; in reality, CRs lose energy primarily through collisions with the thermal plasma; this is generally subdominant to adiabatic cooling in our simulations except for runs with zero (or low) diffusion coefficients, which are not realistic in any case.",
"A more detailed discussion of this point and the shortcomings of the model in general can be found in [51] and [52].",
"When a Type II SNe is formed, it will eject some of its mass and energy back into the surrounding cells.",
"In our model, we assume that some of the energy is in thermal form and some accelerated by blast waves (generally not resolved in the grid).",
"The fraction of energy that is deposited to thermal gas and CR is 0.7 and 0.3 respectively; this CR fraction is somewhat larger than canonical values but within suggested ranges [71], [16], [33].",
"Moreover, we assume a relatively steep initial mass function (our chosen $e_{\\rm SN}$ corresponds to 1 SN per 185 M$_\\odot $ masses of stars produced), and hence a relatively low SN energy injection rate, so the CR energy produced per solar mass of stars formed is within typical estimates.",
"We adopt these values for consistency with previous work [30], [51], where we also explored the impact of varying them."
],
[
"Initial Conditions and Halo Selection",
"We are interested in modeling a number of dwarf galaxies forming out of cosmological initial conditions.",
"Therefore, we use MUSIC [26] to set up the initial density and velocities fields.",
"This is done using the cosmological parameters from the best fit Planck 2013 results [47], i.e.",
"H$_0$ = 67.11 km/s/Mpc, $\\Omega _\\Lambda = 0.6825 $ , $\\Omega _\\mathrm {m} = 0.3175 $ , $\\Omega _\\mathrm {b} = 0.0490 $ ${\\sigma _8 = 0.8344}$ .",
"All the simulations start at $z =100$ and stop at the present day.",
"Table: Simulation parameters for fixed halo, varying CR runs.First we perform a dark matter only run in a (20 Mpc/h)$^3$ box and identify halos that are around $10^{10}$ to $10^{11}$ M$_\\odot $ .",
"All of these halos become a little more massive in the final runs, listed in Table REF .",
"To study the impact of CR feedback, we select relatively isolated halos so that the physical properties of the dwarfs are not affected by nearby massive galaxies.",
"In particular, all the selected systems are at least 0.5 Mpc away from any halos that are more massive than 0.1 times of the target halo.",
"Then we trace the dark matter particles currently in the halo to their positions at the beginning of the simulation.",
"The selected halos are re-simulated at high-resolution with baryons and adaptive mesh refinement (AMR) is performed in the cubical region that encloses all the dark matter particles from the beginning.",
"Our root grid is 128$^3$ and we use three additional levels in the initial conditions for a particle mass of M$_{DM} = 8.3\\times 10^5$ M$_\\odot $ .",
"During the simulation, additional refinement is added whenever the gas (dark matter) mass in a cell exceeds $6.0\\times 10^5$ ($3.3\\times 10^6$ ) M$_\\odot $ .",
"We allow up to 10 levels of refinement, resulting in a smallest cell size of 227 pc.",
"In order to explore the impact of some of the uncertain parameters in our model, simulations with different CR physics, including a no CR run, and CR runs with diffusion coefficients of 0, $3\\times 10^{27}$ , $1\\times 10^{28}$ , and $3\\times 10^{28}$ cm$^2$ s$^{-1}$ are done on the most massive of the selected halos.",
"The results of these simulations are listed in Table REF .",
"We analyze the results from these runs (see below) and choose the coefficient of $3\\times 10^{28}$ cm$^2$ s$^{-1}$ to carry on the simulations with the other selected halos.",
"As discussed below, this choice is somewhat arbitrary, but it is consistent with estimates from recent GALPROP models [49], [1] and with observational measurements [65], [66].",
"It is also found to be consistent with galactic gamma ray emission in [52].",
"The properties of all selected halos simulated with this choice of parameters are listed in Table REF .",
"Table: Simulated dwarf galaxy propertiesFirst, we focus on comparisons of runs with different CR physics, using the most massive of the halos.",
"Second, we discuss the physical properties of the suite of five halos with fixed CR physics.",
"Third, we compare our simulation results to observations, specifically the Baryonic Tully-Fisher relation and HI stellar mass relation.",
"Analysis has been carried out using the yt package [67]."
],
[
"Impact of Varying CR Physics",
"We choose an isolated halo of around $3 \\times 10^{11}$ M$_\\odot $ and study the effect of CR feedback by varying the CR physics and fixing all the other parameters (See Table REF for the simulation parameters).",
"Figure REF shows edge-on images of the gas density, stellar density and CR energy density of the dwarf galaxies produced in each run at $z=0$ , focusing on just the center of the halo where the stellar system forms.",
"Since H1-0 and H1-1 are not disk-shaped, we show them in the direction of their total angular momentum within 10 kpc radius.",
"The run without any CR feedback (H1-0; left-most panels) forms a diffuse, irregular cloud.",
"We find that the energetic thermal feedback is actually remarkably effective in regulating the formation of dense, star-forming clumps.",
"This behavior is quite different than that seen in high-mass galaxies using identical star formation and feedback prescriptions [30] and is more like the behavior observed in very low-mass galaxies [60], [59].",
"We explore this point in more detail in Section REF Moving on to the simulations with cosmic-rays, we see in Figure REF that H1-1, with no CR diffusion, has an almost spherically symmetric gas distribution.",
"This occurs because when the CRs are tied to the gas, they provide complete pressure support and rotational support is unimportant.",
"Clearly this is unrealistic (and probably violates gamma-ray emission from dwarf irregulars – see [52]).",
"Figure: Comparison of the star formation rate history in simulations of the same halo but different CR physics, as noted in the legend.The other runs, which have non-zero CR diffusion, all produce disk-shaped galaxies, generally becoming thinner as the diffusion coefficient increases.",
"H1, which has the highest diffusion coefficient of $3\\times 10^{28}$ cm$^2$ s$^{-1}$ , results in the most extended disk.",
"H1-1, which has the lowest non-zero diffusion coefficient, results in a smaller disk which is surrounded by denser gas.",
"The gas density and CR energy density immediately surrounding the disk also decreases as $\\kappa $ increases.",
"The CRs escape more easily, both because they diffuse out more rapidly and because they drive stronger outflows [51], and therefore they provide less pressure support, decreasing the weight of gas they can support in the diffuse (not-rotating) halo.",
"Figure: Edge-on views of the simulated galaxies with different halo masses but a fixed diffusion coefficient (κ=3×10 28 \\kappa = 3 \\times 10^{28} cm 2 ^2 s -1 ^{-1}.",
"The plot show halos H1, H2, H3, H4, H5 from left to right, and gas density, stellar density, and CR energy density from top to bottom.",
"Each panel represents a 30 kpc ×30\\times 30 kpc area at z=0z=0.Since all the runs with non-zero CR diffusion produce disk galaxies, we can examine the rotational velocity of the gas in the disk.",
"Figure REF shows the rotational velocities of the HI gas in H1-2, H1-3, and H1.",
"To compute these rotation velocity curves, we look at the actual rotational velocities of the HI gas (rather than a simple dynamical measure such as $(GM/r)^{1/2}$ ).",
"To do this, we adopt $10^{20}$ cm$^{-2}$ as the lowest detectable HI surface density in order to make an approximate comparison to observations – the results are not strongly sensitive to this threshold, but disks from runs with low diffusion coefficients start to pick up some CR-pressure supported gas and show falling rotation curves.",
"In particular, in H1-2, the run with a diffusion coefficient of 3 $\\times $ 10$^{27}$ cm$^2$ s$^{-1}$ is affected by this issue: the circular velocity peak at 160 km/s at about 1.5 kpc radius and falls quickly beyond that radius.",
"As can be seen from its edge-on plot, the disk in H1-2 has a small radius, but its surrounding gas is relatively dense, which makes the HI detectable radius even larger than the disk radius.",
"In that run, outside the disk, the gas is actually supported in part by the pressure of the CRs, which explains why the circular velocity curve drops so quickly after the peak.",
"From this figure, we see that, at least for the two higher diffusion coefficient runs, the circular velocity curves are relatively flat, in reasonable agreement with observed systems.",
"For example, in the run with the highest diffusion coefficient (H1), the circular velocity is relatively flat at about 140 km/s out to 6-7 kpc.",
"Next, we examine the impact of CR pressure on the star formation rate.",
"This is shown in Figure REF for all the runs of halo H1 with different CR physics.",
"For all of our more realistic runs, with non-zero diffusion coefficient, the SFRs are quite similar: an early rapid rise follow by a nearly constant rate that falls slowly from 1 to 0.5 M$_\\odot $ /yr.",
"The run without diffusion shows a rate which is a factor of two larger, due to the higher gas densities in its core, while the thermal-only feedback is lower by nearly an order of magnitude and shows considerably more variation with time – consistent with the bursty feedback seen in the gas distribution."
],
[
"Varying Halo Mass with Fixed CR Physics",
"In this section, we fix the CR diffusion coefficient to be $3\\times 10^{28}$ cm$^2$ s$^{-1}$ and study the impact on dwarf galaxies with different halo masses, varying the total mass from about $8 \\times 10^{10}$ M$_\\odot $ to $3 \\times 10^{11}$ M$_\\odot $ .",
"We examine how CR feedback behaves in different scales and also use this suite of runs to compare with the observational scaling relations in Section REF .",
"All of these runs produce dwarf galaxies with extended disk features; Figure REF shows edge-on images of these galaxies.",
"The halo masses decrease from left to right: H1 to H5 as described in Table REF .",
"There is a fairly clear trend of decreasing disk extent in the stellar distribution, with the larger halos having more extended disks.",
"H2 is a slight outlier in this trend with a truncated disk, particularly in the gas and CR components.",
"In addition, the circumgalactic medium is less dense (with a lower CR energy density) for the lower mass halos.",
"Figure: Measured circular velocity of the neutral hydrogen in the disk for halos H1 to H5.",
"As in Figure , we plot the circular velocity only out to a limiting HI surface density of 10 20 10^{20} cm -2 ^{-2}.We plot the rotational velocities of the gas for these five galaxies in Figure REF .",
"Within our adopted HI detection limit, the curves of H3, H4, and H5, stay flat at about 100 km/s, just above 80 km/s, and just below 80 km/s, respectively.",
"The velocities are consistent with their masses, and although we do not make a detailed comparison the curves are similar to observed rotation curves – in particular, they do not posses large bulges with centrally peaked rotation curves, as seen in many simulations without effective feedback.",
"H1 has a higher rotational velocity (not surprising given its larger mass) and shows some mild features at 1 and 5 kpc.",
"On the other hand, the curve of H2 looks quite different: it is highly peaked, and falls sharply at about 4 kpc (again because of partial CR pressure support).",
"As can be seen from the edge-on plot of H2, this dwarf galaxy does have a dense core, larger than the other dwarfs.",
"Figure REF shows the star formation histories of these same halos.",
"There is a clear progression of increasing SFR with increasing mass.",
"Despite some bumps and wiggles, the overall histories are remarkably flat, with each profile showing typically a factor of 2-3 variation over 12-13 Gyr.",
"There is a general trend for slowly falling star formation rate with time, particularly evident in H3.",
"This occurs because of the relatively tight self-regulation – feedback acts to decrease the gas density, slowing star formation.",
"Figure: Star formation histories of our simulated halos H1 to H5 (all with fixed CR diffusion coefficient)."
],
[
"Comparison with Observed Scaling Relations",
"Dwarf galaxies are less efficient in star formation than normal galaxies, and the baryon fraction of galaxies decreases as the total mass goes down [38].",
"In this section, we compare our simulation results with a range of observational probes, including the baryonic Tully-Fisher relation, as well as observed gas-to-star ratios and inferred stellar mass-to-halo mass relations.",
"We begin with the observed relation between the disk mass (the combined stellar and gas masses) and the flat part of the rotational velocity of the disk – the baryonic Tully-Fisher relation.",
"Figure REF shows the simulation results from varying halo mass with fixed CR physics overlaid on the fitted line of the Baryonic Tully-Fisher relation from [38].",
"They provide a fit for disk galaxies in the range of 20 to 350 km/s given by M$_{b} = x\\ \\mathrm {log}\\ {V}_{c} + {A}$ with $x = 4.0$ and $A =1.65$ .",
"In Figure REF , the solid line represents their best fit, and the grey area represents one standard deviation of $M_b$ .",
"We estimated the scatter in the observed $M_b$ as the average for spiral and gas disk galaxies in Table 2 of [38].",
"For the simulations, we determined the baryonic mass as the sum of the stellar and gas masses, with the gas mass calculated as $1/0.74$ times the HI mass.",
"Although the code evolves the HI density, it is underestimated in dense gas due to the lack of self-shielding.",
"To correct for this, the HI density is estimated as follows: when the total H number density $n_{\\rm H} > 2\\times 10^{-2}$ cm$^{-3}$ , we assume that the neutral fraction is nearly unity and take $n_{\\rm HI} = n_{\\rm H}$ ; otherwise we assume the HI density is zero.",
"This is approximately consistent with radiative transfer calculations [50] that take into account the metagalactic radiation background (a local radiation background would decrease the neutral gas fraction even lower, and so our bounds here are conservative).",
"Both the stellar mass and HI mass are calculated within the same HI column density limit we used to determine the radial extent for the rotation curves ($10^{20}$ cm$^{-2}$ ) and extend above and below the plane by 2 kpc.",
"The circular velocity is calculated as the average rotation velocity of the gas at a radius of 2.2 scale lengths of the stellar surface density.",
"To determine the scale length, we fit the stellar surface density curve with a straight line from 2 kpc to 10 kpc.",
"Our simulation results all lie within the grey area and scatter around the observed relation.",
"Although we do not have a significantly large numerical sample to make a detailed comparison, these preliminary results appear to be in good agreement, indicating that the baryonic content of our galaxies is consistent with that observed.",
"We note that we cannot include the purely thermal simulation (H1-0) on this plot as the systems has no systematic rotation.",
"Figure: Comparison between the five dwarf simulations (H1-H5) run with our standard diffusion coefficient and the baryonic Tully-Fisher relation from observations.",
"The solid line is the fit from .",
"The red crosses are simulation data.",
"The grey area shows the region of one sigma error in M b M_b.",
"The dashed line is taken from and indicates the relation between the halo circular velocity and the cosmic baryon fraction of the halo.The baryonic Tully Fisher relation compares the galaxies' baryonic mass with a measure of its total mass.",
"We would also like to compare the two main components of baryons: neutral hydrogen and stars.",
"Figure REF shows the ratio of the HI to stellar components as a function of the stellar mass, all quantities which are directly observable.",
"The observed relation found in [17] (their Eq.",
"5) can be approximated as: $\\begin{split}\\frac{M_\\mathrm {HI}}{3.36 \\times 10^9 \\rm {M}_\\odot } = &\\left(\\frac{M_\\star }{3.3 \\times 10^{10} \\rm {M}_\\odot }\\right)^{0.19}\\\\ & \\times \\left[1+\\left(\\frac{M_\\star }{3.3 \\times 10^{10} \\rm {M}_\\odot }\\right)^{0.76}\\right]\\end{split}$ In Figure REF , the solid line shows this relation.",
"We also estimate the observed scatter around this relation by taking a variation of +/- 0.5 log M$_\\mathrm {HI}$ /M$_\\star $ , based on a visual inspection of Figure 1 of [17]; this is shown as a grey area in our Figure REF .",
"The ratio from the simulations is also plotted, again using the stellar and neutral hydrogen masses within our adopted HI surface density limit, and +/- 2 kpc in vertical extent.",
"As can be seen from Figure REF , all of our simulation results are below the observed relation, indeed most lie below the estimated scatter in the observations (the grey band).",
"This demonstrates that our simulation produces consistently lower HI fraction than observed.",
"There is some indication that the simulated relation shows a rising gas fraction with decreasing stellar mass; however, the sample size is too small to be sure.",
"This mismatch in the gas to star ratio, by approximately a factor of five, when combined with the agreement in the baryonic Tully-Fisher relation (which measures the total baryonic content), implies that our galaxies contain the correct amount of baryons, but are overly efficient in their conversion of gas to stars.",
"One alternative way to examine just the stellar content of our systems is based on the dark matter abundance matching measurements in [17], which provides a relation between the stellar mass and halo mass under the assumption that there is a monotonic relation between halo mass and stellar mass.",
"Indeed, if we compare our results to their inferred M$_*$ -M$_{\\rm halo}$ relation, we find that our $M_*$ values are systematically too large by factors of 2-5, consistent with the idea that the division between stellar and gas mass is incorrect in these simulations.",
"Finally, we note that these relations also imply that the simulations disagree with the observed Kenicutt-Schmidt relation between surface gas density and star formation surface density.",
"Figure: Comparison between this work and the relation of HI mass and stellar mass inferred from observations.",
"The solid line is derived in .",
"The grey area shows an estimated scatter of +/- 0.5 log around the observed relation in the vertical direction.",
"The red crosses are the simulation results."
],
[
"Dark Matter Profiles",
"There are a number of observational indications that dwarf galaxies do not contain the cuspy dark-matter profiles predicted in dark-matter only simulations [11], [69], [3], [4], [55], [34].",
"A number of simulations with strong feedback have found that it is possible for the feedback to decrease the dark matter density in the center of halos [41], [20], [48], [44].",
"Therefore, we examine our simulation to see the impact of thermal vs. CR feedback on the core properties of the systems.",
"Figure REF shows the dark matter density profile of our most massive halo in the simulations with different CR physics.",
"Interestingly, all of the CR runs show very similar results, indicating that although they produce different distributions for the gas and stars, the presence of CR feedback leads to similar dark matter core properties.",
"In all cases, the profile is quite steep, with an inner density profile between $r^{-1}$ and $r^{-2}$ (similar to dark-matter only runs).",
"On the other hand, the run without CR diffusion shows a clear core in the dark matter with a core radius of several kpc.",
"The cored dark matter profile in the no-CR case might be due to the large and rapid motions of the gas, which can disturb the dark matter distribution [48] and is consistent with the observed asymmetric gas distribution and the bursty nature of the feedback in the no-CR run.",
"Figure: Dark matter profiles of simulations with different CR physics.We have performed a preliminary investigation into the impact of CRs on the formation and evolution of dwarf galaxies with rotation velocities in the 70-140 km/s range using cosmological simulations.",
"This complements similar work [53] which looked at a higher mass galaxy (with a rotation velocity above 200 km/s).",
"As in that work, we found that the inclusion of CR feedback had a strong impact on the gas and stellar properties.",
"In both cases, CR feedback led to robust outflows and rotationally supported disks with relatively flat rotation curves, but only if CR diffusion was included.",
"A comparison to observations showed good agreement with the baryonic Tully-Fisher relation, indicating that the total disk mass was reasonable.",
"This mass, converted into a fraction of the total halo mass, is well below the cosmic value, indicating that feedback was important in generating mass-loaded winds.",
"On the other hand, a comparison of the gas content of the simulated galaxies against observations showed that too much gas had been converted to stars.",
"In addition, the gas depletion times of our galaxies (computed as $M_{\\rm gas}/\\dot{M}_{\\rm SFR}$ ), are a few Gyr, well below observed values for low-mass isolated dwarfs [29].",
"Our star formation and feedback model is approximate, with a number of free parameters required to describe physics that is not fully modeled.",
"In section REF , we explored the impact of some of these for halo H1.",
"Remarkably, we find that the CR diffusion rate has a relatively small impact on the stellar mass produced (see Table REF ), or the gas mass.",
"The exception was the run with $\\kappa = 0$ , which produced a substantially higher stellar mass and did not produce a disk (this combination does not generate winds).",
"We also carried out an additional run of halo H1 with the standard diffusion rate (and other parameters), but a reduced CR energy efficiency (10% instead of 30%) and find that relative to the fiducial run, it produces 50% more stars.",
"The resulting disk rotates marginally faster such that the system stays close to the observed baryonic Tully-Fisher relation.",
"However, the HI to gas mass ratio is even lower the fiducial run, resulting in an an even stronger disagreement with the observed relation.",
"In principle, a higher CR efficiency might bring this into agreement with observations, but that would disagree with local observational and theoretical work on CR acceleration efficiencies.",
"Another parameter which can be varied is the star formation efficiency per free-fall time, which we took to be 1%.",
"We carried out another simulation much like H1 but with a star formation efficiency four times lower (with $\\kappa = 3 \\times 10^{28}$ cm$^2$ s$^{-1}$ ; this run was not included in figures and tables above).",
"This had almost no effect on the resulting stellar mass or gas mass, consistent with the idea that the disk is in a state of self-regulation – indeed, we verified visually that the lower efficiency resulted in a higher density gas disk and hence essentially the same net star formation rate at $z=0$ (since the star formation rate increases as $\\rho ^{1.5}$ ).",
"There are, of course, other parameters to vary, such as the IMF, the fraction of energy in SN, as well as other uncertainties in the subgrid model.",
"In addition, there is some indication from H2 and the high-mass halo explored in [53] that an additional form of feedback is required in high-density regions – possibly radiative heating or pressure.",
"Finally, we briefly discuss our results in comparison to [68], who recently carried out simulations of dwarf galaxies with cosmic rays.",
"The implementation differs somewhat in that they adopted a model in which CRs stream along magnetic field lines and heat the gas (rather than isotropically diffuse, as in our model).",
"Although they do not simulate cosmological volumes, their halo mass which most closely compares to ours produces approximately similar results: robust cosmic-ray driven outflows with a modest mass-loading factor.",
"Although we are unable to compute a mass-loading factor, the large stellar mass to halo mass we find implies a relatively low mass-loading factor."
],
[
"Feedback efficiency of the purely thermal run",
"Finally, we turn to a discussion of the thermal-only feedback run (H1-0).",
"Interestingly, the feedback in this case was actually more effective than the CR+thermal feedback, in contrast to the result for the higher mass halo with the same code and CR parameters discussed in [53].",
"One possibility is that the purely thermal feedback is more efficient than in more massive galaxies because the shallower potential well results in low gas densities, lowering the radiative cooling rate and permitting blast waves from SN feedback to accelerate the gas.",
"We explore the relative importance of cooling and heating in Figure REF , which shows a two-dimensional distribution of radiative cooling times and a measure of the possible supernova heating rate for all gas within 30 kpc of the center of this halo as $z=0$ .",
"We compute the SN heating rate in a way similar to that in [60], where we estimate the timescale required for our thermal energy injection mechanism to heat a cell to $T_6 = 10^6 K$ (the precise temperature selected does not affect our conclusions): $t_{\\rm heat} = \\frac{3}{2} \\frac{M_{\\rm cell} k T_6}{\\mu m_H} \\frac{t_f}{e_{\\rm SN} M_* c^2}$ where $M_{\\rm cell}$ is the gas mass cell in the cell, $\\mu \\approx 1$ , $M_* = 10^5$ M$_\\odot $ , our typical stellar particle mass, $t_f = 10$ Myr, the timescale over which a star particle returns its supernova energy to the ISM in our model, and $e_{\\rm SN} = 3 \\times 10^{-6}$ , as defined in the methods section.",
"We stress that not all cells are being heated at this rate, but it is the rate that a single star particle would typically heat its local gas and so is a good measure of the timescale over which feedback heating acts.",
"From this figure, we see that nearly all the gas in the simulation lies above the black line indicating equality.",
"This demonstrates that the gas densities are sufficiently low that pure thermal feedback is efficient in heating the gas and driving outflows – we focus on density because the cooling rate is proportional to the density squared.",
"A close examination of this figure indicates that a substantial amount of gas lies close to the line of equality (in fact, this gas is in the disk), and so we expect that the larger potential depths of higher mass halos will push gas beyond the line into a regime where purely thermal supernovae feedback cannot heat the gas.",
"This is consistent with the finding that our dwarf galaxy simulations do drive outflows while higher mass (e.g.",
"Milky Way mass) systems do not.",
"Note that this statement depends on details of the simulations (e.g.",
"see definition of $t_{\\rm heat}$ ) and so may be a reflection of numerical models rather than physical quantities.",
"This demonstrates that purely thermal feedback is sufficient to drive feedback but it does not answer the question why adding cosmic rays decreases the bursty outflows, as seen in Figure REF .",
"The answer appears to be that cosmic rays actually increase the pressure in the diffuse circumgalactic gas surrounding galaxies.",
"The purely thermal run has a (thermal only) CGM pressure at a radius of 10 kpc of approximately $10^{-14}$ erg cm$^{-3}$ , while even the CR run with the highest diffusion coefficient (H1), has a (mostly CR) pressure of nearly $10^{-13}$ erg cm$^{-3}$ .",
"This larger pressure resists the impulsive outflows driven by bursts of supernovae in the disk (although it does not appear to eliminate the slower gas outflows driven by the CR pressure gradient).",
"Finally, we note a consequence of the effective thermal only feedback, which is that it resulted in bursty star formation at a low-level; and was so effective that the resulting stellar system was “hot\" – that is, it was supported by velocity dispersion rather than rotation (an issue that has been raised in the context of other simulations – see the discussion in [72]."
],
[
"Summary",
"We simulate a series of dwarf galaxies with zoom-in cosmological simulations, and study the influence of CR feedback.",
"First we compare the outcome by varying the CR physics in different runs of the same galaxy halo.",
"The different CR physics include feedback without CR, CR feedback with zero diffusion, CR feedback with $\\kappa _\\mathrm {CR}$ equal to $3 \\times 10^{27}$ , $1 \\times 10^{28}$ , and $3 \\times 10^{28}$ cm$^2$ s$^{-1}$ .",
"Then we fixed the CR diffusion at $\\kappa _\\mathrm {CR} = 3 \\times 10^{28}$ cm$^2$ s$^{-1}$ and simulated five dwarf galaxies with different masses, ranging from 8 to 30 $\\times 10^{10}$ M$_\\odot $ .",
"We summarize the results of this work as follows: Adding CRs and some realistic level of diffusion consistently produced thin, extended disk galaxies with nearly flat rotation curves, in contrast to a case with purely thermal feedback, which produced a hot stellar systems.",
"The star formation rate in the CR systems was relatively constant, compared to the lower (and burstier) SFR in the purely thermal feedback model.",
"Simulations of our five dwarf galaxies with different masses but the same CR model matched well with the observed baryonic Tully-Fisher relation.",
"However, the five galaxies' neutral hydrogen fractions were always lower than what is observed, by a factor of about five and the stellar content was larger than observed (but the total baryonic content matched observations).",
"Simulations with CR feedback produce cuspy dark matter profiles, while our purely thermal feedback case resulted in a cored dark matter profile.",
"These results indicate that the impact of cosmic rays in dwarf galaxies is potentially quite important, a conclusion which is in general agreement with previous work [31].",
"In detail, the results are somewhat dependent on the physical model, but broadly speaking CRs with diffusion produce dwarf galaxies which are rotationally supported and have relatively flat rotation curves.",
"This result appears to be quite robust (and occurs even though 70% of the feedback energy is in the form of thermal energy in these models) and is probably due to the smooth pressure provided by CRs (along with the outflows they drive).",
"The baryonic content of the dwarf systems with CRs is in broad agreement with observations.",
"On the other hand, there are two areas of disagreement with observations in these models: (a) the detailed census of baryons in each system (stars vs. gas) is much too heavily weighted to the stellar side, and (b) the presence of dark matter cores may be a problem (although CR pressure may systematically support the gas at small radius and so bias the observed measures of dark matter on small scales).",
"Clearly, more work is required to explore and refine these results.",
"For example, it would be useful to explore more realistic models including magnetic fields, anisotropic CR diffusion, and the impact of field-CR streaming instabilities."
],
[
"Acknowledgments",
"We thank the anonymous referee for a useful report which improved the clarity of the paper.",
"We would like to thank Cameron Hummels, Mordecai-Mark Mac Low and Mary Putman for useful discussion related to this work.",
"We acknowledge financial support from NSF grants AST-1312888, and NASA grants NNX12AH41G and NNX15AB20G, as well as computational resources from NSF XSEDE, and Columbia University's Yeti cluster.",
"Computations described in this work were performed using the publicly-available Enzo code (http://enzo-project.org), which is the product of a collaborative effort of many independent scientists from numerous institutions around the world.",
"Their commitment to open science has helped make this work possible."
]
] | 1605.06115 |
[
[
"General formulation of coupled radiative and conductive heat transfer\n between compact bodies"
],
[
"Abstract We present a general framework for studying strongly coupled radiative and conductive heat transfer between arbitrarily shaped bodies separated by sub-wavelength distances.",
"Our formulation is based on a macroscopic approach that couples our recent fluctuating volume--current (FVC) method of near-field heat transfer to the more well known Fourier conduction transport equation.",
"We apply our technique to consider heat exchange between aluminum-zinc oxide nanorods and show that the presence of bulk plasmon resonances can result in extremely large radiative heat transfer rates (roughly twenty times larger than observed in planar geometries), whose interplay with conductive transport leads to nonlinear temperature profiles along the nanorods."
],
[
"3 General formulation of coupled radiative and conductive heat transfer between compact bodies Weiliang Jin Department of Electrical Engineering, Princeton University, Princeton, NJ 08544, USA Riccardo Messina Laboratoire Charles Coulomb, Université de Montpellier and CNRS, Montpellier, France Alejandro W. Rodriguez Department of Electrical Engineering, Princeton University, Princeton, NJ 08544, USA We present a general framework for studying strongly coupled radiative and conductive heat transfer between arbitrarily shaped bodies separated by sub-wavelength distances.",
"Our formulation is based on a macroscopic approach that couples our recent fluctuating volume–current (FVC) method of near-field heat transfer to the more well known Fourier conduction transport equation.",
"We apply our technique to consider heat exchange between aluminum-zinc oxide nanorods and show that the presence of bulk plasmon resonances can result in extremely large radiative heat transfer rates (roughly twenty times larger than observed in planar geometries), whose interplay with conductive transport leads to nonlinear temperature profiles along the nanorods.",
"Radiative heat transfer (RHT) between objects held at different temperatures can be many orders of magnitude larger in the near field (short separations $d \\ll $ thermal wavelength $\\lambda _T =\\hbar c/k_\\mathrm {B} T$ ) than for far-away objects [1], [2], [3], [4], [5].",
"Recently, we showed that that the interplay of near-field RHT and conduction in planar geometries can dramatically modify the temperature and thermal exchange rate at sub-micron separations [6].",
"Such strongly-coupled conduction–radiation (CR) phenomena are bound to play a larger role in situations involving structured materials, where RHT can be further enhanced [7], [8], [9], [10], [11], [12] and modified [13], [14], [15], [16], and in on-going experiments exploring nanometer scale gaps, where the boundary between conductive (phonon- and electron-mediated) and radiative transport begins to blurr [17], [18].",
"We present a general CR framework that captures the interplay of near-field RHT and thermal conduction along with the existence of large temperature gradients in arbitrary geometries.",
"We show that under certain conditions, i.e.",
"materials and structures with separations and geometric lengthscales in the nanometer range, RHT can approach and even exceed conduction, significantly changing the stationary temperature distribution of heated objects.",
"Our approach is based on a generalization of our recent fluctuating volume-current (FVC) formulation of electromagnetic (EM) fluctuations, which when coupled to the more standard Fourier heat equation describing conductive transport at macroscopic scales, allows studies of CR between arbitrary shapes, thereby generalizing our prior work with slabs [6].",
"As a proof of concept, we consider an example geometry involving aluminum-zinc oxide (AZO) nanorods separated by vacuum gaps, which exhibits more than an order of magnitude enhancement in RHT compared to planar slabs, and hence leads to even larger temperature gradients.",
"We find that while RHT between thin slabs is primarily mediated by surface modes, resulting in linear temperature gradients, the presence of bulk nanorod resonances leads to highly distance-dependent nonlinear temperature profiles.",
"Coupled radiative and conductive diffusion processes in nanostructures are becoming increasingly important [19], [18], with recent works primarily focusing on the interplay between thermal diffusion and external optical illumination such as laser-heating of plasmonic structures [20], [21], [22], [23], [24].",
"On the other hand, while it is known that conduction has a strong influence on RHT experiments [25], [26], the converse has thus far been largely unexplored because RHT is typically too small to result in appreciable temperature gradients [27], [28], [29].",
"However, our recent work [6] suggests that such an interplay can be significant at tens of nanometer separations and in fact may already have been present (though overlooked) in recent experiments involving planar systems [30], [31], [32], [33].",
"Moreover, since planar structures are known to exhibit highly suboptimal RHT rates [7], we expect even stronger interplays in more complex geometries, such as metasurfaces [34], hyperbolic metamaterials [35], [11], or lattices of metallic antennas [8], [7].",
"Figure: (a) Schematic illustration of two square lattices of nanorods(labelled aa and bb) of thickess tt, period Λ\\Lambda ,cross-sectional area l×ll\\times l, and separation dd, whosetemperature distribution and energy exchange is mediated by bothconductive ∇·[κ(𝐱)T(𝐱)]\\nabla \\cdot [\\kappa (\\mathbf {x})T(\\mathbf {x})] andradiative H(𝐱,𝐱 ' )H(\\mathbf {x},\\mathbf {x}^{\\prime }) heat transfer.",
"(b) Totalradiative heat transfer spectrum Φ(ω)\\Phi (\\omega ) between two AZOnanorods (solid lines) of thickness t=500t=500 nm and cross-sectionalarea A=l 2 A=l^2, separated by d=20d=20 nm and held at temperaturesT a(b) =800(300)T_{a(b)}=800(300) K. The spectrum is shown for differentcross-sections l={10,20}l=\\lbrace 10,20\\rbrace nm (blue and red lines) and in thelimit l=∞l = \\infty , corresponding to two planar slabs.",
"(c) Spatialradiative heat flux in nanorod aa for the case l=20l=20 nm,corresponding to the (i) first, (ii) second, and (iii) SPP plasmonresonances, respectively, annotated in (b).Formulation.— In what follows, we describe a general formulation of coupled CR applicable to arbitrary geometries.",
"Consider a situation involving two bodies (the same framework can be extended to multiple bodies), labelled $a$ and $b$ , subject to arbitrary temperature profiles and exchanging heat among one other, shown schematically in Fig.",
"REF (a).",
"Neglecting convection and considering bodies with lengthscales larger or of the order of their phonon mean-free path, in which case Fourier conduction is valid, the stationary temperature distribution satisfies: $\\nabla \\cdot [\\kappa (\\mathbf {x})\\nabla T(\\mathbf {x})]+\\int \\mathrm {d}^3\\mathbf {x}^{\\prime }\\,H(\\mathbf {x},\\mathbf {x}^{\\prime })=Q(\\mathbf {x})$ where $\\kappa (\\mathbf {x})$ and $Q(\\mathbf {x})$ describe the bulk Fourier conductivity and presence of some external heat source at $\\mathbf {x}$ , respectively, and $H(\\mathbf {x},\\mathbf {x}^{\\prime })$ denotes the radiative power per unit volume from $\\mathbf {x}^{\\prime }$ to $\\mathbf {x}$ .",
"Our ability to compute $H(\\mathbf {x},\\mathbf {x}^{\\prime })$ in full generality hinges on an extension of a recently introduced FVC method that exploits powerful EM scattering techniques [36] to enable fast calculations of RHT under arbitrary geometries and temperature distributions.",
"The starting point of this method is the volume-integral equation (VIE) formulation of EM, in which the scattering unknowns are 6-component polarization currents $\\xi $ in the interior of the bodies coupled via the homogeneous $6\\times 6$ Green's function $\\Gamma $ of the intervening medium [36].",
"Given two objects described by a susceptibility tensor $\\chi (\\mathbf {x})$ and a Galerkin decomposition of the induced currents $\\xi =\\sum _i x_i b_i$ , with $\\lbrace b_i\\rbrace $ denoting localized basis functions throughout the objects ($i$ is the global index for all bodies), the scattering of an incident field due to some fluctuating current-source $\\sigma =\\sum _is_i b_i$ can be determined via solution of a VIE equation, $x+s=Ws$ , in terms of the unknown and known expansion coefficients $\\lbrace x_i\\rbrace $ and $\\lbrace s_i\\rbrace $ , respectively, where $W^{-1}_{i,j}=\\langle b_i,(I+i\\omega \\chi G) b_j\\rangle $ and $G_{i,j}=\\langle b_{i},\\Gamma \\star b_{j}\\rangle $ are known as VIE and Green matrices [36].",
"Previously, we exploited this formalism to propose an efficient method for computing the total heat transfer between any two compact bodies [36], based on a simple voxel basis expansion (uniform discretization).",
"The solution of (REF ) requires an extension of the FVC method to include the spatially resolved heat transfer between any two voxels, which we describe below.",
"Consider a fluctuating current-source $\\sigma _{\\alpha }=s_{\\alpha }b_{\\alpha }$ at $\\mathbf {x}_a=b_{\\alpha }$ in a body $a$ .",
"Such a “dipole” source induces polarization–currents $\\xi _{\\beta }=x_{\\beta }b_{\\beta }$ and EM fields $\\phi _{\\beta }$ throughout space in body $b$ (and elsewhere), such that the heat flux at $\\mathbf {x}_b=b_{\\beta }$ is given (by Poynting's theorem) by: $\\Phi (\\omega ;\\mathbf {x}_a\\rightarrow \\mathbf {x}_b)=\\frac{1}{2}\\langle \\operatorname{Re}\\left( \\xi _{\\beta }^{*}\\phi _{\\beta } \\right)\\rangle $ where “$\\langle \\ldots \\rangle $ ” denotes a thermodynamic ensemble average.",
"Expressing the polarization–currents and fields in the localized basis $\\lbrace b_{\\alpha }\\rbrace $ , and exploiting the volume equivalence principle to express the field as a convolution of the incident and induced currents with the vacuum Green's function (GF), $\\phi =\\Gamma \\star (\\xi +\\sigma )$ , one finds that (REF ) can be expressed in a compact, algebraic form involving VIE matrices: $\\Phi (\\omega ;\\mathbf {x}_a\\rightarrow \\mathbf {x}_b)&=\\frac{1}{2}\\langle \\operatorname{Re}\\left\\lbrace x_{\\beta }^{*}[G(x+s^\\alpha )]_{\\beta }\\right\\rbrace \\rangle \\nonumber \\\\&=\\frac{1}{2} \\langle \\operatorname{Re}\\left\\lbrace (x+s^\\alpha )_{\\beta }^{*}[G(x+s^\\alpha )]_{\\beta }\\right\\rbrace \\rangle \\nonumber \\\\&=\\frac{1}{2}\\langle \\operatorname{Re}\\left[ (Ws^\\alpha )_{\\beta }^{*}(GWs^\\alpha )_{\\beta }\\right]\\rangle \\nonumber \\\\&=\\frac{1}{2} \\operatorname{Re}\\left[ D_{\\alpha ,\\alpha }W^{\\dagger }_{\\alpha ,\\beta }(GW)_{\\beta ,\\alpha }\\right]$ where $s^{\\alpha }$ is a vector that is zero everywhere except at the $\\alpha $ th element, denoted by $s_{\\alpha }$ , and ${D}_{\\alpha ,\\beta }=\\langle s_{\\alpha }^{*}s_{\\beta }\\rangle =\\int \\int d^3\\mathbf {x}\\,d^3\\mathbf {y}\\, b_{\\alpha }^\\ast (\\mathbf {x}) \\langle \\sigma (\\mathbf {x})\\sigma ^\\ast (\\mathbf {y}) \\rangle b_{\\beta }(\\mathbf {y})$ is a real, diagonal matrix encoding the thermodynamic and dissipative properties of each object [36] and described by the well-known fluctuation–dissipation theorem, $\\langle \\sigma _i(\\mathbf {x},\\omega ) \\sigma _j^\\ast (\\mathbf {y},\\omega ) \\rangle =\\frac{4}{\\pi } \\omega \\operatorname{Im}\\epsilon (\\mathbf {x},\\omega ) \\Theta (T_{\\mathbf {x}})\\delta (\\mathbf {x}-\\mathbf {y})\\delta _{ij}$ , where $\\Theta (T)=\\hbar \\omega /[\\mathrm {exp}(\\hbar \\omega /k_bT)-1]$ is the Planck distribution.",
"It follows then that the heat flux emitted or absorbed at a given position $\\mathbf {x}_a$ , the main quantity entering (REF ) through $\\int \\mathrm {d}^3\\mathbf {x}^{\\prime }\\,H(\\mathbf {x},\\mathbf {x}^{\\prime })=\\int \\mathrm {d}\\omega \\,\\Phi (\\omega ;\\mathbf {x})$ , is given by: $\\Phi (\\omega ;\\mathbf {x}_a)&=\\int _{V_{b}} \\mathrm {d}^3\\mathbf {x}_b\\left[ \\Phi (\\omega ;\\mathbf {x}_b\\rightarrow \\mathbf {x}_a)-\\Phi (\\omega ;\\mathbf {x}_a\\rightarrow \\mathbf {x}_b)\\right]\\nonumber \\\\&=\\frac{1}{2}\\text{Tr }_{\\beta | b_\\beta \\in V_b} \\operatorname{Re}\\left[D_{\\beta ,\\beta }W^{\\dagger }_{\\beta ,\\alpha }(GW)_{\\alpha ,\\beta }-(\\alpha \\leftrightarrow \\beta )\\right]\\nonumber \\\\&=\\frac{1}{2}\\operatorname{Re}\\left[\\underbrace{GWD^bW^{\\dagger }}_{\\Phi _\\mathrm {a}}-\\underbrace{DW^{\\dagger }P^bGW}_{\\Phi _\\mathrm {e}}\\right]_{\\alpha ,\\alpha }$ Here, $P^{a(b)}$ denotes the projection operator that selects only basis functions in $a(b)$ , such that $D^b=P^bDP^b$ is a diagonal matrix involving only fluctuations in object $b$ .",
"Furthermore, the first (second) term in (REF ) describe the absorbed (emitted) power in $\\mathbf {x}_a$ , henceforth denoted via the subscript “a(e)”.",
"Figure: (a) Temperature profile along the zz coordinate of a nanorod(solid lines) when it is heated from one side to a temperature of800 K, and is separated from an identical, constant- anduniform-temperature nanrod held at T=300T=300 K on the other side, by agap size d=20d=20 nm.",
"The nanorods have cross-sectional widthl=10l=10 nm and thicknesses t=500t=500 nm, and are made up of AZO withresults shown for multiple values of the doping concentration{2,6,11} wt %\\lbrace 2,6,11\\rbrace \\mathrm {wt}\\% (blue, red, and black lines).",
"Also shownare the temperature profiles of slabs (dashed lines) of the samethickness (corresponding to the limit l→∞l \\rightarrow \\infty ).",
"(Inset:)Temperature distribution throughout the nanorod in the case of11 wt %11\\mathrm {wt}\\%.",
"(b) Temperature profiles of nanorods of widthl=20l=20 nm under various separations d={5,10,20,30}d=\\lbrace 5,10,20,30\\rbrace nm (black,blue, red, and green lines).",
"(c.inset:) The ratio of total radiativeheat flux for nanorods of width l=20l=20 nm to that of the slabs as afunction of dd, in the presence (red dots) or absence (black dots)of temperature gradients induced by conduction and radiationinterplay, with the flux value shown in (c), for nanorods (red) andslabs (blue), also in the presence (solid lines) or absence (dashedlines) of temperature gradients induced by the interplay ofconduction and radiation.Equation REF is a generalization of our previous expression for the total heat transfer between two arbitrary inhomogeneous objects [36] in that it includes both the spatially resolved absorbed and emitted power throughout the entire geometry.",
"In Ref.",
"polimeridis2015fluctuating, we showed that the low-rank nature of the GF operator enables truncated, randomized SVD factorizations and therefore efficient evaluations of the corresponding matrix operations.",
"We find, however, that in this case, the inclusion of the absorption term does not permit such a factorization, except in special circumstances.",
"In particular, writing down the two terms separately by expanding into the subspace spanned by each object, we find: $\\Phi _\\mathrm {a}(\\omega ;\\mathbf {x}_a)&=\\frac{1}{2}\\operatorname{Re}\\left[G^{ab}W^{bb}D^{bb}W^{ab\\dagger }+G^{aa}W^{ab}D^{bb}W^{ab\\dagger }\\right] \\\\\\Phi _\\mathrm {e}(\\omega ,\\mathbf {x}_a) &= -\\frac{1}{2}\\left[ \\operatorname{Re}(D^{aa}W^{ba\\dagger } G^{ba} W^{aa})\\right.",
"\\nonumber \\\\&\\hspace{50.58878pt}\\left.+ D^{aa} W^{ba\\dagger } \\operatorname{sym}(G^{bb}) W^{ba} \\right]_{\\alpha ,\\alpha }$ with $X^{ij} = P^i X P^j$ denoting the sub-block of matrix $X$ connecting basis functions in object $i$ to object $j$ , and $\\operatorname{sym}X =\\frac{1}{2}(X+X^\\dagger )$ denoting the symmetric part of $X$ .",
"Equation , describing emission, can be evaluated efficiently because the matrices $G^{ba}$ and $\\operatorname{sym}{G^{bb}}$ are both low rank ($\\ell \\ll N$ ) [36], in which case they can be SVD factorized to allow fast matrix multiplications.",
"It follows that the total heat transfer, i.e.",
"the trace of (), can also be computed efficiently.",
"Unfortunately, the second term of (REF ) involves both the symmetric and anti-symmetric parts of $G^{aa}$ , the latter of which is full rank.",
"More conveniently, detailed balance dictates that $\\Phi (\\omega ;\\mathbf {x}_b\\rightarrow \\mathbf {x}_a)=\\Phi (\\omega ;\\mathbf {x}_a\\rightarrow \\mathbf {x}_b)$ whenever $T(\\mathbf {x}_a) = T(\\mathbf {x}_b)$ , which implies that $\\operatorname{Re}\\left[ M_{\\beta ,\\beta }W^{\\dagger }_{\\beta ,\\alpha }(GW)_{\\alpha ,\\beta }\\right]=\\operatorname{Re}\\left[M_{\\alpha ,\\alpha }W^{\\dagger }_{\\alpha ,\\beta }(GW)_{\\beta ,\\alpha }\\right]$ , where $M_{\\alpha ,\\alpha }=\\operatorname{Im}\\varepsilon (\\mathbf {x}_{\\alpha },\\omega )$ is a real, diagonal matrix encoding the dissipative properties of the bodies, leading to the following modified expression for the absorption rate: $\\Phi _\\mathrm {a}(\\omega ;\\mathbf {x}_a)&=\\frac{1}{2}\\left[\\operatorname{Re}(M^{aa}W^{ba\\dagger }K^{bb}G^{ba}W^{aa})\\right.\\nonumber \\\\&\\hspace{50.58878pt}\\left.+M^{aa}W^{ba\\dagger }\\operatorname{sym}(K^{bb}G^{bb})W^{ba}\\right]_{\\alpha ,\\alpha }$ where the real and diagonal matrix $K_{\\alpha ,\\alpha }=D_{\\alpha ,\\alpha }/M_{\\alpha ,\\alpha }$ is only relevant to the Plank function $\\Theta (T(\\mathbf {x}_{\\alpha }),\\omega )$ .",
"Noticeably, the symmetrized operator in the second term is full rank except whenever the temperature of object $b$ is close to uniform, in which case $\\operatorname{sym}(K^{bb} G^{bb}) \\approx K^{bb} \\operatorname{sym}G^{bb}$ .",
"While solution of (REF ) is feasible, it remains an open problem to find a formulation that allows fast evaluations of the spatially resolved absorbed power under arbitrary temperature distributions.",
"Given (REF ), one can solve the coupled CR equation in any number of ways [37].",
"Here, we exploit a fixed-point iteration procedure based on repeated and independent evaluations of (REF ) and (REF ), converging once both quantities approach a set of self-consistent steady-state values.",
"Equation REF is solved via a commercial, finite-element heat solver whereas (REF ) is solved through a free, in-house implementation of our FVC method [36].",
"While the above formulation is general, below we explore the computationally convenient situation in which object $b$ is kept at a constant, uniform temperature by means of a carefully chosen thermal reservoir, such that the absorbed power in object $a$ can be computed efficiently via (REF ).",
"Furthermore, absorption can be altogether ignored whenever one of the bodies is heated to a much larger temperature than the other (as is the case below).",
"The power emitted by $a$ (the heated object), obtained via (), turns out to be much more convenient to compute, since the time-consuming part of the scattering calculation can be precomputed independently from the temperature distribution and stored for repeated and subsequent evaluations of (REF ) under different temperature profiles.",
"Results.— As a proof of principle and to gain insights into coupled CR effects in non-planar objects, we now apply the above method to a simple geometry consisting of two metallic nanorods of cross-sectional widths $l$ and thickness $t$ ; in practice, both for easy of fabrication and to obtain even larger RHT [25], such a structure could be realized as a lattice or grating, shown schematically in Fig.",
"REF (a).",
"However, for computational convenience and conceptual simplicity, we restrict our analysis to the regime of large grating periods, in which case it suffices to consider only the transfer between nearby objects.",
"The strongest CR effects generally will arise in materials that exhibit large RHT, e.g.",
"supporting surface–plasmon polaritons (SPP) in the case of planar objects, and low thermal conductivities, including silica, sapphire, and AZO, whose typical thermal conductivities $\\sim 1$ W/m$\\cdot $ K. In the following, we take AZO as an illustrative example [38], [39].",
"To begin with, we show that even in the absence of CR interplay, the RHT spectrum and spatial RHT distribution inside the nanorods differ significantly from those of AZO slabs of the same thickness.",
"Figure REF (b) shows the RHT spectrum $\\Phi (\\omega )$ per unit area $A=l^2$ between two AZO nanorods (with doping concentration $11\\mathrm {wt}\\%$ [39]) of length $t=500$ nm and varying widths $l=\\lbrace 10,20,\\infty \\rbrace $ nm (blue solid, red solid, and black dashed lines), held at temperatures $T_{a(b)}=800(300)$ K and vacuum gap $d=20$ nm.",
"The limit $l\\rightarrow \\infty $ corresponds to the slab-slab geometry already explored [6], in which case the $\\Phi (\\omega )$ exhibits a single peak occuring at the SPP frequency $\\approx 3\\times 10^{14}$ rad/s.",
"The finite nature of the nanorods results in additional peaks at lower frequencies, corresponding to bulk/geometric plasmon resonances (red and blue solid lines) that provide additional channels of heat exchange, albeit at the expense of weaker SPP peaks, leading to a roughly 20-fold enhancement in RHT compared to slabs.",
"More importantly and well known, such structured antennas allow tuning and creation of bulk plasmon resonances in the near- and far-infrared spectra (much lower than many planar materials) that can more effectively transfer thermal radiation.",
"The contour plots in Fig.",
"REF (i–iii) reveal the spatial RHT distribution $\\Phi (\\omega ,\\mathbf {x})$ (in arbitrary units) at three separate frequencies $\\omega =\\lbrace 0.4,0.8,2.3\\rbrace \\times 10^{14}$ rad/s, corresponding to the first, second, and SPP resonances, respectively.",
"As expected, the highest-frequency resonance is primarily confined to the corners of the nanorod surface (becoming the well-known SPP resonance in the limit $l \\rightarrow \\infty $ ), with the fundamental and intermediate resonances have flux contributions stemming primarily form the bulk.",
"As we now show, such an enhancement results not only results in larger temperature gradients but also changes the resulting qualitative temperature distribution.",
"Figure REF (a) shows the temperature profile along the $z$ direction for the nanorod geometry of Fig.",
"REF (a), with width $l=10~$ nm and gap size $d=20$ nm, obtained via solution of (REF ).",
"For the purpose of generality, we show results under various doping concentrations $\\lbrace 2,6,11\\rbrace \\mathrm {wt}\\%$ (green, red, and black solid lines), corresponding to different SPP frequencies and bandwidths [39].",
"In particular, we consider a situation in which the boundary I of nanorod $a$ is kept at $T_\\text{I}=800$ K while the entire nanorod $b$ is held at $T_b=300$ K (through contact with a room-temperature reservoir), and assume an AZO thermal conductivity of $\\kappa =1$ W/m$\\cdot $ K [38].",
"The temperature along the $x$ –$y$ cross section is nearly uniform (due to the faster heat diffusion rate along the smaller dimension) and therefore only shown in the case of $11\\mathrm {wt}\\%$ (inset).",
"In all scenarios, the temperature gradient is significantly larger for nanorods (solid lines) than for slabs ($t \\rightarrow \\infty $ , dashed lines), becoming an order of magnitude larger in the case of $6\\mathrm {wt}\\%$ due to its larger SPP frequency compared to the peak Planck wavelength near 800 K. Furthermore, while slabs exhibit linear temperature profiles (RHT is dominated by surface emission [1]), the bulky and de-localized nature of emission in the case of nanorods results in nonlinear temperature distributions.",
"Figure REF (b) shows the temperature profile at various separations $d=\\lbrace 5,10,20,30\\rbrace $ nm (black, blue, red, and green lines) for nanorods of width $l=20$ nm and $11\\mathrm {wt}\\%$ , illustrating the sensitive relationship between the degree of CR interplay and gap size.",
"Notably, while the RHT and therefore temperature gradients increase as $d$ decreases, the profile becomes increasingly linear as the geometry approaches the slab–slab configuration.",
"The transition from bulk- to surface-dominated RHT and the increasing impact of the latter on conduction and vice versa is also evident from Fig.",
"REF (c).",
"The figure shows the radiative flux rate $H\\times d^2$ as a function of $d$ for slabs (black lines) of thickness $t=500$ nm and nanorods (red lines) of equal thickness and width $l=20$ nm, either including (solid lines) or excluding (dashed lines) CR interplay (with the latter involving uniform temperatures).",
"While the RHT between bodies of uniform temperatures is shown to scales as $1/d^2$ (dashed lines), the temperature gradients induced by CR interplay in the case of nanorods begins to change the expected powerlaw behavior at $d\\approx 15$ nm; the same occurs for slabs but at much shorter $d\\lesssim 5$ nm.",
"These differences are further quantified on the inset of the figure, which shows the ratio of the RHT rate between the two objects as a function of $d$ .",
"While the ratio remains almost a constant for uniform-temperature objects (black dots), it decreases visibly when considering CR interplay (red dots).",
"As shown in Ref.",
"riccardo2016strongly, in the limit $d\\rightarrow 0$ , RHT will asymptote to a constant (not shown) rather than a diverge.",
"Concluding remarks.— As experiments continue to push toward larger RHT by going to smaller vacuum gaps or through nanostructuring, accurate descriptions of CR interplay and associated effects will become increasingly important [18], [17].",
"Future work along these directions could focus on extending our work to periodic structures, which could potentially exhibit much larger RHT and hence CR effects.",
"Acknowledgements.— This work was supported by the National Science Foundation under Grant no.",
"DMR-1454836 and by the Princeton Center for Complex Materials, a MRSEC supported by NSF Grant DMR 1420541."
]
] | 1605.05708 |
[
[
"Effect of the polydispersity of a colloidal drop on the drying induced\n stress as measured by the buckling of a floating sheet"
],
[
"Abstract We study the stress developed during the drying of a colloidal drop of silica nanoparticles.",
"In particular, we use the wrinkling instability of a thin floating sheet to measure the net stress applied by the deposit on the substrate and we focus on the effect of the particle polydispersity.",
"In the case of a bidisperse suspension, we show that a small number of large particles substantially decreases the expected stress, which we interpret as the formation of lower hydrodynamic resistance paths in the porous material.",
"As colloidal suspensions are usually polydisperse, we show for different average particle sizes that the stress is effectively dominated by the larger particles of the distribution and not by the average particle size."
],
[
"Floating films",
"Polydimethylsiloxane (PDMS, sylgard 184, Dow Corning) is mixed and degassed at $0^\\circ $ C to slow down the reaction and to ensure a reproducible viscosity.",
"A silicon wafer is first coated with a 10% wt.",
"aqueous solution of polyvinyl alcohol (PVA, Sigma-Aldrich, $M_w = 10$ kDa).",
"The coating is dried until a uniform color is obtained.",
"Then, the PDMS is spin coated on this wafer and placed in an oven at $65^\\circ $ C for 10 hours.",
"The edge of the coated PDMS is removed to obtain a circular film of diameter 87 mm.",
"A petri dish, painted in black to enhance the contrast on the picture, is partially filled with deionized water.",
"The wafer is dipped in this bath to dissolve the sacrificial layer of PVA and to detach the PDMS film.",
"Once the PDMS film is fully detached and floats at the surface, the wafer is removed.",
"We used a white light spectrometer (OceanOptics USB2000+ used with a LS-1-LL tungsten halogen light source) to measure the film thicknesses, which range between $[15.8,63.9]$ $\\mu $ m. The presence of PVA lowers the surface tension of the bath, which is $\\gamma _b = 55\\pm 5$ mN/m."
],
[
"Visualisation",
"Our experiments of controlled drying of drops are performed in a transparent glove box, which has humidity controller presented in Fig.",
"REF .",
"The relative humidity is set to 50% in all of our experiments.",
"A grid is placed in the glove box on the top of the sample and a light source is placed behind a light diffuser to obtain an even illumination.",
"The deformation of the membrane is visualised by the distortion of the pattern made by the grid.",
"The time evolution of the system is recorded by a Nikon camera (D7100) mounted with a 18-55mm Nikon objective (Fig.",
"REF ).",
"Figure: Detailed experimental setup.A grid is placed between the light and the sample.The reflection of the grid on the bath interface allows the visualisation of the deformation of the elastomer."
],
[
"Dialysis",
"Each suspension was dialysed to ensure that they have the same properties (ionic strength and pH).",
"Dialysis bags (12-14 kDa, Fisherbrand) are partially filled with the colloidal solutions.",
"A solution denoted s1 is prepared from deionized water, the pH is raised to 9.5 by the addition of NaOH and the ionic strength is adjusted by the addition of 5 mM of NaCl.",
"To wash these suspensions, the bags are plunged in large baths of solution s1 for 10 days and baths are replaced twice.",
"Then, the diluted suspensions are concentrated by osmotic stress.",
"This second step consists in preparing a bath in which polyethylene glycol (Sigma-Aldrich, 35 kDa) is dissolved in a solution s1.",
"Mass fractions are measured and the suspensions are diluted with a solution s1 to reach a mass fraction $\\Phi _0=0.25$ , which corresponds to a volume fraction $\\phi _0=0.15$ ."
],
[
"Particle size",
"The particle size distributions are characterized by Dynamic Light Scattering (DLS) with a Malvern Zetasizer Nano ZS instrument.",
"Samples are diluted to a volume fraction of $\\approx 0.001$ .",
"We denote $n(a)$ the distribution function of particle number and $v(a)$ , the distribution function in volume, and satisfy $\\int _0^\\infty n(a^{\\prime }){\\rm d}a^{\\prime }=1\\\\\\int _0^\\infty v(a^{\\prime }){\\rm d}a^{\\prime }=1$ The average particle radius $a_0$ is defined as $a_0 = \\int _0^\\infty a^{\\prime } n(a^{\\prime }) {\\rm d} a^{\\prime }.$ The particle radius $a^\\star $ , representative of the large particles of the distribution is defined as follow.",
"We consider 30% of the tail of the particle distribution $v(a)$ , i.e.",
"$a>a_c$ with $\\int _0^{a_c} v(a^{\\prime }) {\\rm d} a^{\\prime } = 0.7.$ The radius $a^\\star $ is the average particle size of the distribution tail, i.e $a^\\star = \\int _{a_c}^\\infty a^{\\prime } n(a^{\\prime }) {\\rm d} a^{\\prime }.$ The suspension properties are summarized in Table REF .",
"Figure: NO_CAPTION$n(a)$ for the number of particles.",
"Table: NO_CAPTION tableProperties of the colloidal suspensions of silica nanoparticles.",
"For all suspensions, the pH is 9.5, the ionic strength is $[$ NaCl$]$ = 5 mM and the volume fraction is $\\phi _0=0.15$ ."
],
[
"S3. Particle organization",
"For blend suspensions, we define the particle volume ratio as $\\Gamma _{\\cal V} = \\frac{{\\cal V}_{\\ell }}{{\\cal V}_{s} + {\\cal V}_{\\ell }},$ and the particle number ratio as $\\Gamma _{\\cal N} = \\frac{{\\cal N}_{\\ell }}{{\\cal N}_{s} + {\\cal N}_{\\ell }} = \\left[ 1 + \\frac{a_\\ell ^3}{a_s^3} \\left(\\Gamma _{\\cal V}^{-1} - 1\\right) \\right]^{-1},$ where $a_s$ and $a_\\ell $ are the average radii of small and large particles, respectively.",
"SEM images of dried drops of colloidal suspensions were captured with the FEI Quanta 200 FEG Environmental-SEM in low vacuum mode.",
"Samples are prepared in the same conditions as presented in section S1.",
"To image cross-sections, a razor blade is gently pressed on the top of the deposit to nucleate a crack along the diameter.",
"The two pieces are then separated and placed on a sample holder.",
"In this section, we present additional images.",
"In Fig.",
"REF , the cross section of a dried deposit containing only SM particle is imaged and we see that the particle size is smaller than the resolution we could obtain.",
"In Fig.",
"REF , we present cross sections images for $\\Gamma _{\\cal N}=0.05$ %.",
"As for the pictures in the main text, we do not see a segregation of the large particles across the thickness.",
"Also, similar particle organisations are obtained on the top and near the contact line.",
"In Fig.",
"REF , we imaged the surface of the deposits for different $\\Gamma _{\\cal N}$ where we notice the presence of the large particles as their concentration increases.",
"In our experiment, we can estimate the Péclet numbers for small (${\\rm Pe}_s$ ) and large (${\\rm Pe}_\\ell $ ) particles.",
"The Péclet number is defined as ${\\rm } Pe = \\frac{v_e h}{D}$ where $v_e$ is the evaporation speed, $h$ the characteristic thickness and $D$ the diffusion coefficient.",
"We use the Stokes-Einstein diffusion coefficient $D=k_B T/(6\\pi \\eta a)$ , where $k_B$ is the Boltzmann constant, $T$ is the absolute temperature, $\\eta =10^{-3}$ Pa$\\cdot $ s is the solvent viscosity and $a$ the particle radius.",
"For our bidisperse suspensions and considering the average particle size of each species, we have ${\\rm Pe}_s\\approx 1.6$ and ${\\rm Pe}_\\ell \\approx 21$ for $h \\approx 1$ mm.",
"Fortini et al.",
"evidenced numerically and experimentally a segregation effect in drying films with ${\\rm Pe} \\gg 1$ for both species (Fortini et al.",
"PRL 116, 118301 (2016)).",
"However, as shown in our SEM images, we could not observe a significant segregation of the two species in the dried film.",
"Nevertheless, as pointed by Fortini et al., the segregation is a dynamic process that is observed after a transient state.",
"The duration of this transient must be function the specific interactions between the particles and the initial volume fraction.",
"In our system, the material becomes visco-plastic at a volume fraction about $0.3$ and our initial volume fraction is $0.15$ .",
"Thus, a possible explanation is that the system cannot evolve to a segregated state.",
"This observations would deserve further investigations, which are beyond the scope of the present study.",
"Figure: Scanning-electron-microscope (SEM) images in the cross-section of a dried deposit of SM particles (Γ 𝒩 =0\\Gamma _{\\cal N}=0).The different pictures show different magnifications.Figure: Scanning-electron-microscope (SEM) images in the cross-section of a dried deposit for a number ratio Γ 𝒩 =0.05\\Gamma _{\\cal N}=0.05%.Images are taken at different locations and the scale bars represent 1 μ\\mu m.Figure: Scanning-electron-microscope (SEM) images of the colloidal deposit surface after complete drying for different number ratios of particles Γ 𝒩 \\Gamma _{\\cal N}.Only the large particles can be imaged.The scale bar represents 500 nm.In our experiments, we measure the final length of the wrinkles $L_f$ .",
"Then, from models derived by Davidovitch et al.",
"[19], we calculate the inner tension $T_i$ applied by the deposit on the film.",
"Davidovitch et al.",
"predict this tension in two different domains.",
"The first domain is named Near Threshold (NT) limit and correspond to the limit of infinitesimal deformation amplitude $\\zeta $ and large bendability ($\\epsilon ^{-1} = R_i^2 T_o / {\\cal B}\\gg 1$ .",
"In that domain, the tension ratio $\\tau ^{NT} = T_i^{NT}/T_o$ is given by $\\tau ^{NT} = 1 + \\frac{L_f^2}{R_i^2},$ which corresponds to equation (1) in the Letter.",
"The tension ratios for this limit and for all our experiments are presented in Fig.",
"REF (a).",
"The second domain is the Far From Threashold (FFT) limit and it assumes that the deformation $\\zeta $ is finite and $\\epsilon \\rightarrow 0$ .",
"Therefore, the tension ratio is $\\tau ^{FFT} = \\frac{2 L_{f}}{R_i},$ which corresponds to equation (2) in the Letter.",
"Similarly, we plot this prediction for our data in Fig.",
"REF (b).",
"Figure: Tension ratio τ=T i /T o \\tau = T_i/T_o as a function of the bendability ϵ -1 \\epsilon ^{-1}.The points correspond to our experiments (a) where we used the expression from the NT limit to calculate the tension ratio τ NT =T i NT /T o =1+L f 2 /R i 2 \\tau ^{NT} = T_i^{NT}/T_o = 1 + L_f^2 / R_i^2,and (b), we used the expression from the FFT limit τ FFT =T i FFT /T o =2L f /R i \\tau ^{FFT} = T_i^{FFT}/T_o = 2 L_f/R_i.On both plots, the green band represents the Near Threshold (NT) domain predicted by Davidovitch et al.",
".In Fig.",
"REF , we also represent the NT domain predicted by Davidovitch et al.",
"[19] and we observe that both estimations are above the NT limit.",
"If the NT limit would describe our results, we expect that $T_i^{NT}$ values to be in the NT green domain, but the prediction is about at least an order of magnitude higher.",
"Thus, the estimate from the NT limit is not self-consistent.",
"Let us now consider the values deduced from the FFT limit.",
"These values are also above the NT domain (represented in green in Fig.",
"REF (b)), which is self-consistent.",
"Also, we observe that $\\tau ^{NT} \\approx \\tau ^{FFT}$ , which further validates that the NT prediction is not valid as the prediction is significantly above the NT domain.",
"Consequently, for all our measurements, we considered $T_i = T_i^{FFT}$ .",
"Let us remark that equations (REF ) and (REF ) are valid for $R_i \\ll R_o$ .",
"More general equations have been established recently [24] but they do not modify significantly the estimate of the tension in the film in our experiments, which confirms this assumption."
]
] | 1605.05915 |
[
[
"Breaking the disc-halo degeneracy in NGC 1291 using hydrodynamic\n simulations"
],
[
"Abstract We present a pilot study on the nearby massive galaxy NGC 1291, in which we aim to constrain the dark matter in the inner regions, by obtaining a dynamical determination of the disc mass-to-light ratio (M/L).",
"To this aim, we model the bar-induced dust lanes in the galaxy, using hydrodynamic gas response simulations.",
"The models have three free parameters, the M/L of the disc, the bar pattern speed and the disc height function.",
"We explore the parameter space to find the best fit models, i.e.",
"those in which the morphology of the shocks in the gas simulations matches the observed dust lanes.",
"The best-fit models suggest that the M/L of NGC 1291 agrees with that predicted by stellar population synthesis models in the near-infrared ($\\approx$0.6\\,$M_{\\odot}/L_{\\odot}$), which leads to a borderline maximum disc for this galaxy.",
"The bar rotates fast, with corotation radius $\\leq$ 1.4 times the bar length.",
"Additionally, we find that the height function has a significant effect on the results, and can bias them towards lower or higher M/L."
],
[
"Introduction",
"The distribution of dark matter in galaxies has been a source of debate in the scientific community, with one of the main sources of uncertainty being the “disc-halo degeneracy” ([15]).",
"This degeneracy arises because rotation curve decompositions depend critically on the mass-to-light ratio (M/L or $\\Upsilon $ ) which is assigned to the stellar disc component.",
"Rotation curves can be fit equally well with barely any disc contribution, or where the disc contributes the maximum amount possible – also known as the “maximum disc hypothesis” ([15], [13]).",
"There is no conclusive evidence to either prove or disprove the maximum disc hypothesis and in fact there are a number of arguments both for and against it for the Milky-Way (e.g.",
"[13]) as well as for external galaxies (e.g.",
"[2], [16], ).",
"A way to break the degeneracy between the disc and the dark matter halo is by obtaining dynamical estimates of the M/L ratio of the disc.",
"One way to do this is via the non-axisymmetries induced by the bar, which shocks the gas and leads to the well known “dust lanes” observed in barred galaxies ([10]).",
"These shocks can be reproduced in hydrodynamic simulations ([1]), and their morphology is dependent on the M/L of the disc, the bar pattern speed and the height function of the disc.",
"In this study we examine the dust lanes present in the nearby galaxy NGC 1291, and aim to reproduce their morphology using hydrodynamic gas response simulations.",
"The major assumption of this method for obtaining the M/L is that the halo is axisymmetric, and that all the non-axisymmetry in the potential is due to the stellar component."
],
[
"Comparing the models to the observations",
"There are three main free parameters in the models, and we vary these parameters in order to explore the allowed parameter space.",
"This leads to more than 300 simulations, from which the best fit models are obtained according to how well the shock loci in the models match those in the observations.",
"This is done by placing pseudo-slits perpendicular to the shocks, as shown in Figure REF .",
"The three free parameters are varied as follows: M/L.",
"To obtain the potential we use a 3.6$\\mu m$ image of NGC 1291 from the S$^4$ G survey ([14]) to which we apply a M/L.",
"We assume a fiducial value of $\\Upsilon _{3.6}$ =0.6$M_{\\odot }/L_{\\odot }$ for the 3.6$\\mu $ m band ([9], [12]), colour corrected to account for the presence of dust by combining information from the 3.6 and 4.5$\\mu $ m images ([4], [11]).",
"We then vary the M/L from 1.5 to 0.25 $\\Upsilon _{3.6}$ in steps of 0.25 and add a dark matter halo which maintains the total rotation curve flat at a value of 220 km/s (see Figure REF ) as found from the Baryonic Tully-Fisher relation ([8]).",
"Pattern speed (Lagrangian radius).",
"By varying the Lagrangian radius while keeping all the other parameters of a model constant, we effectively vary the pattern speed of the bar.",
"The Lagrangian radius is varied between 1 and 2 times the bar semi-major axis, in steps of 0.2.",
"Height function and scaleheight.",
"We vary the height function of the disc by assuming two functional forms – isothermal and exponential – and we also vary the scaleheight from $z_0$ =0.5 to $z_0$ =1.5 kpc."
],
[
"Results",
"Dynamical M/L: The best fit models obtained in this study have a M/L in the range of M/L$_{3.6}$ =0.6-0.75 which falls within the range predicted by stellar population synthesis models and other approaches (i.e.",
"$\\Upsilon _{3.6}$ =0.6M$_\\odot $ /L$_\\odot $ by e.g.",
"[9] and [12]).",
"It is important to note that our results are independent of systematic uncertainties such as the choice of IMF.",
"The maximal disc of NGC 1291: The best fit models have borderline a maximal disc on average, with the disc contributing (on average) 74% of the total rotation velocity.",
"It is worth noting that we are using the definition of Sackett (1997) for disc maximality, which examines the rotation velocity at 2.2$h_r$ ; in this context, and because the scalelength of NGC 1291 is unusually large (5.8 kpc), we are examining the rotation velocity at a radius of almost 13 kpc.",
"Therefore, even for cases where the disc is not according to this definition “maximal”, the stellar disc dominates the rotation curve in the inner regions of the galaxy.",
"The height function of the disc: It's furthermore important to note that for this galaxy, models with exponential scaleheights lead to sub maximal discs, while models with an isothermal scaleheight lead to maximal discs.",
"This indicates that the choice of height function, especially for such borderline cases, can affect whether we obtain a maximal disc or not.",
"Additionally, the scaleheight affects the results, as was already shown in previous works, such as in [6], where we showed the important effect a boxy/peanut height function can have on the models.",
"Fast rotating bar: The best fit models have fast rotating bars with $\\mathcal {R}$ $\\le $ 1.4.",
"This is consistent with observational estimates of the bar pattern speed for early type galaxies ([5] and [3]).",
"The results are also in agreement with theoretical values ()."
]
] | 1605.05754 |
[
[
"Applications of fidelity measures to complex quantum systems"
],
[
"Abstract We revisit the fidelity as a measure for the stability and the complexity of the quantum motion of single and many-body systems.",
"Within the context of cold atoms, we present on overview of applications of two fidelities which we call static and dynamical fidelity, respectively.",
"The static fidelity applies to quantum problems which can be diagonalized since it is defined via the eigenfunctions.",
"In particular, we show that the static fidelity is a highly effective practical detector of avoided crossings characterizing the complexity of the systems and their evolutions.",
"The dynamical fidelity is defined via the time-dependent wave functions.",
"Focussing on the quantum kicked rotor system, we highlight a few practical applications of fidelity measurements in order to better understand the large variety of dynamical regimes of this paradigm of a low-dimensional system with mixed regular-chaotic phase space."
],
[
"Introduction",
"Generally it is very useful to have a working tool for the characterization of systems and their dynamical evolution.",
"An experimental technique with great value for practical measurement applications is the echo method, in particular known as spin or Hahn echo and much used in nuclear magnetic resonance experiments [1].",
"Echos are necessarily dynamical functions characterizing the quality and the difference between forward and backward evolutions.",
"As a consequence, it was quite natural to suggest the echo in the context of dynamical systems as a measure for the dynamical spreading of trajectories, which was done by Asher Peres back in 1984 [2].",
"This quantity is defined as the overlap of two quantum mechanical wave function developed with two slightly different Hamiltonians, whose difference is quantified by the change of an appropriate control parameter.",
"While it turned out that a categorization of fidelity decays for different dynamical systems, with classically integrable, mixed regular-chaotic or fully chaotic phase space, is generally not simple and may depend on non universal details of the systems, it is well established that several universal decay regimes can be found characterized not so much by the systems' properties but by the perturbation strength.",
"These universal regimes are known as perturbation regime, Gaussian decay and Fermi Golden Rule regime [3] and they are well described in recent reviews [4], [5].",
"In this paper we present a compact overview of the use of the overlap function, vulgo fidelity, in the broader context of complex quantum systems, which range from quantum many-body models [6] to strongly driven dynamical quantum systems [7].",
"We review two concepts of overlap functions relevant for cold-atom experiments and which both are equally relevant for characterizing the response of quantum systems to perturbations as well as their dynamical evolution.",
"The first concept is based on the static overlap of two eigenfunction computed for two slightly different control parameters, hence it is basically a spectral characterization of the system at hand.",
"The second concept is the original fidelity introduced by Peres [2], which represents the overlap of two dynamical wave functions taken at equal times but evolved with two different Hamiltonians up to this time.",
"Both measures are useful to characterize the temporal evolution of a problem, since the latter is connected with the spectrum by the spectral theorem, of course.",
"What we mean is that knowing all static overlaps between eigenfunctions at different control parameters and their corresponding eigenvalues, we can in principle reconstruct the dynamical overlap function.",
"This is true for Hamiltonians which either do not depend explicitly on time or are periodically time-dependent (since then we can use Floquet theory to arrive at an eigenbasis).",
"The advantage of the purely dynamical overlap function is that it can be computed also for Hamiltonians which are explicitly time dependent, and there is no need for computing eigenvalues and eigenfunctions, which can be a true practical problem for large quantum few-body or many-body systems [9], [8]."
],
[
"Static fidelity measure",
"Given a control parameter $\\lambda $ and a perturbation $V$ , we are studying the following class of Hamiltonians $H(\\lambda ) = H_0 + \\lambda V$ .",
"Then the static fidelity between the $n$ -th eigenstates (assuming normalized states, a discrete spectrum, and an ordered spectrum such as $E_{n+1} \\ge E_n$ ), denoted by $|n\\rangle $ , of the two Hamiltonians $H(\\lambda )$ and $H(\\lambda + \\delta \\lambda )$ is defined as $f_n(\\lambda , \\delta \\lambda ) \\equiv \\vert \\langle n(\\lambda ) | n(\\lambda +\\delta \\lambda )\\rangle \\vert \\,.$ Here the change in the control parameter $\\delta \\lambda $ is assumed to be small.",
"Locally in the spectrum, the quantity $f_n(\\lambda ,\\delta \\lambda )$ characterizes the change of the eigenfunctions.",
"If the latter change adiabatically, the value of $f_n(\\lambda ,\\delta \\lambda )$ will stay close to one.",
"On the other hand, if we encounter avoided level crossings, at which the eigenfunctions strongly mix, the value of $f_n(\\lambda ,\\delta \\lambda )$ will substantially decrease.",
"To better detect and characterize avoided crossings for a given quantum level $n$ , we better investigate the fidelity change [6], also known as fidelity susceptibility in condensed-matter literature [10], defined by $S_n(\\lambda , \\delta \\lambda ) \\equiv \\frac{1 - f_n(\\lambda , \\delta \\lambda ) }{(\\delta \\lambda )^2} \\,.$ For most of the spectra so far encountered in practice, as long as $\\delta \\lambda \\ll 1$ , $S_n$ turned out to be largely independent of $\\delta \\lambda $ , i.e.",
"$S_n(\\lambda , \\delta \\lambda ) \\approx S_n(\\lambda )$ , and vanishingly small except in the vicinity of an avoided level crossing.",
"The independence of $\\delta \\lambda $ follows in the limit $\\delta \\lambda \\rightarrow 0$ directly from perturbation theory.",
"The independence of $\\delta \\lambda $ has a simple origin, namely that the first non-vanishing contribution to $f_n$ in the perturbation expansion of the changed state $|n(\\lambda +\\delta \\lambda )\\rangle $ is of second order in $\\delta \\lambda $ , a well known fact from quantum perturbation theory.",
"The fidelity measure (REF ) and the fidelity susceptibility (REF ) had found a nice application in the characterization of ground states of quantum many-body systems [11], [12], [10].",
"$S_n$ detects qualitative changes in the ground state very well, when they are connected to an avoided level crossing.",
"Since most of quantum phase transitions in finite systems are characterized by a gap, $S_n$ acts as an effective detector of these transitions and one can check its scaling with the system size, for instance, to pin down the interesting parameter value at which the transitions occurs.",
"Our definition above generalizes the utility of $S_n$ to gaps, e.g.",
"avoided crossings, lying anywhere in the spectrum.",
"Figure: (a) Energy spectrum of Eq.",
"() for x=0,y=2,z=3x=0, y =2, z=3.",
"All levels are coupled and the spectrum shows two close avoided crossings; (b) fidelity change S n (λ)S_n(\\lambda ) and (c) renormalized curvature C n (λ)C_n(\\lambda ) for the three levels shown in (a).",
"Data adapted from .Of course, the computation of $S_n$ is based on the eigenfunctions, which is a practical disadvantage for large quantum systems, where its simpler to compute just the eigenvalues (and not the eigenfunctions explicitly) with optimized numerical methods.",
"Yet this problem turns into an advantage in the sense that it turned out that $S_n$ detects a large percentage, close to one hundred per cent, of the total number of avoided crossings in either a region of the spectrum or the entire spectrum of complex quantum systems.",
"Hence, compared to other measures, e.g.",
"which are based solely on the eigenvalues and their change with the control parameter, our measure $S_n$ is very effective and reliable for the practical numerical detection of avoided crossings all over the spectrum.",
"Figure: (a) Section of the energy spectrum of the Bose-Hubbard model from Eq.",
"() with F=0F=0 and for U=uU = u and J=1-uJ = 1 - u for a small system 5 particles in 5 lattice sites.",
"One level undergoing several avoided crossings is highlighted by the thick (blue) line.",
"Some crossings are not resolved on the scale of the figure but marked by circles.",
"(b) Individual fidelity measures S n S_n of the different eigenstates (dashed lines) and the highlighted level of (a) (thick solid blue line).",
"In this example all avoided crossings are effectively and easily detected.There is an intrinsic connection between our measure $S_n$ and the local level curvature in the spectrum.",
"In order to see this connection, let us expand the eigenfunction $|n(\\lambda +\\delta \\lambda )\\rangle $ in second order in $\\delta \\lambda $ $S_n(\\lambda ) \\approx \\frac{1}{2} \\sum _{m\\ne n} \\frac{\\vert \\langle m(\\lambda ) | V | n(\\lambda )\\rangle \\vert ^2}{[E_n-E_m]^2} \\approx \\frac{\\vert \\langle n^{\\prime }(\\lambda ) | V | n(\\lambda )\\rangle \\vert ^2}{2\\;[E_n-E_{n^{\\prime }}]^2}\\,.$ In the last step, we reduced the sum near an isolated avoided crossing to the nearest neighbour level $n^{\\prime }$ .",
"Similarly, one can expand the renormalized curvature with the gap function $\\Delta (\\lambda )=|E_{n}-E_{n^{\\prime }}|$ , also in second order to approximate the level curvature by [13] $C_{n}(\\lambda ) \\equiv \\bigg |\\frac{1}{\\Delta (\\lambda )}\\frac{\\partial ^2 E_{n}(\\lambda )}{\\partial \\lambda ^2}\\bigg | \\approx \\bigg |\\frac{2}{\\Delta (\\lambda )} \\sum _{m\\ne n} \\frac{\\vert \\langle m(\\lambda ) | V | n(\\lambda )\\rangle \\vert ^2}{E_n-E_m}\\bigg | \\\\ \\approx 2\\frac{\\vert \\langle n^{\\prime }(\\lambda ) | V | n(\\lambda )\\rangle \\vert ^2}{[E_n-E_{n^{\\prime }}]^2}\\,.$ Within this approximation, which is obviously exact for a two level system, we have the following interesting relation between the measure $S_n$ and the level curvature $C_n(\\lambda ) \\approx 4S_n(\\lambda )$ .",
"This in turn justifies the use of $S_n$ for the numerical detection of avoided crossings, which are intimately connected to changes in the curvature of the eigenvalues as a function of the control parameter.",
"Of course, the last steps in the Eqs.",
"(REF ) and (REF ) are crude approximations.",
"Experience tells that on average over a full spectrum they are quite good though.",
"In any case, our numerical results and computations in general do not make use of them explicitly.",
"From the practical point of view it turns out the detection of avoided crossings by the fidelity is much more reliable than by using the levels.",
"This defines the value of $S_n$ for practical numerical work.",
"We are testing the measure from Eq.",
"(REF ) in the following.",
"We start with a very simple three state model described by the real symmetric matrix $H(\\lambda ) = \\left(\\begin{array}{ccc} 0 & x & y \\\\ x & 0 & z \\\\ y & z & 0 \\end{array}\\right) + \\lambda \\left(\\begin{array}{ccc} -1 & 0 & 0 \\\\ 0 & 0 & 0 \\\\ 0 & 0 & 1 \\end{array}\\right) = \\left(\\begin{array}{ccc} -\\lambda & x & y \\\\ x & 0 & z \\\\ y & z & \\lambda \\end{array}\\right) \\,.$ Figure REF nicely shows that $S_n$ is able to detect and to distinguish two nearby avoided crossings.",
"As an example of a quantum many-body system, we study now the class of Hamiltonians of the form $ \\hat{H}(t) = - J \\sum _{l=1}^{L} (\\text{e}^{i Ft} \\hat{a}_{l+1}^{\\dagger }\\hat{a}_{l} + \\text{h.c.}) + U \\sum _{l=1}^{L} \\hat{n}_l(\\hat{n}_l-1) \\,.$ $\\hat{a}_{l}^{\\dagger }$ and $\\hat{a}_{l}$ are creation and annihilation operators of bosons at the lattice site $l$ and $\\hat{n}_l=\\hat{a}_{l}^{\\dagger }\\hat{a}_{l}$ is the corresponding number operator.",
"For $F=0$ , this model is the well known Hubbard model with onsite interactions (last term) for bosonic ultracold atoms [14], [17].",
"Figure REF shows a scan of the many-body spectrum of a small system of five atoms in five lattice sites as a function of $U\\equiv u$ , fixing $J=1-u$ and $F=0$ .",
"We see that all avoided crossings are detected by our fidelity measure.",
"The next figure REF shows a histogram count of avoided crossings as a function of $U/J$ now.",
"The zone in parameter space with a large number of crossings corresponds to the spectral region which follows a Wigner-Dyson distribution for the nearest neighbour spacings in the spectrum (one possible definition of quantum chaotic system [15], [20]).",
"This is shown by the thick/red solid line in the figure.",
"In ref.",
"[18] further details may be found on the connection between quantum chaos, avoided crossings and also entanglement measures in time-independent Bose-Hubbard systems.",
"Figure: (a) Local density of avoided crossings ρ AC \\rho _{AC} (left axis) for N EL N_{EL} energy levels from the centre of the spectrum for the Bose-Hubbard model from Eq.",
"() as a function of U/J for 8 bosons in 7 lattice sites (F=0F=0): N EL N_{EL} = 10 (dashed), 100 (solid) and ca.",
"400 eigenstates(blue histogram); a χ 2 \\chi _2, see for details, test is also shown (right axis and thick/red solid line) with values close to zero for good Wigner-Dyson (quantum chaotic) statistics of the full spectrum.",
"The central part of the spectrum follows the overall distribution of crossings, but it has a larger local density of crossings especially around the maximum of the distribution.",
"Interestingly, this maximum is found close to the Mott-insulator transition in the ground state of our 1D Bose-Hubbard model , a fact that most likely has do with the reordering of the spectrum around this transition briefly discussed in .The case with finite $F$ represents a paradigm of a many-body Floquet Hamiltonian, which is analyzed in detail in references [13], [6], [8].",
"Ref.",
"[6] shows, in particular, that the transition from the regular to chaotic regime as a function of $F$ can be well captured by the analysis of the width distribution of the detected avoided crossings[19], [6] $P(c) = (1-\\gamma ) \\delta (c) + \\frac{2\\gamma ^2}{\\pi \\bar{c}}\\;\\text{exp}\\left[-\\frac{\\gamma ^2c^2}{\\pi \\bar{c}^2}\\right]\\,.$ We use the fidelity measure $S_n$ to find all avoided crossings and then just compute the difference $c$ of nearest neighbouring levels at the avoided crossings from the spectrum independently.",
"This distribution has a chaotic part of weight $0\\le \\gamma \\le 1$ and a finite regular component, which is visible as a strong enhancement of $P(c)$ close to zero, see the $\\delta (c)$ function.",
"$\\bar{c}$ is the mean value of the width in the ensemble of avoided crossings.",
"$P(c)$ separates better the regular from the chaotic components of a given quantum spectrum than the nearest neighbor level spacing distribution.",
"Using the fidelity change (REF ), the perturbation may or may not preserve symmetries intrinsic to the system.",
"Hence, for an analysis based on our static fidelity measure a splitting of the basis into symmetry reduced subspaces may not be strictly necessary (for symmetry breaking perturbations) in order to characterize the chaotic properties of a quantum system.",
"This is another advantage with respect to standard measure such as the nearest level spacing distributions [15], [20].",
"The static fidelity measure (REF ) is hence a very reliable technique for analysing complex quantum spectra.",
"Its great advantage is that it can easily be automatized and it detects almost all avoided crossings.",
"The measure only fails in cases where two avoided crossings get too close to each other such that their clear separation is not possible any more.",
"Too close means here that their distance in the control parameter becomes comparable to their width in energy.",
"Such close coincidences of crossings of comparable strength are from our experience rather rare in typical quantum systems.",
"Besides in the mentioned applications for detecting avoided crossings (see above) and, in particular, quantum phase transitions [11], the authors of [9] have recently applied our measure to characterize the complexity of an electronic two-body quantum dot systems based on heavy numerical diagonalizations.",
"We finally note that the concept of the static fidelity measure based on Eq.",
"(REF ) is easily extended to matrices of overlaps of all eigenfunctions at given control parameters, see e.g.",
"[21].",
"The disadvantage is then, however, that one obtains too much information, which may be hard to interpret, and one must compute, of course, all the eigenfunctions of a given system in order to get all the non-diagonal elements of this generalized fidelity matrix."
],
[
"Dynamical fidelity function",
"The stability of quantum evolution against parametric changes is a subject of broad theoretical and experimental interest.",
"A widely used concept is the fidelity introduced by Peres [2], and the closely related Loschmidt echo [4], [5], which is built as the interference pattern between wave functions that are obtained by propagating the same initial state under two slightly different Hamiltonians.",
"The behaviour of fidelity, e.g.",
"the quantity $F(t)=\\left| \\langle \\psi _{\\lambda +\\delta \\lambda } (t) | \\psi _ {\\lambda }(t) \\rangle \\right|^2 \\,,$ in time is known to display some universal properties that reflect also the underlying classical dynamics [4], [5].",
"Not much is known, however, for the temporal evolution of $F(t)$ for long times, in particular, for classically nearly integrable and mixed regular-chaotic systems.",
"An optimal platform for experiments on fidelity in low dimensional chaotic systems is provided by the quantum kicked rotor model.",
"It is realized with cold and ultracold (Bose-Einstein condensates) atoms [22], [7] and with light pulses in optical fibers [23].",
"The used techniques to measure fidelity range from interferometric methods in either internal atomic states [24] or in the centre-of-mass motion of the atoms [25] to the time reversal of the dynamics by exploiting the properties of the quantum resonant motion [26]."
],
[
"The quantum kicked rotor and fidelity",
"Experiments on the kicked rotor based on (ultra)cold atoms work with particles moving along a line, periodically kicked in time by an optical lattice, and possibly subject to a constant Stark or gravity field.",
"Neglecting atom-atom interactions, the quantum dynamics is described by the following Hamiltonian in dimensionless variables (such that $\\hbar =1$ ) [27], [7], [28]: $\\hat{H}(\\hat{x},\\hat{p},t)\\;=\\;\\frac{\\hat{p}^2}{2}-\\frac{\\eta }{\\tau }\\; \\hat{x}+ k\\cos ( \\hat{x}) \\sum _{{\\tt t}\\in \\mathbb {Z}}\\delta (t-{\\tt t}\\tau )\\ .$ The kick period is $\\tau $ , the kick strength is $k$ , and ${\\tt t}$ is a discrete time variable that counts the number of kicks.",
"The additional parameter $\\eta $ yields the change in momentum produced by the constant field over one kick period.",
"In the accelerated (by gravity) frame of reference, the total potential is again spatially periodic, implying the conservation of quasi-momentum $\\beta $ .",
"With the chosen units, $\\beta $ can take on allowed values between 0 and 1.",
"Using Bloch theory, the atom dynamics from immediately after the $({\\tt t}-1)$ -th kick to immediately after the next ${\\tt t}$ -th kick is then described by the unitary operator, which is derived in detail in [27]: $\\hat{\\cal U}_{\\beta ,k,\\eta }({\\tt t})\\;=\\;e^{-\\mathrm {i}k \\cos (\\hat{\\theta })}\\; e^{-\\mathrm {i}\\tau /2(\\hat{\\cal N}+ \\beta +\\eta {\\tt t} +\\eta /2)^2}\\ .$ The full evolution operator over the first ${\\tt t}$ kicks is given by $\\hat{\\cal U}_{\\beta ,k,\\eta }^{{\\tt t}}\\;\\equiv \\;\\hat{\\cal U}_{\\beta ,k,\\eta }({\\tt t-1})\\; \\hat{\\cal U}_{\\beta ,k,\\eta }({\\tt t-2})\\dots \\hat{\\cal U}_{\\beta ,k,\\eta }({\\tt 1})\\; \\hat{\\cal U}_{\\beta ,k,\\eta }({\\tt 0}) \\,.$ where $\\hat{\\cal N}= - \\mathrm {i}\\frac{\\mathrm {d}}{\\mathrm {d}\\theta }$ is the (angular) momentum operator with periodic boundary conditions.",
"The time-dependent Hamiltonian that generates the quantum evolution corresponding to (REF ) would be $\\hat{\\cal H}({\\hat{\\cal N}},\\hat{\\theta },\\beta ,t)= \\frac{1}{2}\\left(\\hat{\\cal N} +\\beta +\\frac{\\eta }{\\tau } t\\right)^2+ k\\cos (\\hat{\\theta }) \\sum _{{\\tt t}\\in \\mathbb {Z}}\\delta (t-{\\tt t}\\tau )\\ .$ The temporal evolution of the fidelity can now be studied with respect to changes of one of the parameter in Eq.",
"(REF ).",
"Motivated by the experiments reported in Refs.",
"[24], [29], we consider in the following changes in the kick strength $k$ and the gravity parameter $\\eta $ .",
"For a given quasimomentum $\\beta $ , the dynamical fidelity is defined as $F_\\beta (k_1,k_2,\\eta _1, \\eta _2, {\\tt t}) = \\Big | \\left< \\hat{\\cal U}_{\\beta ,k_1,\\eta _1}^{{\\tt t}} \\psi _\\beta (0) \\Big |\\hat{\\cal U}_{\\beta ,k_2,\\eta _2}^{{\\tt t}} \\psi _\\beta (0)\\right>\\Big |^2 \\,.$ In a typical experimental situations with cold atoms [32], [31], [25], [24], [7], [29], [30], the atoms are non interacting and each of them has in good approximation a well defined initial momentum, meaning that its quasimomentum is essentially fixed.",
"This ensemble is best approximated by a statistical density operator $\\hat{\\rho } = \\int _\\beta \\mathrm {d}\\beta \\rho (\\beta ) \\mathinner {|{\\psi _\\beta (0)}\\rangle }\\mathinner {\\langle {\\psi _\\beta (0)}|}$ .",
"Then the fidelity generalized to density operators [5] gives the slightly more involved formula for our system of cold atoms [33], [28]: $F(k_1,k_2,\\eta _1, \\eta _2,{\\tt t}) =\\bigg |\\int _\\beta \\rho (\\beta ) \\left< \\hat{\\cal U}_{\\beta ,k_1,\\eta _1}^{{\\tt t}} \\psi _\\beta (0) \\Big |\\hat{\\cal U}_{\\beta ,k_2,\\eta _2}^{{\\tt t}} \\psi _\\beta (0)\\right> \\mathrm {d}\\beta \\bigg |^2 \\,.$ A series of experimental investigations has looked closely at the so-called quantum resonant motion of the quantum kicked rotor over the last fifteen years, see e.g.",
"[7], [31] and references therein.",
"This interest is mainly motivated by the type of ballistic motion with fast acceleration which can be obtained in this particular region of parameters.",
"The quantum resonances allow one the realization of ratchet-like dynamics as well, when breaking the spatial-temporal symmetries of the kicked rotor [7], [34].",
"We restrict here, as most of previous works, to the main quantum resonances occurring whenever $\\tau $ is not only commensurate to $2\\pi $ but an integer multiple of it, i.e.",
"$\\tau =2\\pi \\ell $ , $\\ell $ integer.",
"Then, for specially chosen values of quasi-momentum $\\beta $ , the energy of the rotor with this fixed $\\beta $ increases quadratically as ${\\tt t}\\rightarrow +\\infty $ [35], [27].",
"In the presence of gravity, asymptotic quadratic growth of energy at certain values of $\\beta $ is still possible close to resonant values of the kick period[27].",
"From the theoretical point of view, these quantum resonances are convenient since we have analytical access to the wave functions [35], [27] and hence to analytical estimates for the development of fidelity [33], [28].",
"Using pseudoclassical approximation theory [35], [27], [37], [7], this is true also for small detuning from the resonance conditions.",
"We review here just the result of simple analytical considerations, whose details can be found in [33], [28].",
"The fidelity at a main quantum resonance is considered in two cases: first for the perturbation parameter $\\Delta k=k_2-k_1$ and a fixed quasimomentum $\\beta $ .",
"The fidelity is then computed from the overlap $\\left< \\hat{\\cal U}_{\\beta ,k_1,\\eta }^{{\\tt t}}\\psi _\\beta (0) \\Big |\\hat{\\cal U}_{\\beta ,k_2,\\eta }^{{\\tt t}} \\psi _\\beta (0)\\right>= J_0(\\Delta k|W_{{\\tt t}}|)\\ ,$ such as to yield $F_\\beta (k_1,k_2,\\eta ,{\\tt t})= \\Big |J_0(\\Delta k |W_{{\\tt t}}|)\\Big |^2 \\,.$ The argument of the zero-order Bessel function contains the following sum of phases: $W_{{\\tt t}}\\equiv W_{{\\tt t}}(\\eta ,\\beta )=\\sum _{r=0}^{{\\tt t}-1}e^{-\\mathrm {i}\\pi \\ell (2\\beta +1)r}e^{-\\mathrm {i}2\\pi \\ell r \\eta {\\tt t}+\\mathrm {i}\\pi \\ell \\eta r^2}\\ .$ Applying Eq.",
"(REF ) at $\\eta =0$ and resonant $\\beta =\\beta _{res}$ (e.g.",
"$\\beta _{res}=0$ for $\\ell =2$ ) we have simply $W_{{\\tt t}}={\\tt t}$ , which leads to $F_{\\beta _{res}}(k_1,k_2,\\eta =0,{\\tt t})= \\Big |J_0(\\Delta k {\\tt t} )\\Big |^2 \\,.$ This fidelity decays asymptotically, for large times, like a power-law with exponent one [33].",
"Averaging over a full ensemble of quasimomenta in a Brillouine zone, we arrive at a fidelity which asymptotically saturates to a constant value $F(\\Delta k, {\\tt t}) \\equiv \\bigg |\\int _{\\beta =0}^{\\beta =1}\\left< \\hat{\\cal U}_{\\beta ,k_1}^{{\\tt t}} \\psi _\\beta (0) \\Big |\\hat{\\cal U}_{\\beta ,k_2}^{{\\tt t}} \\psi _\\beta (0)\\right> \\mathrm {d}\\beta \\bigg |^2\\stackrel{t \\rightarrow \\infty }{\\longrightarrow } \\frac{1}{(2\\pi )^2} \\left(\\int _{0}^{2\\pi }J_0^2\\left(\\frac{\\Delta k }{2\\sin \\alpha }\\right) \\mathrm {d}\\alpha \\right)^2\\,.$ This asymptotically exact saturation of the fidelity sets in very quickly in practice for typical experimental parameters, after ${\\tt t} > z_0/\\Delta k$ , where $z_0$ is the first zero of the zero order Bessel function [33].",
"An interferometric cold-atom experiment at Harvard [25] nicely confirmed our theory showing this saturation of fidelity for various perturbations $\\Delta k$ and measurements up to about 60 kicks.",
"In the second case, we consider the motion in a gravity field for the forward motion, whilst this field is absent in the backward motion.",
"The kick strength does not change here.",
"This gives for the fidelity, following a similar reasoning as done in [28], $F_ \\beta (\\eta , {\\tt t} ) = \\left| \\langle \\psi _\\beta (0)| \\hat{\\mathcal {U}}_{\\beta , k, {\\eta =0} }^{\\dagger {\\tt t} } \\,\\hat{ \\mathcal {U} }_{\\beta , k, \\eta }^{{\\tt t} } | \\psi _\\beta (0) \\rangle \\right| ^2= \\bigg |e^{-i\\phi ({\\beta }, {\\eta }, {\\tt t})} J_0\\left(\\sqrt{({\\tt t} k)^2+k^2 |W_{{\\tt t}}|^2 -2 {\\tt t} k^2 \\text{Re} (W_{{\\tt t}}) } \\right)\\bigg |^2 \\,.$ For a single $\\beta $ , the additional phase $\\phi ({\\beta }, {\\eta }, t) = \\ell \\pi \\sum _{q=0}^{t-1} \\left({\\beta }+q{\\eta } +/ {\\eta }2\\right)^2$ is superfluous, yet it becomes important when considering contributions from different quasimomenta, see subsection REF for this case.",
"Figure: (left) phase space of the pseudoclassical model for η=0,k|ϵ|=0.04π\\eta =0, k|\\epsilon |=0.04\\pi (green/grey data points) as compared to its pendulum approximation , (solid lines, where the red line shows the separatrix of the pendulum).",
"(right) phase space of the pseudoclassical model with gravity for the parameters τ=5.97,η=0.0257,k=1.4\\tau =5.97,\\eta =0.0257,k=1.4.",
"(Quasi)momentum zero corresponds to the centre of both islands on the y axis, while θ\\theta is the angle parameter of the standard map as classical limit of the kicked rotor problem.",
"The horizon stripe in the right plot marks the typical width of an experimental quasimomentum distribution for a Bose-Einstein condensate."
],
[
"Fidelity revivals",
"Even if we detune the kick period $\\tau $ slightly from the mentioned resonance conditions, we may still be able to estimate the true quantum motion using a pseudoclassical model introduced in [35], [27].",
"In this model, the absolute value of the detuning $|\\epsilon |$ plays the role of Planck's constant, and hence the theory has a semiclassical limit at exact quantum resonance.",
"The $\\epsilon $ -classical dynamics is described by the following discrete map which relates the variables $I$ and $\\theta $ from immediately after the ${\\tt t}$ -th kick to immediately after the $({\\tt t}+1)$ -th one $\\begin{array}{rcl}I_{{\\tt t}+1}&=&I_{{\\tt t}}+\\tilde{k}\\sin \\theta _{{\\tt t}+1}+\\mathrm {sgn}(\\epsilon )\\tau \\eta \\\\\\theta _{{\\tt t}+1}&=&\\theta _{{\\tt t}}+\\mathrm {sgn}(\\epsilon )I_{{\\tt t}}\\;\\;\\;\\textrm { mod } 2\\pi \\end{array}\\,.$ Here, ${I}={J}+\\mathrm {sgn}(\\epsilon )[\\pi \\ell +\\tau \\beta +\\tau \\eta {\\tt t}]$ and $J=|\\epsilon | p$ is the rescaled momentum.",
"Similarly, the new effective kick strength is defined by $\\tilde{k}=|\\epsilon | k$ .",
"Since the effective kick strength is multiplied by the small detuning in this model, the classical phase space is nearly integrable and its centre corresponds to a pendulum-like resonance island [36], [38], [37], [20], see Fig.",
"REF .",
"In this subsection, we continue now to study the case without gravity, i.e.",
"$\\eta =0$ (see left panel of the figure).",
"Within the harmonic approximation, initial states starting at momentum close to zero will oscillate around the island, with an oscillation frequency depending on the coupling strength, $\\omega _i=\\sqrt{k_i |\\epsilon |}$ ($i=1,2$ ) in this case.",
"Hence, while the fidelity initially can decay fast at finite detuning [39], it may recover if the two frequencies – related to the two kick strength – are commensurable.",
"Focusing only onto resonant values of quasimomentum, the fidelity that carries over formula (REF ) to finite detuning is $F_{\\beta _{res}}(k_1,k_2, {\\tt t},\\eta =0) \\sim \\frac{\\epsilon }{2\\pi } \\left(| \\omega _2 \\cos ( \\omega _1 {\\tt t} ) \\sin ( \\omega _2 {\\tt t}) - \\omega _1 \\cos ( \\omega _2 {\\tt t} ) \\sin ( \\omega _1 {\\tt t}) | \\right)^{-1} \\,,$ where the symbol $\\sim $ means that this result is asymptotically true for small $\\epsilon $ and large times.",
"The predicted revivals of the fidelity are typical for integrable motion around nonlinear islands characterized by their winding numbers ($\\omega _i$ measured in number of kicks) in the Poincaré sections.",
"Typical results of fidelity revivals are reported in Fig.",
"REF .",
"Figure: Slightly smoothed version of the fidelity as predicted by Eq.",
"() (thick black line) and numerical data(grey/red curve) for k 1 =0.8π,k 2 =0.6πk_1=0.8\\pi , k_2=0.6\\pi , and detuning ϵ=0.01\\epsilon =0.01.",
"Please note that the fidelity does start at 1 on the yy axis,which is not visible here owing to the fast initial drop after just a few kicks.",
"Data adapted from ."
],
[
"Dynamical tunneling",
"This subsection deals now with the case of a perturbation in the kick strength and one fixed value of gravity $\\eta \\ne 0$ .",
"All we want to say here is essentially contained in Fig.",
"REF .",
"While initially, for short times, the fidelity suffers a fast exponential decay, this decay and the further evaluation is modulated with almost periodic oscillations.",
"This oscillations are the better visible, the more we average over initial conditions (in this case different values of quasimomenta).",
"The initial decay is arising from the part of the initial wave packet that lies in the chaotic component of either of the two dynamics (defined by the two different kick strengths) in the pseudoclassical phase space, see right panel of Fig.",
"REF .",
"While its behaviour seems to be non-uniform and strongly dependent on parameters, we can estimate the oscillation frequency by a similar argument as in the previous subsection (REF ).",
"The simplest form of this estimate gives an oscillation frequency of $|\\omega _1-\\omega _2|+\\Delta \\theta $ , where $\\omega _i$ ($i=1,2$ ) were defined above and $\\Delta \\theta $ is the shift of the island centre in phase space induced by the change in the kick strength at finite gravity.",
"This shift of the frequency can be a understood in the following way: for the elliptic motion in one island the trajectory is ahead, whereas for the elliptic motion in the other island the trajectory is behind, so they get closer to each other and rephase after a shorter period than in the case when both islands lie concentric.",
"The shift, which we estimate by $\\Delta \\theta $ , depends on the precise geometry of the two islands, and the overlapping area.",
"Already the data in the left panel of Fig.",
"REF shows a significant change in slope of the overall fidelity decay after some time, see the solid line at about 100 kick.",
"We can analyze this asymptotic behaviour better focussing on an initial wave packet with just one single initial quasimomentum, see right panel of Fig.",
"REF .",
"The fidelity is – after the initial decay – dominated by the parts of the wave packet which are trapped inside the islands, similar to the one shown in Fig.",
"REF (right panel).",
"We chose our initial states in the form of Gaussian states mostly located inside one island.",
"The decay from the islands into the chaotic sea is a well known phenomenon known as dynamical tunneling, see in this context [40], [30] and refs.",
"therein.",
"We will argue now that this is the main mechanism responsible of the fidelity decay at long times.",
"The most important contribution to fidelity comes from the parts of each factor in the scalar product in (REF ) that are trapped in the respective travelling island.",
"The decay of fidelity is then in turn determined by the decay of each of these two parts, which is determined by its respective tunneling rate $\\Gamma _{i}$ ($i=1,2$ ) into the chaotic sea.",
"Based on this semiclassical reasoning, we came up with the following ansatz for the asymptotic decay of fidelity [28] $F({\\tt t})\\;\\propto \\;{\\mu ({\\cal A}_1\\backslash {\\cal A}_2) e^{-\\Gamma _1 {\\tt t}} +\\mu ({\\cal A}_2\\backslash {\\cal A}_1) e^{-\\Gamma _2 {\\tt t}} + \\mu ({\\cal A}_2\\cap {\\cal A}_1)e^{-(\\Gamma _1+\\Gamma _2) {\\tt t}}} \\ .$ Here $\\mu $ is the classical invariant measure of phase space sets, ${\\cal A}_i$ is the island in the phase space around the fixed point associated with $k_i$ .",
"In our simulations, the phase-space areas appearing in (REF ) were estimated as described in the appendix of [28].",
"The quantum decay rates due to dynamical tunneling were computed numerically for each $k_i$ .",
"To do so we computed the survival probability as shown in Fig.",
"REF by the blue dashed line.",
"Our semiclassical formula (REF ) provides a simple yet powerful estimate for the fidelity decay since it is rather universal and applicable to a large variety of systems.",
"We finish this subsection by stating that first experimental measurements are on there way to detect signatures of dynamical tunneling of states prepared in quantum accelerator mode islands such as shown in the right panel of Fig.",
"REF , see ref.",
"[30].",
"The problem will remain, of course, to reach a high enough experimental stability to get access to the evolution to times longer than just a few dozens of kicks.",
"Figure: (left) Fidelity averaged over a broad ensemble of quasimomenta in [0,1)[0,1) (corresponding to a cold-atom experiment with wideinitial momentum distribution) for the parameters τ=5.86,η=0.01579τ,k 1 =0.8π,k 2 =0.75π\\tau = 5.86, \\eta = 0.01579\\tau , k_1 = 0.8\\pi , k_2 = 0.75\\pi (dashed line) and k 2 =0.7πk_2 = 0.7\\pi (solid line).",
"(right) Comparison between the survival probability (black dots and thin blue line) and the fidelity for one fixed quasimomentum(red/grey thick solid line) for τ=5.86\\tau =5.86, ϵ=τ-2π\\epsilon =\\tau -2\\pi ,η=0.01579τ\\eta =0.01579\\tau , β≈0.49\\beta \\approx 0.49, k 2 =0.8πk_2=0.8\\pi , k 1 =0.7πk_1=0.7\\pi .",
"The upper thin blue line corresponds to k 1 k_1 and gives thetunneling rates Γ 1 =5.1×10 -4 \\Gamma _1 = 5.1 \\times 10^{-4}, whereas the lower black dots corresponds to k 2 k_2 and gives Γ 2 =4.4×10 -5 \\Gamma _2 =4.4 \\times 10^{-5}.",
"The crosses represent an exponential fit with a rate Γ=3.5×10 -5 \\Gamma = 3.5 \\times 10^{-5} to guide the eye.",
"Data in the right paneladapted from ."
],
[
"Thermometry in ultra-cold atoms",
"We now turn back again at exact quantum resonance conditions, more particularly to formula (REF ) from subsection REF .",
"From the phase term $\\phi ({\\beta }, {\\eta }, t)$ in Eq.",
"(REF ) it can be seen that, when quasimomentum $\\beta $ is non zero, then the total phase induced by different values of $\\eta $ depends not only on its magnitude but also on its sign.",
"Experiments to investigate this situation were performed at Oklahoma with a Bose-Einstein condensate of about 40 000 rubidium atoms [29].",
"The atoms correspond to a distribution of quasimomenta with a FWHM of about $0.05$ in our dimensionless units.",
"We must average Eq.",
"(REF ) over these initial conditions in $\\beta $ .",
"The data – guided by our theory – clearly indicate that the asymmetric dependence of the averaged fidelity on gravity may turn out to be useful in determining externally applied accelerations and, in turn, the temperature (given by the kinetic energy) of ultracold atoms in the limit of negligible interatomic interactions.",
"The immediate value of this experimental application is highlighted by the fact that just four kicks were used in Fig.",
"REF , making similar experiments very fast, stable and versatile.",
"While the experimental data from Fig.",
"REF is not yet as stable as one would like for a practically useful thermometer, numerical results shown in Fig.",
"REF confirm the linear correlation between the normalized visibility of the asymmetry and the momentum width of the atomic cloud.",
"We conclude this section by remarking that the quantum kick rotor in the presence of gravity, as describe by the Hamiltonian (REF ), is for generic values of $\\eta $ not reducible to a periodically time-dependent system.",
"Hence, we indeed need the dynamical fidelity in order to characterize the large variety of different dynamical regimes of this paradigm system of quantum chaos.",
"A description by the static fidelity measure of section would instead only be possible for values of $\\eta $ commensurable with $\\tau /|\\epsilon |$ , see ref.",
"[40] for details.",
"The fidelity measure has found a plethora of applications over the last sixty years, from spin and photon echos over to low-dimensional quantum chaotic systems [1], [2], [3], [4], [5].",
"We reviewed here two important specific fidelities, a static overlap of eigenfunctions and a dynamical overlap of generic wave functions.",
"The former has been successfully used to detect avoided crossings in complex quantum systems, ranging from quantum phase transitions in the ground state [11], [10], [12] to the characterization of the chaoticity of the entire or parts of quantum spectra [21], [6], [8], [9].",
"The much more studied dynamical fidelity has been investigated here for the paradigm system of quantum chaos, the quantum kicked rotor.",
"We showed several applications, ranging from characterizing quasi-regular motion close to quantum resonance, over to dynamical tunneling and its indirect use as a thermometer for ultracold atomic gases.",
"We hope that this overview will inspire further searches for interesting applications of the fidelity and generalized quantum echos, possible exploiting much more the phase dependence of the quantum mechanical overlap function for new precision experiments.",
"Figure: Numerical simulations of an asymmetry visibility defined as [F(η - )-F(η + )]/[F(η - )+F(η + )][F(\\eta _-) - F(\\eta _+)]/[F(\\eta _-) + F(\\eta _+)] show an almost linear scaling with the momentum width of the atomic cloud for the experimentally relevant range, see previous figure, Δβ≤0.08\\Delta \\beta \\le 0.08.",
"This linear correlation could be used to precisely define the momentum width on the xx axis via measurements of the asymmetry (on the yy axis).",
"Figure taken from .The author is grateful to Italo Guarneri and Gil Summy for stimulating our work on fidelity related to the dynamics of cold atoms.",
"Many thanks goes to the large number of students and my former postdoc Remy Dubertrand, who all were involved in specific parts of the work reviewed here.",
"Support by the FIL program of the Università di Parma is kindly acknowledged.",
"The author thanks very much the organizers of the international conference Echoes in Complex Systems at the MPIPKS, in particular Arseni Goussev, for support and the kind invitation to Dresden."
]
] | 1605.06135 |
[
[
"Finite intersection property and dynamical compactness"
],
[
"Abstract Dynamical compactness with respect to a family as a new concept of chaoticity of a dynamical system was introduced and discussed in [22].",
"In this paper we continue to investigate this notion.",
"In particular, we prove that all dynamical systems are dynamically compact with respect to a Furstenberg family if and only if this family has the finite intersection property.",
"We investigate weak mixing and weak disjointness by using the concept of dynamical compactness.",
"We also explore further difference between transitive compactness and weak mixing.",
"As a byproduct, we show that the $\\omega_{\\mathcal{F}}$-limit and the $\\omega$-limit sets of a point may have quite different topological structure.",
"Moreover, the equivalence between multi-sensitivity, sensitive compactness and transitive sensitivity is established for a minimal system.",
"Finally, these notions are also explored in the context of linear dynamics."
],
[
"Introduction",
"By a (topological) dynamical system $(X,T)$ we mean a compact metric space $X$ with a metric $d$ and a continuous self-surjection $T$ of $X$ .",
"We say it trivial if the space is a singleton.",
"Throughout this paper, we are only interested in a nontrivial dynamical system, where the state space is a compact metric space without isolated points.",
"This paper is a continuation of the research carried out in [22], where the authors discuss a dynamical property (called dynamical compactness) and examine it firstly for transitive compactness.",
"Some results of this paper can be considered as a contribution to dynamical topology – an area of the theory of dynamical systems in which the topological properties of maps that can be described in dynamical terms.",
"Let $\\mathbb {Z}_{+}$ be the set of all nonnegative integers and $\\mathbb {N}$ the set of all positive integers.",
"Before going on, let us recall the notion of a Furstenberg family from [1].",
"Denote by $\\mathcal {P} =\\mathcal {P}(\\mathbb {Z}_{+})$ the set of all subsets of $\\mathbb {Z}_{+}$ .",
"A subset $\\mathcal {F}\\subset \\mathcal {P}$ is a (Furstenberg) family, if it is hereditary upward, that is, $F_1 \\subset F_2$ and $F_1 \\in \\mathcal {F}$ imply $F_2 \\in \\mathcal {F}$ .",
"Any subset $\\mathcal {A}$ of $\\mathcal {P}$ clearly generates a family $\\lbrace F\\in \\mathcal {P}: F\\supset A\\ \\text{for some}\\ A\\in \\mathcal {A}\\rbrace $ .",
"Denote by $\\mathcal {B}$ the family of all infinite subsets of $\\mathbb {Z}_{+}$ , and by $\\mathcal {P}_+$ the family of all nonempty subsets of $\\mathbb {Z}_+$ .",
"For a family $\\mathcal {F}$ , the dual family of $\\mathcal {F}$ , denoted by $k\\mathcal {F}$ , is defined as $\\lbrace F\\in \\mathcal {P}: F\\cap F^{\\prime }\\ne \\varnothing \\mbox{ for any }F^{\\prime } \\in \\mathcal {F}\\rbrace .$ A family $\\mathcal {F}$ is proper if it is a proper subset of $\\mathcal {P}$ , that is, $\\mathbb {Z}_+\\in \\mathcal {F}$ and $\\varnothing \\notin \\mathcal {F}$ .",
"By a filter $\\mathcal {F}$ we mean a proper family closed under intersection, that is, $F_1, F_2\\in \\mathcal {F}$ implies $F_1\\cap F_2\\in \\mathcal {F}$ .",
"A filter is free if the intersection of all its elements is empty.",
"We extend this concept, a family $\\mathcal {F}$ is called free if the intersection of all elements of $\\mathcal {F}$ is empty.",
"For any $F\\in \\mathcal {P}$ , every point $x\\in X$ and each subset $G\\subset X$ , we define $T^{F} x= \\lbrace T^i x: i\\in F\\rbrace $ , $n_T(x, G)=\\lbrace n\\in \\mathbb {Z}_{+}: T^n x \\in G\\rbrace $ .",
"The $\\omega $ -limit set of $x$ with respect to $\\mathcal {F}$ (see [1]), or shortly the $\\omega _\\mathcal {F}$ -limit set of $x$, denoted by $\\omega _{\\mathcal {F}}(x)$Remark that the notation $\\omega _{\\mathcal {F}}(x)$ used here is different from the one used in [1] (the notation $\\omega _{\\mathcal {F}}(x)$ used here is in fact $\\omega _{k \\mathcal {F}}(x)$ introduced in [1]).",
"As this paper is a continuation of the research in [22], in order to avoid any confusion of notation or concept, we will follow the ones used in [22]., is defined as $\\bigcap _{F \\in \\mathcal {F}} \\overline{T^{F}x}= \\lbrace z\\in X: n_T(x,G) \\in k\\mathcal {F} \\ \\mbox{for every neighborhood} \\ G \\ \\mbox{of} \\ z \\rbrace .$ Let us remark that not always $\\omega _{\\mathcal {F}}(x)$ is a subset of the $\\omega $ -limit set $\\omega _{T}(x)$ , which is defined as $\\bigcap _{n=1}^\\infty \\overline{\\lbrace T^kx: k\\ge n\\rbrace }= \\lbrace z\\in X: N_T(x,G) \\in \\mathcal {B} \\ \\mbox{for every neighborhood} \\ G \\ \\mbox{of} \\ z \\rbrace .",
"$ For instance, if each element of $\\mathcal {F}$ contains 0 then any point $x\\in \\omega _{\\mathcal {F}}(x)$ .",
"But, as well known, a point $x\\in \\omega _{T}(x)$ if and only if $x$ is a recurrent pointA point $x\\in X$ is called recurrent if $x\\in \\omega _T(x).$ of $(X,T)$ .",
"Nevertheless, if a family $\\mathcal {F}$ is free, then $\\omega _{\\mathcal {F}}(x) \\subset \\omega _{T}(x)$ for any point $x\\in X$ and if $(X,T)$ has a nonrecurrent point, then the converse is true (see Proposition REF ).",
"A dynamical system $(X,T)$ is called compact with respect to $\\mathcal {F}$, or shortly dynamically compact, if the $\\omega _\\mathcal {F}$ -limit set $\\omega _{\\mathcal {F}}(x)$ is nonempty for all $x\\in X$ .",
"H. Furstenberg started a systematic study of transitive systems in his paper on disjointness in topological dynamics and ergodic theory [14], and the theory was further developed in [16] and [15].",
"Recall that the system $(X, T)$ is (topologically) transitive if $N_T (U_1, U_2)= \\lbrace n\\in \\mathbb {Z}_+: U_1\\cap T^{-n}U_2\\ne \\varnothing \\rbrace \\ (= \\lbrace n\\in \\mathbb {Z}_+: T^n U_1\\cap U_2\\ne \\varnothing \\rbrace )\\in \\mathcal {P}_+$ for any openeBecause we so often have to refer to open, nonempty subsets, we will call such subsets opene.",
"subsets $U_1, U_2\\subset X$ , equivalently, $N_T (U_1, U_2)\\in \\mathcal {B}$ for any opene subsets $U_1, U_2\\subset X$ .",
"In [22] the authors consider one of possible dynamical compactness — transitive compactness, and its relations with well-known chaotic properties of dynamical systems.",
"Let $\\mathcal {N}_T$ be the set of all subsets of $\\mathbb {Z}_+$ containing some $N_T (U, V)$ , where $U, V$ are opene subsets of $X$ .",
"A dynamical system $(X, T)$ is called transitive compact, if for any point $x\\in X$ the $\\omega _{\\mathcal {N}_T}$ -limit set $\\omega _{\\mathcal {N}_T}(x)$ is nonempty, in other words, for any point $x\\in X$ there exists a point $z \\in X$ such that $ n_T(x,G) \\cap N_T(U,V)\\ne \\varnothing $ for any neighborhood $G$ of $z$ and any opene subsets $U,V$ of $X$ .",
"Let $(X, T)$ and $(Y, S)$ be two dynamical systems and $k\\in \\mathbb {N}$ .",
"The product system $(X\\times Y, T\\times S)$ is defined naturally, and denote by $(X^k, T^{(k)})$ the product system of $k$ copies of the system $(X, T)$ .",
"Recall that the system $(X, T)$ is minimal if it does not admit a nonempty, closed, proper subset $K$ of $X$ with $T K\\subset K$ , and is weakly mixing if the product system $(X^2, T^{(2)})$ is transitive.",
"Any transitive compact system is obviously topologically transitive, and observe that each weakly mixing system is transitive compact ([4]).",
"In fact, as it was shown in [22], each of notions are different in general and equivalent for minimal systems.",
"Recall a very useful notion of weakly mixing subsets of a system, which was introduced in [9] and further discussed in [32] and [33].",
"The notion of weakly mixing subsets can be regraded as a local version of weak mixing.",
"Among many very interesting properties let us mention just one of them – positive topological entropy of a dynamical system implies the existence of weakly mixing sets (see [28] for details).",
"A nontrivial closed subset $A\\subset X$ is called weakly mixing if for every $k\\ge 2$ and any opene sets $U_1, \\dots ,U_k, V_1, \\dots , V_k$ of $X$ with $U_i\\cap A \\ne \\varnothing $ , $V_i\\cap A \\ne \\varnothing $ , for any $i=1, \\dots ,k$ , one has that $\\bigcap _{i= 1}^kN_T(U_i\\cap A, V_i)\\ne \\varnothing .$ Let $A$ be a weakly mixing subset of $X$ and let $\\mathcal {N}_T(A)$ be the set of all subsets of $\\mathbb {Z}_+$ containing some $N_T (U\\cap A, V)$ , where $U, V$ are opene subsets of $X$ intersecting $A$ .",
"The notion of sensitivity was first used by Ruelle [36], which captures the idea that in a chaotic system a small change in the initial condition can cause a big change in the trajectory.",
"According to the works by Guckenheimer [20], Auslander and Yorke [6] a dynamical system $(X,T)$ is called sensitive if there exists $\\delta > 0$ such that for every $x\\in X$ and every neighborhood $U_x$ of $x$ , there exist $y\\in U_x$ and $n\\in \\mathbb {N}$ with $d(T^n x,T^n y)> \\delta $ .",
"Such a $\\delta $ is called a sensitive constant of $(X, T)$ .",
"Recently in [30] Moothathu initiated a way to measure the sensitivity of a dynamical system, by checking how large is the set of nonnegative integers for which the sensitivity occurs (see also [29]).",
"For a positive $\\delta $ and a subset $U\\subset X$ define $ S_T (U, \\delta )= \\lbrace n\\in \\mathbb {Z}_+: \\mbox{ there are } x_1, x_2\\in U\\ \\mbox{such that}\\ d (T^n x_1, T^n x_2)>\\delta \\rbrace .$ A dynamical system $(X, T)$ is called multi-sensitive if there exists $\\delta > 0$ such that $\\bigcap _{i= 1}^k S_T (U_i, \\delta )\\ne \\varnothing $ for any finite collection of opene $U_1, \\dots , U_k\\subset X$ .",
"Such a $\\delta $ is called a constant of multi-sensitivity of $(X, T)$ .",
"Recall that a collection $A$ of subsets of a set $Y$ has the finite intersection property (FIP) if the intersection of any finite subcollection of $A$ is nonempty.",
"The FIP is useful in formulating an alternative definition of compactness of a topological space: a topological space is compact if and only if every collection of closed subsets satisfying the FIP has a nonempty intersection itself (see, for instance [13], [25]).",
"Obviously that a filter (say $\\mathcal {N}_T$ , when $(X,T)$ is weakly mixing), the family $\\mathcal {N}_T(A)$ for a weakly mixing subset $A$ of $(X,T)$ and the family $\\mathcal {S}_T(\\delta )$ when $(X, T)$ is a multi-sensitive system (with a constant of multi-sensitivity $\\delta > 0$ ) have FIP.",
"Since all of these families are also free, actually they have the strong finite intersection property (SFIP), i.e., if the intersection over any finite subcollection of the family is infinite (see Proposition 2.2).",
"In fact we can say more — the FIP is useful in characterizing the dynamical compactness (see Theorem REF ).",
"Theorem FIP All dynamical systems are dynamically compact with respect to $\\mathcal {F}$ if and only if the family $\\mathcal {F}$ has the finite intersection property.",
"We also introduce two new stronger versions of sensitivity: sensitive compactness and transitive sensitivity.",
"Denote by $\\mathcal {S}_T(\\delta )$ the set of all subsets of $\\mathbb {Z}_+$ containing $S_T (U, \\delta )$ for some $\\delta > 0$ and opene $U\\subset X$ .",
"We will call the system $(X, T)$ transitively sensitive if there exists $\\delta > 0$ such that $S_T (W, \\delta )\\cap N_T (U, V)\\ne \\varnothing $ for any opene subsets $U, V, W$ of $X$ ; and sensitive compact, if there exists $\\delta > 0$ such that for any point $x\\in X$ the $\\omega _{\\mathcal {S}_T(\\delta )}$ -limit set $\\omega _{\\mathcal {S}_T(\\delta )}(x)$ is nonempty, in other words, for any point $x\\in X$ there exists a point $z\\in X$ such that $ n_T(x,G)\\cap S_T (U, \\delta )\\ne \\varnothing $ for any neighborhood $G$ of $z$ and any opene $U$ of $X$ .",
"The paper is organized as follows.",
"In Section 2 we recall some basic concepts and properties used in later discussions from topological dynamics.",
"In Section 3 we obtain some general results concerning dynamical compactness.",
"In particular we show that all dynamical systems are dynamically compact with respect to a Furstenberg family if and only if this family has the finite intersection property (Theorem REF ).",
"In Section 4 we discuss two stronger versions of sensitivity: transitive sensitivity and sensitive compactness.",
"It was shown that each weakly mixing system is transitively sensitive (Proposition REF ), and in fact we can characterize transitive sensitivity of a general dynamical system in terms of dynamical compactness (Proposition REF ).",
"Furthermore, all of the multi-sensitivity, sensitive compactness and transitive sensitivity are equivalent for a minimal system (Theorem REF ).",
"Even though each minimal transitive compact system is multi-sensitive, nevertheless, there are many minimal multi-sensitive systems which are not transitive compact.",
"Observe that the sensitivity of a dynamical system can be lifted up from a factor to an extension by an almost open factor map between transitive systems by [17].",
"We prove that the transitive sensitivity can be lifted up to an extension from a factor by an almost one-to-one factor map and that the transitive sensitivity is projected from an extension to the sensitivity of a factor by a weakly almost one-to-one factor map (Lemma REF ).",
"In Section 5 we show that dynamical compactness can be used to characterize the weak disjointness of dynamical systems (Theorem REF ).",
"We also improve the result of Jian Li [27]: weak mixing implies $\\mathcal {F}_{ip}$ -point transitivity in terms of transitive compactness (Proposition REF ).",
"In Section 6 the further difference between weak mixing and transitive compactness is explored.",
"Precisely, there is a totally transitive, non weakly mixing, transitive compact system (Theorem REF ); and in fact any compact metric space can be realized as the $\\omega _{\\mathcal {N}_T}$ -limit set of a non totally transitive, transitive compact system $(X, T)$ (Theorem REF ).",
"As a byproduct, we show that the $\\omega _{\\mathcal {N}_T}$ -limit sets and the $\\omega $ -limit sets have quite different topological structures for a general dynamical compact system $(X,T)$ .",
"At the end of this section we add one more chaotical property of transitive compact systems (in additional to already known from [22]): transitive compactness implies Li-Yorke chaos (Proposition REF ).",
"In Section 7 we consider the dynamics of linear operators on infinite dimensional spaces in relation to the properties studied in previous sections.",
"In particular, we show the equivalence of the topological weak mixing property with a weak version of transitive compactness (Theorem REF ).",
"Some results on sensitivity are also obtained.",
"Acknowledgements.",
"The first and third authors acknowledge the hospitality of the School of Mathematical Sciences of the Fudan University, Shanghai.",
"The third author also acknowledges the hospitality of the Departament de Matemàtica Aplicada of the Universitat Politècnica de València and the Department of Mathematics of the Chinese University of Hong Kong.",
"The first author was supported by NNSF of China (11225105, 11431012); the fourth author was supported by MINECO, Project MTM2013-47093-P, and by GVA, Project PROMETEOII/2013/013; and the fifth author was supported by NNSF of China (11271078)."
],
[
"Preliminaries",
"In this section we recall standard concepts and results used in later discussions."
],
[
"Basic concepts in topological dynamics",
"Recall that $x\\in X$ is a fixed point if $T x= x$ , and an $\\mathcal {F}$ -transitive point of $(X, T)$ [27] if $n_T(x, U)\\in \\mathcal {F}$ for any opene subset $U$ of $X$ .",
"It is a trivial observation that if a family $\\mathcal {F}$ admits an $\\mathcal {F}$ -transitive dynamical system $(X,T)$ without isolated points, then $\\mathcal {F}$ is free.",
"Since $k(k\\mathcal {F})= \\mathcal {F}$ , it is easy to see that $x\\in X$ is an $\\mathcal {F}$ -transitive point of $(X, T)$ if and only if $\\omega _{k\\mathcal {F}}(x)=X$ .",
"Denote by ${\\mathrm {Tran}}_\\mathcal {F} (X, T)$ the set of all $\\mathcal {F}$ -transitive points of $(X, T)$ .",
"The system $(X, T)$ is $\\mathcal {F}$ -point transitive if ${\\mathrm {Tran}}_\\mathcal {F} (X, T)\\ne \\varnothing $ , and is $\\mathcal {F}$ -transitive if $N_T(U,V)\\in \\mathcal {F}$ for any opene subsets $U,V$ of $X$ .",
"Write ${\\mathrm {Tran}}(X, T)= {\\mathrm {Tran}}_{\\mathcal {P}_+} (X, T)$ for short, and we also call the point $x$ transitive if $x\\in {\\mathrm {Tran}}(X, T)$ , equivalently, its orbit ${\\mathrm {orb}}_T (x)= \\lbrace T^n x: x= 0, 1, 2, \\dots \\rbrace $ is dense in $X$ .",
"Since $T$ is surjective, the system $(X, T)$ is transitive if and only if ${\\mathrm {Tran}}(X, T)$ is a dense $G_\\delta $ subset of $X$ .",
"In general, a subset $A$ of $X$ is $T$ -invariant if $TA = A$ , and positively $T$ -invariant if $T A\\subset A$ .",
"If $A$ is a closed, nonempty, $T$ -invariant subset then $(A,T|_A)$ is called the associated subsystem.",
"A minimal subset of $X$ is a closed, nonempty, $T$ -invariant subset such that the associated subsystem is minimal.",
"Clearly, $(X,T)$ is minimal if and only if ${\\mathrm {Tran}}(X,T) = X$ , if and only if it admits no a proper, closed, nonempty, positively $T$ -invariant subset.",
"A point $x \\in X$ is called minimal if it lies in some minimal subset.",
"In this case, in order to emphasize the underlying system $(X, T)$ we also say that $x\\in X$ is a minimal point of $(X, T)$.",
"Zorn's Lemma implies that every closed, nonempty, positively $T$ -invariant set contains a minimal set.",
"A pair of points $x, y \\in X$ is called proximal if $\\liminf _{n\\rightarrow \\infty } d (T^n x, T^n y)= 0$ .",
"In this case each of points from the pair is said to be proximal to another.",
"Denote by ${\\mathrm {Prox}}_T (X)$ the set of all proximal pairs of points.",
"For each $x\\in X$ , denote by ${\\mathrm {Prox}}_T (x)$ , called the proximal cell of $x$, the set of all points which are proximal to $x$ .",
"Recall that a dynamical system $(X,T)$ is called proximal if ${\\mathrm {Prox}}_T (X)= X\\times X$ .",
"The system $(X,T)$ is proximal if and only if $(X,T)$ has the unique fixed point, which is the only minimal point of $(X,T)$ (e.g.",
"see [4]).",
"The opposition to the notion of sensitivity is the concept of equicontinuity.",
"Recall that $x \\in X$ is an equicontinuity point of $(X, T)$ if for every $\\varepsilon > 0$ there exists a $\\delta > 0$ such that $d (x, x^{\\prime }) < \\delta $ implies $d (T^n x, T^n x^{\\prime })< \\varepsilon $ for any $n\\in \\mathbb {Z}_+$ .",
"Denote by $\\text{Eq} (X, T)$ the set of all equicontinuity points of $(X, T)$ .",
"The system $(X, T)$ is called equicontinuous if $\\text{Eq} (X, T)= X$ .",
"Each dynamical system admits a maximal equicontinuous factor.",
"Recall that by a factor map $\\pi : (X, T)\\rightarrow (Y, S)$ between dynamical systems $(X, T)$ and $(Y, S)$ , we mean that $\\pi : X\\rightarrow Y$ is a continuous surjection with $\\pi \\circ T= S\\circ \\pi $ .",
"In this case, we call $\\pi : (X, T)\\rightarrow (Y, S)$ an extension; and $(X, T)$ an extension of $(Y, S)$ , $(Y, S)$ a factor of $(X, T)$ ."
],
[
"Basic concepts of Furstenberg families",
"In this subsection we recall from [1] basic concepts about Furstenberg families.",
"Let $F\\in \\mathcal {P}$ .",
"Recall that a subset $F$ is thick if it contains arbitrarily long runs of positive integers.",
"Denote by $\\mathcal {F}_{\\text{thick}}$ the set of all thick subsets of $\\mathbb {Z}_+$ , and define $\\mathcal {F}_{\\text{syn}}= k \\mathcal {F}_{\\text{thick}}$ .",
"Each element of $\\mathcal {F}_{\\text{syn}}$ is said to be syndetic, equivalently, $F$ is syndetic if and only if there is $N\\in \\mathbb {N}$ such that $\\lbrace i, i + 1, \\dots , i + N \\rbrace \\cap F \\ne \\varnothing $ for every $i \\in \\mathbb {Z}_{+}$ .",
"We say that $F$ is thickly syndetic if for every $N\\in \\mathbb {N}$ the positions where length $N$ runs begin form a syndetic set.",
"Denote by $\\mathcal {F}_{\\text{cof}}$ the set of all cofinite subsets of $\\mathbb {Z}_+$ .",
"Note that by the classic result of Gottschalk a point $x\\in X$ is minimal if and only if $n_T(x, U)= \\lbrace n\\in \\mathbb {Z}_+: T^n x\\in U\\rbrace $ is syndetic for any neighborhood $U$ of $x$ .",
"Hence, for any minimal system $(X, T)$ , the subset $N_T (U, V)$ is syndetic for any opene subsets $U, V$ of $X$ .",
"Recall that a family $\\mathcal {F}$ is proper if it is a proper subset of $\\mathcal {P}$ , that is, $\\mathbb {Z}_+\\in \\mathcal {F}$ and $\\varnothing \\notin \\mathcal {F}$ .",
"By a filter $\\mathcal {F}$ we mean a proper family closed under intersection, that is, $F_1, F_2\\in \\mathcal {F}$ implies $F_1\\cap F_2\\in \\mathcal {F}$ .",
"For families $\\mathcal {F}_1$ and $\\mathcal {F}_2$ , we define the family $\\mathcal {F}_1\\cdot \\mathcal {F}_2:= \\lbrace F_1\\cap F_2: F_1\\in \\mathcal {F}_1, F_2\\in \\mathcal {F}_2\\rbrace $ and call it the interaction of $\\mathcal {F}_1$ and $\\mathcal {F}_2$.",
"Thus we have $\\mathcal {F}_1 \\cup \\mathcal {F}_2 \\subset \\mathcal {F}_1 \\cdot \\mathcal {F}_2$ ; and it is easy to check that $\\mathcal {F}$ is a filter if and only if $\\mathcal {F}= \\mathcal {F}\\cdot \\mathcal {F}$ , and $\\mathcal {F}_1\\cdot \\mathcal {F}_2$ is proper if and only if $\\mathcal {F}_2 \\subset k\\mathcal {F}_1$ .",
"For each $i\\in \\mathbb {Z}_+$ , define $g^i: \\mathbb {Z}_+\\rightarrow \\mathbb {Z}_+, j\\mapsto i+ j$ .",
"Recall that a family $\\mathcal {F}$ is $+$ invariant if for every $i\\in \\mathbb {Z}_+$ , $F\\in \\mathcal {F}$ implies $g^{i} (F)\\in \\mathcal {F}$ ; $-$ invariant if for every $i\\in \\mathbb {Z}_+$ , $F\\in \\mathcal {F}$ implies $g^{- i} (F)\\in \\mathcal {F}$ , where $g^{-i}(F)=(g^i)^{-1}(F)=\\lbrace j-i:j\\in F,j\\ge i\\rbrace $ ; and translation invariant if it is both $+$ and $-$ invariant, equivalently, for every $i\\in \\mathbb {Z}_+$ , $F\\in \\mathcal {F}$ if and only if $g^{- i} (F)\\in \\mathcal {F}$ .",
"As $g^{- i} (g^i A)= A$ and $g^i (g^{- i} A)\\subset A$ for any $i\\in \\mathbb {Z}_{+}$ , it is easy to obtain that the family $\\mathcal {F}$ is $+$ ($-$ , translation, respectively) invariant if and only if $k \\mathcal {F}$ is $-$ ($+$ , translation, respectively) invariant (see for example [1]).",
"And then we have: Proposition 2.1 Let $x\\in X$ .",
"Then $T \\omega _\\mathcal {F} (x)\\subset \\omega _\\mathcal {F} (T x)$ .",
"Additionally, if $\\mathcal {F}$ is $-$ ($+$ , translation, respectively) invariant then $\\omega _\\mathcal {F} (T x)\\subset $ ($\\supset $ , $=$ , respectively) $\\omega _\\mathcal {F} (x)$ .",
"Since the other items are alternative versions of [1] in our notations, it suffices to prove that if $\\mathcal {F}$ is $+$ invariant then $\\omega _\\mathcal {F} (T x)\\supset \\omega _\\mathcal {F} (x)$ .",
"For each $y\\in \\omega _\\mathcal {F} (x)$ take an arbitrary neighborhood $U$ of $y$ , and let $F\\in \\mathcal {F}$ .",
"Then $g^{1} (F)= \\lbrace i+ 1\\in \\mathbb {Z}_+: i\\in F\\rbrace \\in \\mathcal {F}$ as $\\mathcal {F}$ is $+$ invariant, and hence $n_T (x, U)\\cap g^{1} (F)\\ne \\varnothing $ , thus $\\varnothing \\ne g^{- 1} (n_T (x, U)\\cap g^{1} (F))= n_T (T x, U)\\cap F$ .",
"It follows $y\\in \\omega _\\mathcal {F} (T x)$ from the arbitrariness of $U$ and $F$ , which finishes the proof.",
"Proposition 2.2 Let $(X,T)$ be a dynamical system and let $\\mathcal {F}$ be a family.",
"If $\\mathcal {F}$ is free, then $\\omega _{\\mathcal {F}}(x) \\subset \\omega _{T}(x)$ for any $x\\in X$ .",
"Moreover, if $(X,T)$ has a nonrecurrent point, then the converse is true.",
"If $\\mathcal {F}$ is free and has FIP then it has SFIP.",
"(i) Suppose $\\omega _{\\mathcal {F}}(x)\\ne \\varnothing $ and take a point $y \\in \\omega _{\\mathcal {F}} (x):=\\bigcap _{F \\in \\mathcal {F}} \\overline{T^{F}x}$ .",
"Let us show that $y \\in \\omega _{T} (x)$ .",
"Since $\\mathcal {F}$ is free, $\\bigcap _{F \\in \\mathcal {F}} T^{F}x=\\varnothing $ .",
"Otherwise there is $m \\in \\mathbb {Z}_+$ such that $T^mx \\in T^Fx$ for all $F \\in \\mathcal {F}$ , in other words $m \\in \\bigcap _{F \\in \\mathcal {F}}F$ , a contradiction.",
"It means that there is $F \\in \\mathcal {F}$ such that $y \\notin T^Fx$ .",
"So $y \\in \\overline{T^{F}x} \\setminus T^Fx$ , and $F$ is infinite.",
"Therefore there exists an infinite sequence $T^{n_1}x, \\dots , T^{n_i}x,\\dots , $ which converges to $y$ .",
"Hence $y \\in \\omega _T(x)$ .",
"Now, let $(X,T)$ be a dynamical system with a nonrecurrent point $x_0\\in X$ , let $\\mathcal {F}$ be a family and $\\omega _{\\mathcal {F}}(x) \\subset \\omega _{T}(x)$ for any $x\\in X$ .",
"Suppose $\\mathcal {F}$ is not free.",
"It means there is a $k\\in \\mathbb {Z}_{+}$ that lies in each element of $\\mathcal {F}$ .",
"Then obviously that $T^k(x)\\in \\omega _{\\mathcal {F}}(x)$ for any $x\\in X$ .",
"Since $x_0$ is nonrecurrent, $x_0\\notin \\omega _T(x_0)\\ne X$ .",
"It is well known that $\\omega _T(x)$ is $T$ -invariant, therefore $\\omega _T(y)=\\omega _T(x_0)$ for any $y\\in \\lbrace T^{-i}(x_0): i \\in \\mathbb {Z}_{+}\\rbrace $ , and $y\\notin \\omega _T(y)$ .",
"Take a point $x_k\\in X$ with $T^k(x_k)=x_0$ .",
"As we know $x_0 \\in \\omega _{\\mathcal {F}}(x_k)$ .",
"But $\\omega _{\\mathcal {F}}(x_k)\\subset \\omega _T(x_k)= \\omega _T(x_0)$ , a contradiction.",
"(ii) Suppose that for some $F_1, \\dots F_k \\in \\mathcal {F}$ $\\bigcap _{i=1}^k F_i=\\lbrace n_1, n_2, \\dots n_m\\rbrace $ .",
"Since $\\mathcal {F}$ is free for each $k \\in \\mathbb {Z}_+$ there is $G_k \\in \\mathcal {F}$ such that $k \\notin G_k$ .",
"Then $\\bigcap _{i=1}^k F_i \\cap \\bigcap _{j=1}^m G_{n_j}=\\varnothing $ , contradiction."
],
[
"The concept of an almost one-to-one map",
"Let $\\phi : X\\rightarrow Y$ be a continuous surjective map from a compact metric space $X$ onto a compact Hausdorff space $Y$ .",
"Recall that $\\phi $ is almost open if $\\phi (U)$ has a nonempty interior in $Y$ for any opene $U\\subset X$ .",
"Note that each factor map between minimal systems is almost open [5], in particular, for a minimal system $(X,T)$ the map $T:X\\rightarrow X$ is almost open [26].",
"Denote by $Y_0\\subset Y$ the set of all points $y\\in Y$ whose fiber is a singleton.",
"Then $Y_0$ is a $G_\\delta $ subset of $Y$ , because $Y_0= \\lbrace y\\in Y: \\phi ^{- 1} (y)\\ \\text{is a singleton}\\rbrace = \\bigcap _{n\\in \\mathbb {N}} \\left\\lbrace y\\in Y: {\\mathrm {diam}}(\\phi ^{- 1} (y))<\\frac{1}{n}\\right\\rbrace $ and the map $y\\mapsto {\\mathrm {diam}}(\\phi ^{- 1} (y))$ is upper semi-continuous.",
"Here, we denote by ${\\mathrm {diam}}(A)$ the diameter of a subset $A\\subset X$ .",
"Recall that the function $f: Y\\rightarrow \\mathbb {R}_+$ is upper semi-continuous if $\\limsup \\limits _{y\\rightarrow y_0} f (y)\\le f (y_0)$ for each $y_0\\in Y$ .",
"Denote by $X_0\\subset X$ the set of all points $x\\in X$ such that the pre-image of $\\phi (x)$ is a singleton.",
"Then $X_0= \\pi ^{- 1} (Y_0)$ is a $G_\\delta $ subset of $X$ .",
"We call $\\phi $ weakly almost one-to-one if $Y_0$ is dense in $Y$ , and almost one-to-oneHere we use the concept of almost one-to-one following [3], and the concept of almost one-to-one used in [12], [23], [26] is in fact our weakly almost one-to-one.",
"if $X_0$ is dense in $X$ .",
"It is not hard to show that: if $\\phi $ is weakly almost one-to-one, then for any $\\delta > 0$ and any opene subset $U$ of $Y$ there exists opene $V\\subset U$ with ${\\mathrm {diam}}(\\phi ^{- 1} V)< \\delta $ ; and if $\\phi $ is almost one-to-one, then for any opene subset $U^*$ of $X$ there exists an opene subset $V^*$ of $Y$ with $\\phi ^{- 1} V^*\\subset U^*$ .",
"Clearly almost one-to-one is much stronger than weakly almost one-to-one.",
"For example, let $X$ be the closed unit interval, define $T (x)= 2 x$ for $x\\in [0, \\frac{1}{2}]$ and $T (x)= 1$ for $x\\in [\\frac{1}{2}, 1]$ , and then $T: X\\rightarrow X$ is clearly not almost one-to-one but weakly almost one-to-one.",
"For each minimal system $(X, T)$ , the map $T: X\\rightarrow X$ is weakly almost one-to-one [26], and in fact almost one-to-one [23].",
"The following result characterizes the relationship between weakly almost one-to-one and almost one-to-one, which extends [23].",
"Proposition 2.3 Let $\\phi : X\\rightarrow Y$ be a continuous surjective map from a compact metric space $X$ onto a compact Hausdorff space $Y$ .",
"Then $\\phi $ is almost one-to-one if and only if it is not only almost open but also weakly almost one-to-one.",
"Firstly assume that $\\phi $ is almost one-to-one.",
"Let $U\\subset X$ be an arbitrary opene subset.",
"And then we can take $x_0\\in U$ such that the pre-image of $\\phi (x_0)$ is a singleton.",
"From this it is easy to see that $\\phi (x_0)$ is contained in the interior of $\\phi (U)$ .",
"This implies that $\\phi $ is almost open.",
"The map $\\phi $ is clearly weakly almost one-to-one.",
"Now assume that $\\phi $ is not only almost open but also weakly almost one-to-one.",
"Let $U\\subset X$ be an arbitrary opene subset.",
"Since $\\phi $ is almost open, $\\phi (U)$ has a nonempty interior in $Y$ , and then $\\phi ^{- 1} (y_0)$ is a singleton for some $y_0\\in \\phi (U)$ , as $\\phi $ is weakly almost one-to-one.",
"This shows $U\\cap X_0\\ne \\varnothing $ , which finishes the proof.",
"As a direct corollary, we have: Corollary 2.4 Let $\\phi : X\\rightarrow Y$ and $\\pi : Y\\rightarrow Z$ be continuous surjective maps between compact metric spaces.",
"Then the composition map $\\pi \\circ \\phi : X\\rightarrow Z$ is almost one-to-one if and only if both $\\phi $ and $\\pi $ are almost one-to-one.",
"Denote by $X_0$ ($X_1$ , respectively) the set of all points $x\\in X$ such that the pre-image of $(\\pi \\circ \\phi ) (x)$ ($\\phi (x)$ , respectively) is a singleton.",
"Denote by $Z_0$ ($Z_1$ , respectively) the set of all points $z\\in Z$ whose $\\pi \\circ \\phi $ -fibers ($\\pi $ -fibers, respectively) are singletons.",
"All of them are $G_\\delta $ subsets.",
"Moreover, $X_0= X_1\\cap \\phi ^{- 1} (\\pi ^{- 1} Z_1)$ .",
"In fact, $x\\in X_0$ if and only if $\\lbrace x\\rbrace = (\\pi \\circ \\phi )^{- 1} (\\pi \\circ \\phi (x))= \\phi ^{- 1} (\\pi ^{- 1} (\\pi (\\phi x)))$ , if and only if $\\pi ^{- 1} (\\pi (\\phi x))= \\lbrace \\phi (x)\\rbrace $ and $\\phi ^{- 1} (\\phi x)= \\lbrace x\\rbrace $ , if and only if $\\pi (\\phi x)\\in Z_1$ and $x\\in X_1$ .",
"First assume that $\\pi \\circ \\phi $ is almost one-to-one, and then by Proposition REF : $X_0$ is a dense subset of $X$ , $Z_0$ is a dense subset of $Z$ and the map $\\pi \\circ \\phi $ is almost open.",
"Note that $X_0\\subset X_1$ and $Z_0\\subset Z_1$ , we have that $X_1$ is dense in $X$ and $Z_1$ is dense in $Z$ .",
"Hence $\\phi $ is almost one-to-one.",
"Furthermore, as the map $\\pi \\circ \\phi $ is almost open, for any opene $V\\subset Y$ one has that $\\pi (V)= (\\pi \\circ \\phi ) (\\phi ^{- 1} V)$ has a nonempty interior in $Z$ , which implies that $\\pi $ is almost one-to-one by Proposition REF .",
"Now assume that both $\\phi $ and $\\pi $ are almost one-to-one.",
"Then $X_1$ is a dense $G_\\delta $ subset of $X$ and $Z_1$ is a dense $G_\\delta $ subset of $Z$ .",
"Moreover, by Proposition REF both $\\phi $ and $\\pi $ are almost open, and then the continuous surjection $\\pi \\circ \\phi $ is also almost open, which implies that $(\\pi \\circ \\phi )^{- 1} (Z_1)$ is also a dense $G_\\delta $ subset of $X$ .",
"Thus, $X_0= X_1\\cap \\phi ^{- 1} (\\pi ^{- 1} Z_1)$ is a dense $G_\\delta $ subset of $X$ , that is, the composition map $\\pi \\circ \\phi : X\\rightarrow Z$ is almost one-to-one.",
"This finishes the proof.",
"Let $\\pi : (X, T)\\rightarrow (Y, S)$ be a factor map between dynamical systems.",
"If the map $\\pi : X\\rightarrow Y$ is almost one-to-one (weakly almost one-to-one, respectively), then we also call $(X, T)$ an almost one-to-one extension (a weakly almost one-to-one extension, respectively) of $(Y, S)$.",
"The main result of [23] states that a minimal system is either multi-sensitive or a weakly almost one-to-one extension of its maximal equicontinuous factor.",
"This is an analog of the well-known Auslander-Yorke dichotomy theorem: a minimal system is either sensitive or equicontinuous."
],
[
"Symbolic dynamics",
"Let $A$ be a nonempty finite set.",
"We call $A$ the alphabet and elements of $A$ are symbols.",
"The full (one-sided) $A$ -shift is defined as $\\Sigma = \\lbrace x = \\lbrace x_i\\rbrace _{i=0}^\\infty : x_i \\in A \\text{ for all } i \\in \\mathbb {Z}_+\\rbrace ,$ where we equip $A$ with the discrete topology and $\\Sigma $ with the product topology, and the shift map $\\sigma : \\Sigma \\rightarrow \\Sigma $ is a continuous surjection given by $ x = \\lbrace x_i\\rbrace _{i=0}^\\infty \\mapsto \\sigma x = \\lbrace x_{i+1}\\rbrace _{i=0}^\\infty ,$ that is, $\\sigma (x)$ is the sequence obtained by dropping the first symbol of $x$ .",
"Usually we write an element of $\\Sigma $ as $x = \\lbrace x_i\\rbrace _{i=0}^\\infty = x_0 x_1 x_2 x_3 \\dots $ A block $w$ over $\\Sigma $ is a finite sequence of symbols and its length is the number of its symbols (denoted by $|w|$ ).",
"An $n$ -block stands for a block of length $n$ .",
"In general we are only interested in a block $w$ with $|w|\\ge 1$ if without any special statement, and denote by $\\Sigma ^*$ the set of all blocks over $\\Sigma $ .",
"The block $w$ is a subblock of a block $v= v_1 \\dots v_m$ with $v_1, \\dots , v_m\\in A$ if there exists $1\\le i\\le j\\le m$ with $w= v_i \\dots v_j$ .",
"The concatenation of two blocks $u = a_1 \\dots a_k$ and $v = b_1 \\dots b_l$ is the block $uv = a_1 \\dots a_k b_1 \\dots b_l$ .",
"We write $u^n$ for the concatenation of $n\\ge 1$ copies of a block $u$ and $u^\\infty $ for the sequence $uuu \\dots \\in \\Sigma .$ By $x_{[i,j]}$ we denote the block $x_i x_{i+1} \\dots x_j$ , where $0 \\le i \\le j$ and $x =\\lbrace x_k\\rbrace _{k=0}^\\infty \\in \\Sigma $ .",
"The subset $X\\subset \\Sigma $ is called a subshift if it is a closed, nonempty, $\\sigma $ -invariant subset of $\\Sigma $ .",
"A cylinder of an $n$ -block $w \\in \\Sigma ^*$ in a subshift $X$ is the set $C[w] = \\lbrace x \\in X: x_{[0, n-1]} = w\\rbrace $ .",
"The collection of all cylinders forms a basis of the topology of $X$ ."
],
[
"Dynamical compactness with respect to an arbitrary family",
"Recall that a family $\\mathcal {F}$ has the finite intersection property (FIP) if the intersection of any finite subcollection of $\\mathcal {F}$ is nonempty.",
"The following theorem shows that the FIP is useful in characterizing the dynamical compactness.",
"Theorem 3.1 All dynamical systems are dynamically compact with respect to the family $\\mathcal {F}$ if and only if $\\mathcal {F}$ has the finite intersection property.",
"Sufficiency.",
"Suppose that $\\mathcal {F}$ has FIP.",
"Take arbitrary dynamical system $(X, T)$ and let $x\\in X$ .",
"Obviously the family $\\lbrace \\overline{T^{F}x}: F\\in \\mathcal {F}\\rbrace $ also has FIP, and then by compactness of the space $X$ the family has a nonempty intersection itself, i.e., $\\omega _{\\mathcal {F}}(x)=\\bigcap _{F \\in \\mathcal {F}} \\overline{T^{F}x} \\ne \\varnothing $ .",
"Thus $(X, T)$ is dynamically compact with respect to $\\mathcal {F}$ .",
"Necessity.",
"Suppose that the family $\\mathcal {F}$ has no FIP.",
"And then there is a collection $\\lbrace F_1, \\dots , F_k \\rbrace \\subset \\mathcal {F}$ with $\\bigcap _{i=1}^k F_i=\\varnothing .$ Let $A=\\lbrace a_1, \\dots , a_k\\rbrace $ be an alphabet and let $(X, T):=(\\Sigma , \\sigma )$ be the full (one-sided) $A$ -shift.",
"We are going to define a point $x \\in X$ with $\\omega _\\mathcal {F}(x)= \\varnothing $ .",
"Let $x_0=a_1$ .",
"For any $n \\ge 1$ there is $i$ with $n \\notin F_i$ , else the intersection of $F_1, \\dots , F_k$ would be nonempty.",
"Then define $x_n:=a_i$ .",
"Finally, let $x=x_0 x_1 x_2 x_3\\dots $ and the construction is finished.",
"Assume the contrary that we can take $z \\in \\omega _\\mathcal {F}(x)$ , and that $z$ begins with $a_i \\in A$ .",
"Take $G_z=C[a_i]$ .",
"As $z \\in \\omega _\\mathcal {F}(x)$ we have $n_T(x, G_z) \\cap F_i \\ne \\varnothing $ .",
"But if $n \\in n_T(x, G_z)$ , then $x_n=a_i$ and so $n \\notin F_i$ by the construction, a contradiction.",
"As we have mentioned in Introduction, obviously that a filter $\\mathcal {F}$ (in particular $\\mathcal {N}_T$ , when $(X,T)$ is weakly mixing), the family $\\mathcal {N}_T(A)$ for a weakly mixing subset $A$ of $(X,T)$ and the family $\\mathcal {S}_T(\\delta )$ when $(X, T)$ is a multi-sensitive system (with a constant of multi-sensitivity $\\delta > 0$ ) have FIP.",
"Let $\\mathcal {F}$ has the finite intersection property.",
"Then there exists an ultrafilter $\\mathcal {U}$ (in $\\mathcal {P}$ ) such that $\\mathcal {F}\\subset \\mathcal {U}$ .",
"This result is known as Ultrafilter Lemma (see details and proof in [21]).",
"Recall that an ultrafilter is maximal among all proper filters.",
"As a consequence of this fact we have a natural open question: Question A Let $(X,T)$ be a dynamically compact system with respect to a family $\\mathcal {F}$ and $\\mathcal {F}$ has FIP.",
"When $\\mathcal {F}$ is a filter, or at least contains a nontrivial filter?",
"Especially we address this question to the family $\\mathcal {S}_T(\\delta )$ .",
"More precisely, when a system $(X,T)$ is dynamically compact with respect to the family $\\mathcal {N}_T$ and $\\mathcal {N}_T$ has FIP, then, as well known, the systems is weakly mixing and $\\mathcal {N}_T$ is a filter.",
"Now, let a system $(X,T)$ is dynamically compact with respect to the family $\\mathcal {S}_T(\\delta )$ for some $\\delta >0$ and $\\mathcal {S}_T(\\delta )$ has FIP, then the systems is multi-sensitive.",
"But, the following question is still open – when is $\\mathcal {S}_T(\\delta )$ a filter?",
"A collection $\\mathcal {H}\\subset \\mathcal {F}$ will be called a base for $\\mathcal {F}$ if for any $F\\in \\mathcal {F}$ there is $H \\in \\mathcal {H}$ with $H \\subset F$ .",
"We are interested in those families which have a countable base, that is, there exists a base $\\mathcal {H}$ which is countable.",
"Remark that not any Furstenberg family $\\mathcal {F}$ has a countable base, for example, the family $\\mathcal {B}$ .",
"Assume the contrary that $\\mathcal {B}$ admits a countable base $\\lbrace F_n: n\\in \\mathbb {N}\\rbrace $ .",
"We take $k_1\\in F_1$ , and once $k_m\\in F_m, m\\in \\mathbb {N}$ is defined we choose $k_{m+ 1}\\in F_{m+ 1}$ with $k_{m+ 1}> k_m+ m+ 1$ .",
"Set $E= \\lbrace k_n: n\\in \\mathbb {N}\\rbrace $ and $F= \\mathbb {Z}_+\\setminus E$ .",
"Then $E\\cap F_n\\ne \\varnothing $ for all $n\\in \\mathbb {N}$ , and $F\\supset \\lbrace k_m+ m: m\\in \\mathbb {N}\\rbrace $ and hence $F\\in \\mathcal {B}$ , in particular, there exists no $n\\in \\mathbb {N}$ with $F_n\\subset F$ , a contradiction.",
"Not so hard to show even the existence of a family with FIP, but without a countable base.",
"Nevertheless the families $\\mathcal {N}_T$ and $\\mathcal {S}_T (\\delta )$ have countable bases.",
"Indeed, we can consider a countable base $\\mathcal {U}$ of open sets for the space $X$ .",
"Note that $U_1 \\subset U$ , $V_1 \\subset V$ implies $N_T(U_1, V_1) \\subset N_T(U, V)$ and $S_T(U_1,\\delta ) \\subset S_T(U,\\delta )$ .",
"Then $\\lbrace N_T(U, V): ~U, V \\in \\mathcal {U}\\rbrace $ and $\\lbrace S_T(U,\\delta ):~U \\in \\mathcal {U}\\rbrace $ are countable bases for $\\mathcal {N}_T$ and $\\mathcal {S}_T (\\delta )$ , respectively.",
"The following is a general result that will be especially useful for families with countable bases.",
"Proposition 3.2 Let $(X,T)$ be a dynamical system and let $\\mathcal {F}$ be a family such that there exists $x\\in {\\mathrm {Tran}}_{\\mathcal {F}} (X, T)$ .",
"Then ${\\mathrm {orb}}_T(x)\\subset {\\mathrm {Tran}}_{\\mathcal {F}} (X, T)$ .",
"By assumption, given an arbitrary opene $U\\subset X$ we have that $n_T(x,U)\\in \\mathcal {F}$ .",
"Thus, for any $m\\in \\mathbb {N}$ , $n_T(T^mx,U)=n_T(x,T^{-m}(U))\\in \\mathcal {F},$ and we conclude that $T^mx\\in {\\mathrm {Tran}}_{\\mathcal {F}} (X, T)$ .",
"Proposition 3.3 Assume that $\\mathcal {F}$ admits a countable base $\\mathcal {H}$ .",
"Then ${\\mathrm {Tran}}_{k\\mathcal {F}} (X, T)$ is a $G_\\delta $ subset of $X$ .",
"Moreover, the following are equivalent: The system $(X,T)$ is $k \\mathcal {F}$ -transitive, ${\\mathrm {Tran}}_{k\\mathcal {F}} (X, T)$ is a dense $G_\\delta $ subset of $X$ , ${\\mathrm {Tran}}_{k\\mathcal {F}} (X, T)\\ne \\varnothing $ .",
"Let $\\mathcal {U}$ be a countable base of the family of all opene subsets of $X$ .",
"Then the class $\\mathcal {U} \\times \\mathcal {H}$ is countable, and we enumerate it as $\\lbrace (U_i, F_i): i \\in \\mathbb {N}\\rbrace $ .",
"Denote by $T^{- F} U= \\bigcup _{n\\in F} T^{- n} U$ for any $F\\subset \\mathbb {Z}_+$ and each $U\\subset X$ .",
"Then it is easy to obtain $ {\\mathrm {Tran}}_{k\\mathcal {F}}(X,T) = \\bigcap _{i=1}^\\infty T^{-F_i} U_i.$ In fact, given arbitrary point $x\\in X$ , $x\\in {\\mathrm {Tran}}_{k\\mathcal {F}}(X,T)$ if and only if $N_T (x, U)\\in k \\mathcal {F}$ for any opene subset $U$ of $X$ , if and only if $N_T (x, U)\\cap F\\ne \\varnothing $ for any opene subset $U$ of $X$ and each $F\\in \\mathcal {F}$ , if and only if $N_T (x, U_i)\\cap F_i\\ne \\varnothing $ for each $i\\in \\mathbb {N}$ by the construction.",
"In particular, ${\\mathrm {Tran}}_{k\\mathcal {F}} (X, T)$ is a $G_\\delta $ subset of $X$ .",
"Thus $(X, T)$ is $k \\mathcal {F}$ -transitive, if and only if for any $F\\in \\mathcal {F}$ and arbitrary opene subsets $U, V$ of $X$ we have $N_T (V, U)\\cap F\\ne \\varnothing $ and equivalently $T^{- F} U\\cap V\\ne \\varnothing $ , if and only if $T^{- F} U$ is an opene dense subset of $X$ for any $F\\in \\mathcal {F}$ and each opene subset $U$ of $X$ , if and only if ${\\mathrm {Tran}}_{k\\mathcal {F}} (X, T)$ is a dense $G_\\delta $ subset of $X$ by (REF ).",
"Now we assume ${\\mathrm {Tran}}_{k\\mathcal {F}} (X, T)\\ne \\varnothing $ .",
"Let $x\\in {\\mathrm {Tran}}_{k\\mathcal {F}} (X, T)$ .",
"By Proposition REF ${\\mathrm {orb}}_T (x)\\subset {\\mathrm {Tran}}_{k\\mathcal {F}} (X, T)$ , and hence ${\\mathrm {Tran}}_{k\\mathcal {F}} (X, T)$ is a dense $G_\\delta $ subset of $X$ since $x\\in {\\mathrm {Tran}}(X, T)$ .",
"This finishes the proof.",
"Remark 3.4 Observe that when the state space $X$ is a compact metric space without isolated points, $x\\in {\\mathrm {Tran}}(X, T)$ if and only if $x\\in {\\mathrm {Tran}}_\\mathcal {B} (X, T)$ .",
"The family $\\mathcal {F}_{\\text{cof}}$ is clearly translation invariant (and hence $+$ invariant) and admits a countable base, and $k \\mathcal {F}_{\\text{cof}}= \\mathcal {B}$ .",
"Thus by Proposition REF one has: $(X, T)$ is transitive if and only if ${\\mathrm {Tran}}(X, T)$ is a dense $G_\\delta $ subset of $X$ if and only if ${\\mathrm {Tran}}(X, T)\\ne \\varnothing $ ."
],
[
"Transitive sensitivity and sensitive compactness",
"Recall that a dynamical system $(X, T)$ is transitively sensitive if there exists $\\delta > 0$ such that $S_T (W, \\delta )\\cap N_T (U, V)\\ne \\varnothing $ for any opene subsets $U, V, W$ of $X$ ; and sensitive compact if there exists $\\delta > 0$ such that for any point $x\\in X$ the set $\\omega _{\\mathcal {S}_T(\\delta )}(x)$ is nonempty.",
"Sometimes in that cases we will say also $(X, T)$ is transitively sensitive with a sensitive constant $\\delta $ and $(X, T)$ is sensitive compact with a sensitive constant $\\delta $ .",
"The main result of this section is the following Theorem 4.1 Let $(X, T)$ be a minimal system.",
"Then the following conditions are equivalent: $(X, T)$ is multi-sensitive.",
"$(X, T)$ is sensitive compact.",
"There exists $\\delta > 0$ such that $\\omega _{\\mathcal {S}_T (\\delta )} (x)= X$ for each $x\\in X$ .",
"There exist $\\delta > 0$ and $x\\in X$ with $\\omega _{\\mathcal {S}_T (\\delta )} (x)= X$ .",
"$(X, T)$ is transitively sensitive.",
"Before proceeding, we need: Lemma 4.2 Let $\\delta > 0$ and $x\\in X$ .",
"If $T: X\\rightarrow X$ is almost open, then the family $\\mathcal {S}_T (\\delta )$ is $-$ invariant, and the subset $\\omega _{\\mathcal {S}_T (\\delta )} (x)$ is positively $T$ -invariant.",
"By Proposition REF it suffices to prove that $\\mathcal {S}_T (\\delta )$ is a $-$ invariant family.",
"Take arbitrary $F\\in \\mathcal {S}_T (\\delta )$ and any $i\\in \\mathbb {Z}_+$ .",
"Then there exists opene subset $U$ of $X$ with $S_T (U, \\delta )\\subset F$ .",
"As $T: X\\rightarrow X$ is almost open, $T^i: X\\rightarrow X$ is also almost open, and then we can choose opene $V \\subset T^i U$ .",
"One has $g^{- i} (F)\\supset g^{- i} S_T (U, \\delta )= S_T (T^i U, \\delta )\\supset S_T (V, \\delta )$ , which implies that the family $\\mathcal {S}_T (\\delta )$ is $-$ invariant.",
"The following result gives a characterization of transitive sensitivity for a general dynamical system in terms of dynamical compactness.",
"Proposition 4.3 Let $(X, T)$ be a dynamical system.",
"Then the family $\\mathcal {S}_T (\\delta )$ is $+$ invariant for any $ \\delta > 0$ .",
"Furthermore, the following conditions are equivalent: $(X, T)$ is transitively sensitive.",
"There exist a $\\delta > 0$ and a dense $G_\\delta $ subset $X_0\\subset X$ such that $\\omega _{\\mathcal {S}_T (\\delta )} (x)= X$ for each $x\\in X_0$ .",
"There exist a $\\delta > 0$ and a point $x\\in X$ with $\\omega _{\\mathcal {S}_T (\\delta )} (x)= X$ .",
"Firstly, we show that $\\mathcal {S}_T (\\delta )$ is a $+$ invariant family.",
"In fact, take any $F\\in \\mathcal {S}_T (\\delta )$ and each $i\\in \\mathbb {Z}_+$ .",
"We choose opene subsets $U, V$ of $X$ with $F\\supset S_T (U, \\delta )$ and $V\\subset T^{- i} U$ satisfying ${\\mathrm {diam}}(T^j V)< \\delta $ for all $j= 0, 1, \\dots , i$ .",
"Thus $g^i (F)\\supset g^i S_T (U, \\delta )\\supset S_T (V, \\delta )$ from the construction, and then $g^i (F)\\in \\mathcal {S}_T (\\delta )$ .",
"This implies the $+$ invariance of the family $\\mathcal {S}_T (\\delta )$ .",
"Observe that $(X, T)$ is transitively sensitive with a sensitive constant $\\delta $ , if and only if $(X, T)$ is $k \\mathcal {S}_T (\\delta )$ -transitive; and that the family $\\mathcal {S}_T (\\delta )$ has a countable base: let $\\mathcal {U}$ be a countable base of the family of all opene subsets of $X$ , then $\\lbrace S_T (U, \\delta ): U \\in \\mathcal {U}\\rbrace $ is a countable base of $\\mathcal {S}_T (\\delta )$ .",
"Then applying Proposition REF the equivalence of $(1)\\Leftrightarrow (2)\\Leftrightarrow (3)$ follows from the fact that $x\\in {\\mathrm {Tran}}_{k \\mathcal {S}_T (\\delta )} (X, T)$ if and only if $\\omega _{\\mathcal {S}_T (\\delta )} (x)= X$ .",
"Observe that by [17] the sensitivity of a dynamical system can be lifted up from a factor to an extension by an almost open factor map between transitive systems.",
"The following result gives the lift-up and projection property of transitive sensitivity between transitive systems.",
"Lemma 4.4 Let $\\pi : (X, T)\\rightarrow (Z, R)$ be a factor map between dynamical systems.",
"Assume that $\\pi $ is almost one-to-one.",
"If $(Z, R)$ is transitively sensitive with a sensitive constant $\\delta > 0$ then $(X, T)$ is also transitively sensitive.",
"Assume that there exists $z\\in Z$ whose fiber is a singleton.",
"If $(X, T)$ is transitively sensitive then $(Z, R)$ is sensitive, in particular, $\\text{Eq} (Z, R)= \\varnothing $ .",
"(1) We take a compatible metric $\\rho $ over $Z$ and let $\\varepsilon > 0$ such that $d (x_1, x_2)\\le \\varepsilon $ implies $\\rho (\\pi x_1, \\pi x_2)\\le \\delta $ for any $x_1, x_2\\in X$ .",
"Now let $U, V, W$ be arbitrary opene subsets of $X$ .",
"As the map $\\pi : X\\rightarrow Z$ is almost one-to-one, we may take opene subsets $U_Z, V_Z, W_Z$ of $Z$ with $\\pi ^{- 1} U_Z\\subset U$ , $\\pi ^{- 1} V_Z\\subset V$ and $\\pi ^{- 1} W_Z\\subset W$ .",
"Observe that if $n\\in S_R (W_Z, \\delta )\\cap N_R (U_Z, V_Z)$ , then: on one hand, there exist $z_1, z_2\\in W_Z$ with $\\rho (R^n z_1, R^n z_2)> \\delta $ , and so $d (T^n x_1, T^n x_2)> \\varepsilon $ for any $x_1\\in \\pi ^{- 1} (z_1)$ and $x_2\\in \\pi ^{- 1} (z_2)$ , hence ${\\mathrm {diam}}(T^n W)> \\varepsilon $ ; on the other hand, $U_Z\\cap R^{- n} V_Z\\ne \\varnothing $ , and then $U\\cap S^{- n} V\\supset \\pi ^{- 1} U_Z\\cap \\pi ^{- 1} (R^{- n} V_Z)\\ne \\varnothing $ .",
"This implies $S_T (W, \\varepsilon )\\cap N_T (U, V)\\supset S_R (W_Z, \\delta )\\cap N_R (U_Z, V_Z)\\ne \\varnothing $ , as $(Z, R)$ is transitively sensitive.",
"Thus, by the arbitrariness of $U, V$ and $W$ , we have that $(X, T)$ is also transitively sensitive.",
"(2) As $(X, T)$ is transitively sensitive (and assume with a sensitive constant $\\delta > 0$ ), it is clear that $(Z, R)$ is transitive, and then by the refined Auslander-York dichotomy the system $(Z, R)$ is sensitive if and only if $\\text{Eq} (Z, R)= \\varnothing $ (see [6], [17], [2] and the book [1]).",
"Thus it suffices to prove $\\text{Eq} (Z, R)= \\varnothing $ .",
"Let $\\rho $ be a compatible metric over $Z$ , and assume the contrary to take a point $z\\in \\text{Eq} (Z, R)$ .",
"By the assumption that there exists a point of $Z$ whose fiber is a singleton, we may take an opene subset $W$ in $Z$ with ${\\mathrm {diam}}(\\pi ^{- 1} W)< \\delta $ , and an opene subset $W_*\\subset W$ and $\\delta _*> 0$ such that if the distance between a point of $Z$ and $W_*$ is smaller than $\\delta _*$ then the point belongs to $W$ .",
"Since $z\\in \\text{Eq} (Z, R)$ , there exists an open neighborhood $U_*$ of $z$ with ${\\mathrm {diam}}(R^n U_*)< \\delta _*$ for all $n\\in \\mathbb {Z}_+$ .",
"As $(X, T)$ is transitively sensitive with a sensitive constant $\\delta $ , take $m\\in N_T (\\pi ^{- 1} U_*, \\pi ^{- 1} W_*)\\cap S_T (\\pi ^{- 1} U_*, \\delta )$ .",
"Thus $T^m (\\pi ^{- 1} U_*)\\cap \\pi ^{- 1} W_*\\ne \\varnothing $ , and then $R^m U_*\\cap W^*\\ne \\varnothing $ , which implies $R^m U_*\\subset W$ by the construction of $U_*, W_*$ and $W$ .",
"Hence ${\\mathrm {diam}}(T^m (\\pi ^{- 1} U_*))\\le {\\mathrm {diam}}(\\pi ^{- 1} (R^m U_*))\\le {\\mathrm {diam}}(\\pi ^{- 1} W)< \\delta ,$ a contradiction to the selection of $m\\in S_T (\\pi ^{- 1} U_*, \\delta )$ .",
"This finishes the proof.",
"Now we are ready to prove Theorem REF .",
"$(1)\\Rightarrow (2)$ follows directly from the definitions.",
"As the system $(X, T)$ is minimal, the map $T: X\\rightarrow X$ is almost open.",
"Observing that $\\omega _{\\mathcal {S}_T (\\delta )} (x)$ is a closed subset of $X$ for each $x\\in X$ , the implication of $(2)\\Rightarrow (3)$ follows from Lemma REF and the minimality of $(X, T)$ .",
"The implication of $(3)\\Rightarrow (4)\\Rightarrow (5)$ follows from Proposition REF .",
"Since a minimal system is either multi-sensitive or a weakly almost one-to-one extension of its maximal equicontinuous factor by [23], and then $(5)\\Rightarrow (1)$ follows from Lemma REF .",
"This finishes the proof.",
"Clearly each multi-sensitive system is sensitive compact.",
"Observe that each non-proximal, transitive compact system is multi-sensitive by [22].",
"In particular, each minimal transitive compact system is multi-sensitive, as each minimal proximal system is trivial by [4] and all dynamical systems considered are assumed to be nontrivial.",
"Nevertheless, there are many minimal, non transitive compact, multi-sensitive systems.",
"For example, consider the classical dynamical system $(X, T)$ given by $X= \\mathbb {R}^2/\\mathbb {Z}^2$ and $T: (x, y)\\mapsto (x+ \\alpha , x+ y)$ with $\\alpha \\notin \\mathbb {Q}$ (see [15]).",
"As commented in [22], $(X, T)$ is an invertible minimal multi-sensitive system; note that $(X, T)$ is not weakly mixing, since $(X, T)$ admits an irrational rotation as its nontrivial equicontinuous factor and any equicontinuous factor of a weakly mixing system is trivial.",
"Remark that by [22] for a minimal system the system is transitive compact if and only if it is weakly mixing, and then the constructed system $(X, T)$ is not transitive compact.",
"Proposition 4.5 Each weakly mixing system $(X, T)$ is transitively sensitive.",
"Observe that we are only interested a nontrivial dynamical system, and then let $0< \\delta < {\\mathrm {diam}}(X)$ .",
"We choose opene subsets $W_1, W_2$ of $X$ such that the distance between $W_1$ and $W_2$ is strictly larger than $\\delta $ .",
"Now take arbitrary opene subsets $U, V, W$ of $X$ .",
"As $(X, T)$ is weakly mixing, $(X^3, T^{(3)})$ is transitive by [14], and then $N_T (U, V)\\cap S_T (W, \\delta )\\supset N_{T^{(3)}} (U\\times W\\times W, V\\times W_1\\times W_2)\\ne \\varnothing $ .",
"This implies that $(X, T)$ is transitively sensitive with a sensitive constant $\\delta > 0$ .",
"We give a sufficient condition for a dynamical system being transitively sensitive (by Proposition REF ) as the end of this section.",
"Lemma 4.6 Assume $\\omega _{\\mathcal {S}_T (\\varepsilon )} (x)= X$ for some $x \\in X$ and $\\varepsilon >0$ .",
"Then there is $\\delta > 0$ such that for any opene subset $U$ of $X$ and each neighbourhood $U_x$ of $x$ there are $y\\in U_x$ and $n\\in N_T (x, U)$ with $d(T^n x, T^n y)>\\delta $ .",
"If in addition, the map $T: X\\rightarrow X$ is almost one-to-one, then the converse holds.",
"Fix an opene subset $U$ of $X$ and a neighborhood $U_x$ of $x$ .",
"As $\\omega _{\\mathcal {S}_T (\\varepsilon )} (x)= X$ , there is $n\\in N_T (x, U)\\cap S_T (U_x, \\varepsilon )$ , and then there are points $x_1, x_2 \\in U_x$ with $d(T^n x_1, T^n x_2)> \\varepsilon $ .",
"We have either $d (T^n x, T^n x_1)> \\frac{\\varepsilon }{2}$ or $d (T^n x, T^n x_2)> \\frac{\\varepsilon }{2}$ , and then obtain the desired statement for $\\delta = \\frac{\\varepsilon }{2}$ .",
"Now suppose that there is $\\delta > 0$ such that for any opene subset $U$ of $X$ and each neighbourhood $U_x$ of $x$ there are $y\\in U_x$ and $n\\in N_T (x, U)$ with $d(T^n x, T^n y)>\\delta $ , and that the map $T: X\\rightarrow X$ is almost one-to-one.",
"Let $U, W$ be arbitrary opene subsets of $X$ .",
"It is clear $x\\in {\\mathrm {Tran}}(X, T)$ , and so there is $k\\in N_T (x, W)$ .",
"Note that $T: X\\rightarrow X$ is almost one-to-one, the map $T^k: X\\rightarrow X$ is also almost one-to-one by Corollary REF , and then we may take an opene subset $V$ of $X$ with $T^{- k} V= (T^k)^{- 1} V\\subset U$ .",
"Observe that $T^{- k} W$ is a neighborhood of $x$ , and then by the assumption there exists a point $y\\in T^{- k} W$ and an integer $n\\in \\mathbb {Z}_+$ such that $n\\in N_T (x, V)$ and $d (T^n x, T^n y)> \\delta $ .",
"In fact, we may assume $n>k$ , else we can replace $T^{- k} W$ by a small enough open neighbourhood $G_x\\subset T^{- k} W$ of $x$ with ${\\mathrm {diam}}(T^i G_x)< \\delta $ for all $0\\le i\\le k$ .",
"Then ${\\mathrm {diam}}(T^{n- k} W)> \\delta $ as $T^k x\\in W$ and $T^k y\\in W$ , and $T^{n- k} x\\in T^{- k} (T^n x)\\subset T^{- k} V\\subset U$ .",
"In particular, $N_T (x, U)\\cap S_T (W, \\delta )\\ne \\varnothing $ .",
"Thus $\\omega _{\\mathcal {S}_T (\\delta )} (x)= X$ by the arbitrariness of $U$ and $W$ ."
],
[
"Weakly disjointness and weakly mixing",
"Recall that dynamical systems $(X, T)$ and $(Y, S)$ are weakly disjoint if the product system $(X\\times Y, T\\times S)$ is transitive.",
"The following theorem characterizes weak disjointness, which is proved firstly by Weiss [38] in some special class and then is generalized by Akin and Glasner [3].",
"We say that $\\mathcal {F}$ is thick if $\\tau \\mathcal {F}= \\mathcal {F}$ , where $\\tau \\mathcal {F}= \\left\\lbrace F\\subset \\mathbb {Z}_+: \\bigcap _{j= 1}^n g^{- i_j} F\\in \\mathcal {F}\\ \\text{for each}\\ n\\in \\mathbb {N}\\ \\text{and all}\\ i_1, \\dots , i_n\\in \\mathbb {Z}_+\\right\\rbrace .$ Weiss-Akin-Glasner Theorem Let $\\mathcal {F}$ be a proper, translation invariant, thick family.",
"A dynamical system is $k\\mathcal {F}$ -transitive if and only if it is weakly disjoint from every $\\mathcal {F}$ -transitive system.",
"Observe that a dynamical system is weakly mixing if and only if it is weakly disjoint from itself, and then weak disjointness is characterized by [22] in some special case.",
"Now we discuss weak disjointness using dynamical compactness which will be some generalization of [22].",
"We will need the following Lemma 5.1 Let $(X, T)$ and $(Y, S)$ be dynamical systems and let $x\\in X$ .",
"Then the family $\\mathcal {N}_S$ is translation invariant and $\\omega _{\\mathcal {N}_S} (x)= \\omega _{\\mathcal {N}_S} (T x)$ .",
"By Proposition REF it suffices to prove that $\\mathcal {N}_S$ is a translation invariant family.",
"We also suppose that $\\mathcal {N}_S$ is proper (i.e., $(Y,S)$ is a transitive system) since, otherwise, the result is trivial.",
"Take arbitrary $F\\in \\mathcal {N}_S$ and any $i\\in \\mathbb {Z}_+$ .",
"Then there exist opene subsets $U, V$ of $Y$ with $N_S (U, V)\\subset F$ .",
"As the non-singleton space $Y$ contains no isolated points, we can take suitable opene $V_1\\subset V$ and $U_1\\subset U$ such that $U_1\\cap \\bigcup _{k= 0}^i S^k \\overline{V_1}=\\varnothing $ .",
"One has $g^{i} N_S (U, V)\\supset N_S (S^{- i} U_1, V_1)$ , which implies that the family $\\mathcal {N}_S$ is $+$ invariant: in fact, if $n\\in N_S (S^{- i} U_1, V_1)$ then $n> i$ by the selection, and so $n- i\\in N_S (U_1, V_1)\\subset N_S (U, V)$ .",
"Moreover, it is clear $g^{- i} (F)\\supset g^{- i} N_S (U, V)\\supset N_S (U, S^{- i} V)$ , and then the family $\\mathcal {N}_S$ is $-$ invariant.",
"This finishes the proof.",
"Theorem 5.2 The following conditions are equivalent: The systems $(X, T)$ and $(Y, S)$ are weakly disjoint.",
"Both ${\\mathrm {Tran}}_{k \\mathcal {N}_S} (X, T)$ and ${\\mathrm {Tran}}_{k \\mathcal {N}_T} (Y, S)$ are dense $G_\\delta $ subsets.",
"The set ${\\mathrm {Tran}}_{k \\mathcal {N}_S} (X, T)$ is a dense $G_\\delta $ subset of $X$ .",
"Both ${\\mathrm {Tran}}_{k \\mathcal {N}_S} (X, T)$ and ${\\mathrm {Tran}}_{k \\mathcal {N}_T} (Y, S)$ are nonempty subsets.",
"The set ${\\mathrm {Tran}}_{k \\mathcal {N}_S} (X, T)$ is a nonempty subset of $X$ .",
"$(1)\\Leftrightarrow (2)\\Leftrightarrow (3)$ : It is clear from the definition that: the system $(X, T)$ is $k \\mathcal {N}_S$ -transitive, if and only if the systems $(X, T)$ and $(Y, S)$ are weakly disjoint, if and only if $(X, T)$ is $k \\mathcal {N}_S$ -transitive and $(Y, S)$ is $k \\mathcal {N}_T$ -transitive.",
"As both $\\mathcal {N}_T$ and $\\mathcal {N}_S$ are families admitting a countable base, it is direct to obtain the equivalence of $(1)\\Leftrightarrow (2)\\Leftrightarrow (3)$ by applying Proposition REF .",
"The implication $(2)\\Rightarrow (4)\\Rightarrow (5)$ is obvious.",
"To finish the proof, we only need to show $(5)\\Leftrightarrow (3)$ .",
"By Lemma REF the family $\\mathcal {N}_S$ is translation invariant, and then the family $k \\mathcal {N}_S$ is also translation invariant.",
"Thus the equivalence of $(5)\\Leftrightarrow (3)$ follows from Proposition REF .",
"Note that we have a characterization of weak mixing by using dynamical compactness [22].",
"Now we improve [22] as follows.",
"Recall that $\\mathcal {S}\\subset \\mathbb {N}$ is an IP set if there exists $\\lbrace p_k: k\\in \\mathbb {N}\\rbrace \\subset \\mathbb {N}$ with $FS\\lbrace p_i\\rbrace _{i=1}^\\infty \\subset \\mathcal {S}$ , where $FS\\lbrace p_i\\rbrace _{i=1}^\\infty =\\lbrace p_{i_1}+ \\dots + p_{i_k}: k\\in \\mathbb {N}\\ \\text{and}\\ 1\\le i_1< \\dots < i_k\\rbrace .$ Analogously, for each $n\\in \\mathbb {N}$ we define $FS\\lbrace p_i\\rbrace _{i=1}^n= \\lbrace p_{i_1}+ \\dots + p_{i_k}: k\\in \\mathbb {N}\\ \\text{and}\\ 1\\le i_1< \\dots < i_k\\le n\\rbrace .$ Denote by $\\mathcal {F}_{ip}$ the family of all IP sets.",
"By [27], the subset ${\\mathrm {Tran}}_{\\mathcal {F}_{ip}} (X, T)$ contains a dense $G_\\delta $ subset of $X$ for any weakly mixing system $(X, T)$ , while ${\\mathrm {Tran}}_{\\mathcal {F}_{ip}} (X, T) \\ne \\varnothing $ does not imply the weak mixing of the system $(X, T)$ by [27].",
"We will improve that in the following Proposition REF .",
"Before proceeding, we make the following Lemma 5.3 Let $(X, T)$ be a dynamical system and $\\mathcal {F}$ be a family.",
"Let $\\delta > 0$ .",
"If the family $\\mathcal {S}_T (\\delta )\\cdot \\mathcal {F}$ is proper then $\\mathcal {S}_T (\\delta )\\cdot \\mathcal {F}\\subset \\mathcal {B}$ .",
"If the family $\\mathcal {N}_T\\cdot \\mathcal {F}$ is proper then $\\mathcal {N}_T\\cdot \\mathcal {F}\\subset \\mathcal {B}$ .",
"(1) Assume the contrary that there exists an opene subset $U$ in $X$ and $F \\in \\mathcal {F}$ such that $S_T (U, \\delta )\\cap F$ is finite, and so we may choose $m \\in \\mathbb {N}$ such that $n \\notin S_T(U, \\delta )\\cap F$ for any integer $n>m$ .",
"Since $T:X \\rightarrow X$ is uniformly continuous one can find opene $V \\subset U$ small enough such that ${\\mathrm {diam}}(T^i V) < \\delta $ for all $0\\le i \\le m$ .",
"Then $S_T (V, \\delta ) \\subset S_T (U, \\delta )$ , which implies $S_T(V, \\delta ) \\cap F= \\varnothing $ .",
"A contradiction.",
"(2) Assume the contrary that there exist opene subsets $U, V$ in $X$ and $F \\in \\mathcal {F}$ such that $N_T(U, V)\\cap F$ is finite, say $N_T(U, V)\\cap F= \\lbrace n_1, \\dots , n_k\\rbrace $ .",
"As the non-singleton space $X$ contains no isolated points, we can take opene $U_1\\subset U$ small enough such that $V_1:= V\\setminus \\bigcup _{i= 1}^k T^{n_i} \\overline{U_1}$ is an opene subset of $X$ .",
"By the construction we have $N_T(U_1, V_1)\\subset N_T (U, V)$ and then $N_T(U_1, V_1)\\cap F= \\varnothing $ , a contradiction.",
"Proposition 5.4 The following conditions are equivalent: $(X, T)$ is weakly mixing.",
"There exists a dense $G_\\delta $ subset $X^{\\prime }$ of $X$ such that, for each $x\\in X^{\\prime }$ , $ n_T(x, G) \\cap N_T(U, V) \\in \\mathcal {F}_{ip}$ for all opene subsets $G, U, V$ of $X$ .",
"$(2)\\Rightarrow (1)$ : Just observe from the assumption that $\\omega _{\\mathcal {N}_T}(x)=X$ for all $x \\in X^{\\prime }$ , and hence the system $(X, T)$ is weakly mixing by [22].",
"(1) $\\Rightarrow $ (2): Since $(X, T)$ is weakly mixing, $(X^2, T^{(2)})$ is also weakly mixing by [14] and [34], and hence by [22] there is a dense $G_\\delta $ subset $Y \\subset X^2$ such that $\\omega _{\\mathcal {N}_{T^{(2)}}}((x_1, x_2))=X^2$ for each $(x_1, x_2) \\in Y$ .",
"Applying the well-known Ulam Lemma there is a dense $G_\\delta $ subset $X^{\\prime } \\subset X$ such that, for any $x \\in X^{\\prime }$ , $\\lbrace y: (x, y) \\in Y\\rbrace $ is a dense $G_\\delta $ subset of $X$ .",
"Now we show that $X^{\\prime }$ is the desired set.",
"Let $x \\in X^{\\prime }$ and fix any opene subsets $G, U, V$ of $X$ .",
"Choose $y \\in G$ with $(x, y) \\in Y$ and then $\\omega _{\\mathcal {N}_{T^{(2)}}}((x, y))=X^2$ , in particular, $(y, y) \\in \\omega _{\\mathcal {N}_{T^{(2)}}}((x, y))$ .",
"Thus $n_T(x, G)\\cap n_T(y, G) \\cap N_T(U, V) \\cap N_T(V, V) \\ne \\varnothing ,$ and take $p_1\\in \\mathbb {N}$ from this set by Lemma REF .",
"We have $p_1 \\in n_T(x, G) \\cap N_T(U, V)$ and $T^{p_1} y \\in G$ , $T^{p_1} V \\cap V \\ne \\varnothing $ .",
"Define opene subsets $G_1= G \\cap T^{- p_1}G\\ni y$ and $V_1=V \\cap T^{-p_1}V$ .",
"Now we proceed inductively.",
"Suppose that we are given a sequence $\\lbrace p_1, \\dots , p_k\\rbrace \\subset \\mathbb {N}$ with $FS\\lbrace p_i\\rbrace _{i=1}^{k} \\subset n_T(x, G) \\cap N_T(U, V)$ , and opene subsets $G_k=\\bigcap _{s \\in FS\\lbrace p_i\\rbrace _{i=1}^k \\cup \\lbrace 0\\rbrace } T^{-s} G\\ni y, V_k=\\bigcap _{s \\in FS\\lbrace p_i\\rbrace _{i=1}^k \\cup \\lbrace 0\\rbrace } T^{-s}V.$ As $(y, y) \\in \\omega _{\\mathcal {N}_{T^{(2)}}}((x, y))$ , we may take $p_{k+1}\\in \\mathbb {N}$ by Lemma REF from the set $n_T(x, G_k) \\cap n_T(y, G_k) \\cap N_T(U, V_k) \\cap N_T(V_k, V_k)$ .",
"It is not hard to check that $G_{k+1}=G_k \\cap T^{-p_{k+1}}G_k=\\bigcap _{s \\in FS\\lbrace p_i\\rbrace _{i=1}^{k+ 1} \\cup \\lbrace 0\\rbrace } T^{-s} G\\ni y,$ $V_{k+1}=V_k \\cap T^{-p_{k+1}}V_k=\\bigcap _{s \\in FS\\lbrace p_i\\rbrace _{i=1}^{k+1} \\cup \\lbrace 0\\rbrace } T^{-s}V$ are both opene subsets of $X$ , and that $FS \\lbrace p_i\\rbrace _{i=1}^{k+1} \\subset n_T(x, G) \\cap N_T(U, V)$ , which completes the induction.",
"Finally, $n_T(x, G) \\cap N_T(U, V)\\in \\mathcal {F}_{ip}$ with $FS \\lbrace p_i\\rbrace _{i=1}^{\\infty } \\subset n_T(x, G) \\cap N_T(U, V)$ .",
"This finishes the proof."
],
[
"Transitive compact (non weakly mixing) systems",
"Recall that the system $(X, T)$ is totally transitive if $(X, T^k)$ is transitive for each $k\\in \\mathbb {N}$ ; and is topologically mixing if $N_T(U,V)\\in \\mathcal {F}_\\text{cof}$ for any opene subsets $U,V$ in $X$ .",
"Note that $(X, T)$ is weakly mixing if and only if $N_T(U,V)\\in \\mathcal {F}_{\\text{thick}}$ for any opene sets $U,V$ in $X$ by [14], [34], and so any weakly mixing system is totally transitive.",
"It is direct to check that each weakly mixing system is transitive compact.",
"In [22] the authors showed the existence of non totally transitive, transitive compact systems in both proximal and non-proximal cases.",
"We extend it as follows: Theorem 6.1 There is a totally transitive, transitive compact system $(X, T)$ which is not weakly mixing.",
"Take a nontrivial proximal, topologically mixing system $(Y, S)$ and let $(\\mathbb {S}^1, R_\\alpha )$ be the standard irrational rotation on the unit circle $\\mathbb {S}^1=\\mathbb {R}/\\mathbb {Z}$ with $\\alpha \\notin \\mathbb {Q}$ .",
"Note that a dynamical system is proximal if and only if it contains the unique fixed point, which is the only minimal point of the system [4].",
"Denote by $p_Y$ the unique minimal point (fixed point) of $(Y, S)$ .",
"Observe that the system $(Y \\times \\mathbb {S}^1, S\\times R_\\alpha )$ is totally transitive: for each $n\\in \\mathbb {N}$ , the system $(Y, S^n)$ is topologically mixing by the definition and it is standard that the system $(\\mathbb {S}^1, R_\\alpha ^n)$ is minimal, then it is direct to see that these two systems are weakly disjoint.",
"Let $(X,T)$ be the quotient system $Y \\times \\mathbb {S}^1/ \\sim $ equipped with the action $T$ induced naturally from $S\\times R_\\alpha $ , where the equivalence relation $\\sim $ is defined via: given $x, y\\in X$ , $x\\sim y$ if and only if either $x=y$ or $x$ and $y$ both have $p_Y$ in the first coordinate.",
"In other words the space $X$ looks like a cone space, where the vertex of the cone is a point $p$ , each “horizontal” fiber spaces are the space $Y$ , the vertical fiber spaces are the circles (see Figure 1).",
"Clearly, $(X, T)$ is totally transitive.",
"calc [rotate=-90] asin(0.5/3) (h) at (0, 3); (O) at (0, 0); (A) at (-2*cos(), 3-0.5*sin()); (B) at (2*cos(), 3-0.5*sin()); [fill=gray!50] (A) – (O) – (B) – cycle; [fill=gray!30] (h) ellipse (2 and 0.5); (0,-0.2) node p; (1.15, 1.25) nodeY; (0,2.75) node$\\mathbb {S}^1$ ; Figure 1.",
"Denote by $q:Y \\times \\mathbb {S}^1 \\rightarrow X$ the corresponding quotient map, then $q: Y_\\infty \\times \\mathbb {S}^1\\rightarrow X \\setminus \\lbrace p\\rbrace $ is a homeomorphism, where we set $Y_\\infty = Y\\setminus \\lbrace p_Y\\rbrace $ .",
"It is standard that the system $(\\mathbb {S}^1, R_\\alpha )$ is not weakly mixing, and then there exist opene subsets $U_*, V_*$ of $\\mathbb {S}^1$ with $N_{R_\\alpha } (U_*, V_*)\\notin \\mathcal {F}_{\\text{thick}}$ , hence $N_{T} (q (Y_\\infty \\times U_*), q (Y_\\infty \\times V_*))= N_{S\\times R_\\alpha } (Y_\\infty \\times U_*, Y_\\infty \\times V_*)\\subset N_{R_\\alpha } (U_*, V_*)$ is not thick.",
"This implies that the system $(X, T)$ is not weakly mixing.",
"Now let $U, V$ be arbitrary opene subsets of $X$ .",
"We can choose opene subsets $U_1, V_1\\subset Y_\\infty $ and $U_2, V_2 \\subset \\mathbb {S}^1$ with $U_1\\times U_2 \\subset q^{- 1} U$ and $V_1\\times V_2 \\subset q^{- 1} V$ .",
"As $(Y, S)$ is topologically mixing and $(\\mathbb {S}^1, R_\\alpha )$ is minimal, $N_S (U_1, V_1)\\in \\mathcal {F}_\\text{cof}$ and $N_{R_\\alpha } (U_2, V_2)\\in \\mathcal {F}_\\text{syn}$ , and then $N_S (U_1, V_1)\\cap N_{R_\\alpha } (U_2, V_2)\\in \\mathcal {F}_\\text{syn}$ , thus $N_{T} (U, V)= N_{S\\times R_\\alpha } (q^{- 1} U, q^{- 1} V)\\supset N_{S\\times R_\\alpha } (U_1\\times U_2, V_1\\times V_2)$ is a syndetic set.",
"Observe from the construction that the system $(X, T)$ is proximal with $p$ as its unique fixed point, then $n_T (x, U_p)$ is a thickly syndetic subset for each point $x\\in X$ and any neighbourhood $U_p$ of $p$ (see [22]).",
"This implies $p \\in \\omega _{\\mathcal {N}_T} (x)$ for each $x\\in X$ , and then the system $(X, T)$ is transitive compact.",
"The following result is proved independently in [11] and [37].",
"Lemma 6.2 Any $\\omega $ -limit set $\\omega _T(x)$ can not be decomposed into $\\alpha $ disjoint closed, nonempty, positively $T$ -invariant subsets, where $2\\le \\alpha \\le \\aleph _0$ .",
"Before proceeding, we need the following example, for which we fail to find a reference and hence provide a detailed construction, as it is crucial in our arguments.",
"Proposition 6.3 For any given compact metric space $Z$ , there exists a topologically mixing system $(X, T)$ such that, $Z$ can be realized as the set of all its minimal points, furthermore, its each minimal point is a fixed point.",
"The construction is divided into two steps.",
"In the first step we shall construct a topologically mixing system $(Y, F)$ but with two fixed points, which are the only minimal points of the system.",
"Let $\\Sigma =\\lbrace 0,1\\rbrace ^{\\mathbb {Z}_+}$ and $\\sigma :\\Sigma \\rightarrow \\Sigma $ be the full (one-sided) shift.",
"We are going to find the system $(Y, F)$ of the form $(\\overline{{\\mathrm {orb}}_{\\sigma } (x)}, \\sigma )$ for some $x \\in \\Sigma $ .",
"In order to define $x\\in \\Sigma $ , firstly we represent each $W\\in \\Sigma ^*$ with $|W|\\ge 1$ in the following form: $W=a^{i} Q b^{j}$ , where $a^{i}$ and $b^{j}$ (with $i\\ge 1$ ) are the longest segments of equal digits which we can take at the beginning and at the end of $W$ , whereas $Q$ is the rest, possible the empty subblock.",
"Clearly, $j$ may be equal to 0 and then $b^j$ will be the empty subblock, in this case, we treat the digit $b$ as 0; in particular, if $W= a^k$ then we set $Q$ to be the empty subblock and $i= k$ , $b= 0$ , $j= 0$ .",
"Now we are going to define $x\\in \\Sigma $ .",
"Let $A_1=10$ be the first block of $x$ and define inductively the rest blocks $A_2, A_3, \\dots $ , then $x$ will be the limit of the starting blocks $A_k$ .",
"Suppose that we have defined $A_k, k\\in \\mathbb {N}$ .",
"Since $A_{k}$ has finitely many subblocks, there is a finite number of different pairs of these subblocks.",
"For any pair $(W_1, W_2)$ of subblocks of $A_k$ we will define a block $c (W_1, W_2)$ by using their combination.",
"Then we are ready to define $A_{k+1}$ : at the beginning of $A_{k+1}$ we write $A_k0^k1^k$ , and then all possible blocks $c (W_1, W_2)$ of pairs $(W_1, W_2)$ of subbloks of $A_{k}$ in any fixed order.",
"The definition $c (W_1, W_2)$ depends on the structure of $W_1$ and $W_2$ .",
"Let us write $W_1$ and $W_2$ in the form as above: $W_1=a^{i_1} Q_1 b^{j_1}$ and $W_2=c^{i_2} Q_2 d^{j_2}$ , where $a^{i_1}$ , $b^{j_1}$ , $c^{i_2}$ and $d^{j_2}$ (with $i_1, i_2\\ge 1$ ) are the longest segments of equal digits which we can take at the beginning and at the end of $W_1$ and $W_2$ , whereas $Q_1$ and $Q_2$ are the rest, possible empty subblocks ($j_1$ and $j_2$ may be equal to 0, and then we treat the corresponding digits as 0).",
"The combination block of the pair $W_1, W_2$ , i.e.",
"$c(W_1, W_2)$ , is defined as follows: $c(W_1, W_2)=a^{k+i_1} Q_1 b^{j_1+k}c^{k+i_2}Q_2d^{k+j_2}a^{k+i_1} Q_1 b^{j_1+k+1}c^{k+i_2}Q_2 d^{k+j_2} .$ We see from the construction that $A_{k+1}$ is presented as a sequence of blocks with length not longer than $|A_k|$ , which are separated from each other with some sequences of blocks of consecutive 1's or 0's of length not less than $k$ .",
"In fact, for all $m>k$ this property holds for $A_m$ (and hence in any subblock of $A_m$ with length more than $|A_k|+ 2k- 1$ one can find $0^k$ or $1^k$ ).",
"Suppose that $A_m$ may be presented in this form.",
"Observe that $A_{m+1}$ is obtained by adding to $A_m 0^m 1^m$ combination blocks $c(W_1^*, W_2^*)$ of all pairs $(W_1^*, W_2^*)$ of subblocks of $A_m$ glued in a proper way, and note that if the property holds for blocks $W_1^*$ and $W_2^*$ (with beginning and end parts of the blocks possible exception) then the property holds for the combination block $c(W_1^*, W_2^*)$ .",
"In particular, the property holds for $A_{m+ 1}$ .",
"Put $(Y, F):=(\\overline{{\\mathrm {orb}}_{\\sigma } (x)},\\sigma )$ .",
"Now let us check that it has the required properties.",
"Firstly, in any subblock of $x$ with length more than $|A_k|+2k-1$ one can find $0^k$ or $1^k$ .",
"And hence, if we increase the length of a block $w$ of $x$ the length of the biggest subblock of $w$ in form of $0^m$ or $1^m$ increases unboundedly.",
"Moreover, for each $k\\in \\mathbb {N}$ both $0^k$ and $1^k$ appear in $x$ .",
"In particular, both $0^\\infty $ and $1^\\infty $ are fixed points of $(\\overline{{\\mathrm {orb}}_{\\sigma } (x)}, \\sigma )$ , and there is no other minimal sets in it.",
"Recall that the base for the open sets in $\\Sigma $ is given by the collection of all cylinder sets $C[c_0c_1c_2 \\dots c_m]=\\lbrace x \\in \\Sigma :~ x_i=c_i \\text{ for } 0\\le i\\le m \\rbrace .$ Given any two subblocks $W_1$ and $W_2$ of $A_m$ , we write $W_1$ and $W_2$ in the form as above: $W_1=a^{i_1} Q_1 b^{j_1}$ and $W_2=c^{i_2} Q_2 d^{j_2}$ .",
"If $b= c$ , then for each $l\\ge m$ there exists a combination block $c_l (W_1, W_2)$ containing the subblock $W_1 b^l W_2$ , and hence $N_F ([W_1]\\cap Y, [W_2]\\cap Y)\\supset \\lbrace m+ |W_1|, m+ |W_1|+ 1, \\dots \\rbrace $ ; if $b\\ne c$ , then for each $l\\ge m$ there exists a combination block $c_l (W_1, W_2)$ containing the subblocks $W_1 b^k c^k W_2$ and $W_1 b^k 1 c^k W_2$ , and hence $N_F ([W_1]\\cap Y, [W_2]\\cap Y)\\supset \\lbrace 2 m+ |W_1|, 2 m+ |W_1|+ 1, \\dots \\rbrace $ .",
"This shows that the system $(Y, F)$ is topologically mixing.",
"Now we shall finish the construction by the second step.",
"Firstly, we take $(X^{\\prime }, T^{\\prime })$ to be the product system $\\prod _1^\\infty (Y, F)$ .",
"It is ready to check that the system $(X^{\\prime }, T^{\\prime })$ is topologically mixing, for which the middle-third Cantor set $C$ is the set of all its minimal points and its each minimal point is a fixed point.",
"Note that there exists a continuous surjection $h: C\\rightarrow Z$ (see for example [25]), and then we consider the quotient system $(X, T)$ with $X= X^{\\prime }/\\sim $ equipped with the action induced naturally from $T^{\\prime }$ , where the closed positively $T^{\\prime }\\times T^{\\prime }$ -invariant equivalence relation $\\sim $ is defined via $x\\sim y$ if and only if $x= y\\in X^{\\prime }\\setminus Z$ or $h (x)= h (y)$ for $x, y\\in Z$ .",
"Then the system $(X, T)$ has the required properties.",
"The following result shows that in general there is no a topological structure similar to Lemma REF for the $\\omega _{\\mathcal {N}_T}$ -limit sets.",
"Theorem 6.4 For any given compact metric space $Z$ , there exists a non totally transitive, transitive compact system $(X, T)$ such that, $Z$ can be realized as the set of all its minimal points with its each minimal point being a fixed point, furthermore, $Z$ is realized as $\\omega _{\\mathcal {N}_T} (x)$ for some $x\\in X$ .",
"The idea of the proof is very similar to that of the first part (proximal case) of [22].",
"Instead of a nontrivial proximal, topologically mixing system $(Y, F)$ there (main point is “a map with exactly one minimal point!”), we take again a topologically mixing system $(Y, F)$ , but with $Z$ realized as the set of all its minimal points where its each minimal point is a fixed point (for existence of such a dynamical system see Proposition REF ).",
"Then, by the wedge sum construction there, we obtain a non totally transitive system $(X, T)$ such that $Z$ is realized as the set of all its minimal points with its each minimal point being a fixed point.",
"Similar to arguments there, it is not hard to show $Z\\supset \\omega _{\\mathcal {N}_T} (x)$ for all $x\\in X$ .",
"Firstly we prove the following claim: Claim.",
"For $x\\in X$ , if $x= y\\in Y$ and $p\\in Z\\cap \\omega _{\\mathcal {N}_{F^2}} (y)$ then, as a point in $X$ , $p$ belongs to $\\omega _{\\mathcal {N}_T} (x, T)$ .",
"Let $U_p$ be an open subset of $X$ containing $p$ , and clearly $U_p$ may be also viewed as an open subset of $Y$ containing $p$ .",
"Now for any given opene subsets $U$ and $V$ of $X$ : if both $U$ and $V$ can be viewed as opene subsets of $Y$ , then we can take $n\\in N_{F^2} (y, U_p)\\cap N_{F^2} (U, V)$ by the assumption $p\\in \\omega _{\\mathcal {N}_{F^2}} (y)$ and hence $2 n\\in N_{T} (y, U_p)\\cap N_{T} (U, V)$ ; if both $U$ and $V$ can be viewed as opene subsets of $Y_c$ , then both $T^{- 1} U$ and $T^{- 1} V$ can be viewed as opene subsets of $Y$ and hence $N_{T} (y, U_p)\\cap N_{T} (U, V)= N_{T} (y, U_p)\\cap N_{T} (T^{- 1} U, T^{- 1} V)\\ne \\varnothing $ ; if $U$ and $V$ can be viewed as opene subsets of $Y$ and $Y_c$ , respectively, noting $p\\in Z$ and hence $T p= p$ , there is an opene subset $V_p$ of $X$ containing $p$ such that $T V_p\\subset U_p$ , and then by the above reasoning we may take $n\\in N_{T} (y, V_p)\\cap N_{T} (U, T^{- 1} V)$ , and hence $n+ 1\\in N_{T} (y, U_p)\\cap N_{T} (U, V)$ ; it can be treated similarly the other case that $U$ and $V$ can be viewed as opene subsets of $Y_c$ and $Y$ , respectively.",
"Now we continue our proof.",
"As $(Y, F)$ is topologically mixing, the system $(Y, F^2)$ is weakly mixing, and then by [22] we may choose $x^*\\in Y$ such that $\\omega _{\\mathcal {N}_{F^2}} (x^*)= Y$ , in particular, $\\omega _{\\mathcal {N}_{F^2}} (x^*)\\supset Z$ .",
"Thus, by the above Claim, we obtain $Z\\subset \\omega _{\\mathcal {N}_T} (x^*)$ and hence $Z= \\omega _{\\mathcal {N}_T} (x^*)$ .",
"Note that a dynamical system is proximal if and only if it contains the unique fixed point, which is the only minimal point of the system [4].",
"Thus, as a direct corollary of Lemma REF and Theorem REF , we have: Corollary 6.5 There exists a non-proximal, non totally transitive, transitive compact system $(X, T)$ and a point $x_0\\in X$ such that $\\omega _{\\mathcal {N}_T} (x_0)\\ne \\omega _T (x)$ for all $x\\in X$ .",
"Nevertheless is still open the following Question B Let $(X,T)$ be a weakly mixing system.",
"Is any $\\omega _{\\mathcal {N}_T}(x)$ undecomposable into $\\alpha $ disjoint closed, nonempty, positively $T$ -invariant subsets, where $2\\le \\alpha \\le \\aleph _0$ ?",
"At the end of this section let us prove one more chaotical property of transitive compact systems in additional to already known in [22].",
"Recall that a pair of points $x, y \\in X$ is asymptotic if $\\lim _{n\\rightarrow \\infty } d (T^n x, T^n y)= 0$ .",
"Denote by ${\\mathrm {Asym}}_T (X)$ the set of all asymptotic pairs of points.",
"Any pair $(x, y)\\in {\\mathrm {Prox}}_T (X)\\setminus {\\mathrm {Asym}}_T (X)$ is called a Li-Yorke pair.",
"Recall that a dynamical system $(X, T)$ is Li-Yorke chaotic if there exists an uncountable set $S\\subset X$ with $(S\\times S)\\setminus \\Delta _2 (X)\\subset {\\mathrm {Prox}}_T (X)\\setminus {\\mathrm {Asym}}_T (X)$ , where $\\Delta _2 (X)= \\lbrace (x, x): x\\in X\\rbrace $ .",
"Proposition 6.6 Each transitive compact system $(X, T)$ is Li-Yorke chaotic.",
"Clearly $(X, T)$ is transitive.",
"Observe that we have assumed the state space to be not a singleton and in fact a compact metric space without isolated points, then $(X, T)$ is a transitive system with $X$ infinite.",
"Thus, the subset ${\\mathrm {Asym}}_T (X)$ is a fist category subset of $X\\times X$ by [24].",
"It is easy to show that ${\\mathrm {Prox}}_T (X)$ is a $G_\\delta $ subset of $X\\times X$ , and applying [22] to the transitive compact system $(X, T)$ we have that ${\\mathrm {Prox}}_T (x)$ is a dense subset of $X$ for each $x\\in X$ .",
"Thus ${\\mathrm {Prox}}_T (X)$ is a dense $G_\\delta $ subset of $X\\times X$ , and then ${\\mathrm {Prox}}_T (X)\\setminus {\\mathrm {Asym}}_T (X)$ is a second category subset of $X\\times X$ .",
"Now applying the well-known Mycielski Theorem [31] we obtain an uncountable subset $S\\subset X$ with $(S\\times S)\\setminus \\Delta _2 (X)\\subset {\\mathrm {Prox}}_T (X)\\setminus {\\mathrm {Asym}}_T (X)$ .",
"That is, $(X, T)$ is Li-Yorke chaotic."
],
[
"Weak transitive compactness and sensitivity for linear operators",
"In this section we are considering the dynamics of linear operators on infinite dimensional spaces in relation to the properties studied in previous sections.",
"More precisely, we will show the equivalence of the topological weak mixing property with a weak version of transitive compactness.",
"We obtain some results on transitive sensitivity too.",
"One should keep in mind that, for a linear dynamical system $(X, T)$ , where $X$ is an infinite dimensional space, neither compactness nor even local compactness of $X$ is satisfied.",
"In particular, we are interested in the case where $X$ is an infinite dimensional separable Banach space and $T:X\\rightarrow X$ is a continuous linear map (in short, operator).",
"In this framework, we will just write $(X,T)$ is an infinite dimensional linear dynamical system.",
"We recall that $X$ is a Banach space if it is a vector space endowed with a norm $\\Vert \\cdot \\Vert $ such that $X$ with the associated distance $d(x,y):=\\Vert x-y\\Vert $ becomes a complete metric space.",
"It is well known that $T: X\\rightarrow X$ is an operator if and only if $\\Vert T\\Vert :=\\sup \\lbrace \\Vert Tx\\Vert : \\Vert x\\Vert \\le 1\\rbrace <\\infty $ .",
"We refer the reader to the books [7] and [19] for the theory of linear dynamics.",
"Note that all notations and concepts discussed in previous sections can be introduced into linear dynamics.",
"We also introduce a weak version of dynamical compactness.",
"A linear system $(X,T)$ is called weakly dynamically compact with respect to the family $\\mathcal {F}$ if there exists a dense subset $X_0\\subset X$ such that the $\\omega _\\mathcal {F}$ -limit set $\\omega _{\\mathcal {F}}(x)$ is nonempty for all $x\\in X_0$ .",
"In particular, $(X, T)$ is called weakly transitive compact, if there exists a dense subset $X_0\\subset X$ such that for any point $x\\in X_0$ the $\\omega _{\\mathcal {N}_T}$ -limit set $\\omega _{\\mathcal {N}_T}(x)$ is nonempty, in other words, for any point $x\\in X_0$ there exists a point $z \\in X$ such that $ n_T(x,G) \\cap N_T(U,V)\\ne \\varnothing $ for any neighborhood $G$ of $z$ and any opene subsets $U,V$ of $X$ .",
"Theorem 7.1 Let $(X, T)$ be an infinite dimensional linear system.",
"Then $(X, T)$ is weakly mixing if and only if it is weakly transitive compact.",
"Sufficiency.",
"Suppose that $(X, T)$ is weakly transitive compact.",
"Let $X_0\\subset X$ be a dense subset such that, for each $x\\in X_0$ , there exists $z(x) \\in X$ such that $ n_T(x,G) \\cap N_T(U,V)\\ne \\varnothing $ for any neighborhood $G$ of $z (x)$ and opene $U, V\\subset X$ .",
"As $(X,T)$ is obviously transitive, by [18] (see also [19]) to obtain the weak mixing property we just need to show that, for each opene $U\\subset X$ and 0-neighbourhood $W$ , there is a continuous map $S: X\\rightarrow X$ commuting with $T$ such that $ S(U)\\cap W\\ne \\varnothing \\ \\ \\ \\mbox{ and } \\ \\ \\ S(W)\\cap U\\ne \\varnothing .$ Given an opene subset $U$ of $X$ and a 0-neighborhood $W$ , we fix $x\\in U\\cap X_0$ and $z(x)\\in X$ accordingly to the weak transitive compactness of $(X,T)$ .",
"Since 0-neighbourhoods are absorbing, we find a scalar $\\lambda \\ne 0$ such that $\\lambda z (x)\\in W$ .",
"Let $G$ be a neighbourhood of $z (x)$ such that $\\lambda G\\subset W$ .",
"By the hypothesis we can find $m\\in n_T(x,G) \\cap N_T(\\lambda W,U).$ That is, $T^mx\\in G$ and so $\\lambda T^mx\\in W$ ; additionally, there exists $w\\in W$ with $T^m\\lambda w\\in U$ .",
"Now pick $S:=\\lambda T^m$ , we have that $S$ commutes with $T$ and the property (REF ) is satisfied, therefore the system is weakly mixing.",
"Necessity.",
"Conversely, under the assumption of the weak mixing property for $(X,T)$ , we know by [8] (see also [19]) that there exists an increasing sequence $\\lbrace n_k: k\\in \\mathbb {N}\\rbrace \\subset \\mathbb {N}$ and a dense subset $X_0\\subset X$ such that $T^{n_k}x\\rightarrow 0$ for each $x\\in X_0$ and, for arbitrary opene $U,V\\subset X$ , we can find $k\\in \\mathbb {N}$ such that $T^{n_k}(U)\\cap V\\ne \\varnothing $ .",
"Thus, we obtain easily that $(X,T)$ is weakly transitive compact by selecting $z(x)=0$ for every $x\\in X_0$ .",
"Concerning sensitivity, the situation is more complicated and, although we obtain some advances, three related problems are left open.",
"Proposition 7.2 Let $(X, T)$ be an infinite dimensional linear, topologically transitive system.",
"Then $(X, T)$ is thickly multi-sensitive, that is, there exists $\\delta > 0$ such that $\\bigcap _{i= 1}^k S_T (U_i, \\delta )$ is thick for any finite collection of opene $U_1, \\dots , U_k\\subset X$ .",
"Let $U_1, \\dots , U_k$ be opene sets, and let $m\\in \\mathbb {N}$ .",
"Pick points $x_1, \\dots , x_k$ such that $x_i \\in U_i$ and choose $\\varepsilon >0$ such that $B_\\varepsilon (x_i) \\subset U_i$ , where $B_\\varepsilon (x_i)$ is the open ball of radius $\\varepsilon $ centered at $x_i$ , for all $i \\in \\lbrace 1, \\dots , k\\rbrace $ .",
"By a hypercyclic vector we mean that its orbit is dense in the space $X$ .",
"Take a hypercyclic vector $u \\in B_\\varepsilon (0)$ by [19], and let $y_i=x_i+u$ .",
"Then $y_i \\in U_i$ by the construction.",
"Since $u$ is hypercyclic there is $n \\in \\mathbb {N}$ , $n>m$ , such that $\\Vert T^n u\\Vert >(\\Vert T\\Vert +1)^m$ .",
"Then $\\rho (T^{n-j}x_i, T^{n-j}y_i)=\\Vert T^{n-j}(x_i-y_i)\\Vert =\\Vert T^{n-j}u\\Vert >(\\Vert T\\Vert +1)^{m-j}>1$ for all $i= 1, \\dots , k$ and $j= 0, \\dots , m- 1$ .",
"Hence $\\lbrace n, n- 1, \\dots , n-m+1\\rbrace \\subset \\bigcap _{i=1}^k S_T(U_i, 1)$ , and therefore $(X, T)$ is thickly multi-sensitive.",
"Proposition 7.3 Let $(X,T)$ be an infinite dimensional linear system.",
"Then the following conditions are equivalent: For each $\\delta > 0$ , $(X, T)$ is transitively sensitive with a sensitive constant $\\delta $ .",
"There exists $\\delta _0> 0$ such that $(X, T)$ is transitively sensitive with a sensitive constant $\\delta _0$ .",
"There exists $\\delta _0> 0$ such that $S_T (W_0, \\delta _0)\\cap N_T (U, V)\\ne \\varnothing $ for any opene subsets $U, V$ of $X$ and any 0-neighbourhood $W_0$ .",
"We just need to show $(3) \\Rightarrow (1)$ .",
"Indeed, let $\\delta >0$ be arbitrary, and fix arbitrary opene $U, V, W$ of $X$ .",
"We select $\\varepsilon >0$ and $x\\in W$ such that $x+B_\\varepsilon (0)\\subset W$ .",
"Observing $S_T (\\lambda W_0, \\lambda \\delta _0)$ for any scalar $\\lambda \\ne 0$ , and so without loss of generality we assume $\\delta >\\delta _0$ .",
"Let $0<\\varepsilon ^{\\prime }< \\frac{\\delta _0 \\varepsilon }{\\delta }$ , and set $W_0=B_{\\varepsilon ^{\\prime }}(0)$ .",
"By the hypothesis there are $y,z\\in W_0$ and $n\\in N_T(U,V)$ such that $\\Vert T^ny-T^nz\\Vert > \\delta _0$ .",
"Set $y^{\\prime }=x+\\frac{\\delta }{\\delta _0} y$ and $z^{\\prime }=x+\\frac{\\delta }{\\delta _0} z$ .",
"We have $y^{\\prime }, z^{\\prime }\\in W$ and $\\Vert T^ny^{\\prime }-T^nz^{\\prime }\\Vert > \\delta $ .",
"As opene $U, V, W\\subset X$ are arbitrary, $(X, T)$ is transitively sensitive with a sensitive constant $\\delta $ .",
"In this framework the weak mixing property implies transitive sensitivity too.",
"The following result establishes a very close connection of transitivity with transitive sensitivity.",
"We do not know, however, whether every transitive linear system is transitively sensitive.",
"Proposition 7.4 Let $(X,T)$ be an infinite dimensional linear, topologically transitive system.",
"If $(X, T)$ is not transitively sensitive, then there exists a dense open subset $U_0\\subset X$ such that every $x\\in U_0$ has a dense orbit.",
"If $(X,T)$ is not transitively sensitive, by Proposition REF we find opene $U, V$ of $X$ and $\\delta >1$ such that $\\Vert T^nx\\Vert \\le \\delta $ whenever $n\\in N_T(U,V)$ and $\\Vert x\\Vert \\le 1$ .",
"We fix an arbitrary opene $V^{\\prime }\\subset V$ and select an opene $\\widehat{V}\\subset V^{\\prime }$ and $\\varepsilon >0$ such that $\\widehat{V}+B_\\varepsilon (0)\\subset V^{\\prime }$ .",
"Given $u\\in U$ , there is $\\varepsilon ^{\\prime }< \\frac{\\varepsilon }{\\delta }$ such that $U^{\\prime }:=u+B_{\\varepsilon ^{\\prime }}(0)\\subset U$ .",
"Since $T$ is transitive, there exists $m\\in N_T(U^{\\prime },\\widehat{V})\\subset N_T(U,V)$ .",
"That is, we find $u^{\\prime }=u+w\\in U^{\\prime }$ with $\\Vert w\\Vert <\\varepsilon ^{\\prime }$ and $T^mu^{\\prime }\\in \\widehat{V}$ .",
"By the assumption $\\Vert T^mw\\Vert \\le \\delta \\varepsilon ^{\\prime }<\\varepsilon $ .",
"Therefore, $T^mu=T^mu^{\\prime }-T^mw\\in \\widehat{V}+B_\\varepsilon (0)\\subset V^{\\prime }$ .",
"Since $u\\in U$ and opene $V^{\\prime }\\subset V$ are arbitrary, we obtain that the orbit of every element in $U$ is somewhere dense, thus everywhere dense by transitivity of the system.",
"Finally, the open set $U_0:=\\bigcup _{n\\in \\mathbb {N}}T^{-n}(U)$ is dense, and every element in $U_0$ has a dense orbit.",
"There are (very difficult) examples of linear systems $(X,T)$ such that every non-zero element has a dense orbit [35], but it seems to unknown whether every linear system that admits an open set of elements whose orbit is dense is so that every non-zero element has a dense orbit.",
"It is also worthy to mention that there are (also rare) examples of transitive but not weakly mixing linear systems [10] (see also [7]), but as far as we know there are no examples of transitive non-weakly mixing linear systems such that every non-zero element has a dense orbit.",
"Concerning weak disjointness, observe that for each separable Banach space the family of all opene subsets admits a countable base, and then it is a routine to show that Theorem REF holds true within linear systems too.",
"Note that the intersection of finitely many thickly syndetically sets is still thickly syndetic, and that an interesting property is that every topologically ergodic linear system $(X, T)$ (i.e., each element of $\\mathcal {N}_T$ is a syndetic set) satisfies that each element of $\\mathcal {N}_T$ is actually a thickly syndetic set (see the exercises in [19]).",
"Thus any finite family $(X_1,T_1), \\dots ,(X_k,T_k)$ of topologically ergodic linear systems is weakly disjoint and, moreover, the product system $(X_1\\times \\dots \\times X_k, T_1\\times \\dots \\times T_k)$ is topologically ergodic."
]
] | 1605.05851 |
[
[
"Exploiting Equitable Partitions for Efficient Block Triangularization"
],
[
"Abstract In graph theory a partition of the vertex set of a graph is called equitable if for all pairs of cells all vertices in one cell have an equal number of neighbours in the other cell.",
"Considering the implications for the adjacency matrix one may generalize that concept as a block partition of a complex square matrix s.t.",
"each block has constant row sum.",
"It is well known that replacing each block by its row sum yields a smaller matrix whose multiset of eigenvalues is contained in the initial spectrum.",
"We generalize this approach to weighted row sums and rectangular matrices and derive an efficient unitary transformation which approximately block triangularizes a matrix w.r.t.",
"an arbitrary partition.",
"Singular values and Hermiticity (if present) are preserved.",
"The approximation is exact in the equitable case and the error can be bounded in terms of unitarily invariant matrix norms."
],
[
"Equitable Partitions",
"Let $\\Gamma $ be a (multi-)graph and let $\\mathbf {A}$ be its adjacency matrix, whose entries $a_{vw}$ are the number of edges connecting vertices $v$ and $w$ .",
"Let $\\Pi =\\left(c_1,\\ldots ,c_k\\right)$ be a partition of the vertex set of $\\Gamma $ into $k$ cells, inducing a block partition of $\\mathbf {A}$ , i.e.",
"a simultaneous (disjoint and exhaustive) partition of its rows and columns.",
"It is convenient to define an indicator matrix of a partition as Definition 1 $\\mathbf {B}=\\left(b_{vi}\\right)\\in \\left\\lbrace 0,1\\right\\rbrace ^{N\\times k}\\text{ with }b_{vi}=\\left\\lbrace \\begin{array}{ll}1&\\text{, if item $v$ is in cell $i$}\\\\0&\\text{, else.",
"}\\end{array}\\right.$ The partition $\\Pi $ is called equitable if all vertices of $\\Gamma $ in the same cell have the same number of neighbours in any cell.",
"Equivalently, we may call it equitable if each induced submatrix of $\\mathbf {A}$ has constant row sum.",
"The equitable partitions of $\\mathbf {A}$ ordered by refinement form a lattice which contains the trivial equitable partition, in which every cell has size exactly one, as the minimum.",
"From the definition it follows that a partition of $\\mathbf {A}$ is equitable if and only if there exits a matrix $\\mathbf {\\Theta }=\\left(\\theta _{ij}\\right)$ s.t.",
"$\\mathbf {A}\\mathbf {B}=\\mathbf {B}\\mathbf {\\Theta }\\quad \\text{i.e.",
"}\\quad \\forall \\ i,j\\in \\left\\lbrace 1,\\ldots ,k\\right\\rbrace \\ \\forall \\ u\\in \\left\\lbrace 1,\\ldots ,n_i\\right\\rbrace \\ \\sum \\limits _v^{n_j}\\mathbf {A}_{ij,{uv}}=\\theta _{ij}.$ The matrix $\\mathbf {\\Theta }$ is called the quotient of the partition.",
"Its entries $\\theta _{ij}$ are the constant row sums of the matrix blocks $A_{ij}$ induced by the cells $c_i$ and $c_j$ , which are the number of edges connecting a fixed vertex in $c_i$ to vertices in $c_j$ ."
],
[
"Applications in Graph and Matrix Theory",
"The notion of equitable partitions was developed in graph theory.",
"In network analysis the same concept is also known as exact coloration [3] or exact role assignment [20].",
"It is closely related to graph fibration [2] and arises naturally in the context of graph automorphisms problems since every non trivial automorphism induces a non trival equitable partition.",
"As graph invariants which can be searched for quickly using quite efficient algorithms, equitable partitions are useful in attacking graph isomorphism problems.",
"In that context they are also known as 1-dimensional Weisfeiler-Lehman stabilizers [5].",
"Block partitioned matrices s.t.",
"each block has constant row sum are called block-stochastic matrices.",
"In the context of markov chains the technique of lumping exploits equitable partitions in order to reduce the number of states [4].",
"The quotient $\\mathbf {\\Theta }$ is also known as the front divisor [7].",
"Famously, the spectrum of $\\mathbf {\\Theta }$ , called the main spectrum, is a subset of the spectrum of $\\mathbf {A}$ since the columns of $\\mathbf {B}$ span an invariant subspace if (REF ) holds.",
"Therefore, there is a similarity transformation which $2\\times 2$ block triangularizes $\\mathbf {A}$ s.t.",
"one diagonal block is the quotient.",
"Such a transformation can be constructed and applied efficiently in a way provided in [15], [6].",
"The block triangularization method given below differs from that approach in order to fit in a generalized framework of equitability and provides efficient unitary transformations."
],
[
"Aim and Outline",
"We will generalize the notion of ordinary equitable partitions to arbitrary weighted partitions of the rows and columns of complex matrices.",
"According to a given partition we derive an efficient and stable unitary similarity transformation in order to $2\\times 2$ block triangularize the matrix up to an error term, which is minimized w.r.t.",
"to several matrix norms and vanishes if and only if exact equitability holds.",
"The transformation can be computed in $O\\left(N\\right)$ and applied in $O\\left(N^2\\right)$ .",
"It can be further generalized enabling the application to rectangular matrices while maintaining the unitarity property.",
"However the further generalized transformation does only preserve the singular values, but (in general) not the spectrum.",
"Despite offering insides into the structure of objects represented by a graph or a matrix our notion of equitability and its corresponding transformation may be used for compression and for preprocessing eigen and singular value problems.",
"Although describing our transformation as an efficient compression method may seem to suggest that the exploited structure is, in a sense, rare, the concept of equitable partitions, as indicated above, is rather common in various applications, where it is found directly in the studied problem or as an interesting exceptional or ideal case.",
"The usefulness might be increased in particular by the fact that deviations from an exact equitability may be allowed within our framework.",
"In order to get used to the concept and some notation, we briefly discuss in section () the special case of an ordinary unweighted equitable partition of a complex square matrix including the derivation of the associated efficient unitary block triangularization and we give an example.",
"In the main part, section (), we consider weighted not necessarily equitable partitions introducing the deviation matrix and give our main theorem.",
"In section () we consider non exact equitability as an eigenvalue perturbation, give a short overview of several other known generalizations of equation (REF ), and briefly consider the problem of finding an equitable partition.",
"Our further generalized version of the concept applicable to rectangular matrices can be found in the appendix.",
"Throughout the article we use the apostrophe to denote the complex conjugated transpose without distinguishing between real and complex operands and we utilize the following notation Definition 2 Let $n\\in \\mathbb {N}$ .",
"$\\mathbf {j}_n=(\\hspace{1.00006pt}\\underbrace{1,\\ldots ,1}_{\\text{$n$ times}}\\hspace{1.00006pt})^{\\prime }$ and $\\mathbf {f}_n=(1,\\hspace{-5.0pt}\\underbrace{0,\\ldots ,0}_{\\text{$(n-1)$ times}}\\hspace{-5.0pt})^{\\prime }$ ."
],
[
"Indicator Matrix and Quotient",
"Let $\\mathbf {A}\\in \\mathbb {C}^{N\\times N}$ and let $\\Pi =\\left(c_1,\\ldots ,c_k\\right)$ be a simultaneous (disjoint and exhaustive) partition of its rows and columns with indicator matrix $\\mathbf {B}$ as in definition (REF ).",
"Let $\\mathbf {A}_{ij}\\in \\mathbf {C}^{n_i\\times n_j}$ be the matrix block in $\\mathbf {A}$ induced by row cell $c_i$ and column cell $c_j$ .",
"Let $n_i$ be the size of the cell $c_i$ and let $\\mathbf {N}=\\left(\\mathbf {B}^{\\prime }\\mathbf {B}\\right)^{\\frac{1}{2}}=\\operatorname{diag}\\left(\\sqrt{n_1},\\ldots ,\\sqrt{n_k}\\right)$ We introduce the front quotient, the rear quotient and the Rayleigh quotient respectively as $\\mathbf {E}^{-}=\\mathbf {N}^{-2}\\mathbf {B}^{\\prime }\\mathbf {A}\\mathbf {B},\\quad \\mathbf {E}^{+}=\\mathbf {B}^{\\prime }\\mathbf {A}\\mathbf {B}\\mathbf {N}^{-2},\\quad \\mathbf {E}^{\\mathrm {0}}=\\mathbf {N}^{-1}\\mathbf {B}^{\\prime }\\mathbf {A}\\mathbf {B}\\mathbf {N}^{-1}$ We call $\\mathbf {A}$ front equitable (i) and respectively rear equitable (ii) w.r.t.",
"$\\mathbf {B}$ if $\\left(i\\right)\\ \\mathbf {A}\\mathbf {B}=\\mathbf {B}\\mathbf {E}^{-}\\quad ,\\quad \\left(ii\\right)\\ \\mathbf {B}^{\\prime }\\mathbf {A}=\\mathbf {E}^{+}\\mathbf {B}^{\\prime }.$ It is easy to see that for Hermitian matrices row equitability and column equitability imply each other.",
"For the rest of this section we assume front equitability, i.e.",
"$\\mathbf {A}_{ij}\\mathbf {j}_{n_j}=e^{-}_{ij}\\mathbf {j}_{n_i}\\quad \\forall \\ i,j\\in \\left\\lbrace 1,\\ldots ,k\\right\\rbrace .$"
],
[
"Block Triangularization",
"In order to block triangularize $\\mathbf {A}$ according to $\\mathbf {B}$ we utilize the Householder matrices $\\mathbf {H}_i=\\mathbf {I}_{n_i}-2\\frac{\\mathbf {y}_i\\mathbf {y}_i^{\\prime }}{\\mathbf {y}_i^{\\prime }\\mathbf {y}_i}\\ ,\\quad \\mathbf {y}_i=\\mathbf {j}_{i}+\\sqrt{n_i}\\mathbf {f}_{n_i}.$ The following useful relations are easily verified $\\mathbf {H}_i\\mathbf {f}_{n_i}=-\\frac{1}{\\sqrt{n_i}}\\mathbf {j}_{n_i}\\quad ,\\quad \\mathbf {H}_i^{\\prime }\\mathbf {j}_{n_i}=-\\sqrt{n_i}\\mathbf {f}_{n_i}.$ In order to simplify notations but w.l.o.g.",
"we assume suitable indexing which means that $\\mathbf {A}$ and $\\mathbf {B}$ are indexed in such a way that for $u$ in cell $c_i$ and $v$ in cell $c_j$ it holds that $i<j$ implies $u<v$ .",
"Then our proposed transformation of $\\mathbf {A}$ can be written conveniently in matrix form using the matrix $\\mathbf {\\tilde{H}}=\\operatorname{diag}\\left(\\mathbf {H}_1,\\ldots ,\\mathbf {H}_k\\right),$ which is explicitly block diagonal and, according to (REF ), unitary.",
"$\\mathbf {\\tilde{A}}=\\mathbf {\\tilde{H}}^{\\prime }\\mathbf {A}\\mathbf {\\tilde{H}}=\\left(\\begin{array}{ccc}\\mathbf {\\tilde{A}}_{11}&\\cdots &\\mathbf {\\tilde{A}}_{1k}\\\\\\vdots &\\ddots &\\vdots \\\\\\mathbf {\\tilde{A}}_{k1}&\\cdots &\\mathbf {\\tilde{A}}_{kk}\\end{array}\\right)\\quad \\text{with}\\quad \\mathbf {\\tilde{A}}_{ij}=\\mathbf {H}_{i}^{\\prime }\\mathbf {A}_{ij}\\mathbf {H}_{j}.$ By (REF ) and (REF ) there exists a matrix $\\mathbf {E}=\\left(e_{ij}\\right)$ s.t.",
"$\\mathbf {\\tilde{A}}_{ij}\\mathbf {f}_{j}=e_{ij}\\mathbf {f}_{n_i},$ which immediately shows that each $\\mathbf {\\tilde{A}}_{ij}$ is block triangular with the left upper block being the scalar $e_{ij}$ .",
"Therefore, there is a readily available, in general not unique permutation matrix $\\mathbf {\\Omega }$ such that $\\mathbf {\\hat{A}}=\\mathbf {\\Omega }^{\\prime }\\mathbf {\\tilde{A}}\\mathbf {\\Omega }=\\left(\\begin{array}{cc}\\mathbf {E}&\\mathbf {D}\\\\\\mathbf {0}&\\mathbf {F}\\end{array}\\right)$ is explicitly block triangular.",
"Since the applied transformations are unitary, the spectrum and the singular values of $\\mathbf {A}$ are preserved.",
"We will refer to $\\mathbf {F}$ , which in general depends on the indexing of $\\mathbf {A}$ and on $\\mathbf {\\Omega }$ , as a factor.",
"One shows that all factors are unitarily equivalent and that by similarity $\\sigma \\left(\\mathbf {A}\\right)=\\sigma \\left(\\mathbf {E}\\right)+\\sigma \\left(\\mathbf {F}\\right).$ Additionally, if $\\mathbf {v}$ is an eigenvector of $\\mathbf {\\hat{A}}$ then $\\mathbf {\\tilde{H}}\\mathbf {\\Omega }\\mathbf {v}$ is an eigenvector of $\\mathbf {A}$ to the same eigenvalue.",
"One also shows that $\\mathbf {D}$ vanishes if and only if rear equitability holds.",
"The computational costs for the transformation $\\mathbf {\\tilde{H}}$ are of order $O\\left(n_in_j\\right)$ on each subblock for we apply only matrix vector multiplication and matrix addition since $\\mathbf {H}_i$ is a rank one update of the identity.",
"Therefore, the total costs are of order $O\\left(N^2\\right)$ .",
"Since $\\mathbf {\\tilde{H}}\\mathbf {\\Omega }$ is unitary, Hermiticity (if present) of $\\mathbf {A}$ is preserved.",
"Numeric stability is supported by using Householder matrices.",
"Note that in this section we constructed $\\mathbf {\\tilde{H}}$ s.t.",
"$\\mathbf {E}=\\mathbf {E}^{\\mathrm {0}}$ .",
"In the general case those two matrices are unitarily equivalent but not necessarily identical."
],
[
"Example",
"Let $\\mathbf {A}_0=\\left(\\begin{array}{cccccc}1&2&3&3&3&2\\\\2&4&3&1&2&1\\\\3&3&1&4&1&1\\\\3&1&4&0&2&3\\\\3&2&1&2&3&2\\\\2&1&1&3&2&4\\end{array}\\right)\\quad \\text{and}\\quad \\mathbf {P}_0=\\left(\\begin{array}{cccccc}1&0&0&0&0&0\\\\0&1&0&0&0&0\\\\0&0&0&0&0&1\\\\0&0&0&1&0&0\\\\0&0&0&0&1&0\\\\0&0&1&0&0&0\\end{array}\\right)$ One verifies that $\\mathbf {A}_0$ is (unweighted) front equitable w.r.t.",
"$\\Pi _0=\\left(1\\vert 2,6\\vert 3,4,5\\right)$ .",
"Using the permutation $\\mathbf {P}_0$ we can transform it into the suitably indexed form $\\mathbf {A}=\\mathbf {P}_0^{\\prime }\\mathbf {A}_0\\mathbf {P}_0=\\left(\\begin{array}{cccccc}1&2&2&3&3&3\\\\2&4&1&1&2&3\\\\2&1&4&3&2&1\\\\3&1&3&0&2&4\\\\3&2&2&2&3&1\\\\3&3&1&4&1&1\\end{array}\\right),$ which is (unweighted) front equitable w.r.t.",
"$\\Pi =\\left(1\\vert 2,3\\vert 4,5,6\\right)$ with front quotient $\\mathbf {E}^{-}=\\left(\\begin{array}{ccc}1&4&9\\\\2&5&6\\\\3&4&6\\end{array}\\right).$ One may employ $\\mathbf {H}_1=\\mathbf {H}\\left(\\mathbf {j}_1\\right)=-1,$ $\\mathbf {H}_2=\\mathbf {H}\\left(\\mathbf {j}_2\\right)=-\\frac{1}{\\sqrt{2}}\\left(\\begin{array}{cc}1&1\\\\1&-1\\end{array}\\right),$ $\\mathbf {H}_3=\\mathbf {H}\\left(\\mathbf {j}_3\\right)=-\\frac{1}{\\sqrt{3}}\\left(\\begin{array}{ccc}1&1&1\\\\1&\\frac{1+\\sqrt{3}}{-2}&\\frac{1-\\sqrt{3}}{-2}\\\\1&\\frac{1-\\sqrt{3}}{-2}&\\frac{1+\\sqrt{3}}{-2}\\end{array}\\right)$ and $\\mathbf {\\tilde{H}}=\\operatorname{diag}\\left(\\mathbf {H}_1,\\mathbf {H}_2,\\mathbf {H}_3\\right)$ to transform $\\mathbf {A}$ s.t.",
"$\\mathbf {\\tilde{A}}=\\mathbf {\\tilde{H}}^{\\prime }\\mathbf {A}\\mathbf {\\tilde{H}}=\\left(\\begin{array}{cccccc}1&\\frac{4}{\\sqrt{2}}&0&\\frac{9}{\\sqrt{3}}&0&0\\\\\\frac{4}{\\sqrt{2}}&5&0&6\\frac{\\sqrt{2}}{\\sqrt{3}}&0&0\\\\0&0&3&0&\\mbox{-}3\\scalebox {0.8}{\\mbox{+}}\\sqrt{3}&\\mbox{-}3\\mbox{-}\\sqrt{3}\\\\\\frac{9}{\\sqrt{3}}&6\\frac{\\sqrt{2}}{\\sqrt{3}}&0&6&0&0\\\\0&0&\\mbox{-}3\\scalebox {0.8}{\\mbox{+}}\\sqrt{3}&0&\\sqrt{3}\\mbox{-}1&\\mbox{-}6\\\\0&0&\\mbox{-}3\\mbox{-}\\sqrt{3}&0&\\mbox{-}6&\\mbox{-}\\sqrt{3}\\mbox{-}1\\end{array}\\right).$ Using the permutation $\\mathbf {\\Omega }=\\left(\\begin{array}{cccccc}1&0&0&0&0&0\\\\0&1&0&0&0&0\\\\0&0&0&1&0&0\\\\0&0&1&0&0&0\\\\0&0&0&0&1&0\\\\0&0&0&0&0&1\\end{array}\\right)$ we obtain the matrix $\\mathbf {\\hat{A}}=\\mathbf {\\Omega }^{\\prime }\\mathbf {\\tilde{A}}\\mathbf {\\Omega }=\\left(\\begin{array}{cccccc}1&\\frac{4}{\\sqrt{2}}&\\frac{9}{\\sqrt{3}}&0&0&0\\\\\\frac{4}{\\sqrt{2}}&5&6\\frac{\\sqrt{2}}{\\sqrt{3}}&0&0&0\\\\\\frac{9}{\\sqrt{3}}&6\\frac{\\sqrt{2}}{\\sqrt{3}}&6&0&0&0\\\\0&0&0&3&\\mbox{-}3\\scalebox {0.8}{\\mbox{+}}\\sqrt{3}&\\mbox{-}3\\mbox{-}\\sqrt{3}\\\\0&0&0&\\mbox{-}3\\scalebox {0.8}{\\mbox{+}}\\sqrt{3}&\\sqrt{3}\\mbox{-}1&\\mbox{-}6\\\\0&0&0&\\mbox{-}3\\mbox{-}\\sqrt{3}&\\mbox{-}6&\\mbox{-}\\sqrt{3}\\mbox{-}1\\end{array}\\right),$ which is explicitly reducible.",
"Since $\\mathbf {A}$ is Hermitian, the (unweighted) partition $\\Pi $ induces front and rear equitability and we actually obtain a block diagonal form.",
"Note that both blocks are Hermitian but the front quotient $\\mathbf {E}^{-}$ is not.",
"One verifies that $\\mathbf {E}=\\left(\\begin{array}{ccc}1&\\frac{4}{\\sqrt{2}}&\\frac{9}{\\sqrt{3}}\\\\\\frac{4}{\\sqrt{2}}&5&6\\frac{\\sqrt{2}}{\\sqrt{3}}\\\\\\frac{9}{\\sqrt{3}}&6\\frac{\\sqrt{2}}{\\sqrt{3}}&6\\end{array}\\right)=\\operatorname{diag}\\left(1,2,3\\right)^{\\frac{1}{2}}\\mathbf {E}^{-}\\operatorname{diag}\\left(1,2,3\\right)^{-\\frac{1}{2}}.$ Let $\\mathbf {F}$ denote the lower diagonal block of $\\mathbf {\\hat{A}}$ .",
"Let $\\mathbf {V}_{\\mathbf {E}}$ and $\\mathbf {V}_{\\mathbf {F}}$ be the eigenvector matrices of $\\mathbf {E}$ and $\\mathbf {F}$ , respectively.",
"Then one shows that $\\mathbf {V}=\\mathbf {P}_0\\mathbf {\\tilde{H}}\\mathbf {\\Omega }\\left(\\begin{array}{cc}\\mathbf {V}_{\\mathbf {E}}&\\mathbf {0}\\\\\\mathbf {0}&\\mathbf {V}_{\\mathbf {F}}\\end{array}\\right)$ is an eigenvector matrix of $\\mathbf {A}$ .",
"Note that $\\mathbf {A}$ and $\\mathbf {V}$ need more storage than $\\mathbf {\\hat{A}}$ , $\\mathbf {V}_{\\mathbf {E}}$ , and $\\mathbf {V}_{\\mathbf {F}}$ .",
"The transformations $\\mathbf {P}_0$ , $\\mathbf {\\tilde{H}}$ and $\\mathbf {\\Omega }$ follow from $\\Pi _0$ which can be stored as a vector.",
"Due to the small size the blocks of $\\mathbf {\\tilde{H}}$ were given explicitly as dense matrices.",
"For larger problems one would prefer the usual sparse form as a rank one update of the identity given in (REF )."
],
[
"Preliminaries",
"In this section we generalize equitable partitions and accordingly the proposed block triangularization method for square matrices.",
"We introduce the generalized quotient defined for arbitrary partitions of a matrix as a generalization of front and rear quotient.",
"We also introduce the deviation vectors and the deviation matrix and utilize the norm of the latter in order to quantify deviations of a given partition from our generalized notion of equitability.",
"The generalization of the efficient unitary similarity transformation introduced above yields a block triangularization up to an error term due to the deviation from equitability.",
"A further generalization applicable to rectangular matrices preserving only singular values but in general not the spectrum is discussed in the appendix.",
"Note that whenever we invert a matrix explicitly (i.e.",
"not by complex conjugated transposition) this matrix is diagonal.",
"The occasional uses of the pseudo inverse with the property $c^{\\dagger }=\\left\\lbrace \\begin{array}{cc}0&,c=0\\\\ \\frac{1}{c}&,\\text{else}\\end{array}\\right.",
",\\quad c\\in \\mathbb {C}$ may be regarded as merely technical."
],
[
"Complex Householder Transformations",
"This subsubsection aims at the transformation in definition (REF ) and its properties given in (REF ).",
"We consider elementary unitary matrices (EUMs) which are rank (at most) one updates of the identity and necessarily (in order to be unitary) [21] of the form $\\mathbf {U}\\left(\\gamma ,\\mathbf {y}\\right)=\\mathbf {I}-\\frac{2}{1+i\\gamma }\\left(\\mathbf {y}^{\\prime }\\mathbf {y}\\right)^{\\dagger }\\mathbf {y}\\mathbf {y}^{\\prime },\\quad \\mathbf {y}\\in \\mathbb {C}^{n},\\gamma \\in \\mathbb {R}.$ EUMs are a complex generalization of real Householder matrices [16], [19].",
"We observe that for $c\\in \\mathbb {C}\\setminus \\left\\lbrace 0\\right\\rbrace $ and $\\mathbf {P}$ being a permutation matrix $\\mathbf {U}\\left(\\gamma ,c\\mathbf {y}\\right)=\\mathbf {U}\\left(\\gamma ,\\mathbf {y}\\right)\\ ,\\quad \\mathbf {U}\\left(\\gamma ,\\mathbf {P}\\mathbf {y}\\right)=\\mathbf {P}\\mathbf {U}\\left(\\gamma ,\\mathbf {y}\\right)\\mathbf {P}^{\\prime }.$ Let $\\mathbf {x}$ and $\\mathbf {z}$ be non vanishing complex vectors.",
"We seek an EUM mapping $\\mathbf {z}$ into the direction of $\\mathbf {x}$ , i.e.",
"a complex vector $\\mathbf {y}$ and a real number $\\gamma $ s.t.",
"$\\mathbf {U}\\left(\\gamma ,\\mathbf {y}\\right)\\mathbf {z}=\\alpha \\mathbf {x}\\quad \\text{with}\\quad \\alpha \\in \\mathbb {C}\\setminus \\left\\lbrace 0\\right\\rbrace ,$ which implies that $\\Vert \\mathbf {x}\\Vert $ and $\\Vert \\mathbf {z}\\Vert $ determine $\\alpha $ up to a phase factor $\\sqrt{\\left(\\mathbf {U}\\left(\\gamma ,\\mathbf {y}\\right)\\mathbf {z}\\right)^{\\prime }\\left(\\mathbf {U}\\left(\\gamma ,\\mathbf {y}\\right)\\mathbf {z}\\right)}=\\Vert \\mathbf {z}\\Vert =\\left|\\alpha \\right|\\Vert \\mathbf {x}\\Vert .$ Again by (REF ) $\\mathbf {y}$ is a linear combination of $\\mathbf {x}$ and $\\mathbf {z}$ , namely $\\mathbf {z}-\\alpha \\mathbf {x}=\\frac{2}{1+i\\gamma }\\left(\\mathbf {y}^{\\prime }\\mathbf {y}\\right)^{\\dagger }\\left(\\mathbf {y}^{\\prime }\\mathbf {z}\\right)\\mathbf {y}.$ Since according to (REF ) scaling of $\\mathbf {y}$ does not change $\\mathbf {U}\\left(\\gamma ,\\mathbf {y}\\right)$ we may choose $\\mathbf {y}=\\mathbf {x}-\\frac{1}{\\alpha }\\mathbf {z}$ .",
"We are particular interested in the case $\\mathbf {z}=\\mathbf {f}_n$ .",
"Setting $\\alpha =\\beta \\left|\\alpha \\right|$ and using (REF ), we reach in the non trivial case, $\\mathbf {y}\\ne 0$ , $\\gamma \\left(\\mathbf {x},\\beta \\right)=\\left(\\Vert \\mathbf {x}\\Vert -\\operatorname{Re}\\left(\\beta x^1\\right)\\right)^{\\dagger }\\operatorname{Im}\\left(\\beta x^1\\right).$ Thus, the required EUM of $\\mathbf {f}_n$ into the direction of $\\mathbf {x}$ is determined up to a complex parameter $\\beta $ lying on the unit circle.",
"We introduce $\\mathbf {H}\\left(\\mathbf {x},\\beta \\right)=\\mathbf {U}\\left({\\gamma \\left(\\mathbf {x},\\beta \\right)},\\mathbf {x}-\\Vert \\mathbf {x}\\Vert \\overline{\\beta }\\mathbf {f}_n\\right)\\text{ with }\\left|\\beta \\right|=1$ and give an explicit definition.",
"Definition 3 Let $\\mathbf {x}\\in \\mathbb {C}^n$ with $\\Vert \\mathbf {x}\\Vert >0$ , let $x^1=\\mathbf {f}_n^{\\prime }\\mathbf {x}$ denote its first entry and let $\\beta $ be a complex number with $\\left|\\beta \\right|=1$ , then $\\mathbf {H}\\left(\\mathbf {x},\\beta \\right)=\\left\\lbrace \\begin{array}{cc}\\mathbf {I}_n & ,\\frac{1}{\\Vert \\mathbf {x}\\Vert }\\mathbf {x}=\\overline{\\beta }\\mathbf {f}_n\\\\\\mathbf {I}_n+\\frac{\\left(\\mathbf {x}-\\Vert \\mathbf {x}\\Vert \\overline{\\beta }\\mathbf {f}_n\\right)\\left(\\mathbf {x}-\\Vert \\mathbf {x}\\Vert \\overline{\\beta }\\mathbf {f}_n\\right)^{\\prime }}{\\Vert \\mathbf {x}\\Vert \\overline{\\beta }\\left(\\overline{x^1}-\\Vert \\mathbf {x}\\Vert \\beta \\right)} & ,\\text{ else.",
"}\\end{array}\\right.$ Using $\\mathbf {y}=\\mathbf {x}-\\Vert \\mathbf {x}\\Vert \\overline{\\beta }\\mathbf {f}_n$ we may rewrite $\\mathbf {H}\\left(\\mathbf {x},\\beta \\right)=\\mathbf {I}_n+\\frac{\\beta }{\\Vert \\mathbf {x}\\Vert }\\left(\\mathbf {y}^{\\prime }\\mathbf {f}_n\\right)^{\\dagger }\\mathbf {y}\\mathbf {y}^{\\prime }=\\mathbf {I}_n-\\left(\\mathbf {x}^{\\prime }\\mathbf {y}\\right)^{\\dagger }\\mathbf {y}\\mathbf {y}^{\\prime }.$ And we summarize the following properties $\\mathbf {H}\\left(\\mathbf {x},\\beta \\right)\\mathbf {f}_n=\\frac{\\beta }{\\Vert \\mathbf {x}\\Vert }\\mathbf {x}\\quad \\text{and}\\quad \\mathbf {H}\\left(\\mathbf {x},\\beta \\right)^{\\prime }\\mathbf {x}=\\frac{\\Vert \\mathbf {x}\\Vert }{\\beta }\\mathbf {f}_n.$ Since $\\mathbf {H}\\left(\\mathbf {x},\\beta \\right)$ is a rank one update of $\\mathbf {I}_n$ , it can be stored with $O\\left(n\\right)$ and multiplied with a square matrix of size $n$ in $O\\left(n^2\\right)$ .",
"Note that $\\mathbf {H}\\left(\\mathbf {x},\\beta \\right)$ crucially depends on the ordering of the entries of $\\mathbf {x}$ , $\\mathbf {H}\\left(\\mathbf {P}^{\\prime }\\mathbf {x},\\beta \\right)\\ne \\mathbf {P}^{\\prime }\\mathbf {H}\\left(\\mathbf {x},\\beta \\right)\\mathbf {P}\\quad \\text{for general $\\mathbf {x}$ and permutation matrix $\\mathbf {P}$}.$ Although its norm is determined to be 1, the actual choice of $\\beta $ is arbitrary.",
"We may exploit that freedom in order to enhance the numerical properties of $\\mathbf {H}\\left(\\beta ,\\mathbf {x}\\right)$ .",
"Particular useful is a choice s.t.",
"$\\beta x^1\\in \\mathbb {R}$ , implying $\\gamma =0$ by (REF ) and leading to a Hermitian matrix.",
"Furthermore, for real $\\mathbf {x}$ , $\\beta \\in \\left\\lbrace -1,1\\right\\rbrace $ ensures a real matrix.",
"A practical recommendation might be Definition 4 $\\beta _0\\left(\\mathbf {x}\\right)=\\left\\lbrace \\begin{array}{ll}-\\frac{\\overline{x^1}}{\\left|x^1\\right|}&, x^1\\ne 0\\\\1&, x^1=0\\end{array}\\right.\\quad ,\\ \\mathbf {x}\\in \\mathbb {C}^n$ , which supports numerical stability and coincides with the usual recommendation for the numerical construction of a real Householder matrix.",
"In the previous section we applied $\\beta _0$ tacitly."
],
[
"Weighted Partition, Quotient and Deviation Matrix",
"Let $\\Pi =\\left(c_1,\\ldots ,c_k\\right)$ be a partition of $\\lbrace 1,\\ldots ,N\\rbrace $ into $k$ cells with indicator matrix $\\mathbf {B}$ .",
"Let $\\mathbf {w}\\in \\mathbb {C}^{N}$ and $\\mathbf {A}\\in \\mathbb {C}^{N\\times N}$ .",
"We introduce the weighted indicator matrix $\\mathbf {W}=\\operatorname{diag}\\left(\\mathbf {w}\\right)\\mathbf {B}$.",
"Definition 5 Let $\\Pi =\\left(c_1,\\ldots ,c_k\\right)$ be a partition of $\\left\\lbrace 1,\\ldots ,N\\right\\rbrace $ .",
"Let $\\mathbf {w}\\in \\mathbb {C}^N$ and let $w^v$ denote its $v$ -th entry.",
"$\\mathbf {W}=\\left(w_{vi}\\right)\\in \\mathbb {C}^{N\\times k}\\text{ with }w_{vi}=\\left\\lbrace \\begin{array}{ll}w^v&\\ v\\in c_i\\\\0&\\text{, else}\\end{array}\\right.$ A weighted indicator matrix $\\mathbf {W}$ is called admissible if $\\Vert \\mathbf {w}_i\\Vert $ for all vector blocks $\\mathbf {w}_i$ induced by $c_i$ .",
"This implies that $\\mathbf {W}^{\\prime }\\mathbf {W}$ is invertible and ultimately ensures that the complete spectrum of the quotient, to be defined below, is contained in the spectrum of $\\mathbf {A}$ .",
"For the rest of this section we assume admissibility.",
"We call $\\mathbf {W}$ suitably indexed if for $u\\in c_i,v\\in c_j$ it holds that $i<j$ implies $u<v$ .",
"In that case the index set is ordered block wise and $\\mathbf {W}$ is explicitly block diagonal.",
"In order to simplify the exposition, we may w.l.o.g.",
"assume a suitable indexing.",
"Definition 6 Let $\\mathbf {A}\\in \\mathbb {C}^{N\\times N}$ and let $\\mathbf {W}$ be an admissible weighted indicator matrix and let $\\alpha \\in \\mathbb {R}$ .",
"The generalized quotient $\\mathbf {E}^{\\alpha }$ is given by $\\mathbf {E}^{\\alpha }=\\left(\\mathbf {W}^{\\prime }\\mathbf {W}\\right)^{-\\frac{1-\\alpha }{2}}\\mathbf {W}^{\\prime }\\mathbf {A}\\mathbf {W}\\left(\\mathbf {W}^{\\prime }\\mathbf {W}\\right)^{-\\frac{1+\\alpha }{2}}.$ We call $\\mathbf {E}^{0}$ the Rayleigh quotient.",
"The matrix entries of $\\mathbf {E}^{\\alpha }$ are $e^{\\alpha }_{ij}=\\left(\\frac{1}{\\Vert \\mathbf {w}_i\\Vert }\\right)^{\\left(1-\\alpha \\right)}\\mathbf {w}_i^{\\prime }\\mathbf {A}_{ij}\\mathbf {w}_j\\left(\\frac{1}{\\Vert \\mathbf {w}_j\\Vert }\\right)^{\\left(1+\\alpha \\right)}.$ Since $\\mathbf {E}^{\\alpha }=\\left(\\mathbf {W}\\mathbf {W}\\right)^{\\frac{\\alpha }{2}}\\mathbf {E}^0\\left(\\mathbf {W}\\mathbf {W}\\right)^{-\\frac{\\alpha }{2}}$ , all generalized quotients are similar.",
"We distinguish the front quotient $\\mathbf {E}^{-}=\\mathbf {E}^{-1}$ and the rear quotient $\\mathbf {E}^{+}=\\mathbf {E}^{1}$ .",
"The matrix $\\mathbf {A}$ is called front equitable w.r.t.",
"$\\mathbf {W}$ if and only if $\\mathbf {A}\\mathbf {W}=\\mathbf {W}\\mathbf {E}^{-}\\quad \\text{, i.e.",
"}\\quad \\forall \\ i,j\\in \\left\\lbrace 1,\\ldots ,k\\right\\rbrace \\ \\mathbf {A}_{ij}\\mathbf {w}_{j}=e^{-}_{ij}\\mathbf {w}_{i}$ and we call $\\mathbf {A}$ rear equitable w.r.t.",
"$\\mathbf {W}$ if and only if $\\mathbf {W}^{\\prime }\\mathbf {A}={\\mathbf {E}^{+}}\\mathbf {W}^{\\prime }\\quad \\text{, i.e.",
"}\\quad \\forall \\ i,j\\in \\left\\lbrace 1,\\ldots ,k\\right\\rbrace \\ \\mathbf {w}_{i}^{\\prime }\\mathbf {A}_{ij}={e^{+}_{ij}}\\mathbf {w}_{j}^{\\prime }.$ Definition 7 Maintaining the notation above the front and rear deviation vectors are defined respectively as $\\mathbf {t}^{-}_{ij}=\\frac{1}{\\Vert \\mathbf {w}_j\\Vert }\\left(\\mathbf {A}_{ij}\\mathbf {w}_{j}-e^{-}_{ij}\\mathbf {w}_{i}\\right)\\quad \\text{ and }\\quad {\\mathbf {t}^{+}_{ij}}=\\frac{1}{\\Vert \\mathbf {w}_i\\Vert }\\left({\\mathbf {w}_{i}^{\\prime }\\mathbf {A}_{ij}}-{e^{+}_{ij}}\\mathbf {w}_{j}^{\\prime }\\right)^{\\prime },$ and the front and rear deviation matrices are $\\mathbf {T}^{\\pm }=\\left(\\begin{array}{ccc}\\mathbf {t}^{\\pm }_{11}&\\cdots &\\mathbf {t}^{\\pm }_{1k}\\\\\\vdots &\\ddots &\\vdots \\\\\\mathbf {t}^{\\pm }_{k1}&\\cdots &\\mathbf {t}^{\\pm }_{kk}\\end{array}\\right)\\in \\mathbb {C}^{N\\times k}\\ \\ ,\\textrm {i.e.",
"}\\ \\ \\begin{array}{lll}\\mathbf {T}^{-}=\\left(\\mathbf {A}\\mathbf {W}-\\mathbf {W}\\mathbf {E}^{-}\\right)\\left(\\mathbf {W}^{\\prime }\\mathbf {W}\\right)^{-\\frac{1}{2}}\\\\\\\\\\mathbf {T}^{+}=\\left(\\mathbf {A}^{\\prime }\\mathbf {W}-\\mathbf {W}{\\mathbf {E}^{+}}^{\\prime }\\right)\\left(\\mathbf {W}^{\\prime }\\mathbf {W}\\right)^{-\\frac{1}{2}}.\\end{array}$ The entries of $\\mathbf {E}^{\\pm }$ and the deviation vectors have an intuitive interpretation in the framework of ordinary equitability arising for $\\mathbf {w}_i=\\mathbf {j}_{n_i}$ .",
"Then $e^{-}_{ij}$ and $\\Vert \\mathbf {t}^{-}_{ij}\\Vert $ ($e^{+}_{ij}$ and $\\Vert \\mathbf {t}^{+}_{ij}\\Vert $ ) are the mean and the standard deviation of the row (column) sums of $\\mathbf {A}_{ij}$ .",
"Scaling the vector blocks $\\mathbf {w}_{i}$ by $\\mu _i\\in \\mathbb {C}\\setminus \\left\\lbrace 0\\right\\rbrace $ changes the entries of the generalized quotient to $\\mu _i^{\\alpha }e^{\\alpha }_{ij}\\mu _j^{-\\alpha }$ although such a transformation sustains equitability (if present).",
"Note that $e^{0}_{ij}$ and $\\Vert {\\mathbf {t}^{\\pm }_{ij}}\\Vert $ , and therefore the singular values of $\\mathbf {T}^{\\pm }$ , are independent of such a scaling.",
"By definition, $\\mathbf {T}^{\\pm }$ is an all zero matrix if and only if its respective equitability holds.",
"At the end of this section, we will consider suitable norms of $\\mathbf {T}^{\\pm }$ as measures for deviation from equitability."
],
[
"(Approximate) Block Triangularization",
"Let $\\mathbf {W}$ be an admissible weighted indicator matrix of a partition $\\Pi =\\left(c_1,\\ldots ,c_k\\right)$ with weight vector $\\mathbf {w}\\in \\mathbb {C}^{N}$ and indicator matrix $\\mathbf {B}$ .",
"Let $\\mathbf {w}_i$ be induced by $c_i$ .",
"Replacing $\\mathbf {w}_i$ by $\\mathbf {f}_{n_i}$ for all $i$ yields the new vector $\\mathbf {f}$ .",
"Let $\\mathbf {N}=\\left(\\mathbf {W}^{\\prime }\\mathbf {W}\\right)^{\\frac{1}{2}}$ and let $\\mathbf {V}=\\operatorname{diag}\\left(\\beta _1,\\ldots ,\\beta _k\\right)$ be a unitary diagonal matrix of size $k$ .",
"We introduce $\\mathbf {Y}\\left(\\mathbf {W},\\mathbf {V}\\right)=\\mathbf {Y}\\left(\\mathbf {w},\\Pi ,\\mathbf {V}\\right)=\\operatorname{diag}\\left(\\mathbf {w}\\right)\\mathbf {B}-\\operatorname{diag}\\left(\\mathbf {f}\\right)\\mathbf {B}\\mathbf {N}\\mathbf {V}^{\\prime },$ which has the form of a weighted indicator matrix.",
"The actual choice of $\\mathbf {V}$ is a priori arbitrary.",
"This freedom may be exploited in order to enhance the numerical properties of the transformation matrix given in the next definition.",
"Definition 8 Let $\\mathbf {Y}$ be derived from an admissible weighted indicator matrix $\\mathbf {W}$ and a unitary diagonal matrix $\\mathbf {V}$ as above, then $\\mathbf {H}\\left(\\mathbf {W},\\mathbf {V}\\right)=\\mathbf {I}_N-\\mathbf {Y}\\left(\\mathbf {W}^{\\prime }\\mathbf {Y}\\right)^{\\dagger }\\mathbf {Y}^{\\prime }.$ Since $\\mathbf {Y}$ and $\\mathbf {W}$ have the same block diagonal form, $\\mathbf {Y}^{\\prime }\\mathbf {W}$ is a diagonal matrix and $\\mathbf {H}\\left(\\mathbf {V},\\mathbf {W}\\right)$ is block diagonal, hence its numerical properties are comparable to those of a single Householder matrix.",
"In particular, the costs for computing and storing are of order $O\\left(N\\right)$ , and it can be applied to a square matrix in $O\\left(N^2\\right)$ .",
"For suitably indexed $\\mathbf {W}$ the block diagonal form of $\\mathbf {H}\\left(\\mathbf {W},\\mathbf {V}\\right)$ is explicit, $\\mathbf {H}\\left(\\mathbf {W},\\mathbf {V}\\right)=\\operatorname{diag}\\left(\\mathbf {H}\\left(\\mathbf {w}_1,\\beta _1\\right),\\ldots ,\\mathbf {H}\\left(\\mathbf {w}_k,\\beta _k\\right)\\right).$ The diagonal blocks are given in definition (REF ).",
"For $\\mathbf {A}\\in \\mathbb {C}^{N\\times N}$ we consider $\\mathbf {\\tilde{A}}=\\mathbf {H}\\left(\\mathbf {W},\\mathbf {V}\\right)^{\\prime }\\mathbf {A}\\mathbf {H}\\left(\\mathbf {W},\\mathbf {V}\\right)\\quad \\text{with}\\quad \\mathbf {\\tilde{A}}_{ij}=\\mathbf {H}\\left(\\mathbf {w}_i,\\beta _i\\right)^{\\prime }\\mathbf {A}_{ij}\\mathbf {H}\\left(\\mathbf {w}_j,\\beta _j\\right).$ By the properties (REF ) of the $\\mathbf {H}\\left(\\mathbf {w}_i,\\beta _i\\right)$ it follows that Aijfnjfni i,j if and only if A is front equitable w.r.t.",
"W, fni'Aijfnj' i,j if and only if A is rear equitable w.r.t.",
"W. If we consider for a moment front (row) equitability, the first column (row) of each block $\\mathbf {\\tilde{A}}_{ij}$ would be all zero from its second to last entry.",
"This implies an implicit block triangular form of $\\mathbf {\\tilde{A}}$ , which can be made explicit by the following permutation mapping the first index of each cell accordingly into $\\left\\lbrace 1,\\ldots ,k\\right\\rbrace $ .",
"Definition 9 Let $\\mathbf {n}=\\left(n_1,\\ldots ,n_k\\right)$ be a sequence of $k$ positive integers with $\\sum _{i=1}^{k}n_i=N$ .",
"The permutation $\\Omega _{\\mathbf {n}}: \\left\\lbrace 1,\\ldots ,N\\right\\rbrace \\rightarrow \\left\\lbrace 1,\\ldots ,N\\right\\rbrace $ is defined by $\\Omega _{\\mathbf {n}}\\left(m_i+\\sum \\limits _{j=i}^{i-1}n_j\\right)=\\left\\lbrace \\begin{array}{ll}i & ,m_i=1\\\\k-i+m_i+\\sum \\limits _{j=i}^{i-1}n_j & ,m_i\\in \\left\\lbrace 2,\\ldots ,n_i\\right\\rbrace \\end{array}\\right.$ with $i\\in \\left\\lbrace 1,\\ldots ,k\\right\\rbrace $ .",
"We proceed with the general case and give the following theorem, which may be seen as a corollary of theorem (REF ).",
"In order to keep this section self contained, it is proven independently.",
"Theorem 1 Let $\\Pi =\\left(c_1,\\ldots ,c_k\\right)$ be an admissible partition for $\\mathbf {A}\\in \\mathbb {C}^{N\\times N}$ and $\\mathbf {w}\\in \\mathbb {C}^{N}$ with weighted indicator matrix $\\mathbf {W}\\in \\mathbb {C}^{N\\times k}$ , generalized quotient $\\mathbf {E}^{\\alpha }$ and deviation matrices $\\mathbf {T}^{\\pm }$ .",
"Let $\\mathbf {V}=\\operatorname{diag}\\left(\\beta _1,\\ldots ,\\beta _k\\right)$ be a unitary diagonal matrix and let $\\mathbf {\\tilde{H}}=\\mathbf {H}\\left(\\mathbf {W},\\mathbf {V}\\right)$ as in definition (REF ) and let $\\mathbf {\\Omega }$ be the permutation matrix corresponding to $\\Omega _{\\left(\\left|c_1\\right|,\\ldots ,\\left|c_k\\right|\\right)}$ .",
"Let $\\mathbf {\\tilde{A}}=\\mathbf {\\tilde{H}}^{\\prime }\\mathbf {A}\\mathbf {\\tilde{H}}=\\left(\\begin{array}{ccc}\\mathbf {\\tilde{A}}_{11}&\\cdots &\\mathbf {\\tilde{A}}_{1k}\\\\\\vdots &\\ddots &\\vdots \\\\\\mathbf {\\tilde{A}}_{k1}&\\cdots &\\mathbf {\\tilde{A}}_{kk}\\\\\\end{array}\\right),\\quad \\mathbf {\\tilde{A}}_{ij}=\\mathbf {H}\\left(\\mathbf {w}_{i},\\beta _i\\right)^{\\prime }\\mathbf {A}_{ij}\\mathbf {H}\\left(\\mathbf {w}_{j},\\beta _j\\right)$ and $\\mathbf {\\hat{A}}=\\mathbf {\\Omega }^{\\prime }\\mathbf {\\tilde{A}}\\mathbf {\\Omega }=\\left(\\begin{array}{cc}\\mathbf {E}^{\\phantom{-}}&{\\mathbf {D}^{+}}^{\\prime }\\\\\\mathbf {D}^{-}&\\mathbf {F}^{\\phantom{+}}\\end{array}\\right)\\quad \\text{with}\\quad \\mathbf {E}\\in \\mathbb {C}^{k\\times k}.$ Then $\\mathbf {\\hat{A}}$ is unitarily similar to $\\mathbf {A}$ , the upper left block $\\mathbf {E}$ is unitarily similar to the Rayleigh quotient $\\mathbf {E}^0$ and the off-diagonal blocks $\\mathbf {D}^{\\pm }$ have the same singular values as $\\mathbf {T}^{\\pm }$ , respectively.",
"Additionally, any eigenvector $\\mathbf {\\hat{z}}$ of $\\mathbf {\\hat{A}}$ yields an eigenvector $\\mathbf {z}=\\mathbf {\\tilde{H}}\\mathbf {\\Omega }\\mathbf {\\hat{z}}$ of $\\mathbf {A}$ to the same eigenvalue.",
"Unitary similarity to $\\mathbf {A}$ follows from the unitarity of $\\mathbf {\\tilde{H}}$ and $\\mathbf {\\Omega }$ .",
"Considering the matrix blocks $\\mathbf {\\tilde{A}}_{ij}$ of $\\mathbf {\\tilde{A}}$ induced by cells $c_i$ and $c_j$ we have $e_{ij}=\\mathbf {f}_{n_i}^{\\prime }\\mathbf {\\tilde{A}}_{ij}\\mathbf {f}_{n_j}=\\frac{\\overline{\\beta _i}}{\\Vert \\mathbf {w}_i\\Vert }\\frac{\\beta _j}{\\Vert \\mathbf {w}_j\\Vert }\\mathbf {w}_{i}^{\\prime }\\mathbf {A}_{ij}\\mathbf {w}_{j}=\\frac{\\beta _j}{\\beta _i}e^{0}_{ij}.$ By $\\mathbf {\\Omega }$ those $e_{ij}$ are mapped accordingly into the upper left block $\\mathbf {E}$ .",
"Therefore, we may rewrite $\\mathbf {E}=\\mathbf {V}^{\\prime }\\mathbf {E}^0\\mathbf {V}$ , which proofs unitary similarity of $\\mathbf {E}$ and $\\mathbf {E}^{0}$ .",
"In order to show that $\\mathbf {D}^{\\pm }$ is unitarily equivalent to $\\mathbf {T}^{\\pm }$ , we observe that by the properties of $\\mathbf {\\Omega }$ we can write $\\mathbf {D}^{\\pm }$ as $\\mathbf {D}^{\\pm }=\\left(\\begin{array}{ccc}\\mathbf {d}^{\\pm }_{11}&\\cdots &\\mathbf {d}^{\\pm }_{1k}\\\\\\vdots &\\ddots &\\vdots \\\\\\mathbf {d}^{\\pm }_{k1}&\\cdots &\\mathbf {d}^{\\pm }_{kk}\\\\\\end{array}\\right)\\in \\mathbb {C}^{\\left(N-k\\right)\\times k},$ wherein $\\mathbf {d}^{-}_{ij}$ is the first column and ${\\mathbf {d}^{+}_{ij}}^{\\prime }$ is the first row of the matrix block $\\mathbf {\\tilde{A}}_{ij}$ starting from the second entry.",
"We have $\\left(\\begin{array}{c}\\hspace{-5.0pt}0\\\\\\mathbf {d}^{-}_{ij}\\end{array}\\right)=\\mathbf {\\tilde{A}}_{ij}\\mathbf {f}_{n_j}-e^{-}_{ij}\\mathbf {f}_{n_i}=\\mathbf {H}\\left(\\beta _i,\\mathbf {w}_i\\right)^{\\prime }\\mathbf {t}^{-}_{ij},$ $\\left(0,{\\mathbf {d}^{+}_{ij}}^{\\prime }\\ \\right)=\\mathbf {f}_{n_i}^{\\prime }\\mathbf {\\tilde{A}}_{ij}-e^{+}_{ij}\\mathbf {f}_{n_j}^{\\prime }={\\mathbf {t}^{+}_{ij}}^{\\prime }\\mathbf {H}\\left(\\beta _j,\\mathbf {w}_j\\right),$ which shows that $\\mathbf {\\Omega }^{\\prime }\\mathbf {\\tilde{H}}^{\\prime }\\mathbf {T}^{\\pm }=\\left(\\begin{array}{c}\\mathbf {0}^{\\phantom{\\pm }}\\\\\\mathbf {D}^{\\pm }\\end{array}\\right).$ The eigenvector relation can be shown by applying $\\mathbf {\\tilde{H}}\\mathbf {\\Omega }$ from the left to $\\lambda \\mathbf {\\hat{z}}=\\mathbf {\\hat{A}}\\mathbf {\\hat{z}}=\\mathbf {\\Omega }^{\\prime }\\mathbf {\\tilde{H}}^{\\prime }\\mathbf {A}\\mathbf {z}.$"
],
[
"Deviation from Equitability",
"Let $\\Vert \\cdot \\Vert _{U}$ denote a unitarily invariant norm.",
"Corollary 1 $\\Vert \\mathbf {D}^{\\pm }\\Vert _{U}=\\Vert \\mathbf {T}^{\\pm }\\Vert _{U}.$ Corollary 2 Let $\\mathbf {T}^{-}_{\\mathbf {\\Theta }}=\\left(\\mathbf {A}\\mathbf {W}-\\mathbf {W}\\mathbf {\\Theta }\\right)\\mathbf {N}^{-1}$ and $\\mathbf {T}^{+}_{\\mathbf {\\Theta }}=\\mathbf {N}^{-1}\\left(\\mathbf {W}^{\\prime }\\mathbf {A}-\\mathbf {\\Theta }\\mathbf {W}^{\\prime }\\right)$ with $\\mathbf {N}=\\left(\\mathbf {W}^{\\prime }\\mathbf {W}\\right)^{\\frac{1}{2}}$ .",
"Then $\\Vert \\mathbf {T}^{\\pm }\\Vert _{U}=\\min _{\\mathbf {\\Theta }}\\Vert \\mathbf {T}^{\\pm }_{\\mathbf {\\Theta }}\\Vert _{U}.$ The minimum is unique if $\\Vert \\cdot \\Vert _{U}$ is a Schatten norm.",
"Applying $\\mathbf {\\Omega }^{\\prime }\\mathbf {\\tilde{H}}^{\\prime }$ from the left and $\\mathbf {V}^{\\prime }$ from the right to $\\mathbf {T}^{-}_{\\mathbf {\\Theta }}$ yields 'H'T-V'U= A 'H' WN-1V- 'H'WN-1VU =(cE- D-) -(c V'N N-1V 0)U using $\\mathbf {\\Omega }^{\\prime }\\mathbf {\\tilde{H}}^{\\prime }\\mathbf {W}=\\left(\\begin{array}{c}\\mathbf {V}^{\\prime }\\mathbf {N}\\\\\\mathbf {0}\\end{array}\\right)$ .",
"The last term is readily minimized for $\\Theta =\\mathbf {N}^{-1}\\mathbf {V}\\mathbf {E}\\mathbf {V}^{\\prime }\\mathbf {N}=\\left(\\mathbf {W}^{\\prime }\\mathbf {W}\\right)^{-\\frac{1}{2}}\\mathbf {E}^{0}\\left(\\mathbf {W}^{\\prime }\\mathbf {W}\\right)^{\\frac{1}{2}}=\\mathbf {E}^{-}.$ Obviously, the minimization is unique for several $\\Vert \\cdot \\Vert _{U}$ including the Schatten norms.",
"A similar proof applies for $\\mathbf {T}^{+}$ .",
"The idea underlying the last proof is essentially the same as in [8].",
"A particular useful choice for ${\\Vert \\cdot \\Vert _{U}}$ might be the Frobenius norm, which upper bounds the spectral norm.",
"Its square is simply the sum of the squared norms of the deviation vectors.",
"One may also think of other characterizations for approximate equitable partitions which have moderate computational costs, for instance the number of nonzero columns of $\\mathbf {T}^{\\pm }$ , which upper bounds the rank."
],
[
"Relating Equitability Deviation and Spectral Deviation",
"Since $\\mathbf {\\hat{A}}$ and $\\mathbf {A}$ are unitarily similar and by corollaries (REF ) and (REF ) of theorem (REF ), we may in a sense 'measure' the deviation of a partition from being equitable by using a suitable unitarily invariant norm of $\\mathbf {T}^{\\pm }$ , yielding a norm of $\\mathbf {D}^{\\pm }$ , which in turn may serve as a measure for the deviation of the joint eigenvalue sets or the joint singular value sets of $\\mathbf {E}$ and $\\mathbf {F}$ from the respective values of $\\mathbf {A}$ .",
"As an example we consider the spectral norm and the eigenvalue bound of Weyl for Hermitian matrices.",
"Assuming Hermiticity we may set $\\mathbf {D}^{\\pm }=\\mathbf {D}$ and $\\mathbf {\\hat{A}}=\\left(\\begin{array}{cc}\\mathbf {E}&\\mathbf {0}\\\\\\mathbf {0}&\\mathbf {F}\\end{array}\\right)+\\left(\\begin{array}{cc}\\mathbf {0}&\\mathbf {D}^{\\prime }\\\\\\mathbf {D}&\\mathbf {0}\\end{array}\\right).$ Let $\\mu _1\\le \\ldots \\le \\mu _N$ be the joint spectrum of Hermitian $\\mathbf {E}$ and $\\mathbf {F}$ , $\\lambda _1\\le \\ldots \\le \\lambda _N$ the eigenvalues of $\\mathbf {A}$ and let $\\tau _{\\text{spec}}$ be the largest singular value of $\\mathbf {D}$ .",
"We have $\\left|\\mu _i-\\lambda _i\\right|\\le \\tau _{\\text{spec}}\\ ,\\ 1\\le i\\le N$ by the Weyl inequalities.",
"Many more pertubation bounds on eigenvalues and singular values and thier corresponding vectors are feasible, e.g.",
"[10],[9],[11]."
],
[
"Cognate Concepts",
"The notion of quasi-block-stochastic matrices of Kuich [18] as a generalization of quasi-stochastic matrices [14] bears a close resemblance to our notion of equitability.",
"A minor difference is that for quasi-block-stochastic matrices it is required that the first entry of each $\\mathbf {v}_i$ has to be 1.",
"Kuich also describes how to exploit this structure to triangularize a (real) matrix by a (real, in general not unitary) similarity transformation using a theorem of Haynsworth [15].",
"Another similar but less general concept is used by Fiol and Carriga and is called pseudo-regular partitions.",
"It considers a positive eigenvector $\\mathbf {v}$ of binary matrices [12].",
"The partition $\\Pi $ of the matrix is pseudo-regular if $\\mathbf {v}$ and $\\Pi $ induce a (weighted) equitable partition.",
"Since $\\mathbf {v}$ is fixed up to a positive scale factor, the pseudo-quotient (i.e.",
"front divisor) is unique.",
"There are some more techniques in network analysis which can be described as variations of (REF ) and which are used to partition the node set of a graph (=assigning roles) according to structural properties and to derive a smaller graph (the quotient or image graph) which gives a condensed representation of essential relations between the cells (=roles) of that partition.",
"Some of those are without apparent regard to the spectrum.",
"For instance, Kate and Ravindran introduced epsilon equitable partitions for (an adjacency matrix $\\mathbf {A}$ of) a simple graph [17].",
"Let $\\Pi =\\left(c_1,\\ldots ,c_k\\right)$ be a partition of the node set of $\\mathbf {A}$ .",
"Let $\\mathbf {A}_{ij}$ be induced by the $i$ -th row cell and the $j$ -th column cell.",
"Let $\\mathbf {r}_{ij}=\\mathbf {A}_{ij}\\mathbf {j}_{n_j}$ be a column vector of length $n_i=\\left|c_i\\right|$ .",
"If $\\forall \\ i,j\\in \\left\\lbrace 1,\\ldots ,k\\right\\rbrace \\quad \\max \\limits _{1\\le v,w\\le n_i} \\left|\\mathbf {r}_{ij,v}-\\mathbf {r}_{ij,w}\\right|\\le \\epsilon $ then $\\Pi $ is called $\\epsilon $ -equitable.",
"The ordinary equitable partition arises for $\\epsilon =0$ .",
"Another variation of (REF ) can be employed to describe the concept of regular equivalence [3], which is defined by the restriction that for a partition $\\Pi $ any vector $\\mathbf {r}_{ij}=\\mathbf {A}_{ij}\\mathbf {j}_{n_j}$ must have either no zero entry or all entries zero i.e.",
"$\\forall \\ i,j\\in \\left\\lbrace 1,\\ldots ,k\\right\\rbrace \\quad \\prod _v\\mathbf {r}_{ij,v}=0\\Rightarrow \\sum _v\\left|\\mathbf {r}_{ij,v}\\right|=0.$"
],
[
"Finding Equitable Partitions",
"There are several algorithms for finding ordinary equitable partitions of graphs and matrices, for instance [1], [13].",
"We sketch the most often employed top-down approach made suitable to the case of finding an ordinary front equitable partition of a complex matrix $\\mathbf {A}$ .",
"At each step one considers a temporary partition (initially often the single cell partition) and (sequentially) subdivides any cell $c_i$ for any $j$ according to the entries of $\\mathbf {A}_{ij}\\mathbf {j}_{n_j}$ , called colors, s.t.",
"each subcell is induced by a unique color, until this subdivision is non trivial, resulting in a refined partition.",
"One iterates until any feasible subdivision is trivial, i.e.",
"the final partition is the unique coarsest front equitable refinement (w.r.t.",
"to the initial partition).",
"Of course, this can be adapted for the weighted case.",
"However, if the weight vector $\\mathbf {w}$ has no zero entries one may employ the sketched procedure for the unweighted case readily by considering the matrix $\\operatorname{diag}\\left(\\mathbf {w}\\right)^{-1}\\mathbf {A}\\operatorname{diag}\\left(\\mathbf {w}\\right)$ .",
"This follows by left multiplication of $\\operatorname{diag}\\left(\\mathbf {w}\\right)^{-1}$ to the equitability condition $\\mathbf {A}\\operatorname{diag}\\left(\\mathbf {w}\\right)\\mathbf {B}=\\operatorname{diag}\\left(\\mathbf {w}\\right)\\mathbf {B}\\mathbf {E}^{-}.$ In general, the choice of a weight vector $\\mathbf {w}$ may be guided by insights into the problem underlying the considered matrix $\\mathbf {A}$ .",
"In search for $\\mathbf {w}$ , one may also exploit that the columns of the weighted indicator matrix $\\mathbf {W}$ are a basis for the linear span of all eigenvectors of $\\mathbf {A}$ corresponding to eigensolutions of $\\mathbf {E}^{-}$ .",
"As an example, let $\\mathbf {x}$ and $\\mathbf {y}$ be two such eigenvectors for different eigenvalues.",
"Considering them separately using the top down approach above one finds the single cell partition since $\\mathbf {x}$ and $\\mathbf {y}$ are eigenvectors.",
"This may be avoided by using a non trivial linear combination, which lies in the linear span of the columns of $\\mathbf {W}$ but is not an eigenvector.",
"How to find partitions with suitably small but non zero deviation from equitability is out of the scope of this article.",
"tocchapterAppendix:"
],
[
"Generalization as a Singular Value Decomposition",
"Our proposed method for block triangularization can be described as an employment of a one-step singular value decomposition (SVD) of the weighted indicator matrix $\\mathbf {W}$ as $\\mathbf {W}=\\left[\\mathbf {\\tilde{H}}\\mathbf {\\Omega }\\right]\\left(\\begin{array}{c}\\mathbf {N}\\\\\\mathbf {0}\\end{array}\\right)\\mathbf {V}^{\\prime }$ wherein the square diagonal matrix $\\mathbf {N}$ contains the singular values of $\\mathbf {W}$ and $\\mathbf {V}=\\operatorname{diag}\\left(\\beta _1,\\ldots ,\\beta _k\\right)$ is unitary diagonal.",
"In deed, if we interpret $\\mathbf {f}_{n_i}$ and the vector blocks $\\mathbf {w}_i$ as matrices in $\\mathbb {C}^{n_i\\times 1}$ , then $\\mathbf {w}_i$ has the SVD $\\mathbf {w}_i=\\mathbf {H}\\left(\\mathbf {w}_i,\\beta _i\\right)\\frac{\\Vert \\mathbf {w}_i\\Vert }{\\beta _i}\\mathbf {f}_{n_i}=\\mathbf {H}\\left(\\mathbf {w}_i,\\beta _i\\right)\\left(\\begin{array}{c}\\Vert \\mathbf {w}_i\\Vert \\\\\\mathbf {0}\\end{array}\\right)\\overline{\\beta _i}.$ In that view, one may obtain a generalization by replacing the non vanishing vector blocks $\\mathbf {w}_i$ by rectangular matrix blocks $\\mathbf {W}_i$ with maximal column rank.",
"In the remainder of this section we build on this idea and derive an approximate block triangularization of a rectangular matrix $\\mathbf {A}\\in \\mathbb {C}^{m\\times n}$ , using given SVDs of a pair of block diagonal matrices with maximal column rank, acting on the rows and columns of $\\mathbf {A}$ respectively and separately.",
"For notational convenience we define a $2\\times 1$ block matrix with empty lower block and the identity matrix in the square upper block.",
"Definition 10 Let $r$ and $n$ be positive integers with $r\\le n$ .",
"$\\mathbf {I}_n^r=\\left(\\begin{array}{c}\\mathbf {I}_r\\\\\\mathbf {0}\\end{array}\\right)\\in \\left\\lbrace 0,1\\right\\rbrace ^{n\\times r}.$ We may identify $\\mathbf {I}_{n}^1=\\mathbf {f}_{n}$ .",
"As a block diagonal generalization we define Definition 11 Let $\\mathbf {r}=\\left(r_1,\\ldots ,r_k\\right)$ and $\\mathbf {n}=\\left(n_1,\\dots ,n_k\\right)$ be ordered sequences of positive integers, s.t.",
"$r_i\\le n_i$ .",
"$\\mathbf {I}_{\\mathbf {n}}^{\\mathbf {r}}=\\operatorname{diag}\\left(\\mathbf {I}_{n_1}^{r_1},\\ldots ,\\mathbf {I}_{n_k}^{r_k}\\right).$ We will also utilize the following permutation.",
"Definition 12 Let $\\mathbf {r}=\\left(r_1,\\dots ,r_k\\right)$ and $\\mathbf {n}=\\left(n_1,\\dots ,n_k\\right)$ be ordered sequences of positive integers s.t.",
"$\\forall \\ i\\in \\left\\lbrace 1,\\ldots ,k\\right\\rbrace \\ r_i\\le n_i$ .",
"Let $r=\\sum _ir_i$ and $n=\\sum _in_i$ .",
"Then $\\Omega _{\\mathbf {n}}^{\\mathbf {r}}: \\left\\lbrace 1,\\ldots ,n\\right\\rbrace \\rightarrow \\left\\lbrace 1,\\ldots ,n\\right\\rbrace $ is defined by $\\Omega _{\\mathbf {n}}^{\\mathbf {r}}\\left(s_i+\\sum \\limits _{j=1}^{i-1}n_j\\right)=\\left\\lbrace \\begin{array}{cc}s_i+\\sum \\limits _{j=1}^{i-1}r_j&,0<s_i\\le r_i\\\\s_i+r+\\sum \\limits _{j=1}^{i-1}\\left(n_j-r_j\\right)&,r_i<s_i\\le n_i\\end{array}\\right.$ with $i\\in \\left\\lbrace 1,\\ldots ,k\\right\\rbrace $ .",
"$\\Omega _{\\mathbf {n}}^{\\mathbf {r}}$ maps the first $r_i$ elements of cell $i$ into the first $r$ elements.",
"Proposition 1 In the notation of definitions (REF ) and (REF ) above, let $\\mathbf {\\Omega }$ be the permutation matrix corresponding to $\\Omega _{\\mathbf {n}}^{\\mathbf {r}}$ .",
"Then $\\mathbf {\\Omega }^{\\prime }\\mathbf {I}_{\\mathbf {n}}^{\\mathbf {r}}=\\mathbf {I}_n^r.$ Definition 13 Let $\\mathbf {W}^{-}\\in \\mathbb {C}^{m\\times q}$ be a block diagonal matrix with $l$ diagonal blocks $\\mathbf {W}^{-}_i\\in \\mathbb {C}^{m_i\\times q_i}$ of rank $q_i$ and let $\\mathbf {W}^{+}\\in \\mathbb {C}^{n\\times r}$ be a block diagonal matrix with $k$ diagonal blocks $\\mathbf {W}^{+}_i\\in \\mathbb {C}^{n_i\\times r_i}$ each of rank $r_i$ .",
"Let singular value decompositions for the $\\mathbf {W}^{-}_i$ be given by $\\mathbf {W}^{-}_i=\\mathbf {U}^{-}_i\\mathbf {S}^{-}_i{\\mathbf {V}^{-}_i}^{\\prime }=\\mathbf {U}^{-}_i\\left(\\mathbf {I}_{m_i}^{q_i}\\mathbf {N}^{-}_i\\right){\\mathbf {V}^{-}_i}^{\\prime }$ with square unitary $\\mathbf {U}^{-}_i\\in \\mathbb {C}^{m_i\\times m_i}$ and $\\mathbf {V}^{-}_i\\in \\mathbb {C}^{q_i\\times q_i}$ , and with $\\mathbf {S}^{-}_i=\\mathbf {I}_{m_i}^{q_i}\\mathbf {N}^{-}_i$ wherein $\\mathbf {N}^{-}_i\\in \\mathbb {R}^{q_i\\times q_i}$ is a square diagonal matrix with positive diagonal elements.",
"Let $\\mathbf {U}^{-}\\in \\mathbb {C}^{m\\times m}$ , $\\mathbf {S}^{-}\\in \\mathbb {C}^{m\\times q}$ , $\\mathbf {N}^{-}\\in \\mathbb {R}^{q\\times q}$ , and $\\mathbf {V}^{-}\\in \\mathbb {C}^{q\\times q}$ be block diagonal with $l$ diagonal blocks given by $\\mathbf {U}^{-}_i$ , $\\mathbf {S}^{-}_i$ , $\\mathbf {N}^{-}_i$ , and $\\mathbf {V}^{-}_i$ , respectively.",
"Let $\\mathbf {\\Omega }^{-}$ be the permutation matrix corresponding to the permutation $\\Omega ^{\\left(q_1,\\dots ,q_l\\right)}_{\\left(m_1,\\dots ,m_l\\right)}$ s.t.",
"${\\mathbf {\\Omega }^{-}}^{\\prime }\\mathbf {S}^{-}=\\mathbf {I}_m^q\\mathbf {N}^{-}$ .",
"This induces a singular value decomposition of $\\mathbf {W}^{-}$ as $\\mathbf {W}^{-}=\\mathbf {U}^{-}\\left({\\mathbf {\\Omega }^{-}}^{\\prime }\\mathbf {S}^{-}\\right){\\mathbf {V}^{-}}^{\\prime }=\\mathbf {U}^{-}\\left(\\mathbf {I}_m^q\\mathbf {N}^{-}\\right){\\mathbf {V}^{-}}^{\\prime }.$ Let the corresponding relations hold for $\\mathbf {W}^{+}$ and let $\\mathbf {A}\\in \\mathbb {C}^{m\\times n}$ .",
"Define the Rayleigh quotient $\\mathbf {E}^{\\mathrm {0}}\\in \\mathbb {C}^{q\\times r}$ as a block matrix with $\\mathbf {E}^{0}=\\left(\\begin{array}{ccc}\\mathbf {E}^{0}_{11}&\\cdots &\\mathbf {E}^{0}_{1n}\\\\\\vdots &\\ddots &\\vdots \\\\\\mathbf {E}^{0}_{1m}&\\cdots &\\mathbf {E}^{0}_{mn}\\end{array}\\right),\\quad \\mathbf {E}^{0}_{ij}=\\mathbf {V}^{-}_{i}{\\mathbf {I}^{q_i}_{m_i}}^{\\prime }{\\mathbf {U}^{-}_{i}}^{\\prime }\\mathbf {A}_{ij}\\mathbf {U}^{+}_{j}\\mathbf {I}^{r_j}_{n_j}{\\mathbf {V}^{+}_{j}}^{\\prime }\\in \\mathbb {C}^{q_i\\times r_j}.\\nonumber $ The front and rear deviation matrices, $\\mathbf {T}^{-}\\in \\mathbb {C}^{m\\times r}$ and $\\mathbf {T}^{+}\\in \\mathbb {C}^{n\\times q}$ respectively, are block matrices with T-ij= (Aij)U+jIrjnjV+j'- U-iIqimiV-i'(E0ij)'Cmiri, T+ij= (Aij)'U-iIqimiV-i'- U+jIrjnjV+j'(E0ij)'Cniqi.",
"Proposition 2 In the notation of definition (REF ), $\\mathbf {E}^{0}$ , $\\mathbf {T}^{-}$ , and $\\mathbf {T}^{+}$ are identical for all singular value decompositions of $\\mathbf {W}^{-}$ and $\\mathbf {W}^{+}$ which obey the block diagonal form.",
"$\\mathbf {E}^{0}$ and $\\mathbf {T}^{\\pm }$ can be entirely expressed in terms of $\\mathbf {W}^{\\pm }$ since $\\mathbf {U}^{-}\\mathbf {I}^{\\mathbf {q}}_{\\mathbf {m}}{\\mathbf {V}^{-}}^{\\prime }=\\mathbf {W}^{-}\\left({\\mathbf {W}^{-}}^{\\prime }\\mathbf {W}^{-}\\right)^{-\\frac{1}{2}}\\text{and}\\ \\ \\mathbf {U}^{+}\\mathbf {I}^{\\mathbf {r}}_{\\mathbf {n}}{\\mathbf {V}^{+}}^{\\prime }=\\mathbf {W}^{+}\\left({\\mathbf {W}^{+}}^{\\prime }\\mathbf {W}^{+}\\right)^{-\\frac{1}{2}}.$ E0=(W-'W-)-12W-'AW+(W+'W+)-12, T-= AW+ (W+'W+)-12- W-'(W-'W-)-12 E0, T+= A'W- (W-'W-)-12- W+' (W+'W+)-12 (E0)'.",
"Theorem 2 In the notation of definition (REF ) above, let $\\mathbf {\\hat{A}}={\\mathbf {\\Omega }^{-}}^{\\prime }{\\mathbf {U}^{-}}^{\\prime }\\mathbf {A}\\mathbf {U}^{+}\\mathbf {\\Omega }^{+}=\\left(\\begin{array}{cc}\\mathbf {E}^{\\mathrm {\\phantom{f}}}&{\\mathbf {D}^{+}}^{\\prime }\\\\\\mathbf {D}^{-}&\\mathbf {F}^{\\mathrm {\\phantom{r}}}\\end{array}\\right)\\quad \\text{with}\\quad \\mathbf {E}\\in \\mathbb {C}^{q\\times r}.$ and let $\\Vert \\cdot \\Vert _{U}$ be a unitarily invariant matrix norm.",
"Then $\\mathbf {E}$ and $\\mathbf {E}^{0}$ are unitarily equivalent, $\\mathbf {D}^{\\pm }$ and $\\mathbf {T}^{\\pm }$ have the same singular values, respectively, and T-U=AU+IrnV+'- U-IqmV-'U T+U=A'U-IqmV-'- U+IrnV+'' U. Unitary equivalence of $\\mathbf {E}$ and $\\mathbf {E}^{0}$ follows from $\\mathbf {E}={\\mathbf {I}^{q}_{m}}^{\\prime }\\mathbf {\\hat{A}}{\\mathbf {I}^{r}_{n}}=\\operatorname{diag}\\left(\\mathbf {U}^{-}_{1}\\mathbf {I}^{q_1}_{m_1},\\ldots ,\\mathbf {U}^{-}_{l}\\mathbf {I}^{q_l}_{m_l}\\right)^{\\prime }\\mathbf {A}\\operatorname{diag}\\left(\\mathbf {U}^{+}_{1}\\mathbf {I}^{r_1}_{n_1},\\ldots ,\\mathbf {U}^{+}_{k}\\mathbf {I}^{r_k}_{n_k}\\right),$ which uses proposition (REF ), yielding $\\mathbf {E}^{0}=\\mathbf {V}^{-}\\mathbf {E}{\\mathbf {V}^{+}}^{\\prime }.$ That $\\mathbf {D}^{-}$ and $\\mathbf {T}^{-}$ share the same multiset of singular values follows from (c0- D-)= AIrn-IqmE= -'U-'A U++Irn- IqmV-'E0V+ =-'U-'T-V+.",
"The proof for $\\mathbf {D}^{+}$ and $\\mathbf {T}^{+}$ is analogous.",
"Applying the unitary matrices ${\\mathbf {\\Omega }^{-}}^{\\prime }{\\mathbf {U}^{-}}^{\\prime }$ from the left and ${\\mathbf {V}^{+}}$ from the right to the second term in the penultimate equation of theorem (REF ) yields A +'Irn- -'Iqm V-'V+U =(cE- D-) -(c V-' V+ 0)U.",
"The last term is readily minimized for $\\mathbf {\\Theta }={\\mathbf {V}^{-}}\\mathbf {E}{\\mathbf {V}^{+}}^{\\prime }=\\mathbf {E}^{0}.$ The minimum is obviously unique if $\\Vert \\cdot \\Vert _U$ is a Schatten norm.",
"A similar proof applies for the minimum property of $\\mathbf {T}^{+}$ ."
]
] | 1605.05924 |
[
[
"Finite-size scaling of Lennard-Jones droplet formation at fixed density"
],
[
"Abstract We reaccess the droplet condensation-evaporation transition of a three-dimensional Lennard-Jones system upon a temperature change.",
"With the help of parallel multicanonical simulations we obtain precise estimates of the transition temperature and the width of the transition for systems with up to 2048 particles.",
"This allows us to supplement previous observations of finite-size scaling regimes with a clearer picture also for the case of a continuous particle model."
],
[
"Introduction",
"Despite the long-lasting scientific interest in droplet formation, it is still a modern problem with many open questions.",
"This is partially due to the general nature of droplet formation, with relevance ranging from metastable decay over cloud formation to cluster formation in protein solutions.",
"In this work, we consider equilibrium droplet formation which yields a firm basis to study the transition between a homogeneous gas and a mixed phase of a droplet in equilibrium with surrounding vapor [1], [2], [3], [4], [5].",
"The problem is formulated in the canonical ensemble.",
"A common control parameter is the density while the temperature is fixed.",
"This allows one, in principle, to access temperature-dependent quantities such as the isothermal compressibility and the interface tension.",
"At fixed temperature the theory has been verified by several computational studies, including lattice [6], [7], [8] and Lennard-Jones [9], [10], [11] systems.",
"Instead, however, one can fix the density and vary the temperature which yields an equivalent but “orthogonal” finite-size scaling behavior [12], [13].",
"In this case, we recently observed an intermediate scaling regime consistent with finite-size scaling results for polymer aggregation [14].",
"Here, we present new results for the three-dimensional Lennard-Jones system at fixed density which extend and supplement our previous results [13]."
],
[
"Model and Method",
"We consider $N$ Lennard-Jones particles in a three-dimensional box of length $L$ with periodic boundary conditions, see Fig.",
"REF .",
"The self-avoidance and short-range attraction is modeled by the pairwise Lennard-Jones potential $V_{\\rm LJ}(r_{ij})= 4\\epsilon \\left[\\left(\\frac{\\sigma }{r_{ij}}\\right)^{12}- \\left(\\frac{\\sigma }{r_{ij}}\\right)^6\\right],$ where $r_{ij}$ is the distance between particle $i$ and $j$ .",
"As in Ref.",
"[13], we set $\\epsilon =1$ and $\\sigma =2^{-1/6}$ .",
"The computational demand can be reduced by introducing a cutoff radius $r_c=2.5\\sigma $ above which particles do not interact anymore.",
"The potential is then shifted by $V_{\\rm LJ}(r_c)$ in order to be continuous, yielding $V^*_{\\rm LJ}(r) ={\\left\\lbrace \\begin{array}{ll}V_{\\rm LJ}(r)-V_{\\rm LJ}(r_c) & r < r_c \\\\0 & \\mathrm {else}\\end{array}\\right.",
"}.$ This is in accordance with the existing literature and enables the application of a domain decomposition, where the periodic box is decomposed into equally large (cubic) domains.",
"These domains have to be at least of the size $r_c$ .",
"Then, the interaction of each particle is obtained by evaluating only its domain and the adjacent ones (in three dimensions this adds up to $3^3=27$ domains).",
"Especially in the gas phase the simulation benefits from this procedure, where the particles are equally distributed in the full box.",
"Following Ref.",
"[13], we fix the density $\\rho =N/L^3$ and vary the temperature $T$ .",
"This allows the application of multicanonical simulations [15], [16], [17], [18], which are well-suited for first-order phase transitions such as condensation.",
"At a first-order transition two phases are in coexistence with suppressed transition states in between.",
"This is circumvented by replacing the canonical Boltzmann weight $\\exp (-E/k_{\\rm B}T)$ with an alterable weight function, which is iteratively adapted in order to yield a flat histogram in the energy.",
"Each iteration is in equilibrium, sampling the distribution according to the current weight function.",
"This leads to a straightforward parallelization [19], [20] which has been shown to perform very well for the problem at hand [8].",
"In the end, canonical expectation values are estimated by reweighting the data from a multicanonical production run.",
"Figure: Illustration of a three-dimensional Lennard-Jones gas (left) and a dropletin equilibrium with surrounding vapor (right).",
"Shown is a system with N=2048N=2048particles at density ρ=10 -2 \\rho =10^{-2} in a box with periodic boundaryconditions.A crucial aspect is the selection of Monte Carlo updates.",
"While in Ref.",
"[13] we restricted ourselves to local particles shifts, we added here particle “jumps” with a larger update range.",
"The update range for a particle shift was set to $r=0.3$ and for a particle jump to $r=L/2$ .",
"A new position is then selected with equal probability from a sphere of size $r$ around the old position.",
"This simple non-local update significantly enhanced the sampling of the gaseous phase and increased the performance of the simulation drastically.",
"Instead of 512 we are now able to sample up to 2048 particles."
],
[
"Theory",
"Here we only want to briefly recapture the results of many previous works [2], [3], [4], [5], [13].",
"For a supersaturated gas, it was shown that the probability of intermediate-sized droplets vanishes [3], [4], which leaves us with the scenario of a homogeneous gas phase on the one side and a droplet in equilibrium with surrounding vapor on the other side, see Fig.",
"REF .",
"At a fixed temperature, the droplet formation may be achieved by adding more particle excess to the already supersaturated gas.",
"For both sides it is possible to formulate a contribution to the free-energy in terms of fluctuations (entropy of the gaseous phase) and surface tension (energy of the droplet).",
"Importantly, the finite-size dependence could be rewritten in terms of the fraction of particle excess in the droplet as a function of a dimensionless density.",
"This in turn allowed us to expand the results around the infinite-size transition temperature $T_0$ to yield the leading finite-size scaling behavior of the condensation temperature $T_c$ at fixed density [13].",
"Similarly, we showed that an expansion of the free-energy difference at the condensation transition yields an estimate of the leading-order finite-size scaling of the transition rounding $\\Delta T$ , i.e., the width of the transition.",
"For details we refer to the prior literature and give here only the three-dimensional results to leading order: $T_c -T_0 &\\propto N^{-1/4}, \\\\\\Delta T &\\propto N^{-3/4}.",
"$ A crucial observation is now that the size $R$ of the droplet at the condensation transition itself scales non-trivially with system size, namely in three dimensions $R\\propto N^{1/4}$ .",
"Thus, the leading-order scaling may be identified in terms of powers of the linear extension of the droplet itself, $\\propto R^{-1}$ respectively $\\propto R^{-3}$ .",
"This is consistent with the interpretation that the droplet size at transition is the relevant system size.",
"Then, a virtual subsystem around the droplet would lead to a transition between a homogeneous liquid (droplet) phase to a homogeneous gas phase with open boundary conditions.",
"The competition between finite-size contributions from volume ($\\propto R^3$ ) and surface ($\\propto R^2$ ) would in this case give rise to an intuitive finite-size correction of the order $R^{-1}$ [21], [22], [23], [13], [14]."
],
[
"Results",
"Due to the non-local Monte Carlo update, we are now able to extend our previous results [13] for the three-dimensional Lennard-Jones system to $N=2048$ particles with improved statistics.",
"The temperature scale is fixed by setting $k_{\\rm B}=1$ .",
"The finite-size transition temperature $T_c$ is obtained as the location of the largest peak in the specific heat $C_V=k_{\\rm B}\\beta ^2\\left(\\langle E^2\\rangle -\\langle E\\rangle ^2\\right)/N$ and plotted in Fig.",
"REF (left) versus the expected scaling behavior Eq.",
"(REF ).",
"Overall, the data qualitatively shows a linear behavior as predicted.",
"A leading-order fit for $N\\ge 1024$ yields $T_0=0.7032(3)$ with goodness-of-fit parameter $Q\\approx 0.1$ , shown in the figure as dashed black line.",
"Including additional higher-order corrections, i.e., $T_c=T_0 +a N^{-1/4}+b N^{-2/4}$ , yields $T_0=0.6979(2)$ for $N\\ge 64$ with $Q\\approx 0.1$ .",
"Both estimates differ from our previous results outside the error bars, which is expected because the fit error underestimates systematic uncertainties.",
"This is partially due to the fact that we are not yet fully in the asymptotic scaling regime, which we will discuss below.",
"The general trend of the results is compatible with our previous conclusion, where we already noted that the available three-dimensional Lennard-Jones system sizes were too small for clear results.",
"However, it is worth noting that the current estimates are closer to each other, implying an infinite-size limit of the transition temperature around $T_0\\approx 0.70$ .",
"Figure: Finite-size scaling of the transition temperature (left) and rounding(right) as the maximum location and the half width of the specific-heat peak,respectively.",
"Error bars are included but smaller than the symbol size.The distance from the asymptotic scaling regime is best visible in the rounding of the transition.",
"This is here obtained from the half width of the specific-heat peak, i.e., the width for which $C_V(T)>C_V^{\\rm max}/2$ , shown in Fig.",
"REF (right).",
"Since the width vanishes in the thermodynamic limit, a double-logarithmic plot reveals the power-law scaling Eq.",
"() as a straight line, here dashed black.",
"As previously noticed [13], an intermediate scaling regime $\\propto N^{-1}$ is clearly visible and marked with a dashed-dotted gray line.",
"This is consistent with the droplet as relevant system size as in this regime still a large fraction of particles contribute to the transition droplet.",
"However, for $N\\ge 500$ it appears that the asymptotic scaling behavior slowly manifests itself and leading-order estimates become more reliable.",
"This can be tested by performing a direct fit of the power-law ansatz $\\Delta T\\propto N^{-\\alpha }$ , shown in Fig.",
"REF for variable lower bound $N_{\\rm min}$ .",
"Only for $N_{\\rm min}\\ge 640$ the fit yields plausible goodness-of-fit values $Q>0.1$ .",
"Still, the distance to the predicted asymptotic scaling regime is noticeable.",
"This is consistent with results obtained for the three-dimensional lattice gas where decent results were only obtained for $N_{\\rm min}\\approx 2000$ [13].",
"This emphasizes once more that care has to be taken with leading-order finite-size scaling away from the asymptotic scaling regime.",
"While estimates of the thermodynamic limit become better with increasing system size, fit errors are difficult to judge because the underlying ansatz is not accurate enough.",
"The same holds true for additional higher-order corrections.",
"While the results are surely good estimates, the fit errors are a difficult measure, not capturing the uncertainty that comes from an incomplete fit model [24]."
],
[
"Conclusions",
"We have extended our previous results [13] for the three-dimensional Lennard-Jones system to larger system sizes.",
"This allows for more consistent estimates of the thermodynamic limit when considering leading-order and higher-order fits.",
"Still, the current Lennard-Jones system with up to 2048 particles at density $\\rho =10^{-2}$ is quite far away from the asymptotic scaling limit.",
"We notice that the rounding of the transition is a good indicator to visualize this distance from the asymptotic scaling regime.",
"Here, also the emerging intermediate scaling regime is best noticeable.",
"Figure: Results of direct fits of the ansatz ΔT∝N -α \\Delta T\\propto N^{-\\alpha } to the data inFig.",
"(right) with variable lower bound N min N_{\\rm min}.",
"Thedotted line shows the predicted asymptotic scaling exponent."
],
[
"Acknowledgments",
"The project was funded by the European Union, the Free State of Saxony and the Deutsche Forschungsgemeinschaft (DFG) under Grant No.",
"JA 483/31-1.",
"The authors gratefully acknowledge the computing time provided by the John von Neumann Institute for Computing (NIC) on the supercomputer JURECA at Jülich Supercomputing Centre (JSC) under Grant No.",
"HLZ24.",
"Part of this work has been financially supported by the DFG through the Leipzig Graduate School of Natural Sciences “BuildMoNa” and by the Deutsch-Französische Hochschule (DFH-UFA) through the Doctoral College “${\\mathbb {L}}^4$ ” under Grant No.",
"CDFA-02-07."
]
] | 1605.05844 |
[
[
"Suppression of interference in quantum Hall Mach-Zehnder geometry by\n upstream neutral modes"
],
[
"Abstract Mach-Zehnder interferometry has been proposed as a probe for detecting the statistics of anyonic quasiparticles in fractional quantum Hall (FQH) states.",
"Here we focus on interferometers made of multimode edge states with upstream modes.",
"We find that the interference visibility is suppressed due to downstream-upstream mode entanglement; the latter serves as a \"which path\" detector to the downstream interfering trajectories.",
"Our analysis tackles a concrete realization of filling factor 2/3, but its applicability goes beyond that specific case, and encompasses the recent observation of ubiquitous emergence of upstream neutral modes in FQH states.",
"The latter, according to our analysis, goes hand in hand with the failure to observe Mach-Zehnder anyonic interference in fractional states.",
"We point out how charge-neutral mode disentanglement will resuscitate the interference signal."
],
[
"Supplementary Material for “Suppression of interference in quantum Hall Mach-Zehnder geometry by upstream neutral modes”",
"In this Supplemental Material we present some technical aspects which were omitted in the main text.",
"Sec.",
"contains a detailed calculation of the visibility in the general case.",
"The limit $\\min (L_d,L_u)(1/v_c+1/v_n) \\gg 1/T$ is treated in Sec.",
", while the case of equal interferometer arms, $L_u=L_d=L$ is addressed in Sec.",
"; this leads to the results presented in Table.",
"II and Fig.",
"2 of the main text.",
"Sec.",
"presents a qualitative discussion of some general features of the results.",
"Finally, analysis of the almost closed QPC limit (as opposed to the nearly open QPC limit discussed in the main text) is described in Sec.",
"."
],
[
"Calculation of the visibility: The general case",
"Here we present a detailed calculation of the visibility, based on Eqs.",
"(3)–(6) of the main text.",
"This requires the evaluation of the time integral in Eq.",
"(5), which we will do in the complex $t$ plane.",
"We consider general, non-integral values of $\\kappa _c$ and $\\kappa _n$ (the values relevant for $\\nu =2/3$ are listed in Table I, where the singularities of $F(y_d,y_u,t;T)$ appearing in Eqs.",
"(5)–(6) are branch cuts with branch points at $y_d/v_c + i m/T + i \\delta _{d,c}$ , $y_u/v_c + i m/T + i\\delta _{u,c}$ , $-y_d/v_n + i m/T + i \\delta _{d,n}$ , and $-y_u/v_n + i m/T + i\\delta _{u,n}$ , where $m \\in \\mathbb {Z}$ and $\\delta _{d/u,c/n}$ are positive infinitesimals.",
"The second line of Eq.",
"(6) defines $F(y_d,y_u,t;T)$ on the real $t$ -axis {See e.g., Appendix H of Ref.",
"[38], for an explicit calculation.",
"The formulas appearing there require three modifications for our use: (i) the imaginary time $\\tau $ is replaced by $i t$ ; (ii) $x$ is replaced by $-y_d/v_c$ , $-y_u/v_c$ , $y_d/v_n$ , or $y_u/v_n$ , respectively; (iii) $\\sigma \\equiv \\mathrm {sgn}(\\tau )$ should be omitted, since we are interested in the greater Green function $G^>$ , rather than in the time ordered one}.",
"The branch cuts of $F(y_d,y_u,t;T)$ in the complex plane must be chosen in a way consistent with this value of $F(y_d,y_u,t;T)$ on the real $t$ axis but are otherwise arbitrary.",
"For $F(0,0,t;T)$ (i.e., $y_d=y_u=0$ ) the four sets of branch points merge into one set, at $i m/T + i \\delta $ .",
"It is then useful to choose the branch cuts of all the fractional-power functions appearing $F(y_d,y_u,t;T)$ (namely, $z ^{\\kappa _c^2}$ and $z ^{\\kappa _n^2}$ ) to be along the negative imaginary $z$ axis, with the argument of $z$ defined as $-\\pi /2$ and $3\\pi /2$ on the right and left sides of the cut, respectively.",
"We employ the rectangular contour depicted in Fig.",
"REF (a), which includes a “tongue” going around the $m=0$ branch cut.",
"Let us assume that $\\kappa _c^2,\\kappa _n^2$ are small enough so that the small circle around the branch point do not contribute; the final result can be analytically continued to the general case.",
"With the above choices, the part of the contour along the line $\\text{Im}(t)=1/T$ will give $-e^{i 2 \\pi (\\kappa _c^2 + \\kappa _n^2)}e^{-e^* V/T}$ times the part of the contour along the real $t$ axis.",
"We then get: $ j_{\\kappa _c^2,\\kappa _n^2}(0,0;V,T) & =- 2 i e^* e^{i\\pi (\\kappa _c^2 + \\kappa _n^2)} \\sin [2\\pi (\\kappa _c^2 + \\kappa _n^2)](\\pi T \\tau )^{2(\\kappa _c^2 + \\kappa _n^2)}\\frac{1 - e^{- e^* V / T} }{1 - e^{i 2 \\pi (\\kappa _c^2 + \\kappa _n^2) - e^* V/T}}\\int _{0}^\\infty \\text{d}t e^{i e^* V t }\\frac{ 1 }{\\sinh ^{2(\\kappa _c^2 + \\kappa _n^2)} ( \\pi T t ) }\\nonumber \\\\& =\\frac{ e^* \\sinh [e^* V/(2T)] \\sin [2\\pi (\\kappa _c^2 + \\kappa _n^2)]}{\\sin \\left[ \\pi (\\kappa _c^2 + \\kappa _n^2) + i \\frac{e^* V}{2T} \\right]}\\frac{(2 \\pi T \\tau )^{2(\\kappa _c^2 + \\kappa _n^2)}}{\\pi T}B \\left( 1 - 2 \\kappa _c^2 - 2 \\kappa _n^2, \\kappa _c^2 + \\kappa _n^2 - i \\frac{e^* V}{2 \\pi T} \\right)\\nonumber \\\\& =2 e^* \\tau \\sinh [e^* V/(2T)](2 \\pi T \\tau )^{2(\\kappa _c^2 + \\kappa _n^2)-1}\\frac{\\left| \\Gamma \\left( \\kappa _c^2 + \\kappa _n^2 + i \\frac{e^* V}{2 \\pi T}\\right) \\right|^2}{\\Gamma \\left[ 2(\\kappa _c^2 + \\kappa _n^2) \\right]},$ where $B(x,y)$ and $\\Gamma (z)$ are the Beta and Gamma functions, respectively [39].",
"Thus, for $|e^* V| \\ll T$ we have $ j_{\\kappa _c^2,\\kappa _n^2}(0,0;V,T) =2 \\pi \\tau ^2 (e^*)^2 V(2 \\pi T \\tau )^{2(\\kappa _c^2 + \\kappa _n^2 - 1 )}\\frac{\\left[ \\Gamma ( \\kappa _c^2 + \\kappa _n^2 ) \\right]^2}{\\Gamma \\left( 2\\kappa _c^2 + 2\\kappa _n^2 \\right) }\\sim V T^{2(\\kappa _c^2 + \\kappa _n^2 - 1 )},$ while for $|e^* V| \\gg T$ , remembering that $\\Gamma (z+a)/\\Gamma (z+b) \\sim z^{a-b}$ for large $|z|$ , we find $ j_{\\kappa _c^2,\\kappa _n^2}(0,0;V,T) =\\frac{(e^*)^2 V \\tau ^2}{\\Gamma (2\\kappa _c^2 + 2\\kappa _n^2)}\\left( |e^* V| \\tau \\right)^{2(\\kappa _c^2 + \\kappa _n^2 - 1)}\\sim V^{2(\\kappa _c^2 + \\kappa _n^2) - 1 }.$ Figure: Contours of integration in the complex time plane for the calculation of:(a) j κ c 2 ,κ n 2 (0,0;V,T)j_{\\kappa _c^2,\\kappa _n^2}(0,0;V,T), Eq.",
"();(b) j κ c 2 ,κ n 2 (L d ,L u ;V,T)j_{\\kappa _c^2,\\kappa _n^2}(L_d,L_u;V,T), Eq.",
"().Branch points and lines are depicted in red, the contours (traversed in a counterclockwise fashion) in dashed-blue.The infinitesimal shifts δ\\delta and δ u/d,c/n \\delta _{u/d,c/n} of the imaginary parts of the branch points from integer multiples of 1/T1/T are exaggerated for clarity.See the text for further details.For $j(L_d,L_u,t;T)$ (i.e., $y_d=L_d$ and $y_u=L_u$ ) it will be useful to choose the branch cut of the fractional power function $z ^{\\kappa _c^2}$ as before, that is to be along the negative imaginary axis, with the argument of $z$ defined as $-\\pi /2$ and $3\\pi /2$ on the right and left sides of the cut, respectively.",
"On the other hand, the branch cut of the fractional power function $z^{\\kappa _n^2}$ will be chosen along the positive imaginary axis, with the argument of $z$ defined as $\\pi /2$ or $-3\\pi /2$ on the right and left sides of the cut, respectively.",
"With this the function $F(L_d,L_u,t;T)$ will have the following branch structure [cf. Fig.",
"REF (b)]: The branch cuts starting at the branch points at $L_d/v_c + i m/T + i \\delta _{d,c}$ and $L_u/v_c + i m/T + i\\delta _{u,c}$ will go horizontally to the right (left) for even (odd) $m$ , and the branch cuts staring at the branch points at $-L_d/v_n + i m/T + i \\delta _{d,n}$ and $-L_u/v_n + i m/T + i\\delta _{u,n}$ to go horizontally to the left (right) for even (odd) $m$ .",
"We will focus on $L_u>L_d$ , and take for definiteness $\\delta _{d,c} > \\delta _{u,c}$ and $\\delta _{d,n} > \\delta _{u,n}$ .",
"For the right- (left-)going branches the phase of the argument of the corresponding fractional power is $\\mp \\pi /2$ below the cut and $\\pm 3\\pi /2$ above it.",
"Since $\\kappa _c^2,\\kappa _n^2 < 1$ (cf.",
"Table I), the small circles around the branch points do not contribute (by analytical continuation, the final results can be shown to be valid even without this restriction).",
"With the above choices, the part of he contour along the line $\\text{Im}(t)=1/T$ will give $- e^{i 2 \\pi (\\kappa _c^2 - \\kappa _n^2)}e^{-e^* V/T}$ times the part of the contour along the real $t$ axis.",
"We thus find $ j_{\\kappa _c^2,\\kappa _n^2}(L_d,L_u; & V,T) =- e^*\\frac{1-e^{- e^* V / T}}{1 - e^{i 2 \\pi (\\kappa _c^2 - \\kappa _n^2) - e^* V/T}}\\int _{-\\infty }^\\infty \\text{d}t e^{i e^* V (t-L_d/v_c) }\\times \\nonumber \\\\ &\\left\\lbrace e^{-i\\pi \\kappa _n^2}\\left(e^{i 3 \\pi \\kappa _c^2/2} - e^{-i \\pi \\kappa _c^2/2 }\\right)\\left[\\theta (t-L_d/v_c) \\theta (L_u/v_c-t) e^{i \\pi \\kappa _c^2/2}+ \\theta (t-L_u/v_c) \\left( e^{i 3 \\pi \\kappa _c^2/2} + e^{- i \\pi \\kappa _c^2/2} \\right)\\right]\\right.",
"\\nonumber \\\\ & \\left.+e^{i\\pi \\kappa _c^2}\\left(e^{-i 3 \\pi \\kappa _n^2/2} - e^{i \\pi \\kappa _n^2/2 }\\right)\\left[\\theta (t+L_u/v_n) \\theta (-L_d/v_n-t) e^{-i \\pi \\kappa _n^2/2}+ \\theta (-L_u/v_n-t) \\left( e^{-i 3 \\pi \\kappa _n^2/2} + e^{i \\pi \\kappa _n^2/2} \\right)\\right]\\right\\rbrace \\times \\nonumber \\\\ &\\left\\lbrace \\frac{ (\\pi T \\tau )^2}{\\sinh \\left[ \\pi T | t - L_d/v_c | \\right]\\sinh \\left[ \\pi T | t - L_u/v_c | \\right]}\\right\\rbrace ^{\\kappa _c^2}\\left\\lbrace \\frac{ (\\pi T \\tau )^2 }{\\sinh \\left[ \\pi T | t + L_d/v_n | \\right]\\sinh \\left[ \\pi T | t + L_u/v_n | \\right]}\\right\\rbrace ^{\\kappa _n^2}.$ This general expression can be simplified in particular cases, as detailed in the following two Sections."
],
[
"Visibility of a long-arms interferometer",
"An explicit expression for Eq.",
"(REF ) can be obtained for $\\min (L_d,L_u)(1/v_c+1/v_n) \\gg 1/T$ , allowing for arbitrary $|L_u-L_d|$ .",
"In this region, on the cuts corresponding to the neutral propagators the hyperbolic sines appearing in the charge propagators may be replaced by exponentials and vice versa.",
"The integral thus reduces to: $j_{\\kappa _c^2,\\kappa _n^2} & (L_d,L_u; V,T) =-2 i e^* e^{i\\pi (\\kappa _c^2-\\kappa _n^2)}(2 \\pi T \\tau )^{2(\\kappa _c^2 + \\kappa _n^2)}\\frac{1-e^{- e^* V / T}}{1 - e^{i 2 \\pi (\\kappa _c^2 - \\kappa _n^2) - e^* V/T}}\\int _{-\\infty }^\\infty \\text{d}t e^{i e^* V (t-L_d/v_c) }\\times \\\\ &\\left\\lbrace \\vphantom{ \\frac{ e^{-\\pi \\kappa _n^2 T (2t + L_d/v_n + L_u/v_n)} }{\\left\\lbrace 4 \\sinh \\left[ \\pi T | t - L_d/v_c | \\right] \\sinh \\left[ \\pi T | t - L_u/v_c | \\right] \\right\\rbrace ^{\\kappa _c^2} } }\\sin ( \\pi \\kappa _c^2 )\\left[ \\theta (t-L_d/v_c) \\theta (L_u/v_c-t)+ 2 \\theta (t-L_u/v_c) \\cos ( \\pi \\kappa _c^2 ) \\right]\\frac{ e^{-\\pi \\kappa _n^2 T (2t + L_d/v_n + L_u/v_n)} }{\\left\\lbrace 4 \\sinh \\left[ \\pi T | t - L_d/v_c | \\right]\\sinh \\left[ \\pi T | t - L_u/v_c | \\right] \\right\\rbrace ^{\\kappa _c^2} }\\right.",
"\\nonumber \\\\ & \\left.",
"\\nonumber - \\sin (\\pi \\kappa _n^2)\\left[\\theta (t+L_u/v_n) \\theta (-L_d/v_n-t)+ 2 \\theta (-L_u/v_n-t) \\cos (\\pi \\kappa _n^2)\\right]\\frac{ e^{\\pi \\kappa _c^2 T (2t - L_d/v_c - L_u/v_c)} }{\\left\\lbrace 4 \\sinh \\left[ \\pi T | t + L_d/v_n | \\right]\\sinh \\left[ \\pi T | t + L_u/v_n | \\right] \\right\\rbrace ^{\\kappa _n^2} }\\right\\rbrace .$ Each of the resulting integrals (on the intervals $t<-L_u/v_n$ , $-L_u/v_n < t < -L_d/v_n$ , $L_d/v_c<t<L_u/v_c$ , and $L_u/v_n<t$ , respectively) can be calculated exactly and expressed in terms of the hypergeometric function ${}_2F_1(a, b, c; z)$ [39] through an appropriate change of variables and the Euler integral: $\\int _0^1 w^{b-1} (1-w)^{c-b-1} (1-z w)^{-a} \\text{d}w= B(b,c-b) {}_2F_1(a,b,c;z).$ Employing the relations ${}_2F_1(a,b,c;z) = & \\frac{\\Gamma (c) \\Gamma (c-a-b)}{\\Gamma (c-a) \\Gamma (c-b)} {}_2F_1(a,b,a+b-c+1,1-z) +(1-z)^{c-a-b} \\frac{\\Gamma (c) \\Gamma (c-a-b)}{\\Gamma (c-a) \\Gamma (c-b)} {}_2F_1(a,b,c-a-b+1,1-z),\\\\{}_2F_1(a,b,c;z) = & (1-z)^{c-a-b} {}_2F_1(c-a, c-b, c; z),$ to bring the argument $z$ of all the hypergeometric functions to the same form, the result can be recast as: $ j_{\\kappa _c^2,\\kappa _n^2} & (L_d,L_u;V,T) =-\\frac{i e^* e^{i\\pi (\\kappa _c^2 - \\kappa _n^2)}}{\\pi T} (2 \\pi T \\tau )^{2\\kappa _c^2} (2 \\pi T \\tau )^{2\\kappa _n^2}\\Gamma (1-\\kappa _c^2)\\frac{1-e^{- e^* V /T}}{1 - e^{i 2 \\pi (\\kappa _c^2 - \\kappa _n^2) - e^* V/T}}e^{-i e^* V L_d/v_c}\\times \\nonumber \\\\ &\\left\\lbrace \\left[ \\sin (2 \\pi \\kappa _c^2)\\frac{\\Gamma \\left( \\kappa _c^2 + \\kappa _n^2 - i \\frac{e^* V}{2\\pi T} \\right)}{\\Gamma \\left(1 + \\kappa _n^2 - i \\frac{e^* V}{2 \\pi T} \\right)}\\right.+ \\sin (\\pi \\kappa _c^2)\\frac{\\Gamma \\left( -\\kappa _n^2 + i\\frac{e^* V}{2\\pi T}\\right)}{\\Gamma \\left( 1 - \\kappa _c^2 - \\kappa _n^2 + i \\frac{e^* V}{2\\pi T} \\right)}\\right]\\times \\nonumber \\\\ &e^{i e^* V L_u/v_c-\\pi \\kappa _n^2 T (2L_u/v_c + L_d/v_n + L_u/v_n)- \\pi \\kappa _c^2 T (L_u-L_d)/v_c}{}_2F_1 \\left( \\kappa _c^2, \\kappa _c^2 + \\kappa _n^2 - i \\frac{e^* V}{2\\pi T}, 1 + \\kappa _n^2 - i\\frac{e^* V}{2\\pi T} ; e^{-2 \\pi T (L_u-L_d)/v_c} \\right)\\nonumber \\\\ &+\\left[\\sin (\\pi \\kappa _c^2)\\frac{\\Gamma \\left( \\kappa _n^2 - i \\frac{e^* V}{2\\pi T}\\right)}{\\Gamma \\left( 1-\\kappa _c^2+\\kappa _n^2 - i \\frac{e^* V}{2\\pi T} \\right)}\\right]\\times \\nonumber \\\\ &e^{ i e^* V L_d/v_c-\\pi \\kappa _n^2 T (2L_d/v_c + L_d/v_n + L_u/v_n)-\\pi \\kappa _c^2 T (L_u-L_d)/v_c}{}_2F_1 \\left( \\kappa _c^2, \\kappa _c^2 - \\kappa _n^2 + i \\frac{e^* V}{2\\pi T}, 1 - \\kappa _n^2 + i\\frac{e^* V}{2\\pi T} ; e^{-2 \\pi T (L_u-L_d)/v_c} \\right)\\nonumber \\\\ & \\left.- \\left[ c \\leftrightarrow n, V \\rightarrow -V \\right]\\vphantom{\\frac{\\Gamma \\left( \\kappa _c^2 + \\kappa _n^2 - i \\frac{e^* V}{2\\pi T} \\right)}{\\Gamma \\left(1 + \\kappa _n^2 - i \\frac{e^* V}{2 \\pi T} \\right)}}\\right\\rbrace $ rearranging and using the identity $\\Gamma (z)\\Gamma (1-z) = \\pi /\\sin (\\pi z)$ [39] we find $ j_{\\kappa _c^2,\\kappa _n^2} & (L_d,L_u;V,T) =-\\frac{e^*}{2 T} (2 \\pi T \\tau )^{2\\kappa _c^2} (2 \\pi T \\tau )^{2\\kappa _n^2}e^{e^* V/(2 T) - i e^* V L_d/v_c} \\left[ 1-e^{- e^* V /T} \\right]\\times \\nonumber \\\\ &\\left\\lbrace \\frac{1}{\\sin \\left( \\pi \\kappa _n^2 - i \\frac{e^* V}{2 T} \\right)} \\frac{\\Gamma \\left( \\kappa _c^2 + \\kappa _n^2 - i \\frac{e^* V}{2\\pi T} \\right)}{\\Gamma (\\kappa _c^2) \\Gamma \\left(1 + \\kappa _n^2 - i \\frac{e^* V}{2 \\pi T} \\right)}\\right.\\times \\nonumber \\\\ &e^{ie^* V L_u/v_c-\\pi \\kappa _n^2 T (2L_u/v_c + L_d/v_n + L_u/v_n)- \\pi \\kappa _c^2 T (L_u-L_d)/v_c}{}_2F_1 \\left( \\kappa _c^2, \\kappa _c^2 + \\kappa _n^2 - i \\frac{e^* V}{2\\pi T}, 1 + \\kappa _n^2 - i \\frac{e^* V}{2\\pi T} ; e^{-2 \\pi T (L_u-L_d)/v_c} \\right)\\nonumber \\\\ &-\\frac{1}{\\sin \\left( \\pi \\kappa _n^2 - i \\frac{e^* V}{2 T} \\right)}\\frac{\\Gamma \\left( \\kappa _c^2 - \\kappa _n^2 + i \\frac{e^* V}{2\\pi T}\\right)}{\\Gamma (\\kappa _c^2) \\Gamma \\left( 1-\\kappa _n^2 + i \\frac{e^* V}{2\\pi T} \\right)}\\times \\nonumber \\\\ &e^{ ie^* V L_d/v_c-\\pi \\kappa _n^2 T (2L_d/v_c + L_d/v_n + L_u/v_n)-\\pi \\kappa _c^2 T (L_u-L_d)/v_c}{}_2F_1 \\left( \\kappa _c^2, \\kappa _c^2 - \\kappa _n^2 + i \\frac{e^* V}{2\\pi T}, 1 - \\kappa _n^2 + i\\frac{e^* V}{2\\pi T} ; e^{-2 \\pi T (L_u-L_d)/v_c} \\right)\\nonumber \\\\ & \\left.- \\left[ n \\leftrightarrow c, V \\rightarrow -V \\right]\\vphantom{\\frac{\\Gamma \\left( \\kappa _c^2 + \\kappa _n^2 + i e^* \\frac{V}{2\\pi T} \\right)}{\\Gamma \\left(1 + \\kappa _n^2 + i \\frac{e^* V}{2 \\pi T} \\right)}}\\right\\rbrace .$ In the limit $L_u-L_d \\gg \\max (v_c,v_n)/T $ the hypergeometric functions approach unity.",
"We are then left with $j_{\\kappa _c^2,\\kappa _n^2} & (L_d,L_u;V,T) =-\\frac{e^*}{2 T} (2 \\pi T \\tau )^{2\\kappa _c^2} (2 \\pi T \\tau )^{2\\kappa _n^2}e^{e^* V/(2 T) - i e^* V L_d/v_c} \\left[ 1-e^{- e^* V / T} \\right]\\times \\nonumber \\\\ &\\left\\lbrace \\frac{1}{\\sin \\left( \\pi \\kappa _n^2 - i \\frac{e^* V}{2 T} \\right)} \\frac{\\Gamma \\left( \\kappa _c^2 + \\kappa _n^2 - i \\frac{e^* V}{2\\pi T} \\right)}{\\Gamma (\\kappa _c^2) \\Gamma \\left(1 + \\kappa _n^2 - i \\frac{e^* V}{2 \\pi T} \\right)}\\right.e^{i e^* V L_u/v_c-\\pi \\kappa _n^2 T (2L_u/v_c + L_d/v_n + L_u/v_n)- \\pi \\kappa _c^2 T (L_u-L_d)/v_c}\\nonumber \\\\ &-\\frac{1}{\\sin \\left( \\pi \\kappa _n^2 - i \\frac{e^* V}{2 T} \\right)}\\frac{\\Gamma \\left( \\kappa _c^2 - \\kappa _n^2 + i \\frac{e^* V}{2\\pi T}\\right)}{\\Gamma (\\kappa _c^2) \\Gamma \\left( 1-\\kappa _n^2 + i \\frac{e^* V}{2\\pi T} \\right)}e^{ i e^* V L_d/v_c-\\pi \\kappa _n^2 T (2L_d/v_c + L_d/v_n + L_u/v_n)-\\pi \\kappa _c^2 T (L_u-L_d)/v_c}\\nonumber \\\\ & \\left.- \\left[ c \\leftrightarrow n, V \\rightarrow -V \\right]\\vphantom{\\frac{\\Gamma \\left( \\kappa _c^2 + \\kappa _n^2 + i \\frac{e^* V}{2\\pi T} \\right)}{\\Gamma \\left(1 + \\kappa _n^2 + i \\frac{e^* V}{2 \\pi T} \\right)}}\\right\\rbrace .$ In this case, for $|e^* V| \\ll T$ we obtain $j_{\\kappa _c^2,\\kappa _n^2}(L_d,L_u;V,T) = &-e^* V \\frac{(2 \\pi \\tau )^{2 (\\kappa _n^2 + \\kappa _c^2)} }{2 \\pi } T^{2(\\kappa _c^2 + \\kappa _n^2 -1)}e^{- ie^* V L_d/v_c} \\times \\nonumber \\\\ &\\left\\lbrace B(-\\kappa _n^2, \\kappa _c^2 + \\kappa _n^2 )e^{ie^* V L_u/v_c-\\pi \\kappa _n^2 T (2L_u/v_c + L_d/v_n + L_u/v_n)- \\pi \\kappa _c^2 T (L_u-L_d)/v_c}\\right.\\nonumber \\\\ &+B(\\kappa _n^2, \\kappa _c^2 - \\kappa _n^2 )e^{ ie^* V L_d/v_c-\\pi \\kappa _n^2 T (2L_d/v_c + L_d/v_n + L_u/v_n)-\\pi \\kappa _c^2 T (L_u-L_d)/v_c}\\nonumber \\\\ & \\left.- \\left[ c \\leftrightarrow n, V \\rightarrow -V \\right]\\right\\rbrace ,$ whereas for $|e^* V| \\gg T$ {recalling that $\\Gamma (a+z)/\\Gamma (b+z) \\sim z^{a-b}$ for large $z$ [39]} we find: $j_{\\kappa _c^2,\\kappa _n^2}(L_d,L_u;V,T) = &-\\frac{i e^*}{T} (2 \\pi T \\tau )^{2\\kappa _c^2} (2 \\pi T \\tau )^{2\\kappa _n^2}e^{ - ie^* V L_d/v_c} \\times \\nonumber \\\\ &\\left\\lbrace \\frac{ e^{-i \\pi \\kappa _n^2 \\text{sgn}(e^* V)} }{\\Gamma (\\kappa _c^2)}\\left( - i e^* \\frac{V}{2 \\pi T} \\right)^{\\kappa _c^2 - 1}e^{i e^* V L_u/v_c-\\pi \\kappa _n^2 T (2L_u/v_c + L_d/v_n + L_u/v_n)- \\pi \\kappa _c^2 T (L_u-L_d)/v_c}\\right.\\nonumber \\\\ &-\\frac{e^{-i \\pi \\kappa _n^2 \\text{sgn}(e^* V)}}{\\Gamma (\\kappa _c^2)}\\left( i e^* \\frac{V}{2 \\pi T} \\right)^{\\kappa _c^2 - 1}e^{ i e^* V L_d/v_c-\\pi \\kappa _n^2 T (2L_d/v_c + L_d/v_n + L_u/v_n)-\\pi \\kappa _c^2 T (L_u-L_d)/v_c}\\nonumber \\\\ & \\left.- \\left[ c \\leftrightarrow n, V \\rightarrow -V \\right]\\vphantom{\\frac{\\Gamma \\left( \\kappa _c^2 + \\kappa _n^2 + i \\frac{e^* V}{2\\pi T} \\right)}{\\Gamma \\left(1 + \\kappa _n^2 + i \\frac{e^* V}{2 \\pi T} \\right)}}\\right\\rbrace .$ Returning to Eq.",
"(REF ), in the limit $L_u - L_d \\ll \\min (v_c, v_n) / \\max (T,|e^* V|)$ , we may use ${}_2F_1(a,b,c;1) =\\frac{\\Gamma (c) \\Gamma (c-a-b)}{\\Gamma (c-a) \\Gamma (c-b)},$ to obtain: $j_{\\kappa _c^2,\\kappa _n^2}(L_d,L_u;V,T) = &-\\frac{e^*}{2 \\pi T} (2 \\pi T \\tau )^{2\\kappa _c^2} (2 \\pi T \\tau )^{2\\kappa _n^2}e^{e^* V/(2 T) - i e^* V L_d/v_c} \\left[ 1 - e^{- e^* V / T} \\right]\\times \\nonumber \\\\ &\\left\\lbrace \\frac{\\Gamma (1 - 2 \\kappa _c^2) \\sin (\\pi \\kappa _c^2)}{\\sin \\left( \\pi \\kappa _n^2 - i \\frac{e^* V}{2 T} \\right)}\\frac{\\Gamma \\left( \\kappa _c^2 + \\kappa _n^2 - i \\frac{e^* V}{2\\pi T} \\right)}{\\Gamma \\left(1 -\\kappa _c^2 + \\kappa _n^2 - i \\frac{e^* V}{2 \\pi T} \\right)}\\right.e^{ie^* V L_u/v_c-\\pi \\kappa _n^2 T (2L_u/v_c + L_d/v_n + L_u/v_n)}\\nonumber \\\\ &-\\frac{\\Gamma (1 - 2 \\kappa _c^2) \\sin (\\pi \\kappa _c^2) }{\\sin \\left( \\pi \\kappa _n^2 - i \\frac{e^* V}{2 T} \\right)}\\frac{\\Gamma \\left( \\kappa _c^2 - \\kappa _n^2 + i \\frac{e^* V}{2\\pi T}\\right)}{\\Gamma \\left( 1 - \\kappa _c^2 - \\kappa _n^2 + i \\frac{e^* V}{2\\pi T} \\right)}e^{ i e^* V L_d/v_c-\\pi \\kappa _n^2 T (2L_d/v_c + L_d/v_n + L_u/v_n)}\\nonumber \\\\ & \\left.- \\left[ c \\leftrightarrow n, V \\rightarrow -V \\right]\\vphantom{\\frac{\\Gamma \\left( \\kappa _c^2 + \\kappa _n^2 + i \\frac{e^* V}{2\\pi T} \\right)}{\\Gamma \\left(1 + \\kappa _n^2 + i \\frac{e^* V}{2 \\pi T} \\right)}}\\right\\rbrace .$ In the limit $|e^* V| \\ll T$ we now have $j_{\\kappa _c^2,\\kappa _n^2}(L_d,L_u;V,T) = &-(e^*)^2 V \\frac{\\Gamma (1 - 2 \\kappa _c^2)}{2 \\pi } T^{2(\\kappa _c^2 + \\kappa _n^2 - 1)}e^{- i e^* V L_d/v_c} \\times \\nonumber \\\\ &\\left\\lbrace \\frac{\\sin (\\pi \\kappa _c^2)}{\\sin (\\pi \\kappa _n^2)}B (1 - 2 \\kappa _c^2, \\kappa _c^2 + \\kappa _n^2 )\\right.e^{i e^* V L_u/v_c-\\pi \\kappa _n^2 T (2L_u/v_c + L_d/v_n + L_u/v_n)}\\nonumber \\\\ &-\\frac{ \\sin (\\pi \\kappa _c^2) }{\\sin ( \\pi \\kappa _n^2)}B ( 1 - 2\\kappa _c^2, \\kappa _c^2 - \\kappa _n^2 )e^{ i e^* V L_d/v_c-\\pi \\kappa _n^2 T (2L_d/v_c + L_d/v_n + L_u/v_n)}\\nonumber \\\\ & \\left.- \\left[ c \\leftrightarrow n, V \\rightarrow -V \\right]\\vphantom{\\frac{\\Gamma \\left( \\kappa _c^2 + \\kappa _n^2 + i \\frac{e^* V}{2\\pi T} \\right)}{\\Gamma \\left(1 + \\kappa _n^2 + i \\frac{e^* V}{2 \\pi T} \\right)}}\\right\\rbrace ,$ while for $|e^* V| \\gg T$ we get $j_{\\kappa _c^2,\\kappa _n^2} (L_d,L_u;V,T) = &-\\frac{i e^*}{2 \\pi T} (2 \\pi T \\tau )^{2\\kappa _c^2} (2 \\pi T \\tau )^{2\\kappa _n^2}e^{- i e^* V L_d/v_c} \\times \\nonumber \\\\ &\\left\\lbrace \\Gamma (1 - 2 \\kappa _c^2) \\sin (\\pi \\kappa _c^2) e^{-i \\pi \\kappa _n^2 \\text{sgn}(e^* V)}\\left( - i \\frac{e^* V}{2\\pi T} \\right)^{2\\kappa _c^2 - 1}\\right.e^{i e^* V L_u/v_c-\\pi \\kappa _n^2 T (2L_u/v_c + L_d/v_n + L_u/v_n)}\\nonumber \\\\ &-\\Gamma (1 - 2 \\kappa _c^2) \\sin (\\pi \\kappa _c^2) e^{-i \\pi \\kappa _n^2 \\text{sgn}(e^* V)}\\left(i e^* \\frac{V}{2\\pi T} \\right)^{2\\kappa _c^2 - 1}e^{ i e^* V L_d/v_c-\\pi \\kappa _n^2 T (2L_d/v_c + L_d/v_n + L_u/v_n)}\\nonumber \\\\ & \\left.- \\left[ \\rho \\leftrightarrow \\sigma , V \\rightarrow -V \\right]\\vphantom{\\frac{\\Gamma \\left( \\kappa _c^2 + \\kappa _n^2 + i \\frac{e^* V}{2\\pi T} \\right)}{\\Gamma \\left(1 + \\kappa _n^2 + i \\frac{e^* V}{2 \\pi T} \\right)}}\\right\\rbrace .$ If, on the other hand, $\\max (v_c,v_n)/|e^* V| \\ll (L_u-L_d) \\ll \\min (v_c, v_n)/T$ , we return to Eq.",
"(REF ).",
"We first use the relation $\\lim _{w \\rightarrow \\infty } {}_2F_{1}(a,b_0+w,c_0+w;z) = (1-z)^{-a}$ , then expand in $T(L_u-L_d)/v_{c/n}$ to find $ j_{\\kappa _c^2,\\kappa _n^2}(L_d,L_u;V,T) = &-\\frac{i e^*}{T} (2 \\pi T \\tau )^{2\\kappa _c^2} (2 \\pi T \\tau )^{2\\kappa _n^2}e^{- i e^* V L_d/v_c} \\times \\nonumber \\\\ &\\left\\lbrace e^{-i \\pi \\kappa _n^2 \\text{sgn}(e^* V)}\\left(- i \\frac{e^* V}{2\\pi T}\\right)^{\\kappa _c^2-1}\\frac{1}{\\Gamma (\\kappa _c^2)}\\right.e^{i e^* V L_u/v_c-\\pi \\kappa _n^2 T (2L_u/v_c + L_d/v_n + L_u/v_n)}\\left(2 \\pi T (L_u-L_d)/v_c \\right)^{-\\kappa _c^2}\\nonumber \\\\ &-e^{-i \\pi \\kappa _n^2 \\text{sgn}(e^* V)}\\frac{1}{\\Gamma (\\kappa _c^2)}\\left( + i \\frac{e^* V}{2\\pi T}\\right)^{\\kappa _c^2-1}e^{ i e^* V L_d/v_c-\\pi \\kappa _n^2 T (2L_d/v_c + L_d/v_n + L_u/v_n)}\\left(2 \\pi T (L_u-L_d)/v_c \\right)^{-\\kappa _c^2}\\nonumber \\\\ & \\left.- \\left[ c \\leftrightarrow n, V \\rightarrow -V \\right]\\vphantom{\\frac{\\Gamma \\left( \\kappa _c^2 + \\kappa _n^2 + i \\frac{e^* V}{2\\pi T} \\right)}{\\Gamma \\left(1 + \\kappa _n^2 + i \\frac{e^* V}{2 \\pi T} \\right)}}\\right\\rbrace .$"
],
[
"Visibility of an equal-arms interferometer",
"Here we examine in greater detail the case of equal-arms interferometer, $L_d=L_u=L$ .",
"We allow for arbitrary relation between $L/v_c + L/v_n$ , $1/T$ , and $1/(e^* V)$ .",
"Returning to Eq.",
"(REF ), it now becomes $& j_{\\kappa _c^2,\\kappa _n^2}(L,L;V,T) =-2i e^* e^{i \\pi (\\kappa _c^2-\\kappa _n^2)}\\frac{1 - e^{- e^* V / T} }{1 - e^{i 2 \\pi (\\kappa _c^2 - \\kappa _n^2) - e^* V/T}}\\int _{-\\infty }^\\infty \\text{d}t e^{i e^* V (t-L/v_c) }\\times \\nonumber \\\\ &\\left\\lbrace \\sin (2 \\pi \\kappa _c^2) \\theta (t-L/v_c)- \\sin (2 \\pi \\kappa _n^2) \\theta (-L/v_n - t)\\right\\rbrace \\left\\lbrace \\frac{ \\pi T \\tau }{\\sinh \\left[ \\pi T | t - L/v_c | \\right]}\\right\\rbrace ^{2 \\kappa _c^2}\\left\\lbrace \\frac{ \\pi T \\tau }{\\sinh \\left[ \\pi T | t + L/v_n | \\right]}\\right\\rbrace ^{2\\kappa _n^2}.$ Employing the same methods as before, the integral over each of the regimes $t>L/v_c$ and $t<-L/v_n$ can be expressed exactly in terms of hypergeometric functions, resulting in: $ j_{\\kappa _c^2,\\kappa _n^2} (L,L;V,T) = &2 \\pi i e^* \\tau \\left( 2 \\pi T \\tau \\right)^{2(\\kappa _c^2 + \\kappa _n^2)-1}\\times \\nonumber \\\\ &\\left\\lbrace e^{- 2 \\pi \\kappa _n^2 T (L/v_c + L/v_n) }\\frac{\\sinh \\left( \\pi \\frac{e^* V}{2 \\pi T} \\right) }{\\sinh \\left( \\pi \\frac{e^* V}{2 \\pi T} - i \\pi (\\kappa _c^2-\\kappa _n^2) \\right) }\\frac{\\Gamma \\left( \\kappa _c^2 + \\kappa _n^2 - i \\frac{e^* V}{2\\pi T} \\right) }{\\Gamma (2 \\kappa _c^2) \\Gamma \\left( 1 - \\kappa _c^2 + \\kappa _n^2 - i \\frac{e^* V}{2\\pi T} \\right)}\\times \\right.",
"\\nonumber \\\\ &{}_2F_1 \\left( 2 \\kappa _n^2, \\kappa _c^2 + \\kappa _n^2 - i \\frac{e^* V}{2\\pi T}, 1 - \\kappa _c^2 + \\kappa _n^2 - i \\frac{e^* V}{2\\pi T}; e^{-2\\pi T (L/v_c + L/v_n) } \\right)\\nonumber \\\\ &-e^{- i e^* V (L/v_c + L/v_n) - 2 \\pi \\kappa _c^2 T (L/v_c + L/v_n)}\\frac{\\sinh \\left( \\pi \\frac{e^* V}{2 \\pi T} \\right) }{\\sinh \\left( \\pi \\frac{e^* V}{2 \\pi T} - i \\pi (\\kappa _c^2-\\kappa _n^2) \\right) }\\frac{\\Gamma \\left( \\kappa _c^2 + \\kappa _n^2 + i \\frac{e^* V}{2\\pi T} \\right) }{\\Gamma (2 \\kappa _n^2) \\Gamma \\left( 1 + \\kappa _c^2 - \\kappa _n^2 + i \\frac{e^* V}{2\\pi T} \\right)}\\times \\nonumber \\\\ &\\left.",
"{}_2F_1 \\left( 2 \\kappa _c^2, \\kappa _c^2 + \\kappa _n^2 + i \\frac{e^* V}{2\\pi T}, 1 + \\kappa _c^2 - \\kappa _n^2 + i \\frac{e^* V}{2\\pi T}; e^{-2\\pi T (L/v_c + L/v_n) } \\right)\\right\\rbrace .$ For $L/v_c + L/v_n \\gg 1/T$ we then find $j_{\\kappa _c^2,\\kappa _n^2}(L,L;V,T) = &2 \\pi i e^* \\tau \\left( 2 \\pi T \\tau \\right)^{2(\\kappa _c^2 + \\kappa _n^2)-1}\\times \\nonumber \\\\ &\\left\\lbrace e^{- 2 \\pi \\kappa _n^2 T (L/v_c + L/v_n) }\\frac{\\sinh \\left( \\pi \\frac{e^* V}{2 \\pi T} \\right) }{\\sinh \\left( \\pi \\frac{e^* V}{2 \\pi T} - i \\pi (\\kappa _c^2-\\kappa _n^2) \\right) }\\frac{\\Gamma \\left( \\kappa _c^2 + \\kappa _n^2 - i \\frac{e^* V}{2\\pi T} \\right) }{\\Gamma (2 \\kappa _c^2) \\Gamma \\left( 1 - \\kappa _c^2 + \\kappa _n^2 - i \\frac{e^* V}{2\\pi T} \\right)}\\right.\\nonumber \\\\ &\\left.-e^{ - i e^* V (L/v_c + L/v_n) - 2 \\pi \\kappa _c^2 T (L/v_c + L/v_n)}\\frac{\\sinh \\left( \\pi \\frac{e^* V}{2 \\pi T} \\right) }{\\sinh \\left( \\pi \\frac{e^* V}{2 \\pi T} - i \\pi (\\kappa _c^2-\\kappa _n^2) \\right) }\\frac{\\Gamma \\left( \\kappa _c^2 + \\kappa _n^2 + i \\frac{e^* V}{2\\pi T} \\right) }{\\Gamma (2 \\kappa _n^2) \\Gamma \\left( 1 + \\kappa _c^2 - \\kappa _n^2 + i \\frac{e^* V}{2\\pi T} \\right)}\\right\\rbrace .$ In this case, in the limit $T \\gg |e^* V|$ one obtains $& j_{\\kappa _c^2,\\kappa _n^2}(L,L;V,T) =- 2 \\pi (e^*)^2 V \\tau \\left( 2 \\pi T \\tau \\right)^{2(\\kappa _c^2 + \\kappa _n^2 - 1)}\\times \\nonumber \\\\ &\\left\\lbrace e^{- 2 \\pi \\kappa _n^2 T (L/v_c + L/v_n) }B(\\kappa _n^2+\\kappa _c^2,\\kappa _n^2-\\kappa _c^2)+e^{- i e^* V (L/v_c + L/v_n) - 2 \\pi \\kappa _c^2 T (L/v_c + L/v_n)}B(\\kappa _c^2+\\kappa _n^2,\\kappa _c^2-\\kappa _n^2) \\right\\rbrace ,$ and in the limit $T \\ll |e^* V|$ we have $& j_{\\kappa _c^2,\\kappa _n^2}(L,L;V,T) =2 \\pi i e^* \\tau \\left( 2 \\pi T \\tau \\right)^{2(\\kappa _c^2 + \\kappa _n^2)-1} e^{i \\pi (\\kappa _c^2-\\kappa _n^2) \\text{sgn}(e^* V)}\\times \\nonumber \\\\ &\\left\\lbrace e^{- 2 \\pi \\kappa _n^2 T (L/v_c + L/v_n) }\\frac{1}{\\Gamma (2 \\kappa _c^2)}\\left( - i \\frac{e^* V}{2\\pi T} \\right)^{2 \\kappa _c^2 - 1}-e^{- i e^* V (L/v_c + L/v_n) - 2 \\pi \\kappa _c^2 T (L/v_c + L/v_n)}\\frac{1}{\\Gamma (2 \\kappa _n^2)}\\left( i \\frac{e^* V}{2\\pi T} \\right)^{2\\kappa _n^2-1}\\right\\rbrace .$ .",
"For $L/v_c + L/v_n \\ll 1/T, 1/|e^* V|$ we find $j_{\\kappa _c^2,\\kappa _n^2}(L,L;V,T) =2 i e^* \\tau \\left( 2 \\pi T \\tau \\right)^{2(\\kappa _c^2 + \\kappa _n^2)-1}& \\left\\lbrace \\frac{\\sin (2 \\pi \\kappa _c^2)\\sinh \\left( \\pi \\frac{e^* V}{2 \\pi T} \\right) }{\\sinh \\left( \\pi \\frac{e^* V}{2 \\pi T} - i \\pi (\\kappa _c^2-\\kappa _n^2) \\right) }\\frac{\\Gamma (1 - 2 \\kappa _c^2 - 2 \\kappa _n^2) \\Gamma \\left( \\kappa _c^2 + \\kappa _n^2 - i \\frac{e^* V}{2\\pi T} \\right) }{ \\Gamma \\left( 1 - \\kappa _c^2 - \\kappa _n^2 - i \\frac{e^* V}{2\\pi T} \\right)}\\right.\\nonumber \\\\ &\\left.-\\frac{\\sin (2 \\pi \\kappa _n^2) \\sinh \\left( \\pi \\frac{e^* V}{2 \\pi T} \\right) }{\\sinh \\left( \\pi \\frac{e^* V}{2 \\pi T} - i \\pi (\\kappa _c^2-\\kappa _n^2) \\right) }\\frac{\\Gamma (1 - 2 \\kappa _c^2 - 2 \\kappa _n^2) \\Gamma \\left( \\kappa _c^2 + \\kappa _n^2 + i \\frac{e^* V}{2\\pi T} \\right) }{ \\Gamma \\left( 1 - \\kappa _c^2 - \\kappa _n^2 + i \\frac{e^* V}{2\\pi T} \\right)}\\right\\rbrace ,$ which can be shown {using the identity $\\Gamma (z)\\Gamma (1-z) = \\pi /\\sin (\\pi z)$ [39]} to reduce to Eqs.",
"(REF )–(REF ), as it should.",
"Finally, for $1/|e^* V| \\ll L/v_c + L/v_n \\ll 1/T$ we have: $ & j_{\\kappa _c^2,\\kappa _n^2}(L,L;V,T) =2 \\pi i e^* \\tau \\left( 2 \\pi T \\tau \\right)^{2(\\kappa _c^2 + \\kappa _n^2)-1} e^{i \\pi (\\kappa _c^2-\\kappa _n^2) \\text{sgn}(e^* V)}\\times \\nonumber \\\\ &\\left\\lbrace \\frac{1}{\\Gamma (2 \\kappa _c^2)}\\left( - i \\frac{e^* V}{2\\pi T} \\right)^{2\\kappa _c^2-1}\\left[ 2\\pi T (L/v_c + L/v_n) \\right]^{-2\\kappa _n^2}-e^{- i e^* V (L/v_c + L/v_n)}\\frac{1}{\\Gamma (2 \\kappa _n^2)}\\left( i \\frac{e^* V}{2\\pi T} \\right)^{2\\kappa _n^2-1}\\left[ 2\\pi T (L/v_c + L/v_n) \\right]^{-2\\kappa _c^2}\\right\\rbrace .$ From the above results, we can determine the visibility of the interference pattern as function of temperature and source-drain bias, using Eqs.",
"(3)–(4).",
"For $|\\gamma _a|=|\\gamma _b|$ it is the ratio $|j(L_d,L_u;V,T)/j(0,0;V,T)|^3$ , reflecting the fact that a unit charge detected at D$_2$ [Fig.",
"1(a)] involves the tunneling of three $e/3$ quasi-particles, each subject to a different effective AB flux.",
"The results are displayed in Table.",
"II and Fig.",
"2 of the main text."
],
[
"Some qualitative features of the visibility",
"There are some remarkable insights that follow from the results of the previous Sections.",
"First and foremost, the interference signal is exponentially-suppressed with increasing temperature as soon as $\\min (L_d,L_u)(1/v_c+1/v_n) \\gg 1/T$ .",
"This is true even even if $T \\ll |e^* V|$ (!).",
"At the same time, increasing the bias voltage does not lead to exponential suppression even when $|e^* V|$ is larger than any other scale, including the temperature.",
"This can be intuitively understood by thinking about free fermions with velocity $v$ : at zero temperature the voltage defines a window of allowed energies, and hence wavevectors, for tunneling.",
"Since the window is sharp, the corresponding wavepackets decay in real space at large distances $x$ only as $e^{i V x/v}/x$ , hence there is no exponential suppression of the interference.",
"On the other hand, introducing a finite temperature amounts to smearing the sharp energy window.",
"Roughly, this can be thought of as having a distribution of voltage values around some average $V_0$ .",
"Averaging $e^{iV x/v}$ with respect to this distribution will leave us with an oscillatory factor $e^{i V_0 x/v}$ multiplied by a suppression factor.",
"The latter is a function of $T x/v$ , and decays exponentially when $T x/v \\gg 1$ , independently of the value of $V_0$ .",
"Another noteworthy feature is that the two terms in Eq.",
"(REF ), and correspondingly in Eqs.",
"(REF ) and (REF ), originate from domains in the time integration where either the charge or the neutral mode overlap is maximal, but not both (which is the essence of the dephasing mechanism discussed in this work), cf.",
"Fig.",
"REF (b).",
"These two terms have a relative phase between them.",
"This gives rise to oscillations of the visibility as function of the voltage when the latter is high and the temperature small [so that the the two terms in Eqs.",
"(REF ) and (REF ) are comparable], the case described by Eqs.",
"(REF ) and (REF ).",
"This leads to the last line of Table.",
"II and the lobe structure seen in Fig.",
"2 of the main text.",
"Figure: An almost-closed QH MZI (compare with the almost-open case, Fig.",
"1 of the main text).Shown are the downstream chiral edge modes, sources, drains, and the tunneling bridges.",
"The schematic equivalent geometry is essentially the same as the one depicted in Fig.",
"1(b)–(c),with the upper and lower edges corresponding, respectively, to the inner and outer edges of the lower edge + island depicted here, instead of the island and lower edges in Fig.",
"1(a)."
],
[
"The case of electron tunneling",
"In the limit of strong tunneling bridges, the interfering paths are dominated by electron tunneling.",
"It is easy to obtain results for the correlation functions by noting that $L_d$ and $L_u$ are interchanged and the AB phase is flipped as compared with quasiparticle tunneling (cf. Fig.",
"REF ).",
"In addition, the values of $\\kappa _c$ and $\\kappa _n$ are now taken from the fourth, rather than the first row of Table I.",
"Finally, Eqs.",
"(3)–(4) are replaced by $ I_{D2}(V,T) =\\left(\\left| \\gamma _a \\right|^2 + \\left| \\gamma _b \\right|^2 \\right) j(0,0;V,T)+ \\gamma _a \\gamma _b^* e^{-2 \\pi i \\Phi _{AB}/\\Phi _0 } j(L_u,L_d;V,T)+ \\text{c.c.",
"}$ Thus, when $|\\gamma _a|=|\\gamma _b|$ the visibility is $|j(L_u,L_d;V,T)/j(0,0;V,T)|$ .",
"The quantities $j(L_u,L_d;V,T)$ and $j(0,0;V,T)$ were calculated in Secs. –.",
"The resulting behavior is summarized in Table.",
"II of the main text."
]
] | 1605.06060 |
[
[
"Consensus+Innovations Distributed Kalman Filter with Optimized Gains"
],
[
"Abstract In this paper, we address the distributed filtering and prediction of time-varying random fields represented by linear time-invariant (LTI) dynamical systems.",
"The field is observed by a sparsely connected network of agents/sensors collaborating among themselves.",
"We develop a Kalman filter type consensus+innovations distributed linear estimator of the dynamic field termed as Consensus+Innovations Kalman Filter.",
"We analyze the convergence properties of this distributed estimator.",
"We prove that the mean-squared error of the estimator asymptotically converges if the degree of instability of the field dynamics is within a pre-specified threshold defined as tracking capacity of the estimator.",
"The tracking capacity is a function of the local observation models and the agent communication network.",
"We design the optimal consensus and innovation gain matrices yielding distributed estimates with minimized mean-squared error.",
"Through numerical evaluations, we show that, the distributed estimator with optimal gains converges faster and with approximately 3dB better mean-squared error performance than previous distributed estimators."
],
[
"Introduction",
"For decades, the Kalman-Bucy filter [1], [2] has played a key role in estimation, detection, or prediction of time-varying noisy signals.",
"The Kalman filter is found in a wide variety of applications ranging from problems in navigation to environmental studies, computer vision to bioengineering, signal processing to econometrics.",
"More recently, algorithms inspired by the Kalman filter have been applied to estimate random fields monitored by a network of sensors.",
"In these problems, we distinguish two distinct layers: (a) the physical layer of the time-varying random field; and (b) the cyber layer of sensors observing the field.",
"A centralized approach to field estimation poses several challenges.",
"It requires that all sensors communicate their measurements to a centralized fusion center.",
"This is fragile to central node failure and severely taxes computationally the fusion center.",
"Moreover, it also requires excessive communication bandwidth to and from the fusion center.",
"Hence, the centralized approach is inelastic to estimation of large-scale time-varying random fields, like, when estimating temperature, rainfall, or wind-speed over large geographical areas [3], [4].",
"In [5], we proposed the Distributed Information Kalman Filter (DIKF) that is a distributed estimator of time-varying random fields consisting of two substructures.",
"The first is the Dynamic Consensus on Pseudo-Observations (DCPO), a distributed estimator of the global average of the pseudo-observations (modified versions of the observations) of the agents.",
"The second substructure uses these average estimates of pseudo-observations to estimate the time-varying random field.",
"In this paper, we develop a distributed Kalman filter like estimator, the Consensus$+$ Innovations Kalman Filter (CIKF), that instead of using the pseudo-observations uses distributed estimates of the pseudo-state (modified version of a state) to estimate the field.",
"We show how to design optimally the gain matrices of the CIKF.",
"We prove that the CIKF converges in the mean-squared error (MSE) sense when the degree of instability of the dynamics of the random field is within the network tracking capacity [6], a threshold determined by the cyber network connectivity and the local observation models.",
"Figure: Time-scales of operation of dynamics, sensing and communications.We review related prior research on distributed estimation of time-varying random fields.",
"We classify prior work into two categories based on the time-scales of operation: (a) two time-scale and (b) single time-scale.",
"In two-time scale distributed estimators, see Fig REF , agents exchange their information multiple number of times between each dynamics/observations time-scale [7], [8], [9], [10], [11], [12], [13], [14], [15], so that average consensus occurs between observations.",
"In contrast, in single time-scale approaches [6], [16], [17], [18], [19], [20], [21], [22], [23], [24], [5], the agents collaborate with their neighbors only once in between each dynamics/observation evolution.",
"In other words, the dynamics, observation, and communication follow the same time scale as depicted in Fig REF .",
"The two time-scale approach demands fast communication between agents.",
"In most practical applications, this is not true.",
"Further, references [13], [14] developed distributed state estimators assuming local observability of the dynamic state in the physical layer.",
"Such assumption is not feasible in large-scale systems.",
"References [25], [26] assume a complete cyber network, which is not scalable.",
"We are interested in a single time-scale approach.",
"We consider that the time-varying system is not locally observable at each agent, and we assume that the communication network in the cyber layer is sparsely connected.",
"In the single time-scale category, references [19], [20] propose distributed Kalman filters where the agents communicate among themselves using the Gossip protocol [27].",
"Although Gossip filters require very low communication bandwidth, their MSE is higher and their convergence rate is lower than the consensus$+$ innovations based distributed approaches.",
"References [21], [22], [24] introduced consensus$+$ innovations type distributed estimator for parameter (static state) estimation.",
"This approach is extended to estimating time-varying random states in [6], [16], [17], [18].",
"Distributed consensus$+$ innovations dynamic state estimators converge in MSE sense if their degree of instability of the state dynamics is below the network tracking capacity, see [6].",
"In this paper, we derive the tracking capacity for the CIKF.",
"Its tracking capacity is a function of the local observation models and of the agents communication connectivity.",
"The single time-scale consensus$+$ innovations distributed estimators that we introduced in [28], [29], [30], [31] run a companion filter to estimate the global average of the pseudo-innovations, a modified version of the innovations.",
"The DIKF, we proposed in [5], uses averaged pseudo-observations (linearly transformed observations) rather than pseudo-innovations.",
"In contrast, this paper uses estimates of the pseudo-state, a linear transformation of the dynamic state.",
"In centralized information filter, the pseudo-state is directly available from the observations.",
"In the distributed setting, since not all the observations are available to the local sensors, the CIKF has to distributedly estimate the pseudo-state through a consensus step.",
"Using pseudo-state rather than pseudo-innovations [28], [29], [30], [31] or pseudo-observations [5] leads to significant better performance as we show here.",
"The MSE of the CIKF is lower than that of the distributed estimators in [19], [20], [28], [29], [30], [31], [5].",
"Developing distributed estimators for time-varying random fields (distributed Kalman filter) has gained considerable attention over the last few years.",
"The goal has been to achieve MSE performance as close as possible to the optimal centralized Kalman filter.",
"We show in this paper that the CIKF converges to a bounded MSE solution requiring minimal assumptions, namely global detectability and connected network, and not requiring the additional distributed observability assumption, as needed by the DIKF.",
"The distributed optimal time-varying field estimation is, in general, NP-hard.",
"The distributed parameter estimation [21], [32] have shown asymptotic optimality, in the sense that the distributed parameter estimator is asymptotically unbiased, consistent, and efficient converging at the same rate as the centralized optimal estimator.",
"However, estimation of time-varying random fields adds another degree of complication since, while information diffuses through the network, the field itself evolves.",
"So, this lag causes a gap in performance between distributed and centralized field estimators.",
"Numerical simulations show that the proposed CIKF improves the performance by 3dB over the DIKF, reducing by half the gap to the centralized (optimal) Kalman filter, while showing a faster convergence rate than the DIKF.",
"These improvements significantly distinguish the CIKF from the DIKF.",
"The rest of the paper is organized as follows.",
"We describe the three aspects of the problem setup in Section .",
"Section introduces the pseudo-state and presents the proposed optimal gain distributed Kalman filter (CIKF).",
"In Section , we analyze the dynamics of the error processes and their covariances.",
"Section includes the analysis of the tracking capacity of the proposed CIKF.",
"We design the optimal gain matrices to obtain distributed estimates in Section .",
"Numerical simulations are in Section .",
"We present the concluding remarks in Section .",
"Proofs of the proposition, lemmas and theorems are in the Appendix REF , Appendix REF -REF , and Appendix REF -REF , respectively."
],
[
"System, Observation, and Communication",
"The distributed estimation framework consists of three components: dynamical system, local observation, and neighborhood communications.",
"These three parts include two layers: the physical layer and the cyber layer.",
"For the sake of simplicity, we motivate the model with the example of a time-varying temperature field over a large geographical area monitored by a sensor network."
],
[
"Physical layer: dynamical system",
"Consider a time-varying temperature field distributed over a large geographical area, as shown in Fig REF .",
"A first-order approximation and discretization of the temperature field provide spatio-temporal discretized temperature variables $x^j_i, j = 1, \\cdots , M$ , of $M$ sites at discrete time indexes $i = 1, \\cdots , \\;$ .",
"We stack the $M$ field variables in a temperature state vector ${\\mathbf {x}}_i = \\begin{bmatrix} x^1_i, \\cdots , x^M_i \\end{bmatrix}^T \\in \\mathbb {R}^{M}$ .",
"The evolution of the time-varying temperature field, ${\\mathbf {x}}_i$ , can be represented by a linear time-invariant (LTI) dynamical systemEven though the random field ${\\mathbf {x}}_i$ is time-varying, the system is time-invariant because the field dynamics matrix $A$ is time-invariant.",
"For a time-varying dynamical system, the field dynamics matrix $A_i$ would be a function of time.",
"${\\mathbf {x}}_{i+1} = A {\\mathbf {x}}_i + {\\mathbf {v}}_i,$ where the first-order dynamics matrix $A \\in \\mathbb {R}^{M \\times M}$ contains the coupling effects between the $M$ temperature variables, and the residual ${\\mathbf {v}}_i = \\begin{bmatrix} v^1_i, \\cdots , v^M_i \\end{bmatrix}^T \\in \\mathbb {R}^{M}$ is the system noise driving the dynamical temperature fieldThe temperature field is a time-varying random field.",
"Although the field is time-varying, it is generated by a linear time-invariant system with perturbations.. At each of the $M$ sites, the input noise $v^j_i, j = 1, \\cdots , M$ accounts for the deviations in the temperature after the overall field dynamics.",
"The field dynamics $A$ incorporates the sparsity pattern and connectivity of the physical layer consisting of the dynamical system (REF )."
],
[
"Cyber layer: local observations",
"The physical layer consisting of the field dynamics (REF ) is observed by a cyber layer consisting of a network of $N$ agents (sensors).",
"In Fig REF , we see that there are $N$ sensors, where each sensor observes the temperatures of only a few sites.",
"Denote the number of sites observed by sensor $n$ by $M_n, M_n \\ll M$ , and its measurements at time $i$ by $z_i^n \\in \\mathbb {R}^{M_n}$ .",
"The observations of the agents in the cyber layer can be represented by a linear model: ${\\mathbf {z}}_i^n = H_n {\\mathbf {x}}_i + {\\mathbf {r}}_i^n, \\qquad \\qquad n =1, \\hdots , N,$ where, the observation matrix $H_n \\in \\mathbb {R}^{M_n \\times M}$ contains the observation pattern and strength information, and the observation noise $r_i^n \\in \\mathbb {R}^{M_n}$ reflects the inaccuracy in measurements due to sensor precision, high frequency fluctuations in temperature, and other unavoidable constraints.",
"To illustrate, consider that sensor 1 is observing the temperature of 3 sites $\\lbrace x^1_i, x^2_i, x^3_i \\rbrace $ , i.e., $M_1 = 3$ .",
"An example snapshot of the observation model of agent $n$ is $\\underbrace{\\begin{bmatrix} z^{11}_i \\\\ z^{12}_i \\\\ z^{13}_i \\end{bmatrix}}_{{\\mathbf {z}}^1_i} &=\\!\\!",
"\\underbrace{\\begin{bmatrix} 1 & 0 & 0 & \\hdots & 0 \\\\ 0 & 5 & 0 & \\hdots & 0 \\\\ 0 & 0 & 6 & \\hdots & 0 \\end{bmatrix}}_{H_1}\\!\\!",
"\\underbrace{\\begin{bmatrix} x^1_i \\\\ \\vdots \\\\ x^{\\scriptsize {M}}_i \\end{bmatrix}}_{{\\mathbf {x}}_i}\\!+\\!",
"\\underbrace{\\begin{bmatrix} r^{11}_i \\\\ r^{12}_i \\\\ r^{13}_i \\end{bmatrix}}_{{\\mathbf {r}}^1_i}= \\begin{bmatrix} x^1_i \\\\ 5x^2_i \\\\ 6x^3_i \\end{bmatrix} \\!+\\!",
"\\begin{bmatrix} r^{11}_i \\\\ r^{12}_i \\\\ r^{13}_i \\end{bmatrix}.$ Now, for the ease of analysis, we aggregate the noisy local temperature measurements, ${\\mathbf {z}}^1_i, \\cdots {\\mathbf {z}}^N_i$ , of all the $N$ agents in a global observation vector, ${\\mathbf {z}}_i \\in \\mathbb {R}^{{\\sum _{n=1}^{N} M_n}}$ : $\\underbrace{\\begin{bmatrix} {\\mathbf {z}}_i^1\\\\ \\vdots \\\\ {\\mathbf {z}}_i^N\\end{bmatrix}}_{{\\mathbf {z}}_i} &=&\\underbrace{\\begin{bmatrix}H_1 \\\\ \\vdots \\\\ H_N \\end{bmatrix}}_{H} {\\mathbf {x}}_i +\\underbrace{\\begin{bmatrix} {\\mathbf {r}}_i^1 \\\\ \\vdots \\\\ {\\mathbf {r}}_i^N\\end{bmatrix}}_{{\\mathbf {r}}_i},$ where, the global observation matrix is $H\\in \\mathbb {R}^{\\sum _{n=1}^N M_n \\times M}$ and the stacked measurement noise is $r_i\\in \\mathbb {R}^{{\\sum _{n=1}^{N} M_n}}$ .",
"Note that, in general, the temperature measurement model is non-linear.",
"For non-linear cases, refer to distributed particle filter in [33] and the references cited therein.",
"Here we perform a first-order approximation to obtain a linear observation sequence."
],
[
"Cyber layer: neighborhood communication",
"In the cyber layer, the agents exchange their temperature readings or current estimates with their neighbors.",
"In many applications, to reduce communications costs, neighbors communicate only with their geographically nearest agents as shown in Fig REF .",
"Formally, the agent communication network is defined by a simple (no self-loops nor multiple edges), undirected, connected graph $\\mathcal {G = (V,E)}$ , where $\\mathcal {V} $ is the set of sensors (nodes or agents) and $\\mathcal {E}$ is the set of local communication channels (edges or links) among the agents.",
"The open $\\Omega _n$ and closed $\\overline{\\Omega }_n$ neighborhoods of agent $n$ are: $\\Omega _n &= \\lbrace l|(n,l) \\in \\mathcal {E}\\rbrace .",
"\\nonumber \\\\\\overline{\\Omega }_n &= \\lbrace n\\rbrace \\cup \\lbrace l|(n,l) \\in \\mathcal {E}\\rbrace .",
"\\nonumber $ In Fig REF the open and closed neighborhoods of agent 1 are $\\Omega _1 = \\lbrace 2,3\\rbrace $ and $\\overline{\\Omega }_1 = \\lbrace 1, 2,3\\rbrace $ , respectively.",
"The Laplacian matrix of $\\mathcal {G}$ is denoted by $L$ .",
"The eigenvalues of the positive semi-definite matrix $L$ are ordered as $ 0 = \\lambda _1(L) \\le \\lambda _2(L) \\le .",
".",
".",
"\\le \\lambda _N(L)$ .",
"For details on graphs refer to [34].",
"The communication network is sparse and time-invariant."
],
[
"Modeling assumptions",
"We make the following assumptions.",
"Assumption 1 (Gaussian processes) The system noise, ${\\mathbf {v}}_i$ , the observation noise, ${\\mathbf {r}}_i$ , and the initial condition of the system, ${\\mathbf {x}}_0$ , are Gaussian sequences, with ${{\\mathbf {v}}_i \\sim \\mathcal {N}(0,V)}, \\;\\;{{\\mathbf {r}}^n_i \\sim \\mathcal {N}(0,R_n)}, \\;\\;{\\mathbf {x}}_0 \\sim \\mathcal {N}(\\bar{{\\mathbf {x}}}_0,\\Sigma _0),$ where, $V \\in \\mathbb {R}^{{M} \\times {M}} $ , $R_n \\in \\mathbb {R}^{{M_n} \\times {M_n}}$ and $\\Sigma _0 \\in \\mathbb {R}^{{M} \\times {M}}$ are the corresponding covariance matrices.",
"The noise covariance matrix $R$ of the global noise vector ${\\mathbf {v}}_i$ in (REF ) is block-diagonal, i.e., $R=\\textrm {blockdiag} \\lbrace R_1, \\ldots , R_N \\rbrace $ , and positive-definite, i.e., $R > 0$ .",
"Assumption 2 (Uncorrelated sequences) The system noise, the observation noise, and the initial condition: $\\lbrace \\lbrace {\\mathbf {v}}_i\\rbrace _i, \\lbrace {\\mathbf {r}}_i\\rbrace _i, {\\mathbf {x}}_0\\rbrace _{i \\ge 0}$ are uncorrelated random vector sequences.",
"Assumption 3 (Prior information) Each agent in the cyber layer knows the system dynamics model, $A$ and $V$ , the initial condition statistics, $\\bar{x}_0$ and $\\Sigma _0$ , the parameters of the observation model, $H$ and $R$ , and the communication network model, $\\mathcal {G}$ .",
"In large-scale system applications, the dynamics, observation, and network Laplacian matrices, $A$ , $H_n$ , and $L$ , are sparse, with $M_n \\ll M$ , and the agents communicate with only a few of their neighbors, $|\\Omega _n| \\ll N, \\forall n$ .",
"In the dynamics (REF ) and observations (REF ), we assume that there is no deterministic input.",
"The results are readily extended if there is a known deterministic input."
],
[
"Centralized Information filter",
"Although not practical in the context of the problem we study, we use the centralized information filter to benchmark our results on distributed estimator.",
"In a centralized scheme, all the agents in the cyber layer communicate their measurements to a central fusion center, as depicted in Fig REF .",
"The fusion center performs all needed computation tasks.",
"Refer to [5], [35] for the filter, gain, and error equations of the centralized information filter.",
"Figure: Centralized estimator"
],
[
"Distributed filtering and prediction",
"This section considers our single time-scale distributed solution.",
"We start with the introduction and derivation of the key components of our distributed estimator and then present our distributed field estimator."
],
[
"Pseudo-state model",
"In a centralized information filter [35], all the observations are converted into pseudo-observations [5] to obtain the optimal estimates.",
"Following (REF ), the pseudo-observation $\\widetilde{{\\mathbf {z}}}^n_i$ of agent $n$ is $\\widetilde{{\\mathbf {z}}}^n_i &= H^T_n R_n^{-1} {\\mathbf {z}}^n_i = \\overline{H}_n {\\mathbf {x}}_i + H^T_n R_n^{-1} {\\mathbf {r}}^n_i, \\quad \\\\\\text{where,} \\qquad \\overline{H}_n &= H^T_n R_n^{-1} H_n.$ The centralized information filter computes the sum, $\\overline{{\\mathbf {z}}}_i$ of all the pseudo-observations $\\overline{{\\mathbf {z}}}_i &= \\sum _{n=1}^{N}\\widetilde{{\\mathbf {z}}}^n_i = G{\\mathbf {x}}_i + H^T R^{-1} {\\mathbf {r}}_i \\qquad \\\\\\text{where,} \\qquad \\qquad G &= \\sum _{n=1}^{N} H^T_n R_n^{-1} H_n = \\sum _{n=1}^{N} \\overline{H}_n .$ The aggregated pseudo-observation, $\\overline{{\\mathbf {z}}}_i$ , is the key term in the centralized filter.",
"It provides the innovations term in the filter updates enabling the filter to converge with minimum MSE estimates.",
"However, in the distributed solution, each agent $n$ does not have access to all the pseudo-observations; instead it can only communicate with its neighbors.",
"To address this issue, in [5] we introduced a dynamic consensus algorithm to compute the distributed estimates of the averaged pseudo-observations, $\\overline{{\\mathbf {z}}}_i$ , at each agent.",
"In (REF ), we note that the crucial term is $G{\\mathbf {x}}_i$ which carries the information of the dynamic state, ${\\mathbf {x}}_i$ ; and the second term in $\\overline{{\\mathbf {z}}}_i$ in (REF ) is noise.",
"We refer to it as the pseudo-state, ${\\mathbf {y}}_i$ , ${\\mathbf {y}}_i = G{\\mathbf {x}}_i.$ The pseudo-state, ${\\mathbf {y}}_i$ , is also a random field whose time dynamics can be represented by a discrete-time linear dynamical system.",
"The pseudo-observations $\\widetilde{{\\mathbf {z}}}^n_i$ are its linear measurements.",
"We summarize the state-space model for the pseudo-state in the following proposition.",
"See Appendix REF for the details of the proof.",
"Proposition 1 The dynamics and observations of the pseudo-state ${\\mathbf {y}}_i$ are: ${\\mathbf {y}}_{i+1} &= \\widetilde{A}{\\mathbf {y}}_i + G{\\mathbf {v}}_i + \\check{A}{\\mathbf {x}}_i \\\\\\widetilde{{\\mathbf {z}}}^n_i &= \\widetilde{H}_n{\\mathbf {y}}_i + H^T_n R_n^{-1} {\\mathbf {r}}^n_i + \\check{H}_n {\\mathbf {x}}_i.$ The pseudo-dynamics matrix $\\widetilde{A}$ , pseudo-observations matrix $\\widetilde{H}_n$ , and the matrices $\\check{A}, \\check{H}_n$ , and $\\widetilde{I}$ at agent $n$ are: $\\widetilde{A} &= G A G^{\\dagger } \\\\\\widetilde{H}_n &= H^T_n R_n^{-1} H_n G^{\\dagger } = \\overline{H}_n G^{\\dagger }\\\\\\check{A} &= GA\\widetilde{I} \\\\\\check{H}_n &= H^T_n R_n^{-1} H_n \\widetilde{I} \\\\\\widetilde{I} &= I - G^{\\dagger }G,$ where, $G^{\\dagger }$ denotes the Moore-Penrose pseudo-inverse of G. In [5], the distributed information Kalman filter (DIKF) assumes distributed observability, i.e., it considers the case where $G$ is invertible.",
"Under this assumption, $G^{\\dagger } = G^{-1}$ and $\\widetilde{I} = 0$ .",
"In this paper we relax the requirement of invertibility of $G$ , proposing a distributed estimator for general dynamics-observation models under the assumption of global detectability.",
"In most cases $\\widetilde{I}$ is low-rank.",
"In (REF ), the term $\\left( G{\\mathbf {v}}_i~+~\\check{A}{\\mathbf {x}}_i = \\xi _i, \\text{say} \\right)$ can be interpreted as the pseudo-state input noise, which follows Gauss dynamics $\\xi _i &\\sim \\mathcal {N} \\left( \\check{A} \\overline{{\\mathbf {x}}}_i \\; , \\; GVG + \\check{A} \\Sigma _i \\check{A}^T\\right) \\qquad \\qquad \\\\\\text{where,} \\qquad \\Sigma _{i} &= \\mathbb {E}\\left[ ({\\mathbf {x}}_i - \\overline{{\\mathbf {x}}}_i)({\\mathbf {x}}_i - \\overline{{\\mathbf {x}}}_i)^T \\right].$ Similarly, in (), the term $\\big ( \\delta ^n_i = H^T_nR_n^{-1}{\\mathbf {r}}^n_i + \\check{H}_n {\\mathbf {x}}_i \\big )$ is the pseudo-state observation noise at agent $n$ , which is Gaussian $\\delta ^n_i \\sim \\mathcal {N} \\Big ( \\check{H}_n \\overline{{\\mathbf {x}}}_i , \\overline{H}_n + \\check{H}_n \\Sigma _{i} \\check{H}_n \\Big ).$ For the ease of analysis, we express the pseudo-state observation model in vector form by $\\widetilde{{\\mathbf {z}}}_i \\in \\mathbb {R}^{{\\sum _{n=1}^{N} M_n}}$ , the aggregate of the noisy local temperature measurements, $\\widetilde{{\\mathbf {z}}}^1_i, \\cdots \\widetilde{{\\mathbf {z}}}^N_i$ , of all the agents, $\\underbrace{\\begin{bmatrix} \\widetilde{{\\mathbf {z}}}_i^1\\\\ \\vdots \\\\ \\widetilde{{\\mathbf {z}}}_i^N\\end{bmatrix}}_{\\widetilde{{\\mathbf {z}}}_i} &=&\\underbrace{\\begin{bmatrix} \\widetilde{H}_1 \\\\ \\vdots \\\\ \\widetilde{H}_N \\end{bmatrix}}_{\\widetilde{H}} {\\mathbf {y}}_i + D_{H}^T R^{-1} {\\mathbf {r}}_i + \\underbrace{\\begin{bmatrix} \\check{H}_1 \\\\ \\vdots \\\\ \\check{H}_N \\end{bmatrix}}_{\\check{H}} {\\mathbf {x}}_i,$ where, the matrices $\\widetilde{H}, \\check{H} \\in \\mathbb {R}^{NM \\times M}$ , and, $D_{H} = \\text{blockdiag} \\lbrace H_1, \\cdots , H_N \\rbrace $ .",
"We have established in (REF )-() the state-space dynamics and observations model of the pseudo-state, ${\\mathbf {y}}_i$ .",
"The structure of (REF )-() is similar to the dynamics and observations model (REF )-(REF ) of the random field, ${\\mathbf {x}}_i$ .",
"In the following subsection, we develop a distributed estimator with optimized gains to obtain unbiased estimates of the pseudo-state, ${\\mathbf {y}}_i$ , and of the state, ${\\mathbf {x}}_i$ , at each agent with minimized mean-squared error.",
"The pseudo-state ${\\mathbf {y}}_i$ is a noise-reduced version of the sum $\\overline{{\\mathbf {z}}_i}$ of the local pseudo-observations.",
"We demonstrate in Section that, by sharing distributed estimates of pseudo-states, the CIKF achieves 3dB better MSE error performance than the DIKF [5], which shares distributed estimates of the summed pseudo-observations $\\overline{{\\mathbf {z}}}_i$ ."
],
[
"At time $i$ , denote the $n^{\\text{th}}$ agent's distributed filter and prediction estimates of the state ${\\mathbf {x}}_i$ by $\\widehat{{\\mathbf {x}}}_{i|i}^n$ and $\\widehat{{\\mathbf {x}}}_{i+1|i}^n$ respectively.",
"Similarly, its distributed filter and prediction estimates of the pseudo-state ${\\mathbf {y}}_i$ are denoted by $\\widehat{{\\mathbf {y}}}_{i|i}^n$ and $\\widehat{{\\mathbf {y}}}_{i+1|i}^n$ .",
"At any time $i$ , each agent $n$ has access to its own pseudo-observation $\\widetilde{z}^n_i$ and receives the prediction state pseudo-state estimates, $\\widehat{{\\mathbf {y}}}_{i|i-1}^l$ , $l \\in \\Omega _n$ , of its neighbors at the previous time $i-1$ .",
"Under this setup, the minimized MSE filter and prediction estimates are the conditional means, $&\\widehat{{\\mathbf {y}}}_{i|i}^n \\!=\\!",
"\\mathbb {E} \\!\\left[ {\\mathbf {y}}_i \\;|\\; \\widetilde{{\\mathbf {z}}}^n_i, \\lbrace \\widehat{{\\mathbf {y}}}_{i|i-1}^l \\rbrace _{l \\in \\overline{\\Omega }_n} \\right] \\\\&\\widehat{{\\mathbf {x}}}_{i|i}^n \\!=\\!",
"\\mathbb {E} \\left[ {\\mathbf {x}}_i \\;|\\; \\widehat{{\\mathbf {y}}}_{i|i}^n \\right] \\\\&\\!\\!\\!\\!\\!",
"\\widehat{{\\mathbf {y}}}_{i+1|i}^n \\!=\\!",
"\\mathbb {E} \\left[ {\\mathbf {y}}_{i+1} \\;|\\; \\widetilde{{\\mathbf {z}}}^n_i, \\lbrace \\widehat{{\\mathbf {y}}}_{i|i-1}^l \\rbrace _{l \\in \\overline{\\Omega }_n} \\right] \\\\&\\!\\!\\!\\!\\!",
"\\widehat{{\\mathbf {x}}}_{i+1|i}^n \\!=\\!",
"\\mathbb {E} \\left[ {\\mathbf {x}}_{i\\!+\\!1} \\;|\\; \\widehat{{\\mathbf {y}}}_{i|i}^n \\right].$ In (REF ), $\\widehat{{\\mathbf {y}}}_{i|i}^n$ is the filtered estimate of the pseudo-state ${\\mathbf {y}}_i$ given all the pseudo-observations available at agent $n$ up to time $i$ including those of its neighbors.",
"By the principle of recursive linear estimation, instead of storing all the pseudo-observations $ \\lbrace \\lbrace \\widetilde{{\\mathbf {z}}}^n_t \\rbrace _{t = 0, \\cdots , i}, \\lbrace \\widetilde{{\\mathbf {z}}}^{n_1}_t \\rbrace ^{n_1 \\in \\Omega _n}_{t = 0, \\cdots , i-1}, \\lbrace \\widetilde{{\\mathbf {z}}}^{n_2}_t \\rbrace ^{n_2 \\in \\Omega _{n1}, \\forall n_1}_{t = 0, \\cdots , i-2}, \\cdots \\rbrace $ , we need only the current pseudo-observation $\\widetilde{{\\mathbf {z}}}^n_i$ and the pseudo-state estimates (including those of its neighbors) from the previous time instant $\\lbrace \\widehat{{\\mathbf {y}}}_{i|i-1}^l \\rbrace _{l \\in \\Omega _n}$ .",
"The distributed pseudo-state estimate, $\\widehat{{\\mathbf {y}}}_{i|i}^n$ , is the expectation of the pseudo-state, ${\\mathbf {y}}_i$ , conditioned on its own pseudo-observation $\\widetilde{{\\mathbf {z}}}^n_i$ and the prior pseudo-state estimates, $\\widehat{{\\mathbf {y}}}_{i|i-1}^l, l \\in \\overline{\\Omega }_n,$ received from its neighbors.",
"In this paper, we formulate the distributed field estimate, $\\widehat{{\\mathbf {x}}}_{i|i}^n$ , as the expectation of the field, ${\\mathbf {x}}_i$ , conditioned on the distributed pseudo-state estimate $\\widehat{{\\mathbf {y}}}_{i|i}^n$ .",
"The distributed field estimates, $\\widehat{{\\mathbf {x}}}_{i|i}^n$ , are not optimal given all observations, but they are still optimal given only the pseudo-state estimates $\\widehat{{\\mathbf {y}}}_{i|i}^n$ .",
"The filtered estimate $\\widehat{{\\mathbf {x}}}_{i|i}^n$ of ${\\mathbf {x}}_i$ in () depends on the current pseudo-state filtered estimate $\\widehat{{\\mathbf {y}}}_{i|i}^n$ .",
"Similarly, the prediction estimates $\\widehat{{\\mathbf {y}}}_{i+1|i}^n$ and $\\widehat{{\\mathbf {x}}}_{i+1|i}^n$ of ${\\mathbf {y}}_{i+1}$ and ${\\mathbf {x}}_{i+1}$ in ()-(), respectively, are conditioned on the corresponding available quantities up to time $i$ .",
"Theorem 1 The iterative updates to compute the distributed filtered estimates with optimized gains in (REF )-() are: $ \\!\\!\\!",
"\\widehat{{\\mathbf {y}}}_{i|i}^n \\!=\\!",
"\\widehat{{\\mathbf {y}}}_{i|i-1}^n &\\!+\\!",
"\\sum _{l \\in \\Omega _n} \\underbrace{ \\!",
"B^{nl}_i \\!",
"\\left( \\widehat{{\\mathbf {y}}}_{i|i-1}^l \\!-\\!",
"\\widehat{{\\mathbf {y}}}_{i|i-1}^n \\right) }_{\\text{Consensus}} \\nonumber \\\\& \\!+\\!",
"B^{nn}_i \\underbrace{ \\left( \\widetilde{{\\mathbf {z}}}^n_i \\!-\\!",
"\\left(\\widetilde{H}_n \\widehat{{\\mathbf {y}}}_{i|i-1}^n \\!+\\!",
"\\check{H}_n \\widehat{{\\mathbf {x}}}_{i|i-1}^n \\right) \\right) }_{\\text{Innovations}}, \\\\ \\!\\!\\!",
"\\widehat{{\\mathbf {x}}}_{i|i}^n \\!=\\!",
"\\widehat{{\\mathbf {x}}}_{i|i-1}^n & \\!+\\!",
"K^{n}_i \\underbrace{ \\left( \\widehat{{\\mathbf {y}}}_{i|i}^n \\!-\\!",
"G \\widehat{{\\mathbf {x}}}_{i|i-1}^n \\right) }_{\\text{Innovations}}$ where, pseudo-state gain block matrix, $B_i \\in \\mathbb {R}^{MN \\times MN}$ , is $ B_i = \\begin{bmatrix} B^{nl}_i \\end{bmatrix}_{n = 1, \\cdots , N}^{l = 1, \\cdots , N}, B^{nl}_i=0 \\; \\text{if} \\; l \\notin \\Omega _n$ .",
"The state gain block-diagonal matrix is $K_i = \\text{blockdiag}\\lbrace K^{1}_i, \\cdots , K^{N}_i\\rbrace , K^{n}_i \\in ~\\mathbb {R}^{M \\times M}$ .",
"The optimized MSE prediction estimates in ()-() are: $\\widehat{{\\mathbf {y}}}_{i+1|i}^n &= \\widetilde{A}\\widehat{{\\mathbf {y}}}_{i|i}^n + \\check{A}\\widehat{{\\mathbf {x}}}_{i|i}^n, \\\\\\widehat{{\\mathbf {x}}}_{i+1|i}^n &= A \\widehat{{\\mathbf {x}}}_{i|i}^n.$ The proof of Theorem REF is presented in Appendix REF .",
"The update equations reflect the computation tasks of each agent $n$ at each time index $i$ .",
"The gain matrices, $B_i$ and $K_i$ , in (REF )-() are deterministic and can be pre-computed and saved at each agent.",
"We discuss the details of the design of these optimal gain matrices in Section .",
"With these optimal gain matrices, the Kalman type Consensus$+$ Innovations filter and prediction updates (REF )-() provide the minimized MSE distributed estimates of the dynamic states, and hence we term our solution as Consensus$+$ Innovations Kalman Filter (CIKF)."
],
[
"CIKF: Assumptions",
"The Consensus$+$ Innovations Kalman Filter (CIKF) achieves convergence given the following assumptions: Assumption 4 (Global detectability) The dynamic state equation (REF ) and the observations model (REF ) are globally detectable, i.e., the pair $(A, H)$ is detectable.",
"Assumption 5 (Connectedness) The agent communication network is connected, i.e., the algebraic connectivity $\\lambda _2(L)$ of the Laplacian matrix $L$ of the graph $\\mathcal {G}$ is strictly positive.",
"By Assumption REF , the state-observation model (REF )-(REF ) is globally detectable but not necessarily locally detectable, i.e., $(A, H_n), \\forall n, $ are not necessarily detectable.",
"Note that these two are minimal assumptions.",
"Assumption REF is mandatory even for a centralized system, and Assumption REF is required for consensus algorithms to converge.",
"Further, note that in this paper we do not consider distributed observability (invertibility of $G$ ) of the model setup, which is the strong and restrictive assumption taken in [5], [21], [22] and [36] and similar to weak detectability presented in [19]."
],
[
"CIKF: Update algorithm",
"In this subsection, we present the step-by-step tasks executed by each agent $n$ in the cyber layer to implement the Consensus$+$ Innovations Kalman Filter (CIKF) and thereby obtain the unbiased minimized MSE distributed estimates of the dynamic state ${\\mathbf {x}}_i$ .",
"Each agent $n$ runs Algorithm REF locally.",
"[h] Consensus$+$ Innovations Kalman Filter Input: Model parameters $A$ , $V$ , $H$ , $R$ , $G$ , $L$ , $\\overline{{\\mathbf {x}}}_0$ , $\\Sigma _0$ .",
"Initialize: $\\widehat{{\\mathbf {x}}}^n_{0|-1} = \\overline{{\\mathbf {x}}}_0$ , $\\widehat{{\\mathbf {y}}}^n_{0|-1} = G\\overline{{\\mathbf {x}}}_0$ .",
"Pre-compute: Gain matrices $B_i$ and $K_i$ using Algorithm REF .",
"$i \\ge 0$ $\\!\\!\\!\\!$Communications: Broadcast $\\widehat{{\\mathbf {y}}}_{i|i-1}^n$ to all neighbors $l \\in \\Omega _n$ .",
"Receive $\\lbrace \\widehat{{\\mathbf {y}}}_{i|i-1}^l\\rbrace _{l \\in \\Omega _n}$ from neighbors.",
"$\\!\\!\\!\\!$Observation: Make measurement ${\\mathbf {z}}^n_i$ of the state ${\\mathbf {x}}_i$ .",
"Transform ${\\mathbf {z}}^n_i$ in pseudo-observation $\\widetilde{{\\mathbf {z}}}^n_i$ using ().",
"$\\!\\!\\!\\!$Filter updates: Compute the estimate $\\widehat{{\\mathbf {y}}}_{i|i}^n$ of ${\\mathbf {y}}_i$ using (REF ).",
"Compute the estimate $\\widehat{{\\mathbf {x}}}_{i|i}^n$ of the state ${\\mathbf {x}}_i$ using ().",
"$\\!\\!\\!\\!$Prediction updates: Predict the estimate $\\widehat{{\\mathbf {y}}}_{i+1|i}^n$ of ${\\mathbf {y}}_{i+1}$ using (REF ).",
"Predict the estimate $\\widehat{{\\mathbf {x}}}_{i+1|i}^n$ of the state ${\\mathbf {x}}_{i+1}$ using ().",
"Later in Section , we analyze and compare the performance of CIKF with that of the distributed information Kalman filter (DKF) [5] and of the centralized Kalman filter (CKF).",
"The centralized filter collects measurements from all the agents in the cyber layer.",
"The DIKF performs distributed dynamic averaging of the pseudo-observations.",
"The pseudo-state can be perceived as a noise-reduced version of the global average of all the pseudo-observations or pseudo-innovations.",
"The distributed estimates of the pseudo-states reduce the noise of the innovations in the filtering step of field estimation and hence further improve the MSE performance of the distributed estimator.",
"Before going into the numerical evaluation, we do theoretical error analysis of CIKF, derive the conditions for convergence guarantees, and design the optimal consensus and innovations gains in the following sections.",
"We analyze the MSE performance of the CIKF and derive its error covariance matrices.",
"First, we define the different error processes and determine their dynamics.",
"Denote the filtering error processes ${\\mathbf {e}}_{i|i}^n$ and ${\\mathbf {\\epsilon }}_{i|i}^n$ of the pseudo-state and of the state at agent $n$ by ${\\mathbf {e}}_{i|i}^n &= {\\mathbf {y}}_i - \\widehat{{\\mathbf {y}}}_{i|i}^n, \\\\{\\mathbf {\\epsilon }}_{i|i}^n &= {\\mathbf {x}}_i - \\widehat{{\\mathbf {x}}}_{i|i}^n.$ Similarly, represent the prediction error processes ${\\mathbf {e}}_{i+1|i}^n$ and ${\\mathbf {\\epsilon }}_{i+1|i}^n$ of the pseudo-state and the state at agent $n$ by ${\\mathbf {e}}_{i+1|i}^n &= {\\mathbf {y}}_{i+1} - \\widehat{{\\mathbf {y}}}_{i+1|i}^n, \\\\{\\mathbf {\\epsilon }}_{i+1|i}^n &= {\\mathbf {x}}_{i+1} - \\widehat{{\\mathbf {x}}}_{i+1|i}^n.$ We establish that the CIKF provides unbiased estimates of the state and pseudo-state in the following lemma whose proof is sketched in Appendix REF .",
"Lemma 1 The distributed filter and prediction estimates, $\\widehat{{\\mathbf {y}}}_{i|i}^n$ , $\\widehat{{\\mathbf {x}}}_{i|i}^n$ , $\\widehat{{\\mathbf {y}}}_{i+1|i}^n$ and $\\widehat{{\\mathbf {x}}}_{i+1|i}^n$ , of the pseudo-state and state are unbiased, i.e., error processes, ${\\mathbf {e}}_{i|i}^n$ , ${\\mathbf {e}}_{i+1|i}^n$ , ${\\mathbf {\\epsilon }}_{i|i}^n$ , and ${\\mathbf {\\epsilon }}_{i+1|i}^n$ are zero-mean at all agents $n$ : $\\mathbb {E}[{\\mathbf {e}}_{i|i}^n] = 0, \\;\\mathbb {E}[{\\mathbf {e}}_{i+1|i}^n] = 0, \\;\\mathbb {E}[{\\mathbf {\\epsilon }}_{i|i}^n] = 0, \\;\\mathbb {E}[{\\mathbf {\\epsilon }}_{i+1|i}^n] = 0.$ Each agent exchanges their estimates with their neighbors, hence their error processes are correlated.",
"Since the error processes of the agents are coupled, we stack them in a error vector and analyze all errors together.",
"$\\widehat{{\\mathbf {x}}}_{i|i}&\\!=\\!",
"\\begin{bmatrix} \\widehat{{\\mathbf {x}}}_{i|i}^1 \\\\ \\vdots \\\\ \\widehat{{\\mathbf {x}}}_{i|i}^N \\end{bmatrix} \\!,\\widehat{{\\mathbf {x}}}_{i+1|i}\\!=\\!",
"\\begin{bmatrix} \\widehat{{\\mathbf {x}}}_{i+1|i}^1 \\\\ \\vdots \\\\ \\widehat{{\\mathbf {x}}}_{i+1|i}^N \\end{bmatrix}\\!,\\widehat{{\\mathbf {y}}}_{i|i}\\!=\\!",
"\\begin{bmatrix} \\widehat{{\\mathbf {y}}}_{i|i}^1 \\\\ \\vdots \\\\ \\widehat{{\\mathbf {y}}}_{i|i}^N \\end{bmatrix}\\!,\\widehat{{\\mathbf {y}}}_{i+1|i}\\!=\\!",
"\\begin{bmatrix} \\widehat{{\\mathbf {y}}}_{i+1|i}^1 \\\\ \\vdots \\\\ \\widehat{{\\mathbf {y}}}_{i+1|i}^N \\end{bmatrix}\\!, \\\\{\\mathbf {\\epsilon }}_{i|i}&\\!=\\!",
"\\begin{bmatrix} {\\mathbf {\\epsilon }}_{i|i}^1 \\\\ \\vdots \\\\ {\\mathbf {\\epsilon }}_{i|i}^N \\end{bmatrix}\\!,{\\mathbf {\\epsilon }}_{i+1|i}\\!=\\!",
"\\begin{bmatrix} {\\mathbf {\\epsilon }}_{i+1|i}^1 \\\\ \\vdots \\\\ {\\mathbf {\\epsilon }}_{i+1|i}^N \\end{bmatrix}\\!,{\\mathbf {e}}_{i|i}\\!=\\!",
"\\begin{bmatrix} {\\mathbf {e}}_{i|i}^1 \\\\ \\vdots \\\\ {\\mathbf {e}}_{i|i}^N \\end{bmatrix}\\!,{\\mathbf {e}}_{i+1|i}\\!=\\!",
"\\begin{bmatrix} {\\mathbf {e}}_{i+1|i}^1 \\\\ \\vdots \\\\ {\\mathbf {e}}_{i+1|i}^N \\end{bmatrix}\\!.$ We summarize the dynamics of the error processes in the following lemma whose proof is in Appendix REF .",
"Lemma 2 The error processes, ${\\mathbf {e}}_{i|i}$ , ${\\mathbf {e}}_{i+1|i}$ , ${\\mathbf {\\epsilon }}_{i|i}$ , and ${\\mathbf {\\epsilon }}_{i|i}$ are Gaussian and their dynamics are: $&\\!\\!",
"{\\mathbf {e}}_{i|i}\\!=\\!",
"\\left( I_{\\!M\\!N} \\!-\\!",
"B^{\\mathcal {C}}_i \\!-\\!",
"B^{\\mathcal {I}}_i \\widetilde{D}_{H}\\right)\\!",
"{\\mathbf {e}}_{i|i-1}\\!-\\!",
"B^{\\mathcal {I}}_i \\check{D}_{H} {\\mathbf {\\epsilon }}_{i|i-1}\\!-\\!",
"B^{\\mathcal {I}}_i D_{H}^T R^{-1} r_i , \\\\&{\\mathbf {\\epsilon }}_{i|i}= \\left( I_{MN} - K_i \\left( I_N \\otimes G \\right) \\right) {\\mathbf {\\epsilon }}_{i|i-1}+ K_i {\\mathbf {e}}_{i|i}, \\\\&{\\mathbf {e}}_{i+1|i}= \\left(I_{\\!N} \\!\\otimes \\!",
"\\widetilde{A}\\right)\\!",
"{\\mathbf {e}}_{i|i}+ \\!\\left( I_{\\!N} \\!\\otimes \\!",
"\\check{A} \\right)\\!",
"{\\mathbf {\\epsilon }}_{i|i}+ 1_{\\!N} \\!\\otimes \\!",
"\\left(G{\\mathbf {v}}_i\\right), \\\\&{\\mathbf {\\epsilon }}_{i+1|i}= \\left( I_N \\otimes A\\right) {\\mathbf {\\epsilon }}_{i|i}+ 1_N \\otimes {\\mathbf {v}}_i,$ where, $B^{\\mathcal {C}}_i$ is the consensus gain matrix and $B^{\\mathcal {I}}_i$ , $K_i$ are the innovations gain matrices for the pseudo-state and state estimation, respectively.",
"The block diagonal matrices are $\\widetilde{D}_{H} = \\text{blockdiag}\\lbrace \\widetilde{H}_1, \\cdots , \\widetilde{H}_N \\rbrace $ and $\\check{D}_{H} = \\text{blockdiag}\\lbrace \\check{H}_1, \\cdots , \\check{H}_N \\rbrace $ .",
"The symbol $\\otimes $ denotes the Kronecker matrix product.",
"Lemma REF established that the error processes (REF )-() are unbiased.",
"It then follows that the filter and prediction error covariances of the pseudo-state and state are simply: $P_{i|i} &= \\mathbb {E}\\left[{\\mathbf {e}}_{i|i}{\\mathbf {e}}_{i|i}^T \\right] \\\\P_{i+1|i} &= \\mathbb {E}\\left[{\\mathbf {e}}_{i+1|i}{\\mathbf {e}}_{i+1|i}^T \\right] \\\\\\Sigma _{i|i} &= \\mathbb {E}\\left[{\\mathbf {\\epsilon }}_{i|i}{\\mathbf {\\epsilon }}_{i|i}^T \\right] \\\\\\Sigma _{i+1|i} &= \\mathbb {E}\\left[{\\mathbf {\\epsilon }}_{i+1|i}{\\mathbf {\\epsilon }}_{i+1|i}^T \\right]$ Note that the state estimates $\\widehat{{\\mathbf {x}}}_{i|i}$ , $\\widehat{{\\mathbf {x}}}_{i+1|i}$ depend on the pseudo-state estimates $\\widehat{{\\mathbf {y}}}_{i|i}$ , $\\widehat{{\\mathbf {y}}}_{i+1|i}$ .",
"Hence the error process (REF )-() are not uncorrelated.",
"The filter and prediction cross-covariances are: $\\Pi _{i|i} &= \\mathbb {E}\\left[{\\mathbf {\\epsilon }}_{i|i}{\\mathbf {e}}_{i|i}^T \\right] \\\\\\Pi _{i+1|i} &= \\mathbb {E}\\left[{\\mathbf {\\epsilon }}_{i+1|i}{\\mathbf {e}}_{i+1|i}^T \\right] \\\\\\Gamma _{i} &= \\mathbb {E}\\left[{\\mathbf {\\epsilon }}_{i|i-1}{\\mathbf {e}}_{i|i}^T \\right]$ In the following theorem, we define and derive the evolution of the state, pseudo-state, and cross error covariances.",
"Figure: NO_CAPTIONTheorem 2 The filter error covariances, $P_{i|i}$ , $\\Sigma _{i|i}$ , $\\Pi _{i|i}$ , and the predictor error covariances, $P_{i+1|i}$ , $\\Sigma _{i+1|i}$ , $\\Pi _{i+1|i}$ , follow Lyapunov-type iterations ()-(), where, $J = (\\mathbf {1}_N \\mathbf {1}_N^T)\\otimes I_{\\!M}$ and the initial conditions are $\\Sigma _{0|-1} = J\\!\\otimes \\!\\Sigma _0, P_{0|-1} = J\\!\\otimes \\!\\left(G\\Sigma _0 G\\right), \\Pi _{0|-1} = J\\!\\otimes \\!\\left(\\Sigma _0 G\\right)$ .",
"The proof of the theorem is in Appendix REF .",
"The iterations () and () combined together constitute the distributed version of the discrete algebraic Riccati equation.",
"The MSE of the proposed CIKF is the trace of the error covariance, $\\Sigma _{i+1|i}$ in ().",
"The optimal design of the gain matrices, $B_i$ and $K_i$ , such that the CIKF yields minimized MSE estimates, is discussed in Section .",
"Before that in Section we derive the conditions under which the CIKF converges, in other words, the MSE given by the trace of $\\Sigma _{i+1|i}$ is bounded."
],
[
"Tracking Capacity",
"The convergence properties of the CIKF is determined by the dynamics of the pseudo-state and state error processes, ${\\mathbf {e}}_{i+1|i}$ and ${\\mathbf {\\epsilon }}_{i+1|i}$ .",
"If the error dynamics are asymptotically stable, then the error processes have asymptotically bounded error covariances that in turn guarantee the convergence of the CIKF.",
"Note that if the dynamics of the prediction error processes, ${\\mathbf {e}}_{i+1|i}$ , ${\\mathbf {\\epsilon }}_{i+1|i}$ are asymptotically stable, then the dynamics of the filter error processes, ${\\mathbf {e}}_{i|i}$ , ${\\mathbf {\\epsilon }}_{i|i}$ are also asymptotically stable.",
"That is why we study the dynamics of only one of the error processes and in this paper we consider the prediction error processes."
],
[
"Asymptotic stability of error processes",
"To analyze the stability of the error processes, we first write the evolution of the prediction error processes, combining (REF )-(), ${\\mathbf {e}}_{i+1|i}&= \\underbrace{ \\left(I_{\\!N} \\!\\otimes \\!",
"\\widetilde{A}\\right)\\!",
"\\left( I_{\\!M\\!N} \\!-\\!",
"B^{\\mathcal {C}}_i \\!-\\!",
"B^{\\mathcal {I}}_i \\widetilde{D}_{H}\\right) }_{\\widetilde{F}} {\\mathbf {e}}_{i|i-1}+ \\widetilde{\\mathbf {\\phi }}_i, \\\\{\\mathbf {\\epsilon }}_{i+1|i}&= \\underbrace{ \\left( I_{\\!N} \\!\\otimes \\!",
"A\\right)\\!",
"\\left( I_{\\!M\\!N} \\!-\\!\\!",
"K_i \\left( I_{\\!N} \\!\\otimes \\!",
"G \\right) \\right) }_{F} \\!",
"{\\mathbf {\\epsilon }}_{i|i-1}\\!+\\!",
"{\\mathbf {\\phi }}_i ,$ where, the noise processes $\\widetilde{\\mathbf {\\phi }}_i$ and ${\\mathbf {\\phi }}_i$ are $\\widetilde{\\mathbf {\\phi }}_i &= \\!\\left( I_{\\!N} \\!\\otimes \\!",
"\\check{A}\\right)\\!",
"{\\mathbf {\\epsilon }}_{i|i}- \\left(I_{\\!N} \\!\\otimes \\!",
"\\widetilde{A}\\right)\\!B^{\\mathcal {I}}_i \\check{D}_{H} {\\mathbf {\\epsilon }}_{i|i-1}+ 1_{\\!N} \\!\\otimes \\!",
"\\left(G{\\mathbf {v}}_i\\right) \\nonumber \\\\& \\qquad - \\left(I_{\\!N} \\!\\otimes \\!",
"\\widetilde{A}\\right)\\!",
"B^{\\mathcal {I}}_i D_{H}^T R^{-1} r_i, \\\\{\\mathbf {\\phi }}_i &= \\left( I_{\\!N} \\!\\otimes \\!",
"A\\right)\\!",
"K_i {\\mathbf {e}}_{i|i}+ 1_N \\otimes {\\mathbf {v}}_i.$ The statistical properties of the noises, $\\widetilde{\\mathbf {\\phi }}_i$ and ${\\mathbf {\\phi }}_i$ , of the error processes, ${\\mathbf {e}}_{i+1|i}$ and ${\\mathbf {\\epsilon }}_{i+1|i}$ , are stated in the following Lemma, and the proof is included in Appendix REF .",
"Lemma 3 The noise sequences $\\widetilde{\\mathbf {\\phi }}_i$ and ${\\mathbf {\\phi }}_i$ are zero-mean Gaussian that follow $\\widetilde{\\mathbf {\\phi }}_i \\in \\mathcal {N} \\left( {\\mathbf {0}}, \\widetilde{\\Phi }_i \\right)$ and ${\\mathbf {\\phi }}_i \\in \\mathcal {N} \\left( {\\mathbf {0}}, \\Phi _i \\right)$ .",
"The dynamics of the error processes are characterized by (REF )-() and Lemma REF .",
"Let $\\rho (.",
")$ and $\\Vert .\\Vert _2$ denote the spectral radius and the spectral norm of a matrix, respectively.",
"The error processes are asymptotically stable if and only if the spectral radii of $\\widetilde{F}$ , $F$ are less than one, i.e., $\\rho \\left(\\widetilde{F}\\right) < 1, \\quad \\rho \\left(F\\right) < 1$ and the noise covariances, $\\widetilde{\\Phi }_i$ , $\\Phi _i$ are bounded, i.e., $\\Vert \\widetilde{\\Phi }_i\\Vert _2 < \\infty $ , $\\Vert \\Phi _i\\Vert _2 < \\infty , \\; \\forall i$ .",
"Now if (REF ) holds, then the prediction error covariances $P_{i+1|i}$ , $\\Sigma _{i+1|i}$ are bounded; this ensures the filter error covariances $P_{i|i}$ , $\\Sigma _{i|i}$ are also bounded.",
"Further, the model noise covariances $V$ and $R$ are bounded.",
"Then, by (REF )-(), the noise covariances $\\widetilde{\\Phi }_i$ and ${\\Phi }_i$ are bounded if the spectral radii are less than one.",
"Thus, (REF ) are the necessary and sufficient conditions for the convergence of the CIKF algorithm."
],
[
"Tracking capacity for unstable systems",
"The stability of the underlying dynamical system (REF ) in the physical layer is determined by the dynamics matrix $A$ .",
"If the system is asymptotically stable, i.e., $\\rho (A)<1$ , then there always exist gain matrices $B_i$ , $K_i$ such that (REF ) holds true.",
"Hence for stable systems, the CIKF always converges with a bounded MSE solution.",
"In contrast, for an unstable dynamical system (REF ), $\\rho (A)>1$ , it may not always be possible to find gain matrices $B_i$ , $K_i$ satisfying (REF ) conditions.",
"There exists an upper threshold on the degree of instability of the system dynamics, $A$ , that guarantees the convergence of the proposed CIKF.",
"The threshold, similar to Network Tracking Capacity in [6], is the tracking capacity of the CIKF algorithm, and it depends on the agent communication network and observation models, as summarized in the following theorem.",
"Theorem 3 The tracking capacity of the CIKF is, C, $C = \\max _{B^{\\mathcal {C}}_i, B^{\\mathcal {I}}_i} \\frac{\\lambda _{1}}{\\lambda _{m} \\begin{Vmatrix} I_{\\!M\\!N} \\!-\\!",
"B^{\\mathcal {C}}_i \\!-\\!",
"B^{\\mathcal {I}}_i \\widetilde{D}_{H} \\end{Vmatrix}_2 }$ where, $B^{\\mathcal {C}}_i$ has the same block sparsity pattern as the graph Laplacian $L$ , $B^{\\mathcal {I}}_i$ is a block diagonal matrix, and, $\\lambda _{1}$ and $\\lambda _{m}$ are the minimum and maximum non-zero eigenvalues of $G$ , $0<\\lambda _{1} \\le \\cdots \\le \\lambda _{m}$ .",
"If $\\Vert A \\Vert _2 < C$ , then there exists $B_i$ , $K_i$ such that the CIKF (REF )-() converges with bounded MSE.",
"The proof is in Appendix REF .",
"In the above theorem, the structural constraints on the gain matrices $B^{\\mathcal {C}}_i, B^{\\mathcal {I}}_i$ ensure that each agent combines its neighbors' estimates for consensus and its own pseudo-observations for the innovation part of the CIKF.",
"The block sparsity pattern of $B^{\\mathcal {C}}_i$ being similar to that of $L$ implies that the tracking capacity is dependent on the connectivity of the communication network.",
"Similarly, since $\\widetilde{D}_{H}$ is a block-diagonal matrix containing the observation matrices $H_n$ we conclude that the tracking capacity is also a function of the observation models.",
"The tracking capacity increases with the increase in communication graph connectivity and observation density.",
"For instance, the tracking capacity is infinity if all agents are connected with everyone else (complete graph) or all the agents observe the entire dynamical system (local observability).",
"Given the tracking capacity is satisfied for the system, observation, and communication models (REF )-(REF ), the question remains how to design the gain matrices $B_i$ and $K_i$ to minimize the MSE of the CIKF, which we discuss in the following section."
],
[
"Optimal Gain Design",
"The asymptotic stability of the error dynamics guarantees convergence of CIKF and bounded MSE, but here we discuss how to design the $B_i$ and $K_i$ such that the MSE is not only bounded but also minimum."
],
[
"New uncorrelated information",
"In CIKF Algorithm REF , at any time $i$ each agent $n$ makes pseudo-observation $\\widetilde{{\\mathbf {z}}}^n_i$ of the state and receives prior estimates $\\lbrace \\widehat{{\\mathbf {y}}}_{i|i-1}^l\\rbrace _{l \\in \\Omega _n}$ from its neighbors.",
"The CIKF algorithm employs this new information to compute the distributed filter estimates of the pseudo-state and state.",
"Denote the new information for the pseudo-state and state filtering by $\\widetilde{\\mathbf {\\theta }}^n_i$ and ${\\mathbf {\\theta }}^n_i$ , respectively, $\\widetilde{\\mathbf {\\theta }}^n_i = \\begin{bmatrix}\\widehat{{\\mathbf {y}}}_{i|i-1}^{l_1} \\\\ \\vdots \\\\ \\widehat{{\\mathbf {y}}}_{i|i-1}^{l_{d_n}} \\\\ \\widetilde{{\\mathbf {z}}}^n_i \\end{bmatrix}, \\quad {\\mathbf {\\theta }}^n_i = \\widehat{{\\mathbf {y}}}_{i|i}^n$ where, $\\lbrace l_1, \\cdots , l_{d_n}\\rbrace = \\Omega _n$ and $d_n = |\\Omega _n|$ is the degree of agent $n$ .",
"Note the new information, $\\widetilde{\\mathbf {\\theta }}^n_i$ and ${\\mathbf {\\theta }}^n_i$ , are Gaussian since they are linear combinations of Gaussian sequences.",
"However, $\\widetilde{\\mathbf {\\theta }}^n_i$ and ${\\mathbf {\\theta }}^n_i$ are correlated with the previous estimates $\\widehat{{\\mathbf {y}}}_{i|i-1}^n$ and $\\widehat{{\\mathbf {x}}}_{i|i-1}^n$ .",
"So, we transform them into uncorrelated new information and then combine the uncorrelated information with the previous estimates $\\widehat{{\\mathbf {y}}}_{i|i-1}^n$ and $\\widehat{{\\mathbf {x}}}_{i|i-1}^n$ to compute the current filtered estimates.",
"Lemma 4 The new uncorrelated information $\\widetilde{\\mathbf {\\nu }}^n_i$ and ${\\mathbf {\\nu }}^n_i$ for filtering update at agent n are, $\\widetilde{\\mathbf {\\nu }}^n_i &= \\widetilde{\\mathbf {\\theta }}^n_i - \\overline{\\widetilde{\\mathbf {\\theta }}}^n_i, \\qquad \\overline{\\widetilde{\\mathbf {\\theta }}}^n_i = \\mathbb {E} \\begin{bmatrix} \\widetilde{\\mathbf {\\theta }}^n_i | \\widetilde{\\mathbf {z}}^n_{i-1}, \\lbrace \\widehat{\\mathbf {y}}^l_{i-1|i-2} \\rbrace _{l \\in \\Omega _n} \\end{bmatrix} \\\\{\\mathbf {\\nu }}^n_i &= {\\mathbf {\\theta }}^n_i - \\overline{\\mathbf {\\theta }}^n_i, \\qquad \\overline{\\mathbf {\\theta }}^n_i = \\mathbb {E} \\begin{bmatrix} {\\mathbf {\\theta }}^n_i | \\widehat{\\mathbf {y}}^n_{i-1|i-1} \\end{bmatrix}$ that expands to $\\widetilde{\\mathbf {\\nu }}^n_i \\!=\\!",
"\\begin{bmatrix}\\widehat{{\\mathbf {y}}}_{i|i-1}^{l_1} - \\widehat{{\\mathbf {y}}}_{i|i-1}^n \\\\ \\vdots \\\\ \\widehat{{\\mathbf {y}}}_{i|i-1}^{l_{d_n}}-\\widehat{{\\mathbf {y}}}_{i|i-1}^n \\\\ \\widetilde{{\\mathbf {z}}}^n_i \\!\\!-\\!",
"\\widetilde{H}_{\\!n} \\widehat{{\\mathbf {y}}}_{i|i-1}^n \\!-\\!",
"\\check{H}_{\\!n} \\widetilde{I} \\widehat{{\\mathbf {x}}}_{i|i-1}^n \\end{bmatrix}\\!\\!\\!, \\;\\;{\\mathbf {\\nu }}^n_i \\!=\\!",
"\\widehat{{\\mathbf {y}}}_{i|i}^n \\!-\\!",
"G\\widehat{{\\mathbf {x}}}_{i|i-1}^n.$ The uncorrelated sequences $\\widetilde{\\mathbf {\\nu }}^n_i$ and ${\\mathbf {\\nu }}^n_i$ are zero-mean Gaussian random vectors.",
"Hence $\\widetilde{\\mathbf {\\nu }}^n_i$ and ${\\mathbf {\\nu }}^n_i$ are independent sequences.",
"The proof is in Appendix REF .",
"We write the CIKF filter updates (REF ) and () in terms of the new uncorrelated information $\\widetilde{\\mathbf {\\nu }}^n_i$ and ${\\mathbf {\\nu }}^n_i$ from (REF ), $ \\widehat{{\\mathbf {y}}}_{i|i}^n &= \\widehat{{\\mathbf {y}}}_{i|i-1}^n + \\widehat{B}^n_i \\widetilde{\\mathbf {\\nu }}^n_i \\\\ \\widehat{{\\mathbf {x}}}_{i|i}^n &= \\widehat{{\\mathbf {x}}}_{i|i-1}^n + K^n_i {\\mathbf {\\nu }}^n_i$ where, $\\widehat{B}^n_i$ are the building blocks of the pseudo-state gain matrix $B_i$ ."
],
[
"Consensus and innovation gains",
"Here, we present the methods to: (a) design the matrices $\\widehat{B}^n_i$ and $K^n_i$ ; and (b) obtain the optimal gains $B_i$ and $K_i$ from them.",
"These optimal gains provide the distributed minimized MSE estimates of the field.",
"At agent $n$ , we define the matrix $\\widehat{B}^n_i$ as $\\widehat{B}^n_i = \\begin{bmatrix} B^{n l_1}_i, \\cdots , B^{n l_{d_n}}_i, B^{n n}_i \\end{bmatrix}$ where, $\\lbrace l_1, \\cdots , l_{d_n}\\rbrace = \\Omega _n$ .",
"The gain matrix $B_i$ is a linear combination of $B^{\\mathcal {C}}_i$ and $B^{\\mathcal {I}}_i$ , where $B^{\\mathcal {C}}_i$ has the same block structure as the graph Laplacian $L$ and $B^{\\mathcal {I}}_i$ is a block diagonal.",
"The $(n,l)^{\\text{th}}$ blocks of the $n^{\\text{th}}$ row of $B^{\\mathcal {C}}_i$ are: $\\left[B^{\\mathcal {C}}_i\\right]_{nl} = \\left\\lbrace \\begin{array}{ll}- B^{n l}_i, & \\mbox{if } l \\in \\Omega _n \\\\\\sum _{j=1}^{d_n}B^{n l_j}_i, & \\mbox{if } l = n \\\\{\\mathbf {0}}, & \\text{otherwise.}",
"\\\\\\end{array} \\right.$ The $\\lbrace n,n\\rbrace ^{\\text{th}}$ blocks of the diagonal block matrices $B^{\\mathcal {I}}_i$ and $K_i$ are: $\\left[B^{\\mathcal {I}}_i\\right]_{nn} &= B^{n n}_i, \\\\\\left[K_i\\right]_{nn} &= K^{n}_i,$ Hence, once we design the matrices $\\widehat{B}^n_i$ and $K^{n}_i$ , it will provide the optimal gain matrices $B^{\\mathcal {C}}_i, B^{\\mathcal {I}}_i$ , and $K_i$ .",
"Theorem 4 The optimal gains for the CIKF algorithm are $\\widehat{B}^n_i &= \\Sigma _{{\\mathbf {y}}_i \\widetilde{\\mathbf {\\nu }}^n_i} \\left(\\Sigma _{\\widetilde{\\mathbf {\\nu }}^n_i}\\right)^{-1}, \\\\K^n_i &= \\Sigma _{{\\mathbf {x}}_i {\\mathbf {\\nu }}^n_i} \\left(\\Sigma _{{\\mathbf {\\nu }}^n_i}\\right)^{-1}$ where, $\\Sigma _{\\widetilde{\\mathbf {\\nu }}^n_i}$ , $\\Sigma _{{\\mathbf {\\nu }}^n_i}$ are the covariances of the new uncorrelated information $\\widetilde{\\mathbf {\\nu }}^n_i$ and ${\\mathbf {\\nu }}^n_i$ ; and, $\\Sigma _{{\\mathbf {y}}_i \\widetilde{\\mathbf {\\nu }}^n_i}$ , $\\Sigma _{{\\mathbf {x}}_i {\\mathbf {\\nu }}^n_i}$ are cross-covariances between ${\\mathbf {y}}_i$ , $\\widetilde{\\mathbf {\\nu }}^n_i$ and ${\\mathbf {x}}_i$ , ${\\mathbf {\\nu }}^n_i$ , respectively.",
"These covariance and cross-covariance matrices are related to the error covariance matrices, $P_{i|i-1}, P_{i|i}, \\Sigma _{i|i-1}, \\Pi _{i|i-1}, \\Gamma _i$ , by the following functions: $& \\!\\Sigma _{{\\mathbf {y}}_i \\widetilde{\\mathbf {\\nu }}^n_i} \\!=\\!",
"\\left[\\begin{array}{l;{2pt/2pt}l;{1pt/1pt}l;{2pt/2pt}l}\\!\\!\\!\\!",
"P_{i|i\\!-\\!1}^{nn} \\!-\\!",
"P_{i|i\\!-\\!1}^{nl_1} & \\cdots & P_{i|i\\!-\\!1}^{nn} \\!-\\!",
"P_{i|i\\!-\\!1}^{nl_{d_n}} & P_{i|i\\!-\\!1}^{nn} \\!",
"\\widetilde{H}^T_n \\!+\\!",
"\\Pi _{i|i\\!-\\!1}^{nn^T} \\!",
"\\check{H}^T_n \\!\\!\\!\\!\\end{array}\\right] \\\\&\\!",
"\\Sigma _{{\\mathbf {x}}_i {\\mathbf {\\nu }}^n_i} = \\Sigma _{i|i\\!-\\!1}^{nn} G - \\Gamma ^{nn}_i \\\\&\\begin{bmatrix} \\Sigma _{\\widetilde{\\mathbf {\\nu }}^n_i} \\end{bmatrix}_{qs} = \\\\&\\left\\lbrace \\begin{array}{ll}P_{i|i\\!-\\!1}^{nn} \\!-\\!",
"P_{i|i\\!-\\!1}^{nl_s} \\!-\\!",
"P_{i|i\\!-\\!1}^{l_q n} \\!+\\!",
"P_{i|i\\!-\\!1}^{l_q l_s}, & \\mbox{if } q \\le s \\le d_n \\\\\\!\\left( P_{i|i\\!-\\!1}^{nn} \\!-\\!",
"P_{i|i\\!-\\!1}^{l_q n} \\right)\\!\\widetilde{H}_n^T \\!+\\!",
"\\left( \\Pi _{i|i\\!-\\!1}^{nn} \\!-\\!",
"\\Pi _{i|i\\!-\\!1}^{l_q n} \\right)^T\\!\\!\\check{H}_n^T, & \\mbox{if } q < s = d_n\\!\\!+\\!1 \\\\\\widetilde{H}_n P_{i|i\\!-\\!1}^{nn}\\widetilde{H}_n^T + \\widetilde{H}_n \\Pi _{i|i\\!-\\!1}^{nn^T}\\check{H}_n^T + \\check{H}_n\\Pi _{i|i\\!-\\!1}^{nn}\\widetilde{H}_n^T & \\\\\\qquad \\qquad + \\check{H}_n\\Sigma _{i|i\\!-\\!1}^{nn}\\check{H}^T_n + \\overline{H}_n, & \\mbox{if } q = s = d_n\\!\\!+\\!1 \\\\ \\begin{bmatrix} \\Sigma _{\\widetilde{\\mathbf {\\nu }}^n_i} \\end{bmatrix}_{sq}^T, & \\mbox{if } q > s \\\\\\end{array} \\right.",
"\\\\&\\begin{bmatrix} \\Sigma _{{\\mathbf {\\nu }}^n_i} \\end{bmatrix}_{qs} = G\\Sigma _{i|i\\!-\\!1}^{nn}G - G \\Gamma ^{nn}_i - \\Gamma ^{nn^T}_iG + P_{i|i}^{nn}$ where, $\\left[\\Sigma _{\\widetilde{\\mathbf {\\nu }}^n_i}\\right]_{qs}$ denotes the $\\lbrace q,s\\rbrace ^{\\text{th}}$ block of the $(d_n\\!\\!+\\!1)\\times (d_n\\!\\!+\\!1)$ block matrix $\\Sigma _{\\widetilde{\\mathbf {\\nu }}^n_i}$ .",
"The proof is in Appendix REF .",
"By the Gauss-Markov theorem, the CIKF algorithm, along with this design of the consensus and innovation gain matrices, as stated in Theorem REF , results in the minimized MSE distributed estimates of the dynamic random field ${\\mathbf {x}}_i$ .",
"The gain matrices are deterministic.",
"Hence they can be precomputed offline and saved for online implementation.",
"In Algorithm REF , we state the steps that each agent $n$ runs to compute the optimal gain matrices.",
"[h] Gain Design of CIKF Input: Model parameters $A$ , $V$ , $H$ , $R$ , $G$ , $L$ , $\\Sigma _0$ .",
"Initialize: $\\Sigma _{0|-1} = J \\!\\otimes \\!",
"\\Sigma _0, \\; P_{0|-1} = J \\!\\otimes \\!",
"\\left(G\\Sigma _0 G\\right), \\; \\Pi _{0|-1} = J \\!\\otimes \\!",
"\\left(\\Sigma _0 G\\right)$ .",
"$i \\ge 0$ $\\!\\!\\!\\!$Optimal gains: Compute $\\widehat{B}^n_i$ and $K^n_i$ using Theorem REF .",
"Using (REF )-(), obtain $B^{\\mathcal {C}}_i$ , $B^{\\mathcal {I}}_i$ and $K_i$ from $\\widehat{B}^n_i$ and $K^n_i$ .",
"$\\!\\!\\!\\!$Prediction error covariance updates: Update $P_{i+1|i}, \\Sigma _{i+1|i}, \\Pi _{i+1|i}$ using ()-().",
"The offline Algorithm REF along with the online Algorithm REF completes our proposed distributed solution to obtain minimized MSE estimates of the dynamic field ${\\mathbf {x}}_i$ at each agent in the cyber network.",
"CIKF does not require sparsity of the matrices $A$ , $H_n$ and $L$ .",
"However, in most distributed applications these matrices are sparse.",
"For example, consider the temperature distribution over the entire United States as a time-varying random field.",
"It is conceivable that the temperature in a location will be directly dependent on the temperature of a few locations leading to a sparse $A$ .",
"Similarly a weather station will make localized measurements of the temperature (at its own location) rather than measuring the temperature all over the US.",
"Similarly the weather stations will exchange their data with the neighboring weather stations, thereby reducing the communication cost and leading to a sparse graph Laplacian matrix.",
"From a computational point of view, if the matrices $A$ , $H_n$ and $L$ are sparse, they can be stored as lists, for example, and computation of matrix products with them will involve less multiplications and/or additions operations.",
"Since the computation of $B_i$ and $K_i$ involve these matrices, their sparsity will reduce both the computation and storage costs."
],
[
"Numerical Evaluation",
"We numerically evaluate the MSE performance of the CIKF and compare it against the centralized Kalman filter (CKF) and the distributed information Kalman filter (DIKF) in [5].",
"To this objective, we build a time-varying random system, observation and network model that satisfies the Assumptions REF -REF .",
"The Algorithms REF -REF run on these model parameters.",
"First the Algorithm REF computes and save the gain matrices and the error covariance matrices.",
"The traces of the error covariance matrices provide the theoretical MSE trajectory of the CIKF with time.",
"Then, we Monte-Carlo simulate Algorithm REF to compute the numerical MSE of the distributed estimators, CIKF and DIKF, and the centralized estimator CKF."
],
[
"Model specifications",
"Here, we consider a time-varying field, ${\\mathbf {x}}_i$ , with dimension $M=50$ .",
"The physical layer, consisting of $M=50$ sites, is monitored by a cyber layer consisting of $N=50$ agents.",
"Each agent in the cyber layer observes $M_n=2$ sites of the physical layer.",
"We build the field dynamics matrix $A$ to be sparse and distributed.",
"The dynamics $A$ possess the structure of a Lattice graph, where the time evolution of a field variable depends on the neighboring field variables.",
"The structure of the Lattice graph is randomly generated using R library.",
"The degree is considered to be 4 and the non-zero entries are randomly chosen from $(0,1)$ .",
"For illustration, we consider an unstable field dynamics with $\\Vert A\\Vert _2=1.05$ to test the resilience of the algorithms under unstable conditions.",
"We scale the randomly generated $A$ such that its spectral norm becomes 1.05.",
"The observation matrices, $H_n \\in \\mathbb {R}^{2\\times 50}, n=1, \\cdots , 50$ , are sparse $0-1$ matrices with one non-zero element at each row corresponding to the site of ${\\mathbf {x}}_i$ observed by the $n^{\\text{th}}$ agent.",
"The local observations ${\\mathbf {z}}^n_i$ are $2\\times 1$ random vectors.",
"The mean $\\overline{x}_0$ of the initial state vector is generated at random.",
"The system noise covariance $V$ , the observation noise covariances $R_n$ , and the initial state covariance $\\Sigma _0$ are randomly generated symmetric positive definite matrices.",
"We then scale the covariance matrices such that their covariance becomes: $\\Vert V\\Vert _2=4, \\Vert R_n\\Vert _2=8$ , and $\\Vert \\Sigma _0\\Vert _2=16$ .",
"The agents in the cyber layer communicate among themselves following a randomly generated Erdős-Rényi graph $\\mathcal {G}$ with 50 nodes and $E=138$ edges.",
"The average degree of each node/agent is approximately $5.5$ .",
"The communication network $\\mathcal {G}$ is also sparse.",
"For Monte-Carlo simulations, we generate the noises, ${\\mathbf {v}}_i, {\\mathbf {r}}^n_i$ , and the initial condition, ${\\mathbf {x}}_0$ as Gaussian sequences, with ${\\mathbf {v}}_i \\sim \\mathcal {N}(\\bar{0},V), \\;{\\mathbf {r}}^n_i \\sim \\mathcal {N}(\\bar{0},R_n), \\;{\\mathbf {x}}_0 \\sim \\mathcal {N}(\\bar{{\\mathbf {x}}}_0,\\Sigma _0)$ .",
"The sequences $\\lbrace \\lbrace {\\mathbf {v}}_i\\rbrace _i, \\lbrace {\\mathbf {r}}_i\\rbrace _i, {\\mathbf {x}}_0\\rbrace _{i \\ge 0}$ are generated to be uncorrelated.",
"Each agent $n$ in the cyber layer has access to the system parameters $A, V, H, R, \\bar{{\\mathbf {x}}}_0, \\Sigma _0$ , and $\\mathcal {G}$ .",
"This numerical model satisfies Assumptions REF -REF .",
"The pair $(A,H)$ is detectable and the pairs, $(A,H_n) \\; \\forall n$ , are not detectable.",
"The agent communication graph $\\mathcal {G}$ is connected with the algebraic connectivity of the Laplacian $\\lambda _2(L) = 0.7 > 0$ .",
"Hence the Assumptions REF -REF hold true for this numerical system, observation, and network model.",
"Figure: Comparison of MSE performance of the proposed CIKF with CKF and DIKF."
],
[
"Optimized gains and theoretical MSE",
"We run Algorithm REF on the numerical model to obtain the gain matrices and the theoretical error covariances of the CIKF.",
"We compute the gain matrices and the error covariances of the centralized Kalman filter (CKF) and of the distributed information Kalman filter (DIKF).",
"In Fig.",
"REF , we plot the MSE (in dB), trace of the predictor error covariance matrices $\\Sigma _{i+1|i}$ , for each of these cases up to time $i=30$ .",
"The MSE of the optimal CKF is the smallest (recall the CKF, if feasible, would be optimal) and the objective of the distributed estimators is to achieve MSE performance as close as possible to that of the CKF.",
"From Fig.",
"REF , we see that the MSE of the proposed CIKF is 3dB more than the CKF but is 3dB less than the DIKF.",
"From the plot we see that the CIKF converges faster than the DIKF.",
"Hence, the proposed CIKF provides faster convergence and 3dB MSE performance improvement over the DIKF.",
"The performance of the CKF is 3dB better than the CIKF's due to the fact that the CKF has access to the observations of all the sensors at every time steps.",
"In contrast in CIKF, each agent has access to its own observations and the current estimates of its neighbors only; the impact of the observations from the other agents propagate through the network with delay.",
"As the time-varying field ${\\mathbf {x}}_i$ is evolving with input noise ${\\mathbf {v}}_i$ , lack of access to all the observations containing the driving input ${\\mathbf {v}}_i$ , combined with the network diffusion delay, causes a performance gap between the CKF and the CIKF."
],
[
"Monte-Carlo Simulations",
"We empirically compute the MSE of the distributed estimates given by the CIKF Algorithm REF .",
"We implement the algorithms using Matlab in the Microsoft Azure cloud.",
"Given the computation load because of the large system (M = 50) and network (N = 50) models, we run our simulation on Azure DS13 (8 cores, 56 GB memory) virtual machine (VM).",
"The MSE computation for the CIKF, CKF, and DIKF algorithms with 1000 Monte-Carlo runs require approximately 30 hours in the Azure DS13 VM.",
"Once we obtain the field prediction estimates for the three algorithms, we compute the empirical prediction error covariance matrices $\\widehat{\\Sigma }_{i+1|i}$ and then obtain the Monte-Carlo MSE (in dB) from their trace.",
"From the Monte-Carlo MSE plot in Fig REF , we see that the the empirical plots follow closely the theoretical plots in Fig REF .",
"Both the theoretical Fig REF and Monte-Carlo simulated Fig REF MSE performance confirms our CIKF analysis in Sections -.",
"The novel Consensus$+$ Innovations Kalman Filter (CIKF) proposed in Section along with the optimized gain designs in Section provides unbiased distributed estimates with bounded and minimized MSE for the Consensus$+$ Innovations distributed solution.",
"The CIKF achieves nearly 3dB better performance than the DIKF [5]."
],
[
"Conclusions",
"Summary: In this paper, we propose a Consensus$+$ Innovations Kalman Filter (CIKF) that obtains unbiased minimized MSE distributed estimates of the pseudo-states and real-time employs them to obtain the unbiased distributed filtering and prediction estimates of the time-varying random state at each agent.",
"The filter update iterations are of the Consensus$+$ Innovations type.",
"Using the Gauss-Markov principle, we designed the optimal gain matrices that yield approximately 3dB improvement over previous available distributed estimators like the DIKF in [5].",
"Contributions: The primary contributions of this paper are: (a) design of a filter and corresponding gain matrices to obtain minimized MSE distributed estimates at each agent under minimal assumptions; and (b) a theoretical characterization of the tracking capacity and distributed version of the algebraic Riccati equation.",
"Compared to DIKF, the contributions of this paper are: $(a)$ The CIKF is a new algorithm for distributed estimation of time-varying random fields which provides 3dB MSE performance over the DIKF and much closer to the performance of the optimal centralized Kalman filter.",
"$(b)$ The CIKF does not require the strict distributed observability assumption, i.e., invertibility of the matrix $G$ .",
"The DIKF require that $G^{-1}$ exists, which demands that all state variables be observed by at least once agent in the network.",
"$(c)$ In CIKF, we design the optimized gain matrices for the dynamic averaging of pseudo-states using the Gauss-Markov Theorem.",
"Hence, the distributed dynamic averaging step of the CIKF has lower MSE than that of the DIKF.",
"$(d)$ These advantages are obtained by using pseudo-state and this paper demonstrates its usefulness in distributed field estimation."
],
[
"Appendix",
"The Appendices prove the proposition, lemmas, and theorems stated in the paper.",
"Proof of Proposition"
],
[
"Proof of Proposition ",
"First we derive the dynamics (REF ) of the pseudo-state ${\\mathbf {y}}_i$ .",
"Using (REF ) and (REF ), ${\\mathbf {y}}_{i+1} &= G{\\mathbf {x}}_{i+1} \\\\&= G \\left( A{\\mathbf {x}}_i + {\\mathbf {v}}_i\\right) \\\\&= GA \\left( G^{\\dagger }G + \\widetilde{I}\\right) {\\mathbf {x}}_i + G{\\mathbf {v}}_i, \\quad \\begin{small} \\left[ \\text{by (\\ref {eqn:p_identity_mat})}, \\; I = G^{\\dagger }G + \\widetilde{I} \\right] \\end{small} \\\\&= GAG^{\\dagger }{\\mathbf {y}}_i + G{\\mathbf {v}}_i + GA\\widetilde{I}{\\mathbf {x}}_i \\\\&= \\widetilde{A}{\\mathbf {y}}_i + G{\\mathbf {v}}_i + \\check{A}{\\mathbf {x}}_i, \\qquad \\quad \\left[ \\text{by (\\ref {eqn:pstate})} \\right].$ Now, we derive the observations () of the pseudo-observations $\\widetilde{{\\mathbf {z}}}^n_i$ .",
"Using (REF ) and (REF ), $\\widetilde{{\\mathbf {z}}}^n_i &= H^T_n R_n^{-1} H_n{\\mathbf {x}}_i + H^T_n R_n^{-1} {\\mathbf {r}}^n_i \\\\&= H^T_nR_n^{-1}H_n \\left( G^{\\dagger }G + \\widetilde{I}\\right) {\\mathbf {x}}_i + H^T_n R_n^{-1} {\\mathbf {r}}^n_i \\\\&= H^T_nR_n^{-1}H_n G^{\\dagger }G {\\mathbf {x}}_i + H^T_n R_n^{-1} {\\mathbf {r}}^n_i + H^T_nR_n^{-1}H_n \\widetilde{I} {\\mathbf {x}}_i \\\\&= \\widetilde{H} {\\mathbf {y}}_i + H^T_n R_n^{-1} {\\mathbf {r}}^n_i + \\check{H}_n {\\mathbf {x}}_i.",
"\\qquad \\qquad \\qquad \\Box $ Proof of Lemmas"
],
[
"Proof of Lemma ",
"Consider the filtering error definitions (REF )-().",
"We take expectations on both sides, $\\mathbb {E}\\left[{\\mathbf {e}}_{i|i}^n\\right] &= \\mathbb {E}\\left[{\\mathbf {y}}_i - \\widehat{{\\mathbf {y}}}_{i|i}^n \\right] \\\\&= \\mathbb {E} \\left[ \\mathbb {E} \\left[ {\\mathbf {y}}_i - \\widehat{{\\mathbf {y}}}_{i|i}^n~|~\\widetilde{{\\mathbf {z}}}^n_i, \\lbrace \\widehat{{\\mathbf {y}}}_{i|i-1}^l \\rbrace _{l \\in \\Omega _n} \\right] \\right] \\\\&= \\mathbb {E} \\left[ \\widehat{{\\mathbf {y}}}_{i|i}^n - \\widehat{{\\mathbf {y}}}_{i|i}^n \\right] = 0 \\qquad \\left[ \\text{by} (\\ref {eqn:MMSE_yf}) \\right] \\\\\\mathbb {E}\\left[{\\mathbf {\\epsilon }}_{i|i}^n\\right] &= \\mathbb {E}\\left[{\\mathbf {x}}_i - \\widehat{{\\mathbf {x}}}_{i|i}^n \\right] \\\\&= \\mathbb {E} \\left[ \\mathbb {E} \\left[ {\\mathbf {x}}_i - \\widehat{{\\mathbf {x}}}_{i|i}^n~|~\\widehat{{\\mathbf {y}}}_{i|i}^n \\right] \\right] \\\\&= \\mathbb {E} \\left[ \\widehat{{\\mathbf {x}}}_{i|i}^n - \\widehat{{\\mathbf {x}}}_{i|i}^n \\right] = 0 \\qquad \\left[ \\text{by} (\\ref {eqn:epsfn}) \\right].$ Similarly, taking expectations on prediction errors (REF )-(), $\\mathbb {E}\\left[{\\mathbf {e}}_{i+1|i}^n\\right] &= \\mathbb {E}\\left[{\\mathbf {y}}_{i+1} - \\widehat{{\\mathbf {y}}}_{i+1|i}^n \\right] \\\\&= \\mathbb {E} \\left[ \\mathbb {E} \\left[ {\\mathbf {y}}_{i+1} - \\widehat{{\\mathbf {y}}}_{i+1|i}^n~|~\\widetilde{{\\mathbf {z}}}^n_i, \\lbrace \\widehat{{\\mathbf {y}}}_{i|i-1}^l \\rbrace _{l \\in \\Omega _n} \\right] \\right] \\\\&= \\mathbb {E} \\left[ \\widehat{{\\mathbf {y}}}_{i+1|i}^n - \\widehat{{\\mathbf {y}}}_{i+1|i}^n \\right] = 0 \\qquad \\left[ \\text{by} (\\ref {eqn:MMSE_yp}) \\right] \\\\\\mathbb {E}\\left[{\\mathbf {\\epsilon }}_{i+1|i}^n\\right] &= \\mathbb {E}\\left[{\\mathbf {x}}_{i+1} - \\widehat{{\\mathbf {x}}}_{i+1|i}^n \\right] \\\\&= \\mathbb {E} \\left[ \\mathbb {E} \\left[ {\\mathbf {x}}_{i+1} - \\widehat{{\\mathbf {x}}}_{i+1|i}^n~|~\\widehat{{\\mathbf {y}}}_{i|i}^n \\right] \\right] \\\\&= \\mathbb {E} \\left[ \\widehat{{\\mathbf {x}}}_{i+1|i}^n - \\widehat{{\\mathbf {x}}}_{i+1|i}^n \\right] = 0 \\qquad \\left[ \\text{by} (\\ref {eqn:epspn}) \\right].",
"\\qquad \\Box $"
],
[
"Proof of Lemma ",
"We write the pseudo-state filtering update (REF ) in vector form, $\\widehat{{\\mathbf {y}}}_{i|i}\\!=\\!",
"\\widehat{{\\mathbf {y}}}_{i|i-1}& \\!-\\!",
"B^{\\mathcal {C}}_i \\widehat{{\\mathbf {y}}}_{i|i-1}\\!+\\!",
"B^{\\mathcal {I}}_i \\!",
"\\left( \\!\\widetilde{{\\mathbf {z}}}_i \\!-\\!",
"\\left(\\widetilde{D}_H \\widehat{{\\mathbf {y}}}_{i|i-1}\\!+\\!",
"\\check{D}_H \\widehat{{\\mathbf {x}}}_{i|i-1}\\right) \\right) ,$ where, $\\left[B^{\\mathcal {C}}_i\\right]_{nl} = -B^{n,l}_i, \\; \\forall n \\ne l$ , $\\left[B^{\\mathcal {C}}_i\\right]_{nn} = \\sum _{l \\in \\Omega _n}B^{n,l}_i$ , $\\left[B^{\\mathcal {I}}_i\\right]_{nn} = B^{n,n}_i$ , $\\left[B^{\\mathcal {I}}_i\\right]_{nl} = 0, \\; \\forall n \\ne l$ .",
"The block-diagonal matrices: $\\widetilde{D}_H = \\text{blockdiag}\\lbrace \\widetilde{H}_1, \\cdots , \\widetilde{H}_N\\rbrace , \\; \\check{D}_H = \\text{blockdiag}\\lbrace \\check{H}_1, \\cdots , \\check{H}_N\\rbrace $ .",
"Note that $B^{\\mathcal {C}}_i \\left(1_N\\otimes {\\mathbf {y}}_i\\right) = \\mathbf {0}$ .",
"Using this relation and the vector form of $\\widehat{{\\mathbf {y}}}_{i|i}$ , we expand the pseudo-state filter error process ${\\mathbf {e}}_{i|i}$ , ${\\mathbf {e}}_{i|i}&= 1_N \\!\\otimes \\!",
"{\\mathbf {y}}_i - \\widehat{{\\mathbf {y}}}_{i|i}\\\\&= \\left( 1_N \\!\\otimes \\!",
"{\\mathbf {y}}_i - \\widehat{{\\mathbf {y}}}_{i|i-1}\\right) - B^{\\mathcal {C}}_i \\left( 1_N \\!\\otimes \\!",
"{\\mathbf {y}}_i - \\widehat{{\\mathbf {y}}}_{i|i-1}\\right) \\!-\\!",
"B^{\\mathcal {I}}_i D_{H}^TR^{-1}{\\mathbf {r}}_i \\\\& \\qquad - B^{\\mathcal {I}}_i \\widetilde{D}_H \\left(1_N \\!\\otimes \\!",
"{\\mathbf {y}}_i - \\widehat{{\\mathbf {y}}}_{i|i-1}\\right) - B^{\\mathcal {I}}_i\\check{D}_H \\left( 1_N \\!\\otimes \\!",
"{\\mathbf {x}}_i - \\widehat{{\\mathbf {x}}}_{i|i-1}\\right) \\\\&= \\left( I_{\\!M\\!N} \\!-\\!",
"B^{\\mathcal {C}}_i \\!-\\!",
"B^{\\mathcal {I}}_i \\!\\widetilde{D}_{H}\\right) {\\mathbf {e}}_{i|i-1}\\!-\\!",
"B^{\\mathcal {I}}_i \\check{D}_{H} {\\mathbf {\\epsilon }}_{i|i-1}\\!-\\!",
"B^{\\mathcal {I}}_i\\!",
"D_{H}^T R^{-1} r_i.$ The state filtering update (), in vector form, is $\\widehat{{\\mathbf {x}}}_{i|i}= \\widehat{{\\mathbf {x}}}_{i|i-1}+ K_i \\left( \\widehat{{\\mathbf {y}}}_{i|i}\\!-\\!",
"\\left(I_N\\!\\otimes \\!G\\right) \\widehat{{\\mathbf {x}}}_{i|i-1}\\right) ,$ where, $K_i = \\text{blockdiag} \\lbrace K^1_i, \\cdots , K^N_i \\rbrace $ .",
"Using the relation $\\widehat{{\\mathbf {y}}}_{i|i}= \\left(1_N\\otimes {\\mathbf {y}}_i\\right) - {\\mathbf {e}}_{i|i}= \\left(I_N\\!\\otimes \\!G\\right)\\left(1_N\\otimes {\\mathbf {x}}_i\\right) - {\\mathbf {e}}_{i|i}$ , we expand the state filter error process ${\\mathbf {e}}_{i|i}$ , ${\\mathbf {\\epsilon }}_{i|i}&= 1_N \\!\\otimes \\!",
"{\\mathbf {x}}_i - \\widehat{{\\mathbf {x}}}_{i|i}\\\\&= \\left( 1_{\\!N} \\!\\otimes \\!",
"{\\mathbf {x}}_i - \\widehat{{\\mathbf {x}}}_{i|i-1}\\right) \\!-\\!",
"K_i \\left(I_{\\!N}\\!\\otimes \\!G\\right) \\left(1_{\\!N} \\!\\otimes \\!",
"{\\mathbf {x}}_i \\!-\\!",
"\\widehat{{\\mathbf {x}}}_{i|i-1}\\right) \\!+\\!",
"K_i {\\mathbf {e}}_{i|i}\\\\&= \\left( I_{MN} - K^{\\mathcal {C}}_i - K_i \\left(I_N\\!\\otimes \\!G\\right) \\right) {\\mathbf {\\epsilon }}_{i|i-1}+ K_i {\\mathbf {e}}_{i|i}.$ The dynamics of the pseudo-state and state prediction errors, ${\\mathbf {e}}_{i+1|i}&= 1_N \\!\\otimes \\!",
"{\\mathbf {y}}_{i+1} - \\widehat{{\\mathbf {y}}}_{i+1|i}\\\\&= \\left(I_N\\!\\otimes \\!\\widetilde{A}\\right)\\left(1_N \\!\\otimes \\!",
"{\\mathbf {y}}_i\\right) + \\left(I_N\\!\\otimes \\!",
"\\check{A} \\right)\\left(1_N \\!\\otimes \\!",
"{\\mathbf {x}}_i\\right) \\\\&\\quad + 1_N\\!\\otimes \\!",
"(G{\\mathbf {v}}_i) - \\left(I_N\\!\\otimes \\!\\widetilde{A}\\right)\\widehat{{\\mathbf {y}}}_{i|i}-\\left(I_N\\!\\otimes \\!",
"\\check{A} \\right)\\widehat{{\\mathbf {x}}}_{i|i}\\\\&= \\left(I_N\\!\\otimes \\!\\widetilde{A}\\right){\\mathbf {e}}_{i|i}+ \\left(I_N\\!\\otimes \\!",
"\\check{A} \\right) {\\mathbf {\\epsilon }}_{i|i}+ 1_N\\!\\otimes \\!",
"(G{\\mathbf {v}}_i) \\\\{\\mathbf {\\epsilon }}_{i+1|i}&= 1_N \\!\\otimes \\!",
"{\\mathbf {x}}_{i+1} - \\widehat{{\\mathbf {x}}}_{i+1|i}\\\\&= \\left(I_N\\!\\otimes \\!A\\right)\\left(1_N \\!\\otimes \\!",
"{\\mathbf {x}}_i\\right) + 1_N\\!\\otimes \\!",
"{\\mathbf {v}}_i - \\left(I_N\\!\\otimes \\!A\\right)\\widehat{{\\mathbf {x}}}_{i|i}\\\\&= \\left(I_N\\!\\otimes \\!A\\right){\\mathbf {\\epsilon }}_{i|i}+ 1_N\\!\\otimes \\!",
"{\\mathbf {v}}_i.$ Since the state ${\\mathbf {x}}_i$ , pseudo-state ${\\mathbf {y}}_i$ , their initial condition and all the noises are Gaussian, their estimates are also Gaussian making all the filtering and prediction errors Gaussian.",
"$\\Box $"
],
[
"Proof of Lemma ",
"Lemma REF and Assumption REF guarantee that ${\\mathbf {\\epsilon }}_{i|i}, {\\mathbf {\\epsilon }}_{i+1|i}, {\\mathbf {e}}_{i|i}, {\\mathbf {v}}_i, {\\mathbf {r}}_i$ are Gaussian.",
"The error noises $\\widetilde{\\mathbf {\\phi }}_i$ and ${\\mathbf {\\phi }}_i$ are therefore Gaussian as they are linear combinations of the error processes and the model noises ${\\mathbf {\\epsilon }}_{i|i}, {\\mathbf {\\epsilon }}_{i+1|i}, {\\mathbf {e}}_{i|i}, {\\mathbf {v}}_i, {\\mathbf {r}}_i$ .",
"By Lemma REF , we have $\\mathbb {E}\\left[{\\mathbf {\\epsilon }}_{i|i}\\right] = \\mathbb {E}\\left[{\\mathbf {\\epsilon }}_{i+1|i}\\right] = \\mathbb {E}\\left[{\\mathbf {e}}_{i|i}\\right] = \\mathbf {0}$ .",
"From Assumption REF , we know $\\mathbb {E}\\left[{\\mathbf {v}}_i\\right] = \\mathbb {E}\\left[{\\mathbf {r}}_i\\right] = \\mathbf {0}$ .",
"We take expectation on both sides of (REF )-() and apply these relations $\\mathbb {E}\\left[\\widetilde{\\mathbf {\\phi }}_i\\right] &= \\!\\left( I_{\\!N} \\!\\otimes \\!",
"\\check{A}\\right)\\!",
"\\mathbb {E}\\left[{\\mathbf {\\epsilon }}_{i|i}\\right] - \\left(I_{\\!N} \\!\\otimes \\!",
"\\widetilde{A}\\right)\\!",
"B^{\\mathcal {I}}_i D_{H}^T R^{-1} \\mathbb {E}\\left[r_i\\right] \\nonumber \\\\& \\quad + 1_{\\!N} \\!\\otimes \\!",
"\\left(G \\mathbb {E}\\left[{\\mathbf {v}}_i\\right]\\right) - \\left(I_{\\!N} \\!\\otimes \\!",
"\\widetilde{A}\\right)\\!B^{\\mathcal {I}}_i \\check{D}_{H} \\mathbb {E}\\left[{\\mathbf {\\epsilon }}_{i|i-1}\\right] = \\mathbf {0}, \\\\\\mathbb {E}\\left[{\\mathbf {\\phi }}_i\\right] &= \\left( I_{\\!N} \\!\\otimes \\!",
"A\\right)\\!",
"K_i \\mathbb {E}\\left[{\\mathbf {e}}_{i|i}\\right] + 1_N \\otimes \\mathbb {E}\\left[{\\mathbf {v}}_i\\right] = \\mathbf {0}.$ Combining (), () and (REF ), we have $\\widetilde{\\mathbf {\\phi }}_i &= F_1 {\\mathbf {\\epsilon }}_{i|i-1}+ F_2 {\\mathbf {e}}_{i|i-1}- F_3 D_{H}^T R^{-1} r_i + 1_{\\!N} \\!\\otimes \\!",
"\\left(G {\\mathbf {v}}_i\\right)$ where, $F_1 &= \\left( I_{\\!N} \\!\\otimes \\!",
"\\check{A}\\!\\right)\\!",
"\\left( I_{\\!M\\!N} \\!-\\!",
"K_i \\!\\left(I_{\\!N}\\!\\otimes \\!G\\right)\\!",
"-\\!",
"K_i \\!B^{\\mathcal {I}}_i \\check{D}_{H}\\!\\!",
"\\right) \\!-\\!",
"\\left( I_{\\!N} \\!\\otimes \\!",
"\\widetilde{A}\\!\\right)\\!",
"\\!B^{\\mathcal {I}}_i \\check{D}_{H} \\\\F_2 &= \\left( I_{\\!N} \\!\\otimes \\!",
"\\check{A}\\!\\right)\\!",
"K_i \\left( I_{\\!M\\!N} \\!-\\!",
"B^{\\mathcal {C}}_i \\!-\\!",
"B^{\\mathcal {I}}_i \\!",
"\\widetilde{D}_{H} \\right) \\\\F_3 &= \\left( I_{\\!N} \\!\\otimes \\!",
"\\widetilde{A}\\!\\right)\\!",
"\\!B^{\\mathcal {I}}_i.$ Since $\\widetilde{\\mathbf {\\phi }}_i$ and ${\\mathbf {\\phi }}_i$ are zero-mean, the noise covariances are, $\\widetilde{\\Phi }_i &\\!=\\!",
"\\mathbb {E} \\!",
"\\left[\\!\\widetilde{\\mathbf {\\phi }}_i \\widetilde{\\mathbf {\\phi }}^T_i \\!",
"\\right] \\!=\\!",
"F_1 \\Sigma _{i|i-1} F_1^T \\!+\\!",
"F_2 P_{i|i-1} F_2^T \\!+\\!",
"F_3 \\overline{D}_H F^T_3 \\nonumber \\\\& \\qquad \\qquad + J \\!\\otimes \\!",
"\\left(GVG\\right) + F_1 \\Pi _{i|i-1} F_2^T + F_2 \\Pi _{i|i-1}^T F_1^T, \\\\\\Phi _i &\\!=\\!",
"\\mathbb {E}\\!",
"\\left[ \\!",
"{\\mathbf {\\phi }}_i {\\mathbf {\\phi }}^T_i \\!",
"\\right] \\!=\\!",
"\\left( I_{\\!N} \\!\\otimes \\!",
"A\\right)\\!",
"K_i \\Sigma _{i|i} \\!",
"K^T_i \\!",
"\\left( I_{\\!N} \\!\\otimes \\!",
"A^T\\right) \\!+\\!",
"J \\!\\otimes \\!",
"V,$ where, $F_1, F_2$ and $F_3$ are defined in (REF )-().",
"$\\Box $"
],
[
"Proof of Lemma ",
"We first compute the conditional means $\\overline{\\widetilde{\\mathbf {\\theta }}}^n_i$ and $\\overline{\\mathbf {\\theta }}^n_i$ of the new information $\\widetilde{\\mathbf {\\theta }}^n_i$ and ${\\mathbf {\\theta }}^n_i$ from (REF ).",
"The means $\\overline{\\widetilde{\\mathbf {\\theta }}}^n_i$ and $\\overline{\\mathbf {\\theta }}^n_i$ depend on the conditional means of $\\widehat{{\\mathbf {y}}}_{i|i-1}^{l}, \\widetilde{\\mathbf {z}}^n_{i-1}$ and $\\widehat{{\\mathbf {y}}}_{i|i}^n$ .",
"$&\\mathbb {E} \\begin{bmatrix} \\widehat{{\\mathbf {y}}}_{i|i-1}^{l} | \\widetilde{\\mathbf {z}}^n_{i-1}, \\lbrace \\widehat{\\mathbf {y}}^l_{i-1|i-2} \\rbrace _{l \\in \\Omega _n} \\end{bmatrix} \\\\&= \\mathbb {E} \\left[ \\mathbb {E} \\left[ {\\mathbf {y}}_i | \\widetilde{\\mathbf {z}}^l_{i-1}, \\lbrace \\widehat{\\mathbf {y}}^k_{i-1|i-2} \\rbrace _{k \\in \\Omega _l} \\right] \\;|\\; \\widetilde{\\mathbf {z}}^n_{i-1}, \\lbrace \\widehat{\\mathbf {y}}^l_{i-1|i-2} \\rbrace _{l \\in \\Omega _n} \\right] \\\\&= \\mathbb {E} \\left[ \\mathbb {E} \\left[ {\\mathbf {y}}_i | \\widetilde{\\mathbf {z}}^n_{i-1}, \\lbrace \\widehat{\\mathbf {y}}^l_{i-1|i-2} \\rbrace _{l \\in \\Omega _n} \\right] \\;|\\; \\widetilde{\\mathbf {z}}^l_{i-1}, \\lbrace \\widehat{\\mathbf {y}}^k_{i-1|i-2} \\rbrace _{k \\in \\Omega _l} \\right] \\\\&= \\mathbb {E} \\left[ \\widehat{{\\mathbf {y}}}_{i|i-1}^{n} \\;|\\; \\widetilde{\\mathbf {z}}^l_{i-1}, \\lbrace \\widehat{\\mathbf {y}}^k_{i-1|i-2} \\rbrace _{k \\in \\Omega _l} \\right] = \\widehat{{\\mathbf {y}}}_{i|i-1}^{n}, \\quad \\forall \\; l \\in \\Omega _n.",
"\\\\&\\mathbb {E} \\begin{bmatrix} \\widetilde{\\mathbf {z}}^n_i \\;|\\; \\widetilde{\\mathbf {z}}^n_{i-1}, \\lbrace \\widehat{\\mathbf {y}}^l_{i-1|i-2} \\rbrace _{l \\in \\Omega _n} \\end{bmatrix} \\\\&= \\mathbb {E} \\begin{bmatrix} \\widetilde{H}_n{\\mathbf {y}}_i + H^T_n R_n^{-1} {\\mathbf {r}}^n_i + \\check{H}_n {\\mathbf {x}}_i \\;|\\; \\widetilde{\\mathbf {z}}^n_{i-1}, \\lbrace \\widehat{\\mathbf {y}}^l_{i-1|i-2} \\rbrace _{l \\in \\Omega _n} \\end{bmatrix} \\\\&= \\widetilde{H}_n \\mathbb {E} \\begin{bmatrix} {\\mathbf {y}}_i \\;|\\; \\widetilde{\\mathbf {z}}^n_{i-1}, \\lbrace \\widehat{\\mathbf {y}}^l_{i-1|i-2} \\rbrace _{l \\in \\Omega _n} \\end{bmatrix} \\\\& \\qquad \\qquad \\qquad + \\check{H}_n \\mathbb {E} \\begin{bmatrix} {\\mathbf {x}}_i \\;|\\; \\widetilde{\\mathbf {z}}^n_{i-1}, \\lbrace \\widehat{\\mathbf {y}}^l_{i-1|i-2} \\rbrace _{l \\in \\Omega _n} \\end{bmatrix} \\\\&= \\widetilde{H}_n \\widehat{{\\mathbf {y}}}_{i|i-1}^n \\!+\\!",
"\\check{H}_n \\mathbb {E} \\left[ \\mathbb {E} \\left[ {\\mathbf {x}}_i | \\widehat{{\\mathbf {y}}}_{i|i}^n, \\widehat{\\mathbf {x}}^l_{i-1|i-2}, \\widetilde{\\mathbf {z}}^n_{i-1}, \\widehat{\\mathbf {y}}^l_{i-1|i-2}, \\in \\Omega _n \\right] \\right] \\\\&= \\widetilde{H}_n \\widehat{{\\mathbf {y}}}_{i|i-1}^n + \\check{H}_n \\mathbb {E} \\left[ \\widehat{{\\mathbf {x}}}_{i|i-1}^n \\;|\\; \\widetilde{\\mathbf {z}}^n_{i-1}, \\lbrace \\widehat{\\mathbf {y}}^l_{i-1|i-2} \\rbrace _{l \\in \\Omega _n}\\right] \\\\&= \\widetilde{H}_n \\widehat{{\\mathbf {y}}}_{i|i-1}^n + \\check{H}_n \\widehat{{\\mathbf {x}}}_{i|i-1}^n.",
"\\\\&\\mathbb {E} \\begin{bmatrix} \\widehat{{\\mathbf {y}}}_{i|i}^n | \\widehat{{\\mathbf {y}}}_{i-1|i-1}^n \\end{bmatrix} = \\mathbb {E} \\left[ {\\mathbf {y}}_i - {\\mathbf {e}}_{i|i}^n \\;|\\; \\widehat{{\\mathbf {y}}}_{i-1|i-1}^n \\right] \\\\&= \\mathbb {E} \\left[ {\\mathbf {y}}_i \\;|\\; \\widehat{{\\mathbf {y}}}_{i-1|i-1}^n \\right] - \\mathbb {E} \\left[{\\mathbf {e}}_{i|i}^n \\;|\\; \\widehat{{\\mathbf {y}}}_{i-1|i-1}^n \\right] \\\\&= G \\mathbb {E} \\left[ {\\mathbf {x}}_i \\;|\\; \\widehat{{\\mathbf {y}}}_{i-1|i-1}^n \\right] = G\\widehat{{\\mathbf {x}}}_{i|i-1}^n.$ The conditional means of $\\widehat{{\\mathbf {y}}}_{i|i-1}^{l}, \\widetilde{\\mathbf {z}}^n_{i-1}$ and $\\widehat{{\\mathbf {y}}}_{i|i}^n$ shows that the new uncorrelated information defined in (REF )-() expands to the vectors $\\widetilde{\\mathbf {\\nu }}^n_i$ and ${\\mathbf {\\nu }}^n_i$ in (REF ).",
"Note that $\\widetilde{\\mathbf {\\nu }}^n_i$ and ${\\mathbf {\\nu }}^n_i$ are zero mean, by definition, and, are Gaussian since they are linear combination of Gaussian vectors.",
"Now to prove $\\widetilde{\\mathbf {\\nu }}^n_i$ and ${\\mathbf {\\nu }}^n_i$ are sequence of uncorrelated vectors, we have to show: $\\mathbb {E}\\left[ \\widetilde{\\mathbf {\\nu }}^n_i \\widetilde{\\mathbf {\\nu }}^{n^T}_j\\right] = \\mathbb {E}\\left[ {\\mathbf {\\nu }}^n_i {\\mathbf {\\nu }}^{n^T}_j\\right] = {\\mathbf {0}}, \\; \\forall \\; i\\ne j, \\; \\forall \\; n.$ First, we write $\\widetilde{\\mathbf {\\nu }}^n_i$ and ${\\mathbf {\\nu }}^n_i$ in terms of the filtering and prediction error processes using (), (REF )-(), $\\widetilde{\\mathbf {\\nu }}^n_i \\!=\\!",
"\\begin{bmatrix}{\\mathbf {e}}_{i|i-1}^n \\!-\\!",
"{\\mathbf {e}}_{i|i-1}^{l_1} \\\\ \\vdots \\\\ {\\mathbf {e}}_{i|i-1}^n \\!-\\!",
"{\\mathbf {e}}_{i|i-1}^{l_{d_n}} \\\\ \\widetilde{H}_{\\!n} {\\mathbf {e}}_{i|i-1}^n \\!+\\!",
"\\check{H}_{\\!n} {\\mathbf {\\epsilon }}_{i|i-1}^n + H^T_n R^{-1}_n {\\mathbf {r}}^n_i\\end{bmatrix}\\!\\!\\!, \\;\\;{\\mathbf {\\nu }}^n_i \\!=\\!",
"G{\\mathbf {\\epsilon }}_{i|i-1}^n \\!-\\!",
"{\\mathbf {e}}_{i|i}^n.$ Without loss of generality, we consider $i > j$ .",
"The rest of the proof is similar to the proof of Lemma 5 in [5].",
"Here, the only difference is that we should condition on $\\lbrace \\widetilde{\\mathbf {z}}^l_{i-1}\\rbrace _{l \\in \\Omega _n}, \\lbrace \\widehat{\\mathbf {y}}^l_{i-1|i-1} \\rbrace _{l \\in \\Omega _n}$ .",
"$\\Box $ Proof of Theorems"
],
[
"Proof of Theorem ",
"In Lemma REF , we showed that $\\widetilde{\\mathbf {\\nu }}^n_i$ and ${\\mathbf {\\nu }}^n_i$ are independent Gaussian sequences.",
"By the Innovations Property [35], there are $1-1$ correspondence between $\\lbrace \\widetilde{{\\mathbf {z}}}^n_i, \\lbrace \\widehat{{\\mathbf {y}}}_{i|i-1}^l \\rbrace _{l \\in \\overline{\\Omega }_n} \\rbrace $ and $\\widetilde{\\mathbf {\\nu }}^n_i$ , and between $\\lbrace \\widehat{{\\mathbf {y}}}_{i|i}^n \\rbrace $ and ${\\mathbf {\\nu }}^n_i$ .",
"The Innovations Property guarantees that there exists a unique way to get one from the other.",
"$\\widehat{{\\mathbf {y}}}_{i|i}^n \\!=\\!",
"\\mathbb {E} \\!\\left[ {\\mathbf {y}}_i \\;|\\; \\widetilde{{\\mathbf {z}}}^n_i, \\lbrace \\widehat{{\\mathbf {y}}}_{i|i-1}^l \\rbrace _{l \\in \\overline{\\Omega }_n} \\right] ~&\\Longleftrightarrow ~ \\widehat{{\\mathbf {y}}}_{i|i}^n \\!=\\!",
"\\mathbb {E} \\!\\left[ {\\mathbf {y}}_i \\;|\\; \\widetilde{\\mathbf {\\nu }}^n_i \\right] \\\\\\widehat{{\\mathbf {x}}}_{i|i}^n \\!=\\!",
"\\mathbb {E} \\left[ {\\mathbf {x}}_i \\;|\\; \\widehat{{\\mathbf {y}}}_{i|i}^n \\right] ~&\\Longleftrightarrow ~ \\widehat{{\\mathbf {x}}}_{i|i}^n \\!=\\!",
"\\mathbb {E} \\left[ {\\mathbf {x}}_i \\;|\\; {\\mathbf {\\nu }}^n_i \\right]$ By the Gauss-Markov principle, $ \\widehat{{\\mathbf {y}}}_{i|i}^n &= \\widehat{{\\mathbf {y}}}_{i|i-1}^n + \\widehat{B}^n_i \\widetilde{\\mathbf {\\nu }}^n_i \\\\\\widehat{{\\mathbf {x}}}_{i|i}^n &= \\widehat{{\\mathbf {x}}}_{i|i-1}^n + K^n_i {\\mathbf {\\nu }}^n_i$ where $\\widehat{B}^n_i$ are the non-zero blocks of the $n^{\\text{th}}$ row of $B_i$ .",
"Now we expand the term $\\widehat{B}^n_i \\widetilde{\\mathbf {\\nu }}^n_i$ by multiplying the gain blocks $B^{nl}_i$ with the corresponding $(\\widehat{{\\mathbf {y}}}_{i|i-1}^l - \\widehat{{\\mathbf {y}}}_{i|i-1}^n)$ and the gain block $B^{nn}_i$ with $\\left( \\widetilde{{\\mathbf {z}}}^n_i \\!\\!-\\!",
"\\widetilde{H}_{\\!n} \\widehat{{\\mathbf {y}}}_{i|i-1}^n \\!-\\!",
"\\check{H}_{\\!n} \\widehat{{\\mathbf {x}}}_{i|i-1}^n \\right)$ .",
"This gives us the consensus$+$ innovations filtering pseudo-state update (REF ).",
"The pseudo-state and state prediction updates are $\\widehat{{\\mathbf {y}}}_{i+1|i}^n &\\!=\\!",
"\\mathbb {E} \\left[ {\\mathbf {y}}_{i+1} \\;|\\; \\widetilde{{\\mathbf {z}}}^n_i, \\lbrace \\widehat{{\\mathbf {y}}}_{i|i-1}^l \\rbrace _{l \\in \\overline{\\Omega }_n} \\right] \\\\&\\!=\\!",
"\\mathbb {E} \\left[ \\widetilde{A}{\\mathbf {y}}_i + G{\\mathbf {v}}_i + \\check{A}{\\mathbf {x}}_i \\;|\\; \\widetilde{{\\mathbf {z}}}^n_i, \\lbrace \\widehat{{\\mathbf {y}}}_{i|i-1}^l \\rbrace _{l \\in \\overline{\\Omega }_n} \\right] \\\\&\\!=\\!",
"\\widetilde{A} \\widehat{{\\mathbf {y}}}_{i|i}^n + \\check{A} \\widehat{{\\mathbf {x}}}_{i|i}^n \\\\\\widehat{{\\mathbf {x}}}_{i+1|i}^n &\\!=\\!",
"\\mathbb {E} \\left[ {\\mathbf {x}}_{i\\!+\\!1} \\;|\\; \\widehat{{\\mathbf {y}}}_{i|i}^n \\right] \\!=\\!",
"\\mathbb {E} \\left[ A {\\mathbf {x}}_i + {\\mathbf {v}}_i \\;|\\; \\widehat{{\\mathbf {y}}}_{i|i}^n \\right] = A \\widehat{{\\mathbf {x}}}_{i|i}^n.",
"\\quad \\Box $"
],
[
"Proof of Theorem ",
"By Lemma REF and Lemma REF , the error processes, ${\\mathbf {e}}_{i|i}$ , ${\\mathbf {e}}_{i+1|i}$ , ${\\mathbf {\\epsilon }}_{i|i}$ , and ${\\mathbf {\\epsilon }}_{i+1|i}$ are zero-mean Gaussian.",
"The Lyapunov-type iterations ()-() of the filter and predictor error covariances, $P_{i|i}$ , $\\Sigma _{i|i}$ , $\\Pi _{i|i}$ , $P_{i+1|i}$ , $\\Sigma _{i+1|i}$ , and $\\Pi _{i+1|i}$ , follow directly from the definitions (REF )-() and error dynamics (REF )-() by algebraic manipulations.",
"$\\Box $"
],
[
"Proof of Theorem ",
"For any square matrix, $\\rho (\\widetilde{F}) \\le \\Vert \\widetilde{F} \\Vert $ .",
"Hence if $\\Vert \\widetilde{F} \\Vert < 1$ , then it implies that $\\rho (\\widetilde{F}) < 1$ .",
"We derive the tracking capacity with the sufficient condition, $\\Vert \\widetilde{F} \\Vert < 1$ , $\\Vert \\widetilde{F} \\Vert &= \\begin{Vmatrix} \\left( I_{\\!N} \\!\\otimes \\!\\widetilde{A}\\!\\right)\\!",
"\\left( I_{\\!M\\!N} \\!-\\!",
"B^{\\mathcal {C}}_i \\!-\\!",
"B^{\\mathcal {I}}_i \\!",
"\\widetilde{D}_{H} \\right) \\end{Vmatrix} \\\\&\\le \\begin{Vmatrix} I_{\\!N} \\!\\otimes \\!\\widetilde{A}\\!",
"\\end{Vmatrix} \\begin{Vmatrix} \\left( I_{\\!M\\!N} \\!-\\!",
"B^{\\mathcal {C}}_i \\!-\\!",
"B^{\\mathcal {I}}_i \\!",
"\\widetilde{D}_{H} \\right) \\end{Vmatrix} \\\\&\\le \\begin{Vmatrix} GAG^{\\dagger }\\!",
"\\end{Vmatrix} \\begin{Vmatrix} \\left( I_{\\!M\\!N} \\!-\\!",
"B^{\\mathcal {C}}_i \\!-\\!",
"B^{\\mathcal {I}}_i \\!",
"\\widetilde{D}_{H} \\right) \\end{Vmatrix} \\\\&\\le \\frac{\\lambda _m}{\\lambda _1} \\begin{Vmatrix} A\\!",
"\\end{Vmatrix} \\begin{Vmatrix} \\left( I_{\\!M\\!N} \\!-\\!",
"B^{\\mathcal {C}}_i \\!-\\!",
"B^{\\mathcal {I}}_i \\!",
"\\widetilde{D}_{H} \\right) \\end{Vmatrix},$ where, $\\Vert G \\Vert _2 = \\lambda _m$ and $\\Vert G^{\\dagger } \\Vert _2 = \\frac{1}{\\lambda _1}$ .",
"Since $G$ is a symmetric positive semi-definite matrix, its spectral norm is its largest eigenvalue $(\\lambda _m)$ and the spectral norm of its pseudo-inverse, $G^{\\dagger }$ , is the inverse of its smallest non-zero eigenvalue $(\\lambda _1)$ .",
"If there exists $B^{\\mathcal {C}}_i$ and $B^{\\mathcal {I}}_i$ such that $\\frac{\\lambda _m}{\\lambda _1} \\begin{Vmatrix} A\\!",
"\\end{Vmatrix} \\begin{Vmatrix} \\left( I_{\\!M\\!N} \\!-\\!",
"B^{\\mathcal {C}}_i \\!-\\!",
"B^{\\mathcal {I}}_i \\!",
"\\widetilde{D}_{H} \\right) \\end{Vmatrix} < 1,$ then $\\Vert \\widetilde{F} \\Vert < 1$ and also $\\rho (\\widetilde{F}) < 1$ .",
"The bound on the spectral norm of $A$ is, $\\Vert A \\Vert &< \\frac{\\lambda _1}{\\lambda _m \\begin{Vmatrix} \\left( I_{\\!M\\!N} \\!-\\!",
"B^{\\mathcal {C}}_i \\!-\\!",
"B^{\\mathcal {I}}_i \\!",
"\\widetilde{D}_{H} \\right) \\end{Vmatrix} } \\\\&\\le \\max _{B^{\\mathcal {C}}_i, B^{\\mathcal {I}}_i} \\frac{\\lambda _1}{\\lambda _m \\begin{Vmatrix} \\left( I_{\\!M\\!N} \\!-\\!",
"B^{\\mathcal {C}}_i \\!-\\!",
"B^{\\mathcal {I}}_i \\!",
"\\widetilde{D}_{H} \\right) \\end{Vmatrix} } = C.$ Thus as long as $\\Vert A \\Vert _2 < C$ , there exists $B^{\\mathcal {C}}_i$ and $B^{\\mathcal {I}}_i$ such that $\\rho (\\widetilde{F}) < 1$ .",
"By global detectability Assumption REF , there exists $K_i$ such that $\\rho (F) < 1$ .",
"Refer to [35], for the convergence conditions of the centralized information filters.",
"Further, by Lemma REF the Gaussian noises processes $\\widetilde{\\mathbf {\\phi }}_i$ and ${\\mathbf {\\phi }}_i$ have bounded noise covariances.",
"Thus, if $\\Vert A \\Vert _2 < C$ , then from (REF )-() we conclude that the CIKF (REF )-() converges with bounded MSE.",
"$\\Box $"
],
[
"Proof of Theorem ",
"By the Innovations Property and the Gauss-Markov principle [35], the optimal gains $\\widehat{B}^n_i$ and $K^n_i$ in (REF )-() are: $\\widehat{B}^n_i &= \\Sigma _{{\\mathbf {y}}_i \\widetilde{\\mathbf {\\nu }}^n_i} \\left(\\Sigma _{\\widetilde{\\mathbf {\\nu }}^n_i}\\right)^{-1}, \\\\K^n_i &= \\Sigma _{{\\mathbf {x}}_i {\\mathbf {\\nu }}^n_i} \\left(\\Sigma _{{\\mathbf {\\nu }}^n_i}\\right)^{-1}$ that yield minimized MSE estimates $\\widehat{{\\mathbf {y}}}_{i|i}^n$ and $\\widehat{{\\mathbf {x}}}_{i|i}^n$ of the pseudo-state ${\\mathbf {y}}_i$ and of the field ${\\mathbf {x}}_i$ , respectively, at each agent $n$ .",
"The cross-covariances $\\Sigma _{{\\mathbf {y}}_i \\widetilde{\\mathbf {\\nu }}^n_i}$ and $\\Sigma _{{\\mathbf {x}}_i {\\mathbf {\\nu }}^n_i}$ are $&\\Sigma _{{\\mathbf {y}}_i \\widetilde{\\mathbf {\\nu }}^n_i} = \\mathbb {E} \\left[ \\left({\\mathbf {y}}_i - \\overline{{\\mathbf {y}}}_i \\right) \\widetilde{\\mathbf {\\nu }}^{n^T}_i \\right] \\\\&= \\mathbb {E} \\!\\left[ \\!\\left({\\mathbf {e}}_{i|i-1}^n \\!+\\!\\left(\\widehat{{\\mathbf {y}}}_{i|i-1}^n \\!-\\!",
"\\overline{{\\mathbf {y}}}_i \\right) \\right)\\!\\!",
"\\begin{bmatrix} {\\mathbf {e}}_{i|i-1}^n \\!-\\!",
"{\\mathbf {e}}_{i|i-1}^{l_1} \\\\ \\vdots \\\\ {\\mathbf {e}}_{i|i-1}^n \\!-\\!",
"{\\mathbf {e}}_{i|i-1}^{l_{d_n}} \\\\ \\widetilde{H}_{\\!n} {\\mathbf {e}}_{i|i-1}^n \\!\\!+\\!",
"\\check{H}_{\\!n} {\\mathbf {\\epsilon }}_{i|i-1}^n \\!+\\!",
"H^T_n R^{-1}_n {\\mathbf {r}}^n_i \\end{bmatrix}^{\\!\\!T} \\!\\right] \\\\&= \\mathbb {E}\\!",
"\\left[ {\\mathbf {e}}_{i|i-1}^n \\!",
"\\left[ \\!\\!\\begin{array}{l;{2pt/2pt}l;{1pt/1pt}l;{2pt/2pt}l}\\!",
"{\\mathbf {e}}_{i|i-1}^{n^T} \\!-\\!",
"{\\mathbf {e}}_{i|i-1}^{l^T_1}\\!",
"&\\!",
"\\cdots \\!",
"&\\!",
"{\\mathbf {e}}_{i|i-1}^{n^T} \\!-\\!",
"{\\mathbf {e}}_{i|i-1}^{l^T_{d_n}} \\!&\\!",
"\\widetilde{H}_{\\!n} {\\mathbf {e}}_{i|i-1}^{n^T} \\\\& & & \\; + \\check{H}_{\\!n}{\\mathbf {\\epsilon }}_{i|i-1}^{n^T} \\!\\!\\end{array}\\!\\!",
"\\right] \\!",
"\\right] \\\\&= \\left[\\begin{array}{l;{2pt/2pt}l;{1pt/1pt}l;{2pt/2pt}l}\\!\\!\\!\\!",
"P_{i|i\\!-\\!1}^{nn} \\!-\\!",
"P_{i|i\\!-\\!1}^{nl_1} & \\cdots & P_{i|i\\!-\\!1}^{nn} \\!-\\!",
"P_{i|i\\!-\\!1}^{nl_{d_n}} & P_{i|i\\!-\\!1}^{nn} \\!",
"\\widetilde{H}^T_n \\!+\\!",
"\\Pi _{i|i\\!-\\!1}^{nn^T} \\!",
"\\check{H}^T_n \\!\\!\\!\\!\\end{array}\\right], \\\\&\\Sigma _{{\\mathbf {x}}_i {\\mathbf {\\nu }}^n_i} = \\mathbb {E} \\left[ \\left({\\mathbf {x}}_i - \\overline{{\\mathbf {x}}}_i \\right) {\\mathbf {\\nu }}^{n^T}_i \\right] \\\\&= \\mathbb {E} \\left[ \\left({\\mathbf {\\epsilon }}_{i|i-1}^n +\\left(\\widehat{{\\mathbf {x}}}_{i|i-1}^n - \\overline{{\\mathbf {x}}}_i \\right) \\right) \\left( G{\\mathbf {\\epsilon }}_{i|i-1}^{n} \\!-\\!",
"{\\mathbf {e}}_{i|i}^{n} \\right)^T \\right] \\\\&= \\mathbb {E} \\left[ {\\mathbf {\\epsilon }}_{i|i-1}^n \\!",
"\\left( G{\\mathbf {\\epsilon }}_{i|i-1}^{n} \\!-\\!",
"{\\mathbf {e}}_{i|i}^{n} \\right)^T \\!",
"\\right] = \\Sigma _{i|i\\!-\\!1}^{nn} G - \\Gamma ^{nn}_i.$ In the above derivation, using the iterated law of expectation it can be shown that the terms $\\mathbb {E} \\left[ \\left(\\widehat{{\\mathbf {y}}}_{i|i-1}^n - \\overline{{\\mathbf {y}}}_i \\right) \\widetilde{\\mathbf {\\nu }}^{n^T}_i \\right] = \\mathbb {E} \\left[ \\left(\\widehat{{\\mathbf {x}}}_{i|i-1}^n - \\overline{{\\mathbf {x}}}_i \\right) {\\mathbf {\\nu }}^{n^T}_i \\right] = {\\mathbf {0}}$ .",
"Also, $\\mathbb {E} \\left[ {\\mathbf {e}}_{i|i-1}^n {\\mathbf {r}}^{n^T}_i \\right] = {\\mathbf {0}}$ due to the statistical independence of the noise sequences.",
"Similarly, the covariances matrices $\\Sigma _{\\widetilde{\\mathbf {\\nu }}^n_i}$ and $\\Sigma _{{\\mathbf {\\nu }}^n_i}$ are: $&\\Sigma _{\\widetilde{\\mathbf {\\nu }}^n_i} = \\mathbb {E} \\left[ \\widetilde{\\mathbf {\\nu }}^{n}_i \\widetilde{\\mathbf {\\nu }}^{n^T}_i \\right] \\\\&= \\mathbb {E} \\left[ \\begin{bmatrix} {\\mathbf {e}}_{i|i-1}^n \\!-\\!",
"{\\mathbf {e}}_{i|i-1}^{l_1} \\\\ \\vdots \\\\ {\\mathbf {e}}_{i|i-1}^n \\!-\\!",
"{\\mathbf {e}}_{i|i-1}^{l_{d_n}} \\\\ \\widetilde{H}_{\\!n} {\\mathbf {e}}_{i|i-1}^n \\!+\\!",
"\\check{H}_{\\!n} {\\mathbf {\\epsilon }}_{i|i-1}^n \\\\ \\qquad + H^T_n R^{-1}_n {\\mathbf {r}}^n_i \\end{bmatrix} \\begin{bmatrix} {\\mathbf {e}}_{i|i-1}^n \\!-\\!",
"{\\mathbf {e}}_{i|i-1}^{l_1} \\\\ \\vdots \\\\ {\\mathbf {e}}_{i|i-1}^n \\!-\\!",
"{\\mathbf {e}}_{i|i-1}^{l_{d_n}} \\\\ \\widetilde{H}_{\\!n} {\\mathbf {e}}_{i|i-1}^n \\!+\\!",
"\\check{H}_{\\!n} {\\mathbf {\\epsilon }}_{i|i-1}^n \\\\ \\qquad + H^T_n R^{-1}_n {\\mathbf {r}}^n_i \\end{bmatrix}^T \\right] \\\\&\\Sigma _{{\\mathbf {\\nu }}^n_i} = \\mathbb {E} \\left[ {\\mathbf {\\nu }}^{n}_i {\\mathbf {\\nu }}^{n^T}_i \\right] = \\mathbb {E} \\left[ \\left( G{\\mathbf {\\epsilon }}_{i|i-1}^n \\!-\\!",
"{\\mathbf {e}}_{i|i}^n \\right) \\left( G{\\mathbf {\\epsilon }}_{i|i-1}^n \\!-\\!",
"{\\mathbf {e}}_{i|i}^n \\right)^T \\right].$ The rest of the derivation of $\\Sigma _{\\widetilde{\\mathbf {\\nu }}^n_i}$ and $\\Sigma _{{\\mathbf {\\nu }}^n_i}$ in terms of the error covariance matrices is by block-by-block multiplication of the above expressions.",
"$\\Box $ [Figure: NO_CAPTION Since July 2016, he is a Postdoctoral Researcher at IBM T.J. Watson Research Center, Yorktown Heights, NY.",
"His research interests are in statistical inference, distributed estimation over multi-agent networks, model adaptation in dynamic environments and time-series analysis; broadly in the areas of statistical signal processing, machine learning and data-mining with applications in healthcare and smart infrastructure.",
"[Figure: NO_CAPTION He is the Philip L. and Marsha Dowd University Professor at Carnegie Mellon University (CMU).",
"He was on the faculty at IST and has held visiting faculty appointments at MIT and New York University (NYU).",
"He founded and directs a large education and research program between CMU and Portugal, www.icti.cmu.edu.",
"His research interests are on data science, graph signal processing, and statistical and algebraic signal and image processing.",
"He has published over 550 papers and holds thirteen patents issued by the US Patent Office.",
"The technology of two of his patents (co-inventor A. Kavčić) are in about three billion disk drives read channel chips of 60 % of all computers sold in the last 13 years worldwide and was, in 2016, the subject of the largest university verdict/settlement in the information technologies area.",
"Dr. Moura is the IEEE Technical Activities Vice-President (2016) and member of the IEEE Board of Directors.",
"He served in several other capacities including IEEE Division IX Director, member of several IEEE Boards, President of the IEEE Signal Processing Society(SPS), Editor in Chief for the IEEE Transactions in Signal Processing, interim Editor in Chief for the IEEE Signal Processing Letters.",
"Dr. Moura has received several awards, including the Technical Achievement Award and the Society Award from the IEEE Signal Processing.",
"In 2016, he received the CMU College of Engineering Distinguished Professor of Engineering Award.",
"He is a Fellow of the IEEE, a Fellow of the American Association for the Advancement of Science (AAAS), a corresponding member of the Academy of Sciences of Portugal, Fellow of the US National Academy of Inventors, and a member of the US National Academy of Engineering."
]
] | 1605.06096 |
[
[
"Generic Demand Model Considering the Impact of Prosumers for Future Grid\n Scenario Analysis"
],
[
"Abstract The increasing uptake of residential PV-battery systems is bound to significantly change demand patterns of future power systems and, consequently, their dynamic performance.",
"In this paper, we propose a generic demand model that captures the aggregated effect of a large population of price-responsive users equipped with small-scale PV-battery systems, called prosumers, for market simulation in future grid scenario analysis.",
"The model is formulated as a bi-level program in which the upper-level unit commitment problem minimizes the total generation cost, and the lower-level problem maximizes prosumers' aggregate self-consumption.",
"Unlike in the existing bi-level optimization frameworks that focus on the interaction between the wholesale market and an aggregator, the coupling is through the prosumers' demand, not through the electricity price.",
"That renders the proposed model market structure agnostic, making it suitable for future grid studies where the market structure is potentially unknown.",
"As a case study, we perform steady-state voltage stability analysis of a simplified model of the Australian National Electricity Market with significant penetration of renewable generation.",
"The simulation results show that a high prosumer penetration changes the demand profile in ways that significantly improve the system loadability, which confirms the suitability of the proposed model for future grid studies."
],
[
"Introduction",
"Power systems are undergoing a major transformation driven by the increasing uptake of variable renewable energy sources (RES).",
"At the demand side, the emergence of cost-effective “behind-the-meter” distributed energy resources, including on-site generation, energy storage, electric vehicles, and flexible loads, and the advancement of sensor, computer, communication and energy management technologies are changing the way electricity consumers source and consume electric power.",
"Indeed, recent studies suggest that rooftop PV-battery systems will reach retail price parity from 2020 in the USA grids and the Australian National Electricity Market (NEM) [1].",
"A recent forecast by Morgan Stanley has suggested that the uptake can be even faster, by boldly predicting that up to 2 million Australian households could install battery storage by 2020 [2].",
"This has been confirmed by the Energy Networks Australia and the Australian Commonwealth Scientific and Industrial Research Organisation (CSIRO) who have estimated the projected uptake of solar PV and battery storage in 2050 to be 80 and 100GWh [3], which will represent between 30%–50% of total demand, a scenario called “Rise of the Prosumer” [4].",
"Here, the prosumer they refer to is a small-scale (residential, commercial and small industrial) electricity consumer with on-site generation.",
"A similar trend has been observed in Europe as well [5].",
"Given this, it is expected that a large uptake of demand-side technologies will significantly change demand patterns in future gridsWe interpret a future grid to mean the study of national grid type structures with the above-mentioned transformational changes for the long-term out to 2050., which will in turn affect their dynamic performance.",
"Existing future grid feasibility studies [6], [7], [8], [9], [10] typically use conventional demand models, possibly using some heuristics to account for the effect of emerging demand-side technologies, and the synergies that may arise between them.",
"They also assume specific market arrangements by which RES are integrated into grid operations.",
"The challenge associated with future grid planning is that the grid structure and the regulatory framework, including the market structure, cannot be simply assumed from the details of an existing one.",
"Instead, several possible evolution paths need to be accounted for.",
"Future grid planning thus requires a major departure from conventional power system planning, where only a handful of the most critical scenarios is analyzed.",
"To account for a wide range of possible future evolutions, scenario analysis has been proposed in many industries, e.g.",
"in finance and economics [11], and in energy [12], [13], [14].",
"As opposed to power system planning where the aim is to find an optimal transmission and/or generation expansion plan, the aim in scenario analysis is to analyze possible evolution pathways to inform policymaking.",
"Given the uncertainty associated with long-term projections, the focus of future grid scenario analysis is limited to the analysis of what is technically possible, although it might also consider an explicit costing [15].",
"Our work is part of the Future Grid Research Program funded by the CSIRO, whose aim is to explore possible future pathways for the evolution of the Australian grid out to 2050 by looking beyond simple balancing.",
"To this end, a comprehensive modeling framework for future grid scenario analysis has been proposed in [16], which includes a market model, power flow analysis, and stability analysis.",
"The demand model, however, assumes that the users are price-takers, which does not properly capture the aggregated effect of prosumers on the demand profile, as discussed later.",
"Due to the influence of a demand profile on power systems performance and stability, recent studies have attempted to integrate the aggregated impact of prosumers into the demand models [17], [18], [19], [20], [21], [22].",
"The focus, however, is usually on scheduling of particular emerging demand-side technologies, e.g.",
"HVAC [17], [18], [19], flexible loads [20], PV-battery systems [21], and plug-in electrical vehicles (PEVs) [22].",
"Most of these modeling approaches assume an existing market structure, with the impact of prosumers incorporated by allowing demand and supply to interact in some limited or predefined ways.",
"Specifically, this is mainly done via three different approaches: Only the supply-side is modeled physically while prosumers are considered by a simplified representation of demand-side technologies.",
"In [20], flexible loads' effects on reserve markets are analyzed by modeling prosumers with a tank model; however, the reserve market is greatly simplified.",
"In [17], prosumers are represented by a price-elasticity matrix, which is used to model changes in the aggregate demand in response to a change in the electricity price, and are acquired from the analysis of historical data.",
"Demand-side technologies are physically modeled while a simplified representation of supply-side is employed.",
"For instance, in [21], the supply-side is represented by an electricity price profile.",
"Both supply and demand sides can be modeled physically and optimized jointly, as in [18], [22], which can produce more realistic results.",
"For example, the study in [22] integrates the aggregated charging management approaches for PEVs into the market clearing process, with a simplified representation of the latter.",
"Although the above models have shown their merits, they are dependent on specific practical details such as the electricity price or the implementation of a mechanism for demand response (DR) aggregation, which limits their usefulness for future grid scenario analysis where the detailed market structure is potentially unknown.",
"Against this backdrop, the paper proposes a principled method for generic demand modeling including the aggregated effect of prosumers.",
"The model is formulated as a bi-level program in which the upper-level unit commitment problem minimizes the total generation cost, and the lower-level problem maximizes the aggregate prosumers' self-consumption.",
"In more detail, the lower-level objective is motivated by the emerging situation in Australia, where rooftop PV owners are increasingly discouraged from sending power back to the grid due to very low PV feed-in-tariffs versus increasing retail electricity prices.",
"In this setting, an obvious cost-minimizing strategy is to install small-scale battery storage, to maximize self-consumption of local generated energy and offset energy used in peak pricing periods.",
"Similar tariff settings appear likely to occur globally in the near future, as acknowledged in [23].",
"Moreover, self-consumption within an aggregated block of prosumers is a good approximation of many likely behaviors and responses to other future incentives and market structures, such as (peak power-based) demand charges, capacity constrained connections, virtual net metering across connection points, transactive energy and local energy trading, and a (somewhat irrational) desire for self-reliance.",
"A key difference from existing bi-level optimization frameworks is that in our formulation, the levels are coupled through the prosumers' demand, not through the electricity price.",
"In contrast, other models, which focus on the interaction between an aggregator and the prosumers [19] or the aggregator and the wholesale market [22], couple the levels through prices.",
"These approaches essentially define a market structure, that is, a pricing rule to support an outcome.",
"In contradistinction, our proposed model is market structure agnostic.",
"That is, it implicitly assumes that an efficient mechanism for demand response aggregation is adopted, with prices determined by that unspecified mechanism, which support the outcomes computed by our optimization framework.",
"The paper builds on our previous work [24], [25].",
"In [25], we have assumed that a prosumer aggregation represents a homogeneous group of loads; that is, we have assumed that they all behave in the same way and have the same capacity, and are allowed to send power back to the grid.",
"In the absence of an explicit transmission pricing, this can create perverse outcomes, such as power exchange between aggregators located in different parts of the network.",
"In the model proposed in this paper, the aggregators are not allowed to send the power back to the grid, which better models the assumption of self-consumption within an aggregated block of prosumers.",
"The model proposed in [24] is similar to the model in this paper, however, the prosumer battery storage is modeled implicitly, which requires a heuristic search to capture the prosumer behavior.",
"The remainder of the paper is organized as follows: Section II presents the proposed modeling framework.",
"In Section III, the efficacy of the proposed framework is demonstrated on a simplified 14-generator network model of the NEM with a significant RES penetration.",
"Finally, Section IV concludes."
],
[
"Generic Demand Model Considering the Impact of Prosumers",
"Generic demand models are essential for power system studies.",
"They are commonly used to reflect the aggregated effect of numerous physical loads [26], [27].",
"Conventional demand models only account for the accumulated effect of independent load changes and some relatively minor control actions, that is, they are by necessity simplified to not include all details of the loads.",
"In the following, we explain the motivation behind this work and the modeling assumptions."
],
[
"Research Motivation and Modeling Assumptions",
"The main purpose of developing generic demand models is to provide accurate dispatch decisions for balancing and stability analysis of future grid scenarios.",
"Given the uncertainty associated with future grid studies, the modeling framework should be market structure agnostic, and capable of easy integration of various types and penetrations of emerging demand-side technologies.",
"To this effect, we make the following assumptions: The loads are modeled as price anticipators.",
"It is well understood that price-taking load behavior, which simply responds to a given price profile, can result in load synchronization, i.e.",
"all users move their consumption to a low-price period, resulting in an inefficient market outcome.",
"In contrast, price anticipatory loads influence the electricity price by playing “a game” with the wholesale market.",
"The game is captured by the proposed bi-level model.",
"Specifically, the price-anticipating assumption implies that the aggregate effect of the prosumers is to change the market clearing prices and quantities, and, moreover, that the prosumers have a model of this effect.",
"Given this, the prosumer aggregation bids follow an equilibrium strategy, with an accurate expectation about the response of the market; this is the standard reasoning behind a Cournot or Stackelberg game formulation corresponding to a bi-level optimization problem.",
"This equilibrium strategy can be thought of as being generated by the following iterative process: First, the market operator creates a price profile by clearing the market based on the predicted demand.",
"Second, the prosumers (and other price-anticipatory participants) respond by shifting their consumption to cheaper time slots.",
"This gives a new demand profile, and the market operator clears the market again, and the process repeats until convergence.",
"Note that the proposed model does not specify an iterative mechanism, but rather, it encodes the optimality conditions for any price profile for the prosumers in Karush-Kuhn-Tucker (KKT) conditions, and in this way it captures the outcome resulting from the price-anticipatory prosumer behavior.",
"In our previous work [28], [16], we modeled the loads as price-takers, inspired by the smart home concept [29], in which the loads respond to the electricity price to minimize energy expenditure.",
"The study has shown that with large penetration of price-taking prosumers, the marginal benefit might become negative when secondary peaks are created due to the load synchronization (see Fig.",
"REF showing the operational demand with different penetrations of prosumers).",
"Therefore, demand response aggregators (henceforth simply called aggregators) have started to emerge to fully exploit the demand-side flexibility.",
"To that effect, the model implicitly assumes an efficient mechanism for demand response aggregation (an interested reader might refer to [30] for a discussion on practical implementation issues).",
"However, specific implementation details, like price structure or the division of the profit earning by the aggregated collection of prosumers, are not of explicit interest in the proposed model.",
"The demand model representing an aggregator consists of a large population of prosumers connected to an unconstrained distribution network who collectively maximize self-consumption (made possible by an efficient internal trading and balancing mechanism).",
"Aggregators do not alter the underlying power consumption of the prosumers.",
"That is, except for battery losses, the total power consumption before and after aggregation remains the same; however, the grid power intake profile does change, as a result of the prosumers using batteries to maximize self-consumption.",
"Prosumers have smart meters equipped with home energy management (HEM) systems for scheduling of the PV-battery systems.",
"Also, a communication infrastructure is assumed that allows a two-way communication between the grid, the aggregator and the prosumers, facilitating energy trading between prosumers in the aggregation.",
"These assumptions appear to be appropriate for scenarios arising in time frame of several decades into the future.",
"Figure: Operational demand with different penetrations of price-taking (top) and price-anticipating prosumers (bottom)."
],
[
"Bi-level Optimization Framework",
"In the model, we are specifically interested in the aggregated effect of a large prosumer population on the demand profile, assuming that the prosumers collectively maximize their self-consumption.",
"Given that the objective of the wholesale market is to minimize the generation cost, the problem exhibits a bi-level structure.",
"In game theory, such hierarchical optimization problems are known as Stackelberg games.",
"They can be formulated as bi-level mathematical programs of the form [19]: $\\mathop {\\operatorname{minimize}}\\limits _{ \\textbf {x}, \\textbf {y} } \\quad & \\Phi \\left( \\textbf {x}, \\textbf {y} \\right) \\\\\\operatorname{subject\\,to} \\quad & \\left( \\textbf {x}, \\textbf {y} \\right) \\in \\mathcal {Z} \\\\ & \\textbf {y} \\in \\mathcal {S} = \\mathop {\\operatorname{arg\\,min}}\\limits _{\\textbf {y}} \\lbrace \\Omega \\left( \\textbf {x}, \\textbf {y} \\right): \\textbf {y} \\in \\mathcal {C} \\left( \\textbf {x} \\right) \\rbrace $ where $\\textbf {x} \\in \\mathbb {R}^n$ , $\\textbf {y} \\in \\mathbb {R}^m$ , are decisions vectors, and $\\Phi \\left( \\textbf {x}, \\textbf {y} \\right): \\mathbb {R}^{n+m} \\rightarrow \\mathbb {R}$ and $\\Omega \\left( \\textbf {x}, \\textbf {y} \\right): \\mathbb {R}^{n+m} \\rightarrow \\mathbb {R}$ are the objective functions of the upper- and the lower-level problems, respectively.",
"$\\mathcal {Z}$ is the joint feasible region of the upper-level problem and $\\mathcal {C(\\textbf {x})}$ the feasible region of the lower-level problem induced by $\\textbf {x}$ .",
"In the existing market models that adopt a hierarchical approach, the coupling variable $\\textbf {y}$ is the electricity price (e.g.",
"[22], [19]).",
"That is, the upper-level (the wholesale market in our case) determines the price schedule, while the lower-level (the aggregator acting on behalf of the prosumers), optimizes its consumption based on this price schedule.",
"Fig.",
"REF shows the structure of the proposed modeling framework.",
"The demand model consists of two parts: (i) inflexible demand, $p_{\\text{d}}^{\\text{inf},m}$ , with a fixed demand profile, representing large industrial loads and loads without flexible resources; and (ii) flexible demand, $p_{\\text{d}}^{\\text{flx},m}$ , comprising a large population of prosumers who collectively maximize self-consumption.",
"Note that not every bus in the system has a load connected to it, hence the distinction between an aggregator $m \\in \\mathcal {M}$ and bus $i \\in \\mathcal {B}$ .",
"Unlike in most existing studies, the interaction between the wholesale market and the aggregators in our model is through the demand profile of the aggregator, $p_{\\text{d}}^{\\text{flx},m}$ .",
"Note that in contradistinction to the price-taking assumption when the electricity price is known in advance, now the collective action of the prosumers affects the wholesale market dispatch, which is the salient feature of the proposed model.",
"Figure: Structure of the proposed modeling framework."
],
[
"Upper-level Problem (Wholesale Market)",
"To emulate the market outcome, the upper-level problem is cast as a unit commitment (UC) problem aiming to minimize the generation cost: $\\mathop {\\operatorname{minimize}} \\limits _{s,u,d,p,\\theta } \\sum _{h \\in \\mathcal {H}} \\!",
"{\\sum _{g \\in \\mathcal {G}}} \\Big (c_g^\\text{fix} s_{g,h} \\!",
"+ \\!",
"c_g^\\text{su} u_{g,h} \\!",
"+ \\!",
"c_g^\\text{sd} d_{g,h} \\!",
"+ \\!",
"c_g^\\text{var} p_{g,h} \\Big ),$ where $s_{g,h},u_{g,h},d_{g,h} \\in \\lbrace 0,1\\rbrace $ , $p_{g,h}\\in \\mathbb {R}_{+}$ , $\\theta _{i,h}\\in \\mathbb {R}$ are the decision variables of the problem.",
"The problem is subject to the following constraints: $& \\sum _{g \\in \\mathcal {G}_i}{p_{g,h}} = \\!",
"\\sum _{m \\in \\mathcal {M}_i} \\!",
"(p_{\\text{d},h}^{\\text{inf},m}+p_{\\text{d},h}^{\\text{flx},m})+\\sum _{l \\in \\mathcal {L}_i}(p_{l,h}+\\Delta p_{l,h}), \\\\& |B_{i,j}(\\theta _{i,h}-\\theta _{j,h})| \\le \\overline{p}_{l}, \\\\& \\underline{p}_{g} s_{g,h} \\le p_{g,h} \\le \\overline{p}_{g} s_{g,h}, \\\\& u_{g,h}-d_{g,h}=s_{g,h}-s_{g,h-1}, \\\\& \\sum \\nolimits _{g_{\\text{synch}}\\in \\mathcal {R}}({\\overline{p}_{g}} s_{g,h} - p_{g,h}) \\ge p^{\\text{res}}_{r,h}, \\\\& u_{g,h} + \\sum \\nolimits _{\\tilde{h}=0}^{\\tau _g^{\\text{u}}-1} d_{g,h+\\tilde{h}} \\le 1, \\\\& d_{g,h} + \\sum \\nolimits _{\\tilde{h}=0}^{\\tau _g^{\\text{d}}-1} u_{g,h+\\tilde{h}} \\le 1, \\\\& -r_g^- \\le p_{g,h} - p_{g,h-1} \\le r_g^+, \\\\&\\mathop {\\operatorname{arg\\,min}} \\limits _{p_{\\text{d}}^{\\text{flx}}, p_{\\text{b}}, e_{\\text{b}} } \\left\\lbrace \\sum _{h \\in \\mathcal {H}} p_{\\text{d},h}^{\\text{flx},m} \\ \\operatorname{subject\\,to} \\ (\\ref {LC15})\\textrm {-}(\\ref {LC18}) \\right\\rbrace , $ where (REF ) is the power balance equation at each bus $i$ in the systemNote that the flexible demand of each aggregator $m$ , $p_{\\text{d},h}^{\\text{flx},m}$ , couples the upper-level (wholesale market) problem with each of the $m$ lower-level (aggregator) problems., with $\\mathcal {G}_i$ , $\\mathcal {M}_i$ , $\\mathcal {L}_i$ representing respectively the sets of generators, aggregators and lines connected to bus $i$ , and $p_{\\text{d},h}^{\\text{inf},m}$ , $p_{\\text{d},h}^{\\text{flx},m}$ , $p_{l,h}^{i,j}$ and $\\Delta p_{l,h}^{i,j}$ representing respectively the inflexible and flexible demand of aggregator $m$ , line power and line power loss (assumed to be 10% of the line flow) on each line connected to bus $i$ ; () represents line power limits; () limits the dispatch level of a generating unit between its respective minimum and maximum limits; () links the status of a generator unit to the up and down binary decision variables; () ensures sufficient spinning reserves are available in reach region of the grid; () and () ensure minimum up and minimum down times of the generators; () are the generator ramping constraints; and () is the constraint resulting from the prosumer aggregation optimization problem, as explained in Section REF ."
],
[
"Lower-level Problem (Aggregators)",
"The prosumer aggregation is formulated in the lower-level problem.",
"The loads within an aggregator's domain are assumed homogeneous, which allows us to represent the total aggregator's demand with a single load model.",
"The flexibility provided by the battery is only used to maximize self-consumption, with grid supply readily available.",
"This implies that the end-users' power consumption pattern is left unaltered, so that their comfort is not jeopardized.",
"The coupling between the upper-level (wholesale) and the lower-level (retail) problem in the proposed model is through the power demand.",
"This removes the need to define the market in terms of a pricing mechanism or rule, and it follows that the electricity price is not explicitly shown in the optimization problem.",
"This is why it is called a generic model.",
"This approach stands in contrast to other existing bi-level formulations [19],[22], in which the loads and wholesale market are coupled through the electricity price.",
"The prices generated by any market depend inherently on the specific pricing mechanism adopted.",
"However, in practice, several different pricing rules can implement a desired outcome, including the widely-used uniform price reverse auction or nodal pricing mechanisms.",
"For example, the electricity price could comprise the dual variables associated with the power balance constraint (REF ) and power flow constraints () of the upper-level problem, plus retail/aggregator and network charges.",
"Moreover, a range of different pricing rules result in different retail/aggregator and network charges, with all supporting an efficient outcome.",
"However, by instead coupling the upper- and lower-level problems via power demand, our proposed model avoids the need to specify a particular pricing rule, which makes it market-structure agnostic (see also Assumptions 1 and 2 in Section II-A).",
"Nonetheless, it is fair to assume that in any practical system, users will have access to a price forecastNote that in a HEM problem [29], a HEM system is an agent acting on behalf of a prosumer, and the electricity price is known ahead of time, resulting in a price-taking behavior.",
"and other bidders' historical behavior–but these are all market design-specific.",
"In the absence of pricing rule details, the proposed optimization model does not require such information.",
"Specifically, the lower-level problem is formulated as follows for each demand aggregator $m \\in \\mathcal {M}$ : $& \\mathop {\\operatorname{minimize}}\\limits _{ p_{\\text{d}}^{\\text{flx}}, p_{\\text{b}}, e_{\\text{b}} } \\sum _{h \\in \\mathcal {H}} p_{\\text{d},h}^{\\text{flx},m},$ where the decision variables of the problem are flexible demand $p_{\\text{d}}^{\\text{flx}}$ , battery power $p_{\\text{b}}$ , and battery capacity $e_{\\text{b}}$ .",
"The problem is subject to the following constraints: $& p_{\\text{d},h}^{\\text{flx},m} = p_{\\text{d},h}^{\\text{u},m}-p_{\\text{pv},h}^{m}+p_{\\text{b},h}^m,\\\\& e_{\\text{b},h}^m = \\eta _{\\text{b}}^m e_{\\text{b},h-1}^m + p_{\\text{b},h}^m, \\\\& \\underline{p}_{\\text{b}}^m \\le p_{\\text{b},h}^m \\le \\overline{p}_{\\text{b}}^m,\\\\& \\underline{e}_{\\text{b}}^{m} \\le {e}_{\\text{b},h}^{m} \\le \\overline{e}_{\\text{b}}^{m},$ where (REF ) is the power balance equation; and ()-() are the battery storage constraints.",
"Power $p_{\\text{d},h}^{\\text{u},m}$ is the underlying power demand of the prosumers.",
"Note that according to the Assumption 3 in Section II-B, except for battery losses, the underlying power demand does not change, however the power intake from grid can.",
"Finally, the KKT optimality conditions of the lower-level problem are added as the constraints to the upper-level problem, which reduces the problem to a single mixed integer linear program that can be solved using of-the-shelf solvers.",
"Note that because the two levels interacts through a power, not through a price, unlike in [19], no linearization is required."
],
[
"Case Studies",
"To showcase the efficacy of the model, we analyze steady-state voltage stability of a simplified model of the NEM with scenarios reflecting different prosumer penetrations."
],
[
"Model of the Australian National Electricity Market (NEM)",
"The 14-generator IEEE test system shown in Fig.",
"REF was initially proposed in [31] as a test bed for small-signal analysis.",
"The system is loosely based on the NEM, the interconnection on the Australian eastern seaboard.",
"The network is stringy, with large transmission distances and loads concentrated in a few load centers.",
"It consists of 59 buses, 28 loads, and 14 generators.",
"The test system consists of four areas representing the states of Queensland, New South Wales, Victoria and South Australia, and 28 aggregators, one for each load bus.",
"The generator technologies and modeling assumptions follow [24].",
"We consider two RES penetration rates.",
"In the business as usual (BAU) scenario, the generation portfolio includes 39.36 coal, 5.22 gas, and 2.33 hydro.",
"In the high-RES scenario, 40 of the total demand is covered by variable RES.",
"Inspired by two recent Australian 100% renewables studies [6], [10], part of coal generation is replaced with wind and utility PV using wind and solar traces from the AEMO's planning document [32], which results in 28.94 coal, 5.22 gas, 2.33 hydro, 21 wind, and 12 utility PV.",
"Given the deterministic nature of the model, we assume 10 reserves for each region in the system to cater for demand and RES forecast errors.",
"In market simulations, generators are assumed to bid according to their short-run marginal costs, while RESs bid at zero cost.",
"Simulations are performed using a rolling horizon approach with hourly resolution assuming a perfect foresight.",
"The optimization horizon is three days with a two-day overlap.",
"Last, wind and solar generators are assumed to operate in a voltage control mode.",
"Figure: Single-line diagram of the 14-generator model of the NEM with the 16 zones that define wind and solar traces ."
],
[
"Prosumer Scenarios",
"We assume four different prosumer penetrations: zero, low, medium and high.",
"With no prosumer penetration, the demand is assumed inflexible.",
"For the other three scenarios, we assume that part of the demand is equipped with small-scale (residential and small commercial) PV-battery systems.",
"The uptake of PV loosely follows a recent AEMO study [33].",
"The PV capacities are respectively 5, 10, and 20 for the low, medium and high uptake of prosumers (the existing penetration in the NEM is 5).",
"We consider three different amounts of storage: zero, 2, and 4 of storage for 1 of rooftop PV.A typical ratio in the NEM today is 2h of storage [33], however, in the future, this will likely increase due to the anticipated cost reduction.",
"Hourly demand and PV traces are from the AEMO's planning document [32]."
],
[
"Dispatch Results",
"Dispatch results for a typical summer week with high demand (12-15 January) for a few representative scenarios are shown in Figs.",
"REF and REF .",
"The figures show, respectively, generation dispatch results (top row), combined flexible demand of all aggregators (middle row), and a combined battery charging profile of all aggregators (bottom row).",
"Fig.",
"REF shows results for a medium prosumer penetration with, respectively, zero, 2h and 4h hours of storage.",
"Fig.",
"REF shows results for different prosumer penetrations (zero, medium, high) with 4h of storage.",
"Observe that in all six cases peak demand occurs at mid-day due to a high air-conditioning load.",
"After the sunset, however, the demand is still high, so gas generation is needed to cover the gap.",
"In the available generation mix, gas has the highest short-run marginal cost, which increases the electricity price in late afternoon/early evening.",
"The balancing results over the simulated year have revealed that the increased RES penetration in the renewable scenarios requires more energy from gas generation compared to the BAU scenario.",
"This is due to RES intermittency, and the ramp limits of conventional coal-fired generation.",
"An increased penetration of prosumers with higher amounts of storage, however, reduces the usage of gas due to a flatter demand profile.",
"Figure: Dispatch results for a typical summer week with high demand (12-15 January) for a medium prosumer penetration with different amounts of storage: zero (left), 2h (middle) and 4h (right).Figure: Dispatch results for a typical summer week with high demand (12-15 January) for different penetrations of prosumers with 4h of storage: zero (left), medium (middle), high (right).Observe in Fig.",
"REF how an increasing amount of storage increases prosumers' self-sufficiency.",
"Without storage, the load is supplied by PV during the day, and the rest is supplied from the grid.",
"When storage is added to the system, batteries are charged when electricity is cheap (mostly from rooftop PV during the day and from wind during the night) and discharged in late afternoon to offset the demand when the electricity is most expensive.",
"Note that the plots in the bottom two rows show a combined load profile of all aggregators in the system, which explains why storage is seemingly charged and discharged simultaneously.",
"Observe how high amounts of storage (rightmost columns in Figs.",
"REF and REF ) flatten the demand profile.",
"During the day, the flexible demand is supplied by rooftop PV, which reduces the operational demand, while during the night, with sufficient wind generation, batteries are charged, which increases the operational demand.",
"This has a significant beneficial effect on loadability and voltage stability, as discussed in the next section."
],
[
"Loadability and Voltage Stability Results",
"Dispatch results from the market simulations are used to perform a load flow analysis, which is then used in the assessment of loadability and voltage stability.",
"In the analysis, only scenarios with 4h of storage were considered.",
"The prosumer scenarios are thus called, according to the respective penetration rates, zero (ZP), low (LP), medium (MP), and high (HP).",
"Note that the market model only considers a simplified DC power flow with the maximum angle limit set to $30^{\\circ }$ .",
"This can sometimes result in a non-convergent AC power flow in scenarios with a high RES penetration.",
"The number of non-convergent hours is, respectively, 175, 37, 12, and 0, in scenarios ZP, LP, MP, and HP.",
"An increased penetration of prosumers thus improves voltage stability, as explained in more detail later.",
"In loadability assessment, N-1 security is considered, so a contingency screening is performed first.",
"We screened all credible N-1 contingencies to identify the most severe ones based on the maximum power transfer level [34].",
"Twenty most critical contingencies were selected for each hour of the simulated year."
],
[
"Loadability calculation",
"To calculate system loadability (LDB), the load in the system is progressively increased using the results of the market dispatch as the base case.",
"After each increase, load flow is calculated twice; first for the case after the last load increase without a contingency, and then again for each of the preselected critical contingencies.",
"When the load flow does not converge for a particular contingency, the last convergent load flow solution without a contingency is considered the loadability margin.",
"We considered two different load increase patterns, where load and generation are increased uniformly, in proportion to the base case: (i) NEM: only load and generation in the NEM are increased; (ii) SA/VIC: only load in VIC and generation in SA are increased.",
"The results are summarized in Table REF .",
"Comparing the BAU scenario and the renewable scenario with conventional demand (ZP), it can be seen that with the increased RES penetration, the average loadability margin over the simulated year in the demand increase scenario NEM is decreased from 7.8 to 2.5.",
"Similarly, the average loadability margin in the demand increase scenario SA/VIC is reduced from 2.2 to 0.8.",
"With a high RES penetration, conventional synchronous generation is replaced by inverter-based generation with inferior reactive power support capabilityFor synchronous generation, a 0.8 power factor is assumed.",
"For RES, we used the reactive power capability curve for the generic GE Type IV wind farm model [35], in which the reactive power generation is significantly constrained close to the nominal active power generation.",
"In practice, detailed planning studies are required before connection is granted, which can result in an increased requirement for reactive power support., which results in a reduced reactive power margin in the system and hence lower stability margin.",
"With an increased penetration of prosumers, the system loadability improves.",
"Observe that the average system loadability margin in both load increase scenarios, NEM and SA/VIC, is increased from 2.5 and 0.8 for the renewable scenario with no prosumers (ZP) to 7.1 and 1.7 for high penetration of prosumers (HP), respectively, which indicates a considerable improvement in the system loadability margin.",
"This is explained by a demand reduction when prosumer demand is supplied by rooftop PV.",
"In the night hours, however, even with high prosumer penetration, the loadibility can be reduced when prosumers charge their batteries, as observed in Figs.",
"REF and REF .",
"The situation is further illustrated in Fig.",
"REF that compares the loadability margin for the load increase scenario NEM and the results of the modal analysis, discussed next.",
"Table: Loadability results"
],
[
"Modal analysis",
"Using the market dispatch results, modal analysis of the reduced V-Q sub-matrix of the power flow Jacobian is performed to assess voltage stability.",
"The smallest real part of the V-Q sub-matrix' eigenvalues is used as a relative measure of the proximity to voltage instability.",
"Furthermore, the associated eigenvectors provide information on the critical voltage modes and the weak points in the grid, that is, the areas that are most prone to voltage instability.",
"The results are summarized in Table REF .",
"With the increased RES penetration, the average of the minimum of the real part of all eigenvalues, henceforth called the minimum eigenvalue, is reduced from 49.5Np for the BAU scenario to 42.3Np for the ZP scenario.",
"With an increased prosumer penetration, the average of the minimum eigenvalue over the simulated year increases from 42.3Np (ZP) to 48.8Np (HP).",
"Observe in Fig.",
"REF that the results of the modal analysis confirm the results loadability analysis.",
"The general trend remains the same; higher RES penetration with conventional demand reduces the minimum real value of an eigenvalue, which implies a lower voltage stability margin.",
"Table: Modal analysis resultsFigure: Comparison of the loadability and modal analysis results for a typical summer week with high demand (12-15 January) for the load increase scenario NEM for BAU, and low and high prosumer penetration.Another observation that can be made from the modal analysis concerns the location of the weak points in the system, that is, the buses with the highest participation factor in the critical voltage modes.",
"These clearly change with the increased RES penetration.",
"In the BAU Scenario, the weakest points are typically buses with large loads (e.g.",
"306 and 308, representing Melbourne).",
"With the increased RES penetration and no prosumers (ZP), however, the weakest part of the system become buses located close to RESs (e.g.",
"505 and 410, representing, respectively a large wind farm in SA and a large solar PV farm in QLD).",
"Further, we observed that the voltage stability margin in the system improves significantly when there are more synchronous generators in the grid, due to their superior reactive power support capability compared to RES.",
"The participation factor analysis of the renewable scenarios revealed that SA and QLD are the most voltage constrained regions where the penetration of WFs and utility PVs is higher compared to other regions in the NEM, which can be, to a large extent, mitigated with a sufficiently large penetration of prosumers.",
"This clearly illustrates that RESs and prosumers change power system stability in ways that have not been experienced before, which requires a further in-depth analysis."
],
[
"Conclusion",
"The emergence of demand side technologies, in particular rooftop PV, battery storage and energy management systems is changing the way electricity consumers source and consume electric power, which requires new demand models for the long-term analysis of future grids.",
"In this paper, we propose a generic demand model that captures the aggregate effect of a large number of prosumers on the load profile that can be used in market simulation.",
"The model uses a bi-level optimization framework, in which the upper level employs a unit commitment problem to minimize generation cost, and the lower level problem maximizes collective prosumers' self-consumption.",
"To that effect, the model implicitly assumes an efficient mechanism for demand response aggregation, for example peer-to-peer energy trading or any other form of transactive energy.",
"Given that any efficient mechanism for demand response aggregation that aims to minimize generation cost will utilize self-generation first, the self-consuming assumption appears reasonable at the level of abstraction assumed in long-term future grid scenario analysis.",
"Moreover, the model is generic in that it does not depend on specific practical implementation details that will vary in the long-run.",
"To showcase the efficacy of the proposed model, we study the impact of prosumers on the performance, loadability and voltage stability of the Australian NEM with a high RES penetration.",
"The results show that an increased prosumer penetration flattens the demand profile, which increases loadability and voltage stability, except in situations with a low underlying demand and an excess of RES generation, where the aggregate demand might increase due to battery charging.",
"The analysis also revealed that with a high RES penetration, the weakest points in the network move from large load centers to areas with high RES penetration.",
"Also, loadability and voltage stability are highly dependent on the amount of synchronous generation due to their superior reactive power capability compared to RES, which requires further analysis.",
"Our future work will focus on designing market mechanisms for large scale aggregation of distributed energy resources to enable the participation of prosumers in power system operation."
],
[
"MILP Formulation of the Lower-level KKT Optimality Conditions",
"For the sake of completeness, we provide the complete optimization problem below.",
"The upper-level problem (REF )-() remains the same.",
"The price-anticipatory prosumer behavior is captured in constraint (), defined as the solution to the lower-level prosumer aggregation problem.",
"To be able to incorporate it in the upper-level problem, we first need to convert it into a set of MILP constraints.",
"We do that by using the optimality (KKT) conditions of the underlying optimization problem.",
"The KKT conditions follow from the associated Lagrangian: $\\mathcal {L}(\\textbf {x}, \\lambda , \\mu ) = f(\\textbf {x})+ \\lambda ^\\top \\textbf {g}(\\textbf {x}) + \\mu ^\\top \\textbf {h}(\\textbf {x}),$ where $\\textbf {x} = \\lbrace p_{\\text{d}}^{\\text{flx}}, p_{\\text{b}}, e_{\\text{b}} \\rbrace $ is the decision vector of the lower-level problem; $f(\\textbf {x})$ is the objective function (REF ); $\\textbf {g}(\\textbf {x})$ is the vector of equality constraints (REF )-(); $\\textbf {h}(\\textbf {x})$ is the vector of inequality constraints ()-(); $\\lambda ^\\top = \\lbrace \\lambda _{h}^{m,p}, \\lambda _{h}^{m,e} \\rbrace $ is the vector of Lagrange multipliers associated with the equality constraints, and $\\mu ^\\top = \\lbrace \\mu _{h}^{\\text{flx},m}, \\mu _{h}^{m,\\underline{p}}, \\mu _{h}^{m,\\overline{p}}, \\mu _{h}^{m,\\underline{e}},\\mu _{h}^{m,\\overline{e}}\\rbrace $ is the vector of Lagrange multipliers associated with the inequality constraints.",
"The KKT optimality conditions consist of: primal feasibility (REF )-() with the associated Lagrange multipliers given in parentheses: $& \\qquad \\qquad p_{\\text{d},h}^{\\text{flx},m} + p_{\\text{pv},h}^{m} - p_{\\text{b},h}^{m} - p_{\\text{d},h}^{\\text{u},m} = 0, &( \\lambda _{h}^{m,p} ) \\\\& \\qquad \\qquad e_{\\text{b},h}^{m} - \\eta _\\text{b}^m e_{\\text{b},h-1}^{m} - p_{\\text{b},h}^{m} = 0, & ( \\lambda _{h}^{m,e} ) \\\\& \\qquad \\qquad -p_{\\text{d},h}^{\\text{flx},m} \\le 0, &( \\mu _{h}^{\\text{flx},m} ) \\\\& \\qquad \\qquad \\underline{p}_{\\text{b}}^{m} - p_{\\text{b},h}^{m} \\le 0, &( \\mu _{h}^{m,\\underline{p}} ) \\\\& \\qquad \\qquad p_{\\text{b},h}^{m} - \\overline{p}_{\\text{b}}^{m} \\le 0, &( \\mu _{h}^{m,\\overline{p}} ) \\\\& \\qquad \\qquad \\underline{e}_{\\text{b}}^{m} -e_{\\text{b},h}^{m} \\le 0, &( \\mu _{h}^{m,\\underline{e}} ) \\\\& \\qquad \\qquad e_{\\text{b},h}^{m} -\\overline{e}_{\\text{b}}^{m} \\le 0, &( \\mu _{h}^{m,\\overline{e}} ) $ dual feasibility: $& \\mu _{h}^{\\text{flx},m}, \\mu _{h}^{m,\\underline{p}}, \\mu _{h}^{m,\\overline{p}}, \\mu _{h}^{m,\\underline{e}}, \\mu _{h}^{m,\\overline{e}} \\ge 0,$ stationarity (REF )-(): $&\\frac{\\partial \\mathcal {L}}{\\partial p_{\\text{d},h}^{\\text{flx},m}} = 1 + \\lambda _{h}^{m,p} - \\mu _{h}^{\\text{flx},m} = 0, \\\\&\\frac{\\partial \\mathcal {L}}{\\partial p_{\\text{b},h}^{m}} = - \\lambda _{h}^{m,p} - \\lambda _{h}^{m,e} - \\mu _{h}^{m,\\underline{p}} + \\mu _{h}^{m,\\overline{p}} = 0,\\\\&\\frac{\\partial \\mathcal {L}}{\\partial e_{\\text{b},h}^{m}} = \\lambda _{h}^{m,e} - \\eta _\\text{b}^m \\lambda _{h+1}^{m,e} - \\mu _{h}^{m,\\underline{e}} + \\mu _{h}^{m,\\overline{e}} = 0,$ complementary slackness (REF )-(): $&p_{\\text{d},h}^{\\text{flx},m} \\mu _{h}^{\\text{flx},m} = 0, \\\\&(-\\underline{p}_{\\text{b}}^{m} + p_{\\text{b},h}^{m}) \\mu _{h}^{m,\\underline{p}} = 0, \\\\&(-p_{\\text{b},h}^{m} + \\overline{p}_{\\text{b}}^{m}) \\mu _{h}^{m,\\overline{p}} = 0, \\\\&(-\\underline{e}_{\\text{b}}^{m} + e_{\\text{b},h}^{m}) \\mu _{h}^{m,\\underline{e}} = 0, \\\\&(-e_{\\text{b},h}^{m} + \\overline{e}_{\\text{b}}^{m}) \\mu _{h}^{m,\\overline{e}} = 0.",
"$ Complementary slackness conditions (REF )-() are bilinear, however they can be linearized by introducing one binary variable $b$ and a sufficiently large and positive constant $M$ resulting in two constraints per complementarity slackness condition ()-().",
"Given that the KKT conditions are sufficient and necessary for optimality, the bi-level constraint () can be replaced with mixed integer linear constraints (REF )-(): $& p_{\\text{d},h}^{\\text{flx},m} + p_{\\text{pv},h}^{m} - p_{\\text{b},h}^{m} - p_{\\text{d},h}^{\\text{u},m} = 0, \\\\& e_{\\text{b},h}^{m} - \\eta _\\text{b}^m e_{\\text{b},h-1}^{m} - p_{\\text{b},h}^{m} = 0 ,\\\\& 1 - \\mu _{h}^{\\text{flx},m} + \\lambda _{h}^{m,p} = 0,\\\\& - \\lambda _{h}^{m,p} - \\mu _{h}^{m,\\underline{p}} + \\mu _{h}^{m,\\overline{p}} - \\lambda _{h}^{m,e} = 0,\\\\& - \\mu _{h}^{m,\\underline{e}} + \\mu _{h}^{m,\\overline{e}} + \\lambda _{h}^{m,e} - \\eta _\\text{b}^m \\lambda _{h-1}^{m,e} = 0,\\\\& \\underline{p}_{\\text{b}}^{m} - p_{\\text{b},h}^{m} \\le 0, \\\\& p_{\\text{b},h}^{m} -\\overline{p}_{\\text{b}}^{m} \\le 0, \\\\& \\underline{e}_{\\text{b}}^{m} - e_{\\text{b},h}^{m} \\le 0 , \\\\& e_{\\text{b},h}^{m} -\\overline{e}_{\\text{b}}^{m} \\le 0 , \\\\& p_{\\text{d},h}^{\\text{flx},m} - M{^\\text{flx}} b_{h}^{\\text{flx},m} \\le 0, \\\\& \\mu _{h}^{\\text{flx},m} - M^\\text{flx} (1-b_{h}^{\\text{flx},m}) \\le 0,\\\\& p_{\\text{b},h}^{m} - M^{\\underline{p}} b_{h}^{m,\\underline{p}} \\le 0,\\\\& \\mu _{h}^{m,\\underline{p}} - M^{\\underline{p}} (1-b_{h}^{m,\\underline{p}}) \\le 0,\\\\& -p_{\\text{b},h}^{m} -M^{\\overline{p}} b_{h}^{m,\\overline{p}} \\le 0,\\\\& \\mu _{h}^{m,\\overline{p}} - M^{\\overline{p}} (1-b_{h}^{m,\\overline{p}}) \\le 0 ,\\\\& e_{\\text{b},h}^{m} - M^{\\underline{e}} b_{h}^{m,\\underline{e}} \\le 0,\\\\& \\mu _{h}^{m,\\underline{e}} - M^{\\underline{e}} (1-b_{h}^{m,\\underline{e}}) \\le 0,\\\\& -e_{\\text{b},h}^{m} - M^{\\overline{e}} b_{h}^{m,\\overline{e}} \\le 0,\\\\& \\mu _{h}^{m,\\overline{e}} -M^{\\overline{e}} (1-b_{h}^{m,\\overline{e}}) \\le 0,$ which adds the following additional decision variables to the upper-level problem (REF )-(): $& b_{h}^{\\text{flx},m},b_{h}^{m,\\underline{p}},b_{h}^{m,\\overline{p}},b_{h}^{m,\\underline{e}},b_{h}^{m,\\overline{e}} \\in \\lbrace 0,1\\rbrace ,\\\\& p_{\\text{b},h}^{m}, \\lambda _{h}^{m,p}, \\lambda _{h}^{m,e} \\in \\mathbb {R}, \\\\& p_{\\text{d},h}^{\\text{flx},m}, e_{\\text{b},h}^{m}, \\mu _{h}^{\\text{flx},m}, \\mu _{h}^{m,\\underline{p}}, \\mu _{h}^{m,\\overline{p}}, \\mu _{h}^{m,\\underline{e}},\\mu _{h}^{m,\\overline{e}} \\in \\mathbb {R}_+.$ The resulting optimization problem is an MILP that can be solved efficiently with off-the-shelf solvers.",
"[Figure: NO_CAPTION [Figure: NO_CAPTION [Figure: NO_CAPTION [Figure: NO_CAPTION [Figure: NO_CAPTION Prof. Hill is a Fellow of the Society for Industrial and Applied Mathematics, the Australian Academy of Science, and the Australian Academy of Technological Sciences and Engineering.",
"He is also a Foreign Member of the Royal Swedish Academy of Engineering Sciences."
]
] | 1605.05833 |
[
[
"A Robust Measurement of the Mass Outflow Rate of the Galactic Outflow\n from NGC 6090"
],
[
"Abstract To evaluate the impact of stellar feedback, it is critical to estimate the mass outflow rates of galaxies.",
"Past estimates have been plagued by uncertain assumptions about the outflow geometry, metallicity, and ionization fraction.",
"Here we use Hubble Space Telescope ultraviolet spectroscopic observations of the nearby starburst NGC 6090 to demonstrate that many of these quantities can be constrained by the data.",
"We use the Si~{\\sc IV} absorption lines to calculate the scaling of velocity (v), covering fraction (C$_f$), and density with distance from the starburst (r), assuming the Sobolev optical depth and a velocity law of the form: v$~\\propto(1 -\\mathrm{R}_\\mathrm{i}/\\mathrm{r} )^\\beta$ (where R$_\\mathrm{i}$ is the inner outflow radius).",
"We find that the velocity ($\\beta$=0.43) is consistent with an outflow driven by an r$^{-2}$ force with the outflow radially accelerated, while the scaling of the covering fraction ($C_f \\propto \\mathrm{r}^{-0.82}$) suggests that cool clouds in the outflow are in pressure equilibrium with an adiabatically expanding medium.",
"We use the column densities of four weak metal lines and CLOUDY photoionization models to determine the outflow metallicity, the ionization correction, and the initial density of the outflow.",
"Combining these values with the profile fitting, we find R$_\\mathrm{i}$ = 63 pc, with most of the mass within 300~pc of the starburst.",
"Finally, we find that the maximum mass outflow rate is 2.3~M$_\\odot$ yr$^{-1}$ and the mass loading factor (outflow divided by the star formation rate) is 0.09, a factor of 10 lower than the value calculated using common assumptions for the geometry, metallicity and ionization structure of the outflow."
],
[
"INTRODUCTION",
"Strangely, most of the gas within a galaxy is not near stars , [2], , , , .",
"The circum-galactic medium extends to radii greater than 150 kpc, is metal rich, and spans a range of temperatures , , .",
"Further, this reservoir of gas is massive, containing up to three times more mass than gas within discs .",
"Even though the gas extends more than 150 kpc from the stellar disc, the metal enrichment implies that the gas originated within the stellar disc.",
"What could loft so much mass out of discs?",
"High-mass stars inject energy and momentum into the ISM through high energy photons, cosmic rays, and supernovae, together commonly called stellar feedback , [63], [12], [35], , , [23], , [39], [40], [46], [47], [9].",
"If the stellar feedback is highly concentrated, then the combined energy and momentum drives gas out of star forming regions into a galactic outflow [35], [36], , [21].",
"Galactic outflows may enrich the large reservoir of cirum-galactic gas, while also removing metals from low-mass galaxies to create the mass-metallicity relation , [26], , [3], , [17], [15].",
"However, accurate mass outflow rates are required to constrain these relations.",
"Observations of galactic outflows are challenging.",
"First, galactic outflows are multi-phase, with emission from hot X-ray emitting plasma [31], , , to ionized hydrogen [56], [6], , , , [4], and even cold molecular gas , [62], [55].",
"Second, outflows are diffuse, and probing outflows with emission lines is challenging and limited to only the local universe.",
"Recent studies use optical [36], [59], , [11], near UV , [60], , [19], [49], [22], and far UV absorption lines , , , [32], [54], [13], [38], , [14] to study the diffuse gas within outflows.",
"While these studies have hinted at the role outflows play in removing gas from the discs of galaxies, the mass outflow rate calculations are plagued by uncertain assumptions.",
"To calculate the mass outflow rate from absorption lines, an observed column density and velocity are converted into a total mass outflow rate.",
"The first step converts the observed ion density to a total Hydrogen density through an assumed metallicity and ionization correction , [38].",
"The Hydrogen column density is then converted into a total Hydrogen mass through an assumed geometry, typically a thin shell.",
"These studies make four key assumptions: (1) that the absorption lines are optically thin (2) that the measured ion is the dominant ionization state, and there is no ionization correction (3) that the metallicity of the outflow is consistent with the ISM of the galaxy and (4) that the outflow is a spherical shell with a radius of 5 kpc , .",
"Recent observations question these assumptions.",
"Ionization modelling from [14] find that galactic outflows are photoionized, with the dominant ionization state, Si III, at an ionization potential near 25 eV, with only 1-5% of the gas in the neutral phase.",
"For neutral ions like Na I, neglecting the ionization correction can underestimate the total Hydrogen column density by a factor of 4 , [14].",
"Additionally, these photoionization models predict that the outflows are metal rich, and the assumption that the outflow has the metallicity of the ISM underpredicts the total Hydrogen density.",
"These large ionization fractions also imply that the outflow is relatively close to the ionizing source.",
"Stars generate outflows at distances of super-star clusters, 20-40 pc [63], , [12], but metal-enriched gas is also observed out to 10 kpc from the starburst , , , .",
"Even though the outflow extends over this enormous distance, studies typically calculate the mass outflow rates as if the entire mass is in a thin shell with a radius of 5 kpc .",
"Here, we present a new analysis of the mass outflow rate of the nearby galaxy NGC 6090 using Hubble Space Telescope ultraviolet spectroscopy.",
"We calculate the mass outflow rate by first measuring the distance, metallicity, and ionization fraction of the outflow.",
"In data we describe the data reduction, stellar continuum fitting, and measurement of the outflow properties.",
"We then fit for how the optical depth and covering fraction scales with velocity, using a Sobelev optical depth, a $\\beta $ velocity law, and a power-law density scaling (beta).",
"We apply the ionization models of [14] to the outflow of NGC 6090 to determine the metallcity (1.61 Z$_\\odot $ ), total Hydrogen density (18.73 cm$^{-3}$ ), and the ionization fractions of the outflow (ion).",
"In ionstruct we consider the implications of the ionization models and the metal enriched outflows, and then calculate the initial radius of the outflow (63 pc).",
"Using these derived quantities, we examine the covering fraction (cf), velocity (velocity), and density (density) laws with radius.",
"Since these relationships vary with velocity, we find that the mass outflow rate of the galaxy also varies with velocity, with a maximum mass outflow rate of 2.3 M$_\\odot $ yr$^{-1}$ .",
"This mass outflow rate is 10 times smaller than it would be if we used previous assumptions, and only 9% of the star formation rate of the galaxy.",
"In this paper we use $\\Omega _M = .28$ , $\\Omega _\\Lambda = .72$ and H$_0$ = 70 km s$^{-1}$ Mpc$^{-1}$ [43]."
],
[
"DATA",
"NGC 6090 is a massive star forming galaxy at a distance of 128 Mpc (see tab:ngc6090), with no indication of AGN contamination from the optical emission lines (see tab:ngc6090) and the WISE colors.",
"The galaxy is metal rich, with a log(O/H)+12 of 8.77 .",
"The galaxy is in the intermediate stages of a massive merger, as two distinct nuclei and large extend tidal tails are seen in the UV (fig:image) and optical images .",
"The Cosmic Origins Spectrograph [30] on the Hubble Space Telescope observed NGC 6090 during Cycle 18 [54].",
"The full data reductions are given in [13], and here we summarize the steps taken to measure the optical depths, covering fractions, and column densities of the outflow.",
"First, we describe the data reductions (reduc) and the stellar continuum fitting (cont).",
"We then explore the effects of the Si II fine structure and resonance emission lines (finestruct).",
"Finally, the optical depths, covering fractions, and column densities are measured in measure."
],
[
"Data Reduction",
"[54] observed NGC 6090 with the G130M grating for a total integration time of 8096 s. We downloaded the spectra from MAST and processed the data through the CalCOS pipeline, version 2.20.1.",
"The individual exposures were aligned and co-added following the methods outlined in .",
"The spectrum was deredshifted using the redshift from .",
"The flux was normalized to the median flux in a line free region between 1310-1320 Å in the restframe, the wavelength array was then binned by 5 pixels (10 km s$^{ -1}$ at 1400Å), and smoothed by 3 pixels.",
"The set-up of the G130M grating affords continuous wavelength coverage between 1125-1425 Å, in the restframe.",
"Our fully processed data have a median signal-to-noise ratio of 12 per pixel after rebinning."
],
[
"Stellar Continuum Fitting",
"Metal absorption in stellar atmospheres contaminates the ISM absorption profiles.",
"To remove the stellar contribution, we fit the observed spectrum with a linear combination of STARBURST99 simple stellar populations models [51], [53].",
"To match the resolution of the observations, and to avoid stellar libraries with Milky Way contamination, we use the fully theoretical Geneva models, with high-mass loss [64], computed using the WM-basic method [51], [53].",
"A linear combination of different age stellar populations is required to simultaneously recover the Si III photospheric lines near 1295 Å (signatures of B stars) and the Si IV and N V P-Cygni profiles (signatures of O stars).",
"Since O and B stars dominate the FUV stellar continuum, we include STARBURST99 models of ten ages between 1-20 Myr, and use MPFIT [58] to determine the linear combination of these ten models.",
"Five ages contribute to the stellar continuum of NGC 6090 (1, 3, 4, 5, 20 Myr), but the 4 and 5 Myr models account for 84% of the FUV light, for a weighted stellar age of 4.48 Myr (see tab:ngc6090).",
"The stellar continuum fit is shown in the upper panel of Figure 2 in [14].",
"This figure shows that the STARBURST99 models nicely reproduce the stellar P-Cygni features, and provide a continuum level for the spectrum.",
"Both the stellar age and metallicity determine the number of ionizing photons emitted by high-mass stars [51].",
"Accordingly, we fit for both the metallicity and stellar age.",
"STARBURST99 models have five stellar continuum metallicities – 0.05, 0.2, 0.4, 1.0, 2.0 Z$_\\odot $ – and we determine that the stellar metallicity of NGC 6090 is solar using a $\\chi ^2$ test (tab:ngc6090).",
"During the fitting, we include continuum reddening using a Calzetti extinction law [10], and find that the stellar continuum is extincted by E(B-V) = 0.316 mags, after accounting for foreground Milky Way extinction .",
"We cross-correlate the fitted stellar continuum with the observed spectrum to establish the zero-velocity of the stellar continuum.",
"Finally, we remove the stellar continuum by dividing the observed spectrum by the STARBURST99 model.",
"Strong Milky Way absorption and geocoronal emission lines are blueshifted from the metal lines of NGC 6090.",
"However, the Milky Way lines are masked during the stellar population fitting to avoid contaminating the fit.",
"Since the galaxy non-uniformly fills the COS aperture (see fig:image), the spectral resolution is degraded.",
"The COS spectral resolution varies from 20 km s$^{-1}$ for a point source to 200 km s$^{-1}$ for a uniformly filled aperture [27].",
"We fit the line width of the Milky Way absorption features, and find that the effective spectral resolution of the spectrum is 48 km s$^{ -1}$ .",
"fig:profile gives the metal absorption profiles from the fully reduced, stellar continuum normalized spectra.",
"There is a weak C I 1277Å Milky Way absorption line near the N V 1243 Å transition and the stellar continuum is slightly underestimated.",
"While we fit, and remove this weak C I feature, imperfect subtraction creates residual emission, contaminating the continuum near the weaker N V 1243 Å line.",
"Therefore, we do not draw conclusions from the N V profiles."
],
[
"Fine Structure Emission Lines",
"Before we measure the outflow properties from the metal absorption lines, we must understand how resonance emission lines affect the absorption profiles , , , .",
"Continuum photons excite ground state electrons, and the electron then transitions into a lower energy state by emitting a photon with an energy of the difference between the two states.",
"If the electron transitions back to the ground state through the same path as it was excited, it is called resonance emission.",
"Resonant photons have the same restframe wavelength as the absorption and, depending on the geometry of the outflow, the resonance emission can overlap in velocity space with the absorption.",
"This effectively reduces the measured absorption profile.",
"The amount, and the velocity, of the infilling can be estimated if fine structure splits the ground state.",
"The electron's angular momentum splits the ground state into multiple levels, allowing the electron to transition through either resonance or fine structure emission (fine structure emission is normally denoted by a $^\\ast $ , i.e.",
"as Si II$^\\ast $).",
"The probability of transitioning by a specific path is given as $\\mathrm {P}_\\mathrm {i} = \\mathrm {A}_\\mathrm {i} / \\Sigma \\mathrm {A}_\\mathrm {i}$ , where A$_i$ is the Einstein A coefficient corresponding to level i, and the summation is over all possible levels.",
"Therefore, we use the observed fine structure emission to estimate an upper limit on the in-filling of resonance profile.",
"There are four Si II$^\\ast $ lines in the observed wavelength regime at 1194, 1197, 1265, and 1309Å, however the 1265 line is blended with the O I geocoronal emission line.",
"The 1194Å line has the largest probability of fine structure emission (84% of the 1190Å absorption is re-emitted as 1194Å Si II$^\\ast $), therefore this line predicts the maximum amount of resonance emission.",
"The 1194 Si II$^\\ast $ line is the strongest of the four Si II$^\\ast $ lines in the bandpass, but the equivalent width is only $-0.15$ Å, a small fraction of the 1.74 Å Si II 1190 Å equivalent width.",
"Further, the line is redshifted by +132 km s$^{ -1}$ , with emission only extending to -80 km s$^{ -1}$ .",
"Below, we chiefly use the Si IV doublet, which does not have fine structure emission lines in the wavelength regime.",
"The Si II and Si IV profiles have similar velocity profiles [14], allowing for the Si II$^\\ast $ to be used as a proxy for the Si IV emission.",
"To avoid possible contributions from resonance emission, we exclude velocities greater than -80 km s$^{-1}$ when fitting the profiles in beta."
],
[
"Equivalent Width, Optical Depth, Covering Fraction, and Column Density",
"Once we have reduced the data, removed the stellar continuum, and explored the impact of the resonance emission features, we measure the properties of the outflow.",
"Here, we describe the outflow with four parameters: equivalent width (W), optical depth ($\\tau $ ), covering fraction (C$_f$ ), and column density (N).",
"W is measured by integrating $1-F_\\mathrm {o}$ (the observed continuum normalized flux) over the velocity ranges given in column three of tab:obs.",
"As discussed in [14], the maximum velocity of each transition strongly depends on the strength (the W) of the transition, not the ionization state of the transition.",
"The W errors are computed by bootstrapping the observed flux with the flux error array 1000 times, and the standard deviation is calculated from the resultant W distributions.",
"The equivalent width ratios of doublets (and triplets) can be used to diagnose the saturation of the transitions.",
"With an observed doublet ratio of 1.83 (and an expected ratio of 2), the Si IV 1402Å and 1392Å lines are the only doublet that does not suffer from severe saturation effects.",
"We also use the column densities of the weak S II 1250Å, Si II 1304Å, and O I 1302Å lines during the ionization modelling (see ion).",
"C$_f$ is the fraction of the continuum source covered by the foreground absorbing gas (see for a thorough description of the interpretation of C$_f$ ).",
"At low $\\tau $ , C$_f$ and $\\tau $ are degenerate because both impact the depth of the absorption profile.",
"However, this degeneracy can be broken by solving the radiative transfer equation for the C$_f$ and optical depths of the lines [33].",
"Here, we use the Si IV doublet because it does not suffer from strong saturation effects.",
"The covering fraction is calculated in each pixel to give a velocity (v) resolved C$_f$ profile as [33] $C_f(\\mathrm {v}) = \\frac{\\mathrm {F}_\\mathrm {W}(\\mathrm {v})^2-2\\mathrm {F}_\\mathrm {W}(\\mathrm {v}) + 1}{\\mathrm {F}_\\mathrm {S}(\\mathrm {v})-2\\mathrm {F}_\\mathrm {W}(\\mathrm {v})+1}$ where F$_\\mathrm {W}$ is the continuum normalized flux of the weaker doublet line (Si IV 1402Å), and F$_\\mathrm {S}$ is the continuum normalized flux of the stronger doublet line (Si IV 1394).",
"In this equation we use the fact that the Si IV 1402 line has half the oscillator strength of the 1392 line.",
"We set unphysical C$_f$ values that are less than zero to zero.",
"We also calculate the velocity resolved $\\tau $ as $\\tau (\\mathrm {v}) = \\ln \\left(\\frac{C_f(\\mathrm {v})}{C_f(\\mathrm {v})+F_\\mathrm {W}(\\mathrm {v})-1}\\right)$ We bootstrap the errors of C$_f$ and $\\tau $ , similar to the W errors above.",
"Finally, we calculate the integrated column density of each transition using the apparent optical depth method .",
"Since some of the weak lines are singlets, we cannot use eq:cf to calculate the column density, rather we have to assume that the lines are fully covered.",
"Below, we find that this assumption is fair for the line centers, which have most of the column density.",
"The integrated column density is calculated as $\\mathrm {N} = \\frac{3.77 \\times 10^{14}~\\text{cm}^{-2}}{\\lambda [\\text{Å}] f} \\int \\text{ln}\\frac{1}{F_\\mathrm {o}(\\mathrm {v})} \\mathrm {dv}$ where f is the oscillator strength [14], and the total N is calculated by integrating over the velocity interval given in tab:obs.",
"With the measured properties of the outflow, we now study how the Si IV 1402Å optical depth and covering fraction evolve with velocity."
],
[
"PROFILE FITTING",
"In fig:profile the metal absorption lines are broad ($\\sim 175$ km s$^{ -1}$ ) and blueshifted by $\\sim -150$ km s$^{ -1}$ .",
"Under the Sobolev approximation, these profiles are produced by an expanding medium, that has a steep velocity gradient (dv/dr) but a narrow intrinsic line profile, such that the expanding medium broadens the line profile by absorbing and scattering continuum photons , [50], , .",
"The Sobolev optical depth of the Si IV 1402Å profile is given by $\\tau \\left(\\mathrm {v}\\right) = \\frac{\\pi e^2}{\\mathrm {mc}} f \\lambda ~\\mathrm {n}_\\mathrm {4}\\left(\\mathrm {r}\\right) \\frac{\\mathrm {dr}}{\\mathrm {dv}}$ where $f$ is the oscillator strength (0.255 for Si IV), $\\lambda $ is the rest wavelength, and n$_\\mathrm {4}$ is the number density of the Si IV gas at a given radius (r).",
"To simplify the calculation, we introduce normalized quantities by dividing the velocity by the maximum velocity ($w = \\mathrm {v}/\\mathrm {v}_\\infty $ , v$_\\infty = -399$ km s$^{-1}$ for Si IV 1402Å) and the radius by the inner radius ($x = \\mathrm {r}/\\mathrm {R}_\\mathrm {i}$ ), which we measure in ionstruct.",
"v$_\\infty $ is calculated as the velocity at which the absorption reaches 90% of the continuum, similar to the method outlined in [13].",
"We must model how n$_\\mathrm {4}$ is distributed with radius.",
"Previous studies assume that the density is either constant with velocity [60], , or that the density follows the continuity equation , .",
"However, to remain general we assume that the density follows a power-law scaling with radius, such that $\\mathrm {n}_\\mathrm {4}(x) = \\mathrm {n}_\\mathrm {4,0} x^\\alpha $ where n$_\\mathrm {4,0}$ is the Si IV density at the initial radius, R$_\\mathrm {i}$ .",
"Placing this into eq:sobelv gives the relation $\\tau (x) = \\frac{\\pi e^2}{\\mathrm {mc}} f \\lambda _\\mathrm {0} \\frac{\\mathrm {R}_\\mathrm {i}}{\\mathrm {v}_\\infty } \\mathrm {n}_\\mathrm {4,0} x^\\alpha \\frac{dx}{dw} = \\tau _0 x^\\alpha \\frac{\\mathrm {d}x}{\\mathrm {d}w}$ where we have combined the constants into a single constant, $\\tau _0$ , that represents the maximum optical depth.",
"Additionally, we derive the covering fraction of the Si IV absorption (C$_f$ ; see eq:cf).",
"As we discuss in cf, the covering fraction is the proportion, expressed as a percentage, of the starburst area covered by the outflow at a given radius.",
"We assume that the covering fraction scales as a power-law with distance (as physically motivated in cf), such that $C_f(w) = C_{f} (\\mathrm {R}_\\mathrm {i}) \\left(\\frac{\\mathrm {r}}{\\mathrm {R}_\\mathrm {i}}\\right)^\\gamma = C_{f} (\\mathrm {R}_\\mathrm {i}) x^\\gamma $ Now we have related the two observables of the profile to the distance from the starburst.",
"eq:sobelvnorm and eq:cffit are given in terms of distance, while we measure the parameters in terms of velocity.",
"Many of the physical mechanisms for driving outflows scale the velocity with radius as a $\\beta $ velocity law [50], such that $\\mathrm {v} = \\mathrm {v}_\\infty \\left(1-\\frac{\\mathrm {R}_\\mathrm {i}}{\\mathrm {r}}\\right)^{\\beta }$ Substituting for w and x, and inverting this relation gives the normalized radius in terms of the velocity as $x = \\frac{1}{1-w^{1/\\beta }}$ Taking the derivative of this gives the velocity gradient in terms of the normalized velocity and the $\\beta $ exponent as $\\frac{\\mathrm {d}x}{\\mathrm {d}w} = \\frac{w^{1/\\beta -1}}{\\beta \\left(1-w^{1/\\beta }\\right)^2}$ We can also use the $\\beta $ -law to derive the density scaling with velocity as $\\mathrm {n}(w) = \\mathrm {n}_\\mathrm {4,0} \\left(\\frac{1}{(1-w^{1/\\beta })}\\right)^{\\alpha }$ producing an $\\tau $ (eq:sobelvnorm) and C$_f$ (eq:cffit) velocity scaling as $\\begin{aligned}\\tau \\left(w\\right) &= \\tau _\\mathrm {0} \\frac{w^{1/\\beta -1}}{\\beta (1-w^{1/\\beta })^{2+\\alpha }}\\\\C_f(w) &= \\frac{C_f (\\mathrm {R}_\\mathrm {i})}{\\left(1-w^{1/\\beta }\\right)^\\gamma }\\end{aligned}$ We then have five parameters to fit for: the maximum optical depth ($\\tau _0$ ), the covering fraction at the initial radius (C$_{f}(\\mathrm {R}_\\mathrm {i})$ ), the exponent of the beta velocity-law ($\\beta $ ), the exponent of the density law ($\\alpha $ ), and the exponent of the covering fraction law ($\\gamma $ ).",
"Using MPFIT [58], we simultaneously fit for the five parameters from the unsaturated Si IV 1402 Å $\\tau $ and C$_f$ distributions (see fig:beta).",
"At w below 0.2 ($v > -80$ km s$^{ -1}$ ) zero-velocity absorption and resonance emission may effect the distributions (see finestruct); therefore, we only fit the distributions between w of 0.2 and 1.0 (the points outside this range are coloured gray in fig:beta).",
"The C$_f$ and $\\tau $ fits are shown in fig:beta, and the fit parameters are given in tab:beta.",
"In these fits we have binned the flux array by a factor of two (20 km s$^{ -1}$ ) to increase the signal-to-noise ratio, while still Nyquist sampling the observed velocity resolution (see the bar in fig:beta).",
"These fits reproduce the observed $\\tau $ and C$_f$ distributions, and describe the radial velocity and density laws of the outflow through eq:beta and eq:continuity.",
"In discussion we discuss some implications of these trends.",
"To test the covering fraction fits, we can use the Si III and C II profiles.",
"Since these profiles are much stronger, they are strongly saturated at all velocities.",
"This means that we cannot measure the C$_f$ with the previous formula (eq:cf), while $\\tau $ cannot be accurately measured at all.",
"Rather, at large $\\tau $ , the depth of the line is completely set by C$_f$ , and is measured as C$_f(\\mathrm {v}) = 1 - F(\\mathrm {v})$ .",
"In the lower right panel of fig:beta we plot the C$_f$ for Si III (circles) and C II (triangles).",
"The Si IV C$_f$ fit is over-plotted on these measurements and provides reasonable agreement with the data.",
"Table: Table of the quantities derived from the profile fits of the optical depth (τ\\tau ) and covering fraction (C f _f; see eq:cfbeta).",
"The individual parameters are: Column 1, τ 0 \\tau _0, optical depth at line center; Column 2, β\\beta , the velocity power-law index; Column 3, α\\alpha , the column density power-law index; Column 4, C f (R i )_f (\\mathrm {R}_\\mathrm {i}), the maximum covering fraction; Column 5, γ\\gamma , the covering fraction power-law index."
],
[
"IONIZATION MODELLING",
"In measure we calculate the integrated column density (N) for individual metal ions, but these only describe the individual ion and not the total mass of the outflow.",
"While Ly-$\\alpha $ absorption traces the neutral Hydrogen in the outflow, we opt to use the metal lines to describe the outflow because (1) the strong P-Cygni Ly-$\\alpha $ profile requires detailed radiative transfer models to constrain and (2) up to 99% of the Hydrogen in the outflow may be ionized [14].",
"To calculate the total Hydrogen in the outflow, ionization models are required to measure the ionization fraction ($\\chi _\\mathrm {i}$ , or the fraction of the total gas within transition i) and the abundance (N$_\\mathrm {i}$ /N$_\\mathrm {H}$ , or the ratio of an element to the total Hydrogen).",
"In [14] we find that photoionization models describe the ionization structure of galactic outflows.",
"These models require the outflow metallicity (Z$_\\mathrm {o}$ ) to be greater than 0.5 Z$_\\odot $ and the ionization parameter (log(U); the ratio of the photon density to outflow density) to be between -1.5 and -2.25.",
"Additionally, the ionization structure depends both on the spectral energy distribution and the strength of the ionizing source, as well as the density (n$_\\mathrm {0}$ ) of the outflow.",
"We model the ionization structure using CLOUDY version 13.03 [25].",
"The CLOUDY models are not velocity resolved profiles, rather we are fitting the measured integrated column densities to the integrated values from CLOUDY.",
"We assume that the outflow is ionized by the observed stellar continuum, using the best-fit STRABURST99 stellar continuum model from cont.",
"These STARBURST99 models have an age of 4.478 Myr and a constant SFR of 25.15 M$_\\odot $ yr$^{-1}$ (see tab:ngc6090).",
"In the CLOUDY models, we assume an expanding spherical geometry, which is not meant to reproduce the velocity profiles, but to account for back-scattering of radiation.",
"We set the covering fraction to 1.00, the observed value from the Si IV transition, and scale the density with a power-law of -5.72, as measured above.",
"The CLOUDY models are stopped once the simulations reach 3000 K, which is lower than the default criteria to allow for higher metallicities.",
"These lower temperatures require a cosmic ray background to be included [42].",
"We use CLOUDY's default H II abundances, which are similar to the Milky Way values [5], , , , and include an Orion Nebular dust grain distribution [5].",
"The dust grains account for scattering and destruction of photons, as well as depletion of metals onto grains.",
"The abundances are scaled by a constant factor to change the outflow metallicity (Z$_\\mathrm {o}$ ).",
"Similarly, we vary the outflow density at the inner radius (n$_\\mathrm {0}$ ), ionization parameter (U), and the stellar continuum metallicity (Z$_\\mathrm {s}$ ).",
"We use a Bayesian approach to estimate Z$_\\mathrm {o}$ , n$_\\mathrm {0}$ , U, and Z$_\\mathrm {s}$ [44], [7], [24].",
"While [14] uses the W ratios to define the ionization structure, the relatively high signal-to-noise ratio and spectral resolution observations of NGC 6090 allow us to use the measured column densities from four weak transitions: O I 1302Å, Si II 1304Å, S II 1250Å, and Si IV 1402Å (see tab:obs).",
"We do not use the N V column densities due to Milky Way C I1277Å and imperfect continuum subtraction contamination near the N V line (see cont).",
"We create grids of CLOUDY models, with differing input parameters (tab:gridmod), and tabulate the predicted CLOUDY column densities.",
"We assume a uniform prior – each model is equally likely – and compute the likelihood function of each set of parameters as $\\mathrm {L} \\propto \\exp (-\\chi ^2/2).$ where $\\chi ^2$ is the chi-squared function using the observed column density (N), the errors on the observed N, and the predicted N from CLOUDY.",
"For each parameter (Z$_\\mathrm {o}$ , n$_\\mathrm {0}$ , U, and Z$_\\mathrm {s}$ ), the likelihood function is marginalized over the other nuisance parameters and normalized to one to create probability distribution functions (PDFs).",
"We calculate expectation values and standard deviations of the individual parameters from these PDFs as estimates of the parameters, and their uncertainties.",
"Figure: Probability density functions (PDFs) from the CLOUDY ionization modelling.",
"The three parameters are outflow metallicity (top panel), ionization parameter (middle panel), and the total Hydrogen density at the initial radius (bottom panel).",
"The PDFs are single peaked, with expectation values and standard deviations given in tab:ion.We do the fitting in two iterations.",
"The first iteration uses a course grid to determine the best Z$_s$ .",
"Since there are only five stellar continuum metallicities in the fully theoretical STARBURST99 models, we only coarsely estimate the stellar continuum metallicity.",
"Similar to the COS spectrum fitting (see cont), the Bayesian analysis finds that the 1 Z$_\\odot $ model best fits the ionization structure, assigning 100% of the probability to the 1 Z$_\\odot $ stellar continuum model.",
"Table: Grid of CLOUDY models used in the Bayesian analysis of the ionization structure.The second iteration uses a more finely spaced grid for Z$_o$ , log(U) and n$_0$ , while only using the solar metallicity stellar continuum model (see tab:gridmod).",
"In fig:ionpdfs we show the PDFs for the three parameters, with narrow distributions that have expectation values of $Z_\\mathrm {o} = 1.61\\pm 0.08$ Z$_\\odot $ , $n_\\mathrm {0} = 18.73\\pm 2.37$ cm$^{-3}$ , and log(U)$ =-1.85\\pm 0.02$ (see tab:ion).",
"Using these parameters, we create a best-fit CLOUDY model, producing the ionization fractions of the outflow (see tab:ion).",
"As discussed in ionstruct, these ionization models have important implications for the metallicity of the outflow, the inner radius of the outflow, and the mass outflow rate.",
"Figure: The velocity resolved column density plots for the Si IV (black) and O I (red) lines, normalized to their maximum values.",
"The velocity structure between the neutral (O I) and highly ionized gas (Si IV) stays roughly constant over the velocity range, implying that the ionization structure does not vary substantially with velocity.Table: Table of the quantities derived from the CLOUDY photoionization modelling.",
"Row (1) is the ionization parameter; (2) is the derived outflow metallicity; (3) is the stellar continuum metallicity; (4) is the Hydrogen density at the base of the outflow; (5) is the ionization fraction of Si IV, or the per cent of the total Si in the Si IV transition; (6) The best-fit Si to H abundance value; (7) The measured Si IV column density; (8) The total Hydrogen column density calculated from the Si IV column density, Si/H abundance, and the Si IV ionization fraction; (9) The inner radius, calculated using eq:r; (10) the H II temperature of the best-fit CLOUDY model.",
"The errors on χ Si IV \\chi _\\mathrm {Si {\\sc IV}} and log(Si/H) are calculated by producing CLOUDY models of the estimate plus/minus 1σ\\sigma errors of the Z o _\\mathrm {o}, n 0 _0, and log(U)."
],
[
"Ionization Structure With Velocity",
"In the following analysis we use an integrated ionization correction.",
"However, if the outflow is heated as it is accelerated, then the column density ratio of high to low ionization potential lines should increase with velocity.",
"fig:colfrac shows the velocity resolved column densities for the Si IV and O I transitions, showing that there is not a coherent variation of the column densities with velocity between $-400$ and 0 km s$^{ -1}$ .",
"This implies that a majority of the column density arises from a small radius, and the photoionization models are approximately plane-parallel."
],
[
"DISCUSSION",
"Here we discuss various aspects of the profile and ionization fitting from beta and ion.",
"In ionstruct, we first derive important parameters of the outflow.",
"We then study the implications for the covering fraction (cf), velocity (velocity), and density (density) radial scalings.",
"Finally, we combine the various relations to determine the mass outflow rate of NGC 6090 (mout)."
],
[
"Ionization Modelling",
"With the ionization fractions and metallicities derived in ion, we convert the measured Si IV column densities into a total Hydrogen density (see tab:ion).",
"The total (neutral plus ionized) Hydrogen column density is 20.89$_{-0.03}^{+0.04}$ cm$^{-2}$ , where the uncertainties for each parameter are propagated to compute the uncertainties of the total Hydrogen column density.",
"The UV continuum extinction provides a complimentary way to estimate the total Hydrogen column density by relating the extinction to the amount of dust through a dust-to-gas ratio.",
"We use the relation from [52] and [37] to calculate the total Hydrogen column density as $\\mathrm {N}_\\mathrm {H} = \\frac{3.6 \\times 10^{21} \\mathrm {E(B-V)}}{Z} \\text{cm}^{-2}$ where we use a starburst dust attenuation law and the relation for the FUV optical depth from [29] to convert the spectral slope to E(B-V).",
"Using the extinction measured from the UV continuum in [14] and the outflow metallicity from the ionization structure (see tab:ngc6090 and tab:ion), we expect a log(N$_\\mathrm {H}$ ) of 20.85, in agreement with the derived total N$_\\mathrm {H}$ from the ionization modelling.",
"The ionization model predicts that 99.3% of the Hydrogen in the outflow is ionized.",
"These low neutral fractions suggest that the outflow has a H I column density of $5 \\times 10^{18}$ cm$^{-2}$ .",
"Current radio arrays cannot detect this H I column in emission, but the upcoming SKA will detect these densities with 100 spatial resolution and 5 km s$^{-1}$ spectral resolution in 10-100 hours of integration time .",
"The outflow metallicity is 1.61 Z$_\\odot $ , 61% and 34% larger than the measured stellar and ISM metallicities (see tab:ngc6090).",
"Substantial uncertainties on log(O/H) measurements [45] means that the outflow metallicity is at least consistent with the ISM values, and likely enriched compared to the ISM of the galaxy.",
"Additional metals may reside in a hotter phase not probed by these observations (see below for a discussion of this phase).",
"Metal enriched outflows are important constraints for models using galactic outflows to explain the mass-metallicity relation , [26], , [17], [15].",
"For example, to match the mass-metallicity relation of [18] for galaxies with log($M_\\ast $ /M$_\\odot $ ) of 10.7, Figure 8 of suggests that the outflow metallicity is 1.58 Z$_\\odot $ .",
"The same figure also uses Z$_\\mathrm {o}$ to constrain the scaling of the mass outflow rate divided by the star formation rate (mass-loading factor).",
"In future studies, we will determine the scaling of the outflow metallicity with stellar mass and star formation rate to better constrain these studies.",
"The initial radius (R$_\\mathrm {i}$ ) can be determined from the fitted Si IV optical depth (see eq:sobelvnorm).",
"Solving for R$_\\mathrm {i}$ in terms of the observed maximum Si IV optical depth, we find that $\\mathrm {R}_\\mathrm {i} = \\frac{\\mathrm {mc}}{\\pi e^2} \\frac{1}{f \\lambda } \\frac{\\mathrm {v}_\\infty \\tau _\\mathrm {0}}{\\mathrm {n}_\\mathrm {0} \\chi _\\mathrm {Si~IV} \\mathrm {Si/H}} = 63.4~\\text{pc}$ Using the values from the $\\beta $ -profile fitting (tab:beta) and the ionization modelling (tab:ion).",
"We have related the initial Si IV density ($n_{4,0}$ ) to the Hydrogen density at the base of the outflow, the abundance, and the ionization corrections from the ionization models ($n_{4,0} = n_\\mathrm {0} \\chi _\\mathrm {Si IV} \\mathrm {Si/H}$ ).",
"The inner outflow radius and the size of the mass-loading region (maximum extent of about 150 pc; see mout and fig:mout below) is consistent with models of the mass-loading region in M 82 , , and with analytical work from [38], who assume that the inner outflow radius is twice the size of the starburst.",
"However, what is the physical meaning of the inner outflow radius?",
"The shredding of supernovae blastwaves by hydrodynamical instabilities is a possible origin for this inner radius.",
"Supernovae energy builds up in the ISM due to multiple impulsive events , which creates a blastwave that shocks when it encounters the ambient ISM , , , [63], , [46].",
"The shock compresses surrounding ISM into a thin, dense shell of relatively cool gas that travels outward.",
"This phase is called the snowplow phase, and the radius depends on the amount of injected energy, the density of the medium, and how much of the energy is radiated away , [63], [16], , [46], with typical values near 20-30 pc [20], [46].",
"Once the shell forms, Rayleigh-Taylor and Kelvin-Helmholtz instabilities rapidly destroy it, creating many small warm (10$^4$ K) cloudlets [57], [16], [28], [60], , .",
"The destruction of the blastwave allows the hot interior gas to escape and travel into the halo as a hot wind.",
"The hot, high-velocity wind, may then accelerate these clouds through ram pressure to the observed velocities [28], [60], .",
"Therefore, a possible origin for R$_\\mathrm {i}$ is that hydrodynamic instabilities have shredded the dense blastwave, and injected cloudlets into the hot wind.",
"Finally, the CLOUDY modelling estimates the temperature of the H II in the outflow to be 5470 K (see tab:ion), leading to a pressure of P/k$_\\mathrm {b}$ = $1 \\times 10^5$ K cm$^{-3}$ .",
"The hot wind is typically assumed to have a temperature of 10$^7$ K [12].",
"If the observed cloudlets and hot wind are in pressure equilibrium, then the hot wind has a density of 0.01 cm$^{-3}$ at R$_\\mathrm {i}$ , a factor of 10 lower than the density in the [12] model.",
"However, recent models by [9] show that the density and temperature of this hot wind can greatly vary, depending on the efficiency and mass-loading of the outflow.",
"Regardless of the origin of the outflow, it must be evenly distributed across the stellar continuum to produce a unity covering fraction (see tab:beta).",
"However, the external pressure on the outflow decreases as it accelerates out from the starburst.",
"The change in pressure has a dramatic effect on the size of the cloudlets in the outflow, and in turn their covering fraction.",
"In the next section we discuss this pressure change, and how it naturally leads to the observed scaling of the covering fraction with velocity."
],
[
"Covering Fraction",
"In beta, we observe that the outflow initially completely covers the background stellar continuum, but as the velocity increases the outflow covers less of the background stars (C$_f$ drops as $x^{-0.8}$ ).",
"To physically understand this scaling relation, we hypothesize that the absorption arises from cloudlets of gas initially a distance R$_\\mathrm {i}$ from the starburst (see the upper left panel of fig:cfnop).",
"At R$_\\mathrm {i}$ these clouds are large enough to completely occupy the volume along the line-of-sight to the stars: none of the stellar continuum is transmitted (see lower left panel in fig:cfnop for the face on view at the initial time, t$_0$ ).",
"However, the starburst imparts energy and momentum to these clouds, accelerating them radially outwards (upper right panel of fig:cfnop).",
"In the simplest scenario, these clouds retain their size as they move outward, and a gap appears between the clouds (see upper right panel in fig:cfnop).",
"The gap allows the background stellar continuum to be transmitted, reducing the covering fraction of the stellar continuum.",
"This is illustrated by the face on view in the lower right panel of fig:cfnop, where background stars become visible in the gaps between the clouds at higher velocities.",
"This physical picture can be expressed numerically as a ratio of the cloud area to the total surface area at a given radius (r) as [60] $\\frac{C_f(\\mathrm {r})}{C_f(\\mathrm {R}_\\mathrm {i})} = \\frac{\\mathrm {A}_\\mathrm {c} (\\mathrm {r})}{4 \\pi \\mathrm {r}^2} \\frac{4 \\pi \\mathrm {R}_\\mathrm {i}^2}{\\mathrm {A}_\\mathrm {c}(\\mathrm {R}_\\mathrm {i})} = \\frac{\\mathrm {A}_\\mathrm {c} (\\mathrm {r})}{\\mathrm {A}_\\mathrm {c} (\\mathrm {R}_\\mathrm {i})} x^{-2}$ Where A$_c$ is the area of the individual cloudlets and $x = \\mathrm {r}/\\mathrm {R}_\\mathrm {i}$ .",
"Assuming that the area of the clouds remains constant, the C$_f$ in this scenario scales as x$^{-2}$ , significantly steeper than the observed relation of x$^{-0.8}$ .",
"Figure: Similar to fig:cfnop, but the clouds are in an external pressure gradient (orange shaded region).",
"The initial pressure (P e ,0_\\mathrm {e,0}) is greater than the pressure at t 1 _1 (P e ,1_\\mathrm {e,1}).",
"The cloudlets adiabatically expand to remain in pressure equilibrium with the external medium (see the arrow in the upper left cloud demonstrating it's initial diameter).",
"Comparing the bottom right panel with the same panel in fig:cfnop, the expanded clouds now cover more area of the starburst at higher velocities.",
"This slows the drop in C f _f.",
"The scaling of the C f _f with distance depends on how the external medium changes, either isothermally (see eq:cftheoryiso) or adiabatically (see eq:cftheoryadia).However, this simple model of static clouds is not the most physical.",
"These cloudlets are likely in pressure equilibrium with an external pressure (P$_\\mathrm {e} = \\mathrm {P}_\\mathrm {c}$ , where P$_\\mathrm {e}$ is the external pressure and P$_\\mathrm {c}$ is the pressure of the cloud), possibly a hot wind (see fig:cfp and the discussion in ionstruct).",
"Changes in P$_\\mathrm {e}$ change P$_\\mathrm {c}$ , and the clouds expand adiabatically to account for these changes as P$_\\mathrm {c} \\propto V_\\mathrm {c}^{-\\gamma _\\mathrm {c}}$ , where V$_\\mathrm {c}$ is the volume of the clouds and $\\gamma _\\mathrm {c}$ is the adiabatic index (5/3 for monatomic ideal gas).",
"The larger cloud volume reduces the gap between the individual cloudlets [60].",
"The lower right panels of fig:cfnop and fig:cfp illustrate that the larger clouds cover more of the background stars than the static clouds do, and C$_f$ falls more slowly with distance.",
"Approximating the outflow as spherical clouds provides a relation for A$_\\mathrm {c}$ in terms of the volume of the cloudlets as A$_\\mathrm {c} \\propto \\mathrm {V}_\\mathrm {c}^{2/3}$ .",
"Using this approximation, eq:cfchanges, and the observed C$_f$ (C$_f \\propto x^{-0.8}$ ), the cloud volume changes with distance from the starburst as $\\frac{\\mathrm {V}_\\mathrm {c}(\\mathrm {r})}{\\mathrm {V}_\\mathrm {c}(\\mathrm {R}_\\mathrm {i})} = \\left(\\frac{\\mathrm {A}_\\mathrm {c}(\\mathrm {r})}{\\mathrm {A}_\\mathrm {c}(\\mathrm {R}_\\mathrm {i})}\\right)^{3/2} = \\left(\\frac{C_f(\\mathrm {r})}{C_f(\\mathrm {R}_\\mathrm {i})}x^2\\right)^{3/2} = x^{1.8 \\pm 0.4}$ This demonstrates that the cloud's volume increases with distance from the starburst, but what differential pressure is needed to change the volume as we observe?",
"Here, we consider two different pressure laws for a mass-conserving external medium: isothermal expansion and adiabatic expansion.",
"If the external pressure changes isothermally with radius then P$_\\mathrm {e}(\\mathrm {r})/\\mathrm {P}_\\mathrm {e}(\\mathrm {R}_\\mathrm {i}) = x^{-2}$ .",
"Assuming that the spherical clouds and the isothermal external medium remain in pressure equilibrium (P$_\\mathrm {c} = \\mathrm {P}_\\mathrm {e}$ ), the pressure of the outflowing clouds changes as $\\frac{\\mathrm {P}_\\mathrm {c}(\\mathrm {r})}{\\mathrm {P}_\\mathrm {c}{\\mathrm {R}_\\mathrm {i}}} = \\left(\\frac{\\mathrm {V}_\\mathrm {c}(\\mathrm {r})}{\\mathrm {V}_\\mathrm {c}(\\mathrm {R}_\\mathrm {i})}\\right)^{-\\gamma _\\mathrm {c}} = \\left(\\frac{\\mathrm {A}_\\mathrm {c}(\\mathrm {r})}{\\mathrm {A}_\\mathrm {c}(\\mathrm {R}_\\mathrm {i})}\\right)^{-3\\gamma _\\mathrm {c}/2} = x^{-2}$ Solving for $\\frac{A_\\mathrm {c}(x)}{A_\\mathrm {c}(R_i)}$ in terms of x finds that A$_\\mathrm {c}$ changes as $x^{4/(3\\gamma _\\mathrm {c})}$ .",
"Placing this into eq:cfchanges gives the C$_f$ scaling of $\\frac{C_f(\\mathrm {r})}{C_f(\\mathrm {R}_\\mathrm {i})} = x^{4/(3\\gamma _\\mathrm {c})} x^{-2} = x^{-1.2}$ For a $\\gamma _\\mathrm {c} = 5/3$ .",
"If the clouds expand adiabatically while immersed in an isothermal external medium, the covering fraction declines less steeply with height than if the clouds maintain their initial size.",
"eq:cftheoryiso is still too steep to match the observed scaling.",
"Another possibility is that the external pressure changes adiabatically.",
"In this situation P$_\\mathrm {e} \\propto x^{-2\\gamma _\\mathrm {e}}$ , where $\\gamma _\\mathrm {e}$ is the adiabatic index of the external medium.",
"Following a similar process as above, the covering fraction of an adiabatically expanding outflow in pressure equilibrium with an adiabatically expanding external medium is [60] $\\frac{C_f(\\mathrm {r})}{C_f(\\mathrm {R}_\\mathrm {i})} = x^{4\\gamma _\\mathrm {e}/(3\\gamma _\\mathrm {c}) -2} = x^{-2/3}$ where we have assumed that $\\gamma _\\mathrm {e}$ and $\\gamma _\\mathrm {c}$ are equal to each other (i.e.",
"both the clouds and external medium are monatomic ideal gases).",
"Interestingly, since diatomic gas has more degrees of freedom than monatomic gas, diatomic gas requires a larger energy change to remain in pressure equilibrium, as seen by the smaller $\\gamma _\\mathrm {c}$ of diatomic gas (7/5).",
"Using eq:cftheoryadia, we find that molecular clouds expand more rapidly, and consequently have a more gradually declining C$_f$ with distance, as x$^{-0.4}$ .",
"This has an important consequence for the survival of molecular clouds in a hot medium.",
"By increasing the radius of the cloud, it takes longer for a shock wave to propagate across the cloud and dissociate the molecular gas [48], .",
"This may extend the lifetime of molecular gas entrained in galactic outflows, while the increased C$_f$ may explain the presence of molecular gas at high velocities [62], , [55].",
"In beta, we find that the covering fraction scales as $C_f~=~1.0~x^{-0.8 \\pm 0.2}$ , in agreement with the adiabatic expansion of cloudlets in an adiabatically expanding external medium (eq:cftheoryadia).",
"However, it is an interesting theoretical question whether clouds can remain in pressure equilibrium with an external medium and still be accelerated.",
"Ram pressure introduces a secondary pressure term, but if the clouds are shielded from direct interaction with the hot medium then the pressure of the cloud is set by the external thermal pressure [12].",
"Further, ablation and ram pressure stripping create a fractal distribution into smaller cloudlets [48], , which may produce the observed change in covering fraction.",
"The velocity scaling of the covering fraction provides further constraints for simulations of the acceleration of clouds.",
"The acceleration of the cloudlets out of the star forming region decreases C$_f$ , but how are the clouds accelerated?"
],
[
"Velocity Law",
"The velocity law describes the acceleration of the outflowing clouds with distance.",
"In beta we find a $\\beta $ velocity profile of $\\mathrm {v(r)}~=~\\mathrm {v}_\\infty ~(1~-~\\frac{\\mathrm {R}_\\mathrm {i}}{\\mathrm {r}})^{0.43 \\pm 0.07}$ .",
"The outflow initially accelerates rapidly, but the acceleration moderates at large radii (see the black line in fig:vellaw).",
"Using the initial radius calculated in ionstruct, the outflow reaches 50% (90%) of the maximum velocity in 79 pc (291 pc).",
"Acceleration much beyond this is not well constrained by the data.",
"Outflows typically have a \"saw-tooth\" line profile , where the red side of the profile sharply declines and the blue side rises gradually.",
"The velocity and C$_f$ laws produce these profiles.",
"Initially, the velocity sharply increases over a short distance, keeping the C$_f$ near unity because C$_f$ scales as x$^{-0.82}$ .",
"Meanwhile, at higher velocities the clouds travel larger distances per velocity interval, forcing the high-velocity clouds to expand to remain in pressure equilibrium with the adiabatically expanding external medium.",
"This moderates the decline in C$_f$ , and produces the gentler rising blue portion of the profile.",
"The observed Si IV 1402Å outflows do not escape the galactic potential.",
"Following [36], the escape velocity is no more than three times the circular velocity [13], therefore the outflow velocity needs to exceed 651 km s$^{-1}$ for the clouds to escape the potential.",
"There are two possible explanations for why we do not observe gas escaping the potential: (1) the outflow does eventually exceed the escape velocity but the Si IV density drops below the detection limits at high velocities and we do not observe the escaping gas (see density) or (2) the outflow does not actually escape the galaxy, but rather recycles back into the disc as a galactic fountain .",
"The latter is consistent with [13], which finds most galaxies with log($M_\\ast $ /M$_\\odot $ ) greater than 10.5 cannot drive outflows faster than their escape velocity, unless they are undergoing a merger.",
"Observations probing lower density gas, such as Lyman-$\\alpha $ , may constrain whether these outflows are capable of escaping the gravitational potential.",
"Now we study how these outflows are accelerated.",
"There are numerous theoretical ways to accelerate galactic outflows: radiation pressure on dust grains , , cosmic rays [23], , and ram pressure of a hot wind on the cloudlets , [16], [28], [60].",
"Below, we use analytical expressions for how the velocity profile evolves radially to explore which mechanisms could accelerate these outflows.",
"In each case, we give the scaling of the normalized velocity (w = v/v$_\\infty $ ; where v$_\\infty $ is the maximum velocity) with the normalized radius (x = r/R$_\\mathrm {i}$ ; where R$_\\mathrm {i}$ is the initial radius).",
"We then scale the analytical expressions to the observed relations to determine the plausibility of each mechanism.",
"In velsum we summarize the implications for the theoretical profiles."
],
[
"Optically Thick Radiation Pressure",
"Radiation pressure is an attractive way to drive outflows in dusty, vigorously star-forming galaxies: the high luminosity provides a large momentum source , while the large dust optical depth scatters photons multiple times .",
"give the radial scaling of the velocity for optically thick radiation pressure as: $w = \\sqrt{\\frac{4\\sigma ^2}{\\mathrm {v}_\\infty ^2} (\\frac{L}{L_\\mathrm {E}}-1)~\\text{ln}(x)}$ where $\\sigma $ is the velocity dispersion of the galaxy, L is the luminosity of the galaxy, and L$_\\mathrm {E}$ is the Eddington luminosity.",
"We fit for the constant value that best matches the observed $\\beta $ -law using MPFIT [58], while excluding velocities less than 0.2.",
"The fit is shown by the blue dot-dashed line in fig:vellaw.",
"The optically thick radiation model poorly matches the observations."
],
[
"An r$^{-2}$ Force",
"A second appealing driving mechanism is ram pressure.",
"Supernovae thermalize ambient gas into a hot wind, which expands adiabatically out of the star forming region.",
"The hot wind imparts a ram pressure force on the clouds, which depends on the speed and density of the hot wind and the area of the cloud as F$_\\mathrm {ram} = \\rho _\\mathrm {h} \\mathrm {v}_\\mathrm {h}^2 \\mathrm {A}_\\mathrm {c}$ [48], .",
"F$_\\mathrm {ram} \\propto x^{-2}$ in a simplified case of a mass conserving, adiabatically expanding hot wind.",
"Ram pressure driving is appealing because the C$_f$ scaling is consistent with clouds being in pressure equilibrium with a adiabatically expanding external medium (cf), a situation that could lead to ram pressure driving.",
"Including the effects of gravity, calculate the velocity profile of a ram pressure driven outflow as $w(x) = \\sqrt{\\frac{v_\\mathrm {c}^2}{\\mathrm {v}_\\infty ^2} (1-\\frac{1}{x}) -\\frac{4\\sigma ^2}{\\mathrm {v}_\\infty ^2} \\mathrm {ln}(x)}$ Where v$_c$ , the characteristic velocity clouds reach before gravity dominates, is defined as v$_\\mathrm {c} = \\frac{3\\dot{M}_\\mathrm {h} \\mathrm {V}_\\mathrm {h}}{8\\pi \\rho _\\mathrm {c} \\mathrm {R}_\\mathrm {c} \\mathrm {R}_\\mathrm {i}}$ .",
"Defining A as v$_\\mathrm {c}^2/\\mathrm {v}_\\infty ^2$ and B as $4\\sigma ^2/\\mathrm {v}_\\infty ^2$ , we fit for the values of A (fitted value of 1.10) and B (fitted value of 0.03) that match the observed $\\beta $ velocity law, as shown by the red line in fig:vellaw.",
"Similarly, shockwaves from supernovae accelerate cosmic rays (CR).",
"These relativistic particles stream out of the star forming regions along the magnetic fields and exchange momentum with magnetized plasma.",
"Cosmic rays have various advantages over radiation pressure and ram pressure, including: CRs interact multiple times with the gas, magnetic fields confine CRs to the galaxy making it difficult for CRs to escape without imparting momentum, and CR feedback is independent of the distribution of ISM gas [23], .",
"derive the force imparted by cosmic rays as $F_\\mathrm {CR} = \\frac{\\kappa _\\mathrm {CR}}{c} \\frac{L_\\mathrm {CR}}{4 \\pi r^2}$ where $\\kappa _\\mathrm {CR}$ is the cosmic ray opacity, and L$_\\mathrm {CR}$ is the cosmic ray luminosity.",
"This force produces a similar velocity scaling as the ram pressure scaling in eq:ramvel.",
"Additionally, the scaling of the ram pressure and CR velocity profiles are similar to an optically thin radiation profile .",
"In fact, eq:ramvel is a general form for any r$^{-2}$ force that opposes gravity.",
"Therefore, the red dashed line in fig:vellaw corresponds to outflows driven by any r$^{-2}$ force (ram pressure, cosmic rays, or optically thin radiation pressure), and we cannot distinguish these mechanisms from the measured velocity profile."
],
[
"Velocity Summary",
"The ram pressure, cosmic rays, and optically thin radiation pressure velocity profile (any r$^{-2}$ force) matches the observed velocity relation for all observed velocities, while optically thick radiation pressure poorly matches the observed velocity profile (fig:vellaw).",
"In fig:vellaw, the r$^{-2}$ profile begins to decelerate at a radius of 2.5 kpc (r$_\\text{turn}$ ; marked by the vertical line in fig:vellaw).",
"However, at these large radii the outflow is now diffuse, and the deceleration of the outflow is challenging to study with the Si IV line profile."
],
[
"Density Law",
"In beta we find the density to scale with the normalized radius as $\\mathrm {n}(x) \\propto x^{-5.72}$ .",
"In fig:denlaw we show the measured relation for the outflow density with velocity.",
"At low velocities, the density remains nearly constant because the outflow accelerates over a short distance , but at higher velocities the density precipitously drops.",
"Stronger transitions probe lower densities than weaker transitions, causing stronger transitions to probe higher velocities than the weaker transitions.",
"Therefore, stronger transitions have larger measured terminal velocities and larger line widths (because they probe a wider velocity distribution) than the weaker transitions [32], [14].",
"This density-velocity scaling implies that the outflow traced by Si IV does not conserve mass.",
"Fitting the $\\tau $ distribution with a mass-conserving flow produces a poor fit because the $\\tau $ distribution declines sharply at high velocity.",
"In the continuity equation, the mass flux is conserved, such that the density scales as $\\mathrm {n}(w) \\propto \\frac{n_\\mathrm {0}}{x^2v} \\propto \\frac{n_\\mathrm {0}}{v_\\infty }\\frac{(1-w^{1/\\beta })^2}{w}$ In fig:denlaw we compare the mass-conserving density profile (red dashed line) with the observed profile.",
"Compared to the observed density profile, the density of a mass-conserving flow rapidly decreases at low velocities as the outflow accelerates over small distances, while at higher velocities the conserved density decreases more gradually because the velocity gradient is flatter (see fig:vellaw).",
"If the Si IV flow is not mass-conserving, what happens to the Si IV gas?",
"The density could decrease by: (1) increasing the volume of the outflowing clouds while keeping the number of Si IV ions constant or (2) decreasing the number of Si IV ions in the clouds.",
"eq:volume shows that the volume of the outflowing clouds scales as x$^{1.77}$ , implying that the number of Si IV ions in the outflow must decrease as x$^{-3.95}$ to satisfy the observed density relation.",
"One possibility is that the outflowing clouds lose mass through ablation, ram pressure stripping, or conduction from the hot wind [48], , , [8].",
"In this scenario, the hot wind rapidly destroys the clouds and incorporates them into the hot wind, making the mass undetectable through Si IV absorption at higher velocities.",
"A steep density scaling relation is also seen in the nearby starburst M 82.",
"[55] probe the surface density profiles of the molecular, neutral and ionized phases of the outflow from M 82.",
"They find that only the H I follows an n$ \\propto \\mathrm {r}^{-2}$ density law largely because of the large amount of tidal material present at large radii.",
"However, other surface density profiles along the minor axis decrease more rapidly.",
"The 70 $\\mu $ m emission, a tracer of warm dust, and CO emission, a tracer of diffuse molecular gas, scale with the distance from the starburst as n$ \\propto \\mathrm {r}^{-4}$ [55], roughly consistent, with the density scaling found here.",
"The authors use the divergence of the density profile to argue that the outflow of M82 is a galactic fountain, with gas leaving the outflow and recycling back into the disk."
],
[
"Mass Outflow Rate",
"Finally, we combine all of the derived relations to measure the mass outflow rate ($\\dot{M}_o$ = dM$_o$ /dt) of the photoionized galacitc outflow.",
"The photoionized outflow is a single component of the outflow, and other, unexplored, phases likely contribute to the total mass outflow rate.",
"Since the hot phase is such low density, this photoionized phases likely dominates the mass budget of the outflow.",
"Typically the mass outflow rate is calculated assuming the outflow is in a thin spherical shell as $\\dot{M}_\\mathrm {o} = \\Omega {C}_{f} \\mu \\mathrm {m}_\\mathrm {p} \\mathrm {N}_\\mathrm {H} \\mathrm {r} \\mathrm {v}_\\mathrm {cen}$ where $\\mu m_\\mathrm {p}$ is 1.4 times the mass of the proton, $\\Omega $ is the angular covering fraction of the wind and v$_\\mathrm {cen}$ is the centroid velocity of the outflow (132 km s$^{ -1}$ ).",
"Previous studies assume a radius of 5 kpc, a full opening angle of 140$^\\circ $ (and $\\Omega = 3.11\\pi $ steradians), solar metallicity, and no ionization correction , [59], , .",
"Using the values from the Si IV line, and these assumptions, we derive a mass outflow rate for NGC 6090 of 7.32 M$_\\odot $ yr$^{-1}$ .",
"However, previous studies typically use absorption lines of cooler gas like Na I, Mg II and Fe II , [59], , [60], .",
"If we use the Si II 1304 column density, a similar ionization potential to the previous outflow tracers, $\\dot{M}_{\\text{o}}$ rises to 22.6 M$_\\odot $ yr$^{-1}$ .",
"However, with our tightly constrained physical model of the outflow, now we only have to assume an angular covering fraction to calculate the mass outflow rate.",
"Above, we derive a radius, a metallicity, and an ionization fraction for the outflows (see tab:ion for values).",
"If we use the radius of peak optical depth (R$_\\mathrm {p}$ = 72.2 pc, at a w of 0.41), we calculate a total mass outflow rate of 0.81 M$_\\odot $ yr$^{-1}$ , 28 times lower than the $\\dot{M}_{\\text{o}}$ calculated using Si II.",
"While this calculation uses many of our derived values, it ignores the evolution of these quantities with velocity.",
"In velocity we find that the outflow is accelerated over short distances, in cf we find that the covering fraction decreases at large velocities, and in density we find that the outflow is not a mass conserving flow, rather it evolves as x$^{-5.7}$ .",
"All of these impact how mass is distributed in velocity space.",
"Using the scaling of the covering fraction, radius, and density with velocity, we define the mass outflow rate per velocity as $\\begin{aligned}\\dot{M}_\\mathrm {o}(\\mathrm {r}) &= \\Omega C_{f}(\\mathrm {r}) \\mathrm {v}(\\mathrm {r}) \\rho (\\mathrm {r}) \\mathrm {r}^2 \\\\\\dot{M}_\\mathrm {o}(w)&= \\Omega C_f(\\mathrm {R}_\\mathrm {i}) \\mathrm {v}_\\infty \\mu \\mathrm {m}_\\mathrm {p} \\mathrm {n}_\\mathrm {0} \\mathrm {R}_\\mathrm {i}^2 \\frac{w}{(1-w^{1/\\beta })^{2+\\gamma +\\alpha }} \\\\&= 10.1~\\text{M}_\\odot ~\\text{yr}^{-1} \\frac{w}{(1-w^{1/\\beta })^{2+\\gamma +\\alpha }}\\end{aligned}$ where we use the values from tab:beta and tab:ion for the constants and the exponents, and the only assumed parameter is that $\\Omega $ is 3.11$\\pi $ steradians.",
"The $\\dot{M}_{\\text{o}}$ relation is not constant with velocity, as typically assumed (fig:mout): $\\dot{M}_{\\text{o}}$ increases rapidly at low velocities as the outflow accelerates, and declines at high velocities as the density and covering fraction of the outflow decline (fig:vellaw and fig:denlaw).",
"At w of 0.35, the $\\dot{M}_{\\text{o}}$ relation peaks at a value of 2.3 M$_\\odot $ yr$^{-1}$ .",
"This is a factor of ten times smaller than the Si II value calculated with previous assumptions for geometry, ionization fractions, and metallicities.",
"Combining $\\dot{M}_{\\text{o}}$ with the derived outflow metallicity, we find that the maximum metal outflow rate is 0.07 M$_\\odot $ yr$^{-1}$ .",
"This may constrain the impact of outflows on the mass-metallicity relationship , [26], , [3], , [17], [15].",
"It is typical to normalize the mass outflow rate by the star formation rate to produce the \"mass-loading\" factor .",
"The global $\\eta $ is 0.09.",
"This $\\eta $ is smaller than recent measurements from [38], who find an $\\eta $ near 0.3 for galaxies with SFRs near 30 M$_\\odot $ yr$^{-1}$ .",
"However, [38] use the mean ISM metallicity of the sample (0.5 Z$_\\odot $ ), do not calculate ionization fractions, and use a multiple of the half-light radius (typically 0.5-1 kpc) as the radius of the outflow.",
"Other studies make similar assumptions about the metallicity and the ionization state, but assume constant outflow radii between 1-5 kpc , [61], .",
"These studies typically find $\\eta $ between 0.02 and 1 for SFRs and $M_\\ast $ similar to NGC 6090.",
"The method outlined here does not rely on these uncertain assumptions and allows for these quantities to vary from galaxy to galaxy, reducing the scatter in the measurements.",
"The relatively small inner outflow radius implies that the local SFR may be more important than the global SFR.",
"In [13] we calculate the fraction of the total GALEX UV flux within the COS aperture to be 22%, leading to a SFR within the COS aperture of 5.55 M$_\\odot $ yr$^{-1}$ , however this does not account for the significant spatial variation in the IR flux and is only a crude approximation for the local SFR.",
"Using this local SFR we calculate a local $\\eta $ within the COS aperture of 0.42.",
"The local and global $\\eta $ values are a factor of 5 and 22 times lower than the redshift independent $\\eta $ predicted by the FIRE simulation [39], [41], , although the high-mass FIRE galaxies at redshift 0 only have upper limits for $\\eta $ .",
"[34] develop an analytical model that uses the momentum from stellar feedback to regulate star formation and drive outflows.",
"A requirement of this model is that $\\eta $ depends on the gas fraction and the stellar mass of the host galaxy.",
"Making approximations for these scalings with redshift, the analytical model predicts the mass-loading for a log($M_\\ast $ /M$_\\odot $ ) galaxy of 10.7 at $z\\approx 0.03$ to be 0.05, in rough agreement with the observed global $\\eta $ .",
"While the outflow from NGC 6090 is weak compared to the SFR, [34] predict that 10$^9$ M$_\\odot $ galaxies drive outflows with an $\\eta $ of 10 at $z\\sim 0$ .",
"In a future paper we will explore the scaling of $\\dot{M}_{\\text{o}}$ and $\\eta $ with host galaxy properties to better constrain these types of studies.",
"Figure: The total mass within the outflow at each velocity, as given by eq:m. The total integrated outflow mass is 7.5×10 5 7.5 \\times 10^5 M ⊙ _\\odot .The total mass in the outflow (M$_o$ ) at a particular velocity is given by $\\begin{aligned}M_\\mathrm {o} (w) &= \\Omega C_f(\\mathrm {r}) \\mu \\mathrm {m}_\\mathrm {p} \\mathrm {n}_\\mathrm {0} \\mathrm {R}_\\mathrm {i}^3 \\left(\\frac{1}{1-w^{1/\\beta }}\\right)^{3+\\alpha +\\gamma } \\\\& = 1.5 \\times 10^{6} M_\\odot \\left(\\frac{1}{1-w^{1/\\beta }}\\right)^{3+\\alpha +\\gamma }\\end{aligned}$ This relation is shown in fig:m. Like the density, the total mass in the Si IV outflow declines with velocity.",
"The integrated mass between w of 0 and 1 is $7.5 \\times 10^{5}~M_\\odot $ .",
"While the current mass of outflowing gas is significantly lower than the measured H I mass of 10$^{10.2}$ M$_\\odot $ , if $\\dot{M}_{\\text{o}}$ remains constant over the $\\sim 1$ Gyr time scale of the merger, than the outflow will process (and enrich) 15% of the observed H I in NGC 6090, possibly leading to the metal enriched gas seen in the halos of galaxies , , , ."
],
[
"CONCLUSION",
"Here we measure a physically motivated mass outflow rate ($\\dot{M}_{\\text{o}}$ ) of the nearby starburst NGC 6090.",
"To calculate $\\dot{M}_{\\text{o}}$ , we first fit the optical depth with a Sobolev optical depth, and the covering fraction with a radial power-law (see eq:cfbeta and fig:beta).",
"We then calculate the ionization corrections and the metallicity of the outflow using a Bayesian analysis, CLOUDY models, and the measured column densities (see tab:ion and fig:ionpdfs).",
"The main results of this study are: The ionization model estimates the metallicity (1.61 Z$_\\odot $ ), density (18.73 cm$^{-3}$ ) and the ionization parameter (log(U) = -1.85) of the outflow.",
"The estimated H column density is consistent with that derived from the UV continuum extinction.",
"The outflow is at least as metal enriched as the ISM of the host galaxy (see ionstruct).",
"Using the absorption line profile and ionization models, we determine that the inner edge (R$_\\mathrm {i}$ ) of the outflow is 63 pc from the starburst (see eq:r), consistent with the absorption arising from the shredded blastwaves of supernovae remnants.",
"Most of the outflow is constrained within 300 pc of the starburst (ionstruct).",
"The covering fraction scales as C$_f = 1.0 (\\mathrm {r}/\\mathrm {R}_\\mathrm {i})^{-0.8\\pm 0.2}$ (tab:beta).",
"This scaling relation is consistent with predictions of clouds in pressure equilibrium with an adiabatically expanding external medium (cf).",
"This warrents further study of whether outflowing clouds remain in pressure equilibrium with an external medium, and if a fractal cloud distribution could produce a similar scaling relation.",
"The Si IV outflow velocity scales with radius as v$~=~\\mathrm {v}_\\infty (1~-~\\mathrm {R}_\\mathrm {i}/\\mathrm {r})^{0.43 \\pm 0.07}$ (fig:vellaw and tab:beta), where v$_\\infty $ is the maximum velocity.",
"We compare this velocity profile to models of different driving mechanisms, and find that an r$^{-2}$ force law matches the observed profile (see fig:vellaw and velocity).",
"The outflow density decreases with radius as n$ \\propto \\mathrm {r}^{-5.7 \\pm 1.5}$ (fig:denlaw), which declines more rapidly than a mass conserving flow (density).",
"This rapid density reduction could be due to interactions between the outflowing clouds and a hotter wind.",
"Combining all of our measurements, we derive a maximum mass outflow rate ($\\dot{M}_{\\text{o}}$ ) of 2.3 M$_\\odot $ yr$^{-1}$ (see fig:mout).",
"The mass-loading factor (mass outflow rate divided by star formation rate) is 0.09.",
"The $\\dot{M}_{\\text{o}}$ is a factor of 10 lower than the $\\dot{M}_{\\text{o}}$ calculated using common assumptions for ionization state, metallicity, and geometry (see mout).",
"In future work, we will continue this analysis for a larger sample of star forming galaxies, studying how the outflow properties (metallicity, initial radius, mass outflow rate) scale with host galaxy properties.",
"These scaling relations will constrain future models of galaxy evolution."
],
[
"Acknowledgments",
"We thank the anonymous referee for constructive comments that strengthened the paper.",
"Joseph Cassinelli inspired this work with helpful conversations and notes.",
"We thank Bart Wakker for help with the data reduction and discussions on the analysis.",
"Support for program 13239 was provided by NASA through a grant from the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5-26555.",
"Some of the data presented in this paper were obtained from the Mikulski Archive for Space Telescopes (MAST).",
"STScI is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5-26555.",
"All of the HST data presented in this paper were obtained from the Mikulski Archive for Space Telescopes (MAST).",
"STScI is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS5-26555.",
"Support for MAST for non-HST data is provided by the NASA Office of Space Science via grant NNX09AF08G and by other grants and contracts.",
"Funding for the Sloan Digital Sky Survey (SDSS) has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Aeronautics and Space Administration, the National Science Foundation, the U.S. Department of Energy, the Japanese Monbukagakusho, and the Max Planck Society.",
"The SDSS Web site is http: www.sdss.org/.",
"The SDSS is managed by the Astrophysical Research Consortium (ARC) for the Participating Institutions.",
"The Participating Institutions are The University of Chicago, Fermilab, the Institute for Advanced Study, the Japan Participation Group, The Johns Hopkins University, Los Alamos National Laboratory, the Max-Planck-Institute for Astronomy (MPIA), the Max-Planck-Institute for Astrophysics (MPA), New Mexico State University, University of Pittsburgh, Princeton University, the United States Naval Observatory, and the University of Washington."
]
] | 1605.05769 |
[
[
"Computing the n-th coefficient of an algebraic power series modulo p in\n O(log n) operations"
],
[
"Abstract This is an exposition, for pedagogical purposes, of the formal power series proof of Bostan, Christol and Dumas [3] of the result stated in the title (a corollary of the Christol theorem)."
],
[
"This is an exposition, for pedagogical purposes, of the formal power series proof of Bostan, Christol and Dumas [3] of the result stated in the title (a corollary of the Christol theorem).",
"Introduction The generating function $c=c(x)=\\sum _{n\\ge 1}c_nx^n$ of the Catalan numbers $c_n=\\frac{1}{n}\\binom{2n-2}{n-1}=1,1,2,5,14,42,\\dots $ satisfies the quadratic equation $c=c^2+x\\;.$ Can we effectively determine parity of $c_n$ ?",
"Very easily: modulo 2 one has $c(x)=c(x)^2+x\\equiv c(x^2)+x$ (by Lemma C below), and therefore $c_n\\equiv c_{n/2}$ mod 2 for $n>1$ and $c_1\\equiv 1$ mod 2 ($c_{n/2}=0$ if $n/2\\notin \\mathbb {N}_0$ ).",
"Hence $c_n\\equiv 1$ mod 2 for $n=2^m$ , $m\\in \\mathbb {N}_0$ , and $c_n\\equiv 0$ mod 2 else .",
"And $c_n$ modulo 3 or modulo other prime $p$ ?",
"An answer to this is provided much more generally by the Christol theorem, proved by Christol [5] for $q=2$ and by Christol, Kamae, Mendès France and Rauzy [6] for any prime power $q$ (see also Allouche and Shallit [1]): a power series $f(x)\\in F_q[[x]]$ is algebraic over $F_q(x)$ if and only if one can generate its coefficients $f_0,f_1,\\dots $ by a DFAO (a deterministic finite automaton with output that reads the $l+1$ digits of the $q$ -adic expansion $n=(n_l\\dots n_1n_0)_q$ and outputs $f_n$ , we give one example in Concluding remarks).",
"It is immediate from the Christol theorem that one can compute the $n$ th coefficient of an algebraic power series in $F_q[[x]]$ in $O(\\log n)$ arithmetic operations because $n$ has $O(\\log n)$ $q$ -adic digits.",
"This write-up is an exposition of the proof of this result, stated as Theorem A below, that was given by means of power series in the recent preprint of Bostan, Christol and Dumas [3].",
"I promised in my course `Kombinatorické počítání' (taught in this summer semester 2015/16) to supplement my unfortunately not very clear oral presentation of (something like) Theorem A with a written one — here it is.",
"Algebraic power series in $F_p[[x]]$ First we review some notation and notions from algebra.",
"Puiseux series $K((x))_P$ and their order are only used at the end in Proposition H that is not needed to prove Theorem A.",
"$\\mathbb {N}=\\lbrace 1,2,\\dots \\rbrace $ , $\\mathbb {N}_0=\\lbrace 0,1,\\dots \\rbrace $ , $\\mathbb {Z}$ is the ring of integers, $F_p=(\\mathbb {Z}/p\\mathbb {Z},+,\\cdot )$ is the finite $p$ -element field where $p$ is a prime number, and $[x^n]f$ denotes the coefficient of $x^n$ in $f$ .",
"For $p\\in \\mathbb {N}$ , $p\\ge 2$ , the $p$ -adic expansion $m=(m_l\\dots m_1m_0)_p$ of $m\\in \\mathbb {N}_0$ is given by the unique expression $m=\\sum _{i=0}^lm_ip^i$ where $m_i\\in \\lbrace 0,1,\\dots ,p-1\\rbrace $ and (for $m>0$ ) $m_l\\ne 0$ ; $0=(0)_p$ .",
"Let $K$ be a field or an integral domain.",
"For a field we denote by $\\overline{K}$ its algebraic closure (e.g.",
"$\\overline{\\mathbb {R}}=\\mathbb {C}$ or $\\overline{\\mathbb {Q}}$ are the algebraic numbers).",
"We denote by $K[[x]]\\subset K((x))\\subset K((x))_P$ , respectively, the ring of power series and the fields of Laurent series and Puiseux series with coefficients in $K$ .",
"Their elements are the formal series (we treat here all power etc.",
"series formally) $\\sum _{n\\ge k}a_{n/d}x^{n/d}$ , $a_{n/d}\\in K$ , where $k=0$ and $d=1$ for p. s., $k\\in \\mathbb {Z}$ and $d=1$ for L. s., and $k\\in \\mathbb {Z}$ and $d\\in \\mathbb {N}$ for P. s. For $f=f(x)\\in K((x))$ we use notation $f_n:=[x^n]f(x)$ .",
"From the algebraic point of view, $K((x))$ is the field of fractions of $K[[x]]$ and for $K$ of characteristic 0 (e.g.",
"$K=\\mathbb {Q}$ ) $\\overline{K}((x))_P$ is the algebraic closure of $K((x))$ .",
"Thus in $\\overline{K}((x))_P$ every polynomial with coefficients in $K((x))$ has a root.",
"Recall that $K[x]\\subset K(x)$ is, respectively, the ring of polynomials with coefficients in $K$ and its field of fractions ($K(x)$ are the rational `functions' over $K$ ).",
"In fact, $K(x)$ is naturally a subfield of $K((x))$ and $K[x]$ a subring of $K[[x]]$ .",
"By $\\deg a$ , $a\\in K[x]$ , we denote the degree of a polynomial and by $\\deg _y a$ , $a\\in K[x,y]$ , the degree $\\deg a$ when $a$ is understood as $a\\in K[x][y]$ .",
"By $\\mathrm {ord\\,}f$ , $f\\in K((x))_P$ , we denote the order of $f$ , the minimum $n/d$ such that $a_{n/d}\\ne 0$ .",
"We set $\\deg 0=-\\infty $ and $\\mathrm {ord\\,}0=+\\infty $ .",
"For any $f,g\\in K((x))_P$ one has $\\mathrm {ord\\,}(fg)=\\mathrm {ord\\,}f+\\mathrm {ord\\,}g$ and $\\mathrm {ord\\,}(f+g)\\ge \\min (\\mathrm {ord\\,}f,\\mathrm {ord\\,}g)$ , with equality if $\\mathrm {ord\\,}f\\ne \\mathrm {ord\\,}g$ .",
"Similarly, for any $f,g\\in K[x]$ one has $\\deg (fg)=\\deg f+\\deg g$ and $\\deg (f+g)\\le \\max (\\deg f,\\deg g)$ , with equality if $\\deg f\\ne \\deg g$ .",
"We state the basic result on computing coefficients of an algebraic power series in $F_p[[x]]$ (this is essentially proven in $\\cite {chri_al}$ but the formulation below is ours).",
"Theorem A ([6], 1980).",
"Suppose that $p$ is prime, $P\\in \\mathbb {Z}[x,y]$ is a nonzero polynomial, $f=f_0+f_1x+f_2x^2+\\dots \\in \\mathbb {Z}[[x]]$ is a power series satisfying $P(x,f(x))=0$ , and $h=(d+1)(p^d-d+1)\\deg _xP\\ \\mbox{ where }\\ d=\\deg _yP\\ (\\ge 1)\\;.$ Then from given $p,P,f_0,f_1,\\dots ,f_h$ one can construct an algorithm (actually a DFAO) that computes $n\\mapsto f_n\\ \\mathrm {mod}\\ p$ in $O(\\log n)$ arithmetic operations ($n\\ge 2$ ).",
"But which operations exactly?",
"These are arithmetic operations ($+,-,\\times ,:$ ) in the field $F_p$ and the operation of division with remainder in the ring $\\mathbb {Z}$ .",
"By `one can construct' here and below we mean that the task can be performed constructively (i.e.",
"the algorithms or objects not merely exist but exist effectively).",
"Theorem A follows from the following more specific result (again the formulation is ours).",
"Theorem B ([3]).",
"Let $p$ , $P$ , $f$ , and $h$ be as in Theorem A.",
"We denote the mod $p$ reduction of $f$ again by $f\\in F_p[[x]]$ .",
"Then from given $p,P,f_0,f_1,\\dots ,f_h$ one can construct polynomials $a,c\\in F_p[x]$ , a number $e\\in \\mathbb {N}$ , a row $u\\in F_p^{1\\times e}$ , $p$ matrices $A_0,A_1,\\dots ,A_{p-1}\\in F_p^{e\\times e}$ , and a column $v\\in F_p^{e\\times 1}$ such that for any $n\\in \\mathbb {N}_0$ one has $[x^n]f(x)&=&f_n=[x^n](c(x)+a(x)h(x))\\ \\mbox{ where }\\\\&&h(x)=\\sum _{m=0}^{\\infty }(uA_{m_l}\\dots A_{m_1}A_{m_0}v)x^m\\in F_p[[x]]$ ($m=(m_l\\dots m_1m_0)_p$ is the $p$ -adic expansion of $m$ ).",
"Polynomial $P$ in Theorem A in general does not determine unique power series solution $y=f(x)$ of $P(x,y)=0$ , e.g.",
"$y^2-x^2=0$ has two solutions $y=x$ and $y=-x$ .",
"One can introduce restrictions, for example to require that $P(0,0)=0$ , $P_y(0,0)\\ne 0$ , and $f(0)=0$ , producing unique solution, but we want result for completely general $P$ .",
"So to determine $f(x)$ we need to know a few initial coefficients, which is captured by the parameter $h$ in Theorems A and B. Bounds on $h$ are considered in Corollary G and Proposition H. Let us see how Theorem B implies Theorem A.",
"We are given a prime $p$ , a nonzero $P\\in \\mathbb {Z}[x,y]$ , and first few coefficients $f_0,f_1,\\dots ,f_h$ of a solution $y=f(x)\\in \\mathbb {Z}[[x]]$ of the equation $P(x,y)=0$ , where $h$ is as in Theorem A.",
"(We assume this is an honest input, $f_0,f_1,\\dots ,f_h$ are really initial coefficients of a solution.",
"We leave as an exercise for the reader to design an effective check that it is true.)",
"Taking out the largest common factor of the coefficients of $P(x,y)$ we may assume that they are together coprime.",
"Then reducing $P(x,f(x))=0$ modulo $p$ we get the relation $E(x,f(x))=0$ where $E\\in F_p[x,y]$ is the nonzero mod $p$ reduction of $P$ and $f\\in F_p[[x]]$ denotes again the mod $p$ reduction of $f$ — its coefficients $f_n$ we want to compute.",
"We do it by Theorem B.",
"We construct in $O_{p,P}(1)$ operations the objects $a,c,e,u,A_0,A_1,\\dots ,A_{p-1}$ , and $v$ .",
"For given $n\\in \\mathbb {N}_0$ we compute $f_n$ as $f_n=[x^n]c(x)+\\sum _{m=n-\\deg a}^n[x^{n-m}]a(x)\\cdot uA_{m_l}\\dots A_{m_1}A_{m_0}v$ where $m=(m_l\\dots m_1m_0)_p$ .",
"For each $m$ ($>0$ ) in the summation range it takes $l+1=l(m)+1=\\lfloor \\log _pm\\rfloor +1$ operations in $\\mathbb {Z}$ (divisions by $p$ with remainder) to determine its $p$ -adic expansion.",
"It takes $l+1$ matrix times column multiplications, one row times column multiplication, and one scalar multiplication (in $F_p$ ) to evaluate the summand.",
"So we evaluate $f_n$ in $\\sum _{m=n-\\deg a}^n(l(m)+1+(l(m)+1)e(2e-1)+2e+1)=O((\\deg a)e^2\\log _pn)$ operations.",
"Altogether we need $O_{p,P}(1)+O((\\deg a)e^2\\log _pn)=O_{p,P}(\\log n)$ operations ($n\\ge 2$ ).",
"Proof of Theorem B We start with four lemmas.",
"Lemma C is well known and is crucial for the existence of the algorithm of Theorem A, and in fact for the all algebra (and analysis) in characteristic $p$ .",
"Lemma C. If $p$ is prime and $z\\in F_p((x))$ is any Laurent series then $z(x)^p=z(x^p)$ .",
"Consequently, for any $k\\in \\mathbb {N}_0$ one has $z(x)^{p^k}=z(x^{p^k})\\;.$ Proof.",
"Let $z(x)=z_rx^r+z_{r+1}x^{r+1}+\\cdots $ .",
"Since $a^p=a$ for any $a\\in F_p$ and $(u+v)^p=u^p+v^p$ for any $u,v\\in F_p((x))$ (because $\\binom{p}{i}\\equiv 0$ mod $p$ for $0<i<p$ ), we indeed have $z(x)^p&=&z_rx^{pr}+(z_{r+1}x^{r+1}+\\dots )^p\\\\&=&z_rx^{pr}+z_{r+1}x^{p(r+1)}+(z_{r+2}x^{r+2}+\\dots )^p\\\\&\\vdots &\\\\&=&z(x^p)\\;.$ $\\Box $ We define operators $S_r:\\;F_p[[x]]\\rightarrow F_p[[x]]$ , $r\\in \\lbrace 0,1,\\dots ,p-1\\rbrace $ , by ${\\textstyle S_rz(x)=S_r(z(x))=S_r(\\sum _{n\\ge 0}z_nx^n)=\\sum _{n\\ge 0}z_{pn+r}x^n\\;.",
"}$ Thus $S_rz(x)$ arises from $z(x)$ by taking only terms $z_nx^n$ with $n\\equiv r$ modulo $p$ and replacing the $n$ in the exponent with $(n-r)/p$ .",
"Lemma D ([3]).",
"Operators $S_r$ have the following properties.",
"Linearity, $S_r(au+bv)=aS_r(u)+bS_r(v)$ for any $a,b\\in F_p$ and $u,v\\in F_p[[x]]$ .",
"$S_r(uv)=\\sum _{s+t\\equiv r\\;\\mathrm {mod}\\;p}x^{\\lfloor (s+t)/p\\rfloor }S_s(u)S_t(v)$ for any $u,v\\in F_p[[x]]$ .",
"$S_r(u(x)v(x^p))=S_r(u(x))v(x)$ for any $u,v\\in F_p[[x]]$ .",
"If $u\\in F_p[[x]]$ and $n=(n_l\\dots n_1n_0)_p\\in \\mathbb {N}_0$ then $u_n=[x^n]u(x)=(S_{n_l}\\dots S_{n_1}S_{n_0}u(x))(0)=[x^0]S_{n_l}\\dots S_{n_1}S_{n_0}u(x)\\;.$ For any $0\\ne u\\in F_p[[x]]$ there is an $r\\in \\lbrace 0,1,\\dots ,p-1\\rbrace $ such that $S_ru\\ne 0$ .",
"$S_rF_p[x]\\subset F_p[x]$ and $\\deg S_ra\\le (\\deg a)/p$ for any $a\\in F_p[x]$ .",
"Linearity, $S_r(au+bv)=aS_r(u)+bS_r(v)$ for any $a,b\\in F_p$ and $u,v\\in F_p[[x]]$ .",
"$S_r(uv)=\\sum _{s+t\\equiv r\\;\\mathrm {mod}\\;p}x^{\\lfloor (s+t)/p\\rfloor }S_s(u)S_t(v)$ for any $u,v\\in F_p[[x]]$ .",
"$S_r(u(x)v(x^p))=S_r(u(x))v(x)$ for any $u,v\\in F_p[[x]]$ .",
"If $u\\in F_p[[x]]$ and $n=(n_l\\dots n_1n_0)_p\\in \\mathbb {N}_0$ then $u_n=[x^n]u(x)=(S_{n_l}\\dots S_{n_1}S_{n_0}u(x))(0)=[x^0]S_{n_l}\\dots S_{n_1}S_{n_0}u(x)\\;.$ For any $0\\ne u\\in F_p[[x]]$ there is an $r\\in \\lbrace 0,1,\\dots ,p-1\\rbrace $ such that $S_ru\\ne 0$ .",
"$S_rF_p[x]\\subset F_p[x]$ and $\\deg S_ra\\le (\\deg a)/p$ for any $a\\in F_p[x]$ .",
"Proof.",
"1.",
"This is immediate from the definition.",
"2.",
"We have $[x^n]S_r(uv)=\\sum _{i+j=pn+r}u_iv_j=\\sum _{l+m=n,n-1;s+t=r,r+p}u_{pl+s}v_{pm+t}$ ($i,j,l,m,n\\in \\mathbb {N}_0$ , $r,s,t\\in \\lbrace 0,1,\\dots ,p-1\\rbrace $ ).",
"The last sum equals the coefficient of $x^n$ in the sum stated in 2.",
"3.",
"This is a corollary of 2: $S_t(v(x^p))$ is the zero power series for $t\\ne 0$ and $S_0(v(x^p))=v(x)$ .",
"4.",
"Since ($r,s,r_i\\in \\lbrace 0,1,\\dots ,p-1\\rbrace $ ) $S_ru(x)=\\sum _{n\\ge 0}u_{pn+r}x^n,\\ S_sS_ru(x)=\\sum _{n\\ge 0}u_{p(pn+s)+r}x^n=\\sum _{n\\ge 0}u_{p^2n+ps+r}x^n$ and so on, the constant term of $S_{r_l}\\dots S_{r_1}S_{r_0}u(x)$ is just $u_{(r_l\\dots r_1r_0)_p}$ .",
"5.",
"By the definition of $S_r$ , for $r$ one may take the least significant $p$ -adic digit of any $n\\in \\mathbb {N}_0$ for which $u_n=[x^n]u\\ne 0$ .",
"6.",
"Clear from the definition of $S_r$ .",
"$\\Box $ Lemma E. Let $K$ be any field.",
"If $v_1,\\dots ,v_{d+1}\\in K[x]^d$ are $d+1$ $d$ -tuples whose entries have degrees at most $c\\in \\mathbb {N}_0$ , then one can construct coefficients $d_1,\\dots ,d_{d+1}\\in K[x]$ , not all 0, such that $\\sum _{i=1}^{d+1}d_iv_i=(0,0,\\dots ,0)$ and $\\deg d_i\\le dc$ for every $i=1,\\dots ,d+1$ .",
"Proof.",
"Let $M\\in K[x]^{(d+1)\\times d}$ be the matrix with rows $v_1,\\dots ,v_{d+1}$ .",
"We may assume that it has full rank $d$ , its columns are linearly independent over $K(x)$ .",
"(If not, we find by standard linear algebra some maximal index set $I\\subset \\lbrace 1,2,\\dots ,d\\rbrace $ of linearly independent columns and solve the problem for restrictions of the tuples $v_i$ to coordinates with indices in $I$ ).",
"We find $d$ row indices $I\\subset \\lbrace 1,2,\\dots ,d+1\\rbrace $ such that $\\det M_I\\ne 0$ , where $M_I$ arises from $M$ by deleting the row $j$ not in $I$ ($[d+1]\\backslash I=\\lbrace j\\rbrace $ ).",
"The displayed equation can be written as (we replace $d_i$ by $x_i$ ) $(x_1,\\dots ,x_{j-1},x_{j+1},\\dots ,x_{d+1})M_I=-x_jv_j\\;.$ We set $x_j=1$ and solve this linear system for the $x_i$ , $i\\in I$ .",
"By Cramer's rule, the solution is $x_i=\\pm \\det N_i/\\det M_I$ , $i\\in I$ , where $N_i$ arises from $M_I$ by replacing the row $i$ with the row $-x_jv_j=-v_j$ (the determinants are in $K[x]$ ).",
"From the definition of determinant and bound on degrees of the entries in $M_I$ and $N_i$ we get that $\\deg \\det N_i,\\deg \\det M_I\\le dc$ .",
"Thus the desired coefficients are $d_j=\\det M_I$ and $d_i=\\pm \\det N_i$ , $i\\in I$ .",
"Then $\\deg d_i\\le dc$ for every $i=1,\\dots ,d+1$ and $d_j\\ne 0$ .",
"$\\Box $ Lemma F. Let $p$ be prime, $0\\ne E\\in F_p[x,y]$ , $d=\\deg _yE$ , and let $f\\in F_p[[x]]$ satisfy $E(x,f(x))=0$ (so $d\\ge 1$ ).",
"Then from given polynomial $E$ one can construct polynomials $c_0,c_1,\\dots ,c_k\\in F_p[x]$ , $0\\le k\\le d$ , such that we have equation $\\sum _{i=0}^kc_i(x)f(x^{p^i})=0\\;,$ $c_0\\ne 0$ , and $\\deg c_0\\le (d+1)(p^d-d+1)\\deg _xE$ .",
"Proof.",
"We have $a_df^d=\\sum _{i=0}^{d-1}a_if^i$ where $a_i=[y^i]E\\in F_p[x]$ and $a_d\\ne 0$ .",
"It follows that for given $E$ , $0\\le i\\le d$ , and $n\\in \\mathbb {N}_0$ one can construct polynomials $a_{i,n}\\in F_p[x]$ such that ${\\textstyle a_{d,n}f^n=\\sum _{i=0}^{d-1}a_{i,n}f^i,\\ a_{d,n}\\ne 0\\;.",
"}$ Thus the $d+1$ elements $f^{p^i}=f(x^{p^i})$ (Lemma C), $i=0,1,\\dots ,d$ , are linearly dependent over $F_p(x)$ and one can construct the dependency relation ${\\textstyle \\sum _{i=0}^dd_i(x)f(x^{p^i})=0}$ with $d_i\\in F_p[x]$ that are not all zero.",
"Let $j\\in \\mathbb {N}_0$ be minimum with $d_j\\ne 0$ .",
"Using Lemma D.5, we can construct $r_1,\\dots ,r_{j-1}\\in \\lbrace 0,1,\\dots ,p-1\\rbrace $ such that $S_{r_1}\\dots S_{r_{j-1}}d_j(x)\\ne 0$ .",
"Applying $S_{r_1}\\dots S_{r_{j-1}}$ on the dependency relation and using Lemma D.3 we get the desired equation ${\\textstyle \\sum _{i=j}^d(S_{r_1}\\dots S_{r_{j-1}}d_i(x))f(x^{p^{i-j}})=0\\;.",
"}$ We set $c_i=S_{r_1}\\dots S_{r_{j-1}}d_{i+j}(x)$ , $i=0,1,\\dots ,k=d-j$ , and have $c_0=S_{r_1}\\dots S_{r_{j-1}}d_j(x)\\ne 0$ .",
"We derive the upper bound on $\\deg c_0$ .",
"Since $S_r$ does not increase degree, $\\deg c_0\\le \\max _i\\deg d_i$ .",
"We have this recurrence for the polynomials $a_{i,n}$ : $a_{d,n}=1$ and $a_{i,n}=\\delta _{i,n}$ for $0\\le i,n\\le d-1$ , $a_{i,d}=a_i$ for $0\\le i\\le d$ , and, for $n>d$ , $a_{d,n}=a_{d,n-1}a_d=a_d^{n-d+1}$ and $a_{i,n}=a_{d-1,n-1}a_i+a_{i-1,n-1}a_d$ for $0\\le i\\le d-1$ , with $a_{-1,n-1}=0$ .",
"Clearly, $\\max _{0\\le i\\le d}\\deg a_i=\\deg _xE$ .",
"Using the recurrence and induction we get the bound $\\max _i\\deg a_{i,n}=0$ for $n=0,1,\\dots ,d-1$ , $\\max _i\\deg a_{i,d}=\\deg _xE$ , and $\\max _i\\deg a_{i,n}\\le (n-d+1)\\deg _xE$ for $n\\ge d$ .",
"Applying Lemma E on the $d+1$ $d$ -tuples $(a_{i,n}\\;|\\;0\\le i\\le d-1)$ , $n=1,p,p^2,\\dots ,p^d$ , we get $d_i\\in F_p[x]$ in the dependency relation with $\\max _i\\deg d_i\\le (d+1)(p^d-d+1)\\deg _xE$ .",
"$\\Box $ The lemma is inspired by Lemma 12.2.3 in Allouche and Shallit [1].",
"Equations of this type are called Mahler equations, after the German–British mathematician Kurt Mahler (1903–1988) who used them to prove transcendence of various numbers.",
"We start the proof of Theorem B proper.",
"Let $0\\ne E\\in F_p[x,y]$ be the mod $p$ reduction of $P(x,y)$ (with coprime coefficients) and $d=\\deg _yE\\ge 1$ .",
"Since $E(x,f)=0$ , by Lemma F one can construct a Mahler equation $c_0(x)f(x)+c_1(x)f(x^p)+\\dots +c_k(x)f(x^{p^k})=0$ for $f(x)$ where $c_i\\in F_p[x]$ , $k\\le d$ , $c_0(x)\\ne 0$ , and $\\deg c_0\\le (d+1)(p^d-d+1)\\deg _xE\\le h$ ($h$ is as in Theorem A).",
"We change the variable $f(x)$ to $g(x)$ by $f(x)=c_0(x)g(x)$ .",
"Using Lemma C and dividing by $c_0(x)^2$ , we get a Mahler equation for $g(x)$ , $g(x)+c_1(x)c_0(x)^{p-2}g(x^p)+\\dots +c_k(x)c_0(x)^{p^k-2}g(x^{p^k})=0$ — now the new coefficient $c_0$ is 1.",
"The small price we paid is that in general $g(x)$ is not a power series but $g(x)\\in F_p((x))$ .",
"We write $g(x)=\\sum _{n<0}g_nx^n+\\sum _{n\\ge 0}g_nx^n=:g_-(x)+h(x)\\;,$ $g_-\\in x^{-1}F_p[x^{-1}]$ and $h\\in F_p[[x]]$ (this is the $h(x)$ of Theorem B).",
"Denoting the Mahler equation for $g(x)$ as $L(x,M)g(x)=0$ (here $L(x,M)$ is a skew polynomial in $x$ and the operator $M:\\;u(x)\\mapsto u(x)^p=u(x^p)$ ) we get effectively a Mahler equation for the power series $h(x)$ , $L(x,M)h(x)=-L(x,M)g_-(x)=:b(x)$ or more explicitly $h(x)+c_1(x)c_0(x)^{p-2}h(x^p)+\\dots +c_k(x)c_0(x)^{p^k-2}h(x^{p^k})=b(x)\\;.$ Necessarily $b(x)\\in F_p[x]$ because $b(x)=-L(x,M)g_-(x)\\in F_p[x^{-1},x]$ but also $b(x)=L(x,M)h(x)\\in F_p[[x]]$ .",
"We write the equation as $h(x)=b(x)+d_1(x)h(x^p)+\\dots +d_k(x)h(x^{p^k})$ where $b(x),d_i(x):=-c_i(x)c_0(x)^{p^i-2}\\in F_p[x]$ .",
"Let $D:=\\max (\\deg b,\\deg d_i,1\\le i\\le k)$ and $h_{(i)}:=h(x^{p^i})$ .",
"We consider the vector space $V:=\\lbrace \\alpha +\\beta _0h_{(0)}+\\dots +\\beta _kh_{(k)}\\;|\\;\\alpha ,\\beta _i\\in F_p[x],\\deg \\alpha ,\\deg \\beta _i\\le D\\rbrace $ over $F_p$ .",
"Clearly, $h(x)=h_{(0)}\\in V$ .",
"It follows that $S_r(V)\\subset V$ for each of the $p$ operators $S_r$ : using the equation for $h(x)$ and Lemma D.3 (and D.1) we have $&&S_r(\\alpha +\\beta _0h_{(0)}+\\dots +\\beta _kh_{(k)})={\\textstyle S_r(\\alpha +\\beta _0b+\\sum _{i=1}^k(\\beta _0d_i+\\beta _i)h_{(i)})}\\\\&&={\\textstyle S_r(\\alpha +\\beta _0b)+\\sum _{i=1}^kS_r(\\beta _0d_i+\\beta _i)h_{(i-1)}}$ and $\\deg S_r(\\alpha +\\beta _0b),\\deg S_r(\\beta _0d_i+\\beta _i)\\le \\deg (\\alpha +\\beta _0b)/p,\\deg (\\beta _0d_i+\\beta _i)/p\\le 2D/p\\le D$ (Lemma D.6).",
"The $(D+1)(k+2)$ -element set $B:=\\lbrace x^j,x^jh_{(i)}=x^jh(x^{p^i})\\;|\\;0\\le j\\le D,0\\le i\\le k\\rbrace $ is a generating set for $V$ — the linear span of $B$ over $F_p$ is $V$ (but $B$ is not a basis for $V$ , as stated in [3]; $h_{(0)}$ is a linear combination of other elements of $B$ ).",
"We set $e:=(D+1)(k+2)$ and by fixing an ordering of $B$ make it a tuple $B=(b_1,\\dots ,b_e)$ .",
"We set $v\\in F_p^{e\\times 1}$ to be the column of coordinates of $h(x)$ in $B$ (it has all 0s and one 1), $A_r\\in F_p^{e\\times e}$ for $r=0,1,\\dots ,p-1$ to be the matrices of the operators $S_r$ with respect to $B$ (the $j$ -th column of $A_r$ lists the coordinates of $S_r(b_j)$ in $B$ ), and $u:=(b_1(0),\\dots ,b_e(0))\\in F_p^{1\\times e}$ to be the row of constant terms in $B$ .",
"Clearly we can obtain $v,A_r$ , and $u$ effectively.",
"By Lemma D.4 and the definition of $v,A_r$ , and $u$ we have for any $n=(n_l\\dots n_1n_0)_p\\in \\mathbb {N}_0$ that $h_n=[x^n]h(x)=[x^0]S_{n_l}\\dots S_{n_1}S_{n_0}h(x)=uA_{n_l}\\dots A_{n_1}A_{n_0}v\\;.$ Returning to $f(x)$ we see that $f(x)=c_0(x)g(x)=c_0(x)g_-(x)+c_0(x)h(x)$ and set, finally, $c(x):=c_0(x)g_-(x)$ and $a(x):=c_0(x)$ .",
"It follows that $a,c\\in F_p[x]$ .",
"How do we get $g_-(x)$ ?",
"It is the negative part of $g(x)=f(x)/c_0(x)$ .",
"Since $\\deg c_0\\le h$ , we compute $g_-(x)$ from $c_0(x)$ and $f_0,f_1,\\dots ,f_h$ .",
"Let us review the computation.",
"We are given $p,P$ , and $f_0,f_1,\\dots ,f_h$ (initial coefficients of a solution $y=f(x)\\in \\mathbb {Z}[[x]]$ to $P(x,y)=0$ , $h$ is as given in Theorem A).",
"Reducing $P$ modulo $p$ we get a nonzero $E\\in F_p[x,y]$ .",
"From $E$ we compute by Lemma F the coefficients $c_0,\\dots ,c_k\\in F_p[x]$ , $c_0\\ne 0$ , $\\deg c_0\\le h$ , and $k\\le \\deg _yE$ , of a Mahler equation for $f(x)$ .",
"We compute $g_-(x)\\in x^{-1}F_p[x^{-1}]$ as the negative part of $(f_0+f_1x+\\dots +f_hx^h)/c_0(x)$ .",
"Thus we have $L(x,M)=M^0+\\sum _{i=1}^kc_i(x)c_0(x)^{p^i-2}M^i$ (the Mahler equation for $g(x)$ ) and compute $b(x)=-L(x,M)g_-(x)\\in F_p[x]$ .",
"We have $K(x,M):=-\\sum _{i=1}^kc_i(x)c_0(x)^{p^i-2}M^i$ (the relation $h(x)=b(x)+K(x,M)h(x)$ ) and set $D=\\max (\\deg b,\\deg _xK(x,M))$ and $e=(D+1)(k+2)$ .",
"We take the $e$ -tuple $B=(x^j,x^jh(x^{p^i})\\;|\\;0\\le j\\le D,0\\le i\\le k)$ and compute the column $v$ of coordinates of $h(x)$ in $B$ (this is easy), the matrices $A_0,\\dots ,A_{p-1}$ of the operators $S_r$ with respect to $B$ ($S_r$ acts on $x^j$ in the clear way, $S_r(x^jh(x^{p^i}))=S_r(x^j)h(x^{p^{i-1}})$ for $i>0$ , and for $i=0$ we replace $h(x)$ with $b(x)+K(x,M)h(x)$ ), and the row $u$ of constant terms in $B$ (note that the constant term $h(0)$ is the constant term in $(f_0+f_1x+\\dots +f_hx^h)/c_0(x)$ ).",
"Finally, we compute $c(x)=c_0(x)g_-(x),a(x)=c_0(x)\\in F_p[x]$ .",
"Thus we have computed $a,c,e,u,A_0,\\dots ,A_{p-1},v$ , all we need to quickly evaluate $f_n$ modulo $p$ .",
"We are done.",
"$\\Box $ Concluding remarks An interesting problem is to locate the first different coefficient of two different power series solutions of the same polynomial equation.",
"Theorem B yields the following corollary.",
"Corollary G. If $p$ is prime, $0\\ne E\\in F_p[x,y]$ , and $f,g\\in F_p[[x]]$ , $f\\ne g$ , satisfy $E(x,f(x))=E(x,g(x))=0$ , then $f_n\\ne g_n\\ \\mbox{ for some }\\ n\\le (d+1)(p^d-d+1)\\deg _xE,\\ d=\\deg _yE\\;.$ Another bound is given in Proposition H. If $H$ is any field of characteristic 0, $0\\ne P\\in H[x,y]$ , and $f,g\\in H[[x]]$ , $f\\ne g$ , satisfy $P(x,f(x))=P(x,g(x))=0$ , then $f_n\\ne g_n\\ \\mbox{ for some }\\ n\\le \\frac{d^2+d-4}{2}\\deg _xP,\\ d=\\deg _yP\\;.$ Proof.",
"We work with the fields $K=H(x)\\subset H((x))=L$ and the algebraic closure $M=\\overline{L}=\\overline{H}((x))_P$ , the field of Puiseux series with coefficients in $\\overline{H}$ .",
"There is a nonzero polynomial $Q\\in H[x,y]$ such that $Q$ divides $P$ in $H[x,y]$ , $Q(x,f(x))=Q(x,g(x))=0$ , and $Q$ as $Q(y)\\in K[y]$ has in $M$ only simple roots.",
"Indeed, write $P(y)=P_1(y)^{a_1}P_2(y)^{a_2}\\dots P_k(y)^{a_k}$ where each $P_i\\in K[y]$ is irreducible (in $K[y]$ ), $a_i\\in \\mathbb {N}$ , and no $P_i$ divides other $P_j$ , $i\\ne j$ .",
"Irreducibility of the $P_i(y)$ s implies as usual that they have in $M$ only simple roots and do not share roots.",
"By the Gauss lemma (Lang [7]) we may take $P_i\\in H[x][y]$ .",
"Hence we have $Q=P_i(y)$ where $P_i(f)=P_i(g)=0$ or $Q=P_iP_j$ , $i\\ne j$ , where $P_i(f)=P_j(g)=0$ .",
"Let $f_1=f,f_2=g,\\dots ,f_d\\in M$ be the (different) roots of $Q(y)=a_dy^d+\\dots +a_1y+a_0$ , $a_i\\in H[x]$ and $a_d\\ne 0$ , so $d=\\deg _yQ\\in \\mathbb {N}$ (and $d\\ge 2$ ).",
"Let $e=\\max _i\\deg a_i=\\deg _xQ\\in \\mathbb {N}_0$ .",
"From Vièta's formula $(-1)^ia_{d-i}/a_d=\\sum _{1\\le j_1<\\dots <j_i\\le d}f_{j_1}\\dots f_{j_i}$ and properties of the $\\mathrm {ord\\,}$ function we get $\\min _{1\\le i\\le d}\\mathrm {ord\\,}f_i\\ge \\min _{0\\le i\\le d}\\mathrm {ord\\,}(a_i/a_d)\\ge -e\\;.$ We look at the order of the discriminant $D\\in H[x]$ of $Q(y)$ (Lang [7]), $D=a^{2d-2}_d\\prod _{1\\le i<j\\le d}(f_j-f_i)^2=:(f-g)^2E\\;.$ Since $Q(y)$ has no multiple root, $D$ is a nonzero homogeneous polynomial in $a_0,a_1,\\dots ,a_d$ with degree $2d-2$ and coefficients in $\\mathbb {Z}$ .",
"So $2\\,\\mathrm {ord\\,}(f-g)&=&\\mathrm {ord\\,}D-\\mathrm {ord\\,}\\,E\\le \\deg D-\\mathrm {ord\\,}E\\\\&\\le & e(2d-2)-(2d-2)0-2(-e)(d(d-1)/2-1)\\\\&=&e(d^2+d-4)$ and, since $d\\le \\deg _yP$ and $e\\le \\deg _xP$ , the stated bound follows.",
"$\\Box $ Alternatively, to determine uniquely an algebraic power series $f(x)$ , instead of taking first few coefficients $f_0,f_1,\\dots ,f_h$ and the polynomial equation $f(x)$ satisfies, one can represent $f(x)$ by a system of polynomial equations such that $f(x)$ is the first coordinate in the tuple of unique solutions to the system.",
"Then one can effectively do some operations with algebraic power series in such representation (which applies in fact to multivariate power series), see Alonso, Castro-Jiménez and Hauser [2] for this interesting topic.",
"So how do the Catalan numbers $c_n$ behave modulo 3?",
"According to this DFAO (taken from the note of Rowland [8]): $&&a_101b_1,\\,a_12e_2,\\,b_10b_1,\\,b_11c_2,\\,b_12d_0,\\,c_20c_2,\\,c_21b_1,\\,c_22d_0,\\,d_0012d_0,\\,e_20c_2,\\\\&&e_21d_0,\\,e_22e_2\\;.$ It has five states $a_1,b_1,c_2,d_0$ , and $e_2$ , with the output mod 3 residue in the index, and $5\\cdot 3=15$ transitions, with the input ternary digits written between the states.",
"Computation starts always at $a_1$ and follows the ternary digits $n-1=(t_l\\dots t_1t_0)_3$ , read from the least significant $t_0$ .",
"For example, $6-1=(12)_3$ sends us in two steps from $a_1$ to $d_0$ , and indeed $c_6=42$ is 0 mod 3.",
"For a bound on the number of states of the DFAO obtained from an algebraic power series see Bridy [4], and for further $f\\in \\mathbb {Z}[[x]]$ whose reduction $f_n$ mod $p$ (or mod $p^k$ ) can be computed by a DFAO see Rowland and Yassawi [9] (and the references therein).",
"Finally, we quote the two main results of Bostan, Christol and Dumas [3] which quantify the dependence of the complexity bound on $P$ and $p$ .",
"Theorem 5 in [3] states: Let $E$ be a polynomial in $F_p[x,y]_{h,d}$ such that $E(0,0)=0$ and $E_y(0,0)\\ne 0$ , and let $f\\in F_p[[x]]$ be its unique root with $f(0)=0$ .",
"Algorithm 1 computes the $N$ th coefficient $f_N$ of $f$ using $\\tilde{O}(d^3h^2p^{3d}\\log N)$ operations in $F_p$ .",
"Here $h$ and $d$ bound the $x$ - and $y$ -degree, respectively, and $\\tilde{O}(\\cdot )$ is $O(\\cdot )$ with polylogarithmic factors omitted.",
"Proof of this theorem we (roughly) followed in our expose.",
"Another algorithm, considerably more efficient, is presented in Theorem 11 in [3]: Let $E$ in $F_p[x,y]_{h,d}$ satisfy $E(0,0)=0$ and $E_y(0,0)\\ne 0$ , and let $f\\in F_p[[x]]$ be its unique root with $f(0)=0$ .",
"One can compute the coefficient $f_N$ of $f$ in $h^2(d+h)^2\\log N+\\tilde{O}(h(d+h)^5p)$ operations in $F_p$ .",
"Charles University, KAM MFF UK, Malostranské nám.",
"25, 118 00 Praha, Czechia"
]
] | 1605.06087 |
[
[
"Bounding the persistency of the nonlocality of W states"
],
[
"Abstract The nonlocal properties of the W states are investigated under particle loss.",
"By removing all but two particles from an $N$-qubit W state, the resulting two-qubit state is still entangled.",
"Hence, the W state has high persistency of entanglement.",
"We ask an analogous question regarding the persistency of nonlocality introduced in [Phys.",
"Rev.",
"A 86, 042113].",
"Namely, we inquire what is the minimal number of particles that must be removed from the W state so that the resulting state becomes local.",
"We bound this value in function of $N$ qubits by considering Bell nonlocality tests with two alternative settings per site.",
"In particular, we find that this value is between $2N/5$ and $N/2$ for large $N$.",
"We also develop a framework to establish bounds for more than two settings per site."
],
[
"Introduction",
"Entanglement and nonlocality are two manifestations of quantum correlations, both are essential ingredients of quantum theory.",
"They also play an important role in the field of quantum information [1].",
"For instance, entanglement is in the heart of quantum teleportation and plays a crucial role in quantum algorithms [2], [3].",
"Nonlocality, on the other hand, witnessed by the violation of Bell inequalities [13], reduces communication complexity [4], [5] and enables device-independent quantum information protocols [6], [7].",
"These protocols do not rely on a detailed knowledge of the internal working of the experimental devices used, thereby allowing secure cryptography involving untrusted devices [8] or expansion of secure random numbers [9], [10].",
"The quantification of entanglement and nonlocality in the bipartite case (i.e., the case when two parties share an entangled state) is relatively well understood.",
"However, the multipartite case is much less explored.",
"This is mainly due to the rapidly growing complexity of the problem with the number of parties.",
"For instance, for bipartite pure states, there is a unique measure of entanglement, however, for three or more parties, this is not true any more (see e.g., Refs.",
"[11], [12]).",
"Concerning nonlocality, there is a single tight Bell inequality for two binary settings per party, the Clauser-Horne-Shimony-Holt (CHSH) one [13], [14].",
"However, moving to three parties, the number of tight Bell inequalities already becomes 42 [15].",
"In fact, determining all Bell inequalities for growing number parties is an NP-hard problem [16], [17].",
"Bipartite and tripartite quantum nonlocality are different from a fundamental point of view as well.",
"While Gleason's theorem can be extended to the bipartite scenario, however, there remains a gap between the quantum and Gleason's correlations in the tripartite scenario [18].",
"One of the most famous multipartite quantum states is the W state [19] playing a crucial role in the physics of the interaction between light and matter.",
"Up to now, W states have been prepared in lots of experiments, e.g., photonic experiments can generate and characterize six qubit entangled states [20].",
"More recently, genuine 28-particle entanglement was detected in a Dicke-like state (a generalization of the W state) in a Bose-Einstein condensate [21].",
"W states are very robust to losses, hence suited to quantum information applications such as quantum memories [22].",
"For instance, by tracing out all but two parties from an $N$ -party W state, the remaining two-qubit state is still entangled, no matter how big $N$ is.",
"Hence, the W state shows a high persistency of entanglement against particle loss (actually, the highest possible persistency among $N$ qubit states [23]).",
"On the other hand, one may wonder how robust the nonlocality of the W state is with respect to particle loss.",
"In order to quantify this, Ref.",
"[24] introduces the persistency of nonlocality of $N$ -party quantum states $\\rho $ , $P_{NL}(\\rho )$ , which is the minimal number of particles to be removed for nonlocal quantum correlations to vanish.",
"Ref.",
"[24] investigates this measure for various classes of multipartite states including W states up to $N=7$ with two settings per site.",
"In this paper, we bound this value both from above and from below for any $N$ -qubit W states.",
"To do this, we refine the definition of $P_{NL}(\\rho )$ to account for Bell nonlocality involving $m$ settings per party.",
"This quantity will be called $P_{NL}^m(\\rho )$ .",
"Clearly, in the limit of large $m$ , $P_{NL}^m(\\rho )$ tends to $P_{NL}(\\rho )$ .",
"Our main result concerns the case of $m=2$ and we prove the bounds of $2N/5\\le P_{NL}^2(W_N)\\le N/2$ for $N$ large, featuring a relatively small gap between the upper and lower bounds.",
"The lower bound is based on an explicit construction of a class of Bell inequalities.",
"We also give a numerical framework to put reliable lower bounds on $P_{NL}^m(W_N)$ beyond two settings (up to $m=6$ ) and a tractable number of $N$ parties.",
"This numerical study supports that our analytical lower bounds for $P_{NL}^2(W_N)$ are tight.",
"There are recent papers which discuss the robustness of Dicke states [25] (and in particular the W state) to various types of noises [26], [28], [27].",
"Lower bounds also follow from these papers for the value of $P_{NL}^2(W_N)$ .",
"In particular, we obtain considerable improvement over the lower bounds presented in Ref. [28].",
"Notably, the persistency of nonlocality also gives a device-independent bound on the persistency of entanglement introduced in Ref. [29].",
"Other device-independent approaches to quantify multipartite entanglement including W states appeared in Refs.",
"[30], [31], [32].",
"On the other hand, multipartite W states are promising candidates to close the detection loophole [33] in multipartite Bell tests [34].",
"Such Bell violations would complement the experimental loophole-free violations obtained recently in the bipartite case [35].",
"The structure of the paper is as follows.",
"Section introduces notation.",
"Section proves a simple upper bound on the persistency of nonlocality $P^m_{NL}$ for W states and in general for any permutationally symmetric state with two settings per party (case $m=2$ ).",
"On the other hand, section presents lower bound values based on numerical investigations.",
"To this end, we first outline the numerical method, then show results for $m=2$ and also beyond $m=2$ up to $m=6$ .",
"In section , a family of Bell inequalities is presented (valid for any number of parties $N$ ), which allows us to obtain good lower bounds for $P_{NL}^2(W_N)$ .",
"The paper ends with a discussion in section ."
],
[
"Bell setup",
"Let us imagine the following Bell setup [13].",
"A quantum state $\\rho $ is shared between $n$ spatially separated systems, on which the local observers can conduct measurements.",
"We focus on binary outcome measurements in which case we may define the joint correlators by the following set of expectation values: $\\left\\lbrace \\langle \\mathcal {M}_{j_1}^{(1)}\\ldots \\mathcal {M}^{(n)}_{j_n}\\rangle \\right\\rbrace =\\left\\lbrace \\mathrm {Tr}{\\left(\\rho \\mathcal {M}_{j_1}^{(1)}\\otimes \\ldots \\otimes \\mathcal {M}_{j_n}^{(n)}\\right)}\\right\\rbrace $ with $j_l=0,1,\\ldots ,m$ and $l=1,\\ldots ,n$ .",
"We identify $\\mathcal {M}_0^{(l)}=\\leavevmode \\hbox{\\small 1\\normalsize \\hspace{-3.30002pt}1}$ and $\\mathcal {M}_{j_l}^{(l)}$ refers to the $j_l$ th $\\pm 1$ -valued observable of party $l$ .",
"The corresponding real vector of correlators defines a point in the $(1+m)^n$ dimensional space of probabilities.",
"Each member of the set (REF ) has an order, which is given by the amount of parties $o$ involving a non-trivial observable (i.e.",
"not involving $\\mathcal {M}_0^{(l)}=\\leavevmode \\hbox{\\small 1\\normalsize \\hspace{-3.30002pt}1}$ ).",
"In particular, those with $o=n$ are usually called full-correlators, while those with $o=1$ are called one-body correlators or marginal terms.",
"A multipartite (2-outcome) Bell inequality [13] is a linear function of the above correlators (REF ), $\\sum _{j_1=0}^{m}\\ldots \\sum _{j_n=0}^{m}{\\alpha _{j_1,\\ldots ,j_n}}\\langle \\mathcal {M}_{j_1}^{(1)}\\ldots \\mathcal {M}^{(n)}_{j_n}\\rangle \\le \\beta _c,$ where we denote by $\\beta _c$ the bound which holds for any local hidden variable model.",
"These are the correlations which the parties can simulate by merely using local strategies and some shared classical information.",
"The local correlations attainable this way forms a polytope, the so-called Bell polytope, whose extremal points consist of those vectors in (REF ) in which all correlators factorize, that is, $\\langle \\mathcal {M}_{j_1}^{(1)}\\ldots \\mathcal {M}^{(n)}_{j_n}\\rangle =\\langle \\mathcal {M}_{j_1}^{(1)}\\rangle \\cdot \\ldots \\cdot \\langle \\mathcal {M}_{j_n}^{(n)}\\rangle $ and the mean value of each single party $\\langle \\mathcal {M}_{j_l}^{(l)}\\rangle $ , $(l=1,\\ldots ,n, j_l>0)$ equals either $-1$ or $+1$ .",
"Let us define the persistency of nonlocality of a multipartite state $\\rho $ according to Ref.",
"[24] as follows.",
"Let us take the partial trace over $0<k<N$ systems $i_1,...,i_k\\in \\lbrace 1,...,N\\rbrace $ , and denote the $n=N-k$ -party reduced state by $\\rho _{red}$ .",
"The persistency of nonlocality of $\\rho $ , $P_{NL}(\\rho )$ , is defined as the minimal $k$ such that the reduced state $ \\rho _{red}$ becomes local for at least one set of subsystems $i_1,...,i_k$ .",
"In other words, the correlators (REF ) obtained from local measurements $\\mathcal {M}_{j_l}^{(l)}$ on $\\rho _{red}$ do not violate any Bell inequality.",
"If we allow at most $m$ different measurement settings in the Bell expression (REF ), we arrive at $P^m_{NL}(\\rho )$ which provides a lower bound to $P_{NL}(\\rho )$ and for $m\\rightarrow \\infty $ recovers $P_{NL}(\\rho )$ .",
"In this paper, we focus on computing $P^m_{NL}(\\rho )$ for the noiseless $N$ -qubit W state [19], $\\rho = |W_N\\rangle \\langle W_N|$ , where $|W_N\\rangle =\\frac{1}{\\sqrt{N}}\\left(|0\\ldots 01\\rangle + |0\\ldots 10\\rangle + \\ldots + |10\\ldots 0\\rangle \\right).$ We may consider this state as a state of an atomic ensemble, and we assume that $k$ particles are lost from this ensemble.",
"In that case, the reduced state contains $n=N-k$ particles, and the density matrix reads $\\rho (N,k)=\\frac{n}{N}|W_{n}\\rangle \\langle W_{n}| + \\frac{k}{N}|0^{\\otimes n}\\rangle \\langle 0^{\\otimes n}|.$ Since the W state is permutationally symmetric, the reduced state $\\rho (N,k)$ does not depend on the particular set of subsystems removed, which simplifies considerably the analysis of $P_{NL}^m(W_N)$ .",
"In the next section we provide an upper bound of $N/2$ for $P_{NL}^{m=2}(W_N)$ in case of arbitrary even $N$ number of parties, whereas in Sec.",
"we bound this quantity by $2N/5$ from below."
],
[
"An upper bound for the persistency of nonlocality of the W state",
"We first prove the following lemma: Lemma 1 Let us have a $2n$ -qudit permutationally invariant state, $\\rho \\in ({\\mathbb {C}}^d)^{\\otimes 2n}$ , where $d\\ge 2$ and $n\\ge 1$ .",
"By tracing out any $n$ qudits, the remaining $n$ -qudit system $\\rho _{red}\\in ({\\mathbb {C}}^d)^{\\otimes n}$ cannot violate any two-setting $n$ -party Bell inequality with arbitrary number of outcomes.",
"We take $N=2n$ and denote the $n$ -qudit reduced state of an $N$ -qudit permutationally invariant state $\\rho $ by $\\rho _{red}$ and the two measurements conducted on station $l\\in \\lbrace 1,\\ldots ,n\\rbrace $ by $M^{(l)}_{a_l|j_l}$ , where $j_l=1,2$ .",
"By permutationally invariance we mean that the exchange of any two qudits of the $N$ -qudit state $\\rho $ does not change the state itself.",
"Then, the $n$ -particle joint probability distribution reads $&P(a_1,a_2,...,a_n|j_1,j_2,...,j_n)\\nonumber \\\\&=\\mathrm {Tr}\\left(\\rho _{red}\\mathcal {M}^{(1)}_{a_1|j_1}\\otimes \\mathcal {M}^{(2)}_{a_2|j_2}\\ldots \\otimes \\mathcal {M}^{(n)}_{a_n|j_n}\\right),$ It is not difficult to see that the same probability distribution can be achieved in the following way: Let us redistribute the permutationally invariant state $\\rho $ between $n$ parties such that the $l$ th qudit pair $(l,l+n)$ belongs to party $l\\in \\lbrace 1,\\ldots ,n\\rbrace $ (so that each party owns two qudits).",
"Then party $l$ performs measurement $\\mathcal {M}^{(l)}_{a_l|j_l=1}$ on the first qudit and measurement $\\mathcal {M}^{(l)}_{a_l|j_l=2}$ on the second qudit of the $l$ th pair.",
"This generates the same distribution as of Eq.",
"(REF ).",
"Since for any $l$ these two measurements act on different subspaces, they are commuting.",
"However, Bell inequality violation is not possible (with any number of outcomes) if the two alternative measurements for each party are pairwise commuting [36].",
"Since the W state is a permutationally invariant multiqubit state (with local dimension $d=2$ ), Lemma 1 above directly applies to our situation, hence we get the upper bound $P_{NL}^m(W_N)\\le n$ for $m=2$ with $N=2n$ .",
"In other words, $P_{NL}^2(W_N)\\le N/2$ for even $N$ .",
"Some notes are in order.",
"(i) It is straightforward to extend Lemma 1 to more than two settings as well.",
"For multiple settings, we get the general upper bound $P_{NL}^m(W_N)\\le (m-1)n$ with $N=mn$ .",
"However, we conjecture that these upper bounds are not tight in general.",
"For $m=2$ , a gap between the lower and upper bound values for the $6\\le N\\le 20$ -qubit W states are supported by a numerical study performed in Sec. .",
"For $m$ infinite, the trivial upper bound $P_{NL}(W_N)\\le N-1$ follows by plugging $n=1$ in the above formula.",
"If this bound happened to be tight, it would imply that the two-qubit reduced state $\\rho _{red}=(2/N)|\\psi ^+\\rangle \\langle \\psi ^+| +(1-(2/N))|00\\rangle \\langle 00|$ of the $W_N$ state was Bell nonlocal.",
"For large $N$ this is very unlikely, since the weight of the entangled part $|\\psi ^+\\rangle \\langle \\psi ^+|$ goes to zero.",
"In fact, a recent computer study in Ref.",
"[37] suggests that the state $\\rho (p)=p|\\psi ^+\\rangle \\langle \\psi ^+| +(1-p)|00\\rangle \\langle 00|$ is local for $p<1/\\sqrt{2}$ .",
"Similarly, A. Amirtham conjectures in Ref.",
"[38] that the state $\\rho (p)$ is local for $p<2/3$ .",
"(ii) The permutational invariance property of the state is crucial in the above state.",
"If the multipartite state does not possess this high symmetry, e.g.",
"it only obeys translational invariance, the above theorem does not hold true any more.",
"Let us illustrate this with a simple example.",
"We consider a 4-qubit translationally invariant state $|\\psi \\rangle =|\\psi ^+\\rangle _{13}|\\psi ^+\\rangle _{24}$ , where $|\\psi ^+\\rangle _{ij}=(|0\\rangle _i|1\\rangle _j+|1\\rangle _i|0\\rangle _j)/\\sqrt{2}$ .",
"Let us trace out particles 2 and 4 (constituting half of the 4 particles), and as a result we get a maximally entangled pair of qubits, which violates maximally the bipartite CHSH-Bell inequality [14].",
"Nevertheless, translationally invariant systems also impose certain restrictions which can be exploited in Bell scenarios as studied in Ref.",
"[39]."
],
[
"A lower bound for the persistency of nonlocality of the W state",
"We now give a numerical procedure which allows us to get useful (and often tight) lower bounds to $P_{NL}^m(W_N)$ .",
"We note that this procedure with some modification can also be applied to generic permutationally invariant multiqubit states.",
"We consider the Bell violation of the following one-parameter family of states: $ \\rho (n,p)=(1-p)|W_{n}\\rangle \\langle W_{n}| +p|0^{\\otimes n}\\rangle \\langle 0^{\\otimes n}|.$ Notice that the state (REF ) reproduces (REF ) with $p=k/N$ .",
"Hence, given an $n$ -party $m$ -setting Bell inequality which is violated by state (REF ) with a given critical value, $p_{crit}$ , we get the following lower bound on the persistency: $P_{NL}^m(W_N)=N-n+1,$ where $N=\\lfloor n/(1-p_{crit})\\rfloor $ , where $\\lfloor x\\rfloor $ maps a real number $x$ to the largest previous integer.",
"Hence, in order to get good lower bounds to $P_{NL}^m(W_N)$ our task reduces to get good upper bounds to $p_{crit}$ .",
"To this end, we introduce the following linear programming based numerical method.",
"Let us stick to $m=2$ settings (generalization to more settings is straightforward).",
"For simplicity we assume that all parties measure the same qubit observables, that is, $\\mathcal {M}^{(1)}_j=\\mathcal {M}^{(2)}_j=\\ldots =\\mathcal {M}^{(n)}_j$ for $j=1,2$ .",
"Moreover, due to symmetry of the states, we assume these observables are coplanar, lying on the $X-Z$ equatorial plane.",
"Other works (e.g., Refs.",
"[40], [44], [31]) maximizing Bell functionals using the W state rely on the same symmetry considerations.",
"With this simplification, we have two optimization parameters.",
"Hence, for the state $W_n$ and the above measurements, the $(1+m)^n=3^n$ -dimensional correlation point $P_1$ given by the set of correlators (REF ) is defined by two angles.",
"Likewise, we define the correlation point $P_0$ generated by the same measurements and the state $\\rho =|0^{\\otimes n}\\rangle \\langle 0^{\\otimes n}|$ in Eq.",
"(REF ).",
"Geometrically, the correlations accessible in a local hidden variables theory form a polytope, the so-called Bell polytope, with vertices defined by deterministic classical strategies for a fixed scenario of $n$ parties and $m$ settings (see Ref.",
"[6] for a review).",
"The two correlation points $P_0$ and $P_1$ are situated within this space.",
"Since the product state $|0^{\\otimes n}\\rangle \\langle 0^{\\otimes n}|$ is local, point $P_0$ sits inside the Bell polytope, whereas point $P_1$ depends on the two measurement angles and may well fall outside the Bell polytope (see Fig.",
"REF ).",
"For a given $p$ in (REF ), the corresponding correlation point is $(1-p)P_1+pP_0$ , and $p_{crit}$ is given by the intersection of the line joining points $P_0$ and $P_1$ with the boundary of the polytope (see Figure REF ).",
"Given the two measurement angles, standard linear programming allows us to compute $p_{crit}$ and the underlying facet, which corresponds to a Bell inequality.",
"Figure: A schematic view of the Bell polytope (region L) and thespace attainable with quantum systems (region Q).",
"The correlationpoint P 1 P_1 corresponding to the W N W_N state is outside the Bellpolytope, whereas the point P 0 P_0 corresponding to the productstate falls inside the Bell polytope.",
"p crit p_{crit} is fullydetermined by points P 0 ,P 1 P_0,P_1 and the facets of the polytope.Clearly, the above described procedure works for $m>2$ as well.",
"In particular, we have chosen a given Bell scenario ($n$ parties, $m$ settings) and by varying the $m$ angles, we maximized the value of $p_{crit}$ .",
"We note that this search is a heuristic one and as such it is not guaranteed to terminate in a global maximum of $p_{crit}$ .",
"However, the obtained value still defines a lower bound to $P^m_{NL}$ in (REF ).",
"The critical $p$ values obtained in function of $n$ and $m$ are displayed in Fig.",
"REF .",
"We may observe that, as $m$ increases, the affordable number of parties $n$ decreases.",
"For the simplest case of $m=2$ , numerically we could afford $n=14$ .",
"In addition, we had to resort to a symmetrization procedure of the Bell polytope introduced in [41] which considerably reduces the complexity of the problem in order to attain $n=14$ .",
"Figure: For ρ(n,p)\\rho (n,p) states of nn qubits (), the graph shows lower bounds on the thresholdprobability p crit p_{crit} obtained using our numerical method up tom=6m=6 measurement settings.Plugging the $p_{crit}$ values shown in Fig.",
"REF into formula (REF ), we get Table REF , where the computed persistencies $P_{NL}^m$ are shown for $m\\le 6$ .",
"This way we get entries in the table only for $N$ satisfying $N=\\lfloor n/(1-p_{crit})\\rfloor $ , but a slight modification allows us to obtain numbers for any $N$ parties: Fix $N$ , and choose the largest $k$ integer such that $p_{crit,n-k}>k/N$ , where $p_{crit,n-k}$ stands for $p_{crit}$ evaluated by the number of parties $n-k$ .",
"Then a lower bound on the persistency $P_{NL}^m$ for the $N$ -qubit W state is given by $P_{NL}^m=k+1$ .",
"Table: Lower bound values for persistency ofnonlocality P NL m P_{NL}^m for the W N W_N state with m=2,...,6m=2,\\ldots ,6binary-outcome measurement settings per party.",
"The results arebased on the numerical method described in the text.In Table REF , for $m=2$ , the first entries $N=2,3,\\ldots ,7$ match the numbers from previous study [24].",
"However, by fixing $N$ and going to higher number of settings $m$ sometimes we get a higher persistency of nonlocality.",
"For instance, in case of $N=6$ parties, $P_{NL}^5=3$ , which is to be compared to $P_{NL}^2=2$ .",
"For $m=2$ , we extracted the optimal Bell inequalities corresponding to $p_{crit}$ for various $n$ values.",
"For $n$ even, a common structure has been found.",
"In fact, they turned out to be members of a family of Bell inequalities valid for any $n$ even.",
"Details of this class of inequalities are presented in Sec. .",
"We optimized the quantum value of these inequalities in function of the two measurement angles.",
"Fig.",
"REF shows results for $p_{crit}$ up to $n=50$ of this family.",
"We also show results using the ansatz that the two measurement settings are $Z$ and $X$ , in which case $p_{crit}=(2n-4)/(5n-2)$ for $n$ even.",
"According to the figure, for larger $n$ the $(Z,X)$ pair of settings become close to optimal.",
"The $p_{crit}$ values have been also studied in Ref.",
"[28] for specific families of known Bell inequalities from the literature, with the best lower bound values coming from the WWWZB inequalities [42].",
"For comparison, these $p_{crit}$ values are displayed.",
"For $n=50$ , $p_{crit}=0.3142$ , which is suboptimal with respect to our $p_{crit}=(2n-4)/(5n-2)\\simeq 0.3871$ .",
"Figure: For ρ(n,p)\\rho (n,p) states of nn qubits (), thegraph shows lower bounds on the threshold probability p crit p_{crit}obtained using our numerical method for m=2m=2 measurementsettings.",
"These bounds coincide with the bounds coming from ourclass of Bell inequalities for 2 measurement settings up to n=14n=14and we conjecture to be the same for any even nn particles.",
"Weshow results for our Bell inequalities with optimized andnon-optimized settings (in the latter case, the two observablesare the Pauli operators ZZ and XX).",
"The lower curve reproducesthe results of Ref.",
"in case of the WWWZBinequalities .Finally, Fig.",
"REF shows the upper bound values due to section and the lower bound values for $P^2_{NL}(W_N)$ up to $N=50$ due to our Bell family.",
"We conjecture that the exact value for $P^2_{NL}(W_N)$ is defined by the lower curve (that is, no better Bell inequalities than our family in section exist in this respect).",
"For instance, the best lower bound for $P^2_{NL}(W_N)$ so far comes from the WWWZB inequalities analyzed in Ref.",
"[28], which lies below our curve for the lower bound, and according to the conjecture in Ref.",
"[28] it goes to $N/3$ in the limit of large $N$ .",
"On the other hand, our asymptotic lower bound value of $2N/5$ is considerable higher.",
"It is also worth noting that the upper bound value on $P^2_{NL}$ shown in the figure holds true for any permutationally invariant $N$ -qubit state.",
"Hence, Dicke states with more than one excitation, as a notable subset of permutationally invariant states, cannot perform significantly better than $W$ states in the two-setting case.",
"Figure: For WW states of NN qubits, the graph shows upperbounds (marked by a triangle pointing down) and lower bounds (marked by triangles pointing up) on the persistency of nonlocalityP NL m=2 (W N )P_{NL}^{m=2}(W_N).In Ref.",
"[28], the Dicke states (including the $W_N$ state) were analyzed in terms of two decoherence models: loss of particles and loss of excitations.",
"The first one relates to the measure $P_{NL}$ discussed in this paper, whereas the latter one is related to $p_{crit}$ in formula (REF ).",
"Indeed, if we start from an $n$ -qubit W state which is effected by a decoherence where in each mode an excitation has a probability $p$ of being lost, it brings the W state into (REF ).",
"Appendix A shows a derivation of this result.",
"Next section gives all the details, including the quantum and local bounds, of our particular family of two-setting Bell inequalities providing $p_{crit}=(2n-4)/(5n-2)$ for even $n$ .",
"In case of large $n$ , this goes to $2/5$ which we conjecture to be the largest achievable critical value among all two-setting Bell inequalities."
],
[
"A family of two-setting multipartite Bell inequalities",
"Standard form of permutationally symmetric Bell inequalities.– The Bell inequality to be considered consists of correlators (REF ) which are invariant under any permutation of the parties.",
"By imposing the permutational symmetry, one requires that the expectation values $\\langle \\mathcal {M}_{j_1}^{(1)}\\ldots \\mathcal {M}^{(n)}_{j_n}\\rangle $ and $\\langle \\mathcal {M}_{j_1}^{\\sigma (1)}\\ldots \\mathcal {M}^{\\sigma (n)}_{j_n}\\rangle $ are the same, where $\\sigma :\\lbrace 1,\\ldots ,n\\rbrace \\rightarrow \\lbrace \\sigma (1),\\ldots ,\\sigma (n)\\rbrace $ is an arbitrary permutation of the set $\\lbrace 1,\\ldots ,n\\rbrace $ .",
"Let us denote by $S_n$ all permutations of the set $\\lbrace 1,\\ldots ,n\\rbrace $ .",
"Then, it will be useful to define $S^o_r \\equiv \\sum _{\\sigma \\in S_n}\\langle \\mathcal {M}_{j_1}^{\\sigma (1)}\\ldots \\mathcal {M}^{\\sigma (n)}_{j_n}\\rangle ,$ where the sum ranges over all permutations of the set $\\lbrace 1,\\ldots ,n\\rbrace $ .",
"In $S^o_r$ above, $o$ is the order of correlators and $r$ is the number of 1's occurring in the list $\\lbrace j_1,\\ldots ,j_n\\rbrace $ .",
"Let us now define the symmetrized correlation vector by the above ordered real vectors as follows $\\vec{S} \\equiv \\lbrace S^o_r\\rbrace ^{o=1,\\ldots ,n}_{r=0,\\ldots ,o}.$ In a similar way, we define the vector of coefficients associated with $\\vec{S}$ as $\\vec{\\alpha }\\equiv \\lbrace \\alpha ^o_r\\rbrace ^{o=1,\\ldots ,n}_{r=0,\\ldots ,o}.$ As a result, we arrive at the general form of a permutationally symmetric Bell inequality: $I \\equiv \\vec{\\alpha }\\cdot \\vec{S} = \\sum _{\\begin{array}{c}{o=1,\\ldots ,n}\\\\{r=0,\\ldots ,o}\\end{array}}{\\alpha ^o_r S^o_r}\\le \\beta _c,$ where $\\beta _c$ is the local maximum, which holds for any local hidden variable model.",
"It is worth noting that the WWWZB class [42] contains only full-correlator terms ($o=n$ ), whereas our construction of Bell inequalities turns out to contain all different orders, starting from marginal terms ($o=1$ ) up to full-correlation terms ($o=n$ ).",
"There also exist permutationally symmetric $n$ -party Bell inequalities involving only first and second order terms ($o=1,2$ ) [43].",
"The usefulness of these kind of inequalities in the particle loss model is an open and interesting question in our view.",
"As an illustrative example, let us discuss the case of 4 particles ($n=4$ ).",
"In this case, $\\vec{S}$ looks as follows $\\vec{S} = \\lbrace S^1_1,S^1_0;S^2_2,S^2_1,S^2_0;S^3_3,S^3_2,S^3_1,S^3_0;S^4_4,S^4_3,S^4_2,S^4_1,S^4_0\\rbrace ,$ where we used semicolon (;) in order to separate components with different order $o$ .",
"By identifying $\\mathcal {M}_0^{(l)}=1$ , $l=1,\\ldots ,4$ and $\\mathcal {M}_j^{(1)}=A_j$ , $\\mathcal {M}_j^{(2)}=B_j$ , $\\mathcal {M}_j^{(3)}=C_j$ , $\\mathcal {M}_j^{(4)}=D_j$ for $j=1,2$ in Eq.",
"(REF ), we have the following mean values displayed for some particular cases (neglecting the mean value signs): $S^1_1 =& 6(A_1 + B_1 + C_1 + D_1),\\nonumber \\\\S^2_2 =& 4(A_1B_1 + A_1C_1 + A_1D_1 + B_1C_1 + B_1D_1 + C_1D_1),\\nonumber \\\\S^2_0 =& 4(A_2B_2 + A_2C_2 + A_2D_2 + B_2C_2 + B_2D_2 + C_2D_2),\\nonumber \\\\S^3_1 =& 2(A_1B_2C_2+A_1B_2D_2+A_1C_2D_2)\\nonumber \\\\& +2(A_2B_1C_2+A_2B_1D_2+B_1C_2D_2)\\nonumber \\\\& +2(A_2B_2C_1+A_2C_1D_2+B_2C_1D_2)\\nonumber \\\\& + 2(A_2B_2D_1+A_2C_2D_1+B_2C_2D_1),\\nonumber \\\\S^4_0 =& 24A_2B_2C_2D_2,$ where we recall that $o$ denotes the order in the superindex of $S^o_r$ , whereas $r$ is the number of occurrences of subindex 1 in $S^o_r$ .",
"For instance, $S^3_1$ above consists of all length-3 sequences from $A,B,C$ and $D$ letters with the occurrence of a single subindex 1.",
"The specific class of Bell inequalities.– Let us now give the explicit form of a Bell inequality $I$ defined by the vector of coefficients $\\vec{\\alpha }$ in Eq.",
"(REF ).",
"This family has been extracted from the numerical method of Sec. .",
"In particular, we introduce a family of Bell inequalities $I_n$ valid for any even number of particles $n$ as follows $I_n \\equiv \\vec{\\alpha }_n\\cdot \\vec{S}_n = \\sum _{\\begin{array}{c}{o=1,\\ldots ,n}\\\\{r=0,\\ldots ,o}\\end{array}}{\\alpha ^o_r S^o_r}\\le \\beta _{c,n}.$ The vector of coefficients $\\vec{\\alpha }=\\lbrace \\alpha ^o_r(o=1,\\ldots ,n, r=0,\\ldots ,o)\\rbrace $ reads as $\\vec{\\alpha }={\\left\\lbrace \\begin{array}{ll}\\alpha ^{2k}_0 = F_{k,n}, & k=1,\\ldots ,n/2\\\\\\alpha ^{2k-1}_1 = G_{k,n}+2w_n\\delta _{k,1}, & k=1,\\ldots ,n/2\\\\\\alpha ^2_2 = -w_n,\\\\\\alpha ^o_r = 0, & \\text{otherwise}\\end{array}\\right.",
"}$ where $\\delta _{i,j}$ is the Kronecker delta (1 if $i=j$ , 0 otherwise), and the functions entering eq.",
"(REF ) above are $F_{k,n}&=(-1)^k\\frac{n-2}{2}(n+1-2k)\\binom{n/2}{k},\\nonumber \\\\G_{k,n}&=(-1)^k\\frac{n+2}{2}n\\binom{n/2-1}{k-1},\\nonumber \\\\w_n&=\\frac{n(n-1)(n+2)}{2^{4-n}\\binom{n}{n/2}}.$ Especially for $n=4$ , our 4-party Bell inequality looks as follows: $I_4 = 12 S^1_1 - 12 S^2_2 - 6S^2_0 + 12S^3_1 + S^4_0 \\le 128,$ where the middle terms appear explicitly in (REF ) and the local bound is due to formula (REF ) given in the next subsection for the general $n$ -party case.",
"Local bound.– The local maximum, which is the maximum value one obtains using local resources only, is given by $\\beta _{c,n}=n!\\left(w_n-\\frac{(n-2)(n+1)}{2}\\right)$ in case of any even number of particles $n$ , where $w_n=\\frac{n(n-1)(n+2)}{2^{4-n}\\binom{n}{n/2}}.$ We have checked the validity of the above local bound $\\beta _{c,n}$ for $n\\le 16$ ($n$ even) by listing all possible deterministic strategies (i.e., vertices of the Bell polytope).",
"These are the strategies for which all correlators factorize (see eq.",
"REF ) and each single party marginal $\\langle \\mathcal {M}_{j_l}^{(l)}\\rangle $ , $(l=1,\\ldots ,n, j_l=1,2)$ equals either $-1$ or $+1$ .",
"One of the above deterministic strategies, due to linearity of the Bell functional, will provide the local maximum.",
"Notice that $\\langle \\mathcal {M}_0^{(l)}\\rangle $ , ($l=1,\\ldots ,n$ ) equals $+1$ by definition.",
"Below we provide an analytical proof of formula (REF ) for any even $n$ .",
"We checked with brute-force computations that for smaller values of $n$ the inequalities are far from tightness, that is, they do not define a facet of the Bell polytope.",
"We conjecture that they are not tight for larger $n$ as well.",
"Here follows the proof of the local limit (REF ) for even $n$ .",
"First let us fix notation.",
"For a given local deterministic strategy let us define a four-tuple ($a$ ,$b$ ,$c$ ,$d$ ) which counts the number of parties whose marginal expectations $\\lbrace \\langle \\mathcal {M}^{(l)}_1\\rangle ,\\lbrace \\langle \\mathcal {M}^{(l)}_2\\rangle \\rbrace $ are the respective pairs ($\\lbrace 1,1\\rbrace $ , $\\lbrace 1,-1\\rbrace $ , $\\lbrace -1,1\\rbrace $ , $\\lbrace -1,-1\\rbrace $ ).",
"By definition we have $a+b+c+d=n$ .",
"In case of permutationally invariance under party exchange, a four-tuple ($a,b,c,d$ ) represents uniquely a deterministic strategy.",
"Hence, our task is to calculate the Bell value $\\vec{\\alpha }_n\\cdot \\vec{S}_n$ for all possible integer-valued four-tuples $a,b,c,d\\ge 0$ fulfilling $a+b+c+d=n$ , and then pick the maximum value out of this set.",
"This defines the local maximum $\\beta _{c,n}$ attainable using classical resources.",
"In the proof below, all positive integer four-tuples $a,b,c,d$ are understood to sum up to $n$ , where $n$ is even.",
"Let us introduce the permanent of an $n\\times n$ matrix $A=(a_{i,j})$ , $(i,j=1,\\ldots ,n)$ .",
"It is defined as $\\text{perm}(A) = \\sum _{\\sigma \\in S_n}\\prod _{i=1}^n{a_{i,\\sigma (i)}},$ where $\\sigma $ is a permutation over the set $\\lbrace 1,\\ldots ,n\\rbrace $ .",
"Using the above definition for the permanent, $\\vec{S} = \\lbrace S^o_r\\rbrace $ ($o=1,\\ldots ,n$ , $r=0,\\ldots ,o$ ) is given by the components $S^o_r=\\text{perm}(M^{o,r})$ for a given deterministic strategy $a,b,c,d$ , where $M^{o,r}(a,b,c,d)$ is an $n\\times n$ , $\\pm 1$ -valued matrix whose components $u,v=1,\\ldots ,n$ are defined as follows $M^{o,r}_{u,v}(a,b,c,d)=\\nonumber $ ${\\left\\lbrace \\begin{array}{ll}-1,& \\text{if } \\quad n-o<u\\le n-o+r\\quad \\text{and }\\quad a+b<v\\\\-1,& \\text{if } \\quad n-o+r< u\\quad \\text{and }\\\\&\\quad \\quad (a<v\\le a+b\\quad \\text{or }\\quad a+b+c<v)\\\\+1, & \\text{otherwise.}\\end{array}\\right.",
"}$ However, for $S^o_1=\\text{perm}(M^{o,1})$ , $o=1,\\ldots ,n$ , there exists a closed form expression as well (which we will make use of later): $&S^o_1 =(o-1)!",
"(n-o)!\\nonumber \\\\&((a-c)\\sum _{k=0}^n{(-1)^k\\binom{n-a-c}{k}\\binom{a+c-1}{o-1-k}}+(a+2b+c-n)\\nonumber \\\\&\\sum _{k=0}^n{(-1)^k\\binom{n-a-c-1}{k}\\binom{a+c}{o-1-k}}).$ This expression comes from an expansion of matrix $M^{o,r}$ in terms of row $(n-o)$ .",
"Let us now define two auxiliary functions $f_{1,2}$ which later will prove to be useful: $f_1(n,b+d)={\\left\\lbrace \\begin{array}{ll}2,& \\text{if } 2(b+d)=n\\\\1,& \\text{if } |2(b+d)-n|=2\\\\0, & \\text{otherwise}\\end{array}\\right.",
"}$ and $f_2(a+b+c+d)={\\left\\lbrace \\begin{array}{ll}a+b-c-d,& \\text{if } a+c=b+d\\\\a-c,& \\text{if } a+c=b+d+2\\\\b-d,& \\text{if } a+c=b+d-2\\\\0, & \\text{otherwise}\\end{array}\\right.",
"}$ Recall that $a+b+c+d=n$ .",
"Let us also recall that the Bell expression for a particular deterministic strategy looks as follows $I(a,b,c,d)= \\vec{\\alpha }\\cdot \\vec{S}(a,b,c,d),$ where $\\vec{\\alpha }$ defines the Bell coefficients through eq.",
"(REF ).",
"Let us divide the terms appearing in (REF ) into three distinct cases $\\vec{\\alpha }=\\vec{\\alpha }_{I} + \\vec{\\alpha }_{II} + \\vec{\\alpha }_{III}$ as follows: $\\vec{\\alpha }_{I}={\\left\\lbrace \\begin{array}{ll}\\alpha ^{2k-1}_{I,1} = 2w_n\\delta _{k,1}, & k=1,\\ldots ,n/2\\\\\\alpha ^2_{I,2} = -w_n,\\\\\\alpha ^o_{I,r} = 0, & \\text{otherwise},\\end{array}\\right.",
"}$ $\\vec{\\alpha }_{II}={\\left\\lbrace \\begin{array}{ll}\\alpha ^{2k}_{II,0} = F_{k,n}, & k=1,\\ldots ,n/2\\\\\\alpha ^o_{II,r} = 0, & \\text{otherwise},\\end{array}\\right.",
"}$ $\\vec{\\alpha }_{III}={\\left\\lbrace \\begin{array}{ll}\\alpha ^{2k-1}_{III,1} = G_{k,n}, & k=1,\\ldots ,n/2\\\\\\alpha ^o_{III,r} = 0, & \\text{otherwise}.\\end{array}\\right.",
"}$ Using the above formulas, one can show (after tedious but straightforward calculations) that $\\vec{\\alpha }_{I}\\cdot \\vec{S}(a,b,c,d) =& w_n(n-2)!\\left(n(n-1)+4(c+d)(1-c-d)\\right)\\nonumber \\\\\\vec{\\alpha }_{II}\\cdot \\vec{S}(a,b,c,d) =& \\left(\\frac{n}{2}-1\\right)(2^{n-1}\\left(\\frac{n}{2}+1\\right)\\left(\\frac{n}{2}!\\right)^2 f_1(n,b+d)\\nonumber \\\\&-(n+1)!",
")\\nonumber \\\\\\vec{\\alpha }_{III}\\cdot \\vec{S}(a,b,c,d) =& -f_2(a,b,c,d)2^{n-1}\\frac{n}{2}!\\left(\\frac{n}{2}+1\\right)!$ Furthermore, summing up the above three equations (REF ), we arrive at $&I(a,b,c,d) = \\vec{\\alpha }\\cdot \\vec{S}(a,b,c,d) = \\nonumber \\\\&\\left(\\frac{n}{2}!\\right)^2 2^{n-2}\\left(\\frac{n}{2}+1\\right)2\\left(f_1(n,b+d)\\left(\\frac{n}{2}-1\\right)-f_2(a,b,c,d)\\right)\\nonumber \\\\&+ \\left(\\frac{n}{2}!\\right)^2 2^{n-2}\\left(\\frac{n}{2}+1\\right)\\left(\\frac{n}{2}(n-1)+2(c+d)(1-c-d)\\right)\\nonumber \\\\&-(\\frac{n}{2}-1)(n+1)!$ Now we separate the above expression into four different cases according to the numbers $a,b,c,d$ occurring in the auxiliary functions $f_1$ and $f_2$ in eqs.",
"(REF ,REF ).",
"After a bit of manipulation, we arrive at $&I(a,b,c,d)=\\left(\\frac{n}{2}!\\right)^2 2^{n-2}\\left(\\frac{n}{2}+1\\right)x\\nonumber \\\\&+ \\left(\\frac{n}{2}!\\right)^2 2^{n-2}\\left(\\frac{n}{2}+1\\right)\\left(\\frac{n}{2}(n-1)+2(c+d)(1-c-d)\\right)\\nonumber \\\\&-(\\frac{n}{2}-1)(n+1)!$ where $x={\\left\\lbrace \\begin{array}{ll}-4+4(c+d),& \\text{if } a+c=b+d\\\\-a+b+3c+d-2,& \\text{if } a+c=b+d+2\\\\a-b+c+3d-2,& \\text{if } a+c=b+d-2\\\\0, & \\text{otherwise.}\\end{array}\\right.",
"}$ Let us subtract the conjectured $\\beta _{c,n}$ (REF ) from $I(a,b,c,d)$ in formula (REF ).",
"Notice that we end the proof once we find that $\\beta _{c,n}-I(a,b,c,d)\\ge 0$ for all possible positive integers $a,b,c,d$ with $a+b+c+d=n$ .",
"Dividing by $\\left(\\frac{n}{2}!\\right)^2 2^{n-2}$ and after some lengthy manipulation, we get $4\\left(\\frac{n}{2}+1\\right)-\\left(\\frac{n}{2}+1\\right)y\\ge 0,$ with $y={\\left\\lbrace \\begin{array}{ll}4(c+d)+2(c+d)(1-c-d),& \\text{if } a+c=b+d\\\\4c+2(c+d)(1-c-d),& \\text{if } a+c=b+d+2\\\\4d+2(c+d)(1-c-d),& \\text{if } a+c=b+d-2\\\\4+2(c+d)(1-c-d), & \\text{otherwise.}\\end{array}\\right.",
"}$ It is straightforward to check that in each above case the maximum allowed value of $y$ is 4.",
"Substituting this value back into (REF ), we obtain that inequality (REF ) is never violated.",
"This completes the proof of the local bound $\\beta _{c,n}$ expressed by formula (REF ).",
"Quantum violation.– Hereby we give a closed form of the vector $\\vec{S}$ in case of qubit observables and an $n$ -qubit quantum state.",
"In quantum theory, we have the expectation value defined by Eq.",
"(REF ).",
"We specify the $n$ -qubit state $\\rho $ to be the one-parameter family of states given by $\\rho (n,p)$ in eq.",
"(REF ).",
"The qubit observables, on the other hand, are chosen as $\\mathcal {M}_1^{(l)}=Z$ and $\\mathcal {M}_2^{(l)}=X$ for all $l=1,\\ldots ,n$ , where $Z=|0\\rangle \\langle 0|-|1\\rangle \\langle 1|$ and $X=|0\\rangle \\langle 1|+|1\\rangle \\langle 0|$ are the $2\\times 2$ Pauli matrices.",
"Also, $\\mathcal {M}_0^{(l)}=\\leavevmode \\hbox{\\small 1\\normalsize \\hspace{-3.30002pt}1}$ for all $l=1,\\ldots ,n$ , by definition.",
"Borrowing formulas from Ref.",
"[44], we find a closed form expression for $\\vec{S} \\equiv \\lbrace S^o_r\\rbrace $ ($o=1,\\ldots ,n$ , $r=0,\\ldots ,o$ ).",
"Let us write $S^o_r=(1-p) P^o_r + p Q^o_r$ where we have $Q^o_r =& n!\\delta _{o,r},\\nonumber \\\\P^o_r =& (n-1)!\\left((n-2r)\\delta _{o,r} + 2\\delta _{o,r+2}\\right),$ where $\\delta _{i,j}$ stands for the Kronecker delta.",
"Critical value of $p$ .– Our next task is to compute the critical value $p_{crit}$ in function of $n$ for which we have $\\vec{\\alpha }\\cdot \\vec{S} = \\beta _{c,n}$ , where the components of $\\vec{S}$ in Eq.",
"(REF ) contain $p$ as a parameter.",
"Note that $\\vec{\\alpha }$ and $\\beta _{c,n}$ are defined through Eq.",
"(REF ) and Eq.",
"(REF ), respectively.",
"By substitution we arrive at $\\vec{\\alpha }\\cdot \\vec{P} &= \\sum _{\\begin{array}{c}{o=1,\\ldots ,n}\\\\{r=0,\\ldots ,o}\\end{array}}{\\alpha ^o_r P^o_r}=n!\\left(w_n-\\frac{(n-2)(n-1)}{2}\\right),\\nonumber \\\\\\vec{\\alpha }\\cdot \\vec{Q} &= \\sum _{\\begin{array}{c}{o=1,\\ldots ,n}\\\\{r=0,\\ldots ,o}\\end{array}}{\\alpha ^o_r Q^o_r}=n!\\left(w_n-\\frac{n(n+2)}{2}\\right).$ Next, applying the criterion $\\vec{\\alpha }\\cdot \\vec{S} = (1-p)\\vec{\\alpha }\\cdot \\vec{P} + p\\vec{\\alpha }\\cdot \\vec{Q}=\\beta _{c,n}$ , we have the critical value for $p$ as follows, $p_{crit}=\\frac{\\beta _{c,n}-\\vec{\\alpha }\\cdot \\vec{P}}{\\vec{\\alpha }\\cdot \\vec{Q}-\\vec{\\alpha }\\cdot \\vec{P}}=\\frac{2n-4}{5n-2},$ where we used formulas (REF ) above.",
"Note that the formula for the critical value of $p=(2n-4)/(5n-2)$ above goes to $p_{crit}\\rightarrow 2/5$ as the number of particles $n$ goes to infinity."
],
[
"Discussion",
"The multipartite W state is an important state relevant to the interaction between light and matter.",
"We addressed the persistency of the nonlocality $P_{NL}$ of this state both by numerical and analytical means.",
"In case of two-setting measurements ($m=2$ ) we could pin down the value of $P_{NL}^m$ such that there remains only a relatively small gap between the upper and lower bound values for any number of parties $N$ .",
"For $N$ large, this value tends to be within the range $[2N/5,N/2]$ .",
"Moreover, based on a numerical investigation regarding the lower bound value, we conjecture that $2N/5$ is the exact value for large $N$ .",
"In this respect, it would be interesting to improve further the upper bound value.",
"Note that the proof for the upper bound of $P_{NL}^{2}$ in section relies merely on the permutationally symmetry of the state and does not exploit the full structure of the W state.",
"On the other hand, our numerical study indicates that for a fixed but small $N$ , $P_{NL}^m(W_N)$ increases by increasing $m$ .",
"This suggests that for $N$ large the lower bound $2N/5$ on $P_{NL}^{m=2}(W_N)$ increases as well in case of $m>2$ .",
"Finding a general $N$ -party $m$ -setting family of Bell inequalities to lowerbound $P_{NL}^m$ of which the present one is a special $m=2$ member would be most welcome.",
"Let us mention some possible ways to generalize the persistency of nonlocality of multipartite states.",
"The concept of EPR steering [46] lies between entanglement and nonlocality, and EPR steering of multipartite quantum states has been investigated recently [47].",
"Similarly to the persistency of nonlocality, it would be interesting to study the behavior of persistency of steering for the W state or other permutationally invariant states such as Dicke states.",
"Finally, the question of genuine nonlocality [45] of the W state has also been left open.",
"Indeed, instead of studying the persistency of standard nonlocality $P_{NL}$ of the W state, we may ask as well what is the minimal number of parties $k$ to trace out from an $N$ -qubit W state, such that the reduced $N-k$ -party state lacks genuinely multipartite nonlocality.",
"We acknowledge financial support from the Hungarian National Research Fund OTKA (K111734)."
],
[
"Loss of excitations",
"Suppose that a source emits an $n$ -qubit state $\\rho $ , which can be effected by some losses, e.g., noise due to a lossy channel.",
"We treat channel losses in the following noise model, which is called amplitude damping.",
"In each mode (out of $n$ modes), there is a probability $p$ of losing an excitation.",
"The operator sum formalism (see e.g., Ref.",
"[1]) describes the transformation between the initial state $\\rho $ and the final state $\\rho _{noisy}$ in the following way $\\rho _{noisy}=\\sum _{\\vec{k}=(0,\\dots ,0)}^{(1,\\dots ,1)}\\mathcal {K}_{\\vec{k}}\\rho {\\mathcal {K}_{\\vec{k}}}^\\dagger ,$ where $\\mathcal {K}_{\\vec{k}}$ denotes the tensor product of certain combinations of the following two Kraus operators corresponding to the amplitude damping noise model $K_0 =\\begin{pmatrix}1 & 0\\\\0 & \\sqrt{1-p} \\\\\\end{pmatrix};\\quad K_1 =\\begin{pmatrix}0 & \\sqrt{p} \\\\0 & 0\\\\\\end{pmatrix},$ where $p$ stands for the probability that an excitation is lost.",
"With these, $\\mathcal {K}_{\\vec{k}}$ is defined by $\\mathcal {K}_{\\vec{k}} = \\otimes _{l=1,\\ldots ,n}K_{k_l}$ , where $n$ is the number of qubits and $k_l$ can take 0 or 1.",
"Using the W state, $\\rho _W=|W_n\\rangle \\langle W_n|$ as the initial state $\\rho $ in (REF ), the following relations turn out to hold true: $\\mathcal {K}_{0\\ldots 0}\\rho _W\\mathcal {K}_{0\\ldots 0}^\\dagger &= (1-p)\\rho _W\\nonumber \\\\\\mathcal {K}_{0\\ldots 01}\\rho _W\\mathcal {K}_{0\\ldots 01}^\\dagger &= (p/n)|0^{\\otimes n}\\rangle \\langle 0^{\\otimes n}|,$ Further, if we permute the index $\\vec{k}={0\\ldots 01}$ in all the $n$ different ways, we get the same state as that on the right hand side of the second line of (REF ), i.e., the $n$ -partite vacuum state multiplied by $p/n$ .",
"On the other hand, if the number of 1's in $\\vec{k}$ are at least two, we have $\\mathcal {K}_{\\vec{k}}\\rho _W\\mathcal {K}_{\\vec{k}}^\\dagger = 0$ .",
"Substituting into (REF ), we arrive at $\\rho _{noisy}=(1-p)|W_n\\rangle \\langle W_n| + p|0^{\\otimes n}\\rangle \\langle 0^{\\otimes n}|,$ which is the same state as (REF )."
]
] | 1605.05846 |
[
[
"Magnetic hyperbolic metamaterial of polaritonic nanowires"
],
[
"Abstract We show that the axial component of the magnetic permeability tensor is resonant for a wire medium consisting of high-index epsilon-positive nanowires, and its real part changes the sign at a certain frequency.",
"At this frequency the medium experiences the topological transition from the hyperbolic to the elliptic type of dispersion.",
"We show that the transition regime is characterized by extremely strong dependence of the permeability on the wave vector.",
"This implies very high density of electromagnetic states that results in the filamentary pattern and noticeable Purcell factor for a transversely oriented magnetic dipole."
],
[
"introduction",
"Simple wire media, defined as optically dense arrays of parallel metal wires in a host dielectric material, have been in a considerable attention since electromagnetic metamaterials have been introduced.",
"Many applications of such media have been suggested from radio frequencies where the period of wire media is on the millimeter scale, to optical frequencies where this period is submicron.",
"Among them one can mention biological sensing [1], subwavelength imaging and endoscopy with image magnification [2], [3], [4], enhancement of quantum emitters, thermal sources, and classical dipole radiators [5], [7], [6], enhancement of radiative heat transfer [8], [9], etc.",
"More details can be found in overview [10].",
"In all these works, electromagnetic properties of wire media were described in a condensed form – via an effective permittivity tensor: $\\overline{\\overline{\\epsilon }}=\\left(\\begin{array}{ccc}\\varepsilon _{\\perp } & 0 & 0 \\\\0 &\\varepsilon _{\\perp } & 0 \\\\0& 0 & \\varepsilon _{\\parallel } \\end{array} \\right).$ Here, $\\varepsilon _\\perp $ and $\\varepsilon _\\parallel $ are the perpendicular (transversal) and parallel (axial) components with respect to the wires axis.",
"Therefore, standard wire media are uniaxially anisotropic materials (though not usual ones because they possess spatial dispersion – dependence of $\\overline{\\overline{\\epsilon }}$ on the wave vector ${\\bf k}$ [11], [12].",
"At infrared frequencies, the relative dielectric constant of the metal nanowires is highly negative.",
"If the fraction ratio of the metal in the effective medium is low the perpendicular component $\\varepsilon _\\perp $ of the permittivity is approximately equal to the permittivity of the host medium, whereas the parallel component $\\varepsilon _\\parallel $ has negative sign [11], [10].",
"For the dispersion, it implies that the transverse magnetic (TM) polarized wave has an open dispersion surface similar to a hyperboloid [10], [12].",
"This dispersion surface results in the high density of electromagnetic states and enhancement of a subwavelength electric dipole located in the medium orthogonally to the wires [5], [12].",
"The radiation of a magnetic dipole can be also enhanced but this effect is lower because such the dipole produces mainly TE-waves whose electric field is orthogonal to nanowires.",
"These transversely electric (TE) polarized waves weakly interact with the wire medium and their dispersion surfaces are rather similar to spheres like dispersion surfaces of free space.",
"Notice, that for metal nanowires the effective permeability tensor ($\\overline{\\overline{\\mu }}$ ) is practically equal to unity [10].",
"Recently, in Ref.",
"[13] one studied an anisotropic medium with artificial magnetism.",
"This medium is dual to the dielectric hyperbolic metamaterial described by the indefinite tensor in Eq. ().",
"In this magnetic metamaterial the effective permittivity tensor is definite, positive and not resonant.",
"Therefore, the dispersion surfaces of TM-polarized waves in this medium are similar to the spheres, whereas the dispersion surfaces of TE-waves are either hyperbolic or elliptic depending on frequency.",
"However, medium in Ref.",
"[13] is not a wire medium.",
"It is a racemic array of metal helices with both left-handedness and right-handedness.",
"Such media, in accordance to initial works [14], [15] where they were introduced, should be referred as spiral media.",
"Spiral medium of [13] operates at microwave frequencies, where metals are close to perfect conductors.",
"In the present paper, we study wire media at optical frequencies.",
"Figure: A medium of long parallel wires which are set in a square lattice.",
"The wires are made of polaritonic materials whose relative dielectric constant is much larger than unity (ε r ≫1\\varepsilon _{\\rm {r}}\\gg 1).",
"Here, aa and bb are the wires radius and lattice constant, respectively.",
"For simplicity, the host medium is free space.Below, we theoretically demonstrate that a simple wire medium (see Fig.",
"REF ) may have in the infrared range similar electromagnetic properties to those manifested by [13] exhibited at microwaves, however, there are also significant peculiarities which share our wire medium out from all hyperbolic metamaterials.",
"Nanowires of our wire medium should be made of high-index epsilon-positive materials, such as lithium tantalate, silicon carbide or hexagonal boron nitride, and some other polaritonic materials in the range 15–30 THz.",
"The resonance of the axial component of the effective permeability tensor arises because the dynamic magnetic polarizability of the high-index dielectric wire is resonant, and origins from the Mie resonance of a single nanowire.",
"We show that this axial permeability changes its sign at a certain frequency and in a narrow frequency range around this transition is a very sharp function of the wave vector.",
"The corresponding topological transition is related to the resonant enhancement of radiation of a subwavelength dipole located inside the medium.",
"It is an expected effect which is similar to that studied in [6].",
"However, in the present case it holds for a magnetic dipole oriented transversely to nanowires.",
"The sharp dependence of the permeability on the wave vector results in the unusual property of our wire medium – the radiation of the internal magnetic dipole is concentrated at the line which passes through the dipole center in parallel to the optical axis.",
"This property is difficult to detect from the dispersion contours (sections of the dispersion surfaces).",
"In the present case these contours for TE-waves do not differ very much from analogous contours for TM-waves we have studied in [9].",
"Moreover, these contours were analyzed for TE-waves in the spiral medium [13].",
"Therefore, we do not spend the volume of this paper to the demonstration and analysis of the dispersion contours in the region of topological transition and fully concentrate on two effects: pattern of the internal source and its Purcell factor.",
"The paper is organized as follows: In Section , we obtain analytical expressions for the magnetic polarizability of a high-index dielectric cylinder and the effective permeability of the corresponding wire medium.",
"In Section , we discuss the implications of the resonant and non-local behavior of this permeability and show the simulated results for the radiation of an embedded magnetic dipole.",
"In Section , we present the brief summary of the work."
],
[
"Effective permeability of the wire medium",
"Let a plane wave with electric field polarized along $\\mathbf {y}_0$ illuminate a dielectric cylinder, as it is shown in Fig.",
"REF .",
"The incident wave has perpendicular polarization and its magnetic field along $\\mathbf {z}_0$ can be written as $H_{z_{\\rm {inc}}}=H_0e^{-j(k_xx+k_zz)},$ where $H_0$ is the magnitude, $k_z$ and $k_x$ are the wave vector components in the free space.",
"Due to the cylindrical geometry, it is proper to expand the field into cylindrical harmonics.",
"Therefore, $H_{z_{\\rm {inc}}}=H_0\\sum _m\\left[(-j)^nJ_m(h_0R)e^{jm\\phi }\\right]e^{-jk_zz},$ in which $J_m(x)$ is the Bessel function (first kind, order $m$ ), $h_0=\\sqrt{k_0^2-k_z^2}$ ($k_0$ is the free-space wave number) and $(R, \\phi , z)$ are the components of the cylindrical coordinate system.",
"Figure: (a)–An infinite dielectric cylinder illuminated by a TE-polarized plane wave.Taking into account the scattering and penetration of the wave into the dielectric cylinder, the total electromagnetic fields outside and inside the cylinder can be described by $\\begin{split}&E_{z_{\\rm {out}}}=\\sum _m{C_mH_m^{(2)}(h_0R)e^{jm\\phi }},\\cr &H_{z_{\\rm {out}}}=\\sum _m \\left(D_mH_m^{(2)}(h_0R)+H_0(-j)^mJ_m(h_0R)\\right)e^{jm\\phi },\\end{split}$ and $\\begin{split}&E_{z_{\\rm {in}}}=\\sum _m{A_mJ_m(hR)e^{jm\\phi }},\\cr &H_{z_{\\rm {in}}}=\\sum _m{B_mJ_m(hR)e^{jm\\phi }},\\end{split}$ where $h=\\sqrt{k_0^2\\varepsilon _{\\rm {r}}-k_z^2}$ ($\\varepsilon _{\\rm {r}}$ denotes the relative dielectric constant of the cylinder) and $H^{(2)}_m(x)$ represents the Hankel function (second kind, order $m$ ).",
"The other components of the electromagnetic fields ($E_R$ , $H_R$ , $E_\\phi $ and $H_\\phi $ ) can be readily derived by solving the Maxwell's equations.",
"In Eqs.",
"(REF ) and (REF ), the unknown coefficients ($A_m$ , $B_m$ , $C_m$ and $D_m$ ) are related to each other through boundary conditions, i.e.",
"the tangential components of the electric and magnetic fields should be continuous at the radius of the cylinder ($R=a$ ).",
"After determining the fields inside the cylinder, we can obtain the polarization current that is equal to $\\mathbf {J}_{\\rm {p}}=j\\omega \\varepsilon _0(\\varepsilon _{\\rm {r}}-1)\\mathbf {E}_{\\rm {in}}$ .",
"On the other hand, we know that the magnetic moment per unit length of our cylinder is described by $\\mathbf {m}=\\frac{1}{2}\\mu _0\\int _S\\mathbf {r}\\times \\mathbf {J}_{\\rm {p}}dS,$ where $\\mathbf {r}=\\mathbf {R}+z\\mathbf {a}_z$ is the distance vector from the origin to the current element.",
"Therefore, the $\\mathbf {z}$ -component of the magnetic moment can be obtained as $m_z=\\frac{1}{2}j\\omega \\mu _0\\varepsilon _0(\\varepsilon _{\\rm {r}}-1)\\int _SR^2E_{\\phi _{\\rm {in}}}dRd\\phi ,$ in which the $\\phi $ -component of the electric field inside the cylinder is as follows: $\\begin{split}E_{\\phi _{\\rm {in}}}=&\\frac{j\\omega \\mu _0}{h}\\times \\cr &\\sum _m\\left(A_m\\frac{mk_z}{j\\omega \\mu _0hR}J_m(hR)+B_mJ^{\\prime }_m(hR)\\right)e^{jm\\phi }.\\end{split}$ The function $J^{\\prime }_m(x)$ is the derivative of $J_m(x)$ .",
"From the Eqs.",
"(REF ) and (REF ), we can see that the only cylindrical harmonic which is responsible for non-zero magnetic moment per unit length produced by the magnetic field of the incident wave is $m=0$ .",
"Then, based on Eq.",
"(REF ), we should only calculate the coefficient $B_0$ – $A_0$ gives the zero contribution into Eqs.",
"(REF ).",
"From imposing the boundary conditions to determine the unknown coefficients, we can achieve $B_0=\\frac{j2h}{\\pi h_0a\\left[h_0J^{\\prime }_0(ha)H^{(2)}_0(h_0a)-hJ_0(ha)H^{\\prime (2)}_0(h_0a)\\right]}H_0.$ If we substitute Eq.",
"() into Eq.",
"(REF ) and use Eq.",
"(REF ), finally, we derive the magnetic polarizability as $\\begin{split}\\alpha ^{zz}_{\\rm {mm}}=\\frac{m_z}{\\mu _0H_0}&=\\frac{j2k_0^2(\\varepsilon _{\\rm {r}}-1)}{h_0(k_0^2\\varepsilon _{\\rm {r}}-k_z^2)}\\times \\cr &\\frac{\\left[ha J_0(ha)-2J_1(ha)\\right]}{\\left[h_0J_1(ha)H^{(2)}_0(h_0a)-hJ_0(ha)H^{(2)}_1(h_0a)\\right]}.\\end{split}$ Figure: (a)–Real part of the magnetic polarizability versus the normalized frequency for three different incident angles.",
"The relative dielectric constant of the dielectric cylinder is assumed to be ε r =120\\varepsilon _{\\rm {r}}=120.",
"(b)—Real part of the magnetic polarizability versus the normalized frequency for three different dielectric constants.",
"Here, we assume that θ inc =π/11\\theta _{\\rm {inc}}=\\pi /11.To keep only the term with $m=0$ in Eq.",
"(REF ) is the same as to excite our cylinder by only a magnetic field of the electromagnetic wave letting the external electric field be zero at the cylinder axis.",
"Adding a corresponding term to Eq.",
"(), it is easy to see that the excitation of the cylinder by two plane waves incident on it from two opposite sides and forming the standing wave with the node of the electric field at the $z$ -axis results in the same formula ().",
"It is worth to note that retaining in Eq.",
"(REF ) the term $A_1$ we would attribute to the magnetic moment per unit length the dependence on the incident electric field, and the response of the cylinder would be bianisotropic.",
"Keeping only $B_0$ we follow to the approach of [16] that allows one to avoid the seeming bianisotropy in the response of a symmetric scatterer.",
"This bianisotropy is not physically sound and would result in the inconsistencies of the effective-medium model (see e.g.",
"in [17]).",
"Sign minus in the denominator of the second factor in () allows the resonance of the magnetic polarizability if $\\varepsilon _{\\rm {r}}$ is positive.",
"This is the fundamental Mie resonance of the cylinder.",
"Since $k_z$ enters both $h$ and $h_0$ , the polarizability is non-local i.e.",
"depends on the angle of incidence $\\theta _{\\rm {inc}}$ .",
"To show the impact of the incidence angle and that of the dielectric constant $\\varepsilon _{\\rm {r}}$ on the Mie resonance we plot the magnetic polarizability as a function of normalized frequency for several values of $\\varepsilon _{\\rm {r}}$ and $\\theta _{\\rm {inc}}$ .",
"In Fig.",
"REF it is assumed that $\\varepsilon _{\\rm {r}}=120$ .",
"Then the Mie resonance occurs approximately at $k_0a=0.22$ for all angles.",
"For small angles, the resonance is, indeed, stronger than that for large angles.",
"Thereby, we see that the resonant polarizability depends strongly on the incident angle or – more specifically – on the axial component $k_z=k_0\\cos (\\theta _{\\rm {inc}})$ of the wave vector.",
"At frequencies below the Mie resonance band the magnetic polarizability is negligibly small.",
"The location of the resonance on the frequency axis is a function of the dielectric constant of the cylinder, as can be seen in Fig.",
"REF .",
"When $\\varepsilon _{\\rm {r}}$ is not very high, the resonance happens at higher $k_0a$ and vice versa.",
"For the accuracy of the effective-medium model it is better if the resonance occurs at lower $k_0a$ .",
"This model requires that both the wires radius ($a$ ) and the array period ($b$ ) are sufficiently small compared to the wavelength in host medium, in other words, $k_0a<k_0b\\ll 1$ .",
"Materials such as lithium tantalate, silicon carbide, or hexagonal boron nitride which support phonon polaritons would provide such high dielectric constants at the infrared.",
"The only problem with these materials may be optical losses which are obviously noticeable when $\\varepsilon _{\\rm {r}}\\gg 1$ .",
"Figure: The parallel component of the effective permeability versus the normalized parallel component of the wave vector, whenε r =120\\varepsilon _{\\rm {r}}=120 and f v =0.1963f_{\\rm {v}}=0.1963.",
"(a)–Frequencies below the topological transition k 0 a=0.22k_0a=0.22.Solid blue curve–k 0 a=0.2058k_0a=0.2058, solid red one–k 0 a=0.2096k_0a=0.2096, solid black one–k 0 a=0.2135k_0a=0.2135, solid magenta one–k 0 a=0.2173k_0a=0.2173, and dashed blue one–k 0 a=0.2182k_0a=0.2182.",
"(b)–Frequencies above the topological transition k 0 a=0.22k_0a=0.22.",
"Dashed red–k 0 a=0.2201k_0a=0.2201, dashed black–k 0 a=0.2211k_0a=0.2211, dashed magenta–k 0 a=0.2220k_0a=0.2220.Deriving the effective permeability of the lattice we follow to the approach [11], [13] which allows us to take into account the non-locality also in the electromagnetic interaction of cylinders.",
"For the axial component of the effective permeability we have the following formula: $\\mu _\\parallel =1+\\frac{1}{A_{\\rm {cell}}}(\\frac{1}{\\alpha ^{\\rm {zz}}_{\\rm {mm}}}-C_{\\rm {int}}^z)^{-1},$ which is dual to that obtained for the axial permittivity in [11] with the substitution of the magnetic polarizability by the electric one.",
"Here $A_{\\rm {cell}}=b^2$ and $C_{\\rm {int}}^z$ is the lattice interaction constant responsible for the dipole-dipole interaction.",
"It is the same for magnetic and electric dipole moment per unit length of the cylinder and was derived in [11]: $\\begin{split}C_{\\rm {int}}^z\\approx &h_0^2\\displaystyle \\left(\\frac{j}{4}+\\frac{1}{2\\pi }\\ln (\\frac{h_0b}{4\\pi })+\\xi \\right),\\cr &\\xi =\\frac{C}{2\\pi }+\\frac{1}{12}+\\sum _{n=1}^\\infty \\frac{(e^{2\\pi \\vert n\\vert -1})^{-1}}{\\pi \\vert n\\vert }.\\end{split}$ Here, $C$ is the Euler constant.",
"Let us choose $\\varepsilon _{\\rm {r}}=120$ and the fraction volume $f_{\\rm {v}}=\\pi a^2/b^2=0.1963$ (this corresponds to the relation between the period and the wire radius $b=4a$ ).",
"In Fig.",
"REF we depict the axial component of the effective permeability as a function of $k_z$ for these design parameters.",
"For very low $k_0a\\ll 0.22$ , $\\mu _\\parallel (k_z)$ is approximately uniform and close to unity.",
"The increase of $k_0a$ results in increasing $\\mu _\\parallel (k_z)$ especially for the transverse propagation, however it keeps finite and smooth versus $k_z$ .",
"At frequency $k_0a=0.22$ , $\\mu _\\parallel (k_z)$ changes its behavior that corresponds to the topological transition.",
"For $k_0a=0.2201$ the axial permeability is negative in the interval $\\vert k_z/k_0\\vert <0.83$ and becomes positive in $\\vert k_z/k_0\\vert >0.83$ .",
"At $\\vert k_z/k_0\\vert =0.83$ , there is a vertical asymptote.",
"This resonance of $\\mu _\\parallel (k_z)$ implies that the transverse component $q$ of the wave vector experiences the similar resonance at $\\vert k_z/k_0\\vert =0.83$ .",
"It is clear from the dispersion equation for TE-waves in a magnetic hyperbolic medium [13] (for the case when the host medium is free space $\\varepsilon _{\\rm {h}}=1$ ): ${q^2/\\mu _\\parallel (k_z)}+{k_z^2/\\mu _{\\perp }}=k_0^2, \\quad \\mu _{\\perp }\\approx 1.$ In practical cases – when optical losses are taken into account – the resonance of $q$ means that $q$ is a very large imaginary value.",
"As we can see in Fig.",
"REF this effect keeps for propagating eigenmodes within a certain range of frequencies.",
"For frequencies $k_0a\\ge 0.23$ the resonances of $\\mu _\\parallel (k_z)$ still hold but correspond to very high spatial frequencies.",
"For so high spatial frequencies the effective-medium model is not applicable [18].",
"In our plot Fig.",
"REF the region of $k_z$ (of the order of $k_0$ or smaller) is that compatible with the model.",
"Imaginary $q$ for a propagating eigenmode with allowed value of $k_z$ means that the medium eigenmode propagates along the $z$ -axis, being evanescent in the transverse plane $(x-y)$ .",
"Very high imaginary $q$ implies the ultimate subwavelength concentration of the eigenmode in this plane.",
"If we embed a subwavelength source of TE-waves (such as a point magnetic dipole oriented orthogonally to the $z$ -axis) in our wire medium this mode will be excited dominating over all other modes propagating along the $z$ -axis.",
"This domination results from the huge slope of the curve $\\mu _\\parallel (k_z)$ which implies the huge density of electromagnetic states.",
"This feature was noticed for the axial permittivity of some dielectric hyperbolic metamaterials in [19].",
"Our magnetic hyperbolic metamaterial is dual to dielectric hyperbolic metamaterials and the roles of $\\overline{\\overline{\\epsilon }}$ and TM-waves are played by $\\overline{\\overline{\\mu }}$ and TE-waves.",
"Since the dominant mode strongly attenuates in both $x$ - and $y$ -directions, it means that the whole electromagnetic field generated by our magnetic dipole should be concentrated around the line which passes through the dipole center along the $z$ -axis."
],
[
"Localization of magneto-dipole radiation",
"This filamentary localization of magneto-dipole radiation was confirmed by using a 3D electromagnetic simulator CST Microwave Studio.",
"In our simulations we position a strongly subwavelength current loop whose magnetic moment is directed along the $y$ -axis in the center of the wire medium sample as shown in Fig.",
"REF .",
"The medium sample is finite-size with length $L_{\\rm {sample}}=(N-1)b+2a$ where $N=12$ is the number of the wires.",
"It has realistic optical losses: we suppose that the tangent of dielectric losses in the material of our wires is equal to $\\tan \\delta =0.1$ , where the complex permittivity of nanowires is $\\varepsilon _{\\rm {r}}={\\rm {Re}}[\\varepsilon _{\\rm {r}}][1-j\\tan \\delta ]$ , ${\\rm {Re}}[\\varepsilon _{\\rm {r}}]=120$ .",
"This value is the complex permittivity of lithium tantalate at $f=23$ THz.",
"The radius of the wires is $a=455$ nm and $b=4a$ as above.",
"The spatial distribution of the transverse component of the electric field (the $x$ -component in the $y-z$ plane) is shown in Fig.",
"REF for four frequencies.",
"We see that only at the frequency $k_0a=0.2192\\approx 0.22$ , the field so significantly decays in the transverse plane so that it is practically concentrated in one unit cell of the wire medium.",
"This is the maximal possible concentration in a composite material – that restricted only by the granularity of the medium.",
"The excited wave propagates along the $z$ -axis and experiences nearly total internal reflection at the interfaces of the medium sample.",
"Similar pictures correspond to other components of the electromagnetic field.",
"The comparison of different components confirms that the wave is TE-polarized.",
"At three other frequencies the waves are also TE-polarized, whereas their localization is not filamentary and even the cross-like dipole pattern inherent to usual hyperbolic metamaterials [12] can be guessed in Fig.",
"REF .",
"Figure: Distribution of the normal component of the electric-field in the wire-medium sample (consisting of 12×1212\\times 12 nanowires).",
"The magnetic dipole moment located in the center of the sample is along the 𝐲 0 \\mathbf {y}_0-axis.",
"Here, ε r =120-j12\\varepsilon _{\\rm {r}}=120-j12, f v =0.1963f_{\\rm {v}}=0.1963.The reason of the striking difference in the internal dipole patterns for our wire medium and the majority of hyperbolic materials is spatial dispersion.",
"Metal wire media operating at microwaves are also spatially dispersive, and the slope of $\\epsilon _\\parallel (k_z)$ (see e.g.",
"in [10]) at their spatial resonance $|k_z|=k_0$ is as huge as that of our $\\mu _\\parallel (k_z)$ at their resonant $k_z$ in the topological transition frequency range.",
"It is not surprising, therefore, that in microwave wire media the similar filamentary dipole pattern was also observed (see e.g.",
"in [20]).",
"However, in what concerns this pattern our infrared magnetic wire medium and microwave wire media differ qualitatively.",
"The first difference is duality: microwave wire media are hyperbolic dielectric metamaterials and the filamentary pattern in them corresponds to transverse electric dipoles.",
"The second difference is the effect bandwidth: microwave wire media are broadband, and the same dipole pattern is inherent for them at any frequency for which the effective-medium model is applicable.",
"Our wire media possess the dipole pattern which strongly varies versus frequency, that is illustrated by Fig.",
"REF .",
"The filamentary pattern of the magnetic dipole corresponds only to the range of the topological transition."
],
[
"Purcell factor",
"Another implication of the topological transition in our magnetic metamaterial is the strong Purcell effect.",
"Originally, the effect was known as the enhancement of the decay rate of a quantum emitter located in an open cavity [21], [22].",
"However, the role of the cavity is only to extract more power from the emitter, whereas the enhancement of its decay rate is the same as the enhancement of the radiated power.",
"Therefore, the notion of the Purcell effect was extended, first, to any scatterer located in the vicinity of an emitter [23], [24], [25], [26], and, finally to any active radiator whose radiation is enhanced by any environment different from free space [27], [28], [6].",
"Based on this general concept, we can define the magnetic Purcell factor of our wire medium as the increase of the radiated power of a subwavelength magnetic dipole due to the presence of the wire medium around it.",
"According to the “antenna terminology”, the radiated power of the emitter is proportional to the radiation resistance of that emitter [29].",
"Therefore, to calculate the Purcell factor, it is the same as to obtain the ratio of the radiation resistances in presence of the wire medium ($R_{\\rm {r}}$ ) and in absence ($R_{{\\rm {r}}_0}$ ): $P_{\\rm {F}}=\\displaystyle \\frac{R_{\\rm {r}}}{R_{{\\rm {r}}_0}}.$ For the same structure as above we have simulated the radiation resistances.",
"Besides of the case of the transverse magnetic dipole shown in Fig.",
"REF , we have studied the case when the same magnetic dipole is oriented in parallel to the $z$ -axis.",
"Also, we simulated the cases of the transverse and axial electric dipoles of the same subwavelength size, all centered at the same point as in Fig.",
"REF .",
"In Fig.",
"REF which depicts the radiation resistance versus frequency, the blue curves refer to the magnetic dipole in presence of the sample.",
"As one can see, the radiation resistance of the magnetic dipole oriented perpendicularly to the optical axis experiences the strong resonance at $k_0a\\approx 0.22$ , exactly where the effective-medium model predicts the topological transition.",
"The dashed blue curve shows weaker effect for the parallel magnetic dipole.",
"Red solid and dashed curves correspond to the perpendicular and parallel electric dipoles, respectively.",
"For comparison with the case when the host is free space, the frequency dispersion of $R_{{\\rm {r}}_0}$ is given in the same plot for both electric and magnetic dipoles.",
"Figure REF presents the frequency dispersion of the Purcell factor for all four dipoles.",
"It is clearly seen that the strongest resonance and highest $P_{\\rm {F}}$ correspond to the transversal magnetic dipole.",
"An axial magnetic dipole also creates the TE-polarized spatial spectrum, and therefore it also experiences the resonance at the topological transition.",
"However, the resonant values of $P_{\\rm {F}}$ are smaller because it mainly creates the TE-polarized radiation which weakly interacts with the nanowires i.e.",
"is not enhanced.",
"The values of $P_{\\rm {F}}$ for the axial electric dipole are low.",
"It is not surprising.",
"Recall, that in a wire medium of perfectly conducting (PC) wires the radiation of the axial electric dipole is fully suppressed [5].",
"The suppression results from the destructive near-field coupling of the dipole with adjacent wires.",
"The axial current of the dipole induces opposite axial currents in them which cancel the radiation.",
"This effect is similar to the suppression of radiation of a horizontal electric dipole located on the PC substrate.",
"Well, our wires are not PC, they are highly refractive.",
"However, the substrates with high positive permittivity and finite negative permittivity also suppress the radiation of the horizontal electric dipole.",
"This destructive interaction origins from the capacitive coupling which causes the inverse mirror image of the horizontal electric dipole in the substrate with the strong skin-effect.",
"A similar situation holds for the axial dipole source in wire media if the wires possess strong skin-effect.",
"If the wires have high or negative $\\varepsilon _{\\rm r}$ , the values of $P_{\\rm {F}}$ for the axial electric dipole should be low.",
"However, we can see in Figure REF that the radiation of this dipole experiences the resonance (at slightly lower frequency than that of the magnetic topological transition).",
"This is also not surprising, because not all radiation of an axial electric dipole is TM-polarized.",
"It also produces some TE-waves, and this effect is competing with the capacitive destructive interaction.",
"Therefore, we have $P_{\\rm {F}}=2$ at $k_0a=0.21$ , whereas far from this frequency we see $P_{\\rm {F}}\\approx 0$ .",
"As to the noticeable enhancement of the transverse electric dipole, it, definitely, occurs due to the constructive interaction of this dipole with the adjacent nanowires (similarly to a vertical dipole on the high-permittivity substrate).",
"This effect is complemented by the resonant enhancement of TE-polarized waves.",
"However, in the near zone of the electric dipole the magnetic field is weak.",
"Therefore, an electric dipole cannot interact with the magnetic medium as strongly as a magnetic dipole, and the resonant Purcell factor of the transversal magnetic dipole is higher."
],
[
"Conclusions",
"In this work we calculated the effective permeability of a simple wire medium whose wires are made of a material with high positive dielectric constant.",
"Its parallel component $\\mu _\\parallel $ is resonant due to the Mie resonance of a single wire, and is an indefinite function of both frequency and axial wave vector $k_z$ (changes the sign depending on these arguments).",
"This change of the sign results in a transition in the topology of the dispersion surface for TE polarized waves.",
"In the frequency range of the topological transition, the huge (filamentary) localization of radiation occurs for an internal magnetic source, that clearly shares our medium out from other hyperbolic metamaterials, including that introduced in [13].",
"Though a similar pattern is inherent to the medium of perfectly conducting wires operating at microwaves, in our case this pattern corresponds to the magnetic dipole and occurs only at the topological transition.",
"Our wire medium manifests a strong radiation enhancement for an internal dipole source.",
"Unlike dielectric hyperbolic metamaterials, including microwave wire media, and polaritonic media operating at higher frequencies [6] the maximal enhancement corresponds to magnetic dipoles.",
"Unlike spiral media [13] it holds at the topological transition and the maximal radiation enhancement corresponds to the transversal magnetic dipole.",
"What is also important: from the comparison of the present paper with [6] it is clear that the same polaritonic wire media which manifest the resonant effects related to the regime epsilon-near-zero at 35–45 THz [6] may manifest the similar (but different) resonant effects related to the regime mu-near-zero in the range 15–30 THz.",
"This is another argument in favor of this type of hyperbolic metamaterials, which definitely deserve more attention of theorists and an experimental implementation to confirm the claimed effects."
]
] | 1605.05701 |
[
[
"Ultrafast Isomerization in Acetylene Dication: To Be or Not To Be"
],
[
"Abstract Experimental evidence has pointed toward the existence of ultrafast proton migration and isomerization as a key process for acetylene and its ions, however the actual mechanism for ultrafast isomerization of the acetylene [HCCH]2+ to vinylidene [H2CC]2+ dication remains nebulous.",
"Theoretical studies show a high potential barrier of over 2eV for the isomerization pathways on the low lying dicationic states, implying that the corresponding isomerization should take picoseconds or even longer according to transition state theory.",
"However a recent experiment at a femtosecond X-ray free electron laser (XFEL) [Nature Commun.",
"6, 8199 (2015)] suggests that large amplitude hydrogen migration proceeds on a sub-100 femtosecond time scale.",
"In order to resolve the contradiction, we present a complete theoretical study of the dynamics of acetylene dication produced by Auger decay after X-ray photoionization of the carbon atom K shell.",
"We find that isomerization does not occur on the sub-100 fs timescale and is not required to explain the time-resolved Coulomb imaging experiment.",
"This study resolves the seeming contradiction between experiment and theory concerning the isomerization time scale in acetylene dication.",
"This work calls for careful interpretation of structural information from the widely applied Coulomb momentum imaging method but also points out its strengths in mapping out momentum dispersion dynamics even when structural variation is minor."
],
[
"=1 pdfinfo= [pages=1-last]main.pdf [pages=1-last]suppl.pdf"
]
] | 1605.05707 |
[
[
"Optical spectroscopy of Be/gamma-ray binaries"
],
[
"Abstract We report optical spectroscopic observations of the Be/gamma-ray binaries LSI+61303, MWC 148 and MWC 656.",
"The peak separation and equivalent widths of prominent emission lines (H-alpha, H-beta, H-gamma, HeI, and FeII) are measured.",
"We estimated the circumstellar disc size, compared it with separation between the components, and discussed the disc truncation.",
"We find that in LSI+61303 the compact object comes into contact with the outer parts of the circumstellar disc at periastron, in MWC 148 the compact object goes deeply into the disc during the periastron passage, and in MWC 656 the black hole is accreting from the outer parts of the circumstellar disc along the entire orbit.",
"The interstellar extinction was estimated using interstellar lines.",
"The rotation of the mass donors appears to be similar to the rotation of the mass donors in Be/X-ray binaries.",
"We suggest that X-ray/optical periodicity of about 1 day deserves to be searched for."
],
[
"Introduction",
"The rapid and sustained progress of high energy and very high energy astrophysics in recent years enabled the identification of a new group of binary stars emitting at TeV energies (e.g.",
"Paredes et al.",
"2013).",
"These objects, called $\\gamma $ -ray binaries, are high-mass X-ray binaries that consist of a compact object (neutron star or black hole) orbiting an optical companion that is an OB star.",
"There are five confirmed $\\gamma $ -ray binaries so far: PSR B1259-63/LS 2883 (Aharonian et al.",
"2005), LS 5039/V479 Sct (Aharonian et al.",
"2006), LSI+61$^0$ 303 (Albert et al., 2009), HESS J0632+057/MWC 148 (Aharonian et al.",
"2007), and 1FGL J1018.6-5856 (H.E.S.S.",
"Collaboration et al.",
"2015).",
"Their most distinctive fingerprint is a spectral energy distribution dominated by non-thermal photons with energies up to the TeV domain.",
"Recently, Eger et al.",
"(2016) proposed a binary nature for the $\\gamma $ -ray source HESS J1832-093/2MASS J18324516-092154 and this object probably belongs to the family of the $\\gamma $ -ray binaries as a sixth member.",
"The binary system, PSR B1259-63 is unique, since it is the only one where the compact object has been identified as a radio pulsar (Johnston et al.",
"1992, 1994).",
"The nature of the compact object is known in PSR B1259-63 as a neutron star, and in AGL J2241+4454/MWC 656 as a black hole (Casares et al.",
"2014).",
"Although not included in the confirmed list, MWC 656 was selected as a target here despite not having shown all the observational properties of a canonical $\\gamma $ -ray binary yet.",
"It was only occasionally detected by the AGILE observatory at GeV energies and not yet detected in the TeV domain (see Aleksić et al.",
"2015).",
"Nevertheless, the fact that the black hole nature of the compact companion is almost certain renders it very similar to the typical $\\gamma $ -ray binaries.",
"In the other systems the nature of the compact object remains unclear (e.g.",
"Dubus 2013).",
"In addition to these objects, there are several other binary systems ($\\eta $ Car, Cyg X-1, Cyg X-3, Cen X-3, and SS 433) that are detected as GeV sources, but not as TeV sources so far.",
"Here we report high-resolution spectral observations of LSI+61$^0$ 303, MWC 148, and MWC 656, and discuss circumstellar disc size, disc truncation, interstellar extinction, and rotation of their mass donors.",
"The mass donors (primaries) of these three targets are emission-line Be stars.",
"The Be stars are non-supergiant, fast-rotating B-type and luminosity class III-V stars which, at some point in their lives, have shown spectral lines in emission (Porter & Rivinius 2003).",
"The material expelled from the equatorial belt of a rapidly rotating Be star forms an outwardly diffusing gaseous, dust-free Keplerian disc (Rivinius et al.",
"2013).",
"In the optical/infrared band, the two most significant observational characteristics of Be stars and their excretion discs are the emission lines and the infrared excess.",
"Moving along the orbit, the compact object passes close to this disc, and sometimes may even go through it causing significant perturbations in its structure.",
"This circumstellar disc feeds the accretion disc around the compact object and/or interacts with its relativistic wind."
],
[
"Observations",
"High-resolution optical spectra of the three northern Be/$\\gamma $ -ray binaries were secured with the fibre-fed Echelle spectrograph ESpeRo attached to the 2.0 m telescope of the National Astronomical Observatory Rozhen, located in Rhodope mountains, Bulgaria.",
"The spectrograph uses R2 grating with 37.5 grooves/mm, Andor CCD camera 2048 x 2048 px, 13.5x13.5 $\\mu m$ px$^{-1}$ (Bonev et al.",
"2016).",
"The spectrograph provides a dispersion of 0.06 Å px$^{-1}$ at 6560 Å and 0.04 Å px$^{-1}$ at 4800 Å.",
"The spectra were reduced in the standard way including bias removal, flat-field correction, and wavelength calibration.",
"Pre-processing of data and parameter measurements are performed using various routines provided in IRAF.",
"The journal of observations is presented in Table REF , where the date, start of the exposure, exposure time, and signal-to-noise ratio at about $\\lambda 6600$ Å are given.",
"The orbital phases are calculated using $HJD_0= 2443366.775$ , $HJD_0= 2454857.5,$ and $HJD_0= 2453243.7$ for LSI+61$^0$ 303, MWC 148, and MWC 656, respectively, and orbital periods given in Sect. .",
"Emission line profiles of LSI+61$^0$ 303, MWC 148, and MWC 656 are plotted on Fig.REF .",
"Spectral line parameters equivalent width (W) and distance between the peaks ($\\Delta V$ ) for the prominent lines ($H\\alpha $ , H$\\beta $ , H$\\gamma $ , $HeI \\lambda 5876,$ and $FeII \\lambda 5316$ ) are given in Table REF .",
"The typical error on the equivalent width is below $\\pm 10$ % for lines with $W > 1$ Å and up to $\\pm 20$ % for lines with $W \\lesssim 1$ Å.",
"The typical error on $\\Delta V$ is $\\pm 10$ km s$^{-1}$ .",
"It is worth noting that (1) in LSI+61$^0$ 303 FeII lines are not detectable; (2) In MWC 656 on spectrum 20150705 the HeI $\\lambda 5876$ line is not visible (probably emission fills up the absorption).",
"In addition to the Rozhen data we use 98 spectra of MWC 148 and 68 spectra of MWC 656 (analysed in Casares et al.",
"2012) from the archive of the 2.0 m Liverpool TelescopeThe Liverpool Telescope is operated on the island of La Palma by Liverpool John Moores University in the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrofisica de Canarias with financial support from the UK Science and Technology Facilities Council.",
"(Steele et al.",
"2004).",
"These spectra were obtained using the Fibre-fed RObotic Dual-beam Optical Spectrograph (FRODOSpec; Morales-Rueda et al.",
"2004).",
"The spectrograph is fed by a fibre bundle array consisting of $12\\times 12$ lenslets of 0.82 arcsec each, which is reformatted as a slit.",
"The spectrograph was operated in a high-resolution mode, providing a dispersion of 0.8 Å px$^{-1}$ at 6500 Å, 0.35 Å px$^{-1}$ at 4800 Å, and typical $S/N \\gtrsim 100$ .",
"FRODOSpec spectra were processed using the fully automated data reduction pipeline of Barnsley et al.",
"(2012).",
"The typical error on the equivalent width is $\\pm 10$ % and on $\\Delta V$ is $\\pm 20$ km s$^{-1}$ .",
"Figure: Emission line profiles of LSI+61 0 ^0303, MWC 148, and MWC 656.Table: Spectral line parameters"
],
[
"Objects: System parameters",
"LSI+61$^0$ 303 (V615 Cas) was identified as a $\\gamma $ -ray source with the $COS B$ satellite 35 years ago (Swanenburg et al.",
"1981).",
"For the orbital period of LSI+61$^0$ 303, we adopt $P_{orb}=26.4960 \\pm 0.0028$ d, which was derived with Bayesian analysis of radio observations (Gregory 2002) and an orbital eccentricity $e=0.537$ , which was obtained on the basis of the radial velocity of the primary (Casares et al.",
"2005; Aragona et al.",
"2009).",
"For the primary, Grundstrom et al.",
"(2007) suggested a B0V star with radius $R_1=6.7 \\pm 0.9$ $R_\\odot $ .",
"A B0V star is expected to have on average $M_1 \\approx 15$ $M_\\odot $ (Hohle et al.",
"2010).",
"We adopt $v \\sin i = 349 \\pm 6$ km s$^{-1}$ for the projected rotational velocity of the mass donor (Hutchings & Crampton 1981, Zamanov et al.",
"2013).",
"MWC 148 (HD 259440) was identified as the counterpart of the variable TeV source HESS J0632+057 (Aharonian et al.",
"2007).",
"We adopt $P_{orb} = 315 ^{+6}_{-4}$ d derived from the X-ray data (Aliu et al.",
"2014), which is consistent with the previous result of $321 \\pm 5$ days (Bongiorno et al.",
"2011).",
"For this object Aragona et al.",
"(2010) derived $T_{eff} = 27500 - 30000$ K, $\\log g = 3.75 - 4.00$ , $M_1 = 13.2 - 19.0$ $M_\\odot $ , and $R_1 = 7.8 \\pm 1.8 $ $R_\\odot $ .",
"For the calculations in Sect.REF , we adopt $e=0.83$ , periastron at phase 0.967 (Casares et al.",
"2012), and $v \\sin i = 230 - 240$ km s$^{-1}$ (Moritani et al.",
"2015).",
"MWC 656 (HD 215227) is the emission-line Be star that lies within the positional error circle of the $AGILE$ $\\gamma $ -ray source AGL J2241+4454 (Lucarelli et al.",
"2010).",
"It is the first and until now the only detected binary composed of a Be star and a black hole (Casares et al.",
"2014).",
"For the orbital period, we adopt $P_{orb}=60.37 \\pm 0.04$ d obtained with optical photometry (Williams et al.",
"2010), $e=0.10 \\pm 0.04$ estimated on the basis of the radial velocity measurements and $v \\sin i = 330 \\pm 30$ km s$^{-1}$ (Casares et al.",
"2014).",
"For the primary, Williams et al.",
"(2010) estimated $T_{eff} = 19000 \\pm 3000$ K, $\\log g = 3.7 \\pm 0.2 $ , $M_1 = 7.7 \\pm 2.0$ $M_\\odot $ , $R_1 = 6.6 \\pm 1.9$ $R_\\odot $ .",
"Casares et al.",
"(2014) considered that the mass donor is a giant (B1.5-2 III) and give a mass range $M_1 = 10 - 16$ $M_\\odot $ .",
"On average a B1.5-2 III star is expected to have about $R_1 \\approx 8.3 - 8.8$ $R_\\odot $ (Straizys & Kuriliene 1981).",
"From newer values of the luminosity (Hohle et al.",
"2010), such a star is expected to have $M_1 \\approx 8.0 - 10.0$ $M_\\odot $ and radius $R_1 \\approx 9.5 - 10$ $R_\\odot $ .",
"We adopt $R_1 \\approx 10$ $R_\\odot $ for the calculations in Sect.REF .",
"Table: Disc size in different emission lines"
],
[
"Peak separation in different lines",
"For the Be stars, the peak separations in different lines follow approximately the relations (Hanuschik et al.",
"1988) $\\Delta V_\\beta \\approx 1.8 \\Delta V_\\alpha \\\\\\Delta V_\\gamma \\approx 1.2 \\Delta V_\\beta \\approx 2.2 \\Delta V_\\alpha \\\\\\Delta V_{\\rm FeII} \\approx 2.0 \\Delta V_\\alpha \\\\\\Delta V_{\\rm FeII} \\approx 1.1 \\Delta V_\\beta ,$ where Eq.",
"is derived from Eqs.",
"REF and .",
"For LSI+61$^0$ 303 using the measurements in Table REF , we obtain $\\Delta V_\\beta = 1.30 \\pm 0.04 \\, \\Delta V_\\alpha $ and $\\Delta V_{HeI5876}= 1.38 \\pm 0.13 \\, \\Delta V_\\alpha $ .",
"The ratio $\\Delta V_\\beta / \\Delta V_\\alpha $ is considerably below the average value for the Be stars (see Eq.REF ).",
"We obtain $\\Delta V_\\beta = 1.78 \\pm 0.06 \\, \\Delta V_\\alpha $ , $\\Delta V_\\gamma = 1.07 \\pm 0.03 \\, \\Delta V_\\beta $ , and $\\Delta V_{FeII5316} = 1.12 \\pm 0.03 \\, \\Delta V_\\beta $ , $\\Delta V_{HeI5876} = 1.47 \\pm 0.10 \\, \\Delta V_\\beta $ for MWC 148.",
"We use only three spectra for $H\\alpha $ (20140113, 20140217, and 20140218) when two peaks in $H\\alpha $ are visible.",
"The value of $\\Delta V_\\beta / \\Delta V_\\alpha \\approx 1.78$ is very similar to 1.8 in Be stars, the ratio $\\Delta V_{FeII5316} / \\Delta V_\\beta \\approx 1.07 $ is similar to 1.1 in Be stars, and the value of $\\Delta V_\\gamma / \\Delta V_\\beta \\approx 1.07$ is again similar to the value 1.2 for Be stars.",
"We estimate $\\Delta V_\\beta = 1.72 \\pm 0.18 \\, \\Delta V_\\alpha $ for MWC 656 (using six spectra from the Liverpool Telescope FRODOSpec, where two peaks are visible in both $H\\alpha $ and $H\\beta $ ), $\\Delta V_\\gamma = 1.22 \\pm 0.04 \\, \\Delta V_\\beta $ , $\\Delta V_{FeII5316} = 1.01 \\pm 0.10 \\, \\Delta V_\\beta $ , where all three ratios are similar to the corresponding values (Eq.",
"REF , , ) in Be stars.",
"We do not see two peaks on high-resolution Rozhen spectra of this object.",
"However two peaks are clearly distinguishable on a few of the LT spectra.",
"In every case of detection/non-detection of the double peak structure, $W\\alpha $ is very similar $19 < W\\alpha < 25$ Å.",
"The comparison of the peak separation of different emission lines shows that MWC 148 and MWC 656 have circumstellar disc that is similar to that of the normal Be stars.",
"At this stage considerable deviation from the behaviour of the Be stars is only detected in LSI+61$^0$ 303.",
"In this star the $H\\alpha $ -emitting disc is only 1.7 times larger than the H$\\beta $ -emitting disc, while in normal Be stars it is 3.3 times larger.",
"This probably is one more indication that outer parts of the disc are truncated as a result of the relatively short orbital period.",
"Figure: Distance between the peaks of HαH\\alpha emission line(normalized with stellar rotation and inclination)versus W α W_\\alpha on a logarithmic scale.Black empty circles indicate normal Be stars,red pluses indicate Be/γBe/\\gamma -ray binaries.The solid line denotes y=-0.592x+0.165y =- 0.592 \\: x + 0.165 (see Sect.",
")."
],
[
"Disc size",
"For rotationally dominated profiles, the peak separation can be regarded as a measure of the outer radius ($R_{disc}$ ) of the emitting disc (Huang 1972) $\\left( \\frac{\\Delta V}{2\\,v\\,\\sin {i}} \\;\\right)= \\; \\left( \\frac{R_{disc}}{R_1}\\;\\right)^{-j} ,$ where $j=0.5$ for Keplerian rotation, $j=1$ for angular momentum conservation, $R_1$ is the radius of the primary, and $v\\,\\sin {i}$ is its projected rotational velocity.",
"Eq.",
"REF relies on the assumptions that (1) the Be star is rotating critically, and (2) that the line profile shape is dominated by kinematics and radiative transfer does not play a role.",
"When the two peaks are visible in the emission lines, we can estimate the disc radius using Eq.",
"REF .",
"The calculated disc size for different emission lines are given in Table REF .",
"In the $H\\alpha $ emission line of LSI+61$^0$ 303 two peaks are clearly visible on all of our spectra.",
"However in MWC 656 the $H\\alpha $ emission line seems to exhibit three peaks (see Fig.",
"REF ).",
"Two peaks in $H\\alpha $ emission of MWC 148 are clearly detectable on January-February 2014 observations.",
"Two peaks are not distinguishable on the spectra obtained in March 2014 (when the companion is at periastron), which probably indicates perturbations in the outer parts of the disc caused by the orbital motion of the compact object.",
"In the $H\\beta $ line two peaks are visible on all the Rozhen spectra.",
"We take this opportunity to obtain an estimation of the $R_{disc}(H\\alpha )$ using $\\Delta V_\\beta $ ; the ratios $\\Delta V_\\beta / \\Delta V_\\alpha $ (as obtained in Sect.",
"REF ), and Eq.REF .",
"The $R_{disc}(H\\alpha )$ values calculated in this way are given in Table REF and indicated with $(^b)$ .",
"Figure: Histograms of the disc size.The vertical dashed (red) lines indicate the distance between the componentsat periastron and apastron, a(1-e) and a(1+e), respectively.The resonances n:m are also indicated (in blue).",
"For LSI+61 0 ^0303 and MWC 148 only those with m=1 are given."
],
[
"Disc size and $W_\\alpha $",
"The disc size and $W_\\alpha $ are connected (see also Hanuschik 1989; Grundstrom & Gies 2006).",
"This simply expresses the fact that $R_{disc}$ grows as $W_\\alpha $ becomes larger.",
"In Fig.",
"REF , we plot $\\log \\Delta V_\\alpha / 2 $$v \\: \\sin \\: i$ versus $\\log W_\\alpha $ .",
"In this figure 138 data points are plotted for Be stars taken from Andrillat (1983), Hanuschik (1986), Hanuschik et al.",
"(1988), Dachs et al.",
"(1992), Slettebak et al.",
"(1992), and Catanzaro (2013).",
"In this figure, the data for Be/$\\gamma $ -ray binaries are also plotted.",
"The three Be/$\\gamma $ -ray binaries are inside the distribution of normal Be stars.",
"There is a moderate to strong correlation between the variables with Pearson correlation coefficient -0.63, Spearman's (rho) rank correlation 0.64, and $p$ -$value \\sim 10^{-15}$ .",
"The dependence is of the form $\\; \\; \\log \\; (\\Delta V_{\\alpha }/2 v \\sin i ) = -a \\; \\log W_\\alpha + b,$ and the slope is shallower for stars with $ W_\\alpha < 3$ Å as noted by Hanuschik et al.",
"(1988).",
"For 120 data points in the interval $3 \\le W_\\alpha \\le 50$ Å, using a least-squares approximation we calculate $a=0.592 \\pm 0.030$ and $b=0.165 \\pm 0.036$ .",
"This fit as well as the correlation coefficients are calculated using only normal Be stars.",
"Using Eq.",
"REF and Eq.",
"REF we then obtain $\\left( \\frac{R_{disc}}{R_1}\\;\\right)^{-j} = 1.462 \\; W_\\alpha ^{-0.592}.$ Having in mind that (1) the discs of the Be stars are near Keplerian (Porter & Rivinius 2003, Meilland et al.",
"2012); (2) the Be stars rotate at rates below the critical rate (e.g.",
"Chauville et al.",
"2001), and (3) at higher optical depths the emission line peaks are shifted towards lower velocities (Hummel & Dachs 1992), we calculate the disc radius with the following formula: $\\frac{R_{disc}}{R_1} = \\; \\epsilon \\; 0.467 \\; \\; W_\\alpha ^{1.184},$ where $\\epsilon $ is a dimensionless parameter (see also Zamanov et al.",
"2013), for which we adopt $\\epsilon = 0.9 \\pm 0.1$ .",
"The disc sizes calculated with Eq.",
"REF are given in Table REF and are denoted with $(^c)$ .",
"As can be seen, the values agree with those obtained with the conventional method.",
"We estimate average values of the dimensionless quantity $R_{disc} / \\epsilon R_1 = 8.7 \\pm 1.9$ (for LSI+61$^0$ 303), $R_{disc} / \\epsilon R_1 = 43 \\pm 5$ (for MWC 148), and $R_{disc} / \\epsilon R_1 = 18.0 \\pm 1.1$ (for MWC 656), respectively."
],
[
"Disc truncation",
"The orbit of the compact object, the average size of $H\\alpha $ disc, the average size of $H\\beta $ disc, and the Be star are plotted in Fig.",
"3.",
"The coordinates X and Y are in solar radii.",
"The histograms of $H\\alpha $ disc size, calculated using Eq.",
"REF , are plotted in Fig.",
"4.",
"For LSI+61$^0$ 303, we use our new data and published data from Paredes et al.",
"(1994), Steele et al.",
"(1996), Liu & Yan (2005), Grundstrom et al.",
"(2007), McSwain et al.",
"(2010), and Zamanov et al.",
"(1999, 2013).",
"We use Rozhen and Liverpool Telescope spectra for MWC 148 and MWC 656.",
"In all three stars the distribution of $R_{disc}$ values has a very well pronounced peak.",
"The tendency for the disc emission fluxes to cluster at specified levels is related to the truncation of the disc at specific disc radii by the orbiting compact object (e.g.",
"Coe et al.",
"2006).",
"Okazaki & Negueruela (2001) proposed that these limiting radii are defined by the closest approach of the companion in the high-eccentricity systems and by resonances between the orbital period and the disc gas rotational periods in the low-eccentricity systems.",
"The resonance radii are given by ${R_{n:m}^{3/2}} = \\frac{m \\; (G \\: M_1)^{1/2}}{2 \\: \\pi } \\: \\frac{P_{orb}}{n},$ where $G$ is the gravitational constant, $n$ is the integer number of disc gas rotational periods, and $m$ is the integer number of orbital periods.",
"The important resonances are not only those with $n:1$ , but can also be $n:m$ in general.",
"For LSI+61$^0$ 303 (assuming $M_1 \\approx 15$ $M_\\odot $ , $M_2 \\approx 1.4$ $M_\\odot $ , $e \\approx 0.537$ ), we estimate the distances between the components $a(1-e) \\approx 44$ $R_\\odot $ and $a(1+e)\\approx 146$ $R_\\odot $ for the periastron and apastron, respectively.",
"As can be seen from Fig.",
"REF , the disc size is $R_{disc} \\sim a(1-e)$ and it never goes close to $a(1+e)$ .",
"The resonances that correspond to disc size are between 5:1 and 1:1, and the peak on the histogram corresponds to the 2:1 resonance.",
"For MWC 148, with the currently available data ($M_1 \\approx 15$ $M_\\odot $ , $M_2 \\approx 4$ $M_\\odot $ , $e \\approx 0.83$ ), we estimate $a(1-e) \\approx 88$ $R_\\odot $ and $a(1+e)\\approx 951$ $R_\\odot $ for the periastron and apastron, respectively.",
"In Fig.",
"REF , it is apparent that $a(1-e) < R_{disc} < a(1+e)$ .",
"The 2:1 resonance is the closest to the peak of the distribution.",
"We note in passing that the disc size in this star could have a bi-modal distribution (a second peak with lower intensity seems to emerge close to 4:1 resonance radius).",
"For MWC 656 (assuming $M_1 \\approx 9$ $M_\\odot $ , $M_2 \\approx 4$ $M_\\odot $ , $e \\approx 0.1$ ), we estimate $a(1-e) \\approx 137$ $R_\\odot $ and $a(1+e)\\approx 167$ $R_\\odot $ for the periastron and apastron, respectively.",
"In Fig.",
"REF , it is seen that the 1:1 resonance is very close to the peak of the distribution and $a(1-e) \\lesssim R_{disc} \\lesssim a(1+e)$ .",
"The disc size rarely goes above $a(1+e)$ ."
],
[
"Interstellar extinction: Estimates of $E(B-V)$ from interstellar lines",
"There is a strong correlation between equivalent width of the diffuse interstellar bands (DIBs) and reddening (Herbig 1975; Puspitarini et al.",
"2013).",
"There is also a calibrated relation of reddening with the equivalent width of the interstellar line $KI \\lambda 7699$ Å (Munari & Zwitter 1997).",
"Aiming to estimate the interstellar extinction towards our objects, we measure equivalent widths of $KI \\lambda 7699$ Å and DIBs $\\lambda 6613, \\lambda 5780, \\lambda 5797$ .",
"LSI+61$^0$ 303: For this object, Hunt et al.",
"(1994) use $E(B-V)=0.93$ (Hutchings & Crampton 1981).",
"Howarth (1983) using the 2200 Å extinction bump obtained $E(B-V)=0.75 \\pm 0.1$ .",
"Steele et al.",
"(1998) estimated $E(B-V)=0.70 \\pm 0.40$ from Na I D$_2$ and $E(B-V)= 0.65 \\pm 0.25$ from diffuse interstellar bands.",
"For LSI+61$^0$ 303, we measure $0.17 \\le W(KI \\lambda 7699) \\le 0.19$ Å, $0.17 < W(DIB \\lambda 6613) < 0.19$ Å, $0.34 < W(DIB \\lambda 5780) < 0.41$ Å, $0.09 < W(DIB \\lambda 5797) < 0.16$ Å, which following Munari & Zwitter (1997) and Puspitarini et al.",
"(2013) calibrations corresponds to $E(B-V)=0.84 \\pm 0.08$ .",
"MWC 148: For this star, Friedemann (1992) estimated E(B-V)=0.85 from the 217 nm band.",
"We measure $0.14 < W(KI \\lambda 7699) < 0.20$ Å, $0.13 < W(DIB \\lambda 6613) < 0.18$ Å, $0.28 < W(DIB \\lambda 5780) < 0.34$ Å, $0.13 < W(DIB \\lambda 5797) < 0.15$ Å, which corresponds to a slightly lower value $E(B-V)=0.77 \\pm 0.06$ .",
"MWC 656: For this star, Williams et al.",
"(2010) gave a low value E(B-V)=0.02.",
"Casares et al.",
"(2014) estimated E(B-V)=0.24.",
"We measure $0.04 < W(DIB \\lambda 6613) < 0.06$ Å, $0.09 \\le W(DIB \\lambda 5780) \\le 0.10$ Å, $0.03 < W(DIB\\lambda 5797) < 0.05$ Å, and estimate $E(B-V)=0.25 \\pm 0.02$ .",
"Figure: Rotational period of the mass donor (P rot P_{rot}) vs. P orb P_{orb}.Filled (blue) circles indicate high-mass X-ray binaries with giant/supergiant mass donor, open circles indicate Be/X-ray binaries, and red plusses indicate γ\\gamma -ray binaries."
],
[
"Rotational period of the mass donors",
"In close binary systems the rotation of the companions of compact objects is accelerated by mass transfer and tidal forces (e.g.",
"Ablimit & Lü 2012).",
"Adopting the parameters given in Sect.",
", we estimate the rotational periods of the mass donors $P_{rot} \\approx 0.92$ d (for LSI+61$^0$ 303), $P_{rot} \\approx 0.91$ d (MWC 148), and $P_{rot} \\approx 0.86$ d (MWC 656).",
"For the $\\gamma $ -ray binary LS 5039, the mass donor is a O6.5V((f)) star with $R_{1} = 9.3 \\pm 0.6 $ $R_\\odot $ , inclination $i \\approx 24.9^0$ , $v \\: \\sin \\: i$$= 113 \\pm 8$ km s$^{-1}$ (Casares et al.",
"2005).",
"We calculate $P_{rot} = 1.764$ d. Because of the short orbital period $P_{orb} = 3.91$ d, the rotation of the mass donor could be pseudo-synchronized with the orbital motion (Casares et al.",
"2005).",
"Radio observations of the pulsar in PSR B1259-63/LS 2883 allow the orbital parameters to be precisely established: $P_{orb} = 1236.72$ d and eccentricity of e = 0.87 (Wang et al.",
"2004; Shannon et al.",
"2014).",
"For the primary, Negueruela et al.",
"(2011) estimated $R_1 = 9.0 \\pm 1.5$ $R_\\odot $ , $v \\: \\sin \\: i$ $ = 260 \\pm 15$ km s$^{-1}$ , and $i_{orb} \\approx 23^0$ , which give $P_{rot} = 0.689$ d. In Fig.",
"REF we plot $P_{rot}$ versus $P_{orb}$ for a number of high-mass X-ray binaries (see also Stoyanov & Zamanov 2009).",
"High-mass X-ray binaries with a giant/supergiant component, Be/X-ray binaries, and Be/$\\gamma $ -ray binaries are plotted with different symbols in this figure.",
"The solid line represents synchronization ($P_{rot} = P_{orb}$ ).",
"Among the five $\\gamma $ -ray binaries with known rotational velocity of the mass donor, the rotation of the mass donor is close to synchronization with the orbital motion in only one (LS 5039).",
"The four Be/$\\gamma $ -ray binaries are not close to the line of synchronization.",
"They occupy the same region as the Be/X-ray binaries; in other words regarding the rotation of the mass donor the Be/$\\gamma $ -ray binaries are similar to the Be/X-ray binaries.",
"In these, the tidal force of the compact object acts to decelerate the rotation of the mass donor.",
"The spin-down of the Be stars due to angular momentum transport from star to disc (Porter 1998) is another source of deceleration.",
"In the well-known Be star $\\gamma $ Cas, Robinson & Smith (2000) found that the X-ray flux varied with a period P = 1.1 d, which they interpreted as the rotational period of the mass donor.",
"A similar period is detected in photometric observations (Harmanec et al.",
"2000; Henry & Smith 2012).",
"This periodicity is probably due to the interaction between magnetic field of the Be star and its circumstellar disc or the presence of some physical feature, such as a spot or cloud, co-rotating with the star.",
"The optical emission lines of MWC 148 are practically identical to those of $\\gamma $ Cas.",
"All detected lines in the observed spectral range (Balmer lines, HeI lines and FeII lines) have similar equivalent widths, intensities, profiles, and even a so-called wine-bottle structure noticeable in the $H\\alpha $ line (see Fig.",
"1 in Zamanov et al.",
"2016).",
"Bearing in mind the above estimations and the curious similarities between: (i) the mean 20-60 keV X-ray luminosity of $\\gamma $ Cas and LSI+61$^0$ 303 (Shrader et al.",
"2015) and (ii) the optical emission lines of $\\gamma $ Cas and MWC 148, we consider that periodicity $\\sim \\!1$ day could be detectable in X-ray/optical bands in the Be/$\\gamma $ -ray binaries and could provide a direct measurement of the rotational period of the mass donor."
],
[
"Discussion",
"The three Be/$\\gamma $ -ray binaries discussed here have non-zero eccentricities and misalignment between the spin axis of the star and the spin axis of the binary orbit could be possible (Martin et al.",
"2009).",
"The inclination of the primary star in LSI+61$^0$ 303 to the line of sight is probably $ i_{Be} \\sim 70^0$ (Zamanov et al.",
"2013).",
"Aragona et al.",
"(2014) derived $a_1 \\sin i_{orb} = 8.64 \\pm 0.52$ .",
"Assuming $M_1 = 15$$M_\\odot $ , $M_2 = 1.4$ $M_\\odot $ , we estimate $ i_{orb} \\sim 67^0 - 73^0$ .",
"It appears that there is no significant deviation of the orbital plane from the equatorial plane of the Be star.",
"The emission lines of MWC 148 are very similar to those of $\\gamma $ Cas.",
"The emission lines are most sensitive to the footpoint density and inclination angle (Hummel 1994).",
"It means that in MWC 148 the Be star inclination should be similar to that of $\\gamma $ Cas, for which the inclination is in the range $40^0 - 50^0$ (Clarke 1990, Quirrenbach et al.",
"1997).",
"For MWC 148, Casares et al.",
"(2012) estimated $a_1 \\sin i_{orb} = 77.6 \\pm 25.9$ , which for $M_1 = 15$ $M_\\odot $ and a 4 $M_\\odot $ black hole gives $ 45^0 \\lesssim i_{orb} \\lesssim 65^0$ .",
"For MWC 656, Casares et al.",
"(2014) give $M_1 \\sin ^3 i_{orb} = 5.83 \\pm 0.70$ .",
"Bearing in mind the range of 8 $M_\\odot $ $\\le M_1 \\le 10$ $M_\\odot $ , this gives $53^0 \\lesssim i_{orb} \\lesssim 59^0$ .",
"Inclination of the Be star can be evaluated from the full width at zero intensity of the FeII lines $FWZI/2 \\sin i = (G M_1/ R_1)^{1/2}$ .",
"From FWZI of FeII lines (Casares et al.",
"2012) and using $R_1= 9.5 - 10.0$ $R_\\odot $ we estimate $i_{Be} \\approx 53 - 61^0$ .",
"It appears that both planes are almost complanar.",
"There are no signs of considerable deviation between the two planes.",
"The opening half-angle of the Be stars' circumstellar disc are $\\sim \\!",
"10^0$ (Tycner et al.",
"2006; Cyr et al.",
"2015) and it means that the compact object is practically orbiting in the plane of the circumstellar disc.",
"The comparison between the orbit and circumstellar disc size (see Fig.",
"REF and Sect.",
"REF ) shows that in these three objects we have three different situations: In LSI+61$^0$ 303 the neutron star passes through the outer parts of the circumstellar disc at periastron, but it does not enter deeply in the disc; In MWC 148 the compact object goes into the innermost parts of the disc (passes through the innermost parts and penetrates deeply in the disc) during the periastron passage; In MWC 656 the black hole is constantly accreting from the outer edge of the circumstellar disc.",
"In MWC 656, this means that the compact object (black hole) is at the disc border at all times and as a consequence it will have a higher and stable mass accretion rate along the entire orbit.",
"It seems to be a very clear case of disc truncation in which the circumstellar disc is cut almost exactly at the black hole orbit.",
"Because the $H\\alpha $ peaks are connected with the outermost parts of the disc, the above three items probably explain the observational findings: 1.",
"In LSI+61$^0$ 303 the $H\\alpha $ emission line has a two-peak profile at all times (in all our spectra in the period 1987 - 2015),because the circumstellar disc size is relatively small and the neutron star passes only through the outermost parts of the circumstellar disc at periastron; 2.",
"In MWC 148, the big jumps in the $H\\alpha $ parameters, $W_\\alpha $ , full width at half maximum and radial velocity (see Fig.",
"4 of Casares 2012) occur because the compact object enters (reaches) the inner parts of the disc during the periastron passage.",
"3.",
"In MWC 656 the double-peak profile is not often visible because the black hole is at the outer edge at all times and makes perturbations exactly in the places where the $H\\alpha $ peaks are formed.",
"It is worth noting that when the compact object causes large-scale perturbations, distorted profiles, such as those observed in 1A 535+262 (Moritani et al.",
"2011, 2013), will appear.",
"If only a small portion of the outer disc is perturbed then it will appear in the central part of the emission line profile (e.g.",
"in between the peaks or even filling the central dip) because the outer parts of the disc produce the central part of the $H\\alpha $ emission line profile.",
"Similar additional emission is already detected in the $H\\alpha $ spectra of LSI+61$^0$ 303 (Paredes et al.",
"1990; Liu et al.",
"2000; Zamanov & Marti 2000).",
"Gamma-ray emission has been repeatedly observed to be periodic in the system LSI+61303 (Albert et al.",
"2009; Saha et al.",
"2016) and also very likely in MWC148, where its periodic X-ray flares are highly correlated with TeV emission (Aliu et al.",
"2014).",
"A similar situation occurs in the case of LS 5039 (Aharonian et al.",
"2006), 1FGL J1018.6-20135856 (H. E. S. S. Collaboration et al.",
"2015), and PSR B1259-63 (H.E.S.S.",
"Collaboration et al.",
"2013), which has the longest orbital period observed in a $\\gamma $ -ray binary (about 3 yr).",
"This is likely related to the very different physical conditions sampled by the compact companion as it revolves around the primary star in an eccentric orbit.",
"Non-thermal emission (e.g.",
"Dubus 2013) is produced from particles accelerated at the shock between the wind of the pulsar and matter flowing out from the primary star (the polar wind as well as the disc).",
"The X-ray and $\\gamma $ -ray light curve of MWC 148 shows two maxima at orbital phases 0.35 and 0.75 and minimum at apastron passage (Acciari et al.",
"2009; Aliu et al.",
"2014).",
"X-ray and $\\gamma $ -ray fluxes are correlated as mentioned above, in agreement with leptonic emission models, where relativistic electrons lose energy by synchrotron emission and inverse Compton emission (Maier & for the VERITAS Collaboration 2015).",
"The highly eccentric orbital geometry sketched in the central panel of Fig.",
"REF is also in good agreement with a periodic flaring system.",
"In the case of MWC 656, $\\gamma $ -rays have been detected occasionally (Williams et al.",
"2010), but so far never reaching the TeV energy domain.",
"The main differences from previous sources are the fact that the compact object has been shown to be a black hole and its orbit is only moderately eccentric (Casares et al.",
"2014).",
"Based on our spectroscopic observations, the size of the excretion disc is such that the black hole is accreting matter only from its lower density outer edges.",
"As mentioned before, this implies that the accretion rate is stable but at the same time low.",
"Indeed, the quiescent X-ray emission level of the system is as weak as $\\sim 10^{-8}$ Eddington luminosity (Munar-Adrover et al.",
"2014).",
"Therefore, episodic $\\gamma $ -ray flares such as those detected by AGILE, likely require some enhancement of mass loss from the primary Be star or clumps in its circumstellar disc.",
"We speculate here that the physical mechanism responsible for $\\gamma $ -ray emission in the MWC 656 black hole context could be related to the alternative microquasar-jet scenario also proposed for $\\gamma $ -ray binaries (see e.g.",
"Romero et al.",
"2007).",
"The fact that MWC 656 seems to adhere to the low-luminosity end of the X-ray/radio correlation for hard state compact jets also points in this direction (Dzib et al.",
"2015)."
],
[
"Conclusions",
"From the spectroscopic observations of the three Be/$\\gamma $ -ray binaries we deduce that in LSI+61$^0$ 303 the neutron star crosses the outer parts of the circumstellar disc at periastron, in MWC 148 the compact object passes deeply through the disc during the periastron passage, and in MWC 656 the black hole is accreting from the outer parts of the circumstellar disc during the entire orbital cycle.",
"The histograms in all three stars show that the disc size clusters at specific levels, indicating the circumstellar disc is truncated by the orbiting compact object.",
"We estimate the interstellar extinction towards LSI+61$^0$ 303, MWC 148, and MWC 656.",
"The rotation of the mass donors is similar to that of the Be/X-ray binaries.",
"We suggest that the three stars deserve to be searched for a periodicity of about $1.0$ day.",
"The authors are grateful to an anonymous referee for valuable comments and suggestions.",
"This work was partially supported by grant AYA2013-47447-C3-3-P from the Spanish Ministerio de Economía y Competitividad (MINECO), and by the Consejería de Economía, Innovación, Ciencia y Empleo of Junta de Andalucía as research group FQM-322, as well as FEDER funds."
]
] | 1605.05811 |
[
[
"Energy fluctuations of finite free-electron Fermi gas"
],
[
"Abstract We discuss the energy distribution of free-electron Fermi-gas, a problem with a textbook solution of Gaussian energy fluctuations in the limit of a large system.",
"We find that for a small system, characterized solely by its heat capacity $C$, the distribution can be solved analytically, and it is both skewed and it vanishes at low energies, exhibiting a sharp drop to zero at the energy corresponding to the filled Fermi sea.",
"The results are relevant from the experimental point of view, since the predicted non-Gaussian effects become pronounced when $C/k_B \\lesssim 10^3$ ($k_B$ is the Boltzmann constant), a regime that can be easily achieved for instance in mesoscopic metallic conductors at sub-kelvin temperatures."
],
[
"Introduction",
"Physical quantities of an equilibrium macroscopic system are well characterized by their average values.",
"However, random deviations from the average values - fluctuations - are very important since they contain important information on the system.",
"Under most circumstances the distribution of small fluctuations is Gaussian [1], ch. 7.",
"However, this is not the case for small devices, which are currently studied intensively [2], [3], see [4], [5], [6], [7] for a review.",
"Energy and temperature fluctuations in a single electron box [9] were considered in [8]; several works experimental and theoretical, e.g., [10], [11], [12], were devoted to temperature fluctuations.",
"In the present work we will derive the distribution function for a finite Fermi gas and show that its shape is determined by only one dimensionless parameter - the total heat capacity $C$ divided by the Boltzmann constant, $k_{\\text{B}}$ .",
"At $C/k_{\\text{B}}\\gg 1$ we recover the well-known Gaussian distribution, while at finite values of this parameter significant deviations are expected.",
"We will analytically derive the distribution of the energies of a finite sample of the Fermi gas kept at a given temperature and analyze its properties including moments and skewness absent in the thermodynamic limit.",
"Since the heat capacity of a metallic conductor of sub-micron dimensions at standard sub-kelvin experimental temperatures is of the order of $(10^2-10^3)k_{\\text{B}}$ [4], [13], our results have potential impact on sensitive bolometers [14], [4], [10].",
"In future, we plan to use the obtained distribution for analysis of heat exchange between a quantum device and a mesoscopic metallic calorimeter.",
"The paper is organized as follows.",
"We derive general expression for the energy distribution in Sec.",
"REF , which will be analyzed in Sec.",
"REF .",
"Moments of the distribution are considered in Sec.",
"REF .",
"It is convenient to calculate the Fourier transform, $F(\\lambda )$ , of the energy distribution function.",
"It can be expressed in the form $ F(\\lambda ) =\\left\\langle e^{i\\lambda E} \\right\\rangle =\\frac{\\sum _m e^{(i\\lambda -\\beta ) E_m} }{\\sum _m e^{ -\\beta E_m}}=\\frac{Z(\\beta - i \\lambda )}{Z(\\beta )}\\, .$ Here $Z(\\beta ) \\equiv \\sum _m e^{ -\\beta E_m}$ is the partition function, $\\beta \\equiv (k_{\\text{B}}T)^{-1}$ where $T$ is temperature, while $ E_m =\\sum _\\mathbf {k}(\\epsilon _{\\mathbf {k},m} - \\mu ) \\, n_{\\mathbf {k},m} \\, .$ A micro-state is characterized by the set of quantum numbers $\\lbrace \\mathbf {k},m\\rbrace $ where $\\mathbf {k}$ is the quasi-momentum.",
"In the following, concentrating on mesoscopic electronic systems, we will consider a gas of elementary particles characterized only by the quantum number $\\mathbf {k}$ .",
"According to the definitions (REF )-(REF ), we calculate the characteristic function of quasiparticle energy.",
"The energy distribution function, $P(E)$ , can be calculated as the inverse Fourier transform of $F(\\lambda )$ as $P(E)=\\int _{-\\infty }^\\infty \\frac{d\\lambda }{2\\pi } \\, e^{-i\\lambda E} F(\\lambda )\\, .$ For the Fermi gas [1], $\\ln Z(\\beta )&=& \\sum _\\mathbf {k}\\ln \\left( 1+e^{-\\beta (\\epsilon _\\mathbf {k}- \\mu )}\\right) \\\\&=&\\mathcal {V} \\int _0^\\infty d\\epsilon \\, N(\\epsilon ) \\, \\ln \\left( 1+e^{-\\beta (\\epsilon - \\mu )}\\right) \\, .$ Here $N(\\epsilon )$ is the density of states.",
"In the following we put $N(\\epsilon ) = \\text{const} \\equiv N_0$ .",
"This assumption is valid for the two-dimensional gas, while in the three-dimensional case $N(\\epsilon ) \\propto \\sqrt{\\epsilon }$ .",
"One can show that in the limiting case $\\beta \\mu \\gg 1$ the results differ only by renormalization of numerical constants in the expressions for the average energy and heat capacity.",
"For realistic experimental conditions $\\beta \\mu = T_F/T \\sim 10^6$ for a metal with Fermi temperature $T_F \\sim 10^5$ K at $T=0.1$ K. For $N(\\epsilon )= N_0$ we obtain $\\ln Z(\\beta )&=&\\frac{\\mathcal {V}N_0}{\\beta } \\int _{-\\beta \\mu }^\\infty d\\xi \\ln \\left(1+e^{- \\xi } \\right) \\nonumber \\\\&=&-\\frac{\\mathcal {V}N_0}{\\beta }\\, \\mathrm {dilog}\\, \\left(1+e^{\\beta \\mu } \\right)\\, .$ Here $\\mathrm {dilog}\\, (x) = \\int _1^x \\frac{\\ln (t)}{1-t}\\, dt$ is the dilogarithm function.",
"At $\\beta \\mu \\gg 1$ we expand $ - \\mathrm {dilog}\\, \\left(1+e^{\\beta \\mu } \\right) \\approx \\frac{1}{2} (\\beta \\mu )^2 + \\frac{\\pi ^2}{6}-e^{-\\beta \\mu } + \\ldots \\, .$ Then $ \\ln Z(\\beta ) \\approx \\mathcal {V}N_0 \\left( \\frac{1}{2}\\beta \\mu ^2 + \\frac{\\pi ^2}{6}\\frac{1}{\\beta }\\right)\\, .$ Note that we assume that the chemical potential is fixed rather than determined by the normalization of the single-particle distribution to the total number of particles.",
"This assumption is appropriate for an electronic system with contacts.",
"In the case when $N(\\epsilon ) \\propto \\sqrt{\\epsilon }$ we can put $N(\\epsilon ) = N_0\\sqrt{\\epsilon /\\mu }$ to obtain instead of Eq.",
"(REF ) $\\ln Z(\\beta ) \\approx \\mathcal {V}N_0 \\left( \\frac{2}{5}\\beta \\mu ^2 + \\frac{\\pi ^2}{4}\\frac{1}{\\beta }\\right)\\, .$ Therefore, different forms of the dependence of the density of states upon the energy indeed result only in numerical constants defining the average energy and heat capacity.",
"The difference in the numerical factors is compatible with the well known Sommerfeld expansion, see, e.g., [15].",
"Using the expansion (REF ) we obtain $\\ln F(\\lambda ) &=&\\ln \\frac{Z(\\beta - i\\lambda )}{Z(\\beta ) } \\nonumber \\\\&=&\\frac{ \\mathcal {V}N_0}{2} \\left(-i \\lambda \\mu ^2+ \\frac{\\pi ^2}{3} \\frac{i\\lambda }{\\beta (\\beta -i\\lambda )}\\right).$ The first item is responsible for the energy shift of the energy by the Fermi energy $E_0 =\\mathcal {V} N_0\\mu ^2/2$ .",
"In our approximation the specific heat has the standard free Fermi-gas expression (for a given chemical potential) $C=k_{\\text{B}}\\frac{\\pi ^2}{3\\beta }N_0\\mathcal {V}\\, .$ Then the quantity in the parentheses in Eq.",
"(REF ) can be rewritten as $ i \\lambda E_0 + \\frac{C}{2k_{\\text{B}}} \\frac{i\\lambda }{\\beta - i\\lambda }\\, .$ We rewrite $\\frac{C}{2k_{\\text{B}}} \\frac{i\\lambda }{\\beta -i\\lambda }&=&i\\lambda \\frac{C}{2k_{\\text{B}}\\beta }\\left(\\frac{i\\lambda }{\\beta -i\\lambda } +1 \\right) - i\\lambda \\frac{C}{2 k_{\\text{B}}\\beta }\\\\&\\equiv & - i\\lambda E_\\beta +\\frac{C}{2k_{\\text{B}}\\beta } \\frac{(i\\lambda )^2}{\\beta - i \\lambda }.$ Here $E_\\beta = C/2 k_{\\text{B}}\\beta $ .",
"Denoting $\\delta E =E-E_0- E_\\beta $ we obtain in the exponent $-i\\lambda \\, \\delta E -\\frac{C}{2 k_{\\text{B}}}\\frac{\\lambda ^2}{\\beta (\\beta - i \\lambda )}\\, .$ Let us measure the energy deviation $\\delta E$ in units of $\\beta ^{-1}\\sqrt{C/k_{\\text{B}}}$ .",
"Then we put $\\lambda \\equiv \\frac{\\beta }{\\sqrt{C/k_{\\text{B}}}} \\Lambda , \\ \\delta E \\equiv \\frac{ \\sqrt{C/k_{\\text{B}}}}{\\beta } u, \\ c \\equiv \\frac{C}{k_{\\text{B}}}$ to obtain for the second term $-\\frac{1}{2}\\frac{\\Lambda ^2}{1-i\\Lambda /\\sqrt{c}}= -\\frac{1}{2}\\frac{\\Lambda ^2(1+i\\Lambda /\\sqrt{c})}{1+\\Lambda ^2 /c}\\, .$ Therefore, we arrive at the expression for the distribution of dimensionless energies $u$ : $ P(u) = \\int _{-\\infty }^\\infty \\frac{d\\Lambda }{2\\pi }\\exp \\left[ i\\Lambda \\left(\\frac{1}{2}\\frac{i\\Lambda }{1-i\\Lambda /\\sqrt{c}}-u \\right)\\right]\\, .$ This is our final expression, which we have to analyze for different values of dimensionless heat capacity, $c$ ."
],
[
"Energy distributions for different dimensionless heat capacity",
"We compute the integral in Eq.",
"(REF ) by deforming the contour in the complex plane.",
"The integrand is analytic in the full complex plane except for (an essential) singularity at $\\Lambda =-i\\sqrt{c}$ .",
"For large $|\\Lambda |$ the argument of the exponential behaves as $-\\frac{1}{2}\\frac{\\Lambda ^2}{1-i\\Lambda /\\sqrt{c}}-i\\Lambda u =-\\frac{c}{2}-i\\Lambda \\left(\\frac{\\sqrt{c}}{2}+u\\right)+{\\mathcal {O}}(1).$ For thermodynamics it suffices to consider energies above the Fermi energy $E>E_0$ which means $u+\\sqrt{c}/2 >0$ .",
"We may then deform the contour to the lower half plane $\\mathop \\mathrm {Im}\\Lambda <0$ .",
"We get $P(u)=P_{0}(u)+P_\\infty (u)$ where $P_\\infty (u)$ is contribution from the semi circle at infinity in the lower half plane and $P_{0}(u)$ from a circle around $\\Lambda =-i\\sqrt{c}$ .",
"Writing $\\Lambda =\\rho e^{i\\phi }$ , $\\pi \\le \\phi \\le 2\\pi $ we obtain $P_\\infty (u)$ as the limit of $e^{-\\frac{_1}{^2}c}\\frac{i\\rho }{2\\pi }\\int _{2\\pi }^\\pi d\\phi e^{i(\\phi -\\rho b e^{i\\phi })}=\\frac{\\sin (b\\rho )}{\\pi b}$ as $\\rho \\rightarrow \\infty $ i.e.",
"$P_\\infty (u)$ where $ b=\\sqrt{c}/2+u$ .",
"Since we assume $b\\ne 0$ this limit vanishes.",
"To compute $P_{0}(u)$ change variables to $z=\\frac{1}{c}(1-i\\frac{\\Lambda }{\\sqrt{c}})$ to get $P_{-i}(u) =c^{\\frac{3}{2}}e^{-(2c+\\sqrt{c}u)}\\oint e^{\\frac{1}{z}}e^{xz}\\frac{dz}{2\\pi i}$ where $x=c(c+\\sqrt{c}u)$ and the integral is around the origin.",
"Then $&&\\oint e^{\\frac{1}{z}}e^{x z}\\frac{dz}{2\\pi i}=\\sum _{n=0}^\\infty \\frac{1}{n!}",
"\\oint z^{-n}e^{x z}\\frac{dz}{2\\pi i}\\nonumber \\\\&=&\\sum _{m=0}^\\infty \\frac{1}{(m+1)!}\\frac{1}{m!",
"}x^{m}=x^{-\\frac{_1}{^2}}I_1(2x^\\frac{_1}{^2})\\nonumber $ where $I_1$ is the modified Bessel function.",
"As a result, $P (u)& =&\\frac{c^{3/4}}{\\sqrt{\\sqrt{c}+2u}} I_1 \\left(c^{3/4}\\sqrt{\\sqrt{c}+2u}\\right)e^{-u\\sqrt{c} -c}\\nonumber $ Graphs for $P(u)$ for different heat capacities, $c$ , are shown in Fig.",
"REF .",
"One observes that at large $c$ the distribution is essentially Gaussian, while at small $c$ it becomes very asymmetric.",
"Figure: (Color online) Distribution P(u)P(u), where u=βδE/cu= \\beta \\delta E/\\sqrt{c}, where δE\\delta Eis the deviation of the energy of the Fermi-gas from its mean, for a few different values of c=C/k B c = C/k_B (solid lines).Dashed line is the Gaussian distribution.Let us crudely estimate the dimensionless heat capacity, $c$ .",
"The estimate reads as $c \\sim n_e \\mathcal {V} (k_{\\text{B}}T/\\mu )$ where $n_e$ is the electron density.",
"For a typical metal $n_e \\approx 10^{23} \\ \\text{cm}^{-3}$ .",
"Assuming the sizes of a mesoscopic system (in $\\mu \\text{m}$ ) as $(0.3-1)\\times (0.05-0.1)\\times (0.01-0.03)$ $(\\mu \\text{m})^3$ , we get $\\mathcal {V} = (1 .5\\cdot 10^{-16} - 3\\cdot 10^{-15}) \\ \\text{cm}^3\\, .",
"$ and the total number of electrons is $\\mathcal {N} \\equiv n_e\\mathcal {V} =1.5\\cdot 10^7 - 3\\cdot 10^8\\, .$ Putting $T_{\\text{F}} \\approx 5\\cdot 10^4~\\text{K}$ and $T=(0.05 - 0.1)~\\text{K}$ we get $k_{\\text{B}}T/\\mu =T/T_{\\text{F}}=(1-2)\\cdot 10^{-6}.$ Consequently, the range of the quantity $c$ is $15 -600$ .",
"As we have seen, at these values of $c$ the energy distribution is clearly non-Gaussian."
],
[
"Moments of the energy distribution",
"The moments of the energy distribution can be readily calculated either from the Eq.",
"(REF ), or explicitly from the Hamiltonian$ \\mathcal {H}=\\sum _\\mathbf {k}\\epsilon _\\mathbf {k}a^\\dag _\\mathbf {k}a_\\mathbf {k}$ .",
"In particular, $\\langle E \\rangle =\\sum _\\mathbf {k}\\epsilon _\\mathbf {k}\\langle a^\\dag _\\mathbf {k}a_\\mathbf {k} \\rangle =\\sum _\\mathbf {k}\\epsilon _\\mathbf {k}f(\\epsilon _\\mathbf {k})$ where $f(\\epsilon )=\\left[e^{\\beta (\\epsilon - \\mu )}+1 \\right]^{-1}$ is the Fermi function.",
"The second moment reads $\\langle E^2 \\rangle =\\sum _\\mathbf {k}\\epsilon _\\mathbf {k}\\epsilon _\\mathbf {l}\\langle a^\\dag _\\mathbf {k}a_\\mathbf {k}a^\\dag _\\mathbf {l}a_\\mathbf {l} \\rangle =\\langle E \\rangle ^2 + \\sum _\\mathbf {k}\\epsilon _\\mathbf {k}^2 f( \\epsilon _\\mathbf {k})[1-f( \\epsilon _\\mathbf {k})].$ Here we have decomposed the product of four Fermi operators according to the Wick theorem.",
"We observe that only the vicinity of the Fermi level is important for the difference $\\langle \\delta E^2 \\rangle \\equiv \\left\\langle \\left(E-\\langle E \\rangle \\right)^2\\right\\rangle =\\langle E^2 \\rangle -\\langle E \\rangle ^2\\, .",
"$ Therefore, while calculating $\\langle \\delta E^2 \\rangle $ we can assume constant density of states and $\\beta \\mu \\gg 1$ .",
"In this way we obtain $\\langle \\delta E^2 \\rangle =k_{\\text{B}}T^2 C\\, .$ In a similar way, we calculate $\\langle \\delta E^3 \\rangle =2\\sum _\\mathbf {k}\\epsilon _\\mathbf {k}^3 f( \\epsilon _\\mathbf {k})[1-f( \\epsilon _\\mathbf {k})]^2=3k_{\\text{B}}^2T^3C\\, .$ The skewness of the distribution is then $\\gamma =\\langle \\delta E^3 \\rangle /\\langle \\delta E^2 \\rangle ^{3/2} =3/\\sqrt{c}\\, .$ Obviously, the skewness vanishes as $c \\rightarrow \\infty $ which is the thermodynamic limit.",
"For a Gaussian distribution, the relationship between the 2nd and the 4th moments is $\\langle \\delta E^4 \\rangle =3\\langle \\delta E^2 \\rangle ^2$ .",
"In the general case, an additional contribution appears, such that $\\langle \\delta E^4 \\rangle =3\\langle \\delta E^2 \\rangle ^2\\left[1+\\frac{4}{\\sqrt{c}} \\right]\\, .$ The 5th moment, absent in the Gaussian approximation, is $\\langle \\delta E^5 \\rangle = 60k_{\\text{B}}^4T^5 C\\, .$ The results obtained by numerical integration of Eq.",
"(REF ) agree with the analytic expressions given above.",
"In summary, we have shown that the energy distribution of a free Fermi-gas with small heat capacity is non-Gaussian, with a sharp cut-off at low energies.",
"This is a natural consequence of the minimal energy of the filled Fermi sea.",
"We find that the heat capacities demonstrating strong non-Gaussian features can be achieved in standard metallic nanodevices at sub-kelvin temperatures.",
"We thank Ivan Khaymovich and Kay Schwieger for discussions.",
"YMG thanks Aalto University for hospitality during the preparation of this manuscript.",
"The work has been supported by the Academy of Finland (contracts no.",
"272218, 284594 and 271983)."
]
] | 1605.05877 |
[
[
"An Apparent Dissociation Transition in Anharmonically Bound 1D Systems"
],
[
"Abstract For diatomic molecules and chains bound anharmonically by interactions such a the Lennard Jones and Morse potentials, we obtain analytical expressions for thermodynamic observables including the mean bond length, thermally averaged internal energy, and the coefficient of thermal expansion.",
"These results are valid across the shift from condensed to gas-like phases, a dissociation transition marked by a crossover with no singularities in thermodynamic variables for finite pressures, though singular behavior appears in the low pressure limit.",
"In the regime where the thermal energy $k_{\\mathrm{B}} T$ is much smaller than the dissociation energy $D$, the mean interatomic separation scales as $\\langle l \\rangle = R_{e} + {\\mathcal B} (P R_{e}/k_{\\mathrm{B}} T)^{-2} e^{-D/k_{\\mathrm{B}} T} \\left( D/k_{\\mathrm{B}} T \\right )^{1/2}$ for both the Morse and Lennard Jones potentials where $p$ is a pressure term, $R_{e}$ is the $T = 0$ bond length, and ${\\mathcal B}$ is a constant specific to the potential."
],
[
"An Apparent Dissociation Transition in Anharmonically Bound 1D Systems D. J. Priour, Jr Department of Physics & Astronomy, Youngstown State University, Youngstown, OH 44555, USA Christopher Watenpool Department of Electrical Engineering, Youngstown State University, Youngstown, OH 44555, USA For diatomic molecules and chains bound anharmonically by interactions such as the Lennard Jones and Morse potentials, we obtain analytical expressions for thermodynamic observables including the mean bond length, thermally averaged internal energy, and the coefficient of thermal expansion.",
"These results are valid across the shift from condensed to gas-like phases, a dissociation transition marked by a crossover with no singularities in thermodynamic variables for finite pressures, though singular behavior appears in the low pressure limit.",
"In the regime where the thermal energy $k_{\\mathrm {B}} T$ is much smaller than the dissociation energy $D$ , the mean interatomic separation scales as $\\langle l \\rangle = R_{e} + {\\mathcal {B}} (P R_{e}/k_{\\mathrm {B}} T)^{-2} e^{-D/k_{\\mathrm {B}} T} \\left( D/k_{\\mathrm {B}} T \\right)^{1/2}$ for both the Morse and Lennard Jones potentials where $p$ is a pressure term, $R_{e}$ is the $T = 0$ bond length, and ${\\mathcal {B}}$ is a constant specific to the potential.",
"64.60.De,61.43.-j,34.80.Ht,51.30.+i Potentials used to describe interactions among atoms such as the Lennard Jones potential [1] (e.g.",
"Van der Waals couplings among noble gas atoms such as Argon) and the Morse potential [2] (for bonding with significant covalent character) contain attractive and repulsive terms.",
"Whereas the attractive component decays at large distance, the repulsive piece decays even more rapidly in the large separation regime while rising sharply when atoms are in proximity where Pauli Exclusion effects play a role as atomic cores begin to overlap.",
"With the combination of attractive and repulsive terms, there is a potential minimum at the equilibrium separation $R_{e}$ with harmonic (parabolic) dependence in the vicinity of $R_{e}$ and increasingly asymmetric and anharmonic character for significant deviations from $R_{e}$ caused, e.g., by thermal fluctuations in the high temperature regime.",
"Whether due to Van der Waals coupling as in the Lennard Jones model or the sharing of charge represented by the Morse potential, entropic effects drive dissociation despite the attractive component, except at $T = 0$ where thermal fluctuations are absent.",
"Nevertheless, confinement of the system volume for finite temperatures is realized by taking into consideration a pressure term (incorporated in the 1D context as a finite cost per length of elongating the system, such as a chain of identical atoms), which precludes indefinite expansion of the system.",
"In this work, we operate in terms of three distinct energy scales relevant to both the Morse and Lennard Jones potentials; the thermal energy $k_{\\mathrm {B}} T$ , the dissociation energy $D$ needed to completely separate an interacting pair of atoms, and the pressure confinement scale $p R_{e}$ .",
"For high $T$ , where $k_{\\mathrm {B}} T \\gg D$ the competition among $p R_{e}$ and $k_{\\mathrm {B}} T$ dominates, and the state is well approximated as an ideal gas.",
"On the other hand, for low $T$ , where $k_{\\mathrm {B}} T \\ll D$ , there is a more subtle interplay among $p$ , $T$ , and $D$ with a crossover with decreasing $p$ from a condensed state ($ \\langle l \\rangle \\approx R_{e}$ ) to a diluted gas-like state ($\\langle l \\rangle \\gg R_{e}$ ).",
"For a more detailed description of dissociation, we obtain analytical expressions for thermodynamic variables of interest valid across the shift from the condensed to gas-like states in the low to intermediate temperature regimes.",
"With a unified treatment applied to both the Lennard Jones and Morse cases, we find strong qualitative similarities in molecular dissociation phenomena despite distinct bonding physics driving the potentials.",
"With interactions only among nearest neighbors, the system energy is $E = \\sum _{i} P_{i}^{2}/2M + \\sum _{i} V (R_{i+1} - R_{i}) + pL$ with the sum over the $N$ members of the chain, $L$ being the total system size.",
"Since atomic momenta and spatial coordinates are not coupled, apart from a contribution $N k_{\\mathrm {B}} T/2$ to the internal energy (and $N k_{\\mathrm {B}}/2$ to the specific heat), one need only consider site positions in sampling system configurations.",
"We obtain thermodynamic variables of interest from the partition function $Z$ ; though in general the rapid scaling of configuration space with the number of system components precludes an analytical calculation of $Z$ , we decouple the calculation of $Z$ by describing the system in terms of the separation among adjacent sites $\\Delta _{i} \\equiv R_{i+1} - R_{i}$ instead of the absolute atomic coordinates $R_{i}$ [3], [4].",
"Noting that $L = R_{N} - R_{1} = \\Delta _{N-1} + \\Delta _{N-2} + \\ldots + \\Delta _{1}$ , the partition function may be factorized as $Z = \\hat{Z}^{N-1}$ where $\\hat{Z} = \\int _{0}^{\\infty } \\exp ( -\\beta [ V(\\Delta ) + p \\Delta ] ) d \\Delta $ .",
"In this manner, the calculation for a chain of arbitrary length is reduced to the case of a single atomic pair separated by a distance $\\Delta $ (subsequently labeled as $R$ ), subject to an asymptotically linearly diverging effective potential $V(R) + pR$ .",
"It is convenient to operate in terms of dimensionless energy variables $\\varepsilon \\equiv \\beta D$ and $\\eta \\equiv p R_{e}$ as well as the ratio $\\gamma \\equiv p R_{e}/D = \\eta /\\varepsilon $ characterizing the strength of the pressure energy scale in relation to the dissociation energy.",
"Salient thermodynamic variables may then be calculated by differentiating with respect to $\\varepsilon $ and $\\eta $ , such as $\\langle E \\rangle = -\\partial /\\partial \\beta \\ln Z = -( \\varepsilon Z_{\\varepsilon } + \\eta Z_{\\eta } ) \\beta ^{-1} Z^{-1}$ for the internal energy where $Z_{\\varepsilon } \\equiv \\partial Z/\\partial \\varepsilon $ and $Z_{\\eta } \\equiv \\partial Z/\\partial \\eta $ .",
"Though having different functional forms, $V_{\\mathrm {LJ}} = B R^{-12} - A R^{-6}$ (Lennard Jones) and $V_{\\mathrm {M}} = -D + D \\lbrace -1 + \\exp [-a (R - R_{e})] \\rbrace ^{2}$ (Morse), $V_{\\mathrm {LJ}}$ and $V_{\\mathrm {M}}$ are amenable to the same analysis with qualitatively similar results in both cases.",
"When expressed in terms of $D$ and the reduced separation $r \\equiv R/R_{e}$ , $V_{\\mathrm {LJ}}$ and $V_{\\mathrm {M}}$ are both of the form $V(r) = -D + D[ \\chi (r) - 1]^{2}$ where $\\chi _{\\mathrm {LJ}} = r^{-6}$ and $\\chi _{\\mathrm {M}} = \\exp [- {\\mathcal {A}} (r - 1)]$ .",
"In terms of specific parameters, $D = -A^{2}/4B$ and $R_{e} = (2B/A)^{1/6}$ for the Lennard Jones case while ${\\mathcal {A}} = a R_{e}$ (${\\mathcal {A}} = 2.5$ for results exhibited here) for the Morse potential.",
"In terms of $\\chi (r)$ , the partition function for both $V_{\\mathrm {LJ}}$ and $V_{\\mathrm {M}}$ is $Z = R_{e} \\exp {\\varepsilon } \\int _{0}^{\\infty } \\exp [ \\chi (r) -1]^{2} \\exp [-\\eta r] dr$ .",
"Although one could in principle expand $\\chi (r) - 1$ about $r = 1$ , where the potential basin is locally parabolic, the significant departure from harmonic character with the shift from the condensed phase ($r \\approx 1$ ) to the gas-like phase ($r \\gg 1$ ) hampers a perturbative analysis of this kind.",
"For a treatment which offers good convergence with only a handful of terms, and which provides a good description of the condensed and dissociated states alike, we consider the substitution $u = \\chi (r)$ , obtaining $Z = -R_{e} e^{\\varepsilon } \\int _{0}^{\\infty } e^{-\\varepsilon (u - 1)^{2}} e^{-\\eta \\chi ^{-1}(u)} \\frac{1}{\\frac{d}{du} \\chi ^{-1}(u)} du$ Integrating by parts, neglecting boundary terms, and using $v = u - 1$ leads to $Z = \\frac{2 R_{e} \\varepsilon }{\\eta } e^{\\varepsilon } \\int _{-1}^{\\infty } e^{-\\varepsilon v^{2}} e^{-\\eta \\chi ^{-1}(v+1)} v dv$ which may be evaluated as a series of Gaussian integrals by Taylor expanding $\\exp [-\\eta \\chi ^{-1}(v+1)]$ about $v = 0$ with $e^{-\\eta \\chi ^{-1}(v+1)} = e^{-\\eta } \\left[ 1 - \\eta d_{1} v + \\frac{1}{2!}",
"(\\eta ^{2} d_{1}^{2} - \\eta d_{2})v^{2} + \\ldots \\right]$ where the $d_{i}$ are $i$ th derivatives of $\\chi ^{-1}(v+1)$ evaluated at $v = 0$ .",
"For $V_{\\mathrm {LJ}}$ , $d_{1} = -1/6$ , $d_{2} = 7/36$ , and $d_{3} = -91/216$ while for $V_{\\mathrm {M}}$ , $d_{1} = -1/{\\mathcal {A}}$ , $d_{2} = 1/{\\mathcal {A}}$ , and $d_{3} = -2/{\\mathcal {A}}$ .",
"For the sake of a quantitatively accurate description of the dissociation transition, it is important not to neglect the finiteness of the lower integration limit in evaluating the Gaussian integrals, which must be taken into consideration to account for dissociation.",
"In the regime of interest, $Z$ is well represented by $Z = e^{\\varepsilon - \\eta } \\bigg [ e^{-\\varepsilon } (\\eta ^{-1} + d_{1} ) - \\\\\\nonumber \\frac{\\sqrt{\\pi }}{4} \\varepsilon ^{-3/2} (4 \\varepsilon d_{1} + d_{1}^{3} \\eta ^{2} -3 \\eta d_{1} d_{2} + d_{3} ) \\bigg ]$ where we have used $\\int _{-1}^{\\infty } \\exp ( -\\varepsilon v^{2})v dv = \\exp -\\varepsilon /(2 \\varepsilon )$ , $\\int _{-1}^{\\infty } \\exp ( -\\varepsilon v^{2})v^{2} dv \\approx (\\sqrt{\\pi }/2) \\varepsilon ^{-3/2} - \\exp -\\varepsilon /(2 \\varepsilon )$ (i.e.",
"retaining the leading term in the asymptotic series); we have neglected $\\int _{-1}^{\\infty } \\exp ( -\\varepsilon v^{2})v^{3} dv$ , and truncated $\\int _{-1}^{\\infty } \\exp ( -\\varepsilon v^{2})v^{4} dv$ at $(3 \\sqrt{\\pi }/4) \\varepsilon ^{-5/2}$ .",
"For a description valid for both the condensed and gas-like phases, we obtain from $Z$ rational expressions for observables of interest where typically leading and next to leading order terms in $\\eta $ and $\\varepsilon $ are retained in the numerator and the denominator; we calculate and discuss in turn the thermally averaged bond length $\\langle l \\rangle $ , the internal energy $\\langle E \\rangle $ , the specific heat $c_{p}$ , and the thermal expansion coefficient $\\alpha $ .",
"With the mean interatomic separation being $-R_{e} \\partial \\ln Z/\\partial \\eta $ , the normalized thermally averaged bond length is well approximated with $\\frac{\\langle l \\rangle }{R_{e}} = 1 + \\eta ^{-1} \\left[ \\frac{e^{-\\varepsilon } - \\frac{3}{4} \\sqrt{\\pi } d_{1} d_{2} \\varepsilon ^{-3/2} \\eta ^{2} }{e^{-\\varepsilon } -\\frac{\\sqrt{\\pi }}{4} \\varepsilon ^{-3/2} \\eta \\left( 4 d_{1} \\varepsilon + d_{3} \\right)} \\right]$ readily inverted via the quadratic formula to obtain $\\eta $ in terms of $\\langle l \\rangle $ .",
"Asymptotically, for $\\eta \\ll \\varepsilon $ , the residual component of the mean separation (small in the condensed phase but diverging with the dissociation transition) may be represented by the simpler expression $\\tilde{\\Delta }_{l} \\equiv (\\langle l \\rangle /R_{e} - 1) = \\varepsilon ^{1/2} \\exp -\\varepsilon /(d_{1} \\sqrt{\\pi } \\eta ^{2})$ .",
"To more conveniently visualize $\\langle l \\rangle $ , (as well as $\\langle E \\rangle $ , $\\alpha $ , and $c_{p}$ ), we choose $\\varepsilon $ for the abscissa where only $T$ is varied while holding $p$ fixed for a given curve; $\\varepsilon \\gg 1$ and $\\varepsilon \\ll 1$ correspond to the low and high $T$ regimes respectively.",
"Whereas $\\gamma = p R_{e}/D$ remains constant, $\\eta = \\beta p R_{e} = \\gamma \\varepsilon $ varies with temperature.",
"Figure REF displays results corresponding to $V_{\\mathrm {LJ}}$ in panel (a) and $V_{\\mathrm {M}}$ in panel (b) with open symbols indicating numerical results and solid curves representing the foregoing approximations to $\\langle l \\rangle $ .",
"The black traces, calculated with the rational expression in Eq.",
"REF , are in good quantitative agreement with numerical data even for $\\gamma = 1.0$ .",
"On the other hand, the $\\tilde{\\Delta }_{l} + 1$ approximation is represented by lighter (red/blue for $V_{\\mathrm {LJ}}$ /$V_{\\mathrm {M}}$ ) curves, and accurately indicates the location where $\\langle l \\rangle $ begins to diverge significantly from $R_{e}$ .",
"Moreover, despite the simple structure, good agreement with exact numerical results is evident for $\\gamma < 0.01$ .",
"To estimate $\\gamma $ for physically realistic systems, we consider a chain of noble gas atoms (e.g.",
"Argon) in the context of the Lennard Jones Model or a covalently bonded pair of atoms (e.g.",
"H$_{2}$ ) in the Morse potential framework in a 1D conduit.",
"With typical atomic radii on the order of an Ångstrom, we assume a cross sectional area on the order of $(1~\\textrm {Å})^{2} = 10^{-20}~\\textrm {m}^{2}$ .",
"Using $1.01 \\times 10^{5}~\\textrm {Pa}$ for standard atmospheric pressure, find $p \\sim 10^{-15}~J/\\textrm {m}$ .",
"In the case of Argon, one has $D = 0.011~\\textrm {eV}$ and $R_{e} = 3.8~\\textrm {Å}$ [5], [6], [7], [8] and $\\gamma = 2.2 \\times 10^{-4}$ .",
"On the other hand, for H$_{2}$ , $D = 4.52~\\textrm {eV}$ [9] and we take $R_{e}$ to be the measured bond length of $0.74~\\textrm {Å}$ [10]; one obtains $\\gamma = 1.0 \\times 10^{-7}$ , with both $\\gamma $ values deep in the $\\gamma \\ll 1$ range.",
"Figure: (Color online) Normalized mean atomic separations for the Lennard Jones (left panel) and Morse potential (right panel)for various γ\\gamma values;solid traces are analytical approximations while open symbols are numerical data.",
"Black curves correspond to the Δ ˜ l +1\\tilde{\\Delta }_{l}+1relationship, while colored traces represent the rational 〈l〉\\langle l \\rangle expression.The internal energy, $\\langle E \\rangle $ , is well approximated as $\\frac{\\langle E \\rangle }{k_{\\mathrm {B}} T} = \\eta - \\varepsilon + \\frac{e^{-\\varepsilon } \\eta ^{-1} (1 + \\varepsilon ) - \\varepsilon ^{-3/2}\\frac{\\sqrt{\\pi }}{8} (4 d_{1} \\varepsilon + 3d_{3} )}{e^{-\\varepsilon }\\eta ^{-1} - \\varepsilon ^{-3/2} \\frac{\\sqrt{\\pi }}{4}(4 d_{1} \\varepsilon + d_{3}) }$ To strip away trivial dependencies, Figure REF shows the residual internal energies, $E_{\\mathrm {Res}} = \\langle E \\rangle + D - p R_{e}$ , with Lennard Jones results in the main panel and Morse potential results in the inset for a variety of $\\gamma $ values.",
"Solid curves obtained from the rational expression closely coincide with the open symbols representing numerical data.",
"Though $E_{\\mathrm {Res}}$ tends to $k_{\\mathrm {B}} T/2$ for sufficiently large $\\varepsilon $ , the expected dependence where $\\langle l \\rangle \\approx R_{e}$ where anharmonicities are negligible, the residual energy component is non-monotonic.",
"One sees from the main graph and the inset graph that the breadth and height of the peak separating the low and high $T$ regimes scales asymptotically as $\\log _{10}(1/\\gamma )$ Figure: (Color online) Residual internal energy results for various γ\\gamma values with the main graph showing Lennard Jones results and the insetdisplaying Morse results.",
"Solid traces represent analytical results, while open symbols indicate numerical data.The specific heat at constant pressure is $c_{p} = \\partial \\langle E \\rangle /\\partial T$ , which is $\\frac{c_{p}}{k_{\\mathrm {B}}} = \\frac{\\frac{1}{2} \\varepsilon ^{2} \\left( \\varepsilon + \\frac{3 d_{3}}{2d_{1}} \\right)- \\Upsilon \\varepsilon ^{1/2} \\left( 1 + \\varepsilon + \\frac{d3}{4d_{1}} \\right) + \\Upsilon ^{2}}{\\varepsilon ^{-2}\\left( \\varepsilon + \\frac{d_{3}}{2d_{1}} \\right)- 2 \\Upsilon \\varepsilon ^{-3/2} \\left( \\varepsilon + \\frac{d3}{4d_{1}} \\right) + \\Upsilon ^{2} }$ where $\\Upsilon = \\exp {-\\varepsilon }/( \\sqrt{\\pi } \\eta d_{1} )$ .",
"Specific heat results are displayed in the main graph (for $V_{\\mathrm {LJ}}$ ) and the inset (for $V_{\\mathrm {M}}$ ) of Fig.",
"REF for a range of $\\gamma $ values; as in the case of the internal energy, there is good agreement among the analytical (solid curve) and the numerical (open symbols) results.",
"One sees from the analytical $c_{p}$ expression in Eq.",
"REF and the curves in Fig.",
"REF that the specific heat tends to $k_{\\mathrm {B}}$ for sufficiently low $D$ (i.e.",
"higher $T$ ) and flattens to $k_{\\mathrm {B}}/2$ for $D \\gg 1$ (low $T$ ).",
"Whereas the latter is a hallmark of the condensed phase, the former is expected for a dissociated system; the transition between the two asymptotically flat regions is non-monotonic, with a peak separating the $k_{\\mathrm {B}}$ and $k_{\\mathrm {B}}/2$ regimes.",
"The region where $c_{p}$ rises to a peak, representing the dissociation transition, becomes taller and narrower with decreasing $\\gamma $ .",
"Figure: (Color online) Specific heat curves for assorted γ\\gamma values with Lennard Jones results in the main graph and Morse results in theinset.",
"Analytical results are shown as solid traces, while open symbols represent numerical data.The coefficient of thermal expansion $\\alpha = \\langle l \\rangle ^{-1} \\partial \\langle l \\rangle /\\partial T$ is well represented by $\\frac{\\alpha }{k_{\\mathrm {B}}} = \\frac{\\frac{3}{4} \\eta d_{2} \\varepsilon ^{-1} -\\Upsilon \\varepsilon ^{1/2} \\left( \\varepsilon + \\frac{3}{2}+ \\frac{d_{3}}{4 d_{1}} - \\frac{3 \\eta d_{2}}{2} \\right) + \\Upsilon ^{2}}{\\eta -\\Upsilon \\varepsilon ^{-1/2} \\left( \\varepsilon +2 \\eta \\varepsilon + \\frac{d_{3}}{4 d_{1}}\\right) + \\Upsilon ^{2}}$ Juxtaposed analytical (solid traces) and numerical results (open symbols) are shown in Figure 4 for various $\\gamma $ values with $D \\alpha /k_{\\mathrm {B}}$ on the vertical axis.",
"As in the case of the specific heat, the $\\alpha $ curves are non-monotonic, with peak heights increasing with decreasing $\\gamma $ .",
"Choosing $D \\alpha /k_{\\mathrm {B}}$ for the ordinate is in part to show the slow convergence to the low $T$ value of $3 d_{2}/4$ in the $\\gamma \\ll 1$ limit.",
"All of the cases shown correspond to experimentally realistic $\\gamma $ values, and in all but one of the curves shown, $\\alpha $ is appreciably different from the limiting value even for $\\varepsilon $ as high as 10.",
"Figure: (Color online) Thermal expansion coefficient results for a range of γ\\gamma values with Lennard Jones results plotted in the maingraph and corresponding results for the Morse potential in the inset.",
"Solid curves represent analytical results, while open symbols indicate numerical data.The peak locations calculated in the framework of the analytical approximations (solid traces) for $c_{p}$ and $\\alpha $ are in close agreement with the exact numerical results (open symbols) as may be seen in the upper panels of Fig.",
"REF .",
"In statistical mechanics singularities are not in general encountered for single component systems, with non-analytic behavior emerging only in thermodynamic limit as the number of degrees of freedom tends to infinity.",
"Nevertheless, the low pressure regime is atypical in the sense that singular behavior is inevitable as $p$ (or $\\gamma $ ) tends to zero due to the divergence of $\\langle l \\rangle $ for any finite $T$ as $p \\rightarrow 0$ .",
"Hence, the possibility of a sharp dissociation transition for $\\gamma \\ll 1$ must be examined with care.",
"As a measure of the extent to which the dissociation transition is singular, the sharpness of the thermal expansion coefficient and specific heat peaks is quantified in the lower panels of Fig.",
"REF as the relative full width half maximum, $\\Delta T_{\\mathrm {Peak}}/T_{\\mathrm {Peak}}$ .",
"In the case of $\\alpha $ , $\\Delta T_{\\mathrm {Peak}}/T_{\\mathrm {Peak}}$ tends to a finite value common to both $V_{\\mathrm {LJ}}$ and $V_{\\mathrm {M}}$ , indicating the $\\alpha $ peaks cease to become narrower relative to their location with decreasing $\\gamma $ .",
"On the other hand, the specific heat relative peak width appears to tend to zero with decreasing $\\gamma $ , a trend highlighted in the lower right panel inset of Fig.REF showing $\\Delta T_{\\mathrm {Peak}}/T_{\\mathrm {Peak}}$ relative to $1/\\log _{10} (\\gamma ^{-1})$ , The curves are asymptotically linear as $\\gamma \\rightarrow 0$ , with the relative peak width vanishing for $p \\rightarrow 0$ as singular behavior appears.",
"Figure: (Color online) Peak locations for the specific heat (upper right panel) and thermal expansion coefficient (upper left panel).Corresponding relative full width half maxima (i.e.",
"ΔT peak /T peak \\Delta T_{\\mathrm {peak}}/T_{\\mathrm {peak}}) appear in the lower right and lower left panels for thespecific heat and coefficient of thermal expansion respectively.Throughout, open squares represent Lennard Jones results and open circles Morse results.In conclusion, with a nonperturbative treatment of the anharmonicity of interatomic potentials, we provide a theoretical description of the dissociation transition valid for the condensed state as well as the gas-like phase where thermal fluctuations have driven pairs of atoms far from their equilibrium separations.",
"By applying a unified treatment to disparate potentials, we have obtained analytical results for salient thermodynamic observables in good agreement with precise numerical results.",
"Though ascribed to distinct bonding physics, there are striking similarities in results, e.g.",
"with $\\langle l \\rangle /R_{e} = 1 + \\varepsilon ^{1/2} \\exp -\\varepsilon /(d_{1} \\sqrt{\\pi } \\eta ^{2})$ specifying the mean bond length for $\\gamma \\ll 1$ .",
"Useful conversations with Michael Crescimanno are gratefully acknowledged."
]
] | 1605.06082 |
[
[
"Ancilla-free Reversible Logic Synthesis via Sorting"
],
[
"Abstract Reversible logic synthesis is emerging as a major research component for post-CMOS computing devices, in particular Quantum computing.",
"In this work, we link the reversible logic synthesis problem to sorting algorithms.",
"Based on our analysis, an alternative derivation of the worst-case complexity of generated reversible circuits is provided.",
"Furthermore, a novel column-wise reversible logic synthesis method, termed RevCol, is designed with inspiration from radix sort.",
"Extending the principles of RevCol, we present a hybrid reversible logic synthesis framework.",
"The theoretical and experimental results are presented.",
"The results are extensively benchmarked with state-of-the-art ancilla-free reversible logic synthesis methods."
],
[
"=1 pdfinfo= Title=Ancilla-free Reversible Logic Synthesis via Sorting, Author=Anupam Chattopadhyay and Sharif MD Khairul Hossain, Subject=, Keywords= [pages=1-last]paper.pdf"
]
] | 1605.05989 |
[
[
"Algebraically special Einstein-Maxwell fields"
],
[
"Abstract The Geroch-Held-Penrose formalism is used to re-analyse algebraically special non-null Einstein-Maxwell fields, aligned as well as non-aligned, in the presence of a possible non-vanishing cosmological constant.",
"A new invariant characterisation is given of the Garcia-Plebanski and Plebanski-Hacyan metrics within the family of aligned solutions and of the Griffiths metrics within the family of the non-aligned solutions.",
"As a corollary also the double alignment of the Debever-McLenaghan 'class D' metrics with non-vanishing cosmological constant is shown to be equivalent with the shear-free and geodesic behaviour of their Debever-Penrose vectors."
],
[
"Introduction",
"In the quest for exact solutions of the Einstein-Maxwell equations, $R_{ab}-\\frac{1}{2}R g_{ab}+\\Lambda g_{ab} = F_{ac} {F_b}^c-\\frac{1}{4}g_{ab}F_{cd}F^{cd},$ a large amount of research (see for example the reviews in [21], [40] has been devoted to the study of so called aligned Einstein-Maxwell fields, in which at least one of the principal null directions (PND's) of the electromagnetic field tensor ${F}$ is parallel to a PND of the Weyl tensor, also called a Debever-Penrose direction, with main emphasis on the doubly aligned Petrov type D solutions, in which both real PND's of ${F}$ are parallel to a corresponding double Weyl-PND.",
"One of the prominent tools in these and related activities has been the Goldberg-Sachs theorem, which in its original form [18] says that a vacuum space-time is algebraically special if and only if it contains a shear-free geodesic null congruence (in an adapted Newman-Penrose tetrad, `$\\Psi _0=\\Psi _1=0$ if and only if $\\kappa =\\sigma =0$ ').",
"Goldberg and Sachs also proved that, if a space-time admits a complex null tetrad $({k},{\\ell },{m},\\overline{{m}})$ such that ${k}$ is shear-free and geodesic and $R_{ab}k^a k^b = R_{ab}k^a m^b = R_{ab}m^a m^b = 0$ (as is the case when ${k}$ is a PND of the electromagnetic field tensor), then the Weyl tensor is algebraically special, with ${k}$ being a multiple Weyl-PND.",
"While, for a null Maxwell field [29], [38], the Maxwell and Bianchi equations imply that both conditions $\\Psi _0=\\Psi _1=0$ and $\\kappa =\\sigma =0$ are trivially satisfied (also in the presence of a possible cosmological constant), the situation is less straightforward in the non-null case.",
"It still is true [18], [39] that, when a PND ${k}$ of the electromagnetic field tensor is shear-free and geodesic, then the Weyl tensor is algebraically special, but the reverse no longer holds.",
"A key property in this respect is the Kundt-Trümper theorem [26], which says that for an algebraically special aligned non-null Einstein-Maxwell space-time (with a possible non-0 cosmological constant and with ${k}$ the PND of ${F}$ coinciding with a multiple Weyl-PND) one necessarily has $\\kappa (3 \\Psi _2-2 |\\Phi _1|^2)=0 \\textrm { and }\\sigma (3 \\Psi _2+2 |\\Phi _1|^2)=0,$ implying that the exceptional case $|\\kappa |^2+|\\sigma |^2 \\ne 0$ can only occur for Petrov types II or D with $\\frac{3}{2}\\Psi _2=\\pm |\\Phi _1|^2.$ Subsequent research has been concentrated on the doubly aligned Petrov type D cases, in which the Kundt-Trümper relations (REF ) also hold with $\\kappa $ and $\\sigma $ replaced by $\\nu $ and $\\lambda $ .",
"This culminated in the complete integration of the field equations for the Petrov type D doubly aligned non-null Einstein-Maxwell fields, with a possible non-0 cosmological constant: When $\\frac{9}{4}\\Psi _2^2- |\\Phi _1|^4\\ne 0$ , both real PND's are geodesic and shear-free ($\\kappa =\\nu =\\sigma =\\lambda =0$ ) and, following Debever and McLenaghan[10], I will refer to the corresponding set of solutions as the `class $\\mathcal {D}$ ' space-times.",
"They all admit at least a two-dimensional isometry group and count among their most famous members the Reissner-Nordström and Kerr-Newman metrics.",
"Several authors [5], [6], [7], [8], [9], [10], [11], [12], [14], [15], [22], [24], [33], [34] have independently contributed to the determination of class $\\mathcal {D}$ , beginning with Carter's seminal study of the separability of the Hamilton-Jacobi and Klein-Gordon equations and culminating in the discovery [33] of Plebański and Demiański's 7-parameter metric for the non-null orbit solutions and García's construction [14], [15] of a single metric form for both the non-null and null orbit solutions.",
"The case $\\frac{3}{2}\\Psi _2= |\\Phi _1|^2$ was fully integrated in [35].",
"In the resulting `Plebański-Hacyan space-times' ${k}$ and ${\\ell }$ are respectively non-geodesic ($\\kappa \\ne 0$ ) and geodesic ($\\nu =0$ ), while both are shear-free ($\\sigma =\\lambda =0$ ) and have vanishing complex divergence ($\\rho =\\mu =0$ ).",
"The metric is given by $\\textrm {d}s^2= 2\\textrm {d}\\zeta \\textrm {d}\\overline{\\zeta }+2 \\textrm {d}u\\textrm {d}v+[\\Lambda u^2+\\zeta \\overline{F}(v)+\\overline{\\zeta } F(v)] \\textrm {d}v^2,$ which, for an arbitrary function $F(v)$Note1 does not admit any isometries and only has an electromagnetic energy-momentum tensor of the correct sign when $\\Lambda <0$ .",
"The case $\\frac{3}{2}\\Psi _2= -|\\Phi _1|^2$ , which was overlooked in [35], was dealt with in [16].",
"Its unique solutions are the `García-Plebański metrics', $\\textrm {d}s^2 = -\\frac{1}{\\Lambda }\\left( e^{2z}{\\omega ^1}^2+e^{-2 z}{\\omega ^2}^2+\\textrm {d}z^2-4 (\\cosh z)^2{\\omega ^3}^2 \\right),$ with $\\omega ^1+i \\omega ^2 = 2\\frac{e^{iu}}{1-\\zeta \\overline{\\zeta }}\\textrm {d}\\zeta \\textrm { and } \\omega ^3 = \\textrm {d}u-i \\frac{\\zeta \\textrm {d}\\overline{\\zeta }-\\overline{\\zeta }\\textrm {d}\\zeta }{1-\\zeta \\overline{\\zeta }}.$ The corresponding space-times admit a 3 dimensional group of isometries; both ${k}$ and ${\\ell }$ are geodesic ($\\kappa =\\nu =0$ ), shearing ($\\sigma \\lambda \\ne 0$ ) and twisting, but non-expanding ($\\rho , \\mu \\in i \\mathbb {R}$ ).",
"Also this metric can only describe an Einstein-Maxwell space-time with an electromagnetic energy-momentum tensor of the correct sign when $\\Lambda <0$ .",
"In this paper I will have a closer look at the full set of algebraically special cases, both aligned (and hence by (REF ) necessarily of Petrov type II) and non-aligned ones.",
"First we will see that the García-Plebański metrics are the unique members of the class of algebraically special and aligned Einstein-Maxwell solutions for which the null direction ${k}$ is shearing.",
"While doing so, I will correct an error in [25] (cited also in [40] p. 409), claiming that, at least for $\\Lambda =0$ , the case $\\kappa =0\\ne \\sigma $ admits no solutions.",
"This is true indeed but, as will become clear in section 2, the proof requires a much more subtle argumentation than the one presented in [25]Note2 .",
"The error occurs after relation (2.6) when the case $\\overline{\\rho }=-\\rho $ is dismissed by remarking that it ”leads to $\\rho =0$ ”.",
"Most likely this conclusion was prematurely arrived at by inspection of the Newman-Penrose equation corresponding to our GHP equation (REF ): with $\\kappa =\\epsilon +\\overline{\\epsilon }=\\Phi _0=0$ this equation reads $D\\rho = \\rho ^2+\\sigma \\overline{\\sigma }$ , the real and imaginary parts of which only allow one to conclude that $\\rho =\\pm i |\\sigma |$ and $D \\rho =D \\sigma = 0$ .",
"In section 2 a correct proof of Kozarzewski's no-go claim will be provided, generalizing it to the case $\\Lambda \\geqslant 0$ and showing that for $\\Lambda <0$ the only allowed solutions are the doubly aligned García-Plebański metrics with $\\rho = \\pm i |\\sigma | \\ne 0$ .",
"In section 3 I will consider the algebraically special and aligned Einstein-Maxwell fields for which the null direction ${k}$ is non-geodesic.",
"The general solution in this family so far is not known, but, remarkably, the Plebański-Hacyan metrics exhaust the sub-family characterised by the vanishing of the complex divergence of ${k}$ .",
"Finally I prove in section 4 that an algebraically special Einstein-Maxwell solution possessing a shear-free and geodesic multiple Weyl-PND which is not a PND of ${F}$ necessarily has vanishing cosmological constant and I give a characterization of the sub-class of the Griffiths [20] solutions.",
"As a corollary of this theorem it follows that the `class $\\mathcal {D}$ ' metrics[10] are the unique Petrov type D Einstein-Maxwell solutions for which the real Weyl-PND's are both geodesic and shear-free and for which the cosmological constant is non-vanishing: in other words, the double alignment condition of the class $\\mathcal {D}$ metrics with non-vanishing cosmological constant is a consequence of their multiple Weyl-PND's being geodesic and shear-free.",
"Whether this property persists when $\\Lambda =0$ is at present still an open problem, with only the Kundt case ($\\rho =0$ ) so far having been dealt with.",
"In order to study these and related issues, linking kinematic properties of certain invariantly defined null directions (such as being geodesic and/or non-shearing) to algebraic properties of the electromagnetic field tensor or of the Weyl tensor, it is natural to use the Geroch-Held-Penrose (GHP) formalism [17].",
"In this formalism only a pair of null directions is singled out at each point rather than an entire null tetrad, as is the case in the Newman-Penrose [31] formalism.",
"The resulting formalism is covariant with respect to rotations of the spatial basis vectors and boosts of the real null directions and, as such, is `halfway' between a fully covariant approach and the NP spin-coefficient approach and leads to considerably simpler equations with fewer complex variables.",
"Throughout I will use the sign conventions and notations of Kramer §7.4, with the tetrad basis vectors taken as ${k},{\\ell },{m},\\overline{{m}}$ with $- k^a \\ell _a = 1 = m^a\\overline{m}_a$ .",
"However, in order to ease comparison with the (more familiar) Newman-Penrose formalism, I will write primed variables, such as $\\kappa ^{\\prime },\\sigma ^{\\prime },\\rho ^{\\prime }$ and $\\tau ^{\\prime }$ , as their NP equivalents $-\\nu ,-\\lambda , -\\mu $ and $-\\pi $ .",
"For completeness the resulting weights, commutators, GHP, Bianchi and Maxwell equations are presented in an Appendix.",
"Finally note that, at least when the electromagnetic field is non-null, the pair of null directions ${k},{\\ell }$ can always be invariantly defined, aligning for example ${k}$ with a PND of ${F}$ and null-rotating about ${k}$ such that $\\Phi _1$ is the only non-vanishing component of the Maxwell spinor.",
"Obviously this choice is not unique (for example in the non-aligned case it will be preferable to align ${k}$ with the multiple Weyl-PND and to null-rotate about ${k}$ such that $\\Phi _1=0$ and $\\Phi _0 \\Phi _2\\ne 0$ ), but it is important to realise that any ensuing well-weighted GHP relations, such as $\\kappa =0, \\rho =0, \\ldots $ are automatically geometrically invariant statements."
],
[
"Aligned electrovacs with a shearing multiple Debever-Penrose vector",
"Let us first, for the sake of completeness, re-confirm the well-known fact [18], [39] that if a PND ${k}$ of the electromagnetic field tensor ${F}$ is shear-free and geodesic, then the Weyl tensor is algebraically special with ${k}$ being the multiple Weyl-PND: with ${k}$ a PND of ${F}$ ($\\Phi _0=0$ ) satisfying $\\kappa =\\sigma =0$ , it follows from () that $\\Psi _0=0$ and hence ${k}$ is also a Weyl-PND.",
"When ${F}$ is non-null we can null-rotate about ${k}$ such that also $\\Phi _2=0$ .",
"The integrability conditions for the Maxwell equations (REF , ) simplify then with (, ) to $\\Phi _1 \\Psi _1=0$ .",
"It follows that $\\Psi _1=0$ and hence ${k}$ is a multiple Weyl-PND.",
"Now let us consider the reverse situation and assume that the Weyl tensor is algebraically special, with the multiple Weyl-PND ${k}$ being also a PND of the non-null electromagnetic field tensor: by means of a suitable null rotation about ${k}$ the null tetrad can be chosen such that $\\Psi _0=\\Psi _1=\\Phi _0=\\Phi _2=0$ .",
"The Maxwell equations (REF , ') and () become $\\textrm {Þ}\\Phi _1=2 \\rho \\Phi _1, \\ \\eth \\Phi _1 =2\\tau \\Phi _1,\\ \\eth ^{\\prime }\\Phi _1 =- 2\\pi \\Phi _1,$ while the Bianchi equations (REF , ) yield the Kundt-Trümper relations (REF ).",
"In the present paragraph we consider the case where ${k}$ is shearing and hence $\\kappa =0,\\ \\Psi _2 = -\\frac{2}{3} |\\Phi _1|^2 .$ Bianchi equation (REF ) simplifies now to $|\\Phi _1|^2 (\\rho +\\overline{\\rho })=0$ , implying that ${k}$ is non-expanding.",
"From the real part of GHP equation (REF ) it follows that $\\rho $ is related to $\\sigma $ by $\\rho = \\pm i |\\sigma |$ and hence, by (REF , , ) $\\textrm {Þ}\\rho =0,\\ \\textrm {Þ}\\sigma =0,\\ \\textrm {Þ}\\tau = \\rho (\\tau +\\overline{\\pi })+\\sigma (\\overline{\\tau }+\\pi ).$ From (,') one finds then $\\Psi _3 = -\\frac{2}{3} \\frac{|\\Phi _1|^2}{\\sigma } (2\\tau +\\overline{\\pi }),$ and $\\textrm {Þ}\\pi = 2(\\overline{\\pi }\\overline{\\sigma }-\\overline{\\tau }\\rho ) .$ Acting with the $[\\eth ,\\, \\textrm {Þ}]$ commutator on $\\Phi _1$ now leads to an expression for $\\eth \\rho $ , which, together with () and the $\\eth $ derivative of (REF ), yields $\\eth \\rho &=& \\sigma (\\overline{\\tau }+2 \\pi )+2\\rho \\tau ,\\\\\\eth ^{\\prime }\\sigma &=& \\sigma (\\overline{\\tau }+2\\pi ),\\\\\\eth \\sigma &=& \\sigma (3\\tau -2\\overline{\\pi })-2\\rho \\frac{\\sigma }{\\overline{\\sigma }}(\\overline{\\tau }+2 \\pi ).$ The remaining integrability conditions for the Maxwell equations can then be written as $\\textrm {Þ}^{\\prime }\\tau +\\eth \\mu &=& \\rho \\overline{\\nu }+\\overline{\\lambda }\\pi ,\\\\\\textrm {Þ}^{\\prime }\\pi -\\eth ^{\\prime }\\mu &=& \\lambda \\tau -\\nu \\rho -\\mu \\overline{\\tau }-\\overline{\\mu }\\pi ,\\\\\\eth \\pi +\\eth ^{\\prime }\\tau &=& \\rho (\\mu +\\overline{\\mu }),\\\\\\textrm {Þ}\\mu +\\textrm {Þ}^{\\prime }\\rho &=& \\pi \\overline{\\pi }-\\tau \\overline{\\tau }.$ Next we apply the $[\\eth ,\\, \\textrm {Þ}]$ commutator to $\\sigma $ , to obtain an expression for $\\tau $ , $\\tau = \\textstyle {\\frac{1}{3}}\\overline{\\pi }+\\frac{4}{3}\\frac{\\rho \\pi }{\\overline{\\sigma }},$ which together with its $\\textrm {Þ}$ derivative leads to $\\sigma \\pi -\\rho \\overline{\\pi }=0 .$ This suggests the definition of an auxiliary variable $S$ (by (REF ) one has $|S|=1$ , while $\\textrm {weight}(S)=[2,-2]$ ) such that $\\sigma = S \\rho $ and enabling us to conclude from (REF ) that $\\tau =-\\overline{\\pi },\\ \\overline{\\pi }= S \\pi .$ By (REF ,REF –) one has then $\\textrm {Þ}S =0,\\ \\eth S=-S (3 \\overline{\\pi }-S \\pi ),\\ \\eth ^{\\prime }S=3 S \\pi -\\overline{\\pi },$ after which (',REF ,REF ') yield $\\eth \\pi &=& S \\pi ^2 +2 \\rho (\\mu +\\overline{\\mu })-\\frac{3}{2} S \\rho ^2 \\Psi _4 |\\Phi _1|^{-2}, \\\\\\eth ^{\\prime }\\pi &=& -\\pi ^2-2 S^{-1} \\rho (\\mu +\\overline{\\mu })-\\frac{3}{2} S^{-2} \\rho ^2 \\overline{\\Psi _4} |\\Phi _1|^{-2},\\\\\\textrm {Þ}\\Psi _4 &=& \\rho (\\Psi _4 -S^{-2}\\overline{\\Psi _4})-\\frac{4}{3} S^{-1}(\\mu +\\overline{\\mu })|\\Phi _1|^2.$ Herewith () reduces to $2 (\\mu +\\overline{\\mu }){|\\Phi _1|^2}+\\rho (S^{-1}\\overline{\\Psi _4}-S\\Psi _4)= 0,$ while expressing that $\\textrm {Þ}^{\\prime }(S\\overline{S})=0$ (with $\\textrm {Þ}^{\\prime }S$ evaluated from ()) and simplifying the result by means of (REF ), one finds that $\\mu +\\overline{\\mu }= 0$ (i.e.",
"also ${\\ell }$ is non-expanding) and hence $\\overline{\\Psi _4} = S^2 \\Psi _4$ .",
"Furthermore, calculating $\\textrm {Þ}^{\\prime }\\rho $ from () and expressing that $\\textrm {Þ}^{\\prime }(\\rho +\\overline{\\rho })=0$ gives us an equation from which we can obtain $\\Psi _4, \\textrm {Þ}^{\\prime }\\rho $ and $\\textrm {Þ}^{\\prime }S$ : $\\Psi _4 &=& \\frac{|\\Phi _1|^2}{18\\rho ^2} \\left[ 24\\pi ^2+6\\rho (\\lambda -S^{-2}\\overline{\\lambda })-S^{-1} (8 |\\Phi _1|^2+12 \\rho \\mu -R) \\right], \\\\\\textrm {Þ}^{\\prime }\\rho &=& -\\frac{\\rho }{2}(S\\lambda +S^{-1}\\overline{\\lambda }),\\\\\\textrm {Þ}^{\\prime }S &=& S \\left[ -2 \\mu +S \\lambda -S^{-1}\\overline{\\lambda }+\\frac{1}{\\rho }(2 S \\pi ^2-\\textstyle {\\frac{2}{3}}|\\Phi _1|^2+\\textstyle {\\frac{1}{12}} R) \\right].$ Next we solve (REF ', ') together with () and the $[\\textrm {Þ}^{\\prime },\\, \\textrm {Þ}]\\rho ,\\ [\\eth ,\\, \\textrm {Þ}^{\\prime }] S$ commutator relations for $\\textrm {Þ}^{\\prime }\\pi $ , $\\eth \\mu $ , $\\textrm {Þ}\\mu $ , $\\textrm {Þ}\\lambda $ , $\\eth \\lambda $ and $\\eth ^{\\prime }\\lambda $ : $\\textrm {Þ}^{\\prime }\\pi &=& -\\textstyle {\\frac{3}{2}} S \\lambda \\pi +\\textstyle {\\frac{1}{2}}S^{-1}\\rho \\overline{\\nu }+S^{-1}\\overline{\\lambda }\\pi +\\textstyle {\\frac{1}{2}}\\mu \\pi -\\textstyle {\\frac{5}{2}}\\nu \\rho +( \\textstyle {\\frac{4}{3}}|\\Phi _1|^2-3 S \\pi ^2 -\\textstyle {\\frac{1}{8}} R)\\frac{\\pi }{\\rho },\\\\\\eth \\mu &=& -\\textstyle {\\frac{5}{2}} S\\rho \\nu -\\textstyle {\\frac{1}{2}}\\lambda \\pi S^2-\\textstyle {\\frac{3}{2}}S \\mu \\pi +\\textstyle {\\frac{3}{2}}\\rho \\overline{\\nu }+\\overline{\\lambda }\\pi +(\\textstyle {\\frac{2}{3}} |\\Phi _1|^2-S\\pi ^2-\\textstyle {\\frac{1}{24}} R)\\frac{S \\pi }{\\rho },\\\\\\textrm {Þ}\\mu &=& \\textstyle {\\frac{1}{2}}\\rho (S\\lambda +S^{-1} \\overline{\\lambda }),\\\\\\textrm {Þ}\\lambda &=& \\textstyle {\\frac{1}{2}}\\rho (\\lambda +S^{-1}\\overline{\\lambda })-2 \\pi ^2-\\textstyle {\\frac{1}{12}} S^{-1}(R-8 |\\Phi _1|^2), \\\\\\eth \\lambda &=& -\\textstyle {\\frac{5}{2}}(\\mu \\pi +\\nu \\rho )+\\textstyle {\\frac{3}{2}}S \\lambda \\pi +\\textstyle {\\frac{7}{2}} S^{-1}\\rho \\overline{\\nu }+(S \\pi ^2 -\\textstyle {\\frac{8}{3}}|\\Phi _1|^2+\\textstyle {\\frac{1}{24}}R)\\frac{\\pi }{\\rho },\\\\\\eth ^{\\prime }\\lambda &=& \\textstyle {\\frac{1}{2}}S^{-1}(\\mu \\pi +\\rho \\nu )+\\textstyle {\\frac{3}{2}}S^{-2}(3\\rho \\overline{\\nu }- \\overline{\\lambda }\\pi )-3\\lambda \\pi +S^{-1}(3 S\\pi ^2-\\textstyle {\\frac{8}{3}} |\\Phi _1|^2+\\textstyle {\\frac{1}{8}} R)\\frac{\\pi }{\\rho },$ after which an expression for the spin coefficient $\\mu $ follows from the $[\\eth ^{\\prime },\\, \\eth ]S$ commutator relation, $\\mu = \\frac{1}{2}(S\\lambda -S^{-1} \\overline{\\lambda })+\\frac{1}{36\\rho }(24 S \\pi ^2-32 |\\Phi _1|^2+R).$ Of the Maxwell integrability conditions there only remains now the $[\\eth ^{\\prime },\\, \\textrm {Þ}^{\\prime }]\\Phi _1$ relation, namely $6 (S \\lambda +S^{-1} \\overline{\\lambda })\\rho \\pi +\\pi (8 |\\Phi _1|^2-24 S \\pi ^2-R) -36\\rho ^2(\\nu - S^{-1}\\overline{\\nu }) = 0,$ which with GHP equation (') gets simplified to the key algebraic equation $\\pi (8|\\Phi _1|^2-24 S \\pi ^2-R)=0.$ Herewith all Bianchi equations and GHP equations (except those involving the derivatives of $\\nu $ ) are identically satisfied.",
"At this stage the Weyl spinor components are given by, using (REF ) to simplify (REF ), $\\Psi _0 &=& \\Psi _1=0,\\\\\\Psi _2 &=& -\\textstyle {\\frac{2}{3}}|\\Phi _1|^2,\\\\\\Psi _3 &=& \\textstyle {\\frac{2}{3}}\\frac{\\pi }{\\rho }|\\Phi _1|^2 , \\\\\\Psi _4 &=& \\frac{|\\Phi _1|^2}{27 \\rho ^2}S^{-1}(24 S \\pi ^2+4|\\Phi _1|^2+R).",
"$ I now show that the case $\\pi \\ne 0$ is inconsistent, while the case $\\pi =0$ leads to the García-Plebański metrics."
],
[
"$\\pi \\ne 0$",
"When $\\pi \\ne 0$ we can use (REF ) to rewrite (REF ) as $\\pi (S\\lambda +S^{-1}\\overline{\\lambda })-6\\rho (\\nu -S^{-1} \\overline{\\nu })=0,$ while (REF ) becomes $\\mu =\\textstyle {\\frac{1}{2}}(S\\lambda -S^{-1}\\overline{\\lambda })-\\textstyle {\\frac{2}{3}}|\\Phi _1|^2\\rho ^{-1}.$ Taking the $\\eth $ derivative of (REF ) and eliminating $\\pi $ from the resulting equation and (REF ), we also find $3\\rho (S\\lambda -\\overline{S}\\overline{\\lambda })-4|\\Phi _1|^2=0$ and hence, by (REF ), $\\mu =0$ .",
"The $\\textrm {Þ}$ derivative of (REF ) yields then $S\\lambda +S^{-1} \\overline{\\lambda }=0$ and hence, by (REF ), $\\lambda =\\textstyle {\\frac{2}{3}}|\\Phi _1|^2 S^{-1} \\rho ^{-1}$ .",
"Substituting this in the expression () for $\\eth \\lambda $ one obtains $\\nu =\\textstyle {\\frac{2}{3}}|\\Phi _1|^2\\pi \\rho ^{-2}$ , which together with GHP equation (') leads to the contradiction $|\\Phi _1|^2\\pi =0$ ."
],
[
"$\\pi =0$",
"Substituting $\\pi =0$ in (REF ,REF ) one obtains $R \\, (=4 \\Lambda ) =-4|\\Phi _1|^2 $ and $\\overline{\\nu }-5 S \\nu =0,$ whence $\\nu =0$ .",
"While (REF ) proves Kozarzewski's no-go claim for $\\Lambda =0$ , it also generalizes it to the case $\\Lambda \\geqslant 0$ .",
"Furthermore, when $\\Lambda < 0$ the only non-0 Weyl spinor component is $\\Psi _2=R/6$ and we are in the doubly aligned situation with both Weyl-PND's ${k}$ and ${\\ell }$ being geodesic and non-expanding.",
"The field equations and Maxwell equations have been completely integrated in this case by García and Plebański [16] and the resulting metric is given by (REF ), admitting a 3D isometry group."
],
[
"Aligned electrovacs with a non-geodesic multiple Debever-Penrose vector",
"We again consider the case where the Weyl tensor is algebraically special, with the multiple Weyl-PND ${k}$ being also a PND of the non-null electromagnetic field tensor, but now we take $\\kappa \\ne 0$ and hence, by (REF ), $\\sigma =0 \\textrm { and } \\Psi _2 = \\frac{2}{3} |\\Phi _1|^2 .$ Bianchi equation () now reduces to $|\\Phi _1|^2 (\\tau -\\overline{\\pi })=0$ , whence $\\tau =\\overline{\\pi }$ , after which (REF ) implies $\\Psi _3=\\frac{2}{3}\\frac{2\\rho -\\overline{\\rho }}{\\kappa }|\\Phi _1|^2, $ showing that solutions cannot be of Petrov type III or N when $\\rho \\ne 0$ .",
"GHP equations (,,) and Bianchi equation (REF ) yield then $\\eth \\kappa &=& 0 ,\\\\\\eth \\rho &=& \\overline{\\pi }(\\rho -\\overline{\\rho })+\\kappa (\\mu -\\overline{\\mu }),\\\\\\eth ^{\\prime }\\rho &=& -2 \\pi \\overline{\\rho }-2 \\overline{\\kappa }\\overline{\\mu },\\\\\\eth ^{\\prime }\\pi &=& \\pi ^2+\\lambda \\overline{\\rho }-\\nu \\overline{\\kappa }.$ The integrability conditions for the Maxwell equations furthermore give an expression for $\\textrm {Þ}\\pi $ , $\\textrm {Þ}\\pi = \\overline{\\kappa }(2\\overline{\\mu }-\\mu )+2\\pi \\overline{\\rho },$ together with four extra relations $\\eth \\mu +\\textrm {Þ}^{\\prime }\\overline{\\pi }&=& \\overline{\\nu }\\rho +\\overline{\\lambda }\\pi ,\\\\\\eth ^{\\prime }\\mu -\\textrm {Þ}^{\\prime }\\pi &=& \\pi (\\overline{\\mu }+\\mu )-\\lambda \\overline{\\pi }+\\nu \\rho ,\\\\\\eth \\pi +\\eth ^{\\prime }\\overline{\\pi }&=& \\rho \\overline{\\mu }-\\overline{\\rho }\\mu , \\\\\\textrm {Þ}\\mu +\\textrm {Þ}^{\\prime }\\rho &=& 0,$ with which GHP equations (,') imply $\\textrm {Þ}^{\\prime }\\kappa &=& \\kappa (2\\mu -\\overline{\\mu }),\\\\\\textrm {Þ}\\lambda &=& \\lambda (\\rho +\\overline{\\rho })+2\\pi ^2-2\\nu \\overline{\\kappa }.$ So far it has not been possible to complete the analysis of this case.",
"However, it is easy to see that the Plebański-Hacyan metrics completely exhaust the non-diverging subfamily of solutions ($\\rho =0$ ) and have $\\Lambda <0$ , whereas for $\\Lambda \\geqslant 0$ no solutions exist.",
"In fact, when $\\rho =0$ (REF ) and $\\kappa \\ne 0$ imply that also the second PND ${\\ell }$ of the electromagnetic field is non-diverging ($\\mu =0$ ): herewith (REF ) and (') show that $\\Psi _3=\\Psi _4=0$ , so that we are in the doubly aligned Petrov type D case.",
"This suffices already to conclude that the only possible solutions are the Plebański-Hacyan metrics, as this case was completely integrated in [35], modulo the error corrected in [16].",
"However, it is instructive to provide a short and coordinate independent proof of the essential step in [35], namely the part in which the authors prove that $\\Gamma _{423}=0$ , leading to the conclusion that ${\\ell }$ not only is shear-free ($\\lambda =0$ being an immediate consequence of (')), but also is geodesic ($\\nu =0$ ): First observe that the $[\\eth ,\\, \\textrm {Þ}^{\\prime }]$ commutator applied to $\\kappa $ implies $\\overline{\\nu }\\textrm {Þ}\\kappa =0$ .",
"Assuming $\\nu \\ne 0$ leads then to an inconsistency, as we would have $\\textrm {Þ}\\kappa =0, \\ \\eth ^{\\prime }\\kappa = \\overline{\\kappa }\\overline{\\pi }-\\kappa \\pi ,$ with the second relation being obtained from (REF ).",
"Herewith the $[\\textrm {Þ},\\, \\textrm {Þ}^{\\prime }]\\kappa $ commutator relation yields $\\kappa (\\overline{\\kappa }\\overline{\\nu }+3\\kappa \\nu -6\\pi \\overline{\\pi }+\\textstyle {\\frac{1}{6}}R-\\textstyle {\\frac{20}{3}}|\\Phi _1|^2)+2\\overline{\\kappa }\\overline{\\pi }^2=0,$ which, with $\\nu \\overline{\\kappa }=\\pi ^2$ (obtained from (')) simplifies to $|\\kappa |^2(40|\\Phi _1|^2-R)-18(\\kappa \\pi -\\overline{\\kappa }\\overline{\\pi })^2=0.$ On the other hand ('-$\\overline{\\ref {ghp5d}}$ ) reads $|\\kappa |^2(8|\\Phi _1|^2+R)-6(\\kappa \\pi -\\overline{\\kappa }\\overline{\\pi })^2=0,$ which, when added to (REF ), results in $2|\\kappa \\Phi _1|^2 - (\\kappa \\pi -\\overline{\\kappa }\\overline{\\pi })^2 =0.$ The left hand side being positive definite, this shows that the case $\\nu \\ne 0$ is inconsistent.",
"We therefore have $\\nu =\\pi =0$ and hence, by GHP equation (), $R = 4\\Lambda =-8|\\Phi _1|^2 < 0$ ."
],
[
"Electrovacs with a shear-free and geodesic multiple Debever-Penrose vector",
"We now consider algebraically special (non-conformally flat) Einstein-Maxwell fields with a possible non-zero cosmological constant for which the multiple Weyl-PND ${k}$ is geodesic and shear-free ($\\Psi _0=\\Psi _1=\\kappa =\\sigma =0$ ) and for which ${k}$ is not parallel to a PND of ${F}$ ($\\Phi _0\\ne 0$ ).",
"Choosing a null-rotation about ${k}$ such that $\\Phi _1=0$ , it follows that $\\Phi _2\\ne 0$ with $\\Phi _2 = 0$ ${\\ell }$ would be geodesic and shear-free; according to the generalised Goldberg-Sachs theorem we would have then $\\Psi _3=\\Psi _4=0$ and the Petrov type would be D, in which case [13], [42] the only null Einstein-Maxwell solutions are given by the (doubly aligned) Robinson-Trautman metrics.",
"The Maxwell equations (REF ,) and Bianchi equations (REF -) yield then $\\eth \\Phi _0 &=& 0,\\\\\\eth ^{\\prime }\\Phi _0 &=& -\\pi \\Phi _0,\\\\\\textrm {Þ}\\Phi _0 &=& 0,\\\\\\textrm {Þ}^{\\prime }\\Phi _0 &=& -\\mu \\Phi _0,\\\\\\eth \\Phi _2 &=& -\\nu \\Phi _0+\\tau \\Phi _2,\\\\\\textrm {Þ}\\Phi _2 &=& -\\lambda \\Phi _0+\\rho \\Phi _2,\\\\\\eth \\Psi _2 &=& -\\pi \\Phi _0\\overline{\\Phi _2}+3\\tau \\Psi _2,\\\\\\textrm {Þ}\\Psi _2 &=& \\mu |\\Phi _0|^2+3 \\rho \\Psi _2,$ after which the commutators $[\\eth ^{\\prime },\\, \\eth ], [\\eth ^{\\prime },\\, \\textrm {Þ}], [\\eth , \\, \\textrm {Þ}^{\\prime }]$ and $[\\textrm {Þ}^{\\prime },\\, \\textrm {Þ}]$ applied to $\\Phi _0$ give $\\eth \\pi &=& (3\\rho -\\overline{\\rho }) \\mu -2\\Psi _2+\\frac{R}{12},\\\\\\textrm {Þ}\\pi &=& 3\\rho \\pi ,\\\\ \\eth \\mu &=& \\overline{\\lambda }\\pi +3 \\mu \\tau ,\\\\\\textrm {Þ}\\mu &=& \\pi (\\overline{\\pi }+3 \\tau ) +2 \\Psi _2-\\frac{R}{12}.",
"$ Herewith GHP equation (') becomes a simple algebraic equation for $\\Psi _2$ , $\\Psi _2 = \\rho \\mu -\\tau \\pi +\\frac{R}{12},$ the $\\textrm {Þ}$ derivative of which (using (REF ,,REF ,)) results in $\\rho R=0$ .",
"As $\\rho =0$ would imply $\\Phi _0=0$ , this shows that an algebraically special Einstein-Maxwell solution possessing a shear-free and geodesic multiple Weyl-PND which is not a PND of ${F}$ necessarily has a vanishing cosmological constantNote3.",
"The corresponding class of solutions is non-empty: imposing the additional restriction that $\\pi =0$ one can deduce that $\\overline{\\rho }=\\rho $ and $\\overline{\\mu }=\\mu $ , together with a $[-2,0]$ –weighted relation $\\Psi _3=\\rho \\nu +\\mu \\overline{\\tau }-\\tau \\lambda $ .",
"It is then straightforward to construct a Newman-Penrose null-tetrad with the additional restrictions $\\alpha =\\beta =\\epsilon =0$ and $\\mu =2\\gamma $ : the unique solutions in this case are the Griffiths [20] metrics, containing as special cases the metrics [3], [4], [19], [41].",
"At first sight one would expect the class $\\pi \\ne 0$ to admit a larger set of solutions, but this is by no means guaranteed (compare with section 2.1): anyway no explicit examples appear to be known.",
"The only property which is easy to demonstrate –though somewhat tedious to be included in the present paragraph– is that no solutions exist for which ${k}$ is non-expanding ($\\Re (\\rho )\\ne 0$ ).",
"As a corollary of the above result we also obtain a new characterization of the class $\\mathcal {D}$ metrics[10] with non-vanishing cosmological constant: if one of the multiple Weyl-PND's a Petrov type D Einstein-Maxwell solution would be non-aligned with a PND of ${F}$ then $R=0$ .",
"In other words, for a non-vanishing cosmological constant, the double alignment property of the class $\\mathcal {D}$ metrics is a consequence of their Weyl-PND's being geodesic and shear-free.",
"It is tempting to conjecture that this same conclusion also will hold when $\\Lambda =0$ .",
"However it has not been possible so far to prove this, except for the special case of the Kundt space-times (i.e.",
"in which the Weyl-PND ${k}$ has vanishing complex divergence).",
"As the proof for this particular case is again quite tedious and little illuminating, I prefer to postpone this part to a possible later and more general publication."
],
[
"Discussion",
"Most results dealing with exact solutions for (algebraically special) Einstein-Maxwell fields have been obtained in the past introducing special coordinate systems, usually adapted to geodesic and/or shear-free null-congruences, or by imposing, sometimes haphasard looking, restrictions on the spin coefficients of a Newman-Penrose tetrad.",
"This not only has turned their comparison and classification into an awkward procedure, often involving sophisticated computer algebra packages, but also has made it difficult for researchers and students entering the field to recognize the blanks which remain to be filled in.",
"As furthermore the “Exact Solutions book”[40] touches the subject of non-aligned Einstein-Maxwell fields, or of aligned fields with a non-geodesic or shearing multiple PND, only superficially, I present below the known results, together with the ones obtained in the previous sections, schematically.",
"From these diagrams I exclude the conformally flat case, as it implies[36] $\\Lambda =0$ with the only non-null member being therefore[40] the Bertotti-Robinson[28], [2], [37] metric and the null members being given by a special class of plane waves[1], [30].",
"In the figures below labels (1), (2), ...next to vertical arrows refer to results obtained in the corresponding previous sections, $\\nexists $ indicates that no solutions are allowed, while a question mark indicates that no solutions are known to occur in the literature.",
"Roman capitals II, III, N, D refer to the Petrov types and $`1\\times `$ or $`2\\times `=$ indicate singly aligned or doubly aligned solutions.",
"Figure: NO_CAPTIONIn Fig.",
"1 the sub-tree corresponding to the singly aligned Einstein-Maxwell solutions with a shear-free and geodesic multiple Weyl-PND (as well as the null fields, which automatically obey this condition) is not included, as little or no progress has been made in this area since the early nineties[40], [32], with the bulk of the material contained in [40].",
"The double occurence of the Plebaǹski-Hacyan metric in Fig.",
"1 (once as PH and once as PH', the GHP-primed version of PH with the roles of ${k}$ and ${\\ell }$ interchanged) is due to the fact that PH is doubly aligned with only one null-ray being geodesic and both being shear-free.",
"The absence of doubly aligned Petrov type III solutions in Fig.",
"1 has been noticed already in [40] for $\\Lambda =0$ , but can easily be seen to hold also for $\\Lambda \\ne 0$ .",
"[40] also mentions that the Leroy metric[27] is the unique Einstein-Maxwell solution of Petrov type II with $\\Lambda =0$ , in which both the Weyl PND's ${k}$ and ${\\ell }$ are also PND's of ${F}$ and in which the multiple PND ${k}$ is geodesic and shear-free; this is not clear at all, as Leroy's solutions were only obtained in the non-radiative sub-case.",
"It furthermore remains to be checked whether this still holds –for a suitable generalisation of the Leroy metric– when $\\Lambda \\ne 0$ .",
"Figure: Algebraically special non-nul Einstein-Maxwell solutions for which the multiple Weyl-PND is k{k} not a PNDof F{F}.",
"The sub-tree under Griffiths presents in an invariant way the subcases mentioned in ,originally obtained by imposing coordinate restrictions or restrictions on the NP spin-coefficients."
],
[
"Acknowledgment",
"All calculations were done using the Maple symbolic algebra system[45].",
"I also thank Lode Wylleman for a critical reading of the manuscript."
],
[
"Appendix: GHP, Maxwell and Bianchi equations",
"Weights of the spin-coefficients, the Maxwell and Weyl spinor components and the GHP operators: $&\\kappa : [3, 1], \\nu : [-3, -1], \\sigma : [3, -1], \\lambda : [-3, 1], \\nonumber \\\\&\\rho : [1, 1], \\mu : [-1, -1], \\tau : [1, -1], \\pi : [-1, 1], \\nonumber \\\\&\\Phi _0 : [2, 0], \\Phi _1 : [0, 0], \\Phi _2 : [-2, 0], \\nonumber \\\\&\\Psi _0 : [4, 0], \\Psi _1 : [2, 0], \\Psi _2 : [0, 0], \\Psi _3: [-2,0], \\Psi _4 : [-4, 0] ,\\nonumber \\\\&\\eth : [1, -1], \\eth ^{\\prime }: [-1,1], \\textrm {Þ}^{\\prime }: [-1,-1], \\textrm {Þ}: [1,1].",
"\\nonumber $ The prime operation is an involution with $\\kappa ^{\\prime } &=& -\\nu ,\\sigma ^{\\prime }=-\\lambda ,\\rho ^{\\prime }=-\\mu , \\tau ^{\\prime }=-\\pi ,\\\\{\\Psi _0}^{\\prime } &=& \\Psi _4, {\\Psi _1}^{\\prime }=\\Psi _3, {\\Psi _2}^{\\prime }=\\Psi _2,\\\\\\Phi _0^{\\prime } &=& -\\Phi _2, \\Phi _1^{\\prime }=-\\Phi _1.$ The GHP commutators acting on $(p,q)$ weighted quantities are given by: $\\left[ \\textrm {Þ},\\textrm {Þ}^{\\prime } \\right] &=& (\\pi +\\overline{\\tau })\\eth +(\\overline{\\pi }+\\tau )\\eth ^{\\prime }+(\\kappa \\nu -\\pi \\tau + \\frac{R}{24}-\\Phi _{11}-\\Psi _2)p \\nonumber \\\\&&+(\\overline{\\kappa }\\overline{\\nu }-\\overline{\\pi }\\overline{\\tau }+ \\frac{R}{24}-\\Phi _{11}-\\overline{\\Psi }_2)q,\\\\\\left[ \\eth ,\\eth ^{\\prime }\\right] &=& (\\mu -\\overline{\\mu })\\textrm {Þ}+(\\rho -\\overline{\\rho })\\textrm {Þ}^{\\prime } +(\\lambda \\sigma -\\mu \\rho - \\frac{R}{24}-\\Phi _{11}+\\Psi _2)p \\nonumber \\\\&&-(\\overline{\\lambda \\sigma }-\\overline{\\mu }\\overline{\\rho }-\\frac{R}{24}-\\Phi _{11}+\\overline{\\Psi }_2)q,\\\\\\left[ \\textrm {Þ},\\eth \\right] &=& \\overline{\\pi }\\,\\textrm {Þ}-\\kappa \\textrm {Þ}^{\\prime } +\\overline{\\rho }\\,\\eth +\\sigma \\eth ^{\\prime }+(\\kappa \\mu -\\sigma \\pi -\\Psi _1)p + (\\overline{\\kappa \\lambda }-\\overline{\\pi }\\overline{\\rho }-\\Phi _{01})q .$ GHP equations: $\\textrm {Þ}\\rho -\\eth ^{\\prime }\\kappa &=& \\rho ^2+\\sigma \\overline{\\sigma }-\\overline{\\kappa }\\tau +\\kappa \\pi +\\Phi _{00}, \\\\\\textrm {Þ}\\sigma -\\eth \\kappa &=& (\\rho +\\overline{\\rho })\\sigma +(\\overline{\\pi }-\\tau )\\kappa +\\Psi _0, \\\\\\textrm {Þ}\\tau -\\textrm {Þ}^{\\prime }\\kappa &=& (\\tau +\\overline{\\pi })\\rho +(\\overline{\\tau }+\\pi )\\sigma +\\Phi _{01}+\\Psi _1, \\\\\\textrm {Þ}\\nu -\\textrm {Þ}^{\\prime } \\pi &=& (\\pi +\\overline{\\tau })\\mu +(\\overline{\\pi }+\\tau )\\lambda +\\Psi _3+\\overline{\\Phi _1}\\Phi _2,\\\\\\eth \\rho -\\eth ^{\\prime }\\sigma &=& (\\rho -\\overline{\\rho })\\tau +(\\mu -\\overline{\\mu })\\kappa +\\Phi _{01}-\\Psi _1,\\\\\\textrm {Þ}^{\\prime }\\sigma -\\eth \\tau &=& -\\sigma \\mu -\\overline{\\lambda }\\rho -\\tau ^2+\\kappa \\overline{\\nu }-\\Phi _{02},\\\\\\textrm {Þ}^{\\prime }\\rho -\\eth ^{\\prime }\\tau &=& -\\overline{\\mu }\\rho -\\lambda \\sigma -\\tau \\overline{\\tau }+\\kappa \\nu -\\frac{R}{12}-\\Psi _2.",
"$ Maxwell equations: $\\textrm {Þ}\\Phi _1-\\eth ^{\\prime }\\Phi _0 &=& \\pi \\Phi _0+2\\rho \\Phi _1-\\kappa \\Phi _2, \\\\\\textrm {Þ}\\Phi _2-\\eth ^{\\prime }\\Phi _1 &=& -\\lambda \\Phi _0+2 \\pi \\Phi _1+\\rho \\Phi _2.",
"$ Bianchi equations ($\\Phi _{IJ}=\\Phi _I\\overline{\\Phi _J}$ and $\\Lambda =R/4=constant$ ): ${\\eth ^{\\prime }} \\Psi _{{0}} -{\\textrm {Þ}} \\Psi _{{1}}+{\\textrm {Þ}} \\Phi _{{01}} -{\\eth } \\Phi _{{00}} &=& -\\pi \\,\\Psi _{{0}}-4\\,\\rho \\,\\Psi _{{1}}+3\\,\\kappa \\,\\Psi _{{2}}+ \\overline{\\pi }\\Phi _{{00}}+2\\,\\overline{\\rho }\\Phi _{{01}}+2\\,\\sigma \\,\\Phi _{{10}} \\nonumber \\\\&&-2\\,\\kappa \\,\\Phi _{{11}}-\\overline{\\kappa }\\Phi _{{02}}, \\\\{\\textrm {Þ}^{\\prime }} \\Psi _{{0}} -{\\eth } \\Psi _{{1}}+{\\textrm {Þ}} \\Phi _{{02}} -{\\eth } \\Phi _{{01}} &=& -\\mu \\,\\Psi _{{0}}-4\\,\\tau \\,\\Psi _{{1}}+3\\,\\sigma \\,\\Psi _{{2}}-\\overline{\\lambda }\\Phi _{{00}}+2\\, \\overline{\\pi }\\Phi _{{01}}+2\\,\\sigma \\,\\Phi _{{11}}\\nonumber \\\\&&+ \\overline{\\rho }\\Phi _{{02}}-2\\,\\kappa \\,\\Phi _{{12}}, $ $&&3\\,{\\eth ^{\\prime }} \\Psi _{{1}} -3\\,{\\textrm {Þ}} \\Psi _{{2}}+2\\,{\\textrm {Þ}} \\Phi _{{11}} -2\\,{\\eth }\\Phi _{{10}} +{\\eth ^{\\prime }} \\Phi _{{01}} -{\\textrm {Þ}^{\\prime }}\\Phi _{{00}} = 3\\,\\lambda \\,\\Psi _{{0}}-9\\,\\rho \\,\\Psi _{{2}}-6\\,\\pi \\,\\Psi _{{1}}+6\\,\\kappa \\,\\Psi _{{3}}+ (\\overline{\\mu }-2\\,\\mu ) \\Phi _{{00}}\\nonumber \\\\&& \\ \\ + 2\\,(\\pi +\\overline{\\tau }) \\Phi _{{01}}+2\\, ( \\tau + \\overline{\\pi }) \\Phi _{{10}}+2\\, ( 2\\,\\overline{\\rho }-\\rho ) \\Phi _{{11}}+2\\,\\sigma \\,\\Phi _{{20}}-\\overline{\\sigma }\\Phi _{{02}}-2\\,\\overline{\\kappa }\\Phi _{{12}}-2\\,\\kappa \\,\\Phi _{{21}}, \\\\&&3\\,{\\textrm {Þ}^{\\prime }} \\Psi _{{1}} -3\\,{\\eth } \\Psi _{{2}}+2\\,{\\textrm {Þ}} \\Phi _{{12}} -2\\,{\\eth }\\Phi _{{11}} +{\\eth ^{\\prime }} \\Phi _{{02}} -{\\textrm {Þ}^{\\prime }}\\Phi _{{01}} = 3\\,\\nu \\,\\Psi _{{0}}-6\\,\\mu \\,\\Psi _{{1}}-9\\,\\tau \\,\\Psi _{{2}}+6\\,\\sigma \\,\\Psi _{{3}}-\\overline{\\nu }\\Phi _{{00}}\\nonumber \\\\&& \\ \\ \\ +2\\, (\\overline{\\mu }-\\mu )\\Phi _{{01}} -2\\,\\overline{\\lambda }\\Phi _{{10}}+2\\, ( \\tau +2 \\overline{\\pi }) \\Phi _{{11}}+ (2\\,\\pi +\\overline{\\tau }) \\Phi _{{02}} + 2\\,(\\overline{\\rho }-\\rho ) \\Phi _{{12}}+2\\,\\sigma \\,\\Phi _{{21}}-2\\,\\kappa \\,\\Phi _{{22}}.",
"$"
]
] | 1605.05830 |
[
[
"Forward $J/\\psi$ production at high energy: centrality dependence and\n mean transverse momentum"
],
[
"Abstract Forward rapidity $J/\\psi$ meson production in proton-nucleus collisions can be an important constraint of descriptions of the small-$x$ nuclear wavefunction.",
"In an earlier work we studied this process using a dipole cross section satisfying the Balitsky-Kovchegov equation, fit to HERA inclusive data and consistently extrapolated to the nuclear case using a standard Woods-Saxon distribution.",
"In this paper we present further calculations of these cross sections, studying the mean transverse momentum of the meson and the dependence on collision centrality.",
"We also extend the calculation to backward rapidities using nuclear parton distribution functions.",
"We show that the parametrization is overall rather consistent with the available experimental data.",
"However, there is a tendency towards a too strong centrality dependence.",
"This can be traced back to the rather small transverse area occupied by small-$x$ gluons in the nucleon that is seen in the HERA data, compared to the total inelastic nucleon-nucleon cross section."
],
[
"Introduction",
"The production of $J/\\psi $ mesons at forward rapidity in high energy proton-proton and proton-nucleus collisions can provide valuable information on gluon saturation.",
"Indeed, the production of particles at forward rapidity probes the target at very small $x$ , where saturation effects should be enhanced.",
"In particular, the charm quark mass being of the same order of magnitude as the saturation scale, $J/\\psi $ production should be sensitive to these dynamics.",
"The charm quark mass is also large enough to provide a hard scale, making a perturbative study of this process possible.",
"In addition, $J/\\psi $ production has been the subject of many experimental studies at the LHC, both in proton-proton [1], [2], [3], [4], [5], [6], [7] and in proton-nucleus [8], [9], [10], [11], [12] collisions.",
"This provides a lot of data to confront with nuclear effects predicted by various theoretical models, both in the Color Glass Condensate (CGC) framework [13], [14], [15], [16], [17] and as constraints for nuclear parton distribution functions [18] and energy loss in cold nuclear matter [19], [20], [21], [22].",
"In a recent work [14] we studied, in the CGC framework, the production of forward $J/\\psi $ mesons in proton-proton and proton-nucleus collisions at the LHC.",
"We showed that, when using the Glauber approach to generalize the dipole cross section to nuclei, the nuclear suppression for minimum bias events is smaller than in previous CGC calculations such as [13], and much closer to experimental data.",
"In this paper we will study, in the same framework, other observables of interest in this process, such as $J/\\psi $ production at backward rapidity, the centrality dependence in the optical Glauber model and the mean transverse momentum of the produced ${J/\\psi }$ 's."
],
[
"Formalism",
"Let us briefly recall the main steps of the calculation.",
"For more details we refer the reader to Ref. [14].",
"We use the color evaporation model (CEM) to relate the cross section for $J/\\psi $ production to the $c\\bar{c}$ pair production cross section.",
"In this model a fixed fraction $F_{J/\\psi }$ of the $c\\bar{c}$ pairs produced below the $D$ -meson threshold are assumed to hadronize into $J/\\psi $ mesons: $\\frac{\\, \\mathrm {d}\\sigma _{J/\\psi }}{\\, \\mathrm {d}^2{{P}_\\perp }\\, \\mathrm {d}Y}=F_{J/\\psi } \\; \\int _{4m_c^2}^{4M_D^2} \\, \\mathrm {d}M^2\\frac{\\, \\mathrm {d}\\sigma _{c\\bar{c}}}{\\, \\mathrm {d}^2{{P}_\\perp }\\, \\mathrm {d}Y \\, \\mathrm {d}M^2}\\, ,$ where ${{P}_\\perp }$ , $Y$ and $M$ are the transverse momentum, the rapidity and the invariant mass of the $c\\bar{c}$ pair respectively, $M_D=1.864$ GeV is the D meson mass and $m_c$ is the charm quark mass that we will vary between 1.2 and 1.5 GeV.",
"Note that in this work we will focus on ratios where $F_{J/\\psi }$ cancels so we do not need to fix it to any specific value here.",
"The study of gluon and quark pair production in the dilute-dense limit of the CGC formalism was started some time ago [23], [24] (see also [25]) and used in several calculations such as [26], [27], [13], [28].",
"The physical picture is the following: an incoming gluon from the projectile can split into a quark-antiquark pair either before or after the interaction with the target.",
"The partons propagating trough the target are assumed to interact eikonally with it, picking up a Wilson line factor in either the adjoint (for gluons) or the fundamental (for quarks) representation.",
"Since we study $J/\\psi $ production at forward rapidity, where the projectile is probed at large $x$ , we will use the collinear approximation in which the incoming gluon is assumed to have zero transverse momentum.",
"In this approach the cross section for $c\\bar{c}$ production reads, in the large-${N_\\mathrm {c}}$ limit [13], $\\frac{\\, \\mathrm {d}\\sigma _{c\\bar{c}}}{\\, \\mathrm {d}^2{{p}_\\perp }\\, \\mathrm {d}^2{{q}_\\perp }\\, \\mathrm {d}y_p \\, \\mathrm {d}y_q}=\\frac{\\alpha _{\\mathrm {s}}^2 {N_\\mathrm {c}}}{8\\pi ^2 d_\\mathrm {A}}\\\\ \\times \\frac{1}{(2\\pi )^2}\\int \\limits _{{{k}_\\perp }}\\frac{\\Xi _{\\rm coll}({{p}_\\perp }+ {{q}_\\perp },{{k}_\\perp })}{({{p}_\\perp }+ {{q}_\\perp })^2}x_1 G_p(x_1,Q^2)\\\\\\times \\phi _{y_2=\\ln {\\frac{1}{x_2}}}^{q\\bar{q},g}({{p}_\\perp }+ {{q}_\\perp },{{k}_\\perp })\\;,$ where ${{p}_\\perp }$ and ${{q}_\\perp }$ denote the transverse momenta of the quarks, $y_p$ and $y_q$ their rapidities, $\\int _{{{k}_\\perp }} \\equiv \\int \\, \\mathrm {d}^2 {{k}_\\perp }/ (2\\pi )^2$ , and $d_\\mathrm {A}\\equiv {N_\\mathrm {c}}^2-1$ is the dimension of the adjoint representation of SU(${N_\\mathrm {c}}$ ).",
"The longitudinal momentum fractions $x_1$ and $x_2$ probed in the projectile and the target respectively are given by $x_{1,2}=\\frac{\\sqrt{{{P}_\\perp }^2+M^2}}{\\sqrt{s}}e^{\\pm Y} \\; .$ The explicit expression for the “hard matrix element” $\\Xi _{\\rm coll}$ is given in Ref. [13].",
"In Eq.",
"(REF ), $G_p(x_1,Q^2)$ is the gluon density in the probe and is described, in the collinear approximation that we use here, in terms of usual parton distribution functions (PDFs).",
"In the following we use, unless otherwise stated, the MSTW 2008 [29] parametrization for $G_p$ .",
"For consistency we use the leading other (LO) parton distribution since also the rest of the calculation is made at this order.",
"To estimate the uncertainty associated with the choice of the factorization scale $Q$ , we will vary it between $\\frac{1}{2}\\sqrt{{{P}_\\perp }^2+M^2}$ and $2\\sqrt{{{P}_\\perp }^2+M^2}$ .",
"The function $\\phi _{_Y}^{q \\bar{q},g}$ describes the propagation of a $q\\bar{q}$ pair in the color field of the target and reads $\\phi _{_Y}^{q \\bar{q},g}({{l}_\\perp },{{k}_\\perp })=\\int \\mathrm {d}^2 {{b}_\\perp }\\frac{N_c{{l}_\\perp }^2}{4 \\alpha _{\\mathrm {s}}} \\;\\\\ \\times S_{_Y}({{k}_\\perp },{{b}_\\perp }) \\;S_{_Y}({{l}_\\perp }-{{k}_\\perp },{{b}_\\perp }) \\;,$ where ${{b}_\\perp }$ is the impact parameter.",
"In this expression, the function $S_{_Y}({{k}_\\perp },{{b}_\\perp })$ contains all the information about the target.",
"It is the fundamental representation dipole correlator: $S_{_Y}({{k}_\\perp },{{b}_\\perp }) = \\int \\, \\mathrm {d}^2 {{r}_\\perp }e^{i{{k}_\\perp }\\cdot {{r}_\\perp }}S_{_Y}({{r}_\\perp },{{b}_\\perp }) \\;,$ with $S_{_Y}({{x}_\\perp }-{{y}_\\perp },{{b}_\\perp }) = \\frac{1}{{N_\\mathrm {c}}}\\left< \\, \\mathrm {Tr} \\, U^\\dag ({{x}_\\perp })U({{y}_\\perp })\\right>,$ where $U({{x}_\\perp })$ is a fundamental representation Wilson line in the color field of the target.",
"The dipole correlator $S_{_Y}({{k}_\\perp },{{b}_\\perp })$ is obtained by solving numerically the running coupling Balitsky-Kovchegov (rcBK) equation [30], [31], [32].",
"For the initial condition we use the 'MVe' parametrization introduced in Ref.",
"[33], which reads, in the case of a proton target, $S^p_{Y= \\ln \\frac{1}{x_0}}({{r}_\\perp },{{b}_\\perp }) = \\exp \\bigg [ -\\frac{{{r}_\\perp }^2 Q_\\mathrm {s0}^2}{4}\\\\ \\times \\ln \\left(\\frac{1}{|{{r}_\\perp }| \\Lambda _{\\mathrm {QCD}}}\\!+\\!e_c \\cdot e\\right)\\bigg ],$ where $x_0=0.01$ .",
"The running coupling is taken as: $\\alpha _{\\mathrm {s}}(r) = \\frac{12\\pi }{(33 - 2N_f) \\log \\left(\\frac{4C^2}{r^2\\Lambda _{\\mathrm {QCD}}^2} \\right)} \\;,$ where $C$ parametrizes the uncertainty related to the scale of the strong coupling in the transverse coordinate space.",
"The free parameters in these expressions are obtained by fitting the combined inclusive HERA DIS cross section data [34] for $Q^2<50$ GeV$^2$ and $x<0.01$ .",
"Their best fit values (with $\\chi ^2/\\text{d.o.f} = 1.15$ ) are $Q_\\mathrm {s0}^2= 0.060$ GeV$^2$ , $C^2= 7.2$ , $e_c=18.9$ and $\\sigma _0/2 = 16.36$ mb.",
"In the case of a proton target, the dipole amplitude does not have an explicit impact parameter dependence and we thus make the replacement $\\int \\mathrm {d}^2 {{b}_\\perp }\\rightarrow \\frac{\\sigma _0}{2}$ in Eq.",
"(REF ), $\\frac{\\sigma _0}{2}$ corresponding to the effective proton transverse area measured in DIS experiments.",
"The fit only includes light quarks.",
"In particular this leaves the charm quark mass that would be consistent with the DIS data in this model still uncertain, which is why we vary it in a rather large range for the uncertainty estimate in this work.",
"Figure: Nuclear modification factor R pPb R_\\text{pPb} at negative rapidity as a function of YY (left) and P ⊥ P_\\perp (right) at s NN =5\\sqrt{s_{NN}}=5 TeV.",
"Data from Refs.",
", , .Figure: Forward to backward ratio in proton-lead collisions as a function of YY (left) and P ⊥ P_\\perp (right) at s NN =5\\sqrt{s_{NN}}=5 TeV.",
"Data from Refs.",
", .To generalize this proton dipole correlator to the case of a nuclear target we use, as in [33], the optical Glauber model.",
"In this model the gluons at the initial rapidity $Y=\\ln 1/x_0$ are localized in the individual nucleons of the nucleus.",
"The nucleons are then taken to be distributed randomly and independently in the transverse plane according to the standard Woods-Saxon nuclear density profile.",
"An analytical average over the positions of the nucleons leads to the following initial condition for the rcBK evolution of a nuclear target: $S^A_{Y=\\ln \\frac{1}{x_0}}({{r}_\\perp },{{b}_\\perp }) = \\exp \\Bigg [ -A T_A({{b}_\\perp })\\frac{\\sigma _0}{2} \\frac{{{r}_\\perp }^2 Q_\\mathrm {s0}^2}{4}\\\\\\times \\ln \\left(\\frac{1}{|{{r}_\\perp }|\\Lambda _{\\mathrm {QCD}}}+e_c \\cdot e\\right) \\Bigg ] \\; .$ Here $T_A$ is the standard Woods-Saxon transverse thickness function of the nucleus: $T_A({{b}_\\perp })= \\int dz \\frac{n}{1+\\exp \\left[ \\frac{\\sqrt{{{b}_\\perp }^2 + z^2}-R_A}{d} \\right]} \\; ,$ with $d=0.54\\,\\mathrm {fm}$ and $R_A=(1.12A^{1/3}-0.86A^{-1/3})\\,\\mathrm {fm}$ , and $n$ is defined such that $T_A$ is normalized to unity.",
"All the other parameters in the initial condition (REF ), which is evolved using the rcBK equation for each ${{b}_\\perp }$ , are the same as in the proton case.",
"In this model the dipole amplitude depends on the impact parameter and we need to integrate explicitly over it.",
"The impact parameter dependence, which carries over to the centrality dependence, thus appears naturally in this model.",
"Our practical procedure for carrying out this comparison will be discussed in more detail in Sec.",
"."
],
[
"Backward rapidity",
"In our previous work [14] we only considered $J/\\psi $ production in proton-proton and proton-nucleus collisions at forward rapidity.",
"In these kinematics the process can be seen as the collision of a dilute proton probed at large $x$ , which can be described using well known parton distribution functions (PDFs), and a dense target described in terms of classical color fields.",
"The nuclear modification of $J/\\psi $ production was also measured at backward rapidity by ALICE [8], [10] and LHCb [9].",
"In this case the produced $J/\\psi $ is moving in the direction of the incoming nucleus and the physical picture is the same as at forward rapidity, but with the roles of the projectile and the target interchanged, i.e.",
"a dilute proton or nucleus interacts with a dense proton target.",
"The latter is described in the same way as in proton-proton collisions at forward rapidity, while the projectile is described either by a PDF in the case of a proton or by a nuclear PDF (nPDF) in the case of a nucleus.",
"Therefore the calculation is very similar to the case of proton-proton collisions in our previous study.",
"Note, however, that in general nuclear PDFs are less tightly constrained by experimental data than usual proton PDFs.",
"In the following we will use the leading order EPS09 nPDF parametrization [35] which provides additional error sets to estimate this uncertainty.",
"The LO EPS09 analysis uses the CTEQ6L1 [36] proton PDFs as a reference, therefore for consistency we use same parametrization of proton PDFs when computing the nuclear modification factor at backward rapidity.",
"Nevertheless, while it could be sizeable for the cross section, the difference compared to the MSTW 2008 parametrization is very small for the nuclear modification factor.",
"In Fig.",
"REF we show the nuclear modification factor $R_\\text{pA}$ , defined as $R_{\\rm pA}= \\frac{1}{A}\\frac{\\frac{\\, \\mathrm {d}\\sigma ^\\text{pA}}{\\, \\mathrm {d}^2 {{P}_\\perp }\\, \\mathrm {d}Y}}{\\frac{\\, \\mathrm {d}\\sigma ^\\text{pp}}{\\, \\mathrm {d}^2 {{P}_\\perp }\\, \\mathrm {d}Y}} \\; ,$ as a function of $Y$ and $P_\\perp $ obtained in this way at negative rapidity compared with data from ALICE [8], [10] and LHCb [9] experiments.",
"The uncertainty in our calculation is significantly larger than at forward rapidity [14].",
"This is due to the fact that we include in our uncertainty band, in addition to the variation of the charm quark mass and of the factorization scale, the nuclear PDF uncertainty obtained following the procedure described in Ref. [35].",
"In particular, our lower bound for the factorization scale is $Q=\\frac{1}{2}\\sqrt{{{P}_\\perp }^2+M^2}$ which can reach values of less than 2 GeV at small transverse momentum.",
"The nuclear PDFs are not well constrained at such small scales at the moment.",
"Nevertheless the general agreement with data is quite good taking into account the rather large theoretical and experimental uncertainties.",
"In this case deviations of $R_\\text{pPb}$ from unity are entirely due to the nuclear PDFs.",
"Now that we have computed the nuclear modification factor both at forward [14] and backward rapidities, we have access to the forward to backward ratio $R_\\text{FB}$ , defined as $R_\\text{FB}(P_\\perp ,Y)=\\frac{R_\\text{pA}(P_\\perp ,Y)}{R_\\text{pA}(P_\\perp ,-Y)} \\; .$ This ratio can be interesting to study because there may be an additional cancellation of some uncertainties common to the numerator and the denominator.",
"In particular, when determining the nuclear modification factor, experimental studies such as [8], [9] have to use an interpolation for the reference proton-proton cross section since there is no data at $\\sqrt{s}=5$ TeV.",
"This interpolation is not needed to study $R_\\text{FB}$ , but the final statistical uncertainty may be larger if the coverage in rapidity by the detector is not symmetric with respect to 0.",
"Concerning our calculation, we have seen that at negative rapidity the uncertainty on $R_\\text{pA}$ due to nuclear PDFs is rather large.",
"This error will remain in $R_\\text{FB}$ since the computation of $R_\\text{pA}$ at positive rapidity does not involve nPDFs.",
"Indeed, we see from Fig.",
"REF , where we show the forward to backward ratio as a function of $Y$ and $P_\\perp $ , that the uncertainty on this quantity is still quite large.",
"Nevertheless, within this error band the agreement with data is reasonable, although the variation at low $P_\\perp $ seems to be steeper than in the data.",
"In Fig.",
"REF (R) we only show our results for $R_\\text{FB}$ as a function of $P_\\perp $ integrated over the same $Y$ range as ALICE data [8], which is slightly smaller than for LHCb data [9], but our results for $2.5<Y<4$ would be very similar and ALICE and LHCb data are compatible with each other.",
"Figure: Nuclear modification factor Q pPb Q_\\text{pPb} as a function of P ⊥ P_\\perp at s NN =5\\sqrt{s_{NN}}=5 TeV in different centrality bins compared with ALICE data .Table: Average number of binary collisions in each centrality class as obtained in the optical Glauber model compared with the value estimated by ALICE .",
"The values of bb in the last column are solved from the relation N coll,opt.",
"(b)=〈N coll 〉 ALICE N_\\text{coll,opt.",
"}(b)=\\langle N_\\text{coll} \\rangle _\\text{ALICE}"
],
[
"Centrality dependence",
"We have seen that the optical Glauber model contains an explicit impact parameter dependence which can be related to centrality determinations at experiments.",
"In this section we will compare this centrality dependence with its recent measurement at the LHC by the ALICE collaboration in the range $2<Y<3.5$ [12]."
],
[
"Centrality in the optical Glauber model",
"We still need to relate the explicit impact parameter dependence in our model to the definition of centrality used by experiments.",
"The experimental data are usually presented in terms of centrality classes.",
"In the optical Glauber model these classes would be defined by calculating the impact parameter range corresponding to the centrality class $(c_1-c_2)\\%$ using the relation $(c_1-c_2)\\% = \\frac{1}{\\sigma _\\text{inel}^{\\text{pA}}} \\int _{b_1}^{b_2} \\mathrm {d}^2 {{b}_\\perp }p({{b}_\\perp }).$ Here $\\sigma _\\text{inel}^{\\text{pA}}$ is the total inelastic proton-nucleus cross section, given by $\\sigma _\\text{inel}^{\\text{pA}} = \\int \\mathrm {d}^2 {{b}_\\perp }\\, p({{b}_\\perp }) \\; ,$ and the scattering probability at impact parameter ${{b}_\\perp }$ is $p({{b}_\\perp }) \\approx 1- e^{-A T_A({{b}_\\perp }) \\sigma _\\text{inel}},$ where $\\sigma _\\text{inel}$ is the total inelastic nucleon-nucleon cross section.",
"The particle yield in each centrality class is then given by $\\frac{\\mathrm {d}N}{\\mathrm {d}^2 {{P}_\\perp }\\mathrm {d}Y} = \\frac{ \\int _{b_1}^{b_2} \\mathrm {d}^2 {{b}_\\perp }\\frac{\\mathrm {d}N({{b}_\\perp })}{\\mathrm {d}^2 {{P}_\\perp }\\mathrm {d}Y} }{\\int _{b_1}^{b_2} \\mathrm {d}^2 {{b}_\\perp }\\, p({{b}_\\perp })},$ where the values of $b_1$ and $b_2$ are obtained from Eq.",
"(REF ) and $\\frac{\\mathrm {d}N({{b}_\\perp })}{\\mathrm {d}^2 {{P}_\\perp }\\mathrm {d}Y}$ corresponds to the expression of the cross section before integrating over ${{b}_\\perp }$ .",
"In practice, however, this straightforward procedure cannot be used for comparing our calculation with the experimental centrality classes.",
"This can be seen most clearly by calculating the average number of binary nucleon-nucleon collisions for each centrality class.",
"In the optical Glauber model this is computed using the relation $\\langle N_\\text{coll} \\rangle _\\text{opt.}",
"= \\frac{\\int _{b_1}^{b_2} \\mathrm {d}^2 {{b}_\\perp }N_\\text{bin}({{b}_\\perp })}{\\int _{b_1}^{b_2} \\mathrm {d}^2 {{b}_\\perp }p({{b}_\\perp })},$ with $N_\\text{bin}({{b}_\\perp }) = A T_A({{b}_\\perp }) \\sigma _\\text{inel} \\,.$ Table REF shows $\\left< N_\\text{coll} \\right>$ for the optical Glauber centrality classes compared to the values given by ALICE [12] for the experimental ones.",
"For central collisions the average number of binary collisions estimated by ALICE is smaller than in the optical Glauber model, while the opposite is true for peripheral collisions.",
"The impact parameter is not directly observable, so the experimental centrality selection has to use some other observable as a proxy for it.",
"In the case of the ALICE analysis [12] the observable used is the energy of the lead-going side Zero-Degree Calorimeter (ZDC).",
"Due to the large fluctuations in the signal for a fixed impact parameter, the values of $N_\\text{coll}$ vary less strongly with centrality in the experimental classes than in the optical Glauber ones.",
"Thus, for example, a relatively peripheral smaller-$N_\\text{coll}$ event can end up in the most central class in the case of a fluctuation in the ZDC signal."
],
[
"Fixed impact parameter approximation",
"Developing a detailed Monte Carlo Glauber model that would enable us to exactly match the experimental centrality classes would be beyond the scope of this work.",
"Therefore we start from the assumption that the ALICE Glauber model is correct and produces a reliable estimate for the $\\langle N_\\text{coll} \\rangle $ in each experimental class.",
"We then assume that the mapping between this $\\langle N_\\text{coll} \\rangle $ and the impact parameter is accurately enough described by our optical Glauber model, and use the relation $N_\\text{coll.",
"opt}(b)=\\langle N_\\text{coll} \\rangle _\\text{ALICE}$ to determine a mean impact parameter corresponding to the experimental centrality class.",
"The values of $b$ resulting from this procedure are shown in Table REF .",
"We then calculate the nuclear modification factor for the centrality class using this fixed value of $b$ .",
"We note that following this procedure for the 80-100% centrality class would lead to a value of $b$ for which our calculation would not be applicable because the saturation scale of the nucleus falls below the one of the proton.",
"Therefore in the following we will only show results for the five most central classes considered by ALICE.",
"Figure: Left: saturation scale of the lead nucleus in the MVe parametrization as a function of the impact parameter compared with the saturation scale of the proton at x=10 -4 x=10^{-4}.",
"Right: number of binary collisions in proton-lead collisions as a function of the impact parameter in the optical Glauber model.In Fig.",
"REF we show a comparison of our calculation with ALICE data for the nuclear modification factor in different centrality classes, $Q_\\text{pA}$ , defined as $Q_{\\rm pA}= \\frac{\\frac{\\, \\mathrm {d}N^\\text{pA}}{\\, \\mathrm {d}^2 {{P}_\\perp }\\, \\mathrm {d}Y}}{ A \\langle T_A \\rangle \\frac{\\, \\mathrm {d}\\sigma ^\\text{pp}}{\\, \\mathrm {d}^2 {{P}_\\perp }\\, \\mathrm {d}Y}} \\; ,$ as a function of $P_\\perp $ .",
"We observe that the description of experimental data is generally satisfactory in the first four bins.",
"For the fifth bin the value of $Q_\\text{pPb}$ obtained in the optical Glauber model is almost constant and very close to one, while the data still shows a significant variation with $P_\\perp $ .",
"The discrepancy with experimental data for peripheral collisions comes from the fact that in our model the saturation scale of the lead nucleus falls below the one of the proton for a value of $b$ of the order of 6.3 fm, as shown in Fig.",
"REF (L).",
"We see from Fig.",
"REF (R), where we show the number of binary collisions as a function of $b$ , that this corresponds to $N_\\text{coll} \\sim 4.3$ .",
"This is the point where, by definition, $Q_\\text{pA}$ reaches 1 in our calculation and beyond which the validity of the framework we have used is questionable.",
"On the other hand, ALICE data shows that $Q_\\text{pA}$ is still significantly smaller than 1 down to $N_\\text{coll} \\sim 2.1$ (see Fig.",
"REF ).",
"The strong centrality dependence is caused by the value of $\\frac{\\sigma _0}{2}$ extracted from DIS fits being much smaller than the total inelastic nucleon-nucleon cross section.",
"The HERA data, both the inclusive cross section fitted in Ref.",
"[33] and data on exclusive vector meson production (see e.g.",
"[37]), lead to a picture where the small-$x$ gluons that can participate in a hard process in a proton are concentrated in a rather small area in the transverse plane (see also the discussion in Ref. [38]).",
"This small-$x$ gluon “hot spot” is then surrounded by a larger “cloud” that only participates in soft interactions, contibuting to the total nucleon-nucleon inelastic cross section.",
"Our model takes this picture to the extreme, by assuming that at the initial rapidity $\\ln 1/x_0$ the gluons contributing to ${J/\\psi }$ production are concetrated in the area $\\sigma _0/2 \\sim 0.3 \\sigma _\\text{inel}$ inside the target nucleons.",
"Thus, for peripheral collisions, the probe proton can overlap with the soft cloud of $N_\\text{coll} \\sim 4$ target nucleons while still seeing on average only one small-$x$ gluon hot spot in the target, thus leaving a hard process like ${J/\\psi }$ production approximately unmodified.",
"This centrality dependence could probably be mildened by using a larger value for $\\frac{\\sigma _0}{2}$ of the order of $\\sigma _\\text{inel}$ (as is effectively done in [13]).",
"This would, however, lose the consistency of our description of the nucleon from HERA to the LHC.",
"Also, as discussed in Refs.",
"[33], [14], varying these parameters in an uncontrolled way could very easily, depending on how exactly it is done, lead to an excessive suppression for minimum bias collisions, or to an $R_\\text{pA}$ that is very far from unity even at high transverse momentum.",
"Note that the saturation scale (or gluon density) at the edge of the nucleus being smaller than that of the proton is an artefact of averaging over the transverse locations of the dense but small gluon hot spots of the nucleons in the target nucleus.",
"Therefore we do not use the optical Glauber parametrization in this region, but explicitly set $R_\\text{pA}$ to unity.",
"Nevertheless, even an explicit Monte Carlo Glauber procedure with the same parameters would not change the ordering $\\sigma _0/2 < \\sigma _\\text{inel}$ that leads to the absence of nuclear effects for peripheral collisions with $N_\\text{coll} \\lesssim 4$ .",
"Figure: Q pPb Q_\\text{pPb} at a function of P ⊥ P_\\perp at s NN =5\\sqrt{s_{NN}}=5 TeV in the 60-80% centrality class, both when using a fixed impact parameter and when integrating explicitly over bb using the N coll N_\\text{coll} distribution obtained in two different models.",
"Data from Ref.",
".Figure: Nuclear modification factor Q pPb Q_\\text{pPb} as a function of N coll N_\\text{coll} at s NN =5\\sqrt{s_{NN}}=5 TeV compared with ALICE data , both when using a fixed impact parameter and when integrating explicitly over bb.Figure: Mean transverse momentum as a function of YY in proton-proton collisions at s=7\\sqrt{s}=7 TeV (left) and in proton-lead collisions at s NN =5\\sqrt{s_{NN}}=5 TeV (right).",
"Data from Ref.",
".Figure: Average transverse momentum squared〈P ⊥ 2 〉 pPb \\langle P_\\perp ^2 \\rangle _\\text{pPb} (left) and nuclear transverse momentum broadening 〈P ⊥ 2 〉 pPb -〈P ⊥ 2 〉 pp \\langle P_\\perp ^2 \\rangle _\\text{pPb}-\\langle P_\\perp ^2 \\rangle _\\text{pp} (right) as a function of N coll N_\\text{coll} at s NN =5\\sqrt{s_{NN}}=5 TeV compared with ALICE data .",
"For the latter both the result using a fixed impact parameter and when integrating explicitly over bb are shown."
],
[
"Explicit integration over the impact parameter",
"The results we have shown in the previous section were obtained with a fixed impact parameter chosen so that the number of binary collisions in the optical Glauber model is equal to the average number of binary collisions estimated by ALICE in each centrality class.",
"However, the nuclear modification ratio in a given centrality bin receives contributions from a distribution of different impact parameters.",
"One could therefore argue that having a profile in the impact parameter space and integrating over $b$ could lead to different results.",
"To quantify this effect we will here use two different kinds of $N_\\text{coll}$ distributions to obtain distributions in the impact parameter space.",
"The first one is provided by ALICE [39] and is obtained from the Slow Nucleon Model (SNM) [40].",
"Since in this model the average number of binary collisions is not the same as the one obtained in the hybrid method used in Ref.",
"[12], we shift the distributions so that $\\langle N_\\text{coll} \\rangle $ matches the one in the third column of Table REF .",
"It should be noted that, contrary to the hybrid method, this method is biased [40].",
"In addition, this is only one possible way of extracting $N_\\text{coll}$ distributions at experiments.",
"Other methods could yield significantly different distributions.",
"To try to quantify the dependence of our results on the particular shape of the $N_\\text{coll}$ distributions, we will also use, for the 60-80% centrality bin which is the most sensitive to fluctuations, a simple linearly decreasing distribution.",
"The two parameters of this distribution, its height at the origin ($h$ ) and the $N_\\text{coll}$ value at which it vanishes ($N_\\text{max}$ ), are determined by imposing that it is normalized to unity and that $\\langle N_\\text{coll} \\rangle =\\langle N_\\text{coll} \\rangle _\\text{ALICE}$ : $N_\\text{max}=3 \\langle N_\\text{coll} \\rangle $ , $h=2/N_\\text{max}$ .",
"In Fig.",
"REF we show the values obtained for the nuclear modification factor as a function of $P_\\perp $ in the 60-80% centrality bin, both when using a fixed impact parameter and when integrating explicitly over $b$ using the two $N_\\text{coll}$ distributions described previously (SNM and linear).",
"The explicit integration over $b$ leads to a smaller $Q_\\text{pPb}$ at small transverse momentum.",
"The effect is more pronounced with the linear distribution.",
"Similar results are obtained when looking at $Q_\\text{pPb}$ integrated over $P_\\perp $ as a function of $N_\\text{coll}$ , as show in Fig.",
"REF .",
"In particular, we see that the value of $Q_\\text{pPb}$ in the 60-80% centrality bin obtained with the linear $N_\\text{coll}$ distribution is significantly closer to the ALICE data point.",
"On the other hand, the value obtained with the SNM distribution is very close to the fixed impact parameter result.",
"In conclusion, it is not possible for now to directly compare our impact parameter dependent results with the centrality dependent measurement performed by ALICE.",
"Indeed, for this one would need to have access to an unbiased determination of the $N_\\text{coll}$ distributions in each centrality bin, which does not exist at the moment.",
"Here we have tried to estimate the importance of this effect by using $N_\\text{coll}$ distributions obtained in two models.",
"The fact that these two models lead to significantly different results for peripheral collisions while the central bins are much less sensitive to fluctuations means that the variation of $Q_\\text{pPb}$ as a function of centrality is too model dependent to have a reliable comparison with experimental data.",
"Figure: Ratio of the proton-proton cross section at s=13\\sqrt{s}=13 and 8 TeV as a function of YY (left) and P ⊥ P_\\perp (right) compared with LHCb data .Figure: Nuclear modification factor R pPb R_\\text{pPb} at s NN =5\\sqrt{s_{NN}}=5 and 8 TeV as a function of YY (left) and P ⊥ P_\\perp (right).",
"Data from Ref.",
", ,"
],
[
"Mean transverse momentum",
"In Ref.",
"[14] we found that the uncertainty on cross sections both in proton-proton and proton-nucleus collisions was rather large.",
"This uncertainty mostly affects the normalization and therefore quantities such as the nuclear modification factor show a smaller uncertainty.",
"Besides the nuclear modification factor, another observable which is not sensitive to the absolute normalization of the cross section is the mean transverse momentum of the produced $J/\\psi $ meson.",
"In Fig.",
"REF (L) we show this quantity as a function of the rapidity in proton-proton collisions at a center of mass energy of 7 TeV and compare with LHCb data [2].",
"We observe that our calculation is compatible with the data but it is still affected by a relatively large uncertainty.",
"In the collinear approximation on the proton side that we are using here, the mean transverse momentum increases slightly with rapidity, a trend not seen in the data.",
"One must, however, keep in mind that towards central rapidities the intrinsic transverse momentum from also the proton should increase, leading to the opposite behavior.",
"A matching between the collinear and $k_T$ -factorized approximations required to fully quantify this effect is beyond the scope of this paper.",
"In Fig.",
"REF (R) we show the same quantity in proton-lead collisions at a center of mass energy $\\sqrt{s_{NN}}=5$ TeV.",
"The ALICE collaboration has also presented results for $\\langle P_\\perp ^2~\\rangle $ as a function of $N_\\text{coll}$ .",
"On Fig.",
"REF (L) we see that our calculation agrees with this measurement within the rather large uncertainty band (except for most peripheral collisions, where our calculation is not applicable as explained previously).",
"When one considers the difference between $\\langle P_\\perp ^2 \\rangle $ in proton-lead and in proton-proton collisions, as shown on Fig.",
"REF (R), the uncertainty on our calculation shrinks and shows a too strong variation as a function of $N_\\text{coll}$ , both when using a fixed impact parameter and when integrating explicitly over $b$ using the $N_\\text{coll}$ distributions obtained in the Slow Nucleon Model.",
"However, as in section REF , the results obtained when integrating over $b$ depend strongly on the exact shape of the $N_\\text{coll}$ distributions used.",
"In particular, one can see that using a linear $N_\\text{coll}$ distribution leads to a better agreement with data for peripheral collisions."
],
[
"Proton-proton collisions",
"In this work we use the simple color evaporation model to describe the hadronization of $c\\bar{c}$ pairs into $J/\\psi $ mesons.",
"The normalization of cross sections then depends on a non perturbative constant $F_{J/\\psi }$ , see (REF ).",
"The uncertainty associated with this parameter can be eliminated by studying the ratio of cross sections at different center of mass energies.",
"In addition, from the experimental point of view, systematic uncertainties can cancel to some extent in this ratio.",
"Such a measurement has been made possible at the LHC for proton-proton collisions thanks to the recent increase of $\\sqrt{s}$ from 8 to 13 TeV.",
"In particular the ratio $\\sigma _{13 \\, \\text{TeV}}/\\sigma _{8 \\, \\text{TeV}}$ was studied as a function of $Y$ and $P_\\perp $ by the LHCb collaboration [6].",
"In Fig.",
"REF we compare these data with the results that we obtain for this ratio in our model.",
"The resulting uncertainty is rather small and the agreement with data is quite good, in particular at large rapidity and relatively low transverse momentum which is the kinematical domain where our calculation is expected to be the most reliable."
],
[
"Proton-nucleus collisions",
"Thanks to its recent upgrade, the LHC may also perform proton-lead collisions at a higher center of mass energy in the future.",
"Here we study how our results would be affected by a change of $\\sqrt{s_{NN}}$ from 5 to 8 TeV.",
"In Fig.",
"REF we show the nuclear modification factor $R_\\text{pPb}$ at forward rapidity as a function of $Y$ and $P_\\perp $ at these two energies, as well as existing LHC data at $\\sqrt{s_{NN}}=5$ TeV.",
"The values at $\\sqrt{s_{NN}}=5$ TeV shown here differ slightly from the ones shown in Figs.",
"8 and 10 of Ref.",
"[14] because we corrected a numerical problem which was causing the region of large impact parameters (where the saturation scale of the lead nucleus falls below the one of the proton, see Fig.",
"REF (L)) to be neglected.",
"Here we impose $R_\\text{pA}=1$ in this region, as in Ref. [33].",
"As one could expect, the higher center of mass energy leads to a stronger suppression due to the higher densities reached in the target.",
"However, the effect is quite small, in particular compared to the size of the uncertainties.",
"For this reason we only show, for $\\sqrt{s_{NN}}=8$ TeV, our results for the “central” values of the parameters ($m_c=1.29$ GeV and $Q=\\sqrt{{P}_\\perp ^2+M^2}$ ).",
"For $\\sqrt{s_{NN}}=5$ TeV we show both the central value and the uncertainty band corresponding to the variation of $m_c$ and $Q$ .",
"Therefore, while measuring forward $J/\\psi $ production in 8 TeV proton-lead collisions could help reduce experimental uncertainties by getting rid of the interpolation needed for the proton-proton reference, we do not expect significantly stronger nuclear effects at this energy.",
"This is not surprising since we use the same dipole cross sections as in Ref.",
"[33], where a weak energy dependence of the nuclear modification factor was found in single inclusive particle production."
],
[
"Conclusions",
"In this paper we have extended our study of forward $J/\\psi $ production in proton-nucleus collisions in the Color Glass Condensate framework to new kinematics and observables.",
"In particular we have studied the nuclear suppression at negative rapidities by describing the nucleus probed at large $x$ in terms of nuclear parton distribution functions.",
"We achieved a quite good description of experimental measurements of this quantity, even if the uncertainty is larger at backward than at forward rapidities because the nuclear PDFs are not yet very strongly constrained by data.",
"This allowed us to compute the forward to backward ratio, again with a good agreement with data within the rather large uncertainties.",
"We have also studied the centrality dependence of our calculation.",
"While using the optical Glauber model to extend the description from a proton target to a nucleus leads to a better agreement with experimental data for minimum bias observables than previous calculations in the same framework, it is difficult to compare directly the resulting centrality dependence to experimental data.",
"Indeed, using a fixed impact parameter obtained in the optical Glauber model from the average number of binary collisions estimated by ALICE leads to a too strong centrality dependence.",
"On the other hand, to integrate explicitly over the impact parameter one has to use $N_\\text{coll}$ distributions based on various assumptions and our calculation is very sensitive to the exact shape of these distributions.",
"Finally we have studied how the nuclear modification factor would be affected by an increase of the center of mass energy achievable at the LHC.",
"As expected nuclear effects are stronger but the change is too small to be significant given the size of theoretical uncertainties."
],
[
"Acknowledgments",
"T. L. and B. D. are supported by the Academy of Finland, projects 267321 and 273464.",
"H. M. is supported under DOE Contract No.",
"DE-SC0012704.",
"This research used computing resources of CSC – IT Center for Science in Espoo, Finland.",
"We would like to thank C. Hadjidakis and I. Lakomov for discussions on the ALICE data."
]
] | 1605.05680 |
[
[
"Non-linear behaviour of XTE J1550-564 during its 1998-1999 outburst,\n revealed by recurrence analysis"
],
[
"Abstract We study the X-ray emission of the microquasar XTE J1550-564 and analyze the properties of its light curves using the recurrence analysis method.",
"The indicators for non-linear dynamics of the accretion flow are found in the very high state and soft state of this source.",
"The significance of deterministic variability depends on the energy band.",
"We discuss the non-linear dynamics of the accretion flow in the context of the disc-corona geometry and propagating oscillations in the accretion flow."
],
[
"Introduction",
"The microquasar XTE J1550-564 is a low-mass X-ray binary, with a companion main sequence star, and was discovered in its 1998 outburst by the All-Sky Monitor onboard the RXTE satellite [14].",
"The radio-jets that accompany its activity were also detected, similarly to many other black hole X-ray binaries [5].",
"The compact object's mass is estimated at 8-11 Solar masses and thus it is considered to be a black hole [11].",
"The source exhibits all the canonical spectral states typically associated with LMXBs.",
"The source's spectral analysis performed for the RXTE data of XTE J1550-564, that spans the period between September 23 and October 6, 1998, place this object in the very high state (VHS) category [3].",
"Further analysis performed for this source by [8] showed that the temperature of the accretion disk is much lower than expected for the VHS luminosity, and suggested that this should be interpreted as the signature of the disk truncation at the radius of about 30 $R_{\\rm g}$ .",
"The intrinsic luminosity of the source peaks above $10^{39}$ erg/s, and during both very high and soft states it remains well above $2-3 \\times 10^{38}$ erg/s.",
"Therefore the derived accretion rate can be in this source equal to at least 25% of the Eddington limit, and reaching even $0.8 \\dot{M}_{\\rm Edd}$ at the outburst peak.",
"In the standard accretion disk theory, such high values of $\\dot{M}$ lead to the intrinsic thermal and viscous instabilities caused by the dominant radiation pressure.",
"Such instabilities manifest themselves in the non-lienar evolution of the disk temperature and density at very short timescales, and consequently imprint a characteristic, time-dependent pattern in the emitted X-ray luminosity.",
"In this work, we study the properties of the XTE J1550-564 light curves.",
"We implement our method, recently described in [17], and we study the light curves in the X-rays as taken by the RXTE satellite during the fall 1998 and spring 1999.",
"The capabilities of this method were tested numerically in [18] and its dependence on parameters was discussed in [4].",
"Here we focus on the spectral and temporal evolution of the microquasars's properties and its possible non-linear behaviour.",
"We discuss the results in the context of the thermal-viscous instability of a Keplerian accretion disc, as well as coupling of its variability with the behaviour of hard X-ray emitting, quasi-spherical corona."
],
[
"Observational data",
"The overall light curve and hardness ratio of the outburst of XTE J1550-564 shows two separated stages of high luminosity [1].",
"The first one lasted from the beginning of September to the beginning of October 1998 (MJD 51063 to 51120), and the second one spanned from the mid of December 1998 to the mid of April 1999 (MJD 51160 to 51280).",
"From the list of the publicly available observations made by RXTE in NASA's HEASARC arxivheasarc.gsfc.nasa.gov we selected several observations, which cover the two epochs quite evenly and which are long enough for our purpose.",
"The data sample is summarised in Table REF ."
],
[
"Recurrence analysis and non-linear dynamics indicator",
"We study the properties of the light curves obtained in different energy bands by means of the time series analysis.",
"Our aim is to find out if the time variability of the flux contains the information about the nature of the dynamical system behind the emission of the radiation.",
"Mainly, we want to discover if the underlying system behaves according to deterministic non-linear dynamics, which is governed by a low number of equations rather than by stochastic variations.",
"In the former case, the traces of the non-linear behaviour are imprinted into the outgoing radiation.",
"Table: Non-linear dynamics indicator 𝒮 ¯\\bar{\\mathcal {S}} obtained for N ϵ N_\\epsilon different values of recurrence threshold ϵ\\epsilon in different energy bands for three observations 30188-06-05-00 (MJD 51068), 30191-01-12-00 (MJD 51082) and 30191-01-33-00 (MJD 51108).",
"The number of points used is N=32768N=32768In the preceding paper [17] we developed the method, which can reveal such traces hidden in the observational data.",
"The method includes performing recurrence analysis on the observational time series, which is in our case the light curve, and on the set of surrogate data series.",
"The results are then compared according to discriminating statistics.",
"Our null hypothesis is that the data are a product of a linearly autocorrelated Gaussian process, so that the data point is given by a linear combination of the preceding points and a contribution of uncorrelated Gaussian noise, $\\xi (n)$ , hence $x_n = a_0 + \\sum _{k=1}^q a_k x_{n-k} +\\xi (n)$ .",
"The properties of such time series are fully described by their Fourier spectrum.",
"The surrogate data are constructed in such a way that they share the value distribution and the Fourier spectrum with the original series [19], [13].",
"However, if the original time series is a product of a non-linear system, the dynamical attributes obtained by the analysis will differ from the surrogate series.",
"Here we use this method to study the outburst of XTE J1550-564, so we only briefly describe its main points.",
"We normalise the light curve to have zero mean and unit variance, then we construct a bunch of 100 surrogates using the procedure surrogates from TISEANhttp://www.mpipks-dresden.mpg.de/~tisean/[6].",
"The recurrence analysis is performed on the basis of the software package described by [10], [9], which also yields the cumulative histogram of diagonal lines.",
"We compute the recurrence quantifiers for the data and the surrogates, using the educated guess of its parameters time delay $\\Delta t$ and embedding dimension $m$ for a set of recurrence thresholds $\\epsilon $ , such that the recurrence rate is approximately in the range (1% – 25 %).",
"After computing the cumulative histograms of diagonal lines for the observed data and their surrogates, we estimate the Rényi's entropy $K_2$ by the linear regression of the cumulative histogram.",
"We follow the rules given in [17] Appendix B, in particular that the minimal length of the diagonal lines must exceed the time delay and that there must be at least five points for the regression.",
"As the discriminating statistic we choose the logarithm of the estimated Rényi's entropy, which we compute for each threshold for the observed time series $Q^{\\rm obs}(\\epsilon )=\\ln (K_2^{\\rm obs})$ and similarly for the set of surrogates $Q^{\\rm surr}_i(\\epsilon )$ .",
"We find the mean $\\bar{Q}^{\\rm surr}(\\epsilon )$ and the standard deviation $\\sigma _Q^{\\rm surr}(\\epsilon )$ of the set $Q^{\\rm surr}_i(\\epsilon )$ .",
"Then we define the weighted significance as $\\mathcal {S}(\\epsilon ) = \\frac{N_{\\rm sl}}{N^{\\rm surr}} \\mathcal {S}_{\\rm sl} - {\\rm sign}( Q^{\\rm obs} (\\epsilon ) - \\bar{Q}^{\\rm surr}(\\epsilon ) ) \\frac{N_{\\mathcal {S}_K}}{N^{\\rm surr}} \\mathcal {S}_{K_2}(\\epsilon ) , $ where $N_{\\rm sl}$ is the number of surrogates, which have only short diagonal lines, and $N^{\\rm surr}$ is the total number of surrogates, $\\mathcal {S}_{\\rm sl} =3$ and $\\mathcal {S}_{K_2}$ is the significance computed only from the surrogates, which have enough long lines according to the relation $\\mathcal {S}_{K_2} (\\epsilon ) = \\frac{| Q^{\\rm obs} (\\epsilon ) - \\bar{Q}^{\\rm surr}(\\epsilon ) |}{\\sigma _{Q^{\\rm surr}(\\epsilon )}}.$ Finally, we define the non-linear dynamics (NLD) indicator, $\\bar{\\mathcal {S}}$ , as the average of $\\mathcal {S}(\\epsilon )$ over the set of $N_\\epsilon $ different thresholds.",
"We say that the observation shows traces of non-linear behaviour if it exceeds the chosen threshold, $\\bar{\\mathcal {S}} > 1.5$ .",
"The accuracy of the resulting NLD indicator is quite a complicated issue, because the observational data with their uncertainness undergo several subsequent statistical processes.",
"However, we can treat the resulting values $\\mathcal {S}(\\epsilon )$ as a small set of $N_\\epsilon $ measurements of $\\bar{\\mathcal {S}}$ .",
"As such, the uncertainty of NLD indicator is given as $\\sigma = R/(2\\sqrt{N_\\epsilon })$ , where R is the spread of the data set, that is $R={\\rm max}(\\mathcal {S}(\\epsilon )) - {\\rm min}(\\mathcal {S}(\\epsilon ))$ ."
],
[
"Energy dependence of NLD indicator",
"We start our study with the observation 30188-06-05-00 taken on 12.9.1998 (MJD 51068).",
"We extract the data with the time bin ${\\rm d}t=0.025$ s in six energy bands denoted as A – F, which are listed in Table REF .",
"We take the longest available continuous part of the light curve up to $N_{\\rm max}=32768$ points.",
"We set the upper limit of the length of the studied time series in order to keep the numerical demands to a reasonable level.",
"The light curve in band F, the recurrence plot (RP)Recurrence plot is a visualisation of the recurrence matrix, where the recurrence points are plotted as dots.",
"Because the plot is symmetric with respect to the main diagonal, we plot only the triangle below/above the diagonal.",
"of the observation and RP of one of the surrogates are plotted in Fig.",
"REF .",
"Figure: Lightcurves in energy band F and recurrence plots for the observations (in red) and one surrogate (in green) for the three observations listed in Table :a) MJD 51068 b) MJD 51082 c) MJD 51108.We compute the NLD indicator $\\bar{\\mathcal {S}}$ with the parameters $m=9$ and $k=4$ .",
"The highest $\\bar{\\mathcal {S}}$ is obtained for the energy band D, which covers the energy range $7.96 - 12.99$ keV.",
"On the other hand in the lowest energy band A, the NLD indicator is very small ($\\bar{\\mathcal {S}}=0.45$ ).",
"The indicator increases from band A to band D, slightly exceeding our threshold in band B ($\\bar{\\mathcal {S}} = 1.77 > 1.5$ ) and giving quite a high value $\\bar{\\mathcal {S}} = 4.58$ in band C. In the highest energies covered by band E, the value decreases to $\\bar{\\mathcal {S}} = 2.80$ .",
"Figure: Dependence of 𝒮\\mathcal {S} on ϵ\\epsilon for the observation from MJD 51068 extracted in different energy bands.We also provide results for the energy band F, which includes channels 0 – 35 (< 12.99 keV).",
"This band corresponds to the extraction of the data from the binned mode or the single-bit mode without energy filtering, so that such a light curve has the highest count rate.",
"This band can also be compared with the results given in [17].",
"The light curve in this band yields a high value $\\bar{\\mathcal {S}} = 6.69.$ We notice that the uncertainty $\\sigma $ of NLD given in Table REF in bands D and F is very high.",
"The reason for this is that the significance $\\mathcal {S}(\\epsilon )$ is not independent on $\\epsilon $ , but rather grows with the threshold and the spread of the data set is high (see Fig.",
"REF ).",
"This behaviour is not so prominent in any other case, and can be explained if the underlying cumulative histograms are inspected.",
"The histograms of this observation for higher thresholds show an `elbow' around $l \\sim 3.5$ s and the slope of the part after the elbow is about two times smaller than for the shorter lines.",
"A similar elbow is not seen in the histograms of the surrogates, partly because they do not have many such long lines.",
"For higher $\\epsilon $ , the elbow is more prominent in the observation's histogram, lowering the estimated value of $K_2^{\\rm obs}$ and thus making the difference between the observation and the surrogates bigger.",
"This feature can be explained as the presence of recurrence period in the system.",
"The slope of the first part is higher, because it is more difficult to predict the evolution of the trajectory, when we have information about its piece shorter than the recurrence period (see [10] for more details about the properties of RPs).",
"Also a possible interplay between the value of the recurrence threshold and the noise in the data can play a role in this effect.",
"Similar analysis of the energy dependence of NLD was also made for two later observations, 30191-01-12-00 taken on MJD 51082, and 30191-01-33-00 taken on MJD 51108 (denoted as b) and c) in Fig.",
"REF ).",
"The second observation yields similar values of NLD in bands D and C and the overall band F gives the highest value.",
"The third observation also has its highest significance in the band F, however for this observation the NLD indicator in band C is higher then in band D. The values in all bands are much less than for the first two observations, which suggests that the non-linear dynamics in the source fades away during the outburst."
],
[
"Temporal evolution of NLD indicator during the outburst",
"We continue with the study of the temporal evolution of the NLD indicator during the outburst of XTE J1550-564.",
"We extract the light curves in energy bands D and F for each observation from Table REF .",
"In order to use our procedure meaningfully, we need to have a sufficiently high number of counts in each time bin.",
"Hence, the total count rate combined from all working proportional counter units (PCUs) in each energy band is an important quantity for setting the time bin ${\\rm d}t$ .",
"We provide the values and the number of working PCUs in Table REF .",
"The normalised count rate per PCU is given in Figs REF a) and REF b), respectively.",
"We quantify the hardness ratio as the count rate in band D divided by count rate in lower energies, $h = N_{\\rm D} / (N_{\\rm F} - N_{\\rm D})$ , which is given in Table REF and in Fig REF c).",
"Strong spectral transition from hard state, with $h \\gtrsim 0.2$ , to soft state, with $h \\approx 0.05$ , can be seen during the first stage of the outburst.",
"Hence, the count rate in band D drops down significantly in the later epoch.",
"Therefore we do not compute the significance in band D for observations after 1.11.1998 (MJD 51119).",
"During the second stage of the outburst, the hardness ratio $h$ is much lower, with the exception of short time period around MJD 51250 (see [8]), when it reaches $h=0.18$ .",
"We map this time period more closely, choosing several observations between MJD 51240 and MJD 51250.",
"Apart from those observations, the count rate in band D is low, so that applying our method on the light curve in band D is not possible due to the lack of counts in a short time bin.",
"Because of this, we provide NLD indicator only for band F. We use the time bin ${\\rm d}t^F=0.032$ s for light curves in band F and ${\\rm d}t^D=0.025$ s for light curves in band D. We find the educated guess of the embedding dimension $m$ and the time delay $\\Delta t = k {\\rm d}t$ as in [17].",
"The values vary in the range $m \\in [7,9]$ and $k \\in [2,9]$ .",
"For each observation and each energy band, $N_\\epsilon $ indicates the number of different $\\epsilon $ , for which the NLD indicator is obtained by averaging the significance.",
"Figure: Characteristics of the source XTE J1550-564 during both epochs of the outburst 1998/1999.a) Count rate in energy band F. b) Count rate in energy band D. c) Hardness ratio hh.Figure: NLD indicator in band D and F during the outburst of XTE J1550-564.The threshold 1.5 for significant result is indicated by the dashed horizontal line.Our results, which are summarised in Table REF and plotted in Fig.",
"REF , show that the NLD indicator is higher in band D than in band F at the beginning of the outburst.",
"This is in agreement with the result obtained for the observation 30188-06-05-00 (see Table REF ).",
"The traces of non-linear dynamics are most prominent in the light curves consisting of counts in the energy range of 8-13 keV.",
"Hence, the non-linear behaviour of the accreting system is expected to manifest mainly in the regions that emit radiation in energies 8-13 keV.",
"Later, this tendency is smeared and $\\bar{\\mathcal {S}}^F > \\bar{\\mathcal {S}}^D$ in some cases.",
"The non-linear behaviour shifts into lower energies as the outburst continues, while its total importance decreases.",
"This supports the trend, which we presented in section REF for the three selected observations.",
"The obvious exception of the NLD indicator behaviour is the observation 30191-01-02-00, taken on MJD 51075, for which the NLD indicator in band F is slightly above the threshold, but the NLD indicator in band D is non-significant.",
"This observation is special because it covers the peak of the outburst, during which the count rate is three times higher than in any other observation.",
"Remarkably, the hardness ratio does not vary much, having very similar value, $h=0.21$ , like for the other observations.",
"Table: Observations selected for the analysis.",
"Total count rate in energy bands F and D, the number of working PCUs and the hardness ratio h=N D /(N F -N D )h = N_{\\rm D} / (N_{\\rm F} - N_{\\rm D}) are shown, along with non-linear dynamics indicator 𝒮 ¯\\bar{\\mathcal {S}} in energy bands F and D, obtained for N ϵ N_\\epsilon different values of recurrence threshold ϵ\\epsilon and the time bin dt{\\rm d}t, with uncertainty σ.\\sigma .On MJD 51110, the NLD indicator is less than one in both bands and the evidence for non-linear behaviour vanishes.",
"The hardness ratio drops down to $h=0.13$ , while the total count rate $N^F$ remains at similar level $\\sim 14 000$ cts/s.",
"From that moment, not only the hardness ratio decreases further to $h\\sim 0.05$ , but the total count rate also declines significantly.",
"NLD indicator shows no signs of non-linear dynamics.",
"After the source stays at a low luminosity level for about two months, the second stage of the outburst occurs.",
"This epoch spans approximately from the middle of January to the beginning of April 1999.",
"During this epoch, the count rate in band F reaches similar values to the first stage ($\\sim 15000$ cts/s) except for the outburst peak.",
"Almost all observations from this stage including those with higher $h$ provide non-significant values of NLD indicator, with only slight glimpses of non-linear motion on MJD 51245 ($\\bar{\\mathcal {S}}^F=1.74$ ) and MJD 51247 ($\\bar{\\mathcal {S}}^D=1.72$ ).",
"Hence, there are only very weak signs of non-linear dynamics during the second epoch of the outburst and the behaviour of the source is considerably different from the first stage."
],
[
"Discussion",
"During the two epochs of the outburst the source XTE J1550-564 undergoes a complicated evolution of its total luminosity and the spectral and timing properties.",
"Existence and properties of quasi-periodic oscillations (QPOs) were studied, for example, by [12], [16], [7].",
"Detailed spectral study was provided by [15].",
"[8] discussed the spectral fitting of the data with respect to the geometry of the accretion flow.",
"They focus on the first epoch of the outburst, when the evolution goes from a low/hard state through a strong very high state, VHS and a weak VHS to the standard high/soft state.",
"Authors show that the data are hardly compatible with the constant inner radius of the accretion disc.",
"They propose a scenario of the evolution of the accretion flow during the initial state of the outburst, which involves the accretion disc truncated at higher radius than the ISCO, which is immersed in the hot Comptonizing corona.",
"With the elapsed time, the disc propagates closer to the center, eventually reaching the ISCO around MJD 51105.",
"Within next 12 days the remaining corona disappears and the source moves into the standard high/soft state.",
"The connection of the outburst time and spectral variability with the properties of companion star was shown by [20].",
"They proposed a scenario of the outburst, which begins with the accretion of low angular momentum gas released from a magnetic trap near the L1 point created by the companion star magnetic field.",
"At the same time high angular momentum accretion also begins through the Roche lobe overflow forming an accretion disc further away.",
"However, the low angular momentum accretion is much faster, operating near the free fall speed, and the electrons gains high energy.",
"Close to the black hole, where a centrifugal barrier is met, a shock in this component could create and possibly oscillate, showing the non-linear behaviour and radiating in the highest energies.",
"This scenario explains the fast rise of the hard X-ray component versus much slower rise of the soft component, and it is compatible with the previously presented picture of the colder Keplerian disc slowly propagating inward through the already existing hot corona [8].",
"This scenario is also viable in point of view of our results.",
"We showed that at the beginning of the outburst the non-linear behaviour is the strongest in the higher energies, particularly in band D. We note that the drop down of the NLD indicator in the highest energy band E may be caused by a small count rate, and perhaps data from different instrument would be more suitable for the analysis in the highest energies.",
"Later the non-linear dynamics is more prominent in lower energies (band C).",
"The oscillations of the accretion disc caused by the radiation pressure instability occurs typically on timescales of at least five to ten seconds.",
"The exact value depends on the global parameters of the source, such as the black hole mass and viscosity in the accretion disk, and is triggered by the accretion rate being a large fraction of the Eddington.",
"The threshold accretion rate for the instability depends weakly on the system parameters.",
"The truncation of the inner disc, on the other hand, could shut down the instability in XTE J1550-564.",
"For instance, the disc truncated at 30 Schwarzschild radii would be completely stable, if the accretion rate is below 0.7 of Eddington rate.",
"Therefore only the `memory' of the disc's oscillations would be kept in the hot corona, which is dynamically and radiatively coupled with the disc.",
"It should also be noted, that instead of the regular outbursts on large amplitudes and long timescales, which are not observed in XTE J1550-564, a kind of flickering behaviour is possible, in case of a specific configuration of system parameters for which the disc is only marginally unstable (Grzedzielski et al., 2016, in prep.).",
"Here, we link the non-linear behaviour of the high energy emitting regions with the oscillations inside the corona, triggered by the formation and propagation of the accretion disc in the equatorial plane.",
"When the disc is more deeply immersed in the corona, the non-linear behaviour shifts to the lower energies.",
"When the disc reaches the ISCO, and accretion rate is low, it stabilises and emits only thermal stochastic radiation.",
"The oscillations of the corona cease and the corona itself is depleted within several days.",
"The propagation of oscillations through the corona can potentially lead to the evolution of the low frequency QPOs, modelled by [2] within an outburst scenario, that invoked formation of a shock in the inner region.",
"We can expect that during the flare peak (MJD 51075) any non-linear oscillatory behaviour is overpowered by some rapid, very energetic event accompanied by a strong stochastic emission, so that the NLD indicator is quite low.",
"This can be related to the jet launching because strong radio emission, which peaked two days after the X-rays, was observed preceded by a peak in optical band that occurred about one day after the X-ray peak [20].",
"A few days later the apparent superluminal motion of the ejecta in the radio band was observed [5].",
"In the second epoch of the outburst, we detect no strong signs of non-linear behaviour, even in the short period between MJD 51244 and MJD 51250, when the hardness ratio increased significantly.",
"Here, however, the overall evolution of the source is different.",
"We first see the soft state with high luminosity and small $h$ , compatible with an accretion disc spanned down to the ISCO, and then the increase of the high energy flux $N^D$ (and $h$ ).",
"Obviously, the mechanism of the corona creation in this case is different than in the first epoch and the non-linear mechanism leading to the oscillations is not triggered."
],
[
"Acknowledgments",
"This work was supported in part by the grant DEC-2012/05/E/ST9/03914 from the Polish National Science Center."
]
] | 1605.05849 |
[
[
"Cluster-lensing: A Python Package for Galaxy Clusters & Miscentering"
],
[
"Abstract We describe a new open source package for calculating properties of galaxy clusters, including NFW halo profiles with and without the effects of cluster miscentering.",
"This pure-Python package, cluster-lensing, provides well-documented and easy-to-use classes and functions for calculating cluster scaling relations, including mass-richness and mass-concentration relations from the literature, as well as the surface mass density $\\Sigma(R)$ and differential surface mass density $\\Delta\\Sigma(R)$ profiles, probed by weak lensing magnification and shear.",
"Galaxy cluster miscentering is especially a concern for stacked weak lensing shear studies of galaxy clusters, where offsets between the assumed and the true underlying matter distribution can lead to a significant bias in the mass estimates if not accounted for.",
"This software has been developed and released in a public GitHub repository, and is licensed under the permissive MIT license.",
"The cluster-lensing package is archived on Zenodo (Ford 2016).",
"Full documentation, source code, and installation instructions are available at http://jesford.github.io/cluster-lensing/."
],
[
"Introduction",
"Clusters of galaxies are the largest gravitationally collapsed structures to have formed in the history of the universe.",
"As such, they are interesting both from a cosmological as well as an astrophysics perspective.",
"In the former case, the galaxy cluster number density as a function of mass (the cluster mass function) is a probe of cosmological parameters including the fractional matter density $\\Omega _{\\rm m}$ and the normalization of the matter power spectrum $\\sigma _8$ .",
"Astrophysically, the deep potential wells of galaxy clusters are environments useful for testing theories of general relativity, galaxy evolution, and gas and plasma physics, among other things [40].",
"The common thread among these diverse investigations is the requisite knowledge of the mass of the galaxy cluster, which is largely composed of its invisible dark matter halo.",
"Although many techniques exist for estimating the total mass of these systems, weak lensing has emerged as somewhat of a gold standard, since it is sensitive to the mass itself, and not to the dynamical state or other biased tracers of the underlying mass.",
"Scaling relations between weak lensing derived masses, and other observables, including richness, X-ray luminosity and temperature, for examples, are typically calibrated from large surveys and extrapolated to clusters for which gravitational lensing measurements are impossible or unreliable.",
"Since weak lensing masses are often considered the “true” masses, against which other estimates are compared [26], [41], [19], it is paramount that cluster masses from weak lensing modeling are as unbiased as possible.",
"For stacked weak lensing measurements of galaxy clusters, an important source of bias in fitting a mass model is the inclusion of the effect of miscentering offsets.",
"Miscentering occurs when the center of the mass distribution – the dark matter halo – does not perfectly coincide with the assumed center around which tangential shear (or magnification) profiles are being measured.",
"Candidate centers for galaxy clusters are necessarily chosen from observational proxies, and often include a single galaxy, such as the brightest or most massive member, or the centroid of some extended quantity like the peak of X-ray emission or average of galaxy positions [16].",
"The particular choice of center may be offset from the true center due to interesting physical processes such as recent mergers and cluster evolution, or simply due to misidentification of the proxy of interest [22].",
"The miscentering effect on the stacked weak lensing profile can be included in a proper modeling of the measurement, as done in [22], [27], [32], [16], [38], [31], [13], [15], [39].",
"The inclusion of this effect commonly assumes a form for the distribution of offsets, such as a Rayleigh distribution in radius (which represents a 2D Gaussian in the plane of the sky).",
"This distribution is convolved with the standard (centered) halo profile to obtain the miscentered version.",
"Software for calculating these miscentered weak lensing profiles was developed in order to produce results in [13], [15], and has recently been publicly released to the astronomical community [12].https://github.com/jesford/cluster-lensing When many different gravitational lenses are stacked, as is often necessary to increase signal-to-noise for weak lensing measurements, care must be taken in the interpretation of the average signal.",
"The issue here is that the (differential) surface mass density is not a linear function of the mass, so the average of many stacked profiles does not directly yield the average mass of the lens sample.",
"Care must be taken to consider the underlying distribution of cluster masses as well as the redshifts of lenses and sources, all of which affect the amplitude of the measured lensing profile.",
"One approach to this is to use a so-called composite-halo approach [18], [14], [13], [15], [39], where profiles are calculated for all individual lens objects and then averaged together to create a model that can be fit to the measurement.",
"The [style=codeintext]ClusterEnsemble() class discussed in Section REF is designed with this approach in mind.",
"A popular model for the dark matter distribution in a gravitationally collapsed halo, such as a galaxy cluster, is the Navarro, Frenk, and White (NFW) model.",
"This density profile (given in Equation REF below) was determined from numerical simulations that included the dissipationless collapse of density fluctuations under gravity [30].",
"The simpler Singular Isothermal Sphere density model, which only has one free parameter in contrast to the two for NFW, does not tend to fit the inner profiles of halos well and is also unphysical in that the total mass diverges [37].",
"Other models such as the generalized-NFW and the Einasto profile tend to better describe the full radial distribution of dark matter in halos, at the expense of adding a third parameter to characterize the inner slope of the density profiles [8].",
"In the software package presented in this work we only include the standard 2-parameter NFW model, but future work should make alternative models available as well."
],
[
"Description of the Code",
"In this section we demonstrate each of the individual modules available in the [style=codeintext]cluster-lensing package.",
"In Section REF we describe a class for calculating surface mass density profiles directly from NFW and cosmological parameters.",
"Next we outline the functions available for mass-concentration relations in Section REF .",
"Then in Section REF we present the class [style=codeintext]ClusterEnsemble(), and its related functions, which tie together the previously discussed functionality into a framework for easily manipulating and producing profiles for multiple galaxy clusters at once, from common observational quantities.",
"Much of the content of this section comes directly from the online documentation.http://jesford.github.io/cluster-lensing/ Throughout the modules, dimensionful quantities are labelled as such by means of the [style=codeintext]astropy.units package."
],
[
"The [style=codeintext]nfw.py module contains a single class called [style=codeintext]SurfaceMassDensity(), which computes the surface mass density $\\Sigma (R)$ and the differential surface mass density $\\Delta \\Sigma (R)$ using the class methods [style=codeintext]sigmanfw() and [style=codeintext]deltasigmanfw(), respectively.",
"These profiles are calculated according to the analytical formulas first derived by [44], assuming the spherical NFW model, and can be applied to any dark matter halo: this module is not specific to galaxy clusters.",
"The 3-dimensional density profile of an NFW halo is given by $\\rho (r) = \\frac{\\delta _{\\rm c} \\rho _{\\rm crit}}{(r/r_{\\rm s})(1+r/r_{\\rm s})^2},$ where $r_{\\rm s}$ is the cluster scale radius, $\\delta _{\\rm c}$ is the characteristic halo overdensity, and $\\rho _{\\rm crit} = \\rho _{\\rm crit}(z)$ is the critical energy density of the universe at the lens redshift.",
"These three parametersor, in the case of calculating multiple NFW halos at once, three array-like objects representing each of these parameters must be specified when instantiating the class [style=codeintext]SurfaceMassDensity(), via the arguments [style=codeintext]rs, [style=codeintext]deltac, and [style=codeintext]rhocrit, respectively.",
"The units on [style=codeintext]rs are assumed to be Mpc, [style=codeintext]deltac is dimensionless, and [style=codeintext]rhocrit is in $M_{\\odot }{\\rm Mpc}^{-1}{\\rm pc}^{-2}$ , although the actual inclusion of the [style=codeintext]astropy.units on these variables is optional.",
"The user will probably also want to choose the radial bins for the calculation, which are specified via the keyword argument [style=codeintext]rbins, in Mpc.",
"The surface mass density is the integral along the line-of-sight of the 3-dimensional density: $\\Sigma (R) = 2 \\int _0^{\\infty } \\rho (R,y) {\\rm d}y.$ Here $R$ is the projected radial distance (in the plane of the sky).",
"We can adopt the dimensionless radius $x \\equiv R/r_{\\rm s}$ and, following from [44], show that: $\\Sigma (x) = 2 r_{\\rm s} \\delta _{\\rm c} \\rho _{\\rm crit} f(x),$ where $f(x) = $ ${\\left\\lbrace \\begin{array}{ll}\\frac{1}{x^2 - 1} \\left( 1 - \\ln { \\left[ \\frac{1}{x} + \\sqrt{ \\frac{1}{x^2} - 1} \\right]} / \\sqrt{1 - x^2} \\right), & \\text{for } x < 1; \\\\1/3, & \\text{for } x = 1; \\\\\\frac{1}{x^2 - 1} \\left( 1 - \\arccos {(1/x)} / \\sqrt{x^2 - 1} \\right), & \\text{for } x > 1.\\end{array}\\right.",
"}$ The differential surface mass density probed by shear is calculated from the definition $\\Delta \\Sigma (x) \\equiv \\overline{\\Sigma }(<x) - \\Sigma (x),$ where $\\overline{\\Sigma }(<x) = \\frac{2}{x^2} \\int _0^{x} \\Sigma (x^{\\prime }) x^{\\prime } {\\rm d}x^{\\prime }.$ We can rewrite the differential surface mass density in the form in which it is computed in [style=codeintext]nfw.py: $\\Delta \\Sigma (x) = r_{\\rm s} \\delta _{\\rm c} \\rho _{\\rm crit} g(x),$ where $g(x) = $ ${\\left\\lbrace \\begin{array}{ll}\\end{array}\\left[ \\frac{4 / x^2 + 2/ (x^2 - 1)}{\\sqrt{1 - x^2}} \\right] \\ln \\left( \\frac{1 + \\sqrt{(1-x) / (1+x)} }{1 - \\sqrt{(1-x) / (1+x)} } \\right) \\\\ + \\frac{4}{x^2} \\ln \\frac{x}{2} - \\frac{2}{(x^2 - 1)}, & \\text{for } x < 1; \\\\\\right.",
"(10/3) + 4 \\ln (1/2), & \\text{for } x = 1; \\\\}\\left[ \\frac{8}{x^2 \\sqrt{x^2 - 1}} + \\frac{4}{(x^2 - 1)^{3/2}} \\right] \\arctan \\sqrt{\\frac{x-1}{1+x}} \\\\ + \\frac{4}{x^2} \\ln \\frac{x}{2} - \\frac{2}{(x^2 - 1)}, & \\text{for } x > 1.$ Running [style=codeintext]sigmanfw() or [style=codeintext]deltasigmanfw(), with only a specification of halo properties [style=codeintext]rs, [style=codeintext]deltac, [style=codeintext]rhocrit, and radial bins [style=codeintext]rbins, will lead to the calculation of halo profiles according to Equations REF and REF outlined above.",
"from clusterlensing import SurfaceMassDensity rbins = [0.1, 0.5, 1.0, 2.0, 4.0] # Mpc smd = SurfaceMassDensity(rs=[0.1], rho_crit=[0.2], delta_c=[9700.",
"], rbins=rbins) sigma = smd.sigma_nfw() # surface mass density with default units sigma[0] <Quantity [ 129.33333333, 11.64751032, 3.33992059, 0.89839601, 0.23327149] solMass / pc2> # surface mass density with no units sigma[0].value array([ 129.33333333, 11.64751032, 3.33992059, 0.89839601, 0.23327149]) These are the standard centered NFW profiles, under the assumption that the peak of the halo density distribution perfectly coincides with the identified halo center.",
"This may not be a good assumption, however, and the user can instead run these calculations for miscentered halos by specifying the optional input parameter [style=codeintext]offsets.",
"This parameter sets the width of a distribution of centroid offsets, assuming a 2-dimensional Gaussian distribution on the sky.",
"This offset distribution is equivalent to, and implemented in code as, a uniform distribution in angle and a Rayleigh probability distribution in radius: $P(R_{\\mathrm {off}})=\\frac{R_{\\mathrm {off}}}{\\sigma _{\\mathrm {off}}^2}\\ \\mathrm {exp}\\bigg [-\\frac{1}{2}\\bigg (\\frac{R_{\\mathrm {off}}}{\\sigma _{\\mathrm {off}}}\\bigg )^2\\ \\bigg ].$ The parameter [style=codeintext]offsets is equivalent to $\\sigma _{\\rm off}$ in this equation.",
"from clusterlensing import SurfaceMassDensity rbins = [0.1, 0.5, 1.0, 2.0, 4.0] # single miscentered halo profile smd = SurfaceMassDensity(rs=[0.1], rho_crit=[0.2], delta_c=[9700.",
"], rbins=rbins, offsets=[0.3]) sigma = smd.sigma_nfw() sigma[0] <Quantity [ 38.60655298, 17.57285034, 4.11253461, 0.93809627, 0.23574031] solMass / pc2> # example calculating multiple profiles smd = SurfaceMassDensity(rs=[0.1,0.2,0.2], rho_crit=[0.2,0.2,0.2], delta_c=[9700,9700,9000], rbins=rbins, offsets=[0.3,0.3,0.3]) sigma = smd.sigma_nfw() sigma <Quantity [[ 38.60655298, 17.57285034, 4.11253461, 0.93809627, 0.23574031], [ 181.91820855, 92.86651598, 27.34020647, 6.94677803, 1.81488253], [ 168.79009041, 86.16480864, 25.36720188, 6.44546415, 1.68391163]] solMass / pc2> The miscentered surface mass density profiles are given by the centered profiles (Equations REF and REF ), convolved with the offset distribution (Equation REF ).",
"We follow the offset halo formalism first written down by [45], and applied to cluster miscentering by, e.g.",
"[22], [16], [13], [15], [39].",
"Specifically, we calculate the offset surface mass density $\\Sigma ^{\\rm off}$ as follows: $\\Sigma ^{\\rm off}(R) = \\int _{0}^{\\infty } \\Sigma (R | R_{\\rm off})\\ P(R_{\\rm off})\\ {\\rm d}R_{\\rm off}$ $\\Sigma (R|R_{\\mathrm {off}})=\\frac{1}{2\\pi }\\int _{0}^{2\\pi }\\Sigma (r) \\mathrm {d}\\theta $ Here $r = \\sqrt{R^2+R_{\\mathrm {off}}^2-2RR_{\\mathrm {off}}\\cos (\\theta )}$ and $\\theta $ is the azimuthal angle [45].",
"The $\\Delta \\Sigma ^{\\rm off}$ profile is calculated from $\\Sigma ^{\\rm off}$ , in analogy with Equations REF and REF .",
"from clusterlensing import SurfaceMassDensity rbins = [0.1, 0.5, 1.0, 2.0, 4.0] # perfectly centered DeltaSigma profile smd = SurfaceMassDensity(rs=[0.1], rho_crit=[0.2], delta_c=[9700.",
"], rbins=rbins) deltasigma = smd.deltasigma_nfw() deltasigma[0] <Quantity [ 108.78445455, 25.47093418, 10.29627483, 3.71631903, 1.23840727] solMass / pc2> # miscentered DeltaSigma profile smd = SurfaceMassDensity(rs=[0.1], rho_crit=[0.2], delta_c=[9700.",
"], rbins=rbins, offsets=[0.3]) deltasigma = smd.deltasigma_nfw() deltasigma[0] <Quantity [ 0.71370144, 9.35821817, 8.90118561, 3.6475417, 1.23610325] solMass / pc2>"
],
[
"The [style=codeintext]cofm.py module currently contains three functions, each of which calculates halo concentration from mass, redshift, and cosmology, according to a prescription given in the literature.",
"These functions are [style=codeintext]cDuttonMaccio() [8], [style=codeintext]cDuffy() [7], and [style=codeintext]cPrada() [36].",
"Halo mass-concentration relations are an area of active research, and there have been discrepancies between results from different observations and simulations, and disagreement surrounding the best choice of model [8], [24].",
"We do not aim to join this discussion here, but focus on outlining the functionality provided by the [style=codeintext]cluster-lensing package, for calculating these different concentration values.",
"All three functions require two input parameters (scalars or array-like inputs), which are the halo redshift(s) [style=codeintext]z and the halo mass(es) [style=codeintext]m. Specifically, the latter is assumed to correspond to the $M_{200}$ mass definition, in units of solar masses.",
"$M_{200}$ is the mass interior to a sphere of radius $r_{200}$ , within which the average density is $200\\rho _{\\rm crit}(z)$ .",
"The default cosmology used is from the measurements by the [34], which is imported from the module [style=codeintext]astropy.cosmology.Planck13.",
"However, the user can specify alternative cosmological parameters.",
"For calculating concentration according to either the [7] or the [8] prescription, the only cosmological parameter required is the Hubble parameter, which can be passed into [style=codeintext]cDuffy() or [style=codeintext]cDuttonMaccio() as the keyword argument [style=codeintext]h. For the [36] concentration, the user would want to specify [style=codeintext]OmM and [style=codeintext]OmL (the fractional energy densities of matter and the cosmological constant) in addition to [style=codeintext]h, in the call to [style=codeintext]cPrada().",
"The [style=codeintext]cDuttonMaccio() calculation of concentration is done according to the power-law $\\log _{10} c_{200} = a + b \\log _{10}(M_{200} / [10^{12} h^{-1} M_{\\odot }]),$ where $a = 0.52 + 0.385\\ {\\rm exp}[-0.617\\ z^{1.21}],$ $b = -0.101 + 0.206 z.$ The above three equations map to Equations 7, 11, and 10, respectively in [8].",
"The values in these expressions were determined from simulations of halos between $0 < z < 5$ , spanning over 5 orders of magnitude in mass, and were shown to match observational measurements of low-redshift galaxies and clusters [8].",
"This concentration-mass relation is the default one used by the [style=codeintext]clusters.py module, discussed in Section REF .",
"from clusterlensing import cofm # single 10**14 Msun halo at z=1 cofm.c_DuttonMaccio(0.1, 1e14) array([ 5.13397936]) # example with multiple halos cofm.c_DuttonMaccio([0.1, 0.5], [1e14, 1e15]) array([ 5.13397936, 3.67907305]) The concentration calculation in [style=codeintext]cDuffy() is $c_{200} = A\\ (M_{200} / M_{\\rm pivot})^B \\ (1 + z)^C,$ where $\\lbrace A, B, C\\rbrace = \\lbrace 5.71, -0.084, -0.47\\rbrace ,$ $M_{\\rm pivot} = 2 \\times 10^{12}\\ h^{-1} M_{\\odot }.$ Equation REF above corresponds to Equation 4 in [7].",
"The values for $A$ , $B$ , and $C$ can be found in Table 1 of that work, where they are specific to the “full” (relaxed and unrelaxed) sample of simulated NFW halos, spanning the redshift range $0 < z < 2$ .",
"$M_{\\rm pivot}$ can be found in the caption of their Table 1 as well.",
"One caveat with this relation is that the cosmology used in creating the [7] simulations was that of the now outdated WMAP5 experiment [25].",
"from clusterlensing import cofm # default cosmology (h=0.6777) cofm.c_Duffy([0.1, 0.5], [1e14, 1e15]) array([ 4.06126115, 2.89302767]) # with h=1 cofm.c_Duffy([0.1, 0.5], [1e14, 1e15], h=1) array([ 3.93068341, 2.80001099]) The [style=codeintext]cPrada() concentration calculation is much more complex, and written in terms of $\\sigma (M_{200}, x_{\\rm p})$ , the rms fluctuation of the density field.",
"The [36] halo concentration is given bywe use the subscript “p” to distinguish some variables in the equations from [36] from those in the current work $\\begin{split}c_{200} = 2.881 B_0 (x_{\\rm p}) \\bigg [ \\left( \\frac{B_1(x_{\\rm p}) \\sigma (M_{200},x_{\\rm p})}{1.257} \\right)^{1.022} + 1 \\bigg ] \\\\\\times \\ {\\rm exp}\\left( \\frac{0.06}{[B_1(x_{\\rm p}) \\sigma (M_{200},x_{\\rm p}) ]^2} \\right).\\end{split}$ The cosmology and redshift dependence is encoded by the variable $x_{\\rm p}$ , which is $x_{\\rm p} = \\left( \\frac{\\Omega _{\\Lambda , 0}}{\\Omega _{{\\rm m}, 0}} \\right)^{1/3} (1 + z)^{-1}.$ The functions within Equation REF are as follows: $\\sigma (M_{200},x_{\\rm p}) = D(x_{\\rm p}) \\frac{16.9 y_{\\rm p}^{0.41}}{1 + 1.102 y_{\\rm p}^{0.2} + 6.22 y_{\\rm p}^{0.333}}$ $y_{\\rm p} \\equiv \\frac{10^{12} h^{-1} M_{\\odot }}{M_{200}}$ $D(x_{\\rm p}) = \\frac{5}{2} \\left( \\frac{\\Omega _{{\\rm m}, 0}}{\\Omega _{\\Lambda , 0}} \\right)^{1/3} \\frac{\\sqrt{1 + x_{\\rm p}^3}}{x_{\\rm p}^{3/2}} \\int _0^{x_{\\rm p}} \\frac{x^{3/2} {\\rm d}x}{(1 + x^3)^{3/2}}$ $B_0(x_{\\rm p}) = \\frac{c_{\\rm min}(x_{\\rm p})}{c_{\\rm min}(1.393)}$ $B_1(x_{\\rm p}) = \\frac{\\sigma ^{-1}_{\\rm min}(x_{\\rm p})}{\\sigma ^{-1}_{\\rm min}(1.393)}$ $c_{\\rm min}(x_{\\rm p}) = 3.681 + 1.352 \\bigg [ \\frac{1}{\\pi } \\arctan [6.948 (x_{\\rm p} - 0.424)] + \\frac{1}{2} \\bigg ]$ $\\sigma ^{-1}_{\\rm min}(x_{\\rm p}) = 1.047 + 0.599 \\bigg [ \\frac{1}{\\pi } \\arctan [7.386 (x_{\\rm p} - 0.526)] + \\frac{1}{2} \\bigg ]$ In order of appearance above, beginning with our Equation REF , these equations correspond to Equations 14-17, 13, 23a, 23b, 12, 18a, 18b, 19, 20 in [36].",
"The numerical values in these equations were obtained empirically from the simulations described in that work.",
"from clusterlensing import cofm cofm.c_Prada([0.1, 0.5], [1e14, 1e15]) array([ 5.06733941, 5.99897362]) cofm.c_Prada([0.1, 0.1, 0.1], [1e13, 1e14, 1e15]) array([ 5.71130928, 5.06733941, 5.30163572]) The last code example demonstrates the controversial feature of the [36] mass-concentration relation – an upturn in concentration values for the highest mass halos.",
"This is in opposition to the canonical view that higher mass halos have lower concentrations [29], [30], [21], [2]."
],
[
"The [style=codeintext]clusters.py module is designed to provide a catalog-level tool for calculating, tracking, and updating galaxy cluster properties and profiles, through structuring data from multiple clusters as an updatable Pandas Dataframe, and providing an intelligent interface to the other modules discussed in Sections REF and REF .",
"This module contains a single class [style=codeintext]ClusterEnsemble(), as well as three functions, [style=codeintext]masstorichness(), [style=codeintext]richnesstomass(), and [style=codeintext]calcdeltac().",
"The function [style=codeintext]calcdeltac() takes a single input parameter, the cluster concentration [style=codeintext]c200 (e.g.",
"as calculated by one of the functions in [style=codeintext]cofm.py), and returns the characteristic halo overdensity: $\\delta _{\\rm c} = \\left( \\frac{200}{3} \\right) \\frac{c_{200}^3}{\\ln (1 + c_{200}) - c_{200}/(1 + c_{200})}.$ Both input and output are dimensionless here.",
"For example, to convert a concentration value of $c_{200} = 5$ to $\\delta _{\\rm c}$ , you could do: from clusterlensing.clusters import calc_delta_c calc_delta_c(5) 8694.8101906193315 The pair of functions [style=codeintext]masstorichness() and [style=codeintext]richnesstomass(), as their names imply, perform conversions between cluster mass and richness.",
"The only required input parameter to [style=codeintext]masstorichness() is the [style=codeintext]mass, and likewise the only required input to [style=codeintext]richnesstomass() is [style=codeintext]richness.",
"The calculations assume a power-law form for the relationship between these variables: $M_{200} = M_0 \\left( \\frac{N_{200}}{20} \\right) ^ \\beta .$ Here $M_0$ is the normalization, which defaults to $2.7 \\times 10^{13}$ , but can be changed in the call to either function by setting the [style=codeintext]norm keyword argument.",
"The power-law slope $\\beta = 1.4$ by default, but can be set by specifying the optional [style=codeintext]slope input parameter.",
"When these functions are invoked by the [style=codeintext]ClusterEnsemble() class, they are applied to the particular mass definition $M_{200}$ , and assume units of $M_{\\odot }$ .",
"However the functions themselves do not assume a mass definition or unit, and can be generalized to any parameter (or type of richness) that has a power-law relationship with mass.",
"from clusterlensing.clusters import \\ mass_to_richness, richness_to_mass richness_to_mass(50) 97382243648736.9 mass_to_richness(97382243648736.9) 50.0 # specify other power-law parameters richness_to_mass(20, slope=1.5, norm=1e14) 100000000000000.0 The [style=codeintext]ClusterEnsemble() class creates, modifies and tracks a Pandas DataFrame containing the properties and attributes of many galaxy clusters at once.",
"When given a new or updated cluster property, it calculates and updates all dependent cluster properties, treating each cluster (row) in the DataFrame as an individual object.",
"This makes it easy to calculate the $\\Sigma (R)$ and $\\Delta \\Sigma (R)$ weak lensing profiles for many different mass clusters at different redshifts, with a single command.",
"In contrast to using the [style=codeintext]SurfaceMassDensity() class discussed in Section REF , the user only needs to specify the cluster redshifts and either of the mass or richness.",
"If richness is supplied, then mass is calculated from it, assuming the form of Equation REF (which is customizable); if mass is specified instead, than the inverse relation is used to calculate richness.",
"In either case the changes are propagated to any dependent variables.",
"from clusterlensing import ClusterEnsemble z = [0.1,0.2,0.3] c = ClusterEnsemble(z) n200 = [20, 20, 20] c.n200 = n200 # display cluster dataframe c.dataframe z n200 m200 r200 c200 delta_c rs 0 0.1 20 2.700000e+13 0.612222 5.839934 12421.201995 0.104834 1 0.2 20 2.700000e+13 0.591082 5.644512 11480.644557 0.104718 2 0.3 20 2.700000e+13 0.569474 5.442457 10555.781440 0.104636 # specify mass directly c.m200 = [1e13, 1e14, 1e15] c.dataframe z n200 m200 r200 c200 delta_c rs 0 0.1 9.838141 1.000000e+13 0.439664 6.439529 15599.114356 0.068276 1 0.2 50.956400 1.000000e+14 0.914520 4.979102 8612.362538 0.183672 2 0.3 263.927382 1.000000e+15 1.898248 3.886853 4947.982895 0.488377 The above examples also demonstrate that cluster masses are converted to concentrations and to characteristic halo overdensities.",
"This assumes the default mass-concentration relation of the [style=codeintext]cDuttonMaccio() form, or the user can instead specify another of the relations by setting the keyword [style=codeintext]cm=\"Prada\" or [style=codeintext]cm=\"Duffy\", when the [style=codeintext]ClusterEnsemble() object is instantiated.",
"Cosmology can also be specified upon instantiation, by setting the [style=codeintext]cosmology keyword to be any [style=codeintext]astropy.cosmology object that has an [style=codeintext]h and a [style=codeintext]Om0 attribute.",
"If not specified explicitly, the default cosmological model used is [style=codeintext]astropy.cosmology.Planck13.",
"Here is an example of creating a [style=codeintext]ClusterEnsemble() object that uses the WMAP5 cosmology [25] and the [7] concentration: from astropy.cosmology import WMAP5 as cosmo c = ClusterEnsemble(z, cm=\"Duffy\", cosmology=cosmo) c.n200 = [20, 30, 40] c.dataframe z n200 m200 r200 c200 delta_c rs 0 0.1 20 2.700000e+13 0.599910 4.520029 6920.955951 0.132723 1 0.2 30 4.763120e+13 0.702040 4.136873 5678.897592 0.169703 2 0.3 40 7.125343e+13 0.775889 3.851601 4849.836498 0.201446 Instead of using the [style=codeintext]dataframe attribute, which retrieves the Pandas DataFrame object itself, it might be useful to use the [style=codeintext]show() method, which prints additional information to the screen, including assumptions of the mass-richness relation: c.show() Cluster Ensemble: z n200 m200 r200 c200 delta_c rs 0 0.1 20 2.700000e+13 0.599910 4.520029 6920.955951 0.132723 1 0.2 30 4.763120e+13 0.702040 4.136873 5678.897592 0.169703 2 0.3 40 7.125343e+13 0.775889 3.851601 4849.836498 0.201446 Mass-Richness Power Law: M200 = norm * (N200 / 20) ^ slope norm: 2.7e+13 solMass slope: 1.4 # update the mass-richness parameters # and show the resulting table c.massrich_norm = 3e13 c.massrich_slope = 1.5 c.show() Cluster Ensemble: z n200 m200 r200 c200 delta_c rs 0 0.1 20 3.000000e+13 0.621353 4.480202 6784.805438 0.138689 1 0.2 30 5.511352e+13 0.737028 4.086481 5526.615129 0.180358 2 0.3 40 8.485281e+13 0.822406 3.795500 4696.109606 0.216679 Mass-Richness Power Law: M200 = norm * (N200 / 20) ^ slope norm: 3e+13 solMass slope: 1.5 The last example also demonstrates how the slope or normalization of the mass-richness relation can be altered, and the changes propagate from richness through to mass and other variables.",
"Then all the ingredients are in place to calculate halo profiles by invoking the [style=codeintext]calcnfw() method, which interfaces to the [style=codeintext]sigmanfw() and [style=codeintext]deltasigmanfw() methods of the [style=codeintext]SurfaceMassDensity() class, and passes it the required inputs $\\lbrace r_{\\rm s}, \\rho _{\\rm crit}, \\delta _{\\rm c} \\rbrace $ for all the clusters behind the scenes.",
"The value of $\\rho _{\\rm crit}$ is calculated at every cluster redshift using the (default [style=codeintext]astropy.cosmology.Planck13 or user-specified) cosmological model.",
"The user must specify the desired radial bins [style=codeintext]rbins in Mpc.",
"import numpy as np # create some logarithmic bins: rmin, rmax = 0.1, 5.",
"# Mpc rbins = np.logspace(np.log10(rmin), np.log10(rmax), num = 8) # calculate the profiles: c.calc_nfw(rbins=rbins) # profiles now exist as attributes: c.sigma_nfw <Quantity [[ 128.97156123, 62.58323349, 27.01073105, 10.60607722, 3.88999449, 1.36360964, 0.46464366, 0.15563814], [ 132.13989867, 64.10484454, 27.66159293, 10.85990257, 3.98265113, 1.39599118, 0.47565695, 0.15932308], [ 135.62272115, 65.782882, 28.38138702, 11.14121765, 4.08549675, 1.43196834, 0.48790043, 0.16342108]] solMass / pc2> c.deltasigma_nfw <Quantity [[ 105.3190568 , 72.43842908, 43.74538085, 23.44005481, 11.37085955, 5.10385452, 2.16011364, 0.87479771], [ 107.98098357, 74.25022426, 44.82825347, 24.01505305, 11.64776118, 5.22744541, 2.21219956, 0.89582394], [ 110.88173507, 76.23087398, 46.01581348, 24.64741078, 11.95297965, 5.36391529, 2.26978998, 0.91909571]] solMass / pc2> Similar to the direct use of [style=codeintext]SurfaceMassDensity(), discussed in Section REF , the miscentered profiles can be calculated from the [style=codeintext]calcnfw() method, by supplying the optional [style=codeintext]offsets keyword with an array-like object of length equal to the number of clusters, where each element is the width of the offset distribution in Mpc ($\\sigma _{\\rm off}$ in Equation REF ).",
"c.calc_nfw(rbins=rbins, offsets=[0.3,0.3,0.3]) # the offset sigma profile is now: c.sigma_nfw <Quantity [[ 42.50844685, 39.74291121, 32.29894213, 18.50988719, 6.16284894, 1.89335218, 0.62609991, 0.20840423], [ 68.10228964, 63.87901872, 52.56539317, 31.20890672, 11.17821854, 3.5884285, 1.20745376, 0.40574057], [ 95.16077234, 89.48298631, 74.29328561, 45.24074628, 17.06333763, 5.66481165, 1.93408383, 0.65518747]] solMass / pc2> Although [style=codeintext]SurfaceMassDensity() from the [style=codeintext]nfw.py module, and [style=codeintext]ClusterEnsemble().calcnfw() from the [style=codeintext]clusters.py module, are both capable of computing the same $\\Sigma (R)$ and $\\Delta \\Sigma (R)$ profiles, each require different forms of input which would make sense for different use cases.",
"For the studies in [15], [13], and [14], the authors wanted do the profile computations for many clusters at once, while varying the mass and the miscentering offset distribution during the process of fitting the model to the data.",
"What was known were the redshifts and mass proxies (cluster richness in [15] and [13], and a previous mass estimate in [14]), and mass-concentration relations from the literature, so the [style=codeintext]ClusterEnsemble() framework made sense.",
"However, if someone wanted to simply calculate the NFW profiles according to the [44] formulation, then they might prefer to use [style=codeintext]SurfaceMassDensity() as a tool to get profiles directly from the NFW and cosmological parameters $r_{\\rm s}$ , $\\delta _{\\rm c}$ , and $\\rho _{\\rm crit}(z)$ .",
"As an example use case, we take the Canada-France-Hawaii Telescope Lensing Survey [17], [10] public galaxy cluster catalog, which is available on Zenodohttp://dx.doi.org/10.5281/zenodo.51291 [11].",
"This dataset was previously explored using a pre-release version of the [style=codeintext]cluster-lensing software in [13], [15].",
"The W1 field of this survey contains 10,745 galaxy cluster candidates in the redshift range $0.2 \\le z \\le 0.9$ : import numpy as np data = np.loadtxt(\"Clusters_W1.dat\") data.shape (10745, 5) data[0:4, :] # print first 4 clusters array([[ 34.8023, -7.01005, 0.3, 4.435, 10.",
"], [ 34.9425, -7.38996, 0.5, 4.545, 21.",
"], [ 34.8651, -6.69449, 0.5, 3.858, 6.",
"], [ 34.6224, -7.32768, 0.5, 3.619, 8.]])",
"redshift = data[:, 2] sig = data[:, 3] richness = data[:, 4] We select a subset of the lower redshift clusters that were detected at high significance.",
"Then we import [style=codeintext]clusterlensing to create a dataframe of the cluster properties, of which we just print the first several, and calculate the NFW profiles.",
"# select a subset here = (sig > 15) & (redshift < 0.5) sig[here].shape (15,) z = redshift[here] n200 = richness[here] import clusterlensing c = clusterlensing.ClusterEnsemble(z) c.n200 = n200 c.dataframe.head() z n200 m200 r200 c200 delta_c rs 0 0.4 181 5.897552e+14 1.531367 3.966101 5173.016417 0.386114 1 0.3 420 1.916332e+15 2.357815 3.658237 4332.615805 0.644522 2 0.4 176 5.670737e+14 1.511478 3.980218 5213.746469 0.379747 3 0.3 113 3.049521e+14 1.277703 4.341779 6324.420397 0.294281 4 0.4 162 5.049435e+14 1.454129 4.022285 5336.272412 0.361518 rbins = np.logspace(np.log10(0.1), np.log10(10.0), num=20) c.calc_nfw(rbins) Next we import the [style=codeintext]matplotlib and [style=codeintext]seaborn libraries and configure some settings to make our plots more readable.",
"The first plot we create with the commands below presents the $\\Sigma (R)$ profiles for every one of these 15 clusters, and is given in Figure REF .",
"import matplotlib.pyplot as plt import seaborn as sns; sns.set() import matplotlib matplotlib.rcParams[\"axes.labelsize\"] = 20 matplotlib.rcParams[\"legend.fontsize\"] = 20 # strings for plots raxis = \"$R\\ [\\mathrm{Mpc}]$\" sgma = \"$\\Sigma(R)$\" sgmaoff = \"$\\Sigma^\\mathrm{off}(R)$\" delta = \"$\\Delta$\" sgmaunits = \" $[M_{\\odot}\\ \\mathrm{pc}^{-2}]$\" # order from high to low richness order = c.n200.argsort()[::-1] for s, n in zip(c.sigma_nfw[order], c.n200[order]): plt.plot(rbins, s, label=str(int(n))) plt.xscale(\"log\") plt.legend(fontsize=10) plt.ylabel(sgma+sgmaunits) plt.xlabel(raxis) plt.tight_layout() plt.savefig(\"f1.eps\") plt.close() # output is Figure 1 Figure: Surface mass density profiles Σ(R)\\Sigma (R) for all 15 clusters used in this example.",
"These are the most significant (S/N >15>15) clusters detected at low redshifts (z<0.5z < 0.5) in the W1 field of CFHTLenS.",
"See the text for links to download this public dataset.",
"The legend gives the richness values estimated in corresponding to each of these clusters, which are assumed to scale with mass.",
"They are listed from highest to lowest richness, in the same order as the curves.If we had made a stacked measurement of the shear or magnification profile of these clusters, then we would want to know what the average profile of the stack looks like.",
"Since we already have the individual profiles, we just need to calculate the mean across the 0th axis of the [style=codeintext]sigmanfw and [style=codeintext]deltasigmanfw attribute arrays.",
"The plot of these average profiles is given in Figure REF .",
"sigma = c.sigma_nfw.mean(axis=0) dsigma = c.deltasigma_nfw.mean(axis=0) plt.plot(rbins, sigma, label=sgma) plt.plot(rbins, dsigma, \"--\", label=delta+sgma) plt.legend() plt.ylim([0., 1400.])",
"plt.xscale(\"log\") plt.xlabel(raxis) plt.ylabel(sgmaunits) plt.tight_layout() plt.savefig(\"f2.eps\") plt.close() # output is Figure 2 Figure: The average of all the surface mass density profiles Σ(R)\\Sigma (R) for each of the clusters shown in Figure is given in blue.",
"The green curve is the average of all the individual differential mass density profiles ΔΣ(R)\\Delta \\Sigma (R).",
"These curves assume clusters are perfectly centered on their NFW halos.Finally, we may want to investigate whether cluster miscentering has a significant effect on our sample.",
"We would calculate the miscentered profiles, given in Figure REF , which could be compared to the centered profiles in Figure REF to see which is a better fit to our measurement.",
"Below we will assume that the miscentering offset distribution peaks at 0.1 Mpc.",
"offsets = np.ones(c.z.shape[0]) * 0.1 c.calc_nfw(rbins, offsets=offsets) sigma_offset = c.sigma_nfw.mean(axis=0) dsigma_offset = c.deltasigma_nfw.mean(axis=0) plt.plot(rbins, sigma_offset, label=sgmaoff) plt.plot(rbins, dsigma_offset, \"--\", label=delta+sgmaoff) plt.legend() plt.ylim([0., 1400.])",
"plt.xscale(\"log\") plt.xlabel(raxis) plt.ylabel(sgmaunits) plt.tight_layout() plt.savefig(\"f3.eps\") plt.close() # output is Figure 3 Figure: Same as Figure but for miscentered profiles.",
"Cluster centroid offsets are assumed to follow a Rayleigh probability distribution (Equation , discussed in Section ), which is convolved with the perfectly centered profiles to achieve this result.The above example shows a simple application of [style=codeintext]cluster-lensing to a real dataset – a subset of the CFHTLenS cluster catalog.",
"For this example we kept the customizations to a minimum, but as shown in Sections REF , the user can alter the parameters in the power-law relation used to convert richness to mass, choose the form of the mass-concentration relation assumed for the NFW profiles, and specify a particular background cosmology.",
"When fitting a model produced by [style=codeintext]cluster-lensing to a measurement, one could iterate through parameters in this space by setting various attributes of the [style=codeintext]ClusterEnsemble() object [13], [15]."
],
[
"Relation to Existing Code",
"The [style=codeintext]cluster-lensing project offers some unique capabilities over other publicly-available software, most notably the cluster miscentering calculations.",
"Here we attempt to compare the software presented in this work with other open source tools that we are aware of, and show how [style=codeintext]cluster-lensing fits into the larger ecosystem of astronomical software.",
"[style=codeintext]Colossus is a Python package aimed at cosmology, halo, and large-scale structure calculations [3].",
"It was used in work by [4] and is made available under the MIT licensehttp://www.benediktdiemer.com/code/.",
"Much of the functionality of [style=codeintext]cluster-lensing appears to overlap with [style=codeintext]Colossus, including mass-concentration relations (although [style=codeintext]Colossus has the advantage of containing many more relations from the literature) and NFW surface mass density profiles.",
"However, [style=codeintext]cluster-lensing also provides the miscentered halo calculations, which are are lacking from [style=codeintext]Colossus.",
"While [3] has chosen to implement basic cosmological calculations from scratch, [style=codeintext]cluster-lensing instead relies on external modules supplied by [style=codeintext]astropy.",
"The only dependencies claimed by [style=codeintext]Colossus are [style=codeintext]numpy [42] and [style=codeintext]scipy [23], whereas [style=codeintext]cluster-lensing additionally requires [style=codeintext]astropy [1] and [style=codeintext]pandas [28].",
"Fewer dependencies might be seen as a positive feature of [style=codeintext]Colossus; on the other hand, [style=codeintext]astropy could be viewed as possibly a more robust source for standard astronomical and cosmological calculations, since it is maintained by a large community of developers.",
"Another related set of code is provided by Jörg Dietrich's NFW routines, archived on Zenodo [5], and available on GitHubhttps://github.com/joergdietrich/NFW.",
"These [style=codeintext]Python modules calculate NFW profiles for $\\Sigma (R)$ and $\\Delta \\Sigma (R)$ , as well as the 3-dimensional density profiles and total mass and projected mass inside a given radius.",
"[style=codeintext]cluster-lensing goes beyond the functionality of [5] by supplying means for calculating cluster miscentering, and having a built-in framework for handling many halos at once.",
"For [5], the user must provide the halo concentration (along with mass and redshift) to the [style=codeintext]NFW() class, but additional routines are available for converting mass to concentration, including [7] and another by [6] (a partial overlap with the mass-concentration relations provided by [style=codeintext]cluster-lensing).",
"[5] depends on [style=codeintext]astropy for cosmological calculations and units, similar to [style=codeintext]cluster-lensing, as well as the [style=codeintext]numpy and [style=codeintext]scipy packages."
],
[
"Future Development",
"Some of the future plans for [style=codeintext]cluster-lensing include adding support for different density profiles.",
"Currently only the NFW model is provided, and alternative mass density models would make the package more complete and useful.",
"The first priority will be inclusion of the Einasto profile [9], and later possibilities may include the generalized-NFW [46].",
"The default cosmology is currently that of [34], but should be updated to [35], since this is now available as [style=codeintext]astropy.cosmology.Planck15 (the user can currently specify this cosmology, it is just not the default).",
"When surface mass density profiles have to be calculated many times for many clusters, as is the case when iterating over parameters in the process of fitting a model, the processing time can become lengthy.",
"This issue is most pronounced for calculation of miscentered profiles, which require the convolution laid out in Equations REF and REF .",
"One major improvement to [style=codeintext]cluster-lensing will be the option to use parallel-processing in these computations.",
"The likely structure of this parallelism will be to divide the halos in a [style=codeintext]ClusterEnsemble() catalog object among the parallel threads, which will calculate the profiles for each of their assigned clusters.",
"All of these future developments are currently listed as issues on the GitHub repository.",
"This GitHub Issue trackerhttps://github.com/jesford/cluster-lensing/issues will continue to serve as the central place for listing future improvements and feature requests.",
"Users and potential-users alike are encouraged to submit ideas and requests through that URL."
],
[
"Summary",
"In this work we presented [style=codeintext]cluster-lensing, a pure-[style=codeintext]Python package for calculating galaxy cluster profiles and properties.",
"We described and gave worked examples of all the functionality currently available, including mass-concentration and mass-richness scaling relations, and the surface mass density profiles $\\Sigma (R)$ and $\\Delta \\Sigma (R)$ , which are relevant for gravitational lensing.",
"The latter density profiles are not cluster-specific, but apply to any mass halo that can be approximated by the NFW prescription.",
"The structure of [style=codeintext]cluster-lensing is ideal for calculating properties and profiles for many galaxy clusters at once.",
"This “composite-halo” approach [15], is especially useful for fitting models to a stacked sample of clusters that span a range of mass and/or redshift.",
"Compared to existing code, [style=codeintext]cluster-lensing stands out by seemingly being the only publicly-available software for calculating miscentered halo profiles.",
"Miscentering is a problem of great relevance for stacked weak lensing studies of galaxy clusters, where halo centers are imperfectly estimated from observational data or simply not well defined (as is the case for individual non-spherical halos – for example in merging systems).",
"The resulting offsets between the assumed and real centers change the shape of the measured shear or magnification profile and need to be accounted for in the modeling.",
"[style=codeintext]cluster-lensing is released under the MIT license, and archived on Zenodo [12].",
"It being developed in a public repository on GitHub: http://github.com/jesford/cluster-lensing/.",
"Contributions to the code can be made by submitting a pull request to the repository, and we welcome feedback, suggestions, and feature requests through GitHub issues, or by emailing the author.",
"Full documentation (including much of the content of this paper), as well as installation instructions and examples, are available in the online documentation, at http://jesford.github.io/cluster-lensing/.",
"If [style=codeintext]cluster-lensing is used in a research project, the authors would appreciate citations to the code [12] and this paper."
],
[
"Acknowledgements",
"The authors are grateful for funding from the Washington Research Foundation Fund for Innovation in Data-Intensive Discovery and the Moore/Sloan Data Science Environments Project at the University of Washington.",
"This project would not have been possible without packages available in Python's open scientific ecosystem, including NumPy [42], SciPy [23], Pandas [28], matplotlib [20], IPython [33], AstroPy [1], Seaborn [43], and related tools."
]
] | 1605.05700 |
[
[
"Three-dimensional distribution of hydrogen fluoride gas toward NGC6334 I\n and I(N)"
],
[
"Abstract Aims.",
"We investigate the spatial distribution of a collection of absorbing gas clouds, some associated with the dense, massive star-forming core NGC6334 I, and others with diffuse foreground clouds.",
"For the former category, we aim to study the dynamical properties of the clouds in order to assess their potential to feed the accreting protostellar cores.",
"Methods.",
"We use spectral imaging from the Herschel SPIRE iFTS to construct a map of HF absorption at 243 micron in a 6x3.5 arcmin region surrounding NGC6334 I and I(N).",
"Results.",
"The combination of new, spatially fully sampled, but spectrally unresolved mapping with a previous, single-pointing, spectrally resolved HF signature yields a 3D picture of absorbing gas clouds in the direction of NGC6334.",
"Toward core I, the HF equivalent width matches that of the spectrally resolved observation.",
"The distribution of HF absorption is consistent with three of the seven components being associated with this dense star-forming envelope.",
"For two of the remaining four components, our data suggest that these clouds are spatially associated with the larger scale filamentary star-forming complex.",
"Our data also implies a lack of gas phase HF in the envelope of core I(N).",
"Using a simple description of adsorption onto and desorption from dust grain surfaces, we show that the overall lower temperature of the envelope of source I(N) is consistent with freeze-out of HF, while it remains in the gas phase in source I.",
"Conclusions.",
"We use the HF molecule as a tracer of column density in diffuse gas (n(H) ~ 10^2 - 10^3 cm^-3), and find that it may uniquely trace a relatively low density portion of the gas reservoir available for star formation that otherwise escapes detection.",
"At higher densities prevailing in protostellar envelopes (>10^4 cm^-3), we find evidence of HF depletion from the gas phase under sufficiently cold conditions."
],
[
"Introduction",
"The hydrogen fluoride molecule, HF, was first observed in the interstellar medium by [59] with the Infrared Space Observatory [41].",
"While ISO had a wavelength range that encompassed only the $J$ =2–1 rotational transition of HF, the next observatory able to observe HF – the Herschel Space Observatory [69] – covered longer far-infrared wavelengths, and it thus opened up access to the ground-state rotational transition, $J$ =1–0, at 1232.48 GHz (243.24 $\\mu $ m).",
"Herschel has observed HF in absorption along many lines of sight, both inside the Galaxy [56], [84], [85], [67], [42], [51], [18], [46], [27] and in nearby extragalactic objects [74], [39], [79], [52].",
"HF absorption has even been detected with ground-based observatories: [53] have made use of the substantial redshift of the Cloverleaf quasar at $z$ =2.56 shifting the HF 1–0 line into the submillimeter window attainable with the CSO on Mauna Kea, and [40] detect it in the $z$ =0.89 absorber toward PKS1830$-$ 211, using ALMA in the Chilean Atacama desert.",
"Because of its large dipole moment and high Einstein $A$ coefficient for radiative decay, rotational states $J$$\\ne $ 0 of HF only become significantly populated in highly energetic conditions.",
"It is for this reason that HF has been clearly detected in emission in a mere handful of cases: in the inner region of an AGB star's envelope [2], in the Orion Bar photodissociation region [90], and in an external galaxy harboring an actively accreting black hole [91].",
"Atomic fluoride, F, has a unique place in the interstellar chemistry of simple molecules.",
"It is the only element which, simultaneously, (1) is mainly neutral because of its ionization potential $>$ 13.6 eV, (2) reacts exothermically with H$_2$ – unlike any other neutral atom – to form its neutral diatomic hydride HF, and (3) lacks an efficient chemical pathway to produce its hydride cation HF$^+$ due to the strongly endothermic nature of the reaction with H${_3}^+$ .",
"We refer to [57], references therein, and the comprehensive review by [22] for more details on the chemistry of HF and a comparison with other hydride molecules.",
"For the reasons listed above, chemical models predict that essentially all interstellar F is locked in HF molecules [98], [58], which has been confirmed by observations across a wide range of atomic and molecular ISM conditions [84], [85].",
"With recent experimental results by [88] showing that, especially at low temperatures approaching 10 K, the reaction F + H$_2$ $\\rightarrow $ HF + H proceeds somewhat slower than earlier assumptions, chemical models are now able to reproduce HF/H$_2$ ratios of $\\sim $ 1$\\times $ 10$^{-8}$ , measured most directly by [36], and observed to be rather stable across different sightlines.",
"Interferometric observations show that CF$^+$ , the next most abundant F-bearing species after HF, has an abundance roughly two orders of magnitude lower than HF, both inside our Galaxy [45] and in an extragalactic absorber [54].",
"As for destruction of HF, the most efficient processes are UV photodissociation and reactions with C$^+$ , but both of these are unable to drive the majority of fluoride out of HF, due to shielding, already at modest depths of $A_V>0.1$ [58].",
"Because of the constant HF/H$_2$ abundance ratio and the high probability that HF molecules are in the rotational ground state, measurements of HF $J$ =0$\\rightarrow $ 1 absorption provide a straightforward proxy of H$_2$ column density.",
"This has led to the suggestion that, at least in diffuse gas, HF absorption is a more reliable tracer of total gas column density than the widely used carbon monoxide (CO) rotational emission lines, and is more sensitive than CH or H$_2$ O absorption [22].",
"Apart from the uncertain and variable CO abundance, local excitation conditions have a profound effect on the level populations of CO, complicating the conversion from observed line strength of a particular CO transition to H$_2$ column density [7].",
"The greatest gas-phase CO abundance variations occur in dense, cold regions where CO freezes out onto surfaces of dust grains, proven by observed CO abundances decreasing in the gas phase and increasing in the ice phase as conditions get colder [37], [71].",
"In addition, the particular fraction of the neutral ISM that is in the diffuse/translucent phase is inconspicuous in CO [7], but is detectable using hydride absorption lines.",
"Of course, for absorption line studies, one relies on lines of sight with sufficiently strong continuum background, for example those toward dense star-forming clouds.",
"Such restrictions do not apply for emission line tracers.",
"Besides CO rotational lines, fine structure line emission due to atomic C and the C$^+$ and N$^+$ ions has been used as a tracer of (diffuse) gas throughout the Galaxy [43], [94], [23], [28], [29].",
"For all these tracers, however, the conversion to H$_2$ column density depends strongly on physical properties such as ionization fraction and excitation conditions.",
"Based on the above arguments, HF absorption measurements are a good tracer of overall gas column density.",
"However, as addressed for example by [67] and [18], HF itself may suffer from freeze-out effects as occurs with other interstellar molecules.",
"While studies have been done on the interaction of H$_2$ O with HF as a polluting agent in the Earth's atmosphere [26], the density and temperature conditions needed for HF adsorption onto dust grains have not been studied in astrophysical contexts so far.",
"Any freeze-out of interstellar HF will obfuscate the direct connection between HF absorption depth and H$_2$ column density described above.",
"The well-known progression of pre- and protostellar stages for stars with masses similar to the Sun [83] is not applicable for high-mass stars ($\\gtrsim 8$ $M_\\odot $ ).",
"In the latter category, protostellar hydrogen fusion starts while accretion from the surrounding gas envelope is still ongoing [64].",
"In the `competitive accretion' scenario, multiple massive protostars in a clustered environment are fed from the same gas reservoir [8].",
"For high-mass protostars, material can continuously be added to the gravitationally bound circumstellar envelope which provides the reservoir for further accretion onto the protostar.",
"It is therefore important, particularly for regions of high-mass star formation, to study not just the gravitationally bound circumstellar envelopes, but also the dynamical properties of surrounding gas clouds.",
"Especially for the latter component, simple hydride molecules have the potential to reveal gas reservoirs to which emission lines of `traditional' tracer species, such as CO, HCO$^+$ , and CS, are insensitive due to their relatively high critical densities.",
"In this paper, we investigate two envelopes of (clusters of) protostars embedded in the NGC 6334 molecular cloud as well as lower density clouds surrounding the dense complex.",
"The filamentary, star-forming cloud NGC 6334, at a distance of 1.35 kpc [95], harbors a string of dense cores, identified in the far-infrared by roman numerals I through VI [50], with an additional source identified $\\sim $ 2 north of source I, later named `I(N)' [24].",
"The larger scale NGC 6334 filament has an H$_2$ column density of $>$ 2$\\times $ 10$^{22}$ $\\mathrm {cm}^{-2}$ even at positions away from the embedded cores [80].",
"[97] observed the velocity structure of NGC 6334 at 0.15 pc resolution.",
"These authors explain the velocity profile along the filament with a cylindrical model collapsing along its longest axis under the influence of gravity.",
"In this paper we study specifically the region of $\\sim $ 2.4$\\times $ 1.6 pc surrounding the embedded cores NGC 6334 I and NGC 6334 I(N).",
"Source I is host to an ultra-compact Hii region, designated source `F' in a 6 cm radio image of the cloud [77].",
"Based on multi-wavelength dust continuum measurements, studies by [81] and [89] have independently determined that the mass of source I(N) exceeds that of sister source I by a factor of $\\sim $ 2–5, but the ratio of their bolometric luminosities is 30–140 in favor of source I, due to the markedly lower temperature for source I(N).",
"As expected for a warm (up to $\\sim $ 100 K), dense, massive star-forming core, NGC 6334 I is extremely rich in molecular lines, spectacularly demonstrated by the 4300 lines detected in the 480–1907 GHz spectral survey by [96]See the introduction section of this reference for a list of earlier spectral survey work on NGC 6334 I..",
"The differences between the two neighboring cores all suggest that core I is in a more evolved stage of star formation than core I(N).",
"Both cores have been studied with radio and (sub)millimeter interferometer observatories, showing that each separates into several subcores at arcsecond resolution [34], [33], [10].",
"To probe gas clouds in front of the NGC 6334 complex, absorption measurements have been obtained in lines of several hydrides.",
"Spatially extended OH hyperfine line absorption at 1.6–1.7 GHz was observed toward the NGC 6334 filament by [11].",
"The spectrally resolved mapping observations from the Australia Telescope Compact Array allowed these authors to ascribe particular velocity components of the absorption to a foreground cloud close to NGC 6334 and other components to clouds with even larger angular extent.",
"[92] used the Heterodyne Instrument for the Far-Infrared [16] onboard Herschel to study the spectral profile of the rotational ground state lines of CH at 532 and 537 GHz, and found four distinct absorption components overlapping with the velocity range of OH absorption, and one single emission component emanating in core NGC 6334 I itself.",
"At 1232.5 GHz in the same spectral survey, [18] find the exact same four absorbing clouds in the HF rotational ground state, and invoke three velocity components to explain the hot core component.",
"While the hot core component(s) appear in emission in CH, they are in absorption in HF, because CH 1$\\rightarrow $ 0 has a lower Einstein $A$ coefficient than HF 1$\\rightarrow $ 0 (see above).",
"Table: SPIRE spectrometer observations used in this paper.The CH and HF signatures were observed toward NGC 6334 I with the high spectral resolution spectrometer HIFI in single point mode [16]; its single pixel receiver did not provide any spatial information.",
"In a map of CH or HF absorption covering the region surrounding source I, one would expect to see a disentanglement of the different spatial extent of each velocity component as illustrated in Fig.",
"REF .",
"Toward core I, the velocity resolved HF absorption signature, with a total equivalent width, $\\int (1-I_\\mathrm {norm}) \\mathrm {d}V$ , of 16 $\\mathrm {km}\\,\\mathrm {s}^{-1}$ , was modeled with seven components.",
"At positions away from core I, but still on the NGC 6334 filament, the equivalent width should diminish to 11 $\\mathrm {km}\\,\\mathrm {s}^{-1}$ representing the four foreground clouds, while at positions off of the cores and the filament, only the two foreground components that are more extended than the dense molecular cloud should be visible, and the equivalent width should drop to 3 $\\mathrm {km}\\,\\mathrm {s}^{-1}$ .",
"This paper presents results from Herschel SPIRE iFTS spectral mapping observations toward a 6$\\times $ 35 region surrounding cores I and I(N) in the NGC 6334 star-forming complex.",
"The observations are described in Sect.",
"and the resulting map of HF absorption depth is discussed in Sect. .",
"The signal is interpreted in Sect.",
", both in the context of foreground clouds and in that of freeze-out conditions in the dense cores.",
"Conclusions are summarized in Sect.",
"."
],
[
"Observations and data reduction",
"The spectral mapping observations used in this work were obtained as part of the `evolution of interstellar dust' guaranteed time program [1] with the Spectral and Photometric Imaging Receiver [30] on board the Herschel space observatory [69].",
"SPIRE's imaging Fourier Transform Spectrometer (iFTS) provides a jiggling observing mode that uses its 54 detectors to obtain Nyquist sampled spatial maps, covering the entire frequency range of the Spectrometer Long Wavelength (SLW, 447–1018 GHz) and the Spectrometer Short Wavelength (SSW, 944–1568 GHz) bands.",
"The spectral resolution of 1.2 GHz corresponds to a resolving power $\\nu /\\Delta \\nu \\approx 10^3$ , roughly 300 $\\mathrm {km}\\,\\mathrm {s}^{-1}$ at the frequency of the HF 1–0 transition, 1232.5 GHz (243 $\\mu $ m).",
"Three partly overlapping, fully sampled SPIRE iFTS 4$\\times $ 4 jiggle observations were performed in a total of nine hours of observing time, two centered on NGC 6334 I and the third on NGC 6334 I(N).",
"A fourth, sparsely sampled observation, centered $\\sim $ 2 northwest of core I, has considerable overlap with the combined area covered by the three other observations.",
"This fourth observation is treated as an extra jiggle position in the gridding process described below.",
"Finally, a fifth observation, also sparsely sampled, is centered 45 northwest of core I and its footprint therefore has no overlap with our mapped area.",
"At each jiggle position, four repeated scans of the FTS mechanism were executed in high spectral resolution mode.",
"Details of the observations are summarized in Table REF .",
"The placement of the different pointings described here is shown in Fig.",
"REF in the Appendix.",
"After inspection of the initial observations from February 2011, some detectors were found to suffer from saturation due to the bright emission toward source I.",
"The observation toward source I was therefore repeated in `bright source mode' [47] in September 2012 to obtain well calibrated spectra toward the brightest position.",
"The majority of detector/jiggle combinations in the original observation point toward less bright regions and are therefore still useful in constructing the final map.",
"The above SPIRE iFTS observations are processed with the `extended source' pipeline in HIPE 12.1.0 and the spire_cal_12_3 calibration tree, which includes the outer ring of partly vignetted detectors [21].",
"The pipeline is interrupted at the pre-cube stage, before spectra from individual jiggle positions are gridded onto a rectangular spatial pattern.",
"The spectrum for each jiggle position and each detector is visually inspected.",
"Discarding all spectra that show excessive noise and/or irregular continuum shape (resulting from partial saturation) results in filtering 28 of the total of 919 SLW spectra (3%) and 135 of the 1696 SSW spectra (8%).",
"The final processing step is the combination of the individual positions from each of the 19 (SLW) or 37 (SSW) detectors of each of the 55 jiggle positions into spectral cubes with square spatial pixels in Right Ascension and Declination coordinates.",
"Due to the complex frequency dependence of the beam size and shape [48], the native angular resolution of SPIRE iFTS observations varies non-monotonously across its frequency range.",
"We use the convolution gridding scheme, which weighs each input value according to the distance from the target pixel center by means of a differential Gaussian kernel, with the aim of obtaining a cube with a constant effective reference beam of 43 FWHM.",
"The pixel grid is identical for SLW and SSW, with square pixels measuring 175$\\times $ 175.",
"Gaussian convolution only results in a completely Gaussian target beam if the original beam is also well represented by a Gaussian shape.",
"Such is the case for the entire SSW band and for low frequencies in SLW, but not for SLW frequencies between $\\sim $ 700 and 1018 GHz [48].",
"Figure: Top panel: velocity resolved spectral profile of HF absorption observed with HIFI toward core NG C6334 I (gray histogram).",
"The seven individual model components are the same as in and are drawn in dashed green lines, with their sum in solid gray.",
"Middle panel: four foreground components expected to remain visible for lines of sight away from core I, but still on the NGC 6334 molecular cloud complex (solid blue).",
"Bottom panel: two foreground cloud components hypothesized to be unrelated to the NGC 6334 complex (solid magenta).Figure: Illustration of the procedure adopted to fit the spectrally unresolved, Sinc-shaped absorption line feature in the SPIRE iFTS cube.",
"Rest frequencies of the relevant transitions are marked by vertical dashed lines.Top panel: observed spectrum (black) toward one of the pixels in our map, fit to continuum and CO lines (blue), and the resulting ratio spectrum (gray).",
"The rectangular box marks the axes limits of the next panel.Middle panel: zoom of continuum-normalized spectrum (gray, same as top panel) and fit to the combined signature of the HF absorption line and the nearby H 2 _2O emission line (red).Bottom panel: residual remaining after subtracting the fit shown in the middle panel.Figure: Map of equivalent width of the HF 1←\\leftarrow 0 absorption signature measured with Herschel SPIRE iFTS (colored contours).",
"The peak value in the center of the `12.5 km s -1 \\mathrm {km}\\,\\mathrm {s}^{-1}' contour is 16 km s -1 \\mathrm {km}\\,\\mathrm {s}^{-1}.",
"The grayscale represents 250 μ\\mu m dust continuum emission measured with the SPIRE photometer (Herschel observation ID 1342239909, HIPE 13 standard pipeline, color bar stretched from 4 to 250 GJy sr -1 ^{-1}).",
"The thin white contours are from the continuum as measured by the FTS, which closely follow the structure seen in the photometer grayscale map.",
"Beam sizes of the convolved spectrometer cube and the photometer map are shown in the bottom left corner in light gray and dark gray circles, respectively.",
"The scalebar in the top left corner indicates a projected length of 0.3 pc at a distance to the source of 1.35 kpc."
],
[
"Results",
"The spectral cubes, as constructed in Sect.",
", show a smooth dust continuum superposed with spectrally unresolved lines.",
"The dominant line signal in the cube arises from the ladder of rotational transitions of CO ($J$ =4–3 to 13–12).",
"Early versions of the CO intensity maps of NGC 6334, based on subsets of the SPIRE iFTS observations used here, were presented in [55] and [49].",
"To retain the highest possible spectral resolution, we use unapodized FTS data, in which any unresolved spectral line has a Sinc-shaped profile.",
"It is therefore important that one carefully fits and subtracts the Sinc-shaped profiles of nearby bright lines, to avoid any remaining sidelobes of strong lines affecting the apparent profile of the weak absorption line under study.",
"This work focuses on absorption signatures of two hydride molecules, for which we use two spectral sections of the data cubes: 1132–1332 GHz from SSW and 760–935 GHz from SLW, chosen specifically to include the two CO emission lines closest to HF 1–0 at 1232.48 GHz [61] and CH$^+$ 1–0 at 835.08 GHz [65].",
"We construct a script in HIPE [63], derived from one of the post-pipeline analysis scripts provided by the SPIRE iFTS working group [70], to fit a third order polynomial for the continuum simultaneously with the following lines: CO at 1152.0 and 1267.0 GHz (SSW), and CO at 806.7 and 921.8 and [Ci] at 809.3 GHz (SLW).",
"Knowing that the intrinsic width of CO lines in this region is only a few $\\mathrm {km}\\,\\mathrm {s}^{-1}$ [96], lines are spectrally unresolved by SPIRE iFTS, and we adopt for each spectral line a Sinc profile with a fixed peak-to-first-zero-crossing width of 1.18 GHz.",
"We then divide the observed spectrum by the fit of continuum and two/three Sinc lines to obtain a continuum-normalized spectrum, $I_\\mathrm {norm}$ .",
"This process, illustrated in the top panel of Fig.",
"REF , is repeated for each spatial pixel in the cube.",
"In the resulting continuum-normalized cube around 1232 GHz, we fit two Sinc functions to the absorption profile of HF 0$\\leftarrow $ 1 at 1232.48 GHz and the nearby emission line of H$_2$ O 2$_{2,0}$$\\rightarrow $ 2$_{1,1}$ at 1228.79 GHz (Fig.",
"REF , middle panel).",
"In the continuum-normalized cube around 835 GHz, we fit three Sinc functions to the absorption profile of CH$^+$ 0$\\leftarrow $ 1 at 835.08 GHz and $^{13}$ CO emission lines at 771.18 and 881.27 GHz.",
"The maps of equivalent width of the HF and CH$^+$ absorption depth, in units of Hz, are multiplied by the ratio of the speed of light, c, and the observed frequency, $\\nu _\\mathrm {obs}$ , to convert to $\\mathrm {km}\\,\\mathrm {s}^{-1}$ units: c/$\\nu _\\mathrm {obs}$ =2.43$\\times $ 10$^{-7}$ $\\mathrm {km}\\,\\mathrm {s}^{-1}$ Hz$^{-1}$ for HF, and 3.59$\\times $ 10$^{-7}$ $\\mathrm {km}\\,\\mathrm {s}^{-1}$ Hz$^{-1}$ for CH$^+$ .",
"Line fits are rejected if the signal-to-noise ratio is lower than 2 and/or the fitted line center is more than half a resolution element from the line's expected frequency at $-8.3$ $\\mathrm {km}\\,\\mathrm {s}^{-1}$ $V_\\mathrm {lsr}$ [92].",
"The resulting map of HF equivalent width in Fig.",
"REF reveals the spatial distribution of the HF absorption feature, detected in 81% (signal-to-noise$>$ 2) or 72% (signal-to-noise$>$ 3) of all the pixels in the 6$\\times $ 35 map coverage.",
"See also the signal-to-noise map in Fig.",
"REF .",
"Uncertainties are calculated from the spectral rms noise in the continuum-normalized cubes in two 20 GHz ranges surrounding the HF absorption line.",
"For CH$^+$ , the frequency ranges for calculating noise are composed of a 10 GHz section below and a 30 GHz section above the CH$^+$ frequency, to avoid incorporating residual from the fit to the blended CO 7–6 and [Ci] $^3$ P$_2$ –$^3$ P$_1$ lines at 806 and 809 GHz.",
"The 1-$\\sigma $ noise on the equivalent width is obtained by multiplying the unitless spectral rms with the instrumental line width of 1.18 GHz $\\times $ c/$\\nu _\\mathrm {obs}$ .",
"Noise values are variable across the maps, in the range 1.2–3.4 $\\mathrm {km}\\,\\mathrm {s}^{-1}$ for HF and 1.4–7 $\\mathrm {km}\\,\\mathrm {s}^{-1}$ for CH$^+$ .",
"We do not include the following contributions to the uncertainty.",
"Firstly, any multiplicative effects such as those of the absolute intensity calibration [4], [87] are divided out by normalizing the spectra to the local continuum.",
"Secondly, additive uncertainties in the continuum level offsets are $\\sim $ 0.5$\\times $ 10$^{-19}$ W m$^{-2}$ for SLW and $\\sim $ 2$\\times $ 10$^{-19}$ W m$^{-2}$ for SSW [87], [31].",
"These values are negligible compared to the brightness of the continuum in our cube, which exceeds the offset uncertainty by a factor of a few hundred even in the faintest outer regions.",
"Spectra from the `OFF' observation, pointed just northwest of the mapped area shown in Fig.",
"REF , are also inspected at the HF frequency, but no convincing detections are found.",
"The spectra from individual detectors in the OFF observation exhibit rms noise values between 4 and 10 $\\mathrm {km}\\,\\mathrm {s}^{-1}$ , with a median of 6 $\\mathrm {km}\\,\\mathrm {s}^{-1}$ .",
"This noise is considerably higher than that in the cube pixels based on the other four observations combined, in which each pixel encompasses at least four, but typically more than eight individual detector pointings.",
"The lack of HF absorption detections in the OFF position is thus consistent at the 2-$\\sigma $ level with an HF absorption depth $\\lesssim $ 12 $\\mathrm {km}\\,\\mathrm {s}^{-1}$ in the area 3–6 northwest of source I, i.e., absorption depths could be anywhere in the range shown in our mapped area, except the central 40 around source I itself where the strongest absorption is seen.",
"To rule out contamination of the HF signature by other spectral lines within SPIRE's spectral resolution element, we inspect the high spectral resolution spectrum toward the position of the chemically rich core NGC 6334 I [96], observed as part of the spectral survey key program CHESS [12] using the HIFI spectrometer [16], [78].",
"The only spectral lines detected by HIFI in a frequency span of $\\pm $ 2 GHz around the HF frequency are four marginally detected methanol emission lines (A. Zernickel, private communication, Jun.",
"2014) together amounting to $<$ 0.4 $\\mathrm {km}\\,\\mathrm {s}^{-1}$ in equivalent width.",
"The possible methanol contaminations for the measured HF absorption depth are therefore contained within the uncertainty for our HF equivalent widths quoted above.",
"The effect of the H$_2$ O emission line at 1229 GHz (see also above) could be more significant: at the brightest position, toward core I, the water line is as bright as one third of the deepest HF absorption.",
"As described above and shown in Fig.",
"REF , the effects of the water line on the HF absorption profile are taken into account by applying a simultaneous fit of these two lines, separated in frequency by three times the SPIRE instrumental line width.",
"We also detect the signature of CH$^+$ 1$\\leftarrow $ 0 absorption at 835.08 GHz in the spectral map from the SLW array.",
"We refrain from interpreting its signal here for the following reasons.",
"Firstly, the signal-to-noise ratio in the CH$^+$ absorption map is much lower than that in the HF map (see Fig.",
"REF and REF ), resulting in signal-to-noise$>$ 2 detections in only 46% of the mapped pixels.",
"For completeness, the distribution of detected CH$^+$ absorption is shown in the Appendix in Fig.",
"REF .",
"Importantly, around the position of source I, there is no confident detection to be compared with heterodyne observations from [96].",
"Only a few isolated pixels near that position have detections of CH$^+$ , but at a signal-to-noise of $<$ 3.",
"Secondly, the spectrally resolved HIFI spectrum toward source I (A. Zernickel, private communication, Jun.",
"2014) show seven distinct emission line features due to methanol at frequencies within 0.6 GHz of the CH$^+$ line, i.e., half of the SPIRE spectral resolution.",
"The combined intensity of these lines is sufficient to compensate more than half of the CH$^+$ absorption observed by HIFI, and they therefore severely contaminate the spectrally unresolved profile of CH$^+$ absorption in the SPIRE spectrum.",
"In fact, these methanol emission lines could be the cause of the weak detection at the position of core I with SPIRE's modest spectral resolution.",
"Thirdly, the CH$^+$ transition falls in a frequency range in which the beam profile of the SPIRE iFTS is non-Gaussian in shape [48], complicating the map convolution and the interpretation of any spatial structure.",
"Our SPIRE spectroscopic data also show evidence for detections of NH$_2$ at 952.6 and 959.5 GHz, and NH at 974.5 GHz and at 1000 GHz.",
"However, compared to HF, it is more challenging for astrochemical models to explain the observed abundances of N-bearing hydrides [66], and the analysis of line features is complicated by hyperfine structure within rotational transitions [96].",
"For these two reasons, we refrain from interpreting the NH and NH$_2$ absorption depth maps in this paper, but for completeness they are shown in Figs.",
"REF and REF .",
"The OH$^+$ doublet at 909.05 and 909.16 GHz is also within the frequency range covered by the SPIRE iFTS, but the bright methanol emission line at 909.07 GHz apparent in the HIFI spectrum published by [96] would make analysis of the blended OH$^+$ signature in the SPIRE spectrum impossible."
],
[
"HF optical depth, equivalent width, and column density",
"An absorption line is saturated when its depth reaches zero, i.e., absorbing all continuum photons in any specific spectral channel.",
"HIFI observations seem to show that the combined absorption feature of HF due to the NGC 6334 I envelope, hot cores, and the foreground cloud at $-3.0$ $\\mathrm {km}\\,\\mathrm {s}^{-1}$ is saturated between $V_\\mathrm {lsr}$ of $-7$ and $-4$ $\\mathrm {km}\\,\\mathrm {s}^{-1}$ (Fig.",
"REF a).",
"The individual components, however, do not reach 100% absorption.",
"Of all seven components, the cloud at $+6.5$ $\\mathrm {km}\\,\\mathrm {s}^{-1}$ comes closest to having saturated absorption in HF.",
"As already noted by [18], even this component is only marginally optically thick, evidence for which is provided by the line width of the same component in the optically thin tracer CH [92] which is the same (1.5 $\\mathrm {km}\\,\\mathrm {s}^{-1}$ ) as for HF.",
"All other components are believed to be optically thin [18].",
"Absorption line depth is converted into optical depth using $I_\\mathrm {norm}=\\mathrm {e}^{-\\tau _\\nu }$ .",
"With the caution of one of the seven components being marginally saturated, we take the optically thin limit to relate optical depth integrated over the line profile, $\\int \\tau _\\nu \\,\\mathrm {d}V$ , to column density, $N_\\mathrm {HF}$ , following [56]: $\\int \\tau _{\\nu ,\\mathrm {HF}} \\mathrm {d}V = \\frac{A_\\mathrm {ul} g_\\mathrm {u} \\lambda ^3}{8\\pi g_\\mathrm {l}} N_\\mathrm {HF} = 4.16\\times 10^{-13} \\, [\\mathrm {cm}^2 \\, \\mathrm {km}\\, \\mathrm {s}^{-1}]\\ N_\\mathrm {HF} ,$ where $A_\\mathrm {ul}$ is the Einstein $A$ coefficient for spontaneous emission, 2.42$\\times $ 10$^{-2}$ s$^{-1}$ for HF 1–0 [68], $g_\\mathrm {u}$ =3 and $g_\\mathrm {l}$ =1 are the statistical weights of rotational levels $J$ =1 and $J$ =0, respectively, and $\\lambda $ is the wavelength of the transition, 243.24 $\\mu $ m. In addition to the optically thin limit, Eq.",
"(REF ) assumes that all HF molecules are in the rotational ground state, a fair assumption given its high $A_\\mathrm {ul}$ .",
"The conversion from $I_\\mathrm {norm}$ to $\\tau _\\nu $ follows a linear relation for low values of $\\tau _\\nu $ : $\\nonumber \\tau _\\nu & = & -\\ln (I_\\mathrm {norm}) \\\\& \\approx & 1-I_\\mathrm {norm} \\qquad [\\mathrm {for}\\ I_\\mathrm {norm} \\approx 1].$ This relation holds to within $\\sim $ 10% for $\\tau _\\nu <0.2$ (line absorbs up to 20% of the continuum), but $\\tau _\\nu /(1-I_\\mathrm {norm})$ is already 1.2 at 40% absorption, rising to 2 at 80% absorption.",
"The integrated optical depth is therefore systematically underestimated for a spectrally unresolved line that is smeared out over a velocity range wider than its intrinsic profile.The logarithm operator gives a disproportionally higher weight to more strongly absorbed channels.",
"For example, a 20 $\\mathrm {km}\\,\\mathrm {s}^{-1}$ wide feature with constant 80% absorption ($I_\\mathrm {norm}$ =0.20) and a smeared-out version of 320 $\\mathrm {km}\\,\\mathrm {s}^{-1}$ wide at 5% absorption ($I_\\mathrm {norm}$ =0.95) both have the same equivalent width, $\\int (1-I_\\mathrm {norm}) \\mathrm {d}V$ = 16 $\\mathrm {km}\\,\\mathrm {s}^{-1}$ .",
"The latter, however, yields a $\\int \\tau _\\nu \\mathrm {d}V$ that is smaller by a factor 2.",
"The effect is larger yet for line profiles that come even closer to full absorption.",
"Since the HF line is spectrally unresolved in our SPIRE spectra, a column density derived from these observations would merely constitute a lower limit to the true column density.",
"Contrary to the optical depth, the absorption depth integrated over the line profile, i.e., the equivalent width of the absorbed `area' below $I_\\mathrm {norm}$ =1, is conserved regardless of the spectral resolution.",
"This is confirmed by the matching equivalent width values measured by HIFI and SPIRE toward core I: $\\int (1-I_\\mathrm {norm}) \\mathrm {d}V$ is 15.7 and 16.4 $\\mathrm {km}\\,\\mathrm {s}^{-1}$ , respectively, with uncertainty margins of $\\sim $ 1 $\\mathrm {km}\\,\\mathrm {s}^{-1}$ in both cases.",
"In the remainder of this paper, we analyze HF absorption depth measured with SPIRE based exclusively on the conserved quantity, equivalent width, $\\int (1-I_\\mathrm {norm}) \\mathrm {d}V$.",
"When deriving optical depths and column densities, we rely exclusively on spectrally resolved profiles such as that in [18]."
],
[
"Distribution of HF absorbing clouds toward NGC 6334",
"The range of HF equivalent width values in the map shown in Fig.",
"REF can be divided into three regimes: (a) $>$ 12 $\\mathrm {km}\\,\\mathrm {s}^{-1}$ , only occurring toward the position of core I; (b) 8–12 $\\mathrm {km}\\,\\mathrm {s}^{-1}$ , spatially consistent with the larger scale filament in which cores I and I(N) are embedded; and (c) $\\lesssim $ 5 $\\mathrm {km}\\,\\mathrm {s}^{-1}$ , exclusively localized at projected distances $>$ 0.6 pc from the cores and the connecting filament.",
"The non-detection of HF in the `OFF' observation (see Sect.",
"), sparsely sampling the area just northwest of our map, is consistent with regimes (c) or (b)."
],
[
"Distinguishing foreground from dense star-forming gas",
"We interpret the three regimes in the context of the velocity resolved HF spectrum published by [18], who identified seven individual physical components responsible for the HF absorption toward NGC 6334 I: the dense envelope at $V_\\mathrm {lsr}$ =$-6.5$ $\\mathrm {km}\\,\\mathrm {s}^{-1}$ , two compact subcores at $-6.0$ and $-8.0$ $\\mathrm {km}\\,\\mathrm {s}^{-1}$ , and four foreground layers at $-3.0$ , $0.0$ , $+6.5$ , and $+8.0$ $\\mathrm {km}\\,\\mathrm {s}^{-1}$ .",
"The spectral signature of each of these components is reproduced in our Fig.",
"REF a.",
"Regime (a) requires all seven components to explain the total equivalent width of HF.",
"The two other panels in Fig.",
"REF represent adaptations of the model from [18] with progressively fewer absorption components taken into account.",
"In Fig.",
"REF b, regime (b) is explained by the superposition of four absorbing foreground clouds, discarding the components associated with the envelope and subcores of NGC 6334 I.",
"We highlight that the HF absorption depth observed toward the dense star-forming envelope I(N) is consistent with regime (b).",
"Differences in HF content between the dense envelopes I and I(N) are discussed in more detail in Sect.",
"REF .",
"Finally, in Fig.",
"REF c, we show that regime (c) is consistent with a model composed of just two specific foreground clouds, namely those at $+6.5$ and $+8.0$ $\\mathrm {km}\\,\\mathrm {s}^{-1}$ [92], [18].",
"In contrast with the detailed study of the HF profile in [18] and in this work, the spectral survey paper by [96], analyzing $\\sim $ 4300 individual spectral line features, uses a simplified model which explains the HF absorption with only three components.",
"The two approaches are not inconsistent, but the latter paper groups components together as: (1) the NGC 6334 I envelope and two subcores, (2) two foreground clouds that are kinematically close to the NGC 6334 complex, and (3) two other foreground clouds with larger $V_\\mathrm {lsr}$ offsets [18].",
"In this three-component model, regime (a) would be explained by groups (1)+(2)+(3), regime (b) by groups (2)+(3), and regime (c) by only group (3).",
"The combination of our HF absorption depth map with the previous, single-pointing, velocity resolved HF spectrum now reveals a three-dimensional picture of the layers of absorbing gas toward the NGC 6334 complex.",
"Our interpretation of the relative positions of the foreground layers is sketched in Fig.",
"REF .",
"With the exception of the direction toward core I, the HF absorption depth at all other positions in the map can be explained by (a subset of) the four extended foreground clouds.",
"Figure: Sketch of the location and extent of the foreground clouds.",
"Note the broken scale in the horizontal direction.",
"Velocity magnitudes are indicated with black arrows, shifted such that the core NGC 6334 I, at V lsr V_\\mathrm {lsr}=-6.5-6.5 km s -1 \\mathrm {km}\\,\\mathrm {s}^{-1}, is at rest in this frame.",
"From left to right, they represent the components referred to in the text as the +8.0+8.0, +6.5+6.5, +0.0+0.0, and -3.0-3.0 km s -1 \\mathrm {km}\\,\\mathrm {s}^{-1} clouds.",
"The dotted line represents the line of sight from the observer to NGC 6334 I.",
"The dashed lines indicate the angular extent of clouds on the far foreground.",
"The color differential between core I and the filament indicates the elevated temperature in core I, preventing freeze-out of HF molecules."
],
[
"Relation of foreground clouds to NGC 6334",
"[92] used a single-pointing Herschel HIFI observation of CH rotational line absorption, coupled with results from OH hyperfine line absorption measurements at radio wavelengths [11], to suggest that the clouds at $V_\\mathrm {lsr}$ =$-3.0$ and $+0.0$ $\\mathrm {km}\\,\\mathrm {s}^{-1}$ are associated with the NGC 6334 complex, while the remaining two velocity components, at $+6.5$ and $+8.0$ $\\mathrm {km}\\,\\mathrm {s}^{-1}$ , originate in foreground clouds farther away from the NGC 6334 complex.",
"This interpretation is consistent with the first two components being seen exclusively in regime (b) of the HF equivalent width map and the latter two components being spread over a more extended area of regimes (b) and (c) combined.",
"Therefore, the spatial distribution of HF absorption measured with SPIRE iFTS supports the previous hypothesis that the $-3.0$ and $+0.0$ $\\mathrm {km}\\,\\mathrm {s}^{-1}$ clouds are associated with NGC 6334, since they have a spatial morphology that closely follows the dense molecular cloud traced by the 250 $\\mu $ m dust emission (Fig.",
"REF ), i.e., a region roughly 2–3 in east-west extent and stretching along the north-south direction.",
"The vertical extent of the `related' foreground clouds depicted in Fig.",
"REF is drawn directly from the observed east-west extent of these component in the plane of the sky.",
"Besides the diatomic hydride species HF, CH, and OH, a subset of our absorbing clouds also exhibit absorption lines due to H$_2$ O [17], [89], H$_2$ O$^+$ [62], and H$_2$ Cl$^+$ [44].",
"Absorption components in OH$^+$ and H$_2$ O$^+$ detected toward both cores by [35] peak at $V_\\mathrm {lsr}$ =$-2$ and $+3$ $\\mathrm {km}\\,\\mathrm {s}^{-1}$ .",
"The former could be a blend of the $-3.0$ and $+0.0$ $\\mathrm {km}\\,\\mathrm {s}^{-1}$ clouds seen in CH and HF, but the latter is inconsistent with any of our components.",
"These lines were all detected in observations with the single-pixel, high spectral resolution Herschel HIFI spectrometer, in some cases toward both individual protostellar cores.",
"It is interesting that ionized species, H$_2$ O$^+$ and H$_2$ Cl$^+$ , are only detected at $V_\\mathrm {lsr}$ values that match the $-3$ and $+0$ $\\mathrm {km}\\,\\mathrm {s}^{-1}$ clouds.",
"Chemical models tailored to halogen hydrides [57] indicate that cation species become abundant under the influence of strong UV radiation.",
"We hypothesize that the presence of H$_2$ O$^+$ and H$_2$ Cl$^+$ is further evidence for the physical proximity of these two clouds to the massive protostars and the Hii region embedded in one of the dense cores.",
"Another ionized species that has been detected in spectrally resolved observations toward NGC 6334 is CH$^+$ .",
"While a velocity resolved observation of HF exists only toward the position of core I, CH$^+$ 1–0 has been observed with HIFI toward both cores I and I(N), the latter as part of the WISH program [93], [5].",
"Toward source I, the CH$^+$ profile looks very similar to the HF profile, amounting to a total equivalent width of $\\sim $ 20 $\\mathrm {km}\\,\\mathrm {s}^{-1}$ .",
"In comparison, the CH$^+$ observation toward core I(N)Estimate based on the HIFI spectrum downloaded from the Herschel Science Archive, observation ID 1342214306, processed with HIPE pipeline version 13. reveals an equivalent width of only $\\sim $ 13 $\\mathrm {km}\\,\\mathrm {s}^{-1}$ .",
"The majority (85%) of the reduced absorption toward core I(N) relative to core I falls in the $[-15,5]$ $\\mathrm {km}\\,\\mathrm {s}^{-1}$ range, as expected if the missing components are the envelope and subcore components at $V_\\mathrm {lsr}$ =$-6.5$ , $-6.0$ , and $-8.0$ $\\mathrm {km}\\,\\mathrm {s}^{-1}$ (cf. Fig.",
"REF ), i.e., those components that occur only in regime (a) in our HF absorption map.",
"As mentioned above in Sect.",
", the SPIRE iFTS spectral cube has too low signal-to-noise near 835 GHz to make a meaningful comparison with the CH$^+$ signature detected by this instrument.",
"The foreground clouds at $V_\\mathrm {lsr}$ =$+6.5$ and $+8.0$ $\\mathrm {km}\\,\\mathrm {s}^{-1}$ , supposedly unrelated to the NGC 6334 dense filament [11], [92], have a combined HF equivalent width that is entirely consistent with the observation in this work that regime (c) is spatially extended beyond the dense filament.",
"An unrelated set of foreground cloud(s) (the two leftmost clouds in Fig.",
"REF ) are likely to have a larger angular extent than the background continuum source (dashed lines in Fig.",
"REF ), despite a possibly modest linear size.",
"This explains why a minimum level of HF equivalent width of $\\sim $ 3 $\\mathrm {km}\\,\\mathrm {s}^{-1}$ is detected not just toward the NGC 6334 filament, but throughout the extent of our 6$\\times $ 35 map (Figs.",
"REF and REF ).",
"All four foreground clouds detected in HF and CH are redshifted ($V_\\mathrm {lsr}$$\\ge $$-3$ $\\mathrm {km}\\,\\mathrm {s}^{-1}$ ) with respect to the $V_\\mathrm {lsr}$ of the part of NGC 6334 cloud near cores I and I(N) (around $-5$ $\\mathrm {km}\\,\\mathrm {s}^{-1}$ , based on HCO$^+$ observations by [97]).",
"Thus, the absorbing gas clouds are moving toward NGC 6334, instead of following the Galactic rotation, which at $\\ell =351$ yields exclusively negative line-of-sight velocities for sources between Sun and NGC 6334, i.e., approaching the standard of rest of stars in the Solar neighborhood.",
"We therefore conclude that, in addition to the gas flows within the dense gas along the filament's long axis [97], gas may also be accreting onto the filament in the perpendicular direction.",
"We also note that, whereas [35] put the OH$^+$ and H$_2$ O$^+$ absorption clouds toward both cores I and I(N) at the distance of 1.35 kpc of the NGC 6334 cloud, their location along the sight line toward NGC 6334 is in fact poorly constrained.",
"Recognizing that there must be peculiar motions at play, deviating from the `rigid' Galactic rotation curve, these clouds could in fact be anywhere between the local arm of the Milky Way and the Sagittarius arm that harbors the NGC 6334 complex.",
"For the two foreground clouds related to NGC 6334, at $-3$ and $+0$ $\\mathrm {km}\\,\\mathrm {s}^{-1}$ , we calculate their total mass by multiplying the sum of their column densities with a rough estimate of the area covered by this component of 160$\\times $ 270 (regime (b) in Fig.",
"REF ), corresponding to 1.5 pc$^2$ at the distance of NGC 6334.",
"Depending on the choice of fiducial H$_2$ tracer, either taking $N_\\mathrm {CH}$ from [92] and $N_\\mathrm {CH}$ /$N_\\mathrm {H_2}$ =(2.1–5.6)$\\times $ 10$^{-8}$ from [82], or $N$ (HF) from [18] and $N_\\mathrm {HF}$ /$N_\\mathrm {H_2}$ =(0.5–1.4)$\\times $ 10$^{-8}$ from [36], this calculation yields a mass in the range 37–98 $M_\\odot $ or 69–191 $M_\\odot $ , respectively.",
"From symmetry arguments, a similar reservoir of additional gas is expected to lie behind the NGC 6334 dense cloud.",
"This means that a significant total mass of several hundred $M_\\odot $ could be on its way to accreting onto the dense cloud NGC 6334 near ($<$ 0.3 pc) the embedded cores I and I(N).",
"This gas reservoir has escaped detection so far, because it does not appear in traditional gas tracers such as CO, HCO$^+$ , and CS.",
"While there is evidence that the $-3.0$ and $+0.0$ $\\mathrm {km}\\,\\mathrm {s}^{-1}$ foreground clouds are closer to NGC 6334 than the $+6.5$ and $+8.0$ $\\mathrm {km}\\,\\mathrm {s}^{-1}$ clouds, there is no direct metric of the geometrical distance along the line of sight from each cloud to the dense filament and cores.",
"Therefore, we refrain from speculating about accretion time scales of even the `related' clouds, since this would rely on unsupported assumptions on relative distances."
],
[
"Spatial distribution of HF toward other Galactic sight lines",
"The only other published work of a spatial map of HF absorption so far is that toward Sgr B2(M) by [19], using data also obtained with Herschel SPIRE iFTS.",
"A direct comparison of measured absorption line depths is complicated by the choice of [19] to present signal in terms of integrated optical depth, apparently without taking into account the systematic underestimation of optical depths derived from spectrally unresolved measurements, as discussed in Sect.",
"REF .",
"Nonetheless, it appears that the total HF absorption toward Sgr B2(M) is about an order of magnitude stronger than toward NGC 6334 I.",
"[19] find a variation of HF absorption depth of only a factor $\\sim $ 2 across the $\\sim $ 25 mapped area, significantly less variation than in our fully sampled map toward NGC 6334 I.",
"Any intrinsic variation may have been partially smoothed by the interpolation process that was applied in [19] to construct a map from the spatially undersampled SPIRE iFTS observation.",
"More importantly, the line of sight toward Sgr B2, close to the Galactic Center, crosses many more spiral arms than that toward NGC 6334.",
"Evidence of this is found for example by [73], who detect a total of 31 individual velocity components in CH rotational ground state absorption toward Sgr B2(M).",
"The 30 foreground clouds, associated with the various intervening Galactic arms, amount to a total CH column density of 1.7$\\times $ 10$^{15}$ $\\mathrm {cm}^{-2}$ , with Sgr B2(M) itself adding a component of only 0.5$\\times $ 10$^{15}$ $\\mathrm {cm}^{-2}$ [73].",
"This is consistent with many sheets of foreground gas together creating a roughly uniform cover of absorbing gas spanning at least a few arcminutes on the sky.",
"Comparatively little additional absorption is contributed by the massive molecular cloud Sgr B2 and the embedded cores in the background, which explains the lack of variation in HF absorption depth seen toward Sgr B2 by [19].",
"In contrast, our map of HF absorption toward NGC 6334 reveals a mix of components due to foreground clouds and the star-forming envelope and cores within NGC 6334, which we are able to disentangle owing to the complementary, velocity resolved spectra of CH and HF obtained with Herschel HIFI [92], [18].",
"In addition, we highlight the discovery by [46] of a foreground cloud toward the intermediate-mass star-forming core OMC-2 FIR 4, based on single-pointing HIFI spectra of HF and other hydride molecules.",
"Several oxygen-bearing hydrides show absorption exclusively at a blueshifted velocity relative to the background protostellar core.",
"The same foreground cloud was later also identified in H$_2$ Cl$^+$ by [38].",
"The HF profile shows absorption at the same blueshifted velocity, but shows additional evidence for a second absorption component that matches the $V_\\mathrm {lsr}$ of the protostellar core [46].",
"Instead of kinematical and morphological arguments such as those used in this work to determine the physical location of foreground clouds toward NGC 6334, [46] use detailed photochemical modeling to infer proximity of their OMC-2 foreground gas to a source of copious far-UV radiation.",
"With that radiation source assumed to be the trapezium cluster of OB stars, it is concluded that the absorbing slab is physically connected to OMC-1.",
"In an attempt to study the spatial distribution of this absorbing OMC-1 `fossil' slab, we have searched archival SPIRE iFTS data toward OMC-2 FIR 4 (Herschel observation ID 1342214847) for signatures of HF at 1232.5 GHz (and CH$^+$ at 835.1 GHz), but find no detections in any of the 37 (and 19) SSW (and SLW) detectors in the $\\sim $ 3 footprint.",
"The HF equivalent width of 15.9$\\pm $ 1.4 $\\mathrm {km}\\,\\mathrm {s}^{-1}$ measured at the position of NGC 6334 I in our SPIRE map is explained in Sect.",
"REF by invoking the superposition of four absorption components in the foreground and three associated to the dense core I itself [18].",
"A striking feature of our HF absorption map is the lack of additional absorption toward the position of core I(N), where the HF equivalent width of only 10.9$\\pm $ 1.1 $\\mathrm {km}\\,\\mathrm {s}^{-1}$ can be explained by the four foreground clouds alone, adding up to 10.5 $\\mathrm {km}\\,\\mathrm {s}^{-1}$ (Fig.",
"REF b).",
"In contrast, adding even just an envelope component similar to that of core I sums up to 13.7 $\\mathrm {km}\\,\\mathrm {s}^{-1}$ (not counting the two subcore components), which is inconsistent with the observation toward core I(N).",
"Since the envelope of core I(N) is more massive than that of core I, but has a similar size [89], the total gas column density toward core I(N) should be higher.",
"Therefore, the lack of HF absorption associated to the I(N) core is not due to the difference in total (H$_2$ ) column.",
"Instead, we hypothesize that HF is primarily frozen out onto dust grains in core I(N), while HF is in the gas phase in core I.",
"To support the hypothesis of HF being depleted from the gas phase in core I(N), we set up a rudimentary model based on the following ingredients.",
"We take the spherically symmetric physical structure, i.e., radial profiles of density and temperature, of the envelopes of NGC 6334 I and I(N) as fitted to submillimeter dust continuum maps and the far-infrared / submillimeter spectral energy distribution [89].",
"We then calculate, at every radial point, the timescales for adsorption (freeze-out) and desorption (evaporation) of HF molecules onto dust grains.",
"Following, e.g., [76] and [37], we assume that thermal desorption is the dominant mechanism that drives molecules from the grain surface back into the gas phase, and are left with the balance between adsorption rate: $\\lambda (n_\\mathrm {H}, T_\\mathrm {gas}) = 4.55\\times 10^{-18} \\left( \\frac{T_\\mathrm {gas}}{m_\\mathrm {HF}} \\right)^{0.5} n_\\mathrm {H} \\qquad [\\mathrm {s}^{-1}] ,$ and desorption rate: $\\xi (T_\\mathrm {dust}) = \\nu _\\mathrm {vib} \\exp \\left( -\\frac{E_\\mathrm {b,HF}}{k\\,T_\\mathrm {dust}} \\right) \\qquad [\\mathrm {s}^{-1}] .$ Here, $T_\\mathrm {gas}$ and $T_\\mathrm {dust}$ are the temperatures of gas and dust, assumed to be equal as in the modeling of [89], $m_\\mathrm {HF}$ is the molecular weight of HF (20), $n_\\mathrm {H}$ is the density of hydrogen nuclei, $\\nu _\\mathrm {vib}$ is the vibrational frequency of the HF molecule in its binding site, for which we adopt $10^{13}$ s$^{-1}$ , $k$ is the Boltzmann constant, and $E_\\mathrm {b,HF}$ is the binding energy of HF to the dust grain surface.",
"It has previously been inferred by [67] that a density of $\\sim $$10^5$ $\\mathrm {cm}^{-3}$ allows HF to condense onto dust grains, whereas densities of $\\sim $$10^3$ $\\mathrm {cm}^{-3}$ more typical for diffuse gas are too low for HF freeze-out to occur.",
"In our case, the density $n_\\mathrm {H}$ – which incidentally exceeds $10^5$ $\\mathrm {cm}^{-3}$ at almost all radii in the envelopes of I and I(N) – enters directly into Equation REF to govern the adsorption rate.",
"Figure: Radial dependence of adsorption and desorption timescales of HF from various types of grain surfaces.",
"The top half of the figure relates to the envelope of core I(N), the bottom half to that of core I.",
"The small panels above each main panel depict the physical structure (temperature in solid blue, left axis; density in dashed red, right axis) for the envelopes in question from .",
"The horizontal axis is logarithmic cumulative mass increasing from left to right; labels in brackets indicate the percentage of enclosed mass starting from the outer shell of the core.",
"Freeze-out of HF (depletion from the gas phase) occurs in the region where t desorb >t freeze t_\\mathrm {desorb} > t_\\mathrm {freeze}; light shading indicates the case of H 2 _2O ice mantles, darker shading that of CO/CO 2 _2 ice mantles.",
"The desorption time scale lines for CO ice and CO 2 _2 ice in the bottom panel lie to the far right, outside of the limits of the axes.In this work we consider multiple versions of the desorption timescale, because the binding energy in the exponent of Equation REF is heavily dependent on the type of grain surface.",
"Unlike for more common molecular species such as CO [6], [60], the desorption behavior of HF from astrophysically relevant grain surfaces has not been studied experimentally, so we rely on theoretical calculations.",
"Typical interstellar dust grains, especially those embedded in cold, star-forming regions, are covered in one or multiple layers of ice consisting of various molecules, mainly H$_2$ O, CO, and CO$_2$ [9].",
"We collect binding energy values for several types of grain surfaces in Table REF .",
"For CO and CO$_2$ ice covered grains we adopt calculated binding energies from the literature [13], [75], while for hydrogenated bare silicate grains and H$_2$ O ice covered grains, these values result from original ab initio chemical calculations performed for this work.",
"Table: Binding energies for HF onto various surfaces.To calculate the HF binding energies in the first two rows of Table REF (with a bare grain and with H$_2$ O ice), we carry out quantum calculations within the Kohn-Sham implementation of Density Functional Theory using the Quantum Espresso Simulation Package [25].",
"Perdew-Burke-Ernzerhof exchange-correlation functional ultrasoft pseudopotentials are used.",
"KS valence states are expanded in a plane-wave basis set with a cutoff at 340 eV for the kinetic energy.",
"The self-consistency of the electron density is obtained with the energy threshold set to $10^{-5}$ eV.",
"Calculations are performed using the primitive unit cell containing a total number of 46 atoms for bare hydrogenated silica, and 54 atoms for hydrogenated silica covered with one layer of H$_2$ O ice.",
"The geometry optimization is used within the conjugate gradients scheme, with a threshold of 0.01 eV $Å^{-1}$ on the Hellmann-Feynman forces on all atoms; the Si atoms of the bottom layers are fixed at their bulk values.",
"The binding energy of HF with the SiH terminus of hydrogenated crystalline silica is based on calculations for the hydroxylated alpha-quartz (001) surface.",
"The binding energy of HF with one layer of H$_2$ O ice on amorphous hydrogenated silicate is estimated by assuming that the the most common structure in this case is the HF molecule interacting with a H$_2$ O molecule bonded to silanol (SiOH), which is the most abundant surface group in amorphous silica [20].",
"The binding energies of HF with CO and CO$_2$ ice (last two rows of Table REF ) are taken from calculations by [75] and [13], respectively.",
"These authors performed calculations for molecules in the gas phase.",
"We consider the gas phase binding energies of HF with CO and CO$_2$ to be similar to those of HF with CO and CO$_2$ ices adsorbed on an inert surface such as that of hydroxylated amorphous silica.",
"This approximation is based on the weak interactions of these ices with hydroxylated silica and within the CO and CO$_2$ molecular solids, so that the electronic density of CO and CO$_2$ in solid form is not significantly altered with respect to their state in the gas phase.",
"Hence, for the aim of the present work, the binding energy of the HF molecule with CO or CO$_2$ as calculated in the gas phase is applicable for the condensed phase.",
"The situation is notably different for interactions with H$_2$ O in the gas or adsorbed form, because of its stronger interaction with silica and HF.",
"For HF interacting with H$_2$ O ice, we use binding energies from our own calculations described above.",
"With the physical structure of both envelopes, the equations for the adsorption/desorption balance, and the binding energy values, the `freeze-out' region within each envelope is calculated in Fig.",
"REF .",
"Defining $t_\\mathrm {freeze} = 1/\\lambda $ and $t_\\mathrm {desorb}=1/\\xi $ , HF molecules will deplete from the gas phase in the region of the envelope wherever $t_\\mathrm {desorb}$ > $t_\\mathrm {freeze}$ .",
"Numerical simulations by [15] suggest that, within a mixed-composition ice layer, the abundances of CO and CO$_2$ are enhanced compared to H$_2$ O in high-density ($\\gtrsim 10^5$ $\\mathrm {cm}^{-3}$ ) environments.",
"At such densities applicable for the protostellar envelopes studied here, we thus expect the HF binding energy to lie close to, but slightly above that of pure CO or CO$_2$ ice, and the true desorption time scale line in Fig.",
"REF therefore somewhat to the left of the dash-dotted line for CO ice.",
"In this case, HF is expected to stay frozen onto grain surfaces at a broad range of radii in core I(N): cumulative mass $\\sim $ 0.1 to 1, i.e., 90% of the mass, where the temperature is $\\lesssim $ 20 K. Core I(N) is overall colder than core I both in the envelope (blue solid lines in Fig.",
"REF ) and in the embedded subcores [34].",
"For the comparatively warmer envelope of core I, the lines for the desorption timescale of HF from CO/CO$_2$ ice fall off the scale on the right hand side of the axes, leaving no freeze-out zone in this envelope.",
"This could explain why HF is seen in the gas phase in core I, but not in core I(N).",
"In an alternative scenario in which the dust grains are covered in pure H$_2$ O ice – or the desorption characteristics of a mixed mantle are dominated by that of H$_2$ O ice [14] – the binding energy of HF with the ice would be greatly increased (see Table REF ).",
"In this case, the HF freeze-out zone would expand to cover $>$ 98% of the mass for both envelopes, and our observations should have revealed no gas phase HF in either of the cores.",
"Our observations are therefore inconsistent with pure H$_2$ O ice coating on the grains.",
"Instead, our interpretation relies on a significant part of the ice coating to consist of CO and/or CO$_2$ molecules.",
"In principle, the composition of ice coatings do not need to be the same in the two neighboring envelopes.",
"Particularly, given the lower temperatures of core I(N), there is higher probability that a significant amount of H$_2$ O is frozen out in that envelope.",
"This, again, would enhance the binding energy of HF onto the ice-covered dust grains in the envelope of core I(N), which may help to explain the lack of gas-phase HF observed toward source I(N).",
"Conversely, if additional mechanisms for desorption, e.g., induced by (UV) photons or cosmic rays, would be taken into account, $\\xi $ would increase, and the freeze-out region would be pushed to larger radii in the envelopes.",
"Particularly UV photodesorption could have a different effect in one envelope compared to the other, because core I is more evolved and contains an Hii region.",
"An increased total desorption rate would have no effect on HF freeze-out in the envelope of core I, in which HF is already completely in the gas phase, but would reduce the size of the freeze-out zone for envelope I(N).",
"If, however, thermal desorption as expressed in Equation REF is the dominant desorption mechanism and the binding energy of HF onto the ice surface is close to that of CO or CO$_2$ ice (Table REF ), our model predicts significant freeze-out of HF in core I(N), and none in core I.",
"It is important to note that, indeed, the temperature and density conditions under which HF remains frozen onto dust grains depend greatly on the exact composition and mixing of the ice mantle and therefore on the chemical history.",
"It has already been recognized for example for the CO molecule that the freeze-out temperature `threshold' can vary considerably from one object to another [72]."
],
[
"Conclusions",
"In this work we present a map of HF absorption toward the northern end of the molecular cloud NGC 6334, containing two well studied massive star-forming cores I and I(N).",
"Although in the original definition of the observing program it was not anticipated that hydride absorption lines would be found within these data, the discovery space provided by the enormous frequency coverage of the Herschel SPIRE iFTS instrument has made this study possible.",
"Such wide coverage in the far-infrared/submillimeter is only attainable with broadband FTS spectrometers [55].",
"The absorption line of HF is detected in 80% of our mapped area, although it is spectrally unresolved by SPIRE.",
"By complementing the new SPIRE iFTS data with existing, single pointing, high spectral resolution spectra from the Herschel HIFI instrument [92], [18], we construct a three-dimensional picture of gas clouds in front of and inside the massive star-forming filament NGC 6334.",
"We find that our observations are consistent with a scenario of four individual foreground clouds on the line of sight toward NGC 6334 I and I(N), two of which are unrelated to the star-forming complex (Sect.",
"REF ).",
"The other two clouds are posited to be close to the dense molecular filament based on their spatial morphology.",
"Their velocities are such that they are moving toward the star-forming cloud and could be adding several hundreds of solar masses of gas to the dense filament and the embedded cores in which massive star formation is already ongoing (Sect.",
"REF ).",
"This component of gas is detected in rotational lines of diatomic hydride molecules, but had been unseen in studies of traditional dense gas tracers.",
"In fact, using such a tracer, HCO$^+$ , [97] have inferred that the roughly cylindrically shaped NGC 6334 filament is collapsing along its longest axis.",
"Our work now indicates that accretion may also be ongoing in the perpendicular (radial) direction.",
"Future studies of (competitively) accreting high-mass star-forming cores may need to take into account this additional low-density phase of the gas reservoir.",
"Finally, in Sect.",
"REF we explain why HF is observed in the gas phase toward core I, but appears completely absent in core I(N).",
"For this purpose, we use a simple description of adsorption and desorption time scales for HF interacting with dust grain surfaces, depending on the (radially variable) density and temperature.",
"Since interactions of HF with interstellar-like dust grains have not been studied in the laboratory, we adopt binding energy values for different types of grain surfaces from theoretical calculations from the literature as well as from original work first presented in this paper.",
"The conclusion is that the lower temperature of core I(N) compared to core I could lead to freeze-out of HF exclusively in the former, but only if the binding energy of HF onto the grain surface is governed by that of CO or CO$_2$ ice on a silicate surface.",
"In this case, at the densities relevant in the envelope of source I(N) ($>$ 3$\\times $ 10$^{5}$ $\\mathrm {cm}^{-3}$ ), we find that HF freezes out in the region of the envelope where the temperature is below $\\sim $ 20 K, rather similar to the freeze-out temperature often adopted for CO.",
"In contrast, if H$_2$ O is the dominant constituent in the ice mantles, our model predicts that HF should have been frozen out at all radii in the envelopes of both sources I(N) and I.",
"Since we observe a significant amount of HF in the gas phase in source I, this scenario is inconsistent with our data.",
"Summarizing, this work uses HF as a sensitive tracer for (molecular) gas at relatively low densities that may be contributing mass to star forming cores.",
"The HF signature reveals a gas reservoir that is inconspicuous in traditional dense gas tracers such as CO.",
"In addition, we show that gas phase HF in higher density environments ($>$$10^5$ $\\mathrm {cm}^{-3}$ ) is extremely sensitive to interactions with dust grains and will be depleted significantly at low dust temperatures.",
"The research of MHDvdW at the University of Lethbridge was supported by the Canadian Space Agency (CSA) and the Natural Sciences and Engineering Research Council of Canada (NSERC), and at the University of Copenhagen by the Lundbeck Foundation.",
"Research at the Centre for Star and Planet Formation is funded by the Danish National Research Foundation and the University of Copenhagen's programme of excellence.",
"SPIRE has been developed by a consortium of institutes led by Cardiff University (UK) and including Univ.",
"Lethbridge (Canada); NAOC (China); CEA, LAM (France); IFSI, Univ.",
"Padua (Italy); IAC (Spain); Stockholm Observatory (Sweden); Imperial College London, RAL, UCL-MSSL, UKATC, Univ.",
"Sussex (UK); and Caltech, JPL, NHSC, Univ.",
"Colorado (USA).",
"This development has been supported by national funding agencies: CSA (Canada); NAOC (China); CEA, CNES, CNRS (France); ASI (Italy); MCINN (Spain); SNSB (Sweden); STFC, UKSA (UK); and NASA (USA).",
"HIPE is a joint development by the Herschel Science Ground Segment Consortium, consisting of ESA, the NASA Herschel Science Center, and the HIFI, PACS and SPIRE consortia.",
"This research has made use of NASA's Astrophysics Data System Bibliographic Services.",
"The graphical representations of the results in this paper were created using APLpy, an open-source plotting package for Python hosted at http://aplpy.github.com, Astropy, a community-developed core Python package for Astronomy (Astropy Collaboration [3]), and the matplotlib plotting library [32].",
"The authors are grateful to Alexander Zernickel for providing and discussing excerpts of the CHESS spectral survey data, to Raquel Monje for providing the HF spectral model component profiles, and to Floris van der Tak for providing the physical structure models of the two envelopes in electronic table format.",
"We thank Tommaso Grassi, Wing-Fai Thi, Jes Jørgensen, Søren Frimann, and Mihkel Kama for discussions."
],
[
"Complementary figures",
"Maps displaying the signal-to-noise ratio of the detections of the HF and CH$^+$ lines in each spatial pixel of our spectral cube (Sect. )",
"are shown in Fig.",
"REF and REF .",
"The colored contours in Fig.",
"REF show that CH$^+$ absorption is only confidently detected (signal-to-noise $>$ 5) in the northeastern section of the map.",
"In addition, although the signal from nitrogen species is not interpreted in this paper, maps of line absorption depth due to NH and NH$_2$ are shown in Figs.",
"REF and REF .",
"The continuum-normalized cube for these lines is created – analogous to those for HF and CH$^+$ in Sect.",
"– by fitting the continuum and the CO 9–8 line in the 945–1055 GHz section of the SSW cube.",
"Absorption lines of NH (at 974.47 and 999.98 GHz) and two of NH$_2$ (at 952.57 and 959.50 GHz) are then fitted simultaneously with emission lines due to $^{13}$ CO 9–8 at 991.3 GHz and H$_2$ O $2_{02}$ –$1_{11}$ at 987.9 GHz.",
"Figure: Placement of the SPIRE iFTS footprints on the NGC 6334 region (cf. Table ).",
"Solid circles indicate the fully sampled observations, two centered on cores I and one on I(N), while the sparsely sampled `INT' and `OFF' observations are marked by dashed circles.",
"Circles are drawn with a diameter of 31 to include the vignetted outer ring of the detector arrays .",
"The grayscale represents the same 250 μ\\mu m dust continuum map as in Fig.",
", but with the scale bar stretching from 1 to 250 GJy sr -1 ^{-1}.",
"The thin white contours, the beam size indicators in the bottom left and the scale bar in the top left corners are the same as in Fig.",
".Figure: Signal-to-noise maps (color scale, with gray indicating values below 2) of the absorption signature of HF 1←\\leftarrow 0.",
"Colored contours are at the same equivalent width levels as in Fig. .",
"As in Fig.",
", white contours trace continuum dust emission and the beam of the SPIRE iFTS map is shown in the bottom left corner.Figure: As in Fig.",
", but for CH + ^+ 1←\\leftarrow 0.",
"Colored contours are at 22, 19, 16 and 13 km s -1 \\mathrm {km}\\,\\mathrm {s}^{-1} in gray, blue, cyan and magenta, respectively.Figure: As in Fig.",
", but for the JJ=2←\\leftarrow 1 NN=1←\\leftarrow 0 transition of NH at 974.47 GHz.",
"The JJ=1–1 transition at 1000.0 GHz shows a similar absorption distribution, but with detections in fewer pixels.",
"Colored contours are at 16.5, 13.5, and 10.5 km s -1 \\mathrm {km}\\,\\mathrm {s}^{-1} in gray, blue, and cyan, respectively.Figure: As in Fig.",
", but for the 1 11 1_{11}←\\leftarrow 0 00 0_{00} JJ=3/2←\\leftarrow 1/2 transition of NH 2 _2 at 952.57 GHz.",
"The JJ=1/2←\\leftarrow 1/2 transition at 959.50 GHz shows a similar absorption distribution, but with detections in fewer pixels.",
"Colored contours are at 15.0, 11.5, and 8.0 km s -1 \\mathrm {km}\\,\\mathrm {s}^{-1} in gray, blue, and cyan, respectively."
]
] | 1605.06031 |
[
[
"Sharpness of the phase transition for continuum percolation in R^2"
],
[
"Abstract We study the phase transition of random radii Poisson Boolean percolation: Around each point of a planar Poisson point process, we draw a disc of random radius, independently for each point.",
"The behavior of this process is well understood when the radii are uniformly bounded from above.",
"In this article, we investigate this process for unbounded (and possibly heavy tailed) radii distributions.",
"Under mild assumptions on the radius distribution, we show that both the vacant and occupied sets undergo a phase transition at the same critical parameter $\\lambda_c$.",
"Moreover, - For $\\lambda < \\lambda_c$, the vacant set has a unique unbounded connected component and we give precise bounds on the one-arm probability for the occupied set, depending on the radius distribution.",
"- At criticality, we establish the box-crossing property, implying that no unbounded component can be found, neither in the occupied nor the vacant sets.",
"We provide a polynomial decay for the probability of the one-arm events, under sharp conditions on the distribution of the radius.",
"- For $\\lambda > \\lambda_c$, the occupied set has a unique unbounded component and we prove that the one-arm probability for the vacant decays exponentially fast.",
"The techniques we develop in this article can be applied to other models such as the Poisson Voronoi and confetti percolation."
],
[
"Introduction",
"Percolation is the branch of probability theory focused on the study of the geometry and connectivity properties of random media.",
"Since its foundation in the 1950s, with the work of Broadbent and Hammersley [8], the area reached new heights over the decades to come, specially in two dimensions: During the 1980s following Kesten's determination of the critical threshold [23], and at the turn of the century with Schramm's introduction of Schramm-Loewner evolution [37] and Smirnov's proof of Cardy's formula [38].",
"This progress has been well documented in a range of books on the subject; see for instance [24], [18], [10] and [19].",
"Bernoulli percolation on a symmetric planar lattice, e.g.",
"on the square or triangular lattices, is a cornerstone within percolation theory.",
"Loved for its simple yet challenging structure, Bernoulli percolation has become quintessential in the study of phase transitions and other phenomena emanating from statistical mechanics and mathematical physics.",
"In the site-percolation version of this model, each vertex of the lattice is independently declared `open' with probability $p$ and `closed' with probability $1-p$ .",
"A random graph is obtained from the initial lattice by removing the closed vertices and the connected components of this graph are called clusters.",
"As the parameter increases the model undergoes a sharp phase transition.",
"More precisely, in this planar case, it is well-known that there exists a critical parameter $p_c$ , strictly in between zero and one, such that for $p<p_c$ , the probability of observing an open path from 0 to distance $n$ decays exponentially fast in $n$ ; at $p=p_c$ , there is no infinite open connected component, and the probability of an open path from 0 to distance $n$ decays polynomially fast in $n$ ; and for $p>p_c$ , there exists a unique infinite open cluster.",
"The phase transition is said to be `sharp' because of the abrupt change in the decay of the connection probabilities, from exponentially small in the subcritical regime to uniformly bounded in supercritical.",
"Proofs of the above properties can be found in [18] and [10].",
"In this paper we investigate the sharpness of the phase transition for Poisson Boolean percolation in ${\\mathbb {R}}^2$ , establishing results analogous to the ones described above for Bernoulli percolation.",
"Poisson Boolean percolation is an archetype for percolation in the continuum, and shares many features of Bernoulli percolation while posing significant additional challenges.",
"Apart from being continuous rather than discrete, these challenges come from its asymmetrical nature (the `open' and `closed' set have different properties) and long-range dependencies.",
"One particular strength of the present paper is its generality, and its structure has been oriented with this in mind.",
"Although our results are presented for Poisson Boolean percolation, they extend to many other percolation processes in ${\\mathbb {R}}^2$ , such as Poisson Voronoi and confetti percolation, as will be described below.",
"Together with an accompanying paper [6], our results give a precise description of the phase transition for Poisson Boolean percolation.",
"There, very specific properties of Poisson Boolean percolation will be exploited, as opposed to the robust methods developed here.",
"An important ingredient in establishing properties (i)–(iii) for Bernoulli percolation is what we call the dual process.",
"It can be defined by looking at the closed vertices on a modified graph, called the matching or dual graph.",
"This dual graph has a critical parameter $p_c^\\star $ which was proved to satisfy the duality relation $p_c + p_c^\\star = 1.$ This relation is especially useful in the study of self-similar processes, for which the dual process at $p$ coincides with the primal process at $1-p$ .",
"In this case it can be linked with the equality $p_c = p_c^\\star $ , yielding $p_c = 1/2$ , see more examples in Section .",
"Kesten's [23] original proof of the sharpness of the phase transition for Bernoulli percolation is based on the analysis of the crossing probabilities for rectangles; a rectangle is said to be crossed if there is a path from left to right, made out of open vertices.",
"The proof involves three essential ingredients: Finite-size criterion – if for some $n$ , the probability to cross a $n$ by $3n$ rectangle in the short direction is smaller than some small constant $\\theta >0$ , then the two-point connection probability decays exponentially fast.",
"On the other side, if for some $n$ , the probability to cross a $3n$ by $n$ rectangle in the long direction is larger than $1-\\theta >0$ , then there exists an infinite cluster almost surely, see [35], [24] and [2].",
"Russo-Seymour-Welsh theory – relates crossing probabilities for rectangle with different aspect ratios, see [34] and [39].",
"Threshold phenomenon – given a positive constant $c > 0$ and a large rectangle $K$ , there is only a small interval of parameters $p$ for which the crossing probability of $K$ remains between $c$ and $1 - c$ this interval is called the critical window.",
"The first proof of this sharp threshold is due to Kesten in [23] and is based on a geometric interpretation of the derivative of the crossing probability for a rectangle.",
"A better understanding for threshold phenomena has since been obtained in work like [36], [25], [41] and elsewhere.",
"The strategy described above is inherently two-dimensional.",
"As such, the finite-size criterion extends well beyond the product structure of the Bernoulli percolation measure, and require only a minimal assumption on the decay of long-range dependencies.",
"The original Russo-Seymour-Welsh techniques and sharp threshold results rely in a much stronger sense on the product structure of the Bernoulli measure, and do not extend easily to more general percolation models.",
"These will thus be the two foremost challenges in for the present study, in which we study the sharpness of the phase transition for continuum percolation in ${\\mathbb {R}}^2$ , which may present arbitrarily slow decay of dependencies."
],
[
"The phase transition for Poisson Boolean percolation",
"Poisson Boolean percolation was introduced by Gilbert in [15].",
"In this model we start with a Poisson point process on $\\mathbb {R}^2$ with intensity parameter $\\lambda > 0$ .",
"We then independently associate to each of these points a disc of a randomly chosen radius, according to a fixed probability measure $\\mu $ on ${\\mathbb {R}}_+$ .",
"The set $\\mathcal {O} \\subseteq \\mathbb {R}^2$ of points which are covered by at least one of the above discs is called the occupied set, while its complement $\\mathcal {V}:={\\mathbb {R}}^2\\setminus \\mathcal {O}$ is referred to as the vacant set.",
"We denote by ${\\mathbb {P}}_\\lambda $ the measure associated with this construction, and defer a more formal definition to Section .",
"Denote by $[0 \\overset{\\tiny \\mathcal {O}}{\\longleftrightarrow }\\partial B_r]$ the event that there exists an occupied path from 0 to distance $r$ .",
"We write $[0 \\overset{\\tiny \\mathcal {O}}{\\longleftrightarrow }\\infty ]$ if $[0 \\overset{\\tiny \\mathcal {O}}{\\longleftrightarrow }\\partial B_r]$ holds for every $r \\ge 1$ .",
"Analogously we define $[0 \\overset{ \\mathcal {V}}{\\longleftrightarrow }\\partial B_r]$ and $[0 \\overset{ \\mathcal {V}}{\\longleftrightarrow }\\infty ]$ for the corresponding events for the vacant set.",
"Finally, define the critical parameters $\\begin{array}{ll}{\\lambda _{c}}:= \\sup \\big \\lbrace \\lambda \\ge 0 \\; : \\; \\mathbb {P}_\\lambda \\big [0 \\overset{\\tiny \\mathcal {O}}{\\longleftrightarrow }\\infty \\big ] = 0 \\big \\rbrace ,&\\\\{\\lambda _{c}^\\star }:= \\inf \\big \\lbrace \\lambda \\ge 0 \\; : \\; \\mathbb {P}_\\lambda \\big [0 \\overset{ \\mathcal {V}}{\\longleftrightarrow }\\infty \\big ] = 0 \\big \\rbrace .&\\end{array}$ We aim to describe the percolative properties of the model at and around these critical values.",
"That $\\lambda _c$ and $\\lambda _c^\\star $ are finite may be obtained via a comparison with Bernoulli percolation on ${\\mathbb {Z}}^2$ , where a standard Peierls argument may be used.",
"To show, on the other hand, that $\\lambda _c$ and $\\lambda _c^\\star $ are strictly positive is a different matter, and requires a condition on the radii distribution in order to be true.",
"The most fundamental condition in the study of Poisson Boolean percolation in ${\\mathbb {R}}^2$ is that of finite second moment on the radius distribution: $\\int _0^\\infty x^2 \\mu (\\mathop {}\\!\\mathrm {d}x) < \\infty .$ It was observed by Hall [20] that (REF ) is necessary in order to avoid the entire plane to be almost surely covered, regardless of the intensity (as long as positive) of the Poisson point process.",
"Gouéré [16] further showed that this condition is also sufficient for $\\lambda _c$ to be strictly positive.",
"Lower bounds on $\\lambda _c^\\star $ may be obtained from their comparison with $\\lambda _c$ .",
"Under the assumption of bounded support on $\\mu $ it is known that $\\lambda _c^\\star =\\lambda _c$ ; see [32].",
"At the critical point $\\lambda _c$ it is further known that there is almost surely no unbounded occupied component in the case of $\\mu $ having bounded support [32], whereas the analogous statement for an unbounded vacant component is only known to hold in the case of unit radii, see [4].",
"Our goal with the present and forthcoming paper [6] is to extend these results to hold under the condition (REF ).",
"We will in this paper make no further assumption of the above results, as they will be easy consequences of the techniques we develop.",
"We merely acknowledge the observation that (REF ) is necessary in order for the model to present non-trivial behavior.",
"Following Kesten's lead, we shall in this paper focus on the study of crossings of rectangles and build a theory around them.",
"Quantitative estimates on the rate of decay of connection probabilities, in the different regimes, will be obtained as consequences of this.",
"The first result we state will thus relate crossing probabilities to the critical parameter $\\lambda _c$ .",
"Let us first define, for every $r, h \\ge 1$ , the event $\\operatorname{Cross}(r, h)$ that there exists an occupied path inside the rectangle $[0, r] \\times [0, h]$ from the left side to the right side.",
"That is, $\\operatorname{Cross}(r, h) = \\hbox{\\;[scale=1.5]{(0, 0) rectangle (6ex, 4ex);[solid] (0, 2ex) .. controls (2.6ex, 3ex) and (3.3ex, 1ex) .. (6ex, 2ex);(3ex, .2ex) node[anchor=north] {$\\scriptstyle r$};(5.7ex, 2ex) node[anchor=west] {$\\scriptstyle h$};}}$ .",
"We also write $\\operatorname{Cross}^\\star (r, h)$ for the existence of a vacant crossing of the same box.",
"The results of this paper are based on the following theorem.",
"Theorem 1.1 Assume the second moment condition (REF ).",
"Then $\\lambda _c$ is strictly between zero and infinity, and for all $\\lambda > {\\lambda _{c}}$ and all $\\kappa > 0$ , we have $\\lim _{r \\rightarrow \\infty } \\mathbb {P}_\\lambda \\big [ \\operatorname{Cross}(\\kappa r, r) \\big ] = 1.$ for $\\lambda = {\\lambda _{c}}$ and all $\\kappa > 0$ , there exists a constant $c_{\\ref {c:thm_sharp_cross}} = c_{\\ref {c:thm_sharp_cross}}(\\kappa )>0$ such that $c_{\\ref {c:thm_sharp_cross}} < \\mathbb {P}_{{\\lambda _{c}}} \\big [ \\operatorname{Cross}(\\kappa r, r) \\big ] < 1 - c_{\\ref {c:thm_sharp_cross}}, \\text{ for every $r \\ge 1$.",
"}$ for all $\\lambda < {\\lambda _{c}}$ and all $\\kappa > 0$ , $\\lim _{r \\rightarrow \\infty } \\mathbb {P}_\\lambda \\big [\\operatorname{Cross}(r,\\kappa r)\\big ] = 0.$ Moreover, at $\\lambda _c$ there is almost surely no unbounded cluster of either kind.",
"Remark 1 Our proof gives quantitative bounds for the rate of convergence in parts (i) and (iii).",
"The next two theorems are intended to illustrate what can be obtained using the techniques of the present work.",
"They give an overview of the percolative behavior of the vacant and occupied sets in Poisson Boolean percolation.",
"Several of these characteristics resemble the known features of Bernoulli percolation, and these two theorems are general: Similar theorems can be proved also for other percolation processes using the methods of this paper (see Section ).",
"However, their general nature calls for a slightly stronger moment condition than the one in (REF ).",
"These two alternative hypotheses are $\\int _0^\\infty x^2\\log x\\, \\mu (\\mathop {}\\!\\mathrm {d}x) < \\infty , \\text{ and}\\\\\\text{for some $\\alpha > 0$,} \\quad \\int _0^\\infty x^{2 + \\alpha } \\,\\mu (\\mathop {}\\!\\mathrm {d}x) < \\infty .$ The three hypotheses (REF ), (REF ) and () will be useful because they imply some decorrelation inequalities discussed in Section REF .",
"The natural second moment assumption (REF ) already implies some spatial decorrelation properties, and the hypotheses (REF ) and () imply quantitative bounds on the spatial decorrelation.",
"The hypothesis (REF ) will be sufficient for most of the paper, but the hypotheses (REF ) and () will be useful to apply some renormalization methods presented in Section .",
"For the vacant set we have the following.",
"Theorem 1.2 Assume the $2+\\log $ moment condition (REF ).",
"Then $\\lambda _c^\\star =\\lambda _c$ and is thus strictly between zero and infinity.",
"Moreover, for $\\lambda = {\\lambda _{c}^\\star }$ there exists a constant $c_{\\ref {c:poly_decay}} > 0$ such that $\\mathbb {P}_{\\lambda _c^\\star }\\big [ 0 \\overset{ \\mathcal {V}}{\\longleftrightarrow }\\partial B_r \\big ] \\le \\frac{1}{c_{\\ref {c:poly_decay}}} r^{-c_{\\ref {c:poly_decay}}}.$ for all $\\lambda > {\\lambda _{c}^\\star }$ there exists a constant $c_{\\ref {c:exp_decay}} = c_{\\ref {c:exp_decay}}(\\lambda ) > 0$ such that $\\mathbb {P}_\\lambda \\big [ 0 \\overset{ \\mathcal {V}}{\\longleftrightarrow }\\partial B_r \\big ] \\le \\frac{1}{c_{\\ref {c:exp_decay}}}\\exp \\lbrace -c_{\\ref {c:exp_decay}}r\\rbrace .$ Remark 2 Condition (REF ) in Theorem REF is not sharp.",
"In the forthcoming paper [6], we show that the second moment condition suffices to prove all of the above.",
"The fact that ${\\lambda _{c}^\\star }= {\\lambda _{c}}$ is analogous to $p_c + p_c^\\star = 1$ for Bernoulli percolation.",
"There are important distinctions between the vacant and the occupied sets in regard to their percolation properties.",
"An important difference comes from the fact that parts (REF ) and (REF ) of Theorem REF do not hold in general for the occupied set, since the existence of long occupied connections can be triggered by a single disk of large radius.",
"For example, if one chooses the radius distribution $\\mu $ such that $\\mu [x, \\infty ) = x^{-2}(\\log x)^{-2}$ for $x$ large enough, then the hypothesis (REF ) is satisfied but for every $\\lambda > 0$ and every $r$ large enough ${\\mathbb {P}}_\\lambda \\big [0 \\overset{\\tiny \\mathcal {O}}{\\longleftrightarrow }\\partial B_r\\big ] \\,\\ge \\, c \\cdot \\lambda \\int _r^\\infty x \\,\\mu [x,\\infty )\\mathop {}\\!\\mathrm {d}x \\,\\ge \\, c \\cdot \\lambda /\\log r.$ Therefore, in this case the one-arm probability ${\\mathbb {P}}_\\lambda [0 \\overset{\\tiny \\mathcal {O}}{\\longleftrightarrow }\\partial B_r]$ cannot decay polynomially fast at criticality, just because this choice of $\\mu $ implies that the probability that 0 is contained in a ball of radius larger than $r$ does not decay polynomially.",
"Nevertheless, under the stronger moment assumption (), one can prove the polynomial decay of the arm exponent as shown in the next result.",
"Theorem 1.3 Assume that the $2+\\alpha $ moment condition () holds for some $\\alpha >0$ .",
"Then, for $\\lambda = {\\lambda _{c}}$ , there exists a constant $c_{\\ref {c:poly_o_decay}} = c_{\\ref {c:poly_o_decay}}(\\alpha ) > 0$ such that $\\mathbb {P}_{\\lambda _c}\\big [ 0 \\overset{\\tiny \\mathcal {O}}{\\longleftrightarrow }\\partial B_r \\big ] \\le \\frac{1}{c_{\\ref {c:poly_o_decay}}} r^{-c_{\\ref {c:poly_o_decay}}}.$ for all $\\lambda < {\\lambda _{c}}$ , there exists a constant $c_{\\ref {c:poly_decay}}=c_{\\ref {c:poly_decay}}(\\lambda ) > 0$ such that $\\mathbb {P}_\\lambda \\big [ 0 \\overset{\\tiny \\mathcal {O}}{\\longleftrightarrow }\\partial B_r \\big ] \\le c_{\\ref {c:poly_decay}} r^{-\\alpha }.$ Remark 3 The exponent in part (ii) of Theorem REF cannot be improved in general.",
"To see this, consider the radii distribution for which $\\mu [x,\\infty )=x^{-(2+\\alpha )}(\\log x)^{-2}$ for large $x$ .",
"This distribution satisfies (), but ${\\mathbb {P}}_\\lambda \\big [ 0 \\overset{\\tiny \\mathcal {O}}{\\longleftrightarrow }\\partial B_r \\big ]\\,\\ge \\,\\lambda r^2\\mu [2r,\\infty )\\,=\\,\\lambda (2r)^{-\\alpha }(\\log 2r)^{-2}$ for large $r$ .",
"We will now describe the main steps in the proof of the above results, which roughly speaking correspond to the three steps described for the case of Bernoulli percolation above.",
"These are finite-size criterion, Russo-Seymour-Welsh theory and a threshold phenomenon.",
"We describe each of these steps below and emphasize that their corresponding proofs can be read independently of one another."
],
[
"Finite-size criterion",
"In Section we prove that the crossing probabilities converge to 1 as soon as they become close enough to 1.",
"Roughly speaking we prove that there exists $\\theta >0$ and $r_0(\\lambda )$ such that the following are equivalent $\\mathbb {P}_\\lambda \\big [\\operatorname{Cross}(3r, r)\\big ] > 1 - \\theta \\text{ for some $r\\ge r_0$,}\\\\\\lim _{r \\rightarrow \\infty } \\mathbb {P}_\\lambda \\big [\\operatorname{Cross}(3r, r)\\big ] = 1,$ and the analogous statement holds for vacant crossings, see Proposition REF .",
"This motivates us to introduce $\\lambda _0 := \\sup \\big \\lbrace \\lambda \\ge 0 : \\lim _{r\\rightarrow \\infty } {\\mathbb {P}}_\\lambda \\big [ \\operatorname{Cross}(r, 3r) \\big ] = 0\\big \\rbrace ,\\\\\\lambda _1 := \\inf \\big \\lbrace \\lambda \\ge 0 : \\lim _{r\\rightarrow \\infty } {\\mathbb {P}}_\\lambda \\big [ \\operatorname{Cross}(3r, r) \\big ] = 1\\big \\rbrace .$ The first important consequence of the finite-size criterion is that $0\\,<\\,\\lambda _0\\,\\le \\,\\lambda _c\\,\\le \\, \\lambda _1\\, < \\,\\infty .$ These results are proved in Section .",
"Up to now, it is not clear that $\\lambda _0$ is actually equal to $\\lambda _1$ .",
"We call $[\\lambda _0, \\lambda _1]$ the critical regime and we can see that, for every $\\lambda \\in [\\lambda _0, \\lambda _1]$ , $\\inf _{r\\ge 1} {\\mathbb {P}}_\\lambda \\big [ \\operatorname{Cross}(r, 3r) \\big ] > 0 \\quad \\text{and} \\quad \\inf _{r\\ge 1} {\\mathbb {P}}_\\lambda \\big [ \\operatorname{Cross}^\\star (r, 3r) \\big ] > 0.$ These statements say that in the critical regime, rectangles have non-degenerate probabilities of being crossed by both the vacant and the occupied set.",
"However, one should notice that these crossings happen in the easy direction of the rectangles.",
"In order to obtain a more precise description of the behavior in the critical regime we require similar statements regarding crossing probabilities in the long direction.",
"This is precisely the purpose of Russo-Seymour-Welsh theory, as we describe next."
],
[
"Russo-Seymour-Welsh theory",
"In this step we show that (REF ) implies that $\\inf _{r\\ge 1} {\\mathbb {P}}_\\lambda \\big [ \\operatorname{Cross}(3 r, r) \\big ] > 0 \\quad \\text{and} \\quad \\inf _{r\\ge 1} {\\mathbb {P}}_\\lambda \\big [ \\operatorname{Cross}^\\star (3 r, r) \\big ] > 0.$ Note that the only difference between the above and (REF ) is that now the rectangles are crossed in the long direction.",
"This `box-crossing' property for the critical regime rules out the existence of unbounded clusters and provides bounds on arm probabilities.",
"These results are proved in Section .",
"Remark 4 RSW bounds were obtained separately for the occupied and vacant regimes during the 1990s by Roy [32] and Alexander [4].",
"However, these proofs are technical, and pose pose heavy restrictions on the radii distribution.",
"Recently, Tassion [42] found an argument allowing for greater generality.",
"His argument, presented in the setting of Voronoi percolation but extends verbatim, shows that $ \\liminf _{r\\rightarrow \\infty }{\\mathbb {P}}_\\lambda \\big [\\operatorname{Cross}(r,r)\\big ]>0\\quad \\Rightarrow \\quad \\liminf _{r\\rightarrow \\infty }{\\mathbb {P}}_\\lambda \\big [\\operatorname{Cross}(3r,r)\\big ]>0.$ While this is all that is needed in `symmetric' percolation models, such as Poisson Voronoi percolation, it is indispensable for `non-symmetric' models, which we consider here, that crossings of rectangles the easy way imply crossings also the hard way.",
"This extension is not straightforward."
],
[
"Sharp thresholds",
"The final step of the proof is to show that $\\lambda _0 = \\lambda _1$ .",
"The main strategy here is to show that the occupied crossing probabilities grow very fast from zero to one as we increase the density $\\lambda $ of the system.",
"There is a solid theory that predicts the occurrence of threshold phenomena in the setting of Boolean function on the discrete cube $\\lbrace 0, 1\\rbrace ^n$ equipped with a product structure.",
"Poisson point processes is the natural analogue to the discrete Bernoulli measure.",
"It is therefore natural to believe that these techniques should carry over also to the Poissonian continuum setting.",
"This is indeed the case, although doing so is not straightforward task.",
"We will here follow an approach introduced in [1].",
"This part of the argument involves the analysis of Boolean functions and is the least general part of our argument, since it involves representing the crossing probabilities in our Poisson point process as a certain function of discrete Bernoulli random variables.",
"This construction and the proof that $\\lambda _0 = \\lambda _1$ can be found in Section ."
],
[
"Other examples",
"We have chosen Poisson Boolean percolation as the main example to illustrate the techniques of this work.",
"However, the techniques that we develop apply in greater generality.",
"We emphasize this fact in Section , where we apply these techniques to Poisson Voronoi and confetti percolation.",
"We direct the reader to that section for a statement of the results we obtain for these models, and give here just a short description of the scope of our techniques.",
"The finite-size criterion described above only uses the fact that the law of our percolation process is invariant under translations and right-angle rotations, see (REF ), and that it satisfies a mixing condition, see (REF ).",
"The Russo-Seymour-Welsh part of our argument, we solely use the invariance of $\\mathcal {O}$ and $\\mathcal {V}$ under translations, right-angle rotations and reflections in coordinate axes, see (REF ), the mixing property (REF ), the FKG inequality (REF ) and a certain continuity property of the crossing probabilities stated in Lemma REF .",
"The threshold argument is the least general and uses, in addition to the assumptions made above, the fact that process is based on a Poisson point process."
],
[
"Some previous works on the model",
"The reference book [27] provides a general exposition of continuum percolation.",
"The case of uniformly bounded radius distribution has been extensively studied in [44], [45], [29], [32], [28], [4].",
"These works have already established much of our results in this restricted setting.",
"In Lemma A.2 of [17], it is proved that for sufficiently small $\\lambda > 0$ , $\\mathbb {P}_\\lambda [B_r \\leftrightarrow B_{2r}^c]$ goes to zero with $r$ under the second moment condition (REF ).",
"This implies that $\\lambda _0 > 0$ under the second moment condition.",
"In [33], Roy studied a Poisson soup of bounded sticks in the plane, proving a Russo-Seymour-Welsh theorem and establishing that ${\\lambda _{c}^\\star }= {\\lambda _{c}}$ .",
"Uniqueness of the unbounded components (for both vacant and occupied regions) has been established in [26]."
],
[
"Organization of the paper",
"In Section we provide some notation and preliminary results that are needed throughout the text.",
"Sections , and present the three steps of the proof that were described in the introduction, namely: the finite-size criterion, Russo-Seymour-Welsh result and the sharp threshold step.",
"These sections can be read independently of one another.",
"Section proves the continuity of the critical parameter with respect to the law $\\mu $ of the radius distribution.",
"The main results of the paper are then proved in Section .",
"We treat other models and provide some open questions in Sections and .",
"Acknowledgements This work began during a visit of V.T.",
"to IMPA, that he thanks for support and hospitality.",
"We thank the Centre Intradisciplinaire Bernoulli (CIB) and Stardû for hosting the authors.",
"D.A.",
"was during the course of this project financed by grant 637-2013-7302 from the Swedish Research Council.",
"A.T. is grateful to CNPq for its financial contribution to this work through the grants 306348/2012-8, 478577/2012-5 and 309356/2015-6 and FAPERJ through grant number 202.231/2015.",
"V.T.",
"acknowledges support from the Swiss NSF."
],
[
"Notation and preliminary results",
"Throughout the text we let $c$ denote positive constants which may depend on the radius distribution and may change from line to line.",
"However, numbered constants such as $c_0, c_1, \\dots $ refer to their first appearance in the text.",
"We will further write $B^\\infty (r) = [-r, r]^2$ for the ball centered at the origin in the supremum norm and let $B(x,r)$ , on the other hand, denote the closed Euclidean ball with center $x$ and radius $r$ .",
"When $x$ is the origin we omit it from the above notation.",
"Rather informally, a realization of Poisson Boolean percolation is obtained by decorating the points in a Poisson point process in ${\\mathbb {R}}^2$ by Euclidean discs with independent radii sampled from some distribution.",
"There are various (equivalent) ways of making this description formal.",
"We will start this section by describing one way which will be suitable for our purposes."
],
[
"Definition of the process",
"To define our process, let us first introduce a Poisson point process on the following space of point measures $\\Omega = \\Big \\lbrace \\omega = \\sum _i \\delta _{(x_i,z_i)}: (x_i,z_i) \\in \\mathbb {R}^2 \\times \\mathbb {R}_+ \\text{ and } \\omega \\big ( K \\times \\mathbb {R}_+ \\big ) < \\infty \\text{ for all $K$ compact} \\Big \\rbrace .$ We endow this space with the $\\sigma $ -algebra $\\mathcal {M}$ generated by the evaluation maps $A \\mapsto \\omega (A)$ , for $A \\in \\mathcal {B}(\\mathbb {R}^2 \\times \\mathbb {R}_+)$ , the Borel sets on ${\\mathbb {R}}^2 \\times \\mathbb {R}_+$ .",
"We next fix an intensity parameter $\\lambda \\ge 0$ and some probability measure $\\mu $ on $\\mathbb {R}_+$ which will give the radius distribution of our discs.",
"We can now define on $(\\Omega , \\mathcal {M})$ a Poisson point process with intensity $\\lambda \\cdot dx \\, \\mu (dz)$ , i.e., Lebesgue measure on $\\mathbb {R}^2$ product with $\\mu $ and multiplied by $\\lambda \\ge 0$ .",
"The law of this process is denoted by $\\mathbb {P}_\\lambda $ throughout the text and we complete the $\\sigma $ -algebra $\\mathcal {M}$ with respect to $\\mathbb {P}_\\lambda $ .",
"For each point $(x,z) \\in \\mathbb {R}^2 \\times \\mathbb {R}_+$ in the support of the measure $\\omega \\in \\Omega $ , we associate the disc $B(x, z)$ and the occupied region of the plane is consequently given by $\\mathcal {O} := \\bigcup _{(x,z) \\in \\operatorname{supp}(\\omega )} B(x, z),$ while the vacant set is given by its complement $\\mathcal {V} := \\mathbb {R}^2 \\setminus \\mathcal {O}$ .",
"(Below, we will often identify $\\omega $ with its support in order to ease the notation.)",
"From the definition of the model, it is trivial to conclude that the law of $\\mathcal {O}$ is invariant under translations, right-angle rotations, and reflection in coordinate axes.",
"Although the law is invariant under arbitrary rotations, and reflections in axes with other orientation, we will only use the above statement throughout the text.",
"There is a natural partial ordering of elements in $\\Omega $ , namely, $\\omega \\le \\omega ^{\\prime }$ if $\\omega (A) \\le \\omega ^{\\prime }(A)$ for all $A \\in \\mathcal {B}(\\mathbb {R}^2 \\times \\mathbb {R}_+)$ .",
"An event $A\\in \\mathcal {M}$ is said to be increasing if $\\omega \\in A$ implies that $\\omega ^{\\prime }\\in A$ for all $\\omega \\le \\omega ^{\\prime }$ .",
"It is decreasing if its complement is increasing.",
"A useful property of increasing events is that they are positively correlated.",
"The following proposition, known as the FKG inequality, was proved by Roy in his doctorate thesis; see also [27]: If $A_1$ and $A_2$ are increasing events, then ${\\mathbb {P}}_\\lambda (A_1\\cap A_2)\\,\\ge \\,{\\mathbb {P}}_\\lambda (A_1){\\mathbb {P}}_\\lambda (A_2).$ The above also holds when $A_1$ and $A_2$ are decreasing events.",
"We will also use the following standard consequence of the FKG inequality, referred to as the square-root trick.",
"Let $A_1,\\ldots ,A_k$ be $k$ increasing events (or $k$ decreasing events), then $\\max _{1\\le i \\le k}{\\mathbb {P}}_\\lambda (A_i)\\,\\ge \\, 1-\\big [1-{\\mathbb {P}}_\\lambda (A_1\\cup \\ldots \\cup A_k)\\big ]^{1/k}.$"
],
[
"Crossing events",
"Throughout the text we will often deal with crossing events of various types.",
"In particular, we will be interested in the following general definition.",
"Given subsets $A_1, A_2$ and $\\mathcal {C}$ of $\\mathbb {R}^2$ let $A_1 \\overset{\\mathcal {C}}{\\longleftrightarrow }A_2 := \\text{there is a path in $\\mathcal {C}$ connecting $A_1$ to $A_2$}.$ We are now in position to give a formal definition of the crossing event $\\operatorname{Cross}(r,h)$ , for $r, h > 0$ , as $\\operatorname{Cross}(r, h) := \\big [ A_1 {\\overset{\\mathcal {O} \\cap K}{\\longleftrightarrow }} A_2 \\big ],$ where $K$ denotes the box $[0,r] \\times [0,h]$ and $A_1 = \\lbrace 0\\rbrace \\times [0,h]$ , $A_2 = \\lbrace r\\rbrace \\times [0,h]$ stand for its left and right sides.",
"We also define the event corresponding to a vacant crossing as $\\operatorname{Cross}^\\star (r,h) := [A_1 {\\overset{\\mathcal {V} \\cap K}{\\longleftrightarrow }} A_2 ]$ .",
"It is rather straightforward to verify that events of this type are measurable.",
"Notice that occupied crossing events are increasing, and vacant crossing events are decreasing."
],
[
"Decay of spatial correlations",
"The second moment condition given in (REF ) is sufficient to imply a spatial decorrelation, in the sense of the function $\\rho _\\lambda $ introduced in Definition REF below.",
"First set $Y_v = \\operatorname{{\\bf 1}}_{\\lbrace v \\in \\mathcal {V}\\rbrace }$ for $v \\in \\mathbb {R}^2$ .",
"Definition 2.1 Given $0 < r, s < \\infty $ , let $\\rho _\\lambda (r, s) := \\sup _{f_1, f_2} \\operatorname{Cov}\\big (f_1(\\mathcal {O}), f_2(\\mathcal {O})\\big ),$ where the above suppremum is taken over all functions $f_1,f_2:\\mathcal {P}(\\mathbb {R}^2)\\rightarrow [-1,1] $ such that $f_1(\\mathcal {O}) \\in \\sigma (Y_v; v\\in B^\\infty (r))$ and $f_2(\\mathcal {O}) \\in \\sigma (Y_v; v \\notin B^\\infty (r+s))$ .",
"The function $\\rho _\\lambda $ has a nice geometric interpretation.",
"Namely, one can observe that $\\rho _\\lambda (r, r+s)$ is directly related to the probability that there exists one big occupied disk crossing the annulus $B^\\infty (r+s)\\setminus B^\\infty (r)$ .",
"By making this observation rigorous, and computing this crossing probability (see Lemma REF ), we prove the following upper bound.",
"Proposition 2.2 For any $\\lambda > 0$ and $r, s \\ge 1$ , we have $\\rho _\\lambda (r, s)\\, \\le \\, c_{\\ref {c:decorrelation}} \\lambda \\left(1+\\frac{r}{s}\\right)^2\\int _{s/2}^\\infty x^2\\mu (\\mathop {}\\!\\mathrm {d}x).$ In particular, the second moment condition (REF ) implies that $\\text{for any $\\kappa > 0$, }\\lim _{r\\rightarrow \\infty } \\rho _\\lambda (r, \\kappa r) = 0.$ and the $2 + \\alpha $ moment condition in () implies that for every $\\lambda > 0$ and $\\varepsilon >0$ $\\sup \\limits _{r\\ge 1, \\kappa \\ge \\varepsilon } r^\\alpha \\rho _\\lambda (r, \\kappa r) < \\infty .$ For $K \\subseteq \\mathbb {R}^2$ , we write $\\mathcal {O}_K = \\bigcup _{(x,z) \\in \\omega :\\, x \\in K} B(x, z).$ The next lemma says that deleting balls from $\\mathcal {O}$ which are far from a given box do not alter the configuration of $\\mathcal {O}$ inside that box.",
"Figure: An illustration of the sets 𝒪\\mathcal {O}, 𝒪 B ∞ (r+s/2) \\mathcal {O}_{B^\\infty (r + s/2)} and 𝒪 B ∞ (r+s/2) c \\mathcal {O}_{B^\\infty (r + s/2)^c} respectively.",
"Note that in this particular realization we have 𝒪 B ∞ (r+s/2) ∩B ∞ (r)=𝒪∩B ∞ (r)\\mathcal {O}_{B^\\infty (r + s/2)} \\cap B^\\infty (r) = \\mathcal {O} \\cap B^\\infty (r) and also that 𝒪 B ∞ (r+s/2) c ∩B ∞ (r+s) c =𝒪∩B ∞ (r+s) c \\mathcal {O}_{B^\\infty (r + s/2)^c} \\cap B^\\infty (r + s)^c = \\mathcal {O} \\cap B^\\infty (r + s)^c as in Lemma .Lemma 2.3 For all $r, s > 0$ , writing $K_1 = B^\\infty (r)$ and $K_2 = B^\\infty (r + s)$ , we have $\\mathbb {P}_\\lambda \\big ( \\mathcal {O}_{K_2^c} \\cap K_1 \\ne \\varnothing \\big )\\le 8\\lambda \\left(1+\\frac{r+1}{s}\\right)^2\\int _s^\\infty x^2\\mu (\\mathop {}\\!\\mathrm {d}x).$ Also, $\\mathbb {P}_\\lambda \\big ( \\mathcal {O}_{K_1} \\cap K_2^c \\ne \\varnothing \\big )\\le 8\\lambda \\left(\\frac{r}{s}\\right)^2\\int _s^\\infty x^2\\mu (\\mathop {}\\!\\mathrm {d}x).$ Remark 5 The above lemma implies in particular that, whenever $\\int _0^\\infty x^2\\mu (\\mathop {}\\!\\mathrm {d}x) < \\infty $ , if $K \\subseteq \\mathbb {R}^2$ is compact, then $\\mathbb {P}_\\lambda $ -almost surely there are only finitely many balls in $\\mathcal {O}$ that intersect $K$ .",
"We first derive the second bound.",
"Since $K_1$ has area $8r^2$ , an immediate estimate of the expected number of points in $K_1$ with radius at least $s$ gives that $\\mathbb {P}_\\lambda \\big ( \\mathcal {O}_{K_1} \\cap K_2^c \\ne \\varnothing \\big ) \\,\\le \\, 8r^2\\lambda \\int _s^\\infty \\mu (\\mathop {}\\!\\mathrm {d}x) \\,\\le \\, 8\\lambda \\left(\\frac{r}{s}\\right)^2\\int _s^\\infty x^2\\mu (\\mathop {}\\!\\mathrm {d}x),$ hence establishing (REF ).",
"For the first bound, we split the complement of $K_2 =B^\\infty (r+s)$ into the disjoint annuli $A_i = B^\\infty (r + s + i + 1) \\setminus B^\\infty (r + s + i)$ and write $K_2^c = \\bigcup _{i\\ge 0} A_i$ .",
"Using the fact that the area of $A_i$ is bounded by $8(r+s+i+1)$ , we can now estimate $\\begin{split}\\mathbb {P}_\\lambda \\big ( \\mathcal {O}_{K_2^c} \\cap K_1 \\ne \\varnothing \\big )& \\,\\le \\, \\sum _{i\\ge 0} \\mathbb {P}_\\lambda \\big ( \\mathcal {O}_{A_i} \\cap K_1 \\ne \\varnothing \\big )\\\\& \\le \\, 8\\lambda \\sum _{i\\ge 0} (r + s + i+1)\\int _{s+i}^\\infty \\mu (\\mathop {}\\!\\mathrm {d}x).\\end{split}$ Exchanging the order of summation yields the bound $8\\lambda \\int _s^\\infty \\sum _{i=0}^{x-s}(r+s+i+1)\\mu (\\mathop {}\\!\\mathrm {d}x) \\,\\le \\, 8\\lambda \\int _s^\\infty (r+x+1)^2\\mu (\\mathop {}\\!\\mathrm {d}x),$ from which (REF ) follows.",
"Let us now prove Proposition REF .",
"[Proof of Proposition REF ] Recall that $\\Vert f_1 \\Vert _\\infty , \\Vert f_2 \\Vert _\\infty \\le 1$ .",
"Let $K=B^\\infty (r+s/2)$ .",
"By the Poissonian character of the law $\\mathbb {P}_\\lambda $ , the two random variables $ f_1(\\mathcal {O}_K)$ and $f_2(\\mathcal {O}_{K^c})$ are independent.",
"Therefore, $\\begin{split}\\operatorname{Cov}(f_1(\\mathcal {O}), f_2(\\mathcal {O}) & \\,\\le \\, \\mathbb {P}_\\lambda \\big (f_1(\\mathcal {O}) \\ne f_1(\\mathcal {O}_K)\\big ) + \\mathbb {P}_\\lambda \\big (f_2(\\mathcal {O})\\ne f_2(\\mathcal {O}_{K^c})\\big )\\\\& \\le \\, \\mathbb {P}_\\lambda \\big ( \\mathcal {O}_{K^c} \\cap B^\\infty (r)\\ne \\varnothing \\big ) + \\mathbb {P}_\\lambda \\big ( \\mathcal {O}_K \\cap B^\\infty (r+s)^c\\ne \\varnothing \\big ),\\end{split}$ and the proof then follows from Lemma REF .",
"For the last conclusion (REF ), one may use Markov's inequality."
],
[
"An alternative notion of decoupling",
"Note that Definition REF is symmetric for $\\mathcal {O}$ and $\\mathcal {V}$ , in the sense that the roles of the occupied and vacant sets and are interchangeable.",
"In what follows, we will introduce another notion of decoupling that will not be symmetric for $\\mathcal {O}$ and $\\mathcal {V}$ .",
"This distinction will be very relevant to explain the different behavior of these sets, see Remark REF , and explains why we obtain Theorem REF under (REF ) but require (REF ) for Theorem REF with the techniques used here.",
"To motivate this new notion, observe from Definition REF that $\\mathbb {E}_\\lambda \\big (f_1(\\mathcal {O}) f_2(\\mathcal {O})\\big ) \\,\\le \\,\\mathbb {E}_\\lambda \\big (f_1(\\mathcal {O})\\big ) \\mathbb {E}_\\lambda \\big (f_2(\\mathcal {O})\\big ) + \\rho (r, s),$ for every functions $f_1,f_2:\\mathcal {P}(\\mathbb {R}^2)\\rightarrow [0,1]$ such that $f_1(\\mathcal {O}) \\in \\sigma (Y_v; v \\in B^\\infty (r))$ and $f_2(\\mathcal {O}) \\in \\sigma (Y_v; v \\notin B^\\infty (r + s))$ .",
"This will be used in several parts of the text.",
"However, some results will be stronger using the following modified version of the above.",
"Definition 2.4 Let $\\bar{\\rho }_\\lambda :\\mathbb {R}_+^2 \\rightarrow \\mathbb {R}_+$ be defined as the smallest value such that $\\mathbb {E}_\\lambda \\big (f_1(\\mathcal {O}) f_2(\\mathcal {O})\\big ) \\,\\le \\,\\mathbb {E}_\\lambda \\big (f_1(\\mathcal {O})\\big ) \\Big ( \\mathbb {E}_\\lambda \\big (f_2(\\mathcal {O})\\big ) + \\bar{\\rho }_\\lambda (r, s) \\Big ),$ for all decreasing functions $f_1, f_2:\\mathcal {P}(\\mathbb {R}^2)\\rightarrow [0,1]$ satisfying $f_1(\\mathcal {O}) \\in \\sigma (Y_v; v \\in B^\\infty (r))$ , $f_2(\\mathcal {O}) \\in \\sigma (Y_v; v \\notin B^\\infty (r + s))$ .",
"(A function $f:\\mathcal {P}(\\mathbb {R}^2) \\rightarrow [0,1]$ is said to be decreasing if $f(A)\\ge f(B)$ whenever $A\\subset B$ .)",
"Remark 6 Note that the error $\\bar{\\rho }_\\lambda $ in the above definition is being multiplied by $\\mathbb {E}_\\lambda (f_1)$ .",
"This represents an important improvement in some cases, as will become clear later for instance when we compare (REF ) with (REF ).",
"Although the error term in Definition REF is smaller than the one in (REF ), the bound (REF ) can only be used when $f_1$ and $f_2$ are decreasing functions.",
"This restriction was intentionally introduced in the definition of $\\bar{\\rho }_\\lambda $ and it is necessary in our proof that $\\bar{\\rho }_\\lambda $ vanishes, see Proposition REF below.",
"This restriction reflects the distinct behavior that the vacant and occupied sets display at criticality, see Remark REF .",
"We now prove an analogue of Proposition REF .",
"Proposition 2.5 For any $\\lambda > 0$ and $r, s \\ge 1$ we have $\\bar{\\rho }_\\lambda (r, s) \\,\\le \\, c_{\\ref {c:decorrelation}} \\lambda \\left(1+\\frac{r}{s}\\right)^2\\int _{s/2}^\\infty x^2\\mu (\\mathop {}\\!\\mathrm {d}x).$ In particular, if $\\int _0^\\infty x^2 \\mu (\\mathop {}\\!\\mathrm {d}x) < \\infty $ , then $\\text{for any } \\kappa > 0,\\, \\lim _{r\\rightarrow \\infty } \\bar{\\rho }_\\lambda (r, \\kappa r) = 0.$ We write $K_1 = B^\\infty (r)$ and $K_2 = B^\\infty (r + s)^c$ and define $\\check{\\mathcal {O}} = \\bigcup _{\\begin{array}{c}(x,z) \\in \\operatorname{supp}(\\omega ); \\\\ B(x, z) \\cap K_1 = \\varnothing \\end{array}} B(x, z).$ Note that this set is contained in $\\mathcal {O}$ and independent of $f_1(\\mathcal {O})$ .",
"Using the fact that $f_2$ is decreasing, we can write $\\begin{split}\\mathbb {E}_\\lambda \\big ( f_1(\\mathcal {O}) f_2(\\mathcal {O}) \\big ) & \\,\\le \\, \\mathbb {E}_\\lambda \\big ( f_1(\\mathcal {O}) f_2(\\check{\\mathcal {O}}) \\big ) \\,\\le \\, \\mathbb {E}_\\lambda \\big ( f_1(\\mathcal {O}) \\big ) \\mathbb {E}_\\lambda \\big ( f_2(\\check{\\mathcal {O}}) \\big )\\\\& \\le \\, \\mathbb {E}_\\lambda \\big ( f_1(\\mathcal {O}) \\big ) \\Big ( \\mathbb {E}_\\lambda \\big ( f_2(\\mathcal {O}) \\big ) + \\mathbb {P}_\\lambda \\big ( \\check{\\mathcal {O}} \\cap K_2 \\ne \\mathcal {O} \\cap K_2 \\big ) \\Big ),\\end{split}$ where we have used the fact that $f_1,f_2\\in [0,1]$ .",
"In view of the definition of $\\bar{\\rho }_\\lambda $ , all we need to do is to bound the probability above, $\\begin{split}\\mathbb {P}_\\lambda (\\check{\\mathcal {O}} & \\cap K_2 \\ne \\mathcal {O} \\cap K_2) \\,\\le \\, \\mathbb {P}_\\lambda \\Big (\\begin{array}{c}\\text{there is $(x,z) \\in \\operatorname{supp}(\\omega )$ such that}\\\\\\text{$B^\\infty (x,z)$ touches $K_1$ and $K_2$}\\end{array}\\Big )\\\\& \\le \\, \\mathbb {P}_\\lambda \\big (\\mathcal {O}_{B^\\infty (r + s/2)} \\cap K_2 \\ne \\varnothing \\big ) + \\mathbb {P}_\\lambda \\big ( \\mathcal {O}_{B^\\infty (r + s/2)^2} \\cap K_1 \\ne \\varnothing \\big ).\\end{split}$ The proof now ends in the same way as the proof of Proposition REF ."
],
[
"Local continuity",
"Another rather straightforward consequence of the Poissonian nature of the model is the following `local continuity' property.",
"In essence this property says that locally the topological properties of the set $\\mathcal {O}$ are unlikely to be affected by slight change in the radii of the discs.",
"Given a set $A\\subseteq {\\mathbb {R}}^2$ we define its $\\varepsilon $ -interior and $\\varepsilon $ -closure, for $\\varepsilon >0$ , as follows: $\\begin{aligned}\\operatorname{int}(A,\\varepsilon )&:=\\lbrace x\\in {\\mathbb {R}}^2:B(x,\\varepsilon )\\subseteq A\\rbrace ,\\\\\\operatorname{cl}(A,\\varepsilon )&:=\\lbrace x\\in {\\mathbb {R}}^2:B(x,\\varepsilon )\\cap A\\ne \\varnothing \\rbrace .\\end{aligned}$ We omit the proof of the following proposition.",
"Proposition 2.6 Let $K \\subseteq {\\mathbb {R}}^2$ be compact and convex and $A_1, A_2 \\subseteq K$ closed.",
"For every $\\lambda > 0$ $\\lim _{\\varepsilon \\rightarrow 0}{\\mathbb {P}}_\\lambda \\Big ( A_1 {\\overset{\\operatorname{cl}(\\mathcal {O},\\varepsilon ) \\cap K}{\\longleftrightarrow }} A_2\\text{ occurs but not }A_1 {\\overset{\\operatorname{int}(\\mathcal {O},\\varepsilon ) \\cap K}{\\longleftrightarrow }} A_2 \\Big ) \\,=\\,0.$ A consequence of the above proposition that will be used later is the following lemma.",
"Consider a variation of the standard crossing event defined as $\\hbox{[scale=1.5]{(0, 0) rectangle (4ex, 4ex);(3.8ex, 2ex) -- (4.2ex, 2ex);(0, 1.7ex) .. controls (1.3ex, 4ex) and (2.6ex, 1.4ex) .. (4ex, 3.2ex);(2ex, .2ex) node[anchor=north] {$\\scriptstyle r$};(.4ex, 2ex) node[anchor=east] {$\\scriptstyle h$};[decorate,decoration={brace,amplitude=.4ex,raise=.4ex},yshift=0pt] (4ex, 4ex) -- (4ex, 2ex);(4.3ex, 3ex) node[anchor=west] {$\\scriptstyle y$};}\\;}$ := [ A1 O K A2 ], where $K$ is again the box $[0,r] \\times [0,h]$ , while $A_1 = \\lbrace 0\\rbrace \\times [0,h]$ and $A_2 = \\lbrace r\\rbrace \\times [y, h]$ .",
"Lemma 2.7 Given $r, h$ and $\\lambda > 0$ , the function $F:[0,h] \\rightarrow [0,1]$ given by $F(y) = \\mathbb {P}_\\lambda \\Bigg [ \\smash{\\hbox{[scale=1.5]{(0, 0) rectangle (4ex, 4ex);(3.8ex, 2ex) -- (4.2ex, 2ex);(0, 1.7ex) .. controls (1.3ex, 4ex) and (2.6ex, 1.4ex) .. (4ex, 3.2ex);(2ex, .2ex) node[anchor=north] {$\\scriptstyle r$};(.4ex, 2ex) node[anchor=east] {$\\scriptstyle h$};[decorate,decoration={brace,amplitude=.4ex,raise=.4ex},yshift=0pt] (4ex, 4ex) -- (4ex, 2ex);(4.3ex, 3ex) node[anchor=west] {$\\scriptstyle y$};}\\;}}$ ] is continuous."
],
[
"Finite size criterion for percolation",
"The first tool in our study of the phase transition for Poisson Boolean percolation in ${\\mathbb {R}}^2$ will be the following result that bootstraps the probability of crossing a rectangle: from close to one to converging fast to one.",
"This will be fundamental in this paper; as a first indication of its importance we show how it relates to the existence of an unbounded component, and the non-triviality of the threshold parameters $\\lambda _0$ and $\\lambda _1$ .",
"Proposition 3.1 Assume the second moment condition (REF ).",
"Then, there exists a constant $\\theta = \\theta (\\mu ) > 0$ and an increasing function $r_0:[0,\\infty )\\rightarrow [0,\\infty )$ such that the following are equivalent: There exists $r\\ge r_0(\\lambda ) $ such that ${\\mathbb {P}}_\\lambda (\\operatorname{Cross}(3r,r))>1-\\theta $ .",
"$\\displaystyle \\lim _{r\\rightarrow \\infty }{\\mathbb {P}}_\\lambda (\\operatorname{Cross}(3r,r))=1$ .",
"Under the stronger condition (REF ), if either of (i) or (ii) holds, then $\\sum _{k\\ge 0}\\big [1-{\\mathbb {P}}_\\lambda (\\operatorname{Cross}(3^{k+1},3^k))\\big ]<\\infty $ .",
"These statements remains valid for $\\operatorname{Cross}$ replaced by $\\operatorname{Cross}^\\star $ .",
"Fix $\\theta =1/100$ and let $C$ be a sufficiently large constant so that, for all $r\\ge 1$ , $\\rho _\\lambda (5r,r)\\le g(r) := C \\lambda \\int _{r/2}^\\infty x^2\\mu (\\mathop {}\\!\\mathrm {d}x).$ The existence of the constant follows from Proposition REF .",
"Set $r_0:=\\min \\lbrace r\\ge 1:g(r)\\le \\theta /2\\rbrace $ .",
"Since $g(r)$ increases with $\\lambda $ it is clear that $r_0$ does too.",
"Since (ii) trivially implies (i), it will suffice to prove that (i) implies (ii) and, under the stronger assumption (REF ), that (ii) implies (iii).",
"Let $p(r):=1-{\\mathbb {P}}_\\lambda (\\operatorname{Cross}(3r,r))$ and let $B_k:=\\operatorname{Cross}(9r,r)$ .",
"Note that a $9r\\times r$ -rectangle may be tiled with seven $3r\\times r$ -rectangles, four positioned horizontally and three vertically, in such a way that if each is crossed in the `hard' direction, then the $9r\\times r$ -rectangle is crossed horizontally.",
"Consequently, using the union bound, we obtain that ${\\mathbb {P}}_\\lambda (B_r)\\,\\ge \\,1-7{\\mathbb {P}}_\\lambda (\\lnot \\operatorname{Cross}(3r,r))\\,=\\,1-7p(r).$ Let $B_r^{\\prime }$ denote the translate of $B_r$ along the vector $(0,2r)$ .",
"Since the occurrence of either of $B_r$ or $B_r^{\\prime }$ implies the occurrence of $\\operatorname{Cross}(9r,3r)$ , he have $p(3r)\\,\\le \\,{\\mathbb {P}}_\\lambda (\\lnot B_r\\cap \\lnot B_r^{\\prime })\\,\\le \\,{\\mathbb {P}}_\\lambda (\\lnot B_r)^2+\\rho _\\lambda (5r,r).$ Hence, by definition of $g$ , we obtain for every $r\\ge 1$ the bound $p(3r)\\le 49p(r)^2+g(r).$ Now, assume there exists $r\\ge r_0$ so that $p(r)<\\theta $ .",
"Then, via iterated use of (REF ), we find that $p(3^kr)<\\theta $ for all $k\\ge 0$ .",
"Consequently, further use of (REF ) gives, for $\\ell =1,2,\\ldots ,k$ $p(3^kr)\\,\\le \\,\\frac{1}{2^\\ell }p(3^{k-\\ell }r)+\\sum _{j=1}^\\ell \\frac{1}{2^{j-1}}g(3^{k-j}r)\\,\\le \\,\\frac{1}{2^\\ell }+2g(3^{k-\\ell }r).$ Hence, sending $\\ell $ to infinity with $k$ shows that $\\lim _{k\\rightarrow \\infty }p(3^kr)=0$ .",
"Since for every $r^{\\prime }\\in [r,3r]$ we have $p(r^{\\prime })\\le {\\mathbb {P}}_\\lambda (\\lnot B_r)\\le 7p(r)$ , it follows that $\\lim _{r\\rightarrow \\infty }p(r)=0$ , so (ii) holds.",
"Finally, we assume that (ii) holds and pick $k_0$ so that $3^{k_0}\\ge r_0$ and $p(3^{k_0})<\\theta $ .",
"Summing over $k$ in (REF ), with $r=3^{k_0}$ and $\\ell =\\lfloor k/2\\rfloor $ , leads to $\\sum _{k\\ge 1}p(3^{k+k_0})\\,\\le \\,\\sum _{k\\ge 1}\\frac{1}{2^{(k-1)/2}}+2\\sum _{k\\ge 1}g(3^{k/2+k_0})\\,\\le \\,4+2\\int _0^\\infty g(3^{y/2})\\,\\mathop {}\\!\\mathrm {d}y.$ Since $g$ is itself an integral we may use Fubini's theorem to reverse the order of integration, and obtain $\\sum _{k\\ge 1}p(3^{k+k_0})\\,\\le \\, 4+C\\lambda \\int _0^\\infty x^2\\log x\\,\\mu (\\mathop {}\\!\\mathrm {d}x),$ for a possibly larger constant $C$ , which is finite under the assumption (REF ), and (iii) follows.",
"Remark 7 Note that the proof of Proposition REF assumes very little about the underlying percolation model.",
"Indeed, the argument applies to any percolation model on ${\\mathbb {R}}^2$ satisfying the invariance assumption (REF ) and decay of correlations corresponding to conditions (REF ) or (REF ).",
"The following more quantitative statement sheds some light on the role of spatial correlations on the rate of convergence in Proposition REF .",
"Given a function $f:[1, \\infty ) \\rightarrow (0, \\infty )$ we say that it is regularly varying if for any $a \\in (0, \\infty )$ $\\frac{f(ar)}{f(r)} \\text{ converges to a non-zero limit.",
"}$ The following proposition says, in particular, that if the spatial dependence decays polynomially fast, then the convergence in Proposition REF occurs (at least) at the same polynomial rate.",
"Proposition 3.2 Assume the finite moment condition (REF ) and let $f:[1,\\infty )\\rightarrow (0,\\infty )$ be a regularly varying function satisfying $\\lim _{r\\rightarrow \\infty }f(r)=0$ and $\\rho _\\lambda (5r,r)\\le f(r)$ for all $r \\ge 1$ .",
"Then, if $\\lim _{r\\rightarrow \\infty }{\\mathbb {P}}_\\lambda (\\operatorname{Cross}(3r,r))=1$ , then there exists $c<\\infty $ such that ${\\mathbb {P}}_\\lambda (\\operatorname{Cross}(3r,r))\\ge 1-cf(r)\\quad \\text{for all }r\\ge 1.$ The statement remains true if $\\operatorname{Cross}$ is replaced by $\\operatorname{Cross}^\\star $ .",
"With $p(r):=1-{\\mathbb {P}}_\\lambda (\\operatorname{Cross}^\\star (3r,r))$ , we may repeat the proof of Proposition REF to obtain $p(3r)\\le 49p(r)^2+f(r).$ Since $f$ is regularly varying there exists a constant $c^{\\prime } \\in (0, 50)$ such that $f(3r)/f(r) \\ge c^{\\prime }$ for all $r\\ge 1$ .",
"Let $\\theta ^{\\prime }=c^{\\prime }/100$ and pick $r^{\\prime }$ such that $p(r)\\le \\theta ^{\\prime }$ and $f(r)\\le \\theta ^{\\prime }/2$ for all $r\\ge r^{\\prime }$ .",
"Let $c=\\max \\lbrace 2/c^{\\prime },1/f(r^{\\prime })\\rbrace $ .",
"We claim that $p(3^kr^{\\prime })\\le cf(3^kr^{\\prime })\\quad \\text{for all }k\\ge 0.$ The case $k = 0$ is immediate, and the remaining cases follows from (REF ) via an induction step.",
"This proves the statement of the proposition along exponentially growing sub-sequences from which one can easily derive the full statement.",
"Propositions REF and REF show how an observation in a finite region can be sufficient to draw conclusions regarding crossing probabilities over arbitrarily large regions.",
"The next well-known result connects the finite observation to the existence of unbounded components.",
"Proposition 3.3 The condition $\\sum _{k\\ge 0}\\big [1-{\\mathbb {P}}_\\lambda (\\operatorname{Cross}(3^{k+1},3^k))\\big ]<\\infty $ implies that the probability, at $\\lambda $ , that the origin is contained in an unbounded occupied component is positive.",
"The same holds when $\\operatorname{Cross}$ and `occupied' are replaced by $\\operatorname{Cross}^\\star $ and `vacant', respectively.",
"Denote by $R_k$ the rectangle $[0,3^{k+1}]\\times [0,3^k]$ , and by $R_k^{\\prime }$ the rectangle $[0,3^k]\\times [0,3^{k+1}]$ , resulting from a right-angle rotation of $R_k$ .",
"Consider the sequence, indexed by $k\\ge 0$ , alternating between $R_k$ or $R_k^{\\prime }$ , depending on whether $k$ is odd or even.",
"The above condition implies that the expected number of these rectangles that fail to be crossed in the `hard' direction is finite.",
"Hence, by Borel-Cantelli, it follows that all but finitely many contain a crossing in the `hard' direction almost surely.",
"However, all but finitely many of these crossings must intersect as a consequence of how the rectangles are placed.",
"Thus, with probability one, there is an unbounded occupied component.",
"Due to invariance with respect to translations it follows that the origin has positive probability to be contained in this component.",
"We close this section by showing that the (finite-size) phase transition is non-trivial.",
"Corollary 3.4 Assume the second moment condition (REF ).",
"Then, $0<\\lambda _0\\le \\lambda _c\\le \\lambda _1<\\infty .$ Under the stronger assumption (REF ) we also have $\\lambda _0\\le \\lambda _c^\\star \\le \\lambda _1$ .",
"Assume that (REF ) holds.",
"We first show that $\\lambda _0 > 0$ .",
"Let $\\theta >0$ and $r_0:[0,\\infty )\\rightarrow [0,\\infty )$ be as in Proposition REF .",
"Clearly ${\\mathbb {P}}_0(\\operatorname{Cross}(r, 3r))=0$ for all $r\\ge 1$ , in particular for $r=r_0(1)$ .",
"That the crossing probability varies continuously with respect to the intensity parameter is an easy consequence of (REF ) of Lemma REF and the fact that it is unlikely to add any point at all to a bounded region for small enough changes in the parameter.",
"Hence, we may pick $\\varepsilon \\in (0,1)$ such that ${\\mathbb {P}}_\\varepsilon (\\operatorname{Cross}(r,3r)) < \\theta $ , implying that $\\lim _{r\\rightarrow \\infty } {\\mathbb {P}}_\\varepsilon (\\operatorname{Cross}(r,3r)) = 0,$ and thus that $\\lambda _0 \\ge \\varepsilon > 0$ .",
"The fact that $\\lambda _1 < \\infty $ follows from the fact that this holds for the model with constant radius, together with a stochastic domination argument.",
"We next show that $\\lambda _c\\ge \\lambda _0$ and $\\lambda _c^\\star \\le \\lambda _1$ .",
"First, consider $\\lambda <\\lambda _0$ .",
"Then, since a path connecting the origin to $\\partial B(r)$ has to cross one of four rectangles of dimension $r/3\\times r$ in the easy direction, the union bound and the definition of $\\lambda _0$ gives that $\\lim _{r\\rightarrow \\infty }{\\mathbb {P}}_\\lambda (0\\overset{\\tiny \\mathcal {O}}{\\longleftrightarrow }\\partial B(r))\\,\\le \\,\\lim _{r\\rightarrow \\infty }4{\\mathbb {P}}_\\lambda (\\operatorname{Cross}(r/3,r))\\,=\\,0.$ Thus, $\\lambda \\le \\lambda _c$ , which shows that $\\lambda _0\\le \\lambda _c$ .",
"An analogous argument shows that $\\lambda _c^\\star \\le \\lambda _1$ , To prove that $\\lambda _c^\\star \\ge \\lambda _0$ , we make the stronger assumption that (REF ) holds and fix $\\lambda <\\lambda _0$ , in which case we have $\\lim _{r\\rightarrow \\infty }{\\mathbb {P}}_\\lambda (\\operatorname{Cross}^\\star (3r,r))=1$ .",
"Then, Proposition REF shows that $\\sum _{k\\ge 0}\\big [1-{\\mathbb {P}}_\\lambda (\\operatorname{Cross}^\\star (3^{k+1},3^k))\\big ]<\\infty ,$ which by Proposition REF implies the almost sure existence of an unbounded vacant component, and thus that $\\lambda \\le \\lambda _c^\\star $ .",
"Hence $\\lambda _c^\\star \\ge \\lambda _0$ .",
"An analogous argument shows, under the additional condition (REF ), that $\\lambda _c\\le \\lambda _1$ .",
"However, we claim that this additional assumption is unnecessary.",
"Our goal will therefore be to show that (REF ) holds, with $\\operatorname{Cross}^\\star $ replaced by $\\operatorname{Cross}$ , also under the weaker second moment condition (REF ).",
"As before, let $p(r)=1-{\\mathbb {P}}_\\lambda (\\operatorname{Cross}(3r,r))$ .",
"Repeating the first steps of the proof of Proposition REF , observing that the complements of occupied crossing events are decreasing, we obtain the following analogue to (REF ) $p(3r)\\,\\le \\,7p(r)\\big (7p(r)+\\bar{\\rho }_\\lambda (5r,r)\\big ).$ Since both $p(r)$ and $\\bar{\\rho }_\\lambda (5r,r)$ tend to zero as $r\\rightarrow \\infty $ , we may find $r_0$ such that $7p(r)+\\bar{\\rho }_\\lambda (5r,r)$ is bounded by $\\frac{1}{14}$ for all $r\\ge r_0$ .",
"So, for some $k_0\\ge 1$ , we obtain that $p(3^{k+k_0})\\,\\le \\,\\frac{1}{2}p(3^{k-1+k_0}r_0)\\,\\le \\,\\frac{1}{2^k}p(3^{k_0}),$ which is summable.",
"Hence (REF ) holds for $\\operatorname{Cross}$ also under the weaker condition (REF ), and $\\lambda _c\\le \\lambda _1$ follows."
],
[
"Russo-Seymour-Welsh theory",
"In the study of planar percolation, Russo-Seymour-Welsh (RSW) techniques play a central role and have numerous consequences.",
"The original proof for Bernoulli percolation ([34], [39]) is strongly based on planarity and the independence structure of the Bernoulli percolation measure, and does not extend easily to other contexts.",
"Considerable technicalities had to be overcome even in the extension to Poisson Boolean percolation in ${\\mathbb {R}}^2$ with fixed radii, see [32], [4], and in the case of (unbounded) random radii such a result has until this point not been obtained.",
"In the last few years some new arguments have been developed to prove RSW-results for dependent percolation models, e.g.",
"Voronoi percolation [9], [42], [3] and the random-cluster model [7], [13].",
"In this section we develop an RSW-theory applicable for Poisson Boolean percolation.",
"Our method of proof will be greatly inspired by that of [42].",
"However, due to the asymmetry between the vacant and occupied regions, we need a stronger version of the result proved in [42].",
"In that paper, the RSW statement bounds the crossing probabilities for rectangles in the long direction, assuming a bound on the crossing probabilities for squares, see (REF ).",
"Here we assume only that rectangles are crossed in the easier direction.",
"Our proof will be rather general and applies in settings far beyond Poisson Boolean percolation; see Remark REF below.",
"Theorem 4.1 (RSW-Theorem) Assume the finite second moment condition stated in (REF ).",
"Then, for any $\\lambda > 0$ , if for some $\\kappa >0$ we have $\\inf _{r\\ge 1} \\mathbb {P}_\\lambda \\big (\\operatorname{Cross}(\\kappa r, r) \\big ) > 0,$ then the same is true for all $\\kappa >0$ .",
"The same holds for $\\operatorname{Cross}$ replaced by $\\operatorname{Cross}^\\star $ .",
"Remark 8 Throughout this section, we are going to establish the above result and some of its consequences for Poisson Boolean percolation with finite second moments.",
"However, let us emphasize that the same proof works for several different types of percolation measures on the plane.",
"More precisely, the only properties of the random set $\\mathcal {O}$ that we use in this section are: $\\begin{array}{c}\\text{the translation, reflection and rotational symmetries in (\\ref {e:invariance}),}\\\\\\text{the FKG inequality (\\ref {eq:3}),}\\\\\\text{the decay of correlations stated in (\\ref {e:rho_to_zero}) and}\\\\\\text{the continuity of the crossing probabilities stated in Lemma~\\ref {l:cross_continuous}}.\\end{array}$"
],
[
"Consequences of Theorem ",
"The Russo-Seymour-Welsh Theorem stated above, which will be proved in Section REF , has several important consequences that we develop next.",
"These consequences concern `box-crossing' and `one-arm' probabilities in the critical regime $[\\lambda _0,\\lambda _1]$ .",
"However, we first state a `high-probability' version of Theorem REF .",
"All proofs of the consequences listed here will be given in Section REF below.",
"Corollary 4.2 Assume the finite second moment condition (REF ).",
"Given $\\lambda > 0$ , if for some $\\kappa >0$ we have $\\lim _{r\\rightarrow \\infty }\\mathbb {P}_\\lambda \\big (\\operatorname{Cross}(\\kappa r, r) \\big ) = 1,$ then it is true for all $\\kappa >0$ .",
"The same holds for $\\operatorname{Cross}$ replaced by $\\operatorname{Cross}^\\star $ .",
"Recall the definition of $\\lambda _0$ and $\\lambda _1$ in (REF ) and ().",
"Outside of the interval $[\\lambda _0,\\lambda _1]$ the probability of crossing a fixed-ratio rectangle will converge rather rapidly to either zero or one as the side length of the rectangle increases.",
"This is merely a matter of definition and the FKG-inequality.",
"One of the main consequences of the RSW Theorem is that within the interval $[\\lambda _0,\\lambda _1]$ the probability of crossing a fixed-ratio rectangle remains rather balanced.",
"This is sometimes referred to as the `box-crossing' property, which will also be used to provide bounds on the one-arm probabilities.",
"For $0 < r < r^{\\prime }$ , we define $\\operatorname{Arm}(r, r^{\\prime }) := \\big [ B^\\infty (r) \\overset{\\mathcal {O}}{\\longleftrightarrow } \\partial B^\\infty (r^{\\prime }) \\big ].$ As before we write $\\operatorname{Arm}^\\star $ for the above event, where $\\mathcal {O}$ is replaced by $\\mathcal {V}$ .",
"Given $r^{\\prime } > r\\ge 1$ we also introduce $\\operatorname{Circ}(r, {r^{\\prime }}) := \\hbox{[scale=1.5]{(5ex, 3.5ex) rectangle (7ex, 5.5ex);(3ex, 1.5ex) rectangle (9ex, 7.5ex);[rounded corners=.25em](4ex, 6.5ex) .. controls (6ex, 6.5ex + .4ex*rand) ..(8ex, 6.5ex) .. controls (8ex + .4ex*rand, 4.5ex) ..(8ex, 2.5ex) .. controls (6ex, 2.5ex + .4ex*rand) ..(4ex, 2.5ex) -- (4ex + .4ex*rand, 4.5ex) -- cycle;(9.4ex, 3.5ex) -- (9.4ex, 5.5ex);(9.2ex, 3.5ex) -- (9.6ex, 3.5ex) (9.2ex, 5.5ex) -- (9.6ex, 5.5ex);(9.4ex, 4.5ex) node[anchor=west] {$\\scriptstyle 2r$};(2.4ex, 1.5ex) -- (2.4ex, 7.5ex);(2.2ex, 1.5ex) -- (2.6ex, 1.5ex) (2.2ex, 7.5ex) -- (2.6ex, 7.5ex);(2ex, 4.5ex) node[anchor=east] {$\\scriptstyle 2r^{\\prime }$};}}$ := Arm(r, r')c Finally we let $\\operatorname{Circ}^\\star (r,{r^{\\prime }}) := \\operatorname{Arm}(r, r^{\\prime })^c$ .",
"Corollary 4.3 Assume the second moment condition in (REF ).",
"Then, for every $\\kappa > 0$ , there exists $c_{\\ref {c:q-bound}} = c_{\\ref {c:q-bound}}(\\kappa ) > 0$ such that for every $\\lambda \\in [\\lambda _0, \\lambda _1]$ we have ${\\mathbb {P}}_\\lambda (\\operatorname{Cross}(\\kappa r, r)) \\in (c_{\\ref {c:q-bound}}, 1 - c_{\\ref {c:q-bound}})\\quad \\text{for every $r\\ge 1$,}$ and also $\\mathbb {P}_\\lambda (\\operatorname{Arm}(r, 2r)) \\in (c^{\\prime }, 1-c^{\\prime })\\quad \\text{for every $r\\ge 1$,}$ where $c^{\\prime }=c_{\\ref {c:q-bound}}(4)^4$ .",
"As before, the above result also holds for the vacant set, i.e.",
"with $\\operatorname{Cross}$ and $\\operatorname{Arm}$ replaced by $\\operatorname{Cross}^\\star $ and $\\operatorname{Arm}^\\star $ respectively.",
"Another consequence of Theorem REF concerns the decay of arm probabilities at criticality.",
"Unlike previous results of this section, the bounds on the arm events distinguish between vacant and occupied sets.",
"The one-arm probability always decays polynomially fast for the vacant set.",
"For the occupied set, we can prove polynomial decay of the one-arm probability only under the stronger assumption () and we explain why this restriction is necessary in Remark REF .",
"Corollary 4.4 (Bounds on the arm-events) Assume the second moment conditions in (REF ).",
"There exists a function $f: (0,1) \\rightarrow (0, 1)$ such that $\\lim _{x \\rightarrow 0}f(x) = 0$ and for every $\\lambda \\in [\\lambda _0, \\lambda _1]$ we have ${\\mathbb {P}}_\\lambda (\\operatorname{Arm}(r,r^{\\prime })) \\le f \\Big ( \\frac{r}{r^{\\prime }} \\Big ).$ If $\\operatorname{Arm}$ is replaced by $\\operatorname{Arm}^\\star $ , then a stronger conclusion holds: there exists $c > 0$ such that ${\\mathbb {P}}_\\lambda ( \\operatorname{Arm}^\\star (r,r^{\\prime }) ) \\le \\frac{1}{c} \\left(\\frac{r}{r^{\\prime }}\\right)^c.$ Under the stronger assumption () the conclusion of (REF ) holds also for $\\operatorname{Arm}$ .",
"Remark 9 In Corollary REF , under the $2 + \\varepsilon $ -moment condition (), we can choose $f(x) = \\tfrac{1}{c} x^c$ for some constant $c > 0$ .",
"However, if $\\mu ([x, \\infty ))$ is $1/(x^2 \\log ^2(x))$ , then the second moment condition holds but the arm event does not decay polynomially.",
"To see this, consider the event that the origin is covered by a occupied disk of radius at least $r$ , and observe that the probability of this event does not decay polynomially fast in $r$ .",
"An immediate consequence of Corollary REF is the following observation.",
"Corollary 4.5 Assume the second moment conditions in (REF ).",
"For every $\\lambda \\in [\\lambda _0,\\lambda _1]$ there is almost surely no unbounded occupied nor vacant cluster.",
"Consequently, $\\lambda _c^\\star \\le \\lambda _0\\le \\lambda _1\\le \\lambda _c.$"
],
[
"Standard inequalities",
"Before starting the proof of the RSW Theorem and its consequences, let us recall some standard inequalities on crossing probabilities.",
"Lemma 4.6 For every $\\lambda >0$ , $r>0$ , $\\kappa >0$ and integer $j\\ge 1$ we have: ${\\mathbb {P}}_\\lambda [\\operatorname{Cross}((1+j\\kappa )r,r)]\\ge {\\mathbb {P}}_\\lambda [\\operatorname{Cross}((1+\\kappa )r,r)]^{2j-1}$ , ${\\mathbb {P}}_\\lambda [\\operatorname{Cross}(r,(1+\\kappa )r)]\\ge 1-\\left(1-{\\mathbb {P}}_\\lambda [\\operatorname{Cross}(r,(1+j\\kappa r)]\\right)^{1/(2j-1)}$ , ${\\mathbb {P}}_\\lambda [\\operatorname{Circ}(r,2r)]\\ge {\\mathbb {P}}_\\lambda [\\operatorname{Cross}(4r,r)]^4$ , ${\\mathbb {P}}_\\lambda [\\operatorname{Cross}(2r,r)]\\ge {\\mathbb {P}}[\\operatorname{Circ}(r,2r)]$ .",
"The same holds if we replace $\\operatorname{Cross}$ and $\\operatorname{Circ}$ by $\\operatorname{Cross}^\\star $ and $\\operatorname{Circ}^\\star $ , respectively.",
"Part (REF ) is a straightforward consequence of the FKG-inequality (REF ): A horizontal crossing of an $(1+j\\kappa )r\\times r$ -rectangle can be enforced by $j$ horizontal crossings of $(1+\\kappa )r\\times r$ -rectangles and $j-1$ vertical crossings of $r\\times (1+\\kappa )r$ -rectangles overlapping one another.",
"Similarly, part (REF ) is the consequence of a circuit being possible to construct out of four overlapping rectangle crossings.",
"Part (REF ) is a direct consequence of (REF ) since it may be rewritten as ${\\mathbb {P}}_\\lambda [\\operatorname{Cross}^\\star ((1+\\kappa )r,r)]\\le {\\mathbb {P}}_\\lambda [\\operatorname{Cross}^\\star ((1+j\\kappa )r,r)]^{1/(2j-1)}.$ Inequality (REF ) follows from the observation that any circuit in the annulus $B^\\infty (2r)\\setminus B^\\infty (r)$ must cross the rectangle $[-r,r]\\times [r,2r]$ horizontally."
],
[
"A useful circuit Lemma",
"For Bernoulli percolation, the bounds on crossing probabilities provide some bounds on the arm events.",
"This is based on a circuit argument, that uses independence.",
"In our case, the argument need to be adapted because of spatial dependencies.",
"Lemma 4.7 (Circuit Lemma) Assume the second moment conditions in (REF ).",
"Let $\\lambda \\in [0,\\infty )$ .",
"For every $c>0$ , there exists a function $f=f_{c,\\lambda }: (0,1) \\rightarrow (0,1)$ such that $\\lim _{x \\rightarrow 0} f(x) = 0$ and for all $r^{\\prime } \\ge 2r \\ge 2$ , $\\text{if }\\displaystyle \\inf _{r\\le s\\le r^{\\prime }/2} {\\mathbb {P}}_\\lambda [\\operatorname{Circ}^\\star (s,2s)]\\ge c,\\text{ then }{\\mathbb {P}}_\\lambda [ \\operatorname{Circ}^\\star (r,r^{\\prime }) ] \\ge 1- f \\left(\\tfrac{r}{r^{\\prime }}\\right).$ For every $c>0$ , there exists a constant $c^{\\prime }=c^{\\prime }(c,\\lambda )>0$ such that for all $r^{\\prime } \\ge 2r \\ge 2$ , $\\text{if }\\displaystyle \\inf _{r\\le s\\le r^{\\prime }/2} {\\mathbb {P}}_\\lambda [\\operatorname{Circ}(s,2s)]\\ge c,\\text{ then }{\\mathbb {P}}_\\lambda [ \\operatorname{Circ}(r,r^{\\prime }) ] \\ge 1-\\tfrac{1}{c^{\\prime }}\\left(\\tfrac{r}{r^{\\prime }}\\right)^{c^{\\prime }}.$ Under the $2+\\varepsilon $ moment condition (), Item $(\\ref {item:6})$ holds also for $\\operatorname{Circ}$ replaced by $\\operatorname{Circ}^\\star $ .",
"We begin with the proof of Item (REF ).",
"It suffices to prove the statement for $r^{\\prime }\\ge 16r$ .",
"Let $r \\ge 1$ and $r^{\\prime }\\ge 16r$ be given, and assume that $\\inf _{r\\le s\\le r^{\\prime }/2} {\\mathbb {P}}_\\lambda [\\operatorname{Circ}^\\star (s,2s)]\\ge c>0.$ Let $g(s) := \\sup _{s^{\\prime }\\ge s} \\rho _\\lambda (s^{\\prime },2s^{\\prime })$ .",
"By Proposition REF , we have $\\lim _{s\\rightarrow \\infty }g(s) = 0$ .",
"Set $t = \\sqrt{rr^{\\prime }}$ and $\\ell _i=4^it$ .",
"Note that if $\\operatorname{Circ}^\\star (r,r^{\\prime })$ fails to occur, then $A_i=\\operatorname{Circ}^\\star (\\ell _i,2\\ell _i)$ cannot occur for no $i=0,1,\\ldots ,k-1$ where $k=\\lfloor \\frac{1}{2}\\log _4(r^{\\prime }/r)\\rfloor $ ($k\\ge 1$ since we assumed $r^{\\prime }\\ge 16r$ ).",
"The hypothesis (REF ) implies that ${\\mathbb {P}}_\\lambda (A_i^c)\\le 1-c<1$ .",
"Therefore, ${\\mathbb {P}}_\\lambda \\Big [ {\\textstyle \\bigcap \\limits }_{i = 0}^{k-1} A_i^c \\Big ]\\,&\\le \\, {\\mathbb {P}}_\\lambda [A_{k-1}] {\\mathbb {P}}_\\lambda \\Big [ {\\textstyle \\bigcap \\limits }_{i = 0}^{k-2} A_i^c \\Big ] + \\rho _\\lambda (2\\ell _{k-2},\\ell _{k-2})\\\\&\\le \\, (1-c) {\\mathbb {P}}_\\lambda \\Big [ {\\textstyle \\bigcap \\limits }_{i = 0}^{k-2} A_i^c \\Big ] + g(t),$ which by induction gives that ${\\mathbb {P}}_\\lambda \\Big [ {\\textstyle \\bigcap \\limits }_{i = 0}^{k-1} A_i^c \\Big ] \\,\\le \\, (1-c)^k + \\frac{1}{c} g(t).$ By the choice of $k$ , and the fact that $g(t)\\le g(t/r) = g(\\sqrt{r^{\\prime }/r})$ , we finally obtain ${\\mathbb {P}}_\\lambda \\big [ \\operatorname{Circ}^\\star (r,r^{\\prime })^c \\big ] \\,\\le \\,{\\mathbb {P}}_\\lambda \\Big [ {\\textstyle \\bigcap \\limits }_{i = 0}^{k-1} A_i^c \\Big ] \\,\\le \\, \\frac{1}{1-c}\\Big ( \\frac{r^{\\prime }}{r} \\Big )^{\\frac{1}{2}\\log _4(1-c)}+ \\frac{1}{c}g\\left(\\sqrt{\\tfrac{r^{\\prime }}{r}}\\right).$ Hence, (REF ) holds with $f(x)=\\frac{1}{1-c}x^\\alpha +\\frac{1}{c}g(x^{-1/2})$ for any sufficiently small constant $\\alpha >0$ .",
"The stronger moment condition () implies (REF ), which readily gives that $f$ may be chosen of the form $\\frac{1}{c^{\\prime }}x^{c^{\\prime }}$ for some constant $c^{\\prime }=c^{\\prime }(c,\\lambda )>0$ , which also proves Item (REF ).",
"We now turn to the proof of Item (REF ) and reuse the above notation.",
"Set $B_i=\\operatorname{Circ}(\\ell _i,2\\ell _i)$ .",
"As above, by (REF ) it follows that ${\\mathbb {P}}_\\lambda \\big (B_i^c\\big )\\le 1-c$ for every $i=0,1,\\ldots ,k-1$ .",
"By Proposition REF we may fix $r_0$ large enough so that $\\bar{\\rho }_\\lambda (2r,r) \\le c/2\\quad \\text{for every }r\\ge r_0.$ Observe that the events $B_i$ are decreasing.",
"Hence, for $r\\ge r_0$ , $r^{\\prime }\\ge 16r$ and $t=\\sqrt{rr^{\\prime }}$ , we obtain $\\begin{aligned}{\\mathbb {P}}_\\lambda \\big ( \\operatorname{Circ}(r, r^{\\prime })^c \\big ) \\,& \\le \\, {\\mathbb {P}}_\\lambda \\Big ( {\\textstyle \\bigcap \\limits }_{i = 0}^{k-1}B_i^c \\Big )\\\\& \\le \\,{\\mathbb {P}}_\\lambda \\Big ( {\\textstyle \\bigcap \\limits }_{i = 1}^{k-1} B_i^c \\Big ) \\Big ( \\mathbb {P}_\\lambda (B_0^c) + \\bar{\\rho }_\\lambda (2\\ell _0, \\ell _0) \\Big )\\\\& \\le \\,\\prod _{i = 0}^{k-1} \\Big ( \\mathbb {P}_\\lambda \\big ( B_i^c \\big ) + \\bar{\\rho }_\\lambda (2\\ell _i, \\ell _i) \\Big )\\\\& \\le \\,(1-c/2)^{\\frac{1}{2}\\log _4(r^{\\prime }/r)-1}.\\end{aligned}$ By allowing for a larger constant on the right-hand side we may replace the assumption $r\\ge r_0$ with $r\\ge 1$ .",
"This concludes the proof of (REF )."
],
[
"Proof of the consequences of Theorem ",
"We now assume the validity of Theorem REF and prove Corollaries REF -REF .",
"[Proof of Corollary REF ] We will show that if $\\lim _{r\\rightarrow \\infty }{\\mathbb {P}}(\\operatorname{Cross}(\\kappa r,r))=1$ for some $\\kappa >0$ , then $\\lim _{r\\rightarrow \\infty }{\\mathbb {P}}(\\operatorname{Cross}(2\\kappa r,r))=1$ , from which the statement follows via iteration.",
"Partition the right-hand side of the rectangle $R=[0,\\kappa r]\\times [0,r]$ into $n$ equal intervals (of length $r/n$ ), and denote the event that there is a horizontal crossing of $R$ into the $k$ -th of these intervals by $A_k$ .",
"Using the square-root trick (REF ), we find that $\\max _{1\\le k\\le n}{\\mathbb {P}}_\\lambda (A_k)\\ge 1-[1-{\\mathbb {P}}_\\lambda (\\operatorname{Cross}(\\kappa r,r))]^{1/n},$ which by assumption tends to 1 as $r\\rightarrow \\infty $ .",
"By assumption, Theorem REF shows that $\\inf _{r\\ge 1}{\\mathbb {P}}_\\lambda (\\operatorname{Cross}(4r,r))>0$ .",
"Hence, the standard inequalities of Lemma REF show that $\\inf _{r\\ge 1}{\\mathbb {P}}_\\lambda (\\operatorname{Circ}(r,2r))>0$ .",
"Hence, by Lemma REF we conclude that for large enough $r$ , ${\\mathbb {P}}_\\lambda (\\operatorname{Circ}( r/n,\\kappa r/2))\\ge 1-f(\\kappa n/2),$ for some function $f$ such that $f(x)\\rightarrow 0$ as $x\\rightarrow \\infty $ .",
"Finally, we combine the above estimates to obtain that for every $\\varepsilon >0$ ${\\mathbb {P}}_\\lambda \\big (\\operatorname{Cross}(2\\kappa r,r)\\big )\\,\\ge \\,{\\mathbb {P}}_\\lambda \\Bigg [\\hbox{[scale=1.5]{(0, 0) rectangle (6ex, 8ex) (6ex, 0) rectangle (12ex, 8ex);(5.8ex, 4.2ex) -- (6.2ex, 4.2ex) (5.8ex, 4.8ex) -- (6.2ex, 4.8ex);[solid] (0, 3ex) .. controls (2ex, 2ex) and (4ex, 5ex) .. (6ex, 4.65ex);[solid] (6ex, 4.35ex) .. controls (8ex, 4ex) and (8ex, 7ex) .. (12ex, 6ex);[solid,decorate,decoration={brace,amplitude=.7ex,raise=.2ex},yshift=0pt] (12ex, 0ex) -- (0, 0);(5ex, 3.5ex) rectangle (7ex, 5.5ex);(3ex, 1.5ex) rectangle (9ex, 7.5ex);(3ex, 1ex) -- (9ex, 1ex);(3ex, .8ex) -- (3ex, 1.2ex) (9ex, .8ex) -- (9ex, 1.2ex);[fill, color=white] (4ex, .2ex) rectangle (8ex, 1.3ex);[solid,rounded corners=.25em](4ex, 6.5ex) .. controls (6ex, 6.5ex + .4ex*rand) ..(8ex, 6.5ex) .. controls (8ex + .4ex*rand, 4.5ex) ..(8ex, 2.5ex) .. controls (6ex, 2.5ex + .4ex*rand) ..(4ex, 2.5ex) -- (4ex + .4ex*rand, 4.5ex) -- cycle;(6ex, -.4ex) node[anchor=north] {$\\scriptstyle 2\\kappa r$};(.4ex, 4ex) node[anchor=east] {$\\scriptstyle r$};(9.4ex, 3.5ex) -- (9.4ex, 5.5ex);(9.2ex, 3.5ex) -- (9.6ex, 3.5ex) (9.2ex, 5.5ex) -- (9.6ex, 5.5ex);(9ex, 4.5ex) node[anchor=west] {$\\scriptstyle \\tfrac{r}{n}$};(6ex, 1.7ex) node[anchor=north] {$\\scriptstyle \\kappa r$};}\\;\\;}$ ] P(Ak)2 P(Circ(r/n,r/2)) > 1-, $for some choice of $ $ and $ n$, and all large $ r$, as required.$ [Proof of Corollary REF ] Take $\\theta > 0$ and $r_0 = r_0(\\lambda _1)$ so that Proposition REF is in force.",
"Then, ${\\mathbb {P}}_{\\lambda _1}(\\operatorname{Cross}(3r, r))\\le 1-\\theta \\text{ for all $r\\ge r_0$.",
"}$ Otherwise, by continuity, we could find $\\varepsilon > 0$ and $r\\ge r_0$ such that ${\\mathbb {P}}_{\\lambda _1-\\varepsilon }(\\operatorname{Cross}(3r,r))>1-\\theta $ , which by Proposition REF would imply that the probability indeed converges to 1; a contradiction to the definition of $\\lambda _1$ .",
"Analogously, we have ${\\mathbb {P}}_{\\lambda _0}(\\operatorname{Cross}(r, 3r)) \\ge \\theta $ for all large $r$ .",
"Using Theorem REF and its dual version this establishes (REF ).",
"Equation (REF ) follows from the fact that the complement of $\\operatorname{Arm}(r,3r)$ is $\\operatorname{Circ}^\\star (r,3r)$ and the standard inequalities of Lemma REF .",
"[Proof of Corollary REF ] This is an immediate consequence of Corollary REF and Lemma REF , as ${\\mathbb {P}}_\\lambda (\\operatorname{Arm}(r,r^{\\prime }))=1-{\\mathbb {P}}_\\lambda (\\operatorname{Circ}^\\star (r,r^{\\prime }))$ .",
"[Proof of Corollary REF ] The bounds $\\lambda _c\\ge \\lambda _1$ and $\\lambda _c^\\star \\le \\lambda _0$ follow respectively from Item (REF ) and Item (REF ) of Corollary REF ."
],
[
"Proof of Theorem ",
"In this section we prove Theorem REF .",
"We assume throughout the proof the second moment condition (REF ) and that $\\lambda >0$ is fixed.",
"By assumption there exists $\\kappa \\in (0,\\tfrac{1}{3}]$ such that $\\inf _{r\\ge 1}{\\mathbb {P}}_\\lambda (\\operatorname{Cross}(\\kappa r,r))>0$ .",
"Clearly, there is no restriction assuming the upper bound on $\\kappa $ .",
"Hence, we note that either also $\\inf _{r\\ge 1}{\\mathbb {P}}_\\lambda (\\operatorname{Cross}^\\star (\\kappa r,r))>0$ , or we have $\\sup _{r\\ge 1}{\\mathbb {P}}_\\lambda (\\operatorname{Cross}(3r,r))=1.$ The latter implies, via Proposition REF , that $\\lim _{r\\rightarrow \\infty }{\\mathbb {P}}_\\lambda (\\operatorname{Cross}(3r,r))=1$ , in which case there is nothing left to prove.",
"So, we may without loss of generality assume that for some $\\kappa >0$ $\\inf _{r\\ge 1} \\mathbb {P}_\\lambda \\big (\\operatorname{Cross}(\\kappa r, r) \\big ) > 0\\quad \\text{and}\\quad \\inf _{r\\ge 1} \\mathbb {P}_\\lambda \\big ( \\operatorname{Cross}^\\star (\\kappa r, r) \\big ) > 0.$ Moreover, by the standard inequalities of Lemma REF (part (REF ) and its dual version) this is equivalent to assuming the existence of a constant $c_{\\ref {c:easy_cross}}>0$ such that $\\inf _{r\\ge 1} \\mathbb {P}_\\lambda \\big ( \\operatorname{Cross}\\big (r, \\tfrac{5}{4}r\\big ) \\big ) \\ge c_{\\ref {c:easy_cross}} \\quad \\text{and} \\quad \\inf _{r\\ge 1} \\mathbb {P}_\\lambda \\big ( \\operatorname{Cross}^\\star \\big (r, \\tfrac{5}{4}r\\big ) \\big ) \\ge c_{\\ref {c:easy_cross}}.$ In what follows we will explore several consequences of the above assumption.",
"This will be done in a series of lemmas that will culminate in the proof of Theorem REF .",
"Throughout this section we will need to introduce a range of events that are closely related to the crossing event defined in (REF ).",
"We will typically depict these events graphically, since we believe the proof is better explained this way.",
"However, we emphasize that it is always possible to give a formal definition.",
"For instance, recall the definition of $[A \\overset{\\mathcal {C}}{\\leftrightarrow } B]$ in (REF ).",
"We start with the definition of the event of having a vacant crossing above an occupied crossing, both reaching the right-hand side on the upper half: $\\hbox{[scale=1.5]{(0, 0) rectangle (2ex, 6ex);(1.8ex, 3ex) -- (2.2ex, 3ex);[densely dotted] (0, 5ex) .. controls (.6ex, 6ex) and (1.4ex, 4ex) .. (2ex, 5ex);(0, 3.5ex) .. controls (.6ex, 5ex) and (1.4ex, 2.5ex) .. (2ex, 4ex);(1ex, .2ex) node[anchor=north] {$\\scriptstyle r$};(0ex, 3ex) node[anchor=east] {$\\scriptstyle h$};}\\;}$ := q [0, h] Q [ ( {0} [0,q] O K A ) ( {0} [q, h] V K A ) ], where $K = [0, r] \\times [0,h]$ and $A = \\lbrace r\\rbrace \\times [h/2, h]$ .",
"Lemma 4.8 There exist $r_0\\ge 1$ , $0 < c_{\\ref {c:prob_dual}} < 1/8$ and $c_{\\ref {c:box_height}} \\ge 23$ such that $\\inf _{r\\ge r_0} \\mathbb {P}_\\lambda \\Bigg [ \\hbox{[scale=1.5]{(0, 0) rectangle (2ex, 6ex);(1.8ex, 3ex) -- (2.2ex, 3ex);[densely dotted] (0, 5ex) .. controls (.6ex, 6ex) and (1.4ex, 4ex) .. (2ex, 5ex);(0, 3.5ex) .. controls (.6ex, 5ex) and (1.4ex, 2.5ex) .. (2ex, 4ex);(1ex, .2ex) node[anchor=north] {$\\scriptstyle r/c_{\\ref {c:box_height}}$};(0ex, 3ex) node[anchor=east] {$\\scriptstyle \\tfrac{r}{3}$};}\\;}$ ] cREF and $\\sup _{r\\ge r_0} \\mathbb {P}_\\lambda \\big ( \\operatorname{Cross}(\\tfrac{r}{3}, \\tfrac{r}{c_{\\ref {c:box_height}}}) \\big ) \\le \\frac{1}{16} \\quad \\text{and} \\quad \\sup _{r\\ge r_0} \\mathbb {P}_\\lambda \\big ( \\operatorname{Cross}^\\star (\\tfrac{r}{3},\\tfrac{r}{c_{\\ref {c:box_height}}}) \\big ) \\le \\frac{1}{16}.$ We first argue for the existence of $r_0\\ge 1$ and $c_{\\ref {c:box_height}}>18$ for which (REF ) holds.",
"Given $s\\ge 1$ , set $A_0:=\\operatorname{Cross}(\\tfrac{5}{4}s,s)$ and let $A_i$ denote the translate of $A_i$ along the vector $(2is,0)$ .",
"Translation invariance, duality and (REF ) imply that for every $i$ , ${\\mathbb {P}}_\\lambda (A_i)= 1- {\\mathbb {P}}_\\lambda \\big ( \\operatorname{Cross}^\\star \\big (s, \\tfrac{5}{4}s \\big ) \\big )\\le 1-c_{\\ref {c:easy_cross}}.$ If for some $k\\ge 1$ $\\operatorname{Cross}(2ks, s)$ occurs, then $A_i$ occur for all $i=0,1,\\ldots ,k-1$ .",
"Therefore, ${\\mathbb {P}}_\\lambda \\big ( \\operatorname{Cross}(2ks,s)\\big )\\,\\le \\, {\\mathbb {P}}_\\lambda \\Big (\\bigcap _{i=0}^{k-1} A_i \\Big )\\,\\le \\, {\\mathbb {P}}_\\lambda \\Big ( \\bigcap _{i=0}^{k-2}A_i \\Big )(1-c_{\\ref {c:easy_cross}}) +\\rho _\\lambda (s, \\tfrac{3}{4} s).$ Via induction in $k$ we obtain that ${\\mathbb {P}}_\\lambda \\big ( \\operatorname{Cross}(2ks,s)\\big )\\le (1-c_{\\ref {c:easy_cross}})^k+\\frac{1}{c_{\\ref {c:easy_cross}}}\\rho _\\lambda (s,\\tfrac{3}{4} s).$ For $k\\ge 4$ and $s_0\\ge 1$ large enough this expression is bounded by $1/16$ for all large $s\\ge s_0$ .",
"Hence, the choice $r_0=6ks_0$ and $c_{\\ref {c:box_height}}=6k\\ge 23$ guarantees that ${\\mathbb {P}}_\\lambda \\big ( \\operatorname{Cross}(\\tfrac{r}{3},\\tfrac{r}{c_{\\ref {c:box_height}}})\\big )\\le \\frac{1}{16}.$ for every $r\\ge r_0$ .",
"With exactly the same proof, we can show that (REF ) also holds with $\\operatorname{Cross}$ replaced by $\\operatorname{Cross}^\\star $ .",
"Hence (REF ) holds for this choice of $r_0$ and $c_{\\ref {c:box_height}}$ .",
"To conclude the proof, we will show that any choice of $c_{\\ref {c:box_height}}\\ge 23$ implies $\\inf _{r \\ge r_0} \\mathbb {P}_\\lambda \\Bigg [ \\hbox{[scale=1.5]{(0, 0) rectangle (2ex, 6ex);(1.8ex, 3ex) -- (2.2ex, 3ex);[densely dotted] (0, 5ex) .. controls (.6ex, 6ex) and (1.4ex, 4ex) .. (2ex, 5ex);(0, 3.5ex) .. controls (.6ex, 5ex) and (1.4ex, 2.5ex) .. (2ex, 4ex);(1ex, .2ex) node[anchor=north] {$\\scriptstyle r/c_{\\ref {c:box_height}}$};(0ex, 3ex) node[anchor=east] {$\\scriptstyle \\tfrac{r}{3}$};}\\;}$ ] > 0.",
"To prove this, observe that $\\begin{split}\\mathbb {P}_\\lambda \\Bigg [\\hbox{[scale=1.5]{(0, 0) rectangle (2ex, 6ex);(1.8ex, 3ex) -- (2.2ex, 3ex);[densely dotted] (0, 5ex) .. controls (.6ex, 6ex) and (1.4ex, 4ex) .. (2ex, 5ex);(0, 3.5ex) .. controls (.6ex, 5ex) and (1.4ex, 2.5ex) .. (2ex, 4ex);(1ex, .2ex) node[anchor=north] {$\\scriptstyle r/c_{\\ref {c:box_height}}$};(0ex, 3ex) node[anchor=east] {$\\scriptstyle \\tfrac{r}{3}$};}\\;}\\end{split} \\Bigg ] \\,& \\ge \\,{\\mathbb {P}}_\\lambda \\big ( \\operatorname{Cross}(\\tfrac{r}{c_{\\ref {c:box_height}}}, \\tfrac{r}{18})\\big ) {\\mathbb {P}}_\\lambda \\big ( \\operatorname{Cross}^\\star (\\tfrac{r}{c_{\\ref {c:box_height}}},\\tfrac{r}{18}) \\big )-\\rho _\\lambda (\\tfrac{r}{18}, \\tfrac{r}{18})\\\\& \\ge \\, c_{\\ref {c:easy_cross}}^2- \\rho _\\lambda (\\tfrac{r}{18},\\tfrac{r}{18}).$ By (REF ), this shows that the probability on the left hand side of (REF ) is larger than $ c_{\\ref {c:easy_cross}}^2/2$ for $r$ large enough, as required.",
"The next event we will be interested in is $\\hbox{[scale=1.5]{(0, 2ex) .. controls (1.4ex, 1ex) and (.6ex, 3ex) .. (2ex, 3.3ex);[densely dotted] (0, 4ex) .. controls (1.4ex, 5ex) and (.6ex, 3.2ex) .. (2ex, 3.7ex);(0, 0) rectangle (2ex, 6ex);(1.8ex, 2ex) -- (2.2ex, 2ex) (1.8ex, 3ex) -- (2.2ex, 3ex) (1.8ex, 4ex) -- (2.2ex, 4ex);[decorate,decoration={brace,amplitude=.5ex,raise=.4ex},yshift=0pt] (2.2ex, 4ex) -- (2.2ex, 2ex);(1ex, .2ex) node[anchor=north] {$\\scriptstyle r$};(0ex, 3ex) node[anchor=east] {$\\scriptstyle h$};(2.4ex, 3ex) node[anchor=west] {$\\scriptstyle 2a$};}}$ := q [0,h] Q [ ( {0} [0,q] O K A ) ( {0} [q, h] V K A ) ], where $K = [0,r] \\times [0,h]$ and $A = \\lbrace r\\rbrace \\times [h/2, h/2 + a]$ .",
"Definition 4.9 For every $r\\ge r_0$ , let $0 \\le \\alpha _r\\le r/2$ be such that $\\mathbb {P}_\\lambda \\Bigg [ \\hbox{[scale=1.5]{(0, 2ex) .. controls (1.4ex, 1ex) and (.6ex, 3ex) .. (2ex, 3.3ex);[densely dotted] (0, 4ex) .. controls (1.4ex, 5ex) and (.6ex, 3.2ex) .. (2ex, 3.7ex);(0, 0) rectangle (2ex, 6ex);(1.8ex, 2ex) -- (2.2ex, 2ex) (1.8ex, 3ex) -- (2.2ex, 3ex) (1.8ex, 4ex) -- (2.2ex, 4ex);[decorate,decoration={brace,amplitude=.5ex,raise=.4ex},yshift=0pt] (2.2ex, 4ex) -- (2.2ex, 2ex);(1ex, .2ex) node[anchor=north] {$\\scriptstyle r/c_{\\ref {c:box_height}}$};(0ex, 3ex) node[anchor=east] {$\\scriptstyle r$};(2.4ex, 3ex) node[anchor=west] {$\\scriptstyle 2 \\alpha _r$};}}$ ] = cREF 2.",
"Let us briefly show that $\\alpha _r\\le \\frac{r}{6}.$ By Lemma REF , the quantity $f(a)=\\mathbb {P}_\\lambda \\Bigg [ \\hbox{[scale=1.5]{(0, 2ex) .. controls (1.4ex, 1ex) and (.6ex, 3ex) .. (2ex, 3.3ex);[densely dotted] (0, 4ex) .. controls (1.4ex, 5ex) and (.6ex, 3.2ex) .. (2ex, 3.7ex);(0, 0) rectangle (2ex, 6ex);(1.8ex, 2ex) -- (2.2ex, 2ex) (1.8ex, 3ex) -- (2.2ex, 3ex) (1.8ex, 4ex) -- (2.2ex, 4ex);[decorate,decoration={brace,amplitude=.5ex,raise=.4ex},yshift=0pt] (2.2ex, 4ex) -- (2.2ex, 2ex);(1ex, .2ex) node[anchor=north] {$\\scriptstyle r/c_{\\ref {c:box_height}}$};(0ex, 3ex) node[anchor=east] {$\\scriptstyle r$};(2.4ex, 3ex) node[anchor=west] {$\\scriptstyle 2a$};}}$ ] is continuous in $a$ , and by Lemma REF , it satisfies $f(0)=0$ and $f(\\tfrac{r}{6})\\ge c_{\\ref {c:prob_dual}}$ .",
"Hence, there exists $0\\le \\alpha _r\\le r/6$ such that $f(\\alpha _r)=\\tfrac{c_{\\ref {c:prob_dual}}}{2}.$ The next event is what we call the fork and it is crucial in what follows $\\hbox{[scale=1.5]{(0, 0) rectangle (4ex, 4ex);(-.2ex, 1.5ex) -- (.2ex, 1.5ex) (-.2ex, 2.5ex) -- (.2ex, 2.5ex);(3.8ex, 1.5ex) -- (4.2ex, 1.5ex) (3.8ex, 2.5ex) -- (4.2ex, 2.5ex);[solid] (0, 3ex) .. controls (.5ex, 3ex) and (1ex, 2.5ex).. (1ex, 2ex) .. controls (1ex, 1.5ex) and (.5ex, 1ex) .. (0ex, 1ex);[solid] (1ex, 2ex) .. controls (1.6ex, 3ex) and (2.4ex, 1ex) .. (3ex, 2ex);[solid] (4ex, 3ex) .. controls (3.5ex, 3ex) and (3ex, 2.5ex).. (3ex, 2ex) .. controls (3ex, 1.5ex) and (3.5ex, 1ex) .. (4ex, 1ex);[decorate,decoration={brace,amplitude=.7ex,raise=.4ex},yshift=0pt] (0, 0) -- (0, 4ex);(2ex, .2ex) node[anchor=north] {$\\scriptstyle r$};(-.5ex, 2ex) node[anchor=east] {$\\scriptstyle h$};(3.8ex, 2ex) node[anchor=west] {$\\scriptstyle 2a$};}}$ := [ c ( {0} [0,h/2 - a] O K {r} [h/2 + a, h] ) ( {0} [h/2 + a, h] O K {r} [0,h/2 - a] ) ], and analogously for the dual (where the picture is traced using dashed curves).",
"Lemma 4.10 Assuming (REF ), for every $r\\ge r_0$ , one of the following holds $\\mathbb {P}_\\lambda \\bigg [ \\hbox{[scale=1.5]{(0, 0) rectangle (4ex, 4ex);(-.2ex, 1.5ex) -- (.2ex, 1.5ex) (-.2ex, 2.5ex) -- (.2ex, 2.5ex);(3.8ex, 1.5ex) -- (4.2ex, 1.5ex) (3.8ex, 2.5ex) -- (4.2ex, 2.5ex);[solid] (0, 3ex) .. controls (.5ex, 3ex) and (1ex, 2.5ex).. (1ex, 2ex) .. controls (1ex, 1.5ex) and (.5ex, 1ex) .. (0ex, 1ex);[solid] (1ex, 2ex) .. controls (1.6ex, 3ex) and (2.4ex, 1ex) .. (3ex, 2ex);[solid] (4ex, 3ex) .. controls (3.5ex, 3ex) and (3ex, 2.5ex).. (3ex, 2ex) .. controls (3ex, 1.5ex) and (3.5ex, 1ex) .. (4ex, 1ex);[decorate,decoration={brace,amplitude=.7ex,raise=.4ex},yshift=0pt] (0, 0) -- (0, 4ex);(2ex, .2ex) node[anchor=north] {$\\scriptstyle r$};(-.5ex, 2ex) node[anchor=east] {$\\scriptstyle r$};(3.8ex, 2ex) node[anchor=west] {$\\scriptstyle 2 \\alpha _r$};}}$ ] 2-13 or P[ ] 2-13.",
"By duality and invariance under right-angle's rotations, see (REF ), $\\mathbb {P}_\\lambda \\big ( \\operatorname{Cross}(r, r) \\big ) + \\mathbb {P}_\\lambda \\big ( \\operatorname{Cross}^\\star (r, r) \\big ) = 1$ The two cases in the statement of the lemma will correspond to which of the above probabilities is at least one half.",
"Therefore, we assume without loss of generality that $\\mathbb {P}_\\lambda \\big ( \\operatorname{Cross}(r, r) \\big ) \\ge \\frac{1}{2}.$ Using symmetry and then a simple union bound, we get $\\frac{1}{4} \\le \\mathbb {P}_\\lambda \\bigg [ \\hbox{[scale=1.5]{(0, 0) rectangle (4ex, 4ex);(3.8ex, 2ex) -- (4.2ex, 2ex);(0, 1.7ex) .. controls (1.3ex, 4ex) and (2.6ex, 1.4ex) .. (4ex, 3.2ex);(2ex, .2ex) node[anchor=north] {$\\scriptstyle r$};(.4ex, 2ex) node[anchor=east] {$\\scriptstyle r$};[decorate,decoration={brace,amplitude=.4ex,raise=.4ex},yshift=0pt] (4ex, 4ex) -- (4ex, 2ex);(4.3ex, 3ex) node[anchor=west] {$\\scriptstyle \\tfrac{r}{2}$};}\\;}$ ] P[ ] + P[ ], where the event appearing in the last term of the above sum can be rigorously defined as $ \\hbox{[scale=1.5]{(0, 0) rectangle (4ex, 4ex);(3.7ex, 1.3ex) -- (4.3ex, 1.3ex);(3.8ex, 2ex) -- (4.2ex, 2ex);(3.7ex, 2.7ex) -- (4.3ex, 2.7ex);(0, 2ex) .. controls (1.3ex, 1ex) and (2.6ex, 2ex) .. (4ex, 2.2ex);[densely dotted] (2ex, 4ex) .. controls (2ex, 3ex) and (2.3ex, 3ex) .. (4ex, 2.4ex);[decorate,decoration={brace,amplitude=.3ex,raise=.4ex},yshift=0pt] (4.2ex, 2.7ex) -- (4.2ex, 1.3ex);(.4ex, 2ex) node[anchor=east] {$\\scriptstyle r$};(2ex, .2ex) node[anchor=north] {$\\scriptstyle r$};(4.4ex, 2ex) node[anchor=west] {$\\scriptstyle 2 \\alpha _r$};}}$ = q [0, r] Q ( {0} [0,q] O K A2 ) ( A1(q) V K A2 ), $ with $ = [0, r]2$, $ A1(q) = {0} [q, r] [0, r] {r}$ and $ A2 = {r} [r/2, r/2 + r].$$ Recalling that $c_{\\ref {c:box_height}} > 1$ and (REF ), we bound the second term in the above sum by a sum of two terms.",
"They respectively correspond to whether the dual path depicted above stays confined in an $(r/c_{\\ref {c:box_height}})$ -wide rectangle or not.",
"More precisely, recalling (REF ), $\\begin{split}\\mathbb {P}_\\lambda \\bigg [ \\hbox{[scale=1.5]{(0, 0) rectangle (4ex, 4ex);(3.7ex, 1.3ex) -- (4.3ex, 1.3ex);(3.8ex, 2ex) -- (4.2ex, 2ex);(3.7ex, 2.7ex) -- (4.3ex, 2.7ex);(0, 2ex) .. controls (1.3ex, 1ex) and (2.6ex, 2ex) .. (4ex, 2.2ex);[densely dotted] (2ex, 4ex) .. controls (2ex, 3ex) and (2.3ex, 3ex) .. (4ex, 2.4ex);[decorate,decoration={brace,amplitude=.3ex,raise=.4ex},yshift=0pt] (4.2ex, 2.7ex) -- (4.2ex, 1.3ex);(.4ex, 2ex) node[anchor=east] {$\\scriptstyle r$};(2ex, .2ex) node[anchor=north] {$\\scriptstyle r$};(4.4ex, 2ex) node[anchor=west] {$\\scriptstyle 2 \\alpha _r$};}}\\end{split} \\bigg ] \\,& \\le \\, \\mathbb {P}_\\lambda \\Bigg [ \\hbox{[scale=1.5]{[color=gray!50!white](4.5ex, 2ex) -- (6ex, 2ex) (4.5ex, 4ex) -- (6ex, 4ex);(0, 0) rectangle (6ex, 6ex);(4.5ex, 0) -- (4.5ex, 6ex);(5.7ex, 2.3ex) -- (6.3ex, 2.3ex) (5.8ex, 3ex) -- (6.2ex, 3ex) (5.7ex, 3.7ex) -- (6.3ex, 3.7ex);(4.5ex, 1.5ex) .. controls (5ex, 1ex) and (5ex, 3.5ex) .. (6ex, 3.3ex);[densely dotted] (5ex, 6ex) .. controls (6ex, 5ex) and (4ex, 3.5ex) .. (6ex, 3.5ex);[decorate,decoration={brace,amplitude=.5ex,raise=.5ex},yshift=0pt] (6.2ex, 3.7ex) -- (6.2ex, 2.3ex);[decorate,decoration={brace,amplitude=.5ex,raise=.4ex},yshift=0pt] (6ex, 6ex) -- (6ex, 4ex);[decorate,decoration={brace,amplitude=.3ex,raise=.2ex},yshift=0pt] (4.5ex, 6ex) -- (6ex, 6ex);(.4ex, 3ex) node[anchor=east] {$\\scriptstyle r$};(3ex, .2ex) node[anchor=north] {$\\scriptstyle r$};(6.5ex, 3ex) node[anchor=west] {$\\scriptstyle 2 \\alpha _r$};(5.2ex, 6ex) node[anchor=south] {$\\scriptstyle r/c_{\\ref {c:box_height}}$};(6.3ex, 5ex) node[anchor=west] {$\\scriptstyle \\tfrac{r}{3}$};}}$ ] + P[ ] 116 + cREF 2 18, where we also have used (REF ), (REF ) and that $c_{\\ref {c:prob_dual}}<1/8$ .",
"Therefore $\\mathbb {P}_\\lambda \\bigg [ \\hbox{[scale=1.5]{(0, 0) rectangle (4ex, 4ex);(3.7ex, 1.3ex) -- (4.3ex, 1.3ex) (3.8ex, 2ex) -- (4.2ex, 2ex) (3.7ex, 2.7ex) -- (4.3ex, 2.7ex);(0, 2ex) .. controls (1.3ex, 1.4ex) and (2.6ex, 4ex) .. (4ex, 3.4ex);[decorate,decoration={brace,amplitude=.3ex,raise=.4ex},yshift=0pt] (4.2ex, 2.7ex) -- (4.2ex, 1.3ex);(.4ex, 2ex) node[anchor=east] {$\\scriptstyle r$};(2ex, .2ex) node[anchor=north] {$\\scriptstyle r$};(4.4ex, 2ex) node[anchor=west] {$\\scriptstyle \\alpha _r$};}}$ ] 18.",
"To finish the bound in (REF ), we are going to apply the FKG inequality, see (REF ) above.",
"We start by defining the events $A_1, A_2,A_3, A_4$ obtained by reflecting the above square along the vertical and horizontal axis.",
"Of course the above bound will remain valid by the assumed symmetry of the system.",
"Also, using the fact that the law is invariant with respect to rotations by right angles, the probability of observing a vertical crossing of the above box is at least one half, by (REF ).",
"It is not difficult to see that the fork event occurs as soon as there is a vertical crossing of the box, together with the four events $A_1, A_2, A_3, A_4$ .",
"Plugging the above together with the FKG-inequality leads to the bound in (REF ).",
"Lemma 4.11 Let $\\tilde{\\alpha }_{c_{\\ref {c:box_height}}r} := \\alpha _{c_{\\ref {c:box_height}}r} \\wedge \\big ( \\tfrac{2}{3}r\\big )$ .",
"Then, for some $c_{\\ref {c:hit_middle}} \\in (0, c_{\\ref {c:easy_cross}})$ , we have $\\inf _{r\\ge r_0} \\mathbb {P}_\\lambda \\bigg [ \\hbox{\\;[scale=1.5]{(0, 0) rectangle (3ex, 4ex);(2.8ex, 2ex) -- (3.2ex, 2ex) (2.8ex, 2.8ex) -- (3.2ex, 2.8ex);[solid](0, 2ex) .. controls (1ex, 3.5ex) and (2ex, 1ex) .. (3ex, 2.6ex);(1.5ex, .2ex) node[anchor=north] {$\\scriptstyle r$};(.4ex, 2ex) node[anchor=east] {$\\scriptstyle \\tfrac{4}{3}r$};(2.8ex, 2.4ex) node[anchor=west] {$\\scriptstyle \\tilde{\\alpha }_{c_{\\ref {c:box_height}} r}$};}}$ ] cREF and rr0 P[ ] cREF .",
"where the above event is defined as $\\lbrace 0\\rbrace \\times [0, 4r/3] \\overset{K}{\\leftrightarrow }A$ , with $K = [0,r] \\times [0,4r/3]$ and $A = \\lbrace r\\rbrace \\times [2r/3, 2r/3 + \\tilde{\\alpha }_{c_{\\ref {c:box_height}} r}]$ (and analogously for the dual).",
"We only show the first inequality above, since the dual case is completely analogous.",
"For this we split the proof in two cases.",
"Either $\\alpha _{c_{\\ref {c:box_height}} r} \\ge \\tfrac{2}{3}r$ , in which case $\\tilde{\\alpha }_{c_{\\ref {c:box_height}} r} = 2r/3$ and the result follows from (REF ), together with the vertical reflection symmetry of the system and a union bound.",
"We now treat the case $\\tilde{\\alpha }_{c_{\\ref {c:box_height}} r} =\\alpha _{c_{\\ref {c:box_height}} r} < \\tfrac{2}{3}r$ .",
"We first recall that (REF ) gives $\\mathbb {P}_\\lambda \\Bigg [ \\hbox{[scale=1.5]{(0, 2ex) .. controls (1.4ex, 1ex) and (.6ex, 3ex) .. (2ex, 3.3ex);[color=white] (0, 4ex) .. controls (1.4ex, 5ex) and (.6ex, 3.2ex) .. (2ex, 3.7ex);(0, 0) rectangle (2ex, 6ex);(1.8ex, 2ex) -- (2.2ex, 2ex) (1.8ex, 3ex) -- (2.2ex, 3ex) (1.8ex, 4ex) -- (2.2ex, 4ex);[decorate,decoration={brace,amplitude=.5ex,raise=.4ex},yshift=0pt] (2.2ex, 4ex) -- (2.2ex, 2ex);(1ex, .2ex) node[anchor=north] {$\\scriptstyle r$};(0ex, 3ex) node[anchor=east] {$\\scriptstyle c_{\\ref {c:box_height}}r$};(2.4ex, 3ex) node[anchor=west] {$\\scriptstyle 2 \\alpha _{c_{\\ref {c:box_height}} r}$};}}$ ] cREF 2.",
"We now split the interval $\\lbrace r\\rbrace \\times [c_{\\ref {c:box_height}}/2,c_{\\ref {c:box_height}}/2 + \\alpha _{c_{\\ref {c:box_height}} r}]$ into eight equal parts.",
"Then, using the union bound, there must exist some $h_r \\in \\lbrace c_{\\ref {c:box_height}}r/2 + i \\alpha _{c_{\\ref {c:box_height}} r}/8; i = 0, \\dots ,7\\rbrace $ , such that $\\mathbb {P}_\\lambda \\Bigg [ \\hbox{[scale=1.5]{(0, 2ex) .. controls (1.4ex, 1ex) and (.6ex, 3.3ex) .. (2ex, 3.5ex);(0, 0) rectangle (2ex, 6ex);(1.8ex, 3.3ex) -- (2.2ex, 3.3ex) (1.8ex, 3.8ex) -- (2.2ex, 3.8ex);(1ex, .2ex) node[anchor=north] {$\\scriptstyle r$};(0ex, 3ex) node[anchor=east] {$\\scriptstyle c_{\\ref {c:box_height}}r$};(2ex, 3.5ex) node[anchor=west] {$\\scriptstyle I_r$};}}$ ] cREF 16, where $I_r = \\lbrace r\\rbrace \\times [h_r, h_r + \\alpha _{c_{\\ref {c:box_height}} r}/8]$ .",
"Since we are assuming $\\alpha _{c_{\\ref {c:box_height}} r} < 2r/3$ , the length of $I_r$ is no larger than $r/12$ .",
"Thus $\\begin{split}\\mathbb {P}_\\lambda \\bigg [ \\hbox{\\;[scale=1.5]{(0, 0) rectangle (3ex, 4ex);(2.8ex, 2ex) -- (3.2ex, 2ex) (2.8ex, 2.8ex) -- (3.2ex, 2.8ex);[solid](0, 2ex) .. controls (1ex, 3.5ex) and (2ex, 1ex) .. (3ex, 2.6ex);(1.5ex, .2ex) node[anchor=north] {$\\scriptstyle r$};(.4ex, 2ex) node[anchor=east] {$\\scriptstyle \\tfrac{4}{3}r$};(2.8ex, 2.4ex) node[anchor=west] {$\\scriptstyle \\tilde{\\alpha }_{c_{\\ref {c:box_height}} r}$};}}\\end{split} \\bigg ] & \\ge \\mathbb {P}_\\lambda \\Bigg [ \\hbox{[scale=1.5]{(0, 0) rectangle (3ex, 4ex);(2.8ex, 2ex) -- (3.2ex, 2ex);(0, 2ex) .. controls (1ex, 3ex) and (2ex, 1ex) .. (3ex, 3ex);(1.5ex, .2ex) node[anchor=north] {$\\scriptstyle r$};(0ex, 2ex) node[anchor=east] {$\\scriptstyle \\tfrac{5}{4}r$};}\\;}$ ] FKG, (REF ), (REF )(cREF 2)2 cREF 16 := cREF Where above we have made a slight abuse of notation in the second term above: The first two events in the triple intersection need to be translated vertically.",
"This concludes the proof.",
"We now define $p_r:= \\mathbb {P}_\\lambda \\bigg [ \\; \\hbox{[scale=1.5]{(0,0) rectangle (4ex, 3ex);[solid] (0, 1.5ex) .. controls (1.3ex, 2.5ex) and (2.6ex, .5ex) .. (4ex, 1.5ex);(2ex, .2ex) node[anchor=north] {$\\scriptstyle 3r/2$};(3.6ex, 1.5ex) node[anchor=west] {$\\scriptstyle \\tfrac{4}{3} r$};}}$ ] and pr:= P[ ].",
"Lemma 4.12 If $\\alpha _{r/2} \\ge \\frac{2}{5 c_{\\ref {c:box_height}}} \\alpha _{c_{\\ref {c:box_height}}r}$ for some $r \\ge 2 r_0$ , then $\\max \\lbrace p_r, p_r^\\star \\rbrace \\ge c_{\\ref {c:rsw}} =c_{\\ref {c:rsw}}(c_{\\ref {c:box_height}}, c_{\\ref {c:hit_middle}})$ .",
"Using Lemma REF we can assume without loss of generality that $\\mathbb {P}_\\lambda \\bigg [ \\hbox{[scale=1.5]{(0, 0) rectangle (4ex, 4ex);(-.2ex, 1.5ex) -- (.2ex, 1.5ex) (-.2ex, 2.5ex) -- (.2ex, 2.5ex);(3.8ex, 1.5ex) -- (4.2ex, 1.5ex) (3.8ex, 2.5ex) -- (4.2ex, 2.5ex);[solid] (0, 3ex) .. controls (.5ex, 3ex) and (1ex, 2.5ex).. (1ex, 2ex) .. controls (1ex, 1.5ex) and (.5ex, 1ex) .. (0ex, 1ex);[solid] (1ex, 2ex) .. controls (1.6ex, 3ex) and (2.4ex, 1ex) .. (3ex, 2ex);[solid] (4ex, 3ex) .. controls (3.5ex, 3ex) and (3ex, 2.5ex).. (3ex, 2ex) .. controls (3ex, 1.5ex) and (3.5ex, 1ex) .. (4ex, 1ex);[decorate,decoration={brace,amplitude=.7ex,raise=.4ex},yshift=0pt] (0, 0) -- (0, 4ex);(2ex, .2ex) node[anchor=north] {$\\scriptstyle r/2$};(-.5ex, 2ex) node[anchor=east] {$\\scriptstyle r/2$};(3.8ex, 2ex) node[anchor=west] {$\\scriptstyle 2 \\smash{\\alpha _{r/2}}$};}}$ ] 2-13 and then prove that $p_r\\ge c_{\\ref {c:rsw}}$ .",
"If on the other hand the above bound holds for the dual, an identical proof shows that $p^\\star _r\\ge c_{\\ref {c:rsw}}$ .",
"Recall that we are assuming $\\alpha _{r/2} \\ge \\frac{2}{5c_{\\ref {c:box_height}}} \\alpha _{c_{\\ref {c:box_height}} r} \\ge \\frac{2}{5c_{\\ref {c:box_height}}} \\tilde{\\alpha }_{c_{\\ref {c:box_height}} r}$ .",
"With this, we define the intervals $I_j = \\lbrace r\\rbrace \\times [(2/3) r+ (j-1) \\alpha _{r/ 2},(2/3) r+ j \\alpha _{r/ 2}]$ , for $j = 1, \\dots , \\lceil 2 c_{\\ref {c:box_height}}/ 5 \\rceil $ , which cover the interval $A$ defined below (REF ).",
"Therefore, the union bound yields $\\begin{split}\\max _{j \\le \\lceil 2 c_{\\ref {c:box_height}}/5 \\rceil } \\mathbb {P}_\\lambda & \\bigg [ \\hbox{\\;[scale=1.5]{(0, 0) rectangle (3ex, 4ex);(2.8ex, 2.2ex) -- (3.2ex, 2.2ex) (2.8ex, 2.5ex) -- (3.2ex, 2.5ex);[solid](0, 2ex) .. controls (1ex, 3.5ex) and (2ex, 2ex) .. (3ex, 2.4ex);(1.5ex, .2ex) node[anchor=north] {$\\scriptstyle r$};(.4ex, 2ex) node[anchor=east] {$\\scriptstyle \\tfrac{4}{3}r$};(2.8ex, 2.4ex) node[anchor=west] {$\\scriptstyle I_j$};}}\\end{split} \\bigg ]\\, \\ge \\, \\frac{1}{\\lceil 2 c_{\\ref {c:box_height}} / 5 \\rceil } \\sum _{j \\le \\lceil 2 c_{\\ref {c:box_height}} / 5 \\rceil } \\mathbb {P}_\\lambda \\bigg [ \\hbox{\\;[scale=1.5]{(0, 0) rectangle (3ex, 4ex);(2.8ex, 2.2ex) -- (3.2ex, 2.2ex) (2.8ex, 2.5ex) -- (3.2ex, 2.5ex);[solid](0, 2ex) .. controls (1ex, 3.5ex) and (2ex, 2ex) .. (3ex, 2.4ex);(1.5ex, .2ex) node[anchor=north] {$\\scriptstyle r$};(.4ex, 2ex) node[anchor=east] {$\\scriptstyle \\tfrac{4}{3}r$};(2.8ex, 2.4ex) node[anchor=west] {$\\scriptstyle I_j$};}}$ ] 12 cREF / 5 P[ ] cREF 2 cREF / 5 =: cREF .",
"Let $j_o$ be the index attaining the maximum in the left hand side of the above equation.",
"Since the interval $I_{j_o}$ has length $\\alpha _{r/ 2}$ , it can be covered by the small interval appearing in (REF ), such as illustrated in the picture below $p_r\\,\\ge \\, \\mathbb {P}_\\lambda \\Biggl [ \\hbox{[scale=1.5]{(0, 0) rectangle (4ex, 4ex);(-4ex, -3.5ex) rectangle (4ex, 4.5ex) (0, -3.5ex) rectangle (8ex, 4.5ex);(-.2ex, 1.5ex) -- (.2ex, 1.5ex) (-.2ex, 2.5ex) -- (.2ex, 2.5ex);(3.8ex, 1.5ex) -- (4.2ex, 1.5ex) (3.8ex, 2.5ex) -- (4.2ex, 2.5ex);(0, 3ex) .. controls (.5ex, 3ex) and (1ex, 2.5ex).. (1ex, 2ex) .. controls (1ex, 1.5ex) and (.5ex, 1ex) .. (0ex, 1ex);(1ex, 2ex) .. controls (1.6ex, 3ex) and (2.4ex, 1ex) .. (3ex, 2ex);(4ex, 3ex) .. controls (3.5ex, 3ex) and (3ex, 2.5ex).. (3ex, 2ex) .. controls (3ex, 1.5ex) and (3.5ex, 1ex) .. (4ex, 1ex);(-4ex, -1ex) .. controls (0ex, -2ex) and (0ex, 1ex).. (4ex, 2ex);(0ex, 2ex) .. controls (3ex, 4ex) and (5ex, 3ex).. (8ex, 4ex);[decorate,decoration={brace,amplitude=.5ex,raise=.2ex},yshift=0pt] (4ex, -3.5ex) -- (0ex, -3.5ex);[decorate,decoration={brace,amplitude=.7ex,raise=.2ex},yshift=0pt] (-4ex, -3.5ex) -- (-4ex, 4.5ex);(-4.3ex, .5ex) node[anchor=east] {$\\scriptstyle \\tfrac{4}{3}r$};(2ex, -3.7ex) node[anchor=north] {$\\scriptstyle r/2$};[fill, color=white](4.4ex, 1.1ex) rectangle (6.2ex, 2.7ex);[decorate,decoration={brace,amplitude=.2ex,raise=.2ex},yshift=0pt] (4.2ex, 2.5ex) -- (4.2ex, 1.5ex);(4.2ex, 2ex) node[anchor=west] {$\\scriptstyle 2 \\smash{\\alpha _{r/2}}$};}}$ ] FKG2-13 cREF 2 =: cREF , finishing the proof of the lemma.",
"Remark 10 It is important to observe that $c_{\\ref {c:rsw}}$ is strictly smaller than $c_{\\ref {c:hit_middle}}^2$ , as one can clearly see from the definitions of $c_{\\ref {c:hit_smaller}}$ and $c_{\\ref {c:rsw}}$ .",
"This will be used in Lemma REF below.",
"Lemma 4.13 There exists a constant $c_{\\ref {c:tiny_rsw}} > 0$ such that the following holds.",
"Suppose that for some $r \\ge r_0$ , $p_r\\ge c_{\\ref {c:rsw}}$ (respectively $p_r^\\star \\ge c_{\\ref {c:rsw}}$ ) and that for some ${r^{\\prime }}\\ge 30 r$ we have $\\alpha _{c_{\\ref {c:box_height}}{r^{\\prime }}} \\le r$ .",
"Then $p_{r^{\\prime }}\\ge c_{\\ref {c:tiny_rsw}}$ (respectively $p_{r^{\\prime }}^\\star \\ge c_{\\ref {c:tiny_rsw}}$ ).",
"Using the fact that $p_{r} \\ge c_{\\ref {c:rsw}}$ and Inequalities (REF ) and (REF ) in Lemma REF , we can conclude that $\\mathbb {P} \\big ( \\operatorname{Circ}(2 r, 4 r) \\big ) \\ge c_{\\ref {c:rsw}}^{124},$ We now recall (REF ) and our assumption that $\\alpha _{c_{\\ref {c:box_height}}{r^{\\prime }}} \\le r$ .",
"Using symmetry and the FKG-inequality, we obtain $p_{{r^{\\prime }}} \\,=\\, {\\mathbb {P}}_\\lambda \\big (\\operatorname{Cross}(3 {r^{\\prime }}/2, 4 {r^{\\prime }}/3)\\big ) \\,\\ge \\, \\mathbb {P}_\\lambda \\Biggl [ \\hbox{[scale=1.5]{(0, 0) rectangle (6ex, 8ex) (6ex, 0) rectangle (12ex, 8ex);(5.8ex, 4.2ex) -- (6.2ex, 4.2ex) (5.8ex, 4.8ex) -- (6.2ex, 4.8ex);[solid] (0, 3ex) .. controls (2ex, 2ex) and (4ex, 5ex) .. (6ex, 4.65ex);[solid] (6ex, 4.35ex) .. controls (8ex, 4ex) and (8ex, 7ex) .. (12ex, 6ex);[solid,decorate,decoration={brace,amplitude=.7ex,raise=.2ex},yshift=0pt] (12ex, 0ex) -- (0, 0);(5ex, 3.5ex) rectangle (7ex, 5.5ex);(3ex, 1.5ex) rectangle (9ex, 7.5ex);(3ex, 1ex) -- (9ex, 1ex);(3ex, .8ex) -- (3ex, 1.2ex) (9ex, .8ex) -- (9ex, 1.2ex);[fill, color=white] (4ex, .2ex) rectangle (8ex, 1.3ex);[solid,rounded corners=.25em](4ex, 6.5ex) .. controls (6ex, 6.5ex + .4ex*rand) ..(8ex, 6.5ex) .. controls (8ex + .4ex*rand, 4.5ex) ..(8ex, 2.5ex) .. controls (6ex, 2.5ex + .4ex*rand) ..(4ex, 2.5ex) -- (4ex + .4ex*rand, 4.5ex) -- cycle;(6ex, -.4ex) node[anchor=north] {$\\scriptstyle 2 {r^{\\prime }}$};(.4ex, 4ex) node[anchor=east] {$\\scriptstyle \\frac{4}{3}{r^{\\prime }}$};(9.4ex, 3.5ex) -- (9.4ex, 5.5ex);(9.2ex, 3.5ex) -- (9.6ex, 3.5ex) (9.2ex, 5.5ex) -- (9.6ex, 5.5ex);(9ex, 4.5ex) node[anchor=west] {$\\scriptstyle 2 r$};(6ex, 1.7ex) node[anchor=north] {$\\scriptstyle 6 r$};}\\;\\;}$ ] cREF 2 cREF 124 =: cREF , finishing the proof of the lemma.",
"We have obtained in the previous lemma a condition for $p_{r^{\\prime }}\\ge c_{\\ref {c:tiny_rsw}}$ .",
"However, the constant is smaller than the original $c_{\\ref {c:rsw}}$ that lower bounded $p_r$ in the hypothesis of Lemma REF .",
"The purpose of the next lemma is to use the above in order to bootstrap the lower bound of $p_{r^{\\prime }}$ back to $c_{\\ref {c:rsw}}$ .",
"Lemma 4.14 There exists a constant $c_{\\ref {c:boost_rsw}} > 300$ such that the following holds.",
"Suppose that for some $r \\ge r_0$ , $p_r\\ge c_{\\ref {c:rsw}}$ (respectively $p_r^\\star $ ) and that for every ${r^{\\prime }}\\in [30r, c_{\\ref {c:boost_rsw}} r]$ we have $\\alpha _{c_{\\ref {c:box_height}}{r^{\\prime }}} \\le r$ .",
"Then $p_{2 c_{\\ref {c:boost_rsw}} {r^{\\prime }}} \\ge c_{\\ref {c:rsw}}$ (respectively $p_{2 c_{\\ref {c:boost_rsw}} {r^{\\prime }}}^\\star \\ge c_{\\ref {c:rsw}}$ ).",
"We wish to apply the circuit argument of Lemma REF .",
"For this purpose let us fix a function $f=f_{\\lambda ,c}$ as in Item (REF ) of Lemma REF , corresponding to the constant $c:=c_{\\ref {c:tiny_rsw}}^{124}$ .",
"Recall from Remark REF that $c_{\\ref {c:rsw}} < c_{\\ref {c:hit_middle}}^2$ , so that we can choose $c_{\\ref {c:boost_rsw}} >300$ so that $1-f(\\tfrac{300}{c_{\\ref {c:boost_rsw}}}) \\ge \\frac{c_{\\ref {c:rsw}}}{c_{\\ref {c:hit_middle}}^2}.$ We only prove the statement for the dual quantities, the same proof also works for the primal ones (except that the function $f$ need to be chosen according to Item (REF ) in Lemma REF ).",
"Assume that $p_r^\\star \\ge c_{\\ref {c:rsw}}$ and that for every ${r^{\\prime }}\\in [30r, c_{\\ref {c:boost_rsw}} r]$ we have $\\alpha _{c_{\\ref {c:box_height}}{r^{\\prime }}} \\le r$ .",
"Using Lemma REF , we know that for every $r^{\\prime \\prime } \\in [30 r, c_{\\ref {c:boost_rsw}}r]$ , we have $p_{r^{\\prime \\prime }} \\ge c_{\\ref {c:tiny_rsw}}$ .",
"By the standard inequalities of Lemma REF , this implies in particular $\\inf _{60r\\le r^{\\prime \\prime }\\le c_{\\ref {c:boost_rsw}}r/10}{\\mathbb {P}}_\\lambda \\big (\\operatorname{Circ}^\\star (r^{\\prime \\prime },2r^{\\prime \\prime })\\big )\\ge c_{\\ref {c:tiny_rsw}}^{124}.$ Therefore, using the circuit argument (REF ) of Lemma REF , we obtain ${\\mathbb {P}}_\\lambda \\big (\\operatorname{Circ}^\\star (60r,\\tfrac{c_{\\ref {c:boost_rsw}}r}{5})\\big )\\ge 1-f(\\tfrac{300}{c_{\\ref {c:boost_rsw}}}) \\overset{~(\\ref {e:choose_boost})}{\\ge } \\frac{c_{\\ref {c:rsw}}}{c_{\\ref {c:hit_middle}}^2}.$ We can finally proceed as in the end of the proof of Lemma REF , observing that $p_{2 c_{\\ref {c:boost_rsw}} r}^\\star \\ge {\\mathbb {P}}_\\lambda (\\operatorname{Cross}^\\star (3c_{\\ref {c:boost_rsw}} r, c_{\\ref {c:boost_rsw}} r))$ , which can be lower bounded by $\\mathbb {P}_\\lambda \\Bigg [ \\hbox{[scale=1.5]{(0, 0) rectangle (6ex, 8ex) (6ex, 0) rectangle (12ex, 8ex);(5.8ex, 4.2ex) -- (6.2ex, 4.2ex) (5.8ex, 4.8ex) -- (6.2ex, 4.8ex);[densely dotted] (0, 3ex) .. controls (2ex, 2ex) and (4ex, 5ex) .. (6ex, 4.65ex);[densely dotted] (6ex, 4.35ex) .. controls (8ex, 4ex) and (8ex, 7ex) .. (12ex, 6ex);[densely dotted,decorate,decoration={brace,amplitude=.7ex,raise=.2ex},yshift=0pt] (12ex, 0ex) -- (0, 0);(5ex, 3.5ex) rectangle (7ex, 5.5ex);(3ex, 1.5ex) rectangle (9ex, 7.5ex);(3ex, 1ex) -- (9ex, 1ex);(3ex, .8ex) -- (3ex, 1.2ex) (9ex, .8ex) -- (9ex, 1.2ex);[fill, color=white] (4ex, .2ex) rectangle (8ex, 1.3ex);[densely dotted,rounded corners=.25em](4ex, 6.5ex) .. controls (6ex, 6.5ex + .4ex*rand) ..(8ex, 6.5ex) .. controls (8ex + .4ex*rand, 4.5ex) ..(8ex, 2.5ex) .. controls (6ex, 2.5ex + .4ex*rand) ..(4ex, 2.5ex) -- (4ex + .4ex*rand, 4.5ex) -- cycle;(6ex, -.4ex) node[anchor=north] {$\\scriptstyle 3 c_{\\ref {c:boost_rsw}}r$};(.4ex, 4ex) node[anchor=east] {$\\scriptstyle c_{\\ref {c:boost_rsw}} r$};(9.4ex, 3.5ex) -- (9.4ex, 5.5ex);(9.2ex, 3.5ex) -- (9.6ex, 3.5ex) (9.2ex, 5.5ex) -- (9.6ex, 5.5ex);(9ex, 4.5ex) node[anchor=west] {$\\scriptstyle 60r$};(6ex, 1.7ex) node[anchor=north] {$\\scriptstyle c_{\\ref {c:boost_rsw}} r/5$};}\\;\\;}$ ] cREF 2 cREF cREF 2 = cREF , as required.",
"Lemma 4.15 Fix $r \\ge r_0$ and suppose that for some ${r^{\\prime }}\\in [30 r, c_{\\ref {c:boost_rsw}} r]$ we have $\\alpha _{c_{\\ref {c:box_height}} {r^{\\prime }}} \\ge r$ , then $\\max \\lbrace p_{r^{\\prime \\prime }}, p^\\star _{r^{\\prime \\prime }}\\rbrace \\ge c_{\\ref {c:rsw}}$ for some $r^{\\prime \\prime } \\in [{r^{\\prime }}, c_{\\ref {c:rsw_blows}} {r^{\\prime }}]$ .",
"Let us choose an integer $K = K(c_{\\ref {c:box_height}}, {r^{\\prime }}/r) \\ge 1$ such that $\\Big ( \\frac{5 c_{\\ref {c:box_height}}}{2} \\Big )^K r\\ge c_{\\ref {c:box_height}} (2 c_{\\ref {c:box_height}})^K {r^{\\prime }}$ and then set $c_{\\ref {c:rsw_blows}} := 2 (2 c_{\\ref {c:box_height}})^K.$ Observe now that if for some $k = 0, \\dots , K - 1$ we have $\\alpha _{c_{\\ref {c:box_height}}(2 c_{\\ref {c:box_height}})^k {r^{\\prime }}} \\ge \\big ( \\tfrac{2}{5 c_{\\ref {c:box_height}}} \\big ) \\alpha _{c_{\\ref {c:box_height}}(2 c_{\\ref {c:box_height}})^{k+1} {r^{\\prime }}},$ we can apply Lemma REF , replacing $r$ with $(2c_{\\ref {c:box_height}})^{k+1} {r^{\\prime }}$ .",
"This will yield the bound $\\max \\lbrace p_{(2 c_{\\ref {c:box_height}})^{k+1} {r^{\\prime }}}, p^\\star _{(2 c_{\\ref {c:box_height}})^{k+1} {r^{\\prime }}}\\rbrace \\ge c_{\\ref {c:rsw}}.$ By our choice of $c_{\\ref {c:rsw_blows}}$ , we conclude that ${r^{\\prime }}\\le (2 c_{\\ref {c:box_height}})^{k+1} {r^{\\prime }}\\le (2 c_{\\ref {c:box_height}})^K {r^{\\prime }}\\le c_{\\ref {c:rsw_blows}} {r^{\\prime }}$ , as desired.",
"On the other hand if (REF ) does not hold for any $k = 0, \\dots , K - 1$ , then $r \\le \\alpha _{c_{\\ref {c:box_height}} {r^{\\prime }}} \\le \\big ( \\tfrac{2}{5 c_{\\ref {c:box_height}}} \\big ) \\alpha _{c_{\\ref {c:box_height}}(2 c_{\\ref {c:box_height}}) {r^{\\prime }}} \\le \\dots \\le \\big ( \\tfrac{2}{5 c_{\\ref {c:box_height}}} \\big )^{K} \\alpha _{c_{\\ref {c:box_height}}(2 c_{\\ref {c:box_height}})^{K} {r^{\\prime }}},$ so that by our choice of $K$ we would have $\\alpha _{c_{\\ref {c:box_height}}(2 c_{\\ref {c:box_height}})^{K} {r^{\\prime }}} \\ge (5 c_{\\ref {c:box_height}} / 2)^{K} r \\ge c_{\\ref {c:box_height}}(2 c_{\\ref {c:box_height}})^{K} {r^{\\prime }}$ , which is a contradiction with (REF ).",
"Lemma 4.16 There exists $c_{\\ref {c:rsw_max}} > 0$ such that $\\inf _{r\\ge 1} \\big (\\max \\lbrace p_r, p^\\star _r\\rbrace \\big ) \\ge c_{\\ref {c:rsw_max}}.$ We first claim that there exists an increasing sequence $r_1\\le r_2\\le \\cdots $ such that $3\\le \\tfrac{r_{i+1}}{r_i}\\le \\max \\lbrace 2c_{\\ref {c:boost_rsw}}, c_{\\ref {c:rsw_blows}}\\rbrace $ for every $i \\ge 1$ and $\\max \\lbrace p_{r_i}, p^\\star _{r_i}\\rbrace \\ge c_{\\ref {c:rsw}}$ , for every $i \\ge 1$ .",
"We first construct $r_1$ .",
"Since $\\alpha _r\\le r/6$ for every $r\\ge 1$ (see (REF )), there must exist $k\\ge 1$ such that $\\alpha _{r_0(2 c_{\\ref {c:box_height}})^k} \\ge \\big ( \\tfrac{2}{5 c_{\\ref {c:box_height}}} \\big ) \\alpha _{r_0(2 c_{\\ref {c:box_height}})^{k+1}},$ Applying Lemma REF to $r_1:=2r_0(2c_{\\ref {c:box_height}})^{k}$ yields the bound $\\max \\lbrace p_{r_1}, p^\\star _{r_1}\\rbrace \\ge c_{\\ref {c:rsw}}$ as desired.",
"To finish the construction of the $(r_i)$ 's, it is enough to show that for any $r\\ge 1$ such that $p_r\\ge c_{\\ref {c:rsw}}$ , there exists $r^{\\prime \\prime } \\in [3r, \\max \\lbrace 2c_{\\ref {c:boost_rsw}}, c_{\\ref {c:rsw_blows}}\\rbrace r]$ for which $\\max \\lbrace p_{r^{\\prime \\prime }}, p^\\star _{r^{\\prime \\prime }}\\rbrace \\ge c_{\\ref {c:rsw}}$ .",
"We then split the proof of this claim into two cases, depending on whether or not there exists ${r^{\\prime }}\\in [3 r, c_{\\ref {c:boost_rsw}} r]$ such that $\\alpha _{c_{\\ref {c:box_height}} {r^{\\prime }}} \\ge r$ .",
"In this case, Lemma REF shows that the existence of $r^{\\prime \\prime } \\le c_{\\ref {c:rsw_blows}} r$ as in (REF ).",
"On the other hand, if (REF ) does not hold, than we can use $r^{\\prime \\prime } = 2c_{\\ref {c:boost_rsw}}$ , according to Lemma REF .",
"In any case we have proved (REF ), which shows the existence of $r_1,r_2, \\dots $ as above.",
"To end the proof we use the standard inequalities of Lemma REF to interpolate between the values $r_i$ .",
"We now have obtained a statement similar to the RSW result stated in Theorem REF .",
"However, we only know that the crossing probability is bounded below for the primal or the dual and not both at the same time.",
"It could be still the case that only the dual crossings have a bounded probability.",
"The purpose of the following lemma is to show that this cannot be the case.",
"Lemma 4.17 There exist constants $c_{\\ref {c:rsw_both_1}} \\ge 10$ and $c_{\\ref {c:rsw_both_2}}>0$ such that the following holds.",
"Given $r\\ge r_0$ , if $p_{{r^{\\prime }}}^\\star \\ge c_{\\ref {c:rsw_max}}$ for every ${r^{\\prime }}\\in [r, c_{\\ref {c:rsw_both_1}} r]$ , then $p_{c_{\\ref {c:rsw_both_1}} r} \\ge c_{\\ref {c:rsw_both_2}}$ .",
"Moreover, the same holds true if we swap the roles of $p^\\star $ and $p$ above.",
"We wish to apply the circuit argument of Lemma REF .",
"To this end, let us fix a function $f=f_{\\lambda ,c}$ as in Item (REF ) of Lemma REF , corresponding to the constant $c:=c_{\\ref {c:rsw_max}}^{124}$ .",
"We first choose $c_{\\ref {c:rsw_both_1}} = c_{\\ref {c:rsw_both_1}}(c_{\\ref {c:easy_cross}},c_{\\ref {c:rsw_max}}) \\ge 30$ large enough such that $f(\\tfrac{10}{c_{\\ref {c:rsw_both_1}}}) \\le \\frac{c_{\\ref {c:easy_cross}}}{3}$ and $c_{\\ref {c:rsw_both_2}}>0$ small enough that $\\big (1 - c_{\\ref {c:rsw_both_2}}^{1/(2c_{\\ref {c:rsw_both_1}})}\\big ) \\big ( 1 - \\frac{c_{\\ref {c:easy_cross}}}{3} \\big ) \\ge 1 - \\frac{c_{\\ref {c:easy_cross}}}{2}.$ Suppose now, for a contradiction, that $p_{c_{\\ref {c:rsw_both_1}} r} \\le c_{\\ref {c:rsw_both_2}}$ , therefore $\\mathbb {P}_\\lambda \\big [\\operatorname{Cross}^\\star \\big (\\tfrac{4}{3} c_{\\ref {c:rsw_both_1}} r, \\tfrac{3}{2} c_{\\ref {c:rsw_both_1}} r\\big ) \\big ] = 1 - p_{c_{\\ref {c:rsw_both_1}} r} \\ge 1 - c_{\\ref {c:rsw_both_2}}.$ Cover the interval $[0, \\tfrac{3}{2} c_{\\ref {c:rsw_both_1}} r]$ by at most $H =\\lceil \\tfrac{3}{2} c_{\\ref {c:rsw_both_1}} \\rceil \\le 2 c_{\\ref {c:rsw_both_1}}$ intervals of length $r$ .",
"The square-root trick (REF ) implies that there exists $h \\in \\lbrace 0, 1, \\dots , H-1\\rbrace $ , such that $\\mathbb {P}_\\lambda \\bigg [ \\hbox{\\;[scale=1.5]{(0, 0) rectangle (3ex, 4ex);(2.8ex, 2ex) -- (3.2ex, 2ex) (2.8ex, 2.8ex) -- (3.2ex, 2.8ex);[densely dotted](0, 2ex) .. controls (1ex, 3.5ex) and (2ex, 1ex) .. (3ex, 2.6ex);(1.5ex, .2ex) node[anchor=north] {$\\scriptstyle (4/3) c_{\\ref {c:rsw_both_1}} r$};(.4ex, 2ex) node[anchor=east] {$\\scriptstyle \\tfrac{3}{2} c_{\\ref {c:rsw_both_1}} r$};(2.8ex, 2.4ex) node[anchor=west] {$\\scriptstyle I_h$};}}$ ] 1 - cREF 1/H, where $I_h = \\lbrace \\tfrac{4}{3} c_{\\ref {c:rsw_both_1}} r\\rbrace \\times [h r, (h + 1) r]$ .",
"We now recall that $p_{{r^{\\prime }}}^\\star \\ge c_{\\ref {c:rsw_max}}$ for every ${r^{\\prime }}\\in [r, c_{\\ref {c:rsw_both_1}} r]$ .",
"Therefore, by the standard inequalities of Lemma REF , we obtain in particular $\\inf _{2r\\le r^{\\prime }\\le c_{\\ref {c:rsw_both_1}} r/10}{\\mathbb {P}}_\\lambda \\big (\\operatorname{Circ}^\\star (r^{\\prime },2r^{\\prime })\\big )\\,\\ge \\, c_{\\ref {c:rsw_max}}^{124}.$ Therefore, by the circuit argument (REF ) in Lemma REF , $\\mathbb {P} \\Big ( \\operatorname{Circ}^\\star (2 r, c_{\\ref {c:rsw_both_1}} r/ 5) \\Big ) \\,\\ge \\, 1-f(\\tfrac{10}{c_{\\ref {c:rsw_both_1}}}) \\,\\overset{(\\ref {e:choose_both_1})}{\\ge }\\, 1 - \\frac{c_{\\ref {c:easy_cross}}}{3}.$ As in (REF ), we obtain $\\mathbb {P}_\\lambda \\big ( \\operatorname{Cross}^\\star (\\tfrac{8}{3} c_{\\ref {c:rsw_both_1}} r, \\tfrac{3}{2} c_{\\ref {c:rsw_both_1}} r)\\big )\\,& \\ge \\, \\mathbb {P}_\\lambda \\Bigg [ \\hbox{[scale=1.5]{(0, 0) rectangle (6ex, 8ex) (6ex, 0) rectangle (12ex, 8ex);(5.8ex, 4.2ex) -- (6.2ex, 4.2ex) (5.8ex, 4.8ex) -- (6.2ex, 4.8ex);[solid] (0, 3ex) .. controls (2ex, 2ex) and (4ex, 5ex) .. (6ex, 4.65ex);[solid] (6ex, 4.35ex) .. controls (8ex, 4ex) and (8ex, 7ex) .. (12ex, 6ex);[solid,decorate,decoration={brace,amplitude=.7ex,raise=.2ex},yshift=0pt] (12ex, 0ex) -- (0, 0);(5ex, 3.5ex) rectangle (7ex, 5.5ex);(3ex, 1.5ex) rectangle (9ex, 7.5ex);(3ex, 1ex) -- (9ex, 1ex);(3ex, .8ex) -- (3ex, 1.2ex) (9ex, .8ex) -- (9ex, 1.2ex);[fill, color=white] (4ex, .2ex) rectangle (8ex, 1.3ex);[solid,rounded corners=.25em](4ex, 6.5ex) .. controls (6ex, 6.5ex + .4ex*rand) ..(8ex, 6.5ex) .. controls (8ex + .4ex*rand, 4.5ex) ..(8ex, 2.5ex) .. controls (6ex, 2.5ex + .4ex*rand) ..(4ex, 2.5ex) -- (4ex + .4ex*rand, 4.5ex) -- cycle;(6ex, -.4ex) node[anchor=north] {$\\scriptstyle \\tfrac{8}{3} c_{\\ref {c:rsw_both_1}} r$};(.4ex, 4ex) node[anchor=east] {$\\scriptstyle \\tfrac{3}{2} c_{\\ref {c:rsw_both_1}} r$};(9.4ex, 3.5ex) -- (9.4ex, 5.5ex);(9.2ex, 3.5ex) -- (9.6ex, 3.5ex) (9.2ex, 5.5ex) -- (9.6ex, 5.5ex);(9ex, 4.5ex) node[anchor=west] {$\\scriptstyle 2 r$};(6ex, 1.7ex) node[anchor=north] {$\\scriptstyle c_{\\ref {c:rsw_both_1}}r/5$};}\\;\\;}$ ] (1 - cREF 1/H)2 (1 - cREF 3 ) (REF ) 1 - cREF 2 which is a contradiction with (REF ).",
"Remark 11 In the proof of Lemmas REF and REF we have focused on the dual paths instead of primal to emphasize that we only needed the weak decoupling inequality (REF ) that has been derived from (REF ).",
"[Proof of Theorem REF ] We first claim that it is enough to prove that for any $r\\ge r_0$ there exists some ${r^{\\prime }}\\in [r, c_{\\ref {c:rsw_both_1}} r]$ such that $p^\\star _{{r^{\\prime }}}\\ge \\min \\lbrace c_{\\ref {c:rsw_max}}, c_{\\ref {c:rsw_both_2}}\\rbrace $ .",
"Indeed, if this is the case, we can use the standard inequalities in Lemma REF to show that $\\inf _{r\\ge 1} p^\\star _{r} > 0$ .",
"(The case $p_r$ is completely analogous.)",
"So, let us fix some $r\\ge r_0$ and assume that $p^\\star _{{r^{\\prime }}} < c_{\\ref {c:rsw_max}}$ for every ${r^{\\prime }}\\in [r,c_{\\ref {c:rsw_both_1}} r]$ , then by Lemma REF , we would have $p_{{r^{\\prime }}} \\ge c_{\\ref {c:rsw_max}}$ for every ${r^{\\prime }}\\in [r, c_{\\ref {c:rsw_both_1}} r]$ .",
"But then Lemma REF implies that $p^\\star _{c_{\\ref {c:rsw_both_1}} r} \\ge c_{\\ref {c:rsw_both_2}}$ as desired."
],
[
"Sharpness of the phase transition",
"We have so far showed that decay of correlations is sufficient for the two thresholds $\\lambda _0$ and $\\lambda _1$ to be non-trivial; see Section .",
"Moreover, we have shown that a distinctive critical behavior, namely the box-crossing property, occurs throughout the interval $[\\lambda _0,\\lambda _1]$ ; see Section .",
"Our goal for this section is to prove that these two thresholds $\\lambda _0$ and $\\lambda _1$ are in fact the same.",
"Theorem 5.1 For Poisson Boolean percolation satisfying (REF ), we have $\\lambda _0 = \\lambda _1$ .",
"In order to establish an equality between the two thresholds $\\lambda _0$ and $\\lambda _1$ , our aim will be to show that the crossing probabilities grow very fast from close to zero to close to one.",
"More precisely, we have seen that the probability of crossing a fixed-ratio rectangle is bounded away from zero and one for $\\lambda \\in [\\lambda _0, \\lambda _1]$ .",
"In this section, we show that this implies that the derivative of the crossing probability is large throughout the same interval.",
"The phenomenon of sharp thresholds is well understood in the context of Boolean functions on the discrete cube $\\lbrace 0,1\\rbrace ^n$ equipped with product measure.",
"In order to apply the tools from this theory in our context we will following an approach introduced in [1].",
"Thus, the first step of our proof will be to enhance the probability space in which we work to suit this setting, see Subsection REF .",
"After having defined this alternative construction, we will employ Margulis-Russo's formula to relate the derivative of crossing probabilities to influences of these auxiliary random variables.",
"At this point, an inequality of Talagrand will become handy in order to bound the total influence from below.",
"This is done in two steps, where we bound the conditional variance of the crossing probabilities from below and the individual influences from above, see Subsections REF and REF respectively.",
"Subsection REF combines the above bounds to prove Theorem REF .",
"Remark 12 Of the three main theoretical components of this paper, the one presented in this section is the least general one.",
"Although what we will present below does apply to some other models, it strongly uses the Poissonian nature of the process, see Section ."
],
[
"Sharp thresholds for Boolean functions",
"It has been known for some time that monotone events involving a large number of independent Bernoulli variables typically exhibit sharp thresholds, in the sense that the probability of the event increases sharply as the parameter of the Bernoulli variables passes a certain value.",
"The first evidence thereof dates back to the study of random graphs by Erdős and Rényi [14], and the proof that the critical probability for bond percolation on ${\\mathbb {Z}}^2$ equals $1/2$ by Kesten [23].",
"A more general understanding of these phenomena has been pursued since in work of Russo [36], Bollobás and Thomason [12], and Kahn, Kalai and Linial [25] as well as elsewhere.",
"See e.g.",
"[31] or [19].",
"For $p \\in [0, 1]$ let $\\textup {P}_p$ denote product measure of intensity $p$ on $\\lbrace 0,1\\rbrace ^n$ .",
"A Boolean function $f : \\lbrace 0, 1\\rbrace ^n \\rightarrow \\lbrace 0, 1\\rbrace $ is said to be monotone when $f(\\eta ) \\le f(\\eta ^{\\prime })$ whenever $\\eta \\le \\eta ^{\\prime }$ coordinate wise.",
"For monotone Boolean functions, the map $p \\mapsto \\textup {P}_p(f = 1)$ is also monotone as a function of $p$ .",
"Moreover, the rate of change of $\\textup {P}_p(f = 1)$ is related to the concept of influences of the individual variables, as made explicit by the Margulis-Russo formula (see [34]), $\\frac{d}{dp} \\textup {P}_p(f = 1) \\,=\\, \\sum _{i = 1}^n \\operatorname{Inf}_i^p(f),$ where $\\operatorname{Inf}_i^p(f) := \\textup {P}_p\\big (f(\\eta ) \\ne f(\\sigma _i\\eta )\\big )$ and $\\sigma _i$ is the operator that flips the value of $\\eta $ at position $i$ .",
"In case $f(\\eta ) \\ne f(\\sigma _i\\eta )$ , we say that the bit $i$ is pivotal for $f(\\eta )$ .",
"Against one's initial intuition, Russo [36] further showed that a uniform upper bound on the individual influences implies a lower bound of their sum, thus assuring a rapid increase of $\\textup {P}_p(f=1)$ .",
"This mystique has since become better understood with the work of Kahn, Kalai and Linial [25] and its extensions, which shows that not all influences may be too small.",
"The following inequality, due to Talagrand [41], illustrates this fact well and gives a precise formulation of Russo's initial observation: There exists a constant $c_{\\ref {c:talagrand}} > 0$ such that for every Boolean function $f:\\lbrace 0,1\\rbrace ^n\\rightarrow \\lbrace 0,1\\rbrace $ we have $\\sum _{i=1}^n \\operatorname{Inf}_i^p(f) \\,\\ge \\, \\frac{c_{\\ref {c:talagrand}}}{\\log \\big ( \\tfrac{2}{p (1 - p)} \\big )} \\, \\operatorname{Var}_p(f) \\log \\Big ( \\frac{1}{\\max \\lbrace \\operatorname{Inf}_i^p(f)\\rbrace } \\Big ).$ Talagrand stated this inequality for monotone functions, but this restriction is not necessary.",
"From (REF ) and (REF ), the strategy of the proof becomes intuitive: We first give an alternative construction of the process involving independent Bernoulli random variables, then we bound the variance of the crossing probabilities from below and the influence of individual bits from above.",
"These are the contents of the next subsections."
],
[
"A two-stage construction",
"We will for the rest of this section work on an enlarged probability space $(\\overline{\\Omega },\\overline{\\mathcal {M}},\\overline{{\\mathbb {P}}}_\\lambda )$ in order to construct our process in two stages, where the second step employs independent coin flips in order to define the final configuration.",
"In this way we will be able to condition on the outcome of the first step in order to arrive at a situation where tools from the discrete setting applies.",
"Recall that $\\Omega $ denotes the space of locally finite point measures on ${\\mathbb {R}}^2 \\times {\\mathbb {R}}_+$ .",
"We will by $\\overline{\\Omega }$ denote the space of locally finite point measures on ${\\mathbb {R}}^2 \\times {\\mathbb {R}}_+ \\times [0, 1] \\times [0, 1]$ , by $\\overline{\\mathcal {M}}$ we denote the corresponding Borel sigma algebra, and we let $\\overline{{\\mathbb {P}}}_\\lambda $ denote the law of a Poisson point process defined on $(\\overline{\\Omega },\\overline{\\mathcal {M}})$ with intensity $\\lambda \\,dx\\,\\mu (dz)\\,du\\,dv$ .",
"In this section we will use a very dense set of points and the last two coordinates $u$ and $v$ in the above construction will be used to perform a thinning of the Poisson process.",
"Throughout this section we will suppose that $m\\ge 1$ is a large integer and that $\\lambda =2m\\lambda _1$ is fixed.",
"Given $\\overline{\\omega }\\in \\overline{\\Omega }$ we denote by $\\omega $ the projection of $\\overline{\\omega }$ onto $\\Omega $ .",
"For $p\\in [0,1]$ and $m\\ge 1$ we define $\\omega _p:=\\sum _{\\begin{array}{c}(x,z,u,v)\\in \\overline{\\omega }\\\\ u\\le p,\\, v\\le 1/m\\end{array}}\\delta _{(x,z)}\\quad \\text{and}\\quad \\mathcal {O}_p:=\\bigcup _{(x,z)\\in \\omega _p} B(x,z).$ The usual properties of a Poisson point process imply that $\\omega $ and $\\omega _p$ are Poisson point processes on $\\Omega $ distributed according to ${\\mathbb {P}}_\\lambda $ and ${\\mathbb {P}}_{p\\lambda /m}$ , respectively.",
"Notice that $\\omega _p$ is a $(p/m)$ -thinning of $\\omega $ , and therefore it can be seen as a percolation process on $\\lbrace 0,1\\rbrace ^\\omega $ with density $p/m$ .",
"Since $\\lambda =2m\\lambda _1$ , the above construction comes with two parameters, $p$ and $m$ , of which we think of $m$ as large and fixed while $p$ will be varying.",
"We have for this reason suppressed $m$ in the above notation, and apply a subscript $p$ to indicate that the events $\\operatorname{Cross}_p$ , $\\operatorname{Circ}_p$ and $\\operatorname{Arm}_p$ refer to crossings in $\\mathcal {O}_p$ .",
"We will study the conditional probability $Z_p:=\\overline{{\\mathbb {P}}}_\\lambda \\big (\\operatorname{Cross}_p(4r,r)\\big |\\omega \\big ).$ The product structure of the above conditional measure implies that for ${\\mathbb {P}}_\\lambda $ -almost every $\\omega \\in \\Omega $ the variable $Z_p$ can be viewed as the expectation of some Boolean function $f_\\omega :\\lbrace 0,1\\rbrace ^\\omega \\rightarrow \\lbrace 0,1\\rbrace $ .",
"While $\\omega $ has infinite support, it follows from (REF ) that $\\overline{{\\mathbb {P}}}_\\lambda $ -almost surely, only finitely many balls touch the box $[-4r, 4r]^2$ .",
"Therefore the function $f_\\omega $ can be viewed as having only finitely many variables that we index by $i \\in \\lbrace 1, \\dots , i_0\\rbrace $ ($i_0$ is the number of balls touching $[-4r, 4r]^2$ ).",
"In this conditional setting, the tools from the discrete analysis will apply, in particular the Margulis-Russo formula and Talagrand's inequality.",
"With the use of these we will be able to bound the gap between $\\lambda _0$ and $\\lambda _1$ by controlling the conditional variance of $f_\\omega $ and its influences, defined, for $\\overline{{\\mathbb {P}}}_\\lambda $ -almost every $\\overline{\\omega }\\in \\Omega $ , as $\\operatorname{Inf}_i^p(f_\\omega ):=\\overline{{\\mathbb {P}}}_\\lambda \\big (f_\\omega (\\omega _p)\\ne f_\\omega (\\sigma _i\\omega _p)\\big |\\omega \\big ),$ where $\\sigma _i\\omega _p=\\omega _p+(1-2\\operatorname{{\\bf 1}}_{\\lbrace u_i\\le p,\\,v_i\\le 1/m\\rbrace })\\delta _{(x_i,z_i)}$ ."
],
[
"Quenched decoupling",
"In order to handle dependencies in the quenched setting we will need an alternative to the decoupling inequality proved in Proposition REF .",
"Given $\\overline{\\omega }\\in \\overline{\\Omega }$ , this can be done via an estimate on the influence region of each point in the support of $\\omega $ .",
"Since we will only consider events defined in terms of $\\mathcal {O}_p$ on $[-4r,4r]^2$ it will suffice to consider points intersecting this region.",
"For $r\\ge 3n\\ge 3$ , let $G = G(r,n)$ denote the event $G := \\Big [ \\max \\big \\lbrace z:(x, z)\\in \\operatorname{supp}(\\omega )\\text{ and } B(x, z) \\cap [-4r, 4r]^2 \\ne \\emptyset \\big \\rbrace \\le r/3n \\Big ].$ Lemma 5.2 Assume the second moment condition (REF ).",
"For every $n\\ge 1$ , $\\lim _{r\\rightarrow \\infty } \\overline{{\\mathbb {P}}}_\\lambda (G) = 1$ .",
"The proof follows the same steps as that of Lemma REF .",
"A first consequence of the above observation concerns quenched decouplings of events defined in well separated regions.",
"To state this precisely, fix integers $n, J \\ge 1$ and let $D_1, \\dots , D_J \\subseteq [-4 r, 4 r]^2$ be measurable sets such that $d(D_i, D_j) \\ge \\frac{r}{n}, \\text{ for every $i \\ne j$}.$ Having these well separated sets, our aim is to decouple what happens inside each of them.",
"For this, fix some events $A_1, \\dots , A_J$ that only look inside the sets $D_j$ , that is $A_j \\in \\sigma (Y_x, x \\in D_j)$ .",
"Observe that on the event $G = G(r, n)$ the events $A_j$ depend on disjoint subsets of $\\omega $ under the conditional measure $\\overline{{\\mathbb {P}}}_\\lambda (\\,\\cdot \\,|\\omega )$ .",
"This implies that $\\overline{{\\mathbb {P}}}_\\lambda \\big ( A_1 \\cap \\dots \\cap A_J \\big | \\omega \\big ) \\,\\le \\, \\operatorname{{\\bf 1}}_{G^c}(\\omega ) + \\prod \\nolimits _{j \\le J} \\overline{{\\mathbb {P}}}_\\lambda (A_j | \\omega ).$ For us it is important to observe that not only the events $A_j$ decouple on $G(r, n)$ , but also their conditional probabilities $\\overline{{\\mathbb {P}}}_\\lambda (A_j | \\omega )$ .",
"To see why this is true, let us first decompose the point measure $\\omega $ as $\\omega _1 + \\dots + \\omega _J + \\tilde{\\omega }$ , where $\\omega _j = \\sum _i \\delta _{(x_i, z_i)}$ with the sum ranging over $i \\ge 0$ such that $d(x_i, D_j) \\le r/(3n)$ and $\\tilde{\\omega }$ stands for the remainder.",
"Observe that on $G$ , the conditional probabilities $\\overline{{\\mathbb {P}}}_\\lambda (A_j | \\omega )$ depend on $\\omega $ through $\\omega _j$ only.",
"In particular, we may exhibit a function $g_j$ such that $[\\operatorname{{\\bf 1}}_G \\cdot \\overline{{\\mathbb {P}}}_\\lambda (A_j | \\omega )](\\omega ) = \\operatorname{{\\bf 1}}_G \\cdot g_j(\\omega _j)$ , and by the Poissonian nature of $\\omega $ , all the $\\omega _j$ 's are independent.",
"As a consequence, for any $\\delta > 0$ , $\\begin{split}\\overline{{\\mathbb {P}}}_\\lambda \\Big (G \\cap \\bigcap _{j \\le J} \\big [ \\overline{{\\mathbb {P}}}_\\lambda (A_j | \\omega ) \\le \\delta \\big ] \\Big ) \\,& \\le \\, \\prod _{j \\le J} \\overline{{\\mathbb {P}}}_\\lambda \\big [ g_j(\\omega _j) \\le \\delta \\big ]\\\\& \\le \\, \\prod _{j \\le J}\\bigg [ \\overline{{\\mathbb {P}}}_\\lambda \\big ( \\overline{{\\mathbb {P}}}_\\lambda (A_j | \\omega ) \\le \\delta \\big )+\\overline{{\\mathbb {P}}}(G^c)\\bigg ].\\end{split}$ Equations (REF ) and (REF ) give a more precise version of the decoupling of Section and will be important in bounding the conditional variance and influences in the next two sections."
],
[
"Controlling the variance",
"In Section we showed that the probability of crossing a rectangle is non-degenerate throughout the critical regime $[\\lambda _0, \\lambda _1]$ .",
"In the notation used in this section, that is to say that $\\overline{\\operatorname{{\\mathbb {E}}}}_\\lambda [Z_p]\\in (c,1-c)$ for some $c>0$ and all $r\\ge 1$ and $p\\lambda / m \\in \\big [ \\lambda _0, \\lambda _1 \\big ]$ .",
"Recall that $\\lambda = 2 m \\lambda _1$ .",
"Our goal for this section is to show that this holds with probability close to one also for $Z_p$ .",
"Proposition 5.3 Assume the second moment condition (REF ).",
"Then, for every $\\varepsilon >0$ there exists $\\delta >0$ and $c_{\\ref {c:lambda}} = c_{\\ref {c:lambda}}(\\varepsilon ) > 0$ such that for every $m\\ge c_{\\ref {c:lambda}}$ we have $\\overline{{\\mathbb {P}}}_\\lambda \\bigg (Z_p\\in (\\delta ,1-\\delta )\\text{ for all }p\\in \\big [\\tfrac{\\lambda _0}{2\\lambda _1},\\tfrac{\\lambda _1}{2\\lambda _1}\\big ]\\bigg )\\ge 1-\\varepsilon \\quad \\text{for all }r\\ge 1.$ The same holds with $\\operatorname{Cross}_p(4r, r)$ replaced by either $\\operatorname{Cross}_p^\\star (4r, r)$ , $\\operatorname{Circ}_p(r, 2r)$ or $\\operatorname{Circ}_p^\\star (r, 2r)$ .",
"An immediate consequence of the above proposition is that $\\overline{{\\mathbb {P}}}_\\lambda \\bigg (\\operatorname{Var}_\\lambda (\\operatorname{{\\bf 1}}_{\\operatorname{Cross}_p(4r,r)}|\\omega )\\in (\\delta ,1-\\delta )\\text{ for all }p\\in \\big [\\tfrac{\\lambda _0}{2\\lambda _1},\\tfrac{\\lambda _1}{2\\lambda _1}\\big ]\\bigg )\\ge 1/2\\quad \\text{for all }r\\ge 1.$ The first step in the proof of Proposition REF is the following estimate on the effect of observing $\\omega $ in determining whether there is a crossing of $[0,4r] \\times [0,r]$ in $\\mathcal {O}_p$ .",
"Lemma 5.4 For every $m\\ge 1$ and $p\\in [0,1]$ , and every measurable event $A$ , we have $\\operatorname{Var}_\\lambda \\big (\\overline{{\\mathbb {P}}}_\\lambda (A|\\omega )\\big )\\le 1/m.$ Recall that $\\omega _p$ is obtained as a subset of $\\omega $ based on the value of the third and fourth coordinates of each point in $\\overline{\\omega }$ .",
"Alternatively, we could obtain $\\omega _p$ in two stages: First, partition $\\overline{\\omega }$ into $\\omega _p^1,\\omega _p^2,\\ldots ,\\omega _p^m$ and some remainder $\\tilde{\\omega }$ , where $\\omega _p^j$ contains all points with third coordinate at most $p$ and fourth coordinate in $\\big [\\frac{j-1}{m},\\frac{j}{m}\\big )$ .",
"Second, use auxiliary randomness, independent from everything else, to determine which of the $\\omega _p^j$ 's to choose as the set $\\omega _p$ .",
"Since each of the $\\omega _p^j$ 's have the same distribution, the result would be the same.",
"Let us assume that this construction has been made, and that $J$ is the auxiliary random variable, uniformly distributed on $\\lbrace 1,2,\\ldots ,m\\rbrace $ , that determines which of the $\\omega _p^j$ 's that is chosen in the second step.",
"Let ${\\mathcal {F}}$ denote the sigma algebra containing the information of the partitioning $\\omega _p^1, \\ldots , \\omega _p^m$ .",
"Since revealing the partitioning can only increase the variance, we have that $\\operatorname{Var}_\\lambda \\big (\\overline{{\\mathbb {P}}}_\\lambda (A|\\omega )\\big )\\le \\operatorname{Var}_\\lambda \\big (\\overline{{\\mathbb {P}}}_\\lambda (A|{\\mathcal {F}})\\big ).$ Moreover, by conditioning on the outcome of $J$ , $\\overline{{\\mathbb {P}}}_\\lambda (A|{\\mathcal {F}})=\\frac{1}{m}\\sum _{j=1}^m\\operatorname{{\\bf 1}}_{\\lbrace \\omega _p^j\\in A\\rbrace }.$ Since the configurations $\\omega _p^1,\\omega _p^2,\\ldots ,\\omega _p^m$ are independent, we find that $\\operatorname{Var}_\\lambda \\big (\\overline{{\\mathbb {P}}}_\\lambda (A|\\omega )\\big )\\,\\le \\,\\operatorname{Var}_\\lambda \\bigg (\\frac{1}{m}\\sum _{j=1}^m\\operatorname{{\\bf 1}}_{\\lbrace \\omega _p^j\\in A\\rbrace }\\bigg )\\,\\le \\,\\frac{1}{m},$ as required.",
"[Proof of Proposition REF ] Note that $Z_p$ is monotone in $p$ , so it will suffice to consider the extremes of the interval $\\big [\\tfrac{\\lambda _0}{2\\lambda _1},\\tfrac{\\lambda _1}{2\\lambda _1}\\big ]$ .",
"Due to Corollary REF there exists a constant $c_{\\ref {c:q-bound}} = c_{\\ref {c:q-bound}}(4) > 0$ such that $\\overline{\\operatorname{{\\mathbb {E}}}}_\\lambda [Z_p]\\in (c_{\\ref {c:q-bound}},1-c_{\\ref {c:q-bound}})$ for all $p\\in \\big [\\tfrac{\\lambda _0}{2\\lambda _1},\\tfrac{\\lambda _1}{2\\lambda _1}\\big ]$ .",
"So, by Chebyshev's inequality and Lemma REF , for $b = 0, 1$ , $\\overline{{\\mathbb {P}}}_\\lambda \\Big (Z_{\\lambda _b/(2\\lambda _1)}\\notin (c_{\\ref {c:q-bound}}/2,1-c_{\\ref {c:q-bound}}/2)\\Big )\\,\\le \\,\\frac{\\operatorname{Var}_\\lambda (Z_p)}{(c_{\\ref {c:q-bound}}/2)^2}\\,\\le \\,\\frac{4}{mc_{\\ref {c:q-bound}}^2},$ which is at most $\\varepsilon /2$ for $m\\ge 8/(\\varepsilon c_{\\ref {c:q-bound}}^2)$ .",
"Now, the result follows via a union bound."
],
[
"Controlling the influences",
"The next step is to obtain an upper bound on the individual conditional influences, defined as in (REF ), allowing for Talagrand's inequality to give a lower bound on the total conditional influence.",
"The result proved here is the following.",
"Proposition 5.5 Assume the second moment condition (REF ).",
"There exists a constant $c_{\\ref {c:lambda2}}$ such that for every $m\\ge c_{\\ref {c:lambda2}}$ and $\\eta >0$ we have $\\overline{{\\mathbb {P}}}_\\lambda \\Big ( \\max _{i \\ge 1} \\big ( \\operatorname{Inf}^p_i (f_\\omega ) \\big ) > \\eta \\text{ for some }p\\in \\big [ \\tfrac{\\lambda _0}{2\\lambda _1}, \\tfrac{\\lambda _1}{2\\lambda _1} \\big ] \\Big ) \\le \\eta ,$ for all $r$ large enough depending only on $\\eta $ .",
"This will be done in two steps.",
"First we relate the influence of a point to arm probabilities.",
"Then we bound the probability of the latter.",
"Before stating the first lemma, consider a partitioning of the rectangle $[0, 4r] \\times [0, r]$ into $36 n^2$ smaller squares of side length $r/ 3n$ .",
"We write $\\operatorname{Arm}_p^\\ell (\\tfrac{r}{n}, r)$ , with $\\ell = 1, \\dots , 36 n^2$ , for the arm event defined as in (REF ), but centered around each of these boxes.",
"These arm events help us to bound the influences $\\operatorname{Inf}^p_i(f_\\omega )$ as follows.",
"Assume that $i$ is pivotal, and the corresponding ball has radius smaller or equal to $r / 3n$ .",
"Then there must exist some $\\ell $ for which the event $\\operatorname{Arm}_p^\\ell (r/n,r)$ must occur.",
"Therefore, $\\operatorname{Inf}_i^p(f_\\omega )\\le \\operatorname{{\\bf 1}}_{G^c}(\\omega ) + \\overline{{\\mathbb {P}}}_\\lambda \\big (\\operatorname{Arm}_p^\\ell (\\tfrac{r}{n},r)\\big |\\omega \\big )\\quad \\text{for some }\\ell =1,2,\\ldots , 36 n^2,$ for $\\overline{{\\mathbb {P}}}_\\lambda $ -almost every $\\overline{\\omega }$ .",
"Estimating influences will now boil down to estimating conditional arm probabilities.",
"Lemma 5.6 Assume the second moment condition (REF ).",
"There exist $\\gamma > 0$ and $c_{\\ref {c:lambda2}} > 0$ such that for every $m > c_{\\ref {c:lambda2}}$ and $n \\ge 16$ we have $\\overline{{\\mathbb {P}}}_\\lambda \\Big [ \\overline{{\\mathbb {P}}}_\\lambda \\big ( \\operatorname{Arm}_p\\big ( \\tfrac{r}{n}, r\\big ) \\big | \\omega \\big ) > n^{-\\gamma } \\text{ for some }p\\in \\big [\\tfrac{\\lambda _0}{2\\lambda _1}, \\tfrac{\\lambda _1}{2\\lambda _1} \\big ]\\Big ] < n^{-100},$ for all sufficiently large $r$ depending only on $n$ .",
"The proof follows the same structure of that of Lemma REF .",
"For $r\\ge n\\ge 16$ let $\\ell _i=4^i\\frac{r}{n}$ .",
"Note that if $\\operatorname{Circ}_p^\\star (\\frac{r}{n},r)$ fails to occur, then $A_i = \\operatorname{Circ}_p^\\star (\\ell _i, 2\\ell _i)$ must also fail for every $i = 0, 1, \\ldots , k - 1$ , where $k = \\lfloor \\frac{1}{2} \\log _4 n \\rfloor $ ($k \\ge 1$ as we assume $n \\ge 16$ ).",
"By (REF ) we therefore have $\\overline{{\\mathbb {P}}}_\\lambda \\big (\\operatorname{Arm}_p(\\tfrac{r}{n}, r) \\big | \\omega \\big )\\, \\le \\, \\operatorname{{\\bf 1}}_{G^c} (\\omega ) + \\prod _{i = 0}^{k - 1} \\overline{{\\mathbb {P}}}_\\lambda (A_i^c | \\omega ).$ Fix $\\varepsilon >0$ so that $2^k\\varepsilon ^{k/2}\\le \\frac{1}{2}n^{-100}$ .",
"By Proposition REF there exists $\\delta >0$ and $c_{\\ref {c:lambda}}(\\varepsilon ) > 0$ large enough so that for every $r \\ge 1$ and $m\\ge c_{\\ref {c:lambda}}$ we have $\\overline{{\\mathbb {P}}}_\\lambda \\Big ( \\overline{{\\mathbb {P}}}_\\lambda \\big ( \\operatorname{Circ}_p^\\star (r, 2r) \\big | \\omega \\big ) > \\delta \\text{ for every }p\\in \\big [\\tfrac{\\lambda _0}{2\\lambda _1}, \\tfrac{\\lambda _1}{2\\lambda _1} \\big ]\\Big ) > 1 - \\varepsilon .$ Fix $\\gamma > 0$ (independent of $n$ ) so that $n^{-\\gamma } \\ge (1-\\delta )^{k/2}$ .",
"By (REF ) we have, on $G$ , that if $\\overline{{\\mathbb {P}}}_\\lambda (A_i|\\omega )>\\delta $ for at least half of the indices $i=0,1,\\ldots ,k-1$ , then $\\overline{{\\mathbb {P}}}_\\lambda \\big (\\operatorname{Arm}_p(\\tfrac{r}{n},r)\\big |\\omega \\big )\\,\\le \\,(1-\\delta )^{k/2}\\,\\le \\, n^{-\\gamma }.$ In particular, via a union bound, $\\overline{{\\mathbb {P}}}_\\lambda \\Big (G\\cap \\big [\\overline{{\\mathbb {P}}}_\\lambda \\big (\\operatorname{Arm}_p(\\tfrac{r}{n},r)\\big |\\omega \\big )>n^{-\\gamma }\\big ]\\Big )\\,\\le \\,2^k\\sup _{I}\\overline{{\\mathbb {P}}}_\\lambda \\Big (G \\cap \\bigcap _{i \\in I}\\big [\\overline{{\\mathbb {P}}}_\\lambda (A_i|\\omega )\\le \\delta \\big ]\\Big ),$ where the supremum is taken over all $I\\subseteq \\lbrace 0,1,\\ldots ,k-1\\rbrace $ of size at least $k/2$ , of which there are at most $2^k$ .",
"By (REF ) and (REF ), this is bounded by $2^k\\sup _{I}\\prod _{i\\in I}\\Big [\\overline{{\\mathbb {P}}}_\\lambda \\big (\\overline{{\\mathbb {P}}}_\\lambda (A_i|\\omega )\\le \\delta \\big )+\\overline{{\\mathbb {P}}}_\\lambda (G^c)\\Big ]\\,\\le \\,2^k\\big (\\varepsilon +\\overline{{\\mathbb {P}}}_\\lambda (G^c)\\big )^{k/2}.$ For large values of $r$ , depending on $n$ , this probability is at most $n^{-100}$ by the choice of $\\varepsilon >0$ .",
"We have now gathered all the pieces necessary to prove the main result of this subsection.",
"[Proof of Proposition REF ] We start by choosing $n$ large enough so that $36 n^{-98} \\le \\eta /2$ .",
"Then, $\\overline{{\\mathbb {P}}}_\\lambda \\Big ( \\max _{i \\ge 1} \\big ( \\operatorname{Inf}_i^p(f_\\omega ) \\big ) > n^{-\\gamma } \\text{ for some }p \\in \\big [ \\tfrac{\\lambda _0}{2\\lambda _1}, \\tfrac{\\lambda _1}{2\\lambda _1} \\big ]\\Big )$ may via (REF ) be bounded from above by $\\overline{{\\mathbb {P}}}_\\lambda \\big ( G^c \\big ) + \\overline{{\\mathbb {P}}}_\\lambda \\Big (\\overline{{\\mathbb {P}}}_\\lambda \\big ( \\operatorname{Arm}_p^\\ell \\big ( \\tfrac{r}{n}, r\\big ) \\big | \\omega \\big ) > n^{-\\gamma }\\text{ for some }p \\in \\big [ \\tfrac{\\lambda _0}{2\\lambda _1}, \\tfrac{\\lambda _1}{2\\lambda _1} \\big ]\\text{ and }\\ell \\le 36 n^2 \\Big ).$ By Lemmas REF and REF , this is smaller than $\\eta $ by our choice of $n$ , once $r$ is taken large enough."
],
[
"Proof of Theorem ",
"Roughly speaking, in order to prove Theorem REF , we are going to show that for every $\\varepsilon > 0$ $Z_{\\lambda _1/(2\\lambda _1)} - Z_{\\lambda _0/(2\\lambda _1)} \\ge \\frac{\\lambda _1 - \\lambda _0}{\\varepsilon },$ with positive probability.",
"Since $Z_p(\\omega ) \\in [0, 1]$ almost surely, we must have that $\\lambda _0=\\lambda _1$ .",
"[Proof of Theorem REF ] We first give estimates on the derivatives of $Z_p$ .",
"Recall that, for $\\overline{{\\mathbb {P}}}_\\lambda $ almost every $\\overline{\\omega }\\in \\overline{\\Omega }$ , $Z_p$ coincides with the expectation, with respect to the conditional measure $\\overline{{\\mathbb {P}}}_\\lambda (\\,\\cdot \\,|\\omega )$ , of some function $f_\\omega :\\lbrace 0,1\\rbrace ^\\omega \\rightarrow \\lbrace 0,1\\rbrace $ .",
"Moreover, with probability one, only finitely many points in $\\omega $ will affect the outcome of events defined on $[-4r,4r]^2$ .",
"Hence, the domain of $f_\\omega $ is for $\\overline{{\\mathbb {P}}}_\\lambda $ -almost every $\\overline{\\omega }$ finite dimensional, and the Margulis-Russo formula gives that $\\frac{d Z_p}{d p} \\,=\\, \\frac{d Z_p (\\omega )}{d p} \\,=\\, \\frac{1}{m}\\sum _{i \\ge 0} \\operatorname{Inf}_i^p (f_\\omega ),$ where $\\operatorname{Inf}_i^p (f_\\omega )$ is as defined in (REF ), and the additional factor $1/m$ comes from the chain rule.",
"Let us now assume that $\\lambda _0 < \\lambda _1$ and fix $m\\ge c_{\\ref {c:lambda}}(1/2) \\vee c_{\\ref {c:lambda2}}$ .",
"Combining (REF ) with the mean-value theorem and Talagrand's inequality (REF ) we obtain, for some $q\\in \\big [\\frac{\\lambda _0}{2\\lambda _1},\\frac{\\lambda _1}{2\\lambda _1}\\big ]$ , $\\frac{Z_{\\lambda _1/(2\\lambda _1)} - Z_{\\lambda _0/(2\\lambda _1)}}{\\lambda _1 - \\lambda _0}\\, =\\, \\frac{1}{2\\lambda _1} \\frac{d Z_p(\\omega )}{d p} (q)\\, \\ge \\, \\frac{c}{2 \\lambda _1 m \\log (m)} \\operatorname{Var}_\\lambda (f_\\omega | \\omega ) \\log \\bigg ( \\frac{c}{\\max _i \\big ( \\operatorname{Inf}_i^q (f_\\omega ) \\big ) } \\bigg ).$ From Propositions REF and REF , we conclude that for $r$ large enough (depending only on $\\lambda _1 - \\lambda _0$ ) the right hand side of the above equation is larger than $2/(\\lambda _1 - \\lambda _0)$ with positive probability.",
"This contradicts the fact that $Z_p \\in [0,1]$ , thus finishing the proof of the theorem."
],
[
"Continuity of the critical parameter",
"We have in the previous section seen that $\\lambda _0=\\lambda _1$ , and thus coincides with $\\lambda _c$ , under the assumption of finite second moment of the radii distribution.",
"In this section we will investigate the dependence of this critical parameter with respect to the radii distribution $\\mu $ , and write $\\lambda _c=\\lambda _c(\\mu )$ throughout this section in order to emphasize this dependence.",
"Theorem 6.1 Let $(\\mu _m)_{m\\ge 1}$ be a sequence of radii distributions, all dominated by some distribution $\\nu $ with finite second moment.",
"If $\\mu _m\\rightarrow \\mu $ weakly, then ${\\lambda _{c}}(\\mu _m)\\rightarrow {\\lambda _{c}}(\\mu )$ .",
"Remark 13 A weaker version of Theorem REF , proved under the stronger assumption of uniformly bounded support, has been known since long (see [27]).",
"The argument given here is significantly simpler and shorter, but is in contrast to the aforementioned proof distinctively two-dimensional.",
"The convergence may fail when the assumption of uniformly bounded tails is relaxed.",
"For a first example of this, let $(\\mu _m)_{m\\ge 1}$ be a sequence with infinite second moments converging weakly to a point mass at 1.",
"The critical parameter for each $\\mu _m$ is degenerate (meaning that $\\lambda _0=\\lambda _1=0$ ), and cannot converge to the critical value of the point mass, which is strictly positive.",
"In the above example the convergence fails for rather obvious reasons.",
"A second example, which gives further insight to what can go wrong, may be obtained from a cumulative distribution function $F$ of some distribution with finite second moment, when $F_m$ is given by $F_m(x):=(1-\\tfrac{1}{m})F(x)+\\tfrac{1}{m}F(\\tfrac{x}{m}).$ Each $F_m$ has finite second moment, but there is no uniform upper bound on its value.",
"The dilation of $F$ by a factor $\\frac{1}{m}$ corresponds to a scaling of the radii by $m$ , and thus has an inverse quadratic effect on the critical parameter in that ${\\lambda _{c}}(F(\\frac{x}{m}))=\\frac{1}{m^2}{\\lambda _{c}}(F(x))$ .",
"So, although ${\\lambda _{c}}(\\mu )>0$ , we have ${\\lambda _{c}}(F_m)\\,\\le \\,{\\lambda _{c}}\\big ((1-\\tfrac{1}{m})\\operatorname{{\\bf 1}}_{\\lbrace x\\ge 0\\rbrace }+\\tfrac{1}{m}F(\\tfrac{x}{m})\\big )\\,=\\,m\\,{\\lambda _{c}}\\big (F(\\tfrac{x}{m})\\big )\\,=\\,\\tfrac{1}{m}{\\lambda _{c}}(F).$ Although $F_m\\rightarrow F$ weakly, the critical parameters diverge.",
"[Proof of Theorem REF ] Fro the proof, we will write $F$ , $F_m$ and $H$ for the cumulative distribution functions of $\\mu $ , $\\mu _m$ and $\\nu $ .",
"We first observe that if $F_m\\ge H$ for all $m\\ge 1$ and $F_m\\rightarrow F$ , then also $F\\ge H$ and $F$ has finite second moment too.",
"To see this assume the contrary, in which case $F(x)<H(x)$ for some $x\\ge 0$ .",
"Either $x$ is a continuity point of $F$ , or right continuity of $F$ and $H$ allows us to find a continuity point $x^{\\prime }$ of $F$ for which $F(x^{\\prime })<H(x^{\\prime })$ remains.",
"But, in that case $\\liminf _{m\\rightarrow \\infty }F_m(x^{\\prime })\\ge H(x^{\\prime })>F(x^{\\prime })$ , contradicting the assumed convergence.",
"That $F$ has finite second moment in particular implies that ${\\lambda _{c}}(F)\\in (0,\\infty )$ .",
"We will base the proof on Propositions REF and REF .",
"We therefore fix $\\theta >0$ and $r_0 = r_0({\\lambda _{c}}(F) + 1)$ accordingly.",
"Write ${\\mathbb {P}}_\\lambda ^{F_m}$ for the measure at intensity $\\lambda $ and with radii distributed as $F_m$ .",
"To complete the proof, it will suffice to show that for every $\\varepsilon \\in (0,1)$ there exists $r\\ge r_0$ and $m_0$ such that for all $m\\ge m_0$ ${\\mathbb {P}}_{{\\lambda _{c}}(F)-\\varepsilon }^{F_m}\\big (\\operatorname{Cross}(r,3r)\\big )<\\theta \\quad \\text{and}\\quad {\\mathbb {P}}_{{\\lambda _{c}}(F)+\\varepsilon }^{F_m}\\big (\\operatorname{Cross}(3r,r)\\big )>1-\\theta .$ Proposition REF then implies that ${\\lambda _{c}}(F_m)\\in [{\\lambda _{c}}(F)-\\varepsilon ,{\\lambda _{c}}(F)+\\varepsilon ]$ for all $m\\ge m_0$ .",
"Fix $\\varepsilon >0$ .",
"By Corollary REF and Theorem REF , we may choose $r$ large so that ${\\mathbb {P}}_{{\\lambda _{c}}(F) - \\varepsilon }^F\\big (\\operatorname{Cross}(r, 3r)\\big ) < \\theta /4 \\quad \\text{and} \\quad {\\mathbb {P}}_{{\\lambda _{c}}(F) + \\varepsilon }^F\\big (\\operatorname{Cross}(3r, r)\\big ) > 1 - \\theta /4.$ Next, based on (REF ) of Lemma REF , fix $K\\subseteq {\\mathbb {R}}^2$ such that ${\\mathbb {P}}_{{\\lambda _{c}}(F)+\\varepsilon }^H\\big (\\mathcal {O}_{K^c}\\cap [0,r]^2=\\varnothing \\big )>1-\\theta /4.$ And, choose $\\delta >0$ such that the event in Proposition REF has ${\\mathbb {P}}_{{\\lambda _{c}}(F)\\pm \\varepsilon }^{F}$ -probability at most $\\theta /4$ to occur.",
"By weak convergence of $F_m$ to $F$ it follows that $F_m^{-1}(U)\\rightarrow F^{-1}(U)$ almost surely, as $m\\rightarrow \\infty $ , if $U$ is uniformly distributed on $[0,1]$ and $F^{-1}$ denotes the generalized inverse of $F$ .",
"Consequently, as the projection of $\\omega $ onto $K$ , denoted by $\\omega (K)$ , is almost surely finite, we will have $F_m^{-1}(z)\\in (F^{-1}(z)-\\varepsilon ,F^{-1}(z)+\\varepsilon )$ for all $(x,z)\\in \\omega (K)$ for large enough $m$ .",
"We may, in particular, choose $m_0$ so that for all $m\\ge m_0$ ${\\mathbb {P}}_{{\\lambda _{c}}(F) \\pm \\varepsilon }^F \\Big ( \\operatorname{int}(\\mathcal {O}_K, \\delta ) \\subseteq \\mathcal {O}_K^{F_m} \\subseteq \\operatorname{cl}(\\mathcal {O}_K, \\delta ) \\Big ) > 1 - \\theta /4.$ Combining the above estimates we conclude that (REF ) holds, as required."
],
[
"Proof of main results",
"In this section we complete the proofs of the three main theorems stated in the introduction.",
"We first prove Theorem REF , and then deduce Theorems REF and REF .",
"Before starting we recall the content of Corollary REF .",
"Assuming (REF ), we have $0<\\lambda _0\\le \\lambda _c\\le \\lambda _1<\\infty ,$ and the analogous relation holds for $\\lambda _c$ replaced by $\\lambda _c^\\star $ under the stronger condition (REF )."
],
[
"Proof of Theorem ",
"For any $\\lambda >\\lambda _1$ we have, by definition, that $\\lim _{r\\rightarrow \\infty }{\\mathbb {P}}_\\lambda (\\operatorname{Cross}(3r,r))=1$ .",
"The standard inequalities of Lemma REF and monotonicity imply that for any $\\lambda >\\lambda _1$ and $\\kappa >0$ , $\\lim _{r\\rightarrow \\infty }{\\mathbb {P}}_\\lambda (\\operatorname{Cross}(\\kappa r,r))=1.$ Similarly, for any $\\lambda <\\lambda _0$ and $\\kappa >0$ the limit in (REF ) equals 0.",
"Since $\\lambda _c\\in [\\lambda _0,\\lambda _1]$ , then Corollary REF shows that for any $\\kappa >0$ there exists a constant $c=c(\\kappa )>0$ such that $c<{\\mathbb {P}}_{\\lambda _c}(\\operatorname{Cross}(\\kappa r,r))<1-c\\quad \\text{for all }r\\ge 1.$ Moreover, by Corollary REF , there is at $\\lambda _c$ almost surely no unbounded component of either kind.",
"Finally, the proof is completed by Theorem REF , which shows that indeed $\\lambda _c=\\lambda _0=\\lambda _1$ ."
],
[
"Proof of Theorem ",
"Assume (REF ), in which case we by now know that $\\lambda _c^\\star =\\lambda _0=\\lambda _1=\\lambda _c$ .",
"Part (i) is then a consequence of Corollary REF .",
"The proof of part (ii) will involve a truncation of the radii distribution, allowing for a comparison with a highly supercritical 1-dependent percolation process on ${\\mathbb {Z}}^2$ , for which a standard Peierls argument gives the required exponential decay.",
"Fix $\\lambda >{\\lambda _{c}^\\star }$ .",
"For $m\\ge 1$ , let $\\mu _m$ denote the truncation of $\\mu $ at $m$ , meaning that $\\mu _m$ coincides with $\\mu $ on $[0,m)$ but that $\\mu _m([0,m])=1$ .",
"We shall write $\\lambda ^\\star (\\mu )$ for the critical parameter associated with radii distribution $\\mu $ .",
"By continuity of the critical parameter, Theorem REF , we have for large $m$ that ${\\lambda _{c}^\\star }(\\mu _m)\\in [{\\lambda _{c}^\\star }(\\mu ),\\lambda ),$ and assume for the rest of this proof that $m$ is chosen accordingly.",
"Given $\\gamma >0$ , we also assume that $r>2m$ is fixed so that ${\\mathbb {P}}_\\lambda ^{\\mu _m}(\\operatorname{Cross}(3r,r))>1-\\gamma .$ Now, tile the plane with overlapping $3r\\times r$ and $r\\times 3r$ rectangles as follows: For each $z\\in {\\mathbb {Z}}^2$ consider both a $3r\\times r$ and a $r\\times 3r$ rectangle positioned so their lower left corners coincide with $2rz$ .",
"We want to register which of these rectangles that are crossed in the hard direction by an occupied path.",
"Define thus, for each $z\\in {\\mathbb {Z}}^2$ , $\\eta (z,z + (1,0))$ to be the indicator of the event $\\operatorname{Cross}(3r,r)$ translated by the vector $2rz$ , and $\\eta (z,z + (0, 1))$ as the indicator of $\\operatorname{Cross}(r,3r)$ translated by the same vector.",
"By the choice of $m$ and $r$ , the process $\\eta $ is a 1-dependent bond percolation process on ${\\mathbb {Z}}^2$ with marginal edge probability exceeding $1 - \\gamma $ .",
"Therefore, by a standard Peierls argument, there is a constant $c_{\\ref {c:exp_decay}}>0$ such that the event that $\\eta $ contains an open circuit which is contained in $B^\\infty (n)$ and surrounds the origin has probability at least $1-c_{\\ref {c:exp_decay}}\\exp \\lbrace n/c_{\\ref {c:exp_decay}}\\rbrace $ .",
"However, by construction, an open path in $\\eta $ corresponds to an occupied crossing in $\\mathcal {O}$ , and thus that for all $n\\ge 1$ $\\mathbb {P}_\\lambda ^{\\mu _m}\\big [ 0\\overset{ \\mathcal {V}}{\\longleftrightarrow }\\partial B^\\infty (rn) \\big ] \\le c_{\\ref {c:exp_decay}}\\exp \\lbrace -n/c_{\\ref {c:exp_decay}}\\rbrace ,$ from which part (ii) of the theorem follows."
],
[
"Proof of Theorem ",
"Under assumption () Item (i) is a consequence of part (iii) of Corollary REF (see also Remark REF ).",
"So, it remains to argue for Item (ii).",
"However, if () holds, then by Proposition REF there exists $c > 0$ such that for all $\\lambda \\le \\lambda _c$ and $r\\ge 1$ $\\rho _\\lambda (5r,r)\\le cr^{-\\alpha }.$ Now, fix $\\lambda <\\lambda _c$ , so that $\\lim _{r\\rightarrow \\infty }{\\mathbb {P}}_\\lambda (\\operatorname{Cross}^\\star (3r,r))=1$ .",
"Proposition REF then gives that ${\\mathbb {P}}_\\lambda (\\operatorname{Cross}^\\star (3r,r))\\ge 1 - c^{\\prime }r^{-\\alpha }$ for all $r\\ge 1$ and some constant $c^{\\prime }$ , possibly depending on $\\lambda $ .",
"Consequently, for all $r\\ge 1$ , ${\\mathbb {P}}_\\lambda \\big [0 \\overset{\\tiny \\mathcal {O}}{\\longleftrightarrow }\\partial B(r)\\big ]\\,\\le \\,4\\,{\\mathbb {P}}_\\lambda (\\operatorname{Cross}(r,3r))\\,\\le \\,4c^{\\prime }r^{-\\alpha },$ as required.",
"This ends the proof of Theorem REF ."
],
[
"Other models",
"We have in Sections - provided a framework to prove the existence of a sharp phase transition and some interesting properties of the critical behavior for Poisson Boolean percolation.",
"In this section we will show how our techniques can be used to obtain alternative proofs for the facts that the critical probability equals $1/2$ for Poisson Voronoi percolation, earlier proved by Bollobás and Riordan [9], and Poisson confetti percolation, first proved by Hirsch [21] (for unit squares) and later Müller [30] (for unit discs).",
"Our results will in fact apply to some generalized versions of these models that are not self-dual.",
"See Figure REF for a simulation of these processes.",
"The finite-size criterion of Section and the RSW techniques of Section solely rely on three basic facts: That the probability measure in question is invariant (with respect to translations, right angle rotations and reflections in coordinate axis), positively associated, and features a decay of spatial correlations.",
"However, the argument used to prove the sharp threshold behavior in Section used the further assumption of a Poissonian structure.",
"All of these properties are satisfied by Poisson Voronoi and confetti percolation.",
"We shall below illustrate what our techniques give for these models, but emphasize that we make no attempt to sharpen the hypothesises in the results we present.",
"Figure: Simulations of Poisson confetti (left) and Voronoi (right) percolation.In the left picture, black confetti have random radii, while the radii of white ones are fixed.The Voronoi picture to the right uses G 0 =1G_0 = 1, G 1 =2G_1 = 2 and q=0.8q = 0.8.Note that the Voronoi cells above are determined by segments of ellipses."
],
[
"Poisson Voronoi percolation",
"The study of random Voronoi tessellations goes back several decades in time, yet it was only about a decade ago that Bollobàs and Riordan [9] gave the first proof for the fact that the critical probability in Poisson Voronoi percolation in the plane equals $1/2$ ; see also [3].",
"One of the main difficulties faced in studying the phase transition is to derive RSW techniques that apply in this setting.",
"In [9] a weak RSW result was provided, whereas a strong version of RSW was established later by Tassion [42], and was used to prove polynomial decay of the one-arm probability at criticality.",
"The RSW techniques developed here allows us to consider more general variants of the standard Voronoi model, where black and white cells do not necessarily follow the same law.",
"Informally speaking, to each point we associate a random `gravitational pull', which gives a bias to the size of the associated tile.",
"The symmetry between black and white points may be broken by considering different laws for the gravitational pull associated to them.",
"Consider a Poisson point process of unit density on the space of locally finite counting measures on ${\\mathbb {R}}^2 \\times [0, 1] \\times [0, 1]$ , endowed with Lebesgue measure.",
"The first two coordinates of a point in the support of a realization $\\omega $ will mark the location of the seed in the plane, the third and fourth coordinates will determine the gravitational pull of a given point and its color, respectively.",
"More precisely, given a parameter $q \\in [0, 1]$ , we say that the seed $(x, z, s) \\in \\operatorname{supp}(\\omega )$ is black if $s \\le q$ and white otherwise.",
"We also fix two non-decreasing functions $G_0, G_1: [0,1] \\rightarrow (0, \\infty )$ , which will be applied to the $z$ coordinate of $(x, z, s)$ to determine the gravitational pull of a given seed ($G_0$ will be used for black seeds and $G_1$ for white).",
"Given $q \\in [0, 1]$ , define the occupied (black) set as $\\mathcal {O}_q := \\bigcup _{\\begin{array}{c}(x, z, s) \\in \\omega ;\\\\s \\le q\\end{array}} \\Big \\lbrace y \\in {\\mathbb {R}}^2 : \\frac{|y - x|}{G_0(z)} \\le \\frac{|y - x^{\\prime }|}{G_1(z^{\\prime })} \\text{ for all $(x^{\\prime }, z^{\\prime }, s^{\\prime }) \\in \\omega $ with $s^{\\prime } > q$} \\Big \\rbrace .$ As before, let $\\mathcal {V}_q := \\mathbb {R}^2 \\setminus \\mathcal {O}_q$ denote the corresponding vacant set.",
"Notice that the cell corresponding to a given seed may be disconnected.",
"Finally, we define the critical values of the parameter $q$ as $\\begin{aligned}q_c:=&\\sup \\big \\lbrace q\\in [0,1]:{\\mathbb {P}}\\big [0\\overset{\\tiny \\mathcal {O}_q}{\\longleftrightarrow }\\infty \\big ]=0\\big \\rbrace ,\\\\q_c^\\star :=&\\inf \\big \\lbrace q\\in [0,1]:{\\mathbb {P}}\\big [0\\overset{\\tiny \\mathcal {V}_q}{\\longleftrightarrow }\\infty \\big ]=0\\big \\rbrace .\\end{aligned}$ We will here only consider the case when $G_0$ and $G_1$ take values in some interval $[a, b] \\subset (0, \\infty )$ .",
"It would be interesting to attempt to relax these assumptions.",
"When $G_0 = G_1$ , then the gravitational pull of all black and white seeds are equally distributed.",
"In this case the model is self-dual, and we recover the equation $q_c+q_c^\\star =1$ ; see Section REF below.",
"The techniques developed in this paper may easily be adapted to prove the following result for Poisson Voronoi percolation.",
"Theorem 8.1 Assume that $G_0$ and $G_1$ take values in some interval $[a,b]\\subset (0,\\infty )$ .",
"Then, $q_c=q_c^\\star $ and their common value is strictly between zero and one.",
"Moreover, for all $q<q_c$ , there exists $c = c(q)>0$ such that ${\\mathbb {P}}\\big [ 0\\overset{\\tiny \\mathcal {O}_q}{\\longleftrightarrow }\\partial B(r) \\big ] \\le \\exp \\lbrace -c r\\rbrace .$ for $q=q_c$ and all $\\kappa >0$ , there exist $c=c(\\kappa )>0$ and $\\alpha >0$ such that for every $r \\ge 1$ $c < {\\mathbb {P}}(\\operatorname{Cross}(\\kappa r,r)) < 1 - c\\quad \\text{and}\\quad \\mathbb {P} \\big [ 0 \\overset{\\tiny \\mathcal {O}_q}{\\longleftrightarrow }\\partial B(r) \\big ] \\le r^{-\\alpha }.$ for all $q>q_c$ , there exists $c=c(q)>0$ such that ${\\mathbb {P}}\\big [ 0\\overset{\\tiny \\mathcal {V}_q}{\\longleftrightarrow }\\partial B(r) \\big ] \\le \\exp \\lbrace -c r\\rbrace .$ Moreover, at $q_c$ there is almost surely no unbounded cluster of either kind.",
"We will for the remainder of this subsection outline the proof of this result, based on the techniques developed over the previous sections.",
"First of all, we note that the law of the above model is invariant with respect to translations, rotations and reflections.",
"The measure is further positively associated, meaning that is satisfies an FKG inequality similar to (REF ), for events that are increasing in the addition of black points and removal of white points.",
"This can be seen by adapting the proof of [10] for the case when $G_0=G_1$ are constant.",
"What remains in order for the techniques of Sections and to apply is an estimate on the spatial correlations.",
"Lemma 8.2 Fix a bounded measurable set $D \\subseteq \\mathbb {R}^2$ and an arbitrary function $f(\\mathcal {O}_q)$ satisfying $f \\in \\sigma (Y_v; v \\in D)$ .",
"Then, defining the event $G_{D, r} = \\Big \\lbrace \\text{for every $y \\in D$, there is a point $(x, z, s)$ in $\\omega $ with $d(y, x) \\le \\frac{a}{b} r$} \\Big \\rbrace ,$ we have that $\\operatorname{{\\bf 1}}_{G_{D, r}} \\cdot f$ only depends on the restriction of $\\omega $ to $B(D, r) \\times [0, 1] \\times [0, 1]$ .",
"Moreover, for every $\\varepsilon >0$ there exists a constant $c=c(\\varepsilon )>0$ such that for any $D\\subseteq B^\\infty (r)$ we have $\\mathbb {P}(G_{D, \\varepsilon r}) \\ge 1 - \\exp \\lbrace - c r^2 \\rbrace $ .",
"The proof of the first claim follows the same steps as those of Lemma 1.1 in [42].",
"Roughly speaking, the argument goes like this.",
"Given $x \\in \\mathbb {R}^2$ , if a seed in $\\omega $ is located within distance $\\ell $ from $x$ , then no other seed outside $B(x, (b/a)\\ell )$ can influence the state of $x$ .",
"The second claim follows from a simple large deviations bound for Poisson random variables.",
"As a first consequence of this fact we obtain an estimate on the decay of spatial correlations.",
"As in Definition REF , let $f_1, f_2:\\mathcal {P}({\\mathbb {R}}^2)\\rightarrow [-1,1]$ be two functions satisfying $f_1(\\mathcal {O}_q)\\in \\sigma \\big (Y_v;v\\in B^\\infty (r)\\big )$ and $f_2\\in \\sigma \\big (Y_v;v\\in B^\\infty (r+t)\\setminus B^\\infty (r+s)\\big )$ , for some $t>s$ .",
"Then, if $G=G_{B^\\infty (r),s/2}\\cap G_{B^\\infty (r+t)\\setminus B^\\infty (r+s),s/2}$ , we obtain $\\begin{split}\\mathbb {E}[f_1 f_2] \\,& \\le \\, \\mathbb {P}(G^c) + \\mathbb {E}[\\operatorname{{\\bf 1}}_G \\cdot f_1 f_2]\\\\&\\le \\,2\\mathbb {P}(G^c)+\\operatorname{{\\mathbb {E}}}[f_1^{\\prime }]\\operatorname{{\\mathbb {E}}}[f_2^{\\prime }]\\\\&\\le \\,6{\\mathbb {P}}(G^c)+\\operatorname{{\\mathbb {E}}}[f_1]\\operatorname{{\\mathbb {E}}}[f_2],\\end{split}$ where $f_1^{\\prime }$ and $f_2^{\\prime }$ denote $f_1$ and $f_2$ evaluated at the restriction of $\\omega $ to $B^\\infty (r+s/2)$ and $B^\\infty (r+t)\\setminus B^\\infty (r+s/2)$ and are hence independent.",
"In particular, when $s=\\varepsilon r$ and $t=\\frac{1}{\\varepsilon }r$ , which has been the case throughout this paper, the correlation between $f_1$ and $f_2$ decays as $\\exp \\lbrace - c r^2\\rbrace $ as in Lemma REF .",
"Consequently, the techniques of Sections and apply, and we obtain the existence of $q_0$ and $q_1$ , defined analogously as $\\lambda _0$ and $\\lambda _1$ , satisfying $0<q_0\\le q_c^\\star , \\, q_c\\le q_1<1,$ and such that part (i) of Theorem REF holds for $q<q_0$ , part (ii) for $q\\in [q_0,q_1]$ , and part (iii) for $q>q_1$ .",
"To complete the proof of Theorem REF , it remains to prove that $q_0=q_1$ .",
"In order to show that $q_0=q_1$ we again enlarge our probability space as in Section REF , to enable a two-stage construction of our process.",
"We thus assume that $\\overline{\\omega }$ is a Poisson point process on the space of locally finite counting measures on $\\big ({\\mathbb {R}}^2\\times [0,1]\\times [0,1]\\big )\\times [0,1]$ .",
"The law of $\\overline{\\omega }$ (denoted $\\overline{\\mathbb {P}}_\\lambda $ ) is chosen with a large density $\\lambda =m\\ge 1$ and the fifth coordinate will be used to thin the process down to density one, which we then use to define $\\mathcal {O}_q$ as in (REF ).",
"We denote by $\\omega $ the projection of $\\overline{\\omega }$ onto $\\mathbb {R}^2 \\times [0, 1] \\times [0, 1]$ (the first four coordinates) and by $\\underline{\\omega }$ the further projection onto $\\mathbb {R}^2 \\times [0, 1]$ .",
"Define $W_q:=\\overline{{\\mathbb {P}}}_\\lambda (\\operatorname{Cross}(4r,r)|\\underline{\\omega }).$ As in Section REF , we note that $W_q$ for $\\overline{{\\mathbb {P}}}_\\lambda $ -almost every $\\overline{\\omega }$ coincides with the expectation of some Boolean function $f_{\\underline{\\omega }}:\\lbrace 0,1\\rbrace ^{\\underline{\\omega }}\\times \\lbrace 0,1\\rbrace ^{\\underline{\\omega }}\\rightarrow \\lbrace 0,1\\rbrace $ evaluated with respect to product measure with density $q$ and $1/m$ for choosing `color' and `presence', respectively, of each point in $\\underline{\\omega }$ .",
"This function is increasing in the choice of color, so we obtain via the Margulis-Russo formula that $\\frac{dW_q}{dq}=\\sum _{i}\\overline{{\\mathbb {P}}}_\\lambda \\big ( \\,\\text{point $i$ is present and its color is pivotal for }f_{\\underline{\\omega }}\\, \\big | \\,\\underline{\\omega }\\, \\big ).$ Note that the concept of pivotality arising here is with respect to color and not presence as in Section .",
"The crucial observation is that is that switching the color of a point has a larger potential change than switching its presence.",
"That is, for the existence of a black crossing, a black point is better than no point, and no point is better than a white point.",
"Hence, from (REF ), we obtain that $\\begin{aligned}\\frac{dW_q}{dq} \\,&=\\, \\overline{\\operatorname{{\\mathbb {E}}}}_\\lambda \\Big [ \\sum _i \\overline{{\\mathbb {P}}}_\\lambda \\big ( \\, \\text{point $i$ is present and its color is pivotal} \\, \\big | \\, \\omega \\, \\big ) \\Big | \\, \\underline{\\omega }\\, \\Big ]\\\\&\\ge \\,\\overline{\\operatorname{{\\mathbb {E}}}}_\\lambda \\Big [ \\sum _i\\overline{{\\mathbb {P}}}_\\lambda \\big ( \\, \\text{point $i$ is present and its presence is pivotal} \\, \\big | \\,\\omega \\, \\big ) \\Big |\\,\\underline{\\omega }\\,\\Big ]\\\\&=\\, \\frac{1}{m} \\overline{\\operatorname{{\\mathbb {E}}}}_\\lambda \\Big [ \\sum _i \\operatorname{Inf}^q_i(f_{\\underline{\\omega }}) \\Big |\\,\\underline{\\omega }\\,\\Big ]\\\\&\\ge \\, \\frac{1}{m} \\overline{\\operatorname{{\\mathbb {E}}}}_\\lambda \\Big [ \\frac{c}{\\log (m)} \\operatorname{Var}_\\lambda (f_{\\underline{\\omega }}|\\omega )\\log \\frac{1}{\\max _i \\big ( \\operatorname{Inf}_i^q(f_{\\underline{\\omega }}) \\big ) } \\,\\Big |\\,\\underline{\\omega }\\,\\Big ],\\end{aligned}$ where in the last step we applied (REF ), and $\\operatorname{Inf}_i^q(f_{\\underline{\\omega }})$ denotes the conditional probability, given $\\omega $ , that changing point $i$ from present to absent changes the outcome of $f_{\\underline{\\omega }}$ .",
"Controlling the variance $\\operatorname{Var}_\\lambda (f_{\\underline{\\omega }}|\\omega )$ can now be done exactly as in Section REF .",
"Also the influences $\\operatorname{Inf}_i^q(f_{\\underline{\\omega }})$ with respect to presence of a point can be estimated similarly as it was done in Section REF , but requires a slight modification of the decoupling inequalities (REF ) and (REF ).",
"Let $D_1,D_2,\\ldots , D_J$ be well separated sets as in (REF ), and let $A_1,A_2,\\ldots ,A_J$ be events where $A_j$ is determined by the restriction of $\\mathcal {O}_q$ to $D_j$ .",
"On the event $G={\\textstyle \\bigcap \\limits }_{j\\le J}G_{D_j,r/(2n)}$ the events $A_1,A_2,\\ldots ,A_J$ are, by Lemma REF , determined by restrictions of $\\overline{\\omega }$ to disjoint subsets of the plane, and are hence independent.",
"We obtain, therefore, the following variant of (REF ) for the Voronoi model $\\overline{{\\mathbb {P}}}_\\lambda \\big (A_1\\cap \\cdots \\cap A_J\\big |\\omega \\big )\\,\\le \\,\\overline{{\\mathbb {P}}}_\\lambda (G^c|\\omega )+\\prod _{j\\le J}\\Big [\\overline{{\\mathbb {P}}}_\\lambda (A_j|\\omega )+\\overline{{\\mathbb {P}}}_\\lambda (G^c|\\omega )\\Big ].$ Since $\\overline{{\\mathbb {P}}}_\\lambda (G^c)\\rightarrow 0$ fast, also $\\overline{{\\mathbb {P}}}_\\lambda (G^c|\\omega )\\rightarrow 0$ fast with probability tending to one.",
"This suffices for the application of (REF ) in the proof of Lemma REF .",
"In order to obtain a variant for (REF ) we observe that the difference between $\\overline{{\\mathbb {P}}}_\\lambda (A_j|\\omega )$ and the conditional probability (given $\\omega $ ) that $A_j$ occurs with respect to the restriction of $\\overline{\\omega }$ restricted to the set $B(D_j,r/(2n))$ is bounded by $\\overline{{\\mathbb {P}}}_\\lambda (G^c|\\omega )$ .",
"Let $H=[\\overline{{\\mathbb {P}}}_\\lambda (G^c|\\omega )\\le \\delta ]$ .",
"Hence, arguing similarly as for (REF ), we obtain that $\\overline{{\\mathbb {P}}}_\\lambda \\bigg (H \\cap \\bigcap _{j\\le J}\\big [\\overline{{\\mathbb {P}}}_\\lambda (A_j|\\omega )\\le \\delta \\big ]\\bigg )\\,\\le \\,\\prod _{j\\le J}\\Big [\\overline{{\\mathbb {P}}}_\\lambda \\big (\\overline{{\\mathbb {P}}}_\\lambda (A_j|\\omega )\\le 3\\delta \\big )+\\overline{{\\mathbb {P}}}_\\lambda (H^c)\\Big ].$ Since $\\overline{{\\mathbb {P}}}_\\lambda (H^c)\\rightarrow 0$ fast, as a consequence of Lemma REF , the two expressions (REF ) and (REF ) replace (REF ) and (REF ) in the proof of Proposition REF .",
"The proof that $q_0=q_1$ is now a straightforward adaptation of the arguments presented in Section , and this ends the outline of the proof of Theorem REF ."
],
[
"Poisson confetti percolation",
"Confetti percolation, or the `dead leaves' model, was introduced by Jeulin in [22].",
"In this model, black and white confetti `rain down' on the plane according to a Poisson point process, and each point in the plane is colored according to the color of the first confetti to cover it.",
"In the case of circular confetti with fixed diameter, Benjamini and Schramm [11] conjectured that the critical probability for this model equals $1/2$ .",
"This was later confirmed by Hirsch [21] for square shaped confetti and by Müller [30] for circular confetti.",
"Just as in the settings of Poisson Boolean and Voronoi percolation, our techniques allow us to handle confetti of random radii, whose laws may differ between black and white.",
"Consider a Poisson point process on the space of locally finite counting measures on ${\\mathbb {R}}^2 \\times {\\mathbb {R}}_+ \\times [0, 1]^2$ .",
"Here again the first two coordinates will assign a location in ${\\mathbb {R}}^2$ to each confetti, while the third coordinate will denote the fall-time of a confetti and will be used to order overlapping confetti.",
"Finally, the fourth and fifth coordinates will help us determine the radius and color of the confetti, respectively.",
"As before, we fix $q \\in [0, 1]$ and, given a realization $\\omega $ with density one, declare a point $(x, t, z, s) \\in \\omega $ black if $s \\le q$ and white otherwise.",
"We also fix non-decreasing functions $G_0, G_1 : [0, 1] \\rightarrow (0, \\infty )$ that will be applied to the $z$ coordinate, in order to determine the radius of the confetti to be placed at $x\\in {\\mathbb {R}}^2$ ($G_0$ for black points and $G_1$ for white).",
"More precisely, given a realization $\\omega $ of the Poisson point process, let $\\mathcal {O}_q := \\bigcup _{\\begin{array}{c}(x, t, z, s) \\in \\omega \\\\s \\le q\\end{array}} \\left\\lbrace y \\in {\\mathbb {R}}^2:\\begin{split}& |y - x| \\le G_0(z), \\text{ and } t < t^{\\prime } \\text{ for all $(x^{\\prime }, t^{\\prime }, z^{\\prime }, s^{\\prime }) \\in \\omega $}\\\\& \\text{such that $|y - x^{\\prime }| \\le G_1(z^{\\prime })$ and $s^{\\prime } > q$}\\end{split}\\right\\rbrace .$ Finally, let $\\mathcal {V}_q := \\mathbb {R}^2 \\setminus \\mathcal {O}_q$ and define the critical parameters $q_c$ and $q_c^\\star $ as in (REF ).",
"For the sake of simplicity, we assume here rather light tails for distribution of the radii induced by $G_0$ and $G_1$ .",
"If these two functions are the same, then black and white confetti have the same radii distributions, which yields $q_c+q_c^\\star =1$ by self-duality, and together with the following theorem that $q_c=1/2$ ; see Section REF below.",
"Theorem 8.3 Assume that $G_0$ and $G_1$ satisfy ${\\mathbb {P}}[G_i(Z) \\ge r] \\le r^{-100}$ , for $i = 0, 1$ and large $r$ , where $Z$ is uniformly distributed on $[0,1]$ .",
"Then, $q_c=q_c^\\star $ are strictly between zero and one, and for all $q<q_c$ , there exists $c=c(q) > 0$ such that ${\\mathbb {P}}\\big [ 0\\overset{\\tiny \\mathcal {O}_q}{\\longleftrightarrow }\\partial B(r) \\big ] \\le \\frac{1}{c}r^{-10}.$ for $q=q_c$ and all $\\kappa >0$ , there exist $c=c(\\kappa )>0$ and $\\alpha >0$ such that, for every $r \\ge 1$ $c<{\\mathbb {P}}(\\operatorname{Cross}(\\kappa r,r))<1-c\\quad \\text{and}\\quad \\mathbb {P}\\big [ 0\\overset{\\tiny \\mathcal {O}_q}{\\longleftrightarrow }\\partial B(r) \\big ] \\le r^{-\\alpha }.$ for all $q>q_c$ , there exists $c=c(q)>0$ such that ${\\mathbb {P}}\\big [ 0\\overset{\\tiny \\mathcal {V}_q}{\\longleftrightarrow }\\partial B(r) \\big ] \\le \\frac{1}{c}r^{-10}.$ Moreover, at $q_c$ there is almost surely no unbounded cluster of either kind.",
"The proof of this theorem is analogous to that of Theorem REF above.",
"The only distinction lies in the treatment of the spatial dependence.",
"The following lemma will replace Lemma REF .",
"Lemma 8.4 Fix a bounded measurable set $D \\subseteq \\mathbb {R}^2$ and $r \\ge 1$ , $M \\ge 1$ and define $G_{D, r, M} = \\left\\lbrace \\begin{array}{c}\\text{for every $(x, t, z, s) \\in \\omega \\cap B(D, Mr)$ with $t \\le r$, $G_0(z) \\vee G_1(z) \\le r/M$}\\\\\\text{for every $(x,t,z,s) \\in \\omega \\cap B(D,Mr)^c$ with $t\\le r$, $G_0(z) \\vee G_1(z) \\le d(x,D)$}\\\\\\text{for every $y \\in D$, there is $(x, t, z, s) \\in \\omega $ with $t \\le r$, $G_0(z) \\wedge G_1(z) \\ge |y - z|$}\\end{array}\\right\\rbrace .$ Then, $\\lim _{r \\rightarrow \\infty } r^{10} \\mathbb {P}(G_{D, r, M}) = 1$ and for an arbitrary function $f(\\mathcal {O}_q) \\in \\sigma (Y_x; x \\in D)$ , we have that $\\operatorname{{\\bf 1}}_G \\cdot f$ is measurable with respect to $\\omega $ restricted to $B(D, r/M) \\times [0, r] \\times [0, 1] \\times [0, 1]$ .",
"The proof of this lemma follows from a simple large deviations bound and is omitted.",
"As a consequence of the lemma, spatial correlations will decay as $r^{-10}$ .",
"This rate of decay is more than enough to follow the outlined proof of Theorem REF and obtain a proof also for Theorem REF ."
],
[
"The consequence of self-duality",
"For the two models considered in this section, the case when the two functions $G_0$ and $G_1$ are equal stands out due to the self-duality retained in the model.",
"Self-duality refers to the property that the occupied set at parameter $q$ has the same law as the vacant set has at parameter $1-q$ .",
"In particular, there is no unbounded occupied component at parameter $q$ if and only if there is no unbounded vacant component at parameter $1-q$ (almost surely).",
"As the critical parameters are defined as the supremum for which this holds it follows that $q_c+q_c^\\star =1$ , in analogy to the Bernoulli case in (REF ).",
"Together with the equality $q_c=q_c^\\star $ of the two parameters, obtained in Theorems REF and REF , it follows that $q_c=q_c^\\star =1/2$ , and we recover the results of [9], [21] and [30].",
"Remark 14 There are other Poisson percolation processes that could in principle be studied with the techniques we have employed.",
"These include for instance Brownian interlacements (see [40]) in ${\\mathbb {R}}^d$ intersected with a plane, and Poissonian cylinders (see [43]) in ${\\mathbb {R}}^d$ intersected with a plane.",
"A rather general setting for Poisson Boolean percolation but with dependence between the radii assigned to different points has been considered by Ahlberg and Tykesson [5], however that study does not address the sharpness of the threshold.",
"It would be interesting to investigate to what extent the techniques developed here applies in that setting."
],
[
"Open problems",
"We have in this paper developed techniques for the study of the sharpness of the threshold for a wide range of models.",
"The weakest link in this scheme is the Poissonian assumption necessary to deduce that the critical regime is indeed a single point and not an interval.",
"It would be interesting to develop an alternative argument for the existence of a sharp threshold beyond the Poissonian setting considered here.",
"It would further be interesting to pursue optimal conditions to study Poisson Voronoi and confetti percolation.",
"Here we only aimed at illustrating how our techniques may be applied to other settings, but we have not pursued to make these applications optimal.",
"Note also that the techniques we have used do not apply to Poisson Boolean percolation models where the discs are replaced by some other deterministic shape which is not sufficiently symmetric.",
"Besides extending the current techniques to other models, one could be interested in answering several questions that are known for Bernoulli percolation.",
"Examples of such problems would be to define the incipient infinite cluster, study noise sensitivity for $\\operatorname{Cross}(r, r)$ and investigate scaling relations, just to name a few."
]
] | 1605.05926 |
[
[
"Active Learning On Weighted Graphs Using Adaptive And Non-adaptive\n Approaches"
],
[
"Abstract This paper studies graph-based active learning, where the goal is to reconstruct a binary signal defined on the nodes of a weighted graph, by sampling it on a small subset of the nodes.",
"A new sampling algorithm is proposed, which sequentially selects the graph nodes to be sampled, based on an aggressive search for the boundary of the signal over the graph.",
"The algorithm generalizes a recent method for sampling nodes in unweighted graphs.",
"The generalization improves the sampling performance using the information gained from the available graph weights.",
"An analysis of the number of samples required by the proposed algorithm is provided, and the gain over the unweighted method is further demonstrated in simulations.",
"Additionally, the proposed method is compared with an alternative state of-the-art method, which is based on the graph's spectral properties.",
"It is shown that the proposed method significantly outperforms the spectral sampling method, if the signal needs to be predicted with high accuracy.",
"On the other hand, if a higher level of inaccuracy is tolerable, then the spectral method outperforms the proposed aggressive search method.",
"Consequently, we propose a hybrid method, which is shown to combine the advantages of both approaches."
],
[
"introduction",
"This paper studies the problem of binary label prediction on a graph.",
"In this problem, we are given a graph $G = (V,E)$ , where the edges $E$ (which can be weighted) capture the similarity relationship between the objects represented by the nodes $V$ .",
"Each node has an initially unknown label associated with it, given by a signal $f:V \\rightarrow \\lbrace -1,+1\\rbrace $ .",
"The goal is to reconstruct the entire signal by sampling its values on a small subset of the nodes.",
"This is achievable when the signal bears some degree of smoothness over the graph, which means that similar objects are more likely to have the same label.",
"Active learning aims to minimize the number of samples needed by selecting the most informative nodes.",
"This problem arises in many machine learning applications, where there is an abundance of unlabeled data but labeled data is scarce and expensive to obtain, for example, requiring human expertise or elaborate experiments.",
"Active learning is an effective way to minimize the cost of labeling in such scenarios [1].",
"A graph based approach to this problem starts by creating a graph where the nodes correspond to the data points ${\\cal X}= \\lbrace {\\bf x}_1, \\ldots , {\\bf x}_n\\rbrace $ and the edges capture the similarity between them.",
"Typically, a sparse graph which connects each data point to few of its most similar neighbors is used.",
"The unknown labels $f_i \\in \\lbrace -1,+1\\rbrace $ associated with the data points define a binary function on the nodes.",
"In datasets of interest, the signal is often notably smooth on the graph.",
"There are two approaches to active learning on graphs.",
"The first approach focuses on identifying the nodes near the boundary region, where the signal changes from $+1$ to $-1$ .",
"The methods with this approach [2], [3] sample nodes sequentially, i.e., the nodes to be sampled next are chosen based on the graph structure as well as previously observed signal values.",
"The second approach [4], [5], [6], in contrast, utilizes global properties of the graph in order to identify the most informative nodes, and sample them all at once.",
"Such global approaches usually focus on providing a good approximation of the signal, rather than exact recovery.",
"It is also possible to combine the two approaches, for example, as in [7].",
"The first contribution of this paper is a new sampling algorithm, called weighted $S^2$ which takes the boundary sampling approach.",
"The weighted $S^2$ algorithm is a generalization of a recently proposed algorithm called $S^2$ [2], which is defined only for the case of unweighted edges.",
"The purpose of the generalized algorithm is to take advantage of the additional information available in the form of the edge weights, in order to reduce the sampling complexity.",
"We characterize the sampling budget required for signal recovery by the weighted $S^2$ algorithm, as a function of the complexity of the signal with respect to the graph.",
"We explain how this generalization can be useful in reducing sampling complexity, and demonstrate a significant reduction (nearly $25\\%$ in one dataset), when neighboring nodes of opposite labels are considerably less similar to each other than identically labeled neighbors.",
"We further compare the sampling complexity of the weighted $S^2$ algorithm with an alternative state-of-the-art method called the cutoff maximization method [4].",
"Unlike the $S^2$ methods, which aim for a complete recovery of the signal by aggressively searching for the boundary nodes, the cutoff maximization method is focused only on providing a good approximation of the signal by ensuring that the unsampled nodes are well-connected to the sampled nodes [8].",
"This method finds a sampling set by optimizing a spectral function defined using the graph Laplacian.",
"We perform the comparison on three realistic data sets, and observe two interesting results: 1.",
"The cutoff maximization method does not discover the entire boundary (i.e, nodes with oppositely labeled neighbors) unless the sampling set contains almost all the nodes.",
"In contrast, the number of nodes required by the $S^2$ methods to discover the boundary is not considerably larger than the number of boundary nodes.",
"2.",
"When the sampling budget is quite limited, the cutoff maximization method provides a much better approximation of the signal.",
"There exists a threshold in terms of the sampling budget that determines which method offers better accuracy.",
"Conversely, the tolerable degree of inaccuracy determines which of the methods offer lower sampling complexity.",
"Motivated by the second observation, we propose a hybrid approach (similar in spirit to [7]) which samples the first few nodes with the cutoff maximization method to approximate the boundary and then switches to the weighted $S^2$ method to refine the approximation.",
"The experiments suggest that the hybrid approach combines the advantages of both methods."
],
[
"$S^2$ Algorithm for Weighted Graphs",
" The $S^2$ algorithm was proposed in [2] for unweighted graphs.",
"In this section we describe the principle of the algorithm, and then generalize the algorithm to weighted graphs.",
"In the next section we analyze the query complexity of the generalized algorithm."
],
[
"Original $S^2$ Algorithm",
"The goal of the algorithm is to find the signal $f$ .",
"To do this, the algorithm operates by finding the edges that connect oppositely labeled nodes.",
"These edges are called cut edges, and together the set of cut edges is called the cut.",
"The algorithm incrementally identifies the cut, with the rationale that once the entire cut is identified, the signal is completely recovered.",
"To find the cut, the algorithm maintains a copy of the graph $G$ , and each time it samples a node that neighbors previously sampled nodes of opposite label, it removes the newly discovered cut edges from the graph copy.",
"This way, the remaining graph copy contains only the undiscovered part of the cut, and the algorithm can more easily focus on discovering these edges.",
"The main idea of the algorithm is to look for a pair of oppositely labeled nodes, and then to find the cut edge on the shortest path between the nodes, using a binary search procedure.",
"The algorithm begins with a random sampling phase.",
"At this phase the algorithm queries a random node, according to the uniform distribution.",
"After each sample, the algorithm checks whether the sampled node has any previously sampled neighbors of opposite label.",
"If such neighbors exist, then the connecting edges are newly discovered cut edges, and they are removed from the graph.",
"After checking and potentially removing newly discovered cut edges, the algorithm checks whether the remaining graph contains any pair of connected nodes of opposite labels.",
"If no such pair exists, the algorithm proceeds with a random sample.",
"If pairs of connected, oppositely labeled nodes do exist, the algorithm looks for the shortest path among all the paths connecting such pairs, and sample the node in the middle of that path (breaking ties arbitrarily).",
"After each sampling operation, either a random sample or a bisecting sample, the algorithm again removes all newly discovered cut edges.",
"The $S^2$ algorithm is described more formally in Algorithm REF .",
"The algorithm is given a budget, which determines the number of queries to perform.",
"Once the budget is exhausted, the algorithm calls a label completion function, which predicts the labels of the unsampled nodes.",
"Several such label completion algorithms are known, such as the POCS method in [9].",
"The $S^2$ algorithm uses a function called Middle Shortest Path (MSP), which returns the node in the middle of the shortest path among all the paths connecting a pair of oppositely labels nodes in the remaining graph.",
"$S^2$ Algorithm [1] Inputs: Graph $G$ , BUDGET $\\le n$ .",
"$L\\leftarrow \\emptyset $ 1 $x\\leftarrow $ Randomly chosen unlabeled node.",
"Add $(x,f(x))$ to $L$ .",
"Remove newly discovered cut edges from $G$ .",
"$|L| =\\text{BUDGET}$ LabelCompletion$(G,L)$ $x\\leftarrow \\text{MSP(G,L)}$ exists"
],
[
"Generalization for Weighted Graphs",
"The $S^2$ algorithm in [2] is defined only for unweighted graphs.",
"Since many learning scenarios provide a weighted graph, we extend the algorithm to exploit the additional available information by modifying the MSP function in the algorithm.",
"Our modification is based on the assumption that the signal is smooth, which means that high-weight edges connect mostly nodes of the same label.",
"Therefore, the weight of cut edges is generally low.",
"In the unweighted $S^2$ algorithm, each MSP query reduces the number of nodes in the shortest of the paths between any two oppositely labeled nodes by approximately one half.",
"The main idea in our generalization is to take advantage of the low weights of cut edges in order to reduce this number by more than a half with each query.",
"To do this, we first switch our perspective, for convenience, from the edge weights to the distances associated with the edges, which are inversely proportional to the weights.",
"The distance between non-neighboring nodes is defined as the sum of the lengths of the edges in the shortest path connecting the nodes.",
"Since the weights are a decreasing function of the distances, it follows that cut edges are typically longer than other edges.",
"We take advantage of this fact by modifying the MSP function to sample the node closest to the midpoint of the path, where the midpoint is computed in terms of length, rather than the number of edges in the path.",
"With each query, the proposed sampling rule can potentially reduce the number of nodes along the shortest of the paths between any two oppositely labeled nodes by more than half if the cut edges contribute significantly more than the non-cut edges to the length of the path.",
"Thus, ultimately it requires less samples to discover a cut edge.",
"This intuition is demonstrated in Figure REF .",
"In this example, the nodes labeled $+1$ are connected with an edge of length $l/2$ , the nodes labeled $-1$ are connected with an edge of length $l$ and the cut edge is of length $3l$ .",
"Figure: An illustration of advantage of weighted S 2 S^2 over unweighted S 2 S^2.Given the labels of the end nodes of this path, the binary search phase of the unweighted $S^2$ algorithm needs to sample labels of 3 extra nodes to discover the cut edge.",
"The weighted $S^2$ algorithm, on the other hand, finds the cut edge with only 2 samples.",
"This type of situation arises more prominently in an unbalanced data set, where the number of nodes in one class is much larger than the other.",
"The advantage of weighted $S^2$ algorithm in such a case is experimentally verified in Section ."
],
[
"Notation",
"Note that $f$ partitions the vertices of $G$ into a collection of connected components with identically labeled vertices.",
"Let $V_1,V_2,\\dots ,V_k$ be these $k$ connected components.",
"Notice that the first node that $S^2$ queries in each collection $V_i$ is often queried randomly, and not by a bisection query.",
"Define $\\beta \\triangleq \\min _{1\\le i\\le k}\\frac{|V_i|}{n}.$ If $\\beta $ is small, more random queries are required by $S^2$ .",
"Let $C$ be the set of cut edges in $G$ .",
"The length of the shortest cut edge in $G$ is denoted by $l_{\\text{cut}}$ .",
"Let $\\partial C$ be the set of nodes which share an edge with at least one oppositely labeled node.",
"The nodes in $\\partial C$ are called boundary nodes.",
"For $1\\le i<j\\le k$ , let $C_{i,j}$ be the subset of $C$ for which each edge $\\lbrace x,y\\rbrace \\in C_{i,j}$ satisfies $x\\in V_i$ and $y\\in V_j$ .",
"If $C_{i,j}$ is not empty for some $1\\le i<j\\le k$ , then it is called a cut component.",
"The number of cut components is denoted by $m$ .",
"For a pair of nodes $v_1$ and $v_2$ , let $d(v_1,v_2)$ denote the length of the shortest path between $v_1$ and $v_2$ .",
"Define $l_n=\\max _{v_1,v_2\\in V}d(v_1,v_2)$ ."
],
[
"Cut Clustering",
"For nodes $x,y\\in V$ , let $d^G(x,y)$ be the length of the shortest path connecting $x$ and $y$ in $G$ .",
"Let $e_1=\\lbrace x_1,y_1\\rbrace $ and $e_2=\\lbrace x_2,y_2\\rbrace $ be a pair of cut edges in $G$ such that $f(x_1)=f(x_2)$ and $f(y_1)=f(y_2)$ .",
"Define $\\delta (e_1,e_2)=d^{G-C}(x_1,x_2)+d^{G-C}(y_1,y_2)+\\max \\lbrace l_{e_1},l_{e_2}\\rbrace ,$ where $G-C$ is the graph $G$ with all the cut edges removed.",
"Let $H_r=(C,\\mathcal {E})$ be the meta graph whose nodes are the cut edges of $G$ , and $\\lbrace e,e^{\\prime }\\rbrace \\in \\mathcal {E}$ iff $\\delta (e,e^{\\prime })\\le r$ .",
"Let $l_{\\kappa }$ be the smallest number for which $H_{l_{\\kappa }}$ has $m$ connected components.",
"The motivation for the definition of $l_{\\kappa }$ is demonstrated in the following lemma.",
"Lemma 1 Consider a case in which after the removal of an edge $e$ by the weighted $S^2$ algorithm, there exist an undiscovered cut edge $e^{\\prime }$ in the same cut component of $e$ .",
"Then the length of the shortest path between two oppositely labeled nodes in the remaining graph is at most $l_{\\kappa }$ .",
"Consider the connected component of the meta graph $H_{l_{\\kappa }}$ that contains the removed cut edge (as a meta node).",
"By the assumptions of the lemma, there must be at least one meta edge in this connected component that connects a discovered cut edge to an undiscovered cut edge.",
"This meta edge corresponds to a path length at most $l_{\\kappa }$ in the remaining graph, proving the lemma."
],
[
"Query Complexity",
"Theorem 1 Suppose that a graph $G=(V,E)$ and a signal $f$ are such that the induced cut set $C$ has $m$ components with cut clustering $l_{\\kappa }$ .",
"Then for any $\\epsilon > 0$ , the weighted $S^2$ will recover $C$ with probability at least $1-\\epsilon $ if the BUDGET is at least $\\small m\\left\\lceil 2\\log _2 \\left(\\frac{l_n}{l_{\\text{cut}}}\\right)\\right\\rceil +(|\\partial C|-m)\\left\\lceil 2\\log _2 \\left(\\frac{l_{\\kappa }}{l_{\\text{cut}}}\\right)\\right\\rceil +\\frac{\\log (1/(\\beta \\epsilon ))}{\\log (1/(1-\\beta ))}.$ The proof of Theorem REF uses the fact that after a pair of connected and oppositely labeled nodes is found, the number of queries until a boundary node is sampled is at most logarithmic in $(l/l_{\\text{cut}})$ , where $l$ is the length of the path between the nodes.",
"We show this fact in the following lemma.",
"Lemma 2 The cut-edge of length $l_\\text{cut}$ is found after no more than $r = \\left\\lceil {2\\log _2\\left(\\frac{l}{l_\\text{cut}}\\right)}\\right\\rceil $ aggressive steps.",
"Figure: The interval of interest is at least halved after two queries.In Figure REF , let $C$ and $D$ denote the queried nodes, such that $C$ is sampled first.",
"Let $E$ denote the midpoint of the interval between $A$ and $B$ , where there may not be a node.",
"By considering all the cases based on labels of $C$ and $D$ and their positions relative to $E$ , it can be shown that after two queries, the length of the interval of interest (i.e., the interval containing the cut-edge) is at least halved.",
"The details are omitted in the interest of space.",
"After $i$ (where $i$ is even) queries, the length of the interval of interest is at most $\\frac{l}{2^{i/2}}$ .",
"Note that the cut-edge is found when the length of the interval of interest is less than or equal to $l_\\text{cut}$ .",
"If $r$ is the maximum number of queries required to locate the cut-edge, then $\\frac{l}{2^{r/2}} = l_\\text{cut} \\Rightarrow r = \\left\\lceil {2\\log _2\\left(\\frac{l}{l_\\text{cut}}\\right)}\\right\\rceil $ [Proof of Theorem REF ] The random sampling phase follows the same argument as in [2], which gives the term $\\frac{\\log (1/(\\beta \\epsilon ))}{\\log (1/(1-\\beta ))}$ .",
"A sequence of bisection queries that commences after a random query or an edge removal, and terminates with an edge removal, is call a run.",
"In each run, at least one boundary node is being queried.",
"Therefore, the number of runs is no greater than $|\\partial C|$ .",
"In each cut component, after the first cut edge and boundary node are discovered, the rest of the boundary nodes are discovered in $\\left\\lceil {2\\log _2\\left(\\frac{l_{\\kappa }}{l_\\text{cut}}\\right)}\\right\\rceil $ queries each, according to Lemmas REF and REF .",
"For discovering the first cut edge in each cut components, we trivially bound the number of queries by $\\left\\lceil {2\\log _2\\left(\\frac{l_{n}}{l_\\text{cut}}\\right)}\\right\\rceil $ .",
"Since there are $m$ cut components, we bound the total number of bisection queries by $m\\left\\lceil 2\\log _2 \\left(\\frac{l_n}{l_{\\text{cut}}}\\right)\\right\\rceil +(|\\partial C|-m)\\left\\lceil 2\\log _2 \\left(\\frac{l_{\\kappa }}{l_{\\text{cut}}}\\right)\\right\\rceil .$"
],
[
"Experiments",
"We consider the following graph based classification problems: 1.",
"USPS handwritten digit recognition [10]: For our experiments, we consider two binary classification sub-problems, namely, 7 vs. 9 and 2 vs. 4, consisting of 200 randomly selected images of each class represented as vectors of dimension 256.",
"The distance between two data points is $d(i,j) = ||{\\bf x}_i-{\\bf x}_j||$ .",
"An unweighted graph $G$ is constructed by connecting a pair of nodes $(i,j)$ if $j$ is a $k$ -nearest neighbor ($k$ -nn with $k = 4$ ) Due to lack of space, we do not study the effect of $k$ in detail.",
"of $i$ or vice versa.",
"A weighted dissimilarity graph $G_d$ is defined to have the same topology as $G$ but the weight associated with edge $(i,j)$ is set to $d(i,j)$ .",
"A weighted similarity graph $G_w$ is defined to have the same topology as $G$ but the weight associated with edge $(i,j)$ is set to $w(i,j) = \\exp \\left(- {d(i,j)^2}/{2\\sigma ^2} \\right)$ .",
"The parameter $\\sigma $ is set to be $1/3$ -rd of the average distance to the $k$ -th nearest neighbor for all datapoints.",
"2.",
"Newsgroups text classification [11]: For our experiments, we consider a binary classification sub-problem Baseball vs. Hockey, where each class contains 200 randomly selected documents.",
"Each document $i$ is represented by a 3000 dimensional vector ${\\bf x}_i$ whose elements are the tf-idf statistics of the 3000 most frequent words in the dataset [8].",
"The cosine similarity between a pair data points $(i,j)$ is given by $w(i,j) = \\frac{\\left\\langle {{\\bf x}_i,{\\bf x}_j}\\right\\rangle }{||{\\bf x}_i||||{\\bf x}_j||}$ .",
"The distance between them is defined as $d(i,j) = \\sqrt{1-w(i,j)^2}$ .",
"The $k$ -nn unweighted graph $G$ (with $k=4$ ), the dissimilarity graph $G_d$ and the similarity graph $G_w$ are constructed using these distance and similarity measures as in the previous example.",
"In addition to the above datasets, we generate a synthetic two circles dataset, shown in Figure REF , in order to demonstrate the advantage of weighted $S^2$ over unweighted $S^2$ .",
"It contains 900 points in one class (marked red) evenly distributed on the inner circle of mean radius 1 and variance $0.05$ and 100 points in the second class (marked blue) on the outer circle of mean radius $1.1$ and variance $0.45$ .",
"A 4-nn graph is constructed using the Euclidean distance between the coordinates of the points.",
"Figure: A synthetic two circles datasetWe compare the performance of the following active learning methods: (1) unweighted $S^2$ method [2] with graph $G$ , (2) weighted $S^2$ method with dissimilarity graph $G_d$ , (3) cutoff maximization method [8] with similarity graph $G_w$ and (4) a hybrid approach combining cutoff maximization and weighted $S^2$ method.",
"After the nodes selected by each method have been sampled, we reconstruct the unknown label signal using the approximate POCS based bandlimited reconstruction scheme [8] to get the soft labels.",
"We threshold these soft labels to get the final label predictions.",
"The hybrid approach uses the non-adaptive cutoff maximization approach in the beginning and switches to the weighted $S^2$ method after sampling a certain number of nodes $n_\\text{switch}$ .",
"In order to determine $n_\\text{switch}$ , after sampling the $i$ -th node with the cutoff method, we compute $1-\\frac{\\mathinner {\\langle {\\hat{{\\bf f}}_i,\\hat{{\\bf f}}_{i-1}}\\rangle }}{\\Vert \\hat{{\\bf f}}_{i}\\Vert \\Vert \\hat{{\\bf f}}_{i-1}\\Vert }$ , where $\\hat{{\\bf f}}_i$ denotes the vector of predicted soft labels.",
"Once this value falls below below $0.001$ , indicating that the newly added label only marginally changed the predictions, hybrid approach switches to weighted $S^2$ .",
"Table REF lists the number of samples required by each of the sampling methods to discover all the cut edges using the observed labels.",
"It shows that weighted $S^2$ can reduce the sample complexity significantly (by $25\\%$ ) compared to unweighted $S^2$ if the ratio of mean length of cut edges and mean length of non-cut edges is high as is the case in the unbalanced two circles dataset.",
"In rest of the datasets, the gain offered by weighted $S^2$ is negligible since the cut edges are only slightly longer than non-cut edges and as a result, taking lengths into account in the bisection phase does not offer much advantage.",
"We also observe that the number of samples required by the weighted $S^2$ method is close to the size of the cut $|\\partial C|$ in most of the datasets.",
"Table REF also shows that the adaptive methods $S^2$ and weighted $S^2$ are very efficient at recovering the entire cut exactly, compared to the non-adaptive cutoff maximization method.",
"In practice, it is not necessary to reconstruct the signal exactly and some reconstruction error is allowed.",
"Figure REF plots the classification error against the number of sampled nodes.",
"It shows that the classification error of the cutoff maximization method decreases rapidly in the early stages of sampling when very few samples are observed.",
"However, the decrease is slow in later stages of sampling.",
"$S^2$ methods, on the other hand, are good at reducing the error in the later stages, but performs poorly with only a few samples.",
"The figure also shows that the hybrid method performs as well as the better method in each region."
],
[
"Conclusions",
" The paper generalizes the $S^2$ algorithm for the case of weighted graphs.",
"The sampling complexity of the generalized algorithm is analyzed, and the gain over the unweighted version is demonstrated by simulation.",
"Additional experiments identify the region of tolerable reconstruction error in which the $S^2$ algorithms outperforms a graph frequency based global approach.",
"A hybrid approach is proposed with the advantages of both methods.",
"It remains open to analytically characterize of the gain of the weighted $S^2$ method over the unweighted version.",
"Another interesting avenue for future work is to provide a performance analysis for the spectral sampling method which can suggest an optimal switching criterion for the hybrid method."
]
] | 1605.05710 |
[
[
"Declarative Machine Learning - A Classification of Basic Properties and\n Types"
],
[
"Abstract Declarative machine learning (ML) aims at the high-level specification of ML tasks or algorithms, and automatic generation of optimized execution plans from these specifications.",
"The fundamental goal is to simplify the usage and/or development of ML algorithms, which is especially important in the context of large-scale computations.",
"However, ML systems at different abstraction levels have emerged over time and accordingly there has been a controversy about the meaning of this general definition of declarative ML.",
"Specification alternatives range from ML algorithms expressed in domain-specific languages (DSLs) with optimization for performance, to ML task (learning problem) specifications with optimization for performance and accuracy.",
"We argue that these different types of declarative ML complement each other as they address different users (data scientists and end users).",
"This paper makes an attempt to create a taxonomy for declarative ML, including a definition of essential basic properties and types of declarative ML.",
"Along the way, we provide insights into implications of these properties.",
"We also use this taxonomy to classify existing systems.",
"Finally, we draw conclusions on defining appropriate benchmarks and specification languages for declarative ML."
],
[
"Introduction",
"Large-scale machine learning (ML) leverages large data collections for advanced analytics in order to find interesting patterns and train robust predictive models.",
"Traditional frameworks and tools like R, Matlab, Weka, SPSS, or SAS provide rich functionality but—except for dedicated packages—struggle to provide scalable analytics.",
"Due to the data-intensive characteristics, increasingly often data-parallel frameworks like MapReduce [14], Spark [41], or Flink [3] are used for cost-effective parallelization on commodity hardware.",
"However, large-scale computation inherently increases the complexity of specifying ML algorithms, especially with regard to efficient and scalable execution.",
"Large-Scale ML Libraries: Large-scale ML libraries like MLlib (aka SparkML) [30], Mahout [37], and MADlib [12], [23] are currently the predominant tools for large-scale ML.",
"These libraries provide algorithms with fixed distributed runtime plans and often expose the underlying physical data representation.",
"Although such libraries are very valuable tools for end-users, it takes substantial effort to write new or customize existing algorithms because it requires knowledge of ML algorithms, their distributed implementation, and the underlying data-parallel framework.",
"Similarly, improvements often require a modification of all individual algorithms to exploit these improvements.",
"Declarative ML: Declarative ML aims at a high-level specification of ML tasks or algorithms to simplify the usage and/or development of ML algorithms by separating application or algorithm semantics from the underlying data representations and execution plans.",
"Table REF categorizes types of declarative ML and delineates them from ML libraries.",
"Overall, the major benefits of declarative ML are: Simple, Analysis-Centric Specification, Physical Data Independence, Automatic Execution Plan Generation (optimization, platform independence, data-size independence), Ease of Deployment (platform independence, adaptivity of “packaged” applications), and Separation of Concerns (skill sets of users/devs).",
"Over time, systems at different levels of abstraction have been proposed by industry and academia.",
"Example systems range from UDF-centric ML extensions of data-parallel frameworks, to domain-specific languages (DSLs) for ML tasks or ML algorithms.",
"This broad spectrum of systems aiming for declarative ML naturally led to a controversy regarding the scope of declarative ML.",
"Not surprisingly—as many ML algorithms are iterative—the discussion centers around the syntax for specifying loops and control flow in general.",
"Various projects adopt an R/Python-like syntax [1], [15], [22], [40], [42], [43] inheriting the full flexibility of loops, branches, and functions.",
"Others support loops with (1) more restrictive iteration constructs [13], [17], (2) model updates with implicit convergence checks [8], or encapsulating entire algorithm classes as ML tasks [27], [33], [42].",
"We argue that the specific language-level syntax is actually irrelevant if the ML task or algorithm specification conforms to a set of basic properties required for declarative ML.",
"Table: Delineation of Types of Declarative ML.Contributions and Structure: The primary contribution of this paper is a systematic analysis and classification of declarative machine learning.",
"We first define—in an syntax-independent manner—a set of basic properties in Section , that any system for declarative ML should satisfy.",
"Subsequently, we describe the types of declarative ML in Section .",
"Finally, we use this taxonomy to classify existing systems in Section , and draw conclusions for defining appropriate benchmarks and languages in Section ."
],
[
"Basic Properties",
"As a foundation for discussing types of declarative ML, we define essential, basic properties in the three categories of data, operations, and result correctness.",
"We also discuss the implications of the individual properties and provide examples of how Apache SystemML—as a representative system for declarative ML—realizes these properties."
],
[
"Physical Data Independence",
"The most significant goal of declarative ML is data independence because it decouples the high-level specification of ML tasks or algorithms from the underlying data representations and related runtime plans and operations.",
"Property 1 Independence of Data Structures: The data types of inputs, intermediate results, and outputs like matrices or scalars are exposed as abstract data types without access to the underlying physical data representations.",
"In the context of declarative ML, independence of data structures serves two major purposes.",
"First, abstract data types like matrix hide the decision on distributed vs local data representations.",
"Accordingly, specified tasks or algorithms become independent of data size and deployment context (e.g., distributed computation vs streaming, different runtime backends).",
"Second, abstract data types also hide the physical data representation (e.g., dense/sparse matrices, lossless compression), which allow internal improvements of storage and operation efficiency.",
"Property 2 Independence of Data Flow Properties: Data flow properties are not exposed, i.e., the user has, at specification level, no explicit control over properties like partitioning, caching, and blocking configurations.",
"The property of independence of data flow properties further restricts the notion of abstract data types, disallowing the explicit specification of interesting data flow properties.",
"Examples are (1) caching and checkpointing (e.g., local and distributed caching, with certain storage levels), (2) logical and physical partitioning (e.g., row/column/block partitioning and range/hash partitioning of distributed data sets), (3) blocking configurations (row/column block sizes, fixed or variable logical/physical block sizes), as well as (4) data formats (text or binary cell/block formats).",
"Note, however, that it is valid to allow the specification of ordering because it is both a logical and physical data flow property.",
"Example SystemML: SystemML satisfies both properties by exposing only the abstract data types frame, matrix, and scalar without their physical data structures or interesting data flow properties as shown in Figure REF , as an example of a valid specification.",
"The decisions on physical data flow properties are, however, crucial for performance.",
"Hence, the system automatically injects, for example, caching and partitioning directives via rewrites.",
"Overall, data independence allowed us to evolve and rebase SystemML without changing a single ML algorithm.",
"Examples are extensions such as the support for different sparse representations, compression, and additional backends like Spark or GPUs.",
"In contrast, for example, Mahout Samsara [15] does not satisfy the properties of data independence because decisions on dense/sparse and distributed/local matrices (e.g., drmFromHDFS, collect) as well as data flow properties like partitioning (e.g., par) and caching (e.g., checkpoint) are exposed to the user as shown in Figure REF .",
"Figure: Mahout Samsara"
],
[
"Operation Semantics",
"The second major goal of declarative ML is to specify ML tasks or algorithms using domain-specific, high-level operations with well-defined semantics to simplify algorithm usage or development, and enable efficient evaluation plans.",
"Property 3 Analysis-Centric Operation Primitives: Basic operation primitives, common in the target analytics domain, are supported.",
"For ML algorithms, this includes linear algebra and statistical functions, whereas for ML tasks, this includes task-specific primitives and models.",
"In order to allow declarative ML with simple specification, there is a need for operation primitives that closely resemble a natural description of ML tasks or algorithms at a conceptual level.",
"For ML algorithms this includes linear algebra, aggregations, and statistical functions but specific domains like deep learning might require additional domain-specific operations like convolution.",
"Similarly, for ML tasks this includes task-specific abstractions, operations, and models.",
"For example, for specifying a task like classify, common classification algorithms, loss functions, and parameters should be supported to describe candidates and the optimization objective.",
"The same applies for the task of general-purpose optimization optimize, where one would expect a way to specify gradient/loss functions, and termination conditions.",
"Note that this property excludes systems that are declarative but unrelated to ML.",
"Property 4 Known Semantics of Operation Primitives: The semantics of operation primitives used to specify ML tasks or algorithms are known to the system in terms of knowledge of operation characteristics and equivalences.",
"We require operational semantics, where there exists at least one naïve evaluation plan or straightforward mapping.",
"Knowledge of operation semantics is essential for generating efficient evaluation plans—for example, via rewrites and operator selection—from high-level specifications.",
"In the context of ML, operation semantics also cover characteristics like commutativity and associativity, sparse-safeness (correctness of processing only non-zero cells), value repositioning (e.g., reorg operations like transpose, or order), symmetry properties, as well as an understanding of composite operations (e.g., sum-product for matrix multiplication).",
"This meta information might be built into the system or annotated in case of extensible systems.",
"Overall operation semantics allow to reason about equivalences, alternative execution strategies, and costs of these alternatives.",
"Property 5 Implementation-Agnostic Operations: The specification of ML tasks or algorithms is independent of the underlying runtime operations.",
"This property prohibits user-defined execution strategies and parameterization.",
"Specifying implementation-agnostic operations—i.e., independent of runtime backends, distributed vs local operations, and execution strategies—is related to the properties for data independence (see Subsection REF ) but with a focus on operations to ensure the flexibility of alternative or hybrid runtime backends, alternative deployments, and optimizations like rewrites and operator selection.",
"Furthermore, avoiding low level parameterization like degree of parallelism or cache blocking ensures independence of the ML tasks or algorithms from workload characteristics (e.g., data size) and underlying hardware infrastructure.",
"Property 6 Well-Defined Plan Optimization Objective: ML tasks or algorithms specify their expected results unambiguously, using a well-defined (potentially multi-criteria) objective for execution plan optimization.",
"To specify ML tasks or algorithms in an unambiguous manner, the specification must exhibit (implicitly or explicitly) a well-defined plan optimization objective.",
"This property differentiates major types of declarative ML.",
"If the specification relies on unambiguous operations, the implicit optimization objective is efficiency (runtime or resource requirements).",
"In case of multi-objective optimizations, we further need to define a primary dimension and constraints for the other dimensions.",
"For example, if we aim to optimize for both efficiency and accuracy, we might want to optimize accuracy in terms of a quality measure (e.g., L2 loss on a holdout dataset) as the primary dimension along with constraints on efficiency (e.g., time budget, number of models).",
"Example SystemML: SystemML satisfies these properties by minimizing execution time (under memory budget constraints per execution context) of specified ML algorithms, composed of linear algebra and statistical operations with well-defined semantics.",
"The known operation semantics are used to propagate dimension and sparsity information through the entire algorithm, compute memory estimates, apply static (size-independent) and dynamic (size-dependent) rewrites, and eventually decide upon alternative physical operators.",
"Implementation-agnostic operations allow fine-grained optimization decisions per operation.",
"Note that sequences of operations like t(X) %*% X %*% p from Figure REF or specifications like independent foreach loops (parfor) [6] are assertions on semantics and properties of the algorithm rather than imperative execution strategies."
],
[
"Result Correctness",
"The properties of data independence and operation semantics are necessary but not sufficient for declarative ML.",
"We further need to define the notion of result correctness.",
"With regard to practicability for distributed computing, we define operation results as equivalent if they are essentially the same, i.e., they are algebraically (logically) equivalent, which ignores round-off errors (e.g., due to partial aggregation or alternative evaluation orders of operations).",
"Property 7 Implementation-Agnostic Results: The results of ML tasks or algorithms as well as individual operations are equivalent (essentially the same), independent of type and location of underlying runtime operations.",
"This property further qualifies implementation-agnostic operations (P5).",
"In order to produce correct results, independent of optimization decisions, alternative execution strategies need to produce equivalent results no matter if they are executed locally or as distributed operations.",
"To accomplish that for an operation like rand with fixed seed, both local and distributed operations need to consistently generate seeds for fixed-sized blocks from the initially given input seed.",
"Furthermore, this property prohibits, for example, lossy compression to reduce communication overhead.",
"Property 8 Deterministic Results: A given ML task or algorithm yields equivalent (essentially the same) results for multiple executions over the same input data and configuration.",
"Randomized tasks or algorithms achieve this using pseudorandom number generators.",
"Deterministic results of operations and ML tasks or algorithms is an important property, especially with regard to fault tolerance, where the same operation might be executed multiple times.",
"Furthermore, it is also the basis for benchmarking ML systems in a systematic manner.",
"Example SystemML: In SystemML, the properties of result correctness are satisfied via consistent local and distributed as well as deterministic operations.",
"ML algorithms are composed of these operations, lifting the properties of result correctness to algorithm level too.",
"Furthermore, we bound the round-off errors via numerically stable operations (based on Kahan+) [38] for descriptive statistics and aggregations.",
"SystemML also provides configuration knobs to disable rewrites and operator selection to force strict computation.",
"However, other than for debugging, we have not seen data scientists or end-users making use of that.",
"Table: Classification of Existing Systems wrt Declarative ML Algorithms (Type 1).So far we discussed general properties of declarative machine learning, which apply to all types of declarative ML.",
"We now create a taxonomy of types of declarative ML, namely declarative ML algorithms and declarative ML tasks.",
"These types refer to fundamentally different concepts and thus, also differ in their scope of specification."
],
[
"Declarative ML Algorithms (Type 1)",
"Declarative ML algorithms allow data scientists to write and customize ML algorithms in a declarative manner.",
"This scope requires fine-grained semantics including control flow and data flow, where the core operation primitives are often based on linear algebra or statistical functions and the common optimization objective is to minimize execution time but other objectives such as resource consumption are possible.",
"The algorithm-centric specification defines precise semantics but leaves substantial freedom regarding data representations and execution plan optimization.",
"This abstraction level allows data scientists to encode algorithms as they are most naturally expressed and thus, to quickly exploit the latest algorithmic advances.",
"End users also benefit from simply calling these algorithms in terms of automatic optimization, adaptivity, and portability.",
"Example system categories are DSL-centric, SQL-centric, and UDF-centric systems."
],
[
"Declarative ML Tasks (Type 2)",
"In contrast to declarative ML algorithms, declarative ML tasks allow end users (without ML background), to specify ML tasks like classify, factorize, optimize independent of ML algorithm specifics.",
"This coarse-grained scope includes automatic feature and model selection and allows for the optimization of both model accuracy and runtime.",
"Core operation primitives are task-specific, i.e., depending on the task at hand, alternative candidate algorithms, loss functions (measure for goodness of fit), and hyper parameters are supported.",
"Operation semantics are either built-in or annotated at the level of used algorithms or loss functions.",
"The properties of declarative ML need to apply to the core optimization problem of the given ML task not the entire stack of used ML algorithms, as long as they are annotated with relevant properties that allow reasoning about alternative plans and costs.",
"However, relying on declarative ML algorithms provides additional flexibility.",
"Example system categories for declarative ML tasks are general-purpose optimization, model selection, and feature selection."
],
[
"Systems Classification",
"Given the taxonomy of basic properties and types of declarative ML, we now classify existing systems.",
"There has been some related work on similar classifications, most notably, Kumar et al.",
"defined a notion of a model selection management system [28], along with categories of ML systems, but primarily focused on coverage of industrial systems rather than specifics of declarative machine learning.",
"Tables REF and REF classify—in the scope of declarative ML algorithms and tasks—existing systems with regard to the defined basic properties and types of declarative machine learning.",
"This classification also indicates distributed vs local operations and the used optimization objective.",
"Table: Classification of Existing Systems wrt Declarative ML Tasks (Type 2)."
],
[
"Declarative ML Algorithms",
"DSL-Centric Systems: The class of DSL-centric systems focuses on domain-specific languages (DSLs) for ML to simplify the writing of ML algorithms.",
"Early examples of declarative systems are RIOT [43] and OptiML [35], which provide R and Scala DSLs, respectively.",
"Both focus on single-node computation only which makes it easier to adhere to basic properties of declarative machine learning.",
"SystemML [7], [22] covers both single-node and distributed computation (on MapReduce and Spark) and satisfies all eight properties of declarative ML as described throughout this paper.",
"More recent systems like Cumulon [24], [25], Mahout Samsara [15], DMac [40], and TensorFlow [1] similarly aim for declarative, large-scale ML but struggle to satisfy all properties of declarative ML.",
"Cumulon [24], [25] can be seen as a declarative system.",
"However, in a strict sense, the optimization objective is ill-defined (P6) as the hard runtime constraint cannot be satisfied without knowing the number of iterations until convergence.",
"Cumulon still allows users to explore per-iteration trade-offs of monetary cost and runtime in cloud environments.",
"Mahout Samsara [15], Distributed R [39], and DMac [40] are not declarative because they expose physical data structures and distributed operations to the user (P1, P5).",
"Mahout Samsara and Distributed R require the user to decide between local and distributed matrices and expose data flow properties like caching and partitioning (P2), while DMac exposes dense/sparse data structures.",
"Additionally, Distributed R executes arbitrary user-defined R functions—i.e., with unknown operation semantics (P4)—per partition.",
"TensorFlow [1] is a compelling system but focuses more on extensibility than declarative specification.",
"Accordingly, operations are handled as black-box kernels, i.e., with unknown operation semantics (P4).",
"Also, deep-learning-centric optimizations like lossy compression for communication work very well for noisy data but do not satisfy the property of implementation-agnostic results (P7) required for general-purpose declarative ML.",
"SQL-Centric Systems: The class of SQL-centric systems complements the class of DSL-centric systems as both aim for custom analysis algorithms.",
"Common SQL-centric systems are array databases and SQL-like ML (e.g., for Bayesian ML).",
"These systems often follow the compelling argument of integrating advanced analytics with traditional query processing to simplify data pre-processing and leverage well-understood abstractions for data and operations.",
"At the same time, linear algebra operations are either directly supported or emulated.",
"A prime example of array databases is SciDB [9], [34] which indeed satisfies all basic properties for declarative ML.",
"Furthermore, also SimSQL [11]—as an example of Bayesian ML or more broadly stochastic analysis—satisfies all basic properties of declarative ML.",
"UDF-Centric Systems: There is also a class of UDF-centric systems that evolved bottom up from existing data-parallel frameworks like MapReduce, Spark, or Flink as well as compiler frameworks to simplify large-scale ML.",
"Examples are ScalOps [8] (which compiles Scala UDF workflows to datalog programs), Tupleware [13] (which compiles workflows of UDFs in various frontend languages to custom distributed programs of native code via LLVM), and Emma [4] (which compiles Scala UDF workflows to Spark and Flink programs).",
"These systems cover a great variety of use cases, but they systematically fail to satisfy several properties of declarative ML.",
"First, data independence (P1) is not satisfied as the UDFs are implemented against custom data structures which makes it hard to efficiently support dense/sparse or compressed datasets.",
"Second, the operation semantics of UDFs are by definition unknown (P4), miss support of analysis-centric operations (P3), and the focus on large-scale computation requires the UDF workflows to realize distributed algorithm implementations (P5)."
],
[
"Declarative ML Tasks",
"As discussed before, systems for declarative ML tasks mostly target end users (not algorithm developers) and automatically optimize for runtime and accuracy of general-purpose optimization or model and feature selection tasks.",
"General-Purpose Optimization: Bismarck [18] provides in-database general-purpose optimization via incremental gradient decent, where users provide UDFs for initialization, transition, and termination.",
"This abstraction covers many algorithms over existing relational data.",
"Similar to UDF-centric systems, however, it does not satisfy the independence of data structures (P1) and known operation semantics (P4).",
"Furthermore, effective parallelization requires either violations of implementation-agnostic and deterministic results (P7, P8) for the pure UDA approach or modifications of the UDFs (P5) for the shared-memory UDA approach.",
"The latter also does not satisfy P7/P8 due to Hogwild!-style [31] model updates.",
"TensorFlow [1] also provides primitives for general-purpose optimization via different optimization algorithms.",
"Similar to Bismarck, users provide UDFs for loss and gradient computation.",
"However, as TensorFlow provides sufficient data abstraction via so-called placeholders, it satisfies the properties of data independence.",
"Furthermore, the operation semantics of inference, loss, and training are known and UDFs can leverage the existing built-in functions which also provide abstractions for gradient computation.",
"Thus, although TensorFlow did not qualify for declarative ML algorithms, it satisfies the properties at the level of declarative ML tasks.",
"Model and Feature Selection: MLbase [27] with its TUPAQ [33] component for automatic model search allows users to specify candidate model configurations, a quality measure, and runtime constraints (e.g., models considered, number of scans, etc) and returns the best model along with tuned hyperparameters.",
"Disregarding the underlying library of ML algorithms, MLbase can be classified as a system for declarative ML tasks, as the model search problem is well-defined, independent of the underlying data and operations, with annotated algorithm characteristics, and deterministic results (unless the runtime constraint is a runtime budget).",
"Columbus [42] allows users to specify feature engineering workflows in R including data preparations and model building with existing R packages.",
"The optimization objective is to minimize runtime, but an error tolerance allows for more aggressive reuse by leveraging runtime-accuracy trade-offs.",
"Columbus also satisfies the properties for declarative ML tasks.",
"DeepDive [32] enables knowledge base construction via statistical inference.",
"In a first step, users specify SQL queries and UDFs for feature extraction to populate a factor graph.",
"In a second step, candidate mappings are specified via SQL queries to express rules for entities and relations.",
"Finally, marginal probabilities are learned via statistical inference over this factor graph.",
"As data and operations are abstracted via SQL and the actual inference algorithms and details are not exposed, DeepDive can be classified as a declarative system for feature selection.",
"Other System Categories: Finally, there are also declarative systems for more specific ML tasks like time series forecasting (e.g., Fa [16], a skip-list approach [20], or F2DB [19]), which also consider the trade-off between accuracy and runtime (model selection) but are not subject to this classification for general-purpose declarative ML."
],
[
"Benchmarking ML Systems",
"Having discussed the individual types of declarative ML, it is clear that there cannot exist a single benchmark to cover them all.",
"Existing benchmarks for large-scale computation like BigBench [5], [21], SparkBench [2], [29], or HiBench [26] do cover machine learning but often simply refer to reference implementations of large-scale ML libraries.",
"This is fine to evaluate underlying Hadoop or Spark implementations but simply cannot serve as a benchmark for declarative ML.",
"A Case for Type-Specific Benchmarks: We argue that industry and academia is best served with benchmarks specific to types of declarative machine learning (see Section ).",
"In order to properly reflect common workload characteristics, the benchmarks should be further tailored to major subcategories of systems.",
"For example, systems for declarative ML algorithms cover the major sub-categories of DSL-centric, UDF-centric, and SQL-centric systems, which—despite some overlap of operation primitives—all target different primary usage scenarios.",
"This calls for specific benchmarks that allow fair comparisons and foster system advancements via challenging workloads.",
"Interestingly, exactly that already happened for SQL-centric systems as both, the SciDB [9], [34] and SimSQL [11] projects published benchmarks covering the main characteristics of array databases [36] and Bayesian ML [10].",
"Defining simple yet challenging ML benchmarks that also cover common workload characteristics is not easy, so we—as a community—should highly appreciate contributions in this area.",
"Note on Specification Languages: The same reasoning as with system-type-specific benchmarks also applies to specification languages.",
"In order to satisfy the property of analysis-centric operation support (P4), systems need to support the operations of their primary analysis use case, which motivates tailor-made specification languages.",
"However, as the properties for declarative ML are syntax independent, we could establish a common syntax for a specification language to cover multiple types of declarative ML."
],
[
"Conclusion",
"To summarize, we introduced a taxonomy of declarative ML in terms of basic properties of data, operations, and result correctness as well as types of systems for declarative ML.",
"The classification of existing systems has shown that this taxonomy is indeed a useful tool for qualifying systems characteristics in a systematic manner.",
"Fundamentally, this makes a case for a syntax-independent classification of declarative ML, which disqualifies the philosophical argument against loops and control flow in general.",
"We are at the beginning of an exciting era of declarative ML, with a good understanding of various aspects but also lots of open research challenges.",
"As advanced analytics become ubiquitous and technology environments are changing at an increasing rate, a declarative specification of ML tasks or algorithms becomes increasingly important.",
"Accordingly, we encourage the research community to participate in this discussion on basic properties of declarative ML in order to eventually converge to a common understanding."
]
] | 1605.05826 |
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